Continual Means and Boundary Value Problems in Function Spaces [Reprint 2021 ed.] 9783112527801, 9783112527795

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Continual Means and Boundary Value Problems in Function Spaces E.M. Polishchuk

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E. M. Polishchuk

Continual Means and Boundary Value Problems in Function Spaces

Mathematical Research Wissenschaftliche Beiträge herausgegeben von der Akademie der Wissenschaften der DDR Karl-Weierstraß-Institut für Mathematik

Band 44 Continual M e a n s a n d Boundary V a l u e Problems in Function S p a c e s by E. M. Polishchuk

Mathematische Forschung

Continual Means and Boundary Value Problems in Function Spaces by Efim Mikhailovich Polishchuk

Akademie-Verlag Berlin 1988

Autors Prof. Dr. Efim Mihailovich Polishchuk, Leningrad Bearbeiter der englischen Fassung: Dr. Bernd Luderer Technische Universität Karl-Marx-Stadt Sektion Mathematik

Die Titel dieser Reihe werden vom Originalmanuskript der Autoren

reproduziert.

ISBN 3-05-500512-0 ISSN 0138-3019 Erschienen im Akademie-Verlag Berlin, Leipziger Straße 3-4, DDR 1086 Berlin (C) Akademie-Verlag Berlin 1988 Lizenznummer i 202

•

100/504/88

Printed in the German Democratic Republic Gesamtherstellungs VEB Kongreß- und Werbedruck, 9273 Oberlungwitz Lektor: Dr. Reinhard Höppner LSV 1035 Bestellnummer: 763 841 5 02200

(2182/44)

FOREWORD

The fates of important mathematical ideas are varied. Sometimes

they are

instantly appreciated by the specialists and constitute the

foundation

of the development of theories or methods. It also happens,

however,

that even ideas uttered by distinguished mathematicians are surrounded with respectful indifference

for a long time, and every effort of

inter-

preters and successors has to be made in order to gain for them the merit deserved. It is the second case that is encountered in the present book, the author of which, the Leningrad mathematician E.M. Polishchuk, reconstructs and develops one of the directions in functional analysis that originated from Hadamard and GSteaux and was newly thought over and taken as the basis of a prospective theory by Paul Levy. Paul Lfevy, Member of the French Academy of Sciences, whose centenary of his birthday was celebrated in 1986, was one of the most original

mathe-

maticians of the second half of the 20th century. He could not complain about a lack of attention to his ideas and results. Together with A . N . Kolmogorov, A . Y a . Khinchin and William Feller, he is indeed one of the acknowledged founders of the theory of random processes. In the probability theory and, to a lesser degree, in functional analysis his work is well-known for its conceptualization arjd scope of the problems His expressive style, rich in ideas rather than technically which sometimes led to a lack of clarity at first, will be

posed.

polished, remembered

by all who read his papers and books about probability theory and functional analysis. I would like to note that even the gap between these two disciplines, which began to be bridged systematically only in the fifties, had an influence on the general theory of random processes. Measure theory in functional spaces had to become the foundation of the theory of

random

processes as was intended by P. Levy, N. W i e n e r and A.N. Kolmogorov. Later on, such unification actually occurred; however, even today there exist theories in functional analysis connected with probability concepts that are remote from this general direction. One of them, presented in the book of E.M. Poli6hchuk, is the development of LSvy's ideas on the mean value of a functional over a domain in a function space and its applications to boundary value problems with an elliptic functional operator, "L&vy's Laplacian". P. L&vy explained his concepts in his books on functional analysis published in 1922 and 1951. It is common knowledge that Levy's Laplacian as well as the harmonic functionals and averaging processes associated with it are objects of a different nature in comparison with those studied by L&vy and his

5

colleagues in their papers on the theory of random processes. Maybe is the reason why they are not very popular even today.

this

E.M. Polishchuk, a mathematician and historian of mathematics, author of many papers about analysis as well as the scientific biographies of mathematicians "Vito Volterra", "Émile Borel", "Sophus Lie" and others published by the Academy of Sciences of the USSR, is the initiator of the renaissance and further development of the theory of continual

means

in the direction indicated by P. Levy. The peculiar feature of the averaging procedure of a functional in the sense of L&vy-Polishchuk consists in the fact that it, in principle, fails to fit in the scheme of usual integration with respect to measure, although it is related to it. In order to avoid associations

possibly

arising from terms and notations of the same name, the reader should have in mind this circumstance. By comparison, the theory developed by E.M. Polishchuk is closer to the theory of generalized functions of infinitely many variables and to ergodic concepts. This is not unusual we know that the Feynman integral, which became now one of the most important instruments of mathematical methods in theoretical physics, also fails to fit in the common scheme of integration even with to complex and unbounded measure. Summarizing, the difference

respect

between

these and those theories of averaging consists in the choice of normalization in passing to an infinite number of arguments. This difference happens to appear also in physics: one may normalize the total energy, but one can also normalize the energy for one particle. The second method occurred to me when I became acquainted with the approach of P. L&vy advanced in the book of E.M. Polishchuk.

The specifications and far-reaching generalizations of the theory of continual means as well as the connections and possible applications of the theory to diffusion processes, normed rings, control systems governed by differential equations involving functional

parameters,

statistical mechanics and other branches of pure and applied mathematics discovered by E.M. Polishchuk and explained in the present book will be of interest to mathematicians engaged in various fields and, it is to be hoped, will encourage young scientists to further develop this direction of functional analysis, the foundations of which were laid by Paul Lfevy.

Prof. A.M. Vershik Vice-presidsnt of the Leningrad Mathematical S o c i e t y

6

CONTENTS

INTRODUCTION

9

C H A P T E R 1. F U N C T I O N A L C L A S S E S A N D F U N C T I O N D O M A I N S . M E A N HARMONICITY AND THE LAPLACE OPERATOR

IN

VALUES.

FUNCTION

SPACES

13

1. F u n c t i o n a l c l a s s e s

13

2. Function domains

15

2.1. Uniform domains

15

2.2. Normal domains

15

3. Continual means

16

3.1. The m e a n o v e r a u n i f o r m d o m a i n 3 . 2 . The m e a n v a l u e

IfKL

R

F

over

16 the H i l b e r t

sphere

a n d its m a i n p r o p e r t i e s

18

3.3. T h e s p h e r i c a l m e a n of a G S t e a u x

functional

21

3.4. F u n c t i o n a l s as rar\dom v a r i a b l e s 3.5. The D i r a c m e a s u r e

4 . The

22

in a f u n c t i o n s p a c e . T h e c e n t r e

of

a function domain. Harmonicity

23

functional

33

Laplace operator

4.1. Definitions and properties 4.2. Spherical Hilbert

33

m e a n s a n d the L a p l a c e

co-ordinate space

operator

C H A P T E R 2. T H E L A P L A C E A N D P O I S S O N E Q U A T I O N S 5. B o u n d a r y v a l u e p r o b l e m s v a l u e s on the G S t e a u x 5.1. F u n c t i o n a l

Laplace

in

the

lg

39

FOR A N O R M A L D O M A I N

for a n o r m a l d o m a i n w i t h

boundary

ring

41

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

5.2. The fundamental,functional

of a s u r f a c e

41 S

5.3. E x a m p l e s

52 60

5.4. The m a x i m u m p r i n c i p l e -and u n i q u e n e s s 5.5. The e x t e r i o r D i r i c h l e t 5.6. T h e d e v i a t i o n

41

H - F

problem

of s o l u t i o n s

62 63 68

7

6. S e m i g r o u p s of c o n t i n u a l m e a n s . R e l a t i o n s s o l u t i o n s of c l a s s i c a l

to the

b o u n d a r y value p r o b l e m s .

probability Applications

of the i n t e g r a l o v e r a r e g u l a r m e a s u r e

70

6.1. S e m i g r o u p s of m e a n s o v e r H i l b e r t s p h e r e s 6.2. The o p e r a t o r s classical

'Jfll,9. lit

70

and the p r o b a b i l i t y s o l u t i o n s

b o u n d a r y value p r o b l e m s in the space

6.3. R e g u l a r m e a s u r e s a n d the e x t e n s i o n of

of

Em

the G S t e a u x

C H A P T E R 3. T H E F U N C T I O N A L L A P L A C E O P E R A T O R A N D C L A S S I C A L

73 ring

DIFFUSION

EQUATIONS. BOUNDARY VALUE PROBLEMS FOR UNIFORM

DOMAINS.

HARMONIC CONTROLLED SYSTEMS

85

7. B o u n d a r y v a l u e p r o b l e m s w i t h s t r o n g L a p l a c i a n and parallelism

77

their

to c l a s s i c a l p a r a b o l i c e q u a t i o n s

85

7 . 1 . The f u n c t i o n a l L a p l a c i a n a n d the c l a s s i c a l

parabolic

operator

86

7 . 2 . D u a l p r o b l e m s a n d an a n a l o g y 8. B o u n d a r y v a l u e p r o b l e m s

table

90

for u n i f o r m d o m a i n s

99

8.1. F u n c t i o n a l and c l a s s i c a l D i r i c h l e t p r o b l e m s 8 . 2 . The D i r i c h l e t p r o b l e m

99

for o p e r a t o r s

108

8 . 3 . The f u n c t i o n a l N e u m a n n p r o b l e m

112

8 . 4 . P r o p e r t i e s of the P o i s s o n e q u a t i o n

113

9. H a r m o n i c c o n t r o l

systems

114

9.1. N o r m a l c o n t r o l d o m a i n

115

9.2. U n i f o r m c o n t r o l d o m a i n

124

. C H A P T E R 4 . G E N E R A L E L L I P T I C F U N C T I O N A L O P E R A T O R S ON F U N C T I O N A L R I N G S 10. The D i r i c h l e t p r o b l e m

in the s p a c e of s u m m a b l e

functions

and r e l a t e d topics 10.1. F u n c t i o n a l e l l i p t i c o p e r a t o r s of g e n e r a l

131

131 type

131

10.2. C o m p a c t e x t e n s i o n s of f u n c t i o n d o m a i n s . C o m p a c t r e s t r i c t i o n s — 135 10.3. Averaging M^/J-F of a f u n c t i o n a l FtR w i t h respect to a f a m i l y of t r a n s i t i o n d e n s i t i e s of d i f f u s i o n p r o c e s s e s 11. The g e n e r a l i z e d f u n c t i o n a l P o i s s o n e q u a t i o n

137 149

COMMENTS

!51

REFERENCES

!56

S U B J E C T INDEX

1 6 0

8

INTRODUCTION

Ihr n a h t e u c h w i e d e r , Die

früh sich e i n s t

(Goethe. Faust.

The p r e s e n t

book

dimensional

spaces,

is c o n c e r n e d w i t h

tions considered classes x(t)

{ F^r

the t h e o r y of i n t e g r a t i o n

(t)).

FCx(t)3

It is s u p p o s e d

belong

whose

q = a < t •< b that

to s o m e s p a c e

arguments

x(t)

A(q)

and

(A

of

to the p - t h p o w e r ,

functions summable D

referred

tinual means over formulae

On the set pose

D , we define

the b o u n d a r y v a l u e a)

p?l,

L H = 0,

questhe

functions

xk(t),

functions,

etc.).

x(t)

=

k=l,...,s, measurable

the

space

In the s p a c e

to as the f u n c t i o n d o m a i n s a n d d e f i n e

these d o m a i n s . W e e s t a b l i s h for t h e i r

are

to

the

is the s p a c e of

q , the s p a c e of c o n t i n u o u s

fective

infinite-

or v e c t o r f u n c t i o n s

f u n c t i o n s b o u n d e d on we select sets

in

and their applications

b e l o w can be d e s c r i b e d as f o l l o w s . W e c o n s i d e r

of f u n c t i o n a l s

respectively,

gezeigt.

for f u n c t i o n d o m a i n s . To b e g i n w i t h ,

d e f i n e d on the i n t e r v a l

(Xj(t),...,x

Gestalten,

trüben Blick

Zueignung)

i.e. c o n t i n u a l m e a n s ,

boundary value problems

schwankende

dem

A(q) con-

their properties and

ef-

calculation. the e l l i p t i c

functional operator

L . Then we

problems

XCD;

Hi

= F,

F t { F}

,

Id-

b)

(where

LF = $

,

X € D;

$ € { f } ,

is the b o u n d a r y of

D"

F| = 0 ID'

D ) as w e l l as s o m e o t h e r

analogous

problems. By v i r t u e of

the s p e c i f i c c h a r a c t e r of

that a l l c o n s i d e r a t i o n s approach

treatment,

although

there are a l s o m a n y

t h e o r y of c o n t i n u a l m e a n s p r e s e n t e d

basic

research device

g o e s back was almost It w a s

to e a r l i e r w o r k forgotten

a n d the from

by R. G Â T E A U X

for a long

fact

spaces,

our

that

the

in

connections.

in the book w i l l

be u s e d as a

for the p r o b l e m s u n d e r c o n s i d e r a t i o n . [l],[2] and P. LÉVY

This

theory

[l],[2]

and

time.

reconsidered only recently

developed

L

to these p r o b l e m s d i f f e r s q u i t e s i g n i f i c a n t l y

classical The

the o p e r a t o r

are l o c a t e d in i n f i n i t e - d i m e n s i o n a l

from a n e w p o i n t of v i e w a n d

in d i f f e r e n t d i r e c t i o n s . In a d d i t i o n ,

interrelations

further between

9

this

t h e o r y a n d a n u m b e r of o t h e r b r a n c h e s of c l a s s i c a l a n a l y s i s

tions of m a t h e m a t i c a l p h y s i c s , lytical

f u n c t i o n s of s e v e r a l v a r i a b l e s ) ,

measures

in l i n e a r s p a c e s , s e m i g r o u p s )

mechanics,

functional analysis

and applied

theories

(statistical

that the u t i l i z a t i o n of c o n t i n u a l s e n t e d in an e x p l i c i t

equations)

r e l a t i o n s w i l l be d i s c u s s e d u n d e r

a s p e c t s w h e n we s t u d y the p r o b l e m s

various

f o r m u l a t e d a b o v e . It s h o u l d be m e a n s a l l o w s all s o l u t i o n s

form a d m i t t i n g , w h e n

ana-

(B-algebras,

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

w e r e d i s c o v e r e d . S o m e of these

(equa-

trigonometric series and integrals,

noted

to be

required, algorithmic

pre-

cal-

culations. The s t r u c t u r e of marks

the book

to a n y c h a p t e r a n d

is as f o l l o w s

(see a l s o

the i n t r o d u c t o r y

In C h a p t e r 1, a f t e r a p r e l i m i n a r y d i s c u s s i o n of the n o n l i n e a r classes considered introduce cipal

below

(Volterra, Picard, Gateaux

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

function domains

types - u n i f o r m a n d n o r m a l d o m a i n s .

v a l u e of a f u n c t i o n a l o v e r

the d o m a i n

is s h o w n

to r e s u l t

function.

t i o n s of this fact y i e l d the n o t i o n of and the c l a s s of w e a k h a r m o n i c to a D i r i c h l e t

S e v e r a l d e f i n i t i o n s of

the f u n c t i o n a l

studied

since

the L a p l a c i a n

the w e a k D i r i c h l e t

boundary values

the G § t e a u x

Dirichlet problem

in a f u n c t i o n

u t i l i z a t i o n of the i n t e g r a l o v e r a r e g u l a r

these

are

with

assumptions,

the s o l u t i o n of an

this i n t e g r a l

of

these

exterior

r e s u l t s b a s e d on

(orthogonal) is e x p l a i n e d

measure in the

means and

for b o u n d a r y v a l u e p r o b l e m s w i t h

the

equations.

