Weighted Inequalities and Degenerate Elliptic Partial Differential Equations (Lecture Notes in Mathematics, 1074) 3540133704, 9783540133704

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Lecture Notes in Mathematics Edited by A. Oold and B. Eckmann

1074 Edward W. Stredulinsky

Weighted Inequalities and Degenerate Elliptic Partial Differential Equations

Springer-Verlag Berlin Heidelberg New York Tokyo 1984

Author Edward W.Stredulinsky Department of Mathematics, University of Wisconsin Madison, Wisconsin 53706, USA

AMS Subject Classification (1980): 35J 70 ISBN 3-540-13370-4 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-13370-4 Springer-Verlag New York Heidelberg Berlin Tokyo 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 payable to 'Verwertungsgesellschaft Wort", Munich.

© by Springer-Verlag Berlin Heidelberg 1984 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210

TABLE OF CONTENTS

Introduction . . . • . . . . . • • . . . . . • . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter O.

Notation •.•.•...•...•.•••..•.....•...•..••.•••.•.•.•

3

Chapter 1.

Preliminary Analysis .••........•.••••.••.•••••......

3

1.1.0

Calculus in Measure Spaces ••••.•.•.••••.•.•••••••••.

4

1. 2.0

Weighted Hardy Inequalities .•....••••••••••.•••.....

12

1. 3.0

Equivalence of Capacities ..•.•.••••••••••••••••••.•.

25

Chapter 2.

Basic Results ..•.•.•.•.•.•..•••.••.•.•••..•..••••.••

32

2. (.0

Weighted Sobolev Inequalities •...•.•..•.••••....••••

33

2.2.0

Properties of Sobolev Spaces, Capacities and Sobolev Inequalities for Applications to Differential Equations .•••••••...•.••••..••••......•

2.3.0

Chapter 3.

45

Higher Integrability from Reverse Holder Inequalities ..•.•...•••..•....•....•••.••••..•......

89

Applications .•.••...•.••.•...•..•.•....•••••...•..••

96

3.1.0

A Harnack Inequality and Continuity of Weak

3.2.0

Modulus of Continuity Estimates for Weighted

3.3.0

Higher Integrability for the Gradient of

Solutions for Degenerate Elliptic Equations

.

97

Sobolev Functions and V.M.O. Functions ..•••••..•.•.•

126

Solutions of Elliptic Systems and Applications to Continuity of Solutions ••....••.••..•••.•••••.•..

134

Bibliography •••••••••••••••.•.••••••.•.....••..•••••••••••..••••

140

Index .•.•...•....•••..•.......•...•.•••..........•.....•..•.....

143

INTRODUCTION The main purpose of these notes is the analysis of various weighted spaces and weighted inequalities which are relevant to the study of degenerate partial differential equations.

The usefulness of

these results is demonstrated in the later part of the text where they are used to establish continuity for weak solutions of degenerate elliptic equations. The most important inequalities dealt with are certain weighted Sobolov and Poincare inequalities for which the admissible weights are characterized.

Weighted reverse Holder inequalities and

inequalities for the mean oscillation of a function are dealt with as well.

A much larger class of degeneracies is considered than

previously appears in the literature and some of the applications are known only in the strongly elliptic case. Two approaches are taken to the problem of establishing continuity of weak solutions.

The first approach taken involves a

Harnack inequality and the second Morrey spaces. equations of the form

div A = B,

where

A, B

The first applies to satisfy the growth

conditions

A Harnack inequality is proved for weights capacitary conditions.

A satisfying certain

Interior continuity follows immediately from

this, and a Wiener criterion is established for continuity at the boundary. This generalizes work of D. E. Edmunds and L. A. Peletier [EP], R. Gariepy and W. P. Ziemer [GZ], S. N. Kruzkov [K], M. K. V. Murthy and G. Stampacchia rMS], P. D. Smith [SM], and N. S. Trudinger [Tl]. A theory of weighted Morrey spaces is developed which establishes continuity estimates for a wide class of weighted Sobolev spaces Wl,p

with

p > d,

d

the spatial dimension.

This is in turn applied

to prove the continuity of solutions of systems of the form div Ai = Bi, i = l, ••• ,m, where Ai and Bi satisfy growth

2

conditions similar to the above with

p > d -

E.

In the non-

degenerate case this is due to K. o. Widman [WI] and, in more general form to N. G. Meyers and A. Elcrat [MYE]. It is necessary to mention related work of E. B. Fabes, C. E. Kenig, D. S. Jerison, and R. P. Serapioni rFKS], [FJK] which was done independently at the same time as the work presented in these notes. The approach taken and the material covered differ considerably but there is a certain overlap (see comments before 2.2.40 and the introduction to Section 3.1.0.). The following is a brief description of the contents of each chapter.

The reader interested mainly in the applications should

proceed immediately to Chapter 3. Chapter 1 contains the basic analysis needed for Chapter 2.

The

relationship between maximal functions, covering lemmas and Lebesgue differentiation of integrals is reviewed.

A calculus for functions

absolutely continuous with respect to a measure is developed and the admissible weights for several new variations of Hardy's inequalities are characterized.

Finally, several comparability results are proved

for "capacities" and set functions which appear later in the analysis of the weighted Sobolev inequalities. Chapter 2 deals mainly with weighted Sobolev inequalities and properties of weighted Sovolev spaces.

The characterization of

weights for Sobolev inequalities is carried out in a very general setting in the first section and is translated to a more useful form in Section 2.2.0 where, in addition, some examples are developed.

The

main thrust of Section 2.2.0 however, is the development of results relating capacity, quasicontinuity, convergence in weighted Sobolev spaces and weak boundary values for Sobolev functions.

Chapter 2

closes with a result on weighted reverse Holder Inequalities. All direct applications to differential equations are contained in Chapter 3.

These include the Harnack inequality as well as the

interior and boundary continuity results for weak solutions of divergence type degenerate elliptic equations (3.1.0); modulus of continuity estimates for Sobolev functions and functions of vanishing mean oscillation (3.2.0); and the continuity result for weak solutions of degenerate elliptic systems in a "borderline" case (3.3.0). I would like to express my sincere thanks to William Ziemer under whose guidance this work was completed.

I would also like to thank

David Adams, John Brothers, and Alberto Torchinsky for conversations pertaining to this material.

CHAPTER 0 The following is a short list of conventions and notation to be used throughout the text. Sections, theorems, and statements each are labelled with a sequence of three numbers, the first two denoting the chapter and section, the third denoting order within the particular section. The Lebesgue measure of a set

E

is represented as

represents n-dimensional Hausdorff measure. sup, inf

The abbreviations

will be used to represent the essential supremum and infimum

unless it is specified otherwise. radius

Hn

lEI.

r > 0,

centered at

x.

B(x,r)

is the open ball of

The specific space in which

is contained will be clear froM the context.

B(x,r)

Sometimes the notation

Br B(x,r) is used. XE is the characteristic function of the x e E set E, that is, The letter c will be XE(x) = {I 0 otherwise used to represent constants which may differ from line to line but

.

which remain independent of any quantities of particular importance to the specific calculation being carried out. LP(lll,E) is the space of equivalence classes of measurable functions

J lulPdlll Ilu

=


x• Let G Be a covering of E>. consisting of such WlBf B sets and use 1.1.2 to get Fe G, F, an at most countable collection of pairwise disjoint sets with

w(E>.)

clw(U B),

(

so that

F

• Proof of 1.1.4. f

Ll(w,O)

Given

since

f

LP(oo,Q),

1 < P

it follows that

Without loss of generality, assume

00(0)

f > O. Let f>. = X{f>>./2}f so that f ( f>. + >./2 and Mf ( Mf>. + >./2, but then w({Mf > >.}) ( OO({Mf>. > >'/2}) 2c l ( -->.-

f

f

2c

= -->.-l f

f>.dw

{f>>./2}

MfPdw

P

f

and

>.p-loo({Mf > >'})d>' ( 2pc

0

2pc

f dw

l fQ

f

f

2f

0

>.p-2 d>. doo

Proof of 1.1.5.

Let

diam B < a}

Lf(y) ( Mf(y) + If(y)1

so

If

g

a+O

f

B

>.p-2

f

f

{f>>'/2}

0

c

f l Q

gn

continuous such that

+ Lg n = L(f - gn) 2(c + 1) l >.

everywhere.

f

•

and so

If - gnldw

f dw d>'

•

fPdoo •

If - f(y)ldw

and

2(c > >'/2}) (

l

>.

+ 1)

is continuous and integrable, then it is clear that

Choose

(

P - 1

Lf(y) = lim SUP{oo(B)

w({Lf > >.}) (w({Mf > >./2}) + w({lfl

l

gn + f

in

Ll(w,O).

f

Ifldw • Lg

w({Lf > >.}) ( w({L(f - gn) > >.}) +

0

as

n

Thus

Lf = 0

=

O.

