G. Fichera ( E d.)
Autovalori e autosoluzioni Lectures given at the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Chieti, Italy, August 1-9, 1962
C.I.M.E. Foundation c/o Dipartimento di Matematica “U. Dini” Viale Morgagni n. 67/a 50134 Firenze Italy
[email protected]
ISBN 978-3-642-10992-8 e-ISBN: 978-3-642-10994-2 DOI:10.1007/978-3-642-10994-2 Springer Heidelberg Dordrecht London New York
©Springer-Verlag Berlin Heidelberg 2011 st Reprint of the 1 ed. C.I.M.E., Ed. Cremonese, Roma, 1962 With kind permission of C.I.M.E.
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CENTRO INTERNATIONALE MATEMATICO ESTIVO (C.I.M.E)
Reprint of the 1st ed.- Chieti, Italy, August 1-9, 1962
AUTOVALORI E AUTOSOLUZIONI
S. Agmon:
On eigenvalues eigenfunctions and resolvents of general elliptic problems ..................................................
1
A. M. Ostrowski:
Il Metodo del quoziente di Rayleigh .................................... 41
L. E. Payne:
Isoperimetric inequalities for eigenvalues and their applications ............................................................ 107
L. De Vito:
Calcolo degli autovalori e delle autosoluzioni per operatori non autoaggiunti.............................................. 171 Sul calcolo per difetto e per eccesso degli autovalori delle transformazioni hermitiane compatte e delle relative molteplicità ................................... 181
J. B. Diaz:
Upper and lower bounds for the torsional rigidity and the capacity, derived from the inequality of Schwarz ............................................................................ 187
M. Schiffer:
Fredholm eigenvalues and conformal mapping ................... 203
ON EIGENVALUES EIGENFUNCTIONS AND RESOLVENTS OF GENERAL ELLIPTIC PROBLEMS Shmuel Agmon
Introduction
In these lectures we shall describe some recent results concerning the spectral theory of general non-self-adjoint elliptic boundary value problems. We shall be interested in the following problems: (i) Completeness of eigenfunctions. (ii) Angular distribution of eigenvalues. (iii) Asymptotic distribution of eigenvalues. The general plan of the lectures is as follows. In Lecture I we shall introduce the general class of regular elliptic boundary value problems and discuss the growth of certain resolvents in the complex plane. In Lecture II we shall establish completeness results for eigenfunctions of general elliptic problems obtaining also some results on the angular distribution of eigenvalues. In Lecture III we shall discuss some special classes of elliptic problems such as self-adjoint problems and absolutely elliptic problems. In Lecture IV we shall describe a very general result on the asymptotic distribution of eigenvalues of non-self-disjoint elliptic problems. We note that the first three lectures are taken from the author's paper [ 1] which is due to appear shortly, whereas the material of the last lecture on the asymptotic distribution of eigenvalues is new.
1
-2S. Agmon Lecture I Regular Elliptic Boundary Value Problems and Growth of Resolvents
We denote by G a bounded domain in n-space with boundary
d G and
closure G. We let x = (xl" ..• x ) be the generic point in E and use the n
n
notation:
denoting by
a general derivativ.e.Here ex stands for the multi -index
0(
= ( 0( l' ..•• 0( n)
whose length 0/ 1 + .•. +~ n is denoted by 10(1 • We consider complex valued functions u(x) defined in G (or G). For u E Cj(G) we introduce the L norms (p ~ 1): p (1. 1)
II ull "L
(G) = J P
~ fa I uI (L. IC/I~ J 01
D
p
dx
) IIp
G
The completion of Cj(G) under the norm (1. 1) is a Banach space of functions denoted here by H. L (G). If the boundary is Lipschitzian, H. L (G) coinJ. J. cides with the sUbclais of functions in L (G) whose derivatives irfthe distrip bution sense of order~ j are functions belonging to L (G). p
We shall denote byA (x;D) an elliptic linear differential operator in G (variable complex coefficients) of even order 2m. Thus the characteristic 3
- 3polynomial associated with the principal part
S. Agmon
it ofAsatisfies: I
(1. 2)
for all real vectors
§ = ( 5 l' ... '5 n) # 0 and x € G. For n = 2 we shall also
always assume thatJ/. satisfies the ROOTS CONDITION. For every pair of linearly independent real vectors
5, 1) ~ x €
G the polynomial in t:
Ji
I
(x;
'5 + t 1] ) has
exactly
I
m roots with positive imaginary parts. As is well known this condition is always satisfied if n and the coefficients of
~
3 or if n = 2
AI are real.
We shall be interested in boundary value problems of the form: A(x;D)u(x) = f(x)
in G
Bj(x;D)U(x) = 0
on oG,
(1. 3)
where {B j }
j
= 1, ... , m,
7=1 . is a given system of m linear differential 9perators with
coefficients defined on the boundary. We shall use the symbol (cR, {B.}; G) J to denote the boundary value problem (1. 3) (omitting reference to the arbitrary given function f ). The general theory for higher order elliptic boundary value problems of
th~
form (1.3) depends on suitable a priori estimates for the solution u.
For these to hold it is necessary to restrict the class of problems by an alI
gebraic condition. Denoting the principal part of B. by B. this condition is J J the following: COMPLEMENTING CONDITION. At any point x of ClG ~ the normal to
oG and 5 # 0
v denote
a real vector parallel to the boundary. We reI
quire that the polynomials in t, B.(x;§ + tv), j = 1, ... , m, be linearly inJ 4
- 4dependent modulo the polynomial
,
A (x; 5+ t 1»
+ + M(t - tk(g) where t k m
S. Agmon
(§) are the roots of
with positive imaginary parts.
Suppose that the Complementing Condition hol,ds. thatthe Bj are of order m. ~ 2m, and that the domain and the differential operators satisfy the foIl lowing SMOOTHNESS ASSUMPTION. G is of class C2m• The leading coefficients of eft are continuous in
G.
the other coefficients being measurable and bounded. The coefficients of B.. j:::: 1, ... , m belong. to C2m - mj on the bounJ dary. Under the above assumptions the following a priori estimates hold: THEOREM 1.1. Consider the class of functions u in C2m(G) sati-
sfying the boundary conditions: B.u = 0 on J
(1. 4)
oG,
j=1. •••• m.
and let 1 < P < 00. Then:
where C is some constant depending on
JL.
{B j}' G and P. but not on u.
A proof of this theorem in a more general situation is given in [5]. We shall denote by H2m• L (G; {Bj } ) the completion in H2m • L (G) of the p C2m(G) satisfying the boundary con&tions (1.4). class of functions in Clearly Theorem 1.1 holds for all functions u E H2 L (G; {B.} ). . ~ J A boundary system of differential operators is called a ~
{BiJ
system if: (i) The boundary 'OG is non-characteristic to B. at each point. J (ii) The orders of the different operators are distinct. 5
- 5S. Agmon We introduce the following DEFINITION 1. 1. An elliptic boundary value problem
en, {BjJ 7;G)
is called a regular problem if
J.1.. (of order 2m and satisfying the roots condition) together with the boundary system {B j } satisfy the Complementing (i) The elliptic operator
Condition. (ii) of orders
~
7 is a normal boundary system of m differential operators
{Bj } ?m-l.
(iii) The smoothness assumption on the domain and the coefficients introduced above holds. In the following all elliptic boundary value problems will be regular. Let (eft,
{Bj }
number: 1 < p <
00 •
7;G) be a regular elliptic problem and p some fixed
We shall denote by A the linear unbounded operator in
L (G) defined as follows: p
(i) The domain of A is (ii) For u €
15 A'
b A = H2m , L
Au =J1.(x;D)u.
(G; {B j }
7)·
p
The operator A is clearly closed and it follows easily from the a priori estimates (1. 5) that the null space of A is finite dimensional and that its range is closed. If the spectrum of A is not the whole complex plane, i. e. if the resolvent:
RO ;A) = or
(1. 6)
- A)-l
exists for some ).. = )\ , then it follows readily (since R( ), ;A) is compact)
o
0
that R( ). ;A) exists for all :A except for a discrete sequence of points {An} which are the eigenvalues of A. In general, however, one cannot exclude the possibility that the spectrum of A· is the whole complex plane. In the following we shall consider a subclass of regular problems for which it is possible to assert that the spectrum of A is discrete. In addition 6
-6S. Agmon we shall obtain estimates for the growth of R(
A;A) along certain rays in the
complex plane. In this connection we introduce DEFINITION 1. 2 A ray arg ~ = 9 in the complex ). -plane is said to be a ray of minimal growth of R( ). ;A) if the resolvent exists for all ). sufficiently large on the ray, and if, moreover, for all such ). :
\\R( ). ;A)II ~
(1. 7)
ill,
c > 0 a constant. For regular elliptic boundary value problems
(A, , {B j } 7;G)
one can
determine the rays of minimal growth of the associated resolvent. The basic result here is the following THEOREM 1. 2. In order that the spectrum of A be discrete and the ~
arg .A =
e be a ray of minimal growth of R ( ~ ;A) it is s?fficient,
and
in case p = 2 also necessary, that the following two conditions be satisfied:
(i)
for all real vectors
5 #0
and all x E G.
(ii) At any point x of
d G let
v be the normal vector and let 5 # 0 + 5 ; A) the
be any real vector parallel to the boundary at x. Denote by t k( m roots with positive imaginary parts of the polynomial in t:
(-1)
m
r.A
I
(x; § + tv) - ). ,
where ). is any number on the ray arg ).. = 9 . Then the polynomials (in t) I
B.(x; § +tl)),
j=l, .. .,m,
J
7
-7S. Agmon are lin-early independent modulo the polynomial
litm
+ (t - \(
5 ; A )) .
In order to establish the sufficiency part of the theorem one needs to show that (a) Under the conditions of Theorem 1. 2 tal' all functions and all ) sufficiently lar~e on the ray arg ), =
1rl
~
15 A
e , the following inequality
holds:
II
(1. 8)
~
u L (G) p
(b) The range of A -
~
~
1).1
II(A -
AI is Lp(G) for
A) uil L
all
(G) , p
A sufficiently large on the
ray. Proof of (a): The inequality (1. 8) is a special case of a more general result to be proved in [6] . We shall reduce the proof of (1. 8) to a variant of Theorem 1.1 for a regular elliptic boundary value problem in n+! variables. To this end introduce a new real variable t, put Dt = ~ and replace in nt1 variables defined by: D by D. Consider the differential operator
t
x
From condition (i) of Theorem 1. 2 it follows that ,£ is an elliptic operator of order 2m in the closure of the cylindrical domain
r = {(x, t) : x € G,
- 00
.(
t .( oo}. Moreover, it is readily checked that
condition (ii) of the theorem is equivalent to the following: the elliptic operator [ and the boundary system
{Bj } satisfy at each point of 0r the
menting Condition introduced above. Consider the class of functions 8
Compl~
- 8S. Agmon 2m v(x, t) E. C ( r ) such that v :: 0 for
It I ~
B .(x, D )v = 0 on J x
(1. 10)
0
1, and
r
for j = 1, ... ,m •
For functions v in this class the following a priori estimate holds:
where C is a constant. Ttle estimate (1. 11) follows from the localized version of Theorem 1. 1. For a proof see proofs of Theorems 15,1 and 15.2 in [5] which implicitly contain this result (the corresponding result for the Schauder estimates is explicitly stated in [51 as Theorem 7.3). Next, let ~ (t) be some fixed ClJlfunction such that
It I
?: 1,
C(t)
:: 0 for
C(t) :: 1 for I t I ~ 1/2. Given a function u(x) E C2m (G) such that
(1. 12)
oG ,
B.u = 0 on J
j = 1, ... ,m
,
we define (1. 12) I
Denote by
vp. (x) =
. t
C(t)e1f!
u (x),
rr the part of r in It I
~
fJ- a real number .
r. Since, clearly, the inequality (1. 11)
is applicable to vp.' we have:
Now, ,[ vfJ- =
C(t)eif.l-\ eft _}L2meie )u + linear combination of derivatives of ~ 2m-1 with bounded coefficients. I t I ~ 1/2) we obtain readily from
u(x)e ijJot of order
Using this (noting that
v}J- :: uei,ut for
(1.13):
9
- 9-
S. Agmon
(1. 13) 1
[)
~ C1 (II rfl.'"jJ-
I
J
2m i9 2m-l . e )u L (G) + .2: 12m-1-Jllull. p J = 0 jl J, L (G)
I·
p
with a constant C1 independent of
fJ. or u.
Also, we have:
In(ue ip.t) Ip dxdt
=
(1. 14)
for any j
~
2m. From (1.14) and (1.13)1 we get:
10
- 10
~
S. Agmon (2m+1)C 1
2m-1
If I
p: 0
+
2
z= If I
.
m-J
I u I j. L
p
(G)
which gives (2m+1)C1)
(
1f'1
1-
(1. 15)
~
(2m+1) c 1 1! b~
- ,M-2meit9 )ull Lp (Gl.
. '\ 2m i6 2mPuttmg .I\::}J- e it follows from (1. 15) that for all u(x) E C (G) satisfying the boundary conditions (1. 12) and for all
I,MI ~
I:A II/2m::
that
2m
(1. 16)
~ l::O
2(2m+l)C 1• the following inequality holds:
2m-j 2m
IAI
Aon the ray arg ). :: 9 such
Ilul1i. L •
o\\CJt- A)u II L
(G) p
(G) •
p
where Co " 2(2m+l)C 1 is a constant independent of u or
A.
Clearly. by com-
pletion. (1.16) holds for all functions u€ H2m • L (G; {B.}) and in particular p J (1. 8) follows. This establishes (a). The estimate (1. 8) shows in particular that for
A ::
the ray arg u
-+
:A sufficiently large on
(J the mapping:
(A - ~ )u is a one-to-one mapping from
!JA into Lp(G).
the proof of the sufficienty part of the theorem one still mapping is onto. or that the range of A -
ha~
To complete
to show that the
:x I is the whole of L p(G).
The proof
of this fact is somewhat long and will be given in the author's paper on the existence theory for general elliptic boundary value problems [4). We note that under additional smoothness assumptions one can also deduce this result from the existence re!!ults of Schechter [22] 11
(in L 2) and Browder
[81 .
- 11 -
S. Agmon and from the following observation: if (Jt, {B j } ;G) is sufficiently smooth then there exists a formally adjoint problem (,ft., {B~} ;G) which is regular
J
and which, moreover, satisfies conditions (i) and (ii) of Theorem 1. 2 with respect to the conjugate ray arg ). = -
e.
Finally, let p = 2. We shall sketch the proof of the necessity part of the theorem.
(Jt,
{B.}
More precisely we shall show the following: if
;G) is a regular problem such that the estimate (1. 8) holds for
(G; {B.} ~ then conditions (i) and (ii) of Theorem 1. 2 must all u € H J 2m, L2 J 1 hold. To this end we first show that the regularity of (c.A., {B.} ;G) and
J
(1. 8) imply the stronger estimate (1. 16). Indeed for u € H 2m , L2 (G;
and ~ sufficiently large on the ray arg ).. =
{Bj }
e we have by assumption:
~
(1. 17)
where here and in the following C2' C3' ... denote constants not depending on ).. Also, applying the a priori estimate (1. 5) and using (1. 17), we get:
(1. 18)
We shall use the following known inequality (see, for instance, Nirenberg [19]) 12
- 12 -
1-...L
II u II j, L2(G)
(1. 19)
~ c II u II L2(~~
S. Agmon
...L 1\ u II ::, L2(G)
for
f 2m
where c is a constant depending only on G, nand m. Combining (1. 19) with (1. 17) and (1. 18) we find that
for j
~
2m which is the same as (1. 16). Next consider the class offunctions v(x,t) €: C2m(
F ) which are pe-
riodic in t, with period 2'11: , and which satisfy the boundary conditions:, B,v = 0 on dr, j = 1, ..• ,m. For such v let the Fourier series expansion J
in t be:
v(x, t)
rv
~ un(x)e int .
~
-00
2mClearly u (x) E C (G), B,u = 0 on n
expansions:
J
aG for j = 1. ... , m, and we have the
DOl u (.)/,)k x\m eint , x n (1. 21)
Also, by (1. 20), we have: 13
1011+ k
~ 2m ,
- 13 -
S. Agmon
(1. 22)
for In I ~ N. Applying now Parseval's formula to the derivatives of v, it
o
follows readily from (1. 22) and (1. 21) and (1. 5) that the class of functions v satisfy the following a priori estimate in
Ire
(the part of
I
in It I < Tt ):
(1. 23)
A priori estimates of the type (1. 23) in various norms were considered in [5] where it was shown that the validity of the a priori estimates implies that certain algebraic conditions must hold. By modifying somewhat the argument given in [5] one can show that the same result holds in our case and that the followillg conditions are necessary for (1. 23) to hold: (i)' tic in (x, t) in
F.
(ii)'
plementing Condition on
l
L is ellip-
and the boundary system {B j } satisfy the ComSince conditions (i)' and (ii)' are equivalent
'Or.
to conditions (i) and (ii) of Theorem 1. 2, the necessity part of the theorem follows.
14
- 14 -
S. Agmon Lecture II
Completeness of Eigenfunctions and Angular Distribution of Eigenvalues
Let (
ell, {B.1 m;G) be a regular elliptic boundary value problem of J"
order 2m. For better clarity we shall denote by A the associated operator p in L (G) (previously denoted by A). Thus, ) A = H2 ~ L (G; {B.} P J and A u = u for u E: A . Suppose that the ~ectrum of pA is not the
p
tJ
15
vA.
p
whole plane so that, as wasPalready remarked, it consists of a discrete sequence of eigenvalues {:\}. Let Zo be some fixed point not in the spectrum of A. We put p
T = (A -z I)
(2. 1)
p
-1
0
p
Clearly T is a compact operator in L (G) (it maps L (G) into H2 L (G)) p ppm, having the eigenvalues flk = 1/ (). k -zo)' We also have the fOllowing~elation between the resolvents of T
(2. 1)'
p
and A : p
R(~ ;T ) = ( :A II -z P
z )1 - (
o
An element CP E L (G) p
(¢
0
A- z0 )2R();AP),
A F )k
.
F 0) is said to be a generalized eigenele-
.
ment (eigenfunction) of T p corresponding to the eigenvalue fk if (T P -J\)J¢= 0 for some integer j
~
called the index of
¢.
1. The smallest j for which the above relation holds is As is well known the space of generalized eigenele-
ments corresponding to an eigenvalue fLk is finite dimensional and its dimension is called the multiplicity of
f\. 15
- 15 -
S. Agmon
We shall denote by sp(T ) the closed subspace in L (G) spanned by all gene-
p
p
ralized eigenelements of T . P The operator-valued function R(
A;T p)
is a meromorphic function
of II).. with poles at the eigenvalues ;Uk' Let f E Lp(G) and consider the vector valued function R( A;T )f which is analytic in ) except for the orip
gin and the pointsfA' k which are possibly poles. If
A= f\
R( ~ ; Tp)f then in a suffi ciently s mall neighborhood of
i-\
is a pole of
one has the Lau-
rent expansion:
(2.2)
where j ? 1, TpiA =
¢1 ~ 0 (the 1;'s and g's are elements in Lp(G)). Applying
to (3.2) one finds readily that:
(~1"'" (Tp - fLk ) ¢j
of T
p
= ¢j-l' so that
(Tp-fk)~l
¢i
=0, (T p -flk )¢2=
is a generalized eigenelement
of index i. Similarly a function
¢ E: 1)Ap (¢ ~ 0)
is said to be a generalized
eigenelement (eigenfunction) of A corresponding to the eigenvalue .
P
(A - ~k)J ¢ = 0 for some j ~ 1 (one assumes, of course, that = rAp - \ ) ¢, ¢(2) = (A -
A0 ¢(1),
etc. belong to
15 A/
¢
Ak
(1)
if =
The smallest
integer j for which (Ap- ~i ¢ = 0 is 'again called the index of
¢.
Clear-
¢ is a generalized eigenelement of Ap corresponding to the Ak if and only if ¢ is a generalized eigenelement of Tp corre-
ly a function eigenvalue
sponding to the eigenvalue fk = The closed subspace in
~
:z k
. 0
L (G) spanned by all generalized p
eigenelements of A is denoted by sp(A). We shall say that the generalip p zed eigenelements of the elliptic problem (Jt, {BjG) are complete in 16
- 16 -
S. Agmon L (G), if P (2.3)
We shall first study the problem of completeness in L 2(G). We shall need the following result on the growth of R(
A;T 2)
near the origin in terms
of the dimension n of the underlying Euclidean space and the order 2m of the elliptic problem. THEOREM 2. 1. Let T2 be the above defined operator in L 2(G). Then given
E > 0 there exists a sequence of positive numbers
I) I = f\
Pi --+ 0 (i = 1, 2, .. :) such that R( A;T 2) exists everywhere on and
(2.4)
for
I}I=
Pi'
i
= 1,2, ...
The proof of this theorem is given in [1] where it is deduced from a much more general result. We now state the main completeness result in L 2(G). THEOREM 2.2. Let (Ji, {B j } 7;G) be a regular elliptic problem of order 2m. Suppose that there exist rays arg ). = 8j' j = 1, ... , N, in the complex plane such that (a) The angles into which the complex plane is divided by these rays 2m are all les s than -n- TC (b) Conditions (i) and (ii) of Theorem 1. 2 hold for j = 1, .•• , N.
Then, the spectrum of the associated te.
operat~r
e= e., J
A2 in L2 (G) is discre-
Moreover, the generalized eigenfunctions of the elliptic problem are
complete in L 2(G). 17
- 17
~
S. Agmon Remark. We observe that if conditions (i) and (ii) of Theorem 1. 2 hold for some
to
e.o
e = eo then they also hold for all e sufficiently near
From this observation it follows that condition (a) of Theorem
2. 2 could be replaced by the slightly weaker condition: (a) I The angles
into which the complex plane is divided by the rays arg ~ =
~ ~ T(
,
In particular if m
~
e.J
are all
n it suffices that conditions (i) and
(ii) of Theorem 1. 2 be satisfi~d for some number
e= eo in order that
the conclusion of Theorem 2.2 should hold. Proof of Theorem 2.2. The discreteness of the spectrum of A2 follows from Theorem 1. 2. Moreover, the same theorem shows that the
ej
are rays of minimal growth of R( A; A2). That is, the resolvent exists on the rays for ). sufficiently large and rays arg ). =
I 1--
as :A
(2.5)
00 ,
arg ). =
9J..
We shall show now that if f* E L 2(G) is orthogonal to sp(A2) then f* is a null function. This will imply that sp(A 2) = L 2(G) or that the generalized eigenfunctions are complete in L 2(G). Tothis end we may assume without loss of generality that the origin is not in the spectrum of A2, Choosing in (2.1) zo = 0 we let: T2 = A-I. Consider now the function
(2.6)
where f is some element in L 2(G) (( )L2(G) denoting scalar product in L 2(G)), From the properties of R( ~ ; T 2) it follows that F( A) is an analytic function for
~#
).k where {).k} is the sequence of eigen-
values of A2. The pOints A= Ak are either regular points or polar singularities of F. However, since f* is orthogonal to all generalized eigen18
• 18 ."
S. Agmon elements of T2 (we have sp(T 2) = sp(A2)) it follows readily from (2.2) that the singular part in the Laurent expansion of F().) around zero. Thus F is also regular at the points
A=
A= Ak
is
~k and we conclude that
FO) is an entire function in the complex plane.
Next. from (2.6). (2.5) and (2.1)' (with p = 2. z = 0) we obtain
o
the growth relations;
(2. 7)
). -
0')
along the rays arg ). =
j = 1•...• N. Also. applying Theorem 2.1 it follows that for every
there exists a sequence of positive numbers r. -+ 00 1
IF()d I ~
(2.7)'
•
eJ.
e> 0
such that
n
e IA I 2m + E for
1),1= r.. 1
i
=1.2 . . . . .
Consider now the function F().) in the closure of anyone of the angles into which the plane is divided by the rays arg ). =
By assumption the size of the angle is < 2m n
T( •
e..J
j = 1•...• N.
On the sides of the angle
we have (2. 7), and on a sequence of circles with radii tending to infinity the inequality (2. 7)' holds. Choosing the number
e in
(2. 7)' sufficiently
small we are in a position to apply - the Phragmen-LindelHf principle in the angle. It follows that in any such angle and consequently in the whole plane;
I
IF( A)
= O(
/),1)
as:A
-+
00 . This in turn implies that F is a linear
function: F( A) = Co + c1A. On the other hand it follows from (2.6) (using R( ~ ;T 2) = ). I + ).2T 2 + ... ) that in the neighborhood of the origin:
19
- 19 -
S. Agmon This and the linearity of F give: (2.8) Since f is arbitrary while the range of T2 is dense in L2(G), it follows from (2.8) that f * = O. This shows that sp(A2) = L 2(G) and establishes the theorem. Example. Let c.A. be a second order elliptic operator in
G with
principal part:
where
1 is
a real number such that 0
sir I
Consider the regular
boundary value problem (eft, B;G) where B is either the identity operator (Dirichlet problem) or B = o~ + a(x) where
i
is a non-tangential (smoothly)
variable direction and a(x) is a smooth function on
aG.
One checks readi-
ly that in this case condition (b) of Theorem 2.2 holds for every ray arg .A "
e which is outside the angle
r ~ ar~ A s. 0 if r < O.
arg
A= Bj
Ir I <
0 ~ arg ).. S'{ if
r
~ 0; the angle
Hence one can find a finite system of rays
satisfying conditions (a) and (b) of Theorem 2.2 if 2:
. Using the theorem we obtain the following result:
the generalized eigenfunctions of the above second order elliptic boundary value problem are complete in L 2(G) if the number of variables n < ,2;,. The completeness result of Theorem 2. 2 implies in particular that under the same conditions the sequence of eigenvalues of the elliptic boundary value problem is infinite. A better result which gives information ori the angular distribution of eigenvalues is the following THEOREM 2.3. Let
(Jt,
{B j } 7;G) be a regular elliptic problem 20
- 20 • S. Agmon of order 2m. SUEPose that the angle (0 <:
91 <
arg.A
< 82
in the complex plane
92 - 91 ~ 2 n: ) has the following properties:
e= e.,
(a) Conditions (i) and (ii) of Theorem 1. 2 hold for
e1 <:
92 -
(b)
2:
n: .
(c) There exists a number
e0with -
81 <80 < £12
such that either con-
dition (i) or condition (ii) of Theorem 1. 2 ia violated for Then the associated operator A2 has infinitely many the angle
91 <:
i = 1. 2.
1
e= e.o eige~values
in
argA <: 82,
Proof: Using Theorem 1. 2 it follows from (a) that the spectrum of A2 is discrete. Also from the same theorem we get that the rays arg i
.A = 9i ,
= 1,2, are rays of minimal growth of R( ~ ;A2) so that the resolvent exists
on the rays for large ). and
(2.9) IIR( A;A 2) II
~
const.
ill
for arg :A = ei'
i = 1. 2,
Next, let z0 be some fixed point not in the spectrum of A2 and put T2 = (A 2 -ZoI)-I. Applying Theorem 2.1 to T2 and using the relation (2.1)' between the resolvents of A2 and T 2 we conclude readily that given
£ > 0 there exists a sequence of positive numbers r k ->- 00, such that
for
I A-zo I
= rk ,
k = 1, 2,,,.
Suppose now by way of contradiction that the theorem is false. This would imply that there are only a finite number of eigenvalues of A2 in the closed angle sector
L:
91 ~ 91 ~
arg).. ~
92,
arg). ~
92,
Hence R(). ;A2) is analytic in the infinite
I AI ~ 1\ 1 if 1\ 1
is chosen sufficiently
large. Taking note of (2.10) and (b), choosing E> 0 in (2.10) so small 21
- 21 S. Agmon that
+ € ) -1 n., we are in a position to apply the Phrag-
S2 - e 1 <: (2:
men-LindelHf principle to the function the function is bounded on the sides of
AR(
L
A;A 2) in
L..
it follows that
Since by (2.9) AR(
A;A2)
is
bounded throughoutL,. In particular we get that
But by the necessity part of Theorem 1. 2 it follows from (2.11) that conditions (i) and (U) of Theorem 1. 2 hold for
e = e.o
This, however,
contradicts our assumption (c) and establishes the theorem. We shall say that a ray arg
A =~
is a direction of condensation
eigenvalues of some given problem if for any E > 0 the angle
Iarg A - eo I< E contains infinitely many eigenvalues.
From the last
theorem we obtain the following COROLLARY. Let (r/l; {B j } , G) be a regular elliptic problem. Suppose that the ray arg
A=
eo satisfies the following conditions:
(a) Either condit on (i) or condition (ii) of Theorem 2. 1 is violated for
e= eo .
(b) Conditions (i) and (ii) of Theorem 1. 2 hold for all that Ie -
eol
< b,
e f e,o 0 some positive number.
Then the ray arg )
= e
o
e such
is a direction of condensation of eigen-
values of A2. It is not difficult to attend the L2 completeness result of Theorem
2.2 to general Lp and variousbther spaces as is shown in [1]. One gets the following more general result. THEOREM 2. 4. Under the conditions of Theorem 2.2 the
general~
zed eigenfunctions of (eft, {B j } ~;G) are complete in Lp(G) and are also 22
- 22 S. Agmon complete in H2m , Lp (G; {B j ) p < 00
~)
in the \I
11
2m , Lp norm for all
Remark. Since the generalized eigenfunctions belong to L (G; {B.} ) for all p it follows from Sobolev's inclusion relations 2m-l _ m, p l (taking p > n) that the generalized eigenfunctions belong to C (G) H2
and satisfy the boundary conditions B. ~ = 0 on dG in the ordinary sense. l Using Sobolev's inequalities and Theorem 2.4 one can also show that any function f E C2m - 1(G) satisfying the boundary conditions BJ = 0
J
on oG (j = 1, ... I m) can be approximated arbitrarily close in the C2m - 1(G) norm by finite sums of generalized eigenfunctions. Similar completeness results (under possibly additional smoothness assumptions on
ril,
B. and G) could be established in various other norms. l
23
- 23
~
S. Agmon Lecture III Some Special Classes of Regular Elliptic Problems We shall apply the previous results to some special classes of regular elliptic boundary value problems. We consider first the case of formally self-adjoint problems. A regular elliptic problem
(cit,
{B j } ~;G)
is said to be formally self-adjoint if the associated operator A2 in L 2(G) (with domain )) is symmetric. From the symA2 = H2 m, L 2 (G; {B.} J metry it follows that for any u E. A2 and non -real).:
tJ
15
(3. 1)
Let us recall that in the last part of the proof of Theorem 1. 2 we have shown that inequality of the type (3.1) along some ray arg ). =
e (with ar-
bitrary constant) implies that conditions (i) and (ii) of Theorem 1. 2 hold for
e.
