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NEW DEVELOPMENTS IN DIFFERENTIAL EQUATIONS

This Page Intentionally Left Blank

N 0 RTH- H0 LLAND MATHEMATICS STUDIES

New Developments in Differential Equations Proceedings of the second Scheveningen conference on differential equations, The Netherlands, August 25-29, 1975 Editor

WIKTOR ECKHAUS University of Utrecht

1976

NORTH -HOLLAN D PU BLlSH ING COM PANY AMSTERDAM - NEW YORK - OXFORD

21

@ North-Holland Publishing Company - 1976

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

North-Holland ISBN: 0 7204 0466 5 American Elsevier ISBN: 0 444 11107 7

PUBLISHERS :

NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM, NEW YORK, OXFORD SOLE DISTRIBUTORS FOR THE U.S.A. AND CANADA:

AMERICAN ELSEVIER PUBLISHING COMPANY. INC. 52 VANDERBILT AVENUE, NEW YORK, N.Y. 10017

Library of Congress Cataloging in Publication Data

Scheveningen Conference on Differential Equations, 1975. New developments in differential equations. (North-Holland mathematics studies ; 21) 1. Differential equations--Congresses. 2. Nonlinear theories--Congresses. I. Eckhaus, Wiktor. 11. Title.

QA371.S38 1975 515 ' * 35 76-9754 ISBN 0-444-11107-7 (American Elsevier)

PRINTED IN THE NETHERLANDS

P R E F A C E The field of differential equations is an ever flourishing branch of matnematics, attracting research workers who, in the traditional terminology, range from very "pure" to very "applied". New problems continue to arise from applications or just from the mathematicians' curiosity, old problems give rise to new developments through application of new methods of analysis.

In 1973 a group of Dutch mathematicians consisting of B.L.J.Braaksma, W.Eckhaus, E.M.de Jager and H.Lemei, organized a conference on differential equations with the aim of promoting contacts and stimulating the exchange of ideas among a purposely limited, relatively small, number of invited participants. The success of that meeting, the favourable reactions afterwards and the favourable reception of the proceedings (published as North Holland Mathematics Studies Vo1.13), convinced the organizing committee of the usefulness of such conferences. This volume i s an account of the lectures delivered at the Second Scheveningen Conference on Differential Equations, held on Aug. 25-29,

1975. The organization

was in the hands of the same committee and the conference was again made possible through the generous financial support of the Minister of Education and Sciences of the Netherlands. There were 51 participants from 9 countries. (A list of participants can be found on page 249 of these proceedings.)

The emphasis in this second conference was on nonlinear analysis. This is reflected by the fact that approximately half of the volume is devoted

to

nonlinear

problems. However, linear problems in differential equations are still challenging and subject to new developments, as witnessed by the other half of the contributions. It is a pleasure to acknowledge the gratitude to all authors who have contributed such stimulating accounts of their research. Particular thanks are due to Professor J.L.Lions, who has accepted the invitation' to act as principal speaker and

delivered a series of four lectures. His contribution opens this volume. Wiktor Eckhaus, Editor Utrecht.

V

February 1 9 7 6 .

This Page Intentionally Left Blank

C O N T E N T S J.L.Lions

Some topics on variational inequalities and applications

G.Stampacchia Free boundary problems for Poisson's equation

.

H Amann

1

39

Nonlinear elliptic equations with nonlinear boundary conditions

43

H.Brezis

On the range of the sum of nonlinear operators

65

L.A.Peletier

On the asymptotic behaviour of solutions of an equation arising in population genetics

73

Optimal control o f a system governed by the Navier-Stokes equations coupled with the heat equation

81

Secondary o r direct bifurcation of a steady solution of the Navier-Stokes equations into an invariant torus

99

C.Cuvelier G.Iooss

W.A.Harris Jr. Application of the method of differential inequalities in singular perturbation problems

111

P.P.N.de Groen A singular perturbation problem of turning point type

117

.

B Kaper H.-D.Niessen

Asymptotics for a class of perturbed initial value problems

125

On the solutions of perturbed differential equations

135

On certain ordinary differential expressions W.N.Everitt and M.Giertz and associated integral inequalities 0

A.Pleije1

On Legendre's polynomials

161 175

W.Jurkat, D.A.Lutz and A.Peyerimhoff Invariants and canonical forms for meromorphic second order differential equations.

181

C.G.Lekkerkerker On generalized eigenfunctions and linear transport theory

189

A.Dijksma R.Martini

Integral-ordinary differential-boundary subspaces and spectral theory

199

Some degenerated differential operators on vector bundles

213

vii

LIST OF PARTICIPANTS Invited speakers H Amann Ruhr-Universitgt, Bochum, Germany H.Brezis Universitg P.et M.Cu~ie, Paris, France W.N.Everitt The University, Dundee, Scotland W.A.Harris Jr. University of Southern California, USA G.Iooss Universitg de Nice, France J.L.Lions Colllge de France, Paris, France D .A.Lutz University of Wisconsin, USA H.-D.Niessen University of Essen, Germany A.Pleije1 Uppsala University, Sweden A.Schneider Gesamthochschule Wuppertal, Germany G.Stampacchia Scuole Normale Superiore, Pisa, Italy

.

Participants from the Netherlands B.L.J.Braaksma T.M.T.Coolen C.Cuvelier B.R.Damst6 0.Diekmann A. Dijksma W.Eckhaus J.A.v.Gelderen B.Gilding J.de Graaf J. Grasman P.P.N.de Groen P.Habets A.v.Harten M.H.Hendriks A.J.Hermans H.W.Hoogstraten F.J. Jacobs E.M.de Jager M. Jansen B . Kaper E.W.M. Koper H.A.Lauwerier C.G.Lekkerkerker H.Lemei R.Mart ini H.G.Meijer G.Y.Nieuwland L.A.Peletier H.Pij 1s J. W. Reijn J.W.de Roever J.D.Siersma M.Sluij ter 1.G.SprinkhuizenKuyper M.N.Spijker J.Sijbrand N. M. Temme J.v.Tie1 P.J.Zandbergen

Rijksuniversiteit Groningen Mathematisch Centrum Amsterdam Technische Hogeschool Delft Landbouwhogeschool Wageningen Mathematisch Centrum Amsterdam Rijksuniversiteit Groningen Rijksuniversiteit Utrecht Technische Hogeschool Delft Technische Hogeschool Delft Rijksuniversiteit Groningen Mathematisch Centrum Amsterdam Vrije Universiteit Amsterdam Institut MathEmatique, Louvain-le-Neuve, Belgium Rijksuniversiteit Utrecht Landbouwhogeschool Wageningen Technische Hogeschool Delft Rijksuniversiteit Groningen Technische Hogeschool Eindhoven Universiteit van Amsterdam Vrije Universiteit Amsterdam Rijksuniversiteit Groningen Universiteit van Amsterdam Universiteit van Amsterdam Universiteit van Amsterdam Technische Hogeschool Delft Technische Hogeschool Delft Technische Hogeschool Delft Vrije Universiteit Amsterdam Technische Hogeschool Delft Universiteit van Amsterdam Technische Hogeschool Delft Mathematisch Centrum Amsterdam Rijksuniversiteit Groningen Mathematisch Centrum Amsterdam Mathematisch Centrum Amsterdam Rijksuniversiteit Leiden Rijksuniversiteit Utrecht Mathematisch Centrum Amsterdam Rijksuniversiteit Utrecht Technische Hogeschool Twente

viii

W . Eckhaus ( e d . ) ,

New Developments i n D i f f e r e n t i a l E q u a t i o n s

@ N o r t h - H o l l a n d P u b l i s h i n g Company (1976)

SOME

TOPICS ON VARIATIONAL INECUALITIES AND APPLICATIONS

J.L. LIONS College de France, Paris and IRIA-LABORIA, Le Chesnay, France.

INTRODUCTION. We present here some results on Variational Inequalities of stationary (elliptic) type (Chapters I and 11) and of evolution type (Chapter 111). Each Chapter gives a direct approach to me of the results (without any attempt to be exhaustive) and it also presents some open problems. The plan is as follows : .I,

CHAPTER I. INTRODUCTION TO SOME VARIATIONAL INEQUALITIES(.

'

I . An example of a V.I. 2 . Proof of existence by a penalty argument. 3. Some other V.I. 4 . An application of dynamic Programming.

Bibliography of Chapter I. CHAPTER 11. STOPPING TIMES AND SINGULAR PERTURBATIONS. 1.

Optimal stopping times and V.I.

2 . Singular perturbations and optimal stopping time. 3 . A direct proof. 4.

Singular perturbations in Visco-plasticity. Bibliography of Chapter 11.

CHAPTER 111. V.I. OF FVOLUTION. I . Optimal stopping times. 2 . Strong and weak formulations of the V.I.

3. Existence of a weak maximum solution.

Bibliography of Chapter 111.

(!) A programme somewhat similar to what is done here could be completed by replacing "0 timal sto in time" by "0 timal im ulse control" in the sense of this would d a : l to Quaze-V:riational InEqualitiez instead of V.I.. We refer to 1 2 1 ; we do not study these questions here.

[

I ] A. Bensoussan and J.L. Lions, C.R.A.S., 276 (1973), pp. 1189-1192, pp. 13331338 ; 278(1974), pp. 675-679, pp. 747-751 ; 280 (19751, pp. 1049-1053.

A. Bensoussan, M. Goursat and J.L. LIONS. C.R.A.S. 276 (1973), pp. 1279-1284. (21 A . Bensoussan and J.L. Lions. Book to appear. Hermann ed.

1

:

2

J.L.

I.

LIONS

I N T R O C U C T I O N TO SOME VARIATIONAL INEQUALITIES

ORIENTATION

We give an elementary i n t r o d u c t i o n t o of t h e methods of V . I . We prove t h e existence of a s o l u t i o n by t h e u s e of a n a l t y methods. We give some e r r o r e s t i m a t e s which l e a d t o a general an{ e p p a r e n t l y open problem ( c f . (3.13) ). We present i n Section 4 some remarks r e l a t e d t o t h e question of t h e v a r i a t i o n of .the s ~o l u t i o n of a V . I . w i t h resgect t o t h e geometrical domain. ~~

1 . AN EXAMFLE

~

OF A V . I .

r

Let 9 be a bounded open s e t i n Bn with a smooth boundary consicer t h e e l l i p t i c operator A defined by

. In

9

we

(1.1)

where aij8

(1.2)

qi,

.

E

a.

~ ~ 1 9 )

We shall s e t : 1

H (62) = Sobolev space of o r a e r 1 = f u n c t i o n s

v

v , E E L2(Q), dx .

such t h a t " )

provided with t h e usual h i l b e r t i a n s t r u c t u r e , \\vl\

= norm i n

H~(61) 2

= norm i n L ( 9 )

Iv/

1

u , v E H (62)

and f o r

:

, (f,v)

2

,

= s c a l t r product i n L (62)

we define :

(1.3)

We define H i ( B ) = closed subspace of

(1.4)

which a r e zero on

H'(62)

of f u n c t i o n s

v

r.

We assume t h a t ( e l l i p t i c i t y hypothesis)

(1.5)

a(v,V)

5

a i\vi\',

a7 0,

v

.

1

v E ~ ~ ( 6 2 )

The problem we want t o consider f i r s t c o n s i s t s i n f i n d i n g IJ

Au

------

where

f

i s given i n 1

( )

E

1

Ho(Q)

-f

< O

,

u GO

,

(nu-f)u = 0

in

--

L2(6a).

A l l f u n c t i o n s a r e supposed t o be r e a l valued.

62

u

such t h a t

3

SOME TOPICS ON VARIATIONAL INEQUALITIES

a(u.v-u) p (f,v-u)

T

1

v E H~(Q).

v

I

The s e t of i n e q u a l i t i e s ( 1 .7) (or ( 1 . 9 ) ) i s w h a t i s c a l l e d a V . I .

It was proved i n [7] that under the hypothesis ( 1 .5) t h e r e e x i s t s a unique s o l u t i o n of (1.9). I n Section 2 below, we s h a l l give a proof of t h i s r e s u l t using a p e n a l t y argument. Remark 1 . l . I -

The uniqueness of t h e s o l u t i o n ( i f i t e x i s t s ) i s obvious ; indeed i f u , a, a r e two s o l u t i o n s , and i f we choose v = d ( r e s p . u ) i n t h e V . I . s a t i s f i e d by u ( r e s p . h ) , and i f we add up, we f i n d that : a(u hence

u

-h

= 0

- 6. u - a) c o

by v i r t u e of ( 1 .5).

Remark 1 .2. The s o l u t i o n of (1.4) d e f i n e s a weak s o l u t i o n of ( 1 .6). The next problem i s t o study t h e r e g d a r i t L of t h e s o l u t i o n of ( 1 .9). The s i t u a t i o n here i s e n t i r e l y d i f f e r e n t from t h e case o f , s a y , t h e usual D i r i c h l e t u does not belonq problem : even w i t h c o e f f i c i e n t s i n Cm(a) and w i t h f € (?"(a), t o C"(B). I -

-

Remark 1 .3. I _ _

A "general" V.1. a s s o c i a t e d w i t h t h e operat r A i s ( 1 .9) where a r b i t r a r closed (non empty) convex s e t i n V = H ( P ) . I f Y K = V , ( 1 .9) reduces t o t h e M r i c h l e t iroblem.

P

K

The r e s u l t of existence and uniqueness of u i s v a l i d f o r a g e n e r a l V . I . Cf. [TI. We r e s t r i c t ourselves here t o t h e case ( 1 .8). m

4

2.

J.L.

LIONS

PROOF OF EXISTENCE BY A PENALTY ARGUMENF.

The p l a n of t h e proof i s a s follows :

(2.1)

( i ) we introduce t h e penalized equation 1 Au t u+ = f u E Ho(G)

1

E

(where (ij.)

E

,

E

,

vt = sup (v,oy) ;

we prove t h a t , a s

E

+

0. uE

.We s h a l l a l s o give ( i i i ) a n e r r o r estimate f o r

u-u

t e n d s t o a s o l u t i o n of (1.9).

.

(and uniqueness) of a s o l u t i o n of t h e penalized equation

P a r t ( i ) : Existence

Tz7iT:-

The uniqueness i s s t r a i g h t f o r w a r d , s i n c e t h e o p e r a t o r 1.e.

t

t

(u - v , u - v )

v3 v t

i s monotone,

20.

i s monotone from

w i t h i t s dual,

1

= V -+ V ' = dual space of

Ho(Q)

-

v -+ Av + 1 v t 2 V when L (0) i s i d e n t i f i e d

The existence follows from t h e f a c t t h a t t h e operator

and coercive : (Av

(2.2)

+

t

i v ,v) 3 a i\vl\*

+

1vtl2 ;

t h e r e f o r e one can apply g e n e r a l r e s u l t s from Minty and Browder ( c f . e.g.

[6] and

t k e bibliography t h e r e i n ) . A very e l m e n t a r y proof can be given as follows ; one i n t r o d u c e s a f a m i l y of

subspaces

Vh

c V (')

(where

h

i s a parameter which t e n d s t o z e r o ) , t h a t satis-

f i e s t h e following assumptions :

(2.3) (2.4)

I

Vh

i s f i n i t e dimensional

y v

E V

, there

l\v-vh\\--

0

exists

,

vh E Vh

such t h a t

as h-. 0 .

We then consider t h e

equation I + a(uh,v) + ;(uh,v) = ( f , v )

(2.5) Uh

E Vh

.

There e x i s t s a (unique) f i x e d point theorem.

.................... 1

\I v E Vh,

uh

s a t i s f y i n g (2.5),

by applying t h e Brower's

( ) The constructive proof given now i s presented s o as t o cover t h e method of f i n i t e elements, t h e next s t e p being then t o construct spaces Vh (and this i s a c t u a l l y t h e main point of i n t e r e s t ! ) .

5

SOME T O P I C S ON VARIATIONAL I N E Q U A L I T I E S

Taking v = uh

i n (2.5) we see t h a t 2 a\\%\\

1

+

1uJ2 G

(f,%)

I

hence i n p a r t i c u l a r

(2.6) llq c (where here and i n what follows,

C

denotes various c o n s t a n t s ) .

Therefore we can e x t r a c t a subsequence, s t i l l denoted by (2.7)

uh

w

-P

in

V

weakly

as

.

uh

+ w

i n L2 ( a ) ( s t r o n g l y )

u+ h

-+

in ~ ~ ( 0 )

, such

that

h G .

We a r e going t o v e r i f y t h a t w = u Since (Rellich-Kondrachoff) t h e i d g n t i t y mapping i t follows from (2.7) t h a t (2.8)

L$,

V

-

2 (l) L (0) i s compact

and t h e r e f o r e t h a t

(2.9) Let now

v

.

w+

be given i n

V. We introduce vh s a t i s f y i n g (2.4) ; w e

have : a(uh,vh) + ;(uh I + ,vh) = ( f . vh)

(2.10)

and s i n c e

/\vh-v\\

-.

0

, we

can p a s s t o the limit i n (2.10).

We o b t a i n

8

We now l e t

E

-P

0 . We have

hence i t follows t h a t (2.12)

liU&Il

e

c *

-.

____---------__I__

1

( )

One could avoid t h e use of this theorem by using t h e monotonicity of

(2) By vlrtue 01 t n e uniquenesv of t h e liuut, uh u in a c t u a l l y s t r o n g l y ) without e x t r a c t i n g n subsequcniz. 4

V

v

weakly (and

v+.

LIONS

J.L.

6

Therefore w e can e x t r a c t a subsequence, s t i l l denoted by u

(2.14)

-

Therefore, a s i n p a r t ( i ) , u But, by v i r t u e of (2.131, u+

(2.15)

I n o t h e r words, L e t now

v

v+= 0 )

-.

=

0 .

0

u i n L2(Q) € 2 i n L (61) s o t h a t

be given i n

-f,

K.

It follows from ( 2 . 1 ) that =

V-u

-

1 + - ( u E , v-u

-

I + ) = ;(v

, v-u

P a r t (iii) : E r r o r e s t i m a t e f o r

----------

u

= a ( u E I v-uE)

- (f ,v-uE)

A

a(u,u+) = atu

(2.17)

so that (2.18)

We a r e going t o show :

0

6 '

i s a second-order o p e r a t o r

it follows that

2

-u

We now take t h e s c a l a r product of (2.1) w i t h t o the fact that

(2.20)

:

+ u , v-u E )

, hence (AuE-f

(2.16)

u + - + u+ i n L 2 ( Q ) .

and

u E K..

(AU

(since

u+

, such t h a t

V weakly.

in

u

-+

u

+,U +

;

u+ ; we observe that

-

due

SOME TOPICS ON VARIATIONAL INEQUALITIES

Since

u-u

- - u+ E E

(2.21)

u

+ u-

t3

prove

<w .

+u ;"

W ' e ttlke v = -u- i n ( 1 .7) and we take t h e s c a l a r product of ( 2 . 1 ) with ; adding up, we o i t a i n : U)

d(UE'

hence i t follows ( s i n c e ( u:, (2.22

and s i n c e we have ( 2 . 1 9 ) i t s u f f i c e s

= u+u

that

7

1

u

+

u+u-) E

(u:

u+u,'

2 0

u i ) = 0) t h a t

a(u+u-,

since

+ $I l E+,

u,)

,-

u) 3

+ L(u+ E

c ,

I

E

-u) < -a(u+, u+u-)

;

(2.22) i m p l i e s

hence

using ( 2 . 1 9 ) ) ( 2 . 2 1 ) follows.

8

.

---------

Remark 2.1

The e s t i m a t e (2.20) depends on t h e f a c t that A i s a seccnd order operat o r apd a l s o on t h e p a r t i c u l a r s t r u c t u r e of K (one can construct penalized egua-.t i o n s a s s o c i a t e d t o general V . I . ; c f . [ 6 ] ) . 8

---__--__ Remark 2.2.

It follows of course from ( 2 . 2 0 ) that

u

-,u

in

v

strongly a s

E

-

G.

Bemark 2.3. --_--The above e x i s t e n c e proof g i v e s a method which-be used f o r numerical purposes. But a more e f f i c i e n t method c o n s i s t s i n approximating (1.9) by i t s f i n i t e dimensional analoguous.

where

K

h

Vh on

= subset of

Vh of f u n c t i o n s vh

a.e. i n

& 0

Q

I

and where

= space of f i n i t e elements piecewise l i n e a r .

The numerical s o l u t i o n o f (2.23) uses an i t e r a t i v e method with p r o j e c t i o n Kh ( c f . [4]). 8

Remark 2.4. _--___-__

It f o l l a y s from (2.18) t h a t bounded s e t of L ( 8 ) ) s o t h a t : (2 24)

Au

Au

E L2(Q).

If the coefficients

a,. satisfy 1J

= f

--u 1

+

E

E

remains, a s

E

-

0

in a

8

J.L.

LIONS

-g aa.

a . . E W' '"(9) =J

(2 2 5 )

1

u E H (9) i m p l i e s t h e r e g u l a r i t y r e s u l t

then (2.24) t o g e t h e r w i t h

E

(2.26)

E L"(P,

i.e.

j

2(Q).

One cannot i n general o b t a i n an 'H r e g u l a r i t y , no matter how much t h e dataes a r e r e g u l a r . For a complete study of t h e r e g u l a r i t y of t h e s o l u t i o n of a number of V . I . we r e f e r t o [ 2 ] [ 3 ] . rn

(Of course one should assume

$

empty).

< 0 < J12

1

on

r

K

i n order

not t o be

One can introduce a penalized equation a s f d l o w s :

(3.1)

If Jii

(3.2)

,

E H'(9)

A G i E L2(P)

,

then one proves along similar l i n e s t o what has been done i n Section 2 , that

(3.3)

-

1Iu

I

32

[IE

11 Is

w. 1

K = ( v [ vE Ho(9)

E

= set

of 2

o

v C J I a s . on E c $2

I

measure

)

A penalized equation a s s o c i a t e d t o t h e corresponding V . I .

1

AuE + ;(u6-d,)+

(3.4) where

xE

xE

uE € Ho(Q)

I

= c h a r a c t e r i s t i c f u n c t i o n of E.

-.

If we assume t h a t

E i s an open s e t of

-

D , E e

dE smooth and t h a t (3.6)

is

1

= f

One proves again t h a t u u in of t h e corresponding V . I . b u t t h e E e s t i m a t e

1

I

+

E H~(E)

.

9

HA(9) a8

E

+

0,

/Iu-uE1( ig

,

YE

u

being t h e s o l u t i o n seems dubious.

9

SOME TOPICS ON VARIATIONAL INEQUALITIES

we i n t r o d u c e (following H. B r e z i s )

= f

A'p

s o l u t i o n of

'p

F =

in

'p

=

+

on

dE,

Y

=

'p

on

F ,

Then, i f

u

denotes t h e s o l u t i o n of t h e V.I..

Au

=

f

(3.7)

and we d e f i n e

(3.8)

in

u ig 6 on

$on

E. we have

F a

6E,

u=Oonr.

so t h a t according t o t h e maximum p r i n c i p l e ,

(3.9)

u

4

'p

on

F.

Let us i n t r o d u c e

Construction of penalized equations a s s o c i a t e d t o V . 1

(3.13) u

g i v e " t h e best" approximation of t h e s o l u t i o n

and estimation of t h e e r r o r

(')

There i s of course no

t o a V.I. !

-

J/u uEI\.

u

.

whose s o l u t i o n s

of t h e

V.I.,

a

uniqueness of "the" penalized equation a s s o c i a t e d

10

J.L.

4.

LIONS

AN APPLICATION OF DYNAMIC PROGRAMMING.

A general problem i s as follows : l e t ~ ( 6 2 ) be t h e s o l u t i o n of a V . 1 . k t h e domain Q ; assuming t h a t a l l d a t a a r e given i n , say, Bn , we w a n t t 3 s t u dy v a r i a t i o n s of ~ ( 6 1 ) with r e s p e c t t o 62.

This seems t o be an open question, For t h e case of e u a t i o n s (K = V ) i t i s a c l a s s i c a l problem, t h e main r e s u l t being due t o Hadamard-the v a r i a t i o n of t h e Green's f u n c t i a n ( f o r D i r i c h l e t ' s problem). Recent extensions have been given i n [g] [ l o ] ( c f . a l s o t h e bibliography t h e r e i n ) s t i l l f o r equations. For V.I. t h e r e seems t o be two p o s s i b l e approaches : 1 ) t o use " e x p l i c i t " formulaes g i v i n g t h e s o l u t i o n of some V . I . ( c f . t b beginning of Chapter 2 f o r t h e s e formulaes ; we s h a l l not pursue here tb p o s s i b l e u s e of these formulaes f o r t h e v a r i a t i o n o f ~ ( 6 1 ) ) :

2 ) t o use "dynamic programming" : i t was proved i n [ l ] ( c f . a l s o [ 8 ] ) t h a t one can recover t h e Hadamard's formula by this method, i n case of equations. We show here w h a t kind of " f u n c t i o n a l equation" one i s l e a d t o when dealing w i t h V.1..

-------

Notations :

v =normal t o

r,

\/v\\ = I

, directed

toward t h e

e x t e r i o r o f 62 ;

rA =

surface spanned by

x

- AO(x)v(x)

,

x E

r ,

where

.

x

.

61, = r e g i m included i n

+

e(x)

i s a given continuous s m a l l enough ; 62

>

0 f u n c t i o n on

with dQ, =

r,

;

r ,A

-

i s given

>

0

-_---------

-------I

I

( )

All what follows could be extended t o general symmetric coercive b i l i n e a r forms on H,'(Q).

11

SOME TOPICS ON VARIATIONAL INEQUALITIES Let

f

be given i n

I

(4.1)

L2(62) and l e t

- Au - f g 0 u

E Hi(P)

,

u GO

u

, (-Au-f)u

1

a(cp,cp)

-

= 0

i n 6;1

,

;

u i s t h e unique element w.hich minimizes (' (4.2)

be t h e s o l u t i o n of

1 1

(f,cp)

I

cp E H ~ ( P )

I

cp

< 0.

We define (4.3)

We a r e going t o v e r i f y

- i n a formal

manner

- that

Remark 4.1.

I

I n [1][8] one deduces t h e Hadamard's formula from t h e similar equation t o (4.7) which. i n cafle K = H1(Q) ( D i r i c h l e t ' s prqblem) i s :

(pg0 i n

cp=Oonr.

'2.

I f we introduce (assuming

'p

t o be smooth)

A

is symmetric, i n which case t h e s o l u t i o n o f t h e

(4.6) we have n e c e s s a r i l y

____------------1 ( )

:

T h i s i s g e n e r a l whenever V.I. i s immediate.

LIONS

J.L.

12

+

(4.7)

and f o r

therefore

A

2 0.

"small" w e have

(4.5) gives :

But approximately

on

rh

we have 'p =

-he

+

s o t h a t (lqoptimality p r i n c i p l e " ) (4.8) g i v e s

P f t e r s i m p l i f i c a t i o n , we d i v i d e by

hence (4.7) f o l l o w s .

A

and we l e t

A

-

0 ; we o b t a i n

SOME TOPICS ON VARIATIONAL INEQUALITIES

13

REFERENCES (CHAPTER I ) .

[1]

Bellman, R. and Osborn, H . (1958). Dynamic programming and t h e v a r i a t i o n s of Green's f u n c t i o n s . J. Kath. and Mech. 7, Nb, 1 , pp. 81-85.

[2]

B r e z i s , H.

[3]

Brezis, H. and Stampacchia, G. (1968). B u l l e t i n Soci6t6 Math. de France, 96, PP. 159-180.

[4]

Glowinski, R.,

[5]

Hadamrd, J. (1968). N6moire SUT le problbme d'analyse r e l a t i f ?1i' 6 q u i l i b r e des plaques 6 l a s t i q u e s e n c a s t r e e e . Oeuvres de J. Hadamard. C.N.R.S Paris.

2roblkmes u n i l a t 6 r a u x . Thbse. (1972) J. de Math6matiques Pures e t Appliquees, 51, pp. 1-68.

Lions, J.L.,

and TremoLibres, R. (1976). Book. N o d , Ed.

.,

[6] Lions, J.L.

(1969). Sur quelques m6thodes de $ s o l u t i o n des problhmes aux l i m i t e s non l i n e a i r e s . m o d Gauthier V i l l a r s , P a r i s .

[7] Lions, J.L.

and Stampacchia, G. (1967). Variational I n e q u a l i t i e s . C.P.A.M.

[ 8 ] L u r e , X.A.

20.

(1975). Optimal c o n t r o l i n problems of mathematical physics. Moscow. (In Russian).

[g] Murat. F., and Simon, J. To appear. [ l o ] Palmerio, B. and Derviewr, A . (1975). Une formule de Hadamard dans des problhmes d ' i d e n t i f i c a t i o n de domaines. C.R,Ac. Sc. P a r i s .

.

14

J L .LIONS

11.

STOPPING TIMES AND SINGULAR PERTiIRBATIONS

O R 1 ENTATI ON

I n this chapter, we f i r s t g i v e , following [ l ] [TI, an i n t s r p r e t a t i o n of t h e s o l u t i o n of some of t h e V.I. introduced in Chapter I i n terms of t h e o p t i m a l cost f u n c t i h aP a stopping time problem. T h i s permits t o g i v e a r e s u l t of s i n g u l a r p e r t u r b a t i o n s f o r a f a m i l y of V . I . (Section 2 ) . These results a r e taken from [ 6 ] . I n Section 4 we g i v e some preliminary i n d i c a t i o n s on a very i n t e r e s t i n g paper of MOSSOMV and MIASNIKOV [9] about s i n g u l a r p e r t u r b a t i o n s for V . I . of a d i f f e r e n t n a t u r e . Problems of s i n g u l a r p e r t u r b a t i o n s f o r o t h e r (and a c t u a l l y aimp1er)classes of V.I. a r e considered i n [ 6 ] [TI. 1 . OPTIMAL STOPPING TIMES AND V . I .

We s t a t e here without p r o o f s some of t h e r e s u l t s e s t a b l i s h e d i n [ l ] [2] Let

[TI.

be t h e s o l u t i o n of t h e s t o c h a s t i c d i f f e r e n t i a l equation ( I t o ' d

y,(s)

d i f f e r e n t i a l equation) :

(1.1)

where

c1

o(x) i s (1.2)

If

x €

Let

P

i n IF?

I

.

a(x) E

Amn

w(s)

standard Wiener process i n Rn.

=

; R ~ ) a ( x ) + = a(x)

we denote by ~ ( x )t h e s m a l l e s t

be an a r b i t r a r y stopping t i m e f o r

8

c1

g(y) i s also

y

8

(l)

o(x)

> a.

I

s> 0

such that

, eg

T

, yx(s)

$

P.

.

We consider the cost f u n c t i o n

where

a

>

0

,

where

f

i s given i n

P

, and

we d e f i n e t h e optimal cost f u n c t i o n

as

(1.4)

u ( x ) = inf

Jx(e).

S T

One proves [I]

[T]

that

u can be c h a r a c t e r i z e d as the s o l u t i o n of t h e V.D.

(of t h e type introduced i n Chapter I )

I _-_---___-___---( 1 .5)

A U - ~ C O ,

u-oon

( ' ) These hypothesis bliography t h e r e i n .

U Q O , ( ~ u - f ) u = o i n ~ ,

r

a r e unnecessarily s t r o n g . Cf. f o r i n s t a n c e [ I ] and th bi-

15

SOME TOPICS ON '.'APIATIONAL INEQUALITIES where

i s given by

A

where

g ( x ) = IgJxH

and where

I

Remark 1 . l . ----_____

In (1.6) t h e o p e r a t o r A appears i n a non divergence form. This leads t o t h e n a t u r a l problem of extending t h e theory of V.I. t o o p e r a t o r s of type ( 1 .6) with non smooth c o e f f i c i e n t s , say : a. f 1J ,

"

Co(z) (but not i n

el(=))

and f o r convex sets of t h e type 1 K = ( v l v E H ~ ( P )I v g

o

a.e.

in

0

,or

v

<+

a.e.

i.1

Q!,

This i s done i n [ I ] ; it would be i n t e r e s t i n g 1-0 see iP similsr resjlzs can be obtaiced f o r o t h e r s e t s K

.

Oce can of course

-

i n case of hypothesis ( 1 .2)

- write

A

i n divergence

form :

(1.7)

7 -LJ

a i = - g i + C J

One does&

.

J

n e c e s s a r i l y have

2

a ( v , v ) '2 a, i/vI)

v

v E

1 Z0(Q)

,

al

>c ,

but conditions Caij(x)

E~

cj

2 > a o l ~ )

I

a.

,a

> e

a r e s u f f i c i e n t f o r t h e r e s u l t s of Chapter I t o hold t r u e , Remark 1.3. ------_---

m

i n t e r p r e t a t i o n of t h e s o l u t i o n u of t h e V.I. ( 1 .5) i s known, by formuA i s symmetric i n i t s p r i n c i p a l p a r t . m

la ( 1 .4), o n l y whenever

--------

Remark 1.4. Let us r e c a l l a t this point t h a t

16

J .L.LIONS

whose s o l u t i o n can be i n t e r p r e t e d as t h e We r e f e r t o [ l ] f m Q t h e r V . I . cFtimal cost function of a s u i t a b l e stopping time problem. It i s an open question t o determine t h e c l a s s of " a l l " V . I . of e l l i p t i c type whose s o l u t i o n can be i n t e r p r e t e d by a stopping Time problem, I Remark 1.6. -------We a l s o r e f e r t o [ I ] , Volume 2 , an? t o [ l o ] f o r t h e i n t e r p r e t a t i o n of 3.1. of s i m i l a r type with Neumann (or o t h e r s ) boundary c o n d i t i o n s , 0

-

2 . SINGULAR PERTURBATIONS ANG (IpTF.MAL STOPEIN?- TIP2

Let us consider t h e s i t u a t i o n of Section 1 with

u

(2,l)

Let

yE(s)

=

€ 1 .

be t h e s o l u t i o a of t h e corresponding I

differer i s

3's

.

eqLa-

tion : dyE(s) = g(yE)ds

(2.2)

+

E

dw(s),

yE(o) = x

I

acd l e t u s s e t ('I

(2.3) According t o Sect;on

,

1

1

u_ is c h a r a c t e r i z e d by t h e s o l u t i o n of t h e V.I.

2

GO

- f (2.4)

+

Bus

- f)u,

I

= 3

u.
on

,

61

u

E HL(Q)

,

whert hV

(2.5)

+ av.

B u = - cgi(x)

We can now study t h e behaviour of

u a3 E-

Let u s f i r s t con.sidar t h e d e t e r m i n i s t i c

E

0.

-t

E

3888

= 0 ; yx(s) = yo(s) i s the

s o l u t i o n of t h e o r d i n a r y d i f f e r e n t i a l equation

(2.6)

dY

let

z ( x ) = inf {sls> 0

= g(Y)

I

= x ;

Y(0)

,

(2.7)

(')

yx(s)

P

f(yx(s))ds,

TE(x) = e x i t time of

E

for

y , '

L '

)

I

and l e t

0

Cz.

u

be defined a8

17

SOME TOPICS ON VARIATIONAL INEQUALITIES

Then u -

can be c h a r a s t e r i z e d as t h e s o l u t i o n of t h e V . I .

Bu

(2.8)

'J.

-f

GO

,

= 0

for

x E

z(x) = 0

u< 0

r

, (Bu-f)u

,

in 62

= 0

and, such t h a t

.

Remark 2.1. -__------Condition ( 2 . 9 ) shculd ba t r e a t e d a l i t t l e aore c s r e f u l l y ; c f . [ 2 j f o r d e t a i l s . Roughly speaking, (2.9) meam t h s t u should be zero wher.ever g.v 3 C. Remark 2.2, -_------

One can s c l v e d i r e c t l y t h e V . I .

first, by c h s g i n g can always assume t h a t

We then consider

u into

- assuming

exp (- p ( x ) j u

(2.10)

( 2 -11 )

-

p

,u,,)

i t follows (using t h e f a c t t h a t u

(a

D

= 0 if

i s bounded i n

g.v

penalized equation

(Observe t h a t :

=ji2+ u

s u i t a b l y chosen, one

- tb.e

implies

Moreover, s i n c e (Bu?

(2.1 7 )

, with

t o be s a t i s f a

T h i s equaticn admits a unique s o l u t i o n

therefore

as follows :

(2.8)(2.9)

0 ) :

L2(r) as

r)

-.

0.

18

J.L.LIONS

Therefore we can e x t r a c t a subsequence, s t i l l denoted by

u

-t

u

in

2 L ( Q ) weakly.

BU

-t

BU

in

~ ~ ( 6 ,weakly, )

9 9

<

(2.19)

in

L

2

u n

, such

that

( r ) weakly.

It follows from (2.18) and (2.19) t h a t

I <=urn.

20 ,

if g.v

(2 2 0 )

u = o

If

2 i s any f u n c t i o n E L (Q)

v

(Bun- f

.

i .e

, v-u n )

2

,v < 0

a.e.,

we have

0

(Bun,v)-(f, v-u ) 3 (Bun, un) =

(2.21)

rl

and

so t h a t (2.21)

implies

(2 2 2 1

(Bu

-

f , v-u)

5 0

if

v C0

E L2(Q).

v

I

Therefore. i t w i l l be proven t h a t u i s a s o l u t i o n "I of ( 2 . 8 ) ( 2 , 9 ) , Bu E L2(Q) i f we check t h a t u g 0. But for every v E L2(Q) ,

such

u

that

+

(un-v

(2 23)

+

, U - V ) $ O .

rl

Using (2.1 51, (2 2 3 ) i m p l i e s (2

4 v + , u-v) 3 0.

24) Let

i .e

'p

E

L2(0)

and l e t

h be

>

A((u+cp)+

B 'p)

20

((U+h'p)+,

'p)

0

0

.

We take

v

<

u+hv i n (2.24),

. .

We l e t A + 0 , and we o b t a i n (u+ ,'p) 5 0 t h e proof i s completed.

V

'p

€ L2(Q)

1

i .e. u+ = 0

_ I _ -

(')

hence

The uniqueness i s immediate.

I

and

19

SOME TOPICS ON VARIPTIONAL INEQUALITIES

The r e s u l t 3f s i n g u l a r p e r t u r b a t i o n s i s now a s follows :

I

(225)

when

E

u

,

0

4

one has

- i n L2(B)

and

a.e.

u

The proof, given i n f21, r e l i e s e n t i r e l y on the i n t e r p r e t a t i o n of

of

and

u by (2.3) and (2.7). It i s t h e r e f o r e an " i n d i r e c t "

proof ; a " d i r e c t " proof has been obtained by H.BREZIS [:1] using r a t h e r d e l i c a t e a p r i o r i estimates ; we g i v e . i n t h e f o l l o wing s e c t i o n , a d i r e c t proof i n a very p a r t i c a l a r case ; we r e f e r t o H. BREZIS, l a c c i t . and t o [ 8 ] , f o r t h e general case.

7.

A DIRECT PROOF. Q7e consider, t o simplify t h e exposition, a two-dimensional case. We assume

that :

(3.2)

[

61 = 1 0 , l [ x l o , ;

(7.1)

r

I

3U

Bu==. + . The condition (2.9) becomes : u ( 0 , x ) = 0. 2

(3.3)

We assume t h a t

f

(7.4) Then, a s

E

(3.5)

u

, f, 0

.

E L2(61)

L

I

+

u

I

1

Figure 1

,

L ~ ( Q ) weakly

in

every s e t 62'

(3.6)

1

on Figure 1 .

For the p r o o f , we introduce

I

as i n d i c a t e d

w

=

,

w

11

s o l u t i o n of t h e penalized e q u a t i o n .

(3.7)

I

2

1

+

-%Aw+Bw+-w

=f.

r)

w = ~ o n r .

w

We know t h a t

-t

u

1 i n H3(P) a s q

-t

C

and we immediately o b t a i n t h a t (7.8)

Jwj c c ,

E l a m

where t h e C's do not depend on

(7.9)

{.,I

E

wI G

and on

C C, c ( g r a d u b \ G

and i n order t o prove (3.5)(7.6)

l

c

--lw r).

+

1

Q

c ,

Therefore :

C

i t s u f f i c e s t o prove t h e following a p r i o r i

* x1

20

J.L.LIONS

estimates : let then :

'p

l"l

(3.10)

C'(5).

be

dW 1

with support i n a s e t

2

------------

We s e t

;

c .

I'pgj;I

Proof of (3.10)

( a s i n d i c a t e d on Fig.1)

c c '

b

(3.111

-8'

:

g = f-

-I w + , and we

m u l t i p l y (3.7) by

'p2

2 6x 1

9

.

(3.12)

But

gives

+

It i s enough t o prove t h e follcrwing : let with compact s G p p o r t i n l o l l [ ; then

l +d Wx la. 2

(3.1 3)

2

-% A& + "x1

(3.14)

0

.

Ow

+

adw

=s B

To s i m p l i f y t h e w r i t i n g , l e t us s e t

+1

9

b = 3 if

obtain :

x = 3 I

ur if

; applying

+2

2

+ a(+dwJ 2

But

1 9

+-(+dw

2 G ( d w ) dx = 0

+1

= 0 ; t h e r e f o r e we

$OW) = ( O f , + d w ) .

s o t h a t (3.15) reduces t o :

x2 *

t o (3.7) g i v e s

dw.

x = 1 and + ( o ) = + ( 1 ) 1

(3.1 5)

I

B

dw+ = d f .

We t a k e thc s c a l a r product of (3.14) with

But

be a smooth f u n c t i o n of

21

SOME TOPICS ON VARIATIONAL INEQUALITIES

( 3 -16) = ($

af

WW).

I

The f i r s t term i n (7.16) i s equal t o

4. SINGULAR PERTURBATIONS I N VISCC-PLASTICITY. 4.1. POSITION OF THE PROBLEM. We consider i n n n 62 =

c

m

,w

bw= 1362 = i?

an open s e t

a s on Fig. 2

P,

, 66

being a smooth closed curve.

The formal problem t h e following :

('I

we consider i s

r

v given such t h a t

for

v i s "small" a t i n f i n i t y ,

(4.1)

and f o r

E

>

O

, we

figure 2

define

and we a r e looking f o r

inf JE(v)

(4.4)

v = 1 on

,

v s a t i s f y i n g (4.1)(4.2)

an8

r.

I

-Remark -------4.1. Without condition (4.1) t h e problem i s trivial : one would have

id J (v) = o = J ( 1 ) .

(')

It i s formal, f o r t h e time being, since one has .to make p r e c i s e t b statement

(*)

W e denote i n this way t h e product space ( L ( Q )

(4.1).

1

n L2 (66)) 2 .

.

22

J L. LIONS

(4.5) (4.6) then (4.4) i s equivalent t o t h e V . I . e a ( u E , v-u )

+ j(v)-j(uE) 2 0

B v satisfying (4.1)(4.2),

(4.7) u

satisfying (4.1)(4.2).

-_____--__

Remark 4.3.

Problem (4.4) i s considered i n [g] ; i t a r i s e s i n connection with t h e problem of l c n g i t u b i n a l motion of a cylinder i n a v i s c o p l a s t i c medium ; o t h e r a s p e c t s of this question a r e considered i n [g]. The problem s t u d i e d i n [g] c o n s i s t s i n f i n d i n g an asymptotic expamion f o r i d J~(v). 4.2. SOME FUNCTIllN STACES. ( 1 ) We introduce :

-62.

:C

9n the space

(z) = space of

?-(awe

C"

functions i n

-hd

I

with compact support i n

introduce :

(4.8) (4.9) These q u a n t i t i e s are obviously norms. We d e f i n e :

We obviously have : (4.1 1 )

c%

voc v . We a r e g o i n g t o show t h a t

(4.12)

1

v c Lloc(Q)

and t h a t , i n some sense, one has (4.1) y v €

V.

figure 7 .

We do not r e s t r i c t t h e g e n e r a l i t y by assuming t h a t 0 E w . L e t CR denote t h e c i r c l e of center 0 and r a d i u s 51 ; we t a k e

_---_---_----

sE?2&-These considerations a r e not introduced i n [g]. (' )

R large

23

SOP% TOPICS ON VARIATIONAL INEOUALITIES

(4.13)

*

Indeed, u s i n g p o l a r coordinates, G(R,0) =

2 (rIe)

-

'p(xl ,x2) = cp(r.0). one has dr

hence

so t h a t :

hence (4.13) follows. ~ s s u m i n g I?

I

smooth enough, one a l s o shows t h a t ( c f . n o t a t i o n s on Fig. 3)

(4.14)

set

Using ( 4 . 1 3 ) ( 4 . 1 4 ) , i t follows t h a t , f o r any open Q G IT , I? c 0 , one has

s bounded,

jC

(4.16)

lvldCR-%

R

0. a s

-+-

R which makes ( 4 . 1 ) p r e c i s e (and shows

v E

The r e s t r i c t i o n s of which c o n s i s t s of f u n c t i o n s

W

v

spans

V

,

V to 0

span t h e space

E L 1 (Q) such t h a t

One can t h e r e f o r e define

when

p' V). Figure 4 .

Remark 4 . 4 . --_----_

(51 t h a t

1

vlr

v

Ir

dW ax.

W1 'l

E L1(q)

(q)

f

i.

€ L'(I') (by using (4.14) ; i t follows from

spans e x a c t l y ~ ' ( r ) .

I

Remark 4.5.

Since ['p] 3 ([(pIIII

one has (4.11) and, i n p a r t i c u l a r , one has (4.6) y v € V o .

J . L .LIONS

24

v E V t o Q span the space

The r e s t r i c t i o n s of

H1/2(r). C f .

H1(S)

and t h e t r a c e s

v(r

Span

[4] f o r t h e s i u d y of spaces o f this kind.

The p r e c i s e statement of problem (4.4) i s now t o f i n d (4.1 7)

inf J ~ ( v )

For every

>

E

,

v E

(E

>

fixed

In Y

(4.18)

R(Q)

The uniqueness

; let

v be a minimizing sequence, [v,] n 0) and t h e r e f o r e we can e x t r a c t a subsequence such t h a t :

J

1

u i n H (q) weakly,

-t

Q as dn Fig. 4.

f o r every

Let

r.

= I on

0 this problem admits a unique s o l u t i o n ,

follows from t h e s t r i c t convexity of i s bounded

,v

V,

be the space of summable measures

on &\ ; one can assume that

ir

.;;“-

n

dV

R(Q)

in

Gi

weak s t a r ,

and by v i r t u e of (4.18)

Pi

du =ax E

figure 5

1 L (0).

i

W e hzve a l s o :

du€ L2(Q) dX

sc that

and u=i ;n i i s s o l u t i o n of t h e problem.

u

We s h a l l denote by

u

rs

t h e s o l u t i o n of (4.17). We want now t o study t h e

function :

as 4.3.

E

4

JE(uE)

E

4

0.

rn

ASYMPTOTIC EXPANSION.

[9]. We normalize t h e l e n g t h d s on r

We follow now t h e t o t a l l e n g t h being t h e coordinates x

1

’X x

(4.19)

L . We t a k e coordinates

so t h a t 0 sg s

<1 ,

s , n such t h a t (we denote by x, y

J

=

x(s)

+

y’(s)n

,

y = y(x)

- xt(s)n ,

and we assume t h a t everything i s smooth so t h a t , i n p a r t i c u l a r . the curvature k ( s ) of (4.20)

r

satisfies k(s)

>ao

% 0

.

I n t h i s new system o f coordinates, on6 has :

SOHE TOPICS ON VARIATIONAL INEQUALITIES

25

By a method which i s standard i n the theory of s i n g u l a r p e r t u r b a t i o n s . we now keep i n (4.21) only t h e normal d e r i v a t i v e f o t h e boundary. We introduce i n t h i s way : (4.22)

A s i t has been done f o r

of a f u n c t i o n u*

JE

, one

v e r i f i e s the e x i s t e n c e and uniqueness

such t h a t :

*

u

i s small a t i n f i n i t y

u* ( s , o ) = 1

(l)

,

,

H (u") i s minimum, o r , what amounts t o the same t h i n g , u* i s E

E

s o l u t i o n of the V.I.

n

v

v(s,o) = 1 ,

such t h a t

We a r e going t o v e r i f y t h a t define

u*

i s e x p l i c i t e l y given a s follows : we

y = yE(s) by :

(one e a s i l y checks t h a t one uniquely defines

i n this manner) ; then

u*(s,n) =

(4.25)

0

for

Indeed one e a s i l y checks t h a t

n > y .

u

s a t i s f i e s the " l l e r ?quation"

du" (4.26)

(' )

y

Actually

3

u*

has a compact support.

J .L.LIONS

26

I f we now multiply (4.26) by

- u"

, we

obtain :

X denotes t h e l e f t hand s i d e of ( 4 . 2 3 ) . w e o b t a i n

and i f

and

v

X

>0

m

hence (4.23) follows. 4

One i n t r o d u c e s next

w

obtained by, a Taylor expansion in (4.25) :

One v e r i f i e s next t h a t

From t h e d e f i n i t i o n s , i t follows t h a t JE(wE) 3 JE(uE) 3 H , ( t )

3 HE(wE)

+ O(E)

so that JE(uE) = HE(WE)

+ O(E)

and we f i n a l l y o b t a i n ( 1 ) : JE(uE) = L(1

(428)

+

e"

f:,

ds

f

O(E))

.

8

Remark 4.6. -------

I n [g], t h e A. give t h e second term hi t h e expansion of

I _ -

(')

I n [9],

t h e A. f i n d

= 3

i n s t e a d of

4 v 3 .

JE(uE).

8

SOME TOPICS ON VARIATIONAL INEQUALITIES

REFERENCES (CHAPTER i I )

21

. Book t o appear. H e r m a n n Ed. Vol.1

[l]

Bensoussan, A . and Lions, J . L . Vol.2 (1977).

(1976),

[2]

Bensoussan, A . and Lions, J . L . (1975). Problkmes de temps d ' a r r 8 t optimaux e t de p e r t u r b a t i o n s s i n g u l i b r e s dans les i n g q u a t i o n s Quasi Variationn e l l e s . Lecture Notes i n Economics and Mathematical Systems. Springer ( 1 0 7 1 , p p , 567-584. (1973). Problhmes de temps d f a r r 8 t optimal e t Ingquations V a r i a t i o n n e l l e s p a r a b o l i q u e s . Applicable Analysis. V O ~ . 31 PP. 267-294.

[3] Bensoussan, A . and Lions, J . L .

[4]

Deny, J. and Lions, J . L . 305-370.

[5]

Gagliardo, E. (1957) C a r a t t e r i z z a z i o n e ,

[6]

Huet, D.

[7]

Lions, J . L . ( l q T ) . P e r t u r b a t i o n s s i n g u l i h r e s dans l e s problbmes aux l i m i t e s e t e n c o n t r 8 l e optimal. Lecture Notes i n Mathematics. S p r i n g e r . Vol. 323.

[8]

f i g n o t , F. and Duel, J.?. (1976).&chive

[g]

Mosolov, P .P. and Miasnikov, V.P. Boundary layer i n t h e problem of l o n g i t u d i nal motion of a c y l i n d e r i n a v i s c o p l a s t i c medium. P.M.M. 38 ' (1974)r PP. 682-692.

Les espaces du t y p e de Beppo L e v i . 5 (195>1954),

. . Rend.

pp.

Sem.Mat. Padova 27,p.284-305.

(1968) P e r t u r b a t i o n s s i n g u l i h r e s d'In6q. V a r . 267, pp.932-934.

Rat. Mech.Analysis.

..

[lo] Bensoussan, A . and Lions, J.L. (1975) Diffusion Processes . i n Probabilistic Methods i n I Z i f f e r e n t i a l Equations , Lecture Notes i n Mathematics, ' S p r i n g e r , 451 , pp. E 2 5 . [ l l ] Brbzis, H.

P e r s o n a l communication.

28

J .L.LIONS

111. VARIATIONAL INEQUALITIES OF EVOLUTION

ORIENTATION [2], t h e s o l u t i o n of a s t o c h a s t i c optimal stopping time problem i n terms of V . I . of evolution. Section 2 d e f i n e s t h e strong and weak s o l u t i o n s o f t h e V.1. & e v o l u t i o n met i n Section 1 , and i n Section 3. we prove an important r e s u l t of fignot-

W e present i n Section 1. following [l],

Puel [6]. Other r e s u l t s along t h e l i n e s of this ( i n t r o d u c t o r y ) chapter w e given i n r21. 1 . OPTIMAL STOPPING TIMES.

We consider, a s i n Chapter 11, Section 1 , t h e s t o c h a s t i c I t o ’ s d i f f e r e n t i a l equation :

+

dy = g(y)ds

(1.1)

whose s o l u t i o n i s denoted by

, we

t

s

>t

f ( x , t ) i s a given -say continuous

,

+(x, ) is a given -say

- f u n c t i o n i n V x]~
,

P

time

e

2)

denote by

z

xt

f3 such t h a t

f inf(Txt,T) =

A T.

T~~

is t h e c h a r a c t e r i s t i c f u n c t i o n of t h e s e t where

*A<W

>t

6-2.

We consider stopping times

..

,

s

Yxt(S)

where

y(t) = x

t h e smallest

S t a r t i n g from such t h a t :

(1

,

y (9). xt

,at

xE D

a(y)dw(s)

A

<

p.

We then define t h e optimal cost f u n c t i o n

(1.4)

u(x,t) = inf

~ ~ ~, ( ee g )T~~

One can then prove [l] dU

- x + A U - f

(1.5)

(-

dU

+

Au

A T.

[2] that u ( x , t ) i s c h a r a c t e r i z e d a s t h e s o l u t i o n of :

$0

U - $

- f ) ( u - +) = 0

in

C O r

p

= 0 X

]O,T[

( t = C does not p l a y any p a r t i c u l a r r o l e then), where,

A

i s given by

29

SOPIE TOPICS ON VARIATIONAL INEQUALITIES 2

(1.6)

AU

=

dU - C a1.1 . .(x) d X . d X . - c g i w E, l3U

1

J

I

1

with t h e boundary conditions u(x,t) = G

(1.7)

if

x

E

r

= a61

I

and t h e " i n i t i a l " condition

( I .8)

u(xIT) = 0

I

x E 61.

8

Remark 1 . l . _ I -

I n what f o l l o w s , we s h a l l solve d i r e c t l y ( 1 .5)(1 .3)(1 .8) without r e f e r e n c e t o t h e optimal stopping time problem.

We s h a l l change t h e o r i e n t a t i o n of time and we s h a l l use gence" form ( l ) I i .e. we shall consider :

A

i n i t s 'tliver-

(1.9) and we s h a l l lomk f o r

u = u ( x , t ) s o l u t i o n of

dU -+AU-fG@

Bt

11 . l o )

IU-$CG

(x dU + - f ) ( u - c) AU

=

G

in

I

Q

,

w i t h ( 1 .7) & u ( x l o ) = 0.

(1.11)

-------

Remark 1 .2. What we a r e going t o say extends t o o p e r a t o r depend on x and on t

.

A

whose c o e f f i c i e n t s 8

a i j I ai

Remark 1.3.

--I---

e s h a l l give The s e t ( 1 . 1 0 ) ( 1 . 7 ) ( 1 . 1 ~ ) i s a so-called V.I. of e v o l u t i o n . W p r e c i s e s t r o n g and weak formulations of this problem i n S e c t i o n 2 below. Remark 1.4. We r e c a l l t h a t i f we d e f i n e (1.12)

then

w

i s t h e s o l u t i o n of

1 ( ) A transformation which i s not p o s s i b l e (or would introduce c o e f f i c i e n t s d i s t r i b u t i o n s ) i f t h e a . .Is a r e o n l y continuous ; we r e f e r t o [l]. 1J

30

J .L.LIONS

we have t h e r e f o r e a V . I . w i t h a cmvex which depends on

-

-

t.

Moreover and t h i s i s t h e main point i n t h e formulation o f t h e problem ( 1 * 5 ) (1 .4) , t h e r e i s no reason t o assume any smoothness on $ except , say, + continuous.

-

Under t h e s e circumstances, t h e problem does not admit i n g e n e r a l a s t r o n g solution. We t h e n proceed with t h e d e f i n i t i o n of weak s o l u t i o n s .

2.2.

Weak formulation. Let u s d e f i n e :

and l e t us make t h e only assumption t h a t

M # @ .

(2.11)

Let

u

be a s o l u t i o n of (2.8) and l e t

v

be an element of

we have X

=[[

(&I

v-u) + a(u,v-u)

-

(f ,v-u)]

dt

+

x

; i f we s e t

[(.wD v-u)dt

and by v i r t u e of (2.8) w e see that

We now define a

“weak soluticsn”

as a f u n c t i o n

The main r e s d t i s now given i n Section 3 below.

u E

s a t i s f y i n g (2.121,.

31

SOME TOPICS ON VARIATIONAL INEQUALITIES

subject t o w j x , t ) = 17

if

x E

w(x,T) = 3

x

E sa ;

r ,

(1.14)

f ~ r m u l a .so t h a t ( 1 .4) can be thought of a6

(1.12) i s the c l a s s i c a l FeynmaM-Kac an e x t e n s i o n of t h i s formula.

8

Remark 1 .5. ------One can use formula (1.4) t o o b t a i n informations on t h e p r o j e c t i o n of t h e s o l u t i o n of (1.5) ( 1 .7)( 1 .8) f o r i n s t a n c e f x s i n g u l a r p e r t u r b a t i o n s as in Chapter 11. We r e f e r t o [ I ] . 8

-

2.

STRONG ANI! WEAK FORMULATIONS OF THE V.1

2.1.

Strong Formulation.

.

We d e f i n e

i s equivalent t o :

f o r a l y (1.10)(1.7)

where in ( 2 . 2 )

t i s f i x e d a.e.

By a " s t r o n g s o l u t i o n " we mean a f u n c t i o n u = u ( x I t ) such t h e t 2

1

(2.3)

u E L (OIT

;V)s

(2 -4)

-

;V')

du 2 E L (O,T dt

C

(2.5) (2.6)

and

+

V = Ho(S2)

I

,

a.e.

U(X,O) = 0

u

s a t i s f i e s (2.2). The main problem. Let us s e t :

(2.7)

K ( t ) = {vl

then ( 2 . 2 ) is equivalent t o :

1

V E H ~ ( ~ ;I )

v(x) < + ( x , t )

a.e. i n 611 ;

32

7.

J.L.LIONS

EXISTENCE OF A WEAK MAXIMUM SOLUTION.

(3*1)

Step ----

There e x i s t s , i n t h e s e t of weak s o l u t i o n s of (2.12), a maximum s o l u t i o n v w weak s o l u t i o n , one has w g u .

i . e . which i s such t h a t ,

1 : We introduce t h e penalized equation I

I:

(3.3)

u

=

x E r ,

0 if

u (x,o) = 0

and we check t h a t

u

--t

G

in

L~(O,T

;v) , weakly

(3.4) being a weak s o l u t i o n of ( 2 . 1 2 ) .

Step 2 : We

----

show t h a t :

(3.5)

u

Step ----

(3.6)

4

as

3 : We show t h a t given w Upj)W

i .

E

, weak

v E >

s o l u t i o n of (2.12),

one has :

0 .

Of course ( 3 . l ) immediately f o l l o w s , w i t h a c t u a l l y t h e supplementary i n f o r m a t i o n that :

(7.7)

t h e m a x i m u m s o l u t i o n i s t h e limit of t h e

I -

(

1

This r e s u l t i s due t o Mignot a n d Puel [ 6 ] .

u

's

as

E

+

0.

m

u.

33

SOFE TOPICS ON VARIATIONAL INEQUALITIES

Proof of Step 1 . 3ne f i r s t proves t h e e x i s t e n c e and uniqueness of a s o l u t i o n of (3.3)

which

bU

2

u € L (0,T;V)

i s such t h a t

,$

E L*(O,T,V').

I n order t o o b t a i n a p r i o r i e s t i m a t e s , we t a k e t h e s c a l a r product of (3.3) with

-v

u

, vo E x

; we o b t a i n :

3u

1

uE-v ) + a i uE ,uE-vo) + ;((uE-+)+, uE-+

($,

13.8) and s i n c e

,it

+-vo 9 0 (

0

+

dV

s u €-v 0

.iit

= ( f , u E-VO)

follows t h a t

aiu -v ) .E

+ -vo)

+

+ 2

1

6)

;l(Uc-

1

+ a ( uE ' u -v0) +

I <

(f,

G7uE-vo)

+ a(vo

,uE-vo)

+

yo)

hence i t follows by standard arguments t h a t

where t h e

C's

denote constants which do not depend on

We can e x t r a c t a subsequenco, s t i l l denoted by (3.11)

let

v

uE -t

be given

and t h e r e f o r e

in

in

x

2 L (0,T;V)

weakly, and i n

u

E

. , such

Lm(0,T;L2(9))

; i t follows from (3.3) t h a t

that weak star ;

34

J.L.LIONS

.J u i s a weak s o l u t i o n .

In order t o prove t h a t

G<

(3.13)

i t remains o n l y t o check t h a t

4 a.e.

We argue as i n (2.23) (2.24), Chapter 11. We

(7.14)

((V'$)+

JOT

which i s v a l i d f o r every

-

?UE:-d+,

start from t h e i n e q u a l i t y

'a 0

v-u,)dt

; by v i r t u e o f (7.10)

vEL!(Q)

w e can pass t o t h e limit

i n (7.14) ; we o b t a i n

(3.1 5)

((v-+)+, v-G)dt We now take

and we l e t

A

-.

[ ((G 0

,A >

+ A?

v=

-4

tr

20 0

, 'p

+ h ' p ) + , 'p) dt

E

V.

L2(Q)

; we o b t a i n a f t e r d i v i d i n g by A :

30

; i t comes :

-

Proof of Step 2 . LeE E < 2, u = u

, u,

=

G.

We w a n t t o show t h a t

u

s c a l a r prcduct of (3.7) ( r e s p . t h e p e n d i z e d equation r e l a t i v e t o resp. with - ( u

We have +$

2 fl-$

SO

-a)+).

a)

with (u

- a)+

We o b t a i n :

Y 3 0 ; indeed we i n t e g r a t e over t h e s e t where t h a t (u-+)+> (L$)+ an2. Y 3 0.

hence i t follows t h a t

We t a k e t h e

(u-a)"

= 0.

u

Q

; on this set

Consequently (3.16) i m p l i e s t h a t

35

SOME TOPICS ON VARIATIONAL INEQUALITIES

---_-----

Proof of Step 3 . T h i s i s the most i n t e r e s t i n g part i n t h e proof.

Let

w

be a weak s o l u t i o n and l e t us i n t r o d u c e

(3.18)

z = w - u

(3.19)

x,= x -

,

E UE.

There one v e r i f i e s , by an elumentary computation, t h a t

.,

Indeed, f o r e v e r y v

E L we have

and we deduce from (3.3) t h a t

by s u b s t r a c t i o n , we o b t a i n ,

I f we s e t

-u

=

v , t h i s i s (3 2 0 ) .

