Elliptic Equations with Degenerate Coercivity: Gradient Regularity

Acta Mathematica Sinica，English Series

April，2003，Vo1．19，No．2，PP．349 370 Acta Mathematica

@Spdnge~Vedag 2003

Elliptic Equations with Degenerate Coercivity：Gradient Regularity

Daniela GIACHETTI

Dipartimento di Metodi e M odelli M atematici per le Scienze Applicate，Universith degli Studi di Roma

‘'La Sapienza”，via A．Scarpa，16，00161 Roma，Italy

E—mail：giachett@dmmm．uniroma1．it

M aria M ichaela P0RZ10

Facoltd di Scienze M atematiche，Fisiche e Naturali，Universitd degli Studi del Sannio，via Port'Arsa

11，82100 Benevento，Italy

E—mail：porzio@unisannio．it

Abstract In this paper，we prove higher integrability results for the gradient of the solutions of some

elliptic equations with degenerate coercivity whose prototype is

div(a(x，u)Du)=f in D (Q)， f∈L (Q)， r>1

where，for example，a(x，u)=(1+lu1) with ∈(0，1)．We study the same problem for minima of functionals closely related to the previous equation．

Keywords Regularity of solutions，Nonlinear elliptic equations，Functionals of calculus of

variations

MR(2000)Subject Classification 35B65，35J60，35J70

0 Introduction

The main aim of this paper is to study the integrability properties of the gradient of(global or

loca1)solutions of problems like

z1 u)。u)=， m D (Q)

on 0Q． (1)

We recall that in the classical case a(x，s)三1，by means of the Calderon—Zygmund regularity

theorem【1】，we have that if f∈L (Q)，then a(x，u)Du=Du belongs to L (Q)，where m=r if 1< r< N ，otherwise m can be any number bigger than one．

The same happens(i．e．a(x，u)Du∈L (Q))，when a(x，s)=a(s)is a function independent of the variable X．As a matter of fact，performing the following change of variable：

：厂“。( )d ， I，0

Received June 18，2001，Revised April 28，2002，Accepted M ．ay 16，2002

维普资讯 http://www.cqvip.com

350

we obtain that V iS a solution of

一 △ V

V= 0

．厂 in

on

D ( )

．

Giachetti D．and Porzio M ．M

Hence by the quoted Calderon—Zygmund theorem，the function"belongs to wJ ( )，where m is as before，and thus Dv=a(u)Du∈L ’( )if r<N．When the function a depends

also on ．the previous change of variable cannot be applied and the problem to establish the

integrability properties of the gradient of the solutions of(1)is，in general，still open．Here we

will study the case when the coercivity can degenerate when 札is too big．More precisely we

assume that f∈L ( )，r>l，and a(x，s)is a Carath~odory function satisfying

(1+ls1) a(x，s) ， 0 <l

otlce that our cas e includes the clas sical one．

We refer for the existence of solutions of(1)to[2】_

If the problem is

I—div(a(x，u)Du)=一div(F)in D ( )，札= 0 。n ，

in[3]we prove the existence of solutions when F∈L ( )， sufficiently close to 2．In this case the space for solutions is sharp and it depends on ．

On the contrary，the existence result for(1)，proved in[2]，is not sharp．We will recall this in Section 2．

We will first prove here an higher integrability result for a(x，u)Du with respect to【2]，(札

solution to(1))，and then also for Du when the datum f is sufficiently regular (see Theorems 1．1一1．71．

A counterexample of Meyers[4]shows that，just in the easier case a(x，s)=a(x)with

a(x) O／>0，if r<N，there are local solutions札with Du not belonging to Llro*c(Q)，but only to il2~

c ( )，with E depending on o／and r．Thus in the general case(1)the goal is to prove this kind of regularity．

Let US point out that，if we assume a more restrictive condition

(1+ls1) a(x，s) (1+ls1) ’ 0< <1．

he prob em will become easier，since the regularity of a(x，u)Du is equivalent to that of

in which there is no explicit dependence on (see Theorems 1．2一1．3)．

Nevertheless，also in the general case(3)，it is possible to achieve the result by giving a

further assumption on a(x，s)，which is implied，for example，by

Oa(x ,s)= ( Oa(x ,s)，⋯， Oa( x,s))6∈-(州 ))Ⅳ，m>Ⅳ，一 =●_ ．--_ I l，，l、，l l m、，V ＼】 ’ ’ " ／～＼， ’ “ 一一

(see Theorems 1．5 and 1．6)．

The higher integrability on a(x，u)Du is data，to deduce higher integrability on Du．

interesting in itself，but can be used，for regular

In the second part of the paper，we study an analogous problem for minima of functionals

一

一


Regularity Results for Elliptic Equations

( )=．／ n( ， )LD 12dx-．／， d where a(x，S)is a Carath6odory function such that

(1+Is1) 0 a(x， S) 1 0<0< 一 2

351

(5)

Functionals like(5)are differentiable just in some directions and，in any case，their Euler

equations do not look like Equation(1)．

1 M ain Results

1．1 Elliptic Equations

Let Q be a bounded open subset of RⅣ． Consider the following nonlinear elliptic problem

u

- d

：

iv

。

( )D 一 i

⋯

n D

Q

'(fl

，

where a(x，U)：Q×R__÷R，is a measurable Carath6odory function satisfying

(1+lsI) a(x， S)

where Ol and are positive constants and 0 is a real number such that

0< < 1

(6)

We recall that in[2]an existence(and regularity)result for the solution of(6)is proved by using approximation techniques．In more details they prove the following：

Proposition 1．1 Let f be 0 function in L (Q)，with r>N／2．Then there exists 0 function

U in硎 (Q)n o。(Q)which is 0 solution (6)．珏 otherwise，r verifies 2Ⅳ Ⅳ+2一O(N一21 ． N <

