Continual Means and Boundary Value Problems in Function Spaces [Reprint 2021 ed.] 9783112471746, 9783112471739

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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 D D R 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

Autorj Prof. Dr. Efira Mihailovich Polishchuk, Leningrad Bearbeiter der englischen

Fassung:

Dr. Bernd Luderer Technische Universität

Karl-Marx-Stadt

Sektion Mathematik

Die Titel dieser Reihe werden vom Originalmanuskript Autoren

der

reproduziert.

ISBN 3-05-500512-0 ISSU 0138-3019 Erschienen im Akademie-Verlag

Berlin,

Leipziger Straße 3-4, DDR 1086 Berlin (C) Akademie-Verlag Berlin 1988 Lizenznummer: 202

•

100/504/88

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

(2182/44)

FOREWORD

The fate9 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 Gâteaux and was newly thought over and taken as the basis of a prospective theory by Paul Levy. Paul Lêvy, 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 hi6 ideas and results. Together with A.N. Kolmogo rov, A.Ya. 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 and scope of the problems

posed.

His expressive style, rich in ideas rather than technically

polished,

which sometimes led to a lack of clarity at first, will be

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. Lêvy, N. Wiener 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. Polishchuk, is the development of Lêvy'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, "Lêvy's Laplacian". P. Lêvy 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 Lêvy 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 Lfcvy-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

physlós,

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. Lfivy 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 L&vy.

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

6

CONTEN TB

INTRODUCTION

9

CHAPTER 1. FUNCTIONAL CLASSES AND FUNCTION DOMAINS. MEAN VALUES. HARMONICITY AND THE LAPLACE OPERATOR IN FUNCTION SPACES

13

1. Functional classes

13

2. Function domains

15

2.1. Uniform domains

15

2.2. Normal domains

15

3. Continual means

16

3.1. The mean over a uniform domain 3.2. The mean value

m

„f a, R and its main properties

16

over the Hilbert sphere r 18

3.3. The spherical mean of a Gâteaux functional

21

3.4. Functionals as rar\dom variables

22

3.5. The Dirac measure in a function space. The centre of a function domain. Harmonicity

23

4. The functional Laplace operator

33

4.1. Definitions and properties

33

4.2. Spherical means and the Laplace operator in the Hilbert co-ordinate space

12

39

CHAPTER 2. THE LAPLACE AND POISSON EQUATIONS FOR A NORMAL DOMAIN 5. Boundary value problems for a normal domain with

boundary

values on the Gâteaux ring

41

5.1. Functional Laplace and Poisson equations 5.2. The fundamental,functional of a surface

41

41 S

52

5.3. Examples

60

5.4. The maximum principle -and uniqueness of solutions

62

5.5. The exterior Dirichlet problem

63

5.6. The deviation

68

H - F

6. S e m i g r o u p s

of c o n t i n u a l

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

means. Relations

to the

boundary value problems.

probability Applications

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

'Jfll8, H L

boundary value problems

in the s p a c e

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

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

70

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

the G S t e a u x

LAPLACE OPERATOR AND CLASSICAL

EQUATIONS. BOUNDARY

of 73

ring

DIFFUSION

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 parallelism

Laplacian and

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

7.1. The functional

77

85

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. Functional and classical Dirichlet problems 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

113

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

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

systems

114

9.1. Normal control domain

115

9.2. Uniform control domain

124

CHAPTER 4. GENERAL ELLIPTIC FUNCTIONAL OPERATORS 10. The D i r i c h l e t p r o b l e m

ON F U N C T I O N A L RINGS

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

functions

and related topics

131

1 0 . 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 10.2. Compact extensions 10.3. Averaging

M^/j-F

to a f a m i l y of

131

type

131

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 of a f u n c t i o n a l FtR with respect

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

1 5 1

REFERENCES

15

S U B O 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

trüben Blick

gezeigt.

