Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
964 Ordinary and Partial Differential Equations Proceedings of the Seventh Conference Held at Dundee, Scotland, March 29-April 2, 1982
Edited by W. N. Everitt and B.D. Sleeman
Springer-Verlag Berlin Heidelberg New York 1982
Editors
W.N. Everitt Department of Mathematics, University of Birmingham RO. Box 363. Birmingham B15 2TT, England B. D. Sleeman Department of Mathematics, University of Dundee Dundee DD1 4HN, Scotland
AMS Subject Classifications (1980): 34-06, 35-06 ISBN 3-54041968-X Springer-Verlag Berlin Heidelberg NewYork ISBN 0-38741968-X Springer-Verlag NewYork Heidelberg Berlin
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © by Springer-Verlag Berlin Heidelberg 1982 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
D E D I C A T I O N
These P r o c e e d i n g s
are d e d i c a t e d
The U n i v e r s i t y
to
of Dundee
on the o c c a s i o n of the c e n t e n a r y
1882 - 1982
celebrations
PREFACE
These Proceedings
form a record of the lectures
Conference
on Ordinary and Partial Differential
University
of Dundee,
delivered at the seventh
Equations which was held at the
Scotland during the week Monday 29 March to Friday 2 April
1982. The Conference Dundee 1882-1982. Conference,
formed part of the centenary
The University
for this purpose,
celebrations
of W~rzburg was officially
for the University
represented
from more than 25 countries.
The Conference was organised by a Committee with membership:
E R Dawson,
B D Sleeman.
Following Organising
at the
by Professor Dr H W Knobloch.
The Conference was attended by 120 mathematicians
W N Everitt,
of
again the tradition set by the earlier Dundee Conferences
Committee named as Honorary Presidents Professor Jean Mawhin Professor
The Committee
expresses
of the 1982 Conference
(Belgium)
Tung Chin-Chu
its gratitude
the
(P R China).
to all mathematicians
who took part in
the work of the Conference. The Committee Conference;
the Warden,
help in providing Sciences
thanks
accommodation
Officer,
of Dundee for generously
supporting
the
Domestic Bursar and Staff of West Park Hall for all their
for continuing
Accommodation
: the University
for the participants;
support for the Conference; the Information
the Department
of Mathematical
the Bursar of Residences,
the
Officer and the Finance Officer of the
University. The Committee offers special thanks to the Vice-Principal Professor P D Griffiths, and at the University
for representing
funds,
at the opening ceremony,
reception held for all participants.
The Committee records with gratitude and subsistence
the University
of the University,
financial
support,
in the form of travel
from
The Royal Society of London and The European Research Office of the United States Army.
Vl
The Professors
Everitt and Sleeman record special thanks to their colleague
Commander E R Dawson R N whose services, helping to organise accommodation
the Conference,
of all participants
number of the manuscripts Likewise Mathematical
in West Park Hall.
submitted
were generously
for publication
for the
Commander ~awson also edited a in these Proceedings.
special thanks are due to all the Secretaries in particular
in the Department
of
to Mrs Caroline Peters for
contribution.
W N Everitt
offered in
and who carried the main responsibility
Sciences of the University;
her invaluable
from retirement,
B D Sleeman
C O N T E N T S
' V Atkinson: )n the asymptotic b e h a v i o u r of the T i t c h m a r s h - W e y l m - c o e f f i c i e n t and the ~pectral function for scalar second-order differential e x p r e s s i o n s ........
1
V A t k i n s o n and C T Fulton: ;ome limit circle e i g e n v a l u e problems and asymptotic formulae for ~igenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V Atkinson,
28
J R H a d d o c k and 0 J Staffans:
integral inequalities and exponential c o n v e r g e n c e of solutions of lifferential equations w i t h b o u n d e d delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
) L B a r r o w and P W Bates: ~ifurcation of p e r i o d i c travelling w a v e s for a r e a c t i o n - d i f f u s i o n
system ..
69
Nle Dirac e q u a t i o n with an anomalous m a g n e t i c moment II ...................
77
I Behncke:
I Benci, A Capozzi and D Fortunato: )eriodic solutions of a class of H a m i l t o n i a n systems
......................
86
.................................
95
~n a p p r o a c h to the Dirichlet index for operators s a t i s f y i n g minimal :onditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ii0
J Browne: 4ultiparameter problems
: the last decade
C Brown:
I A Burns and E M Cliff and J M Amillo Gil qonlinear neutral functional differential equations in product spaces
.....
118
.................
135
.........................
143
?n the H u k u h a r a - K n e s e r p r o p e r t y for some Cauchy problems in locally convex topological vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
162
• A C l a r k s o n and J B McLeod: k c o n n e c t i o n formula for the second P a i n l e v ~ transcendent ) Colton: Fhe inverse s c a t t e r i n g p r o b l e m for acoustic waves 7 Dubois and P Morales:
X D~ng: Essential s e l f - a d j o i n t n e s s and self-adjointness for g e n e r a l i z e d Bchr~dinger operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
171
Vlll
M S P Eastham: Asymptotic theory for a critical class of fourth-order differential equations .................................................................
179
Elbert: Oscillation and nonoscillation theorems for some nonlinear ordinary differential equations ....................................................
187
D E Edmunds: ................................
213
The singular perturbation approach to flame theory with chain and competing reactions .......................................................
232
Entropy numbers,
s-numbers and eigenvalues
P C Fife and B Nicolaenko:
J Fleckinger: On the singular values of non-self-adjoint
operators of SchrZdinger type ..
251
I M Gali: Optimal control of systems governed by elliptic operator of infinite order
263
V B Goyal and P W Schaefer: Comparison principles
for some fourth order elliptic problems
.............
272
R C Grimmer and W Schappacher: Weak solutions of integrodifferential
equations and applications
..........
280
D B Hinton and J K Shaw: Titchmarsh's
l-dependent boundary conditions for Hamiltonian systems
......
298
F A Howes: 318 Exterior boundary value problems H G Kaper,
for perturbed equations of elliptic type .
C Gerrit Lekkerkerker and A Zettl:
Linear transport theory and an indefinite Sturm-Liouville problem .........
326
R M Kauffman: Non-normalizable eigenfunction expansions for ordinary differential operators .................................................................
362
H W Knobloch: Some aspects and recent developments
in linear control theory .............
368
I W Knowles: Eigenvalue problems and the Riemann zeta-function
.........................
388
IX
M A Kon and L A Raphael: Superposition principles and pointwise evaluation of Sturm-Liouville eigenfunction expansions ..................................................
406
K Kreith: Qualitative theory of hyperbolic characteristic
initial value problems I ...
416
R T Lewis: Applications of a comparison theorem for quasi-accretive operators in a Hilbert space .............................................................
422
L L Littlejohn and A M Krall: A singular sixth order differential equation with orthogonal polynomial eigenfunctions ............................................................
435
S-O LondOn: On the asymptotics of some Volterra equations with locally finite measures and large perturbations ..........................................
445
J Mawhin: ...................
458
eddy current problems
477
.......................
485
Periodic oscillations of forced pendulum-like equations R C MacCamy and E Stephan: A simple layer potential method for three-dimensional A C McBride: Index laws for some ordinary differential operators P A McCoy:
Converse initial value problems for a class of heat equations
.............
494
..................................
503
.......................................
519
J R McLaughlin: Higher order inverse eigenvalue problems A B Mingarelli: Indefinite Sturm-Liouville problems S E A Mohammed: The infinitesimal generator of a stochastic functional differential equation ..................................................................
529
M Nakao: On solutions to the initial-boundary problem for perturbed porous medium equation ..................................................................
539
X
F Neuman: A survey of global properties of linear differential equations of the n-th order ................................................................
548
D Pascali: .....................
564
A further result on the essential spectrum of linear 2nth order differential expressions with complex coefficients ........................
569
Strongly nonlinear evolution variational inequalities D Race:
R Rautmann: On error bounds for nonstationary spectral Navier-Stokes
approximations
...
576
T T Read: Sectorial second order differential operators
.............................
584
H R~h: Spectral mapping theorems for dissipative Co-semigroup
generators
.........
591
D A S~nchez: Some preliminary results on periodic solutions of matrix Riccati equations
598
R Saxton: Solitary and travelling waves in a rod ....................................
603
K Seitz: Investigations in the theory of partial differential equations of infinite order .....................................................................
609
J K Shaw and D B Hinton: Well-posed boundary problems for Hamiltonian systems of limit point or limit circle type .........................................................
614
K Soni: On uniform asymptotic expansion of a class of integral transforms
.........
632
.........
642
R L Sternberg~ M J Goldstein and D Drinkard: On a new numerical method for a new class of nonlinear partial differential equations arising in nonspherical geometrical optics A Z-A M Tazali: Local existence theorems for ordinary differential equations of fractional order ..........................................................
652
Tung Chin-Chu: Application of the theorem of connectivity of M 7 in astronomy
.............
666
Xl
L Turyn: .............................
675
............. . ...............
686
......................................
695
On some conjectures on the deficiency index for symmetric differential operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
703
P e r t u r b a t i o n s of p e r i o d i c b o u n d a r y conditions A Vanderbauwhede: B i f u r c a t i o n problems w i t h triangular symmetry J Walter: On the foundations of thermodynamics A D W o o d and R B Paris:
S D Wray: ......
710
....................
718
On a s e c o n d - o r d e r d i f f e r e n t i a l e x p r e s s i o n and its Dirichlet integral E M E Zayed: An inverse eigenvalue p r o b l e m for the Laplace o p e r a t o r
Lectures given at the Conference which are not represented by contributions to these Proceedings. (Results may be presented elsewhere)
A U Afuwape: On Lyapunov Razumikhin approach to certain third-order equations with delay A H Azzam: Boundary value problems for elliptic and parabolic equations in domain with corners I Bihari: Distribution of the zeros of B$cher's pairs with respect to second order homogeneous differential equations T R Blows: Limit cycles of polynomial differential equations L Br~ll: A new abstract existence theory for non-linear Schr~dinger and wave equations L Collatz: Inclusion theorems for singular and free boundary value problems P A Deift: Some recent results in ordinary differential equations and approximation techniques J D~nig: Positive eigensolutions and the lower spectrum of Schr~dinger operators W D Evans: On the distribution of eigenvalues of Schr~dinger operators W N Everitt: Two examples of the Hardy-Littlewood type of integral inequalities R E Kleinman: Recent developments in modified Green's functions M K Kwong: Second order linear and nonlinear oscillation results A G Ramm: Basisness property and asymptotics of spectrums of some nonselfadjoint differential and pseudo-differential operators
XIIl
B P Rynne: Bloch waves and multiparameter spectral theory B D Sleeman: An abstract multiparameter spectral theory R A Smith: Poincar~'s index theorem and Bendixson's negative criterion for certa! differential equations of higher dimension R Villella-Bressan: Functional equations of delay type in Ll-spaces W L Walter: Generalized Volterra prey-predator systems
Address list of authors and speakers A U Afuwape:
Department of Mathematics, University of Ife, ILE-IFE, Nigeria
J M Amillo Gil:
Universidad Politechnica De Madrid, MADRID, Spain
F V Atkinson:
Department of Mathematics, University of Toronto, TORONTO, Ontario,
A H Azzam:
Canada
Department of Mathematics, King Saud University, RIYADH, Saudi Arabia
P W Bates:
Department of Mathematics, Heriot-Watt University, Riccarton, Currie, EDINBURGH, EHI4 4AS, Scotland
H Behncke:
Fachbereich Mathematik, University of Osnabruch, 45 OSNABR~CH, West Germany
V Benci:
Istituto de Matematica Applicata, Via Re David 200, 70125 BARI, Italy
I Bihari:
Mathematical Institute, BUDAPEST, Realtanoda-U 13/15, H-I053, Hungary
T R Blows:
Pure Mathematics Department, University College Wales, ABERYSTWYTH, Dyfed, SY23 3BZ, Wales
R C Brown:
Department of Mathematics, University of Tennessee, KNOXVILLE, Tennessee, U.S.A.
P J Browne:
Department of Mathematics & Statistics, University of Calgary, CALGARY,
L Brull~
Alberta, T2N IN4, Canada
Mathematisches Institut der Universitat zu Koln, 5000 K~LN 41, Weyertal 86, West Germany
J A Burns:
Department of Mathematics, Virginia Tech University, BLACKSBURG, VA 24061, U.S.A.
P A
Clarkson:
E M Cliff:
The Queen's College,
OXFORD, OXI 4AW, England
Department of Mathematics, Virginia Tech University, BLACKSBURG, VA 24061, U.S.A.
L Collatz:
Eulenkrugstrasse 84, 2000 HAMBURG 67, Germany
D L Colton:
Department of Mathematical Sciences,
University of
Delaware, NEWARK, Delaware 191711, U.S.A. P A Deift:
Courant Institute of Mathematical Sciences, 251 Mercer Street, NEW YORK, NY 10012, U.S.A.
XV
J Donig:
Technische Hochschule Darmstadt, Fachbereich Mathematik, Schlossgartenstrasse
N X Dung:
7, D6100 DARMSTADT, West Germany
Texas Tech University, Department of Mathematics, LUBBOCK, Texas 79409, U.S.A.
M S P Eastham:
Department of Mathematics, Chelsea College, Manresa Road, LONDON, SW3 6LX, England
Elbert:
Mathematical Institute, BUDAPEST, Realtanoda-U 13/15, H-IO53, Hungary
D E Edmunds:
School of Mathematical & Physical Sciences, University of Sussex, FALMER, Brighton, BNI 9QH, England
W D Evans :
Department of Pure Mathematics, University College, P O Box 78, CARDIFF, CFI IXL, Wales
W N Everitt:
Department of Mathematics, University of Birmingham, P.O. Box 363, Birmingham B15 2TT, England
P C Fife:
Department of Mathematics, University of Arizona, TUCSON, Arizona 85721, U.S.A.
J Fleckinger:
Universit~ Paul Sabatier, UER MIG, 118 Route de Narbonne, 31062 TOULOUSE, Cedex, France
C T Fulton:
Florida Institute of Technology, Department of Mathematical Sciences, MELBOURNE, Florida 32901, U.S.A.
I M Gali:
Mathematics Department, Qatar University, P O Box 2713, DOHA, Qatar, Arabian Gulf
R C Grimmer:
Institut f~r Mathematik, Elisabethstrasse iI, A-8OIO GRAZ, Austria
D B Hinton:
Mathematics Department, University of Tennessee, KNOXVILLE, Tenn. 37916, U.S.A.
F A Howes:
Department of Mathematics, University of California, DAVIS, CA 95616, U.S.A.
H G Kaper:
Applied Mathematics Division, Argonne National Laboratory, ARGONNE, Iii 60439, U.S.A.
R M Kauffman:
Department of Mathematics, Western Washington University, BELLINGHAM, WA 98225, U.S.A.
R E Kleinman:
Department of Mathematics, University of Strathclyde, Livingstone Tower, 26 Richmond Street, GLASGOW, G1 IXH, Scotland
XVI
H W Knobloch:
Mathematisches Institut, Am Hubland, D-8700 W~RZBURG, Germany
I W Knowles:
Department of Mathematics, University of Alabama in Birmingham, BIRMINGHAM, Alabama 35294, U.S.A.
M A Kon:
Department of Mathematics, Boston University, BOSTON, MA 02215, U.S.A.
A M Krall:
McAllister Building, Pennsylvania State University, UNIVERSITY PARK, PA 16802, U.S.A.
K Kreith:
Mathematics Department, University of California, DAVIS, CA 95616, U.S.A.
M K Kwong:
Department of Mathematics, Northern Illinois University, DEKALB~ Illinois, U.S.A.
R T Lewis:
Department of Mathematics, University of Alabama in Birmingham, BIRMINGHAM, Alabama 35294, U.S.A.
L L Littlejohn:
Department of Mathematics, Computer Science & Systems Design, The University of Texas at San Antonio, SAN ANTONIO, Texas 78285, U.S.A.
S-O Lond6n:
Institute of Mathematics, Helsinki University of Technology, ESPOO 15, Finland
J Mawhin:
Institut Mathematique, Universit~ de Louvain, Chemin du Cyclotron 2, B-1348 LOUVAIN LA NEUVE, Belgium
A C McBride:
Department of Mathematics, University of Strathclyde, Livingstone Tower, 26 Richmond Street, GLASGOW G1 IXN, Scotland
P A McCoy:
Department of Mathematics, U S Naval Academy, ANNAPOLIS, Maryland 21402, U.S.A.
J R McLaughlin:
Department of Mathematical Sciences, Rensselaer Polytechnic Institute, TROY, New York 12181, U.SgA.
J B McLeod:
Wadham College, Oxford University, 24-29 St Giles, OXFORD, OXI 3LB, England
A B Mingarelli:
Department of Mathematics, University of Ottawa, OTTAWA, Ontario, KIN 9B4, Canada
S E A Mohammed:
Department of Pure Mathematics, University of Khartoum, Khartoum, the Sudan
XVH
P A Morales:
Depart~ment de Math~matiques et d'lnformatique, Universit~ de Sherbrooke, SHERBROOKE, Quebec, JIK 2RI, Canada
M N M Nakao:
Department of Mathematics, Heriot-Watt University, Riccarton, Currie, EDINBURGH, EHI4 4AS, Scotland
F Neuman:
Mathematical Institute, Czechoslovak Academy of Sciences, Jan~ckovo N~m 2a, 66295 BRNO, Czechoslovakia
D Pascali:
Fachbereich Mathematik, Technische Hochschule Darmstadt, Schlossgartenstrasse
D Race:
7, D-61OO DARMSTADT, West Germany
Mathematics Department, Witwatersrand University, Jan Smuts Avenue, JOHANNESBURG, South Africa
A G Ramm:
Department of Mathematics, Kansas State University, MANHATTAN, Kansas 66506, U.S.A.
L A Raphael:
Howard University, Department of Mathematics, WASHINGTON, DC 20059, U.S.A.
K A R Rautmann:
Department of Mathematics, The University, Warburger Strasse iOO, D-4790 PADERBORN, West Germany
T T Read:
Department of Mathematics, Western Washington University, BELLINGHAM, Washington 98225, U.S.A.
H R~h:
Department of Mathematics, Heriot-Watt University, Riccarton, Currie, EDINBURGH, EHI4 4AS, Scotland
B P Rynne:
Department of Mathematical Sciences,
The University,
DUNDEE, DDI 4HN, Scotland D A S~nchez:
Department of Mathematics, University of New Mexico, ALBUQUERQUE, NM 87131, U.S.A.
R A Saxton:
Department of Mathematics, Brunel University, UXBRIDGE, Middlesex, UB8 3PH, England
P W Schaefer:
Department of Mathematics, University of Tennessee, KNOXVILLE, Tennessee, U.S.A.
K Seitz:
Technical University of Budapest, iiii Muegyetem, RAKPART 9, H. EPo V EM. 5, Hungary
J K Shaw:
Department of Mathematics, Virginia Tech University, BLACKSBURG, VA 24061, U.S.A.
B D Sleeman:
Department of Mathematical Sciences, The University, DUNDEE, DDI 4HN, Scotland
XVIII R A Smith:
Department of Mathematics, University of Durham, South Road, DURHAM, DHI 3LE, England
K $oni:
Department of Mathematics, University of Tennessee, KNOXVILLE, Tennessee, U.S.A.
E Stephan:
Fachbereich Mathematik, Technische Hochschule, Schlossgartenstrasse
R L Sternberg:
7, D-6100 DARMSTADT, West Germany
Office of Naval Research, Bld 114 Section D, 666 Summer Street, BOSTON, Mass 02210, U.S.A.
A Z-A M Tazali:
The University of Mosul, College of Science, Department of Mathematics, MOSUL, Iraq
Tung Chin-Ehu:
Graduate School, Chinese Academy of Sciences, P 0 Box 3908, BEIJING, P R China
L Turyn:
Department of Mathematics & Statistics, University of Calgary,
A Vanderbauwhede:
CALGARY, Alberta, T2N IN4, Canada
Instituut voor Theoretische Mechanica, Rijksuniversiteit Gent, Krijgslaan 281-$9, B-9000 GENT, Belgium
R Villella~Bressan:
Istituto de Analisi E Meccanica, Universit~ di Padova, Via Belzoni 7, 35100 PADOVA,
J Walter:
Italy
Institut fur Mathematik, Templergraben 55, D-5100 AACHEN~ Germany
W L Walter:
Mathematisches
lnstitut I, Universitgt Karlsr~he,
Kaiserstrasse 12, D-7500 KARLSRDHE, Germany A D Wood:
School of Mathematical Sciences,
N.IoH.E., Ballymun Road,
DUBLIN 9, Ireland S D Wray:
Department of Mathematics, Royal Roads Military College, F.M.O., VICTORIA, British Columbia, V05 IBO, Canada
E M E Zayed:
Department of Mathematics, Zagazig University, Faculty of Science,
A Zettl:
ZAGAZIG, Egypt
Department of Mathematics, Northern Illinois University, DEKALB,
Illinois, U.S.A.
On the asymptotic behaviour of the Titchmarsh-We[l
m-coefficient and th 9
spectral function for scalar second-order differential expressions. F. V. Atkinson Summary: It is shown that a modified version of the Stieltjes inversion formula for Nevanlinna-type functions, of the Titchmarsh-Weyl
function
together with improved estimates
m(k ), constitute an effective approach
to the problem of finding approximations to spectral functions for the case of
- y" + q(x)y =
CHAPTER I. 1.1.
~ y,
0 ~
x -< b ~
cO .
Preliminary discussion and definitions.
Introduction. We discuss here the asymptotics of two functions, both basic to the
theory of boundary-value problems for - y" + q(x)y = here
q(x)
~ y ,
0
<
x <
h
~
c~9 ;
(1.1.1)
is to be real-valued, and should satisfy certain integral
conditions, specified later in
~3-2.
The fundamental roles of these
functions are illustrated by the variety of ways in which they may be introduced. Partipular ways, presented in outline only, might be the following. We denote by
~(o,~)
:
o, ~,(o,~)
For the Titchmarsh-Weyl one function
m(k),
@, ~
: -l,
solutions of (i.i.i) determined by
¢(o,k)
: i,
m-coefficient,
¢ (o,~)
: o .
(i.l.2)
we claim that there is at least
defined and holomorphic in the open upper half-plane,
with the Nevanlinna property Im
m(~ ) > 0
if
Im ~ >
O,
(1.1.3)
and such that i l@(x' X) + m ( X )~(x,X )I 2 dx o and in particular such that @ + m~
Im m ( i ) / I m k 6
L2(O, h).
,
(1.i.4)
For the spectral
function, we claim that there is at least one non-decreasing function -C(t)
from the real line to itself, such that if
f
~L2(O,b),
g(k)=
If(x)~(x,k)dx,
o then
(1.1.5)
la2 i 11 f
dx :
g21 d ~
o
.
(i.1.6)
"00
The asymptotics of these functions, of and of
-~ (t)
m( ~ )
for large positive or negative
for large complex ~ t , have tended to be
developed separately, at widely different times. In the case of the basic results were o b t a i n e ~ i n
the early
,
~C(t),
50's by Mar~enko (17~, [l~),
and developed by him and by Levitan ( [15],~6~ ). So far as leading terms are concerned, it is known that "U(-~)
~
-00 ,
In the case of
-C(t) -~J 2-~ -I t½
as
t --~
+~.
(i.1.7-8)
m ( k ), the leading term was obtained much later by EveritZ,
in [6], who showed in particular that m(k ) ~ as
k--~Oo
i k -½
(1 1°9)
in any sector of the form 0 -~
~
~
arg k
~
7"~ - 6
.
(1.1°lO)
Earlier order-results of this nature were found by Hills (see [3],LlO3 ) Results of the forms (1.1.7-8) and (l.l.9-10)
are, however, not
independent, being linked by formulae such as (see e.g. [20] ) m(~)
=
~ (t-
in cases of non-uniqueness,
~
)-ld-c(t)
,
Im k ~
O;
we must here have for each
(1.1.11) m
the appropriate
choice of -'C . It is clear that information such as (1.1.7-8) can be fed into the right of (1.1.11), so as to obtained results such as (1.1.9-10). It is ho, ever our purpose here, following a suggestion of Everitt, to explore the opposite path. We show that (1.1.7-8) and certain refinements can be obtained from developments of (1.1.9-10). Stieltjes inversion formula (see e.g. f.
~(t2 ) -
_~L(tl) = ~ - i
lid
As is well-known,
the
~_5],[l~)
I?IIm m(t + i~ ) dt ,
(1.1.12)
valid except possibly at points of discontinuity of -C(t), shows that is essentially determined by
"qD
m • However this formula cannot be applied
for asymptotic purposes in the present setting; the limiting process called for in (1.1.12) is specifically excluded by (1.1.10).
We attack this problem on two fronts. Firstly, we prove a more quantitative vei'sion of (1.1.12), not involving any limiting process. This is the subject of Chapter If,and utilises t~he Nevanlinna property (1.1.3), without specific reference to differential equations. Secondly, in Chapter III, we extend the range of formulae such as (1.1.9) beyond (1.1.10). Applications to the proof of results such as (1.1.7-8), and refinements thereof, will be the subject of Chapter IV. 1.2.
Constructive definiticns of
m
and
~
•
The characterization (1.1.4) of a class of functions such as
m(~ )
is not the only way of introducing them. We rely in the sequel on the following construction. For any
X g
(0, b)
and any
~
we define
m(~,~, X)= - I @(X,~)-~@'(X,X )I/{~(X,~)-~'(X,X)/. For fixed
X, ~
, this will be a merom~rphic function of
poles will be the eigenvalues of the problem In particular, if
~ = tan 2
are the roots of
~(X, ~) c o s #
will be the poles of the residue at
, where
~n
7
is
~
, whose
~(X, ~) - ~#'(X, ~) = O.
is real, and ~ n = ~ n ( ~ ' X)
~'(X, ~) s i n /
m(~ , t a n ~
(1.2.1)
= 0 , then the
A n
, X); a ~imple calculation shows that I i~
}-i
2(x,~n ) dx
.
(1.2.2)
o
One is then led to a representation such as (1.1.11), namely l
m(~ , tan~,
X) =
J (t - ~ )-l d ~ ( t , #
, X),
(1.2.3) l
where
~(t,~,
X)
is a step-function with jumps (1.2.2) at the
An .
Actually, we will make only incidental use of (1.2.2), but will rely heavily on (1.2.1), together with the nesting-circle property. The logical order adopted here will be that for any fixed ~ with
Im~
X g
(0, h), and
O, (1.2.1) defines a map of the real
a circle located in the upper half-plane, to be denoted by where ~(X)
k =
~½
; for any
X
~-axis to C(X, k)
~ (0, b), we can thus define a family
of functions m(~ ), holomorphic in the open upper half-plane,
and such that
m( ~ ) ~ D(X, k)
for
Im~ ~
O, where
D(X, k)
is the
closed disc bounded by
C(X, k). The family
~
(X)
will include the
left of (1.2.3) as special cases, and also many others, such as convex linear combinations of (1.2.3) for various y will include ~ ( X ' )
for any
intersection of all such
X' ~
~(X'),
. More importantly,
(X, b), and so also the common
in particular the unique
associated with the limit-point case. Again for any elements of
~L(X)
~(X)
X ~
m(~ ) (0, b), the
will be functions of Nevanlinna type and will admit
a representation (1.1.11); with the formulation based on (1.1.2) and the hypotheses of ~ 3 . 2 ,
we can use the form (1.1.11) rather than the more
general form discussed in Chapter II (see ~3~, present point of view, as in
p. 91). Th~s, f~am the
~3~, the various spectral functions are
seen as derived objects, defined via the various
m-coefficients, rather
than by means of (1.1.5-6) or (1.2.2). Even though the latter can he used to derive asymptotic information about spectral functions~ as for example i m ~ 4 ~ , we shall here derive such information exclusively from similar information about
m-ceefficients.
1.3. Variation in imitial dqta. The above formulae are of course heavily dependent on the choice (1.1.2) of initial data. However a variation in this choice simply occasions a linear-fractional transformation in the consequent change in
m
(see ~3], P. 66);
"I~ is less easy to visualise. Here we note
only the case when, as in ~ 5 ~ - ~18~ , the data for ~(0, k ) with real finite
h
= l, •
~
are replaced by
~'(0, ~) = h ,
(1.3.1)
The effect on (1.2.1) is then to replace
m/(1 + hm). Assuming (l°l°9) to hold with error
m
by
o(~ -i), which will in
fact be the case, we will then have that
m : i~ -½ + h ~ -I + o(A -1), subject to (i.i.i0), and possibly elsewhere. As will be noted in this leads to an additive term spectral function.
- h
(1.3.2) 4.6,
in certain approximations to the
1.4.
Further remarks. An extensive account of the
m-coefficient, together with a full
survey of the literature, is to be foun~ in the recent monograph E3~ of Bennewitz and Everitt. In the case of the vibrating stning a full theory, dealing also with spectral functions in detail, is given by Kac and Krein in L14~, where the letter
"~"
is used in place of
"m" .
Asymptotics of the spectral function for generalized second-order operators have been developed by Kac
~
such as (1.1.11), asymptotic results for
; these imply, by way of relations m
subject to (1.1.10), though
only so far as leading terms are concerned. Matrix extensions of
m-coefficients are of current interest, importamt
contributions to the basic theory being due to Hinton and Shaw ~ l ~
; see
also their recent paper ~8~ with Everitt. To Everitt is due a remarkable connection between the hehaviour of the m-coefficient and the constants appearing in certain integral inequalities which extend classical results of Hardy, Littlewood and others; we cite i~ this c o ~ e c t i ~ n ~3~, ~9~ , and the recent article ~5] for references. 1.5.
Acknowledgements. It is a pleasure to acknowledge many stimulating discussions on these
matters with Professor W. N. Everitt. Appreciation is also expressed for the support of the Science and Engineering Research Council of the U.K., for the hospitality of the Department of Mathematics of the University, Dundee, and for the continuing support o~ the National Science and Engineering Research Council of Canada, under Grant A-3979. CHAPTER II. A modified Stieltjes inversio~ formula. 2.1. Nevanlinna functions and their representations. In this chapter we are concerned with functions holomorphic in the open upper half-plane, satisfying (1.1.3), but not necessarily arising from constructions such as (1.2.1). We have the general representation (see e.g. ~3~' ~
a~
~3]
for discussion and references)
m(~)
where
= A + B~
A , B
+
~{(t
are real and
- ~ )-l
B ~
t(1 + t2)-lld-C(t) ,
O, and
qS(t)
(2.1.1)
is real-valued and
non-decreasing, and such that oo I (I + t2) -I d-~(t)
The function ~ ( t )
~< c~0 .
(2.1.2)
may be rendered unique by fixing its value at some
point, and by fixing its value at points of discontinuity in some way, for example by right-continuity.
In any case we have (1.1.12), except possibly
at points of discontinuity. If ~(1 the term
)-i d'C(t)
+ itl
t(1 + t2) -1
satisfies the stronger condition
oO
~ < ,
(2.1.3)
in (2.1.1) can be dropped, its contribution being
absorbed into the constant we can set
mD(t)
A ; if additionally
m(i-~ )~-~ 0
as z~-~ c~o ,
A = B = O, thus arriving at the form (1.1.11). The replacement
of (2.1.1) by (1.1.11) when (1.1.9) holds is discussed in ( [3], P- 91). Our purpose in this Chapter is to replace (1.1.12) by a result with specific bounds instead of limits. 2.2.
A quantitative inversion formula.
Theorem 2.2.1.
I Im
Let
-~0
m(~ )dk ~ 7[ 8
~
A 1
"~" [~(~
I Im m ( A l
+ i~)
~
A
2
~
O0
~ ~
~
+ Im m(
O° Then
+i
A2+ i~ )I
"
(2.2.1)
Proof: on the left of (2.2.1), the integral may be taken along any path joining the two limits in the upper half-plane. For the purposes of the proof we suppose it taken along a straight line, so that real, and the
"Im"
"dk "
will he
may be taken under the integral sign, as in (1.1.12).
We prove the result first for the special form (1.1.11), with the additional restriction that -C(t)
should be constant outside some finite
interval. Integrals occurring in the following manipulations will then be absolutely convergent. We have
Im m(1)d~ =
d"C(#)
f(~ ,#)dk,
where
+ & 21 -i . We note that
~0
I
f( k , ~ )dk
= "W
and introduce the function
•A
I
c~
I
Then, as is easily verified,
"I
[K'-I~(A) + ~(A)I
Im m(~ )d k =
•
--I
Thus, in order to complete the proof of (2.2.1) for this special case, it will be sufficient to prove that IF(A) -
&Re
m( A
+ i~) 1 ~
Now
½~
~
im m(
A
+ i~).
(2.2.2)
oO ~e
m( A + i@) =
_]o0(7~ - A )f( A ,/~ )
~ ~(#),
and so the left of (2.2.2) equals
I d-~(#)I
f(k,~) d\ - (#- A )f(A,#)
-
-00
-
)
f(X ,h) d X - ( A
We write, for ~-~ ~
O,
h(O--) :
-/~)~(A ,~
.
(2.2.3)
oo
I
~F-(1 + 0--2)-1
(1 + t2) -1 dt -
~-~ and can then replace (2.2.3) by
tan -I G--
-
~-(1 + ~-2)-i
(2.2.4)
(2.2.5)
We claim next that
0 ~
h(O--) ~
½ l-T(l + ~ 2 ) - i
• The first fact
follows from (2.2.4). For the second inequality we introduce the functio~
j(~-)
=
h(~-)(l + O -2)
j'(O--) ~
0
if
, and observe that
o-- ~
j(O) = ½~-Y , and that
O. Thus (2.2.6) is bounded by
--o0
and th±s, in view of (1.1.11), ccmpletes the proof of (2.2.2), and so of Theorem 2.2.1 for this special case. It is a simple matter to extend the result to the general case. We remark first that (2.2.1) is unaffected if to constant and
m(~)
we add any real
A, and so we can apply the result to the form (2.1.1) if
~C~(t)
is still required to be constant outside some finite interval.
In the next stage we remove the latter restriction. t that with in
~(t)
Tl~
IT1, T ~
T2~
c~D , we define
and constant in
define an associated function (2.1.1) and replacing
"IL(t)
apply Theorem 2.2.1. Making
-U~t; T1, T2)
(- c~, T17 and in m ( ~ ; T1, T2) by
T1- ~
~2
that matter for any real
). We then
A = B = 0
in
-g(t; T1, T2) , and to this function - 0~, T2~-~
+ ~o
then yields the A = B = O, or for
A •
Finally, we must allow for a positive B~
T1, T 2
as equal to-c(~)
,oo
by taking
required result for the general case of (2.1.1) with
unaffected,
We assume now just
is non-decreasing and satisfies (2.1.2). For any
.~o ~
such a term
B = 0
B
in (2.1.1). In fact, adding
to the previous case leaves the left of (2.2.1)
on account of cancellati~n between the first and third terms,
while the right is increased. This completes the proof of Theorem 2.2.1. 2.3. A weakened form of the inversion formula. Although the full form of Theorem 2.2.1 was needed for its proof, our
application will treat the term in
Re m ( ~
+ i~ )
as an error term. If we
combine this with the right of (2.2.1) we get a simpler result, namely Theorem 2.3.1.
Under the conditions of Theorem 2.2.1, we have
I
.<
(7~-2 + ¼) ½ ~
{Im(~l
+ i ~ ) 1 +Im( ~2 + iS)If.(2.3 .1)
A convenient bound for the numerical factor on the right v~ll be CHAPTER III.
Asymptotics
of
0.6.
m(~ ).
3.1. Preliminary remarks. We now deal with the other main component of the investigation, improving
(1.1.9-10);
that of
since (1.1.7-8) imply somewhat more than (1.1.9-10),
it is clear that the latter cannot serve to prove sharpened forms of (1.i.7-8). The reasoning of this chapter is very close to that of modified for the present purpose. ,
and, with D(X, k)
k
:
~+
i
C(X, k), D(X, k)
We write, without loss of generality~
,
where
as in
N ) O ,
> O,
(3.1.1)
~ 1.2, obtain a bound for elements of
hy combining a bound for some element of
for its diameter. Bounds for
~2J,
D(X, k)
D(X, k)
with a bound
will then automatically
be bounds
also for
o(x,k)
D(b, k) = provided that ~ > 3.2.
x
C (0, b),
(31.2)
O.
Hypotheses on
q
.
In addition to the reality of (i)
over
q ~ L(bl, b 2)
for all
q
bl, b 2
, we assume: with
0 ~
hI ~
b2 ~
b,
(i~) the limit Q(x)
exists and is finite, (iii) the function
:
=
lim t-~o
I~ q(s)ds
(3.2.1)
10
Q(X, ~ ) = with "sup"
(3.2.2)
over the domain
0 ~x~X, satisfies,
I e ki(x-t) q(t)dt ,
sup
Im k ~/ 0 , Ikl ~
for each fixed
0
(3.2.3)
(0, b),
X
G(X,~ ) -~
~ ,
as
~ ---> c~D •
(3.2.4)
The above hypotheses will certainly hold under the usual condition q ~
L(0, b')
for all
b'
~
(0, b).
(3-2.5)
However we do not need to assume this; as an example in which (i) hold, but not (3.2.5) we cite the case
(iii)
q(x) = x -1 sin (x -1) .
We verify briefly that (i), (ii) are sufficient to ensure the existence of
@, #
define ~(x)
~
satisfying the initial data (1.1.2). = ¢'/¢, we shall have
= Q(x) - ~ x - l ~ 2 ( t ) d t .
~,
If
= q _ ~ _ ~2
= 0, and then define
does exist, and we
, %(0)
= O~ and so
We can reverse this argument and claim
that this integral equation has a solution ~ ( x ) X
#
~(x) = exp
~(t)dt,
in a neighbourhood of @(x) = - ¢(x)
~ dt/¢2(t)
.
Th~se solutions can be extended to the right by means of standard theorems. 3.3-
An auxiliary function. This will provide an element of
will be based. We assume with however ~
X ~
D(X, k)
on which the approximation
(0, b), and that
O. We denote by
z(x) =
z(x, X)
, k
are as in (3.1.1),
the solution of (1.1.1)
such that z(X) = l, z'(X) = ik . We note that
(z ~' - z' ~)' = 2 i c ~ ~ z ~ , so that
non decreasing. Since this function is negative at have that
z, z'
do not vanish in
(3.3.1)
Im (z ~' - z' ~)
is
x = X, hy (3.3-1), we
~0, X~. We can therefore define
n ( ~ , X) = - z(O)/z'(O)
.
(3.3.2)
Combining these observations with remarks from ~2J, p. 347, we have Lemma 3.3.1.
For fixed
X C
(0, b~ , the function
n ( ~ , X)
is regular
11
and of Nevanlinna type in
I
m-
3-4.
n(~,
Im ~
X) I
O. Any
m C D(X, k)
f S
~/
2<~
1
I ~ i 2 dx
o
Approximatio~ to
satisfies (3.3.3)
.
n( ~, X).
Again as in ~27, we have recourse to a Riccati equation technique. Lemma 3.4.1.
Let there he a number
such that, with ,
o
~
xI ~
k
x2 g
as in (3.1.1), x ,
(3.4.1)
•-~LI and also
~, ~'
< Then
n(~,
X)
n(X,
x) - i~ - ~ -
.
(3.4.2)
is defined, and
~-2
e2~it q ( ~ d t
~
8~2J~1-~
-1 •
(3.4.3)
o Here
~/0,
the
finiteness
instead of ~
of
O
n ( ~ , , X)
as in
is
~ 3.3.
noted
only
since
we n o w a l l o w
As in [27, we write
v = ik - z'/z
and have the integral equation
~<x> ~\~<xt~iv~
~<~< from which, together with (3.3.1), we have
Iv(x)l ~
~,~ ~-
~/~ , o
_~ ~< ~x,
(3.4-4)
and so Iv(O) Arguing as in
Ioe2kitq(t)dtl ×
~
2~2~
-1 •
(3.4.5)
[2~, p. 350, we have
~(k, x) : ~(o~
ik I i
: i~i + v~o)~ 2 + ~,
say, where
i,~t_~/v~
tv~]-~ ~
~,,~>,~
I~
We then obtain (3.4.3) on combining this with (3.4.5-6). We use (iii) of ~ 3.2 to bring about (3.4.2). We can take
~ : 2~(x, I~I)
(346)
and then (3.4.2) will hold if
~(x, (~<1) < In particular,
we may derive
l~i~-~.
(3.4.7)
12
Lemma 3.4.2.
For any
X~"
(0, b)
such that for real ~ * ( X ) real-valued,
there is a real
the function
~ *(X)6
n( ~, X)
(-OO , 0)
is analytic and
and so can be continued into the lower half-plane.
Considered as functions of ~ can be continued analytically
, for fixed
X , z(O, X)
and
z'(O, X)
from the upper half-plane across the
negative real axis, and so this is true also for
n(~ , X)
so long as
it is finite. This is ensured by Lemma 3.4.1 if, as in (3.4.7)~ ~ and
G(X, ~ ) ~ ? / 1 6 .
That
n(~,
X)
is real-valued,
= _ ~2
if finite, when
is real and negative follows from the reality in that case of (1.1.1) and the data (3.3.1). 3.5-
Estimation of
~
•
To deal with the right of (3.3.3) we need a lower bound for Lemma 3.5.1.
¼ e ~/x
_
(ll/8)e -~Fx ,
~f=
It will be sufficient to prove this when is independent
of
remain true when ~
.
Let the conditions of Lemma 3.4.1 hold. Then
l~(x)l ~
express
~
X, and the hypotheses X
~
- 2~,
0 ~x~
X.(3.5.1)
x = X; the definition of
(3.4.1-2), if true for some
# X ,
is replaced by any lesser positive number. We can
in terms of
z
by
DC
~(x) = z(x)/z(O) - z(x)z'(0) ~ d t / z 2 ( t )
$
and so have
X
OIo,Z I L, ot f
From (3.4.4)we
have that ~(x) 1 ~
Re ( z ' / z ) ~ exp ~ / ( X
- ~,
and so
- x)~ ,
(~.5.3)
which deals with the last term in (3.5.2). For the first term on the right of (3.5.2) we set
l
where I 1 = (2ki) -1
=
I 1 + 12
,
2z'(t) dt/z3(t) = (2ki) -1 ~z-2(0) O
and
odt/z2(t)
- l~,
13
12 = (ik) -1 Io v(t) dt/z2(t) In
I1
we use (3-5.3), which yields
l~l/>In
12
.
~-~l~'~)).
½1kl-l(1- exp
(3.5.4)
we use (3.4.4) and (3.5.3), to get
×
I~l ~-~
~ I~-~ I ~x~-~ '~-x~ ~ ~ ~ l -~, o
f
since
~ >/ ~ ~ • Combining this with (3.5.4) we have
× 2 I ii:,z
1-1 12k1-1exp(~yx)
(3.5.5)
We note also that
1,'(o)1 : Iz(o)l I z,(o)/zCo)l ~l~(o~l(Ikl-
Iv(O)l)~ ~ /~lox~(~x),
and so
I~,(o) l dt/z~(t) I ~
, e~p(~'x)- ~ ex~(-~'X) •
o
Using this in (3.5.2) along with (3.5.3) we get (3.5.1) with
x = X,
as r e q u i r e d . We now use this on the right of (3.3.3), Lemma 3 . 5 . 2 .
Under the c o n d i t i o n s
I~rid~ >~ We use the remark that if aI ~
a 2 - a 3 , then
al2 ~
al,
to get
o f Lemma 3 . 4 o l ,
~6 -1
we h a v e , i f
~ X ~ . 4,
ex~ (~'x) .
a2, a3
(3.5.6)
are non-negative,
and
½a22 - a32 . Thus (,3.5.1) yields
x) dx • o
o
Hence~ noting that ~ / ~
,
~fx
~/
3, we have
X
~1/~4) which establishes
ex~ (~'~)~l-
(3.5.6).
(1~3/2)e-61
,
14
3.6. The main result.
x~Co,
In what follows we will assume that
h) and
k
asi~
(3.1.1) satisfy
~ >. For fixed
4
,
~.
X , we can arrange (3.6.2) by taking
(iii) of ~3.2; in the case make
~(~,~1)
X -~c~ and also
b =
oo
(3.6.1-2) ]k I large enough, by
, an alternative method will be to
IkI -~ OO , subject to (3.6.2). Always with the
assumptions of ~3.2, we have Theorem 3.6.1. Im
-
Let (3.6.1-2) hold, and also 0(~0. Then, for any
m CD(X, k)
X 2 k i t ~q~t)dt 1 ~ 32G2(X, Ikl)Ikl-2~ -1 + 104c~-le - 2 # x ik -1 - k -2 ~joe
•
(3.6-3) In particular,
Im i ll
le 2 X.
(3.6.4)
We apply Lemma 3.4.1. It follows from (3.6.1-2) that ~ 2G(X, [k I) ~
¼X -1
~
#/16 ,
(3-6-5-7)
so that the requirement (3.4.2) is satisfied. We then combine (3.3-3) and (3.4.3) to get a result of th~ form (3.6.2), with the right replaced by
i i-2 -1 + 3 - exp where the last term comes from (3.5.6). We now have to replace these two terms by those on the right of (3.6.3). For the first term in (3.6.8) we use (3.6.5). In the case of the last term we note that
2 x÷4 x by (3-6.5-6), and that
38e ~104.
2#X+l This proves (3.6.3).
For the simpler but weaker form (3.6.4) we note that Ik-2 I~ e2kit q(t)dt I ~
Ikl-2 G(X, Ikl) ~
~Ikl-2 X - 1
by (3.6.6), and that the first term on the right of (3.6.3) satisfies the same bound~ again by (3.6.6). This completes the proof of the Theorem. For
~
near to the negative real axis we use the simpler
Theorem 3.6.2.
Under the hypotheses of The Jrem 3.6.1,
15
Im- n ( ~ ,
X) I ~
1 0 4 ~ -1 e- 2 ~ X
.
(3.6.9)
This follows from (3.3.3) by the same arguments. 3.7- Extension of some results of Hille and Everitt. In this section we apply the results of ~3.6 with fixed to obtain bounds which will apply to any
m
X ~ ( 0 ~ b),
g D(X', b), X ~ X' ~
also to points in the limit-disc o~ limit-point
b, and
D(h, k) (3.1.2). We take
first the Hille order result corresponding to (1.1.9), which will serve for the right of (2.3.1). It is convenient to separate the cases of the firat and second quadrants, more precisely those of Re ~ 0 ,
Im ~ ~
0 ,
(3.7.1)
R e ~ ~ O, Im ~ ~ O.
(3.7.2)
All the following statements refer to Theorem 3.7.1.
m C D(X, k) with fixed
X,~-~.
We have, in (3.7.1), m = O(Ik1-1) ,
(3-7.3)
provided that
i~ ~
>i 4x -I .
This requires, of course, that
(3.7.4)
~ should lie above a certain parabola
with focus at the origin and vertex on the negative real axis~ it yields an essentially larger region tham the sectorial requirement (1.1.10). Theorem 3.7.2.
In (3.~.2) we have that (3.7.3) holds, provided that lim inf
(Im ~ )I~1-1 exp (2X I~l½) ~
Thus in the second quadrant
O.
(3.7.5)
~ may come~ in a sense, exponentially
close to the negative real axis; this is linked to exponentially small behaviour of ImPel,
~(t) - -6(- O~).
so that
K-1
= O(I~I~),
in (3.6.4) is neglgible. If again ~-'
=
For the proof we suppose first that
O( I ~ I ½ lira ~I -1) ,
~ 0 ~
I~ ~
, and the last term
Im ~ ~
~ = ~I ½
and so we are led to the requirement (3.7.5)~
l, we have
+ o(1) ,
16 We consider next the problem of extending the range (1.1.10) of the Everitt asymptotic formula (1.1.9). It is only necessary to strengthen (3.7.4-5). Still with fixed Theorem 3.7.3.
X
we have
We have m ~ ik -1
as ~ - ~ O O in the first quadrant Im ~ ½
(3.7.6)
if also -~
O0
,
(3-7.7)
and in the second quadrant if (Ira ~ ) I ~ 1-1 exp (2XI~ 1 ½ ) ~ In the case of
n(~ , X)
(~0
(3-7.8)
.
it is not necessary to stay away from the
negative real axis. We have easily from (3.4-3) Theorem 3.7.4.
As I @ °0 in the region (3.7.2), we have
n(~, 3.8.
The case
b = ~
x) = ik -1 + o(ikL-2).
(3.7.9)
•
As indicated in ~ 3.6, it seems advantageous here to make Ikl tend to and
00
together. We determine X(~)G(X(~),
~ ) ~
For example, we may determine
X(~ )
X(~ )
and
X( ~ )-~ oo
~ •
(3.8.1)
for large ~
The resulting bounds will then apply to limit-disc. In the inequalities of ~
subject to
X
by equality in (3o8.1).
D(O~ , k), the limit-point or
3.6
we take
X = X(Ikl), so that
(3.6.2) holds, and in view of (3.6.1) must have (Im k)X(Ik I) ~ For example, let
q(x)
)
(3.8.2)
4.
be monotone, with
lq(x)/?l
q(0) = 0, so that
•
(3.8.3)
We then may require that (3.8.4-5) More specially, let us take q(x) =
xN ,
N~
O°
(3.8.6)
17
Choosing equality in (3.8.5) then yields X(~ ) = ( ~ ) l / ( N + l )
•
(3.8.7)
Subject to (3.7.4), these choices for
X(Ikl)
can be inserted
on the right in the estimates of Theorems 3.6.1-2, 3.7.1-2, with the result applying to any of
X(Ik I)
m C
D(~
, k). We can also use this choice
on the left in Theorem 3.7-4. We can also modify Theorem
3.7.3 in this sense. For example, in the special case (3.8.6), we would have (3.7.6) as
A "-~ ~
in the first quadrant, provided that
l~I 1/(2N+2) Im
~ ½
-~
~
.
CHAPTER IV. Estimates for the spectral function. 4.1.
Preliminaries. We now put together the results of the last two chapters in order to
arrive at approximations to
~(t),
real line appearing in (1.1.11),
the non-decreasing function on the
(1.2.3) and elsewhere.The bounds of the
last chapter were for arbitrary points
m g
D(X, k), or in
D(h, k)
or
D(OO,k) as the case may be. We will now confine attention to the situation that
m = m(~ )
is an analytic function, from the open upper half-plane
to itself, satisfying such inclusion
properties, with
k = A ½
as in
(3.1.1). As a Nevanlinna function obeying (1.1.9-10), or even only the corresponding Hille-type order bound,
m(~)
will (see (3), P. 91) admit
a representation (1.1.11), essentially determining the associated We use (2.3.1) and estimates for
m(~ )
to estimate
~(t).
~(t).
As in Chapter III, there are two procedures. We may use bounds for m(~)
satisfying
m(~ ) g D ( X ,
will then apply to any
"~(t)
C and also to
any
~(t)
indicated in
X g
associated with an
for some
associated with an D(~
b =C~O • Again, if ~
with fixed
o(x,
m( ~ ) ~ if of course
k)
, k), h =
3.8, in which we make
m( ~ )
X g
m(~)
(0, b); the results satisfying
IX, b),
C4.1
satisfying (4.1.2)
Cx0 , we can use the procedure X
~
c~O, so as possibly to
18
obtain better results for the case (4.1.2). We assume throughout the hypotheses (i) - (iii) of 0 3 . 2
concerning
q(x) . Our results are consistent with the standard ones for the case that
q
satisfies the more special requirement
4.2.
Estimation of
~(t)
- -C(- ~D) as
t -~
(3.2.5). - ~
•
It is a question of showing that this is exponentially small as t ~
- O0
, and in particular that
taken to be zero).
"~(t)
to work with a fixed
(0, b).
-C(t)
is finite
Since much larger error-terms, such as
occur in the estimation of
Theorem 4.2.1.
~(-GO)
X
~
as
t-~
Under the assumptions of
(and could be O(1)
or
o(1),
+ O~ , it will be sufficient
~3.2,
for the spectral function
associated with any Nevanlinna function satisfying (4.1.1), we
have that
-C(-O0 )
-~_(t) -
~
is finite and that, as
(- g~) = 0
=
- O0,
~ Itl exp (- 2Xltl ~ )~
We use Theorem 2.3.1 with
&
t -~
A 2 = t ~
-1 ,
•
~l
(4.2.1) g
(- ~ '
Itl exp ( - 2Xltl ½ ) •
We write
t), and (4.2.2)
t~ f Kl(
A
, t)=
m(~)
Im
d~
,
(4.2-3)
and then have from (2.3.1) that ~(t)
-
-g ( ~ )
=
-f~-l K1 ( A
, t) + 0 ( ~
Itl-½),
(4.2-4)
by Theorem 3.7.2, or directly from (3.6.4). We need to show that Kl( A , t) is bounded as
~
~
- ~
, and that it admits a bound of
the form appearing in (4.2.1). As an approximation to
K1
we use
t+~ K2( A , t) = Clearly
Im ~
n( ~ , X) d ~ .
(4.2.6)
19
tf(~ /~+~ We estimate this by deforming the contour so as to pass through the points
~
+ i , t + i , taking it that
On the segment re(X)-
(t + i,
t + i 6)
n( ~, X) = 0
we write
~ ~
2J~
+ i, t + i)
where
A
~
0
~ = t + i7#
~ ~
, and have (4.2.6)
I, we get an amount of order
( A
+ i&,
we use (4.2.6) with 7.2 = 1
A
+ i). On the segment
and with
t
replaced by
s,
s ~ t, to get an amount of order
Isl½ exp (-2xlsl½)
ds
=0
(Itl
+ X3)
(-
2Xltl½
/~
(4.2.1)
Hence, uniformly in 4
,
t t lox
o
Turning to
K 2 , we deform the contour in (4-2.5) so as to pass along
the real axis from 3.4.2, if
1 .
2Xltl ½ )I ; this order-bound will also apply
to the contribution of the segment ( A
is so large that
I ~2-11tl½ exp ( - 2Xltl½)I ,
by (3.6.9). Integrating over 0 ~lln~IItl ½ exp ( -
t
t
~
to
t ; this is permissible in view of Lemma
is suitably large and negative.
real-valued along
( A
Since
n( ~, X)
is then
, t), this segment of the contour contributes
zero to (4.2.5). Using Theorem 3.7.4, or (3.4.3), we see that the contributions are of orders
of the vertical segments of the resulting contour to (4-2.5) O( ~
[AI-½),
Ki( A It follows that --~
- o<3 . Since
0( ~
Itl-½). Hence, uniformly in
, t) = O( Itl½ exp ( - 2Xltl ½) .
Ki( ~ Im
, t), and so also m( ~ ) ~
El( A
A
, (4.2.9)
, t), are bounded as
0, we have that the integral in
(4.2.3) must tend to a finite limit as
~
--~
~
0<3
, and so, by
20
(4.2-4), the non-decreasing function
-~(~ )
must also tend to a finite
limit as ~ -~ - QO . We then derive the required resu±t (4.2.1) from (4.2.4) and (4.2.8-9). If we apply this result to the case
I
m(~ ) E
lexp ( ~ - Itl ½) d-C (t) ~
For the Case that
q
O0
D(b, k), we deduce that
, if
0-~ ~ 2 b
•
(4.2.10)
satisfies (3.2.5), this is a well-known result of
Mar~enko (see, e.g., the discussion in (15) (English translation), p. 193). The results apply uniformly to any
m(~)
of the Theorem. For an individual function
satisfying the conditions
m(~)
C
C(X , k), of the
form given by (1.2.2-3), the result (4.2.1) would be trivial sinae the set of points of increase of
~(t)
would be bounded below. However the
statement is non vacuous when applied to the collection of such The case
q ~
0
will serve here to show that the left of (4.2.1) can be
of order at least 4.3.
m ( ~ ).
[tl½ exp ( - 2X It I½ ).
Approximation to
-~(t)
as
t
-~
+ oo .
It is here that we meet a recognisable asymptotic behaviour, as g~ven hy (1.1.8). We first separate off the tail of the distribution by writing, for large
t
~
0 ,
-C(t)--C(-O0)
=
~-C ( - t ) -
We choose some fixed
X°
~
IZ(-~)]
+ ~(t)-
~(-t)~
•
(4-3.1)
(0, b), and use (4.2.1) to estimate the
first term on the right, and thus have -~ (-t) -
~(-o0)
provided that this
-c(t)
m( ~ ) ~ For such
= 0
~ t exp ( - 2X ° t ½)~ arises from an
D(X, k),
Xo ~
m( ~ ) X
~
(4-3-2) satisfying b .
(4-3-3)
X , we then proceed to find an approximation to the last term
in (4.3.1), though with a much large error-term than that in (4.3.2). We impose the restrictions Xt ½
~
lO ,
XG(X, t½)
~
~ ,
(4.3.4-5)
21
which will, for given assumptions of Theorem 4.3.1.
X ~
~ 3.2.
~Xo, b), be satisfied for large
t , by the
We have then
Under the above conditions l-~(t) -
-C~(-t)- 2 _~-i t½ I ~
20X-1 .
(4.3.6)
We use Theorem 2.3.1 with =
We note that
lOX -1 t ½ ~ ~t,
~
'
1
= - t,
A2 = t .
(4.3.7)
and take the contour in the integral on the left of
(2.3.1) through the points - t + i~
,
- t + it, t+ it, t + i ~ .
(4.3.8)
We need to bound a number of error-terms by numbers will in turn be bounded by various multiples of
Jl' "''' J7 ; these
X -1 .
The first two of these error-terms arise from the right of (2.3.1). Noting that the numerical factor there is less than Jr = (0.6) ~ I m ( ~ r We write J1
( - t + i ~)½ = O~ + i ~
~
+ i~)l
0.6, we set (4.3.9)
, r = l, 2.
and have, if
~ X
4, by (3.6.3),
(0.6)~ It-J + (4Xt)-l + 108~ -1 e - 2 ~ X I.
(4.3.10)
Now ~2 since
~
= ½ I (t2 + ~ 2 ) 3
t I ~
½ ~2 t-l(2½ _ 1),
(4.3.11)
t, and so I
C~/~/ t-~(0.455).
(4.3.12)
Similarly, ~2
= ~ ~(t2
+ ~2) + tl '
(4.3.13)
and so ~ in particular,
t½
;
(4.3.4) ensures that
(4-3.14) ~X
~
4, as required for (3.6.3).
Proceeding to bound the individual terms on the right of (4.3.10), we have that
~ t -½ = lOX -1
(4.3.4), and that
by (4.3.7), that ~ x - l t -1 = lOX-2t -½ ~
X -1
by
22
-i
e-2~X
~
(0.455)-1 t½ e-2Xt½ ~
(0.455)-i lOe-20 X-1 ,
by (4.3.4) ant (4.3.12). Combining these estimates we get that J1
~
In the estimation of
7X -1
J2
.
(4-3-15)
we set
c~+ i ~
= (t + i 8 ) ½
, and have
to interchange the bounds (4.3.12), (4.3.14), so that O(~t ½ , and again ~-l
~
~X ~
e-2~ X
~
~
~ t-½(0.455)
:
(4.55)X -1 ,
(4.3.16)
4. For the last term in (3.6.3-4) we thus have t-½ e-9 • J2
Hence, from a bound of the form (4.3.10),
~
7X-1 "
(4.3.17)
We now consider the contribution of the main term
ik -1
in (3-6.3-4)
to the integral in (2.3.1); we show that this approximates to the main term in (4.3.6). We write J3 = 2-g-1 I t½ - Re (t + i ~)½ + Re
-t + i ~ ) ½ I.
(4.3.18)
Using (4.3.11) we have IRe ( -t + i ~ ) ½ 1
~
½~
t-½
=
5X -1 ,
by (4.3.7) and, in a similar way, 0 ~
Re (t + i ~ )½
-t½
~
~g2
t-3/2
~
5/(4X) ,
by (4.3.4), (4.3.7). Hence J3 Next we denote by
~ J4
4X-1 "
(4.3.19)
a bound for the error arising from integrating
the first term on the right of (3.6.4) around the contour can take
-1 J4 = (4t)(4Xt)-l-f~
here
4t
(4.3.8). We
~
X -I ;
(4.3.20)
is a bound for the length of the contour, and the factor V~-1
comes from the left of (2.3.1). In the last stage of the proof of Theorem 4.3.1 we write
J5' J6
and
23 J7
for the contributions, in modulus, of the last term in
(3.6.3-4),
when integrated along the three segments of the contour (4.3.8). For we write
~ = -t + is, ~
~
s ~
J5
t, and have, using (4.3.12) with
replaced by
s , and (4.3.14), that T 108 (0.455 ~T )-l I~ tWs - I 1 exp
J5 ~
(-
2Xt ~) ds
76t ½ exp( - 2X t½) In (Xt½/10) . Since
Xt ½
~
lO, J5
Since
u2e -2u
~
~
~ 2 ~
C~g ~
Xt½ ~
O, we find that
~
l, and
Xt½~
lO, we have that
l0 -4 X -1 .
= s + it, -t ~ s ~
½ ~(s 2 + t2)½ + s~
~
(4.3.21)
t, 0( + i ~
= (s + it) ½,
½t(2 ½ - 1), and hence . Hence
(2t)(108~-1)(0.455)-1t-½
exp ~ -
(0.9)Xt ½}
(0.9)Xt½~.
10, and since
u exp ~ -
(0.9)u~
is decreasing for
10/9, we deduce that J6 Finally, for
with
u ~
7600 e-20 x - l ~
152 t½ exp ~ -
u ~
v ~
tw(0.455), and likewise for
J6 ~
Since
if
~ (7"6)x2t exp ( - 2Xt ½)} X -1 .
J6 ' we set
so that
v
is decreasing when J5
For
in v ~
s
~
1520 e-9 X -1 ~ J7
in place of
we put ~
5-1X -1 •
~ = t + is, ~
(4.3.22) s ~
t , and have (4.3.17)
• This gives b
~7 < 1°8~'1 t-½ l~exp~ (09)sXt½~ds 108w~-l(o.9)-lx -1 e-9 Our estimates for
Jl' "''' J4
smaller bounds (4.3.21-23)
foz
~
10-2X -1 • above total
J5' J6
and
J7
(4-3.23) 19X -1, and so with the we have proved (4.3.6).
24
4.4.
Asymptotic
formulae for the case of fixed
We summarize the results of ~ 4 . 2 - 3 procedures indicated in hypotheses of
m(I)
with
using the first of the two
b
~O~.
Let
XO
g
(0, b)
be fixed and, for some
be a Nevanlinna function satisfying
with
Im ~ ~
O. Let
-6(t)
T = T(X)
satisfy
XT½
xQ(x, T½)
.< ~ .
Then, for
>
t ~
-
io,
m(~)
C
X g[Xo,
D(X, k)
b)
for all
be an associated spectral function,
satisfying (i.i.ii). Let
-C(t)
Always with the
~ 3.2, we have
Theorem 4.4.1. let
~4.1~
X •
(4.4.1-2)
T(X), i I
-C (- OO) = 2 ~ -
t x + El(t) + E2(t)
(4.4-3)
,
where El(t)
~
2OX -I ,
(4.4.4)
E2(t) = O(t sxp ( - 2X o t½) •
(4.4.5)
and
Here
(4.4.5) holds uniformly in
particular,
t
---> ~
X 6
[Xo , b ). In
one has -C(t) =
as
X , for
27~-1t ½
(4.4.6)
+ O(1),
.
Looking more closely
at (4.4.3), one sees an
and an exponentially small term numerical factor
"20"
0(1)
term (4.4.4),
(4.4.5)- In the first of these, the
is of course too large; however this factor
cannot be reduced below "2" • We see this by considering the case q ~-_ 0
with boundary condition
#'(X, ~)
= O, when the jumps (1.2.2)
in the spectral function are all, except for the first, equal to For similar results, with an
O(1)
or
o(1)
error-term, obtained
by entirely different methods, we refer to (15) - (18). 4.5.
Asymptotic
formulae with an
We now concentrate on the case
o(1) b =
error-term. oO
2X -1.
, though without any
25
hypothesis as to whether the limit-point or limit-circle case holds; result will apply to an
m(~ )
satisfying
the
(4.1.2), whether this is
unique or not. Theorem 4.5.1. Let the assumptions of X = X(t)
hold, with
b = OO
• Let
he determined so that X(t)-)O0
as
t
for large
t ~
satisfying
(4.1.2) we have
-C(t) as
~3.2
-~
oo ,
0. Then for any
-C(- ~ )
t -~ O O
X(t)G(X(t),
-~Z(t)
= 2~lt½
t½)
~
~ ,
associated with an
(4.5.1-2)
m(~)
+ 0(X -I) + O(exp (- O-'t½)),
, for any ~ 0 .
(4.5.3)
In particular,
-K~(t) - -~l(- OO) = 2~-x-lt ½ + o(1)
as
t --) + C~ .
(4-5.4)
The last error-term in (4.5.3) will almost certainly be negligible compared to the term cases, as in If
q(x)
O(X-I). The latter can be estimated in particular
~ 3.8, however with a variation in notation from (3-8.4-5).
is monotone, with
q(O) = O, we need that 1
X(t) --) oo In the case ~(t) 4.6.
,
lim sup
q(x) = x N , N ~ -
tWX(t)q(X(t))
~
~ .
0, this gives
~(i(-oo)
: 2 ~ - i t ~i + O( t_i/(2N+2) ).
~'(0,
) = h.
The case
~
The same arguments can be applied if the last of (1.1°2) is modified in this way. As indicated in in the approximations to
~
m(~).
i.3, one gets a correction term
When this is fed into the integral in
(2.3.1) there results an additional term -C (t), so that, if
b
~
oo
h~-l
h
in the approximation to
, and (4.1.2) holds~ I
~3(t) - ~ ( - O O )
= 2 ~ - i t w _ h + o(1).
More substantial changes take place if the roles of are interchanged, ~Z (t) as
so that, roughly speaking,
m(X)
@, #
behaves as
in (i.i.i-2) X ½
and
t 3/2. It seems likely that the present arguments can be used
for this case, but the details will be very different.
28
REFERENCES 1.
F. V. Atkinson,
"Discrete and Continuous Boundary Problems"
Academic Press, New York, 1964. 2.
F. V. Atkinson, On the location of the Weyl circles, Proc. Roy. Soc. Edin. A 88, 345-356 (1981).
3.
C. Bennewitz and W. N. Everitt, Some remarks on the Titchmarsh-Weyl m-coefficient, in "Tribute to ~ke Pleijel", Department of Mathematics, University of Uppsala, Uppsala, 1980.
4.
E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations", McGraw-Hill, New York, 1955.
5-
W. D. Evans and W. N. Everitt, A return to the Hardy-Little~ood inequality, Proc. Roy. Soc. London, A 380, 447-486 (1982).
6.
W. N. Everitt, On a property of the
m-coefficient of a second-order
linear differential equations, Jour. London Math. Soc. (2), 443-457 (1972). .
W. N. Everitt and S. G. Halvorsen, On the asymptotic form of the Titchmarsh-Weyl
m-coefficient,
Applicable Analysis, 8, 153-169,
(1978). 8.
W. N. Everitt, D. B. Hinton and J. K. Shaw, The asymptotic form of the Titchmarsh-Weyl coefficient for systems,
9.
W. N. Everitt and A. Zettl, On a class of integral inequalities, Jour. Londom Math. Soc. (2), 17, 291-303,
lO.
(1978).
E. Hille,"Lectures on Ordinary Differential Equations", Addison-Wesley,
ll.
(preprint).
London, 1969.
D. B. Hinton and J. K. Shaw, On Titchmarsh-Weyl
M( ~ )-functions
for linear Hamiltonian systems, Jour.of Differential Equations, 40, 316-342 (1981). 12.
I. S. Kac, A generalization of the asymptotic formula of
V.A.
Mar~enko for the spectral function of a second-order boundary-value problem, Izv.Akad. Nauk SSSR Set. Mat.37,422-436 Mathematics Izvestija USSR 7, 424-436 (1973).
(1973), also
27
13.
I.S.
Kac, R-functions - analytic functions mapping the upper
half-plane into itself, Amer. Math. Soc. Transl. (2), 103, 1-18, (1974), (Supplement I to reference no. 1 14.
I.S.
above, "Mir", Moscow, 1968)
Kac and M. G. Krein, On the spectral functions of the string,
Amer. Math. Soc. Transl. (2),103, 19-102 (1974), (Supplement II to reference no. 1 above, "Mir", Moscow, 1968). 15.
B.M.
Levitan, On the asymptotic behaviour of the spectral function
of a selfadjoint second-order differential equation, Izv. Akad. Nauk SSSR Ser. Mat. 16, 325-352 (1952), also Amer. Math. Soc. Transl.(2) lO1, 192-221 (1973). 16.
B.M.
Levitan and I. S. Sargsyan,.Introduction
to Spectral Theory
for Selfadjoint Ordinary Differential Equations",
"Nauka", Moscow,
1970, also in "Translations of Mathematical Monographs", Vol. 39, American Mathematical Society, Providence, R. I., 1975. 17.
V.A.
Mar~enko, Some questions in the theory of one-dimensional
second-order linear differential operators. I, Trudy Moskov. Mat. Ob~6. l, 327-340 (1952), also Amer. Math. Soc. Transl. (2) lO1, 1-104 (1973). 18.
V.A.
Mar~enko,
"Sturm-Licuville Operators and their Applications",
Naukova Dumka, Kiev, 1977. 19.
E.C.
Titchmarsh, "Eigenfunction Expansions", Part I, 2nd. edn.,
Oxford University Press, 1962. 20.
S.D.
Wray, On Weyl's function
41-48, 1974/75.
F V Atkinson Department of Mathematics University of Toronto Toronto Ontario Canada
m(~),
Prec. Roy. Soc. Edin. A 74,
Some Limit Circle Eigenvalue Problems and Asymptotic Formulae for Eigenvalues'
by F V Atkinson and C T Fulton
§i
Introduction Asymptotic properties of eigenvalues of the second order equation -y" + qy = %y
on the half line
[0,~)
with a regular boundary condition at
obtained by a number of authors under various conditions on q(x) ÷ + ~
as
x-~ ~
We mention,
in particular,
x = 0 q(x)
have been
in the case when
the work of Titchmarsh [24,26,
27], McLeod and Titchmarsh [21], MeLeod [19,20], Giertz [9], K J~rgens [17], Levitan and Sargsjan [18, Chap. 12] and Hartman [11,12,13,14,15].
The work on this case
goes back as far as 1929 when W E Milne [22] obtained an asymptotic estimate for the number of eigenvalues in the interval case when
q(x)
increases to
~
[0,%]
(the so-called Milne formula).
belongs to the limit point case at
The
~ , has a
discrete spectrum, and was perhaps the simplest situation to investigate since no boundary condition at
~
was required.
In contrast to the amount of effort which has been devoted to the asymptotics of eigenvalues in the limit point case, considerably less has been done in the case when
q(x) * - ~
faster than
-x 2
and belongs to the limit circle case at
In 1954, P Heywood [16], was the first to obtain a limit circle analogue of the Milne formula, obtaining density formulae for both the positive and negative eigenvalues.
Heywood's results were rediscovered much later by Belograd and Kostyuchenko
[6] in 1973, who were unaware of his work.
Subsequently, A G Alenitsyn [i] in 1976
improved on Heywood's density formulae, obtaining
'quantum conditions'
for both the
positive and negative eigenvalues in which the parameter indexing the choice of boundary condition at
~
made an appearance.
Alenitsyn's quantum conditions were
2g
sharp enough to permit reversion
for the eigenvalues,
were only good enough to restrict to lie in disjoint Recently,
but his error bounds for them
the large positive and large negative eigenvalues
intervals.
the present authors have studied the asymptotics
of eigenvalues
problems with limit circle endpoints on both finite and infinite intervals,
for
which
cover the following cases: (i)
[0,~),
q(x) ÷ - ~
faster than
(ii)
(O,b],
q(x) = C/x ~ ,
(iii)
(O,b],
q(x) = -i/x ~
C # 0 ,
2
,
1 ~ ~ < 2 ,
~ > 2
The full results of these investigations of three papers [3,4,5].
-x
For problem
will be published
separately
in a sequence
(i) an iterative procedure of Atkinson
[2]
enables lower order terms in the quantum condition for the positive eigenvalues be produced algorithmically, produced
in the asymptotic
the negative eigenvalues,
so that higher order terms, as many as desired, expansion of the nth positive
a piecewise
turning point analysis
improves on the strength of Alenitsyn's locating
as
can be
n ÷ ~
For
is employed which
error bounds, but which still falls short of
the eigenvalues with error term tending to zero.
for improvements,
eigenvalue
to
Here,
there is still room
and possibly a different method of approach will prove more
successful. In the present paper we restrict attention
to the Heywood density formulae,
and
show that his results are already sharp enough to draw conclusions
about the
exponent of convergence
We also show that
of the positive and negative eigenvalues.
the Heywood formula for the positive eigenvalues
can be obtained very easily by
following the approach the authors have given in [3]. the results on the exponent of convergence the order of the entire limit-circle p.56, Equa.
w(%)-function
eigenvalue problem. (2.14)(i)]
The
which it is simply the denominator),
to deduce results on
whose zeros are the eigenvalues
and is known to have order
to regular Sturm-Liouville
w(%)
problems,
of a given
was introduced by Fulton [8,
for the Weyl-Titchmarsh
of Professor W N Everitt to the effect that in contrast
of the eigenvalues
w(%)-function
as a replacement
Finally, we also make use of
m(%)-function ~ i.
(of
A conjecture
can have order larger than 1/2, is answered in the affirmative,
30
and, in fact, it is shown by examples
that all possible values of the order,
1/2 up
to i, occur. One might be inclined to ask why the asymptotics was not investigated much earlier than 1982. problems which have to be overcome, authors make their analysis conditions Heywood,
pass
one matter seems to stand out:
Previous
[26] in his classic book on eigenfunction
for example, was lead to use Titchmarsh's
b + ~
solutions
x = 0 , analyze its distribution
{~%,@%}
expansions.
of the basic
of zeros over [O,b], and then
in order to apply a limit circle boundary condition of the form lim
where
While there are several intricate
rest on the formulation of limit circle boundary
given by Titehmarsh
equation defined at
of limit circle eigenvalues
m(%)
Wx(Y , @% + m(%)~%)_ = 0
is the limit of Titchmarsh's
%~(%)-function
as
b
Since this
÷
limit is not uniquely defined in the limit circle case, it is well known that one must let
b ÷ ~
vary with
b
through an appropriate
in such a way that the limit will exist as
Heywood chose the latter method, Titchmarsh circle
sequence of values of
linking his analysis
m(%)
functions had been parametrized
= -
a(%)cot
being entire functions
Titchmarsh-type
Alenitsyn's
of
boundary condition at
m(%)-functions
to the 1950 paper of Sears and
K ~ [0,7)
equivalent
reliance on the
% .
p.300, Equa.
condition at
Similarly,
~
and
of the limit
to that given by Fulton [8, p.52, Equa.
m(%)-function
a(%), b(%),
A G Alenitsyn used a
~ , using a parametrization
to formulate
seems to have given rise to the less than optimal
the boundary
'O(1)'-error bound,
(1.9)].
condition at cf. [I;
(1.9)].
The work of the present authors, functions
continuously.
'
being the index on the choice of boundary
c(%), d(%)
circle
to
in the form
K + b(%)
c(%)cot K + d(%)
K
B
[23], the only existing piece of literature at that time where the limit
m(%)
with
b * ~
b, or make
in contrast,
dispenses with the use of
to formulate limit circle boundary conditions,
m(%)-
and employs instead the
31
formulation used by Fulton [8, p.57, Equa. condition at
oo being parametrized lim
(2.18)],
the limit circle boundary
in the form
Wx(f, v cos ~ + v sin y) = O ,
~ c [O,w)
,
x-YOO
where
{u,v}
is a fundamental
system of the basic equation for
the same Prefer equation as Heywood, this formulation of the Heywood
of boundary
conditions
at
co affords a considerably
This seems to suggest that results on asymptotics
Titchmarsh
~2
would
the limit circle
The above mentioned work of
to fill a long neglected ordinary differential
gap in the Weyl-
equations.
Heywood Density Formulae Putting
interval
q(x) = -F(x)
we consider
the singular eigenvalue problem on the
[0,oo), I
-y" - Fy = ~y
(*)
(i)
y(O)cos ~ + y'(O) sin ~ = 0 , [lim Wx(Y,V)]COS
{u,v}
~ ([0,7)
(2) y ~ [0,w)
y + [lim Wx(Y,U)]sin ~ = 0,
X->Co
where
simpler proof
of limit circle eigenvalues
in the above form at the outset.
theory for second-order
that
than that given by Heywood.
have been obtained much earlier if Titchmarsh had formulated
the present authors seems therefore
Relying on
the reader will note (Theorem i below)
formula for the positive eigenvalues,
boundary conditions
% = O.
,
(3)
X-~m
is a fundamental
system of the equation
for
= O,
y" + Fy = O,
satisfying Wx(U,V) This formulation
= uv'
-
u'v
=
(4)
.
the limit circle boundary
Fulton in [8], where the equivalence established.
i
condition at
to Titchmarsh's
The only essential difference
conditions makes use of limits of Wronskian the basic equation for a real value of conditions
to apply the above form of boundary
form of boundary
conditions
is
is that the above form of boundary combinations
of
y
~, while Titchmarsh's
makes use of limits of Wronskian
the basic equation for nonreal values of
was the choice used by
combinations
%.
conditions
of
In applications
with solutions
of
form of boundary y
with solutions of it is usually easier
because the solutions
{u,v}
can
32
often be found e x p l i c i t l y for m a n y of the equations of m a t h e m a t i c a l physics.
As we
shall see in T h e o r e m 1 b e l o w the above form also has some theoretical advantages. For the case of the positive s p e c t r u m we make the following assumptions on F(x) ~ 0
for
x e [0,~), and
F'(x) > 0
for
x > 0 ,
F:
(5)(i)
lim F(x) = oo
(ii)
X->Oo
F(x) ~ C2[0, °°)
(iii)
7!
(iv)
ix < ~
/f F'(x)
lim
(v)
- = 0 F(x) 3/2 2
F" F3-~
F"
(F') ' F5/2
,
(vi)
e LI(xO,~)
,
(F') 2 • Ll(O,Xo)
,
0 < x0 < ~
O < xO < ~
(vii)
U n d e r these assumptions we prove: Theorem 1
(Heywood/Belogrud-Kostyuchenko)
Let
denote the number of n o n n e g a t i v e eigenvalues of (*) in
E(%)
[0,%].
Then
co
1
~
E(X) = 7
%
dx + 0(I)
,
as
(6)
% ÷ + oo
~¢77F + ~f
Under these same assumptions we can also prove the following theorems: Theorem 2
(Alenitsyn, & = O)
Let the p o s i t i v e eigenvalues of (*) be ordered b y
0 ~ X1 <~2
< .... < % n
< ....
Then the nth p o s i t i v e eigenvalue satisfies the e q u a t i o n
nv = O
Theorem 3
~ c (0,7),
k~--~-#+~ n
~ # O, and
dx - y + 0(I)
,
as
%
n
+ ~
(7)
7 • [0,~).
The nth p o s i t i v e e i g e n v a l u e of (*) is c h a r a c t e r i z e d implicitly as the unique root of the e q u a t i o n
SS
oo 1
(n+½)~T = f 0
÷0
In the case of involving
i oo
f
I~'i
Indeed, putting
% = %
n
] ,
in (5) gives
(n+½)~
replaced by
formula
term in (6).
incorporates
on the boundary condition at
term.
This
to lie in disjoint
'quantum condition'
intervals
The quantum condition
%
n
= X~'Y n
reversion for the asymptotics
of
%
n
as
on both
n + ~
P Heywood obtained
in
[0,%]
and
7.
x = O
The Y, but the
is still con-
F(x) = x
2g
with error term tending to
~
and ~.
incorporates
7
the
and the dependence
In examples,
gives an asymptotic
a
formula for
n
the large eigenvalues.
(9)(i)
is monotone decreasing
F"(x) = OI(F'(x))7 j
(*) contained
~
(5) is a
assumptions
large and for some
Let
in
is ultimately of one sign
F'(x)/(F(x)) d
Theorem 4
a sharper error
for the case of the example
with error term tending to zero, thus "locating" Under the additional
in Theorem 3.
can be shown to restrict
is seen to be more dominant than the dependence on
F"(x)
and the term
(8) obtained by the present authors
dependence of the nth eigenvalue, y
(8)
the dependence on
g > i, but is not sharp enough to locate the eigenvalues
on
-~ co
The Heywood formula
only the number of eigenvalues
(6), on the other hand,
of the eigenvalues
the eigenvalues
%
n~
of the choice of boundary condition parameters
tained in the O(1)-error
zero.
as
(6), and (7) just provides
O(1)-error
density formula which estimates
dependence
I
in Theorem 2 and Theorem 2 is contained
bound to replace Alenitsyn's
Alenitsyn
[Fi[ 2
deleted.
Theorem 1 is contained
irrespective
_ _ _
(%+F) 3/2 + (X+F)5/2 + %--372
O
~ = O, this holds with
cot ~
cot
dx - 7 +
~ + ~
where
for
x
sufficiently
d < 3/2
(ii)
1 < y < 4/3
(iii)
the following result for the density of the negative eigenvalues:
D = -% , and let [-D,O].
Let
G(~)
be the number of negative
x O = Xo(~)
be the turning point of
eigenvalues F(x)
of
defined
,
34
by
F[Xo(U) ) = ~
for all
U > O.
Then
Xo0~) dx ] + 0(I) , as
~F- dx +
G(U) = -~ [
0
U ÷ ~
(io)
W + W -"'~
x0(~)
Proof of Theorems 1-3: For
~ > 0
we make in (i) the modified Prefer transformation,
Y
r sin @ (y+F)i/4
y' = (%+F)
(ll)
1/4
r cos @ ,
which gives rise to the first order system for
8'
= ~'/'~-~ + ~
F !
r
and
8,
sin 28
(12) r' r Writing
1 F' -cos 20 . 4 (X+F)
(12) in the form
1 = - - e' we m u l t i p l y
[4(1, ~13/2.] sin 28
2
t h e s e c o n d t e r m i n (12) by t h e r i g h t
an integration by parts on the term
4(I+F) F'
hand s i d e o f t h e a b o v e and p e r f o r m
(sin 2e-)
@(_I~F+F]
, to obtain the
equation
(%+F)3/2 cos 28
- ~2 ((%+F)5/2] (13)
A similar manipulation on the right hand side of the r-equation brings the r-equation into the form i Isin 2@ 3-7~] r'/r = ~ ~ (X+F) Equation
sin 2@ 8
+
(F')--2 sin 4e 32(%+F)5/2 "
(13) is the same equation used by Hewood to prove theorem i, cf. [16,
p.459, Equa.
(2.3)].
Heywood integrates this equation over
[O,b]
to obtain an
(14)
35
estimate on the number of zeros of the solution x = O ([16, p.459, Equa. Milne-type Liouville
(2.8)]),
satisfying
[O,b],
condition
at
and then makes use of this result to obtain a
formula for the number of eigenvalues problem on
the boundary
e.g.
in
[0,4]
[16; p.464, Equa.
for a regular
Sturm-
(5.2)],
b 1
I
Eb(%) = ~
( ~
- ~-ff)dx + o(i)
(15)
0 The rest of his analysis
is concerned with passing
(6) for the density of positive fix a choice of boundary [23], who formulated case at
eigenvalues
condition
sufficient
at
~
m(l)-functions
m(l,K)
in the form
= lim
cos ~(b,K) cos B(b,K)
@(b,l) ~(b,%)
=
m(X,K)
in
K
_ a(%) cos K + b(%) sin K with c(1) cos K + d(%) sin K '
in
[O,A]
[O,A].
Once established,
~ , but since the dependence
is already contained any hope of obtaining
b
of the ~imit circle
sin $(b,K) sin 8(b,K)
b ÷ ~
continuously
the number of poles of
.
~B(D,K)(%) Lb
did not differ by more than two from the number of poles of
(15) to give a proof of Theorem i. at
In order to
such that the limit circle
+ @'(b,%) + ~'(b,%)
The basic idea was to show that for large for fixed
interval.
for the occurrence
~ = ~(b,K)
were obtainable
b -~°
on the infinite
he relies on results of Sears-Titchmarsh
conditions
~ , and provided a formula for
b + ~ , so as to obtain equation
in the
The choice of
of
O(I)
Eb(X )
passage K
to the limit
of
on the boundary
ultimately
E(%)
on
b + ~
fixes the boundary conditions
error term in (15), Heywood's
the dependence F"(x)
this permits
K.
in
condition
at
O
and
b
approach rules out
Also, his assumptions,
of one sign ,
(16)(i)
and F"(x)
( ] = O[(F'(x))TJ~
seem to have been imposed primarily for occurrence
for some
to guarantee
1 < T < 4/3 ,
that the Sears-Titchmarsh
of the limit circle case were satisfied,
actually holds under the weaker assumptions
cf. [16, p.457].
listed in (5) above.
(ii) criteria Theorem 1
36
To prove Theorem 1 more efficiently,
and obtain Theorems 2 and 3 at the same
time, we first apply the boundary condition (3) at instead of the Sears-Titchmarsh ~.
parameter
~ , relying on the parameter
y
K, to index the boundary conditions at
Indeed the main advantage of (13) over (12) is that (13) enables us to replace
the boundary condition (3) by an equivalent growth condition on while (12) does not. Claim 1
Let
e(x,%)
as
x + ~
To this end we prove:
% ~ [0,~).
Under the assumptions
in (5), there exists, for each
E (0,~), a unique solution of (13) satisfying x
lim [O(x,%) - f ~ x ~°°
Proof:
Subtracting
(17)
] = y .
0
~
from both sides of (13) and integrating over
[Xo,X]
we
have x @(x,%) - f ~
Xo dx = O(Xo,%) - f ~
0
dt
dt + ~
0
x°
~£f7~ +
x
cos 2e F '3/2 (~+F)
1 + ~
(5)(iv),
(F')2 dt (X+F) 5/2
~
(18)
[ cos 40
x0
(v), (vi) the limit as
x ÷ ~
(F') 2
~ Idt . (%+F) 5/2 )
of the right hand
The fact that (17) defines a unique solution of (13) then follows from
a standard result on asymptotic Claim 2
~ x0
x [ F' ~' f cos 20 j dt + T2 x0 (%+F) 3/2
But under the assumptions side exists.
~2 Xo-
For each
% E (0,~)
integration
(cf. Hartman [I0; p.273, Theorem I.i]).
and each solution
(r,@)
of (13),
(14),
lim r(x,%)
exists. Proof:
An integration of (14) over
[x,~)
obtains the result because of (5)(v),
(vi).
q.e.d.
Claim 3 let
Let
rY(x,%)
corresponding ~,~' c EO,~),
OY(.,%)
denote the solution of (13) which is defined by (17), and
be any corresponding solution of (14).
Then if
yY(.,%)
solution (ii) of the basic equation (I) we have for
denotes the
Y1 = Y2
and
37 Y2
lim Wx(yYl(',X),y x-~oo Proof:
Y
(19)
(',X')) = rYl(°°,X)'r 2(°°,X')sin(xl-T2 )
The result follows by a computation of the Wronskian using (ii) and a q.e.d.
passage to the limit which takes advantage of (17). Definition 1
We now fix the normalization of
condition defining
rT(.,%)
rT(m,%) E i Claim 4
in (14).
for all
Letting the solutions
yT(.,%)
by fixing the initial
For convenience we fix
T c [O,g) {u,v}
and
r(O,%)
so as to have
~ ~ [O,m) .
(20)
in the boundary condition at
m
be chosen
by
,
u(x) := r~/2(x'O)sinO~/2(x'O)
v(x) := rO(x'O)sineO(x'O-~) F(x) l/a
F(x) I/4
,
(21)
we have
lim x'+m
Proof:
=
, for all
~ c (0, °°) .
(22)
(cos
t-Wx(YT(',)v),u)J
Use equation (19) to compute the limits.
It follows from (22) that the solution conditions on
O
(3) at
(under the choice of
x = ~
and
r
at
m
yT(.,%)
obtained by fixing the initial
by (17) and (20) satisfies the boundary condition {u,v} made in (21)).
Having obtained a solution satisfying the boundary condition at
~ , we can
proceed to obtain a 'quantum condition' for the positive eigenvalues by applying the boundary condition at x = 0
x = O.
In terms of
@T(x,%)
the boundary condition at
may be expressed as
- ~
tan C~ , C~ e [O,g),
tan e (0, l) = m ,
c~ = r r / 2
C~ # -~ 2
,
or
i -n~ + A r c t a n ( - ~ @ (0' %n) =
g
l-~-
7
tan @
1T
'
a =~
.
'
~ # -~ 2
(2~)
38
Claim 5
The nth positive
the quantum therefore Proof:
condition
(8).
also the weaker
Put
x0 = 0
eigenvalue
of (*) is characterized
This proves versions
in (18), let
Theorem
stated x ÷
as the unique
3, under the assumptions
as Theorems
root of
(5), and
i and 2.
, use (17) and (23), and replace
the Aretan
using
Arctan(-~
Example
F(x) = x
2~
,
I < g < ~ ,
In this case the quantum
Cllg+i/2g
y + n~ =
i + 0( --~7~ ) .
i tan
tan ~) = - ~ + ~ ( 0 )
q.e.d.
~ = 0 .
condition
(8) becomes
I + O[ 1c+i/2~ )
'
% ~ ~
where oo CI = I
I dx . ~l+x2g + x E
0 A reversion
for the eigenvalues
gives
y + n~ + 0(~) )2g/e+l ~n = (---C I (25) + n~
which shows
that Theorem
In contrast,
the result
% n
which
%
)+
3 locates obtained
0(
the eigenvalues
by Alenitsyn
= [ ~ + n~ + 0(I) )2g/g+l CI
is not sharp enough
I
to locate
in Theorem
% + n~ 2g/g+l ----CI )
the elgenvalues
An iterative
procedure
for obtaining
in the quantum
condition
(8), and thereby higher
expansions
like
and refinements
(25) has been of Theorem
with error
4 for the negative
(I + O ( ) )
more
order
,
term tending
(26)
to zero.
terms of lower order
terms
in asymptotic
in [3], as well
spectrum.
to zero.
2 yields
with error
successively
given by the authors
term tending
as improvements
in
39
§3
The w(X)-Function According
to Fulton's
circle problems
formulation of the basic expansion
in [85, the eigenvalues
zeros of an entire function solutions
w(%), which
of the basic equation,
conditions,
condition at the boundary
This follows
~
condition at
at
there,
= sin y
lim x~Oo
= - cos y , for all
the solution
condition
at
at
the boundary %
and satisfies
defined by the requirements
lim Wx(X%,v) x~+co Wx(XX,u)
of two
conditions
X% = X~(x), which is entire in
, is uniquely
as the
and one which is defined by
~ , so as to satisfy
(27)
from Fulton [8, Theorem 1 and Equa.
Similarly, boundary
The solution,
is defined by initial
condition
or 'end' conditions
(*) are characterized
is defined as the Wronskian
one which
x = O, so as to satisfy the boundary initial
of the p r o b l e m
theory for limit
@% = ~ ( x ) ,
(2.5)].
which is entire
in
and satisfies
x = O, may be uniquely defined by the initial
the
conditions
q~%(O) = sin cz (28) ~i(O) = - COS ~
,
for all
% E ~
.
As in [8] the transformation
u' v' : [u carries
solutions
modified
y(x,%)
Sturm-Liouville
lY'(X,X)
of the basic equation
dX
u,v s L2(O, oo)
efficient m a t r i x posed at
co
Y = Y(x,%)
of the
uv -
%
i
2
-u
Y
-uv
(30)
"
by virtue of the limit circle case occuring
is in
LI(o, °°), and a Caratheodory
Indeed this justifies
identity
initial conditions
([8; Equa.
2.9])
at
co , the co-
initial value p r o b l e m may be
the definition of solutions
equation by means of the "transformed" of the determinantal
into solutions
equation,
dY
Since
I; ~-Wx(Y,U) j
of the basic
(27) for
X% .
By means
40
I Wx(#~,v) Wx(X~,v) Wx(~%,x~) = _Wx(~,u) -Wx(X%,u) ' and passage to the limit
x ÷ ~ , the entire function
w(%), whose zeros determine
the eigenvalues of (*) may be written in the various forms, w(~)
= w~'~(1)
= Wx(~,x~)
,
= WO(~,X~)
x ~
(0,~)
= (sin ~)Xi(0) + (cos ~)X%(O)
(31)
= W (~%,X%) = cos ~ W (~%,v) + sin y W (~%,u) . We have the following theorems. Theorem 5 Proof:
The order of
w(%)
is less than or equal to I.
Let
[wx(}l,v)] Y% = Y~(x, ) =
(32)
l-Wx(+l,u)J be the solution of (30) corresponding For fixed
to the solution
~1
of the basic equation.
b < ~ , Wb(%) = cos y Y~(B,%) - sin ~ Y~(B,%)
is an entire function of order 1/2, since
Wb(%)
are the eigenvalues of a regular Sturm-Liouville of [8] we also have the estimate for all
X ÷ ~
problem on
[O,b].
By Theorem l(i)
x,b c (0,~):
IY~ I < Iy~(b,%)I exp{I%l (X,%) Letting
is the entire function whose zeros
~ (u2+v2)ds} b
.
(33)
we therefore obtain l%Ifb (u2+v2)ds [Y+(~,%)I ~ IY~(B, %) Ie
It suffices to show that for every ity is that
~ e gl%I IY~(b,%)I
c = cb
for
I%I
(34) E > O, the right hand side of the above inequal-
sufficiently large.
has order 1/2 for fixed
b.
To this end we make use of the fact Hence, there exist constants
such that IY~(b,%)I S KeCb IxII/2
for
I%1
sufficiently
large.
Given
g > O, we first pick
f (u2+v2)ds < E/2 . b
b
so large that
K
and
41
I%1
Then for
g 1/2, eb < ~ Ill we have
so large that
IY~(~,x)l ~
Ke 2
. e
This proves the theorem since Y~(~,%)
w(1)
= Ke is representable in terms of the components of
by (31).
q.e.d.
Let the zeros of
w(1)
be ordered according to their absolute value,
o ~ IXll ~ I%21 < I131 ~ ... Suppose that
~
value of (*).
and Then
y
are such that
w(%)
I%1] > O, that is,
Q(%)
order of w(l)
% = 0
%lln+~12(llln)m+...+~l(llln)p ]
is a polynomial of degree less than or equal to
w(1).
is not an eigen-
has a Hadamard factorization of the form
w(l) = eQ(1)[n~~ I (i - ~-l,,)e n where
(35)
The order
p
(36)
p , and
and the exponent of convergence
Pl
p
is the
of the zeros of
are defined by
p = inf{~ I w(1) = O(e I%1~)
as
Ill ÷ ~ }
(37)
co Pl : inf{dl
[ n=l
~n
The following lemmas relating
< co } .
P' Pl
and
(38)
p
(which is independent of
n
in (36))
are well known, cf. Titchmarsh [25, pp.248-254]: Lemma i Lemma 2
pl-
eat integer
Lemma 3 Since
.
If the degree <- Pl "
of
Q(1)
is zero, then
Pl = p' and
p
is the great-
is an integer, then oo [01 if 7 IAn I-pl = co n=l P = ! co -Pl Pl-i if ~, ]%nl < o~ n=l
If
p
p ~ i
If
q 01
is not an integer, then
(39)
Pl = p "
by Theorem 5, it follows from the above that there are three
different possibilities for the Hadamard factorization of
w(%):
42
Since
O ~ 1
by Theorem 5, it follows
different possibilities for the Hadamard Case i
If
0 < p < i, then
from the above that there are three
factorization
Pl = p < i,
q = 0
of
and
w(%):
p = O.
In this case
co
w(1) = w(O)
Case 2
If
p = 1
and
~I n=l
Pl = i, then
(i - ~
)
(40)
n
q
may be
0
or 1
and
p
may be
0
or
i.
In this case
~
where
ql
0+p(l/l
w(%) = eqo+qll[
~ (i - ~ )e n=l n
and
0
p
may be
or
i.
i
if
0
if
) n
]
(41)
The determination
of
p
is governed by
oo
I I%n I-I = ~ n= i
P = J
oo
L Case 3
If
p = 1
and
0 < Pl < I, then
qo+ql I w(1) = e
where
use Heywood's
co ~ n=l
( I - ) n
the basic relation
density formulae
case I may occur w i t h any Examples
to show that
and
q = I.
In this case
] ,
(42)
computing
to find
Pl
holds.
in several
In the next section we
special
cases,
and that Case 2 may occur with
Case 3, or Case 2 with
p = O, may occur,
and show that p = I.
are not known.
of Eigenvalues
of convergence
Pl
of the zeros of
the exponent of convergence
separately.
of the positive
w(%)
can be determined by
and negative
eigenvalues
For if
PI = inf{~
%n
p = max(q,pl)
0 ~ []./2,1)
E__xxponent of Convergence The exponent
where
[
p = 0
ql # O. In all cases
§4
~ I~.n I-1 < ~ n=l
are the positive
I
~ n=l
(43)
< co } n
eigenvalues
ordered by
0 < %1 < %2 < "'" , and
43 co
01 : inf{~
where
~n = -In
I
~
Ic~
i
(44)
are the negative eigenvalues ordered by
0 < ~i
<~2
< ...
, then (45)
Pl = max{Pl,P I} • +
To determine
O1
01
and
we put co
EI(I,F)
= ~-
(46)
dx
X/fT~ + /f
0 and
x° (la)
i
GI(U,F) = ~ [
/
~o ~
dx +
o where
x 0 = Xo(P)
and
x I = Xl(D)
F(x0(~))
= U
are defined by the conditions
for all
~ > 0
F(Xo(I]) + Xl(II) ) = 2U
We then have the following Theorem 6
(47)
dx ] s
f
Xo(~)
for all
(48)
(49)
U > 0
theorems:
Under the assumptions
i - A(1) 23/27
of Theorem 1 we have for all
X > 0,
1
(5o)
< ~ A(%)
-< EI(X,F)
where
Xo(X) /
A(%) = [
/~'dx +
o Theorem 7
Let
~ = -%.
f
---% dx ] .
Xo(X)
(51)
H
Then under the assumptions
of Theorem 4 we have for all
U > O, 1 23/2
BI(D) ~ GI(D,F)
1 N ~ B2(U)
(52)
where
xi(~) B I (U) = [
/
o
~
dx +
f
x~ (~)
- - dUx
e97~7
] ,
(53)
44
B2(>)
=
E
x2(~) f /fdx+
co
D f x2(~) /F~7
0
(54)
dx ] ,
and x2(N) = XO(Z) + Xl(~)
Proof of Theorem 6:
Since
(55)
•
is strictly increasing and positive
F(x)
in
[0, ~)
we
have 0 ~ F(x) ~ i
for
x e [O,Xo(1)]
i ~ F(x) < ~
for
x e [Xo(1),~]
and .
Hence
Xo(X) Et(X,r ) ~ ! E
f
X
dx+
o ~
f
---kx dx ] < 1A(X)
Xo(X) xCNTF
and
Xo(X)
1
oo
EI(~.,F) -> ~ [ £ ~
1
1
dx + Xo(%)f - - ~ dx ]
Xo(X)
1 [ >- 2--~
Proof of Theorem 7:
.
~
f 0
% dx + / 2/~ Xo(% )
Letting
x = Xo(~) + t
1
1 dx ] = 2~F
A(%)
q.e.d.
.
and putting
Fl
for
t e [O,Xl(~)]
,
and ~ Fl(t)
for
t c [Xl(D),~ ) .
Estimating as before we have x0
GI(~,F)
1 -< ~ [
1
(~) f 0
x I (>) ~
dx +
/ t=O
co
dt + /11 + Flit)
f xI(N)
dt ~
+ Fl(t)
45
and
Xl(O)
i ->-i-~ [
GI(U'F)
/ t=O
]J dt + ~ U dt ] U/-U-$-F-I (t ) Xl(U) D~-ll (t )
xi(~) i [ -> 2-~
/
U dt + / ~ dt ] " 2/~ xI(D) ~ i (t)
t=O
i
~> ~
Corollary zeros of
BI(~)
.
q.e.d.
Under the assumptions w(%)
of Theorem i, the exponent of convergence of the
is bounded below by 1/2.
Hence, by lemma I, the order of
w(%)
is
also bounded below by 1/2. Proof:
Since
Xo(%) ÷ ~
as
I ÷ ~
we have
A(%) k % 1/2 Xo(%) ~ % 1/2 whenever
Xo(1) k i.
By Theorem i we have
E(1) - EI(X,F) = 0(i), so applying
Theorem 6 it follows that there exists a positive constant CI I/2 for
I
sufficiently
large.
C
such that
_< E(1)
Letting
%
n
be the nth positive eigenvalue,
we there-
fore have Ci 1-2/ < n , n or C
_
n
<
I _ 11/2 n
_
1/2 Hence the series
that
Z ~=l
+
is always divergent.
We may therefore conclude
n
+ 01 e Pl > 1/2 .
q.e.d.
We now apply Theorems
6, 7 and Theorems
i, 4 to calculate
+
O1 and
p~
for 3
examples: Example 1
F(x) = X 2g ,
1 < s < ~ :
in (1/2, i) except for the endpoints. calculation
+ ~ _ g+l Pl = P = Pl 2g , which assumes all values In this case we have
gives A(%) = (i + i
)%(~+i)/2S
,
XO(% ) = %1/2g, and a
46 BI(~) = [(2 I/2e - i) + - 1
]pg+I/2g i
and
B2(D ) = [21/2E +
Since
E(%) - EI(%,F) = 0(i)
formula for
A(%)
I
] c+i/2g
by Theorem I it follows from Theorem 6 and the above
that there exist positive constants
C1 < C2
such that for
sufficiently large
CI%S+I/2e _< E(%) _< C2.1s+i/2~
For
~ = %
n
(56)
the nth positive eigenvalue, this gives
C %g+i/2s ~ n ~ Cl-%e+i/2e 2n n ' or
2e/~+l
CI 2~/~+i (57) u
for sufficiently large It follows that
n. Z n=l
< ~
if and only if
~ > (~+l)/2g.
A similar
n
argument using Theorem 4 and Theorem 7, together with the above formulae for and
B2(U)
Example 2
shows that
BI(U)
p~ = (c+l)/2g.
F(x) = e 2x : 01+ = pl = 1/2
In this case we have
Xo(%) = 1/2 %n 4, and a calculation gives A(%) = 41/2 %n 41/2(1 +
BI(~) = ( ~
+ ]I-
2
) ,
U1/2
/f
B2(~ ) = i / 2 ( % n 1 / 2
and
Since
E(%) - EI(%,F ) = 0(i)
formula for
A(%)
sufficiently large
+ %n /~ + ! ) . ?f
by Theorem i, it follows from Theorem 6 and the above
that there exist positive constants %
CI < C2
such that for
47
C111/2 %n(1 I/2) z E(%) ~ C2%1/2 %n(1 I/2) . For
(58)
% = I , this gives n C 11/2 11/2 < < C % I/2 %n 11/2 I n %n n _ n _ 2 n n
or, for some constants
0 < K I ~ C I < C 2 ~ K2,
K1 ~n___in <-
1
X--" ~
<-
K2
In___!n ,
(59)
121
for
n
> 1/2
sufficiently large.
since
oo ~ n=l
.
the
It follows that
co Z n=l
~
< oo if and only if n
k ~
is convergent only for
k > i.
Hence we conclude that
+ P1 = 1 / 2
For
negative
eigenvalues
that there exist positive constants
we f i n d
C1 < C 2
similarly
from
Theorems
4 and
such that for sufficiently large
C1 ~I/2 ~< G(~) -< C2 DI/2 %n(p I/2) .
For
P = Zm
(~m = -~m C
where
%m
7
(6o)
is the mth negative eigenvalue) we have
1/2 ~ 1/2 i n ( 1 / 2 ) ipm s m s u2~ m Wm
or (61)
m Dm
for if
m
sufficiently large.
~ > 1/2.
Example 3
It follows that
Hence we conclude that
F(x) = x 2 %n4x :
In this case
Xo(1)
m=l 0~ = 1/2. i
+ Pl = Pl = P1 = i.
is defined by Xo(%) In21Xo(1)) = F~"
Squaring and taking the logarithm gives %n2(x~(%) %n4Xo(%)) = ~n21 . Dividing and applying L'Hospital's rule to get
is convergent if and only
48 2 4 ~n(x 0 ~n Xo) ~n x 0
lim Xo-~O
12 (62)
=4,
we see that
Xo(X) = 4
~
(I +o(I))
as
(63)
)l ->- °°
In 2 % A calculation
gives A(%) = ~
Xo(X)
x
+ In(x~(%))
(1 + o(
1
))
as
%+co
II
=
E4 Zn 2 21.t
1/
(1 + 0 (
= %n~
B2(~)
= ~
• x2(~)
= ~
[ 4
E(%) - EI(%,F ) = 0(i)
result
for
A(1)
sufficiently
large
1
~
))
'
and
.D + gn[x2(D) )
n22--7
2F~
(I + 0(i))
(i + 0 (
~ - i- ~ )) ,
- ~nN ~
Since
/~
by Theorem
] +
as
I, it follows
that there exist positive
constants
~ ÷ uo
from Theorem C1 < C2
6 and the above
such that for
1 (64)
For
t = I
n
this gives X CI
~
< n < C2 ( n
) n
or KI ~ < n ~n n
n
K2 < - , - n ~n n
(65)
49
for
n
sufficiently
follows
that
Z n=l
large, E/#k n
and constants converges
KI, K 2
with
if and only if
K 1 ~ C 1 < C 2 ~ K 2.
~ > i.
For
~ = i
It
we have
(66) n=l
Thus we conclude and application
that
+ 01 = I.
n
Similarly,
of T h e o r e m 4 gives
p~ = i
co
l
n=l
~n
Z
also for the negative Since example mines
Bl(~),
B2(~)
and
(67)
eigenvalues.
therefore belongs p = 1
for
= oo
pl = p~ = 0~ = 1 , we conclude
that
the above estimates
from lemma l that
to Case 2, and the divergence
in the Hadamard
ally, was considered by J Weidmann
factorization
of
also.
This
of the above series deter-
w(1).
[28, pp.219-220].
P = 1
This example,
incident-
Weidmann establishes
that for
the sum over all eigenvalues,
Z
I
I #O n for
n
~ < i, and that it is convergent
results
= oo
1 for
~ > i, cf. [28, p.217,
(66) and (67) answer his open question,
the above sum for possible sequence:
result.
~ = i.
It follows
This divergence
if one considers
for
that Weidmann's ~ = 1
integral operator
(7, p. I092, Lemma 6] it follows
~n (T)
Satz 3.5(a)
differential
: ~
The
that we have divergence
operator,
of
is indeed a best
also has another noteworthy
the H i l b e r t ~ c h m i d t
sponds to the inverse of the self-adjoint Dunford and Schwartz
showing
Satz 3.5].
T
con-
which corre-
then by a lemma of
that
,
n=l where
9n(T)
oscillatory
are the singular values of
T.
Example
3 therefore
limit circle p r o b l e m for which the corresponding
integral operator
does not belong to the trace class.
circle problems where
the spectrum is bounded below,
represents
an
Hilbert-Sehmidt
On the other hand, the corresponding
for limit
Hilbert-
50
Schmidt integral operator always belongs to the trace class, cf. Weidmann [28, p.216, Satz 2.6]. It is possible to determine the exponents of convergence for the positive and negative eigenvalues by a more direct appeal to the behaviour of
F(x)
near
Namely, we have the following theorems: Theorem 8
(i)
Under the assumptions of Theorem i, and for
1/2 < ~ < 1 , the
series of positive eigenvalues
(68) n=i
n
converges and diverges together with ao
f
(ii)
F ½-o~ dx .
For the case
(69)
= i, the above series converges and diverges to-
gether with oo
f
en F dx .
(70)
¢Y Proof:
Since
function of
E(X)
is the number of positive eigenvalues in
X, which increases by 1 at each eigenvalue.
[O,X], it is a step
We may therefore write
A
X-(x = f I<X
n
X-c~ dE(h) + C I + O(A -c~)
I
n
for some constant
C I.
The first term can be integrated by parts to give
EX- ~ E(X)
A A X-~-I ]1 + f o~ E(X) 1
d% .
T Since the integral defining
convergent with respect to
EI(X,F )
X
is bounded by
X J 0
and may be differential under the integral sign to
give d
d--7
1
I
~ f o
1 dx , it is uniformly /g
(X/X-@-- Vg)dx
1
1 i
1
= ~7
Thus, a similar integration by parts gives
dx
xc77g
.
51
A
f
x-~ E{(X,P)dX
1 A A i-~-i = !-c~ EI(I,F)] l + f C~ El()k,F)d)t i
Putting
E(1) = EI(I,F) + 0(i)
in the first result and substituting the above for
the El-part we have
A )-0~ E(X)]~I. + f (X)-(X-I E(~k)dTk 1 A = f 1
A 1-(x E{(I,F)dl + O(A -c~) + O( f ¢~ 1-c~-I dl) . 1
Hence there exists a constant
1
n
M
independent of
-2f--~ f 1
%-c~ f 0
A
such that
~ dx d% ¢1-$F
~< M .
(71)
Interchanging the order of integration we find A 1-~
co
i
1
o
~/fTF
f
f
dx
A
f O
1
f i
ts
(%+F) I/2
r',
= / F(x)½_C¢ A F(x) 0 ~i F(x) Passing to the limit
~
A + ~
d l dx
]
I d~) dx x)C~(i+~))i/2 "
in the above inequality we thus see that the series
converges and diverges together with the integral n
/ F(x) ~ ~ 0
For
~ ¢ (1/2,1)
i/ (x)
1 x)C~(l+x))i/2 d~) dx .
(72)
we have
du < oo , u~ (l+v) I/2 and the inner integral is therefore bounded above and below for
x
sufficiently
52
large.
This proves
(i).
For
~ = i, we may integrate by parts in the inner inte-
gral, obtaining
£n F(x) + O(I) /I + I/F(x)
f i d~ I/F(x) v(l+~) I/2 as
x ÷ ~
Since the dominant
Application before. for
term is
£n F(x)
,
(73)
this proves
of Theorem 8 to the above examples yields
For example
3, in particular,
the integral
~ = i, so we again obtain the result
(ii).
q.e.d.
the same results obtained
in (70) is found to be divergent
that
oo n=l Theorem 9
(i)
of negative
n
Under the assumptions
eigenvalues
of Theorem 4 and for
1/2 < ~ < i, the series
~m = -%m '
1
(74)
m=i converges
and diverges
together with co
f (ii)
For the case
F {--0"
dx
.
(75)
O` = i, the above series converges
and diverges
together with
oo
f Proof:
%n F dx
/g
.
We follow the same steps as before
ence of a constant
M
independent
-o`
i --2~
~ K
Interchanging
m
of
i K
f
ll-et[
f
<Xo(~)
using theorem 4 to establish
dx
d~
(77)
< M
x 0 ]~)
1
in the above integral
dx dLt
the exist-
such that i
7
the order of integration
K
A
(76)
gives
53
rain (A,F (x))
f Xo(~)
=
f
f
Xo(~)
1 ,-7~m
F(x)~[
f
1
K/F(x)
A ÷ ~
d~
vC~(I-v) I/2
lax )
in the above inequality we thus see that the series
converges and diverges together with the integral
f
Xo(U)
For
~ ( F - ~ ) I/2 dU dx
min (A/F, i)
oo
Passing to the limit
1
K
~ ~ (1/2,1)
f
r(~) ½-~
K/F(x)
~)~(l_v)l~
d~
we have 1 f 0
, I d~ < ~ ~ ( I + v ) I/2
and the inner integral is therefore bounded above and below for large. For
x
sufficiently
This proves (i).
G = I, an integration by parts in the inner integral gives 1 f K/P(x)
as
(78)
dx .
x ÷ ~
1 d~ v(l-~) I/2
%n r(x) ~I - K/P(x)
Since the dominant term is
+ O(l)
%n F(x)
,
this proves (ii).
(79)
q.e.d.
54
References
i.
A G Alenitsyn, Asymptotic properties of the spectrum of a Sturm-Liouville operator in the case of a limit circle. Differential Equations 12, No. 2 (1976), 298-305. (Differentsial'nye Uravneniya 12, No. 3 (1976), 428-437.)
2.
F V Atkinson, 71-87.
3.
F V Atkinson and C T Fulton, Asymptotic formulae for eigenvalues of limit circle problems on a half line. Annali di Math. Pur. ed. Appl., to appear.
4.
F V Atkinson and C T Fulton, Asymptotics of Sturm-Liouville eigenvalues for problems on a finite interval with one limit circle singularity, I. Proc. Roy. Soc. Edin., Ser. A, to appear.
5.
F V Atkinson and C T Fulton, Asymptotics of Sturm-Liouville eigenvalues for problems on a finite interval with one limit circle singularity, II. Submitted.
6.
V P Belogrud and A G Kostyuchenko, 227-228.
7.
N Dunford and J T Schwartz,
8.
C Fulton, Parametrizations of Titchmarsh's case. Trans. A.M.S. 229 (1977), 51-63.
9.
M Giertz, On the solutions in Lp(-~,~) of y" + (~ - q(x))y = O, when rapidly increasing. Proc. Lond6n Math. Soc. (3) 1 4 (1964), 53-73.
On second-order
linear oscillators.
Revista Tucuman 8 (1951),
Usp. Mat. Nauk~ 28, No. 2 (170)
Linear Operators,
II.
(1973)~
New York: Intcrscience,
m(%)-functions
in the limit circle
Wiley,
q
is
iO.
P Hartman, Ordinary Differential Equations. reprint of original 1964 edition).
ii.
P Hartman, Differential equations with nonoscillatory Math. J. 15 (1948), 697-709.
12.
P Hartman, On the eigenvalues (1951), 657-662.
13.
P Hartman and C R Putnam, The least cluster point of the spectrum of boundary value problems. Amer. J. Math. 70 (1948), 849-855.
14.
P Hartman, equations.
15.
P Hartman, Some examples in the theory of singular boundary value problems. Amer. J. Math. 74 (1951)~ 107-126.
16.
P Heywood, On the asymptotic distribution Soc. (3) 4 (1954), 456-470.
17.
K J~rgens, Die Asymptotische Verteilung der Eigenwerte singul~rer SturmLiouville problem~. Annales Acad. Scientiarum Fennieae, Ser. A (1963), 3-23.
18.
B M Levitan and I S Sargsjan, Introduction to spectral theory: self-adjoint ordinary differential operators. Providence, R.I., A.M.S. Transl., 1975.
19.
J B McLeod, The distribution of the eigenvalues for the hydrogen atom and similar cases. Proc. London Math. Soc. (3) Ii (1961), 139-158.
of differential
Baltimore:
1963.
1973 (corrected
eigenfunctions.
equations.
Amer.
Duke
J. Math.
73
On the zeros of solutions of second order linear differential J. London Math. Soc. 27 (1952), 492-496.
of eigenvalues.
Proc. London Math.
55
20.
J B McLeod, The determination of phase shift. (1961), 17-32.
21.
J B McLeod and E C Titchmarsh, On the asymptotic distribution of eigenvalues. Quart. J. Math. (Oxford) (2) I0 (1959), 313-320.
22.
W E Milne, On the degree of convergence of expansions in an infinite interval. Trans. A.M.S. 31 (1929), 906-913.
23.
D B Sears and E C Titchmarsh, Some eigenfunction formulae. Oxford (2) 1 (1950), 165-175.
24.
E C Titchmarsh, On the asymptotic distribution of eigenvalues. Oxford (2) 5 (1954), 228-240.
25.
E C Titchmarsh, Theory of Functions. Press, 1939.
26.
E C Titchmarsh, Eigenfunction expansions associated with second-order differential equations, I. (2nd edition) London: Oxford University Press, 1962.
27.
E C Titchmarsh, Eigenfunction expansions associated with second-order differential equations, II. London: Oxford University Press, 1958.
28.
J Weidmann, Verteilung der Eigenwerte fur eine Klasse yon Integraloperatoren in L2(a,b). Journal fur Mathematik 276 (1973), 213-220.
F V Atkinson Department of Mathematics University of Toronto Toronto Ontario Canada
N.B.
Quart. J. Math. Oxford (2) 12
Quart. J. Math.
Quart. J. Math.
(2nd edition) London: Oxford University
C T Fulton Department of Mathematical Sciences Florida Institute of Technology Melbourne Florida 32901 U.S.A.
This research supported by the National Science Foundation under grant No MCS-7902025.
INTEGRAL
INEQUALITIES
SOLUTIONS
AND
OF D I F F E R E N T I A L
EXPONENTIAL EQUATIONS
CONVERGENCE
WITH
BOUNDED
OF
DELAY
by
F. V. A t k i n s o n J.
R. H a d d o c k
0. J.
1.
Staffans
Introduction.
In this p a p e r that,
we
establish
when
applied
to c e r t a i n
equations,
produce
information
of s o l u t i o n s .
Our
results
results
for
functional regarding
are p r i m a r i l y
integral
(delay)
differential
exponential based
inequalities
convergence
on an i n e q u a l i t y
t p(t) <
S
--
m(t,s)p(s)ds
(t > t ) ,
t-r
--
(i.i)
0
where
(i) (ii)
(iii)
r>
0
and
tO ~ 0 ;
P
is m e a s u r a b l e ,
on
[t o - r ,~);
m
is n o n - n e g a t i v e
[t0,T ] x [ t -
r,t]
locally
bounded
and n o n - n e g a t i v e
and and
integrable
for e v e r y
T,t~
on to .
57
In Section
2 we state
the proofs
are given
with
integral
other
properties are given applied
of
even
faster
be m e n t i o n e d
under
the p r o p e r t y
presented
here.
represent
generalizations
2.
We assume
and
(iii)
function
(corollaries)
constancy
with bounded
can be
of solutions delay.
As
of solutions
is
We note
t~
of A t k i n s o n of)
and
the afore-
of this paper by conditions
that a locally
LP[tI,~),
p Ih(t) I dt <
S
in S e c t i o n
and A p p l i c a t i o n s .
accompanied
is said to be in
given
theme.
the r e m a i n d e r
is always i.
results
Such
for the w o r k
(perturbations
is the m a i n
possess
is a solution.
of the m o t i v a t i o n
of
differential
but not always,
function
of recent
throughout
of S e c t i o n
the f u n c t i o n a l
often,
of the T h e o r e m s
(i.i)
h
(i.i)
convergence
how the i n e q u a l i t i e s
some of the examples
the study
equations
Statements
much
Indeed,
[2], where
inequality
that
that each c o n s t a n t
have p r o v i d e d
mentioned
combine
exponential
the c o n v e r g e n c e
consideration
equations
Haddock
theorems
than exponential.
It should equations
of the paper;
Examples
asymptotic
equations
in some cases
These
p(t).
2 to illustrate
exponential
theorems
to yield
and/or
differential
be seen,
3.
inequalities
in S e c t i o n
of functional shall
in S e c t i o n
~ p(s)ds
to obtain
the two m a i n
that (i) ~ (ii)
integrable
(t I £ R,p > 0)
if
2
58
THEOREM
2.1.
continuous
Let
(I.I)
functions
hold
¢
and
and
suppose
~
such
there
exist
positive
that
s+r ¢(s) - I s
¢(t)m(t,s)dt
> ~ (s) --
(s > t ). -- 0
(2.1)
Then ~p e LI[ t0,~) .
A few c o m m e n t s decreasing
on
tive,
then
which such
> t 0
to apply. with
the that
> 0 ,
t /t
~
it r e a d i l y hand,
(2.2)
if
in
(2.1)
follows ¢
is non-
from
is a l l o w e d
can be r e p l a c e d
(2.2)
that
to be n e g a -
by
~ (s)p(s)ds <
be p o i n t e d
out
It a p p e a r s
that
an i n c r e a s i n g
left-hand (2.1)
If
0
it s h o u l d
to b e g i n
then
the c o n c l u s i o n
--
simple
in order.
On the o t h e r
suPT
Finally
are
[t0,~),
p E L 1 [t0,~).
(2.2)
side
that Theorem the m o s t
function
of
is s a t i s f i e d .
(2.1) For
¢,
2.1
is o f t e n
fruitful impose
is p o s i t i v e example,
by
and
quite
approach
conditions
is for
then define
setting
#(s) = e ~s,
we can d e d u c e :
COROLLARY
i.
In a d d i t i o n
to the c o n d i t i o n s
of T h e o r e m
2.1,
suppose s+r l i m s u P s ~ ~ fs m(t,s)dt=
y < l.
(2.3)
59
If
y > 0 , 0 < a
If
let
a
satisfy ]
J
< - (InyJ/r.
y = 0,
let
~
(2.4)
be a n y
(2.5) positive
number.
Then ms
I
e
~(s)ds
(2.6)
<
tO
PROOF. if
Let
y > 0
¢(s) = e and
~(s)-
(2.6)
ms
,
if
where y = 0 .
m
as
mr [i-
e
Since
the r e s t r i c t i o n s
tions
of T h e o r e m
m
2.1 hold.
s+r e as_
S
e a S m ( t , s) dt S
def = ~(s) .
Y]
on
(2.4)
Then
s+r Is ¢(t)m(t,s)dt=
> e
satisfies
imply The
~
is p o s i t i v e ,
conclusion
f
the c o n d i -
eaSp(s)ds
<
to now
follows
Under see
from
directly
shall
(2.6)
be
constant
of the a b o v e
corollary,
it is e a s y
to
that
seen
in d e t e r m i n i n g of c e r t a i n
(2.2).
the c o n d i t i o n s
/ p(s)ds= t
As
from
0 { e -at}
as
in the n e x t
an e x p o n e n t i a l
functional
t ÷ ~
corollary, rate
differential
is a s o l u t i o n .
(2.7)
this property
of c o n v e r g e n c e equations
A prototype
of t h e s e
is u s e f u l
of s o l u t i o n s
for w h i c h equations
each is
60
x' ( t ) =
g(t,x(t))-
g : [0,~) × R n ÷ R n
where
COROLLARY
2.
Suppose
and n o n - n e g a t i v e
on
g(t,x(t-
r))
(r > 0),
(2.8)
is c o n t i n u o u s .
there
exists
[0,~),
such
Ig(t,u) - g ( t , v ) I ~
a function that
g
in
p
, continuous
(2.8)
satisfies
P (t) l u - v l (2.9)
for all
(I" I) is a n o r m
t > 0
in
and
R n)
u , v s R n.
J
If
s+r limsup s ÷ ~ I s p ( t ) d t = y < i,
then
each
(t I ~ 0)
solution tends
of c o n v e r g e n c e
x
tends
Let
x
follows
from
(2.8)
where
be
t O = t 1 + r.
t ÷ ~. If
[t I - r, ~)
Furthermore,
y = 0
(i.e.,
the
limit
that
x'
Thus,
(2.11)
faster
a solution
of
than
(2.8)
any e x p o n e n t i a l .
on
is c o n t i n u o u s for
rate
if
p ( t ) d t = 0) ,
t o its
PROOF.
on an i n t e r v a l
as
is e x p o n e n t i a l . .s+r
x
(2.8)
to a c o n s t a n t
l i m s ÷ ~Js
then
of
(2.10)
[t I - r,~) . on
It
[t0,~) ,
t ~ t O t
Ix'(t) I < p(t) Ix(t) - x ( t - r) I < /
p(t) Ix'(s)l ds. t-r
That
is,
condition
(i.i)
is s a t i s f i e d
for
(2.12)
61
p(t) =
Hence, the
from
Ix' (t) I
(2.13)
conditions
and
m(t,s)
a n d the h y p o t h e s i s
of C o r o l l a r y
1 are
(2.13)
= p(t) .
of C o r o l l a r y We
satisfied.
2, we
see
conclude
that
from
(2.7)
It Ix' s) Ids = 0 {e -st }
is given
where whether and
y > 0
it f o l l o w s
by (2.4)
or
x(t)
Clearly, faster
if
than
for any
from
linear
case
[2, T h e o r e m extension actually limsuPs
= x(~)
depending,
exists
(2.14)
case,
respectively,
on
Ix'I ~ L l [ t 0 ,~)
lim s ÷ S
the
4.1].
of the
that
conditions
The
above
previous
x
for t h i s
the
solutions
each
of 2
solution
corollary
results
case,
to its (2.15)
limit holds
proof.
since
determined. T h a t the c o n s t a n t s+r ~ ~ Ss p(t)dt < 1 is the b e s t
f r o m the e q u a t i o n
converges
of C o r o l l a r y
Also,
(2.15)
t ÷ ~.
since,
completes
known
only, [i].)
as
s+r s p(t) dt = 0,
any e x p o n e n t i a l This
(and is f i n i t e ) .
that
- x (~) : 0 { e - ~
It is a l r e a d y under
(2.5)
that
e > 0.
limits
(2.14)
t ÷ ~,
In e i t h e r
y = 0.
lim t ÷ x(t)
We d e d u c e
or
as
(2.8) (cf.
t e n d to f i n i t e [2] or,
is u n i f o r m l y
represents a rate 1
for the stable
a marked
of c o n v e r g e n c e
in the
possible
condition
can be
seen
is
62
-i
x' (t) = r
where and
x(t)
[x(t)
= ~ + ~t
- x (t - r) ] ,
is
a solution
f or
arbitrary
be
directly
constants
B.
REMARK.
Corollary
2 can
applied
to t h e
linear
system
x' (t) = P (t) [x(t)
where
P
(2.10)
- x (t - r) ],
is a c o n t i n u o u s
becomes,
for
n × n
P(t)=
(2.16)
matrix
IP(t) I
function.
(I'I
a matrix
Condition norm),
s+r limsuPs
Thusfar, regarding we
have
itself
said at
nothing
~.
This
requires
asymptotic
i,
only and
stability
2.2.
suPt
Let
> t --
limsuPt
0
]P(t) Idt : 7 <
concentrated to
zero
at a l l
with
decay
among of
(i.I)
an
its
respect
St m(t,s)ds t-r
of
in p a r t i c u l a r ,
p
behavior
of
2.2 b e l o w ,
where
is g i v e n .
This
modification
of
consequences
is e x p o n e n t i a l
and
the
p a
conditions
equation.
suppose
< ~ '•
+ ~ S~_rm(t,s)ds
results
to t h e
in T h e o r e m
integral
hold
(2.17)
StP(s)ds;
property
a slight
1 •
on providing
of
is r e m e d i e d
exponential
of C o r o l l a r y
Theorem
we have
convergence
"strong" result
÷ ~ Is
=
(2 • 18)
y < i;
(2.19)
63 t+h 0suPt >t0ft m(t+h,s)ds=
limh÷
Then
p
tends
specifically, there such
exists
to
zero
let
0 .
at an e x p o n e n t i a l
(2.4)
hold
a constant
if
y > 0
K(~) ,
(2.20)
rate
and
as
t ÷ ~.
(2.5)
independent
of
if p
More
y = 0. and
Then
t l~
that -e (t-t I ) P(t) ! K ( a ) e
tI I Ipl Itl_r
v def = sup u PIlu <s
An i m m e d i a t e result
for the
(2.21)
p (s)
consequence
functional
(t >_tl),
of T h e o r e m
integral
2.2
is a s t a b i l i t y
equation
x(t) = / ~ _ r G ( t , s , x ( s ) ) d s .
H e r e we take conditions and
tO ,
in o r d e r
solutions
COROLLARY
x
3.
exist
(2.22)
to be c o n t i n u o u s that
the
locally
integral (see,
and r e q u l r e in
e.g.
(2.21)
[4, pp.
G
to s a t i s f y
is w e l l - d e f i n e d 86-87,
98]).
Suppose D
IG(t,s,u) I <_ m (t,s) lul~
with
m
tO = 0 . stable such
satisfying Then in the
that:
the
the
zero
large.
same
conditions
solution That
(2.23)
J
t >_O,t-r <_ s < _ t , u s R n
is,
as in T h e o r e m
is e x p o n e n t i a l l y there
exists
2.2
for
asymptotically
a constant
~ > 0
64
for every for each
> 0
there exists a
t I ~0
and each initial
[t I - r ,t I ]
satisfying
solution
of
x
M(6) > 0
(2.21)
such that,
function
l~(s) I ! 6
%
on
(t I - r < s
the
satisfies
-~ (t-tl) s u P t l - r < s < t 1 I~ (s) I
Ix(t) [ ~ M(~)e
for all
t~t I .
PROOF.
Setting
p(t) =
lows from T h e o r e m
Ix(t) I,
we see that
Corollary
3 fol-
2.2 and the inequality
t Ix(t) I i f -rlG(t's'x(s) [ds ! / t_r m(t,s) Ix(s) Ids"
We close this section
by p o i n t i n g
out that we have
chosen in this paper to place the major emphasis on certain integral
inequalities.
As a result,
been given in the most general
form.
the a p p l i c a t i o n s
This is p a r t i c u l a r l y
true since most examples we have considered ferential
equations
equations
By employing
the usual n o t a t i o n
differential
equations,
by c o n s i d e r i n g (cf. 3"
(2.8),
(2.17).
for functional
one can easily generalize,
an inequality
to simple
(2.16) and
x' = f (t,x t)
dif-
say C o r o l l a r y 2,
If(t,xt) i ~ l ~ _ r m ( t , s ) I x ' (s)Ids
[2] ) . Proofs of the Theorems.
PROOF OF T H E O R E M
2.1.
M u l t i p l y both sides of to
for functional
have a c t u a l l y been r e s t r i c t e d
differential-difference
have not
T
(T > t0)
This proof (i.i) by
to obtain
is not at all difficult. ~(t)
and integrate
from
to
65
T T t St0 ¢ ( t ) p ( t ) d t < S t 0 ¢ ( t ) / t _ r m ( t , s ) p ( s ) d s d t (3.1) T s+r < c + / t p(s)I s ~(t)m(t,s)dtds, 0 tO
s+r p(s)S ~(t)m(t,s)dtds. C h a n g i n g the v a r i a b l e of t0-r s i n t e g r a t i o n in the l e f t - h a n d side of (3.1) to s and a p p l y i n g
where
c = S
(2.1) , we have T
T
s+r (s) - fs ~(t)m(t,s)dt]p(s)ds
$t0~(s) p (s)ds ~<St0[~ Since
(3.2)
proof
is c o m p l e t e .
PROOF
OF T H E O R E M
(whichever
s=
Then
holds
arbitrary
2.2.
Fix
is a p p r o p r i a t e )
(l-ye
0 < s < i,
integer
for
n
~r
and
such
~ > 0 and mr
)/(l+e
so by
T > to,
(2.2)
according
follows
to
(2.4)
or
<_c. (3.2)
and the
(2.5)
define
(3.3)
).
(2.20)
one
can
find
a positive
that
t+h S
t
m(t + h , s ) d s <
.
~
.
(0 < h < r/n; . .
For the sake of n o t a t i o n a l c o n v e n i e n c e , set t b(t) = / t _ r m ( t , s ) d s . F r o m (2.19) and the c h o i c e of (3.3),
we can
find
constants
b(t) <_ (i - s)e ~
B,T~ 0
(t > t 0)
(3.4)
t > t0).
such
in
that
and (3.5)
-~r b(t) <_ ( i - s)e
(t>t0+T)
.
66 -mr
It follows
from
(3.1) that
b(t) < (i - e)e
y + ~ = (i - g ) e
,
which means
(t>t) --
0
(3.5) -~r
b(t) < (i- g)e
Let
0 < h < r/n;
t>tl>t
0
and use
(i.i)
tt + h _ r m ( t + h , s ) p ( s ) d s + / t
_<$
p(t+h)
( t > t 0 +T) "
and
t+hm(t + h,s)0(s)ds
t
(3.4) to obtain
t+h
Ol It+h_ r + ~ I iPi 1t t
< b (t + h) t-r
t+r/n
+ llptl
t
V
II II
where
is defined U
over
0 < h < r/n,
divide by
1- ~
Now, take the supremum t+r/n move the term ~1 iPl 1t to the other side, and employ (3.5) to obtain
t+r/n I IPI It !
B e
in
(2.21).
t I ]pi it_r
(t h t l ) ,
I (3.6)
t+r/n t !
Ilpl First,
consider
non-negative
integer
e-~rl
lpl
the interval k,
+ kr/n-r.
(3.6) d
<e k--
d
k-i
(t!tl + T ).~
tl
For every
define
+ kr/n
Then by
it t-r
(k> I),
(3.7)
87
which
implies Bk d
This
< e k --
d
inequality
(k > i) .
0
combined
with
(3.7)
yields
t1
KillPllti
p(t) !
-r
(3.8)
(tl < t < t i + T),
(l+nT/r) where
K
1
Next,
= e consider
the i n t e r v a l
t+r/n
tl+
(3.6) w i t h
By
(3.6) ,
t (t > t I + T) .
I IOl It+r/n_ r ~ I IPl It_ r
Thus,
T _< t < ~.
t
replaced
by
t + r/n
yields
t+2r/n -ar tTr/n -~r I t I IPl It+r/n ~ e I Ipl I t + r / n _ r ~ e Ipl It_ r. Similarly,
for e v e r y
k~l,
t+kr/n
Ilpll
-~r
t+(k-l)r/n-
t
llpl t-r
< e
(t > t --
+ T) . I
However, t+kr/n
t+r I IPl It
=maxl
< k < n I Ipl t + ( k - l ) r / n '
and so t+r
I Ipl I t In p a r t i c u l a r ,
-~r _< e
choosing
t IIpIIt-r
(t -> t I + T)
t = t] + T + kr,
k = 1,2,3,ooo
(3.9)
68
and solving
(3.9)
recursively
-~kr I
tl+T+kr I Ipllti+T_(k_l)r
For any
t ~ t I + T,
t I + T + kr
~ e
choose
This combined
with
p(t) ! K l e
k
so that
t l + T + (k - l)r ~ t
Ipl Itl < e-e(t-ti+T) tl+T tl+T-r-I IPl Itl+T-r.
3.8) yields
eT -~(t-tl) t1 e I Iplltl_r
(2.21) holds with
The proof of Theorem
K(e) = Kle
of Volterra
applying
results
recent
of
(t>t I ) .
~T
B (i+nT/r ) Kl=e
2.2 can be shortened
the terminology
above proof
tl+T IPl Itl+T_r.
and obtain
p(t) < e -~kr --
That is,
leads to
equations [3].
since it requires
and,
However,
by borrowing
in particular, we have retained
only elementary
from
by the
techniques.
REFERENCES [i]
O. Arino and P. S6guier, Publications Math4matiques, Universit4 de Pau; Pau, France, 1980.
[2]
F. V. Atkinson and J. R. Haddock, Criteria for asymptotic constancy of solutions of functional differential equations, J. Math. Anal. Appl., to appear.
[3]
G. Gripenberg, On the resolvents of nonconvolution Kernels, Funkcial. Ekvac., 23 (1980), 83-95.
[4]
R. K. Miller, "Nonlinear Volterra Integral Equations", W. A. Benjamin, Menlo Park, California, 1971.
Volterra
BIFURCATION
OF PERIODIC TRAVELLING WAVES
FOR A
REACTION-DIFFUSION SYSTEII by
D.L. Barrow
and
P.W. Bates
The second author was supported by an SERC v i s i t i n g g r a n t at H e r i o t - W a t t U n i v e r s i t y . I.
fellowship
INTRODUCTION In a recent paper we r e p o r t e d on the e x i s t e n c e and s t a b i l i t y
steady s t a t e and t r a v e l l i n g equations.
wave s o l u t i o n s of a c e r t a i n system o f r e a c t i o n - d i f f u s i o n
wave s o l u t i o n s b i f u r c a t i n g from a c i r c l e
To show the r i c h s t r u c t u r e
The system ( l . l ) sufficiently
2w-periodic
of degenerate steady s t a t e s o l u t -
o f the s o l u t i o n set o f the system below, and to
set the stage f o r the new r e s u l t s we f i r s t
study w i l l
2~-periodic
Here we give c o n d i t i o n s which imply the e x i s t e n c e o f o t h e r
travelling ions.
of
summarize the contents o f I l l .
was proposed by J.~1. Lasry as a model which may have p r o p e r t i e s
s i m i l a r to the Hodgkin-Huxley equations o f nerve conduction t h a t i t s lead to a b e t t e r understanding o f t h a t system.
Consider (l.l)
I ut = uxx + k
l where
vt
-O)u - ~#(B)v
Vxx + k 1 - O)v + s~(B)u
u = pcos~, v = psinB,
smooth
k
s
-~<
ed towards the u n i t c i r c l e ing on c e r t a i n rays from viz.,
(0,0)
and a r o t a t i o n a l
,
t > O, k > 0
and
#
is a
Thus, the v e c t o r f i e l d
symmetric r a d i a l component d i r e c t -
component which changes d i r e c t i o n ,
vanish-
There are, t h e r e f o r e , constant s o l u t i o n s
and p o i n t s where these rays meet the u n i t c i r c l e .
We seek s o l u t i o n s which are considering first
[0,27].
has a r o t a t i o n a l l y
(u,v) = (0,0).
x <~
are parameters w i t h
2 w - p e r i o d i c f u n c t i o n which changes sign on
determined by the n o n l i n e a r i t i e s
of ( l . l ) ,
and
,
the case where
2 ~ - p e r i o d i c in c = 0
x
and which t r a v e l w i t h t i m e ,
and the speed o f p r o p a g a t i o n is zero.
Then
70 (I.I)
reduces to f
(1.2)
lu'+
k(l-p)u
= 0
Lv " +
k(l - p ) v = 0
,
u,v 2T-periodic.
The f o l l o w i n g lemmas are e a s i l y e s t a b l i s h e d . Lemma 1.I
A l l s o l u t i o n s of (1.2) with
Idea of proof If
and points
(u,v)
Lemma 1.3
satisfy
u2 + v 2 < I .
Suppose not and use the maximum p r i n c i p l e to reach a c o n t r a d i c t i o n .
Lemma 1.2
Idea of proof
k > 0
0 < k < 1 with
then (1.2) has only the constant s o l u t i o n s comprising (0,0) u2 + v 2 = I .
Use Lemma I . I
and a comparison of eigenvalues argument.
The s o l u t i o n set of ( I . 2 ) is i n v a r i a n t under the group
in the uv-plane and in the independent v a r i a b l e . can be obtained from a s o l u t i o n uv-plane and t r a n s l a t i n g in Idea of proof the f i r s t
The f i r s t
(u,v)
with
u
0(2)
acting both
Furthermore, any s o l u t i o n of (1.2) even and
v
odd by r o t a t i n g in the
x.
part f o l l o w s by i n s p e c t i o n .
The second p a r t f o l l o w s from
p a r t and a uniqueness argument f o r the corresponding f i r s t
order system.
There are c e r t a i n e a s i l y found s o l u t i o n s of (1.2) which we c a l l c i r c u l a r and c o l l i n e a r because o f t h e i r image in the uv-plane. is a f a m i l y of c i r c u l a r s o l u t i o n s given by v(x) = (I - j 2 / k ) s i n j x action). at
For each nonzero i n t e g e r
there
u(x) = (I - j 2 / k ) c o s j x ,
(and those generated from t h i s p a i r by the symmetry group
So we have a branch (surface) of s o l u t i o n s b i f u r c a t i n g from
k = j2
j
and e x i s t i n g f o r
k >j2.
(u,v) ~ (0,0)
In seeking c o l l i n e a r s o l u t i o n s , t h a t is
l i n e a r l y dependent, we can in view of Lemma 1.3 assume t h a t
v ~ 0
and
u
u,v
is even.
The system (1.2) becomes (I,3)
U" + k(l - l u l ) u
= O,
u
2~-periodic.
E i t h e r by using a b i f u r c a t i o n argument or a shooting method, as was done in [ l ] ,
one
can prove Theorem 1.4
For each nonzero i n t e g e r
n o n t r i v i a l even s o l u t i o n s [0,27]
and
lim u j ( . ; k ) k÷ j2
{uj(.;k):k = O.
j, >j2}
problem (1.3) has a continuous f a m i l y of such t h a t
uj(.;k)
has
2j zeros in
71
Returning to the c i r c u l a r s o l u t i o n s , i f we r e w r i t e (1.2) in p o l a r coordinates and l i n e a r i z e about a p o i n t on the j t h branch of c i r c u l a r s o l u t i o n s we obtain the operator defined by L(p,#) =
[i"-
2j(k-j2)B'/k
- (k-j2)p]
"+ 2jkp'/(k-j2) Considered as an o p e r a t o r from L
is n o n i n v e r t i b l e ( f o r k > j 2 )
such s i n g u l a r value of range.
k,
L
J"
{p,# E C2(S1): p even, B odd}
iff
into
k = 5j 2 - n 2, n = 1,2 . . . . . 21j I- I .
(C(Sl)) 2, For each
has a one dimensional kernel and codimensional one
I t is easy to check t h a t the other hypotheses of Theorem 1.7 of Crandall and
Rabinowitz [2] are s a t i s f i e d ,
thus o b t a i n i n g secondary b i f u r c a t i o n from the branches
of c i r c u l a r s o l u t i o n s at the above values of Now consider ( I . I )
with
upon the s i n g l e v a r i a b l e
E ~0
and look f o r 2~-periodic s o l u t i o n s depending
z = x - ct (waves t r a v e l l i n g with speed
parameter which may depend upon
(1.4)
k.
c.
c), where c
is a
This gives the system
[u"+
k(l - p ) u + cu' - c¢(B) v = 0
Itv " +
k(l-
p)v + cv' + c¢(B)u = 0 ,
u,v
2~-periodic in z.
Again, changing to p o l a r coordinates and l i n e a r i z i n g about the p o i n t (p,B,c) = (I - j 2 / k , from
jx,O)
on the j t h branch of c i r c u l a r s o l u t i o n s y i e l d s an operator
{p,# ~ C2(SI):B(O) = O} x ~
to
(C(SZ)) 2.
t h i s operator is an isomorphism, so the i m p l i c i t of s o l u t i o n s to (1.4) f o r stability
Iel
For
k # : 5 j 2 - n 2, n = 1 , 2 . . . . . 21j I- I,
f u n c t i o n theorem gives the existence
small, near the c i r c u l a r s o l u t i o n s of (1.2).
The
of these t r a v e l l i n g waves and steady s t a t e s o l t u i o n s can be determined
using l i n e a r i z a t i o n , S a t t i n g e r [5]
the p e r t u r b a t i o n theory in Kato [4] and the s t a b i l i t y
(or Henry [ 3 ] ) .
r e s u l t s of
This has been done in [ I ] .
The question now arises as to the existence of t r a v e l l i n g wave s o l u t i o n s of (1.4) with
c ~0
in the v i c i n i t y
of the c o l l i n e a r s o l u t i o n s of (1.2).
The next section
is devoted to answering t h i s question and these r e s u l t s are not contained in [ I ] . 2.
BIFURCATION FROM THE COLLINEAR SOLUTIONS Recall t h a t f o r each f i x e d i n t e g e r
solutions, (ui(x;k)cos~,ui(x;k)sin~),
n and
k >n 2
~ c [0,2~],
there are
i = 1....
Inf.
Inl
c i r c l e s of
We w i l l
consider
72 one of these c i r c l e s and t r e a t some j , k
and r e c a l l t h a t
~
~
as a b i f u r c a t i o n
i s even and has
parameter.
2j
zeros in
that
B = ~(u,v)
restrictive
solutions,
passing through
u = u.(.;k) J
for
[0,2~].
We would l i k e to use the b i f u r c a t i o n r e s u l t s found in [2] degeneracy of the c o l l i n e a r
Let
but because o f the
( u , v ) = (0,0)
w i t h the f a c t
is not continuous t h e r e : we need an a b s t r a c t theorem w i t h less
hypotheses.
The f o l l o w i n g two theorems are probably known but we have
not found them in the l i t e r a t u r e .
The f i r s t
is an i m p l i c i t
second is a b i f u r c a t i o n theorem proved using the f i r s t
f u n c t i o n theorem, the
theorem in much the same way
as the C r a n d a l l - R a b i n o w i t z r e s u l t [ 2 ] . Let
Y,Z
be Banach spaces and
w i t h domain in Theorem 2.1 on
Y
S×Y
Suppose
for all
f(s,y)
s E S
f(O,O) = 0
(ii)
L
(iii)
N(O,y) = o(rlyll)
(iv)
f o r some
6 > 0
Using ( i i )
IIN(O,y)II
continuous by ( i i )
for
takes
iTyII ~ r .
into itself Z
with
r
Solving
f(s,y) in
Isl < ~,
for
as
llyH ~ r
{y:[iyIT ~ r } .
s --> O.
(O,R] such t h a t is e q u i v a l e n t
This map i s c o m p l e t e l y + L-I(N(O,y)). for
Iiyll ~ r
the c o m p l e t e l y continuous map
if
be Banach spaces;
suppose
i s continuous.
to
L1
i s a map from
Z with dom(Ll(~))
By ( i v ) Is I < ~.
T
and has a f i x e d p o i n t by Schauder's theorem.
Define
x c domL1(~ ).
small in
TIN(s,y)-N(O,y)II
~.
E R and
= 0
there exists y(s) E y
y ( s ) --> 0
L-1(N(s,y)-N(O,y))
X
M I : ~ × X --> Z
Isl < 6
arbitrarily
( p o s s i b l y unbounded) l i n e a r o p e r a t o r from Suppose
u n i f o r m l y f o r liyH ~ R as s --> O.
Furthermore,
Ty ~ L - 1 ( N ( s , y ) )
such t h a t
X and
L - I : Z --> Y.
s
= O.
we may choose
s E S with
{y:iiyll < r } Now, l e t
Assume
and
and may be w r i t t e n as
~> 0
Hence, f o r each
y --~ 0
f(s,y(s))
to f i n d i n g a f i x e d p o i n t of
there exists
small.
such t h a t f o r a l l
and ( i i i )
be a mapping
( s , y ) E dom(f) w i t h N ( s , - ) continuous
R > O, N(s,y) - N(O,y) --> 0
Iiy(s)il ~ R such t h a t
Proof
for
is l i n e a r w i t h compact inverse as
f
Z.
= Ly - N(s,y)
sufficiently
(i)
Then t h e r e e x i s t s with
and range in
S a normed l i n e a r space and l e t
~
[] i n t o the
independent o f
~ ( ~ , x ) = L1(~)x + M l ( ~ , x )
for all
73
Theorem 2.2 and t h a t
!11(I,x) = o(llxlr)
L l ( 1 ) x = Ll(O)x + L21x + !12(A)x
a t o r s from sets.
Suppose t h a t
X
to
Z
and
Suppose t h a t
as
with
M2(1)x = 0(111)
H2(.)x
x --> 0 L2
as
LI(O) -~
X
Finally,
that for all
R(LI(O))
into
suppose t h a t
= 0
and
Proof
and
y = (X,x) ~ R x X ~
y
[l(A)(Uo+X)
s = 0
f(s,y) Let
--~ O, x ( s ) --> 0
if
= I~(A,S(Uo+X))
Ly =- L l ( O ) x + L2lu o
and
if
N(s,.)
and
1 N(O,y) - N(s,y) = y M l ( l , s ( u 0 + x ) ) - - > 0
so
L
Furthermore,
Finally,
L-I
as
span{uo}) is 6 > 0 such
s --> O.
s #0.
if
s = 0
f(O,O) = O,
We have codimR(Lz(O)) = I , L I ( O )
is an isomorphism from
has a
such t h a t
M2(1)(u0+x ) - ~H1(l,s(u0+x))
and
bounded.
LI(O )
N(s,y) - Ly - f ( s , y )
4[ - k 2 1 x
i s continuous.
Suppose
in bounded
define
r - L 2 t x - H2(Z)(Uo+X ) N(s,y)
SO
bounded
c o d i m R ( L l ( O ) ) = 1 and t h a t
l ( s ) ~ ~, x ( s ) ~ X
l(s)
x
Then t h e r e e x i s t s
F(l(s),s(u O+x(s))) s E ~
uniformly for x.
~
bounded l i n e a r oper-
(a t o p o l o g i c a l complement of
L2u 0 ¢ R ( L I ( O ) ) .
Ism ~ 6 t h e r e e x i s t s
For
112(I)
i s continuous f o r each f i x e d u 0 ~ X, t h a t
compact.
and
I --~ O,
one dimensional n u l l space spanned by defined from
uniformly for
Y onto
as is
s ~ O,
N(O,y) = o(HylT)
s --~ O, 1- 1
if
on
uniformly for X
and
as
y --> 0
(l,x)
L2u 0 ¢ R(Lz(O ) ,
Z.
is compact by the assumption on
LI(O) -I
and Theorem 2.1 gives
the c o n c l u s i o n . Returning to the d i f f e r e n t i a l
equations, let
u"+ k(l - p)u+cu' - ~,(B(u,v))v] (2.1) where ¢
F(u,v,c,c) p
~B = @(u,v)
k(1
+ c¢(@(u,v))u
u = p c o s 6 , v = psih~B.
so t h a t
¢(B(',.))
uv-plane and is u n i f o r m l y bounded t h e r e .
k(1-lul)u
= 0
Let
Let
We suppose t h a t
is d i f f e r e n t i a b l e ~[
be the s o l u t i o n of
0,0),
then
F ( w ( t ) ) - O.
We w i l l
write
X = {x = ( u , v , c , 8 ) E (HI(S1)) 2 x[~ 2 : u ± u-', v l u-}
a ± b
whenever
w i t h norm given
-IT
by
on the
mentioned a t the beginning o f t h i s s e c t i o n and l e t
w(X) = ( ~ c o s X , u - s i n l , F a b = O.
p)v+cv'
are given by
i s 2 ~ - p e r i o d i c and d i f f e r e n t i a b l e ,
punctured u"+
and
= (v"+
Ilxll 2 = IlUllH1 H1 + c 2 + ~2, 2 + [Iv112
then
X
is a Banach space.
Define the
74 (unbounded) operator in (2.2)
RxX
to
Z--- (L2($1)) 2
by
F ( X , x ) = F(w(X) + x). ru"+k(1-lul(1 +cos 2 ~ ) ) u - k l u l c o s Z s i n Z v + c u ' c o s Z - c q ~ ( - ~ ( Z ) ) u - s i n X ] L l ( Z ) x = Lv " + k ( l luI(l+sin2z))v k1~IcosZsinZu+c~'sinZ+c#(~(Z))ucosZJ
Let
where -#(~) = I~
if
u-> 0
+ ~ Let
if
-O-<0.
Clearly,
Mz(~,x ) = F ( ~ , x ) - L~(X)x,
to see t h a t
HI : ~ x X
--> Z
dom(Lz(~))
is independent of
~.
then since second order terms cancel, i t is easy
is continuous.
The f i r s t
component of
Hl(~,x )
is
mz ---- k cos ~[ [~I (u cos ~ + v sin ~) +#( I~I - U(ucos ~ + u) 2 + (u- sin ~ + v)2)] + + cusin~[q~(~(~))-q~(8(ucos~+u,usin~+v))]
+ terms which are
in
term can be w r i t t e n (when u ~ 0 )
~,
as
x --> 0.
The bracket in the f i r s t
-2E~I/(I~I +w(
~(ucosZ+vsinZ)[l
and both of these terms are second term of
mI
)2+(
o(llxIl) in
is seen to be
o(llxll), u n i f o r m l y
)2)] +~(u2+v2)/(l~l +/( )2+( L2
o(lixll)
uniformly in in
L2
Dominated Convergence Theorem (the uniformity in
as Z
~
as
ilxll --> 0
llxll --> O.
as
)2) The
by using Lebesgue's
is not difficult to prove but
we omit the proof to save space). The second component of Lemma 2.3
Let
Au = u " +
k(l-2]~l)u
Bv = v " +
k(1-I~l)v
uo >!Jz > u 2 t>- . . . .
u"+
is similar to
A, B : H 2 ( S 1) --> L2(S 1)
Let the eigenvalues of
Proof
Hl(~,x )
A and
be given by
.
B be denoted by
respectively.
X2j_l = 0
and
N(A) = span{u')
(ii)
~2j = 0
and
N(B) = span{u-}.
k ( 1 - j u I)u = O,
u - e N(B)
and since
differentiation
u ( - + h)
shows t h a t
standard S t u r m - L i o u v i l l e theory gives (a) #i > X i ' an eigenvalue f o r both [-~,%).
A and
~o > x l
> X2 > . . .
and
Then
(i)
By d e f i n i t i o n ,
m 1.
is a 2~-periodic s o l u t i o n of
~ ' e N(A). i = 0,1 . . . . .
Since
I]i >0 Also, (b)
B w i t h the eigenfunctions each having
2j
Again using Sturm theory (c) the eigenfunctiens corresponding to
a.e., 0
is
zeros in
/b 12j-l'
~2j-l'
12j' ~2j
Statements (a),
are p r e c i s e l y those which have
2j
zeros in
[-~,~).
(b) and (c) imply
12j < ~ 2 j
= 0 = 12j_l < ~ 2 j _ l
and
12j_l < 1 2 j _ 2
and
P2j+l < ~ 2 j '
which
proves the lemma. We now assume
(HI)
$(0) : - $ ( ~ ) .
To f i n d
N(Li(O))
set
Li(O)x = 0
with
x E X.
This gives
Au + c~' = 0
Bv + e$(~(O))u = O. M u l t i p l y i n g the f i r s t and the d e f i n i t i o n symmetry of
u,
equation with see that
of
X we have
u = O.
there i s , f o r each v ± u.
N(LI(O))
f , g E L2(S i)
equation by u ' and i n t e g r a t i n g gives
then
Thus, i f
u0 ~ (O,#,O,l).
span{uo}
LI(O) : ~ n dom(Ll(O)) --> R(Li(O))
and
M2(I) :X --> Z
v,
By Lemma 2.3
by (HI) and the of the second
is t h a t s o l u t i o n corresponding to
( f , g ) T E R(LI(O)) i f f
pactness of the embedding of
~(~(O))u± u
c, a unique s o l u t i o n ,
V
is spanned by
is a t o p o l o g i c a l complement of
L2
Since
c = O.
g ± u, in
in
we
One can now check t h a t i f so
codimR(Li(O)) = I .
If
X one sees t h a t
is i n v e r t i b l e .
H2(S1)
E = l,
HI(S 1)
Furthermore, because of the comthe inverse is compact.
Define
by
M2(I ) = L1(I) - LI(O) - IL2.
Then, apart from terms which are obviously
bounded,
M2(1 )
has components
o(111)
as
I --> O,
cu($(~(O) I - ~ ( ~ ( 1 ) ) s i n l )
eu($(~(1))cosl
- $(~(0))-I$'(~(0))).
uniformly for
Icl
This ensures t h a t
o(11])
bounded, by Lebesgue's Dominated Convergence Theorem.
$'(0) ~-$'(~). $'(~(O))u ~ u
and so
L2uo ~ R(LI(O)).
l!xll
and
These too are seen to be
able to apply Theorem 2.2 we need to assume (H2)
uniformly for
as
I --> 0 To be
76
We have proved Theorem 2.4
Supposethat
angle such that interval
I
continuous at
~
is d i f f e r e n t i a b l e and 2~-periodic and l e t
~(0) + # ( 8 + ~ ) = O,
containing 0 c I
0
#'(9) + # ' ( 8 + ~ ) 5 0 .
and mappings
9
be any
Then there exists an
~ , c , c : I --~ m, u,v : I --~ C2(Sl)
which are
and take on the value zero there, such that f o r each s c I ,
(1.4) is s a t i s f i e d with
(u,v,c,c) =
(ucos(O+~(s)) + s ( u ( s ) - ~ s i n O ) , u s i n ( O + ~ ( s ) ) + s ( v ( s ) + ~cosG), sc(s), s + s c ( s ) ) The only claims in the theorem not discussed above are the r e g u l a r i t y , which follows by bootstrapping, and that
9
need not be zero which is not r e a l l y a d i f f i -
culty since we can rotate in the uv-plane and t r an s lat e
#.
REFERENCES l°
D.L. Barrow and P.W. Bates, Bifurcation and S t a b i l i t y of Periodic T r a v e l l i n g Waves f o r a Reaction-Diffusion System, to appear, J. D i f f e r e n t i a l Equations.
2.
M.G. Crandall and P.H. Rabinowitz, Bifurcation from Simple Eigenvalues, J. Functional Analysis 8 (1971), 321-340.
3.
D. Henry, Geometrlc Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics vol. 840, Springer-Verlag, New York, 1981.
4.
T. Kato, Perturbation Theory f o r Linear Operators, Springer-Verlag, New York, 1966.
5.
D.H. Sattinger, On the S t a b i l i t y of Waves of Nonlinear Parabolic Systems, Advances in Math. 22 (1976), 312-355.
The D i r a c
Equation with
an Anomalous M a g n e t i c Moment I I
Horst
Though i t
is
well
known,
m a g n e t i c moment, t h i s mathematical
that
make i t
t h e e , l e c t r o n possesses
term has n o t
investigations
has a t most been t r e a t e d literature.
Behncke
of
the
to
study
the
of
This
singularity
leads
Most n o t a b l e among t h e s e Hamiltonian
for
is
almost all
is
are c a r r i e d
out mainly
the
The D i r a c
Hamiltonian
tromagnetic
potential
for
equation [3,4,5,6
]
methods i n
Barut [I]
A and s c a l a r
small
field
rather
sin-
results. of the
distance
be-
spectrum of the Dirac The i n v e s t i g a t i o n s
symmetric in
author [2]
interesting
paper.
spherically
and i t
physical
selfadjointness
The e s s e n t i a l
an e l e c t r o n
term is
and t h e
this
the
due to t h e anomal.ous
essential
of
the
and t h i s
because t h i s
potentials
t h e main o b j e c t for
in
to a number o f
h a v i o u r of the w a v e f u n c t i o n s . Hamiltonian
so f a r
interaction
m a g n e t i c moment n o n p e r t u b a t i v e l y , gular.
Dirac
by p e r l u ~ a t i o n
Recent i n v e s t i g a t i o n s
desirable
been c o n s i d e r e d
an anomalous
potentials.
an e x t e r n a l
static
elec-
B is [2].
H = ~ To Tk ( _ i ~ k - eo A k - e I A A k ) + (1)
e ° Ao + elAA o + mot o + y°B + i i
-
f Z Tk~ k A°
Here e o r e p r e s e n t s f
f ZT°Tk¥ 1 (2 I A k)
the c h a r g e ,
eI
the e l e c t r i c a l
dipole
moment and
t h e anomalous m a g n e t i c moment.
Theoretically w h ich a r i s e
one c o u l d
from r a d i a t i v e
However, s i n c e the
these
terms
same o r d e r as o t h e r
addition
also
we assume t h a t
include
higher
corrections
in
lead to
effects,
terms,
quantum e l e c t r o d y n a m i c s .
nonlocal
we r e s t r i c t
t h e most s i n g u l a r
moment i n t e r a c t i o n
interactions ourselves
terms
in
(i)
and ar e o f to
(1).
In
are s p h e r i -
78
cally
symmetric
corresponds
and due to
to
the
charge
the electric
potential
distribution
of
the
as
(2)
H = m • (-i
V)
With
V e = eoA ° + e l & A o and Vm = fA o s p h e r i c a l l y
a symmetric tions
of
4 by 4 m a t r i x ,
A1 .....
L 2 (R3)
we r e q u i r e
in
for
d o m a i n PH = ~ o
H to (~)
Ve,V s , . . . , ~ 3 A 3
(3.2)
sgn Vm ( r )
symmetric. are
linear
be a s y m m e t r i c
~ ~ 4 , R+3 = ~ 3 \
is
are
real
valued
constant
for
and
small
in
of
l/r,
physical
V e,
Thus H
symmetric, is
o
is
it
Each K ( c , ~ , m )
unitarily
on
be more s i n g u -
satisfied
equivalent
# = 1,2,3 ....
operates
on L 2 ( 0 , ~)
in
all
cases
to
by u s i n g
a direct
(~,~,m)
1
#
\ _ D + E ( _ I ) £ ~£ - Vm
d o m a i n P = t ~o ( ( o , ~ ) )
~
2 ¢
/2
~ ~]2 by
g
Ve + Vs + m
m)
V
;P=d~
po-
sum o f
c = +- 1
D+c(-1) z
[ V e - Vs - m
with
on
{0}
functions
can be s e p a r a t e d
m = -~+ I / 2 , . . . ,
K
operator
K (c.l.m).
H o _.=- zo K ( c , Z , m )
(4)
combina-
r
V s and H 1 n e a r O. T h i s
spherically
coordinates.
operators
s
interest.
Since H is o lar
H1
L 21oc ( ~ )
Here gk ( ~ # ) i s t h e s p a c e o f k - t i m e s d i f f e r e n t i a b l e o R +3 w i t h c o m p a c t s u p p o r t . We s h a l l a l s o assume Vm t o than
(1
addition:
(3.1)
lar
field,
~r Vm + H1 = Ho + HI
whose c o e f f i c i e n t s
~3A3 . In o r d e r
~ ~4 w i t h
ibm
Because AA 0
electric
can bc r e w r i t t e n
+ V e + B(V s + m). -
A o.
d
7g For
simplicity
of
notation
Ve-V s (5)
) -D+V 3
Let
now
D+V 3
K ~(
Theorem
consider
with
((0,~))
DK = C~o
~ ~2
Ve+V s
1:
Ho be as i n lim r÷O
ii
(2).
In
addition
1 > c Ue,
assume
V s = CsV m + U s
+ Ws w i t h
2 2 - c and e s
Us a r e
locally
integrable
near
0 and
= lim W (r) = 0 r÷O s /Vm(r )
limr÷o W e ( r ) / V m ( r )
Then H ° i s
(3)
r Vm(r )_ = ±
V e = CeV m + Ue + We;
iii
to
essentially
selfadjoint.
Proof: a)
Ho i s
Since ally
essentially
K is
limit
selfadjoint
selfadjoint,
point if
at
the
only
(O,b),
one n o n t r i v i a l
for
infinity
solution,
true
theorem
found
Here we s k e t c h
in [1].
transformation
all
5.1 ] , i t
which
is
square
= - {V 1 cos28
for
[6]
the
solutions
another
of
K (e,l,m). is
essenti-
integrable
proof.
this With
= -
+ V2 s i n 2 8
1/2(V I
equation the aid
u = pcosO and v = psin@ + V32 sinO
cos@ }
(6) +(Inp)'
for
over
some b > O. estimates
8'
[6,
is
= 0
The n e c e s s a r y
PrUfer
this
equation
K (~) has
if
V2)
sin
20 + V 3 cos
2e
can be of
the
, one o b t a i n s and
80 Argue now as i n [ 5 ] lim - c sin r÷O s
and show t h a t 2e+(r)-
Then the c o r r e s p o n d i n g p(r) 2 ~ p (ro)2 Thus t h e r e
is
(6)
has a s o l u t i o n
cos 2@+(r)
= ~
p can be e s t i m a t e d
r° exp 2 f r
a solution,
( Jl
which
-c~
+ C2s '
by
2 2 )' - ce + cs V3(t)
is
not
readily
to
e+ w i t h
square
dt
integrable.