Lf ( L(f - gn)

almost

6

1.1.8

Covering Lemmas

The covering lemma 1.1.9 is a direct aeneralization to doubling measures of a standard covering lemma for Lebesgue measure. For nondoubling measures this may be replaced by Besicovitch-type covering lemmas, a very general form of which is proved in [MR], the proof following the basic outline in Besicovitch's original paper [B]. A more accessible reference is [G]. 1.1.9 set

Proposition. n,

Le.

If

00

w(B(x,2r»

is a doubling measure in a bounded open

(Cww(B(x,r»

for all

x, r

such that

B(x,2r) n, then the covering property 1.1.2 holds with being the collection of all balls Ben with y e B.

{Ly}yen

Proof.

Proceed as in rST], page 9.

1.1.10

Calculus for Functions Absolutely Continuous to a Measure.

The basic calculus for functions absolutely continuous with respect to a measure If

00 00

closely resembles that for

00

= Lebesgue measure.

is a finite positive Borel measure on

f: ra,b) + R, respect to 00

then it is said that if

Y£:>0:il6>0 the Ii = [ai,b i) intervals, then

f

[a,b)

and

is absolutely continuous with

L 00(1.1. ) < 6, where i=l are pairwise disjoint

so that if [a,b)

L

i=l

If(b i) - f(ai)1

(1.1.11)

< £: .

As a direct consequence f is left-continuous and in fact discontinuous only on atoms of w. Let

N = {y e [a,b) : wry,x) = 0

for some

x > y, x e [a,b)}.

is a countable union of disjoint maximal intervals of measure zero, and so

N

weN)

o .

(1.1.12)

The results of the previous section will be applied to the measure space [a,b) - N, with BY consistina of all intervals ry,x) with x > y, x e [a,b), so that Mf(y) = sup ([ 1 » f f dw. y 0

gP(t)dv(t) )l/p

and all nonnegative Borel measurable

where

-v

g

iff

(1.2.15)

and sup r(O

( 1. 2 .14)

v(t)-l/(p-l)dt)l/p' r is as in Theorem 1.2.1.

15

If ci' i = 1,2,3, b i, i = 1,2 possible, then b i ( ci (pl/qpll/plbi max{b

- f

o

( c

2,b3}

(p

3 for

g(s)ds

t

throughout.

t

Also

are chosen as small as for i = 1,2 and

1/ qpl 1/ P max{b I

The convention 2,b 3}. is used in (1.2.14) and = 0

< 0

pll/pl

=

1

for

p'

f

t

o

g(s)ds

is used

=

The inequalities dealt with in Theorem 1.2.16 depart somewhat from the structure of the classical Hardy inequalities, but their analysis is similar.

They arise naturally in the study of certain

Sobolev inequalities.

It is somewhat remarkable that (1.2.23) and

(1.2.24) are equivalent since in general their left-hand sides are not comparable unless 1.2.16 Theorem.

(f

fS

t

For

1111
{(1.2.l8), (1.2.2l)} ==> {(1.2.l9), (1.2.22)} ==> (1.2.23). (1.2.23) ==> (1.2.24) is trivial.

Recall 1.2.25.

To show (1.2.24) ==> (1.2.18),

first reduce the interval of integration on the far left of (1.2.24) to

and replace

g

by

to get

J r

(Jr IJ

JSg(a)da

r

for bounded s, t < r.

g

dA(s)lqdp(t»)l/q < c

r

of compact support since

2

'

r

t

J

= 0

s

Take monotone limits to get all positive measurable

a similar way (1.2.24) ==> (1.2.21).

if both q.

{(1.2.l8), (1.2.2l)} ==>

{(1.2.l9), (1.2.22)} by the first part of Theorem 1.2.16.

From

Theorem 1.2.1 it is seen that {(1.2.l9), (1.2.22)} ==> {(1.2.2), . v(t) (1.2.5)}, w1th v(t) replaced by so replacing g ( t ) by

< v(t),

and using that

(J

(J

IJ

t

s

+

g(a)daldA(s»)qdp(t»)l/q

(J

(J

t

t

J

s

t

+

it follows that

(J _00

(J

t

_00

and the proof is complete.

g(a)da dA(s»)qdp(t»)l/q

In

24

Proof of Theorem 1.2.26.

I

II

I

t

s

I

(J

I

g(a)

+

t

00

f(a,t)dp(t)da

-00

for

I

I

Let

).(a,oo] ,

o

f(a,t)

a > t a t a < t

,

{ x r _00, a),

g(a)h(a)da

(1. 2. 38)

h( a) =

for

t

I

+ A[-OO,a)p(a,oo].

EO = (t : h(t) = OJ,

Assume (1.2.27).

E00 = (t : h(t) = oo}.

Replace

of the singular part of

g

by

gX A,

to get

v

where

A

is the support

(1.2.39} From this it is easy to see that

-v

on E 00 so that hP(t) v(t) = 00 on E using the convention for that 00 hP(t) v(t) Also v*(t) = 00 on EO using the -0 = 0 convention for 0 it is not true that

I

g(s)ds
1

let

g(t) = v*(t)-l/(p-l)

in (1.2.40) to get (1.2.28).

p

1

let

gn (t ) = 1 X[t-n,t+n]

For

n

25 in (1.2.40), and let

n +

to differentiate the integrals and

achieve (1.2.28). Assume (1.2.28). hP

= 0

vet)

or

0

o

h(t) g = 0

Also it can be assumed that

implies that

v

0=0

-- < a.e., and so considering the v positive measure

either

h P l/(p-l)

(::-)

The integrability of

convention, off a set of

for a qiven

a.e. on

t

(1. 2 .41)

•

{t: v(t)

since

otherwise (1.2.27) is trivially true.

Considering this and (1.2.41) P lip it follows that g(t)h(t) ( a.e. and so vet) Holder's inequality applied to (1.2.38) gives (1.2.27). 1.3.0 Equivalence of Capacities The set functions which arise naturally in the analysis of the

Sobolev inequalities treated in Chapter 2 are difficult to work with in their original form except in special cases.

In the present

section they are shown to be comparable to more familiar capacities and set functions. Let

(M,F,v)

be a measure space with

be a set of real valued

F

positive, and let

v

measurable functions on

satisfying the following properties. composition with functions bounded, and

f'

) OJ.

: f(O) = 0,

feN = {f e

There is a map

M

is closed under IDI

:

f'

is

LP(v,M)

+

such that (1.3.1) 1, P ( v,M, ) e Wo

for all

oS

If

1 ( p

d-la < b (da

iff

a -d b

h . were t h e notat10n

0


0 .

A(M}

It is clear that this is impossible if

A(M}

In applications

is typically equal to one.

1. 3.5 Theorem.

KH,p (A) CH,p (A)

-d Cn,p (A) -d Cn,p (A) P P

if

I f ( 1. 3.4) holds, then

Remark.

,

i f (1.3.4) does not hold

(1.3.6)

A(M} = 1

(1.3.7)

KH,p (A)

(1. 3.8)

0

Under fairly general circumstances, it is possible to show

that another comparable expression is

e

< 0

H,

inf{t :

)

t}}

on =

O}

A

I} .

Proof of Theorem 1.3.5. (1.3.8) follows from the definition of and (1.3.3). (1.3.7) follows by replacing with 1 and using

f(x}

=

1 - x

KH,p

in (1.3.l).

It can be assumed that there exists a e H with < 0 and > O}} > 0 since otherwise K from its (A) o - H,p definition and the -0 = 0 convention, and C (A) = using the H,p convention inf = if G = ¢, since it would be true that 1.3.9.

on

A

J

o.

dA
1,

-

inf a (a,b)

inf v (a,b) a

then (1.3.15) will be proven for

-

Vi a-

a = 1,

in which

making the sUbstitutions q = g'a and v = and recallinq that o < a < DO on (a,b) , it follows that {1.3.15} holds for :Jenera 1

a.

=

a

Assuming

an inequality in one direction is obtained by

1,

letting q = X v-l/(p-l) (Jb v{t)-l/{p-l)dt)-l as long as b (a,b) a J v{t)-l/{p-l)dt < DO, otherwise a construction virtually identical a

to that in the proof of the first Hardy inequality gives 9 such b b that J g{t)dt = DO and J gP{t)v{t)dt < DO and so, letting a a max{a,-n}, b n mi.n Lb n} , a n i

E

=

n

{g

< n} n (a ,b ) n

n

and

(J

b

9X

a

E

b

(J

dt

)-1

and

1

J

n

b

as

a

a

n +

00

,

the same inequality follows. The opposite inequality is a consequence of Jensen's inequality. The inf is not increased if only 9 G supported in (a,b) are considered. Given such a g, let gn be as above.

(J

b

(v +

a

< (J

spt gn

(v

+

< Jb a

+

31

f

by Jensen's inequality since Let

E

b

gndt = 1. a using the monotone convergence theorem on the left,

+ 0

then

f

b

a

f

a

gPvdt n

(f

b

gPX

b

E n

vdt

f

+

gPVdt

a

9XE dt)P n

a

b

as well by the monotone convergence theorem, and so the opposite inequality holds and therefore equality as well. (1.3.15) will now be proven for smooth

Given

q.

e G,

9

pick

such that bounded and positive with compact support in (a,b) 00 in (a,b). Let om be a C approximate identity with

gn

+9

gn

f

om ) 0, 0m* gn

om

=

1,

and the diameter of the support of

has compact support in

independent of lim f m+OD a

b

m,

gn *

(a,b)

for large

m

om +

o.