Hence we conclude that if (A,
{B.} J
;G) is a regular formal-
ly self -adjoint elliptic problem then conditions (i) and (ii) of Theorem 1. 2 hold for all
e such that
0 <
IeI" 7(.
Using the sufficiency part of
Theorem 1. 2 it follows that the spectrum of the symmetric operator A2 is discrete which in turn implies that A2 is a self-adjoint operator. Summing up we have established that if
(Jt,
{B j } ~;G) is a re-
gular formally self-adjoint elliptic boundary value problem then the operator A2 is a self-adjoint operator having a discrete real spectrum. We shall always assume in the following that
elL'
(which is a real operator)
is normalized so that (3.2)
(_1)m
Ji' 25
(x;
s )> 0
- 24 S. Agmon for all real
5
is violated for hold for all
~
0 and
e =0
:lC
E. G. Since now condition (i) of Theorem 1. 2
whereas conditions (i) and (ii) of Theorem 1. 2
e such that
0 <:
IeI
it follows from the Corollary to Theo-
rem 3.3 that the operator A2 has infinitely many positive eigenvalues. We shall say the problem
(eft. {B j} ;G) is bounded from below if the
associated self-adjoint operator A2 is bOWlded from below. or what aznoWlts to the same thing if it has only a finite number of negative eigenvalues. Using again the Corollary to Theorem 2.3 and Theorem 1. 2 one eees that this will be the case if and only if condition (U) of Theorem
1. 2 holds for
e = TI.
following criterion
fo~
(condition (i) obviously holds). This yields the semi-boWldedness:
(JL.
{B j } 7;G) be a self-adjoint regular elliptic bOWldary value problem normalized by (3.2). Let x O be an arbitrary point of 'dG. V the normal vector at xo • and § ~ 0 a real vector THEOREM 3.1.
orthogonal to
».
Let
.
Fmally. for
'\/I > 0 denote by \( + §;x 0 .1\). '\ k =1, ... , m,
the m roots with positive imaginary parts of the polynomial in t:
With these notations, a necessary and sufficient condition for
(cA, {Bj } 7;G) to be bounded from below is that the polynomials (in t) Bj(xO; 5 + t y), j = 1, ... ,m, be linearly independent modulo th~ polynomial.: m n k=l
~l } > o. xO€
oG
and
+ 0 (toO tk ( 5;x , A))
5 parallel to the boundary at xO.
We note that there exist regular self-adjoint elliptic bOWldary value problems which are not semi -boWlded. This was shown in [3] where 26
- 25 S. Agmon the general problem of semiboundedness was discussed from a different point of view. Next consider a regular elliptic problem 'V
differs from a self-adjoint problem terms. That is the principal parts:
N
(A, {B.}
(cA,
{B.} ~;G) which J ~;G) only in lower order
J",
eft:;: A r
and
:;:
N
Since conJ J ditions (i) and (ii) of Theorem 1. 2 depend only on the principal parts B~
B~.
of the operators it follows from the properties of self-adjoint problems that
(A., {B j } ~;G) satisfies condition (i) of Theorem 1. 2 for all e such that 0 < I eI ::; T( (one assumes (3.2)), and condition (ii) for e such that 0 .( I eI <. n:. Applying Theorem 1. 2, the Corollary to Theorem 2.3, Theorem 2.4 'lnd Theorem 3.1 we obtain the following THEOREM 3.2.
(A,
Let
{Bj } ~;G)
be a regular elliptic pro-
blem which differs from a (normalized) self-adjoint problem 'V N m (rA, {B j } 1,;G) only in lower order terms. Let Ap (1 <: p <: (0) be the unbounded operator in Lp(G) associated with (elL, (i) The spectrum of A
p
{B j} ~;G).
Then:
is discrete, the eigenvalues and genera-
lized eigenfunctions are common to all operators A . P (ii) All rays arg }. :;: with 0 <. €I < IT are rays of minimal
e
I I
growth of R( A;A). In particular there are only a finite number of eigenp
values in any double angle: 0..:: E
:s Iarg AI
~ IT - E •
(iii) The positive axis is a direction of condensation of eigenvalues. (iv) The negative axis is a ray of minimal growth of R(). ;Ap)
J!..
the condition of Theorem 3.1 holds. On the other hand if the condition does not hold then the negative axis is a direction of condensation of eigenvalues. (v) The generalized eigenfunctions are complete in L (G); they p
are also complete in H2m , Lp (G;
{B j } )
in the
We shall call a regular elliptic problem 27
/I
11
2m , Lp ~.
(A. {B j} ~;G)
absolute-
- 26 -
S. Agmon ly elliptic b9undary value problem if the boundary system {~j}
7 has
the property that the Complementing Condition of Lecture 1 is always satisfied no matter what is the elliptic operator
cA,
(subject to the "roots
condition" if n = 2). An example of such a problem is the Dirichlet pro_ blem. More generally let OS+j-1
(3.3)
'Of s+j-1
+ a lower order differential boundary operator,
where s is some fixed integer, 0 S s
~
o
m and j = 1, 2, .•• , m; 3)L
denoting differentiation at the boundary along the nontangential (smoothly variable) direction f '
(~,
{B~S)}
Then, the "oblique derivative" problem
7;G) is absolutely
ellipti~._ ~till another example of absolu-
tely elliptic problem is when B. = (3 J. 1 for j = 1, ••• , m-l· while J (7 JB is a real normal boundary operato~ of order m. In the general case m the algebraic structure of all absolutely elliptic boundary value problems was determined by HHrmander [17] who was also the first to introduce this class of elliptic problems. For absolutely elliptic boundary value problems the statement of many of the results given previously becomes much simpler. The reason for this being that for such problems of condition (ii) of Theorem 1. 2 is always a consequence of condition (i) and thus need not be stated. For instance, Theorem 1. 2 for absolutely elliptic problems takes the following form. THEOREM 3.3. Let (vA., {Bj } 7;G) be an absolutely elliptic regular boundary value problem. Suppose that the following condition holds for some real number
e:
28
- 27 -
S. Agmon (_1)m
(3.4)
cA'(x; s)
# ei9
JA'(x;§)\
for all real
5 # 0 and x E G. Let Ap be the associated unbounded ope-
rator in L (G). Then the spectrum of A is discrete. Moreover, the p p is a ray of minimal growth of R~). ;Ap)' For p = 2 ~ arg :A =
e
condition (3.4) is also necessary for the abQve properties to hold. Suppose that addition
cA.'
(eft,
{B j } ~;G) is absolutely elliptic and that in
is a real operator normalized by (3.2), Then condition
(3. 4) holds for all
e such that
0 '<
IeI~ n:,
Hence, using the above
form of Theorem 1. 2, the Corollary to Theorem 2.3 and Theorem 2.4, we obtain the following results on eigenfunctions and eigenvalues of absolutely elliptic problems. THEOREM 3.4. Let (cA., {B j } ~;G) be an absolutely ell~ptic regular boundary value problem such that cA' is real and normalized by (3.2). Let A
L (G) (1 .:: p .:: 00). - - p be the associated unbounded operator p
Then: (i) The spectrum of A, is discrete; the eigenvalues and generalip zed eigenfunctions are common to all operators A , (ii) All rays arg
A '"
e
P
different from the positive axis are rays
of minimal growth of R( )..;A). In particular there are only a finite nump ber of eigenvalues outside any angle: arg ), < C' € > O.
I
I
(iii) The positive axis is a direction of condensation of eigenvalues, (iv) The generalized eigenfunctions are complete in L (G); they p
are also complete in H2m , Lp (G; {B j }) in the II 112m, Lp ~ Results on completeness of eigenfunctions and distribution of eigenvalues similar to ours were derived by Browder [7 ; 9] in the case of the 29
- 28 -
S. Agmon Dirichlet problem for higher order elliptic operators with a real principal part. The case of certain second order elliptic boundary value problems was considered earlier by Carleman
[111 and Keldys [18] . We note
however that all these special problems belong to the class of problems discussed in Theorem 3.2 and the proofs exploit the fact that the problems differ from a self-adjoint problem only in lower order terms. Theorem 3.4 on tpe other hand is applicable to non self-adjoint boundary value problems which are not obtained from a self-adjoint problem by a perturbation of lower order terms. A typical example of this·kind is the
obliq~e
deri-
vative boundary value problem for a real second order elliptic operator.
30
- 29 -
S. Agmon
Lecture IV The asymptotic distribution of eigenvalues
In the preceding lectures we have discussed the completeness of eigenfunctions and the angular distribution of eigenvalues of general elliptic problems. In this lecture. we spall discuss some results on the asymptotic distribution of eigenvalues. Let (cA, {B j } ;G) be a regular elliptic problem of order 2m and let A be the as@?Ciated operator in L 2(G) with domain
iJA = H2m , L2 (G; {Bj }
and let
{Aj }
). Assume that the spectrum of A is discrete
be the sequence of eigenvalues of A each repeated accor-
ding to its multiplicity. Without loss of generality we shall assume in the following that \ = 0 is not an eigenvalue and we shall put: T=A- 1. We first mention the following rather crude result concerning the distribution of the
A.
J
's but which holds for all regular problems. Namely, for eve-
ry t>O, we have:
(4. 1)
__ n __ 2m
e <
00 .
To prove the theorem one notes that T is a compact operator in L 2(G) with range contained in H2
L (G) and with eigenvah!.es 1/ ). .. m, 2 J By a general result proved in [1] (Th. Al. 1) the series (4. 1) con-
verges for any compact operator possessing the last mentioned property. In order to obtain an asymptotic result on the distribution of eigenvalues it is necessary to restrict the class of regular problems considered. We shall not bother here to give results in the most general situation but shall limit ourselves to an important quite general subclass of problems. 31
- 30 -
S. Agmon
Z
We denote by
the subclass of regular problems such that the principal
part of the elliptic operator is real and normalized by (3.2) and such that conditions (i) and (ii) of Theorem 1. 2 hold for all directions different f:rom the positive axis. For simplicity we shall also assume that the domain G ~
00
is of class Cand that all the coefficients of the differential operators are C iu,!!
{B j}
ctions. It follows from our previou~ results that if (rA, to
Z
;G) belongs
then the associated operator A has a discrete spectrum with eigen-
A.
condensing along the positive axis. It is possible to give an J asymptotic formula for the number of eigenvalues with Re:Aj ~. t . To
values
this end, put:
N(t) =
L
1
Re \. <:: t ,l\J -
and define w(x) =
f
d§
and
w(G) =
(-ireA' (x; s) < 1
fG
w(x) dx
we have: THEOREM 4.1. belonging to the class
Let
Z.
(cA.,
{Bj } ;G) be an elliptic problem of order 2m
Then the following asymptotic formula holds. n
(4.2)
N(t)3 (2
n: )-n w(G)t
2m (1 +0(1)).
t -
+ 00
•
For second order self-adjoint elliptic boundary value problems the asymptotic formula (4.2) is classical (Weyl [23] and Courant [13] ). Carleman [11] has introduced a powerful method which for second 32
- 31
~-
S. Agmon order problems yields the asymptotic formula also for certain non selfadjoint problems. For self-adjoint problems associated with the biharm,£ nic equation (plate problems) the asymptotic formula was established by Courant (12) and Pleijel [21]. F or higher order elliptic problems, but in the special case of the self-adjoint Dirichelet problem, formula (4.2) was established by G~rding method of G~rding
L15]
[15]
(also Browder [10] for
>
2m
n).
The
is applicable to certain other regular self-adjoint
and semi-bounded elliptic problems (see Ehrling [14
J).
Also, as was
pointed out by Garding, if (4.2) holds for a regular self-adjoint and semibounded problem, th:m (4. 2) also holds for the non-self-adjoint problem obtained from the self-adjoint problem by a perturbation of lower order terms.
This follows from general results announced byKeldys [18] . Our Theorem 4. 1. includes the above results as special cases.
It includes a wide class of non-self-adjoint problems which are not obtain-
ed from a self-adjoint problem by a perturbation of lower order terms. Even for second order elliptic boundary value problems one has here a new result in the case of the regular oblique derivative boundary value problem. The proof of Theorem 4. 1 is base on the following result. THEOREM 4. 2.
~ k nel
> n/2m
K(x, y)
Under the conditions of Th.
the operator
Tk
is an integral operator given by a ker-
which is continuous on
G x G.
K (i. e.
is the resolvent integral operator of Tk (I -
ATk) -1),
then
r
4. 1 for every inte-
(x, y; -t)
Moreover, if
r
r
(x, y;
A. )
is the kernel of
exists for all positive
t
sufficien-
tly large and (x, x;-t) dx = (21t)
(4. 3)
where
a
=
n/2mk
<
-n
1.
33
w(G)
7T a a-I . 1\ t (1 + 0(1)), t ... +00 8m a
- 32 -
S. Agmon The deduction of Theorem 4.1 from Theorem 4.2 is easy. Indeed. if D().) is the Fredholm determinant corresponding to the kernel K. then we have the well known relation:
_rJ_
(4.4)
G
r (x. x,. ~ )dx -_.l2.W D ()d
It is also well known that D().) is an entire function of order S 2 (actually
~
1) which in our case will have for zeros the kIth powers of
the eigenvalues of A. Since by (4. 1):
L I A.I
(4.5)
. J
J
<
-k
00
we have by Hadamard Is factorization theorem of entire functions:
(4.6)
A)/D( ). ) -
Since by (4.3) and (4.4): DI(
0 for ). real -- - 00
•
it follows readily that the constants Co and c 1 in (4.6) are zero. Using this it follows from (4; 6) and (4. 3) that
(4.7)
Let
Lj
1
~
).. = /oJ-. + i ).l. J I J J
= (211:)
-n
7t a a-I w(G) sin 7t a t (1+0(1)), t -
+ 00 •
j Since
fj -
it follows from (4. 7) and (4. 5) that
34
+ 00 and Vj /
fj -
0,
- 33 -
S. Agmon
_....;;..1.,..-Ita1t a k = (2 1rI~ )-n w(G) sin t + fj
(4.8)
t a- 1(1+0 ()) 1 , t ~ + 00.
Applying now a Tauberian theorem of Hardy and Littlewood [16]
to (4. 8)
one arrives at the asymptotic formula (4.2,). Concerning the proof of Theorem 4. 2 we note that from the re_
[151 it already follows that for t=O and for t
sults of Garding
I' (x, y;
sufficiently large there exists a kernel
>
0
- t) continuous on
G x G, such that for all domains Go with Go C G and all functions having their support in Go' one has:
r (x,y;-t) f (y)dy.
~
Tk (I +t Tk) -1 f =
o Also from the results of [15 J it follows that for any such interior domain Go one has:
(4.9)
r (x, x; -t)dx = (21'(,)
-n
I
Tta a-1 w(G) - . - - t (1+0(1)), t __ +00. 0 sm7ta
To prove Th. 4.2 one uses (4.9) in conjunction with the following THEOREM 4.2'.. The kernel
I' (x, y; -t)
(for t=O and all t
>0
sufficiently large) is continuous on G x G and one has the estimate:
Theorem 4.2', is this part in the proof of the asymptotic formula which takes into account the boundary conditions. In proving it we use 35
- 34 -
S. Agmon
Theorem 1. ~ applied to the regqlar elliptio boundary value problem
(cA k.
{B j Ai} ; G) (j=l. .... m; i=O~ .... k-1) as well as Sobolev
type inequalities involving a parameter due to EhrUng [14]
36
- 35 S. Agmon BIBLIOGRAPHY [ 1]
Agmon, S., On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems, Comm. Pure Appl. Math., Vol. 15, 1962.
[2] Agmon, S., The angular distribution of eigenvalues of non selfadjoint elliptic boundary value problems of higher
ord~r,
Conf.
on Partial Differential Equations and Continuum Mechanics, The Univ. of Wisconsin Press, 1961, pp. 9-18. [3 J Agmon, S., Remarks on self -adjoint and semi-bounded elliptic boundary value problems, proc. International Symposium on Linear Spaces, The Israel Academy of Sciences and Humanities, Jerusalem, 1961, pp. 1-13.
[4J Agmon, S., General elliptic boundary value problems, to appear. (5] Agmon, S., Douglis, A., and Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math., Vol. 12, 1959, pp. 623-727. [6] Agmon, S., and Nirenberg, L., Properties of solutions of ordinary differential equations in Banach space, Comm. Pure Appl. Math., to appear 1963. [7] Browder, F.E., On the eigenfunctions and eigenvalues of the general elliptic differential operator, Proc. Nat. Acad. Sci. U.S.A., Vol. 39, 1953,pp. 433-439. [8] Browder, F.E., Estirnates and e.xistence theorems for elliptic ~oundary
value problems, Proc. Nat. Acad. Sci. U. S. A., Vol. 45,
1959, pp. 365-372. 37
- 36 S. Agmon
[91 Browder,F. E. ,
On the spectral theory of strongly elliptic differen_
tialoperators, Proc. Nat. Acad. Sci. U.,S. A. Vol. 45, 1959, pp. 1423-1431. [10] Browder, F.E .• Le probleme des vibrations pour un operateur aux derivees partielles self-adjoint et du type elliptique a coefficients variables, C. R. Acad. Sci. Paris 236 (1953), 2140-2142.
[11}
Carleman,
T'.,
Uber die Verleigung der Eigenwerte partie Her
Differentialgleichungen, Ber. Verb. SachS. Akad. Wiss. Leipzig. Math. -Nat. Kl., Vol. 88, 1936, pp. 119-132.
[12}
Courant, R., tiber die Schwingungen eingespannter Platten, Math. Zeitschrift. Vol. 15 (1922). pp. 195-200.
[13}
Courant.
R.,
and D. Hilbert. Methoden der Mathematischen Phys'ik
I. Berlin 1937.
[14) Ehrling, G., On a type of eigenvalue problems for certain elliptic differential operators, Math. Scand. Vol. 2 (1954). pp. 267-285.-
[151 Garding, L.,
On the asymptotic distribution of the eigenvalues
and eigenfunctions of efiiptic differential operators. Math. Scand.
Vol. 1 (1953) pp. 237-255. [16J Hardy•.G. H.. and Littlewood. J. E., Notes on the theory of series (XI): On Tauberian theorems, Proc. London Math. Soc .. (2) Vol.
30 (1930). pp. 23-27.
[17]
Hormander. L., On the regularity of the solutions of boundary problems, Acta Math., Vol. 99, 1958, pp. 225-264.
[18] Ke1dys, M. V., On the eigenvalues and eigenfunctions of certain
classes of non-self-adjoint equations, Doklady Akadr Nauk SSSR,' I
Vol, 77. 1951, pp. 11-14. 38
- 37 -
[191
S. Agmon
Nirenberg, L., On elliptic partial differential equations, Ann. t Scuola Norm. Super. Pisa, Vol. 13, 1959, pp. 115-162.
[20]
A.
Pleijel, Propri~tes asymptotigues des fonctioas et valeurs
propres de certains problemes de vibrations, Arkiv Milt. Astr. • • Fys. Vol. 27 A, No. 13 (1940), 100 pp.
[211
A.
Pleijel, On the eigenvalues and eigenfunctions of elastic plates, i
Comm. pure Appl. Math. Vol.
3 (1950), pp. 1-10.
[22] Schechter, M., General boundary valUe problems for elliptic partial differential equations, Comm. Pure Appl. Math.. Vol. 12. 1959. pp. 457-482. [23J Weyl, H.. tiber die asymptotische Verteilungsgesetz der Eige~ werte. Gottinger Nachrichten. (1911), pp. 110-117.
39
CENTRO INTERNAZIONALE MATEMATICO ESTIVO ( C. I. M. E. )
A. M. OSTROWSKI
IL METODO DEL QUOZIENTE DI RAYLEIGH
ROMA - Istituto Matematico dell'Universita 41
A. M. OSTROWSKI
IL METODO DEL QUOZIENTE DI RAYLEIGH l )
Lezione I Espressione approssimata delle autosoluzioni. Sian una matrice quadrata. Diremo che Ae un autovalore di
n , se esiste un
vettore·
~ non nullo tale che f2 t, =A'S. Se). e
{2 -\ I e singolare. Sia Ao un valore approssimato di A ; chiameremo autosoluzi~ ne approssimata relativa all'autovalore Ae all'approssimazione Ao la soun autovalore,la matrice
luzione
So del sistema lineare
(1)
/
ove
1 e un vettore £lsso scelto in modo generico (in un senso che verra
precisato). Si pub dimostrare che,quando \ vettore relativo a
'A.
-) \
'~o/\C;o\
converge ad un auto-
(2)
Sussiste infatti il seguente teorema piu generale: (1) Mi e sommamente grato esprimere la mia piu viva gratitudine aHa dott. Lucilla Bassotti e al prof. Luciano de Vito che hanno preso appu~ ti delle mie lezioni e mi hanno prestato il loro esperto aiuto nella versione italiana di queste lezioni. (2) efr. A. OSTROWSKI, tiber naherungsweise Auflosung von Systemen homogener linearer Gleichungen, Zeitschrift fur ang., Math. Phys.
VIII, 1957. 43
-2A. M. Ostrowski
I. Sia S(t) una matrice dipendente analiticamente dal parametro t
..!!!...
I
un intorno J di t = 0 , tale che S(o)1 = 0, S(o) F 0 e per t E J - 0 riesca
IS(t)1 f 0; fisaato comunque, fuori della varieta lineare descritta dal vettore S(o)
S al variare di S' !,
detta ~ (t) la soluzione dell 'eguazio-
ne (2)
S(t) ~ (t)=
esiste' 'una 'soluzione
'1, )
If ,divers a da zero, di
S(o) IT
=0 tale
che
Dimostrazione
11 vettore
~ (t)
e funzi one analitica di
t in J - 0, dipendente
razionalmente dagli e1ementi di S(t); per 10 sviluppo in serie di LAURENT, si ha:
S, (t) = t j e quindi
1~(t)1
(U + 0 (t), U
to,
(j finito ed intero)
. =Itld(/UI+ O(t»;
donde U + O(t) = (
lui + O(t) e pertanto
(3)
C; (t) =
11 + O(t),
\ ~ (t) \
44
(-L lui
+ O(t»)
- 3 -
A. M. Ostrowski
essendo
1/ un vettore unitario. Se j < 0 si ha: lim \ C; (t)\ = + 00 ; mpltiplicando la (3) a sinistra
t-to
per S(t), si otiiene :
_ _'Y_ l. _ = S(t) 11 + S(t) 0 (t)
(4)
I~ (t) I e quindi, per t 70, S(o)
Tt
= o.
Se j = 0, si ha lim
Is(t)\ = IuI e quindi passando al limite
ho
per t ->0 nella (4) :
~=s(o)TTlul.
(5)
Se si sceglie ~ in modo che non sia della forma S(o) ~ , la (5) non pub essere verificata. Se inCine j >0, si ha lim o = lim S(t)
l..,o
~
l; (t)
e di conseguenza
=0
b-+o
(t) = S(o) U = 0 f 'Y\ ; cib l
e contro l'ipotesi.
Osserviamo che, per l'ipotesi S(o) degenere, la varieta descri!: ta dal vettore S(o) ~
hadimensione minore dell'ordine
,'11,
di Se pertanto,
scelti arbitrariamente 'YI. vettori linearmente indipendenti, per uno almeno di essi l'ipotesi sUl e verificata. N. B.
Ao e una approssimazione di A, posta Ao - ~ = t, si ha Ao I = .0 - A, I - tIe quindi, assumendo tale matrice come ma-
Se
n-
trice S(t), si pub applicare i1 teorema precedente. Volendo estendere i1 teorema precedente al caso di matrici dipendenti analiticamente da piu parametri, si incontra la difficolta consisten 45
- 4A. M. Ostrowski te nel fatto che non esiste
lim ~
/j Zj I
Ad esempio, se
S • S (u, v) •
l'equazione S(u, v) ~ =
'1.
UvO 00) (o 001 ha, per u v
1 '" (1,1, 1) ) t 0, soluzione
~. (l/u, I/v, 1) e
quindi
~I ~ ( ~ +:+ ~'/0~1 +:.. v'l.' ~~~2+ iN} 1
I
V1•
Se u e v tendono a zero in modo che
u/v ..... Q"
si ha lim ~ /I'~I = (1,0,0);
se invece u e v tendono a zero in modo che v I u ~O, s i ha lim l; ~ ~ I =
= (0,. 1, 0) • Esaminiamo ora il caso in cui 1a matrice S =S(O) non sia nota, ma si conosca una matrice
D.
approssimante S •
Definendo allora la matrice A in modo che S + A
=.fl , si pone il seguente
problema: Fissato il vettore"l e detta ~A una soluzione deU'equazione (S
II
+ A) ~ A =~, studiare il comportamento di 2; A ~ AI quando A tende a
zero, cioe quando lende a zero la norma IAI }, di FROBENIUS della mat rice
;~i
pone
IAI, ~ ~
L lo,l4l ). /4) I}
I
Osserviamo innanzitiltto dile.:proprietA ben note daUa norma ora introdotta, estensioni di proprietA corrispondenti nel caso dei vettori :
46
- 5-
A. M. Ostrowski
/A+BI1 ~ IAI + \BI t .
II)
'I.
In particolare, per un vettore ~ , si ha:
IA~ If IAI2,/2: I
Sussiste il seguente teorema :
II.
Se S
e una matrice singolare,
S + A sia non singolare, posto
A una matrice tendente a zero tale che
' esist e un vettore ITA
~A = (S + A) -11
!!:..
Ie che : S
(6)
11A
= 0,
Dimostrazione. Dimostriamo dapprima il teorema nel caso particolare che S abbia la forma :
(1)
ove Ok indica una matrice quadrat a di ordine k nulla, I n- k indica la matrice unitaria di ordine n-k e Ie sono matrici rettangolari nulle.
°
Sia't un vettore non appartenente alla varieta. descritta da S
S ; nel caso
attuale, bisogna supporre che una almeno delle prime k componenti di , sia diversa da o.
~
Poniamo
r =(x ":>1
1
J •••
JXIt.
J
=(Y1 , ••• , Ytrv )
;
~
+A) ~
=S~
+A~
S"
+ Sot
, con
~ ~ =(0, .•• ,0, x I:tf , ••• , x"""" ).
0, •.• ,0) e
Si ha : (S
=
=
A~ + ~1. 47
•
-6A. M. Ostrowski e pertanto il sistema :
(8)
(S+A) ~
=
'1,
pub mettersi nella forma
Indichiamo con
~
un indice fra 1 e k • Si ha: n
y
x
=
L\1=1
ax v x"'
da cui. per la diseguaglianza di Cauchy-Schwartz, si ricava :
e, ricordando che, per almeno un valore di ~
, Y,
~
1
___ L
Il; I quindi
Dividendo lare1azione (9) per
IsI ' si ottiene :
A~
~e
~
Il;\
P;\
I~\
- - - - + -~- =--=----
I I '- IA 12 I ~ I ' riesce:
e pertanto, essendo A ~
48
:/;0, si deduce
- 7A. M. Ostrowski
111 I ~I
I..
Ne segue
+_1
lsi
I{,J
lim
IAI21~Lc ii+IA/ 2 =0(IAI 2).
=,0,
I ~I
- I ~I lim/ ~41 = +00.
D'altra parte:
~
I~I =
=
~4
+
I ~I I ~i I
~~
~t
P~I
=
~\
+ 0 dAI~ ) =
I z: I
+0 (IAI~),
I ~41 I~ \ /~412 I ~tlt =1 I ~ \1 I ~ \2
2-
=l+Q(/AI1.)'
e quindi:
'S / I~ I ; Posto
ITA
=
l; I ~ , . I
St /1 ~11
+ 0 (IA~ . ' si ha S
ITA =
°e di conseguenza la
teal. Per ottenere il teorema nel caso generale, cioe per un'arbitraria matrice S di rango n - k, basta osservare che,ammesso il ~eorema per una particolare matrice Si' il teorema seguita a sussistere per ogni matrice 8 del tipo S = B 81 C, con B; C arbitrarie matrici non singolario
49
- 8 -
A. M. Ostrowski Lezione II Metodo del guoziente di RAYLEIGH. Sia A una matrice reale lIimmetrica e ~ un vettore-colonna. Introdotto il vettore ~ I trasP:O!to di ~
,si consideri la Corma quadratica
Q (t) = A )
xllx"ove A· (a&A,u)'
~I
':>
$IA~
=
L f'lv
a
JI-~
,
,~
r'
= (xl' • • •• xn) •
Chiamasi quoziente di RAYLEIGH l'espressione :
(10)
Ii qUQziente di RAYLEIGH gode della propriet~ che, se
auto801ulione della matrice A relativa all'autovalore
S e una
l , riesce :
11 quoziente di RAYLEIGH permette. in .generale, la costruzione di
1convergente all 'autovalore ~
una succe88ione [
AII
ne di vettori
convergente ad
Sia
Ao
[fv}
un'approssimazione di A,
il vettore ~0
un'autosoluZione'~
e di una successio"relativa a >v •
• Fissato un ~ generico costruiamo
soluzione del sistema :
(11)
11 quoziente di . RAYLEIGH relativo a 51
~o fornisce un numero
- 9-
A. M. Ostrowski
che protrebbe migliorare l'approssimazione di A. Iterando il
\.
procedimento si ottengono Ie successioni 1~~} e [ { . . } di cui .sopra. Faremo ora vedere che, se Ao e abbastanza vicino a ~ ,abbiamo la con-
l A",} e questa e una eonvergenza quadratica, nel senso che
vergenza· di
esiste rinito il (12)
Poicbe, eome
e evidente,
il quoziente di RAYLEIGH e inva-
riante rispetto alle trasformazioni S ortogonali (SS' = I), e poiche una matrice simmetrica pub essere diagonalizzata con una trasformaZione orto,onale, supponiamo senz'altro A = diag (ft' .•• , flfV)
r
. I numeri
non sono necessariamente distinti; indichiamo aHora con quelli dei
ft =
=" • =
~"'"
fk 6""",
'f'W che sonG fra loro distinti e supponiamo ehe r~ e rIC +6'1 per k~h.
t '" •
6"1
Si ha allora :
(13)
Indichiamo eon
trt
= (y I' ••. , yn) un vettore avente diversa
da zero almeno una delle prime h eomponenti; riesee:
s, ·(r:\
' ' .'
Dalla (13) si deduce:
52
=
- 10 -
A. M. Ostrowski
""
RA (
?r·
~6 ) =
'IYI,
I ~I( I~ (rlr. - ~o)
'"
L
hI
I ~,.IJ, (r .... Xot
k.. 1
L f)~
~
=
r~
10;. - AolZ,
'Iw\.