，

then again there exists 0 function U that is 0 solution (6)and it belongs to硎(Q)n Lh(Q)， where

Finally ifr>1 such that

N

=

Ⅳ+1一O(N一11

Nr(1—0) N ——2r

< r < 2Ⅳ

Ⅳ+2一O(N一21

then there exists 0 solution U in ’。(Q)，where

q= < 2．

More。 e r，ifr>丙N ( 2=--j0 ) ， such。s。luti。礼 e7 es theIoll。t 礼9 7， ul。 t c。礼diti。礼

n( ，u)Du∈(L2(Q))Ⅳ．

(10)

Rem ark 1．1 Notice that when 1< r< 而，then q< 1．Anyway,it is stiu possible

to give a definition of the solution(see[2])，but such a solution doesn’t belong to any Sobolev


352

space．This is the reason why WC will not consider this case


R m ark 1．2 It is easy to verify that the L or Lh regularity on u holds true if u is any

solution in础 (Q)(not necessarily got by approximation)，and that we have

L (【2) Co=Co(1al，N，r，II／IILr(【2)) ( 4)

if r>譬；while if r satisfies (9)then

lLh(n1 Cl(a，N，， (Q))

Moreover．these resu1ts hold true，with obvious modifications，for the following more general

problem：

with P> 1

{ v0‘ 。训p一。

We are interested in the regularity of the gradient Du of a solution of(6)

The case of bounded solutions

Theorem 1．1 Assume(7)and(8)hold and

solution of problem(6)then

f∈Lr(Q)，where r> N．fu∈硎 (Q)is 0

Du∈(Ll2oq-ce(Q)1Ⅳ，

伽he代E is 0 positive constant that depends only on ，，r，N，【QI and【tfttLr(Q)．

Remark 1．3 In genera1．as noticed in the introduction，under the assumptions of Theorem

1．1，if r< Ⅳ，which is the interesting case，we cannot expect that Du belongs to Llro*c(Q)as

it happens for the Lapla~ian equation(i．e．in the particular case a(z，S)=1)．As a matter of

fact，it is not true even if a(x，S)=0( )，as it is showed by the counterexample of Meyers(see

[4】)．

The ease of unbounded solutions

W e start with a simpler case

Carath6odory function satisfying

(1+l s1)

W e have the following results：

when，instead of(7)we assume that a(x，u)is a measurable

the stronger condition

n( ) ，。< <L ( )

Theorem 1．2 Let(18)be satisfied and assume that，∈L (Q)，where

u∈硎 (Q)is a solution ofproblem(6)，then there exists a positive constant E

on N， and (with E r 一2)，such that

a(x，u)Du∈( T2+。 ( )) ．

Finally，ifr> 2N then there exists 0 constant s>2 such that

Du∈( s0。(Q)) ．

r ver／~es(9)．盯

that depends only

(19)

(20)

Remark 1．4 We observe that the result of Theorem 1．2(respectively 1．1)works in general

C：，

D a

．口叽

，



if U∈H0'(Q)is a solution of the following nonlinear problem：

- d

：

iv

。

(。‘ ， '。 =，

。

in

n 弓

353

where a(x．，Du)：Q×R ×RⅣ RⅣis a measurable Carath6odory vector—valued function

satisfying

。(x,u,Du)I (respectively la(x，U，Du)l c LDu L)

a(x，U，Du)Du ~lDul。 0∈(0，1) (22)

Moreover，proceeding aS in[5】1 if aQ is sufficiently smooth，we can prove regularity up to the

boundary．

Theorem 1．3 Let 6e 0 solution of problem(6)that belongs to ’ (Q)，where q is as in

(12)．Assume that a(x，U)satisfies(18)．

。( ，u)Du∈( (Q))Ⅳ (23)

and ，belongs to Lr(Q)，where r>而2N yerifies(11)，then there exists 0 positive constant E

that depends only on N，Q and (with e r 一2)，such that

。(x,u)Du∈( (Q))Ⅳ． (24)

Remark 1．5 We notice that the solutions obtained in[2】satisfy(23)if r satisfies the further

c。nditi。n r> (see Pr。P。siti。n 1．1)．M。reover，the c。nditi。n r> 2N is not restrictire

since if r< we will prove in the~llowing The0rem 1．7 that ∈( (Q))Ⅳjust in

the general case(7)．

Remark 1．6 W e point out that under the assumptions of Theorem 1．3 one has q = h and

it is not possible to improve the summability of Du(see Remark 3．3 at the end of the proof of

Theorem 1．3)．

Moreover the results of Theorems 1．1．1．2 and 1．3 have a local version，as shown by the

~llowing result：

Theorem 1．4 Assume U∈H?oc(Q)or (Q)is 0 local solution problem 0( ， )s0tisfies(18)．可，∈ r0 (Q)，with r>丙干 2N = ，there ezists 0 pos t e

that depends only on N，Q and (with E r 一2)，such that

(6)，where

constant E

a(x，u)Du∈(L2⋯+ (Q)) ． (25)

Moreover r>N／2，then U belongs to (Q)，while r ue es(9)，then U is in ％c(Q)，

where h is as in(10)．

Remark 1．7 We point out that the proof of the L c(Q)regularity of the local solutions of

problem (6)requires the assumption(18)while when we have a global solution，the proof of

the L(Q)regularity needs just the weaker condition(7)．

We return now to the general case when the weaker condition(7)(with respect to(18))is


354 Gi0chetti D．and Porzio M ．M

assumed and we study what happens when u，in generalj is an unbounded solution·We have：

The。rem 1．5 ￡(7)，(8)。nd(9)hold．Assume th。t钆∈日8(Q)is。s。luti。n of problem(6)

that satisfies '

0

u Oa

a

(x ,s)ds 2 + (Q)，

cr>O． (26)

