SEMINARS IN MATHEMATICS V. A. Steklov Mathematical Institute, Leningrad Volume 3
POTENTIAL THEORY AND FUNCTION THEORY FOR IRREGULAR REGIONS Yu. D. Burago and V. G. Maz'ya
Translated from Russian
@coNSULTANTS BUREAU· NEW YORK· 1969
The original Russian text was published in Leningrad in 1967 by offset reproduction of manuscript. The hand-written symbols have been retained in this English edition.
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Library of Congress Catalog Card Number 69-12504 ('!) 1969 Consultants Bureau A Division of Plenum Publishing Corporation 227 West 17 Street, New York, N.Y. 10011 All rights reserved
No part of this publication may be reproduced in any form without written permission from the publisher Printed in the United States of America
FOREWORD This monograph consists of two independent parts which are related to a certain extent in that the methods applied are very similar. The first part is devoted to potential theory in regions with irregular boundaries, and the second, to the study of the dependence on region geometry of the properties of functions whose derivatives are measures.
CONTENTS Part 1 Multivariate Potential Theory and the Solution of Boundary Value Problems for Regions with Irregular Boundaries Introduction
3
Chapter 1. P r o p e r t i e s of t h e C 1 ass of S e t s § 1. Definition and Certain Properties of a Solid § 2. Properties of Sets Satisfying Condition (A). § 3. Properties of Sets Satisfying Condition (B).
B e in g Angle. . ••. . . . . . . . . .
C on s i d e r . •. •. . . • ••. . . . •• . •. . . . ••
e d. •. . . •• . . .
. . . .
. . . .
. . • •
. . . . . • •.
. . . .
. . • •
. . . •
. . • .
6 6 10 14
Chapter 2. Potentials and the Solution of Boundary Value Problems.... § 4. Integral Equations of Boundary Value Problems . . . • . . . . . • • . • . • • • . • . • . • § 5. On the Continuity of the Simple-Layer Potential Generated by the Solution of the Equation :A T•Cf' =0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 6. Fredholm Radius of Operator T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 7. Solvability and Uniqueness. • . • . . . . . • . . . . . . . . . . . . . • . . . . . . . . . . . . . .
17 17
Appendix. On the Approximation of a Solid Angle • . . . . • . . . • . • • • . . . . . .
32
Literature Cited. • • . • • . . • . . • . . . . . . • . . • . . . . . . . . . . . • . . . . . . . . . . . . . .
39
cp-
21 26 28
Part 2 On the Space of Functions Whose Derivatives are Measures Introduction.................................................... § 1.
.......
§ 2. § 3. § 4.
Properties of the Set Perimeter and of Functions from BV(Q). . . . . . . On the Continuation of Functions from BV(Q) onto the Whole Space. . Certain Exact Constants for Convex Regions. . • • . . . . . . . . • • • • • • . The Rough Trace and Certain Integral Inequalities. . . . . . . • . • . . . • . § 5. The Trace of Functions from &V(Q) on the Boundary . . . . . . . . . • .
. . . .
43
. . . .
43 45 50 55 60
Literature C i t e d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
vii
. • •. . . . .
. . •. . . . .
. . . .
PART 1
MOLTIV ARIA TE POTENTIAL THEORY AND THE SOLUTION OF BOUNDARY VALUE PROBLEMS FOR REGIONS WITH IRREGULAR BOUNDARIES
INTRODUCTION In recent years the irregular theory of potentials of simple and double layers have been explored in [1-7]. The solvability of the Dirichlet and the Neumann problems for a wide class of three-dimensional and planar regions has been established by the methods of potential theory in [1-3]. Independently, J. Kral [4-6] has made a detailed study of double-layer potentials and has solved the Dirichlet problem for a planar region. In (7] the same author has announced certain results concerning the theory of double-layer potentials in Q". The present monograph is devoted to potential theory and to the solution of the Dirichlet and the Neumann problems for regions with irregular boundaries under assumptions which in a certain sense are the best possible. By the same token, the theory constructed in [1-3] is extended and refined in various directions. For the reader's convenience the presentation here has been made independent of previous publications, sometimes at the expense of duplicating certain arguments. We assume everywhere that the dimension of the space is n > 2. ; in the case of n = 2. the arguments merely become more simple. The study of the integral equations of potential theory is carried out in the classical way. In this connection, we follow mainly the articles by Radon [8, 9] which were the starting point for [1-3] and for the present volume. Just as in (1-7], the class of sets E c a certain set function c.u the point p ." E
(.?,'B)
R"
being considered here are characterized in terms of
namely, "the solid angle under which the set ~ (\
Boundary value problems are posed inside and outside of the set E.. equations of potential theory under the following assumption:
Cl
E is visible from
They reduce to the integral (A)
for all points p E R" \ 11 E. •* Condition (A) is necessary and sufficient for the operator T occurring in the integral equation of the Dirichlet problem to be bounded in C ( 'Zl E. } • It is also necessary and sufficient for the existence of the limit values of the potential of a double layer with an arbitrary continuous density. The solvability of the integral equations obtained is established under the following stricter requirement on E :
where
I ... (f)={":~t:'f-1'"-~'t.}
, and
c.u,. is the surface area of the unit sphere in R".
Condition (B) sig-
nifies that the Fredholm radius of operator T is greater than unity. We give simple examples which clarify the geometric meanings of conditions (A) and (B). Exam p 1 e 1 .
Consider the curve L shown in Fig. 1. We set ~ ... .,. w. 2._,., t,,:= 2-", \,,...,. .(1 ?.. ->< L satisfies conditions (A) and (B). However, the variation of
It is not difficult to see that the curve
*The solid angle c..>E (
r·
~)
is defined below, in
§ 1.
3
•
4
MULTIVARIATE POTENTIAL THEORY AND THE SOLUTION OF BOUNDARY VALUE PROBLEMS
rotation of curve L is infinite. The curve L is smooth, but the Lyapunov condition is not fulfilled for it. If we se t o..,.. .. K ·• , 0"" = K -~ , In,.. =K -~ , th en the curve L sa tis f ies t h e Lyapunov condi t'wn but has an infinite variation of rotation. Example 2. The curves shown in Fig. 2 satisfy condition (A) but do not satisfy condition (B). The curve shown in Fig. 1 has the very same property when o..... "" K- 4 C.,..= K-~, \,,..., K -~
"=
Example 3 . When 2. , a region bounded by a curve with a finite variation of rotation sa tisfies condition (A) (see [9]). In this connection it is not without interest to note that when n'>2 there exist regions bounded by smooth surfaces with finite area of spherical representation, for which condition (A) is not fulfilled. (This remark is due to A. P. Shablinskii.) As an example, we can consider the surface obtained by rotating the curve shown in Fig. 1 around the axis th, Here we should set o.. .. ~~-K ·K 1 •'\ ·IC t<. > t ..... 2 , n... ~K ~ . EXample 4 . and (B) if ~ normal to
0
A region in
f '-' (~) d~ <. oo r at the point
R"
, where
'-l
bounded by a surface
r
of class
c
I
satisfies conditions (A)
(~) is the modulus of continuity of the vector-function
11(';1:.),
the
llt..
We remark that in [1-7] the solid angle c.>£ was defined for regions bounded by surfaces (or curves) and, moreover, these definitions were based on some multiplicity function or other; in the present monograph, w1 is defined as a functional on K (R"), bounded in C ('a E). The solid angle introduced in this manner is closely related with the Caccioppoli-De Giorgi perimeter [10-13]. This approach has allowed us to consider a wider class of regions (not necessarily bounded by triangulable manifolds). Along with this, it seems to us that it has made it possible to simplify the presentation. For the classes of regions considered in [1-7], our definition of wE is equivalent to the definitions given in the references cited. This is easily shown by using the results of [12-16]. This part of the volume consists of two chapters and an appendix. In the first chapter we study the charge "'e
E is a fixed subset of the n-dimensional Euclidean space
R" ; 'B is an arbitrary subset, and G is an open subset in R" ; ~ B is the boundary of .B ; CI)= = R" \ ~; BJ and i.nt B are the closure and the interior of I) . We shall take it that all the sets being considered are Borel sets. In what follows, ~~ is the characteristic z
function of set f> ; I 't. ("X.) is the ~ -dimensional ball of radius "1:. with center at the point -x. ; 'lf" is the volume of the 1'\ -dimensional unit ball; w,= n v .. is the area of its boundary; '1..., 4 is the distance between the points~
and 'a' ;
"t.
(":ll:.,
B) is the distance from the point
The Lebesgue measure in R" is denoted by sional Hausdorff measure in R" is denoted by \-1 K'
rn.lh., ;
~ totheset
the
1\
B.
-dimen-
We shall use the following standard notation for functional spaces: Fig. 1
c . ctt).
c.t.... >, L.. •
c·: . c(~)and K ( ~) By
functions from, respectively, C( c..) and in
G- •
The space
Gr) with compact support
B V ( ('..) is called the space of functions
locally summable in Fig. 2
c(oo) (
we denote the spaces of
G-
t
which are
and whose gradients v~ (in the sense of distribu-
tion theory) are finite charges in G- • The class of all functions belonging to BV(G-1"1 IR) for any R is denoted by 'B V(tot.) ( Cr) , If }~ e B V('Q"),
INTRODUCTION then the total charge variation v
i" is called the perimeter of set
5
f>
and is denoted by "P( ~).
Constants which depend only on the dimension of the space and whose precise values are not essential, will be denoted everywhere in what follows by the capital letter C (with various indices). The work in this monograph was reported at the All-Union Conference on Geometry in the Large (Novosibirsk, March, 1966).
CHAPTER 1
PROPERTIES OF THE CLASS OF SETS BEING CONSIDERED §1.
Definition and Certain Properties of a Solid Angle Let
E
R" and let r be a point in 'R" . K( R \ p) defined by the equality
be a bounded Borel subse\of
tribution c..>E ( p. ~) on the space
Let us consider the dis-
(n-2.) c.JE(p.~)=-~ Vi(7:.)V"C.;: d~.
(1.1)
E n
Let us assume that c.JE (~, Cf) is the charge* i~ Q ; p, i.e., that the functional (1.1) admits of an extension up to a linear, continuous functional on C ( Q \ p) . The set function c.JE (p,
B) is defined and finite for all Borel sets B such that
LEMMA 1. Let the points
p., ... , Pn+t
functionals c.>E (p;.•
i
Ec R"\ p.
not lie in one hyperplane and let the
be charges.
Then,
P(E)<
oo.
E•, ;;.,;, -t, ..• , n+of, denote the complement to the hyperplane passing through the points p.., ... , PH , Pt+• , ... , p11+4 The sets E, form an open cover of -a E . Le~e1 , ••• , e...1 be the decomposition of unity corresponding to the cover {E:} i.e., e.£ K (E,) e.~ 0, ,Le.: i on 'a E. Since Proof.
Let
!Upp"l~c-aE, it is sufficient to show that the functionals We set '\'
= c;' en+4 where 9' E. K( R").
E.....1 ,
there exist functions
'-
4
..
To be specific, let i= n+i.
'
1 = 'tocr;. Since the vectors e,ae. c
We introduce the notation :r.E.
'
e. V f,E are charges.
" 'a "' ' -'11'CIIlC."' ="""' ,. -L... "'i 'a't.
'
J=1
"t4 ,
•• , ,
"C.,
are linearly independent when
i.· ~ .... 'n.
d
Consequently,
*A selection of the properties of charges sufficient for our purposes is presented, for example, in the book by N. S. Landkof [17]. 6
DEFINITION AND CERTAIN PROPERTIES OF A SOLID ANGLE Hence, keeping in mind that the lemma.
"C.~-"-e,. e ~
7
c- ( E_.) and su.pp '\' c surr ~1 c Ellt-1' we obtain the assertion of
LEMMA 2. If P(E)
the distribut~on ~ Cp.~f)
pER"
is a charge and, moreover, for any Borel set B
such that
pe
~.
(1.2) Proof.
where
Let
\f =~ 'il "t;.:: c- (R"). E
Therefore,
which proves the lemma. COROLLARY!. If
P(E)
and "t(p.&)>O, then
~ (ftB)~t.1-"(p.f>)'IMV~E (&).
-JQ.'t
LEMMA 3 (De Giorgi [13]). If P{E )< oo all
~
(1.3)
then for any point p
and for almost
>0, (1.4)
The next corollary follows from Lemmas 2 and 3. COROLLARY2. If P(E)
p
QEni,Cf) (,, f>)=c.>E(~,80I, Cp))+'"\~cp) (~.
and for almost allt->0,
&fl E),
(1.5)
~ei,cpl\.8)=~ (~, Bnci&Cp))-"\:tcp) (~. BOE) f o r a 11 s e t s
B
COROLLARY 3. If
Proof.
~
and p o i n t s
a) Let
p (E)< oo
{e. K
s u c h t h at 't ( '\• &) > 0 • then
a) we
11
when
b)
(p, 1 E)= (J
when
WE
(1.6)
p€ ;.nt E ~ pe C E.
{R"\.p)·Then from equalities (1.1) and (1.6) we -get that for almost all f.,
l
~ V l 'fl ~: d\IC, = ~4\ (p.dar.)-t" ~ f. d6', Entt,(r) cr,
(2-n)
(1.7)
J.s-
is an element of area of the sphere 'CJIE
The assertion of u;m b) follows immediately from equality (1.1) when in the neighborhood of Q \ p.
9'
e. \.<: (Rn\ r)
, 9' == ~,
We now assume that (1.8)
8 if
MULTIVARIATE POTENTIAL THEORY AND THE SOLUTION OF BOUNDARY VALUE PROBLEMS
B c Rn \ p. Then the charge
"'e (p, B)
can be extended by additivity onto all Borel subsets in R" \
We define the set function <->e ( P• B) at the point
p.
p.
By definition, let
(1.9)
We assume, further, that when
c.>c
fE. f>,
Cr, &)=-wE(p,
&'\p )+r.>E
(p,p).
Now, the charge "'E ( p, B) is defined and is finite for all Borel subsets in Q". From equality (1.9) and Corollary 3, we get
cu., we(p ,R")=
when
r
~ ~'\:
~ when r~ a.
0
when
E., (1.10)
-a E,
pe.c.E:.
In the following lemma w_e establish the connection of We. ( P• of the set E at the pomt p .
S"E { r)
R"'\p) with the
volume density
LEMMA 4. If p E ?l E and condition (1.8) is fulfilled, then the volume density SE
Proof. we get
In equality (1.7), if set ~= 1 in the neighborho~d of
E ()Cit Cp), foralmostall 8>0
Consequently,
~~~
r
cuE (p, CI&)~-~
de.-= ~.. ~ HW\-1 (En o1'-)Je-= {w ~ (I'L ()E).
(1.11)
0
The expression on the left-hand side of (1.11) equals (1.12) By virtue of (1.8),
t.m
't+O
'Ualt. WE
(p,I
Consequently, the integral in (1.12) tends to zero. Now, from (1.11) and (1.12), we get
The lemma is proved.
DEFINITION AND CERTAIN PROPERTIES OF A SOLID ANGLE Let ?J* E denote the "reduced boundary" of set E. i.e., the set of points which there exists a normal (in the sense of Federer) to
E .
9
p E ~ Etat each
We recall that a unit vector
1
of is said
to be normal to E at a point p, if
~ ~
t Vl'\.U)J ~; 4"'En It cr), 1 ;:rr~ > o1= o, -n
If, however, there is no such vector for a point 14]),
dense on
~E
Pro of.
='il
CE
p , then we
'ltalt'V}E
LEMMA 5. Let 11 W. E
1f:
{
c· ~.. l ~ : ~ t:. E nIt
1 tr~ < o} ==-?- · set
1 Cp) =0. t
(1.13)
As is well known (see [13,
(R'\.,•F)=O.
(1.14)
and P(E)
-a•E
is everywhere
• We first prove that for any point
~E."'
E and any
t.
> 0, (1.15)
We assume that for all
~ ~
E0
,
(1.16) According to Corollary 2, the equalities
c.>Eni._l'>)(or,a (It(~)nE))= c.>E(~,It(~))+q~,c ..) ('\--,E)'
(1.17)
(Q ,a(I,_(~)nC E))=wE(1,l 6 (~))+" l!~)('j.-. CE)
(1.18)
(<)
tEnit.{!~)
are satisfied for almost every
e.> o
.Y
t.
anct for all
'\t6 a It(!~) n.., E.
