Linear elliptic differential systems and eigenvalue problems

Lecture Notes in Mathematics An informal series of special lectures, seminars and reports on mathematical topics Edited by A. Dold, Heidelberg and B. Eckmann, ZUrich

8

[_!

Gaetano Fichera University of Rome

Linear elliptic differential systems and eigenvalue problems 1965

The Johns Hopkins University, Baltimore Md, March- May 1965

S p r i n g e r - V e r l a g . Berlin 9 H e i d e l b e r g 9 N e w York

All rights, especially that oftranalafion into foreign languages, reserved. It is also forbidden to reproduce this book, either whole or in part, by photomechanical rues.us (photostat, microfilm and/or microcard)or by other procedure without written permission from Springer Verlag. @ by Springer-Verlag Berlin 9 Heidelberg 1965. Library of Congress Catalog Card Number 65--27796. Printed in Germany. Title No. 7328

These N ot e s c o n t a i n t h e l e c t u r e s of delivering

as V i s i t i n g

P r o f e s s o r at the Department of ~echanics

of The J o h n s Hopkins U n i v e r s i t y Clifford

I had t h e p l e a s u r e

on t h e i n v i t a t i o n

of P r o f e s s o r

Truesdell.

They a r e i n t e n d e d t o be an i n t r o d u c t i o n approach to higher order elliptic

t o t h e modern

b o u n d a r y v a l u e p r o b l e m s and r e l a t e d

eigenvalue problems.

I am d e e p l y g r a t e f u l kind collaboration

t o D r. Warren E d e l s t e i n

for his

i n c h e c k i n g b o t h t h e E n g l i s h and t h e M a t h e m a t i c s

of t h e s e N o t e s .

G. F i c h e r a

B a l t i m o r e s Md. - L i a y 1965.

CONTENTS

Lecture

1.

"Well posed"

Lectnre

29

Existenc e principle

boundar 7 value

problems

.......................

1

~.....................................

~9

11

O

Lecture

3~

The f u n c t i o n

Lecture

4~

The trace

Lecture

6.

Elliptic

Lecture

6.

Existence

Lecture

7o

Semiveak solutions

Lecture

8.

Regularity

at

the

boundary:

Lecture

9.

Regularity

at

the

b oundar]v: tangential

Lecture

10.

Re~ularit~

at

the

boundary:

Lecture

11.

The c l a s s i c a l

12.

linear

Lecture

18~

14~

StronKly

Interior f oi r

elliptic

plates

operators.

problems....... Problems.

Lecture

16~

T h e Wei na.t.ei n-Aron8, s a . j n , m e t h o d

Lecture

IT.

Construction

Lecture

18.

0rtho~onal

Lecture

19.

Upper approxinmtion

elliptic

system

results

...

44

..............

deriv.atives

The R a y l e i K h - R i t z

......

62

.......... 9.......

61

. ....

69

Physics: 80

P.hysics: 9.........

88

Ph]rsics:

method

~ 0. o . o . 9 . . . . .

intermediate of pomitiTe

operators

~

96

of the

O O O O O . . . . . o e e o o e e e .

~

112

0...0 ............

120

o ~~~9 9~~~. . . . . . . . .

130

9 9 9 9 9 9 9 9 9 9 139

of a I~0.

invariants. Green's

. 101

9............

compeer operator8

of .the ei~envalues

of ortho~onal

construction

...........

39

...................................

EiKenvalue

Explicit

...........

G~rdin K inequality.

16.

20.

systems

30

o.ooo.~ .........................

Lecture

Lecture

systems

BVP o f M a t h e m a t i c a l

Ei~envalue

Representation

24

~...............

preliminar 7 le.m~s

final

17

9.............

~ 1 7 6 1 7 6 1 7 6 1 7 6 1 7 .6. . . . . . . . . . . . .

elliptic

i nvariants

elliptic

BVP o f M a t h e ~ t i c a l

i

of the

regularity

...

.......................................

elliptic

elliptic

lethal.

BVP o f M a t h e m a t i c a l

PDE

of thin

...o .....................

and Ehrling

o f BVP f o r

elliptic

The c l a s s i c a l

Hm

solutions

Elastostatics 9

Equilibrium Lecture

systems~

linear

and

Sobolev

of local

The classical Linear

Hm

operator.

2nd order Lecture

spaces

Btrix

.........

~. . . .

99~....

162

~or an

e . o . . e . o . . o o o . . e

e m o . o . m . o .

164

-1-

L e c t u r e

"Well

posed"

The c l a s s i c a l equations

properly

those

expressed

of it.

theory

solutions

conditions. equations

of partial

with

reason~

these

number

a PDE c o n s i s t s

a particular

one ~ich

These conditions

of findsatisfies

are generally

which the unknown functions

b o u n d a r y o f t h e d o m a i n ~ w h e r e t h e PDE i s

For this

differential

a d m i t %o a n i n f i n i t e

problem connected

as complementary on t h e

p r o b l e . ~ s' .

such equations

possible

given auxiliary

value

of view in the

The t y p i c a l

ing amongst all

satisfy

boundary

(PDE) a s s u m e s t h a t

of solutions.

part

point

1

conditions

must

considered,

o r on

a r e known as " b o u n d a r y

c o n d i %i o n s " 9 In spite with

of the fact

applications

m u s t be p o i n t e d

that

o f PDE i n v a r i o u s out that

the

with very

smooth coefficients,

be false,

since

respect

let

a few years

the

where

equation

us consider

point

of view is

branches

assump%ion that possesses might fail

a very

the one that

of applied

mathematics,

a PDE, e v e n a l i n e a r

infinitely

~ the

~ ~ ~ ~'

in the real

cartesian

the Wirtinger

may

In this

e x a m p l e g i v e n b y Hans Lewy [ 4 ]

3-space

with coordinates

PDE

. By u s i n g

i%

one

many s o l u t i o n s

to have any solution.

interesting

agrees

ago.

Let us consider~ X ,'/',t

this

differential

operator

-2-

L.~

~-~ ) , we can write @y

I(X y,~ )

We a s s u m e t h a t

s u p p o s e t o be r e a l as a d e r i v a t i v e

is

valued.

of a real

a function We w r i t e ,

function

in a more compact form

d e p e n d i n g o n l y on

for convenience, ~(t)

t

, w h i c h we

such a function

. The a b o v e e q u a t i o n

can then

be ~ r i t t e n

(1.1)

We s h a l l

prove that

differentiable real

analytic

)~

0"~

"~.

a necessary

solution

=

~

condition

for

~

function

of

seen

real

is

that

~

be a

~ 9 :t

be a p o s i t i v e

t o have. a c o n t i n u o u s l ~

in a neighborhood of the origin

'"

Let

(1.1)

number and s e t

~

{,e

=

e.

. It

is

easily

that -

z

+

rD~

Z

~

oe

Since

0

0

Te o b t a i n

)

(1.2)

~

0

dS. 0

.s From

(1.1) and (1.2),

~

by assuming

~9 and i n t e g r a t i n g

t)ao

~t o

d~

dt

ve get

-3-

Now set ~ ~ ~ti" ~,t

and

Z~

l

s

,:,,~

f

~,g

o Equation

(1.3)

gives

This means that of

~

for

the

O~ ~

function

~ ~

and -~

The f u n c t i o n

~/(~)

is

From this

follows

that

it

and vanishes

for

across

the

on t h e

t-axis

If

real

the

equation the

for

real

with

an holomorphic ~

0~- ~ ~ &

part

of ~r

~/

is

conveniently and vanishes continuous

c a n be c o n t i n u e d

(~>~)-plane.

function

Hence ~

chosen. for

for

~ :0. 0 _~ ~ ~

analitically

- as the trace

o f ~r

function

~

is

a s s m o o t h as we w a n t ,

h a s no s o l u t i o n

in any arbitrarily

but not analytic, fixed

neighborhood

of

origin.

order

which possesses

closed

and l e t

square

another

example of a linear

only one solution

of the

b4(x,y ] and b ~ ( x , y )

b4 (.-4, Y)

(z,y)

first

in a given domain. Let

be two r e a l

,

->0 _>o

C_

PDE o f t h e

~

be

(zj~)-plane,

a s s m o o t h a s we w i s h a n d s a t i s ~ y S n g

Let

~

is

analytic.

L e t u s now c o n s i d e r

the

(

9 Therefore

of the

- is

(1.1)

the

+ b~Tr~

~ t

continuous

~ : 0

~-axis

= ~

~/

be a n a r b i t r a r i l y

the

functions

following

b.~(4,,y)

on

conditions

~ O.

,

smooth real

defined

o.

function

defined

on

Q

and

-4-

negative

at every point

differentiable

of

solution

~

solution

~

solution of (1.4)

i n an i n t e r i o r

point

over of

Let us assume that instance

~ - 0 Q

(X~ - 4 )

~(~oj-~)~y(Xo)-~)~O. Xo:-~

Since it

~

is

cannot

be n e g a t i v e .

if

be p o s i t i v e .

0

) ~O

and

1928. ~ore sophisticated

o r d e r whose o n l y s o l u t i o n these

simple

is

and t h e r e f o r e

~ ~ O

can also

e x a m p l e s show t h a t ,

besides

what a "well posed boundary value

space

~

consider

when we s a y t h a t values space.

of

vector-valued

of By

~

are ~-:

ua

shas b e e n known

like

the above considered

of higher

(see [2],[3]). point

of view,

o n e s , when

problem" is

for

a

PDE.

be a d o m a i n ( i . e .

X ~ . The p o i n t

.~(xQ)-r

t h e maximum o f

the classical

aiming to describe

Let

b4 ( X o ) - 4 ~ x ( ~ o - 4 ) ~O,

be c o n s t r u c t e d

also

system of linear

connected

X ~ will

functions

~ (x)

is

n~-vectors (~4 ,'",

~

an

open set)

be d e n o t e d by uu(x )

defined

Oh-vector

of the ) ) ~

~O;

t~ _: 0 .

one h a s t o i n c l u d e

general

(xo)-4)

examples of homogeneous equations

situations

for

~

follows

T h i s e x a m p l e ~ w h i c h was g i v e n by ~lauro P i c o n e [ 5 ] since

t a k e n on

and t h e r e f o r e

. In any case

of (1.4) Then

is 9

for-t~•

X~:~

of the real

on t h e b o u n d a r y ,

C(X~)-4)t~(xo;~)~_O,it

a solution

~

If f~

f~:

minimum ~ t

b~x(~o -4) = 0

(Xo - ~ ) ~ 0

-b~

t h e minimum

. Then ~ ( X o - I

m u s t be t h e c a s e t h a t

The f u n c t i o n over

~

and

on i t s

PDE

u, ~ o

, then~ obviously~

takes

in the point

In fact,

cannot

Q

~

.

the only continuous

of the linear

+ b~ (.x~y.) Q---~ + c. ( x , y ) ,gy

the trivial

if

9 ~Te ~ a n t t o p r o v e t h a t

in the square

(l.4)

is

~

of the real

g ~ (x4 ,-.-, x~).~%e s h a l l on ~

function,

oh-dimensional = ~-~;

cartesian

9 ~ore precisely, we mean t h a t

the

complex cartesian

we d e n o t e t h e d i f f e r e n t i a t i o n

-5-

p~q ,,~...

~ e c t o r ~ The l e t t e r s integral

components e . g . ,

~ ~ (~

O t h e r w i s e f o r any v e c t o r len~ht

1~1 ~:

If

~

p,~

Dr:

,

i s any p o i n t - s e t

C ~ (~the

clems o f a l l

derivative

of ~

and c o i n c i d e j ~

sub-class By'

up t o t h e o r d e r

of

~

(A)

§ K

~

exists

We w r i t e

Cartesian

[ ~:. ~...~)].

ve shall

functions ~.

denote

9 0

.

that

any

p o i n t of

apt ~

C ~ ~A) ~

b7

b~ p o s s e s s i n g

T h i s means

, we d e n o t e by

of f u n c t i o n s

denote the class

~.

(support of ~ )

will

such that

denote the spt ~ c A.

o f f u n ~ t i o n 8 d e f i n e d i n t h e w h o le s p a c e ~ ~ up t o t h e o r d e r

denote the sub-class The symbols

of

c-

~

, i e.

C ~ consisting (A)

C~176

,

C ~ :

of f u n c t i o n s ,

C

,

6~

explanatory. be an k~

~•

- m a t r i x d e f i n e d on

the matrix differential

L~

--- d~

D ~

We u s e h e r e t h e s u m n a t i o n c o n v e n t i o n , i . e .

) , We

A

operator

. when a v e c t o r - i n d e x

r e p e a t e d t w i c e , a summation must be u n d e r s t o o d T h i c h i s T h o l e d o m ~ n of v a r i a b i l i t y Let

its

o).

points,

in

Pc

~.

represent

a t ~ver 7 i n t e r i o r

A

J~(x)j

consisting

w i t h a bounded s u p p o r t .

d e n o t e by

will

Jpl--

p: -

P,,

X ~ with interior

o f t h e s e t w he r e

: C ~ (~),&Zwill

~(•

J~l

~L~ : ~, ~ . ~ . .

and p o s s e s s i n g c o n t i n u o u s d e r i v a t i v e s

Let

and Te s e t

the (vector-valued-)

of o r d e r

with non-negative

with a f u n c t i o n which i s c o n t i n u o u s i n t h e whole s e t

C, K ve s h a l l

are self

~-~ectors

.... ) p ~ ) ,

i s a ny f u n c t i o n d e f i n e d on

the closure

denote

~..,~),

Dh

of

continuous derivatives

If

~ -: ( ~

I~I-;, I ~" and

~

n,

_-

will

~ ~ k t be a d i s c r e t e

of

is

extended to the

~ 9

s e t o f complex v e c t o r

mean t h e s e t t o be empty, f i n i t e ,

~

or countableo

s p a c e s . By d i s c r e t e Let

Mk

ve

be a l i n e a r

-6-

C ~(A)and with

t r a n s f o r n ~ t i o n defined on

range i n the

vector space S ~ .

~e s h a l l c o n s i d e r t h e f o l l o w i n g problem

(1.~)

L ~, :

0.6)

,

The symbol~ denotes a giTen ~ - v e c t o r a given Tector of the space

H

=

~,

v a l u e d f u n c t i o n d e f i n e d on ~ , ~

5}1.

I n s p i t e of t h e i r e x t r e m e l y a b s t r a c t d e f i n i t i o n ve s h a l l r e f e r to c o n d i t i o n s (1.6) (when

I n the case t h a t a solution

Ix,

~ S~]is

not empty) as bo,un,dar ~ con.,d,.iti,ons,.

~ S~ ] i s empty, t h e problem consists merely i n f i n d i n g of the e q u a t i o n ( 1 . 5 ) .

Let us furthermore suppose t h a t (i.e.

~

i s a bounded r e g u l a r domain of X ~

t h e Green-Gauss i d e n t i t y holds f o r i t )

belongs to

cl~l(A)

the adjoint matrix, i.e. matrix-differential

o~ _~ ((o~ ~K ))

. If

the

,u- :

(~:,(...~]K:1..'~)we denote

by ~

re.x-rim, m a t r i x ( ( ~

a~l

(,-~)

K ~ ) ) . The f o l l o w i n g

D ~,~.

and ~Y both belong to

~

C~ ( x )

o p e r a t o r w i l l be c a l l e d the a.djoint o p e r a t o r of L Li~ -

Suppose t h a t

and t h a t t h e m a t r i x

C ~ ( A ) then t h e f o l l o w i n g

G r e e n ' s formula h o l d s :

A where

~ (~jlr)is a bilinear

~A (4)differential

o p e r a t o r of order '~ -4

(4)Since ~j~r)is d e f i n e d f o r complex v e c t o r Talued f u n c t i o n s t h e term " b i l i n e a r " means t h a t H(%~)is l J n e a e w i t h r e s p e c t to It, , i . e . H (~t~ + bu'~ ~) -:~(%~)+~H( i ) end a n t i l i n e a r with r e s p e c t to ~ , i . e . H(~)~.~ b~') = --o-H (u.,~)~-b H (I~.,~')[Z,~ a r e the complex c o n j u g a t e s of ~ and b ~ .

-7-

in

~

and i n

matrices fJA of

~

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

CL and o f t h e f i r s t

a r e e x p r e s s e d i n terms of t h e

order differential

e l e m e n t s of t h e boundary

. I t i s somewhat t e d i o u s t o w r i t e down e x p l i c i t l y ~ ( ~ W)o lloTever, t h i s ~'e s h a l l

the full

expression

i s n o t n e e d e d f o r our p u r p o s e s 9

c o n s i d e r , i n s t e a d of the general problem (1.5) , ( 1 . 6 )

the

f o l l o w i n g one w i t h "homogeneous b o u n d a r y c o n d i t i o n s "

(1.5)

L~ : ~ ~

When a f u n c t i o n

(1.6.)

,

admissible solutions)

MI~ ~ ~

o.

( b e l o n g i n g t o t h e s p a c e of what we s h a l l d e f i n e as exists

such t h a t

M~. ~ ~ ~ ~

, t h e n , and o n l y t h e n ,

p r o b l e m ( 1 . 5 ) , (I .6) i s e q u i v a l e n t t o p r o b l e m ( 1 . 5 ) , ( 1 . 6 o ) . L e t us d e n o t e by V

the linear variety

valued functions belonging to

C ~ (AI

" H (~,,~,-)d~= "aA f o r any

~

satisfying

If a solution then

~

~

conditions

the integral

of a l l

~-vector

such t h a t 0

(1.6~

of problem ( 1 . 5 ) ,

must s a t i s f y

consisting

belonging to C ~ ( A

(1.6o) exists belonging to

),

C ~'(/~)

equation

(1.7)

for

eve17

9 This i s t h e s t a r t i n g

~s

p o i n t o f t h e c o n c e p t of weak

solution for the boundary problem ( 1 . 5 ) , ( 1 . % ) 9 substituting

the integral

the equations

(1.5),(1.6o).

equations (1.7),

I t consists merely in

written

f o r any

~y~r

In order to make t h e e q u a t i a n s

(1.7)

, for

consistent

Te assume t h e f o l l o ~ r i n g h y p o t h e s i s 9 1~ the

zero-vector

The l i n e a r v a r i e t y 9

"Vr

c o n . r a i n s some v e c t o r i d i f f e r e . n t f r o m

-8-

It is co~enient

to e n l a r g e our problem i n order to i n c l u d e t h e p o s s i b i l i t y

t h a t t h e given f u n c t i o n

~

and t h e unknown f u n c t i o n ~ be g e n e r a l i s e d

f u n c t i o n s . We do t h i s i n a q u i t e a b s t r a c t Tay.

Let

5~

be a complex •anaoh sp~ce ( B - s p a c e ) .

~e ~s~ume that

c o n t a i n s a l i n e a r s u b v a r i e t y t h a t i s l i n e a r - i s o m o r p h i c to w i l l be the

S~

space of t h e a d m i s s i b l e unImolms. Let

We assume t h a t

S{

C ~

S~

(,~), S~

be a second ~ - s ~ a c e .

c o n t a i n s a l i n e a r s u b v a r i e t y l i n e a r - i s o m o r p h i c to

C~

In a d d i t i o n to the h y p o t h e s i s 1 ~ we make the f o l l o l r i n g ones: 2~

There e x i s t too complex ~ - s p a c e s

~

and

I'~.

such t h a t

c o n s i s t s of measurable (complex l ~ - v e c t o r v a l u e d ) f u n c t i o n s and

{~

measurable (~omplex ~ - v e c t o r valued) functions. ~oreover S ~ : ~

S~r~ b~'~* ( ,~*~ and

I~,

of

and

_are . . the. topolol~i-cal . . . dual spaces of I ~ and [ ~

i~;.

respectiTely) 9 3") ~ varies in V

contains

an_~d I ~

c ont_ains the range o_f L ~

S~

~-space

[~

] c o n t a i n s a. l i n e a r s u b v a r i e t ~ Banach-iso,morphic

of m e a s u r a b l e f u n c t i o n s ~ t h e n ! i f

~

[ ~ ] denotes any

f u n c t i o n of t h i s su ,bvari,et~ an.d "~ i s any f u n c t i o n .o_f the scalar function

( (

)

The____nn

9

4") I f to. a

V

~~

[ ~ ~/3

denotes the dua,lit~ b e t . e e n

i~

i s Lebesgue i n t e g r a b l e on

,a

~-space

L I~ ~ ~

and

and i t s topologica,l dual

space) 9

We s h a l l c o n s i d e r t h e f o l l o T i n g problem: A vector

~

of the ~ - s p ~ e

$~ |

vector

of t h e

(1 ~ fo.r any

%~ s V .

~-space

St ~ such t h a t

i s g i v e n . We . ~ t

to find a

),

-9-

Because of h y p o t h e s i s 4 ~

(1.8)

s e n s e , t h e n the s y s t e m The v e c t o r

~

when

w i l l be c a l l e d

(1.5) ,(1.%) , with space

.~

and ~ a r e f u n c t i o n s i n t h e c l a s s i c a l

i

r e d u c e s to t h e system ( 1 . 7 ) . a weak s o l u t i o n

S~

o f t h e boundary v a l u e p r o b l e m

as t h e s p a c e o f " d a t a " and s ~ e

6

as t h e

space of a d m i s s i b l e s o l u t i o n s . Assume t h a t in the variety all

theveexists

"~ 9 L e t

some n o n t r i v i a l

~r~

the linear

solution

of t h e e q u a t i o n ~ ' ~

s u b v a r i e t y o f ~/

consisting

of

t h e s e s o l u t i o n s ~ t h e n a n e c e s s a r y c o n d i t i o n f o r t h e e x i s t e n c e of a

solution

of o u r p r o b l e m i s t h a t

(..~, 'b~~

(1.9)

We s h a l l s a y

:

for

0

~jo

Vo

t h a t t h e b o u n d a r y v a l u e p r o b l e m (B.V.]7.) ( 1 . 5 ) , ( 1 . 6 o )

.well p o s e d b o u n d a r y v a l u e p r o b l e m i n t h e s p a c _ e s -~ ~.. 5 ~ vector

satisfying ~ e ~

the compatibility

satisfying

5~ '

condition

~a

is a

, ~dlen f o r any

( 1 . 9 ) t h e r e e x i s t s some

equations (1.8).

We want now t o g i v e a n e c e s s a r y and s u f f i c i e n t

c o n d i t i o n f o r a B.V.~. Q

t o be w e l l p o s e d .

~ r o be t h e c l o s u r e o f t h e v a r i e t y

space

~

If

i s any f u n c t i o n i n

~/

.

Let

Let us denote by ~

c l a s s - as an e l e m e n t o f ~

~

the factor

, we s h a l l -

-~r

~-space

in the

~

d e n o t e by [~v~ t h e

d e t e r m i n e d by ~ v .

/

~-

V~

equivalence

Set

II L~(~ iiB~ ~,V-~

I n t h e n e x t l e c t u r e we s h a l l 1.I. (1.6.)

Ji E~ l ji

~

prove the following theorem:

A n e c e s s a r y and s u f f i c i e n t

t o be w e l l posed i n t h e s p a c e s

c_o_n_dition f o r t h e B . V . P . S~ ,

S

is that

~

(1.5),

be g r e a t e r

than zero. w i l l be c a l l e d t h e d i s c r i m i n a t o r spaces

S t ,

5~.

of t h e B.V.I:'. ( 1 . 5 ) , ( 1 . 6 o )

in the

-10-

Bibliography,

F_x]

G. FICItERA - L e z i o n i

sulle

Trieste,

[2J

[3]

o f L e.et,ur,e 1

trasformazioni

le

Ediz.

- Roma,1958.

Veschi

G. F I C H ~ A - S u l c o n c e r t o differenziale

equaz.ioni

Ediz.

di problema -

without

proble.mi

"ben posto"

Rendiconti

solution

dei

differenziali

Veschi

-

par~ia.1

- Annals

al

- Corsi

per

di Matematica

- An e x a m p l . e .o,f a s m o o t h l i n e a r equation

M. PICONE

~enerale

c o.ntorno per

66 -

-

1954.

Go FICHI~IA - P r e m e s s , e a d u n a , t e o r i a

I t . LI~#Y

lineari

una -

INAM

e~uazione

19 -

1960.

d i f f , e,r e n t i a l

of Mathematics

-

195"/.

-Ma~iorazi.one

degli

mente paraboliche ordine

-

inte~rali alle

derivate

Ani~ali di M a t e m a t i c a

dell.e .e~uazioni parzialt

del

pura eapplicata

totalsecondo - 1929.

-11-

Lecture

Existence

Let Let

~f

Lhi~

9

~'e d e n o t e by ~

~Ye s h a l l A

principle.

be a complex v e c t o r s p a c e and ~

~1 h ( ~ : 4,~ ) be a l i n e a r

space

2

vector

and ~

two complex ~ - s p a c e s .

t r a n s f o r m a t i o n ~ i t h domain ~r and r a n g e i n t h e the topological

dual space of

~h "

c o n s i d e r t h e f o l l o w i n g problem: ~

of

the

,space ~: is

given;

find a

vector

~/

_of ' ~ :

such t h a t

(2.1) ~'e s h a l l 2.I.

prove the following theorem:

it n e c e s s a r y , and s u f f i c i e n t

. . s o l u t i o n . of problem

. (~.i) , f o r any r

c o n d i t i o n f o r t h e , e x i s t e n c e of a ~

~ ' ~ ~ ~ i s t h a t .a p o. s i t i. v e . c o n s t a n t

e x i s t such t h a t t h e f o l l o w i n g i n e q u a l i t y

S~.fficiency. Let ~Z be any vector in end s e t ~V:

the

h o l d f o r any %~e ~r

range M ( V ) o f M . L e t % - M ~

~/~ = M4~r. 'I~e v e c t o r ~X/4 i s u n i q u e l y d e t e r m i n e d by ~,'~ , s i n c e Iv~'~ ' ' ~'z

implies,

because of ( 2 , 2 ) ,

that

~ M 4-v - M4%v' II -~

V~ il ~ z ~y- M ]Y'l] = O. L e t us d e f i n e on M2. ( V ) t h e f u n c t i o n a l

-12-

Obviously

depends linearly on ~ z 9 On the other hand, l ~ (.~)i ~ i i ~ j t l l M ~ l l

~

il~llllM~rii tional

of ~

-" V~ t t ~ l l [ I ~ II 9

i n such a way t h a t functional

(2.3)

still

in the whole space

M t (V)

and

spaces

holds for the continued functional.

w i l l be a s o l u t i o n

N . e c e s s i t y . We can r e s t r i c t

~4

Mz~V)

and ~ .

there exists

ourselves to c o n s i d e r a t i o n

of

~,

~

and ~

T h i s a l l o w s us t o s a y t h a t

the solution

~--~

such t h a t

is a solution

[M~(V)]

M~ ~ V ) functional I~enceforth the range

, l e t us consider the f ~ o t i o n a l is linear ~i~ w M~(V)

and It ~rv'~r Ii z il ~

M~(V)

and i t

~ 9 T

~ and T ~ - - ~

" V~ ilc~li 9

i s bounded s i n c e

il ~- V, ~ i ~

II .

, the linear

, i.e.

of a n y

con~t~t).

~et us define on ~

is linear.

I t i s bounded s i n c e

has

such t h a t

L e t us is a

a closed Then we

"I.~ : ' l ' . ~ . .

By t h e

A constant ~,

t~

in the range

This

I ~ Q r ~ z ( ~ ) l - ' t ~ v c z ~ l ~ t~ II~[IJl~z[~,

L e t us now c o n s i d e r f o r a n y

functional

~

9 I

(~)- <~ #, ~, ) ,

~(~):.It

H * A l s o l e t us c o n s i d e r t h e l i n e a r

consisting

~

~,

For a n y f i x e d

V~

c~ ~ [ M 4 ( V ) I ~

of ( 2 . 1 ) .

i s a bo~mded t r a n s f o r m a t i o n .

itS{ [I : i l T ~ i

of t h e whole

f or any given

:<'r~,~,M~.), t h e n :
instead

of ( 2 . 1 ) c o r r e s p o n d i n g t o

I n f a c t ~ l e t us s u p p o s e t h a t

" c l o s e d graph theorem"~

Such a

of t h e s u b s p a c e s

respectively,

linear transformation of [ M t ( ~ r ) . l " i n t o graph.

~

of ( 2 . 2 ) .

o n l y one ~ ~ [ M z ( V ) ] ~ t h a t

d e n o t e by ~ -

of

is a bounded func-

I1~ It ~ v, ilq H.

the Hahn-Banach theorem we may continue ~

exists

~

and

(~.3)

~y

Therefore

t~ ,- t ~x~

the f ~ c t i o n a l

~

in

i s bounded subvariety (

t

~ (~) 9 t ,~,

i s a complex

~.

-13-

I .~(~) I

I~

li~ ii and Ii ~ i{- ]i ~V~ {i.

{/~ W ~ ,

"%4(~)

.

defined

on ~ t ( ~ r ) s u c h

exists

that

ii F i} : {i ~/4 i[ j

W~, i{ II F it - II~,

, '~-~: M z b "

~ 6 [~4 (~r)]~

It

ii : Ii~, ~i 9 IIM, ~ II

iiW

there

We have proved so f a r t h a t

~/~ -- ~14~r

for any

Since

(2.2)

~

, F ) { ~ II

IIW,~4..i1 .>~ ii ~z4 II us now assume

It i ii ~4 Ii

By t h e I i a h n - B a n a c h t h e o r e m ,

bounded f u n c t i o n a l

and a l s o :

{i ~ {,

II II w{l o {l~/~

that

follows

that

From t h e p r o o f

a solution

same c o n s t a n t

W~/

(4):

{{*VVvr~/4tt = II'~Vr~/ It .

IIW ~ . II _z K ii~ Ii : K

and

V.

that

of the first

exists

satisfying

appears

in (2.2).

of (2.1)is

obviously

orthogonal

to the range of

part

iiM ~ ~ ii,

the inequality

of

. This space is a closed

shall

equivalence

~]

the

space - determined is

defined

as

following

.1 {{ ~

of (2.3)

inequality

~/

9

with the

are

subspace

~

of

/ Y~

~,&

We

- as an e l e m e n t o f t h i s

S i n c e t h e norm {i [~/-1 i~ i n t h e f a c t o r

follows

{{E ~ as a c o n s e q u e n c e

by

class

that

~'~"

and c o n s i d e r the f a c t o r space

denote by

(2.3)

it

The s p a c e o f t h e e i g e n s o l u t i o n s

composed o f t h e f u n c t i o n a l s M~

of the theorem,

Let us denote i t by Y ~

space

by ( 2 . 1 ) ,

Let

follows. Remark.

factor

Therefore

i[ : i[ v~4 l} .

9 Then we h a v e , follows

a linear

holds

~.

{{~ , ~~ {{

we h a v e t h a t ,

for any solution

of (2.1),

the

-14-

(2 ~

II C',Y]II _-- i< Ii~il.

Inequality

(2.4) rill

be c a l l e d

the dual i n e q u a l i t y

of ( 2 . 2 ) .

It is evident that in order for the existence principle t o be u s a b l e i t i s n e c e s s a r y t h a t tile k e r n e l

V'!

be c o n t a i n e d i n t h e k e r n e l ~ 4 ~v~= 0

(2.5)

of ~ ~,

~

f o r any

~

~

M~r

~

V 4

-0

M~

must i m p l y t h a t

, the given vector

and t h e f o l l o w i n g n e c e s s a r y c o n d i t i o n

~o

for the

must be s a t i s f i e d

M 4 IX~ )

'= 0

for every

We w i s h now t o g e n e r a l i z e and s u f f i c i e n t

of t h e t r a n s f o r m a t i o n

M 4 , i.e.

. In the opposite case, i~

c a n n o t be c h o s e n a r b i t r a r i l y existence

of

j u s t proven

condition satisfying

theorem 9 . I .

for the existence the condition

~X, ~ ~f~.

and t o g i v e t h e n e c e s s a r y of a s o l u t i o n

(2.5).

L e t us c o n s i d e r t h e c l o s u r e of t h e l i n e a r Banach f a c t o r

space ~

= ~

class~W]of ~

i s g i v e n by

of e q u a t i o n ( 2 . 1 )

/M~,.

variety

As u s u a l ,

M4 ( ~ r )

and t h e

t h e norm of an e q u i v a l e n c e

II [ ~] II M~

L e t us d e n o t e by

thst

range in the space of the space

2.II. solution cons t a u t

(2.8)

the linear brings

transformation ~

T i t h domain ~ r

into the equivalence class

and [ M4~r ~

The f o l l o T i n g theorem h o l d s :

~t"

A n e c e s s a r y and s u f f i c i e n t of ( 2 . 1 ) , exist

f o r any

condition

satisfying

such t h a t

II M 4

Ii

v, II

ii ~

for the existence (2.5),

is that

of a

a positive

-15-

This theorem, be c o n s i d e r e d known t h a t ,

which appears

as a p a r t i c u l a r if

i s an

as a g e n e r a l i s a t i o n

case of that

element of

~~

of theorem 2.I.,can

theorem (l). then it

)

In fact;

it

is

admits the following

r e p r e s e n t a t i on:

(2.7) where if

is

~

.~.

an e l e m e n t o f

~4

is such a functional,

vanishing

then (2.7)

identically

defines

on

M (K~Conversely,

an e l e m e n t o f

Q~

9Bec

ause

of the condition (2.5), we may write (2.1) as follows

where

~

is

defined

by ( 2 . 7 ) .

we c a n a p p l y t h e o r e m 2 . 1 .

m~,t,"

Since

=0

is

In fact,

evident

let

introduced ~ 0

then there

exists

(a.8)

(1.5),(1.8o)

a constant

equivalence

considered

in the spaces

Inequality

(2.8)

equivalence

class

6t:

the fact

K

{} { ~ ] {I ~ is the

expresses

= 0

I

of (2.6)

a particular space ~

that

and (2.6)

~

is still

case of theorem 2.II.

is the linear 6

(2.4).

9 6~

variety

"Vr

. Then t h e c o n d i t i o n

holds.

i s well posed in the spaces

S~ , 5 ,

such that

holds:

the dual inequality

~ li.~il.

class ~

is

the vector

1 and that

simply expresses

I f the S . V . ~ .

[~

how t h e o r e m 1 . I

us assume that

in lecture

~qY

T h u s , we g e t t h e p r o o f .