Then the e s￡s 0 p0siti"e c0nstant E that depends only on N and (with E r 一2)such

that

。( ，钆)D ∈( (Q))Ⅳ

M。Te。"er r> 而 2丽N the礼there exists s>2 s钆c ￡。￡

Du∈(LL。(Q))Ⅳ．

Remark 1．8 We observe that assumption(26)is verified if，for example，we have

百Oa(x,s)l ( )a．e．， ∈Q，Vs∈R x 1 where h(x)belongs to L (Q)with m>N

(27)

(28)

(29)

Theorem 1．6 Let(7)，(8)hold and，∈ (Q)with r as in(11)．Assume is 0 solution o／

problem(6)belonging to ’ (Q)and verifying

g(x)= “ ( ))Ⅳ， ⋯ (30)

ere q s。s (12)．Then there exist tw。positi"e c。nst。nts ro。nd E，where ro> 万 N

d印 ends only on N and 0， while E depends only on N，0， and r，such that we assume

then

r0 < r < 2Ⅳ

Ⅳ+2一O(N一2)

。(z，u)D ∈ 盛 (Q))Ⅳ

Moreover 9( )belongs to L2+ (Q)，then there exists 0 positive constant 7 such that

n( )D ∈( (Q))Ⅳ．

(32)

(33)

Rem a】 k 1．9 The assumption r > ro is taken to guarantee that q is“near”2 SO that it is

Do8sib1e to use the techniques related to the Hodge decomposition(see Lemma 2．2 below)．

Rema】rk 1．10 Let US now point out that，in some particular case，the solution can reach a

better regularity．

For example if a(x，U)= a(u)then under the weak conditions(7)and(8)it is easy to

prOve every solution ∈Hi(Q)belongs to ’ (Q)if N／2<r<N and to ’q(Q)if

丽 2而N <r<譬，where q is，as before，given by the following formu1a：

q=q(r) Nr(1—0)

N—r(1+0)

W e notice that when r> 丽 2 N we have q> 2． M。reOVer，we have q r as r_．+N／2

r{l jjIl


Regularity Results soT Elliptic Equations 355

In all the previous results we have assumed that the summability r of the data f verifies

the following condition：

r > 2Ⅳ

Ⅳ + 2 (34)

If condition(34)is violated，it is possible to prove by using the techniques of the following

regularity result：

Theorem 1．7 Let(7)and(8)hold and f∈L (Q)，where r ve es N 2N

Ⅳ+1一O(N一11 <r< Ⅳ+2

Then every solution U∈ ’ (Q)(where q is as in(12))of problem(6)satisfies

1．2 M inima of Functionals

∈ (Lr*(Q))Ⅳ

Let be in ’ (Q)and consider the~llowing functional

J(v) n( ， )I)：(。 )d 一， d

Here a：Q ×R R is a Carath6odory function such that

。( ) 。< < 1，

where 0< 1，and J is a convex function，where J：RⅣ R，j(0)=0，satisfying

J(∈) ( +1)， >0

Moreover，f is a function in L (Q)，where r verifies

r (1一 )]

(35)

(36)

(37)

(38)

(39)

(40)

We notice that the functional in(37)is differentiable just in the directions of ’ (Q)n (Q)

(see[7])，and in this case its Euler equation is not(6)．

By the assumptions(3s)一(40)，J turns out to be defined on the whole space ’ (Q)．We extend|，t。a larger sPace， ’ (Q)with q=NⅣp(

一

1

p

--

p

O)by

=

i

劬

f

。

J

一

(v)

．

蚰 nit

W e recall the following result proved in

Proposition 1．2 Let(39)一(40)hold．Then there exists 0 minimum U of I(v)in ’ (Q)．

Moreover r> N then any minimum dI on ’ (Q)belongs to ’ (f1)NL (Q)；thus

J attains its minimum in ’ (Q)．

／f( ) r< ，舭礼 be2帆9s幻， (Q)n h(Q)，： ! ! 二 !

N —rp

and thus，again J attains its minimum on ’ (Q)．

(42)

fi ～ ——————’—’———— 一

㈠


356

Finally，if (1一 )】 r< r p 、币，

Rem ark 1．11 We observe that P =

of in the two cases above．Moreover．

then h tends to+。。．

Giachetti D．and Porzio M．M

then belongs to ’ (Q)，with

Nr~(1—0)一1】 N—r(1+Op) (43)

As before，we are interested in the regularity of the gradient of solutions depending on the

regularity of the datum f．As in previous sections we have to take into account the upper

growth condition on a(x，s)．More precisely，let US first suppose that，instead of(38)，a(x，S) satisfies the stronger condition

where

al( )02( ) a(x，S) 101(x)a2(u)，

0<Q1 al(x) Q2，

02(s) (1+Isip。

A simple example in which these hypotheses are satisfied is when

(1+lsI)0p Assume that r verifies

a(x，S) (1+lsI)0p’ 0< 1， 0<0< P

r>[p (1—0)1

(44)

(45)

(46)

(47)

(48)

Theorem 1．8 Let(48)，(39)and(44)(46)hold．Then every minimum ∈ ’ (Q)satisfies

【0( ，u)l~lDul∈Lp⋯+ (Q)，

where E is a positive constant depending only on the data．Besides．if r verifies

r >( ) ， then there exists a positive constant such that

RJem ark 1．12

Dul∈Lp⋯+ (Q)

We do not know if a higher regularity for Du holds true， when r

(49)

(50)

r 、 l+Op／『’

Remar k 1．13 W e observe that，in genera1．all the previous results hold for quasi—minima of

(37)．

Remark 1．14 We recall that if a(x，S) general，this equivalence doesn’t hold if

following example：

a(x，S)

a(s)then(44)is equivalent to(38)and that，in

a depends also on the X variable，as shown by the

(1+a(x)ls1)p0’

where，for example，a(x)is bounded with inf【2 Q( )=0

(52)