We now prove that

(3 2 3 ) X ~ E B D .

where

@= (3 24)

(el

e p o Indeed, l e t

v

BE L;?O,T:V),

ae E L2 ( O s T ; V ' ) ,

8(T) = 0

,

I. be a given element of

31, and l e t us d e f i n e

36

J .L.LIONS

We have

vE tll

and we take

v given by (3.25) i n ( 3 . X ) .

We o b t a i n :

but

X = 3 ; indeed

since

e(T) = 9 ;

t h e r e f o r e ( 3 . 2 6 ) implies

rag.

(3 2 7 ) But

and s i n c e

i t comes

We show now t h a t (3.23) i m p l i e s : (3.28)

zg13

which is ( 3 . 6 ) . We d e f i n e

On

as t h e s o l u t i o n of

(7.29)

We have obtain :

0 3 0 so that 11

0 E ri

a

and w e can choose

8 = 9 tl

i n (3.23) ; we

37

SOPE TOPICS ON VARIATIONAL INEQUALITIES

PT

( s i n c e a l l terms a r e

We can now l c t

q

But ( u -+)+ z + = 0

-f

and s i n c e w

<$

one has

0)

C

, so

t h a t (3.30) i m p l i e s

i n (3.31).

since

if

uE = $

.

uE p

We obtain

+

and

z = w-u

2 0

I

then w 3 u

a4

Therefore (3.32) reduces t o :

8

--_----

Remark 3.1 , Other r e s u l t s f o r V . I . of the type s t u d i e d here OM and t h e bibliography t h e r e i n . ([4] uses a r e s u l t of [5]).

be found i n [ 5 ] [4] [7] 8

Remark 3.2. ------The preceding proof uses, i c an e s s e n t i a l manner, t h e f a c t t h a t A i s a second order operator. The study of V.I. with c o n s t r a i n t s of t h e type v Q + ( x , t ) , 5 "non smooth", f o r . say, t h e o p e r a t o r + A 2 , i s ax open question. 8

.

J L. LIONS

38 REFERENCES

( CHAFFTER 111).

[ l ] Bensoussan, A. and Lions, J . L . Vol. 2 (1977).

Book t o appear. Hermann ed. Vol. 1 (1976Ia

[2] Bensoussan, A. and Lions. J.L.

(1973) Applicable Analysis. Vol. 3 pp. 267-294.

b] BrBzisa

H. C.R.Ac. Sc. P a r i s , t. 274 (1972), pp. ?lo-313.

[4] Charrier, ?. and T r o i d e l l o , G.M.

[5] Hanomet, B. and J O l y a J.L.

161

rntignot, F. and P U e l a J.P.

[7] T r o i d e l l o . G.M.

(1975) C.R.Ac.

(1975) C.R.Ac. (1975) C.R.Ac.

To appear.

Sc. P a r i s .

SC. D a r i s .

Sc. P a r i s .

W . Eckhaus ( e d . ) , New Developments i n D i f f e r e n t i a l Equations

@ North-Holland P u b l i s h i n g Company (1976)

FREE BOUNDARY PROBLEhiS FOR P O I S S O N ’ S EOUATION

Cui do

Stxpacchia

Scuola Hormale S u p e r i o r e P I S A

(Italy)

$1 - I n t h i s t a l k we s h a l l g i v e an account o f a j o i n t paper w i t h 3. K i n d e r l e h r e r which d e s c r i b e s t h e f o r m u l a t i o n and s o l u t i o n o f a f r e e boundary v a l u e problem i n t h e framework o f t h e t h e o r y o f v a r i a t i o n a l i n e q u a l i t i e s . Lie c o n f i n e o u r a t t e n t i o n t o a problem i n t h e p l a n e which c o n s i s t s i n f i n d i n g a domain 61 and a f u n c t i o n u d e f i n e d i n 52 s a t i s f y i n g t h e r e a P o i s s o n ’ s e q u a t i o n t o g e t h e r w i t h b o t h a s s i g n e d D i r i c h l e t and Reumann t y p e d a t a on t h e bounda r y r o f C2 . Under s u i t a b l e hypotheses about t h e g i v e n data, we p r o v e t h a t t h e r e i s a unique s o l u t i o n p a i r n , u which s o l v e s t h i s problem and t h a t I‘ i s a smooth curve. itJ

Let z=xltixz=oe , 020<2n , denote a p o i n t i n t h e z p l a n e . L e t us suppose t h a t F ( z ) i s a f u n c t i o n i n C 2 ( R 2 ) which s a t i s f i e s t h e c o n d i t i o n s p-’F(z)

E Cz(RZ)

i n f p-’ F ( z ) RZ

>

0

F(0) = F (0) = 0 P Our o b j e c t i s t o s o l v e , i n some manner, t h e

PhvbLem I - T o 6ind a bounded i2 and a @ w L i o n i n

-Au=p”Fp

R

u

nuch t h a t

(1 -2)

u=o

(1-4)

u(0) = 7 whete I? = an, v LA t h e o&ahd dctected veCtoh and F 4uakh6~e4 ( 1 $ 1), and 7 ~h gcven.

s

4-4 t h e m c l e n @ h on

r,

t o be a s o l u t i o n t o PtobCern I , t h e maxinium p r i n c i p l e f o r Supposing Qu superharrrionics i m p l i e s t h a t u>O i n $2 s i n c e b ~ i 0n ‘ LZ . IJe assume, conseq u e n t l y , t h a t y>O and t h a t u E C ( R 2 ) w i t h n =I z : u ( z ) > O l . F u r t h e r , i f R i s a domain w i t h smooth boundary r and u s a t i s f y ( 1 . 2 ) i n C and ( 1 .3) on r then au

- (z) av

0

for

z E

r

o r t h e c e n t r a l a n g l e 0 i s a s t r i c t l y i n c r e a s i n g f u n c t i o n o f t h e a r c l e n g t h paranie t e r on r . I n t e r p r e t i n g t h i s s i t u a t i o n c l e o m e t r i c a l l y , we conclude id r i n

STAMPACCHIA

GUIDO

L(0

b w o t h and ht4ped t o

u buLbd4e4 ( 1 . 2 ) i n Ci and (1.3) z=O.

r , -then

on

Ci

A b-tmhaped w i t h

We s h a l l s o l v e Pfioblem 1 by means o f a v a r i a t i o n a l i n e q u a l i t y suggested by t h e p r o p e r t i e s o f a f u n c t i o n e ( z ) which s a t i s f i e s gp = -p-lU

.

(1.5)

A c h a r a c t e r i s t i c o f t h e p r e s e n t work i s t h e l o g a r i t h m i c n a t u r e o f a f u n c t i o n g d e f i n e d by (1.5) a t z=O. T h i s d i f f i c u l t y w i l l be overcome by c o n s i d e r i n g an unbounded o b s t a c l e . I n t h e f o l l o w i n g s e c t i o n we t r a n s f o r m o u r problem t o one c o n c e r n i n g a variational inequality.

52 - I n t h i s s e c t i o n we i n t r o d u c e e v a r i a t i o n a l i n e q u a l i t y and determine i t s r e l a t i o n s h i p t o Phublenr I . He b e g i n \r!ith some n o t a t i o n s . Employing usual n o t a t i o i i s f o r f u n c t i o n spaces, s e t Br = I z : I z l < r l , r.0, and v=loc r

and

K = t v E H1(Br):vLlog P i n Er r Define

on

aBr}

the b i l i n e a r form I

where we have depressed t h e dependence Let f E L' (R') 1oc Phublem [*J- T u dind a p&

.

f o r some

p > 2

r > l and

w EIKr

buch t h a t

I

w EM

1. 1 f ( v - w ) d x

: a(w,v-w)

v EM

'Br

and t h e ,5unctiun G(z)

dedined by z E Br

G(2)

=

z

.in i n c ' ( R ' )

F

Br

.

The e x i s t e n c e and o t h e r p r o p e r t i e s o f a s o i u t i o n t o PhubLem I*) can be easw t o BR f o r R > r w i l l be a s o l u t i o n o f ( 2 . 1 ) it BR S i n c e t h i s means t h a t (2.2) w i l l be a u t o m a t i c a l l y s a t i s a l s o a s o l u t i o n t o PtlobLem [ * J ,we s h a l l n o t d i s i s f i e d , s o t h a t R,wl EMr BR t i n g u i s h between w and i n t h e sequel. i l y i n v e s t i g a t e d . We n o t e t h a t t h e r e s t r i c t i o n o f

.

w"

We have t h e f o l l o w i n g THEOREM 1 - L e t S2,u y>O. Suppuhe t h a t r ud PkubLem (*J doh

be a b O & t h n 0 6 PtlubLem I whehe F 4 a t i b d i e n ( 1 . 1 ) and .in a brmoth cuhue. Then t h e h e e x h a a bullLtiun r , w EIKr f(z) = -

1 ^IP

F(z)

buch t h a t

R = (z:w(z)>log P }

and

U ( Z ) = T(1-pw

P

(2))

.

FREE BOUNDARY PROBLEMS FOR POISSON'S EQUATION

441

03 - According t o a w e l l known theorem, t h e r e i s a s o l u t i o n t o t h e v a r i a t i o n a l i n e q u a l i t y ( 2 . 1 ) f o r each r > O . To e s t a b l i s h i t s smoothness i n E r we s h a l l prove t h a t i t i s bounded. F o r once t h i s i s known, t h e o b s t a c l e l o p may be r e p l a c e d by a smooth o b s t a c l e which equals locy when

f u n c t i o n s which exceed \I, and ( 2 . 1 ) may be s o l v e d i n t h e convex IKb o f H'(Er) in B and s a t i s f y t h e boundary c o n d i t i o n v ( z ) = log r , ] z ] = r . The s o l u t i o n t o t h i s r a t t e r problem i s known t o be s u i t a b l y smooth and i s e a s i l y shown t o be t h e solution o f (2.1). By s t a n d a r d methods we have t h e f o l l o w i n g

LEIWA

-

Let

f E Lp(Br)

p.2

604 b o w

f <- O

Then Rhe

w

bOh,tiOk?

v6 (2.1)

log r

-

and b a t i b d y

in

bcLtindiec,

f

6oh

Br.

c w(z) 5 l o g r

cllfll

in

Br

,

LP(Br) lohehe

c=c(r,p)>O On t h e o t h e r hand we have t h e f o l l o w i n g

-

THEOREFI 2

Let

f E Lyoc(R2)

d o t a p>2 bdL566y

sup f R*

0

<

.

In addction, M E HZ > p (Br). r , w E IK t o P h u b f k c i I * ) . r The main s t e p i n t h e p r o o f i s t o c o n s t r u c t a s u p e r s o l u t i o n g ( z ) = h ( p ) t o

Then t h e h e e d L . 6 a

bV&U%fl

the form i

a(w,r) -

f o r some

I

fSdx

' Br

r > l , which s a t i s f i e s

h EKr

(3.2)

h = -1 ~r

(3.3).

I f ( 3 . 2 ) and ( 3 . 3 ) a r e f u l f i l l e d , then, from a we1

w < h i-loreover, s i n c e

Therefore

Br.

l o o pLwLh we conclude from ( 3 . 3 ) t h a t 1 w (z) = f0.r / z I : r: P

and, s i n c e

in

known p r o p e r t y

w=log r

for

r

Iz/ = r ,

w (z) = 0 for /zl = r 8 d e f i n e d by ( 2 . 2 ) i s i n C ' ( R ' ) .

04 - Next s t e p i s t o show t 3 a t t h e s e t where t h e s o l u t i o n t o Paobleni I*] exceeds l o g p i n starshaped under an assumption about f . Indeed we have t h e f o l l o w i n g :

GUIDO

42

-

THEOREM 3

r,w E Mr

Let

f E Lyoc(R’)

L&

STAilPACCHIA

hc~tA6y

denote t h e n o l d o n

06

SUP

and p - l ( ~ ’ f ) ~ ~ O .

f
R2Ptrobtem [*I

52 = I z : w ( z ) > l o g p l

Then SL an n t u h a p e d w i t h h e 4 p e d t o

z = 0

doh

f

and n e t

.

.

-

85 I n t h i s s e c t i o n we s h a l l r e p o r t on t h e smoothness o f t h e f r e e boundary d e t e r mined by a s o l u t i o n t o PtrobLem ( * ) assuming t h a t f E C’(RZ) and t h a t

Then t h e f o l l o w i n g theorems h o l d THEOREM 4 - L e i f E C1(R2) 4&6y t o Phobteni (*) don f Lei

( 5 . 1 ) and L e t

.

a =tz:w(z),log r 06

Then t h e boundahy

in

THEOREH 5

-

denote t h e nolLLtion

.

PI

52 han the. hephebent&on

r when@ p

r,w E K p

: P = p(e)

, 0 9 ~ 2 ~

a cuw%nuow d u n c t i o n u6 bounded

vahiation.

L e t f E C1(R2) 4aLLn& ( 5 . 1 ) . Let r , w E M r denote t h e n o t a t i o n t o Phobteni (*I doh f and r t h e boundany 0 6 fl= t z : w ( z ) > l o g PI . Then r h a a c”7 pahanlethization, O < T < l .

A t t h i s p o i n t we can g i v e an answer i n t h e a f f e r m a t i v e t o t h e q u e s t i o n o f t h e e x i s t e n c e o f a s o l u t i o n t o Phobtem I w i t h t h e THEOREI.1 6

a

-

Let

and a 6unCtion

F E C1 (RZ ) 4 e A h 6 y conditionn (1.1 ) u E tiy:c(Rz ) nuch t h d

-A =

p-’F

P

in

.

Then thehe e . x h & a domLin

a

u(0) = Y

whehe

v

.LA t h e uLLtwatld d h e c t e d M O h n d !

vectotr

and

s

i 4

t h e ahc LenghL

06

and y>O A & w n . Denote by

1 Given F , d e f i n e f ( z ) = - 7F ( z ) z,w EM t h e s o l u t i o n t o ”PtrobLeni U ( Z ) = y(l-pwp(z))

I t can be proved t h a t

and observe t h a t f s a t i s f i e s (5.1). [ * ) f o r f and d e f i n e

z

E R2

.

u ( z ) , so d e f i n e d , s a t i s f i e s t h e c o n c l u s i o n s o f theorem 6 .

REFERENCES

For t h e b i b l i o g r a p h y and more d e t a i l s we r e f e r t o D . K i n d e r l e h r e r and G.Stampacchia. A Free Boundary Value Problem i n P o t e n t i a l T h e o y Ann.1m.t. FowLieh, 25 3/4 (1975) t o appeah.

W . Eckhaus ( e d . ) , New Developments i n D i f f e r e n t i a l Equations

@ North-Hol land Pub1 i s h i n g Company (1976)

NONLINEAR ELLIPTIC EQUATIOIJS WITH NONL I REAR BOUNDARY CONDITIONS

Herbert Amann Department o f Flathematics Ruhr- Uni v e r s i t a t Ecochum , Germany

INTRODUCTION I n t h i s paper we study m i l d l y nonlinear e l l i p t i c boundary value problems (BVPs) o f the form in R , (1.1) on r , where n i s a bounded domain i n R N w i t h s u f f i c i e n t l y smooth boundary r We suppose t h a t A i s a second order, s t r o n g l y u n i f o r m l y e l l i p t i c d i f f e r e n t i a l operator and B i s a boundary operator o f the form Au = f ( x , u ) Bu = g(x,u)

where

6

.

i s an outward pointing, nowhere tangent vector f i e l d on

r .

Problems o f t h i s type a r i s e , i n p a r t i c u l a r , i n the study o f steady s t a t e s o l u t i o n s o f n o n l i n e a r parabolic equations o f t h e form

* at

t AU = f(x,u)

in

n

(o,-)

,

Bu = g(x,u) u = u

on on

-r x

(0,-)

,

0

(1.2)

R

I n t h i s connection, n o n l i n e a r boundary c o n d i t i o n s seem t o b e ' o f p a r t i c u l a r i m portance. For the study o f t h e s t a b i l i t y o f t h e s o l u t i o n s o f t h e p a r a b o l i c i n i t i a l - b o u n d a r y value problem ( l . Z ) , one has t o have a aood knowledge o f t h e steady states, t h a t i s , o f the s o l u t i o n s o f t h e e l l i p t i c BVP (1.1). O f course, the most i n t e r e s t i n g case occurs i f t h e e l l i p t i c BVP has several d i s t i n c t solut i o n s . (For an i n t e r e s t i n g a n a l y s i s o f an i n i t i a l BVP o f the form (1.2) i n t h e case o f one space dimension and i n the presence o f several d i s t i n c t steady s t a t e s cf. [ 6 1 ). Unlike t o t h e s i t u a t i o n where Q i s independent o f u , n o t much seems t o be known i f the boundary c o n d i t i o n depends n o n l i n e a r l y on t h e unknown f u n c t i o n Recently, the theory o f monotone operators has been a p p l i e d t o BVPs o f t h e u form (1.1) ( c f . [ 7,B,10 I ) . However, i n a l l o f these papers t h e boundary c o n d i t i o n i s o f the special form

.

where g i s decreasing and v i s the co-normal w i t h respect t o the d i f f e r e n t i a l operator A Moreover, the theory o f monotone operators does n o t seem t o y i e l d proper mu1t i p 1 i c i t y r e s u l t s .

.

Besides these r e s u l t s , there are some s c a t t e r e d existence theorems f o r n o n l i n e a r Stecklov problems o f the form

4:i

44

HERBERT AMANN

where A i s supposed t o be f o r m a l l y s e l f - a d j o i n t such t h a t the homogeneous l i n e a r BVP possesses a n o n t r i v i a l s o l u t i o n ( c f . [ 9 , 1 2 ] ). This s i t u a t i o n w i l l a l s o be covered by our general r e s u l t s . So f a r , the o n l y general existence theorem f o r t h e BVP (1.1) seems t o be a r e s u l t due t o t h e author [ 2 I (see a l s o [ 261 ), namely t h e r e s u l t t h a t the s 0 existence o f a subsolution 7 and o f a supersolution ^v f o r (1.1) w i t h guarantees the existence o f a s o l u t i o n . I n t h i s paper we g i v e a new (and more e l e gant) proof f o r t h i s theorem by transforming t h e BVP (1.1) i n t o an e q u i v a l e n t This transformation has t h e advantage t h a t i t maf i x e d p o i n t equation i n C(E) kes the problem (1.1) accessible t o the powerful t o o l s o f n o n l i n e a r f u n c t i o n a l analysis. Some o f these tools, namely the theory o f increasing, completely c o n t i nuous maps i n ordered Banach spaces ( c f . [5,15] ) are then used t o enlarge the domain o f a p p l i c a b i l i t y o f t h e general existence theorem by d e r i v i n ? simple s u f f i c i e n t c r i t e r i a f o r the existence o f sub- and supersolutions. I n addition, we can d e r i v e a nonexistence and a general uniqueness theorem. Moreover, i n o r d e r t o dem n s t r a t e the power o f t h i s a b s t r a c t approach, we d e r i v e a m u l t i p l i c i t y r e s u l t , namely a c r i t e r i o n guaranteeing the existence o f a t l e a s t t h r e e d i s t i n c t s o l u t i o n s .

.

I n the f o l l o w i n g paragraph we s t a t e the main r e s u l t s f o r t h e n o n l i n e a r BVP (1) which a r e proved i n t h i s paper. I n Paragraph 3 we e s t a b l i s h a fundamental a p r i o r i estimate f o r the s o l u t i o n s o f the l i n e a r BVP Au = v i n 62 , Bu = w on r , i n v o l v i n g o n l y a L -norm o f t h e boundary term. P I n Paragraph 4 we d e r i v e t h e e q u i v a l e n t f i x e d p o i n t equation and we prove, besides o f the fundamental existence r e s u l t , t h e above mentioned m u l t i p l i c i t y theorem. The l a s t paragraph i s o f more a b s t r a c t nature. Namely, i t contains a d e r i v a t i o n o f the basic s p e c t r a l p r o p e r t i e s f o r p o s i t i v e l i n e a r operators i n an order e d Banach space which map every p o i n t o f t h e p o s i t i v e cone e i t h e r i n t o the i n t e r i o r o f t h e cone o r onto zero. These r e s u l t s generalize t h e known s p e c t r a l prope r t i e s f o r s t r o n g l y p o s i t i v e operators ( c f . [ 141 ). Moreover they are needed f o r the study o f the l i n e a r eigenvalue problem in n

Au =

mu

Bu =

r u on

r

,

.

This eigenvalue problem p l a y s a considerable r61e i n t h e s o l v a b i l i t y theory o f t h e nonlinear BVP (1.1). 2. STATEMENT

OF THE M A I N RESULT

Throughout t h i s paper a l l f u n c t i o n s a r e real-valued and a l l v e c t o r spaces a r e over the r e a l s . I n t h e f o l l o w i n g we s p e c i f y the hypotheses which a r e used throughout t h e remainder o f t h i s paper. We suppose t h a t n i s a nonempty bounded domain i n R N , N h 2 , o f class C3+U f o r some (X E o 1 , t h a t i s , t h e boundary , r , o f fi i s an (N-1)-dimensional compact CJ+A-!anifold such t h a t n l i e s l o c a l l y on one s i d e of r

.

We denote by

A

a l i n e a r d i f f e r e n t i a l operator o f t h e form

NONLINEAR BOUNDARY CONDITIONS

Au :=

N

N

-

aikDiDku

I:

+ au

z aiDiu

t

i.k=l

i=1

.

w i t h symmetric c o e f f i c i e n t m a t r i x ( a i k ) t!e suppose t h a t a i k E C2+a(z) , ai E Cl+a(;;i) , and a E P ( E ) Moreover, A i s suppose t o be s t r o n g l y uniforml y e l l i p t i c , that is, N i k 2 I: aik(x)c 6 2 aolEI i. k = l f o r some constant a0 > o and every x E 0 , and 5 = ( [ l , . . ,c N ) E lR N

.

.

We denote by t o r f i e l d on

CZta(r,IR N )

6E

r , and

:= zsiDiu

.

an outward p o i n t i n g , nowhere tangent vec-

r

denotes t h e d i r e c t i o n a l d e r i v a t i v e on

of

C1(z)

uE w i t h respect t o 6 . I t should be observed t h a t B i s n o t supposed t o be a u n i t vector f i e l d . Then we d e f i n e a ( r e g u l a r o b l i q u e d e r i v a t i v e ) boundary operator B by

BU where

b

E

C1+a(r)

au t as

:=

.

,

bU

-

.

L e t I be a nonempty s u b i n t e r v a l o f R We denote by f : 5 x I R a f u n c t i o n which i s a-Holder continuous i n the f i r s t v a r i a b l e and l o c a l l y L i p s c h i t z i n the second v a r i a b l e . More p r e c i s e l y , f o r every compact s u b i n t e r v a l I' o f I , t h e r e e x i s t s a constant y ( 1 ' ) such t h a t If(X,C) - f(y,n)l I Y ( I ' ) ( l x - y l a + 1 6 - r l l ) f o r every p a i r ( x , ~ ) , (y,?) E x I' . Moreover, we suppose t h a t i s a l o c a l l y L i p s c h i t z continuous function.

z

g :

r

x

I

-R

Then we consider m i l d l y n o n l i n e a r e l l i p t i c BVP's o f t h e form Au = f ( x , u )

in

R

,

Bu

on

r

.

= g(x,u)

(2.11

By a -elution u o f ( 2 . 1 ) we mean a c l a s s i c a l s o l u t i o n , t h a t i s , a f u n c t i o n u E C2(Q) * C1(E) such t h a t u(5) c I , Au(x) = f ( x , u ( x ) ) f o r x E R , and Bu(x) = g(x,u(x))

for

x E

r

.

A f u n c t i o n u i s c a l l e d a s u b s o l u t i o n f o r the BVP (2.1) i f C1(n) , u(H) c I , and Au(x) S f ( x , u ( x ) ) Bu(x)

<

g(x,u(x))

for

x

E

R

,

for

x

E

r

.

u

E

2

C (0) n

A subsolution which i s n o t a s o l u t i o n i s c a l l e d a strict subsolution. S u p e r s o l u t i o n s and s t r i c t supersolutions are defined by r e v e r s i n g t h e above i n e q u a l i t y signs. L e t X be a nonempty s e t . I f u,v : X -,lR are two f u n c t i o n s such t h a t u ( x ) < v ( x ) , f o r every x E X , then we w r i t e u 5 v Moreover, u < v means t h a t u S v but u v F i n a l l y , by the order i n t e r v a l [ u , v J , between u and v , we mean the s e t o f a l l f u n c t i o n s w : X -+ R such t h a t u w I v

* .

.

.

The f o l l o w i n g theorem contains t h e basic existence r e s u l t f o r the BVP

(2.1).

HERBERT AMANN

46

(2.1) Theorem: L e t -V be a subsolution and l e t 0 be a supersolution such that- V 5 'v . Then the BVP (2.1) has a t l e a s t one s o l u t i o n i n t h e order i n t e r v a l [ v,V] . More p r e c i s e l y , t h e r e e x i s t s a minimal s o l u t i o n U and a maximal s o l u t i o n

-

i n t h e order i n t e r v a l

[T,'v]

such t h a t every s o l u t i o n

U E

u s u s i i .

[y,^v] s a t i s f i e s

The above existence thPorem i s e s s e n t i a l l y contained i n an e a r l i e r paper by the author ( c f . 12, Theorem 3 1 ). I n the present paper we give a simple p r o o f by reducing t h e BVP (2.1) t o an equivalent f i x e d p o i n t equation i n the Banach space C(5)

.

I n the f o l l o w i n g we denote by C1-(X) t h e Banach space o f a l l L i p s c h i t z continuous functions on the compact m e t r i c space X , I f X and Y are two nonempty sets and u : X + R and v : Y -c R are two functions, then we w r i t e (u,v) z (0.0) i f u 2 o and v 2 o . Moreover, (u,v) > (o,o) means (u,v) 2 (o,o) b u t (u,v) (o,o)

*

.

The second fundamental r e s u l t o f t h i s paper concerns the l i n e a r eigenvalue problem (EVP) Au = xmu i n R , (2.21 . . Bu = Xru on r , where, as a r u l e , we suppose t h a t the f o l l o w i n g hypothesis (H) i s s a t i s f i e d :

m

E

~ ( 5, )r

E cl-(r)

, and

There e x i s t s a constant (a + , b + u r ) 2 (o,o)

~1 2 0

.

(m,r)

>

(0.0)

. (HI

such t h a t

By means o f the above mentioned reduction t o a f i x e d p o i n t equation i n

C(5)

we

s h a l l prove the f o l l o w i n g important r e s u l t :

( 2 . 2 ) Theorem: Let t h e hypothesis (H) be s a t i s f i e d . Then t h e EVP ( 2 . 2 ) possesses a s m a l l e s t eigenvalue xo(m,r) , t h e p r i n c i p a l eigenvalue, a n d xo(m,r) i s p o s i t i v e i f (a,b) > (0.0) The EVP ( 2 . 2 ) has e x a c t l y one l i n e a r l y independent t o the eigenvalue X0(m,r) , and Uo can be eigenfunction Uo E c2(R) f- cl(T) chosen t o be everywhere p o s i t i v e . Moreover, Xo(m,r) is t h e only eigenvalue of ( 2 . 2 ) having a nonnegative e i g e n f u n c t i o n .

.

L a s t l y , xo(m,r) c i s e l y , suppose t h a t m l Then xo(m,r) > Xo(ml,rl)

i s a s t r i c t l y decre s i n g f u n c t i o n o f (m,r) ~ " ( 5 ) and r l E ~ f - ( r ) s a t i s f y (m1,rl)

E

.

. More pre.

> (m.r)

The next theorem contains some useful r e s u l t s concerning the s o l v a b i l i t y o f the l i n e a r BVP Au - xmu = c i n Q , (2.3) Bu - Xru = d on r , where

h E

R

,

( 2 . 3 ) Theorem:

(H) be s a t i s f i e d and suppose t h a t t h e l i n e a r BvP ( 2 . 3 ) has f o r every X < Xo(m,r) e x a c t l y one s o l u t i o n which i s everywhere p o s i t i v e i f (c,d) > (0,O) L e t t h e hypothesis

c"(z) m) . Then

(c,d)

E

(c,d)

The BvP ( 2 . 3 ) has no p o s i t i v e s o l u t i o n i f e i t h e r 2 (0.0) or = xo(m,r) and (c.d) > (0.0)

.

.

On the basis o f the theorems (2.1)

-

X

> ho(l",r)

and

(2.3), which w i l l be proved i n the

NONLINEAR BOUNDARY CONDITIONS

47

f o l l o w i n g paragraphs, i t i s easy t o d e r i v e the f o l l o w i n g general existence, nonexistence, and uniqueness r e s u l t s . (2.4) Theorem: L e t t h e h y p o t h e s i s (H) be s a t i s f i e d . Suppose t h a t t h e r e e x i s t n o n n e g a t i v e c o n s t a n t s y and 6 and a r e a l number X < xo(m,r) such t h a t f(.,c)

I Y +

x

m

g(.,c) 5 6 + f o r every

6 2 0

, and f ( - , c ) t -Y + g(.,c) t -6 +

f o r every

5

I 0

.

r

x x

c , c

m r

c ,

(2.4)

( 2 . 1 ) h a s a t l e a s t one s o l u t i o n .

Then t h e n o n l i n e a r BVP

P r o o f : I t f o l l o w s from Theorem (2.3) t h a t t h e l i n e a r BVP

Au = Xmu Bu = hru has e x a c t l y one s o l u t i o n l i n e a r BVP-

0

2 o

.

+ +

y

in R

6

on

Hence 7 := -0

Au = xmu Bu = x r u

-

,

r

(2.5)

i s the o n l y s o l u t i o n o f t h e

y

in a .

6

on

r .

7 i s a subsolution and 0 i s a s u p e r s o l u t i o n f o r the BVP (2.1). Hence Theorem (2.1) i m p l i e s t h e a s s e r t i o n . Q.E.D. I t i s obvious t h a t

I t should be observed t h a t the above p r o o f shows t h a t t h e r e e x i s t s a sol u t i o n i n the order i n t e r v a l [ -9.91 , where 0 i s t h e s o l u t i o n o f t h e BVP (2.5).

We emphasize the f a c t t h a t Theorem (2.4) imposes one-sided growth condit i o n s only. For example, Theorem (2.4) i m p l i e s t h e e x i s t e n c e o f a s o l u t i o n o f t h e BVP (2.1) provided (a,b) > (0.0) and f ( x , . ) , g(y,.) a r e nonincreasing f o r every x E Ci and every y E r , r e s p e c t i v e l y , w i t h o u t any growth r e s t r i c t i o n whatsoever.

Furthermore, i t i s i m p o r t a n t t o n o t i c e t h a t the f u n c t i o n s m and r can be chosen independently o f each other. For example, suppose t h a t a 2 o , b t o , and f ( x , . ) i s nonincreasing f o r every x E 5 Then we can take m = o and Theorem (2.4) i m p l i e s t h e existence o f a s o l u t i o n f o r t h e BVP (2.1) f o r every f u n c t i o n g s a t i s f y i n g one-sided estimates o f t h e above form w i t h x < xo(o,r) . These I t f o l l o w s from Theorem (2.2) t h a t A (o,r) > xo(m,r) , f o r every m > o considerations show, t h a t , by u s i n g s t a r p estimates f o r f , i t i s p o s s i b l e t o e n l a r g e t h e c l a s s of admissible f u n c t i o n s g , and v i c e versa.

.

.

C l e a r l y , i f (a,b) 2 (o,o) , then i t i s always p o s s i b l e t o such t h a t m(x) I 11 f u n c t i o n s m and r by constants 7 and f o r a l l x E 5 and y E r , r e s p e c t i v e l y . However, s i n c e xo(m,r) Theorem (2.2), i t may be advantageous t o use nonconstant f u n c t i o n s a given concrete s i t u a t i o n .

replace t h e and -rLy) I 2 ho(u,p) by m and r i n

O f course, t h e r e a r e o t h e r geometric c o n d i t i o n s f o r t h e f u n c t i o n s f and g which guarantee the existence o f subsolutions and supersolutions f o r t h e BVP (2.1). and t h a t t h e r e e x i s t s a p o s i t i v e conFor example, suppose t h a t (a,b) 2 (o,o) s t a n t el such t h a t

Then i t i s obvious t h a t t h e constant f u n c t i o n

x

+

el

i s a supersolution f o r

HEEBERT AMANN

48

(2.1). Consequently, the BVP (2.1) has a s o l u t i o n i f t h e i n e q u a l i t i e s (2.6) and, f o r 5 5 E~ , the i n e q u a l i t i e s (2.4) are s a t i s f i e d , f o r example. We leave i t t o t h e reader t o deduce s i m i l a r existence r e s u l t s by e s t a b l i s h i n g f u r t h e r geometric c o n d i t i o n s o f t h i s form. I n any case, we l i k e t o p o i n t o u t t h a t , up t o r e g u l a r i t y assumptions, t h e above existence theorems contain and considerably generalize most o f the known existence r e s u l t s f o r the BVP (2.1) which have been deduced by means o f the theory o f monotone operators ( c f . Paragraph 1). Next we prove a simple n o n e x i s t e n c e t h e o r e m which i m p l i e s t h a t , i n some sense, Theorem (2.4) cannot be improved upon. (2.5) Theorem: L e t t h e h y p o t h e s i s (H) be s a t i s f i e d . Suppose t h a t (c,d). > (o,o) , and l e t A 2 ho(m,r) .

CECa(z)

and d E C1-(r)-fy

Then t h e BVP (2.1) h a s no p o s i t i v e s o l u t i o n i f

,

f(*,5) 2 c t b m 5

f o r every

for every

(2.7)

g(-.c) 2 d t x r 5 5 2 0 , and ( 2 . 1 ) h a s no n e g a t i v e s o l u t i o n i f

5 I 0

f ( * , 5 ) 5 -c t g(a.5) 5 -d t

.

x x

,

m 5 r 5

Proof: L e t the c o n d i t i o n (2.7) be s a t i s f i e d and suppose t h a t u i s a p o s i t i v e s o l u t i o n o f (2.1). Then i t f o l l o w s from (2.7), t h a t u i s a p o s i t i v e supersolution f o r the l i n e a r BVP A u - x m u = c in R , (2.8) B u - x v u = d on r . Since zero i s a s t r i c t subsolution f o r t h i s BVP, Theorem (2.1) i m p l i e s t h e e x i s t ence o f a p o s i t i v e s o l u t i o n o f (2.8). But t h i s c o n t r a d i c t s Theorem (2.3). The p r o o f f o r the remaining case i s s i m i l a r . Q.E.D.

I n the f o l l o w i n g we g i v e an existence and u n iq u e n e s s theorem which general i z e s the main r e s u l t o f 1 4 1 ( c f . a l s o [11,25] ). I n t h a t paper the uniqueness be dea s s e r t i o n has been proved under the assumption t h a t the f u n c t i o n g(y,.) creasing f o r every y E r

.

Ca@)

x

(2.6) Theorem: L e t t h e h y p o t h e s i s C l - ( r ) s a t i s f i e s (m,r) > (0.0)

lution i f

€or every

5.n E

with

s a t i s f i e d . Suppose t h a t (m,r) E t h e BVP ( 2 . 1 ) h a s e x a c t l y one so-

f(-,5)

-

f ( * , n ) 5 Arn(5-n)

g(*,c)

-

g(*,n) s xr(5-n)

ri

<5

and some

max-lf(x,o)l

X

X E

and

.

< lo(m,r)

.

.

max I g ( y , o ) l Then the hyYEr potheses o f Theorem (2.4) are s a t i s f i e d . Hence t h e BVP (2.1) has a t l e a s t one solution. proof:

L e t y :=

(H) be

. Then

6 :=

n

Suppose t h a t u1 and u2 are two s o l u t i o n s o f (2.1). Denote by n1 Then t h e hypotheses imply t h a t an a r b i t r a r y component o f { x E R u,(x) < u,(x)l t h e f u n c t i o n u := u1 - u2 s a t i s f i e s the i n e q u a l i t i e s

.

49

NONLINEAR BOUNDARY CONDITIONS

A u - x m u r o

in

B u - x r u r o u z o

on on

fil'

,

rnanl

(2.9)

.

finafil

I t f o l l o w s f r o m Theorem (2.2) t h a t t h e p o s i t i v e e i g e n f u n c t i o n uo o f t h e EVP (2.2) s a t i s f i e s t h e i n e q u a l i t y (2.8). Hence, t h e g e n e r a l i z e d maximum p r i n c i p l e 122, pp. 72-76] i m p l i e s t h a t u 2 o i n fi1 Consequently, "1 = 6 and u 1 2 u2 i n n A s i m i l a r argument shows t h a t u1 5 u2 , t h a t i s , u1 = u2 , Q.E.0.

.

.

As a l r e a d y mentioned e a r l i e r , t h e main importance o f t h i s paper l i e s i n t h e f a c t t h a t t h e BVP (2.1) can b e t r a n s f o r m e d i n t o an e q u i v a l e n t f i x e d p o i n t I t w i l l be shown t h a t t h i s can be done i n e q u a t i o n i n t h e Banach space C(n) such a way t h a t t h e c o r r e s p o n d i n g n o n l i n e a r map i s i n c r e a s i n g w i t h r e s p e c t t o t h e n a t u r a l o r d e r i n g o f C(F) T h i s has t h e consequence t h a t ( w i t h m i n o r m o d i f i c a t i o n ) most o f t h e general a b s t r a c t r e s u l t s f o r i n c r e a s i n g maps (e.9. [ 5,15 1 ) can be a p p l i e d t o t h e BVP (2.1). F o r example, i t i s now p o s s i b l e t o g i v e a d e t a i l e d s t u d y o f t h e n o n l i n e a r e i g e n v a l u e problem

.

.

Au =

x f(x,u)

in

Bu =

x

on

g(x,u)

, r ,

R

i n p a r t i c u l a r w i t h respect t o t h e question o f t h e existence o f m u l t i p l e solutions. We s h a l l n o t go i n t o d e t a i l s b u t we l e a v e i t t o t h e r e a d e r t o c a r r y t h r o u g h t h i s program. However, f o r t h e sake o f demonstration, we s h a l l p r o v e t h e f o l l owing e x i s t e n c e theorem f o r mu1t i p 1 e s o l u t i o n s .

-

-

(2.7) Theorem: Suppose t h a t V 1 , V2 -are s u b s o l u t i o n s and 9 1 , 92 a r e s u p e r s o l u t i o n s f o r t h e BVP (2.1) sxcb t h a t V 1 < ^v1< 7 2 < ^v2 , and such t h a t 91 is a s t r i c t s u p e r s o l u t i o n and V2 is a s t r i c t s u b s o l u t i o n . The" t h e BVP ( 2 . 1 ) h a s a t l e a s t t h r e e d i s t i n c t s o l u t i o n s Ui , i = 1,2,3 , such t h a t V1 S U1 < U2 < u3 5

.

The p r o o f o f t h i s theorem w i l l be g i v e n i n Paragraph 4. We c l o s e t h i s paragraph b y means o f a s i m p l e example i l l u s t r a t i n g t h e p r e c e d i n g theorem. Namely, we c l a i m t h a t , f o r e v e r y R , t h e BVP u = o 2

= 4n2 cos u t u c ( x )

in

R

on

r

,

a5 has a t l e a s t t h r e e d i s t i n c t s o l u t i o n s s a t i s f y i n ? -2n s u l < u2 < u3 < TI , p r o v i ded I c ( x ) l 5 1 f o r e v e r y x E r To see t h i s , i t s u f f i c e s t o v e r i f y t h a t t h e constant functions := -271 , C1 := -TI , v2 := o , t2 := TI s a t i s f y t h e hypotheses o f Theorem (2.3).

.

3. AUXILIARY RESULTS L e t A be an a r b i t r a r y nonempty subdomain o f R N , N t 2 , and l e t k be a nonnegative i n t e g e r . Suppose t h a t o < s < 1 , and l e t p > 1 . Then t h e Sob o l e v space WS+s(A) i s t h e v e c t o r space c o n s i s t i n g o f a l l u E w!(~) such t h a t

f o r every multiindex K w i t h nach space w i t h t h e norm

IKI = k

.

I t i s well-known t h a t

h k + S ( ~ ) i s a BaP

HERBERT AMANN

50

Since R i s a bounded domain o f class C3+', t h e r e e x i s t bounded open sets U j , j = 1,...,M , i n R N , covering r , and C3+a-diffeomorphi ms $ j o f U . onto the open u n ' t b a l l B" o f R N p c h t h a t $ j ( U . n n) = B n ix E I xN > 01 =: B+ and $.(U. n r) = B n ( R N - 1 x { o j ) =: fl f o r j = I , ...,M , Moreover, w i hout loss i f i e n e r a l i t y , we can (and w i l l ! ) assume t h a t $j E C3+u(Uj) and +sf E C3+a($)

a

hN

.

J

...,

L e t T I ' , j = 1, M , be a C3ta-partition o f u n i t y on r subordinate t o the coverin4 {U. n r Ij = 1,. ,MI Then every f u n c t i o n u o r can be u = & ( n j u ) , and we d e f i n e + j ( n j u ) on Bw-l by decomposed i n the -1

..

dm

(Observe t h a t

z" =

BN-l

x

.

{oI .)

For every p > 1 and a E [0,3 1 L p ( r ) I $:(nju) E W@(BN-l) , j = 1, ...,M I

Wa(r) P

, we

d e f i n e W,"(r) by M'(r) := {u I t i s well-known ( c f . [!71 ) t h a t

.

E

i s a Banach space w i t h the norm UP

M

Moreover, up t o equivalent norms, W i ( r ) i s independent o f t h e p a r t i c u l a r choice o f t h e l o c a l coordinates {(U.,+.)I and o f t h e p a r t i t i o n o f u n i t y i n j } J J be nonempty sets, and l e t f : X x Y -+ Z be a map. L e t X, Y , and Then we denote by F : Ya Zx the N e m y t s k i i o p e r a t o r o f f , t h a t i s , F asY t h e map F(u) : X Z defined by F ( u ) ( x ) = signs t o every map u : X f(x,u(x)) , x E x ,

.

--

-

w

(3.1) Lemma: L e t g : T X R be l o c a l l y L i p s c h i t z c o n t i n u o u s . Then, p > l a n d e v e r y u E 10.1) , the N e m y t s k i i o p e r a t o r G of g maps

for e v e r y

W;(r)

n Lm(r) i n t o

+

.

W;(r)

Proof: L e t u E W,"(r) n L m ( r ) be a r b i t r a r y and l e t Then there e x i s t s a constant y such t h a t

-

f o r every p a i r

la(x,E) g(y,n)l 5 y(IX-Yl ( x , ~ ) , ( y , ~ ) E r x [-6,6]

By the d e f i n i t i o n o f

..., .

f o r every j = 1, M and t h e p r o p e r t i e s o f

TI

.

Wz(r)

, it

+

6 :=

IE-nl)

l1ullLm(,,)

.

(3.1)

s u f f i c e s t o show t h a t

But t h i s i s an easy consequence o f the estimate (3.1)

j *

Q.E.D.

I n the f o l l o w i n g we denote by t t h e " t r a c e operator" which assigns t o every u E C(H) i t s boundary value u l r I t i s well-known ( c f . 1211 ) , t h a t t can be extended t o a continuous l i n e a r s u r j e c t i o n (again denoted by t ) o f Wk(n) P

.

NONLINEAR BOUNDARY CONDITIONS

.

onto Wi-l'p(r) , f o r every p > 1 and k = 1,2 p > 1 , there e x i s t s a constant y such t h a t

f o r every

u

E ~'(0)

51

I n p a r t i c u l a r , f o r every

.

P We denote by A ' A'u =

N

-

A

t h e a d j o i n t operator o f

N

D.D (a. u) i,k = l 1 k 1k C

- c

Di(aiu)

i.1

, that

+

is

au

A'

I t should be observed t h a t the u n i f o r m l y e l l i p t i c d i f f e r e n t i a l operator a-Holder continuous c o e f f i c i e n t s on 5

.

has

I t can be shown ( c f . 1 2 0 1 ) t h a t t h e r e e x i s t a f u n c t i o n co E C2+'(r) w i t h c o ( x ) > o f o r very x E F and an outward p o i n t i n g , nowhere tangent v e c t o r f i e l d 0 ' E C2+a(r,R ) such t h a t Green's formula takes t h e form

A

/(uAv

n

f o r every

u,v E

-

vA'u)dx = / ( v E'u

C2(x) , where

B'u :=

r

$,+

-

c0 u Bv)do

b'u

(3.3)

and b ' E C1+a(r)

w: L e t g E c ( F ) . Then t h e r e e x i s t s a f u n c t i o n

(3.2) uIr = o

and

satisfying

where t h e c o n s t a n t

BU =

U E

g , such t h a t

i s independent of

y

.

g

cl@)

,

.

N P r o o f : ( a ) We f i r s t consider l o c a l coordinates. The general p o i n t y E R w i l l be denoted by y = (7,t) , w i t h RN-1 and t E IR Moreover,

.

YE

N N Q : = I y E B + U C 1 ;j + t := ( y l Let

J ~ G C(C

N)

be given and d e f i n e v ( y ) := t

2-N

v : Q + R

+

t,.

. . ,yN-l + t ) E

zN1

by

-

- I-y+t *(q)dn Y

w i t h an obvious a b b r e v i a t i o n f o r t h e (N-1)-fold i t e r a t e d i n t e g r a l . Then i t i s e a s i l y seen t h a t v E C l ( Q ) , t h a t v l x N = o , and t h a t (DNv)IzN = JI

.

By means o f H o l d e r ' s i n e q u a l i t y i t i s e a s i l y v e r i f i e d t h a t b-t x+t b It-' f ( c ) d c l d x I1 I f ( x ) l d x a X a

-

f o r every f E q a , b ] , -m < a < b < , and every t h i s i n e q u a l i t y repeatedly, one e a s i l y proves t h a t

/

Q

i = l,.,.,N

, where

y

lDivlP dy 5 y l N I + I P 6 C i s independent o f J,

t E (o,b-a)

,

.

By a p p l y i n g

(3.4)

HERBERT AMANN

52

set of j = 1,

( b ) For each j = 1, U j such t h a t v j n

...,M

,

...,M r

,

= Uj

o?J (-)au a8

$ j 1 ( Q ) . Thfn-Vj FF .Vioi'every u C (n) E

i s an open suband every

N

z

=

k=l

m

k.j

D [ $?(u) 1 k J

and i t i s easy t o see t h a t

.

i k - @( N x 6 Di $j ) mk,j - J i=l ) due t o the f a c t t h a t 8 i s nowhere tangent t o r Hence, mk,j E C 2 + C C ( ~ Nand, does nowhere vanish. the function m j mN.j L e t g E C ( r ) . For every j = 1 M d e f i n e q. E C(CN ) by q j := J 1 T j ( g ) Denote by v t h e f u n c t i o n i n C1(Q) defined i n p a r t ( a ) by means j au N bu. = o f q j E C ( Z ) , and l e t u j := 'j @,j Then u. = o and Bu. = J + J as J J

'=

,...,

.

.

'

au.

A aa =g set

r

,

.

U. n r 3 Let e l , e be a smooth p a r t i t i o n o f u n i t y w i t h respect t o t h e compact i n JRij , s u b o r j i n a t e t o be open covering {vl,..,,vM} , and l e t on

...,

m

M

u :=

.

Then u E C1(z) , u l r = o , and Bu = g on r L a s t l y , by using PoincarG's ineq u a l i t y and t h e estimate (3.4). i t i s e a s i l y v e r i f i e d t h a t u s a t i s f i e s the asserted estimate. Q.E.D. A f t e r these preparations we are ready f o r t h e p r o o f o f the f o l l o w i n g a p r i o r i estimate.

( 3 . 3 ) P r o p o s i t i o n : Suppose t h a t (a,b) Then t h e r e e x i s t s a c o n s t a n t y such t h a t

> ( o , o ) , and

let

1< p

<

.

m

Proof: The maximum p r i n c i p l e together w i t h the r g u l a r i t y theory f o r e l l i p t i c equations implies t h a t the o n l y f u n c t i o n u E Wp(n) which s a t i s f i e s Au = o i n n and Bu = o on r i s t h e zero f u n c t i o n . Consequently t h e L P estimates ( c f . [ 1 I ) imply t h e existence o f a constant y such t h a t

s

2 Suppose t h a t u E W (n) s a t i s f i e s Then, by using t h e well-kno&n f a c t t h a t i t f o l l o w s f r o m Green's formula (3.3) t h a t

.

A'u = o i n n an9 B'u = o on i s dense i n Wp(n) (e.g. [ 2 1 1 ).

C2(z)

[ uAvdx f o r every

v E C2(F)

.

n

=

-

r

t(u)coBvdo

(3.5)

By t h e c l a s s i c a l Schauder theory (e.g. [ 161 ) , t h e BVP

NONLINEAR BOUNDARY CONDITIONS

A v = f

in

53

R ,

B v = g on r , h a unique o l u t i o n u E CZta(Z) f o r e v e r y f E C" n) and g E C1+'(r) Conis dense i n L p , ( r ) , sequently, s i n c e Ca(E) i s dense i n L , ( a ) and C +"(r) 0' := p/(p-1) , t h e r e l a t i o n (3.14) impyies t h a t / ufdx = / t ( u ) g do

.

I-

62

r

.

.

f o r e v e r y f E L p 1 ( R ) and g E L p , ( r ) Consequently, u = o This f a c t , together w i t h the r e g u l a r i t y properties o f t h e c o e f f i c i e n t s o f (A',B') , i m p l i e s t h e v a l i d i t y o f t h e L,i-estimate

-< y 1 I I A ' u l l

II U I I f o r every

u

Let denote b y

( a , . )

E

~'(5) satisfying

L p ' (n)

B'U

V := {u E C2(E) I Bu = 01 the inner product i n

= o

.

and V ' := {u E C2(z) I B ' u = 01 Lz(R) . F o r e v e r y u E C(h) l e t

, and

.

where k = 1,2 Then t h e e s t i m a t e ( 3 . 6 ) and Green's f o r m u l a (3.3) i m p l y t h a t , f o r every u E V ,

Since, by t h e Schauder t h e o r y , t h e BVP A'v = f

in

n ,

B'v = o

on

r

has a u n i q u e s o l u t i o n u E C2+'(5) f o r e v e r y f E C"(5) , i t f o l l o w s t h a t A ' ( V ' ) i s dense i n L p l ( ~ ) Consequently, t h e above i n e q u a l i t y i m p l i e s t h e e s t i m a t e

.

f o r every a constant

u

E V y2

f o r every

u

E

.

Moreover, t h e Lp-estimates ( c f . [ I ] ) i m p l y t h e e x i s t e n c e o f such t h a t

v

L e t E and F be Banach spaces. Then we w r i t e E L F i f E i s c o n t i n u o u s l y imbedded i n F We denote by L(E,F) t h e Banach space o f a l l c o n t i n u o u s l i n e a r o p e r a t o r s T : E --* F , and by [ E,F] we denote t h e " i n t e r p o l a t i o n space t o t h e parameter 1/2" o b t a i n e d by means o f "holomorphic i n t e r p o l a t i o n " ( c f . [19,241 ). k k L e t k = 1,2 Then Ve denotes t h e c l o s u r e o f V i n W,(n) and V i k denotes t h e c o m p l e t i o n o f Cm(n) i n t h e norm II Il-k,p Then t h e a p r i o r i

.

.

-

.

HERBERT AMANN

54

estimates (3.7) and (3.8) imply t h a t that

has a continuous extension, T

A-1

, such

.

T E ~(v;~,L,(n)) n L ( L ~ ( ~ ) , V ; ) Hence, by the theory o f i n t e r p o l a t i o n spaces ( c f . [ 1 9 , 241 ) , T E L ( [v,2,Lp(n) I 1/2 , LL2(”,Vp 2 1 1/2 )

.

(3.9)

Moreover, ( c f . [ 2 4 , Theorem 4.11 ) , [ Lp(n)’V;l

(3.10)

v;

G

w i t h dense imbedding. Hence, by d u a l i t y , 4 [

vp ;’

(3.11)

vp2,LP(n) 1 1/2

-

The r e l a t i o n s (3.9) (3.11) imply T E L(Vp -1 ,VP) 1

.

Consequently, there e x i s t s a constant

y3

such t h a t (3.12)

f o r every

u

E

Vo1

.

(Observe t h a t by

q.$y

I I A U I I _ ~ =, ~ sup V E

V’

WP‘(a) the r i g h t s i d e o f (3.12) has a w e l l - d e f i n e d meaning f o r L e t u E C2(E) uo E c1(3i) s a t i s f y i n g

u

E

V, 1 . )

be a r b i t r a r y . By Lemna (3.2) t h e r e e x i s t s a f u n c t i o n = o

B(u-u,)

on

r

(3.13)

uo = 0 and (3.14)

i s independent o f

where the constant y4 (3.12) implies t h a t

llull

w

(8)

Iy 3 (

u

IIAuII-

. Consequently,

1YP

, and

u - u O E V1 P

+ I I A U ~ I I - )~ , ~

P .t

+

y3( I I A U I I - ~ I

lluoll

y4

141(52 ) P IIBull Lp(r)

+ u T. b; Di v + auvldx i=l such

-

I r

-1,P

1

(3.15)

*

By p a r t i a l i n t e g r a t i o n one f i n d s t h a t

N

+ IIAuoII

co u Bv do

J uAv dx n

=

j{

N C

aikDiuDkv

n i,k=l

. Hence there e x i s t s a constant

y5

such

NONLINEAR BOUNDARY CONDITIONS

f o r every v E C1(E) and w E C2(E) s a t i s f y i n g wlr = o L p t ( r ) , i n e q u a l i t y (3.16) i m p l i e s the estimate

f o r every

w

E

C2(E)

with w l r = o

. Since

E! I - U P ' ( r ) P'

. Consequently,

I I A W I I - ~ ,5~ y6( llwll

b!l(R) P t(w) = o .

1

55

+ IIBw IIL P ( O 1

(3.17)

(a) w i t h P By applying (3.17) t o the f u n c t i o n u and usin? (3.13) and (3.14), we obtain the existence o f a constant y, such ?hat f o r every

wE W

By combining t h i s e s t i v a t e w i t h (3.15) and u s i n g the obvious f a c t t h a t IILull, we o b t a i n f i n a l l y the assertion. Q.E.D. 1 .P 1 I Lull Lp(R) I t should be remarked t h a t the above p r o o f o f the a p r i o r i estimate f o l l o w s the ideas developed i n the a u t h o r ' s paper [ 2 1 Since the e a r l i e r p r o o f contains a number o f inaccuracies, i n p a r t i c u l a r a f a l s e statement o f the a p r i o r i estimate (which, however, does n o t a f f e c t t h e v a l i d i t y o f t h e main r e s u l t s o f [ 2 1 ), we have decided t o include t h e above more d e t a i l e d d e r i v a t i o n o f Propos i t i o n (3.3).

5

.

I t should be observed t h a t we have proved t h e much stronger a p r i o r i

estimate

Moreover, i t should be observed t h a t i t s u f f i c e s t o suppose t h a t t h e c l a s s c 2 + U i f the p a i r (L,B) is f o r m a l l y s e l f - a d j o i n t .

s2

belongs t o

F i n a l l y , we l i k e t o p o i n t out, t h a t the b a s i c P r o p o s i t i o n (3.3) i s i m p l i c i t l y contained i n [ 181 However, i n t h a t paper much stronger r e g u l a r i t y hypotheses have been presupposed. Consequently i t does n o t seem t o be much e a s i e r t o deduce t h e needed a p r i o r i i n e q u a l i t y . f r o m [ 181 than t o prove i t d i r e c t l y .

.

4. THE REDUCTION TO AN EQUIVALENT FIXED POINT EQUATION

.

Suppose t h a t (a,b) > (o,o) Then, by the maximum p r i n c i p l e and the r e g u l a r i t y theory f o r e l l i p t i c equations, t h e l i n e a r BVP Au = v

in

R

Bu = w

on

r

has a t most one s o l u t i o n i n C2+'(5) as w e l l as i n W 2 (n) Schauder estimates and the L,-estimates take the form

(4.1)

, 1 < p<-

. Hence t h e

r e s p e c t i v e l y . Here and i n the f o l l o w i n g y denotes a p o s i t i v e constant ( n o t n e c e s s a r i l y t h e same i n d i f f e r e n t formulas) which i s independent o f the f u n c t i o n s appearing i n these estimates.

AMANN

HERBERT

56

I t i s well-known ( c f . [ 1 6 , 2 0 ] ) t h a t the BVP (4.1) has a unique s o l u t i o n =: S(v,w) E C2+"(3) f o r every (v,w) E Ca(5) t h a t the s o l u t i o n o p e r a t o r S s a t i s f i e s

u

s

E

L(C"(T)

x

C1+a(r)

x

v(

Hence (4.2) i m p l i e s

.

, c*+"(s~))

Since C(2) G L (n) and C ( r ) 4 L p ( r ) f o r every implies the existence o f a constant y such t h a t 5

.

cl+cl(r)

(I,-) , P r o p o s i t i o n (3.3)

p E

l l v l l c ( ~ i ) + llWllC(r) )

(4.4)

.

f o r every (v,w) E C"(5) x Since t h e l a t t e r space i s dense i n C(5) C ( r ) , the estimate (4.4) i m p l i e s t h a t S has f o r every p E (I,-) a unique continuous extension, denoted again by S , such t h a t

S

E L(C(5) x C ( r )

1 , wp(n))

x

.

L e t I be a nonempty s u b i n t e r v a l of IR and l e t f : E x I -. IR and g : r x I -. R be continuous maps. Denote by F and G the corresponding Nemytskii operators, r e s p e c t i v e l y . Recall t h a t by a s o l u t i o n o f t h e BVP Au = f ( x , u ) i n n , (4.5) Bu = g(x,u) in n we mean a f u n c t i o n u E C2 ( a ) n C l ( 5 ) such t h a t u ( 1 ) c I and Au = F ( u ) i n , and Bu = G(u) on r . I n the f o l l o w i n g we l e t

I

:= { u E

c (5)

C(5) I u@)

c I}, and we define

by T(u) := S(F(u),G o t ( u ) ) , where t denotes the t r a c e operator. Here and i n the f o l l o w i n g we denote by p an a r b i t r a r y but f i x e d r e a l number s a t i s f y i n g p > N This i m p l i e s i n p a r t i c u l a r t h a t i s compactly imbedded i n C(5) Here T can be considered as a mapping o f - i n t o C(n) C(n) The f o l l o w i n g lemma i s o f fundamental importance f o r o u r considerations.

.

d

e:

.

.

.

-

is (4.1) Let (a,b) > ( 0 , o ) Suppose t h a t f : 'rz x I -. l o c a l l y r H o l d e r continuous and 9 : r x 1 i s l o c a l l y L i p s c h i t z continuous. Then t h e BVP (4.5) i s e z u i v a l e n t t o t h e f i x e d p o i n t e q u a t i o n U = T ( U ) i n c(5). The map T : 1~6;) + C(n) i s c o m p l e t e l y c o n t i n u o u s , t h a t i s , T i s continuous and maps bounde s e t s i n t o compact s e t s .

of

.

T

Proof: I t i s obvious t h a t every s o l u t i o n o f t h e BVP (4.5)

i s a f i x e d point

.

Conversely, suppose t h a t u E C(5) i s a f i x e d p o i n t o f T Then u belongs t o the range o f S , hence t o Wi(n) Consequently, t ( u ) E Wb-l/P(r) , Since F(u) E C(1) c Lp(n) , and Lemma (3.1) i m p l i e s t h a t G 0 t ( u ) E Wi-'lp(r) 2 and since, by (4.3). S maps L (n) x W1-l/P(r) i n t o b' (n) , i t f o l l o w s t h a t P P P u E Wi(n) I t i s well-known ( c f . 211 ) t h a t Wi(a) is continuously imbedded

.

.

.

.

.

Cl+'(E) , where u := I-N/p Hence u E C1@) n W2(n) This i m p l i e s t h a t F ( u ) E C"(n) and onsequently, by the i n t e r i o r r e g u l a r i l y theory f o r e l l i p t i c equat'ons, u E C5tu(n') f o r every subdomain R' such t h a t n ' C a Hence, u E C (n) n C1(5) , and u i s a s o l u t i o n o f the BVP (4.7). in

h

.

NONLINEAR BOUNDARY CONDITIONS

51

I t i s e a s i l y seen t h a t the maps

F : Ic(a) + C(5) and G : I c ( r ) + C ( r ) := { u E c ( r ) I u ( r ) C I1 . Moreover, t C(r)i n t o C(r) such t h a t t ( I c ( a ) ) C i s a continuous l i n e a r operator from C(a) I c ( r ) . Since S E L(C(a) x C(r),$(a)) , and s i n c e W1(a) i s compactly imbedded

are bounded and continuous, where

i n C(a) f o r p > !I i n t o C(E) Q.E.D.

.

,it

I

follows t h a t

T

P i s completely continuous from

L e t X be a nonempty compact Hausdorff space. We denote by C+(X) the s e t o f a l l nonnegative continuous f u n c t i o n s on X , t h a t i s , C+(X) := { u E C(X)I u 2 o 3. Clearly, C+(X) i s a closed convex p r o p e r cone i n C(X) w i t h nonempty i n t e r i o r . I n f a c t , u E i n t C+(X) i f and o n l y i f u ( x ) > o f o r every x E X Observe t h a t u -< v i f and o n l y i f v - u E C+(X) I n the f o l l o w i n g we w r i t e u >> v i f u - v E i n t C+(X)

.

.

.

.

L e t D be a nonempty subset o f C(X) Then a map h : D + C(X) i s s a i d t o be i n c r e a s i n g i f h ( u ) _< h ( v ) f o r every p a i r u,v E D s a t i s f y i n g u -< v

.

s:

(4.2) L e t t h e h y p o t h e s e s o f Lemma (4.1) b e - s a t i s f i e d and s u p p o s e t h a t f(X,*) and g ( y , * ) a r e i n c r e a s i n g f o r e v e r y X E fl and y E I' , r e s p e c t i v e l y . Then t h e map T : + C(a) i s i n c r e a s i n g . Suppose i n a d d i t i o n t h a t

for some

X E 8

and e v e r y

f o r some y E r for every p a i r

and every U,V

E,n

f(X,S) < f(x,n) 1 w i t h F < n , or

E

!3(YSO < CJ(Y*n) S,rl E I w i t h 5 < n E I c ( a ) such t h a t v << u

.

. Then

-

T(u)

T(v) E i n t C+(?i)

Proof: The maximum p r i n c i p l e i m p l i e s t h a t every f u n c t i o n satisfying A u 2 0 i n n ,

B u r o o n r

,

u E C2(n)nC1(c)

.

e i t h e r vanishes i d e n t i c a l l y r belongs t o i n C+(iZ) Consequently, the s o l u t i o n := operator S maps Cp a ) x CIw(P) \ I ( o , o ) l i n t o i n t C+(b) , here C+(E) n-Ca(a) and C{+a(r):= C+(r) n C l * ( r ) Since C'(H) x $*(I.)i s dense i n C+(s2) x C k ( r ) i t f o l l o w s t h a t S maps C+(H) x C+tr) i n t o C+(a) , t h a t i s , S E L(C(n) x E(f),C(Q)! i s a p o s i t i v e l i n e a r operator. This f a c t e a s i l y i m p l i e s t h a t T i s increasing.

q(E)

.