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

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

boundary value problems

dem

Gestalten,

Zueignung)

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

dimensional spaces,

schwankende

in

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 ,

tions c o n s i d e r e d

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 functionals

x(t)

of

F[x(t)]

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

(t),...,x

whose arguments

q = a< t

TRl

20

a,pk

(4)

to

F

=

together with

expresses

to S l u t s k i i ' s

function

F^

the From

theorem

i=l,...,m,

r(Fj^,...,F

)

exist

such

equation

Fm' )

from w h i c h ,

m

then

in p r o b a b i l i t y .

(see C R A M E R [l3, p. 2 5 5 ) . If

then for a n y r a t i o n a l

holds almost everywhere,

Relation

is s a t i s f i e d a l m o s t e v e r y w h e r e ,

following proposition analogous

on s t o c h a s t i c l i m i t s that

"Fn

d e f i n e d in 3 , 2 . 2 c o n v e r g e s

this we get the and

F

a,R

F

l

in p a r t i c u l a r , w e

W

a , R

F

m

obtain

•

the e v i d e n t a d d i t i v i t y of

the ring p r o p e r t y of the s y m b o l

TO

the

operation

p • a t r\

(ili)

For

convenience

Lemma Si.

3.2.1. n

and

F[X,U]

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

Let

^Cxl

having

exist

the

and

U

we

formulate

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

another

bounded

form

on

proposition.

the

sphere

. Moreover,

be u n i f o r m l y

continuous

on

let

the s p h e r e

U

and

S I .

.

a

Then F [x,u[x]] In

fact,

by L e v y ' s

F[X , U J

3.3. Let

almost

theorem,

everywhere.

The spherical in

F

the s p a c e

=

5

= F [ x, U ]

g

we

.

(5)

have

U=U

In v i e w of

m e a n of a G a t e a u x Lg

2

(q)

t±)

a.e. But

(i),

this

tm)dtm

=

g(X(t),t)dtm

j

(1)

Q

be g i v e n . g ( E

If i t s

.t)

satisfies |g(Z

generating

= g(ti

the

function

Cm:t

tm)

1

U(T)

,t)|
0,

F

( s / b ^ T ^ R-/ 25C

=

.

£

i=

(

Sl8).

condition ICJ

U £ L ( Q_), B=cons t ^ O x s a t i s f i e s t h e e~m' quation

and

=

(5).

functional

Q

w h e re

Ftx.uCxj] equation

functional

the G S t e a u x

x(tm) ; t1

then

implies

: . . W * t

5

J _ m

dt

m

y

2

, then

J _ms

(2)

. the m e a n

m

g f a j f t ^ R l n

-.(t.)^?«.«,

l m

value

f a ,0K

exists

»ait!)*!»?^.

t.),Xp(-'C-)('^ 2R

1 2

'

g j 2 )

(3) with

d S

=

. * li

We would which

like

sm

to n o t e

is a s s u m e d

that

and sufficient

condition

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

if

(LEVY [ 2 3 ,

(2), t o g e t h e r w i t h

to be s a t i s f i e d

111,18).

s=l,

for

throughout

the c o n t i n u i t y

q=(0,l)

The general

, then formula

(3)

the C a r a t h e o d o r y

condition,

the b o o k ,

is a

of

the w h o l e

(1)

provides

(3) c a n

in

necessary

the G | t e a u x

be p r o v e d

in

space formulg

different 21

w a y s and is a c o n s e q u e n c e of the following Borel's lemma. Let let

M ZH

be the sphere

N

M»(

proposition. = NR2

in

EN

, and

be a random point on it (randomly in the sense

of the uniform d i s t r i b u t i o n on this sphere). Then, for

N

+00 ,

all its co-ordinates are a s y m p t o t i c a l l y normal with p a r a m e t e r s and, for an a r b i t r a r y choice of ^

are a s y m p t o t i c a l l y

and fixed

p ,

(0,R)

^

independent.

P This implies

that, if

N

H

on the sphere

^^

n o ,

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

*

as

Y

ln

P

totica

^y

lying

normal w i t h p a r a -

p

meters

(0. ) and a s y m p t o t i c a l l y i n d e p e n d e n t . Taking into account -/s(b-a)' this fact and arguing as in T h e o r e m 3.1.1, we o b t a i n (3).