By virtue of the lemma's hypothesis, the sets I(c~) t
> 0 . We choose cyeolr. (-~)
nv..-l E.
n~t E and
_ I&<~)nCE are nonempty for all
Then, the left-hand side of (1.17) equals w,. (Corollary 3). Fur-
theremore, estimate (1.3) shows that the equality w('j.-,lt(~))=O follows from (1.16). Therefore, (1.17) yields E
Consequently,
Analogously, by choosing
'l'eil ('>)f\CE,
from (1.18) we get
H,\-1 ("~~lE.(.~)")nC E)=\-\,.... (a It(~)), which contradicts the previous equality. Consequently, inequality (1.15) is valid. tAs shown in [13, 14], v}eC~i')=p·H....,
&naE
This allows us to rewrite formula (1.2) in the "classical"
10 MULTIVARIATE POTENTIAL THEORY AND THE SOLUTION OF BOUNDARY VALUE PROBLEMS We now assume that the lemma •s assertion is false. Then we can find a point -'!. number E.> 0 such that Il.(~)n lows that
-a*E =-c/> •
E: 'II
Since the charge ~v ~E is concentrated on
E and a
-o*E,
it fol-
which contradicts (1.15). The lemma is proved. COROLLARY4. If 'C>in.tE='"~>~E,
re.11E, then dense on 'llt.
all
Proof.
pe-a* 'E,
then equalities (1.13) are fulfilled at the point
Remark.
If
p
p.
cJE (
p, p)= 0 for
all
pe -a* E .
If
Adding them we get that the volume den-
equals ~12. and, consequently, by Lemma 4, wE ( p, p)= 0.
P (E)< oo, the charge
Indeed,in(1.5)weset using (1.10), we get
P>=IE.
(p),where CUE (
Hence, as t.- 0 we get C.Ue (~, p) = 2.
for which c..lE(p.p)=O, is everywhere
By virtue of Lemma 5, it is enough to show that
sity of the set E at the point
§
P(E)
the set of points pE~E,
cJE
('\t,
5)
pE()E\cy
does not have point loadings on
R"\cv.
and t. issosmallthatcyE:lE.(p). Then,
cy 'IL Cp')')=- WIE.(j')
(
q,.' E).
0.
Properties of Sets Satisfying Condition (A) LEMMA 6. Let
p. cy E:
~
n-a E ,
p (E)< oo
.
Then for any Borel set
& and for any points (1.19)
where
"C.-
is the smaller of the distances from
p
and
cy to "&n'llE and K
is a constant depending only on n. Proof.
We set
f' (& )=c...lE (p, &)-c.JE('},l)).
By virtue of equality (1.2),
f (&}=~v c't.;:--c.;-:_J'V~E (d«-} e. Consequently,
Obviously,
\v..c.
't;:) ~ (VI-~J"tr"r~o ~;~ 't;: ~
(VIH) (V~-~)i" "tp,..
The lemma is proved. COROLLARY 5. Under the hypothesis of Lemma 5,
\'11<11\.cue Cr.e,)-1rWVWE c~,l))l ~ K 'W(. v~e (&)~ 'tw
(1.20)
THEOREM 1. If '3i..n:t E-='"I>CE , then the equality
~';P 'VM.c.>E rER \<~E is valid.
(p,Rn)= su.p W/t.~ (p, R")+-%peaE
(1.21)
PROPERTIES OF SETS SATISFYING CONDITION (A)
11
The assertion of this theorem will follow from Lemmas 7 and 9.
f E 1l E there holds the inequality
LEMMA 7. If a i.oJ; E ... 11 CE and
~ wE
Cr. R" )+ ~ ~ p,.-p, t..m
'lf(Vt,
cu Cp" • R" ) .
(1.22)
p,.e..,E Proof. £
V..t E.
evE
Let us first assume that
When 't.P"'- <. ~
,
lf• p) ~ 0. We ro nsider the sequence of points
p" ..... p, p..
according to (1.20) we have
\'11111\-cuE(?,Clt(p))-"ltQJI,CJE.(p".Cit(P))\ ~ K ({-J"P(E)
Note that by virtue of (1.10) and Lemma 6,
Therefore,
Hence, because E.- is arbitrary and wE '1/M.W
E
is nonnegative,
Cp, R")+~='I/M.WE (p. R•\oI )+ 1'4!- We (p. R"\p) I~~ '\rQit. ~ ( o., R,.). K~oO I K
In case ~E ( p. p) < 0, we should choose the points argument. LEMMAS. Let pe-aE
and 'VQII,c..>E(p,R")
p..
lying in W..t CE and repeat the preceding
Then for any function ~EK
(R")
we
have (1.23)
(2-nf 1 ~~tV 't~: d~=-~{ (~) (JE Proof.
Atfirstlet
equal to zero in I-lr
y.JE:.K(R.\p) ,
It is clear that
fd<(R")
(p,d~)+
p,"
CE
and ~(p)=O
Let?K'
*
K=~.~ •... , denotefunctionsfromCoo(R"),
Cp) , to unity outside I-t. (p) , and such that \ v}" \ 'C
by the definition of WE
Cp,Cf) we have
(1.24)
j(p).
I(
,
C==- tond: .
Since
12 MULTIVARIATE POTENTIAL THEORY AND THE SOLUTION OF BOUNDARY VALUE PROBLEMS where obtain
C.', C•:::. ~tt$t
that~
• Therefore, keeping in mind
(2- nf 1h,. tv,._ "t,;: d:x. =) ~
K-
oa, we
wE ( p,d-:c.).
(1.25)
R"
1:
Now let ~ be an arbitrary function from
!
K (R"). By virtue of (1.25) we have
(2.- h )- 1 v,. q (:;x:)-{ tp )) v.oe E
'l.;. d:lC.: ~ nc:lC.)-l
e (p,cl~), .i
.
which on the basis of (1.10) is equivalent to (1.23). Equality (1.24) is proved analogously. LEMMA 9. The inequality
s~ ~we(p,R")~s~.~.p \~E
pER
'\'Wt-<JE
peaE
(p,R")+ ~
is valid. P r oaf.
It suffices to consider the case
We introduce the notation:
(n-2)J (p)= ~
v{ v 't~-;
d~,
CE where
{E. K(R")
and
p eU.t E.
It is clear that
J
is a function which is continuous in
Rn
and
harmonic in ,:..t E.. Furthermore,
According to the maximum principle,
\ ~ {<:x.)we R"
Cp,d~:~e.)j ~mwr. \1 (s)\. SEo'iE
Hence, applying (1.24), we get
Since quently,
K ( R~'~)
is dense in C ('0 E}
,
the latter inequality is true also for all { t
The inequality
is proved analogously. The lemma is proved. It is not difficult to verify that the following elementary lemma is valid.
C (-a E) .
Conse-
PROPERTIES OF SETS SATISFYING CONDITION (A) LEMMA 10. Let
I-; •
in the b a 11
a.~,--·,
13
a.,..1 be the vertices of a regular simplex inscribed
Then, for any point
-.c E I~
and any vector (t , (1.26)
LEMMA 11. If Condition (A) is fulfilled, the estimate
'VOJC.v"f.t:(&}i: cC.J~,
dr,
where
is valid for any Borel set Proof. where
llC0
B
is the diameter of Let a,., ... ,
a..,. 1
is the constant in Condition (A),
B.
be the vertices of a regular simplex inscribed in the ball I 2.d (..-c.) ,
~ ~. It is clear that when
of (1.26), for a set
C
and
(1.27)
De. e, and a point
dr,
~
:x:E:
D we have
"ta,_,.. \
~ 3 dr, for all -:c.e- & , -\ = 1,2., ... ,
n+ 1
r.
. Then, by virtue
(1.28) Let us now assume that
JD < S •
Then,
\:1E(D)n~: \~\ ~v~~ v~~Jd~}\+:,~ \v~~:-v-c.::~ \~ v~E (D)~ ~~~V;~~ vf.E (d~JlH~{' S~v ~E (D).
(1.29)
From (1.28) and (1.29) we get
S > 0 be an 't.arbitrarily small number. Then we can find pairwise nonintersecting sets
Let
'll1 , ... ,
D't.
such that
UDc E I
~
and
whence follows the assertion of the lemma in view of the arbitrariness of S . LEMMA12. Let Bc.Q" and
~ -"6
0 <.n-'1, J
)'t.r..:. 'ltatt. v te (d':C.)~
Then
c n-1:
0
8
where
C
'1-.!.. (e> )] ..., '
..1..
c"-1 ['11Wt. v}
is the constant in Condition (A).
E
(1.31)
14 MULTIVARIATE POTENTIAL THEORY AND THE SOLUTION OF BOUNDARY VALUE PROBLEMS Proof.
We introduce the notation
Then,
By virtue of (1.27),
1 (~)H. C (t~)"" 1 (' -1
Consequently,
1
j'tf"'" 'lrill'l. V]·e
~
.1..1 dt
..... ~:;
..:!...
(,h;)"~c"'"' lr.~ )~~' ..~:~ow c~C ...'["VMv~(&~ g
"a'~
'J•·•
v
E
.
The lemma is proved. LEMMA13. For any
o>Yl-4
( )
there holds the inequality
-; '"t
J
~"'" 'V"Qit. V.f'E (h.) ~
e1 1-M-\
c. t
... 1- 1
(1.32)
'
ext
C
is the constant in Condition (A).
P r o of.
We have
Integrating by parts and using estimate (1.27) we obtain
{
,- T -l--1
j~-l~~t~)=)·1~(~)t~1~)
r
~<~)cl~,cC1lE)
Wl·'t-t.
I
e~
d!=o·n-H
Ct.
... l-1
.
E..
The lemma is proved. §
3.
Properties of Sets Satisfying Condition (B) We introduce the notation
Condition (B) signifies that ex > o · LEMMA 14. If Condition (B) is fulfilled, then
~ su.p 'If~ \.-"'m.v..., (IT.< p) nE) :S ~"t-+0
~~~~E
,.'4; ,
(1.33)
(1.34) Proof.
According to equality (1.11), '"{.
r•
~.. U"t.Cp)() E}= ~ ~ ( C.Ir. Cp )ttl-l JE.. 0
PROPERTIES OF SETS SATISFYING CONDITION (B) Therefore, since
WE
15
(p, Cl,
-t-~ 4 -i""~
't.
(I~cp')~ E).,~" ~~(p.l! (p))s."-'de_
By virtue of Condition (B), for all sufficiently small
~
,
Hence,
which proves the lemma. COROLLARY 6. If Condition (B) is fulfilled, ~,.n-I:E='<~CE and
m.U,
('lE.)=O.
LEMMA 15. If Condition (B) is fulfilled, n=J..
tA-m """" -t" ~ vf,e (I-.: Cp)) ~ e'oe. " .
Proof.
Let
r611 E.
(1.35)
t•O fUi!'
• As a consequence of equality (1.4),
Analogously,
Hence, by virtue of the isoperimetric inequality (see [12]),
c~~(tnl-r.J~~V~E (1-r.)+-,olf,V"h-.. (E),
e~~:(cE.nl?.)(,'l/"aJt,V }E(I't.}+[c.>.. 't~'\Hllt'V "'I?. (E)]. Adding these inequalities we obtain (1.36) Since the function :i\~>~.+ (o\-':IC. )~-9. less than
K>
o, decreases when :c
Consequently, (1.37) According to Lemma 14 we can find
'"t•
> o, such that for all
-tE
to,'t,) and for all
r E:" E. we have
16 MULTIVARIATE POTENTIAL THEORY AND THE SOLUTION OF BOUNDARY VALUE PROBLEMS Hence also from (1.37),
This proves the lemma. LEMMA 16. If Condition (B) is fulfilled, then for any Borel set&. 'V'OJ1.
v~r/'D)=H.,_1 (~f\oE).
As is well known (see [13, 14 ]) , for all Borel sets ~
Pro of.
'V-t>.Jvq
1e (&n -o*E)= H"-~ (!)(I'll
It
E)
( 'b*E is the reduced boundary of E ) and, moreover, (1.38) Therefore, it is enough to show that (1.39) Since the set
E
satisfies Condition (B), by Lemma 14 we have (1.40)
In accordance with known density theorems (see [18], Paragraphs 3.3, 3.1), (1.39) follows from (1.38)
and (1.40). The lemma is proved.
CHAPTER 2
POTENTIALS AND THE SOLUTION OF BOUNDARY VALUE PROBLEMS §
4.
Integral Equations of Boundary Value Problems Let
P (E) .(oo,
is a charge for all pEa E (see Lemmas 1 and 2).
i.e., the extension of c.>E (p,<-f)
The function
defined for
p~a E ,
is called a double-layer potential with a continuous density
f (llC.).
From the representation
(see Lemma 2) it follows that
W(p)
is a harmonic function.*
In what follows we assume everywhere that of ~t
E coincides
with the boundary of
CE .
E
is a bounded subset in
R
11
and that the boundary
The latter condition is a natural one if we wish to examine
W
at every point p ~ '<> E the limiting values of potential
from within and from without.
THEOREM 2. Let ?(E)< oo. Then Condition (A) is necessary and sufficient for the existence of limiting values of the double-layer potential W(p) from within and from without for any continuous function {. These limit values equal, respectively, (i)
W (~)=~ (l>)t-'W{s), le)
W (s)=-~(s)+W(s). Proof.
(2.1) (2.2)
Necessity. Weassumethatforacertainsequence ~e,N.\ E (or pKeU.-I:CE),
~
ll•oa
'VG/1, W (
f\t, R")=oo.
(2.3)
Since P (E)<: o<:>, by virtue of inequality (1.3) the sequence { p,.} has at least one limit points on a E . We retain the notation { p.. 1 for those subsequences which converge to s. *The potential
W(~)
can also be represented in the form
(Compare with the footnote on p. 9.)
17
18 MULTIVARIATE POTENTIAL THEORY AND THE SOLUTION OF BOUNDARY VALUE PROBLEMS
For every fixed
K,
is a linear functional on C(?1 E). By hypothesis there exists a finite limit (.i)
w~
(SJ= ~~~
w, (pK) (i)
for any~
C (o E) , which implies the weak convergence of functionals W~ (p") to Wf (s). By virtue of the weak convergence,
where
K
E
I
is independent of the number
I(
•
We have arrived at a contradiction with (2.3).
Sufficiency. Let Condition (A) be fulfilled. Let us prove, for example, the existence of limiting values from within and, in passing, derive equality (2.1). Let pe'"'H:: , se.<~ E • By virtue of (1.10) we have
"':t W(r)= ~ (fc(L)-f Cs)J WE ( p,d'JC.) + ~ [~(;,c) -f Cs)] wE Cp, d:L)+ co,J (s).
(2.4)
I~ (s)
<:I! (s)
We take an arbitrary t. and a 8 so small that when x.E Ill (s) \ ~ (';(_}- ~ ( s) \ ~
~.
'
where C is the constant in Condition (A). We go to the limit in (2.4) as (A) and of inequality (1.20), we obtain
~~
QM., 'vJ
f+!>
p- s,
In view of Condition
Going to the limit in these inequalities as
8- o , in view of the arbitrariness
of ~ we obtain
The theorem is proved. Note that Theorem 1 shows that in Theorem 2 Condition (A) can be replaced by the equivalent
condition c;.u.p
'1/'"0J'(.
WE ( s,
R")
<.
O(J.
(A')
s~,E
We shall consider the Dirichlet problem in the classical formulation. The interior problem consists of seeking functions which are continuous in
E , and harmonic in iwt E
and which take the given
values W<.;) on ... E. The exterior problem is posed analogously, but here we require that the solution converge to zero at infinity. From Theorem 2 it follows that if the solution of the Dirichlet problem can be represented as a double-layer potential W
19
INTEGRAL EQUATIONS OF BOUNDARY VALUE PROBLEMS in the case of the interior problem and the equation
in the case of the exterior problem. Here,
(1~)(s)= ~ t {C~) w(s,Jx.) n~
is an operator in
C(a E) , and
We emphasize that Condition (A') (or, equivalently, Condition (A)) is necessary and sufficient for
T to be a bounded operator in C (~E) • We remark in passing that if Condition (A) is fulfilled, then T is a bounded operator in
Lip.,.. ('a E) • Namely, the next theorem is valid. THEOREMS. If E satisfies Condition (A) and
~E.Lipo(.('21E),o
estimate
is true, where on
p and 'V are points on 'iiE
and
M is a constant dependent
UDl.;r"' (-a E)" Proof.
By virtue of (1.10),
(Tf)Cp)-lH)(~)= ~ Hc~)-fcp)1[4)E
We set
h= "l:P'\'
?1. (~)= 4
in 1~1. ( p),
p.. (~~t)= 0
outside
\(T0(r)-(Tt)(1)\~l ~ ~(:.:)[f(x.)-{Cr~] [c.!E (p,da)-4lr:(~,d~)]~ \l:
+I ~ (4-~ ('.ll.))[t <x)-tcp)][cue (p.~~)- ~ c,_, da:.Jl= 1 + I.\. 1
R"
"
It is clear that
(2.5)
Using inequality (1.19) we estimate
I~
:
12. ~M~h
~
't;;
Cl2.h.(p)
'lta!tVjE
(~~).