L I kc~
E~
(jIe -
~o It,
h
Poniamo ora
Lk = I Iykl2
Pi =
ove ai e posto
f
= Gf
«)
'
e analogamente per P2,···, Pm.
P = PI • Si ha :
(14)
Posto
(15)
A1. - 6''
Poato :
=
m
')
L
c:
£"'
vk-~
L..rk-~~!--
= k=2
IG"I(
-0'1
P
53
• 11 A. M. Ostrowski
daUa (15) si trae che, se
~i
-
6"
I~~-trl~
Ao ~ sufficentemente prossimo a
~ abbastam;a prossimo a
6 ,
L.
Iterando il ragionamento. si avrA aHora lim ~'" = 6'
e
y~~
11m
:; L.
Per quanto riguarda l'approssimazione dell'autosoluzione ;s1
Ila:
0, to., 0);
11 e un'autosoluzione relativa art. Rlesce
(16)
• • • OJ
-r-/)\_~.;;..n6'-).
Quanto sopra detto si estende in modo quasi evidente al caso d!lle matrici hermitiane. Naturalmente in questo caso QA( ~ )
e defini-
to come ~.A ~ e la diagonalizzazione si ottiene usando invece delle matrici
s*
ortogonal~
Ie matrici unitarie ortogonali S definite dalla relazione
S = I (S~e generalmente la matrice coniugata dalla trasposta di S).
54
- 12 -
A. M. Ostrowski
Pili in lenerale, la nostra argomentazione si applica anche al caso dell, matrigi normali. eioe di queUe matrici che possono essere diagonalizZlte per mezzo di una trasformazione unitaria ortogonale; gU autovalori di una matrlct normale non sono pera necessariamente reali. Le matrici normali poI.ono anche essere definite per mezzo della relazione Apt = A* A• •
moatra cbe l'insleme di queste matrici
e un'intersezione delle varietA
quadraUcbe nella pometria hermitiana (nel campo di tutte Ie matriei comple..e). Un'altra caratterizzazione delle matrici normali A. dovuta a I. SawB, con.i.te nella proprietA che ove
/A 12 =
\ \J2 +
I~1 \1.+ ... \ A.i' )
\"'" \.. sono,li autovalori di A. La teoria delle matriei norma-
li. dovuta quasi nella sua totalitAa I. SCHUR, si presenta come un'estensio-
ne elepnU8Ilma della teoria delle matrici hermitiane; lavorando con Ie matrlcl normali. bi,oana avere particolare cura. perche l'insieme di dette matricl non b lineare, fatto che tacUmente si dimentica.
Metodo
1
il vettore gill formato ~ ,).i
cost"",re una .uccessione di numeri unitar.
1 ~,.. ~
[AI J
e una successione di vettori
tali che :
~II = (A - A.• I) ~ V-i •
(17)
Y
55
e possibile
- 13 -
A. M. Ostrowski Questo procedimento presenta una convergenza cubica. La prima discus-
e la seguente:
sione di questo fatto, dovuta a CRANDALL (3), la matrice A diagonalizzata e fissato l'iildice
)I
,
Supposta
si moltiplichi il vet-
tore ~)i per un faft'ore" in modo che Ie prime h componenti non dipendano da Y • Indicando tale vettore con C, II'
(18)
Scriveremo
,
mo con
:
... ,
I I1Y.' •••(yl'• ....IznYf' 0,• .... 0) e indichia-
Introduciamo il vettore ' fra Zh+l (v) 1'I masslmo
t ~
b')
Riesce ovviamente O( ~ ).
Passando all'indice )) + 1 e indicando con xl' •.• , xn Ie componenti di
't "IIH
si ottiene :
Y, X.
1
=
1
per
k :> h,
dicui :
~II+~
11 6' -
~"H
(,,)
( y)
+
(0, ... , 0,
zh+l '
... ,.
~ ~.I -A 11+\
z n
~,., - AYH ) i
(3) Cfr. S. H. CRANDALL, Iterative procedures related to relaxation methods for eigenvalue problems, Proc. Roy. Soc. London, 207, 1951, 421 - 422.
56
- 14 A. M. Ostrowski
~PH per il fattore 6' - A..pi-! si ricava:
moltiplicando il vettore
~"u. = 1T
(19)
+
(0, ... ,0,
('I)
(I»
zh+l
, ••• ,
fltl - ~~.~
~
)(
0 - A\l+iJ·
f""'- ~.t
Rieordiamo ora che riesce \
_d __
R (1" ... ) - 0 = J\.
~r
~+t
~
In (/'-. - \zJv 6' l
l\2
I~y 12
I I
e quindi, essendo Zk.(v) f E , si ottiene :
A, lifo! -6'=
O(£z)
Introducendo questa espressione in (19) si ha :
~ Jlti -IT
(20)
= 0 (
E.' ).
Questa argomentazione e piuttosto un'argomentazione di plausibilitA e non Mette in 1uce che anche 1a convergenza di
tAy J e cubica, cioe
che riesee :
(21) Usando 1a termino1ogia dell'amico WEINSTEIN, bisogna dunque
di~
re che in questo momento il matematico di alt 0 ingegno deve cedere il pas so al '·!farmacista". La discussione rigorosa del metodo permette erfettiva~ mente di provare non soltanto 1a (21), rna (22)
lim )) ... 00
).-,1.,., -
(A y
ID
_ 6' ) 3
=
57
g
anch~ ..che
> o.
- 15 A. M. Ostrowski
La dimostrazione di questo risultato, con le indicazioni di tutte le ipotesi da fare, sara l'argomento delle lezioni IV e V.
58
- 16 A. M. Ostrowski
Lezione III Quoziente di RAYLEIGH generalizzato. (Argomentazioni di plausibilita). Sia A una matrice non necessariamente hermitiana; ad essa si pub associare una forma bilineare , AS' Si deCinisce aHora come quoziente di RAYLEIGH respressione:
(23)
(d'ora in poi i vettori si intenderanno come vettori riga se sono fattori di sinistra e vettori colonna se sono fattori di destra). Si possono stabilire le formule ricorrenti : (24)
(25)
~Vtt=RA( ~y'
S\I = (A -
A"
'v ) I) -1
Sv-1 '
La proprieta estremale delle autosoluzioni per il quoziente di RAYLEIGH, sulla Quale si fondava il principio di massimo nel metodo di calcolociell'autovalore (cfr. lez. I1),non sussiste piu; in questo caso la pl'Oprieta suddetta pub essere sostituita da una proprieta di stazionarie-
~4). (4) Cfr. A. M. OSTROWSKI, On the convergence of the Rayleigh quotient 1teration. for the computation of the Characteristic Roots on Vectors UI. Archive Rat. Mech. and Anal. vol. 3, 1959, 326.
59
- 17 A. M. Ostrowski
Cib. ad esempio. si verifies quando all'autovalore eorrispondono soltanto i divisori elementari lineari della matrice (5),
In effetti. se 6'
e un autovalore di m~lteplieita
k cui
corrispondono saltanto divisori elementari lineari. esiste una matrice S non degenere tale che
:nJ ove Ik
e la matrlce
identica di ordine k 'e Dn -k
e una matrice quadrat a di
ordine n-k i cui autovalori sono tutti distinti de. 6' ,
S (' )un'autosoluzione adestra (a sinistral corrispon-
Sia dentea 6'
S-
poniamo p = IYI
I
(a 1.... a k• 0..... 0).1- (b 1, .... bk• 0..... 0);
L k
S
= a v b)l e supponiamolo diverse da O. v=l
Siano ora ~ 1 e'1 due vettori pros simi rispettivamente a
S e 1)sup-
posti nella forma :
~!. (a1, ..
OJ
\.
0( k+l" .. • 0{ n).
i1- (b 1·, .. • bk •
~ k+l"" fn)'
essendo Ie ri.. .• A . =0 (£ ) per i = k+l, .... n. 5i ha : 1
r
1
(5) Un'altra via conducente al metodo descritto sopra e stata data da M. R. HE5TENE5 • .!!!!ersion of matrices by biortogonalization and related results. J. Soc. Indust. Appl. Math., vol. 6, 1958, 80 - 83,
60
- 18 A. M. Ostrowski
IX.
1
2
8. = p +0 ( C ),
r1
Pertanto: 6'p+O( E;2) = 6' +O( £2). P + O(
£ 2)
In quest'ultima espressione si riconosce una certa "staziona-
riet~" del quoziente generalizzato(6). Mostriamo ora, con un esempio, che, se non
e verificata la
ipotesi precedente, non si ha in generale stazionarieta. Sia:
Si ottiene aHora : 6' (xl Y1+x2 y2)+x2Y1
xl Yl + x2 Y2 In questo caso 6'
= 6"
X
2 Y1
+ ---=-~-xl Y1 + x2 Y2
e autovalore e un'autosoluziorie a sinistra e (0,
(6) Cfr. Nota citata in (4) pag. 327.
61
1) e
- 19 A. M. Ostrowski
e a destra Posto
e (1,
5' = (1,
/
0) . 0( )
e , =(
~
,1), riesce :
Da cib si trae l'asserto. Altre generalizzazioni del quoziente di RAYLEIGH, in una nuova direzione, si ottengono dall'osservare che A-
A I e una partico-
lare matrice dipenciente linearmente dal parametro)" ; sostituendo allora A-A. I con 1a piu genera1e matrice dipendente linearmente da). , si ottiene il problema di autovalori :
Per questa problema si pub sviluppare una teoria analoga a queUa precedente, definendo il quoziente di RAYLEIGH al modo seguente: RA( ~
,'1
"'t A1 S )=---'~-,. Ao
(7)
S
Un'ulteriore generalizzazione si ottiene sostituendo ad una matrice lineare in A., una" matrice del tipo : 'n'
D( )., ) •
A ~ o
'I'll-I
+A A. 1
'\
+ ••• + A
m-
1""
+ A ,ove Ie A, m 1
(i = 0, ••• , m) sono matrici quadrate di ordine n. (7) Cfr. S.H. CRANDALL, ,Iterative procedure ••••• , Proc. Roy. Soc. London, 1951, 417. 62
- 20 A. M. Ostrowski
In questo caso un modo di definire il quoziente di RAYLEIGH,
e il seguente :
dowto a P. LANCASTER(8).
'Yl4 D( ~ ) ~ ,
ovesieposto D'(.\.)=J11Ao
D'( ir¥I-i
A.
~ ) ~ '11\.-2-
+'(m-l)A1 ),
+... +A m _l •
Per mezzo di tale quoziente si pub definire un procedimento iterativo. mediante Ie formule :
~Y D(Ay)
1-; ~Ytt : [.D (A~H)l
L
D'(A y) Sy
-t
SV )
Osserviamo i seguenti casi particolari: per m=l. si ha:
R (
D
~
, /)') ,
I
A ) =., ~ Al S ~Ao S
che rappresenta la prima generalizzazione introdotta in questa lezione; per n = I. A. sono scalari e D (A. 1
) e un pOlinomio; il procedimento ri.,
corrente introdotto si rappresenta con la formula:
(8) Cfr.. P.. L~CASTER., A Generalized Rayleigh Quotient Iteration for Lambda - 'Matrices, Archive Ilat. Mech. and Anal. VIII, 1961, 309.-
310.
63
- 21 A. M. Ostrowski
D ( )..,)
e coincide pertanto con il ben noto procedimento di NEWTON. Pare dapprima che nel caso delle matrici quadrate A qualsiasi, il quoziente di RAYLEIGH generalizzato e il solo razionalmente applicabile. In qualche calcolo eseguito all I Istituto di Cal colo di Ramo Wooldridge Corporation in Los Angeles, il metodo del quoziente classico di RAYLEIGH era applicato anche aUe matrici complesse non hermitiane, con un successo sorprendente. Lo studio dettagliato di questo caso mostra che c'e generalmente una convergenza, ma soltanto quadratica. Questo e perb compensato dal fatto che, utilizzando il quoziente classico, bisogna ad ogni passo risolvere soltanto un sistema lineare anziche due, come nel caso del quoziente generalizzato. Inoltre e molto facile lavorare con il denominatore vece del prodotto
'I ~
;* ~
in-
che pub tendere a zero(9)molto rapidamente.
(9) Cfr. A. M. OSTROWSKI. On the Convergence of the Rayleigh Quotient ..•.. , V, Archive Rat. Mech. and Anal. III. 1959, ed ancora A. M. OSTROWSKI. On the Convergence of the Rayleigh Quotient .••• , VI, Archive Rat. Mech. and Anal., IV, 1959.
64
- 22 A. M. Ostrowski
Lezione IV La lege asintoticaper il metodo del Quoziente di RAYLEIGH Dimostriamo ora il seguente teorema : III. Se S
e una matrice hermitiana,
() un suo
autovalore,~,\, j una
suc-
cessione, costruita per mezzo del quoziente di RAYLEIGH, approssimante
er
(26)
,e tale che
\,* 6"
per ogni
v , sussiste la seguente proprieta:
lim
=
y~OO
¥>O.
Se S e normale e non hermitiana, la (26) sussiste sotto 1'ulteriore eondizione che non esistano due autovalori ~ ~~!
distinti fra lora e da
6" ,aventi da 6"' la stessa distanza. Dimostrazione. (10) Sia A = diag (
distinti di
ri'
~.2.
, ••• ,
f1 ,... , .,) e ~
(lV ; supponiamo
i valori
0, 6'1 , ••• , 6"""" (5
se k > h. Consideriamo un vettore
=
r =••• = i
~f., e
1,/. , (Y1"'"
Y ) aven-
n
te almena una delle prime h eomponenti diversa da zero e poniamo
~
0::
'1'
Mediante il procedimento rieorrente esposto nelle lezioni preeedenti, si pub formare una successione di numeri
!~ v ~ che sono supposti approssi-
manti 1'autovalore 6" e costruire in corrispondenza ad essi la successio-
(10) efr. A. M. OSTROWSKI, On the convergence of the Rayleigh Quotient .••• I, ARCHWE Rat. Meeh. and Anal., vol. l, 1958, 233-241.
65
- 23 -
A. M. Ostrowski ne di vettori
f S'Y}
definiti daile relazioni : ( y)
xk
(27)
fk ~ (-,I) ~v = (xl
avendo posto
(-;)
' .••• xn
( k = l, ••• , n) ,
AY+1
).
Daile (27) si ricava, ricordando che (28)
=
Poniamo : y N
V
f( ( ~ - A ),
= v 1'd.
11 (
N = v,k '<.'-=1
r
riesce ailora : tVYV
2
I~v I =
)
E
INv \2
E~
+L
!N)), k.1~
k=i
L, I'((v
~* )yA ~ y
pO'
=
INy 1°
+
k.=i
Pk. ~
IN y
k,1~
)
denotando Pk opportuni numeri positivi. Ne segue che :
66
- 24 A. M. Ostrowski
h
(Ilk - 0') t
\
1\
=
v+!
I f(/~ \ - G = ------"-----
I ,.). I,}, :
~
v
e quindi
ove si ponga
\I \~rj I,
.
)
:''ffv
K -; = p+
L11.0"'
Pk
'
';
,J
\~
i~ 1;'
i
,":,.'
Ir
I.
I
l~" , ""
i
Facciamo ora vedere che, se si prende \ a 6'
, aHora 1a successione
1 = "2
min
!t 10'1- n' I , ••• , !
t>. ,,\
' converge a 6' . Sia c l =
\ (,," - 6' I! j' \
i
e c un numero positivo mino-
Supponiamo che I () - At \ f c <: c l ' Riesce aHora : \I
='\: t:
abbastanza vicino
I
67
per
t =l, ••. , Y .
- 25 -
A. M. Ostrowski I'(fV
IZy \ ~
1 !y
c1
l(,=l
L
Pk
16" - 6' I =
2»
c1
!((II
c2 =
avendo posto
L
c2
Pk
k={
I6 k - 6' I
DaIle disuguaglianze precedenti e dall'osservazione che
K",
~
p, se-
gue:
L.
c"
e quindi
(29)
IA y +1 -6'1 c
t...
c" pc
I Ny \ ci~"
~
L
--;-; Ct
(
r
-::- . C
2 P ci Scegliamo c in modo che sia positivo e minore dei numeri c,;, - - ; dalla (29), per
Y = 1,2, ... si ricava
68
IA,)ytl - 61~ c
ed-anco;:'
- 26 A. M. Ostrowski
\~ ' ' ' t l -ffi questa relazione
(_eJ
2v
LC J
ei
.
.
e dunque vera per ogni ~
se si verifiea per
v=
1;
cil> prova la convergenza della sueeessione { ~v \ a 6
Osserviamo ora il seguente : Lemma I.
=========
(30)
Si ba infatti dalla (29) :
Oal Lemma I seguono immediatamente i seguenti: f:?I:?J~"t;.i:?=!. Se
(31)
....
lim ~
<(~l'
-
~)
e una eostante arbitraria,
e..
= o.
Si ba infatti :
69
allora
- 27 A. M. Ostrowski
= O(A.
-0).
II
~
Corollario 2. =============
La serie
L IA~-6'1
e convergente.
y={
Si ha infatti :
= 0
ht;,T:PJ~=H.
(Av - 6' ).
Per ogni indice k, esiste un numero
Tk tale
ehe:
1
(32)
=
Dimostrazione. Per quanto proeede, si ha :
v
N
I)k.
:::
11
(
t'=!
POieM 1a serie
to" - f)
L !=J.
v
\ + 6' -
/1.-,:) = ( 6"~
6" -
~t'
-
tf)
Y
1'1'
II
t'::d
\
(1
+
6" - ",; 6'k. - 6"
e asso1utamente convergente
ok - 6
(efr. corollario 2), posto
70
).
- 28 -
A. M. Ostrowski
v
T(
TV,Ie =
6' (1
't':.1
).
At'
f
6'k - "
esiste finito e diverse da zero il limite delle successioni • Posto
~ ... 00
tk = lim
T-.) k ,
si ha :
)
" .... DO
00
t
(33)
k
-
T
-.1,
k
=
L
~~y
( T
~+l,k - T
f'
k)'
Riesce inoltre
donde
ove
c. e
un numero maggiore di tutti i numeri
Dalle (33) e (34) segue:
Per il Lemma I esiste una costante c 6 tale che :
71
- 29 -
A. M. Ostrowski
e quindi, per V abbastanza grande :
/6' - Atot! I
(35)
16' - ~ YH I co
L
Dalla (35) segue la convergep.za della serie tanto esiste una costante c7 tale che :
e per-
~",-1+i
Si ha infine :
e quindi 'I
( 6'k, - 0 ) =
-L + 0 ( Q - \+V . tk
E' cost provato il Lemma II.
72
- 30 -
A. M. Ostrowski
Lezione V
Fine della dimostrazione della legge asintotica. Vari metodi per accelerare la convergenza. Applichiamo ora i Lemmi I e II per ottenere una tappresentazione asintotica di
Z~
Si ha :
m
=fr ove
«It
Z~
y b. V. 1
Consid~riamo
'"
1~-e'I"V
i valori distinti di
= 1••••• m); si ha allora per
(36)
K
• + 0 (
\ I\.'+i,- 6)) y
sono opportune costanti. Indichiamo con
(k
Ol~
P
ora l'espressione
:
73
l'espressione:
1
I
- 31 -
A. M. Ostrowski
poiche riesce
y NJ Ny, k.
~
() - At' = 11 i 1" '0'1 I G'1l .- At'
1
L
si ha : NV ----=0
lim \lot
00
Nvk ,
e quindi :
lim
.,1-;. co
Ky = p•
Possiamo aHora scrivere Zy
Supponiamo prima che la matrice S sia hermitiana. Si ha:
y (37)
ove con c1, .•. ,
=
t
C
1 P
0
(1)) \
I
"C'd
\
G' -
v
"2 r ~ I Ij c1 ~j. +... +
AJ
'
si intendono quei numeri b l' ... , bs non nuHi,
ordinanti in modo che riesca Supponiamo che
+
r-'
£> 0
t4 > tt> ... ). ae .
: sia aHora c1 74
l
O. Possiamo scrivere :
- 32 A. M. Ostrowski
AVtl - 6 I~V
;t a1
rv
- () rv
y
'l
~t r i
IT \6' - ~r \
i
T=! ~-t
~-i
rr
16'-~LI
T:i
2-
e quindi
Ne segue che lim )I~oo
Rimane ora da dimostrare che ~ >0 . Nel caso opposto dalle (37) seguirebbero Ie relazioni :
dalle quali si dedurrebbe
1 J
Z
'iT Itf - ~t I
il che
= 0 (1)
t~f
e impossibile.
Sia ora S una matrice normale non hermitiana. Nella Ipotesi supple-
75
- 33 A. M. Ostrowski mentare fatta in questo caso, i numeri
1
r
10',,_61
fra loro e pertanto si ha r, = s = m ; inoltre i numeri no essere ordinati in modo che
d't > at).
..... >~.,...,
sono tutti distinti
a
~1 , .•• , trtv
posso-
Ripetendo aUora
l'argomentazione del caso precedente si giunge alla tesL Si osservi che per mezzo di una matrice unitaria ortogonale possibile diagonalizzare la matrice S : pertanto il teorema
e sempre
e completamen-
te dimostrato. Esempio (11)
2\
Applichiamo il metodo ora esposto alla malrice S • (: quale ha come autovllori 2 e 7 . Scegliendo come vettore
;0
6 ) ' la i1 vettore
(1, 0), si trovano come approssimazioni del primo autovalore e delle corri-
spondenti autosoluzioni i numeri riportati nella seguente tabella:
,
I
I I
(i)
\
xl
11."
(i)
x2
0
3
1
1
2.076923
1.5
-1
2
2.0 419073413568826
2.0 29803921568627451
-1
4
2.0 15 277
1. 9762747097570
-1
0
(11) Cfr. A. M. OSTROWSKI, On the convergence of the Rayleigh quotient ........ , II, Archive Rat. Mech. and Anal. vol. II, 1959, 427.
76
- 34 -
A. M. Ostrowski
Vari,!!etodi per ac:ceJlrll'e 1& convergenz&,. Consideriamo la formula che esprime un procedimento iterativo del primo ordine ;
'f (~1)
~"+i:
f (),.)
e 8upponiamo che la funzione Si dice che fI
sia derivabile con derivata continua.
e un punta tis so dell' itenzion., .e riesce
6 :
r
(ft) •
Di.tinauiamo i seguenti easi ; 1) sial limo a
fj ,
f' (8' d.(
1. In tal caso, se
il procedimento 'iterativo
A.
e convergente
e abbastanza pros" e si ha :
(S8)
La convergen.a di 2) S1a
l A,,} a 6
e pertanto soltanto lineare,
se
e
fl (6') i o.
l1' I > 1. La successione l~,} non converge a ~ (6')
eceettuato il calo in cui uno dei ~!> diventi uguale a ~ ; pill precisamente,
e pOllibile costruire un intorno di 6' tale che,
se ~"e contenuto in questo
intorno e non coincide con ff , i valori successivi si. allontanano da 6' finche non liano usciti :dall' intorno. 3) Sia
If-l( f) I : 1.
E' questo un cas a indeterminato, eSElendo
1a convergenza possibile oppure no. Se si ha convergenza, questa e molto lenta, Sussiste a proposi .,. 77
- 35 A. M. Ostrowski to il seguente teorema : IV. Se, per \ ~~, riesce
e la successione
iA
y \ converge a 6'
(12)
1
(39) lim
v..
,si ha :
00
Torniamo ora al caso 1), supponendo in particolare = o. Nell'ipotesi che la
lim
AV+i -
6"'
( Av - 6'
1 2=
)
e quindi la convergenza dere se
f (A)
e
sia di classe 2, si pub affermare che
~"(6)
2
e almeno quadrati ca.
e possibile otten;re da
I
'f (Er) =
If (A.,)
Nel caso opposto ci si pub chie-
un'altra funzione
cp (6" ) = O. Ad esempio,
tale che
conoscendo due funzioni
(12) Per la dimostrazione cfr. A. OSTROWSKI, Sur la convergence et l'estimation des erreurs dans quelques procedes de resolution des equations numeriques, nel volum.e commerpo:rativo per D. A. GRAVE, Mosca, 1940, 213-234. Una traduzione inglese di questa memoria' si trova nel Technica.l Report No.7, 1960, degli Applied Mathematics and Statistics Labo:ratories, Stanford, University, California.
78
-36 A. M. Ostrowski
di iterazione :
Lf(6)=O, si pub considerare la funzione
(40)
- p)
r()" ) • I.
1a Quale ha derivata nulla in 6'
particolare, scegliendo
p ().,)
per p =
'0/( 6 (\f (6') . In
'Y ( A. ) = A. , l'espressione di
= p
- P)
e 1a derivata si armulla nel punto 0'
Teoricamente, cioe se
6')
diviene:
>v
per p
~ '(
P(~)
=
e noto,
1
si potrebbe rendere
convergente in modo quadrati co ogni procedimento iterativo dato da una fup. zione che abbia derivata diversa da 1. Dal punto di vista pratico, in generale, sarA nota solo una limiI I tazione per (6') • Sia m ~
f
Sara plausibile prendere p = 1 / ( 1 - M ; m) e quindi
M+m
1 -
2
79
- 37 A. M. Ostrowski Si osserva allora che
I~ I I( fi)
-
M
=
m
/2-M- m
1)
M+m 2
<
1
e
M
2)
M+m 2
>1
e
m
<
1
>1
}
.
80
l
<:
1
se
~
38 A. M. Ostrowski
Lezione VI
I metodi di STEFFENSEN e HOUSEHOLDER pera~celerare. 1a convergen;: (l3)
!!:..
r(
~v, A~+t .A v ), con punt~
SUpponiamo ora di conoscere tre valori successivi forniti dal procedimento iterativo
::I
• Formiamo i1 quoziente di STEFFENS~N 14) :
6' (41)
AM
t
\, ~ IIti, ...; :t~,+ j
A J
Esprimendo
L"H =
A '"
\
2 ~Ht+ ~Y+2-
I\y -
per mezzo di
H
.1\v= .x
y si ottiene
ove si ponga
(43)
~(Z)
=
z
tfLy> (z)]
z -
2
- [
If (,z)l
~
f (z) - f [tp (ztl
Possiamo dimostrare il seguente teorema : V. La funzione di iterazione punto fisso 6"
q) (z}
, introdotta da STEFFENSEN,ha, nel
,derivata in modulo; minore di 1 e pertanto il procedi;..
(13) eli'. A. OSTROWSKI, Uber Verfahren von Steffens.n und Householder zur Konvergenzverbesserung von Iterationen, ZAMP, vol. VII,1956, 218- 229. (14) Cfr. J. F. STEFFENSEN, Skand. Aktuarietidskr. 16,1933,64 - 72. 81
- 39-
A. M. Ostrowski mento iterativo (42) converge almeno linearmente.
t.p '( 0') = 1.
Oimostrazione nell'ipotesi Dalla (43) segue: (44)
q;
(z) - z ,. z -
2~(z) +~ t~(Z)]
4 (z) - z
si verifiea allora immediatamente ehe l'espressione
te per un eambiamento di riferimento rappresentato da z
~
e invarian
z +c •
If (z)~ f (z) + cecil> consente diassumere. per semplicita. f (z).
La funzione
te nell'intorno dell'origine.
r
(z) = z
+
{> (z)
supposta derivabile in certo numero di vol-
avra 10 sviluppo: k
~ z
equindi.posto f(z) = f(z) -
(45)
- z. = -
Z
(
+ •••• •
~
f
O. k intero ~ 2)
siha. daUa(44):
f [~ (z)] - f(z)
(O(z) designa brevemente il denominatore
[f(z)
=
f [ ~ (z)]
(46)
o (z)
=
.-f(z) ).
1(Z)
f'(x) dx
•
z dalla quale. applieando il teorema della media. si deduce ;
82
J2
O(z)
In forma integrale O(z) ha l'espressione :
j
6' = 0 •
- 40 A. M. Ostrowski
D(z) = f(z) f' [z +
(47)
~ f(z)]
Tenendo conto delle seguenti espressioni di f(z) e f'(z) f(z) =
~
f'(z): k
k k+l z + 0 (z ),
~ zk-l + 0 (zk) ,
si ha, daUa (47) : D(z) = k
(48)
~ 2 z2k-l + 0 (z2k) .
Sostituendo la (48) nella (44) si ricava :
~ (z) - z
= -
~ 2 z2k + 0 (z2k+l) k ~ 2z2k-l + 0 (z2k)
:::
z k
e quindi
(49)
~ (z)
= z (1 - _1 ) + 0 (z2) k
La (49) mostra che
q> '( 6) = 1 -
+.(
1, come dovevasi dimostrare.
Si noti che il procedimento iterativo (42) riesce convergente, in base aila (49), anche se il procedimento originarionon 10 Nal caso generale il teorema V
83
e.
e contenuto nel seguente teore-
- 41 A. M. Ostrowski
ma VI. HOUSEHOLDER (15) ha proposto una generalizzazione del metodo di STEFFENSEN. Si posseggano due procedimenti iterativi
\/+~ = tp ( ~\I
).
~v+~ = t (A v ). con 10 stesso punto fisso ff
(50)
e si costruisca la funzione
\JI z "f [If' (z)] - t.p (z) 1 (z) = _....:--.!-_ _
'4' (z)
......l-_--:..-~_-
tf (z) - 'f (z) t 't t tf
Z -
(z)]
sussiste il seguente teorema :
AVH = \f (Ay ). nell'ipotesi che la funzione l'
VI. La iterazione
ab-
bia derivata diversada 1 nel punto rtf • risulta convergente verso 6' e la convergenza
e almena \ quadrati ca.
Dimostrazione. Possiamo. senza ledere la generalita. supporre f5 = O. Nello intorno di
(51)
6' = O. si avranno per
Cf (z) =
0(
z t
't(z) =
0(1
z t
e
k ' .. z t ... ki ~~ z t ...
'If'
gli sviluppi :
~
(15) Cfr. A. S. HOUSEHOLDER, Principles of Numerical Analysis. Mc Graw - Hill, New York 1953. 126-128.
84
A. M. Ostrowski
eon
e
~
r t
diversi da zero eke kl interi maggiori
Supponiamo dapprima
~ '(0) =
0(
t1,
0
uguali a 2 .
t
~ '(0) = 0(1
1.