Suppose t h a t (u,v) E C+(a) x C+(r) and m o l l i f i e r s , i t i s easy t o see t h a t there e x i s t s a such t h a t (u,v) 2 ( u v ) > (0.0) Hence S(u,v) t h a t S(u,v) E i n t C+@O. Hence S : C(H) x C(r) t h a t i s , S y p s C+(n) x C + ( r ) \ I ( o , o ) l into the a s s e r t i o n i s now obvious. Q.E.D.

.

*

.

(u,v) (o,o) Then, by u s i n g p a i r (uo,vo) E P ( h ) x c l w ( r ) 2 Sfuo,vo) >> o and i t f o l l o w s .+ C(a) 1s s t r o n g l y p o s i t i v e , i n t C+(n) The second p a r t o f

.

A f t e r these preparations we are ready f o r t h e p r o o f o f Theorem (2.1) and Theorem (2.7). Proof of Theorem (2.1): Without l o s s o f g e n e r a l i t y we can assume t h a t I equals the compact i n t e r v a l [ m i n max 01 Hence the r e g u l a r i t y assumptions such t h a t f o r f and g imply the existence o f a p o s i t i v e constant w f(x,c) - f(x,n) 2 -u0(c-n) and g(y,s) a(y,n) 2 - W (5-n) !or every x E 5' , y E r , and every c , n E I s a t i s f y i n g II I 6 Let := w0 + maxIllall

.

v.

-

.

C(5)

'

HERBERT

58

llbllC(r) I

. Then the

AMANN

BVP (2.1) i s obviously equivalent t o the BVP

(A

+

W)U = f ( x , u ) + wu

(B

+

W)U = g(x,u)

+ wu

in n

,

r

.

on

Moreover i s a subsolution and F , i s a supersolution f o r the BVP (4.6). Consequently, w i t h o u t loss o f g e n e r a l i t y we can assume t h a t (a,b) > (o,o) , and t h a t f ( x , - ) and g(y,.) a r e i n c r e a s i n g f o r every x E 5 and y E r , respectively. Therefore, by Lemma ( 4 . 1 ) and L e n a (4.2), the BVP (2.1) i s e q u i v a l e n t , where T : I c q + C(K) i s i n t o the f i x e d p o i n t equation u = T(u) i n creasing and completely continuous. Since T E I - and 7 I S subsolution, i t follows that C(Q)

CP)

T(T)

-

v = S(F(j),G(t(v)) -

S(F(T)-

S(Ay,Bv) =

.

Ay,G(t(y))

- BY)

2 o

and, s i m i l a r l y , t h a t T(O) -< 0 Hence T maps the nonempty order i n t e r v a l i n t o i t s e l f . Now t h e existence o f a f i x e d p o i n t f o l l o w s from Schau[7,01cIc d e r ‘ s f i x & ) p o i n t theorem. The existence o f a minimal and o f a maximal f i x e d p o i n t f o l l o w s by an easy i t e r a t i o n argument ( c f . [ 3, Theorem 3 1 o r [ 5, Theorem (6.1) 1).

-

Q.E.D.

Proof of Theorem (2.7):

Mithout l o s s o f g e n e r a l i t y we can assume t h a t

I = [ m i n 71, max 021 . Hence, s i m i l a r l y as i n t h e preceding p r o o f , we f i n d t h a t t h e BVP (2.1) i s e q u i v a l e n t t o the f i x e d p o i n t equation u = T(u) i n C(a) , where T : Ic(E)-+ C(n) i s i n c r e a s i n g and completely continuous. Furthermore, v1 S

-

T(Yl) , T(F1) < F1 , v2 < T(Y2)

T(F2) s G2

.

By Theorem ( Z . l ) , the BVP (2.1) has a maximal s o l u t i o n i3 ic t h e o r d e r i n t e r v a l Hence the [vl,^V11 and a minimal s o l u t i o n u2 i n the order i c t e r v a + [ v g , B ~ ] maximum p r i n c i p l e i m p l i e s t h a t 01 Gl and v belong o he i n t e r i o r o f C(5) , t h a t i s , 01 << G1 and << u These c o n 3 i t i o n s s u f f i c e f o r t h e v a l i d i t y of the p r o o f o f the general m u l t i p ? i c i t y r e s u l t [ 5, Theorem 14.21 (or [ 31 ), which guarantees the existence of a t h i r d f i x e d p o i n t i n [71,82] , The f a c t t h a t there are three d i s t i n c t f i x e d p o i n t s which can be l i n e a r l y ordered f o l l o w s from F 1 the existence o f a minimal f i x e d p o i n t and a maximal f i x e d p o i n t i n I ’ Q.E.D.

<*

-

.

c2 -

.

.

We now consider the l i n e a r BVP (2.3) which, as a special case, contains t h e l i n e a r EVP (2.2). I n t h i s case we l e t w := 1 t p , where p 2 o i s an conThen the l i n e a r BVP (2.3) i s equivas t a n t such t h a t (a + Dm , b + u r ) 2 (o,o) l e n t t o the BVP

.

(A + wm)u

-

( A + w)mu = c

in Q

,

(4.7) (B + wr)u - ( A + w)ru = d on r Denote by S, the s o l u t i o n operator f o r t h e p a i r (A + wm , B t w r ) Since (a t clun , b + wr) > (o,o) , i t f o l l o w s from t h e p r o o f o f Lenma (4.2) t h a t S i s a s t r o n g l y p o s i t i v e compact l i n e a r operator from C(E) x c ( r ) i n t o C(H) Nence t h e equations (4.7) are e q u i v a l e n t t o the equation

.

.

u in

C(5)

. Observe

that

We denote by

-

(A

+

S,(c,d)

w)Sw(mu,rt(u))

>> o

= Sw(c,d)

whenever

(c,d)

Tw the l i n e a r endomorphism o f Twu := Sw(mu,rt(u))

>

.

(0~0)

C(5) defined by

.

i s a compact p o s i t i v e endomorphism o f Then i t f o l l o w s from Lemma (4.2) t h a t T, C(3) Moreover, Tw i s almost s t r o n g l y p o s i t i v e i n t h e sense t h a t

.

NONLINEAR BOUNDARY CONDITIONS

T,(C+(F)\ker(T,))

C

59

.

i n t C+(E)

has the a d d i t i o n a l p r o p e r t y t h a t Moreover, Lemma (4.2) i m p l i e s t h a t T , C+(n))C i n t C+(E) , T,(int that is,

ker(T,)

n i n t C+@)

=

d

. x

F i n a l l y , since we can obviously assume t h a t l i n e a r BVP (2.3) i s e q u i v a l e n t t o l i n e a r equation p~

in

, where

C(5)

p :=

+

(a

-

TU = v and

*

-a

, it

follows that the (4.8)

.

v := pS,(c,d)

I n the f o l l o w i n g paragraph we s h a l l study equations o f t h e form (4.8) i n a r b i t r a r y ordered Banach spaces, where T i s an almost s t r o n g l y p o s i t i v e compact endomorphism l e a v i n g i n v a r i a n t the i n t e r i o r o f the p o s i t i v e cone. Then, due t o the above considerations, Theorem (2.2) and Theorem (2.3) w i l l t u r n o u t t o be easy consequences o f the f o l l o w i n g a b s t r a c t r e s u l t s .

5. ALMOST STRONGLY POSITIVE LINEAR OPERATORS L e t E be a r e a l Banach space. A subset P C E i s c a l l e d a cone i f P i s closed, P + P c P , R+P c P , and P f- (-P) = {o) Every cone induces an o r d e r i n g i n E by s e t t i n g "u I v i f f v u E P" A Banach space E together w i t h a cone P i s c a l l e d an ordered Banach space (OBS) and denoted by (E,P) provided E i s given the o r d e r i n g induced by P Then P i s c a l l e d the p o s i t i ve cone o f E and t h e elements i n fJ := P \ 01 a r e c a l l e d p o s i t i v e . We w r i t e u > o i f u E P a d u >> v i f f u v E , provided, o f course, t h a t P has nonempty i n t e r i o r , I n the f o l l o w i n g , t h e r e a l s , R , a r e always i d e n t i f i e d w i t h the OBS ( R , R + ) whose p o s i t i v e cone, R + := l o , - ) , induces the n a t u r a l ordering

-

-

!.

B

.

.

.

.

L e t (E,P) and (F,Q) be OBSs. A l i n e a r ope a t o r T : E F i s called p o s i t i v e i f T(P) c Q , s t r o n g l y p o s i t i v e i f o T(P) C , and a i m s t s t r o n g l y p o s i tive i f P\ker(T) d and T ( P \ k e r ( T ) ) C Q , provided, o f course, t h a t Q has nonempt y i n t e r i o r .

8

*

F i n a l l y we r e c a l l t h a t the s p e c t r a l radius, morphism

T

o f a Banach space i s defined by

r(T)

-+

,of

a continuous endok l/k r ( T ) := l i m ( I T I I . k-t-

(5.1) Theorem: L e t (E,P) be a n 00s whose p o s i t i v e cone h a s nonempty i n t e r i o r . s u p p o s e t h a t T i s an a l m o s t s t r o n g l y p o s i t i v e compact endomorphism o f

E. Then the s p e c t r a l r a d i u s of T i s p o s i t i v e and a n e i g e n v a l u e of T and, The e i g e n v a l u e r ( T ) , p o s s e s s e s an e i g e n v e c t o r U E P of t h e d u a l o p e r a t o r T' p f T and a p o s i t i v e e i g e n f u n c t i o n a l 0, of T s a t i s f y i n g +,(P\ker(T))Oc

.

R+ . o Proof: By assumption, t h e r e e x i s t s a p o s i t i v e element v such t h a t Hence t h e r e e x i s t s a p o s i t i v e number (Y such t h a t Tv ~ l vE P , t h a t Tv E P This i m p l i e s t h a t t h e s p e c t r a l r a d i u s r ( T ) i s p o s i t i v e ( c f . [ 2, i s , Tv 2 av 14,151 ) . I t f o l l o w s now from the Krein-Rutman theory I 141 t h a t r ( T ) i s an eigenvalue of T and o f T' having an eigenvector uo E P o f T and a p o s i t i v e eigenfunctional 9 of TI,, r e s p e c t i v e l y . Hence uo E P \ k e r ( T ) and, consequently, uo = r(T?-lTuo E P .

.

.

-

HERBERT AMANN

60

.

0

Suppose t h a t u E P \ ker(T) Then Tu E P and, consequent1 > o (cf. [ 5,141 ). Hence <$,,u> = r ( T ) - 1 = r(T)-Y’<@,,Tu> t h a t i s , < Q ~ , U > > o f o r every u E P\ k e r ( T ) Q.E.C.

>

.

r(T)

0,

( 5 . 2 ) Corollary: L e t the hypotheses of Theorem (5.1) b e s a t i s f i e d . Then i s t h e only nanzero eigenvalue o f T possessing a p o s i t i v e eigenvector.

Suppose t h a t t h e r e e x i s t s an element u 1 *. oConsequently, such t h a t Tu = Xu . Then u P \ k e r ( T ) . Hence by applying t h e eigenfunctional Proof:

P and a r e a l number A > o , and u = x-1Tu E q0 , we o b t a i n t h a t = C@~,TU>= <TI$ ,u> = r(T)<@,,u> E

E

x<$o,u> Hence A = r ( T )

0

since <@,,u>

>

o

.

.

Q.E.D.

( 5 . 3 ) Theorem: Let the hypotheses of Theorem (5.1) b e s a t i s f i e d . Then the equation XU

-

TU = v

!5.1)

.

has f o r every h > r ( T ) e x a c t l y one s o l u t i o n , which is p o s i t i v e if V E P rf X < r ( T ) and V E P , then (5.1) has no s o l u t i o n i n P b k e r ( T ) Moreover, if X = r ( T ) and e i t h e r v E P \ k e r ( T ) o r -v E P\ker(T) , then (5.1) has no solution a t a l l .

.

I f x > r ( T ) , then t h e r e s o l v e n t (x-T)-’ i s a p o s i t i v e endomorE ( c f . 1 2 3 1 ). This implies the f i r s t a s s e r t i o n .

Proof:

phism o f

$o

Suppose t h a t x s r ( T ) and v E P t o the equation (5.1), we f i n d t h a t

. Then,

by a p p l y i n g t h e e i g e n f u n c t i o n a l

.

= <@o,v> 2 o ( A - r ( T ) ) <$,,u> Hence t h e remaining p a r t o f t h e a s s e r t i o n i s an easy consequence o f t h e f a c t t h a t g o ( P \ k e r T) C d + Q.E.D.

.

x

o f an endomorphism T o f E i s c a l l e d 1 , t h a t i s , i f t h e v e c t o r subspace i s one-dimensional.

Recall t h a t an eigenvalue

simple i f i t s algebraic m u l t i p l i c i t y equals

u ker(x-T)k o f E k r l (5.4) Theorem: Let t h e hypotheses o f Theorem (5.1) b e s a t i s f i e d . Then r ( T ) is a simple eigenvalue o f T and o f T ’

.

Proof: By t h e Riesz-Schauder theory, i t s u f f i c e s t o show t h a t simple eigenvalue o f T

.

-

r(T)

is a

We f i r s t show t h a t t h e kernel o f r ( T ) T i s one-dimensional. Suppose t h a t o v i s an a r b i t r a r y eigenvector o f T t o the eigenvalue r ( T ) Then, since u E P , there e x i s t s a nonzero r e a l number a such t h a t uo + av E aP Suppose Then r ( T ) ( u o + av) = T(uo + av) = o i m p l i e s t h a t uo & a t uo + av E ker(T) and v are l i n e a r l y dependent. Hence, i f uo and v were l i n e a r l y indepen ent, then uo + av E P\ k e r ( T ) and, consequently, r ( T ) ( u + UV) = T(uo + UV) E , which c o n t r a d i c t s the choice o f a This i m p l i e s tha? k e r ( r ( T ) T) = span(uo)

.

.

.

d

-

.

We suppose now t h a t t h e r e e x i s t s an element v E E such t h a t T)2v = o and w := ( r ( T ) (r(T) T)v o Then Tw = r(T)w and, by t h e f i r s t p a r t o f the proof, we can assume t h a t w = uo Consequently, f o r every a o , r ( T ) ( u o + av) T(u, + av) = u(r!T) T)v = d o ,

-

* .

-

-

which c o n t r a d i c t s Theorem (5.4) since k e r ( r ( T ) - T)2

.

.

uo E P \ k e r ( T )

*

-

. Hence

ker(r(T)

-

T) =

.

NONLINEAR BOUNDARY CONDITIONS

61

- .

F i n a l l y , l e t S := r ( T ) T Then f o r every k 2 2 , the preceding r e s u l t implies that k e r Sk = S-k(o) = S-k+2(S-2(o)) = S- k t 2 ( S -1( 0 ) )

Hence k e r ( r ( T ) follows. Q.E.D.

= S-k+l(o) = k e r Sk-' . T) k = k e r ( r ( T ) T) f o r every

-

-

k t 1

, and

the assertion

(5.5) Theorem: L e t t h e h y p o t h e s e s of Theorem (5.1) b e s a t i s f i e d . Suppose i n addition t h m ker(T) = d and t h a t S is an a l m o s t s t r o n g l y p o s i t i v e compact endomorphism of E such t h a t Tu - SU E P \ k e r ( T ) f o r e v e r y U E P Then r ( S ) < r ( T )

.

.

b

o f S t o the P r o o f : By Theorem (5.1) t h e r e e x i s t s an eigenvector vo E eigenvalue r ( S ) Hence r(S)vo = Svo = Tv - (Tv, Sv,) By applying t h e e i o f the operator T t o & i s equation, we f i n d t h a t genfunctional 0,

.

r(S)<@,,v0>

.

= r(T)

-

<$,,Tv0

-

Svo>

,

that i s since <0,,v0>

-

Tvo

>

0

r ( S ) <~J,,V,> < r(T)<$o,vo> Sv E P \ k e r ( T ) Hence t h e a s s e r t i o n f o l l o w s from t h e f a c t t h a t

. . I.E.D.

be an a r b i t r a r y OBS and l e t e > o be a p o s i t i v e element i n T o f E i s s a i d t o be e - p o s i t i v e if, f o r every u E P , there e x i s t p o s i t i v e numbers a and 5 such t h a t ae 5 Tu I Be This d e f i n i t i o n i s a special case o f the more general d e f i n i t i o n o f an e - p o s i t i v e l i n e a r operator due t o M. A. K r a s n o s e l ' s k i i [ 1 5 1 The reason f o r t h i s r e s t r i c t e d d e f i n i t i o n o f an e - p o s i t i v e endomorphism l i e s i n t h e f a c t , t h a t , besides their importance f o r a p p l i c a t i o n s , these operators t u r n out t o be c l o s e l y r e l a t e d t o t h e class o f s t r o n g l y p o s i t i v e operators. I n f a c t , by endowing t h e v e c t o r subspace Ee := A[-e,e] o f E w i t h t h e order u n i t topology ( t h a t i s , w i t h the eLet

E

(E,P)

. An endomorphism

.

.

norm), i t can be shown (e.g. 15,151 ) t h a t Ee becomes an ordered normed v e c t o r space whose p o s i t i v e cone P has nonempty i n t e r i o r . Moreover, T i s e - p o s i t i v e i f f T(P) c Pe ( f o r more d e t a i l s c f . 1 5 1 ). This f a c t suggests the d e f i n i t i o n o f an almost e - p o s i t i v e endomorphism. Namely, a l i n e a r operator T : E E i s s a i d t o be a l m o s t e - p o s i t i v e , i f P ker(T) d and i f , f o r every u E P \ k e r ( T ) , t h e r e e x i s t p o s i t i v e numbers a and 5 such t h a t ue 5 Tu I Be Then i t can be shown by a c a r e f u l a n a l y s i s o f the proofs i n [ 151 , t h a t almost e - p o s i t i v e endomorphisms o f an OBS have, roughly speaking, the same spectral p r o p e r t i e s as s t r o n g l y p o s i t i v e endomorphisms ( c f . [I31 ). I n a much more elegant way, these r e s u l t s can be e s t a b l i s h e d by u s i n g t h e above i n d i c a t e d connection w i t h almost s t r o n g l y p o s i t i v e endomorphisms.

*

-+

.

Moreover, by modifying t h e corresponding p r o o f i n [ 1 5 ] i t can be shown t h a t the s p e c t r a l r a d i u s o f an almost s t r o n g l y p o s i t i v e endomorphism T i s t h e o n l y eigenvalue o f the c o m p l e x i f i c a t i o n o f T l y i n g on t h e s p e c t r a l c i r c l e . F i n a l l y i t should be observed t h a t i n the above theorems (5.1) - (5.5), the completeness o f E and the compactness o f T have o n l y been used ( v i a t h e KreinHence Rutman theorem) i n order t o guarantee t h a t r ( T ) i s an eigenvalue o f T these theorems can considerably be generalized.

.

Proof o f Theorem ( 2 . 2 ) and Theorem (2.3): A t t h e end o f the preceding paragraph i t has already been observed t h a t the BVP (2.3) i s e q u i v a l e n t t o t h e equation p~ - TU = v

HERBERT AMANN

62

i n C(E) , where P := ( A + W ) -1 , v := PSW(c,d) , and T i s an almost s t r o n g l y p o s i t i v e compact endomrphism o f C(3i) such t h a t i n t C+(E) n k e r ( T ) = @ Hence t h e a s s e r t i o n s f o l l o w f r o m t h e above general r e s u l t s b y o b s e r v i n g t h a t S (c,d) E p r i n c i p l e , t h e BVP (2.3) i s u n i q u e l y Y o l v a b l e

.

BIBLIOGRAPHY

11 1 S. AGNON, A. DOUGLIS, and L. NIRENEERG: Estimates n e a r t h e boundary f o r so-

(2 1 [ 3 1. .

(41 [51 [6 1 17 1 18

I

[9 1 1101

[11 1 [12 I [ 13

I

[141 115 I [16 1 [171 [181

[19 1 [20 1

[21 I

s o l u t i o n s of 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 e q u a t i o n s I..Conun.Pure s a t i s f v i n a aeneral boundarv c o n d i t i o n s .~. Appl .Mith: iII (1959), 623-727. H. AMANN: On t h e e x i s t e n c e o f p o s i t i v e s o l u t i o n s o f n o n l i n e a r e l l i p t i c bound a r y v a l u e problems. I n d i a n a Univ. Math. J., 2 (1971). 125-146. : On t h e number o f s o l u t i o n s o f n o n l i n e a r e a u a t i o n s i n o r d e r e d Banach spaces. J . F u n c t i o n a l Anal., 11 ( 1 9 7 2 ) , 346-384. : A uniqueness theorem f o r n o n l i n e a r e l l i p t i c boundary v a l u e problems. Arch. Rat. Mech. Anal., 44 (1972), 178-181. : F i x e d p o i n t e q u a t i o n s and n o n l i n e a r e i g e n v T u e problems i n o r d e r e d Banach spaces. S I A M Review, t o appear. D.G. ARONSON and L.A. PELETIER: Global s t a b i l i t y o f symmetric and a s y m t r i c c o n c e n t r a t i o n p r o f i l e s i n c a t a l y s t p a r t i c l e s . Arch. Rat. Rech. Anal., 54 (1974), 175-204. H. BREZIS: Probldmes u n i l a t e r a u x , J. F l a t K Pures Appl., 51 (1372), 1-168. H. BRILL: Eine s t a r k n i c h t l i n e a r e l l i p t i s c h e G l e i c h u n g u z e r e i n e r n i c h t l i n e a r e n Randbedingung, Z e i t s c h r . Angew. Math. Mech. t o appear. J.M. CUSHING: N o n l i n e a r S t e k l o v problems on t h e u n i t c i r c l e . J. Math. Anal. Appl., 38 (1972), 766-783. J.-P. D I A Z : Un theordine de S t u r m - L i o u v i l l e pour une c l a s s d ' o p h a t e u r s non l i n e a i r e s m x i m a u x monotones. J. Math. Anal. Appl., 47 (1974). 400-405. J.P.G. EWER and L.A. PKETIER: On t h e a s y m p t o t i c b e h a v i o u r o f s o l u t i o n s o f s e m i l i n e a r parabo1;c'equations. SIN{ J. Appl. Math., 28 (1975), 43-53. K. KLINGELHUFER: N o n l i n e a r harmonic boundarv v a l u e Droblems. I.Arch. Rat. Mech. Anal., 31 ( i m j , 364-371. W. KNOKE: P o s i t i v e Losunaen f u r ZweFPunkt-RandwertDrobleme. D i p l o m a r b e i t . R u h - U n i v e r s i t a t Bochum, 1974'. M.G. KREIN and M.A. RUTMAN: L i n e a r o p e r a t o r s l e a v i n g i n v a r i a n t a c c x i n a Banach soace. Amer. Math. SOC. T r a n s l . . Ser. 1. l o (1962). i99-325, M.A. KRASNOSEL'SKII: " P o s i t i v e S o l u t i o n s o f O p e r a t o r Equations". Noordhoff. Groningen, 1964. O.A. LADYZHENSKAYA and N.N. URAL'TSEVA: " L i n e a r and O u a s i l i n e a r E l l i p t i c Equations". Academic Press, New York, 1968. J.L. LIONS and E. MAGENES: Problemi a i l i m i t i non omogenei (111). Ann. Sc. Norm. Sup. P i s a 15 (1961), 39-101. : Problemi a i 7 i m i t i non omogenei ( V ) . Ann. Sc. (1962), 1-44. Norm. Sup. P i s a : "Non-Homogeneous Boundary Value Problems and A p p l i c a t i o n s I." S p r i n g e r Verlag, B e r l i n - H e i d e l b e r g New York, 1972. C. MIRANDA: " P a r t i a l D i f f e r e n t i a l Equations o f E l l i p t i c Type", S p r i n g e r Verl a g , Berlin-Heidelberg-New York, 1970. J. NECAS: "Les methodes d i r e c t e s e n t h e o r i e des e q u a t i o n s e l l i p t i q u e s " . Academia, E d i t i o n s de 1'Academie Tchecoslovaque des Sciences, Prague, 1967.

-

-

-

16

NONLINEAR BOUNDARY CONDITIONS

[22 1 M.H.

[23 1 [24 ]

[25 ] [261

63

PROTTER and H.F. WEINBERGER: "Maximum P r i n c i p l e s i n D i f f e r e n t i a l Equat i o n s " . Prentice-Hall, Englewood C l i f f s , N.Y., 1967. H.H. SCHAEFER: "Topological Vector Spaces". Springer Verlag, B e r l i n - H e i d e l berg-New York, 1971. M. SCHECHTER: On Lp estimates and r e g u l a r i t y , I. Amer. J. Math. (1963), 1-13. J. SERRIN: A remark on the preceding paper o f Amann. Arch. Rat. Mech. Anal. 44 (1972), 182-186. P. HESS: On the s o l v a b i l i t y o f n o n l i n e a r e l l i p t i c boundary value problems. t o appear: Indiana U n i v e r s i t y Math. Journal.

This Page Intentionally Left Blank

W . Eckhaus ( e d . ) , New Developments i n D i f f e r e n t i a l Equations

@ N o r t h - H o l l a n d P u b l i s h i n g Company (1976) ON THE RANGE OF THE SUM

OF NONLINEAR OPERATORS

H. BREZIS Dept. de Mathematiques U n i v e r s i t e P. e t ) I . C u r i e 4 place Jussieu 75230 PARIS 5"

INTRODUCTLON

-

Let A and B be two continuous functions on R. Clearly we have : R(A+B)

u A~ +

BU

c R(A) + R(B)

t.

u

AV

+ B~

,

V,We(R

M!R

and in general R(A+B) is much smaLler than R(A) + R(B).

However equelity

holds in two simple cases : CASE I : A and B are both non decreasing

-

CASE I1 : A is linear and B is (non decreasing and) bounded. Our purpose is to extend this observation to mappings in infinite dimension a l spaces and t o discuss some applications.

In 9 1 (the extension of case I) we present some results from a joint paper with A. muUX(Image d'une somme d'opErateurs monotones et applications,

to

appear in Israel J. of Math.).

In 02 (The extension of case II) we present a preliminary version of a joint work with L. NIRENBERG,

91.

THE MONOTONE + MONOTONE CASE Let H be a real Hilbert space and let A and B be maximal monotone

-

operators in H. In general,R(A+B) could be much smaller than R(A) + R(B) ; consider for example in H -R2, A

a rotation by + n / 2 and B = a rotation by

- n/2.

Hovewer it turns out that in "many" important cases, R(A+B) and R(A) + R(B) are almost equal in the following sense. We say that two sets Sl and S2 are almost

65

66

H.BREZIS

equal

(S

1

2:

THEOREM 1 .

S ) 2

A = %

Assume

f u n c t i o n s such t h a t Then

-S 1

provided

-

= S

and

2

- a$

B

and

I n t S, = I n t S2

.

are s u b d i f f e r e n t i a l s o f convex 1.s.c.

.

a @ + $ ) = % + a$

R(A+B) = R(A) + R(B).

THEOREM 2 .

Assume

is any maximal monotone o p e r a t o r and

A

B =

a$

with

D(B) = H Then

R(A+B) = R(A) + R(B).

Proof. -

By a w e l l know r e s u l t of R.T. ROCKAFELLAR

We p r o v e f i r s t t h a t

f

E

Av + Bw ( f o r some Let

(I)

R(A) + R ( B ) C

u E E D(A) EU

v

and

+ AuE + BuE 3

a b l e t o conclude t h a t By t h e m o n o t o n i c i t y of (AuE

-

AV

Let

f

E

,A+B

i s maximal monotone.

R(A) + R(B), so t h a t

w).

be t h e s o l u t i o n o f

We are g o i n g t o p r o v e t h a t

(2)

R(A+B).

f

,

E >

& luEl .

f o R(A+B)

we h a v e

A

,u -

v)

>

0

.

.

0

remains bounded as

E +

0

. So,

w e w i l l be

67

ON THE RANGE OF THE S u t l OF NONLINEAR OPERATORS

On the other hand we have (BuE

-

(BuE

- Bw, uE - V)

(3)

-

Bw,uE

(Bw,~) - $(v)

v)

-

JI*(Bw)

so

-

$(u,)

$(v)

(Bw,~) - $(v)

+ (Bw, v

-

uE)

- $*(Bw).

Adding ( 2 ) and ( 3 ) we find (f

-

i.e.

-

f, u

E U ~

-

3

V)

- C, C

4 E / U E [ !vI +

EIUJ2

c

independent of

E.

.

Therefore Next we prove that

Int [R(A) + RCB)]

remains bounded as

E

*

C R(A+B)

.

; we are going to show that the solution uE of ( I )

Let f e Int [R(A) + R(B)]

0. This wilL enable us to infer that us&

(weak convern

gence) with Au + Bu 3 f. By assumption, there is some r > , O such that for a l l h f + h

E

E

H with Ihl< r,

Avh + Bwh (for some vh and wh ) .

Using now ( 2 ) and ( 3 ) with v and w replaced by vh and w we get h - f h , u - vh) > C(h) (f

-

-

where C(h) is independent of

E

(but depends qn h).

Thus (h.UE)

(h,vh)

+

C(h)

1

+

7 cIVhl2

.

Applying the uniform boundedness principle, it follows that uE remains bounded as E

-+

0.

Let B C RN be a bounded smooth domain.

EXAMPLE 1 Given f

E

L2(n),

the equation -Au+

1

T i 3

- €

onB

on

% - O

an

68

H.BREZIS

11

1

h a s a (unique) s o l u t i o n u e H2 (a) i f f

f (x) d x / < 1

n

.

F i r s t o b s e r v e t h a t i f u e x i s t s we have from ( 4 )

-

Next we can w r i t e ( 4 ) i n H Au + Bu

-

Au --Au

-

D(A) with

-

{u e H2(Q) ;

-

D(B)

H

and

2 we s e e t h a t

R(A

B

-

0 on

anl

a$

/= dx)

Applying Theorem 1

f,]l

-

where

with

- J-iT

Bu (Jl(u)

f

L2'(Q) a s

+ B)z R ( A ) + R(B). But, t h e a s s u m p t i o n

I R I i m p l i e s f e I n t [R(A) + R(B)],

dxl

/hlL2 < r

( r s m a l l enough)

f + h

-

[ (f+h)

-

we have

m 1n 1

s i n c e f o r every h e L2(n) with

(f+h)dx] +

[-p!q-

(f*h) dxl E R ( A ) + R(B) 52

By a s i m i l a r argument one can t r e a t the f o l l o w i n g EXAMPLE 2

Given f

8

L2(Q), the equation 1

\

- b ~ - X u + - - f I m

/ u - o where X

on0

onan

is t h e f i r s t e i g e n v a l u e of -A ( w i t h homogenous D i r i c h l e t B.C.)

1

and v1

(> 0 on Q) i s t h e 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 , h a s a u n i q u e s o l u t i o n i f f

T h i s k i n d of e q u a t i o n can a l s o b e s o l v e d by t h e "semi-coercive"

REMARK

methods ( s e e t h e p a p e r s o f SCHATZMAN and H5SS)

-e 11 , - -

Given f

EXAMPLE 3

1I

du + dt

h

L2(0,T), th e equation on ( 0 , ~ )

JIG7

u(T)

u(0)

has a s o l u t i o n i f f

f\ < 1

.

0

PROOF AU

Bu

-

u1

Use Theorem 2 i n H

, D(A)

J i T

IU

D(B)

H(B

a$)

-

L2(0,T) w i t h

e H * ( o , T ) ; U(O)

.

u(T)~

69

ON THE RANGE OF THE Sm4 OF NONLINEAR OPERATORS

THE LINEAR NON MONOTONE + MONOTONE NONLINEAR CASE.

2

§

THEOREM 3.

Let

b e a l i n e a r (unbounded) o p e r a t o r i n

A

H

D(A) = H

with

and c l o s e d g r a p h . Assume N(A) = N(A*)

(5)

f u e D(A)

(6) Let

B

Proof. -

H

+ u2 with

u = u = A

IR(A)

R(A)

-

1

1

f

i n t o i t s e l f . Given

, u2

with

R(B)

R(A) @ N(A)

and

H

- ..

By ( S ) ,

E N(A).

R(A)

and o n t o

6

H

dim N(A) <

H i n t o a d i r e c t sum

u , E R(A)

is

D(A)

i s c l o s e d and

R(A)

and so we c a n s p l i t

R(A) = N(A)I

into

.

R(B)

It f o l l o w s f r o m ( 6 ) t h a t

We write

from

+

R(A+B) = R(A)

bounded. Then

Then

11 i s compact i n H

; l u l + ]Aul G

be a m o n o t o n e h m i c o n t i n u o u s o p e r a t o r from

1 . -A

i s a compact o p e r a t o r

, t h e r e is

E

> 0

+

B,(uE) = f l

a

u

D(A)

E

satisfying

(7)

EU

2E

+ AuC + BuE = f

I n d e e d (7) c a n b e w r i t t e n a s a s y s t e m

1

1 or

1 1

AulE

€u2€ + B2(uE) = f z

-- 1 [ f l

uIc = A

[ f2

u2€

-

B1(uE)I

- B2(uE)1

which h a s a s o l u t i o n by Schauder f i x e d p o i n t theorem ( n o t e t h a t

-

Let

f

,

R(A) + R(B)

E

i s continuous

H). F i r s t we p r o v e t h a t R(A) + R(B)

from t h e s t r o n g t o t h e weak t o p o l o g y i n C R(A+B).

B

f = Av + Bw.

so t h a t

C l e a r l y w e have IuIE/

(8)

C

,

lAulEl

By t h e m o n o t o n i c i t y of

B

C

we o b t a i n (BuE

-

Bw

,u

- W) > 0

ideu

(f

-

c u Z E - Au

-

- w)

f + Av,u

0

Hence

E I u 2~ ~ ~ Q

and t h e r e f o r e

6~

IwI

+

(C

+

IAvI)(C

+

u Z Er e~m a i n s bounded as

Iwl) E

-+

0,Consequently f

E

R(A+B).

70

H.BREZIS

I n t [ R(A) + R(B)] C R(A+B)

Next we show t h a t Let

f

[h/

< r

E

which i m p l i e s u

+

f + h = Avh + Bwh , f o r a l l h

that

2a

- Au 6

( h , u Z E ) 4 C(h)

-h

+ Avh

u

-

Au + Bu

>

0

remains bounded as

2f

n Remark

- Wh)

,u

and t h e r e f o r e

which i s a s o l u t i o n of

u

with

now (- f u

Hence

, so

I n t [ R(A) + R(B)]

. We have

.

E

+

0

.

f

By a s l i g h t m o d i f i c a t i o n of t h e p r o o f , one can show t h a t , under

t h e assumptions of Theorem 3

, R(A) + conv R(B) = R(A+B).

TREOREM 4.

Let

f

E

(9)

Let

H

and

A

B

.

b e a s i n Theorem 3

be such t h a t

lim ( B ( t v ) , v ) > ( f , v ) t++-

for a l l

v

E

Then t h e r e i s a

(lo)

,v #

N(A) u

E

Au + Bu = f

0

D(A)

. s o l u t i o n of

.

Conversely i f (10) h a s a s o l u t i o n . t h e n

l i m (B(tv) ,v) 2 ( f , v ) t++-

(11)

Sketch of t h e p r o o f .

for a l l

exists.

.

i s monotone, t

B

Observe t h a t s i n c e

l i m B(tv),v)

nondecreasing. and

v e N(A)

I+

(B(tv),v)

F i r s t assume t h a t (10) h a s a

t++-

s o l u t i o n . We t h e n have (B(tv)

-

Bu

(B(tv)

-

f + Au

,

tv

- u)

>

0

- u)

a

or

Therefore, f o r (B(tv),v) and ( 1 1 )

v

E

,

tv

.

0

N(A) = N(A*)

a

+ (f,v)

-

(Au,u)

-+

t

follows.

Assume now t h a t ( 9 ) h o l d s . We have

lim

(B(tv),v)

t++-

Since (12)

dim N(A) < 6(v)

-

-

,

<

6(v)

-

Sup ( a , v ) aeconv R(B)

.

(9) i m p l i e s t h e e x i s t e n c e of

(f,v) 3

€1~1

for a l l

v

E

N(A)

6 >

0

such t h a t

.

From (12) we deduce by a s e p a r a t i o n argument ( r e l y i n g o n t h e f a c t t h a t

is

.

THE RANGE OF THE

ON

dim N(A) <

f F I n t [ N(A)'

that

m)

+ conv R(B)].

f o l l o w i n g Theorem 3 t o conclude t h a t Example 4

.

and l e t

N

5 :

Let

Let

f E

Then we a p p l y t h e remark

I n t [ R(A+B)]

b e any e i g e n v a l u e of

X

7:

SUM OF NONLINEAR OPERATORS

-A

.

( w i t h homogenous D i r i c h l e t B.C)

be t h e c o r r e s p o n d i n g n u l l s p a c e . + iR

b e a c o n t i n u o u s , nondecreasing,bounded

function.

The e q u a t i o n

(-

AU

1

-

AU

+ ~ ( u =) f

u - o

on

an

V
B-

s

on

has a s o l u t i o n provided

[! where

v>O] 6 +

6 ?

Proof. -

-

+

I[

'

I,

fv

for a l l

v

f

N

,v f

0

3

l i m B(r).

r+?= Apply Theorem 4

in

H

=

L2(n)

with

Au =

- Au - Xu

, Bu

= 6(u)

and n o t e t h a t ( a s a consequence o f Lebesgue Theorem )

l i m (B(tv),v) =

6-

Remark.

v

*

v<01

t++m

Example 4 can a l s o b e t r e a t e d (even f o r non monotone

u s i n g a Theorem of Landesman-Lazer

L. NIRENBERG , P. HESS e t c

...

6's)

; s e e a l s o t h e subsequent works of

by

This Page Intentionally Left Blank

73

W . Eckhaus (ed.), New Developments in Differential Equations

@ North-Holland Publishing Company (1976)

On the asymptotic behaviour of solutions of an equation arising in population genetics

L.A. Peletier Technische Hogeschool Delft

1. Introduction In this paper we present some results which were obtained jointly with P.C. Fife of the University of Arizona. Consider a population of diploid individuals, living in a onedimentional habitat R . Suppose we c a n distinguish in a gene in this population two different types, called alleles. We shall denote them by A and B. Then the population can be divided into three classes of individuals, called genotypes. Two of these carry alleles which are alike, they are denoted by AA and BB. The third one carries both alleles and is denoted by AB. Let u(x,t) denote the frequency of allele A at place xcn and time t. Then it can be shown that in a simple model, which incorporates the effects of dispersal in the habitat and selection adventages of the genotypes, the function u satisfies an equation of the form

in which the subscripts denote partial differentiation. When the viability of genotype AB lies between the viabilities of genotypes AA and BB, an appropriate choice for f is

This case has been considered by Conley [21, Fleming 131 and Hoppensteadt C41.

74

L.A.PELETIER

In t h e present paper we s h a l l i n v e s t i g a t e t h e c a s e when t h e v i a b i l i t y of AB i s l e s s than t h e v i a b i l i t i e s of AA and BB. I n t h i s case, a p o s s i b l e choice f o r f i s f ( x , u ) = u(l-u)Cu-a(x)I where t h e f u n c t i o n a : R + ( O , l )

(2)

depends on t h e v i a b i l i t i e s of t h e t h r e e

genotypes E l l . For s i m p l i c i t y we s h a l l assume throughout t h i s paper, t h a t f

i s given by ( 2 ) . However, it should be emphasized t h a t our a n a l y s i s a p p l i e s equally w e l l t o many o t h e r choices of f . I t is c l e a r t h a t t h e functions u I 0 and u I 1 a r e s o l u t i o n s of equation ( 1 ) . I f a ( x ) E c o n s t a n t , equation ( 1 ) a l s o possesses a wave f r o n t s o l u t i o n , i.e, a s o l u t i o n of t h e form u ( x , t ) = q f x - c t ) where c i s a c o n s t a n t and q(€,) a monotone function such t h a t q(-m) = 0 and q(m) = 1 [ l , 51. The 1

s i g n of t h e wave speed c i s determined by t h e i n t e g r a l / f ( s ) d s . To s e e t h i s 0

we observe t h a t q s a t i s f i e s t h e equation

I f we now multiply t h i s equation by q

5

and i n t e g r a t e t h e r e s u l t over (--,-),

we o b t a i n

which implies t h a t 1

sgn c = 1

- sgn /

f(s)ds.

0

Thus, f o r example, i f I f ( s ) d s < 0 , then c > 0 and t h e wave moves towards 0

p o s i t i v e x , i . e . f o r each x b I R , u ( x , t ) = q(x-ct) tends t o zero as t

-+

-.

Genetically t h i s means t h a t genotype BB i s dominant and t h a t t h e genotypes

AA and AB a r e swept away. In t h i s paper we s h a l l d i s c u s s a population, l i v i n g i n an i n f i n i t e l y extended h a b i t a t , i n which genotype AA i s dominant f o r l a r g e values of x and genotype BB is dominant f o r l a r g e values of (-x). Thus, t h e function

75

POPULATION GENETICS

1

J(x) = l f(x,s)ds

(3)

0

is positive for large x and negative for large (-x).

2.

The transition layer

By a transition layer we shall mean a monotone equilibrium solution u(x) of equation ( 1 ) such that u ( x )

+

0 as x

-+

--

and u(x)

-+

1 as x

+

-.

Thus, u is a monotone solution of the two point boundary value problem

We shall assume that the finction a, appearing in f, has the following properties:

(i)

1 acC (IR)

j

(ii) a' < 0 on (iii) a(-m)

E

m;

(3,

1 ) and a(-)

c

(0,;).

Without loss of generality we may assume that a(0) = J(x), defined in ( 3 ) is negative on THEOREM 1.

Let f

properties (i)

-

(-a,

;.

Thus, the function

0) and positive on (0, - ) .

be given by ( 2 ) , and let the function a & . ( 2 )

have the

(iii). Then problem I has one and only one soiution. This

solution is monotone. To prove this result we first consider the auxiliary problem u" + f(x,u) = 0 0 < x <

-

(II+) U(O)

= a , u(-) = I ;

where a € [ O , 11. It can be shown that this problem has a unique solution v(x, a ) . This solution is monotone, its derivative at x = 0, v'(0, a ) depends continuously on a and v'(0, 0) > 0, v ' ( 0 , 1 ) = 0 .

L.A.PELETIER

76

Next we consider in a similar vein the problem u" + f(x,u) = 0

--

< x < 0

(11-1 u(--) = 0

, u(0)

= a ,

where ar[O, 1 3 . We denote the solution of this problem by w(x, a ) . It is a l s o monotone, its derivative at x = 0, w'(0, a ) , is continuous on LO, 11 and w'(0, 0) = 0

, w'(0,

1 ) > 0.

It follows that there exists an a c * ( O , 1 ) such that v'(0, a * ) = w'(0, a * ) . Hence the composite function

c::2:

u(x) =

O L X <

-m

-

< x < 0

is the desired solution. Because v and w are both monotone, u is monotone as well.

To prove that this solution is unique, we assume to the contrary that there exist two solutions, u1 and u2. By the uniqueness of solutions of problems II+ and 11-, u1( 0 ) # u2 ( 0 ) . Thus, let us assume that u1 ( 0 ) > u2(0). Then, invoking uniqueness again, it is readily shown that ul(x) > u2(x) on 7R. If we multiply equation ( 4 ) by u' and integrate over IR. we obtain, after subtracting the result for u2 from that for ul:

where y1 and y2

are

However, because u1

the inverse functions of u1 and u 2. >

u2 on IR, y 1 < y2 on (0, 1).

Because fx(x,u) =

= -at(x)u(l-u) > 0 on (0, 1 ) this implies that

It follows that problem I can only have one solution.

POPULATION GENETICS

77

3. S t a b i l i t y We now i n v e s t i g a t e t h e s t a b i l i t y of t h e t r a n s i t i o n l a y e r $ ( x ) , we constructed i n t h e previous s e c t i o n . Thus, w e consider t h e Cauchy problem

U(X,O)

= *(x)

-m<x<m

i n which $ e C ( I R ) , and t a k e s on v a l u e s i n t h e i n t e r v a l LO,

(6)

11. The e x i s t e n c e

and uniqueness of a s o l u t i o n of t h i s problem was e s t a b l i s h e d i n C61; we denote

it by u ( x , t ; $ ) . Consider t h e one parameter family o f functions v(x, h ) = $(x

+

h)

hem.

because f x > 0. Therefore, if h > 0 , v ( x , h ) i s a s u p e r s o l u t i o n of problem I. S i m i l a r l y , i f h < 0, v ( x , h ) i s a subsolution of problem I. This family of sub and supersolutions enables us t o prove t h e following r e s u l t . THEOREM 2.

&

(2),

Let u ( x , t ; + )

be t h e s o l u t i o n of problem (11, ( 6 ) i n which f i s niven

a s a t i s f i e s t h e assumptions ( i ) - ( i i i ) .Suppose t h e r e e x i s t numbers

h , , h2cIR such t h a t hl < 0 < h2,

Then uniformly on 1R.

&

78

L.A.PELETIER

Proof. Since

v(x, h,)

5

$(XI 2 v(x, h2)

-m<x<m

it follows from t h e m a x i m p r i n c i p l e t h a t u(x,t;v(. ,h,))

5 u ( x , t ; $ ) 5 u ( x , t ; v ( .,h2) 1.

However, it can be shown by means of an argument, similar t o one used by Aronson and Weinberger [ l ] t h a t t h e f u n c t i o n s u ( x , t ; v ( . , h i ) )

(i =

1, 2 ) both

tend t o a s o l u t i o n of problem I. Since 4 is t h e only s o l u t i o n of problem I t h e r e s u l t follows. The basic t o o l i n t h e proof of Theorem 2 was t h e family v ( x , h ) of sub and super s o l u t i o n s of problem I. By using more s u b t l e f a m i l i e s of sub and super s o l u t i o n s , we can prove t h e following r e s u l t . THEOREM 3. &J

&

(2) & a

u ( x , t ; $ ) be t h e s o l u t i o n of problem ( 1 1, ( 6 ) i n which f i s Riven s a t i s f i e s assumptions ( i ) - ( i i i ) . Suppose

l i m inf X”

+(XI

> l i m a(x), X”

uniformly on IR.

The proof of t h i s theorem, as well as t h e d e t a i l s of t h e proofs of t h e previous theorems w i l l appear i n a subsequent paper.

POPULATION GENETICS

REFERENCES

El1 Aronson, D.G. and Weinberger, H.F., Nonlinear diffusion in population genetics, combustion, and nerve propogation, Proc. W a n e Progr. in Partial Differential Eqns., Springer Lecture Notes in Mathematics,

(446), 1975. Conley, C., An application of Wazewski's method to a nonlinear boundary value problem which arises in population genetics, Univ. of Wisconsin Math. Research Center Tech. Sumnary Report No. 1444, 1975. C31 Fleming, W.H., A selection-migration model in population genetics, Journal Math. Biology (to appear). C41 Hoppensteadt, F.C., Analysis of a stable polymorphism arising in a selection-migration model in population genetics.

C51 Kanel', Ja.I., Stabilization of solutions of the Cauchy problem for equations encountered in combustion theory, Mat. Sbornik (N.S. ) 59 ( 1 0 1 ) (1962), supplement, 245

- 288.

C61 Kolmogoroff, A., Petrovsky, I., and Piscounoff, N., Etude de 1'6quation de la diffusion avec croissance de la quantit6 de matisre et son application 5 un problzme biologique, Bull. Univ. Moskou, Ser. Internat., See. A, 1 (1937) # 6 ,

1

- 25.

79

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W. Eckhaus ( e d . ) , New Developments i n D i f f e r e n t i a l Equations @ North-Holland P u b l i s h i n g Company (1976)

OPTIMAL CONTROL OF A SYSTEM GOVERNED BY THE NAVIER-STOKES EQUATIONS COUPLED WITH THE HEAT EQUATION. C. CUVELIER university of Delft Delft, The Netherlands.

Introduction.*)

'.

Let 0 be an open set in R The boundary r of n, which is assumed to be regular, is divided into two parts: rl and r2. We consider the following problem of free convection (cf. LANDAU, LIFCHITZ [ I ] ) : Find two scalar functions Te = Te(x,t) = temperature, p-p( x,t )=pressure axid one vector function u=u( x,t ) = [ul(x,t ) ,u2( x,t ) ) = velocity, defined on 'i x [ O , T [ , (x is the space variable, t is the time variable) such that

-

(3)

div u

-

0

,

in Q x ]o,T[ where n = thermal diffusivity, V - kinematic viscosity, u = constant two-component vector, g = heat source6 anf f = external force. The functions T e , u, p boundary conditions:

(4)

Te(x,o)

-

Teo(x),

should satisfy the following initial and

-

. ( X , . )

uo(x),

x E

n,

a denotes the normal derivative at r, directed towards the exterior of 0. The function v is at our disposal and is called the control of the problem. Frequently, in practice, the control is subject to some constraints and we express this fact by requiring that v belungs to a set Uad of admissible controls.

*) For standard notation see LIONS [l], 81

[2]

.

C.

82

CWELIER

We are particularly interested in the temperature distribution at time t-T and our objective is to determine the function v in such a way that the temperature distribution at time t=T is as close as possible (in scme definite sense) to a desired distribution zd(x), x E Q

.

F o r v belonging to the set of admissible controls Uad, the system

(1),..,,(7)

defines Te(x,T) in a unique way: Te(x,T) = Te(x,T;v).

Next, we define a functional v

-

J ( v ) by

-

Zd(x)I2dx

T

J(v)

&(

I

Find

inf 'ad

=

Te(x,T;v)

+ a

/

Iv(x,t)12dr Q O rl where a is a positive constant. The problem of optimal control can be stated as follows: (8)

dt )

J(v).

Any element v+ E Uad such that J(v*)

=

inf 'ad

J ( v ) is termed an optimal

+=

control.

The contents of this paper are as follows:

...,

In chapter 1 we study the well-posedness of problem ( 1 ), ( 7 ) for fixed V. Next we prove the existence of an optimal control and we stqte a necessary cvndition for an element v E Uad to be an optimal control. In order to write this necessary condition in an simpler way we introduce an adjaint system of equations, With the aid of this system we construct an iterative method which provides us, in the limit, with an element of Uad which satisfies the necessary condition for optimality.

...

Because system ( 1 ), ,( 3 ) is not of Cauchy-Kowaleska type (i.e. div u=O does not contain a term it is not easy t o solve numerically.

g),

In order to overcome, in some sensetthis difficulty, we introduce a perturbed system of equations (depending on a parameter e

>

0) which is of

,

Cauchy-Kowaleska type (cf. LIONS [2], TEMAM [ I ] [2]). In chapter 2 we study the well-posedness of this pertuybed system and the existence of an optimal control. As in the unperturbed case we introduce an adjoint system and define an iterative method for the construction o f an element satisfying the necessary condition for optimalily, Chapter 3 is devoted to a convergence theorem in which we prove that the solution (Te , u,, p,) of the perturbed system corresponding to an optimal control veopt converges, in some topological sense, as the parameter E tends to zero, to the solution (Te,u,p) of the unperturbed system corresponding to one of its optimal controls vopt.

83

OPTIMAL CONTROL

In chapter

4

w e g i v e , v e r y b r i e f l y , some of t h e n u m e r i c a l r e s u l t s ,

0PTIi.iAL CONk'ROL OF TKE UlII'ERTUHBEU SYSTEM OF EQUATIONS.

1. 1 .I

THE UlC?ERTURBED SYSTFX. N O T A ' I I O N P ( T e , u , p ; v l .

F3RI~iULATlON O F

B e f o r e we g i v e t h e e x a c t f o r m u l a t i o n o f o u r problem we i n t r o d u c e some f u n c t i o n s p a c e s :

v=Iu E

2

1

d i v u=O) 2 H = c l o s u r e of V i n ( L ) ? 2 V = c l o s u r e of $pin (Ho) (D(D))

E L2, uo E H,

"1

E L2(0,T;L2),

f E L2(0,T;(L2)

2

) , w ,v 0, u = { 0 1 , u 2 } c o n s t a n t v e c t o r , v E L ~ ( c =~ ) L 2 ( o , T ; L 2 ( r l ) ) , we can s t a t e o u r problem as f o l l o w s : Given Teo

g

>

positive canstants, T

Find a s c a l a r f u n c t i o n Te and a v e c t o r f u n c t i o n u s u c h t h a t Te E L2(o,T;H 1 ),

Te' E L 1 ( o , T ; ( H 1 ) . ) ,

u E L2(o,1';V),

U'

**)

T e ( o ) = Teo,

E L~(O,T;V'),

=

4 0 )

uo,

and

(9)

+

Te' +wC1[Te;v]

+ vC,[u]

u' where C1,

C2,

- uTe

B [u,u]

= f

ind'(]o,T[

1 ; (H

i n &(]o,T[

; V'),

I,),

A a n d B a r e d e f i n e d as f o l l o w s :

C1 [Te;v]

C2

+

A [u,Te] = g

[u]

A[u,Te]

B[u,w]

1

: Y(EH )

-

: V(E(H;)~)

: Y(€H1)

:

-

( g r a d Te, g r a d Y )

-

-g -

(v,

( g r a d u, g r a d T )

( LLO P

- ui

i=l

'

YOYIL

au -

Te axi

2

)dx

(r1

***I 1 9

,

We w i l l c a l l t h i s problem: P ( T e , u , p ; v ) .

#=

The f o l l o w i n g theorem d e a l s w i t h t h e well-posedness

of P ( T e , u , p ; v ) .

THEOREM 1.1. Problem P ( T e , u , p ; v )

admits, f o r f i x e d v E L2(C1),

( T e , u ) which i s c o n t i n u o u s from [o,T] u n i q u e element o f L,

*) Pi6

*",

We w r i t e L2, HA,

) N o t a t i o n 7'

-

-

a unique s o l u t i o n

L2 x H. The p r e s s u r e p i s t h e

( o , T ; L ~ ) / Xi a a t i s f y i n g : 1 1 H' i n s t e a d of L 2 ( Q ) , Ho(0), H (a).

; X'

d e n o t e s t h e d u a l s p a c e of

yo d e n o t e s t h e t r a c e o p e r a t o r , yo : H 1 ( Q )

-8

x.

H (P).

C. CUVELIER

84



p d i v p dx and

= 0

<.,.>

0

1 2 denotes t h e d u a l i t y p a i r i n g of (HA)2 and ( ( H o ) )’. PROOF

7

The proof i s c l a s s i c a l and can be found i n LIONS[2],TEMAM[2]or

CUVELIER [ I ] .

4

Concerning t h e dependence of (Te,u) on v we have PROPOSITION 1.1. Let {Te(v), u ( v ) ) be t h e s o l u t i o n o f P(Te,u,p;v).

Then:

00

with 0

<

y

< i.

( f denotes t h e F o u r i e r transform with r e s p e c t t o time).

Let (Te(vi), u ( v i ) )

, TEMAM [23,

s e e LIONS [ 2 j 1.2.

, i=1,2,

be t h e s o l u t i o n o f P(Te,u,p;v

i

1,

then:

CUVELIER [23.

db

THE OPTIMAL CONTROL OF P(Te.u.p;v). The f u n c t i o n v is c a l l e d t h e c o n t r o l , Let Uad be a closed convex s e t

-

i n L2(Gl); by v

Uad

is t h e s e t of admissible c o n t r o l s . The o b s e r v a t i o n i s given

Te(T;v).

Find

inf

-

Let zd € L2 be given; z d is t h e d e s i r e d s t a t e . We d e f i n e

t h e cost-function v

J ( v ) by (8). The c o n t r o l problem then i s :

J(v).

‘ad

THEOREM 1.2.

E Uad such t h a t J ( v o p t ) S J ( v ) , opt i s termed an optimal c o n t r o l .

There e x i s t s a t l e a s t one element v f o r a l l v € Uad.

v

opt

-

*) o j , j = l , 2 , , . . ,

are c o n s t a n t s only depending on t h e d a t a , and n o t on

V.

OPTIMAL CONTROL

85

PROOF Let {vn } be a minimizing sequence in Uad, i.e. J(vn)

-

inf 'ad

Because of the coercivity of the functional J (J(vn) B gal Ivn

J(v), 2

I IL

( 2 %

1)

the sequence (vn} is bounded in L2(Cl) and consequently it contains a subsequence lvnl] such that vn,

(10)

-

w

weakly in L2(C1).

Since Uad is a closed convex set, it is weakly closed. Hence (10) implies w E Uad' Next we claim that Te(T;vnl)

(11)

-

weakly in L2.

Te(T;w)

This result, which is based on compactness theorems, is proved in CUVELIER [ Z ] . From (10) and (11) it is easily seen that the functional J(v) is lower semi-continuous in the weak topology of L 2 ( C l ) . Thus v

-

inf ' a d

J(v)

=

lim nl- m

inf J(vn,) Z J(w) inf ' a d

Hence we must necessarily have J(w)

=

control of the problem P(Te,u pi.):

w = vopt.

A

J(v),

so that w is an optimal

An optimal control can be characterized by the following lemma: LEIm 1.1. A necessary condition for

E Uad to be an optimal control ;-zn element v opt is that it satisfies the following inequality8

for all w E Uad, where TeG(v) [ w ] denotes the Gateaux derivative of Te at the point v in the direction W. PROOF -

This lemma is a direct application of a lemma in LIONS [l].

.sf.

In order to write inequality.(l2) in a more sympathetic form we introduce an adjoint system of equations. l+

1.3.

FORMULATION OF AN ADJOINT SYSTEM. NOTATION P*(B,a.Xzv,Te.u.u~.

C. CUVELIER

86

Let { T e , u , p ) be t h e s o l u t i o n of P ( T e , u , p ; v ) ,

t h e n we d e f i n e t h e

a d j o i n t problem as f o l l o w s : Find a s c a l a r f u n c t i o n 8 and a v e c t o r f u n c t i i l n q s u c h t h a t

where A* and B* a r e d e f i n e d by:

We d e n o t e t h i s problem by P*(B,q,x;v,Te,u,g).

dh

THEOREM 1.3.

Let ITe,u,p) be t h e s o l u t i o n of P(Te,u,p;v). has

a u n i q u e s o l u t i o n {B,q)

Q,,(o,T;L2)/IR

d e f i n e d by:

<-q’(t) + v C2[q(t)] -B[u(t),

- A*[Te(t), PROOF See -

Prohlem P*(B,q,n;v,Te,u,p)

and t h e p r e s s u r e x i s t h e u n i q u e e l e m e n t i n

q ( t ) ] -B*[u(t),q(t)]

d t ) ] + D [n(t)],cp

CWELIER [2].

>

+

1 2

f o r a l l ‘p E ( H o )

= 0

4

Using t h e s o l u t i o n of t h e a d j o i n t s y s t e m we can prove t h e f o l l o w i n g LEMMA 1.2.

E Uad t o b.e a n o p t i m a l c o n t r o l

A n e c e s s a r y c o n d i t i o n f o r a n element v

opt i s t h a t i t s a t i s f i e s the following i n e q u a l i t y ;

i

( w o ~ ( t ; v o p t )+ a v o P t ( t ) , w ( t )

-

(r

Vopt(t))L

2

0

d t 2 0,e)w

E Uad.

1

PROOF

.

T h i s lemma is a d i r e c t consequence o f l e m m a 1.1 and t h e f a c t t h a t G G { e ( v ) , q ( v ) ] is t h e a d j o i n t s t a t e of { T e ( v ) [w] ,u ( v ) [w]]

u

REMARK 1.1. G

We also f i n d t h a t J ( v )

0

wyoB(v)

+ av 9

.

The r e s u l t s may now be summarized i n t h e f o l l o w i n g theorem:

OPTIMAL CONTROL

THEOREM 1,4. A necessary condition for an element vopt E Uad to be an optimal control is that (Te,u,p,B,q, n,vopt ) satisfies (i) [Te,u,p) is the solution of P(Te,u,p;vopt) (ii) [B,q,n) is the solution of P*(8,q,n;vopt,Te,u,p) T

(iii)

J

(Hroe(t) + crvopt(t),

1.4.

w(t)

+

0

-

Vopt(t)lL

( p dt g 0 for all w€Uaa 2 1

ITERATIVEALGORITHM. For the construction of an optimal control we can use,among others,

a gradient method with a projection. We choose v(O) E Uad arbitrarily. Once we know v'~) we calculate [Te(vL:L), u(vopt)] (m) Optby solving P(Te,u,p;v(m)) opt opt which provides us with Next we solve P*;B,q,n;vopt,Te,u,p) (m) The (m+l)-th approximation of an optimal control is

{B(vOpt (m) ), q(vopt)). (m) now given by

where P

-

denotes the projection on the set Uad;p is a positive constant means the restriction to C,.

1

#

THEOREM 1.5. When p satisfies 0

<

p

<

2 (M will be defined in lemma

1,3),

the sequence

)!;:v[

contains a subsequence (v("')) converging to an element opt satisfies the necessary E Uad weakly in L2(C1). The limit v* v* opt G opt h 0 , v w E Uad. condition for optimality, i.e. J (vZpt) [w-v&]

PROOF In the proof of this theorem we use the following two lemmas; the proofs

of these lemmas can be found in CUVELIER [2].

LEMMA

1.3.

The functional v

-

lb J(v) is two times Gateaux-differentiable and for all

v E Uad we have

JGGW [WliW2I

~MllWqllL( C )Ilw211L2(x,)9 2

where J

for all w1w2 E L ~ c1 ()

1

GG (v) [wl;w2] denotes the second derivative of J at the point v in

the directions w1 and w2.

LEMMA l r 4 . If {vn) is a sequence in L2(C,)

+ that converges weakly in L2(x,)

to an

c. CUVELIER

88

element v * E L 2 ( C 1 ) ,

t h e n yo&(vn) converges s t r o n g l y t o yo8(v")

in

4=

L2( z1 1 PROOF OF THEOREM 1.5.

We give t h e Taylor s e r i e s of J ( V ( ~ + ' ) )about t h e element v ' ~ ) up t o and opt opt i n c l u d i n g t h e t h i r d o r d e r term

J(vk:il))

(14)

+

where

vpl)=

pvopt (m)

*

8

JGG(V(m+l) k

(l-p)vopt (m+l)

(m+l)-v(m)* v(m+l)-v )[Vopt opt' opt pt

f o r some p E

[0,11.

The second term of t h e r i g h t hand s i d e of (14) can be estimated by:

This is a simple consequence o f (13). F o r t h e t h i r d term we use lemma 1.3.