20 MULTIVARIATE POTENTIAL THEORY AND THE SOLUTION OF BOUNDARY VALUE PROBLEMS Applying (1.32), we get that 1~ ~ M5 hcL. which together with (2.5) proves the theorem. We shall say that the function t{ E.
1) for any function
E , there
converging to
where
Y
C1 (u.~ E) has an interior boundary flow if:
lA f
K (R") and any sequence of sets E.,., c i.~~ E with smooth boundaries,
exists the limit
is the outward normal too Em;
2) the functional
eu (tt), given on
K ( R"), is bounded in C(oE).
u conditions 1) and 2) of this definition are fulfilled, the extension of e~.~. Cc.t) onto C(<~~)can be represented in the form (2.6)
where ~ ~
(i)
C ("'E) and L:
boundary flow of the function
is a finite charge on () E .
The exterior boundary flow Let
¢
This charge also will be called the interior
IA.l~).
be a finite charge on
<'(e) L... 71 E
of the function u £ c~ (R¥1\ E) is defined analogously. . The integral
is called a simple-layer potential with charge THEOREM4. Let mv..WI(<>E)=O
cp .
Obviously, the function
V(p)
is harmonic in
and let Condition (A) be fulfilled.
simple-layer potential with charge ary flow defined by the equalities
4>
R'\ '11 E .
Then the
has an interior and an exterior bound-
(2.7)
2:::.(e\&)::cp(B)+~
1
WE
c~.B)¢CJ~).
(2.8)
'OlE
Proof. viously,
Let
\E,., 1 be the sequence of sets occurring in the definition of the interior flow.
Since the function v 'i v
V
Ob-
is summable in E. , there exists the limit (2.9)
By Tonelli's theorem (for example, see [19], p. 223),
CONTINUITY OF THE SIMPLE-LAYER POTENTIAL GENERATED BY
21
Applying equality (1.23) we obtain
tv c~) is bounded in(, <.:~E),
Hence it follows immediately that the functional (2. 7).
which validates equality
The existence of the exterior flow is proved analogously. Remarks.
1. With every function
r '{ !~) (k). Since the flow of potential \Itt> is a finite charge for all
c1> , the family of these filnctionJ ~ ~" als is bounded on each finite charge 4> . Consequently, it is uniformly bounded. Setting \..\. = 1:.1'"" in the definition of flow and using the definition of wE., we see that Condition (A) is not only sufficient but also necessary for the existence of a flow of any potential V.p~
2. From (2.6) and (2.9) it follows that the equality (2.10)
is fulfilled for the function '-f
~)
E
K ( R"). An analogous equality holds for Z v .
The interior (exterior) Neumann problem is posed in the following way: find the function harmonic in ~!'lot E (in finite charge on
CE ),
'-~
whose interior (exterior) boundary flow exists and coincides with a given
'tl E .
We shall seek the solution of the Neumann problem in the form of a simple-layer potential. Then, in view of (2.6) and (2. 7}, the interior and the exterior Neumann problem reduce to the equations
*,+.,
,.h
"'" (\)
- 't" -t T 't" = £...
'
where T * is an operator adjoint toT. §
5.
On the Continuity of the Simple-Layer Potential
Generated by the Equation c\>-~T*'C!>=O We shall say that the fine charge
cp
belongs to the class 0
generated by it has equal finite limiting values V~
(!' , v
if the simple-layer potential
Vr:p
from within and from without oE . t
The aim of the present section is to prove the following assertion. THEOREM 5. Let tion ~-:t\.,T•q:,=,O
E
where
satisfy Condition (A). \"'.\
Then, any solution of the equa-
is less than the Fredholm radius of operator
belongs to l!'v. We first state and prove a number of auxiliary assertions. LEMMA 17. Let Condition (A} be fulfilled and let
L 5 (-aE)
,tt
where n-~ < s. < oo.
Then, the potential
V
~t~)
be a function in
of the charge
4> t & )· ~ fcx.) v~M- ~e cd~') e,
t Recall that here and in what follows we have assumed that ttL~ (~E) is the space of functions
'4l E
i ..tE::.,C,E.
which are summable with degree s in measure 'II~ v f.e •
22 MULTIVARIATE POTENTIAL THEORY AND THE SOLUTION OF BOUNDARY VALUE PROBLEMS s a t i s f i e s a H 1:i 1 d e r c o n d i t i on w i t h i n d ex <\- ";~ It is easy to make up an example of a function
4> E.
a n d , c o n s e q u en tl y ,
f 1: L,._, (-a E )for which '\> ~
~v .
C:v and to prove the
exactness of the index ~ ... ";' · Proof.
From the Holder inequality and Lemma 12 it follows that for any point
r
~~
E the po-
tential Vcr)isanabsolutelyconvergentintegral. LetM=~Hl..,(oE.)and '!1 1::: S~l. Then,
Mf< IV()f- V<)J ~ 'K• L)
1 Jt. f ~-" ,_,.,,• 1~ +\~e. )~V}'E(d~)j + ~aML \ \-r.,x.--z.!'.._ 'ITOII.V~Etd~)J = J.+ Ja.
rs'ca-..) ''ct-..) \'t.,l\
l:t~r,l')
Here and below,
K 4 ,K~,, ...
ti1 .. ~"t)
are certain constants. By virtue of Lemmas 12 and 11,
Further,
Hence, using Lemma 13, we get
The lemma is proved. COROLLARY7. Let
P(E)
are defined and c on tin u o us with res p e c t to
.
s
Then the functions
in
R".
To prove this it suffices to make use of the lemma just proven and of equality (1.2). LEMMA18. If the set E satisfies Condition (A) and if the equality
":lt~'liE,p~aE,
then
(2.11)
is valid, where the plus or minus sign is chosen depending on whether the point
p
lies in t.,t E or in
Proof.
We carry the proof out only for the case (
l-11
) -1
r)V't:r~~..., V'toc.~ CI~=J"t~, r r 1....
Hence, recalling the definition of
E thenT*¢€~v
LEMMA19. Let
pe CE.
l"
)
cJE(';(.,d'O
From (1.23) follows cJ,.
t.-n
--rt-p.,..
R"
E
cj:>£ (!"
CE •
~e
, we easily obtain (2.11).
satisfy Condition (A) and letmt~., (aE)=O and the equality 0
0
TV$= VT,.
Then, if
CONTINUITY OF THE SIMPLE-LAYER POTENTIAL GENERATED BY - :XT*= 0 0
Y
is valid, where Proof.
are the limiting values of the potential
Let { be a function or a charge in
metric kernel from
Vq, E. c, (R")
K (R") Since ¢E: O"v '
compactum. In particular,
V..,<s)-VCs)
R" .
h
Let
.
on
-aE •
denote the average of { with a sym-
and, consequently,
uniformly when s E.oE
Y
23
v..,.- v uniformly on any
Therefore, when
re '()E.
(2.12) Since
YY\lbn
(-a E)= 0,
We have
=(2-llt>;J R"
l R"~~:wE Cp,d~')}¢~.Cd~).
Here, by virtue of Tonelli's theorem ([19], p. 223) and of Corollary 7 it is possible to reverse the order of integration. Since for all ? €
-a E the integral in the square brackets is a continuous function of s
(Corollary 7) and since
cp , by recalling (2.2) we obtain
cphweakly
(2.13) On the other hand,
Comparing the last equality with (2.13) and keeping (2.11) in mind, we obtain
'\~q, (p)= (2-n\.i; ~..¢ (d'"')\ ~.. 't-:: wE(p,d$)±5? t~~) = ( WV~) (p)+ V~ ( p), where the minus or plus sign is taken depending on whether terior of
p lies
in the interior of
(2.14)
E or in the ex-
E..
Going to the limit in (2.14) as the assertion of the lemma.
p-oE
and using the jump formulas (2.11) and (2.12), we obtain
The subsequent discussion in this section is mainly a reproduction of the scheme used by Radon in an analogous situation (see [9], pp. 122-123 of the Russian translation) and we carry it out merely for completeness of presentation. Here, without saying so particularly, we shall rely on well-known facts from the general theory of integral equations. LEMMA20. Let
\CIC)
be a sequence of linearly independent operators in
C (oE) satisfying the following conditions:
24 MULTIVARIATE POTENTIAL THEORY AND THE SOLUTION OF BOUNDARY VALUE PROBLEMS 1)
CK* takes the class
2)
if
¢E
(! , then v
0
00
Pro of.
C=~
~
0
0
c. .
Since the function
is uniformly convergent on
in t 0 its elf;
vc,...,... "A-.= etc: v~ ·,
a)
Then, the operator
crv
0
and yc"= cv~
C(-oE)
operates in
' where C\> E.G"".
y~· is continuous, the series
E . By virtue of condition 2), 0
-
0
cv~ = ~ vc...·~.
(2.15)
Hence we get that the series ~ V •"' , whose sum, obviously, equals w=t cl(.,.. converges uniformly in
R" .
Thus, the limiting values of 0
DO
Vc•cp
Vc*J....,...
at every point in R"'\~ E,
exist and there holds the equality
0
Ve• = ~ Ye:4.> ' which in conjunction with (2.15) proves the lemma. Proof of Theorem 5. Let A 0 beaneigenvalueofoperatorl ,locatedinsidetheFred-
S,_ of operator T in the neighborhood of .A. 0 has the form
holm circle. The resolvent
(2.16) where
F!).-~ 0
is the regular part of A. 0 close to S~ , while each of the operators
bination of one-dimensional operators ~~<
tt< ,
I<=~ .... ,
tt< are linear functions in claE) . Both the ~K further, that Suppose that the functionals the equation
el(
p,
1\
K
K
is a linear com-
where the ~t< are functions in C(oE) and the
as well as the tK are linearly independent. We note,
have been generated by charges
4>- 'AT.,.¢= 0 are linear combinations of the
enough to establish that
A
Av.. •
Then all the solutions of
/\K . Therefore, to prove the theorem it is
E. (!' v·
We have
A:=t "~ ~ ~K¢(d~).
(2.17)
-aE
.Consider the set function ~"" ~ ... , p.
If in (2.17) we set the charge
equal to ~, d== 1, ... ,
tem with a nonzero determinant. Therefore, the
A.._
p , then for the A>< we obtain an algebraic sys-
are linear combinations of charges
A. cp.i •
We
CONTINUITY OF THE SIMPLE-LAYER POTENTIAL GENERATED BY -A.T*= 0 note that by virtue of Lemma 17,
cp..
£
25
C"v . Consequently, the lemma will be proved if we establish
* a that the operator A. takes (!'v into C'v. Let the resolvent in the neighborhood of ).. • have the form
(2.18)
Hence from (2.16) it follows that A. is a linear combination of a finite number of the&~. ing (2.18) with the representation of
s"'
Compar-
in the neighborhood of zero,
~ ')...KC,.
S
=
)..
_;l<~=~o- - -
(2 .19)
~:AKlK K=o
where the numerator and denominator are regular inside the Fredholm circle, we obtain (2.20) Here, the series (2.21) converges since
II C,.,.. \\ ~ MR-1'11 , where M"" cons-1:
If we can show that the operators
Cl(
, and
R can be taken larger than I A. I •
satisfy conditions 1) and 2) of Lemma 20, then, by using
the convergence of series (2.21) we can conclude that the operators same token, finish the proof of the theorem.
&~
take
(Yv into ~ and, by the
In the neighborhood of zero the resolvent has the series representation
Comparing this expression with (2.19) we get
~
By virtue of Lemma 19, the operators T, t=~.:l.~ ... , satisfy conditions 1) and 2) of Lemma 20. Consequently' the same is true also for the operators
cl(.
The theorem is proved. THEOREM 5'. Let
E.
satisfy Condition (A).
Then any solution of the equa-
tion (2.22) where '\IE ~v longs to
and IJ-..1
is less than the Fredholm radius of operator T , be-
Since every solution of Eq. (2 .22) is representable in the form
4>= '¥ + f o "¥ + 'Po , 11
26 MULTIVARIATE POTENTIAL THEORY AND THE SOLUTION OF BOUNDARY VALUE PROBLEMS where
~
is any solution of the homogeneous equation, belonging to l!'v by virtue of Theorem 5, by
comparing (2.16) and (2.18) we get that
Fo* is a linear combination of the operators
sequently (see the proof of Theorem 5),
Fo*
0
§
6.
takes
*
&~ and, con-
The theorem is proved.
Fredholm Radius of Operator T The aim of the present section is to prove the following theorem. satisfy condition (A) and let R be the Fred-
THEOREM 6. Let the set E. holm radius of operator T .
Then,
.1.=~ ~ R
c..l"
sup '1tM " \ ~~o p~:"E
(p, l
This equality was first derived by J. Krlil in [4, 7] and proved by him for the case n=The estimate of
R from above,
below is derived here just the same
K ( R")
'1tOJ't-
(.JE (
P· I~ ( p)).
A by the equality
Proof. We define an operator
f
R from
E satisfies Condition (A), then ..L ~ ~ ~ S~tf I( w., ~+0 pE7>E
where the function ct"t. (-:x:.)
in [6 ]. t
which we derive below, reproduces the proof of the corresponding as-
sertion in [6] as applicable to our case. The estimate of way as in [3]. LEMMA21. If the set
~
and, moreover, r::/.7. (-:x:) = {
I
when
0 when
1~ I ~
i
1':1:: 1 >'t.
and io~'t(\lC.)I~ ~for all -;x:.. Let us estimate from above, the norm of the operator T- A
We have
i.e.,
Let us llow show that the operator
A
is completely continuous. To do this it suffices to show
that the function (AO<s) satisfies in the unit ball of the space C (a E) a Lipschitz condition with a constant independent of
f.
Let 'ts,._ < 1~
. Then,
tFor planar regions bounded by curves with a finite variation of rotation, the expression for the Fredholm radius
R
of operator
T
was found by Radon in [9]. In this case,
the rotation of the boundary at the point -x.. .
R= rn:ot("-)
where
oL
(:-c.) is
27
FREDHOLM RADIUS OF OPERATOR T
where K.._ is the Lipschitz constant for the function which we have made use of here.
o£"t(:x.)
and
K
is the constant from inequality {19)
The lemma is proved. Let (}t denote the set of those finite-dimensional operators
(A'Ol£>)=± l a~{
for which all the charges
G.' •6
~
(s)
A'
from C(•E) into
~ ¥<:;) G-~ (ds},
C(~E) , (2.23)
~E
do not have point loadings .
LEMMA22. Let the set E equality
satisfy Condition (A).
l_=i.wf 1\T- A' II R A'£~
Pro of. Since
t
==
Then, there holds the
.
~ II T- A II,
where the greatest lower bound is taken over the set of all finite-dimensional operators, it suffices to prove that
.i..R ~~- 1\T-A'I\. A.' f.IX.
The last inequality will be proven if from any finite-dimensional operator operator
A
we construct an
A' €0L such that
I T- A II 9 Let the operator
1\ T- A' 1\.
(2.24)
A have the form
(AOCs)=~ t(s) ~ { cs) er.i (ds\ b- i u aE u We consider the decomposition
G-.= C...'+ G/0 , where the charge G;'• ! 4
does not have point loadings while
A' be the operator defined by equality (2.23); A'' , T- A , T- A' by~· (s,~) 'h (s, B)' n'(s, B), respectively. Since the operator T-A' operates inc (•E) ' consists only of the point loadings of charge (; .• Let 6
obviously, A
1 E.
~· (s, I·) weakly
&(, • We set A 11 = A'- A and we denote the kernels of the operators
'h' (s, ·) as s,-+ s
. Consequently'
Hence it follows that sl.l.p '!.€:';1
- where
M
E:
'1/'Wl, \,
1
(s, ._ E )= l'.Ll.p SE M
is a subset of ~ E , everywhere dense on '.l E. •
'1f(llt \,' ( s,
~ E),
28 MULTIVARIATE POTENTIAL THEORY AND THE SOLUTION OF BOUNDARY VALUE PROBLEMS Since
1\T-AI\=!.u.p
'11M
h (s,<~E), 1\T-A'D=s!Af
1rWt
h' (s,
SE1>E.
equality (2.24) will be proven if we show that
'\I'(AII.h (s;~>E)
~
'VQII.
h (s;aE)
(2.25)
for all s from some set M everywhere dense on a E. As M we take a* E (see Lemma 5). From Corollary 4 and the remark to Lemma 5, it follows that when H'a.E '
4)!:
(s,r)=O for all f'
E: 'a
E . Furthermore, we can take it that su.pp G~
d·~=~ .... ,'t:.
~
,4..
q
I
.Therefore, !>u.pph (s,.)n supp'h (s,·)='t'· Consequently, when s e Cl• E,
n su.pp e-;::: cp'
Inequality (2.25) is proved, and together with it also the assertion of the lemma. Proof of The or em 6. We shall use the notation introduced in the proofs of Lemmas 21 and 22. Let A' e Oi • Then,
The last equality follows from the absence of point loadings in C...' • Thus, when A' E. 0'(, , 6
IIT-A'II~~ ·~ su.p ¥1
't-O
S1;1>t
'II'MWE
(s,I~(s)).