Thus

and is bounded

so

smodt

f

b

q odt -n

a

and lim f m+ oo a

b (gn*

s III )PVdt

f

=

b

qPvdt -n

a

by the dominated convergence theorem since v is integrable on compact subsets of (a,b) and a is bounded uniformly away from 0,00

°.

on the support of 9 n * m The monotone convergence theorem now leads to lim lim f n+ oo m+ oo a

b

gn *

sm

f

b

go

1

a

and lim lim f n+ oo m+oo a

b (gn*

sm )PVdt

f

a

b

gPvdt

From this can be extracted a sequence that f

f

a

f

f

k b

f

f

e k b

a gPvdt

a

k

b

fko + 1,

i f follows that

fkOdt

and

f

a

b

fk

Uk}, b

f

a

fkOdt

and 1. 3.14 is proven for smooth

=

1

g.

°

=

9n * b k + f gP\idt. a b

and

•

f

a

such Letting +

32 1. 3 .16 Lemma.

If

inf J gPVdt, Proof.

v

the

Pick

s

is as in Lemma 1, then inf

being taken over

inf J gPdV

G n

a support of the singular part of

v

with

lsI

=

0

open such that s £ On' a,b e On' ann IOnl + o. Since collection of pairwise disjoint open intervals, it is easy to On is a construct CWo(a,b) functions . (each . = 1 off of a finite n s i, n, number of the intervals such that . + X( b)-O everywhere on a, n (a,b) with o 1 on (a,b) - On. It then . '1, and and

On

follows for

9

e

n,l 00 G n CO(a,b)

.J..

that

J

gPvdt

(a,b)-On and b

lim

J

J

.adt

i+ CD a

(a, b)-On

gadt ,

so lim lim 0+00 i+ CD

J

lim lim n+ W

J

.)PdV

nsa

and .dt

J

b

a

From this extract a sequence gk

e

G

W

n cO(a,b)

lim k+ w

J

1

b J

.

so that

a

and

J

9 dt

gP'Vdt

and the result is proven.

•

CHAPTER 2 The results of Chapter 2 form the foundation on which Chapter 3 is built. For the most part they involve weighted analogues of important basic tools used in the study of partial differential equations.

33

In Section 2.1.0 the weiahts for several Sobolev inequalities are characterized in a very general setting.

Section 2.2.0 develops the

theory of weighted Sobolev spaces, weighted capacity, and weighted Sobolev inequalities in a setting appropriate for the application to differential equations.

An example is developed in which Sobolev

inequalities are proven having weights of the form class of sets

K

disto(x,K)

for a

including unions of manifolds of co-dimension) 2.

In Sections 2.3.0 a result on "reverse Holder" inequalities is developed which implies higher integrability for functions satisfying a maximal function inequality. 2.1.0 Weighted Sobolev Inequalities Conditions equivalent to two types of Sobolev inequalities are developed involving the dominance of measure by "capacity".

It should

be noted that V. G. Mazya [Ma2] has proved 2.1.7 for v = lebesgue measure and M = Rd and O. R. Adams rAl-3] has done the same for higher order inequalities (as well as two-weiahted inequalities for potentials).

He has also shown that 2.1.9, in the special case

described above, is needed only for

K

which are balls.

After having

discussed my results on Sobolev inequalities with me, Adams found an alternative proof for 2.1.7 and some cases of 2.1.20 using strong type capacitary estimates, the study of which was initiated by V. G. Mazya [MAl] • Let

(M,F,oo)

positive.

Let

and (M,G,v) be measure spaces with Wl,p( oo,v,M ) be a set of real-valued F

oo,v measurable

o

functions satisfying the following properties. is closed under composition with {f e

f(O) = 0, There is a map

J d-IU

1

f'

101

feN =

is bounded and of one sign}.

such that

101 :

+

LP(v,M)

10f • cl>IPdV)l/p < (J If' (cI»IPIOcl>IPdV)l/p

and

(2.1.2)

< d(J 10f • cl>IPdv)l/p

for some fixed

d > 0,

where the notation

IDcI>1 = Inlcl>

is used.

The symbol Inl is only meant to suggest the absolute value of It should be noted that the gradient on the classical wl,p space. l,p( oo,v,M ) are Special cases of Wo 101 need not be sublinear. developed in Section 2.2.0. . ) let be the finite measure defined G1ven cI> e W1, P ( oo,v,M, by

=

respect to

o

10cl>IPdv.

will be the distribution measure of

that is,

*

=

-1

(E»

so that

J

cI>

with

*

34

=J

for all Borel measurable g. density of the ahsolutely continuous part of

let p'

he the will always

represent the exponent conjugate to p, that is + = 1. proofs of the following theorems will be deferred till later. 2.1.3 Theorem.

If

1 ( P ( q
0

and all

=

u

f

feN,

iff b
0 1/

w

sup

and all

q (111 < r ) (J

..

u = f

feN'

t}) l/(p-l)

(_:

)

dt)

iff lip'

< ...

(2.1.14 )

r

If

x (M)

w(M) < .. ,

1,

1

and

e L- (A,M),

then (2.1.15)

for some

c2 > 0

and all

=f

u

f

e Nl

iff (2.1.14) holds.

The conventions 0·" = 0 and. for = 0, = 0 are used. If the c. are chosen as small as possible, then < c < c < b . 1 2 l l If H c Wl,P(w,V,M) and H is closed under composition with feN',

then for

A C M let

CH,p (A) = inf{J )

:

1

on

A

e H n Ll(A,M) , and

j

=

o} ,

=

I}

and

n

C ,p (A)

inf{j < 0

:

on

A

e H n and

j

In Theorem 2.1.17 it will be assumed that if

e H,

then

) t}) < ..

for

t > 0

(2.1.16)

36

CH,p and defined in Section

If this is not the case, then the theorem still holds but

c'H,p

KH,p

must be replaced by the set function

1. 3. O. 2.1.17 Theorem.

1 < P < q
0

and all

u

e H iff

1ll1/q(A) < b e·l/P(A) 1 H,p for some If

(2.1.19)

ep bl > 0 and all sets A = {ep < O}, 1 ).(M) 1, lll(M) < and H c L ().,M),

(J lu(x) for some

- J

c2 > 0

u(y)d).(y)lqdlll(X))l/q < c and all

u

2

(J

e H. then

IDUIPdv)l/p

(2.1. 20)

e H iff

1ll1/q(A) < b el/P(A) 2 H,p for some

b2 > 0

(2.1.21)

and all sets

A

=

{ep < OJ,

ep e H,

iff (2.1.19)

(A) c C (A) < oPC (A). H,p H' ,p H,p i = 1,2, b i = 1,2 are chosen as small as If c i ' i, 3 < < l/q ,l/P'b . = 1, 2• b . possible, then d J. ci "p Pi' J.

holds since

d-PC

2.1.10 is applicable to K ,p , e ,p ,and CH,p as well as H H' and C H,p H,p Theorem 2.1.22 is an example of how the conditions in Theorems

Remark: K

2.1.7 and 2.1.17 can be put into a more computable form when H = COUl), where n c Rn be absolutely continuous with density -v e 2.1. 22 Theorem. If

p = 1,

Let

s... o

for some c > 0, dist(x,A) < Ii}.

-v

is open, and let

v

then condition (2.1.9) is equivalent to

. f "5 1 1ll 1 / q (A) < c I'J.m J.n

If

p = 1.

all

J vdx

(2.1.23)

Cli

A

compact with

C

ee

boundary and

Cli

{x

f A:

is continuous, then this reduces to

1ll1/q(A) < c J

aA

vdH n- l ,

or, in a more suggestive notation,

(2.1.24)

37 If

H

=

G'

Q C G',

open and

then for

p

1

condition (2.1.19) is equivalent to ool/q(A)A(G - A) ( c lim inf ! J Il ...o Il C c > 0

for some

and all ee

compact sets with If

v

C

A,

vdx

(2.1.25)

s

closed relative to

boundary in

G,

which extend to

G'.

is continuous, then this becomes

ool/q(A)A(G - A) ( c J vdH n- l aA G

(2.1.26)

The proof of Theorem 2.1.22 will rely on the following proposition. 2.1.27 Proposition. t} n

that

2

If

, "6 1 J oodx I J.m Il ...o CIl(t)

where

CIl(t) If

=

¢,

O}

=

{x e

and if OOdH n - l

J

and

00

is open,

R

t e R

is such

is continuous, then

00

,

=

(t}

e

n

Q

e Co (G),

t}) ( Il}.

is an integrable Borel measurable

function, then lim inf Il ...o

i

for almost all

oodx

J

Cil (t)

(J

oodH n- l

t e R.