Remark: This
result
As i n [2] dices
can be e x t e n d e d this
method a l s o
allows
o f fro i n more g e n e r a l
to d e t e r m i n e
L =l o c
This
is
In t h e of
(~3)
Then H i s
the d e f i c i e n c y
in-
matrix
coefficients
o f H1
selfadjoint.
shown as i n [ 2 ] remainder
we s h a l l
restrict
ourselves
spectrum
to
an i n v e s t i g a t i o n
o f H.
i:
L e t H be as i n (7)
all
essentially
the spectrum and e s s e n t i a l
Lemma
potentials.
situations.
L e t H° be as above and assume t h a t are
more g e n e r a l
Ai(x),
(2)
Ve(r),
with Vm(X),
Then (-m,m) c c ess(H ) f o r
Vs(X),
H l ( X ) ÷ 0 as
any s e l f a d j o i n t
Ix
= r ÷
extension
H of H
Proof: The f r e e (Hoo) nitely
Dirac
= (-m,m) c.
functions
is
Hoo has t h e
For each ~ ~ (-m,m)
many i n d e p e n d e n t
whose s u p p o r t these
Hamiltonian
compact are
also
essential
there
exists
smooth and ~ - a p p r o x i m a t e and f a r
from t h e o r i g i n .
~-approximate
spectrum therefore
Oes s infi-
eigenvectors, By a s s u m p t i o n
eigenvectors
for
H and
81 Remark: Let
H
be as i n
O
(2 I and assume in a d d i t i o n
Ve, Vm and Vs are monotonic Then i t
follows
from [ 6 ,
H° embedded in H consists
of
§ 8 ]
(-m,m) c
for
large
that
to
(3) and ( 7 ) :
r.
there
are no e i g e n v a l u e s
Thus one would e x p e c t t h a t
(-m,m) c and c o u n t a b l y
a t ~ m. Under p h y s i c a l l y
of
the spectrum of
many e i g e n v a l u e s a c c u m u l a t i n g
reasonable
assumptions
this
is
indeed th
case. For any ~ e C 1 (R 3) w i t h ~D~
ii # II ,
II ~i*[l
one has
~ <
D~ and
c
He
-
#H
=
A#
=
-
i
~
•
with
V~
A~ bounded
Lemma 2: Let H be as in (8)
Ve . . . . .
and l e t
(2) w i t h
~3A3 e L~O c ( ~ i )
% e Cao ( ~ ) "
Then the o p e r a t o r
M}, m u l t i p l i c a t i o n
by ¢,
i s H-compact. Since the p r o o f
of t h i s
Lemma i s
rather
i t . The c o n c l u s i o n s
o f t h e Lemma are a l s o
placed
Stummel c l a s s e s .
by s u i t a b l e
Let H be as in Lemma
2 and
assume
straight true
if
f o r w a r d we d e l e t e L~o c (R3)
lim Hl(X ) = 0 for
is re-
Ixl ÷ ~
Then
H I is H o - C O m p a c t . With
slightly
thening
stronger
of T h e o r e m
i.
assumptions
we can
show a c o n s i d e r a b l e
streng
82 Lemma 3: L e t H be as i n Moreover,
(2)
assume t h a t
rl i,m O Vs(r)/vm ( r)
and
such
lim r÷O
Ir
that
the
Vm i s
potentials
absolutely
satisfy
continuous
= lr÷O i m V e ( r ) /Vm ( r)" = r÷olim Vm '(r)
Vm(r)l
(7)
and
(8).
near 0 with
/Vm2(r)
= 0
= ~
Then Ve and V s a r e
K(c,/,m)-compact
and aes s ( K ( c , / , m ) )
Then V e and V s a r e K ( ~ , l , m ) - c o m p a c t
= (-m,m) c
and
Proof: Considering
the form
Jv'3J
K 2 # V~ This of
estimate
the aid
of
by K on C~(O, ~)
2
~
one sees
12V e V 3 i
together
V e and V s w i t h
With
induced
with
respect
Lemma 2 shows
t o K.
the
relative
compactness
Hence we may assume V e = 0 and V s = 0
the decomposition
method
the
result
is
shown as i n
t61. Remark: Though Lemma 3 i s for
H
o~
because
Thus we have t o
the
Vm i s
for
fixed
estimates
impose
V e and V s a r e (10)
true
the
twice
differentiable
sgn V e + V s + m i s
are
l,
one c a n n o t not
following
differentiable and p o s i t i v e constant
uniform
expect in
smoothness
it
to
be v a l i d
assumptions
n e a r O.
I.
83
t
Ve ' + Vs
Ve
~(r) (r)
:
Ii
+ Vs
ii
:
B(r) 2 r
# 0 and ~ ( r )
< -1
;
V e + Vs+m
(r) Ve + V s +m
Ir
;
(10) V ' m
~(r)
(r)
-
vm with
(r)
m ,# and y c o n t i n u o u s
Though,
{hese
conditions
most
physically
With
these
of
this
seem r a t h e r
interesting
additional
restrictive,
potentials,
assumptions
e.g.
n e a r O.
they
the
hold
for
Coulomb p o t e n t i a l .
we can now show t h e m a i n r e s u l t
paper.
L e t H be as i n (i0)
and ~ ( r )
(2)
and assume t h e
potentials
.satisfy
(7),
(8),
and l i m H l ( X ) = O. Then r~O Oess(H ) = ( - m , m ) c
Proof: Because o f such the
that
Lemma 2 we may assume H I = O. Choose r o > 0 the conditions
potentials
such
Such a change trum
o f H.
With
this
in
transforming
equation will
hold
for
all
r # r o.
Moreover,
V e ( r ) = V s ( r ) = Vm(r ) = 0 f o r
potentials
will
not
affect
the
change
r ~ r o.
essential
spec-
one shows
a(K(e,l,m)) by
the
that
(i0)
so s m a l l
for
= (-m,m) c the
eigenvalue
u as i n [ I
be d e l e t e d .
for equation
] . Since
this
into is
large
1
a single
rather
second o r d e r
tedious,
this
part
84
In
general
the
computation
ther
complicated,
ticle,
however,
of
the
b e c a u s e Vm i s a complete
spectrum
singular
solution
of
at
K(c,/,m)
O.
will
For t h e
be r a -
neutral
par-
can be g i v e n .
Theorem 3: Assume V~ ~ V e ~ 0 o (K(e,/,m))
and t h a t
Vm s a t i s f i e s
(7),
(8)
and
(10).
Then
= (-m,m) c
Proof: Assume ~ ~ o (K)
2 < m2.
satisfies
Then t h e
eigenvalue
equation
fo
u becomes (-D2 + Vm2 _ Vm ' + m2 _ ~2) With
F(r)
= exp
{_(F-1D
Since
the
If
addition
the
in
only
(f~o
Vm(t ) dt)
this
+ (m 2
~2)}
F)(FDF-I)
first
summand i s to
the
eigenvalues
u = 0
can be r e w r i t t e n
positive,
above conditions of
K(e,l,m)
as
u = 0
this
implies
V
m
is
can be +- m.
2 = ~2. m
monotonic
for
large
r,
85 References
1. B a r u t , A . O . , Kraus, J . :
S o l u t i o n o f the D i r a c e q u a t i o n w i t h Coulomb and m a g n e t i c moment i n t e r a c t i o n . J. Mathem a t i c a l Phys% 17, 506-508 (1976)
2.
Behncke, H.:
The D i r a c e q u a t i o n w i t h an anomalous m a g n e t i c moment. Math. Z. 174 (1980) 213-225
3.
J~rgens,
P e r t u r b a t i o n s of the D i r a c O p e r a t o r ; i n : Conference on the Theory of O r d i n a r y and Partial Differential E q u a t i o n s (Dundee 1972), pp. 87-102. L e c t u r e Notes i n Mathematics 280. B e r l i n - H e i d e l b e r g - N e w York: S p r i n g e r 1 9 7 2 - -
4.
Kalf,
5.
Rejto,
6.
Weidmann, J . :
K.:
H.:
P.A.:
A L i m i t P o i n t C r i t e r i o n f o r Separated D i r a c O p e r a t o r s and a L i t t l e Known R e s u l t on Riccati's E q u a t i o n . Math. Z. 129, 75-82 (1972) On Reducing Subspaces f o r O n e - E l e c t r o n . D i r a c O p e r a t o r s , I s r a e l J. Math. 9, 144-171 (1971) Oszillationsmethoden fur Differentialgleichungen. 371 (1971)
Systeme g e w ~ h n l i c h e r Math. Z. 119 , 349-
PERIODIC
SOLUTIONS
OF A C L A S S
HAMILTONIAN
V.
Let
H e C1
0R2n,,R)
nary differential
(i)
BENCI
H. T h i s
(2)
(p,q)
system
- J ~ = H
where
z =
There
(p,q)
study of periodic
the H a m i l t o n i a n
, ~ = Hp
z
(p,q)
• denotes
can be
and J =
are many
- D. F O R T U N A T O
system of
2n o r d i -
equations
p and q are n-tuples,
= gradp
SYSTEMS
CAPOZZI
and consider
p = - Hq
where
- A.
OF
,
d ~-- and Hq = gradq
represented
more
concisely
H, H p = as
(z)
O (I
types
-I O ), I b e i n g
of questions,
solutions
of
such
the
both
systems
identity
local (cf.
matrix
and global, llOI
and
in R n. in the
its r e f e r e n -
ces). Here we are when t--~
concerned
the p e r i o d 1
t,
(3)
and we
about
T = 2~
is p r e s c r i b e d .
of p e r i o d i c
Making
solutions
the change
of
(2)
of variable
(2) b e c o m e s
- J ~. = ~ H
seek
the existence
2~-periodic
z
(z)
solutions
of
(3), w h i c h ,
of course,
correspond
87
to the 2 ~ - p e r i o d i c
solutions
p o i n t s of the f u n c t i o n a l
(4)
f(z)
= /o 2~
If there e x i s t p o s i t i v e
5)
[Hz(Z)]
(+(-J
~,z)
constants
2 n - ~ H(z))
this
f is
~ontinuousl'$
dt.
integrable
"derivative
the f u n c t i o n a l
Fr~chet-differentiable
of 2 ~ - p e r i o d i c of o r d e r
of i n f i n i t e l y m a n y p o s i t i v e
reason
are the c r i t i c a l
k I, k 2, a such that
s p e c t r u m of the l i n e a r o p e r a t o r
consists
solutions
of the a c t i o n
(S I, tR2n) of 2 n - t u p l e s
sess square The
(2). T h e s e
_< k I + k 2 Iz] a
it is easy to see that s p a c e W I/2
of
z~-~-J
functions,
z (with p e r i o d i c
and n e g a t i v e
(4) is i n d e f i n i t e
j2s
[Ujk]2
< +~
w h i c h pos-
1/2". (X) condition~
eigenvalues.
in a s t r o n g
For
sense,
We set L t = L t (S I, ]R2n) , t > i, and for e v e r y ~ erR we shall W s = {u ~ L 2 l j ~
on the
i.e.
set
},
k=l,..,2n
where ~jk W
s
Ujk are the F o u r i e r = e jtJ
~k
is an H i l b e r t
(j E ~
components
~ ~k
with
(k=l ..... 2n)
respect
to the basis
is the s t a n d a r d
space w i t h the inner p r o d u c t
(uIv) W
basis
in ,R2n).
~ .2s s = /-- 3 Ujk Vjk. j,k
88
it is not b o u n d e d
from
nu0us p e r t u r b a t i o n s Using nimax
direct
r > o,
(6)
at
from below,
even modulo
weakly
conti-
I2{ , 151) •
m e t h o d s of the c a l c u l u s
of v a r i a t i o n s
Rabinowitz
if H(p,q)
infinity
(z), z)
proved
in b o t h
2n Z u H tR
(2) p o s s e s s e s
bed period For
or
that
the v a r i a b l e s
p and
and
grows q,
suitable
more
i.e.
mi-
than qua-
there
exists
~ > 2 s.t.
(H z
then
(cf.
arguments,
dratically
above
T
(cf.
(z)
> o
a nonconstant
for
IZ{
T-periodic
> r
solution
for e a c h
prescri-
I9{,[iO[).
"superquadratic"
Hamiltonians
other
results
are a v a i l a b l e
(cf. Ii ] ,
{2{,161,171,181). Consider now
pend
on
time
and
a mechanical
system
suppose
it is e m b e d d e d
riving
from a potential
system
has
(7)
where aij
the
H(p,q)
{aij(q)} do not
V (q) e C 1
following
=~.. aij 13
depend
(q) piPj
on q,
of the
(8)
~U ~x
constraints
which
in a f i e l d
(,Rn,~). The
of
Hamiltonian
do not deforces
of
such
dea
f~rm
is a p o s i t i v e
of n e q u a t i o n s
~ =
that
with
+ V (q)
definite
(i) can
,
matrix
be r e d u c e d
for e v e r y
q e IRn.
to a s e c o n d
order
form
U = U(X)
x E 'Rn
system
89
Therefore, reduced
(9)
in this case,
to the s t u d y of the c r i t i c a l
= /o 2~
f(x)
The f u n c t i o n a l
In this case infinity
(9) is not
H o w e v er,
(6)), then
if aij d e p e n d on q,
(7) is q u a d r a t i c
indefinite,
continuous
for e a c h fixed T > o
is
x e W 1 , ~ = T/2~.
dt
because
perturbation
that if U g r o w s m o r e
(in the s e n s e of
solutions
p o i n t s of the f u n c t i o n a l
"strongly"
the w e a k l y
it is k n o w n
dic s o l u t i o n
of the T - p e r i o d i c
~2 _ ~2 u(x))
(TI
dad from b e l o w m o d u l o
since
the r e s e a r c h
it is b o u n -
2~02~
x~
U(x)dt.
than q u a d r a t i c a l l y
(8) has a n o n - c o n s t a n t
(cf.
T-perio-
141,IIO1).
(i) c a n n o t
in p, it d o e s
at
be r e d u c e d
not s a t i s f y
to
(8). M o r e o v e r
the g r o w t h c o n d i t i o n
(6). H e r e we are c o n c e r n e d
with
the H a m i l t o n i a n
(7), w i t h a. • d e p e n 13
d i n g on q. The
following
THEOREM.1.
result
holds
(cf.
!31).
Consider
the H a m i l t o n i a n
the f o r m
(7). S u p p o s e
C 1 (Rn,~)
that V
(i,j=l,...,n)
c,r
o < ~ V (q) _< (Vq, q) An
~2)
IVq(q~l
i~
aij
H(p,q) and
the f o l l o w i n g
~ c lql ~
u > o s.t.
(q) piPj _> u
Lp~ 2 for e v e r y p,q
having
aij
> o and ~ > 2 s.t.
i I)
th e r e e x i s t s
(i) w i t h
e C 1 (~n,~)
satisfy
i) th~ere are c o n s t a n t s
ii)
system
Rn
assumptions: for
lql >r
90 iii)
if p,q E Rn ~A i j . 13
(q) piPj
where
Aij
iv) Under
the above
~ o ,
(q) = (grad aij(q) , q)~n
Igrad aij(q) I is b o u n d e d
assumptions,
(i) p o s s e s s e s
solution
for each
fixed
In o r d e r
to prove
the T h e o r e m . l
cal point
theorem
(cf.
period
131)
THEOREM. 2. Let X be a real sentation
fl ) f(u)
we need the f o l l o w i n g
Hilbert
12
product
space,
(Lu
on w h i c h
S 1 a~ts.
abstract
e CI(x,R),
a unitar~
criti-
lU) x - V (u) , w h e r e
(.I.) x is the inner
selfadJ0int
are e q u i v a r i ~ n t
with
We suppose
respect
be a
assumptions:
~ (o) = o, is a f u n c t i o n a l
is compact.
repr e -
Let f E C 1 (X,~)
the f o l l o w i n g
in X, L is a b o u n d e d
derivative
T-periodic
:
on X s a t i s f y i n g
=
a nonconstant
T.
Tg of the g r o u p
functional
(i,j = 1 ..... n)
operator whose
Fr~chet
that both L and
to the a c t i o n
and
~'
of the group
S1 . f2 ) O does not b e l o n g f3 ) E v e r y and
sequence
I If'
to the e s s e n t i a l
{u n} c X, for w h i c h
spectrum
of L.
f (Un)+ c e ]o,+
(Un) I l.I lUnl I ÷ o, p o s s e s s e s
a bounded
subse-
quence. f4 ) There
are two c l o s e d
and R,6
subspaces
> o s.t.
a) W i_ss L - i n v a r i a n t , i . e .
LW = W
sl-invariant
~[
V,W c X
91
b) Fix
(SI) ~ cV or Fix
c)
f(u)
d)
f is b o u n d e d
e)
f(u)
< 6 for u ¢ Fix
~ ~
f) c o d i m Umder
orbits ~ than
(dim
of c r i t i c a l
(S I)
f r o m a b o v e on W
for u ~ V s.t.
(V+W)
the a b o v e a s s u m p t i o n s 1 2
(S I) cW
<+ ~ , d i m
I [ul [= R
(V~W)
<+~
there e x i s t s at least
(V~W)
points,
- codim
(V+W))
with critical
values
greater
or equal
6.
ThiS theorem only a weak Outline
is a g e n e r a l i z a t i o n P.S. c o n d i t i o n ,
=
namely
O.i
in
12[, b e c a u s e
(f3). is r e q u i r e d .
of the p r o o f of the T h e o r e m . l
Obviously ~(z)
of the T h e o r e m
H
the continuous
the f u n c t i o n a l (z(t)) dt
(4) s a t i s f i e s
(we shall
selfadjoint
take
operator
assumption
in the sequel
in W I/2 d e f i n e d
(fl) w i t h
X=W I/2,
~ = i), and L is by the b i l i n e a r
form (IO)
(Lu
where
Iv )X = < -J ~' v >
u,v e W I / 2 and
a n d its dual.
<.,.>
denotes
The s p e c t r u m
the c a n o n i c a l
pairing
X
I.= ~, 3 l+lj I
2n and Ker L = IR2n , so also a s s u m p t i o n
is satisfied . N o w we have to v e r i f y
that
(f3)
f(z n) ~ c ¢ ] o,+ - [ and
]If'
(Zn) I ].[ IZnl [ ~ O. The
Fix(sl) = {u ¢ X ~
between
of L is g i v e n by the e i g e n v a l u e s
j ¢ ~ . lj has m u l t i p l i c i t y (f2)
,
If u e X the
is s atisfied.
ITg u = u for e v e r y
"orbit"
of u is the set
Let
{z n} c W I/2 s.t. first
g ¢ S I} {Tq u : g ¢ S I}
step,which
92 will be not given here,
is to prove that, by virtue of (il),
(ii) and
(iii), the sequences (ii)
I Ipnl IL2 , I lqnl [La
are bounded. Hence
I IZnI~L2xL~
is bounded.
The second step is to prove that By
(i2) , (iv) some computations
Then,
I IZnl !WI/2 is bounded. show that
I IHz (Zn) ! ILl
is bounded.
if q > o, by the Sobolev imbedding theorems we deduce that the
sequence I ]H z (Zn) I IW_i/2_n/2
(12)
is bounded. On the other hand we know that -J Zn- Hz(Zn )÷ o in W -I/2, then by
(12) we deduce that the sequence
(13)
-J
n
is bounded in W- 1 / 2 - n / 2 . eigenfunctions
Let
corresponding
u s now s e t
X+ ( r e s p .
to the positive
X-)
I IJ z! IW_i/2_q/2, ded. Moreover
is bounded.
with
of the
eigenvaz+ e X + ,
~ count •
I [Znl IW+i/2_n/2 is boun-
since Ker L is finite-dimensional,
I IZnl IW+i/2_q/2
zO
I Izl IW+I/2_o/2
then by (13) we deduce that
span
(reap.negative)
lues, and we set X ° = Ker L. Then z = z + +z - +z o = ~ + z- ~ X-, zO ~ X O. It is easy to show that
the
we have also that
Hence, as ~ > o is arbitrary,
by Sobolev
embedding theorems we deduce that for every t>l (14)
I IZnl ILtxLt
is bounded.
Now, as < f' (z n) , z n > + o, it is easy to see that for
n e N (15)
I Iz+n II wl/2 -<
Moreover by (16)
(i2) and
count
(I+ ~o 2~
(Hz(Zn), z +n )~2n dt)
(iv) we deduce that for n e
/o/2~ (Hs(Zn)'
z~) ~2n
dt < count --
(i+ l[z If t n L2 t
I Iz~I IL 2 ),
g3
where
t = max
is b o u n d e d . because
{2,~}.
By
(15),
Analogously
(16) and
we p r o v e
that
Ker L is f i n i t e - d i m e n s i o n a l ,
(14) we c o n c l u d e
that
I Iznl IWI/2 is bounded. we have also that
I Tz~IIwl/2 Finally,
I Iz~l IWI/2 is
bounded. In o r d e r
to v e r i f y
technicality. I step)
27
In c o r r e s p o n d e n c e
(f4) we need m o r e
s t a t e the steps of this p r o o f only:
that
• sup
{- H(z)
Iz E PR2n}
to 6 we can c h o o s e f(z) ~ 6 for e v e r y
is the span of the e i g e n f u n c t i o n s lj,
assumption
6 > o be s.t.
~ >
ge in o r d e r
"geometrical"
Here we shall
Let
(17)
the
R> o and j e ,N z e X~
sufficiently
lar-
, I IzT IWI/2 = R , w h e r e
corresponding
X~
to the e i g e n v a l u e s
j> k
+ We set V = X k. II step) quadratic
We set W =
g r o w t h of V(q)
+
i
(Xk+ ~)
, w i t h ~ e IN, 5 ~ i. U s i n g the super-
it can be p r o v e d
that f is b o u n d e d
from above
on W. So the a s s u m p t i o n s by the T h e o r e m . 2 , equal
than
of the T h e o r e m the c r i t i c a l
6. So by
are not c o n s t a n t s . a
~
non c o n s t a n t
Observe,
moreover,
2 are
values,
fullfilled.
Now observe
w h i c h we find,
are g r e a t e r
(17) we h a v e that the c o r r e s p o n d i n g Therefore
we find
critical
points.
that,
(V~W)
critical
- codim
by the fact that ~ is a r b i t r a r y ,
ce also that t h e s e c r i t i c a l n i t e l y m a n y non c o n s t a n t
+(dim
points
T-periodic
are i n finite. solutions
of
or points
(V+W))
we can d e d u -
So t h e r e e x i s t (2).
that,
infi-
g4
REFERENCES Ii
A. A M B R O S E T T I
- G. MANCINI,
class of convex 12
V. BENCI,
Hamiltonian
On the critical
in the p r e s e n c e 3
V. BENCI tonian
4I
zione
- A. CAPOZZI
5!
lineari",
V. BENCI zioni
6
"Metodi
L'Aquila
differenziali
V. BENCI
- D. FORTUNATO,
Ann.Mat.Pura 7
V. BENCI nite
!8
19
P.H.
vey,
of Hamil
Preprint. per u n ' e q u a -
simmetriche,
e topologici
periodiche
relative
Proceedings
in p r o b l e m i
multiple
a sistemi
"Metodi
asintotici
L'Aquila
(1981).
The dual m e t h o d for indefinite
RABINOWITZ, Inv.math.,
Periodic
RABINOWITZ,
RABINOWITZ, Preprint.
Critical 52,
solutions
of P. Rabinowitz,
Pure A p p l . M a t h . , IiO I P.H.
results
functionals,
theorem
solutions
diff.
per equa-
conservativi,
e topologici
in critical functionals,
point
in
theo-
to appear on
e Applicata.
- P.H.
I. EKELAND,
functionals
Amer.Math. Soc.
di m o l t e p l i c i t ~
su variet~
lineari
non lineari",
ry. M u l t i p l i c i t y
Periodic
period,
Soluzioni
of the S y m p o s i u m
diff.
for a
(1981).
non
problemi
for indefinite
to appear on Trans.
asintotici
- D. FORTUNATO,
Proceedings
theory
Un teorema
non lineare
period
Preprint.
- D. FORTUNATO,
- D. FORTUNATO,
of the S y m p o s i u m non
point
with a p r e s c r i b e d
ellittica
of m i n i m a l
systems,
of symmetries,
systems
V. BENCI
Solutions
Periodic
(19791,
theorems
solutions
for indefi-
336-352.
of H a m i l t o n i a n
J. Diff.Eq.,
3__1, (1978), Periodic
point
equations
3_44, (1979),
of H a m i l t o n i a n
and a
523-534. systems,Comm.
157-184.
solutions
of H a m i l t o n i a n
systems:a
sur-
MULTIPARAMETER PROBLEMS
: THE LAST DECADE
Patrick J° Browne
Dedicated to Richard Guy on the occasion of his 65th birthday.
i.
INTRODUCTION Research in the area known generally as multiparameter spectral
theory has been progressing at a steady rate for the last i0 years or so.
This lecture will highlight some of the features of that period.
I make no claims of this being a complete coverage of the decade; rather I have concentrated on areas with which I am most familiar and where open problems and ideas needing further development can be easily identified.
Why concentrate on the last i0 years?
Atkinson's book
[2~
appeared in 1972 and in many ways set the stage for much of the subsequent development. problems for matrices,
The book deals with multiparameter eigenvalue i.e. finite dimensional problems
(except for the
last chapter where problems with compact operators are introduced). is tantalizingly entitled
It
"Vol. I".
Two other references which fall outside our 10-year span must also be mentioned.
Atkinson's survey lecture
for newcomers to the field.
Many of the results of the i0 years in
discussion have roots which can be found here. Fa±erman
is also important
F~] is mandatory reading
The paper
~28] of
(and we shall refer to it later) for it
seems to be the first successful attack on the eigenfunction completeness problem for a multiparameter system of ordinary differential equations.
Another useful reference is the 1978 monograph of Sleeman ~34] where most of the results of the first half of our decade are collected together and presented in orderly fashion.
96
2.
TYPICAL PROBLEMS Our aim here is to describe briefly the standard types of
multiparameter
eigenvalue problems firstly in relation to differential
equations and subsequently in terms of an abstract formulation. Consider the system of ordinary differential equations with unknown functions Yr(Xr), x r E [ar,br],
d2yr (xr) + qr(Xr)Yr(Xr) +
dx 2
r
Here
%l,...,hk
functions
k ~
s= l
i <_ r <_ k :
%s ars(Xr)Yr(X)=
0 ,
i <_ r <_ k.
(i)
are spectral parameters and the coefficient
qr(Xr), ars(Xr) are real valued and continuous.
subject the equations
We
to separated boundary conditions of the form W
Yr(ar) c o s ~
+ Yr(ar)
sin a r = 0 (2)
Yr(br) cos Br + Yr(br) sin Sr = O, Thus we have k Sturm-Liouville parameters,
i ~- r <_ k.
equations,
each with several
linked via these parameters.
We say that a (necessarily real) k-tuple % = (%1 .... ,%k) is an eigenvalue
and y(x) = Yl(Xl)...Yk(Xk),
a corresponding eigenfunction
if the functions Yr(Xr)
the boundary conditions and differential Multiparameter
x = (X I..... Xk) ,
eigenvalue problems
satisfy
equations with the given h . for systems of ordinary
differential equations essentially revolve around the investigation of such equations.
97
An abstract m u l t i p a r a m e t e r
spectral p r o b l e m is formulated as
Let H I,...,H k be Hilbert spaces and in each space consider
follows.
operators Tr,Vrs , self-adjoint,
i <_ s <_ k, such that
possibly unbounded,
T r : D(Tr) ~ H r ~ H r
V
: H
rs
r
-+ H
is
is self-adjoint
r
and
continuous. We work with the operators W ( 1 ) ,
I ( I~k, where
k
w(x)
=
r
+
~
h
~ V rs
S=I and we consider the linked eigenvalue problems w
Q ) x r = o, x r # o,
It turns out to be appropriate k H = ® H r=l r
space
I = (ll"'''hk)
eigenvector
to w o r k in the tensor product
as we shall see later.
is an eigenvalue
We now say that
and x = x I ® . . . ®
in the abstract
is easy to see for if
formulation we take H r = L 2 ([ar,br]),
(xr) = ars (Xr) f (xr) and Tr to be the Sturm-Liouville
operator T /
= f" + q /
boundary conditions,
with domain specified by the above
then we immediately
in ordinary differential
3.
x k a corresponding
of the above equations are satisfied.
The connexion between the two problems
(Vrs f )
(3)
I <_ r <_ I<.
DEFINITENESS
recover the original p r o b l e m
equations.
CONDITIONS
For the standard
Sturm-Liouville
p r o b l e m with a weight
function
viz
y" + qy + fry = 0 a customary assumption hypothesis
is to demand r (x)
for the multiparameter
system
> 0.
The corresponding
(i) is k
A(x) = detIars(Xr) I > O, V x E [a,b] = r~l
[%,b].
:
98
Of course the continuity of the functions a
rs
yields the existence
of y > 0 such that
A(~)
> ~ > 0,
strong right definiteness.
This condition is called Determinants
of the type
contains functions of the r determinants.
V x ~ [a,%].
th
detlars(Xr) I variable
are called Stgckel
They have some curious properties.
under strong right definiteness, cr = ± i~
Xr,
where the r t h - r o w
1 _< r _< k,
For example,
given any k-tuple of sign factors
there exists
~ IRk
a = (a l,...,ak)
such that
k
~r ~ as ars (Xr) > O,
V Xr,
s=l Further,
this sign factor property
right definiteness.
is sufficient
This is a result of F.V. Atkinson.
For a 2 x 2 Stackel determinant
all(X)
A(x,y)
satisfying
al2(X)
= det
>_ a21(Y)
it turns out that the set
CI,C 2
0,
a22(Y)
{(x,y) I A(x,y)
disjoint union of 4 Cartesian product C 1 x C 2 where
to imply strong
= 0} can be expressed as the
sets - i.e. sets of the form
are subsets of the domains of
This prevents e.g. the zero set of rectangle of definition.
For the system
respectively.
from being a diagonal of its
This is an unpublished result of the author.
An open question is to determine k × k Stackel determinants
A(x,y)
x,y
the analogue of this result for
satisfying A(x) ~ 0.
(i) an alternative
to suppose the existence of ~ E ~ k
definiteness
condition is
so that each of the k determinants
^
formed from
[ars(Xr) ]
by replacing row r with v is positive for all x;
this can be compactly expressed as requiring
99 k
hr(X) = where ars* require
denotes
~
8=I
~s ars* (x) > O,
the co-factor
1 <- s -~ k,
of ars in detiars I.
that qr(Xr) > O, x r ~ [ar,br] , we arrive
x E [a,b ],
If we further
at the c o n d i t i o n
known as strong left definiteness. For the abstract several
forms.
problem
For so called
(3) d e f i n i t e n e s s
conditions
can take
strong right scalar definiteness we ask
SRD~ : d e t ( V s Ur,Ur) ~ ~ > O, V u r E Hr, IIurll = i. A weaker
condition,
right scalar definiteness is
RD 6 : det(Vrs Ur,U r) > O, V U r ~ Hr, IlUrll = I, and a yet w e a k e r
weak right scalar definiteness,
condition,
is
WRD 6 : det(Vrs Ur, U r) >_ O, V u r E Hr, llUrll = i. Strong left scalar definiteness could be formulated as SLD 8 : (Tr Ur'Ur) > ~r > O, 3 ~ E ~k
such that
k
X
S= 1 where
~s(WrsUr~U r)
~ y > o,
Vu
r
~ H r,
IIUrll = i,
(Vrs Ur,Ur)* denotes the co-factor of (Vrs Ur,Ur).
LD6, WLD~ take the same form with appropriate We can also Each Vrs induces
formulate
operator
an operator
relaxations
definiteness
V ~ in H by means rs
Vrs t (Xl ~...® x k) = x I ®...® Vrs x r ® . .•® extended
to general x
~ H by linearity
Conditions of the inequalities.
conditions
as follows.
of
xk
and continuity.
We can consider
too
the array of operators
[Vrt8] in H.
Operators
commute and we form the determinantal
from different
rows
operator
A 0 = detIV~s i For
k x = r=l ® Xr"
k Y = r=l® Yr , decomposable
tensors,
(A 0 x~y) = det(Frs Xr,Yr). Our right operator SRD A
definiteness
conditions
(AQx, x) >_ ~ > 0,
:
RD A : (A0x, x) > 0 ,
V x E S,
llxll = l
V x E H,
WRD A : (A0x, x) >_ 0 An early result
run as follows:
V x 6 H.
of Atkinson
showed that in finite dimensions,
SRD~ = SRD A and more recently Binding
[4,5] has proved
this in infinite
dimensions;
in fact he has shown that if (A 0 x,x) ~ yilxll2 for
decomposable
tensors x~ then
Binding
(A 0 x,x) ~ k -I yIIxll2 for all x E H.
in [ 4,5 ] also shows SLD~ = SLD A and provides
and thorough discussion The implication
of these
(and other)
a detailed
definiteness
conditions.
RD6 = RD A remains an open question in infinite
dimensions.
4.
EIGENFUNCTION
EXPANSIONS
The following
result which
expansion
incorporates
theorem and a Klein oscillation
of differential
equations
(i) subject
and strong right definiteness THEOREM i.
an eigenfunction
theorem for the system
to the boundary
was first established
conditions
by Faierman
For each k-tuple of non-negative integers
(2) [28].
p = Op I, .... pk )
there is an eigenvalue X ~ of (i) with corresponding eigenfunction k y~(X) = ~ y~(x such that y ~ has Pr interior zeros in (ar,br). r=l
r
I01
The eigenvalues
h~ have no finite point of accumulation i n ~ k
and form a complete orthonormal basis for the weighted Lebesgue 2 L ([a,k1,
space
A(x)).
Faierman's original approach to this beautiful result was to replace each of the differential equations by difference equations and then use limiting arguments. Browne
The problem was also studied by
[14,18] from a more abstract point of view.
The left definite problem has been successfully Faierman
[29,30,31], Kallstrom and Sleeman
studied by
[32] and Sleeman
[34].
The singular problem in which all or some of the variables x r
may range over a half-line has been studied by Browne spectral function p(t), t ~ ~ k can be produced. monotonic on ~k and the eigenfunction
[15].
Here a
It is positively
series expansion for the regular
case is replaced by an integral with respect to the measure generated by p(t) thus providing a direct extension of the well known Titchmarsh~ Kodaira theory to k × k systems. The doubly singular case when each x
ranges over
(-~,~) remains
r
unsettled although a spectral theorem should involve a matrix of size 2k × 2 k of functions of the type of p(t)
-
see [17] for a discussion ~~
of this. Sleeman
[35] has provided an introductory
study of limit point-
limit circle theory, although there is much left to be investigated and understood here. Abstractly by Browne
for the problem
[16,1§,20,22]
(2), spectral theory has been studied
in a series of papers.
strong right definiteness
Under the condition of
(the easiest case to present in this lecture),
102
the operator
A0
has a bounded inverse A -I 0
by investigating
the operators
F
s
We proceed
= A -I A in H where now we 0 s
use the inner product generated by
[x,y] = (h 0 x , y ) ,
: H ÷ H.
A 0 , viz
x,y
E H.
The operators As, 0 5 s S k, are defined via the equation
k s=o
a 8 A s = det
for arbitrary ~o,...,ak.
sO
a I ... a k
T~
t Vii"
Vf ik
Vki"
V~k
(In the simplest case, T I , . . . , T k are bounded.)
The main step is to extend some earlier finite dimensional of Atkinson to infinite dimensions
(i)
the operators
Fs
results
and to establish
are self-adjoint with respect
to the
[.,.]-inner product (ii)
the operators
(iii)
if Fsx = hsx , decomposable
F
8
are pairwise commutative
1 E s ~ k, then x can be taken to be
and as such is an eigenvalue for the p r o b l e m
Of course when the T r 's are unbounded, must be interpreted
as commutativity
commutativity
E
on the Borel sets of
of the F s' s
of their spectral resolutions.
One can then appeal to the functional calculus commuting self-adjoint
(2).
for several
operators and so produce a spectral measure ~ k which is H - p r o j e c t i o n valued and such
that, for example,
d[E(h) x,x]
[x,x] = ~ k
103
The papers mentioned above attempt weakenings
of the definitiness
this line of analysis under various
conditions.
Results under left definiteness Sleeman's monograph
conditions
can be found in
[34].
The theory of spectral representation
for 1-parameter problems
is detailed by Dunford and Schwartz in [27, pp.1205-1222]. speaking, space
given a self-adjoint
operator
T = I h E(dh) on a Hilbert
H and a non-zero vector x E H , one considers
set of all vectors of the form function.
k(T)x where
Thus is generated a space H
space L2(Vx) , ~x (') =
x
Roughly
the closure of the
k is a bounded Borel
a H which is isomorphic
to the
(E(.)x,x) by means of the correspondence
k(T)x +-~ k(~). This makes T (at least on H x) an isomorphic copy of multiplication
by the independent variable.
spectrum if H
= H for some x ; in general, H can always be expressed
x
T is said to have simple
as a direct sum of such spaces H . x To date there is no satisfactory parallel parameter
case.
It would be desirable
theory for the multi-
to show for example that if each
Wr(~) has simple spectrum for each h then E could be represented
as a
space of functions on~R k.
5.
V A R I A T I O N A L METHODS Here we consider the abstract p r o b l e m
definiteness
and the extra hypothesis
and has compact resolvent.
(3) subject to strong right
that each
T r is bounded below
Then each Wr(~) has these properties
and so has a spectrum consisting entirely of eigenvalues
too
104 pO(h) ~ i:)l(h) _< r r each of finite multiplicity
and accumulating
main tool is the minimax principle
only at ~
for eigenvalues
The
for such
operators: oi(h) r ~
= Max{Min{ (Wr(~ )Ur,Ur) I ilurlI : i, (Ur,yJ) yJ E Hr,
= O} 1
1 ~_ j <_ i}
The first important result to be obtained was an abstract oscillation theorem which states that given p = negative integers,
(pl,...,pk) ,
a k-tuple of non-
there is an eigenvalue h~ so that Pr
Pr (hE)
= 0.
All eigenvalues occur in this manner and so they can be systematically numbered according to the various choices of p. first discussed by Binding and Browne Given Ul,...,Uk,
V(u)
= [(Vrs
Ur,Ur)],
u r 6 Hr, llUrlI =
These ideas were
[~,~9]" i, consider the matrix
of real numbers and the cone C ~ k
C = {a 6 ~ k
I V(u) a ~ 0
for some u I ....
defined by
,uk} .
Here for points ~, B 6 ~k, a ~ ~ means ~r ~ ~r' 1 ~ r ~ k. Descriptions
of the geometrical
features of the spectrum of the above
problem were given by Binding and Browne in [ii] using the cone C. Specifically,
a key result is
i i ~ j and in particular,
=
j
~~ ~ ~~ + C
if Z denotes the totality of all eigenvalues,
then
[_< ~9 +c. ~
The cone C was also used by Browne
[2i] and Browne and Sleeman
when discussing a system of ordinary differential type (i) where the coefficients
[24]
equations of the
are periodic and the system is subject
105
to periodic and semi-periodic boundary conditions
: 2 k possibilities
exist and the interlacing of the 2k sets of eigenvalues can be described using
6.
C.
ASYMPTOTICS FOR ONE EQUATION Important initial work on this topic as it relates to Sturm-
Liouville equations with many parameters was done by Turyn
[36]
and was subsequently put into a more abstract setting by Binding and Browne
[12] and Binding, Browne and Turyn [13]. ~ ~
Abstractly, we consider one equation
k
w(~)x = (T +
~ ~ ~)x=
o,
~ E~,
8=I
in a Hilbert space H
where T
is self-adjoint, bounded below and
has compact resolvent and gl,...,gk are bounded and self-adjoint. As before
W(h)
has a sequence of eigenvalues
po(~) < p1(#)
<
...
~
and we are interested in the nature of the set where p0(h) = 0. The case k = 2 has been given complete treatment in [12,13], and results for k ~ 3 are currently under investigation. concentrate on the case k = 2.
We put
g = { x
po(~)
P={~
= o}
p0(~) > 0} ~
N={a
p0(¢
< 0}
We have that Z is closed, P and N open; G i v e n u E H,
We shall
Itull = 1 , we s e t
and define, c o denoting
v(u)
P is convex.
= ((g 1 u,u),
convex hull,
(v 2 u,u))
E ~2,
106
o,lJull
v = co{~ E(u) l~ > Y
is a convex cone
in ~2.
= i} c m 2
We also introduce ~2
K
= int K if C- # ~ . We introduce
I /~
two conditions:
: 0 ~ V
is the stronger
IS
: C- # ~ .
of the two and under
contained
strictly
in a closed
cone of P
(or the set of asymptotic
0tP : (~ Important
results
space
I P + ~ }
is a convex
in ~2.
cone
The r e c e s s i o n
of P)
is
•
are S = P #
(2)
P ~ ~=
(3)
P # ~ = Z = ~P : 3N
0+ : K
(or I s ) Z is the boundary
asymptotically
V
directions
(i)
Thus under I
has directions
are in terms of the operators These results system
half
it
given by ~K. Notice
are useful w h e n we return directions
in geometrical
the curves must
intersect
may for certain
choices
set and
that these results
V I , V 2 only and hold for any choice of T
for n o w we k n o w asymptotic
and can then discuss
of an open convex
to a linked of the curves
terms conditions
or conditions
under w h i c h
of T 1 and T2, not intersect.
2 × 2 Z 1 and Z 2
under w h i c h the curves This
107
detailed analysis can be found in [13].
Corresponding results
for k ~ 3 are under investigation. Space does not permit a discussion of the various results for non-linear multiparameter problems.
The interested reader
could consult Binding [5,6], Browne [19] or Browne and Sleeman
[~,~,~]
BIBLIOGRAPHY i.
F.V. ATKINSON
Soc., 2.
Multiparameter spectral theory, Bull. Amer. Math.
74 (1968), 1-27.
F.V. ATKINSON
Multiparameter Eigenvalue Problems I : Matrices
and Compact Operators, 3.
P.A. BINDING operators,
4.
P.A. BINDING
Academic Press, New York, 1972.
Another positivity result for determinantal
Proc. Roy. Soc., Edinburgh, 86A (1980), 333-337. Multiparameter definiteness conditions,
Proc. Roy. Soc., Edinburgh, 89A (1981), 319-332. 5.
P.A. BINDING
Multiparameter definiteness conditions II, Proc.
Roy, Soc., Edinburgh, 6.
P.A. BINDING
(to appear).
On the use of degree theory for nonlinear
multiparameter eigenvalue problems, J. Math. Anal. Appl., 733, (1980), 381-391. 7.
P.A. BINDING,
Variational methods for one and several parameter
non-linear eigenvalue problems, Can. J. Math., 333 (1981), 210-228. 8.
P.A. BINDING and P.J. BROWNE,
Positivity results for determinantal
operators, Proc. Roy. Soc. Edinburgh, 81A (1978), 267-271. 9.
P.A. BINDING and P.J. BROWNE,
A variational approach to
multiparameter eigenvalue problems of matrices, S.I.A.M.J.
Math. Anal., 8 (1977), 259-273.
108
Bibliography (Cont'd)
i0.
P.A. BINDING and P.J. BROWNE,
A variational approach to
multiparameter eigenvalue problems in Hilbert space,
S.I.A.M.J. ii.
Math. Anal., 9 (1978), 1054-1067.
P.A. BINDING and P.J. BROWNE,
Comparison cones for multiparameter
eigenvalue problems, J. Math. Anal. Appl., 777 (1980), 132-149. 12.
P.A. BINDING and P.J. BROWNE,
Spectral properties of two-parameter
eigenvalue problems, Proc. Roy. Soc. Edinburgh, 13.
P.A. BINDING, P.J. BROWNE and L, TURYN,
89A (1981), 157-173.
Existence conditions for
two-parameter eigenvalue problems, Proc. Roy. Soc. Edinburgh, 14.
to appear
P.J. BROWNE, A multiparameter eigenvalue problem, J. Math. Anal. Appl., 38 (1972), 553-568.
15.
P.J. BROWNE, A singular multiparameter eigenvalue problem in second order ordinary differential equations, J. Differential
Equations, 122, (1972), 81-94. 16.
P.J. BROWNE,
Multiparameter spectral theory, Indiana Univ. Math. J.,
24 (1974), 249-257. 17.
P.J. BROWNE,
Multiparameter problems, Proceedings 174 Dundee
Conference on Ordinary and Partial Differential Equations, Springer-Verlag Lecture Notes in Mathematics, Vol. 415, 78-84. 18.
P.J. BROWNE,
Abstract Multiparameter Theory I, J. Math. Anal. Appl.
60 (1977), 259-273. 19.
P.J. BROWNE,
A completeness theorem for a non-linear multiparameter
eigenvalue problem, J. Differential Equations, 20.
P.J. BROWNE,
233 (1977), 285-292.
Abstract multiparameter theory II, J. Math. Anal. Appl.,
60 (1977), 274-279. 21.
P.J. BROWNE, The interlacing of eigenvalues for periodic multiparameter problems, Proc. Roy. Soc. Edinburgh, 80A (1978), 357-362.
22
P.J. BROWNE, Abstract multiparameter theory III, J. Math. Anal. Appl., 73 (1980), 561-567.
109
Bibliography (Cont'd.)
23.
P.J. BROWNE and B.D. SLEEMAN,
Non-linear multiparameter
Sturm-Liouville problems, J. Differential Equations, 34 (1979) 139-146. 24.
P.J. BROWNE and B.D. SLEEMAN,
Stability regions for periodic
multiparameter eigenvalue problems, Proc. Roy. Soc. Edinburgh, 84A (1979), 249-257. 25
P.J. BROWNE and B.D. SLEEMAN,
Bifurcation from eigenvalues in non-
linear multiparameter Sturm-Liouville problems, Glasgow Math. J., 21 (1980), 85-90. 26.
P.J. BROWNE and B.D. SLEEMAN,
Non-linear multiparameter eigenvalue problems
for ordinary differential equations, J. Math. Anal. Appl., 777 (1980), 425-432. 27.
N. DUNFORD and J.T. SCHWARTZ,
Wiley-fnterscience, 28.
M. FAIERMAN,
Linear Operators, Part II,
New York (1963).
The completeness and expansion theorems associated with
the multiparameter eigenvalue problem in ordinary differential equations, J. Differential Equations, 5 (1969), 179-213. 29.
M. FAIERMAN,
Eigenfunction expansions associated with a two-parameter
system of differential equations, Proc. Roy. Soc. Edinburgh, 81AA (1978), 79-93. 30.
M. FAIERMAN,
An Oscillation theorem for a two-parameter system of
differential equations, Quaestionnes Math., 3 (1979), 313-321. 31.
M. FAIERMAN,
An eigenfunction expansion associated with a two parameter
system of differential equations, Proc. Roy. Soc. Edinburgh 89AA (1981), 143-155. 32.
A. KALLSTROM and B.D. SLEEMAN,
A left definite multiparameter
eigenvalue problem in ordinary differential equations,
Proc. Roy. Soc. Edinburgh, 74A (1974/5), 145-155. 33.
A. KALLSTROM and B.D. SLEEMAN,
Ark 34.
Multiparameter spectral theory,
. f~r Mat., 155 (1977), 93-99.
B.D. SLEEMAN,
Multiparameter Spectral Theory in Hilbert Space,
Pitman, London,(1978). 35.
B.D. SLEEMAN,
Singular linear differential operators with
many parameters, Proc. Roy. Soc. Edinburgh,
71AA (1973),
19@-232. 36.
L. TURYN,
Sturm-Liouville problems with several parameters,
J. Differential Equations,
38 (1980), 239-259.
AN APPROACHTO THE DIRICHLET INDEX FOR OPERATORS SATISFYING MINIMAL CONDITIONS
Richard C. Brown
§I
Introduction and Notation. In t h i s note we o f f e r an i n t e r p r e t a t i o n of the D i r i c h l e t Index f o r symmetric
d i f f e r e n t i a l expressions with non-negative c o e f f i c i e n t s .
E s s e n t i a l l y we relate the
index to " l i m i t point" properties of a pair of "maximal" and "minimal" vector valued operators
~
and ~o ~ which are introduced in Section 2.
By doing t h i s
several facts about D i r i c h l e t index become easy to state and prove. material is f a m i l i a r (cf.
[I],
[4], [6], [7]).
to a "minimal condition" s e t t i n g .
Most of our
However, the results are extended
Also (Theorem 6) we present a new r e s u l t on the
invariance of the index with respect to perturbations of the associated quadratic form.
Other r e s u l t s ( c f . , Theorem 5) complement recent unpublished work of Hinton
and Kauffman [ 4 ] .
We also hope that our approach may be of some independent i n t e r -
est since i t d i f f e r s from methods previously employed. The main results are presented in Section 3. lined and should not be d i f f i c u l t
Most proofs are at least put-
f o r the reader to reconstruct.
(An extended ver-
sion of the paper with f u l l d e t a i l s w i l l appear elsewhere.) We now f i x notation and introduce some preliminary concepts. In what follows
L2(a, b)
s i g n i f i e s the classical H i l b e r t space generated by
the complex square integrable functions on L]oc(a, b)
and ACn
[a, b)
Let
[a, b)
(we w r i t e
PO' Pl . . . . .
AC0
as
nth d e r i v a t i v e is absolutely con-
AC) .
Pn be real nonnegative functions on
PO > 0 , Pn ~ c > 0 , and Naimark [ 9 ] , §15.
-~ < a < b ~
are respectively the spaces of complex valued functions
which are l o c a l l y lebesgue integrable or whose tinuous on
where
M[f]
be the q u a s i - d e r i v a t i v e
(Note that i f the
Pi
[a, b) f[2n]
such that
as defined in
are s u f f i c i e n t l y smooth, then
IThe author wishes to thank Don B. Hinton for many i n t e r e s t i n g discussions concerning the subject of t h i s paper, e s p e c i a l l y r e l a t i n g to Theorpm 4.
111 M[f] :
n (n-i)(p i f(n-i))(n-i) Z (-i) i=O
and minimal operators ratic fo~
a
We say t h a t
"= o
T(M) , To(M)
Y E D(T(M))
s o l u t i o n s of
in
is D i r i c h l e t
partial
M[f] = 0 .
verifications
[12], and Read [ I 0 ] , ways" true.
L2(a, b) .
if
t(y)
The D i r i c h l e t
[II].
12
< =
~(y)
and t h a t
index (DI) of
n .
denote the quad-
T(M)
is D i r i c h l e t
M is the span of the
DI L n
and a conjecture of
Although the conjecture is s t i l l
have been made by Kauffman [ 6 ] ,
open,
[ 7 ] , Robinette
These r e s u l t s imply t h a t the conjecture is "almost a l
We also note here t h a t i f
M is l i m i t
Kauffman [7] has found a s i x t h order non l i m i t which
Let
I t is known t h a t
Kauffman [6] is t h a t i t is e x a c t l y significant
In the usual way we define the maximal
P i l y ( n - i ) l 2 where the i n t e g r a l s are a b s o l u t e l y convergent.
i f i t s domain is D i r i c h l e t . Dirichlet
) .
p o i n t then
DI = n ; however
p o i n t operator of Euler type f o r
DI = 3 .
i S, S o, S O, S +, and ~+
The ~ e r a t o r s Let
"0
n H: = X L2(a, b) . i=l
Define
S:
L2(a, b) ÷ H by
/2 y/n/ \
i
Z(Y): :
1/2 y ( n - l )
\i12 Pn
Y
/
#
on the domain
9: = {y E L2(a, b) N AC( n - l ) "
Further l e t
~oi
be the r e s t r i c t i o n
compact support in Lemma I .
S
(a, b) .
of
Note t h a t
~Y E H} .
~
to the class I~yll 2 = t ( y )
is a normally solvable operator.
.
goi
of functions w i t h
112
Proof•
A demonstration that
apparent from the structure of
S
S
is closed may be found in [2].
that
S
I t is
has closed range i f and only i f i t is
closed. Remark I.
Lemma 1 is a way of saying that the form
[5], ch. VI). S -I
Moreover since
S
t
is closed (cf. Kato
is normally solvable i t has a bounded inverse
defined on i t s range. Now set
J
~j: = (z I . . . . .
zj) E X L2(a, b) i=l
and define r e c u r s i v e l y the ex-
pressions + ~ ~l(Zl)
=
p~/2
z ,
+ ~ + c2(z2) = - ~ l ( Z l ) ' •
o
+
pl/2
z2
,
+ ~ + I/2 ~n+l(Zj ) = -~n(Zn )' + Pn We assume that the ~+ : H + L2(a, b)
~T(zi)
Zn+l
, 1 < i < n , are absolutely continuous and define
by S (Zn+ I) = ~n+l(Zn+l)
•
We denote the domain of ~+
9+ . Lemma 2 (a Green's formula).
~(Y),
For all
y
in
~
and
Zn+ 1 E
'~ ~ Zn+l] - [Y, S + (Zn+l)] = {Y, ~n }(b-) - {y, ~n}(a +)
where n
{y, Z~n}(S) : i ! 1 ( - I ) ( i - l ) +
Let S o
be the r e s t r i c t i o n
y(n-i)(s)
of S +
c~(~j) (s) . +
to the domain go
such that
{y, Zn}(b-) = {y, Zn}(a ) = 0 , ¥ y E
and S o
be the r e s t r i c t i o n
of S
to the domain go
{y, Zn}(b-) = {y, Zn}(a ) = 0 ,V Zn+ 1 E
such that
.
then
by
113 These d e f i n i t i o n s imply t h a t
y ( J ) ( a ) = ~+(~k)(a) = O, 0 <~j < n - I , 1 < k < n .
The proof of the next r e s u l t t y i n g a l l these operators together may be found in [2]
Proposition I.
SO =S + ,
S
S +*
S~* = S
~, So' S, So, + S+ Thus ~o'
: So
are densely defined operators.
s o l v a b l e and in p a r t i c u l a r Corollary I.
SS
,
S~,S +
and Sc/~0
tion:
See Kato [ 5 ] ,
e.g., for
S+
are normally
are s e l f - a d j o i n t
p. 275.
restrictions
of
T(M) .
Their
and S o . That S ~ c
T(M) , e t c .
f o l l o w s by a computa-
n = 1
:
-
(pi12(_i12 0
w i t h boundary c o n d i t i o n s
13
S o , S +o ,
are onto.
domains are cores r e s p e c t i v e l y o__ff Z Proof.
=S O ,
NO
,
Y ))'
+ p
I12
y ' ( a ) = 0 , (poy'
Y
f)(b-)
= O,
V f C~ •
D i r i c h l e t Index Theory +
Lemma 3.
The set o f D i r i c h l e t functions i n
t h e space of D i r i c h l e t s o l u t i o n s is e x a c t l y Theorem I . Proof.
(i)
is e x a c t l y
D ~ S)
R ~ ) N N ~ +) •
DI = d i m ~ / Z ° .
This f o l l o w s from the e a s i l y v e r i f i e d i d e n t i t y
R ~ ) = R ~ O) ® ( (N~ ) -F N R(Z)) C o r o l l a r y 2.
Z
D(T(M))
is " l i m i t
DI = n
point".
iff
Z
Equivalently
is a n-dimensional extension o f DI = n
Zn+1 E Theorem 2.
DI = dim D~ Z+) / D C ~ o+
) •
if__ff {y, Zn}Ib ) = 0
•o
; i.e.,
YY E ~
,
an__d_d
114 Proof.
I n t e r s e c t both sides of ( I ) by ~+
Remark 2.
A c a l c u l a t i o n shows t h a t i f n-I
{y, ~n}(S) = D(f, g ) ( s ) :
Zn+ 1 E R ~ )
y[i]
=
then
g--[2n-i-I ]
i=O where Z(g) = Zn+l. Consequently C o r o l l a r y 3 and Theorem 2 show t h a t D(y, g ) ( b - )
= 0 ,
Theorem 3.
yy E ~ Let
and
T I , T2
y[i](a)
Proof.
D ~ I = D°
I t is known ( c f . •
and
T2
and ~ c
and
b
so
T2
r e s p e c t i v e l y of
T(M)
to the D i r i c h -
= 0 , 0 < i < n - 1 or
T1
and
D~22 = ~
T2
=
DI and
are s e l f - a d j o i n t
iff
.
Kato [ 5 ] , p. 334) t h a t
T 1 =S+Zo
iff
n ,
Do = D ( ~ o ~ o Z°
and ~
T2 = Z ~
=
and
~+ ~O
are free o f
On the other hand~
so t h a t by C o r o l l a r y 1 ~+~o = T1
and S T
= T2
if
T1
are s e l f - a d j o i n t .
Lemma 4. D(T(M))
then
DI = n
.
y(i)(a)
Further by C o r o l l a r y 2 i f
boundary c o n d i t i o n s at ~+S o c T 1
(i)
= 0 , n < i < 2n - 1
DI = n ; i n which case
= D(~
y E D(j~)
be r e s t r i c t i o n s
l e t functions such t h a t e i t h e r (ii)
+
Fo_[r j = 0 . . . . .
2n - 1 ~
contains a f u n c t i o n of compact support i n
such t h a t
=
Y(k)(a) J
{i
if
k = j
otherwise.
Proof.
See Naimark [ 9 ] , ~17.3, Lemma 2.
Theorem 2. Proof.
Using Lemma 4 we can produce at l e a s t an n-dimensional extension of
Theorem 4. (a, b)
in
DI > n .
DI = n
iff
(i)
D~(: = functions of compact support on
D(T(M)) is a core of the r e s t r i c t i o n
0 < i < n - 1 , __or (i i ) D(T(M))
either
such t h a t
y[i](a)
Do'' (" = f u n c t i o n s
y
So
of
S
such t h a t
of compact support in
= 0 , n < i < 2n - I )
is a core of
Z •
y (i)(a) [a, b)
: O, __in
115
Proof.
If
DI = n
(i)
Friedrichs extension of
follows from the f a c t that
T(M)ID ~
and C o r o l l a r y I.
4 -
S ~ o is the unique
S i m i l a r reasoning using the
u
f a c t that hand
Do'
that
~
~Z
has no boundary conditions at
and
Do'
b yields (ii).
I f on the other
have the stated core property i t is easy to show via Lemma 4
is an n-dimensional extension of ~o "
Remark 3.
Theorem 4 can be extended to show that subspaces of
compact support in
[a, b)
D(T(M)) of
s a t i s f y i n g general symmetric boundary conditions at
(see Hinton [ 3 ] ) are cores of
~
iff
DI = n .
a
A s i m i l a r generalization is pos-
s i b l e for Theorem 3 ( c f . [ 6 ] ) . The deficiency index of
M (DEF) is
I / 2 dim D(T(M))/D(To(M)) .
(See [ I I ]
f o r discussion of the properties of the deficiency index). Theorem 5.
4
Proof. theory.
DI = DEF i f f -
4
T(M)
is D i r i c h l e t . +
-
DI = dim D(~'Z)/D(~'Z o)
Further
Remark 4.
D(Z+~) = D(T(M))
and iff
DEE = D ( T ( M ) ) / D ( ~ o) T(M)
is D i r i c h l e t .
Theorem 4 is a p a r t i a l generalization (to the case of minimal con-
d i t i o n s ) of a unpublished r e s u l t of Hinton and Kauffman [4]• s e t t i n g the e > 0 .
by standard
Pi
1 < i < n may be negative since
However, in t h e i r
[To(M)y, y ]
need only be
Also t h e i r treatment posits the existence of a closed extension of the
closure of
[To(M)y, y ] I c ~ . ~0 I Suppose PO' Pli . . . . . Pni
are real nonnegative measurable functions with
I
I
Pi E L oc(a, b), 1 < i < n . computed r e l a t i v e to {Pi + Pi} "
Then (Po + p~)-I E L oc(a, b) . i
{Pi + Pi } '
0 < i < n .
Let
M[f]
Define ~ , Z o, Z , ~o
be
f[2n]
r e l a t i v e to
Thus f o r example:
(Po + p~)I/2 y(n) 1 (Pn + Pn) I / 2 y
Further l e t
t'(y)
denote the quadratic form
=
is t'(y)
t
bounded i f f
p~ly (n-i)12 . i=O
D ( t ' ) ~ D(t)
< allyll 2 + b t ( y )
.
and there e x i s t
We say that
a
a, b > 0
such that
t,
116 I t is natural to investigate the relationship between the D i r i c h l e t indices of
M and M .
are both
n) .
(Of course, i f the D i r i c h l e t index conjecture is true, the indices We have the following simple results.
Theorem 6.
Suppose t '
is
~ bounded, then the D i r i c h l e t indices of
M and
M are the same. Indication of proof. 9 =~
Since
~'
is positive and
~ bounded one can show that
and 9 o = 9 o , whence the theorem is a consequence of Theorem I . Corollary 3.
~'(y) ~ K ~(y))
l_f_f D(~') D D(t)
,
K > 0
C o r o l l a r y 4.
If the
and
t'
is dominated by
t h e n t h e i n d i c e s of Pii
M and
are essentially
~
(i.e.,
M a r e t h e same.
bounded and
PO Z 6 > 0
then the
%
indices of
M and
M a r e t h e same.
REFERENCES I.
J. Bradley, D.B. Hinton, and R.M. Kauffman. dratic functionals.
2.
R.C. Brown.
On minimization of singular qua-
Proc. ROy. Soc. Edinburgh, 87A(1981), 193-208.
An operator theory approach to inequalities involving D i r i c h l e t
functionals, submitted. 3.
D.B. Hinton. equations.
4.
On the eigenfunction expansions of singular ordinary d i f f e r e n t i a J. D i f f e r e n t i a l Equations, 24(1977), 282-308.
D.B. Hinton and R.M. Kauffman.
On some properties of the D i r i c h l e t index, in
preparation. 5.
T. Kato.
"Perturbation Theory for Linear Operators", (Springer-Verlag, New
York, 1966). 6.
R.M. Kauffman.
The number of D i r i c h l e t solutions to a class of l i n e a r ordi-
nary d i f f e r e n t i a l equations. 7.
R.M. Kauffman.
J. D i f f e r e n t i a l Equations, 31(1979), 117-129.
On the l i m i t - n c l a s s i f i c a t i o n of ordinary d i f f e r e n t i a l opera-
tors with positive c o e f f i c i e n t s .
Proc. London Math. Soc., 35(3) (1977), 495-
526. 8.
R.M. Kauffman, T.T. Read, and A. Z e t t l .
"The Deficiency Index Problems for
Powers of Ordinary D i f f e r e n t i a l Expressions", (Lecture Notes in Mathematics #621, Ed. A. Dold and B. Eckmann, Springer-Verlag, B e r l i n , Heidelberg, and New York, 1977).
117 9.
M . A . Naimark.
"Linear D i f f e r e n t i a l Operators, Part I I " ,
(Ungar, New York,
1968). I0.
T.T. Read. D i r i c h l e t solutions of fourth order d i f f e r e n t i a l operators, "Spectral theory of D i f f e r e n t i a l Operators", Ed. I. Knowles and R. Lewis, (Math. Study Series, North Holland Publishing Co., Amsterdam, 1981).
II.
T.T. Read. The number of the D i r i c h l e t solutions of a fourth order differer t i a l equation, submitted.
12.
J.B. Robinette. in preparation.
On the D i r i c h l e t index of singular d i f f e r e n t i a l operators,
NONLINEAR NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS IN PRODUCT SPACES Jose M, A m i l l o Gil
John A. Burns* Eugene N, C l i f f *
I.
INTRODUCTION In the past few years there has been a rapid development of
approximation methods f o r systems governed by f u n c t i o n a l d i f f e r e n t i a l equations. we w i l l
The l i t e r a t u r e
on t h i s t o p i c is extensive.
In t h i s paper
concentrate on an approximation scheme f o r systems governed
by n o n l i n e a r neutral f u n c t i o n a l d i f f e r e n t i a l retarded systems as a special case).
equations (which include
Moreover, we w i l l
confine our
discussion to a p a r t i c u l a r class of approximations t h a t have proven to be e s p e c i a l l y well suited f o r the numerical s o l u t i o n of optimal control
and parameter i d e n t i f i c a t i o n
problems.
Since i t
is impossible
in a few pages to give a d e t a i l e d review of the r e l e v a n t l i t e r a t u r e , we l i s t
a number of papers in the references ( [ I - 1 0 ] , [ 1 8 - 2 2 ] , [ 2 4 - 2 6 , 2 8 ] )
and r e f e r the i n t e r e s t e d reader to the discussions given there. The usual Lebesgue space, Sobolev space and space of k continuously d i f f e r e n t i a b l e will
be denoted by LP(a,b),
1~n-valued functions defined on [a,b]
S'P(a,b)
and cJ<(a,b), r e s p e c t i v e l y .
We
shall use Schultz's n o t a t i o n (see [27]) and l e t PcJ<'P(a,b) stand f o r the space o f functions x such t h a t x has k-1 continuous d e r i v a t i v e s ,
x [k-l]
and x[k]~LP(a,b). wI ' p , Ck and PCk'p f o r the
is piecewise continuously d i f f e r e n t i a b l e
When a = - r and b = 0 we simply w r i t e 2 , above spaces.
The symbol I " I X denotes the norm in the normed l i n e a r
÷ ~n, then f o r 0 ~ t ~ T we d e f i n e the f u n c t i o n x t : [ - r , O ] ÷ ~n by xt(s) = x(t+s). Note t h a t x t does not represent a p a r t i a l d e r i v a t i v e . space X.
II.
I f T > O, r > 0 and x : [ - r , T ]
EXAMPLES I t has been known f o r sometime t h a t c e r t a i n i n i t i a l - b o u n d a r y
value problems f o r h y p e r b o l i c p a r t i a l d i f f e r e n t i a l be transformed to e q u i v a l e n t i n i t i a l functional differential
equations.
equations could
value problems f o r neutral In [14] Brayton and Miranker
observed t h a t the lossless transmission l i n e with n o n l i n e a r elements This research was supported in p a r t by the National Science Foundation under grant ECS-8109245.
119 at each end could be realized by an equivalent d i f f e r e n t i a l - d i f f e r e n c e equation.- Brayton and Willoughby [15] made use of this equivalence in t h e i r study of a numerical scheme f o r approximating and simulating such c i r c u i t s .
Later, Cooke and Krumme [16] developed a systematic
procedure f o r transforming a general class of hyperbolic problems to neutral equations.
More recently, Reid, Russell and Teglas (see [24-
26,28]) have shown that neutral equations may be viewed as canonical forms f o r large classes of l i n e a r boundary control systems. r e s u l t has a number of i n t e r e s t i n g r a m i f i c a t i o n s .
This
For example, i t
implies that many of these systems can be realized as neutral equations that are "minimal" in the sense that the equations contain the fewest possible parameters necessary in order to completely describe the input-output behavior of the system.
C l e a r l y , such r e a l i z a t i o n s are
ideal f o r parameter estimation problems.
Moreover, as we shall see
below there are problems where the neutral equation r e a l i z a t i o n reduces the complexity of the parameter dependence in the model and hence s i m p l i f i e s the t h e o r e t i c a l and computational aspects of various i d e n t i f i c a t i o n problems. The basic ideas can best be i l l u s t r a t e d by a few simple examples. Consider f i r s t I.
the idealized cable-mechanical system shown in Figure
We assume that the cable of length j has l i n e a l density ~, is
under tension "~ and is connected to a mass-spring-dashpot system at the r i g h t end.
We take the classical e l a s t i c boundary condition at
the l e f t end.
The purpose of the devices at the ends is to maintain
the tension and we assume they provide no impedance to the v e r t i c a l motion of the cable at the ends. A simple model fo r the v e r t i c a l motion of the cable is given by the wave equation
(2.1)
~2
~2
~t 2 ~v(s,t) = m2---y(s,t2)3s
0 < s < I,
t > O,
120
with initial
data
(2.2)
where ~ = T/~.
y(s,O)
At the r i g h t end we have the system
where u(t) i s a c o n t r o l f u n c t i o n . c o n d i t i o n a t the l e f t
(2.4)
~ y(s,O) = h(s),
= g(s),
The c l a s s i c a l
end i s w r i t t e n
~ y(O,t) = x[g(O) +
in i n t e g r a l
e l a s t i c boundary form
It ~t y(O,~)d~], o
where we assume y # O. Note t h a t the i n i t i a l
c o n d i t i o n s f o r (2.3) are s p e c i f i e d by
(2.5)
y(1,0) = g(1), 7t y(1,0) = h(1).
k
I"
! C
(s, t)
1 FIGURE l
121 In order to construct an e q u i v a l e n t neutral system we f o l l o w the general procedure of Cooke and Krumme [16] and f i r s t (2.4) as a system.
If p(s,t)
and q ( s , t )
P = ~s y + (1/~) ~ y ,
rewrite (2.1)-
are defined by
q = ~s y - (1/a)
y ,
then
~y
and since
= [p+q]/2,
~y
= m[p-q]/2
,
-~ y(O,t) = xy(O,t) i t f o l l o w s t h a t
(2.6)
Problem ( 2 . 1 ) - ( 2 . 2 )
(2.7)
~
with i n i t i a l
becomes the system
+
0
~
q
= 0
0 < s < 1, t > O,
data
p(s,O) = g'(S) + (1/a)h(s)
(2.8) q(s,O) = g'(s) - (1/a)h(s).
The boundary c o n d i t i o n (2.3) becomes
~t
(2.9)
{~[p(1, t)-q(1,t)]} = - ca [p(1, t)-q(1, t)] }m - k2--~m[P(O't)+q(O't) ]/~+Ii [p(~'t)+q(p't) ]d~) - ~m p(1, t)+q(1,t)]+u(t) m
122
with initial
condition
(2.10)
~- [p(1,o) 2
Differentiate
(2.11)
- q(1,o)] = h(1)
(2.4) w i t h respect to t to o b t a i n
~{~[p(O,t)+q(O,t)]}
with initial
= ~
[p(O,t)-q(O,t)]
data
I [p(O,O) 2
(2.12)
+ q ( O , O ) ] = ¥g(O)
Observe t h a t the c h a r a c t e r i s t i c s defined by
,
s=-a[t
f o r (2.7) are s t r a i g h t
lines
- ~] + 1 and s = a [ t - ~ ] , where ~ i s c o n s t a n t .
In p a r t i c u l a r ,
the f u n c t i o n s t ÷ p ( - a [ t
are c o n s t a n t .
Therefore, it
- ~]+ 1 , t )
follows that for all
and t ÷ q ( m [ t - ~ ] , t ) t ~ O,
p(1,t) = p(O,t + i/a) (2.13) q(O,t) = q(1,t + 1/~).
Let r = ( i / ~ )
(2.14)
and d e f i n e x i : [ - r , + ~ )
x1(t) = p(O,t+r),
respectively.
÷ R
i = 1,2,
by
x2(t) = q(1,t+r),
Equations ( 2 . 1 3 ) - ( 2 . 1 4 )
p(1,t) = xl(t) ,
imply t h a t i f t ~ O, then
q(O,t) = x2(t)
(2.15) p(O,t) = xl(t-r)
,
I n t e g r a t i n g along c h a r a c t e r i s t i c s ,
q(1,t) = x2(t-r)
it
•
f o l l o w s t h a t f o r 0 ~ s ~ 1, t ~ 0
123 (2.16)
p(s,t)
x1(t + r(s-1)),
=
q(s,t) = x2(t-rs) ,
and hence
(2.17)
IS y(s,t) = [xl(t-r)+x2(t)]/2 Y + ½
o[xl(t+r(v-1))+x2(t-rv)]dp.
Therefore, y is completely determined by xl(t) , x2(t) and t h e i r histories
(Xl)t, (x2) t.
xl(t), x2(t)
first
To determine the governing equations for
note that a simple change of variables y i e l d s
the e q u a l i t y
1 (2.18)
t)+q(p,t)]dv
= r
xl(t+p)+x2(t+v)]d~
"
and f o r - r ~ s ~ O, (s) = g'( s+r ) + r h ( s+r ) = P ( s+r
(2.19) t2(s ) = g,( -s ) -r h ( -r ) = q ( -~ r s r
O) = x2(s)
.
Substituting equations (2.15)-(2.17) i n t o (2.9) and (2.11) y i e l d s the neutral equations
d
d~ [xl(t)-x2(t-r)]
= _ c riLXl(t)_x2(t_r) j G
(2.20)
-kmr{xl(t-r)+x2(t)+~I
[x1(t+~)+x2(t+~)]d~ -r
%r [xl(t) + x2(t _ r)] + 2ru(t) -
m
m
and
(2.21)
d [xl(t-r) d-~
The appropriate i n i t i a l
+ x2(t)] = y [xl(t_r)
_ x2(t)]
.
data for (2.20)-(2.21) is given by
124
[Xl(O) - x2(-r)] = 2r h(1)
(2.22) [Xl(-r) + x2(O)] = 2x g(O)
and
(2.23)
xj(s) = %l(S) ,
Let x(t) = aog(xl(t),x2(t)),
and
~(s)
= aog(tl(S),t2(s)).
x2(s) = t2(s)
-r <- s < 0 .
n = aoZ(nl,q 2) = aol(2rh(1),2~g(O)),
Define the matrices
A!f01 -llO -kr m
(c-~r) m
~
0
B1 =
BO= 0
m
7 C =
K =
0
0
and l e t D, L be the l i n e a r R2-valued operators defined on C(-r,O) by
D~ = Ao~(O)
+ A1t(-r) 0
L~ = B04~(O) + Bl~(-r)
respectively.
The system ( 9 . 2 0 ) - ( 2 . 2 3 )
+ I r Kt(v)dv ,
can now be written in standard
f o rm
(2.24)
d-td [Dxt ] = Lx t + Cu(t)
t >_ 0 ,
125
(2.25)
DXo(.)
= ~ ,
Xo(S)
= ~(s)
-r
<- s < 0 .
Equations (2.24)-(2.25) are equivalent to the i n i t i a l - b o u n d a r y value problem ( 2 . 1 ) - ( 2 . 5 ) in the sense that given the proper i n i t i a l data,
y(s,t)
defined by (2.17) w i l l be a solution (or generalized
solution ) to ( 2 . 1 ) - ( 2 . 5 ) and conversely.
Moreover, as indicated
above, the procedure described above does apply to much more general systems, including nonlinear boundary conditions. As a second example, consider the lossless transmission l i n e shown in Figure 2 with the nonlinear element at the r i g h t end.
The
current through device (in the d i r e c t i o n shown) is given by the
I_ E(t)
T
I
T I
o
s
1
FIGURE 2 nonlinear function f =
f(v(1,t))
of the voltage at s = 1.
equations for the l i n e are
L s ~i(s,t)
= - ~-s- v(s,t)
C s i ~ V(s,t)
= - ~~ i(s,t)
(2.6)
with boundary conditions
The
c
126
v(O,t) + Ri(O,t) = E(t)
(2.7) a ~--~v(1,t) = i(1,t) - f(v(1,t)) ,
where Ls, C s are the s p e c i f i c inductance and capacitance of the l i n e , respectively. I f one follows the general Cooke-Krumme procedure, then as f or the mechanical system above an equivalent nonlinear neutral equation is obtained.
For the lossless transmission l i n e , the neutral
equation has the form
(2.8)
~ t Dxt ] = Lx t + F(t,x(t),x(t-r)) + Cu(t),
where F is determined by f and u ( t )
= E(t).
These two simple examples are given merely to i l l u s t r a t e the fundamental equivalence between certain hyperbolic p a r t i a l d i f f e r e n t i a l equations and neutral systems.
There are many physical systems that
are modeled d i r e c t l y or i n d i r e c t l y by neutral equations (see [ 1 3 ] , [ 1 7 ] , [ 2 3 ] , [ 2 4 ] ) and more recently i t has been observed that certain problems in unsteady aerodynamics involve neutral equations (see [ 7 ] , [ 9 ] ) .
III.
GENERAL RESULTS Let D be a l i n e a r ~n-valued operator with domain D(D) ~ L p
such that ~ ' P ~ s = O.
D r e s t r i c t e d to C is continuous and atomic at
~(D),
In p a r t i c u l a r , we assume that there is a n × n matrix valued
function u of bounded v a r i a t i o n s a t i s f y i n g Vat (~) ÷ 0 as c ÷ O ÷ and sucl
[-~,o]
that i f t~C, then
Dt = t(O) + I~r dv(s)t(s) •
We consider the nonlinear system governed by the neutral functional d i f f e r e n t i a l equation
(3.1)
d Dxt = f(x(t)'xtJ + Cu(t), 0 <- t <- T , d~
127 with i n i t i a l
data
(3.2)
Dx 0 = n
x0 = ~ ,
,
where ( n , t ) ~ z = ~Rn × L~ ucLP(o,T) and the functional f has domain ~ ( f ) s a t i s f y i n g IRn × O(D) ~ ~ ( f ) ~ Z. A solution
( c l a s s i c a l ) to the i n i t i a l
is a function x ( t ) = x ( t ; n , t , u ) ii)
Xo(S) = t ( s ) on [ - r , o ] ;
such t h a t : i ) x ~ S ' P ( - r , T ) ; iii)
absolutely continuous on [O,T]; on [O,T].
iii)
the function y ( t ) = Dxt is i v ) y ( O ) = n and v)
(3.1) holds a.e
A generalized s o l u t i o n (LP-solution) to ( 3 . 2 ) - ( 3 . 3 ) is a
pair of functions ( y ( . ) , x ( . ) ) to C(O,T);
value problem ( 3 . 1 ) - ( 3 . 2 )
such that: i ) y ( t ; n , t , u )
ii) x(t) = x(t;n,t,u)
Xo(S) = t ( s ) a.e on
belongs
belongs to L P ( - r , T ) ;
[ - r , O ] and
i v ) the pair ( y ( . ) , x ( . ) )
s a t i s f i e s the system
y(t) = n +
It f(x(s),xs)dS + 0
x(t) = y ( t )
- I
(3.3)
a.e.
It Cu(s)ds o
o
-y~
d~(s)x(t+s)
,
on [ O , T ] .
In order to obtain existence, uniqueness, continuous dependence and r e g u l a r i t y of s o l u t i o n s , we impose what are now f a i r l y standard (yet c l e a r l y not the most general) assumptions on f . and x ~ S ' P ( - r , a ) ,
define [ ~ a ( X ) ] ( t )
For a > 0
on [O,a] by
[Qa(X)](t) = f(x(t)~x t) .
We shall need the f o l l o w i n g conditions on f :
FI)
For each ae[O,T], Qa defines a continuous operator from w1"P(-r,a)
into LP(o,a) and Q~ can be extended
128
to a unique continuous o p e r a t o r -Qa:LP(-r,a)÷Ll(O,a), F2)
For each bounded s e t M c
LP(-r,a), t h e r e
exists a
continuous nondecreasing f u n c t i o n M such t h a t
IQa(X) -Qa(y )IL1
~ aM(a)Ix-
for
YlLP(-r,a)
x,y ~M.
all F3)
(O,a)
Given 0 < a < r and a bounded s e t N ~
w1"P(-r,a), N
t h e r e e x i s t s a continuous nondecreasing f u n c t i o n a , such t h a t aN(0) = O and
IQa(X)-Qa(Y)ILp < (O,a) f o r a l l x,y~N s a t i s f y i n g F4)
aN(a)Ix-yl 1,p " W (O,a) x(s) = y(s) f o r - r ~ s ~ 0.
For each bounded s e t K ~ w1"P(-r, OJ, t h e r e e x i s t s K continuous nondecreasing f u n c t i o n a such t h a t
for
IQa(X)-Qa(y)I ~ LP(O,a) a l l x,ycK.
The f o l l o w i n g
and [ 2 ] .
Suppose t h a t c o n d i t i o n s
t h e r e i s a T = T(6) > 0 such t h a t i f then ( 3 . 1 ) - ( 3 . 2 )
- YlLP(_r,a)
r e s u l t s may be e s t a b l i s h e d using techniques s i m i l a r
t o those found in [ I ] THEOREM 3 . 1 .
aK(a)Ix
a
FI) - F2) h o l d .
Given 6 > 0,
( % ~ ) ~IRn x Lp and
l(n,~)l
~ 6,
has a unique g e n e r a l i z e d s o l u t i o n on [ - r , T ] .
Let B6 = { ( n , t , u )
~IRnxLP×LQ(o,T(6))
l(n,~)l
~6} and d e f i n e
~:B6÷C(O,T(6JJxLP(-r,T(~)) by [~(q,t,u)](t) = (y(t;%~,uJ,x(t;q,~,uJ), where (y,x) THEOREM 3 . 2 .
×LP(-r,T(~)) THEOREM 3 . 3 .
is the unique g e n e r a l i z e d s o l u t i o n t o ( 3 . 1 ) - ( 3 . 2 ) . If
FI) - F2) hold and 6 > 0, then ¢:B6+C(O,T(6))
i s a continuous o p e r a t o r . If
x(t) = x(t;q,t,u)
FI) - F3) hold and tcW l " p s a t i s f i e s
DO = n, then
i s the unique ( c l a s s i c a l )
to (3.11-(3.2).
solution
L e t H:wT"P÷I~ n be d e f i n e d by H(~) = f ( ~ ( O ) , t ) n o n l i n e a r o p e r a t o r A:D(A) ~ Z ÷ Z by
(3.4)
D(A) = { (D~,~) ~i~n x Lp l ~ W / ' p }
and d e f i n e the
129 and (3.5)
A(Dt~#)
If
= (H(#),~)
.
FI) - F3) hold, ( D t , # J E D ( A )
to (3.1) - (3.2) on [ O , T ] ,
and x ( . )
is the unique solution
then i t follows that z ( t )
= (Dxt,x t)
s a t i s f i e s the abstract equation
d d ~ z(t) = Az(t) + (Cu(t),o)
(3.6) z(O) = z o = (Dr, t) .
Moreover, the solution of (3.6) s a t i s f i e s the integral equation
(3.7)
z(t) = Zo + I~Az(~)d~ + I~(Cu(~),O)d~
.
In order to obtain a representation fo r generalized s o l u t i o n s , the integral equation must be extended.
D(I)
= {zcC(O,T;Z)
Define I : D ( I ) ÷ C ( O , T ; Z )
z(t) = (Dxt,xt) , xEwI"P(-r,T)}
[I(z)](t) =
Az(~)d~
by
and for zED(I)
.
o
If
F4) holds, then one can show that I can be extended to a con-
tinuous operator I : ~ ÷ C ( O , T ; Z ) .
Therefore, (3.7) is replaced by
the operator equation (3.8)
z(t)
= z0 + [I(z)
](t)
+
(Cu(o),O)do
,
o
and we have the f o l l o w i n g r e s u l t . THEOREM 3.4.
Let z 0 = ( n , t ) c Z
and assume that FI)-F4) hold.
I f (y(.),x(..
is the unique generalized solution to ( 3 . 1 ) - ( 3 . 4 ) , then z ( t ) = ( y ( t ) , x is the unique solution of the abstract equation (3.8).
IV.
APPROXIMATINGSYSTEMS We turn now to the development of an approximation scheme f or
nonlinear neutral equations.
The scheme is analogous to the method
Jsed by Kappel [19] f o r l i n e a r equations.
In order to shorten the
t)
130
p r e s e n t a t i o n , we s h a l l the r e s u l t s are v a l i d
assume a g l o b a l L i p s c h i t z if
condition.
However,
one assumes the a p p r o p r i a t e l o c a l c o n d i t i o n s
and p r o p e r l y m o d i f i e s the s t a t e m e n t o f the r e s u l t s .
We s h a l l
con-
c e n t r a t e on the n o n l i n e a r problem
d ~-~ Dx t = h(x(t),x(t-rl) ,...,x(t-rv),x t) + Cu(t)
(4.1) (4.2)
Dx 0 = n
where 0 = r 0 < r I < . . . <
,
x0 = t
,
r v = r and v
~o
D¢ = ¢(0) - j=l~ Bj¢(-r6) - ~-rS(S)~(s)ds .
Moreover, we assume t h a t h i s c o n t i n u o u s l y d i f f e r e n t i a b l e , an n × n - v a l u e d HI)
O 1 f u n c t i o n and h s a t i s f i e s
B(.)
is
the c o n d i t i o n :
There i s a c o n s t a n t K such t h a t
lh~xo'~1"'""'Xv"~)-h(Yo"YI""""'YW~)-<~{Sok~-Y5I+I*-~ILp}, for all (Xo,X1,...,Xv, t) ,(yO,yl,...,yv,t) sZE~(v+I)xL p. Note t h a t ( 4 . 1 ) - ( 4 . 2 )
is a s p e c i a l case o f ( 3 . 1 ) - ( 3 . 2 )
h(x,~(-rl),...,~(-rv),~).
=
Consequently, it
3.4 t h a t z ( t )
= (Dxt,x t)
(4.3)
z(t) = z 0 +
where f ( x , ~ )
f o l l o w s from Theorem
i s the unique s o l u t i o n t o
Az(a)do + o
(Cu(a),O)da o
when z 0 = ( D ~ , t ) s O ( A ) . We s h a l l
use the H i l b e r t
weighted norm l ( n , t ) linear ~
space z = z
Ig = [n12 + -2r0l t ( s )
f u n c t i o n d e f i n e d on ( - r j , - y j _
satisfy
[19]).
It
a . <_ - v ( v ÷ 2 )
IBjl/(rj-r
I)
~ i),
g
= / ~ n × L g w i t h the ( e q u i v a l e n t )
12g(s)ds" by g ( s )
where g i s the piecewise = aj(s+~j_l)+v-j+1
j = 1,2,...,v.
(See [ I ] , [ 2 ]
is easy t o show t h a t in t h i s case the n o n l i n e a r o p e r a t o r A
satisfies
for all
where
z,wsD(A),
where ~ i s a c o n s t a n t (see [ I ] ) .
131 A t r i p l e {ZN, PN, AN}, N = 1 , 2 , . . - ,
DEFINITION 4.1.
is called an
approximation scheme f o r the Cauchy problem (4.1)-(4.2) i f : i)
{Z N} is a sequence of f i n i t e dimensional subspaces of z such that zN ~ P(A} f o r a l l N q
ii)
=
1,2,....
{P~/} is a sequence of orthogonal projections from z onto S v.
iii)
g
{AN} is the sequence of operators on ZW defined by N = 1,2,....
AN =
DEFINITION 4.2.
The approximation scheme {ZN,PN, AN} s a t i s f i e s property
P1 i f f o r each zcZg, PNz ÷ z. The approximation scheme (ZN, PN, AN} s a t i s f i e s
DEFINITION 4.3.
property P2-k i f f o r each z : (Dr, t) with ~cPCk'~,
APNz ÷ Az.
DEFINITION 4.4.
Let {ZN, pN, AN} be an approximation scheme that
s a t i s f i e s P2-k.
We say that {ZN, PN, AN} s a t i s f i e s P3 i f f or each
xcpck'~(-r,T) for
there is a function F(t)
z(t) = (Dxt,xt)
and a sequence cN such that
and 0 ~ t ~ T,
IA~zCt) - A=ct) l ~ r(t)~ . Note that r ( t ) may depend on x. Given an approximation scheme f o r ( 3 . 1 ) - ( 3 . 2 ) , we consider the approximating system in ZN given by (4.5)
zN(t) = PNZo + I~ ANzN(a)d~ + I~ pN(cu(~)'O)d~ "
which is equivalent to the ordinary d i f f e r e n t i a l equation
ddt z~7(t) = ANwN(tJ + F~7(Cu(tJ,Oj (4.6)
zN(O)
=
Sz
0
•
The f o l l o w i n g r e s u l t establishes the convergence of the approximating systems.
Its proof is quite involved and is e s s e n t i a l l y contained
in [ I ] and [ 2 ] .
132
THEOREM 4.1.
Assume that HI) holds, z 0 = (Dt, t)c#(A), u~L2(O,T)
and {ZN, pN, AN} is an approximating scheme s a t i s f y i n g P1, P2-k I f zN(t)
is the solution to (4.5) and z ( t )
to (4.3), then zN(t)
converges to z(t)
and P3
= (Dxt,x t) is the s o l u t i o n
uniformly in t on [O,T] and
in bounded subsets of L2(O,r). The general scheme given above can be realized using spaces of spline functions as the approximating spaces. discrete delay r , p a r t i t i o n [ - r , O ]
For the case of one
by t~ = -jr~N, J
j = 0, I , 2 , , . . N
and
define z N to be the space of a l l elements (D#,t) where ~ is a f i r s t s order, a cubic or a cubic Hermite spline with knots at t { , respectively. Let pN be the orthogonal projection of 2Yn x L 2 = Zg onto ZN. I t is s f a i r l y easy to establish that the r e s u l t i n g approximating scheme s a t i s f i e s PI, P3 and P2-k f o r some k ~ 1 (see [1,2] and [19]). IV.
CONCLUDINGREMARKS Before selecting a p a r t i c u l a r approximation scheme f o r any problem,
i t is important to decide the purpose of the approximation.
I f one is
interested only in simulation, then a high order d i r e c t approach might be best.
However° i f one is interested in numerical methods f o r
i d e n t i f i c a t i o n and control~ then rapid convergence may not be as important as s t a b i l i t y and robustness o f the approximating model.
The spline schemes
described above work extremely well when applied to i d e n t i f i c a t i o n and control problems, Converting i n i t i a l - b o u n d a r y value problems f o r PDE's ( s i m i l a r to the examples in Section 2) to i n i t i a l
value problems f or neutral equations
can in some cases s i m p l i f y the parameter estimation problem.
For example,
in the mechanical system i f the parameter ~ in (2.4) was unknown, then the estimation of ¥ by a d i r e c t approximation of the PDE system is d i f f i c u l t On the other hand, in the corresponding neutral system the parameter x appears on the " r i g h t hand side" of (2.24) and the f i n i t e dimensional part of the i n i t i a l
data (2.22).
numerically and t h e o r e t i c a l l y .
This dependency is easier to handle
133
REFERENCES 1.
J. M. Amillo G i l , J. A. Burns and E, M. C l i f f , Approximation of nonlinear neutral functional d i f f e r e n t i a l equations on product spaces~ p r e p r i n t .
2.
J. Amillo G i l , Nonlinear Neutral Functional Differential Equations on Product Spaces, Ph.D. Thesis, V i r g i n i a Polytechnic I n s t i t u t e and State U n i v e r s i t y , Blacksburg, V i r g i n i a , August 1981.
3.
H. T. Banks, Approximation of nonlinear functional d i f f e r e n t i a l equation control ~ystems, J. Optimization Theory Appl., 29 (1979), 383-408.
4.
H. T. Banks, I d e n t i f i c a t i o n of nonlinear delay systems using spline methods, International Conference on Nonlinear Phenomena in the Mathematical Sciences, U n i v e r s i t y of Texas, A r l i n g t o n , Texas, June 16-20, 1980.
5.
H. T. Banks and J. A. Burns, Hereditary control problems: Numerical methods based on averaging approximations, SIAM J. Control and O p t i m i z a t i o n , 16, (1978), 169-208.
6.
H. T. Banks, J. A. Burns and E. M. C l i f f , Spline-based approximation methods for control and i d e n t i f i c a t i o n of hereditary systems, in Intl. Symp. on Systems Optimization and Analysis, A. Beusousan and J. L. Lions, Eds., Lecture Notes in Control and Info. Sci. Vol. 14, Springer, Heildeberg, 1979, 314-320.
7.
H. T. Banks, J. A. Burns and E. M. C l i f f , Parameter estimation and i d e n t i f i c a t i o n for systems with delays, SIAM J. Control and Optimization, 19 (1981), 791-828.
8.
H. T. Banks and F. Kappel, Spline approximations for functional d i f f e r e n t i a l ~quations, J. D i f f . Eqs., 34 (1979), 496-522.
9.
J. A. Burns and E. M. C l i f f , Hereditary models f o r a i r f o i l s in unsteady aerodynamics, numerical approximations and parameter estimation, AFWAL Technical Report, Wright-Patterson AFB, Ohio, 1981, to appear.
I0.
J. A. Burns and E. M. C l i f f , An approximation technique f o r the control and i d e n t i f i c a t i o n of hybrid systems, Proc. T h i r d AIAA Symposium on Dynamics and Control of Large Flexible Spacecraft, L. Meirovitch, Ed., June 1981, 269-284.
II.
J. A. Burns, T. L. Herdman and H. W. Stech. Linear functional d i f f e r e n t i a l equations as semigroups on product spaces, SIAM J. Math. Anal., in press.
12.
J. A. Burns, T. L. Herdman and H. W. Stech, D i f f e r e n t i a l - b o u n d a r y operators and associated neutral functional d i f f e r e n t i a l equations, Rocky Mountain J. Math., in press.
13.
J. A. Burns, T. L. Herdman and H. W. Stech, The Cauchy problem for l i n e a r functional d i f f e r e n t i a l equations, in I n t e g r a l and Functional Differential Equations, T. L. Herdman, S. M. Rankin and H. W. Stech, Eds., Marcel Dekker, 1981, 137-147.
14.
R. K. Brayton and W. L. Miranker, A s t a b i l i t y theory f o r nonl i n e a r mixed i n i t i a l boundary value problems. Arch. R a t i o n a l Mech. Anal. 17 (1964), 358-376.
134
15.
R. K. Brayton and R. A. Willoughby, On the numerical integration of difference-differential equations of neutral type, J. Math. Anal. Appl., 18 (1967), 182-189.
16.
K. L. Cooke and D. W. Krumme, Differential-difference equations and nonlinear initial-boundary value problems for linear hyperbolic partial d i f f e r e n t i a l equations, J. Math. Anal. Appl., 24 (1968), 372-387.
17.
J. Hale, Theory of Functional Differential Equations, SpringerVerlag, New York, 1977.
18.
F. Kappel, Approximation of neutral functional d i f f e r e n t i a l equations in the state space Rn x L 2 , Colloquia Mathematica Societatis J~nos Bolyai, 30. Qualltative Theory of Differential Equations, Szeged (Hungary), 1979, 463-506.
19.
F. Kappel, An approximation scheme for delay equations, Proc. Int'l Conference on Nonlinear Phenomena in Mathematical Sciences, Arlington, Texas, June 1980, in press.
20.
F. Kappel and K. Kunisch, Spline approximations for neutral functional d i f f e r e n t i a l equations, SIAM J. Nwner. Anal., 18 (1981), I058-I080.
21.
K. Kunisch, Neutral functional d i f f e r e n t i a l equations in LP-spaces and averaging approximations, J. Nonlinear Anal. TMA 3 (1979), 419-448.
22.
K. Kunish, Approximation schemes for nonlinear neutral optimal control systems, J. Math., Anal. Appl., 82 (1981), 112-143.
23.
O. Lopes, Forced occillations in nonlinear neutral d i f f e r e n t i a l equations, SIAM J. Appl. Math., 29 (1975), 196-207.
24.
R. M. Reid, Ph.D. Thesis, Department of Mathematics, University of Wisconsin, Madison, Wisconsin, August, 1979.
25.
D. L. Russell, "Control canonical structure for a class of distributed parameter systems," Proc. Third IMA Conference on Control Theory, Sheffield, September 1980.
26.
D. L. Russell, "Closed-loop eigenvalue specification for i n f i n i t e dimensional systems: augmented and deficient hyperbolic cases," Technical Summary Report #2021, Mathematics Research Center, University of Wisconsin, Madison, August 1979.
27.
M. H. Schultz, Spline analysis, Prentice Hall, Englewood C l i f f s , NJ, 1973.
28.
R. G. Teglas, A control canonical form for a class of linear hyperbolic systems, Ph.D. Thesis, Mathematics Department, University of Wisconsin, Madison, Wiscons, June 1981.
"A C o n n e c t i o n
Formula
Painlev4
for the
Second
Transcendent"
by P.A.
Clarkson,
and J.B.
The Q u e e n ' s
McLeod,
Wadham
College,
College,
Oxford
Oxford
Abstract We c o n s i d e r
a particular
case
of the S e c o n d
Painlev4
Transcendent y" = xy + 2y 3
It is k n o w n
y(x)
where
nonlinear
I.
and c(k) ordinary
lecture
d2(k)
which
if y(x)
~ kAi(x)
as x ÷ + % t h e n
~ d l x l - ~ s i n { ~ I x l 3/2 -!4 d2
d~k)
The
that
are the
= -~-l~n(l
confirms
connection
differential
shows
~n
if 0 < k < i,
Ixl - c} as x
formulae
for this
equation.
that _ k 2)
the n u m e r i c a l
estimates
of A b t o w i t z
and Segar.
Introduction In this d2y
lecture
we
shall
consider
solutions
of
(i)
= xy + 2y 3
dx 2 satisfying
y(x)
÷ 0
as
÷
x ÷ +~
.
(2)
136
(i)
is a p a r t i c u l a r
Transcendent. by P a i n l e v 4 century have
The P a i n l e v 4 a n d his
in an i n v e s t i g a t i o n
the p r o p e r t y
solution
and
that
singularity infinity,
are
transcendents around
the
into w h i c h
independent only
at all e x c e p t
(i) is i n t e r e s t i n g
no
were
first
turn
of the
second
of
order
other
because
14 of
In the
has
any
from
equation
(3)
f = dx dy - y 2 , t h e n u(x,t) Hastings
result,
and McLeod
in c o n n e c t i o n
function,
A~' - xA.
= 0
1
1
=
(3t)-2/3f(x/(3t)i/3).
[3] h a v e
with
The A i r y
at
similarity
u t + 6uu x + U x x x = 0 If
(i)
Ai(x),
proved
and
the
following
(2).
is d e f i n e d
to be the
solution
satisfying
Ai(x) ~ ~-~ix1-~cos(~ixl3/2 Ai(x)
Any KAi(x), unique
as
x
as
x ÷ +~
÷
-~
A.
solution
of
for s o m e k, solution
solution
~Z)
-
~ { ~ - ½ [ x l - ¼ e x p ( - ~2 x3/2 )
Theorem
of
be Y k ( X ) ,
(i) s a t i s f y i n g and c o n v e r s e l y ,
(i) a s y m p t o t i c then
of
[43).
it a r i s e s
Vries
equations
than poles
and the p o i n t
see C h a p t e r
studied
the p a r t i c u l a r
solution
for p o l e s
of the K o r t w e g - d e
Painlev4
on the e q u a t i o n .
transcendent,
(for a s u r v e y
solutions
Second
the s i n g u l a r i t i e s
so d e p e n d e n t
of the s e c o n d
of the
colleagues
any of the s o l u t i o n s
case
case
(2)
is a s y m p t o t i c
for any k,
to k A i ( x ) .
there Let
to is a
this
of
137
If
If
[k]
< i,
Yk (x)
~ d[x[-¼sin{~
where
the
constants
d and
c are
c} as x ÷ -~
dependent
(4)
on k.
[k I : i, 1 [ x)
Yk(X)
has
at a f i n i t e
Yk(X)
~ sgn(k) (x - x0)
Ik[
as x ÷ -~
> i,
Notice y ÷ -y,
a pole
that
hence
takes
Theorem
assume
without
loss
of the
a step
result
constants
the
transformation
interest
d,
that
c and x 0 on k.
direction
k > 0.
to e s t a b l i s h
the
This
by s h o w i n g
- k 2)
was
of the
c o n j e c t u r e d , and
and S e g u r
of T h e o r e m
method
for Yk(X)
in terms and
between
of s o l u t i o n
of
of a l i n e a r Segur
= kAi(X2--~)
[2]
+ ~
ff
upon
(i) and
(3)
leads
integral
show
then
verified
[13.
1 is d e p e n d e n t
relationship
scattering
Ablowitz
(5)
first
by A b l o w i t z
The p r o o f
K(x,y)
by
of g e n e r a l i t y
of some
in that
so
1
numerically,
because
a matter
on k,
as x + x 0 unchanged
d 2 = -~-l~n(l
This
-i
dependent
is left
dependence
lecture
x0,
(i)
It has b e c o m e exact
½
~ sgn(k)(-
Yk(X) If
Ix13/2 - ~4 d 2 Z n l x ] -
that
the
(3),
fact the
inverse
to an e x p r e s s i o n
equation.
if the
integral
K(XlS)Ai(~)Ai(t2--~)dsdt
XX
that,
equation
(6)
138
is c o n s i d e r e d say x > Xl, L2(Xl,~)
for x s u f f i c i e n t l y
then
and
Yk(X)
exists
(depending
a unique
solution
on k) ,
for K in
for x > x 1
= K(x,x)
Hastings hold
there
large
(7)
and M c L e o d , [ 3 ] , s h o w
further
that
(6) and
(7)
for all x if 0 < k < i. Ablowitz
and S e g u r
the d i f f e r e n t i a l
[2] a l s o
show
t h a t K(x,y)
satisfies
equation
2
(Tx + ~ )
K(x,y)
:
for y a x and H a s t i n g s for all
x for
In then
and M c L e o d
[3] s h o w t h a t
(8)
(8) h o l d s
1 v = ~(y
+ x),
- x)
and F(u,v)
= K(x,y)
(8) b e c o m e s
which
(u,v)
= uF(u,v)
is a l i n e a r
if v is a f i x e d To p r o v e asymptotics positive.
ordinary
positive
Theorem
There
differential
as x ÷ -~ w i t h
are two w a y s
equation
(6),
of the d i f f e r e n t i a l
this
the
involves
A comparison asymptotic
equation
for F(u,v)
in t e r m s
asymptotic
is to i n v e s t i g a t e
y - x fixed
of d o i n g
in w h i c h
behaviour
upon
(9)
constant.
is e x p r e s s e d
depends
- v))2F(u,v)
i, the t e c h n i q u e
of K(x,y)
integral
+ 2(Yk(U
behaviour
and
+ 2(K(x,x))2K(x,Y)
0 < k < i.
1 let u = ~ ( y
(8)
~2F ~u 2
the
(x2--~)K(x,y)
this,
case
the
behaviour
and s t r i c t l y
either
through
asymptotic
of k, or t h r o u g h
equation
the
the a s y m p t o t i c
(9), w h i c h
in t u r n
as u ÷ -~ of Y k ( U
- v)
d. of the
behaviour
two
resultant
of K(x,y)
yields
expressions Theorem
i.
for
the
139
II
The A s y m p t o t i c s
of the linear
In the d i f f e r e n t i a l
where Then
equation
- v))2}y(u)
= 0
(i0)
the t r a n s f o r m a t i o n
n(u)
= q%(u)y(u)
~(u)
=
q(u)
(t) dt = -(u + 2y k2(u
- v))
(i0) b e c o m e s d2n d~ 2 + n (where
Using
differential
equation
d2y (u) - {u + 2(Yk(U du 2 make
ordinary
2 q,, _ ( 5q'3)~ 4q 2 16q
= 0
(ii)
' = d/du)
(4) one then
shows
that
d2n + {i - 2d2 (3) i d~ 2 - ~ - cos(2~ + 2 /3v~i/3+
1 d2 ~ ~n[~l- Cl)+
0(I~I-4/3)}~=0
(12) where
c I is a c o n s t a n t
~2 [u 13/2 + ~1 d2~nlul
=where
and
+ c 2 + o(lu1-1)
c 2 is a constant. Then,
since
one can show combination Theorem
in
(12) v is a s t r i c t l y
that ~(~) of cos~
behaves
and sin~.
asymptotically Hence
constant,
like
a linear
we o b t a i n
2
As x ÷ -~ w i t h y - x
fixed
ix+yl%K(x,y ) ~ A ( ~ )-x COS{
for some
positive
functions
and s t r i c t l y
positive
then
52 x+Y2 [3/2
d22 ~ n
x+Y2 [}
+ B ( ~ 2 x) sin{ 2 x+y 3/2 2 1
d2 2 ha
Ix--~ I }
A and B.
(13)
140
III
The Asymptotics
of the integral
equation
If coco
s+t t+ )dsdt = kAi(X--~ )+ ~211 K(x,s)Ai(--~-)Ai(~--~
K(x,y)
(6)
XX
then
K(x,y)
can
K(x,y)
= k
be
expressed
~ n=0
(2k)
which is convergent
series
In(x,y)
for 0 < k < I, where
= Ai(l(x+y))
and
co
In(X'Y)
I
=
co
I X+tl tl+t2 ~ t2n+Y ~n Ai(--~---)Ai(-----~) .... A i ( ~ ) d t l . . . d t 2 n
X
Consider
a Neumann
2n
co
I 0(x,y)
as
X
the operator L
defined by
X
co
(Lxf) (y) = ~
Ai(
)f(s)ds
.
x Then
as
shown
by
Hastings
and
(indeed Hilbert-Schmidt)
McLeod
[3],
self-adjoint
L
X
is a compact
operator
in L2(x, co)
and, at least in a formal sense L --oo 2
=
I
SO co
oo
ifAi(
f(y) = ~ Using
this
it
K(x,y)
can
fAit-~-) s+t f(t)dt}ds
){
--oo
be
shown
- k(2-k2) ~ 2(l-k 2 ) n=0
where A 0(x,y)
.
--oo
1 = Ai(~(x+y))
that
for
y
> x
-k 2 n (I_--~) An(X,y )
and
0
< k
< 1
(14)
that
141 and i i n ~Ai(X~r)Ai' (Xlr-l)-Ai' (Xlr)Ai(Xlr-i 1 An(x,y ) = (-i) n ... Ai(xl n) 1 n 1 r it Ir - Ir-i dXldX 2 . . . d t n 1 with xl 0 = [(x+y)
These An(X,y ) are multiple integrals from -~ to x. Using the known asymptotic expansions of Ai(z) and Ai' (z), as z ÷ -~ Ai(z)
Ai,(z)
~ ~-½1zl
T%
~ ~-½1zl
,2 3/2 1 cost]Izl - 7~)
%
2 3/2 1 sin(] zl - 7~)
one can show that oo
IAi(xl n) {Ai(Xln)Ai'
(Xln_ I)- Ai(Xln_ l)Ai' (xl n)
1 1 sin(21x 3/213/2~ 1 2 Ai(Xln-l)+ ¼ ~~ - I - ~ ) 27 xl In_ 1 The logarithmic cos (Xln) =
e Re Hence
in½ 1 +I In_l -I
term arises from integration of
(i + cos(2Xln) ) .
Then, by induction, "wronskian"
dl
n In-ln-i
this process of multiplying by the
and integration produces terms like
2 IXl 3/213/2 1 xp{i(~ -~)} 1%
½ }] m-i {I_!_Z n (i~+i)+ k Z 2~i k=0 I 2-1
(14) becomes, upon doing the summation after obtaining
a couple of .recurrence relations, exp{i(21xi3/213/2- 47)} "'0 K(x,y) = Re z½1xl%1%
i+i~ I - ~ i £n ( ½) k ½ (l_k 2) i-I 0 (l-k 2)
+ O(Ix1-7/4) 1 (where xl 0 = ~(x+y))
J
142
}zi
in
I+I~
iziln (l-k 2 )
I-I 0
(l-k 2 )
~i~n
t ( 1-k 2 ) as x ÷ -~
Hence K(x,y)
K cos(2 ix_~l 3/2 (x+y)~ ~i 2 ' + 2~ for some constants
x+YI_B} in(l-k2)In T
K and B, d e p e n d e n t
(15)
on v,
as x ÷ -~ and therefore
comparison
of
(13) and
(15) yields T h e o r e m
i,
that is d 2 = -7 -I £n(l-k 2)
References [I]
M.J. A b l o w i t z and H. Segur, "Asymptotic solutions of the K o r t e w e g - d e Vries equation", Stud. Appl. Math. 57 pp.13-44 (1977).
E23
M.J. A b l o w i t z and H. Segur, "Exact solution of a P a i n l e v 4 T r a n s c e n d e n t " , Phys. Rev. Lett. 3_88p p . l 1 0 3 - 1 1 0 6 (1977).
E3]
S.P. Hastings and J.B. McLeod, "A B o u n d a r y Value P r o b l e m A s s o c i a t e d w i t h the Second P a i n l e v 4 T r a n s c e n d e n t and the K o r t e w e g - d e Vries equation", Arch. Rat. Mech. Anal. 73 pp.31-51 (1980).
[43
E.L.
Ince,
"Ordinary D i f f e r e n t i a l
Equations",
Dover
(1944).
The
Inverse Scattering Problem for A c o u s t i c W a v e s *
by David
Colton
A l l i m p o r t a n t d e c i s i o n s m u s t be m a d e on the b a s i s of i n s u f f i c i e n t d a t a - If Y o u M e e t the B u d d h a on the Road, K i l l Him! b y S h e l d o n K o p p
I.
Introduction. In t h i s
open
problems
acoustic
variety
precise
since
applied
recent We
of a p l a n e
shall
harmonic medium
*
was
research
inverse
that
the
is b a s i c a l l y
is m o r e
values
o f the
frequency.
a n y of the
important
Sleeman
for
a survey
our
attention
by a fixed bounded
in p a r t i c u l a r
in p a r t
accurately Hence
new
refer
to the
in
results
the
of t h e s e
reader results
scattering
obstacle
to d e t e r m i n e
by AFOSR
in a c o u s -
between
former
instead
supported
a large
latter
but
wave
scattering
identification
a distinction
for
necessary
the
restrict
and
it is f i r s t
problem"
of B r i a n
in a h o m o g e n e o u s
This
whereas
and discuss
problem
to d e s c r i b e
target
and note
discuss
further
"the
used
first make
diffraction
paper
time
not
by
progress
scattering
proceeding
with
intermediate
shall
to the
We
phenomena
to low and
"inverse
([19]).
concerned
recent
inverse
has been
"diffraction"
frequency
on the
the
before
this phrase
and
this paper we
with
survey
on w h a t w e m e a n
propagation.
"scattering"
shall
However
of p r o b l e m s
tic wave
a high
connected
waves.
to be m o r e problem"
paper we
situated
information
Grant 81-0103o
144
about
this obstacle
of the
from a k n o w l e d g e
s c a t t e r e d wave.
the s c a t t e r i n g of the
Hence we are e x c l u d i n g
of waves
speed of sound
tion of e q u i v a l e n t with
determining
viz.
its shape
a ltho u g h
tion.
and/or
its surface
has
of t e c h n o l o g y impedance
the inverse
in many
composition
not the
scattering The
obstacle
inverse
particularly
(2) i m p r o p e r l y
it is perhaps
the
latter
deed we
shall
see shortly
pattern
in general
and if a s o l u t i o n on the m e a s u r e d
construct
a solution
point
the q u e s t i o n
exists
does exist data.
is of b a s i c
object,
cannot be r e m e d i e d
e.g.
Of these
the most
above,
or
is
two reasons
difficulty.
measured
In-
far field
to the inverse
scattering
it does not depend
con-
Hence before we can begin
the
in
it is
scattering
remark
to
p r o b l e m we m u s t
of what we mean by a "solution". recalling
on the
of w h e t h e r
as d e f i n e d
for a given
to the inverse
it is w o r t h w h i l e
of i n f o r m a t i o n
that
less atten-
or a submarine.
posed.
crea£es
no s o l u t i o n
tinuously
answer
that
from
and
information
for two reasons:
and
sonar,
received
the q u e s t i o n
problem,
(i) n o n l i n e a r
problem,
scattering
that
as a b a s i c p r o b l e m
problem
it gives
obstacle,
of an o b s t a c l e
such as radar,
is a w h a l e
to solve
We note
has
loca-
only be c o n c e r n e d
recognized
since
as
and the
scattering
the shape
answer
scattering
difficult
of the
problem
of the unknown
the case of sonar it can help
medium,
impedance
applications
such topics
we shall
long been
behavior
the d e t e r m i n a t i o n
impedance.
of d e t e r m i n i n g
of areas
Nevertheless
material
Finally,
two b a s i c p r o p e r t i e s
the inverse
importance
obstacles,
in a n o n - h o m o g e n e o u s
far field m e a s u r e m e n t s
tomography,
by m o v i n g
sources.
the p r o b l e m
in a v a r i e t y
of the a s y m p t o t i c
At this
of Lanczos:
by any m a t h e m a t i c a l
"A lack
trickery".
145
Hence
in o r d e r
necessary
to d e t e r m i n e
to i n t r o d u c e
the p h y s i c a l
situation
the q u e s t i o n
of w h a t
actually
construct
o n l y by the fact
that
fact
the
incorporated We scattering discuss
into shall
in this
needed
to c o n s i d e r
unit
continuous the
mentioned
and this
normal
~.
three domain
impedance
boundary
value
formulated
u c C 2 ( ~ 2 \ D ) ncl(~2\D)
of
such
resolved
have
to not
also
information
the has
been
topics
we w a n t
our problem
dimensional
I = l(x)
8D and k>0
smooth
for a c o u s t i c of
are D be
boundary
~D
the
number,
waves
finding
to
Let
> 0 denote
the w a v e
inverse
in ~2 and
that
problem.
in ~2 w i t h let
of the
then
can be
a function
that
- - + iklu r I/2
but
modifications
as the p r o b l e m
A 2 u + k2u = 0
lim
specific
problem
u = ui + US
~U
reflects
is c o m p l i c a t e d
formulation
formulate
If we
then
it is
model.
the n e c e s s a r y full
we
is n o n l i n e a r ,
the
shall
connected
surface
mathematically
a solution,
a mathematical
We
the
to model.
"nonstandard"
and o u t l i n e
simply
impedance
by
the p r o b l e m
appropriate,
outward
trying
solution,
paper.
when
and
Having
are
n o w give
state,
a bounded,
we
the m a t h e m a t i c a l
problem
by a solution that
is m e a n t
that
we m e a n
"nonstandard'information
this
above
what
= 0
in
N2\D
(l.la)
in
~2\~
(l.lb)
on
~D
~uS (~--~---iku s) : 0
(l.lc)
(l.ld)
146
where
the
"incoming
Helmholtz the
equation
radiation
(r,8) when
wave" (l.lb)
condition
are p o l a r
u i is an e n t i r e and the (l.ld)
coordinates.
I is i n f i n i t e ,
i.e.
case we
"soft". (l.ld)
The and
(c.f.
(l.la),
[i0]).
We
F is k n o w n
given
incoming
(l.lc'),
paper
find
know
this
fact m a k e s
far
u i.
(2) g i v e n
only
D.
both
The
are
(l.lc)
case
becomes
obstacle
of a s o l u t i o n
that
is w e l l
as b e i n g
to
(l.la)-
known
if u is a s o l u t i o n
or
inverse
a knowledge We
shall
of the
(l.ld)
then
of
u s has
but
a knowledge
precise
as m e n t i o n e d
inverse
to the
[i0])
to the
problems
we
on w h a t above
scattering with
direct
we
defines
a
condi-
we m e a n in g e n e r a l
inexact
and
problems
is that
scattering
(l.lc') , (l.ld)
shall
of u i, F and D,
are by d e f i n i t i o n
fact we b e g i n
(l.lb),
(c.f.
corresponding
scattering
be m o r e
above
(1.2)
of u i, F and the b o u n d a r y
which
solution
(l.la),
equations
pattern
(i) g i v e n
The b a s i c
of a u n i q u e
of i n t e g r a l
the
(l.lc')
(l.lc'),
field
of F shortly,
posed.
(l.la)-(l.ld)
condition
to G w h e r e
r - i / 2 F ( e ; k ) + 0 ( r -3/2)
F from m e a s u r e m e n t s
improperly
consider
scattering
shortly
= e ikr
as the
by a " k n o w l e d g e "
existence
to the
(l.la) , (l.lb),
wave
in this
tion
respect
~D.
uniqueness
see
u s satisfies
behavior
where
l, or
on
also
of the
wave"
with
(l.lb) , (l.lc') , (l.ld)
uS(r,0)
find
and
shall
(or
asymptotic
discuss
refer
existence
(l.la)-(l.ld) the
shall
shall
the b o u n d a r y
u = 0
In this
"scattered
uniformly We
solution
the problem
by the m e t h o d
(nonlinear)
mapping
147
from
D or i to F.
;ate our
)
tasks
Determine
Hence
from
an a b s t r a c t
point
of v i e w we
can
as follows:
the
range
of T
(denoted
by
R(T))
as a s u b s e t
of
L2[0,2~].
~) E s t a b l i s h
the e x i s t e n c e
uniqueness
of T -I on R(T) , i.e.
of the
solution
a subset
XcR(T)
to the
inverse
show
the
scattering
problem.
;) D e t e r m i n e L210,2~]
such
that
~-i
and
= T-I
^-i T defined
an o p e r a t o r on X and ~-i
on
is c o n t i n u o u s
~
on L 2 [ 0 , 2 ~ ] , (In o r d e r
i.e.
to do this
information
,) Give
at
about
shall
in the
should
be
The M a p p i n g
T and
As m e n t i o n e d ' (l.la)-(l.ld)
the
form
of
uS(x)
we
about
and
a priori
^-i Tx,x~X.
the
above
in a d d i t i o n
progress
Introduction
corresponding
(l.lc'))
is to be
we
the
single
four
give
directions
achieved.
1( ~ ( y ) G ( x , y ) d s ( y )
obtain with
u s of
the
solution
Dirichlet
of i n t e g r a l
solution
layer
can
problem
by the m e t h o d
represent
aD
to a s s u m e
problem
its Range.
a modified = ~1
is k n o w n
sections
in the
can
scattering
for d e t e r m i n i n g
further
(or the
pundary c o n d i t i o n particular
what
following if
inverse
i).
method
examine
taken
the
it is n e c e s s a r y
D or
a constructive
We "oblems
stabilize
eguations.
(l.la)-(l.ld)
potential ~ ; x~
R 2
\~
(2.1)
148
where
G is an a p p r o p r i a t e l y
and @ is the unique
chosen
solution
fundamental
of an i n t e g ra l
solution
equation
([22])
of the
~u i + 315 + IK2 ~ = ~ + lu i
where ~i and 32 are compact dent of I and d e f i n e d Dirichlet
problem
on C(~D).
[22])
1 = ~
~(~)
soluton
33 is a compact (2.2)
and use ship
or
i:
a continuous
The
above
function
we now
behavior
to d e t e r m i n e
problem
(2.3)
of the
form
(2.4)
defined
let r=
of G(x,y)
T:Z÷F
equation
If in
Ixl tend to i n f i n i t y we obtain
or T:$D÷F.
the v a l i d i t y
on C(~D).
the
relation-
In p a r t i c u l a r
of the
following
is an entire
function
such
theorem:
of
8 and
of k for k>0.
the
Theorem
far field p a t t e r n
scattering
form of a double
shows b o t h the n o n l i n e a r
inexact m e a s u r e m e n t s , measured
of the c o r r e s p o n d i n g
i
as the fact that R ( T ) ~ L 2 [ 0 , 2 ~ ] .
not p o s s i b l e
are indepen-
; x~R~\D
operator
far field p a t t e r n
analysis
T as well
(2.4)
establishes
The
= u
integral
and the m a p p i n g
a calculation
Theorem
(2.3),
the a s y m p t o t i c
(1.2)
in the
of an integral
@ + K3~
(2.1),
solution
that
~
~ is the unique
w here
(2.2)
operators
~ 9-~-C~ G(~'~)ds(z)
~D w here
The
can be r e p r e s e n t e d
layer p o t e n t i a l
uS(j)
integral
form
unless
analyticity 1 implies
no solution further
nature
of the o p e r a t o r
Furthermore of a function
that exists
assumptions
in general to the
since from
for a given
inverse
are made.
it is
149
From
(2.2)
on u i , i.e.
The
over
We
dense
it is c l e a r
the
T depends
following
auestion:
pose
all e n t i r e
solutions
of the H e l m h o l t z
fixed,
example
shows
the
operator
therefore
in L 2 [ 0 , 2 ~ ] ,
following
that
can
~ or D is k e p t
patterns
(2.4)
T = T ( u i) .
If u i r a n g e s where
or
is the
corresponding
or m o r e
concisely,
that
this
is n o t
equations
set of does
true
far
field
R---~=L2[0,2~].
in g e n e r a l .
!
Example:
Consider
is the u n i t Helmholtz
J
n
i
u
(r,8)
denotes
convergent S
disk.
in the
=
F(8;k)=
e iz/4
~
for some
.
Hence
is n o t
dense
form
n9 + b
and the
of
sin ng]
(2.5)
~2.
series
Then
for
(2.5) r>l
is u n i f o r m l y
we c a n e x p a n d
series
n cos n9 + b n sin nS]
(2.6)
function
of the
from
of H a n k e l ' s
of the
integer
to cos no8
the
class
D
of the
first
function
kind
we h a v e
and that
the
far
by
~ (-i)nJn(k) [a n cos n9 + b sin ne] (2.7) ~k n=0 H (i) (k) n n
Jn(ko)=0
is o r t h o g o n a l
solution
when
-Jn(k) - [ a H(1) (k) n
for u s is g i v e n
an e i g e n v a l u e
u
subset
Hankel's
If k 2 is o
i
function
convergent
behavior
pattern
u i in the
(l.ld)
n
= - ~ H (I) (kr) n=0 n
asymptotic
u i is an e n t i r e
Jn(kr) [anCOS
Bessel's
H(1)denotes n
field
since
can e x p a n d
~ n=O
uniformly
uS(r,8)
the
we
(l.la) , (l.lb) , ( l . l c ) ,
on any compact
.
where
Then
equation
u
where
problem
of
in L 2 [ 0 , 2 ~ ] .
interior
Dirichlet
n = n o and h e n c e and sin no@
far f i e l d
in this
for all
patterns
problem case
incident
for such
then
F(9;k
o
)
fields
values
of k
150
It is an open q u e s t i o n to determine if similar e x a m p l e s are v a l i d for arbitrary domains D. w o u l d e s t a b l i s h an interesting
If such e x a m p l e s exist this
relationship b e t w e e n the far field
patterns of e x t e r i o r b o u n d a r y value p r o b l e m s e q u a t i o n and the
(interior)
for the Helmholtz
eigenvalue p r o b l e m for Laplace's
equation. The v a l i d i t y of the above example is b a s e d on the fact that the set
Jn(kr) cos ne; n = 0,1,2,... J (kr) sin n~ n is incomplete
in L2(~D)
D i r i c h l e t problem.
if k 2 is an e i g e n v a l u e of the interior
H o w e v e r it can be shown
proof can be b a s e d on the ideas of
[13])
([5]; a simpler
that the set
(b
+ ikl)
Jn(kr)
cos ne
(~
+ ikl)
Jn(kr)
sin nO
; n=0,1,2,...
is complete in L2(~D)
(2.9)
for a r b i t r a r y b o u n d e d domains p r o v i d e d
0
(2.8
This fact implies the following
([5]) :
T h e o r e m 2:
Let I be a c o n s t a n t such that 0
Then R(T)=
L2[0,2~]. A w e a l t h of q u e s t i o n s
concerning R(T)
and this provides a m a j o r m a t h e m a t i c a l
challenge of b a s i c impor-
tance to the inverse scattering problem. tioned one of these questions problem.
remain u n a n s w e r e d
We have already men-
in connection w i t h the D i r i c h l e t
A n o t h e r is the following:
Determine
the compact set of
151
far field p a t t e r n s field and wave in a given
number,
compact
to the p r o b l e m by the
fact that
subset
of the p o s i t i v e
by T h e o r e m
already
solutions
to the inverse scattering
3:
problem
D is u n i q u e l y
positive
axis.
same
Suppose
there
existed
far field p a t t e r n
D2 are disjoint. Helmholtz
Then
equation,
containing
that uSH0
([15]),
F.
and this
the
i.e.
first
determines
radiation
(l.lc')
is given by
of the
of the
A similar
the
the
D1 and
of solutions
to the
a disk by analy-
of the H e l m h o l t z But this
ui=0
implies
on 8D I.
for an i n t e r v a l
since
far
D 1 and D 2 h a v i n g
solution
of the L a p l a c i a n
is discrete.
data
and we can c o n c l u d e
we have
of
D from a k n o w l e d g e
u s outside
condition.
the
consider
the case when
from the a n a l y t i c i t y
from
first
by a k n o w l e d g e
two o b s t a c l e s
is a c o n t r a d i c t i o n
for the L a p l a c i a n
surfaces.
8 and k on any interval
Consider
F uniquely
u i is an e i g e n f u n c t i o n values
can be p o s e d
the u n i q u e n e s s We
of d e t e r m i n i n g
that u s is an entire
satisfying
As o p p o s e d
Introduction,
problem.
D 1 and D 2 in its i n t e r i o r
tic c o n t i n u a t i o n equation
in the
determined
F for all angles
Proof:
lying
is c o m p l i c a t e d
set of closed
that the b o u n d a r y
field p a t t e r n real
real axis.
to e s t a b l i s h i n g
scattering
of ui and F where we assume
Theorem
impedance
A similar p r o b l e m
mentioned
of T -I is e q u i v a l e n t
([17]).
surface
incident
of T-I.
existence
(l.lc')
to a fixed
2, this p r o b l e m
SD lying in a compact
The E x i s t e n c e
inverse
(constant)
it is nonlinear.
As we have
the
that c o r r e s p o n d s
but with
answered
for i fixed but
III.
in L2[0,2~]
Hence of k
set Of e i g e n v a l u e s
argument
holds when
152
D=D,nD 2 is nonempty,
in p a r t i c u l a r by analytic c o n t i n u a t i o n one
can show that the L a p l a c i a n defined in D I \ D or D 2 \ D has a continuous spectrum, w h i c h is a contradiction. We now c o n s i d e r the inverse scattering p r o b l e m of determining
i from a k n o w l e d g e
p r o b l e m is given by
(l.la)-(l.ld).
to Colton and Kitsch
T h e o r e m 4:
of u i, F and D, i.e. the b o u n d a r y value The following theorem is due
([8]).
I is u n i q u e l y d e t e r m i n e d bv a k n o w l e d g e of the far
field pattern F for all angles 8 and any fixed positive value of the wave number k.
Proof:
Let 11 and 12 be two solutions of the inverse
problem corresponding
to the same far field pattern F.
analytic c o n t i n u a t i o n we can conclude that Ul=U 2 and $u2/~
on ~D where u I and u 2 are the solutions of
c o r r e s p o n d i n g to I 1 and 12 respectively. condition
scattering Then by ~Ul/~
=
(l.la)-(l.ld)
Then from the b o u n d a r y
(l.lc) we have
(Ii-12)u I = 0
on ~D.
(3.1)
We now note that u I is not i d e n t i c a l l y zero in any n e i g h b o r h o o d ScUD since if this were the case
(l.lc)
S and hence by H o l m g r e n ' s u n i q u e n e s s But this is a c o n t r a d i c t i o n
implies that ~ U l / ~ H 0
theorem
([3])
UlH0
in
in ~2\D.
since U l = U ls+ U i 1 and u~ satisfies the
radiation c o n d i t i o n but u i does net. 1
Therefore
if xe3D there ~
exists a seauence of points Xn÷X such that Ul(Xn)~0.
From
(3.1)
we have that 1 1 ( X n ) = 1 2 ( X n ) and since i I and 12 are assumed to be continuous we have 11(x)=12(x). ~D this completes
Since x is an a r b i t r a r y point on
the proof of the theorem.
153
We note analytic assume in
that
functions,
from T h e o r e m it suffices
that F is only known
[0,2z]
instead
pattern
in T h e o r e m
of the
we are given
above
theorems
the
scattering
of 8 values
of e in
of c o n s i d e r a b l e
for
4 to contained
[0,27].
interest
if i n s t e a d cross
theorem
3 and T h e o r e m
for an interval
of for all values
An open p r o b l e m the v a l i d i t y
1 and the i d e n t i t v
is to d e t e r m i n e
of the
section
far field
~ defined by
27 ~(k;~)
= I
IF(~;k'~) 12d8
(3.2)
o w here
F(8;k,e)
is the far
incoming
plane wave
IV.
Operator
The
As p o i n t e d problems the
Hence
that
exist
is to restore e xtra
a priori
impedance clude
out in the
in g e n e r a l
task
in this p a p e r
no solution
stability.
XcR(T).
We shall
operator
~-l d e f i n e d
show that
on L2[0,2~]
on L2[0,2~],
of I or D on the far
such
thus
that the term
imply that
such i n f o r m a t i o n
of these
this by a s s u m i n g the unknown
it is p o s s i b l e
to define
such that T - I : T - I
"a priori
data.
problems
lies in a compact
restoring
to conset an
on X and ~-i
the c o n t i n u o u s
dependence
At this p o i n t we w o u l d information"
is i m p o s s i b l e
in
and if a solution
it is p os s i b l e
field pattern.
to m e n t i o n
improperl~7 p o s e d
concerning
that
scattering
on the initial
accomplish
far field p a t t e r n
then
are
treatment
is available
obstacle
the inverse
exists
continuously
We shall
information
the exact
is continuous
Introduction,
in a s a t i s f a c t o r y
or s c a t t e r i n g
that
to the
~-i.
it does not depend
a major
corresponding
u±=exp(ikx.~).
we are c o n s i d e r i n g
sense
does
field p a t t e r n
is not m e a n t
to obtain
like to
from e x p e r i m e n t a l
154
data, but only that this
(in general nonstandard)
information
is
sufficient to stabilize the p r o b l e m under consideration. We first c o n s i d e r the p r o b l e m of d e t e r m i n i n g the impedance from a k n o w l e d g e of the far field pattern F w h e r e the incident wave u i and the s c a t t e r i n g obstacle are assumed known using the compactness of the operators ~i and ~2 in
([8]).
By
(22) it is
possible to show that the m a p p i n g T:I+F is continuous and in particular if ~ is the solution of the integral equation write F=F(e;#,l)
where F is a c o n t i n u o u s
Now let fcL2[0,2~]
(2.2) we
can
function of @,~ and I.
be the m e a s u r e d far field pattern.
Then we
can reformulate our inverse scattering p r o b l e m as an o p t i m i z a t i o n problem,
denoted by Pf, as follows:
Minimize
27 = I IF(@;~'l)-f(£) 12d@ ~o
Cf(~,l)
subject to ~ being a solution of
(2.2)
"control set" to be specified shortly.
(4.1)
and leU where
U is a
We first note the following
facts: (i) The set U contains the a priori i n f o r m a t i o n we are a s s u m i n g about I. (2) If I eU is a solution of the inverse scattering p r o b l e m for
a given
density,
(3) If is
then
(~ ,l ) a solution
Cf(~
far
,X ) > 0
field
pattern
and
~
is
the
corresponding
(~ ,X ) is a solution of Pf since Cf(~
is a solution of Pf and Cf(~ the
inverse
scattering
problem.
then
the
inverse
scattering
problem
I IF-fl IL2[0,2~]
If is
is a best a p p r o x i m a t i o n is minimal.
,~ )=0.
,~ )=0 then
of
solvable for X eU, but X sense that
f
not in the
155
We now define our control set U in such a way that Pf has a solution.
In p a r t i c u l a r
let
U = {%{C+(3D): Jl(x)I<_MI, ll(x)-l(y)l<M21x-yJ a}
where C+(3D)
is the cone in C($D)
consisting of all continuous
functions i defined on ~D such that I~0, and MI, M 2 and ~ , 0 < ~ i , are fixed constants. is compact in C(3D)
The A r z e l a - A s c o l i
Theorem implies that U
and since the functional Cf is continuous
we have the following theorem:
T h e o r e m 5:
Let f(L2[0,2~].
The Pf has a solution.
In general the solution of Pf is not uniaue.
be the
set
of all
solutions
(~ , t
) of Pf.
Then the
of U and the c o n t i n u i t y of T implies the following
Let ~ (f)
compactness result on
continuous dependence:
T h e o r e m 6:
exists
If fn÷f in L2[0,2z],
a convergent
subseguence
(~n,ln)~#
of
{(~n,tn)
(fn) , then there
} and every
limit
point
lies in ~ (f).
Note that if Pf is u n i q u e l y solvable, single o r d e r e d pair ,
f÷(¢
i.e. ~ (f) is a
(~ ,I ) , then T h e o r e m 6 simply says that
w
,I ) is a continuous m a p p i n g ^-i
In p a r t i c u l a r the o p e r a t o r T
from L2[0,2z]
:f÷(}
*
*
,I ) ,
into C(3D)xU.
T U = X p s a t i s f i e s the
^-l conditions on T as set forth in the opening p a r a g r a p h of this section.
If P f is not u n i q u e l y solvable then we must interpret
~-i as a set valued mapping and the criteria on ~-i as set out
156
in this
opening
paragraph
must
For the e x t e n s i o n inverse
scattering The
of a "soft" pattern
surfaces sisted
U.
of those b o u n d a r i e s
such that
~D has
In the case it was
p r o b l e m was
a uniformly
of the three
also n e c e s s a r y
to the origin
a priori
problem,
(i.e.
the c o n t i n u i t y
mine
V.
we m e n t i o n
that D was
no s t a t e m e n t
problem
that
is too weak
far field
the
impedance
starlike
of
set U conannulus
tangent.
scattering
H~ider, remains
there
Methods
The p r e v i o u s
problem
with
respect
that
This
since
in the case to
if
the
to a c t u a l l y
we refer the r e a d e r
deter-
of [2].
^-l TX,xETU. the
in this p a p e r
optimization
etc.).
hope of using
fashion
in
scattering
to be i n v e s t i g a t e d
of this p o i n t
shows
succeeded
of c o n t i n u i t y
logarithmic,
for D e t e r m i n i n g
we are c o n s i d e r i n g nonlinear
the inverse
is little
problems
section
we have
on the type
in a c o n s t r u c t i v e
scattering
Constructive
constrained,
shape
family
continuous
inverse
to stabilize
For a d i s c u s s i o n
inverse
p robl e m s
[12].
the
compact
of the
in a fixed c i r c u l a r
that a l t h o u g h
Lipschitz,
problem
~ or D.
linear
see
inverse
[7] where
H~ider
dimensional
information
is an i m p o r t a n t
optimization
in
contained
to assume
we have made
that results
discussed
bounded
case
of the
as the
manner.
([i]).
In closing using
waves
from a k n o w l e d g e
an a p p r o p r i a t e
~D
to the
of d e t e r m i n i n g
in a similar m a n n e r
we can define
This
results
problem
obstacle
can be h a n d l e d
in an obvious
for e l e c t r o m a g n e t i c
scattering
scattering
problem provided
of the above
problem
inverse
be m o d i f i e d
inverse
scattering
can be r e f o r m u l a t e d
problems.
The n u m e r i c a l
as
157
solution of these o p t i m i z a t i o n p r o b l e m s i n v e s t i g a t e d by A. Kirsch
([16]).
is p r e s e n t l y being
R e l a t e d approaches have been
p r e v i o u s l y c o n s i d e r e d by A. Roger
([18])
and B. Sleeman
N e e d l e s s to say there are many open problems a n u m e r i c a l approach,
particularly
([19]).
c o n n e c t e d w i t h such
in connection with the q u e s t i o n s
of stability and convergence. The c o m p u t a t i o n a l problems Txl,xETU,
a s s o c i a t e d with computing
are c o n s i d e r a b l y s i m p l i f i e d if one has access to low
frequency data.
We first b r i e f l y consider the p r o b l e m of deter-
mining the surface impedance of an obstacle from low frequency far field data
([6],
[21]).
I,et F be the far field pattern
arising from the addition of the s c a t t e r e d waves c o r r e s p o n d i n g to two incoming plane waves striking the obstacle directions.
If we expand F in its Fourier series
F(8;k)
then the
(weighted)
=
~ an (k) e in@ n:-~
(5.1)
low frequency limit of the c o e f f i c i e n t an(k)
determine a sequence of numbers b harmonic
from opposite
n
from w h i c h we can define the
function h(x) ~
=
~ b r-[nle in0 n n=-~
It can be shown that the series
(5.2)
(5.2)
converges
for r>a where
a is the radius of a disk c o n t a i n i n g the scattering obstacle in its interior.
From
[6] we now have that the unknown impedance
I is the solution of the integral eauation of the first kind
h(x) = I I(z)N(~'z)ds(z) ~D
; Ix; = a
(5.3)
158
where
N denotes
the e x t e r i o r known,
posed.
XEU,
a soft
in a g r e e m e n t
on relating
of the
obstacle
can be used
frequency
a>0
(weighted)
then the ~n are
~n = 2~n a
n-i
for n=l,2.,,,o experimental
related
data.
are d e t e r m i n e d
from
limit
to the
(5.3)
Furthermore, and
(5.3)
is not
is hence
of the p r e v i o u s to determine
([3],
of the
improperly a priori
section.
the shade
This m e t h o d
limit of the F o u r i e r
coeffi
of the L a u r e n t
onto
(unknown)
the
the Laurent
scattering
expansion
a2 + --~ +... w
(5.4)
if Pn denotes
of the F o u r i e r
coefficients
of the
form
coefficients,
radius
(5.5)
a can be d e t e r m i n e d
if the c o e f f i c i e n t s
fN is d e f i n e d by
of
far field
[9]).
a n by a relation
the m a p p i n g
(5.5)
of
In p a r t i c u l a r
an_ 1 + lower order
where
side
f that c o n f o r m a l l y
f has
radius.
low f r e q u e n c y
(Inl<~q for some
function
aI = aw + a ° + ~--
is the m a p p i n g
approximately
to the c o e f f i c i e n t s
analytic
in
the use of r e g u l a r i z a t i o n
from a k n o w l e d g e
at infinity
f(w)
the
results
of the unit disk
such that
bn
equation
can be a c h i e v e d by a s s u m i n g
low f r e q u e n c y
(unique)
the e x t e r i o r
w here
of solving
the
of the the
through
left hand
of the far field p a t t e r n
expansion maps
obstacle
at low values
is b a s e d
with
precedure
scattering
pattern
the
stabilization
A similar
to be only
of the c o e f f i c i e n t s
and the p r o b l e m
However
for Laplace's
F is assumed
In p a r t i c u l a r
exactly
cients
Since
function
integer N) must be done
procedures.
that
of D.
the c o m p u t a t i o n
posit i v e
known
the N e u m a n n
from
ao,al,...aN
159
aI = aw + a ° + w- - +
fN(w) then
from
possible
the A r e a to d e d u c e 1 2~
Note the
that
from
that
to a c h i e v e will
be
given
(5.5)
small
guaranteed
if
that
Section
IV and h e n c e a uniformly
the m a x i m u m
that
are
associated
scattering obstacle
above problem
from
by C o l t o n ments
nol~
of this
([14]),
Sleeman
attempt
to e x t e n d
in ~3 has
been
necessary
it is
([20]), this
given
of the
in this
case
This
in a
of
a priori
tangent.
that
This
inverse
sec-
is w i t h
estimates
respect
the
suggests
scattering
to n o r m s
appropriate
approach
the far
shape field
subsequently
and S m i t h
by C o l t o n
and below.
of the p r e v i o u s
of the
with
of
in o r d e r
to be c o n t a i n e d
to a s s u m e
and K l e i n m a n
approach
values
in
different
compact
sets
norms.
of d e t e r m i n i n g
have
errors
Hence
above
continuous
function-theoretic
method
large
of the p o i n t w i s e
stability
these
and C o l t o n
from
results
and d e t e r m i n i n g
with
large
is large.
estimate
is m e a s u r e d
a knowledge
([4])
the
H~ider
the
cause
a priori
error
it is not
continuity
from
The
theory
(5.7)
whereas
n
(5.7)
with
instead
bounded
a
be b o u n d e d
the
of e x a m i n i n g
where
of a can
8D is k n o w n
to the L 2 n o r m
problem
estimate
in a g r e e m e n t
however
the p r o b l e m
values
a must
respect
~D has
function
estimate
coefficients
the e r r o r
stability
Note
L2-error
of the
annulus,
tion.
the
in g e o m e t r i c
(5.6)
12Z " a2 o [ f ( e l ~ ) - f N (oil) 12d~ ! N+l"
computation
a imply
Theorem
aN " . " + --N w
of the pattern
([9]). been
([21]) .
to the
to the
scattering was
first
Further
provided
develop-
successful
scattering
([ii]).
The
given
by H a r i h a r a n
A partially
inverse
and K r e s s
inverse
problem
problem
in
160
this case
is that
instead
rely on the use of level a ppro a c h
touches
The
discussion
the
surface
problem
of the
and it is hoped in this
in the
previously
has
recent
indicated
sufficient fasci n a t i n g
in this
extent
theory
section
computational
solutions
one must
and such an
that the
scattering
to e n c o u r a g e
only in
scattering
yield
new d e v e l o p m e n t s numerical
is the b a s i c
open
linked
to all of the pro-
Our hope
is that this p a p e r
and p o s s i b l e others
inherent
the e f f e c t i v e
problem
mentioned.
progress
inverse
future will Indeed
obviously
problems
to the
field and is i n t i m a t e l y
blems we have
methods
results.
direction.
of the inverse
mapping
in p o t e n t i a l
provided
approximate
and insights solution
curves
yields m u c h w e a k e r
constructing problem
of conformal
new d i r e c t i o n s
to enter
this
to a
important
and
area of research.
Re fe rence s
i.
T. S. Angell, D. Colton, and A. Kirsch, inverse s c a t t e r i n g p r o b l e m for a c o u s ti c Eqns., to appear.
The three d i m e n s i o n a l waves, J. Diff.
2.
M. Bertero, C. de Mol, and G. A. Viano, The s t a b i l i t y of inverse problems, in Inverse S c a t t e r i n g P r o b l e m s in Optics, H. P. Baltes, editor, S p r i n g e f ± V e i i a g , ' Berlin, 1980, 161-214.
3.
D. Colton, A n a l y t i c T h e o r y of Partial P i t m a n Publlshing, L o n d o n , 1980.
4.
D. Colton, The inverse scattering p r o b l e m for a cylinder, Proc. Royal Soc. E d i n b u r g h 84A (1979), 135-143
5.
D. Colton, R u n g e ' s t h e o r e m and far field p a t t e r n s impedance b o u n d a r y value p r o b l e m in a c o u s t i c w a v e SIAM J. Math. Anal., to appear.
6.
D. Colton, Stable m e t h o d s for d e t e r m i n i n g the surface impedance of an obstacle from low fremuency far field data, s u b m i t t e d for publication.
Differential
Equations,
for the propagation,
161
7.
D. Colton and A. Kirsch, Stable m e t h o d s for solving the inverse scattering p r o b l e m for a cylinder, Proc. Royal Soc. E d i n b u r g h 89A (1981), 181-188
8.
D. Colton and A. Kirsch, The d e t e r m i n a t i o n of the surface impedance of an obstacle from m e a s u r e m e n t s of the far field pattern, SIAM J. A p p l i e d Math. 41 (1981), 8-15.
9.
D. Colton and R. Kleinman, The direct and inverse scattering problems for an a r b i t r a r y cylinder: Dirichlet boundary conditions, Proc. Royal Soc. E d i n b u r g h 86A (1980), 29-42
i0.
D. Colton and R. Kress, I n t e g r a l E a u a t i o n Methods Theory, John Wiley, New Y0rk, to appear.
in S c a t t e r i n g
ii.
D. Colton and R. Kress, Iterative methods for solving the e x t e r i o r D i r i c h l e t p r o b l e m for the l{elmholtz e q u a t i o n with a p p l i c a t i o n s to the inverse s c a t t e r i n g p r o b l e m for low frequency acoustic waves, J. Math. Anal. Appl. 77 (1980), 60-72.
12.
D. Colton and R. Kress, The impedance b o u n d a r y value p r o b l e m for the time h a r m o n i c Maxwell equations, Math. Methods in A p p l i e d Sciences 3 (1981), 475-487.
13.
D. Colton and R. Kress, The unique s o l v a b i l i t y of the null field e a u a t i o n s of acoustics, Quart. J. Mech. A p p l i e d Math., to appear.
14.
S. I. Hariharan, Inverse scattering for an e x t e r i o r D i r i c h l e t problem, Quart. A p p l i e d Math., to appear.
15.
G. Hellwig, Partial D i f f e r e n t i a l Equations, York, 1964.
16.
A. Kirsch, H a b i l i t a t i o n Dissertation, in preparation.
17.
P. D. Lax and R. S. Phillips, Press, New York, 1967.
18.
A. Roger, N e w t o n - K a n t o r o v i t c h a l g o r i t h m applied to an electrom a g n e t i c inverse problem, IEEE Trans. A n t e n n a s and Prop. AP-29 (1981), 232-238.
19.
B. D. Sleeman, The inverse p r o b l e m of acoustic scattering, IMA J. A p p l i e d Math., to appear.
20.
B. D. Sleeman, T w o - d i m e n s i o n a l inverse scattering and conformal mappings, IMA J. A p p l i e d Math. 27 (1981), 19-31.
21.
R. Smith, Ph.D. Thesis, U n i v e r s i t y of Delaware,
22.
F. Ursell, On the e x t e r i o r p r o b l e m s of acoustics, Phil. Soc. 74 (1973), 117-125.
Blaisdell,
New
U n i v e r s i t y of GSttingen,
S c a t t e r i n g Theory,
Academic
in preparation. Proc. Camb.
ON THE H U K U H A R A - K N E S E R P R O P E R T Y FOR SOME CAUCHY P R O B L E M S IN L O C A L L Y CONVEX TOPOLOGICAL V E C T O R SPACES Jacques Dubois and Pedro Morales
i.
Introduction In this note we introduce a net version of the important P a l a i s - S m a l e condition
[14]
(see [13] for other a p p l i c a t i o n of this assumption).
from a recent result of Szufla [21],
This allows us to deduce,
the basic L e m m a 2.1, w h i c h gives a topological
c h a r a c t e r i z a t i o n of the set of fixed points of a n o n - l i n e a r operator in a function topological vector space.
In the Banach space case, a more delicate c h a r a c t e r i z a -
tion is given by Lemma 2.2 in terms of the n o t i o n of Eli.
R6-set introduced by A r o n s z a j n
This L e m m a contains as Corollary an interesting result of V i d o s s i c h [22]. As illustration, we apply these Lemmas to d e s c r i b e topological properties of
the set of solutions of some Cauchy problems in locally convex topological vector spaces.
This permits us to unify several results due to A r o n s z a j n El], Astala [2],
D e i m l i n ~ [5], P u l v i r e n t i
E15] and Szufla ([18], [19], [20]), and to show also that
the H u k u h a r a - K n e s e r p r o p e r t y holds for the Cauchy p r o b l e m under the hypotheses,
for
the existence case, e s t a b l i s h e d by Kato [iii]. 2.
Basic lem~as Consider the class
M
of all m e t r i z a b l e spaces.
A topological space
called an a b s o l u t e retract for m e t r i z a b l e spaces, in symbols and, for each closed subset h:F + S
F
of a m e t r i z a b l e space
admits a continuous e x t e n s i o n
is a convex subset of a normed space, gical space and
S
is called an absolute retract,
S E AR(M).
connected.
S e AR(M)
We note that if
S
is non-empty,
A topological space S is called an R6-set if
m o r p h i c to the i n t e r s e c t i o n of a d e c r e a s i n g sequence in r-image of an S ~ AR
and
AR
space is an
h:S ÷ S'
We denote by in
H* = (Hq)q = i
Let
K
K
to
and X
X
Every
S E M AR.
S
S
A topolois compact
compact and and
S
is homeo-
We note that every if
S' c AR.
the Cech c o h o m o l o g i c a l functor w i t h coefficients
(see E17] for more details).
H q (Y) = {0}
f u n d a m e n t a l property:
T
then
~, defined on the category of topological pairs
to be acyclic if
if
space [3, p. i01], and, in particular,
is a homeomorphism,
the reduced cohomology
from
AR
For example,
[8, Corollary 4.2].
in symbols S c AR, if
S e AR, then
is S c M
M, every continuous function
h : M ÷ S [3, p. 87]. then
S
S £ AR(M), if
for all
(Y, A), and we denote by A topological space
q = i, 2, 3, ...
Y
Hq is said
We note the following
R~-set is acyclic [12, p. ii0].
be topological spaces.
will be denoted by
C(K, X).
The set of all continuous functions If
X
is a u n i f o r m space, the symbol
denotes the topology of u n i f o r m c o n v e r g e n c e on every n o n - e m p t y subset of
163
C(K, X).
If
X
is a Hausdorff topological vector space,
the symbol
denotes the vector space of all bounded continuous functions from this case, it is w e l l - k n o w n that
Cb(K , X)
K
to
X.
In
C u ( K , X) = (Cb(K, X), Tu) is a Hausdorff topolo-
gical vector space w h o s e topology has as local base the c o l l e c t i o n of all sets of the form
N(U) = {x ~ Cb(K, X): x(t) c U
b o u r h o o d of If
0
u0
in
is an element of a normed space and
the closed ball of center 2.1
Lemma
Let
II. II, let F:Y ÷ Yo i) ii) iii)
x
K
u0
v)
and radius
F
U
is a neigh-
r > 0, the symbol
Br(U0)
denotes
r.
Y = Cu(K, X)
and let
satisfies the following conditions:
is continuous.
T h e r e exist t O ~ K and For every
x0 E X
and
such that
c > 0, the relations
lit - slI < E
For every net
convergent subnet
(x)
(yB)BcD,
Then the set
F(y)(t 0) = x 0
x, y ~ Y
and
for all
xiK e = y i K
y ~ Y.
imply
K E = K n Be(t0).
For every n e i g h b o u r h o o d
t, s c K
where
be a Hausdorff t o p o l o g i c a l vector space, let
F(x) IK~ = F(Y)iKc, w h e r e iv)
t ~ K},
be a bounded and convex subset of a normed space w i t h n o r m
A s s u m e that F
for all
X.
Fix(F)
U
of
0
in
X, there exists
imply
F(x)(t) - F(x)(s) ~ U
D
Y
in
of
(x)
such that
E > 0
for all
such that
x ~ Y.
xe - F(x ) ÷ 0, there exists a
c D.
of all fixed points of
F
is non-empty,
compact and
connected. Proof.
To prove that
S = Fix(F)
is n o n - e m p t y and connected,
T h e o r e m 2 of [21], to show that, for every implies
0 c (I - F)(V), where
0 ~ (I - F)(V). so
D
e D'
V
F(y$) ÷ F(y).
y E Cb(K , X)
is closed in But
Lemma
I I.II, let that
F
Let x
K
S
c D in
V
Y.
such that
YB ÷ y"
Moreover,
So
since
Since F
Suppose
(I - F)(x~) ÷ 0, (Y~)BcD' YB ~ V
of
for all
is continuous,
y - F(y) = 0, and therefore 0 ~ (I - F)(V).
be a bounded and convex subset of a normed space w i t h n o r m Y = Cu(K , X)
and let
F:Y ÷ Y.
Assume
satisfies the conditions i), ii), iii) and iv) of L e m m a 2.1 and the
(xn)
If
(xn)
is a sequence in
Y
such that
x n - F(Xn) ÷ 0,
has a convergent subsequence.
Then the set Proof.
Y, 0 E (I - F)(V)
follows immediately from (v).
be a Banach space, let
P a l a i s - S m a l e condition: then
(x)
such that
Y, y e V.
yB - F(yB) ÷ 0.
The compactness of 2.2
it suffices, by of
It follows from (v) that there exist a subnet
and a function and
V
denotes the identity m a p p i n g on
Then there exists a net
x~ - F(x~) ÷ 0.
(x)
I
closed subset
Fix(F)
In this case,
L e m m a 1 of [21]
Y
of all fixed points of
F
is an
R6-set.
is a Banach space With the s u p r e m u m n o r m
I I'II~.
By
(see also proof of T h e o r e m i.i of [22]) there exists a sequence
164
(Fn)
of continuous m a p p i n g s from
p h i s m of
Y
Y
to
Y
such that each
I - Fn
is a homeomor-
and
lim Fn(X ) = F(x) uniformly in x ~ Y. Put T = I - F, n-~ for all n = I, 2, ... and S -- Fix(F). Then we may suppose
T
= I - F n n choosing a subsequence of
(Fn)
if
necessary)
that
all
x c Y
and all
set
Tn(S)
is contained in the closed ball of center
Qn = c--°(Tn(S))"
n = i, 2, 3, ..., and therefore,
Then
Qn c AR(M) and, since
M a z u r t h e o r e m E9, p. 416], Qn e AR.
Qn
Since every
(by
][Tn(X ) - T(x)] ]= -< 1
for
for every
n, the compact 1 and radius n" Let
0
is compact according to the Tn
is a homeomorphism,
Tnl(Qn ) = R n ~ AR. It is clear that
co n = I, 2, 3, ... So S c_ n~ 1 k~ n ~ = co x E limnSU p R n = n~i k~n Rk" Then there exists a sub-
S c R --
lim n inf R n.
Let now
for all
n
sequence
(~) of (Rn) such that x c R k n for all n = i, 2, 3, ... Thus n 1 T k (x) ~ Qk and therefore lITk (x)]]~ <- ~ n " Since also ][Tkn(X) - T(x) l] < knn n n n it follows that T(x) = 0. So x c S, and this shows that S = lim n R n . Let V be a n e i g h b o u r h o o d of n > k0
implies
S.
Suppose that there exists a positive integer
it follows that a subsequence
Yn - F(Yn) ÷ 0. (Ykn)
of
(yn)
Then, by the P a l a i s - S m a l e condition, and an element y c Y
lim T(Ykn) = T(y), and therefore T(y) = 0. n*~O because Ykn ~ V for all n = i, 2, 3, ...
Then
V
of
S
there exists a subsequence
n = i, 2, 3, ...
By L e m m a 5 of [4]
[i]) it follows that A function
h
to be compactt if 2.3
Corollary
space, let T u) ii) an
Proof.
Since
X0eAR(~) function
is continuous
y ~ S _c V.
Consequently, (Rn)
there exist
n~°°lim Ykn = y.
So
This a c o n t r a d i c t i o n
for each n e i g h b o u r h o o d
such that
Rkn ~_ V
for every
(which is a correction of the Th~or~me B of
and
X
to a topological space
F:Y0 ~ Y0"
If
F
Y
is said
h(X) is compact. K
be a compact convex subset of a normed
be a closed convex subset of a Banach space
X, let
Y0 = (C(K, X0),
is a compact m a p p i n g satisfying the conditions
of Lemma 2.1, then the set
Fix(P)
of all fixed points of
F
is
Cu(K, X). K
is compact, C b ( K , X) = C(K, X).
Let
Y = C u ( K , X).
Since
and X 0 is a closed subset of the Banach space X, there exists a continuous r:X ~ X 0
such than
m a p p i n g defined by the formula
r(x) = x
for all
= Fix(G).
Since
K
x e X 0.
G(u) - F(r o u).
proof of T h e o r e m 1.2 of [22], we conclude that Fix(F)
such that
,
is an R6-set.
from a topological space h
iii)
R~-set in
S
(Rkn)
([22, T h e o r e m 1.2]). Let
X0
and let and
such that
~ V. For every n = k 0 + i, k 0 + 2, k 0 + 3, .o. choose n-2 Let Yn = Xn+k0 for all n = I, 2, 3, ... Since t[T(Yn)]] ~ < n+k0
x n e R n - V.
of
k0
R
is compact and
G(Y)
Let
G:Y + Y
be the
By the argument used in the G
is a compact m a p p i n g and is
Tu-COmpact,
G(Y) is uniformly
equicontinuous, and therefore the condition iv) of Lemma 2.1 is verified.
To
165
v e r i f y the condition of Palais-Smale,
let
(Un)
be a sequence in
u n - G(u n) ÷ 0.
Since G(u n) • G(Y) End G(Y) is Tu-COmpact,
(Ukn)
and a function
of
(un)
Ukn - G(Ukn) ÷ 0.
2.4
Lemma
So
Let
u { C(K, X)
n-~°lim Ukn : u.
K
such that
By L emma 2.2,
Y
such that
there exist a subsequence
n+ °°lim G(Ukn) = u.
Fix(G)
is an
be a compact subset of a normed space,
let
X0
be a non-
empty subset of a Hausdorff locally convex topological vector space h:K × X 0 ÷ X
be a continuous function.
Then for every
closed, b a l a n c e d and convex n e i g h b o u r h o o d bourhood
V
whenever
v c C(K, X 0)
Proof.
Let
bourhood imply
0
in
s E K.
Ws
of
Ws°
Vs
X
such that
and Since
0
in
h(t, y) - h ( s , Let
in
of
X
u(s))
U
of
h(t, v(t)) -
v(t) - u(t) • V h
0
such that 1 c ~ U.
h(t, u(t)) • U
(s)
u c C(K, X0) 1 implies u(t) - u(s) • ~ V . s K is compact, there exists a n 1 K _c i__Ul B•si(Si). Let V = ~ Assume that
v • C(K, X 0)
Then there exists u(s i) + Vsi and
1 s i ~ n
ns > 0
and
and
and a neigh-
y ~ X 0 N (u(s) + W s)
such that
•s = Min {~s' qs }"
finite sequence i=l Vsi
Since
(si)in I in
K
0
in
X
K c u B E (s) -- s•K s such that
and
"
v(t) - u(t) • V
such that
Therefore
contained
t • K n Bqs(S)
for every t • K.
t • Besi(Si)"
t ~ K, 1 -- V s i -c u(t) e u(s i) + 12
iS°
Let
v(t) : u(t) + (v(t) - u(t)) • u(s i) + ~ Vsi + V _~ u(s i) + ~
1 Vs i : u(si) + Vsi.
t • K
s
there exists Let
for all
6s > 0
be a closed, b a l a n c e d and convex n e i g h b o u r h o o d of
Since
and every
t • K.
there exist
t E K N B~
and let
X, there exists a neigh-
for every
is continuous,
X
u E C(K, X 0)
in
But
R~-set.
h(t, v(t)) - h(t, u(t))
Vsi +
: (h(t, v(t)) - h(s i,
1 1 u(si)) + (h(si, u(si)) - h(t, u(t)) c ~ U + ~ U : U.
3.
Some applications For
tO c R
and
a > O, let
I = Eto, t O + a].
locally convex topological vector space, convex subset of
X
containing
let
Let
x0 c X
X = (X, T)
and let
X0
be a Hausdorff
be a closed
Xo.
Consider the Cauchy p r o b l e m (CP) where u
x' = f(t, x), x(t O) ~ x 0 f
is an
X - v a l u e d function defined on
defined on some n o n - d e g e n e r a t e
is a solution
of
(CP)
u'(t) = f(t, u(t))
if
u
I x X 0-
compact subinterval
tions from
K
to
K
is T - d i f f e r e n t i a b l e on
X0-valued function of
K,
I, containing u(t0) = x 0
to,
and
for all t c K.
It is w e l l - k n o w n that, under mild conditions on of the integral,
An
f
and with a suitable m e a n i n g
(CP) is equivalent to the following e q u a t i o n for integrable funcX0:
166
(IE)
x(t)
which allows
us to define
set
solutions
= x0 + ft
in the space
ii)
and
iii)
By way of illustration I)
Suppose
Let
x(s))ds
gives exactly
C(K, X). of Lemma
we consider
X = R n, X 0 = Br(X0)
subinterval
In this case, Rn-valued
of
Let
is an equicontinuous C(K, X0)}
that
F(C(K,
+ (C(K, X0) , Tu)
the following and
of
in
examples:
r
and let
(IE)
Since
Mb < r
b = min {a, ~}.
is the Lebesgue
!IF(x)(t)
and
C(K, X0). of
Rn
]IF(x)(t)
Since
F(C(K, t c K,
Therefore
X0))[t]
for
< Mlt-s I
for
F(C(K,
= {F(x)(t):
the mapping
and Sol(f) X
that the set
Sol(f)
X~)
x
theorem
F:(C(K,
is an Rs-set
Eli (see also El0]).
is compact
and connected
is a Banach space,
in
In particular,
X0) , Tu)
X 0 = Br(X 0)
and
f
Cu(K , X).
Sol(f)
(Hukuhara-Kneser
¢
property).
satisfies
the
conditions:
For every
x ~ X, f(~, x)
is strongly measurable.
For every
t c I, f(t,
is continuous.
c)
There exists
a positive
-)
function
h ~ el(1)
such that
llf(t, x) II ~ h(t)
(t, x) e I x X0.
By the Lemma of (15], integrable
appearing
integral
the Ascoli
a)
Bochner
Consider
< (t - to).
- F(x)(s)][
b)
for every
- x011
This shows also that
for every
is Tu-COmpaet.
2.3 it follows
Suppose
Carath~odory
trivially
f ~ C(I x X0, X).
F(x) ~ C(K, X0).
is due to Aronszajn
Teorem)
II)
appearing
that
subset
X0))
of all
satisfies
is compact.
By Corollary This result (Peano's
subset
is a bounded
Sol(f)
F
2.1.
x ~ C(K, X0).
s ~ Et 0, t]} <
s, t ~ K, we conclude
the set
We note that
I:K = [to, t O + b~.
the integral
functions.
sup{l If(s, x(s)l]:
implies
f(s,
M = sup{llf(t , x) ll: (t, x) ~ I x X0}
the compact
all
operator
of all fixed points
(CP)
the conditions
f(s, x(s))ds
tO
Fix(F) of
t
t 0 the integral
F(x)(t) whose
f
: x0 +
in
(IE)
for every
with respect
is the Bochner
I, and therefore
any solution
on some non-degenerate
Choose 0 < b ! a
of
such that
measure
on
Integral with respect
function
defined
u ~ C(I, X0) , the function
to Lebesgue
(CP)
is an absolutely compact
"~ t0+b h(t)dt to
I.
to Lebesgue
measure
continuous
X0-valued
subinterval ~ r,
t + f(t, u(t))
is
Then the integral
of
I, containing
and consider
the compact
on
tO .
167
t
subinterval
of
is absolutely
I:K = [to,
continuous
t o + b].
and
Let
x e C(K, X0).
l[F(x)(t)-
XoIl
Since
f
t +
t ~ I llf(s, x(s)IIds
f(s, x(s))
to t ~ I h(s)ds
to for all
t c K, it follows
(C(K, XO),
ru )
that
is continuous,
converges
to
dominated
convergence
x.
Then
F(x) let
(Xn)
f(s, Xn(S)) theorem
c C(K, XO).
+ f(s, x(s))
in
for all
integrals
that
f(s, x(s))ds
for every
F:(C(K,
C(K, X 0) s ~ K.
XO), T u )÷ Tu-
which
Th~n,r by the
E6, p. 45], ~i+oom ] f(s, Xn(S))ds
t = j
~ r
to To prove
be a sequence
for Bochner
ds
t ~ K, and therefore
n~°lim F(Xn)(t)
to = F(x)(t)
for
to every
t e K.
But the set
{F(Xn):
n = I, 2, 3,
...}
is equicontinuous,
because
given e > 0 choose ~ > 0 such that t, t' c K and It - t' I < 6 imply r t' r t' ]| h(s)dsl < e; then llF(Xn)(t') - F(Xn)(t)ll < II lJf( s, x (s))lldsl < i n J t t' t r ]j h(s)ds[ < ~. So the sequence (F(Xn)) Zu-COnverges to F(x). t With a suitable ties of Sol(f). ~)
Suppose
is an R6-set ~)
that
in
Let
t c K.
satisfying 3, . .
Let
we can describe
the topological
is Tu-COmpact. improves
Then,
by Corollary
the Theorem
= x0 +
J to G:Cu(K , X) ÷ Cu(K, X)
that
ii), iii) and iv) of Lemma and
is a continuous 2.1•
improves
lim ~(S n) = 0, where n-~o It can be shown that
earlier
Suppose
the weak topology, We note
Let
n = i, 2,
fix - G(x) ll = < !}.