Dalla (50) si deducono Ie espressioni :
'1'(Z) - z
(52)
[If (z)
=_ z-
- z] . [ 'I'(z) - z]
f (z) + 1 [f (z)] - 'tt (z)
=-
N(z) D(z)
z D(z) - N(z)
(53)
D(z)
ove si ponga
Dalle (51) seguono : ~(z) - Z
= ( 0(
-
1) z + ~
k Z
+.•• ,
'o/(z) - z
=(
~1 - l)z +
ki
~i z +...
e quindi :
t
[
= ()(
"'1 z + ci
1
~
k Z
+
k f){
~i
z
k~
+ O(z
k+l
+z
ki +l
).
Si ottengono pertanto per D(z) e N(z) Ie espressioni : D(z) = (0( - 1) ( N(z) = (
c( -
O(t
-
k k~ 1) z + 0 (z + z )
1::1 ki +l 2 1) ( o(i - 1) z + (0( - 1) ~i z +(
o (zk+2 +~kt+2) e, di conseguenza, per
r
(z) vale la relazione
85
O(i -
1) ~ z
k+l
+
- 43 A. M. Ostrowski z D(z) - N(z) D(z)
(54)
k:l,+1 k z ) = O(z + O(zk t zI<;:)
O(z
=
Z
Ne segue che, nelle ipotesi ammesse (
0(
k+1
~
+
1,
0(1
t 1) ,
kt.
t Z
si ha per 1j'(z)
una convergenza almeno quadratica. Supponiamo ora che la funzione
~
abbia derivata uguale a 1
nel punta fisso dell'iterazione ) f(O) = 1 . Posto: f(z) =
f(z) - z
f(z) =
~
g(z) = (
()(t
,
'r (z) - z
g(z) =
,
si ha : k
(k
z t ... -
1) z
~
2)
+
e 1a (52) si scrive :
1LT
f(z) g(z) 1 (z) - z = - --'-"-'-.,.---- = g [~ (z)] - g(z)
N(z) D(z)
Potendosi anche scrivere, applicando il teorema della media: f(Z) D(z) =
r
g'(x) dx = f(z) g' [
Jz ed essendo k -1 g'(z) = ( 0/1. - 1) t O(z t ) risulta
86
Z
t
~ f(z~
,
)
o <.$' I.. 1 )
- 44 -
A. M. Ostrowski
r(z) - z
g(z)
= - g' [z
t
( Cl(t
=
~ fez)]
= - z to (z
ki
ki -1)z t ~1 z t .•.
=
kC 1 (C1c l)tO(z)
)
e quindi
La (55) mostra che si ha una
convergen~a
almeno quadratica.
n
teorema b aHora completamente dimostrato. La utilita di questo risultato dipende dalla possibilita che esso offre, unito al metodo di STEFFENSEN. di otten ere una iterazione convergente in modo quadratico. partendo da una iterazione della quale non sia
r
assicurata la convergenza. Infatti se
(z)
e la funzione che rappresen-
r'(
ta l'iterazione che ha 6' come punto fisso. e risulta
(0) = 1 • si
pub ottenere con it procedimento di STEFFENSEN una funzione la quale
I'tt'( I< 1 ; applicando alle funzioni (5 )
~
e
't
'If' (z) per
it procedimen-
to di HOUSEHOLDER, si ottiene una iterazione con convergenza almena quadratica. Quanto ora detto pub servire a dissipare Ie perplessita che potrebbe suscitare, a prima vista, la necessita di conoscere due funzioni iterative distinte (si noti che se fosse a coincidere con la funzione
~
=
't . la funzione 1f verrebbe
di STEFFENSEN).
87
- 45 A. M. Ostrowski Lezione VII Altri metodi per accelerare la convergenza. Applicazione al caso dei divisori elementari non lineari. Descriviamo ora un altro metodo per migliorare la convergenza dei procedimenti iterativi. Questo metodo
e piu lento degU altri,
ma pre-
senta il vantaggio di potersi applicare non solo aHe forme di iterazione del
All+{ = f
1'" ordine (tali che
di ordine infinito (tali che
(A~) ). rna anche aHe forme di iterazione AV41 = f (A~ •... )\d). (16)
Consideriamo aHora una funzione
f (x)
definita per x > 0
e verificante Ie seguenti condizioni : 1)
2)
>0
tf(X)
lim
~(x)
=0
x~o
3)
se
X ---to
Una funzione con
, Y--+ 0,
~
Y/ x ....tI.
(x) del tipo indicato
aHora
If(Y) = 1 +0(1:.. - 1).
f(x)
e ad esempio la funzione
x
x
0(
cx> O.
Sussiste il seguente teorema :
(16) In una forma piu speciale questo metodo e stato sviluppato nellibro: OSTROWSKI, Solution of Equations and Systems of Equations, New York. Acad. Press,1960, Appendice 1.
89
- 46 A. M. Ostrowski
VII. Sia
[ZII J
una successione convergente a
1~ = I z v
del tipo suddetto. Posto lim
'Y))I
Irr
l
()=
1"-i
1lI-t e il numero
- z:; I
.
~ ~
f (x)
una funzione
si suppone che
1 (17). Introdotte Ie guantitA positive
_1
1v ~ (IV-i)
(18)
'0.) = segno (z)l+~ - ~ )
. , pomamo
6"" = Max ( E~) [~H ) y. . "f
Suasiste aHara la seguente relazione ;
(56)
con
Z=z
n+l
-
Dimostrazione.
(17) Si osservi che per genza di ordine 0( + 1 .
f (x) = x
0(
questa formula rappresenta la conver-
(18) Per segno del numero complesso z si intende it numero ove (]- denota I' argomento di z.
90
z
Iz I
=e
LJ-
A. M. Ostrowski
Si ha, per ipotesi,
__
-4~~).l_ _ _ = 1 +O(
'y) e
~V-t
quindi:
(57)
IY)
til'\.
=
1
'1.\1
(1 +O( £'1\.)).
'Il-tl,.,d
Si ha ancora:
Sn
=
I(zn - ~ ) -
(zn+1 -
l;
)! z - t, n
e di conseguenza :
(58)
Sn = '1 n
Riuscendo inoltre della funzione
\
(1 + 0 (
0 n~
0,
Yn) ) •
1
n-+ 0,
i - 1
;::
0
e pertanto sussiste la relazione :
(59)
Dalle (58) e (59) si trae :
91
1Sn
n --'> 1 , per le proprieta
A. M. Ostrowski
2
=
'1 't
n (1
+
0 (
1 't'
n-l
(1
n
~n-l ~n-l
=
n-l
'\f n) ),
'Y n-l) ).
+0(
e di conseguenza : 2
(60)
~n '\fin
=
~ n-l
'n-l
(I +0_ ( L1n A ) )•
Dalla tpotesi
222
'1n
= 'tn-I
1
Si ha allora
6'n
8;
If n
0n-l fn-l
_ 6'
-
n
(1 + 0 (£ )). n
n-l
dall~
(60)
1
W n-I 1 n-l
'til
Consideriamo ora Ie seguenti differenze : 2
".+1 - Z
•
If. °n_l
~. Tn
•
6'.
fn-l
92
~
(1 + 0 ( )). Inn
1.-1
rn
"j' •.1
(1
+0
(l~.)).
- 49 A. M. Ostrowski
zn+1 - 'r">
= 00
I·zn+l - ';':)
= 6"
tr}
'\J)
(1
I = 6" +0
IYJ
n lo+1
(/J. ))
n In I n n
= er n 'Ylo \ '\JJ (1 + O( & ) )= In n+1
= Cii. 'Yl 0
Ln-
1 'tV
I n-
1 'IJ) (1 +O(
In
1l.J);
rormando il quozieote :
- t
")
Si ottiene 1a tesi. Ritornando a1 metodo di STEFFENSEN e HOUSEHOLDER, vogliamo mostrare. su un esempio. come questo metodo si possa applicare anche nei casi nei qualt Ie condiziooi date nella lezione precedente non so:. no soddisratte direttamertte. Sia A uoa matrice comp1essa non hermitiana e consideriamo il metodo del quoziente di Rayleigh (nella sua forma origina'Ie) applicato aUa matrice A • quando la successione
[A \I ~
converge ad
un autova1ore 6' di A al quale corrispoodono divisori elementari con esponente massimo L maggiore di 1. Si ha allora ove
(61)
Si mostra facilmente che :
~ ( A.) =). + (). - o)L 93
E( A ).
AY+i=~( Av),
- 50 -
A. M. Ostrowski
E (~) essendo limitato per ). ~ G" ; ma in ~esto caso ~ ( ~ ) euna pub parlare di derivata net senso che la funzione E ( A.)
e.~
1
funzione razionale delle due variabili
abituale~
e pertanto non si
E' perb possibile dimostrare
soddisfa una condizione di LIPS CHIT Z, nel sen;..
so particolare che:
quando
= 1 + 0 ( ~1 - 6" )
(63)
Per mezzo di queste relazioni, l'argomentazione gia appliicata sopra alIa funzione di STEFFENSEN
~ ( Iv ),
pub essere modlficata in modo che
si abbia anche in questo caso : (64)
~(~ )
- l?
= (1 - 1/L)
Cambiando rispondente
~
P(A)
(A)
(A - 6" e
Cf
)
+0 (
A. _6'
)2 •
(A ) e formando la funzione cor;..
di HOUSEHOLDER, e possibil e mostrare anco"
ra, utilizzando Ie (62) e (64) che riesce : (65) cioe che l'iterazione con
If' (A. )
(19) converge in modo quadrati co.
(19) Cfr. A. M. OSTROWSKI, On the convergence of the Rayleigh quotient ..•.. , IV, Archive Rat. Mech. and Analysis, vol. 4, 1959, 154-160, dove 94
- 51 A. M. Ostrowski C'e anche un altro metodo di accelerare la convergenza che utilizza la decomposizione in prodotto infinito : 00
1 1 - x
2 = 1+x+x + ... =
1T (1 - i~ ) lid
e anche le decomposizioni analoghe ove la base 2 e sostituita dal 3. Questo metodo
e importante nel calcolo della serie di LIOUVIL-
LE-NEUMANN e anche per Ia programmazione del quoziente per Ie macchine elettroniche (20) .
./. i calooli conducenti alle formule (61)-(65) sono sviluppati con tutti i particolari. (20) Cfr. A. M. OSTROWSKI, Sur une transformation de la serie de LIOUVILLE-NEUMANN, C. R. Acad. des Sciences, Paris, 203, 1936, 602-605; A. M. OSTROWSKI, Sur quelques transformations de la serie de LIOUVILLENEUMANN, C. R., 206, 1938, 1345-1347.
95
- 52 A. M. Ostrowski
Lezione VIII 11 metodo del quoziente di RA YLEIGHgeneralizzato per i divisori elemen;.. tari non lineari. Sia E f 'V
una matrice quadrata di ardine
avente
n
fA' -sima e alla colonna
l'elemento corrispondente alla riga
V
-sima
uguale ad 1 e tutti gli altri nulli. Si ha aHora (21)
(66)
Sussiste inoltre la seguente regola di moltiplicazione: (67)
Per mezzo delle matrici matrice A •
E /Ai v
possiamo scrivere ogni
(a ~" ) di ordine n nel modo seguente :
1• n
(68)
A·L ar" E~, fJ )I
Si pub allora considerare l'insieme delle matrici quadrate di or dine n come un sistema di numeri complessi con n2 unita E
(21) Con
b"-f
fY
si indica il simbolo di KRONECKER.
97
•
- 53 -
A. M. Ostrowski
In particolare consideriamo ma matrice "unita ausiliaria U "
-m
(introdotta dall'AITKEN) di ordine m, avente la prima diagonale
sopra la diagonale principale composta di elementi 1 e tutti gli altri elementi nulli.
U
m
pub scriversi : m-l
(69)
Urn =
L
E
~/f+1
~ =1
Le matrici unita ausiliarie
godono di una proprieta molto elegante rigu!!-r
dante Ie potenze. Si ha precisamente: m
(70)
(Um)k
=)"
~ ~=1
E
JA;,IA-+k I I
Dalla (70) si ottiene, in particolare :
(71)
(U )m-l = El ,(U )m = 0, m ,m m Consideriamo una matrice A
o
(72)
A =
o
(U)s = 0 m
se
s > m.
del tipo (22)
+
Sussiste il seguente
(22) Questa e Is. forma canonica di JORDAN di una matrice di ordine 1 corrispondente ad un div~sore elementare di ordine 1 •
98
- 54 -
A. M. Ostrowski
LEMMA 1. Sia Ao una matrice di tipo (72).
A t ()"
si ha
1
(73)
=
Ue ---=(e-l)
(74)
(A
o
-
A, 1)2
Dimostrazione : Dalla (72) segue :
1
1
=
(Ao - AI)2
( 6" -
A )2
[ I
+
Ue
G"-A
j
-2
D'altra parte, della decomposizione Newtoniana of)
(1 +x)-2.r
(-:) xk
k=o segue la seguente relazione algebrica : (75)
1
e-!
·L (-:) k=o
99
l
+
Per
1 ~ fi'
!!.
- 55 A. M. Ostrowski
ove
e un polinomio in x.
P (x)
Ponendo nella (75)
x
=
U,t
6->V
si trae, in base aUe (71) :
[I
+
6-,\,
(J
pOiche
Ue
= ( _l)k (k
+1 )
r" LD
(76)
=
e
(6"_>t,)k
k=o
, si ha allora:
t-i
1
Uk
~
L
(k+1) U-e"
Considerando nella (76) i1 termine principale, corrispondente a k =
e-1,
si deduce la (73). D'altra parte, moltiplicando i due membri della (76) per Ut
'
si ottiene":
~
Ut
(77)
(A 0
~I) 2
=L k
=1
e'"
k U
( A. - 6 )k+i
e, prendendo il termine corrispondente a
k=
t - 1,
si ricava la (74).
Il Lemma e cosi completa rrente dimostrato. E' nota che ogni matrice A con autovalore 6'
e
equivalente
ad una matrice somma riemmanniana di matrici canoniche elementari di 100
- 56 -
A. M. Ostrowski JORDAN; esiste quindi una matrice P non degenere tale che
L. m
C = p- l A P =
(78)
. 1
1=
ove A. = f5 Ie. + Uo. da f) m
1
e B
1<"1.-
•
(23)
A. + B 1
e una matrice avente autovalori diversi
La matrice A ha l'autovalore 6'
di molteplicita. uguale a
Li=l e. . Sussiste il seguente 1
LEMMA 2. ~ L = max (E l , .•. , A. 4= 6" ,-e ~ ---+ 6' si ha : 1
(79)
(A -
A1)2
A - 61 (80)
(A -
~
H
:t 1)2
=L
trd e
supposto L > l, ~
H
( A - (5
= (L ,. 1)
+
)L+1
H
( :t - 6'
)L
O((A_1O)L) +0
e una matrice non nulla dipendente dalla matrice
(23) La (78)
CA~dj A e dalla ma-
e nota come riduzione a forma canonica di JORDAN.
e + indicano la decomposizione diagonale.
101
La
- 57 -
A. M. Ostrowski trice P che ri duce A aila forma canonica di JORDAN. Dimostrazione. Sia C la matrice definita daJla (78); riesce (C -
"\
II.
I)
2)
L . (Ai -
=
,2
'I?'
If + (B - ,{ 1)
A,
e quindi
(C -
"\
1\
1)
-2)" =G
. (Ai - \It
r)
-2'
)-2
+ (B - \. I)
.
Per il teorema precedente si ha :
(81)
(C -
A I)
-2
.~
m
~
= /
_I_.~
i= 1
qualora si conglobi in 0 si mantiene limitato se
Pi,. .. 1-
De.
fl.l _ _ _ __ v - ,_ _
+ 0
(
(:1..-S)?;+-i
1
( A- G )L
l' addendo (B - A.. I) - 2 che
A-+ 6' •
Osserviamo ora che, per la definizione di L , il termine preponden rante nel secondo membro della (81) si ottiene per ~. = L . Ricordando aHora che
Ue~
= 0 se s ~
~
L
e. , possiamo scrivere 1
:
1
m
(C -
A 1)-2
=
+0
i=l
Posto H
L- i
U~.
,riesce H toe resta cosl dimostrata la
~
102
- 58 A. M. Ostrowski.
(79) per 1a C. Si ha inoltre :
-L .
ei (A.
-
U
i=l
1
\ -2 - 1\.1)
'\
-2
+ (B - 6'1) (B - 1\.1) ,
1-" 6'
e quindi in.base a1 teorema precedente e aUlosservazione che per 11ultimo addendo a secondo membro si mantiene limitato, riesce:
+O(
1
l· -1
( ~ - 6" )'
)J.
Con un .rl;lgionamentb perfettamente ana1ogo a1 precedente si ottiene:
=l.
C - 6' 1
(C -
L-~ (L -1) _U_e;___
AI) 2
(
+
A- 6' )L
0(1 ) \(.~. _Ei)L-l
e quindi 1a (80) per 1a C. Per dimostrare comp1etamente il teorema occorre ancora far vedere che i1 teorema seguita a sussistere sostituendo C con A. Cia segUe da1 fatto che (A -6'1) (A -AI)-2
= P (G,.0"I) (C-A.I)-2
p- l
Applichiamo aUa matrice A il procedimento del quoziente di 103
- 59 A. M. Ostrowski
RAYLEIGH. Poniamo:
~
\ -1 (A - A ... I) 0 ( ,
':>/ Si ottiene aHora
\Y+1 =
r(
Ay) con
~)J=
\ -1 f(A - Ay I) •
i (A,,) funzione razionale di
~ e piu precisamente :
tf (A ) ora introdotta possiede
La funzione
6' come punto fis-
so e ammette nel 6' una derivata uguale a 1 - IlL. Si ha infatti :
=
e, per
A f 6'
~(A -
I
6' I) (A -
~ (A
-
AI) -2
A 1)-20( 0(
\
e I\, -) 6' in base alle (79) e (80)
=
=
(L-1) (
A. - 6)
§H rX
+0 (
L ~H(x +0(A..-0')
Se aHora si suppone
f.J H oG I
to,
si ha :
104
X- 6" )2
- 60 -
A. M. Ostrowski
L - 1 L
e quindi si pub enunciare il seguente teorema: IX. IL metodo del guoziente di RAYLEIGH generalizzato applicato ad una matrice d6tata di divisori elementari non lineari, converge linearmente se i vettori
~
sono scelti in modo generico/cioe in modo tale che
0(
~HO
105
f '( (9) = 1 - L1
.
CENTRO INTERNAZIONALE MATE MATI CO ESTIVO ( C. I. M. E. )
L. E. PAYNE
ISOPERIMETRIC INEQUALITIES FOR EIGENVALUES AND
THEIR APPLICATIONS
ROMA· Istituto Matematico dell'Universiia
107
Isoperimetric Inequalities for Eigenvalues
!!!2 Their Applications by L. E. Payne University of Maryland 1. Introduction:
The classical isoperimetric inequality - - the one after which all such inequalities are named - - states that of all plane curves of given perimeter the circle encoloses the largest area. This inequality was known alr..eady to the Greeks who had also some knowledge of its analogue in three dimensions. The next isoperimetric inequality to be estabilished was perhaps that of J. Steiner (1836) who made use of an operation called symmetrization. His symmetrization gave a solid at least one plane of symmetry. preserved its volume and diminished its surface area. In 1856 Saint-Venant ~21 made the conjecture that of all cy= lindrical beams of given cross-sectional area the circulafe beam has the highest torsional rigidity. The first conjectured inequality for eigenvalues waS made in 1877 by Lord Rayleigh
(ag; namely. that
of all membranes of given area
the circle has the minimum principal frequency. An isoperimetric inequality for electrostatic capacity was
conjectured by Poincar~
(66] in 1903; i. e••
of all solids of given volume
the sphere hali! the minimum electrostatic capacity. ArQund 1923. Faber [20J and Krahn
[40 gave independently a
proof of the Rayleigh conjecture for the membrane. (In the next section we give a slight variation of their proof.) In 1930 SzegH f9~ gave the first 109
-2L.E. Payne
complete proof of the isoperimetric inequality conjectured by Poincare. The Saint-Venant conjecture was proved by Polya C6~ in 1948. A search of the literature reveals little concentra,"erl work in the area of isoperimetric inequalities prior to 1945. However, about that time new interest in the subject was awakened, principally by the, inves= tigations of Polya and SzegO' ~7, 68, 69,. 95J. Their book in 1951 has given added impetus to research in this field, and during the past ten or fifteen years many new isoperimetric inequalities involving physical quan= tities from various branches of mathematical physics have been obtained (36, 38, 39, 51 - 55, 57 - 61, 70 - 75, 83 - 85, 96, 99 - 101,
106].
We restrict ourselves in this paper to a consideration of iso= perimetric inequalities for eigenvalues. In fact we concern ourselves primarily with those arising in membrane and plate theory. Even then We are able to cover only a few of the most interesting and important isoperi= metric inequalities. In tbe next section we define the various eigenvalues problems which are to be considered. In sections III through VI we prove a number of interesting isoperimetric inequalities. In sections VI and VII, we present two useful applications of eigenvalue inequalities. Finally, we demonstrate in secti on VIII the use of isoperimetric inequalities in deri= ving "optimal" a priori bounds for solutions of boundary value problems. We shall not discuss in this paper the numerous eigenvalue inequalities in one dimensional problems-those of vibrating strings, .vi: brating rOds, etc. - - which have been derived in the literature (see e. g. , [3, 7 - 9; 42, 50, 86, 88] ). Neither shall we present the various
isoperimetric inequalities relating the eigenvalues of (A) - (E) to those 110
-3L. E. Payne of analogous finite difference problems [24, 28, 36, 71, 100, 101,) • The reader will undoubtelly be aware of various other isoperimetric inequalities which are not mentioned in this paper, as well as other applications for those which are mentioned. This paper is not meant as an axha,ll stive survey, but rather as an introduction to a number of fascinating isoperimetric inequalities for eigenvalues, together with a few interesting applications. II. Problems to be Considered: We consider the following eigt;!nvalue problem defined on a bounded N-dimensional region RN with boundary CN: (A)
(B)
(C)
(D)
~u + Au =
0
in RN
u =
0
on CN in RN
Llv+~v=o
!J 2W
+
/j 2,\
!\
(E)
IQh /0 'It
fJ\)'~," =
0
=
0
Aw
row w-= 'ron
on CN in RN on CN in RN
0
- .0. A- 0 1\ )0."x'Ii - 0 6h =0 i'
-
on CN in RN
O'llJ-
9
h
=
on
0
In the above problems
eN •
/j denotes the Laplace operator and ((;/0""
denotes the normal derivative on CN · The constants ~
111
I
y")
-4-
L.E. Payne
A, .Q ,and
q are the eigenvalues of the various problems. Because
of the forms of the operators it is clear that all eigenvalues are real and non-negative.
Ai
In two dimensions the eigenvalues values
~i
of (B )
of (A) and the eigen=
may be interpreted as the squares of the vibration
frequencies of a fixed anp of a free membrane.
In three dimensions both
and ~. have interpretations in acoustics, diffusions theariy, J. 1 ' electromagnetic theory, etc. In two dimensions the eigenvalues of
the
:A.
1
(C) and
A.
I2..1 of
1
(D) arise in the theory of elastic plates. The quantities
A . are proportional to the critical buckling loads for a clamped 1
elastic plate occupying R2, while thell i are proportional to the squares of the vibration frequencies of a clamped elastic plate defined over R2• The eigenvalues
q
of problem (E) are the so-called Stekloff eigenva=
lues [9~. Most of the inequalities which we consider in this paper will involve the£i:rst non-zero eigenvalues of problems (A) - (E).
Regard~ess
of N we shall refer to the problems throughout as membrane, plate or Stekloff problems. We order the eigenva'lues
A. in the following way 1
(2:1)
The other ei genvalues are similarly ordered. Throughout this paper we assume that the bounding surface C is sufficiently smooth so that all of the applications of the di vergence theorem remain valid.
112
-5L. E. Payne
III. The Faber-Krahn Inequality and Its Extensions :, We give in this section a slight variation of the Faber-Krahn proof of the Rayleigh conjecture in N-dimensions (Krahn actually proved the result inN -dimensions) and indicate a way of extending their result. We prove then first the following theorem: Theorem If The first eigenvalue in the fixed membrane problem for R is not smaller than that for the sphere of the isame N-volume. It is well known (see e. g., [15] ) that the eigenfunction IA.!, corresponding to the first eigenvalue \ ' of (A) does not challge sign RN. If the first eigenfunction is taken to be positive, then clearly u.~, can have no relative minimum in RN. For convenience we now drop the subscript on ll.t. Let RN(u) (with boundary eN (u) ) denote that portion of RN over which u
>ii.
At each point on the surface u = U, a curvilinear
coordinate system is introduced, which involves any convenient (N-1 ) dimensional coordinate system in the surface u = U, and t he normal to the surface. 1)
l)By an easy generalization of results of O. D. Kellogg (Foundations of Potential
Theor~,
Dover Publications, Inc. (1953) pp. 273-277) it can be
shown that the introduction of tnis system of coordinates is legitimate. Another proof of the Rayleigh conjecture by L. Tonelli (Sur un probleme de Lord Rayleigh, Monatsh. Math. Phys., Vol. 37 (1930) pp. 253-280) avoids the introduction of such a system of coordinates.
113
-6-
L. E. Payne For any function F defined in R it then follows that n
'u n
(3. 1) a
,
.
,-
!'
\
t
o ..
-
CN(u)
F
,
I gra d
u\
ds du
~
(see e. g. Polya and Szego L76~ ). Here un denotes the maximum value of u in RN .
AI =
(3.2)
We note then toat \ i ' satisfies
IRN igrad uJ ,0
2dv
2
,
u dv
'RN
iun~C
i
I
(tr) grad u ds dU N = -2 ~Un u ~o "CN(IT) \grad trl ds du
6
We introduce the notation
(3.3)
Clearly
If N( u) I
(3.4)
is the volume of RN( u), Then from (3.3) we have d
'f N (u) dIT
j
\,..,
;:
r:
CN(ii)
Hence by Schwarz's inequality
114
ds \grad ii
I
- 7L. E. Payne
(3.5)
where SN (ti)
is the surface area of CN (u). imetrlc inequality we ha.ve
2( N-l) N
J
(3.6)
Here
By the classical isoper-
t.» N
W
2 N
•
denotes the surface area of the unit sphere in N-dimensions.
Inserting (3,6) into (3.5) we obtain
r
~u) q)
du
(3.7)
tl
J grad u I ds ~
CN (u)
[
N
~
T
N-l
(U)j 2(
T)
CD 2 . N
N
Thus (3.8)
CD 2/N =
N
2( N
~l)
fV!')-f
Jo l
where VN is the volume of RN'
1 2(N~1 )r~J2 NJ
did
L
N
tV N
I
Clearly since u (VN) = 0 115
-8L. E. Payne we have that
~2
du I d- - i d
(3 .•• 9)
rNj
where
J
~N /' l ~o
I
6-
2 u d
is the lowest eigenvalue of the problem
d;
(3. 10)
(VN 2 \
>
rr2(N~I)
:n
+
fpOO
The solution to this problem si given by
(3.11)
where
P
J
=
is to be determined so that P(VN) = O.
That is
(3. 12)
.. N-2 J2- /
where
j N;2 is the first zero of the Bessel function
116
J N;2
r
N
J
L. E. Payne
Combining (3.9) and (3. 12). and inserting the result into (3,2), we obtain finally r
2/N
2(V) \
(J N
(3. 13)
~
N
rVN
Jo
2 u
N
j
~ 2(~) IN
2
[-.!L!L IJ d" d fN
IN
~
drt-J
cJ ] 2/N 2
[ ..-!L > "i' NV
V
l!:! 2 •
Since the equality sign holds if RN is an N-sphere thl1 ia the _ired Faber-Krahn inequality and the theorem is proved. In lact it lollowl easily from our prool that t~ equality sip is valid in (3.13) 11 and only if
~
is an N-sphere. We seek now 80me polsible extensions 01 the Faber-Krahn ine-
quality in two dimen.iona. To this end we introduce a cartesian coordinate system in such way that RN Ues in the hall space
~
> O. Then
uL is representable as (3. 1-4)
i
where . E C
in the closure 01 RN and vanishes on Cit the por-
tion (or portion.) 01 C2 In the open region Xl> O. Clearly slies
117
'f saU-
- 10 -
L. E. Payne
(3. 15)
~ =0
The function
I
on C2 .
f may be interpreted as the first eigenfunction in the fixed
membrane problem for a 4-dimensional body symmetric about the Xl-axis (see Weinstein [10~
). By the FaQer-Krahn inequality in
4·dimensions we have then
(3. 16)
But V4 is proportional to 11, the moment of inertia (unit density) of V2 about the x2-axis, i. e. ,
(3. 17)
Iil fact (3.16) may be written as
(3. 18)
Equality clearly holds if R2
is a semicircle.
Similarly, if the coordinate system is chosen in such a way that R2 lies in the quadrant Xl
'> 0, x2 >0,
(3. 19)
118
and u1 is represented as
~
11
~
L. E. Payne then
~
must satisfy
(3.20)
where C~ is the portion of C2 for which xl ') 0 and x2 ' o. Again ~ may be interpreted as an eigenfunctio of a 6-dimensio~ nal membrane with both the xl and x2 ues as axes of symmetry. The volume of this body is proportional to J 12 where
(3.21)
The Faber-Krahn inequality in
~ 1~
(3. 22)
1/2
6~dimension8
then yields the
in~quality
11/3 j2 .
iT [ T2'J
2
12
Finally, let us suppose that R2 lielJ interior to the wedge bounded by
0 ~
G~ %where
u = rn sin .'" 9
(3. 23)
The function
n is an integer. In this case we set
If
r'
satisfies the equation
119
2
2
2
r = x1 + x2
L. E. Payne
In this case C* is the p<)rtion of C which lies in the open region
o< G < IT/Yo
.
From Krahn's result
[41]
in
2(n +
1)
dimensions it follows as before that
(3.25)
where
f~~
Kn=
(3.26)
L2
r
2n
1
2 sinn e d A \ • J
With such results for integral values of n, it is natural to conjecture the following theorem. Theorem II; If R2 lies interior to the wedge; ~ real
01. ~
i then
(3.27)
where fJ
i
(3.28)
Ko( =
j
R r 2
2~
. 2
S10 0(
120
9
dA ,
e! ~~
for any
- 13 -
L. E. Payne with equality if and only if R2 is a circular sector. This result has recently been proved by Payne and Weinberger [58] . We now apply Theorem II to a simple example and note that there appears to be no obvious systematic method of determining the origin to give the best lower bounds. Let G be a right isoceles triangle with equal sides of unit length. If the origin is taken at the midpoint of the hypotanuse we have 0(.