Proof of Theorem 2.1.3. for

E C R

Assume (2.1.4) holds. so that for all

feN

*

Let and

u

=

f

0

( 2.1. 28)

since

f(O) = 0

and the convention

t

J

J

s

t

s

is used.

Also,

(J IDuIPdV)l/p c d(J If' (2.1.29) =

Letting

g

=

follows that

If'l

and recalling that

f'

does not change sign, it

38

(J

00

11 t 0

_00

g(s)dsl

* 1/

q

(2.1.30)

_00

for all hounded nonnegative

*

q < cd(J 00

C

functions

is a finite measure since integrable hy letting g = 1

g.

e Also t q is seen to in (2.1.30). Taking uniformly

be bounded pointwise limits of bounded nonneqative

00

C

functions

g

it

follows that (2.1.30) holds for all bounded nonnegative Borel measurable g. Taking monotone limits then gives (2.1.30) for all nonnegative Borel measurahle functions that

g.

Using

it follows (1.Z-.14)

and

*

But

(2.1.31)

= w(

:>

considering the sign of follows that

Assume (2.1.5).

r}) r

and

*

= w(


;> 1

" b"

on

A

and

4>

considered for all sets

A

H}

= {W

;> I}

W

where

is a function in

H.

Using Theorem 1.3.2 it then follows that (2.1.8) and (2.1.9) are equivalent and d- 3b" c " d 3 p 1/Qp,1/p'h for b, c chosen as small as possible.

•

Proof of Theorem 2.1.12.

As in Theorem 2.1.3,

(2.1.13) reduces to (2.1.32)

" with

g

=

If'l

and

A4>*

* A4>{E)

defined as

=

A{4>- 1 (E»

for

Also, as in the proof of Theorem 2.1.3, the function

E c R. It - sl

has the necessary integrability properties to allow the taking of limits, thus giving (2.1.32) for all nonnegative Borel measurable g.

Using (1.2.23) and arguing as in Theorem 2.1.3 it is seen that

(2.1.13) is equivalent to sup r

•.• 1 I q ( {'" / 'f'

..

r})

J

r

AP ( {4>

*

;> t}») 11 (p-l)

dt

)1I p ,


(t)

comhined with sup r

II III

q({4>;>

r})(J

r

AP(",,, t. ) I/(p-l)

(:

ii 4> (t)

)

dt)

It is easy to see that this is just (2.1.14).

lip'


0

on

is defined essentially the same as before, -1

+

say

(to,t » is a diffeomorphism for all l s < d for some d.

is an integrable Borel function, then

J

42

1

"6 J

0

1

0

x .. 0

otherwise

1

;0

;0

b

h a, b(X)

where it is assumed that a" o .. b in the w1,p(n) case if w(n) = The cases a =

case or in the b = are

included. For each of the pairs

(x+'X{x>O}),

(lxi, sign x l ,

(ha,b(x):X{a 2 ,

and Vcjl

(J'

- Vcjl P l/(p-l) n 2 ml av ) + (J

cjln + cjl 2 m > 1 on Vcjl + Vcjl p I n 2 m, dv. the

lim sup, n,m+ ClO

where

+ Vcjl

Vcjl

I

n 2

P l/(p-l) ml dV)

nAm = min{n,m},

so

CH(K n Am) (

Using this inequality on the above and then taking

recalling that

C(K n) + C(O),

it follows that

lim sup n,m+ ClO for

p > 2.

A similar inequality holds for

cjln

1 < P < 2

The

cjln

converge to some

u

e

which can, by

Proposition 2.2.21, be chosen such that a subsequence to u

u,

and so

u > 1

is quasicontinuous. In addition,

J

so the

II LP(v,n). The inequality (2.2.29) now implies that K=l and so the cjln are Cauchy in are Cauchy in LP(w,n)

are Cauchy in the

d

IvulPdv = lim n+'"

and so (2.2.30) is established • •

cjln.

quasi-everywhere on

converges 0

and

61

Proof of (2.2.31). Using (2.2.30) and arguing as in Proposition 2.2.18, it follows that cH({lul > A}) ( J IVulPdv for all quasicontinuous

AP

e W6'P(O) • •

u

Proof of (2.2.32).

Proceeding as in the first part of Proposition

2.2.19 implies that a subsequence of the sets of arbitrarily small capacity.

un

converges uniformly off

Since the capacity of a set

E

can be approximated arbitrarily closely by capacities of open sets containing

E,

it follows that the exceptional sets above may be

taken to be open.

Let

u

Redefine it on a set of un

+

u

be the

quasi-everywhere.

with

u

un.

such that

+

u

in

W6,P(Q),

Given

E

there exist quasicontinuous E

0,

it can be seen from (2.2.30) that

e W6,P(Q)

un

such that

Un) 1

+

0

as

n

+

•

Using Clarkson's inequalities as before, it follows that the Cauchy and therefore by (2.2.32) there exists u e W6,P(Q) continuous and a subsequence

{nil

and pointwise quasi-everywhere, so quasi-everywhere on

E.

(hO,l(u)'X{O 1

Choose one such

..

is

e

for some

K'

C

£

0,

C;(Q),

then aK' n

is

CH(K). Given

on

= to

C o.

0

aKn

for some compact set

> I}

the Morse-Sard theorem t

K

CH(K).

If

Proof.

is compact, then

K C 0

n'

is open

0'

K

1 -

is a compact set with smooth boundary,

0 < E < 1,

)

and

O} n

with

E,

=

t}

=¢

E .. to .. 1 K

choose

.. CH(K) + E.

By

for almost all so that

> to}

and

to}

65

Sequences If K

can now be easily chosen. Q', K = K' n Q' for some compact set K' argue as above with chosen so that 1 on K, and

..

J

( C-H(K) +

Q'

and with

E,

C H(

.. to})

Q,

J

then = 0

Q' replaced by

.. to} n Q'). • 2.2.41 Theorem.

If

1 ( P ( q

then (2.2.42)

for some

cl > 0

and all

u

e

C;(Q)

iff (2.2.43)

for some

bl > 0

and all compact sets

K C Q with

boundary iff (2.2.44)

for some

and all Borel measurable sets E C Q. and v is absolutely continuous with respect to Lebesgue measure on with density v e Ll(Q), then (2.2.42) holds If

bl

> 0

P = 1

iff (2.2.45) ee

for some b l > 0 and all compact sets K with C C 6 = {x e Q - K : dist(x,K) ( O. If v is continuous, then this reduces to 1ll1/q(K) ( b J -vdH n-l . l dK III (Q' ) < A(Q' ) 1 and 1 ( P ( q < If

(J

Q'

for some

lu -

J

Q'

c2 > 0

udAlqdlll)l/q

and for all

(

c2 (J Q' u

boundary, where

then

IVuIPdv)l/p

e

,

(2.2.46)

iff (2.2.47)

for some b 2 > 0 and all compact set K' C Q, iff

K C Q

such that

K

K' n Q

for some

66

(2.2.48) b2 > 0

for some

P = 1

If

V e Ll(n'),

and all Borel measurable sets E c n'. and v is absolutely continuous with density

then (2.2.46) holds iff

wl/q(K)A(n' - K) ( b

lim sup 1 J vdx "0 C.

2

(2.2.49)

s

for some b 2 > 0 and all K c n' such that K K' n n' for some compact set K' c n, and where C = {x e n' - K dist(x,K) ( Il}.

5

ci ' b i ' i = 1,2, are chosen as small as possible, then ( c (2p 1/ qp' 1/' P b • c ( pl/qp,l/p'b l 1 and b 2 2 2

If

b

l

(

Proof.

Consider Theorem 2.1. 7.

It is claimeCl that (2.1.9), (2.2.43),

and (2.2.44) are all equivalent. both (2.1.9) and (2.2.43).

It is clear that (2.2.44) implies

(2.2.44) follows from each of these in a

similar manner so only one implication will he done explicitly. Assume (2.1.9).

Given a compact set

K

n,

Lerona

supplies a

sequence {K} of compact sets of the type considered in 2.2.40 such that Wl/q(K) ( Wl/q(K ) ( b Cl/P(K ) b Cl/P(K). Given a Borel set nIH nIH E n, use the regularity of 00 to choose a sequence of compact sets

such that

E

and

w(E),

so

and (2.2.44) is verified. The equivalence of (2.2.42) and (2.2.45) now follows directly from Theorem 2.1.22. The second half of the theorem follows in a similar manner to the first using Theorem 2.1.17 instead of 2.1.7 . • It will he shown in Theorem 2.2.56 that weights of the form distQ(x,K) admit Sobolev inequalities of the type (2.2.42) and (2.2.46). These will be used in Chapter 3 to demonstrate the Holder continuity of solutions of certain differential equations which have these weights as degeneracies. It will first be shown that two weighted isoperimetric inequalities hold under the conditions (2.2.51), (2.2.52), and (2.2.53).

The rather technical verification of these conditions for

specific geometries is left to the proof of Theorem 2.2.56. Let B(x O,2RO)

00,

v

d R,

be nonnegative Borel functions defined everywhere on d) 2.