In C h a p t e r 3 a c o m p l e t e l y d i f f e r e n t

to f u n c t i o n a l

10

defini-

for a n o r m a l d o m a i n

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

value problems

are

number

methods.

B e s i d e s , we s h a l l s t u d y the s e m i g r o u p of c o n t i n u a l the f o r m u l a e o b t a i n e d

strong,

space.

Furthermore, we indicate a generalization l i n e a r s p a c e . The notion' of

(weak,

of

class. Under analogous

the P o i s s o n e q u a t i o n a n d

domain

in a na-

boundary value problems

problem

cer-

investiga-

Their properties

the i n t e r r e l a t i o n s

the c o r r e s p o n d i n g

Chapter 2 deals with we a l s o c o n s i d e r

functional

in a s p a c e of a finite

in the book s e p a r a t e l y a n d by d i f f e r e n t

from

formulae

space.

Laplace operator

different

of d i m e n s i o n s . W e do not c o n s i d e r

mean

the c e n t r e of a f u n c t i o n

essentially

shall

prin-

c o n c e n t r a t e d at a

f u n c t i o n a l s . The l a t t e r l e a d s ,

are g i v e n at the e n d of the c h a p t e r .

tions in d e t a i l ,

the

The s u b s e q u e n t

and others)

we

two

the p r o c e d u r e of

p r o b l e m in a f u n c t i o n

from

and consider

is g i v e n a n d e x p l i c i t

in a D i r a c m e a s u r e

tain p o i n t , w h i c h is a g e n e r a l i z e d

tural w a y ,

D

functional

functionals)

The d e f i n i t i o n of

D

for its c a l c u l a t i o n are d e d u c e d . F u r t h e r m o r e , averaging

re-

section):

approach

the

in a

text. compare

probability

boundary

is p r o p o s e d . W e s h a l l c o n s i d e r s t r o n g p r o b l e m s

with

boundary values

from

the G S t e a u x

class and obtain solutions

the a n a l o g y w i t h p a r a b o l i c e q u a t i o n s - i n The e s t a b l i s h e d a n a l o g y is s h o w n tain s e n s e ,

to be of a d u a l

to e x t e n d

In the s a m e c h a p t e r we e x a m i n e

however, not

f a i r l y w i d e l y and,

not very long ago. They

position between problems

for n o r m a l

to e l l i p t i c

to p a r a b o l i c but

the c u r v e s u n d e r c o n s i d e r a t i o n for u n i f o r m d o m a i n s

The relations

equations

from

transformation new

important

results

into p o i n t s ,

(related,

) and

the

the when

Dirichlet

r e d u c e s to the c l a s s i c a l D i r i c h l e t

problem.

problem with

in bound-

the V o l t e r r a a n d P i c a r d c l a s s e s c a n be i n t e r p r e t e d as a of a g i v e n c o n t r o l s y s t e m

into a "harmonic system"

having

properties. that w e s u c c e e d

in e x t e n d i n g

s o m e of

the

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

tional operator.

For

this p u r p o s e we use

The e x p l a n a t i o n

of a g i v e n

is s e l f - c o n t a i n e d . T h e p r e s e n t book

the t h e o r y of c o n t i n u a l

means and

prefunc-

the t h e o r y of d i f f u s i o n

c e s s e s a n d the n o t i o n of a c o m p a c t e x t e n s i o n

with

Em

In the l i m i t c a s e ,

that the f u n c t i o n a l D i r i c h l e t

In C h a p t e r 4 it is s h o w n vious

degenerate

domains in

do-

occupy

to the t h e o r y of c o n t r o l s y s t e m s are a l s o d i s c u s s e d

this c h a p t e r . W e s h o w ary values

for u n i f o r m

boundary value problems

c l a s s i c a l e q u a t i o n s of m a t h e m a t i c a l p h y s i c s . problem

space. in a c e r -

character.

m a i n s , w h i c h a p p e a r e d in the l i t e r a t u r e an i n t e r m e d i a t e

following

the f i n i t e - d i m e n s i o n a l

pro-

function

domain. to

deal

their applications. E a r l i e r ,

this

m a t e r i a l w a s to be found o n l y in v a r i o u s p a p e r s of

is the first the

author.

Notations A l l s p a c e s in q u e s t i o n are s u p p o s e d Euclidean

s p a c e of p o i n t s

n o t e d by

Em

B

y

i=l,...,m

6"

we d e s i g n a t e

E

, while

the s p h e r e

x

s

(t))

»1

Qm

the set

Em

m-dimensional

( ^

will

by

be

de-

:

. . .

= { t= ( ^ , . . . , t m ) | a < E

of a t o p o l o g i c a l

are u s e d to d e n o t e

with co-ordinates

(a,b)

"f =

is the b o u n d a r y of

. Furthermore,

function spaces:

s p a c e of v e c t o r s tions on

in

real. The

is d e n o t e d

the cube

E'

, respectively,

following

Em

J . The c l o s u r e of

d e n o t e d by and

(vectors)

. The norm in

Qm(a,b)

to be

E

space will

. The s y m b o l s

the u n i t ball As(a,b)

[a.b] ,

1^1
5 J JZ a -i

(

II x || . The s p h e r e

will

be d e s i g n a t e d

Functionals pital

on

letters

Ag

C o n t i n u i t y7 o f we have

is

If

s=l

if

Ag

at

unless

x

will

the p o i n t

specifically

in square

to e m p h a s i z e (a,b)

the s e t when

= Cs

a vector

with centre

desirable

over

. be

a

denoted

and

radius R

.

be d e n o t e d ,

l i m F [ x n 3 = f[x] n o

(uniformly

R

on the i n t e r v a l

F[x]

Ag

g

of

the arguments e n c l o s e d

it

depends v a r i e s

£2,

will

p^l

x^2 (t) dt)1/'2

I x - a || = R

by

with

F [x ( t ) ] , . . . . I f x

to the p - t h power,

n

and s o

— * x o

the

caF =

which

b f[x] . a

means t h a t in

by

i.e.

the a r g u m e n t on

, we w r i t e

CL ^ A_ s x

that

stated,

brackets,

if

x, x € Ct , n ' o

t o p o l o g y of

the

space

on). 2

, the f u n c t i o n s p a c e

Ag

will

be d e n o t e d by

A , thus

L

2 means

L^

meter

V

If

Bi

and i s

etc.

If

the

, we w r i t e and

Bg

a strongly

Some o t h e r abbreviated

are

functional

F = f[x|t]

notations will

and

Y

we w r i t e

be i n d i c a t e d

rems a r e numbered s e p a r a t e l y

12

operator,

parallely

also

upon a s c a l a r

para-

.

two B a n a c h s p a c e s

continuous

notations

depends

F

in

acts ^

from

6 { B J —

t h e t e x t . We s h a l l

Bj B

into

2 ^ often

to the c o m p l e t e o n e s . F o r m u l a e a n d

i n each

section.

Bg

" use theo-

CHAPTER

1. F U N C T I O N A L C L A S S E S A N D F U N C T I O N D O M A I N S . M E A N HARMONICITY AND THE LAPLACE OPERATOR

1. F u n c t i o n a l

(for

the time b e i n g

c l a s s e s w h i c h w i l l be c o n s i d e r e d there be g i v e n

respect

(Xj,,..,*)

to the

for e a c h

function

x

on

tial c o n d i t i o n

y(t0)=y0

y

• Varying

€ (X .

functional

F

tra);

m[x:i

=

We s u p p o s e and

f

e E

i ¿ " i

We s h a l l c a l l

m

:

t

€

g(J;t)

be i n d i c a t e d

9m=9m(?;t)

V

functional

to

(t mv 1

and

suppose

satisfying is a

(or

al-

the

ini-

functional

can be d e s c r i b e d as

t

and

the

integral ^A(a,b).

its g e n e r a t i n g

respect

and,

t . Other conditions

C?!'1!^

te Qm

for all

, the

^

i m p o s e d on

(2)

function.

to the p a i r s

for a l m o s t a l l

to

fol-

=( ^ ^ , . . . , ^ m ) ,

m)dtl-"dtm'

gm

hold:

respect

^

, it g^

will

later. d e f i n e d by the u n i f o r m l y

convergent

series

£

w i l l a l s o be c a l l e d a G a t e a u x k

y(t)

x =

("control

(act \ m l

1

—^0).

^ s

t

i

>dvtv m i .

d v

t l

we obtain

. ) ^

...dv t l n

2

n — • oo

Qm

)dv m

Sin

\ ^ ,

nij.,.1. ^

Passing to the limit

*

[•••( J

v

^

™

.

m

which was to be proved. Note that the mean value the interval the points

(a,b) T.

MF does not depend on ttie way of subdividing G wa into intervals ^i-i^j.) ' o n V choosing

and the coefficients

formula, approximating by step or other In particular,

x

not by

8

%

. We would come to the same

(i.e. by piecewise linear)

the formula obtained allows us to calculate the mean s

value of sphere

F

but

functions.

over the ball

T— 1

Z_

(*k(t)-ak(t))

2

2

< r (t)

or over the

I x-aI=r .

In the simplest case, if

s=l

and

G

is the function beam

u(t)(U), U -

>

m llyH-1

is

we s u p p o s e 2 (a,b) .

A (a,b)»L ^

define

2

k(t)x£(t)dt,

m

_f

via

t h e mean o f that

in

the

k£M(a,b),

the $

equaover

present

and

let

Vj>(?)

be continuous for all

. Then

'JHF • ij) ( $ a

b I f

=

( J

3.2.2. The following property of the operation

m

_f

2) F «

»{>(

^ S E |grad H|

and

E | grad H | d 2UE

' is an element of the surface

19

e •

is

known in statistical mechanics as the microcanonical A(p,q)

of the function

. Let

(S)

mean

be the measure of the set of

points of the surface (1) for which I A(p,q) - A I >

nS

.

(2)

It can be shown (see KHINCHIN [l]) that, under natural assumptions cerning the surface

ZZ ^

and the function

A(p,q)

con-

, we have the esti-

mate 0 (

CO (E)

) .

n £

(3)

In statistical mechanics the function quantities of order

n

to be very large if

S

A(p,q)

by

A

A(p,q)

and its mean

and, moreover, the number

nS^

A

are

is reckoned

is small. In view of (2), the substitution of

yields a relative error of order

(3) shows that the equation

A=A

1/n , and the estimate

is "reasonable for practical

purpo-

ses" . This is known as the "representation principle of microcanonical

means

of statistical mechanics". Some of the lemmas used below are based on the functional analogue of this "representation

principle".

3.2.4. Let us return to the consideration of the mean we also represent in an abbreviated form as

a, _F K , which F . First of all, we state

the following obvious facts; (i) If two bounded functionals FjaFg

almost everywhere,

Fj

and

then clearly

exists for one of the functionals,

F2 F

satisfy the equation

i=F2

'

Moreover

'

if

t 1e

'

m

ean

then the mean for the other exists,

too. (ii) If the equation sequence

f

is satisfied almost everywhere,

defined in 3.2.2 converges to

F

then the

in probability. From

this we get the following proposition analogous to Slutskii's on stochastic limits (see CRAMER [ll, p. 255). If and

F^

a.e., then for any rational function

that

r

(Fi',--'Fn) r

r

J1,1

),w( * ?i2

J il

$i2

))dtdv

$il

g(w( 2 f r ( t ) . w ( 2p r («r ) ) ) d t d T

If we a d d s o m e n e w p o i n t s

to the p a r t i t i o n

, constructed according

be a p o i n t of m e a s u r e c o n c e n t r a t i o n .

26

(11)).

write r

w(jn)

in

= FCw(j

(11),

r

)]

then the

. function

to this new s u b d i v i s i o n w i l l This proves

our

assertion.

also

3.5.3.

It is c o n v e n i e n t

of e q u i v a l e n t finitions

3.5.1.

on

is s a i d

[A,B]

a n d sn i n t e g e r have

("almost

hn

3.5.2.

called eguivalent as w e l l

everywhere

on

Equivalence Definition

and

UPCO

means

(A,B)

if

C A , B ] for

of s e q u e n c e s

Lemma

diverges

3.5.1.

Let 0

the i n t e r v a l

lin>

'function (A,B)

w

5

Aiu

o

that

U r 4 = ^J { J

H

n

-$

(t) nv '

—

).

a r e

and

almos t dt

u n i f o r m l y on

j

by

be a f u n c t i o n

and such

w

A

c

^n(t)

^ » h

n

to be c o n s i d e r e d everywhere

w(>oo, where

by

following

11-16).

functions

functions

k?0

will

pp.

the

there exists a sequence

on any

of

functions

recall

t h© p o i n t s of

uniformly

Two sequences on

of if

converges

as an i n t e g e r

The g e n e r a l i z e d

Let

^n(t)

t h s t , st

we

Ell,

fundamental,

such

called a generalized

d e p e n d on

and SICORSKI

to be

uniformly"

Definition

this p u r p o s e ,

The sequence

k^C

dkHn — = dt

( ^ , a^ (t),..., a^ (t) ) D

containing

the parame-

a=(a^,..., a ^ )

be a manifold

corre-

in a function

which can be considered

as the union of D , a € Oi , i.e. V = a 2 (for example, an open set in L considered as the union of all

space

a

D

a

balls

included). We shall say that a harmonic

F

is a functional of class

functional, if, for every

on the set

D g € V, the

V

or

relation

MF = F[0 ] D

a

is valid and point of

Dg

0

g

as an ordinary

function,

i.e. a

.