Therefore we o b t a i n

-

-

l~~~~

J ( V ( ~ + ~ J(vopt) ) ) (m) S !( $){ opt If p s a t i s f i e s 0 < p < t h e sequence ( J ( v ( " ) ) ) is convergent and

4,

OP t

i s bounded i n L2( El). Thus t h e r e E L2(C1) and a subsequence { v ' " ' ) } o f ( v ( " ) ) with

consequently t h e sequence {v:;;) e x i s t v* opt

-

opt

v ( ~ ' ) v* opt

C Uad

opt

weakly i n L2(C1).

opt

Because Uad i s convex and c l o s e d , we a l s o have v+ E Uad. The remaining opt t h i n g t o check i s t h a t v* s a t i s f i e s t h e necessary c o n d i t i o n f o r opt optimality. We remark t h a t f o r a l l w E Uad t h e f o l l o w i n g i n e q u a l i t y holds,

rn

We denote t h e r i g h t hand s i d e of (15) by

Using remark 1.1.

we

transform (15) i n t o

J

.

Opt- 'L2(1'.,)

J

We

-

-

a s s t o t h e l i m i t (m-m'

RcmP

0 as m

. '0

'

Opt"

0Pt'L2{1',)

-

-

0

0

w

, we

-

= ) and due t o lemma 1.4.

obtain

and t h e f a c t t h a t

OPTIMAL CONTROL

89

U

-2 a

lim inf ml

+ m

which is equivalent to. JG(v* ) [w-v;pt] L 0, f o r all w E Uad. opt This completes the proof of theorem 1.5

+

4

2. OPTIMAL CONTROL OF THE PERTURBED SYSTEM OF EQUATIONS. In this chapter we perturb the original system of equations in such a way that we obtain a system which is o f Cauchy-Kowaleska type. This perturbation facilitates the numerical treatment of the problem, because the condition o f incompressibility of the fluid (i.e. equation ( 3 ) ) ’

seriously complicates the numericd calculations. The method of perturbation (often called the method of artificial cdmpressibility) was first introduced by TEi”lAM [I],

[2] and LIONS [2].

db 2.1. FORMULATION OF THE PERTURBED SYSTEM. NOTATION P (Te ,u ,p ;v). Apart from the data of chapter 1.1.

we take po E L2 arbitrarily. We

state the perturbed problem as follows: Find two scalar functions Te,,p, and a vector function u, such that Te; E LI(o,T;(H1)’), 1 2 u; E L1(0,T;((Ho) I , ) , 1 P: E L2(0,Ti(Ho)’),

Tec E L2(0,T;H1), 1 2 uC E L2(0,T;(Ho) 1, P, E L2(0,T;L2),

Te,(o) u

=

Teo,

p = uo,

PJ”)

=

Po’

and

(16) Tey

+ x C1[Tec;v]

+

g Tee D[p,]

A[u,,Tec]=

+ v C2[u,]+ B[uC,uC]-u EP: + D*[uc] = 0

u;

-

ind)’(]o,T[ = f

;(HI)’ ),

ins*(]~,T[;((H:)~)’ in 23’ ( 30, T[ iL2),

where D* is defined by: D*[uc]

: cp(E L2)

-$

cp div ucdx.

P

We refer to this problem as’to PE(Tee,uc,pE;v)

9 THEOREM 2.1. For fixed v E L2(C,), problem Pc(Tee,u ,p,;v) has a unique solution (Te,,u,,p,] which is continuous from [o,T] L2 x (L2)2 x L2.

-

PROOF -

See CUVELIER [2] which uses standard techniques.

#

),

90

C.

Let {Te,(vi),u,(vi),pc(vi)

I,

CTNELIER

i = 1 , 2 9 be t h e s o l u t i o n of Pc(Tee,ue,pE;v i ),

then t h e following e s t i m a t e h o l d s :

PROOF see LIONS [2], 2.2.

TETlAM [ 2 ] , CUVELIER [2].

Q

THE OPTINAL CONTROL OF P ( r e

,u

,p

;v).

The f u n c t i o n v i s t h e c o n t r o l and we impose t h e c o n s t r a i n t t h a t i t belongs t o t h e closed convex s e t Uad by v

of

L2(c1).

Je(v) =

I

$1

Tec(x,T;v)

1

- zd(x)I2dx + a

(1

The c o n t r o l problem can be s t a t e d as:

O

Find

-

The o b s e r v a t i o n i s given

TeE(T;v) and we d e f i n e t h e c o s t - f u n c t i o n v

inf ’ad

J e ( v ) by b ( x , t ) I2dMt]*

rl

Jp).

THEOREM 2.2. There e x i s t s at l e a s t one element v f o r a l l v E Uad.

copt E Uad such t h a t J & ( v Copt) vcopt i s c a l l e d an optimal control.

A necessary c o n d i t i o n f o r a n element vcopt E Uad

is t h a t i t s a t i s f i e s t h e i n e q u a l i t y

Jp)

t o be an optimal c o n t r c l

OPTIMAL CONTROL

91

PROOF See

*’

the proofs of theorem 1.2.

a$ lemma F o r the same reason as we did in chapter 1.2.

we introduce an adjaint

system of equations with respect to P (TeE,u,,pE;v). 7

2.3.

b

E

FORMULATION OF AN ADJOINT SYSTEM. NOTATION P < 8 ,q

,X

;v,Te ,u ,p ).

Let (Tec,uE,pE)be the solution of P (Te ,u ,pE;v). We define the E E E adjoint problem as follows: Find two scalar functions e E , x E and a vector function q BE

E L2(0,T;H 1 ),

8; E L1(o,T;(H7)’),

1 2

1,

9, E L2(0,T;(Ho)

nE E L2(0,T;L2),

1 2

sk E L1(0,T;((Ho) xk

1

B,(T)

1’1,

such that =

TeE(T)-zd,

qE(T) = 0, K (T) = O,

E L2(0,T;(Ho)’),

and -8;

+ X

- E K ~

C1[8,;0]

- D*[q,]

-A[uc,BE] -oq

=

=

in d ( 10,T[; (H1 )’ ),

0

0

This problem we call P:(BE,qE,n,;v,TeE,uE,pE).

=#

THEOREM 2.3. Let (Te,,uE,p,)

be ‘the solution of P (TeE,uE,pc;v). Problem

P:(B,,q,,nE;v,TeE,u~,p~)

has a unique solution @,,q

E

,K

E

).

PROOF The proof is based on standard techniques; see, for example, CUVELIER [2].

4

In the following theorem we give a characterization of an optimal control (cf. theorem

1.4.).

THEORM 2.4. A necessary condition f o r an element v

E Uad to be an optimal control

is that ITe,,uC,pE,SE,qL,KE,vEop~) satisfies: (i) (Fe,,u,,p,) is the solution of P (TeE,uc,pE;v ) EoPt (ii) IB,,q,,nct is the solution of PZ(BE,qE,xE;v,opt,Te,,uE,~E),

2.4.

ITERATIVE ALGORITHM. To construct an optimal control, we use the same iterative method as ( 0 ) E Uad arbitrarily, Let M, be a constant in chapter 1.4. Thus we choose vEopt

( c f . lemma 1.3.)

which is defined by

c.

92

CUVELIER

We have t h e f o l l o w i n g theorem THEOREM 2.5,

<

If p s a t i s f i e s 0

p

< f- , then

t h e sequence (v$At),

d e f i n e d by

P

contains a subsequence

converging (km')) ( Pt

weakly t o an element vZopt E Uad

i n L ( C ). Moreover vFopt s a t i s f i e s t n e necessary c o n d i t i o n f o r o p t i m a l i t y : 2 1 J:(;Z~,) [w-vzoptI 2 o f o r a l l w E uad.

PROOF The proof i s e x a c t l y t h e same a s t h a t of theorem 1.5.

+

3 . PASSAGE TO THE LIMIT. P,(Te,,u,,pt;vEo

t)

-

P(Te,u,p;vo t ) AS

t

-

0.

I n t h i s chapter we prove t h a t t h e s o l u t i o n (Te,(v,opt),u,(v,opt), p,(vtopt ) ) of t h e perturbed optimal c o n t r o l problem converges, i n some t o p o l o g i c a l sense, t o t h e s o l u t i o n [Te(vopt), u(vOpt), p(vopt) ) of t h e unperturbed optimal c o n t r o l problem.

3.1.

PRELIMINARY

RESULTS,

4

We s t a t e some preliminary r e s u l t s f o r a f i x e d c o n t r o l . The proof of t h e s e r e s u l t s can be found i n CWELIER [ I ] ,

[2]

.

THEOREM 3.1. Let (Te,,u,,p,)

be t h e s o l u t i o n of Pt(Tec,u,,pt;v).

converges t o t h e s o l u t i o n {Te,u,p] of P(Te,u,p;v)

This s o l u t i o n i n the following

t o p o l o g i c a l sense: Te,(T;v)

(17)

Te,(v) Te,(v) u,(T;v) .,(V) U,(V)

--

Te(T;v)

strongly i n

Te(v)

weakly-r i n

Te(v)

strongly i n

u(Tiv)

strongly i n

.(V)

weakly-r i n

u(v)

strongly i n 96

3.2.

vco

RANGES I N A BOUNDED SET OF L 2 ( C , j

LEMMA 3.1. Any optimal c o n t r o l vEopt belongs t o a bounded s e t o f L 2 ( C , ) ,

independent

of t. COROLLARY

There e x i s t an element w E Uad and a subsequence

( v t o p t ) such

93

OPTIMAL CONTROL

-

w weakly in L ~ ( c as ~ ) E'OPt PHOOF OF LWINA 3.1. that v

Let j = J ( v ) opt

=

J(v) and j

inf 'ad

=

-

E

0.

JE(vEopt )

=

inf 'ad

JE(v).

Because we have ( I T ) , the equation lim J c ( v )

(18)

=

J(v)

E - 0

holds for all v E L 2 ( C 1 ) . as

E

-

Hence j

=

J E ( v ) d JE(vopt)

inf

-

J( V o p + , ) = j

'ad 0, so that

lim sup

(19)

jES j

E - 0

On the other hand we have j E = JE(VEopt) 2 h a

I lVEopt 1 IL2 2

(C

1 9

which

1

combined with ( 1 9 ) shows that + a [ lVEoptI (2L 2 ( z 1 ) S 3.

-

=#

3.3.

PE(TeE,uE,pE;vEopt) P(Te,u,p;w) as

-

E

0.

THEOREM 3.2. ). There exists Let (TeE,uE,pE}be the solution of PE(TeE,uE,pE;v Copt a triplet of functions (Te,u,z} such that the subsequence {TeEl,uE,,pcl ) of (Tec,uE,pE}, where the sequence { c l ) is taken from the corollary of lemma 3.1., TeEl

UEl

-

satisfies: 1

weakly-w in L,(o,T;L2),

Te

weakly in L2(0,T;H )and

strongly in L~(o,T;L~),

-

weakly-w in Lm( O, T;( L2)2), weakly in L2(o,T;(Ho)1 2 ) and strongly in L2(0,T;(L2) 2),

PPEl - =

weakly-n i n Lm(

0, T;L2 )

.

Where, (Te,u} i s the solution of P(Te,u,p;w) with p determined by theorem 1.1. and w by the corollary of lemma 3.1. PROOF

-

The proof makes use of proposition 2.1. argument8 .

and i s based on compactness

4

3.4. w IS AN OPTIMAL CONTROL OF THE PROBbEM P(Te.U.D:.) LEMMA 3.2.

Let {TeE,uc,p,) be the solution of PE(Tec,u EsP,lvcopt) and (Te,u,p) the solution of P(Te,u,p;w). The subsequence {Tec,(T;vclopt))of {TeE(T;vcopt)), where [ e l } I s taken from the corollary of lemma 3.1., (20)

-

aatiafiee:

~ e ~ , ( ~ ; T~(T;w) v ~ ~ weakly ~ ~ ~ in ) L2

C. CWELIER

94

PROOF

with We take the scalar product of (16),where we replace v by v Copt' X E D ( a ) and we integrate with respect to time from 0 to T. We pass to the O), which is permitted by theorem 3.2. limit ( e l On the other hand we may multiply (9), where v is replaced by w, with X E D ( 3 ) and, again, integrate with respect to time from 0 to T. Comparing the two results completes the proof of this lemma.

-

=b

THEOREM 3, 3L

The element w E Uad, determined by the corollary of lemma 3.1. optimal control of the problem P(Te,u,p;.).

is an

PROOF

It follows from lemma 3.2.

E'

equivalent to

-

-

that lim inf J,,(vclopt) z J(w) which is 0

lirn inf j c l Z J(w) Z j.

(21)

c'

0

Combining (19) and (21) we find that

So it is legitimate to write: w

-

vopt.

=#

3.5. TWO SUPPLEMENTARY RESULTS THEOREM 3.4.

The convergence of v ~ , as ~ stated ~ ~ ,in the corollary of lemma 3 . 1 , vopt strongly in L~(c~). place in a finer topology, via. vc'opt

-

PROOF -

take8

In order to proof this, we introduce the following norm on L2(C1) 2 2 II 31 T~~(T~o) + I V I1L2(c1 j L2

IVII 12

IT~ (T!V)

-

-

II

Te). (when c = O we set Te Using propositions 1.1. and 2.1. we can prove that this norm is equivalent to the standard norm of L2(C1) for E 2 0 (cf. CWELIER [2]). We consider the following equality: 2 l 2c I = Jc(Vtopt) J(vopt) JEW J(0)

I I btopt II -t

-I

-

boptI I I,

(TOc( TiVcopt)-Te(T;vopt)

- (Tet(T;vEopt), We pass to the limit

(E

-

#

-

+

zd)L2+(Tec (TIo)-Te(Ti

Tec(F;o))

0 )

+ ('fe(Tivopt),Te(T;o))

=2

0) via the sqbsequenoe

I

II

-

(E')

II

of

(E)

I lo -

,zd)L2

*

L2 and due to

(17),(18),(20),(22) we obtain IvCloptl I c I I Voptl Combining this result with the w e a k convergence of vc'opt to Vopt concludes the proof of the theorem.

9

+

OPTIMAL CONTROL

95

T

1

Because the trace operator is a continuous map from L2 (0,T;H ) into L 2 ( , q ) where both spaces are equipped with the weak topology, and because V v strongly in L2(C,), it follows that lim X ' = O . This c'opt opt r 1 - O

-

proves the theorem.

i47&

4. A NUMERICAL EXAMPLE. In this chapter we describe, very briefly, a nunierical example to illustrate the procedure proposed in this paper. A complete treatment of the numerical considerations can be found in CWELIER [2],[ 31We describe the physical situation as follows: An non-isothermal fluid is contained in a square closed cavity (length = d ) placed in the field of gravitation (g=gravitational acceleration). The fluid is initially at rest and at constant temperature Te,. are: v=kinematic viscosity, x

3

The fluid properties

thermal diffusivity, p-density, c =specific

P

,

c. CWELIER

96

heat. Our objective is to obtain at time T a temperature distribution equal to Tel (constant). We normalize the equations and the domain by the following constants: = unit of time, Xc = d = unit of length, uc = v.d-' = unit of tc = d2v" velocity, pc = pv2d-' = unit of pressure, ATe = Tel-Teo = unit of temperature, The situation is as follows: p2

direction of the gravitational field

1

r2

r2 is that part of the boundary which is thermically insulated, while on r l heat input or output is possible without constraints (1.e. Uad = Lz (cl 1) The normalized problem reads as follows: R =

aTe at

( (x,,x2]

10

<

X,

<

1, 0

<

x2

<

9=

11,

T t',

1

2

C u i e = L ATe Pr in1 au au = 0 Te + Au grad P + (Gr) at + l : i ui

7

+

-

-

in R

x

lo,%

div u = 0 P cp v

where P r =

7 = Prandtl number,

Cr = B o $

(p

-

Te d3 = Crashof number

V

coefficient of thermal expansion)

The initial conditions are Te(x,o) =0, and the boundary conditions are

u(x,o)

x E

= 0,

aTe_ 0 an

on

r2

x ]o,T[,

aTe an

on

rl

x ]o,'i;[,

on

r

=

v ATe

u - 0

x]o,'i;[

;,

.

We fix the instant 'i; : 1 = 0.075 and the desired state Tel=l (-zd). The normalized cost-function becomes: u rn

I

with = av-'d For the numerical treatment we discretize, by means of a finite difference

OPTIMAL CONTROL

97

method, the perturbed problem PC(Tee,uC,pC;v) corresponding to the problem just described; furthermore we discretize the adjoint problem Pt(BE,q,,nC;v,TeC,uC,pC) ,the iterative algortihm and the cost-function. Considerations about the discretized problems, such as stability and convergence are treated in CWELIER [2]. For some numerical results we choose P r = 0.733 (for air), Gr = 6850,

,T., 0.075,~ = 25 =

Y

At = time-step in finite difference scheme =

0.0005, Ax = space-step = 0.1. In the iterative algortihm we fixed p = The values of the discretized cost-function are listed below for two

7.0.

values of E. number of iterations

'r

=

=

10-5

0.50000

0.50000

1 2

.22660

.22629

3 4 5

17274

.17242

.i4916

.14872

.

-12339 1 1562

o

7

.10891

.I0886

8

-10487 1 0096

.

.lo402

15

09770 -08702

09673 .oe5so

20

,081 05

07971

6

9 10

For a discussion of these results and others

12277

-11491

el0006

see CUVELIER [2],

[3].

= # #

REFERENCES C. CWELIER

G

10-4

[l]

: Approximation numgrique de la solution des h a t i o n s

de

Navier-Stokes coupldesa celle de la chaleur (Application au problbme de convection libre).Fublication at the University of Delft (1973). [ 2 ] : Doctoral thesis, Delft (to appear) [3] : to appear.

L. LANDAU

- E.

LIFCHITZ [l]

: Mdcanique des fluides

(1971)

J.L. LIONS

[l]

R. TEMAM

[l]: Sur l'approximation de la solution des kquations de

: ContrSle optimal de

systhmes gouvernks par des Cquations a w dkrivies partielles (1968). [2] : Quelques mgthodes de rksolution des problhmes aux limites non line'aires (1969).

Navier-Stokes par la rnithode des pas fractionnaires I.

C.

98

CWELIER

Arch.Rat.Mech.Ana1. R.

TEMAI-1

[2] :

2

(1969) p. 155-1531.

Cours Navier-Stokes. Cours B l’universitk de Maryland

(1973).

W . Eckhaus (ed.), New Developments in Differential Equations

9 North-Holland

Publishing Company (1976)

SECONDARY OR D I R E C T BIFURCATION OF A STEADY SOLUTION OF THE NAVIER-STOKES EQUATIONS I N T O AN INVARIANT TORUS.

Gerard IOOSS I n s t i t u t de Mathematiques e t S c i e n c e s P h y s i q u e s Parc Valrose

I

-

06034

[FRANCE)

NICE

-

Statement o f t h e problem. 11 A c l a s s i c a l e x p e r i m e n t a l example. L e t us t a k e as an i l l u s t r a t i o n o f our a b s t r a c t r e s u l t s ,

the classical

e x p e r i m e n t s on t h e T a y l o r p r o b l e m . Between two c o n c e n t r i c c y l i n d e r s , around t h e i r a x i s ,

rotating

t h e r e i s an i n c o m p r e s s i b l e v i s c o u s f l u i d s a t i s f y i n g t h e

N a v i e r - S t o k e s e q u a t i o n s . I t i s known t h a t w h a t e v e r a r e t h e r o t a t i o n r a t e s w1 and

o2 o f t h e i n n e r and o u t e r c y l i n d e r ,

we have a s t e a d y s o l u t i o n c a l l e d t h e

C o u e t t e f l o w , whose s t r e a m l i n e s a r e c o n c e n t r i c c i r c l e s . I n fact, l e t us t a k e

t F . i s s o l u t i o n is o n l y o b s e r v e d i f

w2 < 0 and l e t us i n c r e a s e

c a l value, i n s u i t a b l e conditions,

lull

i s s m a l l enough. Now,

w1 > 0. When i t c r o s s e s a f i r s t c r i t i -

i t can be o b s e r v e d an o t h e r f l o w w h i c h i s

o f c e l l u l a r t y p e and p e r i o d i c i n ' t i m e . I f we i n c r e a s e a g a i n

t h e n a f t e r t h e o c c u r e n c e of

wl,

some more and more

complicated flows, a t r u e t u r b u l e n t f l o w occurs. T h i s i s j u s t an example t o j u s t i f y t h e f o r m a l t h e o r i e s developped by E. HOPF

[2]

i n 1942 and L. LANOAU

i n certain situations,

b i f u r c a t i o n s of s o l u t i o n s , when a p a r a m e t e r ( a s

[I23

i n 1944. To e x p l a i n t u r b u l e n c e

t h e y had f o r m u l a t e d a n e x p l a n a t i o n b s s e d on s u c c e s s i v e

0,)

o f t h e N a v i e r - S t o k e s e q u a t i o n s . becoming u n s t a b l e increases.

21 N a v i e r - S t o k e s e q u a t i o n s . We have i n g e n e r a l a system o f t h e f o r m

+

VP = uAV

+

f ,

G . IOOSS

100

where V is t h e v e l o c i t y o f t h e f l u i d a t t h e p o i n t ( x . t l c Q

x R + , p is t h e

p r e s u r e . f i s a g i v e n e x t e r n a l f o r c e , and a i s g i v e n on t h e boundary 2 ' 3

bounded r e g u l a r domain il o f t R

an

of a

.

or R

I n f a c t , o t h e r phenomenon such as t h e % n a r d c o n v e c t i o n , o r f l o w s w i t h t h e obeys systems o f e q u a t i o n s w h i c h have a

occurence o f e l e c t r o m a g n e t i c f i e l d , s i m i l a r s t r u c t u r e as (11,

( s e e [I] I

. I n t h e example c i t e d above, we have t o

t a k e an R w h i c h i s a bounded domain o f p e r i o d i c i t y o f t h e f l o w i n z ( p e r i o d 2n/(rI,

i

as t h e e x p e r i m e n t s suggest u s . The t h e o r y , t h a t we s h a l l d e v e l o p p , r u n s

w e l l w i t h t h i s s o r t o f domain s l s o . Moreover,

o t h e r boundary c o n d i t i o n s a r e

.

p o s s i b l e [3]

31 B a s i c f l o w ,

perturbed equations.

L e t us assume t h a t we know a s t e a d y s o l u t i o n (Vo,pol I n fact,

c a l l t h i s s o l u t i o n "the basic flow".

u-'

c h a r a c t e r i s t i c p a r a m e t e r such as i t by A and assume t h a t V

v

o f [I]. We s h a l l

f o l l o w i n g t h e problem, we have a

o r w1 i n t h e example o f 11. L e t us d e n o t e A. Now we pose

i s analytic i n

= Vo[A1 + u.

Hence, t h e p e r t u r b a t i o n u s a t i s f i e s a system o f t h e form du dt

[2)

LAu - H r u l = 0 .

+

where we l o o k f o r t w u ( t 1 as a c o n t i n u o u s f u n c t i o n t a k i n g v a l u e s i n t h e domain g o f t h e l i n e a r o p e r a t o r

LA, w i t h

a c o n t i n u o u s d e r i v a t i v e i n an H i l b e r t

space H. I n t h e case o f t h e s y s t e m (11. we t a k e H = { ~ E [ L ~ [ ~ I: - J ~v . u = G .

g = { U E [ H ~ [ S I I ] ~i

x

=

D}# o n t o H.

uEd

L A u = n[-uA u

H(u1

ulan

0.u = 0,

GI,

Il t h e o r t h o g o n a l p r o j e c t i o n * i n [L2(Rl]3

and i f we denote by we have V

u.n/an =

=

+

(u.0)

Vo[A1

+

[Vo[A).V)u]

I

-ll[[u.Vlu].

I t i s known,

following

[Ill , [IS]

, t h a t H I = I u = V(f.(PEH1[Q)}.

SECONDARY

These d e f i n i t i o n s

BIFURCATION

101

ensure us t h e l o c a l e x i s t e n c e and uniqueness o f t h e Cauchy

problem ( 2 1 , w i t h U(OI

= u o ~ . 9 ( s e e [4]1.

The p r o o f i s based on t h e f a c t s t h a t

il

oo i s an h o l o m o r p h i c f a m i l y o f c l o s e d o p e r a t o r s i n H.

o f domain

9 ,

where Oo c C.

V h E Do,

i i )

i t can be d e f i n e d an h o l o m o r p h i c semi-group o f o p e r a t o r s i n

-LXt H : [e

} t 2 0'

iiil There e x i s t s C 2 D such t h a t VXE Do and V u e g / / e- L A t M

[UI/[~S C

t - a l ( u l l . 92,

with

Q

< 1. t E ] O . T ] .

T <

-

( i n t h e case o f (11 we have a = 3/41.

41,Known r e s u l t s a b o u t t h e f i r s t b i f u r c a t i o n i n t o a p e r i o d i c s o l u t i o n .

I f a l l p o i n t s o f t h e spectrum o f L t h a t t h e b a s i c f l o w Vo, stable.

i.e.

L e t u s assume t h a t f o r

p r o p e r t y , and t h a t f o r a n u l l r e a l part, h crosses

X

have a p o s i t i v e r e a l p a r t , i t i s known

t h e s o l u t i o n u = 0 o f (21. i s a s y m p t o t i c a l l y

h = Xo,

t h e s p e c t r u m o f LA has t h e p r e c e d i n g

h < Xo,

t h e r e a r e two s i m p l e e i g e n v a l u e s o f LA w i t h

and t h a t t h e s e e i g e n v a l u e s c r o s s t h e i m a g i n a r y a x i s ;hen

Xo, I n t h e s e c o n d i t i o n s , i t i s now a c l a s s i c a l r e s u l t t h a t i n geneX

r a l , i t occurs a t t h e neighbourhood of u,(X,tl

periodic solution t t i c i n h , near

o f order

[ o n l y on one s i d e 1 a b i f u r c a t e d

YX - ho/1/2, the period being analy-

XO'

= Vo[Xl + u [ X , t l t h i s s e l f e x c i t e d p e r i o d i c 1 s o l u t i o n o f (11, and l e t us assume t h a t t h i s f l o w e x i s t s f o r X > h o .

L e t us denote by t H V , , ( X , ~ I flow,

I t i s known t h a t t h e b a s i c f l o w Vo losses i t s s t a b i l i t y when A c r o s s e s

4 ] and t h a t t h e new p e r i o d i c s o l u t i o n V1 i s t h e n s t a b l e [ [

, [8]

ho

, [g] I .

Now, o u r p r o b l e m i s

il

t o l o o k f o r what happens when t h e s o l u t i o n V1 becomes u n s t a b l e , h c r o s s i n g a c r i t i c a l value

T h i s i s t h e aim o f I

A,.

I1

;

i i l t o l o o k f o r what happens when, i n s t e a d a f t h e p r e c e d i n g assumptions o n t h e spectrum o f

L A , we 0

have f o u r s i m p l e ( c o n j u g a t e d l e i g e n v a l u e g c r o s s i n g t h e

i m a g i n a r y a x i s when h

crosses

A,.

T h i s case i s t r e a t e d i n I 111.

102

G. IOOSS

I n f a c t , t h i s c a s e i s r a r e i n n a t u r e , b u t i t seems a good model, i n t h e c a s e of p r e c e d i n g a s s u m p t i o n s on LA , when v e r y near of t h e i m a g i n a r y a x i s , we have two o t h e r e i g e n v a l u e s o f LA wfich c r o s s t h i s a x i s f o r h n e a r 0

I1

-

Secondary b i f u r c a t i o n i n t o a n i n v a r i a n t t o r u s .

11 T h e P o i n c a r e map.

We a r e i n t h e c a s e when a s e l f - e x c i t e d p e r i o d i c s o l u t i o n of ( 1 1 becomes u n s t a b l e when X c r o s s e s

A,.

t ++Vl(X,t1

Let u s pose V L t l = V I L X . t l + u [ t l

i

t h e n t h e p e r t u r b a t i o n u s a t i s f i e s a system o f t h e form

2

(31 where t

H

=

d ~ t l u+ H ( u 1 ,

3 L t l i s T - p e r i o d i c [we can s u p p o s e T i n d e p e n d a n t o f h I.

B e f o r e s t u d y i n g p r e c i s e l y t h e s y s t e m (31, l e t u s g i v e one e x p l i c i t example i n two d i m e n s i o n s . H e r e we p o s e u = (u,,u21,

d ( t l u = “A

+

h

c o s t l u l - au2. aul

+

~A+costlu21.

It can be shown on t h i s s p e c i a l s y s t e m , t h a t f o r

X

a s y m p t o t i c a l l y s t a b l e , whereas f o r Moreover, f o r

X

> 0 t h e 0-solution

X 5

0 , the 0-solution i s

is unstable.

> 0 we have two d i f f e r e n t s i t u a t i o n s f o l l o w i n g t h e f a c t

t h a t t h e c o e f f i c i e n t a I s o r n o t a r a t i o n a l number. I f a E 4 , t h e n t h e r e e x i s t s

a = P / q . t h e p e r i o d i s 2nql. whereas 2 2 $ @ g i v e s a non t r i v i a l s t a b l e q u a s i - p e r i o d i c s o l u t i o n [ u l + u z i s Z n - p e r i o d i c

a non t r i v i a l s t a b l e p e r i o d i c s o l u t i o n ( i f CY

whereas t h e argument o f (u,,u21

in

I$

is

a t +Ool.

I n t h e f o l l o w i n g , we s t u d y t h e a b s t r a c t system (31 i n t h e aim t o show t h e e x i s tence f o r h near

XI

of an i n v a r i a n t two-dimensional

t o r u s i n a good f u n c t i o n a l

s p a c e , i n s t e a d of t h e e x i s t e n c e o f a q u a s i - p e r i o d i c s o l u t i o n which i s an open p r o b 1em. F o r t h e study of t h e s t a b i l i t y of t h e 0 - s o l u t i o n of

[31. we can c o n s i d e r t h e

l i n e a r i z e d problem :

i n 9 , i s noted t H v [ t l = S A ( t . T ) vo. (SX(t,~ll has t h e same r e g u l a r i t y X E 0,

whose s o l u t i o n , c o n t i n u o u s The f a m i l y of o p e r a t o r s

SECONDARY

BIFURCATION

103

-LA[ t-T {e

properties as

;

by SXlt.~l = S [t,nl. S 1

except the semi-group property replaced

t?? X ( T ~ , T ) for t L

Moreover, the periodicity of

Sf,

q t T

[see [3]

1.

gives the fundamental property

9h(t+T,01 = SX[t,O). ShLT,OI.

(51

Let us denote now t The map uo

H

q[t,h,uo) the solution of (3) satisfying u[O1 = u o E g 1 is well defined in a neighbourhood o f 0 in 9 .

%[T.A,u

H

The derivative at the origin is the linear compact operator SA(T,Ol. Moreover, if

I ~&[T.h,uol I

(61

@[T,h,

%"T,h.uo)]

is small enough, we have = Q[2T.X,uo).

and so on, because o f the T-periodicity of the equation ( 3 ) . These properties show that the knowledge of the position of the eigenvalues of SX(T.O1 with respect to the unit circle is essential for the study of the stability of the 0-solution of [3). In fact we have already 1 as an eigenvalue o f S LT.01, an eigenvector being h

V1[X,OI. This is due to the fact that V GER,t

c-t

Vl(h,t+61 is also solution

of (11, The justlfication of a formal verification is given by the fact that LA.tl H V LA,t) is shown to be analytic f r o m Do x !$ into g([6]1, hence

aat:

I

the function

V,,(h..l

can be considered as a v in [ 4 ) . with

T =

0.

In the aim to eliminate the eigenvalue 1, we substitute the so-called Poincare map. to the previous map u 0 w

4?[T.h.uoI.

,Assumption H.1.

1 is a simple eigenvalue o f Sh[T,O) f o r X near

A,.

Hence, the projection operator P A , which commutes with S X I T , O ) , to the eigenvalue 1 . depends analytically on h that P A uo

= 0 ;

if u

. Let

and corresponds

us consider u o E g

is in a good neighbourhood of 0, we can define

such

104

G . IOOSS

P A u1

where T i s n e a r T and

= 0.

T h i s i s o u r P o i n c a r e map. The g e o m e t r i c

meaning i s i n d i c a t e d on t h e f i g u r e 1.

fig.

1.

21 P r o p e r t i e s o f t h e P o i n c a r 6 map. The d e t e r m i n a t i o n o f

T

, n e a r T , i s g i v e n by t h e i m p l i c i t f u n c t i o n

theorem a p p l i e d t o t h e e q u a t i o n (81

PX@[~.h.uol

f

Vllh,sl

w h i c h can be w r i t t e n flT.h,uo)

af aT

(T.A,O)=

avl 5-(h,OI

function (X,uol

H

- V,[h,Ol] = 0,

=

0,

w i t h flT,h,Ol

# 0 . By t h e a n a l y t i c i t y o f f

=

0,

([ql, we

T ( X , U ~ I d e f i n e d i n Dox n e i g h . o f 0 i n

can f i n d an e n a l y t i c

9 ,such

that

T(X.01 = T . The P o i n c a r 6 map i s t h e n (91

ih(Uo)

Ug

=@[T[h,Uol~h,U~

+

V,,[h,T(XrUo)]

-

vl[X,oI.

Lemma 1. The map ( X . u o l

H

+ h ( u o l is a n a l y t i c : Do x n e i g h . o f 0 i n

and t h e d e r i v a t i v e o f

9 -+9

a t t h e o r i g i n i s t h e r e s t r i c t i o n o f SXIT.OI i n

l ~ ~ o l

b e l o n g s t o a good n e i g h b o u r h o o d o f 0 f o r p 5 n, t h e n 0-1

(1-PA@.

105

SECONDARY BIFURCATION

. Moreover

For t h e p r o o f s , see [6] t

t h e asymptotic behaviour, o f a s o l u t i o n

, since

u ( t 1 o f ( 3 1 , can be s t u d i e d u s i n g @ ~ ( u O l n, + m

H

C

T~

=

+m.

kELN Lemma 3.

I

i

L e t t h e s p e c t r a l r a d i u s s p r [D

4[01]

< 1, t h e n t h e c y c l e V1 i s a s y m p t o t i c a l l y

stable. I n f a c t we have a more p r e c i s e r e s u l t i n [4]

360

't V(O1 such t h a t

> 0.

I l9

1 J v ( ~ -I

V,,(X.~+~~I

:

/ I V [ O l - V1LA,aoll/9 +

-

o

<- 6 0 ,

then

3 a 1 such t h a t

exponentially.

31 B i f u r c a t i o n i n t o a t o r u s .

By assumption, when h crosses

hl,

t h e c y c l e V,

becomes u n s t a b l e ;

t h i s l e a d s t o t h e f o l l o w i n g assumption :

e x i s t s two and o n l y two c o n j u g a t e d s i m p l e e i g e n v a l u e s o f D m o d u l i 1, n o t e d XI,

To

5,.

such t h a t

# 1 f o r n = 1:2.3,4,5.

:S

#,

When

(01. of 1

X crosses

t h e s e two engenvalues c r o s s t h e u n i t c i r c l e .

Remark. I n t h e case when t h e r e e x i s t s o n l y one e i g e n v a l u e ( 1 o r -11 on t h e u n i t

cg

l that c i r c l e , o r two c o n j u g a t e d e i g e n v a l u e s ( ~ o , ~ osuch

= 1. f o r t h e o p e r a -

Dex (01. i t can t h e n be shown t h a t , i n g e n e r a l [ s e e [5] I t h e r e e x i s t s 1 XI a new b i f u r c a t e d p e r i o d i c s o l u t i o n , o f p e r i o d n e a r nT.

tor near

I n t h i s case we have t o l o o k f o r a non t r i v i a l f i x e d p o i n t o f uo where PA

uo

w $(uo)

= 0 ( t h e p e r i o d o f t h e c o r r e s p o n d i n g s o l u t i o n o f ( 1 ) is t h e n

n-I k I: T [ A ~ # ~ ( U ~ I ]= nT

+

All.

O(11 f o r h n e a r

This l a s t problem i s i n f a c t put i n

k-0 t h e c l a s s i c a l frame o f s t a t i o n a r y b i f u r c a t i o n problems. Now, by t h e assumption H.2, o f RUELLE of

-

b4]

TAKENS

and t h e Lemma 1 we a r e i n t h e f r a m e o f a theorem

which says : i n g e n e r a l t h e r e e i x s t s a

X,, such t h a t , f o l l o w i n g t h e s i g n o f

exists,

i n a neighbourhood o f 0 i n ( 1

rx

f o r t h e map

rx

for

+A[A

< All

#,(A

> XI), ;

a certain coefficient,

- PAlg,

neighbourhood either there

an i n v a r i a n t a t t r a c t i n g " c i r c l e "

o r t h e r e e x i s t s an i n v a r i a n t r e p e l l i n g " c i r c l e "

t h e " r a d i u s " of

Th i s o f orderlX

-

A,,

1/2

.

.

G IOOSS

106

T h i s g i v e s us t h e

Theorem.

Let u s assume r e a l i s e d t h e a s s u m p t i o n s H.1 and H.2, t h e n i n g e n e r a l t h e r e e x i s t s a neighbourhood o f a*circle”r i n

X

X 1 such t h a t there i s [ o n l y on o n e s i d e of

[I - p X ] g s u c h t h a t t h e s e t

9-{ V c t l

=

V1[X,tl

+

u[t)

XI) ;

T ( X , U ~ I ] , uo€rX , t H u [ t l i s t h e s o l u t i o n o f (31 c o n t i n u o u s i n 9 , w i t h u ( 0 ) = u 1 i s i n v a r i a n t by t h e dynamical s y s t e m ( 1 ) . Following t h e s i g n of a c e r t a i n c o e f f i c i e n t t h i s t o r u s F o c c u r s f o r X > XI and i s a t t r a c t i v e o r it o c c u r s f o r X < XI and i s r e p e l l i n g . t€[O,

A detailed

p r o o f can be found i n

[I31

f o r t h e RUELLE

For t h e e x p l i c i t c a l c u l a t i o n o f c o e f f i c i e n t s s e e [6]

-

TAKENS theorem.

.

111. D i r e c t b i f u r c a t i o n i n t o an i n v a r i a n t t o r u s .

Let u s c o n s i d e r t h e c a s e when we have

There a r e only 4 s i m p l e e i g e n v a l u e s

( z i w o and + i w , l

of

eigenvalues c r o s s t h e imaginary a x i s w h i l e

LA on t h e i m a g i n a r y X cFosses

Xo.

t h e s p e c t r u m s t a y on t h e r i g h t s i d e of t h e complex p l a n e f o r

11 B i f u r c a t i o n i n t o p e r i o d i c s o l u t i o n s .

Let u s n o t e T t h e unknown p e r i o d , and r e s c a l e t : T = 2n T - I t . u [ t l = :(TI.

then

We have now t h e s y s t e m

w h e r e H m [ T , E ) d e n o t e s t h e Sobolev s p a c e of n e a r l y everywhere 2 n - p e r i o d i c

f u n c t i o n s such t h a t

SECONOARY

%

L

J

H

t o t h e p r o p e r t i e s o f LA it is shown t h a t t h e l i n e a r o p e r a t o r 1 -dT' a d m i t s a bounded i n v e r s e i n H ( T , H ) , i f and o n l y i f qLX i s n o t an e i g e n v a l u e o f L, , V n E Z . T h e n f o r X n e a r Ao, we s h a l l o b t a i n

Thanks du -

ni/q

a bifurcation point f o r f o l l o w i n g we assume

*I

107

BIFURCATION

1

Z p y ,

v

q

near

p/wl o r q/wo f o r a c e r t a i n p o r qEN. I n t h e

o1 > wo > 0 [ n o l o s s o f g e n e r a l i t y ) .

p€N.

We can do e x a c t l y t h e same c a l c u l a t i o n s a s i n t h e c l a s s i c a l c a s e . w h e n

( o r f i w l ) on t h e imaginary a x i s . T h i s l e a d s t o t h e e x i s t e n c e

t h e r e a r e only f i w o

o f two p e r i o d i c s o l u t i o n s :

-

a A-

e

AoI

-

a, e

e a c h b i f u r c a t i o n i s o n l y on one s i d e of

ir

u

(01

..

1/2

i~

u

(1)

f o l l o w i n g t h e s i g n of a c e r t a i n

Ao,

coefficient. Moreover, i t can be shown t h a t these two p e r i o d i c s o l u t i o n s X a r e t h e a n l y o n e s b i f u r c a t i n g from

Xo

(see [7]

1.

I n t h e o t h e r c a s e s we have t h e f o l l o w i n g r e s u l t s (see [7] If,

I

:

=wo E i t h e r t h e r e e x i s t s only t h e s o l u t i o n

(&,

t h e r e e x i s t s 3 s o l u t i o n s : %, and 2 s o l u t i o n s b i f u r c a t i n g from

or

( o f o r d e r l h - Ao11/21,

a2 and

o f o r d e r [X

-

Xo),

ho.

If w , do E i t h e r t h e r e e x i s t s 2 o r 4 o r 6 o r 8 s o l u t i o n s o f o r d e r Ih- Xo(1/2. c a t i n g from If

w:

=

P wo.

Xo

(one o f t h e s e s o l u t i o n s i s p

z

bifur-

8).

4

There e x i s t s two s o l u t i o n s o f o r d e r ( A

x We mean p e r i o d i c s o l u t i o n of ( 2 ) .

-

Ao11/2,

b i f u r c a t i n g from

Ao.

G . moss

108

One o f t h e s e s o l u t i o n s i s %, previous

t h e o t h e r has t h e p r i n c i p a l p a r t a s t h e

"21,.

21 B i f u r c a t i o n i n t o a t o r u s . Let u s assume

iL wl/wo

#Q or,

"1=

wl/woE 0 t h e n

if

> 1

with p

f

q b 5.

wo

Remark. = pw,

w,.

T h i s contains t h e case when

p >I 4. and a l l c a s e s when

w,

# pwo.

WpEN (see t h e r e s u l t s o f 11.

Let u s now c o n s i d e r

~ ~ €i n9a , neighbourhood

o f 0 , a s an i n i t i a l c o n d i t i o n

f o r t h e s y s t e m ( 2 1 . We can t h e n d e f i n e t h e s o l u t i o n t continuous i n g f o r t

c-1 ' & ( t . X , u o l which i s , w h e r e T is chosen a r b i t r a r i l y f o r t h e moment,

E [D,T]

b u t f i n i t e p o s i t i v e . The map (111

C-,

Uo

$5Ix(u0l = @ ~ T , h . u o J

can t h e n be d e f i n e d i n a neighbourhood o f 0 i n 9 and ( h , u o ) w $ h ( u o l

is

a n a l y t i c . Moreover we have t h e d e r i b a t i v e a t 0 :

-LA T

o

(0) =

e

O

0

and t h i s compact o p e r a t o r i n 9 h a s 4 s i m p l e e i g e n v a l u e s o f moduli 1 : +iwoT

e

+iwlT

. e

. The

o t h e r e i g e n v a l u e s a r e o f moduli l e s s t h a n 1.

Hence, we can use t h e " c e n t e r - m a n i f o l d theorem" [ s e e [I31 problem t o a 4-dimensional

I t o reduce t h e

one.

T h e n , i n t h e aim t o u s e t h e work o f R. JDST and E . ZEHNOER [ I D 1 f o r t h e

new map i n a 4 - d i m e n s i o n a l s p a c e , we have t o choose T s u c h t h a t (S1w0

f

S2 w l l T

= 2r

m, S i E E , mEZ,

(121

Is1/ leads t o S

1

=

S

2

=

+

Isz/ <-

4

rn = 0 .

Thanks t o (H.21,

t h i s i s r e a l i s e d i f we choose T such t h a t 2n/T i s n o t

a r a t i o n a l combination of w

and w

1'

SECONDARY

.

[lo]

Now. a d a p t i n g t h e p a p e r

BIFURCATION

109

we o b t a i n a f t e r a r e g u l a r c h a n g i n g o f

v a r i a b l e s , t h e n o r m a l f o r m o f t h e map ( 1 1 ) i n E2 : n o t i n g z k Zk =

k

Rk

where p 0, and

=

=

1,2,

rk e

iYk

,

t h e map b e c o m e s :

X - An,

O:klare

=

are continuous f u n c t i o n s i n a neighbourhood of

d

of order

kj

(bll

2,

+

/r2111, a n d a 1 (0)= uoT, :2[01

=

w 1T .

To w r i t e (131 we h a v e a s s u m e d t h a t t h e t w o p e r i o d i c b i f u r c a t e d s o l u t i o n s

o b t a i n e d i n I), a p p e a r f o r A > A,

[this give the sign [-I

i n t h e s e c o n d member

i n (1311. T h e s e known c y c l e s a r e o b t a i n e d by c o n s i d e r i n g i n v a r i a n t ” c i r c l e s ” o f L13). T r u n c a t i n g ( 1 3 1 by s u p p r e s s i n g t h e

Okkl,

we o b t a i n t h e p r i n c i p a l

p a r t o f t h e two c y c l e s :

We o b t a i n a l s o t h e p r i n c i p a l p a r t o f a n i n v a r i a n t t o r u s :

il

crl(0) a n d a ( 0 ) < - 1 o r i f CY (01. o (01 < 1 a n d a l [ O l a n d a 2 ( 0 ) > -1 1 2 1/22 , r2 = B2 I.11/2 w i t h

if

we h a v e r1 = Blp

i i l i f a l L O l . ca2[Ol rl

-

> 1, a,(01

and

a2[01 > 0 we h a v e f o r I.I < 0 [X

XOl

B , , ( - ~ I ” ~ , r2 = B ~ [ - ~ I ’ / ’ w i t h

i

B: 0;

- cr,[Ol

-

2

B2 = -1 2

ca2L01 B1 = -1.

I t c a n be shown ( u s i n g l 1 1 l 1 t h a t t h e c y c l e cc.(O1 2

<

< -1 I r e s p . f f Z ( O l > -11. t h e C y c l e

aqlOl < -1 ( r e s p . al(0)

> -11.

is s t a b l e [resp. unstable] i f is s t a b l e tresp. u n s t a b l e ) if

1 T h e torus, whsn it e x i s t s , i s s t a b l e [ r e s p .

u n s t a b l e ) if a l ( 0 ) . a2(01 < 1 ( r e s p . a l ( 0 ) . a2t01 > 1 ) .

Now, it c a n b e shown t h a t t h e i n v a r i a n t t w o - d i m e n s i o n a l t o r u s f o r t h e map [ I 1 1 i s a l s o i n v a r i a n t by t h e d y n a m i c a l s y s t e m ( 2 ) (see [7]

1 . H e n c e we h a v e o b t a i n e d

f o r s u i t a b l e c o e f f i c i e n t s a b i f u r c a t e d i n v a r i a n t t o r u s . F o r i n s t a n c e we c a n have t h e two p e r i o d i c b i f u r c a t e d s o l u t i o n s u n s t a b l e , a n d a s t a b l e t o r u s i n

for

>

.

9

110

G. IOOSS

BIBLIOGRAPHY G.

OURANO, These de 3Qme c y c l e , Pub. Math. Orsay n 0 1 2 8 (19751.

E. HOPF, B e r i c h t e n d e r Math.-Phys. 94, 1 - 2 2 ( 1 9 4 2 ) .

K1. Sachs. Akad. W i s s . L e i p z i g

R a t . Mech. A n a l . 2 - 1 ,

3 5 - 5 6 [ 9751.

G.

IOOSS, Arch.

G.

IOOSS. A r c h . R a t . Mech. A n a l . 47, 301-329

G.

IOOSS. Arch. Mech. Stosowanej.

( 9721.

2,795-804

19741.

G. IDDSS, On t h e secondary b i f u r c a t i o n o f a s t e a d y s o l u t i o n o f systems o f N a v i e r - S t o k e s t y p e ( t o a p p e a r ) .

G.

IOOSS, D i r e c t b i f u r c a t i o n o f a s t e a d y s o l u t i o n i n t o an i n v a r i a n t

t o r u s ( t o appearl, V . I . IUDOVICH, P r i k l . Mat. Mek. Mek. 3 6 . 450-459 (19721.

35, 638-655

(19711, and P r i k l . M a t .

-

0.0. JOSEPH and D.H. (19721,

SATTINGER, A r c h . R a t . Mech. A n a l .

R. JOST and E. ZEHNDER, H e l v e t i c a P h y s i c a A c t a .

O.A.

2, 79-109

5, 258-276

(19721.

LADYZHENSKAYA, The M a t h e m a t i c a l Theory o f V i s c o u s I n c o m p r e s s i b l e

Flow. New-York, Gordon and Breach, 1963.

C?21

L. LANDAU, C.R.

[13]

O.E. LANFORD 111, L e c t u r e N o t e s i n Math., No 322, 159-192. B e r l i n H e i d e l b e r g - New-York. S p r i n g e r 1973.

[I41

0. RUELLE and F. TAKENS, Comm. Math. Phys.

[I57

R.TEMAM ,, On t h e t h e o r y and n u m e r i c a l a n a l y s i s o f N a v i e r - S t o k e s e q u a t i o n s . L e c t u r e N o t e s n09, U n i v e r s i t y o f M a r y l a n d , Oept o f Math. (19731.

Acad.

S c i . U.S.S.R.

3,311-314

(19441.

z , 167-192

(19711.

W . Eckhaus ( e d . ) , New Developments i n D i f f e r e n t i a l E q u a t i o n s @ N o r t h - H o l l a n d P u b l i s h i n g Company (1976)

APPLICATIONS OF THE METHOD OF DIFFERENTIAL INEQUALITIES IN SINGULAR PERTURBATION PROBLEMS

W.A.Harris, Jr.

University of Southern CaZifornia 1. Introduction The maximum principle, a very important special case of the method of differential inequalities, is well known for its applicability in singular perturbation problems as exemplified by Eckhaus and De Jager [ 4 ] Dorr, Parter and Shampine ( 3 1 .

and

There is an extensive literature, see e.g.

Jackson [ 71, on the use of differential inequalities for estimating solutions for ordinary differential equations; however, in general, not of singular perturbation type; an exception being the work of Brig [ 11

. Recently, the

technique of differential inequalities has been effectively utilized by Howes (61 to give an elegant, unified and comprehensive treatment of some classical singular perturbed nonlinear second order boundary value problems. In this note, based on results of Harris and Howes [ S ] , we illustrate further the versitility and applicability of differential inequalities in singular perturbation problems by studying the model nonlinear singular perturbation problem Ey" + yy' - y = 0 y(1) = B

y(0) = A

whose solutions exhibit a wide variety of interesting behavior; iqdeed, one or more solutions of the reduced problem uu'

-

u

=

0

may be required to approximate the solution y, and the transition may occur in an interior boundary (shock) layer. 2 . Basic existence and comparison results.

We begin with the definitions of the concepts thatare fundamental to

OUL

approach.

...................... Supported in part by the United States Army under contract DAHC04-74-6-0013. This paper was completed while visiting Rijksuniversiteit Groningen, the Netherlands.

111

112

W.A.HARRIS, JR.

Definition. A function a = a(t) is called a lower solution of the differential equation x" = F(t,x,x'), if a E

C2

F E C( [a,b]X R 2 ) ,

[a,b] and a"(t)

on [a,b]

F(t,a(t) ,a'(+-))

on (a,b).

Definition. A function f3 = B(t) is called an upper solution of the differential equation x" = F(t,x,x'), if B E C2 [a,b]

F E C( [a,b]x R 2 ) , and B"(t)

Q

on [a,b]

on (a,b).

F(t,B(t).6'(t))

Definition. The function F is said to satisfy a Nagumo condition on [a,b] with respect to a pair a,B

E

C[a,b] in case a(t) 6 B(t)

on [a,b]

and there exists a positive continuous function Q on [ O p ) /F(t,z,x')l C and

Q-'

such that

@(lx'l) for all a 4 t 6 b, a(t) 4 x 6 B(t), lx'( <

03

( s ) sds = m.

With these definitions, the principle existence and comparison theorem is Theorem (Nagumo). Let F = F(t,x,x') satisfy a Nagumo condition with respect to the pair a,f3 which are lower and upper solutions of x" = F(t,x,x') on [a,b] respectively, and let a(a) 6 A

<

B(a), a(b)

<

B 4 @(b). Then the

boundary value problem, x" = F(t,x,x'), x(a) = A , x(b) = B has a solution

x E C2 [a,b] with a(t) 4 x(t) 6 B(t) on [a,b]

.

Remark. If al,a2/Bl,B2 are two lower/upper solutions, then the theorem remains valid for the pair a+,BB-(t) = min(Bl(t) ,B,(t)),

where a+(t) = max(cil(t),a21t)),

i.e. appropriate corners

are also allowable.

Since the function ~-l(-yy'+ y) clearly satisfies a Nagumo condition with respect to the pair a,B,

a(t) = t + min(A,B-l),

B(t) = t + max(A,B-l), which

are lower and upper solutions respectively for the differential equation Ey" t yy'

-

y = 0, we have as an immediate consequence:

Theorem. For each

E>O

there exists a solution y = y(t,E) of the

model problem Ey" + yy'

-

y = 0,

y(0) = A, y(l)

=z

B

which satisfies for 0 < t < 1, t + min(A,B-1) 6 y(t,E)

<

t

+

max(A,B-1).

113

SINGULAR PERTURBATIONS

Having established the existence of a solution, we turn now to the construction of other lower and upper solutions of the model problem which yield not only the existence of solutions but their asymptotic behavior as

f

and explicit boundary/transition layer estimates as well.

O+

-t

3 . The model problem.

We confine our attention to f o u r representative cases. Case 1.

A 3 B-1 > 0

a(t)

=

t

B(t)

=

t

+ +

B- 1

L

B-1

B-1 + (A-B+l) exp {(l-B)t/€}

This is the classic case with a boundary layer of width O ( E ) at t = O.,Note the improvement of this upper solution over

the upper solution t

+ max(A,B-l)

+

A , so that we

=

t

D

, and

have

B(t)

-

&& I

y(t,~)= u(t) = t + B-1, t > 0, where u = u(t) is the solution of

a(t) = O(exp{-kt/E)),

the reduced problem u u '

k

- u = 0,

clearly,

u ( 1 ) = B.

Case 2.

O
A > @ ,

O C t C X

0

a(t)

=

t-X

X S t S l

-t&8(t)

=

+fi + Ke- (t-X)/&

A-:)[

= 1-B,

K =

fi + Ae-A& O d t c X

,

t > O

X
OtG)

transition layer (in y'(t,E) of width

O(fi)

at t = 0 and a boundary/ at t =

Note that two solutions of the reduced problem, uu' u E 0 and u = t this case.

= 1-B.

-

u = 0, namely,

+ B-1 are needed to approximate the solution y(t,E) in

114

W.A.HARR1.S.

JR.

In a similar manner, the case 0 < B < A+l < 1

has a solution y(t,E)

for which three solutions of the reduced equation are needed for approximation, i.e. 0 4 t 6 -A

&8+ Y(t,E) =

<

-A

u(t) = t+B-1

t .f 1-B

1-B C t C 1

i

.

Case 3. 0 < IAl

< B-1.

If 0 < A < B-1, this case may be handled in a similar manner as case 1. However, if 1-B < A ,< 0,

the solution of the model problem must vanish on

the interval (O,l), we have therefore an implicit turning point, the coefficient of y' no longer has one sign, and we need a new type of lower solution. The function utilized here is natural in the sense that it is suggested by the method of matched asymptotic expansions for an inner correction to the outer solution u(t) = t+B-I, see e.g.

Cole [ 2

1.

This

illustrates how the method of differential inequalities my be utilized to establish the validity of formal approximate solutions. An appropriate upper solution is as before B(t) = t

+

B-1,

and an appropriate lower solution in this case is a(t) = t

+

(B-1) t a n h ( 3 - y )

where y is determined by the equation a ( 0 ) = (B-1) tanh(-y) = A.

If A < 0, then y > 0, and it follows that the zero of y ( t , E ) , q ( E ) , satisfies 0 <

q(E)

< ZEy/(B-l), and hence

ll(E)

= O(E).

Hence, once again, we have a boundary layer at t = 0 of width $&m+

O(E)

and

t > 0.

y(t,&) = u(tf = t+B-1,

Case 4 . A < 0 , B > 0 , B > A+l,

-(B+l) < A < 1-B.

This is perhaps the most interesting case and indeed our results are incomplete, leading to a conjecture which we state after our discussion. One can show that the solution y(t,iz) has a unique zero, t = r l ( ~ ) ,

for 0 < t < 1, and that n(E)

+

A = $(l-B-A) > 0

as € + O + .

S I N G U L A R PERTURBATIONS

Consider, as i n Case 3 , t h e f u n c t i o n

w(t)

=

t-11

+

K

K tanh[-(t-n)] 2E

f o r which w e have EW"

+

ww'

-

K2

w = (t-q) - sech2[K(t-q)]. 26

2E

n(~)

C l e a r l y , on t h e i n t e r v a l 0 Q t 6

a(t) = t +

B(t)

=

A

(t-q)

+

K

K tanh[z(t-n)]

are s u i t a b l e lower and upper s o l u t i o n s , and on t h e i n t e r v a l q(E) Q t ,< 1

+

a(t)

= (t-q)

B(t)

= t t B-1

K

K tanh[-(t-n)] 2E

are s u i t a b l e lower and upper s o l u t i o n s , provided t h a t B ( 0 ) = -q

+

K tanh[- g]

2E

a ( 1 ) = 1-11

and

2, A

+

K

K tanh[-(l-q)] 2E

<

B,

or

K < f(B-1-A).

I t i s clear t h a t

&@+ Y ( t , E )

=

I t+A

0s t < h

h < t < l

t+B-1

In

b u t t h e boundary l a y e r behavior of t h e t r a n s i t i o n ( s h o c k ) l a y e r a t t = A depends on how q ( ~ approaches )

A

as

E+O+.

Our r e s u l t s are incomplete i n

t h i s respect, b u t w e

Conjecture:

problem A

of t h e s o l u t i o n ,

t h e u n i q u e zero, T I ( € ) ,

~ y "t yy' - y = 0 ,

y(0)

< 0 , B > 0 , B > A+1, - ( B + 1 ) <

for s u i t a b l e p o s i t i v e c o n s t a n t s

A

+

A,

y(1) = B

< I-B,

k,p.

y ( t , E ) , of t h e model

i n t h e case

Satisfies

116

W.A.HARRIS. JR.

References

1.

N.I.Brig, On Boundary Value ProbZems f o r the equation Ey"=f(x,y,y')

for smazz 2.

E.

Dokl.Akad.Nauk,SSSR 95 (1954), 429-432.

J.~.Cole, Perturbation Methods i n AppZied Mathematics. Blaisdell, Waltham, Mass. 1968.

3.

F.W.Dorr, S.V.Parter and L.F.Shampine, Applications of the M a x i m PrincipZe t o SinguZar Perturbation Problems. SIAM Rev. 15 (1973),43-88.

4.

W.Eckhaus and E.M.De Jager, Asymptotic Sozutions of Singuzar Perturbation Problems f o r Linear Differential Equations of E l l i p t i c Type. Arch.Rationa1 Mech.Anal.,

5.

23 (1966), 26-86.

W.~.Harris,Jr. and F.A.Howes, A Model Singular Perturbation Problem. To appear.

6.

F.A.Howes, Singular Perturbations and DifferentiaZ Inequalities. Memoirs, American Math.Soc.

7.

TO appear.

L.K.Jackson, Subfunctions and Second-Order DifferentiaZ InequaZities. Advances in Math. 2 (1968), 307-363.

W . Eckhaus ( e d . ) , New Developments i n D i f f e r e n t i a l Equations North-Hol l a n d Pub1 i s h i n g Company (1976)

9

A SINGULAR PERTURBATION PROBLEM OF TURNING POINT TYPE

P.P.N. de GROEN

1 . INTRODUCTION

We s t u d y t h e two-point boundary v a l u e problems (If)

ETu

* xu'

-

xu = f ,

u(1) = A

u(-1) = B

and

on t h e r e a l i n t e r v a l c - l , + l ? ; T i s t h e second o r d e r formal d i f f e r e n t i a l o p e r a t o r ,

:= ( a u ' ) ' + bu' + c u ,

TU

whose c o e f f i c i e n t s a r e

m

c

- f u n c t i o n s and whose p r i n c i p a l c o e f f i c i e n t a i s s t r i c t l y

p o s i t i v e and s a t i s f i e s a ( 0 ) = 1 ;

E

i s a small p o s i t i v e p a r a m e t e r ,

a complex

" s p e c t r a l " parameter and f a s u f f i c i e n t l y smooth complex v a l u e d f u n c t i o n . We s h a l l g i v e a s u r v e y o f t h e r e s u l t s on convergence f o r

E

+

+O of t h e e i g e n -

which we o b t a i n e d i n C31 and

v a l u e s and o f t h e s o l u t i o n o f problem ( 1 ' 1 ,

C41, s o

we w i l l omit d e t a i l e d p r o o f s . We b e g i n w i t h a s t u d y of t h e spectrum; we f i n d t h a t t h e e i g e n v a l u e s o f (1') converge t o t h e n e g a t i v e i n t e g e r s and t h o s e o f (1-1 t o t h e n o n p o s i t i v e i n t e g e r s f o r E

-t

+O.

The behaviour o f t h e spectrum f o r

x i n t h e c o e f f i c i e n t of u' i n (1'): (2+)

ETU f

v- 1u

x(x/

-

E +

+O depends h e a v i l y on t h e power of

i n t h e problem

xu = f ,

t h e spectrum i s r e a l and i t ' d i s a p p e a r s a t

v

u(i1) = 0

+O i f 0 <

w+

E

< 1 , and it f i l l s

for

E

+

t h e negative p a r t of t h e r e a l axis d e n s i l y f o r E

+

+O i f V > 1 . T h e r e a f t e r we

-m

c o n s i d e r convergence o f t h e s o l u t i o n s o f (1') u s e H i l b e r t space methods, c f . in L*(-l,l)

by ( * , - )

and

II*II

and ( l - ) f o r

E

-t

V

+O.

We w i l l mainly

C51 and C71; we d e n o t e t h e i n n e r p r o d u c t and norm and t h e supremum norm on ( - 1 , l )

by

[.I.

I n t h e p a s t t e n y e a r s a l a r g e number o f p a p e r s h a s been p u b l i s h e d on t h e s u b j e c t . I n t e r e s t i n t h e s u b j e c t was r a i s e d by Ackerberg & O'Malley c11, who

117

118

P.P.N.DE GROEN

made formal asymptotic expansions and d i s c o v e r e d t h a t t h e s e expansions showed "resonance" a t n e g a t i v e i n t e g r a l v a l u e s o f A . By r e f i n e d matching t e c h n i q u e s Cook & Eckhaus 123 a r r i v e d a t b e t t e r e s t i m a t e s o f t h e c r i t e r i o n f o r r e s o n a n c e .

From a " s p e c t r a l " p o i n t o f v i e b t h e phenomenon o f r e s o n a n c e i s caused merely by t h e n e i g h b o u r i n g e i g e n v a l u e : a s o l u t i o n of a boundary v a l u e problem grows

A

beyond bound when

t e n d s t o an e i g e n v a l u e . I n d e p e n d e n t l y and w i t h t o t a l l y d i f f e r

e n t methods Rubenfeld ti W i l l n e r

C61 o b t a i n e d a proof o f convergence o f t h e e i g e n -

v a l u e s t o o ; t h e i r proof i s b a s e d on L a n g e r ' s approximation method f o r t u r n i n g p o i n t problems and it r e q u i r e s a n enormous amount of e x p l i c i t c o m p u t a t i o n s .