In the same way formula validity in the space , „ I l g ( E . t ) | exp(

(3) may be e s t a b l i s h e d for other spaces. For its CgCql

s(b-a)( 2R

be summable over the set The comparison of e q u a t i o n

E

ms

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

--± xQ

I2 + . . . +

ms

that

|2 2—)

.

(3) and the d e f i n i t i o n of a v e r a g e s of sta-

tistical m e c h a n i c s shows that in our c o n s i d e r a t i o n s of the canonical

(3) plays the

role

mean < f>

m i c r o c a n o n i c a l mean

, and

7H

RF

is an analogue

to the

f . For the most important phase functions

f

in

statistical mechanics, the estimate I f - < f > I = 0 (1/N) holds, w h e r e

N

is the d i m e n s i o n of the phase space. Of course, in our

case both averages

coincide.

3.4. F u n c t i o n a l s as random Comparing

formulae

variables

(1) from 3.1 and (3) from 3.3, we see that they are

built up along the same lines. D e n o t i n g

the left parts by

MF , we can

write these formulae in the form MF = J ...J d t l . . . d t m E g ( 0 Q where

0

t

9 1

t

it,

.

are v e c t o r s independent of each other, and

symbol of m a t h e m a t i c a l e x p e c t a t i o n . F u r t h e r m o r e ,

22

tj

(1)

m E

is the

for the case

(1)

from

3.1,

8

are u n i f o r m l y d i s t r i b u t e d over the d o m a i n

t 1

2I] t and, t . i 1

for

O t? t have a n o r m a l d i s t r i b u t i o n law w i t h indethe centre a.. (t.),..., a (t.) and the d i s p e r s i o n

formula (3) of 3.3, pendent components,

s(b-a)' ' In both c a s e s we have M(C1F1+C2F2) and

additivity C

1MF1

+

C

2MF 2

multiplicativity MtFj^)

= MF1MF2

(2)

.

If

Fn F over the c o n s i d e r e d d o m a i n and if there e x i s t s also MF and we have

MF n

exists

for any'

n ,

MF = lim MF„ . n n Thus, p a s s i n g

to the limit, we can c a l c u l a t e

Gâteaux

ring.

Setting

P^®

as

, where

2 DF=0

F

2

in

' D

w e

9et

for all

F

> w h i c h can also be w r i t t e n

F , c o n s i d e r e d as a random v a l u e , (cf. L e v y ' s theorem

the d i s c u s s i o n of the fact just

3.2).

function

Harmonicity

3 . 5 . 1 . Let

Ç

mal d o m a i n

D , and let

be some set of f u n c t i o n a l s d e f i n e d on a u n i f o r m or n o r -

for each

F € J" , we have

F

xQ

MF

be the mean v a l u e of MF = f [ x q ]

, where

xQ

Ft

1

?

over

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

x

€0

o function,

or

x

i.e.

o

b e l o n 3q s xq€CdD

D

in

to some e x t e n s i o n of , where

. If,

does not d e p e n d on

is said to be a goint of m e a s u r e c o n c e n t r a t i o n

this case, e i t h e r e x t e n s i o n of

is

from

mentioned.

3.5. The D i r a c m e a s u r e in a function s p a c e . The centre of a

, then

the

is the d i s p e r s i o n s y m b o l . -

equal to its mean value almost e v e r y w h e r e

domain.

from

2 * (MF)

This s h o w s that the f u n c t i o n a l

Let us c o n t i n u e

MF

C*3

D D

. In v(

x o means a compact

D ).

L a t e r on w e shall see that the e x i s t e n c e of p o i n t s of m e a s u r e

concen-

tration is a c h a r a c t e r i s t i c p r o p e r t y of the c o n s i d e r e d a v e r a g i n g

pro-

cedure. W e return to formula M F = J...J d t m D

(1) of 3.4 and rewrite it in the form

J...5 9 ( 5 ,

tm)p(I1;t1)...p(fm;tm)df

(1)

23

The

term

sity) of If

p

p("|,t)

is c a l l e d

the d o m a i n

D

with

d o e s not d e p e n d on

a d o m a i n of

the law of d i s t r i b u t i o n respect

t , i.e.

