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

(~)- - 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

/~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

~ ~

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

:*

~
_ 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, 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

~
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 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.,P0).

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~(tO

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 -

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

~,,(/,
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 §