If, for instance, the sphere,

can be considered

Dg

is the ellipsoid

then (19) means

|| ^ ^

|| = 1 , in particular.

MF = F [ a ] . °a

If

32

D

is the beam

a

1

then L e m m a 4 . 1 . 3 a l s o

S2F £ { ^ ( a ^ b , )

be a c o n t i n u o u s

(7') e x i s t

with

s

$ (i.e.,

1^1 < B .

the f o l l o w i n g

§F.

(ii) on the s e t

where

and

is b o u n d e d

i

f(t, ^ , ) and

is s o m e c o n s t a n t

F^

if

way.

Lemma 4.1.3'. Assume

with

this d o m a i n

this c l a i m , we h a v e m e r e l y

in the f o l l o w i n g

a

B

but

)

to be twice c o n t i n u o u s l y d i f f e r e n t i a b l e

the ball

(tl)

not for all

if we c o n s i d e r

in a d d i t i o n ,

, where

for

to

the v a l u e s of

L e m m a 3 is also c o r r e c t

on a u n i f o r m d o m a i n x£D

respect

assuming

i J q

'

c

iji

a r e

constants)

,

1

I jiikl ' I T k l ^ * Then If

(analogously)

the f u n c t i o n a l As=Ms

, the

sufficient

hold.

exists and

relation

(9) is

f o r m e r e s t i m a t e s can be w e a k e n e d .

that,

for e a c h of

the q u a n t i t i e s

valid.

In this case

it

is

S^,..., ^Tkl ' we

dif

have K U u

t )

i (

... . where

• ç

j

(

b^,

• ç

V

cik

are p o s i t i v e

As a c o n s e q u e n c e Lemma 4.1.4.

c

i k

y

k

(f

.

k )

are s o m e p o s i t i v e continuous

of L e m m a s 4 . 1 . 1 If

F^

and

and 4 . 1 . 3 w e

$

satisfy

constants and

vp

functions. formulate

the c o n d i t i o n s

of L e m m a s

4.1.1

then

and 4.1.3,

J oo

f

I T | { 1 | 4 ... .

ui€L(aQ,bo),

as w e l l as

a

bij

=

4.2. Spherical

fc J a.

( E i

a

o

0°

f1± • z : k

S

X

2

> «

.

K

m e a n s and the L a p l a c e o p e r a t o r

in the H i l b e r t

co-ordinate

space 4 . 2 . 1 . Let

H

be an a b s t r a c t

some complete orthonormal be the e x p a n s i o n F[xl point

We

= F[

Pn

i.e. ud

Züxkek]

|| x - a || = R

P

"

form

the m e a n

over

respect

Hilbert

space and . e^.e^,...

. Furthermore, to thir

= f(x1(x2,...)

be the p r o j e c t o r

A .

limit

with

H

can be r e p r e s e n t e d

from

p-

basis.

let The

function

x = 51] x k e k k functional

f(x)

of

the

n^a,R

lim the H i l b e r t

:

is g i v e n

of

provided

the

(1)

. The p r o j e c t i o n

by the

£ t)

the estimate

a(t) + c t * . . . ? *

is a s s u m e d to hold, where

a€L(Qm),

(2) a>0,

c » c o n s t > 0 , which

the c o n t i n u i t y and boundedness of (1) on e v e r y ball 2 the space L (q) : sup | F[X] | $ V R

42

J

\

Qm

a(t)dtm + c R 2 m

.

implies

V_: X x U < R

of

(3)

W e c h o o s e an

R

such

that

5.1.2. Theorem 5.1.1. given

=

Is

S

is a s u r f a c e

(1),

t h e n the

of

type

S

arid

F

is

functional

ÎÏ1F[X4 ? x L y l y ]

is h a r m o n i c H

If

by e q u a t i o n

H[X] =

V C V,

on

V

and

(4)

the b o u n d a r y

condition

F[x3

holds . Proof.

F i r s t of a l l , we n o t e

that f u n c t i o n a l

(4) c a n

be w r i t t e n

in

the

form HM or,

= lF[xts[x]y]

(5)

equivalently, HDO

In fact,

= m

since

x > s [ x l

F

S £{s^

TÎ31 ^ x [ y ] = s [ x ]

.

(5-)

, it f o l l o w s

that,

ÇxCy3=sCx]

exists and

for a n y

x€V

, the

almost everywhere

mean

on

the

sphere

llyll =1 . From

(2) w e see

that,

for e a c h

v i e w of L e m m a 3 . 2 . 1

(where

e x i s t s a n d is e q u a l

to

Applying we

the G â t e a u x

x

fixed

x

, the m e a n

is r e p l a c e d

by

y

(1)

(with

'SIL „ f

exists.

), the m e a n

(4)

In

also

(5).

formula

to f u n c t i o n a l

a=x

and

R=s

),

obtain H[x] =

W

X ( S

F

=

by a p p l y i n g

.2

%

f

the w e l l - k n o w n

relation

_ .2

~\Y

•*>l

in the preceding

F',F"£ R(V) , then besides

, too. theorem generate a

the additivity, we tiave

In the following we denote w 1

^This

46

X ( S

F

=

m

follows from

s

F .

the continuity of the superposition

operator.

Let

H= iilSF . From

lows

that

the multiplicativity of

?ES(F,F") =

HSiSF"

2BILSF

the class

Therefore, by

10lS F'

Let

R(V)

3P0(V)

with regard

R

into

R(V)

and

S E (V)

On the boundary

norms

II HII = sup H . V

to the uniform norm from

H£R, S

of

Thus we are able to

R

into

F£ W V

the operator

R(V) f

jSlH,S

and

is con-

(uniform convergence on can be extended

the

to a mapping

functional

H

.

Q

both rings

coincide.

formulate

Theorem 5.1.1' . Let

F£R(V)

7ffllS

. Then the operator

harmonic on the domain

V

defines a

with boundary values

5.1.4. Let us consider the functional Poisson AU

the uniform

:

MSF,

H =

(12)

to these norms. Since

V ), the mapping

from

.

c ? 0 ( v ) , respectively, be the closure of

and

tinuous with respect set

,

it fol-

also qenerates a ring, which is denoted

(V) . We introduce on II Fll = sup I Fl V

the operator

F .

equation

• F[x] = 0 ,

where the Laplace operator = 2

A U

A

is defined

muCx+XyJ-uDO

lim

%

V O We introduce

(13) via

#

the notation sDO

=

I 0

r $ l F [ x + r y ] dr .

Theorem 5.1.2. Let where

S

F[x]

(15)

be a functional defined on the set

is a surface of type

"£ S i

, and let

conditions of Theorem 5.1.1. Then, for any (15) exists and satisfies

F

V=VuS,

satisfy all

x€V

, the

and

g

the

functional

the equation

A"3lF + F = 0 . Proof. We start with the variable

t

the simple case, if

m=l

does not

contain

explicitly. Thus, let

A F =

$

g(x(t))dt .

47

Forming

21F

, we

obtain

S W UiF =

-I

^

r$fljg(x(t) + ry(t))dt.

0 In v i e w of

0

the G a t e a u x

f o r m u l a , we

1

Sl>] ftp

= -^==5

j

This functional will u

[*

+

^y]

J dt J g ( x ( t ) + r | ) e x p ( - | 2 / 2 ) d £ d r

r

be d e n o t e d by

sUAy]

=

r

Jd 0

W

t

s

the proof Applying

2

• X

'Xy(t))exp(-|2/2)d|dr

. (17)

+

Xy(t))exp(-|2/2)d|dr

,

as it is e s s e n t i a l l y

the s a m e

5.1.2.

the G a t e a u x

formula

to

(18), we

obtain

® u [ x + "Xy] =

Since

0 ^

J exp(-

V we

1 W

'

J

( V ^ - « )

exp( P(

2

exP(-«2/2*2)do
0 , the following conditions are satisfied for some value of n onwards : are continuous on the sets

(i) x«.V n , (ii) I U W - U n [ x ] | < £

V = VUS

x€vnv

for

U

and

and

n

(the bar is used to denote the closure in the metric of

L 2 }.

anc

?n"

In order to find the "polar" functionals of the surfaces

S

and

Sn

§ =

from the equations

*

ll£x+ J y ] » 0

U n [x+J„y]=»0 , it is necessary that the derivatives exist and be nonzero for

xt J y i V

and

x +

S

n

€ V

^

n

and n

and *

From the theorem on the existence and continuity of an implicit

function

we conclude that the sequence

^xCy3

with respect to

lim

x

^nx[y]

converges uniformly to

and with respect to

TBlSnxty3 -

y . In this case

7fil$xty]

also converges uniformly, or, what is the same thing, snCx] rr^s[x] , We now suppose that

x£V

Sn

r - II xll ', where the

r* n [x]

(21) it follows that on the set r Let

(21)

are surfaces of type

2

V

. {s}

and, hence,

are harmonic functionals in V ,

^ [ x j Z j P W

sn » Vn

. From

• Consequently,

Is also a harmonic functional. z€S, z n £ S n

and

and the continuity of

z s

p

— z

. Since

on the set

S

s n £ z n 3 » 0 , from equation

(21)

we get

s[z] - lim s n [ z ] - lim s n [ z n ] » 0 . n n In fact, we are able to formulate the following Let

S

and let

result:

be a closed convex surface given by the equation Sn

u[x] • 0 ,

be a sequence of convex surfaces defined via the equations 59

• 0 , which uniformly approximate the surface tional^

U

the

are surfaces of type

Sn

and

U_

are continuous on the sets

V

r

is^

, then

S

S . If the funcand

V_

and if

is also a surface of

type {s}. 5.3. Examples We intend to consider two interesting examples. The first is peculiar in that the solution of Dirichlet's problem is obtained using only one algebraic operation and the solution of Poisson's equation can be found via algebraic operations and quadrature. The second is related to the eigenvalues and eigenfunctionals of the operators

HSL9

and

'8T,

studied above. 5.3.1. Let the Dirichlet problem be posed in the form n

FLx] - TZ

t»0

where d

G j x ] If i ( X ) .

p X*lxR /2 , the

(1) p

are continuous functions for

is the diameter of the domain

tionals continuous on the set {si

V , and the

V • VuS

Gj^

0{

n

tB)(x(tj)+8[x]y(tj)^dt1...dtm

exp^Xjit,

V W t ^ s H l j ) )

•exp(- l j 2 / 2 ) d | 1 . . . d ] g m ] . d t 1 . . . d t m Using the w e l l - k n o w n

.

relation

^ e x p ( i v | )exp(-^2/2)d-f

= -/~2* e x p ( - v 2 / 2 )

.

we may write H[x] =

J...Jk(tl

tm)

PI

exp(iXj(t1

t m )*tm)

Let the f u n c t i o n s S

R

:

2

^ i ^

+

• • • + •

F

=

F

Then

exp(-

5 . 3 . 4 . Let us s o l v e functional defined

ma

t h e

P

cube

Qm

the p r e c e d i n g e q u a t i o n

R2S2/2)

onto

the

takes the

sphere form

.

(5)

the P o i s s o n e q u a t i o n by e q u a t i o n

Al)=F[x]

, where

(3). F r o m T h e o r e m 5 . 1 . 2 we

F[x]

is

get

so: U = - ?fZ.F =

J

r "WlFfx+ryldr -

j

0 2 e x p ( - g - A-iAj V( t1 l •j J % (¿ /tt t VJ { l1 m -y m'

t ))"l m m "

?

If

A

is a p o i n t on "the s p h e r e R2b2 exp(^_) U - - ffiF = ¿-2 R

Equations

(5) a n d

(6) s h o w

is an e i g e n f u n c t i o n a l to the

k(tj Q

- 1

S„

tj

x

m

(—I * I I exp(iA. (t , ..., t )x(t )dt .. .dt m j l m J i 1 J , then

F .

(6)

that the f u n c t i o n a l /

of the o p e r a t o r s

J2dS

F

g i v e n by f o r m u l a

and

1ft

(3)

corresponding

eigenvalues R2S2/2)

exp(-

-R_2(exp(R2s2/2)-l)

and

,

respectively. Equations

(5) a n d

concerning from

(6) a l s o e n a b l e us to d r a w

the f u n c t i o n a l

(3) a n d

(5) or

the f o l l o w i n g

(3). It is p o s s i b l e

(3) a n d

(6). If

S

to f i n d

the " i n v e r s e p r o b l e m of p o t e n t i a l

the d e n s i t y

F

can be

the " p o t e n t i a l "

s w

20l S F

for the m e n t i o n e d e l l i p s o i d , w e - iikir 1 V i - i k x B 2 '

a n d if the f u n c t i o n the s u r f a c e

k S

theory"

(or

"filF ), the s u r f a c e

S

Since,

for a n y

H[x] •

+

problem

xCV

have

,

is n o r m e d ,

i.e.

, the g i v e n

s[x3

deter-

uniquely.

5 . 4 . The m a x i m u m p r i n c i p l e

62

directly

I k x l = l , then 2 in L : knowing

reconstructed.

Actually,

mines

conclusion

is the e l l i p s o i d

we c a n s o l v e

and

s[x]

and uniqueness

, the r e l a t i o n

"3Jff.FD< J x C y ] v l (Theorem 5.1.1")

shows

of

solutions

x+ 3 x t y ] y " z s ^

holds,

the

that the o b t a i n e d s o l u t i o n of the

is the s p h e r i c a l m e a n of

equation Dirichlet

the b o u n d a r y v a l u e s of

F.

on

S

Moreover, according

to

HI = F , we also have 's

sup |h I i. sup I Fl - sup I H| . V S S Therefore, similarly to the classical theory of harmonic functions of n variables, H cannot attain its extremal values inside of the set V . This yields directly the uniqueness of solutions of the boundary lem AU because if

- Q[x] , Uj

U|g = 0 ;

and

Ug

Q€R(V)

,

S 6 {s}

are two solutions, then

prob-

, Uj-Ug

is a solution

of the Dirichlet problem with zero boundary values.