2. THE SPECTRUM I n connection with ( 2 ' )

we d e f i n e t h e d i f f e r e n t i a l o p e r a t o r

V ~ E := u ETU + x / x ~ " - ~ u ~ , f o r a l l u

(3)

D(v~E): =

:= { v

U(,,To) The +dependence

of

THEOREM 1 :

5

E

-t

+O;

PROOF:

If 0

U(

{V

E

LZ(-15)

I

v"

E

Lz(-lJ)

I

xlxlv-lul

E

T ),

E

V E

L*(-I,I) E

& v(+I) =

Lz(-l,l)

01,

if c > 0,

& v(+l) =

T ) i s a s follows:

LIE

v < 1, then the Zargest eigenvaZue of VTE tends to

If v > 1 then

01,

u(

Define W(x) := $

T ) becomes dense i n ( - m , O )

"F

for

E

+

-m

for

+O.

( t l t \ v - l / a ( t ) ) d t . A f u n c t i o n u i s an e i g e n f u n c t i o n

-

0

of

(4)

if and o n l y i f v : = u e x p ( W ( . ) / E ) i s a n e i g e n f u n c t i o n o f t h e o p e r a t o r

V T E ~: = =

exp(w/E) v ~ E exp(-w/E)) { ~ = ETv

-

(ilxlV-'

+ ixlxlv-l

b/a

+ tlxl

2v

/Ea)v.

v

T

E'

p r o v i d e d ReX > - E ( v - ' ) ' ( v + l )

if

E

If

119

PROBLEM

A SINGULAR PERTURBATION

+ C . T h i s proves t h a t t h e spectrum d i s a p p e a r s a t

-m

+O.

-t

w >

D( ?

1 we d e f i n e f o r each v E

v

the function w , E.

~

Analogous t o

(4) t h i s i n d u c e s t h e t r a n s f o r m a t i o n o f wTE i n t o

1w

(5)

v E

= -d2w 7- - P(S,E)W,

w(tE-9)

dc

"tE, D(~F~),

= 0, f o r all w c

which is s e l f a d j o i n t , The " p o t e n t i a l " f u n c t i o n P s a t i s f i e s

+

P(5,E) = o(E$v-; T

uniformly f o r a l l

5

E

E

+

E v - ' I p )

(E

-+

+O)

i

(-E-~,E-'),

s o it i s m a j o r i z e d by t h e "square-well''

potential V,

It i s e a s i l y seen t h a t t h e e i g e n v a l u e s of d 2 / d c 2 for E

-t

-

V

become d e n s e on

(-m,O)

+O and by well-known comparison theorems t h i s is t r a n s f e r r e d t o q.e.d.

and hence t o U("T,)%

REMARK 1 :

The o p e r a t o r a s s o c i a t e d w i t h (2-1 s a t i s f i e s t h e same r e s u l t a s VTE

d o e s ; t h e o n l y d i f f e r e n c e i n t h e proof i s t h a t w h a s t o b e r e p l a c e d by -w i n formula

(4).

From h e r e on we s h a l l d e a l w i t h t h e i n t e r m e d i a t e c a s e drop t h e s u b s c r ip t

v of

T

V E

v =

1

o n l y , so we w i l l

f o r V = 1 . T h i s c a s e i s t h e most i n t e r e s t i n g , b o t h

p a r t s , E T and xd/dx o f t h e o p e r a t o r T

have an i n f l u e n c e o f e q u a l s t r e n g t h on t h e

spectrum o f t h e o p e r a t o r and t h e s e i n f l u e n c e s a r e i n b a l a n c e , such t h a t t h e spectrum n e i t h e r v a n i s h e s nor t e n d s t o a dense s e t :

THEOREM 2 :

Let { A k ( € )

I

k c N} be the s e t of eigenvalues of TE, arranged i n

decreasing order, i.e. Ak+, < X k , then a l l eigenualues s a t i s f y :

(6)

A

k

(E)

= -k + I ) ( € '

uniformly with respect t o k.

k3l2)

(E +

+0)

P.P.N.DE GROEN

120

We w i l l merely s k e t c h t h e p r o o f ; f o r a d e t a i l e d v e r s i o n we r e f e r t o C31 o r C41.

-

i n t o T,

Again we t r a n s f o r m T

nE

operator

:= Ed2/dxZ

-

as i n ( 4 ) and we c o n s i d e r t h e s p e c i a l ( H e r m i t e - )

x 2 / 4 E on t h e same domain o f d e f i n i t i o n . By computing

X

t h e s o l u t i o n s o f Il v = hv we c a n v e r i f y d i r e c t l y t h a t t h e e i g e n v a l u e s o f satisfy

( 6 ) and t h a t t h e f u n c t i o n xn’

(7)

x,(x,E)

:= e x p ( - ~ x 2 / E ) H k - , ( x / ~ ) ,

( H k i s t h e k-th Hermite p o l y n o m i a l )

approximates t h e e i g e n f u n c t i o n o f RE a t hk up t o L)(E-ne-1/2E) f o r E

I n o r d e r t o compare TE and Il

+O.

-+

we connect them by t h e c o n t i n u o u s c h a i n

..

R

:= ( l - t ) I I E + t T E , which s a t i s f i e s

.t

1 IRE,t

(8)

u

uniformly f o r a l l s , t

-R

E,S

u ( / = (‘((s-t)(l/RE,t u / I

Ilu(1))

+

c 0 , 1 1 and a l l u i n t h e domain o f d e f i n i t i o n . By s p e c t r a l

E

C51 c h . 5 , t h i s i m p l i e s t h a t t h e e i g e n v a l u e s o f R

p e r t u r b a t i o n theorems, c f .

since R

E .t

a( (s-t ) / 1 XI ) ,

depend c o n t i n u o u s l y on t and t h a t t h e i r v a r i a t i o n i s o f o r d e r i s (nearly) selfadjoint.

E .t

Next we prove t h a t t h e approximate e i g e n f u n c t i o n s o f

I I (RE,t -

(9)

uniformly f o r a l l k t h e n h (E,O) k

E

k)Xkl

I

Iand t

satisfies

’

= E

3/2

satisfy

I IX,l I ) ,

(E

-+

+o),

i s t h e k-th e i g e n v a l u e o f R

C0,ll. I f \ ( E , t )

(6). By ( 8 )

n

we can f i n d numbers

E

> 0 and t n

E,t’

d/n

2

( w i t h d > 0 and independent o f E ) such t h a t t h e c i r c l e C ( - j , $ ) around - j w i t h

4

radius E E

i s contained i n t h e r e so lv e n t s e t o f R

1 and t

[O,E

E

[O,tnl

E ,t

for a l l j

S

n (j,n

E

U), a l l

and t h a t t h i s c i r c l e c o n t a i n s o n l y t h e e i g e n v a l u e

h . ( E , t ) and no o t h e r . With t h e a i d o f t h e p r o j e c t i o n o n t o t h e e i g e n f u n c t i o n o f

J

X.(E,t)

J and j

5

s a t i s f i e s (6) f o r a l l t E C O , t n l J n . Now we can r e p e a t t h e argument, s t a r t i n g from t h e p o i n t t = t

it f o l l o w s from ( 9 ) t h a t

h.(E,t)

instead

o f t = 0 ; however, s i n c e we d i d n o t prove a n y t i n g a b o u t X n + , ( e , t ) , t h i s p o s s i b l y can e n t e r C ( - n , ; ) satisfies

(6) f o r

f o r some t > t , . So we p r o v e i n t h e n e x t s t e p t h a t h . ( E , t ) J a l l j 5 n-1 and t

choose n s o l a r g e t h a t

7

j=k+l a f t e r n-k s t e p s , q . e . d .

t.

5

d

E

Ct

1

n’

j=k+l

t

n

+ t

n-1

1

and so on. S i n c e we can

l / j 2 1 , we have proved f o r m u l a

(6)

121

A SINGULAR PERTURBATION PROBLEM REMARK 2: From the proof of theorem 2 we obtain the inequality

I lGEu -

(10)

Xu1

I

2

I lul I{dist(A,{-n

I

n

E

Wj)

- DE')

for some constant D > 0. REMARK 3: By taking the adjoint of T we find that the eigenvalues of the E

operator connected with ( 1 - )

converge to the nonpositive integers for

with the same asymptotic estimate as in

E

-+

+O

(6).

REMARK 4: Theorem 2 can be generalized to elliptic problems in a higher dimensional space which degenerate to a first order operator with a (simple) critical point for

E

+

+o, cf. C41.

REMARK 5 :

The spectrum of the limit operator T

U(To) = {?I

E

E

I

ReA

5

-$I;

in L z ( - l , l ) is the set

we see that there is an apparent lack of spectral con-

tinuity in L2-sense. However, in distributional Sense there is spectral continuity: the only (Schwartz-) distributions whose support is contained in ( - 1 , l )

and which

satisfy the equation xu' = Au (in distributional sense), are Dirac's &-distribution

6(n-l) = -n6 (n-1), n

and its derivatives. They satisfy x

E

W; moreover, the

(approximate) eigenfunctions of T converge to them in distributional sense.

3. CONVERGENCE OF THE SOLUTIONS In order to be able to prove convergence of a formal approximation of the solution of (1') Lemma 3:

we can use the folbwing lemma:

Constants C > 0 and K > 0 exist, such that TE

-

is invertible and

E E

C0,ll.

satisfies

for all u

E

D(T ) , A

E

Q with ReA >

-;+

KE and for all

For the proof we refer to C31 o r 141. Let u

be the solution of the full problem (1')

reduced equation xu' 1.e. :

-

hu = f of ( l ' ) ,

and u the solution of the

which satisfies both boundary conditions,

122

P.P.N.DE GROEN

u o ( x , h ) :=

(11)

xilxl I f f"

E

and i f Re1

L2(-1,1)

II(ET + xd/dx

since u ( C 1 , h )

-

-

u (i1,h) =

= L)(E/

then : u

f(I

-

h)uo

t + dt

f(t)(t)

>.3/2,

=

E

IlETflI

$ ( A + B)/xI' + $ ( A

-

A- 1

.

B)x/x/

and s a t i s f i e s

L2(-1,1)

= L)(El(flI + E l I f " ( / ) ,

+o);

(E

-+

(E

+

0 t h i s i m p l i e s by lemma 3:

I / u E - uoII 5 (Reh

(12)

I"

-

+

If1 I

K E ) - ~ I I ( T -~ h)(uE

-

uo)lI =

I

+ E l If") ),

By S o b o l e v ' s i n e q u a l i t y c u l z 5

+o).

+ 21 lull IIu'II we i n f e r from ( 1 2 ) and

lemma 3: 1

(13)

CUE-

provided Reh >

Uol

=

+ \If''(l)),

O(E'(llflI

(E

+o),

-+

312.

From ( 1 3 ) we s e e t h a t ( i n f i r s t a p p r o x i m a t i o n ) t h e r e a r e no boundary l a y e r s i n t h e a p p r o x i m a t i o n , i f Fie1 > 3 / 2 . From ( 1 1 ) however, we s e e t h a t a non-uniformity can be expected n e a r t h e p o i n t x = 0 and t h a t it w i l l grow l a r g e r a s -ReX grows larger. For Reh 5

6

u;(.,h)

a proof o f convergence becomes more d i f f i c u l t , s i n c e

L 2 ( - l , l ) i n t h a t c a s e . T h e r e f o r e we extend t h e boundary v a l u e problem equiped w i t h t h e norm u

t o t h e l a r g e r space H - n ( - l , l ) ,

(1')

Iu[-n' w i t h n

-+

E

fN,

C81 c h . 1.13. I n t h i s s p a c e we can prove t h e analogue o f ( 1 2 ) p r o v i d e d

cf.

Reh > -n + 3 1 2 . By i n t e r p o l a t i o n we can o b t a i n convergence i n s t r o n g e r norms: LEMMA

4: A positive function Ck(X) e x i s t s such t h a t

(14)

1 lxkul I

I Ixk+'utI I

+

+

€1

Ixku"(

k

f o r a l l functions u, for which ( I x u"II PROOF:

Cf.

+

5

k

81 Ix

(ET + xdidx

*, and f o r every k

-

h)ul E

I

+ Ck(h)Iul-k

1.

C41 lemma 3 . 1 3 .

We o b s e r v e t h a t u g ( * , h ) Reh > -n

I

3/2.

E

H i n ( - l , l ) and xnu:(.,h)

With t h e a i d o f lemma

ixk+ i w 1 2

5

E L 2 ( - l , l ) provided

4 and t h e i n e q u a l i t y

(2k+2)[]~~w + I2]) ~/ x k w ) l I [ x k + ' w '

11,

(kEW),

123

A SINGULAR PERTURBATION PROBLEM

we obtain the final result: THEOREM 5: I f n

E [N

and i f h

(a) satisfies

E C \

Reh > -n +

312,

then

We point out that the weight factors xn and xn+' in the norms of the estimates (15a) and (15b) smoothe down the non-uniformity of uE at x = 0. If h is a negative integer, we cannot expect convergence because of a neighbouring eigenvalue.

The solution of problem ( 1 - ) converges also to a solution of the reduced equation -xu'

-

hu = f in the major part of the interval, as the solution of

does. However, in this case we cannot impose boundary conditions at x = f l

)'1(

on the set of solutions of the reduced equation. We have to select the right solution from this set by a smoothness condition at x = 0; this smoothness condition arises in a very natural way from the choice of suitable domains for the operator connected with ( 1 - ) and for its limit (for

E

+

+O). At the points

x = 21 (ordinary) boundary layers of width O ( E ) arise. We will merely state the final result; for a proof we refer to [ b l . For any f

E

cn(-l ,1) we

provided h

E

define the function w

(n

E

W u {O}) by

4 u {O} and Reh > -n. Clearly this function is a solution of the

reduced equation and is smooth at x = 0. It satisfies:

for

E

-t

+O and uniformly f o r a l l x

E

C-1

,I1.

124

P.P.N.DE GROEN

REMARK 6:

The method by which theorem

6 i s p r o v e d , i s i n some s e n s e d u a l t o t h e

one o f theorem 5 . I n t h i s p r o o f we have t o r e s t r i c t t h e boundary v a l u e problem (1-)

t o t h e smaller spaces H

+n

(-1,l)

(n

E

W ) i n o r d e r t o be a b l e t o e n l a r g e t h e

p a r t o f t h e 1 - p l a n e i n which an analogue o f lemma 3 is t r u e and i n which we hence can prove v a l i d i t y o f t h e a s y m p t o t i c formula ( 1 6 ) .

REFERENCES 1. Ackerberg, R.C.

& R.E.

S t u d i e s i n Appl. Math.,

O'Malley, Boundary Layer problems e x h i b i t i n g resonance,

9 (1970),

p.277-295.

a boundary value problem of singular perturbation type, S t u d i e s i n Appl. Math. , 52 ( 1973 ) , p , 129- 139. 2. Cook, L. Pamela

W. Eckhaus, Resonance i n

! i

3. Groen, P.P.N. d e , Spectral properties of second order singularly perturbed boundary value problems with turning points, p r e p r i n t : r e p o r t 39 (May 1975) o f t h e "Wiskundig Seminarium d e r V r i j e U n i v e r s i t e i t " , Amsterdam, t o appear i n t h e J o u r n a l of Math. Anal. and A p p l i c a t i o n s .

4.

Groen, P.P.N.

d e , SingularZy

perturbed

d i f f e r e n t i a l operators o f second

order, Mathematisch Centrum Amsterdam, t r a c t 68 ( t o appear 1976). 5. Kato, T . , Perturbation theory of linear operators, S p r i n g e r V e r l a g , B e r l i n e t c . , 1966.

6 . Rubenfeld, L . A . and B. W i l l n e r , The general second order turning point problem and the question of resonance f o r a singularly perturbed second order ordinary d i f f e r e n t i a l equation, t o a p p e a r .

7 . L i o n s , J . L . , Perturbations singulikres duns lea problPmes a m l i m i t e s e t en Contr6te optimal, L e c t u r e n o t e s i n Math. 323, S p r i n g e r V e r l a g , B e r l i n e t c . , 1973. 8 . L i o n s , J . L . & E. Magenes, Problkmes a m l i m i t e s m n homogPnes, Dunod, P a r i s , 1968.

W . Eckhaus (ed.), New Developments in Differential Equations

@ North-Holland Publishing Company (1976)

ASYMPTOTICS FOR A CLASS OF PERTURBED INITIAL VALUE PROBLEMS Bob Kaper Department of Mathematics, University of Groningen, Groningen, the Netherlands

INTRODUCTION In this paper we are dealing with initial value problems containing asmall nonnegative perturbation parameter

E.

On time intervals initiating the origin we

will approximate the exact solution (provided it exists) asymptotically with respect to

E

as

6

+

0. The asymptotic solutions could be derived from so-called

formal asymptotic 50lutions, i.e., functions which satisfy the differential equation and the initial conditions up to an asymptotic accuracy of certain order. These formal asymptotic solutions should then be compared asymptotically with the exact solution. The question arises whether such a function approximates the exact solution up to an asymptotic accuracy of the same order as it approximates the equation and the initial conditions. Or at least whether there exists a relation between these two orders. This question will be answered in connection with the type of the interval to be considered. As an application we consider a class of perturbed oscillations described by the nonlinear second order ordinary differential equation with slowly varying coefficients w"

+ F(w,~t) + E~(W,W',E~,E)= 0 ,

t

1. 0 ,

( 1 .a)

( I .b) W(0,E) = C I 1 ( E ) , W'(0,E) = a 2 ( E ) . The force term F, onwhich Ef is to be consideredas a small perturbation, is assumed

to be the derivative of apotential function whichhas an absoluteminimum atthe origin. Let u s briefly recall the concepts of (formal) asymptotic solution in dealingwith the vector differential equation in R": x' = f(X,Et,E),

-

x (x

-

t

E I,

(2.a)

x(0, E ) = a(E) A function Ti is called an asymptotic s o l u t i o n of order E C 0) if

-3=

O(K)

(2.b) K

(K(E)

=

o ( i ) as

on I

represents the exact solution of problem ( 2 ) ) . A function u is called a formal asymptotic s o l u t i o n df o r d e r q as E C 0 ) if g and 0 ,

g(t,E) = 'J(t,E) - f(u(t,E), are O ( q ) on I,

Et,E), 8 ( E ) = a ( € ) -

(call g and 8 the residuaZs of u for problem ( 2 ) ) .

125

(q(6) =

U ( ~ , E ) ,

o(l)

126

B. KAPER

An obvious modification leads to similar concepts for second order problems. Note that without loss of generality a slowly varying dependence of f on t is assumed in view of

the application.

The order symbols 0 and o are understood to be related to the limitprocess E

t 0 uniformly in t on I.

I n section I we will give a brief summary of the form of an Nth order formal

assymptotic solution $N ( whose residuals are O(E~+')) of problem ( 1 ) .

For a

complete description I refer to [ I ] . In section 2 we will treat the remainder problem x

-

u in dealing with the vector

problem ( 2 ) . It includes the existence and uniqueness of the exact solution. Connected to this section we will construct an improved Nth order formal asymptotic solution 0, in section 3 . In a final section 4 we will state some results that hold on the infinite interval [ 0 , - ) .

ON FORMAL ASYMPTOTIC SOLUTIONS OF OSCILLATION PROBLEMS

81.

We may expect the solutionsof problem ( 1 )

to be oscillating functions with

slowly varying amplitude and frequency. In order to include these large-scale variations in the approximation we consider for the moment intervals of order E

-1

.

, i.e., intervals of the type

[O,

E

-1

L] where L is independent of

E.

Asymp-

totically we may distinguish two different time scales, a local- ( o r f a s t - ) time

scale

on which the solution is periodic with a period of order one and a slow-

(or stretched-) time scale, characterized by the slow variable

T =

ct, which

accounts for the slow modulation of the oscillations. Both scales are made explicitely in the form of the formal asymptotic solution which technique is known as the two variable method. An Nth order formal asymptotic solution 0 N problem ( I ) is given by

with residuals

Of

N+ 1

gN+](t,E), gN+I = O ( I ) , and E N+ I Bi(~), Bi = 0 ( 1 ) , i = I , ? . contribution Uo,Uo = n + A Q to the expansion of $N is the even,

The @ ( I )

E

0 0'

2n-periodic solution of the nonlinear conservative system

in which

T

is t o be considered as a fixed parameter. The function n is the alge-

braic average of the extreme values of Uo, which makes it possible to introduce

127

A CLASS OF PERTURBED INITlAL VALUE PROBLEMS

an amplitude function A o ( r ) .

The function w follows from the normalization of the

period of Uo to 2n and will therefore depend on A system ( 1 . 2 ) .

in the case of a nonlinear

The higher order contributions U

Uv = A

V'

v

Z*

2

+

QV,

to the expansion

of $N are determined by linear, second order equations whose homogeneous part is

the first variational equation of ( 1 . 2 ) with respect to U

One homogeneous solution, z;(P,T),

0'

follows direcly by differentiating Uo with

respect to p. A second solution, z*(p,r) could be found by the variation of con2

j = 0,..., N - I , follow from houndedness

stants method. Equations for A . and S j ,

. J+l

requirements of U . 277

J

They are of the type

yj+, (p,r)z:(p,~)dp

0, i

=

.. . ., N

I , 2, j = 0 ,

=

(known a s suppression of secular terms in Yj+]).

- I.

A s an immediate consequence of

the determination of ON we have for the residual function g gN(t,E)

=

P

Y ~ + ~ ( P , T )+ O ( E ) ,

=

E

-1

N

S(T,E,N),T = Et.

At this stage we may draw the following conclusions.

- From the expansions (i)

(1.1)

we see that

ON(t9E) = '$G(P,T,E),

P

= E

-I

s(I,E;N),

T =

Et,

where (ii)

$:

is defined o n R + x [O,L]

x

[O,E~],

Let u s call such functions satisfying (it, (ii) and (iii) funceions of the ?eriodic two variable type.

- In consequence gN is of the periodic two variable type. - No equations for AN and SN have been determined yet. We introduced the quantities anticipating a question on the order of asymptotic accuracy of $N conceiving it as an asymptotic solution on

[~,E-'L].

We will

treat this problem in the next section when dealing with the vector differential problem (2) on arbitrary intervals I.

82.

PROOF OF ASYMPTOTIC CORRECTNESS I n this section we consider the vector differential problem ( 2 ) in Rn:

x'

=

X(0,E)

f(X,Et,E), = CY(E)

I

t

E I,

(2.la) (2.lh)

.

B KAPER

128

where I be some finite o r infinite interval, possibly depending on 6 . Let 1. I denote the vector- and matrix norm and llxll = sup Ix(t) 1 , t E I. Let u be a formal asymptotic solution of (2.1) of order n, i.e., the residuals g and 8 are O ( r l ) .

In

order to compare the formal asymptotic solution u with the exact solution x of (2.1),

whose existence and uniqueness should be established, we apply the change

of variables

x=u+p. Then the remainder function

should satisfy the nonlinear vector differential

p

problem (2.2a) (2.2b) where

Let Y(t,s;E), Y(t,t;E) = E , be a fundamental matrix solution of the linear equation 2' =

A(t,E)z,

Y(t,S;E) = VO(t,E) y o- 1

(S,E),

(2.3) Vo(t,E) = 'Y(t,O;E).

The initial value problem (2.2) for

may be transformed into the nonlinear

p

Volterra integral equation

k(t,E) = Yo(t,E) B ( E )

+

t J Y(t,S;E)g(s,E)dS.

(2.5)

0

Provided f is sufficiently smooth we can show by means of a contraction mapping principle the existence and uniqueness of the solution

p

of (2.4)

within a ball B(R) with radius R if Y(E)K(E)

5 t with R

=

2K(E),

where K(E)

=

IIK(t.E)II

,

t Y(E)

=

II/IY(t,s;E)ldsll 0

Let

K

.

be an asymptotic order function, i.e.,

following

K

=

O(1)

as

E

4 0. Then we have the

A CLASS OF PERTURBED INITIAL VALUE PROBLEMS

129

Theorem I . Let u be a formal asymptotic solution of (2.1) with residuals g and 8 of order q . If = o(K-I)

Y(E)

then for sufficiently small

0 problem (2.1) has an exact solution x

E

=

u + 0

with x - u

(uniformly in t on I)

U(K)

=

(Hence u is an asymptotic solution of order The relation between

K

K).

and I-,given by the function k(t,E) defined in

( 2 . 5 ) , depends on the order of magnitude of the interval I and the behaviour of

the fundamental matrix 1 . Let u s assume throughout this section that I=[O,t for some integer m and that I't'(t,s;E)/

2 K,

0

5 s 5 t, t

-m L]

E I.

In this case y = O(E-~) and the condition on y . of ~ Theorem1 impses a minimum order of asymptotic accuracy of $N as an asymptotic solution and hence also as a formal asymptotic solution. The relation between

K

and

I-

is simply

K

=

E - ~ I -which , means a reduction of the

order of asymptotic accuracy of u as an asymptotic solution when the order of magnitude of I(m)

increases. On intervals of order

E

-I

this means a loss of one

in the order of u as a asymptotic solution compared to u as a formal asymptotic solution. For a subclass of initial value problems related to the oscillation problem ( I ) we may improve the order with one

E

by the application of a partial

integration rule and by imposing a condition on the residual function g (hence on u ) . Therefore we need the concept of

- f i r s t order formal asymptotic m a t r i x s o l u t i o n function for which 8

-I

A(t,s)

0 of

(2.3), i.e., a matrix valued

@(t,E) - 8'(t,E) =cG(t,E), G = O ( 1 ) .

det 8

>

0 (hence

exists).

Apply the change 't'(t,s;E) = O(t,E) Y(t,s;E) of z =

-E

@

-1

@-'(s,E),

thenY is a fundamental matrix

(t,E)G(t,E)z.

In view of the oscillation problems we have the following Theorem 2. Let u be a formal asymptotic solution of (2.1) periodic two variable type, u(~,E) = u*(P,T,E), Let

@

of order q , which is of the -I

S(T,E), T = ~ t . be a first order formal asymptotic matrix solution of(2.3) of the periodic

two variable type, O(t,E)

=

O*(P,T,E), p =

E

-I

p

= E

S(T,E),

T

=

et. If

(2.6)

130

B .KAPER

where g* is the two variable counterpart of g,thenproblem ( 2 . 1 ) has an exact solution x = u + p with

Proof. The theorem is almost an immediate consequence of Theorem I except f o r the improved relation between

K

and q. Substitute the above assumptions on u , Y and

d in k (defined in (2.5))

t k(t,E) = YO(t,c)8(~) + @(t,c)

Y(t,s;dO

*- 1 (p(s),Es,E)g*(p(s),Es,E)dS,

0

p(s)

=

E- 1 S ( E S , E ) .

The variable s of integration appears in the integrand in two different ways: via the periodicity variable p and elsewhere characterized by a derivative of order O ( E )

(9 is of

a gain of one

E

O ( E ) since 0 and G are O ( I ) ) .

in the integral which means

K

=

A partial integration rule gives

E.E

-m

n.

Till now we made the assumption of uniform boundedness of Y . In a corrolory we will replace the assymption by a condition on the matrix A in the special case of intervals of order

E-'.

Corollary I . Let U and 0 be formal asymptotic solutions in the sense of Theorem 2 . If

(2.7)

then u is an asymptotic solution of ( 2 . 1 ) of order 11 on intervals of order

E

-I

.

Proof. Onceyo is bounded, condition ( 2 . 7 ) (ii) assures the boundedness of the inverse -I Yo

which means lY(t,s;E)I

5 K, 0 5

s

2

t <

E

-I

L. Since d

= O(1)

the boundedness

of Yo ( and hence of Y ) follows from an application of cronwall's lemma to the -1 equation for Y on intervals of order c

.

A CLASS OF PERTURBED INITIAL VALUE PROBLEMS

53.

131

IMPROVED F O W L ASYMPTOTIC SOLUTION $N OF THE OSCILLATION PROBLEM ( I ) With respect to the formal asymptotic solution $N constructed in Section I

for the class of oscillation problems we already know from Theorem I that $,

is

-1

L] provided N 2 2 . In order -I that 0, be an Nth order asymptotic solution of ( I ) on [0, E L] the conditions

an (N-I)St order asymptotic solution of ( I ) on [ O , E

(2.7) should be verified if we apply the change of variables to a first order 2 system i n n :

formal asymptotic solution $N condition (2.7) (i) implies N

1 . Once $ I is a

first order asymptotic solution the asymptotic correctness of $ asymptotic solution follows from a comparison of $ and

as a zerothorder

It is a straightforward check to see that zi, i = I , 2, =

Zi(t,E)

Zi*(p,T).

-I P=E

S(T,E;N),

T =

Et,

are first order formal asymptotic solutions of the linear variation 1 equ

ion

of ( 1 ) with respect to $N:

Conditions ( 2 . 7 ) (ii) and (iii) follow by a direct verification (note that the trace of the matrix A(t,E)

is formed by the coefficient of y' in ( 3 . 1 ) ) .

F o r the second order problem condition (2.7) (iv) reads as

(c.f. also the conclusions at the end of section I ) . These conditions (suppression of secular terms in y N + l ) are satisfied if AN and SN are solutions of linear first order differential equations (initial conditions follow in the same way as the values for A. and S . ) . 3

3

In consequence of Corollary I we now

have that $N is an Nth order asymptotic solution of ( I )

on intervals of order

E-'.

The above described form of an Nth order asymptotic solution 0N gives rise to a procedure generating asymptotic solutions of all order. This is due to the independence of the truncating index N of the equations for allquantities involved by the expansions for 4,

(c.f. [ I ] ) .

.

132

B KAPER

14.

OSCILLATION PROBLEMS WITH SMALL DECAY Throughout the examination of the relation between

and

K

n

in section 2 we

assumed boundedness of the fundamentalmatrix Y of (2.3). In case of the second order oscillation problems this equation corresponds with the linear equation ( 3 . 1 ) A much better result in the relation between

and the Order of magnitude of the

K , T ~

interval I could be obtained in cases of an absolute integrable fundamentalmatrix on [ 0 , - ) .

For simplicity let us restrict ourselves to the class of weakly nonlinear

oscillation problems. Its variational equation with respect to bN is y"

+ y + Eq I (t,E)Y + Eq2(t,E)y'

=

(4.1)

0

The coefficient functions q l and q2 are determined by the form oft'neoriginal equation as well as in consequence by the formal asymptotic solution $ N' Let u s start with two subclasses, the class of weakly nonlinear oscillation problems with an exponential decay (class I) w"

+ w + 2EY(Et)W' + Ef(W,W',Et,E) = 0,

(4.2)

where y is some positive function, and the class with an algebraic decay and constant coefficients (class 11) w"

+ w + ( w ~ ) ~ ~ + lEf(w,w*,E)

= 0,

m

(4.3)

2I

In both cases positivity conditions should be imposed on the function f. In a forthcoming paper we will go into details about estimates on $n, gN and the fundamental matrix Y of ( 4 . 1 ) . Here we will restrict ourselves to a brief suruey of the results. I n case I the formal asymptotic solution

ON could be estimated as

follows /$N(t,E)I

Mo PN(E+,E) exp[-

f'

y(o)doI,

0

where pN is a polynomial in Et of a degree depending on N and less than N . A similar estimate holds for the residual function gN of $,.

The linear variational

equation is given by y"

+ y + 2Ey(Et)y'

There exists a constant M

+

Eql(t,E)Y

+

Eqz(t,E)Y'

= 0.

such that for the fundamental matrix Y Et

IY(t,s;E)I(

Mo exp [ -

EL y(o)do],

0

5s 5t <

-.

holds

133

A CLASS OF PERTURBED INITIAL VALUE PROBLEMS

A combination of the above results and the partial integration rule gives the

fo 1lowing Theorem 3 .

+ N be an Nth order formal asymptotic solution of (4.2). For N -> 1 problem (4.2) has an exact solution w of the form w = O N + p with

Let

for some constant M

.

A similar result on LO,-) could be derived in case 11. The exponential damping

should be replaced by a algebraically decreasing function of order (2m)-',

the

polynomial in Et multiplying the exponentially decreasing function in case I1 by a polynomial in logarithmically increasing functions.

REFERENCE [I]

Kaper, B., ( 1 9 7 5 ) . Perturbed Nonlinear Osciallations, SIAM Jrnl. on Appl. Maths.

This Page Intentionally Left Blank

W . Eckhaus ( e d . ) , New Developments i n D i f f e r e n t i a l E q u a t i o n s

@ North-Hol l a n d Pub1 i s h i n g Company (1976)

ON THE SOLUTIONS Cr ?k:RlUBBZD DIr’FER!.SNTI

H.

i1.L

- D.

SOUATIONS

Niesscn

Department of X a t t e m a t i c s U n i v e r s i t y of Essen !Issen, Uerrany

INTRODUCTIOX I n t h e l a s t y e a r s a l o t OP r e s u l t s have been o b t a i n e d which g u a r a n t e e t h a t a l l s o l u t i o n s of some p e r t u r b e d 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 a r e i n some s e n s e i n t e g r a b l e , e.g. t o L p , i f t h i s i s t r u e f o r t h e unperturbed e q u a t i o n . Bradley [ 5

3 , dalvorsen

belong

(Compare e.g.

[ 101, P a t u l a and ‘dong [ 13,p.24],

Zettl

[183).

Here a v e r y e l e m e n t a r y theorem on p e r t u r b e d systems of d i f f e r e n t i a l e q u a t i o n s i s p r e s e n t e d which c o n t a i n s t h e above ment i o n e d theorems and some o t h e r s (Bellman [ 2 ] , r 3

] , [ 4 ,p.43],

Walker [16]) as s p e c i a l cases. Although t h e assumptions of t h i s theorem are weaker and t h e a s s e r t i o n i s s t r o n g e r t h a n t h o s e of t h e

e a r l i e r theorems, i t s proof i s much more e l e m e n t a r y (and n e a r l y obvious). I n s e c t i o n s 1,2 and 3 v a r i o u s a p p l i c a t i o n s a r e made t o l i n e a r l y p e r t u r b e d systems, t o 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 and t o l i m i t c i r c l e c r i t e r i a . S e c t i o n 4 c o n t a i n s . g e n e r a l i z a t i o n s of t h e theorem which e.g.

imply r e s u l t s of Cesari [ 6

[ 101, Levinson [ 1 1 1 and Wong [ 171.

1.35

1, H a l v o r s e n

N I ES S EN

136

1 .SOLUTIONS OF PERTURBED SYSTEMS A l l f u n c t i o n s occuring i n t h i s and t h e f o l l o w i n g s e c t i o n s

are supposed t o be measurable and complex v e c t o r - or matrix-valued of some s u i t a b l e size. L e t I be any r e a l i n t e r v a l and A a l o c a l l y i n t e g r a b l e (n,n)

- matrix-valued

f u n c t i o n on I. Consider on I t h e

perturbed system (1.1)

z'=

A(t)z+f(t,z)

a r r i s i n g from t h e unperturbed system

x'= A(t)X

(1.2) by t h e p e r t u r b a t i o n

,

f ( t z )=B( t 1o( 1 1+c ( t 1o( Is, ( t 1z l

(1.3) ( w i t h matrix-valued on Ixc".

f u n c t i o n s B,C,D,O).

1

f i s assumed t o be d e f i n e d

Then w e have (1.4) THEOREM. L e t X be a fundamental m a t r i x OP (1.2)

and suppose t h a t IX''BI

(1.5)

-

* IX-lfllDXl €L'(I).

Then, f o r any s o l u t i o n z of ( l . l ) * z ( t)=X( t ) O ( 1 )

(tEI).

Proof. Defining u by z=:Xu we g e t by ( i . 1 ) s ( 1 . 2 ) u' =X-'

f ( t ,XU)=X'~ BO( 1 ) +X"

and (1.3)

I

C O ( DXut )

.

By t h e f i r s t p a r t of (1.5) i n t e g r a t i o n y i e l d s f o r f i x e d t , E I u o = o ( l ) + o t~o ~Ix-~clIDxllul~ I 1. Applying t h e Gronwall i n e q u a l i t y and u s i n g t h e second p a r t of (1.5) w e get

t to

For a p p l i c a t i o n s of t h i s theorem t h e f o l l o w i n g r e m a r k s

PERTURBED DIFFERENTIAL EQUATIONS

137

are u s e f u l : 11.6) REMARK.

For a l i n e a r p e r t u r b a t i o n

11.5)

f ( t ,z)=A"( t ) z

may be r e p l a c e d by

11.7)

x-lxx€L1 ( I ) .

To see t h i s , choose B:=o, (1.3)

C:=

a,D:=X".

Then f i s of t h e form

since Dz=O( IDzl). Furthermore, (1.5) holds: IX-~C~IDX~=~X-~;~X~EL~(I).

( 1 .8) REMARK. Consider (1.9) jwhere

y'=

*

-

A*(t)y

denotes t h e ad.joining o r t h e t r a n s p o s i n s operator)..

Then (1.5) is e q u i v a l e n t t o b * B I , /y*Clbxl€L1(I)

(1.10) y of (1 .gL,

and (1.7)

f o r a l l s o l u t i o n s x of (1.21,

i s equivalent t o

~ * " A c L ~ ( If )o r a l l s o l u t i o n s x of (1.21,

(1.11)

y of (1.91.

This follows from t h e f a c t t h a t i f Y denotes a fundamental m a t r i x of ( I .9), then X"(t)=KY*(t)

with some c o n s t a n t - and obviously n o n s i n g u l a r m a t r i x K. B y t h e preceding remarks theorem 1.4 i m p l i e s immediately

t h e f i r s t a s s e r t i o n of t h e f o l l o w i n g (1.12) THEOREM. and l e t Z') (1.13)

L e t X be a fundamental matrix of (1.21,

denote a fundamental m a t r i x of

z' = ( A ( t )+x( t ) ) z.

''The assumption of t h e theorem i m p l i e s t h a t grable. Therefore Z exists.

i s l o c a l l y inte-

N I E S S EN

138

and f o r a l l s o l u t i o n s y of (1.91

If f o r a l l s o l u t i o n s x of (1.2)

y 4 L ELl(I),

then

& g -

Z(t)=X(t)O( 1 )

x(t)=z(t)o(l).

To prove t h e second a s s e r t i o n , w e a p p l y t h e f i r s t p a r t of t h e

theorem t o t h e unperturbed system (1.9)

and t o t h e p e r t u r b e d system

(1.14) obtaining (1.15) where W d e n o t e s a fundamental m a t r i x of (1.14). and (1.13),(1.14)

S i n c e (1.2)

,(1.9)

are a d j o i n t systems, we g e t w i t h n o n s i n g u l a r

constant matrices K,,K2:

Together w i t h (1.15)

t h i s implies 2-1 ( t ) = 0 ( 1 ) x - ' ( t )

proving t h e second a s s e r t i o n . S i n c e X"=(det

X)-'Xkd,

where Xad

d e n o t e s t h e m a t r i x of t h e

a l g e b r a i c complements of X , and s i n c e d e t X(t)= d e t X ( t o )

*exp (

f

trA(T)dT),

to

e.g.

( 1 . 7 ) may be Pormulated a s t exp(- J' trA(7)dT) Xkd xXXtL'(1). to

Then theorem 1.4 and remark 1.6 e s p e c i a l l y y i e l d

-

j1.16)

THEOREN.

A be i n t e g r a b l c and

t l31e trA(T)dT be bounded below,

Let

l c t any s o l u t i o n x of (1.2) be bounded. z(t)=X(t)O(1)

€ o r any s o l u t i o n z oP (1.13).

Especially, z i s bounded.

p &

Then

PERTURBED DIFFERENTIAL EQUATIONS

The l a s t a s s e r t i o n i s theorem 6

Of

[4

1,

139

Ch.2.

2.APPLICATIUNS TO 3IFFERENTIAL EQUATIONS Consider on I c R t h e d i f f e r e n t i a l o p e r a t o r (2.1

n

1

Ly:=

c

i=o

pis ( i ) ,

i t s formal a d j o i n t (2.2) and t h e p e r t u r b e d equation

.y-

Here pi denotes throughout e i t h e r pi o r i t s complex c o n j u g a t e . I n t h e f i r s t c a s e €or any m a t r i x M,M*

denotes t h e transposed matrix,

i n t h e second case i t d e n o t e s t h e a d j o i n t matrix. We suppose t h a t p,

i s never zero, t h a t

-

...

Pi (i=o, ,n-l) Pn i s l o c a l l y i n t e g r a b l e on 1 2 ) and t h a t

w i t h nonnegative (measurable) f u n c t i o n s k

j*

Furthermore, d e n o t e by S:=

(51L5=0), S+:=

{qlL+y=O).

Then theorem 1.4 and remark 1.8 imply

211€ t h e pi's are n o t smooth enough, (2.2) h a s t o be i n t e r p r e t e d a s a q u a s i - d i f P e r e n t i a l operator.

140

NIESSEN

Proof. If 1 A :=

and i €

€or t E I , ztC" f(t,z) :=

P, 1 rp(t.z)e, i s equivalent t o

) , t h e n L{=O ( w i t h e n = ( bj.n )n J=I

x'=Ax w i t h x=(C ( i - 1 and (2.3)

A=.,,

is equivalent t o z'=Az+f(t,z)

(2-9)

w i t h z=(C (i-1)

In i=1

Analogously,

L+q=O i s e q u i v a l e n t t o n C (-l)j-i(piq)(j-i) y'= 4 * y w i t h y=( j=i

(2.10) Defining

D :=diag( kl

n )i=l

...,

,

kn)

1 and u s i n g t h e 1 -norm €or vectors we g e t by (2.4) d t , z ) = k o ( t ) O ( l )+O( l D ( t ) z l

1.

Therefore, f (t

,z)=B( t )O( 1 )+C ( t )O( ( 3(t) z l

with

To v e r i € y (1 .lo) (2.10).

Pn en'

k 2 en, c:= 1

B:=

Pn

let x be a s o l u t i o n of (2.8),

y a s o l u t i o n of

Then

x=( r, ( i - l ) )

w i t h gES,

n y=( ..Z. (-l)j-i(p:q)(J-i))

w i t h T$S+.

J=1

Thus by (2.6)

*

Y B=

ko

Pn

*

Y en=

(P,V Pn k,

* )=k,rl

+

1

EL (I),

141

PERTURBED DIFFERENTIAL EQUATIONS

T h e r e f o r e , i€ 6 i s a s o l u t i o n o€ (2.3), s o l u t i o n of (2.91,

i f ~ = ( 6 ( ~ " ) )i s a

i.e.,

theorem 1.4 and remark 1.8 imply t h e a s s e r t i o n : ( c ( i - l ) ( t ) ) = ( s ( ji - l )

(t))0(1).

T h i s theorem i m p l i e s a r e s u l t of Z e t t l : (2.11) THWREM

(Zettl

r181).

Consider on I : = [ a p )

and LC-y ( t ,6 1

(2.12)

-

with pi r c a l v a l u e d , c o n t i n u o u s and IY(t,C)( S k o ( t ) + k l ( t ) 161

(2.13)

where ko

€or -

,

k l arc nonnegative. Suppose S V S + C L ~ ( ,koqEL1 I) (I)

qCS+ and e i t h e r 1/2

kl

(i) (ii)

k , q € L 2 ( I ) € o r ' a l l qES+.

Then a l l s o l u t i o n s of (2.12)

a r e i n L2(I).

-

Proof .Since i n e i t h e r case koq,k15q€L1(I) f o r eES,qES+,

(2.14)

theorem 2.5 e s p e c i a l l y i m p l i e s (2.15)

6=(!,,**4n)o(1)

f o r any s o l u t i o n

6 of

(2.12).

Since SCL2(I),

(2.15)

y ield s 6cL2(I).

Obviously, i n t h i s proof n e i t h e r t h e assumption S + c L 2 ( I ) n o r t h e assumption t h a t t h e pi are r e a l i s needed. Furthermore, t h i s proo€ shows t h a t (2.14) g e n e r a l (2.6))

(and more

i s t h e e s s e n t i a l assumption of t h e theorem. I t may

be i n t e r p r e t e d a s a s m a l l n e s s c o n d i t i o n on t h e p e r t u r b a t i o n r a t h e r

N I E S S EN

142

than an integrability condition on the solutions of the unperturbed equation and its adjoint. Then the square-integrability of all solutions is only transferred by (2.15) from the unperturbed to the perturbed equation. It is clear that (2.15) (and more general (2.7))

gives more information on the structure of the solutions of

the perturbed equation than does the assertion of Zettl’s theorem. and (1.7) o r (l.lO),

Similarly,(l.5)

may be inter-

(1.11)

preted as smallness conditions on the perturbation. A s an application of theorem 1.12

we get analogously to

theorem 2.5 for linear perturbations the following L2.16) THEOREM. Suppose that

z e r o and that

is never

for all ~ E S , ~ C S + VELl (I).

(2.17)

Let cl,...,bn

denote a basis of S and

c1,...,Gn

a fundamental

system’) of solutions of n

c qjp=0.

(2.16)

j=O

Then for

any

solution 6 of (2.18)

$1

Proof.In

(t)=(!ii)(t)

and for any EES

,...,$)(t)

)0(1)

the terminology of theorem 1.12 and of the proof of

theorem 2.5

31Since by (2.17)

P. (j=O,... qn Pn

q

(compare footnote l)),

,n-1) are locally integrable

such a fundamental system exists.

PERTURBED DIFFERENTIAL EQUATIONS

T h e r e f o r e (2.17)

143

implies

Now theorem 1.12 y i e l d s t h e a s s e r t i o n . Theorem 2.16 g e n e r a l i z e s and s t r e n g t h e n s some r e c e n t (and e a r l i e r ) r e s u l t s on l i n e a r l y p e r t u r b e d d i f f e r e n t i a l e q u a t i o n s . A s

an immediate consequence (compare t h e proof o€ theorem 2.11) w e g e t (2.19) THEOREM ( Z e t t l r l 8 l ) . C o n s i d e r

-

on I:=[a,m)

and

Lc=hc

(2.20)

w i t h pi,h

r e a l v a l u e d and continuous. Suppose SUS+cL2(I) and e i t h e r (i)lhl

1 /2

g € L 2 ( I ) € o r a l l !€$US+

( i i ) h q € L 2 ( I ) f o r a l l qES+. Then a l l s o l u t i o n s of (2.20)

belong t o L2(11.

Another c o r o l l a r y i s 12.21) THEOREM (Halvorsen

[lo]).

Consider on a h a l f - o p e n

i n t e r v a l I t h e equations (2.22)

5 Il+P ( t 1!=O,

(2.23)

cll+qo(t )6=0

-

u d t k po, qo r e a l - v a l u e d

and l o c a l l v i n t e s r a b l e .

I q0-po I1’25

(2.24)

2

€ L 2 ( 11

f o r e v e r y s o l u t i o n 5 of (2.22),

t h e n (2.23)

i s of l i m i t c i r c l e t y p e

o r of l i m i t p o i n t t y p e a c c o r d i n q as which i s t h e c a s e for e q u a t i o n (2.22).

I n a d d i t i o n , (2.24) w i l l be s a t i s f i e d w i t h any s o l u t i o n g

Of (2.231.

NIESSEN

144

Halvorsen's theorem p a r t l y g e n e r a l i z e s t h e f o l l o w i n g theorem, which i s a l s o c o n t a i n e d i n theorem 2.16:

u.

(2.25) THEOREM (Bellman r 3

Consider on I:=[O,-)

the

equations (2.22)

S"+p0( t C=O,

(2.23)

6"+qo(t )6=0.

If

q0-poELW ( 1 1

(2.26)

and i f f o r s o r e p , p ' m 1sps2sp*s,

$ + $4

a l l s o l u t i o n s of (2.22) belong t o LP(I)nLP' ( I ) , t h e n a l l s o l u t i o n s of (2.23) belong t o L p ( I ) n L p ' ( I ) .

-

* Proof. Choosing po:=po,

n

we have L= d' +p -L+. d t 2 O-

T h e r e f o r e (2.17)

reduces t o (2.27)

(qo- P0)6rlEL1 ( 1 )

of (2.22).

for a l l s o l u t i o n s

But t h i s i s i m p l i e d by (2.26)

since

5 ,TELP(I ) ~ L P('I )CL2 (I

- + P'

(which i s t r u e even w i t h o u t t h e assumption 1 P theorem 2.16

.,

c= ( 5 ,c 2)0 ( 1 )ELP(1

(2.28) f o r any s o l u t i o n

6

-1= I ) .

Thus by

11

of (2.23).

P a t u l a and Wong r e c e n t l y g e n e r a l i a e d Bellman's

theorem

p a r t i a l l y t o LP-perturbations under t h e a d d i t i o n a l assumption t h a t

a l l s o l u t i o n s of t h e u n p e r t u r b e d e q u a t i o n are bounded: 12.29) THEOREM ( P a t u l a and Wona r1311. Consider on I:=[O,m) (2.22)

!"+P0(t )5=0,

145

PERTURBED DIFFERENTIAL EOUATIUNS

6"+qo ( t )c-0

(2.23)

with

4,-

(2.30)

f o r some p

with

l5pSm

P o m I )

'I.

Suppose t h a t a l l s o l u t i o n s of (2.221

belong t o L 2 ( I ) n L m ( I ) . Then a l l s o l u t i o n s of (2.23)

are i n

L2(I)nLm(I>.

Furthermore, .r.+++.+=1.

P

T h e r e f o r e (2.30)

9

p-1

P p-1

and P,qEL2nLrn imply (2.27).

Then t h e f i r s t p a r t

of (2.28) h o l d s and y i e l d s t h e assertion. The same argument shows t h a t t h e f o l l o w i n g m o d i f i c a t i o n of theorem 2.29 i s t r u e : (2.31) (2.22)

THEOREM.

If (2.30)

h o l d s and i f a l l s o l u t i o n s of

belonq t o L r ( I ) n L S ( I ) , where

xs

gl5ss m ,

t h e n a l l s o l u t i o n s of (2.23)

are i n Lr(I)nLS(I).

E s p e c i a l l y t o prove t h a t (2.23)

i s of l i m i t c i r c l e t y p e ,

i t i s s u f f i c i e n t t o suppose t h a t (2.22) i s OP l i m i t c i r c l e t y p e and

32 t h a t every s o l u t i o n of (2.22)

belongs t o LP"(1).

If p2 i s l o c a l l y a b s o l u t e l y c o n t i n u o u s , t h e o p e r a t o r d d L = E ( p 2 =)+Po

p = m put

may be written i n t h e form (2.1)

& :=I

and i t is self-

1116

NIESSEN

*

a d j o i n t ( i f p;:=p2,po:=po).

T h e r e f o r e theorem 2.5 and theorem 2.16

i n t h i s c a s e e s p e c i a l l y imply t h e f o l l o w i n g theorems: (2.32)

THEORhm (Bradley r 5

1) , Consider

(2.33)

(P25'Y + Pob=0,

(2.34)

-

(P26'Y + pos=Y'(t,c;)

(2.13)

IY(t,6)1

on I :=( a,m)

with p2>0,po,Y continuous and

-

where ko &a

ko ( t ) + k l ( t ) 161,

k l a r e nonncqative.If

5,k:/*5EL2(I) t h e n a l l s o l u t i o n s of (2.34)

f o r every s o l 3 t i o n 5 of (2.331

-

and kogEL1 ( I ) ,

are i n L 2 ( I ) .

(2.35) THEOREM (Bradley 15 L 2 ( I ) f o r a l l s o l u t i o n s of (2.33),

7 ) . I f ) q o - pd1/25 belonqs t o m I q o - pof/26 b c l o n s s t o L2(11

f o r a l l s o l u t i o n s of (2.36)

(P263) +q06=O

and (2.33)

i s of l i m i t c i r c l e type i f f (2.36) i s of l i m i t c i r c l e

tJp=. If p2 i s n o t l o c a l l y a b s o l u t e l y continuous,

(2.33)

has t o

be c o n s i d e r e d as a q u a s i d i f f e r e n t i a l equation. S i m i l a r t o t h e o p e r a t o r s (2.1)

and (2.2)

every q u a s i d i f f e r e n t i a l o p e r a t o r ( f o r

d e f i n i t i o n compare [ 1 5 ] ) and i t s a d j o i n t may b e transformed t o a d i f f e r e n t i a l system and i t s a d j o i n t system, resp.. theorem 1.4 and theorem 1.12 imply e.g.

Therefore,

r e s u l t s on q u a s i - d i f f e r e n -

t i a l e q u a t i o n s similar t o t h o s e s t a t e d i n theorem 2.5 and theorem 2.16.

O f t h e v a r i o u s p o s s i b l e r e s u l t s w e mention o n l y t h e f o l l o w i n g

two, which g e n e r a l i z e Bradley's theorems:

PERTURBED DIFFERENTIAL EQUATIONS

1117

(2.37) THEOREM. Consider on IcR (2.38)

LC=dt,6,6:.

(2.39)

- ,..., where po

pn,l

w i t h ki(i=O,

a r e l o c a l l y i n t e q r a b l e and

.

n+l # z ~ +I k~o ( t ) + X ki ( t ) 1 zil

Pn

Irp(t , z ,

-

* * , P )

,-

1

...,

,

>I

i=l

n+l) nonnegative. Suppose t h a t

ko5,kjt(J-1)V,kn,.,pn~(n)V€L1(I)

€ o r a l l s o l u t i o n s 5 , of ~ (2.38).

51,...,62n

(j=I,...

,n)

Then €or any fundamental system

o€ s o l u t i o n s o€ (2.38) and €or any s o l u t i o n

...,5&)

6 ('1 ( t ) = ( y !i) ( t >, (2.40) THEOREM.

,...,52n & G I ,...,62n

resp..

Then € o r any s o l u t i o n

be fundamental systems of s o l u t i o n s

C

o€ (2.41)

and 5 of (2.381

( t ) 9 . . '52n ( i ) ( t ) ) o(1)

3-

€or i=O,

)... 1 ,. .,ti;) ., c p

s(i)(t)=(c;i)(t)

6 (n) ( t ) = p ( t

5 ( n ) ( t )=(C,(n)( t ) 9 . .

,6$;)(t))O(l

...,

n-1

,

1

P,W

(t)) O ( $ T ) , n qn(t) ( t >) O ( p ( t 7 . ) n

The p r o o f s proceed a s t h o s e of theorems 2.5 and 2.16 transformations:

2

Suppose t h a t € o r a l l s o l u t i o n s 5.q

Let s1

c ( i ) ( t )=(5!i)

of (2.391

..,n).

(i=o,.

( t ) )0(1)

C

except €or t h e

148

NIESSEN

(2.38) i s transformed by

to x '=Ax with

and

- as i t s

- by

(real) adjoint

to )LA*Y.

I n t h e c a s e of theorem 2.37,

(2.39) i s transformed by (2.42)

to

z' =Az+me2n; i n t h e case of theorem 2.40 t h e d i f f e r e n t i a l e q u a t i o n (2.41) is transformed by

to z'=A

1

z

PERTURBED DIFFERENTIAL EQUATIONS

149

The rest of the proofs is then easily carried out. Finally, applying theorem 1.16 to the systems arrising from (2.1)

and (2.181,

we get the following theorem which is a

generalized and strenghtened version of a theorem due to Bellman [ 21 t (2.43)

THEOREM. Let the real part of

Pn-1 (r 1 dr Pn

t0

bounded above and let

If t(i)

is bounded for all solutions 5

(2.44)

and for any i=O,...*n-l,

then for any solution

C of

(2.45)

and for any fundamental system

Especially, c(i)(i=O,...,n-l)

c,, ...*tn

of solutions of (2.44)

is bounded for any solution of (2.45).

3.0~ THE LIMIT CIRCLE CASE FOR WEIGHTED EIGENVALUE PROBLEMS. Let the differential operator

be formally selfadjoint on IcR with a,(t)#O

(tEI), aiEC1(I) and

let w be a nonnegative continuous function on I. Then the (possibly singular) eigenvalue problem (3.2)

l(t)=XN,

considered in the space

150

NIESSEN

i s s a i d t o be i n t h e l i m i t c i r c l e case i f f o r every AEC every

s o l u t i o n of (3.2) belongs t o L2(w,I).

Recently Walker proved t h e

following (3.3)

TH30REM (Wslkcr r161). I f f o r some A o € C

s o l u t i o n s of

=x oWc

1( f 1

(3.4)

and

1 ( c )=X,wf

(3.5)

belong t o L 2 ( w , I ) , thcn (3.2) i s i n t h e l i m i t c i r c l e case. For ur,l and n even E v e r i t t [9 ] p r m e d t h i s thcorem assumi n g only t h a t f o r some x0EC a l l s o l u t i o n s of (3.4)

are square-

i n t e g r a b l e . The same r e s u l t follows € o r a r b i t r a r y n > 2 and for an a r b i t r a r y weight f u n c t i o n w from theorem 9.11.2 forming (3.2)

of [ 1 ] by t r a n s -

t o an n-th order system u s i n g t h e transformations

given i n [ 141. As s t a t e d above, t h e theorem i s a simple consequence of theorem 2.16 applied t o

x

L :=1- ow,

L+ :=1- Tow

(whepe L+ denotes t h e complex a d j o i n t ) pO:=ao-~ow,qo:=ao-~w,pj:=qj:=aj(j=l

,...

,n).

Theorem 3.3 may e a s i l y be generalized t o ei'genvalue problems of t h e form (3.6)

m(f )=In( s 1

w i t h formally s e l f a d j o i n t d i f f e r e n t i a l o p e r a t o r s m,n such t h a t n

i s p o s i t i v e on a s u i t a b l e f u n c t i o n space.

More g e n e r a l l y , w e get an analogous r e s u l t f o r s i n g u l a r r i g h t - d e f i n i t e 6-hermitian eigenvalue problems. Such eigenvalue

PERTURBEI) DIFFERENTIAL EQUATIONS problems (for definition and properties compare e.g.

151

[12])

may be

reduced to systems of the form (3.7)

F1x'+F2x=AGx

with (n,n)-matrix-valued functions F1,F2,G,defined and continuous on some interval IcR such that Fl(t) is nonsingular for tEI.

A

rightdefinite S-hermitian eigenvalue problem (3.7) especially has the following properties: There exists a continuously differentiable (n,n)-matrixvalued function H and a positive-semidefinite- valued continuous function W such that €or every AEC (3.8)

H'=HFT1 (F2-AG)+[Fy1 (F2- yG)]*H,

(3.9)

G=WF;-~H*.

Let K denote the continuous and positive-semidefinite square-root of W and define

u : = K F -~l ~ * , L ( u ,I

:=( x U ~ L~ E ( I) 1

.

Then the eigenvalue problem (3.7) considered in the space L2(U,X) is said to be in the limit circle case if for every AEC every solution of (3.7) belongs to L2(U,I). Then the following theorem holds : (3.10) THEOREM. If for some A o € C all solutions of F~ X' + F ~ X = A.GX

(3.11)

and of

F,x'+F~x= TOG.

(3.12;

belong to L2(U,I),

Proof. Let

then (3.7) is in the limit circle case.

l € C be fixed. Defining

A:= -F;'

( F ~ -x,G),

"A=(A-x,)F;~G

152

NIESSEN

we have

-

A+A= -FYI (F2-kG). T h e r e f o r e , (3.11) and (3.7) (3.13)

x’=Ax

-

and (3.14) resp..

+re equivalent t o

x’=(A+A)x, Furthermore, by (3.9)

and t h e d e f i n i t i o n of K and U w e

obtain

Z=( A-I o ) F;’

KKF;-” H*= ( A -A o ) H-I

Now l e t x and y be s o l u t i o n s of (3.13)

u*u. and

y’= -A*y resp..

2

Then xEL ( U , I ) by assumption, and u s i n g (3.8) i s a s o l u t i o n of

seen t h a t H*-’y

it is easily

3.12) and t h e r e f o r e belongs t o

L2(U,I), too. Thus y*”A= ( A -I o ) uH* -1 y ) * UxE L

Now theorem 1 .I2 i m p l i e s t h a t a l l s o l u t i o n s of (3.141,.

i.e.,

all

s o l u t i o n s of (3.7) belong t o L 2 ( u , I ) . S i n c e t h e e i g e n v a l u e problem (3.6)

may be transformed t o a

r i g h t d e f i n i t e S-hermitian e i g e n v a l u e problem, theorem 3.10 i m p l i e s an analogous r e s u l t f o r (3.6). 4,GENERALIZATIONS The preceding r e s u l t s may be g e n e r a l i z e d i n v a r i o u s d i r e c t i o n s . E.g.,

i P I = [ t o , b ) , i n t h e c a s e of a l i n e a r p e r t u r b a -

t i o n w e may r e p l a c e t h e i n t e g r a b i l i t y c o n d i t i o n (1.7) by t h e assumption t h a t t h e i n t e g r a l of t h e l a r g e s t e i g e n v a l u e of t h e r e a l p a r t of X’%X

i s bounded above. More g e n e r a l l y , w e g e t

(4.1) THEOREM. L e t X d e n o t e a fundamental m a t r i x of (1.2) and z any s o l u t i o n of (1.13).

Then

PERTURBED DIFFERENTIAL EQUATIONS

153

Here

For p r o p e r t i e s of p compare [ 8

3 ,p.41.

E s p e c i a l l y l p (M)j llMl

, and

i f t h e v e c t o r norm i s taken t o be t h e Euclidean norm, then v(M) is t h e l a r g e s t eigenvalue of t h e r e a l p a r t of M. Theorem 4.1 i s an easy a p p l i c a t i o n of a theorem due t o Lozinskil' (compare

[ a ],p.581

to UI

AS

=x -1 kxu.

an example we c o n s i d e r f o r a#O

which a r r i s e s from

611+qc=o

(4.2) by t h e t r a n s f o r m a t i o n

Choosing A:=O

, X :=E ,

and t h e Euclidean norm, theorem 4.1 i m p l i e s

f o r a l l s o l u t i o n s 6 of (4.2). theorem I of [ 111.

For a > 0 and i = O t h i s result i s

154

NIESY EN

Another a p p l i c a t i o n t o d i f f e r e n t i a l equations gives t h e following theorem, which i s an improved v e r s i o n of [ 101, theorem 2:

(4.3) THEOREM.

Let p,q

g r a b l e , and denote by f , ,

a r e a l fundamental system of

c "+p5=0

(4.4) such t h a t f ,

5,

be realvalued and l o c a l l y i n t e -

fd

- 5,'

c 2 = l . Then (4.2)

(and (4.4)) a r e of l i m i t

c i r c l e type provided

-

Proof. Let

( '), 0

A:=

-P

(, ,). 0

"A=(p-q)

0

0

Then (4.4) i s equivalent t o x'=Ax with

(4.5)

x=(zt),

(4.2) i s equivalent t o z'=(A+I

(4.6)

Furthermore,

z with z=(:t).