S i n c e i i ~ II : {{~{{ , t h e d u a l i n e q u a l i t y It

implies

o f (weak) s o l u t i o n s

, S

the fact

of the solutions

o f t h e B.V.P., when

9 that,

when a B . V . ~ . i s w e l l p o s e d , t h e

depends continuously

on t h e

given data

of the problem. The c l a s s i c

Hadamard p o i n t

of view in defining

a well posed B.V.P.

-16-

assumes

existence

data

From o u r

.

consequence

and continuous approach

of the

it

dependence

follows

of Lecture

N.lYJNDFORD-J.SCH~kttTZ - L i n e a r New Y o r k - L o n d o n ,

C 2 .]

G.FXCHHtA - s e e [ ' I ]

, /'2]

G.FIC!t~;~A - 0 p e r a t o . r i Istituto

[4]

Cell.

condition

is

a

, [3]

, in Lecture

di Riesz-Fr_edholm, Batematico

Pabl.

Publ.

Inc.

I.

operatori

Guido Castelnuovo

riduci_b.i.li,

etc

-

-Roma,1964.

A.n.a l T s . i s ' .and S e m i - G r o u p s

- Amer.Math 9

31 - P r o v i d e n c e , 1 9 5 7 .

1953 - English

A.C.ZAAI~t~ - L i n e a . r . J U l a l y s i s

- Interscience

1958.

F . I ~ I E S Z - B . S z NAGY - L e m o n s d ' A n a l y s e Budapest~

[6 ]

second

on the g i v e n

2.

operators

E.tIILLE-R.I~IILLIPS - Functional Soc.

[5]

the

solution

first.

Bibliography

[1]

that

of the

-

North

fonctionelle

- Akademiai

Translation,

Ungr,a

Holland

Publ.

Kiado,

Nev York,1955.

Co. Amsterdam,1953.

-17-

Lecture

3

0

The

Let

~

function

spaces

~

and

be a bounded domain of the real cartesian space X ~. Without

any loss in generality, we may assume that square

Q

H~.

: {x

I~ H

~

is contained in the open

9 Otherwise9 we may use a proper change of coordinates

in order to reduce to this case. Let us consider the vector space C "~(A)

[ C ~ (A) ] of co~ple~

~-vector valued functions, ~ d

for ~ y p~ir ~

of vectors of this space let us define the following scalar product

(~ ~ )

(S.l)

We d e f i n e s p a c e s

m~ ( ~ )

{H

(A) ,t or s i m p l y (when any confusion can

t o be t h e H i l b e r t f u n c t i o n a l c o m p l e t i o n of

~(~)

f u n c t i o n s p a c e s o b t a i n e d by

iC~(~)]Tith

r e s p e c t to t h e s c a l a r p r o d u c t

(a.l) o

I t i s obvious t h a t f o r ~ = 0

H (A): Ho(A)--J-.:2(A).

we h a ~ o

In general, a function

~

will belong to

H,~

(l"l~,)

i f and o n l y i f

o

such t h a t

"~

The f u n c t i o n ~

~

JAI~(~)-~(x)I

~ ~(x)

dx =0

and, m o r e o v e r , t h e r e e x i s t f u n c t i o n s

- which o b v i o u s l y does n o t depend on t h e sequence

- is called the strong derivative

~

of t h e f u n c t i o n

~

0

H.~

( i-1~ ) , o r , s i m p l y , t h e It

is

integration

easy t o s e e t h a t

~-derivative

f o r any

by p a r t s f o r m u l a h o l d s :

~

gl.~

of

~

and any

. ~/~ ~

(A) the

of

-18-

(3.~) In

fact (3.2)

and

~~

then,

holds

when we r e p l a c e

~

by

~

and

~ are the above mentioned sequences with

making

t< t e n d t o i n f i n i t y ,

ve obtain

v~

%

by

~"K ( t'lPCK

6 ~'w'(~)

) and

(3.2).

0

Any f u n c t i o n trigonometrical

~

of

~J

(~)

4

~

c a n be d e v e l o p e d i n a F o u r i e r

series L~x

(3o3)

~(X)

-

C

,< ,<

~

e.

(2~,T>" "'~ where CK :

the development being convergent

~ (x)

D

:

(, ~ .

IA ~

4 in

fi~ >" - -

~ -bwx ~

~

(A)

~

X For J~ I ~ h~t we h a v e a l s o :

.

~.),~

(

dx

A

and then, using ( 3 . 2 ) : (3.4)

D

..(x)

,>".,~

:

(6)

K

c

e

.

o

T h i s means t h a t differentiated

t~ 6 H ~

, the Fourier

t e r m by t e r m p r o v i d e d

does not exceed

There exist

for any

~

the order

series

(3~

c a n be

of the differentiation

9 We h a v e "

tTo p o s i t i v e

numbers

Po

and

~4

such that

~

~ )

-19-

Therefore,

(3.5)

p~ ~

e. I~1

_"

IC~l

9t" ~ a

JAID~'" I*dx

)""

<- p,l) ~. IKI

I CKI.

0

Let us denote The f o l l o v i n g

II t,l, ll~

by

the norm of an element

,leLma o f t h e t ' o i n c a r b

inequality"

H~

of

(or

bl,l~' )-

holds=

0

3.I.

For any

(3.6)

t~ t~ H

II i,,i, II

~

'w%

(.A)

C 7

ID

I

x.

I '~ I =,vii, The

constant It

is

evident

by i n d u c t i o n ,

ICoi :

/

depends onl~ on

C

the

that

general

follows

suffices case

IJ ~ . d * ,~

(~r,) "" 4

:

9

to prove

follows.

(:2.ar ) 't

(3.6)

for

:~

, and t h e ~ ,

We h a v e :

ca.

A

IA I bu.I ~d~ IAI #'12dx

9

v. a-

that:

2.

II t,. II

~

(4i

4

( 2 . ~ ) "~

It

it

~

:

x

.+

I D I'l' I ~ d

t

/v.

+"

IC K I

Pt )-""i< IKI

-c:~

"l

ICol'

+ (4i'1~)

-~

Keo

IKl~'lCv

I" ~ r~

f-

t~.j; ),..

,,a.

I X I z d x .I-

A

-

P<,

A

rD I,t,,< (~) D t b d e n o t e s

the

~•

matrix

9

/'Dx h

,.,

) -

=

-20-

From ( 3 . 6 ) and ( 3 ~ 3.Iio

The norms

it

f o l l o w s that-"

[l ~ ] l ~' and

~

va

the space

H ~

'et

9 f o r any

o,

dom~n

with

t~

are isomorphic in

IC~I

o

us now c o n s i d e r ,

~th

IK I K

~

~

H~ ( ~

C ),

the operators

o

PI ~

and range in

/t,,~

Hr.

(~th

~, < ~

) tha~ a s s o c i a t e s 0

t h e same

~ , b u t c o n s i d e r e d as a v e c t o r o f

H~

. I% i s o b v i o u s

O

that

~

i s a bounded l i n e a r

transformation of

O

~

into

O

t r a n s f o r m a t i o n w i l l be c a l l e d t h e embeddin~ o f

~

. This

O

a~

int____oo H~

9

O

3.1n. ~h~ embeddin~ U ~

o~

H~ into

~t ( ~

< ~

) i~

c ompac t 9 0

Let

U

such t h a t 9,- ~

~

~ I ~ i

I C K I

I

and t h a t

~

[-,

9 Then a p o s i t i v e f o r any

~ If

<

varies ~

is reKular

L~

in

~

.

the last

It

~

exists

follows that

series

is uniforwly convergent (~) This p r o v e s c o m p a c t n e s s o f ~ 6 .

i s a bounded c l o s e d domain of i f t h e Gauss-Green i d e n t i t y

; (~

The f u n c t i o n

~

belongs to

T

f o r any

~

T

6 C~(T)

V,

Q = (Vt, "~ 9z ) i s t h e e x t e r i o r point of Q T

X ~ ~ we s h a l l s a y t h a t

holds in

"T

finite

w 6 ~,

constant

~

-~"

when

H~

be a bounded s e t o f

u n i t normal v e c t o r t o

~T

i n any r e g u l a r

i s t h e measure o f t h e h y p e r s u r f a c e e l e m e n t on / ~ .

i s s a i d %o b e l o n g t o

~ ~-4 (.[)

D ~ ~T )

and t h e r e e x i s t s

(.~

>o ) whenever it

a decomposition of

number o f n o n - o v e r l a p p i n g r e g u l a r domains

~

, ....

~

, T4~

into a (3)

(%)We h a v e been u s i n g t h e f o l l o w i n g lemma- I f 5 i s a s e p a r a b l e H i l b e r t s p a c e and ~%r~,~ a c o m p l e t e o r t h o n o r m a l s y s t e m i n i t ~ t h e s u b s e t l / o f 5 i s compact i f and o n l y i f t h e s e r i e s ~ I(~ %~)12is u n i f o r m l y bounded and u n i f o r m l y convergent in i; 9 (Se~ [ ~ ] ). (~)By "decomposition of T in a finite number o f non-overlapping domains Tt, . .... T,~ " we mean t h a t 7 : 7 4 4 ..... uT.~ and (.T;.-"bTL)n (Ti-~T,~) for L ~ i

-

-21-

~ e C ~ ( T C ) ( : : 4,..., ~ 1. When t~ 6. ~ ( T )

such that say that

~

has piece-wise

to prove that the subset

~pt ~

~ T

of

c

T-~T

set

~A

@1

for any

the closed

" ~ A

, and ~

c ~A

Xo

there

functions

b~

D ~ (T)

such that

regu, l , a r d o m a i n i f

a neighborhood

the following

, IW I ~- d

fJA n ~

~y

follows

matrix

that

has properties

H ~

If

A

( ~

I~ , . . . , Iq

in the condition

I: 14 u....u I~

) is

is homeomorphic to

~ = y (x)

be a n y o p e n s u b s e t

function

to those

be a f i n i t e

which maps .T

set

f o r m an o p e n c o v e r i n g with non-negative

y =

~/ ~ 9

~g = 0 )

homeomorphically

y(x>~ D'C Y ) ]

which is

bounded

set

9

boundary

Let

A-I

~-I

~

.~

o_~f

H~

principle). like

OA

of

those mentioned A

*

Let

be n o n e m p t y ( o t h e r w i s e is contaized

.

onto

y(X)o

selection

A-I

+

w h i c h maps

the embedding

the set

containing of

determinant

of neighborhoods,

cover the

The c l o s e d

of

[ i.e.

• : X (y),

of

regular,

and s u p p o s e t h a t

the proof is simpler1.

space

i s mapped o n t o t h e s e t :

compact (Rellich

), that

Xo ( o p e n

9

is properly

> ~

of

of the cartesian

has a positive

the inverse

analogous

~

~ -- ~ ~ ~

B + has piece-wise continuous f i r s t derivatives

3.IV,

unity

exists

the vecto~-val~ed f u n c t i o n

It

Let

we d e n o t e by

regular;

the set

C~§ " Ym >- 0

~ Qx away f r o m z e r o i n t h e w h o l e J

into

is

), such that

semiball

and t h e j a c o b i a n

,

o

if

c H ~ ( T - ~ T 1,

properly

~

homeomorphism t h e s e t

~1

of those

~ ~(T)

, then

It is not difficult

derivatives,

are satisfied:

contaihing

onto

consisting

A i s s a i d t o be a

d,)

In this

,~h

(T-~T), M o r e o v e r

) c i~

~ ~ (T)

A domain hypotheses

continuous

we shall

The s e t s

So /

in

I4

A.

Let

) "~ ~

I

i ~ (x) r ~ be a p a r t i t i o n of ~:4 h C ~ functions such that ~t ~hLx) (h=~...;~)

~

-22-

i s c o n t a i n e d i n one of t h e s e t s of t h e above c o n s i d e r e d c o v e r i n g . L e t ~ ~ ~ ) ~'e have

~ : 2_- ~. ~

and ,~ •

Let .$~o~ ~9h C i $(~)o Then i f ~f i s a bounded s e t i n of functions

~b ~

( ~ ~ (/)

. e have p r o v e d t h a t , we can e ~ r a c t

for

i s bounded i n

~/~

a subsequence

Ht ( ~ ) , the se~ ~/h

ki 4 ( I ~ ( h ) n

~ )o

bounded, from any sequence i~w

t such t h a t t ~

~v.

Suppose

~ ~

t

I r

~-

is convergent

~o (~ ~{h)r~ /~ ). Then we can o b v i o u s l y suppose~ t h a t ~d~h b~.~ i i s 7. ~ h b~~ v i i i be c o n v e r g e n t c o n v e r g e n t f o r any ~ Hence t ~ ~ ~.~ in

:

9

in

~i~ ( / ~ ) .

to general

,~

This p r o v e s t h e t h e o r e m f o r alad ~ ( ~ w ~ )

Compactness of f o l l o w s from 3 . I I I

is trivial.

~fh

in the space

since in this case

greater than zero 9 Set, for simplicity, set

~ (~)

~ k x (y)]

~l~

~f~ C ~(h)

when ~1~ ~ ~ o ) ~ L e t : ~ " For any

~e have a l m o s t everywhere on

Fa~

It

~*~ = d ~ ~ : O. The e x t e n s i o n

~

"

~ax h

follows that:

x +

l

ly J

L h

1i4 II

/~
be

~ ~ ~1, ~

~)

-23-

where

II

]1

has an obvious meaning.

L e t us no~ assume ~ (y) : ~ , define in the closed ball

The f u n c t i o n

b~

C~)

"

Then

iI Lj z4

-

subsequenc e

~bs

(theorem 3.III). and t h e r e f o r e

~

[x(y)]~ [x(~)iI

= l yi ~- ~

b e l o n g s to we

t~ ~

This means t h a t

~~

~

~

. We have

from any sequence

t such t h a t

that

the folloving

~1 ( ~ )

can e x t r a c t

t

function.

il~ e t t ~

is convergent in t is convergent in

~ is convergent in

B,iblio~raphy

~ i t h ~ ~ ~T L e t us

of

Lecture

=~ ! ~

II~ lt~,~,

-

a

~I~ L ~ ) Ho ( ~ * )

~o (I~j.

3

R.COURANT-D.HILB~RT - Methoden d.er. _Ma.themati.s.chen P h y s i k - v o l . I I Springer, G.FICtI]~AG~

C~

-

Berlin.

see

~1]

in

lecture

1.

see

~2]

in

lecture

2.

~iultipl.e i n t e g r a l .and r e l a t e d

problems i n t h e c a l c u l u s

topics

S.L.SOBOLEV - A p p l i c a t i o n s

- U n i v . of C a l i f .

of v a r i a t i o n s

P u b l . i n Math.neT e e r . 1 9 4 3 .

of Func.tional A n a l y s i s to M a t h e m a t i c a l

Physics - Leningrad,

1950.

-24-

Lecture

The

Let linear

trace

~

4

operator.

Sob..o.lev

U~

N~rl.in~

be a p r o p e r l y r e g u l a r domain o f

t r a n s f o r m a t i o n which t o any

values

and

on / ~

~ Set for ~

le~as.

~ ~9 L e t ~ d e n o t e t h e

~ e C ~ (/~)

associates

its

boundary

~ t :

we want t o p r o v e t h a t :

II ~: ~- il

(4.1)

_L

c Ii u. II ~

w h e r e C i s a c o n s t a n t d e p e n d i n g o n l y on ~ 9 in the case

/u~= ~ . For t h i s ,

l e t us c o n s i d e r t h e open c o v e r i n g ~o~ l ~ , - . . ,

a l r e a d y i n t r o d u c e d i n t h e p r o o f of t h e o r . 3 . I V , partition with

of u n i t y

~(~)

(4.2)

> 0

I

~. ~ ( x ) =

I t i s enough t o p r o v e ( 4 . 1 )

A. Let~esuch

and t h e c o r r e s p o n d i n g that

~t

~_

A

As i n l e c t u r e

~

:

cartesian Let

~

image o f

~n

~(~

./"~ X

R:a

/aA

maps

C

, t h e n (4.1) w i l l f o l l o w from t h e i n e q u a l i t i e s :

( % ~ I~ d~

-g--

~

3 we set,

: ~ ~(~)

and c o n s i d e r t h e homeomorphism t h a t

--

A

onto

(y t ) - s p a c e

be t h e ~ ~ ~A

~~.

We s u p p o s e t h a t

d e f i n e d by

(~-4)-balls

t =0

t 30

'

..~+ i s t h e s e m i b a l l o f t h e

~ +t z ~4. 7 / ~ ' ' " f Y~.~

, i~l 2 -= ~4 V~z L ~

u n d e r t h e homeomorphism o f

~

onto

, ioeo the o Let

-25-

be t h e f u n c t i o n c~ (x)~(;~)

when c o n s i d e r e d i n

~

, by means o f t h e

above m e n t i o n e d homeomorphism. We assume ~ (ty)=- 0 f o r t ~ O ,~+ lyl >4.We have f o r any

t;,O: t Henc e ,

JDIJCv,o) i~01~ D It, f o l l o w s t h a t

From t h i s ,

returning to the

•

~---,~

coordinates,

(4.2) follows;

(4.1)

i s proven.

From ( 4 . 1 )

it

follows that the operator

e x t e n d e d t o t h e whole s p a c e ~

=- ~ ' ~

H ~ (A)~ Then f o r any

( 0 a t ~ i ! /Wt- A

t if

U

~b ~ C ~ ( ~ ) .

t~

~ F-bf~

and

"~ F ~rill be c o n s i d e r e d t h e boundary v a l u e s o f ~ P ~

t i o n of

~]~ C A ) v ~ i s h e s

any d e r i v a t i v e

4.l. ( 0 ~_ ] ~ l

of o r d e r

If

~ ~ ~

be c o n t i n u o u s l y a set of vectors

) i s determined such t h a t

and

sense. It is obvious that

Z

call

~,) implies

in a generalised

T~ : 0

, i.e.

(in a generalised sense) on rM

a

~ ~ C ~ ~ ) ~ H ~ CA

and ~

any f u n c -

together ~ t h

~_ ~ - ~ ,

-~ ~ ' ~ ) i n t h e c l a s s i c a l

.,~)

w i l l be c a l l e d t h e t r a c e - o p e r a t o r

f~

then

~ p~A. : 0

sense.

The p r o o f i s o b v i o u s .

If

@ P ~ ~'{

a r e two f u n c t i o n s o f

t h e Gauss-Green f o r m u l a h o l d s :

A

one'

-26-

where in

~

~ ('~,~") and ~

is a b i l i n e a r

respectively.

From ( 4 . 1 ) i t

and ~r a r e f u n c t i o n s ~

and

~r

generalised Let

o p e r a t o r of o r d e r

folloTs that

(4.3)

still

holds if

~1~ (A) ~ p r o v i d e d t h e b o u n d a r y v a l u e s o f ~ a r e u n d e r s t o o d i n t h e above i n t r o d u c e d

sense. ~ ~ be any p o i n t of

n u m b e r . The s e t

~ )

every point

X~ o f

~

(~ F

a " c o n e s*.

The domain ~

of a l l

sphere

s p h e r e i xi : ~ . L e t

~

and ~

a positive

such t h a t

o z

~o

~

there A

ix-x~

is the vertex

is said to satisfy

Thich is contained in

exists

of the cone.

a cone

C•

, and i s s u c h t h a t

the reader prove that

~

t h e "cone h y p o t h e s i s " i f

~o is congruent to a fixed set We l e t

the unit

X ~

-----Ix-x~l

be c a l l e d

~ ~ and ~

m e a s u r e on t h e u n i t

Cxo ( ~ X-

and

of

~ 0 -~ ~I z ~

be a s e t of p o s i t i v e

will

differential

~

('IxO , R ) o f ~

for

vertex

~.o

i s i n d e p e n d e n t o f X~

on ~ .

a p r o p e r l y r e g u l a r domain s a t i s f i e s

t h e cone h y p o t h e s i s . 4.II.

( S o b o l e v 1emma.). I f

f y i n g t h e cone h y p o t h e s i s , _

,

continuous

(4.4) Ti t h

~a~ G

i s a bounded domaim o f X

the functions

in

(i.e.

I ~-I

d e p e n d i n g o n l y on

It is

/~

z

c

o f an y s p a c e

H

satis-

H , ~ (/~) ) w i t h

(A)c

il ~ Ii

~ .

enough t o p r o v e t h e i n e q u a l i t y

(4.4) i n order to have t h e

proof of t h e e n t i r e lemma. Let satisfying

X ~ be any p o i n t

of

A . We d e n o t e by

@(x)

the following conditions. : ~t

for

I X - X ~ I /-- Z

-

for

i x-

q~(X) o

a

C ~

function

-27-

We have= R

o

/ag

.-.,"1

Integrating

over

I•

on t h e u n i t s p h e r e , we o b t a i n :

d• A.*

~

O

L

._

x

C,~-4 )!

The l a s t

~

o

-- ,~,)

integral

is finite

.

C~o(U, g,) since ~

~ ~ . Thens

/

9

(~,,-a)!~,,~sF Thus,

dx

\ Jc o(i" a)

( 4 . 4 ) h a s been p r o v e d . Let

as v e c t o r

~

'

spaces,

-~2 '

~ 3 be t h r e e complex Banach s p a c e s and assume t h a t ,

they satisfy

the following inclusion

conditions:

b~ c b~ c B3 L e t us assume t h a t t h e embedding

~,~ea~ing ~ ~ exists

a positive

~a z

of 8 4 i n t o ~ a i s compact and t h e

of ~ i ~ o ~ i , oo~ti~uou,, t o , ~ y ~ , ~ conltant

C(s

such t h a t

for any

~ ~ ~4

~ "0 ~ ~ :

-28-

L e t us suppose t h a t ( 4 . 5 ) i s n o t t r u e f o r some ~ >0. Then f o r any positive integer

~

Set

~e haTe~

~

: . . . .

II~ll

(4.0)

, t h e r e must e x i s t some

> ~ 9 .~ I I ~ l I B ~

)

I t f o l l o T s from ( 4 . 7 ) t h a t I I ~ l] L~I

a subsequence

~

c o n t i n u i t y of

~s

But t h i s c o n t r a d i c t s 4.III. For an_y

o~

~~

~s

[I ~

II~. > ~

[I L%~ ~

to some ~

t converges to ~r i n

: ~

~3

0

On t h e

, we can e x t r a c t 9 Because of t h e Then

~ = O,

~hat follows from ( 4 . 6 ) . d o m a i n

t h.ere e x i s t s a p o s i t i v e

,~,d ~

iI ~

~

~ i s compact i n

( F i r s t Ehrling le~na). Let

~ >0

~ , A

~ r

il~ll.~

i s u n i f o r m l y bounded w i t h r e s p e c t

I converging in ,

such t h a t :

(4.7)

to ,'~ . Then, from ( 4 . 6 ) , i t follows t h a t o t h e r hand, s i n c e t h e sequence

~

) such t h a t for ~

~

be any ..bgunded

constant

~ ~

j~

c (~)

of ~

(depending only

CA)the

follo~ng

i nequal.i.ty h o l d s :

(4.s)

11 ~ II

L

~

II ~ l l

~.

~he lemma f o l l o T s as a p a r t i c u l a r

c(.~) il~IIo 9 c a s e of ( 4 . 5 ) , by u s i n g t h e o r .

3.111.

4.IV. domain of

(Second l~lrling l e n ~ a ) . Le~t A ~g,

For any

(depending o n l y on the inequality

E>C

~ ~ /1 and

be an~ p r o p e r l y r e g u l a r

there exists a positive constant

) such t h a t f o r ~ y

c (s

~ ~ H~(A)

(4.8) holds.

The lemma follows as a p a r t i c u l a r

c a s e of ( 4 . 5 ) , b y u s i n g t h e o r . 3 . I V .

-29-

Bibliography

[lj

G. HtRLING - On a t y p e o f di f f e r e n t i a l

of

ei~envalue operators

6 . FICIH~%A- S u l l ' e s i s t e n z a problemi

e sul

al contor~o)

c orpo elastico vol.IV)

Lecture

4

problem for certain - ~ath.

calcolo relativi

elliptic

Scandinavica ,vol .2,1954.

delle

soluzioni

all'equilibrio

dei d i un

- Annali Scuola Norm.Sup.Pisa,s.III

1950.

C 3 ]

ft. F I C H ~ A - s e e L 2 ] o f l e c t u r e

[4]

L . NIR~B~I~G - Remarks on S t r o n g l y Equations

1.

Elliptic

Partial

Differential

- C o m m . on p u r e and a p p l 9 m a t h . v o l . 8 ,

NeT Y o r k , 1 9 5 5 .

S . L . SOBOLLV - On a t h e o r e m o f f u n c t i o n a l

N.S. 4) 1938.

analysis

- Mat.Sbornik

-

-30-

L e c t u r e

Elliptic.

Let

,,syste.ms.

Interior

A be a domain of X m9 Suppose t h a t t h e

~(•

~ i ~ [ _~ ~

differential elliptic

1.inear.

5

) are defined in

~

.

, regularity.

~X.~

complex m a t r i c e s

Consider the linear matrix

operator

J . ~ -: ~ ( X ) ~ , This o p e r a t o r i s s a i d t o be an 4 i n /~ i f ~ f o r any r e a l non z e r o ~ - v e c t o r ~ ~the

operator

following condition is satisfied:

i'~i- v

xeA

at every point

Examples: i )

If

~=

A

. . . .

Q• elliptic

i f and o n l y i f

the operator

L

~ ~ (• 4

of the interval

L

A

and

~ = A

,

-I- ~ ( •

have t h e l i n e a r In this case

~ _~ .~. I n t h e c a s e ~-~- ~ 0%(•

dx

of t h e r e a l a x i s .

O,,.(X)@X,- +G,z(x/"#X.z+(:t~

For i n s t a n c e ,

, we

I.

may be

~= A , e l l i p t i c i t y

means ~4(X) ~ 0 In the case

is elliptic

operator

a t any p o i n t

~ : ~,

onlyThen

for

~

t h e Cauchy-Rieme~n ( o r W i r t i n g e r ) o p e r a t o r

the operator

O,,(X)~(X'#~'O, x +~ y

~x-~/~

is elliptic 9

( t ) A more g e n e r a l d e f i n i t i o n of e l l i p t i c D o u g l i s & N i r e n b e r g (see ~'4 ] , [ 1 0 ] ) 9

o p e r a t o r has been g i v e n by

)

-31-

ii) :

~= ~

If

.,

,

~:Z

, t h e l i n e a r o p e r a t o r i s as f o l l o w s :

~ b.

.+ c

~ <2• '9 • are real,

.

If the coefficients

~,,~')

QX"

L~ i s

elliptic

i f and o n l y i f t h e q u a d r a t i c

O~Lj

form

(x')/~; ~'K

is definite (positive or negative), iii)

Assume ~ : 5

classical

elastostatics:

this

and c o n s i d e r t h e d i f f e r e n t i a l L : bu&/kk + ~ a / & k .

operator is elliptic In lecture

f o r any

o p e r a t o r of

It is easily seen that

~ @ -4.

1 we c o n s i d e r e d t h e weak f o r m u l a t i o n of a g e n e r a l BVP.

The p r o b l e m c o n s i s t e d i n p r o v i n g t h e e x i s t e n c e o f a v e c t o r

~3

-space

~

satisfying

the system of i n t e g r a l

L ~'

(5.1) f o r any

17 6 V

~

9

or i n t e g r a l ) ,

equations:

L e t us s u p p o s e t h a t

B~," B~ " ~ ~CA )" Then ( 5 . 1 ) can be , , ' i t t e n

I f the "boundary o p e r a t o r s "

of the

~>

:

and f o r a g i v e n

~

as f o l l o w s :

a r e a c t u a l boundary o p e r a t o r s ( d i f f e r e n t i a l h ov contains the linear variety ~ ( A ) . No m a t t e r

then ~/

M

what t h e boundary c o n d i t i o n s a r e , a weo/~ s o l u t i o n o f any L"r and unkno~m f u n c t i o n i n

I~(A))

must s a t i s f y

(~ith datlm

t h e s y s t e ~ of i n t e g r a l

equations (5.2). We have t h e f o l l o w i n g i n t e r i o r 5.I

i~

A

If

"

and

satisfying

A

~(.)

regularity

i s i any domain o f

~ C~~

( 5 . 2 ) f o r any

,

~r 6 ~oo

theorem for elliptic

X~

~x)

'

L

i s an e l l i p t i c

~ C~(/~)

with

@r~ %Y6 A

Lo(.,~

.~)

:

belion~s %o CC~

Z a.~(~)D ~ I.~1: v

operator

, then ant f.~ction ~2

Set

systems:

I~i:o

-32-

L ~o ' Cx,~) : (-~)~Z: g~ (• D I~1:~ emy l o s s

We c ~ n s u p p o s e - w i t h o u t

(G:

I•

hall,

with center

be t h e

~).

Let

h&ll

function

th&t is

equal

outside

i~

: ~(•

9 Let ~ ~•

components.

•176be ~ n y p o i n t of

• ~

concentric

~

and radius of re~ius to

~ ;

in generality

i

~ ~ ~

~ Tector

of

r, o

~

different

by

~(X)

complex

: ~(x) L" (• ,O

The %X ~

matrices

From (5.2)

ve obtain,

~

A

e.~P_,

c~)e

X

a

C ~

real

identically Set

from zero with integer

'2-4

dx

~

g + Z-., ~ (x)~; e

.~

d•

.--Z

K

in :

~.e

CI X +

"~

L o ( x ~ C~ )- L Cx, ;.K )]'-f' ,. e--:KXclx .

A

LoCx~

V'.

- ,..K X

I'sl: o

+

' uK X

l'~i : O

- L,Ig, x

Lo(x ~ C,<) 4 . , e

A

Let

~-Tector.

(x) h~ve their supports contained s because of the arbitrariness of ~

I

a~t

9

We h a v e :

L" (•

Since

be a c l o s e d

&nd T a n i s h e s

conste~t

' r

~-

Let

We d e n o t e

~t any point

is

.

, contemned in

,

be an a r b i t r a r y K

A

~cQ

- that

o ( L is ellipticl),

i'~I : o

ve hate-

A

A

~'

-33-

§

(,x~, ~")

Lo

I

Lo (x ~> ~K)- Lo

-~,Kx

' (x,~K)

q)~,e

dx.

Set=

Lo(X~

K) : Z

.

, x ~ ( x ) K "~

q~....TJ.

)

I~,I = ~

C,

Let

be a non v a n i s h i n g

e

-,I I~.1

constant

real

-vector,

We h a v e :

I D" c- Z,Kx dx : A t

-uKX

K~

:-Z~ Ic, i -o

L o (x ,~,~

(x) ~ e [~I

dx +

A

(5.3)

L~K

4-

Lo(X , OK)

q~

e.

I~.I

4-

ILl

Let

Z K 1~1,~,

be l e s s

e,

- 4

than

Lo (x'~ ~K)

J

o ( (x.) U" Cx) e."~K'

-~ Ikl

the distance

betveen

A

and ~ ,

z~

J o((x) IJ'(x) ~- ~xd,( : Jl o~ (x §

We have=

U (x+k)-%(,()U(x)

|

-i~K%

e,

ILl

,x:

A

x+k) Q

T.)'(x+k)-U(x)

-;Kx

e.,

c~x+

I

o(,~ (x+ k )- (%(x)

Q

ikl

-~x

U'(x) e,

clx .

-34Let us denote by

(r~),

~ HI,

CK

t h e F o u r i e r c o e f f i c i e n t of ~ f

I~1~r I r ~o,,,~

+,T_.

Te

r

{

0 ( i t % (~+~.)

Since the

0

Fro~ (5oa), Te o b t a i n ( u s i n g t h e

(~f~)-th

-~

0

and suppose t h a t Landau symbol):

ii u~ Ii

+

If-.!

;) ~0 (iJ

rJ(x,k)-vo,) Ii

d e r i v a t i v e s of

o~ (~, r

a r e bounded, (c~ (X) ~ C ~

d~(x)

!)

have:

o(ll Let

G~

~ (~+~)-~ I~,1

be such t h a t ( f o r any

and a s s u m e

~$ z ~

and

E(~ II ~e ) " 0 ( il~(~)lie, I~i-

;

- s(z~f_|

~oo?, ~t-~ I (V(~,).u(~). + c__.c I~l jQ _~

)-"

It l "(G"

I~1 (5.4)

~'

Ikl

I

F ~J

,(

~

is

chosen

thatx

JQ{U(x+(X)l_~ ~,)-u e"~KxdXl~'+{

_;~ e

such

)

-35-

where C i s a p o s i t i T e c o n s t a n t . This c h o i c e i s p o s s i b l e s i n c e we h a v e :

Ii,x~(•247 T,,r(.,~,)-r.)'(,~) II z c~ L I~1

?" -

dx

IKI:, ~, Q

(Ix,

.-i- C z ,,~ IKI~ {,

where

CI

t

jkl

and

C~ a r e p o s i t i v e

hounded f u n c t i o n s .

c o n s t a n t s and

Then, by u s i n g ( 3 . 5 ) ,

i~,$~: ( X + ~ ) I

the inequality

are

(5.4) follows.

From ( 5 . 4 ) , we get: ~oO

I~1

u(.,~)-u(~) lit, ,. Then, from ( 5 . 3 ) i t

-~ - o o k

+9

i~
l,(

ic,i

:~ §

0( i1~.11

r )

:~0r

el'

e.

dx

By c o n s i d e r i n g t h e f o l l o w i n g s u c c e s s i T e c h o i c e s o f

~ (O)t)-.-,

I

follows ~hat:

ikl

7.

-4

e..

0 ),

.oo ~ -: ( 0 0

...~ ~ )

.

~

,

~ =" ( t ) O ) . . .

end meLking t - > O

we o b t a i n :

,0) j

-36-

-~:~

1.~

We have so f a r p r o v e n t h a t i f and c o n s e q u e n t l y proved t h a t circular

~

for any

~ H

•

neighborhood

t§ A

~

is a solution classical

(['~).

)

then

~

Since ~ s ~ A ~ )

and any p o s i t i v e of

From t h e S o b o l e v lennna, i t

H L (l" 2 ;

~ 6

X~ s u c h t h a t

follows that

of t h e d i f f e r e n t i a l

~

~ ~+~

([~)

Ho (A) s we h a v e

:

integee ~ there exists

H~(Z),

~ belongs to is

~

equation

a

in

, and t h e r e f o r e

~ ~ =

in the

sense.