= 1
and hence
.At~
(3.29)
6.714.
If one of the acute-angled corners is chosen as the origin we have
d..
= 4.
Then we obtain
~ 1~ 6. 775.
(3.30)
Finally, if the right-angled corner is taken as origin we have
~ = 2
which gives (3.31) All of these bounds
~re
better than the lower bound 6.029 obtained
from the Faber-Krahn inequality. The exact value is known to be 7.025. The results for the cases in which the body lies in a half plane or in a quadrant are easily extended to higher dimensions. It appears likely that an&,logues in N-dimensions of the results for various wedge angles can be established, although this has not been carried out. 121
- 14 -
L. E. Payne IV. An Optimal PoincarH Inequality for Convex Domains.
r
IN; 1
The Poincare constant K is any upper bound for
where
)
2 is the first non-zero eigenvalue of (B). The existence of K
for quite general regions is well-known (see e. g., ever, explicit values for
[15, 65J
); how-
K (or equivalently lower bounds for /N 2 )
are in general npt known. In this section we prove the following theorem first established by Payne and Weinberger Theorem III.
[59J .
The first non-zero eigenvalue }4'2
in the free membra-
ne problem for a convex region RN satisfies the inequality (4.1)
where D is the diameter of RN . The inequality (4. 1) is the best bound which can be given in terms of the diameter alone in the sense that
1"2 D2
11 2
tends to
for
a parallelepiped all but one of whose dimensions shrink to zero. We shall prove the result only in two dimensions; however, the same method applied in the appropriate manner will yield the same result for any N; The quantity
y" 2
is defined as the infimum of the quotient
dA
(4.2)
r
among functions which have bounded second derivatives in R and sati· sfy
~
J R2
d A = O. Now for any such function
122
f
we consider
- 15 L. E. Payne the set of lines which divide
R2 into subregions of equal area. It folR2 into two convex
lows by continuity that at least one line divides
subdomains of equal area over each of which the integral of
vanishes.
~
We now divide each subdomain into two additional convex subdomains of equal area over each of which the integral of
f vanishes.
By cintinuing this process we arrive after a finite number of steps at a division of R2
v
into convex subdomains
area AV' Over each of th,
R:
R2 of arbitrarily small
the integral of
~
vanishes.
We introduce a rectangular coordinate system in any following way:
2
Let the x -axis lie along the line of maximum diameter of take xl -axis perpendicular to the
R~
in the
R;
and
x2 -axis and tangent to one end of
R:. Let L" be the length of the projecton of R~ on the x2 -axis. Cl~arly L v .~ D. Let p(y) qenote the length of intersection of R~ with the line x2 = y. Since the area of
R~
can be made arbitrarily
small, then after a sufficiently large number of subdivisions
p(y)
may
be made to satisfy
p(y) ~ ~
(4.3)
V
for any prescribed ~ . Because of the convexity of R2 ) p(y) be a convex function of y. Let (Actually
M be a uniform bound for
~
r
I
its first derivative in
and the integral
t\f~
y \ d..5
if
f c2
Then by the mean value theorem
123
R2.
its first derivative I . in the closure of R2)
M could denote a uniform bound for ~
must
Ii)
- 16 -
L. E. Payne f() f ~ dAJRtf (~) ' ux2
r
(4.4)
2
(4.5)
J
L(
0
p(y)
,-
B2dy ~
_..2
I '-IlP (:o,y)
2 IVr
Ii; fdA - J.Lvp(y)[r(O'y~:y ~ h"A~ t iJ~; i ~A. I. r6) r(OI~) dl ~ ~ Ai ~. J)radfl2 ~ la/ ~\/ ~ [~'
Av~ ,
J
" Lv
(4.6) (4.7)
dA
2
- 2
~
Av
d A rp(y)
2
t ,
Also
and
(4.9)
J, R2
ply)
41 (o,y) dy
I
~
M A,
We require now the fOllowing inequality
124
i. .
Thu.
(O,y)] 2dy_
- 17 -
L. E. Payne However before we prove this inequality let us demonstrate that knowledge of such an inequality would lead to the desired result (4.1).
Note that
(4.10) together with (4,7) - (4.9) would yield
(4.11)
JlI~ \grad ~12d A nr D-2{III' 12 d A - tr Ayt (2+l)}2
- 2
or
Pl'
2
Av
t
t~ Igrad 1\2 d AntD- 2 t ill~ 12 dA! -
(4.12)
2
- ,; Ai
2
~ [2+ rr2 0- 2 (2+ t )]
.
A summation of the inequaUties (4. 12) would now be made over all subdomains R:. The sum of A., is the area of
R~
• Since f-
is arbitrarily small we would then obtain the inequality
(4.13)
i
is any function with bounded seconcl derivatives satisfying
d A = 0 , we would have from (4. 2)
(4.14)
125
- 18 -
L. E. Payne Thus. once the one dimepsional inequality (4.10) is established Theorem II is proved. We prove then the following Lemma: Lemma I.
Let p(y) be a non -negative convex function
on the interval tiable function
(4.
0 {. ~
o~
y defined
y ~ L; then for any piecewise continuously differen-
(y) satisfying
r
15)
PlY)
i (y) dp 0
it follows that
Assume for the moment thllt differentiable. Then the function
(4. 17)
r
ply)
p is strictly positive and twice
~
which minimizes the quotif,mt
h' d) (y)] 2
r
PlY)
[~(y>f ely
among the functions satisfying (4.15) must satisfy
(4. 18)
~
rr
+ Vp
I{ (0 ) =
f'
r·O. 'il(y)f O. (L) = 0
126
o
- 19 L. E. Payne where
V is the minimum value of the quotient
(4.17).
We now intro-
duce the new variable
w
(4. 19)
=
~'
pl/2 .
The function W satisfies the Sturm-Liouville system
(4.20)
W
+
'(W=O
O( Y < L
W(o) = W (L) = 0 •
Because of the convesity of p) the term in square brackets is non-positive. Hence, multiplying (4.20) by W and integrating by parts we find
LL (WI} 2 dy (4.21)
( L
J
2 W
o
dy
Since W(o) = W(l) = 0, the quotient is bounded below by the first eigenvalue of the vibrating string fixed at its ends, i. e. ,
(4.22)
"
~t/L
Thus the lemma is proved if p) Now if
'"~ (y)
0
and twice differentiable in y.
is any function defined on the interval
the fimction
127
0 ~
y ~ L,
- 20 L. E. Payne
will satisfy (4.15). Hence for p ( G c2) .>
0,
(4.16) implies that
Clearly (4. 24) holds for the uniform limit of admissable functiorls p. Hence
(4.24) is valid for any non-negative convex function of y. This
proves lemma I and thqs establishes the desired inequality (4. 16). We were interested in this section only in the lemma that was required for proving theorem II. The proof, however, actually shows that _1 if P ~ is a concave function of y the eigenvalues '( h of (4.18) satisfy the inequality (4.25)
Vh ~
(h - 1)2 1i 2 L -2
1 with equality if and only if p-'2 is linear in
128
y.
- 21 L. E. Payne
V. Additional Eigenvalue Inequalities: In this section we prove a conjecture of Weinstein ~o~ which relates certain membrane eigenvalues to those of the buckling problem for a clamped plate of the same shape. The proof of this conjecture was giveni by Payne
[5D.
Other interesting inequalities, both proved and
conjectured are discussed in the latter part of the section. We prove first the Weinstein conjecture: Theorem IV. The first eigenvalues in the buckling problem for a clamped plate is not less than the second eigenvalues of the membrane of the same shape which is fixed on the boundary, i. e. ,
This inequality relates eigenvalues of problems (A) and (C) defined on the same domain. To prove the theorem we make use of the following minimum principle for
A2
(5.2)
where ~ is any sufficiently smooth function defined in RN which vanishes on
eN (5. 3)
and satisfies the condition
I
RN
U1
~ dv ~ 0 . 129
~
22 L. E. Payne
Let trial functions
f
0(
be defined as follows:
o
wl tV" = a" wl + ~ ,
(5.4)
r"
V\
I.
~
G\
\,1.
= l,2, .... N.
where w1 is the first eigenvalue of problem (C)
J u'1\ w{dv
and the constants assume that by (5.2)
(5.5)
a,. are so chosen that
(5.3)
defined on RN, is satisfied. We may
is not zero since if it were we would have
llN
A2
and the theorem would be established. The second inequality results from an application of the Schwarz inequality to the identity
\J
(5.6)
w1 /J w1 d v.
RN
In view of the boundary conditions satisfied by wl' it follows from the divergence theorem that
(5.7)
130
- 23 -
L. E. Payne Thus from the minimum principle (5.2),
(5.8)
where summation is to be carried out over Latin subscripts but not over Greek subscripts. Adding the inequalities we obtain
N 2 ~ <- aj J: i
~
rt
~ Igrad w \2 dv + 1 D
"It
t
llc/wi fd~'tJi
,(C) dv ",l~l R t;)l(. X' 'Ox-"'7:l)(' 0 '" ~ Cl \ d
a~ r w~ dv + i Igrad w1\2 dv J
JR~
JRw
But
L14wl12 dv
(5. 10)
j\
grad w11 2' dv
Rw
Thus, in view of (5.5) it follows that (5.11)
and the theorem is proved. It can be shown withQut too much difficulty that the only region for which the equality sign holds, is the sphere (see
[5U ). Polya and SzegH [76J have shown that
131
- 24 -
L. E. Payne
(5. 12)
This leads then to the inequality (5. 13)
Payne [5~ has shown further that for
RN
convex
(5. 14)
This inequality is the best possible in the sense that the equality sign holds tn'the limit for a strip region. He showed also that for convex RN (j
(5.15)
...\..:. 1
<
16 3
~
2 1
and (5. 16)
Before passing on to other isoperimetric inequalities, we mention a few conjectured inequalities involving membrane and plate eigenvalues. We consider only the case N = 2.
~:?Jlj~~t'f:~~ =~: The first eigenvalue
.f.. 1
in the vibration problem for a clamped
plate is not less than that for the circle of the same area.
~:?Jlj~~t~~~ =~I~ The first eigenvalue
il
1 in the buckling problem for a clamped
132
- 25 -
L. E. Payne
Plate
is not less than that for the circle of the same area.
~:PpJ~£t,;~~ =PI ~ th \ The n eigenvalue II n in the fixed membrane problem for R2
satisfies the inequality
(5. 17) where A is the area of R2.
~:ppj~£t,;~~ =~¥ ~ The nth eigenvalue IN n in the free membrane problem for R2
satisfies the inequality
yn
(5.18)
f
~
4
rr (n - 1) A-1 .
Conjecture I was made by Lord Rayleigh [81J ; conjecture II is due to POlya and SzegH Poyla
[74J .
[76] , and conjectures III
and IV are due to
The first two conjectures were proved
[76J
under the
hypothesis that in each case the first eigenfunction does not change sign in R. But under what conditions this hypothesis is satisfied, has not been determined. Conjectures III and IV were proved by Polya
[74J
for
special types of "!Space filling" domains. We list also two other interesting isoperimetric inequalities which in a certain way complement the Faber-Krahn inequality. We state the theorems without proof, referring the reader to the original papers.
133
- 26 -
L. E. Payne
Theorem V:
A1
The first eigenvalue
in the fixed membrane problem
for a simply connected region R2 is not greater than that for the annular domain (concentric circular
boundarie~)
of the same area fixed on the
outer boundary,whose perimeter is equal to that of C, and free along the inner boundary. Theorem VI: problem for
The first non-zero eigenvalue
}v 2
in the free membrane
RN is not greater than that for the sphere of the same vo-
lume.
~5],
Theorem V is an extension of earlier results of Makai
r44l ,
and Polya [75] . It was proved by Payne and weinberger'- [
6~J.
Inequalities of a similar nature for membranes defined in a multiply connected region have been obtained by J. Hersch (unpublished). Theorem VI was formulated in two-dimensions by Korhauser and Stakgold [
40]
and proved for simply connected regions by SzegH
[9~.
For general N-dimensional regions the theorem was proved in an extremely ingenious though quite elementary way by Weinberger (99J . From Theorem I and Theorem VI it follows that for general
(5.19)
an inequality first observed by Polya (see Payne
(5.20)
r70 \ _
..I
[51] )that if in two dimensions
.f n+2
It has been shown in fact R2 is convex
n ~ 1 •
134
- 27 -
L. E. Payne Isoperimetric inequalities for the first non-zero eigenvalue of (E) have been given by Weinstock [106J
and Payne [521 . Since these ~2
inequalities yield upper bounds for
they are not helpful in the appli-
cations treated in sections (VI -VIII) and hence will not be discussed here. Inequalities for eigenvalues of elastically supported membranes and plates, for those of inhomogeneous membranes and for those arising in various other physical problems can be found in the literature (see e. g. ,
~9-35, 48-50, 53,57, 78-80, 87] ). We mention finally, before passing on to other considerations that Payne, Polya and Weinberger
[551
An+l
(5.21)
have shown that for N = 2,
~ 3~ n
(5.22)
Q n+l
"
(5.23)
1\.2
~ 31\1'
~
90.
n
The authors have conjectured that the optimal constant in 2.539, the ratio of A2 /
A1
(5. 21) is really
for a circular membrane.
135
- 28 -
L. E. Payne VI.
An Isoperimetric Inequality in Classical Elasticity: We derive in this section a lower bound for the fundamental vibra-
tion frequency of an incompressible isotropic elastic body occupying a region R3
and fixed on its boundary C3' This bound yields an important
criterion for stability of viscous fluid flow (see e. g., Serrin
[S9J
It improves the previous criterion of Serrin result of Velte
!yS] .
[89J )'.
and overlaps a recent
The theorem which we !Ulall prove in this section
is summarized in a recent note of Payne and Weinberger We omit the details of the
d~rivation
[6 iJ .
of the governing equations
(referring the reader to a text on the Theory of Elasticity) and merely state
tha~
the eigenvalue LA) (which is proportional to the square of the fun-
damental frequency) is characterized ~y the following minimum principle:
co ;:
(6. 1)
t~,J=i
min
{to· . i.P .
J~~ I~'d I~'d
j
\ >.¥.
I. .
.1
dv
W; dv
f;\ , I\; )-
The minimum is taken among all piecewise' continuously differentiable vector fields
(6.2)
'1 i
which satisfy
l.
= 0 on C3
i=1,2,3.
It is an immediate consequence of (6. 1) and (6.2) that
w
is
a non-increasing functional of the domain R3 . Thus W may be bounded below by the corresponding eigenvalue of any domain containing R3 . We compute the exact value for G) if R3 is a sphere of diameter d. This value will then be a lower bound for the fundamental frequency of any do-
136
- 29 L. E.
P~yne
main in terms of the diameter d of the smallest circumscribed sphere about R3' We prove in fact the following theorem: Theorem VII.
The first eigenvalue W of a vibrating incompressible
elastic medium fixed on the boundary is not less than the second eigenvalue in the fixed 3-dimensional membrane problem for any circumscribing sphere. Now for the sphere R3
Ui
we let
be the vector field which
minimizes the right hand side of (6.1) subject to conditions (6.2). It is established by the usual arguments of the calculus of variations that
Ui
exists, is twice continuously differentiable in R3
and satisfies
there the set of Euler equations (6.3)
6. U
(6.4)
+ w U . = d hi r() x. 1
Tr v i, i
1
~
0
and (6.5)
U
i = 0
on the boundary of the sphere. In (6.3) From (6.3) and (6.4) it follows that (6.6)
f1
h
=0
We now define the function
137
h is an auxiliary function.
- so L. E. Payne
(6.7)
X. 1
U
Using (6.3) and (6.4) we obtain ~
5'
6V+wv=
(6.8)
x.
''0 hi"';} x.
" t I l
or
IJ (6
(6.9)
+w)v=o
Moreover, since V vanishes on the boundary of th sphere, and the divergence of
U.1
RS it follows that
vanishes throughout
V,
(6.10)
rav
fc);
= 0
on the boundary of the sphere. Thus either V;;: 0 or W is an eigenvalue
A
of (B). But by Theorem IV it follows that either V 0;; 0 or
CD ~ ~ 2. (Actually the eigenfunction corresponding to
(J
cannot be
the first eigenfunction of (B) since V must vanish at the origin in view of (6.7)
).
Thus either V;; 0 or the theorem is proved.
~J h = 0 ~~l 1 '0 Xi in RS' Since h is a regular harmonic function it must then be a conIf V:: 0 it is clear from (6.8) that
stant. Thus, the components T].
1
(6.11)
~T]
G. x· - -
satisfy
+6JU. = 0 1
in RS' and vanish on the boundary. Not all of the
138
U.1
may vanish iden-
- 31 -
L. E. Payne ticaUy. Consequently
U.
must then be an eigenvalue of (A). Since
1
the first eigenvalue of (A) for the sphere is simple it is clear that the divergence condition (6.4) cannot be satisfied if lows then that
I: 2.
A· 2
U l.
= c. u. . It foI1
1
.
In order to show that the eigenvalue w for the sphere is equal to \ 2 we note that the vector field
(6.12)
jr3/2
U1
= x2 J 3/ 2
i]2
.- r) /3/2 = -xl J 3/ 2 (b) r
lJ 3
= 0
(Iw
r)
satisfies (6.4), (6. 5~ and (6.11) provided (6.13)
where p is the lowest positive root of the equation tan p = p.
(6. 14)
In (6.13)
r
denotes the distance from the origin and J denotes the
Bessel function. The (j) of (6. 13) is precisely equal to
A2
for the
sphere and Theorem VII is proved. The results of this section are directly applicable in Stokes flow problems for a viscous fluid. The interested
reader may find the necessary formulation in any standard text on Fluid Dynamics.
139
- 32 L. E. Payne VII.
Application of Isoperimetric Inequalities in the Study of the Nodal
Domains of (A). In this section we discuss some interesting results obtained by Pleijel [64J ,through use of the Faber-Krahn inequality and inequalities (5. 19) and (5.20). Let
~
be a twice continuously differentiable function defined on a
region RN. A connected domain T is called a nodal domain of ~ -f 0 in
T and if T is bounded by surfaces
also portions of the boundary
1=0
i
if
and perhaps
CN. For Simplicity we consider only the
two- dimensional case, and make the further assumption that
R2 be
simply connected. A similar result will hold in higher dimensions. We make use of the Courant nodal line theorem (14) which states that the number of nodal domains of an eigenfunction belonging to the nth eigenvalue of (A) (or (B) ) is less than or equal to n.
From this theo-
rem and inequality (5. 19) it follows that the eigenvalue V2 corresponding to the
r
2 of (B) cannot have a closed nodal line. For if V2 had
a ring nodal line, then it would be the continuation into R2 eigenfunction u1 of the regiotn
P; 2
=
\T
A -\.
where ~I
T
T
of the first
bounded by the ring nodal line, i. e. ,
denotes the first eigenvalue in the fixed mem-
brane problem for T. But, by the well-known monotony principle for the eigenvalues of (A) it fallows that
'A~ ~
l' which would give
(7. 1)
in contradiction to (5.19). Thus
V2 cannot have a closed nodal line.
By (5.10) it follows in the same way that if R2
140
is convex then V3
- 33 L. E. Payne
cannot have a closed nodal line. In fact in this case the nodal lines of V3
will consist of one or two transverses not cutting each other. The
inequality (5.10) shows further that for convex R2'
Vn
can have at
most n-2 "interior" nodal domains. The question as to whether u2 can have a closed nodal line has not as yet been answered. If it could be established that
u2
can not
have a ring nodal line (and there appears to be reason to believe that this is true) then a number of additional isoperimetric inequalities would follow. We prove now the following theorem due to Pleijel Theorem VIII.
For only a finite number of eigenvalues
the number of nodal domains of Un be equal to
each Tithe function
un f; 0
corresponding to I
so that
An
un of (A) will
n.
To prove this theorem we let T11 T21 ...... Til domains of the eigenfunction up.
[64] .
An
be the nodal of (A). In
is the first eigenvalue
of a membrane covering Ti' Hence by the Faber-Krann inequality (3. 13) we have
(7.2) where
Ai
denotes the area of the
ith nodal domain. By adding the
inequalities we have
(7.3)
A/lf .2 ~ G' ~ -1 Jo, n
or
141
- 34 -
L. E. Payne (7.4)
Taking the limit as n ->
An/ n
t;)O
and noting that (Wyl's law [107, 108] )
tends to 4 iT / A we obtain
n -I
<
C' n
lim. sup.
(7.5)
IT -:1: = 0.691 ... , Jo
00
which proves the theorem of Pleijel. In the cases of the square and the circle the maximal subdivision by nodal domains occurs .only for n = 1, 2, and 4. The result of Pleijel has been extended by Peetre
[62J ' [63] '
to Riemanian Manifolds. The result may also be extended in the following - :1 + way (see for instance Bramb~e and Payne L12J ). Let Ti denote any nodal domains of a function
~
for which
denote any ~Odal domain over which area of Ti
and
Ai
~
is positive,
is negative. Let Ai
~
let Ti denote the
the area of Ti. We establish the following theo-
rem: Theorem IX: If for any constant
'A
~f+Af~
(7.6)
an~
;~
0
in R,
<. 0
on C 2;
I'
then
,
\
~ <'A 1
a)
orr7 is empty, if
b)
Tt is not empty if >.. >).. 1 and for each nodal domain
142
'-J
- 35 -
L. E. Payne To prove tnis theorem we note that either 1.0
+
or Ti
+
will be non-empty, and each Ti
~
I
0
throughout
~
will be bounded by a nodalli-
+ ne. Thus for any nodal domain Ti we have
The first integral on the right of (7.7) is non-negative. Thus
\ + \ 1\(T i)
(7.8)
f 2dv
~ \\ /\ ~ 2dv.
r.+ ~
From the monotony principle for
Tt
AI'
I
it follows that
(7.9)
Thus, if
,,< A1 +
tion unless Ti
the insertion of (7.8) into (7. 7) leads to a contradiis empty.
This proves the first part of the theorem. If
A). ~ 1
and T
t
is not empty we obtain by an application of
the Faber-Krahn inequality to (7.7)
(7. 10) This proves assertion b) if it can be shown that for
A.>A
l'
T7 is non-
empty. To prove this we make use of the fact that the first eigenfunction ul
of (A) is positive throughout R2' We assume that
143
f is nowhere
- 36 -
L. E. Payne positive and show that this leads to a contradiction. From Green's identity we have
I ul[6~+A~JdV··~f~:I~.+(A·~I) hUldv .
(7.11)
~t
C),
The term on the left is non-negative while if the terms on the right are non-positive for the trivial solution
r
~~
is nowhere positive in RN
A;. ~ l'
i ': 0, it follows that for
Hence, if we exclude
\ >>. 1 :
must be po-
sitive at some point in R , and the theorem is proved. Similar results
[sol and
have been obtained by Hartman and Wintner [27] ,Protter McNabb
G~
Let
G"
+
denote the number of components Ti ' Then
G"
(7.12)
A ).
2L:. i
+ A.
II
1
Pt;
(7. 13)
C'<....
J.2o G"
A
AA/ IT .2 Jc
This gives an upper bound on the number of nodal domains
144
+
T.
1
- 37 L. E. Payne VIII.
A Priori Bounds: In this section we are concerned with boundary value problems for
uniformly elliptic operators. We frequently wish to obtain upper and lower bounds for the error in the approximation of the solution (or its dervatives) at points inside some region
RN, if the data of the boundary value problem
is approximated in the mean square sense. Often in an L2
w~
may be interested only
bound for the error (or its derivatives) over the region. Methods
for obtaining pointwise bounds and bounds for energy integrals have been known for some time (see Diaz and Weinstein [19J ,Prager and Synge [77J ,Diaz and Grenberg [18] ,Greenberg [25J
,Diaz [16, 17J '
Maple [46] ,Synge [92, 93J ,and others). In these various methods the approximating functions must belong to special classes of functions, and the upper and lower bounds require the use of two different classes of functions. For example, lower bounds may require the use of functions satisfying the differential equation while upper bounds involve approximating functions satisfying the boundary conditions. Considerable labor might be saved if we were able to obtain both upper and lower bounds from a single set of approximating functions, especially if we were able to use as approximating functions any sufficiently smooth but otherwise arbitrary functions. With this in mind, we are lead to seek appropriate a priori bounds. For example, suppose we wished to con~ider
the following boundary value problem for an elliptic operator L(u)
of order 2 m defined pn·a region RN with boundary CN- a) L(u) prescribed in RN,
b) a set of boundary conditions Bi(u) (i = 1, .•• , m-1)
prescribed on
Let uS suppose that we are able to obtain an a priori
CN'
bound of the following type for an arbitrary sufficiently smooth function W.
145
- 38 L. E. Payne
(The functions u and w employed in this section are not to be CODfus-
eel with the eigenfunctions of (A) and (B) ). 2dv,
(8.1)
with explicitly determined constants Kj . Then by tal$g W = u ..
!f
where
f
is any sufficiently smooth approximafing function we would ...
ve an L2 bound for the error in approximating u in terms of mean sqll!
re integrals of the error in approximation of the data. The Rayleigh.Ritz tec)mique could then be used in determining an optimal choice for linear combinations of a given
s~t
f
(among
of functions}.
By the triangle inequality it follows that
11 /2 _ \"\W 2dV11 /2 ~
(8.2)\\ \l!2dv
~.JRI
I.
I..
+
j
'..'
U.hv11/2.
Rt.I
'j
f. \
u.
L'R
2dv'\
'"
~I
1/2~
~" We would thus have upper and lower bounds for the L2 integral of the solution u. This demonstrates that once explicit constants Kj have been obtained, the L 2-bounds follow in a straight forward manner.
If we had in addition an inequality (8.3)
'2 \ w(p)i ~ .
Co (P)
y
2 ) w dv + F ~ L(w) J Rr.J
I
where F denotes some lqtown functional of L(W) and Co(P) is an explicitly determined constant, then (8.3) and (8.1) together would give
146
- 39 -
L. E. Payne u.
pointwise bounds for
The problem of finding a priori bounds is thus reduced to the problem of obtaining the explicit constants. The existence of such constants is assured for quite a wide class of'boundary value problems, but explicit constants are often difficult to compute. The optimal constants are usually just eigenvalues of associated eigenvalue problems. For istance, suppose all of the Bi(w)
vanish on
C. Then if we define for sufficiently
smooth ~ min
~i =
(8.4)
~f
Then (assuming
\t
dV ~I'I Km is lXi-I. The con-
0) the optimal choice for
stant :X I will usually not be known. However, we may use any lower bound for ct 1 as our Km. In particular, if we have some isoperimetric ty, which gives a lower bound for
~
1, we may use this value for our Km'
Consider the special case in which L and
is the Laplace operator
It is well known that
jRt-I (6 r
min 0 on CN
r=
(8.5)
') t
,
This leads then to the inequality for functions
~
(8.6)
~
I
w2dv
~N
.
~ I2" 1 I
\
dlw)2dv P- ~
is any lower bound for
~ 147
I .
)2dv
2
~~
where
inequa~
dv
W which vanish on
~
IOl
I
~I
~ ell w)2 dv ~tl
C
- 40 L. E. Payne Let us now take for L(u) x = (xl' x2'
i j L(u) = (a (x)u'i)' j
(8.7)
... , x
),
where the summation convention is understood, and the comma denotes partial differentiation. The compqp~nts of the symmetric matrix a i j , are assumed to be piecwise continuously differeatiable in RN• It is assumed further that positive real numbers
const~ts
(5! 15 ~ .5 I"
•
tl )
ao and al
exist such that for all
and all x in ~;
(8.8)
The idea of using a priori inequalities for obtaining bounds was used by Fichera [21,
22J
who derived on a priori inequality of the fol-
lowing type (see a1$0 Bramble ane Payne
T~e optimal constant
K
[10J
):
is the reciprocal of the eigenvalue ""t
by the following minimum principle (see e. g. Fichere and Payne
(8.10)
[10] ) t =
min
r=o
on
eN
148
defined
[221 . Brambl(l
- 41 L. E. Payne where
r".,/
1;'/
~)'I
denotes the conormal derivate, i. e. ,
"'0 J U
(J! /
(8. 11)
1
1)1 = a
.
-'() x. n J • 1
Under sufficient smoothness assumptions the minimizing function V satisfies L L (v) :;: 0
(8. 12)
I
v =0
IUV
on CN
L (v) - 't ('J'(' = 0
't is a Stekloff type eigenvalue. Although an isoperimetric inequality is known fQr
A1
' no such isoperimetric inequality is known for 't .
Results of Payne and Weinberger [56] do, however, give a crude lower bound for 1;
which is sufficient for our purposes. Methods which in so-
me cases give very good bounds for "G were also introduced by Fichera. [21J For points P and a function F
on the iqterior of RN
can be obt~ined (see e. g.
an explicit constant Co(P) [ 101
) without as suming
existence of the Green 1s function. The insertion of the explicit constants in (8.3) and (8.9) then yields the appropriate bounds. We should point out that similar results are contained in papers of Fichera [21-23] ,and Payne and Weinberger [56J . We consider now the Neumann problem for L(u) i. e. , prescribed in RN and b)
~;
a) L(u)
prescribed on the boundary eN of
RN' Let us suppose first that the surface
149
CN is convex. We wish to
- 42 L. E. Payne
obtain pointwise bounds for
u
and its derivatives in
RN: Since the so-
lution is determined only up to an additive constant, we fix the latter by the normalization
f
(8. 13)
(Iiv
-
f )dv :: 0 ,
" I'\N
where
~
is a prescribed approximating function.
By inequality (4.13) we have for any sufficiently smooth function W satisfying
(8. 14)
r'R
W
i.
,In "'I'i
2
w dv
~
dv :: 0
D2 1T2
RN
point in
is
RN then t
2
I
\
,1 R N
grad w \ dv
D2
'.
~ 7r2" \ a II a I
~ ~
Rtl
ij
w, ,w)' dv 1 1 u
w2 ds .
'D
We seek now a bound for Since
\
I
N
in K_ -ON
conv~i1t follows that if the origin is taken at some ~
xi ni
>0
at every point on
CN' Consider
then
?: w2dv + 2
(8.15)
i
(;
( x. w w, . dv 1 RN
~N
1
1
By the arithmetic geometric mean inequality Q
(8.16)
~
. Y t w2 ds ~
(N +1'1., )
~.p
Q
\
- J G
""~
:I
~
t "N
N +iI. + 1
'--;;:-r
:
; 1\
2
w dv + D
rJ..