For each

r,

0 < r (2R O'

let

Cr

and

67

Cr will correspond to Dr be Borel measurable subsets of B(X O ,2R O)' sets where v is "small" and Dr to sets where w is "large". Finally, let P z be the projection of Rd onto the hyperplane {x e Rd : x·z = O}, z e Rd, z t- O, Pz(Rd) will sometimes be

casually identified with Rd- l. a(d) measure of the unit ball in Rd. 2.2.50 Proposition. B(x,r)

will be the d-dimensional

Assume that the following conditions hold for all and z e Rd, z t- O.

,2R O)

») < Hd-l( P z ( Cr n B ( x,r

2

a(d) r d-l ,

(2.2.51) (2.2.52)

max B(x,r)-D for some

w < c2IB(x,r)I((d-l)/d)-1/q r

q) 1

and

exists a constant

(2.2.55)

If

X

where

cl' c2

c(d)

and

inf

independent of

(2.2.53) r

r, x, z,

then there

such that

is open and

X

£

w(E) = J wand E

The abbreviations

min B(x,r)-C

max, min

B(xO,RO)'

vd_l(E)

then

J vdH d- l • E

have been used instead of

sup

to emphasize that it is the true supremum or infimum which

is indicated and not the essential supremum or infimum.

The proof of

Proposition 2.2.50 will be deferred till later. Suppose K £ Rd, IRI = 0, d) 2, and a,B e R. Then let w(x) dista(x,K), v(x) = distB(x,K), and d A(t) {x e R : dist(x,K) < t}. 2.2.56 Theorem.

If

1 < q < d

following conditions hold for all

z

t-

a : d ) d + B-1,

l'

B(x,r)

0:

(2.2.57)

If

B > 0,

for some

then £1'

0 < £1
0 = A(E 2r) if a < 0 and Dr = ¢ if a ) O. Assumptions (2.2.51) r (2.2.53) is and (2.2.52) now follow from (2.2.57) and (2.2.58). verified as follows. Let D

There are a number of cases to consider, depending on the relative geometry of K and B(x,r) and the sign of a and Assume

B(x,r)

B(xO,RO)'

and let

rl = dist(x,K).

72

¢,

then

If

i

1,2,

since

and so

max B(X,r)-D r

min B(x,r)-C r

B.

a ;> 0

w"

a < 0 ,

,,;>

j

3r l

(-2-)

e

s .. e

0

> 0 .

If (3r)a max B(x,r)-Dr

a < 0 ,

(3r) min B(x,r)-C

a ;> 0

r

e

(Elr)e

e .. s

0

> 0

The proof of (2.2.53) is virtually identical in each of the cases so only one will be done explicitly. r

r


0

since

Iii = 0,

so let

v

appealing to 2.2.41, but there are interesting cases when this is not the case.

To handle these, Section 2.1.0 is used in conjunction with

a direct proof of the capacitary conditions involved. be proven that ool/q(A) < c R(a/q)-6 R(d/q)-d+li (A) sets

A =

2 K

< a},

H,l

0

It will first for all level

in which case it will follow from Section 2.1.0 and 1.3.5 that (2.2.62) is true in the case p = 1 and t = q. It will then be shown that (2.2.62) holds in general. Given on

A =

< O}

for some

e

H,

pick

We H with

A. ool/q(A)l({w ) t}) < ool/q({w < t})l(B(XO,Ra) - {w < t})

w
weakly if both u(x)

t

then for

weakly if -u(x) ( - t (t weakly and u(x) ;> t

Proposition 2.2.86 shows that under certain conditions this

87

definition of weak boundary values is virtually equivalent to a more conventional definition. 2.2.86 Proposition. u

e wl,P(O'),

in

0 - 0', If

for all

=

flo'

then

v

0' f

(coo

or

I

u, f

0

0,

and continuous

are bounded. then

u(x) = f(x)

Conversely if

(u -

n'

is open and bounded, quasicontinuous in

, vu - 'Vf 0 ,) e w6' p ( 0' ) ,

e ao'.

x

Remark. u(x)

and either

(u -

x e a0' ,

Suppose

e W6'P(0),

f

u (x )

=

f (x )

weakly

weakly for all

I

, X{u* f } ('Vu - 'Vf 0 • )) e w6' p (0 ' ) •

flo.

In the converse it is only necessary to assume that f(x)

ao'.

weakly quasi-everywhere on

This follows from an

argument similar to that in Proposition 2.2.77, where a capacitary extremal is used to remove an open set of small capacity. Proof.

e

Assume

the

in

0-0'.

If

+ Wn + wnl

'Vu,

+ f,

such that

+ 'VW

=

u=v

and

e

Wn

+

n) + 'Vwnl

o' + wnl o ' u,

,'Vu - 'Vfl o') e W6'P(O'). Choose + u - flo' in Consider the Co(O), let v be a quasicontinuous limit of

(u - flo'

such as functions in

(u,'Vu)

o' )

+ 'Vwnl o')

+ +

Wn

so

=

u f

+

f

in

in

quasi-everywhere on W6'P(0),

with

then

'Vu = 'Vv + 'Vf

(2.2.87)

almost everywhere, and

00

v

so

(u,'Vul o' in Wl,p(O'), but (u,'Vu) in Wl,p(O') also, so

almost everywhere .

Xo e ao' , pick K so that continuous in 0 - 0' , there eixsts an

Since f is f(xO) < K. r > 0 such that f(x) < K then n(u - K)+

Given

f! ' ) , B(xO,r) n (0 so i f n e K)+ = 0 n(f quasi-everywhere in

in

2.2.88.

o -

f! ' •

so that 2.2.5 implies that

and so 2.2.9 implies that (n(u - K)+,nX{U>K}'VU + (u - K)+'Vn)

e

W6'P(0) •

Following the proofs of 2.2.5 and 2.2.9, it is clear that is locally quasicontinuous (2.2.20) and continuous.

n(u - K)+

Proposition 2.2.77 now implies that

-) + ,nx{u>O}'Vu - + ((n(u - K u - K)+ 'Vn) \ 0'

e

(u - K)+

is quasi-

Wol,p( 0 ')

88

but (2.2.87) then implies that

since

v

is absolutely continuous to

w.

This is true for all

K > f(xO) so u(xO ({(xO) weakly. In the same manner it is shown that u(xO ) f(xO) weakly and so u(xO) f(xO) weakly for all Xo e aQ'. Conversely if

u(x) = f(x) weakly for all

x e aQ,

then

for Xo e aQ' and > 0, there exists an r > 0 such that l f l x ) - f(xO)! < for x e B(xO,r) n (Q - Q') and n(u - f(x O) )+ e for all n e From the first inequality it follows that -f(x) + f(xO) < 0 on B(xO,r) n (Q - fl') so n(-f + f(x )+ = 0 on Q - Q' O) for n e As in 2.2.88, it follows that n(-f + f(x O) )+IQ' e aQ' is compact since fl' is bounded so a covering of balls such as B(xO,r/2) can be reduced to a finite subcover that n(u - f(x.) - )+ and for ni

1.

e

n

B(xi,ri/2), i = l, •.. ,n, such n(-f + f(x.) )+ are in Wl,p(Q') 1.

Pick

e

and

ni'

ni = 1 Q'

on

n

-

i = l, ... ,n,

such that

B(xi,ri/2),

and

nO

0

e

U B(x i,r i/2), in which case i=l 2E)+ ( 2 )+ e Wl,p(Q) by 2.2.9. Also, f (u f no(u 0 )+ )+ , + (u (-f + f( xi) (u f( xi) so i f cjl f - 2 )+, then

such that

-

-

nO = 1

-

on

-

1.

n

L

i=l Q'

-

)+ + n.(-f + f(x.) -

)+

-

o ( cjl ( n.(u - f(x.) -

in

-

1.

1.

1.

-

n

U B(x. ,r./2).

Let W= nO(u i=l 1. 1. [n.(u - f(x.) )+ + n.(-f + f(x.) -

Q'

-

1.

and

1.

We

is compact. such that

1.

Pick

1.

)+],

0 (

cjl

(

cjl e Wl,p(Q) so cjl1n. e Wl,p(Q') lac' \ .. cjln e n Wl,p(Q') and W e

W

on

since

n

in

and

in

,p(Q' ).

Letting f(x) x+ and using 2.2.4 and 2.2.5, it follows that there exist f m e m = 1,2, ••. , and a subsequence n m such that {fm(cjln) fm(cjln converges in m m m

89

Wl,p(QI)

to

-

o
0}(Vu - Vf»\Q'

By the dominated convergence theorem this converges in «u - f)+,X{U>f}(Vu - Vf»ln" of

u, f

that

shows that

Doing the above for

(-(f - u)+,X{U 0

is doubling, that is, there

< c lwB(x,r!2)

for all balls B(x,r) £ Q. By iterating this inequality it is easily seen that there exists a constant a > 0 such that (2.3.1) for all

x,r,R,

0 < r < R,

B(x,r) C Q.

90 Let Ql QO n be open concentric cubes with sides parallel to the coordinate axes and with side lengths respectively. Also for ease in applying the doubling condition it will be assumed that QO lies at least a distance of 15 Id from

an.