The latter inequality, together with the assertions of Lemmas 21 and 22, proves the theorem.
§7.
Solvability and Uniqueness LEMMA23. Let the set
the potential equality
Vq,
E
satisfy Condition (A) and let m.tJ.>JaE)::O
of any charge
cp
from
(!'
v
belongs to '£;~)
(RYI)
Then,
and the
(2.26) is valid. Proof. We first show that for the average
there holds the relation
-~ (vVJ"d-x=~ ~ (V"\ (s)ct> R"
~E
(ds).
(2.27)
29
SOL VABU..ITY AND UNIQUENESS It is clear that
Since
~ .{~:lt1:.:~n<~-p)\dp ~M (n,hJ
Keeping the identity
in mind, we get
-~ ( v V" )1 J.~ ... 2 ~ V.., c~) ~ ~~~ c:~e-s) cp (ds) d\l'.= 2 ~ C\>(cls) ~ '\ cll'.-s) \ R"
0."
1E
(ll'.)dll'..
R"
'liE
Equality (2.27) is proved. We now prove that
V(~) t '1..: >(R"). 4
Indeed, as h-+ 0
~ ((\, ¢ (cls') -
~ yocp (d,).
-aE
Hence it follows from (2.27) that when
11E
h.< h..o the
quence of averages, weakly convergent in weakly to
V•
{v~,J is bounded in t.:IJ (Fl."}. From the bounded-
set
. ness of {V~,.} and the uniform convergence of V~, to
t~•l ( 1?,"')
V as to
V•
h.---.0 , it follows that there exists a subse-
Therefore, V£
'i:tl (It"') and {VI..~
converges
Now (2.26) is obtained from (2.27) by passing to the limit. The theorem is proved.
In what follows, for the sole purpose of simplifying the formulation, we shall assume that the sets
i.nt E and CE" are connected sets. The changes which have to be made ·when considering the general case are trivial. THEOREM 7. Let the set sertions are valid:
E
satisfy Condition (B).
1 °. The interior Dirichlet problem for any continuous function solution is representable as a double-layer potential. 2°. The exterior Neumann problem for any finite charge a simple-layer potential. Such a solution is unique.
Then, the following as-
wc·ys) has a unique solution.
L (e) (B) has a solution representable as
3°. The exterior Dirichlet problem for any continuous function
wc•l(s) has a unique solution. This
solution is representable as a sum of a double-layer potential with density f(s) and of the function
't:~" ~ ~ (s')'VM. vj.e.IJs). where -aE.
tr
is a fixed point in
\M.t E .
This
3D 11ULUYAIUATE POI'ENTIAL THEORY AND THE SOLUTION OF BOUNDARY VALUE PROBLEMS (\)
2:. (B)
4=. 'l1ae illlerior Neumann problem for any finite charge
with zero total mass has a so-
lution representable as a simple-layer potential. Such a solution is unique with accuracy up to a con-
stant term. Proof.
As has been shown in Lemma 21, it follows from Condition (B) that the Fredholm ra-
dius of operator T is greater than unity and, hence, the Fredholm theorem is valid for the integral equations of the problems being considered. Further, the proof of the uniqueness of the solution of the
4>+ T "¢= 0 is a unique specific moment (as compared with the proof of the analogous theorem
equation
in classical potential theory).
m.v.. (1E)=O
First of all we note that by virtue of Corollary 6, it is easy to see that Condition (A) follows from (B).
Let us assume that there exists a finite charge
and
"Cl
i.t..t
E=-<~CE
.
Furthermore,
"
¢ 0 :f.
0 such that
-T*¢,... ¢ 0 Consider the potential
V<\>o •
Since
2: <e)= 0 , on the basis of the remark to Theorem 4 we have
~ V~ V V~
-
V~.
~ E K (R").
in the metric of
VtfK
Vcp := 0 in E . 0
v Yq, d~= ~ (v 'Vq,.)~d~=o. •
CE
CE and since Vq,.= 0 at infinity, Vq,.-=- 0 in CE . By virtue of Theorem 5, the
in
limiting values of
Let us approximate
:t!'>(R") by functions 'fK ~ K (R"). Then from (2.30) follows:
~ \/q,, = tonst
v~. E: ~~) (R").
According to Theorem 5 and Lemma 23,
CE
Hence,
(2.30)
Q-:lC=O
0
CE for any
(2.29)
Vq,
from within E equal zero. Since 0
Thus,
V~.= 0
in
R" .
(i)
Consequently, ~.
Vq,
is a harmonic function inside
E,
0
= 0 , i.e., {2.31)
By virtue of (2.29) and (2.31), ¢.= 0 , which contradicts the assumption. The assertion is proved. Note that Theorem 7 does not guarantee the uniqueness of the solution of the Neumann problem. It asserts only that a unique solution is representable as a simple-layer potential. Below we prove a
uniqueness theorem for the Neumann problem for a certain class of harmonic functions. THEOREM 8. Let the set E sertions are valid: 1°. For any finite charge
satisfy Condition (B).
~(I) E 0"v
Then, the following as-
with zero total mass there exists a unique (with accuracy
up to a constant term) solution of the interior Neumann problem from the class C(E')
n BV (-\.nt E)
2°. For any finite chare:e L:(e) E C:v there exists a unique solution of the exterior Neumann problem from the class
C( R"\ i.vr~ E') f'l By
zero at infinity.
Pro of. The solvability of the interior and exterior Neumann problems in the classes mentioned follows from Theorems 8 and 5'.* Let us prove the uniqueness of the solution of the interior problem. Let
1.1.
be a solution of the interior Neumann problem from the class
zero boundary flow. Let p denote an arbitrary point in
R" '\ E .
C(E.)n 'DY
(i.nt E) with a
Applying (1.2) and the Gauss -Green
*From Lemma 23 it follows that the solution will have a finite integral of energy.
31
SOLVABILITY AND UNIQUENESS formula (see Theorem 11 in [20]), which is possible by virtue of Lemma 16, we obtain
~u w (p,d:K.)== ~
~
~
E
uv't;: v }E (d~)== hu vt;~t> d~. E
By virtue of the condition :E~i) 0, the last integral equals zero for all p E R" \E.. Hence, by Theorem 2, 1.1- T~o~ = o •But this equation has the unique eigenfunction 1.1•cofls.t. Since 1.4 is a function
=
which is harmonic in E. and continuous in E. , ~ ... c.ons t in
E.
The proof of the uniqueness of the exterior Neumann problem is completely analogous. R em ark.
The fact that the flow .l: l') ('~::. ~·> )
belongs to the class
is not only sufficient
but also necessary for the continuity of the solution of the interior (exterior) Neumann problem. (Recall that by Lemma 17 membership in n- 1
.)
_4
Indeed, it is not difficult to see that equality (2.10) remains valid for
't;;" ,
theright-handsideequals Wy<.p) when
pe.R."\E
andequals-2V
passing to the limit as p• ~ E and using (2.1) and (2.2), we obtain the continuity of Y%u> case of the exterior problem is examined analogously. Fromthisremarkwegetthat for sets ity of '\ of \J;
in
If'
(f) .
By
The
E satisfying Condition (B), the continu-
follows from the existence of the one-sided limiting values
from within and from without E.
APPENDIX
ON THE APPROXIMATION OF A SOLID ANGLE Below we state and prove certain theorems on the approximation of set E by polyhedra under which the solid angles cv11 ~ ( ?• . ) converge to wE ( p•• ) , ~
f'><
We shall say that the charges
'IMlll. J"~eweakiy
1f(Vt
!'·
THEOREM 9. IfP(E) < oo
TIK-+E
,
converge regularly to the charge
.!"'
weakly fl><~
if
f
nK'
and
then there exists a sequence of polyhedra ITK,
such that for any point
p'E<~E reg.
wnK (p,·) -
CJE
Cp. ·).
The proof of this theorem is based on the following three assertions. LEMMA24. If
E... - E
and
"P(E,.,).s;
e < oo, then the sequence of charges
(p.·)' defined in R"\p 'converges locally-weakly to t h i s c o n v e r g e n c e i s a w e a k c o n v e r g e n c e i n R" .
WEK
WE
If
rE:"11E,
Proof. As is well known, it is enough to verify that (3.1) on any compactum F c
R" \ p,
and that
K~ ~~WE" (p.d~)=~ ~c.JE R"
~n
Cp.d\X.)
(3.2)
for any t.f E K (R"\ p). Inequality (3.1) follows from the uniform boundedness of estimate (1.3). Let us prove (3.2). By Lemma 2,
P (EI()
and from
(n- 2) ~ tfWe Cp,d\X.) = ~ Cf V"lt;:. v jE (d-:x.), ~,
R
(3.3)
(ll-2)~~''t<(p,clx.J=~CfV'"t:: ~
Obviously, the vector-function
if
I
(d~).
v}E K
\- n tf v 'tp~ belongs to K (R" \p ).. Furthermore,
V
,.J weakly
/"e;-- v ,.;f'e
(see [·12]).
Therefore, we obtain (3.2) by passing to the limit in the right-hand sides of (3.3). THEOREM 10 (V. A. Zalgaller (21]). Let the sequence of vector charges ~K, given on a compactum
K,
converge regularly to the vector charge j"'
f(-x.) be a vector-function in C(K)
, and let
Then, (3.4)
32
APPROXIlVIATION OF A SOLID ANGLE
33
This theorem has been formulated and proved by V. A. zalgaller in [21] in terms of the lengths of the projections of curves and under the assumption ec-x)= cor~st. However, his arguments are easily carried over to the case of any vector-charges. For the reader's convenience we present the proof at the end of this appendix.
, p (E ... )~ p (E) , t h en f o r any p oint p~ " E
LEMMA 2 5. If EI(- E
Pro of.
The weak convergence of
wE,.
(p, ·) to
cJE
Cp. ·) has been proven in Lemma 24. The
relation ~wE. Cp, ·) wea!,!.y'V'WtcJE
e(x.')=. <;]~~:, f'K==-<:f1E,. • fl =V~E.
Pro of of Theorem 9.
De Giorgi (see [12]) has shown that if
n"' ' n. . --
a sequence of polyhedra E. such that p this sequence satisfies the hypothesis of Theorem 9. We shall say that the sequence of polyhedra 00
and ~
n.._:::l
n\( = m"t E.
-
E., and
( n. .)-- p(E).
n.
P (E)< <><J
,
then we can find
From Lemma 2 5 it follows that
converges to the set E from within if
Analogously, the sequence
nn.cE·
....
nK ~ E'
{lTK} converges to E from without if
...,
THEOREM 11. If Condition (B) is fulfilled for the set E , there exists a sequence of polyhedra TTK, converging to
E
from within (without), such that
for any point pE'"llE,
The proof of this theorem follows immediately from Lemma 25 and from the next theorem. THEOREM 12. Let E a)
be a bounded Borel set.
Then,
if (3.5)
there exists a sequence of polyhedra 11" , c on verging to E that
from within, such
PCnK)-PCE). b)
If, however,
(3.6) there exists a sequence of ITK.
, converging to E from without, such that
p (IlK"\~ O(E'). To prove Theorem 11 we need the following simple assertions.
{U:) be integrable on an interval tl oflength\tl\and,moreover,let~(-t.) ... 1-t.,where t..>O and f isthe average value of ~(t) on A . Then, for any S>O, LEMMA26. Let the nonnegative function
~JtfU:)<~+f) > \Ll\ (1-f). Proof.
Since ( {- ~
't < E, \ [Ht)- ¥t d-t = ~ [ f- ~ (*)y dt < c \ Lll. ~
~
(3. 7)
34 MULTIVARIATE POTENTIAL THEORY AND THE SOLUTION OF BOUNDARY VALUE PROBLEMS Therefore,
which is equivalent to (3. 7). The lemma is proved. LEMMA27. (See[12].) Let tion of
E.
~E L(R")
and let
}E
Further, let
Then for any -\:e[~.~- 0 ],
0 >0,
the estimate
\ iE- lEt \\L(R") ~ t where
E-\:= -\_.;x:.:
Proof.
be the characteristic func-
~
(:lC.)
~
-lJ,
'
is valid.
Obviously,
Since ~ (~)d: when :lC. E- E \ E-t- and
fc'X.) ~ t
when -xe E t \ E,
e "? (~-t) ~n (E \ E~ )+t m.rt." (E-t\E) ?-o 1\-j,E-}Et \IL(R")' The lemma is proved. Proof of Theorem 11.
We carry out the proof only for case a). The proof for case b)
can be carried out with· trivial changes. Note, further, that it suffices to examine the case u,;t E
+ cp.
1°. Let us show that in condition (3.5) the least upper bound can be taken over all :li:.E R" \ wt E. (It is clear here that the constant c1 can vary.) We assume the contrary. Then, we can find a numerical sequence { ~,) ,
o,
> o,
E.,-+ 0 and a sequence of points {x, 1
.~ 'V'~~ l+oo Let ~· be a point on ~ E close to
( " W\l.b"
'X;
(1{.·' (llt)"== 'J
, x, €: R" "- i.r..t E, such that
c
(3.8)
and let ~; be the distance between :lC.; and ~,.
Then,
Therefore,
and, hence, ~i 1 ~,
_..
0 as ;. -
oo . Further,
mv.." (Il, (~,)II E) ~ Y\'\U" (lEi (:lC.,')f\ E}vn.v.."(It;+~,<~,)\lt,C~)). We divide this inequality by v. c~
"
and we pass to the limit as "--
oo •
Then we get
\
which contradicts (3.8). The assertion of item 1 o is proved. 2 °. Let Kt. ( \t 1) be a decreasing function possessing the properties of an averaging kernel. Let us show that for all :x:eR"\W..tE we have the true estimate
35
APPROXIMATION OF A SOLID ANGLE
('h_\ <~) 0 not exceeding a certain small £ 0
(~EJt
•
(3.9)
Since K~ (I\ I) decreases,
(:i) ' (
'l-1 15 t~))l c~),
where the number ~ is such that~~~ lrC-x.)=~n (I£ C~)f\E). Going over to spherical coordinates we obtain
According to 1 o, when
~
' Eo ,
! = "~(Er\1,('1..))' e'
Therefore, from the normalizing condition for K(t) we get e'
('f..E\. (~)~c.>,. ~KH)t"-idt ~c"< ~o 3°. From estimate (3.9) it follows that m.u, 11 (E \ i.M\ E)= 0. Indeed, consider the set J\{~ = {~: (~E\<';lt.)~e·}. By virtue of estimate (3.9), Mr. c iNI,t E.. Since ("4_\._,. "f-E inL (R" ), then by Lemma 27, 'f.Mt-}E in L (\<"' ), i.e., Y't\lh" (E \Mr..)- 0 • Noting that E\u...\ E c E\ Me, we get
~" (E \ ~t £)=0 ·* 4°. As is well known (see [12, 16]),
or, what is the same (for example, see [22]), (3.10)
L~)= \.c.: ("f..e. \ (~)=t1,.
where
empty for all
t
£
Since in.t E -4:
¢ , for sufficiently large
[o, i] . For almost all t the sets
dimensional manifolds bounding the open sets
K the sets
L..~~<)
are non-
C~) are closed, infinitely-differentiable, (
E:)=t~'
Vl-1 )-
(-I.e.\ (x) > -t1 . Below we shall deal only with
such t. 5°. Let m beaninteger, m >2P(E)+~,andlet .11."'= (m-~, +-m-a.). Letusprovethatwecan find a number
K,.
such that when
K
> Km and for all -i-e
.ll.m,
H,... (L~))> m~~ ~ ~tH (L~')dtt-~ ·
(3.11)
All\
Indeed, otherwise there would exist a sequence of numbers K , there would be a level t E t>. for which , "'
K+ oo,
such that for each K
m
*Besides, this fact is a direct consequence of condition (3.5) and of the well-known theorem on the density points of a set (for example, see [23], pp. 281-282.
36 MULTIVARIATE POTENTIAL THEORY AND THE SOLUTION OF BOUNDARY VALUE PROBLEMS (3.12)
L (R"),
by Lemma 27 for all \
€ l\..,
we have
\\~e-1et ~L.(R~) 0, i.e., E~- E rrt
We pass to the limit in (3.12) as
K-+ao
Here we use (3.10) and the inequality
•
> 2 P (E)+~ . Then we get
~ Hn-~<~7" )'~ -a, P(E)-~ <.P(E)
k~oa
yn
which contradicts the weak convergence of V ~·
l")
E,..._
to V
1
1e .
6°, By Lemma 26 and by virtue of 5°, for any integer m greater than mOJX. ~2. P(E)+-'2., 2 (4- e."J1}. we can find a number 1(.., > m such that
Hence, we can find a value
t"' , e • ~ \., < i
such that 4
2. m" r H,_i ( L.:tKm), \.., m(<~-c•) + m'- 2. JH,_1
r'
(L'"'-r )d 't'.