If,

i s a fundamental matrix of (4.5) and t h e r e a l p a r t of X-%X eigenvalues

& ~1 J P - q ) (2C + I 5 22) ' Thus, by theorem 4.1,

implying ~ C L , ( I )

.

f o r any s o l u t i o n z=

has

PERTURBED DIFFERENTIAL EQUATIONS

155

Theorem 1.4 may be g e n e r a l i z e d t o

let

(4.7) THEOREM. X1,X2

Let X be a fundamental m a t r i x of x‘=Ax

&

be l o c a l l v a b s o l u t e l y c o n t i n u o u s and such t h a t X=X1X2.

For f i x e d UE

To,i1 l e t

f ( t ,z )=B ( t ) g (t ,z ) +C ( t ) ID ( t ) z ‘h ( t ,z) , where B is l o c a l l y i n t e g r a b l e and I g ( t , z ) ] , ] h ( t , z ) \ S 1. (4.8)

I x 2 ( t ) x - l (T)C(T)llD(T)X1 ( T I ! ‘5 kq ( t ) k 2 ( T )

-

( T E b O , t l 5 ) jP

where k l , k 2 a r e l o c a l l y i n t e g r a b l e nonnegative f u n c t i o n s , t h e n €or

any s o l u t i o n z o€ (4.9)

z’=A(t)z+f(t,z)

we have a)

(4.13)

51

f o r t c t o , [to,t]:=[t,to]

156

NI ES S EN

then

(4.14) Proof. Using a% l - a + u a f o r a 20,we o b t a i n w i t h u:=X-’z

I

similarly

t o t h e proof of theorem 1.4 I X 2 ( t ) u ( t ) l I I X 2 ( t ) u ( t o ) l+

t ( T ) B ( T ) I dT

IX2(t)X-’

I+

and by a m o d i f i c a t i o n of t h e Gronwall

a

product k e r n e l K) t h i s y i e l d s (4.10).

To prove b) choose kl:=y,k2:=IX;’ holds by (4.11)

CllDx11‘.

X 2 is bounded by (4.11),

and (4.12),

Then (4.8) kl i s bounded

and k2 i s i n t e g r a b l e by assumption. Thus (4.14) follows from (4.10).

As s p e c i a l c a s e s we mention and u=l

1 ) Choosing X1 :=X,X2:=E

1.4.

, theorem

4.7 b ) reduces t o theorem

I n t h i s case, p a r t i a l i n t e g r a t i o n of

(4.10)

with kl :=l, k2:=lX’1C((DXI

)=x( t ) o(e

2( t

If

Ix-’(7)C(7)((D(T)X(7)1

to t

+IS

yields

I f I X-’

+

(s )C (s )/ID(s > X( s >I ds I

I X - ~ ( ~ > B ( ~e ) 7I

d71)-

to 2) Choosing X2:=X,X1:=E

(4.15)

theorem 4.7

THEOREM. If f o r some

(4.16)

IX(t)X-1(7>1s y

(4.17)

ICIIDIU,

b) i m p l i e s 2 0, OE[O,I]

(7Wo,tl),

I B I E L (~I ) ,

then a l l s o l u t i o n s of (4.9)

are bounded.

PERTURBED D I F F E R E N T I A L EQUATIONS

157

(4.16) i m p l i e s t h a t X i s bounded. By F l o q u e t ' s .theory t h e converse is t r u e i f A i s periodic: i 4 . 1 8 ) REMARK.

-

Let A be p e r i o d i c and X bounded. Then f o r

some y 2 0 (4.16) i s v a l i d . Theorem 4.15 may be c o n s i d e r e d a s a g e n e r a l i z a t i o n of t h e

following (4.19) THEOREM ( C e s a r i r 6

I).

Suppose t h a t

fi-aicL1 [ to,m) ,Pi( t )+ a i ( t + m ) , ( i = o , .

..

,n-l)

and t h a t a l l s o l u t i o n s of (4.20) d e r i v a t i v e s are bounded.

t o g e t h e r with t h e i r f i r s t n-I Then a l l s o l u t i o n s of (4.21)

d e r i v a t i v e s are bounded.

t o g e t h e r with t h e i r f i r s t n-I

A:=(

0 1*

-ao (4.20),

(4.21)

(4.22)

) v

...,

,

-a n-1

a r e equivalent t o

x'=Ax,

x=($ ( i - 1 ) 1

and z' =Az+f ( t , z )

resp..

, z=( 6 ( i - 1 ) ) .

By assumption any fundamental m a t r i x X of (4.22)

S i n c e A i s c o n s t a n t , remark 4.18 i m p l i e s (4.16).

i s bounded.

Finally the intc-

g r a b i l i t y of C shows t h a t (4.17) holds. Thus t h e boundedness oP c(i)(i=O,

...

,n-I ) f o l l o w s from theorem 4.15.

N I ES S EN

158

Obviously t h i s proof does n o t need t h e converqence assumption on t h e Pi.

Furthermore, t h e ai may b e allowed t o be

p e r i o d i c f u n c t i o n s . I t may b e mentioned t h a t t h e o r i g i n a l proof of theorem 4.19 took about 16 pages. The proof given h e r e a l s o seems t o be more s i m p l e t h a n t h a t i n d i c a t e d i n [ 2

1.

Finally we remark

t h a t theorem 4.15 i m p l i e s t h e second p a r t of theorem 1.16 and t h e

theorem of [ 2

1.

R e c e n t l y Wong announced t h e f o l l o w i n g (4.23)

THEOREM (Wonq 1171). Suppose t h a t a l l s o l u t i o n s of

(4.24) belong t o L2rO,m)nLm[Op) and t h a t f o r some a € r O . l l Irn(t,C)lS A ( t ) 1 6 I u w i t h X€LprO,m) f o r some p , l 5 p 5 2 , Then, a l l s o l u t i o n s of

L 6 = d t ,6 1

(4.25)

belong t o L2r O,m)nLmr 0.m). T h i s theorem i s a l s o a c o r o l l a r y t o theorem 4.7 b ) : Transforming (4.24) theorem 2.37,

and (4.25) t o systems as i n t h e proof of and u s i n g a s i m i l a r argument

choosing X1:=X,X2:=E

a s remark 1.9 t h e assumptions of theorem 4.7 b) are seen t o be fulfilled i f (4.26)

for a l l solutions

A !5\'q f,q

ELq[ 0 , ~ )

of (4.24).

T o prove (4.26) w e remark t h a t by

2 5 + 1 < m 1- 1 P

-

Thus

PERTURBED DIFFERENTIAL EQUATIONS

159

Therefore

i m p l i e s (4.26).

Now t h e a s s e r t i o n f o l l o w s from theorem 4.7 b ) .

T h i s proof shows t h a t (4.26) h o l d s i f w e o n l y suppose t h a t a l l s o l u t i o n s of (4.24) belong t o L a r b i t r a r y p 2 1).

9 ’-’ (with

ut[O,l]

and

Then by theorem 4.7 b)

f o r any s o l u t i o n 6 of (4.25) and any fundamental system

T1,...,52n

of (4.24).

REFERENCES

[I]

Atkinson, F.V.: Discrete and continuous boundary problems. Academic ?less, New York 1964.

[2)

Bellman, R.: The s t a b i l i t y OP s o l u t i o n s of l i n e a r d i f f e r e n t i a l e q u a t i o n s . Duke Math. J. 10 ( 1 943). 643-647.

[3]

Bellman, R.: A s t a b i l i t y p r o p e r t y of s o l u t i o n s of l i n e a r d i f f e r e n t i a l e q u a t i o n s . Duke Math.J. 1 1 (1944). 51 3-51 6.

[4] Bellman, R.: S t a b i l i t y t h e o r y of d i f f e r e n t i a l e q u a t i o n s . Dover, New York 1953.

[ 51

Bradley, J.S. : Comparison theorems f o r t h e s q u a r e i n t e g r a b i l i t y of s o l u t i o n s of ( r ( t ) y ’ ) ’ + q ( t ) y = f ( t , y ) . Glasgow Math.J. 13 (1972), 75-79.

[ 61

C e s a r i , L. : S u l l a s t a b i l i t d d e l l e s o l u z i o n i d e l l e e q u a z i o n i d i f f e r e n z i a l i l i n e a r i . Ann. S c u o l a Norm. Sup. P i s a (2) 8 (1939), 131-1460

[7] Chu,S.C.

and F.T.

M e t c a l f : On Gronwall’s i n e q u a l i t y . Proc.

AMS 18 (19671, 439440. 181

Coppel, W.A.:

S t a b i l i t y and asymptotic b e h a v i o r of d i f f e r e n -

NIESSEN

160

tial equations. Heath, Boston 1965.

[ 91

Everitt, W.N.: Singular differential equations I: The even order case. Math. Ann. 156 (1964). 9-24.

[ 101 Halvorsen,S.: On the quadratic integrability of solutions of d2x/dt2+f (t)x=O.

Math. Scand. 14 (1 964), 1 1 1-1 19.

[TI] Levinson, N.:

The growth of the solutions of a differential equation. Duke Math. J. 8 (1941), 1-10.

[ 121 Niessen, H .D. : Singulare S-hermitesche Rand-Eigenwertprobleme. manuscripta math. 3 (1970), 35-68. [13] Patula, W.T. and J.S.W. Wong: An LP-analogue of the Weyl alternative. Math. Ann. 197 (19721, 9-28.

[ 141 Schneider, A. : Zur Einordnung selbstadjungierter Rand- Eigenwertprobleme bei gewi5hnlichen Dilferentialgleichungen in die Theorie S-hermitescher Rand-Eigenwertproblcme. Math. Ann. 178 (19681, 277-294. [ 153 Shin, D.: Existence theorems for the quasi-differential equation of the n-th order. C.R. Acad. Sci. URSS 18 (19381, 515-518. [16] Walker, Ph. W.: Weighted singular differential operators in the limit circle case. J. London Math. S O C . ( ~ ) . 4 (1972). 741-744.

[ 171 Wong, J.S.W.: Square integrable solutions of Lp perturbations of second order linear differential equations. Springer lecture notes no. 145 (19741, 282-292. [IS] Zettl, A.:

Square integrable solutions of Ly=f(t,y). AMS 26 (1970), 635-639.

Proc.

W . Eckhaus ( e d . ) , New Developments i n D i f f e r e n t i a l Equations @ North-Holland P u b l i s h i n g Company (1976)

ON CERTAIN ORDINARY DIFFERENTIAL EXPRESSIONS AND ASSOCIATED INTEGRAL INEQUALITIES

W N Everitt aud M Giertz

1.

T h i s paper is concerned with inequalities of the form

for functions f in certain linear manifolds of the integrable2

squsre function space L (a,-) vith the usual norm A,B

and

c

-

11 11.

Here a,B,y,

(with c to be taken as ‘ s m a l l ’ ) are non-negative real numbers,

and the differential expression M is defined, in terms of the positive valued coefficients p and q, by Mfl = -(pf’)’ + qf on [a,-)

( ’ :d/dx).

(1.3)

In C7, Chapter VI, Sections 6 and 81 Goldberg has given certain a priori estimates of the form (1.2) for differential expressions of arbitrary order with bounded coefficients. The inequalities considered in this paper give an extension of this type of estimate to second-order expressions with. in general, unbounded coefficients. The inequality

(1.1 )

is considered by Everitt and Giertz in

C61. The results given here avoid one of the difficulties met in determining conditions on the coefficients p and q for C6; Theorem 31 to hold. Aa in C61 separation results for the differential expression M

follow from inequalities of the form (1.1);

161

for the definition of separation

362

W.N.EVERITT A N D M.GIERTZ

see

C4; Section 11 o r C6; Section 61. Additionally a compete separation

r e s u l t i s obtained here; f o r t h i s concept see t h e definition i n

(5; Section 11 Inequalities of the form (1.1) and (1.2) a l s o y i e l d r e s u l t s i n the theory o f r e l a t i v e l y bounded perturbations of t h e d i f f e r e n t i a l 2

operators generated by M i n L (a,-).

For results i n t h i s direction see

[6; Section 91, which a l s o depend on thc work of Kato, see C8; Chapter

V,

Section 41. There are also connections with i n e q u a l i t i e s considered by Everitt i n C31. In section 2 of t h i s paper there is a statement of t h e r e s u l t s t o be proved; following sections contain t h e proof of these r e s u l t s together w i t h some coments on t h e i r consequences.

There i s a l i s t

o f references.

2.

R and C denote the r e a l and complex number f i e l d s respectively.

For a E R t h e h d f - l i n e [a,-) is closed a t a and open a t

m.

AC denotes

absolute continuity and L Lebesgue integration; 'loc' f o r local, i.e. of la,-).

a property s a t i s f i e d on a l l compact subintervals

2 L (a,-)

denotes the c l a s s i c a l Lebesgue complex function

space, which i s a l s o indentified w i t h t h e Hilbert function apace of' equivalence classes.

Let the coefficients p and q s a t i s b t h e following basic conditions : p, q : [a,-)

+

R and

These conditions on p and q imply that t h e d i f f e r e n t i a l expression M, given by

(1.3), is regtdar a t a l l points of [a,-) but has a singular

163

DIFFERENTIAL EXPRESSIONS AND INTEGRAL INEQUALITIES

p i n t a t -; see C9; Section 15.13.

it i s knovn t h a t

Since q i s bounded below on [a,-)

M is i n t h e limit-point condition a t t h e singular

17.5 and 23.61 and C21.

point -; see C9; Sections

Following t h e notation i n C6; Section 32 we define t h e l i n e a r 2 of L (a,-) by

manifold D1 L D,(p.q)

D~ = D

E

and

: f l c A C ~ ~ ~ C ~ , -M )

L2(a,-)

2

C ~EI L

(2.3)

(a,-)).

2

1

i s t h e domain of the maximal oFerstor T1 generated by M i n L (a,-)

and defined by T,f = N f I

(f

2 i s not synnnetric i n L (a,-);

E

D

1

1;

it is known t h a t T is closed but 1

for these results see C9; Sections 17.4 and 51,

also the remarks i n C6; Section 31.

The basic operator theoretic

definitions a r e given i n C1; Sections 39 and 411. 2 We say t h a t M i s _separated i n L (a,-)

i f (see

C4; Section 11

and C6; Section 61) qf

E

2

L (a,-;

f o r all f

2 and completely separated i n L (a,-) pf", p ' f ' ,

-

qf

E

(2.4)

D1;

i f (see r5; Section 11) 2

L (a,-) f o r a l l f

a r e all i n

E

D1.

Note t h a t both (2.4) and (2.5) are conditions t o be s a t i s f i e d only at

since the basic conditions (2.1) and (2.2) on the coefficients 2

p and q imply t h a t all terms i n (2.4) and (2.5) are i n LlOc[a,-).

We mey now s t a t e ( r e c a l l t h a t

11 (1

denotes the usual norm i n

L2(a.-)) Theorem 1 Let t h e coefficients p and q satisfy t h e basic conditions (2.1)

(2.2); l e t addition-

p

q s a t i s f y the conditions;

(2.5)

164

W.N.EVERITT AND M.GIERT2

l e t K E @,m) -

where

C,

be given;

lat 11

b e chosen so t h a t 0

2 1

< 11 < min(1, 4C q ( a ) )

= max(7, 3K)

(b)

and (2) f o r

let

all

Q

E (0,n) t h e following i n e q u a l i t y i s v a l i d

where t h e p o s i t i v e number A depends only on t h e c o e f f i c i e n t s p A i s d v e n e x p l i c i t y i n (3.17)

9;

below.

Proof This i s given i n t h e s e c t i o n s which follov. Notes The e x p l i c i t dependence of t h e number A is given i n t h e s e c t i o n s devoted t o t h e proof of (2.9). The i n e q u a l i t y (2.9) has some s i m i l a r i t y with t h e i n e q u a l i t y i n

C6; (8.3) of Theorem 31 but t h e condition ( 2 . 7 ) ebove avoids t h e u n s a t i s f a c t o r y nature of t h e condition C6; (8.213 depending as it does on t h e v e r i f i c a t i o n of another inequality.

As i n C6; Theorem 31 t h e c o n t r o l condision (2.6) is necessary t o t h e proof o f (2.9); it prevents t o o much o s c i l l a t i o n i n t h e c o e f f i c i e n t s p and q i n t h e neighbourhood of

m.

165

DIFFERENTIAL EXPKESSIONS AND INTEGRAL INEQUALITIES

Under the conditions of Theorem 1 the differential 2 expression M is separated in L (a,-). Corollary 1

It follows at once from the inequality (2.9) that condition

h-oof

(2.4) is satisfied.

Corollary 2 If in addition to all the conditions of Theorem 1 coefficients p

the

q satisfy

(2.10)

Proof It follows from the inequality (2.9) that (pf')'

E

L2(a,-)

we obtain Ip'f'I f

E

(f E D1).

5 L{pq}1'2f'

Now (pf')' = pf" + p'f' and from (2.10) E

2 I, (a,-) on using (2.9) again and this for nl1

D1. It now follows that condition (2.5) is satisfied.

Note that in general the conditions (2.7) and (2.10) are independent of each other; howevsr if lower bound on [a,-)

the

ccefficient q has a positive

then (2.10) implies (2.7).

Finally we have Theorem 2

Under the basic conditions (2.1)

(2.2) on t h e coefficients

pandq

under the additional conditions (2.6)

and

(2.7) of Theorem 1

following inequality is valid for all

E E

(0,l)

the

166

W.N.EVERITT A N D M.GIERTZ

where t h e positive number B depends only on t h e c o e f f i c i e n t s p

Proof

and q.

This i s given i n t h e sections which follow. Note t h a t the inequality (2.12) should be compared with t h e

-a p r i o r i

estimates given by Goldberg i n C7; Chapter VI, Sections

6 and 83;

see i n particular

Theorem VI 8.1

However i n (2.12) t h e

coefficients are, i n general, unbounded on [a,-).

3.

In t h i s section we give t h e proof of Theorem 1. The l i n e a r manifold D1 i s defined i n (2.3).

Now define

D1,O I D1,O (p,q) as t h e collection of all f i n D1 which vanish i n

some neighbowhood, which may change Kith f, of

D1,O = {f

E

D1

: f o r some X E X ( f )

m,

> a, f ( x ) = 0

h. ( x c CX,-)).

(3.1)

no r e s t r i c t i o n i s placed on t h e values 190 taken by f a t the regular end-point a. Note t h a t i n defining D

The reason f o r t h e introduction of D

1,o inequality (2.9) i s f i r s t established on D1,3.

i s as follows.

The

From known r e s u l t s

i n the theory of d i f f e r e n t i a l operators it may be claimed t h a t given

any f

E

D

1

there is a sequence {f

n

: fn L D

such that t h e sequences {fn) and {MCf,]) t o f and Lfl respectively.

1,O

for n

= 1,2,3, ...I

a r e convergent i n

2

L (a,-)

Furthermore t h e tenas on t h e left-hand

2 side of (2.9), with f replaced by f n , also converge i n L (a,-)

t o the

corresponding terms with f i n D1. I n t h i s way the inequality is established i n D

1’

DIFFERENTIAL EXPRESSIONS AND INTEGRAL INEOUALITIES

This c l o s u r e argument l e a n s heavily on t h e f a c t t h a t t h e

maximal operator T (2.31, is closed. Sf = M C f l

(f

D

E

1

introduced i n t h a previous s e c t i o n , following

I n f a c t i f S : D,,O

+

2 L (a,m) is defined by 2

1to

) then S i s closeable i n L (a,-) ,and t h e c l o s u r e

= T,. This result depends on no r e s t r i c t i o n being placed on t h e values of f E D

1S O

a t t h e rfgular end-point a ( s e e above) and t h e

property o f M being limit-point at following (2.2)).

m

( s e e t h e previous s e c t i o n

Some a d d i t i o n a l a n a l y t i c a l d e t a i l s may be found

i n C6; Sections 4 and 81. We now prove t h a t (2.9) holds f o r a l l f

E

It i s c l e a r l y

D,,O.

s u f f i c i e n t t o prove t h e r e s u l t for real-valued f i n D

1S O

t h e r e is an immediate extension t o complex-valued f i n D

s i n c e then 1 ,O'

For t h e remainder of t h i s s e c t i o n we t a k e f t o be real-valued and i n DISO; s i n c e then f vanishes i n some neighbourhood of

there

are no convergence problems i n t h e i n t e g r a l s concerned.

As i n 16; Section 81 we have t h e following i d e n t i t y obtained on i n t e g r a t i o n by p a r t s (we r e c a l l from (2.1) m d (2.2) t h a t p and q

are p o s i t i v e on C a p ) )

167

168

W.N.EVERITT AND M.GIERTZ

Thus (3.2) may be r e w r i t t e n i n t h e form

(3.3)

+ 2(pqJff') )(a).

W e now f i n d

&1

estimate f o r t h e expression ( p q ( f f ' l ) ( a )

11 N,fl)I2

i n terms of

and

11 f1I2 ; it

i s f u r t h i s reason t h a t we

have t o have a v a i l a b l e t h e condition (2.7) on p and q.

Note t h a t we cannot t a k e q ( a ) = 0 t o avoid t h i s estimate f o r t h e

l a s t term of (3.3), unless w e t a k e q(x) = 0 e s t a b l i s h e d in

( x E [a,-)).

It was

C6; Section 101 t h a t wfienevw a condition of t h e

form o f (2.6) holds then e i t h e r q i s p o s i t i v e o r i d e n t i c a l l y zero on [a,"). q(x) >

o

It i s f o r t h i s reason t h a t t h e condition (x

E [a,m))

forms p a r t of (2.2).

The main d i f f i c u l t y i n e s t a b l i s h i n g i n e q u a l i t i e s o f t h e kind considered i n t h i s paper i s t o &ow

f o r t h e e f f e c t of having

no boundary condition on t h e elements of D1 at t h e r e g u l a r end-point a.

"his d i f f i c u l t y also occurs, for example, i n consideration o f an i n e q u a l i t y of t h e form

DIFFERENTIAL EXPRESSIONS AND INTEGRAL INEQUALITIES

which is closely related to the inequality (2.12) of this paper.

169

This

last inequality is discussed in detail in C31; see in particular the rermrks in C3; Sections 2, 3 and 41. In the analysis which follows we make use of the following inequalities 2ab 5

+ (b/tl2

valid for all positive a, b, t and

On integration by IP' I

5

p a r t s , use

1/2 2 8 p q, and then

% (k =

(3.5 1

,...,n).

1,2

of the condition (2.71,

(3.5) we obtain for f

E

b.

D 1S O

(3.7)

where t is an arbitrary positive number.

170

W.N.EVERITT AND M.GIERTZ

Multiplying (3.7) by (3.8) we obtain, a f t e r applying the first i n eq u al i t y

in (3.6) with n = 3, and then t ak i n g square r o o t s on both

sides (3.9) where we have p u t

Prom t h e i d e n t i t y

'a

ve

Ja

obtain, using (3.9) and with T = t S + R/2t

After

multiplying both s i d e s by t h e f a c t o r 2 and applying t h e

i n eq u al i t y (3.5) t o the last two terms we o b t ai n , f o r any real p o s i t i v e nunbers h and k, ( [ l p 1 / 2 f f ( (- T)2 < T2 + (hR)'

+ (S/h)?

+

k211 ( p f 1 ) ' [ 1 2 + k-*11

ill

'.

Again using t h e first in e q u a li ty i n (3.6), t ak i n g square r o o t s on both s i d es , s u b s t i t u t i n g f i r s t l y T = t S + R/2t and secondly f o r R and S from (3.101,

11 p'/2f11( -< ( 2 t

+ h-l)S + (t-' + h)R + kll ( p f ' ) ' l l

= ( 4 t 2 + 2th-'

+

(t-2+ ht-'

+ k)ll (pfl)'ll +

+ k-l)II

fll

( 2 t + h-l)K(I (pq)'''f'II

+ (t-' + h)KII q1/2flL

(3.13)

DIFFERENTIAL EXPRESSIONS AND INTEGRAL INEQUALITIES

I n t h i s l a s t result t , k and h are, thus far, a r b i t r a r y p o s i t i v e numbers.

We now choose t

&

(0.1) and put h

-' =

k = tli2

I

and estimate t h e l a s t term i n (3.13) by, using again (3.51,

Since s E (0,l) it follows from (3.13) t h a t

and then, with

= s3 and d e f i n i n g C1 = max{'l, 3K1

C2

= 3 + 2K,

we o b t a i n

where 2 2 5 C3 = 4C2 C16C1 q ( a ) l

This l a s t i n e q u a l i t y (3.14) i s v a l i d f o r a l l

E

.

satisfying

s (say)

171

172

W.N.EVERITT A N D M.GIERTZ

and f o r all real-valued f

E

D1,O.

Returning now t o (3.11) we obtain, on using (3.14) and then (3.5) f o r all real-valued f

again,

E

D

1 .O

where t h e positive number A i s defined by

on taking i n t o account t h e upperbound f o r

E

given i n (3.15).

Returning now t o (3.31, with rl chosen as i n the statement of Theorem 1, we s u b s t i t u t e t h e e s t i m t e (3.17) t o obtain t h e required inequality (2.9) but valid f o r all real-valued f

&

D1,O.

To complete

t h e proof of Theorem 1 it only remains t o r e c a l l the extension t o complex-valued f

and then t o invoke t h e closure argument given at 190 the start of t h i s section t o extend t o the maximal domain D1. E D

A detailed e d n a t i o n of the above analysis shows t h a t t h e

order of the term

reduced.

E-~,

for small

L,

i s best possible and cannot be

DIFFERENTIAL EXPRESSIONS AND INTEGRAL INEQUALITIES

4.

In t h i s section w e give t h e proof of Theorem 2. W e outline the proof of the Theorem only and omit t h e d e t a i l s which

determine an estimate f o r the positive number B, since t h i s follows the pattern of t h e analysis i n section 3.

The proof of (2.11) follows from known r e s u l t s f o r t h e d i f f e r e n t i a l expression M; see, f o r example, C2; Section 51.

To prove (2.12) we start with (3.14) but now extended by t h e closure argument t o t h e maximal domain D1; t h i s extension follows from the argument given a t t h e start of section 3 above.

From (2.9) and (3.14) it follows t h a t we may e s t a b l i s h a r e s u l t of t h e form

valid for a l l

L E

(O,l), where B

1

i s positive and depends only on the

coefficients p and q.

Also we have f o r any positive k and f E D,

Turning again t o (2.9), using only the t h i r d term on t h e left-hand side we obtain an inequality of t h e form, putting

E

1 =,rlr

Combining (4.2) and (4.3) we obtain, on chousing k t o be small,

v a l i d f o r all

L E

(O,l), where B is positive and depends only on t h e 2

coefficients p and q.

17 3

174

W.N.EVERITT AND M.GIERTZ

Taken together (4.1) and (4.4) give t h e required i n e q u a l i t y (2.12) with B

=

B1 + B2.

This completes t h e proof of Theorem 2. References 1.

N. I. Akhiezer and I. M. GlRzman, Theory of l i n e a r o p e r a t o r s i n (Ungar, New York, 1961; t r a n s l a t e d from t h e Russian

f i l b e r t space edition). 2.

W. N. E v e r i t t , 'On t h e limit-point c l a s s i f i c a t i o n of second-order d i f f e r e n t i a l operators', J. London Math. SOC. 41 (1966), 531-534.

3.

W. N. E v e r i t t , 'On an extension t o an i n t e g r o - d i f f e r e n t i a l i n e q u a l i t y

of Hardy, Littlewood and Polya', Proc. Royal SOC. Edinburuh (A)

69 ( 1971/72 1, 295-333. h.

W. N. E v e r i t t and M G i e r t z , 'Some p r o p e r t i e s of t h e domains of c e r t a i n d i f f e r e n t i a l operators', FToc. London Math. SOC. ( 3 ) (19711, 301-24.

5.

W. N. E v e r i t t and M. Giertz, 'On some p r o p e r t i e s o f t h e domains of

powers of certain d i f f e r e n t i a l o p e r a t o r s ' , Proc. London Msth. SOC. ( 3 ) 24 (1972), 756-768.

6.

W. N. h v e r i t t and M. Giertz, 'Some i n e q u a l i t i e s a s s o c i a t e d with c e r t a i n ordinary d i f f e r e n t i a l o p e r a t o r s ' , Math. Zeit. 126 (1972), 308-326.

7. 8.

S. Goldberg, Unbounded l i n e a r o p e r a t o r s (HcGraw-Hill

New York, 1966).

T. Kato, Perturbation theory f o r l i n e a r opcrators (Springer-Verlag, Heidelberg, 1966).

9.

M. A. Naimark, Linear d i f f e r e n t i a l operators; P a r t I1 (Ungar, New York, 1968; t r a n s l a t e d from t h e Russian editiczi)

W N Everitt Department of Mathematics The University Dundee, Scotland

.

M Giertz Department of Mathematics The Royal Institute o f Technology Stockholm, Sweden

W . Eckhaus ( e d . ) , New Developments i n D i f f e r e n t i a l Equations @ North-Holland Publishing Company (1976)

LEGENDRE’S POLYNOMIALS

ON

by Ake P l e i j e l .

Su = DpDu

With

1.

Su = A T u

>

p(x)

+

qu,

D = id/&,

i s considered on

I = {x: a

0 , 0 < r(x)

_<

Tu

<

x

ru,

=

< b}

C q ( x ) with a constant

t h e equation

. It

is assumed t h a t

,

and observed t h a t

C

t h e s e conditions a r e f u l f i l l e d f o r Legendre‘s d i f f e r e n t i a l equation. 2 p(x) = 1 - x

This equation i s obtained when

,

q(x)

r(x) = 1

=

,

= -1 , b = +1 . Green’s o r Lagrange‘s formula t e l l s that B J ( s u . V - u . G ) d ~ = o u t - i n t e g r a t e d p a r t . I f Su=r;, S v = r G , J = [a,p],

a

0.

( u , v ) ~= l r u ; J J where

dx, Green’s formula reads

J

B

relation

E(I)

u,;

T q = pDu*v

QT = qT -.:q

satisfy

+

of ordered p a i r s

6

Su =

-.

u * p Dv

U

=

and belong t o

AIIIT

i s t h e subspace of 2

r Iu 1 d x

which

<

A(I)

(u,;),

J

where t h e f u n c t i o n s

A C : ~ ~ ( I ) = A(I)

i . e . have l o -

I.

containing only those

By t h i s condition f o r

00.

T ) = Q ,

It i s v a l i d on t h e l i n e a r

c a l l y a b s o l u t e l y continuous f i r s t order . d e r i v a t i v e s on

2.

T

i - l ( ( : , v ) ~ - (u,;) J J

u

for

and t h e same for

u

i,

I t h e l i n e a r space

E[IlT

QT( U , V ) = qb(U,V) T

-

IT T T Q , q B , q, when

i s c u t out from E ( 1 ) .

For

in

U,V

ELIIT,

where t h e terms a r e t h e f i n i t e l i m i t s of qa(U,V), T J-tI, B+b,

,+a.

J

3.

A(;),

E(?) a r e defined as A ( I ) ,

placed by t h e union

?

f < q . The i n t e r v a l

q

For

= c+ 0.

T = qg(U,V)

-

Jo =

*

U,V E E ( 1 )

T qa(U,V).

conditions that

of

I

only that

{x: a < x _ < 5 )

and

I

i s re-

Ib={x: V s x < b ) ,

{x: h < x < 9 ) reduces t o c i f

, a

E Ia ,

h

T

= c - 0,

0 E Ib one d e f i n e s Q (U,V) J

From E(?) t h e space E e l

( u , u ) ~< m, I- JO

=

E(I)

(c,i)T I- Jo 175

<

a. On

=

is c u t out by t h e El?]

t h e hermitean

176

A.PLEIJEL

form

-

QT = q :

I m e space

4.

qz

Q T , q: and I i s a subspace of E

Su

=

XTu

on

a r e defined a s l i m i t s .

defines t h e subspace

[1IT

EX

of

p o s i t i v e d e f i n i t e on

fIIT.

E X -

.

T c(X) Q

Since

~ w -( u , L I ) <

Ex(I). The request

Eh(I)

Green’s formula shows t h a t

(u,;),

determine t h e elements

I

hu , of t h e 2-dimensional space

<2.

q:

/:?IT.

EIIJT

S o l u t i o n s of

i=

exists.

Clearly

,

0

_< dimlEh

(1lT5 is

c(X) = i - ’ ( L - x ) ,

ctx) = -c(X)

t h e same form i s

negative d e f i n i t e on

5.

According t o t h e preceding r e s u l t s t h e form

QT

i s non-degenerate

I T T It can be on t h e d i r e c t sum ( A non-real) RX = EX[I] 4 \[I]T. T proved that i f U belongs t o E[?] but is not 1 : R X , then QT is degenerate on t h e l i n e a r h u l l l e d maximal r e g u l a r f o r

T {U ,RX }

. Because of ‘ t h i s , R,”

$IT. I n R . Every

QT i n E

I other maximal r e g u l a r subspaces

I

i s cal-

general t h e r e e x i s t a l s o in

U

has a unique

T T Q - p r o j e c t i o n U’ on a r e g u l a r R s o that Q (U- U ’ , R ) = 0 . I f R I T I T is maximal r e g u l a r it can be seen that Q ( U , V ) = Q (U’,V’) under I I T Q - p r o j e c t i o n s on R . m e S y l v e s t e r s i g n a t u r e ( p , n ) of Q~ on R I i s t h e same f o r a l l maximal r e g u l a r show that

p

=

(0,O)_< ( p , n )

6.

I The r e s u l t s of S e c t i o n

dim EL [I]T, n = dim E- [IJT when

5

(p,n)

E~[I)

or

T

= (0)

(a-p)

= E ( I ) for

non-real. For s o l u t i o n s of

integral ly

0)

in the

I

2

(p,n) = ( 2 , 2 )

Su

I

Clearly

occurs when

X. m e l i m i t - c i r c l e

i . e . when = hTu

on

r l u l dx i s co ( b u t f o r t h e case u I A-p case, and i s f i n i t e i n t h e a - c case.

( u , u ) ~= I

occurs when

case over

f o r a l l non-real

I

h

>0,

4,5

(2,2).

( l - c ) case over

x

Im(h)

x

By d e f i n i t i o n , t h e l i m i t - p o i n t

(o,o) =

R.

Eh[I]

T

=

I , the

identical-

177

ON LEGENDRE’S POLYNOMIALS

T

Because of

Q (U,V)

=

I

I t a k e s place i f f

El [I,]’

7.

T

Q I

and

T

Q (U’,V’), S e c t i o n

’on

El

= 0

[IblT

El

a r e defined a s

I

5

dim El [1JT5 2

.

Also

CIIT

and a t

E Ia

1

c a l l y f u l f i l l e d . According t o Weyl’s t h e o r y

1

c a s e over

EL?]’.

t h e i n t e g r a b i l i t y conditions a t

similarly

5, t h e 1 - p

T

7

dim

4, b u t

i n Section

p E Ib a r e automati-

5

_<

dim El [Ib]’

3 (?IT = dim EX [I]T

2

,

and

-k

2

.

The l a s t e q u a l i t y has been g e n e r a l i z e d by Kodaira and is t h e r e f o r e r e f e r r e d t o a s Xodaira’s i d e n t i t y .

T QT-nullspace i f Q ( Z , Z ) = O . I I E [I]T, Green’s formula is v a l i d and i m p l i e s t h e

8.

A l i n e a r subspace

For

U,V

in

n

is a

of

.

symmetry

r-’S

Z

Z

to

T . T ( i , v ) = (u,v) It can be proved that t h e r e s t r i c t i o n of I I Z n E(IJT has a unique maximal symmetric H i l b e r t space ex-

tension provided t h e projection maximal r e g u l a r

R

of

ZI;

has t h e maximal dimension

a c c o r d i n g t o S y l v e s t e r ’ s i n e r t i a theorem

R

since

QT(U’,V’)

i n t o one ( t h e n i n t o a l l )

Z

=

I

T Q (U,V)) I

.

(2;

dim Z;

is a

&‘-nullspace

9.

.l-p

case s i n c e

p=n

in

I i s t h e n a symmetric boundary con-

Z

d i t i o n . The maximal symmetric e x t e n s i o n is s e l f a d J o i n t i n the

min(p,n)

=

i n the . b - c

and

i n these cases.

P a r a l l e l t o t h e previous T - p o s i t i v e t h e o r y t h e r e i s a n S - p o s i t i v e

t h e o r y based upon

(U,V)‘

A Green’; E ( 1 ) with

formula S

J p Dun:

determined by t h e c o n d i t i o n s fined similarly a s

(u,v)

J

1 i- (({,v)‘-

q (U,V) =

( p u ’ ~ ’ + q u ~ ) d x i n s t e a d of

=

J

EL?]’.

T

J (u,;)’)

=

S

[q ( U , V ) ]

=

+ 6.pDv. S

(u,u) I On E[?IS

Qs

EIIIS C E ( 1 )

The space

< co, (A,;)‘

< co.

I t h e formula

El[1IS,

Ej;[IIS

El;]‘

S Q = q: T

t r u e i n which t h e expressions a r e l i m i t s as i n S e c t i o n t h e dimensions o f s o l u t i o n s p a c e s

is v a l i d on

J

J

J -

.

the

-

is

i s deS

qa h o l d s

3. I n terms of t-P

and l-c

178

A.PLEIJEL

c a s e s over

Qs

I

a r e defined. Symmetric boundary c o n d i t i o n s r e f e r t o

T

Q

i n s t e a d of

r

I

.

It i s assumed that

10.

Su

=

p o s i t i v e t h e o r y , and i n t h e

ATu

I-p

is i n the I - c

case i n t h e

ease i n t h e S-positive theory, a s

r ( x ) 5 C q(x) ,

t h e s i t u a t i o n i s f o r Legendre's equation. Because of the inequality E

(u,u)

C(U,U)'

c1

S = S+ o T ,

h o l d s t r u e and shows that

J

[;as

Z= E

To s e e t h a t

Hence,

_<

J

[?IS C E /::IT.

11.

T

a>

is a

T

Q -nullspace i n E

I It follows t h a t

0.

i s p o s i t i v e . The 1 - p

(U,U)'~

T-

[?IT, S

(U,V)'~ =

(u,v)

J

J

J 1-p

positive theory implies t h e theory. For, Su = A T u

I

c a s e over

can be w r i t t e n

in the

SOU= ( A +

.

+ a ( u , v )T

J in the

I

c a s e over

consider

S-

a S -positive

a)Tu, and A , h + a

a r e simultaneously n o n - r e a l . From

(u,u)

a

then V

c1

=

U

T

J

(u,

=

5

C(u,u)'

;+au)

belongs t o

( v , ; + o x ) E E[?ISa,

a qs (U",V')

=

it follows that i f E[?]".

I f also

=

(u,;)

E E$]"

* S V=(V,;)EE[I]

,

a simple computation shows t h a t O

T

qs(V,V)

U

J

+ a q (U,V),

and

Qs (U",V") J

=

T

Q S ( U , V ) + a Q (U,V).

J

J

According t o t h e g e n e r a l i z a t i o n t o a n S - p o s i t i v e t h e o r y o f t h e s t a t e ment a t t h e end of S e c t i o n 6 it f o l l o w s , when

T Q (u,v) I 12.

=

o

on

To show that

z

=

E [ $ ] ' .

Z

=

El?]'

J

R which i s maximal r e g u l a r

(QT)

I Su

ximal dimension. Since positive theory

(p,nj

=

I , that

i s a symmetric boundary c o n d i t i o n i n

t h e T-positive t h e o r y i t remains t o prove t h a t one

tends t o

=

in

E

i n Section

Z i

[?IT,

XTu i s i n t h e I - c

8, f o r

has t h e r i g h t ma-

case i n t h e

T-

( 2 , 2 ) , and t h e r i g h t dimension must be

2 .

According t o Weyl's t h e o r y a s presented i n Titchmarsh's textbook, t h e

179

ON LEGENDRE'S POLYNOMIALS

4-dimensional space

Z' 3 Z n R R

=

R = E L I4 I I T = EX(;)

.* *S E II]a n E X ( ? ) = EX [I] @

.

is m a x i m a l regular. Clearly

The l a s t space i s 2-dimensional

a c c o r d i n g t o Kodaira's i d e n t i t y i n t h e theory, see Section Hence, t h e ory

13.

2

=

*S

E[I]

c a s e of t h e S-positive'

a-p

7. This s u f f i c e s t o conclude t h a t

dim

= 2 .

i s a symmetric boundary c o n d i t i o n i n t h e T - p o s i t i v e

.

An element

Z

satisfies

U = (u,;)

=

E[;]'

i f f the integrals

S

* . S ( u , ~ ), ( u , u ) a r e f i n i t e when extended over i n t e r v a l s

Ia and

Ib.

According t o a w e l l h o w n theorem, t h e spectrum i s d i s c r e t e i n t h e a-c

case. A s o l u t i o n of

u = pu,

and

I

Su = p ' h

determines a n element

( P l U ' J 2 + qIu12)dx

a

and

t i o n t h e H i l b e r t space of t h e t h e o r y i s

, J

with

Z is satisfied i f f the integral

is f i n i t e over neighbourhoods of

Su = p Tu

U

satisfies

Z

=

'S

E[I]

For t h e Legendre equa-

b.

2 L (-l,l),

and U , ( u , p u ) ,

when

( 1 -x2) lu,12 dx

is f i n i t e over neighbourhoods of

-1

and

1

.

This c o n d i t i o n is

known (Akhiezer-Glazman). T t has h e r e been deduced from t h e S - p o s i t i v e t h e o r y of Legendre's equation.

_-A more d e t a i l e d account of t h e s u b j e c t t r e a t e d above w i l l be

published i n Annales Academiae Scientiarum Fennicae, i n t h e volume d e d i c a t e d t o R o l f Nevanlinna, under t h e t i t l e "On t h e boundary c o n d i t i o n f o r t h e Legendre polynomials". Uppsala U n i v e r s i t y , Sweden.

180

Akhiezer,

A.PLEIJEL

N.I., and Glazman, I . M . ,

Theory o f Linear Operators i n

Hilbert Space, Volume 11, Frederick Ungar IZtblishing Co, New York

1963, 218 p. Titchmarsh, E.C.,

Eigenfunction Expansions Associated with Second-

order D i f f e r e n t i a l Equations. Part I, Oxford University Press 1962, 203 p.

W. Eckhaus ( e d . ) , New Developments i n D i f f e r e n t i a l E q u a t i o n s @ N o r t h - H o l l a n d P u b l i s h i n g Company (1976)

INVARIANTS AND CANONICAL FORMS FOR MEROMORPHIC SECOND ORDER DIFFERENTIAL EQUATIONS

W.Jurkat,' Syracuse University D.A.LutzJ2 University of Wisconsin-Milwaukee A.Peyerimhoff, UniversitBt Ulm ABSTRACT Given a differential equation xtt+a(z)xt+b(z)x=O , where a(z) and b(z) are analytic functions in a neighborhood of c. (generally an irregular singular point of the solutions), a complete system of invariants is compute& which characterizes the solutions at m to within linear combinations of meromorphic functions. A tklassicaltl differential equation having these same invariants can be named and a transformation from the given equation to this classical one can be contructed. Thus the solutions of the given equation can be effectively calculated in terms of certain classical functions and computable meromorphic functions. STATEMENT OF THE PROBLEM We consider (scalar) differential equations of the form (1)

XI!+ a(z)x'+

,

b(z)x = 0

m

m

b(z) = % bVz-"

where a(z) = Z aVz-'

converge for I z J large. The point at m is "generally" an irregular singular point of the solutions and the differential equation is said to have PoincarC! rank r-1 at m

.

A linear transformation (2)

y = tl(z)x

+

t2(Z)X1

J

where tl(z) and t2(z) are assumed to be at least meromorphic in a neighborhood of takes (1) into a second order linear differential equation for y whose coefficients are meromorphic at m providd the matrix T(z), defined by

1 2

Supported in part by a grant from the National Science Foundation. Supported in part by grants from the National Science Foundation and the Alexander von Humboldt-Stiftung.

181

JURKAT-LUTZ-PEYERIMHOFF

182

T(z)

=

tl

ha not identically vanishing determinant in a neighborhood Moreover, the transformation*canbe reverted to virld x=ml(z)y +m2 (z)y' , which takes the diffmrential-equation for y back into (1). We are concerned with the question of what simplification can be made in the differential equation through the use of such transformations. G.D.Birkhoff had the idea of isolating the singularity atm by transforming the differential equation into a new one which has only two singular points. Only one singular point would be, of course, not generally possible since that would imply that all solutions would be single-valued in a neighborhood of m Birkhoff treated, however, only a tlmaintl case and his statements are not correct without the introduction of some exceptional cases. In order to identify a certain exceptional case which arises, we say that the solution space of (1) totally splits if thp general solution of (1) can be expressed as a linear combination of solutions of analytic linear homogeneous first order differential eauations.

.

STATEMENT OF THE RESULTS By completing arguments of Birkhoff [l] and Turrittin [ 6 ] , we have the following Theorem. (Birkhoff, Turrittin, Jurkat-Lutz-Pryerimhoff) b ( z ) are Let xtl+a(z)xt+b(z)x=O , where a(z) anal. i c x m and assume that the solution space does not total1 s l i g Then there exists a meromo hic transfo ation x *lyt which takes the different21 em-~klassicaltt differential euuation of the form

=&my

n

A

Note that the differential equation ( 3 ) has only two singular points, 0 and -, and 0 is a resular sinctular point in the sense of Frobenius-Fuchs. The differential equation is called llclassicalll because it can be solved explicitly by classical functions, e . g . , Kummer functions, Barnes functions, and certain elementary functions. This theorem may be viewed as a qualitative statement in that it describes the solutions of the original differential equation as a meromorphic linear combination of certain classical functions. We are mainly interested in guantitative versions of this theorem, namely, we wish to identify which of these classical functions are associated with a given differential equation and how the solution of the original differential equation may be expressed in terms of them. To do this, we must determine the coefficients in thp classical equation ( 3 ) . It turns out that they are almost unique in a sense which will be described below. In order to make this

INVARIANTS AND CANONICAL FORMS

183

discussion precise, it is proper to introduce at this point the concept of invariants, that is, quantities which are uniquely associated with the differential equation and which remain unchanged with respect to certain classes of linear transformations of the differential equation. A comvlete svstem of invariants characterizes the differential equation up to linear transformations from the specified class and the calculation of such a complete system leads to the determination of the coefficients in (3) as well as a linear transformation which reduces (1) to a classical equation. In discussing systems of invariants, it is convenient to specialize the class of transformations under consideration so that the invariants become numbers instead of equivalence classes. Invariants corresponding to more general classes of transformations are then built from these. If tl(z) and t2(z) are analytic at [tl(-)l'-

m

and if

a, tl(-) t~(-)+b~[t2(-)]~# 0, i.e.

,

det T ( - ) # 0

then y=t~(z)x +t2(z)x1 is called an a a ic ra sformation of the differential equation. Such a transfoGa:gn t-another second order linear differential equation with analytic Coefficients at If, in particular, tl(a)=l and tZ(m)=O, then y=t,(z)x+ t2(z)x1 is called a strict transformation. It is for the class of strict transformations that the invariants appear in their simplest form and such invariants are called Birkhoff invariants in honor of G.D.Birkhoff who introduced quantities related to them [l] in connection with the problem we discuss here. Invariants corresponding to the classes of analytic and meromorphic transformations are called, simply, analvtic and meromorphic invariants, respectively.

-.

We now come to the problem of calculating a complete system of invariants and in order to do this we will work with the formal solutions of the differential equation. In computing formal solutions, there are two naturally disjoint cases to consider,depending upon whether a$ #4bo or not, i.e., whether the characteristic polynomial A 2 + a o h + b o has distinct roots or equal roots. Since a. and bo are meromorphic invariants, this distinction is an invariant concept.

.

Let the distinct roots of X 2 + a o A + b o be denoted hl,X2 Then there exist two linearly independent formal solutions of the

are calculated recursively in a well-known manner. The series fi(z) generally diverge everywhere and the meaning usually attached to the formal solutions is that in sectors of sufficiently small angle they asmptotically represent actual solutions as z + - . In this discussion we are not concerned with that interpretation. If a; = 4bo, then A2 + aoA + bo has equal roots X 1 = h2 = A ='v2. In this case the quantity 2a = aoal 2bl is a meromorphic invariant.

-

JURKAT-LUTZ-PEYERIMHOFF

184

If a-0, this leads to convergent formal solutions and the problem of constructing a meromorphic transformation which takes (1) into a classical differential equation is a purely formal problem which is algebraic in nature and which we will not further consider here. (This case of a = O is, however. treated in detail in [ 3 ] , Section 6.1

Case 11:

(a: =4b0 Define

u

, aoal f2b1)

= 2

aoa 1

(

.

- bl)~

112

by selecting a branch of the square root. Then there exist two linearly independent formal solutions of the form f(t )th'ewt+At2

and

f(-t)th'e-wt+At2

=c"

where t 2 = z , A ' =-a'/2 , f(t) fnt-" cients fn are calculated recursively.

, fo

-

1, and the coeffi-

ts are defined with respect to formal transpormal inv tl(z) and t2(z) are just asformations y = t z- x +et 2 rz ) x l , i.e., sumed to be formal series. The calculation of formal invariants is an algebraic problem. A co lete system of formal Birkhoff invariants in Case I consists ofyAl,A2,X:,h:) and in Case I1 coneists of ( A , u , A 1 , 6 ) , where 6 is related to fl and is calculated in terms of ao,al,bo,bl, and b2. Beyond these formal invariants, we now seek invariants which will complete the system of (actual They are related in a transcendental manner to the coefficients of the differential equation. The information containing these invariants comes from the asymptotic behavior of the coefficients in the formal series of the formal solutions in the following wayr In Case I, we have

The numbers y1,y2 are Birkhoff invariants and complete the system, i.e., (hl,h2,h5,A:1y1,y21 is a complete system of Birkhoff invariants. These invariants are free, subject only to the condition h l f A 2 . The invariants y 1 and y 2 are related to the convergencedivergence pro erties of the formal series in that y 1 = O iff fl(z) converges for z l large and y2 = O iff f2(z) converges for J z I large. Moreover, tota splitting of the differential equation occurs iff = y 2 =O. A complete system of analytic invariants is given by ~ ~ 1 1 h ~ , A ~ , h ~ , in y l ycase 2 ) y1y2 # O and if y1y2 = O , we must include the knowledge of which one (s) is zero. (See Theorem V and Section 7, [21.)

P

In Case 11, lim n-r

fn(2w)"/r(n)

= Y

where y is a Birkhoff invariant and completes the system, i.e., ( A l = h 2 , w , h 1 , 6 , y ) forms a complete system of Birkhoff invariants.

INVARIANTS AND CANONICAL FORMS

185

Moreover, y = O iff the formal series f(t) converges for It1 large. A complete system of analytic invariants in this case is given by (A~=A~,I.L,A',Y). (See Theorem 11 and Section 7, [ 3 ] . ) In Case I when y1y2 f 0 (this corresponds to Birkhoff's "mainftcase [l]) and in Case II, it turns out that the differential equation (1) can be transfzrmed gglvtica & into a classical equation (3). The parameters ao, bo and are in these cases equal t-o ao, al, bo and bl, respectively and it only remains to determine b2. The parameter g2 is not generally equal to b2, but is influenced by the invariants y1y2, resp. y and is calculated from them as f01lows.

i

al,

We introduce an auxiliary parameter d which satisfies

-

cos[2rd "(A: respectively,

+A:)]

cos[2rd+nal] = ry

= cos r ( A :

- A:) - 2w2yly2

in Case I, y1y2 f0,

, (2)

in Case I1

and these determine d up to sign and modulo one. Then b^, is given by d2t(al-l)d +C2= 0

.

Every solution d in the above equations is permitted and with each admissible d and corresponding value for g2, the resulting classical differential equations are related by explicit analytic transformations, in fact, they are polynomials in 2 - l . This gives rise to a set of functional equations which relate the solutions of the equivalent classical differential equations and these functional equations are theAonly such ones possible. These discrete changes in the parameter be which do not change the invariants is what was meant when we said earlier that the parameters in the classical differential equations are Ilalmost unique". In the cases other than the ones treated above, i.e., y 1 y 2 = 0 or a = 0 , it is sometimes necessary to utilize meromorphic transformations in order to bring about the reduction of the original differential equation to a classical equation. The introduction of the invariants allows us to keep track of the deviations from the normal cases treated above. These cases are treated in detail in [2] and [ 3 ] . In Case I when y 1 = y 2 = 0 and A:$A: (mod l), then (1) is not meromorphically equivalent to a classical differential equation of the form ( 3 ) , but is equivalent to a second order differential equation with two regular singular points in the finite complex plane and the original singular point at O.

.

Once the invariants have been calculated and a classical differential equation (3) having these same invariants has been named, the construction of an analytic or meromorphic transformation from the given differential equation to the classical differential equation can be accomplished using the formal series in the corresponding formal solutions. This procedure is more natural to discuss for two-dimensional differential systems than for second order differential equations. See Theorems V, VI of [2] and Theorems 11, V of [ 3 ] for a statement of these results. The transformation is given as a certain quotient of formal series and the theory tells u8 that the quotient must, in fact, converge in a

JURKAT- LUTZ- PEYWIMHOFF

186

.

neighborhood of 0 Hence the calculation of a transformation (2) is purely a formal matter. We conclude with an application of the Theorem in Case I1 to see how single-valued functions appear in the actual solutions, even though the formal solutions carry power series in ~ - 1 ' ~

.

Assume now that aE+4b0 and a,al#2b,. Then according to our discussion above, the given differential equation (1) is analytically equivalent to a classical differential equation of the form (3). If we let Y we obtain (4)

?It

-

i ,

=xPr-(:0/2)21

+k9,

+

2

and letting z = t2,

(-^a&

22

+

5)9

q(t) =i(t2) , we

3

0

obtain

Note that in (5) the coeffkcient of ;j- is an even function of t , while the coefficient of dy/dt is odd. This feature likewise holds if we would have applied these same transformations to the general differential equation. We now make two normalizations to further simplify (5). Letting s = ot we obtain

and letting

?=spu

, we

Obtain

Hence we mav select o and P to obtain the normalized differential

which is solved explicitly by the (linearly independent) functions

in case 2c is not an integer. Here oFl(atx) denotes the Barnes function

If 2c is an integer, modifications must be made in these functions to obtain two linearly independent solutions. We do not include that discuseion here since the procedure is well-known for these classical differential equations. (See, for example, [ 51. )

INVARIANTS AND CANONICAL FORMS

187

Hence we obtain (for 2c #integer) two linearly independent solutions for the classical differential equation (3) of the form

We remark that oFl(* :z) is an entire function of order 1/2 and this explains how the singularity of the solutions is built in this case of equal roots of the characteristic polynomial.

[I] ~.D.Birkhoff, ~n a simple t e of irregular singular point, Trans.Amer.Math.Soc.

u 8 9 1 3 ) , 462-476.

[2] W.Jurkat, D.Lutz, and A.Peyerimhoff, Birkhoff invariants and effective calculations for meromorphic linear differential equations, I, J.Math.Anal.App1. (to appear). [3] W.Jurkat, D.Lutz, and A.Peyerimhoff, Birkhoff invariants and effective calculations for meromorphic linear differential equations, 11, Houston 3.Math. (to appear). [4] W.Jurkat, D.Lutz, and A.Peyerimhoff, Effective solutions for meromorphic second order differential equations, Symposium on Ordinary Diff.dqus., Lect. Notes in Math.#312, SpringerVerlag New York, 1973, 100-107. [ 5 ] W.Magnus

and F.Oberhettinger, Fomulas and Theorems for the Special Functions of Mathematical physics, Chelsea, New York, 1949.

[6] H.L.Turrittin, Reduction of ordinary differential equations to the Birkhoff canonical form, Trans. A.M.S. =(1963), 485507.

This Page Intentionally Left Blank

W . Eckhaus ( e d . ) , New Developments i n Differential Equations

@ North-Holland Publishing Company (1976)

ON GENERALIZED EIGENFUNCTIONS AND LINEAR TRANSPORT THEORY

C.G. Lekkerkerker, Mathematisch Instituut, Universiteit van Amsterdam, The Netherlands

ABSTRACT The main purpose of this paper is to formulate and prove a theorem on generalized eigenfunctions. This theorem can be applied in the theory of the linear transport equation. Particular attention will be given to the case that the parameter c occurring in that equation is equal to 1 . 1. INTRODUCTION

The equation meant in the abstract reads 1

(1)

u

+

$(x,1.1) =

/S(x,u')dp'

2 -1

+

g(x,U).

Here, J, and g are real functions of two real variablos x and 1.1 ranging over some interval J in IR and the segment I=[-l,ll, respectively. Furthermore, c is a positive constant. Equation ( 1 ) can be written in a more concise form, as follows. Consider the Hilbert space L2(I). Define two operators A and T by putting (Af)(u) = f(p)

-

1

/f(v')du' -1

(Tf)(U) = uf(u)

(f€L2(I)

, UCI).

Then, if J, and g are conceived as functions of the single variable x, with values in L2(I), and differentation of such a function with respect to x is defined by taking limits in L2(I), ( 1 ) takes the form (1')

T

2

= -A$

+ g

or also

= -T-'A$

+ T-lg.

[By modifying this procedure, more precisely by admitting only functions J, and g with values in the space Lip(1)

of HGlder continuous functions on I, one is

led to classical solutions of ( 1 ) . See [ 5 , chapter 811. Equation ( 1 ) combined with various types of boundary conditions was treated amongst others by Case (see [ 2 1 ) . Also, more general instances of the linear transport equation were investigated (see 1 9 1 ) . A functional analytic treatment of (l),

in the case O < c < 1, was given by Hangelbroek C51. The case

c = 1 was dealt with in "73. In the present paper we bring a general theorem on eigenfunction expansions and show how it can be applied in the discussion of ( 1 ) .

C.G.LEKKERKERKER

190

This means that we restore the rdle of the eigenfunctions of T ' - l A the use of which was suppressed in 153 and [TI. The basis for our considerations is the spectral theorem. It will be applied in the following form. Let B be a bounded self-adjoint operator in a Hilbert spdce H, with spectrum N. Suppose that the spectrum is simple, iLe., there exists a cyclic element eo

E

H for the operator B. Then there exist a

finite positive regular Bore1 measure u on N and a unitary map F from H onto the Hilbert space L2(N,0) such that Feo = l N (the function on N that is identically equal to 1 ) (i)

F diagonalizes B, i.e., FBF-l is the multiplication in

(ii)

L2(N,u) with the function h(u) = v . If we wish to apply this theorem to a particular problem, e.g., a boundary value problem connected with ( l ) , we do this by performing the transformation F associated with a suitably chosen operator B. Then our task consists of two parts, viz. a) to determine F explicitly b) to solve the transformed problem. This program will be carried out in sections 3 and 4. Some details will be left out; for them the reader is referred to C51 and [TI. 2.

GENERALIZED EIGENFUNCTIONS Let H,B,N.e

0

,IJ and

F be as in the introduction. There is a close connec-

tion between F and the eigenfunctions of B. Results describing this connection were given by Berezanskii 113, Foiav C41 and Maurin c 8 , chapter 21. However, these theorems are of no help to us, since in our case the conditions are not fulfilled. [These conditions are that a certain SubSpRCe 8 of H having the property that the dual space contains the eigenfunctions of B is either a nuclear space or the embedding of 0 in H is nuclear]. Here, we choose a different approach in which we presuppose the existence of eigenfunctions. We shall not enter upon the general question when this assumption is fulfilled. Let L be a locally convex space fulfilling the following conditions ( p is the set of all polynomials) lo.

L is a subspace of H which is densely and continuously embedded in H

2'.

the set {p(B)eO: p

E

F'1 is sequentially dense in L, with

respect to the topology of L 3'.

B maps L continuously into itself.

Let us consider the anti-dual space L', i.e. the space of all continuous conjugate linear functionals on L (we prefer to consider the anti-dual rather than

EIGENFUNCTIONS AND LINEAR TRANSPORT THEORY

191

the dual space). The space H can be embedded in L' in a very natural way; on account of lo,

(2)

such an embedding is furnished by the map f

H

uf, where uf(g) =

Furthermore, the operator B can be extended to L' by putting

(f,g)(gtL).

(*EL' , grL).

(B$,g) = (J,,Bg)

Here, we have used round brackets in indicating the values of a functional on L. For each J, of .'3

E

L' , the functional BJI defined by (2) again belongs to L' , on account

We now state one further condition, viz. 4'.

for each v

E

N, L' contains an element J,

such that B$, = v$,

(eigenfunction of B)

and (Jlv,eo)= 1.

We observe that the normalization condition ($",e ) = 1 is not a real restriction. 0 For, if ($",eo) = 0, then also ($",Beo) = (BJlv,eo)= ~ ( $ ~ , e = ~0; ) more generally, (JIv,p(B)e ) = 0 for all p 0

E

P , and so JI, = 0.

Theorem 1. Suppose that the conditions stated above are fulfilled. Then, on the subspace L. the transformation F is given by (3)

(Ff)(v) = ( $ " , f )

(v

E

N).

Proof. The right-hand member of (3) is well defined for each u

let it be denoted by ? ( v ) . On account o f 4'

-

E N and each f and ( 2 ) , (qv,Bf) = v($",f).

Thus, since v is a real number, Bf(v) = v P ( v ) . We also have e o ( v ) = 1 Thus the map f

H

?

(v

E

E

L;

N).

satisfies the properties (i) and (ii) of the transformation F.

It f o l l o w s that it coincides with F, at least on the set of elements f = p(B)eO. We observe that F p(B)eO = p. Now let f such that pn(B)eO

E

+

L be arbitrary. By

2O,

f in the space L, as n

functional on L, it follows that ($v.pn(B)e,) pn(v)

+

?(v)

as n

+

a,

there is a sequence of polynomials pn +

-. Since qV is a continuous -+

f o r each v

On the other hand, it is also true that p (B)eo Therefore, pn = Fpn(B)eo

+

(J,

+

,f) as n E

+

-. Thus

N.

finthe coarser topology o f H.

Ff in the space L2(N,u), as n

-+

m,

Then there is a

subsequence p tending point-wise to Ff a.e.. It follows that, a s elements of "k L2(N,U), Ff and f coincide. Remark. The spectral theorem and theorem 1 remain true if we allow B to be an unbounded operator, but retain the other conditions, in particular condition.'3 Consider the case that H = L2(-m,-) and that B is the operator (l/i)d/dx. Take L to be the Schwartz space S. Then the conditions of theorem 1 are fulfilled, with some choice of eo; apart from numerical factors, the normalized eigenfunctions of B are the locally integrable functions $ (x) = eiEX (<,x E IR). 5 We remark that it has some advantage to take e (x) = e-x2h 0

, to

choose

192

C.G.LEKKERKERKER

F i n such a way

t h a t (Fe ) ( v ) = e-"

t h a t ( J I , e ) = e5 0

".