LP QO

norms on

will be estimated in terms of

p > q.

for some

Lq

To accomplish this a continuous "iteration" will

be carried out on a parameterized collection oD cubes 1 ( t

where

parallel to

norms on

Qt is a cube concentric to and side length SIt) = +

Qt'

with sides

The choice of parameterization is related to the following estimate which can be used to show that a ball centered in some actually lies totally in measure is small enough. Given r < r
0 such that B(s,r). Using (2.3.1) it now follows that

so that (2.3.2) for all B(x,r) QO. The maximal functions to be dealt with are defined as follows for

o

< R (

MRf(x) = sup{lIl- l(B) For convenience let

J

Ifldlll : B = B(x,r) c n, 0 < r < R}

Mf

The super-level sets of functions

g, f will be of central importance in the main estimate for theorem 2.3.3. These are denoted by E*(t) = {x e QO : g > t}, E*(t) n Qt and E*(t) = E*(t) n defined analogously with respect to f. E(t)

=

2.3.3 Theorem.

Suppose

functions defined on

00'

F*(t),

F(t),

F*(t)

are

are nonnegative Borel measurable 0 ( a < 1, b ) 1 and

g, f

(2.3.4) then there exists a constant

PO

such that if

Po > P ) q

then

91

J

1

(

..

9PdW)1/p

(2.3.5)

Q..

(

o

J

00

where c depends only on only on d,q,a,cl,b,R/S... Proof.

d,p,q,a,e,Cl,b,R/S .. ,

and

PO

depends

(2.3.5) will first be proved under the assumption that (2.3.6)

Then either

f 4! rJ'(w,OO)

in which case 2,.3.4 is true or

f rJ'(w,OO), in which case from propositions 1.1.3, 1.1.4, 1.1.5 and 1.1.9 it follows that M(Ml/q(fq » ) Ml/q(fq) a s e , (2.3.4) then implies that (2.3.6) holds for

f

replaced by

Ml/q(fq)

and (2.3.5)

follows form propositions 1.1.3 and 1.1.4 and (2.3.5) with as above.

faltered

Let (2.3.7) where cl'd)

c5 is the constant appearing in 2.3.11 (depends only on and k q = 3q(c5b(1 + a)/(l - a» > 1. The doubling condition

implies that

w(O.. )/w(QO)

is bounded be Low by a positive number

depending only on cl,d so that tS is bounded below by a positive constant depending only on d,cl,a,e,R/S ... Normalize

g

tS«w-l(Qo) J

and

f

by dividing by

gqdw)l/q + (w-l(Qo) J

°0

fPdw)l/p)-l

°0

so that without loss of generality (replacE!

g, f

by these normalized

versions) we may assume that is

•

The remainder of the proof will consist of four parts.

(2.3.8 )

92 Part I (Decomposition) Fix s > k and let

wrtu J Q

S

gqdlll ..

III

t

(QO)

s/k. •

s (QO) . wrn:r III

)

gPdoo ( c l I + k P-q ) J

°0

< c6 qoo(00) + ck P- l J

°0

c

independent of

00(000).

gdoo

J

E (I)

fPdoo + c

J gqdoo ( coo(Ooo) EO)

Reversing the normalization of

f, g

gives

(2.3.5) • 2.3.18 Lemma. Suppose h [1,00) + [0,00) is nonincreasing, right continuous, and Lim h{t) O. Also suppose H: rl,oo) + [0,00) t+ oo is measurable, q > 1, a > 1, k > 1 and p satisfies 1 > akP-l{p - q)/{p - 1) with p) q. If

- J

(t,oo)

sq-ldh{s) < atq-l{h{t/k) + H{t/k))

for

t) k

then

< c Proof.

J

l(- (k ,00)

sq-ldh{s)) + c

J

2 (k ,00)

t p- 2H(t/k)dt + c 3h(1)

An integration by parts gives

Let

(2.3.19)

- J

(k, j]

where

J

J

(k,j)

t p-q- l(_

J

sq-ldh{s))dt.

{t,j]

Combining this with the hypothesis it follows that J

but


0 by hypothesis, and tp-ldh(t) .. kP-lh(l), so the desired conclusion is reached •• J (l,k]

but

-

CHAPTER 3 The theme of Chapter 3 is that of establishing continuity for solutions of degenerate elliptic equations. In Section 3.1.0 both interior and boundary continuity are considered for single equations of the form div A(x,u,Vu) = B(x,u,Vu), where A, B satisfy certain natural growth conditions. As a

97 byproduct of this a Harnack inequality is proven for positive solutions. In Section 3.2.0 estimates are derived for the modulus of continuity of functions in weighted Sobolev spaces, analogous to Morrey's result that functions in Wl,P(Rd), p > d, are Holder continuous.

This is relevant since solutions' of equations with

natural exponent

p

(p = 2

for linear equations) are often contained

in such spaces. In Section 3.3.0, degenerate elliptic systems are considered of div Ai(X,u,Vu) = Bi(X,u,Vu), i = 1, ..• ,N, where Ai' Bi satisfy certain growth conditions. Additional integrability is proven for IVul and this, combined with the results of Section 3.2.0, the form

establishes continuity in certain borderline cases where 3.2.0 does not apply directly. In each section an example is worked using equations with degeneracies of the form

distQ(x,K), for a class of sets K which 2 includes finite unions of c manifolds of co-dimension greater than

or equal to 2 (including co-dimension

d,

i.e. points).

3.1.0 A Harnack Inequality and Continuity of Weak Solutions for Degenerate Elliptic Equations The main results of this section are a Harnack inequality for positive solutions and the interior and boundary continuity for weak solutions. The basic structure of the proof of the Harnack inequality is due to Moser [MEl].

Techniques of Trudinger [Tl], [T2] are used to

replace the John-Nirenberg lemma [IN], which is not of use when the weights are badly degenerate.

The proof of the boundary continuity

essentially follows that of Gariepy and Ziemer [GZ]. Various results have been proven for linear degenerate equations by Kruzkov [K], Murthy and Stampacchia [MS], P. D. Smith [SM] and Trudinger [Tl], [T2] and a degenerate Harnack inequality has been proven by Edmunds and Peletier [EP] for quasi-linear degenerate equations.

The present results allow a more general class of

degeneracies.

The reader should note the related work done

independently by E. B. Fabes, D. S. Jerison, C. E. Kenig, and R. P. Serapioni [FKS], [FJK] (see comments preceding 2.2.40). The equations considered are of the form div A(x,u,Vu) where

=

B(x,u,Vu) ,

(3.2.1)

98

and

are Borel measurable functions satisfying the conditions I A( x , u , w) I
1, A, \I, ai , i = 1,2, bi , i = 1,2,3, ci , i = 1,2, are nonnegative Borel measurable functions on n and OJ = \lPA-(p-l) and A are assumed to be integrable with o < A
2

e

where

q > p

and

(for computational simplicity); (3.1.5)

for all

e Wl,P(B(xo,r»,

and some

r > r;

99

(3.1.6) for all

(t,Vt)

is either

0

e

or

W6,P(B) 1

and

and Fr

0
O. Let F R = RP[cl + c2(K-P(R) + bix-(p-l) + b 2 + b 3 K-(P-l) (R)

+ (al + a2 K-(p-l) (R»P/(p-1)v- P/(p-l)X] and (3.1.6) for

with

If

u

u" M and

is a positive weak solution of (3.1.1) in

o

< a < 1,

then

inf u + (C(R) - l)K(R) sup u" C(R) B(XO,aR) B(XO,aR) where

(3.1.5),

r = R.

3.1.13 Corollary. B(xO,R)

and assume (3.1.4),

,

(3.1.14)

C(R) is as in Theorem 3.1.10. For Theorem 3.1.15 let F r =, rP[(clM + c2)K-P(r) + blx-(P-l) +

+ b3)K-(p-l)(r) + (alM + a2)P/(p-l)K-P(r)v- p/(p-l)X], 2M ( 3. 1 • 4 ), (3 • 1 • 5 ), and (3. 1 • 6) for 0 < r c R.

(b

3.1.15 Theorem. with lui" M/2

and assume

If u is a weak solution of (3.1.1) in B(xO,R) and C(R), K(R). a, M as in Corollary 3.1.13, then

102

u = 0 lim Osc k+ CD B(xO,rk)

L

if

k=O

c-l(rk) =

and

CD

0,

lim C(rk)K(rk) k+ CD

r k = e k R.

where

If

is bounded and

C(rk)

for some

c',

Q'

> 0,

then

These conditions are sharp in the sense of Lemma 3.1.16. In addition, if C(r) is nondecreasing and K(r) is nonincreasing as

r + 0, u .. e

where

g(s)

Remark.

then

-cg(r

k)

R

1 dr = J err> r s

(Osc u + B(xO,R)

C

y(s) = C(s)K(s/e).

and

Semicontinuity results for subsolutions and supersolutions

may be proven as in [Tl] using the calculations mentioned in the second remark after Theorem 3.1.10. Example. Let oo(x) =

K

be as in Theorem 2.2.56 such that 2.2.61-3 hold and = A(X) = distQ(x,K),

Q > -Yo

From this and a limiting

argument (as in Lemma 3.1.7) it follows for some q > p and all B(xO,r) Rd , that (3.1.4) and (3.1.5) hold with s(r) = p(r) = 1 and t(r) = q(r) = O. For simplicity assume that al ' a2 ' cl ' c2 ' b l ' b 2 ' b 3 are bounded by a constant multiple of 00 and choose k(r) = r + r(p-l)/p. This implies that F r " coo, so (3.1.6) is trivially true with for

r < R
O.