乏lp

r

吣讧

∞

印，、八

w 二二／

=

n r

C p

=

p ‰

o h S e w

一



Let US now return to the general assumption(38)．Define

= ， s)) ／Pds

357

(53)

We notice that／／(x，·)：R __÷R is strictly increasing and H(R)=R．Hence the inverse

function is well defined，with respect to the variable 7-，C(x，￡)．We have the following result：

Theorem 1．9

Q．and that

Let(48)，(38)and(39)hold true．Assume that there exists丽OC x，￡)n．e．in

㈤ (Q) ∈ ’ (Q)，

n礼d thnt the distrib ti。礼nf de vatiue—Oa—l ／P

五(-z一,s)is n measurable 几cti。礼uerifying

ds +瓦呲．

7 <菩= ．

(54)

(55)

Then there exists a positive constant E，which depends only on ，，N，IQI and IifiiLr( )，such

that every minimum U∈ ’。(Q)with￡，(u)finite satisfies

al／ x，u)iDuI∈Lp．。+ (Q)． (56)

Moreover,if r ve~ifies(50)there exists a positive constant such that

DuI∈Lp．。+ (Q)

Remark 1．15 Notice that we can weaken the assumption(55)SO that it becomes

ds 百( ) +瓦，a．e．in Q，

where百( )∈L (Q)with +妻< 1

(57)

(58)

Remark 1．16 An example of a functional verifying the assumptions(38)，(54)and(55)is

the model case(52)with the function )positive，bounded in Q and verifying，for example，

the following further regularity condition：

∈ m>Ⅳ (59)

We point out that the model case(52)with ( )=XE(X)(XE(X)characteristic function

of E C measurable)does not satisfy condition(55)．In this case the problem of further

regularity of Du is still open．

2 Notations and Prelim inaries

In this section，we give some preliminary results that will play an essential role in what follows．

We start with the following theorem(see【9】)whose proof is the same as the one by Gehring

(【10])：

Lemma 2．1 Let q be an N-cube and set QR= x∈RⅣ：IXi—XotI<R，i

，

}。．。，■■●ff


358 Giachetti D． and Porzio M ．M

Assume that w∈L (Q)and that we have

Q w(x)dx<70{( d ) + 9d > for each 0∈Q and each R<互1 min{dist(x0，aO)，R0}，where R0 and，y0 are p0siti e c0几stan

，

0<m<1 and we have set南u= 口．Assume that the function g belongs to (Q)for 。me s>1·Then there exists n constant d>1 such that w∈ 宅c(Q)，and we ha e

‘

( ) d 7 JQR／2 {( 叫d ) + 9dd )

for Q2R C Q，R<Ro，where 71 is 0 constant that depends only on the da

We will need the~llowing decomposition theorem see[11】or[12])：

Lemma 2·2 Let Q =Q(x0， )be n几ope几Ⅳ be centered in 0 of side 2a， u∈ ， ( )，

> 1 and let-1^r<7<r一1．Then there exist咖：Q —}R and H ：Q —}RⅣ s ch thnt

日∈( 南(Q ))，divH=0， ∈ ’雨( )and

Dvl"Dv De+H， (62)

L (Q ) c(r1 N)I~I’lidull )

D圳L南 (Q ) (1+c(r，N)DI)IIDull )

Rem ark 2·1 W e notice explicitly that the constant c： c

( 63)and(64)doesn’t depend on x0，neither on o-, nor on r if r

(63)

(64)

which appears in the formulas

belongs to a compact set．

An Ol is the foll0wing lemma of real analysis，whose prOof is very easy and can be

found in『10]：

Lemma 2·3 ，(丁) n no 叼n 的 nded func 。几&fined for 0 Ro ￡ R1 Suppose th

atfor 7_<t Rl we have

，(7_) A(t一7_)一“+B+7f(t)， (65)

训，B，Q，7 are non—negative constants， and 7< 1．Then the e s n c0几s￡n几￡c

． depend一

。几g 0几fY 0几ol and 7 such that for every P，R，R0 P<R R1训e hn e

f(P) c[A(R—p)一。+B] We now state the following regularity theorem

Theorem 2．1

proved in[13] (Theorem 2．1)

Le￡u∈W ,o

’

c (f1)，咖。∈ (Q)，where 1< <Ⅳ n d m sn es

1< m < ．

Assume that for all BRl C C Q the following integml estima匏 0fds

r r cl f／ dx+ l

A啪 (R—P)

fo very k∈ N and Ro P < R R1，where A

，口

(N．，m，，R1，la1)，and is n real p0siti e c0几stan￡．

]7

(67)

(68)

n{Ivl> Here c1= c1

Then we have ∈ sn (Q)，where

—一 ——’—’’_ ●

{



8=( m)

359

Finally,we conclude this section with a technical lemma that can be easily proved by

induction．

Lemma 2．4 Let Yn be a sequence of non—negative numbers satisfying

Yn+1 cb 1 ， Vn= 0，1⋯ ．，

where C，b and 5 are positive constants and b 1．

then Yn__+0 when n__++。。

y0 C {6一壶

3 Proofs of Theorems 1．1—1．7(Elliptic Equations)

(69)

(70)

In this section，we prove the results stated in Section 2 concerning the regularity of Du，when

U is a solution of the Dirichlet problem(6)．

PrD of Theorem 1．1 We recall，as noticed in Remark 1．2，that every solution U∈硎(Q) verifies the following estimate：

ullL。。(Q) Co=c0(fi，N，r，II／IlLr(Q))．

Let QR be an N—cube contained in Q．Choose =(U—Un)n。as a test function in(6)，where