5.5. The exterior Dirichlet

problem 2

We study the exterior Dirichlet problem in the space face

S

it is required to find a functional

H

L

. Given a sur-

w h i c h is harmonic out-

side this surface and takes prescribed values on it. Note that the classical method of inversion for solving the exterior problem in a finite-dimensional space is not applicable

here.

Instead, we use other concepts and show that the solution of this problem also may be obtained with the help of the operator where, however, in the present case long as

H» 331

.F

r

s

[x] '

is an imaginary variable. As

is an even function of

X , S [X]

s

s , the functional

H

remains real. In accordance with this, all averaging will be carried out over a sphere of imaginary radius when we consider exterior problems. This means that the equation W-x

i X

F

- MF[x+iXy]

(1)

holds true and harmonic functionals outside the

S

will-be subjected to

requirement lH[x+iXy]

- H[x]

.

(2)

More precisely, it will be shown that equation

(2) is satisfied for the

corresponding integral representation outside

S

the given values

F

on

is said to be a surface of type face given by the equation

S

U[x] 0

for exterior points

x .

Besides this, with a surface of type s [x]

{si

uniformly continuous on any sphere

we associate a functional ||x||l

), all the

reasoning is analogous. The set of points for which the results derived above hold are characterized by inequality (10), which may be rewritten in the form r e x : < HX«2 *

r M

+

2

.

We denote this "layer" by E q . Comparing (9) and (10), we see that 0 < % < < s'tx] , that is, sup \ Q = - f ? . It should be noted that E

o

this boundary depends neither on the functional

F

nor on the surface

S . This concludes the proof of Theorem 5.5.1. Now let where

S

be a surface of type

Pj^ [x] » T [ x ] + 2

functional

-Cs}

s - s ^ x ] =-/|xU2-l~i [x]

for which

15Six

. Applying the operation

[x]

s

to

the

F , we obtain a functional that is harmonic in the layer

TCx]+2

< Kx( 2
0)

X = H x»2/2

l m ( x ) , X ) , where g ( ^ ^ , . .., "j| m , ^ )

, lj

lm

are

is bounded on the set

. The set of all these functionals constitutes a ring with

respect to the ordinary multiplication law. We denote this ring by — 2 R 0 0 ( L ) . Let R q o be the closure of this ring with respect to the norm

I FII » sup |F| . From the L2 T^djCx)

l m ( x ) . X) = g j l ^ x )

we conclude that, for any the half-line

relation lB(x).X+t)

F £ Rq() ,

is a translation operator on

t >0 .

In this way, we have a simple example of a semigroup that is strongly but not uniformly continuous. The latter results immediately if we consider the restriction of this semigroup the form

F = g(X)

6.2. The operators

to the set of functionals of

(see HILLE [l], Chap.

TSU6.

31

IX).

and the probability solutions of

sical boundary value problems in the space Let a linear elliptic equation in the space

Em

clas-

Em be given. It is common

knowledge that after the construction of the diffusion process by means of the operator occurring in this equation, one can describe

the solu-

tion of the boundary value problem for this equation via integrals the measure of the considered process and via quadrature, tion is not

over

if the equa-

homogeneous.

Let us briefly explain the nature of the corresponding

formulae

and

show that the solutions of the functional Laplace.and Poisson

equations

obtained in Sections 5.1-5.4 can be described in a completely

analogous

form. 73

Let

Xt

be a random process with values in

jectories. As usual, let

CO

En

and

x=xt(u)

be points of the space

events on which a set of probability measures is given. let

A C

associated with CO

f

P

X

(A)

Furthermore,

be the probability measure of this process

the point

is a measurable

x 6 En .

function of

, i.e. a random value;

If the time

t

random value

x , then

Mx(f(*t))

f(x^)

is a function of

denotes its mathematical

as an integral over the measure

tation understood

same

elementary

SI and

If

its tra-

of

is also a random value,

i.e.

f ( x ^ ^ ) (CO ) ) = f (x

and uj

» 0

in S e c t i o n 5 we h a v e o b t a i n e d C' ) D')

S

F

the

=

solutions

H =

m

U =

SkC>3 2([F = HX. ^ r F [ x + r y ] d r

$x[y]y] ; S W j r ?Q(If[x+ry]dr .

=

0 Let

x € V , and assume

that

of this d o m a i n a c c o r d i n g

2=

V T ?

The v e c t o r y

Then

moves towards

the

boundary

law (i)

y , iyH=l

. We p r o c e e d

to the

x

.

the r a n d o m d i s t a n c e w i th the law

0

the p o i n t

, will

from

from

be c a l l e d a r a n d o m d i r e c t i o n

x € V

¡Jx[yJ

to the s u r f a c e to the r a n d o m

S

time

in the *Cx[y]

and

^xCy]

direction in

accordance

(1).

i n s t e a d of C') and D " ) w e can w r i t e C")

H = fflJlF[x+y 2 < t x [ y ] ' y ] ,

D")

U = B.

$

F[x+-V2?y]dt

.

0 We define

the

translation

operator

TtF[x] = FCx+V^t1 yD

.

w h i c h can be c o n s i d e r e d as

the s e m i g r o u p

tional Laplacian

the g e n e r a t i n g

Now

A/2

as

f o r m u l a e C") a n d D " ) m a y be w r i t t e n H =

[

y 3

F

D0

operator having operator

in the

(see

the w e a k

func-

5.1).

form

.

txW U = a n d w e see

M

J

TtF[x]dt ,

0 that a f t e r

y ~C0 ,

x e L

2

the

comparisons

~ x 6 E

n

,

F[x]~f(x),

'Ml ~

Mx

75

they are a c t u a l l y

identical with

the e q u a t i o n s A") and B")

presented

above. In the t h e o r y of M a r k o v p r o c e s s e s boundary

G'

of the d o m a i n

important

role:

m(x) = M y •

, then

n

a n d

"T, n

1

of s e q u e n c e s If

=

(?!»•••• S n )

the p r o d u c t

. If the set

E x Rn

ÎJ

is a c y l i n d e r h a v i n g

is d e f i n e d

by the

the

base

condition

rn, - (AkX

\ then

A

the c y l i n d e r

Z =

R Co • Let

=

J

S

k

^k)'

= ^k

K

tT x R n

B

is s a i d a

P (^ n ^

n o n n e

Tn>dTi.-.d?„

P(Ti

1

The

for w h i c h

P(in)dTn

=

1

•

expression =

J

P ^ n ^ i n

is c a l l e d

the m e a s u r e

function

p .

Thus,

for e a c h

of

the q u a s i i n t e r v a l

E

of

constructed. Simultaneously, the s u b s e t s

can be

E C Rn

f

I

t!J

n , the c o u n t a b l y a d d i t i v e

ring of all B o r e l s u b s e t s

respect

measure

(E)

(the p r e m e a s u r e

the m e a s u r a b l e

as w e l l as the

d

(Un)

Rn

with

in

functions

to

on R

the ) may

u

f(E)

the

given

be on

integrals

fn

defined.

The m e a s u r e

is s u p p o s e d

well-known Kolmogorov this m e a s u r e taining

to be c o n t i n u o u s . C o n s e q u e n t l y , due to the 11 theorem p r e s e r v i n g the c o u n t a b l e a d d i t i v i t y ,

can be e x t e n d e d

all c y l i n d r i c a l

In this w a y , 5 M

in

f("|)

R

w

H C R

1

that if

upon

the c o - o r d i n a t e s

OI n

f

f

R

^ S e e K O L M O G O R O V [l] C h a p . Ill, § 4 .

w

è

J W

functions

the m e a s u r e

rrel elated

R

f

R

con-

u

- the " r e g u l a r m e a s u r e on f(^)

and

R

w

integrals

U

is a c y l i n d r i c a l

n•• d 'r1fnn ="

6"-ring of B - s e t s in

the m e a s u r a b l e

d«. ,

Note

to the

s e t s of

( ^U. is the e x t e n s i o n of

78

in

n

fnt)

is summable with respect to this

because

f J I g($.t)|d

( s C x ] , d |

,x)

-

n

(l,d^,0)

»

-z H

—

= H[X]

.

Then,

any

E

of

the

w

$

(8),

g ( x

m +

,

.

since this

Se-fsV v j

superposition in

we

the

, •

implies

we

x m +

^

operator

measure

w

on

have 8

C

x

hx Q

m 3 — =

u

g(x,t), .

obtain

s [ x j | , t ) d £ L

$

i P ( l . d l . O )

-

5

g(z,t)dw

-

F [ z ]

,

«

theorem.

Condition

(4)

can

be

weakened, oo

| g ( * . t ) | «

u(t) € L ( Q

;

Rco

E

the

, d j

, d | , x + X y )

] ) — » g ( z ( t ) )

= lim

g(z, t)dw

. t)i?

,x)dw

account

m

S

9 ( 1 - 1 ) ^ ( 8 .

( | . t )

n

for

into

Q o

( I

( | . t ) ^

continuity

( t ) +

H[xm]

n

,x+Ay)

n

and,

the

taking

g

n

V ( s | X I , d £

= 0

to

n

S

n E

€ V,

J — • s £z]

have

dt

g

E

g ( $ . t ) V ( S [x+ X y ] , d ^

Eco

5

n

J

g ( ? . t )

x

dt

n

$

S

s [ x

,

n

E C0

Qco

Let

dt

Qn

n

0

u(t)

,w),

•

u?0,

12 k=l

-v k * A

A

**

>0

replacing

i t

by

2 kl

,

.

83

This implies

the b o u n d e d n e s s

sup I F I i VR Therefore,

the p r o o f

J

f

u dw

remains

of

F

84

the c o n d i t i o n s of

the

VR:

II x II 6 R:

+ e

the

same.

The u n i c i t y of the o b t a i n e d s o l u t i o n fying

on a n y ball

theorem

in the c l a s s of follows directly

functionals from

(7).

satis-

CHAPTER

3.

THE FUNCTIONAL LAPLACE OPERATOR AND CLASSICAL EQUATIONS.

DIFFUSION

BOUNDARY VALUE PROBLEMS FOR UNIFORM

HARMONIC CONTROLLED

DOMAINS.

SYSTEMS

7. Boundary value problems with strong Laplacian and their to c l a s s i c a l p a r a b o l i c

W e d e a l here w i t h A F in w h i c h

=» 0

boundary value problems

,

A U

for the

equations

= F , A

the o p e r a t o r

rivative,

parallelism

equations

(1)

is d e f i n e d as

i . e . the s t r o n g

Laplacian

(see

the i t e r a t e d v a r i a t i o n a l

de-

4.1):

b A F -

\

F"

a

*

dt

.

(2)

(O

O u r m e t h o d is b a s e d on the a p p l i c a t i o n of parabolic equations these equations of r e a c h i n g tions of

the time

t

the b o u n d a r y

S

al. This approach seems ing

starting

that it d o e s n o t g i v e a d e e p i n s i d e related

, whose definition

cal m e a n fflft . H e r e

£x[y],

=

A u ,

information on how

T[x}

u^.

=

A u + g(

functional

the u n k n o w n

g i v e n d o m a i n . In s u c h p r o b l e m s

form. At

functional time,

equations

before,

and

classias

only

at i s o l a t e d p o i n t s of

2

the f u n c t i o n a l S

to s e t not

local c o n d i t i o n s , w h i c h

functional

2

T » (R -Rx|| )/2

V x II

R

is a s s u m e d

amounts

to be

to c h o o s e an

exthe

(if w e

is a s p h e r e ) a g a i n p l a y s

time, but in a d i f f e r e n t s e n s e : n o w is c o n s t a n t . To g i v e

spheri-

the s a m e

(3)

(1) it m a k e s s e n s e

to the c a s e w h e n

in

method

7.2.

as in S e c t i o n 5 but a l s o

llxll

This

is e v e n of a d u a l c h a r a c t e r

global conditions

role of

study.

the p r o b l e m s e x a m i n e d

for e q u a t i o n s

able and

method".

elliptic equations

It turns out that

restrict ourselves

under

considered

from the c l a s s i c a l

together with

the table c o n t a i n e d in

the v a l u e s of

mention-

, «C ) ,

cal p a r a b o l i c e q u a t i o n s . T h i s a n a l o g y

press

it is w o r t h

to f i n d the

is taken in a ready

to p o s e new p r o b l e m s w h i c h ,

from

solufunction-

from the v e r y b e g i n n i n g w a s b a s e d on the

reveal a wide analogy between can be s e e n

x ), the

in

time

the d e s i r e d

Trx[y]

the " d i f f u s i o n

the d i f f u s i o n m e t h o d p e r m i t s us, s t a r t i n g " u^.

the p o i n t

however,

involving

replacing

(the m e a n

i n t o the p r o b l e m

to the f u n c t i o n a l s

a l s o f a i l s to p r o v i d e s o m e

T[x]

f u n c t i o n s of

to be p r o m i s i n g ,

Sections 5 and 6 remains outside T

from

theory

space. After

by the f u n c t i o n a l

them are u s e d as g e n e r a t i n g

Everything

the d i f f u s i o n

in a f i n i t e ^ d i m e n s i o n a l

the vari-

initial

reference.

85

7.1. The functional Laplaclan and the classical parabolic operator 7.1.1. We suppose that the surface

S

on which the boundary values of

the unknown functional are given belongs to the type

{s}

(see Subsec-

tion 5.1.1), however, now we require only the conditions (1°), (2°) and (4°) to be fulfilled, where (4°) will be formulated in a slightly different form: There exist9 a functional T[x] continuous on the set 2 ST and S T exist and are of the form (for V » VuS such that s=l, see 4.1): b 5 t

3

S2T b 5 a

•

a

T

Furthermore, AT

b

S - T i(t) u < t ) d t ' a a b b - I A[x|t] - ^ ( t j d t + B[x|t1(t2] ^(ti) rl(t2)dt1dt2 S Ao[xit]il(t)dt -

a

b

; > , t1 , * i 2 ( t ) d t * 1 '

+

(4)

T

a

;(t1)x(t_)n(ti),i(t2,dtidt2

•

(5)

suppose b •

\ T", dt » -1, a x2(t)

x«V,

T = 0,

x € S,

(6)

and let, in addition, T

-

2tb^T

(rw-nxn2),

r

(Note that from the harmonicity of sults

^

m

T

.

(7)

the equation

A T

• -1

re-

immediately.)