I f we l o o k back c a r e f u l l y

a t t h e p r o o f of t h e t h e o r e m , we s e e t h a t

t h e arguments we u s e d h a v e p r o v e n t h e f o l l o w i n g more g e n e r a l theoreml 5.II.

A

i s .any domain i n ,the s p a c e

i s an e l l i p t i c

matrix-dif.ferential

(

,~,

~ ~ 0 -~

);

~ ~ ~ O }.

libt il "~ for ant G

L

•

depends o n l y on From t h i s

the interior

~

L

theorem i t

regularity

+

C ~(A),

({)Now

in the sclueI

and

follows,

l[

~

FJo ( ~ )

)

L b~ = ~

~A-.eak

Ii ~ i n s t e a d 0

C~ ( ~ )

of

~

.

(I)

The const..ent

C ~ (A)

solution

II

II

if

such t h a t

by u s i n g t h e S o b o l e v lemma, t h a t

t h e o r e m h o l d s ~llen

read

~ C s247

• o

means t h a t u n d e r t h e s e h y p o t h e s e s a n y belongs to

a

II t, Ii ~"

~ .of t h e e q u a t i o n and on

~(X)

D 5

~4-~ # s (A) (to

there exists

c., ( il ~ II ~-

4 A -weak s o l u t i o n

L = (~(•

o p e r a t o r ,such t h a t

a f u n c t i o n belonging to For any

X~ ;

if

of

, with

L ~=

~r~ ~ o .

,~

-37-

Bibliography

C1]

of

Lecture

5.

F . E . BR0WDI~t- The D i . r i c h l e t probl.em f o r l i : n e a r e l l i p t i c o f a r b . i . t r a r ~ .even o r d e r with v a r i a b l e

e,~u,a t i o n s

coefficients

-

Proc . N a t .Acad . S c i .U .S ~ 9 ~vo1.38, 1952. F~

BR0WDHt- A s s u m p t i o n o f .boundar~ v a l u e s and t h e G r e e n ' s function

in the Dirichlet

elliptic

[3]

equation -

problem for the general

Proc.Nat.Acad.Sci .USA,vol,39,1953.

R. CACCIOPPOLI - Sui t e o r e m i d i e s i s t e n z a

d i Riemann - A n n . S c u o l a

N o r m . S u p . l ~ s a ~ 1939.

[4]

A. DOUGLIS-L. NIRI~BERG- I n t e r i o r _ _e_s_timates f o r e l l i p t i c of partial

differential

Mathem. , v o l . 8 ,

[5]

syst.ems

e~ua~i.ons - Comm. P u r e A p p l .

1955.

K . 0 . FRIEDRICI-L~ - On t h e d i f f e r e n t i a b i l i t y li.near elliptic

.differential,

of the solutions

of

. e ~ u a t i o n s - Comm. P u r e

Appl ~ M a t h e m . , v o l . 6 , 1 9 5 3 .

F . JOHN - The f u n d a m e n t a l s o l u t i o n

of .linear elliptic . .differential

equat.ions w i t h a n a l y t i c

coeffi.cients

- Comm. P u r e

Appl . M a t h e m . , v o l . 3 , 1950.

[7]

F . JOHN - G e n e r a l p r o p e r t i e s partial

o f . s o l u t i o n s o f lin.ear, e l l i p t i c

differential

Theory and D i f f .

e~uations-

Proc.Sump.Spectral

P r o b l e m s , Oklahoma C o l l e g e , 1951.

-38-

F . JOHN - D e r i T a t i v e s o f c o n t i n u o u s weak s o l u t i o n s elliptic

[9]

equations -

Po LAX - On C ~ u c h ~ p r o b l e m differentiabili~y

of linear

Co~n. Pure A p p l . Math. , v o l .6 ,1953.

for hyperbolic

e q u a t i o n s and t h e

of s o l u t i o n s , o f elli~_ti__c e q u a t i o n . s -

Conml. P u r e A p p l . M a t h . , T o l o S , 1 9 5 5 .

10 ]

L . N I R H q B ~ G - s e e [4 1 o f l e c t u r e

4.

H. IrEYL - The m e t h o d o f o r t h o g o n a l p r o ~ e c t i o n i n p o t e n t i a l Duke Math. J o u r n a l ,

v o l o7, 1940.

theory-

-39-

Lecture

Exi.'stence

of

local

The aim o f t h i s t h e domain

A

there

9 he d i f f e r e n t i a l

L

exists

system

elliptic

is to prove that operator

A0 C A 9

and on

inequality

for any point

L ~ - ~

5,

ix, of

exists. (4)

(L (x)

s_tated i n t h e o r .

There e x i s t s

Ao , such t h a t ,

of t h e e l l i p t i c

5~

Let

a positive

operator

L

A o be a bounded domain

constant

co , d e p e n d i n g o n l y

~- ~ ~ oo (Ao) the f o l l o w i . g

for any

holds:

~.~,Ao

We can c o v e r

( ~ . 4)..~9 ) , t h e

5.II).

t-~.t

A~ by a f i n i t e

II ~7 il (theor.

x ~ of

L , introduced in lecture

+

any

systems.

some n e i g h b o r h o o d i n which a s o l u t i o n

Let the coefficients

the conditions

such t h a t .on

lecture

for

p r o v e some lemmas. 6.I.

satisf~

solutions

, where t h e e l l i p t i c

is considered,

We f i r s t

6

Ao

s e t of c l o s e d b a l l s ~

following inequality

"~ c ;

II

L~

II ~" II

II ~"

~,A o

~..

~

such t h a t , f o r

h o l d s (we assume I)'_: 0 i n

x\

+ I1~11

Since: q

(

I ) P l e a s e n o t e t h a t t h e example o f H~ t o an o p e r a t o r which i s n o t e l l i p t i c .

considered in Lecture 1 refers

A ):

-40-

Ii L ",," II ~-r

li L v" II z

~:

I" k

II ~" ii

,

~-v+ ~,, Ao

~ ~-

II ~ II

cl

(6ol) follovs with

Co : ~

C.

L e t us assume - f o r t h e s a k e of s i m p l i c i t y We l e a v e i t potheses,

- from nov on~ ~ ( ~ ) ~ C ~ ( A ) .

as an e x e r c i s e t o t h e r e a d e r t o r e c o g n i z e t h e more g e n e r a l hy ~

u n d e r which what we a r e g o i n g t o s a y h o l d s .

6~

There e x i s t s

only a finite

number of l i n e a r l y

independent 0

solutions Let

[

of t h e homogeneous e q u a t i o n

L tr -- 0

belonging to

be a n y n o n - n e g a t i v e i n t e g e r ~ For a ~ y s o l u t i o n

C ~(/~o)"

~r o f

L ~r - o>

o

belonging to

~oo (A.)

(8.2)

. the following inequality

I1 " II 2"

_-

Co IIv" II z

~,+~, Ao

holds:

.

e, A o 0

Let

~/

be t h e l i n e a r

variety

of s o l u t i o n s

of

L~

-" 0

belonging to C ~ (/~o).

0

From ( 6 . 2 ) i t

f o l l o w s t h a t i f we c o n s i d e r

"~/o as a 8 u b v a r i e t y of

a n y bounded s u b s e t of V o i s compact ( t h e o r .

3.III).

~

(Ao)>

Then V o i s f i n i t e -

d i m e n s i onal 9 6oili.

0

F o r an~" • ~ A .a p o s i t i v e

t h e open b a l l L~y -- 0 h a s Let ).

~

~

, of c e n t e r

only the trivial

be such t h a t

~

L e t us d e n o t e by

number

X ~ and r a d i u s

solution

'U" -" 0

.

~o e'~:ists such t h a t i n ~o ~ ~he homo~eneous~ e q u a t i o n

belonging to

~ o~ ( ~ . ) ~

C A ( ~

i s t h e open b a l l o f c e n t e r

x~

~Y4 )"" . ~ r

a complete system of l i n e a r l y

radius independent

0

solutions

of

d e n o t e by ~ ) ~

L~- o.

belonging to

t h e open s e t ~

~oo(~

Ix-x~

d e t e rmi n a n t :

Jr'z,Ilr, lZdx ~C~)

). For any p o s i t i v e

~(

~ , we

us c o n s i d e r t h e Grmnian

-41-

~e c a n c h o o s e

such t~at

~o

and is a solution of combination of

Lty : 0

independent in

Then ~Y ~_0

in

o-

belongs

~f ~

, it is expressed in

~Y~ ) "--) ~4~

are linearly

~ i.e.

0

to

C ~

as a linear

~

4)" = ~,
o~f,~o ~ t h i s

K

implies

(~,,)

K

~i~,., ~/,v,

Ct : . . , ~ ~

-" 0

~"

Assign to

6.IV.

y (t,~)"

X ~ and

I n the open ha,l,1 [ " w i t h

t h e same m e a n i n g as i n t h e o r o 6 . I I I .

0 L ~ L ~o

Ii ~ II

(~.3)

~o

{

, the following ine~ualit~

holds:

It L ~'il

c

0

f o r an~

~r ~

~,oo([-, ), 0

In fact, such

suppose

(6.3)

were not

true.

Then there

is

a sequence

that~

L

(6.4)

By theor. 3.1II

Ii L % II :

- ~

II 'o'K II

~

~- 9

there is a subsequence, which we still denote by

[4Yz I >

'.2

which converges i n

H ~ C rE )

furthermore

L ~

c o n v e r g e s to zero

0

in the space

Hl.~,~(r~ ~) From the inequality ( 6 . 1 ) ,

considered for Ao " r,

0

it

follows that

have f o r any

For

~/~

N--) O0 ~ i t

~y~ ~

C~176

converges in

)

H~.~(C~)

"

follows that

I

~

L ~ ~dx

=0

and, c o n s e q u e n t l y ,

-42-

Then

e C~(~,

V

) and s i n c e i t v a n i s h e s i d e n t i c a l l y

in

~

0

is a solution

of t h e e q u a t i o n

t h e p r e c e d i n g lemma, we have the fact that

I[~ll~f~C~

~r =_ 0

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

number

~

~ ~

)

and

L

But t h i s

result

),

By contradicts

L ~ = .~.

system

x~

For any

such t h a t i n t h e b a l l

~ - ?-~

with

"

~

which f o l l o w s from ( 6 . 4 ) .

Cv~ (~) .

be such t h a t f o r any

~o

C~

t o prove t h e e x i s t e n c e theorem f o r t h e l o c a l

some s o l u t i o n of t h e e q u a t i o n Let

belonging to in

~- ~

We a r e now i n p o s i t i o n

6.V. L e t

~L~-_ 0

~ ~ I)9

~

Ix~

there exists

I'~ ~

there

>

a positive exists

I ~ -_ ~ . ~ E C o~ (. F'~ )

r e p l a c e d by

L ~',

inequality

(6.3) holds

Then we deduce t h e f o l l o ~ n g

i nequal i ty-

II ~" Iio, I"~ ~

c

II L*~" II ~

I t f o l l o w s , from t h e e x i s t e n c e p r i n c i p l e an

~ ~ L ~( ~ )

f o r any

'U" e

which s a t i s f i e s

C ~o ( r ' ~ ) ,

In general, if of

L~,-_ ~

the integral

equations

proves t h e theorem.

Ao i s any bounded subdomain of

exists in

(o.5)

sis

of l e c t u r e 2, t h a t t h e r e e x i s t s

il ~ li ~

A

, a solution

~o i f and o n l y i f t h e i n e q u a l i t y

O) Ao

_~ c II L'~ IJ

0 A~

0

h o l d s f o r any principle

r

C ~ (A o ) .

This i s a c o n s e q u e n c e of t h e e x i s t e n c e

of l e c t u r e 2. Of c o u r s e , t h e s o l u t i o n i s supposed t o b e l o n g t o .

0.9

o

-43-

The i n e q u a l i t y

( 6 . 5 ) may f a i l

examples c o n s t r u c t e d by P l i s

t o be t r u e f o r some

[ ~]

~o

, as t h e

p r o v e . When ( 6 . 5 ) h o l d s , we have t h e

following "dual inequality":

%eUo

where

~ 6 s

~A)~ O

~

C ~176

o~A~

and

s o l u t i o n s of the homogeneous ~ u a t i o n

.B.ibliography

Eli

of

-

~o

O, Ao

'

is the linear variety

L~o

Lecture

J.PEETRE - A n o t h e r a p p r o a c h ,to e l l i p t i c

: o

b e l o n ~ n g to

of a l l

~ ~ (Ao).

6

b o u n d a r y problems - Comm.

Pure and A p p l . Math. v o l . 1 4 ~ 1961.

J.PEETRE - E l l i p t i c

P a r t i a l _ D i f f e r e n t i a l E q u a t i o n s of H.i.~her Order -

UniT. of ~ a r y l a n d , I n s t . N~ 4 0 ,

[3]

A.

PLIS

-

F l u i d Dyn. A p p l . ~lath. s e r i e s

1962.

No,n-uni~uenes,s i n Cauch~r's. Probl.em .for D i f f e r e n t i a l E ~ u a t i o n s of E l . l i p t i c t y p e . - J o u r n a l vol.9,

1960.

of Math. and ~;ech.

-44-

Lecture

Semiveak

7

solutions

BYP _ f o r

of

L e t us c o n s i d e r an % X ~ (~>-4)

with

elliptic

L

matrix operator

C ~ complex c o e f f i c i e n t s .

systems.

of order ~

For c o n v e n i e n c e we v r i t e

the

o p e r a t o r as f o l l o w s =

L =

~

o. (x)SD

q

oell,,I-'=~,

, o~lcll*-=-,'~.

The a d j o i n t o p e r a t o r i s : Ipi+iql

L ~ : (-4) If

L

i s s u p p o s e d t o be e l l i p t i c ,

Dq~ D P. the ellipticity

condition is then

wri t t en

(.~.l)

d.,~t Z

o..,~q.(x)'~ P~" q ~. o

( ~ = r e a l non z e r o ~ - T e c t o r ) .

We s u p p o s e , f o r s ~ m p l i c i t y , t h a t t h e c o e f f i c i e n t s d e f i n e d i n t h e w h o l e s p a c e (and b e l o n g t o Let

A be a bounded domain o f

the following bilinear

A

X ~.

O,.pq(X)

of L

C ~ ).

To t h e o p e r a t o r L we u s o c i a t e

form

A

are

-45-

which i s d e f i n e d f o r every p a i r of v e c t o r s

~ ; ~r

b e l o n g i n g to

~(A).

Suppose we c o n s i d e r t h e f o l l o w i n g BVP:

(7.2)

[-t~ - ~

in

This i s t h e c l a s s i c a l first

~

(7.3) ~P ~ - 0

,

"Dirichlet: problem".

want t o f i n d a s o l u t i o n

( 7 . 3 ) w i l l be s a t i s f i e d

We assume

~ ~ H~ (A) 9

~ ~ C ~ 1 7 6sand

we

The boundary c o n d i t i o n s

in a generalised sense~if

~

is properly

0

r e g u l a r and we r e q u i r e t h a t ~ belong t o bb ~ ~

(A)~C~176

and

L~: ~ then

H ~ (A) (see l e c t u r e 4 ) . I f f o r any

~Y s

~.~

(~)

the

f o l l o w i n g i d e n t i t y w i l l be s a t i s f i e d :

(7.4) A

o

I t follows t h a t (7,4) i s s a t i s f i e d

by any

~r e

H

(~),

We a r e now l e d to t h e f o l l o w i n g g e n e r a l i s e d f o r m u l a t i o n of t h e Di r i c h l e t p r o b l era: 0

I ) Find a f u n c t i o n ~ belongin,g ,to is satisfied

f o r an]r

~

6

~

~

(~)

such t h a t ( 7 . 4 )

(~),

O

If

lY ~ C oo ( A ) ,

For t h i s r e a s o n we say t h a t problem.

from (7.4) i t f o l l o w s t h a t (5~ ~

i s a semiweak s o l u t i o n

Theorem 5 . I a s s u r e s us t h a t

bb

sakisfies

~

of t h e D i r i c h l e t

(7.2) in the c l a s s i c a l

s e n s e . We have a l r e a d y observed t h a t ( 7 . 3 ) i s s a t i s f i e d sense i f

in a generalised

i s p r o p e r l y r e g u l a r . Under f u r t h e r assumption on

s h a l l prove t h a t t h e boundary c o n d i t i o n s ( 7 . 3 ) a r e s a t i s f i e d classical

is satisfied.

/~

we

in the

sense by a semiweak s o l u t i o n .

We want now t o g e n e r a l i s e problem I ) i n o r d e r t o i n c l u d e BVP~s different

from t h e D i r i c h l e t problem.

-46-

Let

V

lI)

Find a solution ~ 6 V

If

subvariety)

of t he space

H

A

belonging to

V

such t h a t

(7.4) is satisfied

. is properly regular,

we may c o n s i d e r a s t i l l

more g e n e r a l

problem 9 L e t us d e n o t e by by f u n c t i o n a l

~j~

Let

~

~

C~)

]~.~

~,4(~A),

in

V x V. III)

9A

II~ ~, II

the scalar

~ L~

[ ~ 6 H

(defined in lecture

transformation

The f o l l o w i n g b i l i n e a r

of

~

form

[ ~>

~r

for any

~y s ~

4 ) . We d e n o t e

and r a n g e ] is continuous

,"IRo #

such t h a t t h e f o l l o w i n g

_Find a f u_n__ction b e l o n g i n ~ t o

e q u a t i o n _is s a t i s f i e d

( A ) .l

.~ ( ~ A ) .

w i t h domain -~

,:t,(u,,,~)- E~C~,~)*- [ ~ ' ~ , ~ ]

Set

(7.5)

function space obtained

p r o d u c t o f two v e c t o r s

be a bounded l i n e a r

in

the Hilbert

c o m p l e t i o n from t h e v e c t o r s

by means of t h e norm by

:

~ C~,,~) : f t~~ dx. A

As an example, we want t o s h o t b o y some c l a s s i c a l f o r a 2nd o r d e r e l l i p t i c problem I I I ) o

boundary v a l u e problems

e q u a t i o n can be i n c l u d e d as p a r t i c u l a r

L e t us c o n s i d e r t h e s c a l a r

differential

L~

:

,---

Qx,

t•

r'ax.

+ B. (x; q~ " ~x;

example we c o n s i d e r o n l y r e a l f u n c t i o n s .

A

,gx. ,,"ax. ~

.i

We h a v e :

t cu,~]dx.

+ ~b;

-o.;~ . . . .

9x.

i,

c a s e s of

operator with real

coefficients:

In this

(A)

~ ~ ( A ) c V c H,~ ( A ) . R

such that

f o r any

be a s u b s p a c e ( c l o s e d l i n e a r

-47-

Let

~A

- 9~A u ~ A

the possibility ~K ~

that

,

~KA

~4 ~ ~ ( g~4

on ~ 4 A

Let on

or g - ~

i s n o t empty, we suppose t h a t

of smooth h y p e r s u r f a c e s o Assume t h a t

if

A =~

A ~ ~,.~ .

for

~

) may be empty. However~ i f

/~ ~

~

. We do n o t exclude

i s made up of a f i n i t e

number

i s d e f i n e d by t h e c o n d i t i o n ~

~/.We aJsume ~/=H4 (~) i f

/~A

~O

i s empty, and ~/~ H4 (A)

i , empty. -- ~ ( •

~(X)

where

is a c o n t i n u o u s scalar f u n c t i o n d e f i n e d

Equation ( 7 . 5 ) becomesz

A

Assume t h a t a s o l u t i o n Then we h a t e ~

~ ~(A)

of e q u a t i o n s ( 7 , 6 ) e x i s t s b e l o n g i n g t o ~ 4~A) .

~

,

~

--~

in

A

,

~--o

on / ~

and

moreover

A where ~ - ( ~ ... ~.~) i s t h e i n t e r i o r u n i t normal v e c t o r t o follows that

~

/~ A .

It

i s a s o l u t i o n o f t h e BVP

This BVP i n c l u d e s , by p r o p e r c h o i c e s o f c l a s s i c a l BVP f o r a second o r d e r e l l i p t i c

~

, ~A

, ~

many of t h e

equation.

(l) This h y p o t h e s i s i s assumed now o n l y f o r t h e sake of s i m p l i c i t y , but i n g e n e r a l i t i s n o t s a t i s f i e d ,

(z)

-48-

We w i s h now t o g i v e n e c e s s a r 7 and s u f f i c i e n t existence

and u n i q u e n e s s o f s o l u t i o n s

of problem I I I ) .

w i l l i n c l u d e t h e c a s e of p r o b l e m I I )

] . However,

~

~/ = ~

(A) ~w we need n o t impose on merely that it

linear

are in

~

the condition

)

Of c o u r s e t h i s

and problem I )

[ ~ -= O j

of b e i n g p r o p e r l y r e g u l a r ,

ce~es, but

X ~.

form i s c o n t i n u o u s i n

~/

for the

when we a r e c o n c e r n e d w i t h t h e s e p a r t i c u l a r

continuous transformation

and ~

[ ~ ~ 0 ]

be a bounded domain o f

Since the bilinear

conditions

~/x'V"

,

o f ~/ i n t o i t s e l f

there exists

such t h a t ,

a

whenever

one h a s :

Hence ( 7 . 5 ) may be w r i t t e n :

(~.5)

(. T~)~

From t h e e x i s t e n c e p r i n c i p l e 7.1. arbitrar~ for any

= IA ~ v of l e c t u r e

A n e c e s s a r ~ and s u f f i c i e n t ~ ~ ~ ~ (A) .there exist ~ EV

) is that

1.)

II'~'II ~ c IiT~II o

2e)

The r a n g e

T (V)

I f 1 e) and 2 ~

of

follows that:

a unique

~ ~V

satisfying

g i v e n an (7.5)

one have~

(c >o) T

f o r any

is dense in

hold then the solution

II ~ II Now d e f i n e :

2, i t

condition in order that,

the dual i n e q u a l i t y :

(7.8)

ax.

L_ c IIPr II o

~V*.

V. t~ of t h e problem s a t i s f i e s

-49-

and c o n s i d e r t h e r e a l q u a d r a t i c s a y t h a t t h e q u a d r a t i c form ~11)'i~ ~

in

V

I "Y" ( % ~ ) I

~

( ~ j ~)oWe s h a l l

~ ( ~ 1 ) ' ) i s c o e r c i v e on t h e q u a d r a t i c form

C o II 't," II

TM

C~ > 0

such t h a t :

,

~ ~ ~.

7.11. solution

:~

~ (~/, ~)

, when t h e r e e x i s t s a c o n s t a n t

(7.":)

f o r any

form

A sufficient

of problem I I I )

Let T *

condition in order that there exist a unique i s t h ai

denote the adjoint

llt,"h ~"

"~/(~,~r) be c o e r c i v e on

T

of t h e t r ~ u s f o r m a t i o n

i n V,

which

was i n t r o d u c e d a b o v e , and d e f i n e :

T.i -._ 4 ( T + T * )

~"

T~

,! ( T _ T *

.

__

~.

9

One o b v i o u s l y h a s :

And h e n c e i f

( 7 . 7 ) h o l d s , one o b t a i n s :

I i ~ 1 t 2" 'm

~

•

Co

Thens

(7.8)

li v Ii

_~ L tl T~ Co

II ~

II

II

lIT

~11

9

-50-

which o b v i o u s l y Condition 2~

i m p l i e s c o n d i t i o n 1 e) o f t h e p r e c e d i n g t h e o r e m .

is also satisfied,

because, if

b~6~ r

and s a t i s f i e s J

Since

]-4 and ~

are hermitian operators,

j

one g e t s (T~ ~. ~ )

-- O, w h i c h , A~

i n view of ( 7 . 7 ) ,

means t h a t

~ = O.

L e t us now p r o v e , by means of an e x a m p l e , t h a t

the condition (7.7)

used i n theorem 7.1I i s not n e c e s s a r y , but i s only s u f f i c i e n t . end we s h a l l c o n s i d e r a v e r y s i m p l e p a r t i c u l a r and 2 ~ Let

of t h e o r e m 7 . I h o l d ,

~-_ ~

of t h e

~,"4

X-axis,

and ~

{

--

c a s e i n w h i c h , w h i l e 1 e)

( 7 . 7 ) does n o t .

A

and s u p p o s e t h a t

To t h i s

coincides with the interval is

(-,l, ~1)

L e t us p u t :

~I(A).

Ixt

J

dx

-'1

dx

E q u a t i o n ( 7 . 5 ) i n t h e p r e s e n t c a s e becomes:

(7.9)

I xl -,I

It is readily verified one

~ ~ ~/

(•

~

~

dx

dx

that,

, satisfying

-4

given

~ ~ Z z

, t h e r e e x i s t s one and o n l y

( 7 . 9 ) f o r any ~r ~ ~/

and t h a t i t i s d e f i n e d

by t h e e q u a t i o n :

X

t

-4 T h e r e we have p u t :

X

I -4

-f

4

X

-51-

Therefore,

the

above-mentioned

satisfied.

However (7.7)

is

conditions

not.

In the

1")

and 2*) are certainly

present

case,

(7.7)

may be

written:

(7.1o) Indeed

/ i,I

_

from (7.10)

(-~,-6)and

it

(,,~,4)

folloTs

, where E

f

~-'~,

that

any

t.<

O~

,I

%

L~

which vanishes

and

d~r

d•

+lv'l

on t h e i n t e r v a l s

< Co ~ must satisfy:

z

_>0~

-&

and this

is

absurd

if

~

is

not identically

Bibliography

[I]

G. FICtlERA - A l c u n i

recenti

contorno Atti

per

G. F I C H ~ -

[a]

J.L.

see

sviluppi

J.L.

Trieste

[2]

LIONS - S u r l e s

~I.I. VISIK-

della

1954-

of lecture

te.0ria

alle

derivate Equazioni

Ediz.

94,

(in russian)

alle

-

Derivate

des distributions

-

1955.

of ~athem. v.

BY]? f o r

parziali

al

C r e m o n e s e , Roma 1 9 5 5 .

en t h e o r i e

problSmes, aux limites,

On g e n e r a l

dei problemi

1.

LIONS - P r o b l ~ m e s atLx l i m i t e s

Annals

[53

7

le e~uazioni

Acta ~athem. v.

[4]

Lecture

Convegno Internazionale

Parziali,

[2]

of

zero.

64,

elliptic

du t~pe d~riv~e

oblique

1956.

differential

Trudy ~oscov liar.

0bsc.

equations

1952.

-

-52-

Lecture

Regularity

a~

In the for

8

the

s p a c e we s h a l l

K : 4,,.., -~-4

and by

of

X ~

Let

~

by

(~

~-t ),

if

(~)

(

is ~y i)

. We s h a l l

by

of

fixed positive

~ s ~ ~

following conditions The c l o s u r e

9

X ~ d e f i n e d by:

of

0 ~ t ~ )T

~

and o n l y i f :

Z~

the coordinate

X~

The p o i n t ( ~ t J - - , Y ~ - ~ ) and t h e p o i n t X

t),

(as u s u a l we c o n s i d e r complex ~ - v e c t o r

g ~

~K

s p a c e w i l l be d e n o t e d by ~

be t h e open i n t e r v a l

(K, : 4 , - . .

d e n o t e by

~; t h e c o o r d i n a t e

(~-4 )-dimensional

of t h e

boundary..:, p r e l i m i n a r T lemmaso

;

is satisfied:

~6

(~)

ii)

,~ ~'~ x ~.jj

say that the function

valued functions)

belongs to

n~ber

0 ~ ~ ~ ~T )

such t h a t

~)~- 0

I~/I > ~

in the space

-~

)

when a t l e a s t t

one of t h e

>~,

~(~)will

be d e n o t e d

H ~ Let

~ (X)

following function

he a f u n c t i o n of ~Y~(~) defined in

C ~m ( ~ ) , We c o n s i d e r t h e X ~ by:

~- ,~ ( ~ t )

I L

The s c a l a r s

)~' J

: L

~ . ~r(y - i Z )

are the solutions

for

t ~_ o

for

t~o.

of t h e f o l l o w i n g a l b e g r a i c

system:

-53-

(8.,)

2:

j:~

It is easily that its

J

seen t h a t t h e f u n c t i o n

(8.3)

II ~ *

V ~Y~ ~)

il

it

(~ :

i s e a s y to v e r i f y

z_

LQ

c I l v II

~,

FI ~ ( ~ )

e

( 8 . 2 ) , belongs to H~(~)

and c o n v e r g e s i n

(~)

and

that:

L : o,...~

~.

~

~ d e f i n e d by means of

. In fact if

t o bL , t h e n

C

[ X~ [ ~ )~.

~R

~ the function

1~

,

belongs to

C i s a c o n s t a n t which does not depend on If

(8.1),

,--,

support is contained in the square

By v e r y s i m p l e c o m p u t a t i o n s

where

=4

tLr

{ V~ } 6 C ,v~

} converges

in the space

0

~-~ (Q)

towards

(s.4)

t~ ~ o

I1~ ~ II

We d e n o t e by Dy

--

_~ c l l ~ l l

~Q

t h e symbolic

~,g

y-differentiation

t&

( A

(A)

has t h e

being a bounded domain of

~-strong

derivative

. h e n e v e r a sequence of C~^'('A ) - f u n c t i o n s

converges i n

~2(A)

t o some f u n c t i o n vative

(8.5)

vector

,"',

Let t~e ~ say that

Dy

Moreover ~e h a v e :

DP~

towards

~f F

DPt~

(l~l

{~YK ~ e x i s t s

which Te d e f i n e t o be t h e

i

A

d~ = (-4)

iptl

~

~-strong

dx

: ~

)

such t h a t

t converges i n

|A, and { ~ P ~

of ~ 9 Hence:

~ ~

X ~ ) . We

~R(A) deri-

-54o

for any

C

~

( A ) . I t follows t h s t

i s 9 c l o s e d s u b s e t of ~ the

o~-strong

In th,t

and ~ v a n i s h e s i n

deriTative

come (8~

~

~ - ~

, then, if

, this deriT~tiTe ~nishes

h o l d s f o r a n y ~7 ~ C ~ ( A ) .

F o u r i e r development o f I ~ P ~ ~

~ P does n o t depend on t lY~

Assuming t h a t

i s o b t a i n e d from t h e

by f o r m a l l y d i f f e r e n t i a t i n g

~

~.If r hem

in A-r'.

A C Q , the

F o u r i e r development of

i t by mea~l o f ~ P

From t h i s

remark i t

follows th~t8.1.

l_ff

~ E I"l o ( ~ ) and v~ h~s e v e r y

~ 2-strong derivstive

of

|

order ~

~ then

In f~ct,

~

~ ~ ~'(R)for

has t h e ~

~n~ ~

such t h a t

-strong derivatives

h~s t h e f o l l o w i n g F o u r i e r development i n _

(8.6)

~P~

~

~' < ~'

( IpI --~

) snd i f ~

,

~oO

0

~KX

l

:*

~<

Z

K

cK e

thenz

..i-~o P

,

'i

~.

Ipi L

p K

,~Kx c

e

.

( It follows thst l yJ<

~ ,

I tJ

~

~ H

< ~

(Q).Since

, t h e r e e x i s t s s sequence of C ~ f u n c t i o n s

h~Ting t h e i r s u p p o r t i n t h e c y l i n d e r towsrd

~ ~ i n t h e sp~ce

8.II. h~s t h e

~-strong

deri,~tiTe

~q ~P ~

and

The p r o o f i s s t r i v i s l

~ ~

hss the

deriT&tiwe

~

t 17~ 1

I ~ l < G * , I t J ~ ~ ' snd c o n v e r g i n g

~(~).Then

I_f ~ E H o ( ~ ) 0

~ Ta~ishes outside of the cylinder

~1 ~' ( 1 ~ * ) .

o~-strong =~

d e r i v s t i T e ~Pt~ : @ s n d

, t h e n Lt h s s t h e ~ - s t r o n g

) q DP: ~ . c o n s e q u e n c e o f ( 8 . 6 ) and ( 8 . 7 ) .

-55Whenever a f u n c t i o n simply write 8.III.

for

Dy ~ E

~

has some

~-strong derivative

~PL~

we

~P~ ~ ~er

~

~ E

H~

( R ) ( ' ) ,is

(~, ,.. ,~.~ o ) ....~h t h a t

(~),

K, n,e,cessar~, and s u f f i c i e n t

~-vector ~-

t h a t f o r any n o n - z e r o r e a l

J~.l,: ~ - ~

condition

, t h e f o l l o w i n g inequal,it~r be

s a t i s f i edz

(8.8)

wit h

~

C4

independent of

~

.

Then

I~1 Let

US

c

~R

consider the function

t~

~,R

'

(8.8)

i n t r o d u c e d above 9 From ( 8 ~

it

folloTs that:

.Z (8.1o)

11

>I1

z

CC 4

.

Let

(8.11)

L,I,'x" ( X ) -

Z~ K CK e

be t h e F o u r i e r d e v e l o p m e n t of

"(4')If 44)..;~ r

t~ ~

in

(~

9 From

a r e t h e components o f t h e v e c t o r -vector

~

'~YW

~L K

(~=4j..~-~

(8.10)

and

(8.11)

t~ , t h e n by ~)y tr ; t

K=

4j..~),

we mean

-56-

it

follows that ~,Kh

'4" OO

18.1,]

7',

Z

Ipl :o

-~

k

IC

I

J z constaRt..

Hence t h e p r o o f of s u f f i c i e n c y

f o l l o w s by t h e same argument which was

u s e d a t t h e bottom of p a g ~ The p r o o f of ( 8 . 9 )

f o l l o w s from t h e i n e q u a l i t y

I

I

{,K,t + - - . +

L e t us now assume t h a t Since of

~ e

that

Dy~ ~

vanishes in a strip

b~ ~ ( I pI ~ ~

differentiating

t

K

).

H (IR).Tnen 1)TIz.~ c H,~ (Q). near

~(~

, any d e r i v a t i v e

~i)p ~ y j e

) i s g i v e n by t h e development o b t a i n e d by m e r e l y

, with

~P~y~

, b o t h s i d e s of ( 8 ~

It follows

the series

K

Ic

I

K

+.,-,K

)

IpI--o i s c o n v e r g e n t ~ Hence ( 8 ~ 8olY.

Let

h =-(o~..,h;~..o),

t,t, ~

-

i~176 (8.10) - holda~

H~(R)

0 ~ ih~,iz~

and

])ytt~H

(~,).Set

9 Then:

~Vw The p r o o f i s e a s i l y

o b t a i n e d by u s i n g t h e

F o u r i e r development ( 8 . 1 1 ) .