2 -1 f 0(
.,'
02 al)
a ij w, ~ Ii
150
\
Igrad w \\ 2dv RI'i
.W, 1
.dv,
J
- 43 -
L. E. Payne The optimal choice of
f
(8. 17)
~
2
tw ds ~
gives the result
(~= II)
~
(N + 2 rr) D2
IT
all
ij
a w'l,w'J,dv ~N
GN
It can be shown without difficulty that a lower bound for the first
non-zero eigenvalue
'2
E follows immediately from (8.17) i. e.
of
(8. 18)
This inequality is however, not isoperimetric.
f a ij w,' w, ,dv in terms of the
Let us now obtain a bound for
JQ
1 J Neumann data of W. From Green's identity we have
I
ij JRaw, i w, j dv =
(8.19)
W
L(w) dv •
RtJ
Ii
Then by Schwarz I s inequality we obtain
(8.20)
I
ij
a w, ,w, ,dv 1 J
1 2
1
1 f()
ll/2 [\ 2 \'
~ [r~w dSjt~ ((;): )dSJ
RN
(i~
eN
and from (8.14) and (8. 17), (8.21)
151
1,
+ Jw dv Q. N
2
]1/2
L(w) dj QtJ
- 44 L. E. Payne Inequality (8.21) together with (8.14) and (8. 3) thus yields an upper bound for
W(P). By setting W = IA,
bounds for
- ~
we obtain the desired pOintwise
u. Using similar techniques it is also possible to obtain bounds
for derivatives of u. We have obtained a simple bounds in the Neumann problem for a conveX domain. We wish, however, to treat the Neumann problem for mOre general regions. It is clear that convexity was used only in establishing (8. 14) and
~8.
17). Thus the critical step in the derivation of bounds in the
Neumann problem for a general region is the establishment of the corresponding inequalities (8.14) and (8.17). We show now how a lower bound ~
for ~2
of (E) leads to the desired inequalities.
Si~ce ~ that
is a lower bound for
then for
VI
normalized so
( w ds = 0 (Note that we are now using a different normalization
JtN
than that used previously, i. e. , (8. 22)
92
!j
'\I
N
w = w + constant).
2
w ds ~
C~
f~
we use the arithmetic-geomein terms of
2 ds and
c.tJ
For general regions Bramble and Payne [11J
152
have obtained a lower bound
- 45 L. E. Payne
k for
This gives not only the desired bounds in the Neumann pro-
~2'
blem, but also a lower bound for the P'2 of (B). We have illustrated by some simple esamples how the optimal constants in our a priori bounds are related to the eigenvalues of various problems. 1\iany more esamples could be given but let me conclude by considering a some-what different type of problem. We seek bounds for the solution IAI of the Dirichlet problem for
~ U, + '( \AI
the operator and
A n f.I.
where
V is a constan lying between
An
of (A). Bramble and Payne [121 have computed a priori
bounds of the following type for an arbritary sufficiently smooth function
w . (8.24) where the cqnstants
K1
and ~ are explicit.
If we knew the eigenvalues A . we would then have an a priori 1 bound for w2 dv in terms of the Dirichlet data. If the ~ i are not
J
known it
Suffi~S to have a lower bound for A n+ 1
which is still larger
than V and an upper bound for ~ n which is still smaller than 1/ • The upper bounds are usually obtained from an application of the RayleighRitz techique to the Rayleigh quotient. Upper bounds are somewhat more difficult. However, various methods for obtaining lower bounds are known (see e.g., Weinstein [103, 104J ' Aronszajn [1J ' Temple Kato
[3~
, Bazley [4,
5] ,
[97] ,
Weinberger [102] , Bazley and Fox
[6] , and others [2, 13, 26, 42] . ) These and other methods are discussed in the papers of Weinstein and De Vito which appear in this vo-
153
- 46 L. E. Payne
lume and will not be considered here. It is again possible to derive, for points interior to
RN, the ine-
quality,
IW(P) I 2 ~
(8.25)
C1(P) \ w2dv + F 1 (A w + " w )
~N with explicit
C1(P) and F l' This leads then to pointwise bounds for u, We have considered only a simple example of a fOrced vibration.;ty-
pe problem. Much more general results have been obtained. (see [12J ),
154
- 47 L. E. Payne IX.
Concluding Remarks : In this paper we have presented a few of the most interesing and
most useful isoperimetric inequalities for eigenvalues. The many important isoperimetric inequalities for energy integrals ( torsional rigidity, electrostatic capacity, virtual mass, polarization, etc.) have not been considered. Eigenvalue inequalities have then been used to investigate various properties of eigenfunctions and solutions to boundary value problems. They have been employed finally in the determination of a priori bounds for solutions to various boundary value problems. The bibliography which follows is not complete, but is intended only to be representative. Additional references may be obtained from the bibliographies of the books and papers cited there.
155
- 48 L. E. Payne
Bibliography 1.
Aronszajn, N., Approximation methods for eigenvalues of completely continuous symmetric operators, Symp. Spectral Theory and Diff. Probs, Stillwater, Oklahoma (1951) pp. 179-202.
2r
Aronszajn, N., and Weinstein, A., On a nified theory of eigenvalues of plates and membranes, Amer. J. Math., vol. 64 (1942) pp. 623-645.
3.
Banks, D., Bounds for the eigenvalues of some vibrating systems, Pac. J. Math., vol. 10 (1960) pp. 439-474, see also Pac. J. Math., vol. 11 (1961) pp. 1183-1203.
4.
Bazley, N., Lower bounds for eigenvalues with applications to the helium atom, Proc. Nat'l Acad. Sci, vol. 45 (1959) pp. 144-149.
5.
Bazley, N., Lower bounds for eigenvalues, J. Math. Mech., vol. 10 (1961) pp. 289-308.
6.
Bazley, N., and Fox, D., Truncations in the method of intermediate problems for lower bounds for eigenvalues, J. Res. Natl. Bureau Standards, vol. 65 B, (1961) pp. 105-111.
7.
Beesack, P. R., A note on an integral inequality, Proc. Amer. Math. Soc., vol. 8 (1957) pp. 875-879.
8.
Beesack, P. R., Isoperimetric inequalities for the nonhomogeneous clamped rod and plate. J. Math. and Mech., vol. 8 (1959) pp. 471-482.
9.
Beesack, P. R. and Schwarz, B., On the zeros of solutions of second-order linear differential equations, Can. J. Math., vol. 8 (1956) pp. 504-515.
10.
Bramble, J. H., and Payne, L. E., Bounds for solutions of second order partial differential equations. Contrib. to Diff. Eqtns.
157
- 49 -
L. E. Payne
(to appear). 11.
Bramble, J. H., and Payne, L. E., Bounds in the Neumann problem for second order uniformly elliptic operators. Pac J. Math. (in print. )
12.
Bramble, J. H., and Payne, L. E., Upper and lower bounds in forced vibration and allied problems. (to appear). \
13.
Collatz, L., Eigenwertprobleme und ihre numerische Behandlung, Chelsea Press, New York (1948).
14.
Courant, R.; Ein allgmeiner Satz zur Theorie der Eigenfunktionen selbstadjungierter Differential aus drUcke, Nach Akad. Wiss. G~ttingen (1923) pp. 81-84.
15.
Courant, R., and Hilbert, D., Methoden der Mathematischen Physik, vol. 1, Springer, Berlin (1931). English Edit. Methods of Mathematical Phisics, vol. 1, Interscience, New York (1953).
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Diaz, J. B., Upper and lower bounds for quadratic functionals, Proc. Symp. Spectral Theory and Diff. Probs. Oklahoma A. & M. (1950) pp. 279-289, see also Collectaneae Math., vol. 4 (1951) pp. 3-50.
1.7.
Diaz, J. B., Upper and lower bounds for quadratic integrals, and at a pOint, for solutions of linear boundary value problems, Proc. Symp. Bdry. Val. Probs. Diff. Eqtns., U. S. Army Research Center, Univ. Wisconsin, April (1959) pp. 47-83.
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Diaz, J. B., and Greenberg, H. J. / Upper and lower bounds for the solution of the first biharmonic boundary value problem, J. Math. PhYs., vol. 27 (1948) pp. 193-201.
19.
Diaz, J. B., and Weinstein, A., Scharz's inequality and the me-
158
- 50 L. E. Payne thods of Rayleigh-Ritz and Trefftz, J. Math. Phys. vol. 26, (1947) pp. 133-136. 20.
Faber, G., Beweis, dass unter aller homogenen Membranen von gleicher Flache und gleicher Spannung die KreisfBrmige den tiefsten Grundton gibt, Sitz. bayer. Akad. Wiss. (1923) pp. 169-172.
21.
Fichera, G., Formule di maggiorazione cd!messe ad una classe di transformazioni lineari, Annali Mat. Pura Appl. vol. 36 (1954) pp. 273-296.
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Fichera, G., Methods of functional linear analysis in mathematical physics, Proc. Int. Congo Math., Amsterdam, vol. 3 (1954) pp. 216-228.
23.
Fichera, G., Alcuni recenti sviluppi della teoria dei problemi al contorno per le equazioni alle derivate parziali lineari, Conv. Inter. Equaz. Lin. Alle Deriv. Parz. (1954) Trieste.
24.
Forsythe, G.
Asymptotic lower bounds for the frequencies of
certain polygonal membranes, Pac. J. Math., vol. 4 (1954) pp. 467-480. 25.
Greenberg, H. J., The determination of upper and lower bounds for the solution of the Dirichlet problem, J. Math. Phys., vol. 27 (1948) pp. 161-182.
26.
Gould, S. H., Variational Methods for Eigenvalue problems, Univ. Toronto Press (1957).
27.
Hartman, P., and Wintner, A., On a comparison theorem for selfadjoint partial differential equations of elliptic type, Proc. Amer. Math; Soc., vol. 6 (1955) pp. 862-865.
159
- 51 L. E. Payne
28.
Hersch, J., Equations differentielles et fonctions de cellules, C.
R. Acad. Sci. Paris, vol. 240 (1955) pp. 1602-1604. 29.
Hersch, J., !In principe de maximum pour la frequencefondamentale d'une membrane, C. R. Acad. Sci. Paris, vol. 249 (1959) pp. 1074-1076.
30.
Hersch, J., Une methode pour l'evaluation par defaut de la premiere valeur de la vibration ou du flambage des plaques encastrees, C. R. Acad. Sci. Paris, vol. 250 (1959) pp. 3943-3945.
31.
Hersch, J., Une interpretation du principe de Thomson et son analogue pour la frequence fondamentale d'une membrane, C. R. Acad. Sci. Paris, vol. 248 (1959) pp. 2060-2062.
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Hersch, J., Sur la frequence fondamentale d'une membrane vibrante: evaluations par defaut et principe de maximum, ZAMP, vol. 11 (1960) pp. 387-413.
33.
Hersch, J., Physical interpretation and strengthening of M. H. Protter's method for vibrating nonhomogeneous membranes; its analogue for SchorHdinger's equation, Pac. J. Math., vol. 11 (1961) pp. 971-980.
34.
Hersch, J., and Payne, L. E., L'effet d'une contrainte rectiligne sur la frequence fondamentale d'une membrane vibrante, C. R. Acad. Sci. Paris, vol. 249 (1959) pp. 1855-1857.
35.
Hooker, W., and Protter, M. H., Bounds for the first eigenvalue of a rhombic membrane, J. M. Phys., vol. 39 (1960) pp. 18-34.
36.
Hubbard, B., Bounds for eigenvalues of the free and fixed membrane by finite difference methods, Pac. J. Math., vol. 11 (1961) pp. 559-590.
160
- 52 L. E. Payne
37.
Kato, T., On the upper and lower bounds for eigenvalues, J. Phys. Soc. Japan, vol. 4 (1949) pp. 415-438.
38.
Keller·, J. B."
The shape of the strongest column, Arch. Rat.
Mecfl. Anal.. vol. 5 (1960) pp. 275-285; see also Tad1bakhsh,
I., and Keller, J. B., Strongest columns and isoperimetric ine39.
qualities for eigenvalues,·J. Appi. Mech. vol. 29 (l:962)pp. 159-164. Keller, J. B., Lower bounds and isoperimetric inequalities for eigenvalues in the SchrHdinger equation. J. Math. Phys., vol. 2 (1961) pp. 262-266.
40.
Kornhauser, E. T., and Stakgold, I., A variational theorem for
~2u +),u = 0
and its applications, J. Math. Phys., vol. 31
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Krahn, E., Uber eine von Rayleigh formulierte Minimaleigenschaft des Kreises, Math. Ann., vol'. 94 (1924) pp. 97-100; see also
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Krein, M. G., On certain problems on the maximum apd minimum of characteristic values and on the Lyapunov zones of stability, Amer. Math. Soc. Trans., series 2, vol. 1 (1955) pp. 163-187.
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Krylov, N., Les methodes de solution approchee des problemes de la physique mathematique, Mem. Sci. Math., No. 49 (1931).
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Makai, E., On the principal frequency of a conve,x membrane and related problems, Czech. Math. J., vol. 9 (1959) pp. 66-70.
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Makai, E., Bounds for the principal frequency of a membrane and the torsional rigidity of a beam, Acta Szeged, vol. 20 (1959) pp. 33-35.
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Maple, C. G., The Dirichlet problem: Bounds at a point for the
161
- 53 L. E. Payne solution and its derivatives, Quart. Appl. Math., vol. 8 (1950) pp. 213-228. 47.
McNabb, A., Strong comparison theorems for elliptic equations of
48.
s~.cond
order, J. Math. Mech., vol. 10 (1961) pp. 431-440.
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Nehari, Z., Some eigenvalue estimates, J. Analy. Math., vol. 7 (1959) pp. 79-88.
51.
Payne, L. E., Inequalities for eigenvalue!! of membranes and plates, J. Rat. Mech. Anal, vol. 4 (1955) pp. 517-528.
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Payne, L. E.,
~ualities
for eigenvalues of supported and free
plates, Quart. Appl. Math., vol. 16 (1958) pp. 111-120. 54.
Payne, L. E., A note on inequalities for plate eigenvalues, J. Math. Phys. vol. 39 (1960) pp. 155-159.
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Payne, L. E., Polya, G., and Weinberger, H. F., On the ratio of consecutive eigenvalues, J. Math. Phys. vol. 35 (1956) pp. 289-298.
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57.
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60.
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Polya, G., Torsional rigidity, principal frequency, electrostatic. capacity and symmetrization, Quart. Appl. Math., vol. 6 (1948) pp. 267-277.
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Polya, G., f'ur une interpretation de la methode des differences finies qui peut fournir des bornes superieures ou inferieures, C. R. Acad. Sci. Paris, vol. 235 (1952) pp. 995-997.
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Poyla, G., More isoperimetric inequalities proved and conjectural, Comm. Math. Helvetia., vol. 29 (1955) pp. 112-119.
73.
Polya, G., Sur les frequences propres des membranes vibrantes, C. R. Acad. Sci. Paris, vol. 242 (1956) pp. 708-709; see also Sur quelques membranes vibrantes de forme particuliere, ibid, vol. 243 (1956) pp. 469-471.
74.
Polya, G., On the eigenvalues of vibrating membranes, Proc. London. Math. Soc., vol. 11 (1961) pp. 419-433.
75.
Polya, G., Two more inequalities between physical and geometri-:. cal quantities, J. Indian Math. Soc., vol. 24 (1960) pp. 413-419.
76.
Polya, G., and SzegH, G., Isoperimetric inequalities in mathema_tical physics, Annals of Math. Studies No. 27, Princeton U. Press (1951).
77.
Prager, W., and Synge, J. L., Approximations in elasticity based on the concept of function space, Quart. Appl. Math., vol.
5 (1947) pp. 241-269.
164
- 56 L. E. Payne
78.
Protter, M. H., Lower bounds for the first
eigenva~~e
of elliptic_
equations, Annals of Math.; vol. 71 (1960) pp. 423-444. 79.
Protter, M. H., Vibration of a non-homogeneous membrane, Pac.
J. Math., vol. 9 (1959) pp. 1249-1255. 80.
Protter, M. H., A comparison theorem for elliptic equations, Proc. Amer. Math. Soc., vol. 10 (1959) pp. 249-299.
81.
Lord Rayleigh, The theory of sound, 2nd. ed., London 1884/96.
82.
Saint Venant, B. de, Memoire sur la torsion des prismes, Mem. div. Sav. Acad. Sci. vol. 14 (1856) pp. 233-560.
83.
Sc..,hiffer, M., Sur la polarization et la masse virtuelle, C. R. Acad. Sci. Paris, vol. 244 (1957) pp. 3118-3121.
84.
Schiffer, M., and SzegH, G., Virtual mass and polarization, Trans. Amer. Math. Soc., vol. 67 (1949) pp. 130-205.
85.
Schumann, W., On isoperimetric inequalities in plasticity, Quart. Appl. Math., vol. 16 (1958) pp. 309-314.
86.
Schwarz, B., Bounds for the sums of reciprocals of eigenvalues, Bull. Res. Courc. Israel, vol. 8F (1959) pp. 91-102.
87.
Schwarz, B., Bounds for the principal frequency of the nonhomogeneous membrane and for the generalized Dirichlet integral, Pac. J. Math., vol. 7 (1957) pp. 1653-1676.
88.
Schwarz, B., On the extrema of the frequencies of nonhomogeneous strings with equimeasurable density, J. Math. Mech., vol. 10 (1961) pp. 401-422.
89.
Serrin, J., On the stability of viscous fluid motions, Arch. Rat. Mech. Anal. vol. 3 (1959) pp. 1-13.
90.
Steiner, J., Einfache Beweise der isoperimetrischen HauptsHtze, Werke II, Berlin, (1882) pp. 75-91.
165
- 57 L. E. Payne
91.
Stekloff, M. W., Sur les problemes fondamentaux de la physique mathematique, Ann. Sci. E'cole Norm. Sup., vol. 19 (1902) pp. 455-49Q.
92.
Synge, J. L., Pointwise bounds for the solutions of certain boundary value problems, Proc. Roy. Soc. (AJ, vol. 208 (1951) pp. 170-175.
93.
Synge, J. L., The hypercircle in mathematical physics, Cambridge U. Press (1957).
94.
" einige neue Extremalaufgaben der Potentialtheorie, SzegH, G., U'ber Math. Ziet., vol. 31 (1930) pp. 583-593.
95.
SzegH, G., O~ the capacity of a condenser, Bull. Amer. Math. Soc., vol. 51 (1945) pp. 325-350.
96.
SzegH, G., Inequalities for certain eigenvalues of a membrane of given area, J. Rat. Mech. Anal., vol. 3 (1954) pp. 343-356.
97.
Temple, G., and Bickley, W. G., Rayleigh's principle and its applications to engineering, Oxford Univ. Press (1933).
98.
" ein StabilitMtskriterium der Hydrodynamik, Arch. Velte, W., Uber Rat. Mech. Anal, vol. 9 (1962) pp. 9-20.
99.
Weinberger, H. F., An isoperimetric inequality for the N-dimensional free membrane problem, J. Rat. Mech. Anal., vol. 5 (1956) pp. 533 -636.
100.
Weinberger, H. F., Upper and lower bounds for eigenvalues by finite difference methods, Comm. Pure Appl. Math., vol. 9 (1956) pp. 613-623.
101.
Weinberger, H. F., Lower bounds for higher eigenvalues by difference methods, Pac. J. Math.
166
finit~
vol. 8 (1958) pp. 339-368.
- 58 -
L. E. Payne 102.
Weinberger, H. F .• The theory of lower bounds for eigenvalues, U. of Md. Tech. Note BN 183 (1959).
103.
Weinstein, A.• Etude des spectres des equations aux derivees partielles de la tMorie des plaques elastiques, Memorial des Scien. Math .• vol. 88, Paris (1937).
104.
Weinstein, A., Variational methods for the approXimation and exact computation .of eigenvalues, NBS A!'Jplied Math. Series 29 (1953) pp. 83-89.
105.
Weinstein, A"
Generali:l:ed axially symmetric potential theory,
Bull. Amer. Math. Soc., vol. 59 (1953) pp. 20-38. 106.
Weinstock, R., Inequalities for a classical eigenvalue problem,
J. Rat. Mech. Anal.. vol. 3 (1954) pp. 745-753. 107.
Weyl, H., Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hoblraumstrahlung), Math. Ann., vol. 71 (l912) pp. 441-479.
108.
" die Abhangigkeit der Eigenschwingungen einer Weyl, H'l Uber Membram von deren Begrenzung, J. reine Ang. Math., vol. 141 (1912) pp. 1-11.
167
CENTRO INTERNAZIONALE MATEMATICO ESTIVO ( C.!. M. E. )
LUCIANO DE VITO
1. CALCOLO DEGLI AUTOVALORI E DELLE AUTOSOLUZIONI
PER OPERATORI NON AUTOAGGIUNTI 2. SUL CALCOLO PER DIFETTO E PER ECCESSO DEGLI AUTOVALORI DELLE TRASFORMAZIONI HERMITIANE COMPATTE E DELLE RELATIVE MOLTEPLICITA'
ROMA - Istituto Matematico dell'Universita
169
Cal colo degli autovalori e delle autosoluzioni per operatori non autoaggiunti L. De Vito All'Istituto Nazionale per le Applicazioni del Calcolo, si sono spesso presentati problemi riconducibili alla determinazione di autovalori ed autosoluzioni di equazioni lineari non autoaggiunte in spazi di Hilbert, cioe di equazioni che possono scriversi nella forma (1)
=).
Eu
ove E
u
e una trasformazione lineare, definita in una varietfl. lineare U
di uno spazio !filbert S complesso completo e separabile, tale che E (U) CS e tale che, inoltre, sia (Eu, v)* (u, Ev), u, veU. Ogni volta in cui risultavano soddisfatte Ie seguehti condizibni: 1)
E(U)::
s,
2) insieme /\ ' degli autovalori di (1), privo di punti
d'accumulazione al finito, venjva applicato un metodo di calcolo degli autovalori e delle autosoluzioni, proposto dal Prof. Picone, che consiste nel considerare il funzionale
F(u,
(ove
0J
e
10
) .) = /I
Eu
- ) u /1 2
II u 1/2
u E U- w,
). numero complesso
zero di S), determinarne, per ogni fissato} ,il minimo,
}Ln(). ), nell'insieme 'Un di tutti i punti u di U - CV che sono della forma U ::
I
'Yl.
K=1
C k Uk
171
-2L. De Vito
iu k}
( c k numeri complessi,
sistema, arbitrariamente prefissato, di
punti linearmente indipendenti di ,
I
pun
U - W ,completo in U),calcolare tutti "\ (n) '\ (n) "\ (n) d 1 ' 1 I ' I' Al 'A 2 , ... , Amn e plano comp esso A ' nel qua I
t' I
la funzione fL'n( ))
presenta dei minimi relativi (e subito visto chel'i!!
sieme di tali ptinti non
e vu6to e -contiene un numero finito di elementi), as·2(n), ... , Am~)' COS! costruiti, come appro~
sumere i numeri
)
t), )
simazioni n - esime di altrettanti autov/llori di (1) ec;l assumere i punti u
(n~~)
di
Un' che rendono minima
F(u, Ak(n))
in
Un' come ap-
prossimazioni n-esime di autosoluzioni di (I) corrispndenti all'autovalore approssimato da
Ak(n)
per n -H 00 (1).
I numerosi esperimenti numerici eseguiti, se da un lato rivelavano sempre
~a
bonta del metodo stesso, nel senso che mostravano come ogni au-
tovalore di (1) venisse approssimato da qualcuno dei numeri
)k(n), dal-
l'altro ponevano in luce il verificarsi di una circostanza che, dal punto di vista pratico, pot eva presentare qualche inconveniente: precisamente accadeva che alcuni dei
:l k(n),
al cres cere di
n, convergevano verso numeri
complessi che non avevano nulla a che fare con gli autovalori di (I). Il verificarsi di questa circostanza pub essere controllato, ad esempio, in un caso limite: quello di una trasformazione di autovalori;
COS!,
tamente continue in
E per la quale la (1) sia priva
se si assume come U l'insieme delle funzioni assoluO~
x, if\: , nulle in x= 0
di quadrato sommabile in
e dotate di derivata prima
(0, 'iI), come S 10 spazio di Hilbert delle fun-
(I)Esposizioni del metodo di Picone sono state fatte da diversi Autori: M. Nasta ("Rend. Acc. Naz. Lincei" 6, XII, 1930), W. GrHbner ("Jahresber.d.Deut. Mathern. Vereinigung", 48, II, 1938), T. Viola ("Rend. di Mat. e delle sueappl." 5, II, 1941), L. Collatz (Eigenwertprobleme und ihre numerische Behandlung, Chelsea Publ. Co., New York, 194&, pp. 315-316), H. A. Kramers (Die Grundlagen der Quantentheorie - Quantentheorie des elektrons und der Strahlung; Hand - und Jahrsbuch Chern. Phys. D Bd. I, Theorien des Aufbaues der Materie I, II, Leipzig, 1938, pp. 2CO-201. 172
-3L. De Vito
zioni di un quadrato sommabile in (0, 7i. ), e si pone: E -
si vede che: non
du
u=~
uEU
mn = 1,
Al(n) =0
e autovalore per l'equazione
priva di autovalori. Per cosl
,
k = 1, 2, ...
uk:=' sen kx
per ogni n; e, d'altra parte, 10 zero
-t--
~ire,
=).,u
u E U, la quale, anZl, e
quindi, tra i numeri
) k(n)
si osser-
vavano dei valori "parassiti" e nasceva quindi il problema di stabilire un cri terio di selezione che permettesse di eliminarli. Altro problema che veniva posto dall'applicazione del sudetto metodo di Picone, era quello di chiarire in qual modo dovesse intendersi l'approssimazione degli autova10ri di (1) da parte dei numeriJ k(n) le corrispondenti autosoluzioni da parte dei punti u
(n\~).
e quella delIn effetti, il
criterio in base al quale, all'Istituto del Calcolo, si scegHeva,tra i numeri Ak(n) (k= 1, 2, ... , mn)'
n~esimoapprossimantediundatoauto-
I'
valore di (1), se si rivelava comodo dal punta di vista euristico, non era suscettibile di una giustificazione di carattere genera1e. Precisamente, si ordinavano i numeri dell'insieme ana10gamente si ordinavano i
A:
~' )2""
)1(n), )2 n)
in successione, e
" per ogni fissato n, adot-
tando il seguente criterio di ordinamento: se due numeri avevano modulo diverso, si faceva precedere quellQ di modulo minore, e se due numeri ave·vano 10 stesso modulo, si faceva precedere quello di argomento principale minore; si assumeva, quindi, di
Jk(n)
come approssimazione n-esima
Ik' A questo proposito, il Professor Fichera osservo che, se E
e una trasformazione che possiede due autovalori, ). e
uno opposto dell'altro:
- ') , in generale, per una almeno delle due trasformazioni E e
-E, il procedimento non
e valido.
173
-4-
L. De Vito Un ultimo interrogativo che sorgeva, in relazione al metodo di Picone, era quello di stabilire entro quali ipotesi per la trasformazione E era lecita llapplicazione del metodo stesso; in altre parole, si trattava di fornire una giustificazione teorica di questo metodo di calcolo, entro ipotesi di ragionevole generalita per la E. A tutti questi interrogativi, venne data esauriente risposta dal Professor Fichera, nel 1955, in una Memoria degli "lVmali di Matematica pur a e applicata" (vol. XL, serie IV) (2) . L'ipotesi fatta da Fichera sulla trasformazione E e la seguente: a)
E e invertibile e la sua inversa E -1
e compatta •
Cornie noto, questa ipotesi e, dlordinario, verificata quando E sia un
oper~tore
differenziale lineare dqtato di una funzione di Green
(risp~
to ad una assegnata condizione al contorno) che possa riguardarsi come nucleo di una trasformazione integrale compatta, ove si assumano convenientemente gli insiemi U ed S (quindi, ad esempio, quando E
sia un qual-
siasi operatore differenziale lineare ellittico con coefficienti abbastanza regolari), In tale ipotesi, il Professor Fichera ha dimostrato che, se
f uk}
e un sistema eompleto in U, tale che [E uk} sia completo in S (un sisterna siffatto puo, ad esempio, eostruirsi trasformando, mediante la E -1, un sistema complet 0 in S), fissati comunque due numeri positivi ) ' d' t (n) 1'"inSleme d' (n)' "\ ), d e ~ c.. ' in lea 0 con el numen' ) l /l 2( n ,.,., /I mn(n)
1\
relativi al sudetto sistema {uk} nel senso sopra specificato, si ha, definitivamente al erescere di n:
(2)Dedicata al Professor Picone, in occasione del suo 70-esimo compleanno.
174
- 5L. De Vito (2)
e il cerchio ap~rto del piano complesso e l'involucro aperto di raggio t. dell'insieme C~
ove
sieme dei punti che ~
cia r per meno di
r
)
:
IA1<)
e
J€ ( r )
di tale piano, doe l'in-
£ (si conviene che 1'insi~me
vuoto sia contenuto in ogni insieme del piano); in altre parole, nelle dette ipotesi e con Ie dette assunzioni, fissato comunque un autovalore ) di (1), esiste una successione di numeri [ ) (n~ con (Il\.)
1\ (n), tale che
• Inc,ltre, il Professor Fichera ha mostrato che, indicato
lim )(n) = ) "i't-,>oo
) (n) E
con u )('"-)
un qualunque punta di Un
che renda minimo,
il funzionale
F(u, I(n)), la successione di punti { u
in Un,
(n~(n)IIE
u (n\(n)W 1}
e compatta ed ogni suo elemento di compattezza e un'autosoluzione di (1) relativa all'autovalore
) = lim ),(n)
nel senso di Frechet). Come si
e sopra osservato,
ri "parassiti" tra gli elementi di
appunto a causa della presenza di nume-
J\ (n),
puc essere invertita, e, in generale, non crescere di
(qui la compattezza e intesa
11t->~ 00
in generale Ia relazione (2) non
e vero che,
definitivamente al
n, riesca:
(3)
-
C?
= chiusura di C) .
n Professor Fichera,
a questo proposito, ha perc dimostrato che Ie (2) (3) sussistono simuitaneamente se (E- 1)" s;: U (3) , purcM, in esse, l'insieme
~':
Lt)
(3) (E -1)*
I\. (n)
si sostituisca con il suo sottoinsieme "selezion) ~ j\ (n) costituito da tutti e soli quei C)
X
e Ia trasformazione aggiunta di 175
E -1 .
n
-6-
L. De Vito
per i guali riesce:
(4)
[fon(n))t <)u:~[ W-l 1)(: - ) I) u~( f'n( ) )2]
ove: I
e la trasformazione identica,
)
e il coniugato di
)
,
, u;
e un punto di Un,
Un' di norma unitariJl, che rende minima F(u,)) in E if: [(E- 1)" ] -1 (4) e l'aggiunta di E, ~e riesce definita in U
f
in virtu dell'ipotesi (E- 1),f S ~ U, ed ove il sistema Uk}' guest! volta, devegodt\re deU'ulteriore proprieta che uk E (E- 1) U , in modo cbe abbia senso il calcolo di
(E
* -). I) (~ - ). I)
nel punto u ~ che appartiene
a Un (la possibilita di assoggettare { Uk} a questa ulteriore condizione
e bene evidente),
Dunque, la (4) fornisce il criterio di selezione cui sopra
si accennava; applicando tale selezione ai numeri di ottiene un insieme,
(". C~ j\. (:
-..