Assume (3.1.17)

and (3.1.18)

for

k

0,1,2, .•• ,

where

ak

Then

103

Osc u B(xo,rn)

.

(

n-l

n-l n-l a k) Osc u + 2 I k(r j) I a k k=O k=j B j=O R It

,

(3.1.19)

and if

I

k=O as

k

+

C-1 (r

.,

= ..

k)

and

C(rk)K(rk)

then

Osc u B(xO,r n)

+

0

as

n

+

(3.1.20)

0

+

.

(3.1.21)

(3.1.22) This is sharp because if (3.1.20) does not hold, then there exists u such that Osc u) > O. B(xO,r n) el If C(rk) is bounded and K(rk) .. crk for some c, el > 0, then u .. c'r

el'

n

for some

c' ,

If

C(rk)

is nondecreasing as

If

C(r)

is nondecreasing and

k

+

(3.1.23)

> 0 •

o'

then this is sharp as

.. ,

well. K(r)

is nonincreasing as

r

+

0,

then u .. e

where

g(s)

R

J

s

-g(r ) n (

1 C{"'I'T

dr

r'

Osc u + 2c B(XO,R) y(s)

g(r ) n

J o

C(s)K(s/e)

y

and

0

9- l(t)ec tdt) ,

c =

Proof of Theorem 3.1.10. The fundamental inequality (3.1.29) is proven; then this is iterated to give (3.1.31). The final step is the crossover from LP norms of u with P > 0 to those with p < O. This is accomplished by iterating norms of log u. It can be assumed without loss of generality that u is strictly positive, otherwise let u = u + E, E > O. u is a weak solution of div A B, 'olhere A(x,u,Vu) A(x,u E,VU) and B(x,u,Vu) = B(x,u gives

E,VU),

and since

A,

B

satisfy (3.1.2), the following proof

sup U .. C(R) inf u. (3.1.11) is recovering by B(XO,eR) B(xO,eR) letting E + O. Throughout the proof, c will represent a constant depending only on p, q, e, M, b O ' and will change from time to time.

104

Let

signa u),

=

a

*

0,

applications of Proposition 2.2.2 show tha"t

e

C;(B(XO,R». Several e W6,P(B(X O,R» with

and so

Now multiply by u .. M, and let

signa, use the structure inequalities (3.1.2) and E = exp(b o signa u ) to (3.1.24)

+ P - b

J O

J

+ p + bOM

J

J

.

The second and the seventh terms on the right-hand side cancel. This is in fact the reason for introducing

exp(b O signa u)

in the

test function The following inequalities are proven using Young's inequality.

(3.1.25)

105

< 1

(3.1.26)

P

1

+ (p - 1)

1

p

Applying these to (3.1.24) with

I I

£

= .P_- • lal j5"""'='L 4'"""

1

and

£2 = 4(p 1)' and absorbing the gradient terms into the left-hand side it follows that

-(p-l) + £1 P J

< lal J

+ J

+ (£;(p-l) + 1) J

Considering that

y

=

a

+ p - 1

where

F

R

*

and it follows for

0,

RP[cl +

+ b

2

y

*p

- 1,

Y

*

Y

0,

+

for that

and

c(y) = (1 + lal-p)yp. Now use (3.1.6) with

t

=

and

£-1

= max{1,2cc(y)}.

The

resulting qradient terms may be absorbed on the left to give (3.1.27) (recall BR

=

< w).

B(xO,R),

Using inequality (3.1.4) with

t

=

and

it follows that

3.1.28. This inequality is now iterated. To do this choose e i , i = 0,1,2,3, such that = eO < e l < e 2 < e 3 = 1, and let -k = B(xO,rk)' where r k = R(e O + (e l - e O)2 ). Pi = Rei' let Choose e CO(Bk) such that 0 < < 1, = 1 on Bk+ l and

106

.. 2k+2 /[R(e l - e 2 ) J , and let choices of y it follows that

Y k

YO(q/r)

k

.

with these

where Cl,k (R)

+ (e C(Y k)

and in turn

Cl,k(R)

2k+2 p 8) ] + 1 0 blow up if

Yo < 0, then this is impossible. chosen carefully. 3.1.30.

Given

such that for all

k.

E q

0

1 )

> 0

0

< yo
0,

then

yo

must be

(to be determined in 3.1.38), pick and

I ek I = l r k

With this choice of

YO'

-

If

yo

(p - 1) I >

C(y) ..

- E) 2 q and so after a

crude calculation it follows that

Iterating (3.1.29) it follows that (3.1.31)

with

Cl(R)

that

Cl,k(R).

IT

k=O

Recalling that

Yk -_ yo (q)k P ,

The last step in the proof is the "crossover".

it follows

First an

inequality is derived which in the uniforMly elliptic case leads to the conclusion that

log u

"crossover".

is not a douhling measure, this is not

If

l.ll

B.M.O.

which, in turn, qives the

sufficient, and one further iterative procedure is necessary to get the "crossover". 3.1.32.

Take

e

= 1 - P

at (3.1.24), and proceed as before but

without using inequality (3.1.26) to get

107

,

Choose C;(B(XO,R» such that 0 ( , (1, Iv,1 ( 2/(R - P2) = R- l(2/(1 - 8 ) , so 2

,

1

on

Bp

2

and

where

Using inequality (3.1.5) it follows that (3.1.33)

where

K(R)

= p(R)H(R)

+ q(R)

and

k =

P2

)

l

(log u)w.

P2

To derive the inequality needed in the final iteration let the test function be

where

n

n) 0,

v = log

£'

k

as above, and

e ) 1. Repeated applications of Proposition 2.2.2 show that t W6,P(B(X O,R» with Vt = pnP-lvnul-p(lvle + (p - nPu-PVu(p - 1)( Ivle +

- elv!e-l s i gn v)exp(-bou)

- bonPul-p(lvle + 3.1.34. Substituting this in 3.1.3 and letting follows that

• E

exp(-bOu),

it

108

f

- be +

n Pu

l-PE(lvI 8 + (p

nPu l- PE(lvI 8 + (p

f

.

Use the structure inequalities (3.1.2) and Ivl 8 + ( p8)8 to get p-=-T (p _ 1)2 P

f

n Pu-PIVuI PE(lvI 8 +

.. P

f

nPE(lvl8 +

+ P

f

np-lIVnlul-PE(lvI8 +

+ P

f

np-lIVnIE( Ivl 8 +

+ b M

o

3.1.35.

P - 1

I v 1 8- 1

c

P - 1

P -

1 P -

nPu l- PE(lvI 8 +

f

- be

--l2..L

f

nPE(lvl 8 + (

p8 )8)c p-=-T 1

The fourth and sixth terms cancel.

Eliminate

E,

multiply

by RP, use Young's inequality as before, and recall that 1Vu Vv = uto get

.. c Use

f

(Iv1 8 +

Iv1 8 .. (IvI Y/ p +

right and RP

f

8IvI

8- 1

p---=-T

P -

1

R

Y

.. Ivl 8 + (p

+ RPIVnIPw)

=

8 + P - 1

(8) 1)

on the left to get

nPIVvY/PIP). .. Cy P

f

(IvI Y/ p + (

P8_)8/p)p(n PF + RPIVnIPw) p-=-T R

on the

109

Use inequality (3.1.6) on the F R term with p6 )6/p) and -l = 2cy P > 1 and then cancel p-=-T gradient terms to get = n(lvIY/p + (

Use inequaiity (3.1.4) with

Let Pick

rk = R( 6 1 such that 0 .;

= B( x,rk)'

e

nk

k+2 / (R( 6 - 6 IVnkl .; 2 1 2

with

= nv Y/

»,

-

and let

p

to get

(6 .;

2 1,

l)2-

k),

nk = 1

k+2

2

Recalling the defiition of

6

k = 0,1, ••• on Bk+ l , and

Y = p(q/p)k. k

-0 ( 2 c ( s p (R)Y P( k k sF(R) + e

C2,k(R)

-

e

This gives

)p) + tP(R»)l/Yk.