(E,
I]'II) and
for every
G
denotes
the Kuratowski
satisfies
that Fix(G)
of Szufla
results)
~
is a B a n a c h
space,
the Palais-Smale
= Sol(f)
([18, Theorem
and the Theorem
Also there exists
This
6] and [19, Theorem])
X = (E, T w)
the function
a real constant
of
condition
is an R6-set.
2.3 of Deimling
f e C(I × X, X) such that x E C(I, X),
measure
M > 0
(which
[5].
where
Tw
denotes
f(I × X)
is
tw-COmpact
t ÷ f(t, x(t)) such that
is
l[f(t, x) ll
(t, x) ~ I x X.
K = I
[9, p. 434],
2.2 implies
X0 = X
that,
Tw-COntinuous. for every
Lemma
two theorems
several
III)
mapping
For every n
that
E7], and therefore
contain
[15].
such that g(x) = x for every ftf(s,function g(x(s))ds for every x E C(K, X) and
= {x e C(K, X): x(t O) = x 0
non-compactness.
result
2.3, Sol(f)
II of Pulvirenti
n
Suppose
proper-
be a continuous
It is clear
S
X0))
This result
G(x)(t)
the conditions
•
M
F(C(K,
g:X ÷ X 0
hypothesis
two examples:
Cu(K , X).
x E X0, and define every
additional
Consider
and let
co (f(K × X))
x e C(K, X). is
Since,
by the Krein-Smulian
Tw-COmpact , the function
theorem
t ÷ f(t, x(t))
is
168
Pettis integrable with respect integral appearing measure
on
Let Since M]t'
in
(IE)
to Lebesgue measure
is the Pettis
u c C(K, X). Let
Ix*(F(u)(t) - t]
x*
be anYtelement
- F(u)(t'))I : [x*
for all
F:C
U
of
t ~ K
0 V
Then the
to Lebesgue
whenever
x ~ C(K, X)
0 X
0
X.
such that
ft
I[x, I I : I.
x*(f(s,
u(s)))ds I -<
t' F(u) ~ C(K, X).
that
is continuous
Choose
such that
x c C(K,
Since
in
in
X
X)
be such that
- f(s, u(s)).
E*
let
u c C(K, X)
and let
n
of
in of
of
f(s, u(s))ds I : I
(K, X) + C (K, X) U
neighbourhood
f
t' t, t' { K, it follows
be a neighbourhood
bourhood
K [16, p. 74-75].
m.
To show that W
on
integral with respect
a closed, balanced
U -c- ~a W.
such that and
By Lemma 2.4 there exists a
f(t, x(t)) - f(t, u(t))
x(t) - u(t) ~ V
x - u c N(V).
h(K) E U
and
and convex neigh-
For
F(x)(t)
for every
s ~ K
c U
for all
t c K.
Let
putrt h(s) = f(s, x(s))
- F(u)(t)
= I
(t
h(s)ds
).
t
i
to co (h(K)) E (t - t0)U E a U E W
for all
t c X, we conclude
that
F(x) - F(u) E
N (W). To verify
the condition
convex neighbourhood
of
exists a real number such that F(x)(t')
0
iv) of Lemma 2.1, let in
~ > 0
X.
Since
such that
U
f(K x X)
be a closed,
is (strongly)
co (f(K x X)) E ~ U.
balanced
bounded,
Let
t, t' ~ K
0 < t - t' < ~. Then, for every x c C(K, X), we have F(x)(t) t -t' f(s, x(s))ds ~ (t - t') co (f(K x X)) E t U E U.
and
there be
-
i
=
t'
To verify the condition such that
x~ - F(x ) + 0.
f(K x X)
is
Ilx*j,
grals,
y
s ~ K.
2.1, let Since
Tw-COnverges
for every
to an element
x* c E*, then,r~y
is Pettis integrable
J
and
(x)~e D
(ys)BeD,
y(s) e E.
the dominated f(s, yB(s)ds
tO t e K.
So
F(yB)(t)
f(yB)(t'))I
~ MIt - t' I
t ~ K, the set x 0 + u.
+ x 0 + u(t)
But
{F(y6):
6 e D'}
YB - F(y~) + 0.
By Lemma 2.1 it follows is compact and connected. established IV)
when
Suppose
topology on
E*
X
x* E E*
t e K.
(x)~e D
such that
Ix* f(s, yB(s))I
convergence rt
for Pettis
J
U(t)
y(s)ds
Since
such that
is equicontinuo~s, Therefore
of Since
Cu(K , X) and
Z
inte-
for every
tO
for every
for all
be a net in
f(s, x (s)) c f(K x X)
Tw-COmpact , there exists a subnet
(f(s, yB(s)))B~D, M
v) of Lemma Let
IIx*II so
Ix*(F(y6)(t) = 1
-
and all
u ~ C(K, X)
and
t, F(y B) ÷
YB + x0 + u.
that the set
Sol(f)
z ~(Kato's Theorem
This result generalizes
[ii]) and it
a result of Szufla [20]
is reflexive. E
is a barreled
which is stronger
logy of compact convergence,
normed
space, T
than the
X = (E*, T)
and
is a Hausdorff
w*-topology
and weaker
f e C(R x X, X).
locally convex than the topo-
169
By the Banach-Mackey Jt
-
t0J < 1
norm of
E*.
and
is Riemann integrable
1 K = [to, t O + ~].
x ~ C(K, X).
denotes the
x c C(K, X), where in (IE) is the Riemann integral.
of [2], F(x) ~ C(K, X).
Using Lemma 2.4
of [2] it can be shown that
nuous mapping satisfying = Sol(f)
4
such that
J J'JJ
6 and Lemma i0 of [2] that the function
for every
By Proposition
H ~ 1
Jlf(t, x) J J < M, where
Then the integral appearing
and some results
Fix(F)
imply
It follows from proposition
t + f(t, x(t))
Let
theorem there exists a real number
lJx - x0J j ~ 1
F: C u (K, X) + Cu(K, X) is a contiiv) and v) of Lemma 2.1. Then
the conditions
is a non-empty,
compact and connected
subset of
Cu(K , X).
This
result is due to Astala [2, Theorem 13].
REFERENCES
i.
N. ARONSZAJN,
Le correspondant
gquations diff~rentielles, 2.
topologique
de l'unicit~ dans la thgorie des
Ann. of Math. 43 (1942), 730-738.
K. ASTALA, On Peano's Theorem in Locally Convex Spaces, Reports of the Department
of Mathematics,
University of Helsinki
(1980), 1-14.
3.
K. BORSUK, Theory of Retracts,
4.
F. E. BROWDER and C.P. GUPTA, Topological Degree and Nonlinear Mappings of Analytical
5.
6.
Type in Banach Spaces, J. Math. Anal. Appl.
K. D E I ~ I N G , in Math.
Ordinary Differential
596, ~pringer-Verlag,
New York
Warszawa
(1967).
26 (1969), 390-402.
in Banach Spaces, Lect. Notes
(1977). Math. Surveys 15, Amer. Math.
Soc.,
(1977).
J. DUBOIS and P. MORALES, Cauchy sous les conditions
8.
Equations
J. DIESTEL and J. J. UHL, Vector Measures, Providence
7.
Polish Scientific Publishers,
J. DUGUNDJI,
Structure de l'ensemble des solutions du probl~me de de Carath~odory,
Ann. Sc. Math. Quebec
An Extension of Tietze's Theorem,
(to appear).
Pacific J. Math. 1 (1951),
353-367. 9.
N. DUNFORD and J. SCHWARTZ, Linear Operators,
Part I, Interscience
Publishers,
Inc., New York (1958). i0.
C. J. HIMMELBERG and F.S. VAN VLECK, On the Topological Sets, Rocky Mountain J. Math. i0 (1980),
ii.
S. KATO, On Existence and Uniqueness Differential
12.
Equations
Triviality of Solution
247-252.
Conditions
for Nonlinear
in Banach Spaces, Funkcial.
Ekvac.
19 (1976), 239-245.
J. M. LASRY and R. ROBERT, Analyse non lin~aire multivoque, de la d~cision no. 7611, Paris
(1978).
Ordinary
Cahier de Math.
170
13.
L. NIRENBERG,
Variational
and Topological Methods in Nonlinear Problems,
Bull. Amer. Math. Soc., Vol. 4, Number 3 (1981), 267-302. 14.
R. S. PALAIS,
Critical Point Theory and the Minimax Principle,
Pure Math., Vol. 15, Amer. Math. 15.
G. PULVIRENTI, esistenza
e
Equazioni Differenziali
in uno spazio di Banach.
Teorema di
56 (1961), 281-300.
16.
W. RUDIN, Functional Analysis,
17.
E. SPANIER, Algebraic Topology,
18.
S. SZUFLA,
Solutions
McGraw-Hill
Book Company,
Springer-Verlag,
New York
Sets of Nonlinear Equations,
S~r. Sci. Math. Astronom. S. SZUFLA, Some Properties Equations,
Proc. Sympos.
(1970), 185-212.
struttura del pennello delle soluzioni in ipotesi di Caratheodo-
ry, Ann. Mat. Pura Appl.
19.
8oc., Providence
Phys.
(1966).
Bull. Acad. Polon.
Sci.,
21 (1973), 971-976.
of the Solutions
Bull. Acad. Polon.
New York (1973).
Set of Ordinary Differential
Sci., S~r. Sci. ~ t h .
Astronom.
Phys.
22(1974),
675-678. 20.
21.
S. SZUFLA, Kneser's Theorem for weak Solutions of Ordinary Differential Equations
in Reflexive Banach Spaces, Bull. Acad. Polon.
Astronom.
Phys. 26 (1978), 407-413.
S. SZUFLA, Sets of Fixed Points of Nonlinear Mappings Funkcial.
22.
Ekvac.
G. VIDOSSICH, Appl.
22(1979),
in Function Spaces,
121-126.
A Fixed Point Theorem for Function Spaces,
36 (1971), 581-587.
Sci., Sgr. Sci. Math.
J. Math. Anal.
Essential Self-Adjointness and Self-Adjointness for Generalized SchrSdinger Operators. by NGUYEN XUAN DUNG 3y "generalized
SchrSdinger
Operators",
~
(-i) alD~a
tor of the form T =
we mean an elliptic ~(x) D~+~(x)
opera-
on L2(R n) ,
o<1~[, I B I_<m
co
results.
One concerns
the essential
and the other concerns
self-adjointness
the self-adjointness
D ( q ) = { u s L 2 ( R u) lqucL2(R n) }.
of T on H
We will try to motivate
of T on C O (Rn) , 2m
n ( R ) N D ( q ) , where
our results,
and
~;il! only sketch the proofs. z.
General A s s u m p t i o n s
and Statements
of Theorems.
i/
a 8 complex-valued
2/
~SLloc(R n )_ _
3/
Uniform Strong E l l i p t i c i t y
6/
a B~CI~] (Rn) anC bounded
for
l~I+!~l!2m
5/
a 8 uniformly
for
I~I:IBI==
with a ~ = abe,
, real-valued,
continuous
Theorem i. If q(x)>-q*(Ixl),
where q*(r)
is non-negative,
monotone
nonde-
2
creasing general
in r>0,
assumptions,
Comments. R.G.
and
q*(r)=O(r2m/(2m-l)) as T is essentially
r++~,
self-adjoint
The growth rate q*(r)=0(r 2m/(2m-l))
Keller,
theory of ODE
using different (case n=l)
assumptions
[3].
will
q(x)
was also obtained It is w e l l - k n o w n
by in the
when the negative part
drops as 0(IxI2m/(2m-l) 1 , the classical
"escape to infinity
particle
in finite time."
Before we sketch the proof, result.
n
on C0(R ).
that this growth rate is the best possible.
From the point of view of classical mechanics, of the potential
then under the above
we shall state the self-adjointness
We need to change the general
assumptions
2/,4/,5/
to the
172
following:
2'/
qscm(Rn),
q~l
(or bounded
from below),
4'/
a Bscl~I (Rn) and has bounded derivatives.
Theorem 2. There exists a constant
iD~qi~ Eql + I~I 2m Comments.
with
0
then T is self-adjoint
This condition on q is a g e n e r a l i z a t i o n
second order operators T=-A+q,
E>0, depending on m,n,aa8
ponential
This nonlinear
growth,
exp(explxl),
the kinetic
T=(-A)m+q(x);
if
and the potential
(-£) TM and q(x)
(-£) m the unit i-2m Consequently, The constant
unit.
of ex-
etc .... but does not
(i for length),
D~q and ql +
-A and q(x)
represent,
Consider now
any physical
Heuristically
quantity,
speaking,
then q(x)
if we
must also have
I~[ 2m will have the same unit
E in theorei~a 2 is therefore
We now sketch the proofs of theorems II.
For
s is any small pos-
energy.
are to represent
they must have the same physical
i -2m-l~l
for
potentials.
respectively,
unit i-2m.
[2].
condition on q admits potentials
From the point of view of Quantum Mechanics,
assign to
of a condition
IVql!(2-s)q 3/2, where
for example exp[xl,
handle oscillatory
H2m (RN)nD(q).
on
given by Everitt and Giertz with G=R n
the condition becomes
itive number.
such that if
unit free.
1 and 2..
Proof of theorem I. Theorem 1 will be proved through a series of p r o p o s i t i o n s
lemmas.
There are three main ingredients
inequality,
local elliptic
regularity,
We first state the interpolation Interpolation Given m,n,
in the proof:
Garding's
and an interpolation
estimate.
estimate.
lemma. there exist positive
constants
E,C with the following
properties:
! 1
i/
and
If 0!~(x) sC0(Rn) , and for
I~I=i,
ID~(x)l!E~
--
- -
2m
(x),
173
then for any
j [~
m c (R n) NL 2 (R n) and any USHlo
iYIjm,
2m DYui 12 < C /( I lui I 2 + -
2/
If,
in
\
E
I l~l/2D~ui 12~ ( I . 1 ) .
/
I~l=m
addition,
for
some
m>0,0<~
then there exists K'>0,
one has
depending
Z II~D%l12!~<~-2mll~ll i~[=~
_
only on m,u,
and K such that for all
££m, E
Here,
] i~
2m DTul 12< K,Z-2£1 [u[ 12
I I" [ I
denotes
(I.2).
the L2-norm.
Proof.
i i
First, k
we prove
Ak=I I~ 2m Dkui I2.
for the case n=l. Then,
using
equality and the inequality
Ak < ~
(Ak_ I + A k) + [
We rewrite
M(s)A=
this system
Suppose
integration
IDTI<_g~
by parts,
2m
Let
Schwarz's
in-
12abi
(Ak+ I + Ak_l), in matrix
for
i_
form as follows:
11 j01[Al o I[iA016 [l i ]
k
Am- i J
2
The matrix M=
Am
0
can be easily
2
shown
to have
0
a positive
inverse
the matrix M(c)
(an inverse with positive
will have a positive
coefficients).
for small enough
£.
Therefore, (I.l)
follows
immediately. For general
n, we integrate
by parts one variable
at a time,
and
2,
174
and the proof (I.2)
is similar.
is proved
by making
the change of variable
v(x)=u(sx),
and by applying
Application
of the interpolation
Fix ~sC0(R
as ~.
) with
to
and
I I% IYl/2m' ' DYvJ I2
lemma.
0<~(x)
let T (x)=~2m(sx).
For 0<E
n
(I.l)
~(x)=T(~x)
~(x)=l
if
1 Ixl< ~, and ~(x)=0
For the sake of clarity,
It can be shown by induction
of
IxI_>l.
we will write
that for l<7<m,
6
(i)
IDTTI
< c(s)T
l- I~l 2m !_ 2
and
(ii)
IDY~½J
I~l 2m
< e(s)Y
, where
as ~÷0.
c(s)=0(c)
I
NOW let uSH~oc(Rn)NL2(Rn), ( , ) denote
the usual
If-El m denote
L 2 inner product.
with the interpolation
estimate
the Sobolev Hmnorm.
Applying
Let
(i) , (ii) together
(I.l) , we can prove the following
estimates. Proposition There
2.1.
exist positive
that whenever
c~E,
constants
E, C depending
only on n,m,
such
I~I+iBI~2m,
z I l~½m~ul I2 ~ C( IT½u112 + c(c)flu I 12) 1~L=m I(DBu
, (DY-TY)DYu) j _ c(E)C(
It(D ~-y ~ ½ )DYull I ] (D Local
2< _ c 2 (s)c(]lul
~-y ½ 2£ 2 T )DYul I C(I Jul i +
elliptic
lul 2 +
12
+
(i)
ll~ul I2),
~ 2), II~2Ullm
if 7<~
if y<~
(2) (3)
½ 2 IT ul Jm), if 7!~
(4)
regularity.
Let Tma x be the operator
T with domain
D(Tmax)={ucL21Tu£L2(Rn)
T with domain
D(Tmin)=C0(Rn)
a distribution}. Let Tmi n be the operator
.
as
175
Let T . be the adjoint mln Proposition
of T
min"
2.2.
* Hm (R n) NL 2 (Rn) D(Tma x) : D(Tmi n) c loc Proof.
D(Tma x) = D(Tmi n) is obvious
assumption local
3/ and 4/ enable us to apply a
regularity
G~rding's
result
([I], theorem
(more or less)
6.3),
and obtain
The genewell-known
* m D(Tmin)CHloc(Rn)
inequality.
Consider B(f)
from the definitions.
the quadratic
=
form associated
(a BD~f,D~f) ,
~
with P=T-q defined
by
~ n ). for fsC0(R
0<1~l,Ipl<_~ Because
a B are bounded,
assumptions
B can be extended
3/,4/ and 5/ guarantee
o
to Hm(Rn).
.
that Gardlng's
The general
inequality
holds
for
B, i.e.
l If] E2 m
< C(B(f)
+
--
where C only depends
[[fl i2) on m,n,
*
t
for fgHm(R n)
,
the ellipticity
(G)
,
constant,
and a B.
m
NOW, let usD(Tmin) , then USHloc(Rn) NL2(Rn) by proposition 2.2, so ½ o . that Y usHm(Rn). Proposition 2.1 and Gardlng's inequality immediately yield Proposition
2.3.
There exist positive
constant
E,C,
such that, whenever
8<E, I~l,
I~l<m, I (DBu, (D~-YY)DYu)
2 <__ c(8)C([ lu[ I +B(?u)) , if y<~
I i (D~-Y~½) DYU] I 2 < c2(~)C(I lu [ ]2 + B(y2u)), l[ (D~-Y~½) DYul I 2 < C(] lul] 2 + B<~u)),
if y
if 7<~
(2 ~ ) (3') (4').
176
Proof of essential It suffices Suppose
self-adjointness.
to show ker{T~i n + il}={0}
TminU = ~ilu,
~il(Tu,u)
: =
By comparing B(T2u)
(~u, TminU) Z
(D~(Tu),a
~,8
DBu)
~
+ linear
combination
of
the imaginary
parts
( ( D ~ - Y ) D Y u , a BDBu) ~ ( m B - ~ ½ ) m@u) }(L-I)
(* , we get
II ½ull 2 = IIm {linear combination of ((De-YT)DYu,aBDeu)}I(L-2)
For small
Here the Constant the limiting proposition
2.3 comes enough
into play,
the interpolation and
(L-4)
),
s,
(L-3),
estimate
I lul I=o.
(L-4) o
(L-3)
I lul I2
rate of q(x)).
(L-4)
To prove
(Recall
q
(Ixl)
is
(L-3), we need to use
we need to use the fact that ,
Gard±ng's
inequality,
proposition
2.1 and
(1.2).
then give
I IT½ul 12 ~ constant
o n I nor on s.
two more
(c(~)Ilul i2 + lq*~lul 2)
To prove
(2m-l)
we establish
does not depend on E nor on i~
growth 2.3.
(r)=0(r 2m/
have
of
((De-YT½) DYu,a
of
I (Deu' (De-Y~)DYu) I ! Constant
Ill
(*)
(*) . we deduce combination
IB (~2u) I i Constant
(L-2)
(Tu,qu)
- Re{linear
estimates:
q
+
= -(Tu,qu)
To see how proposition
*
I.
we have
the real parts of
By comparing
I~1
for some non zero real
I lul 12, where
So if we choose
Ill large
the Constant
does not depend
enough and let s÷0, we must
177
III.
Proof of theorem 2. The proof of theorem 2 involves
regularity
theory,
The general holds
and a criteria
assumptions
for p2
.
inequality,
for self-adjointness
3/ and 4'/ imply that G~rding's
(recall P=T-q).
we can prove the following Proposition
o
Gardlng's
global
by H. Sohr
[4].
inequality
Using this fact together with theorem i,
global
regularity
result.
3.1.
Let Pmax be the operator as a distribution}. o
.
Also by Gardlng
P with domain D(Pmax)={usL2(Rn) IPucL2(Rn)
Then Pmax is self-adjoint !
s inequality,
and D(Pmax)=H2m(Rn).
we may assume without
loss of gener-
ality that for usH2m(Rn),
(PmaxU,U)
> Constant
I Iu[ I2 >0.
--
From now on, Criteria
m
for clarity,
( , ).
we will denote Pmax by P.
for self-adjointness
Let A,B be positive Suppose:
--
(Sohr).
self-adjoint dense
operators
on H with scalar product
i/
D(A) ND(B)
in H.
2/
A > I > 0, for some fixed 1
3/
Re(Bx,A-Ix)
> -a(x,x),
for all xsD(B),
where
a is
a fixed constant with 0
on D(A+B)
: D(A) ND(B).
We apply SOhr's criteria by letting A=q, iD~qI
< sq
1 + -~ 2m, then p=q-i satisfies
So, all the estimates n
with usC 0(R ). (Pu,q-lu) where
in section
Computations
B=P. (i) and
We obser that if (ii) in section
II hold with p in place of Y and
show that,
for usC0(R n) ,
= {I} - {II} - [ I I I ] ,
½
{I} = B(p u)
(the quadratic
{II} = linear combination
of
II.
form associated
with P)
(a ~(DB-~p½)D~u, (D~-Yp½) DYu)
178 {III}=
linear c o m b i n a t i o n of
Using the estimates
in p r o p o s i t i o n
(a ~DBu, (D~-Tp)DYu)
2.1, we can show that
½
I{II}l
< c(s)
Constant
(
lull 2 + llp ull2m )
l{III}l
< c(s)
Constant
(
lull 2 + llP=uil2m )
SO that
h
, if c is small enough
Re(Pu,q-lu)
> -a(u,u)
for some 0
0o
Since C0(Rn)
is a core of P and q
(by theorem i), Sohr's c r i t e r i a is
s a t i s f i e d and T=P+q is s e l f - a d j o i n t on H 2 m ( R n) nD(q) .
References i.
Agmon, S. Lectures on Elliptic B o u n d a r y Value Problems. D. Van Nostrand, 1965.
2.
Everitt, W.N., and M. Giertz. S c h r 6 d i n g e r - t y p e Operators." 257-265.
3.
Keller, R.G. "The Essential S e l f - A d j o i n t n e s s of D i f f e r e n t i a l Operators." Proc. Roy. Soc. E d i n b u r g h 82A (1979), 305-344.
4.
Sohr, Hermann. "Uber die S e l b s t a d j u n g i e r t h e i t von S c h r o d i n g e r Operatoren." Math. Z. 160(1978), 255-261.
New York:
"Inequalities and S e p a r a t i o n for Proc. Roy. Soc. E d i n b u r g h 79A (1978),
ASYMPTOTIC
THEORY
FOURTH-ORDER
FOR
in
In
the
on
this
paper
we
asymptotic
[X,~).
The
theory
of
(ry")"
-
as
x-->~ , and
situations ential
only
ssary
to
two
in
that
familiar r(x)
(1.2) and
of
of
the
and an
(3,
,
O-estimate
a
gave
for
all
and
Wood
of and full
non-zero is
terms
of
(1.4)
hold
the
that
and
they
generalized and
~
= O.
(1o2)
a borderline have
a
between
certain as
x~-~
It
is
exponand
not
nece-
real-valued.
= b x O,
q(x)
case
c.
Wood
were
in
terms
(1.3)
of
the
to
referring c >
0
= O ( x -8)
for
any
solution.
both of
to
took the
of
solve
mention
the
= O
work
of
recent
(1. 1)
Paris
(1.7)
work
other-
of
explictly when
(1.3)
obtained
but
solutions
(1.1)
functions the
~
and
to
and
(9)
the
evaluation
0
hypergeometric also
,
attention
still
b <
feature
able
y
(1.4)
special
analysis The
= cx
= 2, drew
y(x)
and
We
(1.1)
character.
r are
considered
asymptotic b
equation
coefficients
Subsequently,
form
Paris
case
k 2
(1.1)
this
~I)
this
index. (5)
critical
where
4,
in
Eastham
(I)
the
p(x)
Fig.2;
deficiency
Anikeeva
of
~- T
information
(1.4),
wise
= x
a
= 0
represents
q and
becomes
Devinatz
lack
it
of
example
+ qy
(const.)(q/p)
have
p,
investigate
where
solutions
solutions
assume
the
~
terms
OF
EQUATIONS
fourth-order
is
that
all
character
the
case
show
where
where
In
we
and
(py,),
(Pq)'/pq
CLASS
EASTHAM
identify
critical
CRITICAL
DIFFERENTIAL
M.S.P.
I.
A
of
in and the
180
same
authors
on a c o r r e s p o n d i n g
class
of h i g h e r - o r d e r
equations
(8).
in
The
asymptotic
methods
appearance
b~t,
tic
theory
formulated
and
which
has
and
apply
(7), is
to d r a w
for
one
(9)
2. a~d
of
been
with
able
our
is
following
by
in
the
components
to d i a g o n a l i z e
The
AI
takes
and ~j
(2.1)
which
are
the
is
Hence, we have
(6),
(1.3)
in
and
(1.4)
and
to o b t a i n
in
cover
this
aim
the
case.
those
of g e n e r a l i z e d asymptotic
Our
Except
of P a r i s
hypergeometric
results
of
(9)
writing
general
y,
y,,
the
of
(6,~92-3)
form (2.1)
ry",
(ry")'
- p y ' . The
A = T - I A I T,
v2
= dg(~1'~2'
are
procedure
= AY,
v3
next
step
where
v4)
(2.2)
~3' ~4)"
eigenvectors
and
(2.3) eigenvalues
of A.
¥ = TZ
(2.4)
into
z, = (^i The ~j
asympto-
and Grudniewicz
(1.1)
quasi-derivative
transformation
then
the
A by writing
and the vj
of
general
coefficients.
results
the
T = (v I
Here
cover
(1.2)
the u s e
to p u t
the
an ad hoc
co~text.
(1.1)
Y has
of
solutions
Y' where
to
general
significance four
develop
by Eastham
shown
(1.1)
to so f a r h a v e
we
recently
avoiding
are
We b e g i n write
paper,
situation,
and,
we
a wider
to
the
delicate
functions, into
it
form
and Wood
this
already
out
asymptotic
in
referred
roots
of
r4 D =
the
_ p2 (p2
- T'1~')z characteristic
+ q = O. _ 4qr)~
(2.5) equation
of A,
181 i
fi2 =
-~I'
(2.6)
fi4
-~3"
(2.7)
~I = {(p + D)/2r}Y' ±
~3 Also,
v. c a n J
be
[(p
=
taken
to h a v e
1, The
matrix
where
r. is J
row
r~j,
r p j2
find
(2.5),
that,
~j
has
m ;I
rows
rj,
1)
m 2 = 2~2D
m 3 = -2~3D, then
in
-I
vector
m I = 2~ID ,
We
q~j
occurs
(q~l and
=
components 2
~j,
T -I , w h i c h the
D)/2r]2'
-
= -ml,
m 4 = -2~4D
in
(2.5),
2
J
= -m 3.
T-IT ' =
2
(2.8)
(tjk) , where
,
2
(2.9)
m j t j k = ½m~ + (~j - ~ k ) r ~ k + (~j - f i k ) ( r ~ k ) ' 3. the
We
have
next
following
(i) (ii)
p,
conditions
q and
qr
to w o r k
r are
= o ( p 2)
(i)
and
on
~r'/r
in
qr/p
(2.6)
~h
as
note
:
(P/r)r{1
~3
=
(q/p)~{1
that
~3
impose
[X,~),
= 0(1).
2
and
where
(2.7),
we
obtain
+
0(~)] ~ ( p / r ) ~,
+
0(~)]
= °(~I)"
(3.1)
1
N
Thus
(q/p)2, ~I
and ~2
(3.2) domimate
~3
and
x-~.
After a ~alculatio= obtain
we
i
~I
1
we
and
r.
are L ( X , ~ ) ,
1
and
detail
i
~q,/q,
(ii)
q and
some
x--~,
g = Using
in
zero
i
S~p,/p,
tjk
o n p,
nowhere
as
I
(iii)
out
from
(2.9)
i~volving
(2.8),
(ii)
a~d ( i i i ) ,
~e
182
_T-IT,
R
where
is
1
L(X,~)
= T 1 + R I,
and
ieI ~ = ~}P!
4.
With
1I~'!
(2.5)
l(p~
now
in
the
Z' we
have
to A I
matrix
carry
out
+ T I. I t
By
(3.2),
(3.3)
~4
do
dominate
same
not order
To
of
(iv)
As
=
x-->~,
the
the
as @ w h e n we
critical
meaning
A I + T I and,
A I + TI,
0
-¼(Pq)'
=
(3.4)
(4.1)
diagonalization,
that
(3.4),
magnitude
IP!
(A 1 + T 1 + R 1 ) Z ,
a second
@ in
~
form
is h e r e
and
diagonalize
!
(3.3)
of
this
case
(1.2)
in f a c t ,
is
they
time
(1.2)
of
arises.
that ~3 have
the
and
the
x-->~.
require
further
conditions
o~
p,
q,
let 1
P'/p
=
I
O[(q/p)2} ,
r'/r
O{(q/p)2],
=
1
(pq)'/pq where ~ is for
a no~-zero
convenience)
(v)
"~ 4 ~ ( q / p ) y ,
constant
(the
factor
o~ly
(4.3)
1
p"r~/p ~,
If
introduced
1 "t { (pq) ' / ( p 2 q % ) ] ' ,
Let
be
4 is
.
1
all
(4.2)
r"/(rp)
~
L(X,~).
in
(4.2)
we
write I
(pq)'/pq then We
(4.3) can
now
where and,
implies
that
write A
after
#'
is
AI
+ TI
= dg(~l,
a calculation
4~(q/p)~(1
=
+
#),
L(X,~). = S-IAs,
~2'
using
43'
44)
conditions
(i)-(v),
(4.4)
r.
183 A 1 = 2V'1 + ~
+ rI
A2=~
- r1
2 +~
A3 =
e +
A4
e - j(2
(4.5)
J
+ 8 2)
(4.6)
S ---> d g
as
x-~,
=
2)
3+8
((io)(i a)) ,
S'
0
I
a
is
L(X,~),
(4.7)
I
(4.8)
r 1 i s L(X,~), a = ~/[1
and
The o n l y (3.2),
two l j
(3.4)
which
and
could
(4.2),
2
The
takes
(4.1)
which, (2,
transformation into
by (4.7)
u, and
(4.8),
Uj(x)
= [ ej
+ o(1)}exp
other
components
four
solutions
are
coordinate zero. yj
with
~3 a n d ~4 b u t ,
possibility
by
by assuming (4.10)
the
- s-ls')u standard
X
j(t)
vector
the
form covered
by
solutions
X j(t)
(i ÷ a ) e x p
j-th
back
asymptotic
(Sx
(1 ~ j
dt
with
Transforming
Y j C x ) "~ e x p
y3(x) ~
th£s
(4.9)
I ~ O.
has
there
the
are
= (A + s - I R i s
Hence
e. is J
+
~ 51.
Z = SU
p.88).
where
+ ¢2)]
coincide
we a v o i d
that
5.
+ J(1
dt
to
~ 4),
component
ur~ity
(I. I) , we
obtain
(5.1) and
forms
)
(j
:
CI
X 3(t)
1,
dt
2)
(5.2)
) (~.3)
y4(x) ~ ( , where
a
is
as
in
(4.9).
a}exp(I~4(t)dt),
184
By
(4.5)
and
(3.4),
we o b t a i n
adjusting
given
by
(4.6)
as
a constant
(3.1)
and ~ 2
multiple
= -~I"
#-2)
(3.2)
at the 6 2
and
and
for e x a m p l e .
term
that and
(4.4)
¢2(pq),/pq again
(5.3)
adjusting
we w r i t e
J('*~')
•
terminating Here
r 2
is
+ r2
+ ¢)-2t]
<e<,. On using
(5.4)
where ~I
w i t h Y3 and Y4'
± J[ 1 + ~-2(1
i ~
(j = ~, z ) ,
in Yl a n d Y2'
To d e a l
~3' ~4 = - ¼ ( P q ) ' [ 1 -,~
(5.2)
C?X~tj(t) d t )
yj(x) ~ [p3(x)/r(x)]¼exp after
from
1 •
the b i n o m i a l
is L ( X , ~ ) .
expansion
Hence,
assuming
is L(X,~)
constant
(5.5)
multiples
i n Y3 a~d Y4'
from
we o b t a i ~ Y3(x) ~
I p ( x ) q ( x ) ] - ¼ { I + ~ ( I + ~ - 2 ) ] exp I(x)
(5.6) Y 4 ( X ) "~ { p ( x ) q ( x ) ] - ¼ { 1 - ~
I(x)
where
If,
further
factors are that
to
(5.5),
in (5.6)
our m a i n
= ¼~-2(I
+ ~-2)-½ Jx ¢(pq)'/pq X
¢(pq)'/pq
is L ( X , ~ ) ,
result
results (5.1)
for
t h a t Re IAji(x ) - Ak(X)}
does
condition
satisfied
if,
we h a v e
noted
is c e r t a i n l y
not
Thus ( 5 . 4 )
(1.1).
requires change
dt.
(5-7)
the e x p o n e n t i a l
c a n be ~ e p l a c e d by u n i t y .
asymptotic
the g e n e r a l
I+~-2)} e x p { - I ( x ) ] ,
It s h o u l d
the f u r t h e r sign
in
and ( 5 . 6 ) be a d d e d
condition
[X,~) , and
for e x a m p l e ,
this
p, q, a~d r
are r e a l - v a l u e d .
6. In the case The v a l u e
of ~
in
(1.3), (4.2)
is
that
(1.2)
then I
= 3(¥ + 1)(b/e) ~ and
(4.10)
becomes
becomes
(1.4).
185 e/b Also
# = 0 in
(4.4)
Y3(X), adjustment
with
(5.4), agree the
ease ~
A quite
where
of with
=
the
x
1) 2 gives
x-[Y+l±J[(Y+l)2+4e/h]]/2
the
multiples.
results
These
of Paris
formulae,
together
and Wood ( 9 ,
Table
1)
= O. situation
x B ,
exp
constants
critical
+
(5.6)
constant
different
r(x)
then
Y4 ( x ) ~
after
in
and
~ -¼(y
p(x)
are
bx ~ exp
=
real
(4.2)
situation
covered
with
y
- ~
x C,
B
is g i v e n
by our
~
0
analysis
q(x)
and
=
ex
B <
C.
is
Y exp
x C,
Then
the
by
= 2(c
1).
1
The
value
of
~
in
(4.2)
is
~ = ½C(b/e)~,
¢(x) Then
(5.6)
and
(5.7)
+
= ½(~
x-¼(#+Y)[l+l/#(l+g-2)]exp/-½{1
y4(x) ~
exp[~[J(1
a final
between involves (5.6),
p,
and
q and
has
reason
for
domil~ated
where
exponential solutions
this
Thus
all
(1.2)
solutions in
terms
an algebraic
the
right-hand
exp
I(x)
i~ terms
nature Y3
that
an exponential
nature
as x - ~ ,
behaviour
have
the
where
algebraic
nature.
situations
that
case
an algebraic
(pq)'/pq
exponential
is
we n o t e
r in b a s i c a l l y in the
+ ~(1 + ~ - 2 ) ] x C ]
_ 1]xC]
comment,
(5.6)
especially
by unity, The
+ ~-2)
general
(5.4)
y)c-lx -c.
give
Y3(X) ~
As
and
and
is
of
(1.1)
of p,
q,
behaviour.
side
of
(5.4)
manner
the
(1.2):
represents
difference
c a n be
of
Y4 w o u l d
real
replaced
coefficients.
had ~3
also
whereas
andS4
have
a borderline
have r and
had
between
asymptotically where
two
the
an
186
References
I. L.I.
Anikeeva,
2.
Coppel,
Stability
and
differential
equations
(Heath,
W.A.
Vestnik
3. A.
Devinatz,
Quart.
4. A.
Devinatz,
J.
5. M.S.P.
Eastham,
6. M.S.P.
Eastham
Mathematics 7. C.G.M. 8. R.B.
846
Proc.
Math. Roy.
a n d C.G.M. (Springer,
and A.D.
London
Univ.
6 (1976)
asymptotic
J. Math.
London
Grudniewicz,
Paris
Moscow
behaviour
Bostom, (Oxford)
Soc.
Soc.
23
Edinburgh
Grudniewicz,
Univ.
Wood,
Proc.
Wood,
Quart.
(1972)
1981)
Roy.
79A
(1977)
51-59.
Notes
in
88-99.
thesis,
Soc.
267-286.
135-146.
Lecture
Ph.D.
of
1965).
7 (1973)
Berlin,
44-52.
1980.
Edinburgh
90A
(1981) 2o9-236. 9. R.B.
Paris
97-113.
amd A.D.
J. Math.
(Oxford)
33
(1982)
OSCILLATION
AND NONOSCILLATION
LINEAR
ORDINARY D I F F E R E N T I A L
A.
In order
and
this
paper
(p(x')
the
n
the
half-linear
second
equations
+ nqx n = 0
nonl
EQUATIONS
ELBERT
we c o n s i d e r
differential
(1)
THEOREMS FOR SOME NON-
near
first
,
d = ~F~ )
('
order
differential
equation
sys-
tems 1 y'
= ay
+ bz
z'
= - c y n + dz
(2)
where
the
number
n
a
,
,
the
,
b
tinuous
c on
d
of
[t °,~)
solutions
of
above
exponent
xn usual
(3)
the
preserves
these
the
condition
on
p(t)
is
for
,
positive,
the
independent some
variable
t°eR
differential
and
x
equations,
in
xn
denotes
sign
of
x
p
is
> 0
functions
,
i.e.
that
the
p,
q
,
t
are
con-
, y
, z
are
and
the
power
xn= Ixl n sign
star functior x
The
188 We e m p h a s i z e h e r e t h a t By t h e
substitution
equation
(1)
we do n o t assume
y=x
and
can be t u r n e d
z=px 'n
into
q(t)~O
in
(1).
the differential
a system of
the
form
(2)
I with
a=d=O , b = I / p ~
always
b>O
in
~b< ~ , too. or
(2)
a=d=O
of
the
of
in [i]
(1)
zeros,
(or
(or i.e.
solutions
possibility
of
and w i t h
(2)) the
separate
x(t)
infinitely
x(ti)=O
study
we assume
the solutions
of
(i)
the restrictions
(see Theorem 1 and Theorem 2) we p r o v e d
a solution
(with
the
by (3)
in [31-[6].
solutions
dependent is
and we a l l o w
can be f o u n d also
perty
Henceforth
A quantitative
In [1] the
(2)
, c=nq
of
have t h e zeros
of
interlacing the
each o t h e r . (y(t),z(t))
many z e r o s
ti
, i=1,2 ....
for
the
system)
in-
Hence i f
(or
Y(ti)=O
pro-
linearly
(1)
of
that
there (2)
such t h a t
and
!im
ti=
~ ,
l~
then every this
solution
has a l s o
case we say t h a t
the system exists finitely terval
(2))
x(t)
many z e r o s ~)
has no z e r o s
on
property
(2)) of
(or it
equation
On t h e o t h e r (y(t),z(t))
has none t h e n
, to=> t o , such t h a t [to,
conjugacy i n t e r v a l system
or
~)
and t h e
This
differential there.
the
other
every
(or is
is
has o n l y
y(t)
called
Then by t h e
there
an i n -
equations
solution
(or
hand i f
there
x(t)
and i n
(I)
) which
interval
i s disconjugate zeros
many z e r o s
the differential
is oscillatory.
a solution
I to,
infinitely
)
dis(i)
(or
interlacing
may have a t
189 most one z e r o rential
on t h i s
equation
(i)
(or
Our main r e s u l t
will
and t h e method
is
applied
linear
tions rion
interval.
to
the
(1)
generalize Wintner
concern
By n e g a t i o n
and ( 2 ) .
t71 . In o r d e r
D. M i r z o v
the
[5]
method
differential
equa-
is
oscillation given
criterion
we make use o f
which yields
a rather
crite-
i n w h i c h we
get a generalized
L8]
case
of Hartman's
oscillation
to
diffe-
nonoscillatory
we o b t a i n
A corollary
the wellknown
case t h e
is nonoscillatory.
(2))
second o r d e r
t h e o r e m o f M. Zlamal J.
system
a generalization
(see [ 2 1 ) . for
In t h i s
due to A.
version
of
a
a method o w i n g to sharp oscillation
criterion. The Example 1 shows how we can deduce t h e ing Mirzov~s
results
from
an e s s e n t i a l
role
played
is
by t h e
(~,~)
bility
depends on t h e c o e f f i c i e n t s our theorems
advance. in
the
results.
pairs
apply
of
our
functions
for
easily
equation
interval
the
system
sake o f
brevity
the system into
p,
(2)
q
to are
(1)
To
pairs
in
pairs. the
Finally
special
periodic.
the
proofs
since
the
the corresponding
in
(2).
will
be c a r r i e d
results
can be c o n -
ones f o r
differential
(I).
First system
The a d m i s s i -
such a d m i s s i b l e
our results
case when t h e c o e f f i c i e n t s
out only
admissible
we must have such a d m i s s i b l e
Example 3 we a p p l y
verted
~(t),~(t)
In Example 2 we g i v e
For t h e
In our treatment
so-called
of
correspond-
we g e n e r a l i z e
(2).
Let
the
I t o , ~)
the
system
~ to~t°
Ricatti
(2)
equations
to
be n o n o s c i l l a t o r y
be a d i s c o n j u g a c y
the and t h e
interval
and
190 (y(t),z(t)) t~t o
be a s o l u t i o n
Let the f u n c t i o n
(4)
Riccati
is
follows
and s a t i s f i e s
+ (na-d)r
+nblrl
immediately
making use of
~
x(t) , u(t)
L
, u(t)
be i n t r o d u c e d
and by
by
Yto b n
differentiable n+l
=,=
x
< = ;
T T÷~limYto ~ = co ; T l i m sup T+~
n+l
fto~ T (Yt
b-~ n+l
o
~)
pairs
by the f o l l o w i n g
are c o n t i n u o u s ,
~,:I
(iv)
(4)
of the a d m i s s i b l e
is continuously
(iii)
n + c = 0
(ha-d))
to
the s e t
o f the f u n c t i o n s X(t)
1+!
t
= exp (j"
Then we d e f i n e
(ii)
the g e n e r a l i z e d
by d i f f e r e n t i a t i n g
x (t)
,
(7)
by
(2).
Let the f u n c t i o n
(i)
for
equation
r'
(6)
be d e f i n e d
y(t)#O
n
continuous
(5)
This
such t h a t
z Y
r
(2)
r=r(t)
r
Then
of
positive on
(~,u)
restrictions: and
[t°,~)
xo ;
191
The e x i s t e n c e efficients
of
the
set
a , b , d
of
that
it
is
not
empty,
that
there
are
functions
Concerning vestigate
the
the
L
depends
the
system
moreover, u
fourth
behaviour
heavily
for
(2) the
such t h a t
the
co-
and we suppose sake o f
convenience,
(Am,u) e L
coefficient
of
on the
c
function
we s h a l l H(T)
in-
defined
by
T t Yt !~(Yt ~ c ) d t (8)
o
H =
o
T /to~
This
function
tion
f(t)=f
can be c o n s i d e r e d t t
~c
It
is
not
as an average
difficult
to
of
the
show t h a t
funcif
the
or
in-
0
relation finite
lim then
later
f(t)=C
as an a v e r a g i n g
(9)
we s h a l l
lu n vl
which
is
ween the
n
(y(t),z(t)) some
the
of
and the
system
be a s o l u t i o n to~ t o
some f u n c t i o n
of
Let
of
may be f i n i t e will
the
inequality
1
+ n+11v]
quadratic is
n+l
inequality
in
form
be n o n o s c i l l a t o r y that
bet-
means.
formulated
(2)
be r e f e r r e d
H(T)
the w e l l k n o w n
such
the ~
n+l
]ul
observation
LEMMA. Let
for
property
a generalization
Our f i r s t
If
= ~
C
property
make use of
<
geometric
where
This
l i m H(T)=~ T+~
Finally
with
holds
y(t)~O
function
r(t)
a pair
(~,~)~
on
of
and l e t [ t o , ~)
be g i v e n L
the
a lemma.
by ( 4 ) .
inequality
192
1+! (10)
;to~blrl
holds,
then w i t h the corresponding
defined
by (8) is bounded on
the l i m i t
lim H(T)=C T÷~
Proof. rential over
equation
(5).
r
the f u n c t i o n If
~=~
satisfies
the R i c c a t i
(5) by
~
diffe-
and i n t e g r a t i n g
t
~']+nY to b ~ I r l
na-b=(x*)
'/ x
*
v = [ ~ ( x * ) ' / ~*_~,] ( ~n-e (n+l)
Put
1+!
t
n ~to ~c-r(t°)~(t°)=O
u = ( - ~ - ( n + l ) b ~ ) n÷[ r
bx) -n/n+1
in
(9)
w ith
then ,
Ir[ x(x":)x °'.
(12)
, then
we have
since by (6)
where
H(T)
and is f i n i t e .
Multiplying
) r ~ + Ytt o r [ ~ ( ~ X~
O
p
[ t o , ~)
exists
The f u n c t i o n
[ t o , t]
(11)
and
n <
7(~):
,
x']l ( ~n)
(n+Z)
<- ( n - c ) b X l r l
n+y(~)
,
n+l
I(bx)n ~~I
n
n+l
I rX+nStobXlrl ~ ~+!n
n+l
hence
+ Sto~C_r(tolX(to) 1< ~
n+l
t t < (n-~)Y t bxlrl n +y(~)Yto o
Stor, I~(~-~'I ~ ~" ~
(xb) n
n+l
<
193 First
we show t h a t
From (12)
it
follows
in
(8)
is
bounded from above.
that
n
r~+aftobXlrl
hence by (7)
(ii),
n+l
I
1+ 1
t
(13)
H(T)
+ ft
o
~c-r(to)X(to) < Y ( ~ ) f t
x
o
(bx) n
(I0)
t f t ~c - C1 ~ x l r l o
(14)
where
(x
Multiplying obtain
(15)
(14)
by
p
I
Ix *)'
co
Cl:r(to)X(to)+y(~)ft
o
x (bx) n
and i n t e g r a t i n g
n+l
over
[t o , T ]
we
by (8)
H(T)
Concerning The f i r s t
- C1 < =
T f t x~Irl o T ft ~ o
the f u n c t i o n is
a simple
L(T)
= L(T)
we s h a l l
consequence
of
need two r e l a t i o n s .
the H ~ l d e r
inequality:
194
I
n+l
T n+l (16)
0 < L(T) < {f to~ bn =
=
T
}
n
. { fT b~Irll+~l~}--
n+l
-t
(fto~)
o
To get the second relation we choose the number Tl>t °
Then again
by the HSlder
T1
inequality
arbitrary
but
we get
i
n+l T1 T n+l T 1+1 fto x~Irl fT 1 xp b n {fT b x l r l n } L(T)g T +{ T -} I ft p (/t p)n+Z 0
Hence by (7)
n
o
(iii)-(iv) 1
n
1+! l i m sup L ( T ) g { f T b x l r t n } T÷~ I
and the c o n d i t i o n by l e t t i n o
TI÷~
(17)
lim T÷~
(10)
L(T)
=
Hence by (15) we have
implies
involved
in
l+n
]t o bn } .{sup T>T 1 T n+l (ftoP)
the wanted second r e l a t i o n
0
l i m sup H(T)~C 1 T+~
To get a l o w e r e s t i m a t e inequality
T
n-~
(12)
for
H(T)
we need the second
195
n+l t
r~+nZt b~ Ir l n+/to~C-r(to)~(to) > -(n-~)/tob~Irl n-y(E)/t =
o
~,='=
0
(b~) n
hence
t Zto~C - C 2 >= Irl where 1+ --1
I k ( k Cn= + ) ' - l~ 'cl : co
C2=r(to)X(to)_(2n_a)itobXlrl
and as above i t
by (17)
x (mx) n
follows
H(T)
thus
n_y(a)fto
C2 ~ -I_(T)
lim inf T÷~
above r e l a t i o n
,
H(T)2C 2 , w h i c h ,
l i m sup H(T)~C 1 m~m
,
together
proves
with
the f i r s t
the LEMMA. In the case
(18)
0
=
~=~
rA m
-
we have by (11)
n
1+!
t n + Yt ~ c - C o
~b ~ mI r l
Yt
where
1+1 C = r(to)~(to)-
n/t
bxmlrl o
n
the
part
of
196 Multiplying
by
~
and i n t e g r a t i n g
T Yt n ~ ( Y t b ~ * l r l o T
IH(T)-CI
over
1+~ n )dt --
{ t o ' T]
T ft ~ o + T
ft0
for
T>t °
Irl
fto
1+! For the function
ftb~"IrY
n
tends to zero as
by the averaging property of the function
t+~
hence
H ( T ) the f i r s t
term on the r i g h t hand side tends to zero, while the second term is
L(T)
by (15) and owing to (17) i t
zero, therefore
also tends to
lim H(T)=C , which completes the proof of T+~
t h e LEMMA. It
is
an i n t e r e s t i n g
a conversion
of
this
THEOREM i . disconjugate
be a d m i s s i b l e
by ( 8 )
fulfils
then
the
lim
the
inf
inequality
tion
H(T)
case
~=~*
is
for
(as a f i n i t e
that
(2)
, and t h e (2).
under simple
conditions
valid.
system
If
be n o n o s c i l l a t o r y pair
the
of
the
function
and
functions H(T)
defined
relation
H(T) > -~
(10)
in
bounded on
l i m H(T) T÷~
exists
the
[ t o , ~)
(~,~)
(19)
LEMMA i s
Let
on
fact
= C
number).
the
Lemma i s
[ t o , ~)
valid
Moreover
and t h e in
the
func-
special
197 Proof. we have a l s o and then
As in
a solution
a continuous
equality
(13).
troduced
the b e g i n n i n g
for
function
the p r o o f
of
the Lemma
of
(2) w i t h
r
which
satisfies
S(T)
and
Let the f u n c t i o n s
T>t °
y#O
on
[t
the
M(T)
o
~)
in-
be i n -
by
T
S =
(y,z)
of
t
fto~(ftobXlrl
1+~ n
) dt
(2o)
T M = ft
By (7) that
(iii)
the
(21)
~
l i m M:~ T÷~
inequality
Our p r o o f
(10)
is
will
not v a l i d ,
be i n d i r e c t .
We assume
i.e.
T 1+1n = ~ . lira Y t o b ~ I r l T+~o
By t h e a v e r a g i n g
(22)
o
property
lim
S(T)
T÷=
~
= ~
of
the f u n c t i o n
H(T)
we have then
'
and
(23)
By the
lim S(T) T÷~
inequality
:
(7)
(ii)
we can w r i t e
t 1+~ r~ + C Y t o b ~ I r l n
where the c o n s t a n t
C1
is
the
~
(13)
t C l _ Yto~C
same as in
as
,
(14).
A multipli-
198
cation
of
this
inequality
by
~
and a q u a d r a t u r e
gives
by
(8) T ft X~r o S(T) < C - m ( m ) M ~ + ~ ~)= 1
According
to
bounded from 1
the assumption above,
c S(T)/M(T)
for
(19)
therefore T>T I
the
right
by ( 2 2 )
with
some
it
T1
hand s i d e will
is
be l e s s
sufficiently
than large.
Thus we g e t
1 ~ c ~S(T)
(24)
lq~ I}
where t h e f u n c t i o n
T Z t ~Irl o
<
= L(T)
M(T)
L(T)
is
for
already
known from
T>T 1
('15).
Since
by ( 2 0 )
S'
= ~(T)
T i+~ ft b~lrl n
M'=~(T)
o
the
estimate
(16)
implies 1
T x n+l n+l L n
(25)
By (7)
(iv)
ft bn ~ o
-
Mn+l
}
we have s u f f i c i e n t l y T
(26)
<
-
ft o T
(Yto~)
S~
large
n+l bn n+l < N
for
T>T I
T1
and
N
such t h a t
199
Combining this
with
(24)
and
n+l
(25)
1
we g e t
n+l
y1 M' M ---if-< N~ S~S ---h-- f o r
(27)
where
Y1
is
a positive
An i n t e g r a t i o n by ( 2 3 )
of
and by (7)
constant (27)
T> T
depending
over
[T,~)
i
,
only
gives
on
for
n
and
T>T 1
(iii)
i i i YI M n < Nn S n
,
hence
S
T~ < N¥1
which is
contradicts
not true,
apply
the
-n
to
i.e.
for
(21),
the
T>T i
and t h e
relation
LEMMA and t h i s
,
indirect
(10)
application
is
assumption
valid.
(20)
Thus we can
completes
the
proof
of
Theorem I . A criterion tions
of
the
sufficient
system
THEOREM 2. system
(2).
If
holds
then
pair
(xm,u)
hold
then
the
Remark. are
Let
for
the
(2)
some
independent
is of
of
the
the
is
oscillatory.
is
solu-
theorem.
pair
relation If
the
next
be an a d m i s s i b l e
for
the
l i m H(T)=~ T÷~ for
an a d m i s s i b l e
l i m sup H(T) > l i m i n f T÷~ T~
H(T)>-~
oscillatory.
not difficult the
in
to~t°
(2)
(2)
oscillation
formulated
relations
system It
is
(x,~)
system the
for
choice
of
to the
show t h a t value
to
the
limits
here
200 Proof.
Suppose t h e c o n t r a r y ,
nonoscillatory. (19)
is
By t h e a s s u m p t i o n s
fulfilled,
hence a l l
satisfied.
Then by v i r t u e
tion
, if
H(T)
i.e.
any,
the
the
on
H(T)
be f i n i t e ,
(2)
be
the condition
conditions
o f Theorem 1 t h e
had to
system
of
Theorem i are
limit
which
is
of
the
func-
the desired
contradiction. The c r i t e r i o n lified
in
for
special
oscillation
cases t o
i n Theorem 2 may be s i m p -
one w h i c h
contains
only
single
integral. Corollary. at
least
Let
~
one f u n c t i o n
be a f u n c t i o n ~
such t h a t
satisfying
(~,~)~L
there If
exists
the
rela-
tion T l im Yt ~c = ~ T÷~ o
(28)
holds
then
the
Proof.
system
Let
same l i m i t
(2)
is
us c o n s i d e r
By t h e a v e r a g i n g the
for
property
for
Following stringent
a remark
criterion
for
u n d e r somewhat s t r o n g e r because we have n o t used i n
the
function
H(T)
, i.e.
Theorem 2 t h e s y s t e m ( 2 ) c a n
t >_t o o-
oscillatory.
of
H(T)
some
the
lim
be o n l y
in [2]
limit
in
H(T)=~
T>t o
(28)
yields
Then by
(see on page 365)
a more
can be e s t a b l i s h e d
restrictions all
for
oscillatory.
nonoscillation
exploited
H(T)
on t h e the
pair
properties
(~,u) of
the
, method
t h e Lemma.
THEORE~zi 3. disconjugate
on
Let
the
[t o ,~ )
system Let
(2)
be n o n o s c i l l a t o r y
( ~ ,~)
be a p a i r
of
and be the
201 functions (iii)
satisfying
and
the
conditions
(7)
(i
and
1 (7)
(iv)'
b
l i m sup t~
Moreover
let
H(T)
convergent,
is
the
relation
T Zt~
(29)
< co
j.t to~ (19)
be v a l i d .
l im H(T)=C T÷~ n+l
ic_yt
t
lim
C
, where
funct
on
is
finite
and
this
case.
First
~='=cl n o
T
Then t h e
= 0
J't ~ o
Proof.
We w a n t
to
we show t h a t
the
Theorem
admissible,
3 is
condition T1
(iv)'
Theorem
(Am,u)
with
i.e.
it
i
the
to
restrictions
fulfils
we have f o r
(7)
sufficiently
of
(iv),
too.
large
N
By and
that
bn
(30)
where =p
(7)
pair
apply
Mn ( t )
the
function
< N
M
for
is
the
all
t>T 1 ,
same as i n
(20).
, therefore
,t
hence
~,
n+l p bn
by i n t e g r a t i o n
<
N Mn [4'
we have
for
for
the
all
t>T 1 ,
function
Since
M'(t)=
202
K(T)
T fto
=
n+l bn
~
that
Mn+I(T)-Mn+I(TI n+l
K(T)-K(T i ) < N
) for
T>T 1 ,
hence
l i m sup K(T) n + T÷~ M l (T) i.e.
the
relation
Thus o f Theorem and t h e r e may r e p e a t the
relation
the
(7) pair
1 ar e
holds.
(Am,B)
is
a finite
again
the
(18)
t [ C- ZtoX:':c
(iv)
satisfied,
exists
< N = n+l
admissible
therefore number
proof
which
'
of
the
and t h e
the
C
relation
with
conditions (I0)
holds
l i m H(T)=C T÷~
We
LEMMA and now we c o n s i d e r
we r e w r i t e
in
n+l n
the
form
n +l I+ ! n I n I
rl=':-nftbX':Ir n+l
Since
the
function
t , [ C -Yt o~' =c l
and t h e n
n+l n
n
Ix
1
is
n+l n < 2 {Irl
convex
n+l
(X~)
n
n +l +n
n
n+l n
1+1 _ ~"
[ftb~"Irl
n
]
,
203
T IC-ft
O~ t°~
(31)
n+l n
t
f
x~cl o
n+l n+l T 1 I r l n ( x~')~ n < 2g Yto~
M(T) T
co
1+_1
.~o
1 n+ln Yto!J[ ftbx"lrl + 2nn
n+l n
n]
dt = MI+M2
M(T)
By the averaging M2
property
tends to zero as
then we have f o r
1 2 nMl=
+
M(T)
all
of the f u n c t i o n
T÷~
Let
T1
H(T)
be as l a r g e
the q u a n t i t y as in
(30),
T~_T2>T1
T2 n_+l , n+l T 1+-1 ! YtoUlrl n (~-) n +/T2b~='=trl n.(~:'=)n_~ M(T)
T2 n+l n+l I T 1+! f t o ~ I r l n (~*) n +Nn r,1(T)YT2b~*ir i n <
M(T)
therefore
_! 1 limT÷~sup 2 nM 1=,, <~nr~ JT2b~ * irl
hence by (10)
1+! n
lim Ml=O=lim M2 T÷~ T÷~
for
, thus
all
(31)
T2>T 1 ,
implies
(29),
which was to be proved. The companion c r i t e r i o n
for
oscillation
is
formulated
204
in the n e x t
theorem.
THEOREM 4. system tion to
(2) H(T)
~
Let
(Xm,~)
satisfying defined
the
by (8)
be an a d m i s s i b l e
relation
(7)
has a f i n i t e
pair
(iv)'
If
limit
C
for
the as
the
funcT
tends
and
T t n+l f t ~ I C - f t x~cl n l i m sup - - o T o T÷~
>
0
,
fto~
then
the system Another
(2)
is
oscillatory.
nonoscillation
criterion
we do n o t assume the r e l a t i o n fine (7)
the s e t o f (i),(iii)
dropped,
Z
of
(7)
the p a i r s
and ( i v ) .
and t h e r e f o r e
can be e s t a b l i s h e d
(ii). (x,u)
To be p r e c i s e
if
we de-
by the c o n d i t i o n s
Hence the r e q u i r e m e n t
(7)
(ii)
is
LeT
Let us i n t r o d u c e (~m)' T (32)
H(T)
=
t
n+l
~
ft o V[ ft o (~c-~(e)
(bX) n
) dt
T fto~
with extend
n
the r e s u l t s
in the f o l l o w i n g THEOREM 5. (y,z)
n
~ ( ~ ) = (~-£-Z-) ( n + l ) - ( n + l ) formulated
for in
O<~
as above.
Now we
the Lemma and in Theorem
1
way. Let the
be a s o l u t i o n
system
such t h a t
(2)
be n o n o s c i l l a t o r y
y(t)#O
on
[ t o , ~)
and l e t Let
205 the f u n c t i o n of
r
a pair
(~,u)E~
corresponding
~
On t h e o t h e r and the (i0)
be g i v e n
the
hand i f
system
holds
the
(2)
again
by ( 4 ) .
If
inequality
function R(T)
for
(10)
H(T)
in
the holds
is
(32)
is
function then w i t h
the
bounded from above.
bounded from
is
nonoscillatory
then
the
and,
consequently,
H(T)
is
below
inequality bounded from
above. Proof. Theorem.
We s t a r t
Similarly
the relation
(13)
with
as in
proving
the
which w i l l
proof
the
of t h e
part
of
this
LEMMA we have a g a i n
be r e w r i t t e n
=---~T--=t 1+~1 t 1~(~:'=)' ~' X rX+~f t bXlrl n+f t (xc-~(~) o o (b~) n
(33)
first
here as
I n+l -r(to)X(to)
< 0 = '
hence by (32)
N(T)
T ft ~ulrl <= oT
r(to)X(to)
= L(T)
fto~ where t h e f u n c t i o n (iv)
and (10)
tion
H(T)
is
To prove direct
the relation
then
is
the
(17)
same as in is
bounded from above, t h e second p a r t
argumentation
not true fy
L(T)
the relations
in
(22),
S(T), (23).
therefore
By (7) the
(iii)
func-
as s t a t e d .
of Theorem 5 we f o l l o w
o f Theorem I .
the functions
true,
(14).
If
the r e l a t i o n
M(T)
given
Then by (33)
the
(10)
by (20)
in-
is satis-
206 T f t o ~ur +¢
S(T) M(T)
M(T)
By o u r a s s u m p t i o n
on t h e
bounded from
above,
the
(24)
relation
dedness o f to
(22).
to
the
~ r(to)X(to)
function
and by t h e
ded a b o v e ,
the
hereby the proof
right
hand s i d e
sufficiently
T~T I
is
is
large
TI boun-
, which
contradicts
(10)
and a c c o r d i n g
inequality function
H(T)
is
boun-
complete.
Theorem 5 we g e t
the
next oscilla-
criterion. THEOREM 6.
and
for
o f Theorem 5 t h e
As a c o n s e q u e n c e o f tion
the
same way we w o u l d have t h e
S/M
Thus we have a g a i n part
H
hence we have f o r
the quotient
first
_ H(T)
O<¢
the
Let
the
pair
(~,~)~L
If
for
some
to~t °
relation
,
T
t
fro
~(ft
lim T+~
x(x~) _x, I
n+l
* X (bX) n
0XC-y(¢)
)dr
T
ft ~ o
holds,
then
the
system
We must remark equality believes
O<~
~=0 , t o o . using
oscillation
here
plays
is that
oscillatory. in
an e s s e n t i a l
t h e above p r o o f s role.
However,
Theorem 5 and Theorem 6 r e m a i n
This
an i d e a
(2)
conjecture
due to M i r z o v
criterion.
is [5]
supported
still
by t h e
fact
the
in-
the author valid that
we can p r o v e a r a t h e r
for strong
207
THEOREM 7. a function
Let
~
be a f u n c t i o n
u
satisfying
34)
~'
- ~(~m)'
35)
t t÷~lim f t o
such t h a t
(X,u)e~\L
~ 0
there
exists
If
for
all
t>t
>t o
O-
b ~--TTn -
(~,_ t
36)
lim f (~c t÷~ to
then t h e system cannot
1
(2)
be r e p l a c e d Proof.
n~
(n+l)
is
oscillatory.
by a s m a l l e r
As in t h e p r o o f
(y(t),z(t))
such t h a t
assume
(13)
hold.
y(t)>O
Since
t t
r~ < r ( t o ) ~ ( t o ) - f =
hence by (36) tl>t °
bXlrl 0
where
o
the system
for
t~t o
non-
of
and the r e l a t i o n s
the g e n e r a l i t y
c=O the r e l a t i o n
(~c-~(o)
(11)
is
the LEMMA we have a s o l u t i o n
(Xm),
--)
(13
~
for
all
is
)
tmt I
t
n
< C3 _ f t
xc 0
for
t>t I
'
with
by (34)
1+ ~
we may
n+l
(b~) n
z(t)
Thus we have from
rX + n f t
i.e.
(~'
r
t
(37)
of
the l o s s
now
1 / ( n + 1 ) n+l
one.
y(t)#O
Without
) = ~
The c o n s t a n t
Suppose the c o n t r a r y ,
oscillatory.
(11),
(~)')n+l ~ n (bX)
,
some
208
tI , = r ( t o ) X ( t o ) + f t r[ ~ ' - x ~ ] o
t l im fto~C=~ t÷~
Since by (36) tends to
-~
thus f o r
, the r i g h t
some
t2>t I
t I+ ~ r~ + n Y t o b ~ i r i n
hand s i d e o f
(37)
we o b t a i n
~ -i
for
all
t~t 2
or
IrI~
(38)
t 1+1/n l+nf t b~Irl o
Multiplying
this
over
[ t 2 , t]
I
1+! n
K
into
t
i
~ nYt2blrl
t2
where
K=l+nfto
bXiri
Iri~1+nf
-
n b i r i 1/n
and t a k i n g
t l+nftob~ir log
by
~
i
we have then by a q u a d r a t u r e account
1+1 n
, hence by (38) t exp(nft2a )
thus
hence
(4)
t ~ t _ n f t 2 b r n = _ n f t YZ-aY = t2 Y = n f t 2 a - n l °
yn(t2) t 1+! n ~ K yn(t) t b~trl o
-z=Irly n ~ K
(2),
t i yn(t2) 7 exp(nYt2a)
for
t~t~_(:
209
z
Multiplying
< -
this
K~
) 1 y(t 2 ~
inequality
t exp (Yt2 a)
by
b>O
we have by (2)
i =
y'-ay
)
b
t
< -K ~ y ( t 2 ~
exp ( / t 2 a )
,
hence 1
t , ) (y exp ( - Y t 2 a ) ) _~ - K n y ( t 2
b
and then t exp ( - i t 2 a ) - Y ( t 2 )
y(t)
By our assumptions
y(t)>O
I t K~ f t 2 ~ i ~b
which c o n t r a d i c t s In o r d e r I / ( n + 1 ) n+l
to
for
< 1
1 t < -Kny(t2)Jt2
t>t o
for
all
b ~i/n for
t>t 2
therefore
t>t 2
(35).
to show the sharpness o f the c o n s t a n t
we c o n s i d e r ~:
(x,n)
~
the d i f f e r e n t i a l
+ ( n
n+l
+--+-in )
I
t +---~n T
equation
~'~
xn
= 0
n
to which the f u n c t i o n ferential tem is
equation
x=t n~I-
is a s o l u t i o n ,
is n o n o s c i l l a t o r y .
hence t h i s
The c o r r e s p o n d i n g
dif sys-
210
y~ = z (39)
z'
Since
T m = I and
b=l
the r e q u i r e m e n t s 1 / ( n + l ) n+l
have in
the system Finally
n y
, the f u n c t i o n
If
be r e p l a c e a b l e
t+~lim nn+l[
i.e.
1 tn+l
o f Theorem 7.
would
then we would
n+l
n = -(~)
(36)
for
would
we g i v e
in
(36)
satisfies
some
all
the c o n s t a n t
by a s m a l l e r
1 (n+l)n+. 1
(38)
x=t n
to>O
number
that
t ft ° dt -~- = ~
r]
be o s c i l l a t o r y .
some a p p l i c a t i o n s .
Example 1. Let us assume t h a t Then by (6) ~
and
~ (t)~l
u=b
an a d m i s s i b l e
It
We choose is
pair
na=d i n the system ( 2 ) . t ~ = ( f t o b ) ~ f o r some c o n s t a n t
not d i f f i c u l t
(x,p)
for
(2)
to show t h a t if
ft
b=~
t h e y form According
to
o
the C o r o l l a r y
a sufficient T
criterion
here
is it
there.
comparable
oscillation
is
t
l im f t o C ( f t b) m d t T÷~ o which
for
with
= ~ ,
Theorem 2 in [ 5 ] .
Let us remark
is
not assumed the r e l a t i o n s a=d~O and m~O t Moreover, the pair ( ( f t b)n'b) belongs to ~
that
as
but
o
not to
L Example 2.
Let us consider the h a l f - l i n e a r
differential
211
equation
(40)
y"ly~l n-I
As in the b e g i n n i n g be t u r n e d
into
yn = 0
was t o l d
this differential 1 , y ' = z ~ , z ' = - n q yn
a system
again
x * zl
6~-I
Then the p a i r
(see Example i ) . Let
Moreover the p a i r s the s t r o n g e r following.
+ q(t)
( t m , t B) ( 1 , t $)
condition
(7)
with (iv)',
Let the sequence o f
i=0,1,2 . . . .
be d e f i n e d If
Zi+l=lOg(Zi )
admissible the
the f u n c t i o n s
£i )n
then
m
to t h i s
Another
by the r e c u r r e n c e
xi=(£o£1...
with
restriction
too.
can
Now we have
, u =t 8
~=t m
is
equation
system.
~-1 pairs
satisfy are the
£i=£i(t)
,
relation
£o=t
,
the p a i r
(x i , 1)~
E~\L Example 3. odic
with
Let the c o e f f i c i e n t s
the p e r i o d
in
p , q
(1)
be p e r i -
~ , q~O , and l e t ~4
Q =7
If
Q~O
then
1 S a(t) o
the d i f f e r e n t i a l
Let us choose the p a i r the e q u i v a l e n t shows, t h i s
system
system i s
we a p p l y Theorem 4. automatically.
(1,p)
dt
equation
(1)
which
admissible
is
is
oscillatory. pair
to
1 ~ *n y~=- z , z'=-nq y As the Example i P oscillatory if Q>O To the case Q=O
The r e l a t i o n
Let n+l w t Q* = So p f o q l o3
5o p
n
dt
(7)
(iv)'
is
satisfied
212 By the assumption
q~O
we have
Q*>O
It
is not d i f f i c u l t
to check t h a t T lim T÷~ hence ( I )
n+l n
t
lop loql
dt
T
/o p is oscillatory.
work because
,
= Q
,
In the case
Q
our method does not
l i m H(T)=-~ T+~ R e f e r e n c e s
[1]
A. E l b e r t ,
A half-linear
equation, Bolyai,
P. Hartman,
[ 3]
J
D. M i r z o v , Appl.
J
for
D. M i r z o v ,
non-linear
Sturm-Liouville
D. H i r z o v ,
[6]
J
D. M i r z o v ,
[7]
A
Wintner,
equation
Zlamal,
of the s o l u t i o n s
system ( i n
of a
Russian),
Mat.
of the s o l u t i o n s
of a d i f f e -
Russian),
Diff.
Uravn.,
transcendents
physics,
Amer. J.
occuring
Maths,
87-97.
Oscillation
Fys.
Vys. U~eb.
1504-1508.
in mathematical
M
Izv.
to
401-404.
On L a p l a c e - F o u r i e r
69 (1947)
[8]
Russian),
e q u a t i o n system ( i n
17 (1981)
and Kneser's
28-32.
Oscillation
rential
XI)
boundary value problem
system ( i n
Zam. 23 (1978)
(Chapt.
systems, J. Math. Anal.
On the o s c i l l a t i o n
differential
John Wiley
418-426.
Zav. 203 (1979) J
153-180.
equations,
N.Y.-London-Sydney,
53 (1976)
a non-linear
[5]
(1979)
On some analogs of Sturm's
theorems
L4]
Szeged (Hungary)
Inc.,
J~nos
t h e o r y of d i f f e r e n t i a l
Ordinary differential
et Sons,
differential
Mathematica S o c i e t a t i s
30 Q u a l i t a t i v e
equations,
[2]
second o r d e r
Colloquia
75 (1950)
criterions, 213-218.
~asop.
pro p e s t .
Mat.
E n t r o p y numbers,
s-numbers and e i g e n v a l u e s
D.E. E d m u n d s
T h a t c o m p a c t linear operators acting in B a n a c h spaces are useful objects will h a r d l y strike anyone these days as a novel observation;
but
w h a t may well arouse more than a flicker of interest are the connections e s t a b l i s h e d in recent years b e t w e e n g e o m e t r i c q u a n t i t i e s related to such operators
(such as entropy numbers, a p p r o x i m a t i o n numbers and n-widths)
and more analytical entities,
such as eigenvalues.
My i m p r e s s i o n is that
these results are not very w i d e l y k n o w n in this country;
and it is b e c a u s e
of this, and w i t h the h o p e of s t i m u l a t i n g some i n t e r e s t in the a p p l i c a t i o n of the work to d i f f e r e n t i a l equations,
that I venture to give a b r i e f
a c c o u n t of those results of this nature w h i c h seem i n t e r e s t i n g to me. I shall also a t t e m p t to indicate some trends in the work, e s p e c i a l l y those d i r e c t e d to the study of n o n - c o m p a c t maps. As we shall to some extent be c o n c e r n e d w i t h spectral p r o p e r t i e s we shall take all spaces to be complex.
Moreover,
if
X
and
Y
are B a n a c h
spaces we shall denote the set of all b o u n d e d linear m a p s from
X
by
B
L(X,Y),
stand for
although
L(X,X)
will be shortened to
L(X) ;
and
to
Y
will
X
{x C X : l]Xl I ~ I}.
E n t r o p y numbers G i v e n any
T 6 L(X,Y)
and any
n e IN,
the
n
th
e n t r o p y number of
T, en(T) , is d e f i n e d b y 2n-i en(T)
=
inf{E
> 0 : T(B x) c
U i=l
{Yi + eBy}
for some
yl,...,Y2n_l
in
Y}.
T h e s e numbers w e r e i n t r o d u c e d by P i e t s c h in his book [203, a l t h o u g h certain functions inverse to them h a d b e e n used earlier b y M i t j a g i n and P e l c z ~ n s k i [173 and Triebel [24].
Since the
e (T) n
are m o n o t o n i c d e c r e a s i n g as
n
increases,
214
their limit,
(T) : = lira e (T), n n-~ exists, and clearly
8(T) = inf{e > 0 : T(B x) can be covered b y finitely m a n y b a l l s of radius
We shall refer to that
8(T) = 0
8(T)
as the m e a s u r e of n o n - c o m p a c t n e s s of
if and only if
T
entropy numbers follow directly: R 6 L(Y,Z), w h e r e
~il
Z
if
S
and
note
C e r t a i n p r o p e r t i e s of the T
b e l o n g to
L(X,Y)
and
is a Banach space, then
l l~ll
and for all
is compact.
T:
~}.
=
e 1 (T) -> e2(T)
_>...-> 0,
m,n 6 ~ ,
(2)
era+n_ 1 (S + T)
S
era(S) + en(T),
(3)
era+n_ 1 (R o S)
S
era(R) en(S).
A c o n n e c t i o n b e t w e e n these rather g e o m e t r i c a l q u a n t i t i e s and an o b j e c t c o n n e c t e d to operators in a m o r e analytical way comes via the notion of the essential s p e c t r u m of a map. s p e c t r u m of spectrum
T, ~
g(T)
e
We recall that if
(T), m a y be d e f i n e d as the set of all those points
of
T
the range ~ ( I I - T )
(ii)
1
(iii)
dim
of
is a limit p o i n t of
ker{(II-T) n}
II-T
10 e • \ 0 (T) e
(II_T) -I E L(X) about
10
for
in the
is not closed;
g(T) ;
=
~.
This is Browder's d e f i n i t i o n [i] of definition,
1
such that at least one of the following conditions holds:
(i)
U n=1
T e L(X), then the essential
g
e
(T) :
he proves that w i t h this
if and only if there exists
0 < If-101 < ~
~ > 0
such that
and the L a u r e n t e x p a n s i o n of
(II_T) -I
has only a finite number of n o n - z e r o coefficients w i t h n e g a t i v e
215
indices.
Note
(i) a n d found
(ii)
in t h e
Kato's
that
if
is a H i l b e r t
are equivalent. literature:
[12] p r o d u c e s
C \ {~ e ~
X
Numerous
Browder's
closed
other
gives
the s m a l l e s t ,
: II-T has
space
and
is s e l f - a d j o i n t ,
definitions
either
I e g
if a n d o n l y if Dreadful o f hope,
is n o t
though
dim ker
spectrum,
r
is a s t r i k i n g
to the e s s e n t i a l
definition
e
e
(T)
set,
may be while
(II-T) (II-T)
spectrum
< m or < m
;
according
to K a t o
semi-Fredholm.
this multiplicity
for w h i c h e v e r
essential there
II-T
belongs
o
being
d i m X/
in o t h e r w o r d s ,
of
r i s e to t h e l a r g e s t
his definition
range and
T
of
of d e f i n i t i o n s m a y be, t h e r e o
e
(T)
is s e l e c t e d
(T) = s u p { l l I : ~ 6 o
connection
between
r
(T)},
e
(T)
is a r a y
the radius
is the same;
a n d the m e a s u r e
of t h e
and moreover,
of n o n -
e compactness
~ (T) :
(4)
r
(T)
lim
e
This was For
established
independently
the c a s e of a b o u n d e d
a simple proof
of
put
r = r
e
let
(T),
E(1)
let
Then
be
= 8(S + C) that
put
=
H
E
(T)
=
8(T)
(E(r+c)
C
and
+
and Schechter
in a H i l b e r t
space
[14]
H
then have
of the i d e n t i t y
:
f[)~l >-r+e l d E
is o f f i n i t e _< r + e;
Assuming,
- E(r-£))H
(5) is o b v i o u s )
acting
in f a c t w e
the r e s o l u t i o n
<- I IsIl
-< r.
and
T
and Lebow
8(T) .
f lll
= B(S)
map
[18]
corresponding
T,
to
and write
I IS If -< r + ~, a n d as
follows
case
e
e > 0
T = fXdE x
B(T)
self-adjoint
r
(5),
by Nussbaum
(4) c a n b e g i v e n ,
(5)
To prove
(8(Tn)) I/n _< 8(T) .
n+~
then
I ITxll
>
rank
=
S + C, s a y .
it is c o m p a c t .
a n d as t h i s h o l d s
without dim H
for all
loss of generality, = ~
(r-e) IIxll
(unless for a l l
Hence ~ > 0
that
d i m H < ~, x 6 H,
since
r e o
it
e
(T),
in w h i c h
216
EI(E(r+E)
- E(r-e))
= 0
if
I < - e.
Thus dim T ( H e) = ~,
and since the unit
sphere in an i n f i n i t e - d i m e n s i o n a l space cannot be c o v e r e d by f i n i t e l y m a n y balls of radius less than i, it follows that
8(T) a r-e, and the p r o o f is
complete. We remark that the notion of the m e a s u r e of n o n - c o m p a c t n e s s of a map, and in p a r t i c u l a r the r e l a t i o n s h i p s b e t w e e n it and the essential spectrum, h a v e p r o v e d to b e very useful:
see, for example, Stuart's w o r k [22] on b i f u r c a t i o n
theory, and the work of Evans and others
(cf., e.g., ~53, [91) on the location
of the essential spectra of o p e r a t o r s arising from elliptic b o u n d a r y - v a l u e problems. N o w let
T
The s p e c t r u m of finite
be a compact linear m a p of a B a n a c h space
X
into itself.
T, apart from the p o i n t 0, consists solely of e i g e n v a l u e s of
(algebraic and geometric)
of all e i g e n v a l u e s of
multiplicity:
let
(I (T)) be the sequence n
T, r e p e a t e d a c c o r d i n g to their algebraic m u l t i p l i c i t i e s
and o r d e r e d so that
Ill(T) I
If
T
has only
m 6 ~ , m > n.
n
(< ~)
~
II2(T) I ~...Z 0.
d i s t i n c t eigenvalues we p u t
1 (T) = 0 m
for all
R e c e n t l y Carl and T r i e b e l [43 e s t a b l i s h e d the following
striking c o n n e c t i o n b e t w e e n eigenvalues and e n t r o p y numbers.
Theorem i T h e n for all
(6)
Let
T c L(X)
be compact and let
1 (T), e (T) n n
be as above.
m, n 6 ~ ,
lln(T) I
-<
n (H j=l
i/n IIj(T) I)
<
(/~) (m-l)/n e (T). m
We shall indicate later h o w to p r o v e this, and even a more general result. N o t i c e that b y taking
m = n + i it follows that for all
IIn(T) I
S
n ~ ~ ,
en+l (T),
an i n e q u a l i t y w h i c h is perhaps rather surprising.
More recently, Carl [21 has
217
been able to show that for all
k, n 6 ~ , i/n
n
(7)
Cj~IIIj(T) I]
This last inequality
-<
(~-)I/n ek(T).
can be used to obtain a result which
shows that in a
sense the entropy numbers may be thought of as a deformation
of the norm
(cf. [27]) :
(8)
lira (e (T k)) i/k
for all
n 6 I~,
(8) is simple: lim sup k-~
where
:
n
k-~
r(T)
given any
r (T)
is the spectral
n E ]4, since
so that
T.
The proof of
e (T) -< I ITII, n
(e (Tk)) I/k- -< lim IITklI I/k = r(T). n k-~=
ill(T) I -< ~n an(T),
radius of
Ill(T) Ik
Also,
(7) implies
Ill(Tk) I < ~n en (Tk)
that
and
r(T)
= II l(T) I -< lim inf (e (Tk)) I/k which completes the proof. n k+~ has also obtained estimates for the entropy numbers of embedding Besov spaces, concerning
and has used these,
the distribution
Before
leaving
a related notion, e > 0, T(B x)
let
K(e,T)
Let
numbers
results
be compact,
we mention so that given
can be covered by a finite number of balls of radius The
E-entropy
of
e:
T(B x)
to be
sense,
=
(e (T)) n
log 2 K(e,T) .
is inverse
to
is to point out that it has a connection
from the standpoint e > 0
N(c,T)
i, to establish
for a moment,
T C L(X)
be the least number of such balls.
In a certain
For any
maps between
of certain maps.
the subject of entropy
H (e,T)
H
with Theorem
of eigenvalues
that of entropy.
any
is defined
together
Carl [33
=
of the distribution
H(E,T).
Our purpose
with quantities
of the eigenvalues
in mentioning
which are of interest 1 (T) n
of
put
[lj (T[]I>_ i,
M(E,T)
=
[lj (!),_>e l°g 2 (Ilj (T)
II~).
T.
218
The
connection
Carl
referred
and Triebel
(9)
[43:
for a n y
N(4e,T)
Quite
apart
simple
from
the
of
(6).
proof
Approximation Given
any
These
numbers
enjoy
various
bounded
M(2e,T)
T 6 L(X,Y)
n
(T)
=
(3);
moreover,
I : X ÷ X
by
y(T) there
relatively
Hence
this,
~(T)
we
to
it c a n b e u s e d
to g i v e
a
let
n £ ~ ,
L ~ L(X,Y),
= 0
sequence
~
L < n}.
numbers
of t h e e n t r o p y if a n d o n l y map,
to
numbers,
a
n
T.
They
notably a n d if
(I) = i.
is m o n o t o n i c ~(T),
of
if r a n k T < n;
then
(a (T)) n
converges,
~
rank
approximation
to t h o s e
an(T)
it t h u s
say.
decreasing Then
(cf.
and
[27]),
~(T) .
= {L ~ L(X,Y) : r a n k L < ~}, ~
~ 8(T) :
(T,~
for if t h e r e
dist
(T,~)
is a c o m p a c t
e > 0
compact
it c a n b e c o v e r e d T(Bx)
so s m a l l
~
exists
see t h a t
~(T)
= dist
= {L ~ L(X,Y) : L c o m p a c t }
that by
: = y(T).
map
K
with
~ + e < 8(T), finitely
can be covered
by
many
finitely
and
since
balls many
]IT-KII
of radius
balls
r
e
the position
(T) -< 8(T)
-< y(T)
as f o l l o w s
-< a(T).
for t h e c a s e
< ~ < 8.(T),
K ( B x)
T £ L(X):
is
e, f r o m
of r a d i u s
~ ~ + e < 8(T) : c o n t r a d i c t i o n .
We c a n s u m m a r i s e
(12)
due
that
(ii)
which
(9),
is t h e i d e n t i t y
The
0;
for all
similar
8(T)
To prove
then
of
the s o - c a l l e d
(i0)
Also,
inequalities,
½H(e,T).
interest
put,
are
T E L(X,Y). below
and note
~
inf{IIT-LII:
properties
and
Let
following
c > 0,
intrinsic
an(T)
(2) a n d
dim X ~ n
~
b y the
numbers
a
(I),
to is g i v e n
6 + e.
219
Note that it may well be that
7(T)
< a(T):
this is as a c o n s e q u e n c e of Per
Enflo's c e l e b r a t e d work on the b a s i s problem,
as a result of w h i c h it follows
that there e x i s t Banach spaces and compact operators acting in these spaces w h i c h cannot be a p p r o x i m a t e d a r b i t r a r i l y closely by maps of finite rank. This k i n d of p a t h o l o g y is impossible in H i l b e r t spaces, w h e n
~(T)
N o t e also that since
~(X)
7(T)
is the norm in
of the e q u i v a l e n c e class c o n t a i n i n g as that p a r t of the spectrum of
(13)
r (T) e
T
L(X)/~(X)
T, then since
(where
~e(T)
= 7(T). =~(X,X))
m a y also be r e g a r d e d
u n c h a n g e d b y compact perturbations, we have
lim (y(Tn)) I/n n-~o
The a p p r o x i m a t i o n numbers have important connections w i t h eigenvalues, c o n n e c t i o n s w h i c h are s i g n i f i c a n t from the p o i n t of v i e w of a p p l i c a t i o n s to d i f f e r e n t i a l equations. space and
T 6 L(X)
ITI, and for all
The crucial r e l a t i o n is that if
is compact,
then
T*T
X
is a H i l b e r t
has a p o s i t i v e compact square root
n 6 ~,
(14)
a (T) n
The eigenvalues of
ITI
~n(IT])...
are o f t e n called the s i n g u l a r values of
seem to have b e e n first c o n s i d e r e d by E. Schmidt. s-numbers, have various i n t e r e s t i n g properties, of a p p r o x i m a t i o n numbers;
T;
they
T h e s e singular values, or
as we have seen in our discussiol
more generally, P i e t s c h
[20]
refers to s-numbers
as b e i n g any sequence of n o n - n e g a t i v e real numbers a s s o c i a t e d w i t h b o u n d e d linear operators and h a v i n g certain p r o p e r t i e s similar to those p o s s e s s e d by the that
a . n
If
T
is, in addition, p o s i t i v e and self-adjoint,
a (T) = ~ (T) n n
for all
n e ~ :
then
(14) shows
this r e l a t i o n is at the h e a r t of m u c h
w o r k on the a s y m p t o t i c d i s t r i b u t i o n of e i g e n v a l u e s of elliptic operators in the s e l f - a d j o i n t case
(cf. [i0], [16]), w h i l e
(14) itself is important for
the n o n - s e l f - a d j o i n t problem. All this is, of course, w h a t h a p p e n s in a H i l b e r t space.
In a general
B a n a c h space, w i t h less structure, one could h a r d l y expect to o b t a i n results
220
of such sharpness.
Nevertheless,
most startling developments
something
can be said, and one of the
is that contained
in the following
theorem due
to K~nig [13~:
Theorem
2
for all
Let
X
be a Banach space and let
T ~ L (X)
be compact.
Then
n 6 IN,
(15)
Iln(T) I
:
lim k-~
(an(Tk))i/k
Remarks I) n 6 ~
In general,
no inequality
can be expected:
for if
(~
~], then
(~
~], which has eigenvalues
while
a2(T)
that if
11 (T) = 2,
With
12(T)
T 6 L(X)
for all
is represented
but since
T*T
3 ~ /5, it follows that However,
for all
by the matrix
is represented by
a I (T) =
Carl and Triebel
lIT1[ g I, then for all
(n-m)/n 16 am+ 1 (T)
~
Other connections
-<
16 al/n(T) n
(3 + /5)i/2 > 11(T), [4] have shown
n • IN ,
(m = 0,1,2 ..... n-l).
(n ~ IN)
between eigenvalues
space case.
is compact,
Thus if
a celebrated
and approximation
H
inequality
numbers are
is a complex Hilbert space and due to Weyl E26] states that
n c IN, n
(17)
n
j=l
n Ilj
(T) I
<
-
n j=l
laj
from which it may be deduced that for all n
(IS)
a (T) ~ (~) I (T) n n
m = n-1 this gives
known in the H i l b e r t T • L(H)
= i;
is compact and
I (T) n 2)
T 6 L(~ 2)
= (3 - /~)I/2 < 12(T).
n i/n ( H I.(T)) j=1 3
(16)
of the form
[I[~j(T)IP -< '=
(T)[
'
n ~ IN
n
[ laj(T)lp. j=1
and all
p 6 [I,~),
221
This implies
that if
(a (T)) • £P 3
then
be used in the study of non-self-adjoint obtain information
similar
Studia Math.
67 (1980),
Turning suppose
elliptic
about the distribution
inequalities
Non-compact
(l (T)) E Z p, 3
of eigenvalues
to (18) but in a Banach
T E L(X)
multiplicity.
and put
A(T)
if there are only
multiplicities,
we put
that it is this part
n
A(T)
of
o(T)
hold for the eigenvalues
in
= {I ~ o(T) : Ill > r (T)}. e
denoted by
then for all
m,n • ~ ,
being repeated
(complex) eigenvalues
Euclidean axes
tl(T)
space
T
T
A(T)
of finite
(n = 0,1,2,...),
to its algebraic including
It turns out as the appropriate for various
is merely bounded
the analogue
of Theorem
earlier
and not
1 holds:
if
i/n IIj(T) I)
-<
of the proof
(/2) (m-1)/n e (T). m
in the case when
set of eigenvalues
E > 0
subspace
Then
according
which can be regarded
which cover
of
X
, ....
1
R2n ;
Ill(T) I , Ill(T) I ,
n
spanned (T).
thus
by the
maps
Let
T(Bx),
Interpret
T
X
is a Hilbert
which are larger than
may help to give the flavour of these ideas. of radius
of
space,
1 (T), in such a way that n
such points
A(T), where
For example,
Some indication
X
be a Banach
of a compact map which we mentioned
compact.
has an infinite
X
part of the spectrum of a compact map,
about eigenvalues
n I X j=l
let
lln+l(T) I = lln+2(T) I =...= re(T).
of the non-zero
of the results
in
For
see H. Konig,
and any point in it is an eigenvalue
Order these points,
multiplicity;
T
to
(cf. [6]).
space setting
compact maps,
Ill(T) I ~ II2(T) I ~...~ 0, each eigenvalue
T • L(X)
problems
157-172.
now to not-necessarily
is at most countable,
necessarily
eigenvalue
maps
that
analogue
a result which can
BI,B2[...,B2m_I
and let
eigenvectors C
n
as
r (T) e
Cn
in modulus be balls
be the n-dimensional
belonging
to
a 2n-dimensional
B x n R2n onto an ellipsoid
II2(T) I, II2(T) I .....
space and
lln(T) I, lln(T) I
the real
with halfand volume
222
n Ilj(T) 12 V2n,
where
V2n
is the volume of the unit ball in
R2n.
But
j=l the 2n-dimensional sets
Bj N R2n
volume of
cover
B j N R2n
is at most
s
2n
V2n'
and since the
T(B X) N R2n ,
2m-I S
2n
V2n
->
n I[
j=l
Ilj(T) I2 V2n,
so that n ]I
Ilj (T) I
-<
2 (m-l)/2 C n .
j=t This completes
the proof.
Carl's inequality Theorem
2, the proofs being natural adaptations
see [27].
Another
E27~ following and
(7) also holds for non-compact
T ( L(X),
interesting
earlier work by Carl [2].
For compact maps (T) + 0
as
n ÷ ~,
raised by Lorentz of eigenvalues boundary
is that proved recently by Zemanek gn(T)
= inf{kl/(2n) ek(T) : k • ~]
1/k
(n e ]N) .
T, much interest has been shown in the manner
in a form, by Weyl,
equations
One question (I (T)) ~ ~P n
the speed at which
about the asymptotic
behaviour
on a b o u n d e d open set with zero Dirichlet to have received a good deal of attention
for some
p
if it is known that
this question was settled by Carl's discovery, Iln(T) I ~ / ~ en+l(T).
in which
a famous example of such matters being the question
of elliptic
conditions.
For non-compact
maps
mentioned T
earlier,
(e (T)) ~ zP: n that
it is natural to enquire into
Iln(T) I ÷ re(T) , and to do this more delicacy is required,
although a preliminary T
as does
of those for compact maps:
With
-< lira (gn (Tk)) k-~
and solved,
is that of whether
that if
maps,
he shows that
n i/n ( ~ l~j (T) I) j:l
n
inequality
continuous
result in this direction
is given by the result in [8]
is a bounded self-adjoint map of a Hilbert
space into itself and
223
0 < p < ~, t h e n
there
exists
c > 0
such
that
I
:n (lln (T) I - r e (T)) p
-<
n-l(en(T)
c
n:l
Various
s-numbers
Other
numbers
may be attached
used being
the Kolmo@orovnumbers
Given
n ( ~,
any
d
Y QN
where
:
canonical
c
n
been
(T)
c (T) X
used
n
the most
the G e l f a n d
(T)
commonly
numbers.
is d e f i n e d
by
: d i m N < n},
onto
Y/N.
Equivalently,
il
x { Bx
extensively of e l l i p t i c
x-Ylt
y 6 N in the s t u d y
operators
of t h e a s y m p t o t i c
(cf.
[I0],
[25]).
The
n
th
by
inf{ilT°J
is the e m b e d d i n g
d
sup
is d e f i n e d
:
n
J
Y
N < n
of eigenvalues
number
of
and
number
o Tit
inf
have
distribution
where
map
NcY,dim
T { L(X,Y),
(or n - w i d t h s )
inf{llQ ~
:
numbers
Gelfand
(T)
n
n
to a m a p
the n th K o l m o g o r o v
is the
d
These
- re(T))P
n=l
II:codimM,
map
from
c
or
M
to
X.
M
Letting
properties
s
stand
n
similar
for
n
d
n
it t u r n s
out
that
the
sn
have
t o t h o s e p o s s e s s e d by t h e a p p r o x i m a t i o n n u m b e r s , n a m e l y ,
IITII : s1(~) >- S2(T) ->...-> 0, Sm+n_l(S Sn(T) Sn(In)
These
: 0 = I
properties
approach these
+ T) if
_< s m ( S ) + Sn(T), rank
if
In
have
conditions.
All
o T)
_< sin(S)
Sn(T),
T < n, is t h e i d e n t i t y
been
to s - n u m b e r s :
Sm+n_l(S
taken
he calls s-numbers
map
by Pietsch s-numbers coincide
on a s p a c e
[20] any when
X
with
as the b a s i s
sequence the
d i m X > n.
for an a x i o m a t i c
of n u m b e r s
spaces
involved
which
satisfy
are H i l b e r t
224
spaces.
Moreover,
c
If we w r i t e
(T) = d T*), n
n
c
(T*)
n
-< d
n
(T).
inequalities
= l i m c (T), d(T) = l i m d (T), t h e n f r o m t h e a b o v e n n n-~ n~ we may obtain inequalities for c (T) and d(T). More
importantly,
Carl
X,Y
and
has
-i
c(T)
c ( S + T)
8(T)
= 0
and
are
this
n
n
result
which
[21]
(~)
Pietsch
d(T)
(T) -< d(T)
-< p S ( T ) .
= 0, a n d if a n d o n l y
= d(S)
= sup{c
map
if
has
n
sense
(T o s)
because
:
T
if
is c o m p a c t ;
introduced
T
them
:
T
is c o m p a c t ;
and
the W e y l
numbers
by means
to g i v e
sufficient
absolutely
that
I
n
I ISII
'
x
n
(T)
:
< i} "
l[sl[
f o r all
< I} '
numbers:
conditions space;
for e x a m p l e ,
for the eigenvalues
a n d he i d e n t i f i e s
(p,2)-summing
(T) = O ( n -I/p)
that
R c L(Z2,Y).
of t h e W e y l
to a L o r e n t z
so-called
property
= Cn(R)
'
it c a n be s h o w n
s • L(~2,x)
an(R)
to b e l o n g the
S • L(12,X)
of P i e t s c h ;
can be proved
the h a p p y
as
maps
a class
(2 < p < ~),
n ÷ ~.
theory
To conclude, and the
[21]
o S)
in t h e
has used
Interpolation
theory
if
+ T)
(T) : = s u p { a n ( T
T • L(X),
have
of
then
results
of a c o m p a c t of maps
p > 0, i n d e p e n d e n t
n+~
following
Interesting Pietsch
p-18
n+~
s-numbers
x
is a n u m b e r
that
-< P S ( T ) ,
d(S
recently,
x
there
(T) = lira (c(Tn)) I/n = lira (d(Tn)) I/n
T ( L(X,Y),
These
such
-< c(T)
e
Quite
that
if a n d o n l y
= c(S)
r
if
shown
T (L(X,Y),
P
Also,
c(T)
we discuss
ideas
we h a v e
briefly
the interaction
mentioned.
The manner
between in w h i c h
interpolation the entropy
225
numbers and the various
s-numbers behave under interpolation
subject of much discussion
(see, for example,
recently been some interesting
developments
A0
and
space~
A1
.
are continuously
Given such a pair,
][alIA0nA I = max
A 0 + A I = {a £ ~
and so is
:
{A0,A 1}
embedded
A0 n A 1
[20] and [25]), but there have
which we describe briefly.
We recall that a pair of Banach spaces if
has been a
is an interpolation
in a Hausdorff
topological
pair
vector
is a Banach space when given the norm
([la[IA0,
IIalIAI),
a = a 0 + a I, a i ~ A i
for
i = 0,i}
when
normed by
Ila[IAO+A 1 = inf{llaoIl~ clearly
° + ]lall[A
A 0 n A I c A. c A 0 + A I 3
algebraically
space with respect to
{A0,AI}.
{B0,B I}
L({A0,AI} , {B0,BI})
T :
A0 + A 1 + B0 + B 1
a
=
(j = 0,i).
A0 N A 1 c A c A0 + A I
we denote by
for
1 :
a 0
a 1,
+
ai
~
A Banach space
and topologically
i = 0,i};
A .1
A
such that
is called an interpolation
Given two interpolation
pairs
{A0,AI},
the set of all linear operators
such that the restriction
of
T
to
A.3
is in
L(A.,B.) for j = 0,I. If A and B are interpolation spaces with respect 3 3 to {A0,AI} , {B0,B I} respectively they are called interpolation spaces of exponent
@
(0 < @ < I)
given any
with respect to
T ~ L({A0,AI},
{A0,AI} , {B0,B I}
{B0,BI}),
the restriction
1-8
e
of
T
if, and only if, to
A
is in
L(A,B)
and
l iTI ILIA,B) -~ I ITI I~.~AO,BO) I ITI I~lh,B1 We shall refer to the restriction L(A,B),
by
TA,B;
TA ,Hi
of
T
to
A, viewed as an element of
will have a similar meaning.
J Several methods of constructing respect to given pairs (cf. [253, p.23)
{A0,A I }
and
interpolation {B0,B 1 }
which leads to the spaces
spaces of exponent
are known,
e
with
notably the K-method
(A0,AI)@,p , (B0,BI)@, p
(I ~ p g ~)
226
defined by
A =
(A0,AI)8, p
= {a ~ ~0 + ~i : IIall~ : [/"
_[t-SK(t,a) ]p t-ldt) i/p < ~}
0
if
p < ~;
(A0,AI)8, ~ : {a • A 0 + A 1 : Ilall A :
A =
sup t-@K(t,a) 0
< ~};
where
K(t,a)
: inf{[la011A 0 + tlIallIA 1 : a : a 0 + a I, a 0 ~ A 0, a I { AI}.
Given any interpolation space
A
{A0,A I}
is said to b e l o n g to
such that
and any
J(8; A0,A I)
Ilall A ~ cIlalIl-811aIl@ A0 A1
K(8; A0,AI) all
pair
if there is a positive
@ E (0,i), an interpolation
if there is a positive
for all
a ~ A 0 n AI;
constant C
it belongs
constant c such that for all
a ~ A
to and
t > 0,
+ tllalllA 1 : a = a 0 + a I, a 0 E A0, a I ~ A I} S ctS{lallA . ' '' '
inf{llaO[IAo
The spaces
(A0,AI)@, p
Now let pairs
A
{A0,A I}
and and
belong to B
J(8; A0,AI)
be interpolation
{B0,B I}
A
to
respectively,
B, inherits any compactness
an element of that if
L(A
B0 = B1
,B ). ll and
and let
compact if either established
if
TA0,B 0
A0 = A 1
or and
TAI,B 0 B0 + B 1 .
that in the general case, with TA0,B 0
is compact.
certain approximation Hayakawa T
property,
Ell], although
{B0,BI}).
T, viewed as a map
which it may possess
EI5~ gave a partial answer,
for some is compact)
8 ~ (0,i), then
Shortly afterwards and
an assumption
as showing
TA,B0
is
a similar result was
B 0 + BI,
To do this he had to assume that
is also compact. AI,B 1
A0 + A 1
T e L({A0,AI},
is whether
properties
Lions and Peetre
A e K(@) A0,A I)
K(@; A0,AI).
spaces with respect to interpolation
A question which has attracted much attention from
and
TA, B B0
Persson
[193 proved
is compact if and
B1
had a
removed in the later work of
at the expense of the additional
assumption
that
A good deal of the subsequent work in this area has
227
been to do with the interpolation ideals of compact operators, book [25]: mentioned
properties
an excellent account being given in Triebel's
a feature of this work is that, above,
either
of entropy ideals and width
A0 = A1
report on recent developments
or
like that of Lions and Peetre
B0 = B 1.
Our object here is to
w h i c h have led to the relaxation of these
restrictions. We begin with a discussion non-compactness
under interpolation.
to prove that if 0 < @ < I,
of the behaviour of the measure of
A
It is comparatively
is a Banach space,
B • J(@; B0,BI)
and
T • L({A,A},
B(TA, B) ~ C(8(T A ,B0)) I-8
A similar result holds if Since
6(T) = 0
{B0,B I}
impose a hypothesis
T
and
is compact,
Given an interpolation approximation
property
given any compact P • L({B0,BI}, (i)
P(Bi)
A0 + A 1
on the spaces,
H1
subset
IPBi,Bil I ~ C
(iii)
]x-PXIIB0
< E
of
Theorem 3 {B0,B I}
Let
these results include as a above.
B 0 + B I,
To deal with maps it is convenient
to
just as P e r s s o n did.
B0
and
constant
c
such that
E > 0, there is a map
(i = 0,i); (i = 0,i); for all
of compact maps.
can be established
{B,B}).
{B0,BI}, we say that it has the
x • K.
This is exactly the condition interpolation
pair,
satisfying:
c B0 n B 1
(ii)
and
T • L({A0,AI},
if there is a positive
K
{B0,BI})
pair
is an interpolation {B0,BI}) , then
special case those of Lions and Peetre mentioned T • L({A0,AI} , {B0,BI}) , where
(cf. [7])
(8(T A ,BI ))0.
A • K(8; A0,AI)
if and only if
easy
imposed by Persson in his work on the
With this condition
the following
result
(cf. [23]):
{A0,AI},
has property
HI,
{B0,B I} let
be interpolation A =
(A0,AI)@,p,
pairs,
suppose that
B = (B0,BI)@, p
for some
228
8 • (0,i)
and
such that
T
p 6 [i,~],
and suppose that
is compact.
T 6 L({A0,AI},
Then for all
{B0,BI})
is
m,n • ~ ,
A0,B 0
(19)
em+n_ 1 (TA, B)
<
2c
-
em(TA0,B0 )
To dispose of the hypothesis strengthened any
e > 0
to
H2 :
and any finite sets
ILx-PxlLB. 1 The
Lp
spaces
has property
for all
x • F.
(i ~ p < ~)
H2, the hypothesis
that
TA0,B 0
(20)
(I,c)
These results have various and denote by
E
the operator
has to be
such that given
there exists
(ii) of
satisfy both
then being
<- 4 max
c
HI
H1
hold,
together with:
(i = 0,i).
T h e o r e m 3, the final inequality
em+n_ I (TA, B)
constant
(i) and
8
is compact,
F. c B. (i = 0,i), 1 l
such that
< e
en(TA1 ,B 1)
TA0,B 0
there is a positive
P E L({B0,BI} , {B0,BI})
(iii,~
that
1-0
H1
and
H 2.
If
{B0,B I}
is compact may be dropped in
(cf. [23]):
(em(TA0,B0))I-@
implications.
(en(TAI,BI))8
For example,
ideal of all linear maps
T
let
0 < p < ~
between arbitrary
P Banach spaces such that
e (T) p < ~: n
n=l terminology
[25].
pi
where
TA.,B ' e E l 1 Pi
is an entropy ideal,
= (i-@)
P° I
(i = 0,i), where
~(TA, B)
-< 4 max
(i,c)
in Triebel's
of T h e o r e m
p0,Pl • (i,~), then
+ 8p ;i
Another result comes on letting
(21)
p
Then in the notation and with the assumptions
(19) implies that if
TA, B • E , P
E
m,n ÷ ~
in
(8(TA0,B0))I-@
(20):
(8(TAI,BI))@.
This leads to information
about the essential
spectrum
Theorem 4
be an interpolation
pair which has property
Let
{A0,AI}
le__~t A = (A0,AI)8, p
for some
8 E (0,i)
and
(cf. [7]).
p 6 [I,~],
and let
H2,
3,
229
T 6 L({A0,AI} , {A0,AI}).
r (TA, A) e
Proof.
From
~
Then
(re(TA0
,A 0
(21) we have for all
n 8(TA, A)
~
4 max
))i-@
(r (T ))@. e AI,A 1
n 6 ~ ,
(l,c) (8(T~0,A0))I-@
(BCT~I,BI))@ "
The Theorem follows immediately on using the identity
Knowledge
of
how eigenvalues
behave
under
r (T) = lim (B(Tn)) I/n e n-~o
interpolation
would
be
available
if results of a nature similar to Theorem 3 could be obtained for the Kolmogorov
or
Gelfand
numbers.
There
seem to
be
some promising
developments
in this direction.
References I.
F.E. Browder,
On the spectral theory of elliptic differential
I, Math. Ann. 142 2.
B. Carl, Entropy numbers, operators
3.
(1960/61),
22-130.
entropy moduli,
in Banach spaces
operators,
s-numbers and eigenvalues of
(preprint).
B. Carl, Entropy numbers of embedding maps between Besov spaces with an application
to eigenvalue problems,
Proc. Roy. Soc. Edinburgh
90A
(1981), 63-70. 4.
B. Carl and H. Triebel,
Inequalities
between eigenvalues,
and related quantities of compact operators Math. Ann. 251 5.
(1980),
entropy numbers
in Banach spaces,
129-133.
D.E. Edmunds and W.D. Evans, Elliptic and degenerate
elliptic operators
in unbounded domains, Ann. Scuola Norm. Sup. Pisa 27 (1973), 591-640 6.
D.E. Edmunds,
W.D. Evans and J. Fleckinger,
7.
D.E. Echnunds and M.F. Teixeira, non-compactness,
to appear.
Interpolation
Math. Nachrichten
theory and measures of
(to appear).
230
8.
D.E. Edmunds and H. Triebel, Entropy numbers for non-compact selfadjoint operators in Hilbert spaces, Math. Nachrichten i00 (1981), 213-219.
9.
W.D. Evans, Semi-bounded Dirichlet integrals and the invariance of the essential spectra of self-adjoint operators, Proc. Roy. Soc. Edinburgh A75
i0.
(1975),
41-66.
J. Fleckinger, Estimation des valeurs propres d'op4rateurs elliptiques sur des ouverts non born~s, Ann. Fac. Sci. Toulouse 12 (1980), 157-180.
ii.
K. Hayakawa,
Interpolation by the real method preserves compactness of
operators, J. Math. Soc. Japan 2 1 12.
(1979),
1966).
141-146.
A. Lebow and M. Schechter, Semigroups of operators and measures of non-compactness,
15.
(Berlin-Heidelberg-
H. K~nig, A formula for the eigenvalues of a compact operator, Studia Math. 65
14.
189-199.
T. Kato, Perturbation theory for linear operators New York: Springer-Verlag,
13.
(1969),
J. Functional Anal. 7 (1971),
J.-L. Lions and J. Peetre, Sur une classe d'espaces d'interpolation, Inst. Hautes Etudes Sci. Publ. Math. 19 (1964),
16.
Bull. Soc. Math. France Mem. 51-52
125-219.
(1966),
366-372.
R.D. Nussbaum, The radius of the essential spectrum, Duke Math. J. 38 (1970),
19.
(1977),
B.S. Mitjagin and A. Pelczynski, Nuclear operators and approximative dimension, Proe. I.C.M. Moscow
18.
5-68.
G. M6tivier, Valeurs propres de probl@mes aux limites elliptiques irr6guliers,
17.
1-26.
473-478.
A. Persson, Compact linear mappings between interpolation spaces, Arkiv Math. 5 (1964), 215-219.
20.
A. Pietsch, Operator ideals North-Holland,
21.
(Berlin: Verlag der Wissenschaften and
1978/80).
A. Pietsch, Weyl numbers and eigenvalues of operators in Banach spaces, Math. Ann. 247
(1980),
149-168.
231
22.
C.A. Stuart, Some bifurcation theory for k-set contractions, Lond. Math. Soc. 27 (1973),
23.
Proc.
531-550.
M.F. Teixeira, Entropy numbers and interpolation, Math. Nachrichten (to appear).
24.
H. Triebel,
Interpolationseigenschaften yon Entropie-
Durchmesseridealen kompakter Operatoren,
und
Studia Math. 34 (1970),
89-107. 25.
H. Triebel,
Interpolation theory,
function spaces, differential operators
(Berlin: Verlag der Wissenschaften and North-Holland, 26.
H. Weyl, Inequalities between the two kinds of eigenvalues of a linear transformation,
27.
1978).
Proc. Nat. Acad. Sci. U.S.A. 35 (1949),
408-11.
J. Zemanek, The essential radius and the Riesz part of spectrum (1980 preprint).
The singular perturbation approach to flame theory with chain and competing reactions
by
Paul C. Fife
and
Basil Nicolaenko
Combustion theory provides an extremely fertile ground for asymptotic analysis.
This is due in part to the fact that the theory combines fluid dynamics
and chemical reaction theory, both in themselves part to the prevalence
rich in difficult problems,
of small and large parameters
and in
appearing in the underlying
equations. One of the most notable large parameters expresses problems, strongly
the rates of some important on the temperature.
laminar flame theory,
chemical reactions
the fact that in combustion typically depend very
In the context of the simplest cases of steady plane
this fa~t results in a problem of the form [2]
LU~f
This research supported by NSF Grant 79-04443 and by the Center for Nonlinear Studies,
los Alamos National
Lab.
233
where
e
is a small parameter.
Here
L
is a (vector valued) second order
differential operator in the space variable and
U+
is some reference vector.
following "layer" feature:
which does not depend on
c,
the derivative U' undergoes an abrupt change with
magnitude order of unity (as e.
x
Tne solution of this problem has the
e + 0)
in a small interval with length order of
This interval is called a flame layer because it is here, and only here, that
the source (reaction) term therefore,
f
has a significant effect.
can be approximated for small
~
2he function
U,
by a function with derivative
discontinuous at one point. This latter piecewise smooth function itself satisfies a boundary value problem (the "reduced problem") in which the parameter
e
no longer appears: the
nonlinearity has been replaced by a condition specifying the jump in point of discontinuity
U'
at its
(i.e., across the flame).
The above description holds when there is only one chemical reaction in the flame process, and it is exothermic with high activation energy.
See [1],[3] for
a comprehensive treatment.
the situation
When there is more than one reaction,
outlined above m~st be modified.
Depending on the specific reaction network and
the other given chemical data, a wide variety of flame configurations become possible.
In this talk we shall explore these possibilities in the case of
sequential and competing reaction pairs.
In the former ease there is some
overlap of our results wi~h those of Kapila and Indford have also been considered in [5-8].
some sense limiting cases of those described here. results will be given in [3].
[4].
Pairs of reactions
Cases were treated in [6-8] which are in
In particular,
A much fuller account of our
the latter paper contains an
extension of our methods to flames with competing-fuel reaction pairs.
They
admit immediate extension to most reaction networks with two or three exothermlc reactions with high activation energy. Our analysis is largely formal, although great care is taken to justify the approximations made. el"
They are based on the smallness of certain parameters
Often, other parameters will be assumed to be "order of unity", or "0(I)".
This concept means these parameters are neither very large nor very small; otherwise it is left without strict definition.
Effectively,
it means that the
error introduced by an approximation should be estimable in terms of some
ei,
or a positive power of it, multiplied by a coefficient which depends on these other parameters.
The other parameters,
therefore, should not be so large or
small as to vitiate the accuracy of the approximation.
234
We begin with a description standard
of the asymptotics
and reduced problem for the
case of a single reaction.
i. Simple flames Without for laminar Here
giving the physical flames with a single
U = [~],
reactant
D
Y
M
reaction
B
A + P.
The problem is to determine
and vectors
U(x)
U(--) = U , Y(=) = 0.
(ib)
E (proportional
this paper, K
the "profile"
of
(la)
are
By integrating
[3]),
K
is the vector
and
m
(2)
to the "activation
energy")
positive
of matrices
constants.
such as
D, D-1,
[3] for more general circumstances).
(la) with use of (ib), it is seen immediately
that
u(®)
T+
=
0Y U(~) = U+ -: [ 0- ] .
(3)
s ~ T+2/E << 1
(4)
0Y_ • We assume
and on the basis of this relation,
approximate
m
by
to
is a
type:
we assume that all elements
0(i) (see
[_~] ,
satisfies
~nus
and
transport m a t r i x which is taken for simplicity
(this is not necessary
and
flames.
and concentration
DU' ' - MU' = -~o(U)K,
rate, w h i c h we take to be of Arrhenius
Throughout
We call them simple
temperature
from the conditions
= BYe -E/T
with
we simply write the basic problem
are dimensionless
is a positive definite
be constant reaction
and
A, respectively.
the mass flux
Here
T
derivation,
235
n
=
Y
(5)
7 (7 exp [(T - T+)lq),
where
n = B s2 exp (-E/T+)
This approximation = (T-T+)/E, t
about
is obtained
by writing the exponent
The smallness
of
g
implies
outside a neighborhood
is now with respect
to
J [$]
=
of (2) as a function of
=
(uu+)l~
(see
-
~ = x~/2, M = Mn-I/2;
~ 1
~*(u)K,
(7)
E
(u) = ~ m(U) = ye
t
(8)
for
U(~)
(assumed
~:
(DE)
DU - MU = 0,
(BC)
U(* ~) = U ,
(CC)
• O+ [DU]o_ = -
Here the scalar
(la) becomes
)
We postulate a "reduced problem" of the following form, and
T+.
and *
continuous)
B
x ~ 0,
(9a) (9b)
BK.
(9c)
in the "connection
condition" (9c) is to be determined a * from the function m . U-U+ variables ~ ~ x/S, u c ' and then neglecting the term
priori by a layer analysis By stretching formally
of order
s,
we write
(7) as
DJ'' + ~ (u)K = 0
This is to be solved for
u = u(~)
under the boundary u(~)
Since,
t
[3]) that it can be neglected
of the maximum temperature
We now perform a scale transformation (differentiation
n
(6)
and taking the first two terms in its Taylor series expansion in
t = 0.
for temperatures
where
.
from (i0), u'' is a multiple
= 0
of D-IK
(i0)
condition (11)
for each
~ , it follows from (ii)
236
that
u
itself must be also:
u(~) = ~(~) D-IK.
Using (12) in (i0) and applying
~"
(12)
(ii), we find a problem for
+ ~*(~D-I~
alone:
= 0, ~(=) = 0.
(13)
This has a first integral
( ~ , ) 2 _ e ( ~ ) = 0, 0 [~(o') = 2
where
*
(sD
Assuming now that
D-IK
see from (8) that
~(-~)
-i
K)ds.
has first component positive and the second negative, exists,
can be calculated,
and represents
we
(o'(-~)) 2.
The matching condition between the flame layer and the "outer" solution is
u'(-~)
From this,
= U(O-).
(12), and the fact that necessarily
identify the constant
6
in (9c)
0
(9).
of
B
completes
What remains is to solve it.
[DO - ~]~3 = BK-
=
U E U+
for
~ > 0,
we may
as
= o'(-=°) = (2 f
The specification
(14)
, -i b ~ (sD K)ds) ~2.
the definition
(15)
of the reduced problem
To do so, we note, by integrating
o
=
(9a), that
D0(0-) - ~ + +~_.
MY_K,
hence
= 8/Y . .
With at
M
known,
~ = 0.
value
U_
(9a) may now be solved for
There is a unique solution, as
~ + -~
This completes
~ < 0,
(1 6)
with known initial conditions
and it approaches
the correct limit
the solution of the reduced problem.
237
2. The effects of other reactions on simple flames Consider a simple flame profile, reaction term
mI(U)K 1
to the quantities species
A1
as in (2),
~, K, ~, E, B, and Y.
Thus
which is depleted at the flame.
adding another reaction, (la).
as described in sec.
i, involving a
(5), in which the subscript
with a concomitant
"i" is appended
YI is the concentration
of the
We now perturb the problem by second term
~2K2
on the right of
Our main result will be a simple criterion for judging whether the second
reaction constitutes with use of
a negligible perturbation
of the given flame.
n I , we obtain the following analog of l~
--" MU
-
1 ~I
=-
* ml(U)
K1 -
Rescaling
(7): 1 -~i ~2(U)K2"
(17)
We assume that
~2 = T~/E2 << 1
is also small,
so that the effect of
~2
for
x < 0
comparison with its effect in the flame and for
There are two important (I)
is negligible,
in
x > 0.
cases to be considered:
No reactant for the second reaction is depleted in the original flame;
(II) A reactant for the second reaction is so depleted. Case I. reactions
This case will apply, for example,
in sec. 3.
In this case,
flame, and may be represented
~2
to our study of sequential
remains strictly positive behind the
by -E2/T
~2 = B2Y2e
We are assuming, first, o(E 2)
for simplicity
is first order: from
T+,
only, that the second reaction,
A 2 ÷ P.
As long as
T
deviates
as well as the
by amount
we have -E2/T + m2 ~ CB2e
for some constant
C = 0(i),
and the last term in (17)
is bounded in magnitude
by
-E2/T+ q ~ CB2e Since the first effect initial
term,
due t o d e p l e t i o n
of the second reaction value
problem
of
2 -El/T+ / B1E I e A 1,
vanishes
(18) behind
the flame,
c a n be g a u g e d by e x a m i n i n g t h e s o l u t i o n
the
of the
238
DU - ~
for vectors
f
Ifl
with
= f(~), U(0) = U+, U(0) = 0,
q.
<
The representation x
U(x) = / 0
exp[MD-1(~-s)lD-lf(s)ds
+ U +
yields
Iu(~)-u+l< for some constants
C
and
X
Cqe X~ ,
= 0(i),
since
M
and
D -I
are 0(i) quantities.
If, therefore,
q = 0(g2) ,
it follows length
that
Iu(7) - u+ I << i
for
restricted
-~0
to any interval of
0(I).
We now define functions
H (T)
i
=-f(~n
B i - E /T),
(19)
i
where
= T+21Min(E1,E2).
It is easily seen that H k V ( T +) = 1,
where
k
H '(T) is at most 0(I) for each i, and 1 is s u c h t h a t Ek = M a X [ E l , E 2 ] . Suppose, now,
H2(T+)
then from
that
(21)
- HI(T+) = -~ < 0.
(18),
q = - ~ce e 1 if (for example)
u
Our conclusion negligible
(20)
-~/'~ < o7-2
-~/'7 e
= o(E2) ,
is an 0(1) quantity.
is that (21) is a sufficient
effect on the original
condition
simple flame in regions
for
m2
a distance
to have a 0(I)
fror
239
the flame
(actually,
Case II.
it is clear that much larger regions could be allowed).
In this case, we assume for simplicity
example will be the branching network
in Sec. 4.
that
Y2 = Y1 = Y"
A prime
The last reaction rate in (17)
becomes
I
1
w2 =
-E2/T
-~1B2Ye ~ =
e
1
=
-E2/T +
1
n--~ B2e --
e
r
ye
oJ * ( u )
2
1
Yt
(22)
'
w h e r e we have set E2 Y =
/El
' m2* = yeTt'
(23)
and -E2/T +
r = B2e We assume U+
for
Y = 0(i) ~ > 0,
condition
/B1e
for this exposition.
, so the stretched variable
u(=) = 0.
-EI/~ ~
~nce
(24) Y
is depleted,
u = (U - U+)/c I
The layer problem is found from (17),
U = const =
satisfies
the limit
(22), and assumes
the form
Du'' + ~1*(u)K1 + rm2*(u)K 2 = O,
u(=) = O.
Reasoning
(25b)
in the same manner as before,
we see any solution must be of the form
u(~) = - O I ( 5 ) D - 1 K 1 - q2(~)D-IK 2
for some scalars
°i ''
c . i
ciD-I Ki )
Substituting
(25a)
(26)
into (25), we obtain the problem
= 0,
(27a)
o2'' - r ~ 2 * ( - Z o i D - i K i) = 0,
(27b)
c~(~)
(27c)
-
~
i *(-E
= 0.
240
The following
result will be useful to us latter, as well as in the present
context.
lemma.
Assume the vectors
D-IK i
have positive
components.
Then the problem
(27) has,
translation
of the independent
variable
satisfies on
r,
- ~ < o.'(-~) 1 and satisfy
< 0.
a2 llm-- = 0 ; r+O al Proof:
The equation
r > 0,
~) positive
The quantities
second
first and negative
for each
a unique
solution
-a. = o.'(-~) I i
(except for
oi(~)
.
It
depend continuously
a2 lim-- = ~ r+ ~ al
(28)
(27 a,b) may be w r i t t e n as a first order system
°l'
= -Pl
T
=
_~
Pl
' @
i
(29)
'
c2' = -P2 ' P2' = -r~2*"
The object is to find a trajectory from the positive quadrant
approaching
stable manifold is one dimensional P.
the critical
P = {c i > O, Pl > 0}.
point 0 as
~ ÷ ~
It may be checked that the
and one branch of it indeed approaches
We follow that branch backward:
in terms of
s = -~
with
... = .d., ds
0 from '
becomes
~I
=
Pl
P~ : ~I* c2 = P2 P2 = r~2* "
let the positive
Then f o r
w =
constants
6
["] Pl °2
P2
@
P'
be defined by
D-iKi.[~ll ] -B2i
(30)
(29)
241
Y = [ i so
~i* > O.
B2ioi > 0 ,
~ e r e f o r e the right sides of (30) are positive in
follows that the trajectory entering
In fact, constants
(31)
w($)
P
P,
and it
stays there.
is monotone increasing, and from (30), (31), we have, for some
Ci, c i oi > Cl(S-l)
y ~ C2(o I + 02 ) ,
t < -c3(O I + o 2) ,
-c3(Oi+° 2) el* < C2(o I + O2)e
-c5(O1+o 2) C4e -c 6s ~C5e
Hence from (30) again,
oo
Pl < C6 + f o
-c 6 S
C5e
ds < C 7 .
Similar reasoning shows P2 ~ C7"
Therefore
Pi(S)
approach finite limits
parts of the lemma.
ai = pi(~).
We skip the proof of continuity in
This proves the first r.
When
r = O, the
stable manifold we are following backwards lies entirely on the plane
242
02 = P2 = 0, limit
so we clearly
relation
(28).
have
a 2 = 0,
For the second,
s = ~ r -I/2 , Pi = Pi # / 2
o i = -6i'
a I > 0.
we transform
This establishes
the first
(30) by setting
then
-61 = PI ' ®
P'I =
]
:
m
(°I* (:i'
m
o2)'
-62 = P2 '
P2 = ~2" "
As
1
r + ~, ~
pl (~) + 0,
+ 0 hence
us the second
and we use the argument
symmetric
pl (~) = r 1/2 pl (~) + 0.
of (28), and completes
Thus
to the above:
it follows
a I + 0, a 2 > 0.
that
This gives
the proof.
From (26), we now have
Du,(-~) = alK 1 + a 2 ~
The matching
condition
is
6(0-)
DU(0-)
= u'(-~);
•
therefore
= al(r)K 1 + a2(r) ~
Ehis enables us to form~alate a reduced problem
(32)
.
for Case (II), analogous
(9): DU - MU = 0,
x ~ 0;
(33a)
U(* ~) = U, ;
and (32).
The solution
is obtained
as follows:
(33b)
from
(33a) we have
to
243
D0(Q-) - N(U+
- U_) = 0 ;
then from (32) and this,
- U
U+
There are actually (a) Given
U+,
(b) Given
r,
through
= al(r) ~
two conceivable
determine
K 1 + a2(r) -~
formulations
M,r, and U(~)
determine
Of course,
-
M, U+, and
either case ignores
K2 - c~IK 1
are solvable.
we first determine
r
c~2K2
•
of the problem here:
;
U(~)
.
the fact that
r
and
T+
(24), and so (24) will have to serve as a consistency
Both problems
+
For example in (a),
are also related check.
~. > 0 i
will be given,
so
from the condition
a2(r)
a2 (34)
By the lemma (in particular this relation. r,
(28)),
there exists a value of
It is not clear at this point whether
we then determine
~ = al/~l,
and hence
We now return to the functions follows
from (24) that
~
Therefore
by the lemma again,
problem
(33),
depends
continuously
Our conclusion,
on
a 2.
then,
which the second reaction
Knowing
that
(21) holds.
It
<< i. << i.
But when
a 2 = 0,
the reduced
to (9), and it is easy to see that the solution
Hence it is justified
to neglect
reaction
(ii).
is the same as in Case I: (21) is the criterion does not significantly
Having developed a general criterion,
3. Reactions
is unique.
so that
a2(r)
(32) is identical
which satisfies
from (33a).
H i (19), and suppose
%n r = -~,
r = e
U(~)
r
r
we proceed
alter the original
flame.
to look at specific networks.
in series
We now consider a combustion
process
series :
A24= 4P
by
involving a pair of reactions
in
244
both e x o t h e r ~ c with h i ~
form
activation e n e r ~ ,
I'-I
U
= YA
is prescribed.
~
me
coneeptualize
unburned state, in the
the possible
flame structures,
we first follow the approach of sec. 2: we imagine a simple flame resulting from only one of the reactions, the other being artificially suppressed. whether the criterion is met, w h e r e ~
suppressed, would have a negligible effect. realizable st~cture.
~ter
We then ask
the other reaction, if no longer If so, then that simple flame is a
this is done, we investigate possible nonsimple
flames. mere
are four basic s i d l e
flames:
FI, produced by reaction (i) only, converting the given unburned state to a partially ~ r n e d
one:
U
From (3) it is seen that
T 1 = 01 YA- + T_.
F2, produced by (ii) only:
U_ ÷ U 2
with
loj
T 2 = T_ + 02 YB F12, produced by (ii) acting on the product (final state) of FI:
U I + U+ ~ where
T+ = T
li+J
+ 01YA_ + 02(YA_ + YB_).
F21, acting on the product of F2. A ÷ P sum
Here the reaction is of the type
with rate identical to that of (i), but with heat release equal to the 01 + 02.
(i), and
B
One can think of this flame as converting
A
to
thereupon being almost immediately converted to
B
P
by reaction
by a faster
reaction (ii). We again define functions (20).
We assume that
Hi(T )
T_, TI, T2,
and
according to (19), with T+
~
given by
are all of the same order of
245
magnitude,
and
T I - T_, T 2 - T_, T
The temperature
scale is adjusted
We consider
the consequences
functions
Hi .
difference
of the two quantities
because
- TI, L
so that
of various
When such an inequality
are all
inequalities
is written,
consequences
concerned
of sec.
If
H2(T I) < HI(T1) ,
2)
If
HI(T 2) < H2(T2) , then F2 is.
Note,
from
quantities.
is
involving
the
0(1).
that the
This is the typical Hi
case,
is of order unity.
2 are:
I)
The next structures
0(i)
it will be assumed
the range of at least one of the functions
Immediate
flames.
- T2'
T+ - T_ = I.
then
F1
is realizable.
to be considered
are possible
(16) and the fact that
M = M n I/2
combinations
of 2
that the mass flux for F1
is M I = CIBIe 2I exp(_El/Tl ) ,
(35)
where
s I = TI2/E 1
and
CI
is some constant,
C I = 0(1).
Hence
~'~,nM 1 = ~(£n B 1 - E I / T 1 + 2 ~,n Sl )) + 0(-~) But
IEn e l l
= %n C1
-ftn In a similar
H 1 = HI(T 1) + 0(~ tn i / ~ ) .
manner, "{ %n M 2 = H2(T2) £n MI2 = H 2 ( L )
7
£n M21 = H I (T+) + o(I)
same way as their velocities.
the two flames flame
+ o(1)
,
.
of these four simple flames are ordered
Thus,
for example,
if
of FI, is slower than F1 itself,
both existing,
F1 precedes,
,
7
It turns out that the mass fluxes
built on the products
+ o(1)
but the distance
and is faster than,
F12.
between
H2(T+)
< HI(TI),
in the
then FI2,
and so one can conceive them ever increasing.
We call this total configuration
of Tne
246
FI,12: 3)
If
H2(T +) < HI(T1) ,
then FI,12 is realizable.
Similarly, 4)
If
HI(T+ ) < H2(T2),
then F2,21 is.
The most interesting flames, however, are seen when neither 3) nor 4) hold. and
This last case is split into two subcases: H2(T 2) < HI(T + ) < H2(T+).
HI(T1) < H2(T+) < H I ( L )
,
It can be easily checked that these two are
exclusive and, excluding cases when equality holds somewhere, exhaust the complement of 3) and 4).
Since 3) and 4) do not hold, the diverging
configurations FI,12 and F2,21 are not possible.
Indeed if such a combination of
flames like FI,12 existed at one time, then the rear flame FI2 would approach the forward flame FI.
As it does so, its effect is to heat up the forward one.
As
this happens, the speed of the forward increases (this is feasible because our expression for M, as in (35), is an increasing function of T).
The question now
is whether this heating from the rear is sufficient to raise the velocity of the forward part to equal that of the trailing FI2. cannot by this mechanism be raised above
T+,
Of course, the temperature of F1 which is the temperature of FI2.
And it seems reasonable that the equality of speeds will be attained at a temperature
TO
such that
HI(To) = H2(T+) .
Therefore our conjecture (to be
verified) is that such a tandem configuration, with F1 (ahead) and FI2 (behind) traveling at a fixed distance apart, is realizable if there is a the above equation.
TO
But this is true precisely in the first subease.
satisfying We call
this tandem flame FI-12. 5)
~
HI(T I) < H2(T +) < HI(T+) ,
then
FI-12 is realizable.
Simllar ly, 6)
If
H2(T 2) < HI(T + ) < H2(T+) ,
then F2-21 is realizable.
It should again be emphasized that at this point, (5) and (6) are just conjectures.
However, they can be verified in the following manner.
If a tandem
structure exists, it should correspond to the solution of a reduced problem in the spirit of (9), and this reduced problem should be uniquely solvable. This is, in fact, the case.
For FI-12, for example, the reduced problem
takes the form
U(~)
= U~
,
247
[DU]0+0- = - ZIKI ' [DU]~~--- = -82 K
Here
62, U±,
determine 62
and
T(0) = TO
~i' %' and U(~).
constitute the given data; the problem is then to The problem has a unique solution.
is known by a layer analysis very like that in sec. 2.
coefficient
BI
The constant
However, the
should also be involved with a layer analysis.
And indeed such
an analysis can he performed; it yields a functional relation
B 1 = F(r,
where
r
is given by (24) with
T+
~(0+)),
in the denominator replaced by
TO .
This
relation can be inverted to yield
r = R(BI,
U(0+)).
(36)
The proper consistency check on the above reduced problem, then, is (I) solve it, and find B 1
and
U(0 +); (2)
determine
(24) (T+ in denominator replaced by
r
To).
from (36); see if this
satisfies
We rewrite (24) in the form
£n r = H2(T +) - HI(To).
Now To
r
(37)
was chosen to make the right side vanish; but if we now reason that this
is only the first approximation to the true value of to obtain a corrected value for
To .
Since
turns out that the correction will only be
To,
then (37) can be used
HI'(T o) = 0(i), 0(T),
and
r = 0(I),
so the original value of
will be correct within our general range of accuracy.
it To
The details of the tandem
analysis c a n be found in [3]. In summary, the possibility of the various flame types 1,2, etc., can be found by testing each of the cases I) - 6) in turn.
Cases 3) - 6) are n~atually
exclusive, but any of the latter could hold in conjunction with i) or 2).
There
result nonunique flame structures; which one is realized in a given observation would depend on how and to what extent the gas mixture is ignited. nonuniqueness is the case when
T 1 < T2,
An example of
and
H2(T I) < HI(T I) < HI(T 2) < H2(T 2) < H2(T +) < HI(T +) • In this case, types I, 2, and 1-12 are all realizable. 4. Reactions in parallel The co[~ustion process now allows the reactant A to be consumed in two
248
competing
fashions:
(i)
P1
A
F2
Again,
each reaction is exothermic with high activation energy, and the
unburned state
dimensionless
U(-~)
=
U_
I~-I L'-J
=
concentration
is prescribed
T 1 = T_ + 01 Y_
of definiteness,
Y
represents
the
of A).
There are only two conceivable temperatures
(here
we assume
and
simple flames:
FI and F2, with final
T 2 = T_ + 0 2 Y_
01 < 0 2 ,
respectively.
For the sake
hence
TI < T 2 •
Temperature
is scaled so that
T 2 - T I, T 1 - T_ = 0(I). replaced
by
T 2.
We define
According
HI(T I) > H2(TI) ,
T 2 - T_ = i.
and
Hi
again by (19) and (20) with
to the criterion
F2
is if
H2(T2)
in sec.
> HI(T2).
Ca)
HICT i) < H2(Ti) , i = 1,2
(b)
HI(T1)
> H2(Ti) , i = 1,2,
(c)
HI(T1)
> H2(TI) , HI(T2)
< H2(T2),
(d)
HI(TI)
< H2(T1) ' HI(T2)
> H2(T2) .
The cases are depicted
Assume T+
2, F1 is realizable
if
Four cases therefore
by these four sample graphs of functions
arise:
H i (dotted
lines t
represent
H I (T), dashed lines
H2(T)).
¢
• /
I"
/ /
,
/
a "
,
i
/
7
/
.s.~/
"
o
.-,7,
(c )
~
-17,
Cb)
4
249
In case (a), only F2 is possible; F1 or F2, and in case interactive, exist.
nonsimple,
To
ToE
(TI,T2).
v r an
there may be an
in the burning process
is
It is clear from the graph that this occurs at a postulate
take part.
of case II in sec. 3.
a flame burning at temperature
This situation
can be fit into the
such a flame would be governed
of that p r o b l e m applies,
T U + = [0°]
by (17), with
since
is given.
We now indicate some
can be significant
We therefore
in w h i c h both reactions
framework
This suggests
flame type in case (d), since neither simple flame can
The only way both reactions
for their H's to be the same. value
in case (b), only FI; in case (c), either
(d), neither F1 nor F2.
0 ~ (0,1)
the solution of that problem.
, T O may be represented
To - T_ = 0Y_Ol + (1-0)y_ 0 2
and since
as
T
First,
it is clear that for
= OT 1 + (i - 0) T 2.
Hence
Ki = [Oi] , we have -i
U+ - U_ = aIKI + a2K 2 ,
where
el = 0y _,
hence
M = al/~l,
a2= (I-8)Y _.
The consistency This equation
1he ratio
and the profile
U(~)
check now requires
To
(24) to hold w i t h
- HI(To)
was chosen so the right side vanishes,
inconsistency.
But the inconsistency
to obtain a more accurate
T+
TO .
also exists
But in that case,
temperature
in case
TO
there appears
To .
to be an
the equation
should be adjusted
(38)
(by an amount
value.
(c), because again,
the mixed flame is unstable
is perturbed
simple flames.
replaced by
(38)
It should be remarked that the "mixed" flame constructed apparently
from (34), and
.
is only superficial;
simply seems that the previous value of 0(~))
is n o w determined
can be written
£nr = H2(To)
Since
r
from (33a).
by a small amount,
Hi
in case (d)
have a common value at
in a certain sense:
if its
it will evolve into one of the two
In case (d), this instability
does not occur.
250
Bibliography
i.
J. Buckmaster and G. S. S. ludford 1982, Theory of laminar Flames, Cambridge University Press, New York.
2.
P. C. Fife 1982, Propagating fronts in reactive media, in: Nonlinear Problems, Present and Future, A. Bishop, D. Campbell, B. Nicolaenko, eds., North-Holland.
30
P. Fife and B. Nicolaenko 1982, Two-reaction flame propagation,
4.
A. Kapila and G. S. S. Indford 1977, Two-step sequential reactions for large activation energies, Combustion and Flame, 167-17 6.
5.
A. ~ a n 1971, A theoretical analysis of premixed flame propagation with an isothermal chain reaction, Tech. Report, Inst. Nac. Tec. Aerospacial "Esteban Terradas", Madrid.
6.
S. B. Margolis 1982, lean flame propagation with competing chemical reactions, Combust. Sci. and Technol., to appear.
7.
S. B. Margolis and M. J. Matkowsky 1981, Flame propagation with multiple fuels, SIAM J. Appi. Math., to appear.
8.
S. B. Margolis and B. J. Matkowsky 1981, Flame propagation with a sequential reaction mechanism, SIAM J. Appl. Math., to appear.
in preparation.
On the singular values of non-self-adjoint operators of Schr~dinger type
J.
Fleckinger
In recent times non-self-adjoint elliptic operators,
and especially
non-self-adjoint Schr~dinger operators, have attracted considerable attention. Here we study the asymptotic behaviour of the singular values of operators of this latter type.
In particular,
operator, with a complex potential,
we obtain results for the Sehr~dinger
on an unbounded domain
~
in
~n
and
with homogeneous Dirichlet boundary conditions. In [23 estimates were established for the spectra of Schr~dinger operators with cemplex potential,
even with Neumann boundary conditions,
by using tesselations by cubes.
Indeed,
with complex potentials.
[2] is an extension of If] to deal
The present paper deals with the spect~]m and the
asymptotic behaviour of the singular values of much more general operators but with Dirichlet boundary conditions only. We are concerned with operators where
A
T
of 'Schr~dinger type':
is a 'nice' elliptic operator of order 2m,
potential tending to operator of order spectrum of
~
an asymptotic estimate for the singular values eigenvalues of the positive
is a differential
square root of
s. 3
of
and the max-min principle.
T, that is, the
These results are obtained
T*T.
with the eigenvalues
is discrete and we obtain
I. 3
of the self-adjoint operator
S = A + q, using the known asymptotic estimates for the [63, [8], [Ii])
R
Under suitable assumptions we prove that the
T, with Dirichlet boundary conditions,
s. 3
is a smooth, positive
at ~nfinity, and the perturbation
k < 2m.
by comparison of the
q
T = A + q + R,
I. (cf. [I], [4], [5], 3
In particular we adapt to our
operators Ky Fan's inequality and various results concerning compact operators
252
([33, [7],
[10]).
A consequence
w e can extend the estimates [8], [iI~)
compact, = n
I.
valid
to some n o n - s m o o t h
in [13 and [ ~
of the theorems for a 'smooth'
ones.
established potential
here
([43, [5],
Since the asymptotic aQ
of the domain
our results are v a l i d for n o n - c o m p a c t
~.
Moreover,
~
when
case Let
Here we recall the results of [4] and [5] that we shall need. be an u n b o u n d e d
domain
in
by
H~(~)
the usual Sobolev spaces of order
Hm(~)
and
and w i t h n o r m
If" II
~
is
term are obtained.
Facts about the s e l f - a d j o i n t
I~n
[63,
results p r o v e d
do not assume that the b o u n d a r y
results w i t h a remainder
is that
and let
If-If
m
be a positive
will stand for the
integer. m
Denote
d e f i n e d on
L 2(~) norm.
H m (n) (I)
Let
a
be an i n t e g r o d i f f e r e n t i a l
a(u,v)
we suppose that a 8
=
as~ ~
--
(2)
n ~
Let
infinity,
a
(~)
q
=
f
[
a 8(x ) D~u(x)
is hermitian, when
I~I +
continuous
181 < 2m;
be a r e a l - v a l u e d
function
bounded below by a positive
Denote by
V(~)
form d e f i n e d on
the c o m p l e t i o n
DSv(x)
by
dx;
and coercive
on
H~(~),
a 8 c C(~)
if
d e f i n e d on
~, tending to
n u m b e r and in of
Hm(~)
C0(n)
and that
I~I = 181 = m.
+
at
I Lloc(~).
with respect to the n o r m i/2
llulTvca) = {fa
CID%Cx)12÷ qCxlluCxl12) dx}
It is easy to prove that e q u i p p e d w i t h this n o r m and that the embedding form
a
q
= a + q
of
d e f i n e d on
aq(U,V)
is hermitian,
V(~)
continuous
=
in
V(~)
a(u,v)
L2(~)
V(~)
is a H i l b e r t
is compact.
by
+
and coercive
f q(x)u(x)v(x)
on
V(~).
dx
space
The i n t e g r o d i f f e r e n t i a l
253
Let us denote by operator (resp. of
in
L2(~),
A
in
a )). q
L2(~)
We deduce
1. + = 3
of
as
N(I,L,~)
L
_< t 2
~
(H~(~), L2(~),
a)
o f the embedding it
of finite multiplicity:
-<..._< 1 . _< . . . . J
j + ~; ~ and we study the a s y m p t o t i c
=
unbounded
is w h o l l y discrete;
sequence of eigenvalues
0 < tl
problem
from the c o m p a c t n e s s
that the spectrum of
consists of an infinite
where
self-adjoint,
a s s o c i a t e d w i t h the v a r i a t i o n a l
(V(~), L2(~),
V(~)
(resp. L) the positive,
i, the number of eigenvalues
behaviour,
as
I + m,
less than or equal to
I.
3 When
A
=
estimate
- ~
for
homogeneous
and
N
2 = n
the result
is o b t a i n e d when
Dirichlet
:
There exists
< D
there exists implies
la ~(x) - a H2 :
There u
that
in
V
q
H3
:
For every
and
for all
(a'
>-
kllulIv 2
+ q) (u,u):
q
I > 0,
with
N sq(x) ~,~
to
I~I = 181 = m
are such that for all (x,y) e ~ × ~
~
,
and
with
[~I = 18I = m.
such that for all domains
m c ~
of elements of
and all V(~)
V(m), we have
=
/ I
a 8 (x) D~u (x) D~u (x) + q (x) lu (x) !2) dx
c~"
~I
: = {x ~ ~ : q(x)
I~ll, and there exist
I k 11,
for
so that the
such that for all
8(y) I N ela ~(x) I k > 0
a 8
(x,~) < e0} a 8)
- q(Y)]
=
is unbounded,
and all the
(m), the set of r e s t r i c t i o n s
with measure for all
q
q > 0
e n d o w e d with the n o r m of
a' (u,u):
q
Iq(x)
is a constant
~
conditions.
~ = {x £ ~ n : diet
(still denoted by
e ~ (0,e0), Ix-yl
to
and
and in [I] an
are made:
~0 > 0 such that
can be e x t e n d e d extensions
A = - A
or N e u m a n n b o u n d a r y
The following h y p o t h e s e s HI
is well-known;
I~iI
s k'I~i/21.
< I}
I1 ~ 0
is Lebesgue m e a s u r a b l e and
k' > 0
such that
254
H4
:
We consider centres
tesselations
x
of
iRn ' (Qr)
n , r~EZ we suppose that
and
sides
q;
k >- ~I'
where
I = {r e 7zn : Q r c ~k }
r for a l l
J = {r £ zzn : Q r N ~
These - A +
hypotheses
(I +
IxI2) r
0.
(r > l)i b u t
Suppose
~
(J\I)/card
I = 0
and
for t h e S c h r S d i n g e r
- A + log
Ixl
operator
([x I > 2)
does not
satisfy
H3.
result:
that
N(k,L,~)
lim card r~-~0
cubes with
~ ~}.
are satisfied
W e h a v e the f o l l o w i n g
Theorem
by disjoint
(I),
/
(2), HI to H4 a r e s a t i s f i e d .
~(x)
T h e n as
k + ~,
(~ - q(x)) n / 2 m dx
~k where
=
(x)
IIo
(2~) -n m e a s { ~
The perturbed Let
R
R
~
a 8(x)
Se+~
< I}.
operator
be a differential
with complex
~n :
operator
of order
k N m,
defined
on
Q
and
coefficients:
=
r D ,
with
D(R)
m D(L) .
-
(3) (4)
that
the c o e f f i c i e n t s there
r
is a p o s i t i v e
(i~ I ~ 0) constant
and all their derivatives 6 E
(0,½)
such that
r ~u q-
-6
are bounded
is b o u n d e d .
T h e n w e have:
Lemma y > 0
I
Suppose and
a E
that (0,i)
(i) -
(4)
such that
hold. for all
Then there u £ D (L),
llRull ~ ~II~ull 1-aIlu[l a
on
exist two numbers
255 Proof.
From
(3) a n d
(4) w e
IIRull 2 = c 1
see
that
I
if
u £ D(L),
llD~ull 2
+
o2
fnlroU]2 dx.
0<[al-
all
~ u ~ C0(~)
llD~ull =
and
~ ~ n
all
I1~11 =
=
J (~
c3
n
c~ (
with
]c~ I
= J
_< k
J ~]Rn (1 + [~[2) j la(~)l 2 d~
<_
c 3
(I + 1~12) m la(~)12d~) j/m
Z
<- m,
]loBull2~ j/=lt~ll
(/ lon lfi(~
[2 dE ) i- [j/m)
)
2/~-~/m>
I~[--= <- c3J llul V2j/m ilult2(~-j/m) (~) Thus
by the
(5)
coerciveness
IIDeu[I 2
of
~
aq,
c 4 (aq(U,U)) j/m
[lull211-j/m)
In addition,
(6)
f
r0ul2d~ -< c s J" q=~lul 2 o= _< c6(I c7
q lul 2 d~) 2~ [I lul 2 dx) t-2~
Ilull 2<1-2~1
(a (u,u)) 2~
q
co But
C0(~)
particular
is d e n s e they
in
hold
and
for all
a
it f o l l o w s
V(D)
q
[u,u)
u
=
so
in
(5) a n d D(L),
and
(6) h o l d since
(Lu,u) ,
that
liRull 2 ~ C[I~II ° llul[ 2-~, which
is e q u i v a l e n t
to
ll~ull2/llull 2 ~ ciIILulI/ilulIl ° with
a 6
(0,I).
Lemma
1 follows
with
i - a = 0/2.
on
for
V(~). such
u,
In
256
Lemma 2
The operator
Proof.
From L e m m a
llRull
Since
e
~
R
cIILull l-a
inequality
~
(7)
T + il
By Lemma 2,
small the result follows.
=
L + il + R
S kIILull
-< II (L + il)u I ]2
We also have the following
(I) t__oo(4), the spectrum of
=
[I + R
+ k'Ilul]
(L + il)-l]
with
L 2 (~)
(T + il) -I =
T
is discrete.
< i.
(L + il).
k < I,
and thus
kllull ÷ k'l 1-1 I1 11 =
I]Lu I ]2 + ill 2 Iiu112
large enough to show that there exists
m a p of
from L e m m a 2 that
I ~ 0,
IIRull
I IR(L + il)-III
e > 0,
+ c a ~(a-l)/aI{u][.
(cf. [93) enable us to deduce
Under the h y p o t h e s e s
For real
we see that for all
c(1-a)elILu]]
is closed and m-sectorial.
i.
Proof.
Ill
llu[l a
m a y be made a r b i t r a r i l y
T = L + R
since
L-bounded w i t h L - b o u n d zero.
i and Y o u n g ' s
S t a n d a r d arguments
Theorem
is
Hence
to itself.
AS
k" •
(I + R(L + il)-l) -I Since
(L + il) -I
(L + il) -1 [I + R(L + ii)-13 -I.
k < i
(k,l)
we m a y choose
such that
exists and is a b o u n d e d
is compact,
so is
Thus the spectrum of
T
is
discrete.
III.
Estimates
for the singular values
A singular value study the asymptotic
Theorem
2
s. 3
of
T
behaviour
of
is an eigenvalue M(I,T,~)
Suppose that the h y p o t h e s e s
satisfied and that
N(I,L,~)
~ cl x
=
of
[ 1 s-
(i) t__oo(4) and
for some
x,
as
(T'T) as
1/2
We n e w
I +~.
(HI) t~o (H4) I + ~.
Then
are
257
M(I,T,~)
~ N(I,L,~) ,
M(I,T,~)
~
that is,
f
p(x)
(I-q(x)) n / 2 m dx
as
I + ~.
nI Examples xi < X2n
~% = - A + for
i = 1,2,...,n-i}
b e h a v i o u r of for
(I + ' '[xI2) k + i ~ x
M(I,TI,~I)
M(t,T2,nl)
,
with
As a consequence
n I = {(xl,x 2 ..... x n) E m
s a t i s f i e s the h y p o t h e s e s above,
is g i v e n b y T h e o r e m 2. T2 = - ~ +
of
d e f i n e d on
Theorem
The same e s t i m a t e h o l d s
(1 + [ x l 2 ) k + i ( 1
2 we c a n
improve
and the
+ IxI2) k/p,
Theorem
0 in
p ~ 2. that
h y p o t h e s i s on the s m o o t h n e s s of the p o t e n t i a l can be weakened.
the
In f a c t we
d e d u c e from T h e o r e m 2 the f o l l o w i n g
C o r o l l a r y i.
Let
q0
be a p o t e n t i a l
exists a p o s i t i v e f u n c t i o n
(H5)
L
=
A + q
(H6)
there exists
q
(2) and s u p p o s e t h a t t h e r e
such t h a t
s a t i s f i e s the h y p o t h e s i s of T h e o r e m 0;
~ 6
(0,½)
such that
N ( I , A + q,~) ~ c i x
T h e n if
satisfying
N ( I , A + q0,~) ~ N ( I , A + q,~)
as
for some
(q-q0) q~o
x
as
is bounded.
I ÷ ~, we h a v e
I ÷ ~.
P r o o f of T h e o r e m 2 H e r e we a d a p t the r e s u l t s for c o m p a c t o p e r a t o r s of [3], [7] p r i n c i p a l l y [11].
N o t i c e that
T = L + R =
(I + RL-I)L.
the j - d i m e n s i o n a l space s p a n n e d b y the e i g e n f u n c t i o n s of the first inequality'
(8)
j
eigenvalues.
and
Denote by L
E. 3
associated with
U s i n g the m a x - m i n p r i n c i p l e w e h a v e a 'Ky F a n
:
2 Sj+k+ I >-
inf
u=j+ k
ll~ull 2 IILull =
ll~ull 2 ' Ilull =
= ~2 j+l
~ inf
ll(~+~-1~ull=
u,E~
Ilnull =
IT<~+RL-II~uIE 2 inf
~uIEk
]2
ll~ul
inf
llLull 2
uiEj llol l =
n
:
258
By Lemma I, for all
v • L 2(~) ,
ll~-ivll
llClvll
I
hence
(9)
llv]i Choose
so large that for all
k0
~-a
<
/1 IL-Iv~ Y~~I
[ha
(10)
Y
that is,
sup VlEk
-< Y
\
' '~i~u(?T)''
k
k ~ k0,
½;
<
½"
For
Lu I
Ek,
(9)
implies that
~,-L. /
_< ¥<' 'L-I (Lu), ,)a
IILul[
-a
-< Y X k '
and so
I I c~+~-1) ~'u] 12
-a
>-
(I - Y ~ k
2
)
11~ull 2 Hence by (8),
2
(11)
Sj+k+l
denote by
Now
eigenfunctions
of
2 Sj+l
X2
->
F. 3 T*T
=
j+l
the j-dimensional
(12)
k
subspace spanned by the
associated with the eigenvalues
j
2 2 slt...,s ~J :
II~u11= llull 2
inf
u~. Choose
, -a
(I + O~X k )).
so large that
2 l-a y s; a < ½,
sup v±G k
< %,
Gk
=
(I + RL-I)*F k
259
Then inf v± G k and hence
II (I+RL-I)Vll Ilvl[
½,
c
sup Iivll v±G k ll(I+Rm-i)vll
(13)
_>
_ = sup {[I (I+~-i)-lwll llwll
: w = (z+~-i)v,
v ~ ~k} ~ 2.
Proceeding as before we see that
2
(14)
>
lj+2k+l _ inf
:
where of
{.LL=,~I12I1~112
inf
ll(I+=)Tull2
inf
u~*%
I1~11 =
u~5+ k
K = - RL-I(I + RL-I) -I,
G k. From
u ~ 5+k' u i ~*%}
We actually have
I1~11= '
which is defined on the orthogonal
complement
L = (I + K)T.
(9) we deduce that for
=
I1~11 2
ll~-l(i+~-l)-lwll, II(i+~-l)-lwll
w ± Fk,
]l(i+RL-1)-lwll Ilwll
y/IIL-I(I+RL-I)-Iwl lha II ( I + R L - I ) - I w ~
k , }~+~-~ 1 1 ) -bll 1
I lwli
'
and hence
(i5)
IIwll
Thus from
(16)
(14),
12j+2k+l
(12) and
>-
(15) we have
2 inf{[i (i+K)w I 12/i iW I 12 : w = Hu ± Gk} _> Sj+k+ 2 Sj+k+l 1 (i + 0 (ska))
Theorem I now follows easily from k/j
tending to
0.
(ii) and
(16) on letting
j
tend to
~
with
260
IV.
Estimates
Theorem
3
with a remainder
Suppose that h y p o t h e s e s
(17)
I. 3
where
term
=
cj r (i + O(j-t))
t < ar/(i + ar)
s.
(i) -
and
a
as
are satisfied and that
j + ~,
is d e f i n e d
.r (j-t)) c3 (i + 0
=
(4)
in L e m m a
as
I.
Then
j + ~.
3 Estimates
like
(17) are o b t a i n e d
to some n o n - s e l f - a d j o i n t
operators
non-smooth
or complex potentials.
Example
Consider
k > n/(n-2) L2
=
- A +
as
I ÷ ~,
and
T2
operators
(1 + Ixl2) k + i(1 + Ixl2) k/p, n.
with
W i t h the same n o t a t i o n
and N ( I , L 2 , ~ n )
in(k+l)/2k
= c
(I + O(I
as above,
-(k+l)/2k
=
that L e m m a
(i + k -I)
cj 2k/(n(k+l))
1 holds with
(I + O(j-i/n))
a = 1 - p
(i - k -1 - 2n-l) -I,
-i
and,
the estimate
as
)
j + ~.
if (18) holds
for the
Proof of T h e o r e m 3 The p r o o f
is the same as in [I0].
If
1
satisfies
(10), then
J
I.
=
. j .
(i + O ( j - t ) ) - 1
(i + O((j+£)-t))
3 j ~
(I + O(j-t))
(i + O(j -t
(I + £)-t)j ).
Hence Aj+~
(19)
with
so that E6]
I. ]
p > ~ +
- A +
we can extend them
and to some S c h r ~ d i n g e r
p > 2, d e f i n e d on
(i + IxI2) k
(18)
We notice
:
in [6] and [83;
(1 + ~ ) r 3
X. 3
We deduce from
2
(i + O(j-t))
=
1 + O(3Z~) + O(j-t).
(16) that
sj+£+k+l
S
12
j+2k+2£+l
-a
(I + O(Sk+£)).
s. 3
also
261 From
(11) we have -2 Sk+ £
<
(I + O(lk a)),
1£ 2
and thus 2 sj+£+k+l
12
(I + 0(1£ a) + O(lka)).
j+2k+2£+l
By (ii) again, 2 sj+£+k+ I taking
£ = k
Using
O(lka));
(I +
we obtain
12j+k+l
(20)
-> 12j+£+l
(17),
(1 + 0(I a))
(19) and
1 + O(k -ar) + O(k/j)
+ O(k/j)
< -
s2 j+2k+l
<-
12j+4k+l
(I + O(Ika))
(20) we can write
+ O(j -t) + O(j -I)
2 2 _< sj+2k+i/1 j
-<
I + O(k -ar)
+ O(j -t) + O(j-1).
-z Choosing
k/j = j
we therefore have
sj+2k+I/1 j
=
i + O(j -9),
where
v
=
min
(t, l~arlar.
References I.
D.E. Edmunds and W.D. Evans, On the distribution Schr~dinger operators,
2.
D.E. Edmunds,
to appear.
W.D. Evans and J. Fleckinger,
distribution
of eigenvalues of
On the spectrum and the
of singular values of Schr~dinger operators with a
complex potential,
to appear.
262
3.
K. Fan, Maximum properties completely
continuous
and inequalities operators,
for the eigenvalues
Proc. Nat. Acad.
of
Sci. USA 37
(1951), 760. 4.
Jo Flec$in~er,
Estimate of the number of eigenvalues
of SehrSdinger 5.
J. Fleckinger,
R~partition
SchrSdinger, 6.
type, Proc. Roy. Soc. Edinburgh
Comptes rendus Acad.
globalement
Ecole Polytechnique, 7.
8.
I.C. Gohberg and M.G. Krein,
L. Hormander,
i0.
To Kato, Perturbation A.G. Ramm,
Seminaire EDP,
Amer. Math. Soc.
in
n
18 (1969).
of the eigenvalues
Ark. F~r Math.
of some non-self-adjoint
in Spectral
of
17 (1979)
296.
(Springer-Verlag, operators
theory of differential
1966) and
operators,
1981).
M. Reed and B° Simon, Modern methods of mathematical 1978).
(1981), 359.
to the theory of linear non
theory for linear operators
Spectral properties
de type
precise du spectre
dans n ,
distribution
Math. Studies 55 (North Holland
Press,
(1981), 355.
expos~ 2.
Transl.
operators
some applications,
II.
asymptotique
Introduction
On the asymptotic
pseudodifferential 9.
Sci. Paris 292
elliptiques
1980-81,
selfadjoint operators,
89A
des valeurs propres d'op~rateurs
B. Helffer and D. Robert, Comportement d'op~rateurs
for an operator
physics
(Academic
Optimal Control of Systems Governed by Elliptic Operator of Infinite
Order
I.M. Gali
Abstract In the present paper, using the theory of J.L. Lions [6,7] we find the set of inequalities defining an optimal control of systems governed by elliptic operator of infinite order.
The questions treated in this paper are related
to a previous result by I.M. Gali; et al.
[5], but in
different direction, by taking the case of operators of infinite order with finite dimension. §i. Necessary Spaces and Differential Operators of Infinite Order Ju.A. Dubinskii
[1,2,3,4]
initiated the investigation
of the theory of problems of infinite order linear differential equations. 2
Let W~{a
,2} = {u(x)~C~(Rn) :
Z
lal=0
a [ID~ull2
< ~}
be a Sobolev space of infinite order of periodic function defined on all of R n.
Here e = (el .... 'an) is a multi-index
Key Words: Optimal Control, Elliptic Operator of Infinite Order. *Current Address
: Department of Mathematics, Qatar University, Doha, P.O. Box 2713.
264
eI for d i f f e r e n t i a t i o n ,
sn
Isl = s I + ... + Sn, D S H ~ l e l / ~ x l
a s _> 0 is a n u m e r i c a l
sequence
norm in the space L2(Rn),
and
! I . 11 2
(all functions
...~x n,
is the canonical are assumed
to be
real valued). The imbedding infinite
problem
for nontrivial
order are i n v e s t i g a t e d
tions e x p r e s s e d conditions
in "algebraic"
in [3]. terms,
Sobolev
spaces of
Imbedding
condi-
in p a r t i c u l a r
in terms of the c h a r a c t e r i s t i c
functions
include of these
spaces. In this case topological
W ~
{as,2}
inclusion,
dual space with respect following
is everywhere
and W-~{as,2} to L2(Rn)
denotes
in L 2 ( R n) with
the t o p o l o g i c a l
and then we have the
inclusion: W~{as, 2} ~
In this article infinite
dense
order
L2 (Rn) ~
W-~{ae,2}
our model A is the elliptic
operator
of
[4] ,
CO
Au =
7
(-i) Isl a s D2SU
; AeT.(Wo{as,2}, ~
W O ~ {a s ,2})
Isl=0 for w h i c h has a s e l f - a d j o i n t set of all functions boundary
F of R n.
closure where Wo{as,2}
of W~{as,2}
which vanish
is the
on the
(i)
265
§2.
Our Main Results To set our problem,
two lemmas w h i c h enable
we need to prove us to formulate
the following
our main theorem.
We introduce ~ (u,~)
=
[~I=IZ ((-i) I~I ae D 2e uCx) ,v(x))L2(R n)
+ (q(x) u(x) ,v(x))L2(R n) where q(x) q(x)
is a real valued
> V , 1 > 9 > 0.
function
(2)
from L 2 ( R n) such that
Then,
Oo
(u, v)
=
E lal =I
I Rn
a s (D~u) (x) (D~V) (x) dx + I q (x) u (x) v (x) dx Rn
(3) Lemma
1 Consider
then
~(u,v)
the continuous is coercive,
form
~{a,2},
(3) on W o
that is
[lull
Cu,u)
bilinear
=
~
ueWo{a
,2}
(4)
Proof It is well known for the coerciveness In fact,
that the e l l i p t i c i t y of
~(u,v)
on W~{a
,2}
of A is s u f f i c i e n t (see
[7]).
266 cO
. (u,u)
E
=
I
lal=l
a (Dau) (x) (Do.u) (x)dx + I
Rn
O.
q (X) U (X) U (X) dx
Rn
O0
>
F.
a ((D°'u) (x) , (DO'u) (X))L2(R n)
-[o.1=1
+ v (u(x) ,u(x) )
O.
co
2
z
2
a~l lD~ul [L2(Rn )
91
+
lull
[o.I--1 oo
2
Z
[o.[=1
a
O.
I IDo.ull
L 2 (R n)
oo
L2 (Rn)
2
z {o.I=l ao. I lDo.u[1L2(Rn )
+ ~
2
- v
L 2 (R n)
-
2
Z ac~ l lDo.u[ IL2(Rn ) + v] [u[ IL2(Rn ) Io.1--1
Wo{ao.,2}
a~I IDo.ul IL2(Rn )
Io.I=i
Then, 2 -
which
proves
Lemma
2 Assume
W~{ao.,2},
the coerciveness
that
there exists
we have
t2
a unique
Au = f , (i).
form.