L
rr)[ (L ~(n)
~n)
J\. (n) n C)I
, si
, per 11 Quale seguita a sussistere la relazione
nC)' ) def initivamente a1 crescere din,
ed
inoltre vale anche l'inclusione
La dimostrazione con la Quale Fichera ha provato tali risultati, fa anche vedere che, nell'ipotesi a).J il funzionale
F(u, ~ ), per ogni fissato
1, e dotato di minimo in U - W , ed il valore di tale minimo, )< ( )), e una funzione continua di A in tutto il piano complesso, che ha come zeri tutti e soli gli autovalori della (1) (la)l ()) gode, 'dunque, di alcune delle proprieta della trascendente di Fredh91m in relazione ad E- 1 , ma, a dif(4)L I ~nvertibilita di
(E -1)'~
e conseguenza di quella di 176
E -1 ,
- 7L. De Vito ferenza di questa, non e analitica, anzi, in generale, non e neppure differenziabile, come si pub, ad esempio, constat are nel caso particolare che E sia autoaggiunta; sotto questa condizione per E, il grafico della
".( 1)
!
per )
reale,
e rappresentato nella figura qui accanto,
ove
i ) k denotano gli autqvalori di (1)).
\
\,'
,/ 1\ II
--+1''-'"',",\""-,-
,I ,
, , I
Ii
\
\/\
\.' /\
~::.~. ._// 1..1
)',
I
\ / I,
/
j\~
/
,/
/
\ , , /1J/ ).
!
Inoltre, la successione
[fon())]
converge non crescendo verso
r ().),
uniformemente in ogni insieme limitato del piano. Procedimenti analoghi a quelli impiegati dal Professor Fichera nella citata Memoria del 1955, hanno consentito ad una sua allieva, la dottoressa Bassotti, nel 1961, di provare che il metodo di Picone pub anche applicarsi al calcolo degli autovqlori e delle autosoluzioni di equazioni della forma: ) T u
(5)
ove T
= u
e una trasformazione lineare compatta di
S in se, esattamente
come nel caso della equazione(1) salvo la sostituzione di U con S , del funzionale F(u,).) =II Eu - ) ul1 2 u 11- 2 definito in U - w con il
II
funzionale
177
-8L. De Vito
F (u,
t)
definito in S - W ,di u (ni(n) inoltre, questa volta, {Uk} che
=
II Eu (n~(n) It 1
con u (~ (n) II u
e un qualunque sistema di punti di
e assoggettato alla sola condizione di essere completo
questo caso eSistono, tra i
)(~),
(n~ (n) /1-1;
S-W
in S. Anche in
numeri "parassiti" agli effetti della
approssimazione di autovalori della (5); l'insieme selezionato L ora costituito pa tutti e soli quei valori
Xf))
(~) e
di /\ (n) per i quaU rie-
sce:
[fon( ).(n) )f~ .up [ 1!)Tu~ + ), T'u~ -1t-1 2T'TU}II\, -)'-n(}))2 ] I~I\) ,f1l(1)~l
ove Tt
e la trasformazione aggiunta di
T , e gli altri simboli hanno 10
stesso significato precisato nel caso precedente, (5) A differenza di quanta accadeva nel caso della (I), l'applicazione del criterio di selezione non richiede alcuna ipotesi aggiuntiva, ne sulla trasformazione ne sul sistema completo {uk} . L 'unica sostanziale differenza tra questa caso e quello precedente
e
costituita dal fatto che, in generale, 10 inf F(u,).) ~ I).€S-w non
e minima di
F(u,l)
n)
in S - W , in corrispondenza ad ogni valore
(5) "Atti della Acc. Naz. Lincei", vol. xxx, Maggio, Giugno 1961.
178
-9L. De Vito di ). (come invece succedeva nel caso precedente); in effetti, per un valore di ). tale che
fo ()) = 1 ,
pub accadere che F(u, f)
non sia dotata
di minimo in S - W, come ad esempio si verifica per Ie trasformazioni T , lineari, F (u, sia
autoaggiun~e,
compatte, definite in segno. Si ha perb che
w , per ogni valore di ). tale che )A ()) <1 ; in questo insieme la .Jl- ().) gode di tutte Ie proprieta f)
e dotat/! di minimo in
S-
che essa verificava nel caso precedente. Se T sitiva, il grafico di ,., ( A), per )
reale,
e autoaggiunta definita po-
e rappresentato nella figura
qui accanto.
/
I
----,-i,\.., . . -, / /
, '"
179
/
/'
/
,
- 10 SuI ealeolo per difetto e per eeeesso degli autovalori delle trasformazioni hermitiane eompatt.e e delle relative molteplicita L. De Vito Nelle applicazioni si presenta, sovente, il problema di dover calcolare gli autovalori di una matrice hermitiana sia "per difetto" che "per eccesso", cioe il problema di d~t~rminare, in corrispondenza ad ogni autovalore )
della matrice, una suecessione di numeri convergente per difetto,
ed una convergente per eccesso, verso il numero;"
.
Sia dunque A una matrice hermitiana e siano ;U l' '#2"'" ftr i suoi autovalQri, disposti in ordine di modulo decrescente e ciascuno ripetuto tante volte quantle la sua molteplicita. Supponiamo che riesca:
... =Ifo PI'I I,MPI +1/ =I~PI +2\ =... =/foPI +P2,•.. si ha, COmle ben noto:
Se indiehiamo con t n
il primo membro di tale relazione, e anche nota elle
la eonoscenza 'dei numeri t n eonsente la determinazione di una suceessio-
1
t
ne {b 1, n (di una suecessione a1, n} ) approssimante per difetto 2(per eecesso) il quadrato del piC! grande autovale, in modulo, di A : ) 1 .
181
- 11 L. De Vito
Basta, per questo, assumere: b
l,n
tn + 1 =-tn
a
l,n
= ~/t;
V
Analogamente si ha che, posta: P2
A22n +2 + P3 )
2n+2 3
+ ...
'\ 2n "'\ 2n P2 /I. 2 + P3 t\ 3 + .••
la successione { b2, n} (l~ successione [ 82, n (per eccesso) verso
f ) converge per difetto
)2' cioe verso il quadrato del secondo autovalore,
in modulo, di A. In modo analogo si costruiscono le successioni approssimanti per difetto e per eccesso i numeri }.3 2 ').4 2 , ... Tuttavia, mentre il risultato relativo alle due successioni [al, n} e utilizzabile praticamente, per il calcolo effettivo di potendosi calcolare numericamente i
t
risultato relativo alle due successioni
tb2, n }, fa2, n}
n
[b t , n}
,
)1 2 ,
in numero comunque grande, il presenta
8010
un
interesse teorico, poiche in pratica, non si dispone ne della conoscenza di PI
ne di quella -esatta- di
il numero intero
A~.
Peraltro, se si ammette di conoscere
PI' ed i termini delle successioni taL n } ,
f bl, n}
si possono subito stabilire delle limitazioni inferiori e superiori rer si ha infatti:
182
•
).2 2 ;
- 12 L. De Vito
p2.n
= max [
lX 2,n ove { mn }
n+l tn+l - PI al, mn+l
=
0.
n
tn - PI bi, mn
n lIn ( tn - PI bi, mn) ~
) 22
e una arbitraria ~uccessione crescente di numeri naturali.
A questo punto, vliene spontaneo di chiedersi se non si possa appro-
[~2'
1 [ex n]
fittare della arbitrarietll di {mn } per fare in modo che n e 2, convergano verso ).. 22 , e possano quindi essere assunte come successioni approssimanti in quadrato di
). 2 per difetto e per eccesso rispettiva-
mente. A questo interrogativo pub d/lrsi risposta positiva, come e mostrato in una Nota dello scrivente (1) , facendo mn = n2 0, piu in generale, assumendo
mn
in guisa tale che
vemente la definizione di
f
lim
~
= (fJ
,
2,n e, precisamente, porre n/n+1
~ 2,n
pur di modificare lie-
m-)"O
tn-l - PI
= max [ 0,
n+l al, mn+l
n
tn - PI bI, mn
].
I caratteri di tali convergenze sono messi in evidenza dalle seguenti relazioni ;
lIn 2n
n
lim (a tn-)co ~ 'k, mn
-
Pk
)k)
lIn = 0, lim (CXk 'l\,-t 00
'
mn
- Pk
lIn 2 lIn )k) = 0,
(1) "SuI calcolo approssimato degli autovalori delle trasformazioni compatte e delh~ relative molteplicita" Rend. Acc. Naz. Lincei, xxx, 1961.
183
- 13 L. De Vito
k = 1, 2 'C{'l ,s = a l ,s
A -b '''l,s - l,s
Considerazioni perfettamente analoghe a queste possono ripetersi in relazione a
).
~,
). ~
etc ... , pur di conoscere, di volta in volta, rispet-
tivamente,i numeri PI' P2 ' i numeri PI' P2' P3 etc .... Per quantoriguarda la vautazione dei numeri Pk' non si conoscono ancora procedimenti di carattere generale; si pub solo dire che la successione {rl, n
f ' con
r
n
=
1, n
b l,mn
approssima per eccesso Pl' la successione
rr2, n} , con n
tn - Pl bl, mn r
2,n
=
approssima per eccesso P2. etc .... , ove { mn } sia scelto nello stesso modo sopra indicato (cifr. 10c. cit. in (1) pag. 456). Si ha, infine, che la conoscenza di un numero separatore ro )
tale che
)2 (A <\
J
d~lla coppia
). l' ) 2 ' cioe un nume-
e perfettamente equivalente alla conoscenza
di Pl' E', intanto, ben evidente, in base a quanto sopra detto, che,dalla
t
n tale che
conoscenza di Pl, si deduce quella di ). ; viceversa, assunto 2 , t tale che 0 <2: 1, t bl, ii (bl, ii _ ). 2 )
»
ed
<
tale che
184
<
- 14 L. De Vito
b l, mno
>
0
no al mn ' 0 no bl, mno
,
(si pub vedere che un indice
PI (cfr~
min[f
~
al ,mno - bl, mno
- 1
siff~tto
,
€ 2
E
<
3rl ,mno b1,fi - ).. 2 r 1. no + 1
]
certamente esiste) si ha:
= parte intera di
r1 ,no
loc. cit. in (1) pag. 457). Le considerazioni teste syolte in relazione al calcolo degli autovalo-
ri di una matrice hermitiana, possono ripetersi, senza alcuna modificazione formale, con riguardo al cal colo degli autovalori di una trasformazione Iineare T
di uno spazio di Hilbert S (complesso completo e separabile)
in se, la quale sia hermitiana, compatta e tale inoltre che Ie $erie
(ove Pk e ).. k hanno 10 stesso significato precisato sopra) risultino convergepti, almeno da un certo
n in poi, e Ie loro rispettive somme siano
numericamente determinabili. E' questo, sostanzialmente, il caso delle trasformazioni hermitiane integrali
= K(y, x) ] della spazio di Hilbert
T(u) = )'D K (x, y) u (y)dy, [K(X, y) =
£,2(D)
in se, aventi un nueleo
K(x, y)
tale che uno almeno dei suoi iterati sia di quadrato sommabile in D x D . Le idee ora esposte, pqssono servire, tra l'altro, per integrare il classico metodo di
.!ill.!
per il calcolo degli autovalori. Si consideri infat-
185
- 15 L. De Vito ti, ad esempio, la trasformazione integrale T dianzi introdotta, con il nucleo K verificante Ie condizioni giA menzionate ed inoltre definito positivo; ove non si conosca la molteplicita di
)1'
il metodo di Ritz non con-
sente di limitare inferiormente il numero ). 2. Se perb si sa, appUcando i procedimenti sopra descritti, che il numero intero m - 1 limita superiormente la molteplicita di ). 1 ' si pub senz 1altro assumere, come limitazioneinferiore per ) 2 ' l' m-esima radice de1l'equazione det (( (ove wI' w2""
)
1 f. Wi \. dy)) D
T(wi) w( dy -
D
=0
son funzioni linearmente indipendenti).
Un'altra possibilita di integrare il metodo di Ritz (per trasformazioni integrali aventi il nucleo verificante Ie dette proprieta) .
(m)
guente osservazlOne, Se Ie lettere P1
(m)
,P2
' ..•
e fornita daUa se ..
"'I (m) "'I (m) I
Al
tn(m) hanno significato analogo a queUo delle Pk' }.k' tn
I
/I.
2
' .,•
I
in relazione
aUa m-esima trasformazione approssimante di Ritz, si ha:
o~ ~
come
)2n k
_/1 (m)j2n /
e immediato constatare.
{) k(m) } m
r
t _ t (m), lim t - t(.m)] = ~ n n 1Yr1->e:'(?L n n
i- k
0
Ne viene che, accanto alIa successiQne
fomita dal metodo di Ritz, che converge a ') k per diret-
to, si pub considerare anche la successione
Y k(m)
= (
I. \Ak. I (m)
2n
I
I
I
che converge a ). k per eccesso.
186
[)) k(m? m con
t (m) n
+ t ) 1/2n n
CENTRO INTERNAZIONALE MATE MATICO ESTIVO ( C. 1. M. E. )
J. B. DIAZ
UPPER AND LOWER BOUNDS FOR THE TORSIONAL RIGIDITY AND THE CAPACITY, DERIVED FROM THE INEQUALITY OF SCHWARZ
ROMA, Istituto Matematico dell'Universita 187
UPPER AND LOWER BOUNDS FOR THE TORSIONAL RIGIDITY AND THE CAPACITY, DERIVED FROM THE INEQUALITY OF SCHWARZ
by J.B. DIAZ (Institute for Fluid Dynamics an'd Applied Mathematics, University of Maryland)
1. Introduction. In many problems of mathematical physiGS it is desired to find the numerical value of a quadratic integral of an unknown function, where the unknown function is a solution of a linear boundary value problem consisting of a linear partial differential equation plus linear boundary condition. The quadratic integral in question is usually the quadratic form occurring in
a
Green's identity for the differential operator involved in the boundary value problem. The present exposition is concerned with two particular instances of this general situation. In section 2, which is based upon references [6] and
[7J
in the bibliography, upper and lower bounds for the torsional rigidi-
ty of a cylindrical beam are derived from Schwarz's inequality. Section 3 is devoted to the estimation of the capacity, and is based upon references
[1]
and
[13]
in the bibliography.
The brief bibliography contains references in which a fuller discussion of the topics mentioned is to be found, and is by no means meant to be exhaustive.
2. Upper and lower bounds for the Dirichlet integral. Let p, q, P, Q be sufficiently smooth real valued functions defined on D + C, where D is a bounded plane domain with a (smooth) boundary C. For r:J.. any real number, one has that 189
-2J. B. Diaz
JJ
Ip+ '" p)2 +(q+
c(
Q)2] dxdy
90
,
from which it follows that
( JD(PP t qQ)dxdy)2
~ !o(p2tq2)dXdY Jo(p 2+Q2)dXdY.
(S)
This last inequality, which will be referred to as Schwarz's inequality,
is
the starting point for all the upper and lower bounds for the Dirichlet integral to be given here, Consider the determination of upper a1'\.d lower bounds for the Dirichlet integral
j
(",2 t v 2)dxdy
D x
Y
of a solution v(x,y) of Neumann's prQblem
fj,v=v xx tvyy = 0
,
on D ,
~v r::;- = v n + v n = f , on C. un x X y Y (Here, nand n denote the components of the outer unit normal to the bounx y dary C). The desired bounds follow at once upon choosing suitably the functions p, q, P, Q which appear in the Schwarz's inequality (S). To derive an upper bound, let p, q be such that px t ~ = 0 ,
pn t qn x y
on D ,
'0
v u n
=~ =f,
on C,
and let P, Q be given by P=v
x
Q=v
y
Then, by integration by parts, and Green's identity, one readily obtains
190
- 3-
J. B. Diaz
\
JO
~ (pv + qv )dxdy D x Y = - r v(p + q )dxdy + r v(pn + qn ids olD x y Je x y
(pP + qQ)dxdy =
rI. v ~ rav
"2
'U
n
ds = ~ (v 2 + v 2 )dxdy D x y
which, together with (S)' yields
2 2 (v + v )dxdy x Y D (This inequality is precisely Kelvin's minimum kinetic energy theorem, see
~
Lamb
Gl, pages 47 and 57J ' and also Diaz and Weinstein [7, page 109J ).
Since the condition p + q = 0 on D can always be replaced by p = u and x y y q = -u , where u is a suitable function (not necessarily single-valued), the x last inequality may be restated as follows: If u is any function (not necessarily single-valued, but such that u and u are single-valued) such that x y t() u
=u
f0 s
x
~
+u
ds
~ = -u
Y ds
n + u n = '0 v x Y y x fc) n '
on C ,
then
~ iJ(u! +
\ (v 2 + v 2)dxdy
,j D x
y
o
U
2 y )dXdY
A lower bound for the Dirichlet integral of v can be derived si-
milarly. This time, let p=w
x
,
q=w
,
Y
with w(x, y) a non-constant real valued function defined on D + C, and also P=v
Q=v
x
y
,
as before. Then, by Green's identity,
i
Jo
i (v w + v w )dxdy Jv x x y y .!() = - j w !1 v dxdy + j. w r0
(pp + qQ)dxdy =
iw
D~
•
l;
/'()n
ds ;
191
l;
v
n
ds
-4J. B. Diaz
which, together with Schwarz's inequality (S), yields
Z
Ut~d')
~
t (w 2 + wY2)dxdy ~D x
~ o (vx2 + vy2)dxdy
.
Combining the two inequalities already obtained, one may summarize the results thus:
(Itj ~ d') 2 , ~ ( 2 2
2
v
...
(w + w )dxdy D x y
D x
+ 2)0 v
Y
d
xy
~
~o(u~x + uy2)dxdy
1~
,
that is to say, any non-constant function w furnishes a lower bounds, and any function u satisfying the boundary condition
=
~:
on C furnishes an
upper bound, for the Dirichlet integral of a solution v of Neumann's problem. (While it will not be derived here in detail, the derivation being similar to that just carried out in the case of the Neumann boundary value problem, it will be remarked that a corresponding result is valid for a solution v of the Dirichlet boundary value problem 6.v=v xx +vyy =0 , v
'* d,t r
=f
on D. on C.
In this instance, one has
(1c
v
2
2
. (w + w )dxdy y D x
~
Jr (v x2 + v 2)dxdy 0
Y
~
~
2 2 (u x + u )dxdy D Y
;
that is to say, any non-constant function w furnishes a lower bOUlld, and any function u satisfying the boundary condition u = v on C furnishes an upper bound, for the Dirichlet integral of a solution v of Dirichlet's problem. The right hand inequality is nothing else but Dirichlet's principle, while the left hand i-
192
- 5J. B. Diaz
nequality contains as a special case a lower bound for the Dirichlet integral of a solution of Dirichlet's problem, given by Trefftz
[16J ' in terms of
an
arbitrary non-constant harmonic function. ) Only a single example of the many possible applications of the upper and lower bounds given for the Dirichlet integral of a solution of Neumann's problem will be indicated. Consider the torsion of an elastic cylindrical beam of cross section D ; and assume that Lame's constant of elasticity,
~
, is taken to be unity. The stiffness, or torsional rigidity, S, of
such a cylindrical beam, is given by the formula (Diaz and Weinstein p.
108] ) S= P -
Jo(vx2 + vy2)dxdy
[7 ,
,
where P is the polar moment of inertia. of the domain with respect to its centroid, and v is the warping function, which is a solution of the Neumann boundary value problem 6v=v
xx
+v
'()v '() ~=fds
'un
.
wIth
f()
"0
s
yy
=0,
[12 2+y)2J -(x
on D ,
on
C,
denoting differentiation along the boundary C . Parenthetically, notice that this formula for the torsional rigidi-
ty implies that for any (simple or multiply) connected section one has that S
~
P
,
with equality if and only if the Dirichlet integral of the warping function is zero, which means that the warping function must be a constant in view of the boundary condition satisfied by the function v , this can only happen if the domain D is either a circle or a circular ring. Curiously, therefore, for domains whose connectivity is more than three, the torsional rigidity is al-
193
-6J.B.Diaz ways less than the polar moment of inertia with respect to the centroid. An alternative way of looking at this is the following "isoinertial" principle: of all domains with prescribed polar moment of inertia with respect to the centroid, the
cir~le
and the circular ring possess the maximum torsional rigi-
dity,
3. Upper and lower bounds for the capacity, The capacity C of a smooth surface ("conductor")
S in three
dimensional space may be taken to be defined by"the equation C =
4
Jo
~
/grad vl 2 dxdydz ,
where D is the region exterior to the surface S, and v is the solution of the exterior Dirichlet problem
~v=v v
xx
+v
yy
+v
zz
0
=1
D
on S ,
lim v(x, y; z) (x, y, z) ->
on
=1
00
From Dirichlet's principle, an upper bound for the capacity C is given by C
~
4
~ JoIgrad w\2 dx dy dz ,
where w(x, y, z) is a sufficiently smooth function (i, e"
continuous, with pie-
cewise continuous first partial derivatives) such that w(x, y, z) = 1 if (x, y, z) is a point of S and also that w
= O(r- 1 ) as r = (i + y2 + z2) 1/2 approa-
ches infinity, The purpose of this section is to show how, using a simple "trial function" w (which seems to be naturally dictated by the symmetry of the domains in question) it is possible to obtain fairly close, readily computable
194
7~
~
J. B. Diaz
upper bounds for the capacity of the regular solids. For definiteness in describing the procedure, let S be the cube, of side' 2 and of vertice (~ 1 , ~
1,
~
1), which is circumscribed about the unit sphere with center at the o-
rigin. The exterior D of the cube S is divided into six congruent infinite pyramidal domains (one corresponding to each face of the cube). The trial function w to be chosen will first be defined on one such (frustrated) pyramidal domain, and then thecl.efinition of the function will be extended, "by symmetry" over the rest of the exterior, D . Consider the pyramidal frustrum D' (corresponding to the face of the cube which contains the point (I, 0, 0 )) consisting of the set of points (x, y, z) satisfying the three inequalities x ~ 1 ,
Iyl ~
II
x, z ~ x, and sup-
pose that the trial function has been defined on D', and then the definition of w has been extended symmetrically to the entire exterior of the cube, as explained above. Then
j~grad wl 2dx dy dz • 5 IJ,grad w\2 dx dy dz ; and if, in particular, one sets w(x,y, z) = f(x) on D', it follows that
I
oCI
jlgrad wl 2 dx dy dz
=:
6
~
(4x 2)
[f'(X~ 2 dx
.
i
"Minimizing" this last integral (with respect to all admissible functions f) leads to the Euler- Lagrange equation
~ f t i [f'(X)j2j- d~ ~ f' that is
2 2x f'(x) + x f"(x) = 0 195
t
x2
[f'(X!]
2) • 0 ,
-8J, B. Diaz
which, since f(1) = 1 and f(x) = O(x f(x)
-1
) for large x, means that
= 1
x
gives the "best" possible choice of the function f . This very particular choice of the function f already leads to a fairly good upper bound for the capacity C of a cUQe, since then
~ r \grad w\2 dx dy dz 4 II J D
= -64 11
JOO(4x) 2 1.
l
d-. dx
-1J - 2 dx = -6) x Tr
and it is known (see below) that the capacity of a cube is of the order of 1. 2. (Notice that the simple trial function just described may be used "a priori" entirely independently of the possible justification of the heuristic variational consideration which led to its discovery. ) An improved upper bound for the capacity C may be obtained by letting w(x,y, z) = on D', where
-+
A and
+ (AI y \ +
t
r (-T z)
x
1 2
x
are real numbers, and minimizing the resulting
A
~ . Notice that the trial function w defined in D by symmetry satisfies the boundary condition w = 1 on
Dirichlet integral with respect to
and
the cube, since the additional terms are just "coordinate functions" (in the terminology of Walther Ritz). The final result of the computation, which will be omitted here, is that (for a cube of side 2) one has C
~
1. .6103 .
Using the volume-radius, i. e. the radius of the sphere of the same volume as the the interior volume of the cube (see
polya-Szeg~
[13 , p. 23J ), gi-
ves the lower bound 1. 240 ~ C . The whole "symmetry" process described above for the cube is carried out in full, for any regular solid, in J. Conlan, J. B. Diaz, and W. E. Parr
L1] . For the capacity
C of the icosahedron cir-
cumscribed about the unit sphere, the upper bound obtained there is that 196
- 9-
c
~
J. B. Diaz
1. 096, while the lower bound obtainable from the volume-radius is
1. 064 $ C . The other regular solids can be treated similarly. At the 1954 conference at the University of Trieste, it was reported that (see Diaz [4
J )for the capacity
1. 30S
< C < 1. 336
C of a cube of edge 2 one has
,
where the lower bound was given by Daboni
ls J and the upper bound was gi-
ven by Payne and Wamberger [16J • Quite recently, W. E. Parr [1SJ ' obtained the upper bound
1. 335, as an application of his extension of Polya-Szeg~'s prescribed level surfaces.
197
[13J
method of
- 10 J. B.Diaz
BIBLIOGRAPHY
1. James Conlan, J. B. Diaz and W. E. Parr, On the capacity of the icosahedron, Journal of Mathematics and Physics, vol. 2, 1961. 259-261.
2. R. Courant and D. H. Hilbert, Methods of mathematical physics, First English edition, New York, 1953.
3. J. B.Diaz, Upper and lower bounds for quadratic functionals, Collectanea Mathematica, Serr..inario Matematico de Barcelona, vol. 4, 1951, 3-50.
4. J. B. Diaz, Some recent results in linear partial differential equations, Atti del convegno internazionale sulle equazioni aUe derivate parziali, Trieste, 1954, Edizioni Cremonese, Roma, 1955, 1-29.
5. J. B. Diaz, Upper and lower bounds for quadratic integrals, and at a point, for solutions of linear boundary value problems, in Boundary Problems in D'ifferenfial Equations, edited by Rudolph E. Langer, The University of Wisconsin Press, Madison, 1960, 47-83.
6. J. B. Diaz and A. Weinstein, Schwarz's inequality and the methods of Rayleigh-Ritz and Trefftz, Journal of Mathematics and Physics, vol. 26, 1947, 133-136.
7. J. B. Diaz and A. Weinstein, The torsional rigidity and variational methods, American Journal of Mathematics, vol. 70, 1948, 107-116.
199
- 11 -
J. B. Diaz
8. L. Daboni, Applicazione al caso del cubo di un metodo per il calcolo per eccesso e per qifetto della capacita elettrostatica di un conduttore, Rend. Acc.Naz. Lincei, sez. VIII. vol. XIV, 1953. 461-466. 9. G. Fichera, Risultati concernenti la risoluzione elelle equazioni funzionali dovuti all'Istituto Nazionale per Ie applicazioni del Calcolo, Memorie dell'Accademia Nazionale dei Lincei, serie VIII, Vol. 3, 1950. pp.59-68. 10. G. Fichera. Methods of linear functional analysis in mathematical physics. Pro<;:eedings of the International Congress of Mathematicians. Amsterdam. 1954, vol. III. 216-228: 11. H. Lamb. Hydrodynamics. Sixth edition. New York. 1945. 12. M. Picone - G. Fichera. Neue funktionalanalytische Grundlage fUr die Existenzprobleme und Lgsungamethoden von Systemen linear partieller Differentialgleichungen. Monatshefte fUr Mathematick. vol. 54, 1950. 188-209. 13. G, P~lya and G. Szegl:l. Isoperimetric inequalities in mathematical physics; Princeton University Press. 1951 (see also the book review in Bulletin of the American Mathematical Society. vol. 59. 1953. pp.588-602), 14. W. Prager and J, L. Smge. Approximations in elasticity based on the concept of function space. Quarterly of Applied Mathematics. vol. 5, 1947, 241-269. 15. J. L. Synge, The hypercircle in mathematical physics, Cambridge University Press, 1957 (see also the book review in Bulletin of the American Mathematical Society, vol. 65, 1959). 200
- 12 -
J. B.Diaz
16. L. E. Payne and H. F. Weinberger, New bounds in harmonic and biharmonic problems, Journal of Mathematics and Physics, vol. 33, 1955, 291-307.
17. E. Trefftz, Ein GegenstUck zum Ritzchen Verfabren, Proc. Second International Congress Applied Mechanics, Zlirich, 1927, 131-137.
18. A. Weinstein,New methods for the estimation of the torsional rigidity, Proceedings of the Third Symposium in Applied Mathematics, American Mathematical Society, 1950, 141-161.
19. W. E. Parr, Upper and lower bounds for the capacitance of the regular solids, Journal of the Society for Industrial and Applied Mathematics, vol. 9, 1961, 334-386.
201
CENTRO INTERNAZIONALE MATEMATICO ESTIVO ( C. I. M. E. )
MENAHEM SCHIFFER
FREDHOLM EIGENVALUES AND CONFORMAL MAPPING
Roma
~
Istituto Matematico dell'Universita
203
FREDHOLM EIGENVALUES AND CONFORMAL MAPPING by Menahem Schiffer
Introduction This paper is mainly of expository nature. Its aim is to give a connected account of various results regarding the Fredholm eigenfunctions and the Fredholm eigenvalues of plane domains. The Fredholm eigenvalues of a curve system are a set of functionals of these curves whose study leads to may useful applications. T;1ey are closely related to the boundary value problem for harmonic functions. They are important in the theory of conformal mapping, in the theory of kernel functions and orthonormal series. They playa role in the theory of the Hilbert transform and in the theory of univalent functions. Their dependence on the curve system is displayed by a very elegant and convenient variational formula. Some applications of these results are given to show the significance of the identities and formulas obtained. For a more detailed development of many points the reader is referred to [10], [11} and
[12J.