7.1.2. Let the function

h(^,u,oc)

I f l

of the variables

'

u

' ("1

u

m>'

be given, which depends, moreover, on the parameter

u6
x( r

|M| » R

(1)

62) 6j)

•

condition:

F[x],

H[x|R]

6X)

l|x|2(R) ,

8. Boundary value problems The Dirichlet problem

for u n i f o r m

domains

for a u n i f o r m d o m a i n d i f f e r s

from

the

functional

D i r i c h l e t p r o b l e m e x a m i n e d in S e c t i o n s 5 - 7 in m a n y a s p e c t s . It c a n c o n s i d e r e d as an i n t e r m e d i a t e

problem between

the f u n c t i o n a l a n d

c l a s s i c a l D i r i c h l e t p r o b l e m . W e s h a l l see

that in a c e r t a i n s e n s e

l a t t e r p r o b l e m is a d e g e n e r a t e d D i r i c h l e t

problem

T h e s a m e is true

for c l a s s i c a l

G • G(ZZt,A

G"( Z H t . A s ( q ) ) denote ® o

"

its b o u n d a r y

a?0

types.

problems

be a uniforji f u n c t i o n d o m a i n a n d

(see S e c t i o n 2 . 1 ) . A s in 4 . 1 , w e

the c l a s s of f u n c t i o n a l s w e a k l y h a r m o n i c

We study Problem

(q)), q » ( a , b )

the

domain.

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

8.1. Functional and classical Dirichlet Let

for a u n i f o r m

be

the

in the d o m a i n

G' »

shall G

by

.

the f o l l o w i n g

problems:

I. F i n d a f u n c t i o n a l

H [x]

satisfying

the

conditions

H[x] € c£q(G), H[x] - Y[x], x 6 G' , where Problem

is

Y

given.

II. G i v e n

F

find a f u n c t i o n a l

H[x] € ? e o ( G ) . Problem

with given

x e

which

G.

$

[x]

I IG 1

fulfilling

the

conditions

= 0

F[x] .

these p r o b l e m s m a y be c o n s i d e r e d

for v a r i o u s xneG,

for

- F[x] .

III. F i n d a f u n c t i o n a l A 4» - F [ x ] ,

All

'G

H[x]

functional classes.

zeG'

and

xn—-z

in d i f f e r e n t

The b o u n d a r y

function spaces

conditions mean

in the t o p o l o g y of

Ag(q)

, then

and

that,

if

H[xn]

— 1 - H[x] . We consider Problems values

in the G S t e a u x

I - I I I on the set

8 . 1 . 1 . Let us b e g i n w i t h Y where

M

x(t)

'

f

the

assumption (1)

•

xs(t))

I under

and

99

F

j

Q

Recall nates

9j

holds provided

that

(Ah

the f u n c t i o n s • Z^A.h), rj *

then,

£ IAI ,

a

(ii) Let

m >1

remark at

the a s s u m p t i o n

.We

point belonging

assume

to

G

the e n d of

consequently,

that

(this

of the

lemma.

the d o m a i n

G

restriction

t

is

this, all a r g u m e n t s

equal

£

are r e g a r d e d as s c a l a r s . As a l r e a d y that

g

is a s y m m e t r i c

(without

stance, with

is s t a r - s h a p e d n e a r

is n o t e s s e n t i a l ,

the p r o o f ) . A t f i r s t , w e s u p p o s e

the d i m e n s i o n of

In a c c o r d a n c e w i t h

l,...,m

^p

(14) is a l s o

d e p e n d on "umber

5J/2 ,

(,«...+«„« v

We take an a r b i t r a r y star-shaped domain, Ax€G

a n d thus

J C ^ Qm

result. W e now w a n t e*

and

(t)x^(t1)...x^(tli)dt*

but fixed function for some i n t e r v a l

f [ ^ x ] » 0 . From

> «,+...

for any

5 C^...,, g m

0 ^(t)x°'

,

x £ G • Then, s i n c e 0 < X
0

let

C2(t) / 0

can s u p p o s e

that

to find an I n t e r v a l

A

t € A

;

x

on a set

C2(t) > 0 c

E ,

a n d to indicate a function

Xj(t) » h ? 0, Now w e

• j J C12(t1,t2)x(t1)x(t2)dt1dt2 q q

^(t) =

ECq

= 0

of

posi-

on this set. In

I.AI = &

such

x^i*)

w i t h the

t e A

.

.

this

that

properties

have ^ C2(t)X(j2(t)dt »

^

q

A

C2(t)x52(t)dt ?

S h. 22 .m

and . I J 5 1 q q where

M »

It f o l l o w s

C

12(tl't2)XS(tl)XS(t2)dtldt2|

sup that,

104

q .

I J 5 •••dt1dtj 6 A A

Mh2S2,

|c1_(t,,t_) I . l-i l ^ ' for

tradicts the fact that where on

*

&

s u f f i c i e n t l y small, 4

>

2t

x

s3

*

0

• Hence

$2[xj] ^ C2(t) = 0

0

'

w h i c h

almost

con-

every-

c

Repeating the same arguments, we can show that everywhere on q x q .

i2^tl,t2' *

0

a

lmo8t

Consequently, the term of second order in (15) is equal to zero. In the same manner we may show that all the other terms in (15) containing reduce to zero. Thus In the case

s>l

. we write

Imgit) g x

g( £

«) = 9 ( ? n

and, after fixing

does not depend on • w e show that

8(t)

» (£

g

x2(t),...,*(t) , . .., £ l 8 )

does not depend on

clusion results also from the symmetry of f

). Thus

m

If

G

?

g » g(t) . l8

, we can show

. Then, fixing

fj, g

J

£

2

that

x3(t),...

etc. (the con-

with respect to

fj,...

g - g(t) .

is star-shaped around the point

translation

x , then with the aid of the /v o f - x , we can transform into a star

•

around zero, and from the established independence of the

function

g(i£+g,t)

is independ-

ent of

of

%

it follows that the function

on the set

metric equations parameters g

g(£,t)

v

c . If the domain 3C,. G is given by the para/w (12), then expanding f^ in a power series of the

and repeating the previous arguments, we can prove

does not depend upon

£

. Since in (12) we can regard

co-ordinates, the requirement of (global) star-shapedness of

u

that

as local G

may be

assumed to hold without loss of generality. The lemma is completely proved. Let for the set 5Z!l (4)) —Theorem — — — 8.1.2. — — 'm, » t (see * \ ir and the function g the conditions of Lemma 8.1.2 be fulfilled. If the system (6) has a unique solution in the class of functions mentioned in this lemma, then

H »

^

h dtm

is the unique functional of GSteaux

type

Qm being a solution of problem In fact, if

F - ^

hdtm ,

(9).

~ «

Q

^

hdtm

and on the set

G'

the equal-

Q

SKI

ity

F • F

holds, then owing to Lemma 8.1.2, the generating

of the functional that

F-F

^p(t)

and we can suppose

ip • 0 . In view of the unique solvability of problem

sumed above, we then have if

has the form h - h

on the set

¿_,

function

^ . Hence

(6) 89F •» F

x £G .

Remark. The analyticity of the function arguments of the vector vious because

h("£,t)

%

g(£.t)

with respect to the

used in the proof of Lemma 8.1.2 is ob-

is a harmonic function of the same arguments

and it is well-known that a harmonic function is analytic as well.

105

B.1.3. Looking back on the considerations explained in 8.1.1 and 8.1.2, we observe that Problem I, (9) is related to the standard classical problem if

m«l . For

m>l

harmonic function on skeleton

ZZ^

ZHm

, we encounter the problem of finding an inby its values on the m(s-l)-dimensional

. In some important special cases problems of this kind G: x 2 ( t ) + y 2 ( t ) < 1

are solvable. Let e.g. the domain boundary values be defined by the

be given and the

functional

F - (j j g ( w ( t 1 ) , w ( t 2 ) ; t 1 , t 2 ) d t 1 d t 2

,

q q where

g » g(J

£2;tj,t2)

respect to

and

£

2

and

g(-)

Is an analytic function with

. Setting

"

? i+ i 1 l i '

' ^2+irl2 *

we get the problem 32h

h

32h

x

t

o , such that on the strip [~1 : E a x q the estimate |f(S.t)| 5 a(t) + b I? I 6 7 * , ffj-O, b'const > 0,

a e L e (q)

holds, then f 6

{L,"(q) —

L. e (q) }

Theorem 8.2.2. Suppose

.

f 6 { ^"(q)

""^(q)}'

0 < oc < ff , and let

I"! . For the existence of the derivative of the opf ' e x i s t on erator f at any point x€ L * it is necessary and sufficient that f; (x(t).t) € { L / t q ) — * " L / - " ( q ) }

.

Theorem 8.2.3. Let Otx - J k(t,u)f(x(t),t)dt , q

and let the linear operators k

J ?

" 5 kj(t,u) Y(t)dt q

belong to { L ^ q ) — r L ^ q * ) } . rems 8.2.1 and 8.2.2 hold, then and x -

j kft.ujf^tx.tldt

< C * 0 . If the conditions of Theo& is a differentiable operator ^ ^ ( q l - ^ L ^ q * ) }

.

q

Now we are prepared to formulate Theorem 8.2.4. Suppose ol>0, /i^O, and let positive constants 6 , t , 6> 2oi, 0 < Y < Ji , c o < Cj^, c2« Cj, c 4 > 0 exist such that the following conditions be fulfilled: I hj(Ç.t)| é b o (t) + c o |Ç| Cr/o( ,

(i) (il) (iii) (iv)

b 0 £L f f (q) ;

£ b 1 (t)*c 1 |Ç| , b x € L & (q) ; 3*k 6-Z* _ , ^ z. « b 2 (t) + c 2 | Ç | , bg e L (q) ; ^ 9 Ik kj(u,t)

are linear operators: kj €

L^iq)

(v) the function f( ijj, . . ., u) » f(lj,u) diffe rentiable with respect to the arguments strip 0 « E N x q , where the estimates |f I < b 3 (u) + c 3 l ^ l ^ .

110

b 3 (u) e l A q * ) ,

•L*r(q*)}' ; is twice continuously . • ., î^^ on the

< b4(u)+c4ltll

,

b4(u)eL^(q«|

hold. Then the operator solution of the functional Oirichlet is of the fori Yx

problem

- f(H1*.....HNx) ,

where H

and

J " 5

h

j(X,t)kJ(t,U)dt

hj( C , t ) k j ( t , u )

everywhere on

lem in the domain the set

ZZ^

considered as a function of

q x q*

£

is almost

a solution of the classical Oirichlet

ZH

prob-

with boundary values . 9j( £ > t ) k j ( t , u )

t

on

.

The proof follows immediately from the properties of continuity and differentiability of the operators involved in Theorems It should be noted that almost everywhere on' q x q

8.2.1-8.2.3.

the derivatives

2

_

3 h „_ J

,

exist. Furthermore, in view of Theorems 8.2.1 and

8.2.2, the Fr&chet derivatives hj at

with respect to -x^

3 h1 . „ J and 9xk

exist on the set

9 V — t9 x— k k

of the operators

3 x

G . Omitting the Indices

x , we can state

I r

1

«{«-.-(q) —

e

Lf-^q) } .

{L.-(q> — ^ ( q ) }

.

Theorem 8.2.3 implies that the operators Hjx - ^ hjixitj.tjkjiu.tjdt q are twice differentiable on the space L

^

t

q

)

—

.

and second derivatives

H

xx •

kh

^(q)

and, in addition.

Again omitting the indices at H^ , H £ x

He

x , for the first

we can write

ix ^ ^ ( q ^ L / ' ^ q ^ L ^ q ) }

.

W i t h regard to (v), we have Y

H N x ; u ) £ -[L 6 T (q*) — / ( q * ) }

- f(H x x

Moreover, again by virtue of (v), the

.

relation

H

/ \ I . AYx. [Cf-ISj)«] AHj5. x

111

holds for x € l_8 . Since set A Y x - 0 .

A Hj • 0

on the set

G , we obtain on

All operators considered here are strongly continuous. Hence |HJX-HJZII^—•O

implies

|| x - z B ^ — 0 n

, i.e.

From (v) we conclude that in (18) Consequently, on the set

this

^

is also a continuous operator.

G' , one has

Hj) -

Y(F1

Fj ) .

which completes the proof of the

theorem.

8.3. The functional Neumann problem The functional Neumann problem for a uniform domain with boundary values in the Gdteaux ring (see 8.1, Problem II) has common traits with the Dirichlet problem considered in Section 8.1. Let

G( H t , C s [ q ] )

,

s >1

, be a uniform domain with smooth boundary

(smooth skeleton boundary) H

"

functional

5 O m

h

G'(

( x (t). t ) d t f l " $

»

t,Cg[q]) at

f.

^ lj?dt

the

. By the derjvgtjvg gf jhg

point

x£G'

we understand the

constructed via the generating

]

*

S h0(x(t).t.T[x])dt

.

(6)

a

The f u n c t i o n

hQ

i s c o n t i n u o u s and t w i c e c o n t i n u o u s l y

with

respect

to

f

(cf.

formula

(9)

of

k

and

f

for

differentiable

t " >O . Hence, due to Lemma 4 . 1 . 3

4.1),

117

S A H

If

° "

x£D

l

3h

^h A T ) d t

^

, then in v i e w of

A H

-

]

(iv"), we can w r i t e

J 2 (s(b-a)

-

1

a

8 x

)dt - 0 .

(7)

3 T

k

In a d d i t i o n , from

(2), (6), (iv'), (iv") we get .. to lim H [x] - \ f u (z(t),t)dt = Y [z] . n n J a

for

x£D, n

(6) s a t i s f i e s the two c o n d i t i o n s

2) Now w e c o n s t r u c t

functional

H[x|t] C g»(D). Let

xCD

equation

and

x

y Q , y^,...

= Y[x|t]

(7) and

(8).

.

be the s e q u e n c e of P i i a r d i t e r a t i o n s of

(1) c o r r e s p o n d i n g

Y,[x|t],... if

H[x|t]|D>

xZjz: n (8)

In this way, f u n c t i o n a l the

z€D',

to this f u n c t i o n . Let,

furthermore,

YQ[x],

be the following s e q u e n c e of i t e r a t i o n s , w h i c h is o b t a i n e d

v a r i e s on the set

D : t

Y

n+l£

x|t

3

+

° *0[x]

S

f(u,Yn[x|u],x(u))du

.

to In p a r t i c u l a r

t

YjCxIt] - YQ[X]

^

+

f(u,Y0[x|u],x(u))du

.

to

We form the

functional t

H-Jxlt] - HQ[X] + where

^

h(u.H0[x],x(u),T[x])du

to

h(t,H , £ , T )

is a s o l u t i o n of the

» (^{h)s";l(b-a)-1.