~

e x t e n s i o n of

and

-57-

8.Y,

Let

Assume that

N/ ~

])y ~

G" H o (R)

I

(y

R

zg'v,,

"~

~+lPl~

~t

tg vr

'

o,g

)

~r~

w/ p

~_ ~ ~

~ §

Let

and

be a n y i n t e g e r

(

not less

K,I,

-

th~n

x K

T~Pv')dx y

II~,'~pIIo,p + IID,/,I-II,R )"

~ + 4

: ~r ( y ) t )

and p u t s

s

for

)

O)

(y,t) ,~+4

Z

for

j.-a

~ p (y,t )

~ ~ O.

for

~

> 0

for

~ ~ O.

Kp ~)+4-g

,w~+.l

L

The

are chosen

~3 (J)

w

(y, -jr )

so thatz

(~ = 0,'I,..,~§ i=4

1

o

Let

E C co(Q),

We c l a i m that=

-- O.

-58-

I::_ ( V ) -

(8o]L3)

~ * "~ v d z ,

,%~,

~K

,at K

DPV)d~,

- o.

y

I n f ~ c t we have:

"~

,in~ ~. 4

E (v): E(v)+ Z )"J ~f4

~+lpI ~ ~

fRv(y,j,) V

f, ~ , 4 (y,

- t ) ax +

V+4-K

P ~p (Y' ]{) ]), < ~ ( y , - t ] r

,1:4

:

R

: E (V)+

I

~ii

j~4

IRV(S,t) V

t~,+4

(y -t] "~)dx )

~+4

+/~

Z

),.j" a

~P

"v, (y,-tj "')a•

fl Se%: - ' n ~ "F 4

J:4

~.J c'i ;" V ( y * _t j-' )

We haves

E(V)= [(V+Vo): r- (,~)

0

"

-59-

and

/~t W

v (y,t)

" 0 t=o

Set~

I

tr (y,t)

where

~ (~)

,.~t~'y,-~-~

= 0

C~

is a

for

~ -~ •

for

scalar

t ~

f u n c t i o n which e q u a l s

and which v ~ n l s h e s i d e n t i c a l l y

in a left

~

an

n e i g h b o r h o o d of

"~

(~0G)

. Since

0

~

~

C ~*' (R)

= ~ ~

.e ha.e

so t h a t

IK -

~/:

Itl

(P~4 ,""

~ ~

~

C ~ ((~)

i s an ~ r b i t r a r y

~tL-4) T ) i s d i f f e r e n t

~I"

dx

~

~

t h e o t h e r hand

L P*4 (-~(X)~

function belonging to

lYl ~ ,

- O.

~

E ~

-

~rr where

which e q u a l s constant

~

is a

~ in the cylinder

~-veetor,

and

from zero. From equation (8.13) we get

2"_

W

(x~ ~

~

dx

o_~Iwl.~ p

There the

T~W~(X)

Let

~nd

C~

respectively.

~K

:

(8.13) is proved.

We now ~ssume t h a t scalar

E ~

are f u n c t i o n s b e l o n g i n g to be t h e F o u r i e r c o e f f i c i e n t s

We hate=

O~ of

~ ~ ~nd ~v/~ (x~

-60-

I~c~l

_~

/c

I + s

Ic~l

I K I :z,v.~-~

I K I .9.v+%

(8.14) IKI ~

])y~'~ ~ %

Since

, it

o<

follows

from the

above inequality

that

the

series ,Y.

i s convergent. From (8~

it

Thus follows

~'~lC

I~

~t thats

z c llW IJ + IIDy *l

Bibliograph~

[ 1]

G. F I C H ] ~ A -

of

S e e [2 ] o f l e c t u r e

K . 0 ~ FRIEDRICHS - T h e i d e n t i t y differential vol.559

L'a]

J.L.

LIONS - Some q u e s t i o n s

L . NIRENBERG - On E l l i p t i c Annali

8

1. o f weak a n d s t r o n g

operators

extensions

- Trans~ Amer. ~athem.

of Society

1944o

of ~ndamental

/4]

Lecture

on e l l i p t i c Research, partial

equations

- Tara Inst.

Bombay~ 1 9 5 7 .

Diff.e.rential

S c u o l a Norm. S u p . P i s a ,

Equations

1959.

-

-61 -

Regularity

at

Lecture

9

the

tangential

derivatives.

X ~ such t h a t

/~A-/~A

boundary:

L e t A be a bounded domain o f that

A is

C~-smooth at the point

x ~ of ~A

.

We s a y

if a!neighborhood [

of

•

exists with the following properties: i)

There e x i s t s a C

onto t h e c l o s e d semiball space

homeomorphism, which maps t h e s e t

/v, + , t > 0

, I~' ~ % z ~ i

of t h e

J:

[ n

Z - d i m e n s i o n a l (y~%)

9

ii)

%:o

9

The s e t

I ~ 9A

i s mapped onto t h e

(~ - 4 ) - d i m e n s i o n a l b a l l

, lyi--, t .

A is c a l l e d C -smooth i f i t Suppose t h a t

is

C ~- s m o o t h a t e v e r y p o i n t of i t s

~ is a solution

boundary.

o f e q u a t i o n (7o4) b e l o n g i n g t o ~/

( s e e p r o b l e m I I of l e c t u r e 7 ) . Under p r o p e r a s s u m p t i o n s f o r ~r , a s s u m i n g is

C~-smooth in

X ~, we s h a l l p r o v e t h a t

any o r d e r we wish i n Of c o u r s e

~

N~ A

has c o n t i n u o u s d e r i v a t i v e s

, where N i s a s u i t a b l e

depends on t h e o r d e r of d e r i v a t i v e s

As i n t h e c a s e of t h e c o e f f i c i e n t s s h a l l s u p p o s e , f o r t h e s a k e of b r e v i t y , r e a d e r as an e x e r c i s e , which t h e r e s u l t same method.

~

n e i g h b o r h o o d o f X ~.

we w i s h t o p r o v e e x i s t .

o f L and t h e f u n c t i o n ~ , we that

~=oo

of

. We l e a v e i t t o t h e

t o d e t e r m i n e t h e more g e n e r a l h y p o t h e s e s u n d e r

which we a r e g o i n g t o p r o v e , can be o b t a i n e d w i t h t h e

-62-

Under s u i t a b l e the results

h y p o t h e s e s on t h e o p e r a t o r

~

introduced in lecture 7,

which we s h a l l o b t a i n f o r e q u a t i o n ( 7 . 4 ) can be e a s i l y e x t e n d e d

to equation (7.5).

However, we s h a l l n o t d i s c u s s t h i s c a s e .

The r e a d e r T i l l of t h i s l e c t u r e ,

be a b l e t o s e e , a f t e r

s t u d y i n g t h e p r o o f of t h e o r .

what h y p o t h e s e s a r e needed on

~

9.I

in order to extend this

p r o o f t o t h e more g e n e r a l c a s e o f e q u a t i a n ( 7 . 5 ) .

On t h e o t h e r h a n d , t h e

p r o o f o f t h e o r e m 1 0 . I o f t h e n e ~ t l e c t u r e works e x a c t l y t h e same way i n t h e c a s e o f e q u a t i o n ( 7 . 5 ) and r e q u i r e s no a d d i t i o n a l It is convenient to collect I e)

The c o e f f i c i e n t s

and t h e f u n c t i o n .~ ~•

here all

(~pq ( •

belong to

h y p o t h e s e s on ~

our hypotheses.

of t h e e l l i p t i c

operator L ~ Dq~q~

~

~ oo. &

i s a subspace of ~

(/~) c o n t a i n i n g

(9.1)

(A) and such t h a t f o r

r O Jt Jl

3.)

A is

~'~-smooth at the point

L e t us d e n o t e by ~ :~(• its

~

the

-- ( ~ ) t h e

• o of its

p o i n t of t h e ( y ~ ) s p a c e

~C~

t h a t maps

i n T e r s e . For any ~

and ~

b0~un.da_ry.

such t h a t

J

o n t o ~ ~ and •

0 < ~ (

~ ~ ~ let

C ~ s c a l a r r e a l f u n c t i o n , which v~n~shes o u t s i d e o f t h e b a l l and which e q u a l s function

4.)

~

in the ball

~

z [~ [ ~ ~ .

~x/ ( ~ ) : ~ ( ' ~ ) ~ y L ~ ( ~ ) ] . W e

Let

The f u n c t i o n s z

.-

and l e t

Let ~ / .

•

)

~ o ( ~ ) be a r ' g .* Consider the

make t h e f o l l o w ~ n g h y p o t h e s i s on ~ / s

O) be a r e a l

- v e c t o r such t h a t

O~ I hl < 4-~.

-63-

xeJ :

belong to

0

x e A-,I"

•

~-: 0

in

such that ~ ~ H

~-'~ (z ~

s p a n n e d by ~ F a(~ )1 w h e n ' ~ b e l o n g s t o V . It is evident that

(t)

, then

5 ~ ) ~K/ = ~ / N C ~ ( Z * )

if

a

C ~

function

(The c l o s u r e must be u n d e r s t o o d i n are satisfied

By c o n s i d e r i n g o n l y f u n c t i o n s

~

is possible

~o(~)is

to write inequality

and i s

~/~ ~ "~/o

The h y p o t h e s e s assumed on ~/

if

I"I~,~ (~)).

"~/ ~ N (A)or V _= H~ (A),

~/ such t h a t ~ ( x ) _9 0 i n

(9.1) i n terms of the ~

e l e m e n t a r y c o m p u t a t i o n s we s e e t h a t

( 9 . 1 ) c a n be w r i t t e n

~-~

, it

coordinates.After as f o l l o w s i n t h e

new c o o r d i n a t e s :

j

2

Z'

where t h e

( ~ ) b e l o n g t o C C ~ r'~) and t h e symbols D ~ bct JI Ii eq ' , must be r e f e r r e d t o t h e new c o o r d i n a t e s . The f u n c t i o n ~ i s nov a n y f u n c t i o n of ~

~,~,

4 - ]"

V.

L e t -y~r be t h e v a r i e t y such t h a t

- o

~

~.

exists

From ( 9 . 2 ) , in the space

by theorem 7 . I I , ~

it

m a i n t a i n e d t h e symbol ~

( ( ) ~ c o r d i n g t o our d e f i n i t i o n , of t h e s p a c e s i n s t e a d of ~ i

o n l y one s o l u t i o n

of t h e s y s t e m

F (~)-~ ['~('~)]1~ I. w, h,,o

~[~(~+.~y+)

follows that

(~"),

But t h i s

to denote

H~ ~ we s h o u l d w r i t e would be q u i t e a p e d a n t r y .

-64-

the

new b i l i n e a r

9.1.

Let

form.

Let

r'~~

such t h a t

be any p o s i t i v e

~

Dy

g

~J ~

y

C4

~

-

The theorem i s t r u e f o r I~I

-- IK@ 4

~o(X )

(II F Ii ~"

-I- II u. II

l~l-,~,,,Z:*

~

' ~ 2:

- onl]r depending on t h e

]~ [ -- 0

(~-) be

We s h a l l

9

of o r d e r Set

~F K

~(X)=

such t h a t

"P) I U'(x§

is

the

C

K

)

c~

p q .' s and ~ 9

l"3J -/-- k .

~o s c a l a r

( s e e lemma 8 o l Y ) | D y

+

prove t h e theorem f o r

f u n c t i o n i n t r o d u c e d above~

Because of h y p o t h e s i s 4*) and t h e i n d u c t i o n h y p o t h e s i s , belongs to

)

~ - (~,,,~r

o,Z (~.s)

, The s o l u t i o n

s u p p o s i n g i t t o be t r u e f o r

Let

0 ~ ~ ,C 4 ,

( ~ ~+ ) , Moreover

.i

- f o r any g i v e n

C oo ( ~. +.).

to

t >_ 0 , l y l 2' + t ~ _z c~ ~. L e t

,: c r,+

belongs

number such t h a t

(~-4)-vector-index.

II D~' ~, II ~"

(9.4) with

~

be the. s e m i b s l l ;

be an a r b i t r a . r ~

F("~.~

The f u n c t i o n

is ant

the function

y-partial

~Lr

differentiation

~ Dy~

O ~

I~[ ~

9 We have f o r ~ / n C

(~nd

any r e a l ~ ( ~ 4 ) - - ) ~

4)0)

4-6

s

,v :(-~) I Dq(f(x+k)=(x+~)-~~

y(~r~-D%)dx = (-4)~I ~PsD 'b'< Z*

u'(x)

~.+

)-r

r

(~) From now on we s h a l l u s e t h e l e t t e r

D,
I~I

X i n p l a c e of ~

9

-65-

(x+ L, ) ~, ( x-t- I,, ) - ~ ( •

X

Ikl

.

O{Ij

~(.xJ

I -~ ~<-~ )..+ i

The m e a n i n g

of

the

symbol

Due t o t h e i n d u c t i o n

9

hypothesis

~,11 I~I~K

Y

~

is self-explanatory.

~ J P':I

Ilwll r~+ e-,

and l ~ a

(~) ~

8.111,

~'tB

6'

the last

integral

,1~-~"

,

:

_ _2.

.

We hayer

(-,)

~

~

Dq

D Ikt

D% ax : Y

Z+ q

q(x+k) D ~.(x+k)-~(x)D ~(x) P~

I

PD~

Ill

i

Y

Gt) ~

il

D~ctx

'D K +

+

y

Ikl |

+

D

I M 9',~"

I~,1

19"1 ~ ' ~

S

pcl

7. +

(3) By s i m p l y v r i t i n g

J]

Jl

we mean t h e norm o v e r ~- +.

D ~D - v~<

•

-66-

J_..9~

ps (' x ' h ) - ~ F'~ ( "J

qo (x+k.) D q

J

+

Ikl

2.-*

4

o( T.. IID

V

(-4)

(x)

l ~ l "; K

o(

II

II 'u-Ii ~

r'+

; .~.

D q ~.(x)

),

Pq

§ I~J ~ K

IIDY~ ','., II ~m,j r + Pq

"~'(•

~(x) ])PD '<

Ikl

clx

4*

):

!1'~ II

(x)D q ~(x) D s

~(,,)

[@(• D ,~

,v'(x-(.)- ~(x) J Y Ikl

\ II D ~. II

+

II ~ II

J/ 9

I~I ~_

Then we h a v e :

~(x-~,)-~'(x))

( fJ"(x+k)-ikl U(x) CO( Z__., lid ~' i61 ~ K

Y

y

r ~ ilv~l~).

II,l

-6?-

It is

~) (~' ~'(•

"u-(x-k)-'b'-(x)). f II~I

'

: o(,,F,, For any

)

I~.1

Y

Z_*

}l ~(~"'-~(~> II

'1)'s

q~F I)K '~-(x-~,)-'u-(x)d•

(

= CO I1 F il

get

,v

: CO

Z

ilD ~II

-t ,Fil )

~

=

)

lhl

we o b t a i n from

c~ ( F. IiD ~

l~l~w

,~-Ii

6'

U' (• ~.~,)-I.)"(,() By assuming

).

,1>",

(9.2}=

+ ii F II

U

Y

~,['+

K-~

~

-I" 4

.5'Vt

From t h i s e s t i m a t e t h e p r o o f f o l l o w s -

Biblio~raph~

of i Lecture

9

i

[i]

F . E . BR0WD]~- s e e [1]p[2]of l e c t u r e 5~ F . E . BR0WD]~- .,A p r i o r i

estimates for solutions

of e l l i p t i c

v a l u e p r o b l e m s , I & I I - I n d a g ~ Math~ T01~

boundary 1960o

-68-

F . E . BROWD~t - E s t i m a t e s and exi,s.tence t h e o r e m s f o r e l l i p t i c .boundary v a l u e p r o b l e m s U ~176

Proc~176176

v o l . 4 5 9 1959.

G~ FICHI~tA- L i n e a r e l l i p t i c

e ~ u a t . i o n s of h i g h e r o r d e r i n two

independen_t v a r i a b l e s equations,

and s i n g u l a r

integral

with application_s to anisotropic

i nhom oge neo u s e l a s t i c i t 7 - P r o c e e d i n g s o f t h e Int.

C o n f e r e n c e on PDE and Continuum M e c h a n i c s ,

M a d i s o n , Wis. 1960 - R . E . L a n g e r E d i t o r . I!

CsJ

L . tlORMXNDE~ - On t h e r e g u l a r i t y

o f t h e solu.t.ions o f b.oundar]v

p r o b l e m s - a c t a Mathem. v o l . L . NIRI~BI~G A.Jo u

s e e ~4J o f l e c t u r e

9 9 , 1958.

4 and [ 4 ] o f l e c t u r e

- Boundary v a l u e problems f o r e l l i p t i c of d i f f e r e n t i a l

systems

equations of higher order in

t h e p l a n e - Dokl~ Akad. Nauk SSSR v . 1 2 7 , (in Russian).

8.

1959

-69-

Lecture

Regularity

In lecture Z~" -strong

result

derivative

])~, x)P~I~

results~

t~ of ( 9 . 3 ) h a s any

in the semiball

We wish now t o c o m p l e t e t h i s

result

of a r b i t r a r y

Given an a r b i t r a r y

a4ny 4

and prove t h a t t~

order in

~)y ~

in F / .

o( pq.., Is and on

~

< ~

positive

and any

~ ,0

A constant

~;.

This

, the

, the solution c~2-strong

C~ exists, depending ,onl]v on

_~

IP

c~

F

I

(j-O,

~t~§ Dy t,L and any

~<

~.

+11~1 i~1+~.~,~§

0 F +

L e t us s u p p o s e t h a t :

f o r any

derivat_ive

such t h a t :

'

i+)

r5 § for

i s g i v e n by t h e f o l l o w i n g t h e o r e m .

of (9.3) has, for an~r ~

t h,.e

boundar~ s f i n a l

~ A-strong derivative

lOoI.

~,~

the

9 we p r o v e d t h a t t h e s o l u t i o n

and a n y IPI - ~ . h a s any

at

10

.,

,~+

"

-70-

e

This i s t r u e f o r

9

~: {

9

Set

~(x) :~(X)~(X)

, where

~~

Is the

0

sc&lar function

used in the preTious lecture.

we haTe~ w i t h an o b v i o u s m e a n i n g f o r PP~I

B (u,

ot~-,

~ )

z~ Jz :~Pq

~-~>

:

~

J

~ Dq~ D

~ C =o(R,)

DqU DP~rdx

~ t ~-'

Dqt/DP

dx --c-4)

-

p _= (0). 9 ,, O)aa~).

~ II

j,~ jl

.

,,~ (~) a summRtion e x t e n d e d Pq ( 0 ~ I p l -~ ~ 0 L I q l _~ ~ )

We d e n o t e by

of Tector-indeces

excluding the pLir

pjq

p~ p.

We h a T e , u s i n g s y m b o l s whose m e ~ n l n g i s s e l f - e r p l a n a t o r T :

B('I/,

~t~. '

~):

~ r~ D ~ D

3t~-4

+

+

~dx ipl~_~ Igl /

~

~ t ~'4 ~-.+

+

Z+

L

t o ~n 7 p a i r

~

"

4-

Let

For

+

~)ax

+

d, ,

-?l-

+ IKI < ~'~

i

Dq~ ~'~

.~t ;'~

I o~r~

~

17 ~x

)

i

(F)q~

D

I)"

II

9 o

.1)qu DP~ d X

=

Z~

et~ D ~ II oIP t

dx

U

PP

9

II ~ II

9

+

~t ~

DqU DP~ dx Z+ @

dx

U

.

Z+ We deduce t h a t 8 I

~t,n,,4- ~ - 4

ra

U 2: t

dX

Fat ~

-

.

Y

o~rj

"~'

-72-

From lemma 8 . V and s i n c e

~e~

~

~ 0

(the operator is ellipticl)

PP

it

follows

~

~

and m o r e o v e r t h a t l

t

2

* IIF II=~_~.1 ' Osp 4t~

L e t us s u p p o s e t h a t we h a v e , t o g e t h e r w i t h i F ) , t h e f o l l o w i n g i n d u c t i on h y p o t h e s i s z

5

i) ~

K

We h a v e p r o v e n t h a t

i~)

f,o,r

I~1

~<4.

and an 7

is

true

for

K =0

. We s h a l l

prove that:

+

/jr ~'+" for

I~ ]

=

V

~§

We have f o r any

D

Y

such t h a t

I/~

I :

I

B(u)

~Pq gt;

])q

K -~

~;-4

c~ Pq ~t;-4

DP

D ~ D~U"DPv dx

Y

+

-73-

,~ ~-"

D~D~

D~ d~, o

z..,~,: - , , , ~

By u s i n g t h e same arguments as b e f o r e

II ;~

~

o.r.

,

~-4 v

9

9 t ~ -~

F

:

Y

~ ~

+

v

~t~-~

§ i"

~.+~-4

f 9~"4 +

(

i JI oi DL

IJ~ II

~t-4

Then we deduce t h a t t h e exists

and moreover t h a t z

~ s

derivatiTe

1" Dy ~C

-74-

oft

~+lr, I _~,~t~+;,

~:i

y

9

(lO.a)

+

fl ~

L

D ~

II

j-i-Itl "~a.t ~, 4-,l-l-J, j _-- ,~+ L-,I

o~r ''t" ~r

Suppose we h a v e a l r e a d y p r o v e n t h a t

.

+ n F H~ i.~1 .-t- ~, -~.

(lO.1) holds for the derivatives

c o n s i d e r e d i n t h e i n d u c t i o n h y p o t h e s i s i 4 ) . Then, by u s i n g prove,

by i n d u c t i o n ,

that

(10~

in the induction hypothesis i~).

( 1 0 . 2 ) we

is true for the derivatives

considered

From ( 1 0 . 3 ) i t

(10.1) is

follow~ that

t r u e i n any c a s e . 10.II.

For an7

I-I~, (I";)

wi.'t.h

~ < ~ ~

Co

C

~

of ( 9 . 3 ) b e l . o n p t o

g i v e n a r b i ~ r , r i . 1 7 and

II~ II~,r+ The c o n s t a n t

the solution

z_. c IJFII

depends o n l y on ~ p q ' S

,~,~

, and on t h e c o n s t a n t

of (9.2).

The p r o o f i s a t r i v i a l

c o n s e q u e n c e of theorems 9 . 1 ,

1 0 . I and o f ( 9 . 2 )

and (9.3). L e t us now c o n s i d e r a g a i n t h e domain C~176

a t a n y p o i n t of i t s

of l e c t u r e holds. it

9 for any

X~s

/~A

A

boundary and ~/" s a t i s f i e s . Suppose t h a t

Then as a c o n s e q u e n c e o f t h e o r e m 9 . 1 ,

folloTs that:

and assume t h a t

inequality

theorem 1 0 . I I

i~ is

conditions (9.1) and o f ( 9 . 1 ) ,

-75-

Q

10olii. inequality

For any

L~ ~ C

~)

.and, f o r a n y g~ven

holds:

II ~ II~,

(10.4)

C

.the c.on.stant

z

c II L.~, II ~"

depends o n l y on

may be e x p l i c i t l ~

s t a t e m e n t of t h i s

theorem, the reader is

r e q u e s t e d t o n o t i c e t h a t i n t h e p r o o f of a n y i n e q u a l i t y for the constants

given~

O.r, ~ i s | i t

~ , ~ , C~ , and on t h e

cal~ulatedo

Concerning the last

ered,

the following

Hence an e x p l i c i t

C 4 ~ G~ ) ....

explicit

e x p r e s s i o n s can be

e x p r e s s i o n c o u l d be g i v e n f o r

T h i s v o u l d be e x t r e m e l y t e d i o u s

Te h a v e c o n s i d -

C

of ( 1 0 . 4 ) .

but c o m p l e t e l y f e a s i b l e .

By t h e Sobolev lemma~ t h e s o l u t i o n o f problem I I ( s e e l e c t u r e 7) belongs to C'~A)p For a n y

provided

~ ~ C ~o(/~)

~ ~ C~176 and a n y

1~ ~

~

(A) the following

Green's formula holds:

A

/~A

where

I~ ( ~ v )

vatives

of ~r of o r d e r n o t e x c e e d i n g

(V)

M

manifold

and o n l y i f : f o r any

~

i)

form i n

~

~-~.

as f o l l o T s . A function

and

1~ c o n t a i n i n g d e r i -

We d e f i n e t h e l i n e a r

~

belongs to

M (V)

u.e Vr~ C~ (/i) ; ii) .[~.AH(~,~.Ia~

if

=o

~ V.

10.1T. func t i on

is a bilinear

I n t h e as.sumed h y p o t h e s e s , f o r

~

,

[

, ~

and

, .the

LL ,is a s o l u t i o n of problem l I of l e c t u r e 7, .i f and o n l y i f

-76-

i t i s a s o l u t i o n of t h e f o l l o w i n ~ problem-

The p r o o f i s obvious b O

I n t h e case where ~ r : ~l

(A)the variety

M ( V ) i s composed of a l l

t h e f u n c t i o n s which b e l o n g t o C ~ (A) and a r e such t h a t

(Ipl

-~ ~ - ~

DP~ -- 0 on /~A

).

L e t us d e n o t e by G ~

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

GreenOs t r a n s f o r m a t i o n o f t h e problem~ For any ~ ~ ~ , l i n e a r t r a n s f o r m a t i o n from

~t-~

(A)into

~

If G

L

Moreover ~

is formally 8elf-adjoint

maps any (4) 9 i . e .

i s a bounded

H~,(A). Assuming

see t h a t we may r e g a r d G as a compact t r a n s f o r m a t i o n of (see t h e o r ~ 3 ~

G is called the

,~: ~'m , we

H o into itself

~oo f u n c t i o n i n t o a C ~ f u n c t i o n . L ~=

L

, t h e n , as i s e a s i l y s e e n ,

is hermitian, i~

L e t us n o t c o n s i d e r t h e f o l l o w i n g problems

00.5) where ~ i s any complex c o n s t a n t . The h y p o t h e s e s on

A

,

L

, ~ ,

V

are

t h e above s p e c i f i e d ones. The f o l l o w i n g problem i s p e r f e c t l y e q u i T a l e n t t o problem (10o5)Z

(lO.8)

~ + ),G,.P ;

q e C~176 ipJ ~-i91

(~ )When we say t h a t

L i 8 f o r m a l l y 8 e l f - a d j o i n t we mean

eq

( x) ~

(-~)

~, C~,).

-77-

in the sense that is

a solution

there

if

of (10,5)

corresponds If

f

.,,

a solution

~

c~-

C t~

C ~ ~ (~),then a n y belongs to

~ ~ ~- ~ G ~ and so o n .

of (10,6)

and c o n v e r s e l y ~

a solution

(10o6) necessarily

then

c~ i s

o~ ~

C ~ (~).

belongs to

H~

Thus we c a n c o n s i d e r

following

~

of (10.5)

of (10.6) o solution

QO o f t h e

In fact~

(A)

theory.

if

equation

c~ ~' ~ ~ ( A )

and hence to H~,~ (A)... (10,6)

in the space

We a r e t h e r e b y

~(A)

led to the

theorem:

IO.V. whenever

We have one and only one solution of problem (10.5)

~

is not an eigenvalu e for this pr.obl.e.m, i.e. whenever

is suc.h..thatt assuming 9 0

t~ -- G c~

to any solution

equation

w h e r e we c a n a p p l y t h e R i e s z - F r e d h o l m

then

.

f -= 0

~ (10.5) has only the t_rivial solution

The set of these eigenvalue,s is a set with no ,limit-points

in ,any bounded set of the complex plane. We w i s h now t o showy as a f u r t h e r to give an existence equations

connected with elliptic

d u c e d by V o l t e r r a theory

and u n i q u e n e s s

by some a u t h o r s

A

These equations

C~176 O~

on t h e c o e f f i c i e n t s

Pq

(X)

and t h a t

possible

were intro-

ago~ i n t h e f o u n d a t i o n

for instance 9 with linear

is

is

integro-differential

and h a v e b e e n r e c o n s i d e r e d

in connection~

L e t us a s s u m e t h a t hypotheses

years

elasticity~

how i t

theorem for linear

operators,

more t h a n f i f t y

of hereditary

application9

of his

also

recently

visco-elasticity, the above mentioned

of the elliptic

operator

L

(OjT)

let

and on ~/" h o l d o For any us consider

X E ~

and any

t

the integro-differential

in the closed

interval

equation

t

(x0.7) L ~ (~,~) ~

Z

~ Ct,..~:)D~u.(x,~ldv 0

-78-

The d e r i v a t i v e s

~d

must be u n d e r s t o o d t o be t a k e n w i t h r e s p e c t t o

the space-like coordinates a r e complex ~ ~ ~ (O~T) •

(OjT).

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

g~ j . . . ) g ~ 9

matrices,

f o r any

f~X,t)

t ~ (O~T)

space-like derivatives

~d

in

i s s u p p o s e d t o be

C ~

as

and c o n t i n u o u s t o g e t h e r w i t h a l l

A

)<

(o~-r),

Of c o u r s e ~ what

we a r e g o i n g t o s a y h o l d s u n d e r more g e n e r a l h y p o t h e s e s f o r and

(~/~ )

continuous over the c l o s e d square

The f u n c t i o n

•

The m a t r i c e s

~

~

~ o The p r o b l e m c o n s i s t s

one s o l u t i o n

t~ ( x ~ ' t )

in proving that there exists

one and o n l y

of ( 1 0 . 7 ) w h i c h , f o r any given

~ > 0

when r e g a r d e d as a f u n c t i o n o f ~ i s a c o n t i n u o u s f u n c t i o n of Set ~tegral

L ~(x)t)=

~

with values in the space

Those v a l u e s b e l o n g t o

~o(x,~).It

H ~ (/~) j

M (V).

is readily seen that the following

e q u a t i on i n t h e new unknown f u n c t i o n

C~ ( X ) ~; )

t

(lO.8)

t•

I)

) 9 7-.

:

d~ 0

i s e q u i v a l e n t to (10.7) with the "boundary c o n d i t i o n " f o r any

t

~

Co, T ) ,

This means t h a t t o any

s o l u t i o n of ( 1 0 . 8 ) , t h e r e c o r r e s p o n d s a s o l u t i o n g i v e n by

G~(~,t)

~(X,~) ~(x,~)

~,(g)~:)

of ( 1 0 . 7 )

there corresponds a solution

of ( 1 0 . 8 ) b e l o n g i n g t o

H~_~

L e t us w r i t e ( 1 0 . 8 ) as f o l l o w s .

t

(10.9) J

o

f o r any

M(V)

Thich i s a

and b e l o n g i n g - as a f u n c t i o n o f )( - t o

C o n v e r s e l y , t o such a : Lt~(x,t)

~()(~t )~

M (V). c~(•

~ ~ ~ .

-79-

.here

~ ~)

with values in

and

~= ( ~

must be u n d e r s t o o d t o be f u n c t i o n s of

~

(A) (K-~-~), N(t,~) i s , f o r any ( t , ~ ) E k E (o~T) :~ ( O ~ T ) , a l i n e a r bounded o p e r a t o r from H K ( A ) in itself, such t h a t [I ~ ( ~ j ~ ) [ I The c l a s s i c a l

-~ C

( C

constant).

s u c c e s s i v e a p p r o x i m a t i o n s method~ used f o r t h e

" s c a l a r " V o l t e r r a i n t e g r a l e q u a t i o n works i n e x a c t l y t h e same way f o r e q u a t i o n (10.9) and proves t h e e x i s t e n c e of one and o n l y one s o l u t i o n f o r (10.8)~ i . e .

f o r our i n t e g r o - d i f f e r e n t i a l

Bibliography

See b i b l i o g r a p h y of l e c t u r e 9.

of

Lecture

10

problem.

80

Lecture

The

classical

11

elliptic

2nd

order

BVP

of

linear

Mathematical

Physics:

]?DE.

L e t us c o n s i d e r a 2nd o r d e r l i n e a r

elliptic

equation with real

coefficients:

L u.

(11.1)

'a

The unknown f u n c t i o n ~i'

b~ , C ~ ~

o. j (x) - .

~ i s now a r e a l - v a l u e d

lecture.

We assume t h a t

L

is elliptic

form

~

~X)

The f u n c t i o n s

C ~ , The bounded demain A i s

These h y p o t h e s e s w i l l

means t h a t t h e r e a l q u a d r a t i c for every

+ c(x)u, = ~(x).

function.

are supposed to belong to

s u p p o s e d t o be C ~ - s m o o t h . this

, b., cx)

be m a i n t a i n e d t h r o u g h o u t

and p o s i t i v e ~

in ~

is positive

consider the Dirichlet

problem for

( 1 1 . 1 ) . As a

c o n s e q u e n c e of t h e t h e o r y d e v e l o p e d i n t h e p r e v i o u s l e c t u r e s , the following existence

and u n i q u e n e s s t h e o r e m f o r t h i s

Under t h e a b o v e - m e n t i o n e d h y p o t h e s e s on

under the further exists

definite

X ~ A0

L e t us f i r s t

ll.I.

. This

assumption that

one and o n l y one

C ( X ) ~_ 0

C co f u n c t i o n

~(X)

f o r an y

we h a v e

problem.

~

and x E A

such t h a t :

A

and

, there

-81 -

(11.2)

L u, .-..~

Let us first

*hA

suppose that

(n .4)

for

C

(11.3)

,

on 9 A

~,--0

X ~ ~

~> O.

9•

s 0

We h a v e f o r

-

~ H.~ (A)"

any

]~ ("" " ' ) :

~

"ax.

0~:

,~. i ~ Ox;

dx

v

c

Coll~ll

>

4'

-

A as f o l l o w s

from the

theorem is

proven under the assumption

C (X)

~ 0

ellipticity

follows

from theor.