1. Boundary Value Problem of Potential Theory and FrEidholm Eigenvalues Let C be a closed curve in the complex z-plane with interior domain D
'" We suppose C to be three times continuously diffeand exterior domain D. rentiable. It possesses at every point
s
€
C a normal
n~,
and it is well
known that the kernel
(1)
o
1 k(z, ~) = on1; log~
is a continuous function of both argument points z and 205
~
on C . We shall
- 2M. Schiffer
always assume the normal n'( to be directed into D. The kernel k(z, ~ ) was introduced by Poincare in order to solve the first boundary value problem of potential theory. Suppose that we are given a continuous function f(z) on the curve C and wish to find a function u(z) harmonic in D which takes on C the boundary values f(z). We try to solve this problem by superposition of elementary harmonic functions and make the "Ansatz" (
1
f
u(z) =-
(2)
Tr
with a weight function
'C ~ (~)
k(z,'S)r(l;)ds" ., still to pe determined for our problem. This is
clearly a harmonic function for zeD. As z -
z
o
!
C, we have the well-
known jump condition
Since we have prescribed the boundary values f(z) of u(z), we obtain for the determination of the weight function
f(
~)
the Fredholm integral e-
quation of the second kind
(4)
f(z) =
f (z) + it1
f
C
k(z,
~ ) f('~ )ds ~ .
This particular inhomogeneous integral equation with continuous kernel established by Poincare led Fredholm to his general theory of linear integral equations. He established his celebrated alternative: Either the integral equation (4) possesses a nontrivial solution f == 0 , or it possesses a unique solution tion f(z).
206
f
(z) for
f (z) to each given integrable func-
- 3-
M. Schiffer
Thus, the entire boundary value theory of harmonic functions in the plane is reduced to the study of the homogeneous integral equation
(J!
( 5)
IV
f
(z) =-))1- . k(z, () 1r'"C
<.P
/)1
If it could be shown that -1 is not an eigenvalue
(1; )ds
A.JI
~
of (5), the unique exi-
stence of a harmonic u(z) with prescribed boundary values f(z) on C would have been established. This was indeed done by Fredholm and the existence problem of the function u(z) was solved. There remains the interesting question to study the eigenvalues
1)/
connected with this important integral equation and to determine their potential theoretical significance. They are called the Fredholm eigenvalues of the curve C ; they may also be considered as functionals of the domain D IV
or of the domain D. It is well known that the lowest eigenvalue (in absolute value)
has the value +1 and belongs to the eigenfunction ther eigenvalues are real and satisfy
IA)l1 : :. 1.
f
0(
J..
o
'S ) = const. All o-
Since we easily can handle
to the kernel k(z, ~ ), we aco tually can reduce the numerical solution of the first and second boundary vathe contribution of the trivial eigenvalue
A
lue problem of potential theory to the process of iteration, i. e. , the Liouville-Neumann series development. The details of this procedure were developed by Gershgorin [6] . It probably is the most convenient numerical method in two-dimensional potential theory. However, the speed of convergence of the Liouville-Neumann serIes depends on the absolute value of the lowest nontrivial Fredholm eigenvalue
[1, 9, 17
J.
Thus, we see the significance to estimate the nontrivial Fredholm
eigenvalues of a given curve C. We propose to study the values 207
~)I(C) as
- 4M. Schiffer
functionals or as "fonction de ligne" in the sense of Volterra and to solve extremum problems concerning them.
2. The Fredholm Eigenfunctions
f)l
We consider the Fredholm eigenfunctions eigenvalues
1)).
(~) connected with the
e.
These functions are defined only on the curve N
We now N
introduce harmonic functions h)l (z) and h)) (z) defined in D and D, respectively, which are closely related to the
f lJ
(~).
We define
1
(6)
h)l (z) =
--;!- J1e k(z, r;)
f (~)ds ~ )I
The right-hand side of (6) represents a harmonic function of z if z ranges N
over D; it also represents a harmonic function if z varies in D. But this harmonic function is not the continuation of h(z) defined in D. For the sake N
of clarity we therefore shall denote it by the letters h(z). The jump conditions for the kernel k(z, ~ ) imply for each point
z
o
(7)
E:
e h (z) = (1+ A.,,)
lim
z- z ) l
o
,
These boundary conditions on
I))
e
0
lim
Z-+Z
h (z) = (1o
)I
1,,)
0
determine the harmonic functions hJ) (z)
and hlJ(z) uniquely. It is also well known from the theory of the double-layer potential that everywhere on
e
ah N
(8)
(z ) ).I 0 =----
'On
208
- 5M. Schiffer
We have thus solved the following interesting problem of potential theory: N
To determine a harmonic function h(z) in D and a harmonic function h(z)
'" with the same normal derivatives at the common boundary and having in D proportional boundary values there. We can also give another interpretation to this result. Let
D[hJ=
(9)
If
(V
f
h)2dt
C
D
h
l!!.. ds 'On
£ h dh ds j
C
on
N
denote the Dirichlet integrals of hand h. It follows from (7) and (8) that
(10)
Since Dirichlet integrals are obviously nonnegative; we read off from (10) that,
IA)l1
2
1 and that
IA. v I = '1
is only possible for
Finally, it is easy to see that ~)l = const implies
~)! ('t: ) = const.
J.,)) = +1 . Thus, the i-
dentities (7) and (8) decide the essential step in the above mentioned Fredholm alternative. Consider next an arbitrary continuous function f(z) defined on C ; determine the harmonic functions h(z) and h(z) in D and
D which have the
common boundary values f(z) on C . Let D [h] and D [h] be their Dirichlet integrals and consider the ratio D [h] It is easily seen that for all f holds
l
(11 )
-1
_1_< ),+1
1
209
jD
[hJ
as a functional of f(z).
- 6M. Schiffer
where
A1 is the lowest positive nontrivial Fredholm eigenvalue. The va-
lues ).)1 +1/
1li
-1 are stationary values of this functional. Because of the N
complete symmetry of D and D, the same values must be the stationary values of the reciprocal ratio or, in other words, the values
.l)J -1/ 1)1 +1
must likewise be stationary values for the original ratio. This is, indeed, the case since with each Fredholm eigenvalue
A)I
r -1
also the Fredholm ei-
genvalue - 1)1 will occur. This is best seen from equations (7) and (8). Let g)l(z) and
glitZ)
be the conjugate harmonic functions of h)l(z) and
N
hy (z). Then (7) and (8) yield by use of the Cauchy-Riemann equations.
= 1- A. v
(7')
an
and N
g" (z ) = g (z )
(8')
.v
0
if the additive constants in g))
II
0
g)J
and
are properly adjusted. Hence, if
we define
*
h)) (z)" 1+11)) gll (z) ,
( 12)
1 N h;y (z) = ~ gJ) (z)
N*
)i
we find (13)
:If
1-1)1
ah*
""II
}I
h)l (z) = 1+ 1)) h)l(z)
'On
oh A!-v
=--
on
which coincides with (7) and (8) if we replace 1)1 by -.lv . Since Fredholm eigenvalues occur always in pairs, we shall from now on understand by l)i the positive eigenvalue of the pair. Likewise, the hy (z) shall be the eigenfunctions belonging to the positive eigenvalues. Their conjugates will then automatically belong to 210
-l)) .
- 7M. Schiffer
We introduce now the analytic functions
v-;; (z) =
(14)
where
oOz =
t
d ra; h.v (z) ,
N
v)J (z) =
a; h.v (z) ;,
N
(}x - i '}y) is the well-known complex differentiator. A
simple calculation leads to the following result :
v:n-)
(15)
---2 d t , (s -z)
Clearly, the trivial eigenvalue
J..
~~
(z) = -
1 fjr ~)
~ jJ
D
2 d 't'
(~ -z)
has to be discarded since its eigenfunc-
o
tion h (z) is constant and is annihilated by differentiation. We understand
o
the improper integrals (15) in the sense of a Cauchy principal value: we exclude first ill the integration a circle of radius E around z and then take the limit value of the integral as E ---+ O. With this understanding (15) represents integral equations for the eigenf)lnctions v)l (z), and we may restrict
Ali> O.
ourselves to the case of positive eigenvalues It is also easily verified that
(16)
~lI (z) =
1)) Tt
(1+ 1).,)
v~ ( " )
ff D
v)) (z) =
l}l 1l (1.-
).yl
(r .1
(~
-z)
2
d't
for z
d't
for z
N
D
€
v)) ( ~)
I!
JJ
IV
D
( '( -z)
2
€
D
These integrals are proper and display clearly the analytic character of the eigenfunctions. Another great advantage of introducing the harmonic eigenfunctions h)J(z) and the analytic eigenfunctions v» (z) comes from the important or211
- 8M. Schiffer thogonalJty relations which these functions satisfy. Observe that the kernel k(z,
~)
is not symmetric in its arguments and hence that the eigenfunctions
~)I(
s)
defined on C are, in general, not orthogonal. On the other hand,
let h)z) and hf'l. (z) belong to different eigenvalues:
)lJ}
r lp.'
We com-
pute the Dirichlet integral
Dlh~,h~l ~
(17 )
If
'V h )).
Vhf
D
Using the boundary relations (7) and (8), we can write
(18)
But both Dirichlet integrals are symmetric in )) and ~- while we know by lr+1 I . )'))+1 assumption that -1 . Hence, we proved I
-r-=T. r ,,) . j)
+f'"/,-
(19)
We may then assume without loss of generality that
(19 1)
for
))
r
r
From definition (14) we conclude next the orthogonality relations for the analytic eigenfunctions
for
(20)
II
r
r
We should like to normalize the eigenfunctions hv(z) in the Dirichlet norm.
Cle~rly,
'V
in view of (18) we shall have to multiply hv(z) and h,)z)
212
- 9M. Schiffer with different normalizing factors. Correspondingly, the v)J (z) and VII (z) go over into analytic functions wl> (z) and w).I (z) with the following properties: (21)
w)I
1)./
(~)
=1't
w')J (~). 2. d'Z' , (~ -z)
II D
N
w (z) J)
).)1
=-. rC
1)1
w (z) =- - ~
'rt-VlJ..1 )J
w)/ ( ~ )
If
(~
N
D
-z)
2 d l'
,
wJ) (z)
=
J..)J
1t h;'-1
ff
W)l ( "
)
D (" -z)
IfD
2 d't'
w)) (~ ) (r -z)
2 dt'
and the orthonormaliz-:tion
ff
(22)
w)J wfl. dr
Jf w» wi
=8
)I~
D
d'l:'
=b)lf
"" D
3. Fredholm Eigenvalues and Hilbert Transforms We can now connect our results with some important general theorems of analysis, Let f(z) be an arbitrary complex-valued function defined in the entire complex z-plane and of class
'£
2,
Define its so-called Hilbert transform
(23)
F(z) =
~
rr
fm2
JJ (~-z)
d~
This will be a new function with the same properties as f(z) and with the same norm
(24)
If IF
IIml 2 =
\2 d-r
If
= I f\2 213
dr
=IIfl12
- 10 M. Schiffer
The Hilbert transformation is an involution, that is, the Hilbert transform of F(z) is again f(z). Finally, wherever f(z) is analytic, its transform F{z) will be analytic too. N
Consider the function f(z) defined a,s wll (z) in D and as 0 in D • . T1 w (z) in D and ~N( Clearly, its Hilbert transform in - ) - w)l z) in ll N
) 1 ) 1
D. We may interpret the eigenfunctions w)J (z) as the eigenfunctions 0f the
Hilbert transformation restricted to D and to the class of analytic functions in D. , Let now g(z) be a real-valued function in D which vanishes on the boundary C of D and whose complex derivative 2, It is easily verified that for z E' D
.:e . (25)
1(r
1
J
1t'
(d g( ~ ) )' d
D
such that all such Junctions
s
*
1
(?;' _z)2
~
is in D of the class
uZ
d'" _, dg( z ) • - '0 z
are likewise eigenfunctions of the Hilbert
transformation with the eigenvalue 1. However, if v(z) is an arbitrary analytic function in D with a finite norm, we have
(,(
(26)
JJ
(*)
v(i)
d l'
::
0
D
Hence, the linear space of all analytic functions with finite norm is orthogonal to these eigenfunctions
~.
The linear space of all complex valued functions in D of class can be split into the two complementary subspaces conSisting of the
'£
~
2 and
of analytic functions. It is evident that the nontrivial part of the theory of Hilbert transforms belongs to the subspace of analytic functions and not to the trivial orthogonal complement where it reduces to the identity transfor214
- 11 -
M. Schiffer
mation. The theory of the Hilbert transform in the subspace of analytic functions was developed by Bergman and Schiffer [3, 5]. The general theory for the
;;e 2-space was first indicated by Beurling
[2, ~ .
We shall see that the Hilbert transformation in the analytic subspace can be reduced to an integral transformation with a completely continuous kernel.
4. The Green's Function and its Analytical Kernels Let g(z, ~) be '.:he harmonic Green's function of D. That is, g(z, >:) is harmonic in both arguments for z
t 'S
, vanishes if either ar-
gument point lies on the boundary C of D and behaves such that g(z, ~ ) +
I I. . is regular harmonic as
+ log z- 'S
z~
S.
It is well known that g(z,
is symmetric in both arguments, We define now the two kernels
s)
[3,15J
1.h
L(z,~) = -1'Cdzd~
(27)
K is hermitian in the two variables, analytic in z and antianalytic in
t: .
It is regular even for z = 't: since the differentiation process which defines
K annihilates the singularity of the Green's function. On the other hand, L(z, ~) is analytic and symmetric in its variables, but it has a double pole at z =~
(28)
and can be written as
L(z,
s )=
1
2 - .2 (z, ~ )
j((z-?;)
Here ), (z, ~) is regular analytic and symmetric in both variables. It is even continuous in the closure of D. If C is an analytic curve, it is even analytic in D + C. 215
- 12 M. Schiffer
From the boundary behavior of the Green's function a simple integration by parts leads to the following identities valid for every analytic function
f (z)
with finite norm over D:
H
L(z,
(29)
~)
fP;)d't = 0
D
This shows that K(z; ~) is the Bergman kernel function which reproduces every analytic function with finite norm; L(z, 1;) annihilates the same function class under the integration considered. We may rephrase the second identity as follows :
rr
1
(30)
) .",/1
'JC
0 ,{(z,~) f(~)dL
D On the left side stands the improper integral which defines according to (23) the Hilbert transform of
f (z).
On the right we have an integral transforma-
tion which Is completely continuous and coincides with the Hilbert transform on the subspace of all analytic functions in D with finite norm. This new definition of the Hilbert transform on the subspace is, of course, of very great convenience. Let us consider an arbitrary complete orthonormal system W.v (z) in the subspace of analytic functions in D. The Bergman kernel K(z, ~) can be developed into a Fourier series in the system, and we have by virtue of (29)
'-.---, 0'>
(31)
K(z,~) =
LJ
w),'(z)
-W,v
(s)
Y =1
This was,
ind~ed,
Bergman's original definition of his kernel function. It is
easy to see that the Fredholm eigenfunctions w); (z) defined by (21) and (22) 216
- 13 -
M. Schiffer form a complete set and may be used in the representation (31). Moreover, we have for this particular choice of the orthonormal system in view of (21) and (30) (32)
w)-'
(z) =
1))
II ).
(z, '( ) w>,
(~ ) d 7:'
D
..l (z, '()
We can express
for z fixed as a Fourier series in the w,ll (~ )
and (32) yields us the Fourier coefficients. We have then
(33)
.l (z,
-
_V
~)- ~
w)J(z)w)l(~)
1
)) =1
))
We are led next to an important and beautiful identity for the
I-kernel
by using the concept of the Hilbert transform. Let f(zj be analytic in D and of finite norm. We may conceive it as a complex valued function of class ;£ 2 in the entire plane if we define it as identically zero in
D.
Its Hilbert
transform F(z) can be written as follows:
(34)
F(z) =
if 1
(z, ~ ) f( ~ ) d'L
D
1
F(z) = rr
II D
f( 1;) d 1:' (~ _z)2
if zED'"
The identity of norms (24) for Hilbert transforms yield thus (35)
D (r
+
JJ N
D
217
- 14 M. Schiffer
A standard argument leads, therefore, to the identity
J(
J
(36)
1
~(z, 'S) )(z,s)d1' +-2 Tr
D
Jj'i D N
:d1:'.
2'-2
(z- ~) I('z_n ) I
Observe that the second left-hand integral is regular analytic for seD and regular anti-analytic for
1 Eo
D. It can be computed by integrations and is,
therefore. more elementary than the kernels K and ..e which depend on the Green's function of the domain, that is, on the solution of a boundary value problem for harmonic functions. We shall call the expression
(37)
a geometric term in contradistinction to the more trascendental kernels K and
,t .
Clearly,
r is hermitian and a positive definite kernel. If we insert
into (36) the Fourier developments (31)and (33), we find the Fourier development for the geometric kernel
(38)
This representation may serve as a basis for calculating the kernels
.t
and K. The basic idea is as follows. All numbers
the interval 0
<(1 - ~) S -; Xl
~ 1.
~,l) = (1 - ~) 00
In this interval
verges absolutely and uniformly. Thus,
(39)
218
~ =~
n-O
111
(1-
lie in
~)n
con-
~
15
~
M. Schiffer can be approximated uniformly and arbitrarily by polynomials P N( ~) =
.t y=o
aN
~)J. If r (~ )(~ ,~)
)..' ~ ~) = r (1)(~ ,1)
n,
denotes the )) th iterate of the kernel
it is clear that
Thus, the kernel K and, as is easily seen, the kernel 1, can be approximated arbitrarily in t"'rms of iterates of the elementary geometric kernel
[(>!,~) [3, 1~ , 5, Fredholm Eigenvalues and Univalent Functions Let us suppose that f(z) is analytic and univalent in the unit circle and maps
Iz I <1 onto a domain D in
the w~plane, Because of the
conformal invariance of the Green's function we have for the L-kernel of D the identity
(41)
with W = f( ~) and L(z, ~ ) being the L~kernel of the unit circle. Since the Green's function of a circle is well known, we have
(42)
1
L(z, ~ ) =
rc (z- s)
and hence 219
2
- 16 M. Schiffer
(43)
o I I 1 ~D(w, w)f (z)f (?:) = 1T
Hz) - f(
log
z-
~ )
~
Now, the function log f(z) -f( ),: )
(44)
z-
~
plays a central role in the theory of univalent functions, Indeed, a necessary and sufficient condition that f(z) be regular and univalent in
Iz I <1 is the
regularity of this function in two complex varia-
bles in the product domain
I z I < 1,
IsI < 1.
This important
formulation of univalency was observed a long time ago, but Grunsky [7] was the first to draw important conclusions from it. Using the preceding relations between the
.{ - and K-kernel (in particular (36)),
we can derive the Grunsky inequalities
(45)
for arbitrary complex vectors x v
as the necessary and sufficient
condition for the univalence of f(z). From these conditions many elementary estimates for the coefficients of univalent functions may be obtained; the most startling one is and Schiffer
t 5] . 220
Ia41
~
4 proved
by Charzynski
- 17 M. Schiffer We can bring every quadratic form Q(x, x) with symmetric matrix Q into the normal form
(46)
Q(x, x) =
I:. ~1I y~ ,
where the y yare obtained from the Xv by a unitary transformation. In other words, every simmetric matrix Q can be brought into the Schur normal form
[16J
Q = UT
D.
U where U is a unitary matrix,
UT is its transposed and L:::.. is a diagonal matrix with positive elements
1,V • There arises then the question: Given a univalent function f(z) in
I zl <
1 and its corresponding symmetric matrix c~v
by (44), what are the positive numbers normal forw.? The answer is : The
1))
defined
..t,V which appear in its Schur are simple expressions in the
Fredholm eigenvalues A.).> of the image domain D. This surprising relation can be read off from (33), (43) and (44). It shows clearly the great significance of the Fredholm eigenvalues for the general theory of analytic functions and conformal mapping.
6. The Variation of Fredholm Eigenvalues The most powerful tool in the study of Fredholm eigenvalues are the variational formulas which show how the
.A. v behave as func-
tionals of the curve C and how they vary with continuously changing boundary. To find the functional derivatives of the
A).J with respect
to C we transform the integral equation (21) for wv(z) by partial integration into
221
- 18 -
M. Schiffer
lJi
{
w (z) = -
(47)
))
2ni
J
C N
We select an arbitrary but fixed point z
o
z
(48)
•
=z +
D and consider the variation
€
E z-z
o
I EI
The mapping zAc (z) is, for small enough
'
a regular analytic
function of z in D + C and maps D into a new domain D t with boundary C
*,
eigenvalues
A.:
and eigenfunctions
asymptotic formula for
A.: '
w;~
(z). If we can give an
we shall have achieved our aim since
the most general variation of C may be obtained by superposition of elementary variations (48). We write down the corresponding integral equation (47) for But putting z ~ = Z M- (z),
~*=
S*( ~),
J.. ~ •
we may refer the integration
back to the original curve C and obtain an integral equation for the now depends on z0 and fixed domain D whose kernel ,
(49)
I
) C (s
-z)
(1- (z-z ~z: -z ))
Let us· define the new unknown function in D
222
t . We find
o
0
- 19 M. Schiffer
(50)
and bring (49) into the much simpler form
We thus have for m)) (z) an integral equation whose kernel depends analytically on
E • We may also return to the domain integral form
(52)
We may replace the first term
.!. (s
J(
_z)-2 in the kernel of (52) by
1, (z, ~ ), Thus, this integral equation falls well into the pattern of the Rellich theory [8] for eigenvalues of variable kernels. We may assert that
(53 )
admits a power series development in E • For the: sake of simplicity, let us assume that ).)) is nondegenerate, i. e., it has only one eigenfunction w11 (z). In this case we also have a series development
223
- 20 M. Schiffer
(54)
We now multiply equation (52) by Wy (z) and integrate over D. We use integral equation (21) for in
c
(55)
(z) and observe the asymptotics
w~
as given in (53) and (54). Thus,
If
m.v (z) w;; (z) d'r
=
J..*"
II
)J
1).>
D
~ )w.lJ ( " )d l'
D
+ E.
By our assumption z
o identity (21), we find
D
m,)J (
~
l)) 1r
w» (~ )
Jf D
(~
2
-z ) 0
d~ff D
W.v
(z) 2 dr
+ O( t 2)
(z-z ) 0
,..
D, and hence using again the corresponding
D
Take the real parts in (56), use (54) and the normalization of w).> (z) to find
(57 )
This is the desired variational formula for ). y • A similar formula may be given if
.A..» is a degenerate eigenvalue~ Had we taken a
224
- 21 M. Schiffer
variation (48) with z £ D, we might have reasoned in the same way by
o
starting with the integral equation for
w)J (z)
over
D.
We would have
found the analogous formula
S Ay
(58)
= - Re
for the variation of a nondegenerate eigenvalue. To illustrate the power of these variational formulas, we quote one extremum problem which has been solved by using them. Let f(z) be univalent and regular in the circular ring r r
<1 <
R. Let C be the image of
Iz I
~
IzI
~
R with
.
= 1 by this map. We call C
a uniformly analytic curve with the modulus (r, R). The concept of uniform analyticity is an obvious sharpening of the usual assumption of analyticity of a curve. We have the theorem: The lowest ·nontrivial eigenvalue). 1 of a uniformly analytic curve with the modulus (r, R) satisfie s the inequality
(59)
This estimate is the best po!!!sible. It shows the importance of the concept of uniform analyticity in the numerical procedures of conformal mapping.
7. Fredholm Eigenvalues for Multiply-Connected Domains It is easy to extend the preceding considerations to the case
that
N
D is a multiply-connected domain which contains the point at 225
- 22 M, Schiffer
infinity and is bounded by N ') 1 closed curves C ~ , which shall again be three times continuously differentiable. Its complement D will then consist of N disjoint simply -connected finite domains D)J' Let C= L; C» N
denote the common boundary of D and D, We may then discuss the Fredholm eigenvalue problem (5) with respect to the curve system C , We may extend again the eigenfunctions of this problem which (1/
are defined only on C into harmonic functions in D and in D. As before, we can interpret the eigenvalues
1).' also as eigenvalues of integral
equations for analytic functions of the Hilbert transform type, We can define a function w» (z) defined and analytic ill eacb component D.v
of
D such that
(60)
w)'
Ali
dy.
(z) =-
1r
ZED
and one single analytic function W).> (z) in the connected domain
'" D
such that
(61)
WJ.I
(z)
1/
d 1"
,
zeD
~
D
These functions are related to the Fredholm eigenfunctions
~).>
(z) of
(5) in the same way as in the case of simple connectivity, There is, however, one important difference. The eigenvalue
.1 = 1
occurs in
(5) in (N - 1) st order degeneracy, The integral equation (60) does not possess this eigenvalue at all while
.1
= 1 is an eigenvalue of (61)
of order N - 1. The corresponding eigenfunctions are the derivatives 226
- 23 -
M. Schiffer N
of the harmonic measures of the multiply-connected domain D. Since
l
= 1 leads to simple and well-known eigenfunctions, we still shall
call it the trivial eigenvalue and assume in all subsequent discussions
1/1. It is easily seen that for
l)1 > 1
still
i (s
iJ
(62)
W)J (
D
hl ,....
w~
S)
N
d'r ,
Z (
D
_z)2
{s )
( S - z)
zeD
2 dt
D
and that the w y (z) and
VI"
(z) form orthoflormal systems in their
respective domains. Finally, we can extend the entire theory of the Hilbert transform by means of
,e (z,
N
~
) to the case of the connected region D. However,
if we wish to do the same thing for the set of domains D)I' we first
have tor give a proper definition of a Green's function g(z,
~
~ (z,
s ).
We start with defining
) for the disconnected region D, namely
if z, 'I;
lie in same Dy
(63 ) if z,
227
S
lie in different Dp
- 24 M. Schiffer
Here g)l (z,
S ) is
the ordinary Green's function of the simply - connec-
ted domain D).I • We define next L(z, ?; ) from (27) and
-e (z, s ) by ~ (z,
(64)
z: ) by
means of
means of (28). Thus
S )=
r
)~(z,
~
S)
if z,s
lie in same
D}>
1
I
L -/.)) (z, ~ ) is the
g(z,
rc
2 (z- ~ )
if z, ~ lie in different D,).>
E -kernel of the simply-connected domain D,. •
With this definition, (30) remains obviously valid, and the Fourier representation (33) of
~ (z, ~ ) in terms of the analytic Fredholm eigen-
functions is preserved. The variational formulas of the preceding section can be carried over without change since we did not use anywhere in our calculations that C consists of one single curve.
8. Fredholm Determinants and Conformal Mapping Having enumerated many definitions and identities, we shall now show their usefulness and interest by particular applications. An important concept in integral equation theory is the Fredholm determinant
~2, (65)
141 D(
1. ) =
n
12
(l)
(1 - -
))=1
\2
)
A. y
where the product is extended over all nontrivial eigenvalues observe that the eigenvalues +..Lv
and -.A.. v
228
.A..).>. We
occur always in pairs in
- 25 M, Schiffer
our problem; this accounts for the quadratic factors. Consider D( 1
) for
fixed
J..
as a functional of C and ask
for its variation if C is varied by the standard variation (48) with z (
o
D, By virtue of (58) we find
(66)
This formula can also be justified if some
A))
are degenerate eigen-
values, The result simplifies considerably in the case
A.
= I, Indeed,
in'. view of (33) we can write
(67 )
Thus the important function
-e.
(z, z) has been identified as the funtional
derivative of the logarithm of the Fredholm determinant, A surprising result occurs in the case of ;a multiple connectivity N
>1 ,
We can speak of the eigenvalues of the curve system C and
their Fredholm determinant; we may also consider the eigenvalues
).~)
of the single curve Ck and their Fredholm determinant D(k) (1),
If z0 €" Dk we have by (64) the identity
(68)
That is, under a standard variation (48) which is regular analytic outside of Dk , the ratio Dk(1) / D(1) has zero variation, By reasoning typical for variational theory, we can then extend this result to arbitrary finite conformal maps in the exterior of Dk , 229
- 26 M, Schiffer
Theorem, Let D be a set of disjoint finite simply-connected .
domains D t
.
(t ) (1)
wlth the boundary curve system C, Let D(l) and D
be the Fredholm determinants of the curve system C and the single ~
Ct
exterior of
r ; let minants of
'
respectively, Let w = fez) be a conformal mapping in the
De. .6(1)
It will carry the curve system C into a curve system
and
rand
.6. (1
re
)(1) be the corresponding Fredholm deter-
(the image of C.t ), Then
(69)
It seems difficult to prove the conformal invariance of this ratio
in a nonvariational manner, We recall the fact that if C is a circle, all its eigenvalues are infinite and that the Fredholm determinant of each circle has the constant value 1. In every other case the definition (65) clearly indicates that D(1)
< 1,
Hence, suppose that we start with an arbitrary curve set C
and map the exterior of C {.
conformally onto the exterior of a circle.
By (69) we can assert for the Fredholm determinant
6. (1)
of the new
curve system
D (1)
(70)
Equality in (76) holds only if
C.e
>
D (1)
already happened to be a circle.
This remark throws light on a well-known procedure to map a multiply-connected domain onto a circular domain. One starts with the curve C1 and maps its exterior onto the exterior of a circle. Then one 230
- 27 M. Schiffer
takes the image of C2 and maps its exterior onto the exterior of a circle. One continues this procedure indefinitely taking care to run through the images of all starting curves in fixed order. The limit of this map transforms all initial curves C y
into circles. We see that
in this procedure the Fredholm determinant D(1) is steadily increased. One can base on this observation a convergence proof for this method of iteration. We also draw the following conclusion: Theorem. Among all conformally equivalent domains the circular domain has the largest Fredholm determinant D(1). This theorem was originally proved by variational methods
[12J.
The present derivation explains more clearly its significance.
9. Conclusion.
The close relation between the Fredholm eigenvalue
problem and the theory of analytic functions of one complex variable has been evident throughout the whole exposition, Hence, it will be expected that the potential theory in more than two dimensions will lead to Fredholm eigenvalues with a less elegant and elastic theory. However, many results can be preserved even in this transition. However, one very significant result shows the great difference in the nature of the eigenvalues for different dimensions. Theorem. Let D be a domain in space and let ). 1 be its lowest positive nontrivial Fredholm eigenvalue. Then
(71 )
Equality holds only in the case that D is a sphere 231
[13].
- 28 M. Schiffer
Thus, the Liouville - Neumann series development, which solves the boundary value problem in three-dimensional potential theory, will never converge better than a geometric series with ratio
i.
Another significant difference comes from the fact that the concept of conjugate harmonic: functions fails in more than two dimensions. Hence, we cannot assert that with each Fredholm eigenvalue its negative -
A.)i>
1.
);
also
will occur as an eigenvalue.
The study of Fredholm eigenvalues in more than two dimensions is thus still an open and promising field of research.
232
- 29 -
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