1

it follows that

Use Minkowski's inequality to get (

1

Y l/y l v ] k+1 w ) k+l.; cC

f

B

k+l

Iterate this to get

2'

k(R)

1 W\D R ,

f

B

k

Yk l/Yk Ivl) + Yk]

110

n IT k=O'

C2,k(R) ) 1 CY n

n

L (

so

j=O

IT C 2,k (R) ,;; k=O Let

Given

inequality,

R, B

P2

n n IT. C 2 k(R))Yj c jl Yj C2,k(R) o k=J ' + 1) + t(R))q!(q-p). + 1) + t(R))q!(q-p),

C2(R)

s ) p,

W\D

then

Y s < Y for some n+ l n';; W(Rn+l)!w(B R)';; 1 and

n) O.

,;;

then

Use Holder's

(3.1. 36)

Expanding

eX

in a Taylor series, it follows that

Use Holder's inequality on the first [p] terms and (3.1.36) on the rest to get

L

The series b

< e -1 ,

s=O

b

S

ST (x + s)S

converges and is bounded by

so

if

since (3.1.33) holds.

a
0 a = b

n

c"a(a-b)n

a - b < 0

To prove sharpness first assume that

B

c'r

since 1

n-l

(

C(rk) is not bounded so that C(r k) t 00. Now pick u such that

1'15

If

n-l asc u = IT Br k=O n a asc u < cr n' B rn n-l

Pick

(

then

CanaR a asc u B R

IT

ak
0

al

e L{P/{p-l»+a{ n) loc u, ,

b

e

where

2 s

w which is doubling, that is,

< c w{B{x,r». w such that

w{B{x,2r»

bl

It is also assumed that

e LPs/{s-p)+a{ n) loc u,

(3.3.3)

Ls/{s-I)+a{ Q) loc u, is given in (3.3.4), (3.3.5).

In some cases these conditions can be weakened by using other Sobolev inequalities in combinations with those in (3.3.4), (3.3.5).

135

1 W loc in 2.2.1. A pair of (3.3.1) i f

Let

N IT Wl,p ( i=l l o c (u,Vu) e Wl,p( l oc

=

Lf

)

as defined is said to be a weak solution

+

i

e Wol,p (

for all for

),

0

)

It will be assumed that the following Sobolev inequalities hold p, q, s with 1 < q < p < s.

for all

and all balls

B

with diameter

r,

BC

Q:

e loc ' Q) and all balls B with diameter r, B C Q. It is of course only necessary to assume (3.3.4), (3.3.5) for functions and then the usual limit procedures allow general

for all

..

C

Sobolev functions. Conditions for Sobolev inequalities of the form (3.3.4), (3.3.5) are discussed for scalar-valued functions in Section 2.2.0. The vector-valued case is an obvious corollary. For certain s > p,

inequality (3.3.5) follows from inequality (3.3.4) as in the

proof of Theorem 2.2.56.

A simple consequence of inequality (3.3.5)

is that if e then e The analysis of the equations will produce a "reverse Holder" type of maximal function inequality from which the higher integrability of the gradient will follow. 3.3.6 Theorem. I f (u,Vu) is a weak solution of (3.3.1) in a d, bounded open set Q C R then there exists > 0 so that e depends only on d, p, q, s, aO ' IVul e a, sl ' s2· Remark. As in [MYE], if (u,Vu) e assumptions are made about 3Q, then

IVul

e

and certain weak LP+ (Q).

°

Q with side 0 .. , °1 , °0 be concentric cubes, 0 lengths S , 2S , 3S respectively, as in Section 2.3.0. Estimates will be made over balls B' = B(x,r), x e °1 , r c S.../2, using test functions of the form is a function such that = - k) where Proof.

Let

...

...

...

136

cjl

e

0 .. cjl" 1,

cjl = 1

on

B',

IVcjll

c 2/r.

(3.3.7)

These calculations will yield the inequality (3.3.8 ) for y 2.3.0,

e

and

01

Ivul

g

and on

q, F

R = 5 •.12

e

M are as defined in Section where MR, q = p/q and c depends only on aO' sl ' s2 ' l+ a ' (M,Ol) L for some a' > 0, a' dependent only

a, s, p , Using propositions 1.1.3, 1.1.4, 1.1.5 and 1.1.9 to show that

F .. M(F)

a.e. and letting

f = Fl/q

MR(gq) .. CMq(g) + M(f q) +

M(gq)

it follows that a.e. in

Applying Theorem 2.3.3 it is clear that IVul E > 0 dependent only on d, p, q, s, aO , c w p+E (II, Q). I Vul e L l oc To prove 3.3.8 let K = w-l(B) J udw, B B and = cjlPv, cjl as above. Take = pcjl

I; L

..,

e

LP+E(O )

,

a, sl

B(x,2r),

for some

,

$2 v

so

=

u - k

+ cjl P Vv

e

so that

i

p-e I

01 .

by Proposition 2.2.2, and

J [ pcjl p-l v.Vcjl·A. 1.

1.

] + cjl P sv . -x . + cjl p v.B. 1. 1. 1. 1.

o .

Rearranging terms and using the structure conditions (3.3.2), it follows that

Younges' inequality implies that cjlp-llVcjll Ivl IVul p- l .. EcjlPIVul P + E-(p-l)!VcjlIPlvI P Ivl IvulP-lb Applying these with

l

.. EIVul P +

. {I E = m1.n 4' (4aoP) -I} ,

. absorbing the gradient

terms into the term on the left-hand side, using Holder's inequality on three of the remaining terms and recalling (3.3.7), it follows that

137

J

< J B

B'

+ cr-P J B

+ cr-l(J B

for some

t

< s

+

£

such that

B

_s + a and t < s __ + a s-=-T 't"'"='l s - 1 Finally, use inequality (3.3.4) on the second term, inequality (3.3.5) on the third, followed by an application of Younges' inequality and recall that w(B) < cCwW(B') to get that

Taking the supremum over q IVul = g, q = p/q and

_t


Po - £', with £' > 0 dependent only on PO' aO ' a, d, and the measure w. For simplicity only the borderline case p = Po will be considered. Corollary. weB)

f

B

Suppose -

u, Vu,

n,

£

are as in Theorem 3.3.6 and

f

) and all balls B with l oc x, y are Lebesgue points of u with respect to Ix - yl B(XO,R) c S1 for Xo = and R 2 2 for all

e

(3.3.10)

w\B, B

Wl,p ( 1J,1J,S1

Bc

n,

w,

such that

If

then (3.3.11)

Remark. Conditions for (3.3.10) to hold are discussed in the remark after Theorem 3.2.11. Example.

In the example developed after Theorem 3.2.11, it is easy to

see that the critical exponent is d if a < 0 and d + a if a ) O. Also, since w(B(x,r» - r d maxa{r,dist(x,K)}, a > -y ) d, it follows from Corollary 3.3.9 that Holder continuity can be established for solutions in the borderline cases.

139

Proof of Corollary 3.3.9.

L Iui(x)

- W1B) p

< K(J I Vu. I 0 dW) B

Given a ball

x,y

u = (ul' ..• ,ud)

L ui(y)dw(y)ldw(x) lip

].

e

B(XO,R)

n,

0

Lebesgue points of

such that

so

B(XO,R)

u

nand

with respect to x,y

e

B(XO,R),

00,

and

use the

geometry described in the first remark after Theorem 3.2.5 and apply the second part of Theorem 3.2.5 with f(x) = K(J B(

"o ,R)

to conclude that

Now let remain in

Xo

+

B(xO,R)

and

R

+ Ix ; yl

in such a way that

so that (3.3.11) is verified • •

x, y

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INDEX Absolutely continuous functions, condition, 64 Boundary continuity, Boundary values,

3, 6-8, 10-12, 129-131

97, 116, 118

45, 83, 86, 87, 116

Capacitary extremal,

45, 51, 58, 59, 79, 80, 82, 85

Capacity, 3, 25, 33, 35, 45, 46, 51, 55, 56, 58, 59, 61-63, 85, 118 Capacitable, 51, 52, 62 Clarkson's inequalities, Continuity estimates,

57, 61-63, 81

96, 97, 102, 116, 118, 126, 132, 134, 138

Countable SUbadditivity, Covering properties,

45, 51, 54, 55, 62

3, 4, 6, 64, 71, 94, 127

Differentiation of integrals, Doubling measure,

3, 130, 131

6, 89-91, 128, 129

Elliptic equation,

96, 97

Hardy inequalities,

3, 12, 15

Harnack inequality,

97, 100

Higher integrability,

33, 89, 97, 134, 135

Isoperimetric inequality, Maximal function, Quasicontinuous,

66

3, 4, 33, 89, 90 45, 51, 54, 56, 58-63, 82, 83-87, 118

Quasi-everywhere,

56-63, 80-84, 86, 87, 118

Reverse Holder inequality, Sobolev inequality,

89, 135

12, 15, 17, 25, 33, 45, 51, 59, 63-66, 79, 80, 83, 98, 99, 118, 126, 132, 134, 135, 138

Sobolev space, Subadditive, Systems,

25, 33, 34, 45, 46, 80, 97, 98, 126, 132, 135 45, 51, 54, 55, 62

97, 126, 134

Vanishing mean oscillation, Weak L l estimate, Weak solution,

126, 127

4

98, 100, 101, 103, 117, 135

Wiener condition,

118

Wirtinger inequality,

79, 83

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