(3.13)
D
By virtue of estimate (3.9),
> E
neighborhood of -a E . Consequently,
(
m.fh.,
3°, yru}.," E \ -iNI.t E ) =0. Consequently, in (3.13) as m-. oo , we obtain
Furthermore, the set (~e..,))
((-t..-1: E \E.-~;
Y\'UAn
(1(::) ) ( E.\ E-~;,
proximate each of the sets
t.
-1
is contained in the 1<,.,
"'
0 as "' .... oo . Furthermore, according to (I<..,) - 0' i.e., E~ - E . Passing to the limit
-+
The converse inequality follows from the weak convergence of v
.....) E\-1:,.. by the polyhedron
L(tc..)
n,., c ~.,-\. E •
~
(><..,) to
v~E .
It remains to ap-
Et..,
The theorem is proved. We proceed to the proof of Theorem 10. LEMMA28. If the finite vector charges ~ .. in and if
'1/"Q/t
~(It)- '1t(ll(.~
(R",),
R"'
converge weakly to
then reg.
~~~~r-
p roo f.
Let the set
f>
be such that
'\I"'ft,
~ (~ ~) =0 . Then,
'lNJ!t~(&~l(~ 'II'QI(.r..C&), Hence,
'VIVI.f
(cB)$1(~ ~rl( (c&).
f4
37
APPROXIMATION OF A SOLID ANGLE Consequently, in inequalities (3.14) the equality signs hold. This proves the lemma. LEMMA29. Let .1"1 be a finite vector charge in
Rn.
Then,
Proof. Let p be a constant unit vector. By the Gauss -Green formula, for the half-ball \el ~1} we get
te:ep>O,
0= \
~~ YI'\V:,n(de)=~epdse- ~H,_i(de).
e~>o,le\~i
ep>o,le\=i
ep:o,leiH
Consequently,
~ \ ep\ de-e= ~ ~ 11 \el"~ e~
Since the charge
11 _1
(de}= 21r~-i
.
e~:o, le\41
is absolutely continuous with respect to
'VW'~
Hence,
The lemma is proved. LEMMA 30. Let
Rn•
S' be a finite vector charge and let Q
Then, for any set u schitz condition:
the function ~(e)='liW(,(ef)(&)
e. be a unit vector in
satisfies the Lip-
Indeed,
· LEMM A 31 . Let)"~~~. b e f inite vector measures 1n for any constant vector e.
Rn
an d l et
~ ..
-reg.
~·
Th en
reg. e~ ... -e~.
Pro of. 1 ~=(f,
charges
...
) ,~".
foe.•
,... ,
It is clear that it is enough to examine the case \ e I= i . Let ~I(= ( r~ fl~ ) and s·mce ;. weakly i. ,,=1, • h'l d 1' b' t• ... ,n,wiee1"'« an eS" are mearcommawnsof th e
and
r-c"-r
f! o , respectively,
weakly
e ~"- e ~ . Consequently,
we have
(3.15)
Let us assume that for at least one vector e
0
there are found infinitely many values of
~
for which (3.16)
38 MULTIVARIATE POTENTIAL THEORY AND THE SOLUTION OF BOUNDARY VALUE PROBLEMS
Then, by virtue of Lemma 30, for vectors e. such that \e-e.\<~1'.' where K =sup ('liM.~" (R"), 'lTQtl,
~ ( R" )) , we have
Together with (3.15) this gives
In accordance with Lemma 29 the latter inequality can be rewritten as
~ 'VOlt.~,. (R.. )>
K-.oa
rwut.r (R"),
which contradicts the regular convergence of ~" to l". Consequently, (3.16) is false, i.e., equality holds in (3.15) for all vectors e . The assertion of the lemma is obtained from this equality and from Lemma 28. 0
weakly
We go on directly to the proof of Theorem 10. It is clear that ~~~"- t~ bitrary positive number. Consider a countable partitioning of R" into the sets small that
. B,
Let E.. be an arof diameter so
This partitioning can be constructed so that 'ltOit !:" (-a B,) = 0. Then, by virtue of the regular convergence of ~" to $4• ~.. (B,)-+f (B;) as K•oo . Furthermore, by virtue of the preceding lemma, for any constant vector t.,
Let oc, be a fixed point of
We pass to the limit as
&,
and let~,= t (-:x,) Then,
K? OCI:
ti-
'VM.
K-.oo
(t~ ..')(B,')~'IIW\, Ct;r) (B)H'lJW(,~ (&.).
The right-hand side, obviously, does not exceed 'II'CUt(trXB)+.2~'lf(llt.r (&) By summing over all the sets in the partition we get
~
K-.oo
l\fO.It-
<e~ ..)(R")' 'VQ/t.(t~) CR.,)+2 e ~r (R" ),
which in view of the arbitrariness of S yields
The reverse inequality for the lower limit follows from the weak convergence of l~ .. to '1M/(, (
trl\ ){R")- 1N.IJ(_ (£r) (Rh ), which, by Lemma 28, ensures the regular convergence of The theorem is proved.
tr-' . Thus, f" to r .
LITERATURE CITED 1. 2. 3.
4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.
Yu. D. Burago, V. G. Maz'ya, and V. D. Sapozhnikova, "Double-layer potential for irregular regions," Doklady Akad. Nauk SSSR, 147(3): 523-525 (1962). [In English: Soviet Math. Doklady, 3(3): 1640-1642 (1962).] V. G. Maz'ya and V. D. Sapozhnikova, "The solution of the Dirichlet and Neumann problems by potential theory methods for irregular regions," Doklady Akad. Nauk SSSR, 159(6): 1221-1223 (1964). [In English: Soviet Math. Doklady, 5(3):1681-1683 (1964).] Yu. D. Burago, V. G. Maz'ya, and V. D. Sapozhnikova, "On the theory of double-layer and simplelayer potentials for regions with irregular boundaries," Problems of Mathematical Analysis. Boundary Value Problems and Integral Equations (in Russian], Leningrad State Univ. (1966), pp. 3-34. J. Krlil, "On the logarithmic potential, • Comment. Math. Univ. Carolinae, 3(1): 3-10 (1962). J. Kral, "On the logarithmic potential of the double distribution," Czechoslovak. Math. J., 14(2): 306-321 (1964). J. Kriil, "The Fredholm radius of an operator in potential theory," Czechoslovak. Math. J., 15(3-4): 454-473, 565-588 (1965). J. Kral, "Double-layer potentials in multidimensional space," Doklady Akad. Nauk SSSR, 159(6): 1218-1220 (1964). [In English: Soviet Math. Doklady, 5(3): 1677-1680 (1964).] J. Radon, "Uber lineare Funktionaltransformatoren und Funktionalgleichungen," Sitzungsber. Akad. Wiss. Math. Naturw. Kl., Vol. 128, S.B. ITa (1920). J. Radon, "Uber die Randwertaufgaben beim loga.rithmischen Potential," Sitzungsber. Akad. Wiss. Math. Naturw. Kl., Vol. 128, S. B. IIa (1920). R. Caccioppoli, "Misura e integrazione sugli insiemi dimensionalmente orientati. I," Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat., Ser. 8, 12(1): 3-11 (1952). R. Caccioppoli, "Misura e integrazione sugli insiemi dimensionalmente orientati. II," Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat., Ser. 8, 12(2): 137-146 (1952). E. De Giorgi, "Su una teoria generale della misura (r -1)-dimensionale in uno spazio ad r dimensioni," Ann. Mat. Pura Appl., Ser. 4, 36: 191-213 (1954). E. De Giorgi, "Nuovi teoremi relativi alle misure (r -1)-dimensionali in uno spazio ad r dimensioni," Ricerche Mat., 4:95-113 (1955). H. Federer, "A note on the Gauss-Green theorem," Proc. Am. Math. Soc., 9:447-451 (1958). J. H. Michael, "Integration over parametric surfaces," Proc. London Math. Soc., (3), 7(3): 616640 (1957). W. H. Fleming, "Functions whose partial derivatives are measures," Illinois J. Math., 4(3): 452458 (1960). N. S. Landkof, Fundamentals of Modern Potential Theory (in Russian], Izd. "Nauka," Moscow (1966). H. Federer, "The (tp, k) rectifiable subsets of n space," Trans. Am. Math. Soc., 62:114-192 (1947). E. Kamke, Das Lebesgue-Stieltjes Integral. B. G. Teubner Verlagsgesellschaft, Leipzig (1956). Yu. D. Burago and V. G. Maz'ya, "On the space of functions whose derivatives are measures," Part 2 of this monograph. V. A. Zalgaller, "The variations of curves along fixed directions," Izv. Akad, Nauk SSSR, Ser. Matern., 15(5): 463-476 (1951). H. Federer, "Curvature measures," Trans. Am. Math. Soc., 93:418-491 (1959). H. Hahn and A. Rosenthal, Set Functions, University of New Mexico Press, Albuquerque, New Mexico (1948).
39
PART 2
ON THE SPACE OF FUNCTIONS WHOSE DERIVATIVES ARE MEASURES
INTRODUCTION We shall consider the space in the open set Q c
R" .
BV ( Q)
of functions whose generalized derivatives are measures
We shall study the connection between the properties of functions from
f:, V (Q ') and the geometric characteristics of the boundary of Q.. In terms of "isoperimetric inequalities" we shall give conditions relative to Q. which are necessary and sufficient for any function from
BV<.Q) to admit of continuation from BV (R"} and for the continuation operator to be bounded. The concept of a "trace" on the boundary for a function from 5 V ('>').) is defined and conditions for the summability of the trace are given. Certain results are derived concerning the relation between the "isoperimetric inequalities" and the integral inequalities (of the imbedding theorem type} for
BV<..Q). * In the case of a ball, exact constants which have a geometric meaning are found in these inequalities. The question on the validity of the Gauss-Green formula for functions from BY<..Q.) also is considered. t The classes of sets Q introduced in this paper are in a natural way characterized in terms of a perimeter in the sense of Caccioppoli [3, 4] and De Giorgi [5, 6]. Below we widely use the known
BV (Q) (see [5-10]}. A summary of
properties of sets with a finite perimeter and of functions from the background needed is presented in § 1. Throughout the paper we use the following notation: Q Euclidean space
R" ; E
CE= RYI \ E ; }E
is the characteristic function of set
and
B
are arbitrary measurable subsets of
us ~ with center at the point -x. ~ R"; is the area of its boundary. §
is an open subset of the n-dimensional
E ; I~
(::c.)
't"n is the volume of the
It ;
is an
'
E is
the boundary of
E;
n -dimensional ball of radi-
n-dimensional unit ball and c.v",. VI'V""
Properties of the Set Perimeter
1.
and of Functions from e,y(Q). The space of functions { , locally summable in
Q , whose gradients VQ.
t
(in the sense of
BV (Q) . We denote the variation of the The perimeter of a set E relative to Q is de-
distribution theory) are charges in Q , is called the space charge VQ. ~ on the whole region Q.. by \\~II e.~ (Q.). fined by the equality
[if }EnQ.
4 BV(Q.}, we assume Pn (E.)=
oo
].
--
(.l)
w
*Results of this kind have been obtained previously by one of the authors [1 ,2] for the spaces P (Q} of S. L. Sobolev. tit was only after this paper was written that A. I. Vol'pert kindly informed us about his own new results concerning the concept of trace for functions.from!) V and the Gauss -Green formula [A. I. Vol'pert, "The space !) V and quasilinear equations," Mat. Sb. (N.S.), 73(2): 255-302 (1967)]. 43
SPACE OF FUNCTIONS WHOSE DERIVATIVES ARE MEASURES
44
We introduce further the perimeter of E relative to the closed set
CQ ; namely,
R_g_lE.)=i.~ PQ-(E), G-=>C!it
Cr
where
is an open set.
Pll." (E.)
If
<(X) then, obviously,
PQ. (E ):'1/'0)1. VR" }E (Q), PCQ. (E )=VOlt. VR,-j.E (CQ), (1.1) Note, however, that if
E\ nQ
=E.~(\ Q.' then
PalE,)= PQ. (E. a'>.
(2.1)
In what follows, wherever it causes no ambiguity, we shall write v { instead of VQ. for VR"
f, and
P(E.')
instead of PR" (E). The following isoperimetric inequality holds: .!!:.\.
!l:.l.
-1;
\
• mu, ( ~:E.~~" CE/~tv.. P (E). ........ . .. ... . ........ ... ..... •. We shall say that the sequence of 8"ets .. E~.... converges to the set E (denoted: EK~ E ) if -~·
'
.
.
.
{3.1)
'•
~E._- "f.e
in L (R•). The sequence of finite charges
function
f ~ C(Q)
, finite in
converges to a charge t' locally weakly in Q. if for any
~"
Q., there is fulfilled the equality
PROPOSITION 1. (See [5].) If E. ... ~ E, then
P~ lE ) 10 fMn Pa (E.>~-'),
----
and if s~ H~ (E .. )
j
..
Joe. weak
Vg_ f"e -
PROPOSITION2. (See [5].) If P(E.) hedra*
n~
such that
n. . --. E
<. oo
and
,
'~~st
.J /"E
then there exists a sequence of poly-
P(n ..}-PCE.).
PROPOSITION 3. (See [6].) The collection of sets perimeters
P (E .. )
Ece. with uniformly bounded
is compact.
As was shown in [8], the equality (4.1)
where
Et={ ";)(.: f ('JC.) >-\:} ., is valid for any function f
locally integrable in Q .
*An open set with a polyhedral boundary is called a polyhedron.
CONTINUATION OF FUNCTIONS FROM BV(!l.) ONTO THE WHOLE SPACE Let
H,_, (E.)
45
( n- <~)-dimensional Hausdorff measure in 'R".
denote the
We shall say that the unit vector it (':le.) is a normal (in the sense of Federer) to the set E if
bo i-m.u.J.~: (\£E.()I~c:x.), -ni..~ >o1= o, ~ ~"mt.~~.. { ~: ~£CEf\ I~Cx), -n-t~a < o \ = o, where I~ (.x) is a ball of radius ~ with center at x.. The set of those points of ~ E at which normals to E exist is called the reduced boundary -atrE of set
E.
PROPOSITION 4. (See [6, 7].) The set Cl"E is measurable with respect to and, moreover, '1101tV~E
and '11011.V}E
(-aE\"a11 E)=o
and for any set
Hn_ 1
Bc1lE,
v1E (B)=~ it('>t')H ...1 (d-x.), '1/Wt,v}e (B)=HM- 1 (B). r,
H~~- 1 -almost all-JCe.<:>"E ,t
For
(5.1)
§2.
On the Continuation of Functions
from BVCn) onto the Whole Space For the set
E. c
Q we introduce the following characteristic tt
~(E)""W Pen (B). MQ=E
It is clear that A (E)= A. (Q
THEOREM 1. a)
\£).
If for any function ~e:BV(Q)
there exists the continuation
~"-BV,(~")such that
1\ 1111!1V{R")' cII 'lllW(Q) ' where
C
is a constant not depending on
l ,
(1.2)
then for any set EcQ, (2.2)
b)
Conversely, if for any set
constant
C
not depending on
the continuation ~ E
BV (Q")
Proof of Item a).
E. ,
EcQ. inequality (2.2) is fulfilled with a
then for any function {e.lW(Q) there exists
for which estimate ( 1. 2) is valid.
Inequality (2.2) is trivial if
esis there exists a continuation
~
Pn (E)=o<:~.
of the characteristic function
Let
Jh (E)<.()().
h such that
By hypoth-
II}~: \l~v{R") ( CP~ (E). tNamely, equality (5.1) is true if at a point x e. ~.
ttCompare with the set function
0(E)
1111
E the derivative ~ v i.E. 1 (';1t) exists and is a unit vee-
introduced by Fleming in [9].
~VE
SPACE OF FUNCTIONS WHOSE DERIVATIVES ARE MEASURES
46
Hence, from formula (4.1) follows 00
4
-""
0
c~(E)~ ~ ~/\:K:·~e.,\. l)dh ~ ?R.. n~.h >t J)Jt. Since\. :x: :~e.(~~~:)
>t} oQ= E
when te(o,i), taking (1.1) and (2.1) into account we get
whence follows (2.2). We shall preface the proof of item b) by several auxiliary assertions.
BcQ,
LEMMA 1. If
EcR"
such that
:X.(B)
and EflQ=f)
PCQ (E)= A.( B). p r 0 of.
Let
(3.2)
\E, 1 be a sequence of subsets of R" such that E in Q::= p, and (4.2)
By virtue of (4.2), ~r-P PCQ. (
E,) < C><:l
and, since ~ (E,}=~ (B)< oo, we have s~ P~,.
(E 1 )
< oo.
Hence, from Proposition 3 it follows that there exists a subsequence (for which we retain the notation
\.E,} ) which
converges to a certain set E . As a corollary to Proposition 1,
P(E_)~t.m
P(E,).