JI,(x) = (2n)-'eiSx.

and t o n o r m a l i z e t h e J15 by r e q u i r i n g

0

Then t h e c o n c l u s i o n o f theorem 1 a g a i n h o l d s t r u e , whereas A t any r a t e , t h e r e s u l t i s t h a t t h e t r a n s f o r m a t i o n F d i a -

g o n a l i z i n g B i s t h e F o u r i e r t r a n s f o r m a t i o n . T h i s i s a f i r s t and v e r y s i m p l e a p p l i c a t i o n o f theorem 1 ; it s e r v e s t o s t r e s s t h e analogy between t h e F o u r i e r t r a n s f o r m a t i o n and t h e t r a n s f o r m a t i o n F a p p e a r i n g i n o u r d i s c u s s i o n o f t h e l i n e a r transport equation.

3. THE OPERATOR TiA.

DETERMINATION OF F

F i r s t l e t O < c < 1 . Then A i s i n v e r t i b l e , and so we may c o n s i d e r t h e o p e r a t o r A-lT a s w e l l . This o p e r a t o r i s bounded. It i s s e l f - a d j o i n t p r o v i d e d t h e space L 2 ( I ) ( h e n c e f o r t h d e n o t e d by L'(1.A))

i s endowed w i t h t h e i n n e r p r o d u c t

( f , g ) A = ( f , A g ) . We s a y a few words on how t o d e t e r m i n e t h e spectrum N o f A-lT (see 151). ! L e t E d e n o t e t h e i d e n t i t y o p e r a t o r i n L 2 ( I , A ) and l e t C b e d e f i n e d by

Cf =

1

1

f(u')du'.lI.

(A-~T

-

A simple c a l c u l a t i o n y i e l d s XE)(T

-

A E ) - ~= E

+

CT(T

-

(x d

~ ~ 1 - 1

SP(T) = I ) .

The o p e r a t o r on t h e r i g h t d i f f e r s from t h e i d e n t i t y by a n o p e r a t o r of r a n k 1 . We can i n t r o d u c e t h e d e t e r m i n a n t of t h i s o p e r a t o r ; it i s c a l l e d t h e p e r t u r b a t i o n d e t e r m i n a n t of t h e p a i r A'lT, A(A) = 1

(4)

-

T. We d e n o t e t h i s d e t e r m i n a n t by ( l - c ) - l

:I--&

A (A).

Then

1

c

+

kf*

dA = 1 t 2 -1 u-1

(A

c

I).

-1

We a l s o d e f i n e A(v) = 1

(5)

+

y&'"

P-V

(-1 < v < I ) ,

where we have t a k e n t h e Cauchy p r i n c i p a l v a l u e of t h e i n t e g r a l . Using f u n c t i o n t h e o r e t i c arguments one can p r o v e t h a t t h e f u n c t i o n A ( & ) h a s t w o s i m p l e z e r o s

-+ uo,

w i t h vo > 1 , and no o t h e r z e r o s i n t h e extended complex p l a n e c u t a l o n g t h e

segment I . It i s t h e n e a s y t o show t h a t

(6)

N = I u {zeros of A ( x ) )

= I u I+ vo}.

The spectrum N is s i m p l e ; a s a c y c l i c element f o r t h e o p e r a t o r A - l T , eo

one may t a k e

= (I-C)~~. By t h e f o r e g o i n g , t h e s p e c t r a l theorem may b e a p p l i e d t o A-lT c o n s i d e r e d

a s an o p e r a t o r in L2(I,A). T h i s g i v e s rise t o a u n i t a r y map F from L 2 ( I , A ) o n t o some space L 2 ( N , 0 ) such t h a t F d i a g o n a l i z e s A-lT,

w h i l e Feo = 1"

The c o n d i t i o n s

o f theorem 1 are f u l f i l l e d i f one t a k e s B = A-lT and L = L i p ( 1 ) ( c o n f e r chapter

63.

Here, t h e space L i p ( 1 )

i s d e f i n e d a s f o l l o w s . For 0 < a

5

C5,

1, let

L i p a ( I ) denote t h e l i n e a r s p a c e o f f u n c t i o n s f on I t h a t a r e ( u n i f o r m l y ) H6lder c o n t i n u o u s w i t h exponent a . I t i s a Banach s p a c e i n t h e norm

193

EIGENFUNCTIONS AND LINEAR TRANSPORT THEORY

We p u t L i p ( 1 ) = u o < a s l L i p a ( I ) and g i v e t h i s s p a c e t h e i n d u c t i v e l i m i t topo1og-y [observe t h a t L ip a ( I )

Lip6(i) if a

3

61.

We have t o t a k e some c a r e i f we want t o e x t e n d B = A-lT

t o t h e space

L i p ( I ) ' , s i n c e A-lT i s s e l f - a d j o i n t i n L 2 ( I , A ) , b u t n o t i n L 2 ( I ) . For any $ L i p ( I ) ' , l e t u s d e f i n e a new f u n c t i o n a l

($,flA =

(f

($,Af)

( $ , a )

E

by p u t t i n g

A

Lip(1)).

E

Then t h e e x t e n s i o n o f A-lT t o L i p ( 1 ) '

i s g i v e n by

= ($,A-lTf)A.

(A-'T$,f),

[We would have o b t a i n e d t h e same e x t e n s i o n i f we had t a k e n t h e second a d j o i n t where (A-lT)* i s t h e a d j o i n t o f AdT

((A-lT)*)',

i n L 2 ( i ) and t h e second a d j o i n t

, )I.

i s taken with r e sp e c t t o (

Applying theorem 1 we f i n d t h a t , on t h e subspace L i p ( i ) , t h e t r a n s f o r m a t i o n F i s g i v e n by ( F f ) ( v ) = ?(")

,

($u,Bf)A = v($v,f)A

($v,eO)A = 1

(u

N).

E

t h e n we g e t ? ( v ) = ( @ , , f ) , where

If we d e f i n e @ vby (@",f) = ( $ v , f ) A = ( $ " , A f ) ,

t h e $,

satisfies the relations

= ( $ , f ) * , where $

are d e t e r m i n e d by

(7)

(@,,Bf)

,

= v(@v,f)

From ( 7 ) we deduce ( $ , ( T - v ) f ) =

4" = aSv

- 52 -k , where p-v

-

2

(gv,eo) = 1 . ( T f 1 ) . T h i s r e l a t i o n i s s a t i s f i e d by ' I

a i s a constant, 6

denotes t h e delta-function centered

&y

a t t h e p o i n t v and stands f o r t h e functronal with values f(u)dp. u -v -1 P-V Taking i n t o account t h e second r e l a t i o n ( 7 ) we g e t a = X ( v ) . The r e s u l t o b t a i n e d may b e w r i t t e n a s ?(v) = ($",f) =

(8)

T h i s i s a n expansion of For v =

2

?

<X(v)Gv

- c2 u

p-v

,f>

(u

E

N;u#+

1).

= Ff i n t e r m s of f and t h e e i g e n f u n c t i o n s o f B = A-lT.

vo, X(v) = h ( v ) = 0 .

Conversely, one may e x p r e s s f i n

? by

c o n s i d e r i n g t h e o p e r a t o r FTF-l

in

L2(N,o); t h i s o p e r a t o r i s d i a g o n a l i z e d by F-l and s e r v e s as t h e analogue of A-lT. A s i n 151 one f i n d s

(9)

f(u) =

(h(u)6,,

+

$ A,?),

(?

E

Lip"),

-1

< 1).

where t h e s u b s c r i p t a means t h a t t h e v a l u e s a r e t a k e n w i t h r e s p e c t t o t h e measure a . Thus, more e x p l i c i t l y ,

(9')

f(u)

=

~(u)?(u)

+

$

194

C.G.LEKKERKERKFR

The measure a is absolutely continuous with respect to Lebesgue measure on I, and its Radon-Nikodym derivative on I

i5

Halder continuous. [To compute u it is

necessary to write ( 8 ) and ( 9 ’ ) as contour integrals]. Another observation is that F maps Lip(1) one-to-one onto Lip(N).

Next, let c = 1. Then the operator A is no longer invertible. So we work with the unbounded operator T-lA. It ha5 spectrum 1-l u { O ] , where we have written I-’ =

(-m,-ll

u [l,-)

(observe that l / v o

+

0 as c t 1).

There are two

difficulties which are closely related to each other, viz. T-lA has a double eigenvalue 0

(i) (ii)

T-lA no longer is hermitian.

To overcome these difficulties we proceed as follows (see [ T I ) . The space L2(I) (or L2(I,A))

can be decomposed as a direct sum G @ G1 such that G and G, are 0 0

invariant under T-lA and that

,

u(T-’AIG~=1-l

a(T-lAIG,) = { O } .

It is even possible to give an explicit formula for the projection onto G

1

along Go (see C4, p.1783 or 16, section VII.33). One finds (lo)

{Gl

= the set of linear functions

G~ = {f

E

L*(I)

:

m,(f) = m2(f) = 01,

where mi(f) = LLif(p)du. It may be observed that Go and G1 are not orthogonal and that G1 has dimension 2, in accordance with the fact that 0 is an eigenvalue of multiplicity 2. On G1, the operator T-lA behaves badly, in the sense that it

is not hermitian; in fact, with respect to the basis {lI,dl, where d denotes the function d(p) = p , it is represented by the matrix(::).

On Go, T-lA is invertible,

since 0 does not belong to the spectrum. We put Bo = (T-’AlG,)

-1

; this is the

analogue of the operator A-lT considered in the case 0 < c < 1. Concerning Bo the following can be said. First of all, B

0

is hermitian with respect to the inner product (

, )A.

Next, there is a cyclic element for Bo. It is obtained as follows. An elementary calculation yields

Therefore, B sends polynomials into polynomials, the degree of a polynomial 0 increasing by 1 under the map Bo. Now the coefficients of the polynomials in Go satisfy two independent homogeneous linear relations, on account of (10). So Go contains a polynomial do of degree 2, and by applying repeatedly Bo to this polynomial and taking linear combinations one gets all polynomials in G 0’

195

EIGENFUNCTIONS AND LINEAR TRANSPORT THEORY

From this fact one may conclude that do is a cyclic element. One has do(u) =

(12) Of course, d

0

-

3u2

+

9/5.

is only determined up to a nonzero constant factor. We have chosen

this factor in such a way that ( $ ,d ) = 1 , where the 0, are the eigenfunctions

v o

of Bo. These eigenfunctions are obtained from the functions 0

considered earlier

by simply substituting c = 1. Finally, Bo has spectrum I. The conditions of theorem 1 are fulfilled with H =

Go,

B = B

0’

L = Lip(1) n Go. Hence, there exists a (unique) unitary map F from G onto some 0 space L2(I,o) such that F diagonalizes Bo, while Fdo = ll. On the subspace Lip(1) the transformation F is given by

n G o , the anti-dual of which contains the $”,

4. APPLICATIONS We indicate very briefly how the foregoing results can be applied to solve boundary value problems connected with the differential equation (1).

We

restrict ourselves to the discussion of the homogeneous equation for which g = 0. Moreover, we take c = 1 . For simplicity, we suppose that 0 d J. Let us recall that an arbitrary h c t i o n f E L2(I) can be decomposed as

(14)

f(u) = a + bu + fo(u)

, with

f

The function f satisfies the relations m,(f) 0

l u d p = l p 3 d y = 0 and (15)

LT

a = -3 m (f) 2 2

p2du =

,

-, 2 3

0

E

Go.

= %(f)

= 0. Hence, since

the coefficients a and b in (14) are given by

b = $ml(f).

We further recall that T-’A acts on G1 as the matrix(::)

and that Bo = (T-’AlGO)-’

is diagonalized by F. Thus the homogeneous equation ( 1 1 , or rather the equation d$(x)dx = -T-’A+ is equivalent with the system

-

The relations (16)imply that m,($(x)) linear function of x, and $,(x)

is a constant, m2($(x))

is a

Tv

has the form $,(x)(v)

= e - x / u w . Now suppose

that one has imposed boundary and/or growth conditions on the function J, in such a way that $ ( O ) is determined by these conditions. Then ml($(0)),

m,($(O))

and

196

C.G.LEKKERKERKER

v)

are known. Then also m , ( J , ( x ) ) , m2($(x),)

and

are known. Applying

F-l we get $ o ( x ) . Thus *(x) is determined. In particular, let J = CO,-). Write I+ = [o,I~,I-= [-1,01.

Then $,(x),

does not vanish on I-, whereas and so Jl(x), is exponentially increasing if 0 $,(x) is exponentially decreasing in the opposite case. In realistic problems it is asked whether one can solve th6 differential equation ( 1 ) under conditions of the following type hj

a) $,(o) E L'(I+,o) b) $(O)lI+ is a prescribed function in L2(I+) (half-range boundary condition). The answer to this question is given in the so-called half-range theory (see r71). It turns out that there is a unique solution J, if one imposes one extra condition of the form m,(Jl(x))= a, where a is a given complex number. We give some details. Put G

P

= F-'L2(I+,~) and denote by P+ the projection in L 2 ( I ) that

projects onto L2(I+) along L'(1-j.

One can prove that P+ maps Gp one-to-one onto

a subspace H+,O of H+ = L 2 ( I + ) of codimension ,,and one can derive an explicit formula for P = (P+IGp)-'. The subspace H+,O does not contain the constants # 0. Now, by a ) , $ ( O ) must have the form a + bd + g , where a and b are constants, d

is the function d(v) = v and g So if we require that m l ( J , ( O ) ) For this value b =

5

2 Gp. Since G is a subspace of G o , m , ( J , ( O ) ) = ~ P has a given value a, this fixes the constant b. E

a, there exists exactly one number a such that the function

f+ = ( J i ( 0 ) -a -bd)lI+

belongs to H+,O. Having found a and f+ we obtain $(O) in

the form $(O) = a + bd + Pf+. The Milne problem concerning the density of light emanating from a star is the particular case J , ( O ) I I + = 0. In [ T I , a l s o the inhomogeneous equation (1)

is considered. In

[?I,

the case 0 < c < 1 is treated.

b

.

E I G E N F U N C T I O N S AND LINEAR TRANSPORT THEORY

197

REFERENCES

111

BEREZANSKII, JU.M., Expansions in eigenfunctions of selfadjoint operators, TranslationsMath. Monographs XVII (1968).

121

CASE, K.M. - ZWFJFEL, P.F., Linear Transport Theory, Addison-Wesley Publ. Co, Reading 1967.

[31

DUNFORD, N. - SCHWARTZ, J.T., Linear Operators I, IntersciPnrp Publ., New York 1958.

chi

FOIAg, C., Dicompositions en op6rateurs et vecteurs propres I et 11, Rev. Romaine Math. Pures et Appl. 7 , 241-282 et 571-602 (1962).

r51

HANGELBROEK, R.J., A functional analytic approach to the linear transport equation, Thesis, Groningen 1973.

161

KATO, T., Perturbation theory for linear operators, Springer, Berlin 1967.

171

LEKKERKERKER, C.G., The linear transport equation. The degenerate case c = 1. I. F u l l range theory; 11. Half-range theory. To appear in Proc. Edinburgh Math. Soc

.

181

MAURIN, K., General eigenfunction expansions and unitary representations of topological groups, Warszawa 1968.

C9l

~c CORMTCK, N.J. I., Singular eigenfinction expansions in neutron transport theory, Advances in Nuclear Science and Technology 7, 181-282 (1973).

- KUGER,

'98 This Page Intentionally Left Blank

W . Eckhaus (ed.), New Developments in Differential Equations 1 3

North-Holland Publishing Company (1976)

INTEGRAL-ORDINARY DIFFERENTIAL-BOUNDARY SUBSPACES AND SPECTRAL THEORY Aalt Dijksma 1.

Introduction. In recent years a number of papers have appeared that deal with a class of linear integral-ordinary differential-boundary (eigenvalue) problems. See A.M. Krall's paper [ 9 ] where a long list of references can be 'found. In vector notation such problems are associated with formal expressions and side conditions of the form (Sy)(x)

(1.1)

=

P,(x)y'(x) + P0(x)y(x) b + K(x) F(t) y(t)dt a

+

H(x)[MIy(a)

+

L(x)c,

=

0.

+

NIy(b)l

+

b

(1.2)

My(a)

+ Ny(b)

+

J

Gft) y(t)dt

a Here y is a vector-valued function arid PI,Fo,H,K,F,L and G are matrix-valued functions on a compact interval [a,b] r: iR, Y , N I , !1 and N are constant I matrices and c is an arbitrary parameter vector ( c f . [ I ( ? ! ) . Some of these papers are concerned with finding a suitable formulation of the problem adjoint to the given one. Such formulations should deal with formal expressions and side conditions similar to ( 1 . 1 ) and ( 1 . 2 ) except that in ( 1 . 1 ) P D + Po,is to I be replaced by -DP * + P *, D = d/dx. The adjoint is then used to determine I the conditions under whiFh the original problem is selfadjoint and to calculate the eigenvalues. Restricting ourselves to problems in the Hilbertspace H = L 2 ( i ) of vectorvalued functions on I = (a,b) c R , we shall discuss a method of classifying the operators associated with such problems along with their adjoints. and eigenfunction expansion results for the case that the operators are selfadjoint. In H operators associated with side conditions of the form ( 1 . 2 ) are usually not densely defined and therefore the adjoints of such operators do not exist as operators. This is the reason why we shall llse the notion of subspaces (= closed linear manifelds) in the Hilbertspace ?f = H 0 H, which is a generalization of the notion of closed operators by replacing them by their graphs in H 2 . The subspace ideas have their beginning in a paper of R. Arens 1 1 1 and have been further developed by E.A. Coddington [ Z ] , [ 3 1 , [ 4 1 . The report that follows is on work done in collaboration with Prof. Coddington. Part of this work has been done in the academic year 1 9 7 3 - 7 4 during which I had the opportunity to work at the University of California at L o s Angeles. This is an appropriate place to acknowledge my gratitude to the Department of Mathematics of UCLA and Prof. Coddington for providing this opportunity.

2. Subspaces. Let H be a Hilbertspace over the complex numbers 4, and H2 = H UI H, considered as a Hilbertspace. The domain D ( T ) and ranee R(T) of a subspace T in H 2 are defined by

E H

{f,g} E T, some g E HI,

Ig E H

{f,g} E T, some f E HI.

D ( T ) = {f

R(T)

=

For f E D(T) we let T(f) = { g E H

{f,gl E TI. 199

200

A . DI JKSNA

The s u b s p a c e T i s an o p e r a t o r i f T(0) = { O } and t h e n we w r i t e T ( f ) = T f . We c o n s i d e r s u b s p a c e s a s l i n e a r r e l a t i o n s w i t h

I

n T = {{f,crgl E H 2 T + S T-'

H21

=

{{f,g+hl E

=

{ { g , f l E h2

I

If,g}E T

1, a E

4,

{f,g} E T, {f,h} E S}, { f , g } E T}.

The a l g e b r a i c sum i s d e f i n e d by T

S

{{f+h,g+kl E

=

H21 { f , g } E

T, {h,k} E S}.

I t i s c a l l e d d i r e c t i f T ll S.= [{O,O}l and o r t h o g o n a l i f T I S in H2. I n t h e l a s t c a s e we w r i t e T + S = T d S . I f T C S ' t h e n t h e o r t h o g o n a l complement of T i n S i s d e n o t e d by S e T. I f S = H2 we w r i t e H2 63 T = T I .

In

H2

x

H2 w e d e f i n e t h e s e m i - b i l i n e a r form

< {f,g}

, {h,kf

> =

,

(g,h) - ( f , k )

<

, > by

{f,g}. th,kl

E H2.

It i s n o n - d e g e n e r a t e . For s u b s e t s A , B of H 2 , < A , B > = 0 means t h a t < { f , g } , { h , k } > = O f o r a l l { f , g } E A and a l l I h , k } E B. The a d j o i n t of a s u b s p a c e T i s t h e s u * s p a c e T* d e f i n e d by T*

It i s e a s y t o s e e t h a t T** = T , S c T i m p l i e s t h a t T* c S*, e t c . Two s u b s p a c e s T > = 0 , i . e . , T I c T2* o r T2 C T T I , T 2 a r e s a i d t o be f o r m a l l y a d ' o i n t i f < T I*. A s u b s p a c e S i s c a l l e d symmetric i f < S , S > = b ' o r ; e q u i v a l e n t l y , S c S * . A s e l f a d j o i n t s u b s p a c e H i s one which s a t i s f i e s H = H*. We have t h e f o l l o w i n g r e s u l t (cf. [21, I 7 1 ) . Theorem 2 . 1 . K S

b e a symmetric s u b s p a c e i n H 2 and l e t

MS(L) = { { h , k } E S* Then ( i ) dim M S ( L )

$'

=

( i i ) S*

=

i s constant f o r L E

E $ (Im S

G

M S (9.)

L

i MS

I

E

4.

4-, where

4'

and f o r L E

4'

( d i r e c t sum),

01,

(a),

L E

( i i i ) there e x i s t selfadjoint extensions H H & H2 which s a t i s f y S c H = H* c S* L E

, .P

k = Lh}

of S & H 2 ,

i . e . , subspaces i f and only i f dim M S ( L ) = dim MS(L)

4+,

( i v ) t h e r e a l w a y s e x i s t H i l b e r t s p a c e s R c o n t a i n i n g H a s a s u b s p a c e and RL s u c h t h a t S c H . s e l f a d j o i n t subspaces H For any s u b s p a c e T i n H2 we may w r i t e T = Ts d T_, where T_ = { { f , g } E T

I

f=o

1,

T,

=

T

e

T_

The s u b s p a c e T , i s c a l l e d t h e o p e r a t o r a r t o f T. It is a c l o s e d o p e r a t o r i n dense i n T * ( E ) i and R(Ts) c T ( 0 ) l . H with D ( T ~ = ) D(T) The f o l l o w i n g theorem i s d u e t o R . Arens [ I ] .

INTEGRAL-ORDINARY DIFFERENTIAL-BOUNDARY SUBSPACES

Theorem 2.2. If H = H d Hm is a selfadjoint subspace in H 2 , H is a dens" defingd selfadjoint operator in H ( 0 ) l .

201

then

Let L be a system of n first (a,b) c IR:

3 . The basic linear ordinary differential operators.

order ordinary differential expressions on L = P D 1

where P. is an det P 0,x the edtries of Otherwise L is

(a)*

+

I =

Po , D = d/dx,

nxn matrix whose entries belong to CJ(i), j = O,I, and E 1.The system L is called regular if 7 = [a,b] is compact, P. belong to C J ( i ) , j = O , I , and det P,(x) 0 , x E 7. called singular. The Lagrange adjoint of L is defined by

*

+

L

-DPI* + P

=

0

*

= 0 D 1

+ Q

0'

where 0 = -PI*, Q = P * - (P I ) * . Thus if L is regular then so is L+ and L+ = IL. The sys?em Lois caljed formally symmetric if L = L+. +

Let H = L 2 (I), the Hilbertspace of nxl matrix-valued functions on with innerproduct b g*f , f,E E L2 ( 1 ) . (f,g) = a

I

I

In H 2 we define the minimal operators T

and T

+

associated with L and L+ by

To = {{f,Lf}l f E C~(I)]~, To+ = {{f+,L+f+ll f+

E C;(I)}',

where c denotes the closure in H2. It is easy t o see that = 0, i.e., that T and T are formally adjoint. Their adjoints T 2ndoT+ are called tge maxigal operators and are given by +

T

=

(T~+)* = {{f,Lf} If

T+ = (To)*

=

E m

AC

{{f+,L+f+} If+€

loc

Hn

(I), Lf E H},

ACloc (I), L+f+E H 1

By Green's formula the semi-bilinear form on H2 x H2 restricted to T deals with the behavior of the functions f E D ( T ) , f+ E D ( T ' ) at the end points of I: x+b + + + Pl(x) f(x)l x-ta ( 3 . 1 ) <{f,Lf) , {f ,L f }> = (f+(x))* =

lim(f+(x))* x+b

~,(x) f(x)

-

lim (f+(x))* x+a

. x

T+

~,(x) f(x).

If L is regular then the functions f E D (T), f+ E O(T+) can be continuously extended to all of 7 and ( 3 . 1 ) reads (3.2)

<{f,Lf}

,

{f+,L+f+}> = (f+(b))*

Pl(b) f(b)

- (f+(a))*

Suppose that the subspaces S and its adjoint S* (3.3)

T c S

and

T

+

Pl(a) f(a).

in H 2 satisfy

c S*.

Then S c T, S* c T+, S and S* are operators and it can be shown that there exist finite dimensional subspaces C and C+ of T and T+ such that

2n2

A .DIJKSMA

S = T n(C+)* and S* = T+

n c*.

It f o l l o w s from ( 3 . 1 ) t h a t S i s a r e s t r i c t i o n of T and S* is a r e s t r i c t i o n of T+ by means of a f i n i t e number of c o n d i t i o n s a t t h e two e n d p o i n t s of I . C o n v e r s e l y , i f S i s a r e s t r i c t i o n of T by a f i n i t e number of s u c h two p o i n t boundary c o n d i t i o n s , t h e n ( 3 . 3 ) h o l d s . F o r example, l e t L be r e g u l a r . I f S = T t h e n w e have u s i n g ( 3 . 2 )

S = {{f,Lf)ET and i f S = T

I

f(a)

=

f f b ) = 01,

t h e n we h a v e

+ +

+

S*= { I f .L f ) ET'

1

f + ( a ) = f + ( b ) = 0) = To+.

4 . I n t e g r a l - o r d i n a r y d i f f e r e n t i a l - b o u n d a r y subspaces. I n o r d e r t o o b t a i n subspaces w i t h s i d e c o n d i t i o n s o t h e r t h a n two point-boundary c o n d i t i o n s w e make T and T + s m a l l e r . We i n t r o d u c e two f i n i t e d i m e n s i o n a l s u b s p a c e s B, B+ i n H2 agd d e f i n e t h e f o r m a l l y a d j o i n t o p e r a t o r s So = Ton (B+)*

So

and

+

= To

+

I l B*.

I t can be shown t h a t t h e a d j o i n t s of So , So+ a r e g i v e n by S

*

T+

=

4

B'and

(So+)* = T

G B.

Without l o s s of g e n e r a l i t y w e may and d o assume t h a t t h e s e a l g e b r a i c sums a r e d i r e c t . We want t o c h a r a c t e r i z e a l l s u b s p a c e s S and t h e i r a d j o i n t s S* t h a t satisfy So c S

, dim

S

I3 S o = d ,

I f ( 4 . 1 ) h o l d s t h e n i t c a n b e shown t h a t we must have t h a t d + d + = dim T 0 To

+ dim

and t h a t t h e r e e x i s t s u b s p a c e s C , C+ of T dim C+ = d+ s u c h t h a t S = (T

B)n(C+)*

4B

B + dim B+

and T+

and S* = (T+

4

,

B+ w i t h dim C = d ,

B+) nC*.

The s u b s p a c e s C and C+ r e p r e s e n t t h e s i d e c o n d i t i o n s which now a r e a m i x t u r e of two p o i n t boundary and i n t e g r a l c o n d i t i o n s . We want t o make t h e s e c o n d i t i o n s more e x p l i c i t . I n o r d e r t o do so we s h a l l make u s e of t h e f o l l o w i n g n o t a t i o n s . To i n d i c a t e t h a t a m a t r i x A h a s p rows and q columns ve w r i t e A ( p x q ) . Thus f E H i s o f t h e form f ( n x I ) . The p x q z e r o m a t r i x and t h e and I By p ( n x p) E H , p x p i d e n t i t y m a t r i x a r e d e n o t e d by 0 6 ( n X p) E D(T) i s l i n e a r l y i n d e p e n d e g t mod Dp(T ) o r 'm(n x p) i s a b a s i s f o r S(0) w e mean t h a t t h e p columns of t h e s e m a t r y c e s have t h e s e p r o p e r t i e s . For F(n x p ) , G(n x q) E H we d e f i n e t h e " m a t r i x i n n e r p r o d u c t " (F,G) t o be t h e 4 x p m a t r i x whose i , j - t h element i s b n

'

.

203

INTEGRAL-ORDINARY DIFFERENTIAL-BOUNDARY SUBSPACES

If C(p

r), D ( q

x

x

s ) are constant matrices, then

,

(FC,G) = (F,G)C If I S , L6} ( n

p) E T , ( 6 + , L+6+1 (n

x

+ + +

< { S , L ~ I, ( 6 ,L 6

(F,GD) = D* (F,G).

E T+ then + +

x

q)

=

(LS,~+)-(~,L6

)

is well defined. We denote by (F:G) the matrix whose columns are obtained by placing the columns of G next to those of 5 in the order indicated. We shall also make use of the following definitions:

and

s 1 = T~ n (B,+)* , s I += T ~ +n B]* It is not difficult to see that S

0

c S

1

BI =(So+)*

c(S + ) * = T 1

=

T iB

4

1

((So+)*),

(direct sums),

+.

and that a similar result-holds starting with S Yoreover, it is easy + are operators, Do ( S I ) is dense in H, to see that B , S I * = T+ dim ( S +)*(O)l= dim B2, e:cfllThe following result is a special case of a theo?em that will be stated and proved elsewhere. Theorem 4 . 1 . (4.2)

(4.3)

S , S* satisfy ( 4 . 1 ) and suppose that

I (hp :

1

(n

2to)

2tn+)

: ' 0 (

(n

x x

( s l + s2))

+

+

(sl + s2 ))

is a basis for

S(O),

is a basis for

S*(O),

i s a basis for

(So+)*(0),

is a basis for

So*(0).

Then there exist u

= (y, ~y

I E T ,a

v (4.5)

=

V+ =

( 6 , L6

+ + +

( 6 ,L 6

,8

1 E T

1 E T+ , 6'

and a constant matrix E ( s 2 (4.6)

E B]

= { l a , 1,

}

= {'a,

} E

,

(n

x

s2+),

, (n

x

(d-sl

+ + +

(4.4)

x

2~

= {20+,2T+]

Bl

E B','

,

+

s 2 ) such that

,

6

+ 2u

is linearly independent mod D ( S 1 ) ,

I

6'

+

is linearly independent mod D (S,'),

20+

(n

+

(d

-

- s,+)) , s1

+

-

s2)),

204

A . DIJKSMA

and if (4.8)

(4.9)

then (4.10)

h E D (T),

S = {{h + a, Lh + T + ID]

(h + a, (h + a,<+)

-

B I , (0 E ( S o + ) * ( 0 ) with

Oil+,

'(2)=

<{h + a,Lh +

{ u , T ~E

T),

+ +

v +B

1 C . O =l(oc+

'?[(h

where c( s

x

I

(4.11)

+ a, Y ' )

I) E

S* = iih+ + o+,L+h+ +

4 T+

'1

+ $1

-

'

> = 01+ d - s +-S

<{h + a, Lh

+

+T),

2

+

u +a

>I,

is arbitrary),

1

h+E D(T+),{o+,T+)

E Bl+,m+ E So*(0)with

Conversely if the bases in ( 4 . 3 ) and the matrix E ( S 2 x s 2 + ) are given, the elements of ( 4 . 4 ) and ( 4 . 5 ) exist satisf in ( 4 . 6 ) & ( 4 . 7 ) & I , I , + 5 and 5' are defined by ( 4 . 8 ) and ( 4 . 9 ) , &fined by ( 4 . 1 0 ) satisfies ( 4 . l l ) , and ( 4 . 1 ) & ( 4 . 2 )

If L is regular then in Theorem 4.1 we may replace the elements mentioned in ( 4 . 4 ) , ( 4 . 5 ) by constant matrices and fixed bases of BI and B I + , and express ( 4 . 6 ) , ( 4 . 7 ) , ( 4 . 8 ) and ( 4 . 9 ) in terms of these matrices and bases. The subspaces S, S* are then determined by these matrices, the values of the functions h E D(TJ,+ L h. h E D(T+) at the endpoints of I and certain integrals involvinp h, Lh, h'and Example 4 . 2 . If L i s regular and B I = B + = { { O , O } } then S in ( 4 . 1 0 ) of Theorem 4.1 involves an expression and side coniitions of the form ( I . I ) , ( 1 . 2 ) and so does S* in ( 4 . 1 1 ) . The parameter c occuring in ( 1 . 1 ) is the same parameter c that occurs in the description ( 4 . 1 0 ) of S. It's presence stipulates the subspace (non-operator) character of S . Example 4 . 3 . (Interface conditions). Let L be arbitrary and let c E I be fixed. Let B be spanned by { a l , ~ l }(n x n) where 0' = T' = 0 on (a,c), a1 E C1[c,bf, ol(c) = -P-' (c). a 1 vanishes in a neighborhood o € b 2nd T' = Lo1 on [c,b). Similarly,llet B+ be spanned by { o + l , ~ + l l (n x n) where a+' = T + ~= 0 on (a,c), d' E C 1 [c,b), u+'(c) = (P *(c))-', u+' vanishes in a neighboFhood of b = $.+a+' on [c,b). Then TilB1= TtflB+ = B2 = B2+ = {{O,O>j and hence and B 1 = B, B I = B+. Let S and S* satisfy ( 4 . 1 ) with d+d+ = dim T 0 T + 2n. Then S and S* are operators and by Theorem 4 . 1 can be described as follgws.

205

INTEGRAL-ORDINARY DIFFERENTIAL-BOUNDARY SUBSPACES

S = {{f,Lf)

I

(6+(x))*

f E H fl ACloc (I\{c)), P,(x) f(x)

S*= {{f+,L+f+l

I

f+ E H

P,(c) f(c-0)

n ACloc (idc}), L+f+

x+b + 6*(c)P1*(c) Pl*(x) f+ (x) /x+a

6*(x)

-

f(c+o) = 0'+1, d

PI(c) + (D+)*]

[(S+(c))*

Lf E H , and

x+b /x+a +(s+(c))*

E H and

f+(c-o)-[(6*(c))Pl*(c)-D*lf+(c+o)

= 0;).

+ + +

Here {6,L6} E T, 16 , L 6 ) E T+ and the constant matrices D(n are such that {6,L6)+{01,~llD 20'

,

+ + +

x

d), D+(n

{6 , L 6 ) + {o+1,,+1}D+satisfy(4.6)with

X

d+)

20=01D,

D+ and

=

which is (4.7) of Theorem 4.1. Selfadjoint integral-ordinary differential-boundary subspaces. In this section we assume that L is formally symmetric and B = B+. Then So = So+ is a symmetric operator and we want to characterize the selfadjoint extensions H of S in H 2 . As to the existence of such extensions we remark that it can be pro8ed that (k) =

dim M,

(P.)

+

dim B, P. E

$'

0

Hence as a consequence of Theorem 2.1 (iii) we have the following result (cf. t 5 1 ) . Theorem 5.1. There exist selfadjoint extensions H & H2 of S = T CI B* H where T is the symmetric minimal operator associated with Lo= &"'L i f d %ly if To has selfadjoint extensions in H'. We now also assume that dim M

(P.)=

dim M

(e)= d, say, k E $+.

Using Theorem 4.1 we obtain the following characterization of the selfadjoint extension H of S in H2 which is Theorem ( 3 . 3 ) of [ 5 ] applied to differential operators.

H satisfy

Theorem 5.2.

(5.1)

S cH

=

H* c S o * ,

and suppose that

(5.2) (5.3)

('CO

"0

(n

x

s,)

:2cp)

(n

x

(s,

is a basis for H ( O ) ,

+ s2))

is a basis for S *(O).

Then there exist (5.4)

u = {Y,LY} E T,

a = {'o,~T) E B , ,

(n

x

s2),

(5.5)

v = {6,L6]

8 = { * o , * T } E B 1 , (n

x

(d-sI-s2)),

E T,

206

A. DIJKSMA

and a constant matrix E = E*

(s2 x s )

2

such that

(5.6)

6 + * u is linearly independent mod D ( S ) ,

(5.7)

I

, v+p

= 0

,

d-SI-2 d-sl- B 2

and if (5.5) (5.9)

then __ (5.10)

I

H = IIh + u , Lh + T + (9)

(h

h E D(T) +

a,

'(9)

,

{ u , ~ }E B I , (0 E So*

( 0 ) with

,

= Osl

I

(h + o , ~ )=

{h + a, Lh +

'(2c +'(?[(h

where c ( s I

+ 0,Y)

I) E

x

$

T},

v+5>

- <[h + u , Lh +

=

Od- s - s2 '

T},

u+a>],

S

1 is arbitrary).

Moreover, if the bases in ( 5 . 2 ) , ( 5 . 3 ) are orthonormal then the operator part H of H is given by s (5.11) H (h + u ) = Lh + T - 'tO(Lh + T , '(0) + S

+

'C?[(h

+ o,Y)

- <{h + a, Lh + T\,u+a>

.

Converse1 , if ( 5 . 3 ) and the constant matrix E = E*(s x s ) are iven, the 2 elements fn ( 5 . 4 ) , ( 5 . 5 ) exist satisf in ( 5 . 6 ) , ( 5 . 7 f and $,r; afe defined by ( 5 . 8 ) , ( 5 . 9 ) , then H defined by ( 5 Y I 0 7 satisfies ( 5 . 1 ) e ( 5 . 2 ) . If the basis in ( 5 . 3 ) is orthonormal then H is given by ( 5 . 1 1 . ) . Proof. Let H satisfy ( 5 . 1 ) , '-to+

= lo,

b+= %,

sl+ =

( 5 . 2 ) and let ( 5 . 3 ) hold. By Theorem 4.1 with s I , s2+ = s 2 , d+ = d, and S = H = H* = S*,

there

exist elements as in ( 4 . 4 ) , ( 4 . 5 ) satisfying ( 4 . 6 ) , ( 4 . 7 ) and a constant matrix E ( s 2 x s 2 ) such that if Y,Y+,t,,t,+ are defined by ( 4 . 8 ) , ( 4 . 9 ) then H = H* is described by ( 4 . 1 0 ) as well as by ( 4 . 1 1 ) . Thus H satisfies ( 5 . 1 0 ) . We shall show that v+5 satisfies ( 5 . 7 ) , T i n ( 4 . 8 ) can be written as in ( 5 . 8 ) for some hermitian matrix E, and 5 in ( 4 . 9 ) can be written as in ( 5 . 9 ) . The operator S, i s densely defined (and symmetric) in H and therefore there exist a w(n x ( s I + s 2 ) ) E s 1 such that

207

INTEGRAL-ORDINARY DIFFERENTIAL-BOUNDARY SUBSPACES

Then by ( 4 . 1 0 ) , ( 4 . 1 1 ) of Theorem 4.1 and the facts that S = S*, S c S * and 1 1 u+a, u++a+, v+B E S l * , we have that for all {h + a, L h + r + co} E S = H <{h + a, Lh + T +

<{h +

Lh +

0 ,

+ +

+ (01, u +a

T

to},

-

u

v+B + WA + { O , s } >

= 0

d-s

-S

1

- a + WB + {O,Y+} - { O , Y } >

2

'

= Osl,

2

and hence (5.12)

v+B + WA + { O , C j

(5.13)

u +a

+ + -

u

E H,

- a + wB + {O,Y+} -

{O,Y} E H .

From ( 5 . 1 2 ) it follows that d-s = which is (5.7). From ( 5 . 1 3 ) and the description of Y+ - Y

+ +

and from ( 4 . 8 ) it follows that

I+ - T

= 2cn

Since the elements of (5.14)

E

-1

[E*

-

E

-4

to

d-s

-S

1

2

'

in (5.10) we deduce that

+ +

+ w~,u+a>= -2m ,

I.

are linearly independent it follows that

'Q

+ +

=

+A

E*

=[E

+ +



+ +

-1

]*.

=

'tfl[F,-i ] ,

Now, Y in ( 4 . 8 ) can be written as Y = 'Y,

[E

-4

+ +

>I

+ +

where F = E - 4 c u + a , u +a -u-a> = F* by ( 5 . 1 4 ) . This proves ( 5 . 8 ) where w$ have written E instead of F. From ( 5 . 1 2 ) and ( 5 . 1 3 ) it follows that = , which shows that 5 in ( 4 . 9 ) can be written as in ( 5 . 9 ) . The orthonormality of the bases in ( 5 . 2 ) and ( 5 . 3 ) and (5.10) clearly imply ( 5 . 1 1 ) . The second part of Theorem 5.2 i s a trivial consequence of the second part of Theorem 4 . 1 . This completes the proof. Example 5 . 3 . Consider Example 4 . 3 , where we suppose that L = L+. Then P I = P * and therefore we may and do assume that B = B+. 1

Suppose also that T has selfadjoint extensions. Then all selfadjoint extensions H in H2 of S = T O n B* are operators described by 0

0

H = {If, Lf}

I

f E

H

ll

ACLoc (I\{c}),

Lf E H and

where d = 4 dim T 0 T0 +. n and { A , L6}(n x d) E T,D (n x d) are such that 6 + 0' D i s linearly independent mod D (S 1 ) = D (S ) and

A . DIJKSMA

208

x+b (x+a -D*S(c)

6*(x)Pl(x)6(x)

t

D + D* P -'(c)

6*(c)

D = Od

d '

1

Example 5 . 4 . Let L = L+ be regular. We define the symmetric operator S by

D(S )

=

b

I

I

I f E 17 ( T o )

a

dp*. f = 0, j = I , '

...,

=

If E D ( T o )

1

( I )=

n, L E

,4:

0

T.

It can be

(Tof,a)=Okll.

Thus, if B is spanned by {o,Ol dim HT

c T

kl,

where i~ are n x 1 matrix-valued functions of bounded variation on shown '(cf. [ 5 ] ) that there exist i o , O l (n x k) E H such that O(So)

0

then So = T

il

B*. Since dim If

selfadjoint extensions H of S

in H 2

(L) = TO

exist and are

0

characterized by Theorem 5.2. 6. S mmetric subspaces and generalized spectral families. This section i s a

czntinuation of section 2. Let S be a symmetric subspace in H 2 and let H = H B Ha be a selfadjoint extension o f S in R2, where R i s a Hilbertspace contasning H as a subspace (cf. Theorem Z.I(iv)). By Theorem 2.2 and the Spectral Theorem for selfadjoint operators we have that m

A d Es(X),

Hs = -m

where E = { E ( A ) I X E El is the unique suitably normalized s ectral family of przjectigns in H ( 0 ) l = R 0 H(0) for Hs. The resolvent R 0: Hs can be written as

For X E IR let the linear operators E ( h ) on R be defined by

E(h) f

=I

f E H(O)l,

Es(X)f

f E H(0).

0

Then E = [ E ( h ) I X E I R 1 is called the spectral family of projections in P for H. The selfadjoint subspace extension H of S in RT i s called minimal if the set {E(X)f I f E ff} L' H is fundamental in R . The resolvent R of H defined by H RH(L) where I For L E

is

4-

=

(H - L I ) - '

,

L E

4' ,

the identity operator on R , is an operator valued function on RH(L) is bounded on R and can be expressed in terms of E :

.4:

209

INTEGRAL-ORDINARY DIFFERENTIAL-BOUNDARY SUBSPACES

Let P be the orthogonal projection of R onto H and let R (L)f = P%(L)f

,

F (X)f

, f E H,

=

PE(X)f

f E H. L E C?, h E IR

Then R is called the generalized resolvent and F the generalized spectral family for S corresponding to H. We remark that all generalized spectral families for S can always be constructed using the above method by starting with a minimal selfadjoint extension of S. For f E H it f o l l o w s from (6.1) that

and an inversion of this equality yields for f E H

- F(u))f,f)

((“(A)

=

limn

IX

ESO

I

!J

Im(R (v+iE)f,f)dv,

where A, p are continuity points of F. For more details we refer to [ 4 ] , [5] and [7]. In the next section we shall give a method of calculating all generalized spectral families for S = T n B* as considered in section 5. 0

0

7. Eigenfunction expansions. As in section 5 we assume that L = L+ and B = B+ and hence that S o = So+ is symmetric in H 2 . We do not assume that dim M ( a ) = dim M

( i ) ,P.

E

4

.

Let H = H 8 H_ be a fixed minimal selfadjoint extension of S

in R2

and F = i F ( h ) I X E IR 1 the corresponding generalized spectral family as described in the previous section. IJe shall indicate how one can obtain a suitable expression for F which immediately leads to eigen function expansions. Let c E I be fixed. Let p = dim B and { u , T ~ (n x p) a basis for B. Let s1 (x,i) (n x n) and u(x,L) (n x p) be the unique solutions of

where k E

4.

(L-L)

S’(L)

(L-L)

U

=

(k) =

0, &O

S’(C,k) = I

n’

-

T,

U

(C,k)

=

onp.

Let s 2 ( L ) = u(k) + u and s(L) = ( s ’ ( L ) : s 2 ( L ) ) .

One can easily verify that for all h E {PRs(i)h,k

PRs(k)h

H(0)

+ P h l E So*

= =

T

0 H ( 0 ) and I! E

:4

B.

Hence there exist a unique element. {f,Lf} E’T and a unique vector a(Pxl) E such that .{PRs(L)h, We define

LPRs(P.)h

+ Ph} = {f.Lf}

+

{U,T}

a.

4‘

210

A . DI JKSMA

It can be proved that the linear map hl+ r(PR (L)h) from (H(0))i into #n+p is continuous. The Riesz Representatioh Theorgm implies that there exist Gn+,(L)), Gi(L) E ( H ( O ) l , j = l , . . , n+p, such that G(L) = (G,(L)

,...,

T(PRs(I1)

,

h E (H(0))’

h) = (h,G(L)),

¶.

E $‘

,

We define

-

t(i) = (H - to)(Es(X) - Es(0)) G(Lo),X where

E

¶.

4%

E IR,

is fixed, and put P(X)

=

T(Pt(h))

=

((Hs - Lo)t(h),

A E IR

G(Lo)),

I

Theorem 7 . 1 . The (n+p) x (n+p) matrix-valued function p 0” IR is hermitian, non decreasing and continuous from the right, and for a11 h E H, a,B E IR

(7.1)

(F(8)

-

x

8

I

F(a)) h =

s(X)dx

(h,

a

If h E C

(1)

then

(7.1)

(FfB) - F(a))

s(v)dp

(11)).

0

can be written as B

.f

h =

i,

s(v) dp ( v )

(v),

a where -

b {(u)

=

(h, s ( v ) ) =

S*

(x,v) h(x) dx.

a The matrix p is called the spectral matrix (properly normalized p is unique). It can be shown that if H has a pure point spectrum then p consists of stepfunctions o n l y . Let

f^,

be (n+p)

1 matrix-valued functions on

x

IR and define

-m ^

^

Since p is non decreasing we have (f,f))Oand Let H be the Hilbertspace defined by

i

= L*

(P)

=

t

;I 1 1

we can define

i: 1 1

<

m

II f II

^

=

^

(f,f)l.

}.

The eigenfunction expansion that follows from Theorem 7 . 1 take the following form Theorem 7 . 2 . For f E H b

f(v) =

.f

S * ( x , u ) f(x)dx

a converges in norm in H

, and m

F(m) f =

s(u) dP (u) f(u), -m

211

INTEGRAL-ORDINARY DIFFERENTIAL-BOUNDARY SUBSPACES

1.-

ml 1 .

where this inte ral conver :e in norm in H._Yoreover, (F(m)f,g) = ( f , g ) for all f,g E H , and tht map V: H H iven by Vf = f is a contraction, 1 1 Vf 1 I It is an isometry, Vf I I = 2 f E H n H ( o ) l = H o pH(&) and

f

rl

$m

The map V implies a splitting of H and V H c If:

H

=

H

@

H, @ H 2

,

VH

=

VHo

@

VH2.

Here Ho i s the maximal subspace of H on which V is an isometry, i.e.,

Ho = If E H I / / Vf Once can prove that

I/ H~

=

=

11

f 111 , H I = i f E HI V f = 0 ) and H2

H n (H(o))'=

HI = H n H(0)

=

{f E H

1

F(-)

f = fl,

if E H

1

F(m)

f = 0).

=

H B(H1

@ H2).

.

If D ( S ) is dense in H or if H is an operator then H = H But th&e exist examples of subspaces S and extension Ho for which H and H are I 2 non trivial. Theorem 7.3. We have VH = H if and on1 if F is the spectral family for a selfadjoint subspace extensyon of S in H2 ftself. 0 -

The proofs of the Theorem stated here can be found in [51. References (including references to related work). 1.

R. Arens, Operational calculus of linear relations, Pacific J. Math., 1 1

2.

E.A. Coddington, Extension theory of formally normal and symmetric subspaces, Mem. h e r . Math.Soc. No. 134 (1973). E.A. Coddington, Selfadjoint subspace extensions of nondensely defined symmetric operators, Advances in Math. 14(1974), 309-332. E.A. Coddington, Selfadjoint problems for nondensely defined ordinary differential operators and their eigenfunction expansions, Advances in Math. 14(1974). E.A. Coddington and A. Dijksma, Selfadjoint subspaces and eigenfunction expansions for ordinary differential subspaces, to appear in J.Diff.Equations. A. Dijksma, and H.S.V. de Snoo, Eigenfunction expansions for nondensely defined differential operators, J.Diff.Equations 17(1975) 198-219. A. Dijksma and H.S.V. de Snoo, Selfadjoint extensions of symmetric subspaces, Pacific J . Math. 54(1974) 71-100. A.M. Krall, Differential-boundary operators, Trans. Amer. Math. SOC. 154

(19611, 9-23.

3. 4.

5.

6. 7. 8.

(1971) 429-458. 9. 10.

A N. Krall, The development of general differential and general differentialboundary systems, to be published. problems, H.J. Zimmerberg, Linear i n t e g r o - d i f f e r e n t i a l - b o u n d a r y - p a r a m e t e r to appear in Ann. di Mat. Pura ed Appl. Department of Mathematics * Rijksuniversiteit Groningen Groningen, The Netherlands.

This Page Intentionally Left Blank

W . Eckhaus ( e d . ) ,

New Developments i n D i f f e r e n t i a l Equations

@ N o r t h - H o l l a n d P u b l i s h i n g Company (1976)

SOME DEGENERATFD DIFFERENTIAL OPFPAI'OQC: OM VECTOR BUNDLES

by R. Martini

1.

Introduction I n t h i s p a p e r we s h a l l d e a l w i t h d i f f e r e n t i a l o p e r a t o r s

0

, acting

on

a v e c t o r bundle E o v e r a compact Cm d i f f e r e n t i a b l e manifold w i t h houndary. The d i f f e r e n t i a l operators

8 = a0

+

0 1

a-'Y

a r e of t h e type

+ E ,

where

i)

a i s a c o n t i n u o u s r e a l - v a l u e d f u n c t i o n d e f i n e d o n M such t h a t a i s smooth and

a > 0

i n t h e i n t e r i o r of t h e m a n i f o l d M and such t h a t a v a n i s h e s a t

t h e boundary of M.

ii)

0 i s a s t r o n g l y e l l i p t i c d i f f e r e n t i a l o p e r a t o r of t h e second o r d e r .

i i i ) Y is a d i f f r r e n t i a l o p e r a t o r of a t most o r d e r one. iv)

E i s a d i f f e r e n t i a l operator of order zero.

O u r a i m i s t o prove t h a t w i t h an o p e r a t o r s

0

of t h e t y p e above t h e r e

can be a s s o c i a t e d a n i n f i n i t e s i m a l g e n e r a t o r o f a s t r o n g l y c o n t i n u o u s semi-group, d e f i n e d on c e r t a i n weighted L2-spaces o f s e c t i o n s of P I o v e r E. Let us summarize b r i e f l y t h e c o n t e n t s o f t h i s p a p e r . I n s e c t i o n 2 w e g i v e t h e n o t a t i o n of t h e m a t h e m a t i c a l c o n c e p t s we s h a l l u s e i n our i n v e s t i g a t i o n about t h e d i f f e r e n t i a l o p e r a t o r s o f t h e t y p e d e s c r i b e d above. I n s e c t i o n 3 and sone a u x i l i a r y H i l b e r t a h l e s p a c e s a r e c o n s t r u c t e d . S e o t i o n 5 and

6

4

contain our

b a s i c r e s u l t s and i n s e c t i o n 7 and 8 we i n v e s t i g a t e r e l a t i o n s between t h e c l a s s of d i f f e r e n t i a l o p e r a t o r s c o n s i d e r e d by us and i n f i n i t e s i m a l g e n e r a t o r s o f semigroups o f o p e r a t o r s . F i n a l l y , i n s e c t i o n

9 some e x a n p l e s a r e g i v e n .

2, N o t a t i o n s M w i l l d e n o t e a paracompact Cm d i f f e r e n t i a b l e m a n i f o l d of dimension n , p o s s i b l y w i t h boundary aM. So p a r t i t i o n s o f u n i t y w i t h r e s p e c t t o any open cover i n g of M may be c o n s t r u c t e d . T ( M ) and T*(M) w i l l d e n o t e t h e t a n g e n t s p a c e of bl and t h e c o t a n g e n t space of M r e s p e c t i v e l y . T ' ( M ) is t h e subbundle of T'(M)

of non z e r o c o t a n g e n t

v e c t o r s . All v e c t o r b u n d l e s over F4 a r e supposed t o h e complex v e c t o r b u n d l e s , u n l e s s mentioned o t h e r w i s e e x p l i c i t l y , and a r e u s u a l l y denoted by E, F,

213

... .

214

R.MARTIN1

Wow l e t U be a n open s u b s e t o f !I, t h e n r(TJ,E) i s t h e s e t of a l l C" s e c t i o n s o f E over U. T o ( U , E ) i s t h e s u b s e t of r ( U , E ) c o n s i s t i n g of a l l s e c t i o n s whose s u p p o r t s a r e compact i n U and d i s j o i n t from t h e boundary X I . A p o s i t i v e smooth measure on M i s a B o r e l measure

u

such t h a t f o r each

c h a r t c = (U,@) f o r En t h e measure O(p) h d u c e d by B on t h e n-algebra o f t h e Borel s e t s of O(U) i s a b s o l u t e l y c o n t i n u o u s w i t h r e s p e c t t o t h e Lebesgue measure and h a s m

a s t r i c t l y positive C

d e n s i t y n . Thus

1 odx

u(A) =

O(A)

f o r each B o r e l s e t A 0 f . U . As u s u a l , Wp(E)

(-

i

p <

m )

d e n o t e $he p-th o r d e r Sobolev s p a c e .

I n g e n e r a l , t h i s i s a H i l b e r t a b l e s p a c e , b u t i f E4 i s p r o v i d e d w i t h a f i x e d p o s i t i v e smooth measure we may i d e n t i f y Wo(E) w i t h t h e s p a c e o f a l l measurable squarei n t e g r a b l e s e c t i o n s of E o v e r M. Hence Wo(E) i s a F i l h e r t s p a c e . Let U be an open s u b s e t of M. Then w i t h U'CCIJ we d e n o t e t h a t U' i s a r e l a t i v e l y compact open s u b s e t o f U. I t f o l l o w s t h a t t h e c l o s u r e

of U' i n U i s

compact. Let 0 he an open s u b s e t of t h e n-dimensional u,v b e two mappings of 0 i n t o t h e r-dimensional

E u c l i d e a n s p a c e I? and ? let

u n i t a r y s ? a c e C'.

(s,t)

Then I s I ,

a r e d e f i n e d by / s / ( x )= / s ( x ) j , ( s , t ) i x ) = ( s ( x ) , t ( x ) ) . Here

11

and (

,)

d e n o t e t h e u s u a l norm and i n n e r DrOduct i n Cr r e s p e c t i v e l y .

By . 1 : we d e n o t e t h e h a l f - s p a c e c o n s i s t i n g o f a l l ( x , , such t h a t xn > U and Bn+ d- e n o t e s t h e h a l f - b a l l ,

IR:

i s t h e u n i t b a l l i n IR".

defined h j r: B

d e n o t e s t h e c l o s u r e of IR:

,

= En

.., xn)

n W:,

E

E?

where Bn

i n IR".

Let V b e a f i n i t e - d i m e n s i o n a l v e c t o r s p a c e and l e t 52 h e a non-empty open s u b s e t i n some n-dimensional

Fuclidean space

IR". Then C r ( R , V )

...;

( r = 0, 1,

m )

i s t h e space of a l l r-times 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 f u n c t i o n s , mappingR i n t o V.

Cr(Q,V) d e n o t e s t h e s u b s e t of Cr(R,V) c o n s i s t i n g p r e c i s e l y of t h o s e e l e ments of C r ( R , V )

which have compact s u p p o r t i n 0 .

O c c a s i o n a l l y , we u s e t h e s p a c e

( a , V \ which a s a s e t e q u a l s T ( n , V ) b u t

which i s p r o v i d e d w i t h i t s n a t u r a l i n d u c t i v e l i m i t t o p o l o m . I t s t o p o l o g i c a l d u a l , t h e space of V-valued d i s t r i b u t i o n s , i s denoTed b y d ( 0 , V ) ' . o v e r a manifold M we have t h e c o r r e s p o n d i n g spacesQ(M,E)

For a v e c t o r b u n d l e E and&(M,E)'.

Let E and F b e two v e c t o r b u n d l e s w i t h t h e same b a s e s p a c e M . A l i n e a r d i f f e r e n t i a l operator ro(EI,E)

-t

0

from E t o F i s by d e f i n i t i o n a l i n e a r mapping

(M,F) which is l o c a l : i . e . ,

0

:

for each open s u b s e t TI of b4 and each s e c -

t i o n s c r o ( M , E ) such t h a t t h e r e s t r i c t i o n SIU = 0 we have

(3

s J U = 0. According

215

DEGENERATED DIFFERENTIAL OPERATORS

t o a theorem o f P e e t r e we know t h a t f o r any c h a r t c = ( l J , @ ) o f F4 s u c h t h a t F and F admit t r i v i a l i z a t i o n s (nF,wE)

Q

and (nF,vF) t h e l o c a l e x p r e s s i o n

d e f i n e d by

i s a l i n e a r d i f f e r e n t i a l o p e r a t o r i n t h e o r d i n a r y sense. Conversely, i r l i n e a r mapping o f

r

(V,E) i n t o

6

zations t h e l o c a l expression r y s e n s e it f o l l o w s t h a t

r

is a

0

(M,F) such t h a t f o r any c h a r t and any t r i v i a l i -

is a l i n e a r d i f f e r e n t i a l operator i n t h e ordina-

i s l i n e a r d i f f e r e n t i a l o p e r a t o r . Thus t h e l i n e a r

0

d i f f e r r n t i a l o p e r a t o r s a r e t h e only l o c a l l i n e a r o p e r a t o r s . With a d i f f e r e n t i a l o p e r a t o r

t h e r e i s f o r each n a t u r a l number k a s -

0

s o c i a t e d a k-th o r d e r symbol u k ( A ) such t h a t f o r each 5, into F

a l i n e a r map from t h e f i b e r F:

. The

E

'"'(M)ok( 0 ) ( E x ) i s

s e t of a l l d i f f e r e n t i a l o p e r a t o r s 0

from E i n t o F such t h a t o ( 0 ) v a n i s h e s f o r any m > 1.. i s d e n o t e d by n i f f k ( E , P ) .

A d i f f e r e n t i a l operator Diff (E,F) but

k

0

i s s a i d t o he o f o r d e r k i f

0

0 belongs t o

d o e s not b e l o n g t o D i f f k - , ( E , F ) .

By d e f i n i t i o n , a k-th o r d e r d i f f e r e n t i a l o p e r a t o r x

E

M i f f o r each

Cx

t

x

into F

k

he s t r o n g l y e l l i p t i c i n x

M i f f o r each 5

A

and t h a t F: i s p r o v i d e d w i t h a

Then a k-th o r d e r d i f f e r e n t i a l o p e r a t o r E

X.

i s e l l i p t i c i f it i s e l l i p t i c a t

M. I n a d d i t i o n , suppose t h a t F =

Hermitian s t r u c t u r e <,>.

e

is e l l i p t i c i n

T ' ( M ) o ( 0 ) ( E ; ) i s a l i n e a r isomorphism o f E

k-th o r d e r d i f f e r e n t i a l o g e r a t o r from E i n t o each p o i n t x

0

E

x

is said t o

A

""(V) and f o r e a c h e x

x

E

E

x

with

# 0 we have

A k-th o r d e r d i f f e r e n t i a l o p e r a t o r i n s t r o n g l y e l l i p t i c i f it i s s t r o n g -

l y e l l i p t i c i n each p o i n t . For more d e t a i l e d i n f o r m a t i o n about d i f f e r e n t i a l oper a t o r s on v e c t o r b u n d l e s s e e e.g. PALAIS C41, chn. I'I o r RAFAXWAN C31, chp. 3.

3.

Some a u x i l i a r y s p a c e s Throughout t h i s and t h e n e x t s e c t i o n

M

w i l l d e n o t e a f i x e d compact Cm

d i f f e r e n t i a b l e m a n i f o l d of dimension n 2 1 , p o s s i b l y w i t h boundary aY and E i s a m

fixed C

v e c t o r bundle o v e r M o f r a n k r and w i t h p r o j e c t i o n n, n : E + M. We sup-

pose t h a t M i s equipped w i t h a s t r i c t l y p o s i t i v e smooth measure

.

equipped w i t h a Hermitian s t r u c t u r e <

>

t i n u o u s f u n c t i o n d e f i n e d on M such t h a t

CL(X) >

CL

u

and t h a t E i s

i s supposed t o b e a real-veluerl con0 when x

k

aM.

Now we have t h e f o l l o w i n g d e f i n i t i o n s . D e f i n i t i o n 1. -------_--

L 2 ( E ) i s d e f i n e d t o b e t h e c o m p l e t i o n of t h e s e t o f a l l s e c t i o n s s E

T(M,F) under t h e norm

R .MARTIN I

216

-__-_-____H(E,cI)

i s t h e completion o f t h e s e t of a l l sect i ons s

Definition 2.

r

(M,E)

under t h e norm

1 Is1 l H

(3.2)

------

Remark.

= {

< s, s >

%I;.

M

L2(E) i s a H i l b e r t s p a c e w i t h t h e i n n e r Drodust

and may be c o n s i d e r e d a s t h e s p a c e of a l l s e c t i o n s o f M o v e r E which a r e squarei n t e g r a b l e over M w i t h r e s p e c t t o t h e measure p . H ( E , a ) i s a H i l b e r t s p a c e w i t h t h e i n n e r product

(3.4)

(s,t),

=

J

< s,

M

t

dU '-

and may be c o n s i d e r e d a s t h e s p a c e of a l l measurable s e c t i o n s o f W o v e r E which du

a r e s q u a r e - i n t e g r a b l e over M w i t h r e s p e c t t o t h e measure Let c -________-

Definition.

(71 , V )

section s

E

-.

= ( U , 4 ) b e a c h a r t for M such t h a t t h e r e i s a t r i v i a l i z a t i o n o f F o v e r U and l e t

-

r (M,E) c o r r e s u o n d i n g

7

= Us4

-1

he t h e l o c a l exuression of t h e

t o t h e c h a r t c and t h e t r i v i a l i z a t i o n ( n , v ) .

Suppose U ' c c t a n d p u t N = a 4 - l . Then by

we i n t r o d u c e a semi-norm on t h e l i n e a r s p a c e

r

(M,E),

r 0 (M,E) .

Wow l e t T be t h e c o a r s e s t l o c a l l y convex t o p o l o g y on

such t h a t

f o r a l l c h a r t s c = (U,@), a l l U'CCU and a l l t r i v i a l i z a t i o n s ( n , v ) t h e semi-norm

P ~ , ~ , i~s, c, o n t i n u o u s . 'Then by V ( E , a ) we d e n o t e t h e c o m p l e t i o n o f F (M,E) w i t h r e s p e c t t o t h e l o c a l l y convex t o p o l o g y

-------

T.

