Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZUrich
268 Christian G. Simader Mathematisches Institut der Universit~t M~Jnchen, MLinchen/Deutschland
On Dirichlet's Boundary Value Problem An LP-Theory Based on a Generalization of G&rding's Inequality
Springer-Verlag Berlin-Heidelberg
New York 1972
A M S S u b j e c t Classifications (1970): 39 A 15, 35J 05, 35J 40
I S B N 3-540-05903-2 S p r i n g e r - V e r l a g B e r l i n • H e i d e l b e r g - N e w Y o r k I S B N 0-387-05903-2 S p r i n g e r - V e r l a g N e w Y o r k • H e i d e l b e r g • B e r l i n This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payabie to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin - Heidelberg 1972. Library of Congress Catalog Card Number 72-85089. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
Contents
Outline
I
A priori estimates for solutions of linear elliptic functional equations with constant coefficients
~3
8 I.
Some definitions. Formulation of & basic a priori estimate
13
8 2.
Construction of certain "testing functions"° Analytic tools
21
83.
Proofs of local and global a priori estimates and regularity theorems
41
Chapter I
: A representation for continuous linear functionals on ~ ' P ( G ) (I < p < ~ ) and its &pplications: A generalization of G~rding's inequality and existence theorems
84
84.
A representation for continuous linear functionals on w~'P(G) (I < p < ~)
85
85.
Bilinear forms and a generalization of the Lax-Milgram-theorem
97
86.
Some coerciviness inequalities generalizing G~rd!ng's inequality
101
Existence theorems in the case of uniformly strongly elliptic Dirichlet billnear forms
121
Chapter II
Chapter III
: Regularity and existence theorems for uniformly
133
elliptic functional equations
§8.
Some properties of the spaces
§9.
Differentiability theorems
137
§ 10.
Fredholm's alternative for uniformly elliptic functional equations
163
§11.
Further regularity theorems
186
wk'P(G)
134
IV Appendix
I
200
Appendix 2
228
List of notations
230
Bibliography
234
At this point I want to thank my academic teachers Prof. E. Heinz and Prof. E. Wienholtz.
Further I want to thank the editors of the
"Lecture Notes" Series,
Prof. A. Dold and Prof. B. Eckmann,
Springer-Verlag
and
for publishing my manuscript in this series. Last but
not least I thank Mr. G. Abersfelder for typing the manuscript.
Outline
The starting point of the study of elliptic boundary value problems was Dirichlet's p r o b l e m for the Laplacian
A .
lized as follows.
be a uniformly elliptic par-
Let
L =
7-
as(x)D s
This question is genera-
tial differential
Isl<2m operator o T order 2 m (m c ~ ) (see Definition
list of notations,
p. 230 )
with complex valued coefficients
a bounded open set G c ]R n (n ~ 2 ) j
where
8G
1.3 and
defined in
e Cm-1. The classical
problem is then Problem
(C):
Let f e C°(G) and let
@j 6 C ° ( ~ G ) ,
j = 0 .... ,m-l,
be given. Then
the classical Dirichlet problem consists in finding a
u ec2m(G) N
N cm-1(~) such that Lu
= f
(~)
J u
in G
=
and
@j
on
~G,
j = 0 ..... m-1.
Such a u is called a classical Here,
~-~
normal of
solution.
denotes differentiation in the direction of the outward ~G
.
Since the problem is linear, neous boundary value problem
it is sufficient
(that means
@
to treat the homoge-
= 0 for j = O , . . . , m - 1 )
, if
J the
@j
have certain differentiability properties.
In the case of m = I this problem was very successfully treated by Schauder [55], ri
compare e.g.
[47]. His method is based on sharp a prio-
estimates which are derived from pointwise
solutions with the aid of singular integrals. (and more general boundary value problems) available only when
F.John
[27]3[28],[30]
representations
of the
For higher order equations
the necessary tools were in 1951 had constructed the
fundamental
solution and when Agmon - Douglis - Nirenberg
[4]
in 1959
discovered their Poisson -kernels.
Up to that time a very successful Hilbert It is based on S.L.Sobolev's is
calculus of generalized derivatives
and
intimately connected with the classical Dirichlet ~s principle.
(compare cients of
[15], [16],[21]). L
It can be described as follows:
are sufficiently often differentiable,
partial integration a bilinear form B L (1)
=
holds f o r If
space method was used.
u
every
=
@,~ c Co(G
=
BL[U,@]
(f'@)o
}-
(C) for
for every
we get by means of
such that
(compare Remark 9 . 1 3 ,
is a solution of Problem
If the coeffi-
@j = O,
p. 158) j = O, .... m-l,we get
@ ~ Co(G).
This identity gives rise to the following w e a k e n e d problem: Problem
(We) :
Let f ¢ L 2(G), where G c ~ n
is a bounded open set. Then the weak
Dirichlet problem in L 2 with generalized homogeneous lues consists in finding a
(2)
BT,[U'~] Then
u
=
(f,~)o
is called a weak
u c ~'2(G)
boundary va-
such that
holds for every
@ e C~(G).
L 2 - solution.
The essential tool in solving this problem is G~rdin~'s
inequality
(compare Definition
[20]:
If
L
is uniformly stronsly elliptic
1.3), then there are constants
Cz > O, C 2 ~ 0
that
(3)
Re
BL[U,U] > c~ llull~,2
C~ Ilull~
for
every
u ~ um'2(G)
such
From this inequality one easily deduces with the aid of the representation theorem of P.D. Lax and A.N. Milgram that equation
[37] and with Rellich's lemma,
(2) is equivalent a Fredholm equation in the Hilbert space
~,~'2(G). Therefore,
Fredholm's alternative
L is uniformly strongly elliptic
(see e.g.
(W2), if
applies to Problem [3],[17],[49]).
A far more difficult question is to know whether a weak solution u of Problem
(W2) solves
Problem
(C), that is, if u has the classical
differentlability properties demanded in the statement of Problem This question, (i)
naturally of local type, is divided into two parts:
regularity in the interior of
boundary.
(C).
G
and (ii)
regularity up to the
The first question seems to be simpler in comparison with the
second and was earlier positively answered.
Beside F.John's method
for proving interior regularity of weak solutions,
[29]
there are mainly two
different methods applicable both for proving regularity in the interior and up to the boundary
(and for wider classes of boundary
value problems). One method is based on pointwise representations
of the weak solu-
tions and their derivatives by means of integral equations,
where funda-
mental solutions and Poisson kernels are used. Application of theorems on singular integrals,
mainly the Calderon-Zygmund
then yield the desired estimates.
By this method,
of weak solutions was proved by L. Schwartz E. Wienholtz
theorems
[13],[14],
interior regularity
[58], F.E. Browder
[66](for systems in [67]) and others.
For extensive biblio-
graphy we refer for this and other topics to C. Miranda's book (and to [38]-[41],[49])
. With the same method,
[8],
E. Wienholtz
[47]
[66] proved
regularity of the weak solutions of second order equations up to the boundary. We point out that this method was applicable to higher order equations only when F. John [27],[28] had constructed his fundamental solutions
(interior regularity)
and S. Agmon, A. Douglis and L.Nirenberg
succeeded in the construction of their Poisson kernels
(regularity up
4
to the boundary). The preparation of these very sharp auxiliary tools is very complicated and the estimates of the resulting integrals are not simple
(compare [23],[4],[52]). Roughly spoken, this method is
based on the explicit construction of the solution of elliptic differential equations and boundary value problems
(in a half-space), where
the operator has constant coefficients. Another method is more consistent with the
L 2- approach and
applicable for proving both, interior regularity and regularity up to O
the boundary. It is solely based on Garding's inequality and on some calculus theorems of Sobolev spaces. With this method, Friedrichs (under additional use of his mollifiers), Browder [51]
[19]
[9] and L.Nirenberg
(up to the boundary) achieved regularity results. Roughly spoken,
this method consists in skilful estimates of certain difference O
O
quotients with the aid of Garding's inequality. Since Garding's inequality holds true if and only if the operator under consideration is uniformly strongly elliptic
(compare S.Agmon [I], [3]; equivalence holds
if some mild smoothness properties of the coefficients are satisfied), this method is restricted to that class of operators. To get classical solutions, the conclusion is as follows: If of the bilinear form member
f
u ¢ ~'2(G)
8G
and the coefficients
B L are sufficiently smooth and if the right hand
of equation (2) satisfies N w2m+k'2(G)
. If
f E wk'2(G) , then one verifies
n + 2m+ I k > ~
by a lemma of Sobolev, the solution a set of measure zero : u ¢ c2m(~)
u
and
~ G c c2m+k, then,
satisfies after correction on
. Further, it assumes the boundary
values in the classical sense (compare the text books of S.Agmon [3] and A.Friedman [17]). The advantage of this method is, that it is mainly based on an a priori estimate
(Garding's inequality) being rather simple to prove,
and on elementary properties of
wk'2(G) -
spaces. But the great dis-
advantage may be seen in the fact that the assumptions on the boundary
~G , on the coefficients
a~
of
BL
and on the right hand member
increase with the dimension of the underlying domain, prove classical differentiability properties. striction which arises from the use of the S.L. Sobolev.
On the other hand,
[8],
[66],
L 2 - calculus theorem of
the method u s i n g integral equations
m=1.
If
fo c C~(G) , but
xo ~
F
8G
f { L2(G)
Fo(¢ ) then
The right hand member
has to be a continuous linear functional on
we cannot apply our considerations. and let
as we know from the
[67].
There is a further restriction: := (f'@)o
if we want to
This is an unnatural re-
has no need of such dimension depending assumptions, proofs in [2],
:=
and if
fo(X)
. Moreover,
(fo,~)o
for
linear functional on
right hand member
fo
W~'2(G)j else
:= I X - X o I-(~),
Let n > 2
then
if
q
with
W~'2(G).
2 < q < ~ , Fo
wJ,q(G).Therefore,
catch a solution being defined in the whole of L 2- method nor by Sehauder's method.
:=
® ~ C~(G) ,
for every real number
mines a continuous
F(@)
To give a simple example:
determines no continuous linear functional on
is easily proved,
G
But as deter-
we cannot
, neither by the
On the other hand, we consider this
to be "reasonable".
This is exactly the same
restriction as we have observed when u s i n g Dirichlet's principle pare
f
(com-
[15]). As mentioned above,
elliptic operators,
the method is restricted to uniformly strongly
or by S.Agmon's
investigations
(compare
[3]), to
operators being after multiplication with a suitable complex number of absolut value one uniformly strongly elliptic.
There is, however,
a lot
of operators being uniformly elliptic without having this property.
As these
restrictions
analogous
theory
covering
uniformly
in the spaces elliptic,
operators.
A further
f c LP(G)
for a
then
p >n
but not necessarily
I ~G
whereby
no dimension
= 0
for
depending
are necessary.
up to the boundary. the b o u n d a r y
assumptions
lished
L 2 - theory.
For the reasons established,
e.g. by 0.V.
and F.E. Browder
[10],
[11],
some
[22], A.I.
of order 2m, u c ~ ' P ( G )
Dirichlet's other
boundary
is possible
Lu
to establish
would
imply regularity
L p - theories Koshelev
[35],
a priori L
analysis,
I) we want to establish is rather
have been S. Agmon
estimates
[2]
in L p
is a u n i f o r m l y
n w2m'P(G)
We want to investigate a
In this case,
dates of the problem
= f , where
value p r o b l e m which
L p - approaches:
has the boundary
a rather delicate
[12]. They prove
elliptic
paper
and
is not so easy to be estab-
of the equation
In the underlying
If
u ~w~'P(G),
Theorem 9.16).
demands
above,
Guseva
fact:
that the proof of regularity up to
for the solutions operator
seen, u
on the remaining
Lp - theory
mentioned
~
elliptic
has a solution
(compare
L 2 - solutions
that such a
with
strongly
in that case existence
we can imagine as the
(2)
I). As is readily
If we remember
of weak
p
an
n I - ~ , by a lemma of Sobolev
0<~<
lal ~ m - 1
Therefore,
to establish
m a y be seen in the following
and if equation
(compare Appendix
D~ u
values
it is desirable
W~o'P(G ) for arbitrary
advantage
u E cm-I+~(G),
Kondrashov
indicate,
and
a
f c LP(G)
L p - theory of
different
from the
the question w h e t h e r
it
L p - theory m a i n l y based on a suitable
o
generalization
of Garding~s
to the L 2 - theory.
inequality,
Instead of Problem
and which
is built up similar
(W2) we want to solve
I) Except
§§ 10 and 11 this paper is essentially
lation of
[59].
an English
trans-
.
Problem ,(Wpi: Let
G c IR n
and let
be a bounded open set, let
f e LP(G)
I
~
be a real nuJnber
.
Then the weak Dirichlet boundary value problem in geneous boundary values consists in finding a
Lp
with homo-
u e ~'P(G)
such
that
BL[u'*]
=
(f'~)o
holds for every • e C~(G)
.
This question admits an immediate generalization.
First, we m a y
consider a uniformly elliptic Dirichlet bilinear form of the order (compare Definition
1.4)
(4)
:=
with coefficients all such that for
functional on
Problem Let
aa~ e
(I) holds.
@ e ~'~o'q(G)
m
>__
L~
(G)
cr(G)
There may be no elliptic operator at
Secondly,
(whereby I < q <
~o'q(G)
,
if
f ~ LP(G), by
F(@)
:= (f,@)o t
~ , ~I + I = I)~ a continuous linear
is defined.
Therefore,
we consider
(Fpi:
G c IR n
be a bounded open set, let
I
~
be real num-
bers with hal on
~ + ~ = I and let F be a continuous linear functioP q ~o'q(G) . If B denotes the uniformly elliptic Dirichlet
bilinear form (4) of order m, then Problem a
(5)
u e T¢~o'P(G)
B[u,®]
such that
= F(m)
holds for every @ e ~o'q(G) equation.
(Fp) consists in finding
. We call
(5) an elliptic functional
Considering the
L 2 - theory of Dirichlet's boundary value problem,
we find out that the principle tools for solving Problems
(W2) and
(F2)
are: Riesz's representation theorem for continuous linear functiona!s on a Hilbert space, the Lax-Milgram theorem, of compact operators
the Riesz-Schauder theory
and, specifically for the problem u n d e r considera-
0
tion,
o
Garding's inequality.
With the aid of Garding's inequality and
some calculus theorems~higher solutions spaces
are proved.
~o'P(G)
differentiability properties
If we want to establish a similar theory in the
, which overcomes the restrictions
we have to solve the following problems denote real numbers
such that
of the L 2 - theory,
(let in the following
linear function~is on
]~o'q(G). If
G
of continuous
is a bounded open set and if
by
(6)
7-
:=
a continuous
1,m'q Jo
linear functional on
prove the converse:
If
is defined.
Further,
In
§ 4
F e W~o'q(G)* ( = the topological
Wom'q(G)), then there is a u n i q u e l y determined (6) holds.
I < p,q <
! + ! = I ) : P q
(a) We have to prove a suitable representation
f c ~'P(G),
of weak
the norms in
~'q(G)*
and
will
dual of
f ¢ ?~o'P(G) ~,~o'P(G)
we
such that
are equivalent.
(b) We have to generalize the Lax - Milgram theorem. (c) We have to find an
~
- analogue of Garding's
which applies to u n i f o ~ n l y elliptic,
inequality
but not necessarily uniformly
strongly elliptic Dirichlet bilinear forms. We will prove in § 6 that there are constants
(7)
e wP,q(e) I1{ I1=,~= 1
Cl > O, C~ > 0
qllu-
such that
,
-
C tl llo,
9
holds for every
u c w~'P(G). We only assume
~G
~ Cm
and the "leading O
coefficients"
to be uniformly continuous.
equality applies to B , then
If
p = 2
and Garding's
(7) is a trivial consequence.
in-
This was the
formal reason to the generalization above. (d) We have to show, that Fredholm's Problem
alternative
applies to
(Fp)
(e) We have to generalize the differentiability
In 1958, A.!. Koshelev
[35] stated
theorems.
Lp - estimates
for uniformly
strongly elliptic differential operators with real coefficients.
They
are of the type
"-
(8)
c
I1 11o, ,)
w2m, p (G).
for
His proof is done with a method due to 0.V. Guseva on certain "kernels" being constructed
relative to a constant coeffi-
cient operator L
by means of Fourier transforms.
Mikhlin's
([45],[46]) on the multipliers
theorem
he got the desired estimates rator).
[22] which is based
With the aid of of Fourier integrals,
(performed in the case of a special ope-
Kosheiev's method is very elegant and convincing.
construction of the kernels, far-reaching modifications
But in his
there are some errors which force us to
of the construction,
mainly when kernels
with certain prescribed boundary values in a half-space are constructed (compare Appendix 2) . In a note
[36]~published one year later,
improved his method in some points, With the aid of fundamental
Koshelev
but not in that mentioned above.
solutions
and Poisson kernels,
S. Agmon
[2] established the basis for a complete treatment of Dirichlet~s dary value problem in even order. Lp
Further,
boun-
Lp for uniformly elliptic operators of arbitrary he stated some important regularity theorems in
of the type of Weyl's lemma
[65]. In [4] there are indicated a
10
priori estimates
in
Lp
for equations of t~/pe (2). In 1962, M.Z.Solom-
jak [62] claimed the representation
(6) , but he gave no proof, he only
refers to the methods of [35]. The most far-reaching direction G c IR n
are due to M. Schechter he considers
[56],[57].
In a bounded open domain
a uniformly elliptic operator
complex valued coefficients
a s c clSl(~).
in the case of the homogeneous
results in our
L = ~ - ~
Then he assumes
with
( compare
Dirichlet boundary value problem)
[56]
as
follows: o
(1)
Garding's
inequality
(3) holds for
B L , that is, L is uni-
formly strongly elliptic in G. (ii)
There is a constant
the analogous v c wm'q(G) O
inequality holds with
N w2m'q(G)
L*v
Lu
a solution
:= W~o'q(G ) N w2m'q(G)) with
where
J
(iii) The problems (g ~ Lq(G))
C > 0
= 0
(L*v
for every
equivalent
tion have to be uniformly
least
N N2m'P(G)
u c D
L
:=
[2] is needed.
strongly elliptic.
(v c D*:=
Lu
~-
v c D*
= 0 ). (7). Further,
(_~)~i~z~ J
(ii) and
Schechter's
results
L
(b) There must be a diffe-
smooth coefficients, by (I).
such that
(c) To prove the esti-
demanded in (ii), the boundary must satisfy at
8 G ~ C 2m. This seems to be an unnatural
the representation
f c LP(G)
for every
with
applied to
with sufficiently
L*
have for
L .
(a) The bilinear forms under considera-
the bilinear form is connected with and
L* for a!l
(6). But to verify assumptions
are limited in some directions:
L
is satisfied and
M. Schechter proves inequality
(iii), the whole theory of S.Agmon
mates for
= g)
if and only if (f,v) o = 0
he proves under these assumptions,
L
replaced by
u c D := w~'P(G)
Under these assumptions,
rential operator
L
(8)
L* denotes the formal adjoint of
= f
((g,U)o = 0
a representation
such that
theorem and inequality
(7).
restriction
for both,
11
In our paper we go into quite the opposite direction: We will o
understand the generalization of Garding's inequality to be a basic a priori estimate. With the aid of this inequality we prove then existence and regularity of the solutions of Problems
(Wp) and (Fp). Especially~
we get as a result, what Schechter assumes for inequality
(7) to hold.
Chapter I is mainly concerned with the proof of certain a priori estimates
(Theorem 1.6) and a regularity result
(Theorem 3.6). The
proof is done with a modification of the method of 0.V. Guseva and A.I. Koshelev.
For this aim we construct in § 2 by means of Fourier trans-
forms certain "testing functions". for these functions Evidently,
The cumbersome proof of estimates
(Theorems 2.8 and 2.9) is performed in Appendix
I.
the proof of the a priori estimates can also be done with
the methods of [4]. In Chapter II we apply these a priori estimates first to the proof of the representation boundary
(6) under low assumptions on the regularity of the
( 8 G c cm). In Remark 4.8 we show, that some regularity assump-
tions of the boundary are necessary.
An easy consequence of (6) is the
generalization of the Lax - Milgram theorem
(Theorem 5.4). Again we use o
our a priori estimates for the proof of the generalization of Garding's inequality. We emphasize that the proof is performed under very low regularity assumptions: a~
~ L~(G) otherwise.
We demand
c C°(F) for
I~I = I~l =m,
This is essential for the application of our in-
equality to nonlinear problems. how estimate
~ G c Cm, a ~
In Remark 6.4 we give some devices
(7) may be extended to the case of singular lower order
coefficients.
In the case of uniformly strongly elliptic bilinear forms
we prove by means of sharper estimates derived in § 6 that Fredholm's alternative proofs
applies to Problem
(Galerkin's method)
(Fp). W. Wendland
[64] gave constructive
for the existence of solutions of Problem
in the case of uniformly strongly elliptic Dirichlet bilinear fo~as, based on our results.
(Wp)
12
Chapter III is mainly devoted to the proof of regularity results for w e a k solutions.
It turns out that Nirenberg's method
carried over to the underlying case, Garding's inequality
if we use our generalization of
(compare § 9). The most important application of
these regularity results is the proof of Fredholm's Problem
[51 ] may be
alternative for
(Fp) in § 10. Much cumbersome w o r k has to be done to prove that
the index is zero. An easy consequence alternative
applies to
Problem Let
(Theorem 10.10) is, that the
(Stp.)_:
G c IR n
ber. Let
be a bounded open set, let
V. := ~ - - ~ s
I
be uniformly elliptic in
be a real numG
f c LP(G). Then Dirichlet' s boundary value problem nuous boundary values) in the strong finding
u ~ ~o'P(G)
is called a strong In § 11
N w2m'P(G)
and let (with homoge-
L p - sense consists in
such that
Lu
= f . Such an
u
Lp - solution of the boundary value problem.
we apply the existence theorems of § 10 to the p r o o f of further
regularity theorems of the type of Weyl's lemma
[65]. It is interesting
to observe the correlation between existence and regularity theorems, as may be seen in § 10 and § 11 . This connection we also observe in S. Agmon's paper
[2].
In this paper we were especially concerned with two aims: First, to make the presentation rather self-contained and not to use too m a n y references Secondly,
to other papers
(except,
in using well-known theorems).
we especially cared to supply detailed proofs,
even at those
points where this implies cumbersome labour and one is tempted to "leave the proof to the reader".
Chapter I
: A priori estimates
for solutions of linear
elliptic functional equations with constant coefficients
The aim of this chapter is the proof of an a priori estimate
(Theorem
1.6). All the subsequent theory is based on it. In § I we give some notations and definitions and present an application of S.G. Mikhlin's theorem
[46] on multipliers of Fourier integrals.
is concerned with the construction of certain
The second paragraph
"testing functions" with
respect to an elliptic operator and homogeneous
Dirlchlet boundary con-
ditions. W i t h the aid of these functions we can prove in § 3 b y use of Mikhlin's theorem and an estimate of S. Bochner mates of Theorem
1.6 at first locally
standard methods globally.
[6]
the desired esti-
(Theorem 3.2 ), then by use of
A simple consequence of the proof of Theorem
3.2 is a kind of regularity theorem presented
in § 3 (Theorem 3.6).
The subsequent chapters may be read without detailed knowledge of chapter I, only Theorems
1.6, 3.2 and Lemma 3.5 will be used later.
Some
notations had been listed on page 230.
§ Io
Some definitions.
Formulation of a basic
a priori estimate
Definition
1.1
Let
G
real number with de~otes
be a bounded open subset of I < p < ~ and
the closure of
page 23o). w °'p
m ~ 0
C o (G) in
W m'p
~qn ( n ~ I ). If
an integer, (G)
then
p is a
w~'P(G)
(see llst of notations,
(~) = = LP(G).
With the norm 4
(1.1)
~'P(G)
is a Banach space;
space with scalar product
in the case
p = 2 t wmo ,2 (G) is a Hilbert
(I .2) i~l=~m
1.2
Remark
Instead of the norm defined by (1.1) the following equivalent norms are used by m a n y authors
(~.3)
II ~ I1~'~ ,=
( YIc(l~.,rn 11~ ~-I1~,~ ) ~
II ~ I1%,~ :--
( ] [ I~-~)l ~- ÷ ~-- I~ ~)1 ~] ~ d, ) ~
or
(1.4)
I~I=T~
G
For the general method of defining norms in S.L. Sobolev
Definition
Wm'P(G)
see
[61].
1.3
Let integers.
~oo'P(G) and
G
be an open subset of
For every s ¢ ~ + n
functions
with
as(. ) be defined in
G.
3R n , let Isl
n > 2 and
m > I be
< 2 m let the complex valued
Then,
tSl~,Z'rn
is a partial differential operator of order 2m defined in G. I )
(a)
(1) there exists a constant
holds (tl)
for the
c IR n - 1
1)tlnless
every "root
,in G
L is called u n i f o r m l T e l l i p t l c
x ~ G
and e v e r y
~--I~c I & |=')..~
x
E > 0 such that
and e v e r y
condition"
is
1 c ~ q n and i f
satisfied:
~ G the polynomial
)I =
0
if
For every in
T ~ C
l'
= (l~,..,ln.
1)
15
(1.7)
"P('~il'ix')
has exactly
m
:
=
>
a-,~,}Z '~'
~
~'
,
s = ( s', s~)
roots with positive imaginary part and
m
roots with
negative imaginary part.
(b)
L
is called uniformly strongly elliptic in
G I~~ , if
L
is uniformly elliptic and if in addition there exists a constant E' > 0 such that
Ill
I$1=1~
holds for every
x c G
and every
called the elllpticlty constants of
Definition
G
be a bounded open subset of ~ n
For every
measurable
L .
1.4
Let integer.
1 c ~qn. E and E T respectively are
l~I ~ m,
functions
a~(.)
n ~ 2, and m ~ I an
I~I ~ m let the complex valued, be defined in G. For
~,@ ¢
bounded
C o (G) let
I ~l~'wn
and
(1.10)
Z~
'=
(--1..) ~" TM
i~l~i~L=~a
Then
B
is called (a) ~ i f 0 r m l ~
a elliptic Dirichlet bilinear form in G, if
~
de-
fined by (1.10) is uniformly elliptic in G in the sense of Definition
1.3.
(b) unlformly stron61y elliptic Dirlchlet billnear form in G, if
~
is uniformly strongly elliptic in G.
I ) "uniformly" with respect to x, that means E does not depend on If the a s are constant, the phrase "uniformly" is void.
x.
16
m is called the order of occuring in Definition
Remark
B
and the numbers
E
and
E' respectively,
1.3, are called the ellipticity constants of
B.
1.5
(I) The "root condition"
(ii) is for n ~ 3 no condition at all,
which follows from the fact that the set ~ n - 1
\
[0} is connected for
n>3. (2) We have chosen the name "Dirichlet" bilinear form, since the quadratic form
B[~ ,~ ] is a generalization of the classical so-called
Dirichlet's integral (3) If and
F
B
D[ u,u]
is a
: = ~ Ivu(x)12dx.
uniformly elliptic Dirichlet billnear form in G
a continuous linear functional on
~'~(G),
u e ~'P(G)
(~I + ql = I ) and a relation of the type
(1.11)
.]~,[ ~, ~]
=
for all ~ e W~ "q (G)
holds, we call (1.11) a
functional equation. lity properties,
T (~)
If the coefficients of
uniformly elliptic B
have no differentiabi-
we cannot construct an elliptic differential operator
L B such that the equation
holds for all ~,~ ¢ the often so-called
C ~o G( ) .
Therefore it makes no sense to introduce
"formal differential operator"
L~
•~
(- I)
>
_Dm
D
I~I -~-r. and to call a solution of (1.11) a
"weak" solution of the equation
" LB u = f "
~f for instance
fined and
needs not to have higher derivatives.
u
F(~) : = ( f , ~
The aim of this chapter is the proof of
because
L B is not de-
17
Theorem 1.6 Assume
(I)
that
open set of
IR n
w i t h boundary
with
I < p,q < (2)
with
that
=
D
D c ~r
+
and
~I
=
m > I
(3)
are integers and
3G e Cm ; p I
and
is compact
(r>1)
that for every
k e D
and every
( ~
k eD
a bounded
are real numbers
and for every
the complex valued functions
. For every
q
G
.
~ [ ~, ~] • = >
(1.12)
is a
~I
and
JaJ = J~J = m
tinuous on
n >2
a~B (.)
m,B g Z~
+n
are con-
~,@ e C o ( G )
define
c",,~F ' ~ , m ~ ~')o
the bilinear form
Bk
defined by (1.12)
uniformly elliptic Dirichlet bilinear form v~ith el!ipticity con-
stant
E
not depending on
(4)
that
are functions
u e Co(G ) fa eLP(G)
k eD
.
and that for every
given,
~
such that for
with a
k eD
JmJ = m
there
and all
~ ~o(G) (~.15)
holds.
Then there exist 1,2,3)~
independent of
constants k,u,
and
C i = C i (n,m,p,G,E, s u p J a ~ ( k ) I ) keD ~ f~
with
C I > 0, C 2 ~ O, and
such that i
l~,l='rm
and
•
qU4o.
(i = C 3 ~ 0,
18 If furthermore linear
Bk
is a
form for every
pendent of
k cD
k eD, and if
p~2
uniformly strongly elliptic Dirlchlet hiwith ellipticity constant
E~ > 0
inde-
, then
o
If
p = 2
instance
,
1.16)
S.Agmon
is a special case of Gardlng's inequality
[3]
, L. Bers
Friedman [17] , K. Yosida [68]
-
F. John
and others
easily obtained by Fourler-transforms: n ~(1)
:=
~ e -i(l'x) u(x)dx G we get for every k e D
(1.17)
(2~) - ~
M. Schechter
( see for [5]
, A.
). In this c a s ~ the proof is
Let
u e C~(G)
. We denote by
the Fouriertransform of
u. Then
"~e B % [lTu,l.~] I~t=l~t =+'m
G
= ]Ze~
A i~(x) ~ ])~u(1)~-G~u(i)Ei
= J (~
l~e a.~r~ (~,) 1~1 r~) [~z(l)[ z &l
Z
where
c(n,m) > 0
we get from
depends only on
(1.13)
and
(1.18) ~' c(~,~)U~ll~,~ and
therefore
(1,16)
n
and
m . By Schwarz's inequality
(I..17)
~ a~B~[~,u] in
the
case
p = 2
~ (7-11{~11~,~ .
lul[~,~
19 This m e t h o d t o t a l l y fails in the case where elliptic
and also in the case
p + 2.
Bk
is not strongly
As we m e n t i o n e d
in the outline
w e p r e f e r for the p r o o f of T h e o r e m
1.6
a Fourier integral method.
see that this m e t h o d is p r e f e r a b l e
to the m e t h o d of singular integrals,
we give a simple p r o o f of the following theorem. knovaqproofs
Most of the w e l l -
are done with the aid of f u n d a m e n t a l
Calderon - Zygmund lheorem.
Theorem
To
solutions
and the
For the p r o o f belovT we refer also to
[33].
1.7
Assume that
n > 2
and
m> I
are integers
and that
ISl=/Ta is a
u n i f o r m l y elliptic partial d i f f e r e n t i a l
constant coefficients
Then for every C(n,m,p,E)
> 0
operator of order 2m
with
as 6 C .
p,
I < p < ~
, there exists
a
constant
C =
such that 4
c II L Uo,p
IL]>" o, is valid for all
~ e Co(]R n)
.
Proof: With the n o t a t i o n u s e d above for the F o u r i e r - t r a n s f o r m , we have
-~j
(1.19)
-i(l.x)
,Sl
For every
~
with
S
I~I : 2m we observe the i d e n t i t y
-i (£,×)
of
20
Therefore,
{~.2o}
combining
~
(1.19) with the last formula~
1~
(z) =
~
for
1 ~ 0 is
_L ~ (z}
is
l$1aZ~.1
Since degree
~=2 aslS and i ~ are homogeneous polynomials in I of Is m 2m and since L is uniformly elliptic, we immediately prove
the existence of some constant with
]~[
= 2m
for
every
7-- ~i s
i
1 ~ 0 , 1 c IR n,
the quotient arbitrary for Since
> 0
such that for all
the estimate
,121 I holds
M = M(n,m,E)
~
r ~ I l l -'~t
and e v e r y
~
with
I = 0 , we get a measurable
L~ c C~(IR n) , the estimate
IL@(1)l
< I + cJlJ n+1 --
Therefore, we m a y apply the Fourier inversion formula
The integral at the right converges absolutely. we m a y apply Mikhlin's theorem Therefore,
[~]
(see Appendix
there exists a constant
< n . Defining
function of is valid.
to (1.20):
(1.20) guarantees
that
I) to the last identity.
A = A(p,n) > 0
independent of
such that
ll~llo,~ for every
~-
1 .
z~ A llL~llo, p
~ ~ C~(~R o" n), . From this the assertion follows immediately. q. e. d.
21
§ 2.
Construction Analytic
As we will the type
of certain
see in
(1.14)
§ 3 , it is sufficient
, where
the
"right
m - I , and the
of the derivatives in
of
m - th
G~ or the intersections G
.
restrict
ourselves
identity
(1.13)
struction
G
of unity.
of
u
of
~
and its derivatives
in some sense - , only depending that the desired
estimates
and the Poisson
A.P.
Calderon
0.V.
Guseva
Throughout
and A. Zygmund
[22]
and
A.I.
[14]
of
"global
Therefore,
estimate" we can
are admissible
in the
(1.13)
some calculations fa
and
in the con-
- or of functions
on the data
are possible.
- kernels
The
the last case
in identity
. Furwe get
"nearby"
, and of such kind
As we m e n t i o n e d
this could be done w i t h the aid of F.John's
solution
of boundary points
of proof consists
functions'~ w h i c h
them instead
u
L p - norms
is a ball or a half-ball
they must have the p r o p e r t ~ that after
representations
line,
of half-balls.
The method
"testing
put
the
of
of
totally contained
of coordinates,
by a p a r t i t i o n
to the case where
of certain
contains
of open nelghbourhoods
is satisfied.
sense that we m a y ther,
"left side"
estimates
derivatives
order taken over balls
to the consideration
is then p e r f o r m e d
to get "local"
side" m a y involve
Up to a local transformation
is equivalent (1.14)
functions".
tools
up to the order
with
"testing
[30]
in the outfundamental
[4] , using the theorems
of
. Here we will follow an idea of
Koshelev
[35]
this p a r a g r a p h we make the following
Assumvtio~ (A,) Assume that n>2
D
are integers.
is a compact For every
given complex valued functions the family of partial functions
of
(2.t)
variables
E(k)
that every
s e ~+n
Lk
> 0~ such that
~r
with
a s c C°(D)
differential
• r_> I , and Isl = 2m
there are
. Denote w i t h
operators
of order
m_> I ,
2m
[Lx}x e D defined
for
by
L~ := ~ Assume
stant
n
subset of
~(~)D
s
is u n i f o r m l y E
:= inf X~D
elliptic
E(k)
> 0 .
w i t h elllpticity
con-
22
Lemma
2.1
Assume that Assumption and
T eC
(A)
is satisfied.
define the polynomial in
(2.2)
L:~ (l' ~)
;-
>
T
For
of order
k ~ D , l' ¢ ~R 2m
n-1
by
a..~ (:~) z'"'~ ~"
II~I= Z ' m
Then there exist
2m
functions
Tk + (l';k)
and
k=1,.
Tk-(l';k),
..,m, w i t h the following properties: (I)
For every
l'e IR n-1 and every
the roots of the polynomial
k eD
the
Tk±~(l';k)
are
(2.2),
+
(2) in
For every fixed
IR n-1
every
a' (3)
the functions If
l' ~ 0
(2.3)
and Im Tk
C i = Ci(E,m,n ) > 0
Tk~-~ (. ; X) are analytic functions
the
and positive homogeneous
Im Wk + (l',k) > 0 , stants
k eD
of degree
Dl'' T~+~(l';k) k ~ D ~ the
I . Furthermore,
are continuous
in
for
~n-1
x D.
w~--+~ satisfy the inequalities
(l',k) < 0 , k = I .... ,m. There are con(i = 1,2)
such that
11= ~+~
and
is valid for every
i' e IR n-1 w i t h
Ii' I = I , k e D , Idl < n
and
k : 1,...,m .
The elementary proof of Lemma
2.1
is given in A p p e n d i x
I (see p. 200).
23 Lemma
2.2
Assume that every
m> I
i = (l',in)
(~.5)
n>2
e ]R n-1 × C
a(~)
Then
and
h(e;1) -~ I (E -~0)
(ii)
For every fixed
red as a function of of order
(iii)
e>O
and
let
has the following properties:
(i)
i n = - ~£
For every
[(~+ ~" l~'l~)(~-i~) ~]
~ =
h(e; " )
are integers.
uniformly on every compact subset of ]R n.
l' e IR n-1
lne C
and
is meromorphic
2(n+2m+1)
e > 0, h(e;l',.) and has a
conside-
unique pole at
in the lower half-plane.
There exists a constant M = M(n)~
independent of
e > 0 r such
that
(2.6) for all
I~ ~ h (~il)l Ill '~' ~- M I~I < n , i e ~qn
Proof: in A p p e n d i x
Lemma
(i)
and
and every
(ii)
are trivial.
The proof of
(iii)
is given
I, page 202.
2.3
Assume that the assumptions G
s > O.
is a bounded open subset of
of Lemma 2.2 ]R n. Assume
are satisfied and that I < p ( ~
Let
^
-~
-i(I,~)
&×
G
and
±(I,×) (2.7) IR~
and
f ¢ LP(G).
24 Then~for every
e > 0, H(a;f;.)
E LP(G)
and
~¢×) g (x)d, for every
g g C~(G) .
Proof:
Since
-m ~-i,, -~ (2~) ' )~(G) ~ II f IIo,~,
I~ (1)l
(2.9) From
(2.5)
(2.~o)
f ~ LP(G),
follows for every
lh(~L)
i e ]R
n
I = [(~÷~'I~'1")(~+~'l~)] ( ~
--
÷
~iI~)
-~'~'~
-~'~
By Fubini's theorem and (2.9), 3(. ) is measurable and by
(2.7),
(2.9),
and (2.10) -"'
IH
[
~-~"
{i×)l
(~
~[I
R~
i s v a l i d f o r every ~(a)
< -
x ~ G
the a s s e r t i o n
Now let
e > O.
H(~f;.)
g c Co(G ) .
G
Since
and
e
C~(G)
is integrable over
(2.9),
and by
]R n
×
c LP(G)
i s proved.
Then, by definition,
G
g
Hence by F u b i n i ' s theorem and
~
(2.10),
G. Therefore,
lh(s;1) ~(!) ei(l'X)g(x)l by Fublni's theorem,we may
change the order of integration which gives
25
where
-m
i (l,~)
denotes the i n v e r s e F o u r i e r - t r a n s f o r m o f
exists a constant
I for every
(1)1
"
c (~ + Ill)~+t
by (2.9) and
~R n . Now let
g ¢ Co(G), there
such that
1 ¢ ~R n . From the definitions
f(1) ~ therefore over
C = C(g) > 0
g . Since
q >0
(2.12)
be given.
immediately
f(.) ~(.)
follows
~(1) =
is absolutely integrable
Then there exists a
R° > I
such
that
Since
If(l)
(1)l < const
and
[Ii] < R O] , there exists an
for all
e<~o
" Since
hie;l) -- I (e -~ 0)
eo > 0
uniformly on
such that
]1-h(e;l)l < 2
for
1 s ~n
and
e> 0
we have
m,~
"7 Co(G )
is dense in
e Co(G )
such that
LP(G).
Therefore,
]If- fkIlo,p-*O _ ( k - ~ ) N
by
(f-
fk),
there exists a sequence
we see that
~l)-~
.
[fk )
Replacing in (2.9)
N
f(1)
uniformly and by
(2.12)
f
26
l for
all
='
C (Gi 9,)II { II~p (4
k >_ k o.
Therefore,
other hand, by P a r s e v a l ' s
~ fk(1) ~(i) dx -~ ~ 7(i) ~(i) dl. On the
formula
(see
~ ~k(X) g(x) dx.
2 ~k (1) ~(i) dl
=
i 7k(X) g(x) dx
-* j? ~(x) g(x) dx J? f(1) ?(I) dl =
Then,
K. Yoslda [68] ), T h e ~ b y H61der's inequallty
and therefore
2 g(x) g(x) dx
(2.13) implies (2.8). q.e.d.
Now we are in
~
position to construct the first kind of
"testing
functions" and to prove their properties. They will be used in
§3
to obtain the local "interior" estimates.
Theorem 2 . 4 Assume that Assumption (2.4)
and
C2
k ~D
let
L~ (I) : = 7 -
%s(x) l ~
i" (I',I~) 6 TR~'~×C
I&l,, l'~m
For every
be defined by
let
Further, for
(2.15)
(A) ist satisfied. Let
e >0
and
z E~n
let ~"~ r
i (z,z)
L~ (i)
27 n
where
(1, z) :=
j~llJ
Then for every Gk(e;. ) ~ c2m(]R n)
zj.
e >0
and every
k eD
by
(2.16)
is defined and all derivatives up to order 2m
may be performed "under the integral sign". For every I~l ~ 2m
D ~ Gk(e~z )
E ]R n xD.