, .... 00
Hence, taking into account that
E (\Q= B and equalities (1.1) and (2.1), P,..(E) c... ~.tun ,....... P~,.. . _" (()=Jo.(B).
E. 4
(5.2)
?I.(B)
We obtain (3.2) by comparing (5.2) with the definition of LEMMA2. Let
we get
and E~ be measurable subsets of
R"
Then,
~n(E/'\Ea.)+Rg_ (E~UE~} ~Pen ( E4)+Pen (E.~). Proof.
Let
Gr
(6.2)
be an open set, Gr=>CQ. Then, by formula (4.1),
Pa- (E~)+ P~ (E~') ~ 1\ ( ~ 6 ,+ }'f.~}\\ r,v(~) =:~Po.(\-:('., ~E/ }E~>*-1 )d t= ~ Pc. ({~,+ 1e., -,t J}d-l+ -
i
+~ Pc. ( \ }e,+ 'f,E.,>-\. 1) J·b: Per (t 1 VE, ')+P~ ( E~ f\ E.). Consider the sequence of open sets
G,
such that
<;.,H c c...,
pen(EK\= ~ Pa.., (EK), J ... oo
and
0Gr,-= err.
(7.2) Then
K•-1,2.
Hence, we get (6.2) on the basis of (7 .2). LEMMA 3. Let
Pca(E.K) <(X) •
K-=1\, 2.
•
Set
Pen (EK')=:A.(&,.), then
Pen (E, (\ E2.)=Ptn (E"'),
BK-=E.. n Q Koc -1, 2.,
PtQ (E U E~,)= Ptn (E2.). 4
(8.2)
CONTINUATION OF FUNCTIONS FROM BV(Q} ONTO THE WHOLE SPACE p r
0 0
f.
nQ
E/\ E2.
SinCe
47
n
= B1 and ( E.~u E :~,J Q. = I):t ' by the definition of A' we have
A(B 1 )~Pc9.(E~f1E:t.J, -A(B:~.)*P{n(E~UEt}
(9.2)
By virtue of (8.2), inequality (6.2) can be rewritten as
PcQ. ( EJ\ E~~. )+ PcQ. ( E~U E:t.)~ ~ (B
1)
+A. (
B~ ),
which together with (9.2) proves the lemma. Proof of Item b) of Theorem 1. E~ ={'X' ¥('Zit.) ~
-\.
1 we construct the family
PcQ. CB~) =A. ( E-~: ); B-~: c B'l'
1°. Plan of the proof. Starting from the sets
B-1:
of sets
satisfying the conditions
1\JI Q
=
E.t: ;
when -t. > 't.
Bt
We first construct the sets
\ t,} which is everywhere dense on (-
for a countable set
o0,
o0)
(item 2°), and then for all the other t . (item 3°). Finally, in item 4° we construct the function {(-,c.)=
Bt}
S'"'f \ t: :1t..:
2°. Since
and we prove that
fe BY (Q),
f<:t)
satisfies the conditions of Theorem 1.
by virtue of formula (4.1),
, \-=f.\
we can choose a countable set { t,}
We construct a sequence of sets B-~:. 1)
l\..n Q = E-~:.. ,
2)
PcQ. ( &-~:,) = :>.. { E-t), B
3)
B
c
t,
B-~:
Bt 1 , ... ,
when t · > t · .
\
~
'
B-\:
According to Lemma 1, there exists a set have already constructed the sets
satisfying conditions 1) and 2). Suppose that we 1
such that conditions 1)-3) are fulfilled when i,a=·l,. ,
n-.1
By virtue of Lemma 1, there exists a set
.B<,.)
Bt.";:) Since
B~.,
Bt". 15(n) (1 Q = E. t
t*
' n
Lemma 3 to the sets
P..
u... (\ Q = E-~: ~ Jt ~
e:; .
B~, when
::l
L '
Jr
Q
t
n Q =E..... c
and \)t ' and next, to the sets
"
.
It is clear that
t,, 1.=~ •.. ,1'1-1• Q B-~; (\ =Et • Applying "' "
tn>-1:,. and!\:.,";:) B-~:, when -t,. <.
and u •
.,
be the smallest of those
(B("Jil Bt)U t\..
We set E\~
t, .
<
l\" c
Consequently,
J
-tn
n-1.
satisfying conditions 1) and 2). Let t * be the
smallest of those numbers -\:,, 1.~ ~, .. , n- 1 for which -\:, < tn, and let numbers i:, , i.-= ~, ... , n-1 for which
oo, oo) , such that
<><:>.
i.=~,2, ... , such that
,
'
<<X) for almost all-\:.. Therefore,
if i. ,f. ~ , everywhere countable on (-
From (2.2) it follows that A. ( E-~,1 ) <.
P2 ( E.t,} < 0<:1.
P.Q (E.1)
~ ~
'&
E...'" ' we have
n Bt . and
Bt" ' we
get
pcQ. (&~J=:>-..(E~J.
Thus, for the collection of sets l)t,, ... , B-~:.,, conditions 1)-3) are fulfilled for i,~= ~ , ... , n. 3 °. Let -1: ~ \ t , such that
c:l...
\
< -\:
l.
1.
From the set { t, } we select two monotone sequences \ ot, 1 and { ~; }
o.'J,Jt. and \...,.oo ~ o(, == C.i.m i.~oo
t .
P., =
y,
According to Lemma 1 there exists a set (•)
Consider the sequence of sets It is clear that
troduce the notation other hand,
Bt. = Bt n Bo
~k)
B-e ('\ Q = E-c ~
oo
B= n -t
Kc-1,
~t(\Q= Et ..
(.C)
5~-,;;
(o)
.
and
Since
(•)
Consequently,
(K)
J.)t
c
"-
K . . . {\ >:
(o)
such that B~
(\ Q. = E-t
and
PcQ.
(
(.o))
\)-t
( = A. E-\:).
' K =1, ~, .•..
(K+I)
Bt
(•)
5t
~
.
(o(K))
T")
. By virtue of Lemma 3, ttQ L\
as K~ 00
~
'
P,.,... ( &... )
:A(Et}~PeQ.(B-~:}
'-::n~.
'"l;
Thus,
~
.
b
tc.-+ao
p,.,
C:::.c.
(
( )
=:A. Et . We in-
()<)) =:A.(E t ) •
'E)-t
PcQ.(Bt)=.A.(Et).
On the
SPACE OF FUNCTIONS WHOSE DERIVATIVES ARE MEASURES
48
Now consider the sequence of sets lk)
ct =&... . u Bt.. ,
K== 4. 2.
In exactly the same way as when we considered the sets
&~)
, .... we get that the set
B-~;=
QC~K)
is meas-
urable and satisfies the conditions
Now let -1: and
B""~Q= E-~:,
z>
~Q. ( B-~:)= ~ ( E~),
be arbitrary numbers and, moreover, let t
't'
"'t:::>~.
4 °. Consider the function
t"X:f
1>
t (.':1<.)
defined by the equality
<'l:
•
Then it follows from 3) that
f tox. )= "u.f' \-\: : ac. E. ~t) . We set
and C.t=\~: ftox.)>t) • Obviously, A,.:. ~t ::>Ct. The sets At \Ct
are pairwise nonintersecting and, consequently, vvw.,. (At'\.
tt ') = 0
At ..
forvarioust
for almost all t.
Hence, the sets A,. and C. t are measurable for almost all -1: and, furthermore, Pll" (AI:")=
PR.. <s"' )= ?~~... t ct;.
Let us show that the function
f tx.)
is locally summable. As is well known, the inequality
(VWUA,EJ!lft\- ~C(R,l~\\ (E)
(10.2)
is valid for the subset E. of the ball I~ such that mu.,. E. <. I'YW>,. l~- f. • (In particular, this follows from Lemma 9 below.) Let the closed ball i 1 be contained in Q (10.2) it follows that for any set E c I,. ,
and let l~~, be a ball containing
15
•
Then from
M.CA..E'C (R,s)tl\ (E)+~.. (E. IH1 '))~C. (R,~) t PR" (E)+mv., (E. f\ 15)). By setting E.=- '&t ()I~~, in this inequality when t ~ 0 and equality
PcQ. (
Bt) = ~
(
E... Ill\
~"'-when -t < 0 , and by using the
E t) and estimate (2.2), we obtain
~.. ( Bt n I~~.),C{R,!)tC~ (E,. ~+ww.,. (E/) 18 )when t ~ o,
~ (IR\ I\)~C(R,8)[C~ (~t )+VI'I.t\J(Q\E~n\)when ~
t , from
the latter two inequalities we conclude that
which is equivalent to the inequality
whence follows the local summability of the function "'~ . Applying (4.1) and (2.2), and recalling that
~.. (C-~:}=PI(
( Bt)
for almost all
t, we get
49
CONTINUATION OF FUNCTIONS FROM BV(n) ONTO THE WHOLE SPACE
i.e.,
~E. BY (R") and
estimate (1.2) is fulfilled.
The theorem is proved. The theorem we have proved can be rephrased in terms of the continuation operator, i.e., the operator
An. : f ~
f , which associates with every function
{~
BV (Q)
its continuation
f
E
&V
(R").
As a preliminary we introduce the notation \ .Q I for the greatest lower bound of those numbers 1< for which
~(E)~ 1< PQ.. (E)for all E c Q. THEOREM1'. The operator
AQ_
exists and is bounded if and only if
\QI <<>o.
Here, for any continuation operator An,
IIAQ\1 ~H\OI AQ
a n d t h e r e d o e s ex i s t a n o p e r a t o r
s u c h t h a t \\ AQ.l\-= ~ + I Q I .
Condition (2.2) in Theorem 1 is of a global nature. It is not satisfied, for example, by any nonconnected set Q . This phenomenon can be removed if we relax the requirements on the continuation operator. Namely, the following theorem is valid. THEOREM 2. Let .Q
{E BV(Q)
be a bounded open set.
to have a continuation
f
E
Then, for any function
f>Y (R") satisfying the inequality
llf 1\&v (R•)-$ K (II He.v(Q.) + II~ 1\ L(Q.)}, where
K
does not depend on
f ,
(11.2)
it is necessary and sufficient that there be
found ~> 0 such that for every set there be fulfilled the condition
EcQ whose diameter is less than 8,
~(E)~ CPQ. (E), where the constant Proof.
C
does not depend on E. •
Necessity. Let
the continuation of
j.,E
Ec .Q
and let ~ E be the characteristic function of
E,
while
"' j,E
is
satisfying (11.2). Then, ~
K(PQ(E)+mv...,E) ~~~~EIIl'lvtRl)~ P (\~,1-~>t 1)dt. A
Since
{'X.: -}E > t 1(\ Q. = E when h ( o, 1), from this we get 1-< ( Pu (E)+ mu.., E)~~
D(B).
BnQ.:E
Since
f>::) E , from this and from
the isoperimetric inequality (3.1) we get (12.2)
We set d =~ . Then, from inequality (12.2) under the condition di.o...m
E.
<. ~ , there follows
SPACE OF FUNCTIONS WHOSE DERIVATIVES ARE MEASURES
50 Therefore,
~K~(E)~~ P(B)~:A(E.). BnQ=E
Sufficiency. Consider a decomposition of unity clt(-:.:), i=i, ...,N,
dia.lrl .,urpo(; Since d\o.m
'and \~
E-BY(Q.).
N
-
such that ~ su-pp o<; ~ Q ,
Set ~:.f,=-1 aLi and M-l-:::\~:\<.f;\?
t}
M~ <.a_ for all t"1=-0, )..(M-l) ~CPQ (M~). Therefore, by a verbatim repetition of the proof
of sufficiency in Theorem 1 we obtain the function
<{
E !) V(R") such that
<{ =Cfi
in
.D.
and
II~ II~~)~ (CH)I\ tf.\1~~~ (C+~(IHII&V'!rl.)+ d llf lll.(Q.)\ ~ i)• "'
A
A
We set ~=L. C{,. It is clear that ~-:: {in Q and
The theorem is proved. §
3.
Certain Exact Constants for Convex Regions
Above we saw that the norm of the continuation operator is an exact constant in a certain "isoperimetric inequality." In some special cases this constant can be found. Thus, for planar convex regions this constant has a simple geometric meaning (Corollary 2 of Theorem 3). If ·Q' is an ndimensional ball, the constant is easily computed (Lemma 8). At the end of this section we have considered yet one more extremal problem for a ball, which we shall subsequently need. LEMMA4. Let
~" (E. ) < 00
•
Q
be a finite convex region in
Then the r e eX is t s a seq u en c e
boo PQ(fl)=Pn(E), Proof.
0
Rn
f p 0 l y he d r a
and let
EcQ,
Ill( such that
n" ~
boePcQ (ll")=Pcn(E).
E and (1.3)
Let Qt. be a region obtained from Q by a similarity extension with coefficient 1 +C.
and with center at a fixed point in region Q. By E t. we denote the image of E under the same similarity transformation. It is clear that
f?.l (Et)= (h ?-')•H PQ (E), T1Q/Et)=(1H)0 • 1PcQ (E). Hence it is not difficult to get that (2.3) For almost all
E. ;;. 0 (3.3)
Let the number t
polyhedra n... e.
satisfy this condition. According to Proposition 2 there exists a sequence of J weakly J such that '\rM. 'iJ -~11 'VOlt 'iJ h. • In view of condition (3.3) follows K,f.
(...
We select a sequence of numbers t, satisfying condition (3.3), such that E;, Then from (2.3) and (4.3) follow
~
0 as
l.~
oo.
51
CERTAIN EXACT CONSTANTS FOR CONVEX REGIONS The lemma is proved. We shall need the following elementary lemma later on. LEMMA 5. Let Q polyhedron. Then,
LEMMA 6. Let
be a finite convex region in
n" K and let
n
be a finite
be a finite convex region in R~'~ and let EcQ, q~
Q
The n t h e r e eX i s t s a s e q u en c e
0
f p 0 1y he d r a
n~
.
n. -.
s u c h that
E.
(E)<
o
and
l(~ PQ(n,JK~)=Pn(E'), ~••Ym (nJIQ )=~Q (E_). pro of. By virtue of Lemma 4 there exists a sequence of polyhedra equalities (1.3). It is clear that
nK. nK- E ' satisfying t
According to Lemma 5,
Consequently,
boo ptQ_ ( n" nQ) ~ h PcQ (nK)= RQ (E), ~ p.Q ( nK{IQJ =bo0~ (nK}=Pn (E). Since
nK nQ -
(5.3) (6.3)
E' (7 .3)
From (6.3) and (7.3) follows
which together with (5.3) proves the lemma. LEMMA7. Let P(Q)
Proof.
Then, for any set
PcQ (E)t Since }Q.'=1E+1Q\E, "tt.Jt
exist
Hn_ 1 -almost
every-
E cQ,
qQ (n\ E):H"_
1
(aQ).
v1Q (CQ)~ 'ltatt 'V~E (C0.)t'V'MV1mE(CQ): Pcn(E)+PcQ (Q'\E).
Since a normal to Q exists
H...~ -almost everywhere
on oQ, by virtue of Proposition 4,
~v }n ( CQ)=-PRn (Q)= lill- 1 (aQ} Consequently,
H.~A '3Q) ~pcQ (E)+ PcQ (Q \E). Let us prove the reverse inequality. Let and 0 \E. The sets A* and B.. are and
&"(\ ?J* Q
A""
and & ""denote the reduced boundaries of the sets
Hn-1 -measurable
(Proposition 4). Note that the sets
do not intersect. Indeed, let there exist a point :x: ~ o" Q
common to
E
A•n a*Q
A* and B• . Then
SPACE OF FUNCTIONS WHOSE DERIVATIVES ARE MEASURES
52
the volume density of each of the sets E. and Q \ E at the point ~equals ~/:~.. The latter is impossible since x
E ~"Q. •
Consequently,
It remains
to make use of the equalities
H,_~ (A -n()"Q }= H,_ 1 ( A*f1-o.Q}= PcQ (E)} H.,.. 1 (B·n~·Q)=Hn_, (P>*r\'()Q)=PcQ (Q\E) (see Proposition 4). The lemma is proved. THEOREM 3. If r-.. ~r. is a finite convex region in
Rn,
then the equality
~(E)=~ LPcQ(E.), ~Q.(Q\ E)] is fulfilled for any set E c.Q., P r o of .
P(E)<
(8.3)-
oo.
Let, for definiteness,
B:n (E)~ PcQ. (0 \E) and let the set E:> be such that & (1 Q= E, PcQ( &)= :11.. (E). At first we assume that m,u.11 & < oo. According to Lemma 4, we can find a sequence of polyhedra
Since ~"
&<
00 ' the polyhedra nK are finite.
nK I n. .-
B ' such that
By virtue of Lemma 5,
Hence, from (9.3) we obtain
~ Pc~:t(nKnQ)'PcQ (B). Because
B() Q
nwo. nQ -
' we have
P ( BnQ) ~ ~- P( fiJ' Q), which, together with the first equation in (9.3), yields
Hence, from (10.3) follows
~Q (E')~ Pen (B)= :>..(E.). Thus,
~Q. (E)=:>.. (E). that
Nowleti'\'UJ.)