.

(9)

problem

h(t.H0, i , 0 )

- f(t.H0.f ) .

Because of Lemma 4 . 1 . 3 , w h i c h is o b v i o u s l y a p p l i c a b l e here, one AH, - AH

•

o

I

( ±

t0

Taking into a c c o u n t

+

_ ^

A

H

o +

^

A T ) d t

AH

t

=

\

s

A Hq = o

t0 Moreover,

118

if

2

(Z:

xeD'

1

, the

.

k

(iv") as well as the e q u a t i o n

we qe t t

has

- S(b-a) 4 r ) k relation

d u

=

0

•

for

xeD

,

2

M

!

1

]

"

H

oCz3

+

J

f(u.H0[z].z(u))du

t « Y0c*] •

S t.

f(u.Y0[2].z(u))du - YjCzlt]

holds true. If we repeat the same, then in the next i t e r a t i o n we o b t a i n t H 2 [ x l t J - H o [x] +

JJ h i u . H j C x l u l . x i u J . T t x J J d u . t„

where 3 h ( t , H ,Ç,T) gi - 8 For

h(u,Hj,Tp,T)

and

Açh.

h|T

regarded as a function of

tions of Lemma 4 . 1 . 3 are Repeating

(b-a)

o

u

- f(t.H1#ï) and

"Ç

.

all

condi-

satisfied.

this process, we obtain two functional s e q u e n c e s

H n [ x | t ] , n - 0 , 1 , 2 , . . . , such

Yn[x|t]

that

t H

1 t

n +

x

H Q [X] +

I -

J

h(u,Hn[x|u],x(u),T[x3)du

,

(10)

t„ Hne^(D).

Hn[x|t] - Yn[xlt]

if

x e 0',

Hn[xlt0l « Ho[x] .

A s a result of the existence and unicity theorem of a s o l u t i o n of e q u a tion

(1), the limit Y [ x | t ] - lim Y n [ x | t ] , n

e x i s t s for dY

x€D

t e [ a , b ] , where

Stlt]

- F(t.Y[x|t].x(t)),

Y[x|t0] = YQ[X]

Now we want to show that the sequence From

Yn

.

c o n v e r g e s u n i f o r m l y on

D .

(i) and (ii) we deduce t

lYl"Yol i


l,2,..., we obtain

1 Yn4.1Cx|t3-VnCx|l:l| *

MnKR/'^'o'" * nI

Yn[x|t]

X60 .

.

Whence Y[x|t] .

This implies that the sequence Hn

£ H,

xeo

also converges uniformly:

Hn[x|t]

. This fact can be Justified by establishing an explicit

relation between the generating functions als

Yn

and

f

and

h

of the function-

H n , respectively, . »(b-a)IC-nl 2

s/2 r r - (^Ift ) )'••) E.

T

1

h(..£.-. >

f

l , see formula (3) from

both in the strong and the weak

sense, the relation H n [x|t3 holds, where

a

7(ilYn[x|t] •0« is the Hilbert sphere with centre at

V 2 T [ x ] . This implies |Hn+1[x|t]-Hn[x|0| -

| m(Vn+1[x|t]-Yn[x|t])| IU

If

x CO*

, * then

n • H„n

| Hn+1-Hn | .

M

"

and hence r

k

/

I

•

Consequently Hn

h

if

xed .

From Lemma 4.1.2 it follows that, for 120

xeo ,

x

and radius

lint A H n » n In addition, for

A lim H n » n

A H - 0 .

x n e D, z € D ' , lim x R - z . we gat n

lira H[xnl t] - lim lim H ^ C x ^ n m 3) Now we turn to the expression b Y

M

-

- lim Urn Hffl[xn3 - lim H m [ z ] - Yfz]

(4) and form the functional

$ h 1 (t,H[x|t] f x(t);T[x])dt a

(11)

.

where ah^t.H.ç.T) ± - s 1 (b-s)

l

A

ç

Taking into account the equations

h

h1(t,H,Î.O) - g(t,H.Ç) .

1 (

AH

« 0

and

Hi » Y , conditions ID' ( i V ) and (iv n ) as well as Lemma 4.1.3, after repeating the previous

arguments, we obtain A Y -

0

if

x€ D

(i.e.

Y «

36(D)) .

b Y W

•

5 9(t.H[z|t],z(t))dt - 4» [z] if z e 0 ' . a Thus (11) is the desired functional and the theorem is proved.

9.1.2. If we restrict ourselves to the Dirichlet problem with boundary values

Y[x|t] . It seems to be natural to consider the domain

consisting of functions defined on the interval

a< u it

D{

. In this

case all arguments remain the same as above, however, instead of the b T • T[x] a

functional

rj m t T » T[x} a

we have to use the functional

sa-

tisfying the conditions T >0

if

x C Dt ,

T - 0

The solution of the problem uniform limit

if

AH

M |t],x(t))dt , Q

whore A

t

y(t.

y Following

= g(t, r c x i o . Ç )

if

if

£eZ!

£e S ;

= 0,

x e G ;

U « $

,

*eG'

t

.

the same arguments we show that

AU

126

rfcxifl.Ç) - o

Yq I""*

and

Yn

is the

below.)

converges uniformly on the set

Lemma 4.1.2, the limit functional

, respective-

,

;

G

and, due to

is also harmonic.

.

-vhere $ was defined at the beginning of the present section. proves the first part of the theorem. P

We now intend to show that the obtained functionals

and

U

This

are

weakly harmonic. First of all, we remind the reader of two facts: (i) The mean of the functional S

t:

l*-^

ls

"

of

the

E

'

*(»„)• y i t j f . i t !

^/.(t^

S-S St

St

l

tm)d&

J

o

1

ferential of (1) this from measure ( E 3.1). is some cube of Q m ; see Section

Q_m

*^ 1 to any vector

=*m £

1'

, then

t

m .

(3)

m St

h ( £ , , . . . , "C_;t,,...,t )

" W ^ i ^ i i

d€Tt

is the measure of the sphere

(ii) If

over the sphere

form

" S dt v-- dt ™-^7777^TtJ

r

where

g(x(t),t)dtm

F « ^

and

d frt

the dif-

or a measurable

is a harmonic function with

respect

' m' H - J

subset

^ h(x (tj), . . ., x (t m ); tj, . . ., t m ) d t m

is a

E harmonic functional both in the weak and the strong sense:

HۤP(G),

H € 3 B 0 (of G ) all, . First we discuss the case that the initial functional constant.

Y_o

is a

From (2) and (3) we have t Ml^CxIO

.

^ vj>(u,y 0 .x(u))du + y Q -

^[xlO

.

to

The proof of the analogous fact for

T

2

ta similar but slightly

complicated. The conditions imposed on the function function

ip(u, T j , ^ )

the variables

u,

more

ensure that the

is also continuous on the considered domain of

r ^ , f . For

x€G

, the set of values of

obviously bounded and closed. Therefore sented in the form

f

ip(u, (""^."C)

lim p^fu, T , , ^ ) , where

pN

Tj^

is

can be repre-

are polynomials

in

N

I j

and the sequence

pN

converges to

vj>

uniformly with respect to

all arguments. Moreover, it is not hard to understand that all coefficients of

pN

may be assumed to be harmonic functions of

Thus the sequence of

.

functionals

t PN[x|t] =

^ pN(u, P j x M . x f u n d u

t„ converges uniformly to

TgCxIt]

• yQ

on the set

G . Furthermore, all the 127

PN

have t h e

form

PNC*IO Replacing

P^ N

J

and

PN

. as

follows:

u \ u>( + Xq

aNl>(u, £ )

u

i'v

0

'

x

(

(as w e l l

u

i))

as

d u

i}

d u

j

*

p N ) a r e "harmonic

•

since

it

(ii)

just

i s possible

M , we g e t S

fails

rj^txluldu • yc

we can r e w r i t e

•N1>c«.*(«))-[ n l w

the o p e r a t i o n Yo

value,

from t h e remark

sequently,

If

by i t a

^(u^y, £)

follows

\ ^„(u.xfu))

v

•o

f u n c t i o n s of It

r*

t

**

where

N H

-

M T2

t o be a c o n s t a n t ,

mentioned t h a t

to pass S

r

2

P N e cJ?Q(G)

to the l i m i t

[*|t] o

then f o r

t i.«.

under T2

£

• Con-

the s i g n

§P0(G)

of

.

t PQ »

J

ipo(u,x(u))du

,

a

the r e l a t i o n

M

functional

f~ 0 •

P

0

[*]

t "

^

holds.

Furthermore,

approximating

the

0

•

J

Vp(u,

r

0

M.x(u))du

t. by f u n c t i o n a l s

of

the

form

r 0 • zz r0v \ where

b

l v

are able

bw(Ufx(u))du,

t.

1

(u,£)

to prove

a r e some h a r m o n i c

functions with

that

.

G

(G)

I n t h e same manner we can show t h a t ,

At l a s t ,

repeating

t h e arguments J u s t

inclusion

U[x] e cf0(G)

9.2.2.

n — » • oo, we h a v e

If

f o r each

used,

. The theorem i s

n ,

respect

P

R

to

£ ctQ(G)

f

, we

and

we s u c c e e d i n v e r i f y i n g

completely

the

proved.

t r & o o .

r

w

0

•

J

vp(t,

r[xiti,x(t))dt

a and,

setting ^

-

Ttxlt]

- v(t)

lp(t.v(t).x(t))

The o b t a i n e d d i f f e r e n t i a l system" 128

whose s t a t e

, ,

v(tQ)

aquation

function

v(t)

(4)

-

ro[x]

.

(4)

d e f i n e s a "harmonic

coincides with

control

the s t a t e

function

y(t)

related to the input system on the boundary

domain. Unlike the values of stant

t

the maximum principle holds: v(t)

values in the Interior of the domain of the representation of

P[xlt]

of the control

v(t)

at each in-

cannot attain its extreme

G . This fact is a consequence

(of the functional

tively) in the form of the mean value of the boundary

G'

y(t) , for those of

Y[xlt]

U[x]

(of

, respec-

[x] ) over

G' .

9.2.3. Here we deal with the representation of a solution of the tional Dirichlet problem as the mean over the boundary

func-

G' . In order

to obtain the mentioned representation, we need some generalization of the definition of a continual mean over a uniform domain stated in 3.1. We suppose that, for any measure with density

t e C a . b ] , on the set

p(f,t)

¿Z ^

the probability

is defined.

Let < f n >' be the meani value of the function f n x(see Section 3.1) ' over the set JZ x ... x S corresponding to this distribution. m 1 If, under n — » oo and max — 0 , there exists the limit M P F - lim G n

< f

>

,

then it will be called the mean value of respect to the measure form on

F

over the domain

with

G , we return to the definition from 3.1.

As in 3.1 it may be proved that for the Gâteaux functional M*5 G

in this section the mean mPf = J . . . J a t l . . . i t m

G

G

p . In particular, if the distribution is uni-

\

C ... [ g(

A

(1) described

exists and has the form £

m tm) r i p i C i . t ^ d v ,

m i t l

(5)

A

X

T 1 m provided that its generating function

g

is continuous on the set

(2Z t x ... x ) x Qm (and, of course, under the assumption that 1 m the integral exists). Moreover, the mean M*5 ha6 all the properties Q indicated in 3.1. Now we return to the Dirichlet problem for the domain Let, for each main

v

teQa.b] ,

w(£.*J>t)

G(XZt.Cg[a,b^) .

be the Green function of the do-

associated with the operator

A ^

, and let

^

its derivative in direction of the exterior normal to well-known that J

^>0

on

j(C.tj.t)dn6"

Xj j

t

- i.

and,

=

dw

t e

be

• It is

clearly,

( t , J ) t E

t

,

V ^ i

•

129

Hence t

^

and

generates a family of probability measures with parameters £ , and the solution

for the domain

ZUt

ip

of the interior Dirichlet problem

can be written in the form

tf(f.t) -

^

?«t)d,6'

9( »t-t) 9 (

City Let

G'

be the boundary of the domain

fined on

G

and

F

be a functional de-

G' .

We construct the operator x0(u) ?x(u) 0 F - M F . E ° G i where the averaging on the right side is taken with respect to the distribution

p(?,g(u),u),

Let the functions

f

f eE

and

f

u

,

g(u) « G .

in the differential equation (1) meet

the conditions of Theorem 9.1.1, and let

Y . Y, , Y_, ... be the seO 1 £ quence of iterations related to this equation. g Applying the operator

E

to the functionals

mula (5) (where only boundaries e

E j

§(u)

and

and

G

Yn

and utilizing

for-

are to be substituted by their

G' , respectively, we get

vntxit] -

rncgio

and, passing to the limit, x(u) E °v Y[xio

- rrg 10 •

By the same token, for the functional Utg] • E

8

$ DO

Since the operator tionals

130

Y

and

$

Ex

U [x]

we obtain

. refers only-to the boundary values of the func-

, the desired representation is proved.

CHAPTER 4.

GENERAL ELLIPTIC FUNCTIONAL OPERATORS ON FUNCTIONAL RINGS

10. The Dlrichlet problem in the space of 9ummable functions and related topics

10.1. Functional elliptic operators of general type Our results concerning the homogeneous and inhomogeneous equations LF • 0

and

LF • Q

in function spaces(Sections 5-7) were related to

the simplest case if operator

L

is the strong or the weak functional Laplace

b

L and

C F" dt xz(t) a

L • 2 lim X-o

« F ^ A V J v*

- F[x]

respectively. In accordance with this, the associated classical parabolic operator used in the diffusion method was of the form

d2

9 ^'

t

T?7

In the present chapter we show that we succeed in extending the results obtained above to much more general equations.

10.1.1. First of all, we want to explain how general elliptic functional operators can be obtained, using the procedure of continual averaging. Let

x • (Xj

x9)

as well as

y • (y 1 ....»y e )

with components belonging, say, to the space let

at[xlt3

C[q] :

Ogixltl

Q^xlt] £ {C8 — C } .