Furthermore,
is valid. If
G
for every
=
e > O, k e D
and
and every
a >0
z ~R n
(z,k)
the identity
(- ¢)"' (z ~)- " [ h ( , t ~ l ) ei(l"z)4.1 W"
is a bounded open subset of
(I < p < ~), for every
e>0
is a continuous function of both variables
L~ G-~, (~i z )
(2.17')
a function
and every
IR n
x e ZR n
and
f ¢ LP(G)
we have
(observe(2.7))
(-±)~ I-I (~,i {i ×), G
where
denotes application of
~,y
with respect to the
y - variab-
les.
Proof: By (2.3), zeros of
~(1)
serve that for
(2.19)
are in
~
for
k ¢D.
]l' I < I
ll'lh I
or on
~
for
Lk(l',~ )
If
and
in ~ ~c
2
l n ~ ~C2
:--
2.~n
[~ ~ ~: I~I
-- 2c2, Im ~ > o}, then
we have by (2.4)
no
ll'l ~ I. 0b-
1 e ~n
I L , (1) I ~ ~ 111
for all
For
(2.4) and the homogeneity of the roots of
28 n
1 7 T (1 n - T ~ : + ( l ' ; k ) ) k=l m = a2men(k ) ~ k=1 and
i n e~C2
"%:-(l';x))l
(l n -
(in - T +(l';k)) g
>_ c2 2m . S i n c e
(in - T -(!';k)) g "
n)
=
ll' I < 1 --
for every
we get
lLx (r, i, )l
(2.21)
because
la2men ( x ) l h
(2.22)
i ~z e
Z
(2.19)
by
I -
I
I
• Further, for
I1~ - I~
From (2.10), a
(2.23)(a)
(2.19) - (2.22)
with
l~l !
2m
i(i,z)
L~(i',z~)
-~
I~ Z~
the following estimates are derived for
and every
t
a>0
III ( ~ + g~lll~) "n+z'" ÷ i if
(b)
I1'1 ! 1, i n ¢ ~C2
e
"-- (2 q )~'~ e.
every
Lk(l',l
1 6 ~n
"
if
ll' I <_ I, In~
~C2
and
z z , . . , C ~,~-Z-~- ~. ( ~ z ~ , i ) (c)
.
zd~
l~.,,,I
~-
if
ll'I -< I' i n e ~C 2 , Z n <
Now the differentiabillty properties of (a) - (c)
zn > 0
Gk(e;
.)
0 .
follow from
(2.23)
with the aid of elementary theorems of integral calculus.
29
By the properties proved above and since
L,,~
..~,
G~ ( ~ z )
Since the domain for
e >0
=
C-{) ~
L,(1)
=
(-4]~(1~) -~
~ := [ i n c~ : I!nl <
no pole of
term at the right of
Lemma
iCl,z)
~ h(£il) {wi-~ ~ : ~
h(E,l) e i(l'z)
is valid.
(2.24)
(2.18)
+
2 Ca, Im I n >_ 0 )
contains
(considered as a function of
by Cauchy~s theorem we may perform the
(2.17)
e
(2.16)
we get from
(2.24)
i (1,z,)
~
in),
in - integration in the second
over the entire real axis. Therefore,
is an imm@diate consequence of (2.17)
and
2.3. q.e.d.
Remark
(a)
2.5
Identities
(2.17)
and
(2.18)
have the following analogous
one in the method of using fundamental solutions: If damental solution for the operator
(2.25) where
L~
6
} < ~ (Z)
(G)
(2.26) G
is a fun-
L k , then
0f(?~)
denotes the Dirac distribution.
g@
e co
=
Kk(z)
(2.25) implies for every
3o
On the other hand,
in
put
(2.18) . Then, after a partial inte-
gratlon we get
(2.27) G
(b)
Let us consider
L~,
(- l )
GBk (~Q ,z)
and define
Theorem
- -
1,6,
(1.12), page
~" >
a.~.~
u ¢ C~(G) •
¢e
Let
Define
(~) 3)~ ~a
according to Theorem
in a neighborhood of the support of function
17.
2.4. By assumption
~ G) , 0 ! ~(y) CO(
!
1
and
u . Then for every fixed
~(y) := ¢ (y) GBx(~; x - y)
is
(4)
of
~(y)
~ I
x, the
"admissible" for identity
(1.13), since this holds by continuity for every
u e Cmo (G) .
By the pro-
perties of ~ we have
(2.28)
~[4,~%,c~;,-)]
-
~, [ ~ , % , ( s ; × - ) ]
After a partial integration we get, using (2.21),
~-~) (~
or by identity
(1.13)
and (2.28):
(2.29)
If we can estimate the of
m-th
Lp - n o r m of the right side and its derivatives
order uniformly in
E
by terms involving only
Ilf~IIo,p
and
31
II(D~ ) f~ II~ , mates of
then by
IIH(~;
D~u;
x)II
implies w e a k convergency, Uo
Essentially
of
~
side"
terms involving
decreases,
u . But such estimates have to construct as
without
application
GX( ~
an~ts
derivatives
normal
of
x-y)
our estimates
functions
(with respect to the
where
the support
part of the boundary. tion of functions
of
Kk( ~ ~ x - y )
gether w i t h its derivatives
space
to the m a y be
Assume,
we
having the same pro(1.13)
In this case,
in the direction
~k( s i x - . )
of the outward
must vanish.
But as we m e n t i o n e d to consider
m a y be arbitrarily
with respect
Such an at the
the case of
near the flat
as
up to order
y - variables)
. Without
to some half - space,
Gk( ~ ~ x - y ) , and vanish tom- I
at the boundary
loss of generality,
[ x ~ ~R n , x n _> 0 ]. This construction
that the family
[LkS
( with
this half -
w i l l be done now.
satisfies Assumption
(A). Let
(2,30)
Problem: Construct
u
Now we have reduced the p r o b l e m to the construc-
which have there the same properties
respect
procedure.
sufficient u
Therefore
for identity
y - variables)
it is
of the support of
for our purpose.
is n e a r l y impossible.
of this paragraph,
a half-ball,
m- I
Since the gradient
would depend on the support of
, but being admissible
up to order
w i l l h o l d for
there will appear at the
Kk( s ~ x - y)
of any "cut-off"
construction
beginning
above,
and its derivatives.
are not suitable
testing
perties
explicit
the same estimates
(2.8)
~ 3 • But if we p e r f o r m our esti-
discussed ~
we will have esti-
6 > 0 . In this case
in the same m a n n e r as the distance
to the boundary
G
in
and therefore
the same manner
increases
= D ~ H(~;u;x)
uniformly
this w i l l be done in
mates in quite "right
H(~;D~u~x)
for every
~ > 0
testing
functions
KX( a~ x, y )
de-
32
defined for e > 0
(x,y) 6 H + x H + with the following properties
and
for every
k ~ D
(i)
KX(~;
(ii)
Lk, y K X ( ~ ; X ,
x, y) g
C 2m ( H + x y) =
H +)
Lk, y G k ( e ; x - y ) ,
for
(x,y) ¢ H + X H + ,
Xn > O, Yn > O. (iii)
re n Dy K k ( e ; x , y ) l yn= o = o
Conditions
(ii)
independent @
conditions.
Gk, j ( ~ 1 x , y )
(2.}1)
(iii)
are linear.
Therefore,
we put
x n > o, r = O , . . . , m - I .
Further,
(iii)
contains
m
with certain functions
to be determined l a t e r
K~(ci
(i) - (iii)
and
for
×,5)
:=
implies the
Conditions
for
G x,J * (~;x,y):
For every
8 > 0, k e D,
and
j = O,...,m-1
(i)'
GX, j ( e ; x , y )
(ii)'
~ , y G k,j * ( 8 ~ x, y) = 0
(iii)'
ren Dy for
¢ C 2m (H+x
G *
k,J
H +) (x,y) e H + x H +,
for
J
( ~ ; x, y)j Yn =0
r = 0,..., m-1
.
6
rj
there must be
=
ren 6rj Dy
denotes Kronecker's
[4]
O, yn > 0
GX( s; x , y )
The following lemma is the basis for our construction. in the outline, we use an identity of
Xn>
(see below,
ln=0
delta.
As we mentioned (2.36)).
33
Lemma
2.6 +
Assume that Assumption
(A)
is satisfied.
k = 1,...m, be the functions according to lemma i' e IR n-l,
w
e
Let
•k¢--)(l';k ), and define for
2.1
C
~n
I0, ~. +
Then there are the properties: homogeneous D1. a ~
m + I
For fixed
of degree
(l';k)
functions k ¢ D
they are analytical
k . For every
are continuous
a~ -~ (l'; k), 0 ~ k ~ m , with
in
a' ~ ~ +(n-l) IR n-1 × D;
functions of
l',
the functions
f u r t h e ~ they satisfy
-fn
(2.33) ~0
Define for
j = O,...,m-1
and
I _
where
a := ½
(2.4)
respectively.
continuous ll'I = I
(16 C2 2 Then
differentiable and
2
CI ) J-
and
defined by
(2.3)
and
may be described as a piece-wise
closed
k ~ D~ all the
C I , C2
Jordan curve in the lower halfplane. If
Tk (l';k)
in the bounded domain with boundary
J
(k = 1,...,m) .
Furthermore,
are contained the identity
34 (see [4]).
2~i
T~f- ( l ' z i %)
] is valid for every
i'
e [Ii'I = I} , k e O
and
0 i J,k i m-1
.
Proof: The properties from
(2.32)
(2.4) Case
and
+ a k~--) follow after an elementary
Lemma 2.1
the second assertion I :
in the
Let
is trivial.
of
< m-2
the integral
of
It remains
T~
and b y
to prove
J- , we m a y choose instead
R ~ 2 CI . The degree
m-l-j+k
Case 2 :
. By the homogeneity
calculation
and the degree
vanishes
Let
for
R-~
of the polynomial of
M-
(l',T;k)
J~-l-j
is
m ;
lay
a circle (l',~;k)~ therefore
.
k ~ j . Then by
li,~ (i'~ ~) =
of
(2.3),
(2.36):
0 < k < j . Since all zeros of the denominator
"interior"
w i t h radius is
of
(2.33)
and
(2.34)
Zn-i
3"4
The first integral
above is equal to
I
• if
k = j ,
and equal to
0• if
k > j . To see that the second term on the right is zero•
again
J-
is
k - I
by a circle
of radius
R ~ 2 C2.
< m - 2 , that of the denominator
gral vanishes,
if
R ~
~
The degree is
m
replace
of the numerator
. Therefore
the inte-
. q.e.d.
Now we are ready to construct scribed behaviour
the desired
at the boundary.
testing
functions
with pre-
35 2.7
Theorem
Assume that A s s u m p t i o n k e O
0 < j < m - I
and
(A) is satisfied•
Iz'
For
> I , yn >_ O,
let
=, , =
1
x)g i~.. I1'1
:bl-
l~i
3where
J- ' ~ - 1 - j
s > 0,
(x,y) e H + x H +
[1, x-y ] n - 1
=
and
n-1 7v=]
M-
are defined a c c o r d i n g
and
lv(xv
0 < j < m-
I
Moreover
- y~)
G *
X, j (8~x,y) e C 2m (H +
for
• .,m-1 ) . Further,
(2.39)
~,y
the functions
Qx, j ( ~ ; x , y )
= o
for Let
.])~'e~ G *
r = 0,..•,m-1
[
.
for
11'1 i
k e D, 8 > 0
Dy D x# Gk,* j (s}x,y)
m a y be p e r f o r m e d
and
C2.~o)
x H +)
I~l i 2m , 1131 < 2m,
and all d e r i v a t i v e s
For
we define w i t h the n o t a t i o n
Lx(1) Then,
to Lemma 2.6.
GX, j
for
e C°
"under the i n t e g r a l (J=0, .. •,m-1 )
(x,y)
(H+x
sign"
H + XD)
(j = 0 , . .
satisfy
e H + x H + , Xn>O , Yn>O
36
then
E C ° (H + x H + x D) ,
D ~y D x~ I ~ ( e ; x , y )
(2.39)'
L&,~
~
(%
×,~')
=
L~
K~, (~., x, ~) I~ = o
=
I,:,1 i 2m , I~1 i m , e > 0 ,
G x ( s ; ×,b)
×,,>o, ~,,>o
and
(2.4.o),
I]~e"
T-
0
0,.-
"r,~-I
Proof: For e v e r y
r > 0
we have
e
(- i 11'~) ~ 'u
=
e_..
and
since
Im T ~ 0
rentiability considered stant
of
for Kj
T ¢ J(l'; • ;k)
as functions
C = C(m, ITk
and
l' ( ~
Yn
" ll'l ~ 0 .
and the c o n t i n u i t y
of all variables. ; k)l
, J-,
r )
This
implies
diffe-
of the d e r i v a t i v e s
Further,
there
exists
a con-
such that
",z"
for
ll'I > I , k e D
and
(2.41)
(2.42)
and
0 < r < 2m
. Moreover,
by
(2.10),
(2.19)
37
Because
I~I + I~I i 4m , j + ~n ! 3m -I , Ill h ~, the right side of is estimated by
(2.42)
C
E ~ ~",' [ ~ + ~ I:~'l')(~.,t/.~)]
'''~
Since the integrand is continuous with respect to all the variables x,y,k, 1 , by the estimate above the first assertion is proved. proof of
(2.39)
(2.43)
L~,~ [ ~-i[iis]~_ ~ ~
consider
(i'~ ~ ; x)] s I
=
'
For the
1 ~ri
--
....
~
'
],~
By homogeneity
F
la|-2.*n'~
%(~ (-~r)~'(-klm~) ~ = (_~)~t~,l ~
for l'
11'I Z I . Since and
~
, (2.43)
M+
and
~,~
= (~) ~
(~,,~
M m _ 1 _ j are polynomials
implies
3" This proves
(2.39) ;
(2.39)'
is then trivial
.
:I~ I i ~ ) in
T
for fixed
38 To p r o v e second
Then
(2.40)
t e r m at the
~R
closed
right of
(2.16)
m a y be c o n s i d e r e d Jordan
since by
(2.4)
re n Dy G k ( s ; x , y ) . At first,
, we c a l c u l a t e
. Let
ll'l !
and
(2.5)
the i n t e g r a n d
in the i n t e r i o r
the
same m a n n e r
as
of the b o u n d e d we p r o v e d
Ilnl =
circle
~n
tends
to zero,
ning integral
converges
this
is zero.
integral
Now let
The i n t e g r a l
function
~R
of
" In quite
C2 by R)
we get
over
~R
m a y be
[ ~ ~ ~: J~l <_ R ) and over the half-
if
By
R -* ~ and
to the i n t e g r a l
over
(2.45)
, the i n t e g r a l
zn > 0 , w h i l e ~P
. Therefore,
over
the
remai-
by
(2.44),
From this we get the formula
i(Az) &l'&l~ i~ z~_O, L~ (I)
C-~ C ~-; ~. }
x ¢ H+ ,
Then
"R
+~
(2.46)
zn ~ 0 .
(replace
[ ~ g C : 171 = 2 R , Im ~ > 0 } .
the h a l f - c i r c l e
differentiable
is a h o l o m o r p h i c
co'nst
R, Im i n > 0 .
split in an integration over
continuously
and
(b)
~-
and
I
the
let
domain with boundary
(2.23)
(2.45) zn > 0
R > 2 C2
as a p i e c e - w i s e
curve
in
if
For
.
consider
xn > 0 o
(2.47) ~..r, = 0
By
(2.46)
,
39 h(~ii) i[i,.-~]~.~ e ii.X~(_ii~)~-
(i;
z, On the other hand, by
(2.38)
ai' ai~
and the differentiability properties
proved above,
G~,,~ (~ ×,9
(2.48)
=
~=0
-
If we compare
(2.40)
holds
(2.47)
(2.48)
with
(2°#0)'
true.
i I
,
i.i J ]'i (iT,i,z;
L~ (I) I 1'1 ~
{ il'l-~l~ k'-"
.
and observe
(2.36)
~r
~)
we see that
is trivial. q.e.d.
The following two theorems are of great importance for our estimates in
§ 3 •
Both proofs need cumbersome calculations,
the following considerations. Appendix
Therefore,
uninteresting
we present these proofs in
I.
Theorem
2.8
Assume that Assumption (A) is satisfied. Let Gk(~jx, y ) @ fined by (2.16) and Gk, j(a,x,y) by (2.38) . Let
(2.49)
for
for
i<~(.~.;~,,~)
e > 0 , k c D
and
Then for every = K(6jm, n,E, CI,C2,
:=-
6
G~(%>~-~)
- Y--
be de-
r---a,a ~-- (~) ×' ~)
(x,y) ~ H + x H + .
with
0 < 6 < S
sup I a ~ ( k ) I ) k ~D
there exists a constant
> o , independent of
~ > 0 , (C i
K(6) = de-
4o
fined by
(2.3) , ( 2 . 4 ) )
(,,)
I~I ~
, such that for e v e r y
I:D
I ,
n -
-
k ~ D .
x,y
¢ ~qn
If furthermore
estimates
(2.50)
Proof:
if
Gk
if =
=
m
Then a constant
either
Ixl < R ]
Sk = K k . l~I
and
1~1 > O
-
2m +
t~t
+
1#t
e > 0
< 0
and for all
x n ~ 0 , Yn h 0 ,
is r e p l a c e d by
Let
of T h e o r e m
Gk(ESx, y ) if
or
I < p < ~
2.8
Kk,tS~x,y)
S k = G k , and
Kk .
are satisfied.
Denote
.
let
For
R > 0
G R := Ix : Ixl < R, x n > 0 ]
and define
for
g ¢ LP(G R) , l~I =
,
F(.,e$g;~;~)
~ LP(GR)
for all
g c LP(GR)
M = M(p,n,m,E, Cz,C2, R) > 0; i n d e p e n d e n t
g ¢ LP(GR) ~ such that for all
(2.51)
+
2.9
Sk(s$x,y )
G R := [x :
I~t
I , page 204.
Assume that the a s s u m p t i o n s with
n
Ix-Yl ~ I , for all
Ixl ! I , Iyl ~ I
are valid,
Appendix
Theorem
with
2m +
K"
if
are true for
I~I i2m
and
k: l~ -sl if
(b)
2m
It F(" , S; g;~
~ and
~ ~) lisp (GR)
and there exists
of
I~l =
E > 0
~
with
IBI = m
_<
M It g IlT,PcGR )
and
41
is valid
for every
e > 0 , k e D
Proof: Appendix
Remark (a)
LP(%)
.
2.9
Just the fact that the constants in e > 0
(2.50)
and
is the main result of
(2.51) m a y
Theorems
2.8
2.9 . E x a c t l y this fact makes it possible to derive from an identity
of type of
g
I , page 204.
be chosen independent of and
and
(2.29) , as considered in Remark
2.5 ,
estimates independent
s > 0 . (b)
Theorem
2.8
is quite the analoguous to that one used in the
method based on fundamental
solutions and is proved in this theory with
the aid of the Calderon - Zygmund theorems. based on Mikhlin's
§ 3.
In our theory the proof is
theorem.
Proofs of local and global a priori estimates and regularity theorems
Definition Let
3. I
H R :=
k >_ I be integers ~ cm(~) (i) that
Then
C
m o,k(HR)
Ixl < R , x n > 0 ]
and
let
m >_ k,
denotes the set of all functions
with the following properties: there exists a real number
~(x) = 0 (ii)
.
[ x s IR n "
for
R-
D ~ @(X) = 0
8 for
<
Ixl < R ,
8 > 0
depending on
xn > 0
Ixl < R , x n = 0
Co,~k (HR) :=
N m>k
Comk(HR)
, such
and and all
I . Further let
~
.
~
with
lel <
42
First we w i l l prove need later
on
our basic
local
estimates
in a special
form we w i l l
.
Theorem
3.2
Assume (S)
that
real number. < R }
Let
I
GR
and
that
c ~ +n
D c IR r
with
are c o n t i n u o u s
]~[ =
in
n ~ 2
denote
or the h e m i s p h e r e (2)
~,6
m ~
are integers
either
the
and
sphere
0 < R ~
K R :=
l~I = m
( r ~
I)
the c o m p l e x
D . For e v e r y
k c D
and that valued
is a
Ix c ]R n :
H R := [ x ¢ ]R n : Ixl < R , x n > 0 is c o m p a c t
I
$,~
<
} ;
for e v e r y
functions
and e v e r y
Ix]
~ Co
a~6(.)
(GR) let
I~a-I~l='m (3)
that
for every
elliptic
Dirlchlet
dependent
of
(4) there exists
are given
(5)
bilinear
B k ~ defined
form wlth
as above,
ellipticity
is a
uniformly
constant
E > 0
v ~ C~(G R) functions
R' < R
with
and that g6 ¢ LP(G)
suppg~
c
for e v e r y with
~ c ~ + n
I < p < ~
[ x : ]x[ < R'
with
161 ~ m
such that
there
] ;
that
(3.1)
is v a l i d C
TM
for e v e r y
o,m(HR),
in-
k E D ;
that
an
k ¢ D
if
~ ¢ C m ( K R ) , if
GR = H R .
G R = K R , and
for e v e r y
• c
43
Then for every
6
= M(p,n,m,E, aa~ ) > 0 v, g~
and
~3.2~
R
with and
0 < 6 < I
there exist
K i = Ki(8,M ) > 0
(i=1,2)
constants
M =
independent of
such that
~-- ~:D~,,IIo,~ -~ ~
~
Ir=l=-n~
I~I='t~
II~%Uo,~
÷ Ig >
T~"~-~-~- II~Ilo,~ ÷
Proof: Let
By assumption
(3) the family
every
denote w i t h
or
E > 0
~(~;x,y)
if
ding to Theorem [~]k
G R = H R , where 2.4
< ~(x) < I
and Theorem
properties of
KR Sk
and and
:= ~(y) • S k ( ~ x , y )
satisfies Assumption
either Gk
2.7
and
Gk(a~x,Y) Kk
Further, ~(x) = I
GR = K R ,
are constructed accor-
Ro
there exists
with
R' < R o < R
q e C~(KR) , 0 <
in a nelghbourhood of
q , for every
satisfies
if
(A) . For
w i t h respect to the family
(4) there exists an
supp v c KRo . in
c D
Sk(E;x,y)
~ D " By assumption
such that
[Lk}k
x ~ GR
KRo . By the
the function
m (KR) ' if ~ ¢ Co
GR = KR
$(y) and
:= •
C m (HR) if G R = H R Therefore, @ is "admissible" for (3. I) o,m " and we put @ in this identity. Since ~ - I in a neighbourhood of the supports of
v
and
gD
we get
4z~
By assumption
(4)
on
v
and the different(ability properties
of
Sk
the left term above becomes after partial integration and application of
(2.17),
(3.~)
(2.39)
( v,
and
(2.7)
~ - , ~ ~---
=
Let
(3.5)
I~I
= m .
Since
~)~ H(~iv~×)
~(~~&~(~,.))o
(-~)~
v ~
=
H
C~ivi×).
m
C O (GR)
-
we get the identity
H C~i~vix)
By the different(ability properties of identity (3.3)m - times
-_
with respect to
Sk(~x,y) x ¢ GR .
we m a y differentiate Therefore,
(3.5)
(3.6)
Now we estimate By Theorem
IIH(~; o°v~ • )llTp(ap
2.8
I}l=~
by
(3.3)
45
To get an estimate
for the remaining
terms,
let
(3.8)
Now observe
Theorem
2.7 , (2.50):
Choose
6
with
0 < 6 < I , then
( J i,~ - %~'.-~+~r~'+~-
G~
if
n-
m
+
161 > 0
if
n-
m
+
161 < o
or
°
%
We estimate
and by
as follows:
H8ider 's inequality
Because than
(3.9)
GR c
{ y :
Ix - Yl ~ 2 R ]
K(n,m) R m - 1 6 1 -
the first integral
above is smaller
6 • from w h a t follows
] By F u b i n i ' s
theorem, we change t h e o r d e r o f i n t e g r a t i o n
the right and we gain for the same reason as above
in
t h e t e r m on
46
I l×-~i~-~-~I-@dx
~- K(~,m)T< ~-~-~
~a Therefore,
with
K(6)
K(n,m)
=:
K' (6)
k' (5) I<"~-t~-# (I ~ lll~e(G~)
-~
if To e s t i m a t e
Ga
(3. IO)
~~
and t h e r e f o r e
observe
~
with
=-m+
that w i t h
K =
I~I > 0 .
n
(GaI ~)~ (Gaif~ I
~-
K''
, we have
= K(6)
~~)~
• K
(3.12)
if From
(3.7)
(3.13)
, (3.11)
and
(3.12)
n-~+
there
follows
IIH(~)~v~ )IIL~(G~ -~ M' ~ I~I= "n'1
Let
[en ]
be a sequence
[H(s n ; D q v ;
• )]
with
sn > 0
and
is a b o u n d e d sequence
I~I < o
II~II~
¢n ~ on
0
(n ~ ~)
LP(GR)
.
By
. Furthermore
(3.13), ,
.
4 (n.-*~)
for every
g e ~o(GR).
Since
Cj(GR)
i s dense i n
LP(GR) ,
~7 H( s ~ D e v ,
• )
converges weakly to
De v
and therefore
(3.13) holds with the left side replaced b~ proves
IIzp(GR)
lJDev
estimate .
This
(3.2) • e. d.
q.
In an analogOUS manner we prove the following local regularity theorem:
Theorem
3.3
Assume (1) number.
that Let
m ~ 1 GR
denote
the hemisphere (2)
and
H R :-
n ~ 2 are
either
the
integers
sphere
and
KR : =
(x ¢ IR n : ]xl < R,
0 < R ~ 1 is [x ¢ ~ n
a real
: l x l < R}
or
x n > 0}.
that
(3.14)
is a
uniformly elliptic Dirichlet bilinear
CI(GR) (3)
that
g~ c LP(GR) R' < R
1~1
for
f~1
I < p < ~
for
such that
(4)
=
form in
GR
with
aa~
: m ; is a real number,
that
v e W~o'P(GR) ,
l~I ! m - I, and that there exists a real number supp v c KR, ,
supp g~ c KR,
for
l~l! m - I ;
that
I~ ~-~-~ is valid for every GR
=
HR
•
• ¢ ~o(KR) , i f
GR = KR , and
~ e CommCHR)
,
if
48
-,p' v ~ ~"o (GR)
Then p'
is arbitrary,
if
"
1 I p--F = p
where
I
2n '
if
p < 2n , and
p > 2 n
Proof: Let
Observe
xo e
that by a s s u m p t i o n
elliptic
differential
satisfies rem 3.2 As in
D be a r b i t r a r y .
Assumption consider
(3.3)
(2)
operators
(A)
By
(3.1~)
and
the f a m i l y defined
with
D :=
(3.15)
(Lxo)x o e D
we have
of u n i f o r m l y
by
G R . As in the p r o o f
in the same n o t a t i o n
Sxo
and
e > 0 . Observe
that
~(y)
:=
of
Theo-
~(y)
S x o ( e ; x , y ).
- (3.6) we d e r i v e
(3.16) I~1.1..151~-,'~
for
x e G R , I~I ~ m
v a l i d in the case v
by
holds
Co(GR) for
fore, rem 2.4
v ¢ ~n~P(GR)
functions.
x = xo .
(3.16) and
and
for e v e r y
2.7
and
(3.6)
are
as m a y be seen b y a p p r o x i m a t i n g
(3.16)
On the o t h e r hand,
is v a l i d Theorem
Since
too,
(3.5)
holds x o ~ GR
x = xo c GR .
for e v e r y
x c G R , it
was a r b i t r a r y .
There-
Furthermore,
by Theo-
49
Theorem 2.8
guaranties the estimates
(observe assumption
(3.17)
] for
~-
(2))
~,, i~_~ol - ( ~ D
Iol = 181 = m , Xo, Y e G R
and
- ~ - ~)
for where
K
is defined as in Theorem 2.8
Ioi ! m ; i~l ! m - I and
, Xo, y e G R ,
K' = K'(max]V a~8(x)[,
K) .
Therefore
By
Sobolev's
lemma
at the right of
(1) fore
if
( see Appendix
(3.19) p < 2n,
F(x o)
has the properties then
F c LP'(GR)
with
~, =
p = 2n , then
F e LP'(GR)
for every
I
I
P
2n
p'
with
(there-
p' > p ) (il)
if
The same is then true for formly bounded in a
I , page 224), the function
w ~ LP'(GR)
fore, D ° v
LPT(GR )
H( E ~ D ° v ;
.) . Since this sequence is uni-
(with respect to
s > O) , it converges to
. On the other hand it converges w e a k l y to
~ Lp'(GR)
,
I
lol ~ m . q. e° d.
D ° v . There-
5o This proof implies the following coz011ary to Theorem 3.2.
Theorem 3.4 Assume (I)
that assumptions
(I) - (3) and
(5)
of Theorem 3.2
are
satisfied;
(2) and
that
I < p < =
g~ ¢ LP'(G R)
(3)
that
for
and
I~l !
V e
~'P(GR)o ' g~ ~ LP(GR)
there exists a real number
c KR , , supp g~ ~ KR,
for
I~I =
for
m,
s I I m - I, where ~, = ~ + 2n "
R' < R
such that
supp v
I~I J m .
Then the assertion of Theorem 3.2
is true, if
(3.2)
is replaced
by
>,, II~=~
(3.20)
l¢l='m
o,W
Proof: Consider the proof of Theorem 3.2. in the case where
I~I < m -I
and
Ioi < m . Since
rem 2.7 , (2.50) , we get the estimates and
(3.9)
with
8
We have only to consider R < I , by
(unifying in this case
:= ½)
(3.21)
k
J
GR Now, by Sobolev's lemma
i~_~
~-
(see Appendix
I, page 224 )
we get
(3.8)
Theo(3.8)
51
for
K' = K'(K,R)
I~I i m - I , I~I ! m , w h e r e
.
q.e.d.
To p r o v e
and
Theorem
Lemma
3.5
Assume
that
m
1.6
G
are i n t e g e r s
is a a r b i t r a r y
lemma:
is a b o u n d e d
open
subset
with
and
0 < J < m - I
m > I
of
~n
(n ~
I) , that
and t h a t
j
I < p <
real number.
For e v e r y (i=1,2)
we need a w e l l k n o w n
s>
0 there
exist
constants
Ci(e)
= Ci(
~; n , m , p , G , J )
> 0
such that
(3.23)
Ilull j,p
<
--
~ Ilullm, p
+
c~(~) tlull~,.~(9:,
and
(3.24)
ilullj,p
for all
u e ~oP(G)
In the case forms
p = 2
(see e.g.
gant p r o o f was This p r o o f ~m~P(G) proof
<_ s Ilultm, p
see
in
.
Lemma
3.5 is e a s i l y p r o v e d
K. Y o s i d a
given
+ c~(~) IlulIL~.(G )
by
[68]
b y means
) . For a r b i t r a r y
Ch. B. M o r r e y
jr.
(see
p > I
and depends
on the fact that
wJ~P(G)
is c o m p l e t e l y
c o n t i n u o u s for
[17]
.
trans-
a v e r y ele-
[48] , T h e o r e m
is i n d i r e c t
A. F r i e d m a n
of F o u r i e r
3.6.9)
the e m b e d d i n g
of
m > J ° For a n o t h e r
•
52
H e r e we give
the p r i n c i p l e
on M i k h l i n ' s
theorem.
1
Proof
of
It suffices the case
Lemma
0 < j < m - I , there
IIullj,p i C'
lity,
see e.g.
1 c ]R n
IIUlIm_1,p
[16],
and
Let an a r b i t r a r y
of two direct
proofs,
where
one is b a s e d
3.5
to p r o v e
that
If
Lemma
ideas
3.5
in the case
is a c o n s t a n t
for all
j = m - I . Since
C' = C ' ( G , J , m )
u e Wm~P(G)
. (Poincare's
in
such inequa-
[3]).
p > O~ let
~ > 0
be given.
T h e n there
is a
p = p(~)
> 0
such
that
I]~ ~ (I ~ 1 ~ + ~ ( i ) ) I
(3.25)
for all
]el = m - I ,
induction, (21oI
- I)(2n
But then l~l
:
for e v e r y
m
+ 4m)
(3.25) -
Ioi < n , v = 1,...,n lol ! n
-
Ill '~I ~W
I•]
follows
(if
there
I
~C
As is e a s i l y p r o v e d
is a p o l y n o m i a l
Iol > 0 ; let
immediately.
.
@o(i)
9 O := I),
For e v e r y
• e CO
of degree
such that
(G)
by
and
53 and t h e r e f o r e
-
=
(3.26)
&l+
(i) ~ = 4j,.
,(z,r)}X The first fin's
t e r m on the
theorem
theorem
are
right
of
(see A p p e n d i x
satisfied.
1{¢i)1
je
&l
(3.26)
I) o By
is e s t i m a t e d (3.25)
For the e s t i m a t e
according
to Mikh-
all a s s u m p t i o n s
of the
second
of this
term observe
~ (z~) ' II~llL~m)
and
!
I
Ctl)~
(K = K(n,m)
> 0)
and M i k h l i n ' s
I iI ~-I K III z(~÷z~), 3 ° But this
function
£eT
llJ~ i
K"±I II-zc~÷z~)
is i n t e g r a b l e
£oT l l l - ~ i
over
IR n . By
(3.26)
simple
estimates
and
a trivial
conse-
theorem
II m~ ~ II o,~ ~- A m,-r, ~ ~
II~
~IIo,~ i
Now
(3.23)
q = q(£)
follows
chosen in
quence by H ~ l d e r ' s u ~
for
every
an o b v i o u s inequality.
•
e Co(G )
manner.
after
Then
(3°2#)
By c o n t i n u i t y
is
Lemma 3 . 5
holds
for
all
W'moP(G) o q.e.do
Proof
2
of Lemma
3.5 , ~3.24)
'
1 Let For
q > 0
~ c C : ( ~ ~) , 0 < ~(t) < let
~q(t)
:=~(~)
I
and
. Assume
that
~(t)
=
0
for for
Itl<1 Itl > 2
for all x = (x',Xn)
e G
54
IXnl~
we have
Then for every
o
a
for some u ¢ C~(G)
a > 0 . Let
we h~ve
o
xn
be arbitrary,
but
JXnl <
~(x', ~n) ~On(Xn - ~n) ~ C~'(a) "
Therefore
By HSlder' s inequality x~
4
II ~-~.(~',4 ' ~ ) ~
I -Q.
Now X~
t~ z.
J I~C~}l ~'~ = I I~c~)l 9d~ =
,~ ~ o ~ , ~
and
-m¢x I'¢'(e)1
'~ .}
:
d£
~o'ns~ ~-ff
R
Therefore
(3.27)
I ~(<',~,~ I"I (~'- ~')IP +ec
.
C~
"~
--0~
The right
~e
~f(3.27)
is independent
of
and
x
n
. Choose
a°
55 O
O
in (3.27)~ the assertion. bounded. of
G
then
~q(x n - ~ )
= S .
- As this proof for
(3.24)
Integrating shows,
G
(3.27) proves
needs not to be
It suffles that after a rotation of coordinates
on the coordlnate-axls
xn
the projection
is bounded. q. e, d.
Proof of
Theorem
1.6
The proof is performed in three steps (i)
local interior
(ii)
local estimates up to the boundary
(iii)
global estimates.
(i) and (ill)
estimates
are proved by local application
of Theorem 3.2
and
(iii)
R
with
is a simple consequence. (i)
loca! interior estimates:
Let
x° e G
0 < R < R o := min
and let R(x o) := dlst(x o, 8G) > 0 . For every (R(Xo) , I)
exists a function
(3.28)
q(x)
~ ¢ C ~ ( ~ n)
define with
K R := [ x : I X - X o l 0 ! ~(x) ! I
and
I
I
for
Ixl i
0
for
3 Ixl h ~ .
=
! R} .
satisfying
(3°29)
ID~ ~(x)l < K <
for
x e ]R n
and
I~l < i~m .
Let
(3.3o)
for
0<
R<
Ro o
There
56
Then by
(3.28)
and
(3.29) f
nRCx)
(3.31)
=
for
I
x e KR
~-
0
for
x ~ K3
yR
and
(3.32)
Let
1~~ ~(~)1 i
K
.
~-t~t
%~ ¢ Com(KR) o Therefore
for
all
X ~ ~,
~R " % ¢ Cm(KR), and by
I~1 ! ~
•
(1.13) and
(1.12) (3.33)
We want to calculate (3.33)
B[U~R, ~ ].
Application of Leibniz's rule to
leads to
(3.34)
where
(3.35)
and
We want to apply Theorem 3.2
to
(3.34) , where the notations
(3.35)
57 and
(3.36)
are used. But the second term on the right of
tains
derivatives of order
Since
I~I
that
= m ~ I
~io ~ I . If
that vector
ei
m
of
u , but only of order
there exists at least one ei = (81i ' 62i ' " ' "
with
i = io
and
~.
10
i o,
8ni )
(3.35) < m-I
conof
~.
I ~ i ° ~ n , such then denote by
ei(~ )
> I . After partial integration
(3.37)
All assumptions lation with (3.36) and
of Theorem 3.2 are satisfied
x o -~0 (3.37)
) . Now the right side of (3.34), with notations used,
is of the type considered in
rearrange the terms above with respect to the terms w h i c h
(up to the trivial trans-
occure now
(3.1).
If
we
D~ ~ , ]~I ~ m , and identify
with the corresponding
g~,
then by (3.2),
(3.32) and Schwarz's inequality
(3.38)
c (R) 7-II÷ II .c I¢I---,~
where
Ci(R ) : Ci(R~ M; Ki,K2,n, m3 p ,
K2
are defined by Theorem 3.2
in
KR/2 , (3.38)
Y-IoT-:
lid ° vllr, p m
,
l~l--Tt~
is valid,
and
K, s u p [ a ~ ( k ) ] ) K
> 0 . Here
by (3.32). If we observe
M, K~, ~R =I
if the left side is replaced by
(Ka/2 )
A more careful application of (3.2) would show, that we can prove the
58
existence
of c o n s t a n t s
independent
of
fact
CI(R)
= ( R-6
C2(R)
=
0 < 6 < I later
on
nates.
is d e f i n e d
local
of a p o i n t
This new i d e n t i t y
are g a i n e d
suitable
are
8G
neighbourhood on
e cm( =
[z :
stant
Theorem
this
theorem
this
3G of
xo
and
JJKz(x)]j
<_
I
is r e a d i l y
I~l < m
(1.13) with
, but in the new coordirespect of
. Then
Theorem
and the e s t i m a t e s
global
R ) 0
[z :
Izl < =
such that R
,
0 . Further,
for all
proved
which
exists
there of
x ¢ Uxo
by i n d u c t i o n
an o p e n z
of
z ¢ cm(Uxo ), z-1 ¢
z n > 01,
for the J a c o b i a n
3.2
estimates.
and an o n e - t o - one m a p p i n g
with
N G) = H R =
formula
to
H R o Then we
e C m , by definitlon~ there
m > 0 , such that we have
m <_
we map a suitable
on a h e m i s p h e r e
is a p p l i c a b l e
Izl <_ R , z n = O} , z (Xo)
g e Cm(HR )
. But we do not need
up to the boundary,
to get the d e s i r e d
. Since
z(Uxo
The f o l l o w i n g
C2'
up to the boundary:
KRR = [-z : Izl < R}
(3.4O)
3.2
w i l l be p r e p a r e d
Uxo c ~ n
) and
supla~s(k)l ) > 0 k eD
and
+ R- ~ + n - m )
corresponding
that
Uxo
n, m, p, K,
) CI'
on the b o u n d a r y
the i d e n t i t y
xo c
by
estimates
in such a way,
Let
K2,
°
To achieve
calculate
+ Rn-m
( R- I - 6
(ii) ...1..ocal . estimates
neighbourhood
= C i' (M, KI,
R , such that
(3.39)
where
C i'
Z(Uxo
0
exists
~G) a con-
z
o
on
m
.
Let
=
59
Ix
whe re (~ (~ i~...i~,(×)
(3.42)
"=
and
(3.43)
P6~ (x)
The
are certain
P~
~!a)
=
0
for
IBI h l~I
polynomials
, of order at most
and
I~I = 0
in derivatives
.
D ° zi(x )
(i=1,...,n;
IGI +I -IBI
Now define
v(z)
I
(3.44)
Then
:= u(x(z))
v, h a
functions. ~o(H~)
h~(z) :=
f~(x(z))
~(z)
$(x(z))
and
:=
W
Further,
have
mu
~ ¢ Co( Xo )
the same properties
by the properties of
defined by
the transformation
for
(Z$)(z) rule and
:=
$(x(z))
as the corresponding
zj the mapping
Z : ~o(Uxo) -*
is
and onto. By
(3.41) - (3.44)
one-to-one
and (1.13)~ and an obvious
ordering
(3°45)
BKu,$]
=
BEv, W]
+ ~ K v , ~]
6o
with
(3.~6)
_~ I:~, "
:
:I) "~ "1-) o 111'I. I'I:;I.. "m
and
(3.4r)
3
[ v,
"~1 IRI_~ -,u
I'{'b- l'C I,= 2.-~n-~
(3.48)
Y-- ( ~,~)o l~l="m
where because of assumption
(3.49)
and
(2)
b.¢6- ~- C . ° ( M ' a × ~ )
Therefore, by
(3.5o)
~ G ¢ CTM , (3.42), the properties of
(1.13) , (3.45)
~I:~, Y]
is valid for every
~
,
and
= -~[~,-,:I
~ c ° (g~,:D)
(3.48)
~
7-
(o~ h~,~)~
• ~ Com(H R)._ , because of the properties of the
61 mapping
~.