Since
ll
"· lJ =
C>0 •
Pc,...~L (B)
53
CERTAIN EXACT CONSTANTS FOR CONVEX REGIONS Hence, according to what we have proved,
:A.(Q\ E)= r{~ (Q\ E)
and hence, in view of (8.3),
~(E)= PcQ. (Q\ E)~ Since, obviously, ~(E.)~Pc.Q.
H:Q. (E)·
(E.) , we get ~(E.)=P~Q (E).
The theorem is proved. COROLLARY 1. Let Q
E
where
be a finite convex region.
Then,
is any subset of Q. satisfying the condition
The proof ensues immediately from Theorem 3 and Lemma 7. COROLLARY 2. Let Q
be a finite convex region in
lUI= ;h H
1
R2. .
Then,
('~>.n),
where \, iS the minimum length of the segment whose endpoints separate aQ into arcs of equal length. Pro of.
We take an arbitrary f.> 0 . Let E be a measurable subset of Q such that
o < ~(E)
\Q 1-
Pcs.(E)
n lnl- 8nCnnn) <~c:.
(see Corollary 1). According to Lemma 6 we can find a polygon
such that
Pn (n)
A and B be the points of intersection of ~Q with some component of the boundary of n . The points A and B can be chosen such that the segment of the component of C) n being considered, Let
bounded by the points
A
two sets Q. and Q'. Let
and
& , lies wholly in n. . The straight line segment A~ divides
~n(Q.)~ ~Sl. (0.').
AB
Pcn (n nQ)
>
f.k (n)
"
In\- RQ (eJ) A&
< 2 e.
If ~n (
Let ~n ( Q) <.
B'(:a.Q.
into the
It is clear that
PCQ (Q) and, consequently,
n.
(11.3)
A&~ h
and of the arbitrariness
PeQ ( Q'). We shift the segment Ae, parallel to a new position A' B' ( A'e 'li.Q, ) so that~ (Q1 }= Pc.Q. ( Q:) , where Q" and Q,' are the regions into which the segment A'B'
divides .Q . Elementary calculations show that
R.n(Q,) A'&' which together with (11.3) proves the corollary.
~
Put (a) AB
SPACE OF FUNCTIONS WHOSE DERIVATIVES ARE MEASURES
54
LEMMA 8. Let
A !It:
be the unit ball in
R" .
Then the quantity
equals the area of the spherical part of the boundary of the spherical segment whose base has the area p.
E c. Q , Pn (E) .... p . By virtue of Lemma 6, there exists a sequence of polyhedra n~ n~- E, such that Pn (nO\)-. p, and ~n(n ..nn.)-.PCQ.(E). Proof.
Let
We carry out the spherical symmetrization of the set n~t 1\ Q relative to some ray
with ori-
, symmetric relative to e, , with a piecewise smooth
gin at the center of Q. We obtain the set n~ boundary and such that
n: (\
QK denote the spherical segment, the spherical part of whose boundary is '(I 'a Q . It that p.Q (Q ...)-' Pg_ (~) and ~Q ( ~) = qQ. (n~)' from which follows the assertion of the lemma.
Let is clear
.t
The following assertions are derived from the lemma by simple calculation. COROLLARY 1. If Q
2.
is the unit ball in
R",
then
\Q\:3_· 2"11'",._,
("\ If lc.
is the unit circle in R~ , then for any Ec.Q the inequality
is valid. 3.
If 'Q
is the unit ball in
R3
, then for any E. c Q the inequality
is valid. Let .,)
(V)
denote the greatest lower bound of the quantity
for which mv.., (E.)= V~ ~. LEMMA 9. Let Q be the unit ball in
R".
11, ( E.) on the set of those E c Q
Then
v(V)=I1_(s) ,
where S
is a
ball orthogonal to .Q and such that
mv.., ( Proof.
s1\ Q) =V _it
We restrict ourselves to the case
V< 'II::i: .
examine only the set E. which is the intersection of Q symmetrization of metric relative to
E relative to ray
As in the preceding lemma, it suffices to
with a polyhedron. We carry out the spherical
~ with origin at the center of Q
and we obtain a set
E' sym-
t , such that Y"tl.fhn (E')=rnth (E) and~ (E')' \\l_ (E.)~ 11
Let 2 beaballsuchthat~(811Q.)=~n<E') and81"\'I>Q.=oE'C\'aQ. Byvirtueofthe isoperimetric property of a ball,
11_ ( s') ~ Pn (E'). Elementary calculations show that among all the balls smallest value of the quantity
Pn ( 8 ')
tCompare with [11], Appendices A and C. ttin case V= ~, the ball S degenerates into a halfplane. tCompare with [11], Appendices A and C.
2
such that VY~V;"
(
8 () Q )= eon'!.t , the
is given by the ball orthogonal to Q., which proves the lemma.
55
THE ROUGH TRACE AND CERTAIN INTEGRAL INEQUALITIES It is not difficult to verify that for balls
c::,(R}
of radius
R , orthogonal to the ball
Q , such
that~.(S(R)f\Q)< ~ , the function ~ (<;, (R))
vn.v.: (s(R~IW£) decreases. Hence, we instantly obtain the following corollary.
~(~
COROLLARY. The function any set
E c Q.
decreases on
(o, l-)
In particular, for
there holds the inequality
" " rnM-~[mv., .. (E), ~.. (Q\E)]~ ~ 'li-= [PQ (E ))l=ii,
(12 .3)
where, moreover, the constant is exact. §4.
The Rough Trace and Certain
Integral Inequalities On the reduced boundary of R
we define the "rough trace" ~If of a function ~E. BY (.Q). We
set
where x.e e>*Q • t It is clear that if the function ~ has a limiting value at a point Q:.E:-'b*Q, then ~*(-:c)=~~
¥C'a).
LEMMA 10. Let H,..1 (Q) <. oo. Then for any set Ec:.Q , from PQ(E)
nK • n14c.Q 'niC.- Q Furthermore,
H..-1 ('
' such that
EC'\ n... -
c..
< 00
COROLLARY. If H,._~ (..,Q)<<>Oand ~e-&V(Q), then P(E+)
Hn_ 1-measurable
and ~t:B\i(Q).
Then the rough trace f'<~) is
function on i>"".Q. and, moreover, for almost all
t, (1.4)
Proof. the definition of
Led:besuchthatP(l:\)
r it follows that
Bt::>Cl.. E..J'11l"Q . As is known (Proposition 4), the set ·a"E..,("I (J"Q
is measurable relative to H...1 • The sets and, consequently, for almost all 1: ,
Bt \ l:
for various values of
t
do not intersect
H..., ( 5t\ [-a•E-t rl1l11 Q]')=o. Therefore, the set measurable. THEOREM 4. Let
Bt
is measurable for almost all t. and, consequently, the function
P (Q) <.
oo and let a normal to Q
where on 11Q..
tThe least upper bound of an empty set is taken to be equal to --.
exist
f" (~)
H..... -almost
is
every-
SPACE OF FUNCTIONS WHOSE DERIVATIVES ARE MEASURES
56
Then, in order that the estimate (2.4)
~ ~ \r- c \H •.• (d~1~ Ku~ llj!,V(n) ,
t aQ. where K is in dependent of f , h o 1 d for any function ~ E &V (Q) , it is necessary and sufficient that the inequality
be true for any set Proof.
~Sl (0\ E)}"KP.Q (E)
R.n (E),
MMI \
(3.4)
Ec:Q..
Necessity. Let EcQ, ~(E)
P(E) < 00 •
Then by Lemma 10,
Let }E be the characteristic function of set
E.
Then
Wf ~ \'}.: ('le)- c\H,.~, (d~~:.}=nt'\ \~-ciH,_, (a"E f'I?J•n.)+ e~
[The last equality is valid since, by hypothesis, On the other hand, ~-~-E~ Sufficiency. Let~
E
=PC'- (E).
&V(Q)
H,_,(a.Q\'71"Q)=0 .]
Applying (2.4) we get (3.4).
IW (..Q) . It is clear that H,_~ (oQ {) 7J*t:t}
, as
a function oft , does not
increase. Indeed, let -:c. e r." Q (\a" Et and let te < t . Then,
1= ~ ~'11"., ~" mu..., (Q (\I,)~~ 2'1T., ~ .... 0
~ .... 0
f
~JE_ (\ 1.. ) ~ ~ ~'IT. t"·"vnv. .. (L (\ 1 .. }=~I •
>
~-.·o
n :1
"
'
i.e., -x. ~ <>,. .Q. f\ o'" E-c. Analogously,
H... l<~Q \
o"
Et)
is a nondecreasing function of t .
By virtue of formula (4.1) we have K
11~~&~(Q.) =KT PSl.(E.tjdlc ~Ll.H. .,(-aQn~·s,>. H..J~Q\'ll"Et11Jt. J J -oo
We set
t. .... sl.l.D\t: P(E.J
-~=.
00
'
Then, t.
OQ
K\H\1 fiV(Q..')~ () Hn-, ('llQ lh E'1:')J\:)l 1-1,.., (<JQ \'li"Et:') dt= 10
to
1:
-<>e>
\H,., U.'lt: ~'\-x.)~ t 1)dt. -~-~ H,_, ( \\lC' f'<x.) ~t 1) dt: -oo
'to
~ \ t~\x.)-t.fH... ,(J~)+~tt*(\lC)-tJH,Jd ~ .. ~ \ f<:~t)-t.lH,.., (d"). ~.n
• aR
1Sl.
Consequently,
which proves the theorem. From Corollary 1 of Theorem 3 we get that for a convex region Q the exact constant in (2.4) equals
I.Q.I.
In particular, for a planar convex region this con-
s t a n t c o i nc i d e s wi t h t he r a t i o of
t H" (~ Q)
to t h e 1 e n g t h o f t h e s m a 11 e s t c h o r d
dividing ()Q. into arcs of equal length (Corollary2ofTheorem3).
57
THE ROUGH TRACE AND CERTAIN INTEGRAL INEQUALITIES
for
AccordingtoCorollaryl of LemmaS, the exact constant in (2.4) equals the unit ba 11. Let
Ac Q.
Let S (A) denote the greatest lower bound of those
K for which
PCQ. (E) ~K p~ (E) for all sets E c Q satisfying the condition m.rAn (
Ef'l A)+ \1n-~ (A tl ~·E)= 0.
THEOREM5. Let P(Q)
fV\Ih•.Q)=o,
Then, for any function
~t:BV(Q')
(4.4)
exist Hn-~-a1most every-
such that
{(M~)=O
and
the inequality
~
\+*\ 1-\n-~ (d-x.) 'er (A) 1\HeN(Q.)'
(5.4)
"l)Q. where, moreover, the constant G'(A) is exact, is fulfilled. Proof.
Wehave
By virtue of formula (1.4) the first integral in the right-hand side equals
Note that
V"IUhn
(AI'I E.t:}+\-\n-1 (Ati~·Et)=O
for almost all
t l'O.
Consequently, by the definition of~ (A),
Analogously we obtain
Finally we get
In order to show that the constant ~ (A) is exact, it is enough to set ~ ... ~E in inequality (5 .4), where E is a set satisfying condition (4.4). We introduce the notation
<;(.'~):::sup\ 6 (A):
Ac. ?>Q , 1-\n-•
(~Q\ A) ,-r}.
From the last theorem we obtain the following obvious corollary.
P(Q)
Hn_ 1-almost everyThen, for any function ~E;.lW(Q) such that H._(\"JC::=~*<.:.:)-:#:o))~'t',
COROLLARY. Let
where on<~Q. there is valid the inequality
exist
'
58
where, moreover, the constant S("=) is exact. From Lemma 8 it follows that for a ball, S(-.:) coincides with the ratio of ~ to the area of the base of the spherical segment, the spherical part of whose boundary has the area~. In particular, when " ... ~
when
,,..
n=~
~("1:) = THEOREM 6. Let where on,.Q..
P (Q) < oo
lf'JI"-'t"
and let a normal to Q. exist
Hn_ 1 -almost
every-
Then, in order that the estimate (6.4)
where the constant
K. is independent of
~ , hold for any function
fE BV (Q)
, it
is necessary and sufficient that there exist $ >0 such that for every measurable set cc.Q whose diameter is less than ~ there be valid the inequality (7.4)
where the constant K" is independent of E.. The necessity of condition (7 .4) is easily obtained by setting f• ~E in (6.4) and by applying the isoperimetric inequality. The sufficiency can be obtained from Theorem 5 if we make use of a decomposition of unity (compare with the proof of Theorem 2). Remark. If each of the sets Q1 and Q a. satisfies the conditions of Theorem 6, then their union also satisfies these conditions. The proof follows from formula (6.2). THEOREM 7. Let where on -.Q..
P(.n) < oo
and let a normal to Q. exist
H~~-1 -almost every-
Then for any ~Eo &V(Q) there holds the inequality
(8.4) -.I..
where, moreover, the constant ""'" " is exact. Proof. We have
where M~= ceed
\. "¥:.: \f I ~ 1:.}.
Since the function ~" Mt does not decrease, the last integral does not ex-
59
THE ROUGH TRACE AND CERTAIN INTEGRAL INEQUALITIES Therefore,
t.) H~~ J~1~ ~ ~c~. M~TH~ ~(~JJdt+ 1(,,,,\(Q\Erdt, -
°
2
where
Et= \ ~:
~('X.) ~
-
t} .
0
0
--
According to isoperimetric inequality (3.1),
(~ .. E0~~ n'lr:-k P(Et)=n~~[Pn.(Et}+H ..-1 (~"~n?tQ)]. Since for almost all
t.
H... , (a"Etfl'll"Q~=H._,\{:x:T~\:.})
(Lemma 11),
Keeping in mind that by Lemma 7, P\~="'+R (Q\E~=P(Q'), cQ·""1:} CQ: 0
0
0
-.a
-oa
-oo
we obtain, analogously,
~ (~.(Q\EJfdbn;.,~[ ~ PQ(E0Jt+ ~1-\,Jt:x::r~q)dtl.
Consequently,
[\ \~\;_'LJ·~·~ ",;:\_\ PQ(E~}J t ~\ ~ ..) -~
Q
{~ ·.lf'l ~t J)d~=nY~* (II~ llr.v(Q) +) \tl H...,Cd~)}. ~Q
0
The exactness of the constant in (8.4) follows from the fact that (3.4) becomes an equality when
r=~"4:
where I~ is a ball in Q. . The theorem is proved. In conclusion, we derive one integral inequality which does not use the concept of a trace. Let
Ac. Q
•
Let ~ (A) denote the greatest upper bound of those K for which
Wllh~ E .:s K Pg, (E.) for all sets
E c. Q. satisfying the condition
rnl.b,.
(En A)· 0 .
THEOREM 8. For any function .f£BV(Q) such that the inequality
wh e r e , m o r e o v e r , t h e c on s t a n t Pro o.f.
f-' (A)
i s e x a c t.
As was shown at the beginning of the proof of Theorem 7, 00
11 ~t;l!:r d~ ~r )(t!IV)., M~!;/- dt1~ Since
wn.v.." (M~n A)=o when t> o,
Consequently,
tCA):O, there is fulfilled
SPACE OF FUNCTIONS WHOSE DERIVATIVES ARE MEASURES
60
In order to prove the exactness of the constant ~
(A) it suffices to set
t= } in (9.4), where E 6
is a subset of Q. , satisfying the condition rnv..n (E !'I A)=O . The theorem is proved. We introduce the notation
t('t)=sup{~(A): Ac.Q,
1'1'\U>n
(~\A)-'~}.
From the last theorem we obtain the following obvious corollary. COROLLARY. For any function ~E. there is valid the inequality
f>V(Q) such that~ (\.~·.\f<-x.)\ > o})"'t', "'
where, moreover, the constant tC'l:) is exact. From Lemma 9 it follows that for the ball Q, wh en
v,.
O<'l:
~T,
where v(-t)=Pn (S), where 5 is a ball orthogonal to Q
such that
when o <
5,
' 1: ~'ITn.
The Trace of Functions from e,y(Q.)
on the Boundary Let Q be an open set in
R"
and let the function
l
be summable in the neighborhood of a point
-.c.~'ClQ.
We give the names the upper and the lower trace of function fat the point to the numbers
'X.,
respectively,
~('¥:.)=~ ~n.(l~~)"Q) ~~('a)~~' Isn~;t
where I~ (.x..) is a ball of radius ~ with center at x.. If~
('JC:.) ..
~
('Jt.) , then their common value is called the trace f
LEMMAl2. Let ~~&V(Q), f~o
~ 11"R
and
lr('Jt.)\
Hn_, (h.)
of the function ~ at the point ""'~ 'liQ.