] t ~ ( X ) l ~- ~ • l~(X)] • 9A smooth solution (see,

for

Let

instance~

[7]

on

~

9

for

the

operator

(11.5) Let

)

another

IO.V a n d f r o m t h e

The f o l l o w i n g

L~,-

"~-: ( ~ 4 J ' " , ~

in the

case

inequality

knoTn - holds L~t -- 0

when

classical

BVP f o r

the

r,egula r oblique

be a r e a l

when c o n s i d e r e d

L

The p r o o f

Thus t h e

for

any

C { x ) -~ 0

~ p. 4-5).

the so called

~ ~ ( ~ls,..~ G ~

(11.4).

of the homogeneous equation

(11.1)~

which is

a n d f r o m lemma 3 . I .

which - as is well

L e t u s now c o n s i d e r equation

of

unit

vector

as a f u n c t i o n

derivative defined

for

of the point

BVP i s known as t h e

elliptic

oblique

problem. any X

•

varying

derivative

problem

:

in

A

,

) be t h e i n t e r i o r

(11~ unit

--:0 O~

normal to

on

/~A

9A,

9 Under the

-82-

further

assumption

~

>0

C~•

~ 0

, the problem ( 1 1 . 5 )

( 1 1 . 6 ) i s said to

be r e ~ u l a r . 11oli. and o n l y one

I__f

for

X~ ~

~ then there exists

~ ~

s o l u t i o n of t h e r e g u l a r q b l i q u e d e r i y a t i v e

~i

be a r b i t r a r y

one

problem

Ol.~) (l~.~). Let such t h a t

~i

(ll.l')

functions belonging to

~i $ . The o p e r a t o r

Lu.

L

can be w r i t t e n as f o l l o w s :

~;i

:

C~A)

""

+c~

,

where*

I t is p o s s i b l e to choose the f u n c t i o n s

where ~

is a positive

Q ~

scalzr

~i

i n such a way t h a t :

f u n c t i o n d e f i n e d on

o r d e r t o p r o v e t h i s ~ l e t us f i x a r b i t r a r i l y Let

~

~...~ ~

the point

~A.

•

In

on ~ A .

he an o r t h o n o r m a l s y s t e m of v e c t o r s such t h a t

coincides vdth the interior

normal v e c t o r

may s u p p o s e t h a t t h e f u n c t i o n s and b e l o n g t o

~

(~).

~lj..

,~

~

at

~ ~

~A

in

X.

We

are defined throughout

There i s no l o s s i n g e n e r a l i t y i n

a s s u m i n g t h i s ~ s i n c e we can always c h o o s e

~

to

t h e above ~ f ~ f u n c t i o n s

~ ~ (~)

throu~ou~

since,

and t h e r e a f t e r A

.

continue all

We can a l s o s ~ p ~ o s e t h ~

by t h e r e g u l a r i t y

consider the functions

condition~ it ~hk

)...~ ~

~'

is positive

d e f i n e d as f o l l o w s i n

belonging

is positive

on ~ A . A

:

i~

L e t us

-83-

4

CL--

~).

I

,.~ --,) ~

~).

- o,,.. v'.

( fo~

R k~

I - 0 -

:

~ : ~ ~.a.

4
( fo~"

Rkh

I

K

~).

~ ,~ k )

~- k )

( fo~ ~>k).

The f u n c t i o n s = h #

.

k

,

9

J

satisfy

equations

(11o7) w i t h

given by~ 4

I

J

#

~v 4

L e t us c o n s i d e r f o r

~)~

E ~t ( ~ ) the bilinear

form=

/Jx;

Let

p o ) P, ) ra~

be p o s i t i v e

n u m b e r , such t h a t

for

every

X ~

we have=

(~ : %2,...,=) It follows thats

, -c

_> e~

-84-

-Ov-

+

Ivl )ax.

A Then f o r

PZ l a r g e

enough,

-!5(v-,v,) >_ Co iIwll~ for any

~" ~

of a unique

for ~y

]'-]t ( A ) 9

function

From t h i s

Lt ~

V r H~(A)

inequality

~ 4 (A)

satisfying

Since

=- ~ / .

is a solution of ( l l . 5 ) , ( 1 1 . S ) . in the present

case,

the boundary conditi?n o n l y show t h a t unique if using results Since of

C K O

on

theor,10oV, on l i n e a r C~ 0

L~ : o

(11,6),

A .

elliptic

in the

follows

(11.9) that

In other words, the class C c~ f u n c t i o n s

M (V)

which satisfy

In order to prove our theorem,

of the regular In fact)

The d e s i r e d

, it

, i t follows from ( I I . S ) ,

composed of the

the solution

by ( 1 1 . 7 ) ,

have,

OA

B e to the arbitrariness of ~

is)

existence

the equation:

~ C~(A)we

~

A

follows the

the

oblique existence

uniqueness

is

derivative will

we ~ e e d BVP i s

f o l l o w by

a consequence of well-known

PDE's o f s e c o n d o r d e r , that

(connected)

f o r a n y s m o o t h non v a n i s h i n g domain

A

:

[ ~ (~) I (

solution

~6tX xr ')A

[ t~ [

-85-

for any

• E ~ .

such a solution

On t h e

o~her hand, if

ut

~A

on

mentioned uniqueness

If direction

~i

direction

~

i n s u c h a way t h a t g i v e n by ( 1 1 . 7 )

and t h e r e s u l t i n g

of

/~

BVP i s c a l l e d

such that,

the neighborhood

~

/~4 A

if

vector defined on

/~

~

The f o l l o w i n g

(11.10)

L ~

= ~-

~ ~.

the regularity

of

=

(K = ~,g

stated

~ ~ /~/~

Let ~

condition

in

A

,

~:

0

We u s e t h e r e p r e s e n t a t i o n (11.7).

on / ' ~ A

in the space

composed of the functions

satisfying

/~K ~

at the beginning

~K~,

C

be the

C ~ unit

~)

> O

,

/~

on / J A .

H~ ( A )

which vanish

(11.1')

of

L C0

on

-0

on

/~4(A).

with the coefficients

By t h e same a r g u m e n t u s e d i n t h e o b l i q u e

p r o b l e m , we s e e t h a t : 2

%r ~ ~/

, when

have a unique solution of ~

9

-C

~ ~

of (11.8),

with

p~

corresponding

~,zA.

of the subclass

c o Ii ll for any

) )

BVP i s known as t h e m i x e d BVP:

As s p a c e ~/~ we t a k e now t h e c l o s u r e C I (~)

a r e two d i s j o i n t

(already introduced i n the oblique d e r i v a t i v e

problem) which satisfies

derivative

~

X~ i s any p o i n t

~

) then

t h e Neumann BVP.

of x ~ e n j o y i n g t h e p r o p e r t i e s

~oreover, we assume t h a t

o~ ~

and

: c~j~

~]

i s known as t h e c o n o r m a l

8, c a n be c h o s e n i n s u c h a way t h a t

of l e c t u r e

of

a maximum (minimum) f o r

) by a t h e o r e m o f G i r a u d ) we m u s t h a v e

L e t u s now s u p p o s e t h a t subsets

is

follows.

we c h o o s e t h e

the oblique

•

large

enough.

to the present

Then we choice

-86-

The c l a s s lectures

V satisfies

the conditions for the regularisation

9 and 10 i n t h e n e i g h b o r h o o d of any p o i n t o f /~4A

t h e o r y of and o f ~ A ,

Hence, t h e s o l u t i o n of t h e mixed BVP has t h e f o l l o w i n g r e g u l a r i t y properties =

If

i)

belongs to

C ~ ( A ) n H~(A).

ii)

belongs to

c

~A

-- ~)t A u ~ A

, then ~

,).

( A is

o t h e r w i s e t h e o n l y p o i n t s where ~

~ ~ i n the c l o s e d domain

A,

c o u l d n o t be C c~ a r e i n t h e s e t

I t has been p r o v e n t h a t ~ i s c o n t i n u o u s i n t h e c l o s e d domain

(see ~3 ]

)*

B i b l i o g r a p h y of L e c t u r e

Eli

G.BOULIGAND-G.GIRAUD-P~

-Le

en t h ~ o r i e du p o t e n t i e l

11.

probl~me .de l a d ~ r i v 6 e o b l i q u e -

Actual. Scient.

Industr.

Hermann~ P a r i s ~ 1935.

E2]

G~

- . S u l p r o b l e m a d e l l a d e r i v a t a obl.igua e s u l p r o b l e m a m i s t o p e r l ' e q u a z i o n e di L a p l a c e - B o l l e t t ~ U n i o n e btat. Ital.

Is]

1952.

GoFICtlERA- A l c u n i r e c e n t i

sviluppi della teqria

contorno, etc -Atti 1954-

Cony~ I n t e r n .

Cremonese~ Roma~ 1955.

dei Pr0blemi al

sulle

l~q. Der~ P a r z ~

-87-

[4]

G.FICIIEI~- Analisi

esistenzial.e

contorno misti

E53

G.GIP~UD

-

S c u o l a Norm~ Sup~ P i s a ,

principales

-Ann.

1947.

E c . Norm.

51, 1934.

-Ann~

closes

s o c . P o l o n . Mathem., 1932.

C.~IP~NDA- E~uazioni a l l e

deriva.t.e p a r z i a l i

Ergeb~ S p r i n g e r ,

[8]

dei problemi al

Probl~mes m i x t e s eL p r o b l ~ m e s s u r des v a r i e t ~ s etc.

LT]

etc~ -Annali

- E~uations a int~rales Sup. t .

GoGIP~UD

pe.r l e s o l u z i o n i

di t i p o e l l i t t i c o

-

1955.

C.~IItA~DA- Sul pro, blema m i s t o p e r l e e~uaz.i.oni l i n e a r i

ellittiche-

Ann. d i Matem. , 1955~ G.STAJ~PACCHIA- P r o b l e m i a l c o n t o r n o e l l i t t i c i

c,on ,dati d i s c o n t i n u i ,

I!

dotati

di s o l u z i o n i

holderiane-

Ann. di b~atem~ 1960o

-88-

Lecture

12

The cl.as.s.ical e . l l i ~ t i c ~ P

of ~ a t h e m a t i c a l P h [ s i c s :

9i n e a r E l a s t o s t a t i c s .

We s h a l l now c o n s i d e r t h e c l a s s i c a l t h e c a s e o f an

lnhomogene~us a n i s o t r o p i c

BVP of l i n e a r elastic

elasticity

in

body. I t i s c o n v e n i e n t

t o s t u d y t h e BVP c o n n e c t e d w i t h t h e e q u i l i b r i u m problems i n t h e s p a c e X ~ i n o r d e r t o i n c l u d e b o t h t h e c a s e s of p l a n e and 3 - d i m e n s i o n a l e l a s t i c i t y . Set-

'~x k L e t us c o n s i d e r t h e e l a s t i c 41~

The ( r e a l - v a l u e d )

~(~§

potential 4,,~

f u n c t i o n s ~ ~k.i~(X) a r e s u p p o s e d %o b e l o n g t o

and t h e q u a d r a t i c form ~ r in the

'gx;

variables

(X,6) s

We can s u p p o s e t h a t ~

(x)

i s s u p p o s e d t o be p o s i t i v e (4~L~

~ ~ ) f o r any

~g~

C

definite ~

-89-

L e t us now d e f i n e f o r

arbitrary

v a l u e s of t h e

i n d e c e s 4 ...) 9 :

: (X)

CL

~h~,,jK ( g )

for

~ > i~)

-. ~ h , K i (X)

for

~ ~_ k s J > K

._ O~h~,Ki (X)

for

~ > i~ ,

i >~

"ZO~;h,j ~ ( X )

s

i,= ~ ,

i " K.

We have f o r t h e e l a s t i c

potentialz

4_ C~ c ~,i

(12.1) V q ( •

j "- K

6. ~

~

f. ~

=_ zct~,J

~ ~x~ ~ x

I t must be p o i n t e d o u t t h a t t h e q u a d r a t i c form

~;h,j~

i s .not p o s i t i v e

definite,

as a f u n c t i o n of t h e

real variables

~ ~h

(~ j h 9 4 j . . - ~ ~ ) .

i n t h e s u b s p a c e of t h e the conditions Let

A

~

but o n l y s e m i d e f i n i t e

"-

be t h e

~-dimensional ~h~

~Lh ~ K

It is positive s p a c e of t h e

definite ~h~s

only

d e f i n e d by

d

C ~ - s m o o t h domain c o n s i d e r e d i n l e c t u r e

11.

The e q u a t i o n s o f e q u i l i b r i u m i n A a r e t h e f o l l o w i n g :

(12.2)

~

~

-

in

.

We have t h r e e k i n d s o f b o u n d a r y c o n d i t i o n s c o r r e s p o n d i n g t o t h e t h r e e main p r o b l e m s o f e l a s t i c i t y . boundary c o n d i t i o n s .

We c o n s i d e r h e r e o n l y homogeneous

-90-

1st

BVP

(12.3)

L~ = 0

2nd

(12.4)

( ~

(body f i x e d a l o n g i t s

BVP

on

/~/~.

(body f r e e a l o n g i t s

"~,~ (u.) - v k

is the unit innrd

boundary)

boundary)

W ' (x, c ) -- o on

normal t o /~A

)

ard 8VP (.dxedBVP) (12~

where

t~:O

/~4A

in lecture

on /~4 ~

and

/~zA

,

(12,,6)

"~ (1~,)= 0

a r e t h e s u b s e t s of ~

on f~DA,

a~ready introduced

11.

Other BVP's c o u l d be c o n s i d e r e d , components o f

~

shall

o u r s e l T e s t o t h e t h r e e aboYe c o n s i d e r e d c a s e s and leaYe

it

restrict

and

~-~

f o r i n s t a n c e t h e ones a s s i g n i n g p

components of

t(~)

on / ~ A .

HoweTer Te

as an e x e r c i s e t o t h e r e a d e r t o s t u d y o t h e r BVP's f o r ( 1 2 . 2 ) .

Equations

(12.Z) c a n be w r i t t e n :

(12.2')

Cl,~k,i x (X)

~X k

In order to prove t h e e x i s t e n c e BVP ( 1 2 . 2 )

(12.3),

"F ~;,

: O,

~x~

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

Te need t o p r o v e i n e q u a l i t y

case is the follo~ng:

(9.1),

of

which i n t h e p r e s e n t

-91 -

f

(1~.7)

~'A

~

J i,~

~c gxk

(x)

~nxj ,9 xK

dx >_ c

Ilu'llt o

0

f o r any

%r s

H 4 ( A ) , B e c a u s e of ( 1 2 . 1 ) i n e q u a l i t y

as t h e 1 s t Korn~s i n e q u a l i t y

J"'(

(12.s)

~

( 1 2 . 7 ) - which i s known

- i s e q u i v a l e n t t o t h e f o l l o w i n g onel

~'~ + ~ ~"

d•

> C,I IIv II

(c, > 0 )

T h i s i s i m m e d i a t e l y o b t a i n e d by u s i n g t h e F o u r i e r d e v e l o p m e n t s o f t h e functions

~r. and P a r s e v a l ' s

theorem.

I t must be o b s e r v e d t h a t , t h a t f o r any

• ~ A

as a c o n s e q u e n c e of ( 1 2 . 7 ) , i t

and any n o n - z e r o r e a l

~

follows

I

>0

for every non-zero real t h e o r e m which w i l l particular,

~-vector

be p r o v e d l a t e r

the ellipticity

~

.

T h i s i s a c o n s e q u e n c e of a g e n e r a l

(see theor.

1 4 . I I of l e c t u r e

of system (12.2) follows.

14).

In

We h a v e t h u s t h e

following theorem.

12.I.

Given

~ ~ C oo ( ~ ) , t h e r e e x i s . t s one and o n l y one s o l u t i o n

of the BVP (12,,2), (12.8) ,_ _wh_ich belonE~ t._.__o C " ~ ( A ), I n o r d e r t o p r o v e t h e e x i s t e n c e t h e o r e m f o r ( 1 2 . 2 ) ~ ( 1 9 . 4 ) ~ l e t us c o n s i d e r t h e systems

(12.9)

~ rbx~

cL~, j, ~ ( ' ~

/~;,(,r

- po-.

, ~

:

o

-92-

where

~o

(12o4).

is any positive constant~ We wish first solve problem (12.9)

I t i s e a s i l y seen t h a t the i n e q u a l i t y to be proven i n t h i s

case is the folloxing

~-.

+

x +

dx

~

C~.ll,~ I1~

A

q~ E H 4 ( A ) . S e v e r a l

the original

proofs

of ( 1 2 . 1 0 ) have been g i v e n a f t e r

one due t o Korn [ 5 ] ( s e e [ 2 ] ~ [ 6 ] ~ [ 3 ] ) .

one c a n be o b t a i n e d ( s e e [6]) i f We r e f e r

)

I~

A for any

(IJ.

(2nd K o r n ' s i n e q u a l i t y

A is

A rather

~-homeomorphic

simple

to a closed ball.

the reader to the quoted papers.

From (12o10) i t which i s

~

adjoint,

it

in

follows that

(12,9)

( 1 2 . 4 ) has o n l y one s o l u t i o n

A o S i n c e our d i f f e r e n t i a l

follows that

a

C~

solution

system is formally self-

of t h e f o l l o w i n g d i f f e r e n t i a l

system:

(12o11)

w i t h t h e boundary c o n d i t i o n s

(1 .12)

,

~

(12.4),

fA

;

exists

when and o n l y when:

o

, m ,

(4) A c t u a l l y t h e 2nd K o r n ' s i n e q u a l i t y

is the following:

(% ",o) f o r any qY such t h a t :

, However i t

is easily

seen that this

;0

inequality

i m p l i e s and i s i m p l s

by ( 1 2 . 1 0 ) .

-93-

~Z i s a n y

C ~ solution

the only

C ~ solution

There

and B~i

~,

12.II. if rith

~

(12.4) Tith

: 0.

I n t h e c a s e ~ ~ Po

constants

such t h a t

b~j = - ~'.

~ P (12.2) (12.4) has solutions belon~in~ to C ~ C ~ ) C~

function

.~

satisfies

c.onditi,ons

(12.12)

g i v e n by (12.1.3).

For g e t t i n g BVP ( 1 2 . 2 ) ,

t h e e x i s t e n c e and u n i q u e n e s s of t h e s o l u t i o n

(12.5),

(12~

r e assume t h a t

composed of t h e f u n c t i o n s v a n i s h i n g on inequality any

~

of t h e homogeneous s y s t e m i s :

are arbitrary

and on,ly i f t h e ~

of ( 1 9 . 1 1 )

(considered

~ ~ ~/

for any

~ inequality

V

/~

~r~ V

of t h e

i s the subspace of H 4 ( A ) . From the second Kornts

) it

is easy to derive~ for

( 1 2 o 8 ) . A r g u i n g as i n t h e p r e v i o u s l e c t u r e

f o r t h e c a s e of t h e mixed BYP f o r a 2nd o r d e r e l l i p t i c

equation~ re

deduce t h e f o l l o w i n g t h e o r e m : 12~

For

~ ~ C ~ 1 7 6 ) t h e r e exist.s one and o n l y one . s o l u t i o n

of t h e BYP (1.2.2). (12..5) It

is rorthrhile

(12o6)tT.hich b e l o n g s t o C ~ (A u g 4 A u ~ A ) n

t o remar~ t h a t

the obtained solutions

l s t ~ 2nd and 3rd BVPts a r e t h e ones r e q u i r e d of e l a s t i c i t y 9

s i n c e t h e y m i n i m i z e t h e ener~T i n t e g r a l

H 4(A) ,

H4(A)

and "v"

of t h e

by t h e m a t h e m a t i c a l t h e o r y

O

in the classes

~4(A).

respectively.

-94-

Bibliography

[1]

G. F I C I I E R A - s e e [ 2 ]

of

of lecture

Lecture

12.

4.

K.O.FRIEDRICttS - .On t h e . B o u n d a r T - v a l u e ' P r o b l e m s o f t h e T h e o r y o f El a s t i c i t ~

v.

[3]

J.

GOBERT - U n e

48,

A. KORN

-Annals

of Math.

1947.

in~alitd

Bull.

[4]

.and K o r n ' s i n e q u a l i t y

f o n d a m e n t a l e de l a t h ~ o r i e

Soc. Roy. des Sci.

- Solution ~enerale du

de l ' ~ l a s t i c i t ~

de L i b g e - 3-4 - 1962.

pro bleme .d'e~uilibre dan.s, la

theorie de l'~lasticitd dans le cas ou les efforts sont donnds & la surface - A n n .

A. K01~

Toulose Univ. 1908.

- U.eber einige UnMleiclmngen welche in der Theorie der

elastischen un_d_ele~trischen Schwin~un~en ein.e.Rolle spielen - Bull. Inst. Cracovie Akad. Umiejet, Classe des sci.

math. et nat.

, 1909.

L.E.PAYNF~-H.F.WEINBERGER - On. I{o..rn's Inequality - Arch. for Rat. Mech. & A n a l .

8,

1961.

-

-95-

Lecture

The

classical

elliptic

Equilibri~

The c l a s s i c a l the solution two v a r i a b l e s

13

BVP of

of

~athematical

thin

Physics:

~lates.

t h e o r y of t h e e q u i l i b r i u m of t h i n p l a t e s

of c e r t a i n x,y

BVP's f o r t h e i t e r a t e d

Laplace operator in

:

~4

4 ~ xZ,~y z

9 x~

w i t h s e v e r a l k i n d s of boundary c o n d i t i o n s . bounded ( c o n n e c t e d ) p l a n e

,9 y~

Let, us

is a

C~176

plates

c o n s i d e r s t h e f o l l o w i n g boundary c o n d i t i o n s

suppose t h a t

domain.

~ : o,

(ta.3)

(13.~)

~ ~

~,~

(13.4)

on ~ A

-- o ,

: o,

9~ ~

A

~

~ + (4- e )

~

ov

0

A

The t h e o r y of t h i n

f'j~

(13.1)

requires

-96-

Here

~

i s t h e u n i t innward norma~ t o ~

denotes differentiation

w i t h r e s p e c t t o t h e arc ( i n c r e a s i n g c o u n t e r - c l o c ~ l s e ) c u r v a t u r e of

~A

~

i s a c o n s t a n t such t h a t

~

is the

- t < 6-~ 4

is the Laplace operator, The d i f f e r e n t i a l and

~ /

e q u a t i o n t o be c o n s i d e r e d i s t h e f o l l o T i n g (

real valued functions).

A4

(13.5)

The BVP ( 1 3 . 5 ) ,

(13.1),

:

(13.2) corresponds to the e q u i l i b r i u m p r o b l e m

f o r a p l a t e clamped a l o n g i t s

b o u n d a r y . The boundary c o n d i t i o n s

(13.3) express the f a c t t h a t the p l a t e is supported along i t s boundary c o n d i t i o n s

is free.

to the consideration

ourself

( 1 3 . 2 ) and t h e mixedBVP f o r a p a r t i a l l y and ( 1 3 . 2 ) a r e s a t i s f i e d the remaining part. i n l e c ~ a r e 11.

e d g e . The

( 1 3 . 3 ) and ( 1 3 . 4 ) mean t h a t t h e p a r t o f t h e b o u n d a r y

There t h e s e c o n d i t i o n s a r e s a t i s f i e d We r e s t r i c t

(13.1),

on a p a r t

~4~

and

~

of t h e BVP ( 1 3 . 5 ) , ( 1 3 . 1 ) ,

clamped p l a t e ,

~4 A

o f HA

i oeo when ( 1 3 . 1 )

and ( 1 3 . 3 ) ,

a r e the s u b s e t s of ~A

( 1 3 . 4 ) on considered

The r e a d e r i s r e q u e s t e d t o c a r r y out t h e p r o o f s of t h e

e x i s t e n c e and u n i q u e n e s s t h e o r e m s i n t h e o t h e r c a s e s , f o r i n s t a n c e i n t h e BVP c o r r e s p o n d i n g t o a p l a t e p a r t i a l l y s u p p o r t e d and p a r t i a l l y

As b i l i n e a r (13.2)

form

clamped on ~

, partially

free. b(~,~r)

c o r r e s p o n d i n g t o t h e H~P ( 1 3 . 5 ) , ( 1 3 . 1 ) ,

we a s s u m e "

(

§

~x ~.

)oixol

-97-

The s u b s p a c e imequality

V

H~IA)

of

0

(9.1) reduces to inequlity

6(,,,,,-)

FI~ (A).

t o be c o n s i d e r e d i s (3.6)

for

: 2

~

ii~-, &~

~ ~o

In this

case

Thus Te h a v e

[,,,- ~ H,(A)~ ]

Then-

13.i.I.G.r

~ ~ C ~ C~,) , ~ r

(13.5), (18.1), (!3.a)

I n o r d e r t o c o n s i d e r t h e above m e n t i o n e d mixed BVP ( i . e . on

07 A

and c o n d i t i o n s

convenient to observe that~ for

o.e

C ~ ( ,~ ) ,

and, on,l~r one s,o l u t i o n b e l o n g i n g t o

(13.1) , (13.2)

has

~

(13.3),(13.4)

and ~

on ~

belonging to

conditions ) it

C ~ (A)~

Te h a v e :

0 9'~

.

io [ 9~

2

+~"

'g,7 :~

(

tabu"

ma:~u, Ov '3r

]

al~

-

I ~u. )]ol~

:

OA

A

~gx ~ tax=

Ou. +

~x 2 Oy ~

9,/:~ Oys

qlx 9 y "ax~y

,9 ,,/z '3x :L

u. d x d y .

dxdy A

is

-98-

L e t us now assume as s p a c e of t h e f u n c t i o n s which s a t i s f y

V t h e s u b s p a c e of

conditions

H~ ( A )

( 1 3 o l ) , ( 1 3 o 2 ) on

+ (~- ~

+

~z

t~

~2t~

i

Because o f t h e a s s u m p t i o n

C (~)

9xgy

~ztt

@7:L + 9 yz Qx z

-4

~• ~7-

"~ ~ 9 ~] ~ "are h a v e :

Ipl:~ The c o n s t a n t

~4/~, S e t

+

~•

A

composed

IAIDr~I~ c~x dy.

depends o n l y on ~ .

I n o r d e r t o p r o v e ( 9 . 1 ) ve n e e d o n l y t o show t h a t t h e r e e x i s t s C4> 0 such t h a t ~ f o r any ~ - E ~ r

(z3~)

~

I I D%I =axdv

~- e~

I1~,~.

I A Suppose ( 1 3 . 7 ) t o be f a l s e .

(13.8)

]j ~

Jl~. = '{

)

Then t h e r e e x i s t s

(13o9)

Z~ Ipl= ~-

such t h a t =

I ID~ l=olx oly ~t

A

-99-

We can suppose t h a t

for (13.9),

t qY~ ~ converges in

converges in

(A) and the li

has strong second derivatives vanishing on

that in

~ ~

i s a p o l y n o m i a l of d e g r e e .

This contradicts

(theor.

~ 4(A)

3 . I V ) . Then ,

t function

A . It follows readily

~ . Because

tr

belongs to

%r

~r-O

(13~

Since (9.1) has been proved~ there exists one and only one

solution

~

of t h e e q u a t i o n s :

belon~ng

to

V.

By u s i n g

(13.6),

The f u n c t i o n it

folloTs that

~

belongs to ~

C co ( A u Q ~ A u Q ~ A ) .

is the solution

of our mixed

p r obl em. 13.II. ~

The mixed ~ P ,

(13.5);

~ E C~176

(13.1),(13.2)

on ~ A

~

(18.3),

, has..one..and o n l y one s o l u t i o n

b e l o n g i n ~ ~,0 C ~ 1 7 (6A t~ ~4A t) ~#.A) t~ H I (~). From len,na 4 . 1 I i t

follows also that

Bibliography

[13

Lecture

G. FICHERA - Teorema d'e_sisten.za p e r i l Rend. Acc.

[23

of

~t

G. FICHERA -

Naz. Lincei~

belongs to C~

),

13

p_roblema b i - i p e r a r m o n i c o -

1948.

On some g e n e r a l i n t e ~ r a t i . o n methods employed i n connection with linear Jour.

differential

e~uations -

of Math. and Phy~ voI.XXIX ~ 1950.

-100-

G.

FICHERA

-

Esistenza calcolo

[4]

d e l minimo i n u n c l a s s i c o delle

variazioni

problema di

- Rend. Acc. Naz. Lincei~

1951.

K.0.FRIEDRICIIS - Rie RandTert- und Ei~enwert Probleme aus der Theorie der elastischen Platten - b~th. Annalen,

G. ~ D I N I

- I1 p r i n c i D i o

d i minimo e i t e o r e m i

per i problemi alle

derivate

al contorno parziali

di

relativi

di

esistenza

alle

ordine pari

1928.

e~uazioni

- Rend. Circ.

Matem. P a l e r m o ~ 1 9 0 7 .

A.E.H~LOVE - A

Treatise

on t h e ~ [ a t h e m a t i c a l T h e o r 7 o f E l a s t i c . i t y

Cambridge at the UniT. Press

-vol.I~

1893.

-

-101-

Lecture

Strongly

elliptic

14

o

operatorso

.Garding

inequality.

Eigenvalue . problems.

The e x i s t e n c e t h e o r y d e v e l o p e d i n t h e p r e v i o u s l e c t u r e s on i n e q u a l i t y

(9.1).

We p r o v e d t h i s i n e q u a l i t y

lar cases considered in lectures a general operator.

11, 12, and 13.

Of c o u r s e t h e p o s s i b i l i t y

on t h e c h o i c e of t h e s u b v a r i e t y

for all

~r

any c a s e , b e c a u s e o f t h e a s s u m p t i o n

(A)r

of the p a r t i c u -

We wish n o t c o n s i d e r

o f p r o v i n g ( 9 . 1 ) depends

of t h e s p a c e H

i s founded

H

~

(A) . HoTever, i n

, inequality

(9.1)

consider here only this

case,

O

must be t r u e Then H

(A)-V.

We s h a l l

which c o r r e s p o n d s t o t h e D i r i c h l e t The m a t r i x d i f f e r e n t i a l

operator

i s s a i d t o be a s t r o n g l y e l l i p t i c any r e a l n o n - z e r o vector

~

Te

~-vector

problem.

~

~ ~ (~5(x)~

~

operator at the point

X , if

for

and f o r e v e r y n o n - z e r o complex ~ -

have:

.~

i~I:v

It is evident that strong ellipticity converse is not true,

'9• 4

implies ellipticity.

as t h e example o f t h e W i r t i n g e r o p e r a t o r

'~ x~.

( 4 ) F o r a more g e n e r a l d e f i n i t i o n of s t r o n g e l l i p t i c i t y , From n o t on . e s h a l l o ~ t parentheses Then ~iting ( Z

s e e ~4 ] .

~ (~)?.

The

-102-

For t h e o p e r a t o r we s h a l l point

L =

assume t h a t t h e s t r o n g e l l i p t i c i t y

x~

~

('o

])Po.,P<~(.x) I)q

~IPl_~~ ,

o ~l"Ile ~ )

c o n d i t i o n holds a t any

, where ~ i s a bounded domain of

c o n n e c t e d , we can suppose t h a t t h e q u a d r a t i c

X~. S i n c e

~

is

form i n t h e complex v e c t o r

P Ipl,l
is definite,

for instance positive,

I -- '~

for

L e t us s u p p o s e t h e c o e f f i c i e n t s operator

k

function

~

~ ~ O~

(Ipl=lql=~)

A.

If

eu ~ 0

of the i s t h e minimum

t h e compact s e t s X~ A , I~1

Q(X,~, ~)on

we have f o r a n y

and a n y

~p~(X)

t o be c o n t i n u o u s f u n c t i o n s i n

of t h e p o s i t i v e I'~ I : 4 ~

• ~ ~

and a n y

~

•- ~j

-"

IpI j l q l -,m.

We c a n n o t e x p e c t t h a t strongly

elliptic,

( 9 . 1 ) h o l d s f o r any k

since this

would i m p l y - f o r

and u n i q u e n e s s f o r t h e D i r i c h l e t What we want t o prove i s t h a t , for the operator We s h a l l

C '~ ~ i

~ even one t h a t

~r = H ~

(A)

problem f o r t h e e q u a t i o n f o r some

~

, inequality

is

- existence

~-~ = ~ . (9.1) is true

( I = identity)~ O

9

p r o v e t h e f o l l o ~ r i n g theorem c o n c e r n i n g G a r d z n g t s i n e q u a l i t y

14.1..There

e x i s t two p o s i . t i v e .cons.rants

~

an_dd ~ o

such that

Q

.fp,r any

Lr ~ ~ ] ~ ( A ) t h e f o l l o w i n g i n e q u a l i t y

?o II

(-4)

The c o e f f i c i e n t s

C~

ll

holds-

~t

- loilvli

(X) a r e supposed c o n t i n u o u s i n

A

o

for

9

Ip~ = I~1 = ~v~

-103-

and m e a s u r a b l e and bounded i n

~

IpI

for

+ ICll <

~ .

Set

=

I~, I ~ ~. (- 4 ) O~

e~

for

t~

Then ve can w r i t e :

D

In the first to

integral

IpJ : J q J = ~ t ~ i n

t h e summation must be u n d e r s t o o d t o be r e s t r i c t e d the second to

From now on, i n t h e c o u r s e of t h i s c a s e where

J ~ l t" J ~ J ~ 2 ~ j

O -~ J ~ J ~ _ ~ t ) O ~ J ~ J ~ t .

p r o o f , we w r i t e

~pq

1 s t case=

CLpq

three cases.

- constant,

~

-= O .

We s u p p o s e , w i t h o u t any l o s s i n g e n e r a l i t y ,

that

A c ~

9 ). 0

By u s i n g ( 8 . 3 ) and ( 3 o 4 ) , Te h a v e f o r

~) ~'IY

_

"J"

-I'~

Z,

i,

IpI

,~ ~ H

P

cKe

~w

(A)

~.Kx

~KX

0.,

only in the

J~J:J~J=~Tt.

The p r o o f i s o b t a i n e d by c o n s i d e r i n g

~.:,~,...

dx.

lY :

K

0.,

C

( (~ ' I X ~ l ~ ~

-i04-

Then, by P a r s e v a l 0 s t h e o r e m , ( 1 4 . 1 ) , and t h e o r o 3 . 1 1 ,

b~.~ Q. - oo

§ ~o

In this

particular

X~

~ o II ~ Ii

z

P

K

q

C

C

I~/

2~

~.

ceme ( 1 4 . 2 ) h o l d s w i t h

2nd case8 Let

K

I I ~ II

~o : ~'o

,

and

~ o : O.

d i e m e t e r s p t ~ ~ o9 o (where ~ o i s a p r o p e r p o s t t i T e n u m b e r ) . spt ~

e,

. We havez

o..

A

I?