By continuity
(3.50)
holds for every
Y e Comm(HR)
since every
• ~ Comm(HR) may be approximated in the ~ ' P ( H R ) - norm by functions of class Cmo (HR) for every p with I < p ( ~ . Let us define
]~o [ v, ~ ]
(3.51)
I'~=ITA.-rn
If
qR
is defined by
e Comm(HR) (i)
.
Then by
(3.31) , m R • • ¢ Comm(H R) (3.50)
and
(3.51)
for every
we get analogous
• to part
by Leibniz's rule 5 :
for every
(3.53)
Y ¢ C m (HR) o,m
where
T~ T_~ v, ~3 IVI:l~(:'~n
(3.54)
T~ [ ~ v , ~ ] I%'I:11;I'=-,~ @
These terms arise when we calculate Calculation of
(3.56)
B[v, qR w ]
~F4 [~.~ v , ~ ]
leads to
:: l:|e m
and
~o[(VqR), • ]
from
~[v, (qR • )] .
62
(3.5?) 1"51 -~T.-~L
and finally
(3.58)
G [R~ ~,,~]
Now we will prove that Consider the derivative
Bo
satisfies the assumptions of Theorem 3.2.
~'(Xo)
at a point
x o ¢ Uxo
in matrix - re-
presentation and let
\" ~L~o~ If
I e ~R n, then consider
(3.59) If
>
M[Z(Xo) ] 1
--~'~
~
• We immediately verify
(M
L~o)]].)a
Z o = Z(Xo) , consider the operator
(3.60)
L~,~o I~t=t'r4='.'L
Let for
i c ~n
b ~ (m, ~o) I ~ Iz
(3.61)
Then by
(3.46),
]
~o~
(3.59) - (3.61)
(3.62) led=~[',W'.~
63
Because
(3.40)
(3.63)
holds ,
c~ Ill !
for all
1 e ~n
there exist constants
Ci > 0
such that
In[z(x)] i l !c2 Ill
, x c Uxo . If
Bk
is uniformly strongly elliptic,
by
(3.4o), (3.6o) (3.6~)
{-~)~ ~
L~,~o (I)
=
I~I~l~--~
~- ~n C~ K
for all If
iI ~
i c ]R n , zo e H R , k e D .
Bk
is uniformly elliptic,
(3.65)
then for all
I L k , zo(1) I > m • Cl " E
That the roots-condition is valid for root of the polynomial b y the mapping
~,Zo
(l' ~)
R < I ) .
analogous to part rivatives of
is trivial,
v
since every ~
(~' ,T)
and conversely.
~o satisfies therefore the assumptions
All other assumptions are satisfied too. we m a y assume
Ill 2m .
~ , z o ( l ' , ~ ) is mapped on a root of
( ~' T ) := M [ z (Xo)]
Especiall~
i ¢ IR n , z° e H R , k e D
of Theorem
(Without loss of generality,
But if we w o u l d apply Theorem 3.2. to
(i) , the terms up to order
"at the right"
m , as we see from
de-
(3.53) , (3.55) and make no trouble,
as we will see later on. In
(3.55)
of
. We would llke to throw one derivative
from
•
up to the order v
to
~
m-I
b y partial integration.
not to be differentiableo Jacobian
J[x(z)]
(3.57)
(3.52)
would contain
(3.57). The terms arising from the estimate of (3.53) and
3.2.
occur only derivatives
But the coefficients
by~
Inspection shows~that every one contains
. In the case
m = I
this is only a continuous
need the func-
64
tion.
To overcome this difficulty,
cedure:
By the Weierstrass
we make the following smoothing pro-
approximation theorem,
> 0 , Jy] = m , ]~ ] < m - I , polynomials
there are for every
by(~e ) (k;x)
with
i..(.%) Therefore
(3.66)
]=%[ ~ R
; v,
"~'] v
--
,
~Dei(r]("t~')
])~'~))('I
and
_
(~._e;i~)V ' ~)e;(~-).,"~%-3--, bly6" ...u .~-~[ "?R~)~c"~'))o~
Now we are ready to apply Theorem 3.2 to (3.67)
llke in part
that the constant of
R .
observing
M
defined by Theorem 3.2
there exists a constant
], R ) > 0
~ (R]
:=
~no.× I%'I= ll~I~'rr,
C(~)= C(n,m,p,E,
such that for every
and
(3.69)
<(%)
:
the following estimate holds~
~× "~ 6
H'~.
FI,
(see (3.2)) is independent
0 < R < I , ~ > 0,
and with the notations
(3.68)
(3.66),
(i) . We observe when estimating the terms of
By Theorem 3.2
max I by6] , max ] c ~
(3.52),
[ bro-(~o)- br0-(~)l
65
M~(~.) 2__11~(~)11~.~..) + ccm~ >I_ II~ll~,c,.) ~ l'~'l=-r~l
ITl='m
I'CI~'m-i
By
(3.68)
~o < ~
and t h e c o n t i n u i t y
of
such that ~ ( % ) ! ~ .
(3.71) ~
by8 i n D x ~RR ' t h e r e
exists
~ere~ore
~ ~ (~7~o)II~
-~
I~l=.m-'&
Since
qR °
m
in
I
an
l~I=.f~
HRo/2 , the left side of
(3.71)
may be replaced
by
I¢l=-m
NOW let
~xo :=
transformation = 1,2,3)
z -I(KRoI2 )
and
Wxo :=
rule there are constants
W~xo O ~ o Again by the
C i' = Ci' ( 8G, n,m,p) > 0
such that
~_ c:>
II~"~II~
I~,I='m
(3.74)
7 - ll~Is~C~Oo) ~ q Y - l l ~ l l L ~ ° ~ T
(3.?5)
=
"rn~i
o'~
~ZZ I h~U~(H~o ~ I~I--'m
T"
=~
~- CL ~--II ~II~.~C~o) I~,I=nm
(i=
66
If we observe
(3.76)
l¢l~--m-i~ ~ Z#V~L~(I'I~)
-
and
(3.?7) then
~- c ( ~ . ~ ) (3.71) - (3.7?)
lead
II~II.~,,,(~
to
(3.78) ~-- II~II
which
holds true for every
(iii) By such that of
3G
s > 0.
Global estimates
(ii) , for every (3.78) and
xl,...,x N e ~ G
3G
xo ~
8G
is valid. Since
there exists a neighbourhood WXo O
~G
is open in the topology
is compact, there exists a finite number of points N N such that ~GCi=IU Wxi . Let Gz := i=IWxiU , I
K o(s) := N • max Kxi(e) i=1, ..,N
,
KI
:=
N • max K(R~ i ) " Then i=I, ..,N ~'
N
(3.79)
Wxo
7---ll~llL~c~
~-
~--
~
67 Let
M :=
G\
Gl
For a subset
.
A c ~n
N
Then
M
=
~
~Gl
G~ = i~I Wxi = i~I ~x i N N ~GI = CG u i ~ 1 ~ x i , M = G n i ~ 1 ~ x i
n
and therefore open, M ~G =
.
Since
is closed and because of
c Gz , d > O.
~G ~ M = ~
~A
:=
~n
M c ~
n
~
.
it is compact.
=
]
n
Since
then by part
(1) of
_ A . N N (i~iWxi).
~xl
is
Because
M is a compact subset of G. Therefore
If R~ : = ½ m i n [ d , 1),
(3.80)
let N
dist(M;~G)=
the proof
,=,--~.Z~ll])=
for every G. Since N'
Jl. llr,p
x o ¢ M . Here
denotes norms taken over the whole of
M is compact there exist
Xl,...,XN,
N' C~(m) ~
¢ M
such
that
M c
I1~ I1o,]~
I¢ I~'~
Combining
(3.82)
(3.79)
and
II ~-I1-~,~ --
(3.81)
we have with a constant
k: 2(ll~-n "[,Pdr~.~) ~-ll:~"~-II~.,,,,J I{I--"m
~ Yll])~llz,,.c~) Igl=~
~
was arbitrary.
~ k~ KII~II~
Now we choose
K = K(n,m,p)
~
~o :=
K(~oC~m ÷N'c~cm))ll~ll.~.~.,p
I 4 K • K~
.
Then
Ko(Eo)
+
is de-
tez~TcLned. Let := [ 4 K ( K o ( e o )
KI
+ N' Ce(Rz) ] - I
!
let Cl be
the constant by
by Lemma 3.5
Lemma 3.5 , (3.23)
.
Then we get from
(3.82)
68
(3.83)
~-~ II ~lt~,p
But this proves
~
(1.14)
q'll ~-lls~(G) o
(~ ~__ ~ ÷,~11o,,~ I~t= vn
*
(1.15) is a trivial consequence by H~Ider's
inequality.
If
Bk
is uniformly stronsly elliptic, G~rding's inequality (1.18)
applies. Since
wm~P(G) c wm#2(G),
if p h 2 , we have
I~l=~
this is
(3.84) By
C., II u. I1.,,,,,~.
~- ( }--
HGlder's inequality with
II ÷,~Iio,==) ~ ~ :=
2•,
I I [:= I -
and therefore
(3.85)
( I~ll='vn F_ u{, k:) {
i.-z ½ -"
2--U+~kp r
1~4=',r,'~
Z
~4
L~l=-,~
Since
I
IlutlL= (G) we get
o, p
from
(3.84)
<
#(G)~" 11=11 o,2 and (3.85)
<
C(G,m) llUllm,2
,4
69 4
(3.86)
with
K
=
(3.86)
K = K(p,n,m, GjE) > 0 . If we combine
and
(1.14),
(1.16)
is proved. q. e. d.
In quite a similar way we prove the following global regularity theorem.
Theorem
3.6
As sume (I)
that assumptions
(2)
that
Bk
(3)
that
u e wm~2(G)
f~ ¢ C~(G)
(I) - (3) of Theorem 1.6
is uniformly
such that for
strongly elliptic for
and for every a
are satisfied;
k e D
IGI = m
and all
k e D ; there are fh/nctions
@ e ~o(G)
holds true.
Then
u e ~P'(G)
for every
p'
with
I < p' <
Proof: We have only to prove that G
is compact~
u e wm~P(G)
it suffices to prove
open neighbourhood
Uxo
that for every
p > 2 . Because
xo e G
there is an
I e LP(Uxo ) , I~I < m . This Uxo local property we will prove with the aid of Theorem 3.3 and by use of the p r o o f of Theorem
(i)
1.6
such that
for every
in two steps.
Pro0 f for the interior 0 f
Consider part
D~u
G
(i) of the proof of Theorem
1.6. Let
x° e G
and
?0
R o be d e f i n e d (3.36) right
for of
as there.
Take
. After
a partial
R~
(3.35)
(3.88)
Rz
and in the right
3~ [ ~ ,
~]
:=
½ Ro o
We derive
integration of
(3.87)
again
in the first
(3.33)
-
t e r m at the
we get
-- ~ [~.~;~,~3 . T~[R~, }~, ~]
where
and
Then all the a s s u m p t i o n s
of the local
regularity
with
3.3~
e Wm~PI(KRI ) ,
p = 2 . By T h e o r e m
I 2n , m the
2n n -I
. Therefore,
. Now we a p p l y
this a r g u m e n t
Pl =
kth
step w i t h
2n Let Pk = n - k "
k < n k
gives
I < p' < ~
Is
(ii)
and
defined
generality
1Kin/2~ Lp~
repeated
D~u
(KRI/2)
(induction)
IK R I / 2 n
I Pk
I 2n
= 0
e L p' (KRi/2n)
satisfied I
where
I
p-T
= ~ -
for all
I~I
: So we get a f t e r
IKRk c L p k (KRk ) , w h e r e
I . Then
D~u
< m
D~ u
are
"
~
R~ 2~
=
and t h e r e f o r e
for e v e r y
p
, the
with
.
Proof up to the b o u n d a r y
In the following, we use part
:
= n-
n-th application
HR
(u q R 1 )
theorem
we
suppress
(ii) of the p r o o f as there. that
Since
the a r g u m e n t
of T h e o r e m f~ ¢ C~(G)
1.6.
k , which
Let
we m a y assume
x° e
is fixed. 3G
without
Now
and Uxo , loss
of
71
(3.91)
Uxo
n
All identities
supp fa
=
~
(3.44) - (3.57)
and for every
R' < R .
'
I~l= m .
are valid with
replaced by zero
ha
Consider 5 i-2.
for all
~ E C o,m(HR) m
, with
Fi
by (3.54) - (3.57).
But now we cannot apply Theorem 3.3. The reason is, that the coefficients of continuous.
E
and
Fi
need not to be differentiable,
Therefore, we cannot perform a partial integration
and (3.56) to d~minish the order of derivatives hand,
they are only
of
in (3.54)
W . But on the other
the assertion we want to prove gives no reason to assume that any
differentiability lity properties
properties
are necessary.
of the coefficients
son only in the technique
of
B
in Theorem 3.3
applied in proof.
quality , but not of the "quantity"
That we need differentiablhas its rea-
There is only need
of the derivatives.
of the
Therefore we
overcome this difficulty in the following way: (a) forms B (~)
We approximate
B (~)
and
F~ (e)
is uniformly (b)
B(~)
~
sides"
with smooth coefficients
Fi
uniformly by
in such a manner that
Ro < R
small enough,
u ¢ wm~2(HRo ) _
then
Re B[u,u] h Cllull~,2
and the analogous
inequality
for
. (c)
u s
and the"right
strongly elliptic.
If we take
will hold for all
B
Then~by the well-known
Wmo2(HRo )
such that
L2 - theory~ we get the existence of
B(~)[uE, • ] =
~
Fi(~)[~R o, v, ~ ]
.
i-----~ (d) in
From this solutions we will show that they converge to V~R °
Wmo 2(HRO)
- norm
i f
e -~ 0 .
72 (e) - (g)
Furthermore,
O ~ u s IHR . s L p' (HR*)
have
D~ u s IHR .
we will show that for some for every
Da v IHR .
converges to
p'
I=I
and
in
Lp' -norm.
IYl
< m ,
R* < R
we
~ m~ and that Then the theorem
is proved.
(a)
Since
strass approximation b~ )
c°(~)
byT ~
ITI <
theorem there are for every
m , by the Weier-
g > 0 polynomials
such that
max
(3.93)
Iby,~(z)- b$~)(z)I <
max
Jy; < m z cH~ i~l<m Let
for
~(s)
and
Fi(S)
(3.54) - (3.57)
(i =2,..,5)
with
byT
be defined analogous to (3.46),
replaced by
b$~ ) . Then for all
B(~) is uniformly strongly elliptic with ellipticity constant
(b) constant
(3.94)
Let
Bo
be defined by (3.51). Then, by (1.17),
C = C (n, m, E)
Ro < R
and
max
IYi:l~J:m
C(n,m)
max
;Yt=ITi:m Then for
.
there is a
byx
for all
g
Wmo2(HR )
maxlbyT(o ) - byT(Z)IC(n,m)llull2m, 2
z~H%
there is a constant
max IbyT(O ) - byT(Z)l
<
zcR%
- ~
u ~ Wmo 2(HRo)
@
U C Wm~2(HRo ) . Then
I~o[U,U] - ~[u,u] I <
By the continuity of
~
such that
Re ~o[~,~] >_ c ll~II~,2
NOW let
s ! so , E
we have
Ro < R
C
"
such that
73
(3.95)
Re Y[u,u] >__ Re Yo[U,U] -
l~[u,u] - Yo[U,U]I
>
.
On the other hand, by
IY[u,u] - B(e)[u,u]l
(3.96)
Re
for every
(c) for
there is an
<_ ~ llull~, 2,,~
for
that
~ < cz • Therefore
every
s ! el .
and
Now consider
(3.92) with R' := Rz . Since D ~ v e L2(HRo ) 5 Iml ! m ; Ge(~) := ~ Fi(S) [ qRo; v, W ] is a continuous linear wm~2(HRo )
theorem applies For every
(3.97)
for
~ ! el .
( see e.go S.Agmon
0 < ~ ! el
there is
~ (e) [us,C]
alZ
~> ~ wZ&2(~z~o) "
(d)
By (3.93)
we have
=
By (3.96), the Lax-Milgram
[3] , or
[68], [5] , [17] ):
uze I ~ 2 ( H R o )
~__ Fi(S) [ i=2
max
Ii~l<m IZm for
ez < e o , such
2 ,2 >_ ~c IIu llm
Y (~) [u,u]
u ¢ wm~2(HRo )
functional on
for
(3.93)
x
such that
"qRo; v,
~ ]
max Ib (~) (x)l < M < ~ R ° '~ -
0 < e ! Ez . Therefore, S~p ~G w.",'(~Ro)
for
0 ! ~ i ~
with
C~ = C~(n,m, Ro,m) i a l _~ m
By
tlDa v IlL2 (xRo )
(3.96)
and (3.97)
74
tluEll2m, 2 -<
~2 c~. llu~llm, 2
but that is
(3.98)
II u s
for every
lira, 2
< -
2 C~. c
~ ! el . On the other hand we have for every
(3.99) ~ [ v ~ ° - ~ , { ]
=X[v~.,{]
=
,
¢ ¢ ~2(HRo):
~ [ ~ , , { ] -~
-
-
b~ ) Hrl= tr.l.="rn
~ith
a constant
y = y(n,m, Ro)
this leads by (3.93) and (3.97) to
I~l="m
If
• :=
v qRo - ue ' we get by
II v v ~ . - ~-~ll..,~
~-
We note that this implies
(3,100)
for every
(e)
(3.94)
(since
qRo m I
liD(~ u E - O ~ v il L2(HR, ')
R'' <
Ro --
and every
on
~
0
HRo/2
)
(~ -- O)
I~I < m .
For the subsequent proof we need two inequalities:
?5 &sstlr~e (i) and
1 < p' < ~
that
I < p"
< ~
(ii)
that
and
~
E ! E~ • that
every
such that
1
p,
1
2n ' if
I ~-~ ~ R*
•
z ~ wm'P'(HR*)~
1
=
I p-~ =
arbitrary• if 0 <
1
!
p,
1
2n
> 0
'
0 ;
Ro
~-
and
w E ¢ Wm'P"(HR*),
If
~R w~ ~ Wmo p (H~)
and
~
~
e Wm~P'(H~) for
R ! R* 1 (iii)
that for every
@ ~ ~o(HR*)
and
every
R ~ R* s
(3. ~o~ )
'~o ~'~ [ we "~,~ ~ ~] i--1
holds ,
where
Bo (E)
Then there is a
is defined analogous to (3.51).
R** < R*
> 0 , but independent of
Further, there exists wE.
0 < E < Ez
Ee < ez
~(R)
-
such that
~)
l¢l~"m
s =
such that
if
max (E',E") .
f o r every
0 < s'
E"
>
~ < R
< e2
and
(ii) and (ili) , then there are
= Ki(R•by~•n•m, p') > 0 ( i = 1,2)
(3. 103)
where
K(R) = K ( R , b ¥ t : , n , m • p ' )
and a constant
, wE,. are functions satisfying
constants
&-~
such that for every
~-
I-P'-~J
÷ U
)
76
Proof: Let Then by
R ! R$ and
h ~ wm'P' (H~), such that
Sobolev's lemma
and there is a constant
Inspection
of
(see Appendix I) ,
h q~ e wm~P'(H~).
hlH3/4 R ¢ wm-I'P"(H3/4 ~)
C(R) = C(n,m,p',R) > 0
such that
(3°53) - (3.57) shows that all assumptions of Theorem 3. 4
are s~tisfied. Let
tl(~) :=
(3.1o5)
max l'~l=ITl=m
(E) (z)l, max Ib(~:) (o)- byT z~H--~
max 0<~<s~
m
Let N< ~be a constant such that for all
z c HRo , 0 _< E _< Sl
and
IYl <_ m , I~I <_ m
(3.106)
I b#~ ) (x) l
Then~ by that
<
(3.20), there is with a constant
supp ~
N H~ c H 3 ~
N
.
Cl (R) = Cl (R,n,m)
) :
~R
l~l=mn
l~'l_.-t ~
+
N
7-
(observe
77
By the continuity of so small that
byT
q(R**) ~
and by (3.93) it is possible to choose R I ~-~ o If we take R ! R*@ , then we transfer
the first term at the right side of (3o107) to the left. Now we estimate the second term at the right by (3.104)
If we observe the trivial ine-
quality (which holds by (ii))
(3.108)
7-
11:~('.,~ ~-~)ll~p,,(..~)
inequality ( 3 . 1 0 2 )
To prove
+- c(-~,~,~.')~
~:~(.,,. ~)llz.,..c.~ ~ ,
is proved.
(3.103) we define
z~q--tr,i=-~n and
(F (~ ' )
-F i
(~ ,, ) )
analogous. Then, by the assumptions above
(3- 101 ),
5
I"I S
and therefore
- ~i L ~ i
w~,'', ~]~
+-
and By
78
(3. t09)
~J~')E(w~,- ~t,") ~ ,
{]
---S"
s
4-
S
Again we apply (3.20)
Theorem 3.4
to (3.109)o With the notation used above,
leads to
(3.~1o)
~
11~ C,,,,,,~, - ~,,., 4I¢~l=wn
+ .q
First we choose e~
such that
S ~ ~ Rg-~ such that q(~) !~-~ • then we choose an e2 ! I e2 ! I~ and put the first and the last term to the
left side. To all the remaining terms at the right, where norms appear, we apply (3.104). If we observe (3.108) with ced by
(f)
(wE, - we,,) , (3.103)
In the following let
Lp" (H3/4 ~ ) we
repla-
is proved.
0 < E < a2
and
~ < R** . In quite
m
the same way we calculated (3.52), we calculate now ~(e)[uE ~RR ' ~ ]" Ro If we observe R ! Rg-~ ! -~and ~R ° 1 on HRo/2 , then by (3.97)
79
the result is for ~ ~
C~(H~): S
The left side satisfies the assumptions of Theorem 3.3. But we have to ( ) e) contain derivatives prepare the right side, since Fa "e" and F4(
of m - th order of
• . Since the coefficients are smooth
(but this was
quite one reason for the smoothing procedure above), we transfer in F2( e )
and in
F# (e)
in the case
s = T
tegration one derivative from
$
ged terms we denote by
and
F~ (e)
Jr ~ "
This is true
for every
and
= m
to the other functions~ F~ (s)
z
by
the so chan-
o Then we have
u?(t)F
R < R** , 0 < e --< ea , and
We observe at this point, that (3.101) is valid if uE~ and
by partial in-
v , with the same restrictions on
$ ~ Com(H~) .
wE ~ ,
is replaced by e and
$
above.
(g) We
will prove by induction:
Let
0 e< _
k < n,and
_< e 2 ,
I < Pk <
(2)
D c~ U~IH(Rk )
~
:= 2 ~
and
n-k p~ i " = - f~ .
=# if k > n .
k < n
Then for every (1)
k ~ ~,
e
LPk(HRk )
There are constants
Ci(k ) > 0
(i=1,2)
such that
,
as
8o (~)
7- ilD~(u~, - ~,, )ILP k i~I < m (HRk)
i
c~ (k)
÷ a C~(k7 where
(b)
(3)
iiu~, - u E,,1,~o 2
(HRo )
_____IIDC~v ilL2(HRO ) I~1 <m
~ = max(e',g") , and
7- iD % s , l~7-!m ILPk(HRk)
D e v [H(Rk)
¢
LPk (HRk)
(4) I(z~--7IIDC~Vm IILPk(HRk
< m C~v IL2 (HRo ) -- C2 (k) IG~
_< Ca(k)
TMT-
Io~T~ IIOC~v IIL2(HRo)
Proof: Let
k = I . In (3.112) let
R := R g'~. Then all assumptions
of Theorem 3.3 are satisfied. Therefore, I= p~
n-I 2n
"
We apply
iuslm, 2
and therefore by
< -
(e)
2_ 7-10 c ]c~l <m
p' = 2
c~viL2
with
By (3.98)
for
(HRo)
o<s<s=
(3.102)
This proves (I 7 and (2b7, if rive from (3.103)
with
(us q2Rl ) e wm~PI(HR **)
2
C2(I) := K(RT( U + 17 . By (3.98 7 we de-
81
c~(a) ~__ i1:~, ~.r.,c~, ) with is
C~(1) :=
K~(R), which proves
ue ~ L p~ (HEa)
such that
lID~ue - uglILp ~
pl > 2, llD°ue - u(;IIL2(HI%) -* D e v IHR ~
=
(2a). By the last inequality,
0
(~
~ O)
U O ¢ LP~(HR~) , for every
(H~)
-~
0
and therefore
I~I
there
(~ -~ O) . Since
be (3.100)
< m . This is (3) , and (4)
is valid by continuity.
Assume now that the assertion is proved for some
k ( n o Then the
right side satisfies all assumptions of Theorem 3.3 with defined above), apply
(e)
T
with
if
~
R :=
~
. Therefore
P = Pk
m, P k + 1 (H
(Pk
) . We
u~ ~Rk c W o
P' = Pk" By the induction hypothesis
il :~'
~
~
~:D~(~-,.~,~.)I~.~,(~, D
i~l~_-,r,1
To prove (2a)~apply to (3o103) with
R := ~
and
P' = Pk
the induc-
tion hypothesis
I~-~
II :D~(~,., - ~.,,)IL,~,~. "1,
(,
l-I-~s. ).
÷ ~ (-~). ~. c~ c~) ~
II ~"vll
-F
82
This proves
(2a)with
C2(~) (Kl ( ~ )
and
+ 2K2(~)). i
in the case
C!(k+1)
_
Pk
:= K ! ( ~ ) C I ( K )
The same argument
I 2n j 0 )
C2(K+I)
(if we choose
as above in the case
k = I
:=
Pk+1 > 2
proves
(3)
(4).
Now let
Wxo
:= z-1(HRn ). Then the properties
the transformation
z -I
Therefore~
D° u I
'
and
and
of
v
are preserved by
e LP'(Wxo)
for I
WXO
I~I < m .
So we proved the existence arbitrary tion of
x o ¢ ~G u
of an open neighbourhood
(open in the topology of
to this neighbourhood
belongs
G)
to
Wxo
of an
such that the restric-
wm'P'(Wxo ). q.e.d.
Remark
3.7
Careful inspection of the proof of Theorem is not necessary to assume that
G
is bounded.
that after a rotation of coordinates ordinate
axis
(see below).
xn
(ii) of Theorem
8G
, considered
(~) All these mappings continuity and~ stants
Ci
It is only necessary, G
on the co-
(3.24),
is valid.
Part
(i) and
1.6 need not to be changed, we have only to replace
L p - norms are taken over z
it
has further properties
the right side of (3.78) by terms of analogous
mappings
shows that
the projection of
is bounded and that
Then Lemma 3.5, estimate
1.6
Uxo . We assume,
m
independent
(3.4o) and
Uxi
have the same modulus
a countable
of
(3.63) are valid with con-
of the special function covering of
(G~ := U Ux. ~ of the same "magnitude" ~h~ corresponding
that the locally defined
and their derivatives
estimates
Then we may construct
where the
at the beginning of (ii), have the properties:
(~)
and
structure,
8G
z
considered.
with n e i g h b o u r h o o d s ~ i ,
in such a manner,
has a non-empty intersection
that every of
with the remaining
83
Uxj only for at
most
a fixed number of indices.
that the projection of than a fixed number
G
on the
x n- axis does not become
d' ~ 0, then we may
ing of the "remaining interior set" radius and the same finite global
Theorem u
construct
M := G \ G ~
Poincare's inequality
1.6 , estimate
by a function
a countable cover-
with balls of uniform
(iii). Observe,
that in this
(3.77) is valid . Therefore,
(1.15), holds then. If we replace in Theorem 3.6
v := u ~
(0 ~ ~ ~ I, ~(x) = I
"smaller"
intersection property as above. Then the
estimates are performed as in part
case considered,
If we further assume
for
where
is a "cut-off" function
Ixl ~ m, q(x) = O
assume that an identity of type Theorem 3.6 is valid for
mr
v .
for
(3.87) is valid,
Ixl h r + I )
and
then the assertion of
Chapter Ii
: A representation for continuous nals on
~o'P(G)
linear functio-
(I < p < ~ ) and its applications: o
A generalization of Garding's inequality and existence theorems As we emphasized in the introduction, the L 2 - Theory of Dirichlet's boundary value problem for uniformly strongly elliptic forms is at one hand o
based on Garding's inequality.
On the other hand essential use is made
from the fact, that the function space, under consideration are Hilbert spaces and therefore Riesz's theorem gives a representation of linear functionals. in § 4
Following the ideas outlined in the introduction,
a natural representation of the continuous
w~'P(G)
we prove
linear functionals on
(Theorem 4.6). These proofs are based on two results of chapter
I : The a priori estimate of Theorem
1.6 and the regularity result of
Theorem 3.6 . As a first application of the representation theorem we prove the correspondence of a continuous bilinear form on -~'P(G) x × ~o'q(G)
with a pair of continuous linear operators
(Lemma 5.2). A
further easy consequence is a generalisation of the Lax-Milgram-theorem (Theorem 5.4). The next section is devoted to the proof ofageneralisation of G~rding's inequality to uniformly
(not n e c e s s a r ~ strongly) elliptic
Dirichlet bilinear forms defined on
W m'p(G)
x ~oo'q(G) , where
I < p,q <
O
and
! + ! = I (Theorem 6.3) . We suppose only that the "leading coeffiP q cients" are uniformly continuous and the other coefficients are bounded.
These low assumptions the proof,
are the reason for some technical complications
in
since we have to apply a smoothing procedure to the coeffi-
cients. As usual,
the estimates in the case of variable coefficients
are
derived from the corresponding estimates in the constant coefficient case
(Theorem 6.1) with the help of a partition of unity. We observe
that part (i) of the proof of Theorem 6.1 may be done without use of Lemma 5.2 ~ only Lemma 4.2 is needed. Therefore, Theorem 6.3 is independent of Theorem 3.6 . We did not go this w a y paper°
because we wanted to save
In Remark 6.4 we give some hints how the assumption on the boun -
dedness of the lower order coefficients may be weakened. gives very important local a priori estimates
Theorem 6.5
for uniformly strongly el-
liptic Dirichlet bilinear forms. At this point we have enough material to prove in § 7 existence theorems for functional equations in the case of uniformly strongly elliptic Dirichlet bilinear forms. based on the fact, that in this case
The proofs are
Problem (F~)is locally u n i q u e l y o
solvable
(Theorem 7.2). From this we m a y conclude with the aid of Gar-
ding's inequality and with Theorem 6.3 that there is a
x ° ~0
such that
the half-plane Re x ~ x o does not contain generalized eigenvalues
(Theo-
85
rem 7.3). It is then very easy to prove that Fredholm's holds for the problem under consideration
alternative
(Theorem 7.5). A further easy
consequence of Theorem 7.6 is that the solutions of the problem B[u,@] = O 7 * E C~(G)~ satisfy u c ~,~'P(G) for every p with I ~ p < ~ . This is a kind of regularity result. More general results we will prove in § 11 . We point out, that chapter Ii may be read independent of chapter !, only the knowledge of Theorem
§~.
1.6 and Theorem 3.6 is necessary.
A representation
for continuous
nals on
(I < p <
~'P(G)
linear functio-
~)
We will construct a reflexive Banach space
I~'P(G) in such a man-
ner that we can prove an explicit representation of the continuous linear functiona!s on suitable subspace. § I
Lm'P(G) and that we can consider
Primarily with the aid of our a priori estimates of
(Theorem 1.6, (1.16)) and by use of the Riesz representation theo-
rem for continuous
linear functionais in a Hilbert space and application
of the regularity theorem of § 3 red characterization of Definition Let Let
T~'P(G) as a
I
G c IR n
tupels
(Theorem 3.6)
we can prove the desi-
w~'P(G) * .
4.1 ~ be a real number and let
be a bounded measurable
~ = (a ,...,an)
m~1
and
n~2
be integers.
set. Consider the set of all
with non-negative integers
n-
a i ( i = I..... n) and
1
lal = m . Choose now an arbitrary,
but in the following fixed numbering
of this finite set and denote it with
M m := [aI
,a r]
r = r(m)
Let
Lm, P(G)
=
be the set of long
to
LP(~).
[f : f =
(fal .... ,far),
f~
~ %(G),
(~ = I, .... r)~
r - dimensional vector functions, whose components be-
86
Further, we define for
(,~.~)
({),,:=
f,g
{~.,
¢
, ,,-
Lm'P(G)
and
~,y e C
~., .... ,~
(4.2) 4 T
As we immediately see,
Lm'P(G)
is with addition and scalar mul-
tiplication defined by (4.2)
a vector space over
fined by
Lm'P(G) such that
(4.3) is a norm on
C . Further, Lm'P(G)
~
de-
is a Banach
space. Let us denote
(4.~) If
~{I~,~
p = 2 ,
~(£)
:=
Lm'2(G)
,
~ ~ L~'e(G)
is a Hilbert space with scalar product T
{,~ Lemma4.2 Let m>
I
(c-)
I)
I
, I
be an integer.
be real numbers with
Assume that
Then~for every fixed
G c IR n
~S
+ I = I
and let
is bounded and measurable.
g e Lm'P(G),
is a continuous linear functional on
i)
~
Lm'q(G).
Conversely~every
For the sake of completeness we present the simple proofs of this and the following assertions.
87 F* ~ Lm'q(G)* g e Lm'P(G).
may be put in this form with an uniquely determined Furthermore ~ llgJlm,p = JlF*ll~,q •
ting elemen~Iof
g
is called the" genera-
F* .
Proof : Applying H8ider's
inequalities
side of (4.6), we get for every
for integrals
g e Lm'P(G)
and sums to the right
and every
f c Lm'q(G)
T
1t %, which proves the first part We define
Tv
,=
is continuous
for
of the lemma.
Tv : Lq(G) -~Lm'q(G)
(4.8)
f
by
,
and linear.
f = (f~z .... ,f~r)
1I
Now let a
¢ Lm'q(G)
F* e Lm'q(G)*
be given. Then,
we have
T
(~.9}
T'~
Since
Tv
: and
¢ Lq(G)*
for
isomorphic
to
>-- T * ~ % ~ . F*
are linear and
continuous,
v = I.... ,r . If we observe that LP(G),
then for every
of an uniquely determined F W T v f = (g~v,f)o By (4.9) we have with
g~v c LP(G) for every
it follows that Lq(G) *
v = I, .... r
is isometrically
we get the existence
with f ¢ Lq(G).
g := (g~z,...,g~v)
c Lm'P(G)
'r
The uniqueness ~)
of this representation
~. ~,,~ denotes the norm in
will be proved later.
W o-m~q ( G ) *
F*Tv e
88
To prove II~ll~,q = llgIim,p , let define f o = (fo~i, .... fo~r) by ~ ©O: =
I ~ ( ~o° ( × ) P - ~
~1 + ~1 = 1
Since by the relation we conclude
I
f° ~ ~ ' q ( G )
g ¢ ~'P(G)
~
be given. ~ t
us
(p-l) q = pP -- II p =
P;
~,~
we have
~
and
(4.11) ~=~" G
On the other hand
(4.~2) I~o I -~ II~'*~ II~°II~ = IIv~IIl~(~ZI~oI(~'~)~
By (4.11)
(4.7)
and
anG
(L~.12)
(4.1o)
lead to
IIF*IIm*q <_ IIgII~,p , from what we conclude
by (4. ~3) (4.~4)
From
(4.14)
If there are ~ Lm'q(G),
we derive the uniqueness of the representation gl, g2 ~ Lm'P(G) then
such
(gl-g2,$) m = 0
that and by
(g~'~)m = (g2'~)m (4.14)
(4.10). for all
gl = g2 = 0 .
q.e.d.
89
Lemma 4.3 For every real number
p
with
I
the spaces
Lm'P(G)
are
reflexive. Proof
:
We have to show that the natural imbedding is onto. We define for every -+Lm'P(G) *
by
(Xqg) f
Then~by Lemma 4.2~ Tq
, g c Lm'q(G)
where
F* = T q g
linear, every
~
~Tq
g ~ Lm'q(G)
~
a map
f ~ Lm'P(G)
and maps
is antilinear,
. Let
I
for
is isometric
s Lm'P(G)*
we have
with
:= (g'f)m
As is readily seen, Tq ~ C
q
J : Lm'P(G) -* Lm'P(G) ** ,
.
Lm'q(G)
on
Lm'P(G)*.
that is, Tq(~g) = ~Tq g
~ Lm'P(G) **. Then
with an
Tq , Lm'q(G) -~
g c Lm'q(G).
¢ Lm'q(G) * , where
for
F**F* = F * * T q g
Since
(F**Tq)g
. According to Lemma 4.2 there is a
Tq
is anti-
:= F * * T q g
for
h ~ Lm'P(G)
such that
for every
F~ ~ Lm, P(G)*.
On the other hand there is by definition
(4.16)
(3h)~
But from
~
(4.15)
=
~mh
and
for every
(4.16)
of
J
F* e Lm'P(G) * and
we conclude
Jh
= F *4~. ~e
Lemma
&
4.4
Assume that
G c Xq n
jm, p : W~o,P(G) ._,. Lm, P(G) (4.
h eLm'P(G).
( a
:
is a bounded open set. Let us define
by
,
9o
Then Lm'P(G)
jm, p
is an isometric linear mapping from
and therefore
jm'P(w~'P(G))
~'P(G)
in
is a closed subspace of
Lm'P(G).
Proof : The first part of the assertion follows from the second from the fact that
jm, p
(1.1)
is isometric and
and
(4.3),
~o'P(G)
complete.
q.e.d. In the sense of Lemma
4.4 we may consider
~o'P(G)
as a subspace of
Lm'P (G).
Theorem Let G
4.5
I
< ~
be a real number,
be an open, bounded subset of Then the spaces
W m'P(G) o
let
m > I
be an integer and let
~R n
are reflexive.
Proof : Since
jm'P(W~o'P(G)) is a closed subspace of the reflexive
Lm'P(G) , it is reflexive and therefore, space
~o'P(G)
the topological
space
equivalent
is reflexive too. q. e. d.
Theorem Let m ~ I
4.6
I
~
(Representation theorem) , I
be real numbers with
be an integer. Assume that
boundary
3G ¢ CTM .
Then for every
g ¢ ~o'P(G)
G c IR n
~I + 1 = I
and let
is open and bounded with
91
is a continuous F* g W~o'q(G)*
linear functional on there is a
F*f Furthermore,
W~o'q(G ) . Conversely,
u n i q u e l y determined
= (g'f)m
for all
there is a constant
g c ~'P(G)
f ¢ ~o'q(G)
K = K(n,m,p,G)
laL, g
(i)
such that
. > 0
such that
I ,e
is called the generating element of Proof
for every
F*.
:
The first assertion is trivial by H~ider's inequality
(see
(4.7)). (ii) For the proof of the second assertion, case
I < q~2
. We prove the following assertion:
K = K(n,m,p,G)
(4.20) where
> 0
such that for every
~ II~ll~,p
uc
we first consider the There is a constant
~o'P(G)
~-
Sq := { • eW~o'q(c- ) :
II~llm, q
< 1 3.
Proof : Assume Then of
u c C o~(G)
F* ~ ~ ' q ( G ) * . Lm'q(G).
F* ~ Lm'q(G) *
and define
F* • := (u,@)m
By Lemma 4.g we may consider
Then by the Hahn-Banach with
~*I p~o,q(G) = F*
By Lemma g.2 there is a
f c Lm'P(G)
for
@ c
wm'~@)aSo a subspace
extension theorem there and
such that
'q(G)
is
a
92 for
all
•
e W~o'q(G)
and
(4.23) II ll ,p = II *II , Combining (4.21) - (4.23) second cas~ for
we derive
u e C~(G). Since
is proved for every
u c Wm'P(G)
(4.20) with the aid of Theorem 1.6,
C~(G) by
is dense in
Wm'P(G)~o (4.20)
approximation.
O
Let now a
F* ~ ~ ' q ( G ) *
above we get:
There is a
F*~ Since with
We
C~(G) f~)
=
f e Lm'P(G) such that
(f'~)m
is dense in
e C~(G)
be given. Analogous to the conclusions
for
for all LP(G),
@ e w~'q(G) .
there is a sequence
v = I..... r,
and all
n c~
if(n)] c Lm, P(G)
such that
define
:=
Then
for every
I
n:
j
~. e W ~ , 9 (G),
Fn* e wm, q ( G ) o " Since
inequality that there is a constant
)IIm' P for every
"n ~ IN
p_> 2, we prove by HGlder's
C = C(n,m,G, )> 0
such that
[If(n)IIm,2 <
c IIf(n
¢ ~oo'2(G)*.
By the F.Riesz representation theorem there is for every
n tin a
u~e ~ ' 2 ( G )
By Theorem 3.6 we get a dense subspace of (4.20) and (4.26)
n e]N. Therefore,
Fn*
I ~,2(G)
c
such that
un ~ ~'P(G)
. Since wm'2(G) may be considered as
W~o'q(G), (4.26) holds for all we have
$ ~ wm'q(G). Now by
93
-~ Therefore
there is a
~-~ H~ (~' - ~(~ ",~
u ~ ~o'P(G)
If we pass to the limit
n-~
such that
in
(~,~-~.o).
~ o
flu - UnIlm, p -~ 0 ( n - ~ ) .
(4.26) and observe
(4.2~) and
(4.22),
we get (u'@)m = (f'$)m = F * $
for every
To prove the uniqueness are
u i e w~'P(G)
c ~o'q(G).