For a given u in Wf{a
operator
equation
feW °-~ {as, 2 }
f in
,2} such that
= (f,v)
the following
where A has the form
}
of the bilinear
(4) is satisfied.
(u,v) Then,
W ~o {ao.
267
Proof If L is a continuous
linear form on W ~0 {as , 2 }
then
uoo
L(~)
= (f,~)L2(Rn)
From the coerciveness
;
f£Wo{a
condition,
a unique element us W ~o{a
,2}
,2}
for given f, there exists
such that co
~(u,~) Hence,
from
=
(f,~)
(2), equation
for all ~6Wo{a~,2} (5) is e q u i v a l e n t
(5) to an operator
equation Au = f For this purpose,
We may now formulate our theorem.
we have
L2(Rn)
the s t a t e m e n t of the control problem.
The space
b e i n g the space of control,
A system w h i c h is
is given.
g o v e r n e d by the model A is given. For a control u the state of the s y s t e m y(u)
is given
by the solution of Ay(u)
= f + u
We are also given an o b s e r v a t i o n
equation
Z (u) = y(u) Finally,
we are given N e L ( L 2(R n) ,L 2(R n)) .N i s H e r m i t i a n
p o s i t i v e definite;
(Nu,U) L2(Rn)
i.e., 2 _> YI lul IL2(Rn)
Y > 0
(6)
268
With every control u~U we a s s o c i a t e
J(u) where
2 I ly(u) - Zdl IL2(Rn ) +
=
z d is a given e l e m e n t Let Uad
the cost:
(NU, u) n 2 ( R n )
in L2(Rn).
be a c l o s e d c o n v e x set in L2(Rn),
it will be the set of a d m i s s i b l e The control p r o b l e m
(7)
in the sequel,
controls.
is to
minimize
the q u a d r a t i c
f u n c t i o n a l J(u). Theorem Assume that by
(4) holds.
(7), a n e c e s s a r y
an o p t i m a l
control
The cost function b e i n g given
and s u f f i c i e n t
condition
for u to be
is that the following e q u a t i o n and inequa-
lities be satisfied:
UeUad,
(P(u) + Nu,
where P(u)
= F + u
AP(u)
= y(u)
condition
If N = O and Uad (iO) admit
(8) - zd
If N s a t i s f i e s
(6), then the o p t i m a l
(i0) the p o s i t i v e
control
is unique.
is b o u n d e d then the s y s t e m of e q u a t i o n s at least one solution.
Proof Let J(u)
(9)
~-U) L2(Rn ) >_ O for every V C U a d
is the adjoint state.
definiteness
(9) and
Ay(u)
be w r i t t e n
in the form:
(8),
269
J(u)
= I lY(U)
- y(O)
+ y(O)
2 - Zdl In2(Rn ) + (Nu, U) L2(R n)
If we set (u,v)
=
L(v)
(y(u)
- y(O),
= (z d - y(O)
~he form ~(u,v)
y(v)
, y(v)
is continuous
- y(O))L2(Rn)+
(Nu,V) L2(Rn)
- y(O))L2(Rn )
bilinear
form on L2(R n) and
we
Lave J(v) n d from
f Lions
-2L(v)
2 + i z d _ Y(O) I IL2(Rn )
(6) we have
~(v,v) herefore,
= ~(v,v)
_> vllvl l~2(Rn )
for every
we have reduced
the p r o b l em
[6] can be applied,
i.e.,
vEL2(Rn) to a form where
there
exists
theorems
a unique
element
EUad such that J(u) nd this element
ince A is a canonical ,2}
by (ii)
for all VeUad
isomorphism
from W:{a
we may write y(u)
~ence
inf J(v) VEUad
is c h a r a c t e r i z e d
J' (u) (v-u) _> 0
-'{a
=
= A-l(f
y' (u).~ = A-I%
+ u)
,2} onto
270 and hence y' (u)(v-u) Therefore, (y(u)
= A-l(v-u)
(after division
- zdty(v)
= y(v)
by 2),
-y(u))L2(Rn
- y(u)
(ii)
is e q u i v a l e n t
) + (Nu,v-u)L2(Rn)
to (12)
_> O 00
For a control
veL2(Rn)
the adjoint
state P (v) ~Wo{ae, 2 } is
defined by AP(v) Let us now t r a n s f o r m
(12);
= y(v) from
(13)
- zd
(13) we deduce
(AP(u) ,y (v) -y (u) ) L2 (Rn) = (y(u)-z d, y (v) -y (u) ) L2 (Rn) =
(P(u) ,Ay(v)-Ay(U))L2(Rn)
and hence (P(u)
(12)
is e q u i v a l e n t
+ Nu, v-u) L2(Rn)
which completes
=
(P(u) ,v-u) L2(Rn )
to
>_ 0
for all VEUad
the proof.
Acknowledgement The author w o u l d Professor critically
J.L.
Lions
reading
for valuable
like to express for suggesting
the m a n u s c r i p t ,
discussion.
his gratitude
the p r o b l e m
to
and
also to P r o f e s s o r
W.N.
Everitt
271
References
1]
Ju.A.
Dubinskii•
behavior
of solutions
unbounded Math.
Sobolev
increase
USSR Sb.
2]
27
spaces
of some boundary
Math.
(1975)
3]
29
of Sobolev
of infinite
(1978)
1271-1274.
~]
I.M.
equation•
space and atours.
theorems
order. Soviet Math.
, About one m e t h o d rential
spaces of
(1976).
• Some imbedding spaces
value problem with
143-162.
order for a full E u c l i d e a n
USSR Sb.
order and the
of the order of the equation.
• Non triviality infinite
of infinite
Dokl.
Gali and H.A.
Dokl.
governed by a self-adjoint
Optimal
elliptic
infinite number of variables. Systems
Vol.
19
for solving partial
Akad Nauk SSSR 258
Ei-Saify,
Functional-Differential
for Sobolev
(1981)
control
diffe-
780-784.
of systems
operator with an
International
Conference
and Related Topics
II,
3-10 May 1981 Poland.
;]
J.L.
Lions•
differential J.L.
Optimal
control
equations•
Springer-Verlag
Lions and E. Magenes,
problem
of systems
and applications•
governed by partial Band 170
Non-homogeneous Springer-Verlag,
(1971).
boundary
value
Vol.
(1972).
III
COMPARISON PRINCIPLES FOR SOME FOURTH ORDER ELLIPTIC PROBLEMS
Vinod B Goyal and Philip W Schaefer
i.
INTRODUCTION
Many authors have contributed to the development of Sturmian type comparison and o s c i l l a t i o n theorems f o r e l l i p t i c equations and systems (see [ I ] ,
partial differential
[ 7 ] , [8] and references cited t h e r e i n ) .
These theorems are obtained as a consequence of a v a r i a t i o n a l p r i n c i p l e for a suitable quadratic f u n c t i o n a l . global-type comparison theorem (u < v
Our concern here is with the in
~) which one deduces immedi-
i a t e l y from a maximum p r i n c i p l e in the case of second order e l l i p t i c p a r t i a l d i f f e r e n t i a l equations (see [ 2 ] , [ 5 ] ) .
We extend t h i s l a t t e r
kind of comparison p r i n c i p l e to fourth order e l l i p t i c
problems (where
there is no maximum p r i n c i p l e on the solution) by introducing a s u i t a b l e functional of the solution to the fourth order problem and obtaining a maximum p r i n c i p l e f o r t h i s functional.
This approach has been used
recently to deduce other r e s u l t s of i n t e r e s t f o r higher order e l l i p t i c problems such as uniqueness of the solution [ 6 ] , nonexistence of a solution [ 3 ] , and integral and pointwise estimates [4].
The comparison
theorems we obtain are useful in approximating the solution to boundary value problems since they allow for the determination of a bound on the error in the approximation ( e . g . , see [ 5 ] ) . In section 2 we prove a simple lemma for the class of d i f f e r e n t i a l inequalities
A(pAU) - (C + pr~Au + cru > 0 ,
where
c
is a nonnegative constant,
p
(I.I)
and
r
are s u f f i c i e n t l y smooth
273
functions, that
and
u ~0
A
in a bounded domain
parison result. (I.I)
is t h e n - d i m e n s i o n a l L a p l a c i a n , whereby we conclude
dictates
Although a certain
p
~ .
and
This leads d i r e c t l y
r
to the com-
c o u l d be c o n s t a n t s , t h e form o f
relationship
between the c o n s t a n t c o e f f i c i e n t s
Thus we m o d i f y the argument in o r d e r to o b t a i n s i m i l a r
results
for
A2u - aAu + bu ~ 0 ,
where
a
briefly
and
b
(1.2)
are c o n s t a n t s such t h a t
i n d i c a t e some e x t e n s i o n s o f t h i s
0 ~ b ~ I ~ a2 .
We then
approach and theorem to more
general o p e r a t o r s and h i g h e r o r d e r o p e r a t o r s in s e c t i o n 3.
2.
RESULTS
In the f o l l o w i n g c a l c u l a t i o n s partial
differentiation,
i.e.,
we use the comma n o t a t i o n to denote
u,i
convention for repeated indices,
= ~u/~x i
i e. •
'
, and adopt the summation
u ii ,
= Au = ~n
~2u/~x i2 .
i=l
Our b a s i c lemma may be f o r m u l a t e d as LEMMA:
Let
c
be a n o n n e g a t i v e c o n s t a n t and
f u n c t i o n s d e f i n e d in a bounded domain p ( x ) > P0 > 0 q
and
r E C(~) .
~ ~_ Rn
p
and
such t h a t
r
be
P E C2(~)
If there exists a positive
,
function
such t h a t
q -1 q i
'
q -1Aq - r , are bounded on e v e r y c l o s e d b a l l
in
fi ,
(2.1)
274
Aq - rq S 0 ,
then any f u n c t i o n
in
(2.2)
~ ,
u C C4(~) n C2(~)
which satisfies
A(pAU) - (c + p r ) A u + c r u # 0 ,
u > 0
i s such t h a t
,
u > 0
Proof:
Au < 0
in
For any
u
,
(2.3)
in
on
(2.4)
9~ ,
E . satisfying
(2.3),
let
w = pAu - cu q
where
q
i s as y e t
(2.m)
unspecified,
and compute
(pAU - C U ) , i W •
,1
=
w ..
q,i(pAU
q
q
= A(pAu - Cu)
,ll
_
2q,i(pAu
q
q
follows
- CU),i
2
2q,iq,i(pAU
(Aq)(pAu-z_~_ 2 q
It
- CU) 2
q
that
AW +
2qi
w,i q
+
(Aq)W_ q
=
AZP_~_: c u l q
3
- CU)
275
and by ( 2 . 3 )
that
&w + -2 q- ' i
w'i
+ (Aq - r q ) w > 0 . q -
q
Thus i f
q
w ~ 0
satisfies
in
~
and ( 2 . 2 )
and
u
by t h e H o p f maximum p r i n c i p l e
minimum p r i n c i p l e , From t h i s of
(2.1)
we c o n c l u d e
that
t h e o r e m one e a s i l y
satisfies [5].
u > 0
in
(2.4),
By ( 2 . 5 )
we have
and t h e
~ .
deduces a c o m p a r i s o n
result
for
operators
the form
Lu m &(pAu)
- (c + p r ) A u + c r u
.
Thus we have
THEOREM I : positive u,
Let
c,
function
p, and q
v E C4(~) fl C2(~)
r
satisfying
u > v
in
We o b s e r v e i)
u > v ,
that
(2.3)
includes
- c)u
ii)
(A - c ) ( A
- d)u > 0
iii)
(A - c ) ( A &(pAu)
and ( 2 . 2 )
exists.
If
Au <
Av :
on ~
,
~.
(A - c ) ( A
iv)
(2.1)
a
satisfy
Lu > Lv , i n ~ ;
then
be as i n t h e lemma and assume t h a t
p =l,
> 0
+ c)u > 0 c2 - - p- u -> 0
the special
p=l ,
cases r=
c , q = l
, r = d > O , d # c , q = l
p = 1 , r = -c r -
,
C
P
,
, Aq + cq < 0 ,
&q+C
~-q <_ 0 ,
276
where t h e e x i s t e n c e o f a f u n c t i o n The need f o r
some c o n d i t i o n s
the eigenvalue situation
on
q q
is trivial relative
to
,
in ~
u = Au = 0
,
on
which changes s i g n i n t h i s
~ .
q
which s a t i s f i e s
case o f
used i n t h e p r o o f o f t h e lemma.
and
b
u = sin x sin y
(2.3),
and ( 2 . 2 ) .
we m o d i f y t h e f u n c t i o n a l u
,
,
in ~
determine conditions
under which a comparison r e s u l t
(2.1)
t h e non-
for a constant coefficient
Suppose t h a t
Mu z A2u - aAu + bu > 0
We s h a l l
solution
Hence here the theorem i m p l i e s
function
o p e r a t o r which i s n o t a s p e c i a l
.
by
3~ .
In o r d e r t o o b t a i n a comparison r e s u l t
~
is exemplified
= ( 0 , 2~) × ( 0 , 2~)
This boundary v a l u e p r o b l e m has t h e n o n t r i v i a l
and ( 2 . 4 ) on
~
and i i ) .
(case i i i ) )
A2u - 4u = 0
existence of a positive
in cases i )
satisfies
(2.6)
on t h e c o n s t a n t s
is possible.
Let
w = Au - ku ,
where
k
i s an u n d e t e r m i n e d c o n s t a n t .
(2.7)
Then by ( 2 . 6 )
277
&w = A2u - kAu
h (a - k ) ( w + ku)
If
we choose
k = (a + a ~
z~w -
Now i f well,
w < 0
we c o n c l u d e
- 4b)/2
w
on
that
takes ~
u > 0
in
~ on
0 < k = ½ (a ,+ a ~ - _
we see t h a t
the constants
a > 0
0 < b < a2/4
THEOREM 2:
If
u,
Mu > Mv ,
for
constants
M , then
u > v
a .
maximum on
;
~ .
k > 0 , as
Consequently,
since
applied
k L 0 , to
(2.7).
k , namely,
4b)
and
< a
b
v E C4(2) A C2(~)
in e
2 ,
~Q , and i f
by t h e minimum p r i n c i p l e
Formally,
a = 0 , b = 0 in
its
and hence i n
From t h e r e q u i r e m e n t s
and
, then
(a - k)w > 0 .
k < a , then then
- bu .
u > v ,
or
can b o t h be z e r o o r such t h a t we have satisfy
Au <
Av ,
on 22,
a > 0 , 0 < b < a2/4
in
the operator
278
3.
REMARKS
We note t h a t one c o u l d r e p l a c e t h e L a p l a c i a n i n e i t h e r (2.6)
by a g e n e r a l
Au ~ ( a i j ( x ) y > 0
uniformly
u i),j
and a l l Finally,
where
x E ~
elliptic aij
and r e a l
(2.3)
or
o p e r a t o r o f t h e form
= aji
and
n-tuples
a i j ( x ) ~ i ~ j ~ ¥I~I 2 , f o r ~ .
one can e x t e n d t h e approach used here t o c e r t a i n
higher
o r d e r problems such as
A3v - aA2v - b A v
+ abv > 0
,
v < 0 , Av > 0 , A2v < 0 ,
where
a
inequality
and
b
in
on 3£ ,
are n o n n e g a t i v e c o n s t a n t s .
may be r e w r i t t e n
fi ,
Since t h e d i f f e r e n t i a l
as
~2(Av - av) - b(Av - av) ~ 0 ,
one can a p p l y s p e c i a l to conclude that This i n e q u a l i t y order problem.
case i i i )
Av - av > 0
with in
~
u = Av - av , p = 1 , and
c = /b-
and hence t h a t
~ .
v < 0
then l e a d s one t o a comparison p r i n c i p l e
in
for this
higher
279 REFERENCES
[i]
G.J. Etgen and R.T. Lewis, The Oscillation of E l l i p t i c Systems, Math. Nachr. Bd 94, 43-50 (1980).
[2]
D.Gilbarg and N,S. Trudinger, E l l i p t i c Partial Differential of Second Order, Springer-Verlag, Berlin (1977).
[3]
V.B. Goyal and P.W. Schaefer, On a Subharmonic Functional in Fourth Order Nonlinear E l l i p t i c Problems, J. Math. Anal. Appl. 83, 20-25 (1981).
[4]
. . . . Payne, Some Remarks on Maximum Principles, J. Analyse Math. 30, 421-433 (1976).
[5]
M.H. Protter and H.F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall Inc., Englewood C l i f f s (1967).
[6]
P.W. Schaefer, Uniqueness in Some Higher Order E l l i p t i c Boundary Value Problems, ZAMP, 28, 693-697 (1978).
[7]
C.A. Swanson, A Generalization of Sturm's Comparison Theorem, J. Math. Anal. Appl. 15, 512-519 (1966).
[8]
C.A. Swanson, Comparison and Oscillation Theory of Linear Differentia Equations, Mathematics in Science and Engineering vol. 48, Academic Press Inc., New York (1968).
Equations
WEAK SOLUTIONS
OF INTEGRODIFFER~NTIAL
EQUATIONS AND A P P L I C A T I O N S
R. C. Grimmer
i.
W, Schappacher
Introduction In this paper we consider
Volterra
integrodifferential
(VE)
x' (t)
- s)x(s)ds
x(O)
in a Banach space X, is the generator B(t)
+ f(t),
It is assumed throughout
With the aid of various
a.e.
t > 0.
to this
additional
initial value problem. to initial
assumptions
Also
it
of a unique
data one can define
there
is
a resolvent
of
t + .r B(t - s)x(s)ds, O x(0)
and that
concerning
If in addition
R(t) on X by R(t)x 0 is the solution
x' (t) : Ax(t)
that A
or f E LLI((0,~),X).
A and B(t) one can show the existence
continuity with respect operator
this paper
of a C O semigroup which has domain D(A)
shall be assumed that f E C([0,~),X)
solution
t >_ O,
xo
--
is closed with domain at least DCA),
the operators
for the
equation
t + or B ( t 0
: Ax(t)
the initial value problem
t ~ 0,
: x0,
(cf. [3], [6], [7], [8], [9], [10], [12], [13], [14], [17], [18],
281
[23].)
This resolvent
of parameters
operator
formula
(VP)
x(t)
t = R ( t ) x 0 * .r R ( t 0
which gives the solution of When X = ~n, existence
is then used to obtain a variation
Grossman
- s)f(s)ds
(VE) when it exists. and Miller,
of a r e s o l v e n t R(t,s)
[15],
[16], showed the
for the non convolution
type
equation t
x' (t) : A(t)x(t)
+ .r B(t,s)x(s)ds 0 x(O)
under quite mild assumptions variation tative
of parameters
information
case
When B(t)
x0
on A(t)
formula
and B(t,s).
then enables
about the solutions
about the resolvent. convolution
:
of
When X is infinite
is examined
z 0 so that
+ f(t)
one to obtain quali-
(VE) if enough dimensional
in [5],[12],
(VE) reduces
The associated
and
is known
the non-
[13].
to the differential
equation (I.I)
x'(t)
= Ax(t) x(0)
R(t) will be the semigroup R(t + s) # R(t)R(s) family
{R(t)},
semigroup
=
+ f(t) x 0,
generated by A.
so that R(t)
In general,
is not a semigroup
however,
although
t > 0, will have many of the same properties
the as the
generated by A.
Differential extensively
equations
and a great deal
in a Banach space have been studied is known
(cf.
[19],
[21],
[25]).
One
282
of the most useful
concepts
(1.2)
concerning
x'(t)
C O semigroups. if A generates R(X,A)
relating
That is,
= (XI - A) -I satisfies
[IRn(X,A)II j M/(ReX [21].
Theorem,
tion of when
- w)n,
(VE)
indicates
Laplace
the extension difficult,
In particular,
the nth derivative as a power of
transform).
218].
is the already noted
characteriza-
if M = 1 this - ~).
of the Hille-Yosida-Phillips
This makes
A further
of
(hl - A - B(X)) -I
(Xl - A - B(X)) -I
of the Hille-Yosida-Phillips
[8, p.
for some w and M,
as it gives an algebraic
the extension
which cannot be expressed
if and only
known as the Hille-Yosida-
to IIR(X,A)II ~ i/(ReX
involves
(1.2) with
the condition
(DE) is wellposed.
condition may be reduced
Theorem to
of
if and only if the resolvent
ReX > w, n = 1,2,...
is useful
Unfortunately,
xo
the wellposedness
This last characterization,
Phillips
R(t)
=
(DE) is uniformly w e l l p o s e d
a C O semigroup
equation
= Ax(t)
x(O)
is the circle of ideas
the differential
(here
the implementation
of
Theorem often very
difficulty
in dealing with
fact that R(t + s) # R(t)R(s).
(VE)
That is,
is not a semigroup. It thus seems necessary
of wellposedness useful
of
(I.I) or
characterization
acterization
denote
(1.2) which
of wellposedness
of wellposedness
In particular,
to seek a different
of
the adjoint
X and its dual space X* by < , >.
(i) or
characterization
can be extended of (V£). (2) was
to a
Such a char-
given by Ball
[2].
of A by A * and the pairing betweel A function
x E C([0,T],X)
is
283
a weak solution of function
(x(t),v)
(i) if and only if for every v E D(A*) is absolutely continuous
~t(x(t),v) for almost all t E [0,T]. that A generates
on
the
[O,T] and
= (xCt),A*v) + (£(t),v) With f E LI((0,T),X),
a C O semigroup
Ball
[2] p r o v e d
if and only if for each x 0 E X
there exists a unique weak solution x(t) of
(I) with x(0) = x 0.
( I n H i l b e r t space the "only if" part of this theorem was obtained in B a l a k r i s h n a n extend to
[I].)
(VE).
wellposedness
It is this c h a r a c t e r i z a t i o n which we will
We shall then apply this c h a r a c t e r i z a t i o n of
of
(VE) to a number of problems
involving resolvent
operators.
2.
Preliminaries
If A is the generator of a C O semigroup on X we know that D(A)
is dense in X, A is closed,
some B-
and
(A - wl) -I exists
if w > ~ for
Hence, we may consider the equation t
y' [t)
where y(t)
= (A - ~ I ) y ( t )
= e-~tx(t)
considering questions generality,
+ ,~ e - W ( t - s ) B ( t
- s)y(s)ds
and g(t) = e-~tf(t) of wellposedness.
rather than
(VE) when
Thus, without
loss of
we may assume that A has a bounded
implies also that A s also has a bounded Throughout
+ g(t)
inverse A
inverse and
-i
~A*) -I = (A-I) *
this paper we shall assume the hypotheses
(H1)- (H3). (HI) A is the generator of a C O semigroup on X and 0 E p~A),
the resolvent set of A.
This
284
As A and A * are closed we denote by Y the Banach space formed from D(A) with the graph norm and by Z the Banach space formed by D(A*) with the graph norm. (H2)
D(B(t)) D D(A)
and D(B*(t))
~ D(A ~) a.e.
and B*(t)v strongly measurable Further,
t h 0 with BCt)x
for x E D(A)
and v E D(A*).
there exists b E LLI(0, ~) so that
[IB(t)Yll ! b(t)IIyIIy and {IB*(t)zll ! b(t)IIzIIz for y E Y and z E Z. (H3)
There exists
a set D ~ D(A .2) N D(A*B*(t))
N D(B*(t)B*(s)) is weak*dense
a.e.
p. 43]),
satisfied (H3)
in t,s > 0 with the property
= a(t)A with a E LLI(0, ~)
and,
as D(A *) is weak*dense
is satisfied by D = D(A*2).
somewhat more complicated
however. ,
Dirichlet boundary differential
that A*D
in X*.
We remark that if B(t) matically
N D(B*(t)A *)
conditions,
operator.
du E X, u(0) (u ~ X : ~-~
B(t)
Say B(t) = 0].
(H2)
is auto-
in X*
([19,
Frequently
D may be d2 If X = L2(0,1), A = with dx z could be a first order
= a(t)~x with D(B(t))
Then
(H2) and
=
(H3) are easily
seen
to be satisfied with D = C O . Two lemmas will be critical Lemma
2.1.
([2],
[II, p. 127])
Let x,z 6 X satisfy
for all v E D(A*).
The operator A-IB(t) t > 0 such that A < l ~ t ) x
for each x E X.
in our work.
Then x ED(A)
and z = Ax.
has a bounded extension, is locally
Bochner
A-IB(t),
integrable
285
Proof. on X* a.e.
First note t h 0.
Also,
X** we see that A-IB(t) Thus
that B*(t)(A*) -I is a bounded operator considering
Restricting
desired extension As D(A) x E D(A)
Also,
it follows
x
of A-IB(t)
to X**, the
on X.
is measurable
is measurable
= lIB* ( t ) (A~)-lll and,
on
to each other
to X we obtain
as an operator
in X and A-IB(t)x
that ~
ll[B*(t)(A*)-l]ell
extension
this operator
of A-IB(t)
is dense
as an operator
and B*(t)(A*) -I are adjoint
[B*(t)(A*)-I] * is a bounded
[20, p. 167].
A-IB(t)
for each
for each x E X.
f o r x E X*,
lIB*(t) (A*)-lx[I _< b(t){llxll + ll(A*]-Ixll} ! Mb(t)llx!I"
Thus,
llA-1B(t)xll i s l o c a l l y
follows
that A ~ I B ~ ) x
integrable
is locally
f o r each x E X and i t
Bochner
integrable
for each
x EX. 3.
Weak Solutions
and Resolvents
A number of possible been given.
definitions
We are motivated
of a differential
equation
of a solution
here by the concept
in semigroup
of
(VE) have
of a solution
theory when
f C C([0,~),X). Definition
3.1.
A solution of
x E C([0,~),Y) satisfies
(VE) on
N CI([0,=),X)
(VE) on
with x(0)
= x 0 E D(A) which
[0,~).
In a similar manner our concept modelled
[0,~) is a function
after a C O semigroup.
of resolvent
operator
is
286
Definition
3.2.
operator the
A resolvent
valued
following
operator
function
R(t)
for
(VE) is a b o u n d e d
E B(X),
0 < t < ~, h a v i n g
properties:
(a)
R(t)
is s t r o n g l y
(b)
R(t)
~ B(Y)
continuous
and R(t)
for t > 0 w i t h R(0)
is s t r o n g l y
continuous,
= I.
t > 0,
on Y.
(c)
For each x E Y, R ( t ) x
is c o n t i n u o u s l y
differentiable,
t > 0, w i t h R' (t)x = A R ( t ) x
t + .~ B(t 0
- u ) R ( u ) x du
and R' (t)x = R ( t ) a x
F r o m the p r e v i o u s resolvent
operator
of
In a d d i t i o n ,
(VE).
(cf.
[13]
Theorem
or
t + .F R(t 0
definitions
- u ) B ( u ) x du .
it is clear
that
if R(t)
is a
and x o E Y and f ~ 0 then R ( t ) x 0 is a s o l u t i o n one can e a s i l y p r o v e
the
following
result
[14]).
3.3.
Suppose
R(t)
x 0 E Y and f E C ( [ 0 , ~ ) , X ) .
is a r e s o l v e n t If x(t)
operator
is a s o l u t i o n
for of
(VE),
(VE) then it
is g i ven by
(VPR)
t = R ( t ) x 0 + .~ R(t 0
x(t)
Conversely,
if x 0 E Y and
is a s o l u t i o n
of
It is c l e a r changed
f E C([0,~),Y)
then x(t)
given by
(VPR)
(VE). that the c o n c e p t
in D e f i n i t i o n
shall be i n t e r e s t e d solution.
- s )f(s)ds.
of s o l u t i o n
3.1 to a c c o m o d a t e
in a more
c o u l d be s l i g h t l y
f E LLI((0,~),X)
general weakening
but we
of the c o n c e p t
of
287
Definition (VE)
3.4.
A function
if and o n l y
absolutely
x E C([0,~),X)
if for e v e r y
continuous
on
v E D(Ae),
[0,~)
= (x(t),
almost
t > 0.
all
now i n
3.5.
equation
Let
(VE) h a s
given
by
solution.
yields
to
+ .F ( x ( s ) , B e ( t 0
state
f E LLI((0,~),X). a resolvent
Assume
for x E D(A),
x n E D(A),
is
a unique
s ) v } ds + ( f ( t ) , v )
our main theorem. The i n t e g r o d i f f e r e n t i a l
operator
R(t)
if
and only
if
for
weak solution.
The w e a k s o l u t i o n
has
operator
(VPR).
Proof.
weak
Aev)
a position
e a c h x 0 E X (VE) h a s is
(x(t),v}
that
R(t)x
(VE)
a resolvent
is a s o l u t i o n
If x 0 E X,
n ~ i, w i t h
of
as D(A)
x n + x 0.
(VE)
if f ~ 0 and
is d e n s e
For
each
R(t). thus
in X we m a y
(R(t)Xn,V)
= (Xn,V) t s
- u)v)duds
.
0 0 from Dominated
Convergence
that
t
(R(t)x0,v
so that
~ = ( x 0 , v > + .~ ( R ( s ) x 0 , A * v ) d s 0 t s + J' .F ( a ( U ) X o , B * ( s - u)v>du as 0 0
R ( t ) x 0 is a w e a k
solution
of
a
n > i, an i n t e g r a t i o n
t + .~ ( R ( S ) X n , A * v ) d s
+ .~ .~ ( R ( U ) X n , B e ( s
Then
choose
for v E D(A*)
It f o l l o w s
of
t
~t-f( x ( t ) ' v )
We a r e Theorem
solution
and
d
for
is a w e a k
(VE) w h e n
f ~ 0 for
all
288
x 0 E X.
Now s u p p o s e
As R(t)x
is a w e a k
d t ~[y .f < R ( t 0
-
f E C([0,~),X)
solution
s)f(s),v}ds
and x(t)
is given by
(VPR).
for e v e r y x E X, t + f
=
-
s)f(s),A~v}ds
t t-s + .~ .~ (R(u)f(s),B~(t - s - u)v}duds, 0 0 It now f o l l o w s
from Fubini's
(x(t),v)
=
d
Theorem
(a(t)Xo,V)
= (R(t)x0,A~v) + (f(t),v) t s + .~ (.~ R ( s 0 0 so tha't x(t)
is a w e a k
f E LLI((0,~),X), bounded
xCt)
;
R(t)x
= R(t)x
that x n ÷ x i n
(
R(t
-
s)f(s)ds,v)
t + ,~ ( R ( s ) x 0 , B ~ ( t 0 t + (.~ R ( t - s ) f ( s ) d s , A ~ v > 0 - u)f(u)du,
Be(t
-
[0,T].
On
If
[0,T]
fn ÷ f in L I ( ( 0 , T ) , X ) t 0 + .~ R ( t 0 t 0 + .f R ( t 0
C([0,T]
,X) .
-
s)v)ds
s)v)ds
for each x 0 E X.
let T > 0 and t E
Xn(t)
it follows
+
solution
so if fn E C ( [ 0 , T ] , X )
that
iIR(u)II is and
S)fn(S)ds
- s)f(s)ds
As
t t < X n ( t ) , v ~ ~ ( x 0 , v } + .~ ( X n ( S ) , A ~ v ~ d s + .~ (fn(S),V} ds 0 0 t s + .~ .f < X n ( U ) , B * ( s - u)v>duds 0 0 it f o llows
that x(t)
is a w e a k
solution
of
(VE).
a.e
289 To show uniqueness, solutions
suppose
of (VE) with Xl(0)
that Xl(t)
= x2(0).
and x2(t)
We then see,
are weak
for
v E D(A*) ,
<xl(t
)
t > = <.~ ( X l ( S ) 0
- x2(t),v
x 2 (s)ds ,A'v)
t
t
+ .F < X l ( S ) 0 Define
t = ,~(Xl(S ) 0
z(t)
(z'(t),v)
On D(A*),
x2(s)
x2(s),.~B*(u s
- s)vdu)
ds and integrate
by parts
+ ,F ( z ( s ) ' B ~ ( t 0
- s)v>ds.
= (z(t),A*v)
ds.
to get
B*(t) = B*(t)(A¢)-IA ~ andB*(t)(A~) -I is a bounded
a.e. Further, by Lemma
2.2 A -1B(t) has a b o u n d e d extension
is the restriction
[B*(t)(A*)-I] * to X so z satisfies
of
(z'(t),v>
From is
in
= (z(t)
Lemma
2.1
we
see
D(A)
and
z' (t)
that
+ .r 0
t i ~
~(t)
= Ae(t).
= z(t)
~(t)
has a unique A-IB(t)
= z(t)
solution
maps Y into
to X which
)z(s)ds,A*v).
+ .~ 0
-
)z(s)ds
Now t
(3.1)
operator
+ .~ 0
M
B(t
z(t) 6 C([0,~),X)
-
s)z(s)ds
by Lemma
2.2.
However,
itself and for x E Y
flA is(t)xIIy = t[s(t)xll + flA-Is(t)xll <_ b(t)IlXIIy + MIIxllX
and so A-1B(t) is measurable
E 8(Y) on Y.
and as B(t)x Thus A-iB(t)x
is measurable is locally
in X, A-IB(t)x
Bochner
integrable
290 in Y so
(3.1)
has
a unique
solution
in Y c X.
That
is,
z(t)
E Y
so that z'(t)
Thus,
z(t)
= R(t)z(0)
The p r o o f (VE)
for
each
that
of two w e a k neous
of
There
Denote
= x 0 and d e f i n e
that
R(t)
R(t)
by
existence We
first
is a w e a k
is thus the
of a u n i q u e
the
(VE)
by x(t,x0)
x(0)
weak
existence
difficult.
solutions
equation.
f ~ 0.
the
more
- s)z(s)ds.
= 0 and Xl(t ) - x2(t ).
x 0 E X implies
is t e c h n i c a l l y
t + .] B(t 0
= Az(t)
no
loss
unique
weak
solution
of a r e s o l v e n t
note
that
solution
the
weak
solution
R(t)x 0 = x ( t , X o ) .
operato
difference
of the
of g e n e r a l i t y
of
homoge-
in a s s u m i n g
of
(VE)
It then
with
follows
is linear,
R ( t ) x 0 is c o n t i n u o u s in t and R(0) = I. t Now let x 0 E D(A) and z(t) = .~ R ( s ) x 0 d s . As R ( t ) x 0 is a 0 s o l u t i o n of (VE) one o b t a i n s for v E D(A*)
As b e f o r e , Arguing
z'(t)
x 0 = A~(t)
as b e f o r e ,
continuous t
.~ R ( s ) x o d s
using
where
Lemmas
2.1
w(t)
is g i v e n
and
2.2 y i e l d s
z(t)
(3.1).
that
is
Hence,
t
A .~ R ( s ) x 0 d s 0
t s
to show shown
z(t)
by
= .] R ( s ) A x 0 d s 0
t
+ .~ .~ R ( s 0 0
can be
by
Y so that Az(t) and z(t) are c o n t i n u o u s . t E D(A) and A .£ a ( s ) x o d s is continuous.
t
We w i s h
- s)v>ds.
into
Now d e f i n e
z(t)
t + .~ < z ( s ) , B * ( t 0
- x0,v > =
z(t)
that
- u)B(u)x0duds
~ 0.
z(t)
Noting
is a w e a k
s
.~ B ( t 0 that
z(0)
solution
- s) .f R ( u ) x 0 d u d s . 0 = 0 we
see
that
of the h o m o g e n e o u s
if it
291
equation
(VE) it will
follow from the uniqueness
solution that z(t) 5 0. D is given by
This can be shown by choosing v E D where
(H3) and d i f f e r e n t i a t i n g
of calculations
involving triple
A e B ( t ) v and B*(w)B*(u)v,
rearrange
(3.2)
etc.,
(z(t),v).
integrals
After a number
and terms of the form
one shows t
~t for v E D.
of the weak
(z(t),a~v>
+ .F ( z ( s ) , B ~ ( t 0
- s)v)ds
Now integrate both sides of this equation and to get
-1
t
z(t)-.~
t. SA_ IB
z(s)ds-
~ ~ 0 0
0 for v 6 D.
(s-
u) z ( u ) d u d s
,Aev)
= 0
Given t > 0 and v E D(A *) one may approximate A~v by
elements of the form Aem where w E D as A*D is w e a k * d e n s e One sees that
(3.2)
solution and z(t)
is valid for all v E D(A ~) so z(t)
~ 0.
Thus,
in X ~.
is a w e a k
for v E D(A),
0 = d~
(R(t)x0Aev)
t + C
t a(t
- u)B(u]x0du,v
> -.~
0
- S)Xo,B"(s)v>as
0
or
t
( R ( t ) A x o + .F R ( t 0
- u)l~(U)XodU,v)
< a ( t ) x o + .~ 0
a-
B(u)a(t
-
U)XodU,a~v>.
This says
t re(t)
= R ( t ) x 0 + .~ 0
~ A-1B(u)R(t
- U)XodU E D(A)
and t
Aa)(t)
= R ( t ) A x 0 + .~ R ( t 0
- u)B(U)XodU.
292
Arguing hence
as above,
R(t)
this
: Y ÷ Y).
implies This
R ( t ) x 0 is c o n t i n u o u s
in Y
(and
implies t
A R ( t ) x 0 + .~ B ( u ) R ( t - u ) x 0 d u 0 t = a ( t ) A x 0 + .~ a ( t - u ) B ( u ) x 0 d u . 0 Now
consider
x 0 E D(A)
and v E D(A*).
in Y and R ( t ) x 0 is a w e a k
solution
Then
of
as R ( t ) x 0 is c o n t i n u o u s
(VE),
t
x0,v)
.~ ( R ( s ) x 0 , A e v > d s 0 t s + .f .~ R ( u ) x 0 , B ~ ( s - u ) v ) d u ds 0 0 t s <.f { A R ( s ) x 0 + .f B(s 0 0
for
all v ~'D(A*).
It now
follows
from
t
R(t)x 0 so that
x 0 = .~ 0
u)R(u)x0du}ds,v),
Lemma
2.1 w i t h
x = 0 that
S
{ A R ( s ) x 0 + .f B ( s 0
R ( t ) x 0 is s t r o n g l y
differentiable
t R ' ( t ) x 0 = A R ( t ) x 0 + .~ B(t 0 t = R ( t ) A x 0 + .f R ( t
u)R(u)x0du}ds with
- u)R(u)x0du
- u)B(u)x0du.
0 To show and c o n s i d e r Given
R(t)
is a b o u n d e d
the map
x E X,R(t)x
@
operator
: X + C([0,T],X)
for
each
defined
t > 0, fix by
@(x)
T > 0
= R(.)x.
satisfies t
{R(t)X
for
each
- x,v)
v E D(A*).
= .~ < a ( s ) x , A * v ) d s 0 t s + .F .F
Suppose
now
that
xn
s
-
u]v>duds
x and @x n ÷ y.
Then
293
(y(t),v)
and
y(t)
y(t)
is
a weak
= R(t)x
bounded.
solution
This
completes
to u n i f y m u c h
solutions
of
Corollary
3.6.
that
with
and O is closed.
y(O) This
of the w o r k
Suppose
]Ix(t,x0)
® and,
the f o l l o w i n g
on e x i s t e n c e
there
exists
a dense
solution
is a c o n s t a n t
x(t,Xl)]I ~ M(T)]]x 0
(VE) has a r e s o l v e n t Proof.
By u n i q u e n e s s , thus,
R(t)
is
result w h i c h
and u n i q u e n e s s
can of
(VE).
for each T > 0 there
implies
= x. implies
we o b t a i n
for x 0 E X 0 (VE) has a u n i q u e
assume
- u)v>duds
the proof.
As an easy c o r o l l a r y be u s e d
t = .~ ( y ( s ) , A * v ) d s 0 t s + .~ .~ < y ( u ) , B * ( s 0 0
(x,v)
Without
subspace x(t,x0).
M(T)
X 0 ~ X such Further,
> 0 so that x0,x I E X 0
XlI I for 0 ~ t ~ T.
Then
operator. loss of g e n e r a l i t y
l[x(t,x0)]] <__ M(T)]]x0II,
x 0 E X,
we may
0 <_ t <_ T.
assume
f - 0 so that
For x 0 E X and v ~_ D ( A ~)
we have t
(3.3)
CxCt,xo)
- Xo,V> = .r < x C s , x 0 ) , A ~ v ) d s 0 t
s
+ .F .F ( x ( U , X o ) , B ~ ( s
- u)v~duds.
0 0 It is easy to see that
for x 0 E X there
x n + x 0 and x ( t , x n) c o n v e r g e s f u n c t i o n w h i c h we
It is clear x(t,x0)
Theorem
3.5 one can show
(VE).
on
label x ( t , x 0) and that
(3.3).
fying
uniformly
exists
The r e s u l t
is a w e a k
there now
[0,T],
this
T > 0, to a
f u n c t i o n must
solution.
is at m o s t
follows
{x n) c X 0 w i t h
Arguing
one w e a k
as in
solution
from the p r e v i o u s
satisfy
satis-
theorem.
294
Another which
interesting
characterizes
differential is p r o v e d Theorem
3.7.
o p e r a tor.
in a B a n a c h
semigroup
initial
is c o n t i n u o u s l y Proof.
However,
This
Thus,
(VE).
For
the c o r r e s p o n d i n g
[25, p.
result
Iii].
and s u p p o s e
-
is a w e a k
(VE) has
(VE) has
a resolvent
a solution
for every
s)f(s)ds
as ~(t)
solution
of
(VE) w i t h x 0 = 0
for v E D(A*), t + .~ ~ w ( s ) , B ~ ( t 0
= ~w(t),A~v)
is d i f f e r e n t i a b l e ,
f(t),v>
= <w(t)
- s)v)ds
(~'(t),v}
+ ~f(t),v>. and
= d(~(t),v)
t - F ' ~ + o~ A - I B ( t - s ) w ( s ) d s , A e v } • 0
says
(3.4)
g(t)
is in D(A)
and Ag(t)
of T h e o r e m
3.5 this
w'(t%
Thus
of
differentiable.
3.5.
<~'(t)
space
problem
t = ,F R ( t 0
We k n o w w(t)
d{w(t)~,v}
a solution
theorem
if ~(t)
by T h e o r e m
theory,
value
if and only
is the f o l l o w i n g
(VPR) y i e l d s
Let f be c o n t i n u o u s
The
x 0 E D(A)
when
equations
using
application
the s o l u t i o n
R ( t ) x 0 + w(t).
A number
= w(t)
= ~' (t) implies
- f(t)
of
This
A-IB(t-
- f(t) ~(t)
= Aw(t)
s)~(s)ds
is c o n t i n u o u s .
is c o n t i n u o u s t + .r B(t 0
is just
converse
paper.
applications
The
in the proo:
- s)w(s)ds.
exists
(VPR).
As
into Y and so
(VE) w i t h x 0 E D(A)
of other
in a f o r t h c o m i n g
+ .r 0
are p o s s i b l e
and is g i v e n by is clear.
and w i l l
appear
295
ACKNOWLEDGMENT
The results
in this paper were obtained while the first
author was visiting sity of Graz.
the Institute for Mathematics
The first author gratefully
at the Univer-
acknowledges
the
support of the University of Graz.
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tions, McGraw-Hill, 12.
R. C. Grimmer,
13.
Proceedings
R. C. Grimmer
16.
8(1970),
457-474.
S. I. Grossman
K. B. Hannsgen,
and R. K. Miller,
equation
J. Diff.
in Hilbert
space,
equations
equations
in a
in abstract
on Differential June,
Equations
1981.
Analytic
resolvent
theory for
J. Diff.
Nonlinear
opera-
(to appear).
Perturbation
systems,
The resolvent
131-179.
(to appear).
in Banach space,
and R. K. Miller,
integrodifferential
systems with Ll-kernels, 17.
Soc.
of Conference
Volterra
in
Theory and Applica-
for integral
and A. J. Pritchard,
S. I. Grossman
126(19671,
for integral
equations
equation
1966o
Graz, Austria,
tors for integral 15.
Sot.,
operators
Resolvents
and Applications, 14.
New York,
Trans. Amer. Math.
R. C. Grimmer, spaces,
Math.
integral
Linear Operators:
Resolvent
Banach space,
Volterra
Equations,
integrodifferential
Equations,
13(1973),
551-566.
kernel of an integrodifferential
SIAM J. Math. Anal.,
7(1976),
481-490. 18.
K. B. Hannsgen,
Uniform L 1 behavior
equation with parameters, 19.
E. Hille and R. S. Phillips, groups,
20.
Rev.
New York, 21.
ed., Amer. Math.
T. Kato, Perturbation
for an integrodifferential
SIAM J. Math. Anal., Functional Soc.,
8(1977),
Analysis
Providence,
626-639
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Theory for Linear Operators,
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1976.
S. G. Krein, Amer. Math.
Linear Differential Soc., Providence,
Equations
RI, 1971.
in Banach Space,
297
22.
R. K. Miller, Nonlinear Volterra Integral Equations, Benjamin, Menlo Park, Calif., 1971.
23.
R. K. Miller, Volterra integral equations in a Banach space, Funkcial, Ekvac,, 18(1975),
24.
163-193.
R. K. Miller and R. L. Wheeler, Well-posedness of linear Volterra integrodifferential spaces, Funkcial. Ekvac.,
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21(1978),
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279-305.
A. Pazy, Semi-groups of Linear Operators and Applications Partial Differential Equations, University of Maryland,
to
1974
EXTERIOR BOUNDARY VALUE PROBLEMS FOR PERTURBED EQUATIONS
OF ELLIPTIC TYPE
F. A. Howes
I.
INTRODUCTION Many phenomena
in mathematical
to certain boundary value problems regions of Euclidean were developed equations.
space.
physics can be formulated for partial differential
Indeed,
in terms of solutions
equations
in unbounded
several important areas of classical analysis
to treat such problems
in the special, but important case of linear
Many transform techniques
derive their efficacy from the unboundedness
of the region which permits otherwise
formidable
integrals
to be calculated,
often
in closed form.
However most, if not all, of these classical methods are applica-
ble only because
the problem is linear.
There are simply no general solution strate
gies for nonlinear boundary value problems which give even a qualitative
description
of the solution and its properties. In this paper we treat briefly a class of nonlinear elliptic boundary value problems
in unbounded regions,
suits are available.
for which some general existence and comparison
These problems
can be viewed as simplified models
types of flow phenomena described by the Navier-Stokes large Reynolds number.
Our detailed analysis
equations
re-
of various
in the case of
of this specific application will be
published separately.
2.
STATEMENT OF THE PROBLEM Consider
tive values
the following singularly
of
perturbed Dirichlet
problem for small posi-
s 2 CV u = A(x,u)-Vu + h(x,u),
x in g,
(ee) u
where x = (Xl,...,xN)
=
p(x),
x
on
F
=
Be,
• is the usual Euclidean inner product, 2 V = (~I,...,BN) is the N-dimensional gradient, V = V.V is the Laplacian, and ~N A (al(x,u),...,aN(x,u)). The region g is assumed to be an open subset of which contains
is a vector in ~ N ,
the point at infinity and whose boundary
sional manifold.
For ease of exposition, we assume,
scribed in terms of a smooth scalar function
F
F
is a smooth
in fact, that
in the sense that
e = {x: ~(x) < o];
g
(N-l)-dimencan be de-
319
whence,
F = ~g = F-I(0) and VF(x) is the outer normal
assume that the functions ate subdomains
to
F
at
x.
Finally we
a.z (I < i < N) and h are sufficiently smooth in appropriso as to guarantee the existence of a classical solution
of g X ~,
of (Pc). The task before us is then to prove the existence of a strong solution u = u ( x , ¢ ) of ( P ) within a class of functions having a prescribed growth at infinity,
and to
c
study the limiting behavior of
u
allows us to use the techniques
of perturbation
differential
inequalities
It turns out that the smallness analysis
to give sufficient conditions
existence of such solutions. estimates
as ¢ ~ 0+.
Before presenting some general results, linear problems where the asymptotic
e
¢
coupled with the theory of on
A,
h
and
At the same time, our approach provides
on these solutions which improve as
of
decreases
F
for the
rather sharp
to zero.
let us examine several representative
phenomena
reveal themselves
in a straightfor-
ward way.
3.
LINEAR EXAMPLES For ease of exposition,
examples
is the exterior of the unit circle S I in ~ 2
(Xl,X2) as (x,y). boundary
let us assume that the region
F
of
Then 6
g
in the following four
and let us write the coordinates
is the set [(x,y): F(x,y) = ½(l-x2-y 2) < 0}, and the
is simply S I.
we look for solutions
g
Moreover,
in all of the problems
in this section,
of class C(2)(g) A C(g) which decay to zero at infinity.
given boundary data is always assumed to be continuous
on
The
r.
Our first example is 2
S (Uxx+Uyy) = m u,
(x,y) in g,
(El) u = ~(x,y),
where
m
is a positive constant.
action-diffusion the problem
(x,y) on F,
It may be thought of as a simple model of a re-
system with no convection present.
(El) has a unique solution u = u(x,y;¢)
which satisfies
An easy calculation for each sufficiently
shows that small ¢ > 0
the estimate i
(3.1)
for (x,y) in
lu(x,y;¢)l _< l]~011ooexp[mlF(X,y)/se]
g
and 0 < m I < m.
(Here and throughout
supremum norm restricted
to the boundary F.)
in any closed subset of
g,
while near
F
Thus
u
the paper I!'II~ denotes the converges
to zero as s ~ 0+
there is a thin layer (boundary
of width O(¢ e) in which
layer)
u differs from zero by the amount ll~II~" The identically 2 zero function is the solution of the so-called reduced equation m u = 0 obtained by formally setting
¢
equal to zero in the differential
equation.
320
The second example shows the influence that convective havior of solutions,
terms may have on the be-
namely
~(Uxx+Uyy)
= -(x,y).Vu + u,
(x,y) in 6,
(E2) u = ~(x,y),
Let us proceed a little differently
(x,y) on F.
than in Example
(El), and first set
~
equal to
zero so as to obtain the reduced equation (3.2)
(x,y).Vu = u,
which is Euler's ing a solution
relation for homogeneous u
functions
of degree one.
of (E2) which is zero at infinity.
Since
¢
Now we are seek-
is small, we antici-
pate that such a solution must be close to a solution of (3.2) throughout most of g,
and so we select the solution u 0 m 0 of (3.2) as our candidate for the limiting
value of
u
in
of differential regions
g
as e ~ 0+.
inequalities
(cf. for example
tion u = u(x~y;¢)
In fact, a straightforward
application
of the theory
for elliptic boundary value problems in unbounded
[9] or [6]) shows that the problem
for each sufficiently
small ~ > 0.
(E2) has a unique solu-
Moreover, we obtain simultan-
eously the estimate
(3.3)
lu(x,Y;~) I _< ll~olI==exp[(l-6)F(x,y)/~]
for (x,y) in
~
and 0 < 6 < I, since the function on the right is a barrier function
for (E2) as ¢ ~ 0+.
As in the last example,
away from the boundary timates
F,
g
but let us note an important difference between the es-
In the former estimate the boundary layer (region of non1 has thickness of order O(e~), while in the latter the boundary
layer is thinner, with thickness
of order O(e).
course due to the presence of the gradient that the reduced equation
as characteristic tially,
to zero in
(3.1) and (3.3).
uniform convergence)
implies
the solution converges
curves.
The thinning of the layer is of
terms in the differential
(3.2) has the family of straight
These characteristics
exit
g
through
equation, which
lines y = (const.)x F
nontangen-
that is, -(x,y)-vF(x,y)
for (x,y) on
= x2+ y2 = I > 0
F.
It is essential ary layer along
F.
that the characteristics Consider,
the related differential
exit
as an illustration,
g
for there to be such a bound-
the Dirichlet
equation ~(Uxx+Uyy ) = (x,y).Vu + u in
tic curves of the reduced equation
problem g.
(E2') for
The characteris-
(x,y).Vu + u = 0 are again the family of straight
lines y = (const.)x, but now these curves enter
g
through
r.
There is n__ooboundary
321
layer along
F; indeed, since
F
is not itself a characteristic
curve,
u = Uo(X,y ) of the reduced equation is determined by the requirement ~(x,y)
for (x,y) on
F.
If the data
~
is such that u 0 decays
then one can show that u 0 is a uniformly valid approximation in any subset of
the solution
that Uo(X,y ) =
to zero at infinity,
of the solution of (E2')
g.
We consider next an example in which the differential terms, and yet the solution behaves
¢(Uxx+Uyy)
equation contains gradient
like the solution of Example
= (y,-x).Vu + u,
(El), namely
(x,y) in g,
(E3) u = ~(x,y),
(x,y) on F.
As in the last example, we look first at the reduced equation (3.4)
(y,-x).~u + u = 0.
Its characteristic
curves are the family of concentric
and so the boundary possible,
F
in general,
is itself a characteristic.
circles x 2 + y2 = (const.)2,
This means
that it is not
to find a solution of (3.4) which satisfies
ary data anywhere along
F.
the given bound-
Since we are looking for a solution of (E3) which de-
cays to zero at infinity, we select u 0 m 0 as the solution of (3.4) which we think will serve as the limiting value in closed subsets of (y,-x)'vF(x,y) (that is,
F
is a characteristic
curve),
g.
In addition, we see that
~ 0
and so an estimate
like (3.3) cannot hold
for the solution of (E3). Instead we can show that the function w(x,y;¢) = i II~!l exp[mlF(X,y)/¢2] (0 < m 1 < I) is a barrier function for (E3), and so the solution u = u(x,y;e)
satisfies lu(x,y;¢)l < w(x,y;¢)
for (x,y) in
g
as ¢ ~ 0÷.
These four examples reveal that the behavior of solutions lem ( P )
for small
of the corresponding
¢
reduced solution with the boundary of
istics either are nonexistent
(cf. (El)) or are tangent
boundary (cf. (E3)), then there is a boundary l O(¢2). While if the characteristics exit g is a boundary enter
g
layer with thickness
consult
[7], [2] and
If the character-
layer along F' with thickness
through F', then there is no boundary
for the problem
g.
curves
to a portion F' of the
through F' nontangentially,
of order O(¢).
solution u 0 is chosen by the requirement established
of the general prob-
is governed by the interaction of the characteristic
of order
then there
Finally if the characteristics
layer along F'; indeed,
that u 0 = ~ on F'.
(Pc) in bounded regions;
the reduced
These results have been
the interested reader can
[4] for details and further references.
In the next section
322
we prove some similar statements
4.
about the exterior problem.
GENERAL RESULTS Guided by the examples and observations
a consideration
of two fairly representative
in the last section, results
2
s~7 u = A(x,u).Vu + h(x,u),
let us turn now to
for the general problem
IRN
x in g c
,
(P) u = ~(x),
where the various
x on F,
functions and sets have the properties
order to study the behavior of solutions
of ( P )
solutions u = u0(x) of the corresponding
reduced problem
described
In
in Section 2.
as ¢ ~ 0+, we first examine the
A(x,u).vu + h(x,u) = 0,
x in g,
(Po) u = ~(x),
Here F
x on F
= [x on F: y(x) = A(x,u(x)).vF(x)
ary points at which the characteristic
c F°
< O} is the (possibly empty) set of bound-
curves of (Po) enter
or the theory for bounded regions in [7] and [4]). ed, we must select solutions
of (Po) from a class
a restricted growth at infinity. reduced solutions belonged to infinity.
For instance,
g
(ef. Example
Since the region ~
g
(E2')
is unbound-
of smooth functions which have
in the examples
of Section 3 the
to the class of functions which approach zero as Ilxll tends
In general, we may define
~
as the class of smooth functions v = v(x)
such that
~
lim L R ~
where
U
is a smooth positive
A(x,U)-?U + h(x,U) (cf.
[9],
~
in
g.
v_~]
sup
U(x)J = 0,
function in
g
having the property
Such a function
U
is called an anti-barrier
[6]), and these references
show that the problem
using barrier functions which themselves belong these results,
that cV2U at infinity
(Pc) may be studied by
to the class
M.
Before discussing
let us indicate briefly some of the earlier work in this area.
The exterior Mauss
x in g,
I!xll = R
linear Diriehlet
[8] and Eckhaus
problem in two dimensions has been studied by
[I] for the particular
equation ~(Uxx+Uyy)
= -Uy in regions
g
contained in the half-plane y > YO (fixed) > O. They used as an anti-barrier at r rr sin ~ infinity the function U(r,@) = I 0 ( ~ ) e x p [ ~ ] , where I 0 is the modified Bessel function of the first kind, of order zero, and (r,@) are polar coordinates with respect to an arbitrary reference point
r sin @.
(x0,Y0) defined by x - x 0 = r cos 8, y -
For large positive values of its argument,
is exponentially
unbounded at infinity.
I0(z )
eZ/(2~z) ~, and so
Thus the solutions constructed
Y0 = U
by Mauss and
323
Eckhaus were allowed to be large at infinity, provided they grew slower than r ~ ~.
The particular regions
g
U
as
included the upper half-plane and nonconvex sets
like gl = [(x,y): y > 0 for x ~ 0, y > I for x < O} whose boundary has a "step" at x = 0, and the exterior 6 2 of the unit circle in ~2.
The nonconvexity of such sets
leads naturally to the occurrence of detached boundary layers called free boundary layers, as well as the usual boundary layers illustrated in Section 3.
In the case
of gl' there is a boundary layer along the segment [(x,y): x = 0, 0 ~ y ~ I} which becomes a free boundary layer along the vertical line x = 0, y ~ I. These layers & have thickness of order G ( ~ ) . The free layer arises from the fact that, in general, the corresponding reduced solution u = u0(x ) is discontinuous along the positive y-axis.
Consequently the solution u = u(x,y;s) of the Dirichlet problem in 61 satis-
fies an estimate of the form
lu(x,y;¢)-u0(x) I _< L exp[-x2/(2¢)]
for (x,y) in gl' where L = I~(0+,0) - p(0-,l) I is the magnitude of the difference in the boundary data at x = 0; cf. [5] for some related results in bounded regions. The line x = 0 is of course a characteristic curve of the reduced equation u
= O. Y In the case of the region g2' it turns out that there is a boundary layer on the lower semicircle which becomes a free boundary layer along each of the lines [x = ~I, y > 0}.
The boundary layer on the semicircle has thickness of order O(e)
away from the points (±I,0).
In neighborhoods of these points and along the lines
x = ±I, y ~ O, this layer and its continuations as the free layers are fatter with thickness of order 0(¢2), owing to the fact the lines are characteristic curves of the reduced equation which are tangent to
F
at (~i,0); cf. Example (E3).
In order to deal now with the general problem ( P ) ,
let us introduce the
functions y(x,u) = A(x,u).VF(x) and H(x,u) = A(x,u).~u0(x) + h(x,u), where u = Uo(X ) is a smooth solution of the reduced problem (Po).
Let us also de-
fine the domain ~(u0) = g × [u: lU-Uo(X) l <__ d(x)}, where IIp-UoIl~< d(x) <. IIp-UolI= + 6 for dist(x,F) < 6/2 and d(x) _< 6 for dist(x,F) _> 6 (with
6
a small positive constant).
solutions of (Pc) for small exiting
g
everywhere along
¢, r.
It is in this domain ~(Uo) we will look for
in the case that the characteristic curves are The first theorem treats curves which exit non-
324
tangentially
(cf. Example
Theore~ 4.1. class
M
(E2)).
Suppose that the reduced problem (Po) has a solution u = u0(x) in the
such that
(I)
there exists a positive constant ~6(u0)
(2)
k
for which y(x,u) ~ k(VF.VF)(x)
i__nn
(= ~(u0) with dist(x,F) < 6);
there exists a positive constant
m
for which Hu(X,U) ~ m 2 > 0 in $(u0).
Then there exists an c o > 0 such that the problem (Pc) has a solution u = u(x,¢) i__nn of class C(2)(g) n C(~) whenever 0 < c ! c o .
Moreover,
for
x
in
g
we have
g
tangen-
that {u(x,C)-u0(x)l ! l!~'Uol]~exp[klF(X)/¢] for 0 < k I < k and
c
+ c~,
a positive constant depending o_~nu0,
The next theorem deals with characteristic
k
and
m.
curves which may exit
tially.
Theorem 4.2.
Suppose t h ~
the reduced solution u 0 is such that the assumptions of
Theorem 4.1 hold with assumption
(i) replaced by
~(x,u) __> 0
in
~6(u0).
Then the conclusion of Theorem 4.1 is valid with the term exp[klF(X)/c] 1 exp[mlF(X)/c~] fo___rr0 < m I < m.
Thus the boundary layer is thicker wherever the characteristics
replaced by
exit
g
tangen
tially, as we have noted already for our simple linear examples. The idea behind the proof of these two results is the observation that the function w(x,¢) = n0(x) + II~-u011m p(x,¢) + cKm -I (with p(x,¢) = exp[klF(X)/C] 1 exp[mlF(X)/C~] ) is a barrier function for the problem (Pc) in the sense that lq0(x) l __<W(X,e) and in
on
or
r,
g, 2
cV w _< A(x,w).Vw + h(x,w), ¢V2(-w) _> A(x,-w).v(-w) for appropriately
chosen positive constants k I, m I and
small, say 0 < c ~ c 0. this range of
c,
+ h(x,-w),
Then a theorem of Kusano
K,
and for
~
[6] allows us to conclude that for
the problem (Pc) has a smooth solution u = u(x,c) in
that lu(x,C)-u0(x) I ! II~-u011~ p(x,c) + ce.
sufficiently
Similar calculations
E
such
for the case of a
bounded region were performed in [4] and [5], where the interested reader can find
32S
complete details. We close with two remarks.
The first one is that our results in [5] on the
existence and properties of free boundary layers for problems in bounded regions can be applied mutatis mutandis to the problems in unbounded regions discussed here.
If
the reduced solution u 0 under consideration is either discontinuous or nondifferentiab!e along certain (N-l)-dimensional manifolds in
g,
then this irregularity of
u 0 gives rise to free boundary layers in neighborhoods of such manifolds which serve to smooth out the function u 0 there.
Nonsmooth reduced solutions are present in many
of these problems in unbounded regions, as we have seen above in the two examples of Mauss and Eckhaus. The second remark is that we can apply the theory in [3] to problems like (Pc) in which the Dirichlet boundary condition is replaced by a boundary condition of Neumann or Robin type on all or part of
F.
Such problems occur frequently in the
study of transport phenomena.
ACKNOWLEDGMENTS The author gratefully acknowledges the support of the National Science Foundation under grant no. MCS 80-01615.
He also wishes to thank Mrs. Ida Zalac for her
usual superb typing job.
REFERENCES I.
W. Eckhaus, Boundary Layers in Linear Elliptic Singular Perturbation Problems, SIAM Rev. 14(1972), 225-270.
2.
W. Eckhaus and E. M. deJager, Asymptotic Solutions of Singular Perturbation Problems for Linear Differential Equations of Elliptic Type, Arch. Rational Mech. Anal. 23(1966), 26-86.
3.
F . A . Howes, Robin and Neumann Problems for a Class of Singularly Perturbed Semilinear Elliptic Equations, J. Differential Equations 34(1979), 55-73.
4.
F . A . Howes, Some Singularly Perturbed Nonlinear Boundary Value Problems of Elliptic Type, Proc. Conf. Nonlinear P.D.E.'s in Engrg. and Applied Sci., ed. by R. L. Sternberg, Marcel Dekker, New York, 1980, pp. 151-166.
5.
F . A . Howes, Perturbed Boundary Value Problems Whose Reduced Solutions are Nonsmooth, Indiana U. Math. J. 30(1981), 267-280.
6.
T. Kusano, On Bounded Solutions of Exterior Boundary Value Problems for Linear and Quasilinear Elliptic Differential Equations, Japan J. Math. 35(1965), 31-59.
7.
N. Levinson, The First Boundary Value Problem for CAu + A(x,y)u x + B(x,y)Uy +
8.
J. Mauss, Etude des Solutions Asymptotiques de probl~mes aux Limites Elliptiques pour des Domaines non Born~s, Compte Rendus Acad. Sci., Ser. A 269(1969), 25-28.
9.
N° Meyers and J. Serrin, The Exterior Dirichlet Problem for Second Order Elliptic Partial Differential Equations, J. Math. Mech. 9(1960), 513-538.
C(x,y)u = D(x,y) for Snmll ~, Ann. Math. 51(1950), 428-445.
LINEAR TRANSPORTTHEORY AND AN INDEFINITE STURM-LIOUVILLE PROBLEM* Hans G. Kaper, C. Gerrit Lekkerkerker,** Anton Zettl***
Abstract Linear transport processes occur whenever p a r t i c l e s move in a host medium, carrying mass, momentum, and energy from one point of the medium to another. Mathematical models of such transport processes involve two operators, one accounting for free streaming of the p a r t i c l e s , the other for interactions between the p a r t i c l e s and the atoms or molecules of the surrounding host medium. We investigate a time-independent electron transport problem, where the free s2treaming operator is the m u l t i p l i c a t i v e coordinate operator in L - ( - I , I ) and the i n t e r a c t i o n operator is the Legendre d i f f e r e n t i a l operator.
I.
Introduction
Transport theory.
Linear transport theory
is the study of equations that
describe linear transport phenomena in matter.
The phenomena may relate to
neutron transport in a nuclear reactor or a nuclear scattering experiment, radiative transfer in a s t e l l a r or planetary atmosphere, electron transport in a metal, penetration of x-rays or y-rays through scattering media, and similar processes.
In each case, the transport mechanism involves the migration of
particles (neutrons, photons, electrons, etc.) through a host medium.
As long
as a particle is not subject to any force, i t moves with a constant velocity -- that is,
in a straight line with constant speed.
The particle motion may
*This work was supported by the Applied Mathematical Sciences Research Program (KC-04-02) of the Office of Energy Research of the U.S. Department of Energy under Contract W-31-109-ENG-38. **Permanent address: Amsterdam, Netherlands.
Department of Mathematics,
University
of Amsterdam,
***Permanent address: Department of Mathematical Sciences, Northern I l l i n o i s University, DeKalb, IL 60115.
327 be affected (gravity,
along the way by external
electric field,
etc.).
accelerative
More important,
or decelerative
forces
however, are the c o l l i s i o n s
that take place between the p a r t i c l e s and the atoms and molecules of the host medium.
A c o l l i s i o n is an i n t e r a c t i v e process governed by the laws of c l a s s i -
cal or quantum mechanics. field
of
results
physics; of
The study of individual c o l l i s i o n s belongs to the
transport
individual
theory
collisions,
begins
with
given
laws
governing
and considers the s t a t i s t i c a l
the
problem of
determining the r e s u l t of a large number of c o l l i s i o n s governed by these laws. The object of investigation p a r t i c l e s in phase space. space and velocity
in transport theory is the d i s t r i b u t i o n
Phase space is the d i r e c t product of configuration
space;
in
general,
it
is
six-dimensional.
A transport
equation is a balance equation for the (expected) number of p a r t i c l e s infinitesimal of
a
volume element of phase space.
point in phase space,
the position number
of
in v e l o c i t y
particles
(x,~) at time t.
in
of
x
in an
Let (x,~) denote the coordinates
being the position in configuration space and
space, and l e t f ( x , ~ , t ) d x d ~ denote the (expected) an
infinitesimal
volume
element dxd~ centered at
Let ~ be the region in configuration space where the trans-
port processes take place, and l e t S be the v e l o c i t y domain of the p a r t i c l e s involved in the transport processes.
Then a transport equation is an equation
for the phase space d i s t r i b u t i o n function f of the form ~f
The f i r s t
~
= _ ~--~. ~ f -
~
~
• af +
~f
(Ti~]coll
(I.I)
•
term in the r i g h t member represents the e f f e c t of free streaming; i t
is the spatial
divergence of the p a r t i c l e f l u x vector.
The second term repre-
sents the e f f e c t of external forces; i t is the divergence of the p a r t i c l e f l u x vector in velocity space, a is the force per u n i t mass. The l a s t term represents the rate of change of f due to c o l l i s i o n s ; i t depends l i n e a r l y on f. and for all t > 0 say.
Eqn.
I.I
in l i n e a r transport theory,
is supposed to hold for a l l
(x,~) ~ ~xS,
328
Simplifying assumptions.
Having stated the general form of
equation, we shall now introduce various simplifications.
a transport
To begin with, we
shall ignore the effects of external forces; this eliminates the second term in the right member of 1.1.
Next, we shall assume that the transport system
is in a state of equilibrium, so the rate of change of f , member of 1.1, is identically zero.
i.e.,
the l e f t
Furthermore, i f a l l particles involved in
the transport process have the same speed ( i . e . , the same magnitude of the velocity
vector,
i~l) and this speed is not changed in a c o l l i s i o n , then 0~I
becomes a parameter of the problem; consequently, the velocity variable ~ is effectively replaced by the unit vector m = ~/i~I as independent variable. There are, in fact, many systems for which a one-speed transport equation i s entirely r e a l i s t i c . A d i f f i c u l t y with the transport equation in arbitrary domains is the coupling between the spatial and the velocity variable through the operator (@/@x).~ in the streaming term. This coupling can only be broken in onedimensional plane slab configurations, where ~ = AIX~2, A1 some open interval on the real
axis, and the data (material properties of the host medium,
boundary data,
etc.)
transport equation is circumstances,
are independent of x2 and x3. then also independent of
assuming a one-speed approximation,
(@/~x).~ reduces
to (B/BXl)(m.el) , where
direction of increasing x1. polar coordinate m-eI = cos e.
is
is
x2 and x3.
Under these
we find that the operator the
unit
vector
in
the
Now, i f we l e t eI define the polar axis in a
system, and ~ has It
eI
The solution of the
customary to
the
polar
angle e (0 < E) < x), then
denote the cosine of the polar angle of
by ~: ~ = cos e; this variable ranges over the interval [-1,1].
Note that a
positive value of u represents movement in the direction of increasing Xl, a negative value of ~ movement in the direction of decreasing x1. Of course, ~ is determined not only by i t s polar angle, but also by i t s azimuth,
so,
in
general,
the
distribution
function f w i l l depend on this
329
variable as well.
However, we shall ignore this dependence and assume that
our transport system has azimuthal symmetry, i . e . ,
that the system is invar-
iant under a rotation about the polar axis e1. With these assumptions we have achieved that the phase space is essentially
two-dimensional,
one coordinate to represent the position inside the
slab (measured perpendicularly from some fixed plane of reference), another coordinate to represent the direction of motion (measured by the cosine of the angle between the velocity vector and the direction of increasing depth inside the slab).
We shall
(normalized in
use the l e t t e r
x to denote the position coordinate
some convenient way); the direction coordinate is u, as we
observed earlier.
In terms of these variables, the free streaming operator is
of the form (B/@x)~. So far we have not said anything 1.1.
about the term (Bf/Bt)coll in Eqn.
In stationary transport problems, t does not enter at a l l ,
but f s t i l l collision
changes because of collisions.
of course,
The exact expression for the
term depends on the particular physical
problem under discussion.
In general, collisions are regarded as instantaneous and localized events, so they affect only the dependence of the distribution function on the velocity variable.
With all the simplifying assumptions introduced earlier, collisions
affect only the dependence of f on ~, not i t s dependence on x; the position variable is simply a parameter.
Boundary value problem. The particular problem that we shall discuss in this a r t i c l e comes from electron transport theory.
I t goes back to an a r t i c l e by
Bethe, Rose, and Smith [1938], although the differential equation can already be found in an earlier a r t i c l e by Bothe [1929]. @~ p ~ ( x , p ) where
A = (O,T),
-
~~
The equation is
(1-~ 2) ~~¢ - (x,~) : 0 ,
J = (-1,1).
The unknown f u n c t i o n
(x,~) c AxJ ,
(1.2)
~ represents the electron
330
distribution
in
phase space.
One recognizes
the
first
term as the free
streaming term, the second term as the c o l l i s i o n term. With Eqn. 1.2 are prescribed boundary conditions at the endpoints x = 0 and x : ~.
The nature of these boundary conditions is somewhat unusual from
the mathematical point of view, but easy to understand i f we keep the physics behind the equation in mind. plate;
the
surfaces.
boundary
Eqn. 1.2
conditions
describes what happens inside the
embody what we can prescribe
Physically, we expect that we can prescribe the
at flux
the of
free those
electrons that move into the plate; the f l u x of electrons moving out of the plate should follow as part of the solution. given by the vector ~f in Eqn. i . I ; to the quantity u@.
The electron f l u x used to be
in the s i m p l i f i e d model 1.2 i t corresponds
Recall that p o s i t i v e v-values represent motion towards
increasing values of x, negative v-values motion towards decreasing values of x.
The following
specification
of
the boundary conditions
should now be
understandable: lim u@(x,u) = g+(v) , x+O
0 < v < I ,
(1.3-i)
lim u@(x,u) = g_(v) , x+T
-1 < u < 0 ,
(1.3-2)
where g+ and g_ are given functions, which are defined on [0,1] and [ - 1 , 0 ] , respectively. The boundary value problem 1.2,1.3
was considered by Bethe, Rose, and
Smith, who used formal expansion techniques.
A few years ago, Beals [1977]
proved existence and uniqueness of solutions for t h i s problem in a suitable weak formulation. the
development
monograph
on the
In the l a s t decade, a considerable e f f o r t has been spent on of
spectral subject,
[1982], w i l l appear s h o r t l y .
methods for written
linear
by Kaper,
transport
Lekkerkerker,
equations.
A
and Hejtmanek
331
Functional formulation.
The approach i s based on a r e p r e s e n t a t i o n o f Eqn. 1.2
i n terms o f v e c t o r - v a l u e d f u n c t i o n s and operators on a H i l b e r t space. Let J = ( - 1 , 1 ) ,
and l e t
H be the usual
H i l b e r t space L 2 ( j ) .
In H we
define the m u l t i p l i c a t i v e coordinate o p e r a t o r T,
Tf(p) = u f ( u )
Note
that
T
is
,
u ~ J,
injective,
f ~ H .
bounded
and
(i.4)
selfadjoint;
its
inverse
T-1
is
unbounded and defined on im T. Let p = [ i - ~ 2 : p c J ] , and l e t the expression - ( p f ' ) ' , Mf = - ( p f ' ) '
i.e., ,
f c dom M ,
where dom M = { f ~ H: pf' J, ( p f ' ) '
¢ H}.
M be the maximal operator associated with
absolutely
continuous
We define the c o l l i s i o n
i n H as the r e s t r i c t i o n Af = Mf ,
(1.5) on compact s u b i n t e r v a l s
operator (or:
of
s c a t t e r i n g operator) A
of M to dom A, f c dom A ,
where dom A = { f c dom M: l i m u + l f ( ~ )
(1.6)
and l i m u + _ l f ( u )
That i s , A is the well-known Legendre d i f f e r e n t i a l
e x i s t and are f i n i t e } .
operator.
Furthermore, l e t H± = L 2 ( j ¢ ) , where J+ = ( 0 , i )
and J_ = ( - i , 0 ) ;
can be i d e n t i f i e d with proper subspaces of H, and H = H+ ~ H_.
H+ and H_
Let P± be the
(orthogonal) p r o j e c t i o n which maps H onto H± along H~. Then the boundary value problem 1.2,1.3 can be formulated as an equation in H f o r the vector-valued function ~: [ 0 , z ] ÷ H, (T~)'(X) + A~(x) : 0 , P+ lira T~(x) = g+ , x+O
x ~ (0,z) , P_ l i m T~(x) = g_ . X+T
(1.7) (1.8)
332
In Eqn. 1.7, ' denotes differentiation with respect to x. tive,
we can
also
Because T is injec-
formulate the equation in terms of the function 4, where
4(x) = T@(x) for a l l x ~ (0,~), 4'(X) + AT-I¢(x) = 0 , P+ lim ~(x) = g+ , x+O
X ~ (O,T) ,
(1.9)
P lim ~(x) = g_ X~T
(1.10)
Thus, the rate of change of ~ is determined by the operator AT- I , to which we shall refer as the transport operator. T - l f ~ dora A}.
Note that dom AT-1 : { f ~ H: f E im T,
The function @is commonly known as the angular density, the
function 4 as the current density.
The (macroscopically observable) total
electron
current
density
and
electron
density
are
obtained
by
integrating @and 9, respectively, over a l l directions u. In the following Section I I we investigate the operator A in some detail.
In Section I I I we study the operator AT-1 and present what is known in
transport theory as the full-range theory. spectral representation of AT-1. construct
The objective here is to give a
In Section IV we use the results of the
full-range
theory to
the
general solution
equation.
Section V contains further details about an indefinite Sturm-
L i o u v i l l e boundary value problem that problem for T-1A.
one half
of
the d i f f e r e n t i a l
equivalent with
the eigenvalue
In Section VI we discuss what is known in transport theory
as the half-range theory. that
is
of
The objective here is to show, roughly speaking,
the eigenfunctions
are s u f f i c i e n t to expand a function
defined on one half of the range of the independent variable.
The discussion
is very speculative and meant to stimulate research on this challenging and open problem.
333
II.
Scattering Operator
In this section we summarize some relevant properties of the scattering operator A.
This operator was defined in Eqn. 1.6 as a restriction of the
maximal operator M.
Note that the equation - ( p f ' ) ' = 0 is singular at both
endpoints of J, as p-1 is not integrable in a neighborhood of either of these points.
The equation - ( p f ' ) ' = 0 has two fundamental solutions, namely f l = 1
and f2 = [ ~ n ( ( l + t ) / ( 1 - t ) ) :
t c (-1,1)], which are both in dom M.
Hence, the
maximal operator M is l i m i t - c i r c l e at both endpoints and M is not selfadjoint. To obtain a selfadjoint
realization of the expression - ( p f ' ) ' ,
condition
each endpoint.
is
needed at
a boundary
The boundedness condition in
the
definition of dom A serves this purpose. Notice that i t eliminates the solution f2 from the domain, i . e . , f2 ~ dom M, but f2 ~ dom A.
I t turns out that
there are several equivalent definitions of the operator A.
THEOREM 2.1.
Suppose f c dom M.
Then the following conditions are equiva-
lent: (i) (ii) (iii) (iv)
PROOF.
f is bounded on (-1,1); lim~÷1 f(~) an__~dlim~+_l f(~) exist and are f i n i t e ; lim~ 1 p(p)f'(~) = l i m ~ _ l p ( ~ ) f ' ( ~ ) = O; ~f'
c H.
The equivalence of ( i i ) ,
and Glazman [1981, Vol. follows similarly.
(iii),
2, Appendix 2].
and (iv) is established in Akhiezer The equivalence of (i)
and ( i i )
///
The next theorem shows that the boundedness condition imposed on the elements of dom A is
the "right"
restriction,
i.e.,
adjoint realization of the expression - ( p f ' ) ' in H.
it
determines a self-
334
THEOREM2.2. ~(A) is
The
(unbounded) operator
discrete
n=0,1,...};
and consists
of
A is selfadjoint in H; i t s spectrum
simple
eigenvalues:
o(A)
=
{n(n+l):
the eigenfunction corresponding to the eigenvalue n(n+l) is the
Legendre polynomial of degree n, i dn ( 2 _ 1 ) n ' pn(u) - 2nn! dun PROOF.
n=0,1 . . . .
(2.1)
See, for example, Akhiezer and Glazman [1981, Vol. 2, Appendix 2] or
Kamke [1971, Part C, Section 2.240].
///
For a comprehensive discussion of the Legendre operator, using both the "right
definite"
and the
"left
definite"
approach of the Sturm-Liouville
theory, we refer the reader to the recent a r t i c l e by Everitt [1978].
This
a r t i c l e also contains a discussion of half-range expansions, i . e . , expansions of elements of H+ or H_ in terms of Legendre polynomials. Notice that A has a nontrivial kernel, ker(A) = sP(Po). Because PO has the constant value 1 on J, we prefer to write 1, instead of PO, so ker(A) = sp(1).
THEOREM 2.3. if
The equation Au = f i s solvable _ i _nnH i f and only i f
the solvability
condition
is
(f,1) = O;
met, then the solution u is of the form
u = Kof+cl, where c E ¢ i s arbitrary and K0 is the integral operator i Kof(~) = f k(p,~')f(p')d~' , -1
~ c J,
f c H;
(2.2)
the kernel k i s given by the expression k(u,u')
= -1~LnI(1+u>)(1-~<) ] ,
w i t h u> = m a x ( ~ , ~ ' ) ,
~< : m i n ( u , u ' ) .
(u,~')
< j2
,
(2.3)
335
PROOF. Any element f i n the range o f A s a t i s f i e s the i d e n t i t y converse statement,
as well
from a d i r e c t computation.
Notice
that
= O.
The
as the second a s s e r t i o n o f the theorem, f o l l o w s ///
k ( . , u ' ) ~ L2(j)
for each fixed u' c J,
for each fixed u c J, so k is a Hilbert-Schmidt kernel. and s e l f a d j o i n t in H.
(f,1)
Obviously, ~(KO)
=
and k(~,.) c L2(j) Hence, K0 is compact
{(n(n+1))-1:n=1,2 . . . . } with KoPn
= (n(n+l))-lPn for n = 1 , 2 , . . . , which shows that K0 is ( s t r i c t l y ) positive on the orthogonal complement of 1 in H. 1:K01 = -2(~n2-1/2)I.
The vector K01 is a negative multiple of
Consequently,
(Kof,1) = (f,Kol) = -2(~n2 _ i ~ ) ( f , 1 )
,
f cH ,
(2.4)
which shows t h a t K0 maps the orthogonal complement o f i i n t o i t s e l f . We use the o p e r a t o r K0 to d e f i n e a new o p e r a t o r K in H,
Kf = Kof + 2(~n2 _ I ~ ) ( f , 1 ) 1 , The o p e r a t o r K, operator, {(n(n+l))-l: (strictly)
is
being a p e r t u r b a t i o n
also
compact
and
(2.5)
f cH .
of K0 by a one-dimensional symmetric
selfadjoint
in
H.
n=1,2 . . . . }, w i t h KPn = ( n ( n + l ) ) - I P n f o r
Furthermore,
n= l, 2 . . . . ,
p o s i t i v e on the orthogonal complement of I i n H.
is a positive multiple of I:
KI = 2 ( ~ n 2 - I ~ ) i .
Hence, K is
o(K)
so K is
=
also
Observe t h a t KI (strictly)
posi-
t i v e on the e n t i r e space H. For f u t u r e reference we summarize the r e l a t i o n s h i p between A and K: KAf : f - 1 ~ ( f , 1 ) l
,
f
AKf = f ,
c dom A ,
(2.6)
f c H ,
(2.7)
f c H ,
(2.8)
and note the i d e n t i t i e s (Kf,1) = 2 ( ~ n 2 - 1 ~ ) ( f , 1 )
,
336
(Kf,T1) = 1 ~ (f,T1) ,
f ~H.
(2.9)
Full-Range Theory
III.
Reduction of A1-'1.
Full-range theory is the study of the transport operator
AT-1.
step consists
The f i r s t
of
a reduction
of AT-1
to
isolate
the
generalized eigenspace associated with the eigenvalue at the origin.
THEOREM 3.1. that
the
The Hilbert space H admits a decomposition H = H0 ~ H1, such
pair {Ho,H1}
reduces the operator AT-1.
In
particular,
H0 =
sp(T1,T21) and H1 = { f ~ H: (f,1) = (f,T1) = 0}.
PROOF.
Because ker(A)
= sp(1) and T is
injective,
sp(T1).
A straightforward calculation shows that ker((AT-1) 2) = sp(T1,T21)
and ker((AT-1) n) = ker((AT-1) 2) for n=3,4. . . . .
Note that H0 c dom AT-1.
Now consider the subspace H1, as defined in the theorem.
2(f,i)
= (T-If,A1)
= O, so AT-If ~ H1.
H0 n H1 = {01 .
=
Thus, H0 is the generalized
eigenspace associated with the eigenvalue 0 of AT-1.
then (AT-If,I)
we have ker(AT-I)
= 0 and (AT-If,TI)
I f f~ Hlndom AT-1,
= (T-If,AT1)
= 2(T-If,TI)
T h a t is, H1 is an invariant subspace.
=
Clearly,
///
The projection operator PO which maps H onto H0 along H1 is easily found, POf = ~ (f,T1)T1 + ~ (f,1)T21 ,
f EH.
(3.1)
We denote the projection operator which maps H onto H1 along H0 by P, Pf = f - ~ (f,T1)T1 - ~ (f,1)T21 ,
f EH .
(3.2)
337
THEOREM3.2.
(i)
The r e s t r i c t i o n AT-IIHo is defined on H0 and is represented
by the matrix I 0 2] r e l a t i v e to the basis (T1,T21) o f H O. (ii)
The r e s t r i c t i o n AT-11H1 is i n j e c t i v e and (AT-11H1)-1 : PTKIH1 .
PROOF. (ii)
(i)
Immediate consequence of the i d e n t i t i e s A1 = 0 and AT1 = 2T1.
Take any f ~ H1.
Then TKf c dom AT- I and
AT-1pTKf = AT-I[TKf-PoTKf] = AKf-3(TKf,1)T1 . Because AK is the i d e n t i t y on H, we have AKf = f . (TKf,1) = (Kf,T1) = O.
Furthermore, Kf ~ H1, so
Hence, AT-1pTKf = f f o r a l l f E H1.
Next, take any f ~ H1 n dom AT-1.
Then
PTKAT-If = P T [ T - l f _ l ~ ( T - l f , l ) l ]
= pf _ l / 2 ( T - i f , 1 ) P T 1 .
But Pf = f and PT1 = O, so PTKAT-lf = f f o r a l l f c HI n dom AT-1.
///
Let the operator B be defined on H1 by the expression Bf = PTKf ,
f c HI .
According to Theorem 3 . 2 ( i i ) ,
(3.3)
B coincides with the inverse of AT-1 on HI .
K is compact and P and T are bounded on H, B is compact on H1.
As
The study of
AT-1 has thus been reduced to the study of the compact operator B on HI .
Structure of B.
The operator B is c l e a r l y not symmetric with respect to the
inner product of H.
However, B can be made i n t o a symmetric operator i f
introduce a new inner product ( ' " ) A (f'g)A = ( ~
f' ~
g) '
on H, f'g ~ H .
(3.4)
The A - i n n e r product and the corresponding norm I[.H A d e f i n e a new t o p o l o ~ H.
we
on
On the f i n i t e - d i m e n s i o n a l subspace H0 t h i s topology i s e q u i v a l e n t w i t h the
338 topology
induced
by the
Hence, the l i n e a r
ordinary
inner
product,
but
on H1 i t
space.
We obtain a
space by completing (H,u-ttA) with respect to the A-inner product;
denote t h i s H i l b e r t space by HA . inner
weaker.
space H endowed with the A-inner product, which we denote
by (H,IL-HA) , is an inner product space, but not a H i l b e r t Hilbert
is
product
space which,
Similarly,
upon
completion
product, y i e l d s the H i l b e r t space HI, A. The p r o j e c t i o n
operators
we
the l i n e a r space (Hl,ti. IIA) is an with
respect
to
the
A-inner
C l e a r l y , HA = H0 ~ HI, A,
PO and P = I-P 0 are continuous in the A-norm.
They can t h e r e f o r e be extended by c o n t i n u i t y by the same symbols, PO and P.
to HA; we denote the extensions
The formula P = I-P 0 remains v a l i d in HA, but
PO and P are, of course, no longer given by the expressions 3.1 and 3.2. The f o l l o w i n g
lemma shows that T can also be extended by c o n t i n u i t y
to
HA •
LE~3.3.
PROOF.
The operator T is bounded on (H,li-tIA).
We already know t h a t the Legendre polynomials Pn, n=0,1 . . . . , are e i -
genfunctions Pn = ( n ( n + l ) ) - I
of
~/~
Specifically,
f o r n=l,2 . . . . .
~Pn
= Pn Pn'
where
PO = 2 ( ~ n 2 ~ )
and
Furthermore,
TPn : (2n+l)-1[(n+l)Pn+l+nPn_l ] ,
n=0,1,...,
where P-1 = O. Let f c H be a r b i t r a r y .
Then f =
and Itfll 2 : ii KI /2flt 2 :
Z (n+1/2 n=O
® ~ pn I ant2 , n=O
,ITfH 2 = LIKI/2Tf,,2 =
~ enPnlan 12 n=O
anP n, where a n = ( f , P n ) ,
339 The constants 0 n are uniformly
bounded and p o s i t i v e .
In the d e r i v a t i o n of the
expression f o r liTfil~ we have used the f a c t t h a t the r a t i o Pn+l/Pn is uniformly bounded above and below, f o r any a,b ~ ¢.
and
the elementary i n e q u a l i t y 2iRe(ab)1 < iai2+Ibl 2
The lemma f o l l o w s .
///
We extend the operator T to HA by c o n t i n u i t y , the
same symbol
T.
The
extension
of the operator K to HA is t r i v i a l ,
llKfll A = I[K(~~ f)ll ~ IKI II~~ fll = IKI llfll A f o r extended operator
also by K.
denoting the extension by
every
f ~ H.
We
denote
as the
We now extend the operator B to HI, A by the
expression Bf = PTKf ,
Note t h a t B maps H1, A i n t o i t s e l f .
LE~3.4.
(3.5)
f c H1, A • A c t u a l l y , we have a stronger r e s u l t .
The operator B maps H1, A i n t o H1.
PROOF.
Let
(HI,,.IIA).
{fn:
n=1,2 . . . . } ,be a Cauchy sequence in the i n n e r product space
Then { ~ f n :
n=1'2 " ' ' }
is a Cauchy sequence in the H i l b e r t
HI , so there e x i s t s an element g ~ HI ,
such t h a t ll~~ fn-g, ÷ 0 as n ÷ ~.
cause P T ~ i s
bounded in L(H),
i t f o l l o w s t h a t the sequence
converges to
the element P T ~
g.
Be-
{Bfn: n = 1 , 2 , . . . }
This element belongs to HI , so the
tended) operator B maps H1, A i n t o H1.
space
(ex-
///
The argument used in the proof of the lemma also shows t h a t the operator B is i n j e c t i v e .
As B- I coincides with AT- I on dom AT- I
extends AT-IIHI , with dom B- I = im B. contain
elements
dom B- I c i m PT.
of
HI, A which
are
n HI, we see t h a t B-1
Note t h a t dom B- I not
in
H1.
It
c HI , but im B- I may is
not c l e a r
whether
340
THEOREM 3.5 (ii)
(i)
The operator B is compact and selfadjoint on H1,A.
The function P1 is a cyclic vector for B.
PROOF.
(i)
Because 1~KfllA = nK(~/2f)ll for every f ~ H, the compactness of ~/2
on H implies the compactness of K on HA. B is compact on H1,A.
Furthermore, PT is bounded on HA, so
Using the expression 3.1 for PO one readily v e r i f i e s
that the product operator POT is selfadjoint on H.
As PT = T-PoT and T is
selfadjoint on H, the product PT is also selfadjoint on H. (f,Bg) A for a l l f,g ~ HI . (ii)
Hence, (Bf,g) A =
The i d e n t i t y extends to a l l f,g ~ H1, A.
To show that P1 is a cyclic vector for B, i t
action of B on the Legendre polynomials.
suffices to consider the
Notice that P1 = -2P2; furthermore,
BP2 = (I/10)P3, and BPn
=
-l[n-lPn+ 1 +
(2n+1)
(n+l)
-1Pn_1]
Hence, P1 is a cyclic vector for B in H1.
,
n = 3,4,....
Because the A-norm is weaker than
the usual norm, any cyclic vector in H1 is a f o r t i o r i H1,A•
a cyclic vector in
///
Because B is compact and selfadjoint on HI, A i t s spectrum consists of a countably i n f i n i t e sequence of real eigenvalues with an accumulation point at the origin.
As B is injective, the origin i t s e l f does not correspond to an
eigenvalue.
The existence of a cyclic element implies that the spectrum of B
is simple.
Ei.genfunction expansions. Because B is selfadjoint on H1,A, we can use the Spectral Theorem to obtain eigenfunction expansions.
We write the eigenvalue
equation in the form Bxn : X~ixn ,
n = ±1,±2. . . .
(3.6)
341
THEOREM 3.6.
The eigenvectors {×n: n = +_1,+_2. . . . } form an orthogonal basis in
H1,A; the eigenfunction expansion f =
Z (f'×n)A n=-~ ti×nliA 2 - xn '
f ~ HI'A '
(3.7)
(n#O) converges in the topology of HA.
PROOF. Immediate consequence of the Spectral Theorem.
///
The eigenfunction expansion 3.7 can be interpreted in the framework of H1.
The operator B maps H1, A into H1, so the eigenvectors Xn belong, in f a c t ,
to H1.
Let the vectors An be defined by Cn = Kxn '
n = ±i,±2 . . . . . 2.9
we see that
(3.8)
From Eqns. 2.8
and
K maps H1 into i t s e l f ; hence, each An
belongs to H1.
On H1, A and K are each other's inverses.
Therefore, the
d e f i n i t i o n 3.8 i s equivalent with x n = A@n , The @n'S s a t i s f y
the
n = ±1,±2,.... equation KPT@n = ~nlAn •
(3.9) We observe that
KPT is
the
adjoint of the r e s t r i c t i o n of B to H1 considered as an operator in I_(H1). Thus, i f we denote this adjoint by B', we have the i d e n t i t i e s B'@n = ~nlAn ,
n = _+1,±2,....
(3.10)
That i s , the @n'S are eigenvectors of B' in HI . Because P1 is a cyclic vector for B in H1, the inner product (P1,An) is nonzero for each n.
We can therefore normalize the An'S by the condition
(PI,A n) = 0 ,
n = +_1,±2. . . . .
(3.11)
342
Note t h a t Xn ~ dom AT-1 and Cn ~ dom T-IA f o r each n, w i t h AT-Ix n = knX n ,
THEOREM 3.7. a
T-IA@ n = knT-ipT@n ,
(3.12)
n = ±1,±2,....
The e i g e n v e c t o r s {×n: n=±1,±2 . . . . } and {@n: n = ±1,±2 . . . . } form
biorthogonal
system
in
(×n,@n) # 0 f o r each n.
HI,
in
the
sense t h a t (Xm,@n) = 0 i__f_fm ¢ n and
The e i g e n f u n c t i o n expansion 3.7 can be w r i t t e n
in the
form f =
(f,@n)
Z ~ × n n:-~ (×n,@n) (n#O)
'
f c
(3.13)
HI'A "
PROOF. The theorem f o l l o w s from Theorem 3.6 and Eqn. 3.8.
The eigenvalues Xn have a c e r t a i n
///
symmetry which i s most e a s i l y
detected
when one uses the s o - c a l l e d switch o p e r a t o r S, Sf(u) = f ( - p ) Clearly,
S is a unitary
,
and B'Sf
f ~ H .
o p e r a t o r on H.
K, and anticommutes w i t h PT. BSf = -SBf
p ~ J ,
= -SB'f
(3.14)
I t maps H1 onto i t s e l f ,
commutes w i t h
Hence, S anticommutes w i t h B and B' on H1, i . e . , for
all
f ~ HI .
It
follows
that,
if
Cn i s
an
e i g e n v e c t o r of B' at the eigenvalue k~ I , then S@n i s an e i g e n v e c t o r of B' a t the
eigenvalue -k~ I .
The
eigenvalues
and
eigenvectors
can
therefore
be
indexed such t h a t
~-n = -~n '
×-n = S×n'
with kn > 0 for n=1,2,....
@-n = S@n'
,
(3.15)
We s h a l l assume t h a t the p o s i t i v e eigenvalues are
ordered such t h a t ~n < Xn+l f o r n=l,2 . . . . . each n.
n = ±i,±2,...
Note t h a t (X_n,~_n) = (Xn,¢n) f o r
343
Diagonalization.
Theorem 3.6
and
3.7
imply
that
the
topologically isomorphic with the sequence space z2 of a l l
space H1, A is square summable
(with respect to the weight ~) sequences c = [Cn: n = ±1,±2 . . . . ],
Cn c { ,
where the weight ~ is given by
~±n = ( X n ' ¢ n ) - 1
'
(3.16)
n = 1,2 .....
The isomorphism F which maps H1, A onto z2 o and i t s inverse are given by Ff
= [(f,@n):
n = ±1,±2,...]
F-ic = Z OnCnXn n=-= (n~O)
f c HI, A ,
,
(3.17) (3.18)
c ~ ~2 d
The transformation F diagonalizes the operator B on H1, A, FBf = [~l(f,@n):
We can write
this
n = ±1,±2 ....
]
result more succinctly
f c HI,A .
,
if
(3.19)
we introduce the (unbounded)
o p e r a t o r A on ~2: Ac = [~nCn: n = ±1,±2 . . . . ] , where dom A = {c ~ ~2: d
Z
n= _ ~
c ~ dom A ,
OniknCnl2 < ~}.
(3.20)
Then
(n40) FBf = A-IFf ,
f c HI,A .
(3.21)
The normalization condition 3.11 implies the i d e n t i t y FP1 = 1N, where 1N c ~ is the vector a l l components of which are equal to 1.
IV.
Solution of the Transport Equation
Reduction
of
the equation.
In
this
section we consider the problem of
constructing the general solution of the transport equation 1.9.
In view
of
344 the results of the f u l l - r a n g e theory we extend the equation to an equation in the space HA = Ho~HI, A.
We recall that the operator AT-1 is defined on a l l of
HO, and that B- I is the extension of AT-IIH 1 in HI, A.
Thus, the extension of
Eqn. 1.9 consists of the following pair of d i f f e r e n t i a l
equations:
(Po~)'(x) + AT-1Po~(X) = 0 ,
(4.1)
(P¢)'(x) + B-Ip@(x) = 0 .
(4.2)
The solution of Eqn. 4.1 is of the form (4.3)
Poe(X) = ml(X)T1 + mO(X)T21 ,
where the scalar quantities mO(x) and ml(x) s a t i s f y the pair of d i f f e r e n t i a l equations m~(x) = 0 .
ml'(x) + 2mO(x) = 0
(4.4)
The solution can be written down immediately, ml(x) = ~+26(1~T-x) ,
differential infinite
(4.5)
Thus, the problem has been reduced to the construction of the
general solution of the d i f f e r e n t i a l
Decomposition
= 6 ,
(The constant1~T has been inserted for l a t e r
where ~,6 ~ ~ are a r b i t r a r y . convenience.)
mO(x)
of
H1, A.
equation 4.2 in HI, A.
The construction
of
equation 4.2 is rather delicate,
number of positive
which accumulate at +~ and
and an i n f i n i t e -~.
We
the general
solution
of
the
because of the presence of an number of negative eigenvalues
are only interested in solutions that
remain bounded as x ÷ ±~.
For that reason we shall t r e a t vectors in the space
of
with
the
eigenvectors
×n
positive
index
(for
which ~n is
separately from those in the span of the eigenvectors ×n with
positive)
negative
index
345 (for which Xn is negative).
We shall do so by decomposing the space H1,A,
H1,A = H1, p 8 H1,m ,
(4.6)
where H1, p = cl sp(xn: n = 1,2 . . . . ) and HI, m = cl sp(x_n: n = 1,2 . . . . ), the closure being taken in the A-norm.
Clearly, the pair {Hl,p,Hl,m} reduces B,
with a(BIHI, p) = {Xnl: n = 1,2 . . . . } and a(BIH1,m)
= {-xnl: n = 1,2 . . . . }.
That is, the decomposition 4.6 reduces B to an accretive operator in H1, p and a dissipative operator in H1,m.
This reduction will enable us to write down
the general solution of Eqn. 4.2 in a Form suitable for our purpose. Let Pl,p denote the projection operator which maps HA onto H1, p along H0 e Hl,m, and Pl,m the projection operator which maps HA onto H1,m along H0 8 H1,p.
The representations of Pl,p and P1,m are Pl,p f = Z an(Pf,@n)Xn , n=l
f c HA ,
(4.7)
f c HA .
(4.8)
me
Pl,m f = X an(pf,@_n)X_n , n=l
The differential
equation 4.2 for P~(x) is equivalent with the following
pair of differential equations for the components Pl,p~(X) and Pl,m¢(X): (Pl,p~)'(x) + B-1PI,p~(X) = 0 ,
(4.9)
(P1,m¢)'(x) + B-iPl,m~(X) = 0 .
(4.10)
We solve these equations by semigroup methods.
Solution operator.
The accretive operator BIHI, p defines
the exponential
operator exp(-zB-1) on H1, p for z c {, Rez > O: exp(-zB-1)f = F-le-ZAF f ,
f ~ HI, p ,
Rez > 0 .
(4.11)
346 This operator is holomorphic in Rez > O, uniformly bounded f o r Sarg zl < l ~ x - c (E > 0), limit
and strongly continuous
coincides with the i d e n t i t y
(within
the ~-sector)
on HI, p.
at z = O, where i t s
The general solution of Eqn. 4.9
can be expressed in terms of this exponential operator, Pl,p¢(X) = exp(-xB-1)Pl,ph . Here Pl,ph c H1, p is arbitrary.
(4.12)
The solution is such that Pl,p~(X) remains
bounded (in fact, decays exponentially) as x + -, and its l i m i t as x+O exists in H1, p and is equal to Pl,ph. The switch operator S defined in Eqn. 3.14 and extended continuously to HA maps H1,p onto H1,m and vice versa.
We use i t to define the exponential
operator exp(-zB-1) on H1,m for z c {, Rez < O: exp{-zB - 1 ) f = Sexp(zB -1)Sf , This operator I~_~ its
(~ > 0), limit
is
holomorphic
in
f ~ HI, m ,
Rez < O, uniformly
Rez < 0 . bounded for
(4.13) largz-~l
and strongly continuous (within the ~-sector) at z = O,
coincides with the i d e n t i t y
on HI, m,
<
where
In terms of this exponential
operator, the general solution of Eqn. 4.8 is Pl,m@(X) = exp((~-x)B-1)Pl,mh where Pl,m h E H1,m is
arbitrary.
It
, is
(4.14) such that Pl,m~{X) remains bounded
as x + -~, and i t s l i m i t as x + T exists in HI, m and is equal to Pl,m hThe general solution 4.12 and 4.14,
of Eqn. 4.2 is obtained by adding the expressions
and the general
solution
of the f u l l
adding the expression for Po~(X) obtained e a r l i e r can
be
formulated
transport
to i t .
equation by
The final
result
succinctly i f we introduce the family of operators {U(z):
0 < Rez < ~} in HA thus:
347 U(z) = expC( ½T-z)AT-I]PO + e x p ( - z B - l ) P l , p + exp( (z-z)b - l ) P l , m
THEOREM4.1. • },
(i)
uniformly
(4.15)
U(z) i s holomorphic in the open s t r i p S = {z c { : 0 < Rez <
bounded f o r
z ~ S~ = {z ~ { :
larg zl <1/2T-~,
larg(T-z)-~I
1 ~ _ ~ } , ~ > 0, and strongly continuous ( w i t h i n S~) a t z = 0 and z = ~. (ii)
The function ~(x) = U(x)h s a t i s f i e s the p a i r o f equations 4.1,4.2 f o r
0 < x < • f o r any h E HA; conversely, each s o l u t i o n o f the p a i r o f equations 4.1,4.2 has the form ~(x) = U(x)h f o r some h ~ HA.
PROOF.
On HO, U(z) is represented by the matrix
i 2(1/2T-Z) CO i ) relative
to the basis
(T1,T21);
this
matrix
is clearly
z c {.
The sum exp(-zB-1)Pl, p + e x p ( ( T - z ) B - I ) P I , m is
holomorphic
for all
holomorphic
in
S,
uniformly bounded in S~, and strongly continuous as z = 0 and z = z. (ii)
The image of U(z)
h c HA, then ml(x)
is contained
in H0 ~ dom B- I .
If
and mo(x) have the form 4.5, where POh = ~TI + ~T21 and
Pl,p~(X) and Pl,m~(X) are given by 4.12 and 4.14, respectively.
V.
~(x) = U(x)h,
///
An Indefinite Sturm-Liouville Boundary Value Problem
General remarks. equivalent
with
The eigenvalue problem T-lAy = ~y for the operator T-1A an i n d e f i n i t e
~turm-Liouville
boundary value problem.
boundary value problem consists of the d i f f e r e n t i a l My = - ( p y ' ) '
= xry ,
is The
equation (5.1)
348
where p = [ 1 - t 2 : t c j ] ,
r = [t:
t c J ] , and the boundary condition
y bounded on (-1,1) .
(5.2)
This problem e x h i b i t s a number of i n t e r e s t i n g features.
F i r s t l y , the d i f f e r -
e n t i a l equation is singular at both endpoints of the i n t e r v a l (-1,1), as p-1 i s not integrable in a r i g h t neighborhood of -1 or in a l e f t neighborhood of +1.
Each endpoint is
a limit
circle
type of s i n g u l a r i t y :
all
solutions
of My = xy are in H for any ~ c ~, the same is true f o r a l l solutions of Eqn. 5.1 - - see Bradley [1972].
Secondly, the "weight" function r is p o s i t i v e on a
subinterval of J and negative on another two fundamental solutions of (l-t)). 5.2.
Eqn. 5.1
subinterval. are
Yl(t) = 1
Thirdly,
when ~ = O,
and Y2(t) = ~ n ( ( l + t ) /
Both solutions are in H, but only Yl s a t i s f i e s the boundary condition Thus, Yl is an eigenfunction belonging to the eigenvalue x = O.
In the l i t e r a t u r e on Sturm-Liouville problems associated with the general equation
-{py')'
(5.3)
+ qy = x r y ,
such problems are called " r i g h t d e f i n i t e " when the weight function r i s positive,
"left
d e f i n i t e " when p > 0 and q > 0 but q not i d e n t i c a l l y zero.
For
the l e f t d e f i n i t e case, the additional assumption Ir(t)I ~ clq(t)l
,
c constant ,
(5.4)
i s often made, e.g. in E v e r i t t [1974], Daho and Langer [1977]. The fact that x = 0 is an eigenvalue of the boundary value problem i s a n o n t r i v i a l complication.
In the r i g h t d e f i n i t e case the eigenvaue X = 0 can
be "removed" by a s u b s t i t u t i o n of the type ~ ~ X+~.
In the present case, such
a s u b s t i t u t i o n would y i e l d an equation of the type 5.3, where both q and r change sign.
349
Orthogonality.
I f the vectors @n and @mare eigenvectors of T-1A in HI at the
eigenvalues Xn and Xm' respectively, then A@n = XnT@nand A@m = XmT@m.
Taking
the inner product of the f i r s t equation on the l e f t with @m' and of the second equation on the right with @n' and subtracting the two resulting equations, we obtain the identity (Xn-Xm)(T@n,@m) = O. distinct
if
n
#
m.
Hence, the
The eigenvalues Xn and Xm are
eigenvectors @n and
@m satisfy
the
"orthogonality relation" 1 f U@n(P)~m(~)d~ = 0 , -1 Obviously,
n # m,
(5.5)
this is not a true orthogonality relation,
does not define a weight function.
because the factor
Nevertheless, Eqn. 5.5 is what used to be
called a full-range orthogonality relation in transport theory.
We now show
how i t is related to the biorthogonality relation, (Xn,@m) = 0 ,
n ~ m,
(5.6)
which we established e a r l i e r . From
the
eigenvalue
equation ×n = XnB×n and
the
r e l a t i o n ×n = Ken we
obtain the i d e n t i t y
Xn = XnPT@n . Combined with
the
(5.7)
biorthogonality
r e s u l t (PT~n,~ m) = 0 i f n # m. = (T~n,~m).
relation
5.6,
this
identity
yields
the
But ~n and ~m both belong to HI , so (PT~n,~ m)
Hence, (T~n,~ m) = 0 i f n # m, which i s p r e c i s e l y Eqn. 5.5.
It is
thus evident t h a t the occurrence of the i n d e f i n i t e weight f a c t o r ~ in Eqn. 5.5 i s r e a l l y due to the e l i m i n a t i o n of the eigenvector ×n from the biorthogonali t y r e l a t i o n 5.6.
The occurrence of a b i o r t h o g o n a l i t y r e l a t i o n which involves
the eigenvectors of both B- I and i t s a d j o i n t should not come as a s u r p r i s e , as we are dealing with operators which are not s e l f a d j o i n t . simple
relation
like
5.7,
which
enabled
us
to
The existence of a
e l i m i n a t e Xn, i s
f o r t u i t o u s and has no obvious analogue in more complicated models.
entirely
350
Co~leteness.
We now show that the set of eigenvectors of the boundary value
problem 5.1,5.2 is con~lete in the entire space H in the usual topology.
The
proof is a modification of a proof of Kamke [1939,1942], who derived a similar result for l e f t - d e f i n i t e Sturm-Liouville problems.
The details are rather
straightforward and w i l l be omitted. Kamke considered the general boundary value problem Ny = - ( p y ' ) ' + qy = Xry ,
(5.8)
U(y)
(5.9)
= 0
,
on a compact interval [a,b].
The coefficients p, q and r are assumed con-
tinuous on [a,b], with p s t r i c t l y positive; r changes sign on [a,b] at least once, but no more than a f i n i t e number of times.
The (two-point)
boundary
condition 5.9 is such that the boundary value problem described by Eqns. 5.8 and 5.9 is selfadjoint in L2(a,b).
Furthermore, i t is assumed that x = 0 is
not an eigenvalue and that ~b yNy > 0 for every function ycC2([a,b]) which Q
satisfies the boundary condition 5.9. Kamke showed that the boundary value problem 5.8,5.9 has an i n f i n i t e number of positive eigenvalues, as well as an i n f i n i t e number of negative eigenvalues, which can be ordered such that X-n-1 ~ X-n < 0 and 0 < Xn < Xn+l for n 1,2,... .
Each eigenvalue has a f i n i t e m u l t i p l i c i t y , and ZIXnT-1 = ~. Furn thermore, there exists an i n f i n i t e system of eigenvectors {¢n: n = ± 1 , ± 2 , . . . } , which can
be normalized such that
~b r~m~n = ~mnSgn(~n). u
The series
Z IXnl-2~R(t)- converges for all t c [a,b], and the sum S(t) satisfies the n inequality S(t) < G ( t , t ) , where G is the Green's function of the boundary value problem 5.8,5.9 when r ( t ) = I .
Finally, each function f ~ C2([a,b])
which satisfies the boundary condition 5.9 has an eigenfunction expansion of the form b
f = Z
n=_~
(n~O)
Cn~n ,
where c n = sgn(x_)f
n a
f~nrdt ,
351 and t h i s series expansion converges absolutely and uniformly on [ a , b ] . Kamke presented two proofs Hilbert's
of
theorem on polar integral
the above results;
one proof based on
equations can be found in Kamke [1939,
Section 3], another proof based on the calculus of variations in Kamke [1942]. The boundary value problem 5.1,5.2 does not f i t
d i r e c t l y in the framework
of Kamke's results, so some modifications and extensions of Kamke's proofs are necessary. Firstly,
the boundary condition 5.2 is not of the type 5.9; U(y) repre-
sents a l i n e a r specific
combination
of
the values of y and py'
at a and b.
The
form of 4.9 allows Kamke to get the boundary terms to vanish upon
integration
by parts.
conditions
However, as we have seen in Theorem 2.1, the boundary
5.2 are equivalent with several other boundary conditions;
using
these equivalences, we can apply Kamke's proof almost verbatim. Secondly, boundary
value
the boundary problem
significant
as
function
5.1,5.2
of
it
might is
value problem 5.1,5.2
5.8,5.9
is
regular.
seem, however, still
is
This
because
a Hilbert-Schmidt
singular, difference
the kernel.
whereas the is
generalized
not
as
Green's
The generalized
Green's function is given by
~(t,s) : k(t,s) I t l ~ Is~~ , where k is
the
generalized Green's
t,s c J , function
operator, which was given in Eqn. 2.3. uniform convergence on ( - i , I ) ,
of
the Legendre d i f f e r e n t i a l
Kamke's proof applies, but instead of
we only obtain uniform convergence on compact
subintervals of ( - i , I ) . T h i r d l y , ~ = 0 is an eigenvalue for the boundary value problem 5.1,5.2, with the eigenfunction ~0 = i . proof.
This case is excluded by assumption in Kamke's
Nevertheless, by considering the boundary value problem 5.1,5.2 in the
space G = { i } &, rather than in the f u l l eralized Green's function
k,
space H = L 2 ( j ) ,
and using the gen-
we recover a l l of Kamke's results in the closed
352
subspace G,
including the completeness of the eigenvectors {~n: n=±l,±2. . . . },
and all that needs to be done is supplement this system by the single vector ~0 = 1 to achieve completeness in the entire space H. The above three points refer to both methods of proof of Kamke's.
To see
how Hilbert's theorem on polar integral equations is relevant to the boundary value problem 5.1,5.2, we observe that the nonzero eigenvalues and corresponding eigenfunctions can be characterized as eigenvalues and eigenvectors of a polar integral equation.
LEMI~ 5.1.
The function y ~ G = {1}~ is a solution of the boundary value
problem 4.1,4.2 at the eigenvalue ~ i f and only i f the function z defined by z(t) = I t ~ y ( t ) ,
t ~ J, is a solution of the polar integral equation 1
z(t)
: ~ f
k ( t , s ) V ( s ) I t l l ~ I s I~ l z(s)ds ,
t cJ ,
(5.107
-1 where V(s) = 1 i f 0 < s < 1, V(s) = - i i f -1 < s < O.
PROOF. The proof is straightforward and w i l l be omitted.
Hilbert's
theorem on polar integral
///
equations yields that Eqn. 5.10
admits a system of solutions {Zn: n = ±1,±2. . . . } which is complete in G.
The
eigenvectors of 5.1,5.2 which correspond to the nonzero eigenvalues are then given by Yn(t) = I t T - l ~ z n ( t ) , n = ±I,±2 . . . . .
THEOREM 5.2.
The system of functions {Yn: n = ±1,±2}, supplemented by the
function YO = 1, is complete in H.
PROOF. Suppose h c G = {1}~ and (h,y n) = 0 for n = ±I,±2 . . . . . 1 h(t) Jtl - I ~ Jt11~yn(t)dt
0 = (h,y n) = -1
= f h(t) ItJ-1
Zn(t)dt •
Then
353
Hence, h(t) Itl-16 = O, a.e., consequently, h = O.
Asymptotics.
III
Asymptotic estimates of the eigenvalues ~n and the normalization
constant (×n,¢n) were obtained recently by Veling [1982].
His results can be
summarized as follows: ~n = A(n+1/2)2 + B + O(n-1) ,
n÷ ~ ,
(5.11)
.3/2 (Xn,~n) = (L/2~ + o(1))A n ,
n÷ ~ ,
(5.12)
where the constants A, B, and L involve certain e l l i p t i c integrals; numerical values are A = 6.87518590, B = -0.91184985, L = 1.19814023.
I t follows from
Eqns. 5.11 and 5.12 that the spectral measure ~n decays l i k e n-3 as n ÷ ~.
Wumerical results.
Using the code "SLEIGN" developed by P. Bailey [1978] at
Sandia Laboratory we computed the f i r s t ten positive eigenvalues ~n of the boundary value problem 5.1,5.2.
Although this
code was developed and
j u s t i f i e d only for Sturm-Liouville problems with a positive weight function, we believe that i t is also effective in our case. Veling [1982] independently computed ~1 and ~10 and his results agree with ours.
The computed eigenvalues
are l i s t e d below.
Xl
= 14.527
,
X2 = 4 2 . 0 5 0
,
~3 = 8 3 . 3 0 4
,
X4 = 1 3 8 . 3 1
,
~5 = 2 0 7 . 0 6
,
~6 = 2 8 9 . 5 6
,
~7
= 385.82
,
~8 = 4 9 5 . 8 1
,
x9
= 619.57
,
~i0
Properties of the eigenfunctions.
= 757°07
.
In general the eigenfunctions of a Sturm-
L i o u v i l l e problem on an interval (a,b) have the property that the eigenfunction belonging to the smallest eigenvalue has no zero on (a,b), the eigenfunction belonging to the next smallest eigenvalue has exactly one zero on
(a,b),
354
and so on.
The corresponding property of the eigenfunctions of the boundary
value problem 5.1,5.2 was established recently by Kwong [1982].
THEOREM 5.3 (Kwong). For each n (n=1,2,...) the eigenfunction @n has exactly n zeros on (-1,1), which are a l l positive.
As a consequence of the symmetry relation 3.11, Theorem 5.3 implies that, for each n (n=1,2. . . . ), the eigenfunction @-n has also exactly n zeros on (-1,1); these zeros are all negative.
VI.
Half-Range-Theo~
Heart of the matter.
As we have seen in the Introduction, boundary conditions
for transport equations are of the so-called half-range type.
They involve
only one half of the range of the independent variable u at each endpoint. Although as a result of the full-range theory the structure of the general solution of the d i f f e r e n t i a l equation is transferred immediately to
known, this knowledge cannot be
the half-range subspaces H+ and H_, where the
boundary data are specified.
The basic reason is,
of course, that the
transport operator AT- I does not commute with the projection operators P+ and
P_. The half-range theory is meant to deal with this d i f f i c u l t y .
Here, the
objective is to extend vectors g+ c H+ and g_ ~ H_ with components in H and H+, respectively, in such a way that the extended vectors, E+g+ and E_g_ say, belong to subspaces of H where the structure of AT-1 is known. Whether this objective can be achieved for the electron transport operator under consideration is s t i l l an open problem. The physics suggest that i t must be possible, and that the extension operators
E+ and E are well-defined at least for some
355
classes of vectors g+ and g_. best.
Also,
But this type of evidence is suggestive at
there is an important class of transport problems for which a
half-range theory has been developed successfully.
These transport problems
involve collision operators A that are close to the identity, e.g., contract perturbations of the identity; see the recent monograph by Kaper, Lekkerkerker and Hejtmanek [1982].
I t would be interesting to see whether these theories
could be extended to include unbounded collision operators A. At this
point we could stop our exposition,
but we prefer to take a
somewhat unorthodox approach. The development of a half-range theory for the electron transport operator (and similar operators) poses a challenge that we don't intend to monopolize, and we believe that we have some ideas worth presenting.
However, we emphasize that most of the following
is
purely
speculative and that further research is certainly called for.
Eigenvectors revisited. namely that
We begin with the observation of an interesting fact,
the set of eigenvectors {Xn: n = ±1,±2,...} of the operator B
constitutes a basis not only in the topology of the A-inner product, as shown in Theorem 3.6,
but also
in the stronger topology
of the ordinary
inner
product.
LE~A 6.1.
The vectors {Xn: n = ±1,±2. . . . } form a basis in the space H1 i n
the topology induced by the usual inner product on H.
PROOF.
We alrea~
dom AT-11H1.)
know that
each ×n belongs to
Suppose (f,Xn) = 0 for n = ±1,±2,...
H1. for
(pTf,xn) A = (f,PTKxn) = (f,B×n) = x~l(f,× n) = 0 for each n.
(In
fact,
some f ~ H1.
Xn c Then
The conTpleteness
of the set of eigenvectors {×n: n = ±1,±2,...} in HI, A implies that PTf = 0. Because f c HI, i t follows that Tf is a multiple of TI, i . e . , f is a multiple of 1.
But now f c H1 implies that f = O.
This proves the lemma.
///
356
Decon~osition of HI . limits In
The lemma shows that we can represent vectors in HI as
(in the ordinary topology) of linear combinations of the vectors Xn.
particular,
we can use
the
two
sets of vectors {×n: n = 1,2 . . . . } and
{×n: n = -1,-2 . . . . } to define the subspaces H1, p and H1,m of H1, H1, p = cl sP(×n: n=1,2. . . . ),
HI, m = cl sP(×n: n=-1,-2,...)
where the closure is taken in the ordinary topology.
(6.1)
Thus we obtain the
following decomposition of HI: HI = H1, p $ H1,m .
(6.2)
Note that H1, p and H1,m are mapped onto each other by the switch operator S defined in Eqn. 3.14. The decomposition 6.2, although not orthogonal with respect to the usual inner product, nevertheless induces an interesting "orthogonality property" on dom AT-11H1.
The orthogonality
is
with
respect to the
indefinite
inner
product { . , . } , which can be defined on dom T-1 {f,g} = (f,T-Ig) ,
f,g ~ dom T- I
(6.3)
One verifies that {f,g} = 0 for any pair f C H l , p n dom AT-I,g ~ H1,m n dom AT-1 " Moreover, { f , f } ~ 0 for
all
f c H1, p n dom AT-1
and { f , f } ~ 0 for
all
f c H1,m n dom AT-1, with equality i f and only i f f = O, so { . , - } is positive on H1, p n dom AT-1 and negative on HI, m n dom AT-1.
These considerations w i l l
guide us in the decomposition of the two-dimensional subspace HO.
Decon~osition of 14D.
We now decompose H0 into two subspaces HO,p and HO,m
such that HO,p and HO,m enjoy the same properties as H1, p and Hi,m: they have a trivial
intersection, they are mapped onto each other by the switch operator
S,
are
they
orthogonal
H0 c dom AT-l),
with
respect to the indefinite inner product { - , . }
(note
that
and { . , . } is
positive
HO,m.
Such a decomposition is indeed possible:
on HO,p and negative on
357 (6.4)
H0 = HO, p ~ HO, m , where HO,m = sp(eT1-T21) .
HO,p : sp(eT1+T21) ,
(This result shows that H0 is a Krein
Here, e can be any positive number. space;
cf.
Bognar [1974, Section
(6.5)
V.1].)
The constant e is
determined
uniquely, for example, by requiring that
(6.6-1)
Ho,p = HO n (H+ ~ H1, m) Then one also has Ho,m = HO n (H_ ~ H1, p)
(6.6-2)
This choice of c) is, in fact, most convenient for the following analysis; cf. the discussion in Kaper, Lekkerkerker and Hejtmanek [1982, Section 9.1].
D e c ~ o s i t i o n of H.
We now combine the decon~ositions 6.2 and 6.4 to obtain a
decon~Dosition of H.
We define
Hp = HO,p ~ H1, p ,
Hm = HO,m @H1,m
(6.7)
Then, because H = H0 ~ H1, H = Hp ~ Hm . It
is this
(6.8)
deco~osition 6.8 that we want to connect with the direct sum
decomposi t i on H = H+ ~ H_
(6.9)
Notice that {f,g} = 0 for any pair f c Hp n dom AT-1, well as for any pair
f c H+ n dom T-1,
g c Hm n dom AT-1,
g c H_ n dom T-1.
as
Moreover, { . , . } is
358
positive H
on Hp n dom AT-1 and H+ n dom T -1, and negative on Hm ndom AT-1 and
n dom T - I .
common i s
Another property
that
that
the decompositions
6.8 and 6.9 have in
the component subspaces are mapped onto each other
by the
switch operator S. The idea is bijections imply
the p r o j e c t i o n s
Such a r e s u l t would
sum decompositions H = H+ ~ Hm and H = H_
We could then define E+ on H+ as the inverse of P+IHp and E_ on H_ as inverse
of
P_IH m.
Trivially
extended
operators would then be the p r o j e c t i o n H_ and onto Hm along H+,
Or,
operator
AT- I ,
the
entire
so t h i s
would
space H,
viewed d i f f e r e n t l y ,
the element E+g+ c Hp and E
element g_ ~ H_ to the element E_g_ cHm. the
to
these
operators which mapped H onto Hp along
respectively.
extend an element g+ c H+ to
of
P+ and P_ define s u r j e c t i v e
of Hp onto H+ and of Hm onto H_, r e s p e c t i v e l y .
t h a t we also had the d i r e c t
Hp. the
to prove that
E+ would
would extend an
On Hp and Hm we know the s t r u c t u r e
solve
the
problem of
the half-range
theory. Of
course,
it
may be t h a t
subspaces of Hp and Hm. restricted
P+ and P_ are
In that case the d e f i n i t i o n s
to the corresponding
ranges of P+ and P_.
these ranges may be a major problem. continuously
injective
only
on c e r t a i n
of E+ and E_ have to be The i d e n t i f i c a t i o n
of
Also note t h a t P+ and P_ do not extend
to HA .
Connecting transformations.
Let Pp and Pm denote the projection operators
associated with the decomposition 6.8.
The projection operators associated
with the decomposition 6.9 are P+ and P_.
The pairwise connections of the
subspaces in the decompositions 6.8 and 6.9 are established by means of the so-called connecting transformations, V = P+Pp + P-Pm '
V# = PpP+ + PmP- '
(6.9)
W = P+Pm + P-Pp '
W# = PpP- + ProP+ "
(6.10)
359
They satisfy
several
inner product { - , . } .
interesting identities with regard to the indefinite Using the abbreviations
(f'g)B = {(Pp-Pm) f ' g } '
(6.11)
(f'g)T = {(P+-P-)f'g} '
(6.12)
whenever these expressions are meaningful, one verifies that (Vf,g) T = (f,V#g) g ,
(Wf,g)T = -(f,W#g) B ,
(6.13)
which suggests that V and V# are, in a sense, each other's adjoints, W and W# each other's anti-adjoints.
In
all
Furthermore,
(Vf'Vf)T - (Wf'Wf)T = ( f ' f ) B '
(6.14)
(V#f'V#f)B - (W#f'W#f)B = (f'f)T "
(6.15)
these identities,
f
and g must be restricted to the appropriate
domains.
Shadow operators.
theory
with
operators
We conclude this speculative discussion of the half-range
another
E+ and E_.
suggestive Basically,
remark about the half-range theory
role
of
the
extension
is concerned with the
analysis of the (full-range) operator AT-1 on the half-range subspaces H+ and H.
The way this is achieved is by extending elements of these half-range
spaces to elements of the full-range space.
That is,
by means of E+ an
element of H+ is extended to an element of Hp; on Hp, the action of AT-1 is known; having applied AT-1 to the extended element, we then project back into H+ by means of the projection operator P+. for H_.
A similar procedure is followed
As a result we have the decomposition AT-1 =
P+AT-1E+ + P_AT-1E_
(6.16)
360
The operators P+AT-1E+ and P_AT-1E_ might be called the shadows of AT-1 on H+ and H_.
References
Akhiezer, N. I.
and I.
M. Glazman [1981], Theory of Linear Operators in
Hilbert Space, Pitman, London. Bailey, P. B. [1978], SLEIGN, An Eigenvalue-Eigenfunction Code for SturmLiouville
Problems, Sand 77-2044, Sandi a National
Laboratory,
Albuquerque, New Mexico. Beals, R. [1977], On an Equation of Mixed Type from Electron Scattering, J. Math. Anal. and Applic. 58~ 32-45. Bethe, H. A., M. E. Rose and L. P. Smith [1938], The Multiple Scattering of Electrons, Proc. Am. Phil. Soc. 78_, 573-585. Bognar, J. [1974], Indefinite Inner Product Spaces, Springer-Verlag, New York. Bothe [1929], Zeitschr. f. Physik 54___,161. Bradley, J. S. [1972], Comparison Theorems for the Square I n t e g r a b i l i t y of Solutions of ( r ( t ) y ' ) ' + q(t)y = f ( t , y ) , Glasgow Math. J. 13, 75-79. Daho, K. and H. Langer [1977], Sturm-Liouville Operators with an Indefinite Weight Function, Proc. Roy. Soc. Edinburgh 78A, 161-191. Everitt,
W. N. [1974], Some Remarks on a Differential Expression with an
Indefinite
Weight Function,
in:
Spectral
Theory and Asymptotics of
Differential Equations, E. M. de Jager (Ed.), Mathematics Studies, Vol. 13, North-Holland Publ. Co., Amsterdam. Everitt,
W.
N.
[1978],
Legendre Polynomials
and Singular
Differential
Operators, in: Ordinary and Partial Differential Equations, W. N. Everitt (Ed.), Lecture Notes in Mathematics, Vol. 827, Springer-Verlag, New York.
361
Kamke, E. [1939], Zum Entwicklungssatz bei polaren Eigenwertaufgaben, Math. Z e i t s c h r i f t 45, 706-718. Kamke, E. [1942], fiber die definiten selbstadjungierten Eigenwertaufgaben bei gew~hnlichen linearen Differentialgleichungen, I , I I , I I I , I V ,
Math. Zeit-
s c h r i f t , 45 (1939), 759-787; 46 (1940), 231-250 and 251-286; 48 [1942), 67-100. Kamke,
E.
[1971],
Differentialgleichungen, L6sungsmethodenund L~sungen ,
Chelsea Publ. Co., New York. Kaper, H. G., C. G. Lekkerkerker and J. Hejtmanek [1982], Spectral Methods in Linear Transport Theory, Birkh~user, Basel. Kato, T. [1966], Perturbation Theory for Linear Operators, Springer-Verlag, New York. Kwong, M. K. [1982], personal communication. Veling, E. J. M. [1982], Asymptotic Solution of the Eigenfunctins of a Linear Transport Equation Arising in the Theory of Electron Scattering (to appear).
Non-normalizable
eigenfunction expansions
for
ordinary differential operators Robert M. Kauffman 0. Introduction In physics,
one frequently wishes to represent arbitrary waves in
terms of standing waves.
This is especially true in quantum mechanics,
where the standing waves are eigenstates of the Schroedinger operator. It is important
that this representation be a pointwise representation,
rather than a representation
in some abstract
space, because,
be illustrated in this note,
facts of mathematical
as will
and physical
sig-
nificance may be easily deduced from such a concrete representation. At a point of the continuous
spectrum of an ordinary differential o p e ~
ator in L2(R), where R denotes the real numbers, square-integrable
eigenfunctions;
there need be no
hence one must in general deal with
eigenfunctions which are not square-integrable. What should be the asymptotic behavior at infinity for the eigenfunctions of the representation?
If we ask our eigenfunctions
to be
too well-behaved at infinity, we will not have enough eigenfunctions to complete the representation. of growth of the eigenfunctions it is well-known
However,
some restriction on the rate
at infinity seems reasonable,
in physics that not all solutions
equation are acceptable
eigenfunctions,
because
to the differential
and physicists
in many ex-
amples use requirements of "smallness at infinity" to separate the acceptable
from the unacceptable
eigenfunctions.
In this note, we state such a representation
theorem for self-
adjoint ordinary differential operators in L2(-~,~) We then apply the representation seem to have qualitative physical
of arbitrary orde~
to prove new results, significance,
some of which
about such ordinary
differential operators and their associated unitary groups.
The
growth rate at infinity we shall use for an eigenfunction f is the condition that f = (M~)', where M = (x2+l) 3/4+E and ~ L I ( R ) .
We do not
363
prove the representation
theorem,
as the proof is technical and will
appear elsewhere. Our starting point is the work of Gelfand and Kostyuchenko, which appears in Gelfand-Shilov wise representation,
[2].
This work does not give a point-
but does guarantee that eigenfunctions with the
growth rate given above may be used to calculate cients.
the Fourier coeffi-
There are some points in the proof given in [2] which are
difficult to follow, particularly
the assertions
involving the
differentiation of a Hilbert-space valued measure with respect to another measure.
For this reason, we outline briefly another approach
to their theory, built around the Radon-Nikodym theorem.
(The details
will appear elsewhere.) Another pointwise representation by Bennewitz
[i].
theorem has been recently given
This interesting theorem, which also applies to
the more general situation of pairs of differential operators,
is
based on a completely different construction from the one we use. His method does not give a rate of growth at infinity for the eigenfunctions, but may give more information than ours does about other aspects of the problem.
In applications
(such as the ones given below)
which do not involve the rate of growth of the eigenfunctions
at in-
finity, Bennewitz's representation may be used in place of ours. relationship between the two representations
The
deserves further study.
While this note deals only with the case of an ordinary differential operator,
the author plans to discuss the case of an elliptic
partial differential operator in a later paper. i. The representation. Notation I.i.
Let M = (x 2 + i) 3/4+E, for any positive real num-
ber s which will remain fixed throughout this note. closure of C0(R)
in the norm given by Iieii¢ =IIMS'JI~,
Let ¢ denote the where R denotes
•
the real numbers.
Let L = Z
a.D l bean Nth order formally symmetric z
ordinary differential expression, where D = d/dx, C~(R), with aN(x)
¢ 0 for any x in R.
and each a i is in
(L is said to be formally
364
symmetric if L = L +, where L + is the formal (Lagrange) adjoint of L.) Let H denote a self-adjoint extension of the minimal operator T0(L) in L2(R).
Let S(e) denote the closed linear span in L2(R) of {Ute},
where e is in L2(R) and U t is exp(iHt).
Let E(A) denote the spectral
projection for H corresponding to the Borel set A° = (E(A)e,e), for e in L2(R).
Let E X = E(-~,X].
orthogonal projection in L2(R) on S(e).
Let ~e(A)
Let PS(e) denote the
If ~ is in L2(R),
let C#,e(X)
denote the Fourier coefficient of PS(e)~, which is defined by the relation PS(e)~ = f C~,e(X) dExe. C~, e exists and is well-defined by standard functional-analytic arguments u.Sing the spectral theorem. Theorem 1.2.
(Gelfand-Kostyuchenko).
For any e in L2(R), there
exists a family {fX,e } of C ~ complex-valued functions on R such that, for almost every real X with respect to o e, the following hold: a) fX,e = (M~x,e) ' for some C a function ~X,e in LI(R); b) Lfx, e = XfX,e; c) for any ~ in ~, C~,e(X) = f~ fX,e(X)~(x) dx. Outline of proof: U~,e(A)
For any ~ in ~ and any e in L2(R) define
to be (E(A)~,e) for any Borel set A.
rem of Kostyuchenko
(see Gelfand-Shilov
It follows from a theo-
[2], Theorem IV.6.1) that
there exists a positive constant K such that for any partition {Ai} ~ of (-~,~),
E~ lIE(Ai)ell%, ! K.
(i)
where we identify E(A)e with the element F of }' given by F(%) = (%,E(A)e). Since u~, e is absolutely continuous with respect to °e' it follows that du%, e = h~,edO e for some element h%, e of Ll(Oe). fX,e(%)
to essentially be h%,e(l),
We define
although it takes some work, using
(I), to show that this definition makes sense for almost every I with
365
respect to oe and gives an element of ~'
It is then possible to show
that fX,e(H#) = XfX,e(~) for all ~ in C0(R).
By distribution theory,
it follows that fX,e is a C ~ function and L =~X,e = Xfx,C
Since fX,e is
in ~', one may fairly easily obtain the desired asymptotic behavior of fx,eat ~.
That fX,e may be used in the desired way to calculate
the Fourier Coefficients
follows directly from the relation
PS(e) ~ = /C~,e(X)dExe,since
(Ps(e)~, E(A)e) = fA h~,edO e for all A.
The details of the proof will appear elsewhere. Remark:
We now state the main theorem of the paper.
The proof
will appear elsewhere. Theorem 1.3.
Suppose that, for some positive real number s,
(Hf,f) ~ g(f,f) for all f in domain H. H I/N , and ~ is in S(e).
Suppose ~ is in the domain of
Then C#,e(X) fX,e(X)
is in Ll(O e × R), and,
for almost every x in R, ~(x) = f C#,e(X)fx,e(X)
dOe(X).
Also, for each real t, Ut~(x) = f C~,e(X)eiXt fx,e (x) dOe(X) for almost every real x. Remark:
The above theorem leads fairly easily to the following
corollary, which gives a decomposition
of an arbitrary member of the
domain of H I/N. Corollary 1.4.
Suppose that (Hf,f) ~ s(f,f) for all f in domain
H, where ~ is a positive real number.
Then there exists a set {ei} ~
of members of L2(R) such that k < N and a) S(e i)IS(ej)
if i ~ j;
b) E~ S(e i) = L2(R); c) o
(A) = 0 if and only if E(A) = 0; e1 d) Oe. is absolutely continuous with respect to Oeo for i <j; j i e) Oej(A) = 0 iff E(A) PS(em ) = 0 for all m e j; f) if A = {Xi{fx, e } is linearly dependent}, i I/N g) any ~ in domain H is given by
~(x)
=
Z ki=l
SC~,ei(X)
fX,ei(X)
then E(A) = 0;
d°e.(X)l for almost every real x;
366
h) if 9 is in the domain Utg(x)
of H I/N
= Zki=I fe i%t Cg,ei(%)
f%,ei(X)
and t is any real number daei(%)
for almost
every
real x. Remark:
We now apply
the representation
to obtain
information
about Utg. Corollary
I.S.
Let H be as in Theorem
the domain
of H I/N,
in (a,b).
Then 9(x) = 0 almost
Proof:
and that,
(Utg,0)
= f~ f~,9(x)
most
every
subset
w i t h respect
f~,9(x) almost
every x
that,
1.6.
Borel
linear
for almost
Since
Thus,
every
= 0 for al% with
1.3,
to Lebesgue
this
implies
every
Since Lf~, 9 to 09,
that 9(x)
= 0 for
measure). 1.3.
Suppose
that,
= 9 for some 9 in L2(R).
that the restriction since
dense
for almost
% with respect
every x in (a,b).
A is bounded.
Then 9(x)
Suppose
= 0 for all x.
of H to E(~)L2(R) Hence
continuous.
for
is a
such a 9 is neces-
Therefore,
9(x) = 0
But Ut9 = E0(itH)ng/n! , since the restriction
is bounded.
is clear that Hng(x)
every
= 0 for all x in (a,b).
set &, E(A)9
for every x in (a,b).
that the Fourier
that p(X)
for almost
in L2(a,b).
in domain H, and is therefore
of H to E(A)L2(R)
d~9(X).
Let H be as in Theorem
operator,
1.3,
that
this implies
Hence,
By Theorem
First note
= i, Theorem
0(x)dx = 0 for all 0 in a countable
to o 9 , f%,9(x)
= 0 for almost
Proof:
such x.
to 09 .
every x (with respect
that 9(x)
= 0 for almost
is zero, where
However,
is dense
= 0 for all x.
some bounded
sarily
which
it follows
Corollary
bounded
~(x)dx.
to 09, f~ f%,9(x)
= %f~,9'
9 is in
for all t, it follows
p(~)dog(~)
% with respect
of C0(a,b)
guarantees
fx,9(x)0(x)dx]
on (a,b)
of the measure
p(%)
respect
that
everywhere.
theorem,
If~
= fe i~t
if Ut9 vanishes
transform
for all t, Utg(x)
Suppose
For any ~ in C0(a,b) , since C9,9(~)
together with the Fubini
Hence,
1.3.
Since H is a differential
= 0 for all x in (a,b).
operator,
Hence Utg(x)
9 is in the domain of H I/N , the previous
it
= 0 for all
corollary
367
guarantees that ~(x) = 0 for all x. Remark:
The last two corollaries,
in the case where H is a one-
dimensional Schroedinger operator, provide a qualitative explanation of a physical phenomenon known as tunnelling. that,
Corollary los shows
if the wave function ~ is in the domain of H I/N , Ut~ spreads
throughout one-dimensional
space at some past or future time.
No po-
tential barrier except a singularity is high enough to contain Ut~ for all t.
However,
Corollary 1.6 tells us that this is not surpris-
ing, since either the wave function is spread over all space at time t = 0, or else the wave function has a positive probability associated with arbitrarily high energy values, because E(A)~ is the component of with energies
lying in the Borel set A. References
i. C. Bennewitz,
"Spectral theory for pairs of differential
operators," Ark. Matematik.
15(1977)
2. I. Gelfand and G. Shilov, Academic Press, New York,
1977.
33-61.
"Generalized functions, vol.
3,"
SOME ASPECTS
AND
RECENT
CONTROL
DEVELOPMENTS
IN L I N E A R
THEORY.
H.W.Knobloch
Introduction. A typical theory
situation
which
is d e p i c t e d
a given
underlies
in t h e d i a g r a m
many
system which
reacts
in two w a y s
with
its e n v i r o n m e n t
namely
through
pair
of
input variables
of o u t p u t control, v
variables i.e.
represents
controlled precise
the portion
which
of t h e
The output
and the variable
In g e n e r a l
terms
the p r o b l e m
To f i n d
servations
(i.e.
upon evaluation upon
the of
of
v
Our
considerations
choice y(s)
time
will
of Fig.
input
and assuming
relations
and
It w i l l appear
u
cases
at o u r d i s p o s a l .
system which
cannot
are n o t a c c e s s i b l e
be
to
of t h e o b s e r v e d
variable
y
instantaneous
situation
of
the
u
which
with
and which
namely
can then
is b a s e d
in a p a r t i c u l a r
s~t)
the
z.
be p h r a s e d
on past
moment
reduces
ob-
t
the
of the
depends influence
matrices
In o t h e r
words,
for t h e
is f i n i t e
dimen-
we establish
a state variable
model
z = Dx
.
a x
(1.1)
of a p p r o p r i a t e
the role
be convenient
in the d i s t u r b a n c e
this model
form
y = Cx,
plays
a state-space-model
that
by i n t r o d u c i n g
same unquestioned
sometimes
upon
linear.
and output
are c o n s t a n t
state variable
not have
of
be b a s e d
= A x + B u + Gv,
the
o n the
is c o n c e r n e d
for
for
we have
is to be c o n t r o l l e d ,
I. W e a s s u m e
invariant
link between
A,B,C,D,G
> z
z.
configuration sional,
one
a strategy
> y
Fig.1
consists
which
I
plant . __
input which
about
represents |
u ~ v
a pair
acting
in modern control
"plant"
~
a
in m o s t
is a v a i l a b l e
the plant)
I. T h e
is t h e
and which
measurement.
as f o l l o w s :
and
u
the disturbances
directly
(information
(u,v)
(y,z).
considerations
of Fig.
dimensions.
of an a u x i l i a r y
physical
meaning
to i n c o r p o r a t e and h a v e
variable
as i n p u t in
nothing
x
Note
that
and does
and output.
elements
to d o w i t h
which
the u s u a l
369
interpretation
of t h e
If o n e w o r k s
with
of a c o n t r o l
action
to t h e v a l u e turbance
0
and
wish
x
tion
call
class
objective
this
is
v
zero
for
has
observer
of p r o b l e m
To be m o r e
identified
v
with
with
dynamics.
find
a feedback/observer
been
i d e a of
extensively
model
prinicple."
whole
system
time
from
objective
a difference
signals
in t h e
0. W e
disturbance,
until
to
the
The equilibrium if t h e
system.
disturbance
will
feedback/
A new type
a n d we h e r e
all d i s t u r b a n c e
which
given
are
is n e e d e d
(i.e.
the
one assumes
- as
this
should
choice
design
This
as
is t h e n
of
principle "learn"
it n e e d s
and solution
"internal
is t h a t
the
in t h e c o u r s e about
depend
in
the
the
z ). It m a k e s
it is d o n e
to
effective
control 8);
called
of c o u r s e
we
is a g e n e r a l i z a t i o n
Chapter
is o f t e n
will
words,
belong c a n be
of f u n c t i o n s
integral"
([I],
as m u c h
How much
In o t h e r
space
is s i m u l t a n e o u s l y
plus
behind
signals
this
to a c l a s s
class.
problem
controller)
that
of r e g u l a t o r
by Wonham
synthesis
and
system.
"proportional
upon
as
of d i s t u r b a n c e
loop
belong
objective
The philosophy
whether
that
of a l i n e a r
its o w n r e a c t i o n
of d i s t u r b a n c e . design
assume
studied
(plant p l u s
regard
situation.
strategy
all d i s t u r b a n c e s
of the c o r r e s p o n d i n g
the
The
considera-
exponentially
of t h e t i m e
s p a c e of f u n c t i o n s
the output
that
of t h e r e g u l a t o r
to w a i t
the p r o b l e m
closed
u = -Fx,
to the v a l u e
to do is t h e n
course
We
t y p e of a n
to t h e
(or d e c a y s
stabilization.
should
observer,
is a m o m e n t a r y
t~t °
a stable
we will
a known
has
of
all d i s t u r b a n c e
of the c l a s s i c a l
close
is a p e r s i s t e n t
this
with
against
paper
of d i s -
words: law
aim
close
itself
and we also will z
that
for
in t h e
creates if
synthesis
v
a problem
action
special
the c a s e
thing
specific
that
only
and then
arises
the
in t h i s
the o u t p u t
zero
absence
dynamical
rejection.
to a f i n i t e - d i m e n s i o n a l
assume
called
then the
variable
In o t h e r
This more
ourselves
automatically
concerned
u,y.
disappeared
structure
The
structure.
The natural
becomes
be r e s t o r e d
x = 0).
of d i s t u r b a n c e
to
(1.1)
in t h e
of a s u i t a b l e
strategies
consider
t~).
rejection
mainly
of
is e q u a l
disturbance
input
to b r i n g
form
controlled
in the f o r m of a c o n t r o l
is u s u a l l y
the problem
us f i r s t
that
state
state
restrict
the
is the o u t p u t
strategy the
strategy
We will
of this
design
the
of the
to b r i n g
0
initial
system with
admissible problem.
Let
the
represents
is a l i n e a r
space model
is m o s t l y
by a feedback/observer
to d e s c r i b e
where
a state
of a p l a n t .
(note t h a t
for
be g e n e r a t e d
state
of
the nature upon
the
therefore
[1]-that
the
370
whole
controlled
the problem em in its o w n play
variable
z
of a c q u i r i n g right
no r o l e
c a n be o b s e r v e d
information
as w e do it here.
in o u r
further
about Hence
or w h e t h e r
one
the disturbance the controlled
considerations.
We
considers
as a p r o b -
variable
start with
will
a s y s t e m of
the f o r m = A x + B u + Gv, and assume equation
in a d d i t i o n v = A'v
functionals
kTv
that
with
unknown state
evaluated
input
This
the c l a s s i c a l
concept
approach
lies
in the
duce
Sec.
2 and
in
(see e.g.
[2]
the observer cuss
at the
happens
(i.e. depend
to r e c o g n i z e or r e f i n e
was
wrong
data.
state
of
some
which
structure
with
the
seems more
4 three
specific
(plant p l u s
novelty
we will
dynamical
intro-
observer
to a n s w e r
questions,
questions
namely:
model?
dis-
What
if t h e d e s i g n Is it p o s s i b l e
Is it p o s s i b l e
interruption
of
we will
as a w h o l e
on
of t h i s
the p e r f o r m a n c e
disturbance
without
linear
c a n be
system with
As an example
o n the o u t p u t ?
model
which
a new angle
which
does
controller)
on a wrong
modelling
The
suitable
How much
is to f i n d
linear
standard
upon good modelling?).
the disturbance
aim
differential-
model
presents
of t h e o b s e r v e r s
the question:
based
Our
reconstruction.
structure
system
of t h e o b s e r v e r
the
by a method
of s t a t e
e n d o f Sec.
to the
initial
of a l i n e a r
of t h e d i s t u r b a n c e
3. C o m p a r e d
) this
of r o b u s t n e s s
using
is d o n e
(1.2)
is s o l u t i o n
on the
asympotically (u,y).
v
y = Cx
to a d j u s t
of t h e c o n t r o l
action?
The material
in t h e
following
three
lemmas
at the e n d of t h i s s e c t i o n w i l l
simple
beyond
textbook
small
(big)
vectors
Latin
greek
B(i.e.
a number
tial
BT the
stands
denotes linear
from now on the
complex
etc.
spanned
contained;
and
notation
signifying Im(B)
Re(...)
and
with
column
the range
by the columns
imaginary
the
be a l l w h a t w e n e e d matrix
(A,B,C etc.)
numbers; and
self
standard
the t r a n s p o s e
abbreviation
" and we will write e(t)~
to z e r o
adopt
space
for t h e r e a l
is a l m o s t
of
B).
~m(...)
part respectively
of
or a v e c t o r .
equation
if t h e
We will a,p,x
letters d e n o t e
respectively
We use
letters
(matrices).
of a m a t r i x Small
knowledge.
sections
funktion for
t~.
e(t)
"d.e."
fur t h e w o r d
"differen-
for s h o r t n e s s
0
(scalar or v e c t o r - v a l u e d )
(I .3) decays
exponentially
371
Lemma
1.1.
equation
If
y(t)
y = -ATy
the d i f f e r e n t i a l Proof.
then
y(t)Tx
equation
1.2. e(t)~
If 0
Dx(t)~
0
for e a c h
t h e n we h a v e
also
d.e.
Proof.
We m a k e
a transformation
system
and
D
eigenvalues 0
Hence
Dx(t)
But
solution
Dx(t)~0
of
of the
assumes
~ = (A- 0 ) x, 0 A+ the
is
D =
the
e(t)~0
Lena
1.3.
and
have
where
this
is a d d e d
Let
integral
d.e.
~ = Ax
solution
state
such
(nonnegative)
d.e.
implies
is s o l u t i o n
remains
to the d.e.
a be an e i g e n v e c t o r
e-ataTx
according
= const,
to L e n a
Given
a linear
of
differential
of
unchanged
i=l,...,p
space
there
j = 1 , . . . , r i,
real
parts.
D + = 0. ~' = A x'
if an i n h o m o g e n e o u s
AT
and
let the
corresponding
integral
p = ~m(a).
of the d.e.
~ = Ax,
system
a sequence
state
the
equations.
of
(2.1).
Cp which
exists
and ~i
y = Cx
subspaces
Eo = of the
that
[]
= Ax + Bu, We c o n s i d e r
of the
o
is a f i r s t
1.1.
2. R e c o n s t r u c t i n g
and
,
negative
x' (t)
property
variable
e i g e n v a l u e a h a v e n e g a t i v e real part. L e t p = Re(a) or T Then p x(t)~ 0 for any s o l u t i o n of the d.e. ~ = Ax. Proof.
of
form
(D-,D+)
A - ( A +)
= D x' (t),
x' ( t ) ~ 0
of a
for e a c h
for any solution of the h o m o g e n e o u s
terme
a first
~ = Ax + e(t)
the m a t r i x
Dx(t)~
= const,
of the d i f f e r e n t i a l
[]
inhomogeneous
and
solution
x = Ax.
Straightforward.
Lena if
is a n o n t r i v i a l
has
following
a set of l i n e a r l y
complex
= Ci-1
the
12.2)
numbers
si,j
property. independent such
each
vectors
ki,i,.
that
+ s p a n { k i , 1 ..... ki,r. }' 1
(AT-ai,jI)ki,jC
For
Ci_1
+ Im(C T) ,Re(~i, j) < 0
(2.3)
372
Theorem
2.1.
(2.3).
Then
Given one
i-I - j:1 E
xi = A x i
such
that
Let,
for a g i v e n
and
let
y(t),
a sequence
can
the
find
xi(t),
of
subspaces
coupled
having
linear
the p r o p e r t i e s
systems
of the
(2.2),
form
L i,3.K Tl,j (xi-xj ) - L i ,O ( C x i - Y ) + B u
following
where
p
is true.
control
function
i=I, .... p
y(t)
(2.4)
be the
is the o u t p u t
xi(0)
u(t),
x(t)
solution of
be a s o l u t i o n
of
(2. I))
(2.4) with
(with
initial
of
(2.1)
U ~ u(t),Y ~ value
= 0.
(2.5)
Then kT (x (t)-xi (t)) ~,~0 1,3 Proof. given (i'
We c o n s t r u c t a certain
instead
It f o l l o w s
the d.e.
for
(2.4)
i, the e q u a t i o n s
of
i)
then
such
i=I ..... p,j=1 ..... r i
that
recursively. (2.4)
be r e p r e s e n t e d
(we d e n o t e
by
as a l i n e a r
ko, j
representations
the rows
Since such
the
defined hold
for
true
that,
i'
(with
of the
made
ki, j
combination
i~i').
(2.7)
of c e r t a i n cT),
i.e.
k{,j,
i'
we h a v e
form
out of c o l u m n s
are
j=1 ..... r i,
of the m a t r i x
(A T - a i , j I ) k i , j = ~uj,
is a m a t r i x
been (2.6)
is we a s s u m e
that (AT-ai,jI)ki,j,
can
have
the r e l a t i o n s
That
(2.6)
linearly
j=1 ..... r i,
ki,
,J
(2.8)
,i'
independent
one can
find
a matrix
L
that uj = Lki, j, j=1 ..... r i
It f o l l o w s matrix infer
then
AT-EL
from and
from Lemma
(2.7),
(2.8)
the c o r r e s p o n d i n g
1.3 this
that
ki, j
eigenvalue
On the o t h e r columns
of
hand ~
$ =
it f o l l o w s
that
we
is
si,j"
of the
Hence
we
statement:
kT . e ( t ) ~ 0 j=1 for e a c h l, 3 ' ,...,r i , of the d.e.
is e i g e n v e c t o r
solution
(A_LTK~T)e. from what
can w r i t e
(2.9) was
said
above
about
the
373
LTK~T
where
for f i x e d
of the f o r m (2.6)
h
kT h,j
are a s s u m e d
the rows of " Since,
xj,u,y x(t)
respectively
solution
of
of
(2.4)
(2.1)
It f o l l o w s
tion of a d.e.
of the f o r m
~
of i n d u c t i o n ,
with vectors
the r e l a t i o n s
we have
(2.11)
h-
for the g i v e n
i
and s u b s t i t u t e
xj (t) ,u(t) ,y(t)
for the i n p u t f u n c t i o n
i~j).
thus o b t a i n e d .
(2.1~
can be i d e n t i f i e d
i-i'
the f u n c t i o n s
(2.4)(with
+ L. C l,O /
by h y p o t h e s i s
(t)-xh(t)) ~ 0 ,
n o w the d.e.
is s o l u t i o n
K Ti,h
to be true if
t
i-I ~ T = Li,hKi,h h=1
Let
xi(t)
then from
= Cx(t) ,j
be a s o l u t i o n
(2.11)
that
for
xi(t)
Here
is
of the d.e. is then solu-
i-I - j=IE Li, 3KTl,j (xi-x(t))-Li, o ( C x i - C x ( t ) )+Bu(t)+~(t)
~(t) ~ 0. B e c a u s e
of
(2.10)
this e q u a t i o n
can n o w be r e w r i t t e n
in the f o r m xi = Axi Hence
e(t)
- LT~T(xi-x(t))
:= x(t) =
and the d e s i r e d
- xi(t)
+ Bu (t) +e (t).
is s o l u t i o n
of the d.e
(A-LTK T) e+~(t) ,
result
k. . e ( t ) ~ 0 1,3
follows
from
(2.9)
and L e m m a
1.2. []
Remarks.
(i) We call
(2.4)
a reconstructing
d.e.
for the f a c t o r
space
Let us r e w r i t e
xi =
the d.e.
(A-LTK~T)xi +
(2.4)
as f o l l o w s
i-I T ~ L i , h K i , h X h + L i , o y+Bu" h=1
(2.4')
F r o m w h a t was
said a b o u t the rows of the m a t r i x KT it is c l e a r i,h T t h a t the r o w v e c t o r K i , h X h can be e x p r e s s e d in terms of the l i n e a r T f u n c t i o n a l s k h,jXh_ , h
matrix (2.4')
AT-~L
to the e i g e n v a l u e
a s y s t e m of d . e . ' s ~l,j
e. .. H e n c e one can d e r i v e 1,] for the l i n e a r f u n c t i o n a l s
:= kT1,jxi
from
(2 12)
374
simply by m u l t i p l i c a t i o n
from the left with
kT : 1,]
i-I ~i,j = ~i,j~i
'
j +
~
El:
1=I
Hence the reconstructing
k l']'
differential
by a system of differential functionals. values (iii)
l,k~l,n
The system
,u
(2.13)
(2.4) can be replaced
for the r e c o n s t r u c t e d
is triangular
linear
and has all eigen-
in the left half plane. Let
~1
be the subspace of
]Rn which is spanned by the ( c T , A T c T , (A T ) 2 c T , . . . . ) . T h e n
columns of --the matrix
is a sequence having the properties ~1,j'
T
o,i jy+bi,3
equations
equations
matrix
+IT
j=1,...,n I = Dim
( ~i )
(2.2),
be inferred by standard arguments
form
For later purposes we wish to describe (2.13)
we do not require Corollary
(cf.[2]~Th.
1.35).
the r e c o n s t r u c t i n g
Re(~)<0
and let
of the k
This can
if one assumes
in a special case in more explicit
linear independence
2.1. Let
arbitrarily.
(pole placement),
that the system is given in normal
tial equation
(2.3). The numbers
can be prescribed
differen-
terms.
Here
k. .. 1,3
be such that
r.,
(AT_~I)k =
Pi,j
pE 44+cTIE Z1 i=I j=1 Pi'jki'j
are scalars
and
1
~p +Im TC. {)
(2.14)
o
a vector of appropriate
dimension.
Then the linear functional := kTx can be r e c o n s t r u c t e d
from the scalar differential
equation
r.
= ~
+
p l ~ [ p + 1Ty + k T B u i=lj=1 i,j ~i,j
(for the d e f i n i t i o n
of
~i,j cf.
(2.12)).
.
If in particular
ri (AT-~I)k = k' : ~=I Pi'jki'j
then
~
can be r e c o n s t r u c t e d
from the equation
(2.15)
375
= e6 + ~', ("can be r e c o n s t r u c t e d " of
~' =
means:
(k')Tx. 1 {(t)-kTx(t)~
0,
~(t)
being
solution
{2.15)).
Proof.
Let
x(t)
be a s o l u t i o n
kTx(t) and hence,
in v i e w
of
of
(2.1) . T h e n
= k T(Ax(t)
+ Bu(t))
(2.14),
ri d__ (kTx(t)) dt
Since
Cx(t)
kTx(t) the
We
= y(t)
satisfies
statement
now turn
model form treat
and
of the
plus
x,v
of the
The more
dimension
state
variable
[]
which
which
v = A'v.
0. T h e r e f o r e
1.2.
systems
are
include given
a
in the
It is c o n v e n i e n t and
that
to w o r k
to
with
one
of
x',x" the
portion
of
of
of this
such
from
spaces
the
(2.16)
.
x
in the
form
(x',x")T
f o r m of the d.e. (2.16).
n',n"
(n=n'+n")
Y = ClX' will
system
is k n o w n
that
the
r ) 11
the
by
+ BlU, section
for
H e r e we m a k e use of the fact c a n be w r i t t e n in this f o r m AT =
(C I 0)x
Further-
system
= A11x'
C i
y =
x. We w r i t e
is d e n o t e d
reduced
subspaces
a sequence
u,
0
is o b v i o u s
~,
that
same
term ~
Lemma
systems
x +
introduce
The r e m a i n i n g
d.e.
A22 ]
is the d i m e n s i o n
of f i n d i n g
i.e.
it is c l e a r
form
the p a r t i t i o n i n g
we
from
of l i n e a r
additional
as p a r t s
+ iTcx(t)+kTBu(t)+ekTx(t).
up to an e r r o r
follows
disturbances,
0 n
(2.15)
corollary
the
x(t)
k? .x(t) T 1,3 -- k i , j x i ( t ) ~ 0
x =
where
Pi,3.kTl,j
to the c o n s i d e r a t i o n
of the
Again
[ j=1
the d.e.
for p o s s i b l e (1.2)
d.e.
: i=I
0
(2.16r)
be d e v o t e d (2.16)
to the p r o b l e m
under
for the r e d u c e d
transpose
of the
the a s s u m p t i o n system system
(2.16r). matrix
376 !
Hence let there be given which
are contained
in
spect to the equation
sequence of subspaces
a ~n
,i=1,...,p
and have the properties
(2.3) with re-
(2.16r) :
C' C' =
i-I + Ekl
,I
. . . ." k~
,r i
1
'
(2.17) (A11-~i,jI)
m,j 6
We wish to construct
-I + Im(C
for the original
), Re( ai,j ) < 0 . system
(2.16)
a corresponding
sequence
which we call the canonical
extension
This extension
according
of the given sequence
C s. 1
is possible,
if all eigenvalues The canonical P'ri'
~i,j
A22
k[l,j appearing kl, v
below
lie in the closed right half plane.
should
satisfy the relations
we require that the first component of
such that the following numbers
described (2.3) with
being the same numbers as those appearing
Furthermore vector
of
extension
to the scheme
in
(2.17). Finally the
two relations
(depending upon
i,j)
in
ki, 5 ~
ki, j~
(2.17). equals the
should be chosen
hold true with suitable real :
i-I
T
k~
-
(A11-ai,j I) l,j
[
Z Xl,v"k'l,v ~Im(C~) '
l=Iv (2.18) i-I (AT- ~ i , j I ) k i , j -
~
T ~ kl,vk I ,v
1=1 v Note that this requirement (2.17).
into account condition
(cf.
follows
from the corresponding
If one takes the special (2.16))
"
is in accordance with the second condition
(2.3). That it can be fulfilled condition
61m(CI) 0
one sees that
(2.18)
is equivalent with a
of the form T
,,
(A22-~i,jI)ki, j : li,j, where the expression i'
part of
form of the system matrix
on the right hand side depends upon
(2.18') kit~
This condition can always be met by a proper choice of T (A22-~i,jI) is non-singular.
since the matrix
for k:' ltj
'
377
Let
i
be f i x e d for the m o m e n t .
Theorem
column vectors
k i,,j, i'
T (A11 - ~i,j I) k :1,j using
As we have
2.1 t h e r e e x i s t m a t r i c e s
(2.18)
involving
seen in the p r o o f of
where
K'
consists
of
such that
uj : Lk i ,j , j=1 ..... r i
= K'uj,
this r e l a t i o n
the
K',L,
can be e x t e n d e d
to an a n a l o g o u s
relation
k. .: 1,3 K'
The r e c o n s t r u c t i n g
equation
K'
for the f u n c t i o n a l s
kT .x 1,3
assumes
then
the form i-I 11 AI .= xl
where
I LijKij (xi xj)
consists
0
of h o m o g e n e o u s
F r o m the f o r m of t h i s e q u a t i o n
If
)
(2.19)
x iA22]
P. i
diately
+ Li,o(CXl-y)+BIU+P i
linear
functions
the f o l l o w i n g
of
statement
x[' , j
inferred.
xh(t)
=
(2.19) (with x~' (t) = 0
(x~(t),x h i~h)
(t)),h=1 ..... i
satisfying
for all
t
and
is a s o l u t i o n
the i n i t i a l
condition
h=1,...,i,
and
of the s y s t e m x~' (0) = 0,
x' (t)
then
is s o l u t i o n
of
1
the r e c o n s t r u c t i n g r
Xl = A 1 1 x i
--
equation i-I [ LijK.lj (xi-xj) i
i
j=1 belonging
to the r e d u c e d
In o t h e r words:
s y s t e m and the
Canonical
differential
equations,
constructing
d.e.'s
We s u m m a r i z e
w h a t we f o u n d
Theorem
2.2.
extension
provided
according
General
negative
real parts.
sequence
of s u b s p a c e s
,
to
C'4
< <
space
preserves
/
-I"
the r e c o n s t r u c t i n g
one n o r m a l i z e s
the
states of re-
(2.5).
so far as
hypothesis: Assume
(2 "201
- Lio (Cxi-Y) +BI u
The e i g e n v a l u e s
furthermore
that there
with properties
(2.17)
of
A22
have non-
is g i v e n a and let
378
i-I [ LijKij (x[-x<) j=1 1 3
x{i = A11x'i -
be the r e c o n s t r u c t i n g
differential
,ha, these
d.e.'s
spaces
Ci
Let
k~,j)
equation
are based
then be the c a n o n i c a l
(kl,j,
ki, j =
and
- Lio(Cxl-y)
<
for
+ B1u
/
-I
on the r e d u c e d
extension
&'l
of the
be a set of g e n e r a t o r s
(2.21)
"
Note
system
(2.16r).
and let
satisfying
the c o n d i t i o n
(2.18). Claim.