叩∈ (QR)，0 叩 1，叩三1 on QR／2，Io71 c／R，and UR= u．Using(7)and the

fact that U∈ o。(Q)we obtain

(1+llullL。。(Q))。

Thus we can conclude that

厂IDul。叩。 I，Q

厂IDuI。叩。 JQ

2 ]U--URI+ f(u-UR)叩2

1+ IL。。(Q) (2 ]U--URI+ <c0( lU--URl+f(u-uR)叩。) P，、2、八u—un)n。l ，

(72)

(73)

where Co= max{2／~，1)，with Co as in(71)．From now on，the proof is the same as that

of the uniformly elliptic case(see[10])and we have

[ ( ) + ( ) whe e c1 c(Ⅳ_I~l

⋯

,r，IIIIILr(Q))·Applying Lemma 2．1 with叫( )=IDu(x)l。，m = and g2N

= ( l，1 + )丙2 l，l we get the result． ⋯ 。

Remark 3．1 It is easy to see that Theorem 1．1 holds true if f is supposed to belong to

W ， (Q)where P>N．

Pr0 of Theorem 1．2 We point out that under the assumption(18)the regularity result(19)

r j

1 ．


360

is equivalent to the following

Define

v(x)

(1+ )。 ∈L (Q)

1

sign(札)((1+⋯ ～一1)

Let QR c Q and 7 be a cut—ofr function as in

function W=( 一(v)R)7 ，where is as in(75)．

厂ID l 77 JQ

f

— jQ


(74)

(75)

the proof of Theorem 1．1．Choose as a test

Using assumption(18)and(75)we obtain 72 JD J

(1+lu1) 。

／ 271DTIIDvllv一( )RI+／，77 ( 一( )R) JQR JQR

(76)

that is，(73)with札replaced by V．Thus proceeding as in the proof of Theorem 1．1 we obtain

that there exists a positive constant E such that Dv∈L (Q)，i．e．(74)holds true． Assume now that r> 2N

． It remains to prove the regularity stated in(20)．To

do this，we will use the regularity stated in(74)．We notice that proceeding as in the proof

of Lemma 2．3 in【2】we can prove that there exists a constant C0，depending only on the data． SElch tbat

厂ID札 _1) c。， t，n

(Ⅳ一2)(1—0)r 2(N一2r) (77)

Let P一2+E and s=2A+(1一 )p>2，where E is as in(74)and ∈(0，1)to be determined later．Using Young’S inequality we obtain

厂ID ：厂 JQR jQ

Du[(1一A)p

R (1+l札I) (1一 )

厂Iou J (1+Iu _1) JQ

Du[ (1+lu1) (1一 )

+／Q ]Duly where el is a constant that depends only on the data

that is，A 一百羽 O(p 而． Notice that we have A < l

r> 丽 2 N ．

(78)

and where we have set 1- A = p，

as it is equivalent to the assumption

Remark 3．2 We observe that it is possible to show that the thesis(19)of Theorem 1．2 js

also true if we replace the assumption，∈ (Q) r verifying(9))with a weaker hypothesis

that f belongs to W ， (Q)，where P>2．

Proof of Theorem 1·3 Let ( )be as in(75)．The function W=v— R)?7 ∈Hi(QR)，by the assumption(23)and the fact that 2(1一 )<q ．Thus it is possible to proceed exactly￡LS in

the proof of Theorem 1．2 and to conclude that(24)holds．

Remark 3．3 If f∈ (Q)with r verifying(11)then，as just noticed in Section 2，we cannot expect a better regularity for Du as shown by the following model Case：

- div

U= 0

( +lu1)。，in D (Q) on 0Q


Regularity Results yor Elliptic Equations 361

We have IDul=IDuI(1+Lu1) LD ⋯ul(1— )+1] ，where u= ．By the Calder0n—

Zygmund theorem，the sharp regularity for u is W2：(Q)n ， ’ (Q)which gives Du∈Lq(a)．

Pm of Theorem 1．4 The proof of(25)is exactly the same as that given in Theorems 1．2 and

1．3．Hence it remains to show that if u∈日 (Q)and r>N／2 then u∈Llo~Cc(Q)，while if f

verifies(9)then札belongs to (Q)，where h is as in(10)．

。。loc‘regularity W e notice that our aim is equivalent to prove that V∈Llo

~C

c(Q)，where V

is as in(75)．Denote by Br the ball centered at X0 and of radius r．Let B2P cc Q，Pn=p+参 and define B几= Bpn ． We have B几] B几+1．Moreover，let be a number bigger than one that

will be chosen later and denote = 一嘉．We introduce the following sequence of cut—off function／z in (Q)verifying：

0 几 1， ID 几l 二一P

9n 2

，几= 0 。utside B几，几

+

Choose (u—kn+1)+as a test function in(6)．We obtain

Q ／ IDul。／ n( ，u)DuDv#2~ t，Bnn{v>kn+1) JSnn{v>kn+1)

= 一 2／， n( ，u)DuD# (u—kn+1)++／， (u—kn+1)+ t，Bn n{v>k +1) JB

Moreover，using assumption(18)and Young’S inequality we have

2 fBnfn{v>kn+l}IZna( ，札)DuD#n(” 2 ／ lDvllz ID． I(v一 +1)+ dB n{v>k +1)

詈厂 + )

I )”I。 2dS n{v>k

+ ap J

厂S { > + )

(”一 + ) n n+1) ‘ nn{ > +1，、。。

(80)

Putting together all the previous estimates we obtain

詈 > + l。”l。 2 兰：兰；箬!兰 > + (”一 + ) + ， (”一 + )+．(82)

Now we show that from the integral estimate(82)it follows that u in Bp for a suitable

choice of ．Using the Sobolev inequality and(82)we have

Cs

Cs

c0

( ( 。 V--kn+l 、萼 ID I2 2+厂

n{v>k +1) Js

22(n+2)