~ ~A~

+r215176

(~)D~D~ •

,,~

c~]

Dq~ D~ a~

Pq

-~ IA~-~ e,9

x~

e~l

P~

,~

sp~

i

i

9

(2) D u r i n g t h i s p r o o f and t h e p r o o f o f t h e nex~ t h e o r e m , denote positive constants"

~4 J ~

, ~3 J"

_

-105-

Let

diam. s p t ~ r ~

~o

,'~.a.x ~ x ~= ~ , u "

where

~

i s so s m a l l t h a t :

I ccM(• ~

r,s

Then we haves

-z

From t h i s

~~ ij .u- Ii ~ it

"_ (-~ )

B (%

~e,

"o- )

+~

II 't,-' I1,~ II 't,,- I1,~_.~

follows that:

~

)

:~ls, - 4

From lemma 4 . I I I , 3rd c a s e :

inequality

(14.2)

folloTs.

general case.

L e t us c o n s i d e r t h e f o l l o w i n g p a r t i t i o n

with for

4 h (•

Ix'-x~'l-~ g

Z P~l

~,,o and 0

of u n i t y i n

diem spt ~P (X)~ ~o , where

A : ~

~o

~2(~)-

6=I 6 is such that

one has-"

Io., ~.x~.~.-o,

-~o"

~

Pq

~e hate,-

-- T.

-

Z

c~

I/~h

P~

"

IA

~ D~

-106-

From the 2nd case considered above we have:

A Thus,

,~From t h i s

inequality,

we s e e t h a t

(14.2)

~lltYll

-~911"v'li Iluil

by u s i n g t h e same argument as i n t h e 2nd c a s e ,

follows.

Remark~ R e t r a c i n g t h e s t e p s of t h e p r o o f of ( 1 4 . 2 ) Te s e e t h a t and

~ o depend on Po ~ on t h e modulus of c o n t i n u i t y

u p p e r bound f o r

I(Zpq j

and I ~ t ~ i

possible to express explicitly 14o1Io and on

and on t h e domain A 9 and ~

U,nde,r t h e ,s,ame h ~ p o t h e s e s on

~ ~

~ ~(A),

~

of Cbpq ~ on an It is

i n t e r m s of t h e s e o b j e c t s . c~pq

( IpI =IcJl ~- ~ )

as in the previous theorem~ if (14.2) holds for an~r

then the operator

I

~.

is stron~lr elliptic in

Suppose the theorem is not true. 'l~en (14o2) holds and there exists a real

~

~ 0

and a complex

has

Ipl.,Iql : m~. <)

For

~ ~ N

~vt

(A)

~ ~ 0

such that for some X~

/A

one

~

-107-

~o I1~ I i z_ (.~) J~ ~ (14.4)

"4

Let

~o

be such t h a t for [•

~ ~o :

0

Pq Let

Ix-•176

I o be the ball

(14.5)

(14.6) Let ~ ( •

and lemma 4 . I I I ,

spt~rc

Anl o

)

then from (14.4)~

we deduce t h a t ;

I1~1i be a

P~

f C oo

r e a l v a l u e d f u n c t i o n v h i c h i s not i d e n t i c a l l y

zero and which has i t s support in

~

-- q C x )

e

A ~

Set:

?

~cith ~ r e a l . We have:

q'C~) i~"dx lpl --'~

A

x +

A

-108-

Because of ( 1 4 . 3 ) , disproves

(14.6).

t h e r i g h t hand s i d e of ( 1 4 . 6 ) i s

, such t h a t

Q

From ( 1 4 . 2 ) i t

~r

( 1 4 . 2 ) h o l d s f o r any

Lo = ~ § ( - 4 ) ~ o

the operator

This

Thus we have r e a c h e d an a b s u r d i t y .

L e t us now c o n s i d e r a n y s u b s p a c e ~1~ (~)

~ ( ~ ' ~ .

(~,~)

of H ~ ( A ) ~ ~ E ~.

L e t us c o n s i d e r

~ 9 The c o r r e s p o n d i n g b i l i n e a r

= B(~,~)+{-~)

folloTs that

containing

for

%r~

r

form i s :

~o C ~ , ~ ) o ,

J

II 11 Then t h e r e e x i s t s

one and o n l y one ~ E ~

such t h a t :

A f or any ~ G

~(A).

the solution

L e t us d e n o t e by ~ - - ~

is the Greenls transformation

introduced earlier.

do n o t want n o t t o c o n s i d e r any r e s u l t

arising

of ( 1 4 . 8 ) .

However we

from t h e r e g u l a r i s a t i o n

theory'. From (706) i t from

~(A)

from ~

into

follows that ~

G

i s a bounded l i n e a r

. Then, i f c o n s i d e r e d as a l i n e a r

(A) into i t s e l f ,

transformation operator

G is compact due to Rellich's l e n a .

We w i s h t o g i v e some more i n f o r m a t i o n c o n c e r n i n g t h e f u n c t i o n a l structure

of t h e o p e r a t o r

G

.

L e t us i n t r o d u c e t h e bounded l i n e a r T h i c h was a l r e a d y c o n s i d e r e d i n l e c t u r e

operator

T

of t h e s p a c e

7 , and f o r which Bo C ~ r ~ _ -

-109-

: (~Ttr)

.For any

V~

@w

i s c l o s e d and T o t h e r hand~ i f follows t h a t of

T

maps V w

We have f o r any ~ E V

be t h e embedding of adjoint operator, i.e.

~14o

~

H.

(A) into

an o p e r a t o r from

,~

~o

~

, and

~o

o

~ ~

into

o:

~o 9

is formally self-adjoint,

and so a r e

~o ( A )

Let

H~

and ~

.

i.e.

i n t h e d e f i n i t i o n of

denote i t s

-~,

twto

, is given by,

(3)

L ~ L ~

, then

"7 i s h e r m i t i a n

I t i s worthwhile t o remark t h a t

a k i n d of Greents o p e r a t o r f o r a d i f f e r e n t i a l

constant coefficients,

~o

such t h a t f o r

, t h o u g h t as an o p e r a t o r of ~ ' ( A )

G-

itself

~ ' ~ be t h e i n v e r s e o p e r a t o r

: (~-17)--(~t,%r)..

~14,~,(

and

the operator

L

Thus T ( ~ / ~ ) : ~ . Let

, t h e n , from (14o7)~ i t

i s bounded because of ( 7 . 8 ) . So i s t h e a d j o i n t o p e r a t o r

of ~ .

If

i n a o n e - t o - o n e f ~ h i o n o n t o T ( V ) . On t h e

is orthogonal to T(V)

"~g/~O.

;

W , inequality (7.8) is satisfied. Hence T ( V )

~ ~n o

is

o p e r a t o r with

whose l e a d i n g p a r t , a f t e r i n e s s e n t i a l changes ( ) )~

, i s t h e i t e r a t e d Laplace o p e r a t o r

A ~.

We want now to consider the eigenvalue problem:

(14.9)

5 ~ ( ~ , ~ ) - (-4)

~

~

dx-

o

When t h e r e s u l t s of t h e r e g u l a r i s a t i o n t h e o r y can be a p p l i e d , t h e n (14.9) c o r r e s p o n d s t o t h e e i g e n v a l u e problems f o r t h e d i f f e r e n t i a l operator

L. ~

- (-4)"")~

~.

- o

~. ~. M ( v ; ,

o

( ~ ) I t , ~ s t be pointed out t h a t when ~'e ,,,'rite L "- L ~) we mean i , e . ~(~,~r) i s h e r m i t i a n (see a l s o p a g . 7 6 ) .

:(-i) P~

Ipl+lql

~,

-ii0-

I n any c a s e ,

(14.9) i s e q u i v a l e n t to the f o l l o w i n g e i g e n v a l u e problem

in the space

~ ~(A):

(14.10)

(-4)

Suppose t h a t

L

X

: o,

is self-adjoint.

Then from t h e H i l b e r t s p a c e t h e o r y

of e i g e n v a l u e p r o b l e m s f o r compact h e r m i t i a n o p e r a t o r s i t there exists

follows that

a c o u n t a b l e s e t of r e a l e i g e n v a l u e s f o r problem ( 1 4 . 9 ) .

These e i g e n v a l u e s a r e p o s i t i v e o p e r a t o r ~ The c o n s t a n t

~o

since the operator

G

is a positive

i s a l o w e r bound f o r t h e e i g e n v a l u e s of

problem ( 1 4 . 9 ) . In the ne~

l e c t u r e we s k a l l d e v e l o p a t h e o r y f o r c o m p u t i n g lower

and u p p e r bounds f o r t h e e i g e n v a l u e s of problems which i n c l u d e , particular

as a

case, problem (14.10)~

Remark.

The way as 'we d e r i v e d t h e s t r u c t u r e

transformation

G

problems~ In f a c t ,

s u g g e s t s a more a b s t r a c t the differential

structure

of t h e Green's

approach to e l l i p t i c of t h e o p e r a t o r

only needed in order to consider the integro-differential

~-

form

is

Bo ( ~ r ) .

]1owever, what i s o n l y n e e d e d i n t h e e x i s t e n c e t h e o r y i s t h a t

~o(~r)

be c o n t i n u o u s o v e r

w i t h 7"

linear

V X ~r , i . e .

that

~D ( ~ , ~ r ) : ( t ~ T ~ r )

and b o u n d e d . The q u e s t i o n n a t u r a l l y

general bilinear larisation

forms such t h a t

t h e o r y can s t i l l

arises

of c o n s i d e r i n g more

( 1 4 . 7 ) h o l d s and s u c h t h a t t h e r e g u -

be c a r r i e d o u t .

For a s i m i l a r a p p r o a c h i n t h e t h e o r y of harmonic d i f f e r e n t i a l forms on a R i e m a n n i a n compact m a n i f o l d , s e e [ 2 J ~

-11i-

Bibliography

[i]

of

I~. ~ 0 ~ S Z A J ~ - On c o e r c i v e

Lecture

14.

integro-differential

Conference

on PDE, U n i v .

quadrat.ic

of Kansas 1954,

forms -

Techo

Rep. No 14. G. FICHL~A - T e o r i a

assiomatica

delle

forme differenziali

armo-

niche~ Rend. di ~latem.~ Roma vol.20~ 1961.

[3]

Lo G~EDING- D i r i c h l e t t s differential

problem for equation~

[4]

L . NIR~,~-I~]~G- s e e

[4] of lecture

4.

[5 ]

L . NIRENBL~tG- s e e [ 4 ] o f l e c t u r e

8.

linear Bath.

elliptic

partial

Scand. vol.I~

1953.

-112-

Lecture

15

Problems . The

Raylei~h-Ritz

Eigenv_alue

method_.

We assume t h a t t h e r e a d e r i s a c q u a i n t e d w i t h t h e e l e m e n t a r y t h e o r y of h e r m i t i a n compact ( i . e . space.

completely continuous) operators in a Hilbert

HoTever we s h a l l l i s t

we s h a l l n e e d i n t h e s e q u e l .

h e r e some d e f i n i t i o n s

and t h e o r e m s which

For t h e p r o o f s of t h e t h e o r e m s we r e f e r t o

[2], [3], [4]. We s h a l l always c o n s i d e r o p e r a t o r s which a r e l i n e a r i n t h e whole o f t h e complex H i l b e r t s p a c e 5 t o be h e r m i t i a n of

S

if (T~,~):

.

and a r e d e f i n e d

An o p e r a t o r

T

( ~ , T l r ) f o r any p a i r o f v e c t o r s

is said

~t

and %r

.

15.I.An~ h e r m i t i a n o p e r a t o r i s bounded. The o p e r a t o r 15.II.

is positive

A (linear)

We w r i t e

f o r any

~

S,

posi.tive if it is positive

and ( T ~ ) - - 0

~ = Oo 1~0

sequence of p o s i t i v e

f o r any p o s i t i v e

tive eigenvalue for

compact o p e r a t o r .

A

e i g e n v a l u e s c o n v e r g i n g towards z e r o .

e i g e n v a l u e has f i n i t e

(geometric) multiplicity~ T

~ the subspace of

of t h e homogeneous e q u a t i o n

dimensional 9

(T~j ~)~0

compact ope_r_at_or i s bounded.

i s s a i d t o be s t r i c t l y

only holds for

solutions

if

A (linear_)_posi_tive operator is hermitian.

15.III. T

T

~

i.e.~

if

PC0 has a Each p o s i t i v e

~

is a posi-

composed o f a l l t h e

T~4, - ~ ~ : 0

is finite

-113-

15.IV.

A

tCO

T

is strictly

pos.i.tive i f

and o n l y ,if

~ : 0

,is n o t an , e i g e n v a l u e f o r T . Let:

be t h e s e q u e n c e o f a l l many t i m e s as i t s

t h e e i g e n v a l u e s of t h e 1~0

multiplicity.

of e i g e n v a l u e s o f a 1~0, i t

each r e p e a t e d as

From now on when we m e n t i o n t h e s e q u e n c e

shall

be u n d e r s t o o d t h a t t h i s

ordered according to the criterion

bt'.i

~-

,. ~ . 1 , }

j u s t now s p e c i f i e d .

9 " ,

t'Ck

~ ""

sequence

is

Let:

"

be a corresponding sequence of eigenvectors, i.e. T ~ K- ~ K ~K : o . ~'e may assume t h a t 15.V.

~~

~

The o p e r a ~ o r

i s an o r t h o n o r m a l s y s t e m . ~

admits t h e followi'n~ " s p e c t r a l

decompo-

si ti on""

(15.1)

Tu. - 7 , F,, ( ~ " ~ ' )

~'K "

K

15.vl. if

and o n l y i f t h e tCO It

S

T

} is

i s a c o m p l e t e system in t h e sp ce

strictly

positive.

f o l l o w s t h a t a s t r i c U y p o s i t i v e compact o p e r a t o r can e x i s t

o n l y when S on

he. system

is separable.

and~ moreover~ t h a t 15.VII.

Is

~K~

~.eal n ~ b e r s c o n ~ e . ~

From now on we s h a l l S

has i n f i n i t e

assume t h i s h y p o t h e s i s

dimension.

i s any n o n - i n c r e a s i n g s e q u e n c e of n o n - n e g a t i v e

towards -ero and t ~K } is any o r t h o ~ o ~ l system,

then the operator, .defined by 4.15.,I) is .a ICO. having ~he values and the

L~K

as ei.~envectors.

The value

~ =0

~z

as ellen-

.i.s an ellen-

-114-

v a l u e f o r T - even though i t

i s not i n c l u d e d i n the sequence { ~ l - i f

and o n l y i f t h e System of e i g e n v a l u e s c o r r e s p o n d i n g t o a l l p o s i t i v e

eigen-

values is not complete. The f o l l o w i n g 1 ~

i s of f u n d a m e n t a l i n t e r e s t

in the theory of

e i g e n v a l u e s . I t e x p r e s s e s t h e s o - c a l l e d mini-max p r o p e r t y f o r e i g e n v a l u e s . 15~

Let

( ~ ~_ ~ ) be

~ ,---

( T ~ , ~ ) on t h e u n i t s p h e r e

~ e ~-4

(~)

~

.vary a r b i t r a r i l y If

~

:~

be t h e above c o n s i d e r e d 1~0.

~ - 4 v e c t o r s of t h e H i l b e r t s p a c e

t h e maximum of complement

T

o_ff ~v~, 4 , where

.

The m i n i ~

i n t h e space

,

~ "

~ ~-4

~

Let

~...~ ~rK. 4

~ . Denote by M ( ~ , . . . ,

~.~ )

II ~ II : 4 of t h e o r t h o g o n a l

~-4

is the linear variety

of N ( ~

~

spanned

)when ~ , - . - ,

is the eigenvalue

, then

~

~

of T

.

H(~,

t<-4

"'"

9

K-4

K

The sequence of operators ~T~ t is said to converge weakly (strongly, u n i f o r m l y ) t o t h e o p e r a t o r T whenever-

/ W. - - ~

oO

-~

II T

,~,~

ilT-T

a - T~

11 : o

f o r -.-y

~ ~ S

il : o ] .

~ t - - ~ OO

In the latter

case~ t h e norm must be u n d e r s t o o d t o be t h e norm i n t h e

space of operators. Therefore :

T

and

i

a r e nov supposed bounded.

z

(~) I n t h e c a s e

(~)

K = 4 , ve assume

~ O~r

~ 5. K-4

C h a r a c t e r i z a t i o n of a l l p o s s i b l e c h o i c e s of ~r~ ~ . . . ,~.-4 such t h a t M (~4)-'- , ~ _ 4 ) : ~K has been done by W e i n s t e i n [ 5 ] .

-115-

For Y e a ~ s t r o n g ~ u n i f o r m c o n v e r g e n c e of o p e r a t o r s we s h a l l respectively

t h e symbols:

T~ 15olX. to t h e PCO

use

-T

Le....~_~ ~ "

T

;T

T~,

~T

be a s e q u e n c e of PC0ts. c.onvergin~ uniforml~r

T

be t h e s e q u e n c e of e i g e n v a l u e s of

I

Y~e have:

uniforml~ with respect to

k 9

The p r o o f i s a c o n s e q u e n c e of t h e i n t e r e s t i n g

I!~ K-pK I -'- I I T - T which i s proven If

t h e n f o r any g i v e n v e c t o r of t h e s u b s p a c e

~r

is called

the

there exists

~

~rojector~

A projector

onto i t s

is hermitian Suppose

subvariety

for every

V

o Let ~ ~k]

projection

from

ii)

is finite

p

M.

V~ ~ V

S

)7

(projection

as t h e one s a t i s f y i n g o

The v e c t o r 9

on ~/r i s c a l l e d

is characterised ;

of

one and o n l y one v e c t o r

( o r t h o g o n a l ) p r o , ~ e c t i o n of t~ on ~ r

which maps

for V

t~ E 5

closed linear

This v e c t o r i s c h a r a c t e r i s e d (~r~ ~x/) : ( t ~ V g )

P

(i.e.

which has minimum d i s t a n c e

equations

i)

II,

as a c o n s e q u e n c e of t h e mini-max p r i n c i p l e .

~/~ i s a s u b s p a c e

theorem).

inequality:

the qY-= P t ~

The o p e r a t o r

P

an ( o r t h o g o n a l )

by t h e f o l l o w i n g two p r o p e r t i e s - "

is idempotent, i.e~

d i m e n s i o n a l and ~

)....~

P ~ - "~, is a basis

be t h e i n v e r s e m a t r i x o f t h e Gramian m a t r i x t ~ r ~ j lY~

( ~,~- ~,.-.,~ ), i . . . ~

(~,u~):

g~. Since ( ~ , ~ k ) = ( ~

,~ )

-116-

~,~

: ~~

or 5

The projector

"

onto V

has t h e f o l l o w i n g

r e p r es e n t a t i on:

p~

(15.2)

- ~

( ~, v~ ) ~ .

If the basis is orthonormal, then (15.3) If

~

5

projector (which projects will be called the If

, and P

i s any s u b s p a c e of

T

is

onto ~r ), t h e n t h e operator

~r:c.ompen.ent o.f T

a 1~0~ t h e n

is the corresponding

PTP

PT P

*

is also a I~0.

The f o l l o w i n g theorem h o l d s : 15.X.

Let

to the operator linear to

p

(which i s n e c e s s a r i l y

a projector).

Let

]-

.be a

compac, t o p e r a t o r . . T h , en t h e sequence t P ~ ' T ? w t c o n v e r g e s u n i f o r m l ~

PT P, If

if

} be a sequence of project,ors, c o n v e r g i n g s t r o n g l y

~ ~

I4

T.~- T~

and

T~

a r e l i n e a r h e r m i t i a n o p e r a t o r we s a y t h a t T 4 > T~

is positive . I_~f TI

15.~.

an.d T &

are PCO and

L" ~

values

of

[--~

and

T, ~ ~-~

, then we have:

K

I~ 9

This t h e o r e m i s a c o n s e q u e n c e of t h e mini-max p r i n c i p l e . 15.XIIo

Let

T

be a 1~0 w i t h . e i g e n v a l u e s

t~k

t

' and l e t

~r

-117-

be a subspace o f V-component

S

of

.

T

If

{ ~K } i s t h e sequence of ei,~envalues o f t h e

, then

~

~ ~K

( K: 4,2,-.. ) .

This t h e o r e m a l s o f o l l o w s from t h e m i n i - m a x p r i n c i p l e . Note t h a t t h e o r e m 15.XI1 i s n o t a p a r t i c u l a r in general~ it

c a s e of 15.XI s i n e e ~

"T > ~ [ P ~

is not true that

We a r e n o t i n a p o s i t i o n t o f o r m u l a t e t h e R a y l e i g h - P d t z method f o r t h e a p p r o x i m a t i o n o f e i g e n v a l u e s o f a 1~0 Let

~%r~ ~

in the space

~r

.

.

~

L e t us d e n o t e by

..., lY

o

-~r

We d e n o t e by

the ~m

The R a y l e i g h - R i t z method c o n s i s t s 9

.

t~)

~4t~) , -- ,~K , " - of

~4 , ' " ,

.

be a c o m p l e t e s y s t e m o f l i n e a r l y

~

. p a n n e d by

~-

~K)'-

of

~ T

~ ~ .

independent vectors

~t-dimensional the pro~eetor of

subspace S

onto

in considering the eigenvalues

as a p p r o x i m a t i o n s o f t h e e i g e n v a l u e s We s h a l l p r o v e t h a t :

15oXIII 1)The e i K e n v a l u e s o f

P,,~ T" ~,,A, a r e t h e r o o t s o f t h e

f o l . l o w i n g de t e r m i n a n t a l e ~ u a t i o n :

(15.4) p l u s t h e ei~envalu,e

~ = 0 ;

ii) The sequence ~ i.eo ~K

Z

~(

(~)

f o r any

does n o t d e c r e a s e when ~

~,

inc reases

;

uni,f,ormly T i t h r e s p e c t %o ~< 9 I% i s i m m e d i a t e l y s e e n t h a t t h e o p e r a t o r ~ - [

~)~ has t h e e i g e n v a l u e

-118-

I.=

and t h a t t h e c o r r e s p o n d i n g e i g e n s e t

0

of t h e homogeneous e q u a t i o n t h e subspace

~ ~ ~

.

P

TP

U,-pU, = 0

(subspace of t h e s o l u t i o n s for

p- 0

) contains

The remaining e i g e n v a l u e s and e i g e n v e c t o r s are

o b t a i n e d tteough t h e e q u a t i o n s :

(15.5)

P~ T P,,,, u. - p, P,, u. : o

, u.: 'p.,, u..

Due to ( 1 5 . 9 ) , e q u a t i o n ( 1 5 . 5 I} can

be w r i t t e n i n t h e form-

(15.8)

, '~'~, )- F d i, K (u"V"I~):

ol

o~ . (~., ",,'% ) (T'~j

0

(k: 4,...,~). Since

d e , t t (~ h~

(15,7)

J~

0

, setting

(IA,'I.~) "-. C~

K~,j ( T ~ j ,'t,"~ )c;, - p c k : 0

~uation

, we o b t a i n ,

( k: ,(,...,,~.).

(15.7) ana (,15.5) a r e e q u i v a l e n t .

PUt

~"i

.=c~jG~.Then C~-(~ ,~)~i.Equation ( 1 5 o 7 ) i s equiTalen~

to t h e following=

(15.8)

(Tv-] ~v"k )~j - ~ (v-j ,~)~j

I t follows t h a t

~

:

O.

i s an e i g e n v a l u e f o r (15.5) i f and only i f i s a

r o o t of e q u a t i o n ( 1 5 . 4 ) .

If

~4 ) ' " ~

~

i s an e i g a n s o l u t i o n f o r

t h e a l g e b r a i c system (15.8) c o r r e s p o n d i n g t o t h e e i g e n v a l u e

p

,

-119-

then

t~ = ~ J l/~J

of t h e e i g e n v a l u e -~(~ of

) ~rj)~olf ~

as a n

~

i s an e i g e n s o l u t i o n ~

for (15.5).

of ( 1 5 . 5 ) i s t h e n u l l i t y

~ ~ O

, then this

e i g e n v a l u e of

The m u l t i p l i c i t y

of t h e m a t r i x

nullity

is also the multiplicity

P TP~ In fact, P~ r ~

S t a t . e n t i i ) follo.~ fro. theore~ 1 ~ . ~ I .

is the

V-component of

~

P

T P

converges strongly to the identity

Biblio~raph~

[1"]

,ince

to the oo~pleteness of the s y s t e .

15.X and 1 5 . I X , s t a t e m e n t i i i )

~T~v) ~i )-

operator.

P P ~ ~}

- P.

p.-

lm

,the projector p ~

Therefore,

from t h e o r e m s

follows.

of

Lecture

15

N. ARONSZAJN - The R a y l e ~ h - R i t z and t h e W e i n s t e i n Methods f o r a p p r o x i m a t i o n of E i g e n v a l u e s -

(1. Operators in Hilbert

Space) - D e p t . of Math. Oklahoma A g r i c u l t u r a l Mechanical College, Stillwater

and

O k l a h o m a - T e c h . Rep.

Ne 1~ 1949. [2]

G~ FICII~tA -

see [ 1 ] of

[31

S . t t . GOULD- V a r i a t i o n a l

lecture

Methods i n Ei.~envalu%

of Toronto P r e s s , - B. S z . NAGY -

1.

[4]

F. RItZ

[51

A. ~I~INSTKIN - The i n t e r m e d i a t e

Toronto,

see [5]

1957.

of l e c t u r e

2.

Problems and t h e biaximum-Minimum

T h e o r ~ of E i ~ e n v a l u e s - J o u r . I n d i a n a UniT. v o l .

Problems - U n i T .

12, 1963.

of Math. and Mech.

-120-

Lectur_e

The

16

Weins.tein .- Aronsza~n

method.

The R a y l e g h - K i t z method p r o v i d e s a s t r o n g t o o l f o r t h e l o w e r a p p r o x i m a t i o n of t h e e i g e n v a l u e s o f a 1~0.

Much more & i f f i c u l t

is the

problem c o n s i s t i n g i n t h e u p p e r a p p r o x i m a t i o n of t h e s e e i g e n v a l u e s , i . e. the construction eigenvalue

~K

of a s e q u e n c e c o n v e r g i n g by d e c r e a s i n g t o e v e r y g i v e n of

T

.

A method f o r s o l v i n g t h i s problem was p r o p o s e d by A . W e i n s t e i n . His t h e o r y ha~ by d i f f e r e n t

been e x t e n d e d by A r o n s z a j n and a p p l i e d t o s e v e r a l problems

authors.

Weinsteints idea is the following. To

, whose e i g e n v a l u e s

~ ~K t

a r e known, and which i s such t h a t constructs i)

ii)

a sequence If

to

he c o n s i d e r a ~CO o p e r a t o r

and c o r r e s p o n d i n g e i g e n s o l u t i o n s ~

> ~

f o r any

~

t~X/K

. Then he

of Pc01s such t h a t z

~ a r e t h e e i g e n v a l u e s of

The e i g e ~ r a l u e s o f

e i g e n v a l u e s of

If

~~

T.~

First

~

T~

, t h e n ~-~ ~ ~-~

>.

can be computed i n terms o f t h e

TO 9

~ T~ t c o n v e r g e s u n i f o r m l y t o

K ( t h e o r . 1 5 . I X ) by d e c r e a s i n g .

T

,

then

~-(z)

c onverges

(4)

We u s e t h e name " d e c r e a s i n g s e q u e n c e " f o r a s e q u e n c e

6

(~)

such t h a t

-121-

The o r i g i n a l

method of W e i n s t e i n , which was c o n c e r n e d w i t h t h e

e i g e n v a l u e s problems a r i s i n g

T

as o p e r a t o r V

in the

a I~CO which was

the

b e i n g a g i v e n s u b s p a c e of

which p r o j e c t s 5 E9 V where

~

onto

t h e o r y of t h i n p l a t e s ,

5

V ~

~0-~

P~t

is the projector

The above s t a t e d 15.XII.

-component of a n o t h e r PC0 ~-~ /

and by

t Lr~ T~

for

T~

the projector

a complete system in

a r e g i v e n by ( I - P . ~ ) T o ( I - P ~ ) ,

on t h e s u b s p a c e spanned by

condition i)

P

Then, d e n o t i n g by

, the "intermediate ~ operators

considered

~r4 ) " "

)~r

.

i s t h e n a c o n s e q u e n c e of t h e o r o

Convergence f o l l o w s from 15~

Condition ii)

w i l l he proved

later. A r o n s z a j n p r o p o s e s t o c o n s i d e r as To > ~- ( f o l l o w i n g i n t h i s he c o n s t r u c t s which s a t i s f i e s

a sequence

respect {

T,,t~,}

To

any o p e r a t o r such t h a t

a s u g g e s t i o n of H~ W e y l ) , and t h e n such t h a t

To >

T,,w

> -~+4

) ~-

condition ii).

I n o r d e r t o i n c l u d e i n our t r e a t m e n t b o t h t h e methods of W e i n s t e i n and of A r o n s z a j n , i t However

we must f i r s t

i s c o n v e n i e n t t o g i v e some g e n e r a l t h e o r e m s .

consider the resolvent

operator for the following

e q u a t i on--

(16.1) where

Tu.- F~, T 1)

: v"

i s a I ~ 0 . L e t us d i s t i n g u i s h F

i:s n o t an e i g e n v a l u e f o r

(1G.1) h a s one and o n l y one s o l u t i o n

(16. z)

(:F ~ o )

u. ~. R v : -

Y',

two c a s e s . T

9

In this

case equation

g i v e n by:

( ~ , ~.,, ) u.. - v

.

-122-

This r e s u l t

follows easily

2)

~

and o n l y i f

i s an e i ~ e n v a l u e f o r (~r/~)

to the eigenvalue of ( | ~ . ~ )

from ( 1 5 . 1 ) ~

_- ~) , where ~

9 If this

T

~

9

Then ( ] 6 . 1 ) h a s s o l u t i o n s

if

i s any e i g e n v e c t o r c or r e s pondi ng

condition

is satisfied,

then every solution

i s ~ ' T e n by

(:G.3)

where

~/~ (~)

means t h a t

all

t h e terms c o r r e s p o n d i n g t o i n d e c e s

which

~K : ~ were s u p p r e s s e d from t h e summation.

The

a r e a c o m p l e t e s y s t e m of e i g e m r e c t o r s i n t h e e i g e ~ e t the eigenvalue also easily

~

.

The

C~

are arbitrary

k

t~ (4) ~ . , . ~

for

t4,(~)

corresponding to

constants.

This result

is

o b t a i n e d by u s i n g ( 1 5 . 1 ) .

The f o l l o w i n g t h e o r e m c o n s t i t u t e s

the theoretical

f o u n d a t i o n of

W e i n s t e i n t s method. 16.1.

Let

of PCO's s u c h t h a t :

ii)

~ T. ~

v,,alue

To

..and T

i) :

= T o - D ~ where

oonTerges ~ i f o ~ X ~

(~) -o-f ~ K

~-

be two I ~ 0 ' s .

Let ~-~

D~

be .a s e q u e n c e

is a degenerate operator, )

_to the operator T

9 Then ~ y ei~en-

can be computed i n t e r m s of t h e eig~enTalues

and of t h e e i g e n s o l u ~ i o n s o f

~o

and

e~ _~ oo

f ~21W e say that ~ dimensi onal 9

is a degenerate operator if its

range is finite

-123-

The o n l y t h i n g . e h a v e t o p r o v e i s t h a t

if

T~

i s ~n h e r m i t i a n d e g e n e r a t e o p e r a t o r ~ t h e n ~ g i v e n t h a t

i s a 1~0 and the eigenvalues

{ ~ ~ ~ of To and the corresponding eigenso~utions ~ ~ it

is possible

t o compute t h e e i g e n v a l u e s of

L e t us s u p p o s e t h a t ~4 ~ . . . /

~

t h e r a n g e of ~

be a b a s i s i n t h i s

e~ists a n o n - ~ i n ~ l a r ~ •

~o

i s ~v~ - d i m e n s i o n a l .

range. It

he~tian

~-

} are ~ o ~ , Let

i s easy to prove t h a t

matri~ ~ d ~

there

such that

admits the following representation:

(~)

We ~dsh t o f i n d t h e n o n - z e r o e i g e n v a l u e s of t h e f o l l o w i n g homogeneous e q u a t i on-"

(i~.4) L e t us f i r s t

suppose that

~

Then , by u s i n g ( 1 ~ . ~ ) , . e s e e t h a t

O

i s n o t an e i g e n v a l u e f o r

e q u a t i o n (1G.4) i s

To 9

equivalent to

the following equations:

(l~.s)

(16.e.,)

cl,,.: ( ~ , v ~

1.

(~s) Let ~)~ ~C~(~)~'j. Wehave(D~v~)=cjCu~)(~Yd~r~).ThenC)~)-O(~](D~%V~)= It

f o l l o w s t h a t Dt~

is hermitian,

so i s

:dLj

(t~,~)~)

t d~i 1.

.here

dLj :o(~j

fh~

9 Since D

-124-

By i n s e r t i n g

(16.5)

into

(16.6)

(R "o-. v'k ) ol

(16:)

It

is

obvious that

~

t h e n we c a n T r i t e

eigenvalue

(16.7)

6

that for

To-D

lar

by t h a t

to the spectrum of

~-o )

i s the i n v e r s e

it

of {4

is

7 0

a solution

) is an

of the equation.-

~r (~)

author

is the Weinstein

in connection

determinant,~hich

was

with the above mentioned particu-

case. L e t us now s u p p o s e t h a t

d e n o t e by

t~

corresponding defined

(16.9)

j -to

j 6-.

by ( 1 6 . 6 )

,JjJ

form:

to the spectrum of

and o n l y i f

n~tri,

ol~t { ( R

-

The d e t e r m i n a n t introduced

if

: o.

only i f the algebraic system ( l ~ n )

in the equivalent

(not belonging

W (~)

(16.8)

if :d

solutions. If {C~i~ I

has non t r i v i a l

follows

c,- c

(not belonging

is a. eigenvalue for T o - D

It

we g e t

6-~

0

is

an e i g e n v a l u e

for

a complete system of eigenvectors I f b~ s a t i s f i e s

must satisfy

(16.4),

the constants

T o 9 We of C~

the equations:

d ~i (~j ' ~ (Pj) c~ = o

p: ~)-..,~.