If
x(t)
=
(x' (t),x"
(x~ (t),...,xi(t)) T asymptotic
(t))
is a s o l u t i o n
is a s o l u t i o n
relations
hold
of the
d.e.
of
(2.16)
s (2.21)
and if
then these
true:
(kl,j)T(x' (t) - x'. (t))l
+
(ki, j)'' T x "
(2.22)
(t)~0
or e q u i v a l e n t l y k Tl,j x(t)
, - {i,j (t)~ 0,
, ~i,j (t)
:=
(2.22')
(kl,j)Tx 'i(t) ,
j = 1 , . . . , r i. Remark.
If
kC ~ n
be a s y m p t o t i c a l l y
and
x(t)
evaluated
a solution
of
means
It follows
then f r o m
kTx(t)
We f i n a l l y
-
wish
Z i,j
kTx(t)
~' (t) i,j of the form
that we have a r e p r e s e n t a t i o n k =
then
w i t h the help of the
,'9
k6 ~ p
(2.16)
can
. Indeed
.Z. 1 .k 1,3 i,3 i,j
(2.22') li,j
that
~l (t) = Z i,j i,j
to have a c l o s e r
1
.kT (x (t) -xi (t) ) ~ 0. i, 3 1,j
look on those
elements
kC ~
[]
which
P
are of the form k = The linear
functional
just r e m a r k e d ~i,j
(cf.
Find
further
(k")Tx" a rather follows: y" =(k"
onecan
(2.22')).
(0,k")T.
kTx =
(k")Tx"
reconstruct A natural
functionals
on
x"
problem which
observer
equations
obvious
solution
which
Construct
observers
formally
depends
x"
the f u n c t i o n a l
using
)Tx"
(2.23)
for
roughly
is then
As we have f r o m the
the f o l l o w i n g
one:
can be r e c o n s t r u c t e d
from
x"
only.
speaking
for the s y s t e m
as o b s e r v e d
only.
(k")Tx"
variable.
x"
This p r o b l e m
can be d e s c r i b e d = A22x"
A more
has as
regarding
precise
version
379
is given in the following
theorem.
Theorem 2.3.
Hypothesis:
and P vectors
is the linear subspace of the x " -space spanned by the
~
(A~2)Pk" Furthermore Claim.
k
let
C T2
K2
~is
ponding
be a matrix
eigenvalues
the following
x" (2.16)
Proof.
~ IT .k' i,j 1,3 i,j °
(of appropriate
have negative
t)
(t) - ~ "
dimension)
of
(2.24) with this proper-
(A22-K2C2)T
real parts.
and the corres-
Then for any
~" E~
(t))~0,
is the second component
and
~" =
A matrix
(t)
(A22-K2C2)~" K2
of any solution
x(t)
of the
is any solution of the d.e. (2.25)
+ K2i{jli,j~i, j (t) .
with the property m e n t i o n e d
the theorem does always exist of Theorem
0 =
such that
statement holds true
5" Remark•
and
IT 1,j
eigenvectors
([")T(x" where
of
(kl" '" .., k "s) .
be the matrix
spanned by
v=1, .... s, are elements
p=0,1,...,n " -I
(i) There exist row vectors
(ii) Let
d.e.
(0,k")T,
, v=1,...,S,
T [ IT " . C2 = i,j l'jki'3
ty:
=
(cf. remark
(iii)
in the statement of
following
the proof
2.1).
(i) is clear,
kvE ~ and hence can be written as a P k. In order to prove (ii) note that 1,j" we have, in view of (2.24), (2.22')! T C 2 x " (t) - i,j Z i.1,3.~[l,j (t) = i,j Z i i ,J (ki,j . . .x. (t)- ~i,j ' (t))+(iE,j i..k[ i,] 1,3.)x'~ linear combination
since
of the
= The difference
e"
(t)
e"
=
E li,j(k i,j
:= x " ( t )
,
,jx(t)-~i,j(t)),~0.
-.~"(t)
is
hence
solution
of a d.e.
of the form (A22-K2C2)e"
It follows then from Lemma is eigenvector
of
negative real part.
1.2,
(A2~-K2C2)T o
+ ~(t), ~ ( t ) ~ 0 .
1.3 that
kTe"(t)
~
and the c o r r e s p o n d i n g
0,
whenever
eigenvalue
k has
380 3. A u x i l i a r y We r e t u r n
observers.
to a s y s t e m of the f o r m
(2.1)
and c o n s i d e r
a sequence
~. 1
and r e c o n s t r u c t i n g (2.2)-(2.4)). not e x c l u d e
differential
equations
as e x p l a i n e d
It s h o u l d be n o t e d t h a t we a l l o w
the p o s s i b i l i t y
before
(see
~i-I = ~-i' i.e. we do
that
ki, j 6 C i _ I , j=1 ..... r..1 The r e a s o n
is tha t a r e l a t i o n
t h a t the c a n o n i c a l theoretic
-I = i does not n e c e s s a r i l y i m p l y Ci ! ~! ~ i - I ' __~i are also equal. The s y s t e m -
extensions
interpretation
(3.1)
of this
fact is r a t h e r
obvious.
If
(3.1)
holds, t h e n kT .x. (t) is a s y m p t o t i c a l l y equal to a l i n e a r c o m b i n a tion of k Ti',j l,xi, 3 l(t)' i'
is the o u t p u t
ment however
the r i g h t hand In t h is
longer
s e c t i o n we go a step f u r t h e r
of c o u r s e
extension
(see
(2.18)).
Definition
3.1
Let
kl,v
Furthermore
let
~
Define
k"
as the P
( AT2 2 -
We call t h e n
{ ~
k :=
n o t zero.
=
(kl,v,
-
This
x = Ax + Bu.
introduce
k' = 0 We a d o p t
w h a t one m a y
to a v e c t o r formally
the c a n o n i c a l
k£,v)6 %.
state-
is a d d e d to
the p r e v i o u s
extension
Assume
(0,k")
of {~i}.
that
6 Im(~).
be a n u m b e r
I)k"
and w i l l
} denotes
P E ~ i ' 1=1 v 1,vkl,v
:=
(2.1). signal
equation
of the v e c t o r
is in g e n e r a l
definition
then
equation
true if a p e r t u r b i n g
side of the d i f f e r e n t i a l
call a canonical which
of the u n p e r t u r b e d
is not
(3.2)
such that
solution
Re(~)<0.
of the l i n e a r
equation
r. E A I , v W'1,v = 0. l=Iv
(0,k")T
an e x t e n s i o n
(3.3)
of
0
and the d i f f e r e n t i a l
equation P = ~{ + E 1=I a reconstructing
~ l I v~,v(t)± v ,
differential kTx =
The c o l u m n v e c t o r
equation
1=I
for the l i n e a r
(3.4) functional
(k")Tx.
p is d e f i n e d P
T - p y
, E ll,vkl, v : v
in t e r m s of the r e l a t i o n T ClP
(3.2')
381
which
The
c a n be d e r i v e d
following
sions
of
results
from
theorem
states
0. N o t e
Theorem
independent 3.1
Let
have
k
the
section
:=
~(t)
x(t)
(0,k")
=
We
first
note
with
the
single
structed
Ci/ C
i-I"
sequence
as has
been
rential
theorem
In the
sequel
in D e f i n i t i o n speak
auxiliary
we r e f e r
can be c o n s t r u c t e d
C'
from
in
auxiliary
be
2.1
let
(3.4)
and
(cf.
Let
(2.22)).
of the d.c.
Then
(2.16)
= output
(3.3),
of
(3.2')
and
(2.16)).
are
equi-
p
~ = kTx
.
can be r e c o n -
T
(3.5)
we
reconstructing
however
have
of T h e o r e m
to a l i n e a r
observer Note
that
to
the
upon
(3.2')
equation
absence
=
extension i=I ....
0,
Hence and
the
above
the
statement
of the
More
type
p,
diffe-
as d e s c r i b e d
precisely
we w i l l
to the p a i r
(~,k),
differential
equation
the d i s t u r b a n c e f r o m any
(2.16r).
of
(t)
[]
x2(t)
of
where
model,
sequence
Nothing
for the r e c o n s t r u c t i o n
In the
for
1
(3.4)
functional
belonging
d.e
canonical
x['
2.2.
observer.
not d e p e n d
according
is the
therefore
2.1.
observers
-
x"(t)- p y .
from C o r o l l a r y
does
T (C1) 0
i,v
that
with
(3.2).
equation.
0
equation.
k
i,~
identical
for the u n p e r t u r b e d
unperturbed
i
of the
as an a u x i l i a r y
observer
PZ E i= I ~
{ ~i }
and
follows
as
longer
equation
in the p r o o f
of an a u x i l i a r y
is d e f i n e d
=
is s o l u t i o n
is in fact
3.1
not
the
i,~
sequence
noted
of
y÷y(t)
two r e l a t i o n s
,
{ ~ i;}
equation
of the
k
the
(with
(k~ x ~ ( t ) + k "
(t)
The
of the
exten-
into
relation
i,~
(xi(t) ' x "i
need
2.2
is a s o l u t i o n
from Corollary
[ ~
i=I Here
that
the d i f f e r e n t i a l P
~=~+z
for
(t) ~ 0
(3.4)
(AT-eI) ( O , k " ) T
therefore
from
k[ z,j
as in T h e o r e m
(t))
of the d.e.
Proof.
It f o l l o w s
2.2
theorem
differential
(k")Tx"
(x' (t),x"
valent
=
the
this
be an e x t e n s i o n
same m e a n i n g
a solution
(AT-eI)k
since
reconstructing
(t) whenever
of T h e o r e m
include
vectors.
be the c o r r e s p o n d i n g i ~i,j (t)
the a n a l o g u e
t h a t we c a n n o t
of the p r e v i o u s
linearly
(3.2).
of
of an but
spaces
can be g a i n e d states
every
for
solution
the (3.4)
382 is a s y m p t o t i c a l l y
4. S y s t e m s In this
with
(this
is a c o n s e q u e n c e
disturbances:
section
X
0
we c o n s i d e r
o
(3.2') !).
reconstruction.
a system
=
State
of the r e l a t i o n
of the
form
u,
x +
y=
(C',0,...,0)x
(4.1)
According
to the p a r t i t i o n i n g
x = We a s s u m e (i)
All
(x I , ... ,xN)T , that
the
of the m a t r i x
xi
following
eigenvalues
of
A
.
A
being
of d i m e n s i o n
general
assumptions
are
in the c l o s e d
we w r i t e n l, n = iZ n.l hold: right
half
plane, i>I.
ll
A.. 11
and
A.. 33
have
no e i g e n v a l u e s
in common,
l
ii)
There of
exists
subspaces
a sequence for
!
Co = [ 0 ] ~ <
the r e d u c e d
i ~ j, i,j >I. !
E''" E
~p
system
Xl = A11x1 + BlU, y = C ' x 1 having
the p r o p e r t i e s
instead iii)
There
of
exist
Im (C')
(2.17).
k~ v=1 l,j' '''" M
linear
For
and
e
v
(4.1 r) simplicity instead
combinations
we w r i t e
of
k'
~. .. 1,3
of the
k~
~:1 .....
M.
which
belong
to
: i~ ~)
N Z
:=
l~(~)k~ EIm(C'),
(4.2)
~=I
Note
that
linear
we do not
combinations
assume (4.2)
C'iby' a cu~Lc o r~d i~nIg~
We d e n o t e spaces of
A22
denote
Let
is n o w p l a y e d by
1 (~)
k
that
the
k'
are
independent
.... ~
~p
the c a n o n i c a l
to the d e f i n i t i o n by the m a t r i x
diag
given
i (~) =
extension in Sec.
(A22,...,ANN).
the c a n o n i c a l e x t e n s i o n of e a c h N 1 (u) := Z l(U)k , u=1 ..... M.
be p a r t i t i o n e d
or t h a t
the
are n o n - z e r o .
k'
2
of the (the r o l e As b e f o r e
we
and put (4.3)
as (i~ ~) , .... i ~ ) ) T
(4.4)
383
and let (4.2)
~
be a complex number with
and Definition
Re(a) < 0.
It follows then from
3.1 that the vector
~(~) (~) := (0, (A~2-~I)-ll~ ~) , ...., (ANN_~ I T ) -I~±N(~) ) T is an extension of
0 belonging
Finally we introduce
to the pair
the following matrices
(i)
CI" := (l~ I)'''''I(M))T'i
(ii)
~i
(4.5)
( e , l]~ ) ) . and spaces:
i=1,...,N (4.6)
Im('C T A T C T
=
( i'
ii
i ' (A
i i(~) is the i-th component dimensional column vector. a subspace of ~ ni Theorem 4.1
T
i )
2C T
i ....
with the following
1 (~)
C. l
is hence a matrix of type
reconstructing
(ii)
M
reduced
(a ,12 ~))
(iii) A sufficient
y
and an
n.-
(4.1r),
(k I, .... ,kN)T
l
kiC ~ i
t
observer
It consists of
b a s e d on t h e
(4.2)).
(M,ni), ~ i
using a dynamical
(of the type
observers
(cf.
(4.3), (4.4))
kTx, k=
from the output d.e.'s
system
auxiliary,
(cf.
1
structure.
(i)
i=I .... N .
of
Any linear functional
can be reconstructed
')) '
for
(4.1r),
a
is
(2.4) of
sequence
,...,
belonging
to pairs
an a r b i t r a r y
number of additional
(2.13))
for the
number w i t h
"integrating
Re(a
actions",
)<0.
provided
by d.e.'s of the form =
<~,o
~
+
<~,p
eZ{~,p
satisfies
to the pair
,p-1'
the d.e.
n=1,...,M,p=1,2, ....
for the auxiliary
observer
(4.7) belonging
(~ ,i#~)).
(iv) a standard observer of the form Xl :
(A'i-K 1 i C')x{ I 1 + qi (< 1,1' ~ 1,2 . "". . . .'~M,I' . . .....
for each Remarks.
(4.8)
i=2,...,N.
(i) The reconstructing
the auxiliary observers (iii)
qi
d.e.'s
for the reduced
system and
do not depend upon the disturbance model.
(ii) The number of "integrating turbances.
)
(cf.
actions"
does depend upon the dis-
(4.8)) depends upon
Ajj
for
j ~ i
only.
384
In particular,
if a disturbance model XN+I,N+ I = AN+,,N+ I XN+i,N+I
is added to the existing (4.6)
ones,
then the reconstructing
structure
are obtained
necessary)
and if d.e.'s
(for i=N+1)
for the reduced
is defined according
for the "enriched"
of the form
actions
(4.8).
(if
and by adding a standard
(iv) The reconstructing
system as well as the standard observers
can be replaced by a triangular
to
disturbance
from the existing ones by increasing
the number of integrating
observer
CN+ I
system of the form
d.e.'s
of type
(4.8)
( 2 .13) with
Re(~i,j)<0. Proof.
Let u be a fixed integer with
i,jh2,i ~ j, the two matrices eigenvalues
in common and their respective
are hence relatively hypothesis }j (s)
I~M.
(A~i-a I)-I
prime.
For each pair
i,j
with
and
(AT.-~ I) -I have no 33 P characteristic polynomials
This is a consequence
(i). There exists hence for every
of our g e n e r a l
j=2, .... N
a polynomial
with these properties
~j ((A~j-~ I) -I) = A~3J - a I, ~j ((AT.-~ Zl D 1) -I ) = 0, i~ j, i,j~2 One simply has to choose is divisible
49 s~j (s)-1
0
according to these conditions:
by the characteristic
We denote by
~u
the matrix
(cf.
(4.10))
equation
(4.1)
(A3j-~ I)-I
of any of the matrices polynomials
(of type
(4.5)
and
~ j(~ )~(u) (~)" On the other hand,
(ATi-e I) -I
rule.
(4.9) =
is divisible
by
s.
(n,n))
that
taking into account the special
one sees immediately
then from Corollary
((A-~)PY~I~~ ~ "
))Tx,p=0,1
. .• ,
form of the matrix
matrix A appearing
in the given
that
=
It follows
r
(4.10)
(0 , .... 0 , i!] n) ' 0 .... , 0).
and of the coefficient
11
Note
(ATi-e I)-I
T -I T I) -I (0, (A22-~ I) , .... (ANN-~ ~ ).
diag It follows then from
of
according to the Chinese remainder
is not an eigenvalue
hence none of the characterisitc
A
polynomial
is divisible by the characteristic polynomial of
This is always possible that
~j
(4.9)
2.1 that the functionals
can be reconstructed
recursively
from
385
~)(~ by
~x;
the
reconstructing
differential
equations
are
just
given
(4.7).
Hence
we are
~vO = ties
arrived
at this
[0]~ £ I C . . . c- ~ p , ~ ~ n stated
in Sec.
such
result.
One
for~the
system
that
( i ~ ~) ,
~p,
* .... ~
can
contains
),
~=I,
find
a sequence
(4.1)
having
all v e c t o r s
....
M
all p r o p e r of the
form
,
and ^ ~ (~) (0 ..... O,±j ,0 ..... 0),j=2 .... ,N,
Furthermore
the
are of the theorem.
corresponding
type
reconstructing
as d e s c r i b e d
It is t h e n
an o b s e r v e r
of the
clear, form
u=1 ..... M.
in the
in v i e w
(4.8)
differential
statements of T h e o r e m
in o r d e r
equations
(i)-(iii)
of the
2.3 h o w one
to r e c o n s t r u c t
the
can
find
functionals
kTx, where k =
We c o n c l u d e possible
this
(i) The
theorem on the
aspect
lator
nals
emphasizes
What
which
(in c a s e
(ii) ready
will
by a d d i n g
the r a n g e
reconstruction
well
into
and
one
func-
state
itself.
frame
of r e g u -
of c e r t a i n
funtional,
reconstructs
special
of a c o n t r o l l e r
evaluation
only
which
of
of the
the g e n e r a l
for the d e s i g n
is b a s i c a l l y
additional
further
about
the
namely
full
is
functiou=-Fx)
state
(and
exist).
structure
Hence
[]
reconstruction
control
not
<.
remarks
asymptotic
an o b s e r v e r
remarked.
simply
fits
provides
observer
than
is n e e d e d
of a s c a l a r
in g e n e r a l The
some
cj6
theorem.
explicit
rather
of our m e t h o d
necessarily
which
with
of the
state
synthesis.
an o b s e r v e r
not
section
applications
tionals This
(0 ..... 0,cj,0 ..... 0),
a recursive
disturbances
observers
without
scheme,
c a n be t a k e n
changing
the
as al-
care
of
existing
structure. (iii) (ii)
The
stability To m a k e Take
reconstructing
following
the p r o o f
under
this
a system
certain
remark which
periodic
character
We g u e s s
that
design
more
with
part
can a s s u m e d
of T h e o r e m
2.1)
transparent to
(unknown)
they
s t a b l e (see r e m a r k preserve
basis are
of a c o n t r o l l e r
their
of the d i s t u r b a n c e
let us c o n s i d e r
(scalar)
frequencies
to be
and
c h a n g e s of the d a t a
is s u b j e c t
the b a s i c
the o b s e r v e r
d.e.'s
a special
disturbances
frequencies
~
~2,...,WN
and
based
model. case.
of q u a s i ..... ~
6
accordingly
on a m o d e l
of type
386
(4.1)
2
where Aii =
In order this
to meet
I
requirements
wj
if
i ~ j.
then as before
the p r o p e r t y
pend upon the choice
stated
(iii).
of the m..
M
i,j • I.
elements
These
to
It follows
C.
(4.3)-(4.6).
(4.12)
C i ~ 0, i = 2,...,N.
and
(4.13)
represent
structure
satisfied
This
last theorem.
- provided
system will
coefficient from
follows
of m.. 3
and
be a m o t i o n
cies are numbers will d i s a p p e a r
feedback/observer motions
matrix.
(4.13)
what are
from the set
{~2
structure
of this
arbitrarily given
., w~}
in the
in the proof
~ has been 3 - the output of the
type and the basic . A certain
if the o b s e r v e r
is
frequencies
of the
satisfied
of q u a s i - p e r i o d i c
in the output
with
The e i g e n v a l u e s
choice
for
If these c o n d i t i o n s
~. and can be p l a c e d 3 from the c o n s t r u c t i o n
So r e g a r d l e s s
(4.12)
specification
section.
by q u a s i - p e r i o d i c
independent
left half plane.
frequen-
frequency
is a d j u s t e d
~[
such that
3
(iv)
Certain
changes
in the e n v i r o n m e n t
can be taken care of by the observer. disturbance second
in the first
the basic
in advance.
~3
ding
to T h e o r e m model
4.1 and
in w h i c h
Assume
of the form
(4.1).
substructure
of the o b s e r v e r
tically will
take over
frequency
structure
~2'
will
operate
forequency
(4.11)
has been
which
equally
~i
is a s s o c i a t e d
with
the others will
accora dis-
included
In the first e n v i r o n m e n t
responsibility,
in the
are k n o w n
part has been c h o s e n
if for every p o s s i b l e
with a m a t r i x
that the
and that these data
if the o b s e r v e r
system r e p r e s e n t a t i o n
the system o p e r a t e s
for example
has basic
etc.,
Then a f e e d b a c k / o b s e r v e r
in all e n v i r o n m e n t s
turba n c e
environment
frequency
well
the
the choice
(4.13).
in this
then the c o r r e s p o n d i n g
system d r i v e n
matrix
3
depend upon
a kind of r o b u s t n e s s
described
and a stable
of the
defined
hold
~I,...,~N''
made
These m a t r i c e s
real,
are
do not de-
be the m a t r i c e s
1
Ci
a linear
of course
state
a r g u m e n t s and in v i e w of the special type ~ i = ~ ni ( = ~ 2) (cf. (4.11)) that p r o v i d e d these
Aii
the o b s e r v e r are
of the
now by standard
of m a t r i c e s conditions
of
(4.12) i#Z)"
e le m e n t s
Let then
1
according
at the b e g i n n i n g
that
> 0 , ~i ~
Let there be given space h a v i n g
(4.11).
the general
section we assume ~3
1
0
A22
e.g.
in
the
automa-
operate
also
387
but
in such a way that they do not impede
tribution
to state r e c o n s t r u c t i o n
the first one;
their con-
will
actually
die out
in the course
Linear M u l t i v a r i a b l e
Control.
Lecture
Notes
of the time.
References
[I]
M.W.WONHAM: Economics
and M a t h e m a t i c a l
Systems,
Vol.
101.
Springer
in
Verlag
1974. [2]
H.KWAKERNAAK, Interscience
R.SIVAN: 1972.
Linear
Optimal
Control
Systems.
Wiley-
EIGENVALUE
PROBLEMS AND THE ZETA-FUNCTION
RIEMANN
IAN K N O W L E S
i.
Introduction For complex
function,
numbers
~(s),
s = o + it
is d e f i n e d
above
series
regular order
(see e.g. has
[6])
for all v a l u e s 1.
that
an a n a l y t i c of ~
=
Euler
F
denotes
the a n a l y t i c
except
product
formula
~(s)
The
latter
interest the
zeros
[3, 6]).
in t h e of
zeta
vertical
strip
=
~ p prime
the
and
that
0 < o < 1
,
~(s) ~(s)
has
when
the
so-called
(1.3)
the prime
placed
much
study -4,
numbers
of the
the precise
s = -2,
infinitely
symmetrically
(1.2)
explains
of m u c h
is
is a p o l e o f
o > 1.
In p a r t i c u l a r ,
= 0
~) t h a t
equation
we have
theory,
subject
by the
%(l-s)
Finally,
i-
by
there
functional
function
the
defined
(also d @ n o t e d
on the one hand
has been that
function
relating
function.
It is k n o w n zeros")
{ :
complex
~(s)
"trivial
for
identity,
and on the other,
function.
zeta-
series
s = 1 , where
satisfies
the gamma
Riemann
(i.i)
(S) = 2 s ~ s - l s i n ( 7 1~ s ] F ( 1 - s )
where
the
~ i__ S n=l n
continuation
s
Furthermore,
~ > 1 ,
by the Dirichlet
%(s)
It is k n o w n
with
location
of
(see e.g. -6 . . . .
many
zeros
in t h e
with
respect
to
(the
389 1
o = [
and
trivial
t = 0 ,
zeros.
that
these
much
is k n o w n ,
above
It w a s
complex
ticians
zeros
axis
remains
during
theorem
Theorem.
Let
kernel
a
of t h e
Thus, a unique
Although (see
eigenvalue partial
first
81,000,001
Although zeros
i o = ~ , [2], t h e
line
the best
o = } .
299]
efforts
of many mathema-
123 y e a r s .
lecture
is to o u t l i n e region
real
f;
+
bounded
has
for
a proof
of t h e
~ :
number.
Then
there
exists
k(r,t,s)u(t,s)dt
solution
a unique
a
(1.4)
u(r,s)
~(s+(?)
~ 0
idea
analytic
s
in
s
for
of the
first
contiguous
hypothesis with are
to
problem,
form
~N(S)
is t r u e
Re(s+@)
sub-region Re(s+(?)
> 1 ,
= 0 ,
(1.4)
has
complicated
a suitable
EN ,
if
>
somewhat
produce
value
function
in a n y
sub-region.
of t h e p r o o f is to
solution
~ 1
Riemann
for all
of a b o u n d a r y
~f the
in t h a t
the
solution
equation
bounded
1 [ < Re(s+(?)
strip
the d e t a i l s
product
line
3, p.
the
> 1 ;
[4]), t h e b a s i c
construction
despite
= l
in p a r t i c u l a r ,
bounded
[5; see a l s o
than
equation
unique
(1.4)
then
the
other
such that
Re(s+(?)
if
region,
lie o n t h e
and on the
zero-free
u(r,s)
(ii)
fact that
be a fixed
integral
has
the
of t h i s
0 > 1
the
all
simple
on the
k(r,t,s)
(i)
are
this
by Riemann
~
intervening
The o b j e c t i v e
outside
of
unproven
the
following
zeros
conjectured
including
the real
conjecture
a n d no
on
where
explicit
[0,~) ~N
with
is t h e N - t h
390
-
]I
<[2(s+8))
@ > 1
is an a r b i t r a r y ,
denotes
the
integral
sequence
equation by
equations
of t h e s e
let
a formal
limiting
As
in
{pn}
Q
be a f i x e d
arbitrary
let
positive
real
J
number,
and
beginning
{Pn : n = 1,2 .... }
with
Pl = 2 .
f r o m the d i f f e r e n t i a l
procedure
E
involving
boundary
value
The
equation
the d i f f e r e n t i a l
problems.
n
denote number
number.
real
derived
approximating
Problem
(1.5)
+
Pn
numbers,
is t h e n
The E i g e n v a l u e §i,
fixed,
of p r i m e
(1.4)
obtained
2.
but
I
n=l
the
sequence
greater
We define
than the
of p r i m e
one.
sequence
Let
numbers, also
{c(n)
6
and be an
: n = 1,2 .... }
by
6(1)
and
for
= 1
,
1 = --~ , Pl
6(2)
(2.1)
n a 1
~(2n+k)
01
-
6(k)
,
l_
(2.2a)
6(k)
,
1 + 2 n-I -< k -< 2 n .
(2.2b)
Pn+l _
1 +@6 Pn+l
One
can
Lemma
then
2.1.
(i)
deduce
For
integers
n -> 1
and ]
g(l + 2n-I + q . 2 n+l)
= -~
q >- 0
s(l+q.2
n+l)
Pn (ii)
Next,
s(l+ 3.2n-l+q.2n+l
for
n -> 0
and
) = 1+6 % Pn
0 -< r -< n
e(1 + 2.2 n-I + q . 2 n+l)
define
.
391
w (r) n
and
set
w '0'(~ = w n n
2n-r_l ~ k=0
=
Using
Lemma
~(l+k'2
2.1,
it
r)
is
(2.3)
then
not
difficult
to
prove
Lemma
2.2.
For
3 -< n < ~
w(r) n
and
r =
0,
l + _ _ ~1n _],l (r)
=
+ _~:
Pn ) Also,
for
n
-> 1 ,
> 1
and
i
n
-
(n) gl
=
0
B(n) 1
=
-Wn
B(n)
_
~(n) r
=
Using
(2.4)
one
can
Lemma
2.3.
For
n
a(n) r
-
define
~(n)
-
and
^ (n)
r
by
br
(2.6a)
(2.6b)
'
w(1)w r-i n Wr_ 1 '
2 -< r -< n + l
[W(1) (i) " r -Wr-l)Wn w -w r r-i
,
,
2 _
(2.6c)
.
(2.6d)
deduce
-> 1
^(n) ~r
6r-i
w
=
for
1 s r s n+l
o(n) Pr+l
-
~(n) ~r
n
8 W r _ 1 [Wr_ 1 + 6 ( W r _ I - W r _ 2) ]
(PlP2"''Pr_I)
(ii)
(2.5)
1 + -Pr ) "
<
< r < n+l
-
(i)
(2.4)
"
Pn
r=l
n
w(r)) (r) [Wn-I - n-2
n[ w
For
1 ,
,
while
for
6 r-I =
1 s r s n ,
Wn 8
(PlP2"''Pr)
Wr_lWr
392
We
Lemma
also
have
2.4.
For
n
>- 1
and
~(n) - Dr+l r (n) ~(n) mr - Dr
l-
~(n) (i)
Br +(l n- )~(n) br
(ii)
(iii)
(iv)
Let
are
b (n)
r -Wr-i :
(n) (n) mr - ar+l (n) ~(n) ~r - Dr
=
(n) ~r+l-
~(n) ~r
(n)
^(n)
wr 1 Wr-i
now
in
Wr+l
Pr+l
Wr+l
to
be defined
(n)
=
br (n)
2 .3
noting (i),
an+l(n) =
we
~
that
~r(n)
define
and,
-Wr
state
the
=
br~(n) n
~(n)
+
Lemma
2.5.
For
later
n
E
~(n)
n
~(n)
~(n)
(2.8)
~r
n
> 0 , -> 1 ,
(2.7)
~r
-> 1 ,
1 -< r -< n + l
,
a 0(n)
=
0 -< r -< n + l
:
(n) + ar_ 1
by
by
Lemma
0
£n(P r ) (n)
mr for
problem
1 -< r _< n ,
(n) ar
Finally,
eigenvalue
by
r
a (n) , r
for
1
r
er
Also,
-Wr
Q
a position
c (n) r
0 Pr+l
1 + 6 w r -Wr_
- ~r
and
r
w
(n) ^(n) mr - 5r
mr
We
Wr-i = --; wr
-> 1 ,
use
we
^ (n)
"
- 5r
note
1-< k-< n ,
akn O[PlP2 Pkll ]
(2.9)
393
where
the
For
fixed
partition c
n
(x)
order
of
on
constant
does
not
n ,
set
"fla$ n) :
the
the
interval
[0,~)
ar_l(n)_< x < a r(n) , of
the
with
Consider (2.12b)
now
On
Lar-i
qSn(X'S)
where
the
by
functions
b
forms n
(x)
a
and
n
(x)
= b (n) r
(2.10a)
(x)
= c (n) r
(2.10b)
-sb
n
(x)y'
+ s2c
= 0
y(x)
~ e x p ( B (n) n+l
the
eigenvalue
problem
En
n
(x)y = 0
(2.11)
(2.12a)
solution, a(n)]
'
The
conditions
y(0)
it
sx)
as
}n(X,S), is c l e a r
x ÷ ~
of
(2.12b)
(2.11)
satisfying
that
r
= A(n)r (s)exp[~(n)sx)r
functions
determined
clearly
equation
boundary
r (n)
•
0 s r s n}
.1 <.r < n+l
differential
the
n
n >. 1 .,
y"
together
n .
by
c
consists
on
Define
[0,~) .
b
for
depend
A(n) (s) r
(2.12b).
We
and
set
(2.13)
+ B(n)r ( s ) e x p ( B ; n)sx)
B(n) (s), r
A n(n) + 1 = 0
1 ~ r ~ n+l
and
_(n) Sn+l
are
= 1 '
uniquely and
let
2k-i A (n) n + 2 - k (s) =
~ i=l
C (n) k,i
: (n)s] exp[~k,i j
(2.14a)
D(n) k,i
expIv(n)is )
(2.14b)
2k-i B(n) n + 2 - k (s)
for
certain
1 < k < n+l -
-
constants C (n)
'
i,i
C (n) k,i
= 0
=
~ i=l
'
D (n) k,i
D (n) '
i,i
(n) ' qk,i
= 1
and '
'
and
q(n) i,i
v(n) k,i '
= ~(n) i,i
where
= 0
The "
394
condition that
s
be an eigenvalue of
A(n) 1 (s) + B1(n) (s) = 0 " the coefficients {s(n)}
En
is now
~n(0,s) = 0 ,
or
Our next task is to determine formulae for
in (2.14a-b)
in terms of the members of the sequence
defined earlier.
Consider then the solution boundary point
a (n) , r
¢n(X,S)
1 sr sn .
and its derivative at a
We have
A(n) expr (n) a(n)) + (n) exp(o(n) a~n)) r+l Ler+ 1 s r Br+l Pr+l s : A(n)r exp(c~(n) sa(n)]r + B(n)r exp(6~ n) s a r(n) 1 (2.15) (n)~ (n) expr (n) (n)) + g(n)B(n) -o(n) (n)) ar+iAr+l [ar+l s a r ~r+l r+l exP[Pr+l s a r (n)A(n) exp(~J n) s a (n)) + ~(n)B(n) exp(^(n) s a (n)) . =
ar
r
r
Dr
r
~r
r
(2.16) From these equations it follows that (n) _ (n) ~r+lAr -- - - ~ ~r
^(n) Dr Fr (n) ~ e x P L [ ~ r + l - °r Dr
(n) I sa(n)l(n) r Ar+l
+
sr+l ( n _) Dr ^(n)
[ (~(n) ~(n) 1 S ~"(n)]~(n) r J~r+l (n)- 8(n) exp [Pr+l- r ~r r
(n) (n) (n) ~r - ~r+l [f~(n)~(n)] Br - ~ (n) exp [ r + l - 5r ~r Sr +
(2.17a)
sa(n)IA(n) r+l
~(n) ^(n) r - 5r+l exp[[Br(+) ~(n)] sa(n)IB(n) (n) ~(n) - Dr r J r+l " ~r - Dr
Thus if 2k r (n) s A(n)n+l-K"(s) = [ Uk+l-(n) ,i exP[qk+l,i ] i=l
(2.17b)
395
2k (n) (s) = Bn+l- k
t h e n it f o l l o w s
c(n) k+l,i
from
i=l
(2.17a-b)
i + 6 = ~ Pn+2-k
-
1 0 Pn+2-k
-
(n) ~k,i
_(n) " Ck,i '
D(n)
1 -< i -< 2 k-I
2 k-1 + 1 -< i -< 2 k
(2.18)
k , i _ 2 k-I ' Wn_ k
• c(n)
-
k,i '
1 < i < 2 k-I
-
-
Wn+2-k- Wn+l-k
Wn-k
D
Wn_k+ 1
•(n) k+l,i
2.4 that
Wn+2-k -Wn+l-k
Wn+l- k
=
exp [~ (n) ] [ k+l,i s
and Lemma
Wn+l-k -Wn-k
Wn+l-k-Wn-k
(n) Dk+l, i
D(n) k+l,i
(n)
2 k-I + 1 -< i -<2 k
(2.19)
k , i _ 2 k-I '
[ (n) + [en+2-k
-
(n) ] (n) ~n+l-kJan+l-k
i _ < i _ < 2 k-I ' '
[^(n) (n) ] (n) ~(n) k , i _ 2 k - 1 + [~n+2-k - ~ n + l - k ] a n + l - k
2 k-I + 1 < i < 2 k --
--
r
(2.20)
~(n)
(n) + k+l, i = ~ k , i
[o(n)
_ R(n)
[ n+l-k
= ~(n) k,i_2k-i
]a(n)
~n+l-kj
n+l-k
i _ < i _ < 2 k-I '
r^ (n) - B(n) 1_ (n) (~n+2-k n+l-kJ~n+l-k
+
2 k-I + 1 -< i -< 2 k . (2.21)
Define
Then,
E (n) k,i
from
and
(2.18)
F (n) k,i
and
by
C (n) = W n - k + 2 - W n _ k + 1 E(n) k,i w k,i n
(2.22)
D (n) _ W n - k + l k,i ~ n
(2°23)
F(n) k,i"
(2.19) we have,
E(n) k+l,i
i + 6
-
8 Pn+l-k
E(n)
k,i '
i~i~2
k-I '
396
2k-i + i _
= F (n)
(2.24)
k,i ' (n) Fk+l,i
_
1 8 Pn+2-k
E (n) k,i '
1 < i < 2 k-1 - ' 2 k - I + 1 < i < 2k
= F ~ n)
we n o w d e f i n e
Equations
L (n) (s) r
and
M (n) (s) r
(2.25)
by
A (n) (S) = e x p r s ~ (n) ] wr Wr-i L r(n) (s) r [ n+2-r, 2 n+l-rj Wn
(2.26)
B (n) (s) = e x p [ s ~ (n) ]Wr-i (n) r L n+2-r,2 n+l-r ~n Mr (s) .
(2.27)
(2.14a-b)
now become
2n+l-r L (n) (s) = r
E (n) exp S [ ~ n + 2_r, i n+2-r, i
i=l
(2.28) n+2-r, 2 n + l - r
2n+l-r M (n) (s) = r
F (n) n+2-r,i
i=l
Using
the p r o p e r t i e s
v(n) k,i
one can f i n a l l y d e d u c e
Lemma
2.6. (i)
L(r) (s)
r
of the c o n s t a n t s
E (n) k,i
'
F (n) k,i '
(n) nk, i ,
1 +
(s)
:
(2.29)
and
(2.30a)
= 1
L(r+l) r
_v(n) ]] e x p [ s r~ (n) [ n+2-r,i n+2-r,2 n+l-r "
1 + ~
(2.30b)
,
Pr+l and
r
for
n _> r+2
I1+ Pn
ILnl r
[Lrn-I s,
r
s]
;
Pn (2.30c)
397
M "r'( ~ (s) = i
(ii)
(2 31a)
r
'
r Mr+l(s)
and
°
i_!_
= 1 +
for
(2.31b)
@+s Pr+l
n ~ r+2
M (n) (s)
=
1
~ ] Pn
r
(s)
+ ~ L Pn
r
(s)-
r
(s)
.
(2.31c)
The small
recursion
enough,
partial
the
products
H (n) (s) , r
formulae functions of
n >_ r+l ,
H (n) (s)
r
(2.30c)
and
L (n) (s) r we
and
the
type
by
H (r+l) (s) = 1 r Pn
are
( s ) - r,
indicate
M (n) (s) r
seeking.
=
Pr+l
(2.31c)
are
More
that,
for
essentially
precisely,
define
and
(s)
r
n
Pr+2 j
-> r+2
.
Pn-I ) (2.32)
Then,
using
(2.30c),
one
can
show
by
induction
6 L (n) (s) = U (n) (s) r r
1 +
:
that
n-i
8+s~ Pr+l I 8+s~ 1 + 1 + 6] (i + P r + 2 )
6 H (j) (s) r
+ j:r+2
[
~+s~ (i + ~+s~ (i + pj ] Pj+l j (2.33)
where
=
i+
0+s,
l+
Pr+l )
Moreover,
from
(2.30c)
(2.34)
l+
Pr+2 ]
Pn
again,
L H ~nj (s) r
-
(n-i
(s)
r
(n-i U r
6 H,n-l,C~ (s) +
(s)
r
@+s 1 + -±Pn-~
) "
398
6 =
1 +
+
Pr+l~ 6+s~ i + I + 6] (1 + P r + 2 )
] + 6 H (n-l) (s) r
6 H r(J) (S)
n-2
@+s~
X
j=r+2
8+s~ (1 + 8+s~ (1 + pj ) Pj+l)
J
@+s
1 + Pn-1 (2.35)
on using
(2.33).
n _>r+l ,
by
Similarly,
relative
K (r+l) (s) = 1 r
K (n) (s) : r I
from
(2.31c)
M (n) (s) r
define
K r( n ) ( s ) ,
and
8+sF. (n) M (n-l) (s)] Pn [nr (s) - r [ (
Pr+l ) Then,
to
Pr+2 J
,
n ->r+2 .
]
(2.36)
Pn-1 )
we have that
M (n) (S) = V (n) (S) 1 +
r
r
n-i [
j=r÷1
6 K (j) (S) I ...... @+S~
(i
2.37)
+pj+ jj
where n
(n) (s) = ~ Vr j=r+l
and,
from
(2.31c)
and
[i + pj
2.38)
) ,
(2.37),
M (n-l) (s) 6 K (n-l) (s) K (n) (s) = r + r 8+s~ r v(n-l)r (s) (i + P n _ l )
=
1 ~
n-2 [
6 K (j) (s) ] 6 K (n-l) (s) r j + r @+s~ il 7 @+s~ 1 + Pn-i @+s (i + p j ) Pj+l j
j=r+l
2.39) From
(2.35)
and
(2.39)
one can deduce,
by a s t r a i g h t f o r w a r d
induction
argument,
Lemma
2.7.
6 = 6(00)
For each real number and a constant
C
o0
with
such that
o 0 + e > [1
IH (n) (s) [ -< C r
there and
is a value
399
IKr(n) (s) I -< C numbers
s
with
Thus, given and
3.
for
the
all
n
Re(s+8)
-> o
solution
explicitly
by
and
r ,
all
complex
+ 8 .
0
Cn(X,S)
(2.13),
and
n -> r+l ,
of
(2.11)
(2.26),
satisfying
(2.27),
(2.33),
(2.12b)
(2.34),
is n o w
(2.37),
(2.38).
The
Differential
As
e > 1 ,
partial
sums
it
(2.3)
Equation follows
converge
E from
as
(2.5)
n÷co
that
for
fixed
r ~ 0
the
Define
co
w ! r)
=
[
g(l+k'2r)
•
(3.1
k=O
It is c l e a r
from
sequences
~r(n)}
converge and
ar
(2.6a-d)
to ,
'
finite
,
and
(2.7-9)
{~r(n)} , limits,
respectively.
c r = ~rBr
and
that
{b~ n) } ,
which
(2.9).
we
Observe a0 = 0
Furthermore,
it
for
{c~ n) } ,
denote
also
fixed
by
that
mr br
=
r ~ 1
the
{a~ n) }
all
, 8r
, br
~r
Br
+
, cr
,
,
and
ar = a r - i
from
that
Z n ( P r) + ~r - Br
follows
,
(3.2
r -> 1 ,
from
Lemma
6r-i
2.3
that
for
r ~ 1
w
r - Dr :
(3.3
0 W r - I [Wr- 1 + ~ (Wr_ 1 - Wr_2) ]
(plP2 ".. P r _ l ) and
6r-i
w~
Br+l - Br = (plP2 ''' pr ) One
can
also
prove
(3.4
e Wr_lW r
that
w!r) ÷ (w!rl wlr÷l))
6r-1 B r + w (I) =
(3.5
8
(plP2 "'' pr )
Wr- 1
400
Finally,
from
Lemma
2.5 we
a
have
=
O
r
Setting and
~
y r
-w (I)
÷ 7
Define c(x)
as
it
on
differential
the
equation
(2.11)
boundary
we
condition
at
of d e p e n d e n t
this
(3.9)
may
(3 5)
stage
= b(x)
be r e w r i t t e n
£1exp[
-s
to
that
mr
and
C(x)
b(x)
letting
n ÷co
= b
and
r
in the
= 0,
(3.7)
x÷~
(3.8)
becomes
as
(3.7-8)
= eYSXz(x,s)
+ s2C(x)z
~ i ,
by
equation
transform
y(x,s)
- s B(x) z'
- 2y
(0,o0)
the
~ exp(Tsx)
Z(X,S)
B(x)
on
+ s2c(x)y
(2.12b)
variable
z"
where
and
Formally
obtain
-sb(x)y'
y(x,s)
change
(3.3)
c(x)
r -> 1 .
(ar_ 1 , a r ] ,
It is c o n v e n i e n t
from
and
b(x)
y"
while
follows
(3.6)
r ÷ ~.
functions
= cr
(plP2 "'" P r _ l )~] @r-I
as
= c(x)
by m e a n s
of
the
:
(3.9)
= 0
(3.10)
x÷~
- 7b(x)
+ y
2
Equation
as
10XB]d~]
+ s 2 C ( x ) e x p ( -s
10XB] • z = 0 •
(3.11)
Set
r(x)
=
(3.12)
B(u)du
and u(r,s)
= z(x(r),
s) .
(3.13)
401
Then, we
on
changing
obtain
the
the
independent
variable
from
Lemma
r
in
(3.11)
e-Sru(r,s)
(3.14)
= 0 .
now
4.1
u(r,s)
to
equation
d----re-srB(x(r))~] + s2 ar E
Consider
x
[i].
÷ 1
satisfies
If
and the
u( r, s)
u(r,s)
is a s o l u t i o n
e-SrB(x(r))d~
integral
=
÷ 0
as
of
(3.14)
r + ~ ,
satisfying
then
u(r,s)
equation
~e -s (v-t) Jt
1 - s2
B ( x C (x.(v)) (t)JB[x(v))
u (v, s) d v d t
" (3.15)
Furthermore,
(3.15)
s = ~ + iT
such
has
a unique
The Neumann
series
[r 0 , ~), The
former
r0
solution
for
. C(x.(v) ) B [ x ( t ) ) B ( x ( v) ) d v d t
< ~
values
that
e-d(v-t) Io
bounded
It
latter
assertion
expansion suitably
assertion
follows
in t h e large,
may
be
space
easily of
equipped
from
continuous
with
the
by
twice
established
the
(3.16)
relevant
functions
usual
supremum
integrating
on norm. equation
(3.14). Observe
that,
on
C n = ~n6n
(an_ 1 , a n ] ,
-
C(x)
y( a n+ 6 n) + y
= Cn ,
where
2
= (6 n - 7) 2 + (6 n - y) (a n - B n)
=
(6n - Y ) ( ~ n - Bn)
~
( ~ n - Y) C a n -
Bn)
Bn - ~ ] 1 + an _
Sn--j
as
n ÷ ~
(3.17)
402
by
(3.3)
and
(3.5).
Also,
on the same
interval
B(x)
= B
,
where
n
Bn = 2(Sn-y)
( ~ n - Sn)
+
(c~n - ~n ) [ i + 2(~n-Y)]c--~~n J N
by a s i m i l a r
argument.
(a n - Sn)
By u s i n g
as
(3.18)
n ÷ ~ ,
(3.17-8)
directly
in
(3.16)
one can
deduce
Lemma
4.2.
u(r,s)
The i n t e g r a l
whenever
Re(s+8)
In addition, abaolutely theorem
(3.15)
has a u n i q u e
bounded
solution
> 1 .
as the d o u b l e
convergent
that
equation
for
integral
Re(s+@)
for t h e s e v a l u e s
of
on the r i g h t of
> 1 , s ,
it f o l l o w s
(3.15)
(3.15)
is now
by F u b i n i ' s
may be r e p l a c e d
by
(1.4) w i t h
k(r,t,s)
= _s2e-St
We now r e t u r n ~n(X,S)
constructed
~
Irt
to a d e t a i l e d in §2.
e sv
(r -< t) .
B (x(v)7 dv
consideration
(3.19)
of the s o l u t i o n s
Set
L (n) (s) $(n) (s) - r _ _ r ~ (s) r
(3.20
Y
and M (n) (s) #(n) (s) = r r V (n) (s) r It follows Pn
n inn
I~r(n) (s)} Re(s+8) small
from as
(2.33), n ÷
,
are c o n v e r g e n t
1 > o 0 + 8 > [,
enough.
Set
(2.37),
Lemma
2.7,
that the f u n c t i o n as
provided
n + ~ that
(3.21)
and the fact that sequences
~ n)(s)
and
in any fixed h a l f - p l a n e the c o n s t a n t
6 = 6(00 )
is c h o s e n
Also.
f o r fixed
= g ( s )
and
From (2.21)
NOW.
r
and
Re(s+B)
l i m v l n ) ( s ) = Vr(s) n+m
> 1.
by ( 1 . 5 )
w e have
lim~:)(s) n-t-
where
w e have
by L m a s 2 . 3 and 2 . 5 ,
p[:;:
-Bp)]ap)
= ~(k-')
and therefore
404
n
[~(n) _ 8(n)]
Z k=r where
the o r d e r
argument
using
[~k+i
constant (3.6)
(n)
]a k
= o(i)
is i n d e p e n d e n t
shows
t h a t we
of
also
n
(3.28)
,
and
r .
A similar
have
oo
We First in
are n o w
note
n .
that
in a p o s i t i o n for
fixed
to let
r ,
if
integer
N
x • lar_l ,
(2.26),
and
= ~ (x,s)
it is c l e a r
n ÷ ~
in the
sequence
a(n)]r
that
for
formula
~a!n)~
is fixed,
(at_ 1 , a r]
_[(n)
that
(2.27)
x •
the
Consequently, such
(3.29)
( Bk+ 1 - B k) a k <
k=l
for all Re(s+@)
(2.13).
is d e c r e a s i n g then
there
n a N .
From
> 1 ,
lim
is an (2.13),
~n(X,S)
where
~(x,s)
= Ar(S)exp(<~rSX)
+ Br(S)exp(6rSX)
(3.30)
;
here
Ar(S)
1 exp @+s Pr
-
s
~ ~ k=r
] (Bk+l - 6 k ) a k
"
W r - i + 6(Wr_ 1 - W r _ 2) w Ur (s) Yr (s)
] (3.31)
and
I B r (S) = e x p
oo
s k=r
and the
functions
s ~i)
are
always
that
define
the
Yr(S)
analytic the
and
function
Cr(S)
in a n y r e g i o n
constant
that
equation
(3.9),
combination end-points
the
function because
is e s t a b l i s h e d
= e -~sx
~(x,s)
on e a c h
of e x p o n e n t i a l
> ~0 + ~ >
is c h o s e n
(3.32)
Vr(S)¢r(S)
small
also
1 ~
~(x,s)
'
,
provided
enough.
We
by
~(x,s)
Observe
.
(and t h e r e f o r e
Re(s+@)
6 = 6(00)
%(x,s)
Wr_i " w
(Bk+l - ~k)ak
¢(x,s)
is a s o l u t i o n
interval
solutions, by l e t t i n g
and n
(3.33)
.
to the d i f f e r e n t i a l
(ar_ 1 , a r]
~
is a l i n e a r
the c o m p a t i b i l i t y tend
to
infinity
at the in the
405
identities
(2.15)
and
tion
~(x,s)
÷ 1
that
(2.16). and
One
can
also
[fxi
exp
-s
B
s h o w by a d i r e c t + 0
as
computa-
x ÷ ~
Thus,
0 from Lemmas bounded when
4.1
and
solution
Re(s+8)
solution
4.2,
of the
> 1 .
of equ.
we
see
that
integral
To p r o v e
(1.4)
~(x(r),
equation
part
be d e n o t e d
(ii) by
s]
(1.4)
of the
is the u n i q u e with
kernel
Theorem
u(r,s).
Then
(3.19)
let the
f r o m the
above
argument
u(r,s)
for
Re(s+9)
> 1 .
into
a sub-region,
from
(3.30-2)
a priori) Re(s+9)
the
Assume G,
that
(provided
1
> o 0 + 8 >[
tinuation, for t h e s e then
follows
for
s e G ,
of
s]
s ,
immediately
c a n be a n a l y t i c a l l y
~ < Re(s+8)
~(x(r), from
@
s 1 .
is c h o s e n
is a n a l y t i c
G .
satisfies
(3.34)
1
strip
~(x(r),
s)
u(r,s)
s)
the c o n s t a n t
containing
~(x(r), values
that
of the
function
= ~(x(r),
s]
(3.24-5)
for
and
be
suitably
by a n a l y t i c
s E G .
cannot
It is c l e a r small
in a h a l f - p l a n e
Consequently,
(1.4)
continued
In p a r t i c u l a r ,
identically
(3.30-2)
con-
that
zero. 6(s+8)
It ~ 0
as r e q u i r e d .
References [i]
F.V.
Atkinson,
private
communication.
[2]
R. Brent, On the zeros of the R i e m a n n c r i t i c a l strip, Math. Comp. 33 (1979), M R 80g: 10033.
[3]
H.M. E d w a r d s , 1974.
[4]
I. K n o w l e s , preparation.
[5]
B. R i e m a n n , U e b e r die A n z a h l der P r i m z a h l e n u n t e r e i n e r gegebenen Grosse, Monatsberichte der Berliner Akademie, N o v e m b e r 1859.
[6]
E.C. T i t c h m a r s h , Oxford University
Riemann's
On the
zeros
Zeta
zeta f u n c t i o n in the 1 3 6 1 - 1 3 7 2 ; see a l s o
Function,
of the
Riemann
Academic
Press,
zeta-function,
The T h e o r y of the R i e m a n n Press, L o n d o n , 1951.
New
in
Zeta-Function,
York,
Superposition Principles and Pointwise Evaluat, ion of Sturm-Liouville Eigenfunction Expansions Mark A. Kon*
Louise A. Raphael t
§1. Introduction The problem of correctly summing eigenfunction expansions ill pointwise and function space topologies has been considered classically [10,11], as well as in relatively recent papers (see, e.g, [3,9]). Current intercst in this subject in the Soviet literature has arisen from its relation to the theory of stable summation of expansions, which falls into the area of ill-posed problems [2,6]. The case of expansions in a continuous spectral parameter (arising, for example, from singular Sturm-Liouville problems) has been studied less extensively [7], and techniques which are useful in the case of discrete spectrum often do not directly translate to the continuous case. In this paper we present an approach which is useful in both situations, and two applications which subsume a broad class of summability methods, including the Abel, resolvent, and Gauss-Weierstrass methods. We begin with some definitions. Let ¢(),) be a function of a real (or complex) variable with ¢(0) -~- 1 and
f(x) ~ ~ a,~un(x)
(1)
n
be an expansion in eigcnfunctions of a self-adjoint operator A. The ~ is generic and may represent summation or integration. The expansion (1) is ¢-summable in a given topology if lim ¢(,A)f(x) "= lim ~
a,~¢(ek,~)un(x) --~ f(x),
*Partially supportcd by the NSF under grant MCS800-3407 tResearcla supported by the A ~ O , grant DAA-G29-81-G-0011
407
where k,~ in the spectrum
o(A) of A corresponds to the (generalized) cigenfunction
u,.(x). Abel, Gauss-Weierstrass, and resolvent summability correspond to ¢(X) = e--X~
e
__k 2
~ ~nd
1 1~-~, respectively.
We consider the following Sturm-Liouville problem on [0, oo): : -~- q(x) ) u(x, X) • ku(x, k); Au(x, X) -~ ( --~J~5
u(O,k)eosfl-]-u'(O,X)sinfl--~ O,
u(co, X) < co
with /3 G [0, 2~r) fixed. The eigenfunctions u(x,X) are normalized by u(0, X) ~- sinfl, u'(0, X) ~--- --cosfl. We assume the real potential
q(x) E L'(R) to be continuous and
bounded (see [7]). The operator A is then self-adjoint and bounded from be]ow, its negative spectrum is discrete and finite and its positive spectrum is continuous.
f(z) E L~(R+),
If
the expansion f(:) ~
C
F(X)u(x, X)dp(X)
holds, where
F(>,)~ :Jo°° f(x)u(x,X)dx. Above, b is a lower bound of a(A) and ~ denotes convergence in L ~. If ¢(X) is continuous at O, then formally
¢(~A)f --=-
b
¢(~x)r(×)~(~, X)dp(X)~Z0 :(x).
(a)
The pointwise convergence of (3) in the cases of Riesz and resolvcnt summability has been studied in [7] and [5], and the discrete case has been considered much more extensively (see, e.g., [10,111). All of the following results hold also for expansions on finite intervals, and the proofs simplify in some cases. For brevity, however, most proofs here will be sketched or omitted.
408
§ 2. Use of the Superposition Principle Our aim in this and the following section is to show that the above-mentioned class of results on ¢-SUmlnability in the case of continuous (as well as di::crcte) spcctruxn follows from a supcrposition principle for summability: if an expansion is ¢ l - s u m m a b l e a i d ¢2-summable, and a l and a2 are complex constants whose sum is 1, then it is a1¢1 q- a2¢2 summable.
This fact extends to infinite or continuous sums through
regularity arguments. Hccall x is in the Lebesgue set L ( f ) of f if lim f n
'1--'0 J--r~
]f(z q- rl') -- f(x)ldq' = O.
THEOREM 1 (Analytic multipliers): Let "y > 0 and ¢(z) be analytic in a neighborhood
P of the sector /?w = {z E C : larg zl _< 7}. For z E G ,
lot
¢(z) = O(Iz1-6) as
z ---* oc in P, for some 6 > O. Then for f C L2(R +) and larg e] < y~ < ~/: (i) ¢ ( c A ) f ( x ) ~ o f(x) for all x in the Lehesgue see o f f (and hence almost everywhere) O i ) ¢ ( e A ) f --~ f in LP(R + ) for f E L ~ A L p, 1 < _ p < co. ¢--~0
We sketch a proof of Theorem 1. Initially) let ¢(k) ~- ~: + x " Then
¢ ( ~ A ) f ( x ) = (1 + c,A)-~ f(x) -~- a1
ao(~, ~')f(~')d~',
(4)
where G~ is the Green function of A-+- ~,. 1 We define the mixed p, q norm
Ilg(x, z')llp,q -=
Ig(x, x')lPdx
/
dx'
,
with the natural definition when p or q is infinite. Let G~ denote the Green function d2 of --~-~ -I- ~.1 The identity
R,~(x,x') ~ a~,(x,x')-c~,(x,x*
') = -
/0 ~(~*'x,
x")q('x"G) o["x", x')dx"
409 yields Iln.llp,q _< sup {11(;;(~, ~")llp,~llC~(x", x tl _ ~ 0
z')llq,~,lIqH1},
(5)
where subscripts on the right denete the variables with respect to which norms are taken. A series of estimates of terms on tile right side shows in [argc~[ < 7,
IlR,~llp,%~ ° 0
uniformly
7 <: % for 1 _< p,q <_ oo.
This togethcr with tlSlder's inequality shows tile remainder / ~
to be negligible
in the relevant asymptotic L p and oointwise estimates of (4). The a--~ 0 limit is thus unchanged when G~ is replaced by G~, and the latter can be computed: • ' ) = vFd~---~- + v/~ a~(~,.~
sin fl ~- V~COS fl ~ 2(sin ~ - 4-d cos ~)
~
=- a~')(~,~') + G(g)(~,~'). (6)
This is essentially a sum of convolution kernels, and the proof in the case q~(X) = ~ - x is completed by use of a standard theorem in harmonic analysis [8]: THEOREM: Let ¢(x) E L'(R'~), fl~. ¢(x)dx ~--- 1, and define fl(x) ~-~ sup [¢(t)l. For Itl_>lzl e > 0, let ¢,(z) = e - - " ¢ ( ~ ) . Xt~(z) C L~(R ") a n d y ~ L~(R~), the,~ l i m ( f * ¢,) = f(x) ¢--*0
whenever x belongs to the Lebesgue set ot" f , and f * ~b, ~
f in f p
(1 ~
p <
OO).
~---* 0
The convolution kernels in (6) are essentially L L n o r m preserving dilations (in the parameter a) of functions with L l radially decreasing bounds ~(x) as defined above, so that the Theorem applies to them. We now only require arguments proving uniformity of convergence in arg ~ as ~ --* 0, which will be omitted. For general ~b(z), we use the Cauchy representation of q~(),) as a superpositiou of dilations of ¢(X) = i-~x:
¢(,x) = ~ 1 f r _ ~ (
1-
~ ) - 1 d~
(7)
1 where the contour F contains a(A), and coincides with the boundary of the set ~-B7
for Iz[ large. Let the operator ¢(eA) be defined by (3). Then (3), (4) and (7) together
410 wiLh some regularity arguments show that the integral kernel of ¢(cA) is 1
I~o(~, ~') = ~Z fJr
¢(~z)C--~,(~,~')dz,
(8)
and hence
1
¢(cA)f = 27r~
~(.'.) f ~ ( _ _ ~ ) G _ ~ ( z , z , ) f ( x , ) d z , d z . z
a0
(9)
We now proceed formally: as e -+ 0, ~he inncr integral converges to f(z); since ¢(n) = 1, the proof is completed by use of Cauchy's theorem.
Remark: Although the above mcthod also works for discrete expansions on finite intcrvals, these results in the discrctc case follow more directly from estimates of kernels such as those in [3].
§ 3. Non-Analytic Multipliers Proof of poifitwise summability with non-analytic multipliers ¢ can be accomplished by use of Fourier transforms and the superpo~ition of exponentials. Lct ¢(),) E LI(R) be diffcrcntiable and satisfy
(i) ~$'(() E LI(R)
(ii)~¢(~)i ~i_-~oo0 (iii) ¢(0) ---~ 1 where ¢ is thc Fourier transform of ¢. We denote the above conditions by [D]; note t h a t thcy are essentially differcntiability conditions. We continue with our assumptions on the Sturm-Liouville operator A. TItEOREM 2: If f(x) E L2(R+), ~hen the expansion of f in eigenfunctions of A is
¢-summable on the Lebesgue set of f. Proof: Let x be a Lebesgue point of f . We have
411
¢(ck)f'(k)u(x, k)dp(k) =
¢(cA)f ~-
¢(~)ei~C~×)d(F(k)u(x, k)dp(k)
b
-b
oo
v'~
oo e \ e ] J - - b
The interchange of integrals can bc justiticd if the inner integral is interpreted as an L 2 limit. Let t~(,~)
=
- -
b
~"F(x)~(:~, ×)alp(X).
(lo)
We define ~0 to be a generalized Lebesgue point of h(~) if -
h((o + ( ) d (
--, h((o).
The following lemma is due to M. Freedman. LEIVlMA: Let f ( ( ) E L~(R), with ~o a generalized Lebesgue point of f. Let ¢(k) satisfy
p]. Th~.
Thus to prove Theorem 2 it suffices to show t h a t (i) 0 is a generalized Lcbesgue point of h(() (ii) h(0) =
f(x).
We note that
h(~)d(=
(h~) e'~-F(X)uCx')')dP(k)"
(11)
Let F bc a contour which encloses R + , and for Izl large coincides with { z : lira z I = ½ In (Re z)}. For z e F, ]zl large, we have ~ - e ' ~
(~)
= 9(×.) = 2~--;
4lzl--] ; hence
~-
x~
412 Through interchange of integrals and use of (4), we have
S
b
g(X,DF(x)~(x, x / ; ( x ) .
.1
2~ri
.
.
z tao
-z a r/
, (z, x r ) f ( z ' ) d z '
--~
}
dz.
(12)
\Ve have G , = G~ + R ~ , with [~<~defined by (5). If 17~ can be appropriately bounded, we are reduccd to dealing with G~ and the q ~ 0 problem. To this end, according to (7), we calculate Ila*llv, = _--_( f la'(~, z')l;dz) ~ explicitly. The i 1 and L °° norms are
IIC;(x,x')ll~,: = o
~--
(1~1-~ 0; largal < ~ < ~);
II .11~,~ = O ( v ~ .
By interpolation,
IlaL(~,x')ll~,= = o I~1 ~
,~-½
(13)
uniformly in larg a I < "71. On the other hand,
Ila~ll,,~- Ila;ll,,~ _< I I a ~ - aLIla,~ _< Ilqll~Cllaolh,=, where C is chosen such that
IIG;( ~, z')lh,: N C Hence
IIGolI,,:
(z' E R + ) .
(14)
_< 1--c~qH~'c A similar argument shows
IIG; -- G~II~ = O(lal), and thus Ilaoll~ = o ( I v ~ ) ,
(15)
so that (13) holds as well for O~.
Some calculation shows that along the contour F, for Iz[ large, ~_ ~
G_} p,z
By (5) and (16)
_~+~ __~ " ~.(lnl~l)
= o(n~+', I~1
').
(16)
413
Thus by }lSlder's inequality
27ri
/r
z
/o7 -:" r/
-- :
p
o0,°)
'),z ,1-.o -. o (17)
To prove (i) it now remains to consider (12) with G ' replacing G. In this case u(x, X) --~ CI(X)e ~'/£'* + C,~(X)e ~x/X~, and b := 0. Some substitutions show t h a t the left side of (12) is a convolution:
/7
~(x~)F(x)~(~, x),zp(x) = x 4 ~ ) *
/7
F ( x > ( . , , x)ep(x)
where
'F
x,(~) = G
oo
~(k2V)e~k~dk'
By the harmonic analysis theorem in §2 it now sufl:ces to prove t h a t Xn(Z) has an L 1 radially decreasing bound, which can be done by explicit computations. Thus
/7
g(Xq)F(X)u(z, X)dp(X) n~o f(x),
(18)
from which (i), (ii) and Theorem 2 follow.
§4. Some R e m a r k s on Stable Summability Much of the motivation for study of summability theory in recent years has arisen in the context of stable summability [4,5,6,91. This is used in the study of stability of pointwise values of eigenfunction expansions under perturbations of coefficients. Let ¢(¢A) denote the operator in (3). T h e n qi-summation is pointwise stable if there exists a non-trivial scaling 7(¢) such t h a t for any net {f~}~>0 of L 2 functions such t h a t
]]f'~ -- f]12 <-- % ¢(cA)f'~(~)(x) ~ o f(x) on the Lebesgue set of f . If error causes the generalized Fourier coefficients of f to be measured as those of f n (the L 2 error in Fourier transform space is still Ilf n - - f][2), then pointwise error can
414
be made to vary continuously with L 2 error, (and in particular be small when L 2 error is small) through use of a summability method ~ and an appropriate scaling -~(e). With probabilistie estimation of L 2 error '7 [1], this procedure can be applied, for example, in the recovery of local properties of signals from global properties. The following theorem translates statements on summability into ones on stable summabilily; it is proved in [41. TI1EOREM : Let ¢(k) be a real-valued function and f , f n E L2(R + ) (or L~[a, b]).
Let (0
Ill- - f~ll~ ~ ~y
(ii) f o
:*(×)d), v~ (co
(iii) ~(eA)f(Z) ~ o f ( z ) uniformly on a bounded s,lbset E of R + (or [a, b]). Or) e be a function or'7 such that as "7 ~ O, a ~ 0 and ~
a4
--~ O.
Then q~(eA)f~(~)(z) --~° f(x) uniformly on E; in particular i r e can be an arbitrary point in the Lebesgue set of f , q~(eA) is a stable summation method.
The scaling in Or) is
the best possible. In Theorems 1 and 2 hypothesis (iii), a priori summability, is verified, giving the following: COROLLARY : If ¢(>,) satisfies the hypotheses of Theorems 1 and 2, and condi¢ion Oi) above, then ¢ is a stable summation method with optimal scaling -1x- -+ O. ¢14
This result was proved for resolvent summability on intervals in [9].
REFERENCES
[1]
Arsenin, V.I., Optimal S u m m a t i o n of Fourier Series with Approximate Coefficients, Soviet Math. Dokl. 9(1968), p. 1345.
[2]
Arsenin,V.I., and A.N. Tikhouov, Solutions of Ill Posed t'roblems, V.H. W i n s t o n Sons, 1977
415
[:~]
Benzinger, II.E., Perturbation of the Heat Equation, Jnl. Diff. Eq. 32(1979), p. 398.
[q
Diamond, It., M.A. Kon, and L. Raphael, Stable Summation Methods for a Class of Singular Sturm-Liouville Expansions, PAMS 81(1981), p. 279.
[!;1
Diamond, II., M.A. Kon, and L. Raphael, A Regularizatiou of the Pointwise Summation of Singular Sturm-Liouville Expansions, Proceedings of the International Conference oll I11 Posed Problems, to appear. Krukovskii, N.M., On the Tikhonov-Stablc Summation of Fourier Series with Perturbed Coell~cicnts by Some Regular Methods, Vcstnik Mosk. Univ. Mat., 28, No. 3, p. 17.
[7]
Levitan, B.M., and I.S. Sargsjan, Introduction to Spectral Theory: Self-Adjoint Ordinary Differential Operators, AMS, 1975.
[81
Stein, E. and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 1971.
[0]
Tikhonov, A.N., Stable Methods for the Summation of Fourier Series, Soviet Math. Dokl., 5(t964), p. 641.
[10] [11]
Titchmarsh, E.C., Eigenfunction Expansions, Part 1, Oxford, 1962. Zygmund, A., Trigonometric Series, Cambridge, 1959.
QUALITATIVE THEORY OF HYPERBOLIC CHARACTERISTIC INITIAL VALUE PROBLEMS I)
Kurt Kreith
For ordinary differential equations there is an extensive literature dealing with the oscillation and asymptotic behavior of solutions of u" + f(t,u) = 0.
Despite
physical interpretations which suggest their importance~ corresponding questions for hyperbolic equations have received surprisingly scant attention. The nonlinear telegraph equation b2u ~x~y + f(x,y,u) = 0
(I)
;
x ~ O, y ~ 0
describes the small displacements of an infinite string of unit tension and density° A unique solution of (i) is determined by characteristic initial data of the form
(2)
u(x,O) = p(x)
;
u(O,y) = q(y),
where p(x) and q(y) are assumed continuous on [0,~) and satisfying p(0) = q(0).
If
f(x,y,u) is continuous in R + × R + X R and satisfies uf(x,y,u) > 0 for u ~ 0, then
f
can be regarded as a nonlinear restoring force.
When this force is "sufficiently
large" then solutions of (i), (2) can be expected to exhibit an oscillatory form of behavior, even though p(x) and q(y) are of constant sign for x > 0 and y > 0. Alternatively, if If(x,y,u) l is "sufficiently small", then (I) can be regarded as a perturbation of the wave equation
(3)
u
xy
= O,
whose solution p(x) + q(y) retains certain of its qualitative features when (3) is replaced by (i). In the linear case, oscillation properties of
(4)
u
xy
+ f(x,y)u = 0
have been established by the author [2], [3] and Pagan [7], [8] using Sturm-type
l)Research supported by a grant of the National Science Foundation, NSF MCS 80-02130.
comparison theorems.
Specifically, i f f(x,y)
2
~ ( x , y ) i n a domain D and
h a s a s o l u t i o n s a t i s f y i n g v = 0 on t h e s p a c e l i k e p a r t of a D , t h e n s o l u t i o n s of (4) s a t i s f y i n g p r e s c r i b e d c o n d i t i o n s on t h e c h a r a c t e r i s t i c o r t i m e l i k e p a r t of 8 ~ c a n b e Pagan [ 7 ] a p p l i e d t h i s t e c h n i q u e t o c h a r a c t e r i s t i c i n i 2 t i a l v a l u e problems w i t h f ( x , y ) 2 F(x,y) = k > 0, i n which c a s e t h e s o l u t i o n 2 v = sin(Lx k y / ~ )i s z e r o on t h e s p a c c l i k e boundary of shown t o change s i g n i n
D.
+
5
pl(x)
.
2 and n = 1 , 2 , . . . I t f o l l o w s t h a t i f f ( x , y ) 2 Ic > 0 and f f 0 i n R X R , t h e n e v e r y s o l u t i o n of (4) must change s i g n i n each
>0
for arbitrary L 0, q l ( y )
5
Dn(L). To a d d r e s s t h e c o r r e s p o n d i n g q u e s t i o n f o r f u n c t i o n s f ( x , y ) which a r e
not
bounded
away from z e r o , i t i s n a t u r a l t o c o n s i d e r F ( x , y ) = G(xy), i n which c a s e s = xy t r a n s forms ( 5 ) i n t o
Assuming ( 5 ' ) t o b e o s c i l l a t o r y ,
the zeros s l
<
s2
<
of V(s) d e f i n e a sequence
of unbounded n o d a l domains f o r v ( x , y ) = V(xy) such t h a t v ( x , y ) v a n i s h e s on t h e spacel i k e boundary of
However, t h e unboundedness of t h e D
p r e c l u d e s a d i r e c t a p p l i c a t i o n of t h e t e c h n i q u e
of 171. T h i s d i f f i c u l t y can b e overcome by n o t i n g t h a t f o r a r b i t r a r y c o n s t a n t s
>
0, t h e f u n c t i o n w(x,y) = v(x+t;,y+lj)
satisfies w XY
+ G((x+<)(y+'ll))w
5 >
0,
= 0 and t h a t
t h e c o r r e s p o n d i n g n o d a l domains
a r e bounded. ing
T h i s o b s e r v a t i o n e n a b l e d Pagan and t h e a u t h o r t o e s t a b l i s h t h e f o l l o w -
.
Theorem 1.
Suppose t h e r e e x i s t c o n s t a n t s
E; >
0
g ( s ) such t h a t (i)
d -(s ds
dV -) ds
+ g(s)V
= 0
&
oscillatory,
& T( >
0
a_ c o n t i n u o u s f u n c t i o n
418
(ii)
f(x,y) > g((x+~)(y+~)) ill R + X R +.
If u(x,y) is a solution of (2), (4) with positive characteristic initial data satis lying p'(x) j 0, q'(y) < 0 in R+, then u(x,y) changes sisn in e a c h ~ n.
A different approach to nonlinear hyperbolic oscillations was introduced by yoshida in [I0]o Defining o t U(t) = J u(t-y,y)dy 0
(7)
for a solution u(x,y) of (i), where f(x,y,u) is assumed to satisfy (8) and
f(x,y,u) ~ h(x+y)~(u) > 0 ~
is convex in
u,
in R + × R + X R
it is possible to impose hypotheses on
h
which assure
that U(t), and therefore u(x,y), changes sign. Such "averaging techniques" have recently been applied in more general situations by Kusano, Yoshida~ and the author. (9)
u
xy
For example, if (7) is applied to
+ f(x,y,u) = F(x,y),
subject to characteristic initial data (2), one obtains the ordinary differential inequality n t (tU)" + th(t)~(U) ~ p(t) + q(t) + ~0F(t-T'T)dT
to which recent oscillation criteria of Kusano and Naito [5] can be applied, yielding results such as the following.
Theorem 2.
If for large limt-~~inf
T
~I~(1 - ~)[p(s)
s
+ q(s) + ~[0F(s-~,cy)dc]ds = -¢0
and
l %sup f (1 -
s
+ q(s) + ~[0F(s-o,~)do]ds = q~
then every solution of (9), (2) has a zero in each qp ~ [ (x,y) E R21p < x+y < ~}.
Similar techniques can also be applied to boundary value problems associated with
(iO)
Utt - Au + f(t,x,u) = F(t,x)
419 in a cylinder
(0,=) X G c R n+l.
Imposing the boundary condition
~v g(t,x) !~u=
(11) the inequality
on ~G
(8) and the transformation
(12)
U(t) = ~
I
;
~Gu(t,x)dx
IGI = volume of G
yields
u + h(t)~(u) _< T I~ ~ G g ( t , x ) d x +
i
~GF(t,x)dx '
to which the previously cited theorem of Kusano and Naito [5] can also be applied, yielding analogous oscillation criteria for (I0), (ii). In case (0,~) X G is replaced by a smooth non-cylindrical
domain H c R + X R n,
similar techniques apply in case (II) is replaced by
(13)
u = 0
on ~H
and ~H is timelike for all t > 0.
Defining G
!
= [(t,x) E H l t =
~}, the transforma-
tion
U(t) = *
(12')
r u(t,x)dx ~Gt
and an n-dimensional version of Leibniz's rule yield analogous oscillation criteria for (i0), (13) in
H.
Turning to questions of asymptotic behavior, we recall n
th
order homogeneous
ordinary differential
[9] that a disconjugate
equation %n u = 0 has an asymptotically
ordered fundamental system of solutions Ul(X),...,Un(X ) with the property that lim x ~ ~
ui(x)
- = 0 fer i = l,.o.,n-i Ui+l(X)
equation ~ u + f ( x , u ) =
For n = 2, the question of when the perturbed
0 has solutions with corresponding asymptotic properties goes
n
back to Atkinson
[I] and Nehari
[6]; very general results of this sort (for function-
al equations of order n) have recently been given by Kitamura and Kusano A natural hyperbolic analogue author) includes
[4].
(currently being studied by C. A. Swanson and the
(I) when the initial data p(x) and q(y) are bounded on [0,~).
In
this instance the wave equation u = 0 has the bounded solution u(x,y) = p(x) + q(y) xy and it is natural to seek conditions on f(x,y,u) which assure that the solution of (I), (2) is also bounded for large values of Writing
(i), (2) as an integral equation
x
and y.
42O
u(x,y)
suggests
=
one form of boundedness
differential
Theorem 3.
ry
I ~ ~f(g,~,u(g,~))dg ~0 ~0
p(x) + q(y) -
criterion analogous
d~
to those which exist for ordinary
equations.
If f(x,y,u)
i_~ssuper- or sublinear in
u
and if there exists a constant
c > 0 such that f(x,y,c) > 0 in R + X R + and
~= ~ f ( x , y , c ) d ~ J
d~ < ~,
then there exist constants X > 0, Y > 0 and a solution u(x,y) of (I) which is bounded i__n_n[(x,y)Ix ~ X, y ~ Y}.
The proof consists
of showing that
~[u] = 2c -
;x;;
f(~,~,u(~,~))dg
d~
X has a fixed point in a Banach space of functions which are continuous {(x,y)Ix > X, y __>Y} and sufficiently
istence of such a fixed point follows from the Schauder theorem when chosen sufficiently
large.
These techniques
in
close to u0(x,y ) --- c in the sup norm. X
case also involves
some new difficulties
tions, which are readily established Furthermore~ the existence a hyperbolic
Y
are
to - + f(x,y,u) = 0, in ~xm~y n
of a solution bounded
However,
the
insofar as the necessity condi-
for ODEs, seem to he considerably more difficult.
the existence of bounded solutions
generalization
The ex-
~m+nu
allow a natural generalization
which case more varied forms of asymptotic behavior must be considered. hyperbolic
and
for all
of the asymptotic
new problems which will he interesting
x
"for large x and y" does not insure and
y.
Such questions
suggest that
theory of [ 4 ] may lead to a number of
in their own right.
References I.
F. Atkinson, 643-647.
On second-order nonlinear oscillations,
2.
K. Kreith, Sturmian 22(1969), 277-281.
3.
~ Sturm theory for partial differential Amer. Math. Soc. 81(1981), 75-78.
4.
Y. Kitamura and T. Kusano, Nonlinear oscillation of higher order functional differential equations with deviating arguments, J. Math. Anal. and Appl., to appear.
theorems for hyperbolic
Pacific J. Math. 5(1955),
equations,
Proc. Amer. Math. Soc.
equations
of mixed type, Proc.
421
5.
T. Kusano and M. Naito, Oscillation criteria for a class of perturbed Schrodinger equations, to appear.
6.
Z. Nehari, On a class of nonlinear second order differential equations, Trans. Amer. Math. Soc. 95(1960), 101-123.
7.
G. Pagan, Oscillation theorems for characteristic initial value problems for linear hyperbolic equations, Rend. Accad. Naz. Lincei 55(1973), 301-313.
8.
_ _ , An oscillation theorem for characteristic initial value problems in linear hyperbolic equations, Proc. Royal Soc. Edinburgh 77A(1977), 265-271.
9.
W. Trench, Canonical forms and principal systems for general disconjugate equations, Trans. Amer. Math. Soc. 189(1974)~ 319-327.
I0.
N. Yoshida, An oscillation theorem for characteristic initial value problems for nonlinear hyperbolic equations, proc. Amer. Math. Soc. 76(1979), 95-100.
APPLICATIONS
OF A COMPARISON THEOREM FOR QUASI-ACCRETIVE
OPERATORS
IN A HILBERT SPACE
Roger T. Lewis *
I.
Introduction A considerable
qualitative space.
amount of activity
properties
of the spectra of differential
The study is included
mathematical analysis,
problems
theory
of the spectrum--purely
etc.--in
When the differential
oe(T)
[19, p. 36].
operator,
therein for many selfadjoint
in this lecture,
of the differential
is selfadjoint,
[18], Schechter
of finite multiplicity
h~s recently been investigated
We refer
[ 7 ], Muller-Pfeiffer
[19], and the references
The nonselfadjoint
received a limited amount of attention
[14],
contained
problem, which has
in the past (see the references
by Evans
T = T ,
that belong to either the essential
[ 2 ], Glazman
results.
is
purely continuous,
case has received the most attention.
[15], Reed and Simon
[ 3, 4, 5], Evans, Kwong,
above),
and Zettl
[6 ],
[i0], and Race [16].
In the study of the nonselfadjoint the extensive
spectral
discrete,
T: H ÷ H,
or the set of isolated eigenvalues
This particular
Knowles and Race
in a Hilbert
In defiance of an
terms of the coefficients
to the books of Dunford and Schwartz Naimark
[17]).
the work discussed
the spectrum consists of only real numbers spectrum
selfadjointness,
(Reed and Simon
which includes
characteristics
length of gaps, bounds, operator.
operators
many important aspects of the study remain open.
A typical problem, to determine
to the study of the
in what has been said to be the three basic
in quantum mechanics:
and scattering
impressive history,
has been devoted
literature
problem,
it is natural
that already exists for selfadJoint
lecture we present a comparison
to hope to use
problems.
In this
theorem, which is proved in [i3], that establishes
lower bounds on the real or imaginary
parts of
%
in the essential
spectrum of
423
a closed quasi-accretive operator Re(Tu,u)
or
operators,
Im(Tu,u)
T
in a Hilbert space.
By comparing
with quadratic forms associated with selfadjoint
the real or imaginary parts of
% E o (T) e
are shown to be bounded
below by the least limit point of the essential spectrum of the selfadjoint operators.
For the sake of completeness, we outline the proofs here.
In [13],
the result is illustrated by applying it to certain second-order partial differential operators with Dirichlet boundary conditions. tion depended on a lemma of Wolf
There,
the applica-
[23, Lemma 2.4] that requires Dirichlet
boundary conditions, but it resulted in sharp criteria that are quite accessible We pursue here the problem with more general boundary conditions. tions in [13] and here are related to recent work of Evans
The applica-
[ 5] and the author
[11,12].
2.
A Comparison Theorem Let
domain
T: H ÷ H ~(T)
be a linear operator in the Hilbert space
that is dense in
O(T) = {(Tu,u): u E D(T), {~ E C: Re ~ ~ ~}
accretive.
(If
X = 0,
by
lluJl = i},
T
T
of
T
is a subset of a half-plane
X,
is said to be
is closable
with a
If the numerical range,
for some real number
As a consequence, T
H.
H
then
T
accretive
[ 9, p. 268].
quasi-
is said to be and
-T
is
dissipative.)
We denote the closure of
T. If a sequence
{u } E O(T)
that is normal in
H
has the property that
n
{u n }
has no convergent subsequence and
to be a
singular sequence
of
T.
llm Tu = 0, n-~o n
We take the
then
{u } n
essential spectrum
is said o (T)
of
e
a closed, densely defined operator %
for which
T - %1
T
to be the set of all complex numbers
has a singular sequence.
Kato [9, p. 2431 defines the
essential spectrum to be the complement of the semi-Fredholm domain of which implies that [9-, p. 233].
T - ~I
has a singular sequence for
h
in that set
T,
424
If
T
is selfadjoint,
T = T , then the spectrum, which contains only real numbers,
of the union of isolated eigenvalues see Schechrer
[19, p. 36].
of the essential GustafsOn
and Weidmann
with domain
in
H
For a good discussion
spectrum and how they compare,
Let
THEOREM i.
H
of finite multiplicity
D(T)
and
Oe(T)
of the different
definitions
the reader should consult
[8 ].
T
be a closed linear operator in the Hilbert space
that is dense in
that is bounded below with
H.
Let
A If
D(T) C ~(A).
Re(Tu,u) ~
be a selfadjoint operator
(Au,u)
,
u e D(T),
(i)
then Oe(T) ~ {% 6 ~: Re % h inf oe(A)}. I~
then
Oe(A) = ~
REMARK. D(T) ~
D(B)
Oe(T) = ~.
Let
B
be a bounded-below,
and
Im(Tu,u) ~ (Bu,u)
given below can be adapted
selfadjoint
for all
operator
u 6 D(T).
in
H
such that
The proof of Theorem i
to show that
oe(T) C {~ ~ ~: Im % ~ inf oe(B) }. We refer the reader to [9 , p. 330 and 336] for a discussion that are related
to Theorem i (and Theorem 2 below).
Theorem i follows easily from Theorem 12 of Glazman theory from Lemma 3.5 of Scbechter
PROOF OF THEOREM I. {u.} j C -- D(T) case of
of
a (A) = ~, e
let
will follow.)
dimensional
subspaee in
a
is selfadjoint,
[7 , p. 15] or via form
then there is a singular sequence
a = inf ~ (A). e
be an arbitrarily
For a given H
T
[20].
@ o (T) e Let
T - hi.
(T) = ~
e
If
When
of results
c > 0,
(In the large number and the fact that let
G(A)
be the finite-
that is spanned by the eigenfunctions
to the finite aumber of eigenvalues
of
A
in the interval
corresponding
(-~,a-c)
[ 7, p. 15].
consists
425 For each
uj,
let
u. = u! I) + u (2) J
J
where
u (2) 6 G(A)
J
the orthogonal complement of
G(A)
i = aujn 2 = ilu~l)ll2 + itu!2)il 2,.]
has a convergent subsequence.
in
then
in a finite dimensional subspace
and
J
H.
Let
of
{u~}
J
,
Since
{u~ 2)}
G(A)
u (I) 6 G(A)i
is a bounded infinite sequence H,
which implies that
be a subsequence of
{u~ 2)}
{uj} such that
{u! 2)'} converges. Since {u'.} cannot be Cauehy in H then for some 3 3 6 > 0 there is a subsequenee, which we still name {u'.}, such that 3 v u] - Uj+l[l > ~ for all j. Using {u~} we form the normalized sequence {vj} by letting vj = (u' - u' u'. - u' J J+I)/~ J J+Ill with v (I) = (u! I)' u(1)')/lIu '. - u' j " J - j+l J j+l ~ E G(A) j- and v!2) (u<2), (2) ,I {u!2) 3 - u'j+IIL 6 G(A) Since '} J = . j - uj+ 1 J/NU'. J Uv!2)~ < ~u~ 2)' - u J+l ( 2 ) "il / ~ -~ 0 J -(T
- ~I)vj -~ 0 Let
~
as
as
J -+ co. Also , either
converges then
(T - Xl)vj ÷ 0
or
J -~ •.
he a lower hound for
A,
then by the spectral theorem for
A
and (I) ,(T - )OvjR' + Re X >
I((T-
X)vj,vj} I + Re
_> Re(Tv.,v.) J J > (Av ,v )
-.) J = (A(v~I) + v!2)),v! I ) 3 3
+ v!2))j
Av(I ) v(1) ) + 2(Av(2 ) v(1)) + ,Av(2) (2)~ = ( J ' J J ' J £ j ,vj ) > ( a - e)(v(1),v~l))+ U ( v l 2 ) j ,v j( 2 ) ) + 2(Avl2),vi(1)')
, (2)(2)
(c~ - e) + (~ - a + c)kvj
,vj
) + 2(Av
~2)
,v
~1)).
k v (2) E G(A) then for some k, v (2) = ~ ci ~i where each ~i j ' i=l is an eigenfunction in G(A) corresponding to an eigenvalue li < ~ - e
Since
of
A.
j
Hence, (Ave2)
,vj
(i)) =
k . (i) i~ I c i ~i($i,vj ) = 0.
426
(T - A)v: + 0 as j + ~ Now, v!2)" ~ 0 and J J Therefore Re % > a since e is arbitrary, which completes the proof. In applying Theorem I to differential operators below, it is sometimes easier to establish the required inequality (i) for a closable operator
T.
The next theorem shows that with proper adjustments we can still make the ~
conclusion concerning the location of We let
Tt
denote the
sesquilinear form
t
oe(T).
m-sectorial operator associated with the sectorial
as guaranteed by the first representation theorem
[ 9, p. 322].
LEMMA I.
[13]
Let
T
be a closable operator with closure
be a closed symmetric form that is bounded below. Re(Tu,u) ~ S[u] and
(Im(Tu,u) ~ 6[u])
Re(Tu,u) ~ S[u]
THEOREM 2.
and domain D ( T )
Let
for all
(Im(Tu,u) ~ S[u])
T
If
u E D(T)
for all
D(T) C then
that is bounded below with domain
H. D(S).
Let If
S
and let
D(~)
and
D(T) C
D(S)
u 6 V(T).
be a closable linear operator in
which is dense in
T
with closure
H
be a closed, ey~netric form
D(T) C D ~ )
Re(Tu,u) > S [u] ,
and u • D(T),
then ~e(T) ~ {l e ~: Re I ~ inf Oe(Ts)}.
If
ae(T s) = ~
then
Oe(T) = ~.
The Remark following Theorem i holds in a similar way here. The proof of Theorem 2 follows from Lemma ] and the fact that a core of
S
[9 , p. 322].
D(T 6)
is
Some of the ideas used in the above proofs were
used in work of Evans [3 ] for certain special cases.
427
3.
An Application Let
fl be an open, connected set in with
n
that is the union of bounded,
open, connected sets
{~k }
~.
Assume that each
flk lies on one side of its boundary
C1
(as a submanifold of
~n).
~k C ~k+l'
Let
F
denote the boundary of Fk,
which is
(This requirement allows an application of
the Rellich compact imbedding theorem, which we need--see Showalter [21, p. 49] As an alternative, Agmon [I , p. 30] only requires that
flk be bounded and
have the segment property.) The Sobolev space cm(~) = { f ~ :
Hm(fl)
f E C~(lln)}
is defined to be the completion of
with respect to the norm
ll.l[Hm(fl )
induced by
the inner product (f,g)Hm(fl) = [ { Here, Let
fl~ m
w(x)
denotes the restriction of
be the usual measure on
~n
f
In addition, assume that
section we l e t
H = i2(~), w
DOg :I~I ! m}. to the closure of the set
ft.
and let the real-valued weight function
be measurable on the measure space
x 6 g.
u(x)
f D=f"
w(x)
the Hilbert
(~,m)
and positive for almost every
is bounded on each space
of all
~k"
In this
complex-valued
functions
satisfying f w(x)
The n o r m If
I1"11 a n d i n n e r f
is
product
a real-valued,
the essential range o f
f
lu<x) I2 dx < ~. (.,')
measurable
will
refer
function
to
L~(~).
on t h e m e a s u r e
is the set of all real numbers
t
space
(fl,m)
such that
m{x E ~: If(x) - t I < e) > 0 for all
s > 0.
We denote the essential range of
multiplication operator
Mf: L~(~) + L~(~)
~(Mf) = {u E L~(~): fu E L~(~)}. Reed and Simon [17, p. 229] •
by
f
by
Re(f).
Define the
Mfu = fu with
The next lemma can be found in the book by
428
LEMMA 2.
The spectrum of
Mf
is the essential range of
f;
i.e.
o(Mf) = Re(f). The proof of Lemma 2 follows easily by showing that only if
%
is in the resolvent set of
A sesquilinear form
T(',')
v E H.
A related
on
Let
H.
T
c > 0
H
is said to be
l~[v] I ~ clivllH2
such that
for all
notion is the idea of relative compactness of operators
and
for any sequence {Au } n
if and
Mf.
on a Hilbert space
H-coercive if there is a number
% ~ Re(f)
A
be operators on
{un} C D(T)
H
such that
with {Un}
contains a convergent subsequence.
~(T) C ~(A). and
Then,
A
It is well-known ] 9 , p. 244] for closed operators Oe(T) = Oe(T ÷ A)
when
A
is
T-compact.
{TUn}
Suppose that
are bounded,
is said to be T
and
A
T-compact.
that
We use this fact in the next
lemma. LEMMA 3. on each
~k
Let
p(x)
be a real-valued, measurable function that is hounded
and bounded below on
~.
Define
p(x) - inf R e ( P l ~ k ) , Pk(X) = Define
Mk: [ ~ ( ~ ) +
T: L~(~) + L~(~) u E D(T)
L~(~)
Mk
(Un} C D(T)
x ~ gk"
MkU = - p k u D(T),
u E Hl(~k )
Hl(~k)-coercive then each If
by
have a domain
implies that
PROOF.
0
is
x @ ~k
with
D(M k) = L~(~).
which is dense in
for each
k.
Let
L~(~),
If the form
such that
(Tu,u)
is
T-compact.
then for each
k
there is a constant
ck > 0
such that
CkJlUnli~l(~k ) _< l(TUn,Un)I _< ;ETUnllllUnJl. If {Un}
and
Kl(nk).By {u~}
{TUn}
are bounded, then
{u1~k}
is a bounded sequence in
the Rellich compact imbedding theorem there is a subsequence
such ~hat
{u~I
} ~k
converges in
62(~).
Since
w(x)
and
p(x)
429
are hounded on Mk
is
~k'
then
{~u~}
converges in
Therefore, each
T-compact.
A consequence of Lemma 3 is the fact that each
L~(£).
k
assuming that
T
Oe(T) = Oe(T + M k)
for
is closed and satisfies the hypothesis of Lemma 3.
This fact produces the following corollary which is central in our applications.
COROLLARY TO THEOREM 2.
operator with closure that
D(T) k H'(~ k)
p(x)
and
each
~k
q(x)
T
Let
and domain
and that
be a closable linear
T: L~(~) -+ L~(q)
T
O(T)
is
that is dense in
L2(~). Assume w
Hl(~k)-coercive for each
k.
Let
be real-valued measurable functions that are bounded on
and bounded below on
~.
Re(Tu,u) ~
If
f w(x) p(x)
]u(x)[ 2 dx,
u • D(T),
(2)
g
then
% • ae(T)
implies that
-- k-~o
Im(Tu,u) ~
then
~ • oe(T)
implies that
Let
~
f w(x)q(~) [u(x)[2dx,
Im % > lim inf Re(q~
for
pk(X)
x @ ~k
).
f w(x) lu(x)12dx+f
is defined to be
w(x)p(x)lu(x)I2~x,
~k
p(x)
for
x E ~ ~ ~k
By Theorem 2 and Lemma 2, is
follows.
and
inf R e ( p J ~ k)
then
Re(~T + ~)u,u) ~ (~k(x)u(x),u(x)),
Mk
(3)
By (2)
~k i.e., if
u • D(T)
~k
be defined as in Lemma 3.
Re((T+Mk)U,U ) ~ inf R e ( P 1 ~ k )
If
~k
-- k-~o
PROOF.
).
Re % > lim inf Re(Pl
u • ~(T)
Oe(T + M k) ! {~ • ¢: Re % ! inf Re(Pk)}.
~ T-compact then by Lemma 3
o e (T) = Oe(T + ~ )
Since
and the conclusion
The proof in the case of (3) is similar.
As an example of an appli~ition of the above corollary we consider a nonselfadjoint, second-order differential operator required for domain elements of
T
T.
The boundary conditions
include the Dirichlet, Neumann, and mixed
430
Dirichlet-Neumann
boundary value problems as well as a certain weighted
boundary condition.
Let n
1 T = w~
{-
where the coefficients derivatives
and each iet
are complex-valued
ak
f o r each
k.
Let
(A(x) + A (x))/2
The coefficients
on the boundary w(x)
~ ~k
F
both zero for any s 6 r
s 6 r. define ~u(s) ~A
s 6 r.
Note that each
of
Below, we let
(''')E
for vectors in Euclidean
(4)
has partial
c(x),
each
aij(x),
A(x) = / a i j ( x ) ) PR(X)
and
be the minimum
be the minimum eigenvalue w(x) ~
for each
k.
of
may become unbounded if
~
is unbounded).
All such singularities For each
k
we denote
let
S.
smooth, nonnegative
For
F
function on
and assume that
~(s) =
y(s)
(Ul(S),...,Un(S) 1
and
F. ~(s)
Let
6(s)
are not
the unit outward
n n ~u I ( ~ aij ~j ) ~ = (vu)t A" i=l j=l l
for some
L2w@) ;
u E D(T)
Let
includes
x 6 Fo
We n o w define the domain of
Tu E
Pl(X)
( F
S C F ~ Fk
D(T) = {u: u = ¢Ja
T.
~
may vanish for some
be a piecewise
A(x).
and the weight
of
be a neighborhood
y(s)
of
and let
assume only the value 1 or 0 on
for
a..(x) l]
a r e bounded on
of
normal at
Each
Also,
and we require that
Let
~.
k.
x
Nk ~ ~ n
in
f o r each
Similarly, S
N ~X
ak
(A(x) - A (x))/2i.
by
~
i,j=l
a..(x)~~-~-- + [ b . ( x ) ~ + c(x)} 13 Ox. 1 OX. 3 i=l l
be the c o n j u g a t e - t r a n s p o s e
eigenvalue
for some
~
that are bounded on
bi(x)
A (x)
n
to be
~ E Co(• n a, Nk )
and some
k;
~u(s) + y(s)u(s) = 0}.
and
~(s) ~
vanishes and
T
in a neighborhood
I.IE
n-space.
of the singularities
(5) of
denote tile usual inner product and norm
431
LEMMA 4. Re(Tu,u) _>
If
u E ~T),
then for any
-i n I) [VUI2E + (Re c(x)- e [ i=l
f[(PR(X)-e max[bi(x) fl
~ > 0
i
Ibi(x)I)[u[2]dx
(6)
and n > f[ (Pl(X) - ¢ maxlbi(x)])Ivul 2 + (Im c(x) - e-I [ --~ i i~l
Im(Tu,u)
PROOF.
By the usual Stokes-type formula (see Treves
Ibi(x)I)]u[2]dx. (7) [22, p. 78 and
p. 360])
= for
+
u, v ~ ~(T).
Since
nlbi _Z (x) i-
u v+c<x)u
v
~u(s)
dx-f
i
v(s) ds
F
i * Re(AVu,Vu) E = ~{(Avu,vu) E + (Avu,vu)E}= I ( ( A + A )VU,VU)E,
then
f [~R(X)IvuI~+
Re(Tu,u) Z for all
u E D(T).
If we let
Re
n ~ bi(x) ~ u u + Re c(x)lul i=l ~x i
u I = Re u
and
u 2 = Im u
2]
dx
then calculations
show that Re(hi ~x3i u u) = Re bi(u I --~x i Ul + u2 ~X.l u2) - Im bi(u I ~ l Since b,
2ab ~ ~a 2 + s-lb2
for any
~ > 0
u2 - u2 ~xi Ul).
and arbitrary real numbers
a
and
then
--
Re b i ~x. u u Z -
(I Re
bil + I I m
$ u[ 2 bil (~ t7f7~.
1
Sinee
2-1(IRe bil + t~m hi)
Inequality
Laplacian on
~2 ] ~.
! IbiE'
~hen ~nequality
(6) ~ow follows.
Let
g E C2(£)
be a real-valued function with a nonzero
Then
f lAg(x)[
I+(x) I2 dx<_ 2 f I$(x)I _<
Ivg(x)Ie Iv¢(X)Ig dx
4 f Lag(x) 1-1 ~2
~(x) 6 Co(•n )
q = 0
if
Ag > 0
i~g(x)12 i~,(x)i2
dx
(8)
that satisfy
(-i) n f Vg(s) • v(s) F where
~-iIu]2)/2.
(7) can be established by an identical procedure.
LEMMA 5.
for all
+
l
on
£
and
[+(s)I2 ds
q = i
if
i 0
Ag < 0
(9) on
~.
432
For Dirichlet boundary conditions for any
g(x) • C2(~)
~(s) = 0
with
Ag¢
0
on
~.
s • F.
For any
the boundary product s • F.
More generally, we need either
or
(-i) n vg(s)'~(s) for
(9) is satisfied
(~(s) ~ 0, y(s) ~ i),
F
g E C2(~)
into subsets
(-i) ~ Vg(s)'~(s) For example, if
• CI(~)
< 0
with nonTero Laplacian on F+ g
and
Fg
~
we partition
according to whether the dot
is positive or nonpositive, respectively, at g(x) = ~(IxI )
for some positive-valued function
with positive-valued first and second derivatives,
then
(-i) n vg. ~(s) = IsI -I W(isi)(sl,...,Sn).~(s). + s•F g
Hence,
(Sl,...,s n) inequality
if, and only if the angle and the unit outward normal
between the point vector
~(s)
at
s • F
satisfies the
~ > 0 > ~/2.
THEOREM 3.
Let
be a real-valued all
0
u E O(T)
T
be given by
(4) and (5).
function with a nonzero Laplacian on
u(s) = 0
that there is an
~(T)
and
> 0
for all
+ s E F g
~.
on
F+). g
Re c(x) - - i
Suppose
such that
xE
(~R(X) - g maxlbi(x) l) > Ivg(x)i2/Igg(x)l , and that
g • C2(~)
Assume that for
~(s) ~ 0
(i.e.
Let
n
Ibi(x) I
is bounded below on
~.
~,
If we define
i=l n
p(x) -= [IAg(x) I/4 + Re c(x) - - i
then
X e Oe(T) PROOF.
and 5.
implies that
~ [bi(x){]/w(x ) i=l
Re X > lim inf Re( ~ --k-~oo ~k
).
The proof follows from the corollary to Theorem 2 and Lemmas 4
Note that since
~R(X) - ~ max Ibi(x) I must have n a positive lower bound on each ~2k. If Re c(x) - c -I ~ Ibi(x) I has a i=l ~ positive lower hound on each ~k then the coercivity of T follows from (6) g @
C2(~)
then
While this may not he true, our assumptions do require that
433 n ~ ]bi(x) [ be bounded below--say by B. The theorem would then i=l p r e p l a c e d by p + 6/w and T r e p l a c e d by T + g/w f r o m w h i c h
Re c(x) - e -I follow for
the conclusion above could be d r a ~ . Obviously,
results similar to Theorem 3 related to the imaginary parts
of the coefficients for choices of
of
g(x)
T
hold as well.
We refer the reader to [11,12,13]
which lead to more specific applications
of Theorem 3.
REFERENCES i.
S. Agmon, Elliptic Boundary Value Problems, D. Van Nostrand
Princeton, 2.
Co., Inc.,
N. J., 1965.
N. Dunford and J. Schwartz,
Linear Operators, Part If, Interscience,
New York, 1963. 3.
W. D. Evans,
Or! the essential spectrum of second order degenerate
elliptic operators, J. London Math. Soc. (2), 8(1974), 463-482. 4. , Math. Ann.,
On the spectra of Schrodinger operators with a complex potential, 255(1981),
5. ,
57-76.
On the spectra of non-self-adjoint realizations of second-order
elliptic operators, Proc. Royal Soc. Edinburgh, 90A(1981), 71-105. 6.
W. D. Evans, M. K. Kwong,
differentia~ operators. 7.
and T. Zettl, On the spectra of 2nth-order
Manuscript.
I. M. Glazman, Direct Methods of Qualitative Spectral Analysis of
Singular Differential Operators, Israel Program for Scientific Translations, Jerusalem, 8.
1965.
K. Gustafson and J. Weidmann,
Analysis and Applications, 9. Verlag, i0.
25(1969),
On the essential spectrwn, J. Math.
121-127.
T. Kato, Perturbation Theory for Linear Operators, 2nd ed., SpringerBerlin, Heidelburg
and New York,
1976.
I. Knowles and D. Race, ~z the point spectra of complex Sturm-Liouville
operators, Proc. Royal Soc. Edinburgh, 85A(1979), 263-289.
434
ii.
R. T. Lewis, The spectra of some elliptic operators of second order,
Spectral Theory of Differential Operators,
(Math. Studies Series No. 55),
North-Holland Pub. Co., Amsterdam, New York, Oxford, 1981.
, Singular elliptic operators of second order with purely discrete
12.
spectra, Transactions A. M. S., in press. , The essential spectrum of quasi-accretive operators,
13.
Manuscript (submitted for publication). 14.
E. Muller-Pfeiffer, Spectral Theory of Ordinary Differential Operators,
Ellis Horwood Limited, Chichester, England, 1981. 15.
M. A. Naimark, Linear Differential Operators,
Part II, Frederick
Ungar Pub. Co., New York, 1968. 16.
D. Race, On the essential spectra of linear 2n th order differential
operators with complex coefficients, Manuscript. 17.
M. Reed and B. Simon, Methods of Mathematical Physics, Part I:
Functional Analysis, Academic Press, New York, San Francisco, London, 1972. 18.
,
Methods of Mathematical Physics, Part IV: Analysis of Operators,
Academic Press, New York, San Francisco, London, 1978. 19.
M. Schechter, Spectra of Partial Differential Operators,
North-
Holland Pub. Co., Amsterdam, New York, Oxford, 1971. 20.
, Hamiltonians for singular potentials,
Indiana University
Math. J., 22(1972), 483-503. 21.
R. E. Showalter, Hilbert Space Methods for Partial Differential
Equations, Pitman Pub. Co., New York, 1977. 22.
F. Treves, Basic Linear Partial Differential Equations,
Academic
Press, New York, San Francisco, London, 1975. 23.
F. Wolf, On th~ ecsangial spdctr1~Tz of partial dij~r.a~tia~ boundary
problems, Comm. Pure Appl. Math., 12(1959), 211-228. *The author was partly supported by NSF grant MCS-8005811.
A Singular Sixth Order Differential Equation with Orthogonal Polynomial Eigenfunctions by Lance L. L i t t i e j o h n 1 and Allan M. Krall 2
Abstract In this paper, we develop the eigenfunction
expansion
theory of a selfadjoint
operator generated by a symmetric sixth order differential
equation
This differential
x =
equation has regular singular points at
the limit-5 case at each end. needed at
±i
This means that two boundary
of polynomial
solutions
that are orthogonal
distribution
i W(x) = ~ 6 ( x + l )
i + ~ 6(x-l)
on
Not many examples
equations.
we give is interesting because the eigenvalue problem [-i,i]
L6(Y ) = ky
State College,
The example that has a sequence
with respect to the weight
+ C.
l)Department of Mathematics, Computer Science and Systems Design, Texas at San Antonio, San Antonio, Texas, 78285. 2) 845 North Thomas Street,
and is in
conditions are
to ensure a well-posed boundary value problem.
are known of such higher order singular differential
l,
L6(Y) = Xy.
PA
16801
The University
of
436
Singular
Sixth Order Differential
O rthogonal
§i
Polynomial
Equation
with
Eigenfunctions
Introduction
In [8], we found a set of polynomial
L6(Y ) = ( 2 _ l)3y(Vi)(x)
solutions
to the sixth order equation
+ 18x(x 2 _ l)2y(V)(x)
+ (x 2 - I){(3AC + 3BC + 96)x 2 - (3AC + 3BC + 36)}y(iV)(x) + x(x 2 - I)(24AC + 24BC + 168)y" (x)
(l)
+ {(12ABC 2 + 42AC + 42BC + 72)x 2 + (12BC - 12AC)x (12ABC 2 + 30AC + 30BC + 72)}y~(x) + {(24ABC 2 + 12AC + 12BC)x + (12BC - 12AC)}y'(x) = knY(X)
where X
n
= (24ABC 2 + 12AC + 12BC)n + (12ABC 2 + 42AC + 42BC + 72)n(n - i) + (24AC + 24BC + 1 6 8 ) n ( n - l ) ( n - 2 ) + 18n(n-l)(n-2)(n-
To see how we derived
(i), the reader
The polynomial
Kn(X)
where
Q(n,j)
=
3)(n-4)
solution
n Z
+ (3AC + 3BC + 9 6 ) n ( n - l ) ( n - 2 ) ( n - 3 )
+ n(n-l)(n-
is referred
of degree
n
2)(n-B)(n-4)(n-5)
to [8] or
is given by
(_i) [j/2] (2n_ j), Q(n, j )xn-J "
= i + (-i) j {(n 4 + ( 2 A C + 2 B C _ I ) n 2
[9].
-
-
2 + 4 A B C 2) + 2 j ( n 2 + n + A C + B C ) }
1 - (-1) j + .... 2 (4BC - 4AC) and
['] denotes
polynomials polynomials,
the greatest
after
integer
function.
the many contributions
satisfy
the orthogonality
These polynomials,
by H. L. Krall relation:
named
the Krall
to the theory of orthogona:
437
7_i1 Kn(X)Km<X)W<x)d~ = (2) {n 4 + (2AC + 2BC - l)n 2 + 4ABC2}{
C 6
2(2n + l)(n 2 + n + A C + BC) 2
where
w(x)
and
0 < A,
B s ~,
the classical When and called
1 : ~ 6(x+l)
C > 0.
orthogonal
that these polynomials
1
6(x-l)
Various
polynomials
A = B = 2, the Legendre
+~
C = e/2,
other properties are discussed
(x 2 - 1 ) 2 y ( i v ) ( x )
and their
were previously
known,
In this case, H. L. Krall
the fourth order differential
+ 8 x ( x 2-1)y'"
connection
with
in [8].
the polynomials
type polynomials. satisfy
+ C, -i ~ x _<- i
[6],
[7] showed
equation
(x) + (4 + 1 2 ) ( x 2 - l)y"(x)
+ 8~xy'(x)
(3) [San+ (4u+12)n(n-l)
However,
§2
when
A # B,
The Appropriate Since
be applied.
x = ±i
Kn(X)
Boundary
does not satisfy
a fourth order equation
singular
the indicial
points
of (i), Frobenius'
equation
p(p-l)2(p-2)(p-3)(p+l)
This implies
that there are five solutions
[-l,1] with respect necessary
at
x = ±i
to the weight [3].
+n(n-l)(n-2)(n-
like
3]y(x)
(3).
Value Problem
are regular
At either point,
+ 8n(n-l)(n-2)
w(x).
can
is
= 0.
to (i) that are square Therefore,
method
two boundary
integrable
on
conditions
are
438
Note that if
y
is a polynomial, then
L6(Y)(l) = 24BCy"(%) + (24ABC 2+24BC)y'(1)
and
L6(Y)(rl) = 24ACy"(-I) + (~24ABC 2-24AC)y'(-I}.
Thus, it is apparent that one boundary condition at
x = -i
is the
k-dependent
boundary condition
24ACy"(-I) + (-24ABC 2-24AC)y'(-I)
Similarly, one of the boundary conditions at
24BCy"(1) + (24ABC 2+24BC)y'(1)
= ky(-l~.
x = 1
is
= ky(l]
We now discuss the other boundary condition at By introducing the notation
x = ±i,
W.. = z(~i)y(j) - z(J)y (i), ij
we see that;
zL6(Y ) _YL6(z ) = ( 2 _ I ~ 3 W 0 6 + 1 8 x ( x 2 _ I ) 2 W 0 5 + (x2 -1){(3AC + 3BC + 96)x 2 - (3AC + 3BC + 36)}W04 + x(x 2 - I)(24AC + 24BC + 168)W03 + {(42AC + 42BC + 12ABC 2 + 72)x 2 + (12BC - 12AC)x (30AC + 30BC + 12ABC 2 + 72)}W02 + {(12AC+I2BC+24ABC2)x
Denote the right side of this equation by yields the identity:
+ (12BC-12AC)}W01
[y,z]l.
Also, Lagrange's identity [i
439
zL6(Y)~ - YL6(z)
dly, z] 2 dx
where
[Y,Z] 2 = {(12ABC 2 + 6 A C # 6 B C ) x 2 + (12BC-12AC)x + (-12ABC 2-18AC-18BC-24)}W01
+ {(12AC+I2BC+24~x 3
+ (-12AC-12BC- 48)X}Wo2 + {(3AC+3BC+36)x 4 + (-6AC-6BC- 60)x 2 + (3AC+3BC+24)}W03 + + 12x(x 2-I)2W04 + (x2-I)3W05 + {(-3AC-3BC-6)x 4 + (6AC+6BC+24)x 2 + (-3AC-3BC-18~}WI2
- 6x(x 2-I)2W13
_ (x2_ I)3W14 + ( 2_ I)3W23.
Hence
Cfll_ (zL6(Y) - YL6(z))dx = C lim ly,zJ 2 - C lim [y,zj 2 x~l x-+-i
if these limits exist.
Similarly,
1 1 f_lI (zL6(Y) -YL6(z))[ ~ 6(~{+i) + ~ 6(x-l~]dx
= ~1
lim [y, z] I + ~1 Jim [y,zJI x-~-i x~-i
if these limits exist.
Combining these latter two equations yields
f_lI (zL6(Y) - YL6(z))w(x)dx = = lim {y,z} 1 + lim {y,z}2 x-~l x-+-1
440
where
1 {y,z}l = C[y,z]2 + ~[y,z] 1 and {y,z}2 = -C[y,z]2
It appears
then that the other boundary
and the other boundary
These lim x-~-i
conditions
then,
a)
L6(Y)
b)
i) ii)
i) ii)
(zL6(Y)
write
is
necessary.
-i
is
lim x~-i
lim {Y,K0} 1 = 0, x~l We believe
- YL6(z))W(x)dx
the appropriate
-i<
= ky
24ACy"(-I) lim x -~-1
{y,N0} 2 = 0
where
K 0 = K0(x).
that the conditions are sufficient
to
x<
boundary
= 0.
value problem
is:
1
+ (-24ABC 2 - 2 4 A C ) y ' ( - I )
= ky(-l)
{Y,K0} 2 = 0
24BCy"(1)
+ (24ABC 2 + 2 4 B C ) y ' ( 1 )
= ky(1)
lim {Y,K0} 1 = 0 x-+l
Alternatively,
L2[-I,I] w
1
at
that
In summary,
operator
at
condition
{z,K0} 2 = lim {Y,K0} 1 = lim {z,K0} I = 0 x~l x-~l
/~i
c)
conditions
are certainly
{Y,K0} 2 = lim x-~l
guarantee
1 + ~[y,z] 1
we can describe
has a domain not depending indicates
on
the real Hilbert
Y = (y(x),Y_l,Yl) T.
Define
the boundary \
[5].
value problem
Let
space generated
the inner product
so that the
H = L~[-I,I] ~ ~ ~ by
w(x).
of
Y = ( Y ( x) ' Y - I ,Yl )T
If
Y ( H,
where we and
441
as:
Z = (z(x),z_l,Zl)T
i + i (Y,Z) = C/If_ y(x)z(x)dx + ~ z_lY_l ~ zly I
Clearly, with this inner product, A
as follows:
let
DA
H
is a Hilbert space.
consist of all elements
We define the operator
Y = (y(x),Y_l,Yl)T
satisfying
the conditions:
i.
y(x) ( L2[-I,I] w
2. y,,y,,,y,,,,y(iV),y(V)
For
y(V)
is absolutely continuous on (-i,i)
exists a.e. and is in L2[-I,I] w
3.
L6(Y)
4.
y(-l) = Y-I
5.
y(1)
6.
lim {Y,K0} 1 = 0 x-~l
7.
lim {Y,Ko} 2 = 0 x-~-1
Y EDA,
exist and
= Yl
define the operator
L6(Y) AY = (L6(Y)(-I)
A
by:
~2
L6(Y) 4ACy " (-i) + (-24ABC 2 - 24AC)y'(-I) I
)
!
(iV
k
24BCy" (i) + (24ABC 2+24BC)y'(1) /
442
Theorem 1
The operator
A
is self adjoint in
H.
The proof is by standard techniques. We now turn our attention to the eigenfunction self adjoint operator
A.
Becuase of the weights assigned to ±i t
of the Krall polynomials, immediately
obvious.
expansion theory of the
represented by
Let
H
(Kn(X) ,Kn(-I),Kn(1))T, _ -
the completeness in
H
is not
be the subspace spanned by the set 0
{(Kn(X)'Kn(-l)'Kn(1))T}~=0n
Lemma 1 Then
Let
We show that
F = (f(x),f_l,f I)
(F,X n) = 0,
Proof.
"
is dense in
be orthogonal
_to _
H 0.
H.
Let
Xn
=
(~ n ,
(rl)n,l).
n = 0,i ....
This follows by taking appropriate
0 = (F,Kj) = Cfll- f(x)Kj ( x ) d x + l
Lemma 2
H0
The measure
~
generated
by
linear combinations
f-IKj (-I) + B
~(E)
=
f
flKj (I)
f(x)w(x)dx
of
j = 0,i .... n.
is of bounded
E variation.
Proof.
By Schwarz'
J~l(1,1)
Lemma 3
inequality,
the total variation
fll]f(x)[w(x)dx_ &
The Fourier transform of
[@[
satisfies
(fll[f(x)[2w(x)dx)i/2(fl I _. _
%,
~(k),
is
w(x)dx) I/2 < =.
0.
Proof ~(k) =
i Ci
1
e-i~x
f(x)w(x)dx =
1 ~2T--~
1 i f lim g2-~ 1 N -~°
f
_i
~
Z i n= 0
(-i~x)n
n-----q--f(x)w(x)dx
N (-. ikx) n 1 i N (_ikx) n Z n! f(x)w(x)dx = lim f Z n!-- f(x)w(x)dx n= 0 g2~ N -~° 1 n= 0
443
1
lim N -~
I/2~
N
(-ik) n
Z n=O
n!
f!l xnf(x)w(x)dx = 0,
from Lebesgue's dominated convergence theorem and Lemma 2.
Theorem 2
Proof.
{Kn(X),Kn(-l),Kn(1)}n= 0
Recall that
H0
in (-i,i)
and
Theorem 3
If
forms a complete set of vectors in
is the span of this set of vectors.
F = (f(x),f_l,fl)T E H the Fourier transform
and ~,
F
is orthogonal to
we see that
f-i = fl = O.
~ = 0.
F =(f~x),f_l,fl )T E H,
and by using (2).
then
=
F =
f(x) = 0 a.e.
cn
Z Cn(Kn(X),Kn(-l),Kn(1))T n=0 are given by
If
and
K (-i) = ( - I ) 2 A C ( n 2 + n + 2 B C ) n 2 n +n+AC+BC
•
f E L~(-I,I)
C/II f (X)Kn(X) dx c n
Hence
By inverting
We note that the formula may be further simplified by using
then '
where
(see [4])
F = 0.
(/i I K2n(X)W (x) dx) 2
= BC(n 2 + n + 2 A C ) 2 n +n+AC+BC
Corollary 1
then
We show that if
[cfll f(x)Kn(X)dx + ~1 f_iKn (-I) + ~1 flKn(1)]
The proof is standard. vrl~
H O,
H.
We now have our expansion theorem:
where the generalized Fourier coefficients
c = n
•
K<xw<x)dx
f(x) =
Z CnK (x) n n=0
'
444
References [i]
R. H. Cole, Theory of Ordinary Differential Equations, Appleton-CenturyCrofts, New York, 1968.
[2]
W. N. Everitt, "Fourth Order Singular Differential Equations", Math. Ann., 49(1963), 320-340.
[3]
and V. Krishna Kumar, "On the Titchmarsh-Weyl Theory of Ordinary Symmetric Expressions I: The General Theory", N. Arch. ¥. Wisk. 24(1976), 109-145.
[4]
W. Feller, An Introduction to Probability Theory and Its Applications, Vol. II, John Wiley and Sons Inc., New York, 1966.
[5]
C. Fulton, "Two Point Boundary value Problems with the eigenvalue parameter contained in the Boundary Conditions", Proc. Roy. Soc. Edin., 77(1977), 293-308.
[6]
A. M. Krall, "Orthogonal Polynomials Satisfying Fourth Order Differential Equations", Proc. Roy. Soc. Edin., 87(A), 271-288.
[7]
H. L. Krall, "Certain Differential Equations for Tchebycheff Polynomials", Duke Math. Jour. 4(1938), 705-718.
[8]
L. L. Littlejohn, "The Krall Polynomials: A New Class of Orthogonal Polynomials," Quaestiones Mathematicae, to appear.
[9]
and S. D. Shore, "Nonclassical Orthogonal Polynomials as Solutions to Second Order Differential Equations", Can. Bull. (to appear).
i0]
S. D. Shore, "On the Second Order Differential Equation which has Orthogona Polynomial Solutions", Bull. Calcutta Math. Soc. 56(1964), 195-198.
On the a s y m p t o t i c s with
locally
finite
of
some
measures
and
Stig-Olof
I.
Volterra large
equations perturbations
Londen
Introduction
In this nonlinear
(1.1)
lecture
Volterra
x'(t)
+
we d i s c u s s
the
differential
f
asymptotics
equation
g(x(t-s))d~(s)
of the
scalar,
of c o n v o l u t i o n
= f(t),
real
type
t E R + = [0,~),
x(0)
= x 0.
[O,t] Here
g
and
measure
on
on the
case
some
are
R+
is o n l y for
f
and
when
a transformation finite
measure
to this
real
x(t)
is the
both
locally p < ~,
given
finite
does of
and
transformed
~
not
(1.1) on
functions,
~
unknown
solution.
and
f
are
large;
and
f
vanishes
necessarily
showing
version
that
of
the
(1.1)
Our
of the
limit
has
that
We
real
but
analysis same
type
equation
sufficiently
Borel
concentrate
is on the
at i n f i n i t y
hold.
to an e q u a t i o n
is a g i v e n
case
when
f E LP(R+), is b a s e d
on
but w i t h
a
corresponding small
(£ L2(R))
solutions. Throughout
the
lecture
the
following
basic
assumptions
will
be
made:
(i)
g E C(R),
(ii)
~
(1.2)
on
Observe cally
is a real,
locally
f E LI (R +) loc
(iv)
x 6 (L ~ n L A C ) ( R + ) ,
we do a s s u m e
absolutely
positive
definite
measure
a.e.
R +.
R+ ,
(iii)
that
finite,
the
continuous
x
satisfies
existence
solution
of
(1.1)
of a u n i f o r m l y (1.1).
on
bounded~
lo-
448
u,v
For
functions,
the convolutions
a measure,
(u,v)(t),
all supported
(u*~)(t)
t : f u(t-T)V(T)dT, 0
(u*v)(t) Also
v
on
R +, we define
by u(t-T)d~(T),
(u*~)(t):
t C R +.
[O,t]
let ~(z)
: f+ e-Ztu(t)dt, R
~(z)
: ~+ e-Ztd~(t),
^
and write
u(~)
: ~(i~),
The classical is to define satisfying
%(~)
: ~(i~),
way to analyze
the quadratic
(1.2ii),
form
obtain
(1.3)
< ~.
sup Q(g(x(t)),u,T) T>0
It is well-known
to take
f
Q(~,u,T)
for
of solutions
of (1.1)
~ £ L~oc(R+),
T > 0, by
and then to try to somehow
g(x(t))
the asymptotics
T = f ~(t)(~*u)(t)dt 0
Q(~,B,T)
of
~ £ R.
[6, 7] that a detailed
can be obtained very small,
many interesting
i.e.
applications
from
knowledge
(1.3).
f C LI(R+), where
f
of the asymptotics
But to get which
(1.3) one needs
of course
(in addition
excludes
to (1.2iii))
only satisfies (1.4)
However, least)
lim f(t)
= 0.
by taking
~
~+ dJ~l (t) < ~
small enough,
and by working
corresponding
to (1.1),
(1.5)
+ ~+ g(y(t-s))d~(s)
y'(t)
one may obtain
in particular
asymptotic
results
= 0,
by assuming
(at
with the limit equation
t E R,
on bounded
solutions
of (1.1)
even
447
if
f
satisfies
But (with
the f
only
question
[2,
as to w h a t
satisfying
An obvious
(1.4),
only
difficulty
one
8].
happens
(1.4))
that
3,
when
remains
now
~
and
is o n l y
is m u c h
encounters
locally
harder
is that
finite
to answer.
(1.5)
is
meaningless. A first which
partial
essentially
somewhat
answer
is due
differently,
to this
question
to G r i p e n b e r g
in p a r t i c u l a r
is the
[1],
following
although
note that
~
Theorem
he f o r m u l a t e s
is t a k e n
finite
it in
[1].
Theorem
(1.6)
I.
r'(t)
Let
(1.2)
+ (r~)(t)
hold
= 0
and a s s u m e
a.e.
on
the
solution
R +, r(O)
r
of
= I,
satisfies
(1.7)
r' E
(L I N N B V ) ( R + ) .
Define
v
-r'(t),
t > 0, and
(1.8)
f+ t d l v I ( t ) R
to be the m e a s u r e
corresponding
to
-r',
thus
v([0,t])
=
let
< ~.
Suppose ^
(1.9)
I~(~)
and
the
let
most z
set
denumerable
be the
(1.10)
Then
< ~,
if
locally
z'(t)
lim t÷~
Z
w ~ 0,
defined
and
such
z(t)
that
absolutely
+ (z~)(t)
= z(~)
by
Z = {w I w ~ 0, Re ~(w) Im ~(w)
= 0, for
continuous
solution
= f(t),
exists
t E R +,
(and
z(0)
= 0}
w E Z. of
= x 0.
is finite)
one has
be at
Finally
let
I
448
lim
[x(t+d)
-
x(t)]
=
O,
Vd
O,
>
t+~
lim
[r(~)x(t)
+ (1-r(~))g(x(t))]
= z(~).
t+~
If in a d d i t i o n lim ess sup t ÷~ s>t
Thus, tends
z' E L~(R+),
Ix'(s)l
if the
solution The
of the
Iz'(s)l
then
: 0
= 0.
solution
to a limit
lim ess sup t+~ s>t
then
z : x0r
so does,
nonlinear
transform
under
equation
p(~),
+ r*f
of the
certain
(1.1)
equation
hypotheses,
having
~ ~ 0, is d e f i n e d
linear
as
the
a bounded
same
lim
(1.10)
data.
[(s)
where
s÷iw
Re ~(s)
= f+ e - S t d p ( t ) . One R this is w e l l d e f i n e d ,
that
assumption
(1.9)
Theorem Also
I has
observe
assumptions
that
the
to check.
hard
to find
classes
this
assumption.
to be f i n i t e
= a(t)dt
One
with
a(t)
is c o n s e q u e n t l y
(1.8),
(1.9).
result
was
Theorem
This
was
finite
note
= t-1/2cos
recently
~
apart
that
excluding
motivated
a short
d~(t)
but
Let
(1.2)
be
satisfied
lucid
on
g.
(1.8)
from
Theorem
interesting
proof. The
on the
second
in g e n e r a l
= a(t)dt this
measures
in
[4] w h e r e
example
which
cases
it is
do s a t i s f y
1 requires
~(~)
like
and
let
the h y p o t h e s e s the
is
with
t.
obtained.
2.
and
which
to try to r e m o v e done
The
possibility.
of
if
follows
~ ~ 0.
condition
resolvent
also
for
(1.7)
is i m p o s e d
a moment
locally
~ ~ 0, thus
this
of h a v i n g
k = 0,1,2,3; of only
(1.6),
infinite,
It is s a t i s f i e d
for
from
continuity
include
One m i g h t
for
that
exclude
but
of the d i f f e r e n t i a l
> 0
d~(t)
possibly
advantage
nothing
(-1)ka(k)(t)
show
however
do h o w e v e r
derivative difficult
does
can
s>0
following
449
(1.11)
Let
g(x)
be l o c a l l y
r 6 L A C ( R +)
Lipschitzian,
satisfy
(1.6),
x 6 R.
(1.7)
and
l i m r(t)
suppose
:
t÷~
lim
(r.f)(t)
= 0.
Also
let
t+~
(1.12)
Im ~ ( ~ )
:
0
for
(1.13)
xg(x)
(1 .14)
lim inf x-lg(x)
~ E {~1
~ ~ 0,
Re ~ ( ~ )
= 0},
> O, x ~ O,
> 0.
Ixl+0 Then
l i m x(t) t÷~
lira ess sup t+~ S>/
Note Certain g(x).
(1.14) limit
that
and
p
overly
the
case
[5].
and a s s u m e
that
l×-~el ÷° Let
instead
If(s)l
then
= 0
restrictive.
measure
that
Let
easily The
then
added,
in p a r t i c u l a r
(1.13), checked
relation
even
like
p
on
does
on g
at l e a s t
(1.13)
(1.11),
however
2 c a n be a p p l i e d . (1.13)
definiteness if
(1.14) and
Theorem
nothing
the p o s i t i v e
of
is n e e d e d p
and
is n o n f i n i t e .
(1.7)
That
this
by
(1.2)
g(x)
~
assumptions.
been
to w h i c h
(1.13)
f r o m the
(1.11),
of b e i n g
is s h o w n
0 < l i m inf
absent
assumptions
us to a b o l i s h
3,
(1.15)
have
of n o n l i n e a r i t i e s
suggested
Theorem
are n o w
advantage
is a f i n i t e
allow
is i n d e e d
0}
the
class
it w a s
should
additional
are not
In case
(1.9)
conditions
have
l i m ess sup t ÷~ s>t
: O.
(1.8),
other
the
If in a d d i t i o n
Ix'(s)l
These
clearly
= O.
be s a t i s f i e d ,
< lim
sup
-- l~-×el ~°
g<x) ×-~e
define
< ~
X = {x C R I g(x) =
for
x
E X.
e
4bU
^
(1.16)
and
inf Re ~(~) ~ES
suppose
(1.17)
Then
r,f
lim
for
satisfy
r(t)
lim dist
> 0
= lim
compact
(1.6),
(r,f)
(x(t),X)
any
(1.7)
Sc
set
and
= 0.
: O.
If in a d d i t i o n
lim f(t)
t÷~
we
large that the
condition
need
for
x
Outline
near
It is a p p a r e n t
X. (1.16)
Ig(x)I
m a y be
How much
of p r o o f
x E R; a(t)
(1.1)
has
obviously
is n e i t h e r from
is still
z(t)
sequence
t
by
= x0r(t) and
+ ~
n
(L ~ N LAC)(R)
small
an o p e n
weakened question;
r
and use
t C R +.
This
nor
too
of T h e o r e m without current
3
altering work
there such
by
h £ C(R) exists that
Also
where
y(t)
y
h(x)
is
= g(x) - x ,
t E R +,
(1.17)
satisfies
we t h e r e f o r e
a subsequence x(t+t
) ÷ y(t)
satisfies
+ f+ h ( y ( t - s ) ) a ( s ) d s
define
gives
= z(t),
+ (r*f)(t) as
(1.6).
nK
(2.1)
too
but
3
t + f h(x(t-s))a(s)ds 0
a C L I ( R +)
sets
disappeared
the p r o o f
significantly
of T h e o r e m
= -r'(t),
x(t)
y E
g(x)
to this.
Convolve
As
on
that
conclusions.
where
(1.13)
to a s s u m e
the r e l a t i o n
devoted
2.
= O.
sign
still
= 0, t h e n
t÷~
l i m x'(t)
The
R,
= 0,
t E R.
have t
nK
lim z(t) t+~ that g i v e n
= 0. any
and
uniformly
on c o m p a c t
451
Define
Y = {Yl y E (LAC N L~)(R), x ( t + t n) ÷ y(t)
Note that
if
y E Y
there
uniformly
then
y
on c o m p a c t
not o n l y
+ R~+ h ( y ( t - s ) ) d a ( s )
tn + ~
exist
sets}.
satisfies
a.e.
(2.1)
y'(t)
where
e([0,t])
(2.3)
Re ~(~)
= ~2Re
p(~)li~
(2.4)
Im ~(~)
= ~2Im
~(~)I i~ + ~(~)1-2 + ~I~(~)I 2 li~ + G(~)1-2
Straightforward
on
but also
(2.2)
= a(t).
= 0,
such that
R,
computations
give
+ ~(~)I -2, ~ ~ 0,
~ ~ 0
(2.5)
a(o)
Thus,
by
: o.
(1.2ii)
Note however ~ 0, does
and
(1.7)
~
is a p o s i t i v e
that a l t h o u g h
(1.16)
not n e c e s s a r i l y
hold.
Ip(~0) I = ~
then
Re ~(~0 ) = 0.
~O-Values
have
Im ~(~0)
makes
we
(2.2)
We b e g i n by s h o w i n g
Lemma
I.
Indeed,
then
if
Furthermore This
finite m e a s u r e .
Re ~(w)
w0 ~ 0
is such that
observe
fact,
> 0,
that
for such
together
with
(2.5),
to handle.
that
T h e r e does
is a s s u m e d
= ~0 ~ 0.
quite difficult
definite
Y
does
not e x i s t
contain
y E Y
constants.
such that
for some p o s i t i v e
6,T
(2.6)
t f t-T
Iy'(s)ids
P r o o f of L e m m a that
(2.6)
and o b s e r v e £ Z
1.
> 6, t c R.
Suppose
is s a t i s f i e d that
U {0}.
as
From
the L e m m a
for some
is false
6,T.
Define
[ = i~a, ~ ~ 0; a(0) (1.7)
and
(2.4)
and take
such
Z = { w ~ 0 I I~(~)I =~}
= I, then
f o l lows
y C Y
~(~)
that the set
= 1 Z
for is
4bZ
bounded. Choose 11-~(~)I
!
~ > 0 s'
arbitrarily,
Iwl ! ~0'
take
let s
~O
> 0
be
(depending
such
on
~)
that sufficiently
smal
and d e f i n e
(2.7)
V1 = {~1 d i s t ( 5 , Z ® )
(2.8)
V
j s, I1-£(~)1 ! 8, ~0 ! I~1 ! - 1 }
= V I U [-~0,~0].
Write
vt( "r)
Then
one
can
y(T),
0 < T < t
0
. . ; zt(T) . T < 0; T > t
: ,
s h o w by e s t i m a t e s
ciently
large
(2.9)
R~vf 10t (~)]2d~
where
the
constant
of
t. Define
part
of
F
such
sequence to
y
translate
w(t)
ac I ~
that
if
0
0 < • < t
, ~ < 0;
T
is f i x e d
F(t)
that
w
= y(t)
8
y
but on
can
show
lim sup n ÷~ T61
not
T > t
suffi-
t,T,s
and
T
t 6 R +.
that
IF(T)I
n converges
t ~ T,
on
c, ~
- (a*y)(t),
one
m(In) ÷ ~' of
on
depending
there = 0.
(uniformly
and w h e r e
is i n d e p e n d e n t
Using
the
second
exist
intervalS
Consequently on c o m p a c t
some sets)
of
- ~+ w ( t - s ) a ( s ) d s
y)
l~t(~)l 2d~ + c 2,
depends
(2.7)-(2.9)
(and of
!
although
by
a solution
as
cI
of t r a n s l a t e s
w(t)
But
c2
(1.15),
In c R+
[4]
t.
then
constant
the
of
[ h(y(x)), :
therefore
= @, t E R.
by the
is a s o l u t i o n
of
+ f+ h ( w ( t - s ) ) a ( s ) d s
translation (2.1)
we
invariance
also
: 0, t E R.
have
of
(2.1)
any
453
Consequently o(w + h(w)) w(t)
the s p e c t r u m m {~I
+ h(w(t))
a(~)
~ 0
~
of
= 0}
w(t)
which
and so
+ h(w(t))
is empty.
g(w(t))
~ 0.
satisfies
One c o n c l u d e s
But this
clearly
that violates
(2.6). The next Theorem
lemma
is r a t h e r
technical
2.
Define
Y0 : {y C YI h ( y ( t ) ) (may d e p e n d
y)
on
£ L2(R)~
sup -~<s
lh(y(v))12dv
_
n
i_!f n
sufficiently t
n
g2(y(T))dT
some e s t i m a t e s
which
Once
< 8
long) of
let
Y £ Y0"
larAe,
n
f -t
f i r s t part
3.
Y
[4]
h2(y(T))dT.
of L e m m a
in p a r t i c u l a r part of
it is not d i f f i c u l t
size of the s o l u t i o n s
at i n f i n i t y .
(1.15)
2 m a kes
and of the s e c o n d
2 is p r o v e d
on the
well
of
proof
[3],
Lemma
result
behave
L emma
and
n
(fairly
following
lh(y(v))12dv},
n
s > 0
n
(1.16).
÷ ~,
-t
T h e n for a n ~
f -t
c > O
n
fn
< c
exist
t
t
f s-2t
t
there
such that for some
S
The
to the p r o o f of
3.
Lemma
--
but e s s e n t i a l
The p r o o f
is e s s e n t i a l l y
(1.15)
to o b t a i n
of the limit
of L e m m a
use of and of the
equation
3 is w h e r e
the
needed.
Define
= {y C Y 1 Y l
def = lim y(t), t÷-~
Y2
def = lim y(t) t÷~
both exist}.
Then g(y(t))
Suppose
(£ L2(R),
for a m 6 m e n t
(y(t)
yl ) E L2(R-),
that we have:
y(t) - Y 2
C L2(R+).
454
Lemma 4. defi =
G(x)
Let
y E y~
g(u)du.
with
g(y(t))
~ 0
and let
Then
0
(2.10)
G(y 2) < G(Yl).
The proof of T h e o r e m lim sup dist(x(t),X) t÷ ~ functions,
3 may then be completed
~ 0.
Let
(2.11)
G(y c) = inf G(y). YEY c
Yc
Yc E Y c
(1.15)
d > 0.
the set of constant
limit
y E Y).
By Lemma
I
Yc
is not
be such that
of translates
uniformly
denote
Yc = {y ERI
empty.
to
Y
Suppose
c
i.e. define
Take a sequence
Let
as follows.
on compact
Take
Xn(t)
: x(t+t n)
sets.
6 £ (0,d)
Define
of
x(t)
converging
d = dist(Yc,Yc~Yc ).
sufficiently
By
small so that
def Tn
inf{TI
Let
~n(t)
T ~ 0, IXn(~)
6}
= Xn(t+Tn) , thus for some
Lemma 1 there exists that
ycl
a subsequence
is well-defined v > 0
{n k}
for all
Ign(~n(0)) I ~ v.
of
{n}
and
y E Y
n. By such
~
Y(-~)
(t) ÷ y(t) uniformly on compact sets, g(y(t)) ~ 0 and nk = Yc" But then from (2.10) G(Yc) > G(y(~)) which violates
(2.11)
and completes
We are thus
the proof.
left with proving
(2.10).
We sketch the proof of
this in what follows. Let
rl(t),
sponding (2.12)
to r~(t)
~ > 0, t E R +, be the differential
~,
thus
+ l(rl~)(t)
We have the following
Lemma
5. [4].
resolvent
Let
: 0
a . e. on
result concerning
~
satisfy
R +
, rl(0)
= I.
the size of
(1.2ii)
and (1.16)
rl.
and let
corre-
455
r~ E LI(R+).
Then
r~ £ LI(R +)
r I' [ NBV(R +)
then
rl E NBV(R +)
Convolve
(1.1) by
x(t) +
where easily
z
for
~ > 0.
for
as
t ÷ ~
This gives
- x(t-s)]a~(s)ds
: x0r ~ + rl~f; ai : -rj. z~(t) ÷ 0
~ > 0.
rx, use (2.12).
f [g(x(t-s)) [0,t] ~
If in addition
for
= zx(t)
From Lemma 5 and (1.17) follows ~
fixed.
Thus
y E y
also
satisfies
+ [g(y(t-s))
(2.13)
y(t) +
Define
el([0,t])
: a~(t), differentiate
y(t) + YI' t < 0;
fl(t) : l-lg(y(t))
y(t) + Y2' t ~ 0
where by
y'(t)
h~(t)
el(R + ) = 0.
+ f+ f l ( t - s ) d ~ ( s ) d s R = 0, t < 0; hi(t)
fl, integrate over
and apply Parseval's
G(Y2)-
R
This gives : hl(t), t E R,
: (y2-Yl)al(t),
t ~ 0.
(can be done as by Lemma 3
relation.
G(Yl)+
: 0, t E R.
(2.13), define
fl(t) = ~-lg(y(t))
and use the fact that (2.14)
y(t-s)]al(s)ds
Multiply
(2.14)
fl C L2(R))
This gives
2 ~ (Y2
2 Yl 7
Y(0)Y2 + y( 0)Y.i )
(2.15) + I R[ Ifl(w)i2Re ~l(w)dw
Note that arbitrarily affect
h A 6 LI(R +)
arbitrarily
and that for fixed
small by translating
I~x(~)I ). small.
= X ~R+ fl(T)hl(T)dT.
y.
1
we can make
sup If~(r) TER + (This translation does not
Thus the right side of (2.15) can be made We therefore have, without
loss of generality,
456 lim
[G(y 2) - G(y 1) + X ~ I%A(co) I 2Re ~A(o~)d~]
X+0
R
and so, if
(2.16)
(2.10)
not hold,
lima
I Ifx(~o)t2Re ~ ( ~ ) ~
k+0
R
Substitute that
does
(g(~)
the e x p r e s s i o n
def=
= 0,
< 0.
for
fl
into
(2.16).
We w i s h to show
)(~))
I 1~(~)I2x-IRe ~X(~)d~ R is the d o m i n a n t
part
is b o u n d e d
f r o m zero.
away
A-IRe ~i(~)
and so, if Re ~(~). and
~ ~ 0
in the r e s u l t i n g
that the c o n v e r g e
set
K m R
is u n i f o r m
= ~})
that
sufficiently
A
= 0.)
^
right
on
As a l s o
lim
(2.17)
for any
E m K, m(E)
s > 0
< s, such
(1.7)
X-1Re ~ A ( ~ ) £ 0, V~, VX, we have
small I
^
Y12(t)
(def p)
t hat
is s t r i c t l y
K,E
are such that the
positive.
that
X f 1~12(~)12 Re ~X(~) =
A+0
Finally
that
K~E
side of
relation
have
lira X - 1 R e ~I (~) : I+0
(Note that by
Re ~ > 0, ~ E R, we m a y a s s u m e
(2,18)
then
e x ists
K~E.
^
N e x t one shows
Here
IF (m) I < ~,
there
R
But as
term
~{1-2
t h e o r e m we t h e r e f o r e
m({~ I I ~ ( ~ ) 1 for
+
and such t h a t
for any c o m p a c t
and that this
We have
= ~2(Re ~)liw
By E g o r o f f s
expression
0.
R : y(t)
(2.18)
- yl,
follows
one o b s e r v e
that
t < 0; Y12(t)
after
= y(t)
- Y2'
some c a l c u l a t i o n s ,
(2.17),
(2.18),
p > 0
t ~
from violate
0.
(1.7)
The and
(2.16).
(1.16). Thus
457
Lemma 4 and hence T h e o r e m 3 are proved.
References
I.
G. Gripenberg, kernels,
2.
On n o n l i n e a r V o l t e r r a equations with n o n i n t e g r a b l e
SlAM J. Math. Anal.,
S-0. Londen,
668-682.
On a Volterra i n t e g r o d i f f e r e n t i a l
p e r t u r b a t i o n and n o n c o u n t a b l e J. Integral Eqs.,
3.
11 (1980),
S-O. Londen,
I (1979),
e q u a t i o n with L ~-
zero-set of the t r a n s f o r m e d kernel,
275-280.
On an integral e q u a t i o n with L~-perturbation.
To
appear, J. Integral Eqs.
4.
S-0. Londen,
On some integral equations with locally finite
measures and L~-perturbations.
5.
S-O. Londen, M a n u s c r i p t
6.
O. Staffans,
0. Staffans,
8.
(1976),
in preparation.
Trans. Amer. Math.
T a u b e r i a n theorems
with applications 218
SIAM J. Math. Anal.
Positive definite measures with applications
Volterra equation, 7.
To appear,
Soc.,
218 (1976),
to a
219-237.
for a positive d e f i n i t e form,
to a Volterra equation,
Trans. Amer.
Math.
Soc.,
239-259.
O. Staffans, perturbation,
On a n o n l i n e a r integral e q u a t i o n with a n o n i n t e g r a b l e J. Integral Eqs.,
I (1979),
291-307.
PERIODIC OSCILLATIONS OF PORCEO HhNOULUM-LIRL L~UAIIUI~b +++++++++++++++++++++++++++++++++++++++++++++++++++++++
Jean MAWHIN Introduction. The influence of the problem of the pendulum on the developement of mathematics,
mechanics and technology has been enormous.
Most
part of the history deals with the study of the free oscillations of the pendulum with Galil6e,
Merci, Torricelli,
Oescartes, Huyghens,
Newton, Eule~ Poisson, Bessel, Jacobi andothers and its link with the theory of elliptic functions is well Known. pendulum equation,
The case o% the forced
and in particular the existence of its periodic
solutions has started much later, with Ouffing [9] in 1918 who started with the equation [0,1] andwas
x"+ a sin led
to
the
so-called
x = e[t]
Dulling
[a>O]
equation
x " + a x + cx 3 = e[t] by retaining the two first terms in the Taylor series of sin x. This analysis was therefore only valid for small.
Hemal [11],
[I] when
Ix] is sufficiently
in 1922 wrote a very interesting paper on C1)
with f[t] = b sin t, where
he
first proves,
by a variational method
in the line of the work of Hilbert and Caratheodory,
that the
equation [0.2]
x"+a
sin x = b sin t
has a 2z-periodic solution for every a > O
and b 6 R .
works for 8 sin t replaced by a cl-function with mean
This approach value
zero.
After some remarks on the Ritz method for approximating the solutions, he
considers the
existence of odd 2z-periodic solutions of [2) by
reducing it to the nonlinear
integral
equation
x[t) = -aS ~ K[t,~] sin x[~] d T - b sin t
459
where K(t,~] solution
= t[ Z-
of
the
[0.3]
x[-t] (up t o
first x"+
followed
1]
a sin
translations
of
the
theory
[2],
application
rather
general
to
if
x(0]
t > T,
i.e.
= x(~]
to
the
= 0
on [ - ~ , 0 ]
by t h e
existence
of
when a < l ,
that
on % i x e d p o i n t
ordinary differential
the
1)
problem
t,
solution
proves 2~]
Notice
value
x = b sin
the
He t h e n
approximations.
t < T and T[ ~ -
boundary
by e x t e n d i n g
= -x[t].
if
by t h e
famous
published boundary
successive
and K e l l o g g ' s
contains th problems for n
applied t o
the existence of a solution for every a.
of
solution
same y e a r ,
value
equations w h i c h ,
method
8irkhoff
the
relation
a unique
paper an order
[0.3] would imply
Hammerstein [12] obtained
in 1930 the existence of a solution for x"+
(0.4] [0.5]
a sin x = e(t]~
x[0]
= x[~]
= 0,
{or arbitrary a and arbitrary continuous e as an application of his variational method for nonlinear intezral equations and the uniqueness when a < 1 .
Another proof of those results was given by Iglisch [ 13]
the same year by a method of continuation.
In 1944, Lettenmeyer [16]
proves the existence of a solution for [0.1] with the boundary conditions
(0.5]
or
(o,6]
x[0]
by a s h o o t i n g fying
the
method.
boundary
solution.
He t h e n
conditions
= x'[~]
= 0
shows t h a t (0.8]
solutions
can be e x t e n d e d
Those results are rediscovered
of
[0.4]
to
a 2~-periodic
satis-
by Marlin [17] as special
eases of existence results %or symmetric periodic solutions for systems o% the form x"
= fit,x]
where % is suf%ioiently smooth, periodic in t and x and has some symmetries.
Those results are extended by topological degree
arguments in [19] to systems of the form
460
x " + grad FIx'] + f(t,x,x'] [see also [ 2 3 ] ] .
= 0
When specialized to the scalar case, one obtains
for the p r o b l e m x " + g[x'] + a sin x
e[t], x[O] - x(2w]
= x'[O] - x'[2~]
the existence of a solution satisfying the relation x[t+w)
= O,
= -x[t]
when g is an arbitrary odd continuous function and when e[t+w] = -e[t], generalizing
Lettenmeyer's
result to some d i s s i p a t i v e casss.
Let
us m e n t i o n also some results, more in the direction of numerical analysis,
by Borges,
Cesari and Sanohez [ 4 ] ,
Cesari and Bowman [ 8 ] ,
Bononcini [3] and B. Schmitt and 8rzezinsK± [ 2 4 ] , which all deal with v~rious types of symmetric periodic solution for
[I] or special
cases of it.
All the above results deal with the first and second boundary value problems or with some symmetric periodic solutions if the forcing term e[t] and the d i s s i p a t i v e term g[x'] properties.
In the abstract setting,
have some symmetry
they correspond to nonlinear
perturbatidns of an invertible linear operator whose solutions are a priori bounded so that existence follows easily.
The general case of
periodic boundary conditions without symmetry is of a different nature because a necessary condition for the solvability of [0.1] with periodic boundary conditions is that
dtl
a.
a
as it
follows
easily
range of the operator
by i n t e g r a t i n g ( 0 . 1 ] o v e r one p e r i o d . So t h e d2 d - - ~ + a s i n ( , ] on t h e s p a c e o f 2 ~ - p e r i o d i e
functions is no more the w h o l e space.
The same conclusion holds
for the homogeneous Neumann boundary conditions.
M a y b e the first
paper dealinz with the general situation is that of Knobloch [14] i n t r o d u c i n g the use of upper and lower solutions in the study of periodic boundary value problems.
As an example,
he considers the
problem x " - I x , lx, + sin x = -sin t, x[O] - x[2w]
= x'[O] - x'(2w]
= 0
and proves the existence of a solution; the interest of his method is that it holds when - sin t is replaced by any continuous e[t] such that
le[t]l ~ I.
He also shows that the p r o b l e m
461
x " + ax '2+ sin x = -sin t + c, x(0] - x(2~] has no solution if c > 2 for sufficiently
= x'I0) - x'[2~]
= 0
and that for each c E ] 0 , 2 [ , it has a solution
large a.
We shall see in section 2 end 3 how the
use of lower and upper solutions techniques can give further information.
In 1968, Marlin and Ullrich [18]
proved by the use of the
P o i n o a r 6 operator that the problem x"+a
sin x = e{t], x[0] - x(2~]
= x'(0] - x'(2~]
= 0
where [0,7)
]2~ Bit) dt = 0 0
has a s o l u t i o n
if
[0.8)
a +
max
[e(t]l
1
~.
<
tE[0,2~] When
[0.7] holds,
(0.8] was improved to
(0.9]
a < I
by Castro [ 5 ] , using a min-max m e t h o d ~ a n d
[0,9] was shown to be superflous
by W i l l e m [251 and i n d e p e n d e n t l y by 0ancer [ 7] using a v a r i a t i o n a l approach.
We deal with it in section 4.
not necessary hold, DrabeK
[8]
For cases where
C0.7] does
(0.8] was s u c c e s s i v e l y improved by Koneckny [ 1 5 ] ,
and Zanolin [28]
(who allows some d i s s i p a t i v e terms)
and we shall describe results of this type in section 1. d2 The p r o b l e m of c h a r e e t e r i z i n g the range of d--~+ a sin
(.] was
considered by Koneckny [ 1 5 ] , Castro [5] , Bates [I] and Oancer [7]
d2 and we describe extensions of their results to ~ + in section 3.
Finally,
f(.]
+~
sin
(.)
in the naturally associated c y l i n d e r space,
the equation x " + cx' + a sin x = 0 has two singular points of different nature
Ca center or focus and
a saddle point], that means at least two periodic solutions.
We
present in section 2 a g e n e r a l i z a t i o n of this result to the forced ease, allowing m o r e o v e r the p r e s e n c e of d i s s i p a t i v e terms of rather general nature.
Of course most of the described methods work in situations much more general that ones described here and some of them will be developed in subsequent papers.
462
1. EXPLICIT CONDITIONS OF PERTURBATIONS TYPE FOR THE EXISTENCE OF MULTIPLE PERIOOIC SOLUTIONS
Let us consider the problem x ' + f(x)x' + a [1.1)
x[O) - x[2~)
where f
: ~
~P
=
is continuous,
sin x = e[t),
x'[O)
- x'[2~)
a>O
= 0
and e E L I ( 0 , 2 ~ ) .
The results of this section will unify and extend in various direction previous ones by Marlin and Ullrich [18], Konecny [15], Drabeck [ 8], Nagle and SingKofer [ 22] and Zanolin [26] . Except for the multiplicity results, they follow essentially [28].
If g E L l ( 0 , 2 ~ ) ,
we shall write
if g E L P [ 0 , 2 ~ ) ,
= C2#)_ 1 f 2 ~ g ( t ) 0 [p~l), we write
dt,
g[t)
ig(t)lp
Iglp =
= g(t]
- ~,
dt)I/p
0 and if g E L~(0,2~], then
Igl
=
ess
t
TflEOREM I .
Iglt)l.
sup
E[o,2~]
Assume that
-a<~
[1.2)
2~a+ [~11 < 3,
and that
(1.3)
arc sin Ikl < a
Then [ 1 . 1 )
2
[2 a+ ,,la11
6
has at least two distinct solutions
Proof. We want to apply
such that x E [ - ~ , - - ] .
x
the continuation theorem of [10] and we
refer to the current literature for the usual reduction of [1.1) to a semi linear operator equationbetwen
the spaces CI[0,2~]
satisfying the regularity conditions of [10].
To
and LI(0,2~]
check the a
priori bounds and degree conditions of the continuation theorem, X610,1[
and x be a possible solution of
[1.4)
x " + ~f[x]x' + Xa sin x = Xe[t),
[1.5)
x[O) - x(2~)
Multipyingboth tions,
let
we get
members o f
= x'[O] [1.4)
- x'[2~)
= O.
by 2 and u s i n g
the
boundary
condi-
463
al~l~ .l'2~lsin
12w R'2[t] dt ~< o 0 so that, by the well Known inequality [1.6]
c~.7~
xct~l
I~1~1~11,
dt+
see e.g. [23] ),
I~1~ < ~~,'6 I~' 2"
we o b t a i n
[1.8)
I~1~ < ~ c2~a + ~1~.
Define
: x satisfies (1.5~, l~l~<~
o = {xcc~0,2~
I~[1 ]
(2wa+
and
~c]-~,7[ }. If
x£9~
and s a t i s f i e s
[1.4)-(1.5),
then, using (1.4),
we g e t
[x"ll "< cllx'l=+ c~
at one point,
c1.~
as x' necessarily
I×'1-
for
Therefore,
we have
~10 = { x £ C 1 { 0 , 2 ~ ) :
(1,4]-(1.5),
x satisfies
Now d e f i n e
I~l
{ft.5),
I x l < Z2 and Ix'l~
is
an open bounded s e t
{1.6]-(1.5)
such t h a t -
of
x£~
1T
C1(0,2w]
1,
-
[1.5],
x is
then necessarily,
1T
we must have x = - ~ o r x = ~ . and u s i n ~
and i f
a solution
by
New, i n t e g r a t i n g
(1.8)
(1.4)
and
of [1.g)
on [ 0 , 2 7 ]
we g e t
[I,10)
1S2~
sin(x+~(t))
dt = ~.
0 But,
if
x = 2"
then,
by [ 1 , 2 ] ,
(1.3)
and
[1.8),
we have
s i n [ v + R[t)) > ae for all t E [ 0 , 2 ~ ]
and hence
!2~ I2~ sinC2+RCt?l Similarly,
if x = - 3"
1 2~ 2--~ S sin[D and we c o n c l u d e
that
the other
by
hand,
dt
> ee
0 then
x ~ [1.2)
1 for
~--+~[t]) dt < e 2 - -a
every solution
t h e mapping F : x-~ a sin x-
e
of
(1.4)-[1.5).
On
464
has a B r o u w e r
degree
equal to +1.
Thus
xE]-
dB[F,
]-
- ~ , ~ [ , O] w i t h
[1.4)-[1.5]
has a least
respect
to ]-
~,~[
one solution x such that
~ , ~ [ . One can proceed similarly with 0 {xECI[0,2~] ~2 =
: x satisfies
[1.5],
~_ 3 ~ r } " and X E ] - 2 , 2 .
I~Ioo<-@~2=a+
-_.IT
3~.
and obtain the existence of a solution x such that x & ] - ~ , ~ - [ , so that the proof is complete.
2.
DEGREE THEORY AND THE EXISTENCE OF MULTIPLE SOLUTIONS FOR THE P E N D U L U M - L I E N A R O E@UATION.
For c # O, if the p r o b l e m x"+
[2.1]
cx' +a
x[O) - x[2~] has a solution,
sin
= x'[O]
x = e - x'[2~]
= O.
then, m u l t i p l y i n g the equation by x' and i n t e g r a t i n g
over [ 0 , 2 ~ ] , we get e S 2~ x'2[t] dt = 0 0 and hence x is constant, i.e. x[t] = x, with a
Consequently,
[2.1]
sin
x = e.
has t w o d i s t i n c t
one solution in [ 0 , 2 ~ ]
Ill
if
solutions
in
[0,2~]
if
= e and no solution if
>
eE]-a,a[, a.
we
@hell consider the extension of this elementary result to problems of the form x " + z(t,x,x']
[2.2]
x[O] - x[2~) where g : [0,27]
x~ 2 ~
= x'[O]
for
every
solution
[t,x]
E[0,2~]
: there of
[2.2]
-x'[2~]
and e : [ O , 2 z ] g[t,x,O]
condition
+ a sin x = e(t]
x~,
exists
~
are
C>O
the
following
such t h a t
verifying
is such that Ix' Too < C. Example of admissible g are the following ones a]
g depends
only
on x '
[see [20])
continuous,
= 0
g satisfies
a constant
= 0
:
Nagurno # y p e
every
possible
465
b~ Igct,x,y~l ~ hClyl) for Ct,x,yl c [0,27] x[--~,-~] x~ where h is positive continuous and such that
[see e.g. [I0] ].
f ~ s ds = + ~ hCs) 0 In particular, g[t,x,y]
= f(x]lyly
continuous f satisfies this condition if y ~ O ,
~0
with an arbitrary and y + 6 ( 2.
We have now the following result w h o s e existence part for one solution at
least was already given Knobloch [14] when g[t,x,y]
=
ylyl If the above conditions on g are satisfied then the problem
T H E O P ~ M 2.
(2.1) has at least two solutions not differing by a multiple of 2~
if C2.3]
and at least one solution if [el~
Proof.
By a s s u m p t i o n ,
we h a v e ,
= a.
for
all
KEZ,
e[t] - a s i n [ T + 2k~] - g[t, 22+ 2K~,O] for all t E [ 0 , 2 ~ ] ,
= e[t] + a < 0
with a strict inequality if
(2.3) holds.
Similarly,
for all 1 6 Z,
e(t]
- a sin(~+21~)
-g{t,~+21~,O)
with a strict inequality if (2.3) holds.
= e(t]
-a
Taking just k = i = O, the
existence of at least one solution such that - - ~ ( x [ t } < ~ tE[0,2~]
when
~> 0
for
lel~ ~< a then follows from classical results
e.g. [10] and [20] ].
Now i£ we
dom L = { x E C I[0,2#)
x(O]-x[2~] L : dora L r - C I [ o , 2 # ]
define : x is of class C 2 = x'[O]-x'[2~]
~ C°[0,2~),
= 0},
x -~ x "
[see
466
N : CI(0,2~) -~ C ° ( 0 , 2 ~ ) , Ok,1 = { x E C l ( 0 " 2 ~ ]
Ix']
x ~ e-a
sin x-g(.,x,x'),
• ~+2k~<x{t]<
+21~, t E [ 0 , 2 ~ ]
(l>k~
where C i s the c o n s t a n t g i v e n by the Nagumo c o n d i t i o n , from a s l i g h t
modification
it
follows
of the p r o o f o f Theorem V.5 i n [ 10]
t h a t the c o i n c i d e n c e degree D[ {L,N , ~o,o ] i s d e f i n e d when (2.3) h o l d s , and t h a t -
D[ (L,N), no, ° ] = d B [ e -
-~
e sin{.),
3~-
]~,~-[,0]
= 1.
We find similarly that if (2.3] holds, we have -
D[ (L,N), ~_I _1]= d B [ e - a s i n [ . ) ,
1-
~
~
,-~-[,
O] = +1
and O[ (L,N), 8_1,0 ] :
= +I.
dB[e-asin(.),]-~,~[,O]
But
~0,0 c ~ - 1 , 0 "
~-1,0"
~-1,-1 c
and hence, by the e x c i s i o n p r o p e r t y of degree, we get 1 = D[ ( L , N ) , ~ 1,0 ] = D [ CL,N), ~ - I , - 1 ] +
D[ [L,N]
•
~ - 1 , 0\ (~ - 1 , - 1
D[ [ L , N ) , ~_.1,0 \
+ D[ ( L , N ) , ~0,0 ] +
U ~0,0 ) ] = - 2 +
(~_1 _ I U ~ o , o ) ] ,
and hence O[ ( L , N ) , £_1,0 \ [ ~ - 1 , - 1 U~O,O] ] = - 1 . Therefore there exists i.e.
a solution
a solution
_
for all
of [2.2]
tE[0,2~],
x[T)>-
3~
2
<
xCt)
<
3__~
2
1T
~ f o r a t l e a s t oneT E [ 0 , 2 ~ ]
x(r']<
2 f o r a t l e a s t one T' E [ 0 , 2 ~ ] .
differ
from the one i n £0,0 by s m u l t i p l e
contrast
i n £_1,0 \ ( ~ - 1 , - 1 U~O,O)
such t h a t
the s o l u t i o n s
one from another.
Thus t h i s
and
solution
o f 2~.
i n £0,0 and ~ - 1 , - 1 may j u s t
cannot
Notice that in be 2 ~ - t r a n s l a t e s
467
3. UPPER AND LOWER SOLUTIONS AND THE RANGE OF D I F F E R E N T I A L EQUATIONS OF THE P E N O U L U M - L I E N A R D TYPE
We first recall that a lower
[rasp. upper)
solutlon of the
differential equation (3.1)
x"[t)
where # : [0,2~] x P x ~ : [0,2~] ~ R
~
for
all
tions
t£[0,2~]
1.
is a C 2 - f u n c t i o n
such that
~ f[t,~[t],~'[t)]
< f(t,~(t),B"[t))) ,
[Special
have o c c u r e d i n
section
cases o f 2),
constant
The f o l l o w i n g
lower or lemma i s
upper
solu-
elementary
.
but useful
LE~
is continuous,
(resp. 8 : [ 0 , 2 7 ] ~ ) a"[t)
(resp, ~ " ( t )
= f[t,x[t],x'[t)),
Assume that there exists some T > O f(t,x,y)
for all (t,x,y) E [0,2~] X R
such that
= f[t,x+T,y]
xP.
Then, if (3.1) has a lower solution
~o and an upper solution Bo, it also has a lower solution ~ and an upper solution B such that act] ~ B[t) for all t E [ 0 , 2 7 ] .
Proof.
Take ~ = m and B = 8 + KT w i t h o o
[ ~o(t)
- Bolt) ] •
We now c o n s i d e r [3.2]
KE1N* such t h a t
the problem x"+ f[xJx'
+ a sin
x = e[t)
x[O] - x[2~] - x'[O) - x'[2~) where a > O ,
f : P ~,
k > T -1 max t6[0,2~]
27-periodic,
= 0
and e : [0,27] * P
are continuous.
The results described in this section unify and extend in various directions previous ones of Koneckny [ 1 5 ] , Castro [ 5 ] , Bates [1] and Dancer [ 7 ] .
We first establish a series of lemmas.
468
LE~
2.
If
Iml>a,
then (3.2) has no solution and, for
Iml =
a,
9nly constant solutions ~ ~ are possible.
Proof.
using
[3.2]
IT
has a solution, then, i n t e g r a t i n g over [ 0 , 2 ~ ]
the boundary
conditions a
f
we g e t
2~
sin
x(t]
dt = e
0
I~I <
and hence
If x # t E + 2k~,
a
oq p o s i t i v e m e a s u r e and
and
Isin x[t)l <
then
1 on a set
le[ < a.