To every

xcCs

q " Ca,b] , and

be operators acting from

C8[q]

into

.

we assign the domain

the distribution density

be function vectors

C8[q] ,

D[ylx]

p(fj

of functions

CljCxIt]

y

having

O t ^ O l t])

(

(X

are the parameters of this distribution; see Section 3.5). Let, furthermore,

a^[x|t],...,a 8 [x|t]

tribution

p

be the vector of first moments of the dis-

(the "centre of the domain

D[y|x]") and

the Matrix of its second moments (the existence of all

b^CxIt]

be

a

is

and

b

supposed). Consider a functional

F[x3,

x e C 8 , having the first and second vari-

ation : F[*+ £ y ] - FIX) • 5 f • I S 2 F • o( £ 2 ) ,

(D 131

w h e r e s S

f

-

£

E I

J

^ [ x i t l v ^ t j d t

w i t h

u

^

.

f

;

q

•

2

£

^

.

(

)

2

ol

J

%

[

s

x

|

t

]

y

(

«

t

)

F

x

t

y

)

d

t

( 3 )

w i t h

V A

L e t

u s

F

'

d e f i n e

x

( t

o t

t h e

) x

1

/ i

( t

2

M F [ x +

i * ¡imo -

2

-

L j

l i m

a n d

( t ) x

o l

l_2

y i

v i a

( t )

•

t h e

r e l a t i o n s

£ y ] - F [ x ]

2

£

M F [ x + F

•

o p e r a t o r s

LF

L

)

d e n s i t y

a b o v e ,

U^

,

o f

t h e

W ^

a n d

i . e .

e

%

d

-

i s

! * )

f u n c t i o n s ,

C

î i

( 6 )

C

{

s ^

C

2

}

'

%

C

{

C

s - "

C

}

'

o a n d we

t h e

e s t i m a t e

g e t

o (

£

)

i n

( 1 )

i s

u n i f o r m

w i t h

r e s p e c t

t o

y

,

t h e n

s 4 F

-

Y Z

\

u j x l t l a ^ x h l d t

.

( 7 )

q

L

2

F

"

Z

I

J S

V

C x I v t ^ . J x I t ^ C x I t ^ d t i d t g

Q2 •

I n

132

t h i s

w a y ,

z

:

t h e r e

J

w

^

r e s u l t s

c

x

i

t h e

o

b

^

c

x

i

g e n e r a l

o

d

t

.

s e c o n d - o r d e r

( 8 )

o p e r a t o r

LF

-

LJF

L2F

+

-

Z I

J

04

+

F

z : n

xo.(t1)xa(t2)

«•P Q 2 +

g

1

"

î

^ ( T J ^ ^ I O D T

q

A

2

^(.îx^t)

(Remark. If the a v e r a g i n g

^[«i^^cxitgidtjdtg

Vx|t]dr

.

in (4) and (5) is a c c o m p l i s h e d o v e r

d o m a i n s having d i f f e r e n t d i s t r i b u t i o n laws, cients"

b^

differ

from the q u a n t i t i e s

then the " d r i f t

a^

occurring

different

coeffi-

in the

latter

expression.) In p a r t i c u l a r , » U(x(t),t)

if

Ot C* 11]

are s u p e r p o s i t i o n o p e r a t o r s ,

a^Cxlt]

and

, then

(6) take the form

b^yjCxlt]

a ^ » a Q i (x(t),t)

and

in the f o r m u l a e

b ^

= b ^

Ot[xlt]

i.e. (7)

and

(x(t) , t) , respec-

tively. Below we restrict o u r s e l v e s

to the d i s c u s s i o n of o p e r a t o r s of the

form

s 5 LF

•

1ç s --A(x 0

•

Thus we deal with a family of standard diffusion processes (see e.g. DYNKIN [l], Section 5.6). Henceforth

T

plays the role of time, while

t , the argument of the considered functions

x(t) , is a parameter of

the studied family of processes. Let

F

S g(X(t),t)dt m e R . we define the averaging operator

•

M

T*F

=

I dt m J g l S . t l f l p f r . C ^ J V d r E ¿«1 Q ms

.

(2)

Under the conditions formulated above, the integral (2) exists and, for x e V , the operator If

Fn £ R

and

uF = M F I x ,c

Fn

The functional

F

Let

X ,T o

"r .

"r , then we set

H[x]eS(V)

x Q e v , there exists a M

on

is well-defined in

is called harmonic in

X =

V

such that, for

if, for every 0 < T

< X[xQ] ,

H = Hfx "I . L 0J

oiP(V)

be the set of functionals harmonic on

the following functional Dirichlet problem: Given tional

H [xj

such that: (i) H e 31? (V) ,

z£V,

llz-xnll

»-0

imply

Theorem 10.3.1. Suppose

V . We formulate F£R(V)

find a func-

(ii) the conditions

x R e V,

lim H[x n ] « F [z] . n T[x]

to be a functional on

V

fulfilling

the conditions: (i) T[x] e R ( V ) ; (ii) T[x] > 0 (iii) T[x] = 0 (iv) M x

t

o'

if

x6V ; if

x e V' ;

T = T[x 0 l - t

if

xq€ V .

Then the desired solution can be written in the form H = M and

H

,, , F x, T [x]

is the unique solution in the class

R(V) .

Remark. The mentioned conditions (i)-(iv) are correct. In fact, below we shall establish the existence of domains for which (i)-(iv) are valid, moreover, explicit forms of

138

T

will be indicated.

P r o o f . 1 . L e t us b e g i n w i t h two Lemma 10.3.1. For

F€R

lemmas.

and

T > 0 , the i n c l u s i o n

Proof. Obviously,

it is s u f f i c i e n t

T h e c o n t i n u i t y of

.g(E,t)

p l i e s the c o n t i n u i t y of

»(A.r.t)

E

with respect

the

J

to p r o v e

M

FtR

the l e m m a for

to

F«R(V)

(a.e. on

2

holds.

Qm

.

) im-

function

g( Z

.t)

» l i K ^ I me

fl P ( T . C i ' ¡«1

ms

» i * l . . . . . m ; o( » 1 , . . . ,8 ) on the s a m e ( A * ( ''I l • ' * •' Tl m^ " D u e to r e l a t i o n (12) from 10.1 a n d the k n o w n e s t i m a t e p(T.S.Hlt)


mh

. We form the

is impossi-

functional

- ^ r VTa n O?

„ and consider it on the set

h ^V . In doing so, we have

Y

- M h • o( .

«* > 0 .

w

M , h-mh ^ m^ + — = —

"VMh ¡» — ^ —

V

max Since

that

•

[x] = 1 T | + 1 - T[x] Y w

h

H

hy,

T

LT « - 1

and

• =nnp

. é

M

LH = 0 , from (12) the

• inequality

>°

results. In the case under study, if

c=0 , then

tained relations contradict Lemma 10.3.5. Thus

M

L = LQ h

=

m

and the ob-

h '

Having regard to the equations 145

lim h-0 we

max H[x] hv

get

sup H = sup H V V lemma.

the

From what w a s said posed

functional

ating

operator

in

the

Suppose

that

, we

the

the

lutions

of

and

, respectively,

it m a y

F^RjV),

the s o l u t i o n related

case,

to

proves

of

the

the

gener-

if we s t u d y

, the a b o v e

the

considerations

are

on

IH - H Is I n n+pl that

H = 0

on

V'

. Then,

since

on

V

neither

a positive

on

the s e t

V

LH = 0

. Hence,

in

supremum the

class

established.

x € V m

,

z € V

. However,

v(13)

'

the s a m e

0 F - F n II—>-0 . T h e n ,

the D i r i c h l e t

be s e e n

have

has b e e n

,

FCR(V)

let

sup

, which

relation

Indeed, Fn

the g e n e r a l c f 0

H = 0

cannot Thus

= F [ zJ]

for

of

for a d e n s i t y

. In

and

H

theorem

(see 3 ° )

was derived

+ c

that

infimum.

lim H [ x J m m

the u n i c i t y

problem

R(V)

i£f H a inf H V V

follows.

unicity

5°. A b o v e

max H [x] » s u p H [x] , hv. v.

analogously,

the a s s u m p t i o n

L » LQ

discover

nor a negative R(V)

class

as

lim h^O

a b o v e we d e d u c e

(1) u n d e r

to be m o d i f i e d

and,

Dirichlet

functional

operator

V

» s u p H [x] , y

problem from

the

holds

if

associated

H

for a n y and

with

Hn

F£R(V) are

the b o u n d a r y

two

. so-

values

F

inequality

s u p IF_ -F„I y " I n+p nl

the s e q u e n c e

Hn

converges

to

H

uniformly

on

V .

Furthermore |F[z3-H[xm]| ^ Using F €R

the

the c o n v e r g e n c e

to p r o v e

the

theorem

N o w we ing

the

focus

unicity

the

in of

latter

inequality

R

once

more

is u n i f o r m ,

the s o l u t i o n

is c o m p l e t e l y

functional

Let a g a i n 146

- method,

• |Fn[z3-HnCxj| implies

+ | H n fx J - H [ x j | . (13)

for

every

.

Since

The

|FCz]-FnCzl|

the g e n e r a t i n g

is a s t r a i g h t f o r w a r d

the w h o l e

class

on

the c o n d i t i o n s

R(V)

matter

.

proved.

our attention

T[x]

it

in

(see

the s t a t e m e n t

operator

of T h e o r e m

(i)-(iv) 10.3.1).

concern-

A(t) - Z I

a

8 2

(g.t)

•

c

+

•oc considered

for almost all

tribution density let

t€q

be given, which is related

p(t) = p ( r , t . ^ | t )

and

Vj/(£.t)

be two functions satisfying on

the Carath&odory condition and, besides, for almost all following 1) u

|y(£.t)| 6

Ul(t) c

*

lj>>0,

i*

c

• =

2

|h(£.t)| *

Cl c o n s t

u2(t) • c 2

, the

< h

in some domain

Now we construct

respect to all

(in particular, as

h

we can

constant).

the

Q[*] = ^

,

G c Eg ;

\ p ( t , ? , +J | t)h( £ , t)d| s - h( ^ , t)

take some positive

|£|P

> 0

is twice continuously differentiable with

~%gL, and 3)

t cq

Es x q

requirements:

i' " a 6 1 - ^ )

2)

to the dis-

of the diffusion process, and

functionals

lf)(x(t).t)dt

and rtx]

= 5

h(x(t).t)dt

q and define the sets V: Q[x]
U 1 L U 2 + UgLUj is valid; (ii) the ring II u l s

Rj^V)

is closed with respect to the norm

= sup I Ul + sup I LU I . V V

149

Lemma 11.2.

P

is a linear o p e r a t o r on

P £ {R(V)— and

|P|

-

RJCV)}

|T|

R(V)

, i.e.

.

.

The lemmas Just m e n t i o n e d yield d i r e c t l y the

following

Theorem 11.1. The s o l u t i o n of equation (1) w i t h the boundary condition U| » 0 is given by e q u a t i o n (2). T h i s s o l u t i o n is unique IV" on the ring R«(V) .

150

COMMENTS

To the

Introduction

The mean value and the Dirichlet problem

for the sphere in a Hilbert

space were first considered by R. GATEAUX in his remarkable published posthumously

(GATEAUX

papers

[l] , [2] ). The further more general

re-

sults on means and boundary value problems in a Hilbert space were obtained by P. LEVY and presented in Part III of his book LEVY Cll . This book contains interesting material about analysis in the Hilbert

space,

however, concerning boundary value problems LEVY restricted himself searching solutions. Questions of their existence,

to

the problem of how

the solutions might be effectively constructed, unicity theorems, applications of the theory and its interrelations with other branches are completely

neglected.

As far as I know, during almost 35 years until the author's papers ing with continual means and their applications

(in particular,

ary value problems), no new publication in the directions

deal-

to bound-

mentioned

above appeared.(The second edition of LEVY'S book [1], the book LEVY [2], refines some of the results of LEVY [13, however", it does not contain any new

theorems.)

To Section 1 Functionals

Y £x |t]

generated by differential equations

functional parameters of our century

x

were introduced

containing

by VOLTERRA at the

lated to the linear y" + x ^ t W

of the corresponding

+ x2(t)y = 0 . formulae

for functional

(nonlinear) functional

derivatives

Ytx^x^t]

Some nonlinear equations involving functional parameters were by FANTAPPIE [l]. In particular, he considered

R' • R 2 » x(t) ,

- |

their integrals

functionals,

re-

equation

He has obtained some interesting

and called

beginning

(see VOLTERRA [l]). VOLTERRA"s considerations were

R [x 11]

( ¡p) and

2

studied

the equations

=• x(t)

S[x|t]

the Riccati and S c h w a r z

respectively.

In the paper POLISHCHUK £4] it was shown that, under some general conditions imposed on

f.

and

f_ , the equation

151

y(n,) - f i i t . y . y

yC-^ixft)

generates a harmonic functional POLISHCHUK

Y [x 11]

• f2(t.y.y.....y(m-1)) in the sense of P. LEVY. In

[10] it was noted that this functional is closely connected

with the harmonic functionals investigated by G. SHILOV Equation

(i)

(1) as well as the Picard b

[l,l].

functional

f[*] -

^ g(t.x(t),Y[x|t] )dt a play an important role in the mode rn theory of control systems» The term "Gâteaux functional" for the nonlinear functionals of Integral type considered here was proposed by SHILOV.

To Sections 2 and 3 This notion as well as the classification of function domains normal) were suggested by the author (see POLISHCHUK

(uniform,

Cll.l]).

General explicit formulae for continual means were derived by the author in POLISHCHUK [10]. They represent a far-reaching

generalization

of the Gâteaux formula (3) mentioned in Section 3.3. Another

particular

case (formula (2) from 3.3) was discussed in detail in the paper POLISHCHUK

[l].

The notion of the law of distribution of a functional domain has been introduced by the author (see POLISHCHUK

[lo], where also the corre-

sponding existence theorem for a general function domain can be found). The notion of the centre of a function domain and its relation to the harmonicity of functionals was presented in the paper POLISHCHUK

ClO],

which contains also a brief exposition of the D i r a c measure in a function space based on the distribution law of a function domain. The connection of the Lêvy theorem quoted in 3.2 with the

representation

principle of statistical mechanics was pointed out in the

introduction

to the paper POLISHCHUK

[9].

To Section 4 The Laplacians

A.

AQ,

definition of

A00

and

A°

were introduced by GÂTEAUX and LEVY. The

A°°

essentially the paper POLISHCHUK 152

is due to the author. Section 4.1 is [ll.l].