ÏÏSiO 2Î Ît?5 S S 2 2 D Î k i n d . T h u s ,

Formulae

are d o m a i n s of (1)

from 3.1 a n d

n e r a l f o r m u l a of

the beam

in a d d i t i o n

distribution

functional

density

Note

that

formula

Thus,

form

the g e n e r a l

(1). The

(1),assuming

. We suppose



Lemma 5.1.1. y

such

that

that on a n y , H = H [x]

X ,A

true.

sphere

x+ A y

lies

lying

in the d o m a i n

satisfied.

F o r any f i x e d v a l u e s of

l s [ x 4 y ] holds

is

x

in the d o m a i n

= V s 2 C x ] - X2'

and V , the

and for formula

variable

V

Proof . O w i n g

s £x] = - / P C*1 -II xll 2

to the f o r m u l a

s2£x+ A y ] =

r[*+*y]

-

II x + A y II 2

=

r[x+Ay]

-

||x||2 - 2 * < x , y )

Because

I""

tional

is a h a r m o n i c

(""[x+Ay] mr[x

+

xy] = rt>]

The s c a l a r p r o d u c t with

respect

Finally, Wls

2

to

taking

= ( f s [

. m

(s[x + "Xy] V 2 F )

L e m m a 5 . 1 , 2 . The ?0lH[x+Ay]

x

the

func-

y . Thus

"3Jt(x,y)

2

= rW-Hx||

2

-X

functional

= 0 .

"33111 yll2 = 3JI/1 = 1 , w e

that ] )

y

thus, a h a r m o n i c

= (x,0)

2

=

obtain

S2[x]-X2

that

, -1

-

a

to

proved.

(6) it f o l l o w s

" [ » y -

X

II VII2 .

x , for f i x e d respect

is a l i n e a r and,

. Therefore

into account

i>Ay]

2

write

.

(x,y)

y

and the l e m m a is From

f u n c t i o n a l of

w i l l be h a r m o n i c w i t h

- \

, we c a n

dt"1 J g ( | , t ) e x P ( -

I 0 "fTl

following

IS -x(t)-*Y(t)I

2

}

, g m^

(?)

2s[x+Ay;r

relation

-

holds:

dtl...dtm ^

J...J g d ,

tj-

Qm I I exp k=l

(

2 S^

g

)d J j . . , d | m

,

(8)

w h e re s

= 'SUlsCx.'Xy] = - / s 2 [ x ] - X 2 '

a

Proof. We shall fixed)

first s h o w

the f u n c t i o n a l

In fact,

as

for e v e r y

V

fixed

that

for

Htx+Ay]

is

is a d o m a i n , x

then,

the f u n c t i o n a l

x€V

Replacing functional

44

(7), this e n s u r e s in

(7)

and

x+^y€V

(here

x

is

in v i e w of P

Cy]

the d e f i n i t i o n

has a l o w e r b o u n d

x+ A y = x ,

and

of

J^Cyil .

different

denoting

obtain

s [ x + A y ] = six,] = In v i e w of

(9)

bounded.

from z e r o on the u n i t s p h e r e . P u t t i n g inf P C y ] = m , we lly( = l J x

.

s[x+Ay]

[ y ] >/ m > 0 the b o u n d e d n e s s by the n u m b e r

. of s^

H[xt^y] . (see

(9)), w e get

the

H

(

* =

) m

iT77p

S

^

which is also bounded when is equal that

to

H^

s^

l«-x(t)-^2Y(t)l2)df-

i 9(f.t,.xp(-

x

and

x+^y

lie in

almost everywhere, comparing

is equal

to

H[x+ X y ]

V . Since

(7) and

(10)

,

s£x+Ay]