To

,

-125-

t~oreover

U+ m u s t be g i v e n by

(1(;.lO)

~ :

see

(16.3)]

:

d. ~i c ; R~. ~; + Y_., o. u, p.4

P

Inserting (16.1o) i r t o (IS.S) we get-

(

It

follows

)

that

if

the

:o

eigenvalue

of To

tl, en t h e i,omogeneous s y s t e m ( 1 6 . 1 1 ) , C4 )

. .),

Conversely, easily

(16.11),

if

seen-

It

follows

if

and o n l y i f

(16.12)

~) 0,~

C

the

that

~...

~ O. 4

06.9)

(16.9)

is

an e i g e n v a l u e

h a s non t r i v i a l

for

solutions

,

has non t r i v i a l

solutions,

t h e n - as i s

eigenvalue

~

of

T o is

an e i g e n v a l u e

for

an e i g e n v a l u e

~

of

T o is

~n e i g e n v a l u e

of To-~

it

is

d~t

a solution

(16.4))

"1-~-2,

of the equation:

,

:0.

0 J

The m e a n i n g o f t h e } is obvious.

(~

6)

~ (.Tvt+5)

matrix

between the brackets

-126-

In conclusion, first

i n o r d e r t o compute t h e e i g e n v a l u e s o f

to solve equation (16.8).

Ro o ts o f t h i s

spectrum of T o a r e e i g e n y a l u e s f o r To- D

9 Then we s o l v e e q u a t i o n ( 1 6 . 1 2 ) .

e q u a t i o n which a r e e i g e n y a l u e s f o r

for

It

9

is easily

seen that

T o - ~ i s g i v e n by t h e n u l l i t y

immediate that the original

T=

operator

this

of the eigenvalue of

proved,it

T~ i s s a t i s f i e d

follows that

if

condition i8 satisfied

To-T~ i s

condition ii)

degenerate.

for the intermediate

It

for is

operators

of

Weinstein method. Sinces

( I - P ~ ) T o (I-P~.) : To - (P~,To+ ToP~- P,,.~ToR~),

The a b i l i t y

t o a p p l y t h e o r e m 16.1 r e s t s

i m a t i n g t h e compact o p e r a t o r i s known t h a t

I f we f u r t h e r

this

To-T by a

degenerate. on t h e p o s s i b i l i t y

many ways i n a H i l b e r t

t h a t t h e a p p r o x i m a t i n g ~ e q u e n c e be s u c h t h a t

is satisfied,

w h i c h h a v e b e e n p r o p o s e d by d i f f e r e n t

a g e n e r a l p r o c e d u r e which i s p a r t i c u l a r l y c o n n e c t e d w i t h BVPIs f o r e l l i p t i c

the intermediate

authors.

for

T~

suitable

operators

L a t e r we s h a l l

give

for eigenvalue problems

systenm.

t o show how t o overcome one d i f f i c u l t y

applying Weinstein's

space.

t h e n we c a n a p p l y W e i n s t e i n t s m e t h o d .

T h e r e a r e s e v e r a l methods f o r c o n s t r u c t i n g

We w i s h f i r s t

of approx-

sequence of d e g e n e r a t e o p e r a t o r s .

c a n be done i n i n f i n i t e l y

require

the condition i)

T~

, are also eigenyalues

the multiplicity

the operator ~ t : P-~ To § To P~-PTo P~is

It

To

o f t h e m ~ t r i x ( 1 6 . 8 ) o r (16018) r e s p e c t i v e l y .

From t h e t h e o r e m we h a v e j u s t the intermediate

, ve h a v e

e q u a t i o n w h ich a r e n o t i n t h e

Root s o f t h i s To-D

To-D

which a r i s e s

in

method.

We have s e e n t h a t

the eigenvaluee

c a n be found by s o l v i n g e q u a t i o n s

(16.8)

of the and

intermediate (16.12).

operators

This p r o b l e m

-127-

a p p e a r s t o be a v e r y d i f f i c u l t consists

one, in the general case, since it

i n d e t e r m i n i n g t h e z e r o s o f meromorphic f u n c t i o n s .

TTo p r o c e d u r e s have been p r o p o s e d i n o r d e r t o overcome t h i s The f i r s t

difficulty.

of t h e s e was d e v e l o p e d by W e i n s t e i n h i m s e l f and, l a t e r ,

Bazley [3].

by

I t i s c a l l e d t h e method of s p e c i a l c h o i c e s . The s e c o n d i s

due t o W e i n b e r g e r [ 7 ] and i s known as t h e method of t r u n c a t i o n . The s p e c i a l c h o i c e s c o n s i s t operators

To-~

( r e omit t h e i n d e x ~

t h a t t h e r a n g e "V" o f f o r any

j

simply in constructing the intermediate

D

from ~

i s s p a n n e d by v e c t o r s

) i n such a way

~r~ ,..., ~

, t h e s c a l a r p r o d u c t (~/K, ~ri ) i s d i f f e r e n t

for a finite

s e t o f v a l u e s of t h e i n d e x

instance,

satisfied

operator

To (Ba~ey),

~ . (4)

when ~V" has as a b a s i s o r Then ~/

has

~

from z e r o o n l y

This c o n d i t i o n i s ,

cm a

such t h a t

for

e i g e n v e c t o r s of t h e

basis

~

vectors belonging

t o a s y s t e m which t o g e t h e r w i t h t h e s y s t e m o f e i g e n v e c t o r s

t ~

} of To

forms a p a i r of b i o r t h o g o n a l s N s t e m s . I f t h e above c o n d i t i o n i s s a t i s f i e d , it

t h e n , from ( 1 6 . 2 ) and ( 1 6 . 3 )

f o l l o w s r e a d i l y t h a t s o l v i n g e q u a t i o n s ( 1 6 . 8 ) and ( 1 6 . 1 2 ) c o n s i s t s

i n f i n d i n g z e r o s of p o l y n o m i a l s . The method of t r u n c a t i o n c o n s i s t s

in replacing the operator

~-o

by t h e t r u n c a t e d o p e r a t o r

T

(4)

{~

,,,,,

Z. %

K"4

t is the system of eigenvectors

,~1,.t. 4

[ ,.-f.

of T O .

K.-,I

].

-128-

We d e n o t e - as u s u a l - by ~ ~ o p e r a t o r s we t a k e

~ the e i ge nva l ue s of

7 - ( ~ ) : T C~) D

The s o l u t i o n

-@'~

: "L/ i s

if

does n o t b e l o n g t o t h e s p e c t r u m of

~

]-o ~ As i n t e r m e d i a t e

of t h e e q u a t i o n

i

(~)

g i v e n by-

In the case

(~ " ~K ( ~ : ~ ~J " ' , ~ )

~(~).

t h e most g e n e r a l s o l u t i o n

of the

above e q u a t i o n i s : .4-

J

where

7.

h a s t h e u s u a l meaning and t ~

system of e i g e n v e c t o r s

( ~- ~ / ' ' ' ' ~ ) i s a c o m p l e t e

of ~-~ c o r r e s p o n d i n g t o t h e e i g e n v a l u e

(~ .

By u s i n g t h e same arguments as i n t h e p r o o f of t h e o r e m 1 6 . 1 , i t e a s y t o s e e t h a t t h e e i g e n T a l u e s of ~-, t ~ )

~

satisfying

is

the condition

~t

~ (%If -~ ii)

a r e o b t a i n e d b~ 8 o l T i n g a l g e b r a i c .~-~(~'~) greater

of t h e o r .

_~ ( ~ g ( ~ " ~ ) ~ ' " than

~4

16.I is satisfied,

e (~ p

, it

equations

a r e t h e e i g e n v a l u e s of

i8 easy to prove that

~-~)-

if condition

-129-

Bibliography

of

Lecture

16

[11

N~ARONSZAJN - see [I] of lecture 15.

[2]

N~Alt0NSZAJN - Approximation me tho.ds for ei~envalues continuous

symmetric

of completely

operators - Prec. Symp. Spectral

Theory and Diff. Problems - Stillwater,

[3]

N.W.BAZLLq/

- Lower Bounds vol.10,

[4]

N.W.BAZLL~L-D.W.FOXProblems

for Eigenvalues

0kla. 1951.

- Journ. ~lath.& Mech.

1961.

Truncations

in the Eethod of Intermediate

for Lower Bounds to F~igenvalues - Journ.

of Research of the N.B.S. - Math. and Math.Phy. vol.

[5]

G.FICtiEP~

-

[6]

S.tI.GOULD

-

[7]

H.F.WEIN~ERGER

see

see

56 B, 1 9 6 1 . [1]

[3]

of lecture

1.

of lecture

15.

- A theor~ of lower bounds for e i~envalues

Tech. Note BN-183

-

(Inst. Fluid Dynam. & Appl. Math.

Univ. of Maryland) 91959.

Is]

A.WEINSTEIN - E t u d e s ~artielles

des spectres - M4morial

des d~uations aux d~riv~es des Sci.

Mathem. N0.88~

1937.

-130-

Lecture

Construction_

of

17

t_he

intermediate

o~erators.

The W e i n s t e i n - A r o n s z a j n method d e s c r i b e d i n t h e p r e v i o u s l e c t u r e requires

the construction

Hbase o p e r a t o r " operators

of " i n t e r m e d i a t e

T O and t h e g i v e n o p e r a t o r

are required to satisfy

( ~ - k l "= eigenvalues of

iii)

( I - P ) To ( I - P ) : d prove t h a t ,

conditions:

).

T ,

if

To

> T

>T

~.~4

>T.

W e i n s t e i n method, where ( s e e p r e v i o u s l e c t u r e )

this monotonicity condition, shall

The i n t e r m e d i a t e

is d e g e n e r a t e .

Condition i) is satisfied

T:

between t h e

(~,~ ~ )

converges uniformly to

The o r i g i n a l

9

~-

T o , ~ 6"z("~')~ e i g e n v a l u e s o f T ~

{ F ~ t =" eigenvalues of T

To-T~

T

the follo~ng

(,w.)

ii)

operators"

T~ : ( I - P,~ ) To (I - ~)do~s not sat,,,~ b u t does s a t i s f y

even i n W e i n s t e i n t s c a s e ,

condition i).

However, we

one can always r e d u c e t o t h e

above m o n o t o n i c i t y s i t u a t i o n . Set

Q = I- P

,~d

_G , ~ : ~ - P

. L e t us f i r s t

prove t h a t :

-131-

Any n o n - z e r o

17.I.

ei~env,~lue of the

operator

Q

is an

To Q

eigenvalue for the follo,win~ problem:

L,* -

(17.1)

P~-

= p~,

and, conversel~., an~ non-zero, ei~envalue for .(17.1) is an ei~envalue

ToQ.

Q

for

F i 0

Let

a n d L~ be an eigenvalue and eigenvector f o r

QT O Q.

From t

QToQ~, : p Q - .

Oz.,2) it

follows

Q TO a

satisfy

that

t~ = ~ I,~ 9 T h e r e f o r e ,

(17.1),

Of c o u r s e ,

(17.1).

, i.e.

: p a

then

L~ : L I~

ve can Trite

Conversely,

(17.2)

~ ~

if

as f o l l o w s : and

0

(~Tot~ and hence, t~--(~.Thus

the theorem still

holds

L e t u s now s u p p o s e t h a t 17.1I.

+ p P~.

Any e i g e n v a l u e

if

TO

we r e p l a c e

is

strictly

P

by

~

positive.

for the problem (17.i)

is

C4~ ~

O

~T o(~=

~-

. Then we h a t e =

an e i g e n v a l u e

for: L

(i~.3)

T ou-

Conversely,

To~ P T o ~z - ~ ' u "

any non-zero ei~env.a.lue for (17o3). is an ei~envalue for (1.7.1), I

If we let qY : Tox a

, then equation (17 .I) becomes:

T ~~- - PTo ~- : p u . . Operating

('T/ >0).

on b o t h s i d e s

by

L TO~

, we g e t :

4

T o~ -

T/PT

4

~ qY O

=

p qr

-132-

Conversely~ (17.3)

let

we d e d u c e t h a t

17e c a n ~ r i t e

~ ~ 0

us a s s u m e t h a t there

exists

and

tr

a vector

satisfy(17

bC s u c h t h a t

. 3 J . From s qY = __To~

( 1 7 , 8 ) as f o l l o w s : •

[ "l"~u, _ PTo ,. -

]_- o

I

Since

T~ ~

is strictly

positive~

From t h e a b o v e t h e o r e m s i t m e t h o d we c a n s o l v e

a)

To~

P~

-

P

To~

1

"I- u, - T o ~ p T o ~ u .

Actually 9 in his

m u s t be s a t i s f i e d . that

in Weinstein's

original

problem any of the following:

)u, -p.u.

=o

- p ,.,. - o

•

c)

follows

as i n t e r m e d i a t e

( i- P.~, ) To ( I-

b)

(17.1)

work~ W e i n s t e i n

-~-0.

considered

p r o b l e m s b) as i n t e r m e d i a t e

p r o b l eros.

I

I f we c o n s i d e r

~ d replace

r

problem c),

by T o - Toz r To ~

theorems 17.1 andlV.II)~ is clearly

and she, l l

T

the operator

To-ToAP

T ~

(this is feasible because of

then the above mentioned monotonicity

condition

satisfied.

In considering operators~

assume as

we s h a l l replace

a general assume t h a t

condition

i)

Our c o n s t r ~ z c t i o n i n c l u d e s

methed for constructing the base operator

the intermediate

To i s

greater

than

T

by t h e f o l l o w i n g :

as p a r t i c u l a r

cases the procedures

given

)

-188-

by W e i n s t e i n and A r o n s z a j n and some of t h e methods c o n s i d e r e d by B a z l e y ~nd B a z l e y & Fox.

(;)

L T o- T and u s u m e t h a t

Set

L e t us ~ s s o c i ~ t e w i t h strictly

L.~ a l i n e a r

p o s i t i v e *~We d e n o t e by

by t h e c o m p l e t i o n o f

The o p e r a t o r its S

S

(,t~)~

for ~

.We hate

M~

S~ t h e H i l b e r t

positive S

[ (~t~')~

in

and ~

~ 9 ~r 6 ~ .

On t h e o t h e r hand

u s e t h e same symbol

which i s s u p p o s e d t o be

spe~e which i s o b t a i n e d

~

and i t s

~

For

{~'1~

~

i n such a M y t h a t range still

S

, where

(~).

, there

exists

I

!

belongs to

( I

(Lt,'~)~ i s a l i n e a r

M~

such t h a t ( g , ~ r ) ~ -

M . ~r

i s t h e d e s i r e d e x t e n s i o n of

M~ t o d e n o t e t h e

extension

I~, ) denotes

Since

~ r ~ ~ we h a v e o b v i o u s l y

~Y)~. : ( N . ~~Y , ~ L ~ r ) a n d

~ ML ~

follows that

S~

~ ~ C/, ~

bounded f u n c t i o n a l i n t h e s p a c e and

L~ i s a 1~0.

S~ , l e t us c o n s i d e r t h e s c a l a r p r o d u c t

t h e l e n g t h of a v e c t o r i n t h e space

qY- 0 . I t

operator

M~ can be e x t e n d e d i n t h e s p ~ c e

In fact,

: (t ~ r )

~ ~, x h e r e each

w i t h r e s p e c t t o t h e f o l l o w i n g new s c a l a r p r o d u c t :

extension is strictly .

~ : ~,

:

~

M,

= 0

implies

M-~ .

We s h a l l

M ~ of the operator

under considerationL e t us nov i n t r o d u c e a new H i l b e r t s p ~ c e in

H~

viii

be d e n o t e d by

[

~ ~

,

.

By

~.

HL . The s c a l a r p r o d u c t we d e n o t e a

compact

(4) I t must be o b s e r v e d t h a t t h e s e a u t h o r s c o n s i d e r e x g e n v a l u e p r o b l e m s f o r more g e n e r a l o p e r a t o r s t h a n ~CO~ However t h e main i d e a s do n o t d i f f e r s u b s t a n t i a l l y from t h e PC0 ceme,

-134-

linear

S~ and r a n g e i n t h e s p a c e

o p e r a t o r w i t h domain

i s bounded s t h e r e e x i s t s i t s linear

adjoint

bounded t r a n s f o r m a t i o n of

and any

lr~

HL

H; into

]

, ~'tr

suppose t h a t

.

:

spaces

such t h a t f o r any t~ ~ ~

%r

x-

P(;J

S;

.

.

L ~ admits the following decompositionz

9 : where

~ ~ ~ that is to say, a

:

[R We s h a l l

operator

9 Since ~

~

(;)

M. R~. P

i s any g i v e n p r o j e c t o r

R~,

of t h e s p a c e H~ o n t o one of i t s

~ F . . The d e c o m p o s i t i o n ( 1 7 . 4 ) i s a d m i s s i b l e s i n c e

r e p r e s e n t e d by ( 1 7 o 4 ) , i s an o p e r a t o r which maps

S

L~

into

, if ~

as

f o l l o w s from t h e d i a g r a m :

S. (

Rt

V~ c FI.

On t h e o t h e r hand we hate*.

(M.~ P,"~. P~R~, u . , ~ )

P(:~R~ u, , R~ v

= ( R ~ p(~ IR:~, ' 'u')~ :

=

PC~)R ~,

R v

=

sub-

-135-

From t h e s e e q u a t i o n s i t

L e t us now s u p p o s e t h a t of l i n e a r l y

a complete system of

~"

9 Let

variety

.~

17.Ill.

V~,

r

i s a 1~0.

i s s e p a r a b l e and d e n o t e by t~O ( / ) }

independent vectors

be t h e p r o j e c t o r

spanned by

L ~,

follows that

of

i n t h e s u b s p a c e ~r~

H ~ onto t h e

.~ - d i m e n s i o n a l

(~) ~ ... ~ 6~ .

(4)

The s e q u e n c e :

T

R:

: T o - ~'. M~,R~, P

i s a se~uenc e of i n t e r m e d i a t e

operators~ i.e.

conditions

i)~ii)~iii)

are satisfied. We h a v e f o r a ny

t~ E S j

[ "'

(To..~) ~_ ( T o . , - ) >_. (T

-,~,1

This proves t h a t

P'~'R..].

[ P ...-k ; u , ,

> (ToU,,u,)- Y'.

c o n d i t i o n i I) i s s a t i s f i e d .

The o p e r a t o r :

T o- T

J~,

:

L L=4

M .bR .

b

P

(~)

R~

p..

: (\.,.)> ]

R~ u, ~,

:

,

.).

-136-

P (~) p r o j e c t s

is degenerate since

I n order to prove i i i ) ,

'~,~

(1T.5)

The u n i t s p h e r e ~'~

such t h a t

P c~)~ = ~

if

to shov that~

---

~ of

~

i s mapped by

R.

i n t o a compact

. L e t i%Yt~)l~, be an o r t h o r m a l c o m p l e t e s e t i n t h e s p a c e

" ~--" I [ R~ ~13" ~

suffices

dimensional subspace.

II P o:~R, - P ~'~ R. Ii o.

i~['~

s u b s e t of

to

it

onto a f i n i t e

(1~ 1~

).~

. Then

(~I] % I tends to zero, f o r

I~l~: ~

).

From t h i s ,

II P

R . ~ - P~

V..

R. ~ II

.-

.~f~-, c~ , m~iformly ~ i t h respect

(1~.5)

follows,

and t h e p r o o f of t h e

t h e o r e m i s c o m p l e t e . (~) I f we assume t h a t

L~ , R

M 4 :

"~-

~

is strictly

positive,

~ = 4 , ~ = H

-- pt~)= ~ , t h e n ve h a v e ;

T - T - L P where

P

projects

S

i n t o t h e s p a c e s p a n n e d by

i s any complete system i n the space operators constructed

space

~--4 ~

and

/~

intermediate i

i

T

and i ~ )

are the intermediate

]- ~

I

P T ~z

~ where

P

, t h e n t h e above g e n e r a l p r o c e d u r e s

case the intermediate

i

These

~ cJ

by A r o n s z a j n [1 ] . 4

If

S~ o

~)~ ,

i s any p r o j e c t o r

of t h e

g i v e us as a p a r t i c u l a r

problems c) which a r e t o be e q u i v a l e n t t o W e i n s t e i n ' s

problems.

ml

( ~ ) See f o o t n o t e

(~)

of pag.20.

We have d e n o t e d by I[

II; t h e norm i n t h e s p a c e

[4. 9

(~) The p r o o f of i i i ) c a n be c a r r i e d out by a s i m i l a r p r o c e d u r e i f ve r e p l a c e t h e h y p o t h e s e s of compactness o f R~ by c o m p a c t n e s s of M~ .

t

-137-

s If we assume

~tt)

~

C~ : ,4

s P'R4 : ~

and

5= .~t: Ht s

~4

s

= ~-2'

( L . "z > 0 ) ,

~ we have t h e f o l l o w i n g k i n d o f i n t e r m e d i a t e o p e r a t o r s :

~-~ --

4

~" % -

~ ~ ~ ~-i ~ h e r e 2

by t h e f i r s t

~

q~

is the projection

onto t h e s u b s p a c e s p a n n e d

% v e c t o r s of any c o m p l e t e s y s t e m i n t h e s p a c e

S

*

It is now evident how to construct as many examples as we wish,

starting

from t h e g e n e r a l p r o c e d u r e .

Bibliography

of

Lecture

17

[1 ]

N~kRONSZAJN - s e e [2 ] of l e c t u r e

16.

Is]

N . W . B A Z L E Y - D . W . F O X - Lower Bounds to ~.~genvalues. using operator Decompositions of the form B*B -

Arch. for Rat. Mech.

and A n a l . v o l 10, 1962. N.W.BAZLEY-D.W.FOX-

Improvement of Bounds to Eigenvalues of

Operators of the form T*T - The Johns Hopkins Univ.

A p p l . Phy. Lab. ( R e p o r t ) 1964.

~4.]

N.W.BAZLEY-D.W.FOX - Comparison Operators for Lower Bounds to Ei~envalu.e.s - Battelle Centre de recherche de Geneve-

( R e p o r t ) 1963.

~5]

N.W. BAZLLY-D.W.FOX - ~ e t h o d s f o r Lower Bounds t o F r e q u e n c i e s of Continuous E l a s t i c

S~stems - The J o h n s Hopkins U n i v .

A p p l . Phy. Lab. ( R e p o r t ) ,

1964o

-138-

C61

G.FICUERk - S u l c a l c o l o sulle

degli

applicazioni

Cagliari-Sassari

E71

G,FICHI~t~ - A p p r o x i m a t i o n s Proc,

S.T~URODA-

dell lAnalisi

alla

del Convegno Fisica

Matem. -

1964. a n d Es.t.imat.es f o r E i ~ e n v a l u e s

of ~aryland

On a G e n e r a l i z a t i o n

o f BVP, -

Papers

of Tokyo-

(to appear).

Determinant

of the College vol.

of

of the Weinstein-Aronsza.jn

Formula and the Infinite Sci.

-Atti

o f t h e S y m p o s i u m on t h e N u m e r i c a l S o l u t i o n

PDE - U n i v .

C83

autovalori

- R e p . from

o f Gen. E d u c a t i o n ,

11 - N ~ 1 , 1 0 6 1 .

Univ.

-139-

Lecture

18

Orthogonal . .invariants

of . p o s i t i v e

The method d e v e l o p e d i n l e c t u r e s after

the essential

contributions

B a z l e y , as a v e r y e f f i c i e n t v~lues of a I~0.

Hoverer

the requirement that the entire

s e t of i t s

a serious

by A r o n s z a j n , W e i n b e r g e r and

limitation

to its

applicability

this

Tu, - I

T O must be known t o g e t h e r w i t h

p o i n t ~ l e t us suppose t h a t

K

sp~ce

T

i s an

Z ~ (Oj 4 ) g i v e n by :

(x,y) ~,(y)dy.

i s supposed t o b e l o n g t o

K(x,y)

to be hermitian, i . e .

~

[ (0,4)X (0,4)J

i

f

4

/

K (X~y)l,l,(x)IA,(.y)dx ~ y

> 0

o g ~ E Z ~ ( 0 ~ t ) , I f we w i s h t o a p p l y t h e a b o v e - m e n t i o n e d

method f o r t h e u p p e r a p p r o x i m a t i o n o f t h e e i g e n v a l u e s of t h e k e r n e l (•

Y)

,

K (x,y) : IK (y,~') and to be of "positive t ~ e " ,

iee.-

f o r every/

is

e i g e n v a l u e s and e i g e n v e c t o r s ~

operator in the Hilbert

The k e r n e l

16 and 17 must be c o n s i d e r e d ,

t o o l f o r t h e u p p e r a p p r o x i m a t i o n of e i g e n -

a "base operator"

In order to clarify integral

to it

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

, we have t o know a h e r m i t i a n k e r n e l

K

(x~Y) O

such t h a t

-140-

any of its

eigenvalues

(•

~oreover

of the kernel kernel

is

greater

corresponding

we m u s t know e v e r y e i g e n v a l u e

"-o (Xj y) -

In general~

eigenvalue

of

and every eigenvector

we do n o t know %o c o n s t r u c t

the

k o (• We w i s h now %o d e v e l o p

as an a l t e r n a t i v e requires

a further

condition ~

method will

no~ r e q u i r e

ator

TO .

On t h e

under

special

the second

a different

on t h e

the

other

conditions

one to these

while

more general a 1~0

T

we s h a l l

f r o m now on t h a t

by h i m s e l f

a strictly

positive

the

modifications

slight

be made i n o r d e r

%o i n c l u d e

is

5

-

ex%ension of

(which is

operators

and that

The r e a d e r

following

a separable

sake of simplicity,

positive

compact operator.

positive

the

For the

strictly

of the

of a base oper-

i% n o t y e t k n o w n .

Hilber% space). T

h o w e v e r t h e new

m e t h o d c a n be a p p l i e d

operators,

cases

must belong to

existence

in the space

dimensional

("~

applicability

later),

first

- to non-compact

complex infinite suppose

of the

the

Its

T

be d e f i n e d

assumption

hand,

one.

operator

, which will

Let us consider

for

m e t h o d , w h i c h m u s t be c o n s i d e r e d

to the Weinstein-Aronszajn

one of the classes

stands

than the

will

results

PCO

notice

which must

which are not strictly

positive. We s h a l l the

where

6

vectors

~

is

denote ~

by

~(~)('~,..%,"~

, "'" , ~4

a positive

integer.

)the

with respect

In other

Gramian determinant to the scalar

words,

we s e t ,

of

product

by d e f i n i t i o n ,

-141-

(T ~,,~.,) ...... ( r G

~,,~,~)

('~)

(T ~ , ~ , ) ..... (1" ~,- ~,-~)

Let

~ ~Yk ~ ( k = 4 .9 ...

) be a c o m p l e t e o r t h o n o r m a l s y s t e m i n t h e s p a c e

We p u t : A%

(is .1)

~

( T ) -- ,J. O

and f o r any p o s i t i v e

08.2)

'~

~

The s u m m a t i o n Since

the

integer

4

('r)-

is

terms

of

the

Y'. _

9

extended

~

to

G `') ( ~

any set

multiple

series

how t h e summation i s c a r r i e d o u t . be f i n i t e

or i n f i n i t e .

system

The v a l u e of

~y~ }

operator T

~ i..,e.

~ are

positive

).

integers

non-negative,

Of c o u r s e

t h e v a l u e of

K~ j . . .

it

does

~

not

(T)

~ ~

,~

matter

could

(T) (T)

(T). doe,s, n o t depend ' on, ,ghe o r t h o n o r m a l

i s an orthon~r~nal i n v a r i a n t

of t h e

9

In order t o prove this important theorem we need first the

f o l l o w i n g 1 emma.

, K~ .

It is evident that-

~ (T'~): 18 ~

of

,...,~

-142-

18.II.

If

) . . . , ~ ,~. ) i s

G(,I~ 4

v e c t,,o r s , i n a H i l b e r t

s~,,ace

following

holds:

inequalit~

G,. ( i),,, 4

The p r o o f

) --.

is

~

trivial

if

~4 ),

~

subspace

~)...,

s p a n n e d by ~

K

~ and

with

respect

co-(~.,,..., ~.~ )

where

~

(-4)

= 4 +---§

t

)...

,I

.......

X

.......

X

bells

.-. / 1-~',1~ )-

dtpendent by

X in

vectors.

S~t

the

be t h e c o o r d i ~

.We h a v e :

'1

:

.....

K~4~-..~

4

X

~

~

and

the determinant:

X ........

,1~

X.

J4

4

. . . . . . .

Jr,

•

o f q~

, .t.h. e. .n. t h e

and denote

to an orthonormal

~ X 4~''''~

X

0

. Let

"X

f_,

are linearly

~)>

~

X

=

0 ~-- t,< z: ~

C7" (I~L'K4.4)

G (~,...,

of

Gram~an de t b r m i n a n t

,

Let us suppose thtt

nates

the

~C.

Jr,

~~

'

. . ...... .

~" -,,.

denotes

-143-

The s u m m a t i o n i s rots

e x t e n d e d t o any s u b d e t e r m i n a n t

of the matrix

determinant).

It

t •

contained

t ( ~ J : q~"'~ ~ ) (Laplace development of a

follows that:

~ ( ~ , . . . , ~

:

)

s

Z

X

~ ....

o(~,,.-.,~,)

z (x

~

o

( , . +, , . . . ,

We go n o t t o t h e p r o o f o f t h e o r e m 1 8 . 1 . spectral

in the first

decomposition of the operator

2 .,,

.....

,~

~,

).

L e t us c o n s i d e r

the

T ~ :

O0

We h a v e

-~

( '~ > 0

):

4,q

~lq

4jq

41~

4~,-~oo K4j-,,K, ~ c~->~o

S

k4 ~ kt

h4

h~

~

~ h

-144-

L e t us d e n o t e by spanned by

9

the projection

%r4,... , ~rn~

d e t e r m i n a n t of (~,~)

~ /

~4}...

;

G

~/~

on t h e v a r i e t y

of

(~4,...~/~

i s t h e Crramian

Tith respect to the scalar

product

We h a v e :

A~.--~q

-~! ~ 0"): ~,"~ ~ ' ~

"

L e t us now suppose t h a t t h e m u l t i p l e v h e r e the, s - ~ y y t i o n i s e x t e n d e d t o a n y s e t indices

is

convergent.

Let

~

denote its

(P9, m ~,h4

There

is a positive S i n c e , by lemma

~4 J " - , ~B

~---p~

we h a v e :

,,,).

of d i s t i n e t

sum and a s s u m e

r e a l number g i v e n a r b i t r a r i l y . 18 . I I ,

Pn,~

series

i s such t h a t :

~- Z

J"'/

~

-145-

G

it

)

follows

"

.

.

)

"P ','-

)~-IP,,,.,.,,i

~"

IP.,,,~,

thatz

z~ t"j,,

P'h,

h 4 ,.., h s

.

.

.

)~

.

h~ ,.., I~

4j., cJr

~'h- ['I.-G(P

,r

~,

) ] +.2.,6..

P ~

Thus~

(z8.3)

~ L

(r)

Fh,

Suppos e t h e r i g h t - h a n d s e r i es i n ( 1 ~ . 3 ) let

~i.i

I ~"

be s u c h t h a t

9 fie,,

is

divergent.

I"h '

t~,

Given

H '> 0

--

%"'~H ~t

pN.,, > H

,,

Sincez

,...,

P

,.,.

)>_

-146-

> /;,~ it

~

p~-..l~ '~

follows that

in this

: ..I-~ .

~,

,.-.,P ~ ) > H

This means t h a t

(18.3) also holds

case.

The i n d e x ~

(T)

G(P

(T)

~

w i l l be c a l l e d t h e o r d e r o f t h e o r t h o g o n a l i n v a r i a n t

and ~he i n d e x

Ig . I I Z .

~

We have

t h e degree of t h i s T ) < + 0o

invariant.

i f .and, o,nl E i f

~t

The p r o o f i s a c o n s e q u e n c e of t h e f o l l o w i n g i n e q u a l i t i e s ~

~

(Igo4)

(T)

~

5!

~

(. T)

5

~:4

S i n c e (lemma 18 . I I ) :

G (18.4)

~(~

foiloTs

from

,--., "~'.~) ~ (iX.2).

P,--P~-4 From t h i s

i

inequality

1~0

G(~)

)

(~,)

(~'~) . . . G~

Zn o r d e r t o p r o v e

(18.5)

Te o b s e r v e

tha~:

P~+F,~,, + .... ) ~ "~ ~ - s ( T ) ' (18.5) folloTs readily.

is said t o belon~ t o t h e class ~ ~ ,i,f ~ 4

It is evident that

C '~.~

if

~ n , ~ m,.

(T)z+~,

-147-

18

,IV,

The s e q u e n c e of p o s i t i v e

.i,s a c o m p l e t e s y s t e m of i n v a r i a n t s of two ~ O ' s

of t h e c l a s s

numbers

{ ~'(T)

} (~:4,~,..

)

T i t h r e s p e c t t o t h e unitary., . e ~ u i v a l e n c e

~

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

and ~

a r e tTo o p e r a t o r s of ~ z

such that.*

~(T then a unitary

) - ~(

operator

(18.8)

T -

of t h e s p a c e

U-'R

e x i s t s such t h a t :

equivalent.

Let us denote - as usual - by T

~

(~: ,I,.~, ... )

br

i . e . t h e tTo o p e r a t o r s a r e u n i t a r y

of

R)

{ ~ ~ }

( e a c h r e p e a t e d as many t i m e s as i t s

the s~uence

of e * g e ~ a l u ~ s

multiplicity).

The i n f i n i t e

product: ~t

c o n v e r g e s u n i f o r m l y on any compact s e t of t h e complex ~ - p l a n e d e f i n e s an e n t i r e

function

~(~)

of t h e complex v a r i a b l e

and

~,

Let

us d e f i n e :

Let

T

t t ~ } be a c o m p l e t e o r t h o n o r m a l s e t o f e i g e n v e c t o r s of t h e o p e r a t o r

, ~th

T a

= ~,%.Venote

by

~

the projector ~ich

o n t o t h e ~ - d i m e n m i o n a l m a n i f o l d s p a n n e d by

L~4 j . - - ~ ~

.

projects We have.*

5

-148-

(~X)= Z (-~) J~ ('P,~T) 'xs.

For any

~

, let

us consider

the power seriess

(4)

It

converges

follows

in the

entire

~-plane

with respect

t o rn~. T h i s

from the inequalities:

9+~

9 +~

~

I ?"

(-'

(PT)

;~

I z s

:~=9+.t

J~ (P~T)I:Xl ~ ~-

,~-. 9+4

,~ C --

U

(T)I),I

z_ ?.