But
$ e wm'q(G).
of this representation
( i = 1,2) with
F*@
(4.20) applied to
= (u1,@)m = (u2,@)m
u := u! - u2 gives
Theorem 4.6 is therefore proved in the case is trivial by (4.20) and H~lder's
(iii) show that
Assume now that
2~q<
f e Lm'q(G)
for every
ul - u2 = e.
I
since
(4.19)
inequality.
~,
(4.20) holds in the case
there is for every
we assume that there
I < p~2
and
~I + I = I.
We will
2 ~ q < ~ too. As we proved in (ii),
(observe:
q > 2)
a
w ¢ ~o'q(G)
such
that for every
By (4.20) and HGlder's
¢ c $~o'P(G).
inequality applied to the right side of (4.27)
we conclude
(4.28) Let
K II ',~11~,,~
u ~ ~?'P(G)
w ~ ~o'q(G)
"-
with
11 ~: 11.~,,~ u # O
f ¢ Lm'q(G)
Then, by (4.27) ~e ~l~ould get
il ~ 11.,.,.,,~
given°
be the uniquely determined
Take an arbitrary
diction.
be
Therefore,
element
such that
f c Lm'q(G)
such that
let
(4.27) holds.
(u,f)m ~ O. Suppose that w = O .
O = ( ~ , U ) m = (f,u) m @ O, ~Jhat is contra-
w @ 0. Further,
II ~: t1.,,.,,~
For every
by (4.28) we get
~: II ~ 11.,.,,,~
K
,,,, ~ ~;,,
94
If
f~0
and
(u,f)m=0,
then
_ ~. ~pl(u~,w]~l
I(~$)~I II~II~a
is trivially
satisfied. We get therefore by Lemma 4.2
(4.29)
II -II
-II~:II~,,I-'-i
AS we have shownj by (4.18) a continuous tilinear map
(4.30)
~p : ~oo'P(G)-* W~o'q(G)*
(
is defined, if we put for
--
~p(W~'P(G))= there is a
~'q(G)*
is closed. Suppose
G~-~ e ~ ' q ( G ) * *
F* c ~p(W~'P(G)) . Since such that
(and as it is readily seen) an-
G** = J g
canonical embedding
with
G** ~ 0
wm'q(G)o
but
is reflexive,
~
G'F* = 0
~'q(G)*.Then for every
there is a
g c~'q(G)
with
IIgljm,q = llG**II**m, > 0 , where J denotes the q J : wm'qfG~ - ~ ' q f G ) * * Now for every f g --~^#n'P(G) O
0 = G** ~pf
~p(W~o'P(G))
"
"
O
~
= (Jg) (~p f) = (~p f)g
-
•
O
= (f'g)m"
By (4.20) k IIgIlm,q Therefore
<
sup I(g,f)ml = o, f ~Sp
~p(~'P(G))
=
~o'q(G) * .
which contradicts
IlgIIm,q > 0.
This completes the proof. q.e.d.
The proof
of Theorem 4.6 admits the following geometric interpretation:
Theorem
4.7
Let the assumptions of Theorem 4.6 be satisfied. Then for every
p
P : Lm'p (G) -~ Wm'P(G), p o
with
I < p < ~ there exists a projection operator
that is a continuous linear operator satisfying
95 pp2 = pp,
with the additional properties f c Lm'P(G)
@ ~ ~'q(G)
(f'@)m = (Pp f' ~)m
(2)
Pp (Lm'p(G)) = Wm'P(G)o
(3)
Pq
(4)
if ± + ± : I. P q I - Pp is a projection operator and (I- Pp)Lm'P(G)
(5)
for
and all
(I)
is the adjoint operator of Pp with respect to the form
to
W~O'q (G)
If
p' > p
then
Pp Lm, P'(G)
=
(' )m
is orthogonal
Pp,
Proof: Let Thenj ~
f e Lm'P(G)
F*$
:= (f'$)m
e ~o'q(G) * and by Theorem 4.6 there is a
u ¢ W~o'P(G)
for
• c ~o'q(G).
uniquely determined
with
(u,$)m = F*$ Define
and let
= (f'@)m
Pp f := u,
for every
which gives
• c~!.om'q(G).
(I). It is immediately seen that
Pp
is linear. From (4.19) we get llUlIm'P <
I
sup
~
IF*S1
$CSq
I llfllm,p ,
<
which proves the continuity. Since by the definition Pp f = u = PpU
= Ppf
=
= u
for every
u c W~o'P(G), we have
•
From (I) follows for = (Pp(f- Pp f)' $)m
PpU
f c Lm'P(G), $ ¢ W~o'q(G):
(f-Pp f' ~)m
=
(Pp f- PP f'~)m = 0 , which proves (4), since
(i-Pp)2 = i-Pp. Let
h ¢ Lm'P(G)
and
g c Lm'q(G). Since
h = Fph
+ (h- Pph)
logous for g), we get the relations (~)
(Pph, g)m = (Pph, p q g ) m + (Pph, (g-Pqg))m--
(8)
(h, P q g ) m =
(Pph' Pqg)m
(Pph'Pqg)m
+ ((h-Pph)' Pq g ) m : ( P p h '
Pqg)m
'
(ana-
96 which proves
(3). Properties
(2) and
(5) are immediately seen from the
definition. q. e. d. Remark
4.8
At first sight it seems astonishing that for the proof of the representation theorem dary (SG E Cm)
(Theorem 4.6) regularity assumptions
are necessary.
In the case p = 2 Theorem 4.6 is a trivial
consequence of the F. Riesz representation theorem, boundary is needed.
on the boun-
no quality of the
But the assertion of Theorem 4.6 implies as well
existence as regularity up to the boundary of weak solutions of Dirichlet's p r o b l e m f o r certain elliptic differential operators.
But not eve-
ry domain G admits a classical
The following
solution of such problems.
example shows that the assertion of Theorem 4.6 is in general not true if we don't demand some quality of the boundary: Let
G = Ix ¢~R n : O < IxJ < ~]
Find a
v c C°(D) fl C~(G)
with
and
Av = 0
v(0) = 0 . If a solution v would exist, hood of
n~3. in G
Consider the problem: and v(x) = I
if
v w o u l d be bounded in a neighbor-
x = O. Therefore, at x = O~ v would have a removable
as we know from well-known theorems monic in the whole unit ball be equal to one in [JxJ ~ I]
JxJ = I ,
singularity
[53]. But this means that v
[JxJ < I] . Since v(x) = I
for
by the m a x i m u m principle.
is har-
Jxj =I , v must
This contradicts
v(0) = o. On the other hand, assume that the assertion of Theorem 4.6 holds for this G in the case m = 1
for
g c LP(G).
:= (g'@)o
Let
F$
p>n
. Let for
g(x)
@ c w~'q(G)
IJgIIo,p II@jlo,p ~ IlgIlo,p IJ@JJl,q , where By our assumption,
there is a
=
u c w~'P(G)
for
every
:= 2n
for
. Then
x cG~ then JFSJ
I I I < q < ~, ~ + ~ = I . such that
~-1,q(G )
"
J
From the Sobolev-Kondrashev-theorem
follows that
u c C°(G)
and
97
(4.32)
Since
I~('31
~
u
cIl~ll~,p
m a y be approximated in
it follows from Since
~
(4.32)
UnlSG = 0
that
for every
we conclude from
(4.31)
u ~ C~(G),
g ~ C~(~)
since
the function
v(x)
W 'P(G)
u
by a sequence
is approximated uniformly by the (Un)o
n , we have
u 1 3 G = 0 . On the other hand
by Weyl's Lemma
(see for instance
. But then we have
:= u(x) + Ixl 2° Then
= - 2n + 2n = 0 , v(x) = I
(~n) c Co(G),
for
Au =
v ~C~(G)
-2n.
[17]) that Consider
N C°(~), Av(x) =
Ixl = I, v(O) = 0 . But this is a contra-
diction to the fact noticed above,
that this Dirichlet problem has no
solution.
95.
Bilinear forms and a generalization of the Lax-Milgram-theorem Definition Let
m> I
I
5.1 ~, I < q <
be an integer.
Let
~
be real numbers with G ~ IR n
B[.,..] :'~o'P(G) x ~ o ' q ( G ) - ~ C
~I + I = I and let
be a bounded open set. A map
is called
a continuous bilinear form,
if (i)
for all
~,~ ~ C , u i ¢ W~o'P(G )
and
vie
~n'q(G)o ( i = 1 , 2 )
and
(ii) c
there is a constant
wm'P(G)
x ~o'q(G)
C > 0
such that for all pairs
(u,v)
98
(5.2)
I~[~,
Lemma Let m > I
~- Cll~l~.,p~ll~.,~
5.2 I
~ , I
be an integer.
boundary
8G e Cm
form on
be real numbers with
Assume that
G c ~n
and assume that
B[., .. ]
is a continuous
bilinear
W~o'P(G ) X ~F~O'q(G) .
Then there are two continuous -~o'r(G)
(r=p
for all > 0
1_ + I_ = I and let P q is a bounded open set with
linear operators
Tr :~oo'r(G)
or q )
such that
(u,v) e W m'P(G) O
x ~o'q(G)
With the constants
and
by
defined by
(5.4)
(4.19)
IIT~II~.,~-
K
Let u e ~o~P(G)
be
C > 0
(5.2)
-*
K = K(n,m,p,G)>
is
,
Proof: (i)
tinuous linear functional uniquely determined Define
TpU
on
fixed. Then, ~¢'q(G).
z c %~o'P(G)
:= z . Then
Tp
which proves (ii)
By Theorem
such that
is linear.
(5.5) KnT~II~.,p ~-
G#v
v ¢ P~o'q(G)
there is a
by
(4.19),
~l~E~,~]l
(5.4) .
Take a fixed
4.6
is a con-
B[u,v] = GuWV = (z,v) m.
Further,
=
:= B[u,v]
and define
~- c,~II~,F
99
Fv*U := ~
for
As in Fv@ e W m'P(G)*. o
Then we see
FvWU = (TqV,U)m
(5.3)
u c Wom'P(G) .
, which gives
(i)
there is a
Tq
such that
B[u,v] = Fv*i = (U, TqV)m ~ what proves
.
q.e.d.
Remark
5.3
The operators ('''')m " But
Tq
well defined by
Tp
and
(Tp*F*) u = ~ ( T p U )
~q : ~ ' q ( G )
=
defined by ~q Tq ~q
we distinguish
Theorem
for all
correspondence
-~'P(G)* Tp*
Therefore,
are adjoint with respect to the form
is not the adJoint operator
is only a one - to - one map
Tq
Tp*
-I
from
5.4 (Generalization
¢5.6)
C i > 0 (i= 1,2)
C. I1 -II ,p
Tp , which is
F* e ~'P(G)*.
between
%*
and
There
Tq
by the
(4.30) , page 94 , namely
. Tq
by the notation chosen above.
of the Lax-Milgram-Theorem)
Let the assumptions of Lemma 5.2 are constants
Tp* of
be satisfied. Assume that there
such that
for every
~
u
e
Wm'P(G) o
and
for every
v e ~o' q(G)
h"~ S~, Then the operators mappings of e
~o'r(G)
on
Tr
~o'r(G)
~oo'q(G)* (G* c ~ ' P ( G ) * )
(v ¢ w~,q(G)
)
such that
defined by Lemma 5.2 (r = p or q ).
are topological Further,
there is one and only one
for every u ¢ W~o'P(G )
1 O0
]~[~,~]
(5.7)
= ~*¢
for n l
® ~
~o'q(~)
for all • ~ ~mo'P{a)) and (5.8)
II~-II~,~ ~-
~~q
llm"II<~
( (5.8) '
1 ~ II~,~
~ ~m
1I~11~,~ )
where
K > 0
~
is defined by
(4.19).
Proof (i)
B[u,¢] = (TpU, ¢)m = (u, Tq ~)m
Because
(u,~) c w~'P(G) x ~ O' q ( G )
~
we conclude from
(5.6)
for all pairs and
(5.6)'
re-
spectively
c~ll~ll~,~
(5.9)
~-
~-~I(T~,~)~I
~
IIT~-II=,~
and
Fromthese inequalities we conclude immediately that
Tp Wom'P(G)
(h w~,q(~) )are closed su~spaces of ~,~o'P<~) (~o'q(~)) ~ s s ~ e that with
% Wom'P(~) ~ ~ o ' P I o ) .
llF*l[m*p > 0 , but
F*u=
0
~ e ~ there is a ~
for every
u c Tp }~o'P(G). By the re-
presentation theorem (Theorem 4.6) there is a
w ~ T4'q(G )
llWllm,q > 0
such
u = Tp@.
0 = ~(Tp~)
= (w, Tp@)m = (Tp@,W)m = B[@,w]
But from
(5.6)'
that
~u
we conclude
= (w,U)m. Let
~ ~¢P(o)~
for every
with
Then • c Wom'PCG).
llWl[m,q = 0 , what is a contradiction.
101
In quite the analogous way we see: Tq ~o'q(G)
(ii)
To prove the representation
observe Theorem 4.6: Let is a
f ~ ~o'P(G)
¢ W om'q(G)
(G*Y
= (g'W)m
we have only to
(G* ~ ~o'P(G) *) . Then there
such that
F*@
= (f'@)m
for all
for all ~ ¢ Wm'P(G))o . But by part
the proof there is an u c Wom'P(G) (Tq v =
properties,
F* c ~o'q(G)*
(g ¢ W~o'q(G))
= W~o'q(G ) .
( v ~ W~o'q(G))
such that
TpU=
f
g) . From this we conclude
B[U,$] = (TpU, $)m = (f'¢)m = F * $
for every
• c wm'q(G)
B[~,v] = ( ~ , T q V ) m
for every
~ ~ Wom'P(G)).
(5.8)
(ii) of
and
= (~'g)m = G * ~
(5.8)'
(and
are trivial. q. e. d.
Remark
5.5
It is possible tions of Lemma 5.2 rily
~
u c w~'P(G)
uniqueness.
see
§6.
are satisfied and if (5.6)' holds,
(5.6), we can prove in the same way,
there is
no
to weaken Theorem 5.4 in some sense:
Further,
v ¢ W~o'q(G)
such that
there may be
satisfying
(5.7)
If the assump-
but not necessa-
that for every holds.
F~- c ~o'q(G)*
But we cannot prove
G* c W m'P(G)* o
such that there is
(5.7)'. For abstract
theorems
of this type
T.L. Hayden [23] .
Some coerciviness O
inequalities
generalizing
.
Gardlng' s inequality As we will see in the forthcoming is basic for two important properties bilinear forms.
sections,
the following
of uniformly
The first is that Dirichlet's
theorem
strongly elliptic
problem is always locally
solvable in the weak sense. The second is that there is a non-negative
102
number
C
such
that the halfplane
tain generalized eigenvalues u ¢
'P(G), u ~ 0
when
Re z > C .
[z ~ @ : Re z > C ]
does not con-
(see Theorem 7.3), that is, there is no
such that
B [u,~] = z(u,~) o
for every
• ~ Co(G )
Theorem 6.1 Let
I < p < %
m ~ I be an integer. dary
I < q < ~ Let
(s,~)
with
~ + ~ = I and let P q be open and bounded with boun-
G c ~q n ( n ~ 2 )
~G ~ Cm. Assume that
ry pair
be real numbers with
D c IR r ( r ~ 1
J~l = 16J = m
nuous in D. For every pair
) is compact and that for eve-
there are functions a~6(.)
(@,W) c C:(G) x C:(G)
and every
contik ~ D
let
(6.1)
Further we assume that
BX[$,Y]
is for every
k e D
a
uniformly el-
liptic Dirichlet bilinear form such that the elliptic constant be chosen independent of
(-4"J '~ 1~ ~"
k c D, that means there is
;&,.~,(.,'X)~~ 1~r~
~- ~..Igl 2"'''
E > 0
E
may
with
for every ~ ¢ IRn
and
I~l= II~l='m
every
k ~ D. Then for every
k c D, BX[@,Y ]
is a continuous
W~o'P(G ) × W~o'q(G ). There are constants independent of
for every
(6.2), for every
C i = Ci(n,m,p,E,a
~,G ) ( i = 1 , 2 )
k c D with C~ > 0 and C2 > 0, such that
u c ~'P(G)
and every
k cD
C llvll. , v e
bilinear form on
w~'q(G)
and respectively
e,.llv 11o, and every
k c D.
If furthermore
Bk
is uni-
~o3 formly strongly elliptic, Ca may be chosen equal to zero.
Proof: (i)
There is a constant
max I~l=l~l--m
max la~8(k)l , n,m and p such that by HGlder's inequality x ~D
t~ [{,v]l
(6.3)
~ C, ll~tl~,,,, 1I~'11~,~
for all
({,~) ~ ~ ' P ( G )
theorem.
Let
(6.4)
(6.5)
from
× ~'q(~),
u ~ Co(G ) and let
]~'X [ u~, ~ But
C 3 ~ 0 depending only on
]
==
(T~~ u~,
(5.5) and (6.3),
~hich proves the f i r s t
part of the
k cD be fixed. Then by Lemma 5.2
{).,,,~ (6.4)
for every
• e Co(G ) .
we conclude
KII ,~
Since
D ~ (T;Du) e ~oo'P(G) c Lm, P(G) for
I~1
=m,
all assumptions
of Theorem 1.6 are satisfied. By 0.15) there are constants Cz' = C # ( n , m , p,G,E) > 0 and
(6.6)
C2 ~ 0 independent of
c" I/"11~,~
Combining
(6.71
~
[I ~ 2 ~ ~-II~,~
k
such that
~
C~ I1~11o,~
(6.5) and (6.6) we have
c~u~ll~,~ ~- ~-~ ~ l ~ ~
[~, ~]
which is (6.2) with Cl := K Cl' , C2 := K C2' . TO prove
(6.2)',consider
~[~,~I
: ----
+
104 and apply the same conclusions. (ii)
From now we assume that
Bk
is strongly elliptic. Without
loss of generality we may assume 2 ~ p < ~ Otherwise,
consider
Bk
and therefore
I < q ~ 2 .
which has the same properties as
the same considerations as in
part
B k . We make
(i), but now we apply (1.16) in-
stead of (1.15) and get (6.7) with C2' = 0. This proves that one-to-one.
(iii) In the following, we take a fixed k e D and write only Tq, suppressing the argument ~o'P(G)
on
Wm'P(G). In the special case
Tp ,
Tp maps (the proof
p.18 ).
C' IIVllmY2
-< Re Bk[v,v ]
C' = C'(n,m,G,E).
~ w m ' 2 ( G ~ satisfies is one and only one
Bx[U,@]
Let
for every
u e ~o'2(G)
=
(w,@)m
But from Theorem 3.6 we get
v e C~(G)
w e Co(G), then
F e ~o'2(G) ~.
0
(6.8)
p = 2 we conclude
0
was given on
where
k . Now we want to prove that
is
T~ X)
for
F(@)
and every k eD, :=
(w,¢) m ,
By the Lax-Milgram-Theorem there
such that
{ e W~o'2(G)
u e w~'P(G)
and fixed
k eD.
and (6.8) holds for every
e Wom'q(G ) . (iv)
Let now
{Wk] c C~(G) k e ~
Since
such that
there is an
(6.9)
Bk[Uk,@ ]
sup
~c Sq
c llu k-ulllm,
w ° e w~'P(G) be given. Then there is a sequence llwo - WkIlm, p -* 0
u k e Wm'P(G)o
=
(Wk,@)m
IBx[u k - u 1 , ~]1 p
<_liw k-wlltm,
(k -~ =). By
(iii)~ for every
such that
for every
~ e wm'q(G)'o
<_ IIwk - w!llm, p , we get f r o m p~o
(k,Z~-).
(6.2)
Io5
Therefor%there
is a
u o e W~o'P(G ) such that
continuity we conclude from
(Tpu.o, ~ ) ~
=
for every
@ ¢ ~o'q(G)
and therefore
Let
(k -~ ~). By
(6.9)
(6.10)
(v)
Uk-*U o
]5 [ U . o , ~ ]
@ e wm'q(G). o
=
(~o
il~).m
Tp u o = w o.
Then by Theorem 4.6
(6.11) e, Since
Tp
such that
~
w~P(G.) II u.II.,,,l ~
is on-to, there is for every u = %Y.
wlth C2' = 0 follows ~-
From
a
(TpY, ~ ) ~ = B[Y,@] = (u,@)m
Cz'IlWllm,p i IlUllm,p"
Su.~o
Y c %~o'P(G) and from (6.6)
By (6.11) is
=
~,~,,~ which proves
u e Wm'P(G)o
<11~11~m
--
c2
~;,~
(6.2)' with Cz := K C~' q. e. d.
For the sake of simplicity we consider in the following sections only bilinear forms having the following properties:
Definition 6.2 n Let G c IR be a bounded open set and let
be an uniformly elliptic Dirichlet bilinear form of order m. We call B[@,~]
regular, if for
sume that the
a~
with
tc~i ~ m, If31 ~ m aaj3c I~I
= l~I = m
L~(G). Further~we as-
are continuous in
~, or what
~06
is equivalent,
aG6
Theorem 6.3 Let m > 1
uniformly continuous
(Generalization
I< p < ~,
G
for
of G~rding's
lsI = 181 = m .
Inequality) ~I + I = I
I < q < ~ be real numbers with
be an integer. Assume that
boundary
in
G c ~n
~G e Cm and assume that
and let
is a bounded open set with
B[$,~]
is a
uniformly elliptic
re-
gular Dirichlet bilinear form of order m. Then
B[$,~]
is continuous
there are constants E,a~6) ~ 0
for
(¢,W) ~ ~ ' P ( G )
x wm'q(G)o
and
C i = Ci(n,m,p,G,E ) >0 (i= 1,2), Ci' = C i' (n,m,p,G,
(i = 1,2) such that for every
s .p
u c ~'P(G)
,ll
-
q' II -Ilo,,
or equivalently
Proof: (i)
The continuity of
B[., .. ] is trivial by Definition
6.2.
(ii) For technical reasons we will need differentiability ties of the coefficients.
Therefore we apply Friedrich's
proper-
mollifier
(see e.g. [3] , [17]) to the coefficients.
Let
j c Co(IR n) such that
j(x) ) 0
Ixl >I_
and
for a!l x ¢IR n, j(x) = 0
for
(Take with a suitable constant C > 0 Ixl < I Let
and Je(x)
j~(x) = 0 then for
J(x)
:= 0
f(e)(x)
]xl > ~
for E
and
> 0. Then
Js e Co(]Rn),
~ J~(x) dx = I. If
:= ~ J e ( x - y )
IIf - f(e)IILP(iRn ) -~ 0
:= C exp
= I . for
otherwise).
:= a -n j(x)
for
j(x)
~j(x)dx I - I - Ixl m
(e -~ 0).
f(y) dy
we have
If furthermore
Je h 0 ,
f c L P ( m n) ( l i p < ~ ) , f(e) f
~ C~(]Rn) C°(~Rn),
and then
~o7
max x~K
If(x)
- f(~)(x)l-~o
(6.14)
f(°)(x)
We have a ~
e Lm(G)
for
(~--0)
::
f(x)
for every
for
Let
~ : o.
I=I i m, I~I ! m
for
K C.C IR n.
and therefore
a~8 e LP'(G)
1 < p' < m. Then the mollified coefficients satisfy
q](P'i~) :=
(6.15)
for every
p'
(ill)
with
~×,~,:.II~
- ~~,<~
Let
]~ [~
(~
~o)
S < p' < m. u c w~'P(G), • e ~o'q(G)
an &rbitrary real valued function. Then, by
(6.~6)
~ o
~]
:
~[~,
{]
and let
w ¢ C~(IR n)
be
Leibniz's rule
+ ~o[~ ~,~]
where
(6.17)
ct~
, :
-
II%1~"m
for
B >__ 0
with
our
~greement
( 6 , Iz~).
Let
(6.~8)
Then, with a constant
C = C(n,m)
we get
from (6.17)
(6.19) q-
I~l'tr=-rn-± li~t~_,~
108
c('ncm) k(w) >--
+
~
By t h e S o b o l e v - K o n d r a s h o v - t h e o r e m 1~1 < m - 1 , w h e r e q, = n n~ > q -q
× 11( ~ -
~& ~ ~ ' ~ 11o,~
(see Appendix if
q
I),
and
DY •
tl{ 11~,,
e L q' (G) for
q < q' < ~
if q>_n.
-
By H~ider' s inequality d
q'
2~z3
@
-~ tl ~,~-
with
K~ by
@
~ll~.,co ~ II ~11o,~,
Theorem A.2, where "n
(6.20)
9'
&
i÷
9 '~'~
9 '9 t , ~
I
In quite the analogous way we see
~,
for I ' l l
< m- I
p u
cl>"n
II(a~l 3 - a(~))DT
u
lira,p
<__ K~ 11%6- a(~)IIL~(G) Ilullm,p , where (6.21) with
/~, P' = nnp-p
=
e~ ~,_~
if p < n
we get from (6.19) with
(6.22)
and (6.15),
p' > p
arbitrary,
(6.20) and (6.21)
if p_>n.
Therefore~
i 09 Consider
(6.17)
in the case
for
e > O. In the
lel = m one d e r i v a t i v e
apply H 8 l d e r ' s
inequality
first
t e r m at the
right we t r a n s f e r
from u to the o t h e r
to the result,
functions
and
such that we o b t a i n
where
It was
Since
the aim of our
for fixed
smoothing
u ~ Wm'P(G) o
on
,
procedure
~[u,. ]
to get
(6.23).
is a b o u n d e d
linear
--wm'q(G), we have - -
o
(6.25)
I~[ ~.,w~] I
If we observe
(6.26)
Ii"¢II-,.,~
and combine
(iv)
(6.16),
Tor
~- K¢w) II~II~,~ (6.22),
(6.23)
and (6.25), we get
v ~ Wom'P(G), • e wm'q(G)
and e > 0 let
functional
[,k~A]g-
[A'^]'%~
+
[&'4]<~
({{'9) - (6~3"9) =
[,ix ' ^ ] °x~
£q pu'e ({{'9) uet~
°(&~jql'/'~([( ( ' ' m ~
-(°x)~]~))
----<
----: ]Zax'A']'(°x~
(~{'9)
pu'e
°(a ~q: ' ^ , <
(0)
e ~ TM)
o B,~M ~ ~
ua-=l~l=l~l <
, (O)d,~M o
%e~ o~ (6~'9) mo=j ( ~ ' 9 )
b'~g,~lla"~~ll,,ll','~
(~{'9)
~ a
aog set ptre O ~ °x
%a~
%o9 a~ s~ aaturem a~res eq% e%TUb uI
(~'~)3
(6B'9)
~
seAy~ qoyq~- ",£q.]:I'euf3euT s~,zep'[o.I-I .£Idd'e pu'e
asqq.o
suo$%out'~
eq% 0% n u~o~; aAT%~ATaep auo =a~su'e=% e.t4. "m = I~oI 5I
" 0 < a
%~I
OLI,
111 (v)
Since
B
is a regular uniformly elliptic Dirichlet bilinear
form (see Definition 6.2), the family
[Bxo} x ~
satisfies all assump-
tions of Theorem 6.1. Therefore, there are constants independent of
(6.35)
xo ~,
C~llvll.,,,,~, v := u w .
Let
such that for all
v ¢ ~o'P(G)
"--
+
Then by (6.26),
C~' > 0
and C~' ~ 0
and every
xo ~
c .ll,'llo, ,
(6.27) and (6.34) we get from (6.35)
(6.36)
Since the functions there is a
p
>0
a~8
are uniformly continuous on
~
if
l~l=l#l---m,
such that I
CI
c(n,m)
max
max
I~I=I~I~ for every
if
Ix-xil x c G
lac~B(x) - a ~ ( x O) I
IX-Xol
x o c~. Therefore, by (6.32) and H5lder's inequality
supp v c Ix ~
Since
[x~:
: IX-Xol
< p] •
G is compact, there are < ~. and
such that
Further there are
xl,...,x N ~ ~ such that w i c C o ( ~ n) with
wi(x ) > 0
fo~
Ix-~il
wi(x) = o
for
Ix - xil h
N ~__ wi(x)
=
I
for
<
x c G.
p,
N ~ c [J [x : i=I
0 < w i ( x ) for
112
In (6.37)
let
v := u w i
and put the result in (6.36), which leads
to
{
(6.38)
Let
K "= ~_Imax
there is an
Then
+ c(~,~)~)ll~ll~.~,e~
K(wi). We have the free parameter
E
By (6.15)
eo> o, such that
M(~o) is determined.
By Lemma 3.5 there is a
y > 0 such that
(6.4o) (6.38)- (6.4o)
But by
N
NeS~ that is C~!
~_
which proves
~"
(6.13). From this follows
(6.13)' 7 if we observe Lemma 3.5. q. e. d.
Remark 6.4 (i) assumption
As section a~
(iii) of the proof of Theorem 6.3 indicates,
e L~(G)
may be w e a k e n e d in the case
if we observe the S o b o l e v - K o n d r a s h o v = Bo[u,~]
+ Bz[u,~]
, where
theorem
.
Let
the
l~l+I~I<2m- I , B[u,~]
=
113
Bo[U,~]:=
7---
(a~sD~ u, D ~ ~ )o
is
uniformly elliptic regular
a
I~I=I~I=m Dirichlet bilinear form and
B~[u,~]
:=
(%~ D~ u, D ~ ) o
7--
"
l~l<_~,l~l<_m I~I+I~I±2~-I Then, by Theorem 6.3, inequality (6.13) is satisfied for stants C° > O, C~ > 0. To ensure that
B
B°
with con-
satisfies an inequality of
type (6.13), we have to take such coefficients
as~
for
I~I+ISI
2 m - I, that we can prove an estimate of the type
(6.&~)
I~ ~ ~, ~]I
~- ~ I ~ II~,~II~ II~,~ o
with a constant B~
if
a~8
y < C~ . Let B~
is replaced by
(~)
a- II~IIo,~II~ II~,~
be the bilinear forms resulting from
a ~ ). Then analogous (6.23)~ after a proper
partial integration we get with
(6.42)
~
K = K(n,m) > 0
I:B~"[,.,.,~] l --. K"'-'->ll'-'-II.,,,-~.,~ll~ll,,v~
where
(6.43)
N(~.)
:= I~-',~
For the difference
(6.44) I ~ , ~ l -
Bl - Bl (e)
~
('~') " ~L.
i- ¢,..,N
G
analogous
(6.19)
the estimate
-,~1
~ L ,~]I
holds. Consider any term on the right side of (6.44). Suppose e.g., I~i < m,
I~I ~ m . Then, repeated application of the Sobolev- Kondra-
shov- %heorem
gives
D ~ u ¢ LP~(G),
where np
(6.45)
p'
p~
n-(m- I~I)P > 0
if
with
arbitrary
I < p' < ~ if n - ( m - I ~ I ) P
if
Further, placed and
D~
cLq1(G),
by q
and
n-(m-l~l)q
Kondrashov
where
ql is defined
a is replaced > O, then
n-(m-l~l)p
by
< 0
analogous
(6.45) w i t h p re-
IB. If for instance
by H ~ l d e r ' s
inequality
= 0
n-(m-l~l)p
< 0
and by the S o b o l e v
-
- % h e o r e m we get
3"la~-a(;)I Io~IIo~Id~
< IID~UIIL~ IIa~-a(~)llo q~ IID~®IIo,q~ < --
w
' q 1 ' l
--
_< K(~, ~,p)lla~6- ~(S)llo, ql~-~ llullm, p ll$11m,q • Therefore,
in
k =
=
ql ql - I
F r o m this we
this
case we have t o demand
see that
it is s u f f i c i e n t
is d e t e r m i n e d
(6.44)
an estimate
of the type
(6.46)
I~[u-~g2]
- B~)[~.~]I
By the p r o p e r t i e s
the constant
of F r i e d r i c h ' s
on t h e r i g h t
C o m b i n i n g (6./1-2) and
-~
to suppose
in the m a n n e r
_
(6.47)
, where
nq (n+m- I~ III)q - n
k = k(~,~,m,n,p)
that
a~6 e Lk(G)
a~
above.
~ Lk(G),
where
Then we derive
from
(.~-)
mollifier~ we can choose
side of
(6.46)
is
smaller
an e > 0
than
such
-c? T
(6./t-6) we h a v e
~ II~II~,~II~II~,~ +
k N~o)II~II~-~,~II~II~,~
115
By Lemma 9.5, there is a + 6 llUllo,p with
such that
KN(So)IlUlIm_1, p <_ ~-~ IlUIlm,p +
Combining the last inequality with
(6.47), we get
inequalities ~'P(G)
on
< 2m-I
this estimate
is, that we may derive
for uniformly elliptic bilinear
a
of such a type,
belong to certain
n and p. Such estimates
that the coefficients
L k ( G ) - spaces, where
may have
a~G
k
for
lal+l~l
depends on G,B,m,
are useful if we consider nonlinear forms.
the other hand, we shall understand Theorem 6.3 in the sense, is a regular Dirichlet bilinear form,
estimate
priori
forms defined for a certain
x W~o'q(G), where the lower order coefficients
"singularities"
B
(6.41)
y = C-A-~° Z "
The advantage of
P
8 >0
for any
p with
On
that if
I < p <
(6.13) holds.
(ii)
Let
B[U,~]
G c IR n be a bounded open set and let
:=
------ (a~B O a u ' O ~ # ) o
for
u , ~ E Co(G )
J~j<_m I~l<_m be a b i l i n e a r
form w i t h c o e f f i c i e n t s
aa~ e L~(G) o t h e r w i s e . x wm'q(G)o
aa~ e C°(g)
Then, B i s a c o n t i n u o u s b i l i n e a r
for every pair p,q with
furthermore
as S. Agmon
I < p,q < ~
Re B[u,u] _> CiIlUIlm,2
holds with C~ > 0 [I]
-
and C2 > 0, then proved
(compare
ming paper we will prove that (1.6) holds for
the "root condition"
c211UIIo,2 B
and
is uniformly
p
with
t~l=l~t=m
~I + I =~I
is satisfied~if
3G ~ Cm.
. If u ~ wm'2(G)o
strongly elliptic, assume that
I < p < ~.
(6.13)
In a forthco-
B is then uniformly elliptic,
L B defined by (1.10),
and
form on W~o'P(G)x
for every
[3],pp.86-90).Now
holds for this bilinear form for some
estimate
for
that is,
and in the case
n=2
1t6
Theorem 6.5 Let the assumptions that
B[~,@] Then,
is uniformly
for every
in the topology of is a constant
of Theorem 6.3 be satisfied and assume further strongly elliptic.
xo ¢~
there is a neighborhood
G~ and having a boundary
~Xo
Gxo
of Xo~ open
¢ C m, such that there
K = K(p,n,m,G, Gxo, a ~ , E ) > 0 with
for every
u E ~,~o'P(Gxo )
for every
,7m, q v c ~o (Gxo)"
and
K llvll ,,
(6.48),
=-
s .p
Proof: (i) for every (ii)
If p If
x ° cG, with
then there is a
0< P
x ° ¢%G,
is
Po = Po(Xo"
Kp(Xoi
then there is a
is mapped by an one-to-one
U c [z ~]Rn:
zn > 0], such that
IX-Xol < Rj
is part of
for
H~ := [z c]R n", Izl < p, zn > O]
we may assume that for Hp, c H c
Hpo ,
and with boundary
we calculate
B[@,Y]
proof of Theorem Form
B[@,Y]
I .6 a
z -I
G .
G n [x :
mapping of class Cm on a set
Further, there is with
0 < p < Po
V be
=
=
p ', zn H
uniformly
~
0}~
the image of x wm'q(v) o
getting like as in the
strongly elliptic Dirichlet bilinear
with the same regularity properties
as
U.
with
Then 8V c Cm. For (@,Y) ¢ Wm'P(v)
in the new coordinates,
Ix :
we have ~ p
there is a convex set bH ~ C m" Let
~G fl
Po' such that
'
of the"edge ~r E := {z : Iz[
0 < p' < Po'
under the inverse mapping
such that
Z(Xo) = 0 and the image of
{z • zn = 0}.
After a suitable deformation
:= Ix : IX.Xo I < p ] c Ro > 0
IX-Xol < Rj
~G) > 0 such that
B[@,~]
.
~
de-
117
notes the function serve that the map
~(z):= Sp(H)
$(x(z))
: #-~
Further, there are constants transformation
z(x)
c ~4'P(H )
if
maps W~o'P(v ) Kl > O, K2 > 0
~ e W~o'P(v). We ob-
one-to-one on W~o'P(H ).
only depending on the
and H , such that
(iii) In the following let "smoothed" half-ball
H
G' be either the ball
and let
B' be either
B
Kpo(Xo) or the
or
B
defined on
the respective spaces. Then, by Theorem 6.1 there is a C > 0 such that
(6.5o) cllll..,
for every
u e ~'P(G')
for every
v e ~o'q(G'),
and
where
B O'[u,¢]
&~
=
' --
We consider the map for
w i t h
ac~(Xo)
if
G' = Kp(x o)
K~B(o)
if
G' : ~
Yr : ]Rzn
-~ IRyn
.
defined by
yr(Z)
:= ( z - zo) r +z o
0 < r < I, where I xo Z0
Then
,0 Dc~ , D~ (ac~~ u ~)o
7-
:=
0
if
G' = Kp(Xo)
if
G' =
Yr is one-to-one and of class C .
G' was konvex, (s = p orq)
G r, c G'
for
be defined by
0 < r _< I •
Let Let
G'r := Yr(G'). Since p~r) : W m , s(G ,) . w ~ , s ( G ~ ) o
118
(Ps (r) u)
(y) :=
U(Zr(y))
, where
As immediately
seen by application
maps
one-to-one
W~o's (@)
transformation
z r :=
yr -I .
of the transformation
rule,
Ps (r)
on ~o' s (G~). From this fact and again by the
rule we derive from
(6.50) and
(6.50)'
(6.51)
for every
h
~ ~'•P(GAJ m
.
and
.
o
~u_~o
Observe that the constant where
l~]~J[ ~,~[[I
C in (6.51) and
(6.51)'
for every g EN~o'q(Gr, ).
is independent
of
0 < r < I. At first sight we could not derive this fact from
Theorem 6.1, but this is the essential point in the following part of the proof. (iv)
First we need a sharp form of an inequality
(compare(3.77}):
Let
@ c C ~ ( $G ) .
Then for
already used
x = (x',x n) c G r'
we
have @(x',x n )
Xn = ~
Since
~$ ~n
~ (x' • ]R I
(x',t) dt
&~ n
diam G r' , where
inequality
diam G~ :=
sup
Ix - YJ
and
x,y ~ G rT
Gr
diam G' < r diam G' r
and by H61der's
we get after integration
--
c-; Iterated application
T
of this inequality gives with a constant
y = y(G',m)
(?%)a , ~ o
(,'f~) ~S
(~%'9) mo,~J % ~
o~ snogol~tr~f
•6
IIHII 5
(~'9)
a~ojo,~oq~ p~
(0%) a , ~ t o
~ q
z~o~
zo; (~'9)
pu. (~'9)
"(~)o o ~ g%
'(~'9) oout~
mo~; %o~ a~ uo~S
'~TqIssod
sI q o t q ~
ox
,o~
~1~1=t~1 l ~ O~
~ 0800~0 @~ ~0~
[& '~] °~[ I (ff~'9)
OAWq O~t!t~qg~n~ 'd
pu-w
~
$o Suopuodoput
( u ' w ' ~ ) , £ = ,X
sI
T-~-~~l~Ji+l~ ,~.~ I~|
o~oq~
~.-~'~~I'~I +I~I
19)**~~
,~
~
I~,~
({c]'9)
( ~o)b o
~
o
~,IoAO ZO~ go~ o~ oS "Sta~OU - ~q[ aq~ .los sploq .<%I/~nbeu$ sno~ol~u'e oqo]
6~
120
~C ll~ll~,ac~, 0
(6.55),
If
G' denotes
ved with
I
_
Kpo(Xo),
then
Gxo = KroPo(Xo),
every g~?~o'q(Grs).
G'ro = KroPo(Xo)
and the theorem is pro-
since B' = B. If G' denotes
c H . Now we transform back and denote the image of
G-~o~,
a%o~C m. ~f (h,~) ~'P(%o
transformed functions (i= I•2)
(6.56)
(compare
h,~
o ) ×~'q
with constants
H
•
then G'r o
G'ro with
(Gro) '
Gxo. Then
we get for the
K i = Ki(n,m, S G , p ) > 0
(3.73), (3.74))
II~II~,~(~.~
-~
~II ~ II~,~(%~
II ~ II~.~¢~,o~
~
~II ~ ll~,~co~.)
and
(6.57) Combining
(6.55) with
z
(6.56) we get
w2,PC~.)
-
l~'Eh, W]l
Since
B'[h,~] = B[h,~]
and the
map
~-* ~ maps wm'qo (Gro)'
on ~m'q(Gxo)o '
(6.57) and the last inequality imply
which proves
(6.48) with
K = ~C K2 K-~ ;
(6.48)'
is
proved
analogous
q. e. d.
121
§7,
Existence theorems in the case of u n i f o r m l y strongly elliptic Dirichlet bilinear forms
Now we are in the position to state necessary and sufficient conditions for the existence of solutions of uniformly strongly elliptic functional equations under homogeneous to prove that Fredholm's
boundary conditions.