＼

ID． l。(”一kn+1) 1 I

厂 ( 一 + ) +厂 t，Bn n{v>kn+1) 。 JB礼

＼

1．厂l (”一 + )+1 I (83)

where co=c [吾( + )] and cs：。 (Ⅳ) h。s。b。-。v 。n t ant．we n。w dis gui h


362

the two cases：r 2 and r < 2，where

l +1)+


1／r+1／r ：1．If r 2，we have

lI，lIL ) 一，三 V--]~n) (84)

We observe that when we aSsume r>N／2 we have r <7 <2 ，and thus，using(83)and

(84)we derive that

+ (Lie [ 22(n+2) (v--kn+l llL唧州一 +lI卜

where c1：c and An+1=B +1 n{”>kn+1)．Then

y忆

from which it follows

厂 ( + JB n{v>k l’

that

— 1 =

lA +1I IB n v>kn+1)I

Notice that r 2 is equivalent to r 2．Then using

where C2

with c=

is equivalent to the assumption r

2P

B n IU>kn+1]l，

．

丁 ‘

(86)we have

> 喾 ∥，

(85)

(86)

and(86)we deduce(being k 1)that(69)holds true

b=22(r 十 )and =(1一 r、l 2一乒．We notice that >0

，(1一 rI 1

(c1 max@， Lr(f2)) ／2 ( )) b ．

f70)is also satisfied．Hence we can apply Lemma 2．4 and conclude that y忆__+0 as n-_++。。，that

is k in BP．To prove also ko，(ko>0)，we notice that if瓦=一 then 三J = 一 "．Hence it is sufficient to apply the previous result to the function西 = 一"．As a matter of

fact，面verifies a problem of the same kind of u．When r <2 we set y忆=J ("一 )车and the proof is similar to that of the previous case r 2，an d SO we omit it．

L C--regularity Let US now assume that r verifies(9)．Let Bnl CC Q and 0 R0 丁<

t R1 1 be arbitrarily fixed．Choose叩( 一Tk( ))as a test function in(6)，where V is as

defined before in(75)and叼is a cut．off function in c (Bt)such that 0 叼 1，叼三1 on

B ，lD叩I c(t一下)一．Notice that when r<N we have that f=一div(f1)with fl∈L (Q)．

Hence we cIbta．jn

t

a( )。 D[叼(”一 ( 】= [叩(”一 ( 】

We estimate now the integrals in(87)．Applying(18)we deduce that

。( )。uD【一 ( 】驯一／3C Dv 一 (")

(87)

r一——■■·—— i ㈠ i



。一

where Ak．

t is as in Theorem 2．1 and

Young’S inequality we can increase the

／flD[Tl(v一 ( ))]

“

一 2 (88)

is a positive constant to be determined．Besides，using

right—hand side of(87)as follows：

where c3=譬(1+譬)，c4：吉(1+{)and 0= ．Notice that 0 belongs to mc(Q)with m =r ／2．Now we want to eliminate the first term on the right—hand side including Dv·To

do this，we proceed exactly as in the proof of Theorem 5．1 of【13]．Hence choose 5 such that —y： 2A

Q

< 1 and let P，R be arbitrarily fixed with Ro p< R R1·Thus we can deduce that

for every t and 7_such that P 7．< t R，we have

厂ID I 7厂ID I + l，Ak

l，Ak，

C3

⋯

V2+c4 。 (90)

where c is the constant given by Lemma 2．3，that is，a constant depending only on 7．Thus

V verifies the assumptions of Theorem 2．1 of Section 3 with = =2．The condition(67)is

equivalent to r<N／2，which is true by assumption．Hence V∈L c(Q)，where s=(r ) = N T

，which implies乱∈ h0。(Q)with h as in(10)．

P伯o1 of Theorem 1．5 Let US define

( )= u。( ，s)ds

Then

Dv(x) + “ Oa(x,s)ds (93)

Let QR C Q and叩be a cut—off function as in the proof of Theorem 1．1．Choose as a test

function =(V—vn)n ．We obtain

|a(x u)DuDvT]2dx 2|a(x，u)lD~llv—vnlnlDnl+|f(v—vn)n2． JQR JQR J R

W e now estimate the terms in the previous inequality．Then we have

)DuDwl2= 。 I2T]2-]- a 乱)DuT12 “ ds． (94)

and for the last integral in(94)，using the assumption(26)，we have

)DuT]2 “ ds + ， (95)

啊l一㈡


364

where we have set

Besides，we have

g(x) e ds

2

Qn

a( ，乱)I。u --VRI叼I。叼

2 27／2~去


D叩1 Iu—uR1 1， (97) j where is a p。sitive c。nstant t。be ch。sen．Putting together the preVi。us estimates we deduce

that

( 一 ) 叼2 叼2夕( ) +丢 I。 I”一” + ，(”一 R

Let US choose ： 1． From (9s)and(93)we obtain

2 (0( I27／2+2／Q g㈤叼2 6 9( )2T]2~ 32c2 l

V--VRI +8 ，(”一”R

(98)

Now。bserving that，by the assumpti。n(26)，Ig(x)l bel。ngs t。 (【2)，where = is bigger

than one；we can proceed as in[10]and conclude that there exists a positive constant E0 such

that D ∈L2 +E。(【2)which implies(27)with E=min{a，E0)．Thus proceeding exactly as in h

proof of Theorem 1．2 we can conclude that(2s)also holds．

f D =1D (u—uR)]I。一 D 一uR)]+日， l div(H)=0，

HIIL ，(Q ) c(q，N)lq一2111D[w (u—uR)]l1qL-- (1Q )，

D IIL ，(Q ) (1+c(q，N)lq一21)IID[~ (u—uR)】IIqL-- (1Q )

Choose as a test function in(6)．We obtain

from which，using (93)，we

where g is as in(30)．From

，

(104)，

u)DuD~= ，

using Young’S inequality,we obtain

(101)

(102)

(103)

(104)