If the problems
LEMMA 3.
x"
[3.3]
+ f[x]x'
x[0]
+ a sin
- x[2~]
x = e,z + ~ [ t ]
= x'[0]
- x'[2~)
[i
= 1,2]
by
[3.6),
= O
have solutions and if, say [3.4]
- a < e I < ~2 ~< a ,
then the problem x"+f[x]x'
[3.5]
x[0] - x[2~]
has a solution for every
e
[3,6)
sin x = e + ~ [ t ]
= x'[0]
- x'[2~]
= 0
such that
el < & < ~2'
Proof.
If
have f o r
so t h a t
(.
1
all
(1
satisfying
is
a solution
of
solution
[3.3]
[i
= 1,2),
then,
we
t6[0,2=], (#[t]
<-f[(1)(~-a
sin
~[t]
~> - f [ ~ 2 ] ~ -
a sin
[rasp,
~2 ] i s
the periodic
an u p p e r boundary
27, we deduce t h e e x i s t e n c e
dary
+a
6 such t h a t
conditions.
m[t)
~2 + e + ~ [ t ]
[rasp.
lower)
conditions.
o# a l o w e r
As o u r e q u a t i o n
(l+e+~[t],
clearly
of
[3.5]
By lemma I w i t h
solution
[tE[0,2~]]
solution
T =
m and an u p p e r
which satisfy satisfies
t h e boun-
a Nagumo c o n d i t i c
469
on t h e s e t { t , x , y of a solution
: ~(t)<x
of
(3.5) ~(t]
follows
from c l a s s i c a l
y6P},
~ x[t]
< B(t),
t 6[0,27]
[see e . g . [ 1 0 ] ) .
c o n t i n u o u s and such t h a t
~2~ @[t) dt = O, 0 = {eEIR such that [3.5) has a solution}.
let R[~]
3, R[~) is a [possibly
LEMMA 4.
empty]
For every x 6~,
~"+f[x+~)~'
the existence
such t h a t
results
For ~ : [ 0 , 2 7 ] - * l q
t610,27],
interval
contained
By lemmas
2 and
in [-a,a] .
the problem a -~-#
+ a sin [ x + ~ )
2~
sin
Ix+ ~[s))
ds = @(t)
0
[3.7) ~[0)
-~(27)
=
~'[0)
-~'(27)
=
o,
~2n ~ [ t ) d t = 0 0 has at least one solution.
Proof.
Let, for K 6 ~ , cK[0,27)
with the induced
= {~6CK(0,27)
: ~2~ k[s] ds = O} 0
norm and let
H : C ° ( 0 , 2 7 ) ~ C1(0,27) be the mapping
defined
(HR][t)
by
= ft k[s] 0 N o t i c e t h a t Im H c { x E C I [ o , 2 ~ ] define
N
: CI[0,2~)
ds-2~
~ cO(o,2~)
f 2~ f t R[s) ds d t . 0 0 : x[O) - x [ 2 7 ) = 0 } , and c l e a r l y
if
we
by
X
(N~)Ct) x
= ~(t) - f(~ +~(t))'~x'(t) - a [ s i n ( ~ * ~ ( t )
[t 6[o,27]
),
- 2~- 12~ s i n ( ~ + ~ ( s ) 0
ds] ,
470
then
(3.7] is equivalent to the fixed point problem
[3.8]
£ = H2N [2) X
in C1[0,2~],
Now H2N
is easily
shown t o be a c o m p l e t e l y c o n t i n u o u s
X
mapping on C 1 [ 0 , 2 ~ ]
so t h a t
(3,7]
to Leray-Sehauder's theory solutions
£ = XH2N ( ~ ] , x
is a priori for
bounded i n d e p e n d e n t l y
if
according
the set of all
possible
4E[0,1]
o f 4.
some 4, then 2 E C 2 ( 0 , 2 ~ ) ,
conditions [3.10)
have a s o l u t i o n ,
o~
(3.9)
(3.9]
will
[see e , g . [ 2 1 ] ) ,
If
satisfies
2 is a fixed
point
the periodic
of
boundary
and t h e e q u a t i o n 2"+ 4f[x+~)2.
=
Multiplying
dt = ~Xa f2w £ [ t ] 0
sin(~+ £[t)
by Schwarz and Wirtinger
in turn ,
dt- ~
we get
i2w £ ( t ] ~ ( t ) d t , 0
inequalities,
1~'12
ds]
k~(t],
(3.10] by £ and using the boundary conditions,
I i 2 ~ 2, 2 [ t ] 2~ 0 and hence,
- ~1 r2~ s i n [ x + 2 ( s ) ) 0
+ka[sin[x+~]
(~'= Z ~ a )
by [ 1 . 7 ] ,
I~1~ < {6 ~- Ca'+ Im12~" But t h e n ,
integrating
]x"l
t a k e n from [ 3 . 1 0 ) ,
we g e t
I~"11 < c where C depends o n l y on a and ~ (and net o f x ) , sarily
vanishes
I*'1~ and the possible solutions
LEMMA 5. [3.11]
hence, as 2 '
neces-
at one point, this gives <
2~c
of [3.9] are a priori bounded in CI(0,2~].
For each e E C [ 0 , 2 ~ ] ,
2~ j-2~ sin[x+ £ (t)] dt : R[~) = { a._~_ 0
x
satisfies x
[3.7)
471
and x E]R} ~ ¢ .
Proof. That the second set is nonempty follows from lemma 4. 6R(b);
Let
then there exists a solution x of [3.5) and we can write
it x = x + 2. But [3.5) is equivalent to ~] _ 1
k" + f[~ + k)k' + a [ s i n ( ~ +
Z~T
f27 s i n [ ~ + 2 [ s ) ) 0
a__f2~ s i n [ x + R [ s ) ) 2~
ds] = ~(t),
ds = e
0
which shows that [3.12)
e E {a f27 sin[x + k It)) dt : 2 satisfies 27 0 x x
[3.7]
and x E ~ } . New let i satisfy
[3.1~).
Then there is x E ~
-
a
[3.7] and is such that e = ~-~ f
and 2
which verifies x
27
sin(x + R_[t)dt. Letting x = x + 0 x we see that x verifies the periodic boundary conditions and the equation x"+f[X]X'
a
+a sin x = ~
~
2~ 0
so that e E R ( ~ ) ,
sin(x+ R [s)) x
ds + ~ [ t )
= e+ ~ [ t )
and the proof is complete.
Summarizing and completing the above 3emmas, we obtain the following
THEOREM 3.
For each ~ 6 C ( 0 , 2 ~ ) , R[~) is a non-empty closed sub-
interval of [-a,a] characterized by the relation (3.11).
Proof. sequence
If remains only to prove that R[~) is closed.
Let [e K) be a
in R[@) which converges to e and let x k be a solution of x ' + f ( x ) x ' + a sin x = e K + ~ [ t )
[3.13)
x(O]-x[2~)
= x'(O)-x'[2~)
= O,
[k6]I~).
472
As X k + 2m~ is solution together with x k for every m E Z , assume without
loss of generality
that X k E [ 0 , 2 ~ ]
we can
(hE,hi.
From
i3.13), we see like in the proof of lemma 4, that R k satisfies
the
fixed point equation
i3.14)
R k = H 2 N _ i 2 k]
x
and that
l~kl~ < ~ By the compactness
if+ I~12~, I~I
< 2~C, I~11 < C.
of [0,2~] and Ascoli-Arzela
theorem,
there exists
a subsequenoe
ix. ) such that ix. ] converges to some x E [ 0 , 2 ~ ] Jk JR and JR. ] converges in CliO,2~) to some R. From i3.14) we deduce JK then that
2 = H2N JR), X
and from the relation,
deduced from [3.13),
a 2~ 2--~fO
sinixjk+kjkit]]
dt = e~K ,
we obtain
a $2~
2~ Consequently,
x = x+x
sinlx +xit)] dt = ~. 0 is a solution of (3.5] and the proof is
complete.
Remark
I. If we apply theorem 2 to problem
leI~
lel + lel
<
i3.2] and notice that
, we see that for each e with lel~ <
a,
we
have
473
4. VARIATIONAL
METHODS AND PERIODIC SOLUTIONS OF THE
PENDULUM EQUATIONS WHEN THE FORCING TERM HAS MEAN VALUE ZERO.
We d e s c r i b e Dancer [ 7 ]
here the independant
work of Willem [ 2 5 ]
and
on t h e p r o b l e m x"+ a sin
(4.17
x = e{t]
x(O] - x{2~)
when e E L I ( o , 2 ~ ) ding results
and e = O.
and { o r
= x'(O]
We r e f e r
- x'(2~)
to [25]
considerations
for
= 0
abstract
on t h e s t a b i l i t y
of
corresponsolutions.
If eELl(0,2~] and ~= O, then (4.1) has at least one
THEOREM 4.
solution which is a minimum of the functional m : H~ (0,2%) ~ ,
Proof.
Recall
x ~ f2~ [ (x'(t]) 2 0 -
2
-
+ a cos x [ t ]
+ e[t]x(t]
] dt
that
H 1 [ 0 , 2 ~ ] i s t h e space o f a b s o l u t e l y continuous 27 {unctions such t h a t u ' E L 2 ( 0 , 2 ~ ] , with the inner
and 2 ~ - p e r i o d i c product
[u,v?HZ
= S2~ [ u [ t ) v C t ) + u'[t)v'[t]] dt. 0 It is easily shown that ~ is weakly lower semi-continuous.
Now,
as e = O, we have, using will known inequalities, ~(x)
f2~ =
0 [4.2]
> $2~ o
1 and hence ~[x) ~ + ~ i { for ~.
As ~{x]
generality
[~'(t]] 2
Ck'Ct]] 2
2
~
+ a cos x ( t ]
+ e[t)R(t)
-2~a-Telllkl
>
=lell
Let (x k) be a minimizing
{ x E H ~] , ~
that x k E [ 0 " 2 ~ ]
and, by [4.2]
[2k ) is bounded in
[x K) has the same property.
satisfied
Thus
[x K) has a
to some x and by the weak lower
of ~, x is a minimum of ~.
that the Euler equation of
sequence
we can assume without loss of
subsequence which weakly converges semi-continuity
dt >
I~lH~,
IRIH I ~ ~.
~ ( x + 2~)
H~ [0,2~), so that
2
It is easily checked
by x implies that x is a solution
[4.1].
Remark.
The above theorems
showing that,
completes
for f ~ O, O E R [ ~ )
Know o{ any nonvariational
the results o{ section 2 by
for every ~6C(0,2~).
We do n o t
proof of this result which would make
possible an extension to the not necessarily
conservative
case.
474
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[s]
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DRABEK P.
Ahmad and Lazer ed., Academic
149-160).
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by the alter-
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[8]
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CESARI L. and BOWMAN T.T. native method
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21 (1960) 155-160).
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[11]
HAMEL G,
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Equations
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[12]
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[Lecture Notes in Math. n ° 568,
HAMMERSTEIN
A.
bei endlichen Amplituden
1-13].
Nichtlineare
Integralgleichungen
nebst Anwendungel
[Acta Math. 54 [1930) 117-176), [13]
IGLISCH R.
Zur Theorie Schwingungen
[Monatsh.
Math, Phys.
37 (1930) 325-342],
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KNOBLOCH H.W.
Eine neue Methode zur Approximation
L8sungen nicht-linearer Ordnung
[15]
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M.
177-197].
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solvability
Carolinae
[16]
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Differentialgleichungen
[Math. Z. 82 (1963)
ordinary differential
periodischer
equations
zweiter
of nonlinear
(Comment.
Math. Univ.
18 [1977) 547-582).
F,
Uber die yon einem PunKt ausgehenden
Kurven einer Dif~erentialgleichung
Integral-
zweiter Ordnung
[Deutsche Math. 7 [1944) 5 8 - 7 4 ) .
[17]
MARLIN J.A.
Periodic motions of coupled simple pendulums with
periodic disturbances
[Intern. J. Nonlinear Mech. 3
[1968) 4 3 9 - 4 4 7 ] . [18]
MARLIN J.A. and ULLRICH D.F.
Periodic
order nonlinear differential [SIAM J. Appl, Math.
solutions of second
equations without damping
16 [1968]
998-1010].
476
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MAWHIN 5.
Une g6n~ralisation
de th@or~mesd~ 5.A. Marlin
lint. J. Non-linear Mech. 5 [19Z0]
[2o]
MAWHIN J.
Boundary value problemsfor
vector differential
equations
335-33g].
nonlinear second order
(J. Oifferential
Equations
16 [1974] 257-26g).
[21]
MAWHIN J.
Topological
value Problems
Degree Methods in Nonlinear Boundary
[CBMS Regional
Amer. Math. Soe., Providence,
[221
NAGLE R.K. and SINGKOFER equations
K.
ROUCHE N, and NAWHIN J. Stability
[24]
Nonlinear
11 [1980]
Ordinary
and Periodic
SCHMITT B. and BRZEZINSKI p~riodiques
[Equadiff
n ° qO,
ordinary differential varying nonlinearities
137-149].
Oifferentiel
solutions
R.
in Math.
1978].
at resonance with slowly
[Applicable Analysis
[231
Confer.
[Pitmann,
Localisatien
Equations. Boston,
1980].
num6rique de solutions
78, Conti, Sestini and Villari ed.
F i r e n z e , 1978, 9B-108).
[2s]
WILLEM M.
Oscillations
[Publications
[261
ZANOLIN F.
hamiltoniens
de Besangon,
RemarKs on multiple solutions
nary di~qerential Ital,,
forc@es de syst@mes
de l'Universit@
1961].
for nonlinear ordi-
systems of Li~nard type
[Boll. Un. Mat.
to appear].
Author Adress
; Institut Math~metique Universit~
Catholique de Louvain
Chemin du Cyclotron,
2
8-1348 Louvain-L~-Neuve Belgium.
A SIMPLE L A Y E R P O T E N T I A L M E T H O D F O R T H R E E - D I M E N S I O N A L E D D Y C U R R E N T PROBLEMS
R.C. M a c C a m y
i.
-
E. S t e p h a n
INTRODUCTION
Let
S
be a closed analytic surface w h i c h divides 3
d i s j o i n t domains, an i n t e r i o r is to r e p r e s e n t air and
~'
~'
(bounded)
into simply connected
and an e x t e r i o r
~
(unbounded).
a m e t a l l i c conductor. Thus the exterior d o m a i n
is c h a r a c t e r i z e d b y c o n s t i t u t i v e p a r a m e t e r s
So,~ °
d e n o t i n g p e r m i t i v i t y and
p e r m e a b i l t y and is a s s u m e d to have zero conductivity. The interior domain c h a r a c t e r i z e d b y constants
~,~,~
The total e l e c t r o m a g e n t i c field O0 (E ,~ )
SS (~ ,~ )
and scattered
originate in
w h e r e the c o n d u c t i v i t y
(~,H)
~
~'
is
may be infinite.
will consist of the sum of i n c i d e n t
terms w h e r e the i n c i d e n t field is a s s u m e d to
~ . All fields are a s s u m e d to be time h a r m o n i c and monochromatic,
i.e. to have a time h a r m o n i c d e p e n d e n c e tutive p a r a m e t e r s are r e l a t e d to wave k = ~/s ~ co
,
e
-i~t
. The f r e q u e n c y
numbers a p p r o p r i a t e to
~ ~
and constiand
~'
by
k' = w (~e + i 0 ) ~
(I)
and the time h a r m o n i c M a x w e l l equations are curl ~ = i ~ o ~ curl ~ = i ~
, H ,
curl H = - i ~ S o ~
in
~
(2)
curl H = (-i~e + ~)~
in
~'
(3)
It is n o t e d that the i n c i d e n t field satisfies
(2) almost everywhere in
R 3
and the field q u a n t i t i e s are i n f i n i t e l y d i f f e r e n t i a b l e except at source points in and
~'.Across the interface
total field
(~,~)
(~ = i~Zo~, ~ = ~
S = ~
= ~'
the t a n g e n t i a l component of the
m u s t be continuous. T h e r e f o r e after an appropriate scaling in ~ ; ~ = i m ~
, ~ = ~
in ~')
the eddy current p r o b l e m is
given b y curl ~ = ~
,
curl ~ = e2E
in
curl E = ~
,
curl ~ = i ~
in
and Here = ~
2
(nxE) ~ ~ + = (nxE)- ' (nxH)+ m ~ = (nxH)-m m
= ~ 2 eo~ O , ~ = > 0
~'
(~O
- i~ 2 s )
(P ~) on
S
are d i m e n s i o n l e s s p a r a m e t e r s and
if d i s p l a c e m e n t c u r r e n t s are n e g l e c t e d in m e t a l
(e=O). The s u p e r -
478
scripts plus and minus denote limits from normal on the surface
the p e r f e c t conductor approximation. (2) in
~
and ~'
where
~
is o u t w a r d d i r e c t e d
~
is u s u a l l y large and this leads to
S .
A t h i g h e r c o n d u c t i v i t y the c o n s t a n t
equations
~
F o r m a l l y this means
solving only the M a x w e l l
for the s c a t t e r e d field and r e q u i r i n g that the t a n g e n t i a l
component of the total e l e c t r i c field vanishes on curl ~s = ~s
,
curl ~Hs =
2 s
S (nXE % % = O on S)
,
i.e.
in (P~)
(~;x~S) + = -(~]x~O) +
on
S
If in a d d i t i o n the s c a t t e r e d field satisfies the S o m m e r f e l d radiation condition the following uniqueness theorem holds: T h e o r e m i: There exists at most one solution of
(PB)
for any
~>O
and
O
(P S)
(with ~ O
(P)
can be f o u n d in [5] and the c o r r e s p o n d i n g one
in ~') in [3, p. 592].
In o r d e r to a v o i d a d d i t i o n a l difficulties we need the t e c h n i c a l a s s u m p t i o n curl ~ = ~ , curl ~ = 2 i.e. ~
2
in ~'
, %nXE~ = O on S
implies ~ = ~ = ~ on ~'
(4)
is not an eigenvalue of the interior D i r i c h l e t p r o b l e m for M a x w e l l ' s
equations.
2.
Our simple layer p o t e n t i a l p r o c e d u r e In the f o l l o w i n g we give new b o u n d a r y integral e q u a t i o n methods for s o l v i n g
both
(P 8) and
(P)
. To this end we introduce the simple layer
V
w i t h conY
tinuous d e n s i t y
v
~
on the surface
(~)(x) =
S
by
¢(y)%w(Ix-y])dS
Y
(5)
x c m Y
Here (6)
%(Ix-yl) = is the fundamental s o l u t i o n of the H e l m h o l t z e q u a t i o n S o m m e r f e l d radiation c o n d i t i o n for
2 Aw = -y w
Re y ~ 0 . For a v e c t o r field
s a t i s f y i n g the v
on
S
we
479
define
Vy(~)
by
(5) again with
~
replacing
~ . There hold the following
w e l l - k n o w n p r o p e r t i e s of the simple layer p o t e n t i a l L e m m a 2: (i)
V
For any complex
(~)
y,O <- arg y -< 7/2
(see [4~,[5]) and any continuous
~,~
o_~n S :
3 19
is continuous in
Y (ii)
2 AVy(~) = -y Vy(~)
(iii)
V
(iv)
__in ~ u ~'
y (~)(x)= O(Ixl-le
(-~-~ V y ( ~ ) (x)
)-+
i¥1xl) 1
= ¥ ~ ~(x)
-a-s
+ f Ky(X,y)~(y)dS
~n
(v)
Ix[-~- m
S
(n x c u r l
V y ( y ) (x)
w h e r e the function
= -+ " x - y ( x )
(matrix function)
on Y
Ky
+
S
--
.~,,~,~Ky(X'Y)Y(y)dSy
(K_) is _
O(Ix-y1-11
_
-o-n
S
as
y÷x
Y Our m e t h o d s are b a s e d on the S t r a t t o n - C h u r e p r e s e n t a t i o n formulas of electrom a g n e t i c fields from [7] y i e l d i n g for the s c a t t e r e d fields = V
(~×H) - curl V
(~×E) + grad V
(E,~)
If
nXH , nXE
(P)
- but
but sets field
and
only
nXH
(n×H) - curl curl V
n.E
were all k n o w n on
~,nXE'~ i s
and
known.
~.E
L (see [23)
(P)
:
(n.E) in
= curl V
in
~
(7)
(nXE) S
The s t a n d a r d
then
(7) w o u l d y i e l d a solution of
treatment
equal to zero and replaces
of
-nXE
(Pc~)
starts
from
(7)
by an u n k n o w n tangential
:
E = curl V
(L)~
~ = curl curl V
I m p o s i t i o n of the b o u n d a r y condition in of second k i n d for
~
(P)
(L)
(8)
then yields an integral equation
in the tangent space to
S . The m e t h o d
(8) is analogous to
solving the D i r i c h l e t p r o b l e m for the scalar Helmholtz e q u a t i o n w i t h a double layer. It has the drawback,
for the eddy current problem, that h a v i n g found
is hard to determine
~x~
on
derivative of
on
S .
V
(~)
Our m e t h o d for
(P)
simple layer
(see [i])
unknowns
and
J
it
S . The latter involves finding a second normal
is analogous to s o l v i n g the scalar p r o b l e m s with a
. We set
~XE = O in
M . Thus we take
(7)
and replace
~%nXH
and
~~n'E
by
480
= V Having deterimed
(~) + grad V ~
(M) ,
~ = curl V
holds in
free. But from
(9)
we can use Lemma 2 (v) to determine
A n easy c o m p u t a t i o n shows that the ansatz div E = O
(~)
~ ; because in
(9) only
S .
(9) gives a s o l u t i o n of ~
(P
)
if
is a u t o m a t i c a l l y d i v e r g e n c e
in
~ .
T h e r e f o r e b y uniqueness of the e x t e r i o r D i r i c h l e t p r o b l e m the b o u n d a r y condition
e x t e r i o r domain constraint
on
(9) and Lemma 2 follows
A div ~ = -~2div ~
equation)
~XH
div ~ = O
on
S
guarantees
~ . Imposing the b o u n d a r y conditions of
div E = O
on
S
(for the H e l m h o l t z
(P)
div ~ E O
in the
in (9) and the
we o b t a i n a c o u p l e d system of p s e u d o d i f f e r e n t i a l
equations on the b o u n d a r y surface
S
for the unknown densities
V e ( ~ ) T + gradTV
(M) = _(~×~o)
V
(M) = O
~,M
:
(E) where
V
(~) T
(divT~) - ~2V
denotes the tangential component of the vector function
Here we have used the relations with the surface divergence face g r a d i e n t
grad T
on
S
for any
yeC
div V~(Z) = V y ( d i v T ~ ) , Our p r o c e d u r e for
(P ~)
with
div
O N arg y ~ ~
divTgra%Vy(M)
(9) in
~
(~) .
and the sur-
T
:
2 = -y Vy(M)
p r o c e e d s as follows. This time we let
denote the total fields and use again
V
and its analog in
~ ~'
and . Thus we
put,
E = E° + V
(J) + grad V
(M) ,
H = H ° + curl V
(J) in
~
,
(1o) E = V
(~) + grad V
A g a i n the c o n s t r a i n t in
(P ~)
layers
give a
(J,M,~,m)
div ~ z O
(m)
in
~u~'
,
H = curl V /
(~)
in
~'
t o g e t h e r with the b o u n d a r y conditions
coupled system of p s e u d o d i f f e r e n t i a l equations for the unknown on
S :
481
V(~) T + gradTV(M)
- V
(~) - gradTV
i/j V~(diVT~)
- ~2V
(M)
o
(m) = -E
d7
~
= O
(E~B) g + ~ (0) + ~- K
(%) =-2(~xH °) /iT
V
(divT~)
- iB V
•~ 7 Equation
(E ~) 1
equations U ~' dition
(m)= O
d7 is caused by the interface
(E B) 2 , (E ~) 4
whereas equation (~×~1+ =
(~XH)-
condition
are consequences (E 8) 3
on
S
(~XE)- = (nXE) +
of the constraint
is resulting
on
div E = O
from the second interface
S , in
con-
together with Lemma 2 (v) if we define
Ky(V) (x) = 2 ~ Ky(x,y)v(y)dSy
Collecting
the above results there holds the following representation
Theorem 3 continuous (ii) continuous
3.
(E)
[4] then
If
:
(i)
Existence,
(~,M)
solve
(E)
(9) yields a solution of
(~,M,~,m)
then
If
solve
(E B)
with
(IO) yields a solution of
uniqeness
and regularity
In the half-space
case
is reduced on
x3 = O
(P) {,j
with
J
theorem.
differentiable
and
. differentiable
and
M,m
(P 8)
of our integral
~ : { x 6 ~ 3 ]x3>O}
equations
(~(x) : ~3
on
S = m 2 )
the system
to ~
: -4~(~ 3 x o ) (II)
~e* div ~ - ~2¢ *M = O where the star denotes convolution.
$~(~) and the explicit
Via Fourier
transformation
we find in [4]
-I/2 =
(l~l
2 -
solution of
M = 4 V
~)
(11) reads with
(12)
E~To
= ~3 x E o :
(div E o ) , ~ = -gradT(4 V div E o) ~T %T
2 o + 2(A+~ )V ( E T )
(13)
482
Let of
Hr(~ 2 )
denote the Sobolev
C ~ (~2) o
and let
Hr(192 )
Now, V
is
on
2
, i.e. the completion
r
I (i+ I~12) 15(~)12d~ 2
be the space of vector
a pseudodifferential
with symbol
(12) and maps
The operators
div
H r-l(19 2)
it indicates Theorem
operator
H r-l(l~ 2)
continuously
respectively.
Though
the result
4 [4] :
For any real
r
the mapping
(E)
is bijective
from
Hr-I(S)
4 given
on the analytic
)seudodifferential
operators. e
~ (r)
i~r r
system
(E)
Hr(l~ 2 ) .
S
together
(:JR 2))
Hr(lq 2), Hr(l~2)
(13) is only formal,
in the general
A
: (~,M) ÷
x Hr(s)
onto
in [4] uses partition
surface
in
Hr+l(]~ 2) (,~ Hr+l
into
are of order plus one and take
T
of the
The proof of Theorem
with components
o f o r d e ~ m i n u s one i n t h e s e n s e o f [ 6 ]
Hr(192 ) ( r(lR 2))
and grad
T
functions
the m a p p i n g p r o p e r t i e s
by the system
nate systems
r
under the norm
1[*]12r
into
space of order
case.
(-E~ ,O) Hr(s)
defined
x Hr-I(S)
.
of unity and local coordi-
with standard
techniques
for
The expansion ~ Z
1 ~
~j rj ~.i ,
~e~,
(14)
r = Ix-y[
3 induces
the decomposition V ~ = Vi~ + W ~
where and
the Bessel potential W
is a smoothing
ment t o g e t h e r
space)
since
V. z
the d e c o m p o s i t i o n bation
argument,
is
is a pseudodifferential
Vi
of order
shows t h a t
since
+
elliptic
the bijectivity
(E)
-3 . A standard
of
operator
=
Green's
formula
operator
pseudodifferential (E)
of order minus one
of index zero
is proved in
differs by compact operators
grad T Vi(M)
Vi(div T J) + Vi(M)
operator
the Bessel potential
is a Fredholm
a self-adjoint
(15)
Vi (~) T
V. 1
operator
w i t h Lemma 2
tire. On the other hand
(15)
V. 1
arguis
injec-
(in any Sobolev
operator.
Due t o
[4] by a pertur-
from
F
(16)
= G
483
The system Vi,(Vi) T
(16) can be reduced and
(AT-I)
to a Riesz-Schauder
are bijective
t J, M)
system for yEC
and for any
since
, O ~ arg y ~ ~ , there
holds diVT Vy(~)T = V y ( d i v T ~) + ] y ( ~ )
with a continuous
map
]
The obtained
Riesz-Schauder
by potential
theoretic
from
Hr(S)
system
arguments
into
for
Theorem
i and our technical
problem.
the relation
system
(E)
from
(E B)
Hr(s)
solvable
system
due to
+ g
+ J = K
- L l(J) + H ° n,T
for the eddy current
onto
to a Riesz-Schauder
= J + DI(M,J,j)~ ~ (j-J) i / ~ ,,., ,b
to a Riesz-Schauder
(4).
V
(E B)
which follows
I (together with jump relations).
in F4] the system of
(17) we reduce
solvable
(16). This again is uniquely
assumption
Using the bijectivity
, r£~
is uniquely
from Theorem
of
In a similar way we treat
Hr+I(s)
(~,M)
In a second step we reduce our original making use of the bijectivity
(17)
Hr+I(S)
(re~)
and
Hr-1 ~ (S) x Mr(S)
system on
(18)
m - M = L~(M,J,j) + f z rb ,-u
ibm + ~2M = D2(M,J,j) where
~ E Hr-I(S)
Here
and
LI,L 2
and minus two, respectively. guaranteeing
the following
Theorem system
5 [43
(E B)
(P B)
6 :
:
E o ' ~T
are pseudo-differential
operators
of order minus one
(18) turns out to be uniquely
existence
and regularity
For any real
Hr-i ~ (s)
r
and given
;
mm
~ Mr(s)
differentiable
of the solution
only on the smoothness
result for
~C
6 Hr(S)
solvable (E B) .
Hr o e ~ (S), ~TH
Hr+1 E ~ (S)
the
solution •
Since the field quantities
are infinitely
regularity
~ Hr+l(S)
for given data
Again
has a unique
~,~ ~ Remark
~C
; f,h E Hr(s)
DI,D2,K
+ h
(J,M)~
except of
of the boundary
in
(2),
(3) and hence
at source points
(E B) surface
and S .
(J,M,j,m)~ %
in of
in
(P B)
~ and ~' (E B)
and the
hinges
:
484
References [i]
:
Hsiao, G. and MacCamy,
R.C., Solutions of boundary value problems by integral
equations of the first kind, SIAM Review 15 (1973), 687-705. [2]
Knauff, W. and Kress, R., On the exterior boundary value problem for the time harmonic Maxwell equations,
[3]
Koshlyakov, Mathematical
[4]
MacCamy,
N.S., Smirnov, M.M. and Gliner, Physics,
North-Holland
R.C. and Stephan,
Eddy Current Problems, [5]
Mfiller, C., Foundations Springer-Verlag,
[6]
Journ. Math. Anal. Appl.
72 (1979), 215-235.
E.B., Differential
Publishing Company,
E., Solution procedures
Equations of
Amsterdam,
1964.
for three-dimensional
to appear. of the Mathematical
Berlin - Heidelberg
Seeley, R., Pseudo-Differential
Theory of Electromagnetic
- New York,
Operators,
Waves,
1969.
CIME, Cremonese,
Rome
(1969)
(coordinated by L. Nirenberg). [7]
Stratton,
I.A., Electromagnetic
R.C. MacCamy Carnegie-Mellon University Pittsburgh, PA 15213
Theory, McGraw-Hill,
New York, N.Y.,
1941.
Ernst Stephan Technische Hoehschule Darmstadt Federal Republic of Germany
Index Laws for some Ordinary Differential Operators Adam C. McBride
§i.
Much work has been done in recent years on developing general theories of
fractional powers of operators.
Many of these theories have considered various
classes of operators in Banach spaces and have made extensive use of the Dunford integral;
see, for instance,
[I] and [5].
However, when the operators concerned
have a more concrete form, ot~er approaches are possible. pseudo-differential
operator in L2(R n) with symbol g.
For example, let T be a
Then, for appropriate functions
~eL2(Rn), ( ~ (r~))(x)=g(x)(~)(x)
(xgR n)
(i.i)
so that, formally, T= ~-ig ~ where
~
(1.2)
denotes the Fourier transform.
From examination of positive integral
powers of T, we are led to define T ~, for suitable complex numbers ~, by T ~= 3-1g ~ 3
(1.3)
provided that this is meaningful. As a particular case, we can define fractional 22 _ ~2_~_ powers of -5=- ---~ -... 2' for which g(x)=Ixl 2, and study their properties [3]. ~x I ~xn Similarly, we can consider multiplier
transforms T in weighted versions of the spaces
LP(o,~) which take the form (]~R(T~)) (s)=g(s) ]115(s) where
(1.4)
~ l i s the Mellin transform defined formally by
"m.~(s)=
f
xS-l~(x)dx
(1.5)
0
and s i s an a p p r o p r i a t e l y r e s t r i c t e d
complex v a r i a b l e ;
see [8],
[9].
Again, the
formal definition T~=~-Ig~DI is suggested.
(1.6)
Although
(1.3) and (1.6) seem simple enough, there may be problems in
defining g~, with the possibility of branch points arising.
In contrast, we propose
to study transforms which obey a rule similar to (1.4) but which do not lead to the same difficulties
of interpretation.
The operators T above may be thought of as
486
operators from a space of functions into the same space, whereas the operators which we shall consider can be thought of as mapping one space into a different space.
It is perhaps a little paradoxical that we can sometimes make more progress
in the second case than in the first.
However, we hope to illustrate this clearly
by examining a general class of ordinary differential operators on (O,~). §2.
We consider operators T which satisfy the formal relation h(s)
(~(T~)) (s)= h(s+~------y ~(s+x) where
(2.1)
~f~ is defined via (1.5), y is a fixed non-zero complex number and h is a
fixed function defined on some domain in the complex plane. shall argue formally.
For the moment, we
We can give two examples of operators which are of the form
(2.1). Example 2.1 (i)
For any fixed complex number y, let T
be defined by Y
(Ty*)(x)=xY*(x) Then, trivially,
(x~(O,=)).
(2.2)
(1~l(Ty~))(s)=~l~(s+y) if both expressions are meaningful and
(2.1) is satisfied with h(s)-l. (ii)
For the differentiation operator D, integration by parts gives
xS-l(D~)(x)dx= O
Ex
s-l~(x)
(s-1)xS-2~(x)dx. O- 0
I f ~ is such that the integrated terms vanish, then (]~t(-D~))(s)=(s-l)~(s-l)=
r(s)
r(s-'l) ~ ( s - l )
(2.3)
and (2.1) is satisfied with y=-I and h(s)=r(s). From (2.1), it follows easily by induction (]]l(rr~)) (s)= ~
~
that, for r=l,2,...,
(s+ry) .
(2.4)
If we assume that T -I exists and we replace ~ and s in (2.1) by T-I~ and s-y repsectively we might be tempted to say that (2.4) holds for r=-I also and hence, by induction again, for all integers r (the case r=O being trivial). following definition.
This suggests the
487
Definition 2.2
For suitable complex numbers a and with T as in (2.1), define T a by
(m(Ta~))(s)=
h(s)
~(s+~y)~P&(s+ay).
(2.5)
Continuing with our formal manipulations, index laws of ordinary algebra a and B is no great surprise.
carry over. However,
we can ask whether or not the normal
That TaTB=Ta+B=TBT a for complex numbers
it is of much more interest to observe that
we can form powers of T a since, by (2.5), T a satisfies a relation of the form (2.1) with the same function h but with y replaced by aY. a and ~ in (2.5) by T a, B and ay respectively,
(~l [(Ta) ~ ) ( s ) = for
if we replace T,
we obtain
h(s) TR~ (s+Bay) = (~T~(TaB~)) (s) h(s+Bay)
c o m p l e x n u m b e r s a and B.
considerable
In particular,
problems
in other
This suggests theories
that
(2.6)
(Ta)B=T aB, a l a w w h i c h p r e s e n t s
such as that
b a s e d on t h e D u n f o r d i n t e g r a l
Ell] Without further proof, we state the following results for completeness. Theorem 2.3 (Index Laws)
Let ~ and B be complex numbers.
(i)
TaTB=Ta+B=TBT a ;
(ii)
(T~)B=T~=(TB) ~.
§3.
It is natural to ask at this stage if there are circumstances
formal calculations
T°=I
;
Then
(Ta)-I=T -~ .
(2.8)
in §2 can be justified.
examine one particular instance. Definition 3.1 (i)
(2.7)
Fortunately,
in which all the
there are and we shall
Firstly, we describe suitable spaces of functions.
Let ~ be any complex number.
For lsp<~, F
={~cC~(O, ~) : xkDk(x-~)¢eP(o,~)
for k=O,l,2 .... },
P,~ 7~'~(~)=1] xkDk(x-~)[I p (ii)
(k=O,l,2 .... ).
F , ={~¢C~(0, ~) : xkDk(x-~)->O as x~O+ and as x-~o for k=O,l,2 .... }~ Yk
(~)=II xkDk(x-~¢)I[~
(k=O,l, 2 .... )'
Here, D~d/dx and II lip is the usual LP(o,~) norm (l~p~=).
For each p and ~, Fp,~
I
is a vector space and a Frechet space with respect to the topology generated by the collection { ' P ' ~ fk=O k of seminorms.
Analytic and topological properties of these
spaces are described in detail in ~6, Chapter
2]
488
Secondly,
we describe a suitable class of operators.
Let n be a positive
integer and let al,...,an+ 1 be complex numbers such that n+l a= E ak k=l
is real
(3.1)
m=la-nl>O.
(3.2)
Let T denote the n th order ordinary differential expression T= x a iDx a 2 ...xanDxan+l
(D~d/dx).
(3.3)
Any attempt to get a manageable expression for T r (r=1,2,3,...) directly from (3.3) seems doomed to failure.
However, by writing
D_-mxm-ld/dxm with m as in (3.2), progress can be made.
Equivalently, although T is the composition
of 2n+l operators of the types in Example 2.1, the lack of commutativity prevents the immediate derivation of a manageable expression for T in the form (2.1) so that transcription in terms of m is again helpful. n+l bk=( Z a.+k-n)/m i=k+l l
For convenience, we shall write
(k=l ..... n).
(3.4)
The following result is proved in [7]. Theorem 3.1
For l ~ p ~
and any complex number p, T is a continuous linear mapping
from Fp,~ into F p,~+a-n . Re(mbk+~)@i/p
Further ' T is a homeomorphism from F p,p onto F p,~+a-n iff
(k=l ..... n).
In attempting to obtain an expression of the form (2.1), we must study the Mellin transform on Fp, . then
From the work of Rooney [8], we can say that, if ~eFp,p,
~l~(s) is well-defined provided that l~p~2 and Re s=I/p-Re p°
In practice,
the restriction l~p~2 can often be removed at a later stage, as we shall see below. As the theory unfolds, it is necessary to distinguish between the cases an (a=n being excluded by (3.2)).
For simplicity, we consider only the case a
From
~ ] , we quote the following result. Theorem 3.2
Let l~p~2, ~CFp,~, m=n-a>O, Re s=i/p-Re ~+m.
n F(bk+2-s/m) (Tg(T~))(s)--mnk=l E F(bk+l_s/m ) ~t~(s-m) provided that Re(mbk+~)+m#I/p-m% y=-m and
(~=O,I,2 .... ).
Then (3.5)
This is of the form (2.1) where
489
h(s):mnS/m[ ~ F(bk+l_s/m)] -I. k=l
(3.6)
(We notice in passing that (3.5) includes (2.3) as a special case.)
We can now
apply Definition 2.2 in conjunction with (3.6). Definition 3.3
Let T be as in (3.3), l~
We define Te~ by n P (bk+l+~-s/m) (p(t(T~)) (s)=mn~k~ I= r(bk+l_s/m ) ~l~(s-m~) provided that Re(mbk+H)+m#I/p-m%
(3.7)
(~=O,I,2,...).
Since T is a continuous linear mapping from F p,~ into F p,~-m (Theorem 3.1), we might expect that T e would be a continuous mapping from Fp,~ into Fp,~-m~ under appropriate conditions and this is indeed so.
Under the hypotheses of Definition 3.3,
the mapping properties can be deduced from the work of Rooney in [8], [9]. However, as hinted
above, the range of possible values of p can be extended from 14p~2 to z
l~p~.
One way to achieve this is to express T ~ in terms of the so-called Erdelyi-
Kober operators which we will discuss briefly. §4.
t
For concrete representation of the Erdelyi-Kober operator Imn'Y
in easy stages.
•
we proceed
With m real and positive and ~EFp,~ throughout, we first consider
complex numbers q and y such that Re(mq+~)+m>I/p and Re ~>O. (i~,Y~)(x)=[F(y)]-II~ (l_tm)y-itmn~(xt)d(tm) where d(tm)--mtm-ldt.
We define Iq'Y~m by
(O<x<~)
(4.1)
The definition is then extended to values of y with Re y~O
by means of the formula iq,Y~=(n+y+l)i~,Y+l¢+m-Iiq,Y+l~ m m m where (~¢)(x)=xd¢/dx.
(4.2)
Finally, if Re(mq+~)+m
we let k be the unique positive integer such that I/p-mk
(4.3) The
justification for the various formulae above can be found in [6]. From the original results of Kober ~] and from Definition 3.1 above, we find
490
that the operator Imq'Y is a continuous linear mapping from F p,~ into itself provided that l ~ p ~ and Re(mn+~)+m#I/p-m% (~=0,I,2,...).
Further, when l~p~2 we
have, from [8J, (]~k(l~,y~(s)=
F(q+l-s/m) r(n+y+l-s/m) (Tg~)(s)
where ~EFp,~ and Re s=I/p-Re ~.
(4.4)
It follows easily from (3.7) and (4.4) that under
these conditions, n ibk,-~ (T~) (x)--mn~x-m~ ~ ~ (x) k=l m
(4.5)
with ~ any complex number and bk (k=l,...,n) as in (3.4).
Bearing in mind that
Co(O,~) is dense in Fp,~ for every ~ and l~p~= [6, p.18], we see that Kober's results enable us to show that (4.5) holds, not merely for i~p$2 but for I g p ~ (and any ~) provided only that Re(mbk+~)+m#I/p-m%
(%=O,i,2,...) for k=l,...,n.
In the case Re ~
which is to be found in [7]. If m=n-a>O, Re(mbk+~)+m>i/p (k=l,...,n) and Re ~
(T~)(x)__mn~x-m~-mfx ~n,O.tm ~n,n~ ~O where Gn'O is Meijer's G-function. n,n
m bl-~,...,b -~ n )~(t)d(t ) bl''" 'bn
(~eFp,~)
(4.6)
For the case n=2, (4.6) becomes
(T~)(x)=[r(-2~)]-Im2~x-mblf~ (xm-tm)-2~-12Fl(b2-bl-~'-e;-2~;l-xm/tm)
(4.7)
xtmbl+m~+(t)d(tm ) where 2FI denotes the Gauss hypergeometric function. We mention that our results for the case n=2 are in agreement with those of Sprinkhuizen-Kuyper [i0] who made an extensive study of operators of the form D2+~x-ID
(4.8)
corresponding to the values al=-i , a2=2-~ , a3=~-I in our notation.
Operators of the
form (4.8) turn up in many places e.g. in GASPT and many relationships exist between these operators and the Erd~lyi-Kober operators. L =D2+(2q+l)x-iD q then T~'Y~ -~ Tq'Y under appropriate conditions. -2 -q--~+y-2 the operator L
For instance, if we write (4.9) When we use (3.2) in the case of
we obtain m=2 and hence the appearance of the number 2 in the formula
491
is a natural consequence under conditions 2
of our theory.
of great generality,
~mnj =~Ln+y~
We can extend the result by proving,
that
L2
(4.10)
~7] along with other similar results.
Whether or not (4.10) is of any practical use
remains to be seen. §5.
We have not so far dealt with the index law in Theorem 2.3(ii)
of the operator T given by (3.3).
The sight of the G-function
this law might be related to some results merely indicate one aspect of this. analysis,
involving
in the context
in (4.6)
special functions.
suggests that We shall
At this stage we shall return again to formal
bearing in mind that rigour can be supplied by working
in the spaces F P,~
and imposing conditions
on the parameters
In (2.7), let ~=r, a positive
similar to those in §4.
integer.
Then for complex numbers
B, we nave
(rr) 6=T rB .
(5. I)
On the one hand, we can write down an expression n (rr6~) (x)--mnrBx -mrB ~
ibk,-rB
for T rB directly from (4.5), namely,
~ (x).
(5.2)
m
k=l
On the other hand we can examine T r in a similar manner to that in which we examined T.
By stringing out r copies of T, we obtain an expression of the form (3.3) in which
the n+l numbers al,...,an+ I are replaced by a set of rn+l numbers related to them in a simple way. calculation
Similarly b I .... ,bn are replaced by a set of rn numbers and an easy
shows that m has to be replaced by rm.
calculation,
After another completely routine
we find that
n (Tr)B~=(mr)nrBx -mr6 ~
r ~
k=l~=l
(bk-r+~) /r ,-6 I mr
(5.3)
Comparison of (5.2) with (5.3) suggests that r rB
r 1I ~=1
I
(bk-r+~)/r ,-B mr
as operators.
=ibk , r6 m
Equivalently,
In'Y=r -~ m
r ~ ~=i
if we write y=-r6, n=b k, we might expect that
l(q-r+~)/r'~/r mr
We can show that (5.4) is true by using Mellin transforms, the result
(5.4) whereupon we discover
can be thought of as being a version of the classical
formula
that
492
r
F(z+(~-l)/r)=(2~)(r-l)/2rl/2-rZF(rz)
(r=2,3,4 .... )
(5.5)
%=1 for the gamma function due to Gauss and Legendre
[2, p.4].
Indeed, from (4.4), the
Mellin transforms of the two sides of (5.4) are formally equal if and only if F(q+l-s/m) F(q+y+l-s/m)
=r-y ~ F((q+£-s/m)/r) %=1F((q+y+£-s/m)/r)"
However, by (5.5), the right-hand
(5.6)
side can be written as
(2~)(r-l)/2rl/2-n-l+s/mr(q+l_s/m)r-Y (2~)(r-l)/2rl/2-n-Y-l+s/mF(q+y+l_s/m) and (5.6) follows at once so that (5.4) is established formally.
In the special case
r=2, an alternative proof of (5.4) can be obtained from scratch via the formula ~,
p. i12~ 2Fl(a,b;a+b-I/2;z) r 1/2 =(l-z) -I/2 ~/2+(l-z)
7 l-2a /2J 2Fl(2a-l,a-b+I/2;a+b-i/2;
(l-z)I/2-1~ (l_z)I/2+l ~ '
which is not surprising in view of (4.7). (5.4) does not seem to appear in the books on fractional calculus.
Whether it
is new of not, it would be fair to say that it emerges naturally from our second index law (2.8) without the need for complicated manipulations functions.
involving special
It seems that our theory could suggest similar results closely related
to properties of special functions and these we hope to discuss at some future date. References I.
A.V. Balakrishnan,
Fractional powers of closed operators and semigroups
generated by them. 2.
Pacific J. Math. 10(1960), 419-437.
t o A. Erdelyl et al., Higher Transcendental
Functions, Vol. 1 (McGraw-Hill,
New York, 1953). 3.
R. Johnson, Weighted estimates for fractional powers of partial differential operators.
4.
H. Kober, On fractional 11(1940),
5.
Trans. Amer. Math. Soc. 265(1981), integrals and derivatives.
Quart. J. Math.
(Oxford)
193-211.
H. Komatsu, Fractional powers of operators. 285-346.
511-525.
Pacific J. Math. 19(1966),
493
6.
A.C. McBride, Fractional Calculus and Integral Transforms of Generalized Functions.
7.
(:Research Notes in Mathematics No. 31, Pitman, London, 1979).
A.C. McBride, Fractional powers of a class of ordinary differential operators. To appear in :Proc. London Math. Soc.
8.
P.G. Rooney, On the ranges of certain fractional integrals.
Canad. J. Math.
24(1972), 1198-1216. 9.
P.G. Rooney, A technique for studying the boundedness and extendability of certain types of operators.
I0.
Canad. J. Math. 25(1973), 1090-1102.
I.G. Sprinkhuizen-Kuyper, A fractional integral operator corresponding to negative powe~:s of a certain second-order differential operator. J. Math. Anal. Appl. 72(1979), 674-702.
II.
K° Yosida, Functional Analysis.
(Springer-Verlag, Berlin, 1971).
CONVERSE
INITIAL VALUE PROBLEMS
FOR A CLASS
OF HEAT EQUATIONS
Peter A. McCoy Department
of Mathematics
United States Naval Academy Annapolis,
Maryland
21402
INTRODUCTION.
In the classical
(i)
initial value problem for the heat equation
{~t - ~x[(l-x)~+l(l+x)~+l~x]}T(x't)
with parameters
~ ~ B and B ~ - 1/2 or ~ + B > 0, "arbitrary"
~(x,O) is specified
The w e l l - k n o w n
One is by taking the Weierstrass in L p (p ~ 2); another
Specific problems with symmetric Bochner
independent
transform of the Jacobi series expansion
Weierstrass
With ~ = B = (k-3)/2,
the solutions
Moreover,
summability
The connectiol by S.
represent
(-i, +i) for k = -i, and in h i g h e r - d i m e n s i o n a l symmetric
solutions with L p initial data represent Jacobi
[5].
of data
[3-6] as an
(-i, +i) was established
under the Chebyshev map to conically
These problems data.
by function theoretic methods
on the segment
means solve boundary value problems equations
solving eqn (i) for
data.
Poisson processes
of the segment
E k, correspond
~(x,t)
are connected with ~ and its initial values.
[2] when ~ = B > -i.
temperatures
function
solution can be found in a number of ways.
is representation
transform of analytic
initial data
: f(x), -I < x < + 1
and one seeks the temperature
which ~ ÷ f as t ÷ 0 +.
integral
= 0 , -i < x < I , t > 0
related
spaces
temperatures.
Time
series whose Abel
to the GASP and E u l e r - P o i s s o n - D a r b o u x
the initial value p r o b l e m arises in the theory of [I].
are direct;
This paper considers
the solution is determined
from specified
the converse p r o b l e m of identifying
initial
the initial data *
from the solution.
It maintains
that the solution exists uniquely
in the space H
495
of initial data, hyperfunctions the space H
on [-I, +i], by constructing
and the space H of temperature
functions.
equation are identified with their generalized feature characterizes of continuous
between
Thus, solutions of the heat
boundary values.
An essential
the temperature as a series whose terms depend on a sequence
functions
satisfying a growth condition on [~I, +i].
The characterizaticn methods
isomorphisms
combines
function-theoretic
[3-6] and special function
[i] to extend the converse Dirichlet problem for harmonic
disk developed by G. Johnson
[8] to these parabolic
of his work to elliptic partial differential J. Kelingos & P. Staples
[9] and A. Zayed
equations.
equations,
functions in the
For the extensions
refer to Saylor in [7],
[13].
THE INITIAL VALUE PROBLEM. The normalized
Pn
(~,5) (x)/Pn(~'8)(1)
Jacobi polynomials
Rn(X):= R(~,5) n (x) =
for n [ N:= {0,1,2 .... } are orthogonal +i ~i Rn(S)Rm(S)d~5(s)
functions
= a}nl6n '
dz 5(s) = (l-s)~(l+s)~ds
°~n
(2n+~+5+l)F(n+~+B+l)F(n+c+l) F(n+5+I)F(n+I)~(~+I)F(B+I)
They form a set of eigenfunctions corresponding
to
the
eigenvalues
A(~'S)R (x) = InRn(X) x n X = n(n+a+5+l) n
A~ ~ ,5) = which is complete
for
the
, -i < x < 1 operator
~x[(l_x)a+l
in the space X:= LP(-I, +I) L;C(-I, +i) for p ~ 2.
as the Jacobi series
f(x) =
with Fourier-Jacobi
Z ani~inRn(X), -i < x < i n=O
coefEicients
an = an(f):: /l]f(S)Rn(S)d;~5 + (s) , n ~: N
Expand f s X
496
and form the W e i e r s t r a s s m e a n s
[i] on -i < x < i, t > 0 as -~ t
$(x,t) = n ~ 0 a n ~ n ~ n ( X , t ) ,
These m e a n s are W e i e r s t r a s s
~n(X,t):= e
~ n g N
n Rn(X)
.
summable to f [i] and the functions ~n are p a r t i c u l a r
solutions of the heat equation.
We summarize now.
T H E O R E M i. Let f ~ X.
The solution of the initial v a l u e p r o b l e m
[~t - A(~'B)]1~(x't) = 0 , -i < x < i, t > 0 [ ~(x,0) = f(x)
J
, -i < x < 1
is the u n i q u e t e m p e r a t u r e function c~ ~(x,t):=
E an(f)0~n~n(X,t) n=0
.
This reformulates in terms of the W e i e r s t r a s s t r a n s f o r m as ~(x,t) = ~ i K ( x , s , t )
f(s) d ~ 6 ( s )
w i t h the W e i e r s t r a s s - J a c o b i heat kernel -~ t K(x,s,t)
=
E e n= 0 n
n
R (x) Rn(S), -I < x, s < i, t > 0 . n ---
THE CONVERSE TO THE INITIAL VALUE PROBLEM. A structure theorem, functions is given.
analogous to the one found by J o h n s o n for h a r m o n i c
It shows that aspects of the initial v a l u e p r o b l e m h a v e as
antecendents, b o u n d a r y v a l u e problems for elliptic equations, b e g i n n i n g w i t h the w o r k of P o i s s o n - S t i e l t j e s
[7] and Faber
[12].
Apparently,
W e i e r s t r a s s s u m m a b i l i t y and Abel s u m m a b i l i t y correspond. converse p r o b l e m in terms of the u n i f o r m n o r m
llhll
the key roles played by W e next v i e w the
:= sup{lh(x) I: -i < x < I}
497
THEOREM 2. The function P is a temperature function [St - A(~'B)]'~(x't) = 0 , -i < x < i, t > 0 x
if, and only if, there exists a sequence {gn } of continuous functions on [-i, +i] such that lim (2n! I Ignl I) I/n = 0 and
~(x,t)
~ i.A(~,B))n +l x -/i
=
K(x,s,t) gn(S) dp B(s)
n=0 where K is the Weierstrass-Jacobi heat kernel. PROOF : Consider a sequence {gn}~C[-l,
+1] with the specified growth condition.
Expand gn in a Jacobi series gn(S) =
E ak, n Rk(S) , -i < s < 1 , n s N k=0
and form the Weierstrass transforms +i -/i K(x,s,t)gn(S ) d~ B(s ) = for -I < x < i, t > O.
~ ak,n~k~k(x,t) k=0
, n ~ N
If we now set
+i ~n (x't) = (Ax(~'8))n-fl K(x,s,t)gn(S) dp ~(s) it follows that the series 1'(x,t) =
Z ~ (x,t) n n=O
is a formal solution of the heat equation as a direct calculation shows that ~n(X,t) =
~ ak,n~k(-%k )n 9k(X, t) • k=O
It remains to establish uniform convergence of the series ~ by the method of majorants.
The functions ~ are bounded by the series n n -%k t l~n(X,t) I ! C IIgnl I k~ 0 Wk(~ k) e
found by estimating the Fourier-Jacobi coefficients ak, n.
Applying the fact that
~k ~ 0(k2~+l) to the tail, T n, of this series gives T (t) _< C llgnl I E k 2~+I [k(k+a+$+l)] n e-Xkt n k=n Working with the larger term of this inequality shows that
498
Tn(t) ~ B 4n ]Ignl I where 0 < a(t) < i.
~ k[2~]+2n+l ak n< _B 4 k=n
For sufficiently
]Ign] I ([2~]+2n+I)!
large B, the estimate applies termwise and
majorizes the series ~(x,t) _< B
Thus, it is (subuniformly)
~ 4 n I Ignl I ([2e]+2n+l)!. n=O
uniformly convergent on compacta of [-l,l]X(t > 0)
because limsup (4n IIgn] I ([2~]+2n+i)!) I/n = D limsup (2n!IIgn[]) I/n = 0 . And, ~ is Weierstrass
summable to the initial data.
Having proved sufficiency,
the next step reverses the argument.
Set ~(x,t) =
E an~n@n(X,t), n=0
limsup fan II/n j 1 .
Given ~ > 0, reasoning as in (8,]3) quid pro quo finds a constant B(c) and sequences {ak, n} such that k B(~)a[ 2~]+2n+I k 2n a k = E ak, n, lak,n[ J , 0 < k < n . n=O ([2~]+2n+i)! Define
n (x,t) = ~n ~(-i) ak,n_[2~]_l~k(x,t )(xk)-n n k>_n-[2~]-i
n ~ N
The next estimate shows the series to be subuniformly convergent. 2n+l B(C)~n~
k2(n-[2~]-l)e-~k t -n ~: Xk ! k>n-[2e]-i
l~n(X't) l J 2n! B(~)~ s 2n+l n
E k -2, -i < x < i, t > 0 . I~=i
2n! Consequently,
~>n(X,O):= gn(X) ~ C[-I, +1] for n c N, and
lim sup (2n!IIgnIl) I/n _< g .
From the earlier representation and
(&~'$))n~n(X't)
= ~n E a~,n_[2e]_l k>n-[2~]-I
@k(x,t)
all that remains is to check the sum (A~'B)) n ~n(X,t) = n~[2~]+l
~0n n~[2~]+l
E ak,n_[2a]_l ~k(x,t) k>_n-[2~]-i
Finally, we obtain the form that completes the proof, ~(x,t) =
E an~n~n(X,t), n=O
-i < x < i, t > 0 .
•
499
THE INITIAL VALUE P R O B L E M V I A HYPERFUNCTIONS. W e n o w identify the space of h y p e r f u n c t i o n s as the space of initial values c o r r e s p o n d i n g to temperatures.
The space H
of h y p e r f u n c t i o n s on [-i, +i] is the
strong dual of the space H of analytic functions on [-i, +i] w i t h the topology in [i0].
The term g e n e r a l i z e d function refers only to Schwartz d i s t r i b u t i o n s so that
each g e n e r a l i z e d f u n c t i o n is a hyperfunction.
To c h a r a c t e r i z e h y p e r f u n c t i o n s
in a
w a y c o m p a r a b l e w i t h W e i e r s t r a s s summability, we derive a r e p r e s e n t a t i o n of elements in H
generalizing
[13].
T H E O R E M 3. For i n t e g e r
parameters
a > B > O, t h e
Jaoobi
series
to a h y p e r f u n c t i o n on [-i, +i] if, and only if, lim sup
f:=
2 a w R nnn n=O
converges
]an ]I/n _< 1 .
PROOF: First, c o n s i d e r an analytic function ~ a H.
n
/P (~'B)(1) ~ O(n2~+l), n
B e c a u s e of the r e l a t i o n
the expansion t h e o r e m [ii] applies,
#(x) =
, iLm sup
Z bn~nRn(X) n=O
]bn[1/n
and
< I
converges to a function in H w h o s e m - t h partial sum is d e s i g n a t e d as ~ . m Turning
to (f'
If f h a s
a Jacobi
f
a H , an
product
is
defined
+i (l-s)~(l+s)B~):= -/i f(s)#<s) series (},
However,
inner
expansion,
by
(l-s)~(l+s)Bds,
~ > B > 0 .
then
(l-s)e(l+s) B %m ) ÷
(?,
(1-s)e(l+s) $ }), m ÷ ~ .
the series are o r t h o g o n a l and we infer from the c o n v e r g e n c e m E a n b n w n ÷ E anbn~ n , m + n=O n=O
that lim sup
lanbnWnllln _< i.
Conversely,
given
Simple reasoning gives lim sup
a sequence
for
which
m f (x):= E an~nRn(X) m n= 0 To show that f
m
the
limit
fan[lln _< 1
condition
is
, -i < x < i, m ~ N
met,
.
define
.
converges to a h y p e r f u n c t i o n i!, it suffices that the sequence
{(fm,~)} converges for each ~ a H.
This follows as H and H
are F r e c h e t - M o n t e l
500
s p a c e s and all M o n t e l
spaces
are r e f l e x i v e .
Thus, w e a k and s t r o n g c o n v e r g e n c e
coincide. Consider
the f u n c t i o n
~ g H as a J a c o b i
((l-s)~(l+s)Bfm,~)
exists since
lim sup
as is f b e c a u s e
=
T h e limit
~ anbni~ n + E a n b n ~n n=O n=O
l a n b n ~ n I 1 / n < 1.
~ and
series.
Therefore,
B are i n t e g e r s .
This
, m ÷
(1-s)C~(l+s) ~ f is a hyperfuncti(
completes
the proof.
* The c h a r a c t e r i z a t i o n
~alue set
just
of temperatures.
convolution
mapping
*: H
derived
shows that
To p r o c e e d X H
÷ H
by the H a d a m a r d ~
a b ~ nnn
n=O {H , + , * }
is
a C-ring
with
corresponds
to the initial
(f * g ) ( x ) : = the s p a c e
H
identity
value
to the initial-
problem,
define
product R
n
(x)
.
element
co j(x)
=
~
~n Rn(X)
"
n=O the W e i e r s t r a s s
means yield
elements Jt(x)
for w h i c h
Jt(x) ÷ j(x)
of H,
= K(x,l,t),
g H * as t ÷ 0 +"
t > 0
The i d e n t i f i c a t i o n
theorem
follows.
r H E O R E M 4. The function
~ is a t e m p e r a t u r e
function
[~t - A(~'B)]'~(x't) x if, and o n l y if, a u n i q u e h y p e r f u n c t i o n ~(x,t) &nd, ~(x,t) ÷ f(x)
s H • as t ÷ 0 +
= 0 , -i < x < 1 , t > 0 f exists
= (Jt * f)(x)
on [-i, +i]
s u c h that
.
.
PROOF: Let l i m sup
~ be a temperature fan ll/n _< i.
function
and c o n s i d e r
its J a c o b i
expansion
with
Set oo f(x) =
t h e n ~(x,t)
= (jt*f)(x).
~eierstrass
summability
E an~ n R n ( X ) , -i < x < 1 . n=0
The argument gives
of an e a r l i e r
~ + f e H * as t ÷ 0 +"
theorem, And,
together with
if f s H * t h e n
the
501 m
anW n Rn(X),
lim sup
lanII/n_< 1
n=0
so that ~ = (f * jt ) is a temperature
function.
The proof is complete.
CONCLUDING REMARKS. The roles of the parameters
(a,B) may be reversed because of the symmetries
R(C*, B) (_x) = R(6'CO(x), Several corollaries provided
-x
x
may be derived for polynomial
the growth conditions
to a broader class
A(c~,6) = A(8,a)
n
n
are satisfied.
sequences,
or Radon measures,
Also, the preceeding
results extend
of heat equations by means of generalized Weierstrass
bility and fractional
differential
operators as defined
summa-
in [i].
REFERENCES [i]
H. Bavinck,
Jacobi series and Approximation, Centrum, Amsterdam, 1972.
[2]
S. Bochner,
Strum-Liouville and Heat Equations whose Eigenfunctions are Ultraspherical Polynomials or Associated Bessel Functions, Proc. Conf. on Diff. Eqns., ed. J. B. Diaz & L. E. Payne, Univ. Maryland, 1956.
[3]
D. L. Colton, Partial Differential Equal:ions in the Complex Domain, Research Notes in Math., vol. 4, Pitman Publ., San Francisco, 1976.
[4]
D. L. Colton, Solutions of Boundary Value Problems by the Method of Integral Operators, Research Notes in Math., Pitman Publ. Co., San Francisco, 1976.
[5]
R. P. Gilbert, Function Theoretic Methods in Partial Differential Equations, Math. in Science and Engineering, vol. 54, Academic Press, New York, 1969.
[6]
R. P. Gilbert, Constructive Methods for Elliptic Equations, Lecture Notes in Math., vol. 365, Springer-Verlag, New York, 1974.
[7]
K. Hoffman,
Banach Spaces of Analytic Functions, Cliffs, N. J., 1962.
[8]
G. Johnson,
Jr., Harmonic Functions in the Unit Disk, Math., vol. 12 (1968) 366-396.
[9]
J. Kelingos & P. Staples, A Characterization of Solutions to a Perturbed LaPlace Equation, Illinois J. Math., vol. 22 (1978) 208-216.
[I0]
G. Kothe, Topological
[ii]
G. Szego, Orthogonal Polynomials, Providence, 1939.
Math. Centre Tracts, Math.
Prentice-Hall,
Vector Spaces I, Springer-Verlag, Amer. Math.
Englewood
I&II., Illinois J.
New York, 1969.
Soc. Publ., vol. 23,
502
[12]
M. Tsuji, Potential Theory in Modern Function Theory, Maruzen Publ. Co., Tokyo, 1959.
[13]
A. I. Zayed, Hyperfunctions as Boundary Values of Generalized Axially Symmetric Potentials, Illinois J. Math., vol. 25 (1981) 306-317
Higher
Order
Inverse
Joyce
Eigenvalue
Problems
R. M c L a u g h l i n
ABSTRACT The p r o b l e m
to be d i s c u s s e d
model
for a g i v e n
value
problem
physical
of t h e
is as f o l l o w s .
problem
(iw(1)) (1) + B w
4
e
eigenvalues
which
- lw : 0
w
A, B a n d p o s s i b l y
for f i n d i n g
eigen-
4
i=l
but
in a s e l f - a d j o i n t
form
w (4) +
Suppose
results
Suppose amathematica
li,
eij'
@ij'
i = 1,2,...
the unknown
can be required
i = 1 ..... 4, j : 1,2
are k n o w n .
coefficients
known
are u n k n o w n
A constructive
is p r e s e n t e d .
are t w o n o r m a l i z a t i o n
technigue
Additional constants
data
for e a c h
of the eiaenfunctions.
Introduction Suppose
we consider
a set of p o s i t i v e asymptotic once).
the
numbers
seek
i.e.
coefficients
1,2,
j = 1,2,3,4
...
13 <
any
given
number
a fourth
order
self-adjoint
so t h a t
+
the
at m o s t
eigenvalue numbers
sequence
{li},
(Ay(1)) (I) + B y
-ly = 0,
0 < s < i,
given
a particular
can b e r e p e a t e d
given
problem,
Mij, Nij,_ i =
i = 1,2 ....
is t h e
for
--
--
4
4
F M y(j-l) (0) = 0, J = i ~i~interested
coefficients
We are
(satisfying
(1)
W e are
problem.
A(s) , B(s) , 0 _< s < i, and r e a l
set of e i g e n v a l u e s y(4)
inverse
l I <_ 12 <
form and where
Then we
following
in d e v e l o p i n g
as w e l l
(normalization
~ N
j:l
.y(j-l)(i)
a method
as d e t e r m i n i n g
constants?)
=
0,
i :
1,2.
i]
what
as~ociatea
of c o n s t r u c t i n g other
the u n k n o w n
spectral-type
~,ith t h e t o b e a e r i v e d
data
504
self-adjoint p r o b l e m can
(or must)
he u t i l i z e d to d e t e r m i n e a s o l u t i o ~
Before c o n t i n u i n g with this fourth order p r o b l e m it may provide some p e r s p e c t i v e to review w h a t is known for second order self-adjoint inverse e i g e n v a l u e problems.
The second order p r o b l e m has been
stu'died more e x t e n s i v e l y than the fourth order problem.
In the second
order case the s e l f - a d j o i n t e i g e n v a l u e p r o b l e m to be c o n s i d e r e d is y(2)
(l-q)y = 0, 0 < s < 1 --2 2 jEI~jy(j-I)= (0) = 0, ]ElSjy(j-l)= (i) = 0.
(2)
+
The sequence of distinct numbers 11 < 12 < ... s a t i s f y i n g a certain a s y m p t o t i c form is given and the coefficients q(s),
~j, 8j, j = 1,2
are sought so that the given sequence of l.'s contains all the eigen1 values for
(2).
done by Borg
Historically,
[4], M a r c e n k o
G e l ' f a n d and Levitan
[6].
extensive work in the p r o b l e m has been
[15], Krein
[9,10], L e v i n s o n
It is known that additional
can be u t i l i z e d to d e t e r m i n e the coefficients. shown by B o r g solution,
[4] that additional
[9],
[i0],
related) 1 s = [,
it was first
are to be u n i q u e l y determined.
[15] where b o u n d a r y conditions are different
to those in [7],
[8].
(2).
p o s i t i v e sequence
Or, alternatively,
(but
unique solutions have been [6], w h e n k n o w l e d g e of a
pl,P2,..., which are shown to be n o r m a l i z a t i o n con-
is also assumed.
The e x i s t e n c e t h e o r e m o b t a i n e d by a con-
structive process by G e l ' l a n d and Levitan,
[6], can also be applied
to show e x i s t e n c e when k n o w l e d g e of two sequences of e i g e n v a l u e s assumed.
This is done by a p p l y i n g the results of Levitan
which state that given the two sequences of e i g e n v a l u e s in [i0],
[12],
[4],
One could assume that q is symmetric about
shown to exist, by G e l ' l a n d and Levitan
stants,
In fact,
i n f o r m a t i o n could be a second set of e i g e n v a l u e s
[12],
[4],
information
i n f o r m a t i o n must be given if the
that is the coefficients,
Such additional
[12], and
is
[13],
[14]
[4],
[9],
[15], the n o r m a l i z a t i o n constants a s s o c i a t e d with either
sequence of e i g e n v a l u e s may be constructed.
505
Results for the fourth order inverse e i g e n v a l u e p r o b l e m have been o b t a i n e d by B a r c i l o n author
[17],
[18],
[2],
[19].
[3], M c K e n n a
[ii], and the
B a r c i l o n follows the approach of M. Krein
a s s u m i n g k n o w l e d g e of three distinct a s s o c i a t e d b o u n d a r y conditions. when it is known,
[16], L e i b e n z o n
a priori,
sequences of e i g e n v a l u e s
A constructive technique
and
is given
that the given sequences are e i g e n v a l u e s
for eigenvalue problems which contain the given c o r r e s p o n d i n g sets of b o u n d a r y conditions.
L e i b e n z o n also proves a uniqueness t h e o r e m when
three given sequences of e i g e n v a l u e s and three given c o r r e s p o n d i n g sequences of n o r m a l i z a t i o n constants are given for e i g e n v a l u e p r o b l e m s w i t h related b o u n d a r y conditions. The author,
in
[17],
[18], and
[19] has d e v e l o p e d a m e t h o d for
finding the solution of a w e l l - p o s e d inverse e i g e n v a l u e problem.
It
is assumed that all eigenvalues are simple and that two positive sequences Ii < 12 < "'" and pl,P2.., asymptotic
forms.
are given and satisfy certain
A c o n s t r u c t i v e technique
finding c o e f f i c i e n t s
is then d e v e l o p e d for
in terms of the e i g e n v a l u e s and L 2 norms of the
corresponding eigenfunctions
for a known e i g e n v a l u e problem,
in terms
of solutions of the d i f f e r e n t i a l e q u a t i o n in the k n o w n e i g e n v a l u e problem,
and in terms of the given p o s i t i v e numbers.
The given
sequences of Ii. 's and Pi ! s are the e i g e n v a l u e s and L 2 norms of the c o r r e s p o n d i n g e i g e n f u n c t i o n s of the d e r i v e d problem. c o e f f i c i e n t s are unique
in the sense that they define u n i q u e l y an
e i g e n v a l u e p r o b l e m w h i c h has eigenvalues, zation constants,
The derived
Pi' i = 1,2,...
li' i = 1,2,...
and normali-
and for which an a s s o c i a t e d non-
s e l f - a d j o i n t p r o b l e m has spectral data identical with that of a known non-self-adjoint eigenvalue problem
(same d i f f e r e n t i a l e q u a t i o n as in
known self-adjoint e i g e n v a l u e problem). e x p l a i n e d in Section i.
This t e c h n i q u e will be
It should further be n o t e d that in addition
to the results m e n t i o n e d above there is a c o n t i n u i t y result, This result gives a b o u n d on the derived c o e f f i c i e n t s
[19].
in terms of the
506
differences constants
of the e i g e n v a l u e s
for the d e r i v e d
In this ideas the
paper we
in a more
"new"
concepts
eigenvalue
values data
eigenvalue
associated
with
may be given.
numbers the
should h e l p m o t i v a t e
I i,
Pi'
of the p r o p e r t y
be used
a multitude
that
about
indicates
of
i = 1,2, . . . . all
the
problem mentioned
eigen-
spectral
above.
The
that more "spectral"
in d e t e r m i n i n g
In order
the d e r i v e d
(with unknown
The kernel
kernel
K)
equation
q, 8j,
interpreted
~j,
constant)
eigenvalue
to be given
problem
technique
will
satisfy
required
problem
is given.
in the
asymptotic
the v a l u e
A known
and solutions problem.
of K.
for the d e r i v e d
with L 2 n o r m equal constants,
of the e i g e n f u n c t i o n
of
and the u n k n o w n
in terms
a set of n o r m a l i z a t i o n
Two
of the
eigenvalue
sequences
j = 1,2 are d e t e r m i n e d
eigenfunetions
con-
relationship
solutions
problem
derived
by the two given
forms.
An integral
eigenvalue
(to be)
seem natural,
of G e l ' f a n d - L e v i t a n .
is the set of e i g e n v a l u e s
sequence,
to y i e l d
an additional
the c o n s t r u c t i v e
of the work
in the k n o w n
corresponding
then the other
work,
self-adjoint
is shown to exist b e t w e e n
equation
sequences
If we select
derived
description
K is d e t e r m i n e d
coefficients
and w h e r e
of the inverse
type of n o r m a l i z a t i o n
the a s s u m p t i o n s
eigenvalue
the d i f f e r e n t i a l
solutions
of that of G e l ' f a n d - L e v i t a n .
are given w h i c h
(base problem)
eigenvalues
(another
(to be)
to make
following
differential
of c o n s t r u c t i n g
As in p r e v i o u s
be a g e n e r a l i z a t i o n
the given
the above m e n t i o n e d
in the sense
can be o b t a i n e d
allows m u l t i p l e
of real
sequences
is limited
non-self-adjoint
we seek a m e t h o d
problem which
the
same data
problems.
problem.
Thus,
sequence
have the
(or should)
eigenvalue
shows how to instruct
and b e c a u s e
solutions could
i) present
This p r e s e n t a t i o n
of solutions
for an a s s o c i a t e d
data p o s s i b l y
sider
which
must be simple
fact that many
way.
of the n o r m a l i z a t i o n
self-adjoint
(Section
and b a s i c a l l y
problems
This c o l l e c t i o n
and known
first
general
and d i f f e r e n c e s
One of problem. to i,
can be
(or its derivative)
507
at one
e n d point,
boundary a full the val,
conditions
set of
function
and
that
constants
illustrated assumed
three
papers
form
such seek
Yi' that
the
given
y(4)
of the
of
inter-
assumed
normalization
l] ~ 12 ~
number
so that
+
...
they
and,
will
be
problem.
It is
satisfies
a particula:
satisfy
at m o s t
a given
~i > 0,
and B(s)
eigenvalue
once.
asymptotic
i = 1,2 . . . . .
E CI[0,1]
and real
problem
(Ay(1)) (i) + By y(1)
be g i v e n
can b e r e p e a t e d
if Yi = 0 then
the
will
of s o l u t i o n
special
A(s) ( C3[0,I]
that
and
define
(the v a l u e
it is not
numbers
the m e t h o d
following
sequence
7i ~ 0 and
such
eigenvalues
of real
author,
a given
coefficients
a,b,c
are
derived
completely
eigenfunction
that
the
problem.
by the
~i are
that
at ene of the e n d p o i n t s
sequences
and that
constants
for e a c h
sequences
the p o s i t i v e
sequences
constants
given
to n o t e
a l s o be e m p h a s i z e d
by c o n s i d e r i n g
asymptotic
We w i l l
conditions
It s h o u l d
paper
that
f o r m and
the n o r m a l i z a t i o n
an e i g e n v a l u e
as in p r e v i o u s
It is i m p o r t a n t
its d e r i v a t i v e )
the
for
In this
plus
initial
say x = 0.
a priori
The
say x = O.
= y(1)(1)
- ly = 0 : 0
(3) y(2) (0) + ay(1)(O) y(3) (0) + has
eigenvalues
associated
with
li'
by(O)
(b+A(0))y(1)
i = 1,2, . . . .
I i and h a v i n g
And
L 2 norm
= 0
(0) + c y ( 0 ) =
if Yl. is the e i g e n f u n c t i o n 1 i, t h e n Yl. has n o r m a l i z a t i o n
i
constants
yl
with
1 assigned
the
this
inverse
(0) = Yi and yl data
eigenvalue
1
(0) = ~i"
1 for the problem
inverse
of the
data
eigenvalue
required
problem.
It s h o u l d
eigenfunction, is d e t e r m i n e d
[ Y l (j) (0), j = 0,1,2,3] of e a c h 1 this sense the a s s i g n m e n t of the alization
0
once all
eigenfunction given
for the
data
be o b s e r v e d
of t h e
are
seems
solution
the
solution initial
determined. a "natural"
of t h e
that
second
to values In
generorder
508
The
result
uniqueness given.
result.
It will
K) e x i s t s
and
assumption
will
addition,
of the
an i n t e g r a l
of the
of a k n o w n
2 is an e x i s t e n c e solution
relationship
(to be)
derived
differential
and
will
be
(with k e r n e l
differential
equation.
Under
this
are unique.
will
that
in S e c t i o n
of c o n s t r u c t i o n
be no a p r i o r i
it w i l l
sense
stated
that
solutions
constants
sequences
be
assumed
solutions
normalization
in the
be
solutions
There
will
A method
between
equation
These
which
and e i g e n v a l u e s
only
be
assumption
be r e q u i r e d
shown
that
some
the c o n s t r u c t i v e
that
the
given
sequences
for an e i g e n v a l u e to
satisfy
sets
technique
problem.
asymptotic
of s e q u e n c e s cannot
be
are
forms.
In
are not
allowed
applied
to find
a solution.
Section
1
In this
section
ing s o l u t i o n s eigenvalue 1,2,...
as g i v e n
problems
are
we w i l l in
[17],
where
given.
The
are
eigenvalues
solution
eigenfunction, problem and has occur be
[19]
technique
of f o u r t h
0 < I 1 < 1 2 < ...
will
consist
order and
for
find-
inverse Pi > 0, i =
of c o e f f i c i e n t s
and real
constants
a,b,c
such
(3) and each
Pi is the
L 2 norm
that
of the
the i th
, n o r m a l i z e d so that yl. (0) = i. The e i g e n v a l u e 1 1 (3) is d i f f e r e n t f r o m the e x a m p l e c h o s e n in [17], [18], [19]
been
chosen
so as to i l l u s t r a t e
in n o n - D i r i c h l e t
(ll' 's a n d
will
Pi'S,
self-adjoint
constants
We b e g i n z (4)
has
as a g i v e n
I z =
a known O,
the
product
and
same
changes
The
eigenvalue
as g i v e n
exactly
(the L 2 inner
possible
conditions.
unique
i = 1,2,3,...)
with -
boundary
be the
problem
eigenfunctions)
(4)
iterative
yl
constructed
tion
for
an
[18],
sequences
A(s) ~ C3[0,i] , B(s) ( C 1 [0,i] li's
describe
solution
problem for w h i c h
eigenvalues
of the
non-self-adjoint
(base)
self-adjoint
=
z ( 2 ) (0)
=
z(1)
has
will data
a related
non-
and n o r m a l i z a and
eigenvalue
eigenvalue :
can
which
which
eigenfunction
(base)
z ( 3 ) (0)
which
z ( 1 ) (i)
adjoint problem.
problem =
O.
509
The
eigenvalues
the
ith
for
this
eigenfunction
problem
with
are
zl~(O)
.
= 1 then
zx~ is Pi' i = 1,2 . . . . . Let l solution of z (4) - 1 z = 0 which
value
problem
(4A)
z(0) = i,
01'
the
state
a result
coefficients,
sequences
0 ~
of
if
the
zl* is 1 L 2 norm
satisfies
let
zl, 8 d e n o t e
the
initial,
the
boundary
will
show
how
a n , b n,
I n < An+ 1 < In+ 2
<...< are
and
(for e a c h
the
eigenvalues
and
8)
c n such
and
...
and
fixed
normalization
for
(5)
+
(Any(l)) (i)
= y (2) (0)
UlY
+
y(1) (for
initial,
a given
boundary
y(4)
+
n, value
(0)
= 0 + CnY(0)
=
0
= 1
= y(1) (i)
and
ly = 0
- bnY(0)
(bn+An(0))y(1) y(0)
case
+ BnY_
+ any(l)(0)
~2 y = y (3) (0)
of t h e
and
square
and
An(S,B) , Bn(S,8)
Pn+2'''"
Y (4)
In e a c h
B ~ ~,
alse
which
0 < I 1 < 12
P 2 ' ' ' ' ' 0 n ' 0n+l'
constants
the
< ...
z (2) (0) = 0 = z (3) (0) , ces ~ z(1) + sin 8 z (I) (i) = 0.
first
to c o n s t r u c t that
< 12
1
of
We will
0 < 11
fixed
: 0
8)
n
solutions
yl,B(x),
for
I > 0,
and
An ,
problem
(Any(1)) (i)
+ Bny
_ ly = 0,
(5A)
= 0
u ly = 0,
= i, c o s B y(1)
+ sin B y
i
y(0) are
shown
to be
related
yl,8(x)
to = zl
Bn,
K n is d e t e r m i n e d
an,
state
bn,
a result
asymptotic each
c n are
B(x)
+ Ix K n ( x , t , 8 ) z l , 8 ( t ) d t .
8 we
forms will
then
have
0
as a s o l u t i o n
determined
showing
= 0
zl, 8 b y
'
where
(i)
that
of
in t e r m s if t h e
integral
equation
of K n ( x , t , 8 ) .
11 's a n d
Kn,An,Bn,an,bn,C
constructed
an
Pi'S
n converge
a soltution
to
the
We w i l l
satisfy to
certain
K,A,B,a,b,c
inverse
then
and for
problem.
510
The sense possible
in w h i c h
each
extensions
Before
solution
is u n i q u e will
of this r e s u l t
s t a t i n g the t h e o r e m s
also be d i s c u s s e d
and
for ~ < 8 < z w i l l be given.
we i n t r o d u c e
the
following notation.
Let n
(7)
fn,B (s,t)
:
(s)z I
zI
i,5
E
(t)
zl*
i,5
i:l
i,5
(t) (s) zl* i,B
Pi
Pi
and let zl (8)
fB(s,t)
=
Z
(s)z i,8
i=l Further
observe
zl* (s) zl* (t) i,@ * i,$
Pi
Pi
zl, (s) is i n d e p e n d e n t of B and is the e i g e n 1,$ c o r r e s p o n d i n g to I~ for (4). A c c o r d i n g l y we w i l l w r i t e
function, zl*
(t) li,B
that
1
(s) : zl*(s).
i,~
Then,
we h a v e
the
following
theorems•
i
Theorem
i:
Suppose
Kn(S,t,5)
is c o n t i n u o u s
for 0 j t j s j 1 and
fixed
B, 0 < 6 < ~. Then {y~ }n U {Yl S , as d e f i n e d in (6), --- z Ai,B i=l i,8 i=n+l is a c o m p l e t e o r t h o g o n a l set in L2(0,1) w i t h n o r m a l i z a t i o n c o n s t a n t s PI'P2 ' • "''Pn ,0" ~ n+l
* Pn+2'''"
iff Kn(s,t,~)
is the u n i q u e
s o l u t i o n of the
integral
equation
(9)
f n , 6 ( s , t ) + ix0 Kn(S,U,B) fn,6 (t,u)du + Kn(S,U,5)
In a d d i t i o n ,
Also,
the r e s u l t a n t
n
(s,t,B)
tl,t2,...,tn,tn+l,tn+2,..,
eigenvalues
is a n a l y t i c
in s and t, 0 < t < s < L
and p l , . . . , p n , P n + l , P n + 2 , . . ,
and n o r m a l i z a t i o n
eigenfunctions
(iO)
K
= 0.
constants
in
are the
(5) w i t h c o r r e s p o n d i n g
y 1 2 '5 , i = i ..... n , y l ni,B , i : n + i, n + 2 ....
An(s,B)
= - 4 £
Kn(S,S,B)
sn(s,B)
: _ An(s,B)
~
Kn(S,t,@)It=s
+ 2 !(~2 ~t'~ 2 I< - - Kn) I ds n ~t 2 t=s
d3 -
2
- -
ds 3
K n (s
s)
iff
511 an : _ K n ( 0 , 0 , 8
)=
~ i=l
1 _ l
n
(ll)
bn=
[2Kn,s+Kn,t-
Cn = -
zl.
(Kn)2] I ==3 Y~ t=s=0 i:l L
0)
zI
(0)
i,$ Pi
Pi
+ (an)2
I- 3Kn, sKn - 2Kn,tKn + 3Kn,ss + 3Kn,st + K n ,tt ] I t=s=0
zx'*(°) zx
n
_ k[
: + i0 a n i=l Remark:
The k e r n e l
Kn(S,t,8)
'
Pi
i,8
-3
Pi
in the
(zx 0)) 2
n __Z
(Zxt(O)) 2
.
1
J
i 1
integral
can be d e t e r m i n e d
homogeneous
(0)
equation
Pi
has
simply by s o l v i n g
finite
rank
and thus
a set of linear,
non-
U s i n g this notion it can be shown that n n n n Yl* (s)zl* (t) Yl (s)zl. (t) I
(12)
Remark:
equations.
It can be shown
that
we c a n n o t
have b o t h An(S,8l)
81 ~ 8 2 .
This can be seen
for each
inner p r o d u c t eigenfunction]
from the Yli,~l
8 the e i g e n v a l u e s
of an e i g e n f u n c t i o n are the
sequences
li,Pi,i
-= An(S,$ 2) and Bn(S,8 I)
s = 0 for the e i g e n f u n c t i o n s Further,
for fixed
same
fact: that the
and n o r m a l i z a t i o n
with
rv!
(0) : z
cos B z(0)
!
(0) , z (0) = 1
+ sin 8 z(1) (i) = 0
and y(4)
+
(Any(l)) (i) + BnY
_ ly = 0
!
y(0)
= 0, y
(0) = 1
(14) ~y
n
= U2Y = 0 I
cosSy(0)
+ sinSy
constants
its c o r r e s p o n d i n g
for the two p r o b l e m s
= 0 = z
values
at
and Yl. are not the same. z,~ 2
I!
z(0)
-- Bn(S,8 2) for
initial
z (4) - Iz : 0 (13)
= 1,2 .... n
(0) = 0
adjoint
[L 2
512
iv!
where
the
adjoint
eigenfunctions
and U2Y = 1 r e s p e c t i v e l y . can
show
that
the
data
the n o n - s e l f - a d j o i n t uniquely mined
determine
uniquely
Remark: observing
that
is the o n l y values
problem
Remark:
The t h e o r e m
plus
z
(0) : 1
in L e i b e n z o n eigenvalues]
self-adjoint
we from
problem together
the coefficients
by an
the u n i q u e n e s s
problem
problem
and normalization of
Hence
satisfy
developed
constants
f r o m the
characterize
self-adjoint
to t h o s e
(14)
are d e t e r -
sense.
the e i g e n v a l u e
(4A)
and
and
techniques
A n , B n , a n , b n , C n.
also
(13)
[normalization
in this
We can
Using
for
coefficients
of the t e r m
constants, integral
with
where
to
(3) w i t h
solutions
relationship
m a y be e x t e n d e d
of the
solution
An,Bn,an,bn,C n the
of
given
eigen-
(5A) are r e l a t e d
as d e s c r i b e d
include
by
by
the values
(6).
of
B,~= < B < z for w h i c h
zl. can be d e f i n e d , i = i, .... n, and for w h i c h l,B the set of f u n c t i o n s J = {zl. }n U {zl, } are d e f i n e d and c o m l,B i=l i i=n+l p l e t e in L2(0,1). The e x c e p t i o n a l c a s e s t h e n i n c l u d e t h o s e v a l u e s of I 1• 's , i = I, ... ,n is an e i g e n v a l u e
for w h i c h one of the self-adjoint particular
problem value
of
(13) li"
w~
for the
non-
for then The
zl. c a n n o t be d e f i n e d for that 1,8 r e m a i n i n g e x c e p t i o n a l c a s e s are t h o s e for
lw!
which
z l ~ ( 1 ) c o s B + z 1 * ( 1 ) s i n B = 0 for some i = i, .... n. In this 1 1 c a s e the inner p r o d u c t (z1.,zl,) = 0 for j = 1 ..... n and t h e g i v e n 3 i and h e n c e the set of f u n c t i o n s J is not c o m p l e t e . The derived have
next
in T h e o r e m
the
norms
theorem
given
of the
Pi
i = 1,2,...
1
Pi
_
Let
$ be
i, +R±~ w i t h
Pi
Pi'S
corresponding
2:
sufficient
1 to c o n v e r g e
11.'s and
Theorem
> O,
gives
Z
i=l
the e i g e n v a l u e s
eigenfunctions
(li)~
for the
as n ÷ ~ to e i g e n v a l u e
as all
fixed with
satisfy k oa
conditions
0 <
of the
~ <_ 2"
Let
and
derived
solutions
problems all
the
which L2
problem.
0 < 11 < 12 < ... a n d
+ Pi and k co
(li)4-1Pil < ~,
Z (li)41Ri I < ~ ,
i=l
i
k: 0,1,2,3,4.
513
Then,
there
exists
(15)
a unique
f(s,t,~)
w i t h the p r o p e r t y
solution
K(s,t,f{) ~ C4[0
+ Is f(t,u,~)K(s,u,B)du 0
+ K(s,t,8)
= 0
that
~j+k lim ~ Kn(S,t,B) n÷~ ~s3~t k j,k = 0 : i : 2 , 3 , 4 ,
< t < s < i] of
0 < j + k < 4.
~j+k
:<(s,t,B)
~s3~t k
Further
i = i~2,...,
Yl
as definerS.
i,B by
(16)
Yl
(s) = z I i,8
forms a c o m p l e t e , pl,02,..,
orthogonal
iff K(s,t)
The sequences zation constants 1,2 ....
i,B
iff A(s),
set in L 2(0,I) with n o r m a l i z a t i o n c o n s t a n t s
satisfies
11,12,...
for
(s) + I s K ( s , t , B ) z I (t)dt 0 i,8
and
(3) w i t h
B(s),
a,b,c
=
-
are e i g e n v a l u e s eigenfunctions
are d e f i n e d by =- 4 d~K(s,s)
2
+ a
c = 10 a K t ( 0 , 0 , 8 ) -
Similar
uniqueness
~
---
-.
i=l
Pi
Pi
following
Theorem
l, about
of this t h e o r e m to the case w h e r e
for the l i m i t i n g
c ase
(as n÷~)
as well.
2 s e c t i o n we p r e s e n t
additional
data,
the e i g e n f u n c t i o n s given
(zx[(0))2
z x'1 (01]2
as those
and a b o u t e x t e n s i o n
In this where
[( 3
statements,
< @ < s c a n be m a d e
Section
Yl. ,i = 1,8
_~£u,u,~)
b = 3 Kt(0,0,B)
Remark:
and normali-
d3 + 2(Kss-Ktt) tlt= s - 2 ds 3 K(s,s,8)
A(S)Ks(s,t,G)lt=s a
01,P2,...
corresponding
i(s) B(s) = -
(15).
is t h r e e
that
new existence
is a d d i t i o n a l
of the d e r i v e d
seauences.
and u n i q u e n e s s
normalization
p r o b l e m m a y be given.
An e i a e n v a l u e
results
constants
for
W h a t w i l l be
p r o b l e m will be d e r i v e d
for
514
which
one
given
sequence
is the
functions
will
be a s s u m e d
sequences
will
be v a l u e s
derivatives assumed
at one
a priori
problem.
the
Following existence how
the
the
the
for the
The
will
The
remaining
and their
interval.
are t h o s e
sequences
hypothesis
will
further
of the p r e v i o u s
solution in the
known
formally)
how
to t r e a t
sequences
can
differ
stating
again
i.
of the
sequence
in two parts.
corresponding
Consider
L 2 norm
eigenfunctions
given
forms
format
of e l e m e n t s
Before
of the
of e i g e n v a l u e s .
eigen-
two
first
It w i l l from
be
not be
an e i g e n v a l u e
assumed.
restrict
In
the
relation-
sequences.
theorem
to o b t a i n
number
that
a completeness
ship b e t w e e n
to h a v e
of the e n d p o i n t s
Asymptotic
addition
seauence
First
of the given
eigenvalue the
we w i l l
state
the
we p r e s e n t a t h e o r e m
which
shows
inverse
sequences
from those
problem
case when
then
all
some
(4) and
a finite in a
explain
elements
in the c o r r e s p o n d i n g
problem
only
from t h o s e
We w i l l
3, we e s t a b l i s h
the e i g e n v a l u e
when
differ
problem.
limiting
Theorem
section,
of the
known
problem.
additional
let wl*(s)
(less
notation.
be t h e
i th
1
eigenfunction *
~i = wl?
(i)
with
L 2 norm equal
to
I.
Let
0 < Yi = wl~(0)
and
l
(0),
i = 1,2 . . . . .
Further,
for g i v e n
real
numbers
0
=
l
Yi ~ 0, wl
(s)
~i'
i = i,. .. ,n and p o s i t i v e
satisfy
the
initial
l
value
numbers
I l.
>
,
i
1 ....
,n,
let
problem
w (4) - l w = 0, 1 !
w(0) where
Fn
=
Let,
H
= Y i , W (0) = ~i - Yi n *2 2] j=l ~ [ (Yi) - (yi) •
for
0 < s,t
3:
that
Let
i = 1
0 < Ii !
{w I i }ni=l U {wl* }~ i i=n+! Let K (s:t) be c o n t i n u o u s n
(0) = 0,
n E [ w l * ( s ) w % * ( t ) - w l . (s)w I (t)] i=l l l l i
Let Yi ~ 0, ~i'
~i > 0 if Yi = 0.
wi!
(0) = 0, w
< i
Fn(S,t ) = Theorem
Fn'W
.... n be 12 ! - - - ~
is a c o m p l e t e for
given
In < In+l
set of
0 < t < s < i. --
--
--
real
numbers < ....
functions Define
such
that
Suppose
in L2(0,1).
515
•x
I+
.
= wl. (s) z
Is K n ( S ' t ) w l . (t)dt' 0 z
i = 1 ..... n
and V~, : wl,(s) z z Then K
n
the
(s,t)
set
{V
<
}n i i=l satisfies the
+ I s Kn(S,t)w1*(t)dt, 0 z
n2<
U
{V 1
is a c o m p l e t e i+n+l integral equation
Fn(S,t ) + i s Kn(S,U)Fn(t,u)du 0 The
function
the
functions
Kn(S,t) {V
i = 1,2 . . . . .
is a n a l y t i c
in s a n d
U {VI*} i i=n+l
=i
have
orthonormal
+ Kn(S,t)
set
iff
= 0.
t for
0 ~ t ~
initial
values
s ~ i. ~I
i
(0)
Then =
n(1) (0) = ~i' i = 1 ..... n, a n d J ~ * ( 0 ) = Yi' (0) = ~i + Y i F n Yi' V l i 'i i i = n + i, n + 2,... a n d are t h e e i g e n f u n c t i o n s corresponding to the eigenvalues
1 1 , 1 2 .... , I n , l n + 1 ....
d An(S ) = - 4 ~
= - K
n
n
(3)
iff
Kn(S,S)
Bn = - A n ( S ' @ ) K n , s ( S ' t )
a
for
(0,0)
I + 2(Kn'ss-Kn'tt)tl t=s
= - F
d3 - 2 3ds~
Kn(S,S)
t=s
n
n , , b n = 3 E [~iYi-~iYi] + [Kn(0,0)]2 i=l n * , n * 2 * 2yi ] c n : - i0 K n ( 0 , 0 ) E [~iYi-~iYi] - ,3 E [(~i ) y i - ( ~ i ) i=l i:l where
~i = g i - Y i r n ' i = i, .... n.
Sketch iff
the
then be and
of
Proof:
We
sequence
observe
solved Kn(s,t)
of
that
simply can
show
functions
the by
be
first
expressed
Kn(S,t)
=
the
integral
equation
is s a t i s f i e d
n }~ is c o m p l e t e . We }n U {VI~ l = n + l i i=it equation has finite rank. T h u s it c a n
{V<
integral solving
that
a set
of
linear
nonhomogeneous
as
n E IV , ( s ) w l * ( t ) - V 1 .n (s)w I (t)]. i=l i i 1 1
equations
516
We t h en v e r i f y
{V<. }n U {V<. }~ are the e i g e n f u n c t i o n s i i=l i i=n+l the s t a t e d e i g e n v a l u e p r o b l e m .
Remark:
that
The a s s u m p t i o n
L2(O,1)
automatically
li's can o c c u r
at m o s t twice.
that a certain
a constant
which
on L2(0,1). only
assumption
multiple
also i n s u r e s
o f wt
i+l
is not
(wlj,wl,)± , i, j = l,...,n.
det Mn ~ 0 d e p e n d s
not
the c o m p l e t e n e s s
determinant
minant with components
is c o m p l e t e
The c o m p l e t e n e s s
wA i s i
We can r e p l a c e
oo
{w I ] U{wl,} is a c o m p l e t e set on i i=l i i:n+l that a given number in the sequence of
implies
that if Ai = t i + 1 t h e n Remark:
n
that
for
zero.
hypothesis
That
by the
is, let M
n
assumption
be the d e t e r -
are the L 2 inner p r o d u c t
of w~,~ and wl*,_ 3 l {w li}ni:l U {wl*} ~ i i=n+l
T h e n det Mn / 0 iff
Further, on the
it can be shown that w h e t h e r
values
of
the
ti
and
on the
or not
ratios
of
Yi to ~i' i = l,...,n. A nontrivial satisfied
example where
can be o b t a i n e d
wr!
iff
"
the c o m p l e t e n e s s
easily.
assumption
S i m p l y c h o o s e n :i.
r
(wl*(1)wl.(1) ~wl*.tl)wl.(1))= 0. H e n c e we c h o o s e 1
1
1
equation
is
Remark:
Without
Then (w,,,w I ) = 0 zi i yl,~l so this last
l
satisfied.
show existence
stating
a formal theorem,
for t h e c a s e w h e r e
t h a t the i t e r a t i o n s ,
that is the
3 converge
to a s o l u t i o n
n÷~.
is, we assume,
That
is not
n÷ ~.
we w i l l e x p l a i n
This can be done by s h o w i n g
solutions
of the d e s i r e d in a d d i t i o n
one w a y to
given
for each n in T h e o r e m
inverse eigenvalue
to the a s s u m p t i o n s
problem
of T h e o r e m
as 3,
that
{wl.} ~ is c o m p l e t e and t h a t we h a v e the a s y m p t o t i c forms li=l * 2 - (Yi) 2 = Pi' and ~i : i~i [ - Y i + Q i {exp [_(li)~]} ] (Ii)% _ (I~)% = Ri' (Yi) k ~ k ~
where
Z . .IRillT < ~, i=l
i W
(and w h e r e
to s h o w t h a t K ponding
=
~i
E . ,IPill4 < ~, i=l
}
*
i *
~
function
a n d it's
k = 0,i,2,3,4, '
*
l ~ [ - Y i + Q i { e x p [ - ( l i )h}])" n
' iIQi-Qi]14 ' < ~
i=l
first
This
four d e r i v a t i v e s
K and it's d e r i v a t i v e s .
is c e r t a i n l y converge
Further,
there
sufficient
to a c o r r e s exist
517
A(s) ~ C I, B ( s ) (
C 3, a,b,c
such
that
A
b n ÷ b, Cn ÷ c and the c o r r e s p o n d i n g solution
of our
It s h o u l d det M only
inverse
be o b s e r v e d
~ 0 for all n.
n
select
N such
[ (I * ) ¼ _ ( l i ) ~ We t h e n
requmre
that
Remark:
The
the o n l y
solution
solution
set
It is c l e a r that
derived
for w h i c h
to the
by
and a
n
÷a,
problem
is a
do not
need
asymptotic
is a "small and
forms
enough"
applying
we n e e d
L so that,
[ IQi-O *I ] <
i
-
-
forms
then
the c o m p l e t e n e s s
is u n i q u e
corresponding
{wl. }~ b y the 1 i=l
in the
eigenfunctions
integral
yield
result
sense
L
i
that
in
it is
{VI" ~ i i=l
relationship
(s) = wl. (s) + I s K ( s , t ) w l . (t)dt. i 0 l that
in g e n e r a l
the
Dirichlet
conditions
at
the b o u n d a r y
conditions
at s = 0 are not.
The n e c e s s a r y
and
conditions
s : 0 is c l e a r l y
at
assumed
c a s e we
abow_~ a s y m p t o t i c
above
the
eigenvalue
limiting
-
in L2(0,1)
l
but
of the
'
V1
served
for the
The
÷ B uniformly
n
problem.
i > N there
det M N ~ 0.
}~ complete 1 i=l [2O] .
Remark:
for
defined
* 2 -(Yi ) 2 1 < - -L, I (Y i) i
{wl
related
that
Because
< L -
are
eigenvalue
÷ A, B
n
sufficient
conditions
for p r e s e r v i n g
s = 1 are
pre-
the b o u n d a r y
a = 0, b = 0, c : 0.
References [ l]
V. B a r c i l o n , I t e r a t i v e S o l u t i o n of the I n v e r s e S t u r m - L i o u v i l l e P r o b l e m , J. Math. Phys., 15 (1974), pp. 287-298.
[ 2]
V. B a r c i l o n , h i g h orders,
[ 3]
V. B a r c i l o n , On the u n i q u e n e s s Ibid., 38 (1974), pp. 287-298.
[ 4]
G. Borg, Eine U m k e r u n g der S t u r m - L i o u v i l l e s c h e n Acta. Math., 78 (1946), pp. 1-96.
[ 5]
E. A. C o d d i n g t o n and N. L e v i n s o n , T h e o r y of O r d i n a r y D i f f e r e n t i a l E q u a t i o n s , M c G r a w - H i l l B o o k Co., N e w York, 1955.
On the s o l u t i o n of i n v e r s e Geophys. J. R. Astr. Soc.,
eigenvalue 39 (1974),
of i n v e r s e
p r o b l e m s of pp. 1 4 3 - 1 5 4 .
eigenvalue
problems,
Eigenvertaufgab~
518
[ 6]
I. M. G e l ' f a n d and B. M. Levitan, On the D e t e r m i n a t i o n of a D i f f e r e n t i a l Equation from its Spectral Function, Izv. Akad. Nauk SSSR Ser. Mat.r 15 (1951), pp. 309-360; English transl., Amer. Math. Soc. Transl., 1 (1955) pp. 253-304.
[7]
O. H. Hald, Potentials,
[ 8]
H. Hochstadt, The Inverse S t u r m - L i o u v i l l e Appl. Math., 26 (1973), pp. 715-729.
[ 9]
M. G. Krein, On a M e t h o d of Effective S o l u t i o n of a Inverse B o u n d a r y Problem, Dokl. Akad. Nauk SSSR, 94 (1954), pp. 987-990.
[10]
M. G. Krein, S o l u t i o n Ibid., 76 (1951), pp.
[ii]
Z. L. Leibenzon, The Inverse P r o b l e m of the Spectral A n a l y s i s of O r d i n a r y D i f f e r e n t i a l O p e r a t o r s of Higher Order, T r u d y Moskov. Mat. Obsc., 15 (1966) pp. 78-163.
[12]
N. Levinson, The Inverse S t u r m - L i o u v i l l e B., 25 (1949), pp. 25-30.
[13]
B. M. Levitan, G e n e r a l i z e d T r a n s l a t i o n O p e r a t o r s and Some of Their A p p l i c a t i o n s , Fizmatigz, Moscow, 1962; English trans. Israel P r o g r a m for Scientific T r a n s l a t i o n s , J e r u s a l e m and Davey, New York, 1964.
[14]
B. M. Levitan, On the D e t e r m i n a t i o n of a S t u r m - L i o u v i l l e E q u a t i o n by Two Spectra, Izv. Akad., Nauk SSSR Set. Mat., 38 (1964), pp. 63-78; Amer. Math. Soc. Transl., 68 (1968), pp. 1-20.
[15]
V. A. Marcenko, C o n c e r n i n g of the Second Order, Dakl. 460.
[16]
J. McKenna, On the Lateral V i b r a t i o n Appl. Math., 21 (1971), pp. 265-278.
[17]
J. R. M c L a u g h l i n , An Inverse E i g e n v a l u e P r o b l e m SIAM J. Math. Anal., 7 (1976), pp. 646-661.
[18]
J. R. M c L a u g h l i n , An Infinite Case,
[19]
J. R. McLaughlin, Fourth Order Inverse E i g e n v a l u e Problems, Spectral T h e o r y of D i f f e r e n t i a l Operators, I. W. K n o w l e s and R. T. Lewis (editors), North H o l l a n d P u b l i s h i n g Co. (1981), pp. 327-335.
[20]
Sz. Nagy, B@la Math. Journal,
The Inverse Acta Math.,
Sturm-Liouville 141 (1978), pp.
of the 21-24.
Inverse
the T h e o r y Adad. Nauk
P r o b l e m with 263-291. Problem,
Comm.
Sturm-Liouville
Problem,
Symmetric
Pure
Problem,
Mat.
Tidsskr.
of a D i f f e r e n t i a l O p e r a t o r SSSR, 72 (1950), pp. 457-
of Conical
Bars,
SIAM J.
of Order Four,
An Inverse E i g e n v a l u e P r o b l e m of Order Four SIAM J. Math. Anal., 9 (1978), pp. 395-413.
de, E x p a n s i o n T h e o r e m s of P a l e y - W i e n e r 14 (1947), pp. 975-978.
Type,
-
Duke
INDEFINITE
STURM-LIOUV]LLE
Angelo
i.
The
main
forgotten first
stimulus
paper
actual
of
this
Ralph
paper
(cf.,
problems
over
1
Mingarelli
arose
Richardson
investigation
Sturm-Liouville the
for
B.
PROBLEMS
[]2].
[3])
of
a finite
out
of
an
His
paper
Indefinite
interval;
old
and
somewhat
contained or
that
the
non-definite is,
problems
of
form -(p(t)y')'
where case
p,q,r arises
~ C[a,b] when
are
(I.i),
+ q(t)y
real-valued
under
"left-"nor
"right-definite"
conditions
which
and
suitable
in
determine
= ~r(t)y p(t)
> 0.
boundary
current
the
(i.I) The
conditions,
terminology.
eigenvalues
in
indefinite
The
(l.l)
will
is n e i t h e r
boundary be
of
the
form
where
0 ~ ~,B
in
possible
the
may
appeal
usual
to
cos
~ y(a)
sin
~ y'(a)
= 0
(1.2)
cos
~ y~b)
+ sin
~ y'(b)
= 0
(1.3)
< ~.
One
existence some
is
is r e q u i r e d
now to
complex-valued
of
of
the
indefinite
eigenvalues.
Hilb
[7]
To
wherein
see
(l.l)
case
lies
this
we
takes
the
that
(1.4-5)
eigenvalues
with
no
results,
= ~y and
a
(1.4) (non-trivial)
solution
of
(1.4)
satisfy
showed
].
work
+ q(t)y
y(a)
other
features
form
q
Hilb
the
of n o n - r e a l
earlier
-y" and
of
their
This r e s e a r c h is Research Council
= 0 = y(b). admits
finite
a denumerable
point
asymptotic
supported of C a n a d a
(1.5)
of
infinity
accumulation
representation
by the N a t u r a l Grant U0167.
and as
of
gave,
Ill
Sciences
÷ ~.
and
complex among So
if we
Engineering
520
choose
q(t)
eigenvalue satisfy
of
and
case.
of
(i.I),
[6],
[4],
eigenvalues
about
(
valued 2,
shown
eigenfunction
p(t)
found
that
even
eigenvalue
in
in
[1,§12.8].
singular case
in the
when
will
~ I, q ~ P-ql"
a non-real
m a y be
arise
of)
sequel
paper
).
L2(a,b)
identically
q(t)
~ 0,
(HI)
function
an e i g e n v a l u e
be
(i.I)
with
turning
(cf.,[2],
called
a
points
[I0]).
in the
denote
of
will
the n o n - r e a l
the
on
spectrum
indefinite space
functions
inner-product
subinterval
X = 0 is not
study
continuous
The
sign
(e.g.,
(1.1-3)
will
its
form
A C l o c ( a , b ) will
absolutely
on a n y
the
measurable
The u s u a l
As u s u a l
locally
of
is to
form
complex-valued
L2-norm.
changes
of a p p l i c a t i o n s
of the
ASSUMPTIONS:
(H2)
have
now
admit
work
may
r(t)
Problems
aim of this problems
,
which
[9].
variety
In the
by
of H i l b ' s
non-real
where
would
v ~ 0 is a n o n - r e a l
[s]).
The
finite
(1.1)
non-real
a wide
classes
(i.i)
investigations
turning point,
boundary
form
Hence
extension
X = ~ + iv,
corresponding
recent
A point
have
the
of the
X = i. An
Some
+ i q 2 ( t ) so that
(1.4-5),
an e q u a t i o n
r H ~-q2 this
= ql(t)
L2(a,b)
denote
the
functions
in q u e s t i o n
case.
of all
over
(equivalence
[a,b] will
space
on
of
with
be
denoted
of all
complex
(a,b).
does
not
vanish
[a,b]. of the b o u n d a r y
problem(s)
in
question. We of
(1.1-3).
with of
should
minor
(1.1-2)
long
first
This
can be
that
seen
not
by
modifications
so as
is an e n t i r e
function
as r(t)
~ 0 over
resulting
equation
such
no
with
note
finite
for
[a,b]. the
point
every
adapting
to
show
~ c ~ can be
an e i g e n v a l u e
the
in
that
argument
a fixed
of k of o r d e r
Thus
if y(t,X)
eigenvalues
solution exceeding
satisfies
yields
of a c c u m u l a t i o n .
not
(1.3)
a countable
In f a c t
[1,§8.2]
it was
y(t,~) 1/2,
as
the
number
of
shown
by
521
Richardson
[12]
that,
infinitely
many
positive
Non-real have for in
eigenvalues
turning fixed
the
for
3.
the
case
some
and
so
infinitely
many
the
not To
exist
see
even
this
for
some
and
~ = ~ = 0,
say,
is
real.
will
are
be
eigenvalues. q
spectrum
eigenvalues
Then
difficult
derived
cos
r and
(1.1-2-3)
Sufficient
is
conditions
to
formulate
below.
f"cL2(a,b)
~ f'(a)=0,
admits
if b o t h
let
f,f'~ACloc(a,b),
~ f(a)-sin
negative
a constant.
conditions
cos
(1.i-2-3)
K ~ 0 is
of n o n - r e a l
necessary
and
~ f(b)+sin
~ f'(b)=0
0~,6<~}.
f ~ S, cot
if
0 < a,B
-y"
= ~y
g
If(b) ] 2 + cot
< g.
where
Wirtinger's in
Here y is
~0 i s
subject
inequality. (3.1)
disappear).
Theorem
Assume
a)
b)
r
changes
smallest
to
of
sign
in
eigenvalue
(1.2-3).
When a
that
~0'
(1.1-2-3)
in
this
of
admits
(a,b).
In
[8,
the
follows
a non-real
addition,
if r satisfies
turning
point
in
(a,b)
q(t)
has
at
least
one
turning
point
in
(a,b)
There
exists
Under
(H2)
at
> 0 for
some
a of
[a,b]
a subinterval
the
least
<
-k0'
boundary -y"
has
q(t0)
one
t o c over
e J,
problem
+ q(t)y negative
t
first
is two
the
eigenvalue.
one
and
(3.1)
p 6.4]).
least
< ~/2
problem
from
at
~ a,~
the
= 8 = 0,
case
(3.1)
(cf.,
of
has
q(t) d)
the
r(t)
0
c)
that
(3.1)
If(a)] 2 + f ~ [ f ' 12 a k o f ~ I f l 2
The p r o o f
characterization
1:
~
(We n o t e
variational
Then
case,
(a,b).
S = {f~L2(a,b):
for
terms
in
indefinite
or m a y
r = Kq w h e r e
existence
Let
Then
q let
and
may
points
polar
though
in the
(1.2-3),
= ~y, eigenvalue.
(H1) the~
u n d e r (H1) i f
[a,b]. which
522
Proof.
a)
Let
y(t,X)
Multiplying
(I.I)
find
we m a y
(since cot
~ y be
by y
integrating
suppose
~ly(b) I2 + cot
Assume,
on
the
q is real
p(t)
over
non-real
[a,b]
eigenfunction.
and u s i n g
(1.2-3)
that
from
r does
(3.2)
not
change
sign
in
(3.2)
(a,b).
since
unless
Im k ~ 0, the
r(t)
~ 0.
b)
The
If p o s s i b l e
q(t)
k 0
in
(a,b)
real
and p o s i t i v e
account
of
(a,b).
Under
the
let
whereas This
(HI)
An
part
must also
q have
by h y p o t h e s i s .
(3.3).
c)
integral
other
the
no
from
turning
point
application
of
(3.1)
the
of
and
to the
is i m p o s s i b l e this. in
left
(3.2)
a turning
(a,b).
side
equals
so
of
Then (3.2)
zero
q changes
is
on
sign
in
point.
left
side
of
(3.2)
yields
estimate
f~(Xo+q)]y[2 Assuming
q(t)
y = K~0,
K e ~,
now
then
right-side
have
which
follows
But
then
(3.3)
vanish
is a c o n t r a d i c t i o n
q must
Since
that
f rlyF 2 = 0 and
we
~ i),
~ l y ( a ) I2 + f~{ ly' 12+q]yl 2} = ~ f ~ r l y l 2.
contrary,
it f o l l o w s
a corresponding
follows
[8])
The
that
we r e p l a c e
real
eigenfunction
i.e.,
f" by
the p r o b l e m
(a,b)
has
implies
which
eigenvalue
of
the
left
f"
- qf the
since at
least
left
~ -X 0 .
(3.2-3)
corresponding
now
to X O.
yields There
is i m p o s s i b l e .
side
in the
(3.4)
q(t)
> of
y c S O where
y makes
~ < 0
(3.4)
a 0 an e i g e n f u n c t i o n
smallest
functions
Hence
t,
E 0 in
the m i n i m u m
(normalized)
y c SO.
all
~o(t)
~r(t)
d) (cf.,
> -~0
< O.
(1.4-2-3) of
S O is
(3.2) similar
earlier side
of
v = 0 is not one n e g a t i v e
is g i v e n
taken
over
those
to S a b o v e
definition. (3.2)
by
Now
vanish.
an e i g e n v a l u e eigenvalue.
except the n o n -
Moreover of
(1.4-2-3),
523
Lemma
i.
Let
~'i ~ ~''3
Let
~i
~ ~j
cot
Xi, ~i'
~j be
Proof.
This
+ cot
omitted
(cf.,
that
pairs,
also
(3.5)
i.e.,
real.
[Ii,
that
such
following of
negative
eigenvalues
of the
by S t u r m - L i o u v i l l e
Theorem
2
q,r be
(1.1-2-3)
gives
satisfied
for
rise
when
then
We may
Then
(1.1-2-3)
has
least
one n e g a t i v e
is ~,
denote while
real
and
indefinite
at
be
(3.2)
occur
coefficients of
denote
(1.4-2-3).
are
"pairs"
the
in
of
number
That
of
N <
in
[a,b] Assume
and
assume
that
that
(Hi)
is
Then
that one
N
(3.6)
M > i, non-real
eigenvalue
else
(theorem
eigenfunctions < ~n
~n
If we
i,
corresponding "'"
(3.6)
(~f, f)
(f, f)
(1.4-2-3)
(d)),
denote
''" we know
= inf
is trivial.
eigenvalue
(1.4-2-3)
<
the nol
it will
(I.i-2-3)
as the
case.
40.
< ~I
of
the n u m b e r
continuous
eigenfunction
-~ < >0
lines
A glance
N will
corresponding and
between
theory.
(1.4-2-3).
at
(3.5)
i = j.
corresponding
to the
suppose
usual
4.2.2]).
so
M will
M ;
~here
with
= 0
relation
eigenvalues
(1.1-2-3)
follows
follows
lemma
non-real
if ~ is one
Let
proof
valid
eigenvalues
least
(1.1-2-3)
eigenfunctions.
orthogonality
p 213,
is also
In the
an
the
non-real
Proof.
of
a ~j(a)~i(a ) + f~(~j+q~i~j)dt
As
It is e v i d e n t
all
corresponding
is e s s e n t i a l l y
eigenfunctions.
shows
two
eigenvalues
and
B ~j(b)~i(b)
real
Xj be n o n - r e a l
has
Since at
say >0 with
the
by >i'
eigenvalues
of
~i r e s p e c t i v e l y
that
(3.7)
524
where
Af E -f"
for w h i c h denote that
+ qf and
(f,~i)
= 0 for
a collection
X i ~ ~.
for
M corresponding
the
Since
M
which
we
of
i,j
assume
that
system
now
fix.
We
now
implies
that
that
(1.1-2-3)
mentioned to q,r
The u p p e r
209
That r(t)
The
boundary
so that
any f i n i t e
N
This
=
0,1,2
linear
f
~0'
f c SO
X1 .... ' XM-I in such
@l ....
a way
~M-I
~
said,
let
are
chosen
.....
denote
f
so that (3.g)
in
solution and
= ~ ci~ i
N-I
equations
0
f(t)
E
M
unknowns.
c0,cl,...CM_ SO .
1
Furthermore
~N
~ I f ( a ) I2 + f b ( I f , 1 2 + q l f l 2) = o. a < 0 on a c c o u n t
Thus
at
from that
least
M < N,
N+]
and
a finite
the
bound
appearing
in
relation
use
and
(3.9)
(HI).
But
eigenvalues, point
this
which
spectrum
set. of
(3.7)
for
the
requirements
any
class
of
on q,r m a y
essential
change
in the
is s h a r p
in the
case
(3.6)
(cf.,
(3.7)
negative
continuity without
of
so the n o n - r e a l
the v a l i d i t y
(a,b)
[ii,
p.
206
theorem
be proof.
of a
4.1.6
and
4.1.1]).
= Kq(t)
for
problem
M = 0.
Let
non-trivial
~ L
M = N is,
let
i
that
has
recurrence
example
o,
a
is n e c e s s a r i l y
functions
p.
note
(1.4-2-3)
It f o l l o w s
three-term
labelled
0 ~ i ~ M-I,
of
has
that
Remarks.
weakened
~0'
f ~ 0,
f satisfies
is a c o n t r a d i c t i o n . of
=
Blf(b) I 2 + cot
follows
M > N.
ci,
system
this
cot There
a
> N
shows
~ M-I.
functions
Let
eigenvalues
0 ~ i,j
complex
describes
(3.5)
all
eigenfunctions.
generally
(3.8)
is o v e r
i = 0,1,2 ..... n-l.
(f,~i) Now
"inf"
of n o n - r e a l
all
If p o s s i b l e where
the
in g e n e r a l , each
(1.1-2-3)
On the
number
t and
other
false.
K ~ 0 real. in this
hand
N of n e g a t i v e
For
q may
case be
eigenvalues
example,
for
given
Set ~ = B = 0 in has
real
chosen of
spectrum
q(t)
(1.2-3). only,
so as to g e n e r a t e
(1.4-2-3).
525
4.
The
fundamental
regarding
the
problems when
[12,
p 302]
Ca,b) part
eigenfunctions
do not
case
theorems
the
appear
show
that
and ~ = ~ = 0 in of
a non-real
Theorem
3.
non-real points
Let
(a,b)
then
~i'
Let
so that
~i < ~2 < ' ' ' <
= f(b)
contrary, in an see
that
eigenfunction Thus
(3.2)
f(ci)
= 0,
among
[a,cl],
We m a y
that
the
and
one
without
Since
Y(tl)
above.
the
f(a)
at m o s t turning
t ~
'
at,
Ca,b). and
say,
Cl,
Restricting
Y(Ci)
= 0.
associated
precisely
of
n turning in
Ca,b).
r arranged
Hence
by
c. and l
there
loss
so
f(a)
at
f(tg)
= 0.
(arranged
[a,ci]
~ over
implies
least
one
we
[a,cl]
is a p o i n t = 0.
is
tI c
Continuing
= 0. in
point-free. an
(a,cl)
similar
t2 ~
way
to
(a,tl)
Repeating
this
such
y satisfies
an a r g u m e n t
so y(t 2)
that
such
Since
is a p o i n t
[a,ci].
interval
is t u r n i n g
that
= 0 = f(t I) t h e r e
= rCt2) lY(t2) ]2 and
c n,
(3.3)
= 0 by
follows
c2,...,
to
the
y is a n o n - r e a l
interior
= f(c I) = 0, t h e r e
on
eigenvalue
exists
of g e n e r a l i t y ,
assume,
y(t,~)
Thus
to the n o n - r e a l
b replaced
we h a v e
imaginary
(n-l)-times
If p o s s i b l e
f(tl)
argument
in
(4.1)
= 0 there
previous
point
[ab '].
Y(tl)
0 = f'(t 2)
and
and
points
Hence
so that
real
has
= r(tl) lY(tl) I2
Since
turning
In the sequel let r satisfy (HI)
If r(t)
[ci,c2] ..... [Cn,b ] w h o s e
0 = f'(tl)
(1.2)
in
i = 1,2,..,n.
assume,
interval.
be
n times
(1.2)
with
of the
indefinite
Richardson
one
eigenvalue
vanish
3.3)).
corresponding
holds
precisely
in the
Let
(cf.,
y satisfies
However
zeros
separation,
Sturm-Liouville
counterpart
interlace.
may
~n"
order),
has
(1.i-2-3).
y vanishes
increasing
that
of
oscillation,
non-real.
the
~ fat r l y l 2
= 0
simple
be a n o n - r e a l
i = 1,2,...,n
f(t) f(a)
(1.2-3),
y(t,X)
Proof.
Then
are
eigenfunction
eigenfunction
in
any
if r(t)
~, y ( t , ~ )
viz.
of non-indefinite
to h a v e
eigenfunetions
did
of Sturm,
the
we o b t a i n
526
an
infinite
compact
sequence
there
is
a
tn
y(c)
v(c)
Furthermore
set
h
= tnk
continuous v'(c)
there
= 0.
Corollary
i.
turning The
result,
point
due
Proposition
of
any
Let
above
precisely
y(c) and
to
vanish
~
turning
non-real
as then
can
Consolidating
some
the
p
302,
eigenfunction
so
at
be
the
we
most
If
to
separate
X
the
one
results
of
[12]
problem
(1.1-2-3)
I.
u'
= If
we
is
shows
[a,b].
r(t)
prove
If,
of
u(c)
that This
is
has
exactly
(a,b).
theorem
zeros
is
(n-1)-times.
~ 0 in
(1.2-3).
Since
0 over
theorem.
used
have
argument
y ~
[a,cl]
By
0 ~ @ <
O.
=
Since
[a,cl].
+ iv(t),
similar
y(t,X)
= B = 0 in point
A
now
[12,
c ~
- c))
and
in
= 0.
+ h u'(c+@h),
= 0.
y can
(a,b)
c where
(tnk
= 0,
corollary
Let
u'(c+@
be
y ( t n)
= u(t)
= y'(c)
y(t,X)
which
= u(c)
u'(c)
Richardson
I. one
that
so
in
y(t)
u(c+h)
find
on
tnk÷
Writing
follows
Thus
a contradiction
one
= 0.
c we
-
[a,cl]
subsequence
continuity = 0.
~
the
following
]. in
real
(a,b), and
r(t)
has
imaginary
part
another.
with
the
above
we
conclude
with Theorem an
4.
infinite
X± + n
±~
total
finite number
negative
boundary
number
of
respectively
eigenfunctions most
The
vanish (though
real and
for
n
times
possibly
(necessarily
eigenvalues
eigenvalues
of
n
X ± with n
sufficiently in
(a,b).
empty)
even) the
has,
does
set not
corresponding
in
the
the
large
property
non-real
exceed problem
case,
that
corresponding
Furthermore of
indefinite
there
is
an
eigenvalues
twice
the
number
(1.4-2-3).
at whose of
527
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of
oscillation equations
283-316.
KIN 9B4.
of
lhe Infinitesimal Generator of a Stochastic Functional Differential Equation
S. E. A. Mohammed
§0
Introduction
Trajectories of stochastic retarded functional differential to possess a Markov property (See ~l below). formula for their weak infinitesimal
§i
equations are known
In this article, we prove a general
generator.
Summary of Known Results Let r > O, J = I-r, o] and C ~ C(J, R n) stand for the separable Banach space of
all continuous paths ~ :
J ~
R n, given the supremum norm II~IIC = sup !~(s)l, where
l-I represents the Enclidean norm on R m.
Let (~,~,
m-dimensional Brownian motion W : ~ x R ~° ~
{W(.,s)
: o ¢ s ~ t},
t ~ o.
P) be a probability space carrying
R m, with induced filtration (~t)t~o ~ t
If x : ~ x I-r,
~) ~
Rn i s any continuous ( ~ t ) t ~ o
- adapted process on (£,q~ , P), d e f i n e the sl
(s) = x(w, t+s)
for a l l
s { J and a.a. w ~ ~.
Suppose
L(R m, R n) is the Banaeh space of all linear maps R m --~ R n with the uniform operator
norm II MI[ = sup { t M (×)I = x ~ R m, Ixl < l}, M ~ L(R m, Rn).
Then, for Lipschitz "coefficients" that the
(SFDE)
t
Y~ajecto4y d lx(t)
lx °
H : C ~
Rn
and G : C ~
L(R m, Rm), it is known
{ ~x t : t ~ o} of the stochastic FDE
= H ( l x t) dt + G(Ix t) dW(t)
t>o
=
gives a continuous time-homogenous C-valued Markov process.
Indeed, f o r any B o r e l
subset B of C, one has P(~xt2~
BlOt])
=
P(~xt2e
B I ~zztl) a . s . ,
whenever o .< t 1 ,< t 2 •
=
530
For further details, examples and proofs, the reader may refer to [6] and [7].
~2
The Semi-group In this and the next section, we shall assume, in addition to the Lipschitz
condition, that the d r i f t and diffusion c o e f f l e i e n t s H and G in (SFDE) are globally bounded on C with a common bound K > O.
Let C b be the Banach space of all bounded
uniformly continuous functions ¢ : C ---->R given the supremum norm II ¢I[ = sup {I¢(~)i : ~ ~ c}.
For ¢ E C b, t ~ o, define Pt(¢) : C ~
R by setting Pt(¢)(q) =
E ¢(Zxt(.)) for a l l ~ E C. From the uniform continuity of ¢ and the map C ; R, ~-~ E II t ( . ) [ I 2 , - i t fo.llows t h a t ? t ( ¢ ) i s u n i f o r m l y c o n t i n u o u s . Moreover,
~x
II Pt(¢) II ~ I]¢ II" By the Markov property and time-homogeneity, it is easy to see that PL : Cb -----~ Cb' t ) a, is a contraction semi-group on Cb; viz Po = idc b' P t l + t 2 = P t l o Pt2 , II Pt I ~ 1, f o r a l l
tl,
t2,
t ~ o
([4]
PP. 1 9 - 4 6 ) .
Theorem I: The contraction seml-group {Pt}t~ ° is never strongly continuous on C b.
In
0
particular, its domain of strong continuity C b = {¢ : ¢ ~ C b, lim Pt(¢) = ¢} is a t--->o+ proper closed subalgebra of Cb, independent of H, G and W.
Pnoof : For each ~ e C and t >/ o, d e f i n e '~tE qt(s)
=
~
t
2(0) ( t + s)
D e f i n e the s h i f t It
t+
s >o,
C by
s ~3
-r~< t + s < o, s ~ J
semi-group St
: cb ~
Cb, t > o, by S t ( e ) ( ~ )
i s easy to see t h a t { S t } t ~ ° i s a c o n t r a c t i o n
=
semi-group on Cb.
¢ ( ~ t ),
~ e C, ¢ E Cb.
By the s a m p l e - p a t h
continuity off the trajectory {Ix t : t ~ o} in C, we have lim Pt(¢)(~) = ¢(~) = t--> o+ lim St(e)(~) for every ¢ £ C b, Moreover, the Martingale inequality for stochastic t-e o+ integrals ([5] P. 70) yields
Ell lxt(.) - ~t II z-< E sup {I ~
t+s O
< 2K 2 t 2 + 8K 2 t ,
H(TXu(.)) du+ (.)f
t+s
0(~×u(.)) dW(u)I~t+s~o, s
0
t >~ o.
Therefore, lira EII ~xt(.) - ~t II2 = 0 uniformly For ~ ~ C. t-e o+
Using this and the
531
uniform c o n t i n u i t y o f ¢ ~ Cb, i t i s not hard to see t h a t ] i m [Pt(¢)(~) uniformly f o r ~ ~ C. t--~o+
- S t ( ¢ ) ( ~ ) ] =0
Writing
Pt(#)(~)
- ¢(~) = [ P t ( ¢ ) ( ~ )
- S t ( ¢ ) ( l ) ] + ESt(¢)(" ~) - ¢ ( l ) ] ,
we see t h a t i i m P t ( ¢ ) ( ~ ) = ¢(~) uniformly f o r ~ E C i f t--~o+ St(c)(" ~) = ¢(2) u n i f o r m ~ f a r ~ E C. Thus i t i s s u f f i c i e n t not strongly
c o n t i n u o u s on Cb.
~(7) :11 (So)
Define 9 : C ~
and only i f
lim t-*o+ to prove t h a t { S t } t ~ ° i s
R by
II1 il.<1
T~ ~(s°)
lll II >
for a fixed So¢[-r , o).
Clearly 9 is bounded and globally Lipschitz on C; so ~ E C b.
Fix v E R n such that Ivl = i and construct a sequence {~n} in C with the graph of each n
looking like
Rn U i
.
I
~3
.
.
.
.
s + 1O
Obviously lim St(~)(~n) = lim t-~o+ t~o+ not uniform in n. Hence 9 ~ C b.
.
R
s
-r
.
O
0
n
~ n ( t + So ) = n ( s o )
= 9(n),
and the convergence i s
The Weak Infinitesimal Generator
Give C b the weak topology induced by the bilinear pairing <.,.>: C b x M ~
R,
<~,#> = ~C¢(~) du(~), ¢ E Cb, p E M, the Baoach space of all finite regular Bore1 measures on C with the total variation norm.
If Ct' ~ ~ Cb' t ~ o, then say the weak
limit w-lim ¢t = {~ whenever lim <¢t,p> = <¢, p> for all # e M. This is equivalent t~o+ t-*o+ to saying that {II ~II: t > o} is bounded and Ct(~) ~ ¢(2) as t--~o+~ for each ~ e C. ([4] PP. 36-43).
The
weak infinitesimal generator
A : D(A) C Cb ~ l ~ [Pt(¢) - ¢~ whenever the limit exists.
is defined by A(¢) = w-lim t-~ o+ D(A) is the set of all ¢ E C b for which the last weak limit exists.
C b of [Pt}t~ ° Its domain
Note that
532
w-lim P t ( ¢ ) = ¢, and D(A)E C° b" t-->o+ o Cb - ~
Let S : D ( S ) E
Cb be the weak i n f i n i t e s i m a l
g e n e r a t o r o f { S t } t ~ o.
To
o b t a i n a f o r m u l a fior A, we need to augment the s t a t e space C by a t t a c h i n g a c a n o n i c a l n-dimensional direction o p e r a t o r along F n.
istic
F n.
f u n c t i o n of {o}.
I12ii+
I~I, ~ c ,
Then A equals S p l u s a second o r d e r p a r t i a l
Take Fn = { V ~ { o } :
v E Rn}, where ~ ( { o } :
Form the complete d i r e c t
differential
J - - + R i s the c h a r a c t e r -
sum C~) Fn normed by ll ~ + vE{o}ll_. =
v ~ Rn; i.e. C ( ~ F n is the space of all bounded functions ~ : J ~
which are continuous on E-r, o) but with
at most only a jump discontinuity at o.
In order to establish our formula for the generator, we require the following sequence of lemmas.
Lemma I: Each ~ E C
has a unique weakly continuous extension ~ e [c~)Fn]
i.e. if
is bounded in C and ~k(s) ---~(s) as k --->oo for alls E J and some ~ G(I k) ~
Proof
~(1) as k ~
oo
The map G ~ ,
{~ }k=l
C~Fn,
~ is a linear isometry into.
:
Take n=l.
By Riesz's representation theorem, write
o G(~)
=
~
2(s)
du(s)
~ ~ C
r where U i s a f i n i t e
r e g u l a r B o r e l measure on J.
~(~+ V~{o }) = ~(2) + vp{o}
Define ~ E EC(gFI]* by
Z~c, v~ R
For higher dimensions n > i, note that C(J, Rn) @ F
n = [c(J,
R)¢
Fi]x . . . . . . . .
x[C(a, R)¢
F i]
(n c o p i e s ) Write each ~ E [C(J, Rn)] * in the form n i=l
~ ~ c(J, R), i ( [ ) i=l,...,n.
= ~ ( o , . . . , o , ~,o . . . . . o),
Finally define ~ E [c(3, Rn) @ rn]
C(J, R) occupying the i t h place, by
then
RI
533 n
~(]+ VZ{o }) :
i:i
" ~(~
+
i v ~{o}),
Z~ C(J, Rn),
. v : (v i ....
R n"
v n)
Lemma 2:
Every continuous bilinear form B : C x C - ~ •
bilinear extension
R has a unique weakly continuous . k o ~ koo
8 : [C ~) Fn] × [C (~) Fn] --~ R : . ~ . i f {~ }k=l' {~ ]k=l are bounded
in C with ~k(s) ---~(s), 7k(s) ---~(s) as k ---~ k k ~ ~ 8( I '7 ) ---> 8({,~) as k ~ o0.
or all s ~ J , I , ~ C ( g F n ,
then
Proof: First, take n:l. Since C
Regard 8 : C x C ~
R as a conYginuous l i n e a r
map 8 : C
is weakly complete, then B is weakly compact; hence there is a unique
> C . C ~ -
valued regular Borel measure i on J (with finite semi-variation) such that 0
([3] P. 494).
Now extend 8 to a continuous linear map 5 : C ~ g I ~
C
as in Lemma i,
by using the dominated convergence theorem for vector-valued measures ([3] P. 328).
Define 8 : C~) F1 ~
[C ~ F1]
by
It is easy to check that 8 is weakly continuous as a bilinear form on C (~F I.
When
n>l, use co-ordinates to reduce to the one-dimensional situation above (el. Lemma i).
For uniqueness, observe that 6 is uniquely determined on F n x F n : approximate .ko~ koo vl~{o}, v2 ~{o} E F n weakly by sequences {]o}k=l, {?o]k= 1 in C respectively; note that any weakly
continuous extension ~ of 8 must satisfy
r(vlX{o},
v2X{o}): lira
B({ko, 7to) :
v2X{o;).
k-~co
Lemma 3 :
If ~ @ C , then lira t--~ o+
1
EG(~Xt(')
- 7t )
=
~(H(~) ~/((o } )
for each ~ e C.
Proof :
E denotes expection for C (or R n) - valued random variables.
Since evaluation
534 at each s 6J commutes with expectation in C, then, For each t>o,
[~
~t )
=
E(H(~Xu(.))
--
du
o-°
t+s~o
-r~t+s
s ~ J.
Therefore, lim [~1 E ( T x t ( . ) - ~t)](s) = H(~) ~ { o } ( S ) For all s E J. Moreover, t-~o+ {I~1 E('Txt( " ) - ~t)II ~ K for all t > o; so the result follows From the weak continuity of ~.
Lemma 4:
For each t e (o, r] and a.a. w 6 [B, define Wt(w) ~ C(J, Rm) by F
.
l.~[W(w,
t+s) - W(w, o)]
-t~s~o
Wt(w)(s) = -rss<-t Let B be a continuous bilinear form on C.
1
lim t->o+ Proof
)
[~ E
-
t'
(
.) -
Then, for each ~ .
C, .
t ) - E B(G(7)oWt(.) ' G(2)oWt(.)) ] = 0
:
Let o
E sSUpEJ Ii ~ (~×t - ~t )(s) - [G('?)°Wt ( ' ) ] ( s ) 1 2 2 E < -~
<
sup
s ~ I-t, o]
I (t+s H(qXu)dU 12 2 jo * -t E
~ ' t E I H ( q ' X u ( ' ) ) 1 2 d u +- ~ ' i
s e sup [-t,
l;t+s o] o
[G( ~x u) - G(~z) ]dW(u)i 2
EHc(1%<')>G(~Z)II2du -
0
Letting t--> o+ in the above inequality gives
lira
E
i t ( " j~ i "~x
~-
-
~t ~ -c(2)oWt( " )If~ = 0
t --~ o+ But E II G(~)°Wt(-)ll
x< II G(~)II 2 for all t ~ (o, r], so by continuity of ~ and H'older~s
535
inequality we obtain
I -{ f 6(qxt(.) - I t , qxt(.) - ~t ) - E 8(G(1)oWt(.) , G(])oWt(.)) 1
< lIBIfEII~ <~t(.) -It) - m(1)oWt<.))II2 + 2]I B IIEEII (Ixt(.) -2t ) - C(1)oWt(.)ll2]} EEIIG(1)oWt(.)l[2]½ We get the required result by letting t--~o+ in the above inequality.
Lemma 5: m Let {ej}j: I
be any basis for R m.
i
Then
~
(*) t-)o+lim~ E B(~xt(.) - ~t'
m
~xt(') - ~ t ) : j--~__l~(G(1)(ej))~{o} ' G(?)(ej)X{o} ) for
each ~ ~ C.
Proof:" Observe that the right-hand-side of (*) is the trace of the bilinear map R m x Rm ~ R, (vI, v2) ~ ~(G(~)(vl)}({o}, G(~)(v2)~{o}), and is therefore •
.
.
m
m
independent of the choice of basls in R . So we can assume that e i~ : (6ij)i:l, j=l,...,m, is the canonical basis for R m.
Moreover, using co-ordinates and the
independence of the real Brownian motions {<W(.,t), e > : t ~ R~°}, j:l,...,m, (*) may J be reduced to the one-dimensional case m=n:l, viz.
(**) lim
E B(W[<.), w[(.)) I
t - ~ o+
for one dimensional Brownian motion W, where we have used Lemma 4. To prove (**), form the complete projective tensor product C ~ P Ii h II®w = inf { ~ i=l
iI I iII II
ill
: h =
C with the norm
P i i i i ~___ ~ (9~ , ~ , ~ ~ C, i=l ..... p} i=l
where the infimum is taken over all possible representations of h in the algebraic tensor product C @ C .
Let L2(~, C) denote the Banach space of all measureable maps
$ : ~ ---> C such that the function w ~
II$(w)ll 2
L2(O, C) is given by II #II F [f£II$ (w) l[2 dP(w)] ~"
is P-integerable; But C ( 9
the norm on
C is separable, so for any
536 91' ~2 { L2(o' C) the~ Bochner integral E[WI(.)®92(.)] = ~ well defined in C ~ C. Think of 6 as a member of [C
C
([8] PP. 434-445).
6(E9](.)(9~2(. Z) for all Wl' ~2 ( L2(~' C). define K t( C (9 C by Kt
:
#l(W)(9}2(w) dP(w) is
Fix t ((o, r].
Then E B ( g l ( . ) , ~2(.)) = Then W[ ~ L2(S, C); so
(.)®Wt(.)]
E[W t
By the covariance property of Brownian motion, one gets 1
Kt(s,s~) = [ ] + ~ min ( s , s ' ) ] ~ [ _ t , o ] ( S ) ~ [ _ t , o ] ( S ' ) , _ _
s,s'e
J
Using the last relation, the eigen value problem
[o
Kt(s,s~)
(s')ds ~ = X
I(s), s ~ J
r
can be solved directJy in C, thus obtaining, via Mercer's Theorem ([1] P. 138), the series expansion (***)
Kt
=t k:o
8 ~2(2k+i)2
t,k
i t,k
~
where
i,k (s) = ~ [ _ t , o ] ( S )
cos {(2k+l)2t ~s.}
s ~ J, k=O,l,2,...
The series (***) converges in the norm of C & C uniformly with respect to t E(o, r] due to the fact that II It'k@]it'k II2 .< II It'k i 2 ~< l, k=O,1,2 ..... and the convergence of
Kt(o,o)
= 1 =
2(2k+1)2 k=o
Therefore, lim
t~o+
=
8
k=o
~2(2k+1)2
E 8(Wt(.) , Wt(.)) = lim
t--~o+
lim
t---.e+
8(K t)
X{o )
We may now state the main theorem describing the weak generator A of {Pt}t~o •
537 Theorem
2:
Let ¢ ~ D(S) be C 2 with De, D2¢ globally bounded and D2¢ Lipschitz on C. # ~ D(A) and if {e.}m 1 is any basis for R m then '
j
Then
J:
A(¢)(~) = S(¢)(~) + D$(~) (H(~) ~_{o}) + ~i £j=l D2¢(]) (G(l)(ej),~{o} ' G(7)(ej)~/...{o}) for all ~
C.
Proof : Apply [aylor's Formula ([2] P. 186) to the C 2 function $ getting
(*) ~] E[$(~z t) - $(2)] : ~i [st(¢)(# ) - ¢(?)] + #l E D$ (~t)(~xt _{t ) i + ~ E R2(t)
t>o
- ~ xt _ft)du where R2(L) :]0i (l-u)O2~[~t+ u(Ixt- ~t)] (Ixt-~t'
a.s.
I Since $ ~ D(S), lim ~ [St(¢)(~) - ¢(2)] : 8(¢)(7). t--} o+ For all t > o and 2 ( C, we have I ~l ED¢('~L)(Izt - {t ) _ ~i ED¢(~)(Ixt -~t ) I .< KIi D$(I- t ) -Om(7)il because of temma 3. T1 ED$()
lim
Letting t-e o+ in the above inequality /ields
- 9 t) : D¢(7)(H(~ ) k{o} ')
t--> o+ An easy calculation of fourth-order moments in (SFDE) gives us the estimate
E il ~t
-7t Ll~< 8K4 (t4 ÷ 14~t2)
for all t)o, ~ E C ([5] P. 87). Therefore - + u(IXt ._~t)](~xt - ~ xt - ~t ~ ) - ~i ED2¢(1)(]Xt_ ~t' ~xt _~t) i I ~1 ED2%[~t - It' ,< 2 ~ K2(t2 + 144)½ {ElI O 2 C[~t + u(~xt -~t)] - D2¢(~) II 2}½ ~ as t--> o+, uniformly in u ~ [0,i], because D2¢ is Lipschitz.
0
By Lemma 5, we get
538 lim ~1 ER2(t) = I 1 t --~ o+ o
(i-u) lim ~1E02¢(~ ) (Iztt--> o+
= 2j = l~l
D2¢(7)(G(2)(ej ) ~ { o } '
- , I x t -~t) 7t
du
G(2)(ej)~{o})"
As ¢4 Dis) and De, D2¢ are bounded on C, a l l terms on the right-hand side of (*) are bounded in t a n d s .
Hence ¢ E P(A) and r e s u l t holds.
References i.
Courant, R. and Hilbert, D.:
Methods of Mathematical Physics I, Interscience,
New York (1953). 2.
Dieudonne, J.:
Foundations of Modern Analysis, Academic Press, New York (1960).
3.
Dunford, N. and Schwartz, J.T.:
Linear Operators I:
General Theory,
Interscience, New York (1958). &.
Dynkin, E.B.:
Warkov Processes I, Springer-Verlag, Berlin (1965).
5.
Friedman, A.:
Stochastic Differential Equations and Applications I, Academic
Press, New York (1976). 6.
Mohammed, S.E.A.:
Markov Solutions of Stochastic Functional Differential
Equations, Proceedings of the Second International Conference on DifferentialDelay Systems and Related Topics, Poland (1981), (To appear). 7.
Mohammed, S.E.A.:
Stochastic Functional Differential Equations, Research Notes
in Mathematics, Pitman Publishing Ltd., London, (To appear). 8.
Treves, F.:
Topological Vector Spaces, Distributions and Kernels, Academic Press
New York (1967).
Acknowledqemen~:
I am g r a t e f u l to K.D. E1worthy and K.R. Parthasarathy for h e l p f u l discussions during the preparation off t h i s paper.
ON
SOLUTIONS FOR
TO
THE
PERTURBED
INITIAL-BOUNDARY
POROUS
MEDIUM
MITSUHIRO
PROBLEM
EQUATION
NAKAO
Introduction
§i
In and
this
article
asymptotic
equation
we
are
behavior
with
blowing
of un
~-~u - &u p+
u(x,O) where
~
is
nonnegative When
p : 0
investigated [6]
to
by
the
and
i
[5]
to
give
the
the
in is
nonlinear
u(x,O)
solutions
to
: O
on
concept
with
existence,
nonexistence
the
medium
porous
arbitrarily
as
is w e l l
13 ~ = O
fixed
u ~ O ~
and
positive
was
extended
are
time.
problem by
p
(I)
was
M. T s u t s u m i
equation:
u a+l
~xi ~xi
]
,
IZ ~ = O
u(x,t)
'potential global
~,
the
result
(i)
and
known,
'-~-~u = ~ u]
of
~ × (o,T)
boundary
an
parabolic
~
on
u(x,t)
Rn
His
= u0 (x)
criteria
the
(given),
[3].
~
used
~+i
~ > O,
~-~u - i:~ ~
Tsutsumi
- u
T
Fujita
t~pical
with
term:
domain
and
H.
global
= u0 (x)
a bounded numbers
concerned
well'
O
and
on
~ × (O T)
u ~
O.
introduced
existence
and
by
D.
nonexistence
(2)
Sattinge: of
solutions. The for
(i)
first as
delicate
estimate e >
similar
p
and
technique
The ~ p
[6].
is
problem open.
of
We
difficulty
precise when
in
object
for
note
to The
article
that
is
the
that
in
(2).
the
rate
of
decay
solution
in o u r decide
previous the
estimation
discuss
the
same
-Au p+l
Droblem
causes
second
object
is
of
IIVuP+l(t)IIL 2
as
t -->
u(t)
is
For
this
paper
[4]
the
case
of
admitted.
give
a
a
is u s e d .
solutions
Ifu(t)llr ~
to
more
The
behavior of
to
nonlinearity
than
a global as
this
for
the
for
case
~ > p
as
540
in p o r o u s
medium
L.
Peletier
§2
Global Our
Existence
the
reads
Case
as
admits
that
u 0 (t) I> O
a global
solution
~t ( u P / 2 + l ( t ) ) and
[ i],
D.
Aronson
&
open.
for
here
Alikakos
p >
follows:
1
Suppose (i)
(see N.
[ 2] ) is a l s o
result
Theorem
equation
the
equation T
is
u(x,t)
C L2 ([O,T] ; L2 (~)),
satisfied
in the
10 ~ - { -uu ( x~, t )+ ~ tl( x ,~t ) (
sense
Then
the
problem
u P + l (t) E L ~ ( [ O , T ]
;HO (~)) 1
that
x+ v ,( u Pt+ I ))v ~ }( x ,dt ) x
lu(x,O)~(x,O)dx
-
E HO (~). 1 such that
u 0 p+l
and
d
t
.
(3)
: O
J
for
V % C C] ([O,T] ; H 0) ]
estimate
holds: t+l _ ~
with
%(T)
2
(uP/2+I(s))
~s
ds
: O.
+ flup+
1
Moreover,
(t)II
following
~< C(IIu0P+lll
H0 1
L2
t for
2
the
H0 1
O ~< V t ~< T. For
the
proof
of
Th.l
and
also
{.
for
later
use
we
rewrite
(i)
as
fo i lows :
where
we
~ t (IUi-p/(p+l) u) - A u -
~(u)
= 0
U (x,O)
and
U ~> O
= U 0,
set U : U p+I
O
if
U I3Q, : 0
(4)
, U 0 = u0 p+I
%' (U) = U ( ~ + l ) / ( p + l )
and
if
U >I O,
U ~<0. TO
avoid
the
singularity
[~t(B~(u)+~u)
- Au-
in
(4) we
~(u)
introduce
=o,
the
modified
~ >o,
(~)
!
: Uo ,
[u(x,0) where
we
uI~
: o,
u>o
set ~e(U)
=
8](q)dq
and
B'(U)
=
(p+l)
equat/or
-I (IUI + £)-p/(p+l)
541
Now,
we
begin
{wj
j=l
be
Let
differential
the
proof
a basis
of T h . l .
of
H 0 (Q) ]
We
and
the
Galerkin's
consider
the
system
method.
of ordinary
equation
((~(Um,s(t)) + s)~tUm, c (t),wj)
t
employ
U m, s(O) : U o,m c [wl,
+
(VUm,s ( t ) , V w ) 3
(~(U m , E (t)),wj)
-
= O,
(6)
j = 1,2 ..... m
i
"'''w m ]
m
where way
Um,s(t)
that
= j~iv lj,m(t)w.3
U
~ U --~
0,m unique by
ij
local tm
in
(t)
H0 1
0
solution
U
and
integrating
I:!I
j
t E [O,T
mr
J(U)
s]
-
,
Since
p > ~, J(U)
we
p
The
on
[O,T
be
chosen
system
mt~
in s u c h
(6) a d m i t s
].
a
a
Multiplying
(6)
= J ( U m, s(O))
(7)
get dxds
+ J (Um,c (t))
set
P
+
l
+
~
+
~(u)u
dx.
(8)
2
defined
< c +
should
o
is w e l l
~(U)Udx
, say,
we
IlVgll L2
U0, m
m --~ ~
--~ ~t U m , s
where
.
as
(t)
m,s
8 ~ ( U m , C) + s
for
and
and
it h o l d s
that
HVUII
(9) H 0
From
(7)-
II
9) w e
(V
see
that
(t)
mrs
+ gll
(t))IT m,S
U
L2
exists
(t) ~t
m,S
on
[O,T]
d t +IIu L2 )
and
(t)ll m,s
~< c < ~ H0
(iO) '
I IIv (t)ll ~< c s - p / 2 (P+I)IIVU (t)ll ~< c < ~ m,c H0 m,S L2 g 1 U where
we
Applying
V
standard
converges ively,
set
as
m,S
Iore's<
(n)dn
compactness
m -~. ~
in v a r i o u s
=
(along
topologies
ll-~-,Vg(t)ll
+s
argument
we
see
a subsequence)
to
and
it h o l d s
II
that
U
Us
and
Vs,
(t)ll
+ IIVU
L2
S
~
u Us
and
V
are
connected
by
Vs
and
m,s
V
m,¢
respect-
that
dt
L2
where
(ii)
=
/~
(n)dn
.
<~
(12)
542
U
8
becomes
vergence
a solution
of
U
V for
each
t.
2
h =
observe
with
- V
I
(h 1
we
problem
@ :
(5).
To
show
the
con-
that
(p+ 2)/2(D+
i)
and
q = 4(p+l)/(p+2)
In f a c t ,
lV~(x+h,t)
~< c
the modified
8 --~ O
(t) • L i p ( @ , q )
I for
as
8
of
IUc(x+h't) ..,h
'"
n
)
(x,t)14(p+l)/(P+2)dx
2 2 2 - U c (x 't) I d x 4 cIIu s (t)ll 0 lh] H1 (We d e f i n e U (x + h , t ) = O if x + h ~ ~2), 8
and
hence rlvs(t)IF
~< Lip (8,q)
Noting
that
elude
from
L2(~))
and
Lip(@,q)
(12)
and
a.e.,
(13)
and
U
where
ogies
U(x,t)
that (t)
V
(t)
--~ 3 V ( t )
converges
to
strongly
U(t)
(131 con-
in L 2 ( [ O , T ] ;
in a p p r o p r i a t e
topol-
u 1 [0 /p%T
v
becomes
nonnegativity and
is
C IIV II + IIUsII (p+2)/2(p+l)] < c 8 q H0 1 compactly imbedded in L 2 (~),
setting
In]-P/2(p+l)d~
a desired
of u
u(x,t)
solution
lu-(t)
(14)
of
(4).
--- I U I - P / ( P + I ) u .
= min(u,O)
p -~ 2
" It r e m a i n s Noting
to p r o v e
u'(t)
the
• L~([O,T]:H -I )
we h a v e IP+2dx
=
<
lu-lPu->
u,
H -I x
H0
2
= -IlV(lu-[Pu-)llh2 implies
which §3
Global For
assume
u-(t)
Existence
the
it
for
the
Under
is e a s y
i.e.,
Case
J(U)
potential ~ :
>/ O.
p <
to b e w e l l - d e f i n e d
2) + 4 ) / ( n -
2)
if
this
assumption
we
set
to
well
n >~ 3
for and
U C H0 1 O < ~ < ~
SuP J(IU). I~>0
see
~ > d i> c ( ~ - p ) / 2 ( p + ~ The
u(t)
(p(n+
d --- i n f uE H 0 1 U~O Then
,
functional
O < ~ ~<
n = 1,2.
~ 0
< O
associated
{U • H01
i) > O. with
0 ~< J ( I U )
< d
(i) or for
(4)
is d e f i n e d
V
I • [O,i] }.
by
we if
543 Lemma
i (Tsutsumi
[5]).
W. where
"~P". : {U e H 0 lit VUII2 ] The
purpose
solution Lemma
if
of
] ~ (U)Udx
-
> O
and
J(U)
< d).
L2 this
U 0 ~ u0 p + I
section
is
to
prove
that
(i)
has
a global
@FO~.
2 Let
U 0 C~.
through
(6)
Then
exist
V t @ [O,T],
on
and
IIUm,c(t)ll
H0
the
approximate
[O,T]
(for
large
solutions m)
and
U
U rn,[ (t)
m,~
(t)
C
constructed
~
for
consequently
+
/~
(Urn s ~
(t)
C ~t/ f o r
U
,
+ +u~ urn,+(s)il2)ds ~
HL2
m,C
] Proof.
If
Um
,c
d > J(U
(t)
>~
together
assertion U By
we
(7)
assume
(t)
m,s
Lemma
with
it w a s
m,<
the
(15)
false.
and of
[O,T]
I[VU
yields
~
1 either
t C
a - p 2(p+a-2)
m,s which
V
U
rn,~
(t)
J(U m
~(t
we
) = d
or
(ii)
cases
IIVU
can
(t)IE
1 that
(16)
To
O ~< V
must 2
m,~
Lemma
prove
choose
for
.
,
from
(t)[l 2 L2
~
.
(i)
see
immediately. Then
following
we
t* t <
:
former
> O such
t
that
"
hold: ~
L2
the
.
j~(U m
,ff
.
(t))U
m
,~
(t
dx.
,9, Both
cases
are
easily
Using
the
estimate
Theorem
(15)
to we
contradiction. can
obtain,
as
in
§2,
u0 ~ 0
solution
uP+!(t)
and such
e~and
Uo p + I
C~.
Then
the
problem
followzng:
IIvuP+I (t)ll
for
t ~
~4
Decay
of
Solutions
Here
we
shall
Th. 2.
(1)
admits
a
that 2 +
N
(t)
L2
in
the
2
Let global
reduced
II dt L2
< c <
[O,T] .
prove
for
the Case
the
following
~ > p. decay
property
for
the
solutions
544
Theorem
3
The s o l u t i o n s
in Th.
t+ll]~s u P / 2 + 1
[l2 ~ L2
t where
c1
Proof. ity, that does
is a c o n s t a n t
It suffices
however, u(t)
the e s t i m a t e
+ IlVUP+I(t)IILz L_ C t ( [ %
depending
to d e r i v e
we treat
on
solution
the e s s e n t i a l
(17)
d - J(U 0) but i n d e p e n d e n t
the e s t i m a t e
the e q u a t i o n
is a smooth
not change
2 satisfy
for
U m,E (t).
(i) d i r e c t l y
on
[O, ~)
feature
under
with
T.
For s i m p l i c -
the a s s u m p t i o n
uP+l(t)
of the proof.
of
E~,
which
First we o b s e r v e
that lIVup+l (t)ll 2
>~ [
d
] (~-P)/2 (P+I) Iu P + ~ + 2 d x
(18)
L2 for
O < V £0 < d - J(u0P+l).
u(t) p+l C ~ ,
the i n e q u a l i t y
multiplying
(i) by
-dE
J(uP+
On the other
up+ 1
~[ (t) =
hand,
This (16)
is d e r i v e d
from the fact
and the d e f i n i t i o n
of
d.
Next,
we have
(p + i) uPl
multiplying
u[ dx
(2) by
llVup+l (t)ll 2 = - ; u p + l u t d x L2
(19)
.
u p+I
, we have
+ Iu P + e + 2 d x
+
IIVu p+I (t)II
(by
(18))
L2 and hence,
with
the aid of S o b o l e v ' s
IIVU p+I (t)l] L2 2 ~<
Cl [(- d-~
Lemma
and
(19), (p+2) /2 (p+l)
J(uP+l(t))) ] ½llvuP+l(t)ll L2
or d j (uP+l (t) /> Cl[[ Vu p+I (t)II (3p+2) / (p+l) >/ClJ(uP+l(t)) (3p+2)/2(p+l) dt (20) where (20)
c1 and
denotes (16) y i e l d
constants (17).
depending
on
d - cO.
The
inequalities []
545
§5
Local
Existence
In t h i s if
section
J ( u 0 P+I)
Theorem
< 0
we
and
u(t)
be
t E [O,T] ~(u ~t
Then
Blow
up
shall
show
u0 ~ O
we
T
of
C L2([O,T]
;L 2)
we
can
if
not
the C a s e
u0 p+I expect
~ ~, the
~ > p. more
global
precisely
solutions.
(i)
such
that
J ( u p+l (t))
~< O
for
and
u P+I
@ L~([O,T]
;H0). I
(21)
-p
flu011L +2
have Iiu(t)IiLP+2
Remark
The
J(u0 p+I)
~> c(T0 - t) - I / ~
condition
~< O
(see
of Th.
4.
~< 0
~c
> O.
follows
formally
from
(7)) .
Multiplying
I
implies,
for
J ( u p+l (t))
1 p+2 p T 2 l!u(t)l[LP+2 which
that
for
satisfy = p ~ ~+ ( ~2_ ~+)
Proof
Solutions
and
P/2+I))
must
a solution
T < TO and
of
4
Let a.e.
and
by v i r t u e
(1) by
- llu0[lp+2 LP+2 of
u p+l (t)
and
integrating
Is0 +
tlIvuP+l (s)[l 2ds L2
J(uP+l(t))
we
get
uP+~+2dxds : 0 f~
~< O, ]-(p+2)/~
Ilu(t)II p+2 LP+2 Finally
we
give
~
IiuolI-~ LP+2
a local
- ~(a - P) t p + e + 2
existence
Theorem
of
solutions
which
satisfy
J ( u p+l (t) ~< J(uo p+l) .
Theorem
5
Let
us
O < ~ < ~ exists
assume if
n = 1,2.
T I ------T 1 (u 0) > O
interval
[ O , T I] 4(P+I) (p + 2) 2
for
O < ~ <
a.e.
which
{ (n + 2)p
Then, such
for that
satisfies
V u 0 >i 0 (2) (21)
(uP/2+l(s))II
t E [O,TI] .
+ 4}/2 ( n - 2)
ds L2
admits
if
with
n ~> 3 Uo P+I
a solution
and
@ H0 l u (t)
there on
the
and
+ J(uP+l(t)
~< J ( u o P + l )
(22)
546
Proof.
Let
Sobolev's
U
m,E
Lemma
(t)
be a p p r o x i m a t e
(the a s s u m p t i o n
( B / ( U m 8 ) ÷ 8) I
on
Um 8 ( t )
,
s o l utions. ~
is used here)
2
from
(7) and
we have
IIVUm,a(t)ll L2
I~dx +
,
C II VUm, a (t)II
Then,
+ 82 ] (p+2Ct+2) / ( p + l )
L 2
which
shows 2
IIVu
m,8
(t)ll
{
(2~_ p)
~
L2
ttVu
Thus we can c o n c l u d e
J0 VS 8 (Um ,8) + 8
m,c
(o)11 2
that
~t
+
c
~2
~2
for some
IT
Um, 8 L 2
} - 2 ( I + P )/(2~-P) -
- 2V~lT
8
2
t
T I ~ T1 [IIVu~+IIIL2
dt + [IVUm,a(t)N
L2
~ c
for
t e
[O,TI]
, (23)
from w h i c h we can p r o v e To s h o w t
the e x i s t e n c e
(22) we first n o t e
of local
solution.
that
! "/ ' (s)) + < ~ ~ s U8 (s)''2 ds + J(U (t)) ~< J(U o) 9 @c (Us L2 8 This in
follows
from
(7), (23)
L (p+e+2)/(p+l)
Next,
il~t~ 8 (U 8(t))I[
Lq([O,TI] ; L q) J(U(t))
U
and the fact that by
(5) and
H 0 is c o m p a c t l y I (23) we can o b t a i n
(24)
imbedded
< c < ~ and [I88 (U 8(t) )il < c < Lip (i/ (p+l) ,2 (p+l)
H- i F r o m this we see
, a.e.t.
c
with
(t) i/(p+l)
__~ UI/ (p+l) (t)
q = p + ~ + 2
~< lira J(Us(t))
for
strongly
in
and a.e.
t @ [O,TI] .
[]
References i.
N.D.
A l i k a k o s , L P - b o u n d s of s o l u t i o n s of r e a c t i o n d i f f u s i o n e q u a t i o n s , Comm. P.D.E. 4(8) (1979), 827-868.
2.
D.G. A r o ~ s o n & L.A. P e l e t i e r , L a r g e time b e h a v i o r of s o l u t i o n s of the p o r o u s m e d i u m e q u a t i o n in b o u n d e d d o m a i n s , J. Diff. Eqs. 39 (1981), 378-412.
3.
H. Fujita, On some n o n e x i s t e n c e and n o n u n i q u e n e s s t h e o r e m s for n o n l i n e a r p a r a b o l i c e q u a t i o n s , Proc. Symp. in Pure Math. 18, Amer. Math. Soc., P r o v i d e n c e , R.I. (1970), 105-113.
547
4.
M. N a k a o ,
On s o l u t i o n s
~--[ u - £8(u) 5.
D.H.
6.
M.
Sattinger, equations,
= f,
to t h e
initial-boundary
to a p p e a r
in J. Math.
value Soc.
problem
fo:
Japan.
On g l o b a l s o l u t i o n of n o n l i n e a r h y p e r b o l i c A r c h . Rat. Mech. A n a l . 30 (1968), 1 4 8 - 1 7 2 .
T s u t s u m i , E x i s t e n c e and n o n e x i s t e n c e of g l o b a l s o l u t i o n s for n o n l i n e a r p a r a b o l i c e q u a t i o n s , Publ. R . I . M . S . K y o t o Univ. 8 (1972/73), 2 1 1 - 2 2 9 .
A Survey of Global Properties of Linear Differential Equations of the n-th Order Po Neuman
I.
Introduction Investigations
of linear differential equations started in
the last century. In 183~ E. E. ENmmer ~ l ~ about transformations
published his results
of second order equations,
papers appeared devoted to transformations their canonical forms and invariants.
and then several
of higher order equations,
From many authors let us menti-
on at least E. Laguerre, A. R. Forsyth, F. Brioschi, G. H. Halphen ~l~ , P. St~ckel ~33~, S. Lie, and E. J. Wil~ynski ~34~. Among their results there is, e.g., the so called Laguerre-Forsyth canonical form of linear differential
equations of the n-th order characterized
by vanishing coefficients by the (n-l) st and the ~n-2) nd derivatives. However as lately as in 1893 P. St~ckel E 33~ proved that the transformation considered by Kummer is the most general transformation that converts any linear homogeneous differential
equation of the
n-th order (n~ 2) into an equation of the same kind. Anyhow all their investigations were of local character that was pointed out by G. D. Birkhoff ~l] already in 1910. He gave an example of the third order linear differential be transformed into the Laguer~e-Forsyth f o ~
equation that cannot
on its whole interval
of definition. Of course, the local nature of results is not suitable for investigations of global problems, solutions,
like boundedmess or periodicity of
their convergence to zero, their asymptotic properties
and oscillatory behaviour,
or whether they belong to L 2 or L p, etc.