Our aim is
alternative applies to the problems under con-
sideration. In the case of uniformly strongly elliptic Dirichlet bilinear forms this can easily be done: An immediate consequence of Theorem 6.5 is that Dirichlet's problem is locally u n i q u e l y solvable in the weak L p sense
(see Theorem 7.2) . From this fact and our generalisation of Gar-
dings's inequality
(Theorem 6.3) we get the existence of a real number
k o such that for every B[u,~] has a
+
k(u,$)o
unique solution
are satisfied
=
and every
F($)
F ¢ ~ o ' q ( G ) * the problem
for every
u e <'P(G),
$ e ~o'q(G)
if the assumptions
of Theorem 6.3
(see Theorem 7.3, Corollary 7.4). The same is true for
the adjoint problem. Fredholm's
k ~ k°
By Nikolski's
theorems we immediately conclude that
alternative holds for our problems
(Theorem 7.5).
In the case of uniformly elliptic Dirichlet bilinear forms which are not strongly elliptic, tors T
P
defined b y (5.3)
we will easily prove in § 10 that the operafor
I < p < ~ are Fredholm operators
(Theorem
10.1). But the proof that the index is zero, that is, Fredholm's native holds, section
is rather cumbersome
and we need the results of the next
(differentiability theorems).
Roughly spoken,
arises from the fact that in this case equation have a non - t r i v i a l
solution for
alter-
every
k c C.
this difficulty
(7.1) with
F m 0
may
122
But first we need the following compactness theorem of F. Rellich.
Lemma 7. I Let 0<
G ~ ~R n
be a bounded open set and let
j,m
be integers with
j <m. Then for every
p
with
1
and from every bounded sequence sequence which converges in
u c I*~'P(G)
(u~) c )~'P(G) O
implies u ¢ wJ'P(G)
we can select a sub-
W OJ'P(G)
Proof: (i)
A simple proof may be found in the book of Ch0 B. Morrey Jr.
[48]. Another proof is presented in A.Friedman's book [17]. (ii)
Here we sketch a proof based on integral
I -(n-2)w
K(x,y)
f2-n n Ix - y
if
n>2
if
n=2
operators. Let
:=
2 ~ log I x - Yl
be the fundamental solution for the Laplacian A. As is well known from potential theory, for • e C~(G)
° Let <
const
I
&
Ki(x'Y) :=
-ix_ytn-1
is
For
g
~Yi h(x,y) , i = I .... ,n. Then f e LP(G) let us define
As it is well known, K i maps
(Kif)(x) := ~ Ki(x,y ) f(y) dy. G LP(G) completely continuous in LP(G).From
the representation (7. I ) we conclude that the bounded sequence m&
is proved for
IKi(x,y) I <
(uv) C Co(G ) N WIo'P(G);
1Lmma
is true for every
by approximation, the
lem-
j = 0 and m = I. The general case follows by induction.
q.e.d.
123
Theorem 7.2 Let
m ~ I be an integer and let
set with boundary
3G
G c ]R n ( n ~ 2 )
~ Cm. Let B[~,Y] be a uniformly strongly elliptic
regular Dirichlet billnear form of order numbers with
I < p , q < ~ and
Then for every
determined
x° cG
(7.2)
and denote with p,q
there is a neighborhood
F c ~'q(Gxo)*
u c ~,~o'P(Gxo )
m
real
~ + ~ = I. P q
which is open in the topology of ~ that for every
be a bounded open
=
(H c W~o'P(Gxo)* )
T(#)
~[~,v]
----- H ( ~ )
of
there Is a
xo
m C~
and has a boundary 3 G x o ~
(v c ~o'q(Gxo ))
B[u.,~]
Gxo~G
such
uniquely
wlth
for every
~ ~ ~o' q (Gxo)
for every
W c ~'~o'P(Gxo ) ,
and
(7.2)'
Further, there is a constant
(7.3)
C = C (Xo) > 0 such that
CIl ll ,
nt: IIw , CG,
C I1,,
I1HII w CG o7
and
(7.3)' If
B has constant coefficients
I~I + 1 6 1 ~ 2 m -
I, then we may choose
aa~ E C satisfying Gxo=
a~6 = 0
for
G.
Proof: From Theorem 6.5 follows for every
x o e ~ the existence of a Gxo
such that the assumptions of Theorem 5.4, applied with respect to Gxo , are satisfied, which proves the first part of the theorem. The second
124 part is an immediate
consequence
of Theorem
6.1, second case,
and
Theo-
rem 5.4. q. e. d.
7.3
Theorem
Let the assumptions
Then there C(k,p)
is a
k
of Theorem
7.2 be satisfied.
> 0 and for every o--
X> k ~
For
there
is
every
u
k ¢]R
a
let
constant
o
> 0 such that
(7.5)
C(x,p){I~U.,.,,,~ ~-
su.p {l~,~'l],.,..,,i,]{for
(7.5)'
C(),,p)~,,II.,,,,~ -~
s~e I~°'E,,,,]{
for every
c k~'P(G)
v c ~,~'q(G).
Proof: Without
loss of generality
are constants
we assume
2~p
< ~. By Theorem
6.3 there
C~ > 0, C2 >_ 0 such that
~- c, II~II~
c~n~Uo,I, for every u cWom'P(G) O
Since
B
is u n i f o r m l y
with constants
(7.7)
elliptic,
Garding' s inequality
holds
Cl' > 0, C2' > 0 :
~e~ _~:)[w, w]
Let
strongly
~-
c~'llwll~,, -
c~'IIW L ~ for every w ~ o ' 2 ( a ) .
k o := Ca'. We w i l l show that k o has the desired properties.
Let
k ~]R
be given with k ~ k o and assume,
that
(7.5) holds
true.
Then there
that there
is a sequence
is no C(k)
[uv} c w~,P(G)
such with
125 llu.,ilm, p = 1
and.
(7.8)
for every
-
v e ~.
Since
(7.9)
and. with
(7. lo)
...~['~,{]
= ]~'["~,{]
I},(~,~)ot
"-
--
X("~,gp')o
lXlC(e)ll"~ll~ll{ll~,,~
C(G) > 0 we get from (7.6),
(7.8) and (7.9)
c,,. II '-'.,, - ~,,.11.,..,,,~,
±,, , ~ By Rellich's Theorem
(Lemma 7.1)/there
converging in LP(G). But by (7.10),
~- ( x coG) ÷ q)11 ~.~ is a subsequence
{uv'}
te the limit with u o. Then IIUoIlm,p= I
-
~.~11 o , p
{uv'] c [uv}
converges in
P~o'P(G). Deno-
and by continuity we get from
(7.8) (7.11)
Since
B%[uLo,
p_> 2
~]
=
O
for every
• e l~o'q(G) .
we have u o cwm'2(G)o c k~o,q(G )
and therefore
Bk[Uo,Uo ] =0.
But (7.7) implies 0
=
Re Bk[Uo, Uo ] =
h Since
k>_ X O = Ca'
I~I = m and therefore
k(Uo, Uo) + Re B[Uo,Uo]
(X-c2'ltluollL2 +c~ we have
!
>_ 2 tlU ollm, 2
llUoIlm,2 = 0, that is
IIUollm,~ = o, which contradicts
fore there is a constant
Da u° = 0
a.e. for
IIUoIlm,p = I. There-
C(k) > 0 such that (7.5) holds.
126
T k : ~o'r(G) -*W~o'r(G) (r = p or q) be defined according r to Lemma 5.2 with respect to B k such that Let
Bk[u,~] = (T~u,~)m = (u, for every pair
Tk$)mq
(u,$) ~ ~o'P(G) × w~'q(G).
C(k) llUIlm,p
!
llTk,p u Ilm,p
Then by (7.5) we have
for every
u c ~o'P(G).
In a simillar way as in the proof of Theorem 6.1 we will show: T~,p(W~(~D =Wo~"'P(G). Let Bk[*'~]
=
Tk, 2
be the operator satisfying for all
(Tk,2 $" W )m
$,~ c ~o'2(G)
.
By (7.7) we have
(7.127 Let
-
II~
f ~ W~o'P(G ) be given. Since
:= (f'$)m
for
• ~ ~o'2(G)
ph2
for every
and
@ ¢ ~n, 2(G). o
f c ~o'2(G), by
a continuous linear functional on
is defined. Since (7.12) holds, the L a x - M i l g r a m - T h e o r e m 5.4) ensures the existence of
for every
u ~ ~'2(G)
u c ~o'P(G)
x° ¢ ~
and
p, .'= ~ 2 n
Pl := min [p',p]. Let
if p > 2. For this aim we make use of
Gxo
> 2
if n > 2
R > 0 such that
(Theorem 7.2).
and p' = p
be the neighborhood of
properties described in Theorem 7.2, applied to there is a
(or Theorem
such that
the fact that Dirichlet's problem is locally solvable
Let
~,~o'2(G)
~ ~ ~,~o'2(G) is satisfied.
We want to show
Let
F($) :=
Bk
and
Ix : IX-Xol
if
n=2.
x o with the p~ . Then
Let
127
~ Co(]Rn), IX-Xol >
0
R.
with
q(x) = I
for
Ix-xol <-~
Then, u ~ CW~o'2(Gxo ) and for every
and
~(x) = 0 for
• ~ Co(Gxo )
is
where by Leibniz's rule
|l~L~'rn
By (7.13) we get
We will show that the right hand side of (7.16) defines a continuous linear functional on ~~o' ql(Gxo ), where
1 < q~ < %
Iql_ + p-7 I = I"
Let us
2n estimate F. Consider the case n > 2 and p' = ~ . With q' := 2n ~ I :- n + 2 is + ~ = I " If $ c ~ 'o q ' (Gxo), then by the Sobolev Kondrashov - theorem (see Appendix I), and
c L2(Gxo ) for
IIDY@IIL2 < const II@llm,q , . On the other hand,
IYI < m - I the
DY@
a~
by the same theorem with
IIDYu [[Lp'
are bounded, we get with a constant
DYu
IYI < m - I c LP'(Gxo ) for
< const IIUIIm,2 . Since
C = C( IIaa~IIL%n,m, ~) by
HGlder's inequality
I'~lm.va-~.
With the inequalities above, this leads to
In the case
n = 2 we get the same estimate with arbitrary
I < q' < 2,
128 since
• ~ ~o 'q' (Gxo)
the definition of
implies
DY •
e L 2(Gxo )
IYl
for
_< m - I . By
pz, F is a continuous linear functional on ~o'qZ(Gxo )
and also by H6Ider's inequality
Therefore, by Theorem (7.2) there is a
(7.19)
~'[w,~
1
If we observe
~
(,{,T~{).,..
' o'2(aXo )
'qz (C,Xo)
• ] = 0
for every
w c ~o'Pl(Gxo ) such that
-- :F:['~i cb,~,] since
for @
eW~o'ql(Gxo ).
qz ~ 2~ we get from (7.16)
and (7.19) Bk[w - u ~ ,
that isj Re B k K w - u ~ , w - u q (7.12), since a.e.
which gives
and?since D G W ¢
0 by
Ilw-u~Ilm, 2
( w - u ~) ¢ ~o'2(Gxo ) c ~,~o'2(G). Therefore,
for l~I ~ m
what implies
] = 0
• ¢ %~'2(Gxo)~ ,
Da(un)
~D~w
LPZ(Gxo),We have
(uu) c ~o'PZ(gxo ). Since
~ml
D~(u~) ~ LPl(Gxo ) •= R O~ in Gxo . {IX-Xol <~}
is
D ~ u IG , ¢ LPI(G~o ) for I~l ~ m . x O was arbitrary and ~ compact. xo This implies u ¢ ~o'Pl(G). in the case Pz = P we are ready. In the the case
Pz < P we repeat this arguments and go on by induction llke as
in the proof of Theorem 3.6 and arrive after a finite number of steps at
u ~ ~,~o'P(G). The proof of (7.5)' is derived from (7.5) and the fact that
maps
~o'P(G)
on
~o'P(G)
k Tp
with the same arguments as used in part
(v) of the proof of Theorem 6.1. q. e. d. Corollary 7.4 Let
k o be
defined as in Theorem 7.3
Then for every
F ~ wm, q(G) * o
and let
there is a
k ~ k o.
uniquely
determined
129
u ¢ ~o'P(G)
such that
BX[u,~] =
F(~)
for every • ~ ~o'~(G)
and analogously for every
H ¢ ~,~'P(G)*
there is one and only one
O
v ~
~'q(G) O
such that ~
for every Y ~ ~ ' P ( G )
.
O
Proof: The proof is trivial by Theorem 5.4. q. e. d. Now it is easy to prove
Theorem 7.5 (Fredholm's Alternative) Let the assumptions of Theorem 7.2
be satisfied. Let
Np := [w ¢ ~'P(G)o :
B[w,~] = 0
for every
• ¢ ~J~'q(G)]o
Nq := [z ¢ k~o'q(G ) :
B[~,z] = 0
for every
~ ~ ~o'P(G)}
Then
dim Np
= dim Nq
=
d < ~ .
For
and
let
P
F c l~o'q(G)* the functio-
nal equation BEu,*] = has a solution
u ~ ~'P(G)o
analogousl~ for
If and
d =0
for every
~
~ c ~'q(G)
if and only if
H c w~'P(G)*
B[~,V] = has a solution
F(¢)
F (z) = 0
for every z ~Nq;
the equation for every ~ ~ ~o'P(G)
v c ~,~o'q(G) if and only if
H(w) = 0
for every w ~Np.
both equations are uniquely solvable for arbitrary F ~ ' q ( G ) *
H ~ ~'P(G)*
respectively.
Proof: Choose
ko
as described in Theorem 7.3
and consider
by (7.4). Then estimates (7.5) and (7.5)' hold true.
B k° defined
130
Let
Tkr° : W~o'r(G) -~wm'r(G)o (r = p or q) be the operators defined by B k° [u, • ] X° qk° Lemma 5.2 such that = (Tp u,$)m = (u,T ~ )m for every (u,$) ~ W~o'P(G ) x ~o'q(G) . By Theorem
5.4
Tp °
has a continuous in-
verse. Let Then
H[u,$] : = - ko(U,$)o
(u,$) ~ Wm'P(G) × ~oo'q(G) o is a continuous bilinear form and admits by Lemma 5.2 the re-
H
presentation
H[u,@] = (A~,$)m
-*Wom'P(G). Further we have
for
with a continuous operator
I ( ~ u , ~)m I = l(koU,~)ol < ko[lUlIoll~llo <
< k o C(G) IIUllo ll@llm and therefore 7. I
this implies that
~
~:Wm'P(G) -~
II~ u llm,p < const llUIILp . By Lemma
is completely continuous. With
by Lemma
5.2
¢ ~'P(G)
x wm'q(G), what implies
Tr
defined
such that B[u,~] = (TpU,¢)m = (U, Tq@)~ we derive from ko (7.4): (TpU,@)m = (Tp u,@)m + ( ~ u , @ ) m for all pairs (u,@)
(7.2o) Since
T~ Tp
and of
T~ ° ÷ ~
is the sum of an operator
~
theorems
=
)to Tp being continuously Invertible,
being completely continuous, we derive from Nikolski's (see
[50] or Kantorovitsch - Akilov [31 ] ) that
Tp
is a
Fredholm operator (with index zero), that is, for the problems TpU
= f
T~pF = H
(u,f ~ Wm'P(G))
and
(F,G ~ W~o'P(G)* )
Fredholm's alternative holds. If we observe the dence of of
W~o'P(G)* and
Tp* and
~'q(G)
(Theorem 4.4)
one-to-one corresponand the correspondence
Tq (Remark 5.3), the Theorem is proved. q.e.d.
A further consequence of Theorem 7.3 is a kind of regularity theorem.
131
Theorem
7.6
Let the assumptions real number with
(7.21)
I < p < ~
_~[u.,{]
Then
of Theorem 7.2 be satisfied.
=
and
u c W~o'P' (G)
for every
p
be a
such that
u E W~o'P(G )
f o r every
0
Let
• ¢ C~(G)
p'
1 < p' <
with
Proof: Let (7.4)
and
be the real number defined by Theorem 7.3. Then, by
(7.21)
]5~'°[u.,g2]
(7.22)
and
ko ~ 0
=
- ~(~,~]o
@ ~ C~(G)
for
• ~ Co(G)
let
Since
u ¢ ~o'P(G)
u e LP~(G) Let
with
, by the Sobolev-Kondrashov-theorem Pl
I < q~ < ~
7 4
there is a
arbitrary~if
p ~ n, and
I + q-qI = I . Then p-~
with
v ¢ wm'~l(G)
•
such that
Pl
.
(see Appendix
n np - p ~if
_
p
<
I)
n
F ~ W~o'q~(G ). By Corollary
BX°[v,~]
= F(~)
for every
o
~ W~o'ql(G ). Since Bk°[u - v,$] : 0
Pz h P , v e ~o'P(G)
for every
llu-Vllm, p = 0 , that is In the arbitrary case is
for
case (the
p < n
arbitrary
we g e t
(h-1)p
we r e p e a t if
Theorem
(7.22) 7.3
is
implies
u = v c ~o'PI(G)
p > n
case
$ e wm'q(G), o
and by
we a r e
Pl
< P this
Pl ~ P
< p , hp ~ p
is
ready, trivial,
argument and
since since
and get
P2 = n -n~z p~
and at
p~ ~ p
the
G
is
may be chosen bounded).
u c W~o'P2(G )
= n_n-~2p ° A f t e r h-th
step
therefore
In the
where (h-l~ the
P2 steps asser-
132
tion. q.e.d.
Remark
7.7
The last theorem
is in some sense a regularity
p' ~ n. Then the weak solution
u
by the Sobolev-Kondrashov-theorem
result:
Choose
of the homogeneous
problem satisfies
u ~ cm-S+m(G)
(~)hu
and
I~G = 0 !
for fore
h = 0,1,...,
m-S
0 ( G ~ I~ since
prescribed
(see Theorem 8.6) p'~ n
with
was arbitrary.
That is,
dates at the boundary in the classical
Further we may derive from Theorem 7.6 ty results of the following type u ¢ ~o'P'(G)j Then
0 ~ ~(
B[u,~] = (f'$)o
: Let
for every
and
I-~, u
and there-
satisfies the
sense. Theorem 7.3
I ( p' ~ p ( ~ • ¢ C~(G)
with
regulari-
and let f ~ LP(G).
u ~ w~'P(G).
But we will prove in
§ 11
elliptic but not necessarily
more general theorems of this type for strongly elliptic bilinear forms.
133
Chapter I I l :
Regularity and existence theorems for uniformly elliptic functional equations
Let
B[W,@]
form of order
be a
uniformly elliptic regular Dirichlet bilinear
m . Let for
some
f c LP(G)
(I ( p ( ~ )
u c Wm'P(G) be o
a solution of for every
B[u,¢] = (f'~)o If the coefficients ty properties
of
B
and the boundary
(assume for simplicity
has higher order derivatives ly elliptic operators proved by
and
L.Nirenberg
¢ ~ C~(G)
in
have higher regulari-
: C~), the question arises, if
LP(G)
p = 2
8G
u
. In the case of uniformly strong-
such regularity properties had been
[51]o His proof is based on skilful estimates o
of difference quotients with the aid of Garding's inequality. mon gives in his book
[3]
S. Ag-
a refined and very good readable presenta-
tion of this method. As we will see, Agmon's proofs may be carried over word by word to the case under consideration,
if we use the generalized
O
O
version of Garding's inequality,
inequality
(Theorem 6.3)
instead of Garding's
HSlder's inequality instead of Schwarz's
our representation of continuous stead of Riesz's representation
linear functionals
inequality and (Theorem
in Hilbert space. At some points it is
possible to simplify some proofs and to make some theorems a preparation we list in
§ 8
refer to S. Agmon
[3]
[3].
if generalization to
some of the regularity theorems
tic equations where first given by partially sharper.
The differen-
§ 9 • Mainly we sketch the proofs and I < p < ~ is simple.
when we have changed some things, we give exact proofs. I < p < ~
sharper. As
some theorems of the calculus of w ~ P ( G ) -
spaces. For the proofs we take reference mainly to tiability theorems are given in
4.6 ) in-
S.Agmon
0nly
In the case
for weak solution~of ellip[2]
, our theorems are
S. Agmon's proofs depend on the explicit construction
of the solution of Dirichlet's problem for an elliptic operator with
134
constant coefficients
in a half-space with the aid of
mental solution
and of the Poisson-kernels of Agmon- Douglis -
Nirenberg Fredholm's
[30]
[%] •
We use
our
results in
§ 10
F.John's funda-
for the proof that
alternative holds for uniformly elliptic fh/nctional equa-
tions. In this section some cumbersome work has to be done. Most of the difficulties
arise from our aim to demand low regularity properties of
the coefficients
and of the boundary.
theorems we derive in
§ 11
As a consequence of the existence
theorems of the type of Weyl's lemma
u n i f o r m l y elliptic Dirichlet bilinear forms and for u n i f o r m l y elliptic operators
for
(Theorem 11.1, Theorem
11.2)
(Theorem 11.4, Theorem 11.8) which
are sharper then the corresponding theorems of S.Agmon
[2] .We prove
our regularity theorems globally°
But it is easy to derive with the
aid of our theorems local results:
one has only to apply a suitable
"cut-off" procedure.
§ 8.
S o m e properties of the spaces
We consider the spaces
ltutlk, p
wk•P(G) p
:=
Let
equipped with the norm
I
(7- IIDC~ulILP(G)) ~" lc~l<_k
Theorem
wk'P(G)
for
~ < p < ~
8.1
G c ]R n
be a bounded open set. Then for every integer k > I
and every
1 < p <
(8.1)
W ~ ' P ( G -)
w
If furthermore
(8.2)
.
-----
8G e C k
w~(~)
=
then
c k ( ~ ) w~,,,~,.~
~P(@)
135
If
G
has the segment property,
open covering
[0 i]
that is,
3G
and there are vectors
0 < t < I , x + ty (i) ~ G there is a sequence
for
has a locally finite
[y(i)]
such that for
x ¢ G 0 0 i , then for every
(Ul) ~ C~(]R n)
such that
u cwk'P(G)
flu -Ull~llk, p - * O
(1-*~).
Proof: (i)
The proof of (8.1)
and J. Serrin (ii) Theorem
[44] , see
was originally performed by
e.g.
A.Friedman
[17] , Theorem
This stronger result may be found in 7.1. A sharper version
(W~(@)
=
A.Friedmau C~(@)
N.G. Meyers 6.3. [17] ,
w~'~L~) is easily
proved under these conditions. (iii) If
G
has the segment property
G ¢ C I )l a proof is presented [3] with
in the case
(it is sufficient p = 2
in
that
S.Agmon's
book
, Theorem 2.1. But this proof may be carried over to every I < p < ~ word by word.
Compare
[2]
, Theorem
p
3.1.
8.2
Theorem As sume
(I) integers (2)
that
G c IR n
and that that
Then
is a bounded open set and that
j,k ~ I
are
I < p < ~ is a real number,
u e wJ'P(G)
and that
D ~ u c wk'P(G)
for
lal = j •
u ~ Wj+k,p(G).
Proof: The proof is trivial by Theorem 8.1. Compare for this Theorem S. Agmon
[2]
, Lemma
Definition Let
3.2.
8.3
e i = (61i,62i,...,8ni)
is defined in the open set
G c ]R n
~ ~R n
and
0 ~ h e ~R I. If
u(.)
and if (x + hei) ~ G, x ¢ G, let
136
Theorem Let k ~
8.4
G c IR n
be a b o u n d e d
I , I < p < ~ . Then,
0 < h < dist
(8.4)
o p e n set and let
for e v e r y
G' c ~
G
u ~ ~,~'P(G)
and e v e r y
h
, where
with
(T F, ~ G )
,
II
.....
Proof: S.Agmon
(
[3]
, p.42-43)
The g e n e r a l i z a t i o n
to
A.Friedman,
, p.46,
Partially
[17]
the c o n v e r s e
Theorem Let k > 0 that there
G' c c
is for an D
Lemma
of
theorem
in the case p = 2 .
is trivial.
Compare
in the case p = 2
15.1.
Theorem
be a b o u n d e d
I < p < ~
for every
Then
I < p < ~
this
8.4
is
8.5
G ~ IR n
and
proves
ei
u
. Assume G
W k, P
(G)
set and let
that there
and that
i ¢ [1,...,n] ~
open
~ wk'P(G),
is a c o n s t a n t
for every
:
u
h
with
C > 0
where such
0 < h < ho(G')
IIS#UlIk, p(G,)
< C
.
Proof: In the case Lemma 15.3.
3.3,
I < p < ~
p.410-411),
this t h e o r e m
if p = 2 ~ e o m p a r e
is due to S . A g m o n
[3] ,p.44-45
s nd
(
[2]
,
[17] , Lemma
137
Theorem Let
8.6
G c ZR n
and let that
be a bounded open set, let
~ G ¢ CTM . If
u ¢ cm-I(G)
u ~ ~o'P(G)
is satisfied,
then
m > I
for a
p
be an integer
with
Dau I 3G = 0
I
for
lal
such
~ m-I.
I
Proof: In the case
m = I
and
p =2
this theorem was proved by S.Agmon
[3] ~
Lemma 9.1, p. I04 - 105. For another proof,
[66],
p.
§ 9.
175.
Differentiability
For
see E. Wienholtz
0 < R <
theorems
let
(9.1)
Remark (i)
9. I
Let
0 < R' < R
easily verifies, and SR
k_> I
there is an open set
HR' ~ HR',R c H R . For or the half-ball (ii)
and let
R> 0
H R . If
~R',R
II~l~.in
such that
let us denote with
0 < R' < R
Up till now we have used in
ii~II~, : = ( ~
be an integer.
As one
8~R,,R ~ C W GR
either
let
R'':= I(R + R').
~'P(G)
the norm
this section often it is simpler to use the A
norm
II~lII~,,p :_- ( ~ - -
ii~1~)~
With a constant
k = k(n,m, diam G)_~
!
> I
we have
IIU]Im,p <
k IIullm,p
fore~ t h e n o r m s a r e e q u i v a l e n t .
<
k IlU]m,p
Therefore,
if
(compare
(6.13)
(6.52)). There-
holds
with
138 CI > O, C2>__0
+
C2 = C2 the n o r m
in the norms in the
II.
!
IlUlIm,p
Lemma
%n
9.2
If.lira,p • t h e n it holds
lJm ,P
- norms.
~c'P(G)
(compare
For this
suppressin~
Agmon
[3]
' Cz
with
1 := Ci ~ E ,
r e a s o n we use in
§ 9
the dash.
, p. I07,
Lemma
9.2)
As sume (I)
is a
that
for
uniformly
R > 0
elliptic,
~rhose c o e f f i c i e n t s condition (2) such that (3)
in
GR
that ~u that
aa6 with
for a
and
regular satisfy
Lipschitz p ~ ~q
e ;,~o'P(GR ) there
m > I
Dirlchlet for
I#I = m
constant
with
is a c o n s t a n t
f o r m in
a
uniform
there
is a
GR
Lipschitz
L ,
I
for e v e r y
bilinear
u c Wm'P(GR )
( ¢ C Oo0(GR) 2 C > 0 , such that
for e v e r y
~, e ;,,~o'q(GR)
(9.2)
where
I < q < ~ Then
D
ei
u
~1 + 1 =~t
, ~
Wm, P
(GR')
. for every
R
< R
f
y = y (n, m, p, a~#, L, R, R ) > 0
where
such that
either (I)
i = 1,...,n
if
GR
=
SR ,
(2)
i = I,...,n-I
if
GR
=
HR •
or
and there
is a c o n s t a n t
139
Proof: Without loss of generality assume
a~8 = 0
for
161 < m - I .
Other-
wise is
itl = "~
where fied
il~l !'i,~-i
C~ -- C ~ ( n , m , l l & ~ l l ~ )
and t h e r e f o r e
all
ass~ption~
are
satis-
for the bilinear form at the left side of the above estimate. Let
HR' ~R
0 < R' < R. If
such that
let
G R = H R then by Remark 9.1
3HR' ~R ~ Cm
and
G = H R ' ' R and otherwise
HR,, c
there is a
~HR' ~R ~ H R " In this case
~ = SR, , .
co
Let = 0 and
~ ~ Co(SR) , O < ~ ( x ) < 1
outside
SR'+ -R-R' ~
i = 1,...,n
if
. Then
such that ~uc
G R = S R, i = I,...,n-I
_=
,:
h . For this
pu~ose
and transfer first the operator
on
Let
" if
SR, O
and R-R' 4
G R = H R . Then
•
Our aim is to derive estimates
pendent o f
~o~P(G)
~ = I
IIv~ ttm,p(~)
for
we c o n s i d e r &hm
then
B[v~,~] ~
that we get an expression of such a type that All terms which arise are estimated by
which
ar~ ± n d e -
- B[&~ (~),~1
to the other side, (9.2)
HGlder's
so
is applicable.
inequality.
We do this
in exactly the same w a y as S.Agmon in [3S, p. I09-110. So we get
I~E<',~:ll
(9.4)
for every 48_ from
(9.5)
•
=- I~[~,
@ g Co(GR) , where
¢ Co(GR, + R-R') 4 (9.2) and (9.4)
l~[v~,~]l
=-
~ ~-~ ~]l
~ k II ~11.,~11~,~<~
K = K(n,m,p,~,R,R',L,a~)
and
> 0 .
GR, + R-R' C C G we get by Theorem 4 for every • c Co(G )
k,(c
+ ll<
Since 8.4
~40
for every and
3G
• e Co(GR) ~ CTM
where
Theorem
K' = K'(K,{,m,n)
6.3
applies
. Since
and there are
v
~ Wm'P(G)
Cz > O, Ca > 0
such
that
c,,
(9.6)
II
By Theorem
-
8.4
there is a constant
K"
= K " (~,R',R,m,n)
> 0
such
that
Since
G ~ GR
of (9.5)
we may estimate the right side of
and the last inequality,
(9.7) for
÷
0 < h < R-R-4 " where
Dei(~u) by
~ ~m,p(~)
•
DeZ(~u)
(9.3)
and
. Since
~ m I
on
with the aid
which leads to
ll ll , co )
y = y(Cz,Ce,K',K"
(9.7)
(9.6)
remains GR.
) > 0
By Theorem
8.5
still valid if we replace
we have
D
ei
u
¢
Wm, p
(GR.)
Vhi and
holds. q. e. d.
Definition Let
9.3
G c ]R n
(compare Agmon
[3]
, p. 120, Definition
be a bounded open set, let
m> I
9. i )
be an integer and
w
let
be a
uniformly elliptic
an integer. for ck(G)
Then
181 + j - m
> 0
B
regular Dirichlet bilinear form.
is called and
see notations ).
j - smooth
a~8 ~ L~(G)
in G
otherwise
if
Let
J >0
be
aa8 ¢ C,~BI+J-m(G)
(for the definition
of
141
Lemma
9.4
(compare Agmon
[3]
, p.120,
Lemma
9.5)
Assume (I)
that
I
m~ I
and
J ~0
are integers with
are real numbers with that for & 0 < R < ~
that
B is a
u c F~'P(GR)
and that
! + ~ = I , P q uniformly elliptic,
regular Dirichlet bilinear form of degree (3)
j ~m
and that
(u
m
in
j - smooth
GR ,
c ; ~ ' P ( G R)
for every
~ Co (SR), (4)
that there is a constant
C > 0
such that
(9.8) holds for every
Then,
@ c W m'q(GR) °
for every
R' < R , u ¢ w m + J ( G R ~)
y = y(m,n,p,j, aa~,M,E,R,R')
> 0
(9.9)
such
(c
For technical reasons, two parts:
and there is a constant
that
+
the proof of Lemma
9.4
First we prove the lemma in the case
is divided in
GR = SR
and derive
from this further results. With their aid we prove the lemma in the case
GR = H R •
Proof of Lemma
p.12o-122
9.4
in the case
)
As in the proof of Lemma I~I + j - m
G R = S R. (compare Agmon [3] •
9.2, we assume,
~ 0 . The proof is by induction on
as#(. ) = 0
for
j . The lemma is trivial
142
for
j = 0 .
replaced b y
Let
I <j <m
and assume that the lemma holds if
is
j- I . R ' " : = ~ I (R + R" )
Let and therefore
~-
gives
by Lemma
Assume now
9.2
is defined in Remark
=- cll~ll~_c~.~,~>
u ¢ wm+J-I'P(GR,,, ) . If and also
(9.9)
j = I , then
holds.
J > I . By the inductive hypothesis we have then
u ¢ ~m+J-I'P(GR,,,) ~ wm+I'P(GR,,,). Therefore i = 1,...,n
9.1
Since
cll~ll~_~,~co.~
the inductive hypothesis u ¢ ~+I'P(GR,)
R"
where
0 < R' < R " < R"'< R .
t~[~,~]1
D eiu
e Wm'P(GR,,,) for
and
-~ ~ ( c
(9.10) Let
j
@ E Co(GR, . ) .
ll~ll~,~co~)
+
(2)
Then, b y assumption
and
a~6
= 0
for
I~I + J - m < 0 , integration by parts gives
(~
(9.11)
D ~i ~ ~, ~ ~)o
~-~< I p ~
"~-~ < I F I ~
C o n s i d e r any t e r m i n t h e second sum on t h e r i g h t " We may t r a n s f e r t~t-m+j-
1
by p a r t i a l from
I~I + l ~ l - m + j - 1 ei a ~ further, D
integration
@ to the other
r~De i a ~
differentiations
functions;
this
Da u , D ~ @ ) o.
of order
is prossible
since
(lalim, l~l<__m) and u ¢I~+J-I"P(GR,,,); I ~I +J-m-I . After we have done this, we estimate C
<_m+j-1 ~
the second sum on the right of
(9. I I) by
H8lder's
with a constant
> 0
(9.11)
K = K(n,m,j,M)
from
inequality and get
143
(9.12)
for
I
¢ e Co( R,,, )
1
and
i = I
"
(9.13)
for every
@ ~ Co(GR,,, )
the inductive hypothesis which is
n
Using
(9.8)
this gives
(c ÷ KII~II~.~_~G~.,))II~II~+~,9(e~. )
i = I, .... n. Since D ei u e Wm, P (GR,,,) , ei gives D u e ~m+J-I'P(GR, , ), for i = 1,...,n, and
U e wm+J(GR ,, ) . Combining
tive hypothesis
.
(9.10)
and
(9.13) , the induc-
gives
lib "* ~II~.,_,.~G..)
~
~(G . kll~ll....~_,.~.....~ + II~-II~,~(G4)
~-
~"(c
~- I1~-I1~,~,~,,~)
and therefore
q.
The next theorem is due to S.Agmon
(see
[2]
, Theorem
here a different proof based on our existence theorems
Theorem
e. d.
6.1). We give of
§ 7.
9.5
Assume (I) I
that ~
re>l,
j>1
are integers with
are real numbers with that
O< R< ~
and that
j <_2m
and that
~1 + 1 = 1 , L :=
~_
a s(.) D s
is uniformly
ts l<_2m elliptic
in
SR
with coefficients
a s s cISI+j-2m(SR)
for
Isl+j-2m> O,
14-4
such that
D Y a s c L~(SR )
IyI< Isl + j - 2 m ,
for
and
a s ¢ L~(SR )
for
tst +j -2m< o , (3)
that
u ~ LP(SR)
that for every
$ c Co(SR)
I(~., L{)ol
(9.14)
Then,
for every
C > 0
such
~- C II { II,.-.,_~,~~s.) R' < R
y = y(m,n,j,p,L,R,R ~) > 0
ll,~ll~,~.cs~.,) ~
(9.15)
and that there is a constant
is
u c wJ'P(SR)
and
there is a constant
such that
~'(e
÷
tl~-llo,~c~.~)
Proof: Consider the partial differential operator
(9.~)
A
For every
i ~ 11:{n
(9.17) Therefore
:=
(-~)"
D~i
i.=~.
is with a constant
(-4)~A(1) = BA
~
yl~l
~~
E = E(n,m) > 0
~
E~,lll''
defined by
(9.18) for
W,$ ¢ C ~S o(R)
is a
uniformly strongly elliptic Dirichlet bilinear
form satisfying the assumptions there is a
v c ~'P(SR)o-
of Theorem
such that
7.2 , second case. Hence
1~5 (9.19)
~& Iv, {]
Further,
By
there
=
(u~, ~ ) e
is a constant
f o r every
C' > 0
{ e ~o'q(SR }
independent
of
v , such that
(9.19)
I~,~,~]1 The assumptions
~ I1~11o,~11~11o,~
of Lemma
C := IIUlILP(SR) . Choose
9.4 now
are satisfied
f o r every with
0 < R' < R . By Lemma
and Ilvtl2m, p(SR, ' ) <_ X'(tlUltLP(sR) + IIVIIm, p(SRI) respect o f
(9.2~)
(9.20)
with
11~11~,~ ~s.,,)
For any = (u,¢)o
.
Since
and with
9.4 , v c w 2 m ' P ( S R ,, )
, what gives w i t h
y " := y ' ( 1 + C')
~-
@ ¢ Co(SR, , )
j = m
{ c Co(SR)
~"1I"11o,~=~
we get therefore
oo
Co(SR, , )
from
is dense in
(9.19)
BA[V,@]
Lq(SR, , ), t h e l a s t
= (Av,~)o=
equality
implies
(9.22)
Since
for
(9.23)
For
(Av,~)o
-~
(~,w) o
@ ¢ Co(SR, , )
(Av, L~)o
~
for every
w ~ Lq(SR,,)
we have
L • ~ Lq (SR, ' ) , we
(~,
for every
L~)o
Y ~ w 2 m ' P ( S R . ) , • c w 2 m ' q ( S R ,, )
get
~ S R" ) @ c Co(
let
(9.24) =
~
7-- ~-
(~S~ ~
~, ~ ) ~
146
For
i c ~R n
is
I-~lm~.'m
i=I. "11
because of
(9.17)
satisfied,
since
and a s s u m p t i o n
for
l'
nomial under consideration is
a root
of
c ZR n - l , for
(2)
. Too,
T c C,
(1',I:)
every fixed
one o f t h e p o l 3 ~ o m i a l s
the root is
or
and o n l y
~--
uniformly elliptic Dirichlet biliDear form of order j - smooth by assumption
Since
IBo[V,@]I <__ C
for every
assumptions of Lemma By Lemma
9.4
llvtl2m+j,p(SR,) (92s) where
there is a
B ° is a
2m, which is regu-
for every
and
and
@ c Co(SR, , ) .
y," = y'" (n,m,p,L,R,R")
<_! ~'"(C + tlvtl2m, p(SR, ' ))
(9.24)
by (9.15)
are satisfied, which gives
II
v c w2m+j'P(SR,).
> 0
and
by
in
SR,. Since
All
such that
(9.21)
r (c
y = y'".max [I,¥" ] . Now,
and
9.4
(9.23)
¢ ~ Co(SR, , )
1t¢ll2m-j, q (SR,, )
it
(2).
v c w2m'P(SR,,,) , we get from
Bo[V,¢ ] = (u,L~)o
if
~s(×)l'S'~ ~
But these polynomials satisfy the root condition. Therefore,
lar and
is
a zero of the poly-
x c SR , i f
A(I',T)
condition
A
(9.22)
implies
is of order
u = Av
2m, we have
a.e.
u ~ wJ'P(s R, )
llullj,p(SR,) = liar IIj,p(SR ,) < Ilvll2m+j,p(SR,) proves
v c w2m+j'P(SR,)
The inequality
together with
(9.15) . q.e.d.
(9.25)
147
Lemma
9.6
(compare S.Agmon In the case
[3]
j = I
, p. 112-113, Lemma 9.3. see S.Agmon
[2]
, Lemma 6.1)
As sume (I) I
(9.26)
are integers with
are real numbers with 0 < R < ~ and that Jei D u c LP(HR) for
u c Lp(HR)
~
c
and that
and has generalized
i=1,...,n-1
that there is a constant
ICva.,])'me"~)o[
m>j
1 + I_ = I , P q
that
derivatives (3)
m > 1, j >_ I
that
C > 0
,
such that
n@ll~-s,~¢.~)
oo
for all
• ¢ Co(HR)
Then for
every
y = 7(n,m,j,p,R,R')
.
R' < R > 0
u e wJ'P(HR, )
such
and there
is
a constant
that
(9.27)
Proof: The first part of the proof is quite analogous p.113-18:
As is immediately verified~
cm m ( H R )
(compare Definition
to Agmon
[3]
(9.26) holds for any
3.1, p. 41
,
• s
) by continuity.
By a
N
method due to Lions we construct an extension a manner that x : (x',x n)
(9.28)
u ¢ let
~ (,<', ,<-~)
C 2m-I (HR)
would imply
u
of
u
to
SR
u ~ C 2m-I (SR) . For
in such
~48 where the
(9.29)
~K
are real constants
~'K (- I<)- s =
F
The determinant determinant has a
uniquely determined
we easily prove that rivatives
for
there is a constant
(9.31)
II or
_ ~I,
~ ~ LP(SR )
D jei ~ e LP(sR)
s = 0
2m + 1
equations
(kl,...,k2m+1)
and that
u
i=1,...,n-1
K = K(n,m)
j , i = I,...,n-I
is a Vandermond
k = I, .... 2m + I . Thus,
solution
Further,
where
i
of this system of
for the quantities
satisfying
(9.29)
~ IR 2m+I .