(10), we

observe

almost everywhere. Consequently,

means of these functionals over the sphere

llyll=l

the

are equal to each

other. Thus the lemma is proved. We shall now apply the Gâteaux W"D«*Xy]

= (

i

-^

)

5 p

m

(-

T 7

|

F

formula once more to equation m

)

J

(10):

«•«•J.-.J

tm>

Q

n e t»1

x

p




d s«

)

^

d ^ . . ^

this expression by applying C

1

j

2

2 exp

—

K

? * x

the well-known

relation

}

T ^

=

1 ; .. •fzie ( & i *

mn[x

+

Ay]

, .. i exp()

1 = ( V 2 7T («X • -

f =

2(6'12+&'22)

,)m (...f d t m f ... Ç g ( | 1 J Q „ J J Jav?l' / -

J21

, -e

2 (9),

2

s^ + ^

I 1

t ). m

... , ,2

2(sA2+

i=l

By virtue of

2

X2)

2 =s [x] . Therefore

mHLxf^y] = J-, m (sV2i?)m

J

J

dt m J g ( | , t ) exp(J

< g - x | t ) ) 2 ) dtm 2s

which proves the first part of the It still has to be proved

that

lim x—

does not contain

t

explicitly,

.

for

z£S

theorem. H [ X ] = F[z]

plicity we shall suppose to start with that g

= HCx]

m=l

. For sim-

and that the

function

i.e.

45

F =

J

g(x(t))dt

0 Then

1

H[X]

In

the

z

in

some

= —

case the

under mean,

sequence

quence

also

sequence

for

any

of

g(z(t))1>

.

From

has

also

under

=

Thus,

the

%

(see

II x - z H — • O

in m e a s u r e x

functions

been

converges

bounded.

sLx]—>-0

that

n

said

it

in m e a s u r e , using

integral

and

Lebesgue's

sign

in

LxnD

that

5 g(xn(t) + | 9 [ x n l )

the

/2)

from as

df

.

that

x

5.1.1).

s|~x n 3

(2) w e

to be

this to

in m e a s u r e

to

in m e a s u r e

se-

zero, z(t) to

sequence

see

we

xn

tends

converges

(-f2/2)

theorem,

tends

Let

is w e l l - k n o w n ,

converges

the

exp

from

2

. Because

g(xn+fs£xn^)

follows

f

signifies (5°)

z s

(-

. Then,

to

( * )+ ?

. Hence

exp

x — * • 7.

consideration,

of

(t)

n

\ g(x(t)+|s[x])

where

such

value

f

dt

converges

the

what

\

^

df

that

may

this

proceed

sequence to

the

is

limit

equation

1 H[X ] and

= — J -V 2 *

dt j

(-|2/2)

g ( x n ( t )+ | s [ x n ] ) e x p

df

0

obtain 1 lira H [ x n :

which

was

to be

In c o m p l e t e

= — J •V^F

\ exp

2

(-|

/2)df

= F [z] ,

(11)

0

proved.

analogy

lira H [ x n l n

9(z(t))dt

with

this"result

we

may

prove

that

the

relation

= 2

n

n

"

-«"J

r L

Q is v a l i d 5,1.3.

for

The

the

functionals

ring

R(V)

also

F ' F " £ R(V)

In

the

TT 'This

46

: If

x

.

s

F

follows

=

(1)

in

considered

F',F""£R(V)

following m

functional

the in

, then

case

the

m>l

the

d

%

m

-

F

i

z

i

nJ

preceding

besides

)

,

too. theorem

additivity,

generate we

have

. we m

from

denote s

F .

the c o n t i n u i t y

of

the

superposition

operator.

a

Let

H= 3!l S F . F r o m

lows

that

the m u l t i p l i c a t i v i t y

"J(ts (F ' F" ) = Therefore, SE

by

'®lSF"mSF" ?0L S F

the clas.s

(V)

. We

introduce

II Fll = sup I F= 1 , V Let

R(V)

3£> 0 (V)

with

regard

tinuous with set

o£0(v)

and

from

R

into

R(V)

. respectively,

from

R

H£R, of

T h u s we are able

to

formulate

V

5 . 1 . 1 ' . Let

functional

H

A U + F[x] the L a p l a c e =

is

the u n i f o r m

be

the c l o s u r e

of

the o p e r a t o r

denoted norms

R(V) "JJtS

(uniform convergence can be e x t e n d e d

2

and is

con-

on

the

to a

mapping

coincide.

