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English Pages 164 Year 1989
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
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(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] |