Further,
has the generalized which
de-
satisfy
> 0
such that
for
S = O,...,2m
.
oo
For
~ c Co(SR)
(9.32)
, let for
~'~'(~) =
x c ~R
"u((~',×-n)se n
Then, plies
by
(9.29)
D
W*(x) I
= 0
Xn=O
2m y* ~ Co,2m(HR) c ~o, m (HR)
Further
which im-
(~'D2men ~)o,S R = oo
= (u,D 2men Y*)o,H R
(9.33)
which implies by
I
(9.26)
for every
• ~ Co(S R)
149
where the last inequality follows by estimates analogous to (9.31). For
I < i < n-1
we get from
(9.34)
for
:
every
Then
(9.31)
~ ~ Co(SR)
(9.33)
(9.35)
and
"-
. Consider
(9.34)
now t h e
operator
A
defined
by
(9.16).
imply
I( ~"' &~)~,SRI -i_=~.
for every and by
~ ~
C I)o
o(SR)
. Now from Theorem
(9.15) , (9.35)
and
9.5
we get
~ c wJ'P(SR,)
(9.31)
(9.36) ~.=~.
Since
I
u HR ' = u iHR '
IIuOIwJ,P(HR,)
,
u c wJ'P(HR,)
;
by the trivial inequality
i JI~IIwJ,P(SR,), (9.36) implies
(9.27) q. e. d.
Remark
9.7
Clearly, the statement of Lemma 9.6
is independent of the theory
of elliptic differential operators.We used Theorem of Lemma
9.6
only for the sake of simplicity.
9.5
in the proof
15o
Lemma
9.8
(compare Agmon
Assume that
[3]
u ¢ wm+I'P(HR )
c C o ( SR ) , where
such that
0 < R < ~, I < p <
~
and
~u
Lemma 9.4) c ~,~o'P(HR )
for all
m >~ I • OG
~ D ei u ¢ W~o, P (H R )
Then
, p.118-119,
for all
~ c Co(SR) , i = 1 , .... n-1
.
Proof: Agmon's proof may be carried over word by word to the case under consideration. W~o'P(HR)
We observe only that the reflexive Banach space
is w e a k l y sequentially compact.
is constructed in such a manner that
HR
defined by
supp ~ n H R c H R
Remark
9.1,
J
q.e.d.
Now we are in the position to perform the Proof of Lemma
9.4
in the case
Compare the proof in the case for
181 + J - m < 0
G R = HR:
G R = S R . Again, we assume
and again the proof is done by induction on
the following we will only note where the proof in the case
a~(.)=o j . In
GR = H R
has to be changed or something has to be added such that it works too in the case
GR = HR .
Let us consider the induction step. If
De i u ~ I ~ ' p(HR, , ) -= Nm-j-1 (HR, , ) . If and therefore by Len~ma any
~ ¢
CO
Co(SR,, )
inductive hypothesis
and
9.8
j > I, then
~ D ei u ~ ~o'P(HR,,,),
(9.13)
this gives
j = I,
u ~ ~,~+I'P(HR,,,) i = I,...,n-I,
for
is proved in the same manner. B y the D ei u e Urn+j,
l , p (HR,, ), i = I,...,n-I,
and
It remains to prove
~ ) D en u ~ W re+j- 1,p (HR. ) . For every • c C o(HR''
151
(9.38)
"~-~ < I~I~'~ We e s t i m a t e
the
i = i(~),
I
second
sum at
the
, such that
right.
Since
6i ~ 0 and
u ~ w m + I " P ( H R ,, ), we get by p a r t i a l
D6$
integration
I~I = D
> 0 ,
there
ei D~-e i
is
an
@ . Since
for any term of this sum
~ . J 0 t ~R I'
Now we t r a n s f e r
161 - m + j - I
other functions.
In the case
differentiations J61 <m-l,
I~+ ell + 161 - m + j - I ~ m + j inductive
(9.40)
with
hypothesis.
I( ~m'~ ] ) ~ "
Further,
B
is
~(~)~,~,, I
~
K
K = K(ac~,n,m,j)
In the case
=m
D ~-ei •
this is p o s s i b l e and
to the
since then
u ~ W m+j-~(HR, , ) by the
j - smooth.
Then we get
ll -s,u
> 0
we observe
i(~) ~ n . Then we derive
Thus j
I
from
(9.37) , (9.40) and
from
(9.41)
that the c o n d i t i o n (9.39)
imply
~ me n
implies
~52 Therefore,
by
(9.38)
and assumption
(~)
1
(9.~2)
I CO
for every
@ c Co(HR, , ) .
Let
v ::
7 - a~,me n D ~ u . lal<_m
Then, ve LP(HR. , ).
From the inductive hypothesis
tiUllm+j_l,p(HR, ' ) (_ y' (C + ilUiim, p(HR)). Therefore,
by (9.42)
(9.43) ICv.~)o,~,,1 for every
~ w'(c~-II~ll~,~)n~ll~_~,q(~..)
@ c Co(HR, , ). For i=1,...,n-1,
D el u ~ wm+J-I"P(HR ,, ) implies
D jei v c LP(HR~, ).So, the conditions of Lemma 9.6 are satisfied in HR, , . Hence, v ~ ~,~+J'P(HR,)
tlvlt~,~.~,)
(9.44)
Since
B
and with y'" > 0
~"'((C.ll~ll~.~.,,,) ~- }
is uniformly elliptic,
lamen, men(X)l > E > 0 for every Dmen
is by (9.27) and (9.37)
=
II~)a~Ho.~cs~.)
we derive from (1.6) with x c HR .
Therefor%by
definition of
[amen, me n ]- I (v - 7 as, me n D ~ u ) i~l <m ~@me n
i = (0,..,0,1) v,
153 Where the right is an element of wJ'P(GR, )." From (9.44) azld (9.37) follows
lidmen u IIj,p(HR,) <_ V (c + Hullm,p(HR ) which completes the proof. q.e.d. Theorem
9.9 (compare Agmon [3], p.124- 125, Theorem 9.6)
Assume (I)
that the assumptions (I),(2) and (3) of Lemma 9.4 are satis-
(2)
that
fied,
(9.45)
f ~ LP(GR)
]~[~'~]
=
Then, for every
and that
(~, ~)o
R' < R
y = y(n,m,j,a~,EjR, R') > 0
for every
u c ~
• c Wom'q(GR )
O'P(GR, ) and there is a constant
such that
(9.46)
*
Proof: By (9.45), for
• ~ w~'q(%)
IB[u,$]I = l(f,¢)ol ! IIflILP(GR) II$11Lq(GR)
! IIflILP(GR) ll$11m-j,q(GR) •
Therefore, assumption (4) of Lemma 9.4 is satisfied, too. This lemma applies and the theorem is proved. q. e. d.
Theorem 9.10
(compare Agmon [3] , p.125 - 128, Theorem 9.7)
As s u m e (I)
that assumptions
(I), (2) and (3) of Lemma 9.4 are satisfied,
(2)
that for some integer k ~ 0
B
is
(m +k) - smooth,
15#
(3)
that
f c wk'P(GR )
such that
B[u,¢] = (f'@)o
for every
~ Wom'q(GR) • Then, for every R' < R, u ~ w2m+k'P(GR , ) y = y(m,n,k,p,a~,R,R')
(9.#7)
> 0
ll~-ll,...,.~,ec%,~
and
there is a constant
such that
"-
r(ll{ll~,~.c~.0
~ ll~-ll-~,~co~)
Proof: The proof is by induction on k. By Theorem 9.9 the conclusion holds for k = O . Then
(9.48) Let
Let
k > I and suppose that the theorem is true for
u c w2m+k-I'P(GR ,, )
II ~-II ~-~,~-~,~cc.~) i = l,...,n-I
c Co(GR, , )
k - I.
and
~
if G R = HR,
~" ( II ~ll~,~c~ and
÷ II ~-ll~,,~.co~J
i = l,...,n if G R = S R.
For
we get by partial integration
where
(9.49)
A;_
A i is a well
:~-
defined differential operator of order 2m, since
and by assumption
and therefore
(2)
k~ I
am6 c CI21+k(GR ) c C ~ I + I ( G R ). By assumption
(3)
155
oo
(9.5o)
Since
~[
S)'~'', ~ ]
~-
(]9"~ ~ - A I ~ , ~ ) ~
u ~ w2m+k-I'P(GR, , ), D ei f - A i u
for every ~ ~ Co(GR, , )
e wk-I'P(GR, , )
and by (9.48)
(9.51)
If
GR,,=HR,, ,
i = Ij...,n-1
~D e i ~ ~,#o'P(GR,,)
by Lemma 9.8;
if
for any
case
if GR, =HR,
and
we get from (9.50) and (9.51)
i=1,...,n
GRt = SR, this completes the proof.
L :=
7-
~d
GR, , = SR,,, this is trivial for i=1,..,n.
Therefore, by the inductive hypothesis ei w2m+k-l,p D u e (GR,) and
where i=1,...,n-1
~ ~C~(SR,,)
If
(-I) I~I D8~aB(. ) Dm). Since
if GR, =SR, . In the
GR, =HR,
consider
u ¢ w2m'P(GR, , ), by assump-
l~l<_z,lmlAm tion (2)
(9.53)
Lu
= f. Therefore
(_ &)~m ~ e ~
me~
At the right side of (9.53), x n - variable are of order right side belongs to D 2men u ¢ wk'P(G ')
(9.54) l l ~ ~ I ~ ) Since
all differentiations
~2m-I.
Therefore,
wk'P(GR, ). Since
with respect to the
by the results above, the
lamen, men(X)l ~ E > 0,
and by (9.52) and (9.53)
~_ ~"(II÷II~,~(~ * II~II~,~)
D eL u ¢ W2m+k,P(GR, ) (i=I ..... n-l)
and
D 2men u e wk'P(GR, ) ,
156
Theorem 8.2 implies
u ¢ w2m+k'P(GR , ). Estimate
(9.47) is an immediate
consequence of (9.52) and (9.54). q.e.d.
Now we are in a position to prove
global dlfferentlabillty theorems.
Theorem 9. S I Assume (I)
that m > I and j > 0 ( w i t h J < m ) a r e 1 1 are real numbers with ~ + ~ = S,
integers and that S < p , q < ~
(2)
that G c I R n is a bounded open set with boundary
(3)
that
B
is a
uniformly
let bilinear form of degree (4)
(9.55)
that
m
F e W~o-J'q(G)*
B[~,~
=
in and
T (~)
for
Then u e P~o'P(G) N T¢~+J'P(G)
~ G eC re+j,
elliptic, j - smooth regular DirlchG, u ¢ ~'P(G)
all
such that
$ e Co(G ) .
and there is a constant
y = y(n,m,j,
p,B,G) > 0 such that
(9.56)
II
(II
"-
* II IIo. )
Proof: For every Uxo N G
xo
c ~G
there is a neighborhood
is mapped by a mapping of class
Cm+j
As we have shown in the proof of Theorem 1.6, B
Uxo such that
on
HR with some R > O .
is transformed in a
uniformly elliptic regular Dirichlet billnear form, which is again jsmooth. Further,
all assumptions of Theorem 9.11 ren~in still valid for
the transformed quantities.
Therefore,
by Lemma 9.4 the transformed
157
function belongs to
~f~+J'P(HR)"- and inequality (9.9) holds. TransforE ming back and denoting the intersection of ~ with the image of SR i under the inverse mapping by U~o we get
II ~
(9.57)
II ~ a , ~ c ~ , )
T(~o)(~ ll~,~cm. Since
8 G is compact, there are
G c
~t=l --Uxi " Then the set
xj ~
G-
~G, j=I,...,N
~ J .~U"i
each
such that
is compact and contained in G.
Therefore, there is a finite number of balls ver i%~
÷ ll~ll~,s~,)
Kj , J=I,...,M
which co-
of them having the property that the ball
same center but twice the radius of
Ki
K'. with the J is contained in G. For every Kj
is u ~ W m ' P ( K j ) a n d t h e r e is a Y.I~ such that an inequality analogous to (9.57) holds. From this follows with
y' = (N + M) max[y(U' (xi) , yj } i,j
N
(9.58)
M
II~II~.~,~(~) ~- ~-II~II~.~,~(~) + ~-II"II~.~,~(~)~II~II~)
~' (lit ll~,~c~)~ + From (6.13) and (9.55) follows
(9.59)
G~ II~ II~
~
II~ II~.~c~
Combining (9.58) and (9.59) proves
~
C~ll~IIo,~
(9.56) q. e. d.
Theorem 9.12 As sume (1)
that
numbers with
m~_l a n d k~_O a r e 1
~
+
1
~ = I ,
integers
and that
1
~
are
real
158 (2) 8G
that
G E ~n
is a bounded open set with boundary
B
uniformly elliptic,
¢ C 2m+k , (3)
that
is a
Dirichlet bilinear form of degree (4)
that
(5)
that
m
in
(m + k ) -
smooth regular
G,
f ¢ wk'P(G), u ¢ W~o'P(G )
and
that
B[u,¢] = (f'~)o
for every
¢ ~ Co(Q). Then
u e ~'P(G)
N w2m+k(G)
and there is a constant
y=7(n,m,k,p,
O
G,B)
such that
Proof: With the same considerations
as in the proof of Theorem 9.11
the
theorem above is derived from Theorem 9.10. q. e. d.
Remark 9.13 Let
G ~ IR n be a bounded open set and let
be uniformly
elliptic
isl'm(G)
a s ~ C,
for
in
G. Assume that
Isl >m.
For
Isl ~ m
L :=
a s E L~(G) + I
let
7Is l ~ 2 m for s = sl
as(.) D s
Isl ! m + s2)
and where
Is~l= m. Let
(9.60)
for
~,~ [-~,,]
(~a-~,,~)o ~- y-~:,)'"'y- ({J(~"%,~"~,~'e)o
W,~ c Co(G ) . Then
(9.~) ~ [ m ~ ] and
.= ~
BL
is a
=
uniformly
(L~)o elliptic
for ~,~%(a) regular Dirichlet bilinear form in G,
159
which is for
m - smooth.
Isl ! m
Assume
If furthermore
and a s ~c~Sl+k(G)
for
and let B *L
a s E C sl+k(G)
k ~0
is an integer and
Isl > m ,
then
a s ¢ C~(G)
B L is ( m + k ) - smooth.
* BL[U,V ] := B L [v,u],
be defined by
which is B~[U,
v]
-7-
=
(~s u,
Ds
V)o +
tsl im Then, B L
and
Z
m+1 ! tsl<__m BL
are
(s$) (Ds~_ - T~
(_l)lslz
DTu,DSlV)O
TJs2
(m + k) - smooth.
Theorem 9.14 Assume (I)
that assumptions
(2)
that
(I),
(2)
and
(4) of Theorem 9.12 are satis-
fled, L =
~
as(. ) D s
Isl<
coefficients (3)
Then constant
for
u ¢ ~oo'P(G)
@ e Co(G), where
BL
elliptic
in G with
2m
k a s ~ C.(G)
that
is uniformly
Isl <m, and that
is defined by
u g W~o'P(G ) 0 w2m+k'P(G) y = y(n,m,k,p,G,L)
a s g c,~sl+k(G)
for
Isl >m,
BL[U'$] = (f'$)o for every (9.60). and
Lu
= f. Further,
there is a
> 0 such that
Proof: The theorem is a trivial consequence 9.12
of Remark 9.13 and Theorem
. q.e.d.
Theorem 9.15 As sume (I)
that assumptions
(2)
that
L =
7Is l<2m
(I) and
a s (.) D s
(2) of Theorem 9.12 are satisfied, is uniformly
elliptic
in
G
with
16o
coefficients
a s ~ c.~SI+k(G).
Then there is a constant
for every
y = y(n,m,k,p,G,L) > 0
u ~ W~o'P(G ) n w2m+k'P(a)
and
I1 ,, 11 for every
such that
.
1I ,,
I1o, )
v E wm'q(a) n w2m+k'q(a). o
Proof: Since for
u E Wm'P(G) N w2m+k'P(G)
BL[U,~] = (Lu,~)o
for every
• c Co(G )
BL[V,*] = (L*v,~)o
for every
• e Co(G )
and
the assertion follows from Theorem 9.14 and Remark 9.13. q.e.d.
Theorem 9.16 Assume (1) with a
that assumptions p>n
and
(I) and (4) of Theorem 9.12
are satisfied
k=S,
(2)
that
G c IR n
(3)
that assumption
is a bounded open set with boundary (2)
with
k = I and that assumption
8G (4)
~ Cm, of
Theorem 9.14 are satisfied. _, ' L u Then u c cm-I+~(G) N c2m+ loc ~ (G) n k=0,!,...jm-1, where 0 < ~ < I - ~ .
= f
in G and ( ~3) k u I 8G = 0,
Proof: Since
u ~ W~o'P(G )
and p > n, u ¢ cm-1+~(~) after correction on a
set of measure zero, by the Sobolev - Kondrashov - T h e o r e m
(Appendix I,
161
p. 225 where
I - ~n . Further,
0<~<
there is a constant
c I1{
By definition of
w~'P(G)
I~ I ~ m - 1 .
Since
By (9.62)
D~u n
the normal derivatives
for every
, there is a sequence
flu -unJlm, p -* 0 (n-*~).
= 0 on of order
D~u=
~ m -
1
SR ~
e Wmoo'P(G)
•
such that
uniformly on
0 on3G
vanish
.
is trivial by the Sobolev - Kondrashov - theoremand to any ball
such that
(Un) c C~(G)
D~ Un-+D~u
3G ~
C>O
for That u
~
for
I~I ~ m
and
e
c2m+~(G loc ~ ~'
Theorem 9.10
applied
G . q.e.d.
A very simple consequence
of Theorem 9.10 is the following approxima-
tion theorem.
9. t~
Theorem
As sume
(i)
that m > I and k > m
are integers
and that I ~ p <
~
is a real
number, (2)
that
Then
is open and bounded and that
8G
e
Ck
and every 0 < ~ < I there u g Wm'P(G) 0 wk'P(G) o (my) n c k ' I + ~ ( W ) D w k ' P ( G ) r~ C~(G) such that D ~ u vl 3 G = 0
for every
is a sequence
for
G c ~{n
I~1 < m - 1
and
Itu-uvllk, p(G)-* 0 ( v - + ~ ) ,
Proof: The bilinear form
B[~j~]
tic. Since
u c wk'P(G),
such that
flu - vvllk, p -+
p~ such that
max
[p, i11 -
:= ( ~ ) m
by Theorem 8.1 0
(v -+ ~). Let
]
is uniformly
there is a sequence 0<~<
uv ¢ W m'p' (G) such that
I
. Then p' >n.
vv e ~ 'p' (G) . By Theorem 7.2 and Theorem 5.4 is a
strongly ellip-
be given. Since
for every
(vv)c
C~(g)
Choose
vv e ck(G), v e ~
there
~62
(9.63)
( ~,
If m < k ~ 2 m
~)~
I
for every • c
v~ ~ ~ ' P ' ( G )
(9.65)
(V~, ~ ] ~
for every
• E Co(G ) .
we get after a partial integration
I(
(9.6@)
=
Co(),
II
where I
n wk,p'(G).
(v~, ~ ] ~
II
If
----
= I. By Theorem 9.11,
k>_2m, then
( L - ~ ) ~ --- ~a~v~, (~)o l~l='am
for every
• E C~(G). Since
(-1)m ~
D2~vv e wk-2m'P'(G), by Theorem
l~l--m 9.12
u~ ~wk'P'(G). By the Sobolev-Kondrashov-theorem
since
~ < I -~,
Ix - Xol < R ] c
G.
If
x o ~G, then there is
Since
vv ~ C~(G)
(-I) TM '
R>0
for every
therefore by the Sobolev - Kondrashov- theorem i e IN, which is
uv ¢ C~(G). Since
such that
G R := (x :
7- D 2~ vv ~ wI'P'(G) loc
every 1. By Theorem 9.10, u~ ew2m+l'P'(G SR)
every
uv ¢ck-I+~(G)
1 e IN
for and
uv e c2m-l'1(G ~)
p'~p,
uv gwk'P'(G)~
for by
(9.6}). (9.66)
[~-~v~
~)~
=
(~-vv, ~)~
for every
@ e Co(G )
and therefore by Theorem 7.7
(9.67)
If
CII~-
m
~vll~,e
~_
II~- v~II~,e.
the right side of (9.66) defines a continuous linear
functional on wm-(k-m)'q(G)o
with norm at most
flu - v~llk,p. By (9.56)
and (9.67)
11 ~ - ~
II~,~
"-
~ (H ~ - v~ L ~
~
U ~-
~ lio0~)~
that is
163
II
-
.)II
a'(
- v D ,e
The same inequality is derived in the case
>
o
k > 2m from (9.66) with the
aid of Theorem 9.12. q.e.d. Remark 9.18 If the assumptions of Theorem 9.14 every solution of Problem (Wp)
are satisfied with
k=O
, then
solves Problem (Stp).Furthermore, under
the conditions of Theorem 9.16, every weak solution is a classical solution.
§ 10.
Fredholm's alternative for uniformly elliptic functional equations
The proof of Fredholm's alternative in the case of uniformly strongl y elliptic, regular Dirichlet bilinear form~ B that there is a + ko(U,¢)o
ko > 0
such that for
k> k o
was based on the fact, and
Bk[u,~]
:= B[u,~] +
the problem
Bx[U,~] has for every
=
F (~})
,
for every
F ~ wm'q(G)*
• ~ Co(G )
an uniquely determined solution
u E ~o'P(G). In the case of uniformly elliptic regular Dirichlet billnear forms, being not strongly elliptic, this needs not to be true. As we will see, the only cumbersome part in the proof of Fredholm's alternative is the proof of this under certain ficients of
B
"dim N(Tp) = dim N(Tq)". First we will prove
regularity properties of the boundary and the coef-
in the case
p =q =2
(Lemma ]0.2). Then with the aid of
the differentiability theorems of §9, this result will be extended to the case of arbitrary I ( p < ~
under the same
regularity assumptions
(Lemma 10.3). With the aid of an approximation procedure and some well
164
known stability theorems
for FredhoLm
the regularity assumptions
operators we are able to weaken
(Lemma 10.4,
10.5,
10.6, Theorem
we are in the position to consider very general uniformly tional equations
(Theorem 10.8, observe
Remark
elliptic
func-
10.9). A further simple
consequence
is a complete treatment
Dirichlet's
problem for uniformly elliptic operators
Theorem
10.7). Then
of the existence
of solutions of (Theorem 10.10).
10.1
As s l i m e
(1)
that
such that
m ~ 1 i s an i n t e g e r
(2)
~ + ~ = I, P q that G C ] R n
(3)
that
near operators
(%~,Y)m (4)
and t h a t
1 < p,q < =
are r e a l numbers
is a bounded open set with boundary
T r : ~o'rCG) -~W~o'r(G)
(r=p
or
8G
¢ Cm ,
q) are continuous
li-
satisfying
= (~,Tq~)m
for
every pair
that there are constants
(lo.1)
C. ll ll ,
for every
@ ~ ~w o'r~G), " "
C~ > O, C2 ~ 0
II%mll , where
r=p
(~,~)
g ~o'P(G)
x c'q(G)
such that
C li o,= or
q.
Then (I)
the nullspaces
finite dimensions
(II)
N(Tr)
(r =p
or
:= [~ e wm'r(G) o
= N(Tq) , R(Tq)
("')m
= N(Tp), where
~o'r(G)
O}
of T r
have
(r=p
or
q)
and
orthogonality with respect to
is meant.
Proof: In the following let
=
q),
the ranges R(Tr) are closed in
R(Tp)
: Tr~
r = p or q.
,
165
(a)
We want to prove
(I). Since T r is continuous, N(Tr) is closed.
Suppose, dim N(Tr) = ~ . Then there is a subsequence that ]l¢kllm,r =1
(see e.g. [54] ,p.218 or subsequence
I
tJ~k - ¢lllm, r >
and
[@k,] ~ [@k ]
[31]
) .
for all
[@k] c N(Tr) such
k, 1 c IN with
k~l
By Rellich's theorem, there is a
such that
]J@k' - @l'JJLr(G) -* 0 (k',l' - * ~ ).
But (I0. I) implies
c~n~, - ~ , 1 1 ~ , ~ what is a contradiction to
~-
C,.ll~,,,
~ o (~',~'~).
- ~,11o,~for
II~k' - ~I' ]Jm,r > 1
k' ~ i'.
There-
fore, dim N(Tr) < ~ . (b)
We want to prove: R(Tr)
c R(Tr )
and
Since N(Tr)
f ~ wm'r(G)o
Let
such that
I1 ~
-
h k ¢ N(Tr)
~ ll~,=
~-
Let Let
(fk) c fk = Tr Uk'
h~o'r(G), for every
k
such that
II ~
- h~.~,~
for every
h e N(T r )
v k := u k - h k. Then (I0.2) implies
f o r every
and
q.
J]f - fkJJm,r -~ 0 .
is a finite dimensional subspace of
there is an element
(lo.2)
is closed~ r=p or
T vk
h ~ N(Tr)
= T u k = fk " There are two possibilities:
First, suppose that the sequence
(Vk) is bounded in
Rellich's theorem implies that there is a subsequence verging in
Therefore, tinuity of
Lr(G)
and by
there is a T r,
TrV
Wom'r(G). Then
(Vk,) c (Vk) con-
(10.1)
v ¢ Wo'r(G)
such that
= f, which proves
Vk, -~ v
f ~ T< (Tr).
and by the con-
166 Secondly, that
suppose that there is a subsequence -* ~.
llVk'llm,r
Let
flyk, - lJVk,llm,r " hllm,r
(~o.4)
h
h ll~w
~-
IIWk,IIm,r = I
and
II w~,
Further,
Wk,
-
llVk,llm,r
i
for
TrW = O, that is
(c) If
. Since by (10.3)
for every
every
h E
h ~ N(Tr),
N(Tr).
Again Rellich's theorem and
w e k~o'r(G)
such that
IlW-Wk,Ilm, r-* O.
w e N(Tr). But this contradicts
We want to prove w c R(Tp), then
~ N(Tq)
Vk' llVk' llm,r
T r Wk, -+0.
(I0. I) imply, that there is a Then
:=
(Vk,) c (Vk) such
R(Tp) % w = Tp~
(I0.4).
= N(Tq). with
u ~ ~'~o'P(G) and for every
is (w,$)m = ( T p U , $ ) m = (U, T q $ ) m
= 0
and therefore
w ~ N(~q). Let pose, that
v ~ ~n'P(G) such that (v,~)m = 0 for every ~ c N(Tq). Supo v ~ R(Tp). Since R(Tp) is closed, d := inf llv-wll > 0 .
w ~ R(Tp) Therefore,
(10.5)
there is a
~F(w) = 0
for
F E ~'P(G)*
every
By Theorem 4.4, there is a for every every
g c N(Tq). stLmption on
w ¢ R(Tp)
F(v) = I v
was
and
F(v) = I,
m*
IIFII , p
g ~ ~,~o'q(G), g ~ 0
• ~ ~o'q(G). By (10.5),
u ¢ ~o'P(G) is then
such that
=
I
such that F($) = (g'$)m
(g,W)m = 0 for every
w ¢ R(Tp). For
0 = ( g j T p U ) m = ( T q g , U ) m , that is,
implies
(g,V)m = I. On the other hand, our as-
(v,~)m = 0 for every
• c N(Tq), especially for
= g, what is a contradiction. q. e. d.
167
Lemma IO. 2 Assume (1)
that
m> 1 i s
set with boundary
(2)
an i n t e g e r
and that
G c ]R n i s
a bounded open
8G ~ Cm,
that
is a
uniformly
ents
aa~ c C
elliptic
Dirichlet
bilinear
form i n
G with
coeffici-
TM('@),
(3)
that V : ~ ' 2 ( G ) -~ ~ ' 2 ( G ) is completely continuous, o o that the bilinear form B is defined by
(4)
B["F,¢]
:=
BE['{'g~] + (V~t' 'g~)o
Then, B
"?,@ e Wmo'2(G).
for
is a continuous bilinear form on
W~'2(G) × ~ ' 2 ( G ) . O T : wm'2(G) -~ W~'2(G), satis-
For the continuous linear operator
O
fying
B[~ ,~]= ( T ~ , ~ ) m
for ~,~ e Wm'2 o (G),
Fredholm's alternative
holds: dim N(T)
~(T)
=
dim N(T*)
N(~) ~
=
J
<
R(~)
and
~ =
~(T) ~
Proof: (i) that
Let
TE
~[~,~]
(1o.6)
Since elliptic
T
B~[@,Y]
be the continuous linear operator on
= (TE~,~)m
=
T~
:=
and r e g u l a r ,
such that for every
for
l*~o'2(G) such
~,~ e %~'2(G). Then
~ V
7-
( a - ~ D ~ @ , D ~ Y )o
by Theorem 6.3
u ~ W~'2(G)
there
and
BE[@,~]
are constants
are uniformly Cz ) 0, C2 ~ 0
168
and
(~o.77,
C, l l ~ , ~
where
~
Further,
~- l l r ~ l l ~ . ,
÷
denotes the adjoint of
c,n~o,,
T E.
we prove that for every
~ >0
11 v ~ l l ~
*
there is a constant C(~) > 0
such that
(lO.87
"- ~11-11~,
Assume the contrary. = l~o'2(G),
c(~)~llo,
Then there is an
such that
U eW~o'2(G)
for every
s o > 0 and a sequence
llUkIlm,2 = I and
(Uk) c
IIVuk llm,2 > e o + k llUklIO,2.
This implies
(10.97 Since
~.~o,~ < K-~IIV~.II~,.
(Uk) is bounded in
converges weakly theorem,
that
This implies diction to
to
-- ~-~llVll~,zll~@l~,=
~o'2(G),
there is a subsequence
u o e ~o'2(G).
(Uk,) converges
in
u O = O. But then
Further,
Choose now
Cl e = - ~ . Then
(10.9)
-* V u O
IIV Ukllm, 2 > e O • Therefore,
(k)~)
(Uk,) which
we may assume by Rellich's
La(G). By
V Uk,
~o
llUk,llL2(G) -*
= 0, which
0.
is a contra-
(10.8) holds.
we get from
(I0.7) and (10.8)
what gives
(lO. lO)
~ U~II~,,
-
IIT~II~.,
*
c'll~llo, for every
With the same arguments we prove the analogous Therefore,
Theorem
10.1 applies,
< ~ , ~(T)
= ~(T*) ~ , R(~)
and we get
= N ( T ) ~.
inequality
UE~O'2(G). for
T*.
dim N(T) < ~ , dim N(T*) <
159 It remains to prove (ii)
Let
T*~
(Io.~1)
dim N(T) = dim N(T*).
for every
==
T*~,
%*
u.
:=
T~
u E ~o'2(G), where the dash means the complex conjugate. The
following proof is essentially based on tMe property, that a completely continuous linear operator, which Let
TE - ~ i s
is proved as follows.
u,~ ~ ~o'2(G) ; then
Therefore
Let
u,@ E C o~ (G ) .
Then, by partial integration
and
After substracting both equations and further partial integrations we get
(~r~
~)o
- (
~
~
~=~)o
=
170 If we observe
I~-YI
< m - I, I ~ - ~ I
rule with a constant -
< m - I, then we get by Leibniz's
C' =C'(n,m,a~B)>
C'll~ll~,~ll~ll~,~
0 !
. By ( l o . 1 2 )
( ~
~
this
implies
~,~ ~)o -
~-
( ~ , ~ ) o l
with
C">
0
which gives
(lO 44)
T~II~,,
II T, ~ -
By Rellich's theorem, V*
~-
(TE - TE* ) is completely continuous.~ince
are completely continuous,
is completely continuous on
(iii)
<'2(G)
T* u = g
N(T) = [0}
dim N(T*) = O. By part
implies R(T*) = W~o'2(G).
has for every
g ¢ W~o'2(G) a solution
which is uniquely determined if we additional require T*IN(T*)A : N ( T * ) & - +
= (~I~(~)~
wm'2(G)o
has a continuous inverse. Let
for every
u e N(T*) J"
for every
g e W~o'2(G) .
(10.16)
T~
%
--
Let
(10.17)
u c N(T*) i . So,
? Then u.
~;=
T~
and
.
Suppose, dim N(T) = O. We will prove
the equation
V
(V - V*) is it too. Therefore,
({) of the proof, the assumption Therefore,
c" 11~ 1 1 ~ , ~
for every g ~ o ' 2 ( o ) .
and
171
Then
(10.16) and (10.17) imply
TT
*
=
La
for every
u ¢ N
=
~
for every
g
We will show that for e v e r y
g e ~'2(G)
and
(IO.18)
T u
= g. B y ( 1 0 , 1 5 )
this
is
by (10.18) we have for
T~
(lO.19)
u ~
Since
and
T g
<
Then
equivalent
>
and
to
=
V : ~-*
N
Tu
T'~u
-
implies (I-V)v
h ~ ~
a
v = 0 .
m W~
<
g e ~ '2(G),
= 0 <
v = 0 .
So
But let >
v e N
v = - TKV
T maps
N on
(iv)
By part
Suppose that
lity let us assume d* = dim N(T*) and let H :
~,2
m~
*
>
(I-V)
u
=
(I - V) u
u c ~, if and such that
Further,
Tg
Since
(a) of the proof,
=
O
= K v < ,>Tv (10.15) ~ c wm'2(G) O
= h
has
only if ( I - V ) ~ = O
(I-V)v
O
R(T) = ~'~o'2(G) .
Therefore,
this is an equation of Fred-
< > T-~v (10.18)
;~'t(G).
+ g.
is completely continuous.
=g
unique solution
Ku
the equivalence
holm's type in the Hilbert space N. So, equation for every
u e N such that
= -
g e ~'2(G)
~
g ~o'2(G):
¢ N, for every
there is a
O
u e N
&
V := - T K .
Let for
=
~o'2(G)
. Then =
0
<
we h a v e p r o v e d '
this implies
N(T*) =
[O
dim N(T) ~ dim N(T*). Without loss of genera-
0 < dim N(T) < dim N(T*).
and let
[fl, .... fd ]
Let
be an orthonormal basis in
[gl,...,gd,] be an orthonormal basis in (G) -* ~o'2(G)
d := dim N(T), let
N(T*).
N(T)
Let
be defined b y
cL (lO.2O)
N~
The operator
,=
H
is completely continuous.
Therefore~the
billnear form
>
172
B[Y,#]
:=
Lemma
~[~,~]
10.2. Let
u e ~o'2(G)
+
((V-H) ~,#)m
Zu
:= T u
such that
Zu
satisfies
- Hu,
= 0 .
then
all a s s u m p t i o n s
B[u,$]
= (Zu,~)m
of
" Let
Then
a.
(lO.21)
= i=i
that is, u
Tu
¢ N(T*)
has the r e p r e s e n t a t i o n
with
So T u = O and therefore u g N(T). d = ~--- ( f j u ) m fj . Multiplying (10.21) j=1
= R(T) ± . u
g j, we get
But this means this p r o o f
u =
O . Therefore,
dim N(Z*) = O .
Z*v which
gives
N(Z*)
=
=
T*,,
Z* gd+1 =
(O]
dim N(Z) = 0
and by part
(ill) of
On the o t h e r h a n d &
-
y gd+1 +
O , but
O.
This is a c o n t r a d i c t i o n
to
.
q. e. d.
10.3
Lemma
Assume (I)
that a s s u m p t i o n s
and that furthermore
(2)
that
~G
I < p,q < ~
(I) and
(2) of Lemma
10.2
are satisfied
~ C m+1, are real numbers with
~I
+ I~= I
.
Then for the subspaces
N(Tp)
:=
[u ~ ~'~o'P(G)
: B[u,*]
= 0
for every
$ ~ l~o'q(G)
)
N(Tq)
:=
(v e ~oo'q(G)
: B[v,~]
= 0
for every
• ¢ ~,~'P(G)
]
and
follows
dim N(Tp) = dim N(Tq)
are defined
according
< ~ . Further,
to Lemma 5.2
such that
if the o p e r a t o r s
Tp , Tq
173
B[W,$] then
=
(TpW,$)m
=
(W, T q $ ) m
for
(W,$) ~oo'P(G) XW~o'q(G)~
Fredholm's alternative holds for the problems TpU
=
f
, where
u,f ¢ Wm'P(G)o
TqV
=
g
, where
v,g c wm'q(G)o "
and
Proof: Since
B
is a
uniformly
elliptic,
regular
we conclude with the aid of Theorem 6.3
Dirichlet bilinear form,
that all assumptions of Theo-
rem 10.1 are satisfied. From this follows, the only one prove is
dim N(Tp)
=
dim N(Tq).
we have to
To prove this, we use our dlffe-
rentiability theorems and our knowledge in the case
p = q = 2. For this
purpose, let B[s,$] = 0
for every
$ ~ ~,~'2(G) }
B[~,t] = 0
for every
W ¢ wm'2(G)
0
and N(T2*) :=
It ~ W~o'2(G)
Our aim is to prove N(T2)
:
}.
0
= N(Tp)
When we have done this, the r e s u l ~ o f
and
N(T2*) = N(Tq)
Lemma 10.2 trivially imply
dim N(Tp) = dim N(Tq). Without loss of generality (i)
If
~ ~o'q(G)
we may assume
u c N(Tp), then and by continuity,
2 ~p < ~ .
u ¢ ~.~o'2(G) and for every
B[u,$ ]= 0
for every
$ ~ ~oo'2(G), which implies
N(Tp) c N(T2). (ii) for every
Let
s ¢ N(T2), what means
~ c ~.~o'2(G). All assumptions
satisfied, which gives shov-theorem and,
pt =
s ¢ wm+I'2(G)
we conclude
s c W~o'2(G)
and
B[s,~] = 0
of Theorem 9.11 with
j = I are
0 P~o'2(G) . By the S o b o l e v - K o n d r a -
s ~ ~o'P'(G) where p.
is arbitrary if n = 2 ,
n if n > 2 . From this we get B[s,~] = 0 for every n-2' ~ c WOm'q'(G), where I
174
induction. After a finite number of steps we arrive at which gives Combining
s c ~,~o'P(G),
N(Tp) c N(T2). (i) and (ii) proves
" N(Tq) = N(T2 @) "
N(Tp) = N(T2)0 The proof of
runs simi!iar. q.e.d.
Lemma 10. 4 Assume (I)
that m > 1
is an integer,
I
are real numbers with
1+t=1, P q
(2)
that
G c ]R n (n>2)
is a bounded open set with boundary
G ¢ C m+1
(3)
that
is a, uniformly elliptic Dirichlet bilinear form with coefficients
aaqBcCm(G)
for tc~iim, I # t i
Then, for the operators
TM-
T r : Wom'r(G ) -* ~o'r(G)
fined according to Lemma 5.2 by
( r = p or q) de-
B[~,@] = (TpT,{) m = (W, Tq@) m
for
~,@ ~ Co(G)a Fredholm's alternative holds.
Proof: Let
BE[~,@ ] :=
~--
(a~# O~,O#@) o
B E satisfies the assu~mption~of BE[~,~ ] = (Ts, pW,¢)m
for
Lemma 10.2. Let TE, p be defined by
(~,{) c Wom'P(G) x ~@'q(G)
and Nikolski's theorems
(see e.g.
Wp : W~o'P(G ) -~ ~'P(G)o
being
continuous
(10.22)
operator
~ E,~,
=
for ~,@ ¢ Co(G ) . Then
Vp :
[31]
there is an operator
continuously invertible,andacompletely
Wmo'P(G)
W p ÷ V#
) ,
. By Lemma 10.3
-~ -owm'P(G) such that
]75 Further, let
Since
a~
c cm(~)
for I~I < m,
integrations with a constant
(so.23)
{]1
---
I~I
< m
we get after suitable partial
y>0
II"t'll..,_,,,. II e
for
(~,~) c ~ ' P ( G ) X ~,~o'q(G)
by
B'[W,~] = (Tp' ~'$)m
4.4
we get with respect of (10.23) the estimate IIT~Wllm,p ! Y' llWllm_S,p
for
W ¢ ~o'P(G).
=
is defined according to I,emm& 5.2
for (W,~) ~ ~'P(G)o ¢ ~o 'q(G)' by
Therefore, by Rellich's theorem,
continuous. By (10.22) Tp
If ~
Wp
+
T5
Theorem
is completely
we have (Vp + TS)
The operator in brackets is completely continuous, % invertible. Therefore, by
is continuously
Nikolski's theorems, Fredholm's alternative
holds for the operators Tp, Tq , where we have again observed correspondence between
Tp
and
Tq
(Compare
the
Remark 5.3) q. e. d.
Lemma 10.5 Assume (I)
that assumptions
(2)
that
B[Y,@] :=
(1) , (~) of Lemma 10.4 are satisfied, 7-
( a ~ DG~ , D ~ @ )o
is a
uniformly
l l<m, l l<m elliptic, regular Dirichlet bilinear form.
Then the assertion of Lemma 10.4
holds.
Proof: First we want to show that
B
may be approximated by bilinear
forms satisfying the assumptions of Lemma 10.4 . The technique we apply
176
is similiar to the one we used in the proof of Theorem 6.3 , but in the case under consideration a further difficulty arises: We have to smooth the coefficients of
B
in such a manner,
forms are uniformly elliptic.