. Then

7Bl S

the o p e r a t o r

the d o m a i n

V

with

the f u n c t i o n a l P o i 9 s o n

defines

boundary values

equation (13)

operator m u [ x

a F ,

= 0 ,

l i m

the

rings

F€R(V)

A +

is d e f i n e d

^ y ] - u M ' X

Vie i n t r o d u c e

(V)

.

Q

both

h a r m o n i c on

5 . 1 . 4 . Let us c o n s i d e r

A U

SB

into

F£ W

S

where

and

|l H II = sup H . V

On the b o u n d a r y

Theorem

fol-

§F : o

ZXi S F,

H =

a l s o g e n e r a t e s a ring, w h i c h

on

it

(12)

to the u n i f o r m n o r m

V ), the m a p p i n g

"9S,s

the o p e r a t o r

.

to these n o r m s . S i n c e

respect

of

via

#

2

notation

s M Hi F =

$ 0

Theorem 5.1.2. where

S

rMlF[x+ry] Let

F[x]

is a s u r f a c e of

conditions

.

(15)

be a f u n c t i o n a l type

{SJ-

of T h e o r e m 5 . 1 . 1 . Then,

(15) e x i s t s a n d s a t i s f i e s ATHF

dr

the

d e f i n e d on

, and let

for a n y

F

the set

V=VuS,

s a t i s f y all

x€V

, the

and

g

the

functional

equation

+ F = 0 .

Proof. We start with the v a r i a b l e

t

the s i m p l e

explicitly.

case,

Thus,

if

m= l

d o e s not

contain

let

A

F =

J

g(x(t))dt .

47

Forming

"¡Lf , we

obtain

SDO 7SL F =

i

^

r2flll$g(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

sCx] Hf

This

= — \ "Z2*

r i

J V g(x(t) + r| 1

dt

be d e n o t e d by

=

J

USjJ

J

r

w h e r e , as a b o v e ,

the proof

W

Applying

Jd o

t

s - =-\%

(16)

. Then

. (17)

holds:

Jg(x(t) +r |

M

- X

+

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

,

(18)

2

this lemma is not g i v e n ,

of L e m m a

.

0

0

The proof of

u[x]

following equation

—

) e x p ( - | 2 / 2 ) d f dr

r j d t J g ( x ( t ) + r £ • X y ( t ) ) e x p ( - | 2 / 2 ) d fdr

0 L e m m a 5 . 1 . 3 . The

get

o

functional will

u[x+Xy]

.

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

the s a m e

as

5.1.2.

the G a t e a u x

formula

to

(18), we

obtain

2&u[x+ Ay] =

Since

o0

L o J

exp(

1

we

_

1

( V * ( * ? - oe)2)

exp(-

/ i ?

- / V ^

fu[x

t

.

0 l

2

/ 2

^2

) d o t

_

ll^LLUlf) 2(X2+ r2)

5

'

have

- J Vise

Xy]

=

i d t f ^ S e x p , - ! ! ^ ) J J V ^ + r * J 2 (X + r )

a n d a f t e r a c h a n g e of v a r i a b l e s

.

In our c a s e

(see

(16))

(X

2

J dt

j d/s j

A

SC*]

fld^jdiL L

,

U

2 2 +r =J3 ), vve get

c

1

48

e x p (

2

r

exp(-

)g(n )dij .