61
THE TRACE OF FUNCTIONS FROM BV(O) ON THE BOUNDARY ~ea•Q.
Then for any point
there holds the inequality (1.5)
tion
f
Proof. By virtue of Theorem 7, the function is defined.
f C<>c.)
r (1)= 0.
Inequality (1.5) is trivial if
is summable in .Q and consequently, the func-
Let us assume o o
and a t such that o < ~·(:lt.)- t < t. and ~(E.\)< oo . Then,
-a"
-:x;. E:
Et, where Et = {~: t (~) ~ -\.1,
It is clear that the normal to E.t at the point -x.. coincides with the normal to Q. • Consequently, we can find -e. ~ "J<.) > o such that when o <"t. ~"t. lox.) ,
(2.5) Since
..
~c'a) d~ =~ ~" (~ f\ It. <x.)J d't',
\
0
from (2.5) we get
11\L\.,(I'-nn)
~ tc1)J~ )
t
111&11
"
~nn
{~ nS1) ~t'IUI.I" (E'I!tH,)d
.
t
""4"
(~nit.) ~ (4-t)t
r1114,.(Q nit.)
,
which proves inequality (1.5) when ~·c"") < oo. In case { •
(oat.) .. oo the arguments are carried out in similar fashion.
THEOREM 9. Let P(Q.)<. oo and a normal to Q on '1)Q. •
If
fco f>V (.Q)
exist
H.1-almost everywhere
and
\ \r\H., (~oat.)< oo,
-.a then the trace ~ of the function coincides with the rough
f
trace~
II·
exists
H""'
-almost everywhere on -aQ
.
P r o of • A ccord!.ng to Theorem 7, the function upper and lower traces • and i are defined.
t
is summable in Q. and, consequently, the
We first examine the case of a nonnegative function
t.
Then, by Lemma 12, for all
'X.
E. 71*.Q,
.! t~)) ~·('X.). Let us now prove that if ~ E
+~ o
, then the
inequality
f (~) s. t• (-:a:.).
-aQ..
We assume that
Then we can find c."> o such that
H,._ 1 ( Q)
U= \ -x.:
~ 0, where
'l(,
and
E: ?)•.Q.
l (:lt.) >{ .. C::~t.}tt. ~.
Recalling the definition of ~ <.-:a:.) we have when ""~ Q. ,
holds for H,._,-almost all
62
SPACE OF FUNCTIONS WHOOE DERIVATIVES ARE MEASURES
Since ':lC.
4: ?>• .Q.
(3.5) Therefore,
As a consequence of equality (3.5), Y\'W)"
(E~n I~<.:x.))~ ct~ ~ [mu" (Etf'\1!(~)),
where o(.f does not depend on
t
and oe.f - -1 as ~
+
m.v.." (I~<~)\ Et)J,
o.
Applying the isoperimetric inequality in Lemma 9, we get (5.5) By noting that
and by integrating inequality (5.5) with respect to
t , we get
1 OUA;\Etl) I~(~)) dt.~ a~;(~J~ v~~~{ Tvt\11. vf.E..(l~rwi)dh v
r
+~Hn-l (.:Q.()-GE~:)d~=~(!t v~:. tWttvf (~')+~ ~· (~)~t)d~)l. o
(6.5)
-a"QIU~
Comparing (4.3) and (6.5) and taking into account that
a~.._-.
1 as
~-+
o , we
c1-f c~h~:!~~.o tM.. ~-"'l1flK.v~(~)+~ { .. ~ t*<~ )Hn-i(d~)}. ~~o
obtain (7.5)
'll"2nlrt:x.)
According to formula (5.1), for f-\ 11• 1 -almost all -x. E: ~· Q. ,
~~ ~~" But, 'liM. 'V ~st. \l..)=H""1 written in the form
'lrM
v~R (I~<.x.)J='lTn-1 ·
(,"Q ill~. Therefore, for 1-\"_1 -almost all
4
~ Q , inequality (7.5) can be re-
The integral l ( E') = ~ ~ * ( '3) 1-\"_ 1 ( d~) is absolutely continuous relative to the measure 1-\"_1 (E.') . Therefore, the derivative
!:.
..,HI
" .,..,
exists for
H,.. 1-almost all
c.'X.)"'~ !-+ o
H ~ ~ (rc'a)H"_,(c{~)=-t·c~) ~>-~ (;tQ f\ I,(x.') ~ ')•n n I~t"-)
':t.~ ~·Q (for example, see [12], p. 290).
THE TRACE OF FUNCTIONS FROM BV(Q) ON THE BOUNDARY Therefore, for
Hn-l -almost all~ E- ~ , C '1Tn-l
inequality (8.5) can be rewritten as
~ fM,., (~ '1rM
V
~...,.o
Since
'\li)1\.
v ~ ( R"') <00 , and
'1TM
vf (U J=O,
63
f ( l~ J.
(9.5)
on the basis of the well-known theorems on densities (see
[13], Para. 3.3 and 3.1), from (9.5) it follows that 1-1~-• ( Q)= 0. The assertion is proved. Now let ~ be an arbitrary function from B'J (Q"). Then, the functions{+= 1; ( ~t\fi), also belong to
BV lQ.) .
f=HifH')
By virtue of what we have proved above, the equality (10.5)
is valid
H,.. 1-almost everywhere on<~" Q.
• Consequently, the trace
f of the function
~ exists
H,_,-
almost everywhere on CI*Q and, moreover, (11.5) On the other hand, it is clear that
so that always (12.5) By comparing equalities (10.5)-(12.5) we see that ¥(1<)=-f*<~). The theorem is proved. The hypothesis of Theorem 9 may be weakened somewhat for a characteristic function. Namely, the following lemma holds. LEMMA13. LetP(Q.)
}e
exists for
1-1.,_1 -almost all
;JC..
t<~•.Q
Then the trace
and coincides with
1-r:. '1-e*.
of
We set 5K={:x:.:'JC.E:'Il*Q, _e j. f..x),1,4.f'e>~)~<·~.\ .... Since ,
Proof.
};= J i
when
-;x:.~: -a*Q r\ -a•E,
\ 0 when x. (, a... n.'\ 11•E, the functions
1te
and J c• coincide on the set 71"Q" ~~
U ~K
~~
•
H.,. 1 ( S,.) = 0 , K = 2.,:. , . . . • Since S,.. c. <~* .Q , when ~ ). ~ ~"(Enlrc~)) R. ~ -n l"e<~)= ~ ... o ~" (Q..n lrt~)) = 'II;. ~-..o ~ 1'1'111.1" (Enii'Cx.))~ f.
It remains to prove that
By virtue of Lemma 10,
E:
S" , (13.5)
P (E) < oo. Therefore, it follows from Lemma 9 that ~n (Ef\I~C"JC.))~C.('\1Qfi.VR.,}E (I~t>(.'))]~'.
Comparing this with (13.5) we get (14.5)
64
SPACE OF FUNCTIONS WHOSE DERIVATIVES ARE MEASURES Since
<:i" (\~,. E= cp,
'lrOI\.
v}'£ (S,.') =0.
Therefore, by applying the well-known density theorems
(for example, see [i3], Para. 3.3 and 3.1), from (14.5) we obtain
H""" ( s.. )• 0
THEOREM 10. Let P{Q.)< oo and let a normal to Q. where on 'IIQ .
, which proves the lemma.
H•. 4 -almost every-
exist
Then,
if the inequality
1)
(15.5) where K is independent of E , is fulfilled for any measurable set, then the
f
trace
exists for any function te~'V(.R) and, moreover,
~ ~ \ f-c\Hn-• (d-x.')'K HRIW(A.)'
(16.5)
~R
2)
if inequality (16.5}, with a constant
for any function ~e:BV(Sl) having a trace
f
K independent of
on ~Q. , then estimate (15.5) is
true for any measurable set E c .Q. Pro of.
~, is valid
1) By virtue of Theorem 7 the rough trace of the function
f
+is summable on 'l>Q.
Consequently, by Theorem 9 the trace exists H,._,-almost everywhere on 'liQ. and coincides with Therefore, inequality (16.5) follows from inequality (2.4} of Theorem 4. 2) Let
trace t~~E
;;E
E be a measurable subset of
of the function
'f.E
Q. such that
Pa (E)<
OQ •
exists H,.., -almost everywhere and equals
in(16.5)weget
\
t•.
By virtue of Lemma 13, the
jE•.
Therefore, by setting
~ ~ ~=- e\H (d~) ~ Pn (E.), iln
11• 1
K
which is equivalent to inequality (15.5) (compare with the proof of necessity in Theorem 4). The theorem is proved. The following assertion was actually obtained in [8]. We present it here, with its proof, for the reader's convenience. LEMMA 14. For any function ~eB 'V{Q) and any measurable set & ca holds the equality
there
00
v ~ c& ) .. ~ v f.E
-....
..,
(f>) H ,
{17 .5}
where E~={~: '(-x.):d. ). It will suffice to prove (17 .5) under the assumption function with a compact support in .Q. Then,
f ~ 0.
Let ~ be an infinitely differentiable
-
- ~ttv ~Cdx.)·~ !v~(~':lC.}= ~ ~i.E <~)Jtvct
-~ Jt. ~ } o
St
E..,
(-x.) 'Cot
65
THE TRACE OF FUNCTIONS FROM BV(Q) ON THE BOUNDARY We note further that for almost all
1: , -
00
~ ~v !Cdx) .. ~ Jt ~qv}E (~~')=~f{Q~~v}E Jt. .st.
st
0
Sl
t
0
(18.5)
\:
Here it is permissible to reverse the order of integration because the finiteness of the integral 00
\ H ~ \1:{\'l!Wt.v}e.t(~~') St
follows from equality (4.1 ). Equality (17 .5) follows immediately from (18. 5). THEOREM 11. (The Gauss-Green formula.) Let P(Q)
H,_ 1 -almost
everywhere on-a.Q.
Then, for any function
n_,
rough trace is summable on the boundary of equality
tE
~VCQ.) whose
there is fulfilled the
v~ (.Q')=-) ~\x.)~e.x:.)r\"'_1 Cd:x.), 7151 where1ic:~)
is the normal to Qat the point
"01:,..
Since v~E(R")=o foranyset E. suchthatP(E.)<.oo, byvirtueofLemma14,
Proof.
-
""'
v-\-(Q)=-~ V }E (CQ)dt=-\ V }E --
Since
11,_1 ( ~Q. \ 11"Q) = 0
t
--
(-aQ ~--a•Et)J-t. t
and since the normals to E~ at the points of a"Q II<~* Et coincide with the
normals to Q ,
v }Et('ll Q () ')* Et')= ~ rt (-x.) 1-\11-1 (d:x.') = v ~2. (ll*E.t), ')•n 1\'i e. t where
n <:x..)
v ~(Q.)
is the normal to .Q . Thus, Qro
=-)
0
v }n (()·E~)d-t- ~ vf.n ('ll*Et')Jt
o
-oo -
0
0
-oo
c:>o
0
0
-oo
=-) v ~st(-r.•E.._)dt-+ ~ v}st C~*Q\ -a•E..)Jt=
=-~v~J~~: t\t })dt+~ v~n({~:(' t})dt=-js;~Jd~):·!tcx.)nl~)H_~~x). 'iiR
'!ISl.
The theorem is proved. Simple examples show that the condition tion of the trace.
H,_
1
("'I~\ u• .Q ')-== 0 cannot be removed under our defini-
Let us cite some more assertions concerning functions from ~'J(Q). Here we assume everywhere that
P (Sl) < oo
and
H.,_, (-aQ\~*Q) "'0·
The following corollaries are obtained immediately from Theorems 11 and 9. COROLLARY 1. If f o r any s e t
~
Ec Q
,
U\nC\:.), Pen (Q\t.)}:f
66
SPACE OF FUNCTIONS WHOSE DERIVATIVES ARE MEASURES
1<.
w h ere
do e s no t de pend on
E. , the n for any function { E ~ V(Q.) t h e r e ex i s t s
the trace {t.,._) and there holds the Gauss- Green formula:
v 1 Cn) =
~ ~ (':IC.) ~ex) 1111_, cd~).
,n. We introduce the notation 0 e
T ('X.):
{'<~~t.)when ~e:Q, t. when
':lt. E
tQ.
COROLLARY 2. There holds the equality (19.5) Indeed,
It is clear that
....
~ P~ U~ \t-c.l >t1) dt= Hll&v<StY
(21.5)
=
Q
Further, since 1-\ n-l
(
~ Q. '\ ~ •
Q) =0 ,
00
Oo
0
0
~ ?ta(t~= H-c.J>t})dt·~ H.... (\:~: Cf-c.)*>t})~t+
0
+~
t\lt-1 ({.~: (t-c.)*d= nd~= ~
-~
l
a~
an.
which together with {20.5) and {21.5) proves equality (19.5). Let
BV (n) denote the subset of BV( .Q) , consisting of functions for which IHo ~ fi'IC..") = II f B.. <'"")' o'l o
Then it follows from {19.5) that for the class of regions being considered, { ~ ~V
n
~·- 0 . Hence, if the region
traces
f
Q satisfies condition (22.5), then the classes of functions having equal are the elements of the factor-space &V (.Q) /&0V(~).
We obtain the next corollary from formula (19.5) and Theorem 5. COROLLARY 3. If for any set
Ec
.Q
~ [~n. (E), ~a ln\E)]' KP.a (E),
(22.5)
where the constant K is independent of E. , then we can find c:.
\\{~~ll,'~
such that (23.5)
Conversely, if for every function ~E ~V (Q) we can find c. such that inequality (23.5) is fulfilled, where
k
is independent oft , then inequality (22.5) is valid for any set
Ec
Q..
THE TRACE OF FUNCTIONS FROM BV(n) ON THE BOUNDARY
67
COROLLARY 4. In order that the inequality (24..5)
with a constant K
independent of
t
be fulfilled for any function fE&YUl), it
is necessary and sufficient that there exist a number CS>O such that the inequality
where K~ d\o..m E < ~.
is independent of E , is valid for any measurable set E. c.Q. ,
The necessity follows instantly from equality (19.5) and from the isoperimetric inequality. The sufficiency follows from (19.5) and from Theorem 6. We obtain the next corollary from formula (6.2). COROLLARY 5. If each of the open sets ~, and Q,_ satisfy the conditions of Corollary 4, then their union also satisfies these conditions. Hence,inparticular,itensuesthatfor sets representable as the finite union of regions with Lipschitz boundaries, any function from l)V(U) can have a zero continuation onto the whole space in such a way that inequality (24.5) is fulfilled.
LITERATURE CITED 1.
2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
V. G. Maz'ya, "Classes of regions and imbedding theorems for function spaces," Doklady Akad. Nauk SSSR, 133(3), 527-530 (1960). [In English: Soviet Math. Doklady, 1(4): 882-885 (1960).] V. G. Maz'ya, Classes of Sets and Imbedding Theorems for Function Spaces, Diss., Moscow State Univ., Moscow (1962). R. Caccioppoli, "Misura e integrazione sugli insiemi dimensionalmente orientati," Atti Accad. Naz. Lincei, Rend., Cl. Sci. Fis. Mat. Nat., Ser. 8, 12(1): 3-11 (1952). R. Caccioppoli, "Misura e integrazione sugli insiemi dimensionalmente orientati," Atti Accad. Naz. Lincei, Rend., Cl. Sci. Fis. Mat. Nat., Ser. 8, 12(2): 137-146 (1952). E. De Giorgi, "Su una teoria generale della misura (r -1)-dimensionale in uno spazio ad r dimensioni," Ann. Mat. Pura Appl., Ser. 4, 36: 191-213 (1954). E. De Giorgi, "Nuovi teoremi relativi alle misura (r -1)-dimensionali in uno spazio ad r dimensioni," Ricerche Mat., 4:95-113 (1955). H. Federer, "A note on the Gauss-Green theorem," Proc. Am. Math. Soc., 9:447-451 (1958). W. H. Fleming and R. Rishel, "An integral formula for total gradient variation," Arch. Math., 11(3): 218-222 (1960). W. H. Fleming, "Functions whose partial derivatives are measures," illinois J. Math., 4(3):452458 (1960). K. Krickeberg, "Distributionen, Funktionen beschrankter Variation und Lebesguescher Inhalt nichtparametrischer FHichen," Ann. Mat. Pura Appl., Ser. 4, 44:105-133 (1957). G. P6lya and G. Szeg5, Isoperimetric Inequalities in Mathematical Physics, Princeton Univ. Press, Princeton, New Jersey (1951). H. Hahn and A. Rosenthal, Set Functions, The University of New Mexico Press, Albuquerque, New Mexico (1948). H. Federer, "The (cp, k) rectifiable subsets of n space," Trans. Am. Math. Soc., 62: 114-192 (1947).
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