For this purpose we have two types of ar-
guments whoses resources are identical. I~I = 161 = m
and
~
there is for every variables Second: ~a6e
is compact, I~I = 161 = m
C ° ( • n)
we smooth
First:
Since
as6 ~ C°(~) for
by Weierstrass's
approximation theorem,
a sequence a ~ )
of polynomials
Xl,...,x n , approximating Since
that the resulting billnear
as6
uniformly in
G.
a~6 c C°(F), by Titze's extension theorem, such that
a~61~
=
as6
for
laI=161=m.
there are This
a~
with the aid of Friedrich's mollifier and get a sequence
~a~e) ~ C~(G)
converging uniformly to
a~6
is the method of elliptic continuation
on
~ . Essentially this
of elliptic bilinear forms and
operators to sets larger then the original set of definition. see in the following parts of the proof turbation arguments,
As we will
with the aid of certain per-
it is in most problems unessential,
differentlability properties have
in the
are preserved or not.
if the original
In both cases we
to check that the resulting bilinear form having the smoothed
coefficients
in the principal part
( that is the part with
is uniformly elliptic. By continuity holds with E replace~by
~ 2
for the form wlth coefficients
the original sufficiently enough. condition"
mollifier.
1.6
approximating "root
of a sufficiently high rate of
But this follows from the continuous dependence of the
roots of polynomial The remaining
estimate
It remains to show, that the
still holds for all forms
approximation.
it is trivial that
laI=161=m),
(1.7) from its coefficients
L~(G) - coefficientswe
Denoting with
(see Appendix
I).
smooth with the aid of Friedrich's
B (E) the bilinear form with smoothed coef-
ficients, we derive from the fact of uniform approximation of the coefficients with
I~I = 161 = m ,
of Theorem 6.3, formulas
and
for the same reasons as in the proof
(6.1R) - (6.23)
177
(~o.24) t2,[~,{1f o r ev ery p a i r Since
a~
~')[~,¢'].1
(~,{)
~ L~(G)
"-- c(~)II"Vll.,,,,~, lltStl.,,,,~
~ ~o'P(G) for
x
~o'q(G),
I~I <m, 161 <_m,
where
(I 0.24) implies
(lo.25)
I~,"[~, ~&l ~- ~ I1~1I~,~11{11~,~
(W,$)
Since
c ~o'P(G)
B (s)
× ~o'q(G)
and
s -~ 0.
and by the smoothing method
applied,
for
C(s) -~ 0 f o r
every
0 < s < s°
is uniformly elliptic for
.
O< s < s o , Lemma 10.4
ap-
plies to it. Now we use a well known perturbation argument (see e.g. [26], Satz 5.8
or [32] ).
operators connected with ~(Tp)
Tp and Tp(s)
Let B
and
B (s)
respectively
be the
by Lemma 5.2 and let
:= dim N(Tp) - dim N(Tq) ,
~(T (s)) := dim N(T p(s))-dim N(Tq(s))
for
By Theorem 10.1 , the numbers ~(Tp), ~(T; s))
0 < s < s° . (O<S<So)
Therefore by the theorem cited above, there is a number
are finite. y(Tp) > O, such
that for every continuous linear operator Bp : ~o'P(G) -~ 1'~o'P(G) with llBpllm,p < y(Tp)
we have with
~(Tp + Bp) = ~ ( T p ) . llTp - TpCS)llm,p
~(Tp +Bp) := dim N(Tp + Bp)
By Lemma 5 . 2 and ( 1 0 . 2 4 ) we have
--< C(e~)K . Then
IITp - T(S)llm, p ~< y(Tp) p
for
there is an
0 < s < sz.
0 < sz _< so Since
~(Tp( s ) )
= 0
and t h e r e f o r e
such that
T; s) = T p + ( T ; s) - Tp)
we conclude w i t h t h e theorem mentioned above and g e t By Lemma 10.4,
dim N(Tq+Bq):
~(Tp) = ~ ( T ( S ) ) .
0 = ~(Tp) = dim N(Tp) -
- dim N(Tq) . q.e.d.
178
Lemma 10.6 Assume (I)
that m > I is an integer and that
with --I + _I = I P q ' (2) that G ~ ]R n ( n ~ 2 ) G
I
~
are real numbers
is a bounded open set with boundary
~ Cm , (3)
that
form of order Then,
B
is a
m
in
G
uniformly
elliptic,
with coefficients
the assertion of
Lemma
regular Dirichlet
bilinear
am# ¢ Cm(G).
I0.~ holds.
Proof: Let
BE[~,~]
:=
7-
Then, B E satis-
D~ ~'D~)o
(a~
l~l=l~l--m fies assumption B'
[~,~] :=
(2) of Lemma 10.2. Let
7(a~# I= t<_m, I#i<m
D~
~ 'D# ¢ )o
for
• ,¢ ~ c o(G)
.
1~t+t~t<_2m-1 After a partial integration we get
IB'[~,~]
<
c
[IVttm_l,21J{llm, 2
Therefore,
by Lemma 5.2
continuous
linear operator
B'[~,$]= Since
(V~,$)m
which gives
W,~ ¢
Co(G) .
and Rellich' s theorem there is a completely V : ~o'2(G)
for
-~ }~o'2(G)
such
that
~,~ ¢ t~o'2(G).
B[~,~] = BE[~,~ ] + B'[~,{]
satisfied,
for
,
all assumptions
of Lemma
dim N(T2) = dim N(T2*) < ~ , where
10.2
are
17'9
Without
loss of generality,
we may assume
2 !p < ~
and
I < q !2.
If
N(Tp) , N(Tq)
are defined according to Lemma 10.3, then we will prove:
N(Tp) = N(T2),
N(Tq) = N(T2)*
which immediately gives
dim N(Tp)
=
dim N(Tq) < ~ . Since p ~ 2
we immediately
let u E N(T2).
get
If we can prove
N(Tp) ~ N(T2).
u c },~o'P(G), then from (10.26) follows
u ¢ N(Tp). We will show this locally. proof of Theorem 7.3.
Let x o e ~
x O , such that ~Xo N ~
is mapped on
transformed a V
e
'2(G)
(I0.27)
t{ ]
R' < R and
R',R From
Cm+1
=
~ e Co(SR).
O
R > O. Let
the image of
Since
u g N(T2)
of
B be the
u. Then
billnear form.
f o r every
( ~ Co( ~ S R' )
and let
Uxo be a neighborhood
H R for some
B
is
Further, we get
@ e Co(HR) .
HR,,R chosen such that be
HR, c HR,,R c H R such that
~=I
with
for
Ixli ~
.
(10.27) we get
(1o.28)
where
=
#
arguments
is defined analogous as on page
(1o.29) where
and let
regular Dirichlet
for every
~[v
Let
The proof is similiar to the
bilinear form and let v be
uniformly elliptic,
On the other hand,
127
to formula
(7.15), p. 127. With the same
we prove
I P' = n -2n2
such that
for every
for +
Now we observe,
~
n>2 =
that in
and
p' > p
arbitrary for
=
dim N(Tp, )
Co( R,,R )
n=2,
I (q' <
I. HR, R'
all assumptions
of Lemma 10.5 are satis-
fied. This gives dp.
• ~
=
dim N(Tq, )
(
~
and
180
d2 Since
=
dim N(T2)
p ' > 2, we h a v e
N(~2*) S N(~q,) subspace of
=
dim N(~2*)
1
<2
<
and therefore
Suppose, N(Tp,) ~ N(~2)
•
=
N (Ta), this would imply
• N(Tp,)
•
~ N({a)
,
Since N(~p,) is a closed
dp , = dim N(Tp,)
< dim N(~2) = d2.
On the other hand, de = dim N(T2*) ! dim N(Tq,) = dp, , what is a contradiction. Therefore
N(p,) Let
Then, by
)
,)
,
=
.
?~O'q' (HR, R' )
for @ ¢
(1o.3o)
=
T({)
,=
(10.29), F
is continuous on
diately from the definition of is continuous.
W m'q' (HR, R, ). As follows immeF to W m, o 2 (I~R' R' )
~, the restriction of
Since the equation
[w,{] has the solution Fbjom, 2(~R,R, ) (t)
=-
({)
":FI
w = (v = 0
for every
• ¢
~o'2(G)
¢ ~o '2 (HR, ~ R,), by Theorem 1 0 . 1
must be
for every
N(T2 @) =
t ¢ N(T2@).
Since
N
=
N(Tq,) , this means
F(t) = 0
Again by Theorem 10.1, there is a
(10.31)
~KS, { ]
Since p' > 2 ,
= ]:({)
s c
for every s e
t c N(Tq,)
.
'P'(HR, R, ) such that
for every • ~ ~o "q'(HR,~ R' ) "
R)
and we conclude from (10.28),(10.30) and
(10.317: for every that is (~.v - s.) ¢ N(~2)
. The fact that
Y c W m'2(G) , N(~2) = N(~p,) c Wom,p'(HR,R, )'
181
and
that
s e ~o'P'(HR, R,), gives
{re
we iterate this arguments r if p' < p : ~z = 1 in a neighborhood
of
R' . Ixl _< --~
W om'p' (HR, ~ R' ) " Let
As often done,
~z e C ~ ( S ~ ), such that
Then,
~i v
Wr~o ' p ' (~, ~ ) and
E
~
we ma/<e the same procedure will find that
~v
¢ ~¢~o'P(HRo ) j for every
R o < R. Transforming (~u)
x ° eG
S R e C~
and
for every
p=P(Xo)
for some
and we may apply the same kind of arguments
Therefore
N(T2) = N(Tp).
=
> 0 such that
h ¢ C~([IX-Xol < p]).
S R := [IX-Xol < R]CC G
Pc ~ ~; u E ~ ' P ( S p o ) .
N(Tq)
~ e C~(SRo ) , with a
back, we see that there is a
c ~ o ' P ( G n [IX-Xol < p]) If
a
in H~.~ After a finite number of steps we
Now a compactness
R > O, then as above to get
argument gives u C~o'P(G ).
In quite the same manner we prove
N(T~*) q. e. d.
Theorem
10.7(main
existence theorem)
Assume (I)
with
that m ~ 1
l+l= P
q that
(2) ~G
is an integer and that
I
~
are real numbers
I. G c iR n (n ~ 2 )
is a bounded open set with boundary
eCm , (3)
form of order Then,
(~o.32)
B
that m
is a in
uniformly
elliptic
regular Dirichlet bilinear
G.
if
N(T,~)
,=
[ U. ~ W ~ ' e ( G )
N (m 0
• --~
[ V e W~'9(@) ,
" B[~,@]
= Ofor
every • ~Co(G)]
and (10.33)
we have equation
~[~,v]
dim N(Tp) = dim N(Tq) < ~ , and for every
= oforevery
~ ~C~(G)]
F ~ Wom'q(G) * the
182
(10.3#)
B[
~
~ ]
=
V(~)
for every
~ ~ wm'q(G) o
has a solution v ¢ N(Tq). h ~
Further,
Wr~o'q(G)
(10.35)
~ c W~o'P(G ) given
if and only if
---- H ( ~ )
H(u) = 0
there is a solution
( or
for every
W ¢ Wm'P(G)
u ¢ N(Tp).
In the case
(10.34) and (10.35) are
"right sides".
u ¢ N(Tp,)
where
H c ~o'P(G)*,
for every
= dim N(Tq) = O, equations
then
F(v) = 0 for every
of
]~[%, h i
for arbitrary
if and only if
Further,
uniquely solvable
if u ¢ N(Tp)
v E N(Tq,))for every
I
N(Tp,) , N(Tq,) are defined analogous
dim N ( T p ) =
(or
v ¢ N(Tq)),
(or I
(10.32),(10.33)
),
.
Proof: The first part of the theorem states that Fredholm's alternative holds for the problems under considerations.
But this result we derive
from Lemma 10.6 with quite the same smoothing procedure of the coefficients and the same perturbation arguments,
as we derived Lemma 10.5
from Lemma 10.3. The second part of the theorem is proved with the same arguments as used in the proof of Lemma 10.6: for some
p with
I
. Let I (p' ( ~
Assume that
be given. If
u ¢ N(Tp)
p' < p
,
then
u ¢ ~p'P'(G) by HGlder's inequality and the definition of l~o'P' (G) in this case, u c N(Tp,) follows from (10.32). If
p'>p
(and there-
fore q' (q), then the first part of the theorem applies to (p,q)
and
(pT,q'). This gives
%
:
dim N(TI~ ) -- dim N(Tq)
<
~
%,
=
dim N(Tp,) = dim N(Tq,)
<
~ .
and
N(Tp, ) is a closed subspace of N(Tp). Suppose dp,
= dim N(Tp, ) < dim N(Tp) = dp .
But then
N(Tp, ) ~
N(Tp) . Then,
183
dp, = dim N(Tp,) < dim N(Tp) = dp = dim N(Tq) ! dim N(Tq,)
= dp,
what is a contradiction. q. e. d.
As an immediate consequence we prove Theorem 10.8 As sume (I)
that the assumptions of Theorem 10.7
(2)
that
Cp : Wm'P(G)o
-~ "v~'P(G)o
are satisfied,
is a completely continuous
linear operator. Then Fredholm's alternative holds for the problems ]5[u%~] where
(Cp~l~)~,
=
u c ~,~o'P(G) , F ~ ~ ' q ( G ) * ]~[~V]
where
+
+
V c ~o'q(G),
for every
• ~ Co(G)
for every
v ~ Co(G)
and
(CpV, v ) ~
H-~ ~n'P(G)*o
~(~)
~
G(~)
"
Proof: Let
B[u,~] = (TpU,~) m
for (u,~) e Wm'P(G)
× ~'q(G).
By Theorem
10.7, for the operators Tp, Tq Fredholm's alternative holds. From Nikolski's theorems follows that
+ V , where Wp has a bounded P P inverse in ~,~o'P(G) , and Vp is completely continuous. Then, again by
Niko!ski's theorems, alternative
Tp = W
for the operator
Tp +c P = w P + (vp + c P )
the
is true. q. e. d.
Remark I0.9 (i)
Let
B[u,$] = BE[U,@] + Bc[U,$]
a continuous bilinear form on
Wm'P(G)
be (for some
x Wom'q(G)
p with I
with the properties:
184
B E is uniformly elliptic and regular; rator Cp, corresponding with continuous.
Then Theorem
of Definition 6.2 ties of
and the ope-
B C according to Lemma 5.2, is completely
10.8
Observe that in this case
B C is continuous,
assures that Fredholm's
alternative holds.
B itself needs not to be regular in the sense
(compare Remark 6.4). Further,
N(Tp) stated in Theorem
the regularity proper-
10.7 need not to hold. Therefore we
considered this case separately. (ii) a
Consider the case
uniformly elliptic
is continuous satisfying
regular
on ~ ' P ( G )
x ~o'q(G)
such that the operators
= (U, C q ~ ) m
are completely continuous
may be rather arbitrary:
in the
B'
Cp, Cq
for (u,~) ~ ?~o'P(G) x respective
spaces.
Then
We have only to handle with linear integro-
differential b i l i n e a r forms of m o r e
general type. This is a second
reason~treat the case considered in T h e o r e m
Theorem
B E is
Dirichlet biiinear form, and where
B'[u,~] = ( C p U , ~ ) m
x %~o'q(G) B'
B[u,~] = BE[U,~] + B'[u,~] , where
10.8 .
IO. I0
Assume (I)
that m > I and k > 0 are integers,
(2)
that
G c ]R n is a bounded open set with boundary
(3)
that
L = ~ - a s (.) D s
coefficients Then,
(i)
is uniformly elliptic in
8G G
~ C 2re+k,
and has
a s ¢ CI~I+k(G).--
for every
p
with
I
the following statements are true:
If N(L)
:=
[s ~ Wom'P(G) N w2m+k'PcG)
: L s =
0
]
N(L*):=
{t c W 'q(G) 0 w2m+k'q(G) : L*t = O } , o then dim N(L) = dim N(L*) < ~ . Further, s c N(L) implies s e Wom'p'(G)
for N(L*)
0 W 2m+k'p'(G)
for every 1
185 (ii)
For
f ¢ wk'P(G)
u ¢ W~o'P(G ) 0 w2m+k'P(G) For
a
g c wk'q(G)
N w2m+k'q(G) case
the equation
Lu
if and only if
the equation
= f
has a solution
(f,t)o = 0 for every t cN(L*)
L*v
= g
if and only if (g,S)o = 0
has a solution v ~,~o'q(G)0
for every
s e N(L). In the
dim N(L) = dim N(L*) = 0 the equations are uniquely solvable for
arbltrary
f ~ ~'P(G)
and
g ~ wk'q(G)
(iii) There is a constant solutions
respectively.
y = y(n,m,p,k,L,G)
> 0 such that the
satisfy
tlult2m+k, p <__ ~'(llr, u llk, p + tlullo, p)
for
u eWom'P(G) 0w2m+k'P(G)
Ilvtl2m+k,q <_ Y(llL*vtlk, q + tlVllo, q)
for
v e~o'q(G) 0w2m+k'q(G)
Proof: Let
BL, BL. be constructed
relative to
vely (compare Remark 9.13). Then for every
(10.36)
Bz[~,~]
Therefore,
Lu
= f with
=
(L,,,{)o
u e wm'P(G) o
L
and
L*
respecti-
u C~o'P(G ) 0w2m+k'P(G)
forevery 0 w2m+k'P(6)
~CCo(G)
and
.
f e wk'P(G)
im-
plies for every
Conversely,
if (10.37) is satisfied for
by Theorem 9.14
u ~ Wm'P(G)
under these conditions BL[U'~] = (f'$)o
for every
if u ~ Wom'P(G) n w2m+k'P(G)
u c Wm'P(G) and
0 w2m+k'P(G)
the equivalence: ~ ¢ Co(G )
• ~ C:(G) .
and
Lu
u ¢ ~o'P(G) with a
f ¢ wk'P(G),
= f . So we have satisfies
f ~wk'P(G)
. The analogue is true for
if and only
BL..
If we
observe BL.[V,~] = BL[W,v] we conclude N(Tp) = N(L), N ( L * ) = N ( T q ) , we have used the notation of Theorem 10.7. But from this theorem and (ii) follow,
(iii) was proved in Theorem 9.15. q.e.d.
where (i)
186 § 11.
Further
regularity
theorems
In this section we prove then the results cates,
if
the power p
of the function
fore e.g. the following tic regular Dirichlet
bilinear of
(Theorem
arises:
in Theorem
sharper.
11.7).
Our proofs
of § 10. Therefore,
p>p~.
be a in
Is then
we may derive
G, let
a
• ~ Co(G ), where
u ¢ Wm'P(G) o
11.2.
11.10),
? We
The following these theo-
but our theorems
on the existence
But as is immediately
local
ellipu c
is an approximation
are m a i n l y based
indi-
There-
uniformly
Lemma in L p. Partially,
consequence
type
of the smoothness
for every
(see Remark
Again we see the correlation
Theorem
= F(~)
they are globally.
cut - off procedure,
B
I I. I and Theorem
[2]
A simple
Let
form of order m
B[u,~]
rems were proved by S.Agmon
and Kondrashov
and its lower order derivatives.
are of the type of Weyl's
are a little
of a more general
is a measure
~I + ~I = I , and
with
treat this problems theorems
u
question
W~o'P' (G) be a solution F ~ ~oo' q(G)
results
of § 9. As the theorem of Sobolev
u c W~o'P(G),
properties
regularity
theorem
theorems seen by
results.
of existence
and regularity
theorems.
11.1
As sume (I)
that m > 1
and j > O
are integers
with j < m
and that
I
are real numbers with (2) ~G
that
! + _I = I , P q G c ~n ( n > 2 ) is a bounded open set with boundary
c C re+j, (3)
bilinear
that B
is a
form of order
uniformly m
in
that
F ¢ ~¢u'J'q(G)* o
(5)
that
u ~ Wm'P'(G) o
~B [~., ~ ] ~- ~: ( ~ )
j - smooth
regular Dirichlet
G ,
(4)
(11.1.)
elliptic,
, where
I
f o r every
, such that
@ e Co(G )
187 Then
u c ~-W~o'P(G) n m-~'+J'P(G)
and there is a constant y > 0 such
that c11.2)
"
Proof: If p' ~ p p'
If I
II~llm_j,q ~ to
then the assertion is trivial by Theorem 9.11. Assume now that ~ , + ~ , = I , then
C(G)ll~llm_j,q ,
W~o-JJqiG )
for
on W~oJq'(G ).
B[u,~] = Fq,(~)
for every
By Theorem 10.7
this implies
Since
N(Tq,) = N(Tq)
F(v) = 0
for every
and
By
~ ¢ wm'q'(G)o
F
Since
• c C~o(G), the restriction
is a continuous linear functional
therefore it is continuous
q
Fq,(V)
Fq t of F
on %~o-j'q' (G)
assumption
and
(5) the equation
has a solution u c w~'P'(G).
=
0
for every
is continuous
v E N(Tq). Therefore,
on
v ~ N(Tq,).
w~Jq(G)
there is a
we have
w c w~'P(G)
such
that
(11.3)
foreve
=
•
By Theorem 9.11, w E %~o'P(G) O wm+j'P(G). B[u-w,~] = 0
for every
N(Tp,) =N(Tp), Since u
=
~
.
From (11.1) and (11.3) follows
• ¢ W~o'q'(G), that is,
h E N(Tp) and,
by
u-w
=: h c N(Tp,).Since
Theorem 9.11, h c W~oJP(G ) O%~+J'P(G).
belongs to the same space, we get
h + w
E
of Theorem 9.11
~oo'P(G) n
wm+j'P(G)
.
(11.2) is atrivial
consequence
. q e. d.
Theorem 11.2 Assume (I)
that m > I and k > 0 are integers and that I < p , q < ~
numbers with
--I + _I = I , P q
are real
188 (2) G
that
G c IR n ( n ~ 2 )
is a bounded open set with boundary
c C 2m+k , (3)
that B
is a
uniformly elliptic,
richlet bilinear form of order (4)
that
f e wk'P(G)
(5)
that
u c ~o'P'(G),
(11.4)
] ~ [ u., { ]
Then
=
m
G ,
, with
(~,{)o
u ¢ ~o'P(G)
in
(m+k) - smooth regular Di-
for
0 w2m+k'P(G)
1
every •
~ C~(G)
and there
.
is a constant
y>O
such that
I/utl2m+k, p !
Y (ltfllk, p
+
lluIILP(G )
)
•
Proof: Let
F(@)
:= (f'¢)o for ~ c C~(G),
Then,
F is continuous on
Lq(G). Theorem 11. I applies with j =m, which yields N w2m'P(G).
u ~ ~'~'Po(G) 0
The second part of the theorem is a trivial consequence
of Theorem 9.12
. q. e. d.
1 1.3
Lemma
Assume (I) with
that m>_1 is an integer and that I < p , q
i + i _-
P (2)
q that
(3)
that
< ~
are real numbers
I G c ~R n L
=
is a bounded open set wit boundary
_____ a s D s
is uniformly elliptic in
8G
e C 2m,
G and has
Is [<_2m coefficients (4)
(11.5)
a s cL~(G)
that
I({,L
for
f e LI(G)
)ol
ISI
<m
and
a s E C Isl-m(G)
for
[sl > m
and that there is a C > 0 such that
189
for every
_Tg~.,p(o) rn
u E
Then, f ¢ Lq(G) pendent of
C
and
fl w2m'P(G)--
with
Lu
¢ L~(G).
and there is aconstant y = y(n,m,p,L,G)>0 indef , such that
Proof: At first
we will construct an operator
w~'P(G) fl w2m'P(G)
on
LP(G).
By Theorem 10.7 , ...
If1' .... fd }
of order
m
satisfying
for every
we have
N(Tp)
=
N(T2)
,
N(Tq)
= N(T2 *)
denotes the operators corresponding with
be a basis in
the skalar product in in
BL
u ~ ~,~'P(G) 0 w2m'P(G) and every o (compare Remark 9.13). By Remark 9.13, B L is m - smooth.
~ C~(G)
Tp , Ta
which maps
For this aim consider the uniformly
elliptic regular Dirichlet bilinear form BL[U,~ ] = (Lu, ~ )o
~
N(Ta)
BL .
, where Let
being orthonormal with respect to
La(G), and let
[gl .... 'gd }
N(Ta*). By Theorem 10.7 , fi' gi c w~'r(G)
be such a basis
for I < r < ~
. Therefore,
it makes sense to define d
hcKU,~] := Since Cp, Cq
- ( ~- (fi,U)o gi, e)o
for
(u,~) c ~'P(a)
i=I d IBe[U,@]l ! ( ~ IIfiIIo,q IIgiIlo,p) IIUlJo,p II@IIo,q
,
~'q(G) " × ~o the operators
corresponding with B C are completely continuous in the respecti-
ve spaces. Therefore, the bilinear form
(I~.7)
~[~-,~1
:=
]~[~,~]
for (u,~) ¢ ~o'P(G) x ~o'qCO) 10.8
~
B~[~,~]
satisfies the ass~ptions
of Theorem
which states, that Fredholm's alternative applies to
~Tp, ~q be the operators corresponding with
B
B . Let
by Lentma 5.2. Our aim
19o
is to show: Assume u ~ N
N(%)
I
(¥p).
= N(Tq) = [O} . For this we first show N(Tp) = N(~2). Then
N(32) c N(Tp). To show the converse,
let
• ~ C O~(G)
Then for
11
I:%[,-,., e:li
,. ~ 1({~,,-,.)o11('~,~),1 -'-
(~ 11~llo,, II ~11o. ll~llo,Ollello,~ By Theorem 11.1 we c o n c l u d e N(~p) = N(T2) , t h a t
is
u ~ ~o'2(G)
N({p)
: N(T2)
n w2m'2(G) .
, w h i c h proveg
The case
2
~
is trea-
ted analogously. Let
u ~ N(Tp) =
N(~2).
Then
(11.8)
for every
@ e Wom'2(G).
o:
(u, 7 ~ * g j ) m
Put
• = gj , j=1,...,d
-- ( ~ u ,
Then we conclude from (11.8) that is , u ¢ N(T2
gj)
:
(T2 ~ d ~ ) m
(fj,
. By (11.8)
u) o
.
= 0 for every
~ ~ wm'2(G)'o
) . Therefore u =i~L_1(fi,U)ofi = O. From this we con-
clude
N(Tp) = N(T2) = [0]
and by Fredholm's
Since
~p
inverse,
has a continuous
alternative
there is a constant
N(Tq) = [0}. y' > 0 such
that
(11.9)
Ilu-ll.,.,,,~ ~
"¢ ~,-,.p 1-~,[~,,~],t
for every
u g ~o'P(G)
(11.9),
llvll~,~
~' ~ l ~ [ ~ v ] ]
for every
v ~ W~o'q(G ).
For
h ~ LP(G)
lution of
~
let
~Sp
u ~ W~o'P(G )
B[u,$] = (h,$)o
for every
be
the uniquely determined so$ ¢ W~o'q(G ). By definition of
we have
~[~,~]
=
(~.
~(~,~)o~,~)o
for
every
L u
=
h
$ e ~#n'q(G). By Theorem 9 . 1 4 , u ~ W~o'P(G) O w2m'P(G) d o + 7 - (fi'U)o gi " Therefore, the operator i=I a. for ~ 6
&=& maps W~o'R(G) 0 w2m'P(G) a constant
y"
on
LP(G) . Further,
W~'P(-G')~WZ~'V'CG)
by Theorem 9.14 there is
such that
i=& From this we conclude with the help of (11.9)
For every
I c]N
hl(X) Then, the Let
we define
f
:= hI
{(x)
If(x)iq-2
if
0
If(x)l
i 1
otherwise
are bounded and measurable,
and t h e r e f o r e
h i e LP(G)
!
u I ~ W~o'P(G ) n w2m'P(G)
~ ~o'q(G). Since
Then
gi ~ W~o'r(G)
be the solution of B[Ul,~] = (hl,~)o for d Lu I = h I + ~ ( f i , U l ) o g i . for S < r < =
lev-Kondrashov-theorem
we derive with the aid of the Sobo-
gi e L~(G)" Since
L u I e L®(G) . Then, by assumption
h I e L=(G)
we conclude
(4) &
&
-~ ( ¢ * II ~ll~(~) ~
i=l
and by
(11.11) with
y = y(f
,g
I1{111o,~II~IIL~,o ,) I1=,I1~,~
,~,,', ) > 0
r ( c . fl ~ 11~.(~))II
h~llo, p
192
From the definition
b,: ")'l,o<,.~-'~l "r"c"~;q cl,
of
hI
,,
follows
li' ( c.. + ii )fE llL, _ cc.~ _ .i~{,,
and therefore
By B. Levi's theorem follows
f ~ Lq(G) and (11.6) .
q.e.d. Lemma 1 1.4 Assume (I)
that assumptions
(I) - (3) of Lemma 11.3 are satisfied,
(2)
that
and that there is a constant C > 0 such that
f ¢ L ~ (G)
(11. 2)
L,-,-)ol
for every
"
Cll ll.,,,,r
u ¢ Wm'P(G) A w2m'P(G)
with
Lu
~
L °G (G)
O
Aw2m, P(G) Then, f E ~ ' q ( G ) , (11 • 12) holds for every u ¢ W m'P(G) o o and there is a constant y -- y (n, m, p, L, G ) > 0 independent of C and f, such that
Ilfllm, p
<
_
"r ( C + IlfllT?.(a ) ) .
Proof: consider i B
By Lemma 11.3, f ¢ Lq(G). Again we corresponding with (i) u ~ ~'P(G) Co(G )
~ (see ( 1 1 . 1 0 ) ) .
First We will show that (11.12) O w2m'P(G), For such an
is dense in
llw - WlIlo,p "*
u
is satisfied for every
let
LP(G), there is a sequence
0 (1-*~). Since
there is for every
defined by (11.7)
1 an
~
w := ~ u
~ LP(G). Since
(Wl) c Co(G )
such that
maps ~o'P(G) O w2m'P(G) on LP(G),
u I ~ ~o'P(G) O w2m'P(G)
such that
~u I = w I •
193
By (11.11),
l l u - Ultl2m, p
<
y"
ltw- Wlllo, p "
O (l--®).
Since
d L u I = w I + 7 - (fi' Ul)ogi and since gi ~ L~(G)' Wl ~ L~(G)' we get i=I L u I ¢ L~(G) for every i ~ ~ • Further, IIL u I - L u Iio,p ~ o ( i - ~ )
Since
(11.12) is s~tisfied
for every
1 ¢ ~I,
(11.12) holds for
u
by
continuity. (ii)
Then,
@ e W~o'P(G ) O w2m'P(G)
for
Let
by ( 1 1 . 1 2 )
[ c . II { I1o,, >
II f111o,~ II ~,11o,~] II { I1~,~
l=i
for
every
y"
= y" (fi,gi,~)>
1~(~)1
(11.13) for every a
@ e Co(G ) . O b s e r v i n g
(11.14)
a constant
f~
~- ~',(o + ll~llv,(e))llWll~,~
. Since
v ~ ~o'q(G) ~[~,
we g e t w i t h
0 independent of C and
@ e Co(G ) . Therefore,
F ~ ~o'P(G)*
termined
(11.6)
v]
it may be extended by continuity to
N(~p) = N(Tq) = {O} , there is an uniquely desuch that
=
~-(~r)
for every
~t ¢ ~oo'P(G)
and by ( 1 1 . 9 ) ,
(11.15) Since
tlvll~,~
~-
B[~,v] = ( ~
the definition of (f - V , Z ~ )0 = 0 Wom'P(G) n w2m'P(G) f = v ¢ W~o'q(G ) .
~'~"(c
,V)o F
for every
and
(11.14)
for every on
. II{II~.~G~).
LP(G)
~ ¢ ~o'P(G) (~2,V)o
Q w2m'P(G)
= (~W,~) o
~ e Wom'P(G) n w2m'P(G). we conclude
f - v
, that is
Since
= 0,
, we get from
~
maps
that is
(11.13) follows then from (11.15). q.e.d.
194 11,5
Theorem Assume
(1)
that m >
1 and k > 0 are integers and that 1 < p , q < ~
are real
numbers with (2) G
! + ! = I , P q that G c IR n ( n ~ 2 ) is a bounded open set with boundary
e C 2m+k, (3)
that
L =
~
as Ds
is uniformly elliptic in
G
with
I sl<2m coefficients
a s e c~SI+k(G)
(4)
that
h ¢ wk'q(G) ,
(5)
that
f ¢ LI(G)
(11.16)
for every
(1~, Lu...)o
=
u ¢ W~o'P(G )
and that
(. ~, u..)o
n w2m'P(G)
with
Lu
Then, f ¢ W~o'q(G ) N w2m+k'q(G) , L*f y = y(n,m,p,L G) > 0
= h, and there is a constant
independent of h and
(11.17)
(llhll ,
~L~(G).
÷
f, such that
llfll ., o )
Proof: Since l(h,U)o I ~ llhllo,q llUIio,p ~ c Ilhllh,q llUIlm,p , the assumptions of Lemma 11.4 are satisfied. Then, f ¢ w~'q(G) . according Remark 9.13
with respect to
BL.[f,~] = ( f , L ~ ) o = (h'$)o (m+k)- smooth
for every
(compare Remark 9.13) ,
Theorem 1102, (11.17)
L*
Let
BL* be defined
.Then, by (11.16)
• ¢ C~(G).
Since
BL.
f e w~'q(G) N w2m+k'q(G)
is by
is a consequence of Theorem 11.2 and (11.6). q. e. d.
195 As an immediate
consequence we get
Corollar~ 1 1.6 Let assumptions
(I)- (3) of Theorem
11.5 be satisfied for k = O
and
let
u ~ L I(G).
L*u
Then, u E wm'q(G) N w2m'q(G) for every q with I < q < ~ and o = 0 a.e.,if and only if there is a p with I < p < ~ such that (u, L W )o
with
LW
=
0
~ L~(G)
Theorem
for every
W ¢ ~o'P(G)
.
11.7
Let m > I and k > 0 be integers set wit boundary DaU I 8G = 0 every
N w2m'P(G)
I
~ G ¢ C 2m+k.
for
and let
Then,
G ~ IR n
])~,,K,q- . , ,
I~I < m - I] is dense in
be a bounded open
[~G C~(@) ~CZ~÷x'I(~nWT"~*X'P(@):
Wm'P(G)
N w2m+k'P(G)
for
.
Proof: With respect of Theorem 9.17 it remains to prove that implies
u ~o'P(G).
Let
elliptic
in G. Let I < q < ~
U := (-i)~"~ with
~I + 1q=
By Theorem 9.17 there is a sequence
u ED m'k'p
. L is uniformly
strongly
I a n d l e t v ewm'q(G) ow2m+k'q(G).
(v~) 6 D m'k' q such that
fly - vv II2m+k' q-+ 0 (v-* ~ ). Partial integration gives
Since
Lu
E wk'P(G),
u ~ T¢~o'P(G) 0 w2m+k'P(o)
by Theorem
11.5 .
q.e.d.
Theorem
11.8
As slime (I) that assumptions (2) that
(I) - (4) of Theorem 11.5hold with
f E L I (G) such that
k>_1
196
LU)o
(11.18)
(f,
=
(h'U)o
for every
u c c2m(~) with
D~ u I 8 G = 0
for
l~I~m-1
.
Then, the assertion of Theorem 11.5 holds true.
Proof: Let v ~ o ' P ( G )
0w2m'P(G) with g :=Lv ~L~(G). Let ~ be defined by
(11.10) and let Hw := - ~ ( f i , W ) o g i
. fi, g i ~ck(G) by assumption (I),
Theorem 10.10 and Corollary A.4. Therefore, Friedrich's mollifier p' ~max(p,n)
~v = g + H v ~ L~(G). Apply
(compare p. I06) to g . Then g~ ~ C~(G). Let for
v~ C~o'P'(G) O W 2m+I'p'(G)
Then, v~ cc2m(G), D~v~ J ~ G = 0
be the solution of ~vs=g~+Hv.
(I~l~m-1).
By (11.11),
IIVa-VIl2m, p - ~ O
(s-~O) and therefore Hv -~Hv uniformly. This implies for a proper sub-
sequence Lv~--~Lv a . e . in G. Since IILvEIIL~! IigsllL~.+ tIHVtlL~ ! ! IIgllL~ + IIHVIIL~ we conclude by Lebesque's theorem -~
fFLv
holds
for
dx. Since by ( 1 1 . 1 8 ) ~ L v ~ v.
So, t h e a s s e r t i o n
follows
~Lv~
dx -*
dx = ,U~Vs dx - * ~ v d x ,
(t1.18)
from Theorem 11o5o
q.e.d.
Theorem 11.9 As sume (I)
that m ~ I and 0 < j ~ 2m
are integers and that I < p, q < ~
real numbers with (2) G
I + ! = I P q ' that G c IR n ( n ~ 2 ) is a bounded open set with boundary
¢ C 2m , (3)
that
L =
~-
as Ds
is l<_2m coefficients
a s ~ C~ sl(G) ,
is uniformly elliptic in
G and has
are
197
(4)
that f ¢ L~(G)
(11.19)
I ( f-, T'~),,I
for every
~
u ~ ~'P(G)
C 11'-'-II,..,,,_~.,~
n w2m'P(G)
Then, f ~ wJ'q(G) m<j<2m
and that there is a constant C > 0 such that
if 0< j <m, and
and there is a constant
(11.2o)
I1:f: I1~,,~ ~
with
Lu
E L~(G).
f e W~o'q(G ) n wJ'q(G)
if
y > 0 such that
~' (c ÷ II~llL, c~)
Proof: (i)
Consider the case m < j < 2m.Then, 2m - j < m
f e -W~'q(G) u by Lemma 11.4. Further, there is a
(11.21) If
and
therefore
y' > 0 such that
I1~: I1,,,,,,~ ~- ~(.q "~ ll:t II~.,c~)
BL, is the m - smooth uniformly elliptic regular Dirichlet bill-
near form defined by (9.60)~
By Theorem 1 1 . 1 ,
I1f ll~,q
(11.19) implies for every
f ¢ ~o'q(G) r] wJ'q(G)
~
@ c C~(G)
y"> 0 such that
and there is a
~ C c , II ~ llo,q)
Combining this inequality with (11.21), we get (11.20) . (ii) (11 19)
Consider the case O <j <m. By Lemma 11.3, f e Lq(G) and holds for every
u ¢ Wm'P(G) 0 w2m'P(G)
"
O
as will be seen '
like as in part (i) of the proof of Lemma 11.4. Let ~ (11.10) and a
B
by (11.7). Let
be defined by
A := (-])J ~ D 2jei 3= uniformly strongly elliptic operator of order 2j. Let
Then, A
is
198 e w~'P(G) n w2j'P(G) ; there is
a
then
AW¢
uniquely determined
v = A~
LP(G) . By the properties of
v ¢ ~,~'P(G) ~ w2m'P(G)
~,
such that
Therefore,
(11 .22) for every
@ ~
C~(G)
and therefore
From (9o19) follows then
(11.23)
I1"I1~,~ ~ ~"~11~,~
From the definition of
B
and &
(I1.22)
we derive
( ~ll~Uo,,~llo.~)ll~llo,~n~llo,,
+ I1"~11~,~@ I1~,,
Observing (11.23) we get (11.24) I ~ [ ~ , ~ I
-~ ( ~ II~IIo,~II~IIo,~)(~'~)U~II~,~II~II,,~ C II'~lli,~ll@tl~,~
=:
From Theorem 11.1 we deduce 0 w2m-j'P(G)
and with a
(observe j = m - ( m - j ) )
v E~o'P(G) n
y" > 0 we get from (11.2) and (11.24)
From this follows together with(t1.23)
(11.25) ]Iv I1~,_~,~
=
-~ "d'"' n,~ll~,~
t99
Then
(11.19), (11.23),
(11.25)
imply
,~ ~'"(Cllvi1~
~ II~:llL, c~,,llo,~, )
"¢ ( c ÷ II Y II~,c~ )11 ~ I1~,~ with a suitable to
A
y'" > O. Then,
and leads to
the first part of the proof applies
f e w~'q(G)
and
(11.20)
. q.e.d.
Remark
(i)
11.10
Theorems
by S. Agmon
[2]
of the type considered here were originally proved in the general case
slight generalization
I < p < ~ . So, Theorem
of [2] , Theorem 8.2'
liar to[2] , Theorem 8.1.
The fact
and
Theorem
[u e c2m+k(G)
0 w2m+k'P(G)
k = 0 in [2] ,Theorem
8.3 . The advantage of our Theorems f ¢ L~(G).
Theorems
11.9
is a
is simi-
: D~u I 8G = 0
I~I ! m - I) c ~o'P(G)
is, that we treat the case
for I < p < ~
11.8
for
was proved in the case 11.5 and 11.8
11.1 and 11.2 seems not
to be known under this general assumptions. (il) We have proved all the theorems
globally.
But it is easy to
derive them locally. We have only to apply a suitable cut-off procedure. (lii)Conslder Theorems
11.5,
11.8,
11.9 in the case q >n.
with the aid of the Sobolev - K o n d r a s h o v -theorems differentiability
properties
of weak solutions.
Then,
we derive classical
Appendix
I
Proof of Lemma 2. I For every i' ¢ C n-1
k eD
and • e C let 2.-v~
>
=
l $ I = l'~v*
P(= 0
where I S,'l,zl+Tn-
The coefficient of tion
(A)
K
T 2m is therefore
and (1.6)
a2men(k) and satisfies by Assump-
la2men(k)l > E > 0
o
o
o
for every
. For i = 1 , . .
2
.,n-1 and fixed (1 i..... li_1,1i, li+j, .... in_l) consider
keD
o
e cN~
and fixed
k sD,
for (li,~) e C 2
Q
(i,
O
....