549
In the first half of the century some isolated results of global character occurred,
as e.g., G. Sansone's example of the
third order equa~ion with all oscillatoric
solutions, ~32].
A systematic study of global properties of second order linear differential equations started about 30 years ago by O. Bor~vka. He summarized his deeply developed theory in the monograph [2] and in further works ~3, #, 51 • In the last lO years there was also found enough general approach to global problems of linear differential equations of arbitrary order, This investigation is based on 0. Bor~vka's methods, it discovers what is common for the second order and higher order equations, it shows what is specific for the second order, and introduces methods suitable for equations of arbitrary orders. Here we give a survey of main definitions, results of the approach to global properties
methods and some
of linear differential
equations of the n-th order, n ~ 2. Algebraic,
topological and geometrical tools together with the
methods of the theory of dyn~uical systems and functional
equations
make it possible to deal with problems concerning global properties of solutions by contrast to the previous local investigations. For example; the structure of the set of all global transformations of linear differential
equations is described by algebraic
means (theory of categories-Brandt
and Ehresmann groupoids),
con-
struction of global canonical forms is given by methods of differential geometry (including E. Cartan's moving-frame-of-reference
met-
hod). The theory includes also effective methods for solving several special problems,
e.g., from the area of questions concerning
550
distribution of zeros of solutions of linear differential equations.
I I.
Basic definitions and notations Let Ck(I) be the class of real scalar or vector functions with
continuous derivatives up to order k =~ 0 inclusive on an open interval I¢ ~. Consider a linear homogeneous differential equation of the n-th order, n @ 2, with continuous coefficients on I, i.eo
y(n)+Pn_l(X)y(n-1)+...+Po(X)y
= O,
where pi ~ C°(1), Ia ~, i = 1,..., n-1. Denote by Pn(Y;I) this differential equati on. Similarly Qn(Z;J) - z(n)+qn_l(t)z(n-1)+...+qo(t)z = O, qi ~ C°(J), i = 1,..., n-lo Sometimes we write briefly Pn and Qn instead of Pn(Y;I) and Qn(z;J), or simply P and Q onlyo Let Y l ' " ' ' Y n be n linearly independent solutions of the equation Pn(Y;I). Denote by ~: = (yl,...,yn)T the column vector with coordinates formed by the solutions Yi" Then ~ 6 cn(I) and the Wrons~ian
w[~(x)], : det (Z(x), Z'(x),. .. ,z (n-l) (x)) is different from zero everywhere on I. Analogously for Qn(Z;J) we have z & cn(j), W[z(t)] ~ 0 on J. We say that Pn(Y;I ) can be globally transformed into Qu(z;J), if there exist 1 ° a function f~ cn(j), f(t) ~ 0 on J, 2 ° a bijection h e C n ( j )
of J onto I,
551
such that dh(t)/dt ~ 0 on J J 3 ° a constant regular n by n matrix A, such that 2(t) = A . f(t) . z(h(t))
for all t~ J.
(I)
Due to Pw St~ckel [33], the form of the transformation (1) is the most general one that keeps linearity, homogenity and order unchanged. The bijectivity of h guarantees the globality in our definition. If the coefficient Pn-1 of Pn(y;I ) is identically zero on I, we shall express it by the superscript o as P~(y;I). Consider
P~(y;I)
~ y',+po(x)y = o ,
(2)
P o e cn-2(I). Let Yl and Y2 be its two linearly independ solutions. Then Yl' Y2 e cn(I), and n functions defined by zi(x): = y
(x) .
(x),
i = 1,..o,n,
belong to the class cn(I), their Wronskian being nonzero everywhere on I. Hence zi, i = 1,...,n, can be considered as n linearly independent solutions of the so-called iterative equation of the n-th order with respect to (2). Let Itn[Po ] denote the differential operator of this equation normalized by unit leading coefficient. It can be shown (see, e.g.,[13~ ) that Itn~Po~ = z(n)+(n~l)po(X)Z(~'2)+~(n~l)p~(x)z(n-3)+ . . . .
III.
Structure of globa I transformations The relation of global transformability is an equivalence re-
lation. Hence the set of all linear differential equations of any order n, n ~ 2, is decomposed into classes of globally equivalent equations.
552
Let D be one of these classes, P, Q and R be equations from the class. Since they are globally equivalent, there exist global transformations, say~-1 and
~-2' such that ~ l
transforms P global-
ly into Q, and ~'2 transforms Q into R. Denote the fact shortly as
~P
: Q
and
~ Q : ~.
We may also define the composition of the transformations ?'2 @ ~'l" It is again a global transformation converting P into R. As the identity of P we define the transformation (i) with I = J, A being the unit matrix, f(t) = i for all t ¢ I ,
and h = idi, i.e.,
the identity of the interval I. Since each transformation ~ has its inverse @--i, we may conclude ~26] that Theorem I.. Linear differential equations of all orders n, n ~ 2, as objects and their global transformations as morphisms, form
&~
Ehresmann groupoid. Each class of globally equivalent equations is a Brandt groupoid. The basic structural notion of every Brandt groupoid D is the so-called stationary group S(P) of its element P formed by all global transformations ~ t h a t
s(P):: { ~
transforms P into itself, i.e.
~P=P}
It can be shown [12, 26] that stationary groups of two globally equivalent equations are conjugate. Moreover, it holds Theorem 2. There are 5 possible types of classes of globally equivalent equations with respect to their stationary groups. Stationary group of a (then any) element of D
(TI)
consists of identity only
(T2)
is an infinite cyclic group
(T3)
is a one-parameter group
(T4)
is a two-parameter group
(TS)
is a three-parameter group.
553
The cases (TS) and (Ta) occur exactly when D contains an (then only ) iterative equation It~Po]. Then (TS) holds for bothside oscillatoric
equation (2), and (Ta) when (2) is one-side oscil-
latoric, or of the special kind. The case (T3) occurs just when D is neither of typ (TS) nor (T~), and a linear differential equation with constant coefficients belongs to D. D is of the type (T2) if and only if it is not of types (TS), (T¢) and (T3), and D contains a linear differential equation with periodic coefficients of the same period° The proof of the theorem is based on investigations tain vector functional equations
of cer-
[25] and on the latest results
of O. Bor~v~a [5] about one-parameters groups of transformations of reals. For the definition of the second order equation of the special kind, see [2]. There are several results describing structure of global transformations Theorem ~.
[26]. E.g.: Let ~- transform an equation P globally into Q.
Then all global transforaations
of P into Q form the set
where S(P) is the stationary group of P.
IV.
Canonical forms
Some linear differential
equations of a special form are called
canonical. E. go, the Laguerre-Forsyth canonical form is characterized by vanishing coefficients near the (n-1)st and the (n-2)nd derivatives. A given canonical form is called global, if there is a linear differential
equation of the canonical form in each class of
554
globally equivalent equations. A canonical form is called unique, if there is at most one linear differential equation of the canonical form in each class of globally equivalent equations. We say that a canonical form is effective, if each linear differential equation can be transformed into the given canonical form effectively from its coefficients. The effectivity must be specifie@
by listing operations which are considered as effective.
In this paper quadratures and compositions of known functions are considered effective. It is known [1, 31] that the Laguerre-Forsyth form is neither global, nor unique, nor effective. We shall describe one of the possible constructions of a global canonical form [20]. Consider a linear differential equation Pn(Y;I) and one of its n-tuple of linearly independent solutions forming a column vector function ~. All such vector functions ~ for all equations globally equivalent with our Pn(Y;I) are obtained from formula (1). For getting a special form of differential equations we need to specify f and h in this formula. We shall do it in the folloving way. We consider the vector function ~ as a curve in the n-dimensional euclidean space E n with the euclidean norm
I.i~
We take the central projection ~ of ~ onto the unit sphere Sn_ 1 in En, i.e. ~:= ~/I ~i • Then we introduce the length parametrization into ~ to get ~ (t); t 4 J. It can be shown that ~ c n ( j ) , W [~(t)] ~ 0 on J. Hence coordinates of ~ can be considered as n linearly independent solutions of a certain linear differential equation that may be called canonical° This construction can be done without any restriction on smoothness of coefficients or length of the interval of definition.
555
For the explicit form of equations in our canonical form, we utilize methods of differential geometry. The Frenet Formulas for our curve ~(t), t ~ J, in the group of orthogonal transformation~are
U_l =
£2
u2 = --~i
+ kl(t)~ 3 -kl(t)~ 2
+k2(t) ~
o@o
Un_l:
-kn_3 (t)~n_ 2
+kn_2(t)~ n -kn_2(t)~n_ 1
,
where (~l,...,~n) is the moving orthogonal f r a m ~
I = ~, curvatures
ki(t) ~ 0 on J, k i E cn-i-l(J), i = 1,o..,n-2. By evaluating ~l = ~ from the system we get our canonical form explicitely. E.g., for n = 2 and 3 we have Theorem 4. u" + u = 0 on J
is a global canonical form for linear differential equations of the second order (it depends on the length of J).
u"'- kl/kl.U" + ( l + k ) u ' -
kl/kl.U = 0 on J,
k I = kl(t) ~ 0 on J, kl~ CI(j) is a global canonical form for linear differential equations of the third order (it depends on one function, kl). There is also another way how to get global canonical forms. It can be proved [31] Theorem 5o
Let y" + p(x)y = O, p ~ cn-2(I),
be both-side oscillator~¢o Then n
u (n) + (n~l)p(x)u(n-2)+
~__~ rn_i(x)u(n-i) = o i=3
556
on J ~ I ,
p fixed, rn_i~ C°(J) for i = 3,...,n being n-2 arbitrary
functions,
is a global canonical form for linear differential
equa-
tions of the n-th order with coefficients of the class C n-1 by the (n-1)st derivative and of the class Cn-2 by the (n-2)nd derivative. Especially me have CorollarFt
u(n) + u(n-2) + rn-3 (x)u(n-3)+.. on I ¢ ( - ~ , ~ ) ,
"+
ro(X)U = 0
(3)
is a global canonical form.
If Laguerre and Forsyth had required the first three coefficients l, 0, 1 instead of their l, 0, O, they would have got our global form (3) instead of their local one. All introduced here global canonical forms are not, however, unique. Using Cartan's moving-frame-of-reference for the case of centroaffine
method [6, 28 ]
(and not only orthogonal) transforma-
tions, we get unique canonical forms: u" + u = 0 u'"+ 2 W v ' +
for n = 2 (@'+l)v
= 0
u '~ - io( ~ a + o~'+/~ )u" - io( ~ at
+(9(~2+
~-+>)2
for n = 3
~'+/~ )~'-
- 3(~2+ ~'+/3)")u
5u"
+ 3/Su = 0
f o r n : 4,
/3 In fact we get special types of differential
equations similar
to those already considered by G. H. Halphen Ill]. It can be shown that the types are local [31].
V.
Asymptotic behaviour of solutions Global canonical forms enable us to "coordinate"
differential
each linear
equation. We can do it in the following way.
557
Consider an equation Pn(Y;l ). It belongs to a certain class D of globally
equivalent
equations.
Since our canonical form is glo-
bal, there exists an equation Kn(U;J) Thus K n can be globally transformed Z(x)
= A.f(X)o~(h(x)),
of the canonical form in D.
into Pn,or explicitely
x ~ I.
Hence each linear differential
equation Pn can be determined by a
differential
equation in a global canonical form, Kn, and by a cou-
ple
of a factor f and a change of parameter h that form
h~
the transformation
converting X n into Pn"
Using this approach, equivalently solutions
some global properties
of solutions were
expressed in terms of En, f, and h. Thus
[15, 17, 30], asymptotic properties
ction with the distribution
of solutions in conne-
of their zeros were studied in El6,
A deep study of square integrable linear differential
periodic
21].
solutions of the second order
equations P~(y;I) in connection with the so-call-
ed limit circle and limit point classification
was done by W. N.
Everitt [8, 9]. Let us mention an application
of our approach to these area of
problems. We have n = 2. Fix K 2 ~ u"+u = 0 on J; cf. Sect. IV. To each P~(y;I)
there correspond
a couple < lhSl'l/2,h >
dh(x)/dx ~ 0 on I, h(I) = J~ cf.
, where h ~ C3(I),
the notion of phase in [2]. Now,
we can prove that a. each solution of P2o is square integrable if and only
if~Tlh'(x)l-Idx<~
,
o is bounded b. each solution of P2 lh'(x) l-I is bounded Hence i~ holds
on I if and only if on I.
558
Theorem 6.
There exists an equation P2 (y;I) with all square
integrable solutions, i.e., which is in the limit circle case, and still with an unbounded solution. For more details about these problems and f~rther results, see [18, 19, 23, 29]. VI.
Criterion of global equivalence A criterion for global equivalence of two given linear diffe-
rential equations is based on the following theorem. Theorem 7-
Let n~ 2 be an arbitrary integer. Consider linear
differential equations Pn(y;I ) and ~(z;J) with Pn_2 ~ cn-2(1) and qn-2 g cn-2(J) ° Then we can write pO(y;i ) - Itn[Pn_2/(n~l)] + rn_3(x)y(n-3)+''" ...+ ro(X)y = O, and Q~(z;J) ~ Itn[qn_2/(n~l)] + Sn_3(£)z(n-3)+... ...+ So(~)z = O. The equation P~ is globally equivalent to ~
if and only if
there exists a function h ~cn(j), dh(t)/dt ~ 0 on J, such that rn_3(h(t)).h'3(t ) = Sn_3(t )
on J,
and rn_~(h(t)).h'$(t) = Sn_~(t) on Jn_~:= ~t~ J; Sn_3(t) = 0 } , and rn_5(h(t)).h'5(t) : Sn_5(t) on Jn_5:= ~t~Jn_$;
Sn_$(t) =0}, etc.,
559
and moreover,
the differential
y" ÷
equation of the second order
: o
on I
is globally transformable by the transformation z" + qn_2(t)/(n~l).z = 0
< l h ' l - I / 2 , h > into
on J.
Remarks. Except the case of iterative Sn_ 3 = Sn_ ~ . . . . quadratures,
equations,
i.e., when
= s o = 0, the function h can be expressed by
hence except the case the criterion is effective.
Sufficient and necessary condition for global equivalence of two second order linear differential o. Bor
VII.
equations was found by
vka [2].
Zeros of solutions Perhaps one of the mostly considered proble~n for linear diffe-
rential equations is the question of distribution of zeros of solutions (their oscillatoric behaviour,
disconjugacy etc.)
The essence of our approach to the problem of distribution of zeros of solutions can be expressed in the following theorem, first given in [20], and then used in recent literature, Theorem 8.
e.g. [loj.
Let coordinates of a curve ~ in the n-dimensional
vector space be linearly independent solutions of a linear differential equation of the n-th order, Pn(y;I). There is a 1-1 correspondence between all solutions of Pn(Y;I) and all hyperplanes passing through the origin, in which parameters of intersections of the curve ~ with a particular hyperplane are zeros of the corresponding solution,
and vice versa,
occur as the order of contacts.
counting multiplicities
that
560
If morever, ~ is considered in the n-dimensional euclidean space, the central projection ~ of ~ onto the unit sphere Sn_ 1 is taken (without any change of parameter, then the same situation is in the compact space, Sn_l, and we have to our disposal strong tools of topology as well. Using this approach several open problems were solved and the essence of some theorems with complicated proofs became clear, [7, 2~, 29]. Constructions of linear differential equations with some preseribed properties of zeros of their solutions can be sometimes seen from drawing a curve on the sphere without lengthy calculations. As an example let us draw a picture of a curve in 3-dimensional space, ensuring the existence of a linear differential equation without oscillatory solutions, but still having solutions with arbitrary (finite) number of zeros.
{o,o, o)
(o,o,I)
561
References 1.
Birkhoff, G. D.: On the solutions of ordinary linear homogeneous differential equations of the third order, Annals of Math. 12 (1910/ll), 103-124.
2.
Bor~vka, 0.: Linear Differentialtransformationen 2. 0rdnung, VEB, Berlin 1967; Linear Differential Transformations of the Second Order, The English Univ. Press, London 1971.
3.
Bor~vka, 0.: Teorija global'nych svojstv obyknovennych linejnych differencial'nych uravn~nij vtorogo porjadka, Differencial'nyje uravn~nija 12 (1976), 1347-1383.
4.
Bor~vka, 0.: Algebraic methods in the theory of global properties of the oscillatory equations Y" = Q(t)Y. Equadiff IV - Proceedings, Prague 1977, Lecture Notes in ~athematics 70~, 35-45.
5.
Bor~vka, 0.: Sur une classe des groupes continus & un param~tre form6s des fonctions r~elles d'une variable, Ann. Polon. Math., to appear.
6.
Caftan, ~.: La theorie des groupes finis et la g6om6trie diff6rentielle traitees par la m6thode du repute mobile, GauthierVillars, 1937o
7o
Dolan, J. M.: On the relationship between the oscilatory behavior of a linear third-order differential equation and its adjoint, J. Diff. Equations ~ (1970), 367-388.
8°
Everitt, W. N.: A note of the Dirichlet conditions for secondorder differential expressions, Canad. J. Math. 28 (1976),
312-32o. 9.
lO.
Everitt, W. No: On the transformation theory of ordinary secondorder linear symmetz&c differential equations, preprint. Guggenheimer, Ho: Distribution of zeros and limit behavior of solutions of differential equations, Proc. ~ S 61 (1976), 275-279.
llo
Halphen, G. H.: M6moire sur la r6duction des 6quations differentilles lin6aires aux formes integrables. M6moires pr@sent6s par divers savants & l'academie des sciences de l'institut de France 28 (188~), 1-301.
562
12.
Hasse, M., Michler, Lo: Theorie der Kategorien, VEB, Berlin 1966o
13.
Hust~, Z.: Die Iteration homogener linearer Differentialgleichungen, Publ. Fac. Sci. Univ. Jo E. Purkyn6 (Brno), %49 (1964), 23-56.
14o
Kummer, E. E.: De generali quadam aequatione differentiali tertii ordinis. Progr. Evang. K~nigl. & Stadtgymnasiums Liegnitz 1834o
15°
Neuman, Fo: Criterion of periodicity of solutions of a certain differential equation with a periodic coefficient, Ann. Mat. Pura Appl. 75 (1967), 385-396.
16.
Neuman, F.: Relation between the distribution of the zeros of the solutions of a 2nd order linear differential equation and the boundedness of these solutions, Acta ~ath. Acad. 8ci. Hangar. 19 (1968), 1-6.
17.
Nemuan, F.: An explicit form of the differential equations y" = q(t)y with periodic solutions, Ann. Mat. Pura Applo 85 (1970), 205-300.
18.
Neuman, F.: Linear differential equations of the second order and their applications, Rend. Mat. ~ (1971), 559-617.
19.
Neuman, F.: A note on Santal6"s isoperimetric theorem, Revista Mat. Fis. Teor. Tucuman, 2_~1(1971), 203-206.
20.
Neuman, F.: Geometrical approach to linear differential equations of the n-th order, Rend. Mat. ~ (1972), 579-602.
21.
Neuman, F.: Distribution of zeros of solutions of y" = q(t)y in relation to their behaviour in large, Studia Sci. Math. Hungaro ~ (1973), 177-185.
22.
Neuman, F.: On n-dimensional closed curves and periodic solutions of linear differential equations of the n-th order, Demonstratio Math. ~ (1973), part I, 329-337.
23.
Nemnan, F.: On a problem of transformations between limi~-~ircle and limit-point differential equations, Proc. Roy. Scc. Edinburgh, Sect. A. Z~ (1973/74), 187-193.
24.
Neuman, F.: On two problems about oscillation of linear differential equctions of the third order, J. Diff. Equations 15 (1974), 589-596.
563
25.
Neuman, F.: On solutions of the vector functional equation ~ ( # (x)) = f(x)oA.$(x), Aequationes Math. 16 (1977),
245-257. 26.
Neuman, Fo: Categorial approach to global transformations of the n-th order linear differential equations, ~asopis P~st. Mat. 102 (1977), 350-555.
27.
Neuman, F.: Limit circle classification and boundedness of solutions, Proc. Roy. Soc. Edinburgh, 81 A (1978), 31-34.
28.
Neuman, F.: Invarian~y linejnych differencial'nych uravnenij 3-go porjadka i metod podvi~nogo repera E. Kartana, Differencial'nyje uravnenija XIV (1979), 398-40~.
29.
Newman, F.: Global theory of linear differential equations of the n-th order, Proceedings of the Colloquium on Qualitative Theory of Differential Equations, August 79, SzegedHungary, Seria Colloquia ~a~hematica Societatis J~aos Bolyai & North-Holland Publishing Company, 777-794.
30.
Neuman, F.: Second order linear differential systems, Ann. Sci, ~cole Norm. Super. (Paris) 13 (1980), 437-4~9.
31.
Neuman, F.: Global canonical forms of linear differential equations, Math. Slovaca, to appear.
32.
Sansone, G~: Studi sulle equazioni differenziali lineari omogenee di terzo ordine nel campo reale. Revista Mat. Fis. Teor. Tucuman 6 (1948), 195-253.
33.
St~ckel, P.: Uber Transformationen yon Differentialgleichungen. J. Reine Angew. Math. (Crelle Journal) ~ (1893), 290-302.
34.
Wilczynski, E. J.: Projective Differential Geometry of Curves and Ruled Surfaces, Teubner, Leipzig 1906.
Author's
address:
Mathema~ical Institute of the Czechoslovak Academy of Sciences Branch Brno Jan~6kovo nim. 2a 66295 Brno, Czechoslovakia
STRONGLY NONLINEAR EVOLUTION VARIATIONAL INEQUALITIES DAN PASCALI
Certain
free
boundary
l e a d to the e v o l u t i o n
problems
variational
)
related
<~-6 ~u + p ( t ) u , v - u> + fQg(t, x,u) ( v - u) in an o p e n Q in
~N
cylinder
and
J:s d e t e r m i n e d a caloric
cive operator
function
solution
ing the
equations same
tions
[2],the
approach
Let in
V
and
and obtain
norms
in H, V
tween
V and V*,be
a n d V*
An operator D(A)2V
extensions
and in
is b o u n d e d
here
g
2
(Q))
is
the c o e r -
while
no g r o w t h
g. an existence
result
inequality,based [ 5 ]
to s t u d y
of perturbations.Although [3]
,which
for s t r o n g l y
is in a m o r e
domain
and
condition.Here
author
as B r o w d e r
for
on extendevolution
we use
the
is e s s e n t i a l l y
a
n o n l i n e a r parabolic equa-
abstract
setting,
thus
assumptions. Banach
space
V* b e t h e d u a l V cH aV*.
A :V ~ V *
maximal By
"
and
o f the
is m o n o t o n e
monotone >
compactly
space
Denote
respectively
an e x t e n s i o n
V × V*.
P(t)
variational
lemma
be a r e f l e x i v e
H = ~(Q)
its d u a l
u,vC
criterium
of A u b i n ' s
to d i f f e r e n t
semi-permeable w a l l s
a sign
is to g i v e
by the
this kind
case
leading
ded
employed
involving
compactness
special
set K in L ( 0 , T ~
convex
on the p e r t u r h s t i o n
o f the a b o v e
treatment
wv c K
to a b o u n d e d
through
report
z
closed
satisfying
of t h i s
processes
corresponding
of pseudo-monotone type
is i m p o s e d
The purpose a weak
T >0.The
by the heat f l u x
body
restriction
Q =]0,T[×Q
to a f i x e d
to d i f f u s i o n
inequality
by let
of V.
['[2
"
Identify
[I "[[
(.,.)
and
if
H
with
I] "]1.
the pairing
inner product (Au-Av,u-
if it a d m i t s
" and
a n d densely imbed-
on
the be-
H .
v) ~ o
for all
no properly
monotone
< " we denote
the
strong
and
*)This work was done with the support of an Alexander yon Humboldt grant while the author worked at the T H Darmstadt.
565
weak in
convergence V
implies
respectively.
Au
~ Au
n
Then
in V ~ w h i l e
(Au, u) ilull Let
C be
a closed
P : C ~ V ~ is in
C
said
for which
are
u~--~u
valued)
given
For
from
V
There
term
=<
g (t,u) j)
g : [0,T] × H ~
V x v
because
the
constant for
all on
(single-
P(t) :V ~
cI > 0 for such and
and
V ~ is
p > 2
all
uCV
so t h a t ;
that
a.a.
strongly
H
the no
is c o n t i n u o u s
t E ]0,T[.
nonlinear
, u)
-> 0
perturbation
growth
a t u in H f o r
in t h e
perturbing
and
a.a.
g
not
induce
does
restriction
a.a.tClo,T[;
sense
Vu6H
is m a d e .
g(t,u) gn (t'u) and
the
ity
yields
semilinear
element
t E ]0,T[. a mapping
In t h i s
on
account,
we
Gn(t)u
form
in
if
Ig(t,u) 12 < n
,
= ~ g(t,u) otherwise L n Ig ~ - , ~ I
lan(U~V) I < H.
an(U,V)
=
n
and
tvl 2
Since
V
(gn(.,u),v). so
The
an(U,.)
is c o m p a c t l y
H~iderinequal-
determines
imbedded
in
H
an the
(P+G n) (t) :V ~ V e i s a coercive bounded mapping,for almost all t C ]o,T[. Furthermore,we
C([0,T];H) spaces
and
consider
H=
with
<.,->
the
the
usual
pairing
is t h e
=
the
vector-valued
function
spaces
L2(0,T;H)
V = LP(0,T;V)
dowed
which
v C C.
the truncated perturbation
introduce
sum
all
that:
uEV the
that
of nonlinear
operator
c2
the Sign condition
(g(t,u) that
such
{u n}
follows
u) < 0 it
for
operator
sequence
imposed:
g fulfils
Note
V
if
for every
I t E [0,T]}
constants
hypotheses
are
if
>u n
A demicontinuous
c I (I + IIull p-I )
_> c 2 iEUllp
following
j j)
are
is a p o s i t i v e
(P(t)u,u) The
into
if u
.
(PUn,Un-
t C ]0,T[,
there
il P ( t ) u l l
ii)
o f V.
( P U n , U n- v)
{P(t)
every
and
]]uil ~ ~
limsup
< liminf
almost
pseudo-monotone
and
a family
operators
i)
set
coerciue
is
pseudo-monotone
to b e
(Pu,u-v) We
A
as
convex
is d e m i c o n t i n u o u s
A
and
= L 2 ( Q ) as w e l l a s t h e d u a l r e f l e x i v e I 1 V e = L p (0,T; V ~ ), w i t h --+ --.= I, a l l e n -
norms.
between
We
also
have
V and
~,
that
(u(t),v(t)) dt 0 inner product on H.
for
V c H p V~ P a n d
denote
by
is
all
Suppose,in
uEV
and
v E V~
addition,that:
,
566
iii) F o r e v e r y V ~- m e a s u r a b l e
uE H, t h e f u n c t i o n
on
on
is s t r o n g l y
]0,T[;
jjj) F o r a n y measurable
t ~ P(t)u(t)
u , v C V, t h e
]0,T[
while
function
gr(t)
=
t ~
sup
(g(t,u(t),v(t))
Ig(t,u(t)) 12
~r , wher~ llui[ = e a s
is
defines
a
IFull
LI(0,T) - function is t h e n o r m Set fined
in
for
0 ~ r < ~
L~(0,T;H).
n o w the n o n l i n e a r
for a l m o s t
all
t C ]0,T[.
is a p s e u d o - m o n o t o n e llPull
the
P:V ~ V
operators
sum
and The
and
Gn:H
~ H
de-
assumptions
defined
and
c] > 0. T h e
(GNU) (t) = G n ( t ) u ( t )
above
operator
~ c1( ] + llull~ -I)
a constant
on
same
guarantes
that
P
V such that
~ c2 ,,ull~
conditions
are
for all
also
u 6 V,
satisfied
by
P + G
n Further,let pact
lu(t) 12
by (Pu) (t) = P ( t ) u ( t )
with
sup
t~]0,T[
in the
sense
After
appropriate
whose
measure
by parts
V ~}
be a B a n a c h
space with
resdv
dv IIV[Iw = I[vll V + !I~-~11V ~ w h e r e the d e r i v a t i v e
to t h e n o r m
is t a k e n
dv {v C V i ~ C
W =
of
V e- valued
modification
is zero,
of
we have
distributions
v 6 W
over
on a subset
v C C([0,T];H)
]0,T[.
of
]0,T[,
and the i n t e g r a t i o n
formula
It { (~dv , w)
+
(v , ~)dw }dt =
(v(t) , w(t)) -
(v(s) , w ( s ) )
0 < =s
t
wS
is v a l i d
for a l l
is c o n t i n u o u s
sociate
C with
K =
{v 6 V
and denote
set
B
As usual, fulfiled:
w E W
convex
with
C
a.e.in
assume
that
w(0)
set c o n t a i n i n g B,i.e.,
B :V ~ V ~ w i t h
the p e n a l t y
To e v e r y
{w n} 6 K ' s u c h
for
in~ecti~n A :V ~ V e
of
W into C([0,T];H)
defined
by
= 0
V × V ~ ,(see e . g . [ 1 , p . 6 5 ] ) .
C/ c o n s i d e r
I v(t)C by
operator
the penalty operator
C
operator
to the
. Moreover,the
linear
be a c l o s e d
ed monotone dition
the
A w = d w : w" dt m o n o t o n e in
is m a x i m a l Let
v,w 6 W
and
that
the
v(0) = 0}
operator
the there
w n -~v
in
V
convex
K'= K n W to
compatibility
corresponds and
and
bound-
I Bv = 0 } . I n
two c l o s e d
corresponding
following
v C K
in V . A s -
a demicontinuous
C = k e r B = {v C V
following
]0,T[,
the o r i g i n
a smoothing
ad-
sets ,
K .
conjecture sequence
l i m sup (AWn,W n - v) $ 0.
is
567
We are n o w
in the p o s i t i o n f be a g i v e n
T H E O R E M . L£t
to state
result
e l e m e n t of V* and a s s u m e t h a t
hypotheses
h o l d . Then t h e r e
(g(',u),u)
EL I (O,T) w h i c h s a t i s f i e s
(1)
our m a i n
exists
++
uE K the
all
above
g ( - ' , u ) ~ L I (O,T;H)
With
variational
all
inequality
(g(t,u(t)),v(t)-u(t))dt->-
VvEK'.
0 Proof.
U s i n g an e l l i p t i c
well-known the
surjectivity
approximante
penalisation
method,it
result in m o n o t o n e
type
follows
operator
from a
theory,that
equation
Au +
( P+
Gn)U
+ !Bu
= f,
u(0)
= 0
E
admits
at l e a s t
one
solution
N o w we m u l t i p l y ~I l U n s ( t )
by
- v ( t ) I 22 +
u
n£
ft
u - v
f W for e v e r y n E ~ and s> 0. ns and i n t e g r a t e o v e r [0,t] to o b t a i n
0(v" (T) , U n g ( < ) - v ( T ) ) d <
ft
+
f~
(P (~-) Ung (r) ,Un£ ('r) - v(T))d~c +
0
for each
0 < t
and
v E K " .By t a k i n g
Taking at l e a s t §6),we (2)
c 2 IIUn liP +
llUngl!L~(0,T;H)
is a b o u n d e d
sequence first
the
0 (gn(T'
infer
that
limit
u
n
for e v e r y
n E 1~ . Due
have
further
ma
assures Let
and and such
that
now
IGnUnl2 so t h e r e that
{(gn(-,Uns),Ung
s ~ 0, we have by a s t a n d a r d
-'u in VnL~(0,T;H), n6 n a r g u m e n t ([4],Chap.3,
and
to the w e a k l y
> = > lower
} r e m a i n s a bounded
V n L~(0 ,T;H)
a.e.in
3 is ~ f u n c t i o n
for# any p a i r
]0,T[,
by
@ const.,
as
n- ~
jjj)
7 :JR+ -- JR+, w i t h
(s,t)
It l]Un(<)II~ d~ <
in
[0,T],
y(t - s)
VvEK',
semicontinuity
{(gn(.,Un),Un) in
)}
u
+ l!Un[IV
(t)
>_ 0,
s > 0.
IlUnI!L~(0,T;H)
u n---%u <- gc
then
as
E K
(BUna,Uns)
< Itfll *llu rl v
and,
+ < ( P + G n ) U n ' V - U n
we
since
L I (0,T)_-_, for e a c h
for a s u b s e q u e n c e ,
can
and
dT
-v(~r))dT,
'un
+llUnsl[V--
v=Oand
;t
we d e r i v e
1,1Uns(t) !22+
n(T, Un~ (T)) ,Una (T)-v(r)
= I to ( f ( T ) ' U n ~ ( ' r )
rt + ~I tO (13un~('c)'un~(T)-v(~c))dT
whence
+
while
sequence in . Since
we get 7(r)
with
for all
of norms, Fatou's
lUnl 2 =< c 3
GnUnEL1 -
0
s < t, n El~.
lem-
LI(0]T).
as
(0,T;H) r - 0,
568
By the above mentioned quence
compactness
{Un} is contained
L2(0,T;H)
= L2(Q).
such that
Un-~U
Applying
in a compact
Thus,there a.e.
criterium
in
([2],theorem
set in LP(0,T;H)
is a subsequence,again
Q = 30,T[x~
and,
3),the
se-
and also in
denoted by {Un},
consequently
gn(',Un(-))--->g(',u(-)) a.e. in ]0,T[. again the Fatou lemma,we deduce that (g(.,u),u) C LI(0,T),
while the Lebesque
dominated gn(-,Un)
Finally,the
passage
isfies the variational
convergence ) g(-,u)
in
to the limit in
inequality
theorem yields LI(0,T;H). (2) concludes
(I), which completes
that u C K satthe proof of
the theorem.
REFERENCES [1] V.BARBU - TH.PRECUPANU,
s paces,
C o n v e x i t y and O p t i m i z a t i o n i n Banach
Edit.Academiei Bucuresti - Sijthoff & ~rdhooff, 1978. J
[2] H.BREZIS
Strongly nonlinear parabolic initial- b o u n d a r y v a l u e d p r o b l e m s , Proc.Natl.Acad.Sci. USA 76 (1979),38-40.
- F.E.BROWDER,
[3] F.E.BROWDER, S t r o n g l y
nonlinear parabolic problems,(abstract)
Tagungsbericht
49/1979,
Oberwolfach.
[4] J.L.LIONS,
Quelques M~thodes de R ~ s o l u t i o n des Probl~mes aux
[5] D.PASCALI,
L i m i t e s Non L i n ~ a i r e s , Dunod& ~uthier-~llars, Paris, 1979. V a r i a t i o n a l s o l u t i o n s of s t r o n g l y n o n l i n e a r e v o l u t i o n equations, Workshop on "Applied Functional Analysis'~ (Variational methods and ill-posed problems), Berlin, September 7-12, 1981.
West-
A
FURTHER
LINEAR
RESULT
2nth
ON
ORDER
THE
ESSENTIAL
DIFFERENTIAL
COMPLEX
OF
SPECTRUM
EXPRESSIONS
WITH
COEFFICIENTS
D. Race
1.
Introduction
we shall
0e c o n c e r n e d
w i t h the expression
n Tw(t
where
a >
=
-~
We assume
[a, ~ )
.
W~ define T ' f o
=
Pk
"
are locally
the o p e r a t o r
Tf
,
f
T ' o
e
t))
the c o e f f i c i e n t s
i / ~ , Pl . . . . ' Pn
that
y(n-k>( (Pk(t)
and the c o e f f i c i e n t s
functions. i.e.
n-k
~ (-i) k=O
~(T
0 ~ k ~ n
t •
[a, ~ )
(i.i)
are c o m p l e x - v a l u e d
satisfy the m i n i m a l
conditions,
Lebesgue
on the interval
integrable
L 2 [a, ~)
in
o
,
by
')
where 9(T ') = o
{f • L2[a, ~) : f[k]
is locally
3 < k < 2n- 1 , T f
• L2[a, ~) ,
in
in which
f[k]
denotes
the e s s e n t i a l
spectrum
for which there
exists
satisfying
associated
o
[ 3, 5,.7
] .
support
T
Y
f
associated
is the closure
A ,
of an o p e r a t o r a non-compact
=
sequence
Eo(A)
of
with We
T ' o
,
to be the set
{fn] C ~(A)
with
0 .
theory of o p e r a t o r s If
with
of
the c o n d i t i o n
lim (A - II) fn n + o~
The general
has compact
kth q u a s i - d e r i v a t i v e
also define
llfnll = 1 ,
T
the
operator
.I E C
f
continuous,
(a, ~) }
The m i n i m a l
of
absolutely
1
so it is c l e a r l y of value
associated
with
is any closed e x t e n s i o n to locate the
set
Eo(T
of
o
)
T T
o
m a y be found then
in
Eo(T ) = Eo(T ) 1 o
The s i g n i f i c a n c e
of
570
this and r e l a t e d
results
is d i s c u s s e d
in
We n e e d the f o l l o w i n g
[6].
result here.
Lemma 1.1
(c.f.
[2, p.75],
If
16, 31])
K(T ') o
is the closure of
the set of complex numbers { (To 'f , ~ then
: f C ~(To' )
llfll = i)
Ea(T ) c K(T ') o o
In a p p l y i n g L e m m a
i.i it is useful
to observe
that by i n t e g r a t i n g
by parts
repeatedly,
n
IT 'r, ~
inequality,
Ck
where
is a constant,
We use
Re(~)
respectively, real,
,
and
which
is v a l i d
for any
f C AT
o
')
(lif(k>ii + iifll) 1 < k < n
Im(~)
(~-
(1.3)
.
to denote the real a n d imaginary p a r t s
to denote
the n e g a t i v e
p a r t of
f
,
when
of f
~
,
is
i.e.
(~
when
(1.2)
[2, p.83])
iif'll < ~
2.
Pkl f
k=0
W e also n e e d the f o l l o w i n g (c.f.
la
~
o
=
max(- f ,
Known
results
There
are few results
the coefficients
coefficients. to c o m p l e x
result
are real-valued,
coefficients to
coefficients from
n > 1 ,
concerning
[6].
in the case
n > 1 and
in n > 1
[6], where
in
a brief
found.
of
Eo(T ) o
for c o m p l e x - v a l u e d
for real c o e f f i c i e n t s
n = 1 ,
m a y be
the location
and e v e n fewer
M a n y of the k n o w n results
then generalized complex
for
0)
[4]. review
were extended
M o s t of these w e r e of o t h e r
The following
results
for
is a t y p i c a l
571
([6, Theorem 3.1]).
Theorem 2.1
and for each
t • IN, ~ ) ,
(Im
I f for some
N < ~ ,
Po(t)
~ 1
for
1 <~ k <~ n ,
pk )
(2.1)
<
and
f Re Pk
~
- Dk
D k >~ 0
for some
J
of length
then there is a constant Eo(T )
c
and for all intervals
< 1 for which
A > -~
{I • < : Im I ~ 0
and
Re I ~> A} •
o
A value for
A
is given in
[6, (3.17)] , though it is observed there that it From Theorem 2.1, the following may be
is not the best possible value. deduced.
Corollary 2.2 t •
([6, Corollary
[N, ~ ) , and for each
real consto~t
(i)
ak ,
/
3.2])
1 ~ k < n ,
such that
(Re(Pk - ak))-
Re Pk
<
Im Pk
, t+w i (Re(Pk-ak))t
= o
for some
w~o a
{(Re(Pk - ak)
<
then i
EO(To)
f
C II 6 C : Im i ~> 0 %
(2.1), and there is a :
[a,~) : (Re(Pk(t) -ak))- >~ 6} ;
(and consequently
;
for some
~ 1 ,
where
)_}rk (iii)
Po(t)
one of the following
~ > 0
M6, k = { t E
lira t÷oo
N < ~ ,
satisfies
satisfies
for any
M6,k
(ii)
If for some
r k >I 1 ;
for any)
572
3.
Main Theorem The p u r p o s e of the p r e s e n t w o r k is to give a g e n e r a l i z a t i o n of
[4, T h e o r e m 4~6] and and the latter for
[6, T h e o r e m 3.4]. n > 1 .
only,
and hence obtain a more p r e c i s e b o u n d
Eq (g ) . o
If there are real constants
T h e o r e m 3.1
for some
N < ~
and
i> a °
>
0 ,
la{(me + m Ira)(Pk(t) - ak)}
E<~(T ) o
Proof
ak ,
i <~ k <~ n ,
t C
(3.1)
[N, oo)
< oo
i <~ k ~< n n ~
c {h 6 { : (Re + m Im)h 7> m i n O~<x<~
IDyll = 1
with
(Re + m Im) ( % ' y ,
y)
=
akxn-k}
and h a v i n g compact support in
n[ k=0
~N (Re + m
Im) Pk "
(3.2)
k=0
In the light of L e m m a i.i and (1.2) we consider
y 6 ~(T ') o
such that
m 6 ~i,
(Re + m Im)Po(t)
then
n = 1
In addition to, e x t e n d i n g the latter, we give
a more direct p r o o f than is u s e d there, for
The former is for
(T 'y, y) o (N, ~ )
for any
.
[y(n-k) 12
oo
n
IN (Re + m Im)Po . ly(n)l 2 +
+
(Re + m Im)
k k=l
aklly(n-k)ll2
(Pk-ak)ly(n-k)l 2
k=l
k
aolly(n)ll2
-
+
7q ~ k=l
) 2
aklly(n-k Jf
(3.3)
{(Re + m I m ) ( P k - a k )
}-" lY(n-k) 12
k=l using
(3.1).
for each
Now from (3.2), we m a y choose
1 ~ k < n
IN
,
{(Re + m I m ) ( P k -
ak) }-
<
a
N
sufficiently large so that
573
where
a
will be chosen later.
Also,
for
1 < k ~ n
,~-~)~),~ .<~(f;,.~ <~-~,,~]~,~[~,~ r
=
by a method used in [ 1 ] . (Re + m I m ) ( % ' y ,
y)
>
Hence in
ao lly(n)ll2
t e
[N, co)
(3.3) we have
+
ak 11y(n-k)ll2
[
k=l n
2ally (n-k) II . lly
(n-k+l)
k=l
n >I
a
IIy(n) Jl2
+
(liy(n)ll
2a ~
k=l using
(1.3) with
f = y
(n-k)
akJJy(n-k)JI2
~ k=l
o
Thus for any
.lly(n-k)
0 < C < a
n (Re + m Im)(T ' y , o
y)
>
+
o
>-
by completing
(a - ~ ) , , J n ) , , 2
(a o
- ~)HCn>ll
the square.
~
~ {<_,,Jn),,2_ k=l
+
ll +
Ily(n-k)ll2)
o
2C~k,jy(n)jj.,,y(=-k)jj
+ (ak -
2a C/~k)lly(n-k)jJ
n ~
2 ~
(=k
-
-
=wc~= -I) lIJn-k> i,=
k=l We now choose
2}
(3.4)
a = min { ~ E
Ck- ~, E(2nCk)-~ }
so that
2c( ~kk + n a2 Ck E-1 <<" In (3.4) we now have
n
(Re + m Im) (To 'y, y)
for any
0 < C < a
proof of
[2, T h e o r e m i, p.130] we have
o
and for
i> ~ (ak-E]ll k=0 N(£)
y(n-k) 2
sufficiently
II
large.
Hence,
as in the
574
(Re + m Ira) (T 'y, y) o
Applying
Lemma
any with
{l • C
c = 0
,
ao = 1
to complete
,
the factor o f
the conditions Re I + m Im I
4.
stated, =
Concluding
p
o
a
o
> 0
1 < k < n - 1
~ .
the e s s e n t i a l
o
{h • {
then it is k n o w n
o
:II(T
- h/)fll
useful
of T h e o r e m
T h e o r e m 4.1
and
[5])
that
(3.2)
(3.5)
holds
that
for
(3.5)
holds []
guarantees is
~
o
[6, T h e o r e m
the b o u n d g i v e n
spectrum
Theorem
> -~
The
3.4].
case
The above
in C o r o l l a r y
2.2,
3.1 says t h a t under
lies to the right of the line
the r e g u l a r i t y
~
klilfll
z(T ) o
In the theory
a basic pre-requisite
consequence
(3.1)
(see
E~(T ) .
It is t h e r e f o r e
this gives
remarks
~(r ) o
T ,
since
by c o n t i n u i t y
to have
field o f
and for all
is the c o m p l e m e n t of J - s e l f a d j o i n t
is that
z(T ) o
the following
T
to be
o
for some p o s i t i v e kl
with
[2, 5])
in the c o m p l e x plane.
we define
of the set
say.
(3.5)
Geometrically,
If, in the usual way,
=
'
the proof.
m e t h o d of p r o o f may also be u s e d to refine by removing
(see
Finally,
we m a y conclude
that the c o n d i t i o n ak = 0
principle
6 < ~o
small,
as r e q u i r e d
and
Dc
k=0
: (Re + m Im) l ~> VS}
since
sufficiently
We observe when
C
DS > - ~
C > 0
~ (ak- [)xn-k
rain 0<x<~
i.i and the d e c o m p o s i t i o n
EO(To) We note that
~>
f • 9(T o) }
in the c o m p l e x plane,
operators
be n o n - e m p t y
result,
constant
which
associated
(see
[3, 7]).
is an immediate
3.1.
(c.f.
[6, T h e o r e m
4.1])
then
~(T ) is non-empty. o
If the coefficients
Pk
satisfy
575
References
[i]
W.N. Everitt, M~ Giertz and J. Weidman.
Some remarks on a
separation and limit-point criterion of second-order, ordinary differential expressions.
[2]
Math. Ann. 200 (]973), 335-346.
Direct methods of qualitative spectral ~alysis of
I.M. Glazman.
singular differential operators. Scientific Translations,
[3]
I. Knowles.
(Jerusalem : Israel Program for
1965).
On the boundary conditions characterizing J-selfadjoint
extensions of J-symmetric operators.
J. Differential Equations
40 (1981), 193-216.
[4]
D. Race.
On the location of the essential spectra and regularity
fields of complex Sturm-Liouville operators.
Edinburgh Sect. A [5]
D. Race.
D. Race.
85 (1980), 1-14.
The spectral theory of complex Sturm-Liouville operators.
(Ph.D. thesis
[6]
Proc. Roy. Soc.
: University of the Witwatersrand, Johannesburg,
On the essential spectra of linear
operators with complex coefficients.
2nth order differential
Proc. Roy. Soc. Edinburgh Sect. A,
to appear.
[7]
1980)
N.A. Zhikhar.
The theory of J-symmetric operators
Ukrain. Mat. Z.
ii (4) (1959), 352-364.
!Russian).
ON ERRORBOUNDS FOR NONSTATIONARYSPECTRAL NAVIER-STOKES APPROXIMATIONS
Reimund Rautmann
Summary:
For spectral Galerkin approximations
Stokes solutions on a smoothly bounded suitable Hilbert space are established, in certain H61der spaces. results
of nonstationary
3-dimensional
domain,
local strong Navier-
error estimates
in
which lead to uniform pointwise error bounds
The estimates follow by application of Fujita's
and Kato's
[4].
Since Kiselev's and Lady~enskaya's work [13] the local (in time) existence of unique strong solutions of the Navier-Stokes i n i t i a l - b o u n d a r y value problem in threedimensional regions has been proved in several contextes [ 9 ] , [ 1 8 ] ,
[12,4] ,
[23, 22] , [2] . As recently Heywood has shown in [5], the Galerkin procedure based on the eigenfunctions of the l i n e a r Stokes problem offers a very natural way to proving local existence, Coo - r e g u l a r i t y and asymptotic decay of strong solutions. T:he question, how good the spectral Galerkin approximations are, has been settled in [20] and [21] with respect to error estimates in the L2 - and Dirichlet-norm. Heywood and Rannacher then established related estimates in the framework of f i n i t e elements [8] ,[19] , and Heywood [7] with an additional s t a b i l i t y assumption inquired estimates uniform in time. In this note we w i l l strengthen the e a r l i e r estimates [2o,21] to bounds in s u i t a b l e H i l b e r t spaces, which lead to pointwise uniform error estimates in certain H~Ider spaces ( l o c a l l y in time, c.p. Theorem 2.1. below). The essential tool in the f o l l o w i n g is the a p p l i c a t i o n of F u j i t a ' s and Kato's results [4] . I. Local strong Navier-Stokes solutions and spectral Galerkin approximations. Let ~ be a bounded open set in the ( x l , x 2 , x 3 ) - space~3, the boundary ~Q being a compact 2-dimensional C3- submanifold of R3. The v e l o c i t y vector u ( t , x ) = (ul,u2,u 3)
577
and the pressure f u n c t i o n p ( t , x ) of a n o n s t a t i o n a r y incompressible viscous flow in at times t ~ o solve the Navier-Stokes i n i t i a l - b o u n d a r y value problem (I.I)
ut-Au +Vp = -u.Vu , q.u = o, t > o, u[9 S = o;
u = Uo, t = o,
i f we assume the c o n d i t i o n of adherence on 9~ and f o r short neglect e x t e r n a l forces, and i f
distance and time are measured in the a p p r o p r i a t e u n i t s .
We w i l l
consider t h i s problem in the usual framework of the m i l b e r t spaces Hm con-
t a i n i n g the vector functions (defined
almost everywhere) on
a, which belong to
Lebesgue's class L2(~) together with t h e i r s p a t i a l d e r i v a t i v e s up to the order m. The norm on H is m
1 nl+n2+n3 ~ f(x)
/
m 2 z I] --I dx , \nl+n2+n3=oQ (~x1)n1(~x2)n2(gx3)n3 1 3 2)2 where I f ( x ) I = ( i ~ 1 ( f i ( x ) ) . For the m2(q)-norm we w i l l w r i t e [fIH m :
IIfl]
= I f [ m • By
~m we denote the closure in Hm of the l i n e a r space of the d i v e r g e n c e - f r e e
o
Coo-vector functions having compact support in ~. F i n a l l y l e t P be Weyl's p r o j e c t i o n of H° on ~ o and A F r i e d r i c h ' s extension of the p o s i t i v e d e f i n i t e , -PA
symmetric operator
i n ~ o, DA (or DA~) denoting the domain in ~ o of the operator A (or Aa ,
r e s p e c t i v e l y ) . We r e c a l l that for any ~ E ( o , I ] ,
DAm is a H i l b e r t space equipped
with the norm ( ~ I A m f [ 2 d x ) #2, because the inverse A-~ : ~ o ~ DAm e x i s t s as a bounded operator on ~]~o" Since P commutes with the strong time d e r i v a t i v e ~t' with these notations the Navier-Stokes i n i t i a l - b o u n d a r y value problem can be transformated
in the e v o l u t i o n
equation (1.2)
~ t + A ) u = -P(u.Vu) u = Uo,
, t > o, t = o
f o r the ~ o - v a l u e d f u n c t i o n u ( t ) = u ( t , . ) . We i n q u l r e Galerkin approximations f o r u in terms of the eigenfunctions of the l i n e a r Stokes boundary value problem Av=f f o r the unknown vector f u n c t i o n v E ~ C 1 n H2 and a given f 6~.__ Since the s e l f a d j o i n t operator A is p o s i t i v e d e f i n i t e and possesses a compact inverse on S o, i t has the sequence ( l i ) with k ~
of eigenvalues ~ i , o < ~ 1 ~
~2~"~Ik
corresponding to the sequence ( e i ) of eigenfunctions e i E : ~ 1 n H3
in any f i x e d order. Thus the equation
~
578 (1.3)
A e. = I. e. 1
l
holds, and ( e i ) , ~1'~1
1
(eili-1/2),
(eil~.1) represent complete orthonormal systems in ' ~ o '
N H2 equipped with the norms, which are induced by the inner products
S f - g d x,
S ( v f ) ( V g ) d x or
S(A f ) ( A g ) d x , r e s p e c t i v e l y , c . p . [ 1 4 ]
p.46 .
Let ~o,m be the m-dimensional subspace of ~£o' which is spanned by the f i r s t
m
eigenfunctions e 1 . . . . . e m of A. By Pm we denote the orthogonal p r o j e c t i o n of Ho on ~o,m'
i . e . we have m Pmf = i=1~ e i ~ fei dx
(1.4)
f o r any f EH O. From (1.3) we get (1.5)
Pk A f = A Pk f
f o r f E DA.
We r e q u i r e , t h a t the k th Galerkin-approximation k uk(t ) = ~ aki(t)e i i=1 of u with the unknown c o e f f i c i e n t s a k i ( t ) solves the p r o j e c t i o n of (1.2) in
~'o,m"
Since Pk also commutes w i t h the strong time d e r i v a t i v e and because of PPk=PkP=Pk, we get f o r (1.6)
u k the e v o l u t i o n equation
(~t+A)Uk = -Pk(Uk.VUk), t > o, uk = PkUo ,
In terms of the k functions a k i , system of k o r d i n a r y d i f f e r e n t i a l
t = o. (,1.6) represents the i n i t i a l
value problem of a
equations. Due to the energy i n e q u a l i t y ,
the unique
s o l u t i o n uk(t ) e x i s t s on [o,o~), uk belonging to Coo[o,o~) and (with the e i ) to H3 f o r any t_>o [5, 21]. For short we assume UoE~ I n H2. Then on a ( p o s s i b l y small) time i n t e r v a l T > o, the unique strong s o l u t i o n u ( t ) £ ~ 1 wing p r o p e r t i e s :
[o,T],
N H2 of (1.2) e x i s t s and has the f o l l o -
The norms [IA u ( t ) II and, the H~Ider seminorms [Aau] = sup {If Amu(t)-A~u(s)III t - s l -I~} t,sE[o,T] t~s
are u n i f o r m l y bounded, i . e . we have
(1.7) (a) IIA u(t)11 < c, (b) [A~u] ] ] < c with a bound c > o depending" only on lUolH2 ~, T, ~ and ~ f o r a l l t E [o,T] and any o < ~ + u < 1, ~ > o. In a d d i t i o n , f o r any Galerkin approximation uk from ( 1 . 6 ) , the i n e q u a l i t i e s
,
579
(1.8)
(a)
IlA uk(t)
l]S c ,
(b)
[ASUk(t)] u -<
c
I)
hold uniformly with respect to k = 1,2 ..... tE [o,T], [5,p.666], [4, p. 302]. From the moment inequality for fractional powers of the operator A [3,p. 159] and Cattabriga's estimate [i], using (1.7 a),(l.8 a)
we find
(1.9) ]IAau (t)l I~ c(~) , IIA~Uk(t) II~ c(~) for any t 6 [o,T] , k = 1,2 . . . . .
w i t h a bound c(~) > o depending on
a E [o,I).
In the f o l l o w i n g from the estimates ( 1 . 7 ) - ( 1 . 9 )
we d e r i v e a bound f o r
firstly
denote p o s i t i v e c o n s t a n t s , the
in the case a < 1. By Co,C 1 . . . .
value of which may be d i f f e r e n t
we w i l l
in d i f f e r e n t
]IAa(U-Uk)]l
sections.
2. E r r o r bounds in DAm, m < 1. Let u be the l o c a l strong Navier-Stokes s o l u t i o n w i t h i n i t i a l G a l e r k i n - a p p r o x i m a t i o n , k = 1,2 . . . . . which ( 1 . 7 ) - ( 1 . 9 )
value uo, u k i t s
and [o,T] any e x i s t e n c e i n t e r v a l
k th
o f u, on
hold.
We prove Theorem 2.1:
Assume uo 6 ~
1 n H2 •
Then the error estimate
IIA~u(t)-A~Uk(t)ll~o c Xk+ 1 holds with any a E (~,I),_ a + c < 1, o < ~, uniformly on [o,T] . The sequence (Uk) converges tov u also in the HUlder smace• Cu(Q ) with the error bound C~hk+l-e for
v 6 ( o , 2 ( ~ - ~ ) ) uniformly on [ o , T ] . The p o s i t i v e numbers c ~ and c. depend only on lUo[ H , ~,T,o , u , ~ . 2 Cv(~ ) denotes the Banach space of the continuous v e c t o r f u n c t i o n s f on ~ , which have a f i n i t e ]f[c
H~Ider norm = sup I f ( x ) l + xE~
sup{If(x)-f(Y)l x,yEQ x%y
Ix-Yl-U}.
For the p r o o f we w r i t e the e r r o r of u k from (1.6) in the form U-U k = Qku + w,
where Qk denotes the orthogonal p r o j e c t i o n P-Pk in H° and w = PkU-Uk. Because of
1)In o r d e r to v e r i f y
(1.8 b ) , f i r s t l y
from (1.8 a) a uniform bound f o r .
.
lnquallty
<
lUIL~_ColUlH
using C a t t a b r i g a ' s e s t i m a t e [1] we d e r i v e
l u k ( t ) l H , which l e a d s , by means o f Sobolev's 2
to a uniform bound f o r [IP(Uk.VUk)ll . Then we a p p l y
Lemma 2.11. and 2.12. o# [4] on ( 2 . 3 ) below.
580 (1.5) the operator function AE commutes with Pk and Qk f o r any c E ( o , 1 ] .
Since A ~
is again a p o s i t i v e d e f i n i t e symmetric operator in ~ o ' Ac having the eigenvalues k~l and the eigenfunctions e i , i = 1 , 2 , . . . ,
we get error bounds for the Fourier
approximations of a function f E DA~a~ in terms of the Stokes eigenfunctions e i from Lemma 2.1: [ i o , p. 38]: In a Hilbert space H with inner product < . , . > defining the norm l . I H
,
let
A* be
a symmetric operator which has the complete orthono~al system of eigenfucntions
(e~) corresponding to the sequence ( ~ ) with
of eigenvalues hi, 0 < ~ < k2 < . "'_ < h ~.l ~
i ~ ~. Then the error estimate k
-1
If -
Z < f , eT] > eTlH] ~ ~k+l* A*flH i=l holds for any f E DA.. From this using (1.9) we v e r i f y Corollary 2.1: The estimate
(2.1)
qku[l
II A~
IIQkA%II ~
c(a+c)'Ik+-c I
holds.
On our condition (1.9), the term P(u.vu) in (1.2) is on [o,T] strongly continuous in ~ o '
as we see from [ 4 ] , Lemma 1.2. Therefore u from (1.2) and uk from (1.6)
are the unique solutions of the i n t e g r a l equations (2.2) and
u ( t ) = e-tAu
o
- ~ e- ( t - s ) A
P(u.Vu)(s)ds
o
t e_(t_s)A uk(t ) = e-tAPku ° - ] Pk(Uk.VUk)(S)ds, o where e -tA denotes the ( f o r Re t > o) holomorphic semigroup which is generated by (2.3)
the p o s i t i v e s e l f a d j o i n t operator A in ~ o " The evolution equation (~t +A)Pku = -Pk ( u ' v u ) ' t > o , Pku = PkUo,
t = o
results from (1.2) by the projection Pk commuting with ~t and A. The projection Pk being orthogonal in ~ o '
the estimates in [ 4 ] , Lemma 1.2. hold with Pk instead of P.
Thus Pku is the unique solution of the integral equation t PkU(t) = e-tAPku ° - ~ e - ( t - s ) A Pk(U.VU)(s)ds. o Subtracting (2.3) we get (2.4)
(2.5)
t e_(t_s)A w(t) = - ~ Pk(U'vU-Uk.VUk)(S)dS. o
In order to show w E DAa and to find a bound f o r
I[A%II,
we use [ 4 ] , Lemma 2.1o.
581 and 2.12. Recalling that Pk = PkP represents an orthogonal projection in~;~o and using [4] , Lemma 1.2. again and (2.1), by a short c a l c u l a t i o n we establish the ine quality
(2.6)
I]Pk (u'vu-uk'VUk)l[
~ c1"( x-~
k+l
+ ~)
with the function
1
~(t)
= II A7 w(t)ll
+ [[ AYw(t)ll
,
being strongly continuous On [o,T] , ~ E (#,1). Equation (2.5) f o r w, (2.6) and the lemmata 2.1o. and 2.12. of [4] j u s t mentioned r e s u l t in the integral i n e q u a l i t y t ~(t) ~ c 2 x-c + c2 f # ( s ) _ k+l o (t_s)~
ds
f o r any ~ C (~,1), ~ + ~ < i . Because of the monotonicity of the kernel [27,p.21] , the solution formula [29,p. 152] f o r the corresponding (weakly singular) Volterra integral equation leads to the bound (2.7)
#(t) ~
c 3 x-c k+l
This i n e q u a l i t y together with (2.1) v e r i f i e s the f i r s t second part follows from [4] , Lemma 1.2. and 2.6.
part of Theorem 2.1. The
582
References 1
Cattabriga, L., Su un problema al eontomno relativo al sistema di equazioni di
2
Foias,C., Stafistical study of Navier-Stokes equations I, Rend.Mat.Sem. Univ.
Stokes, Rend. Mat.Sem.Univ. Padova 31 (1961), 3o8-34o.
Padova 48 (1972 219-348. 3
Friedman,A., Partial differential equations, Holt, Rinehart and Winston, New York 1969.
4
Fujita,H., Kato, T., On the Navier-Stokes initial value problem, I. Arch. Rat.
5
Heywood, J.G., The Navier-Stokes equations: On the existence, regularity and
6
Heywood,J.G., Classical solutions of the Navier-Stokes equations, Proc. IUTAM
Mech. Anal. 16 (1964), 269-315.
decay of solutions, Indiana U. Math. J. 29 (198o),839-681.
Symp. Paderborn (W.-Germany) 1979, Springer Lecture Notes in Math. 771, 198o, 235-248. 7
Heywood, J.G., An error estima{e uniform in time for snectral Galerkin approximations of a Navier-Stokes problem, Preprint Univ. of British Co]umh~a, Vancouver 198o.
8
Heywood, J.G., Rannacher,R., Finite element approximation of the nonstationary Navier-Stokes problem I, preprint no.385, Sonderforschungsbereich
72,
198o Universit~t Bonn. 9
ItS, S., The existence and the uniqueness of regular solution
of non-stationary
Navier-Stokes equation, J. Fac. Sci. Univ. Tokyo, Sect. IA,9(1961) lo3-14o. io
J6rgens, K.,Rellich,F., Eigenwerttheorie gewShnlicher Differentialgleichungen, Springer Berlin 1976.
ii
Kato, T., Perturbation theory for linear operators, 2.ed. Springer Berlin 1976.
12
Kato,T., Fujita,H., On the Non-Stationary Nav~er-Stokes System, Rend. Sem.Math.
13
Kiselev, A.A., Lady~enskaya,O.A., On the existence and uniqueness of the solutio~
Univ. Padova 32 (1962) 243-26o.
of the nonstationary problem for a viscous incompressible fluid, Izv. Akad. Nauk SSR Ser. Mat. 21 (1957) 665-68o. 14
Lady~enskaya,O.A., The mathematical theory of viscous incompressible flow,
15
Lady~enskaya,O.A., Solonnikov,V.A., Ural'ceva,N.N., Linear and quasilinear
Second Edition, Gordon and Breach. New York 1969.
equations of parabolic type, Amer. Math.Soc.Transl. Providence, R.I. 1968. 16
Masuda, K., On the stability of incompressible viscous fluid motions past
17
Oskolkov,A.P., On the Asymptotic Behaviour of Solutions of Certain Systems with
objects, J.Math.Soc. Japan 27 (1975) 294-327.
a small Parameter Approximating the System of Navier-Stokes Equations, Proc. Steklov Inst. Math. 125 (1973) 137-153. 18
Prodi,G., Teoremi di tipo locale per il sistema di Navier-Stokes e stabilit~ delle soluzione stazionarie, Rend.Math. Sem. Univ. Padova 32 (1962)374-397.
583
19
Rannachem, R., On the finite element approximation of the nonstationary Navier-Stokes problem, Proc. IUTAM-Symp. Paderborn (W.-Germany) 1979, Springer Lecture Notes in Math° 771 (198o 4o8-424.
2o
Rautmann, R., Eine Fehlerschranke f~r Galerkinapprox~mationen lokaler NavierStokes-L~sungen. Proc. Conference Oberwolfach 1978, ISNM 48, Birkh~user Basel (1979) llo-125.
21
Rautmann,R., On the convergence-rate of nonstationary Navier-Stokes approximations, Proc. YLFFAM Symp., Paderborn (W.-Germany) 19'79, Springer Lecture Notes in Math. 77~ (198o) 425-449.
22
Shinbrot, M., Lectures on fluid mechanics, New York 1973.
23
Shinbrot, M., Kaniel, S., The initial value problem for the Navier-Stokes equations, Arch° Rational Mech. Anal. 2! (1966) 27o-285.
24
Temam, R°, Navier-Stokes Equations, North-Holland Publ. Comp., Amsterdam
25
Temam, R., Behaviour at Time t = o of the Solutions of Semi-Linear Evolution
26
Velte, W., Eigenwertsohranken mit finiten Differenzen beim Stokesschen Eigen-
27
Walter, W., Differential and integral inequalities, Springer Berlin 197o.
28
yon Wahl, W., Analytische Abbildungen und semilineare Differentialgleichungen
rev. ed. 1979.
Equations, MRC Technical Summary Report 2162, Univ. of Wisconsin, Madison ]98~
wertproblem, ISNM 43, Birkh~user Basel (1979).
in Banachr~umen, preprint 229, Sonderforschungsbereich 72, Universit~t Bonn 1978. 29
Yosida, K., Lectures on differential and integral equations, Interscience Publ. New York 196o.
Sectorial
Second Order Differential Thomas
We shall be concerned tors generated
= -(py'
on the half-line
some properties second
[0,~) under hypotheses
q are complex-valued.
order
Here p is a positive
For convenience
we shall assume
C 1 and that r and q are piecewise
not unique.
tain properties,
or with
Our investigation of the formally
of (i) is based
= -(py'
+ ~y)'
We have
operator
the following
H 0 generated
L in the
of our results
of the existence
are
w i t h cerof one.
upon the properties
second order expression
which
is the real
the expression
+ uy'+ qy
where now both p and q are real valued of u.
expression
largely
Thus we begin by considering M[y]
that p and s
of a representation
the consequences
symmetric
and r, s, and
continuous.
Indeed most
either with the construction
that these
function,
(i) is, of course,
part of L.
expression
which guarantee
of a given differential
concerned
opera-
(i)
The representation form
of differential
+ sy)' + ry' + qy
are sectorial.
are piecewise
T. Read
by the nonsymmetric
L[y]
operators
with
Operators
(2) and u is the complex
characterization
by M is positive
of when
definite.
conjugate
the minimal
This extends
Theorem
2.3 of [9]. THEOREM
i.
(i)
These
(H0f,f)
functions
two properties
are equivalent.
> 0 for all functions
with compact
support
(ii) M has a representation
f in the class C0(0,~)
in the interior M[y]
= -(py'
of C
of (0,~).
+ ulY)'
+ ulY'
+ qly o_n_n
(0,~) where ql ~ lUl 12/p" Proof.
It follows
a representation CO .
for which
For the other
z satisfy
easily
from an integration (3) holds,
direction,if
the system
(3)
M[y]
then
by parts
(H0f,f)
that if M has
> 0 for all f in
= 0 and z = py' + uy,
then y and
585
y' = -(u/p)y + (i/p)z z' = (q - luI2/p)y + (u/p)z. It follows from (i) and Theorem
14, p. 61 of Coppel
has a solution for which y > 0 on (0,~). Riccati
[3] that the system
Then w = z/y satisfies
the
equation w' - (2Re u/p)w + w2/p = q - lul2/p
on (0,~) and if we set u I = u - w, ql = lu - wi2/P
it is easy to see
that ql = lU112/p" Remark.
It is evident
w i t h the inequality
from the proof that the result is still true
in (3) replaced by equality.
above is far more useful,
however,
construct
in specific
representations
as we shall see below, stronger condition
both because cases
representations
The formulation it is far easier
satisfying
where
to
(3) and because,
(3) is replaced by the
ql ~ (i + c) lUll2/p are very easy to w o r k with.
The r e p r e s e n t a t i o n
(2) of M may be rewritten M
= N+N
as
+ P
where N[y]
= /py' + (~/~p)y
; P[y]
and N + is the formal adjoint tion by a n o n - n e g a t i v e M is such that
= (q-
of N.
Note that P involves multiplica-
function precisely when the r e p r e s e n t a t i o n
operators
explicitly
generated by L.
the domains
of the Dirichlet
v e l o p e d by Kauffman
Hinton,
Suppose
adding a constant
[7] and Bradley,
of m-
We do this by applying
real part of L the following variant
Notation.
index theory de-
and K a u f f m a n
that M has a r e p r e s e n t a t i o n
to the
satisfying
[2]. (3).
if n e c e s s a r y we may assume q - luI2/p ~ I.
By We then
set
{f: f~ AC loc and f~(IN[y]l 2 + yP[y]) DEFINITION. NP-minimal lies in 9.
of
(3) holds.
We shall w i s h to characterize sectorial
lul2/p)y,
If M, N, and P are as immediately if the equation M[y]
< ~}. above, we say M is
= 0 has exactly one solution w h i c h
586
For a g i v e n M, al u p o n choice
the p r o p e r t y
the r e p r e s e n t a t i o n of N and P.
of M b e i n g
of v a r i o u s
their
square roots
considered,
This w i l l be i l l u s t r a t e d
W h e n M is N P - m i n i m a l , mains
of b e i n g N P - m i n i m a l
the e x p l i c i t
operators
defined
in E x a m p l e
i.
characterizations
just
in gener-
that is u p o n the
as F r i e d r i c h s
can be e s t a b l i s h e d
depends
of the do-
extensions
as in
[2] or
and of
[7].
In par-
t i c u l a r we h a v e THEOREM richs
2.
The e x p r e s s i o n
extension
M is N P - m i n i m a l
H 0 of the m i n i m a l
operator
d o m a i n H 0 = (fc dom M a x : Moreover
the d o m a i n
that f(0)
f(0)
if a n d only if the F r i e d g e n e r a t e d b_z M has
= 0 and fgN}.
of H I/2 is the set of all e l e m e n t s
Let M[y]
T h e n for any real M[y]
=-(x4y
W i t h qc(X)
= - ( x 4 y ' - ix3y) ' + ix3y '
constant ' +
= (3cqc -
x2y,
(c-i)x3y) ' +
l)x 2, Uc(X)
(c+i)x3y ' + (3c-l)x2y.
= (c+i)x 3, we have
lUcl2/p = (2 - c)(c
and c = 2 and qc and P
c
Nc[Y] The
solutions
- l)x 2.
lUc 12/p > 0 o t h e r w i s e .
c
= x2y ' +
of M[y]
o n l y for c = i.
ceptional N 1 Ix -l+i]
(c-~)xy;
The c o r r e s p o n d i n g
Pc[y]
for c = 1 expres-
= (2-c)(c-l)x2y.
= 0 are x -l+i and x -2+i c
It is easy to v e r i f y
for e a c h c, 1 < c _< 2, w h i l e
Thus M is N P - m i n i m a l c c
case c = 1 c o r r e s p o n d s
for 1 < c < 2.
to the f a c t o r i z a t i o n
x -l+i is in The ex-
M = NI+N 1 where
= 0.
We n e x t
develop
for our p u r p o s e s .
the c r i t e r i o n
Niessen
of an e q u a t i o n
our s i t u a t i o n
becomes
are D i r i c h l e t w h i c h w h e n
the f o l l o w i n g
simple p r o o f . )
for N P - m i n i m a l i t y
that is s u i t a b l e
[8] has g i v e n a c h a r a c t e r i z a t i o n
all s o l u t i o n s
for a v e r y
equality
are
that x -2+i is in the space D D
x _> i.
c w e h a v e also
Thus qc - lUc 12/p -> 0 if a n d only if 1 _< c _< 2 w i t h
c
f in P such
= 0.
E x a m p l e i.
sions N
domain
result.
of w h e n
specialized
(See also B e n n e w i t z
to [i]
587
THEOREM 3.
(Niessen)
M is NP-minimal if and only if p-lexp(2Re/u/p)
and (q - lul2/p)exp(-2Re/u/p)
are not both in LI(0,~).
From this we can obtain the result we want. THEOREM 4. Moreover,
I_!f q 2 (i + g)lul2/p for some g > 0, then M is NP-minimal. in this case
= {faAcl°c: Proof.
f~(plf'[ 2 + qlfl2)<
~.}
We recall that we have assumed just before the definition of
NP-minimality that q is bounded away from 0. Now suppose that both p-lexp(2Re/u/p) in LI(0,~).
In particular,
q£Ll(0,~).
and (q-luI2/p)exp(-2Re/u/p)
lie
Then q - lul2/p ~ E]u]2/p implies that also
(lul2/p)exp(-2Re/u/p)
e LI(0,~).
Then by the Schwarz inequality
u/p ~ L 1 (0,~), since [/~lu/pl]2
~-i
~0 p
exp(2Re/u/p)/~(lul2/p)exp(-2Re/u/p)<
It follows from this that exp(-2Re/u/p) has a finite limit as x ÷ and thus that q - ]ul2/p g LI(0,~). this is impossible.
But q - lul2/p ~ (~/(l+~))q so
Thus M must be NP-minimal.
The last assertion follows from the inequality q - lul2/p 2 (c/(l+g))q ~ elul2/p since then f in P implies that f~qlfl 2 is finite and also, since /oplf'l 2 < 2
/01N[f]I 2 + /o(lulR/p) Ifl 2
that /~plf'l 2 is finite. Remark.
In practice, whenever a representation for M can be con-
structed with q ~ lul2/p it is generally true that in fact q ~ (l+~)lul2/p.
In Example I, for instance,
this inequality is valid
forl
As See
Kato [6, p. 310] for the general definition of the real part of a form. In our context this becomes the following. DEFINITION.
The real part, L R, of L is the symmetric expression
LR[Y] = -(PY'+ ~y)' + uy' + (Req)y where u = (r + s)/2.
588
It is easy to verify We recall
that for f in C~(0,~),
that if the minimal
operator
(LR[f],f)
T O generated
sectorial,
then it is possible
to adapt
Friedrichs
extension
an m-sectorial
to define
domain of T is a dense
subset
form h0[f,g]
where
= (H0f,g)
minimal
operator
generated
p. 322]
for details.
operators
second of these Molchanov's
a representation
by the real part of L.
constants
T of T O .
The
of the
and H 0 is the See Kato
on the numerical
by an expression
on discreteness
range
of the form
The proof
of the essential
consists
essentially
of L so that the hypothesis
the real part of L.
5.
of the
[6,
of m(i).
The
(Theorem 6) is in the spirit of the generalization
in [9, Theorem 4.1].
THEOREM
extension
f and g are in C0(0,~)
generated
theorem
representation
the construction
by L is
of the domain of the closure
We shall give two main results sectorial
= Re(L[f],f).
spectrum
given
of constructing
of Theorem
In the first result we shall assume
has already
4 holds that
for
such a
been found.
Let L be as in (i).
Suppose
that there are positive
B, a, k I and k 2 such that
(a) Re q + B ~ (l+g) Ir + s12/4p, (b)
Ir - s12/4p
~ kl(Re
(c)
IIm ql S k2(Re
Then the minimal extension
q + B),
q + B)°
operator
T O is sectorial
T of T O has the following
(ii) Re(Tf,f)
and the m-sectorial
properties.
(i) domain T = {f ~ domain Lma x : f(0)
= 0 and
I~(plf ' 12 + (Re q + B)Ifl 2) < ~}. 0 ~ (~ inf Re q - B)/(I + e).
(iii) The numerical
range
of T is contained
IYl ~ ([kl(l + s)/s]i/2
in the sector
+ k2(l + s)/s)(x + B).
Here the inf in (ii) is to be taken over all x in [0,~). Remark. essential
This
theorem
spectrum
can be converted
into a result
of T simply by replacing
of
about
inf by lim inf.
the In
589
particular Example where
the e s s e n t i a l
2.
Let L[y]
spectrum
= -y" + (x 2 + ( l + i ) k
~ < ~ or ~ = ~ > k.
for L b y l e t t i n g q(x) derivative H e r e q(x) THEOREM
of
= x
We can o b t a i n 2
, r(x)
Let L[y]
imply
if Re q ~ ~.
x ~ sin x S ) y
= s(x)
a suitable
representation
= F(x) w h e r e
F is an anti-
(I + i)k x e sin x 5 such that
= x 2 + ~ does
6.
is e m p t y
F(x) = O ( x a - B + l ) .
that the e s s e n t i a l
= -y" + qy.
Suppose
a > 0 and A > 0, a n d some b o u n d e d (a) for each x > 0 there
region
that,
spectrum for
is empty.
some c o n s t a n t s
D containing
the origin,
is z, x < z ~ x + a such that
(1/a)Re /~q ~ A, and (b) Then
(I/a)f~q
the m i n i m a l
s D whenever operator
t e n s i o n T o f T() has
x < w ! x + a.
T O is s e c t o r i a l ,
the f o l l o w i n g
(iii) The n u m e r i c a l
ex-
properties.
(i) d o m a i n T = {f a dom Lmax: (ii) T ~ A - (Aa + M I ) 2 / 4
and the m - s e c t i o n a l
= 0 and f0(If'l 2+q 1 Ifl 2) < ~}.
f(0)
= x 0.
r a n g e o f T is c o n t a i n e d
in the set
{x + iy: x ~ x0,
M2(MI+2A)
lyl
M2(Ml+2Aa) Here
ql
is
constants
+ 2M2(X-Xl )1/2, +
M2/M2 + M ( X _ X l ) ,
a mean value
x - x I > (M2/M)2.}
o f R e q so t h a t
M, M1, M2, x 1 o f tan
x - x 1 5 (M2/M) 2,
(ii)
and
in
(iii)
particular are
bl = s u p { a r g
z:
z ~ D, R e z = Aa}
-M 1 = s u p ( a R e
z:
z c D},
ql
defined
as
~ A,
and
the
follows.
M 2 = s u p { a i m z: z ~ D}, x I = A a + MI/2. R e m a rk.
In g e n e r a l
it is n o t p o s s i b l e
of T to a s e c t o r w i t h
semi-angle
hypotheses
= ei0f(x)
a l l o w q(x)
to r e s t r i c t
smaller where
e.
Remark.
(b) m a y be r e g a r d e d
complex valued
function.
0 = a r c t a n M,
f is any real v a l u e d
such that f £ A / c o s Hypothesis
than
the n u m e r i c a l
as a B r i n c k
range
for the function
condition
for a
590
COROLLARY
2.
I f M2(x) ÷ 0 as x ÷ ~ where M2(x)
above for the restriction essential line.
of L to [x,~)
spectrum of T is contained
I__n_nparticular
is defined as M 2
of the real line,
in the subset
[x0,~)
then the of the real
this is true if f0 q exists as an improper
inte-
gral with -x 0 = sup[Rerb q]2/4. a,b ~a As a corollary of the proof we also obtain COROLLARY
3.
If f01Reql
of T is contained {x+iy
= k I, f01Imql
= k 2, then the numerical
range
in the set
: x ~ - k~/4 = x0;
IYl J klk2/2 + k2(x-x0 )I/2}
This result was obtained by Evans
[5] with k I and k 2 replaced by
2k I and 2k 2 respectively. References I. C. Bennewitz, rion, Proc.
A generalization
Roy.
2. J. S. Bradley,
Soc. Edinburgh
quadratic
87A(1981), 193-208. 3. W. A. Coppel, Disconjugacy Springer-Verlag,
Berlin,
4. M. S. P. Eastham,
5. W. O. Evans,
functionals,
Proc. Roy.
(Springer-Verlag, 7. R. M. Kauffman,
Soc. Edinburgh
in Mathematics
72A(1973),
Ann.
255(1981),
operators with a com-
57-76.
Theory for Linear Operators,
Berlin,
Second edition
1980). solutions
linear ordinary
differential
J. Differential
tions 31(1979),
117-129. A necessary
for left-definite
Springer-Verlag, 9. T. T. Read, differntial 388-406.
equations,
and sufficient
eigenvalue
ferential Equations,
opera-
9-16.
The number of Dirichlet
8. H. D. Niessen,
No. 220,
second order differential
On the spectra of SchrSdinger
Perturbation
On the minimiza-
1971).
Soc. Edinburgh
Math.
crite-
81-90.
(Lecture Notes
Semi-bounded
Roy.
plex potential, 6. T. Kato,
78A(1977),
limit-circle
D. B. Hinton and R. M. Kauffman,
tion of singular
tors, Proc.
of Niessen's
problems,
to a class of
limit-circle
Equa-
criterion
Ordinary and Partial Dif-
Lecture Notes in Mathematics
415 (Berlin:
1974).
Factorization expressions,
and discrete
spectra for second-order
J. Differential
Equations
35(1980),
Spectral
Mapping
Theorems
for
Dissipative
Harald
i.
Formulation
Let
G
= exp(Gt) inner of
,
of
the
: D(G)
C H ~ H
x(t)
determine
mapping
=
not
for or
hold
in
G
and
2.
t ~> O.
Re
T(t)
:
to
of
of
separable describe
the
,
which
a C0-semigroup
Hilbert
the
abstract
x(O)
G
asymptotic
Cauchy
: x0
•
space
T(t) H
with
behaviour
problem
D(G)
guarantee
that
the
spectral
x)
~
a normal,
holds
([ii]).
(e~g.
([5] , [ 3 ] ) .
In
for
self-adjoint
continuous
this
dissipative
0
(!)
paper
eq. (i)
consider i.e.
bounded
differentiable
However, we
operator,
or
the
does
case
(G - I ) D ( G )
= H
x C D(G).
results
that
Hilbert
G
is
space
H
c~(G)
C
dim
D(G)
a densely and
:
{ x •
D(G)
forward
to p r o v e
that
G -1
resolvent.
] Re
/ m 0 (G)
Do (G)
Hence
defined
maximal
dissipative
operator
satisfies
{ z @ e
where
a compact
is
uniformly
eq. (1)
a maximal
( Gx,
e x p ( J (G) t)
G
is
general
Abstract
the
order
for
If
then
is
Assume on
if
analytic),
where
generator
a complex
G x(t)
conditions
the
theorem
operator or
on In
<5(T(t) )
holds
Roh
be
= T(t) x0
:~(t)
we
,
(-,.).
solutions
Generators
problem
0 ~< t < ~
product
C0-Semigroup
=
I Re is
z < 0
no
(2)
<
( Gx , x) a compact
the
}
spectrum
(3)
= O
}.
operator ~(G)
It
is
on of
H, G
straight i.e. consists
G
has of
592
at m o s t
countable
multiplicities
many ~n"
the
(a)
Let
Hilbert
the
(b)
the if
<
G
be
a densely
their
only
if
if
imaginary
in
and
z
{H n}
of
the
of
H
%
are
uniformly
Theorem
1 remains
G
a nuclear
Theorem dim
Hn ~
nl
Let < ~
n
- %
m
resolvent.
operator
Then
the {z
that
unconditional
whole }
n
{IRe
-~
as
Hilbert
of
subspace
space
where
Hilbert points
space in
Zni11R(Zn,G) n ~
vector
{H n} for
all
an n
that
H
the
ll}
is
~.
H
form
x E H
has
the
series
e
valid
for
if
strip
in
the
operator
i.e. all
instead
that
span
an
(uncondi
a unique converges
this whole
(4)
parallel
there n ~ m
of
is
(3)
case
to
exists
the some
([9])
we the
space
satis-
assume
that
generalized
H,
because
([7]).
unconditional Let
condition
a bounded
separated,
Note
nuclear
be
show
in
I ~
automatically
a dissipative
2:
(4)
dissipative
(5).
span
x n @ Hn,
lie
n
3:
is
~
n
calculations
Remark
G -I
k
H.
II
G
G
if e a c h
with
that
of
G Re
such
eigenspaces
of
and
e > O
has
that
hull.
such
eigenvalues
axis
algebraic
(5)
an
of
Elementary
the
all
form
p(G)
x = Z xn
2:
linear
G
-
a sequence
basis
(unconditionally)
Remark
such
maximal
exists
Subspaces
expansion
finite
n
there
bounded
subspace
all
(2)
of
eigenspaces
uniformly
tional)
for
defined
satisfying
closed
set
l:
with
6 > 0
0
for
eigenspaces
generalized and
~
H
resolvent
Remark
fied
V
space
of
exists
>
<
generalized
basis
In
i
n
on
there
6
n n#k
i:
Assume
eigenvalues
i %k- In I
II
Theorem
isolated
G
be
subspace a closed
basis operator
of
H on
with H
593
such
that
Hn
tion).
If
C D(G)
G
for
generates
o(T(t) )
all
n.
Define
a C0-semigroup
=
e x p (o (G) t)
Gn
T(t)
=
:: G I H n
= exp(Gt)
(restricon
H
, then
O e x p (o (Gn) t ) n
An
immediate
Corollary spaces
of
Remark
4:
then
i:
Let
G
span
If
only
consequence
the
G
be
the
the
of T h e o r e m
as
in T h e o r e m
whole
Hilbert
eigenspaces
point
zero
of
must
be
1 and
1.
If
space
G
the
H,
do n o t
added
2 is
to
then
span
the
generalized eq. (I)
holds.
the whole
right
side
eigen-
space
H,
of eq. (i)
([9]).
Remark left, O C
5:
If
i.e.
Re
p(T(t))
generates
Theorem
Let
all
their
generates
G
eq.(1)
for
all
holds
Proofs
of
Theorem proof. The
operator
the
class
we T
as
form
and
for
all
n,
Thus
t E
~
in
,
on
an u n c o n d i t i o n a l
t @
the
operator
on
actually
H.
are
]R ,
to
implies G
of
multiplicities
(i)
case
on
eigenvalues
t E
then
this
defined
1 is b o u n d e d
H,
G
are
H
uniformly
uniformly then
subspace
with
nuclear separated
bounded.
the
If
G
generalized
basis
of
H
and
IR.
Theorems
define is
G
terminology Coo
the
T(t) ,
1 is p r o v e d
First
eigenspaces In
the
in C o r o l l a r y
a densely
that
a C0-group of
3.
be
algebraic
eigenspaees
G
t ~ O. T(t),
G
Assume
of
~ > - ~
a C0-group
3:
that
spectrum
In ~
for
resolvent. and
the
(3)
in [9] , the
so w e
Cayley
a contraction with of
respect [i0] ,
implies
H
its
condition that
outline
transformation
on to
only
the
and
has
the T
the
eigenvalues
idea
::
of
the
(G+I) (G-I)
same
generalized
an = (In+l) (In-l)
(2)
implies
that
defect
indices
of
T T
-i
is of and
T*
-i
594
both
equal
exists
a minimal
functions of
no < ~
(2),
mT(1)
Moreover,
is the n - t h condition
two d i s j o i n t
subspaces
and the
Hn
that
the
If T
condition
G
we
prove of
that
H.
First
stated
recall
denotes
to p a r t
there
contraction
that
the
T
Theorem
2 (G*-I)-I
M
T
in
where (6).
that
s u m of
that
we
and
u1
proves
only
fact
remark
c = O
eigenspaces
its C a y l e y
for a n y
of all
this
theorem, to the
H
hull
of
in T.
transformation
_ 4(G*-I)-I(G-I)-I
Co.
(2)
T
Hence
is of c l a s s we
can
f o r m an u n c o n d i t i o n a l
Coo,
as a b o v e
subspace
basis
3.
2 note
because
is a n y
,
Ha-interpolating
linear
of the
As by
that
it is s u f f i c i e n t
o(exp(Gnt))
e x p ( o ( G ) t) C o ( T ( t ) ) if
to the
appearing
seen
the
(6)
because
is of c l a s s
Remark
,
then
operator.
of
c ~> o
one p r o v e s
closed
completeness
([i0]),
is a n u c l e a r
to p r o v e
Because
[ mn(T) x = O}
algebraic
is e q u i v a l e n t
-2(G-I)-1
and
the
(b) of
justifies
= O.
is a
([2]),
As
C U O(exp(Gnt))
C e x p ( a ( G ) t)
~
{a n }
It is e a s i l y
that
there
analytic
corresponding
product
that
theorem of
Oi
the e i g e n s p a c e s
In o r d e r o(T(t))
states
resolvent,
This
T
of t h e B l a s c h k e
a nuclear
conclude
Thus
la[ : i ,
n C oi . i.
=
cl_ 1 x , e +___!
:= { x E H
has
I - T*T
again
H
n
Hn
the
I - T*T
mT(T)
with
characterizes
is a w e a k
i.e.
satisfying
of
oi , o2
with
Co.
(bounded
eigenspace
corona
Here
(a) of T h e o r e m
which
mT(~ ) C H a
disk)
(which
subsets
of c l a s s
form
factor
part
(6),
the
(4)
is d i r e c t .
O2
unit
coincides
(5)
is e v e n
an - ~ Iv ~ IT n n lanl 1 anX
a
an
T
function
the g e n e r a l i z e d
sequence),
H
(inner)
has
:
eigenvalue
Using
hence
on the o p e n
mT(X)
mn(l)
,
invertible
to p r o v e
= e x p ( o ( G n ) t)
automatically
hold.
k×k
then
matrix,
C
595
IIM-IIL
where
Yk
define
~<
Yk
is i n d e p e n d e n t Tn(t)
ILTn(t);l
"= T ( t ) I
is u n i f o r m l y
llM[Ik-i I d e t M
of
M
Hn
([6],
in
(7)
chapter
Clearly,
bounded
I-I
I, eq. (4.12)).
Tn(t)
n.
Next
= exp(Gnt)
Assume
and
I ~ U o(exp(G
t)). n
Regarding
Tn(t)
as a m a t r i x
It(II -Tn(t))-lll I ~ o(T(t))
generates as
is u n i f o r m l y
and T h e o r e m
To p r o v e
Theorem
a C0-group
t ~ ~
equivalent
Then
3 we
assume
T(t)
such
there
([8]).
bounded
exists one,
y)o
in
n
without
implies
for f i x e d
that
t,l.
Hence
:=
loss
of g e n e r a l i t y
that
G
that
llT(t)][ d e c a y s
exponentially
a new
inner
(',')0
with
It is g i v e n
( x,
ineq. (7)
2 is p r o v e d .
to the o r i g i n a l
dissipative
operator,
product
respect
to w h i c h
G
on
H,
becomes
by
( T(t)x
, T(t)y)
dt
o
Theorem
3 now
easily
and
3.
4.
Applications
Consider
utt
the
+
follows
following
7 Utx
from Corollary
parameter
+ u XXXX
+
i,
dependent
B u XX
=
O,
(l,t)
= u
observing
Remark
damped wave
O < t < ~,
2
equation
O < x < 1 (8)
u(O,t)
= Ux(O,t ) = u XX
with
real parameters
f o r m on t h e H i l b e r t < f ,g > :
space
f f(t)g(t)
utt Af
8,y ~ 0
: f ....,
-
dt
yCu t + D(A)
(l,t)
=
O
XXX
(Ill).
L2(O,1)
We
with
rewrite
inner
(8)
in o p e r a t o r
product
as
(A + B B )
u
: {f • H 4 I f(O)
=
O = f' (O) = f" (i) = f"' (i) =
O }
596
Bf
= f"
,
C f : -f'
(where
Hk
on t h e
,
denotes
Hilbert
D(B)
: { f C H 2 I f(O)
= f' (O)
D(C)
=
= 0
the
{ f E H I I f(O)
Sobolev
space
space
H = D ( A ½)
o i]
positive The
-A
D(G)
,
= D Go)
where
product
< A½f
compactness
of
H
A -1
,
a compact
resolvent
of
G
shows
real
axis,
upper
are
complex
An
It
is e a s y
satisfies no
= l
half
it is for
+ y GI
-y
+
to c h e c k
all
the
plane,
system
form
+ B G2
Note
conditions
(2)
an
compact
O
is a
resolvent.
D ( A ½) : D ( B ) .
forward
of is
to prove
eigenvalue
G
lie
large
enough,
the
G = G(8,T) analysis
symmetric
to
and,
the
in t h e
representation
i )2
i ~ 2 (n - 7
dissipative
Using
that
An
asymptotic
l ~ i +
,
-B
A = A* > 0
with
and
modulus
(5)
io o1
=
by
- ~
the
G2
that
L2 (0,i)
eigenvalues
y2 i
'
B,Y ~ O.
admit
that
C
straight
if t h e i r
operator
+ O(n
G(O,y)
in p a r t i c u l a r
(3)
-i)
(9)
= Go
holds
+ yGI
with
because
Re (G(O,7)(~)
(~)
,
:
'
By Theorem subspace
0
on
is g i v e n
that
simple
=
in
as
io o]
:
> : < f " , u" >
has
([9])
Go
X D(A½).
operator
on
, A½u
GI
= D(A)
self-adjoint
inner
and
t
Go
with
=
}
]
Hk(o,l))
X L 2 (0,i)
G t t
: 0
i,
basis
eigenspaces
of
the of
G(O,y)
iv(l, I2
for
($)@
D(G)
2
eigenspaces their
- ~
of
G(O,y)
form
closed
linear
hull.
in
follows
by
H
an u n c o n d i t i o n a l
The
simple
completeness perturbation
of
the
argu-
597
ments,
either
by d i r e c t l y
Theorem
1 (b)
or by noting
resolvent.
Hence
G(O,y). C0-group
H
perturbation C0-gr o u p
T(t) ,
Thus T h e o r e m note
that
G
(9)
(Remark
of
that
the spectral
Moreover, on
verifying
G(O,y) mapping
implies 5).
that
As
we conclude
t @ ~ ,
on
is a d i s c r e t e
has
H
theorem
(i) holds
G(O,~)
actually
that
G
spectral
+ BG2
G
and that
to
condition
G
also has
and proves o p e ra t o r
in
in fact a nuclear
G = Go + yGI
G(O,y),
3 can be applied
the c o m p l e t e n e s s
for
G =
generates
a
is a b o u n d e d
generates
a nuclear
eq. (i).
a
resolvent.
Finally
we
([4]).
REFERENCES i.
A.K. Bajaj, P.R. Sethna and T.S. Lundgren, Hopf b i f u r c a t i o n p h e n o m e n a in tubes carrying a fluid, SIAM J. Appl. Math. 39 (1980), 213-230
2.
L. Carleson, I n t e r p o l a t i o n by b o u n d e d a n a l y t i c functions corona problem, Ann. of Math. (2) 76 (1962), 547-559
and the
3.
E.B. Davies, 1980
London
4.
N. D u n f o r d and J.T. Spectral operators, Wiley-Interscience,
5.
E. Hille and R.S Phillips, AMS C o l l e c t e d p u b l i c a t i o n s
6.
T. Kato, Springer
7.
V.B. Soc.
8.
A. Pazy, On the a p p l i c a b i l i t y of L i a p u n o v ' s t h e o r e m space, SIAM J. Math. Anal. 3 (1972), 291-294
9.
H. R~h, D i s s i p a t i v e operators with finite d i m e n s i o n a l Proc. Roy. Soc. E d i n b u r g h Sect. A 91 (1982), in print
One-parameter
semigroups,
Schwartz, Linear Pure and applied New York, 1971
Academic
Press,
operators, Part III: Mathematics, Vol. VII,
F u n c t i o n a l analysis and semigroups, no. 31, Providence, R.I. 1957
P e r t u r b a t i o n theory Verlag, Berlin 1976
for
linear
Lidskii, Nonselfadjoint operators Transl. (2) 47 (1965), 43-46
operators,
with
second
a trace,
edition,
Amer.
Math.
in H i l b e r t
damping,
10. B. S z . - N a g y and C. Foias, H a r m o n i c analysis of operators H i l b e r t space, N o r t h - H o l l a n d , A m s t e r d a m 1970
on
ii.
Sci.
J. Zabczyk, A note on C 0 - s e m i g r o u p s , Ser. Sci. Tech. 23 (1975), 895-899
Bull.
Acad.
Polon.
D e p a r t m e n t of M a t h e m a t i c s Heriot-Watt University Riccarton, Currie E d i n b u r g h EHI4 4AS / S c o t l a n d
Some Preliminary Results on Periodic Solutions of Matrix Riccati Equations
Oavid A. S~nchez
The question
to be considered
is: given the autonomous
matrix Riccati
equation X
(1]
= BX + ~B T + X C X ,
where 8 and C, are nxn reel constant matrices
and C is symmetric,
together
with
initial conditions
~(o)
under what conditions
systems
solutions
symmetric,
solutions?
Also,
are of interest
- for instance
(2j
what is the structure
in {ilt~ring
theory
Knowing the constant
as
solutions
will bifurcate from them if [I] is subjected
forcing term.
Using a formula earlier results
previously
by R. Redheffer
co > o, O real and symmetric,
will satisfy
These questions
of periodic
one can ask whether periodic to e periodic
@, r e a l ,
does it have periodic
of its constant iso&utions? well as the control
=
developed
by the author in
[I ] and W.T. Reid
for which the solution ~(t]
the relation
emB@
T - @e -~BT + @(e -~BT Io etB cetBdt]O
[3 ], and derived from
[ 2 ], one sees that a pair
= O.
of (I ] satisfies
[~,@],
599
As it s t a n d s exponential matrix
Case
1:
last
formula
in series,
C out
algebraic
the
so one m a k e s
of t h e i n t e g r a l
equations
is f a i r l y
even
some
assumptions
Then
the
sign.
are f o r m i d a b l e
useless,
but not
case
expandin Z the matrix about
B and
C so as to m o v e
n = 2 is e x a m i n e d
in d e t a i l
impossible.
B symmetric
If B is s y m m e t r i c ~(O)
= ~(@]
and ~ [ t ]
= O, then
is a s o l u t i o n
~ and
[a~} d e f i n e d
for o
and
satisfying
@ set~{y
e~S8
- @e -~B + @ ( f ~
cosh
e~B@
- 8e -~B + w Q e - ~ B c o
tBdt)C8
=0
if BC = CB and
= 0
if Be = -CB. Using
these
relations
one can
show
for t h e
case
n = 2 that
there
are
no p e r i o d i c
solutions.
Case
2: B s k e w
symmetric
If B is s y m m e t r i c
and ~[t]
and s a t i s f y i n Z ~ [ o ]
if BC = CB,
= ~(~}
is a s o l u t i o n = @, t h e n
o{
@ and
(1) d e f i n e d
for
e
@ satisfy
emB@
- @e -wB + ~ 8 e W B c @
e~S8
- Oe ~B + @ { ~
= 0
and
cssh
tBdt}C8
= 0
if BC = -CB. Using
these
formulas
relations
~or
the case
sin a~
if BC = CB, a n d
-
a
[BO
where
-
OB
+
C = cI,
~OCOBO]
=
B has
eigenvalues
--C B I
+
cos
-
9B
at0[t0cO 2 ]
=
and
sin -
Be
n = 2 when
become:
-
i{
the
since
-
aw a
[B8
+
@C8]
=
0
O
tie,
a /
o, t h e s e
600
For the case BC = CB a s t r a i g h t f o r w a r d solutions,
analysis shows that there are no periodic
whereas for the case 8C = -CB a possible period is w = ~/e,
that the s o l u t i o n a exist for o < t < ~ / a
The condition
imposes a restriction on the choice of initial
values
The result is:
If B is skew symmetric, n = 2, with eigenvalue ~ia # o, and BC : -CB then periodic solutions
[possibly constant]
of period ~ = ~/a will
exist whenever
1 -Ta[ClX + -~[c2x
+ 2c2Y
- 2cfly
- ClZ]Sin
- o2z +
2at
[c# + c ~ ] [ y 2 - x z ] ] s ± n 2 a t
# o,
wlth
C =
Example : B
=
The only constant solutions will be the one p a r a m e t e r family
~ =
2x
'
x / 2
0
and periodic solutions will exist whenever x,y,z satisfy
1 +
sin 4t +
- ~ + ~- + ~
All other solutions will be unbounded.
- -~ xz sin 2 2t / o.
This example was only cursorily examined by
T. Sasegawa in I f ] . /
The solution of
Z[t]
[1] with initial conditions
[2] is given
by the expression
= A[t]O[e-2pt I _ e-2pt[fnt e s S T c e S B d s ] @ ] - I A [ t ] T
601
~here A[t] = ( cos at sin at
sin at] cos at
Now one computes the expression
ftesBTceSBds
= e2ptF[B,C,sin
2at,cos
2at)
- F[B,C,o,1]
0
and one concludes that the only possible period i s
~
= ~/a,
and a unique periodic
solution exists if and only if
F[B,C,sin 2at,cos 2at)
exists {or o< t <%/a,
in which case
~(o)
However, and
a more
-I
detailed
hence is a constant.
= O =
analysis
-F[B,C,o,1)
shows
-I
that
such
e
solution
satisfies
This last assertion is proved by showing that it is a
necessary and sufficient consequence of the relation
end then doing some careful bookeeping.
If
B =
-a
,
e,p
/
o,
C
We conclude that:
=
, then there are no periodic
solutions, ~[t] ~ 0 is an isolated constant solution and there is an -I additional isolated constant solution i~ F[B,C,o,I] exists.
The above result holds even in the case where C is not symmetric.
Example
:
2a'
la1
3,
C= [ o
.
A change of variable converts B and C to
B =
e
X(t]
a) "
~o
-
"
and {or lal < 3 the only nentrivial constant solution is 7[t) _- [4a --
I
o
o ] 4a
"
602
There are no periodic solutions.
This corrects a second example found in the paper
by Sasegawa
[ ~]. / The analysis of the structure of constant solutions in the case n = 2, as well
as the question of the existence of periodic solutions when periodic
[possibly nonlinear)
(I) is subjected
to a
small disturbance will be analysed in a f o r t h c o m i n g
paper.
Mathematics Department University of New Mexico
Ref erenc es
I . R. Redheffer,
On s o l u t i o n s
Jour. of Rational Mech.
of R ~ c c a t i ' s
and Anal.
equation
as a f u n c t i o n
2. W.T. Reid, S o l u t i o n s o f a Riccati m a t r i x d i f f e r e n t i a l values,
of initial
values.
5 (1956), 835-848. as f u n c t i o n s
of initial
differential
equations,
Jour. of Math. and Mech. 8 (195@), 224-230.
3. O.A. S~nchez, C o m p u t i n g p e r i o d i c
solutions
of Riccati
Appl. Math. and Comp. 8 [1880), 283-287. 4.
, Periodic equations:
and constant
5. T. Sasegawa, A n e c e s s a r y Riccati
584 -588.
solutions
of m a t r i x R i c c a t i
differential
n = 2, to appear.
equation
and sufficient
to b e periodic,
condition
f o r the s o l u t i o n
of the
I.E.E.E. Trans. Auto. Control, AC-25
(1980],
Solitary and Travelling Waves in a Rod
Ralph Saxton
We are concerned with the existence and partial analysis of longitudinal travelling waves in a thin rod, with a later application to a growth condition in incompressible elasticity. solutions
u_(× t b ~
~6~,
~ = ~-ct
We consider only
, which have the form of
steady travelling waves for the equation of motion, which is derived most simply by means of Hamilton's principle T
([7]; cp. [4],[5],[6])
applied to the functional
I
(1)
o
Here, the total kinetic energy is given by
<(t5
ZJ
t
where ~(.) is a measure of the lateral deformation undergone during any elongation or contraction of the rod which is assumed to remain straight at all times,
u (x,t)
is the longitudinal
displacement at time t of a material point at x
measured from some
chosen point along the rod, which is now taken to be of infinite length.
We have taken the density in the current and reference
configurations to be unity.
In terms of u X
of incompressibility (Poisson's ratio ~ + ½ small deformation)
~
B (u x)
, by adopting the limit in the special case of
is given here by I
~ - i
(1 + Ux)~
although we need not assume this.
(3)
'
To avoid material inversion or
flattening in general, it is only necessary that
~
~
~ _
I, the total strain energy, is of the form
lit~ where
W(.,.)
ik~i(~×C~,t~(~
(~'tS~
~
is the strain energy.
(4)
604 On carrying out the variations, we obtain from (i) the Euler equation utt
-
(B'B"
u2
xt
+
B'2
Uxtt) x
- (W'
+ W~ 6( )x = O ,
(5) where subscripts denote differentiation with respect to x and t, and superscripts variables
~
t ,~ denote diffentiation with respect to dependent
and
~, respectively.
Further details of this equation
appear in [7]. We now consider the substitution in (5) of
g lying
(7)
On further substituting
~=
~
integrating (7) with respect to Z
~ = c~'(W~
~Y
and
, we obtain
(8) where we have omitted the constant of integration. At some point Zo
the initial data for (8) is given by (9)
and on multiplying (8) by ~
and integrating with respect to
we have the energy equation
- c~ ~ z
+ , qi z
%Jc~, ~ ( ~
)
= ~
where E is a constant depending only on the values of ~o, ~
(io)
and c.
Our analysis of equation (7) is in the phase plane. (For a related discussion in another problem, c.p. Ball [I], Calderer [2]).
We therefore use (8) and (9) to obtain the equivalent
first order system
605
(ll) where
=
)2
,
(12)
(14) and
We now present a result on local existence of trajectories Proposition
i
Assume
~(q<$(~ is locally Lipschitz continuous
on its domain of definition.
Then the initial value problem
(12) - (15) has a unique maximally defined solution (%~E for some
Proof
C~I~
~'~i~
where
~ a I~ • The solution
For
~ ~qJy ~
i
I Z~-~o
~
~
x (z) satisfies
to be locally Lipsehitz,
it is enough for
B (.), W (.,.) and W (.~.) to be locally Lipschitz continuous in each variable,
and for B (.) to be sign definite.
require the existence of some that for any
and~ > o
(,,..[.,,~ ,
such that
"~ , ~% ~
\~
0 <
P
~ ~
"~ ¢_ I _ ~ ,
This implies we ~=~(~,~ such ~
I-~
e I ,
r._'~c
I ~ ~u~
We may then apply standard results in ordinary differential equations (e.g. Priedrichs through
C~
[3 ]) to obtain local existence and uniqueness
~ .
606
Referring
to equation
(i0) we define
(16) and d e n o t e J.~V"
(17) We may characterize
the equilibrium points for (ii) - (14) by the
following proposition. Proposition ~z
2
Let c be given.
~ ~6) ~(~:~
function
~ ze ~
c~dd-~-LW~
Further,
for~
and k / ( ~
Proof
= c~d£ + ~j[~ #
cI< ~-'($~ , ( ~ o) W h e n ~ satisfies
is a centre c ~ 7~ ~r~(W~, ~ , o ]
is a relative maximum.
Follows from (14) and its linearisation
in a neighbourhood Remark
~-cqi~
is a relative minimum.
is a saddle point and ~ i ~
solutions
of (Ii) are the zeros of the
, where
satisfying
Then the equilibrium
of
(~
0"),
In addition to the above cases we may also have
which corresponds
~_~ = ~ ( ~
to a point of inflection of V.
We now assume that for each c, there exists a finite number,
n, of
zeros
(18) c a @ - ~-(.q~") •
to the function Further we consider
two possible hypotheses
(HI) As
k~ _~ + ~
~
(H2) As
dJ
where ~ > O
-~ +~
,
~L__--~%
~o 1
~CW~
~ q~
:-
(HI) and (H2) are compatible with the Lipschitz
assumptions made earlier. the region ~
for ~-
o
We use these conditions
to investigate
of the phase plane and obtain the next two results.
607
Proposition 3
Let ~
0
throughout, and let ( ~ , ~ , )
be chosen so
that
Then a) when cz< ~ and either (HI) or (H2) hold, the trajectory through ( ~ ) i s
periodic with period 7,
(19) where 7 o < 7 L
and
(]o,o),
(~o)
are the intersections of the
trajectory with the axis~< = © b)
When ¢~>q
and (sz) holds, we again have the result of a).
c) When az> I and (HI) holds, no periodic orbits exist through any point (*~o) with
~>
$~ ,
Proof From (i0) and (18). Corollary ~
When (~o ~,) are chosen so that ~
=
~C~
~
then either A) (~o,~i) = ( ~
for some of { = c , ,. ~ 0 ) is a centre, or when c 2 < 1
and (HI) and (H2) hold or c 2 < 1 and (H2) holds, then B) i) for ~/~z O we have
where ~[ < ~o is the closest saddlepoint to the left of (4~,O). ii) for
where~ Remark
<~, ~
W~
O
is the closest saddle point to the right of (~o,O).
The case B) above is that which provides for the existence
of solitary waves.
Note that since centres and saddle points
alternate ( w e a v e excluded the case c Z = ~ ' ~ > >
, given the same
~o in B i) and ii) above implies ~6 = ~ * z We conclude by looking at a case for -~ < ~ & by (3).
0
, with ~(q~) given
608
Hypothesis
(H3)
Proposit, io n 4
For
~ 6
~-~
Under hypothesis
_
as
~_--~
(H3) , a sufficient condition preventing
flattening of the incompressible Proof
[
rod is
~>
O.
With ~ given by (3), (i0) becomes
d z
c ~
and so as
.-~
QT--> -I
C2
/Z ~'~
~
(~÷ ~ )~ ( ~ ~ ~-~ ) - % (l+ ~
and the result follows since
~ > o
)~(,- Y/z ]
(21)
and it is impossible for
References [i]
J. Ball, Discontinuous
Equilibrium Solutions and Cavitation
in Nonlinear Elasticity, [2]
C. Calderer.
Preprint.
The Dynamic Behaviour of Nonlinear Elastic
Spherical Shells, J. Elasticity, [3]
K.O.
Friedrichs,
to appear.
" Advanced Ordinary Differential Equations",
Gordon and Breach Science Publishers, [4]
W.A.
Inc., New York (1965).
Green, "Dispersion Relations for Elastic Waves in a Bar",
in Progress in Solid Mechanics I, Ed. Sneddon and Hill. [5]
W. Jaunzemis,
[6]
A.E.H.
"Continium Mechanics",
Love, Theory of Elasticity
New York MacMillan (4th ed.)
(1967).
278, Cambridge
Univ. Press (1927). [7]
R. Saxton, "Global Existence for the Nonlinear Pochham~erChree Equation",
submitted to J.D.E.
INVESTIGATIONS
IN THE THEORY OF PARTIAL
D I F F E R E N T I A L EQUATIONS OF INFINITE O R D E R
K. Seitz
Technical U n i v e r s i t y of B u d a p e s t
I. I N T R O D U C T I O N Consider the following partial d i f f e r e n t i a l
e q u a t i o n of in-
finite order
~) where
2~
and
A~
are complex variables.
The domain of where
~
-Ou~.~[~i/4]~]1(~=Oi...1~is__.
is an a r b i t r a r y pozitiv constant.
In this short lecture we shall i n v e s t i g a t e the structure of the ~
O< ~
# ~ A~
C~% 6' ~' ~
C~]
and
~'=
are complex numbers
we suppose that
~ ~ * ~
~o~-
t r a n s f o r m a t i o n s w h i c h map
~, ~o)
C4] to ~A =O
•, , . = o
'
~--
e
where
(,4'1
F:
,
=
F
o~m+
0~"
,
%m+&&
•
This i n v e s t i g a t i o n has some ~ p o r t a n t c e up a Lie-like theory of partial d i f f e r e n t i a l
for b u i l d i n g
equations of in-
610
finite
order.
For reader
II.
the
study
is r e f e r r e d
of
the
field
treated
~t
to the w o r k s
in this
paper,
the
[2] I [5] ,
THE T R A N S F O R M A T I O N It is easy
to
see
that,
from
(~)
[2_I
and
[]~
we have
and •
BF~ - :
-
~' - _
~
+
~,
- -
"~
where
a = o
~-a,
(~)
and (G)
÷o
it f o l l o w s
~¢ ~ _ , ~ . ~
,~, ,~,
that,
am, A
~'" F
~,:o
gp.l' ~ ~
I"
~
A
where
19)
c y l &-a1&= (y = 0 t 4 \ , .. \
and
I t i s easy t o s e e , t h a t
s i n c e o w e have
f r o m which
(2)
maps
(4) t o
h)
(3)
iff,
612
Thus we have obtained Theorem
If
Cb~_2~
ables with domain pozitiv
number,
following
®
the following
(~%~0-)
theorem:
are functions
IZ~< ~I ~rl<~
wheFe
~
then the transformations
partial
differential
(~)
equations
O" ~(~,~)
"'
,,.=o ,%.--0
of complex
which map the
of infinite
=
vari-
is an arbitrary
0
order
-to
8 ~ ~" 9,~u"~'-
2: 2:: ~.,~.,.c~,~1 ,,
are determined
^F (~,,~1
,,
by the following
~o
relations
/ru
(~I and
It is well
known that there
the groups
of 2 ~ 2
~ @ | ~% ~
are complex
group os denoted field.
terize
the
~1
(~)
Example. rential
/~v
~=0
~=0
where
and
~~
see that there
partial Consider
~
where
and a subgroup
equations
oo
numbers
by ~ L (2, ~)
In this way we can tion between
is a one to one map between
type matrices
of
differential the following
of~inite
~ ~
where
~ 0
. This
is the complex is a one to one rela-
~ L (~|~], equation
which charac-
ofi~inite
Euler-type
partial
order. diffe-
order
l
S~,~.~ I < ~ = 01 41 9-i' "- I /nJ) In this case we have the following A
and the set of all matrices
subgroups of GL (2 L<]
[ o~0
~]
are c o n s t a n % numbers. transformations
which
is an abelian
613
REFERENCES
Ill Einar Hille,
"Ordinary
Complex Domain", 2] Nathan Jacobson,
Differential
Wiley-Interscience, "Lie Algebras",
Equations New York,
Interscience,
in the 1976. New York
1962. 3] V. S .Varadar~ an, representations",
"Lie Groups,
Lie Algebras,
Prentice-Hall,
Inc.,
and their
1974.
WELL-POSED BOUNDARY PROBLEMS FOR HAMILTONIAN SYSTEMS OF LIMIT POINT OR LIMIT CIRCLE TYPE by J. K. ShawCI) and D. B. Hinton (2)
INTRODUCTION.
We c o n s i d e r a 2n × 2n H a m i l t o n i a n system o f d i f f e r e n t i a
equations
J~' : [ ~ A ( t ) + B ( t ) ] ~ ,
(l.l)
-~ < a < t < b ~ =,
where A ( t ) and B ( t ) are 2n × 2n H e r m i t i a n m a t r i x f u n c t i o n s l o c a l l y i n t e g r a b l e on [ a , b ) ,
A(t)
complex parameter, ~ ( t )
r e a l and nonnegative d e f i n i t e ,
is a 2n x 1 v e c t o r f u n c t i o n and 0
-I
J = In
We assume A t k i n s o n ' s
(l .2)
x is a
n
(I n = n × n i d e n t i t y ) .
0
"definiteness condition"
([I,
p. 253])
I~ ~t)A(t)~(t)dt > 0
(I)
Research supported in p a r t by NSF Grant No. MCS-8101536.
(2)
Research supported in p a r t by NSF Grant No. MCS-8101712.
615 whenever a _< c < d < b and ~ is a n o n t r i v i a l solution of ( I . I ) . To say that ( I . I ) solution of ( I . I )
is of l i m i t c i r c l e type at b means that every
-~ 2 is of " i n t e g r a b l e square" on [ a , b ) , w r i t t e n YELA[a,b),
which is to say f ~
Ay < =.
I t is known ( [ I ] ,
has only L~[a,b) solutions f o r any one
[8]) that if
(I.I)
x, then t h i s is so f o r a l l x.
The Titchmarsh-Weyl M(x) functions associated with ( I . I )
are
c e r t a i n n × n matrix valued a n a l y t i c functions of x which generalize the c l a s s i c a l m(x) c o e f f i c i e n t ( [ 2 ] ,
[I0]).
The present authors
developed a theory of matrix M(x) functions f o r " l i m i t point" systems (I.I)
in [ 4 ] , [ 6 ] .
To continue our study here, we introduce the M
functions in the t r a d i t i o n a l way by s e t t i n g up a boundary problem on a compact i n t e r v a l [ a , d ] , d < b, and considering the l i m i t i n g case of the problem as d ÷ b. Thus consider a compact subinterval [ a , d ] , and l e t ~l(d) and 52(d) be n x n matrices which may depend on d and which s a t i s f y f o r all d
(1.3)
+
= I,
=
Let E~ = Es(d) be the 2n x 2n matrix
(I .4)
By i t s d e f i n t i o n E~
E6
]
satisfies for all d < b
616 (l .5)
E~S1 = E~,
E~JE~ : E6JE ~ = I .
A matrix in the form ( I . 4 ) , whose n × n block components s a t i s f y (1.3), w i l l
be termed admissible.
At the l e f t endpoint t = a we shall need a fixed admissible matrix
E
C~
I I
and a 2n × 2n matrix solution Y (t,~) of ( I . I )
Ya(a,~) = E
We recall from [ I , each fixed t.
(l'6)Ya(t'~)
with i n i t i a l
value
for all x.
p. 257] that Y(t,~) is an entire function of h for
We p a r t i t i o n
~ a(t'~)
Y into n × n blocks by w r i t i n g
~a(t,~)]:[em(t'~)@m(t'~)]
' ~c~ :
~
(~(~
We frequently use the notation of (1.6) for 2n × k matrix q u a n t i t i e s , ~ = (~),
where f and f w i l l
components, respectively. instead, f -
f2
stand for the f i r s t
and second n × k block
For complicated expressions we use subscripts
617 I t is not d i f f i c u l t
to prove that the n × n block components
[ E ~ m ( d , k ) ] 1 and [E~#~(d,X)] 2 are i n v e r t i b l e Im(k) # O. ([I,
f o r a < d < b and f o r
Indeed, by Atkinson's g e n e r a l i z a t i o n of Green's formula
p. 253]), and by (1.5) and (1.3),
(1.7)
Sda
:
: #*(d,~)EaJE~#~(d,~)-[-=2,~ ]J[-=~]sT = ( E*÷ 6 ~ ) z*( E ~ #÷ ~ ) l
-
"(E*~ a ~'1 ~* (E~#m) 2,
where the l a s t expression is evaluated at d,k. v ~ 0 could be found such that [ E ~ ( d , x ) ] i v imply ( ~ - ~ ) v * ( S ~ A ~ ) v
= O, c o n t r a d i c t i n g
I f an n x 1 vector
= 0 then (1.7) would (1.2) since Im(~) ~ O.
In the same way [E~m] 2 is seen to be i n v e r t i b l e ,
and the same can
be said f o r the components of E ~ (d,k). Because of i n v e r t i b i l i t y
of these components, a unique l i n e a r
combination
#~(t,X) : ~ ( t , ~ ) + #~(t,~)M ,~,d(~)
satisfies
at t = d, and for Im(x) ~ O, the boundary condition
(I .8)
[ E*÷ 6 ~ ( d , k ) ] 1 = 61(d)~(d,;~)
+ ~2(d)~(d,k)
: 0 ,
618 and in fact
(i .9)
M~,~, d(x) : - [ ~ l ( d ) ~ ( d ' ~ ) + 6 2 ( d ) $ ~ ( d ' x ) ] - l [ ~ z ( d ) e ~ ( d ' x ) + 5 2 ( d ) e ~ ( d ' x ) ]
*~ (d,k)] 1 , = - [E~ ~ ( d , k ) ] -I [E6e 1
The n x n matrix quantity (1.9) extends the classical
Im(x) ~ 0
Weyl c i r c l e
formula (see [2, p. 226]). We prove in [ 4 ] , [6] that the terms M~,5,d(X) are uniformly bounded on compact subsets of the half planes Im(x) # O, using no assumptions whatever on the number of independent L~[a,b) solutions of ( I . I ) . However, in the " l i m i t point" case we show that the l i m i t
(I.I0)
M~(X) = d+blim M ,~,d(X)
exists for all
Im(~) ~ O, and is independent of the values of 5(d).
We call
(I.I0)
the Titchmarsh-Weyl M function, corresponding to
initial
data E , for ( I . I ) .
I f the l i m i t c i r c l e case prevails at b, the l i m i t in ( I . I 0 ) to e x i s t ( [ 2 ] , [ I 0 ] ) . of the terms (M
fails
Nevertheless, the subsequential l i m i t s as d + b
~,d(~ )} play an important role in spectral theory.
Thus i t is of i n t e r e s t to characterize the set of all such subsequential l i m i t s .
In [3] C. T. Fulton gave such a characterization
for second order scalar equations.
One of the aims of this paper is
619 to extend Fulton's results to systems of equations ( I . I ) , doing we w i l l
and in so
have the necessary s t r u c t u r e to generalize the theory
of boundary problems at l i m i t c i r c l e endpoints due to M. H. Stone ([9, ch. I 0 ] ) .
2.
PARAMETERIZATIONOF THE M FUNCTION. The classical
limit
circle m coefficients
(defined as l i m i t s
of sequences of the defining quotient of [2, p. 226]) are meromorphic. The analogous r e s u l t f o r Hamiltonian systems w i l l
THEOREM 2.1. limit,
Assume ( l . l )
is l i m i t
c i r c l e at b.
now be proved.
Let M(~) be the
uniform on compact subsets of the h a l f planes Im(~) ~ O, of
a sequence of matrices M ,Sk,dk(~) from (1.9).
Then M(~) is a n a l y t i c
in the complex x plane except f o r poles on the real axis.
PROOF: As a function of
X, the r i g h t side of (1.9) is a n a l y t i c
except f o r the isolated points where the determinant of [E~# (d,~)] l vanishes; i . e . ,
M ,~,d(~) is meromorphic.
Hence M(~) is a n a l y t i c
at least for Im(x) ~ O. Let Mk(~) = M ,~k,dk(~) and +~ k ( t , ~ ) so that [E~k#k(dk,~)] l = 0 f o r all
k and
= ÷0 ( t , ~ ) + #~(t,~)Mk(~) x.
Fix a complex number
XI , Im(~ I) ~ O, and use the Green's formula again (cf. obtain
(1.7)) to
620 ¢d,÷
(2.1)
(Xl-X)
_
Ja%~(t,X)A(t)%(t,Xl)dt
+. dk : [~k(t'X)J%(t'Xl)]a -
for Im(x) ~ O.
+
The r i g h t side of (2.1) at t = d k is 0 and at t = a
i t is Mk(Xl) - ME(S) .
Now ME(S) = Mk(X) by [4].
Putting MK(XI)-Mk(X)
on the r i g h t of (2.1), s u b s t i t u t i n g # k ( t , # ) = ~ ( t , ~ ) + #~(t,~)MK(S ) into the l e f t ,
(2.2)
and solving for Mk(X) results in
"k
Idak ÷" % ( t , x -) A ( t ) ~ k ( t , X
-
l)dt}
: Mk(Xl)+(,\-Xl)Idak~*(t,~)A(t)~k(t,Xl)dt, or with obvious notation (2.3)
Mk(X)Dk(&) = Nk(X),
Im(x) # 0 .
Since the l i m i t c i r c l e case prevails, the integrals in (2.2) all converge absolutely as k ~ ~.
Thus Dk(X) and Nk(X) approach entire
functions D(x) and Nix) such that M(x)D(x) = N(X), Im(~) ~ O. DCxI) = I, neither D(~) nor N(X) is i d e n t i c a l l y
zero.
Since
The determinant
of D(x) can vanish at j u s t countably many points, and thus M(x) can be continued a n a l y t i c a l l y onto the real axis, save for those countably many points where a pole w i l l
occur.
This completes the proof.
We now discuss boundary conditions.
We continue to assume the
l i m i t c i r c l e case. Let Yo(t,X) be a fundamental matrix solution of
621 (I.I)
satisfying
at t = a the condition Y~(a,x)JY0(a,x) = J, and
l e t Y0(t) = Y0(t,x0) where x 0 is real. Y~(t)JY0(t)
= J f o r all
shows that
t , since x 0 is real and the i n i t i a l
condition
Let E~ be an admissible m a t r i x (~1 and
Y~(a)JY0(a) = J holds. satisfy
A differentiation
2
(1.3)) and define Y~(t) = Y0(t) EB, a ~ t < b.
Let #~L~[a,b) and l e t ~ ( t ) s a t i s f y J~' = [XA + B]~ + A#.
By
v a r i a t i o n of parameters using some fixed ~, a ~ T < b,
and consequently
Under the present l i m i t
c i r c l e hypotheses, the i n t e g r a l
above converges
absolutely as t ~ b, and consequently lim Y ~ l ( t ) ~ ( t ) e x i s t s . We t÷b choose the f i r s t n x 1 block component, whose l i m i t also e x i s t s , and set as a boundary condition at t = b the requirement
(2.5)
(Y~I~)I(b)D = lim [ YI~ l ( t~) ~ ( t ) ] t÷b
We could use the other component j u s t as w e l l . -I÷ (Y~y)l(b)
:
= 0 Since
B~(Y01~) l ( b )+ ~
o
=
, then
(Y01y)2(b)
which extends the analogous formula of Fulton [3, eq. ( 2 . 6 ) ] . Reasoning much as in ( I . 7 ) , =I ~
[Ya ¢a]l (b) is i n v e r t i b l e
we can show that the n x n matrix
~ for Im(x) } 0 (¢~ =¢~(t,x)).
define a matrix function M
(x) by the equation
Thus we can
622 Mm,~(~) : -[YB- I ÷# m ( ' ' ~ ) ] ~ l ( b ) [ y i l ~ m ( ' ' ~ ) ] l ( b ) '
(2.6)
Im(~) # O.
This is the matrix analog of the representation formula of [3, eq.
(l . 9 ) ] . Our aim is to show that (2.6) characterizes the set of functions which appear as l i m i t s of sequences of terms ~I.9).
That i s , i f M(x) ~
s a t i s f i e s the hypothesis of Theorem 2.1, then M(X) = M ,~(~) for some E6, and vice-versa.
In t h i s way, the admissible matrices EB
may be said to "parameterize" the M functions at a l i m i t c i r c l e endpoint. We w i l l
not give f u l l
d e t a i l s in every instance, but r e f e r the
reader instead to the forthcoming a r t i c l e
[5].
The f i r s t
r e s u l t says
that (2.6) introduces no new M functions.
THEOREM 2.2.
Let M ,6(~) be given by (2.6).
Then there e x i s t admissible
~
matrices E6 = Ea(d), a < d < b, such that M ,~(~) is the uniform l i m i t on compact subsets of Im(x) #0 of a sequence of terms M ,B,d(X) from (1.9). PROOF:
For a < d < b l e t M d ( ~ ) : - [ Y ~ l ( d ) #
so that M 6(x) = lim Md(~). '
(d,X)]il[y~l(d)~
(d,~)] 1
Also
d÷b
-l÷ -l~ [Y~ o ] l ( d ) + [Y# e ]l(d)M d : 0 .
(2.7)
Now YO1 may be calculated from the r e l a t i o n Y~JYo = J and s u b s t i t u t e d into y~l = EilYol to y i e l d
y;l#
=-S2
623 ^* * ^ ) + #2(-6~¢~+e~$m )](d) and s i m i l a r l y and so [Y~-1÷emil (d) --[ 61(¢0¢m-~0¢~ for [Y~I~ ] 1.
Substituting these into (2.7), re-arranging and
introducing at(d) =
(6z$~- B28~)(d )
and
a2(d) = (-61¢~+~2e~)(d),
(2.7) now becomes (2.8)
(ale £ a2e ~) + (oz¢ ~ + o2$~)Md = 0
By i d e n t i f y i n g
the terms 01 and o 2
can show that Ola ~ = o2d[.
B
in the relation YilYB = I, one
While i t may not be true that alo [ + 0202
this matrix is c e r t a i n l y positive d e f i n i t e ,
so we can make a "change
of variable" 61(d) = c(d)Oz(d), 62(d ) = c(d)a2(d , where c(d) is a nonsingular matrix, in which
-q E6 = E~ (d) : 52 is admissible.
Multiplying
61
(2.8) on the l e f t by c(d), and then taking
an inverse, we have
(2.9)
Md(k) = [61(d)¢~(d,Z)+62(d)$a(d,X
which agrees with (1.9).
The r i g h t side of (2.9
]-l[61(d)e
(d,x)+62(d)6#d,~)]
is uniformly bounded
on compact subsets of Im (~) ~ O, so a sequence of i t s terms converges to an M function.
Since the l i m i t must be M
To complete the characterization,
the proof is complete.
we have to take a l i m i t M(Z) of
a sequence of terms (1.9) and show that i t can be represented as in (2.6).
F i r s t , we require a preliminary r e s u l t .
624
LEMMA 2.1.
Let {d k} be a monotone sequence, dk ÷ b, l e t E~k be a
sequence of admissible matrices and let Ykl(~) = 5kl00(dk,~) + ak260(dk,~), Yk2(~) = akl¢O(dk,~) + ak2~o(dk,~), where
~kl' ~k2 are the components
of E6k as in (1.4) and eO,# O, etc. are the components of in (1.6).
Yo(t,~) as
Then there exists a real XO and a subsequence {kj} of integers
such that either -I y kj2(XO)Ykjl(X O)
(2.10)
or
y-I kjl(XO)~kj2(XO)
is convergent. PROOF: Define M~O)(~) = -y-~ 2(~)Ykl(~).
Then M O) is of the form of
(1.9), so a sequence M~! ) of i t s terms converges uniformly on compact subsets of Im(X) # 0 to a l i m i t
M(O)(x),
which is meromorphic by
Theorem 2.1.
I f ~0 is a regular point of M(0) then M~! ) is also J regular there for k. large enough. This says the l e f t member of J (2.10) converges. Since [ M ~ ) ( ~ ) ] - I
-I (~)Ykj2(x) and [M~I)]-I : -Ykjl .
[M(O)]-I
which is also meromorphic, the right member of (2.10) w i l l converge i f ~0 is a regular point of [M(O)] - I .
This completes the proof.
Thus in the lemma, ~0 may be taken as a regular point of M(0) or i t s inverse.
In the scalar case this is no r e s t r i c t i o n on XO' for a
pole of m(~) is a zero of i t s inverse and vice-versa. system this need not be the case, and we i l l u s t r a t e with an example.
For a Hamiltonian
this important fact
We can encompass two independent Sturm-Liouville
problems - y " : [ ~ a l l ( t ) + b l l ( t ) ] y
and -z ''=[~a22(t)
+ b22(t)]z in a single
625 4 x 4 Hamiltonian system by w r i t i n g
oOOO l Oo i o°Io i] [iI I [a!a22 o°o°°il 11oo o°°b22 o°o°l i]I[iJ Setting ml = m2 = 12' we see that Y = Y ( t , 1 )
is a 4 x 4 matrix whose
entries Y21 : Y41 = YI2 : Y23 = Y32 : Y34 = Y41 = Y43 vanish i d e n t i c a l l y because of the i n i t i a l
e
values they take.
=
Oy, #y and
Titchmarsh ( [ I 0 ] )
and ~
0
~z
e and
#
have the forms
#m = ~z
where
Thus ~
° z ' #z are the classical
solutions of
f o r the separate equations in y and z~ r e s p e c t i v e l y .
I f we take ~kl = 12'
ak2 = 0 then M(0) in the lemma becomes
M(O)(~): Imy(~) 0 ] 0 where my and mz
are subseqeuntial l i m i t s
and (Sz/#z) r e s p e c t i v e l y .
mz(X) of the quotients
(ey/my)
That is, my and mz are scalar m - c o e f f i c i e n t s .
I t is easy to produce an example where my has a zero and mz a pole at a common point 10 . :
Thus M(0) w i l l
experience a pole at XO' but [M(O)] -I
diag[m~l ,mz I ] has a pole there as w e l l .
626 THEOREM 2.3. (1.9).
Let M(~) be the l i m i t
of a sequence of terms of the form
Then there e x i s t s a real ~0 and an admissible matrix E6 such
t h a t M(X) : - lim[Y~-I ( t ) ¢+ ( t , x ) ] - I E y : #l ( t ) e÷ ( t , ) ] I " t~b ~ ~ I PROOF:
We w i l l
sketch the proof; d e t a i l s may be found in [ 5 ] .
M(x) = k-~°lim M ~k,d k ( l ) :
Let
from (1.9) and define Sk(l) = E*Sk YO(dk'l)"
Denote the upper n x 2n p o r t i o n of Sk(X) by [Sk(X)] 1 = [ Y k l ( l ) , Y k 2 ( X ) ] . Arguing as in Theorem 2.2, we can find a nonsingular matrix Ck(l) such t h a t i f Fkl = CkYkl, Fk2 = CkVk2 then F k l r [ l [Sk(X)]-I
: Yol(dk,l)E~k.
+ rk2r[2 = I.
Note t h a t
I f one writes out the components of S k l S k : l ,
* - ¥k2Ykl * = O, so i t follows t h a t F k l r [ 2 - Fk2r[l one f i n d s t h a t YklYk2 = c kYk1"k2 ~,* c*k - CkYk2VklCk * * = O-~ i . e . ' ETk(~) is admissible. I f we introduce the fundamental matrices Y k ( t , x ) then a c a l c u l a t i o n (2.11)
using the above i d e n t i t i e s
Using Yk as the fundamental matrix in ( 2 . 4 ) ,
p u t t i n g # = O, we have for an a r b i t r a r y
(2.13)
leads to
M ,~k,dk(X ) : _[y~l ( d k , ~ O )~{ ~ ( d k , l ) ] l- l [ y ~ l ( d k , ~ O ) ~ m ( d k , k ) ] 1
f o r any real XO"
(2.12)
= Yo(t,x)EFk(X),
and
solution ~(t)
Yk l ( t , x O ) ~ ( t ) = E~Ik(XO)~(t,,~,~ ) ,
where
: ¥o I
Setting t = d k in (2.12) gives (2.14)
l-Ykl (dk '~O)~(dk)] 1 = Fkl (Xo)V(dk'>') + I~k2(Xo)V(dk'~)
"
627 I f we put ~k(~) = ~(dk,~,# ( - , ~ ) ) and Wk : ~ ( d k ' ~ ' ~ m ( ' ' ~ ) )
then (2.11)
and (2.14) give (2.15)
-I M ,~k,dk(X)=-[rkl(XO)Uk(X)+rk2(~o)uk(x)][rkl(XO)Wk(X)+rk2(~o)wk(x)]
Because of the l i m i t c i r c l e hypothesis, the i n t e g r a l s in (2.13) converge absolutely as t ÷ b, and t h i s means that Uk(X) and Wk(~) approach f i n i t e limits,
say ~(~) and ~ ( ~ ) , as k + ~.
-I and r k Fk2 = ¥kl~k2 , the lemma t e l l s Since ? k rkl = Yk2Ykl -
us that
~0 can be chosen in (2.15) in such a way that e i t h e r ri~(~O)rkl(XO) or rkl(~O)Fk~(~ O) has a convergent subsequence.
Suppose rk~<xO)Fkl(~ O) has a
convergent subsequence, whose running index we w i l l k to s i m p l i f y notation.
Letting r be i t s l i m i t ,
M(x) = - [ r u ( x )
(2.16)
M(X) : - [ ~ u ( ~ )
Let Y~(t) : Yo(t,~o)E~. where ~ is defined above. ~zV(t,~,~) + ~2 ~ ( t , X , ~ ) .
+ G(x)]
i t can be shown that an
admissible matrix EB exists for which B = B r. i 2 into (2.16) there follows (2.17)
we have by (2.15) that
+ G(x)]-1[rw(x)
By making another "change of v a r i a b l e " ,
continue to denote by
Substituting this
+ B2G(~)]'l[~w(z)
+ ~2~(~)]
.
Repeating (2.4), we get Y~l(t)~(t)- :E~-l+v(t,~,y), This expression has f i r s t Putting f i r s t
(n × I) component
~ = # , then ~ = ~ , and taking
an inverse, we obtain on l e t t i n g t + b lim[y~Ict)7~Ct,~)]~l[y~l~ t+b
~
( t , x ) ] 1 : [# u(~)+~2u(x)]-l[~lw(~)+B2w(~)]. l
628
Comparing t h i s with (2.17), the proof is complete in t h i s case. remains the p o s s i b i l i t y vergent subsequence.
that r k ~k2' instead of r k
There
r k l , has a con-
The proof f o r t h i s case i s not s u b s t a n t i a l l y
d i f f e r e n t , and we omit the d e t a i l s .
3.
BOUNDARYVALUE PROBLEMS. The boundary condition (2.5) generalizes
the Wronskian codition of M. H. Stone ( [ 9 ] ) used in connection with second order scalar equations.
We now consider boundary problems on
[a,b) which, in the l i m i t c i r c l e case, employ (2.5) as r i g h t endpoint conditon. By " l i m i t point case" at x = b we mean that ( I . I )
has f o r a l l
nonreal ~ e x a c t l y n l i n e a r l y independent solutions of class L~[a,b). We have r e c e n t l y shown in [7] that t h i s d e f i n i t i o n of l i m i t point is equivalent to the condition lim y ( t ) J z ( t ) = 0 f o r a l l solutions y , z t÷b of ( I . I ) belonging to L~[a,b). The l a t t e r was the l i m i t point hypothesis of [ 4 ] ,
[6].
Hence the r e s u l t s of [ 4 ] , [6] may be brought to bear on
the present paper. For the remainder of t h i s paper, the symbol M (~) w i l l f o l l o w i n g meaning. (I.I0),
have the
I f x : b is l i m i t p o i n t , M ( x ) w i l l be defined by
the existence of the l i m i t being a consequence of [ 4 ] .
x = b is l i m i t c i r c l e , Ms(h) w i l l
stand f o r one of the c l u s t e r points
of the c o l l e c t i o n {M , ~ , d ( ~ ) } ; i . e . , of terms f r o m ( l . 9 ) . + ~(t,~)M
(~).
If
M (~) i s the l i m i t of a sequence
Whichever the case, we set # m ( t , ~ ) : # m ( t , ~ )
629 I f i t is the l i m i t c i r c l e case which p r e v a i l s , select via Theorem 2.3 a real 10 and an admissible EB such that i f YB(t) : Yo(t)E~, Yo(t) : Yo(t,Xo), then Ms(x) = - lim [ Y ~ l ( t ) ~ ( t , X ) ] l l [ y i l ~ t÷b d e f i n i t i o n of ¢ , we have in t h i s case
(3.I)
lim [ Y i l ( t ) # ~ ( t , ~ ) ] l t-~b
which is to say ~m s a t i s f i e s
(t,X)]l .
By
: 0
(2.5).
Following [6] we define the Green's function for Im(~) # 0 by
O~(X~)~m(t,~),
x < t
~m(x,X)¢~(t,x),
x > t
+*
G(x,t,X) :
and i f f~L~[a,b],
l e t (Gx~)(x) = I ~ G ( x , t , x ) A ( t ) ~ ( t ) d t . 9
point case holds,~ ~ L ~ [ a , b ) ( [ 4 ] )
I f the l i m i t
and so the integral defining Gx
converges. Now l e t #¢LR[a,b) and consider integrable square solutions of the nonhomogeneous equation (3.2)
J~' : [ ~ A ( t ) + B ( t ) ] ~ + A~ ,
THEOREM 3.1.
+YEkA[a,b), 2
Im(x) ~ 0
In the l i m i t point case the boundary problem consisting
of (3.2) together with ~ly(a) + ~2#(a) = u has a unique s o l u t i o n given by ~ ( t ) : ~ ( t , ~ ) u + ( G ~ ) ( t ) .
In the l i m i t c i r c l e case, the
boundary problem consisting of (3.2) together with ~ly(a)+~2Y(a) = v, -It
(Y~ y ) l ( b )
^
: v has a unique s o l u t i o n given by ~ ( t ) : Y~(t,x)~+(Ga~)(t)
630 where w = ( ~ ) ,
w = v and ~ = ( Y i l ~ ) i l E v
PROOF: We proved G k ~ L ~ [ a , b )
- (Yil~)lV].
in [4] in the l i m i t point case.
Since
G(x,t,~) has exactly the same form in the l i m i t c i r c l e case, the proof in [4] carries over to that case, too.
Certainly the forms for ~ in
the statement of the theorem then belong to L~Ea,b). forms s a t i s f y the d i f f e r e n t i a l
Moreover, these
equation J~' = [kA+B]~ + A~ by the
argument of [ 6 ] . In e i t h e r case, g = Gk~ has the form g(x) : (Gxf)(x) : @m(x,~) +
and so
(G~f)(a)=
~I , \( ~1"
j~
(t,S)A(t)f(t)dt ,
jx
b+ a ~ ( t ' k )-A ( t ) # ( t ) d t "
i b~
= (-~zm~ + mlm~)Ja~Af = O.
Thus~g(a)+m2Q(a)
As for the r i g h t endpoint condition in
the l i m i t c i r c l e case, (Ya g ) l ( b ) : (Y~ on account of (3.1). ditions.
l(b)
Thus g s a t i s f i e s a l l homogeneous endpoint con-
Consequently in the l i m i t
~ly(a)+m2y(a)
#~Af = 0
point case for ~ ( t ) = # ~ ( t ) u + g ( t ) ,
= mL(~ - m~M )u + ~2(m~ + ~M )u = u by conditions (1.3)
In the l i m i t c i r c l e case, the function ~ ( t ) = Y ( t ) ~ + g ( t ) s a t i s f i e s ~lY(a) + ~2#(a) : ~z(~w - ~ )
+ ~2(~w + ~ )
= w by (1.3). At -l+ the r i g h t endpoint, (YB- I +Y)I (b) = (Y~ I~ ) 1 (b)w + (Y~ ¢~)i ~ : v i f is as specified in the statement.
631 Finally,
f o r uniqueness, we must show that i f J~' : [hA + B]~
and ~ s a t i s f i e s
the homogeneous end conditions,
l i m i t point case
then ~ = O.
In the
t) = ~m(t)c, where c is n × l ; but 0 = ~ y(a) 1
+ ~ y(a) = c as above.
Similarly,
in the other case ~ ( t ) = Y ( t ) ~ ,
2
where ~ is 2n × l , and 0 = alY(a) + a2y(a) = k as before. -I~ 0 = (Y~ y ) l ( b )
= (Yil~)m(b)k
Lastly,
-I~ + (YB ~ ) I (b)~ implies k = 0 since
-I ~
k = 0 and (YB #m)l (b) is i n v e r t i b l e .
REFERENCES 1.
F. V. Atkinson,
"Discrete and Continuous Boundary Problems,"
Academic Press, New York, 1964.
2.
E. A. Coddington and N. Levinson, "Theory of Ordinary D i f f e r e n t i a l Equations," McGraw-Hill, New York, 1955.
3.
C. T. Fulton, Parameterizations of Titchmarsh's m(x)-functions in the l i m i t c i r c l e case, Trans. A.M.S. 229 (1977), 51-63.
4.
D. B, Hinton and J. K. Shaw, On Titchmarsh-Weyl m(h)-functions f o r l i n e a r Hamiltonian systems, J. D i f f . Eqs. 40 (3) (1981), 316-342.
5.
D. B. Hinton and J. K. Shaw, Parameterization of the m(~)-function f o r a Hamiltonian system of l i m i t c i r c l e type, submitted.
6.
D. B. Hinton and J. K. Shaw, Titchmarsh-Weyl theory f o r Hamiltonial systems, in "Spectral Thoery of D i f f e r e n t i a l Operaotrs," pp. 219231, I. W. Knowles and R. T. Lewis ( e d i t o r s ) N o r t h - H o l l a n d , New York, 1981.
7.
D. B. Hinton and J. K, Shaw, On boundary value problems for Hamiltonian systems with two singular points, submitted.
8.
V . I . Kogan and F. S. Rofe-Beketov, On square-integrable solutions of symmetric systems of d i f f e r e n t i a l equations of a r b i t r a r y order, Proc. Royal Soc. Edin. 74A (1974), 5-39. M. H. Stone, "Linear Transformations in H i l b e r t Space and t h e i r Applications to Analysis," Amer. Math. Soc. Colloq. Pub., vol. 15, Amer. Math. Soc., Providence, RI, 1932.
9.
I0.
E. C. Titchmarsh, "Eigenfunction Expansions Associated with Second Order D i f f e r e n t i a l Equations," Part I, 2 nd e d i t i o n , Clarendon Press, Oxford, 1962.
ON U N I F O R M A S Y M P T O T I C E X P A N S I O N OF A CLASS OF INTEGRAL TRANSFORMS
K SONI
ABSTRACT A s y m p t o t i c expansions for the integrals of the type a F(x,a)
w h i c h hold uniformly in 6 > 0, are obtained. origin and general,
K(t)
a
= f K(xt) 0
f(t)
w h e n either
% > 0
f
or w h e n
~ ~ a < ~
F(x,a)
some cases, a similar situation may arise as K
for some
has an algebraic singularity at the
when
directly from the c o r r e s p o n d i n g expansion w h e n
i.
x ÷
is locally absolutely integrable in
the asymptotic expansion of
incomplete M e l l i n transform of
,
0 ~ a ~ ~
It is assumed that
t X-I ,
dt
a ÷ 0+
a
[0, ~).
In
cannot be o b t a i n e d
is b o u n d e d away from zero.
a ÷ ~
In
A n a l y t i c c o n t i n u a t i o n of the
provides a unified approach to this problem.
INTRODUCTION Let a
(i.i)
F(x,a) = f 0
K(xt)
f(t) dt ,
0 < a < ~ ,
x > 0 .
In recent years a number of techniques have b e e n developed that provide an asymptotic e x p a n s i o n of
F(x,a)
when
x ÷ ~
such expansions are v a l i d w h e n
and
a
a e 6 > 0
is a fixed p o s i t i v e number. but not w h e n
a ÷ 0+.
It has also b e e n
observed that in some cases, even under appropriate conditions on e x p a n s i o n of
F(x,~)
(see, for example,
does not follow d i r e c t l y from that of
[5, p.228]).
In general
F(x,a)
f, the asymptotic as
a ÷ ~ ,
A n i n v e s t i g a t i o n of these phenomena results in a
very simple a p p r o a c h to the p r o b l e m of u n i f o r m asymptotic expansions of a certain class of integral transforms.
(1.2)
f(t) ~
The basic idea is as follows.
~ k=O
a k tk+~-I
and the incomplete M e l l i n transform of the kernel
t ÷ O+
K, namely
Suppose that
633 a
(1.3)
g(K,s
has an asymptotic and
; x,a) = f K(xt) 0 x ÷ ~
expansion as
t s-I dt
Then under reasonable
assumptions
on
f, the formal expansion co
(1.4)
F(x,a) ~
~ a k g(K, k+u k=0
invariably provides
an asymptotic
an expansion holds uniformly
in
may depend on how we represent Our results
NOTATION,
0 ~ a ~ ~
F(x,a)
for some
the function
that
(2.1)
x ÷ oo
as
Whether
or in
6 > 0
such
0 < @ -< a < co
g. stated in section 2.
These are
In section 4 we give some examples.
ASSUMPTIONS
We assume
expansion of
are summed up in two theorems
proved in section 3.
2.
; x,a)
AND STATEMENT OF RESULTS
f(t)
belongs
f(t) = fm_l(t)
to
cP(0, ~)
+ Rm(t)
,
and write
m = 1,2,...,n
,
where
(2.2)
fm_l(t)
We also assume (2.3)
=
m-I ~ k=O
tlk ak
,
-I < 10 < 11 < "'" < %n "
that R(k)(t)
l-k = @(t n ) ,
t * 0+ ,
k = O,l,...,p
.
n
Thus the behaviour effectively F(x,a)
of
f(t)
provided by and
g(K~s
ever these integrals
and its successive
t s-I
C 2 : For some
(2.4)
converges.
is
; x,a)
are defined by (i°i) and
(1.3) respectively
when-
converge.
is locally absolutely ~,B
M[K,s]
Furthermore,
can be continued
near the origin
fn_l(t)"
We will need to refer to the following C 1 : K(t)
derivatives
, the Mellin
= f K(t) 0 M[K,s]
analytically
conditions integrable
transform of
t s-I dt ,
in
[0, ~)
the kernel for
K(t)
Re s > O.
K,
~ < Re s < ~ ,
is an analytic
as a meromorphic
regarding
function of
function
s
in this strip and
into the whole s-plane.
634
K(-m)(t)
,
m ~ 0,i,...
, denotes
need to refer to the following C 3 : For
conditions
m Z 1 , K(-m)(t)
C 4 : There exist numbers
G(K,s
; x,a)
(2.5)
b Z 0
THEOREM
1
conditions
C2
functions Let
and
,
[0, ~)
respectively, n ~ m=l
=
The above expansion
x
a ÷ 0.
as
formly
and
in
,
K(t)
isolate
= exp(it)
containing defining origin as
g(K,s
Whether
; x,a).
,
If
this expansion as
x ÷ ~
the expressions K(-m)(t)
the simplest
x ÷ ~
which
integral
Apart
powers
K(-m)(t) 0 s a s @
satisfy and
the
x > 0,
and for
÷ 0
0 < c s x s d
uniformly expansion of
with respect
which holds
K(-m)(xa)
uni-
and
is that we can identify behaviour.
to
For example,
and if
In this case we obtain an expansion
with the same critical
points
as the integral
holds uniformly
in
a near the
1 in section 4).
; x,a)
Re s > ~
Now by the relation
x
cause nonuniform
2, we observe
in
are
.
on the behaviour
Before we state Theorem is analytic
of
F(x,a)
from that the expansion
(see Example
; x,a)
x,a)
;
of this expansion
= (-i) m K(t).
g(K,s
R~m-l)(a )
is an asymptotic
depends
and
@ > 0,
(~)m g(K(-m)'~
numbers,
and
K(t)
then for any
is in decreasing
The main advantage
F(x,a).
m-> O,
a > 0 .
; x,a)
(-i) m-1 x -m K(-m)(xa)
are any positive
0 s a ~ @
g(K(-m),~ possibly
d
.
such that for
+ (-x)-n i K(-n) (xt) R (n) (t)dt n 0
c
We will
~ < Re s < ~ .
0 < p s i.
n-i [ am(-x)-m m=0
where
K(t).
t ÷ ~
G(K,s
at least in the strip
C3
F(x,a)
in
tS-ldt
is satisfied,
%k = k + p - 1 ,
C1
(2.6)
= O(t mY+b)
of
K(-m)(t).
y < 1
co ; x,a) = f K(xt) a
the condition
complementary
and
integral
is defined by
G(K,s
Whenever
for
is continuous
K(-m)(t) Finally,
an mth iterated
and
that the condition G(K,s
; x,a)
C2
implies
is analytic
in
that Re s < 8.
635
(2.7)
g(K,s
; x,a)
+ G(K,s
both of them can be continued complex s-plane. by parts,
Moreover,
it follows
analytically
by applying
2
Let
in + 1 > p.
and the conditions
(2.8)
as meromorphic
the condition
that each of the functions
also can be continued analytically THEOREM
; x,a) = x -s MEK, s] ,
If
C 1 - C4
F(x,a)
=
to the whole M[K,s]
a x m P
G(K(-m),s
then for
to the whole
to integrate ; x,a),
G(K,s
; x,a)
m = 1,2 . . . . .
s = lk + i, k = O,l,...,n-l,
a ~ 6 > 0,
x > 0,
-~ -I m MEK,%m+I ]
(-I) m-] x -m K(-m)(xa)
Z
functions
s-plane.
is regular at
are satisfied, n-] ~ m= 0
C4
~ < Re s < B ,
f(m-l)(a)
+ E
m=1 where
(2.9)
6 f K(-P)(xt) 0
E = (-x) -p
+ (-x) -p
)
R (p) (t) dt n
K(-P)(xt)
f(P)(t)
dt
n-I (-x) -p
The expansion approximated E
(2.8)
am(%m+l-P) p G[K (-p),Im+l-p,x,6)
~ m=0
is asymptotic
as
by t h e finite sums on the right,
is not necessarily
of the same order as
E
decreases
~ a ~ A bounded,
for any
a ÷ ~ ,
in
x e N
a e 6
as
p
increases
A > O.
then indeed
as
as
If
k = 0,1,...,p-l, for some
x ÷ ~
Sometimes
N
as
as
the iterated
x ÷ ~
are sufficient
in a slightly
the error.
However,
is valid uniformly of
K(t)
to this, if
in (2.9) converges
in
f(k)(a) ÷ D uniformly
(2.8) is valid uniformly
form,
the
are
but not always necessary.
different
is
It is also possible
integrals
In addition
F(x,a)
represents
sum may be of lower order.
a ÷ ~ , then the expansion
is written
E
x + ~
and the second integral
These conditions E
as
and the expansion
K(-m)(t),
E = O(x -p)
in the sense that if
the function
x -p
that some of the terms in the approximating order of
x ÷ ~
.
in
636
(2.10)
E = (-x) -p
i K(-P)(xt) R(P)(t) dt n 0
- (-x)-P
n-i ~ am(%m+l-P) p G(K(-P),Im+I- p ; x,a) m= 0
or, when the integral representation
for
G(K(-P),%n_I+I- p ; ×,a)
co E : (-x) -p ~ K (-p) ( x t ) R (p)(t)dt - ( - x ) -p f K (-p)(xt) f(P)l(t)dt . n
(2.11)
0
3.
can be used,
a
PROOF OF THEOREMS We start with the well known technique in asymptotics.
can write
F(x,a)
in terms of the functions
g
and
By (2.1) and (2.2) we
R . n
n-I (3.1)
F(x,a) =
am g(K,%m+l m=0
; x,a) + ~ K(xt) Rn(t)dt. 0
Next, we apply integration by parts and use (2.3) together with the conditions and (3.2)
C 3.
For
C1
p < In + i, we obtain
fa K(xt) R (t)dt = p~ (-I) k-I x -k K(_k ) (xa) m(k_l) (a) n n 0 k=l + (-x) -p
~ K(-P)(xt) R(P)(t)dt
.
n 0
Similarly, by the conditions
(3.3)
G(K,s ; x,a) =
C I - C 4 , for every positive integer
q ~ (-x)-k(s-k+l)k_l k=l
K(-k)(xa) a s-k
+ (-x)-q(s-q)q G(K(-q),s-q
For
q,
; x,a) .
Re s < -b , (3.3) follows by using integration by parts and for
analytic continuation. consequently,
This provides an asymptotic expansion for
by means of the relation
useful only when
a
g(k,s ; x,a).
is bounded away from the origin.
Theorem l, an expansion for and is valid when
(2.7), for
a ÷ 0
g(K,m+~ ; x,a)
Re s > -b, by
G(K,s ; x,a)
However, this is
Under the conditions of
which is in decreasing exponents of
can be given as follows.
and
We use integration by parts.
x
637
(3.4)
(_l)k_ I x -k K(-k)(xa)
g(K,m+u ; x,a) =
am+U-k(m+~-k+l)k_l
k=l
+ (-x)-m (~/)m i K(-m)(xt)
tU-I dt .
0 This is a finite term expansion. near
a = 0
depends on whether
PROOF OF THEOREM i
(3.5)
In general, the behaviour of the last integral xa
approaches
We start with (3.1).
zero or moves away from zero.
By (3.4),
n-I
n-I ~ m=O
am(-x)-m(D) m
a m g(K,m+u ; x,a) =
~ K (-m)(xt)tU-Idt 0
m=O
+ E1
where
(3.6)
n-I ~ m=l
E 1 =-
m (-x) -k K (-k) (xa) a
am
m+u-k
(m+u-k+l)k_ I •
k=l
Change the order of summation in
(3.7)
f(k-l) (a) - f(kll)(a) = n-I
(3.8)
E 1 =-
E1.
Since
n-i ~ m=k
am(m+u-k+l)k
n-i
i am+U-k ,
{fi~l- 1)(a) - f(k-l) k-i
(-x)-k K(-k)(xa)
(a)}
I -
,
.
k=l
Now let
1
m
= m + ~ - i,
use (3.5) with expansion
E1
(3.2).
p = n
and write
The expansion R(k-l)(a)n
PROOF OF THEOREM 2
(2.6) follows by observing that for
R( = kk-l)(a)
-
We start with (3.1) again.
n-I
n-I a m=0
m
g(K, lm+l ; x , a )
=
~
+ "(k-l)(a) ~k-i
By the relation
m= 0
-I -I
a x m
where n-i (3.11)
f(k-l)(a) n-]
S,
valid for all
For the finite sum,
as given in (3.8) above and replace the integral by its
" (3 9)
(3.10)
(3.1) as follows.
E2 =
a m G(K, Im+I ; x,a) m=0
.
m
M[K, tm+l ] _ E2
i N k < n-l,
"
(2.7) which is
638
By the condition is regular
at
s = % + 1, m
G(K,s ; x , a ) (3.3),
E2
(3.12)
CI, g(K,s ; x,a)
and
has no singularity in
m = O,1,...,(n-1)
(s-p)p G(K(-P),s-p
can be w r i t t e n
E2 = ( - x ) - p
n-[ ~' m=0
; x,a)
, by ( 2 . 7 )
Re s > 0. and ( 3 . 3 )
also are regular
Since it
M[K,s]
follows
at these points.
that By
as follows.
am(%m+l-p) p G(K(-P),%m+I- p ; x , a )
+ [ (-x)-k K (-k)(xa) f(k-l) n-i
k=l
(a) .
The last sum in (3.12) above is obtained by changing the order of summation and then using (2.2).
Now rewrite (3.1) by using (3.2),
(3.10) and (3.12).
The result can
be simplified by observing that n R(k-l)(a)
= f(k-l)(a) - f(k-l)(a) n-i
This provides the expansion (2.8) with
E
K (-p)(xt) f(P)(t)dt = n-i
as given in (2.10). n-i ~ am(%m+l-P) p m=O
To obtain (2.9), use
% -p ~ K (-p)(xt)t m dt
and K(-P)(xt)tS-]dt
4.
= G(K(-P),s
; x,6) - G(K(-P),s
; x,a)
.
EXAMPLES EXAMPLE i.
(4.1)
Let
K(t) = e it
f(t) e ixt dt = 0
and
K(-m)(t) = e-imp/2 e it
By Theorem I,
~ x -m e imp/2 e ixa R m-l)(a) m=l
a n-1 imp/2 + ( / e ixt t p-I dt) ~ x -m am (~)m e 0 m=0
a
+ x-n
e
in~/2
f 0
eiXt R(n) n (t) dt.
The relation (4.1) actually holds for all complex setting
x, x # 0.
In particular, by
x = iz, we may obtain an asymptotic expansion for the integral
639
co co a / e -zt f(t) dt = / e -zt f(t) dt - f e -zt f(t) dt a 0 0
(4.2)
which is valid uniformly Erde'lyi
in
a
as
a + 0.
Such an expansion was first given by
[i] in 1974 by a technique based on fractional
that the two expansions
integrals.
It can be shown
are identical.
The asymptotic expansions
for the Fourier
terms, are well known (see, for example,
integrals,
[3] and [4]).
complete with remainder
Theorem 2 provides
still
another way of deriving such expansions. EXAMPLE 2.
Let
K(t) = t-~/2 J ( ~ ) , K(-m)(t)
The Mellin transform of By using E2, p.326,
converges
in the strip
M[K,s] = 22s-~ F(s)/F(~-s+l)
M[K,s]
has no singularity
m = 0,i, . . . .
0 < Re s < (2 Re ~) + 3)/4.
in
Re s > 0 .
.
In particular,
it is regular at
m
+ l,
Finally,
(4.4)
for
= (-2) m t-(~-m)/2 j _m(/~').
(i)],
(4.3)
where
K(t)
Re V > -i, and let
K(-m)(t) b = -(2v+I)/4.
= O(t (m/2)+b)
Thus
K(t)
,
satisfies
t ÷ ~ , the conditions
C 1 - C 4.
By Theorem I,
0 -< a -< 6, a
(4.5)
f f(t) (xt) -~)/2 J (x#~-{)dt 0 n (2/x) m (xa) -(~-m)/2 J _ m ( ~ )
Rm(m-l)(a)
m=l n-i m=0
a (2/x)m am (~)m ( f (xt)-(~)-m)/2 J~;-m (x~x~)t~-Idt) 0 a
+ (2/x) n f (xt) -(~-n)/2 J)_n(x~-~) R(n)(t) 0
Although the above expansion is valid for all it is not very attractive.
dt .
n
a
near the origin,
In contrast to the expansion
computationslly
(4.1), where the approx-
imating sum contains only one integral whose behaviour changes with the magnitude
640
of
xa, each term of the second sum in (4.5) contains
kernels,
including the Bessel
function kernels,
technique by applying an appropriate When
For certain
this can be avoided if we modify our
differential
operator to integrate by parts.
a e 6 > 0, by Theorem 2,
i (4.6)
such an integral.
f(t)
n-I % +i ~ a 2-v(4/x) m F(%m+l)/F(~_%m ) m= 0 m
(xt) -V/2 J (xg'xt)dt =
0
2m x-(V+m)/2
a(m-~)/2 j _m(x~a)
f(m-l)(a)
+
E
m=l where by (2.9),
6 (4.7)
E = (2/x) p
/ (xt)-(~-P)/2 0
+ 2 p x-(~)+P)/2
J ~-p
(xgT~) R(n)(t)dt n
t-(~-p)/2 J
6
(x~)
fP(t)dt
~-P
n-1 + (-x) -p
~ m=0
am(%m+l-P) p G(K(-P),%m+I- p ; x,6)
The second term on the right in (4.7) indicates of
x -p
as
x + ~
the second integral large as
If
a-(2~+I)/4
that
E
uniformly
in
x
(2.5) although it can be represented with such an integral.
REMARK.
expansion
However,
it is of order
in decreasing exponents of
x -I
in
x a e ~
, and
sufficiently as
x ÷
as an integral of the form
as
x ÷ ~
Recently Wong [5] obtained a uniform asymptotic
(4.9)
g(x) = x -I f(x) .
expansion for the integral
h ÷
such integrals arise in the study of small-angle By a change of variable, we can write
and has an
x (see (3.3)).
I(h) = i g ( 4 ~ - q2 ) sin yh dy , q
membranes.
for all
m = 1,2,...,p
as the sum of a finite number of terms together
(4.8)
Presumably
for
a ÷ ~o , then the above expansion holds uniformly
The last sum in (4.7) cannot in general be represented
asymptotic
is not always of the order
+ m/2 f(m-l)(a ) + 0
in (4.7) converges
.
x-ray scattering
from
641
l(h) = (~h/8) 1/2
(4.10)
where
a = z
2
- q
2
and
~ t-i/2 f ( ~ ) 0
2 b = q .
(t+b)-i/4 Jl/2(h t~-~)dt ,
It is no longer of convolution type but it is
similar to Example 2 and the analysis presented in this paper is still applicable. In particular, under appropriate conditions on to (4.6) and (4.7), holds uniformly in
f, the expansion which corresponds
a = z2 - q
2
~ 6 > O.
REFERENCES i.
A Erde'lyi, Asymptotic evaluation of integrals involving a fractional derivative, SIAM J. Math. Anal., 5 (1974), pp. 159-171.
2.
A Erde'lyi, W Magnus, F Oberhettinger and F Tricomi, Tables of Integral Transforms, Volume I, McGraw-Hill, 1954.