= { c k = ( U 8 ) I b e an a t l a s f o r !I c o n s i s t i n g : o f a f i n i t e numk' k b e r o f c h a r t s such t h a t f o r e a c h k E a d m i t s a t r i v i a l i z a t i o n (n v ) 'k and l e t ( U i j he an open c o v e r i n g of V such t h a t f o r each k U F c U k ; d e n o t e

Theorem 1 . over U k-

-1

.& -"Z k = 'I+'@k ak =

oak- .

t h e l o c a l exnression over U of t h e s e c t i o n s k

Then an i n n e r product f o r

r (M,E) such

E

ro(M,E) and p u t

t h a t t h e a s s o c i a t e d norm g e n e r a t e s t h e l o -

c a l l y convex t o p o l o g y T i s g i v e n by

-----

Proof.

Obviously, (

,

) i s an i n n e r p r o d u c t .

Now suppose t h a t c = (IT,@), F = ( U , @ ) a r e two c h a r t s w i t h t h e same domain

217

DEGENERATED DIFFERENTIAL OPERATORS

for M and (n,w), (n,;)

two trivializations of F: over IT. Let u l C c ~7 ~=,

-1

s

= CsZ - I , 2 = aQ and = a8 functions. Then it follows that = Drl(s(x(x)))

D:(X)

, = w-v -1

-'.

,,s~

-1

,

Let = i', be the transition 5 = nzx. Hence by the chain rule we get

.Ds(x(x))

*

Dx(x).

Therefore

for each x x

E

6

Now U'CCU. Hence there is a constant k 1 such that for each

Q(U').

$(U') we have

where J i s the Jacohian of the transition function

x-'.

Hence there exists a constant k2 such that

=

From

that for each x

n s X , U'CCU it f o l l o w s a l s o that there exists a constant k such 3

U' we have

Now let p

he a semi-norm. Then from the estimates made above we see that C,W,U' there exists a constant k such that

4

Thus the norm

1 I IIv

generates the locally convex topology

T.

Since M is compact and locally compact there exists an atlas

a

having

218

R. MARTINI

t h e p r o p e r t i e s s t a t e d i n theorem 1. Thus r e c a l l i n g t h a t a H i l b e r t a b l e s p a c e i s a complete l o c a l l y convex s p a c e such t h a t t h e l o c a l l y convex s t r u c t u r e i s g e n e r a t e d by an i n n e r p r o d u c t we have t h e f o l l o w i n g c o r o l l a r y o f theorem 1 .

-___----

Corollary.

------

Remark.

V (E,a) i s a H i l b e r t a h l e s p a c e .

Obviously, we have V o ( E , a f ) c H(E,a) c L 2 ( F ) w i t h c o n t i n u o u s i n c l u s i o n maps. C o n s i d e r i n g t r i v i a l b u n d l e s o v e r open s u b s e t s o f a n F u c l i d e a n s p a c e t h e

d e f i n i t i o n s i n t h i s s e c t i o n may b e changed i n an obvious way t o t r i v i a l b u n d l e s over n o n - n e c e s s a r i l y bounded open s u b s e t s o f a n E u c l i d e a n s p a c e Bn

. In

t h i s case,

by t h e e x i s t e n c e of a c a n o n i c a l g l o b a l t r i v i a l i z a t i o n , V ( F , a ) is a H i l b e r t space.

If a i s non-vanishing on V it f o l l o w s t h a t H(E,a) and V ( E , a ) a r e i s o 1 morphic t o t h e Sobolev s p a c e s l J o ( E ) and W ( E ) r e s p e c t i v e l y .

4.

The embedding J of V (E,a) i n t o H(E,a)

"

I n t h i s s e c t i o n we s h a l l d e a l w i t h t h e c o n t i n u o u s l i n e a r embedding J of

Vo(E,a) i n t o H(E,a). The quest,ion whether t h i s embedding i s compact ( c o m p l e t e l y c o n t i n u o u s ) i s now o u r main i n t e r e s t . C o n s i d e r i n g f i r s t m a n i f o l d s w i t h o u t boundary we have t h e f o l l o w i n g consequence of R e l l i c h ' s theorem. Theorem 2. --------

Let M b e a compact C

m

manifold w i t h o u t boundary. Then t h e embedding J

o f V-(R,n) i n t o H ( E , a ) i s compact. u

-----

Proof.

M i s supposed t o be a compact m a n i f o l d and

c1

i s a continuous p o s i t i v e

f u n c t i o n d e f i n e d on M. Hence i s bounded and bounded away from z e r o on M. Thus Vo(E,a) i s t h e same as t h e Sobolev s p a c e W ' ( E ) and H(E,,)

i s isomorphic t o

t h e Sobolev space Wo(E). Now from R e l l i c h ' s embedding theorem ( s e e e . a . tion

4) it

PALAIS

C41,

ch. X , sec-

1

f o l l o w s t h a t t h e embedding of Wo(E) i n t o WO(F) i s compact. Hence t h e

ernbedding of V ( E , a ) i n t o H(E,a) i s compact t o o .

For m a n i f o l d s w i t h boundary t h e s i t u a t i o n i s more c o m p l i c a t e d . I n t h i s c a s e we have a r e s u l t of which t h e proof i s d i v i d e d i n t o a number of p a r t s , t h e

l a s t one o f which g i v e s our r e s u l t .

-------

Lemma 1 .

Let a b e a p o s i t i v e c o n t i n u o u s f u n c t i o n d e f i n e d on ,:RI on t h e l a s t v a r i a b l e ; i . e . ,

u

d e f i n e d on lR+ such t h a t a(x) = a ( x ) when x = ( x , ,

I n a d d i t i o n , suppose t h a t

depending o n l y

t h e r e e x i s t s a p o s i t i v e continuous f u n c t i o n

..., xn- )

c RS.

219

DEGENERATED DIFFERENTIAL OPERATORS

for some a > 0. Then for all

s E 'I

(R:

x

Cr,a)

we have the estimate

----Proof.

Suppose x =

Cm(Py,Cr) and let

s E

(X',Xn),

x'

E

IRn-l, x

IR,

E

Then we have

Using the inequality of Cauchy-Schwarz we see that the last expression is not greater than

Since for fixed a > 0 both sides of inequality functions of

s E

V (1R:

x

Cr,a)

we have established inequality

(4.1) in

case s

E

(4.1) are continuous as

is dense in Vo($+

and the set Cz(lP,:,(?)

V (IR" o

+

x

x

Cr,a)

rr,a), 'Phis completes

the proof of lemma 1.

-------

Lemma 2. For each and all y

Proof. -----

> 0

E

E

there exists a 6 > 0 such that for all

s E

1

W (R"

x

Cr)

IR"with IyI < 6.

Let E > 0 and suppose that s theorem that

E

Cm(IRn,Cr).Then it follows from Plancherel's

220

R. MARTINI

where 5 d e n o t e s t h e F o u r i e r t r a n s f o r m of s d e f i n e d by

E v i d e n t l y , t h e r i g h t hand s i d e of ( 4 . 3 ) e q u a l s

and it can e a s i l y be e s t a b l i s h e d t h a t t h e r e e x i s t s a such t h a t for a l l

Thus f o r all y

6

5

E

IR" and a l l y

E

F

> 0 , depending

only on

E,

B n w i t h ( y ( < fi

En w i t h ( y l < 6 it f o l l o w s t h a t

For f i x e d y f u n c t i o n s of s

6

(4.4) i s

dominated by

R n b o t h s i d e s of i n e q u a l i t y ( 4 . 2 ) a r e c o n t i n u o u s a s

W'($

Cr) and C:(l#,f)

lhr'(18x

is dense i n

IT1(#

X

Cr). Fence f o r a l l

Cr)

i n e q u a l i t y ( 4 . 2 ) i s v a l i d . T h i s com-

uniformly on t h e c l o s e d u n i t b a l l 1: o f Vo(R:

Cr,a); i'.e., u n i f o r m l y on t h e s e t

y

E

Finwith lyl < 6 and a l l s

6

p l e t e s t h e proof of lemma 2 .

K c o n s i s t i n g of a l l s

-----

Proof.

6

Vo(B:

Cr,a) such t h a t

We may r e s t r i c t o u r s e l v e s t o t h e c a s e where t h e l i m i t i s t a k e n over a l l y = (y,,

..., y n )

iRn w i t h y

2 7,

Now f o r any a > 0 we o b t a i n

0.

In

t h i s c a s e we have

221

DEGENERATED DIFFERENTIAL OPERATORS

Let

for a l l y

> 0 b e Riven. By lemma 1 we may f i x a n a > 0 and a 6 1 such t h a t

E

I R n with y

E

Thus for a l l y

E

t 0 and IyI < 6 ,

R n w i t h yn 2 0 and

yI < 6 1 t h e r i g h t hand s i d e of ( 4 . 5 ) is n o t

greater than 2dx.

Next i t follows t h a t

1

Using t h e f a c t t h a t a-' {x = ( x , , with 0 <

..., x n )

E

i s u n i f o r m l y c o n t i n u o u s and hounded on t h e s e t

$ 1 ~> 8~. 1 ~ 1 5

< 6 , such t h a t f o r a l l y

1+6 E

IF?

1

>

t h e r e e x i s t s a c o n s t a n t c and a 6 w i t h IyI < 6 2 and y

e x p r e s s i o n i s dominated by

E v i d e n t l y t h i s e x p r e s s i o n is n o t g r e a t e r t h a n m

m

t 0 the last

2

222

R . MARTINI

Vow by lemma 2 t h e r e e x i s t s a 6 y

E

IRn

with y

Thus f o r a l l y t i m a t e d by

E.

Bn w i t h y

E

3

with 0 < 6

<

3

ri2 such t h a t f o r a l l

0 and IyI < f i 3 t h e second t e r m o f ( 4 . 1 0 ) i s s m a l l e r t h a n

2.

t 0 and

$E.

we s e e t h a t e x p r e s s i o n ( 4 . 8 ) i s es-

Iyl <

T h i s completes t h e proof of lemma 3.

I n t h e next p a r t o f our proof we need a c h a r a c t e r i z a t i o n o f r e l a t i v e l y compact s e t s in t h e space L 2 ( R n

x

Cr)

= L2(1Rn,Cr).

This characterization i s

c o n t a i n e d i n t h e f o l l o w i n g theorem, due t o Frgchet-Kolmogorov.

Theorem ---------3.

A s u b s e t I' o f Lp(lRn, C r ) ( 1

Lp(IRn

Cr)

X

fied: -

(i)

sup S

i.e.,

E

< p

<

m)

i s r e l a t i v e l y compact i n

i f and o n l y i f t h e f o l l o w i n g t h r e e c o n d i t i o n s a r e s a t i s -

-1

1 Is(x)/'dxjP

{

<

m;

IRn

K

t h e s u b s e t K i s u n i f o r m l y bounded i n L P ( I R n , C r ) ;

(ii)

1, Is(x+y) -

lim

uniformly i n s

(iii)

= 0

s(x)\'dx

IR

y + 0

K;

E

1

lim

I s( x ) lPd x = 0 uniformly i n s

E I(.

A -+ m / x \ 2 A

For a proof o f t h i s theorem s e e e . g . DUNFORD-SCHWARTZ [11, p. 301. We c o n t i n u e o u r d i s c u s s i o n w i t h t h e f o l l o w i n g lemma, Lemma 4.

-------

Let a be as i n lemma 1 . Then t h e c o n t i n u o u s and l i n e a r embedding of

v~(B: P roof. -----

x

c r ) into

H ( E ~x

cr)

i s compact.

A s i n t h e proof of lemma 3 we d e n o t e hy $ t h e c a n o n i c a l e x t e n s i o n of a n t o L2(IRn,Cr)

s E L2(By,Cr)

Vo(B:

and we l e t P h e t h e c l o s e d u n i t b a l l o f

Cr,cr).

X

Applying t h e theorem of Frgchet-Kolmogorov we s e e t h a t it i s s u f f i c i e n t t o prove t h a t t h e f o l l o w i n g c o n d i t i o n s a r e s a t i s f i e d ;

uniformly i n s

(iii)

E

K;

lim

A + - 1x1

1 > A

N

I (%)(XI a2

I2dx u n i f o r m l y i n s

E

K.

223

DEGENERATED DIFFERENTIAL OPERATORS

Evidently, condition

(1) and

( i i i )a r e s a t i s f i e d .

$0

we o n l y have t o

v e r i f y c o n d i t i o n ( i i ) . Rut t h i s f o l l o w s immediately from lemma 3. T h i s completes t h e proof of lemma

4.

We may now s t a t e o u r r e s u l t

---

-

T'neorem -----4 .

Let M b e a compact Cm d i f f e r e n t i a b l e manifold w i t h houndary aM a n d l e t d b e t h e m e t r i c on h4 induced by a Riemannian s t r u c t u r e g on M.

Denote by p ( x ) t h e d i s t a n c e from t h e p o i n t x

E

EI t o t h e houndary aM o f M . I n ad-

d i t i o n . suppose t h a t t h e r e a l - v a l u e d f u n c t i o n a i s o f t h e form a = p p ( O

-----

Let c . = ( L J . , 0 . ) ( i = 1 , 2 , 1

oi(ui)

1

-

=

nt(i =

1,

2,

..., P.)

..., L),

p < 2).

i s compact.

Then t h e c o n t i n u o u s l i n e a r embedding of V ( E , a ) i n t o H ( E , a )

Proof.

5

be c h a r t s f o r M such t h a t

P.

aM =

u

i=1

oi

-1

(B

n l

-

($)),

where

Choose a d d i t i o n a l c h a r t s ci = ( U i , a i ) ( i = L+1,

..., m )

such t h a t

where

Let { X i } ( i = 1 , 2,

..., m )

he a Cm p a r t i t i o n of u n i t y f o r I.4 w i t i !

( i = e+l, and l e t I l l . } ( i = 1 , 2,

..., m)

..., m )

h e Cm r e a l - v a l u e d f u n c t i o n s d e f i n e d on P4 such

that

( i = e+l, and such t h a t p . e q u a l s one on supp ( X . ) ( i F i n a l l y , l e t t h e c h a r t s c . ( i = 1 , 2, admits f o r any i = 1 , 2 ,

= 1 , 2,

..., m) h e

..., m a t r i v i a l i z a t i o n

Now we d e f i n e t h e f o l l o w i n g mappings:

..., m )

..., m , .

chosen such t h a t t h e v e c t o r b u n d l e (a,w.)

over U . .

224

R .MARTIN I

where

I

and where t h e mapping P i s given by

.

.

It follows t h a t Q i s a c r o s s s e c t i o n f o r P. Indeed, we have

s i n c e {Xi) i s a p a r t i t i o n of u n i t y and P i e q u a l s one on su?p (

( i = 1, 2,

..., m).

m

I t can e a s i l y be seen t h a t P i s a continuous mapoing when

r

a r e equipped with t h e r e l a t i v e topology o f V(E,a) and @ V ( R . , 1=1 where

and

;. 1 =

BY

x

( i = 1, 2,

C'

..., a ) ,

E~ =

B"

x

cr(i

( M , E ) and

@ r ( xn , C I )

-a . ) r e s p e c1 t i v e l y , 1-

m

E~ =

.

= L+I,

..., m)

..., m).

a@. ( i = 1 , 2, 1

t o unique continuous l i n e a r mappings

m

P

: V(E,a)

-c

&, V ( E i , a i ) 1= 1

and

. r e s p e c t i v e l y . Moreover, t h e p r o p e r t y t h a t Q i s a c r o s s s e c t i o n o f Hence we have t h e commutative diagram

'P

i s preserved.

225

DEGENERATED D I F F E R E N T I A L OPERATORS

iJ where J and

?

a r e continuous i n c l u s i o n mappings.

Since t h e property being a compact l i n e a r mapninp; i s preserved by composit i o n with continuous l i n e a r mappings and s i n c e t h e d i r e c t sum of compact mappings

..., m t h e map-

i s compact again it i s s u f f i c i e n t t o show t h a t for any i = 1 , 2 , pine,

aM ( i =

is compact. Recallinp; t h a t (fi n

..., m )

follows t h a t a i ( i = 9.+1,

,.., m t h e

spaces f ( B n

i s empty, a i s s t r i c t l y posi-

i s bounded and bounded away from zero. Hence f o r

spaces H ( E i , a i )

C r ) and W,)Rn

x

. . ., m )

of M and p i s a s t r i c t l y p o s i t i v e smooth measure on M it

t i v e on t h e i n t e r i o r

i = 9.+l,

~+1,

x

and V ( E i , u i )

a r e isomorphic t o t h e Sobolev

Cr) respectively.

Fow by R e l l i c h ' s theorem it follows t h a t t h e i n c l u s i o n mapping of W1(Bn( 1 )

X

-

i n t o T,fo(Bn

Cr)

w

x

C r ) i s compact. Thus t h e i n c l u s i o n mapping of

V(Eiyai) i n t o H(Ei,ai) i s compact too. Pence what remains t o be proved i s t h e f a c t t h a t f o r i = 1 , 2,

H(B~x

cr,Z.)

..., 9. t h e i n c l u s i o n mapping of V ( R n

into

Cr,a.)

x

i s conpact.

Let t h e i n t e g e r i ( i S 9 . ) h e f i x e d and l e t R be t h e d i s t a n c e of $iand t h e complement : U Evident& to 4.

-1

6 > 0. Denote by D t h e s e t of a l l p o i n t s x

n

is.

E

M , which have a d i s t a n c e -1 n

Denote by S t h e s e t of a l l p o i n t s x

with d i s t a n c e p ( x ) t o t h e boundary aM n o t z e a t e r t h a n $ 6 . Define -

y

E

Ry

(B+)

5=

$.(D),-

-

= $.(S), g t h e Riemannian s t r u c t u r e on lRy induced by + i , asso&ated

(B+)

of Ui.

(B+) not g r e a t e r than Y

;1

t h e metri:

Y

with g, p ( y ) t h e d i s t a n c e with respect t o t h e metric

2

of

d

on i $

point

t o t h e boundary of IR:. Now by t h e s p e c i a l choice o f D and S it follows t h a t T ( y ) = p ( $ - ' ( y ) ) f o r Y

a l l points y

E

S.

Ve a s s e r t t h a t t h e r e e x i s t constants A > 0 and A ' > 0 such t h a t u

p(y)

S

Ayn and yn

5

A'F(y) f o r a l l y = (y,,

...( y n )

- L

6

S. This f a c t can be proved

-

as follows. Let I be t h e canonical m a p p i n e f t h e tangent space ):?I( r e i o n t o t h e tangent space ?(HI:) x

E

IRy l e t A

tinuous posit:ve

E

f o r any

(I?"+).Then A i s a conIRy and by t h e compactness of % it follbws t h a t

be t h e norm of t h e s s t r i c y i o n Ix of I t o f u n c t i o n of x

with Riemannian s t r u c t u -

with Euclidean s t r u c t u r e e.JMoreover, T

R. MARTINI

226

(A Ix 51 i s f i n i t e . Denote by L ( c ) , L ' ( c ) t h e l e n g t h s of a C ' c u r v e i n -AD =w isup th r e s p e c t t o t h e s t r u c t u r e s and e r e s p e c t i v e l y . PVidently we have L ( c ) s E

for a l l y

5 R e ' ( c ) . Hence

y,.

5

E

5,

t a k i n g l e a s t upper hounds, it f o l l o w s t h a t p ( y )

s

Changing g and e i n t h e x a s o n i n g made ahove we s e e t h a t t h e r e e x i s t s a l s o

a c o n s t a n t A ' > 0 such t h a t for a l l y

y,

E

E

Al;(y).

Now it follows t h a t

and

YE for a l l y

Vo(B:

x

E

2 IA'F(y)lP = ( A ' ) C i ( y )

5.

y = (yl,

Thus Vo(B: and F(B:

Cr,;)

..., y n )

bedding of Vo(B: a

I

x

and H ( B y x C r , z . )

Cr,ii)

f,;)

a r e isomorphic t o

r e s p e c t i v e l y , where a ( y ) = y:

f o r each

RY. Hence what remains t o b e proved i s t h e f a c t t h a t t h e em-

E

x

x

i n t o F(By x Cr,a) i s compact. However, by a s s u m p t i o n

?,;)

dYn Y n B <

m.

0

Thus by a p p l y i n g lemma

5.

4 we

complete t h e p r o o f of t h i s theorem.

form B

The s e s q u i - l i n e a r

x

In t h i s and t h e f o l l o w i n g s e c t i o n s

w i l l d e n o t e a f i x e d compact Cm

d i f f e r e n t i a b l e manifold o f dimension n 2 2 w i t h boundary a?4 and E w i l l b e a f i x e d m

C

v e c t o r bundle o v e r M w i t h p r o j e c t i o n

TI:

w i t h a s t r i c t l y p o s i t i v e smooth measure Hermitian s t r u c t u r e h = <

,>

.a

F

-f

' 4 . DI i s supposed t o b e equipped

and F i s suuposed t o h e p r o v i d e d w i t h a

denotes a real-valued continuous f u n c t i o n d e f i -

ned on M such t h a t a ( x ) > 0 when x

1 aM, a ( x ) =

0 when x

E

a V and such t h a t i t s

r e s t r i c t i o n t o t h e i n t e r i o r M \ a N i s Cm.

In t h i s s e c t i o n our a i m i s t o i n t r o d u c e a f a m i l y of c o n t i n u o u s s e s q u i l i n e a r forms ( B ) ( A A

E

C ) on t h e p r o d u c t s p a c e V ( E , a )

x

Vo(E,a). These s e s q u i -

l i n e a r forms a r e connected w i t h a g i v e n d e g e n e r a t e d d i f f e r e n t i a l o p e r a t o r 0 , act i n g on t h e v e c t o r bundle C . Given a f i x e d norm f o r V(F,ci) we s h a l l p r o v e t h a t t h e s e s q u i - l i n e a r form Bx i s c o e r c i v e for a l l complex numbers X w i t h r e a l p a r t Re X sufficiently large.

Now l e t 0 b e a d i f f e r e n t i a l o p e r a t o r , a c t i n g on t h e v e c t o r b u n d l e F, of t h e form

227

DEGENERATED DIFFERENTIAL OPERATORS

where a.

0 i s a s t r o n g l y e l l i p t i c d i f f e r e n t i a l o p e r a t o r of t h e second o r d e r .

b.

Y i s a d i f f e r e n t i a l o p e r a t o r o f a t most o r d e r one.

c.

Z i s a d i f f e r e n t i a l o p e r a t o r of o r d e r zero o r vanishes i d e n t i c a l l y .

Thus 0 i s a l o c a l l i n e a r mapping of

r

(M,F) i n t o

r

(M,E). V i t h t h e d i f f e r e n t i a l

o p e r a t o r 0 we a s s o c i a t e a family of d i f f e r e n t i a l o p e r a t o r s ( 0 )(A h

E

C ) d e f i n e d by

0x = h - 0

(5.2)

and with t h e h e l p o f t h e family of d i f f e r e n t i a l o n e r a t o r s (0 ) ( A h

ce a family of s e s q u i l i n e a r forms ( ^ R x ) ( A

E C ) on t h e space

E C)

we i n t r o d u -

G(E,a) x ?(E,a),

d e f i n e d by

?he f i r s t p r o p e r t y of t h e family of s e s q u i - l i n e a r forms

( gh ) ( h

E

C ) we

s h a l l prove i s s t a t e d i n t h e following lemma A

-

-Lemma ----- 5. The s e s q u i - l i n e a r form BA defined on v ( E , a ) any complex number A.

x

? ( R , a ) i s continuous f o r

Proof. ----- Let c = (U,+) be a c h a r t for 14 such t h a t E admits a t r i v i a l i z a t i o n over U and l e t U'ccU. ?hen f o r a l l s',t

E

IT,^)

ro(EI,F) it f o l l o w s t h a t

and we have t h e f o l l o w i n 8 e q u a l i t i e s and e s t i m a t e s :

-

Y

Here s and t denote t h e l o c a l e x p r e s s i o n s of t h e s e c t i o n s s and t r e s p e c t i v e l y and a denotes t h e d e n s i t y of t h e measure + ( u ) w i t h r e s p e c t t o t h e Lebesque measur e on

+(u). By t h e r e l a t i v e l y compactness of U' i n U and s t r i c t l y p o s i t i v i t y of

t h e r e e x i s t s a c o n s t a n t k5 > 0 such t h a t ( 5 . 5 ) i s dominated by

CI

228

R . MARTINI

(t).

Using matrix n o t a t i o n we have.for a l l s , t

/

(5.7)

< Os,t > dp =

U'

/

 dp =

U'

here t h e prime denotes m a t r i x t r a n s p o s i t i o n and 3 i s a c e r t a i n Wemitian matrix function. If t h e l o c a l expression of 0 with r e s p e c t t o t h e c h a r t c and t h e t r i -

v i a l i z a t i o n (n,v) i s given by

..

.

where a l J , b J , c

E

C"(+(U),L(Cr,Cr))(i

= 1, 2,

..., n ) then

( 5 . 7 ) equals

By i n t e g r a t i o n by p a r t s we s e e t h a t t h e l a s t expression equals

+ I: j p

1

F'(hbj

( u.' .) .

-

1

XD.(haij))D.sdx +

i 1

Now h , a l J ,bJc a r e Cp) en

+ (U) and

;'

Q (U') i s continuous on Q (U), i n a d d i t i o n U ' c c U .

JIence t h e r e e x i s t constants k6 > 0 , k

7

(5.10)

T'hcs.

> 0 and k8 > 0 such t h a t

DEGENERATED D I F F E R E N T I A L OPERATORS

Similarly, we obtain for all s,t

E

229

To(M,E) the evaluations

If the local expression of the differential operator Y' with respect to

the chart c and the trivialization ( n , v ) is given by (5.12)

w

ys =

-

i zd D.s + es,

i where dl,e

E

1

C"@ (U), L(Cr,Cr)) ( i = 1 , 2 ,

..., n) then the last expression equals

230

R.MARTIN1 .

1

Thus by t h e c o n t i n u i t y o f h,dl,e,%' t h a t t h e r e e x i s t c o n s t a n t s kq > 0 and k

10

on

0 (U) and

s i n c e U'ccU it f o l l o w s

> 0 such t h a t

E s t i m a t i n g t h e l a s t t e r m of t h e r i g h t hand s i d e of

( 5 . 4 ) we o b t a i n f o r

some c o n s t a n t k l l > 0

*

where 5 t =

$ and f

E

C"(+(U),

Now choose an a t l a s t h a t for any i = 1 , 2,

{U!} ( i = 1 , 2 ,

L(C

a=

..., m E

..., m) b e

r

,C

r

1).

{ c . = (Ui,pi)}

(i = 1 , 2,

..., m) f o r M

such

a d m i t s a t r i v i a l i z a t i o n ( n , v i ) over U. and l e t

a c o v e r i n g of M such t h a t IJ!ccUi.

Then it f o l l o w s from

231

DEGENERATED DIFFERENTIAL OPERATORS

t h e e s t i m a t e s made above t h a t t h e r e e x i s t c o n s t a n t s {ITi}

( i = 1 , 2,

..., m )

such

t h a t f o r any complex number

5

max KiCPc. , v . ,u!( s ) i=1,2,. ..,m 1 1 1 1

fPCi,". ,U! ( t ) .

*

1

1

A

Thus BX i s a c o n t i n u o u s s e s q u i - l i n e a r form on ; ( F , u )

T'(P,@).

T i s completes t h e

proof o f lemma 5. A

Lemma 5 i m p l i e s t h a t f o r any complex number A . nuous e x t e n s i o n B

X

t o V (E,o) O

Bx h a s a unique c o n t i -

V (8,~).

x

The next p r o p e r t y we want t o p r o v e about t h e f a m i l y (B ) ( A A

E

C) i s the

following Lemma 6. -------

Let

1 1 ILv

f i x e d norm f o r Vo(F,o).

be a

d e f i n e d on Vo(E,a)

x

Then t h e s e s o u i - l i n e a r form R X V o ( E , a ) i s c o e r c i v e f o r a l l complex numbers X w i t h

r e a l p a r t Re X s u f f i c i e n t l y l a r g e ; i . e . ,

t h e r e e x i s t c o n s t a n t s Ic > 0 and X 1 > o

such t h a t (5.17)

f o r a l l X w i t h Re X 2 P r o o f . Let

-----

KI ( s I ;1

Re BX ( s , s ) 5.

,x=

X, and a l l

{ck = (U ,$

k k

f o r any k = 1 , 2 ,

s E Vo(E,o).

) }(k = 1 , 2,

..., m

U

k

..., m ) h e a f i x e d a t l a s f a r M such t h a t

a d m i t s a t r i v i a l i z a t i o n of F o v e r Uk. I n a d d i -

t i o n , l e t b e g i v e n a m o d i f i e d Cw p a r t i t i o n o f u n i t y w i t h r e s p e c t t o t h e a t l a s

b

such t h a t Ew2(x) = 1 f o r each x E 11. Denote by ok t h e d e n s i t y of t h e measure $ ( p ) k k k w i t h r e s p e c t t o t h e Lebesgue measure on 4 (U). Then d r o p p i n g f a r a moment t h e

k

index k we have t h e f o l l o w i n g e q u a l i t i e s and e s t i m a t e s f o r an s

E

ro(M,E)

232

R.MARTIN1

S i n c e @ i s a s t r o n g l y e A & i p t i c d i f f e r e n t i d o p e r a t o r o f t h e second o r d e r we may a p p l y G k d i n g ' s i n e q u a l i t y , f o r m u l a t e d f o r s y s t e m s , and t h e n w e obt a i n t h e f o l l o w i n g . There e x i s t c o n s t a n t s k,2 > 0 and k

13

such t h a t

DEGENERATED DIFFERENTIAL OPERATORS

233

Using t h e i n e q u a l i t y (5.20)

21x1 Iyl

1

5 ~ 1 x 1 ' + -$y12.

which i s v a l i d for any

E

> 0 , we s e e t h a t t h e e x p r e s s i o n ( 5 . 1 9 ) i s n o t smaller

than

.. . Now h , a l J ,bJ

,;

a r e Cm on $ (U) and

li s

s u p p o r t e d i n 4 (U). Thus t h e r e

e x i s t s a c o n s t a n t k,4 such t h a t

Using t h e i n e q u a l i t y (5.20) a g a i n we s e e t h a t for e v e r y

E

> 0 the last

e x p r e s s i o n i s e s t i m a t e d by

.. S i n c e a l J ,h,c,w a r e Cm on 0 (U) and e x i s t s a constant k

15

i s supported i n 9 (IT) t h e r e

such t h a t

Using s i m i l a r arguments we s e e t h a t t h e r e e x i s t s a c o n s t a n t k16 such t h a t t h e exp r e s s i o n (5.21) i s g r e a t e r t h a n

R.MARTINI

234

Combining (5.18) - ( 5 . 2 5 ) , ( 5 . 2 3 ) w i t h a s u i t a h l y chosen i s bounded o n $ ( I T ) we s e e t h a t t h e r e e x i s t c o n s t a n t s 1.. bering t h a t Ly

t,

17

and remem-

> 0 and k , 8

such t h a t

For t h e p a r t o f t h e s e s q u i - l i n e a r

form B A which c o n t a i n s t h e f i r s t or-

d e r d i f f e r e n t i a l o p e r a t o r '? we o b t a i n

a

S i n c e '3,h,e a r e C

-

on Q (U) and a i s hounded on 4 (LJ) t h e r e e x i s t s a con-

s t a n t k l g such t h a t

Using t h e i n e q u a l i t y ( 5 . 2 0 ) a g a i n we s e e t h a t f o r any E > 0 t h e l a s t e x p r e s s i o n is e s t i m a t e d by

w,h,e a r e Cm on 4 ( U ) and such t h a t

- i s bound-ed o n 8 (U). Thus t h e r e e x i s t s a c o n s t a n t 0 .

k.20

DEGENERATED DIFFERENTIAL OPERATORS

(5.30)

i1w2

< Ys,s >

41 <

:k

M

19

IwT).slpdx + k

Z

.

J

0 (lJ)

For t h e p a r t o f t h e s e s q u i - l i n e a r

20

235

1 1s12F .

b

("1

form FA which c o n t a i n s t h e d i f f e r e n -

t i a l o p e r a t o r 5 we have: There e x i s t s a c o n s t a n t k 2 , such t h a t ,

.

Now from ( 5 . 2 6 ) , ( 5 . 3 0 ) and ( 5 . 3 1 ) it f o l l o w s t h a t t h e r e e x i s t c o n s t a n t s

A

k

> 0 (k = 1 , 2 ,

...,

m ) and R

k

( k = 1 , 2,

..., m)

such t h a t

Hence we o b t a i n

(5.33)

Re B

A

s , s ) = Re

1<

OXs,s >

t Re

XI

< s,s

M

1

+ XA X \wkDjgk12dx k k j 4 (U,)

-

XBk

k

a+ 0-

1 1Xl2 $ .

4(Uk)

k

Choose a number C such t h a t t h e s e t s {U'} (1. = 1 , 2 ,

k

U k = {x

t

Mlwk(x)

>

..., m)

r}

form a c o v e r i n g o f M. Then t h e r e e x i s t c o n s t a n t s k > 0 and L s u c h t h a t

d e f i n e d by

236

R.MARTIN1

Hence t h e r e e x i s t constants X , > 0 and K > 0 such t h a t f o r each s

6

ro(M,F) and

all complex numbers X with Re X t A, we have

By t h e c o n t i n u i t y o f B

X

and t h e Vo(E,a)-norm it follows t h a t ( 5 . 3 5 )

holds f o r a l l complex numbers A with R e X 2 X , and a l l s c V ( E , a ) . This completes t h e proof of lemma 6. Summarizing lemma 5 and lemma 6 and usinn t h e theorem o f Lax-Milgram we have e s t a b l i s h e d t h e following theorem.

- 6A- be

Theorem ---------5. L e t

t h e sesqui-linear form ( 5 . 3 ) defined on ?(E,a)

where V ( E , a ) i s t h e space

of V ( E , a ) . Then $ i s 1 unique extension B

I I I I V for Vo(E,a)

A -

a continuous s e s q u i - l i n e a r form on < ( E , & ) -

of B

A -

t o Vo(E,a)

x

?(E,a),

ro(M,E) provided with t h e r e l a t i v e topology x

?(E,a) and t h e

V ( E , d i s with r e s p e c t t o any f i x e d norm

x

coercive f o r Re X s u f f i c i e n t l y l a r g e ; i . e . , t h e r e e x i s t

constants K > 0 and 1, > 0 , dependinK on Re B A ( s , s )

2

K( I s \

1 I IIv

such t h a t

IV

f o r a l l complex numbers A with Re X t X Moreover, f o r any f i x e d norm continuous l i n e a r maps ( d c , ) ( X

E C),<

f o r a l l s , t c Vo(E,a) and such t h a t

1-

and a l l s

E

I I I I v for V

: Vo(E,a) +

(E,a).

( P y a )t h e r e e x i s t a family of

Vl(Y,a) such t h a t

4 i s a l i n e a r isomorphism for Re

6. Bi.jections a s s o c i a t e d with t h e d i f f e r e n t i a l operator

A 2

0.

In t h i s s e c t i o n t h e space V ( E , a ) i s provided with a f i x e d norm and r e l a t e d i n n e r product (

,

Consider t h e diagram

)".

X,.

1 1 1 IV

237

DEGENERATED DIFFERENTIAL OPERATORS

where J i s t h e i n c l u s i o n map of V o ( E , a ) i n t o H ( E,n ) and J* i t s a d j o i n t d e f i n e d by

(6.1)

( J s , t ) H= ( s , J * t ) V

for a l l s

Vo(E,a) and a l l t

E

H(E,a).

E

Since T o ( M , E ) c Vo(E,a) c H(E,a) and ro(EI,E) i s dense i n H(E,a) it f o l l o w s t h a t J has dense range. This i m p l i e s t h a t J x i s i n j e c t i v e . Moreover, it can e a s i l y be seen t h a t t h e range R ( J * ) t h a t t h e map u +

( S , U ) ~i

of J * c o n s i s t s of a l l s

E

Vo(E,a) such

s continuous on V (E,a)with r e s p e c t t o t h e r e l a t i v e

topology of H(E,a). Mow from (5.36) and ( 6 . 1 ) it follows t h a t f o r all complex

X

numbers

i f s,t

E

Vo(E,a) a n d $ s

E

Fence ( 6 . 2 ) h o l d s for t h e s e t DX of a l l

R(J*).

u

s E V ( E , a ) such t h a t t h e map

B X ( s , u ) i s continuous when V o ( E , a ) i s equipped

-f

with t h e r e l a t i v e topology o f H ( E , o ) and a l l t

Vo(E,a).

E

Now it i s a consequence of t h e d e f i n i t i o n of t h e s e s q u i - l i n e a r form B

X

that the set D

h

i s independent of A . So we may drop t h e index A.

Define t h e l i n e a r map L we know t h a t Hence L

X

dX

X

o f D i n t o H(E,a). From theorem 5

= (J*)-'o<

i s a l i n e a r isoinorphism f o r a l l complex numbers h with Re X L X 1.

has f o r Re X 2 X 1

as i n v e r s e t h e continuous l i n e a r map GX =ZX1oJ*w i t h

domain H(E,a) and r a n g e D . Summarizing t h e above d i s c u s s i o n we have t h e f o l l o w i n g lema

Lemma

1. Let

J * be t h e ad,joint of t h e i n c l u s i o n map J o f V o ( E , a )

l e t D be t h e s e t c o n s i s t i n g o f a l l s

E

H(E,a)

&

Vo(E,a) such t h a t t h e m a p

u + B ( s , u ) i s continuous when Vo(E,n) is provided w i t h t h e r e l a t i v e topology of

x

H(E,a). Then t h e l i n e a r map LA = ( J * ) T b 4 , of D i n t o H ( E , u ) i s for Re h 2 X 1 b i j e c t i o n of D onto H(E,

01).

Re X L X , t h e i n v e r s e GX

Here

of LA

4

i s continuous.

We want t o e l u c i d a t e t h e s t r u c t u r e of t h e o p e r a t o r s {LA] t o give

5

X, a r e g i v e n i n theorem 5 . Moreover, f o r (X

E

C ) and

a more c o n c r e t e d e s c r i p t i o n of t h e s e t D . This we s h a l l do i n t h e f o l l o -

wing lemma Lemma ------8. -

Let

L 2 ( E ) be i d e n t i f i e d a n t i - l i n e a r l y w i t h a s u b s e t of t h e t o p o l o g i c a l

dualb)(M,E)'

(6.3)

OX(X

t

( O s ) ( t ) = s( O X t )

f o r each s E&(M,E)'

by t h e i d e n t i f i c a t i o n s

of.&(M,E)

Extend t h e domain 0

E

(t + (t,s)o).

-+

C ) topb(M,E)' by t h e d e f i n i t i o n s

t ( O p ) ( t ) = s( O X t )

and each t E$)(M,E).

j o i n t i n L2(E) of t h e operator 0

Here '0

t

0

a

denote t h e formal ad-

O x r e s p e c t i v e l y . Then

R .MARTIN I

238

D = and L x

s

{ s e Vo(R,a)

!

e H(E,a)>

0s

= 0 s for each s e D and each c6mplex number A . A -

R e c a l l i n g t h e d e f i n i t i o n o f B i n (5.3) it f o l l o w s h

Proof. F i x a complex number A .

-----

that for a l l s,u

6A ( s , u ) =

(6.4)

To(M,E)

(s,toA(yO.

=

(AAS,U)H

A

Therefore s i n c e B

for a l l s

E

x

i s t h e c o n t i n u o u s e x t e n s i o n of BA t o V o ( E , u )

Vo(E,a),u

E

Bh (s,u) = ( L X ~ , ~ =) H E

D and a l l u

(oAs)(;)

= (LASH:).

Thus f o r a l l s

V o ( E , d we have

ro(M,E)

From ( 6 . 2 ) we o b t a i n t h a t for a l l s

(6.6)

x

E

E

D and a l l u

E

ro(E,a)

= (LAs)(t)*

ro(F4,E) we have

Remembering t h a t LA maps i n t o H ( E , a ) t h i s i m p l i e s t h a t f o r a l l s we have L s = 0 s and t h a t

x

x

s

Conversely, suppose t h a t s

Oxs

E

(6.7)

E

D

= Xs - O A s b e l o n g s t o H ( E , d . E

V o ( E , a ) such t h a t 0s

H ( E , a ) and from e q u a l i t y ( 6 . 5 ) we o b t a i n f o r a l l u

--

B A ( s , u ) = ( 0X s () (Y ~ =) (%,OAs)o = (Oxsy~)o=

E

H(E,a). Then a l s o

r0(M,E)

(Oh~,U)H.

S i n c e ro(M,R) i s d e n s e i n V o ( E , u ) we may e x t e n d t h e e q u a l i t y between b o t h ends of ( 6 . 7 ) t o : For a l l u

BA ( s , u ) =

(OAs,u)

E

Vo(E,a) we have

H'

Hence t h e map u + B ( s , u ) i s c o n t i n u o u s on V ( S , d when Vo(Ey a) i s p r o v i d e d w i t h

x

t h e r e l a t i v e t o p o l o g y o f H(E,ar). proof of lemma

This implies s

E

DA = D , which c o m p l e t e s t h e

8.

Combining lemma 7 and lemma 8 w e o b t a i n t h e f o l l o w i n g theorem Theorem -----6. Let L 2 ( E ) he i d e n t i f i e d c o n j u g a t e l i n e a r l y w i t h a s u b s e t of t h e t o p o l o -

_---

239

DEGENERATED DIFFERENTIAL OPERATORS

g i c a l d u a l b ( M , E ) ’ of&(M,E) by t h e i d e n t i f i c a t i o n s domain of 0

X

(A

E

C ) &&(M,E)’

t (os)(t) = s( O X t ) f o r each s c & ( M , E ) ’

+

( t + ( t , s ) ). Extend t h e

by t h e d e f i n i t i o n s

t

( O X S ) ( t ) = s( O X t )

Here

and each t E & ( Y , E ) .

and

t h e r e s t r i c t i o n o f OX t o t h e

OX respectively.

i n L 2 ( E ) of t h e o p e r a to r 0

t OX d e n o t e t h e f o r m a l a d j o i n t

set D = {s

E

V (E,a)

I

0s

E

H(E,a)l

i s a b i j e c t i o n of D o n t o H ( E , a ) f o r a l l complex numbers h l a r g e , s a y f o r Re h t A 1 .

t inuous

7.

X

Moreover, f o r Re

.

2

with Fe

X 1 t h e inverse

0

h sufficiently

of 0 A - h

(ni s con-

R e l a t i o n s w i t h semi-groups of o p e r a t o r s

I n t h i s s e c t i o n we s h a l l connect t h e r e s u l t s o f t h e p r e c e d i n g s e c t i o n s w i t h t h e a n a l y t i c t h e o r y o f semi-groups o f o p e r a t o r s . Our c o n n e c t i o n w i l l b e made by means of t h e theorem o f Hille-Yosida, Banach s p a c e s . See YOSIDA

f o r m u l a t e d in t h e v e r s i o n we need; f o r

C51, p . 248-249.

Theorem _--------7. Let X b e a Banach space and l e t A b e a l i n e a r o p e r a t o r w i t h domain

D ( A ) X and r a n g e R ( A ) X:Then

A i s an i n f i n i t e s i m a l K e n e r a t o r o f a

u n i a y e l v determined s t r o n g l y c o n t i n u o u s semi-Krour,

T of c l a s s

(C.

satisfying

I I T ( t ) l \ 5 exD ( B t ) f o r a l l tcCO,m), where p, t 0 i f and o n l y i r t h e f o l l o w i n g two conditions a r e f u l f i l l e d .

i)

D ( A ) i s d e n s e i n X.

ii)

The r e s o l v e n t R ( X : A )

= ( X I - A)

-1

exists for all r e a l X sqfficientlx

l a r g e and s a t i s f i e s

We s h a l l p r o v e t h e f o l l o w i n g s t a t e m e n t

Theorem _-------8.

Let 0 b e t h e d i f f e r e n t i a l o p e r a t o r d e f i n e d i n ( 5 . 1 ) and l q t L b e t h e o p e r a t o r induced by 0 on D ( s e e theorem

6 ) . Then L i s t h e i n f i n i t e s i -

mal g e n e r a t o r o f a s t r o n g l y c o n t i n u o u s semi-group of c l a s s ( C o ) on tQe H i l b e r t space H(E, a ) .

R .MARTINI

240

Proof, ----- From t h e r e s u l t s of t h e preceding s e c t i o n it f o l l o w s t h a t t h e domain

D ( L ) = D i s dense i n H ( E , a ) and t h a t A

X

f o r a l l r e a l numbers A with

2 A,.

-L

i s a b i j e c t i o n of D o n t o H(E,a)

Hence it i s s u f f i c i e n t t o prove t h a t t h e r e

e x i s t s a c o n s t a n t B such t h a t

for a l l u

E

D and a l l A 2 A t .

For when t h i s i s t h e c a s e a l l t h e c o n d i t i o n s i n t h e

theorem o f Hille-Yosida a r e f u l f i l l e d . Now from t h e e s t i m a t e s of t h e preceding s e c t i o n , e s p e c i a l l y (5.34) it follows t h a t t h e r e e x i s t s a 6 2 0 such t h a t

-

1BX(s,s)I 2 ( A

(7.1) for a l l s

E

fi)I/slIi

Vo(E,a) and a l l A 2 0. From t h e remarks around ( 6 . 2 ) we o b t a i n

B X ( s , s ) = ( L A ~ , ~ =) H( ( A

(7.2) when s

E

D. Hence, when s

(7.3)

E

l(LXs’s)Hl 2 ( A

-

L)s,s)H

D and X 21 A , it follows from (7.1) and ( 7 . 2 ) t h a t

-

8 ) l Is1. ;1

Then by Schwarz’ i n e q u a l i t y we o b t a i n t h a t f o r a l l s

E r)

which completes t h e proof. I n t h e c a s e of l i n e bundles we have a s s o c i a t i n g with t h e d i f f e r e n t i a l o p e r a t o r 0 an o p e r a t o r i n t h e H i l b e r t space L2(F,) i n s t e g d of i n t h e F i l b e r t space

H(E,a) t h e following r e s u l t

---------

Theorem 9. Let E be a l i n e bundle and l e t 0 be t h e d i f f e r e n t i a l o p e r a t o r d e f i n e d i n ( 5 . 1 ) and l e t L be t h e o p e r a t o r induced by 0 on D ( s e e theorem

6).

Moreover, l e t

be bouded on M \

aM. Here da denotes t h e d i f f e r e n t i a l of a. Then w e assert

i)

L i s c l o s a b l e i n t h e H i l b e r t space L2(F).

ii)

The s m a l l e s t c l o s e d e x t e n s i o n

E

of

L i n L 2 ( E ) i s an i n f i n i t e s i m a l gene-

rator of a s t r o n g l y continuous semi-group of c l a s s

( c 0 ) on

L~(E).

241

DEGENERATED DIFFERENTIAL OPERATORS

Proof -------i . We have t o p r o v e ( s e e e.R.

YOSIDA r51, p . 77-78) t h a t for each sequence

...)

(n = 1, 2,

i n t h e domain T)(L) of L w i t h l i m f n+= g it f o l l o w s t h a t g = 0. (f,)

l i m Lf nT h e r e f o r e suppose t h a t f n

... ),

l i m f n = 0 and l i m L f n = g n+nuo i n t h e s e n s e of convergence i n L 2 ( E ) . Then f o r e a c h u E r (EI,F) we have

E

L2(E)

t ( u , L f n ) o = ( Lu.fn)o,

(7.5) where

E

D ( n = 1 , 2,

= 0 and.

t

L

.

IS

t h e f o r m a l ad,joint i n L 2 ( E ) o f L . By t a k i n g l i m i t s a t b o t h s i d e s of

e q u a t i o n ( 7 . 5 ) we o b t a i n ( u , g )

= 0 f o r each u

Proof ii. Analogous t o t h e proof o f theorem --------

E

r

( h 4 , F ) . Fence p

= 0.

8 it s u f f i c e s t o show t h a t t h e r e

e x i s t s a c o n s t a n t 6 ' such t h a t

for all s

E

D and a l l X 2 1,. For i n t h i s c a s e it f o l l o w s from t h e f a c t s , H ( F , o )

i s dense i n L 2 ( E ) , A any h 2 h 1 X

for a l l s

E

-

-

=

X-L and t h e c o n t i n u i t y of t h e norm 1 1 I ,I t h a t n(L) o n t o L 2 ( P ) s a t i s f y i n g t h e i n e q u a l i t y

for

i s a b i j e c t i o n of

D ( L ) . Now t h e p a r t of t h e proof which remains t o h e proved p r o c e e d s

i n p r a c t i c a l l y t h e same d i r e c t i o n as t h a t of lemma

6 i n s e c t i o n 5 . So we u s e t h e

same n o t a t i o n a s i n t h a t proof. Then we have t h e f o l l o w i n g e q u a l i t y for any

s

E

r

( M , E ) (compare 5.18)

Now 0 i s a u n i f o r m l y s t r o n g l y e l l i p t i c d i f f e r e n t i a l o p e r a t o r of t h e second o r d e r a c t i n g on a l i n e b u n d l e . Hence t h e r e e x i s t s a c o n s t a n t kZ2 > 0 such t h a t f o r t h e f i r s t term of t h e r i g h t hand s i d e o f (7.7) w e have

R. M A R T I N I

242

Using i n e q u a l i t y

(5.19) we s e e t h a t t h e r i g h t hand s i d e o f (7.8) i s

not s m a l l e r t h a n

By t h e assumption d . it f o l l o w s t h a t t h e r e e x i s t s a c o n s t a n t k t h a t t h e second term o f t h e r i g h t hand s i d e of

and u s i n g i n e q u a l i t y t e d by

( 7 . 7 ) i s e s t i m a t e d by

(5.20)a g a i n it f o l l o w s t h a t f o r any

E

23

> fl( 7 . 1 0 ) i s domina-

Also t h e r e e x i s t s a c o n s t a n t kZ4 such t h a t t h e t h i r d t e r m o f

(7.7) i s

e s t i m a t e d by

Combining

f o r any s

E

r

(M,E).

(7.7)-(7.12), ( 7 . 1 0 ) w i t h a s u i t a b l y chosen

Here k

25

Similar t o equality

and k26 are c o n s t a n t s such t h a t

(5.27) we have for any

s c

r

such

E,

k

25

(ff,Z)

yields

' O*

243

DEGENERATED DIFFERENTIAL OPERATORS

such t h a t f o r and s i m i l a r t o ( 5 . 3 0 ) we s e e t h a t t h e r e e x i s t c o n s t a n t s kZ6 and k 27 any s E To(M,E)

Obviously, t h e r e e x i s t s a c o n s t a n t k28 such t h a t f o r a n y s

E

To(M,E)

we have

Wow from ( 7 . 1 3 ) , ( 7 . 1 5 ) and ( 7 . 1 6 ) it f o l l o w s t h a t t h e r e e x i s t c o n s t a n t s E

k

> 0 ( k = 1 , 2,

..., m )

and F ( k = 1 , 2 , k

..., m )

such t h a t

"ow l e t C be a number such t h a t t h e s e t s {U') (k = 1 , 2,

k

..., m )

d e f i n e d by

Uk = {x form a c o v e r i n g of

8' such t h a t

E

Mluk(x) > C ) ?I.

Then from ( 7 . 1 8 ) it f o l l o w s t h a t t h e r e e x i s t s a c o n s t a n t

R.MARTIN1

244

f o r a l l U c D and s i n c e elements o f D can b e approximated with r e s p e c t t o t h e topology of V (F,a)by f u n c t i o n s belonging t o r o ( V , F ) we s e e by t a k i n g l i m i t s a t both s i d e s of ( 7 . 1 9 ) t h a t (7.20)

Re ( L X s , s )2 ( A

for a l l U

E

8.

-

f i ' ) l l s l I o2

D and a l l X 2 0. This completes t h e proof of theorem

9.

A method f o r t h e c o n s t r u c t i o n of semi-groups.

We r e c a l l t h a t i n s e c t i o n Vo(E,a) into H(E,a).

4 we

have i n v e s t i g a t e d t h e embedding of

The q u e s t i o n whether t h i s embedding i s compact was our main

i n t e r e s t and a p o s i t i v e r e s u l t concerning t h i s p o i n t was s t a t e d i n theorems 2 and

4. In o r d e r t o g i v e a method f o r t h e c o n s t r u c t i o n of semi-groups we suupose t h a t t h i s indeed i s t h e c a s e . Thus we suppose t h e embedding J of V ( E , a ) i n t o H ( E , a ) t o be compact. I n s e c t i o n

5 we proved t h a t t h e o p e r a t o r LX = 0 X de-

f i n e s a b i j e c t i o n of t h e s e t D = { s

V (E,a)lOs

Re

E

s u f f i c i e n t l y l a r g e , l e t say f o r Re X t X 1 .

nuous i n v e r s e of

L f o r Qe X A

E

H ( E , a ) } onto H ( E , a ) f o r

By GX we have denoted t h e c o n t i -

2 A,.

Now consider t h e f o l l o w i n s diagram

It follows t h a t t h e o p e r a t o r G i = J o GX, G i : H(E,(r) + H ( E , a ) ,

i s compact a s a

composition of a compact l i n e a r and a continuous l i n e a r o p e r a t o r . I n a d d i t i o n i f t h e o p e r a t o r @ i s symmetric on D i n H ( E , a ) with r e s p e c t t o some norm

I I I IH;

i.e.,

(8.1)

( o s , t ) H= (s,Ot&

for a l l s , t

E

D , it follows t h a t f o r any r e a l number X w i t h X t X 1 t h e o p e r a t o r

G i i s s e l f a d j o i n t . Thus i n t h i s c a s e

it i s p o s s i b l e t o apply t h e s p e c t r a l t h e o r y

f o r compact s e l f a d j o i n t o p e r a t o r s t o G;. Then G i has t h e r e p r e s e n t a t i o n

245

DEGENERATED D I F F E R E N T I A L OPERATORS

Tor any s

H(E,a), where

E

{p

x1 k

i s t h e s e t of e i g e n v a l u e s o f G{ and where {Skk}

( E c I ) i s a n orthonormal set of e i g e n v e c t o r s b e l o n g i n g t o t h e e i g e n v a l u e p x

k

k'

Now G' c o i n c i d e s w i t h t h e r e s o l v e n t R ( X ; L ) and between t h e semi-group h

{ T ( t ) } ( t t 0) c o r r e s p o n d i n g t o t h e i n f i n i t e s i m a l g e n e r a t o r L and t h e r e s o l v e n t

R ( A ; L ) t h e r e e x i s t s t h e r e l a t i o n ( s e e ( 2 . 5 ) i n c h a p t e r I)

m

(8.3)

R(h;L)f = Je-ItT(t)f 0

dt.

Hence

m

(8.4)

G X f = /e

-At

T ( t ) f dt.

0 So by i n v e r s i o n o f

(t

9.

2

(8.4) we o b t a i n a r e p r e s e n t a t i o n f o r t h e semi-Rroup { T ( t ) l

0 ) c o r r e s p o n d i n g t o t h e i n f i n i t e s i m a l g e n e r a t o r L.

Examples.

I n t h i s s e c t i o n we s h a l l g i v e some examples t o i l l u s t r a t e t h e r e s u l t s of t h e p r e c e d i n g s e c t i o n s .

-

Example -------- 1 . Let M b e t h e u n i t - h a l l Bn.i n t h e n-dimensional E u c l i d e a n s p a c e lRn l e t E be t h e t r i v i a l complex bundle Bn x C'. operator 0 : rO(M,E)

+

and

Define t h e d i f f e r e n t i a l

r o ( M , E ) by

(Os)(x) = (x,(1

-

IX~~)~AS(X))

if s ( x ) = ( x , Z ( x ) ) and where A d e n o t e s t h e Laplacean ID: and where [ x I 2 = Cx?. i i 1 Then 0 s a t i s f i e s t h e assumptions of theorem a(x) = ( 1

-

8 i n case p

> 0 . Moreover, t a k e

1xI2)' and l e t p ( x ) be t h e d i s t a n c e o f a p o i n t x

o f M. Then s i n c e f o r each x

E

E I4

t o t h e boundary

M we have

it f o l l o w s from theorem 4 t h a t t h e embedding o f V o ( E , a ) i n t o l I ( E , a ) i s compact p r o v i d e d t h a t p < 2.

Also u s i n g G r e e n ' s f o r m u l a it can e a s i l y be s e e n t h a t t h e e x t e n s i o n o f 0 d e f i n e d i n theorem 6 i s symmetric on D . Thus i n c a s e 0 < p < 2 t h e c o n s t r u c t i o n method of s e c t i o n 8 c a n b e a p p l i e d .

246

R. MARTINI

Example --_-_-_-2. Let M be the cylinder R 1

x

I embedded canonically in the 3-dimensional is the 1-dimensional unit-spli~reand

Euclidean space I R 3 . Here 5'

I = r0,ll. Let E be the trivial complex vector bundle over

'1.

Define the linear differential operator 0 : r o ( M ,F ) + r o ( M , E ) by (Os)(x) = (x,a(x)A;(x)) if s(x) = (x,$(x)) and where in cylinder-coordinates a(x) = xp(l - x )' and where 3 3 A is the Laplacean given in the same coordinates by

Then0 satisfies the conditions of theorem and the conditions of theorem 8 in case p

2

7 in case p

> 0 and q > 0

2 and q 2 2. Fvidently, we can define

a similar differential operator on the cylinder Pn

X

I embedded in the (n+2)-

Of course, +* Sndenotes . the n-dimensional unitdimensional Euclidean space I!? sphere. Example -_- ------3. Let M be an oriented n-dimensional Riemannian compact manifold with boundary 3E.I and let

be the vector bundle of complex-valued differential forms on M. As usual, d : r(M,F)

Let

*

+

r ( M , E ) denotes exterior derivation.

be the real linear automorphism of E such that if e l , e2,

is an oriented orthonormal basis for T*(M) and 1 , 2,

...,

A

e2

*1 = e

if

0

A

A

... e2

A

A

e

...

= 1; A

e n ''

is even and

*eO(l) U

..., e

a permutation of the integers

n it follows that *el

if

0

A

eo(2)

is odd. Obviously,

A

..*

A

eo(p) = -eo(p+l)

* maps AP(T*(M))C

the automorphism of E which on AP(T*(M)), It follows that the operation

A

*.*

into An-P(T*(EI))C

and

**

=

w, where

is multiplication by (-l)p(n-p).

* induces a conjugate linear real automorphism of

r(M,E), which also will be denoted by

*.

is

247

DEGENERATED DIFFERENTIAL OPERATORS

The Riemannian s t r u c t u r e f o r

M

d e f i n e s a s t r i c t l y p o s i t i v e smooth

If t h e n o t a t i o n for t h e Riemannian s t r u c t u r e i s (

measure p on h4.

measure has w i t h r e s p e c t t o t h e c h a r t c = (U,$) a l o c a l d e n s i t y

0

,

)

M , then t h i s

which i s g i v e n

by

where g . . ( x ) = (dxOi, d x + j ) h n .

LI

I t a l s o i n d u c e s a Hermitian s t r u c t u r e <, >E f o r E . T h i s we s h a l l d e s c r i b e now. Let e

,,

e2,

...,

s t r u c t u r e <,

e

b e an o r i e n t e d orthonormal b a s i s f o r T*(M).

Then t h e F e r m i t i a n

i s d e f i n e d so t h a t t h e s e t

>

E

{ei 1

he. l2

...

he. ) ( I s p s n , t s i , < i 2 < P

A . . .

< i s n ) P

form a n orthonormal b a s i s f o r A ( T * ( M ) ) . Let To(M,F) b e p r o v i d e d w i t h t h e i n n e r p r o d u c t , g i v e n by

Then, if w i t h r e s p e c t t o t h i s i n n e r p r o d u c t t h e f o r m a l a d j o i n t o f t h e e x t e r i o r d e r i v a t i o n d i s d e n o t e d by t d , it f o l l o w s t h a t

td=;*d

*w,

where w is t h e automorphism of E which is m u l t i p l i c a t i o n by ( - 1 ) ’

on AP(T*(M))

C

( s e e PALAIS C41, chp. IV, p . 76-77). Now l e t a b e a c o n t i n u o u s r e a l - v a l u e d f u n c t i o n d e f i n e d on M such t h a t m

t h e r e s t r i c t i o n of a ( x ) = 0 when x

E

t o t h e i n t e r i o r E4 \ aM i s C

ad : r o ( M , E ) + r0(M,E)

it f o l l o w s t h a t i t s f o r m a l a d j o i n t s a t i s f i e s t

(ad) = a ;

*

d

*

Hence some c a l c u l a t i o n s y i e l d

and

and p o s i t i v e and such t h a t

aM. Then f o r t h e d i f f e r e n t i a l o p e r a t o r

w +

*

(&)

A

*

w.

R.MARTINI

248

So i f t h e Laplacean A i s d e f i n e d by

A =

-

{d(td)

+ (td)d}

it f o l l o w s from t h e e q u a l i t y

t h a t t h e d i f f e r e n t i a l o p e r a t o r 0 : To(M,E)

-

0

t

{a3 ( a d )

-f

r

(M,F) d e f i n e d by

+ P(a3))ndj

equals 0

= a2A + aY,

where Y i s t h e d i f f e r e n t i a l o p e r a t o r o f t h e first o r d e r g i v e n by Y =

-

{(a")

A

w

*

d

*

W

+ d;

*

(da)

A

*

w

+

2 ;

*

(dn)

A

*

wd}

S i n c e A i s s t r o n g l y e l l i p t i c t h e d i s c u s s i o n above shows t h a t t h e t h e o r y developed i n t h e p r e c e d i n g s e c t i o n can b e a p p l i e d t o t h e d i f f e r e n t i a l o p e r a t o r 0 j u s t defined.

RCFmENCES

'11

DDNFORD, N . and J.".

c21

IIARTINI, R . , D i f f e r e n t i a l O p e r a t o r s d e g e n e r a t i n g a t t h e Qoundary a s I n f i n i t e s i m a l G e n e r a t o r s o f Semi-groups, m e s i s , D e l f t ( 1 0 7 5 ) .

r31

PiARASIMHAN,

C41

PALAIS, R., Seminar on t h e Atiyah-Singer Index Theorem, Ann. o f Math.,

141

YOSIDA, K . ,

Sf'Fl'ARTZ, L i n e a r O p e r a t o r s , Vol. 1 , I n t e r s c i e n c e P u b l i s h e r s ( 1958).

R., k n a l y s i s on Real and Complex M a n i f o l d s , North-Holland Publ. Comp. ( 1968). Study 57, P r i n c e t o n ( 1 9 6 5 ) . F u n c t i o n a l A n a l y s i s , Grundlehren d . Vath.

Iliss.,

SpringerV e r l a g ( 1968).