。一”R)]。 l。[(772-1 一”R)]I I。 I+ Ig(圳I。 I+ I，

T ■— — —— — —— — — ___ _-’’ 1

j， 1 i{ l

i{ l } } l ；{ {

，

厂

+

D

厂

<一

-暑 D

厶


／o 。[2 圳一 +(1 ) IDol+ Ig( )11 l+／Q l 1．(105J J ) Q ＼Q Moreover，using inequality(102)，we have

／ [277lD77llu—uRl+(1—7／2)IDo1]lD l+／ l_9( )llD l JQ ＼Q JQ

ll277lD77llu—uRl+(1—7／2)ID~IIIL (Q ＼Q )IID~IIL ，(Q )+IIg(x)ll~ (Q )IID~IIL ，(Q )

c。f 2C 一 lLq(Qa＼Qv)~II IILq(Qa＼Qp)+II )11州 I × 。(V--VR) q--(1

l ·V--VR- + ㈤-。I + ／lD【77。(u—uR) (106)

Q

： 1+ ，Ⅳ)，： c~￡。

2

一

q3

】

q

where CO max el1 3]c(q C1 ．and 5 is a positive constant that will be chosen =1+ q ，，Ⅳ)， = q一1。 a

later．Using(100)，(105)and(106)we get

／o Iv【77。V--VR)】l。= D[7"／2 V--VR) 一 D[7"／2 V--VRJ )】日 Q

{ iV--VRIq-'~ IDvlq+ ㈤·。} +／ I， I+25，n ID 。一uR)1I。+ ( )I1日。，(Q )， (107)J J Q Q

．

⋯

where Cl is as before，i．e． it is a positive constant that depends only on 5，N and q in a

c。ntinu。US way,and ( )= 击．Let US choose s= =((q ) ．Notice that r>s

since r<N／2．Thus using the HSlder and the Young’S inequalities and(100)again，we have

／Q -， ·! (／Q ·，- ) (／Q ·c · ) ! c (／o -，- ) (／Q - ) ·。 ) ’ s( - )i(／Q I嘲V--VR)】l。 -。 )

<(2 (／Q l ) + ／Q I D【一)】l。， (108) where is as in(107)，Cs is the Sobolev constant and"71(6)=61-q．Using in(107)the estimates

(108)，(101)and(102)we obtain

LO-I嘲⋯ { 。 + IDvlq+Ig(x)lq+( Ills) } +36+( c(qjⅣ)。 Iq_2lqI ]／o I D【772 V--VR)】l。， (109)

where C2一max{c1，2q'71(6)c~}．Choose =1／6 and observe that if r 而壬 then


366

q=q(r

q=q(r


_}2．Then it is possible to choose r0 such that if r satisfies(31)then the corresponding

verifies

丢)+ 。 that is，if we choose，for example，T0 satisfying q(ro) 3／／2 such that

[ 。(丢)+丢]ma {( 。ccq，Ⅳ )‘ ， )-qcr。一2· 1 驯。

+ IDvlq+ 刮。+( Adding to both sides the quantity c2

。

IDvl。，and applying Lemma 2．3，we have

⋯ {cq f~ 。+ g(x)iq+ ( where C3 IS a positive constant that depends only(in a continuous way)on q and／Y

that when q=( qN) ，by the Sob。1eV inequality we get

V--VR-。 ( 一 ) ， where Cs is again the Sobolev constant．W e point out that，since q 2，

c ( ) ，p= (see for example【14】)，we deduce that C C4 dividing by RⅣ

，we obtain

]f ID I。 JQR／。

where

(112)

Notice

(114)

=c4(N)=( )。．Using(114)in(113)and

q Cqc4( ) + 圳。+

㈦ - ) + 刮。+ )j

Dv∈ Lq⋯+ (Q)， a(x，u)Du∈ (Q)

(115)

1—8／q

E(q，N，，r)and =q+min{e， )．It remains to show that ifg(x)is more regular

that is，if it belongs to L2+a(Q)， >0，then(33)also holds true．As a first step we will prove

that Dv is in c(Q)and hence by(93)a(x，u)Du also belongs to 20c(Q)．To do this，it is sufficient to show the following equality：

{m∈ 。)，2】： ∈ (Q)) 【q(r0)，2】

]■—————■’—_ l 一 f

l} l

J a ．暑

∞

=

F

㈣

w d 姐

肌

m ∞ 岫

a 盯

n L

耐

伽

y r

拈

n ／

．量

吼

c F

铝鲫一


Regularity Results for Elliptic Equations 367

This last result can easily be proved by following thus showing that A is an open and closed

subset of【q(r0)，2】．Finally to prove(3a)，as now we know that Dv∈L2o (Q)，it is possible to proceed as in the proof of the previous theorem．

Remark 3．4 We notice that if r> ，it is proved in【2]2 that any distributional

solution found by approximation can be used as a test function in(6)and，in particular，

a(x，u)Du∈L。(Q)．So if we suppose a better regularity on ds，that is，that this term

belongs to + (Q)，盯>0，then we can avoid the use of the Hodge decomposition technique

and prove by following the outline of the proof of Theorem 1．5，that the results of Theorem

1．6 hold for every solution obtained by approximation as in『2】．

Now in order to prove the quoted regularity results it is sufficient to apply the following result

that can be easily proved by using the techniques in【6】：

Proposition 3．1 Let W be a measurable function such that Tk(w)∈ ’p(Q)，1<P<Ⅳ，

，0r every positive k．IfF∈L (Q)and we have

where 1<m<(P )

·

N p， the佗IDwl 一 ∈Lm"(Q)

4 Proofs of Theorems 1．8——1．9(Minima of Functionals)

In this section we prove the results stated in Section 2 concerning the regularity of Du，where

U is a minimum of the functional(37)．

Proof of Theorem 1．8 By assumption(44)and the minimality ofU we have

。Q (1。uln；1(u)) 一，u Q (1。切l。；1( )) 一f伽， for every W∈ ’ (Q)with (伽)finite．Let US define