O
By the fundamental theorem of Algebra we get for every i. e C
2m roots
i
xj = Tj(l~, ....I£°_~L li~ I~,~.....I"~,~- ~ ), j =1 ..... 2m Go(k;li, T ) in
7. As we know from the
of the polynomial
theory of algebraic functions o
(see e.g. Knopp [33]
), for every
k eD
o
o
and every (1 i, .... li_1,1i+1,..
o
.,ln_s) ~ C N-S
these roots are the values of 2m analytic
,l~_~,li,~.I, .....,I~_~
zj(lii~ )- Tj(I~, .... a2men(k) ~ 0 parameters
for o
kcD o
(11'''"li-1'
ment for every
. For fixed 1 o
,
i = I, .... n-S
wj(l~,...~l~_¢} ~ )
.
o
j" ~ )~ J = I, ...,2m, since
keD
i+I " ~ln-1)
.
they depend on the ( n - 2 ) Now we may repeat
So we get for every
defined on
C n-S, each of
e.g.S.Bochner
- W.T.Martin
k eD
this argu2m functions
them having the proper-
ty that it is an analytic function of one variable maining variables are fixed in
functions
i i e C, if the re-
C n-2. By the theorem of Hartogs
[7] , Kap IV, § #), every
xj(l'i k) (J =I,.
• .,2m) is an analytic function of the (n- I) complex variables ..,ln_ 1 , if
k ~D
(see
1 S ....
is fixed. Since the roots of a polynomial depend
continuously on the parameters if the below from zero (see e . g . K . K n o p p [ 3 3 ]
"leading coefficient" or M. Marden),
the
is bounded
Tj(I';X) are
201
continuous on the
as(. )
D x C n-l, since
[a2men(k)[ _> E > 0
are uniformly continuous on D. Let
be closed Jordan arcs and with boundary
Ji "
let
Then for
Ii
for every
Ji ~ C i
k cD and
(i = I..... n-S)
be inside of the bounded domain
I ~ j ~ 2 m and every
~, ~ ~ - I ~
every
X~D
(A.~)
D ~' ~
(~,
.... , 1 ~ _ , ~ )
....~ ! ' - - ~ . ~
=
}
(~)~-~
b~
:~
~.~
( ~,- 1~)~,'" . . . . .
(see e.g. [7] ). This immediately implies
(~.,
- 1~.~) ~''" *
c C ° (DX Cn-1 ).
D ~' T j ( l J ~ )
To prove the homogeneity of the roots,consider for (r,l') ~C n the polynomial in o c C
(for fixed k c D )
~(r,l',o)
As proved above, there are exactly 2 m
:= 7--
;~s(~,)l~S~TIs'l~S~
functions oj(T~&JI~)
(j=1 ..... 2m)
being analytic in cn, which are for every (r,l') the roots of ~(r,l;c). On the other hand, for every (r,!') ccn} Tk(3~.IS~) is a root of ~(r,!',o). =
F~rther,
for
j:l
.....
2m we have
0-
r ~
~s'i'=~,(~l'"~J32,')'~
}-- ~,(k~ (Tl')S'(r~(l'iA))S~that is rTj(l'i~) is also a root. Since the 2m Isl=2.q~l
roots oj are uniquely determined, proper j ~ [I, ...... ,2m} such that For r = 1
Tk(~l'~)=rTj(~}~ ) for every (r,l') cC n.
follows Tk(l'f~) ~ ~j(l~ik) for every i' ~C n-l, what implies
Tk(rl';~) = rTk(l'}~) for every Assertion
for every k = I,...,2m there must be a
(r,l') ¢C n. This proves
(I)and
(2).
(3) follows trivially from the ellipticity of L k and the con-
tinuity of the roots. q.e.d.
For the application of the theorems of
Bochner and Mikhlin it is neces-
sary to prove decreasing properties of certain functions. For this puTpose it is useful to know the following formulas,which are easily proved by induction.
202
A. I
Lemma
(I)
Let
~ c ¢
be a n open set and
be an open set and ~ e Cm(G)
w ¢ cm(f~) (re>l). Let
with values
in ~. Then,
for
I Yl
G = ]IR and
im
x~G
where
~(k)
(2) for
e zz+n
Let
G c IR n be an open set, let
x e G, and let
~
be homogeneous
of degree
there is a
e ck-I~I(G),
and for every
i~I ! k
gree
, such that
Icl(p-1)
(3) Then,
for
Leibniz~s
~ ¢ ck(G)
rule:
Y
Let G c ~R n
(k~1),
p ~ I. Then,
~(x) $ 0 -I
homogeneous
be an open set and
e ck(G), of de-
f,g eCru(G).
IYI < m
(A.4)
Proof of Lemma 2.2, assertion Let
Y = (Y',Yn)
¢ ~Z +n • IYl_
y' (n+2m+ I Ol,(1+ell' 12) . ~(i')
:=
(iii.)..
S+e~ll'I 2 . For
We apply
(A.2)
1~_i,j(_n-1
with we have
w(r)
:=
r- (n+2m+ I )
and
203
%ii and
=
iIi
D a'
~(i')
sider only
ki
,
~li z for
with
I < k. < 2
the t e r m in b r a c k e t s one
ki
is g r e a t e r
~
~li~l~
Is' I > 3. T h e r e f o r e
= 0
--
=
and
kI +
l--
vanishes,
since
t h e n two.
Further,
then
k. is e i t h e r one or two, 1 (2v - IY' I)
of the
(A.2) we have to con-
+ kv =
IY' } • If v <
-
"'"
if
kz +...+kv
i% z ,
If
in
l~,~i) I
and if
x~ +...+
k i are e q u a l
I
and
=
2
IT' I • at l e a s t
= Z x~ = Ix' I (IY'I
-v)
l'f'l 2
(vh o f the
k. 1
are e q u a l 2.
case
(i).
As sume
I v ' I = 2k,
considerations
k = 1,2,...
C = C(y')
above~
1K
C 7~.~
l ~
5z.v
Z.(~,- ~)
l l'l (C" ll'l~) ~'~ ( , . t - 11,1~)~z'~*~-~ ~
(4 + £ ~ l r l ~ ) ~ Further,
k ctl~ I
> 0. By ( A . 2 )
(~ - i ~i~) - ~ ( ~ z ~ )
=
and t h e
,
20#
r.,, which
implies
Since
2k + Yn =
conclude from
(A.8)
n
(A.7) and
I~h(~.;i)llil
The case
Ivl !
'~''
and
( 1+~1 ? E~ )(1+~ ~li'
t ~ ) > ( 1 + ~ 1 i l 2)
we
(A.5)
_-.
C"
~_
s~'~*r" li(~'l
IT'I = 2k + I, k = 1,2,..
C m
C'(y) > 0 is treated quite in the
same manner. q. e. d.
Next we are concerned with
Proof of Theorems2.8 and 2.9 for the Kernels (i)
We prove Theorem 2.8 in the case
the differentiability properties I~I J m for the kernel of
(A.9)
I ('x;~i ~., i~x.,%) :=
Gk(e;x,F)
Sk(e;x,y ) = Gk(s;x,y ). From
and (2.16) we derive for
Dx~Dy~Gk(e;x,y)
£ (i,
(Z'rr)'' h(~.', i) ~,~ (ii ~) e..
where
i~+P Z~(1)
~-~)
I~I j m ,
205
Let and
Let
9 e C~(]R n)
such that
o
O!
9( I ) ! I
£oT
for every
I~I -~
~
+
1 e IR n
and
+ ~.c,.~)l
(~
R 0-)
,i
Then
(A.11)
~U
2
In (A.11)
4
we may estimate the terms in brackets
constant
C = C(E;C2)
grands are
independent
uniformly bounded.
we derive from of Bochner's
(2.6)
theorem.
that
e,
since by
Ix-yl < I
by
a
(2.22) the inte-
For the treatment of the first integral
K(1)
Further,
of
for
:= h(e;l)IE= I
by (A.3) and
satisfies
assumption
(~)
(A.4) there is a constant
M = M(n,m,E, C2,~,a s) > 0 such that
holds for every hood of
l=O.
k eD. The functions By Bochner's
<
C(6)Ix-y1-6,
where we have estimated
O < 6 < I. Trivially,
bounded by a constant, are estimated der consideration,
vanish in a neighbor-
theorem the assertion of Theorem 2.8 is
satisfied by the first integral, I+ lloglx-yll
9(.) faB(.)
the other functions~
in the same manner.
Theorem 2.8 is proved.
In the case un -
206
(ii)
For the proof of Theorem 2.9 we consider
We denote the term in brackets with
Let
g e LP(G),
I
~ . Then,
for
~k(e;x;y).
I~1
+
t#l
formula
As mentioned
= 2m
(A.11). above,
is
&
The
LP(G)
- norm of the second term at the right side of
estimated with the aid of
(A.13) and HSlder's
(A. 14)
is
inequality:
!
(A. 15)
Next we want to treat the first term at the right side of (A. 14) ; we denote it with iz(x). We derive from and (2.10) that for every grable over Fubini's
~n
e >0
× G, since
(A.12)
(with
I~(i) f ~ ( k ; l )
g e LP(G)
and
IoI =0,
l~I+I~l=2m)
h(e,l) g(Y)l
is inte-
~(G) < ~ . Therefore,
by
theorem
_-n
(A.16)
i(Zj~) ^ e"
@
We derive from h(e;l)
(2.6) and (A.12) with the aid of Leibniz's
q(1) fa~(l;k)
satisfy condition
Mn
rule
for every e > 0 and
that k cD
207
with a constant e > 0 and
k cD.
Together with
Mi = Mi (n,m,E;C2, q, [[asHL=(G)) > 0 By Mikhlin's
theorem
(see page
independent
of
226)
(A. 15) the last inequality proves the assertion of Theorem
2.9 in the case
S k = G k , with
M := C(E;C2) ~(G) + % , n
M~
.
q. e. d. Proof of Theorem (i)
2.8 for the kernels Kk(~;x,Y)
It suffices to prove the assertions
(0~j~m-1).
for
But this case is rather complicate,
D y ~D X ~G~~,j (E;x,y)
since it is not possi-
ble to apply the theorems of Bochner and Mikhlin to derivatives
in the representation
the fact, that there is no riables
G*k,j
and its
(2.38). The difficulty arises from
term of the type
x n and Yn appear separated in terms
e iln(xn-yn),
but the va-
e ilnxn
e -iTynll' I
and
So it is necessary to transform the representation
(2.38). For every
Ii'] ~ I we calculate the integrals of
b y the residue
theorem.
Let
ll'l > I
and
k cD
i n and
w
be fixed. Then,
-¢rl
-
I'
I'
--
M
]--7 ~o "I- ,,a ,,I
Here for this fixed r = 1,...,po , the
"~k-
]i'
] >_
I
this
and
k eD
and with
I <mr<m -
k) '
their multiplicity,
K&(I',~.;X)
----
So,
-
m I +...+ mpo = m . Then the residue theorem ~,n e.vl
~r -( ~ Ti' ~'
po <_m, we have denoted the pairwise disjoint rootes of
( I i' m' I ' X)
(A.~8) ])~
with
leads to
( -i'l'~l)'~'~ >--c,~-~)'. (~-EJ
•
~ ~,
208
where
% $,1-~
Let
(A.19)
( 1 " , ~,)
"Tz,p,,,
z f we a p p l y
T,eibni~.'s
..~,a..,e.
:=
z"ale t o
X)
I'
(A.18),
-
+'I'~T""L
po t
"=
J- defined by
(A.21)
Then,
"-T'r : =
(2.35)
= 1
contained in that subdomain surrounded by
J+. Let
choose a proper
4
which implies
let
and Gj+
0 < p < I
keD,
-
Tf
(t;'.;k)
lk=l,
of the upper half-plane, be arbltraryj
p). Then
e..
%""
~. "~ ¢ ¢ : -"~ e J ' ] i
I'~'1
for every
I l'l')"'" -
- 1.
~m.: o
z':&
For
we g e t
8
but fixed
....
m)
are
which is (later, we will
209
(A. 22 )
e
----"
1'
I
d~"
We observe
(A.~)
( i ~ . , Irl) ~"'-~'~ ~ t ~ l ~ ' ~
and integrate senting
this equation over
• = 2R e i@
usual integral.
(A.a~)
,0J~<
Applying
= ~--/( ~ ~"~'~" K R := IT : I~I =2R,
R~Ca]
Repre-
2~ , the line integral is transformed
in a
(mr-1-~) -times partial integration we get
(_iT, ir~)~,,~_~_~ } ei.~a,l~., ~-i.~: ~. ~~.,I.~ (~,)
r
By Cauchy's
theorem,
rive from (A.22) and
(A.25)
e~:a'~"
in
(A.24) we may replace
(A.24)
after application
~ll'l
K R by
J+. Then we de-
of Leibniz's
rule
(-~ ~ Iz'I) ~ - ~-,~- ~
J
Again with the aid
of Leibniz's
rule, the right side
(RS) of (A.25) has
the representation
(A. 26
If we put this in (A.20) and replace in (A.19) Leiniz's
rule
•
by
~r, we get by
210
(A.27)
~
k ~ ( l ' , ~ . , , I ~,) =
(-
=
x-~°
{
It. a
I
y~._~l*ei~m~,,.. £ @-~l~,,c~,,,~,,,) &'l; L
o
"~
_
':r
l ~
~1
,~
"~I~
x"
We will show that the right side of (A.27) may be considered as a sum of residues of a proper meromorphic tegration over the sum over
T
function.
In (A.27) we change the in-
with differentiation with respect to
r.
~
and with
Consider the function lI -
IL
Next we construct a closed--Jordan arc J'- in the lower half plane, that for every F(~)
:=
by
-
x c J + , ll'l > I
and
k cD
~ (u~,k;l') , which a r e c o n t a i n e d
J' , are
exactly the zeros of
cially
ar~ - T
(A.28)
"3 '~
in
Gj,-
l'
such
all the poles of in the domain
M-(I-~77 ,~;k)
Gj,-
bounded
. This implies espe-
Let
----
I
where Then
b :=
¼ (4 Caa
- C#)~ 2
J'- has the desired properties.
and
(2.4),
in
Gj,-
1
(
(A.27)
leads to
Wk-(7~77; k )
. Then,
Ii'I ~I,
According to the inequalities X~D)
k = 1 ..... m,
(2.3)
are contained
211
II'i) ~'' ~)~.
(- i
~+~
e ~%+
~
"
3~~
MConsider +~
Ix(-~i*~iri[~"')
(A. 30)
For fixed LX(1),
Iz'l h~
put
'=
I
L~(1)
i n = If' t
(4-izl-.) &(~n+z"~+a) d l ~
and observe the homogeneity of
then
(A.3~)
Let
x n ~ 0 and
ll'l Z I be fixed :
(compare page 38)
Similiar to the proof of (2.40)
we see, that the
t-integral
over~
is equal to
the limit of the integrals over
for in
R -~
, R>C2
. Since all zeros of the denominator
are contained
Gj+ , we get
Changing the order of integration,
(A.33)
I ~ (~-', ~ r . , ~ )
:b~,~
we get from (A.29) and
~<~ (l',~,~
-
-
(A.32)
212
=- (-~.)~"l!'l~'~-~'~"~P " ~(a~i) z #
e ~:~x'~(~~*sx'4
C1 For all the following considerations let 0 < ~ < mln(1,6-~2 )
real number. If
llm ~
i 3 C~.
t e J+, then
Im t > ½ C~ > O.
If
X
be ~ fixed
~-e J'- ,
then
This implies
Therefore, we get for
(%tr)
I~I i 2 m
..(~rri.)~ ~
,
161 i
2m
(.~+ ~ l l , l~).,,+r,~÷~ "
~i, lz.,~_~_~_/~. ~
dl ~
where
(A.36)
~T ( m~ l'~×~; ~ ;
~)
:=
= z6r ~e~'- eta• (=~) L~i~L'~)(~-i~U'l~)Z~z~) 1,
~ ~) ~ Z T (~, ~ r .....
dvd6d~
Next we will prove two estimates. Combining both we will easily perform the proofs of Theorems 2.8 (ii)
Let
(A.37)
Let
and 2.9 .
l(E;x,y;l';k)
l denote the integrand in (A.35); then
213
(A.>8) {tt't ~ i~
brt~3 We w i l l
estimate
the first
integral
by B o c h n e r ' s
theorem.
Let
Yo
n-1 the s m a l l e s t integer greater than 2m + - T h e n we w i l l p r o v e 2 there is a M = M(n, y o , C : , C a ) such that for e v e r y s > 0 and e v e r y
X eD
, for
x n~0,
Yn~O
and e v e r y
i°'l ! Y o
where
(A.40)
For
t l' I > 1
(A.4~) By ( A . 2 )
I ~ l + l[~} - 1 ~
k'. -
+±
is
i ~ ~ 1, ~
I~:,
l l q z~-~- ~-I ~
]
~,,~'~
I ~'1 ~ - ~'~
is l~I,q
×> Since
im T > C--k > 0 --
2
N ~
1
]TI < 2 C? , Yn + --
-
I 'Jl .... ISn:'t I P Xn
> 0
-
~! (~C~ v
C,~ Il'Lv
J
be that
2124-
(A.~) if
l~lm'll
we o b s e r v e
~_ Ii'l
~t
~;~) and
I~'1 <-'~''
I~'I i Y o
we g e t f r o m
~
(A.42) -
(A.44)
I
and a n a l o g o u s l y
(A45
~
I'#
Since
D
to
, and is u n i f o r m l y
i
, T
I ~ c'(~o,~,e~,C¢l~t -~''
(l';k)
is h o m o g e n e o u s continuous
of degree
I - IY'
for
c [Ii'
(I',X)
with
=I]
respect
XD,
L, (~,, ~) M-(m,~m ]3¢'
=
~-
for every
k ~D,
..........
C"(m.~,
Ik' I > I ,
c., cD lI'l -w~
t ¢ J+
_
By
(A.47) I ~ ' < for e v e r y with
,
~
a'
,
l~'[<_yo
.
(A.2) we get
E > O.
respect
(A.39)
~-~,~)-~*~I
to
After T , ~
I ~_ o,(~o,~,<,m)l~,l -~',
application and
t
of Leibniz's
we derive
from
rule
and i n t e g r a t i o n
(A.41) - (A.47)
estimate
2~5 By (A.39), the first integral at the right of
(A.38) satisfies all as-
sumptions of Bochner's theorem. Therefore we conclude for
Ix'-Y'l < I
c~ C~_ + I ~o~1 ~'-~'il) C~ where
Ez = ~z(n,m, Cz,C~).
According to (A.39) a constant
C2
there is a
C(6)
Ix'-y'l
the second integral in (A.38) may be estimated by
depending only on such that
Cz,C~,n,m.
Since
for every 0< 6 < I
I + lloglx'-y' II ! C(6)
Ix'-y' l-~for
< I, we conclude from the last estimate
Co C¢) n + I~1 + I ~ 1 - 2 m < 0
if
for every
~ > O, k ¢D, j = O,...,m-1,
(iii)
Consider
(A.35). Since
where Co(8 ) = max (Cz-C(8), C2) •
Im T > C--A-~ if - -
integer
2
k >0
e
e,,
T ~ J+ , for every
216
if
Yn +
PXn > 0 . Let
0<_#< I . Then
what implies
i
for every
integer
(A.45)- (A.47),
(A.50)
k ~ 0 and every
(A.49)
0~#<
I . From (A.37),
(A.41) ,
follows
I ( ~ ~,~i.~,)~*z~*~ Ir~-~-~--r T M
In the case
2m-l~I-
I#I > n
converges. In the case arbitrary,
and put
put
k=#=O.
2 m - l a l -I~I i n
k=n-2m+
Then, the integral in (A.35)
put
#=6
with
O< 6< I
l~I + I~I ~ 0 else.(A.50) implies that
the integral in (A.35) converges. Further, we get with
C A = C~(CI,C2,
~,~,n,~)
Co'I ~ if
~ ~e
Co',if
(iv) Since
Combining
0
I
and
~ ~ ×~I - C ~ - ~ ÷~' ~l~+g~ n + J~l ÷ I~I- 2m > 0 n+i=i+l~i-2m<
o
(A.48) and (A.51) we get the desired estimates: Xn>O
, yn>_O , yn + PXn >_ p (Yn + Xn)~
217
O< p < m i n ( 1
where Case I: Ix - yl
Let
CI
)
Ix' -Y'I ! Ixn + Yn I • Then
! Ix'-Y'I
+ Ixn - y n I !
21Y n + Xnl !
2 p-11y n + PXnl
and
therefore
Case
2 :
Let
Ixn + Yn I !
Ix - Yl ! Ix' - Y'I +
Ix' - Y'I
. Then
Ixn - Yn I ! 21x' - Y'I , and therefore
(A.54)
I
Let
Observe,
that
O< p<
min(1 ~ ) is a fixed number (see page 212). ' oC2 Then in case I the assertion of Theorem 2.8 is derived from (A.51) and (A.53). In case 2 , the assertion follows from (A.48) and (A.54). In both cases, C
is defined by
(A.55). q.e.d.
Proof of Theorem 2.9 for the kernels Kk(e;x,y) (i)
Again it suffices to prove the assertion for
D y aD x ~ G k; *j(E;x,y)
for
T c J+
" Let
Multiplying
I~I = I~l = m
in (A.33)
and
the
and denoting the resulting function with
let
N> 0
T -integrand
by ~(~; ll' I,T)
SCs;N;l';Xn, k ) _ Yn,
we get
218
~ {,~;~;~."~-.,,~',~,) .... uniformly keD
for
> I~(~,~i~'~)~
l'e K, K compact,
for every
~ K~ (f
~)
e > O, Xn>_O, Y n > O ,
fixed
. Let
(A.56)
Wa
( ~% a21 x,~% ~t,~)
:=
-- (z~)'~ (~* ~h~'lD-{~÷~'*~)e~[~'" a~'~ S C~WI ~'~,~ ~) al' Since the l'-integral converges absolutely,
w~ C~,~i~,~,~) uniformly
for
~ ~
(x,y) ~ H + x H +
For
ore J'-,
is analytic
in
Replacing
[i'I > I and --
:D~ Gx~3 ~. (~;~,~)
and every
suffices to prove the assertion for q > 0
[T ¢ C: Im r k O] D
we get
Wk Z(T)
fixed
transformation
e > O. Therefore,
independent of :=
e
eiTll' l(Yn+pXn) T
+
o~
and
it
q .
h(~;l';T)
~Gj+
J+ by
then the integral over the half circle vanishes integral of ]R
(~o)
converges. in T :=
Ill
for
R-*~
In (A.56) we perform for every
and the ll'f
>1
the
Then
~
~
]--~--~---
"d~dta~
219
where (A.58)
c-~)'~'+~
i '~'+~'i'"+~'O~
(~',~., ~,i ~)~
~;,-,-~
e~ m
~,, { ~ , ~ )
Further, (A.59)
~- llq
+(~r
= i~ + u= ll'I
I ~ - u,~~ ll'~~
l~li,
[ We put Let
-
(A.59) in (A.57) and change the order of integration.
for
and let
~r l l ' l ~
]li
m
i~- ~l'I ~
R > 0
f
¢ Co(Q R+ )
Then,
f@
(y) := f(y' , - y n )
~ e Co(Q R- ) . Let
(A.61) Q~ and p e r f o r m the t r a n s f o r m a t i o n (A.57) and
y' = z' , Yn = -Zn " Then we get from
(A.59)
(A.62) il~
×~
X
220
u~e
-ctu ~" [I].'1_~.~ 1,,,-~
-u'~lr[z
X
where ~j
(ii)
Next we will show
that there are
Yi(~,.;N)
e LP(zR n)
(i= O~ ..... n-l) such that
(A.63)
~o
(~,l~)
=
l¢-~%rl
~
Ik
and that
Yi ( .... ~)
(i=O,...,n-1)
and that there is a constant
are measurable
K = K(p,~,J'-)
for
(~r,t) e J'- × IR~
independent of
f
and
such that
(A.63) and (A.64) are an immediate
consequence of Mikhlin's
if we can prove that the multipliers
of
For this aim we will show that there is a
(A.65)
11¢-
for every
IrL I
I ¢ ZR n,
we
c
J'- .
•
3@(1)
theorem,
satisfy condition
C = C(J'-) > 0
~.
such that
221
We ha~e
Case
l~n - ~ll~l
: lln -~'II"
maxlRe~l
~
Ilnl
i:
=
k ~ c~
C~
z~ "~--' ~ - ~ - ' ~ 1 "
l $
~rther,
mintIm~'J = ¼ Cz
,
tl'l,
lln + :ll'II .
o.
Then
=-
(~=~*~ m , ~ I r l )
~" * ( , x ~ ) ~ l l ' l
-~
(11,,I - ~ I ~ 1 )
~ + ( ~ c.)~l~l ~ ~
_~
i~.l~.k
+
( % c,)~l~'l ~
~ ~-
_~ ~[%(~c.)~izl
llnl ! 3 c~ Iz'l . Then
Case 2 :
1~.1 ~ =
15+1~1
~" ~- ( s c b ~ b 1 1 ' l ~.
Since
we get
, From both cases follows Further,
~ 1~
(A.65)
ll'lZ
and
- , . . ~ 1 1 ' 1 ~"
With the aid of Leibniz's (A.65) fore,
We put
rule,
that the multipliers the
~i
l~Ik 1.. ~ -
infinitely often differentiable
in
have the desired
(~e£]
with
'~'11'1
and are homogeneous
are for i @ 0 ~
of degree
it follows from (2.6), (A.63)
0
~
(A.3) and
satisfy condition
M n. There-
properties.
(A.63) in (A.62) and p e r f o r m the integration over
i.
i n. Then
(A.66) i
~
222
where
~: -
O~
4~,..., ~ - ~
~ C~(]R n-l)
Let
with
0<~(!')
~(l,)
for every for
11'1 < g
for
Ii'l
l ~ ~ ~R n-1
and
= > 2
•
Then
(A.67)
~ (~i~ixi{
]
=
i-o o~'where
0..~.
From
~ a ~ [il'L-~i~
:{-
(A.58) and estimates
(A.45) -(A,47)
and
(A.5
follows
I]3¢' for every m, ~)
]
w ~ J~- , t ~ J+, E > O, k ~D, x n > 0
independent
~' l i I
of
E > O,
Mi =Mz(J~-,J+,n,
and
tg't
Again by Mikhlin's
with
;
theorem and by Fubini's
~- -- i, - " ; ~ - i
theorem we conclude:
/
t"t2, (-n)i
There
223
are functions properties:
@r(~;q,t,~,x',Xn)
(i)
For every e > O, q > 0
of (t,~,x) g (J+ x J'- x IRn). of
x'
, r=O,...,n-1
for fixed (e,q,t,~,Xn)
(ii)
, with the following
they are measurable ~r
belong to
considered as
^ (n-l) ~i" respect to l')
where
r=0,1,...,n-1
(A.69)
~,--.,-.-i
~- M IIw= C~>. } S~;~.BT.~CR,-,)
and where M=M(MI,M2,p,n)
We put (A.68) in (A.67) and perform the
T(~i
i-
and
1{ % C~;R;~;~.,~.~)IIs,(~_,)
q .
~ (~')~ ~
denotes the (n-l) -dimensional Fourier transform with
(~.6s)
and
functions
LP(IR n-l) and satisfy
~,~
(where
functions
~-~ x]~)
is independent of i'-integration.
=
(A.68) and (A.64) is
By
By H~lder' s inequality and Fubini's theorem we get with a constant ~i
=
~1{J'-,J+,n,P)
i=o
~ . ~ 3 ~"
+,e~÷
+.,e~÷
Then
22~
(A.68) and
(A.6#)
imply
II II L ~ C ~ )
~P
By
(A,63),
I < Iz'l
for
there
is
a constant
~ = ~(J'-,J+)
> 0
such t h a t
for
< 2
xn~O,
with
p
K II BL
a~e J'-. Therefore,
M o =Mo(P,n, Cz,C2 )
(A.69)
implies
independent of
s
and
q , where Cz,C2
are
defined by (A.22) and
(A.23). Therefore the assertion of Theorem 2.9
holds,
is dense in L~(QR)
since
Co(QR)
q. e. d.
Theorem A.2 Let c LP(G).
(Sobolev's inequality)
G c ~q n Let for
be a bounded open set, let
I
and let
0< v
T(×) := G~ I x~-
~I v
~
Then, there is constant
C = C(G,v,n,p)
(1)
F is bounded
~1 + ~v - 1 < 0
(ii)
F ~ LP'(G)
if
for arbitrary
such that and
I
x s eu pGl F ( x ) l if ~1 + ~v
< --
-
1 =
ctl¢llsp( G) 0
and
225
tlFltLP' (G)
<-- CP'H~IJLP(G)
(iii)
F c Lql(G)
ilFiiLq±(G )
and
For a p r o o f
Cp,=
C(p'~ C).
~1 + ~v _ 1
if
> 0 , where
I = P 1 + ~v - 1 q-T
i c II~IITP(G)
compare
Theorem
A.3
e.g.
[5] .
(S.L.Sobolev
Let G c IR n with
,
[60land V . l . K o n d r a s h o v
be a b o u n d e d
open domain.
Let
u
[34]
) .
c wJ'P(G)
for a
p
I
there
are c o n s t a n t s
Ki=Ki(m,p,G),
i = 1,2 , such that u
satisfies: (i)
If p < n
IlullLP~(a)
i K~
(ii)
Pl
:= n -np p
" then
u c L pl (G)
and
Ilultl, p
If p = n ,
ll~ilLP'(a )
then
u c L p'(G)
for every
p' with
~ ~ C°(G)
which
I
and
! x~ ILulll, p
(iii) uniform
and
If p >n,
HSlder
m a y be c h o s e n For every
then there
condition
arbitrary,
~ with
in and
is an G ~=u
I - ~P
O
with
exponent
a.e.
there
in
satisfies
~ ,where
G. With
is a c o n s t a n t
a
0 < ~ < I -~ P
other words
:
K 3=K3(n,p,G,a)
such
that
×~jX"G@
X ~: r~
For the p r o o f in
[17]
We derive
are v e r y
A.3
compare
I
e.g.
[5],
[17],
[49],
the proofs
simple.
from T h e o r e m
Corollary Let
of T h e o r e m
I wl -
A.3 by i n d u c t i o n
A.4
G ~ ~R n
be a b o u n d e d
open
set.
Let
I< p
< ~
and let
s
be
225
the smallest integer such that
Then there is an
a
is satisfied.
[ = u
a.e.
Let
k > s be an
u ¢ W ko ' P ( G ) .
integer and let
every
p > ~n
~
with
and
~ ~ ck-S+~(~)
for
with O < m < s - (~) .
Corollary A.5 If
8G
holds for If A.4
e CI
and
u e wI'P(G)
, then the assertion of Theorem A.3
u. 8G
¢ Ck
holds for
Proof:
and
u e wk'P(G),
then the assertion of Corollary
u.
Compare e . g .
Theorem A.6
[17] • [ 4 9 ] .
(S.G. Mikhlin [45]; compare
[63], [25] )
Assume (I)
that n > I is an integer and ] < p < ~
(2)
that
(m)
@ c Cn
(~)
there is a
(JR n
Ixl
ID
(3)
that
Then, F(1)
satisfies
•
is defined on ]R n and satisfies condition M n , that -
[0]
M> 0
¢(x)l
)
such that for every
i
M
I~I ~ n
and every
x ~ 0
,
f c LP(IRn).
defined b y
F • LP(]Rn).F~rther,
there is a constant A = A(p,n)
that IIFIILP(]Rn)
is a real number ,
!
A M IlflI]p(iRn)
such
227
Theorem A.7
(S.Bochner
[61 )
As sume (I)
that
(~)
Let
6
r>
5 +~+ 2
9 c C~(]Rn),
be a real number and let
There is a
Ro> 0
(y)
There is a
M>O
such that
be an integer such that
for
Ixl ~
x c ~n
Ro .
with
Ixl ~ R o
and
lal < r
(2)
that
There is an
(b) o~ ( ~ ) K(0~
=
= 1
<
A > 0
Ixl 6-1=t has the following properties:
such that ~ A
for
x ¢ (~n
O(,~I -f-~-*-
_ [0]) and
0~lal~r
~
.
Then the function
is well defined (i)
M
K e C=(IR n - [0] )
llxl I~I h a K(x)l
(c)
~(x) = 0
such that for every
ID~ ~ ( x ) l
(a)
r >-2
I
(~)
every
n>_2, has the following properties:
F(1)
F(e;I) defined by
and has the following properties: :=
lira F(E;I)
exists for every
1 ~ 0
E-~O
(ii) uniformly
For every compact set to
F(1)
(iii) For every
as
E
R~ > 0
K c IRn with
0 { K
converges
goes to zero. there is a constant
B = B ( n , 6, r, Ro, RI)
such that for every0
N1B
MI5
F(~il)
if
&+~ = o
i f
8"+'~ < 0
Appendix
(i)
For
n >_ 2 ,
2
i' e ~R n-1 , ! n c ~R , e > 0 and
1 := (l',ln)
~ 0
let
(~i l l ' l , l ~ . ]
(A.70)
~ompare
:=
A.I.Koshelev[35,b]p.
says that the functions M
independent
of
defined by
with
i' = O. But for Koshelev's
tion
Mn
~I]
At page (A.70)
131 of his paper,
satisfy condition
e > O. But this is not true.
are not differentiable
C=([ll'l
114).
x ~ql)
.
respect purpose
to
would hold independent
ll,...,in_ I
it w o u l d
But condition of
First,
suffice
Koshelev
Mn
with
this functions at points
that
1
with
~(e;ll'I,l n)
M n is not satisfied.
If condi-
~ > O, there must be a constant
such that
for
every
Therefore,
o
with
la] ~ n
, ll'l ~ I
in the case n > 3 and --
i'
, i n E ]R
~ 91~}11
'
-
Since
(A.71) would
imply t
Since
4
, i n
must have
•
and every
:= (1,1,0,...,0)
0
a
e > 0 -I := e we @
M
229 i
there must follow for
every (ii)
e > 0 , what is a contradiction. In [35]
Koshelev defines
certain functions
At page 136, line 4, he states that this functions If one wants
to verify this,
one has to prove:
Pj,s(e; Ii'I ,in).
satisfy condition M n-
There is a
M > 0 such
that
(A.72)
I~~
for every
~
1 = (l',in) + 0
too% with positive treats (2.1)
I111,~'+I
in [35]
m M <
and every
Iol i n , where
imaginary part of the polynomial
Tk+(l ' ) is a
(2.1). Koshelev
only the case, where all the roots of the polynomial
have multiplicity
I
for every
i'
his considerations
apply to the Laplacian
root with positive
imaginary part is
with
Ii'I = I. Therefore,
A . In this case the only
~(i') = ill'I
By (A.72) we have
to show (A.73)
4 l~-÷~,t1'l I 111 I~+I.
I ])@
for every
I~0
~
bi
-:
witha proper
M, and
lol
But there are two objections:
First,
ll'I
I' = 0. But this is not essential, (A.73)
Let
for
ll'I >I
%1~%Iz
l~+i ll'l
.
is not differentiable
again it would suffice to prove
Second,
llq~
i~ := (1,1,0, .... O)
and
I= (I:,I.1 what contradicts
in
(A.72).
ll't~ let
i
n
c ~R
%1'I
(l~+i }l'l) ~
be arbitrary.
Then,
for
List of notations
=
the set of all integers
> I
=
the set of all integers
=
the set of all nonnegative
=
n - dimensional Euklidlan space
+
IR n x
=
(x I, .... Xn)
integers
are the points in
(n ~ I)
IR n.
(x,y) :=
1 x,y
c m n
For
G c IR n
of
G
~G
= G\G
G~ c c Let
we denote with
and with .
G2
G c IR n
C°(Q)
8G
~G:=
means
G
the closure of
the set - t h e o r e t i c a l IRn~G
(k ¢ ~ )
G-~
denotes the set G
compact
all
and
f ~ C°(G)
~
c G2 •
bounded in
having continuous
k - t h order in
denotes the set of all
with all partial derivatives
G.
f c ck(G) which are together
up to the
k - th order
uniformly
G.
denotes the set of all
f c ck(G),
partial derivative of order
C~(G),
G, that is
resp.)
denotes the set of
(k e ~ +)
on
boundary of
of all complex valued continuous
partial derivatives up to the
ck(G-~
the open kernel
be a bounded open set, G + ~ . Then
functions in G (in
C~(G)
G, G
.
: G i c ]R n, Gi open,
(resp. C°(G))
Ck(G)
for
:= (x,×) 2
Ixl
and
~___ xiY i i =1
< k
such that
f
has a continuous
and
every
extension
G °
0<~
denotes the set of all HSlder continuous
f ¢ C o. ( G) ,
that is (x', x") ~ ~, ~ c'..-
If
ck+~(G),
~--1
k c ~
,
f
is
, 0<~!I
Ix'-
~'~1 ~
called
Lipschitz
continuous.
denotes the set of all
every partial derivative of order
< k
belongs to
f e C~(G) such that Ca(G)
231
If
f
is defined
tes the support C~(G)
~n of
C[(G)
= ~
C~,k(HR)u
xo ~ 3 G
3.1
be open.
For
z(x)
= (z1(x) ..... Zn(X))
(i)
Z(Uxo)
(ii)
Z i ¢ ck(Uxo ),
(iv)
z
--
3G
z : G
.
k ~ ~,
:
~G
U--xo
R >0
-+ K z =
y >0
n -U x o
-
[z : I z l <-R
' zn : 0
-~
[z : Izl < R
, Zn > 0 ]
if it exists
means the surface
K c IR n
If
f,g
is measurable,
are measurable
if the integral index K . If
For every
and a topological
exists.
J[z(x)]
satisfies
x ¢ U-xo
(e.g.
with if
~=
If
:
[z : Izl < R} such that
such that the Jacobian
in
]
v(x) = (v I (x), .... Vn(X)) 3G
~ ~i(x)~L~(x) means differentiation i=i the outward normalL~ ×
n
means
x i := (Z-1)i ~ C k ~ z ) , i--I .... ,n ,
G N Uxo
the outward normal,
c Ck
Uxo , a
is a bounded open set, we denote
(~)(x)
whose support is a compact
Kz
for ever~
:
deno-
"testing functions"
there is an open neighborhood
(iii) there is a
If
f ~ ck(G),
the set of
see Definition
IJ [ z(x)] I > Y
[x ~ G : f(x) ~ O}
f.
C~(G)
:
(f) :=
G.
~ ~ G c ~R n
map
supp
denotes the set of all subset of
Let
G, then
E C I) in the direction of
of the unit sphere in Euklidean
~(~)
denotes
K , we denote
n -space.
the Lebesque measure of
K.
:= ~ f(x) g(x) dx~ K If no confusion is possible, we suppress the
l i p < ~ , f ¢ LP(K),
(f'g)o,K
llfllo,p(K) = [IflILP(K)
denotes the
232
Lp - norm
of
f.
If
measurable
in
G
with
fIK
s LP(K)
, where
G is open,
LPoc(K)
the p r o p e r t y fIK
I
means
that
is the
set of all
for e v e r y
restriction
of
K ~
f
f
being
G
to
K .
I
Often we denote
with
~,~,...,s,t
a multiindex
~ =
(m1,...,~n)
with
n
nonnegative
x
:=
-< ~ ~<~ czI
integers
xI
Gn ...x n ~n
:~
> ~i
:<
~
~I!
:=
for
<
I~I := ~ - ~i i=I
"
IR n
x
for e v e r y
~i
Then
i=1,...,n
~ and for at least
....
( )
~i ' i = I ..... n.
one
i o is
eio < 8io
(Znl
°'
for
8 <
~! (~- ~)t := ~-x~) .... \ ~ x ~ ] If a f u n c t i o n
f(x,y)
we often write to the For
k=1,...,n
Let
Then, I ~p<
,
for
=
w ~ LP(G)
,
(x,y)
Dff(x,y)
x -variables
delta.
is c a l l e d
-derivative,
~ IR n x IR k
to indicate
that
I~I
its order.
is considered,
~ c ~+n
differentiation
with
,
respect
is meant. e k :=
Je ~4
and
a
:
let
is called
(61k,...,Snk) ])3e~
G ~ IR n a
=
,
where
8ik
is K r o n e c k e r
s
~ )
be an open
D~ -derivative
set,
of
u
and let
u
c LP(G).
if
G
holds
for e v e r y
we denote
with
vatives
Dau
linear
space.
By
• ~ C~(G) ~'P(G)
¢ L~(G)
. Then,
the
we
denote
set of all
for e v e r y
~
with
w
by
u ¢ LP(G) I~I ! m .
4
D~u having
. For m = 0 , D~-deri-
Wm'P(G)
is a
I,2,..
233
a norm is defined on a
Wm'P(G). Equipped
Banach space. Often we write
wm'2(G)
is a Hilbert space
with this norm, Wm'P(G)
is
llUllm,p - llUllm,p(G) -= liuj}wm,P(G )
with scalar product
(~'g)m := ~- (D~f'°Bg)o lal<_z ' P G( ) W~loc flK
is the set o~ all
~ Wm'P(K)
w°'P(G)
:=
f ~ LPoc(G)
such that for
every
K ~CG
.
LP(G).
For the properties of these spaces see e.g. [3], [17] ,[5], [49], [61 ].
WP(G)
•
is the closure of
Co( )
o
with respect to the
(Observe that we use another norm in For
I < q < ~
~,~o'P(G) , see (1.1))
we use the abreviation
Sq := (u E ~oo'q(G) : jlUllm,q = I} to denote the unit sphere in
Wm'P(G)-norm.
w~'q(G) , (compare p.91).
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equa-
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