(Remark. Suppose

F » F[x] ,

the first two variations

x e A(q) , and let, as above,

SF[X,»[]

S2FCX,^1

and

F

having

possess the form

b 5 F

•

F

5

;(t)

v

(

)

d t

'

"b b s 2 p

or

•

b

SJ a a

^ t . w t . i W i w t i ^ *

[

*

F

a

: 2 i1t ^ * 1 >

2 ( t ) d t

•

briefly " S S q q

B

\Z^i'X2) V * l > 1(t2>dtldt2

V. VOLTERRA proposed AF

=

$ q

+

B 1 J L (t) ^ 2 ( t ) d t

S q

the following definition of a functional

B12(t,t)dt

.

Laplacian:

.

(3)

This definition was criticized by LEVY [l3,[23 on the strength of iS

fact that if value on the line definition

a

^

&2F

q , then

concentrated definition Starting

summable over

B^

quite

functions

can be considered as a generalized

at the line

l

2

=

•

Ttlis

its

consequently,

by DALETSKli tl] , if we consider

to infinitely differentiable

the

q x q , then

t„ = t 1

However, as was first emphasized support

unction

is not essential for (2) and, r2 (3) does not correspond to O F .

restriction of

(2)

circumstance makes

^

the with

function Volterra's

reasonable.

from these arguments, DALETSKlif suggested a new definition of

an elliptic operator

L

in the Hilbert

space:

LF • T r F"Lx] , where

F" is the second Fr&chet derivative of the operator

the functional) He considered 3F

Tr

is the trace of the

the evolution equation involving

operator.

this

operator

LF

Tt

and expressed ate measure

F , and

(especially,

its solution as the continual integral over an

appropri-

(for related questions, see also DALETSKIÏ and FOMIN

[l]).)

To Section 5 This section was written In analogy with

following the papers POLISHCHUK

the idea of approximating

given

[2j,[9j.

functionals of

general 153

type by

polynomials Pn[x]

where

= Xn

+

X » )lxl|2/2

HjCx] X n _ 1 and

classical Weierstrass

H

+ ... • H n [ x 3

are harmonic

theorem) employed

the a u t h o r h a s d e v e l o p e d

.

functionals

the h a r m o n i c a n a l y s i s of

crete and continuous spectra

(see a l s o P O L I S H C H U K

The Laplace and Poisson equations

(the a n a l o g u e

in 5.2, in P O L I S H C H U K functionals

in a f u n c t i o n s p a c e a n d t h e i r

i n s t e a d of

Section 3.3. Another approach w a s p r o p o s e d by S H I L O V NAROQITSKIY

Ll.ll

[l], S I K I R Y A V Y Ï

Elliptic operators

wh

using

the F o u r i e r

the G â t e a u x

to the s a m e p r o b l e m s (see a l s o D O R F M A N

[l] , S O K O L O V S K I Ï

in g e n e r a l

(see a l s o D A L E T S K I Ï and F O M I N

in these s p a c e s .

a n d V I S H I K s t a r t e d from d e f i n i t i o n s

of c o n t i n u a l

by F O M I N

different

their r e s u l t s

f u n c t i o n a l o p e r a t o r s of v a r i o u s

means and their relations

to b o u n d a r y

boundary value problems

a n d B. O E S S E N . The a p p l i c a t i o n of

these

results

lems in a f u n c t i o n s p a c e w a s p e r f o r m e d by the

154

was

to H. S T E I N H A U S

to b o u n d a r y v a l u e

prob-

author.

7

This section was written along [8].

analogy

author.

T h e c o n c e p t i o n of a r e g u l a r m e a s u r e u s e d in 6.3 g o e s back

and

kind

value

of p r o b a b i l i s t i c s o l u t i o n s w i t h c l a s s i c a l

To S e c t i o n

pro-

C143•

by the a u t h o r . The

by the

by

FOMIN,

p r o b l e m s in f u n c t i o n s p a c e s w e r e e x a m i n e d also emphasized

[£] by

Different

essentially

book. Naturally,

in m y p a p e r P O L I S H C H U K

Semigroups

space

[l3,

directions.

between elliptic

6

in the H i l b e r t

[l]. In their p a p e r s D A L E T S K I Ï ,

that c o n s i d e r e d in the p r e s e n t

To S e c t i o n

in-

(3) of

elliptic operators were obtained

BEREZANSKIÏ

have been d i s c u s s e d

formula

[l] ).

[l] and V I S H I K

Some connections

gener-

[.l] ) from an i n t e r e s t i n g p o i n t of v i e w

t r a n s f o r m s of m e a s u r e s

in o t h e r

dis-

u s e d the

[l], K A L I N I N

BEREZANSKIÏ

vide developments

°

linear spaces were considered

r e s u l t s on i n f i n i t e - d i m e n s i o n a l

from

for

[2]).

a l i z a t i o n s w e r e a l s o e x a m i n e d by M . N . F E L L E R D O " M < tegral over a W i e n e r measure

to the

[3],[7]

the l i n e s of

the p a p e r s P O L I S H C H U K

[5]

To Section 8 Boundary value problems for uniform function domains were not considered by other authors.

To Section 9 This section was entirely written on the basis of the papers POLISHCHUK [11.I] and

[ll,Il].

To Sections 10 and 11 The brief explanation of Section 10 was formerly published

in POLISHCHUK

[l2]. Some rings of spherical means and their maximal ideals were studied by SHILOV

[l,Il], The opportunity to employ compact extensions in

the theory of equations with functional derivatives was communicated me by G. SHILOV in the summer of 1968. The idea of applying Steklov's functions unicity theorem

to

t*h ^x(t)

x(u)du

to the

t-h for the considered equations is due to M. FELLER.

In the paper POLISHCHUK

[13] , on the basis of the employment of contin-

ual means, a construction of the continual Cauchy integral for different domains in complex linear spaces was described. Several

results

concerning the theory of infinite-dimensional elliptic operators of P. Lfevy type (closure of operators, their self-adjointness,

application

of the Gauss measure in a Hilbert space) were obtained by FELLER which, being related to the text of the present book, go beyond

[4-7], it.

I also want to indicate some connections of the theory of continual means presented in the present book (generalized Gateaux formula, semigroups, functional elliptic operators) with probabilistic of T. HIDA and his colleagues (HIDA [1,2], HASEGAVA [l])

investigations (generalized

white noise, functionals of Brownian motion, causal calculus). The analysis of these connections would be very desirable.

155

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D y n k i n , E.B. ClJ

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Fomin, S.V.

(®OMHH,

C.B.)

0 HeicoTopbix hobhx npoöjieMax m p e 3 y n t T a T a x HeJiHHefiHoro $thkumoHaJibHoro aHariM3a, ö e c T H W K MIY, c e p . mst., M e x . 2 5 U 9 7 0 ) 2 , 57-65 ( E n g l i s h t r a n s i . : M o s c o w U n i v . M a t h . B u l l . 2 5 (1970) 1 - 2 ,

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81-86).

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R.

[lj

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White noise and Lêvy's functional analysis, Proc. Oberwolfach C o n f . 1 9 7 7 on M e a s u r e T h e o r y A p p l . S t o c h . A n a l . , L e c t . N o t e s M a t h . 6 9 5 (1978) 1 5 5 - 1 6 3 .

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Brownian motion, Springer-Verlag, New 1980 ( T r a n s i , from the J a p a n e s e ) .

Hille, [l]

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Hirschman, [l]

I.I. a n d D . V .

v.v.

transform, Princeton Univ. Press,

(KanMHMH,

Princeton,

di-

B.B.)

L.V.

^

-

and G.P.

Akilov

( K a H T O p O B M H , Jl.B. M T.II. Akm/IOb)

F u n c t i o n a l a n a l y s i s in n o r m e d s p a c e s , M a c m i l l a n , N e w Y o r k ( T r a n s i , from the R u s s i a n ) .

Khinchin, A.Ya.

(XüHIMH,

1964

A.H.)

M a t h e m a t i c a l f o u n d a t i o n s of s t a t i s t i c a l m e c h a n i c s , D o v e r , Y o r k 1 9 4 9 ( T r a n s i , from the R u s s i a n ) .

Kolmogorov, A. Cl]

Widder

/paBHSHMH Jlannaca ;HHX n p o n 3 B o ; i H a x b ra.nfcö6DT0B0M n p o c T p a H C T B e , M s n e c T H H BVSOB. MaTeMaTHKa 3 ( 1 1 6 )

Kantorovich,

[l]

1949.

B.

(1972 7 2 0 - 2 2 .

[l]

Dover, New York

The t h e o r y of i n t e g r a t i o n in a s p a c e of i n f i n i t e n u m b e r of m e n s i o n s , A c t a M a t h . 63 (1934) 2 4 9 - 3 2 3 .

Kalinin,

[1]

analysis and semigroups,

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Jessen, [l]

York-Heidelberg-Berlin

(KoflMoropoB,

New

A.H.)

Grundbegriffe der W a h r s c h e i n l i c h k e i t s r e c h n u n g , S p r i n g e r - V e r l a g , B e r l i n 1 9 3 3 ( E n g l i s h transi.: C h e l s e a P u b l . C o . , N e w Y o r k 1 9 5 0 ) .

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M.

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1971.

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Paris

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159

SUBOECT

INDEX

Borel's lemma boundary of a function domain Brownian movement Carath&odory conditions characteristic equation of a s u r f a c e centre of a f u n c t i o n domain compact e x t e n s i o n of a space compact r e s t r i c t i o n of a functional domain c o m p l e t e l y r e g u l a r space continual means

22 15 74 13 54 33 135 137 135 9

D e r i v a t i v e of the functional H distribution density d o m a i n of the first k i n d d o m a i n of the s e c o n d k i n d

112 24 24 24

Equivalent sequences exterior Dirichlet problem

27 63

Function domain T u n c t i o n of class {A]r functional of c l a s 9 9B fundamental functional fundamental sequence

15 65 32 42 27

GSteaux formula GSteaux f u n c t i o n a l GSteaux ring generalized function g e n e r a t i n g function

21 13 14 27 13

M e a n over the H i l b e r t s p h e r e mean value of a functional mean value w i t h respect to a measure m e a s u r e of a q u a s i i n t e r v a l m i c r o c a n o n i c a l mean monotonic majorant

39 16 129 78 20 49

Normal domain normal measure

15 79

Picard functional point of m e a s u r e c o n c e n tration

14 23

Quasiinterval

78

Random direction "random d i s t a n c e representation principle Riccati functional

75 75 20 151

Schwarz functional star-shaped domain strong f u n c t i o n a l Laplace operator strong h a r m o n i c i t y in 12

151 103 34 40

strong L a p l a c e o p e r a t o r in 1,, 4 0 strongly harmonic functional 34 s u r f a c e of type {s)42, 63 Uniform function domain 15 u n i f o r m i t y of a family 137 u n i f o r m l y c o n v e r g e n t s u r f a c e s 59 u n i f o r m l y dense basis 40 Volterra functional

13

Weak f u n c t i o n a l Laplace operator weak h a r m o n i c i t y in lg

34 40

62

weak Laplace o p e r a t o r weak Laplace o p e r a t o r in

35 40

Oessen measure

79

weakly harmonic functional

K-property

79

Law of d i s t r i b u t i o n L & v y ' s theorem

24 19

H a r m o n i c c o n t r o l system Harmonic functional 32,

128 138

¿nitial functional inverse p r o b l e m of tential theory

115

160

po-

34,35

Einführung in die klassische Mathematik I Vom quadratischen Reziprozitätsgesetz bis zum Uniformisierungssatz von Helmut Koch

Mit 25 Abbildungen Die Grundidee des Buches besteht darin, Ergebnisse der M a t h e m a t i k im Geist ihrer Entstehungszeit darzustellen. Die Vorteile einer solchen Darstellung sind neben dem Gewinn der historischen Dimension das direkte Vordringen zum Wesentlichen ohne den Ballast vieler Kapitel an Vorbereitungen, der gewöhnlich moderne Lehrbücher der Mathematik charakterisiert, sowie die direkte Motivierung des Lesers durch die Hauptproblemstellungen in dem jeweils b e t r a c h t e t e n historischen Moment, die am Anfang von Kapiteln und teilweise auch von Abschnitten dieses Buches erklärt werden. Die Nachteile einer solchen historischen Darstellung liegen ebenfalls auf der Hand. Die F o r m der ursprünglichen Darstellung weicht o f t so weit von den heutigen mathematischen Denkgewohnheiten ab, d a ß ein zusätzlicher Aufwand f ü r das Verständnis des Stoffes erforderlich ist, der n u r bei historischer F o r s c h u n g gerechtfertigt erscheint, die in diesem Buch aber nicht beabsichtigt ist. Der F o r t s c h r i t t in der Mathematik besteht auch in der Vereinfachung ursprünglich kompliziert erscheinender Ergebnisse, indem m a n sie in den ihnen a d ä q u a t e n R a h m e n stellt (der d a n n jedoch o f t den oben genannten Ballast an Vorbereitungen erfordert). Der Ausweg aus dieser Situation, den wir in diesem Buch gegangen sind, besteht darin, d a ß wir uns grundsätzlich der heutigen mathematischen Sprache bedienen und an einigen Stellen Beweise zurückstellen, bis sie zu einem später im Buch zu behandelndem historischen Z e i t p u n k t durch Einbringung wesentlich neuer Ideen die heutige Einfachheit gewonnen haben.

B e s t e l l n u m m e r : 763 4 7 3 0 B e s t e l l w o r t : Koch, M a t h e matik 6 9 0 3 / 1 DDR 43,00 M Akademie-Verlag Berlin 1986

RANDOM M E A S U R E S Olav Kallenberg Department of Mathematics Chalmers University of Technology and University of Göteborg

My aim in writing this book has been to give a systematic account of those parts of random measure theory which do not require any particular order or metric structure of the state space. The main applications are of course to random measures on Euclidean spaces, but since most proofs apply without changes to the case of arbitrary locally compact second countable Hausdorff spaces, I have chosen to work throughout within this more general framework. B y a random measure on a topological space © is meant a measurable mapping from some abstract probability space into the space 9K of locally finite measures ¡1 on 5 , where the c-field in 2Jt is taken to be the one generated by the mappings fi -> fiB for arbitrary Borel sets B in The most convenient way of treating simple point processes on