3.
F W J Olver, Error bounds for stationary phase approximations, Anal., 5 (1974), pp. 19-29.
4.
K Soni, On uniform asymptotic expansions of finite Laplace and Fourier integrals, Proc. Royal Soc. Edin., 85 A (1980), pp. 299-305.
5.
R Wong, On a uniform asymptotic expansion of a Fourier-type integral, Quart. Appl. Math., 38 (1980), pp. 225-234.
SIAM J. Math.
ON A NEW NUMERICAL METHODFOR A NEW CLASS UF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS ARISING IN NONSPHERICAL GEOMETRICALOPTICS
Robert L. Sternberg
Marvin J. Goldstein and Orew Drinkard
The surfaces S:
z = z(x,y) and S':
z' = z ' ( x ' , y ' ) of a three-dimen-
sional, aperture or volume extremized and e l l i p t i c i z e d , bifocal, nonspherical singlet optical, radar, or sonar lens antenna satisfy--by Snell's law and the optical path length condition of Fermat's p r i n c i p l e - - d i f f e r e n t i a l equations of the form
~
~Z ~ =F,
S:
= F' ~
,
S': ~z _ Oy G,
(i; ~)(x',z') = G' ~(x , y T
symmetry conditions of the form z(-x,y) = z(x,y),
z' (-x' ,y' ) = z ' ( x ' ,y' ), (2)
z(x,-y) = z(x,y~,
z ' ( x ' , - y ' ) = z ' ( x ' ,y' ),
and boundary conditions of the form z(x,y) = 0
and
z'(x,y) = O,
on the ellipse F:
(x ~bo2COS2~o ) + (y2/b2 o) = i •
Here we are given the lens focal length ~Zo] > O, i t s m~imum diameter 2b > O, i t s ind~ of refraction no > i , and i t s o f f - ~ i s or bifocal angle O
{3}
643 ~0 ~ O, while x and y are independent variables and x ' , y ' , z, and z' are dependent variables such that x ' = x ' ( x , y ) , z' = z ' ( x ' , y ' )
= z'[x'(x,y),
are the intercepts on S:
y'(x,y)].
y' = y ' ( x , y ) ,
z = z ( x , y ) , and
Geometrically, x, y, z a n d x ' , y ' , z'
z = z(x,y) and S': z' = z ' ( x ' , y ' )
ray R passing from the primary f i n i t e ,
of an arbitrary
perfect or stigmatic,
focal point F at
Xo' YO' Zo where 2 secz @o]1/2 sin @o' Yo = [b~ + zo
x o= O,
Zo = -IZol '
to the i n f i n i t e perfect focal point Foo located in the direction -@o relative to z-axis and at i n f i n i t y in the y,z-plane as in the i l l u s t r a t i o n shown in Figure 1.
By virtue of the symmetry conditions (2), the lens has, in
addition, two symmetrically displaced secondary, perfect or stigmatic, focal
Y
I SI / E~' ( O [
' b°' O)
/&\ W \ / / ~
F~ ~
,'
/ / !i ,,
(x, y, z
:
_z
F'(Xo,- YO,Zo) /
/
~
)
)
~
/
"
/'
~...L~._W/(0,O, a>
/'t?, no I / k ..f",. / ~ F X',-,V/(x'.y'.z')/ ) , ~ . " ~
/E'(o,-bo, ~l ~
/ /
D""
I#
'v
- d/O
1 Figure i i points, one f i n i t e , F' at Xo, -Yo' Zo, and one i n f i n i t e , Foo, located in
the direction + @o relative to the z-axis and also at i n f i n i t y in the y,z-plane.
The secondary focal points F' and F~ are dual to the primary loci
F and F:o. The coefficient functions F, G, F', and G' in ( i ) are of the form
(5)
644 n
no(X' - x)p - x p ' F = (z - Zo) p' - no(Z' - zip ~
i
i
o(y - y)p - (y - yo)p G = (z - Zo)p' - no(Z' - z ) p '
(6) no(X' - x)
no(Y' - y) + p' sin @o
F' = p, cos @o - no(Z' - z) '
G' = p, cOS@o - no(Z' - z) '
where p and p' are the path length elements p2
X2 +
(y - yo )2 + (z - Zo )2,
p,2 = (x' - x) 2 + ly' - y)2 + (z' - z) 2.
(7)
Here we have suppressed x o since, as in 11), we assume zhat x o = O. E s s e n t i a l l y e q u i v a l e n t to e i t h e r the equations f o r S or those f o r S' in (1)--and capable of being s u b s t i t u t e d f o r one of those sets of e q u a t i o n s - - i s the o p t i c a l path length c o n d i t i o n t h a t
P ~ P +noP' + P" = Po'
(8)
where Po i s a constant independent of x and y for a l l rays R as above going from F to F~when the path length P is measured along R from F to a momentarily fixed wavefront W e~iting from the lens orthogona] to R and propagating toward Fm.
Thus here consequently, p" in (8) is of the form
p" = (a - z ' ) cos @o + y'
sin @o '
(9)
where a i s the z - a x i s i n t e r c e p t of the f i x e d wavefront W while the constant Po in (8) has the value
PO = [b~ + z20 sec2@o ] I / 2 + a cos @o " The geometrical arrangement showing a typical ra~v R and i t s corresponding wavefront W i s perhaps further c l a r i f i e d by the i l l u s t r a t i o n . For f u l l d e t a i l s of the genesis of the problem and the equations (1) through (i0) and information on previously developed and applied methods of t h e i r solution, see Sternberg [1, 3 and 4]. In the following--assuming always that the F-number,~" = I Zol/2bo i s s u f f i c i e n t l y large and that the bifocal angle @o is s u f f i c i e n t l y small, in
(I0)
645
f a c t t h a t they are s u b s t a n t i a l l y l a r g e r and smaller, r e s p e c t i v e l y , than the absolute bounds ~='= I z J / 2 b o > --~ sin @o cos @o
and
@o < ~/2,
(II)
noted in the existence conjecture of Sternberg [ 3 ] since the inaex no here i s assumed f i n i t e - - w e o u t l i n e a new numerical method of solving approximately the problem c o n s i s t i n g of the equations (1) and she symmetry and boundary c o n d i t i o n s (2),
(3), and (4) by what may be described as e l l i p t i c i z e d
and
constrained s p l i n e approximations f o r the surfaces S: z = z(x,y~ and S':
z' = z ' ( x ' , y ' ) .
The new method is based on a new transcendental
e i g e n v a l u e - l i k e process s i m i l a r t o , but d i s t i n c t
from, the r e l a t e d techniques
described i n Sternberg [ 1 , 4 ] . To keep matters simple we l i m i t our c o n s i d e r a t i o n s f o r the present to s o l v ing the problem approximately as described above, f i r s t ,
only l o c a l l y in two-
dimensional neighborhoods of the points Eo and E' o located at O, +b o, 0 at the ends of the p r i n c i p l e diameter of the lens and, second, to extending the partial
s o l u t i o n thus found to solve the complete problem, s t i l l
only l o c a l l y
and approximately, but now in a three-dimensional neighborhood of the complete lens boundary curve
F
given by (4) which passes through the p o i n t s Eo and E' o.
To these ends we begin by assuming t h a t the surfaces S: S':
z' = z ' ( x ' , y ' )
z= z ( x , y ) and
can be approximated in two dimensions near the p o i n t s E o
and E' o at O, +b o, 0 in the y , z - p l a n e by polynomials of the forms
S:
22 2) + y2(y2 _ bo ) , z = y l ( y 2 _ bo (12)
S':
'
2
'
z' = y l ( y '2 - bo) + " ' " + Y4(y
24
,2 -
b O)
, I
and seek to determine the c o e f f i c i e n t s ~k and Yk" known.
The f i r s t
p a i r are
In f a c t
YI = - ~
i
tan ~o I
and
' YI =
J
i tan 2b0 ~o'
(13)
!
where {o and {o are the angles made by S and S' with the y - a x i s at E° and Eo and
646 are readily computed as in Sternberg [ i ,
4].
To evaluate the remaining
c o e f f i c i e n t s in (12) we proceed as follows. Write Y2 ~ u and t r e a t the l a t t e r as an eigenvalue to be determined. With reference to Figure 2, pass a ray R in the y,z-plane from the focal point F at Xo, Yo' Zo as in (5) to the orthogonal wavefront W via pointsx=O, y, z on S andx'=O, y ' ,
z' on S' in the y,z-plane, bending the ray at S so as
to s a t i s f y S n e l l ' s law and computing y' = y ' ( y ) , z' = z ' [ y ' ( y ) ] on S' by the path length condition (8},
Eo(O, bo, O)
,
F~- - ~
F(Xo, Yo, Zo) '). (x, y, z)/~ R./
Z
7i
q
~o
~
/
\/,o o V./
~
~.~-
V/o
F'(xo, - YO, Zo) //
/
//
~^1W
'
E~)(O,- bo, O)
Figure 2 Here y i s an independent variable and f o r fixed values of y , the quantity z = z(y) i s to be thought of as a function of ~ and is to be computed by the f i r s t of equations (12). By the process of passing the ray R through S and S' we also f i n d t h a t y ' = y ' ( y ) and z' = z ' [ y ' ( y ) ] can be computed in terms of ~ and can be expressed as elementary transcendental functions of the l a t t e r and the several constants of the problem.
By applying S n e l l ' s law at y ' = y ' ( y ) ,
647
z'
= z'[y'(y)]
given
fixed
on S ' ,
the derivative
dz'/dy'
can be s i m i l a r l y
expressed.
For
y let
Z(~),
Y'(~),
Z'(~),
Z I
and
y,(~)
(14)
denote these elementary algebraic and transcendental expressions. Similarly, we trace a ray R' in the y,z-plane from F' at Xo, -Yo" Zo to a corresponding orthogonal wavefront W' via the same point x=O, y, z on S and through a point x*=O, y*, z* on S'.
Similarly as before we find that y* =
y*(y) and z* = z * [ y * ( y ) ] can be computed in terms of ~ and also that the derivative dz*/dy* can be similarly evaluated.
Y*(~),
Z*(~),
and
For yiven fixed y l e t
Z~.(~),
(15)
denote these elementary algebraic and transcendental expressions. Keeping in view the fact that each of the quantities Z(~), Y'(~), 2'{~), Zy,(~), Y*(~), Z*(~), and Zy.(~) could now be computed e x p l i c i t l y for fixed y as a function of ~ i f ~ were known, we set up the system of equations
(y,2 - bOJ 2 . 2 Y2 ' + "'" + (y,2 4Y' (y.2
2 2' + . . . (Y , 2 _ bo)Y _ ~2~2 ' UoJ Y2 + " ' "
4Y*(Y*2
- b2)y'o2 + " ' "
i
i
2,4 _ DO~
' : z' Y4
'1(Y '2 - b l,
+ 8Y' (y,2 - b~,3y'4 = Z', Y
2y' 1Y '
2 . 4 ' = Z* ' y,2 2 _ bo ) Y4 - YI ( - bo)'
+(y,2
+ 8Y*(Y*2
- UL2~3 o l Y4' = Z*y *
(16)
- 2YIY*'
i
for the unknowns x2, Y3' and Y4"
These are four equations in three unknowns
with coefficients entirely expressible for fixed y in t e m s of elementary algebraic and transcendental functions of the eigenvalue ~ and known constants. i
I
i
To solve the system (16) for the unknowns Y2, Y3, x4, we have first to detemine the value of ~ by solving the eigenvalue-like equation
648 {y,2-
b~)2
...
2 4Y'(Y '2 - bo)
( y . 2 - b2o)4 2
23 - bo)
(y.2
b2 4 0)
... 8Y'(Y'
yl{Y '2 - b ) - Z' I
- Z'y'
ZyIY'
~(~)~
=0. (y.2
b~)2 -
...
-
(17)
I
yl(Y .2 - b~) - Z* I
4Y*(Y.2 - b~)
"'" 8Y*(Y*2 - b2)3o
2Yi Y* - Z;.
The determinant a(~) simplifies to the expression a(m)
m 8Y'Y*R'R*(R'
-
R*)
[S'R*3(R *
-
2R')
-
S*R'3(R '
-2R*)]
(]8)
+ 2R'2R*2{R* _ R,) 2 [T,R*2y* + T*R'2y'], where R' = y,2 - b2 o~
R* = y.2 _ bZ o '
S'
S*
I
= Z'
- "{~R', I
I
= Z*
-
y1R*,
i
T' = Zy, - 2YiY .
(19)
I
T* = Z;. - 2YIY*.
in which each capital l e t t e r is now a known elementary algebraic or transcendental function of the eigenvalue ~. The solution of the determinantal equation (17) using (1~) and {19) i s not d i f f i c u l t computationally and ~ = Y2 in (12) is reasonably easily computed i
I
following which the remaining c o e f f i c i e n t s Y2, " " ,
Y4 therein can be
found in theory by solving any three of the equations (16J, assuming the appropriate lesser minors therein are nonsingular.
In practice, in an attempt to
reduce round-off errors, we solve each t r i p l e of equations (16) and average. An approximate solution for the surfaces S: z' = z ' ( x ' , y ' )
z = z(x,y) and S':
valid in a three-dimensional neighborhood of the boundary curve I
r g i v e n by (4) which passes through She points Eo and Eo at O, ~bo, 0 can now be written down in the fom
S:
z = yl[(X2/COS2@o) + y2 - b~] + y2[{x2/coS2@o) + y2 _ bo ]22,
(2o) S':
' ' Z' yl[(X ' 2/coS2@o ) + y,2 _ b~ ] + . . . + y¢[(X'2/COS2~ O) + y,2 - b~] 4,
649
by e l l i p t i c i z i n g the two-dimensional solution (12); that is to say by constructing the e l l i p t i c i z e d solution (20) from (12).
Note that here x' and
y' can be thought of as simply new additional independent variables i f we wish, their dependenceon x and y being of no consequence in using (20) to specify the shape of the surfaces S and S'
To determine the quantities x '
and y' as functions x ' = x ' ( x , y ) and y' = y ' ( x , y ) , i f these expressions are actually desired, however, is a simple matter by ray-tracing once the solutions for S and S' have been found. The ray tracing process also provides a simple check on the accuracy of the computed solution in terms of any given wavelength
~0" To improve the solutions obtained for S and S' the value of y used in the computation can be taken closer and closer to bo or higher degree polynomials can be used in (12) or, alternatively, completion or correction terms, as they have been called, can be added, i f desired, to each of the expressions (20) for S:
z= z(x,y) and S':
z' = z ' ( x ' , y ' ) by using a Cauchy-Kovalevskytype
process, or even by t r i a l and error, as described in Sternberg [~]. Finally, the entire computational process can be repeated with suitable modifications to step the two-dimensional solution out of neighborhoods of the points Eo and E' o at O, ~bo, 0 and the three-dimensional solution out of the neighborhood of the boundary curve F thereby determining the f u l l form of the lens antenna to high accuracy. As a numerical example consider She case in which the focal length IZo~, maximum diameter 2bo, index of refraction no, and off-axis or bifocal angle @o have the values
lzol = 9,
2b0 = 18,
no = 1.594
and
@o = 20°,
(21)
as in the example discussed in Sternberg [4]. For this case, proceeding as in Sternberg [1, 3, 4] we find for the angles
650
~o and ~'o and the coefficients ~1 and Y'I in (13), the values ~o = 4.8738440 ,
~o = 55.743568 , (22)
i
Y1 =
-4.7372426x 10-3 ,
y 1= -8.1574583x i0 -2,
whence, applying our new numerical process with the given value of y set at Y = bo - 6 b o, we obtain for the eigenvalue ~ = Y2 and the remaininy J
I
coefficients ~2' " " '
Y4' the values given in the following tab]e
for d i f f e r e n t assumed values ofthe s t e p - s i z e - f a c t o r 6 .
Table of Results i
6
i
I
~ x 105
Y2 x 104
~3 x 106
Y4 x 108
0.i
-1.336
-3.071
-4.094
0.01
-1.231
-2.952
-4.705
....
0.001
-1.215
-2.950
-2.655
....
-9.612
In comparison to these results the true values of ~ = Y2 and the coefficients Y2 and y3--as calculated by a Cauchy-KovalevsKy l i k e process carried out some time ago in connection with Sternberg I t ] - - a r e I
~2 = - 2 . 936 x =
lO-i (23)
V2 = -1.220 x i0-~5 i
v3 = - 2 . 0 3 3 x
10 -6 .
I
Although no check on ~4 i s a v a i l a b l e , that coefficient
since no a c c u r a t e c a l c u l a t i o n s
have e v e r been made, i t
r e s u l t s compare q u i t e w e l l w i t h t h e o l d . borhoods o f the boundary c u r v e t ,
of
may n e v e r t h e l e s s be seen t h a t t h e new Hence, i n s u i t a b l y
restricted
neigh-
t h e antenna should perform w e l l even a t v e r y
small values of the wavelength ~o and by extending the method step-by-step or by using some of the other refinement laethods as described previously, the solutions for the complete surfaces S and S' should be able to be computed to reasonably high accuracy.
651 Physical and engineering a p p l i c a t i o n s of the lens antennas studied here and related devices are discussed in Sternberg [ i ,
3, 4], Sternberg and Goldberg
[ 2 ] , Corbett, Middleton, and Sternberg [5], and Sternberg and Anderson [6] and in the therein referred to additional references and t e x t books. F i n a l l y , note that our r e s u l t s are computational only, no complete existence, uniqueness or convergence theory having ever been established f o r the problem. References i.
R . L . Sternberg, "Successive Approximation and Expansion Methods in the Numerical Design of Microwave D i e l e c t r i c Lenses," Journal of Mathematics and Physics, Vol. 34, 1955, pp. 209-235.
2.
R. L. Sternberg and H. B. Goldberg, "Off-Axis Short Focal Length D i e l e c t r i c Lens Antennas," Proceedings National Aerospace Electronics Conference, Dayton, Ohio, 1961.
3.
R. L. Sternberg, "Surface Series Expansion C o e f f i c i e n t s f o r Numerical Design of Scannable Aspherical Microwave and Acoustic L e n t i c u l a r Antennas," Proceedings of the IEE (England), Vol. 121, 1974, pp. 1351-1354.
4.
R. L. Sternberg, "Optimizing and Extremizing Nonlinerar Boundary Value Problems in L e n t i c u l a r Antennas in Oceanography, Medicine and Communications:
Some Solutions and Some Questions," in Nonlinear Systems and
A p p l i c a t i o n s , V. Lakshmikantham, Editor, Academic Press, New York, 1977.
5.
K. T. Corbett, F. H. Middleton, and R. L. Sternberg, "Nonspherical Acoustic Lens Study," Journal of the Acoustical Society of America, Vol. 59, May 1976, pp. 1104-1109.
6.
R . L . Sternberg and W. A. Anderson, "Frequency Independent Beamforming," Naval Research Reviews, Vol. 33, No. 1, 1981, pp. 34-47.
Local Existence
Theorems
for Ordinary Differential Fractional
Equations
of
Order
by Ahmad Zain - Alabedeen
Mohammad
Tazali
ABSTRACT In this paper, we prove two local method and the Schauder g
fixed-point
(~)
existence
theorem,
(x) = f(x,g(x))
theorems,
by using both the Picard
for the following
initial-value
problem:
(almost all x s[a,a+h])
with
(A) g
where g
(~)
denotes
(~-1) (a)
the derivative
0 < ~ < I ,
= br(~),
of order ~ of a real-valued
the Gamma function where ~ > 0 ; b is a real number,
function
and under
g ; F(~)
is
suitable conditions
on the function f. If ~ = I in the initial-value corresponding Coddington
problem
to this problem are known
and Levinson
above;
(sometimes)
(1955) and Hale
theorem of the maximum and the m i n i m u m
(A), then the existence
(1969).
when e = I, this theorem reduces
as the Carath~odory
Finally,
solutions
theorems see
we prove a local existence
for the initial-value
to Theorem
theorem,
1.2 of Coddington
problem
(A)
and Levinson
(1955). I.
Preliminaries Throughout
£,6 denote numbers,
and definitions
of this work the symbols are:
the Gamma and Betafunctions
non-negative
real numbers;
which are Lebesgue-integrable absolutely
continuous
functions
is the set of all real-valued subinterval
of I;
on interval
I _ R;
respectively;
on I
C(I)
respectively;
L(1) R;
R,R+ are the sets of all real
is the class of all real-valued AC(J)
denotes
defined on any compact
absolutely
continuous
for the set of all real-valued subinterval
functions
is the set of all real-valued
the function
J of R;
ACloc(I)
defined on any compact
continuous
No,N + are the sets of all non-negative,
f = f(.) denotes
functions
functions
positive
f, and f(x) denotes
defined
integers
the function value
653 at the point x;
{fn(.)
: n s N+} denotes a sequence of functions fn;
read as "almost everywhere"; 2.
a.e.
is to be
(ppx ~ A) is to be read as "almost all x cA)".
D e f i n i t i o n s and lemmas For the following definitions 2.1, 2.2 and the proof of lemma 2.2 see Barrett
(1954). D e f i n i t i o n 2.1
Let f be a real-valued function which is L e b e s g u e - m e a s u r a b l e and is
defined a.e. on [a,b[.
For real ~ J 0, define b I a/b(b i °" f = F(~)
provided that the (Lebesgue)
t)~-1 f(t)dt,
integral exists.
A n extension of d e f i n i t i o n 2.1 is: D e f i n i t i o n 2.2
If ~ < 0 and if n is the smallest positive integer such that
+ n > 0, then define an b a provided
that
f = Dn i~+n f at x = b, x a
x I a+n f and a nth
its
(n -
derivative
1) d e r i v a t i v e s
( h > O) a n d
the
exists
at
Lemma 2.1
Let p,q s R such that q > -I.
exist
in a segment
If x > a, then
(n s N+) , p + q =-n
The proof is straightforward. Let ~ > 0 and f ~ L(a,b). x F (x) = I s f a
Define (x ~ (a,b)).
Then the following results hold: (i) (ii)
(iii)
F a
S L(a,b) and exists a.e on [a,b] when ~ ~ (0,1), Xl+ ~ I x F (t)dt = I f(.) ~ a
F (x) = Dx
~l+a
f(.)
Ib - x[
x = b.
0
L e m m a 2.2
)
(x - a) P+q r(p + q + I) ' p + q ~ -n
x i p (t - a) q a r(q + I)
Proof:
(Dn =
(x ~ [a,b])
(ppx e [a,b]).
a (iv)
F (.) c AC[a,b] w h e n ~ ~
(v)
xt x I p I q f(.) = I p+q f(.) as a
I or f ~ AC[a,b],
(p,q > 0 and ppx ~ [a,b]).
< h
654
Definition
2.3
Let ~ a (-~,I);
containing
the point a;
D(~)(I)
= {y
let I be a compact
interval
of the real line R
then define e L(I)
: x +
ly(.)l(x e I)} s AC(1). a
Note
Clearly D(O)(1)
Definition
2.4
= L(I).
For y ~
y
D(~) (I),
(~)
we define for 0 < ~ < I,
day
d x = - - 11-~y(.) dx dx a a
(x) =
(ppx a I)
and for ~ < 0, X
y (a)(x) provided
x I-my(.)
that
= I-my( ) a
(x e I),
exists.
a
Definition
2.5
as
of
the
set
Let all
a ~ (0,1)
m e L(I
)
and
such
I
= [a,a
we define
+ p];
the space LT ( I ) ap
that:
X
(i)
(ii)
{x ÷ I~Im(.)l a x
li~+
(x s (a,a + D]} a C(a,a + p],
l~Im(.)E = 0.
x÷a
Lemma 2.3
Proof
(i)
If ~ g (0,1),
then L~(a,b)
(ii)
If ~ = I, then L~(a,b)
Clearly LT [a,b] _ L(a,b).
following
LT [a,b]
L(a,b).
LT [a,b] = L(a,b).
To prove L T
# L(a,b),
we consider
the
example:
Let m : [a,h] + R be defined by m(x) Obviously m e L(a,b),
= (b - x) -~ for all x s [a,b] and 0 < ~ < I.
but m £ LT [a,b]
since
lim x÷b-
a
fX(x _ t)~-1(b - t)-~dt > lim_ x÷b
in~ - xa does not exist. Now we prove that L~(a,b) _ LT [a,b] and to do this let m ¢ L~(a,b),
define
g : [a,b] ÷ R by: g(x) = ~ I Then m a LT [a,b] and the proof Next, we show that L~(a,b) m(x)
= inb I_-~
m s L(a,b);
(x a [a,b)).
a fX(x - t)~-1[m(t)Idt
(x s (a,b]).
is straighforward. ~ LT [a,b].
Let m : [a,b] ÷ R be such that
Then, by using integration by parts, we obtain that
let further ¢~ : [a,b] ÷ R, where ~ s (0,1), be defined by ~ (x) = afX(x _ t)~-11nb _I t dt
(x s (a,b]).
Thus ~ ~ (x a a)a In( -
I /x (x - t) ~ -{ dt ) + ~ a b
(x s (a,b]),
655
and hence 10.8,
¢ (x) exists
(x s (a,b])
11.7 of Titchmarsh
definition m(x)
2.5 implies
= In(bI _ - ~ )
(1939),
and
lim+ ¢ (x) = 0o Then, by using sections x÷a one can show that Ca c AC[a,b]. Consequently the
that m c LT [a,b].
+ ~ as x + b-.
Furthermore,
Clearly
(ii) holds
m ~ L~(a,b)
since
and this completes
the proof
of lemma 2.3 In the following the elements Theorem
theorem,
we prove a result which gives
some information
about
of the space LT [a,b].
2.1(a)
Let a E (0,1) and ~ : [a,b] ÷ R be a Lebesgue
measurable
function;
let m : [a,b] ÷ R+ be such that [~(t) I ! m(t) (b) depending
(almost all t e [a,b]),
For any Xl, x 2 c [a,b]
then ~ c LT [a,b].
such that x I < x2, there exists
6 = 6e
only on ~ such that x2 Xlf
Proof
of (a)
(x 2 - t)a-lm(t)dt
Let y,z
< g whenever
: [a,b] + R be defined
x 2 - x I < 6.
by
_ r(~) I a/X(x _ t)a-1[+(t)[dt
y(x)
(x e (a,b])
and
(2.1) I a/X(x _ t)~-Im(t)dt z(x) _ r(a)
Now,
if Xl,X 2 e (a,b]
(x s [a,b]).
such that x 2 > x I (the case x 2 < x I is similar),
then from
(2.1) we have x9
JY(X2)-Y(Xl)I J
IZ(Xl) - z(x2)[ + r(~ Xl/ -(x2 -
t)~-Im(t)dt
(2.2)
But I Xlf x2 (x2 - t)~-Im(t)dt F(~) - F(~)I
a/Xl
(x I - t)~-Im(t)dt
_
I F(~)
a/
x2
(x 2
I a/X1{ _ t)~-1 - --F(@ (x2
t)~-Im(t)dt
- (Xl - t)~-1}m(t)dt
x < iZ(Xl) --
_ Z(Xl)l
_ F(G) I a / I {(Xl _ t)e-1
- (x 2 - t)G-1}m(t)dt
(2.3)
But ~ c (0,1), x 2 > x I imply that {(×I Therefore, lY(X2)
from
- t)a-1
(2.1),
- x2 - t)a-1}
(2.3) and
_ Y(Xl) I _< 3[z(x2 ) _ Z(Xl)i
Since m s LT [a,b],
it follows
< (Xl - t)~-1
(2.4)
(2.4), we obtain a / x 1(x I - t)~-Im(t)dt. + ~(~-~ 2
that z e C[a,b]
and so the integral
(2.5)
656
aa /
xI
(Xl - t)~-Im(t)dt
convergence
exists
(t c [a,xl]) , then by the Lebesgue dominated
theorem of Titchmarch
(1939), we have
X o
lim f ~(x I - t) ~-I - t)~-1) }re(t)dt x2÷x I x I - (x 2
0.
x2 lim f (x 2 t)~-Im(t)dt = 0. Now, by using (2.2), we see x2*x I x I y ~ C(a,b[. Furthermore m ~ LT [a,b] implies that lim z(x) = 0 and therefore Thus
x~a
li~ y(x) = 0 since
ly(x) l ! z(x)
(x g (a,b]).
Consequently,
by definition 2.5,
x~a
it follows that ~ ~ LT [a,b]. Proof of (b)
From part (a) we have x2 lim f (x 2 x2÷x I x I
t)~-Im(t)dt = 0. x2
By using (2.3),
(2.4), it is easy to show that the integral xlf
converges uniformly on a J x I < x 2 J b. Note
(x 2 - t)e-lm(t)dt
Hence (2.1)(b) holds.
It is of interest to note that if in the definition of LT [a,a + ~] we
replace condition
(i) by
(i)
fX(x - t)~-11m(t)dt
{ X ÷ ~ aI
(x s (a,a + ~]} e ACloc(a,a + ~]
then it is not known whether theorem 2.1 remains true. Definition 2.6' E(~)(I~)
Let ~ s (0,1] and Ih'
= {g c C(I~)
:
(a,a + h]
h > O.
Define
: li~ (x - a) 1-eg(x) exists and is finite}.
For any
x~a
g s E(~)(I~),
let the norm
II.II~ be defined by
IIgll a = ~ { ( x
- a)~-IIg(x)]}.
Lemma 2.6 (i)
((E(~)(Ih),
II-l[) is a Banach space.
(ii)
A sequence {gn ( •) : n s N+} in E(a)(I~)
only if the sequence {(x - a) I-~ gn(X) (x - a) 1-ag(x) on I~. 3.
Local existence
converges
to g e E(~)(I~),
: n s N+} is uniformly convergent
if and
to
The proof is straightforward. theorem for ordinary differential
equation of order a, where
~ (0,1]. Theorem 3.1
(The Picard method)
Let l,J _ R and ~ s (0,1];
let (a,b) s IXJ and choose M,Y ~ R+ so that the set
G ,y : = {(x,y)
+ ~,
: a < x ia
Suppose f : IXJ + R satisfies:
l(x - a) 1-~y - bl J y }
IXJ.
657
(i)
f(x,.)
e C(J)
(ii)
f(.,y)
is Lebesgue-measurable
(iii)
there exists m : I
(x ~ I M ) and
(x e I),
: = [a,a + p] ÷ R+ such that m ~ LT ( I ) ~ M
M
If(x,y) I _< m(x) h1-~
((m,y) I
su
xc[a,~+h] r(cO then there
exists
(y e J),
~ G ,y).
a/X(x - t)~-Im(t)dt
at least one function
g g C(a,a + h) but g ~ C[a,a + h],
(2)
g g n(~)[a,a
(3)
g
(4)
g
> 0
0 < h _< ~ such that < y,
g : (a,a + h] ÷ R which
(I)
(~)
Define
and m(x)
satisfies:
+ h],
(x) = f(x,g(x))
(almost all x c [a,a + h]),
(~-1) (.) s AC[a,a + h],
(5) g (m-l) (a) = br(~). Note
Before
Lemma 3.1
giving
the proof
Let the conditions
following
X(x - a) c~-1 =
the following
the
X
Then
(3.1)
g(~)(x)
Proof
of Lemma
true,
then
(x s (a,a + h])
is equivalent
= f(x,g(x))
g(~-1)(a)
with
= bF(~)
=
problem:
(3.2)
(0 < ~ < I).
F(a)
Let 0 < ~ < I and suppose
(3.1)
is
~ f(.,g())}; +
a
a
lenmlas 2.1 and 2.2 we obtain
X
initial-value
(ppx E [a,a + h])
~i-~{x(~:~)~-I
I1-~g(')
(3.1)
to the following
The case a = I is obvious.
3.1
x
that
X
11-e g(.) a and this proves
= ~ + 11 f(.,g(.)) a x1_ ~ that I g(.)
f e L(a,a + h) (from (ii) and
= A +
a
/xf(t,g(t))dt
is absolutely
(x s [a,a + h])
continuous
(iii) of theorem 3.1),
on [a,a + h].
it follows
g(~)(x)
= d'~ ~ 1-~ g ( . )
= f(x,g(x))
a
g(a)(a)
and of the converse
is similar.
Now we give the proof of theorem 3.1.
= X = bC(a).
Since
from definition
that
The proof
lemma:
(ii) and (iii) of theorem be given and consider
+ I s f(.,g(.)),
F(e)
~ : = bF(c~).
applying
to prove
equation
g(x) where
of theorem 3.1, we need
(ppx ~ [a,a + h]
2.4
658
Let Ihv =
Proof of theorem 3.1 {gn(.)
(a,a + h] and Ih = [a,a + h];
define the sequence
: n ~ N+} as follows: ~-1 g0(x) :
and gn(X) = go(X) + ~ I Clearly, by induction,
l(x - a) F(~)
afX(x_t)~-If (t,gn_1(t)dt , we have (x,g n) c G ,y
Next we prove gh(.) s C(I~) conditions
(x s I') h'
(n E No )"
(3.3)
(x s lh, n e N+).
(n s N+). Obviously go s C(I~), by using the
(ii) and (I) of theorem 3.1 and applying these in Theorem 2.1(a), we
obtain that f(.,gn_1(.)) (n c N+).
(x s I~)
s LT (I h)
(n ~ N+) and this implies that gn(.) ~ C(I~)
I To prove that lim gn(X) = go(X) + F--~-~ aJX(x - t)~-1 f(t,g(t))dt,
we define the sequence {H (.) : n ~ N ] as follows: n o
Hn(X) = (x - a l-a gn(X) Hence H0(x) = b
(x s lh,'
n c No) , where gn is as given in (3.3)
(x s I~) and
H n (x) = b + (x F(~) - a) I-~ a/X(x - t)a-lf(t,gn _ 1(t))dt, Since gn(.) ~ C(I~)
(n c No) , we have Hn (') s C(I~)
(x s I~, n ~ N+)
(n ~ No )"
(3.4)
Also from (3.4),
we have
li~ Hn(X) = b (n s N+), and thus by defining Hn(a) = li} Hn(X) = b x÷a x+a (n ~ N+), we find that Hn(.) ~ C(I h) (n E N+). Furthermore, by using (3.4) and
theorem 2.1(b), one can show that the sequence {Hn(.), and equicontinuous Levinson
(1955)),
on Ih.
Therefore by the Asceli lemma,
(see Coddington and
there exists at least one subsequence
converges uniformly i.e.
n ~ N+} is uniformly bounded
{H n (.) : s s N+} which s on Ih to some limit function H(say) as s ÷ ~.
lim - a) 1-a gn (x) s-~° H n (x) = lim(x s-~° s s
(x s I~).
Now define g(x) = lims -~ gn (x) = s-~°lim(x - a) a-1 Hn (x) s s = (x - a) ~-I
~
H n (x) = (x - a ) e - I H ( x )
(x s I~).
s
This means that the sequence {gn(.)
: n s N+} converges
hence from this fact and the hypothesis
to the function g on Ih' and
(i) of the theorem 3.1, we obtain that the
sequence {f(t,g n (t)) : s e N+} converges to the function f(t,g(t)) for any fixed s ' Then by using the condition (iii) of theorem 3.1 t s (a,x] and for all x s Ihand the Lebesgue dominated convergence that
theorem,
see Titchmarch
(1939), we obtain
659
g(x) = xlim -~° g(x) = go (x) + ~ I Some properties (i)
(x e I')
a ix (x - t)e-lf(t,g(t))dt
of the limit function
g ~ C(I~) but g ~ C(lh);
(3.5)
h "
g
this follows
from the fact gn (') s C(I~)
(s e N+)
S
and the uniform
convergence
of the sequence
{gn (')
: s s N+}, but g ~ C(I h) because
s
go ~ C(lh)" (ii) g s D(~)(Ih ).
In fact g s C(I~)
implies
that g s L(lh);
furthermore
from the
x
proof of lemma 3.1, we have
I1-~g(.) is absolutely continuous on Ih, thus by a (2.3), we find that g ~ D(~) (lh). Finally from the proof of lemma 3.1,
definition we deduce
the following
properties:
(~) (iii)
g
(x) = f(x,g(x))
(ppx s Ih) ,
(~-I) (iv)
g
(v)
g(~-1)(a)
4.
(.) ~ AC(Ih) , = bY(s).
~I and S 2 continuities
Definition
4.1
Let 0 < ~ < I and D be an arbitrary
G h : = (a,a + h]XR.
A function
any c > 0, there exists
4.2
~ = oe > 0 such that for all s whenever
(x,Yl),(x,y o)~ s D we have
4.1;
on D, if for any e > 0, there exists
(x,Yl),(x,y 2) s D we have
on D, if for
ly I - y2](x - a) I-~ < o.
Let ~,D be as given in definition
to be S2-continuous
in the strip
f : D ÷ R is said to be S1-continuous
If(x,y I) - f(x,Y2)[< Definition
region
[f(x,y I) - f(x,Y2)[(x
a function o = o
f : D ÷ R is said
> 0 such that for all
- a) I-~ < e provided
that
[Yl - Y2 [(x - a)1-~ < o. Note
When ~ = I, S I and S 2 continuities
Lemma 4.1 functions
Let ~,D be as given
reduce
in definition
to ordinary
4.1;
consider
f : D ÷ R such that
M] = {f : f is uniformly
continuous
M 2 = {f : f is S1-continuous
on D},
M 3 = {f : f is S2-continuous
on D}
then the following
hold:
(1)
MI
M2
(2)
M2
M 3 but M 3
(3)
M2
M I.
relations
M 3 # %, where M2,
on D}
% denotes
the empty set,
continuity. the following
sets of
660
Proof of (I)
A constant
Proof of (2)
Obviously
function M2
f = C (say) belong
M3;
to prove M 3
to M I
M2
M 2 we consider
M 3. the following
example: Let f : D + R be defined
by f(x,y)
and these can be checked
easily.
Proof of (3)
Define
= y for all
I
let E and 6 = 6 E be arbitrary
If(x1'Yl)
((x,y)
positive
numbers
If x 2 g (a,a + h] such that
- f(x2'Y2) l = (x I
Now, by choosing
~ D.
x 2 close
Ix2 - Xll a)(x 2 -
enough
-
e D) and choose x I ~ (a,a + h] such that
Ix2 - Xll < 6,
I x2
-
a
to a, we then find that Ix I - x21 < 6,
Since this is true for any 6 > 0, so we have f ~ M I.
Definition
ly 2 - yll < 6, we have
~ a) < (x I - a) (x 2 - a)
If(xl,Y I) - f(x2,Y2) I > e although
we have M 2
Then f e M 3 but f ~ M 2
f : D * R by: f(x,y) -- x - a
x I = a + 6.
(x,y)
IY] - Y21 < 6.
Clearly
f ~ M2, consequently
M I. 4.3
Let ~,D be as given
in definition
4.1;
let
A : [a,a + h] + R+ be such that A c C[a,a + h]. We say f satisfies the following
the Lipsehitz
condition
condition
on D and briefly we write
- a) 1-elyl
Lemma 4.2
If f s Lip(D), then f is St-continuous a,A The proof is straightforward.
Theorem 4.1
(Carath~odory
Let all the assumption the given function the conclusions Proof
Let
- Y2 I"
in y ~ R.
the Schauder method).
of theorem 3.1 be given except
in the condition
(i) we assume
in y 6 R for each fixed x ~ (a,a + M].
of theorem 3.1 hold.
% = bF(~) Clearly
theorem,
f is S2-continuous
(E(~)(I~),
II.I[ ) be the space as given
and go(X)
go s E(~)(I~).
if
holds:
[f(x,y I) - f(x,Y2) I J A(x)[(x
Proof
f e ~(D),
-
%(x - a) e-1 F(~)
Define a closed
F(~)(Ih ) = {g ~ E(e)(Ih)
,
in lemma 2.6;
let
(x s I~).
subset F(a)(I~)
: Ilg - goll~ j Y} •
of E(~)(I~)
as:
Then
661
Thus F(~)(I~),II.II~)
is a Sanach space, since every closed subspace of a Sanaeh
space is also a Banach space and hence F (~)(I~) is a convex set. g c F(~)(I~),
Now, for any
define an operator T as follows:
(rg)(x) = rg(x) = go(X) + ~
I
aYX(x - t)~-If(t,g(t))dt,
Clearly T : F(~)(I~) ÷ F(~)(I~). gl,g 2 s F(~)(I~),
(x c I~)
(4.1)
Furthermore T is continuous on F(~)(I~)
for if
then from (4.1) we have
IITgl - Tg211~ --< xs~Su(x - a) 1-~afx(x - t) (~-11f(t,g 1(t) - f(t,g2(t) l(t - a) 1-~dt Hence, by using the S2-continuity of f in y for each fixed t ~ (a,x], it follows from (4.2) that llTg I - Tg211
< c xSuc ~ --
(x F(~) - a) I-~ afX(x - t) ~-1(t - a)~-Idt,
where ~ is a positive number. llTg I - Tg211~ = s S u ~ ( x
Then by lemma 2.1, we see that
x I - a) 1-~F(~) Is (t - a) eh~F(~) r(~) ! s r(2~) a
'
and this proves T is continuous on F (~)(I~). Next we prove T(F (~)) is relatively compact and to do this, we define Hn(X) = ( x -
a) 1-~Tg n (x)
' n a N+) (x c lh,
where Tgn(X) as given in (4.1). It follows that the sequence {Hn(.) equicontinuous corresponding
: n c N+} is uniformly bounded and
on [1,a + h] and for the proof of this, see the properties of the sequence defined in (3~4).
Consequently,
by the Asceli lemma, we
I-~ find at least one subsequence {(x - a hence T(F (~)) is relatively compact.
Tg n (x) : m s N+} is convergent in F(~)(I~), m Finally by the Schauder fixed~point theorem,
(see Hale (1969)), there exists g c F(~)(I~)
such that Tg = g°
Then from (4.1) we
obtain Tg(x) = g(x) = go(X) + ~
I
a/X(x - t)~-If(t,g(t))dt
(x ¢ I,')
For the rest of the proof of this theorem, see the proof of theorem 3.1. 5.
A local existence theorem for the maximum and minimum solutions of fractional
differential
equations.
Definition 5.1
Consider the following initial-value problem: g
with
(~)
(x)
= f(x,g(x))
(almost all x c [a,a + hi)
662
g where
(a-l)
(a) = bF(~),
(5.1) is as defined in equation
a ~ (0, I]
(5.1)
(3.2) if gM is a solution of (5.1) on
[a,a + hi, i.e.
g ~'~)(x) = f(x,gM(x))
(almost all x £ [a,a + hi with gM(~-1)(a) = bF(~);
if
g is any other solution of (5.1) such that g(x) ! gM (x) with g(a-1)(a)
(x e (a,a + h])
= gM(a-1)(a) = bF(~),
(5.1) on ]a,a + h].
Similarly,
then we say that gM is a maximum solution of
gm is called a minimum solution of (5.1) on
[a,a + h] if any other solution g of (5.1) satisfies the following g(x) ~ gm(X)
relation:
(x 6 (a,a + h])
with (~-I) g Theorem 5.1
(~-1)(a) (a) = gm
Let the hypothesis
br(~) =
"
of theorem 3.1 be satisfied.
Then there exist
maximum and minimum solutions gm' gm of (5.1) on [a,a + h] respectively with (~-I) (a-1)(a) bF(~). gM (a) = gm = Proof
From lemma 3.1, we have that any solution g of (5.1) must satisfy the
following
integral equation:
g(x) = go(X) + F ( ~ where go(X)
y(x - a) ~-I F(~)
a/X(x - t)~-If(t,g(t))dt, and y = bF(~).
(x - a)
1-a
go(X) = b
(x e (a,a + h])
(5.2)
Thus (x e (a,a + h])
and (x - a) 1-~g(x) = b + F(--~ a/X(x - t)a-1(x - a) 1-~f(t,g(t))dt, Let H(x,t)
(x ~ (a,a+h])
: = (x - a) 1-~g(t)
and
a < t < x < a + h P(t,H(x,t))
then using
(5.3)
(5.4)
= (x - a) 1-af(t,g(t))
(5.4)m we see that (5.3) is equivalent Ho(X,X)
: = H (x) = b o
to the following equation: (x ~ (a, a + h])
and H(x,x)
: = H(x) = b + F - 1- ~ a/X(x - t) ~ - I
F
(t,H(x , t))dt (x ~ (a,a+h])
(5.5)
663
It follows from the proof of theorem 3.1 that H(.) ~ C[a,a + h]; uniformly continuous on [a,a + h].
Thus for any c > 0, there exists o = o(E) > 0,
independent of x and H (because of the equicontinuity proof of this see the proof of the corresponding in equation
of set {H(.)}, and for the
sequence {Hn(.)
: n s N+} defined
(3.4)) such that
]H(x 1) Let
hence H is
Z(x)
-
H(X2) I < G whenever
: = sup{H(x)
]x 1 - x2[ <
= (x - a) 1-~g(x)
taken over all solutions g of (5.2) with g
(~-1)
(5.6)
O.
(5.7)
: x ~ lh},
Then Z exists and is
(a) = bF(~).
continuous on Lh and so it is uniformly continuous
on Ih.
Hence, given c > 0, there
exists a = o(s) > 0 such that, no only (5.6) is true for this a but also for all Xl,X 2 E Ih, IZ(Xl ) - Z(x2) I < s whenever
(5.8)
Ix I - x21 < o
For a given c > 0, choose a = a(c) > 0 so that (5.6) and (5.8) hold.
Divide the
interval Ih into n intervals by the points a = Xo < Xl < x2 < "'" < Xn = a + h in such a way that max(xi+ I - x.)l < o, (i = 1,2,...,n-I). For x i ~ Ih (i = 1,2,...,n-I),
choose a function H.l from (5.7) such that
0 < Z(x i) - Hi(x i) < ~ and for i >_ I Hi(x i) - Hi_1(x i) > 0 ; this is possible from the definition of Z. function W
c
Now, for the given ~ > 0, define the
: Ih ÷ R by: W (x) = Hn_1(x)
(x ~ [Xn_1,Xn]).
If Hn_1(Xn_ I) > Hn_2(Xn_ I), define Wc to the left of Xn_ I as Hn_ I up to the point an_ 2 (if it exists) Hn_2(an_2).
in (Xn_2,Xn_1)
nearest
to Xn_ I such that W (an_ 2) = Hn_1(an_ 2) =
Ifan_ 2 does not exist, define W~ on [Xn_2,Xn_ I) as Hn_ I.
Hn_1(Xn_ I) = Hn_2(Xn_1) , define W (x) = Hn_ I (x) on[Xn_2,Xn_1 ).
If
Continuing
this
way, we can define W g on Ih~ having the property 0 ! Z(x i) - W~(x i) < ~,
(x i s Ih, i = 1,2 ..... n).
Since the variation of Z and W C in each interval (5.6),
(5.8) the results from (5.9) give us
(5.9)
[xi,xi+ I) is less than E, by
664
0iZ(x)
- w (x) = Iz(x) - w (x) I
i Iz(x) - Z(xi) l + IZ(x i) -w(xi) l + lw(x i) -W(x) I < 3~
(x s Ih).
(5.10)
Letting s = ~, (m ~ N+), we obtain a sequence {W~(.)
: m s N+} of functions
defined on Ih, which by (5.10) converges uniformly to z on Ih.
Thus from this fact
and using the same orguments as given in the proof of theorem 3.1 and an application of the Lebesgue dominated convergence theorem to (5.5) with H replaced by W !. it m follows that Z(x) = b + ~
afX(x - t)a-IF(t,Z(x,t))dt
(x e Ih)
(5.11)
Then from the definition of Z and H, there exists a function gM : (a,a + h] * R such that Z(x) = (x - a) l-~gM(x)
(x E (a,a + h]
(5.12)
and gM(x) ~ g(x), where g satisfies (5.2). Consequently, from (5.4), (5.11) and (5.12) we obtazn I afX(x - t)~-1 (x - a ~ 1-~f.t,g.t~.dt (x - a) 1-agM(x) = b + F--~ ( ( ~
(x c (a,a + h).
Thus y(x - a) a-1 r(~) + F(--~) afX(x - t)~-If(t,gM(t))dt,
gM (x)
(x ~ (a,a + h]).
Finally Lemma 3.1, implies that gM" (~)(x) = f(x,gM(x))
(almost all x ~ [a,a + hi)
with
(a-1)(a) gM
= br(~)
Similarly, we can prove the existence of the minimum solution gM of (5.1) with g~-1)(a) Note
= br(a).
If ~ = I in theorem 5.1, then it reduces to theorem 1.2, pp. 45-46 of
Coddington and Levinson (1955). Remark
The uniqueness, global cases and the extension for ~ > I of Theorem 3.1
have also been proved;
they will be published later.
This paper is a part of a Ph.D. thesis submitted to the Graduate School of the University of Dundee.
The author wishes to express his sincere appreciation for
the guidance of this work by Professor W.N. Everitt, Baxter Professor in the
665
Department of Mathematical Sciences, The University of Dundee. REFERENCES Barrett, J.H., 1954.
Differential equations of non-integer order. Cand. J. Math., 6, 529-541.
Bassam, M.A., 1961.
Some properties of the Holmgrem-Riesz transform, Ann. Scuala, Norm. Sup. Pissa, 15, 1-24.
Coddington, E.A. and Levinson, N., 1955. Theory of ordinary differential equations, McGraw-Hill, New York. Hale, J.K., 1 9 6 9 .
Ordinary differential equations, John Wiley & Sons.
Titchmarsh, E.G., 1939.
The theory of functions, second edition, Oxford
University Press.
APPLICATION
OF THE THEOREM OF
CONNECTIVITY
OF M 7 IN ASTRONOMY
TUNG CNIN-CHU
My talk today is about the application This theorem is a preliminary
of the theorem of connectivity
part of an answer to the G D Birkhoff's
problem, which is concerned with the problem of three bodies.
of
seventh
The theorem was
obtained by myself and by R W Easton independently with minor differences~ idea of the application,
however,
M 7.
The
was suggested to me by my teacher Chin Yuan-Shin.
The problem of three bodies is too long and involved a story to tell, and so I can only restrict myself to a very narrow aspect of it. opinion that the problem of three bodies everything ~ewton's
is Newtonian here:
in English Whittaker's It is well-known
should be called Newton's problem,
the equations
force of attraction.
I have always been of the
My interest
came from Newton's
because
law of motion with
in the problem was aroused after reading
treatise on analytical
dynamics
[]5].
that Sundman [i0] solved the problem of integrating
the
equations
of motion by means of a power series in a fractional power of the time
variable
t; and in 1907 he obtained
further certain qualitative
resul~s about the
motion from an inequality:
-
where
i ~21-i
~
I = mlr I + m2r 2 + m3r 3 Siegel [8] observed
-
c 2 i-I
-y
-
in his Vorlesungen
As a concrete problem in dynamics, actually behaves.
>
0
, (see Siegel [8], p.51).
are decades old, they are still the newest.
particles
2E
Although
(1955) that, though Sundman's
And I think this is still true today.
one wants to know how a system of
Sundman's
series gives no indication
general about this, it certainly gives a description quantitatively
as well as qualitatively.
results
in
of double collisions
It is intuitively
clear that when two
667
bodies collide,
their trajectories will suffer a change comparable to a cusp on a
curve given by a two-thirds or two-fifths power of a variable at the origin. Sundman's series gives the one-third but not any other power of the time variable. It shows that the singularity of double collisions triple collisions is irremovable.
is removable, while that of
Here we can see that Siegel's contribution [7]
in showing that triple collisions correspond to essential singularities marks a further milestone in the solution of the problem. The qualitative analysis of motions by means of Sundman's inequality is a method reminiscent of Poincar~'s arc-without-contact method or of Lyapunov's in the qualitative theory of differential equations. time of Poincar~ is to use non-integrals, about the motion,
function
The basic idea initiated at the
i.e. inequalities,
in obtaining information
instead of trying to find integrals, which are equalities.
And it
is, I think, due to the dynamic meaning of the quantity used, i.e. the moment of inertia,
that Sundman's inequality still occupies an outstanding position in the
general aspect of the problem.
Although it is not an integral,
it gives an eleventh
item of information in addition to the ten classical integrals.
One might try to
obtain a twelvth item of information by forming an inequality for the evaluation of the rate of change of an angle formed by the three particles.
Unfortunately,
the
expression for such a quantity turns out to be so complicated as to be beyond normal comprehension. If we consider the motion as a rhythmic variation of a triangle formed by the three particles, increases,
the size of the triangle will either increase indefinitely as time
or it will decrease in the case of large and positive total energy.
Birkhoff therefore made the observation that, qualitatively, we may regard the problem as solved in such a case.
In the same sense perhaps we can say that the
problem has also been solved for the case of a large negative energy with a nonvanishing angular momentum,
since now, by the theorem of connectivity,
two of the
bodies will always remain closer to each other than to the third. Let integrals,
M7 let
denote the totality of all motions permitted by the ten classical E
denote the total energy and
theorem can then be stated as follows:
c
the angular momentum.
The
668
Theorem (Tung-Easton) : For from
E < O O
and
F_or ..E ~ o, and
E ( 0
c # O, the connectivity of
to a large negative value,
M7
with
M7
c = O,
M7
is as follows.
is connected.
As
E
decreases
will change from one connected piece to
three mutually disjoint pieces; and the critical value of
E
for such changes
correspond to constant distance co-linear Lagrangian motions. The set
M7
mentioned in the theorem is not a manifold unless certain
considerations regarding double collisions are taken into account.
Easton [3-6]
has made the surgery explicit by referring Sundman's result to the sub-manifold of planar three-body motions in order to turn it from a set into a manifold. surgery is a mathematical
This
reasoning analogous to an operation which turns a set of
papers glued along certain lines, such as the binding - rim of a book, into a whole plane sheet of paper. Before applying the theorem,
let us make clear what Birkhoff's seventh problem
actually is [2,11]. A natural generalization to higher dimensions of Poincar~'s arc without contact joining singular points is Birkhoff's surface of dimension
n-i
surface of section.
in the n-dimensional
That is, to find a
flow, such that every trajectory
intersects the surface many times as the time increases or decreases.
The
correspondence between consecutive intersections defines a map of points of the surface into itself. of dimension
For a closed surface of section, the boundary will be a surface
n-2, which consists of trajectories.
The ten classical integrals define a manifold phase space [13].
M8
of states of motion in the
The equations of motion can be further reduced one order, so
that the manifold can be reduced to a manifold
M7
of dimension 7.
If the surface of section is to be used here, we have to find out such a surface
S.
Birkhoff made the observation that the sub-manifold
motions might be such a surface in connectivity of
M7
M 7.
of planar
He remarked in 1927 that not even the
had anywhere been discussed.
problem of investigating the topological
M5
Two years later, he raised the
structure of
M7; this is his seventh
problem. The structure of
M5
as given by Easton shows such a complexity that, even if
669
it can be used as a surface of section (which is unlikely,
I think), it is still not
easy for us to get from it any information concerning the motion of the three bodies. In case
M7
has three components, notions in one particular component are
characterized by the fact that one of the bodies remains relatively remote from the remaining two.
If we have an example of motion with a non-vanishing angular momentu~
and a sufficiently large negative energy as well, and if we are asked to show that some two of the bodies remain as a nearer pair throughout their motion, all that we have to do is to calculate from the data of the example to make sure that the motion corresponds to a trajectory of the flow belonging to a suitable component of
M 7.
So that the theorem of connectivity can be applied to actual astronomical problems. Let us now make a survey of two examples carried out for the Sun-Neptune-Triton system [12] respectively,
to show that the satellite in each system remains as a
satellite in the future as well as in the past. Let the three masses be denoted by
ml, m2, m 3.
The real motion as observed in astronomy will be denoted by has already definite values for the ten classical integrals. possible motions of the same masses will be denoted by
Mf.
M . r
This motion
Other ficticious They are to have the
same values for all the classical integrals except the energy. The proof will be along a line which depends on the statements: (I)
If an
Mf
is a constant-distance collinear Lagrangian motion,
can be evaluated.
Let the energies be
respectively.
Er
containing these be
M
If r
of
is such that
will have three components,
for three such motions
Er < Ei, for
i = 1,2,3,
then the
in accordance with our theorem.
M7
Let
MI2 , M23 , M31.
(II)
If, for instance,
In order to prove that coincidence of component.
m3
Mr
Mr
and
Mr is in
is in
M31 , then
m3
and
M31 , we take a particular
ml; and show that
Mr
and this
Mfo
mI
will remain closer.
Mf, say
Mfo , with
are in the same
Whether a component contains both of them depends on whether it contains
their initial values, If
Mr
El, E2, E 3
its energy
so that we need consider only their initial values.
has a syzygial position,
then we may take it as its initial position
670
and compare is further
it with
the initial
collineation
of other
Mf's,
and so the consideratiol
simplified.
Proof of (I) The value of calculation
E
and
r
is a matter
r
c
Ef
as
c
for a constant-distance is calculated
r
i i ~3 T - U = ~ I~0T - --a - -
where
I =
3 ~ miri 2 ," and i=l
a
m3ml + ~
and
w
rotating
line
for three such
containing Mf's.
Lagrangian
Mf
with
the same
O
] + mlm 2
with
= E
0 < O < I
m3(l-P)
are determined
from
+ m2
m3(l-P)
-2 m3P + ml m3P + ml
2 3 T a
is the angular velocity,
T
collinear
-2 + mI
m3P + m I
~
data or of a
from
-2 m3P
where
of the astronomical
from such data.
The value value of
c
+ m2
(ml+m2+m3)
measured
the three bodies.
And all of these
,
with a certain
time scale
Three va lue s
El,
E2, E 3
are greater
than
E . r
E. 1
T, of a are obtained
Proof of (If) A fundamental
inequality
for the positional
coordinates
of the bodies
is that
c2D2(~'~IXl ..... ~ 2 z 2 ) ~ 2(U+E) for motions
determined
homogeneous
rational
by the reduced
function
of
equations
(Xl,...,z2).
of motion,
by using new variables ~i = /~ixi .... of points
satisfying
D
It can be transformed
c2D2(~ I ,...,~2 ) ~ 2(U+E)
The projection
where
(i) on the unit sphere
is a certain into
(1)
671
~12 ~22
S5 :
+
+ ~1
2
+ ~2
2
+ ~1
2
+ ~2
2
= 1
is determined by U O2 + 2E c2D O2 ~ O
L0 : where
UO
and
DO
are values of
The connectivity Consider eollineation m2m3m I
of
the totality
and
QO
of
which has a corresponding
point in
S5
respectively.
by the connectivity
M
LO
corresponding
has a syzygial
r
of
L0
on
S 5.
to the initial
initial position
QO.
can happen only in the invariable plane, we cant set
for such initial values.
:
~I 2 + H I
It can be shown that
on
points on Suppose
~i = ~2 = O
D
is determined
of the three masses.
Since collineation
S3
M7
U
(2)
DO = 1
on
Thus for 2
+
~22
QO
+ N2
2
the sphere
= 1
S5
becomes
.
S 3, so that (2) simplifies
to
U 02 > -2Ec 2 . Let
rc2
the distance
be the distance from
m3
to
m I.
(3)
from the centre of gravity
to
m2; and let
r31
be
Take the ratio
= ~U2rc2 / ~ i r 3 1 as a parameter.
Let the value for
Now consider how the value of Writing down the expression of a function in
UO
Mr, as calculated UO
from the data, be
s O.
is affected by varying the parameter
~.
and setting all the available data in it, we get
~ : UO
This is an increasing
=
Jl + 2
(~imBml
function of
the inequality holds for such
~.
~.
+
...
) ~ F(~)
.
As its value does decrease
And the value of
~
for increasing
at infinity corresponds
~, to
Mfo' For a non-planar motion of the three masses, position exists.
it may happen that no syzygial
Such a case can be turned into a case with a syzygial position by
the following consideration.
672
Let the angle b e t w e e n the vector are in the invariable plane. initial time.
and
cm 2
be
B
This instant of time, say
Consider the set of all motions
condition that the centre of gravity O
m3m I
c
of
M
and
r
m3m I
w h e n the three masses
tl, is taken to be the Mf
with the initial
is in the invariable plane.
be the angle b e t w e e n the same vectors as above for all such motions.
existence of a syzygial p o s i t i o n means that numerical data into
U O, w e get a function of
does not decrease for monotonic v a r i a t i o n of 0
or to
~ = $ ~ O
0 = ~
to
~ .
0 = ~
means a collinear case.
@ : @
for
M
r
U 0 = G(0).
.
Let
Non-
S u b s t i t u t i n g the This function
G(0)
in a suitable direction from
So that (3) holds in such a process.
But
0 = 0
or
In conclusion let m e m a k e a remark as follows. The c a l c u l a t i o n has n o t h i n g to do w i t h the differential equations
except that
sometimes we argue w i t h the trivial fact that the phase space of a flow contains a p a t h if it contains its initial point, or conversely. reality,
For motions of a system in
the energy and angular m o m e n t u m are determined values.
calculated from the astronomical data.
But now they are
A set of data from simultaneous o b s e r v a t i o n
at an instant is quite enough, since we need only a set of initial values. o b s e r v a t i o n w i t h o u t error is most desirable. available in reality. of the same quantity,
value too small,
But such an ideal set of data is not
W e have to m a k e use of data from various sources. say the distance b e t w e e n two celestial bodies,
different time, m i g h t differ greatly. such as the mass ratio
On the other hand, m~/m@
.
come up with a new mathematical proble,
i.e.
m~/m@
Two values
observed at
some quantities may have a
We shall be helpless if we always
stick to astronomical errors in a m a t h e m a t i c a l treatment of a problem. consider the mass ratio as negligible,
An
as negligible,
If we
then we shall
the restricted p r o b l e m of three bodies.
Therefore we have used the data as if they w e r e exact values. Still another point that I'd like to call a t t e n t i o n to is concerned w i t h the d i f f e r e n c e of a m a t h e m a t i c a l model and the physical reality.
The m o o n will not
escape f r o m the earth if it is a member of an ideal three-body system; just as we may say that the earth's orbit is an ellipse if the earth is a member of an ideal two-body problem.
673
In conclusion,
I wish to express my sincere thanks to the Organizing
for inviting me to attend this notable Conference,
Committee
and to the Royal Society of Great
Britain and to the Academy of Science in the People's Republic of China for their financial
support which enabled me to do so.
References
I.
Birkhoff,
G.D.
: Dynamical
2.
Birkhoff,
G.D.
: Einige Problem der Dynamik,
Mathematikers 3.
Easton,
R.W.
Vereinigung
systems, Amer. Math. Soc. Coll. Publ. IX (1927). dahresbericht
38, Heft 1/4 (1929), i-Ii.
: Some topology of the 3-body problem,
der Deutschen
Collected works 2778-2793 J. of Diff. Equations
i0, 2 (1971). 4.
Easton, R.W.
: Some topology of n-body problems,
J. of Diff. Equations
19,
2 (1975). 5.
Easton, problem,
6.
Easton,
R.W.
: The topology of the regularized
J. of Diff. Equations R.W.
12, 2 (1972).
: Some qualitative
No 2, An international
integral surfaces of the 3-body
aspects of the 3-body flow, Dynamical
symposium
Systems,
(1976).
7.
Siegel,
C.L.
: Der Dreierstoss,
Ann. of Maths.
8.
Siegel,
C.L.
: Lectures on celestial mechanics
9.
Siegel,
C.L. and Moser,
42 (1941), 127-168. (in Germ.)
1955.
: Lectures on celestial mechanics,
Springer-Verlag,
1971. iO.
Sundman,
K.F.
: M~moire sur le probl~me des trois corps, Acta. Math.
36 (1912),
105-179. ii.
Tung, Chin-Chu
: A brief account of G D Birkhoff's
bodies, Appl. Math. 12.
Tung, Chin-Chu
Tung, Chin-Chu body problem,
(Engl.
ed. i, No 2, Dec 1980).
: The motion of Triton as a satellite of Neptune,
Beijing Astron. 13.
& Mech.
problem in the problem of 3
Observatory,
Publ. of the
No 15 (1978).
: Some properties Seientia Sinica,
of the classical
17, No 3 (1974).
integrals
of the general
3-
674
14.
Tung, Chin-Chu : A property of the motion of the moon, Publ. of the Beijing Astron. Observatory, No I (1979).
15.
Whittaker, E.T. : A treatise on the analytical dynamics of particles and rigid bodies, Cambridge University Press, 4th ed., 1937.
Perturbations
of Periodic
Boundary
Lawrence
Section i:
Liapunov-Schmidt;
Conditions
Turyn
Linear Perturbations
We consider boundary value problems w h i c h are perturbations boundary value p r o b l e m with periodic interested
boundary
conditions.
of a linear
Specifically,
we are
in
TX
defn =
-
X"
+ q(.)x
=
Xr(.)x
+ fo(.,~,%,x,x')
(1.1)
Mx
=
= ~L(~,x)
x'(O)
where L(e,.)
and classical
linear perturbations, tage of discussing We assume
In Turyn
perturbation
+ (f2(~:,)~,x,x')J
is linear and fi is non-linear
the method of Liapunov-Schmidt. discriminant
- x'(1)J
in x, x', i = 0,1,2.
We will apply
[8] we also were able to use Hill's
theory of Rellich
i.e. fi z O, i = 0,1,2,
to achieve
but Liapunov-Schmidt
both linear and non-linear
results on
has the advan-
perturbations.
that %0 is a double eigenvalue
of the linearisation
Tx = %r(.)x (i.2)
MX = 0 , i.e. there are two linearly value p r o b l e m
(i. 2) .
independent
solutions xl, x 2 of the periodic
Without loss of generality x I, x 2 are normalised
For the linear perturbation
x1(0)
= i,
~(o)
=
o,
X 2 (0)
= 0
x~(O)
=
1.
we take
1 [ x(t)] (i)
L(~,x)
= 7~
+
ix'(1))
Io w(t)
+ o(1~1), [x' (#)3
boundary
to satisfy
676
where H is a constant ~2×2 valued
2x2 real matrix,
function.
In Turyn
i.e. H (iR 2x2, and W(-)
[8] W z 0 although
is a continuous
we did allow distributed
terms in the non-linearities. Define Banach
llxll = Ixl. +
spaces
Y = C2[0,I],
Ix'l. + Ix"l.,
IxI~ = SUpo~t~iIx(t) I. (1.4)
Z = C[0,1] xiR 2 with norms
ll(v;c,d)ll = Ivl~ + Tcl + Idl, respectively,
Then
(i.i) is equivalent
to the problem
(B-.A-~a)x = O(~,XO+.,x) + 0((I'I + l~l)211xll)
where v = %-hO' x ~ Y, Bx = (TX-hOr(.)x;Mx),
cx=
0;H
+
w(t)
G(e,h,x) tinuously
,
£=
[x'(t)j
= (fo(.,c,~,x,x'); Fr&chet
Ax = (r(.)x;O)
, o=
~x'(1))
o((I~I
where
f2
f(e,~,x,x')).
differentiable
, and
We assume
and furthermore
that G(-,.,-)
satisfies
is twice con-
G(S,hO+V,x)
=
+ Ivl)211xJl + JlxIl2). Denote Y0 = N(B),
Z 1 = R(B).
By assumption
YO = span{xl'x2}"
One can
show that
R(B) = {(v;c,d):
~j(v;c,d)
1 where
1
f~l(V;c,d) = - d + iVXl, u
Let Cl = -
1 4XlX2
=-
d2,
= 0 for j = 1,2}
f~2(v;c,d) = c + vLvx2 1 2 Xj,
aj= ~
are linear
and zl = aTl(xl;Cl,0),
functionals
on Z
z2 = a21(x2;O,d2).
Then
Q: Z ~ R(B): is a projection.
Define another
P: Y
and corresponding The method
÷
z~z
projection
Y0: x ~ al -I
subspace
- ~l(z)zl - £2(z)z2
(41XXl
-i I )xl + a2 (4 xx2)x 2
Yl = (I-P)Y"
of Liapunov-Schmidt
consists
of replacing
(1.4) by the system
677
(1.5)
Q(B-~A-~C) ( P ~ ( Z - P ) x )
(1.6)
(I-Q)(B-vA-sC)(P~(I-P)x)
= QG(~,XO+V,x) + O((Is I+l~l )211xll) = (I-Q)G(e,IO+~,x)
Rewrite Px = UlX 1 + u2x 2 for real Ul, u 2.
+
o((l~l+l~l)2rl=ll).
Since Bx i = 0 for i = 1,2 and
QB: Yl ÷ R(B) has a bounded inverse, the "auxiliary equation" (1.5) can be solved by (I-P)x = w*(g,w,~) for all sufficiently small I~I, lwl, I~I, where ~ = (Ul,U2),
I~1 = lull + lu21.
Furthermore, w* = o ( ( I c r + l w l ) l ~ ]
+ I~12) as IEI, IvI,
I~1 ÷
o.
Substitute into (1.6) to arrive at the "bifurcation equation"
(I-Q)(B-vA-eC)(UlXl+U2X2+W*)
- (I-Q)G(e,%O+V,UlXl+U2X2+W *)
(1.7)
= o((1~1+1~1)21~[ R
standing for "remainder".
deJn R,
+ (l~l+lvl)l~13)
Note (I-Q)B ~ O.
Since Zl, z 2 are linearly indepen-
dent, (1.7) separates into the system
(i.8)
IrE - s(JH + /oJW(t)X(t))]u + g(e,~,u)
= R
where X ( t ) =
(x~ 1 :~] is the principal fundamental matrix for the linear o.d.e.
system
= x'
q-~o r
x'
, o = (
g(~,~,~) - 0((l~l+Ivl)2]~l÷ I~12), and
~ij)i,j=l,2' H,W(.)
°ij
=
rxixj, J =
-i
0
,
are as in (1.3).
We remark that W(-) need not be continuous for all of this to make sense. One could include finite jumps at interior points, leading to perturbations of the boundary conditions such as
[x'(o) x'(1)]
L [x'(i))
z
+ o(I~l 2) co
as long as [
IHil < ~. i=0
i=l
[x'(ti)]
[llx,(t)jj
678 For linear p e r t u r b a t i o n s of (i.i), i.e. fi E 0 for i = 0,1,2,
(1.8) can be
rewritten as (1.9)
0 = o~ 2 + yev + ~a 2 + ...
where
... represents terms of degree three or higher.
Always o = det E.
Denote
1
= H + ~W(t)X(t)
= (~ij)i,j=l,2.
Then
= det ~, y = - ~iio12 + ~22o12 + ~12Oli - ~21o22
•
E is always symmetric; upon a s s u m i n g r(.) ~ r 0 > 0 it is also p o s i t i v e definite.
T h e o r e m i.i:
For linear perturbations,
i.e. f i ~
0 for i = 0,1,2, a sufficient
condition for local s p l i t t i n g of the double eigenvalue h 0 into two C 1 curves p a r a m e t r i s e d by ~ is y2 _ 4~0 > 0.
R e m a r k 1.2:
y2
_ 4~
Proof
A s s u m e fi E 0 for i = 0,1,2 and r(-) > r 0 > 0.
If tr ~ = 0 then
~ 0.
(of the remark):
implies J ~ symmetric,
Z is p o s i t i v e definite and symmetric. the solutions B 0 of 0 = det(Z-BJg)
Since tr ~ = 0
are all real. []
For W = 0 the b o u n d a r y conditions are self-adjoint iff 1 = det(l+sH+O(Isl2)), from C o d d i n g t o n & L e v i n s o n adjointness.
[3, p. 297].
Thus tr H = 0 is n e c e s s a r y for self-
For the m o r e general b o u n d a r y conditions w i t h W(')
of s e l f - a d j o i n t n e s s
~ 0, any concept
should imply that e i g e n v a l u e s h are real; if tr ~ # 0 then
for some p o s i t i v e r(.) the solutions 60 of det(Z-BJ~) will be neither real nor zero, leading to ~0 s p l i t t i n g into curves w h i c h are not real.
Thus, for the more
general b o u n d a r y c o n d i t i o n s w i t h W(.) ~ 0 a n e c e s s a r y c o n d i t i o n for selfa d j o i n t n e s s w i l l be tr ~ = O.
Example 1.3: ger
n ,
W h e n r = i, q = 0, tr ~ = 0, ~ ~ 0 and h 0 = 4~2n 2 for some inte-
1 -i 2 -i 2 w e calculate o12 = 0 and X 2 - 480 = ~((Cal2H-10 ~21 ) +)tO ~11 ) > 0, so
T h e o r e m i.i is applicable.
679
Lu use o f t h e H i l l ' s
discriminant
1 = (1~ ± 2 a r c s i n ( ¢ / 2 ) ) for
2.
(see T u r y n
1.4:
= 0, ( = 1. resistance".
If
(see e.g. 8zi{ik
an~-~ay f o r
Consider a ring the joining
Appropriate
k/h
transfer
boundary
See Holman
conditions
h is large.
drop across
conditions
= Oli > 0, Theorem In fact,
-
u'(1)
=
o
conditions
this phenomenon
being well-known
on the causes
are self-adjoint
for all ~, with H = (~
the existence
Nonlinear
to the nonlinear
if the heat
have as a consequence
of contact
of two curves
for the example r E i, q ~ 0 one can see that v+(e)
2:
u(~) are then
u(1) = cu'(1)
for more details
i.i guarantees
in practise. resistance.
~].
Since
I = i 0 + ~(s).
= 0 for all e, since
and further one can calculate
that D
e
v*(0)
allowed system
(2.1)
us to reduce (1.8).
0
the nonlinear
Re-write
boundary
value problem
the latter as
(~L~ + ~2L2)~ + £(~) + ~(~,~) = 0 ~ 2 ,
where ~ = (~i,~2),
I~l = I~i I +
R(~,~)
+
= 0(I~121~I
I~13).
~(~)
I~2 I , L 1 = Z, L 2 = ~, ~i = ~' a2 = e, and is assumed
to he a homogeneous
quadratic,
fically
(2.2)
= -21
Perturbations
Liapunov-Schmidt (i.i)
for the temperature
These boundary
the join,
the second row of H is trivial,
Section
be some " c o n t a c t
= (Biot number) -I can be taken to be small and positive
[4, pp. 45-48]
These boundary
then there will
the endpoints
[6, p. 283])
coefficient
a temperature
do n o t r e m a i n s e l f - a d j o i n t
I£1 ~ 2.
is net perfect
u'(o)
e =
curves
o f m e t a l o b t a i n e d by j o i n i n g
u(O) -
where
gives explicit
These b o u n d a r y c o n d i t i o n s
c # 0, b u t t h e c u r v e s e x i s t
Application
[8])
~(~)
=
2 2 ,r (q2oUl+2qllUlU2+qo2Z{2 02 u2+2rl 1
IUlU2 +r2
0u~)"
speci-
680
~I = ~2~i "
Then (2.1) is equivalent to
(2.3)
~(~,~l,a2)
de~n (BILI+L2)~ + ~(w) - 0(I~2[) = 4"
Examination of (2.3) will produce bifurcation curves in the a-plane for the unscaled system (2.1). We remark that using (H2) in practise relies on distinguishing between L 1 and L2:
Since LI~0 + ~(~0) = O always has at least the solution ~0 = 4, (H2)
requires that the 2×2 matrix L 1 be invertible. The bifurcation curves will be approximately
straight lines al = B~a2
where 8 I0 is such that there is a ~w0 # ~0 satisfying { ~0(~,8~ ) dean ($~LI+L2)~ + ~(~) = (2.4) deft= A(~,B~)
~0 det L~-~ (~,B~
= 0 .
0 To ensure that such B I do give rise to bifurcation
curves one must assume some
further generic hypotheses. Mallet-Paret
[5] and Chow, Hale, and Mallet-Paret
[2] discuss a problem
similar to (2.1), and Chow and Hale [i, Chapter 7, Section 4] discuss generality. 1.4:
(2.1) in
Our view will now be narrowed to the matrices LI, L 2 from Application
Take L 1 =
~
, L2
.
For this specific choice of LI, L 2 we will
denote the corresponding nonlinear system by (2.1'), so as to distinguish the general system (2.1).
it from
A prime after an equation number will always indicate
this specific choice of LI, L 2.
Corresponding
to system (2.4) is
0 = h~(w,Bl) :
Blwl+q2Ow~+2qllwlw2+qO2w~
0 = h~(W,~l) =
XW2+~o2wl+2rllWlW2+D20w 2
0 = A(w,B I)
BiX+2(xq20+Blrll)W 1 +
(2.4') =
2
+
2
2
+ 2(Xqll+Blr20)w2 4d21Wl+4d22WlW2
+
2 4d12w 2
where × = 2 + ~71si
(2.5)
d21 =
q20rll_qllr02,
d22 =
q20r20-qo2r02 ~
dl 2 =
qllr20-qo2rll"
681
Note that X depends on Bl; given 6~, we denote X~ = 2 + l~iB~ . The immediate task is to find the solutions (~0,$~) of (2.4').
Remark 2.2:
(a)
If q02 ~ 0 # r02 then the only solutions(~0,6~) of (2.4')
satisfying w~ = 0 or w~ = 0 are (2.6)
(£,0), (b)
(£,-210).
The only solutions (~0,6~) satisfying (2.4') and w~ # 0 # w~
are those for which w 0 satisfies
0 = 2WlW2+rO2w~+(2r~-~lq )w~w 2+(r20-21~lql i )WlW ~ i~ 0 20 (2.7)
4+
-x-lqo w~ 0 2 3 +
2 2
0 = q2oro2Wl 4qllrO2WlW2 (-q2or20+4qllrll+3qo2ro2)wlw2 +4q
r w w3+q r w ~
02 ii 1 2
02 20 2
with correspondingly
B~
(2.8)
The p r o o f s calculations, Remarks tence
2.2
as
of
= -
these
found
- 2.4
of bifurcation
(w~)-1 ( q 2 0 ( W lo) 2+2 q l l W l0wo+ w0 2 2 q02 ( 2 ) )
in Turyn
and other curves.
For any solution
results
are [8].
relatively Lemma 2 . 1 ,
calculations The r e s u l t s
(w0,8~)
of
(2.4')
simple, the
below will are it
but
discussion all
summarised is
"
important
serve
sometimes which
tedious, followed
to prove
the
it,
exis-
in Theorem 2.5. t o know t h e
Jacobian
matrix a (h0,A) ~(w 0 ,B~) defn
~(GBI) (~0,~)
because we will use solutions (~0,6~) of (2.4') to obtain the bifurcation curves ~i = 6 ~ i + °(la21) for the original system (2.1'). We calculate ~(w,B I)_~_ =
682
We call ~ the "solution"
and ~ the "parameters".
which values of ~ there are non-trivial solutions changes as ~ is varied.
solutions ~ # O and how the number of
Scaling and "generic bifurcation"
the techniques we will use to address Scaling will effectively
Our goal is to find out for
methods are
these questions.
reduce the number of parameters
from two to one.
To do so one makes generic hypotheses which insure that no qualitative are overlooked
features
in the process.
Lemma 2.1 (Scaling):
Assume
(HI)
Q(~) = O implies ~ = O.
Then there is a neighbourhood
of (~,~) = (O,O)
E/~% and a constant c in which all
solutions of (4.1) satisfy
Proof:
See Chow, Hale and Mallet-Paret
result,
o
If ~ = ~I~' ~2 = ~IB2 then solving
[2, p. 222] for the proof of a similar
(2.1) in a neighbourhood
of (0,0) is equivalent
to solving
(L~ + B2~2) £ for small e.
+ £(£1 = o(l~I)
We assume
(H2)
LI~0 + Q(£0) = 0 implies det
i + ~
(£0
# 0.
Since (H2) holds, every solution of LI~0 + ~(~0) = O corresponds (2.1) which remains simple for IB21 , I~11 sufficiently We will say that ~0 is a "bifurcation
to a solution of
small.
point" if the number of solutions
of (2.1) is not locally constant as a function of e near s 0.
A curve of bifurca-
tion points will be called a "bifurcation
(H2) assures that
there are no bifurcation with a neighbourhood
curve".
curves in the intersection
of the origin in/? 2.
Hypothesis
of a double cone {I~21 ~ I ~ i I }
As for what remains in a neighbourhood
of O in the ~-plane we introduce a different
scaling for (2.1):
Let ~ = e2~'
683
8l+2q2oWi+2qilw2 I 2qllWl+2qO2w2
[ E 2ro2wl+2rllW2 2(Xq20+81rll )
I
WI
J
I x+2r I IWI+2r20W2
h~im2
II 2(Xqll+61r20)+
2+21~181 +
+8d21Wl+4d22w2 I +4d22Wl+8dt2w2 I
+2(x~lq20+rll)Wl +2(t~lq11+r20)w2
When (~0,B~) is a solution of (2.4') for which det {(~0,B?) ~
0
the
Implicit Function Theorem implies the existence of a local curve 81 = 6~ + o(i), = ~0 + o(i) of solutions of (2.3') for la21 small, with corresponding curve of solutions al = B ~ 2
+ °(la21)' ~ = a2~0 + °(la21) of (2.i') for le21 small.
Furthermore, al ~ B]~20 is in fact a bifurcation curve since in a neighbourhood oJ = 0 it separates regions between which the number of solutions u changes by at least two.
The proof of this can be found in Chow and Hale [i, Chapter 7,
Section 4]. It is interesting when (w0,8~)~ solves (2.4') but det ~(w0,6~)~ = O. fact, the two solutions (2.6) both have det }(~0,B~) = O.
In
But, by analysing
higher-order terms one can establish the existence of bifurcation curves 0 + 0([~212) for (2.1') when 810 = 0 and ~i0 _210, upon assuming some ~i = ~ia2 generic hypotheses.
Those values of B 1 are exactly the "simple eigenvalues" for
(L2;LI), as exposed by Mallet-Paret
Remark 2.3:
[5, p. 390] and Chow and Hale [i].
There are solutions of (2.4') given by
Z = -m (~1-8~)S ° + 0(I~21 + (r~21+r~l~l) 2) 0 for all la21,
18]-6~I
(a)
m 0 = r20
(b)
m0
= q20
suffieiently small, where
'
~
,
0 '
~
'
,
assuming r20 # 0 assuming q20 # 0.
684
Moreover, ~3
those solutions are unique in some neighbourhood
for each small, fixed ~2 # 0.
of ~ = ~, B I = B~ in
Hence, upon assuming r20 # 0 # q20 there are
bifurcation curves a I = B~a 2 + 0(I~212)
for (2.1'), since on such a curve there
is one less solution ~ for (2.1') than for ~ nearby, but not on, the curve. Define I c (2.9)
= q20r02 , c 3 = 4qllr02 , e 2 = -q20r2~4qllrll+3qo2r02 , o I = 4qo2rll , c O = qo2r20,
b 3 = r02, b 2 = 2rll-%olq20, b I = r20-2%~lqll , b 0 = -%~lqo 2.
Remark 2.4:
Assume that q20 # 0 # r20 and q02 # 0 # r02 both hold.
can be at most four solutions ~0 of (2.4') satisfying w~ @ 0 # w~:
Then there W~ = ~
where
m ( {m: c~m4+o3mB+o2m2+clm+o 0 = O, m # 0 # b3m3+b2m2+blr~-~ 0 } and
W~ = -2m(bsm3+D2m2+blm+bo). We can now state the main result of this section.
Theorem 2.5:
For the system (2.1') let us assume
(HI), (H2), and two other
hypotheses:
(H3)
If (~O,B~) satisfies
(H4)
(2.7 - 2.8) then det ~(wO,B~) # 0
q20 # 0 # r20
and
q02 ~ 0 # r02 .
Then there are only a finite number of local bifurcation curves in the ~-plane and they are of the form ~I = B ~ 2 The curves corresponding
+ °(la21) where B~ is given in (2.6) and (2.8).
to B~ given in (2.8) separate a neighbourhood
of the
origin in the ~-plane into regions between which the number of solutions ~ of (2.1') changes by at least two; if the B~ are distinct then "at least two" can be replaced by "exactly two".
The curves corresponding to B~ in (2.6), i.e. 8~ = 0
and -2%0, separate regions with the same number of solutions, but since the number of solutions is one less on the curve itself the number of solutions is not locally constant.
685
The simplicity of the eigenvaluesof of the results of Chow and Hale
(-J~;E)
is required
[i] for the general problem
for the application
(2.1).
concept of simple eigenvalue was motivated by considerations
The general
of transversality;
in our particular problem we recall that y2 _ 460 > 0 implies that the linear perturbation
problem has locally two distinct
through h 0 = 0, s = 0.
when the boundary conditions [7] discusses
curves of eigenvalues In Turyn eigenvalues,
available
are self-adjoint
curves will usually be the case
for e # 0.
(2.1) when the linearisation and the nonlinear
and some numerical
has locally two distinct
problem has symmetry or symmetries. some further remarks on simple
examples.
I would like to thank Professors
Chow and Hale for making
to me portions of their book, which will be published
like to thank Professors Professor
of two such distinct
[8] one finds more of the proofs,
Acknowledgements:
Rostamian
passing
From Remark 1.2 we know that y2 _ 4~q ~ 0 when tr H = 0,
so that the local existence
Shearer
curves of eigenvalues
Binding,
Browne,
soon.
I would
and Hale for helpful discussions
for asking about distributed
and
terms in linear perturbation.
Bibliography: [i]
S.-N. Chow and J.K. Hale, Methods of Bifurcation Berlin and New York (to appear).
Theory,
Springer-Verlag,
[2]
S.-N. Chow, J.K. Hale, and J. Mallet-Paret, Applications of generic bifurcation. II, Arch. Rational Mech. Anal., 62 (1976), pp. 209-235.
[3]
E.A. Coddington and N. Levinson, McGraw-Hill, New York, 1955.
[4]
J.P. Holman, Heat Transfer,
[5]
J. Mallet-Paret, Buckling of cylindrical Appl. Math., 35 (1977), pp. 383-400.
[6]
M.N. Ozi§ik, Boundary Value Problems of Heat Conduction, book Company, Scranton, Pennsylvania (1968).
[7]
M. Shearer, Secondary bifurcation near a double eigenvalue, Anal., ii (1980), pp. 365-389.
[8]
L. Turyn, Perturbation of periodic boundary submitted December, 1981.
Theory of Ordinary Differential
McGraw-Hill,
Equations,
New York, 1976 (fourth edition). shells with small curvature,
conditions,
International
Quart.
Text-
S.I.A.M. Math.
S.I.A.M. Math. Anal..
BIFURCATION
PROBLEMS
A.
WITH
TRIANGULAR
SYMMETRY
VANDERBAUWHEDE
]. I N T R O D U C T I O N In this blems
contribution
having
of such
the
we w a n t
symmetry
a problem
is the
Av + ~ f ( z , v , ~ )
to d i s c u s s
generic
of an e q u i l a t e r a l following
= 0
triangle.
nonlinear
,
z • ~
,
z •
bifurcation
for pro-
A prototype
eigenvalue
problem
:
,
(i) v(z) where
= 0
we make
the
following
(i) ~ c m 2 is open, equilateral is the •
6-element
LOR 2) and
(ii) (iii)
C);
a • ~
and
f
: ~xRxA
f(yz,v,~)
Now tically
fix some zero.
of 0(2)
given
is s u f f i c i e n t l y ,
X 0 • A; we w i l l
For e a c h
~ near
symmetry
each
of an
y • A3, w h e r e
by
the
Vz • m 2
identify ~2
Banach
operators
-, with
(2) the
complex
space;
smooth,
f(0,X)
= 0 for all
~, and
Vy • A 3
assume
A3
: ,
we
for
the
generated
by
Tz = z
an a r b i t r a r y
= f(z,v,X)
and has
~ = y(~),
convenience,
X • A, + ~
LOR 2) ,
(for n o t a t i o n a l plane
i.e.
,
:
bounded,
subgroup
T •
exp(~i)z
~z
assumptions
smooth,
triangle,
~
that
~0 we w a n t
(3)
Dvf(Z,0,X0)
to s t u d y
is not
iden-
the b i f u r c a t i o n
of
687 nontrivial
solutions
(1) from the line (v,e) = (0,e) of trivial
of
solutions. Bifurcation linearized
can only take p l a c e
problem
at those
G-values
for w h i c h
the
:
Au + e D v f ( Z , 0 , 0 ) u
= 0
,
z
E
~
,
z
E
~
(4)
u(z)
=
0
has n o n t r i v i a l a countable
solutions.
set of e i g e n v a l u e s
the c o r r e s p o n d i n g symmetry values.
We k n o w
{~j I J e ~ } ,
eigenspaces
some of these
from s t a n d a r d with
eigenvalues
may generically
2 we will
use g r o u p
fy the e i g e n v a l u e s
according
to the s y m m e t r y
eigenfunctions.
Crandall-Rabinowitz [I] to o b t a i n in s e c t i o n
In s e c t i o n
theorem
the b i f u r c a t i o n
to classi
of the corres-
a generalization
from a simple
We discuss
of the
eigenvalue
the s o l u t i o n
set
PROBLEM
Let ~ e ] 0,1[ , W = C0'~(~) 3Q}.
theory
eigen-
4.
2. THE L I N E A R
Vze
equations.
of the
be m u l t i p l e
properties
3 we a p p l y
(4) has
Because
representation
for b i f u r c a t i o n
that
lejl ÷ ~ as j ÷ ~;
are f i n i t e - d i m e n s i o n a l .
In s e c t i o n
ponding
theory
Define M
and V = C 02,~ (~)
{ v e C 2 '~(~) I v(z) = 0,
: V x A x N ÷ W by M ( v , X , e ) ( z )
Vz e ~, and let L(e)
= DvM(0,0,e),
Ve e ~ .
= Av(z) + e f ( z , v ( z ) , X ) ,
Then p r o b l e m
(I) takes
the form
M(v,X,e) while N(L(e))
the l i n e a r p r o b l e m
(4) c o r r e s p o n d s
to the d e t e r m i n a t i o n
of
.
We can d e f i n e the space W,
a representation
as f o l l o w s
(r(y)w)(z)
Then
(s)
= 0
representation,
of the group
A 3 over
:
= w(y-Iz)
the m a p p i n g s
F : A 3 ÷ L(W)
M and L(g)
i.e. we h a v e
,
Vy
e
A3
.
are e q u i v a r i a n t :
(6)
with
respect
to this
688
M(F(y)v,X,a)
= F(y)M(v,X,a)
,
VT ~ A 3
(7)
,
Vy ~ a 3 •
(8)
and L(~)r(y) It
is
clear
operators
be
from
F(y),
N(L(oj)) will
= r(y)L(~)
will
have
aj
is
that
the
L(U)
an
wing
three
U be
(I)
dim U = I and
(2)
dim
(3)
dim
symmetry
for
the
= u
F(y)u
r(y)0(z)
±1;
=
U has
~(x+iy)
~(~z)
u 0 ~ 0 be
bility or
invariant
gives
us
that the
U.
E A
3
domain
maintained
Then
, Vue
U
cases
to a or
will eigen-
and one
r
of
: 43 +
the
follo.
;
is
it
~
the
A3
)u0, is
e
that
c
F(T)
To~
=
that
the
F(T).
Since
U = s p a n { ~ 0)
or u0
= 0.
(2),
depending
will
have
s = ±1.
~-1 or
and
and
define
(10)
F ( T ) u 0 = su 0, since
if we
.
on U,
immediate
r(6)
(9)
:
identity
that
;
such
, Vz
under
(I)
{ u l , u 2}
v¥
st(-1
either
¥u ~ U
,
then
such
then
= u0 + F ( 6 ) u 0 + r ( 6 2 ) u 0 ;
it f o l l o w s
is
the
vectorspace,
= xu I + Y U 2 , ,
u0
on U,
, Vy
a basis
=
of U is
of
happen
:
F(z) 2 = F(T 2) let
should
corresponding
4 3 over
= -u
r(~)r(¢)Uo = r(6-1)r(~)Uo s p a n { u 0}
of
= u
r(~)u
and
: G ÷ U by
eigenvalues
real
N(L(aj))
:
,
U = 2,
Since
which
symmetry
i.e.
N(L(aj))
triangular
occurs
U = I, a n d
P r o o f.
If
the
generically
action,
perturbation
representation
possibilities
r(~)u
group
a finite-dimensional
irreducible
under then
small
a. in several eigenvalues J spaces are irreducible. Let
this
subspace.
split
Lemma 1.
invariant
an eigenvalue,
an arbitrary
such
is
under
invariant
then f
If
N(L(a))
irreducible
no proper
function
that
y E ff 3. be
reducible,
the
(8)
= 62o~.
Let
subspace
F is The
Then
irreducible
first
on w h e t h e r
possi-
c = +I
c = -I. Now
suppose
u0
= 0.
Let
v0 =
(¢7)-1(r(8)u0-r(sZ)u0).
Then
r(T)u 0
689
= eu 0, r(T)v 0 = -¢v0,
This
and
I ¢~ r(~)u 0 =-~u 0 + ~-v 0
,
shows
that
space
of
follows
that
U,
invariant
that
if ~ = +I, Under
v 0 ~ 0 and under
/J r(6)v 0 =-~-u 0
£.
From the
U = span{u0,v0}. and u I = -v0,
the h y p o t h e s i s
an i r r e d u c i b l e eigenvalues
To o b t a i n
three
can use
disjoint
sub-
hypothesis
we t a k e
u 1 = u0,
~ = -I.
u2 = v0
[]
I j ~N}, { o(i) j
where
for each
i = 1,2,3 the e i g e n s p a c e the s y m m e t r y 3. THE Since
group
BIFURCATION L(o~ i))
Schmidt
can be done can a pply
the f o l l o w i n g
pectively
W.
(k0,a0)
(iii)
theorem
Let M
way
= 0 for all
([3],[4 ]) w h i c h
of C r a n d a l l
of M this
More p r e c i s e l y
we
is a g e n e r a l i z a t i o n
and R a b i n o w i t z
F : G + L(V)
(X,o),
spaces, and ~
and w h i c h
= T(~)M(v,~,o)
~ Ax~ be
is
such is
irreducible
DoDvM(0,X0,~0).u
Then all
(see [4]).
: V x A x N ÷ W be a m a p p i n g
L = DvM(0,~O,a0) N(L)
of the e q u i v a r i a n c e
[I] on b i f u r c a t i o n
and G a c o m p a c t
: G ÷ L(W)
of class
C r+2
is e q u i v a r i a n t
over V,
res-
(r>0)
such
with
respect
:
M(r(y)v,~,~)
(ii)
theorem
representations
(G,r,~)
(i)
(5). B e c a u s e
the L i a p u n o v -
eigenvalues.
that M ( 0 , X , o )
Let
we can apply
2. Let V, W and A be real B a n a c h
group with
to
operator,
in a s y m m e t r y - c o m p a t i b l e
from simple
under
EQUATION
on e q u a t i o n
of the c l a s s i c a l
Theorem
U! i) c o r r e s p o n d i n g to o!i) t r a n s f o r m s J J k 3 a c c o r d i n g to case (i) of lemma I.
is a F r e d h o l m
method
it
o- of (4) c o r r e s p o n d s to J I te d i v i d e the set of
lemma
classes
a two-dimensional
irreducibility
that each e i g e n v a l u e we
(11)
0} i s
(10)
u 2 = u 0 in case
eigenspace,
into
span(u0,v
I ~v 0 .
solutions
,
that
under
of
(5)
~
G .
(t2)
:
a Fredholm
¢ R(L)
vy
the for
in
operator
with
representation
index
zero
;
F ;
some u e N ( L ) .
a neighbourhood
of
(0,~0,~ O)
belong
69O
to a cr+2-submanifold M
where
=
M of the form
((~(u,x,~),x,~)
I (u,X,~)E~}
9 is a symmetry-invariant
and P~(u,Z,a) Moreover, an equation
= u, where
=
0
r(0,~,~)
(b)
DuF(0,X0,~0)
(c)
F is
,
for
(c)
above are
v(x,~)
with
and F defined
Also,
respect
sufficient.
by (6)).
~!i)
of studying
~ 0),
(G,F) ,
to
(iii)
:
(4)',
2 are As f o r
[]
definition
which
then
follows
the properties
problem
(a)-
(1) we r e p l a c e
and the hypotheses
satisfied;
the hypothesis
(14)
o f t h e m a p p i n g F,
L = L(@ i))
immediately
(we t a k e
(iii),
G = A3
we h a v e
then =
and
Dvf(Z,o,xo)u(z)
=
-(o~i))-lAu(z)
,
:
=I also
properties
;
Vy e G .
an e x p l i c i t
~!i)j of
of theorem
I Consequently
(5) is equivalent
= I on N(L)
For our particular
(D~DvM(O,Xo,o~i))u)(z) that
in V onto N(L).
and has the following
to
discussion
if u e N(L(c~i))),-
(remark
= I on N(L),
;
to give
the qualitative
and (ii)
C r+2,
-- r ( 7 ) F ( u , X , e )
~0 b y a n y e i g e n v a l u e (i)
projection
M the equation
= 0 , D DuF(0,X0,~0)
possible
but
Du~(0,~0,a0)
@(F(y)u,X,a)
(13)
equivariant
also
(i.e.
:
is of class
= 0
F(r(¥)u,X,c) is
(X,o),
in
,
F : C + N(L)
(a)
It
= 0 for all
P is an equivariant
of the form
of (0,~0,a0)
C r+2 and equivariant
on the submanifold
F(u,X,(~)
where
O(0,X,o)
,
neighbourhood
N(L) xA×]R, @ : ~ ÷ V is of class F(y)@(u,X,a)),
:
0 is
the bifurcation
satisfied, equation
and our problem (13)
for
(u,X,e)
reduces near
to that
(0,X0,e~i)) J
in N(L(~i)))xAxIR;"
in the
next
section
we a n a l y z e
how t h e
solution
set
:
691 of (13) depends on the value of i ~ {1,2,3}. 4.
THE S O L U T I O N
When
i
= 1
SET
then
N(L(o(1))) j
,
from the properties
F(p-u(1),~j ,o) where F I ( O , X o , O equation tion
1)]
for
J
some
u (1) J
= 0.
It
foltows
(a) and (b) of F that
= oF l ( O , X , o )
,
(16 3
( i ) ] # O. C o n s e q u e n t l y ,
I)i
Fl(P,Z,{
= span{u~
= 0 and D F I ( O , X o , d j
] = 0 can be s o l v e d
for
the
~, and the b i f u r c a t i o n
has for each Z near ~0 a unique branch of nontrivial
equa-
solutions,
given by
{(pu~l),d*(P,X))
I IPl < Po } ,
(17)
where o* is of class C r+l and d*(0,X0)
= ~I).
Moreover,
since
F(T)U! 1) = u! I) Vy ~ A3, all solutions on the corresponding solution J 3 ' branch of (I) will have the full &3-symmetry of the original problem. We conclude
that near each eigenvalue
~!I) of (4) there bifurcates, J for each X near X0' a branch of symmetry invariant solutions of (I). Also in the case i = 2 the bifurcation of nontrivial
solutions
of the form (17)
equation will have a branch (just replace u!1)J by u~2)).
However,
since F(T)u!2J"" = -u! 2), we will have o* (-p,~) = o* (p,X) (i.e. ] J the bifurcating branch is' symmetric in p), and F(T)@(pu~2),~,o*(p,X))" J =
~(-pU~2),~,d*(p,~)).-
(4) there bifurcates, trivial solutions
We conclude
that near each eigenvalue
o~2)- of
for each X near ~0' a symmetric branch of non-
of (1), which are invariant under F(6), and where
the two symmetric parts of the branch are obtained one from the other by application Finally,
of F(T). come to the case i = 3. We have dim N(L(o~3)))
we
= 2,
and we can use the isomorphism ~ given by lemma 1 to identify XU 1
+ y u 2 e N(L(o~3)))
can write
with the complex number z = x+iy e C. Then we
the bifurcation
F(z,~,o)
=
0
,
equation
(13) in the form (18)
692
where
F
: CxAxR
space),
÷ ~
F(0,X,~)
F(yz,X,~) Lemma
3. Let
to A 3. T h e n such
(i)
F(z)
(ii) The
g(yz)
meters,
Therefore
where
also
that
G(z)
=
This
implies
where
h
= Im ~F(z);
G](6z)
=
application
= h(z)
Im ~3
and e q u i v a r i a n t g
,
: C ÷ ~
consequently
C ~+3.
From
and
smoothly
on para-
parameters. and
G(~)
= -G(z).
,
G(6z)
= G(z)
consequently
same
(21)
:
(2i)-1(z-~)G2(z), the
: C +
, Vy e A 3
= G(z)
Vz e ~
respect
(20)
on these
G(Bz)
,
of
and h
C ~', if F d e p e n d s
then
with
;
Vz e C
smoothly
and
l(z)
is of class
that
: ~ ÷ ~
vector-
(19)
Vz ~ ~
= h(z)
(2i)-1(6z-Bz)G](6z),
Another G(z)
,
of class
= 0 if z = z,
GI : ~ + C
C z+2.
, h(yz)
= (2i) -I(z-E)G
C z+4
functions
g and h d e p e n d
G(z)
G(z)
G(z)
unique
+ h(z)z 2
= g(z)
Let
C as a r e a l
(we c o n s i d e r
Vy ~ A 3 .
of class
g and h are
then
P r o o f.
exist
= g(z)z
functions
,
: C ÷ ~ be
there
:
C r+2
= 0 and = yF(z,X,~)
F
that
is of class
G](6z)
where
argument
it then
= 0 if z = z.
G2(z)
gives
us
follows
is of class finally
:
,
(22)
is of c l a s s
C ~+I
and
clearly
satisfies
h(yz)
= h(z)
Vy ~ A 3. Let n o w F(z) which
= F(z) - h ( z ) ~ 2", then
immediately
C ~. The of F.
invariance
This
Using in the
proves
this
form
implies
that F(z)
property the
result,
lemma. we
can
Im F ( z ) z = g(z)z,
of g f o l l o w s
then
= 0 for with
g
from
all
: ~ ÷m the
z e ~, of class
equivariance
[] rewrite
the b i f u r c a t i o n
equation
(18)
:
g(z,X,o)z
+ h(z,X,o)~ 2 = 0 ,
(23)
693
where
the f u n c t i o n s
it also
follows
g(0,10,o0)
g and h are
from the p r o p e r t y = 0
,
z ~ 0 we can m u l t i p l y
and
imaginary
parts,
system
and s y m m e t r y
invariant;
(b) of the b i f u r c a t i o n
Dog(0,~0,o0)
For
equivalent
real v a l u e d
= I
(23) by z, w r i t e
and d i v i d e by
function
p2,
tha (24)
z = pe i@,
respectively
split
into real
p3, we o b t a i n
the
:
g(z,~,o)
+ ph(z,h,o)
h(z,/,o)
sin3e
aos3e
=
0
,
(2s)
Assuming first
enough
equation
0 for all equation
of
=
0
differentiability, (25)
bifurcation branches
can be s o l v e d
@, and ~ ( p , ± @ + ~ , l )
is s a t i s f i e d
(13),
of s o l u t i o n s
of
(13),
{(pr(6J)u1,~(p,0,~) The
three b r a n c h e s
the s y m m e t r y solutions example, similar
, k ~ ~"
this
gives
I Ipl < p 0 }
under
us for each
the
with 5(0,8,10 )
= ~(-p,@+~,l).
returning
that
The s e c o n d
to the o r i g i n a l I near
h 0 three
,
(26)
j = 0,1,2.
from each o t h e r by a p p l i c a t i o n
F(6), w h i l e
are i n v a r i a n t
(24)
of the f o r m
are o b t a i n e d
operator
from
for o = ~ ( p , l , o ) ,
= ~ ( p , e ,h)
for @ = ~
equation
it follows
along
of
each of these b r a n c h e s
an a p p r o p r i a t e
reflection
along
the b r a n c h
j = 0 the s o l u t i o n s
satisfy
result
then holds
for the c o r r e s p o n d i n g
the
in A3;
for
F(T)u = u. A
solution
branches
of
(1). All
other
g(z,l,o)
solutions = 0
A g a i n we can solve
,
of
h(z,l,o)
the first
o*(0,10)
= o 0 and o~(yz,l)
solution
of
Bringing
the s o l u t i o n
we o b t a i n
(25) h a v e
to s a t i s f y
o*(z,l)
of these
o*(z,~)
(27) equations
for o = o*(z,I),
Vy ~ A 3. Of course, = ~(p,O,l),
with
equation
with
for each
z = pe
of g = 0 in the s e c o n d e q u a t i o n
the final b i f u r c a t i o n
:
= 0 .
= o*(z,l),
(27) we will h a v e
the e q u a t i o n s
i0 h = 0,
694 p(z,x) with
= 0 ,
p(yz,X)
(28)
= p(z,X),
p(z,~)
= ~(X)
if ~(~0)
¥y •
~ 0, then
solutions
small "circle"
This
closed
scaling
if a(~)B(0)
of solutions
which we found be 6 points
It results
already;
implies
that
(29)
solutions.
argument
> 0, while
shows
If a(X0)
that
= 0 and
(36) has no non ±
if ~(X)B(0)
< 0 there
is a
of the form ,
of
(I), which
so, if ~(~0)
of secondary
invariance
z = pe i@
= IB(9)I -I/2 and px(±@ + ~ k )
Curve of solutions
~(x)~(0)
,
(27) has no small
p = px(@) = I a ( x ) l l / 2 n ( 0 , x ) with n(9,0)
symmetry
+ ~(X)p 2 + O(p 3 )
B(X 0) P 0, then a simple trivial
A3.
p~(@). connects
This gives
the three branches
= 0 and B(~0)
bifurcation
for those
us a
= 0, there will
~ near
~0 for which
< 0.
is
interesting obtained
to
compare
by golubitsky
these
last
results
and Schaeffer
[2]
with using
some of
the
singularity
theory. REFERENCES
[I] Crandall
M.G.
eigenvalue. [21Golubitsky including
and P.H.
J. Funct.
applications
Instituut Krijgslaan
Vol.
voor Theorctische Gent
281
Gent, Belgium.
London.
with O(3)-symmetry
problem.
846,
Preprint
1980.
from multiple
1981,
eigen-
356-365.
and symmetry.
To appear.
Mechanica
from simple
321-340.
and bifurcation
in Math.
Pitman,
Bifurcation
D., Bifurcation
A., Local bifurcation
in Math.,
Rijksuniversiteit B-9000
Notes
8 (1971)
to the B6nard
A., Symmetry
Lecture
[4] Vanderbauwhede Notes
Anal.
M. and Schaeffer
[3] Vanderbauwhede values.
Rabinowitz,
Research
On the Foundations of Thermodynamics Johann Walter
Wet kann ~ s
ansehen, ohne leise zu pfeifen?
1. Technical Remark: Since the present paper is rather closely connected with our previous paper [30] (and also because of lack of space) we take accross the denotation and terminology of [30] without explaining i t again. The only mo d if ication is
that in the discussions to f o l l o w a certain postulate occurring in the
d e f i n i t i o n of the set ~
of a l l thermodynamical systems (cf.
w i l l be replaced successively by one of the "axioms" ( I ) ,
[30; p . 9 0 , ( i i i ) ] )
(2), (3), (4) corre-
sponding to the "lower corners" of a certain tetrahedron to be described below. We also use the additional abbreviation
~A
;=
~ ~
÷ ~
J ~ ~ )
and
ask the reader to replace the "A" a f t e r "denotated by" in the seventh l i n e of [30; p.88] by "A". 2. Aim of the Paper: I t is a commonplace that even today thermodynamics has not reached yet a generally accepted mathematical status as, e . g . , electrodynamics. In spite of numerous widespread complaints concerning this point there does not seem to e x i s t much plausible knowledge as to the deeper reasons f or the obstinac i t y of this state of a f f a i r s . "The recent tendency to apply more and more r i g orous mathematical methods to physical i n v e s t i g a t i o n s " [22] rather tends to aggravate than to improve the s i t u a t i o n - as w i l l become clear in the f o l l o w i n g . I t is the aim of the present paper to pick out a small sector of this bunch of problems, v i z . , the problem of d e r i v a t i o n of the Fundamental Theorem from some appropriately chosen mathematical representative of the Second Law of classical equilibrium thermodynamics and to t r y to give a "geometric" overview of a certain set
~
of such derivations to be found in the l i t e r a t u r e . Even in this re-
s t r i c t e d area of research we have not succeeded in f in d in g a simple Venn-diagram l i k e that used in [29]. We have developed instead a three-dimensional "graphtheoretic" representation. Besides serving i t s main purpose, v i z . , providing "a c r i t i c a l analysis of various axiomatic systems" [18; p.129] as well as a sound basis f o r the choice of an "optimal" element of
~
, this geometric model
works in a way l i k e the "Periodic System of Elements" in bringing to l i g h t new aspects overlooked so f a r . I t is in this way that we have i d e n t i f i e d a new v a r i ant of such a sis
d e r i v a t i o n which from the point of view of elementary vector analy
might be of independent i n t e r e s t (cf. Remark 2 below).
3. The "Foundational" Tetrahedron: The main assertions or axioms used in the derivations belonging to
~
w i l l be enshrined into the corners (one exception)
of a three-dimensional tetrahedron in such a way that every element of
~)
may
be represented as a certain sequence of such "corners" whose "edges" correspond
696 to the theorems making up that special derivation:
(4)
(3)
4. The Corners:
(1)
(i)
where
= {
~
~--
~-} 0~_ CAI ~Z
and for any
/~ ~ r ~
with
At
> o
and
~-(A) ~ Z , ~ denotes the set of all solution curves of (~)A ~ O whose projection onto the a-space is closed but which are not closed themselves. 1) Bringing into play the "composition" of a special "Hilfsk~rper" E (cf. postulates ( i ) , ( i i ) in [30; p.88,89]) with any A ~ L we get our own version of axiom (1) (which is bound to replace (i) in our derivation 2) being especially wellsuited for applying elementary vector analytic methods), v i z . ,
(1)
where
[×--
and for each (~4
[
~
A~ ]
Yr[
with
~(A)
~ 4
[~(A3
denotes the set of solution curves of
~ 2] QE gA = O
I) .... closed" but not closed " in the terminology of [18]. 2) I t is interesting to note that we have, in fact, ( i ) ~ ( I ' ) . But we w i l l not make use of this equivalence and, therefore, omit its proof, I t is further interesting that the very formulation of (1) or (1') renders pointless an objection raised in [2; Aufgabe 34 (p.84) and L~sung (p.232)].
697 whose projection onto the (a,t)-space is closed but which are not closed themselves [the set of closed isothermal curves
~
with
~
(~A
~
O
].
The p l a u s i b i l i t y of ( i ) or (1') as a mathematical representative of the Second Law derives from i t s high degree of f a l s i b i l i t y : "~
=~= ~{
)
~
4~ ~
or
~Z ~
{
each one of the r e l a t i o n s amounts to the existence of a
perpetuum mobile of the second kind! Remark 1: In t h i s connection by indulging in some rather strange terminology (already hinted at in footnote i ) the author of [18; p.122] (cf. also [17; p.329] and [30; p.90, Remark 2]) puts forward a Postulate whose meaning is that of the F i r s t or the Second Law of thermostatics according as to which of two possible i n t e r p r e t a t i o n s of certain quotation marks occuring in i t is preferred (compare the use of the graphems . . . . c y c l i c . . . . [18; p.122], r e s p e c t i v e l y ) . For every (2)
A~TYL with
d e f i n i t i o n of
A
can be joined with
and . . . . closed . . . .
A~ ~ O
and every
no neighborhood of K
in [31; p.237] or
/~ ×
~
in the domain of
exists a l l points of which
by a s o l u t i o n curve of
(~A =
O
There is a l o t of controversy connected with t h i s axiom of Carath~odory [9]. On the one hand i t has been hailed as "more elegant and at the same time more r a t i o n a l " [25; p.39], " l o g i c a l l y more economic" [15; p.37], "Das Axiom von Carath~odory beschr~nkt die axiomatischen Aussagen auf das logische Minimum" [28; p.96], "Tats~chlich r e i c h t die Erfahrung, dab es Uberhaupt gewisse unausfUhrbare Prozesse g i b t " [5; p.172], whereas on the other hand i t s physical p l a u s i b i l i t y has been depreciated as "not based d i r e c t l y on experience" [32; p.915], "There has always been some d i f f i c u l t y
in motivating Carath#odory's
p r i n c i p l e convincingly" [20; p.485] "no obvious physical i n t e r p r e t a t i o n " [18; p.125]. The persistence up to the present day of t h i s controversy seems to be due to the fact that there are no s p e c i f i c reasons available to kick the beam in favor of one of the two groups of these rather general judgements. I t i s , in the opinion of the present w r i t e r , the second group that can be made precise; the reduced degree of f a l s i b i l i t y
of (2) (as compared with that of ( i ) or ( 1 ' ) )
w i l l become evident in the following way: Since (2) cannot be made to assert the emptiness of a set each element of which embodies some immediately desirable physical process, at the f i r s t
sight i t might seem for the task of f a l s i f y i n g
(2) to be necessary "Versuche anzustellen in der Absicht, a l l e Nachbarzust~nde irgendeines
bestimmten Zustandes auf adiabatischem Wege zu erreichen" [23; p.619
and w i l l become clear a f t e r using an additional mathematical theorem only (cf. [20; p.485],[21; p.42]) 3) that again the existence of a single perpetuum mobile of the second kind w i l l do the job. 3) This theorem corresponds to the edge (1) (2) of our tetrahedron, cf. next section.
698 I t seems to be a rather i n t e r e s t i n g question which are the reasons f o r the choice of an axiom of t h i s kind.
-
This axiom looks reasonably compact and simple at the f i r s t sympathetic is i t s tendency to synonym f o r " ~
s i g h t . Also rather
exclude some widespread misuse of " 4 Q
" ( [ 4 ] , however, s t i l l
writes
~A
~
" as a
). But on closer
inspection i t reveals i t s e l f as h i g h l y redundant and we have been able to dig out an unexpectedly small basis of (3), v i z . ,
%CA)= A Since no use w i l l be made of the equivalence (3)~(3') we omit i t s proof (cf. footnote 2) which, however, is quite simple algebra. Owing to i t s lack of redundancy (3') f i t s more r e a d i l y into the logical texture as precursor as well or as successor than does (3). Most important in t h i s respect: For " i n t e g r a b i l i t y condition"
(~AA
~ ~E~
A :
O
~w(A):
4
the
already guarantees the
existence of an i n t e g r a t i n g denominator (in f a c t , the "absolute temperature") depending on t only and being given as a solution of a simple ordinary d i f f e r e n t i a l equation (cf.
[30; p . 9 1 , ( 2 ) ] ) . Replacing (3) by (3') thus amounts
to the complete avoidance of a theorem of Frobenius 5) which is d i f f i c u l t
to prove
even in the most simple n o n t r i v i a l case (for such proofs c f . [ 1 4 ; p . 3 1 8 ] , [ 1 9 ] , [24; p.225],[26; p.346]). Another advantage of (3') which I suppose w i l l be welcomed not by the least part of my readers is that i t can be w r i t t e n in the move f a m i l i a r form + A Y-Or ~ E + A
(3")
:0
A i ~ exact
thus throwing out e x t e r i o r calculus altogether. -
/
(5)
~C
exact
%(t~ ~ ( A ) ~ 4 The function ~
is uniquely determined by t h i s property up to a constant f a c t o r
and is " a f t e r normalization in the usual manner" [17; p.329] denoted by T and 4) " A "
or " ~ "
denotes the e x t e r i o r product or d e r i v a t i v e , respectively.
5) We do not share the point of view expressed by [1; p.220] in a s i m i l a r cont e x t : "To t r y to avoid using some theorems of mathematics is to impose an absurd handicap on one's technique". The theorem of Frobenius, by the way, is represented as the edge (3) (4) of our tetrahedron, cf. next section. *) See additional note at the end of the paper.
699
called "absolute temperature". Theorem (5) is the natural endpoint of every sequence of corners corresponding to an element of called the Fundamental Theorem of thermostatics. (4)
---
"~'(A)z ~
~
and i s , therefore,
-
exact
V
.
This assertion (not being chosen very often as the i n i t i a l
point) seems to be
affected by a rather strong tendency to appear as the endpoint of an element of : The i n t e g r a t i n g denominator
~
although by no means uniquely defined
by (4) (compare, e . g . , the d i f f e r e n t positions of the two q u a n t i f i c a t i o n s occurring in (5) or (4)) is f r e q u e n t l y denoted by
"I-
true, can be denoted ad l i b i t u m - but then m
too. Bound variables, i t is
cannot be called "absolute"
temperature any longer as is done in [4; p.401],[25; p.41],[31; p.191]. This equivocation does'nt give a f a i r chance of understanding to the u n i n i t i a t e d because certain r e l a t i o n s which look rather f a m i l i a r at the f i r s t
sight (cf.
[4; p . 4 0 4 j ( 6 ) ] , [ 1 5 ; p.26, ( 3 . 1 3 ) ] , [ 3 1 ; p.191, (4.44)7) (but are nevertheless v a l i d f o r every i n t e g r a t i n g denominator!) do not enjoy t h e i r fundamental physical meaning except f o r the appropriate choice of
T
• This fact has already
been c r i t i c i z e d more than a hundred years ago by R. Clausius [10; p.3677 in a s i m i l a r context (cf. also [ 8 ] ) . 5. The Edges: ( i ) (2): This edge has already been mentioned in footnote 3. (2) (4): This is Carath~odory's famous " a c c e s s i b i l i t y theorem". I t is i n t e r e s t i n g to note that Carath~odory's proof of t h i s theorem e s s e n t i a l l y f a l l s into two parts containing the f i r s t or second edge of the sequence (2) (1) (4), respect i v e l y . On the other hand (2) (4) is equivalent (modulo the t r i v i a l edge (4) (3)) to (2) (3) which on i t s part is equivalent to (i .e. contraposition of) a rather d i f f i c u l t theorem of Chow [117(cf. [13; p.947/948],[16; p.249],[19]; a sketch of (2) (3) is also given in [15; p.38/39] where, however, (2) (4) is asserted). Under these circumstances i t does not seem to be astonishing that together with (2) i t s e l f the edge (2) (4) o r i g i n a t i n g in (2) has provoked disagreement too: "encumbered by the emphasis which is l a i d on an abstract mathematical theorem which is otherwise of l i t t l e
use in physics" [12; p.246], "This
"simple" proof, however, is not simple f o r the ordinary p h y s i c i s t " [27; p.7817, "The usual proofs of which are often so unpalatable to the p h y s i c i s t " [6; p.417 (proofs for the case n=3 in [25;p.35 and 37] and for the general case in [37, [7]). (1') (5): This edge is given in [30] where, however, only (1')1 is used.
700 6. The new sequence (1') (3") (5): (3") (5): Existence of T as in [30; Section 4, First Step]. Exactness of
Q^
-Tby inspection. ( i ' ) (3"): Let A be a system with
q~ (AI = 4
{3")1 is violated at
[:=
~
Setting
and ~
a point such that
C 4 , 0 , 0 ] we have ~ + A ( ~
: E # O.
According to our Lemma (cf. Appendix) on a suitable cylinder parallel to the e-axis there exists a solution curve of ( ~ E t A = O contradicting ( 1 ) 1 (note that (1')1 is used for the case .~ ( A ) : 4 only). Let now /-~ be a system with m CA)>_~ and ~ a point such that (3")2 is violated at N
Then two indices
t', ~
exist such that
~"
~
such that
/~,
)(~)
3&O •
c4 -plane there exists a closed curve J~ near )
Ac~c,,
--
t"
Q
Applying Gauss'theorem in the
C /-~
O
k
which according to the First Law contra-
dicts (i')2. 7. Appendix: Let ~' in ]:{3 such that around
~
be a point, ~ a vector and %~(×)
12 C ~ ) ' ~ 9= O . Let •
orthogonal to
q~ ( - ~ >
allel to tion of
E "~
. Then
D (~');-
be a circle of length
, let
arclength be a parametrization of ~
a vectorfield
, S ~[O,~]
C(S)
and ?
,
S =
the cylinder through ~
par-
c('<~)@r~vt£- I ~n. e ['~ is a parametriza-
and
~(P i c ¢6>
os,
): =
-
a parametrization of the directional
field induced in
The solution of (6) with
O
" f l a t screw" on ~ Lemma
~ t ( ~ ) = --
/~L Co) :
with "pitch" ~ ~I/~' ~ l [
3
by %~(×).o{× : O.
then corresponds (for small ~
.i~L C ~ ) IJL
to f i r s t order as ~ ~
The proof follows by simple estimations from the identity
_
) to a
,
.
.
.
.
.
0
701 A
Remark 2:
According to (7) the " c i r c u l a t i o n " of
%J~(×)
along
~
can
be interpreted to f i r s t order as the "pitch" of the above-mentioned "screw". A rather special r e s u l t of this type is contained in [26; p.352, Anmerkung i ] .
References
1
Arens, R.: J. Math. Anal. Appl. 6 (1963), 207.
2
Basarov, J.P.: Thermodynamik, VEB Deutscher Verlag der Wissenschaften, Berlin 1974.
3
Bernstein, B.:
4
Bleuler, K.: Reports on Math. Phys. 12 (1977), 395.
J. Math. Phys. 1 (1960), 222.
5
Born, M.: Ausgew~hlte Abhandlungen 1, Vandenhoeck & Ruprecht, G~ttingen 1963.
6
Buchdahl, H.A.: Am. J. Phys. 17 (1949), 41.
7
Buchdahl, H.A.: Am. J. Phys. 17 (1949), 212.
8
Budde, E.: Wied. Ann. 45 (1892), 751.
9
Carath~odory, C.: Gesammelte mathematische Schriften 2, C.H. Beck, MUnchen 1955.
10
Clausius, R.: Die mechanische W~rmetheorie, Vieweg, Braunschweig 1876.
11
Chow, W.L.: Math. Ann. 117 (1940), 98.
12
Eisenschitz, R.: Sci. Progr. 43 (1955), 246.
13
Flanders, H.: Bull. Amer. Math. Soc. ~ (1979), 944.
14
Fleming, W.: Functions of several variables, Springer, New York-Heidelberg-Berlin 1977.
15
Finkelstein, R.J.: Thermodynamics and s t a t i s t i c a l physics, W.H. Freeman, San Francisco 1969.
16
Hermann, R.: D i f f e r e n t i a l geometry and the calculus of v a r i a t i o n s , Academic Press, New York 1968.
17
Jauch, J.M.: Found. Phys. 2 (1972), 327.
18
Jauch, J.M.: Found. Phys. 5 (1975), i i i .
19
Kirchhart, K.J. und Walter, J.: Eine elementare Studie Uber die S~tze von Carath~odory, Chow und Frobenius in der Thermodynamik, I n s t i t u t fur Mathe matik der RWTH Aachen 1982.
20
Landsberg, P.T.: Nature 209 (1964), 485.
21
Leontowitsch, M.A.: EinfUhrung in die Thermodynamik, Deutscher Verlag der Wissenschaften, Berlin 1953.
22
MrugaJa, R.: Reports on Math. Phys. 14 (1978), 419.
702 23
Planck, M.: Physikalische Abhandlungen und Vortr~ge 2, Vieweg, Braunschweig 1958.
24
Petrovski, J.G.: Ordinary d i f f e r e n t i a l equations, Prentice-Hall, I n c . , Englewood C l i f f s (N.J.) 1966.
25
Sneddon, I . N . : Elements of p a r t i a l d i f f e r e n t i a l equations, McGraw-Hill, New York-Toronto-London 1957.
26
Stepanov, W.W.: Lehrbuch der D i f f e r e n t i a l g l e i c h u n g e n , VEB Deutscher Verlag der Wissenschaften, Berlin 1956.
27
Turner, L.A.: Am. J. Phys. 28 (1960), 781.
28
Waals, J.D. van der: Lehrbuch der Thermostatik, J.A. Barth, Leipzig 1927.
29
Walter, J.: J. Math. Anal. Appl. 59 (1977), 587.
30
Walter, J.: Proc. Royal Soc. Edingburgh 82 A (1978), 87 (cf. also Autorr e f e r a t in Z e n t r a l b l a t t f . Mathematik 39~--(-~979) 35074).
31
Westenholz, C. von: D i f f e r e n t i a l forms in mathematical physics, North-Holland, Amsterdam 1978.
32
Zemanski, M.W.: Am. J. Phys. 34 (1966), 914.
Additional Note: A f t e r d e f i n i t i o n of T r e l a t i o n (3")1 can be looked at as an ordinary l i n e a r f i r s t
order d i f f e r e n t i a l equation in t for each component of A.
From t h i s we deduce that
[T(o).~(~,t)
exact for each t. Thus there is s t i l l
-
~-(~).~Cq,~)~
~
some redundancy l e f t in (3").
is
On some conjectures symmetric
on the deficiency
differential
index for
operators
A.D. Wood and R.B. Paris
Introduction We consider the symmetric differential expression m M [y] = E (-I) r (p(x)y(r)) (r) x ~ [X,~), X > 0 r =0 where ~ are sufficiently -r functions
continuously
and o (x) # 0 on [X,~). -m
(the minimal
symmetric
. o~erator
differentiable
The deficiency L 2
T O d e f i n e ~ in
(I)
real-valued
indices of
(X,~) from) M are
2 the dimensions
of the linear subspaces
satisfying M[y] well-known operator
[6] that the deficiency
given certain conditions
(N,N) where m < N < 2m.
x e [x,~].
that Pm(X)
to lie in the minimal
by R. Kauffman
in 1976 [4].
certain necessary was not possible
condition
maximal
I
For positive
by McLeod.
0 < r < m, for all -
that this would
force the deficiency
position N = m, but this was disproved The method depended
for the minimal
on showing that a
position was broken and it The following conjecture
[3]. coefficients
case N = 2m are possible.
order operator
We shall be
conjecture.
to say how large N might become.
was then made by J.B. McLeod Conjecture
thought
differential
D of M. "S
coefficient
> 0 and m (X) > O, -r -
It was originally
It is
on the Dossible values of N,
on the coefficients
The first of these is the positive
indices
indices of a symmetric
with two recent conjectures
It assumes
(X,~) functions y
= ly for iml > 0 and im I > 0 respectively.
of even order 2m are
concerned
of L
all values of N except
the
We shall construct b e l o w a sixth-
for which N = 2m - I, the largest value suggested
704
The second is the zero coefficient Grudniewicz
conjecture made by E a s t h a m and
at the end of a long paper
that if some of the coefficients
~
[I] in 1981.
were
They suggested
identically
zero, this
*S
restricted
the scope for moving away from the minimal
Conjecture
2
If exactly K of the coefficients
D
case N = m.
in M are
"S
identically
zero in [X,~], then m < N # 2m - K.
b e l o w a sixth-order
operator of form (I) with P1(X)
which N = 5, thus showing conjecture
2.
Transformation We c o n s i d e r
to a generalised
the
We shall exhibit
interval
[1,~)
m = 3 and
equation ~
(x)
= a
"r
in (I), where
Then M[y]
5 > O, a 3 = I,
If we write @ ~ x ~d, we may express x
5
(~+8+I) W (~+8+2) Then the equation M[y]
(8+Br)y
If we set becomes
+
(@+I)
denote (w+1)
Multiplying
mxBy
5 (6 - r ) 8
(@+5+3)
(2).
given in Watson
[8,109]
(@+2)y
the zeros of the polynomial
(w+B+3)
(~+2)
-
a.
in (3)
= %y may be written as
%y
=
z = ~8-6x -B and 6 6 ~ r=1
2r + B
(x6+Sy(3)) (3) as
(8+8+I) @ (@+8+2)
8 6 ~ r=1
+
x r
a 0 = ~ ~ 0.
by standard manioulations
Let -8 r (r = 1,2,..,6)
x
~2 = ~I = O,
_(x6+Sy(3))(3)
~ 0 for
2 to be incorrect.
hypergeometric with
= P2(X)
Y + zy
0
(4) d z ~--~, then @ = -86 and =
(4)
0
(5)
this equation by z-51/B and noting zV6y = (6-v)(~Vy)
for a r b i t r a r y v w e 6(6 - 82-81 5
obtain
) (6 - 53-51)... 5
where Y = z-51/Sy.
(6 - B6-BI)Y 5
This is the generalised
+ zY
=
O
hypergeometric
(6)
equation
givel
705
by Luke
[5, p~81] which may be solved
functions
oFp_1(z)
[2, p.182].
whose asymptotic
Reverting
one solution of M[y] x
are given in
variable x, we see that
= 6y is ~2-~I B
oFD_I(I
asx
%x
Assuming
properties
to the original
-f~1 ~1(x)
in terms of the hyper-Bessel
.... '
I
B6-B I ----~--- ;
X - ~6xB
)
÷~ .
that a can be chosen so that the roots -$
of (3) are r
all distinct repeat
and no two differ by an integer multiple
the above process, multiplying
of B, we may
(5) successively
by
-gk/5 z
for
M[y]
k = 2,3,...,6
to
obtain
a fundamental
set
for
= Xy as ~k
~k(X)
=
x
0Fo_I(I
BI-Bk - ~ , .
%
~6-5k
, ......
I ;
x -$k where * denotes
as x ÷ ~,
the omission of the factor
r
of the polynomial
(in the comolex ~ plane).
B6xB)
k = 1,2,...,6
The L 2 (I,~) nature of these solutions of the zeros -B
-
Zeros
I
Br-Bk 6
with r = k.
is governed by the disposition
(3) with regard
to the line re ~ = -½
lying to the left of this line will
give rise to L2(I, ~) solutions.
3.
The location
of the zeros -B
If we define a new variable condition
s = ~ + ½B + ½ and write
for L2(I, ~) solutions becomes
characterises G(s)
r
=
the asymptotic
behaviour
o ~ ½5.
s = o + it the
The polynomial
of the solutions
(s 2 - a2)(s 2 - b2)(s 2 - c 2)
where a = ½(B+I), b = ½(B+3), c = ½(B+5).
(3) which
becomes
(7)
706
W h e n a = O, the c h a r a c t e r i s t i c polynomial G(s) has the real zeros s = ±a, ±b, ±c.
It is clear that all the positive zeros are greater
than ½6 and that there are only 3 L 2 solutions of M[y]
= %y.
We n o w consider the locus of these positive zeros in the s-plane as a is allowed to increase from O.
We shall show that for suitable
values of $ two of these zeros move across the line o = ½6 to give rise to two additional L 2 solutions.
We note that the zeros of G(s) are symmetric
in the o and t axes and
observe that G(s) has zeros when
H(s)
~
(s 2 - a2)(s 2 - b2)(s 2 _ c 2)
ae 2~ik
(k=O, ±I,±2 .... ).
The locus of these zeros consists of those points s w h i c h satisfy arg H(s)
z
arg(s 2 - a 2) + arg(s 2 - b 2) + arg(s 2 - c 2) = 2~k
This can only be satisfied for real s if a ~ s ~ b or s > c.
(8)
By
c o n s i d e r i n g e = IH(s) I we see that, as a increases from O, the zero initially at c moves outwards along the o-axis while those initially at a and b move towards each other, eventually m e e t i n g and b r a n c h i n g off the axis as a c o m p l e x conjugate pair.
Does this b r a n c h enter
the strip O ~ o ~ ½6 c o r r e s p o n d i n g to L 2 solutions of M[y] = %y?
For s = it we find arg H(s) = 37, hence the locus cannot cross the imaginary axis. monotonically.
Also,
for non-real
s, we see that as o ÷ ~, arg H(s) ÷ O
Thus, for given t, there is a unique value o = o(t) at
w h i c h arg H(s) = 2~.
The locus of such Points is obtained by setting
k = I in (8), taking tangents and separating real and imaginary parts to give . 2 c 2 +c 2 a 2 3(02 - t2) 2 - 4o2t 2 - 2(a2+b2+c2)(o2-t 2) + a 2b 2 +b
=
O.
This quartic crosses into the strip 0 ~ o ~ ½B if the quartic in t resulting when we substitute o = ½B has real roots.
This condition will
(9)
707
be satisfied if (9), regarded as a quadratic positive roots.
in t 2, has distinct
S u b s t i t u t i n g for a,b and c in terms of 6 from (7), this
will occur when B4 - 18B 3 - 2352 + 725 + 112 > O w h i c h holds if 8 > 18.99495 with a c o r r e s p o n d i n g ~ value of 1.68112 x 106 Since the locus has asymptotics args = ± ~/3, it must cross over the line o = ½~ for some higher value of ~.
Recalling the symmetry in the
o-axis, we obtain a range of values of ~ for w h i c h there are two additional L 2 solutions of M[y]
t
= %y, a total of five.
j //!
O
b
c
I
The root-locus d i a g r a m for (3). a, b and c are the roots when = O. The arrows denote the direction of ~ increasing.
O"
708
4.
Concluding
Remarks
This single example attainable
shows both that the case 2m-I of Conjecture
and that Conjecture
2 is incorrect.
I is
Since k = 2, Conjecture
2
would require N = 3 or 4.
These results have been extended
in [7] to equations
of general
even
order 2m of form
(-l)my (2m)
+
where m > q > O, ~ -
qE (_])r r=O
(~ x2r+By(r)) (r) r
real and B > O.
=
%y
on
It is shown that when a
r
to choose
B in such a way that the deficiency
N = m + 2j for all values of j between 0 and the integral (q+1)
index
part of
inclusive.
In conclusion
we remark that it has yet to be shown that the maximal
case 2m-I of McLeod's of sixth-order. above,
= I and q
> 0 it is Possible r -
4
(0,~)
conjecture
For the type of 2mth. order positive
the deficiency
positive
coefficient
be interesting operator,
can be attained by operators
indices are of form m+2j. requirement
is it possible
to know if this is a feature
or is it possible
2m-I with coefficients
other than
symmetric
Only if we relax the to attain m+2j+1.
It would
imposed by the symmetry of the
to obtain every value between m and
of a different
operator
structure?
709
References
I.
M.S.P. Eastham and C.G.M. Grudniewicz, Asymptotic theory and deficiency indices for higher-order self-adjoint differential equations, J. London Math. Soc. (2), 24 (1981), 255-271.
2.
A. Erdelyi (Ed.), Higher Transcendental Functions, Vol. I, (New York: McGraw Hill, 1955).
3.
W.N. Everitt, On the deficiency index problem for ordinary differential operators 1910-1970, Proceedings of UDDsala 1977 International Conference on Differential Equations, 62-81.
4.
R.M. Kauffman, On the limit-n classification of ordinary differential operators with positive coefficients, Springer Verlag Lecture Notes in Mathematics, 564 (1976), 259-266.
5.
Y.L. Luke, Mathematical functions and their approximations, (New York: Academic Press, 1975).
6.
M.A. Naimark, Linear Differential Operators, Part II (London: Harrap, 1968).
7.
R.B. Paris and A.D. Wood, On the L 2 nature of solutions of nth. order symmetric differential equations and McLeod's conjecture, Proc. Roy. Soe. Edin., 90A (1981), 209-236.
8.
G.N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University Press, 1944).
On a s e c o n d - o r d e r and
differential
its D i r i c h l e t
expression
integral
S. D. Wray
Introduction We consider defined
by,
(i)
for
M[f]
where
the
symmetric
suitable
second-order
functions
of the real
line
R.
We also
expression
M
f,
= w-l(-(pf') ' + qf)
p, q and w are r e a l - v a l u e d
differential
on [a,~)
('~d/dx)
coefficients
on the
consider
the
quadratic
, interval
[a,~)
form
{Plf'l 2 + qlfl 2} , this b e i n g
the D i r i c h l e t
on the D i r i c h l e t listed
integral,
will
[4],
be imposed
and w, of M that will (see the next
M is considered, integral, Use
see
associated
with M.
[8] and the other
For recent
work
references
therein.
Conditions
at ~
integral
on the c o e f f i c i e n t
put M in the
section).
A self-adjoint
as is a "spectral"
from w h i c h we o b t a i n
is made
so-called
of a u n i t a r y
change
limit-circle operator
i d e n ti t y
information
functions,
of independent
condition
derived
involving
about
p, q
from
the D i r i c h l e t
its lower bound.
variable.
Notation AC denotes
absolute
fied on all compact symbol
such as
continuity
sub-intervals
'(fC~(T))'
and
'loc'
of the
is to be read
a property
interval as
to be
satis-
in question.
'for all f in the
A
set
P(T)' Statement
of results
We work
on the h a l f - l i n e
[a,~),
with aCR.
Throughout,
the
co-
711
efficients p, q and w are real-valued,
Lebesgue m e a s u r a b l e
on [a,~)
and satisfy the basic conditions:
(i) (2)
p(x)>0
(ii)
(almost all xc[a,~)
and p-l
qCLloc[a,~);
(iii)
w(x)>0
o The space L~[a,~),
(almost all x~[a,~)
where w satisfies
of L e b e s g u e m e a s u r a b l e
functions
andwCLloc[a,~).
(2) iii),
is the Hilbert
space
f:[a,~) ÷ C (the complex field) that
satisfy
Sa wlfl2~" We consider the linear d i f f e r e n t i a l (3)
M[f]
= ~f
on
equation
[a,~),
where M is defined by
(i) and ~ is a complex parameter.
singular case because
[a, ~) is an infinite interval.
fundamental
solutions of = sins
,
p(a)~'(a,~)
e(a,~)
= cos~
,
p(a)0'(a,~) = sins
XCC,
general
solution of
Let ~, 0 be the
(3) d e t e r m i n e d by the boundary conditions
~(a,~)
for all
M is in a
= - cos~
, ,
where a is a real number in the interval
[0,w).
The
(3) is of the form 6 + ~¢; if c is any point in
(a,~), we consider those solutions which satisfy the boundary condition (e(c,~)
+ ~(c,~)~(c,~)}cos~
+ {p(c)e'(c,~)
for all XCC,
where
+
+ ~(c,~)p(c)~'(c,~)}sinB
B is a real number in [0,w) again.
and B, this c o n d i t i o n defines
~(c,') as a m e r o m o r p h i c
poles are all on the real axis. suppose that X has positive
: 0 ,
For given c function,
whose
For the r e m a i n d e r of this paper, we
imaginary part.
Now ~(c,~) depends upon B: for fixed c and X, ~(c,X) d e s c r i b e s a
712
circle in the complex plane as 8 varies.
H. Weyl
c÷~ these circles converge either to a limit-point circle.
Let m(~) be the limit-point
[7] showed that as or to a limit-
or a point on the limit-circle.
Weyl also proved that the solution ~(x,X)
= O(x,X)
+ m(X)O(x,X)
of (3) is in L2Ea, ~) as a f u n c t i o n of x, for fixed ~ (Weyl has w~l, W
but his method extends to the present For this theory, Now,
situation without difficulty).
see [6,Ch.2].
it is known,
see [5], that there is a real n o n - d e c r e a s i n g
function o defined on R w h i c h induces a unitary t r a n s f o r m a t i o n from L~[a,~)
onto
L 2, which is the space of all Lebesgue m e a s u r a b l e
functions F:R÷C
such that
f IFI2d<. The unitary
transform,
F, of f is given by S
(4)
F(t) = him I
f(x)@(x,t)w(x)dx
Ja
where the limit is in L 2 norm.
We shall call a a spectral f u n c t i o n
of M. Finally,
we introduce a self-adjoint
a s s o c i a t e d with M. at ~.
operator and a quadratic
Suppose that M is in the l i m i t - c i r c l e
Let ~(T) C L ~ [ a , ~ )
be given by f C D ( T ) ~
fCL~[a,~),
form
condition f,
H
Pf'CACloc[a,~),
M[f]CL~[a,~),
f(a)cos~+(pf')(a)sin~=0
and f satisfies the boundary conditions
and Z i m { f ( c ) p ( c ) O ' ( c , ~ ) - p ( c ) f ' ( c ) @ ( c , ~ ) ) = 0 . C+~
Here,
c÷~ through a special sequence:
see [5] or [ 2 , C h . 9 , T h e o r e m 4.1].
It is assumed h e n c e f o r t h that c always tends to ~ t h r o u g h this sequence.
We then define the operator T by
713
~f = M[f]
(fCO(T)).
It is known that T is self-adjoint
[2,p.246],
our results below that its domain ~(T) this is also a known result,
B=0.
f(a)=0
if ~=0,
In the one case B=z/2,
dropped.
The quadratic T[f] = ~a
~:~ f ~ L
the requirement
integral
(cotB)
(fCD(T))
term if a=0
,
(S=0).
integral
expression
and let its coefficients (2).
+
The
(absolutely
associated
with M.
of this paper as
Let the differential
ditions
and f(~)=0 if
that f(~) exist is
_ ]f(a)]2cot~
is the Dirichlet
Now we state the results Theorem
p2f,,
form T is then defined by
+ [f(~)[2cotS
convergent)
of I (although
[a,~), f ~ A C l o c [ a , ~ ) ,
f(~) = ~im f(c) exists,
{p]f,]2 + q]f]2}
where we omit the cot~
is independent
see [3]).
Let ~(T) be given b y f C D ( T ) [q]~CL2[a,~),
and it will follow from
M on [ a ~ )
be defined by (I)
p~ q and w satisfy the basic con-
Let the operator
T and quadratic
form T be as
above. Suppose
(5)
p-1
additionally
q, wCL[a
that p~ q and w are so chosen that
~)
Then (i) ~i)
M is in the limit-circle the operator
T has discrete
to ~ and having no finite a sub-set
(iii)
spectrum
i0
cluster point~
tending
its domain ~(T) is
of D(T);
f_CD(T) if and only if transform
condition at ~ (Amos[l]);
of f (see
T[f] =
2
[t[½F(t)CL2,
w h e r e F C L 2 is the unitary
(4)); and if f C D ( T )
tlF(t)12d~(t)
;
then we have
714
Civ )
if fCP(T)
then
f(x) = ~ the
F(t)¢(x,t)da(t)
convergence
s÷~£im~a
being
uniform
plf, _ f,}2s
(x{[a,~)), on
[a,~),
and
= 0 ,
where
fs(X) = Proof
See the next section.
Corollary
Under
the
conditions
of
T[£] h ~o/j'a will 2
Proof
(xcEa,~), sCR).
F(t)¢(x,t)dd(t)
the
The©rem~
we h a v e
(feD(T)),
with equality
if and only if f is an ei$enfunction
corresponding
to the least eigenvalue,
This follows
of T
10.
the lines of the proof of the Corollary
in
[4,§6]. Proof of the Theorem We outline the method following
of proof here.
change of independent
x(x) =~ax P-1 ^
The principal
variable. (x~[a,~))
tool is the
We define x(x) by ,
^
so that xC[0,b),
where oo
=
p
-1
<
co
^
Since p(x)>0 continuous
(almost a l l x c [ a , ~ ) ) , the function
and strictly
inverse x:[0,b) ÷ [a,~).
increasing, Then,
x:Ea,~) + [0,b) is
and hence has a continuous
any function
f on [a,~) is trans-
715
formed to a f u n c t i o n f on EO,b) via the r e l a t i o n A
f^ (^x )
=
The t r a n s f o r m a t i o n f ~ f f o l l o w i n g sense.
fEL2~[O,b)
(XC ^ [O,b)).
f ( x ( x ) )
is one-to-one;
Zf f C L ~ [ a , ~ )
it is also unitary in the
and f is its transform,
then
and
]0 Under this transformation, differential
=J/a
2
expression.
w x)'f<x 12 dx
the coefficient p is removed from the If primes on letters without
caps denote
d i f f e r e n t i a t i o n with respect to x, and on letters with caps with ^
respect to x, then the original d i f f e r e n t i a l
equation
(3) on [a,~)
is t r a n s f o r m e d to ~-I(~){_~,,(~)+~(~)}(;)} = l}(;)
(7) ^
^
^
where q, w E L [ 0 , b ) tral parameter,
on [ O , b )
,
^
and
(almost all x { [ 0 , b ) ) .
l, is used in both equations
The same spec-
(3) and
(7); it
follows that both equations are a s s o c i a t e d with the same spectral f u n c t i o n o and the same a s s o c i a t e d unitary t r a n s f o r m a t i o n (4)).
The equation
(7) is a c t u a l l y in the regular ease because of ^
the i n t e g r a b i l i t y of its c o e f f i c i e n t s function,
f(0)cos~ ^ ^
(7) and the boundary
+ ~'(O)sin~ ^
conditions
= 0 ,
^
f(b)cos6 + f'(b)sin8 = 0 ,
where we use the same ~ and 5 as formerly is denoted by Oc ). ^
^
q and w, and ~ is a step
derived from the S t u r m - L i o u v i l l e p r o b l e m on the closure
[0,b] of [0,b) defined by
(8)
(as in
(see [8,Section 2] where
The usual c o r r e s p o n d i n g
self-adjoint operator ^
T is t r a n s f o r m e d to T via the t r a n s f o r m a t i o n x ~ x; it is straightforward to check most of this; we show here only that the boundary c o n d i t i o n at ~ (for T) is equivalent to the c o n d i t i o n
(8)
(for T).
716
Let c E [ a , ~ ) ~(T),
and let @c(X,k)
= e(x,l)
If f is in
+ £(c,l)@(x,l).
then 0
=
£im{f(k)p(k)~'(k,k) k÷~
-
(pf')(k)~(k,t)}
(9)
= him ~ i m { f ( k ) p ( k ) ~ ( k , k ) k÷~ c÷~
(i0)
= £im{f(c)p(c)*~(c,l)
(pf')(k)~c(k,k)}
- (pf')(C)@c(C,l)}
C + ~
= him - {f(c)cos8 if f is the transform ^
+ (pf')(c)sinS}
of f then it now follows
that
^
f(b) and f'(b)
(Here,
we rely on the Lemma of [I].)
above is justified
(8) holds ^
define
to be the limits
The step from
by a uniform convergence
if we
^
(9) to
(i0)
argument. is a sub-set
But both T and T are self-adjoint,
domains must be in one-to-one
^
of f(~) and f'(c) as c +
It is clear now that the transform of D(T) domain of T.
;
correspondence.
of the
and so their
For each f E D ( T ) ,
^^
the transform of Tf is Tf. The results
(ii) - (iv) of the T h e o r e m now follow from their
analogues
in the regular
tion 3].
Note that
case on the interval
[O,b];
see [8,Sec-
oo
{plf' 12 + qlfl2} --l{ J0
and that in part
(iii)
_f%
co
IF(t)l 2 do(t) =
Z
1 Ifn 12
n= 0
n
where oo
fn =~a
Wf~n
~n being the normalised value
~.
(nhO
,
eigenfunction
corresponding
to the eigen-
- .
717
REFERENCES I
2
a~
Amos, R. J.: On a Dirichlet second-order
ordinary
Mathematicae
3 (1978),
Coddington,
McGraw-Hill,
New York,
Oxford
Everitt,
W. N. and Wray,
(2) 14 (1963),
S. D.: A singular
involving
the Dirichlet
expression.
transforms
Quart.
J. Math.
(2) ~ (1954),
University
E. C.: Eigenfunction Differential
Press,
Oxford,
Weyl, H.: Uber gew~hnliche Singularit[ten Funktionen. Wray,
Utilitas
Equations,
identity
of an ordinary
theory
(I).
47-58. associated
with
Part I, 2nd Edn. Oxford
1962. Differentialglei~hungen Entwicklungen
in press.
mit willk~riicher
220-269.
involving
integral of an ordinary Mathematica,
spectral
integral
Expansions
68 (1910),
S. D.: An inequality
Dirichlet
Quart.
and eigenfunction
und die zugeh~rigen
Math. Ann.
domains.
To appear.
D. B.: Integral Oxford
1955.
41-45.
Sears,
Second-order
Quaestiones
N.: Theory of Ordinary Differ-
J. Math.
Titchmarsh,
for
53-65.
W. N.: A note on the self-adjoint
differential
criterion
expressions.
Everitt,
and inequality
6
differential
E. A. and Levinson,
ential Equations.
4
and limit-circle
a conditionally
differential
convergent
expression.
AN INVERSE EIGENVALUE
PROBLEM FOR THE LAPLACE OPERATOR
E M E Zayed
§I
Introduction Let
We wish
D c IR2
be a smooth bounded
to consider
the oscillations,
and clamped along the boundary It is well known that
~D
domain with finite connectivity
@(x,y;t),
of
of a membrane
@(x,y;t)
~y2
satisfies
the wave equation +
together with the Dirichlet boundary
If we make the substitution
D x
(1.1)
~D .
(1.2)
in
~t 2
~(x,y;t)
D
across
m.
~2~ + $2~ _ $2~ ~x 2
stretched
i _< p
condition
= 0 ,
on
= u(x,y)e ikt , we arrive at a consideration
@(x,y;t)
of
the problem
where
£
Au + k2u = 0
in
D ,
(1.3)
u = O
on
SD ,
(1.4)
is the Laplace operator
problem for which non-trivial
in xy-plane.
solutions
~n
This is a well known eigenvalue (eigenfunctions)
exist provided
k2 =
(eigenvalues). These eigenvalues
form an increasing
0 < ~I ~ ~2 ~ "'" ~ At the beginning ing the asymptotic these eigenvalues
An ~
sequence
"'" ÷ ~
as
of this century the principle
behaviour
of the eigenvalues
%
as
% ÷ ~
N(T<) ~ ~
% + O(1 ~ log l)
problem was that of investigat-
{~ }. n
less than or equal to a bound
N(%) ~ < D ~
n ÷ ~
If
N(%)
is the number of
I~ then
(H Weyl 1912)
,
(1.5)
and L
where
IDI
is the area of the domain
D.
as
% ÷ oo
(R Courant
1920),
(1.6)
719
For either whether,
(1.5) or (1.6) the presence of the quantity
for example,
geometry of
D
the order term in (1.6) contains
and its connectivity.
ID]
leads one to ask
further information
about the
Such a question was discussed and put rather
nicely by M Kac [i] who simply asked "Can one hear the shape of a drum?". this question,
one studies certain functions of the spectrum.
To explore
The most useful to
date comes from the heat equation or the wave equation. Our study in this paper is concerned only with the heat equation method. e
-tA
denotes
the heat operator,
then we can construct oo -% t ~ e n n=l
@(t) = tr(e -tf~) =
which converges
for all positive
for the Laplace operator,
attendant boundary
that some progress asymptotic
@(t)
@(t)
[D[
Furthermore,
is the area of
D
of smooth convex holes,
then
domain
then
(without cusps),
ao = ~ I
to the angle
~. i
~D
expansion
It has been shown by Kac [I]
t, viz
!
and aO
i3DI
as
t ÷ ~ ,
(1.8)
is the total length of its boundary
has geometric significance, D
is permitted
a 0 = (l-r)~ b
and
e.g., if
D
is a piecewise
8Dr K(g)d~ + il c(~ i) ,
c(~i)
at the ith corner of
A simple method for calculating
D
to have a finite number
and finally if
where the summation is taken over all the corners of ture of the boundary
and its
the first three terms of the
+ a 0 + O(t 2)
1 a 0 = ~ , if
then
D
8 (~t) z
the constant term
smooth and convex,
is the spectrum
from the asymptotic
t ?
for small positive
13DI
{% } n
can the shape of the domain
is possible on establishing
expansion of
(1.7)
suppose that
for small positive
4~t where
,
conditions be uniquely determined
of the trace function
the trace function
t.
Now the following question is pertinent: of eigenvalues
If
SD. is "r" smooth
(1.9)
D.
K(O)
denotes
the curva-
may be called the corner number corresponding D. c(~),
0 < ~ < 27
is due to D B Ray and is
720
described by McKean and Singer [2] and the explicit result is 2 c(~)
~
2
~ ~ 24~
(i. i0)
The result
(1.8) is also deduced
determined
the first four terms of the asymptotic
t
by the iterative
from an analysis by Pleijel
solution of an appropriate
form of
[3], who essentially
@(t)
F r e d h o l m integral
We merely note that aspects of the question of Kac, namely, shape of a drum?" have been discussed with the impedance boundary
recently
condition
for small positive equation. "Can one hear the
in [6] for the equation
~u -~ + %u = 0
on
~D, where
y
(1.3) together is a positive
constant. Our objective
of this paper is to determine
string and its Sturm-Liouville frequencies
of vibration.
end constraints
The problems
the length of a u n i f o r m vibrating
from a complete knowledge
of its
to be discussed here can be formulated
as
follows: PI Suppose exactly
the eigenvalues
for the Sturm-Liouville
%1 ~ %2 ~ "'" ~ %
Determine and
~ "'" ÷ ~
as
for
are known
x ~ (O,a)
(I.ii)
cos ~ - y'(O)
sin ~ = 0 ,
~ E [0,~)
'I
y(a) cos B - y'(a)
sin B = 0 ,
~ c (0,7]
.
the unknown
n ÷ ~
problem
d2y~= + ~y = 0 dx 2
y(0)
n
length "a" of the v i b r a t i n g
(1.12)
J
string and the unknown angles
~ .
PII Suppose the eigenvalues exactly
where
for the T-periodic
%
%1 ~ %2 ~ "'" ~ %n ~ "'" ÷ ~
p r o b l e m comprising
y(a) = Y(O)eI~T. '
1
J
period "a" and the parameter
such that T .
n ~ ~
(i.ii) and the boundary
y'(a) = y'(0)e I~T , is a real parameter
as
T E (-i,i]
are known conditions
(1.13)
and
i = ~
.
Determine
the
721
These results extend to eigenvalue problems
for rectangles
and cuboids.
fact we obtain the Roe result [4] easily and certain generalizations the results extend easily to n-dimenslons. analysis
In fact
are attacked by a careful -% t oo of the asymptotic behaviour of the trace function @(t) = n~ I e n for
small positive
§2
of it.
In
These problems
t.
Construction
of the trace function for problem I
Kac [I] has shown that the trace function
@(t)
can be expressed
ix terms of
the simple formula a
@(t) = f G(x,x;t)dx 0 where
G(x,~; t)
(2.1)
,
is the Green's function for the heat equation 22
3 - - ~ ) G(x,~;t)
(~
(2.2)
= O ,
ax subject to the boundary conditions lim t->O where
6(x-~)
(1.12) and the initial condition
G(x,~;t)
= ~(x-~)
(2.3)
,
is the Dirac delta function located at the source point
Let us write G(x,~;t)
= Go(X,~;t)
+ X(X,~;t)
(2.4)
,
where
Go(X,~;t)
is "the fundamental "regular" conditions
= (4~t)
2 exp{ -
(2.s)
4t
solution" of the heat equation
solution chosen in such a way that
(2.2), while
G(x,~;t)
satisfies
X(x,~;t)
is a
the boundary
(1.12).
From (2.1),
(2.4) and (2.5) we deduce
@(t)
=
a
~ +
that
K(t)
,
(2.6)
(4~t) ~
where a
K(t) = f X(X,x;t)dx 0
(2.7)
722
The problem now centres of the integral respect
to
on providing
term (2.7).
t, with
s
2
G(x,~;s 2)
as the Laplace
of the Laplace
satisfies
estimate,
for small positive
In what follows we shall use Laplace
G(x,~;s 2) = f 0 An application
an asymptotic
transform
2 e -s t
transform
parameter.
G(x,~;t)dt
transforms
t,
with
That is define
.
(2.8)
to the heat equation
(2.2) shows
that
the string equation ~2 (
_
2 ) ~(x,~;s 2) = -6(x-~)
Dx 2 The asymptotic directly
expansion
(2.9)
s of
K(t),
from the asymptotic
for small positive
expansion
of
K(s2),
t, may then be deduced
for large positive
s, where
a
K(S 2) = f 0 Lengthy but otherwise
straight
X(x,x;s2)dx
forward
(2.10)
•
calculations
show that the following
cases
can be considered (i)
~ = O, B = ~
0(t) ~
(ii)
a = O,
(Dirichlet
problem)
1 a ! - ½ + O ( t 2) (41Tt) 2
as
(2.11)
t -~ 0
Biz 0(t) ~
. a
! + cot pB t ½ + O(t)
as
t-~ O.
as
t ÷0
(2.12)
(41Tt) 2 (iii)
~+o,~=~ @(t) ~
(iv)
~ ~ O,
$ + ~
@(t)
a- T - -cot - ~ t½ + O(t) (4wt) 2
.
(2.13)
(Mixed problem) a t )½ , + ½ + (cot ~ - cot c~)( (4~t) ~ + ½(eot2B
+ cot2~)t
+ O(t ~)
as
t + 0 .
(2.14)
723
(v)
(~ = B = ~
(Neumann problem)
O(t) ~
a ! + ~ + O(t½) (4~t) 2
An e x a m i n a t i o n of the results determines
(2.11) and
as
t + O .
(2.15)
(2.15) shows that the coefficient of
the length "a" of the v i b r a t i n g string, while the sign of
whether we have a D i r i c h l e t or a N e u m a n n problem.
In the cases
½
t ~
determines
(ii) and (iii) we
observe that the c o n s t a n t term is zero. If
~ # O,
B + ~
to determine the angles
§3
! t2
then the coefficient of ~
and
and that of
t
can be solved
8.
C o n s t r u c t i o n of the trace function for p r o b l e m II As we have done in §2 we deduce that the trace function for the p r o b l e m II
can be w r i t t e n in the form
oo @(t) =
a ! (4~t) 2
~ cos(m~T) m=_ ~
e
2 2 m a 4t ,
t
>
O
.
(3.1)
There are two cases to b e considered. (i)
T = O
(periodic problem)
oo O(t) =
a (4~t) ~
e
2 2 m a 4t
for
t > O ,
(3.2)
m =-oo
w h i c h is an example of the classic summation formula, viz, the c e l e b r a t e d PoissonJacobi inversion formula (ii)
~ = 1
(see [5]).
(semi-periodic problem) 2 2 m a
oo @(t)
=
a
i
(4~t) 2
§4
Applications
~
(-i)TM
e
4t
for
t > O
m=_Oo
to eigenvalue problems for r e c t i l i n e a r domains
A convenient point for an anlysis of rectilinear domains is to examine the structure of
O(t)
for
724
(i)
a rectangle
D
of sides
a, b
where
the boundary
conditions
(1.12) are hold-
where the boundary
conditions
(1.12) are hold-
ing on x-axis and y-axis respectively, (ii)
a cuboid
ing on x-axis,
§§4.1
If
D
of sides
a, b, e
y-axis and z-axis respectively.
~ = O,
8 = ~
For a rectangle
(Dirichlet problem)
D
of sides
a, b ,
ab 2(a+b) G(t) % 4~t + ¼ 8(~t) 2 while for a cuboid
D
of sides
abc @(t) ~
§§4.2
If.
~ = O,
D
of sides
O(t)
ab ~%-~
D
of sides abc
G(t)
a+b+c +
~
+
2(a+b) 4~
+
c°t2B t cot B + - - ~ '
~ + O,
2(ab+bc+ca) 8~ ~ t½
D
of sides
ab Q(t) % 4~t
@(t) %
t ÷
O
.
(4.1.2)
as
t ÷ 0 ,
(4.2.1)
(a+b+c)cot2B 2~~
cot ~ +
as
t->O
t½
(4.2.2)
.
~ =
For a rectangle
while for a cuboid
as
a, b~ c ,
t )'~ c°t38
If
~
a, b ,
+ ( ~
§§4.3
-
! 8(~t) 2
~ +
For a rectangle
while for a cuboid
(4.1.1)
a, b, c ,
2(ab+bc+ca) 16~t
.~ (4~t) z
t -> O ,
as
D
of sides
abc (4~rt) 2
a, b ,
2(a+b) 4~"
cot cot ~ + - -
2 t
as
t ÷O
(4.3.1)
a, b, c ,
2(ab+bc+ea)
cot ~ +
t~ as
t÷O
(a+b+c)cot 2~ t ½ _ ( ~t )~ cot3~ 2~ ~ .
(4.3.27
725
§§4.4
If
~ ~ O,
B + 7
For a rectangle
D
(Mixed problem) of sides
a, b,
@(t) ~ Zab - ~ + 2(a+b) -! + ¼ + 2(a+b) (cot B - cot ~) 47 8(~t) 2 +
{2(a+b)(cotlB+cot 2 2~)
+ (cot B-cot ~)
8"TT2
+ {(Cot B-cot ~) 2
+
Cot2B + eot2~} ~ t
+ i 4(cot 2B+cot 2~) 2 t 2 D
of sides
t½
7 2
(cot B-cot ~)(cot2B+cot2~)
while for a cubo~d
}
as
t~
(4.4.1)
t -> 0 ,
a, b, c ,
abe 2(ab+bc+ca) f2(ab+bc+ca)(cot B-cot~) a + b + c~ i @(t) ~ (47t)~ + 167t + ~ ~ + 87 ½ f -Tt ~ + i + 2(ab+bc+ca)(cot 2 B+cot 2~) + (a+b+c)(cot B-cot ~) 167 2~ i(a+b+c)(cot B-cot ~)2 +~. ~~'~
+
(a+b+c)(cot2~+cot2~) : ~ i 4~ 2
+ ~(a+b+c)(cot B-cot2~ ~)(c°t2B+c°t2~)
+
3(cot B-cot ~) }t ½ :~i 4~ 2
+ 3(cot 2~B-c°t ~)2
+ 3(e°t2B8c°t2~) } t
+ {(a+b+c)(cot2B+cot2~) + 3(cot B-cot ~)(cot2B+cot2~) 1 272
872 +
(cot B-cot ~)3~ t~ ~
" J
3(cot B-cot ~)2(cot2B+-.ot2(~)~ t 2 + ~3(cot2B+cot2CL) 2 L8 " -+ 27 j
as
t ÷ O. (4.4.2)
726
§§4.5
IT ~ = B = ~
If
(Neumann problem)
For a rectangle
D
of sides
a, b,
ab 2(a+bq_)+ @(t) ~ 4--~+ 8(~t) 2 while for a cuboid
@(t) ~
D
of sides
of
abc 2(ab+bc+ca) a + b + c 3 + 16~t + ! + ~ (4~t) ~ 8(vt) 2
-I t 2
(4.5.1)
as
t + O
(4.5.2)
(4.1.1) and (4.5.1) shows that the coefficient
determines the area of the rectangle
determines
t -~ 0 ,
a, b, c,
An examination of the results -I t
as
D
and the coefficient of
-~ t ~
the total length of its boundary, while the sign of the coefficient of
determines whether we have a Dirichlet or a Neumann problem. We note also that the first three terms in (4.1.2) and (4.5.2) are in agreement
with the result of Roe [4] and show that the coefficient of volume of the cuboid the coefficient of
D, the coefficient of t-~
determines
the sign of the coefficient of
t -I
t -1
t -~
determines
the
determines its surface area, and
the sum of the total length of its sides, while determines whether we have a Dirichlet or a
Neumann problem.
References I.
M Kac, Can one hear the shape of a drum?, Amer° Math. Monthly, (1966), 1-23.
73
No 4 part II
2.
H P McKean and I M Singer, Curvature and the eigenvalues of the Laplacian. Diff. Geometry ! (1967), 43-69.
3.
A Pleijel, A study of certain Green's function with applications in the theory of vibrating membranes. Arki Matematik 2 (1952), 553-569.
4.
G M Roe, Frequency distribution of normal modes. No 1 (1941), 1-7.
5.
T Sunada, Trace formula for Hill's operator, Duke. Math. J. 47 No 3 (1980), 529-546.
6.
B D Sleeman and E M E Zayed, An inverse eigenvalue problem for a general convex domain. J. Math. Anal. Applics (to appear).
J.
J. Acoust. Soc. Amer. 13,