SW

u[x] » 31F -

o and,

-t

2

j djl j dt j

)exp(-

o

'

therefore, $ l u [ x + A y ] - U[x] -

I0 dt

- v f c

J ^ 0

g (

J

T

5

exp(

-

•

/

Consequently ?iftu[x+Ay] - u[x]

In order to justify the limiting procedure as

A—»•O

, we

introduce

the notation

0 A . the function

For any self

(the function

'

is

ipf^.x)

the function y

monotonic majorant for it-

is said to be a monotonic majorant of I Cf>( ^ , x ) I

if

x , the function

a

x

increases for

y i>x In fact, the function

x 2

exp(-

an arbitrary value of

)

)

and

sesses this property for every

A

possesses this property for ,x

)

x

- p

)

d

n

-

-

x

r

2

^0 W

i

e x p (

ex P (-oc 2 /2)y3 dot

J d/i J

also pos-

• On the other hand, A

j W v ^

fixed

and decreases in case

. Hence the function

Ji

for

-

{

\ p

t ] )

) d

r

- 1 .

Consequently, by the theorem about a monotonic majorant (see NATANSON fl], Chap. 10), for almost all values of X±mo

^

t

(0N

9 (

U

we can select a number

ft.(1'x(t))d,l

I


1

, analogous

rea-

equation

v - - M " - 5

9

(

1

u

V ' I

v -

„ 2 . n

N o w we i n t r o d u c e

2

the

r

,

S

the f u n c t i o n

cube

Qm

lim

i

to s e e

m;xl

x

_

tk"x(tk))

, ' n i -

t€Q

\m' tl

d

1 .

-

*«>

1

•

is s u m m a b l e

over

the

m

x

'J fVll

^ m ^ l

By v i r t u e of L e b e s g u e ' s

AU

(1

g(x(^),...,x(tm);tj,...,tm)

J . . . J g(tli

sign, w e

H

that

, w e h a v e for a l m o s t a l l

" S^t«!)

dfi

•

4 V V

Since

exP(-

n

x

v

It is n o t d i f f i c u l t

,

notation

C n-(m-l), ^2 p-^^-n 0 .

lfl(1l where

A,a>0,

Under these conditions, ed on any ball

llxll < R

F

1

*

1

• b(t) .

is continuous on the whole

C[q]

and

bound-

. The means described above exist, and all the

lemmas used above remain valid. Finally, it is not hard to understand that the boundary conditions are also

fulfilled.

5.2. The fundamental functional of a surface

S

Although

the formulae obtained which yield solutions of the functional 2 Dirichlet problem and Poisson's equation in the space L were written above in an explicit form but without knowledge of the functional

s£xj

(i.e., in essence, without knowledge of

to

P [ x ] ), it is impossible

regard these equations as completely solved. Meanwhile, until now the only values of

s[x]

that have been studied have been those for the

ellipsoid and, more particularly,

for the sphere. This gap will be

filled up now, and a general method will be demonstrated finding S

the functional

s , starting

below for

from the equation of the surface

. In particular, it will be shown that both of the problems

ered are effectively solvable in the space

L

if

S

is an

consid-

"algebraic"

surface of any order. The ideas are contained

in the following. First of all, we establish

the validity of conditions when the

U

(4°) and

(5°) from Section 5.1 for the case

which appears in the equation

is a polynomial

in

it will be proved

llxll

u[x]=0

of the surface

with harmonic coefficients. At the same

that the problem of finding

the functional

by reduces to the solution of an algebraic equation a general formula is derived

(for any

s

S time,

there-

x ). Then

that can be used to approximate

function-

als by similar polynomials. The fact that this approximation

is possi-

ble enables us to verify conditions dental"

(4°) and

(5°) even for "transcen-

surfaces.

5.2.1. Let again

S

part of the space

2 L

be a closed convex surface situated , and let

assume that the functionals

V

be the set of its inner points. We

§xCy]

and

s[x]

have the same

as in 5.1 and that they satisfy the same conditions ditions

(4°) and

We shall use

V

(5°), which will be proved to denote any domain in

perty that, for arbitrary x+s[x]y

belongs to

x

and

V . Because

in a finite

y ( x£V

L

2

meanings

(apart from

con-

here). which possesses the proand

HyM>l

| s[x] |