(7_) ㈦ds

(117)

where 71 is the positive constant that appears in the assumption(46)．Since耳(7_)is strictly

increasing and百(R)=R by(46)，the inverse function百 ( )is well deftned on R and the

following estimate holds true：

耳-1(￡)} lG一 (圳， where G(7-)= (1+IsI) ds

The estimate(119)follows from the fact that l百(丁)l Ia(r)1．Notice that that

l百 (￡)l<【m一 )+1】 _1_

(119)

by(119)we deduce

(120)


368

Moreover，using assumption(46)we have

I ) ㈤I ( + ㈤ Denote百(叫)=面．Let US point out that百 is bijective between

=W∈ ’ (【2)：J(叫)<+∞)

Gi0chetti D．and Porzio M ．M

and ， (【2)as we can easily see by differentiating叫=百一 (面)that

I=1 ) (面)1 Ij from which，using(121)

D面

(122)

(123)

(124)

Indeed．fr。m (124)、 see that D叫 ∈Lq(【2)if D面 ∈L (【2)，and then W∈ ’。(【2)si＼- -- -] ．．v

nce

万一 (0)=0．M。 er，J(叫)<+∞ since D面= 1。21 p(w)Dw∈L (Q)·We can rewrite

(117)aS follows：

floa,71 ·。一，百 c” max z · [ ·。面· 一，万一c面)] where u：百(札)．Hence u is a quasi—minimum in ’ (【2)of

即 )= IDol (矾

(125)

(126)

and the regularity(49)follows from Theorem 6．7 of[15】．Now we show，using(49)，that if r

s is舶s(50)then(51)also holds true．To do this we notice that，proceeding as in the proof of

Theorem 1．4 in it follows，by using(50)，that there exists co(co >0)，depending only on

the data，such that

1 I)( 如 co

where > 0 is defined by the formula —N— —r~v(1-O)-、1](r-1)． (．Ⅳ一rp) 一1．Let =tp+(1一t)(p+E)

where t∈(0，1)is to be determined later．Using Young’s inequality we obtain

p(1 1)( IDu l(1-t) (p+e)

Ip(1 I)(X--O)p q_ ID ulp+e

where Cl> CO is a constant that

o(p+E)，that is，t= (p+∈) ( 一日)p+8(p+E)

Cl， (127)

depends on1y on the data and where we have set(~l

-

一

o

t)pt：

∈(0，1)．Hence(51)holds true with = 一P．

of The。代m 1．9 Our first goa1 is to prove that the maP旦：一 ’ (【2)defined by

：旦(叫)=H(x，叫( ))，where H is as in(53)and A as in(122)，is well defined and bijective·

Indeed，using(55)we have

。：。／ ( ，叫)。叫+／o” O x ds c Lp(【2)， (128) for every叫∈A．Moreover，for every互∈ ’ (【2)there exists z∈A，such that互旦(z)·As

，

¨

+

一

h

< 一

训

毛吾



a matter of fact，if we define z(x)：C(x，兰( ))，then

Dz= ( )+一OC(It( )D兰， a．e．in Q d

Let US observe that using assumption(38)we have

I when lH(x，7-)l l

—

lD_zl — lD_zl

一一

G(7-)l，where G is as in(119)，we have

Ic(x，t)I lG (t)I

(1+IC(x，兰)I) IDz

369

and thus

I ( )。兰I< ( +【1 1 ) where the right—hand side belongs to L (Q)．We can now conclude，using(54)，that Dz belongs

to L (Q)．

Using the definition of旦 and the minimality of u we have

where we have set

F(x，，∈)

笪，，型，，

∈一／oc )0(a1／v(x，

Yv_ ∈ ’ (Q)

s)) Ox

p

一，C( ，Y)

(130)

and笪=H(x，u(z))．In order to apply the quoted result in【15】it is sufficient to verify the

growth conditions of F．We have，using(55)，that

c。 +cl IV(舢 )l + Iflv+ +c2 ， ) 、一～

>c3 _c 4l∞ 一一一cs，

where <r is to be chosen．Notice that，by(129)，we have

，y)l ([1yl(1— )+1】 +1)仲 c6lyl~+c7 where < P ，by the assumption on 7．Besides，we have

IC(x，y)l f[1yl(1— )+1】南+11 c8l l禹+c9．

We notice that南 <P ，by(48)，and thus we can choose <r such that <P ．Hence it follows that

IC(x，Y)l +IC(x， )I CloI +Cll，

where ：max{ ，尚 }<P ．Thus the hypothesis of the quoted theorem is satisfied and then the sta，ted s1】】ts 111，nw

It remains to prove that under the assumption(50)the regularity result(57)holds true but

we omit such a proof as it is similar to the proof of(51)in Theorem 1．8．

RJeferences

f1】Calderon，A．P．，Zygmund，A—On the existence of certain singular integrals．Actn Mnth．．88．85 139 (1952)

一 —1●——————————’—— ● ：． }

； }


370 Giachetti D．and Porzio M ．M

f21 Boccardo，L．，Dall’Aglio A．，Orsina L．：Existence and regularity results for some elliptic equations with

degenerate coercivity．Atti Sere．Mat．Fis．Univ．Modena，46，51 81(1998) f31 Giachetti，D．，Porzio，M．M．：Existence results for some non uniformly elliptic equations with irregular

data．preprint

f41 Meyers，N．G．：An Lp—estimate for the gradient of solutions of second order elliptic divergence equations．

Ann．Scuola Norm．Sup．Pisa，17，189_206(1963)

【5]Giachetti，D．，Schianchi，R．：Boundary higher integrability for the gradient of distributional solutions of

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Elliptic Equations with Degenerate Coercivity: Gradient Regularity

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