-~!

,t

(T)

I),I.

~=9+ ~

,s=9,4

On t h e

uniformly

other

hand,

for

any given

~

we h a v e s

cl

(P T)), ~: ~ (-.4)~(T)), ~. /vw --~ cQ

Thus,

for

any complex

.,~',~ 7. C-4 4 ~ --, oo

"$:O

,,$:O

"$=O

(~)The operator P~ T in the ~t -dimensional

~

~'('P

T )

Z.. "&"O

i s c o n s i d e r e d as a s t r i c t l y positive s p a c e s p a n n e d by ~t4 j "'" ~ ~ ~ '

operator

-149-

Sets

(,~) = Z.

(-~)',~"(r)),',

~'"(t). 7'. (-,)~,, ('P,,,.T)),'.

"S:O

Given

F.. > 0

, let

~&(~)

s!

We &ssume t h a t

such t h a t f o r

~ "~ ( T )

~ > C~(~)

I>,l ~

~ ( ~ ) i s l a r g e enough t h a t f o r

One has:

Thus, f o r

~

> q8 ( I )

,

I t f o l l o w s that=

oo

(18.~)

5

(T)~.

E.,.

c ~ > ~&

)

-149-

Sets

(,~) = Z.

(-~)',~"(r)),',

~'"(t). 7'. (-,)~,, ('P,,,.T)),'.

"S:O

Given

F.. > 0

, let

~&(~)

s!

We &ssume t h a t

such t h a t f o r

~ "~ ( T )

~ > C~(~)

I>,l ~

~ ( ~ ) i s l a r g e enough t h a t f o r

One has:

Thus, f o r

~

> q8 ( I )

,

I t f o l l o w s that=

oo

(18.~)

5

(T)~.

E.,.

c ~ > ~&

)

-151-

Bib_liofra~hy

C1]

G. FICHER& - F g n z i o n i

of

Lecture

analitiehe

18

di una Tariabile

com~lesea -

E d i z . V e s c h i - Roma~ 1 9 5 9 .

[2]

E . GOUBXkT - C.ours d*Anal]vse ~M~.t h 6 m & t i q u e - v o l ~ Villars

- Paris~

- Gauthier-

199-4~

- U_eber d i e I n t e g r a l e

d e s H e r r n H.e!.linKer und d i e

O r t h o g o n a l i n v a r i .an~en d e r q u a d r a t . i s c h e n T..on u n e n d l . i c h v i e l e n

I!

Varandlichen

Formen

- tionat~

P h y . Bd~ 2 3 , 1 9 1 2 .

[4j

E. H~LLINGER - Di 9 0 r t h o g o n a l i n v a r i a n t e n

quadratiaehen

yen undendlich Tielen Yariablen vl

Gottingen,

1907.

Formen

- Dissertation

-152-

Lecture

Upper

approximation Representation

19

of

the

of

eigenvalues

orth.ogona 1

L e t us c o n s i d e r an a r b i t r a r y

of

_a PC0.

invariants o

c o m p l e t e s y s t e m { ~v4K} of l i n e a r l y

independent vectors in the space

~.

Let and

~r

be t h e

P -dimensional

m a n i f o l d spanned by

~/4 ~-.- , ~/~

Pe the projector

As we saw i n l e c t u r e

15 9 t h e a p p r o x i m a t i n g e i g e n v a l u e s g i v e n by t h e

R a y l e i g h - R i t z method c o i n c i d e w i t h t h e p o s i t i v e operator

on ~ ) ~ r

eigenvalues of the

V~ "1" V~ . I f "1" - as we have assumed - i s s t r i c t l y

then the determinantal

positive,

equation:

c~et I(-]-W'.,w'i)-~(W'~ ~W/j)I = 0 (v) has ~ p o s i t i v e

roots

~4

(v) -~ ~ z -~

"'" ~

(~) ~V "

to Deno~,e hy ~ v~(t<)

L e t us c o n s i d e r t h e e i g e n v e c t o r s of P ~ T P~ t h e above e i g e n v a l u e s 9 s a y the

(~-4)-dimensional

~/(~)

(v}

~/d (w) ~ . .. , ~ )

s u b s p a c e of r ~ Let

corresponding

~(~

spaan~ed by

be t h e p r o j e c t o r

of

The f o l l o w i n g t h e o r e m h o l d s :

19.I.

Suppose ~ G ~ o

Fixed

t~>O

and

~ >0

, s e t f o r V~>~:

-153(') 4

m

~t

(IOn) 6-~(~): t ~-4

~,

~,

We have:

(v) K

where

~

--

}

K

- as ,usual - de,notes ,the se~uenc,e

of

)

,.the eigenvalues o f -/'.

~e have for ( 1 8 . 3 ) : 4j.~ 1%2 "1%

(1~ .5)

~-~

P~"'T P v~ ~

:

z~ ~"'

h~.., h

p~ ' ! ~ . ,

,

tj .-)

where

~ 9(~) k ~ . . ~ h~.~

set of ~-4 I% f o l l o w s

meA.-A t h a t t h e summation i s e ~ e n d e d t o any

increasing

integers

chosen amongst

~j...

K-4 ) K+4; . - - ; ~ .

that-

c~)

~

[ , k~--~-~

~'" - ~ ' ~ 1

~4< --'c h~. 4 (t)

E~ (~} a l s o d e p e n d s on ~ and on ~ , b u t we do n o t n e e d t o p u t i n t o e v i d e n c e t h i s d e p e n d e n c e s i n c e we c o n s i d e r ~ a n d % %o be f i x e d . For the definition of the orthogonal invariants o f P~-r P~ and PvIK) T p(~l see footnote (4) o f lecture 18.

-154-

4 ) .. ,v*~

+

I~,I~:--~' h~s

_.

_

~"

15,,

- - 9 ~,,

Since (see le~na IS~

4,..,1,0

L~)

4).. ) ~-)r4

L~)

~'~' [~,,..i~,~., ] ~,4<.-z h

Z

In.14-- ,~ Inr

,t

(].~ .6) (~)

(:v)

J

L

,'it

I~ ~-.!

.~

4) . . , ~ §

Y----'"' [ I*~,"" I',,~ ]

J

(19.2) f o l l o v s . From i n e q u a l i t y (19 ~

we hate for ~

TM

,~,,,, (pC,,) T pO,)) > ~'- ( P'")T P'")). On the other h~nd, one has (see len~a 15.X and 18. VI)s

Hence, by ~heorem 15.XIII, (19.3) f o l l o v s .

~heorem l ~ . I pro.ides a sequence { ~ ~ ) ~ K

decreasing to

~K

~ich

c o ~ e r g e s by

, provided some orthogonal i n , a r i a n t ~ ( T ) o f

T

-155-

~t

i s known.

In fact,

is expressible

on t h e r i g h t

hand s i d e of ( 1 9 . 1 )

i n t e r m s of t h e R a y l e i g h - R i t z

e v e r y t h i n g but ~

4

(T)

approximating eigenvalues

(see (19.4) and ( 1 9 . 5 ) ) . Theoretically,

(18.2) gives

the invariant

the v a l u e of

can be computed~ However, i t series

(T)

~

g i v e a l o w e r bound t o

c o u l d be c o n s i d e r e d as known s i n c e as t h e sum of a s e r i e s

must be p o i n t e d o u t t h a t ~

whose t e r m s

partial

sums o f t h i s

~ T ) , w h i l e form-dla ( 1 9 . 1 ) r e q u i r e s

an u p p e r bound f o r s u c h an o r t h o g o n a l i n v a r i a n t ~

In c o n c l u s i o n ,

(18*2) c a n be u s e d o n l y when one knows how t o

e s t i m a t e t h e r e m a i n d e r of t h e s e r i e s

on t h e r i g h t

hand s i d e of t h i s

formula 9

In any c a s e ,

theorem ( 1 9 . I ) c a n

be c o n s i d e r e d as a remarkable

advance i n t h e problem of t h e upper a p p r o x i m a t i o n of T p~

e.

, s i n c e t h e upper a p p r o x i m a t i o n

of t h e e i g e n v a l u e s

of a s e q u e n c e o f numbers - t h e

ts - i s r e d u c e d t o t h e u p p e r a p p r o x i m a t i o n of a s i n g l e number, i .

~ ~ (T)

(with ~ and ~

In numerical applications,

chosen a r b i t r a r i l y . t h e u p p e r bound

c a n be i m p r o v e d i f we know an o p e r a t o r for any

K

, ~ne,e

t~

~

such t h a t

for

~g

'~k's

~ ~h ] the corresponding eigenvectors.

for

Let

P

~P

-~ ~ K

h-"'l,...,p

and c o n s i d e r t h e f o l l o w i n g o p e r a t o r s :

R(K) v~

~

~ i s t h e s e q u e n c e of e i g e n v a l u e s o f ~ .

Suppose we know t h e n u m e r i c a l v a l u e s of t h e

We d e n o t e by

Cr( ~ )

(~J

J

~

K"

K ~- ~ ~

(p.~-i-oo).

-156-

w

t

" The v a l u e

"

1

~ ~

t

(R v p )

,&-.~

furnishes a better

u p p e r bound t o

~,~

than

6 (~) K

~

We w i s h noT, by r e p r e s e n t i n g t h e s p a c e S as a H i l b e r t f u n c t i o n space, to establish if

T

some f o r m u l a s which p r o v i d e t h e c a l c u l a t i o n

i s e x p r e s s e d as an i n t e g r a l Let

~

is the Itilbert space Since i.e.

~ the

p

s p a c e where a

has been i n t r o d u c e d . Suppose t h a t

~ ~ (A,~)

o f complex v a l u e d f u n c t i o n s on t h e s e t A .

i s s e p a r a b l e , ~e must s u p p o s e t h a t t h e measure ~ -ring

"~(r)

operator.

be a m e a s u r a b l e s e t o f a g i v e n measure

r e a l n o n - n e g a t i v e measure

of ~

~

is also j

of t h e m e a s u r a b l e s e t s i s g e n e r a t e d by a c o u n t a b l e s

s e m i - r i n g o f s u b s e t s of t h e s p a c e . O f c o u r s e t h e a s s u m p t i o n t h a t t h e space

j~ ~ ( A , ~ )

is the space

separable infinite space

S

is not restrictive,

s i n c e any a b s t r a c t

d i m e n s i o n a l H i l b e r t s p a c e i s H i l b e r t - i s o m o r p h i c t o some

~ Z (A, ~).

The s c a l a r p r o d u c t i s n o t g i v e n by:

A L e t us s u p p o s e t h a t

T ~t

t h e s e q u e n c e of e i g e n v a l u e s o f T following integral

T

representation:

u.

: A

, then the operator T ~

admits the

-157-

where the kernel

K

(X,'y')

is

the

function

Z

if,,

,~]

of ,~2., [ A •

d e f i n e d by t h e d e v e l o p m e n t :

(~) :

(x,y)

K

~

,~ ( x ) ~

(y).

K=4

Here,

{ I.,t,K (X) j

of t h e o p e r a t o r

i s an o r t h o n o r m a l c o m p l e t e s e q u e n c e of e i g e n f u n c t i o n s T.

Then m ~t~

u, =

where

has t h e i n t e g r a l

k

(x,y)

representation:

u,(y)a~y

:

(zg.~)

F~

(x, y) :

P'K u.~ ( x ) u. K K-4

L e t us c o n s i d e r t h e f u n c t i o n : (~)

cx,,x~ ) . . . . (x, , . . . ,

It

is

summable

x~ ) :

on the

~

(x,,x)

9

9

=

9

9

9

9

9

9

t

9

9

9

9

e

$

9

9

~*

r

set

A

x

A

)r

9 - " xA,

In

fact

we have:

-158-

Rr(X' ,... x ). ~ ~'"~" q~- .q,,

(X4, X q~ )...~

k

Cx 'I) x

q~

).

Then:

...fl K

I ~ (x,,..., x, )1 -~ -~! I I

~{p~

A

A /Yt

19.II.

(•

The .ortho~on~l, i n v a r i ~ n t

( T ) has r

following

integral representation:

..., x, ) ~l~x ... C~ PX,& " By u s i n g (19 .'/) one hms;

~(x 4 ) u,~,(k~)

u.. (x 4 ) ~, (x~) ,,$

I.,4

'{j..;.~ J

9

4

) .... u~(x~) u~C(x ~) 4

{.4~~,4 9 .-

~.#, (x,)...ut,{, (x 4

'{

uc, (x,)...u~. (x~)

-159-

Integrating

A x A X--" x A

over

the ~

(this Remark~

previously

and e m p l o y i n g t h e r e s u l t s

p.

my

Then by d e n o t i n g by (~)

~

which we s h a l l

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

and

~ :

t h e v a l u e of

and, a c c o r d i n g l y ,

4

.2', 4..,p

k". . ~ ( •

R:,~, Cx, , • )

~Xt...~ X.

A I~ (~}

Assuming i n ( 1 9 . 1 ) t h a t in the particular [4 ] 9

case

fr~.-~ p = ~

I~)

,

,

t•

,

~= ~

X ) ....

t~

(X

and i n s e r t i n g

&l

into it

, one g e t s a f o r m u l a a l r e a d y

He, however, even i n t h i s

n o t p r o v e t h e c o n v e r g e n c e of Theorems

by

I".., I A

known t o T r e f f t z

t~(X~)

Y Jj ) , f o r m u l a (19o8) becomes:

K,~, ~,' (x,, x,, ) .....

(19.9),

~

u n i t masses on t h e i n t e g e r s

~,~X)

~,2,..,p

~

.

(•

(i~.9)

in

Fredholm

i s t h e p r o d u c t of t h e o r d i n a r y Lebesgue measure i n

x ~ A c X ~

the kernel

(]9.8)

formula

and of an atomic measure composed of

for

of t h e c l a s s i c a l

i n v i e w of t h e a p p l i c a t i o n s

t o w r i t e down e x p l i c i t y

~j s ~ - . - j

~ (~)

equations.

It is wor~b~ile,

~

~ith

c o u l d be o b t a i n e d by u s i n g t h e

c o n s i d e r e d d e v e l o p m e n t of t h e Fredholm f u n c t i o n

t h e o r y of i n t e g r a l

measure

(19~

the integral

we get (19o8)o

is obviously possible),

Representation

a peter series,

consider,

and i n t e r c h a n g i n g

particular

case, did

~ 6- K

l ~ . I and 19.II s o l v e c o m p l e t e l y t h e problem

of t h e u p p e r

-160-

approximation

o f a 1~0

representation problems

for

arising

boundary value matrix

is

belonging

we a r e n o t

in the classical

the

theory

~ case

~

of integral

able to compute the 19 . I I ,

since

of the

elliptic

corresponding

the'kernel"

eigenvalue

equations Green's

e v e n i f we know t h a t

- as i n t h e c a s e

, when an i n t e g r a l

for the

problems in which the corresponding

exists

theorem

t o some

known. This is

is known. Unfortunately,

formation

using

T

T

this

or for

function Green's

boundary value

orthogonal

transproblems,-

invariants

of the transformation

or

is

by not,

in

general ~ known. The f o l l o w i n g this

theorem will

be u s e f u l

in attempting

to overcome

difficulty. be a , d e c r e a s i n g

uniformly

T

to

.

o f ,I~O's , converging

Set:

(P T P.)

(q) ~)

(19.1o)

sequence

K

We h a v e :

and:

(19,12) The p r o o f

of (19.11)

follows

easily

if

we o b s e r v e

that

~

k

> -

tt

-161-

(lemma Ig.V ) and t h a t ,

(19.~),

by t h e same arguments u s e d i n t h e p r o o f of

~K {~'~j > ~K (~'~) "

(le=a

~8.V~

(zg.12) folto~s from ~

~ K (~'~J ~ Kc ~

~

"-

(~.~).

) and from

Remark. I f we r e p l a c e i n t h e f o r m u l a ( 1 9 . 1 0 ) t h e o p e r a t o r

by T~ and . ~ the space

~

~K(

(i.e. tl, e ~ y l e i ~ - m t - approximation, in

, of t h e e i g e n v a l u e

an u p p e r bound by

by p ~ CYK

for

Hoverer,

T

~g

~ (~)

of

~-~

) we s t i l l

obtain

, which i s b e t t e r t h a n t h e one g i v e n

i n o r d e r t o compute ~ ( ~ ' ~ )

Te m u s t , f o r an../ ~

,

compute t h e R a y l e i ~ l - R o t z a p p r o x i m a t i o n s o f t h e e i g e n v a l u e s of T ~ . I f a "base operatbv"

To

i s known, t h e n as o p e r a t o r

may u s e t h e ones c o n s t r u c t e d i n l e c t u r e that,

17.

T~

we

However i t must be remarked

now we do n o t n e e d t o know t h e e i g e n v a l u e s and t h e e i g e n v e c t o r s

f o r To

, but the orthogonal i n v a r i a n t

~

(T~)

j which e n t e r s i n t h e

formula (19.10). We wish t o remark t h a t

orthogonal invariants

can

be u s e d i n

several other topics connected with eigenvalue problems, for instance in the still

partially

of t h e m u l t i p l i c i t y

o f each

orthogonal invariants, to the multiplicity

u n s o l v e d problem c o n s i s t i n g eigenvalue

~ K of

of each

I n f a c t ~ by u s i n g

~ K " I t i s h n T e v e r n o t y e t known h o t t o

This would d e t e r m i n e t h e m u l t i p l i c i t y For an i n t e r e s t i n g

this

T.

it is possible to construct sequences converging

g i v e u p p e r and l o T e r bounds t o t h e m u l t i p l i c i t y , than 1.

in the computation

application

w i t h an e r r o r l e s s completely.

of o r t h o g o n a l i n v a r i a n t s

to

problem see [ 1 ] . A n o t h e r a p p l i c a t i o n T h i c h can be madej c o n c e r n s t h e mini-max

principle

(see IS.VIII).

-162-

We l e a v e t o t h e r e a d e r t h e p r o o f of t h e f o l l o w i n g t h e o r e m (where t h e same n o t a t i o n as i n t h e o r e m 1 5 . V I I I 19 oIV.

Let,

T

15 u s e d ) :

b e l o n g ,to

~.

A n e c e s s a r y and s u f f i c i e n t

condition for the e~uality sign to hold in the following relation:

is t h a t =

I~, ~'~

(R)]

where t h e ol~_er,a t o r

~

-

: t~,,,

is the following= K-4

K-4

R~-- T.

~"

h=,2"(T~,~-h)v.

_ ~~,

( ~ , , ' v ' ~ , ) T vW

4,..j~-4

9+

T__.,

(Tp u ,p~)(~.,p~,)p~.

Another approach t o t h e t w o - s i d e d approximation of t h e e i g e n v a l u e s of a I~0 i n i n t e g r a l highly interesting

form i s due t o L . De V i t o [ 2 ~ . His method i s from a t h e o r e t i c a l

p o i n t of v i e w and does n o t r e q u i r e (2) t h e u s e of t h e R a y l e i g h - R i t z a p p r o x i m a t i o n s . However t h e i t e r a t i v e

( ~ ~' U n f o r t u n a t e l y t h e m a t h e m a t i c a l i n t e r e s t o f De V i t o ' s r e s u l t s has escaped researchers working in this area, probably because of a quite incompetent review of De Vito's paper published in Mathematical Reviews.

-163-

technique rather

needed for

impractical

the

application

procedure

point

of view.

from the numerical

Bibliography

[i]

of his

of

MoP. COLAUTTI - S u l c a l c o l o

Lecture

dei humeri

differenziabile~ atlante

[ 2 ]

L . DE VIT0

-

Sul calcolo

G. FICIIEP~

[4 ]

v.. TREFFTZ -

-

out to be

19

di Betti

di una varlet&

n o t a p e r mezzo d i u n s u o

R e n d . d i M a t e m . - Roma~ 1 9 6 3 .

approssimato

trasformazioni

[3]

turns

compatte

plicit&

- Nota I & II

see [1]

of lecture

Ueber Fehlersh~tzun~

degli, autoya, lori e delle

relative

de,lie molte-

- Rend. Accad. Naz. Lintel,1961.

1.

b e i B e r e c h n u n g y o n F.igenwe.rt, en -

M a t h . JLnnalen B d . 108~ 1 9 3 3 .

-164-

Lecture

~plicit

construction .for

an

20

of

elliptic

~x,'~,

L e t us c o n s i d e r t h e

L (•

the

Greents

matrix

s~stem.

matrix differential

o p e r a t o r of o r d e r ~ u

- D~p cx~ D '~

(o'Ipl',,~_ _ )~

Suppose t h e c o e f f i c i e n t s in the entire

(~ ( x ) t o be complex ~ x ~ Pq X ~ c a r t e s i a n s p a c e and b e l o n g i n g t o

matrices defined ~ oo.

We make t h e f o l l o w i n g h y p o t h e s e s ; i)

The o p e r a t o r

L (x)~) is elliptic

for every

• E X ~ , i.e.

(~ real ~0);

(Ipl:lql: '~) ii)

L (x~D)

is

formally

O.pq(X) : (-4) iii)

self-adjoint,

tpt-t-lCl t

Consider the bilinear

(u,,~)

:

connected with the operator

i.e.,

O..qp(X) ;

form:

(-4)PfAO.p, ~D#u. D~'~ fix, L (•

in

the properly/~dom~in

A

~egulae (4)for the definition

of p r o p e r l y r e g u l a r

domain s e e

lecture

3.

-165-

The c o r r e s p o n d i n g q u a d r a t i c form

~ ( ~ , ~ ) i s such t h a t :

(-~)- ~ (~,~) _> c

/ ID~'u, l~dx

~ Ipl--~

f o r any

~ 6 ~ ~

J

where

A

C i s a p o s i t i v e c o n s t a n t i n d e p e n d e n t of

o

A further hypothesis will require that:

iiii)

A fundamental 'matrix in the large for the operator L (• D)

e~sts.

This m t r i x -

say F(X,F) - is defined as follows: F(x,y) is a ~ x ~

matrix defined for (X , y)E (X ~x X )-~,where ~ cartesian

~(x,y)

2)

F(•

3)

I) '~ F ( x , y ) x

is

and i s such t h a t :

C ~" i n t h e s e t

;

~

: 0

I•165

~o~ I •

belonging to

,,

Z 2,CX ~) and v a n i s h i n g o u t s i d e of

the function:

u(x) = #x~~(y; F(x,y)dy

(zo.1)

Z ~ - w e a ~ s o l u t i o n of t h e d i f f e r e n t i a l From t h e t h e o r y of e l l i p t i c

follows that the function

(~)

(X~xX~)-~

: F (y,x)

For any

bounded s e t ,

i s an

X ~ • X ~

1)

4) a

product

is t h e diagonal of t h e

See l e c t u r e

5.

bb(X)

equation

linear differential has

~

strong partial

L b~ : ~ 9 operators

,

{2)

derivatives

it up

-166-

to the order

~m~

i n any bounded domain of t h e p l a n e .

D e r i v a t i v e s of order not exceeding ~m~-~ can be computed by differentiating h y p o t h e s i s 3) If

(20.1) under t h e i n t e g r a l

9

~ s C ~ ~ then

differential

equation

~(g)

F(•

~pq

Let

Ca~

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

in the c l a s s i c a l

operators Tith constant c o e f f i c i e n t s 9 the

CLpq(X) ~ 0

for

Ipl§

c o n s t a n t m a t r i c e s such t h a t

L (~) : ~ ~qI ) F D q ~ . By L ( ~ )

and denote ( f o r IpI = Iql = ~ )

z~

we

~ (~> : ~et

Let us denote by

dimensional c a r t e s i a n s u r f a c e element on ~

5(•

~

t h e u n i t sphere

apace and by d ~ . Define (

:

(~T~,)~-' (~.4)! for

S

/~

:

(A y)

~ O.

q

and t r a n s p o s i n g

J~J , ~ i n t h e

;O -

t h e measure of t h e h y p e r i s t h e Laplace o p e r a t o r )

( ay)

~ odd~ and

O~pq~ P ~

s h a l l denote the matrix obtained

by t a k i n g t h e m a t r i ~ of t h e c o f a c t o r s of (L q ~ ~ it.

sense.

can be given i n c l o s e d form. For i n s t a n c e p l e t us

suppose t h a t by

is

L t~ ~- f

In t h e case of e l l i p t i c matrix

s i g n . This f o l l o T s from

I G~('~)

-167-

for

~ even (see [ ( ~ ]

).

F (x,y)

Then

i s d e f i n e d as f o l l o w s :

(3)

F (x,y)

L (:D) 5 ( •

In t h e g e n e r a l case of v a r i a b l e fundamental solution - Aj that

~=

~.

.

coefficients

the existence

i n t h e l a r g e h as been p r o v e n by G i r a u d

F or ~a

arbitrary,

see

[ 3 ~ .

of a

~ 4 ] for

The method d e s c r i b e d i n

p a p e r ca n be e x t e n d e d t o t h e c a s e of s y s t e m s , L e t us c o n s i d e r i n t h e s p a c e

functions with scalar

~

H~

strong derivatives

( A ) ( s p a c e of v e c t o r v a l u e d

up t o t h e o r d e r

~

) the net

product:

The s p a c e o b t a i n e d from

~

(~)by

functional

to this

net scalar

product Till

by

the finite

dimensional vector

~

of degree functions

_~ ~ - ~ of

~

belonging to

~.

, such t h a t

(A)

completion with respect

be d e n o t e d by ~

(A),

If .e denote

s p a c e composed of p o l y n o m i a l s ~ /

B (~/p~/) : 0

, Te must c o n s i d e r tTo

as c o i n c i d i n g when t h e y d i f f e r

The s p a c e

~ ( A ) i s none o t h e r s e x c e p t

isomorphism, than the quotient L e t us d e n o t e by ( ~ )

by a p o l y n o m i a l

space

H~

the scalar

(A)

for a Hilbert

/ r.

product in

~

(A)and consider

the operator

IA ~ (y~ I:"(x,y) ay. (3)

~

~

I f L~i (.~') is t h e e l e m e n t o f L.('~'), t h e n by m a t r i x Those e l e m e n t s a r e ~ . .~j ( ] ) ) S ,

/~

L('D)5

.e

mean

the

-168-

Since for

~

~ Z ~

(A),

R U.

I--i~

belongs to

as an o p e r a t o r w i t h domain

~ ~ (A)

It is eanil 7 seen that

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

has

~ (A)

~

R

, we can c o n s i d e r

(A)

and r a n g e i n t h e s p a c e

a dom~n and range in

~ (A).

The a d j o i n t

Z ~ ( A ) . For

~ ~ H

operator

~ ~

(A) i t i s

e x p r e s s e d as f o l l o w s

R 'u-- (-~)

DP v(,)) x

p (x)

F(v,x)ax

Dq x

A and r e p r e s e n t s

a function belonging to

H

i n a n y compact s e t o f t h e

plane. L e t u s nov c o n s i d e r t h e f o l l o w i n g BVPs

L (•

(20.2)

: (-J)~

A ,.

in

(~o.8)

DP~ --0

on

@A

o_~ I p I -~ m r

Suppose we w i s h t o r e p r e s e n t -- R ~13

containing t h e spa~e

b7 a p r o p e r c h o i c e of A ~z

in its (Ao-A

interior.

~ 9 Let

Lot

t~

~/' F_ A o- ~

is

H~

(A)

i n t h e f o l l o w i n g wayz

A o be a bounded domain

( x ) j be a c o m p l e t e s y s t e m i n

) . The b o u n d a r y c o n d i t i o n s

(in the sense of functions of ered for

the solution

( 2 0 . 8 ) w i l l be s a t i s f i e d

) if the f~nction

R x~r

consid-

such t h a t l

/

0,0.4)

~

@,~ (,/) R * ~ d y

= o

Ao-A

Sets /

(x) : |

co

q~K (Y)F(X'y)cly'

K

o-,6,

C o n d i t i o n s ( 2 0 . 4 ) can be w r i t t e n :

( K: ~,:~,- )

-169-

(.-0.5)

(K=

V

L e t us c o n s i d e r t h e m a n i f o l d equation

L (x , ~ ) ~ = o

A

in

of solutions

Let

H~

P

,

('A).

~

since it

This

is a closed sub-

(A). be t h e p r o j e c t o r

PIz - O

is satisfied

if

the function

~--

For

(A) ~

1~ ~ kl

(A)

J .

o f t h e homogeneous

, ~hieh belong to

manifold i s a c l o s e d subspace of space of

~,2,.-.

4%

of

9 It

(A)

onto ~ /

follovs that

, Condition (20,5)

f o r any

~- ~

FI~ (A),

R * (17-PIT;satisfies t h e boundary c o n d i t i o n s ( 2 0 , 3 ) . FI

2~

( A' ) ( f o r every A'

such t h a t A' C A

) we havel

b~

as c a n y p r o v e d e a s i l y .

It

follows that the function:

i s t h e s o l u t i o n o f t h e BYP ( 2 0 . 2 ) ,

(20~

We have t h u s c o n s t r u c t e d e x p l i c i t l y

G--

R R-

This construction is perfectly by u s i n g r e s u l t s In fact,

of lecture

the Green's transformation:

R*P R . iJ

suitable

f o r a p p l y i n g t h e o r e m 19.111

17.

l e t us t a k e a b a s i s i n t h e s u b s p a c e

~/

, say

~ ~

~ ,

-170-

~Q

and d e n o t e by

by

~4

, ..~j~

- Iq ~ P~ ~ (5

,

the

projector of

~ (~)

From theorem 17.]II

converges unifermly to

and t h e o p e r a t o r s

~

the subspace spanned

it follows that

G.

~

: ~

~ -

On the o t h e r hand~ the o p e r a t o r ~n,

belong to

onto

f o r any

~t

such t h a t :

>

This f o l l o w s from p r o p e r t y S) o f The o r t h o g o n a l i n v a r i a n t s be c a l c u l a t e d

by u s i n g t h e r e s u l t s

e x p r e s s e d as an i n t e g r a l

F(X,~/)o of

G~

corresponding

of l e c t u r e

19~ s i n c e

t,o such ~t R* R

o p e r a t o r and t h e same i s t r u e f o r

can

can be

~ ~ ~

~ ]

which i s a d e g e n e r a t e o p e r a t o r . I t f o l l o w s t h a t we may c o n s i d e r as s o l v e d t h e e i g e n v a l u e p r o b l e m s c o n n e c t e d w i t h t h e boundary v a l u e p r o b l e m ( 2 0 . 2 ) ,

C-V-p

(20o3), i.e.

=o.

L e t us c o n s i d e r some p a r t i c u l a r

cases corresponding to classical

e i g e n v a l u e problems of m a t h e m a t i c a l P h y s i c s . For t h e s e p r o b l e m s we s h a l l construct explicitly L e t us f i r s t f o r an i s o t r o p i c space

~

I

or

the approximating sequences for the eigenvalues. consider the classical

o p e r a t o r of l i n e a r

elasticity

homogeneous body~ which we w r i t e as f o l l o w s i n t h e

X 3 :

-171-

with

the

boundary

assume the

condition

L~ = 0

on ~ A

.

As b i l i n e a r

f o r m we may

following:

1~ (' u,,~')

,:/l, 'v':/~,,, ~;/~ %,/~, ) a• A

(we c o n s i d e r

f r o m now on o n l y r e a l

vector-valued

functions).

Let us assume that: -4

(--~(t) I The f u n d a m e n t a l

F,:j (x-y)

:: ),x:)~tt "4 matrix

~.:2,~ =.5 .

for

- as given

by Somigliana

- is

the

following:

9z (x_y i zcp(,Ix_ y j )

g~ (4+,~)

,"~x; '3x i

Set.

"f~j (x,y) ---

IIF:,,/h

j,~/h

~,,(/,<

~/h

A Consider

a complete

system

[ ~ PJ

equation Lbl.-0 , suchthat - [~) (r

(4)For

the

L~ = 0

construction of complete s e e [ ~ ] chap. |I~,

of solutions

of the

coq) = c~pc I .

systems

(4)

of solutions

homogeneous

Set,

of the

equation

-172-

./~

(t)+~ F

(x-t)~P

~I~

)

(~)~dt,

l~li~

A For t h e e i g e n v a l u e s

~ ~K ~

L~+~X~ =o

of the f o l l o T i n g problem:

in

A)

OA

bl,=O

,

we h a v e =

~---,~-> oo

-- ~:~ ~ ~t (K')

:~X~.

~ -~, oo

(,;)

The

where

are t h e roots of t h e f o l l o w i n g d e t e r m i n a n t a l e q u a t i o n :

~ ~v/~ t is

any complete system of f u n c t i o n s v a n i s h i n g on / ~ .

The 17K(v~ are given by the f o l l o w i n g formula:

,.

p~

~

'

A

~j(~,y~ (,)~(y~d~dY -E~ [r~ ]

- ~

AA

'

-173-

It

is

easy for

~

and

- 2

~ : ~

t o d e r i T e from t h e aboTe f o r m u l e ~

the approximations for the eigenT~lues for

a membrane f i x e d a l o n g i t s

boundary. As a s e c o n d e x a m p l e , l e t u s c o n m i d e r t h e T i b r a t i o n s clamped along i t s

boundary, ioeo the two-dimennion~l eigenTalue problem

A s A s ~. -

In this

.),~

: o

in

A

c ~ s e , t h e l o w e r bounds

"~

u.:

o

on

ere e x p r e s s e d , by means o f t h e

I .,i

l•

12'

1

h

A

,,--4

~ G3~ t is an orthonormal symtela of harmonic polynolaials i n A

/'aA.

follows:

:

"iIl

h:4

,

K

Rayleigh-Ritz approximations, ~

4~ ~

of a plate

~ z )!/A,

is supposed simply connected. As a l ~ s t

e x a m p l e , l e t us c o n s i d e r t h e e i g e n v a l u e p r o b l e m c o n n e c t e d

w i t h t h e b u c k l i n g o f a c l a m pe d p l a t e :

~.A

u. 'r ~ / ~ u .

- 0 in

A

U,-

~

9~

:

0

on

9A,

A f t e r c o m p u t i n g t h e R ~ y l e i g h - R i t z a p p r o x i m & t i o n , we h a t e f o r t h e l o w e r approximation of

~ ~ :

-174-

~)

~.(t) A

J

J:4

A

4j~

+7. h

j

;.:4

AA

The

~

h

h a v e t h e same meeming a s i n t h e p r e v i o u s

example.

Ix-t l dr

dx §