On Dirichlet's Boundary Value Problem: LP-Theory based on a Generalization of Garding's Inequality

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 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= ~+~ 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

e0

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 0

=

C-{) ~

L,(1)

=

(-4]~(1~) -~

~ := [ i n c~ : I!nl
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 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 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

_
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

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


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
0 is c o m p a c t

I

$,~



q e C~(KR) , 0
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 O) , it converges to

. On the other hand it converges w e a k l y to

~ Lp'(GR)

,

I 0 . Let

we h~ve

o

xn

be arbitrary,

but

JXnl
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)

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])= ~ 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 -
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 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 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
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


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
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 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 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

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~

- ~~, 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 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 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 .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
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

0 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 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
__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 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 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 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, ' ) ~ 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 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 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,) 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~lm.

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 I and k > m

are integers

and that I ~ 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 _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

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


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




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 1

is an integer,

I 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 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 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 0 independent of C and f, such that

Ilfllm, p


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 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__

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 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

01

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 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

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


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 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

:=

-< ~ ~ ~i

: