REGULARITY OF H1∩Lnγ-12-γ WEAK SOLUTIONS FOR NONLINEAR ELLIPTIC SYSTEMS

10 (1990), 2, 173-184

1-1

REGULARITY OF HlnLn~ WEAK SOLUTIONS FOR NONLINEAR ELLIPTIC SYSTEMS· He X 'lJikJrrbg (1PJ Jtg ~ ) Dept. ot Math.,

Ohen Baouao

(~* ~ _)

Zh0'n(J8han U niv., GfJ,Qngshoo, China

Abstract Let us consider the following elliptic systems of second order -Da(.A?(x,

U,

Du»)==B4 ($,

U,

(1)

i==1, •••, N, xEQcBn, n~3

D·u),

and suppose i) j.A?($, tt, Du) 1 ~L(1+ I Dul); ii) (1+lpl)-1A?(x,u,p)are Holder-continuous functions with some exponent uniformly with respect to p, I.e,

a

on "QXBN

lim sup .A~($+O', u+-r, 1') -.A..?(aJ, u, 1') ~K(u) < +00; (1+ 11'1) (Ia l + l-rj)~

Ccr,'f)-+(O,O)

iii)

A?(aJ, u, 1') are differentiable functiona in p with bounded and continuous derivatives \Afp.i (aJ,

iv) .A~";($,

U, p)£U~~AI£12, \f£E RnN;

v) for all uE Hloc(Q, EN) 8(11'1"+ lul T ) +b(x), where

U,

1') I ~L;

A>O;

')'-1

nL~ (Q, BN),

st»,

U,

D·u) is measurable and IB($, U, p) 1 <;

1+.!<1<2, 't'~max( n+2, 'Y-l n n-2 2-'Y 2",

,"2

b(x) E L n:;:;' ,;;:;::;'

Remarks. Only bounded open set LP,).,«(J)=={UELP«(J):

Q

+8

(Q),

-8),

\7'8>0,

\7'8>0.

will be considered in this paper; for all p~l,

~U~LI',,.«(J)=[ sUR a;E

O
p-).,J

D(e,p)

A~OJ

IU 1Pdx]lIP< + oo}

which is called a Morrey Space. Let assumptions i)-iT) hold, Giaquinta and Modica t1,2 ] have proved the regularity of both 1+.! and the H 1 n L oo weak solutions of (1) n

smallness condition

2aM
n+2

IBI ~a(!pl"+ lul~ +b, O<'Y~ under natural growth condition IBI ~ajpI2+b with a

the Hl weak solutions of (1) under controllable growth condition

which implys that the H 1 n Loo weak solutions have the same

regularity in the case of 1+~ <'Y<2. In the case of 'Y=2, many counterexamples (see [2J) showed n ')'-1 that u must be in H 1nL-, while in the case of 1+.!<'Y<2, we consider the H 1 n Ln 2=Y weak n ,,-1 solutions of (1), weaken the integrability conditions upon them (from L" to L-M) and obtain the

* Received May 16, 1988. Revised Dec. 26, 1988. This work is supported in part by the Foundatio.n of Zhongshan University, Advanced Research Center.

174

ACTA MATHEMATICA SCIENTIA

~::::~

same regularity-results. Finally we show that the exponent n

Vol 10

can not be decreased anymore

for the sake of the regularity results. Definition 1.

f

.jQ

We call uEH1 nLn 2=Y(Q, BN) be a weak solution of (1), provided that ~-t

f

-y-l

Al(x, u, Du)Da¢>'lk=- oBi-ere, u, DU)¢>'dre,

V¢E Han Ln-::Y(Q, BN),

(2)

a==l, •••, n } wheres==(!lJb ••."xn)E·Q, u=(u1, .•••, uN),Du- { DciU ' , .•. N • 1==1, •••, We use the convention that repeated indices are summed- i, j go from 1 to N and a, f3 from 1 to n.

§ ·1. Introduction Under natural growth conditions for the regularity of elliptio equations or systems of second order, authors, in general, assumed that ruE H1 nU", But 0_ A. Ladyzhenskaya and Ural'tseva[3J showed that for single equations, if we want to prove tho Holder continuity of weak solution u, it is suffioient to restrain u in

H1 n L

2

n 1'-1 2 - 1'

.1-1--<1'<2. n In this paper, we oonslder the completely non-linear elliptio systems (1) under natural growth condition v), generalize furthermore the technique in [2J, which was 'employed to treat the case B=O, and obtain two more sophisticated LP-

n

n 1'-1

estimates for the derivatives of the Hl L 2-')' weak solutions of (1). Using these est.imates and the method of perturbation we prove the Holder continuity for the 2

derivatives. Finally, with an example, we show that in the case of 1+-<1'<2 if , n we consider the H?

nY

weak solution of (1), then it follows that r>» 1'2 -1

other-

-I'

wise the local uniqueness principle .for the Dirichlet problem would be Violated. Hence the integrability condition for the weak solutton can not be decreased from 1'-1

L"~. Still more, as what concerns with the growth condition of

we have made our best effort.

Remarks.

9'b+2 1'-1 1) 7:~max( n-2' 2- . I'

n+2

) = m-2' 8 1'-1 { <2-1"

2) Suppose b(x) ELU(D), CT>n, then b(;r) EL W e use the following notations:

f

Q

U

if

&= TQT

Q

U

ax, UR='U"".R=

r +s

2 n

f

•

b(a;) and B on u,

1+.!<'Y~.£.+l:-; 3

n

1

2

n

2+;-<,,<2. (D).

2n 2' 2.= n+2' 2m n; uda; 2* = n-

BR=BB(WO)CR: stands for the ball with radius R and centre a;o, 0 represents the constant at various oocaslons and s,a,() the sufficiently small positive constants.

§ 2. Two Lp-Estimates for the Gradient Let QcRn be an ~-oube, QR=QR(XO) ={xERn:

Iwa-zoal
0;=1,

n},

!,

9 be nonnegative and gE Lfl(Q), Lemma 1. Suppose

fbr

175

WEAK SOLUTIONS FOR NONLIN.ELLIPT. SYS.

No.2

IE Lr(Q)

with tr>g>l, then we.have

( if OO~Ol( ( 11 oo)q + ( jq eJ,a;+() ( if drt JOB JQIl~ JQIB JQIB each aioEQ and each B< ~ dist(a;o, aQ) ARo, where u; 0 1 , and

() are.eenstants

with 0 1>1, R o >0,0,9<1. Then gELi1oc(Q) for pE [q, q+s), and

(t. g"eJ,;rriP ~02 {(to..

Ylq + (t./peJ,a;Y/P}

ifOO

for QJBcQ, R
,

Proposition 1. Let i), ill), Iv), v) be sablsfied w.lbh 'li'o;;;;;max

~(a;)E L 2••';-2.

(),

(n+22' 7-1) 2-1' . '1l.-

(D) and uE H 1 n~, 1+1..<1'<2, be a weak solution of (1), then n

there exist some Ro>O andp>2 s.t, uEHf~(D, RN) andforeaohaioED, R
. {t./t (1+ IDuI)'OO riP t., (1+ IDuIJl)OO, <;,,0

-

'Y-l

where O=O~n, 'A, a, Ilbll L2., -j-2. (D)), R o is a constant depending on the L"2-'Y -norm of u and L2- n or m of Du, Proof. From i), iii) and iv) it follows

..4.~(IlJ, u, p)p~";;;!;~ IplJl-k, L 1>O. Let aioED. R
~

LIDu/

JI-rI'OO<;"OB"+2L{t.

f + t. Ib.(a;) /·IU-UB/-rI'da;

IDuI·lu-UBI711.D7]1da;

+ B. IU-UB /71 ID71 Ida;} +afa. (I Du I'Y+ IU1'1") IU-UBI71 2OO and by the Young inequality, the Sobolev-Potnoare inequality and the Holder inequality, we get

2L

JBB IDul·lu-UBI77I D77ldai

<;" : 2L

1..

jDu 1 -r1'OO + 0 B2

2

(1..

It»

1+ 2

la.eJ,a;)

lr

t. lu-uBI71ID71IOO<;"B"+OB-JI (JB.1DulJl.OO )

GfB. 1.Du17·lu-UBI~daJ

1+ 2 If.

176

r (t.

ACTA MATI-IEMATICA SCIENTIA

..;;;;a(t. IDuro! th;Y/2[(JB. Iu-u.l 2·th; «o

(J

luI nl=~ dz)-n 2-'Y

B.

J IDu/ B.

2

/2 '

Vol. 10

22 Iu-uBI"~da:Y- / ' r-"l'/ll

dz = h1 ( /I u IILn ~~~

(B s

»)

f IDuI

2

. BR

dZ,

where h1(t) and h,(t) , t=2, 3, 4, afterwards are strictly increasing functions with h,(O) =0, 1,=1, 2, 3, 4. 2 3 1 . n+2 In the case of 1+<"Y '-2 +-, it IS sufficient to assnme 'r=--2-. n n n-

a

t. /u ,... ·lu-uBI7J' d
r· 1

da:

By the Sobolev embedding theorem (see [5J, Lemma 5.10), there exists constant

o independent of'U and R, s.t,

(t. lul

";;;;O/GR/G

y+Olc(t. IDuI y, 1

2

&

1

2

k:» n(n-2) ~ 2(n+2) •

&

Hence by the boundedness of Q we have OJ

J Iu \ BR

'U •

IU -

U,R 11]2

r

2 ";;;;0R" 2(t . 1U & 1

I

da: ·1

~O~+ [h.2(IIU~LI(B.»)

17+?

n:7

(t. IDul:J& y+0 (t. 1

+ha(I/DuI/LI(B.»)]

3 1 ')'-1 For -2 +-<"Y<2, 'r<-:;--,we have n ~-I' a

J'B

R

2

IDul 2dx )n=2t., IDuI 2d4l

IDuj2 dai,

t. IU, .... /U-uBI7J'&..;;;;a(t. lu/ 20",&Y/2·(fa.lu-U t! l " th;Y/2' ";;;;OR"/lI(t.

OB. IDullIth;r

lu/ .....dxr/..

~O~+h4(lluIlLn ~~: (B

s»)

1

J

Bit

IDul.2dz.

For b(z)EL2·,i 2* ({J) , we have

t.

Ib

(x) 1·1 u-u. I rio! dx..;;;;(JB.

,,;;;;OCllbh··.;-.·,o)).R"/2

«otr-vo

J IDu B.

1

r (t. r

I b 1 • do:

(t. IDul

2

2

da:

I

u- UB 12* dxrlS"

1

\2dx, 0>0 sufficiently small.

(3)

177

WEAK SOLUTIONS FOR NONLIN. ELLIPT. SYS.

No.2

By the absolute continuity theorem of Lebesgue we get

!UIL"*
~D'UIILI(B.)~O, when R~O, hence hr-+O, ~==1, 2, 3,

and

4, thus there exist someRo s.t, for all R<:Ro

t."

IDuI2dai";;;;0.R"+OR-ll(t.IDttI2.ck)

1+ 2

Or rewritten as

t.,O

1r

+1/2"+1l

1+ 2

(1+ IDuI)2 dai ";;;; 0 (fB. (1+ IDuI)ll.tko)

1r

+

By Lemma 1 there exist some p>2, s.t.

(tR"

~

t.

t.

(1+ IDttlll)tko

(1+ IDul)lltko.

(1+ IDul)flck)ll/fI<;;0ts. (1+ IDuF')da;.

0

Set ¢(R) = ( iDu- (Du)Blllda;, we have

JBR

Proposition 2. Let uEL"E} be a weak solution of (1) and assumptionS.i)v) hold. Then there exist some GliE (2, p), 8>0 and constant 0 s.t; for every zoE Q, R
(t."

IDu- (Duh/41"da;

r/

« <;.0

where h (t) is an increasing function. Proof. Define

t.

IDu- (Du)Bllldai 1

+h(luBI + I (Du)BI-1-¢(R)~)Jl8

Gr(Qj, y) = Af(y, u(y), Du(x)) -Af(Qj, u(Qj), DU(Qj)) I

J:

G(ai.

lI)={~i1[Gf(ai, lI)]ll}i,

A~t= .A~.j(lI,

U(lI), l'+t(Du(ai) -I'»dt,

1'=

(v~).

From iii) and i v) we get

.

11~f! ,L, A:fe~~~AI~12, VeER"N. Z=!.

Hence, for every cf>E Htn L" ~-'Y (D, RN) and eyery yE D we have

tlf!(DsUi-v~)D..¢ida;= Let p
p= :

t

p and

7]1

be a standard test funotion on

EOo(B p) , O~'71~1, ID'711 ~O/p, 7}1=1 on B p / 2. Inserting Du(y) (a;-a;o))7)~ and ')'==Du(y) in (4) we get

to,. I

Du(ai) -Du(lI) Illda;

.,;;;;0 {llpll

JB3

(4)

Gf(ai, lI)D..¢'da;+tB'(ai, u, Du)¢'ck.

IU(ai) -~-Dtt(lI) (ai- :To)

1

2

dai +

t:o

Gll(ai, 1I)da;+

Bep) ,

7]1

ep=eplC= (u(~)-~-

t.. IBI·I¢lldai}

178

ACTA MATHEMATICA SCIENTIA

«o {llpaJ .

lu(w)-uz-

~

Vol. 10

J GfJ(w, Y)dX+f'

(Du)2I(w-wo)ada;+

~

IBI··:I'4>~ldx

~.

+pnIDu(y)-(Du)~llI }. By the Sobolev-TotnoareJnequalfty and taking nheaverage in yon: B:a we have

tP/I dytP/I 1.Du(ai) -.Du(y) 11Ida;
~

IDu(y) - (Du),.llIdy+f tlyf' ~

G2(i, Y)dy+f

~.

~

dyf

~

;/Bl oI4hlda; }'.

(5)

.

In order to estimate the second term on the right hand side of (5), inserting ep

-=¢a = (u(ro) - u p - (Du) p(a;:--. xri)7Ji and' ')'= (D'll) p in (4) ,where TJaE 0 0 (B p) , O~7]:a'

:1., IDTJal "alp,

tp

and 'YJa=l on B~ we obtain

jDu(ai) - (Du)plllda;

,0 {.llpaJ ',I Bp

U-U p-

(Du) p(a;-mo) 12aa,

+J

Bp

(P(w,

y)dQi+J

Bp

IBI ·1
By the Bobolev-Potnoare inequality and taking the average in 'Y on B; we get (

JB 2S

IDu(ai)-(Du),llIdai

<:0 {( ( IDU(ai) - (Du)p IlIodai) J~

1+* + J~ ( dyf· ~(x, ~

Y)dai+f

~

dye J~

\BlolcPlildaf. (6)

Let's estimate the last term of (5) and (6),

of

BZ

IBllcP1Idx
Bz

IB/·I (Du(y) -

(Du)p) (W-aJo) 100

+fB3IBloIU-'l1,,- (Du)~(ai-aio) Idai}.

(7)

Estimating the first term on the right hand side of (7), we get

fB3IBloIDu(y) - (Duh! ·lai- aiolcl4; "OpIDu(y)-(Du)pl·[f

f

Bz

B:a

1.DuI"dQi~f

IDuI"dai+fBz IUI'Vdx+fBz l(bx)lox]

B~

(1+

IDul)Pd~==F(p).

Obviously F (p) ,OF (p) •

For 1+~ <1' ~ 3 2 n

+1.., 'Ti= n

n+ 2 , by the choice of s; and (3) we have

n- 2

r/l!"

fB3 luiTdai< OlP-nT/lI*GB" Iu /1I*da;

y 1

<;'OllP-

n 7i

/

2

*

[pn/lI* (t~ luillda; +(fB~ IDu12da;

rr 1

No.2

WEAK SOLUTIONS. FOB NONLIN. ELLIPT. SYS.

~O [1+p'fG1I» IDuljda;

179

r/J 1

r

l 0[( (1+ IDul)'da; ' .,.;; J1I» { o[ 1+pl»o!1I» IDul'da; ]

for

71~P

<:OF(p). 3 1

For

2+-;-<"1<2, we have

t1l» lul'fda;"';;Op-a{S1l]lIul nr=;da;y,/n "';;Op-·', where 0:1 =

2-'>'

2 <1. Hence, for 1+-<')'<2, we obtain 1 "1n

'11-

-

t1l]lIul'fda;"';;0(J~

(8)

lulJ.'fda;r/J*.o;;;O(F(p)+p-.').

For b(a;) E y*,n2*/2+s(D)

f

B3

ibIda;~( ( Ib IJ.da;1 /a..o;;;Op-<1+n/J)pn/J+8/J* = Op-"

(9)

JEAS

where (12=1-8/2.<1. Thus we have (

JB:a

IBI·IDu(y) - (Du)pI·Ia;-a;o]da;

~OIDu(y)

- (Du)pl (pF(p)

+ pl-tJl+pl-al)

<; IDu(y) - (Du)pI2+0 p2aF2(p)

where o:=min(l-tIl, 1-0::a) >0. Now let's estimate the second term on the right hand side of (7). By (8) and (9) we have

o JB2S ( (lul-'-+lb(a;)I)/u-u,,-(Du)p(a;-a:o)/da; ",0

[(fa" Iula.-'-da;ria. +(t"IbIJ·aa;r/J. ](t,,1 u-u

p-

(Du)P(ai-a:o)

1

.,.;;0 [F(p) +p-a'+p-a']p(t"

",OF

J(p)p:l4+

and O(

JB2S

",0

~(:r'

t"

Io«: (Du)pIJdr.r)~

IDu-(Du)pIJdw

IDul'Ylu-u p - (Du);;(a;-a;o)

Ida;

(fa" IDu I' da;'YI'[ pi (Du);; I + (f1l]l lu- u"I,I<'-'Y)da;y-" I'].

Note that for 2" l+'Y-p < 1..=.!.. P-'Y . 2-')"

80

we have

rIP

IJ*d,a;

180

Vol. 10

ACTA MATHEMATICA SCIENTIA

fB'1l I

(fB]j IU- tl]jI,.da;)9/91'/<'-'1') d'l:'''Op-"",

u

U-


JB 2

(J

B21

1+')'-1' )"" Iu In P=::ydo:

IDuI" dai,

_prp-2) . ( ) >0. Thus we have 1'-1) P-I'

where aa- (

o fB'1lIDul'l'ju-u

- (Du) ('l:'-(2)o) Id'l:' )(")'+1)/1

(

"Op ( h'1l (1+ IDui)'d'l:'

( (

+Opo, h'1l (1+ IDui)'d'l:'

)

~/2), ~= (P-2)/(1'-1) >0.

"Op!J"F!J(jJ) for some 0<6"min (;,

Hence we obtain (

JB2

IBI·lu-u.-(Du)p«(li-(lio)ldz

~ (:

"Op!J"F!JCjJ) +

fB]J IBII
Y" t'1l

IDu- (Du)pl!Jd(li for some 8>0,

+OIDu(y) - (Du)pI!J+

~

(10)

(:Y" t:a I

Du((li) - (Du) , l!Jdz.

(11)

Since we have

( IDU(Qi) -

JB 2 setting it

(Du)

12dai~2f IDU(Qi) -

Du(y)

R2

IJ dQi + 21Du(y) -

(Du) p 1 2 ,

in (11) and taking the average In y, we get

of

R2

dye

J R2

IBII
+ 41 JB~ ( dye IDu«(li)-Du(y) l!Jdx. JB A Analogous to (1.1) wo have (substitute p with p)

o JB(

p

IBIIc/>2/

dZ<:OpfJ8F2(p)

+

Since now for every p>l

fs; IDU(Qi) -

f

s;

(Du)pl'dQi~fB

1 /Du(ai)- (DU)p/2dai= 2

p

f

1 2

fB p

/Du-(Du)plfJdz.

(13)

JDu(Qi) -Du(y) l'dQi,

(14)

B3

dyf

Bp

(12)

dye

JB

p

(15)

IDu«(li)-Du(y) l:ldx,

by (5), set in (12), (6) and (13), finally by (14) withp=2. and (15) we obtain

t""

I

dyt"" DU«(li) -Du(y)

+ JB (

p

dye

JB

1!Jda;<;;o{(t" dyfB.o IDU«(li) -Du(y) I2-d!/' 1

p

(GfJ(z. y) +p!JaF!J(p»da;} + 2

f

Bp

y+i

B IDu(Qi) -Du(y) 1

dyf

VQioE D, p
p

2 d ai ,

(1.6)

181

WEAK SOLUTIONS FOR NONLIN. ELLIPT. SYS.

:No.2

Therefore, applying Lemma 1 in DxQ with g= IDu(a;) -Du(y) )2*, f= [a!" (G 1

+p·F)]ll., q=1+

u;/\disb (wo, ea,

~

and ()= ; , we get for some

~E (2,

p), p<1+')', VXoE D, R<

(t.,. (Zyt." IDu(x) -Du(y)/a
IDu(x) -Du(y) jllth (17)

By Proposition 1 we have F(R/2)

=1../. (1+ 1.DuI)'th~o(t. (1+ IDul)llth )'/

1

~O(l+

1

I CDU)BI +ef> (R)2)'.

It remains only to estimate the second term on the right hand side of (17). Since we have established Proposition 1, by the standard Giaqutnta technique ([1, 2]) we have

(t..

dyt.,. e"'(x, y)dx

y/'" ~h (luRI + I (Du)BI +~(B)~)B-.

Then from (14) and (15) Proposition 2 holds

§ 3. Holder Continuity of the Gradient In this section we'll use the above V-estimates and the method of per burbatton to prove the Holder continuity for the gradient of weak sol.ublon of (1). At first, let's establish the following Lemma. 1'-1

Lemma 2. Let uEL"2=Y(D, RN) be a weak solution of (1), and assumptions

i)-V) hold. Then there exist positive constants 0 and 8 s.t, for every woE D and O
t.o

IDu-

t.

(DU)p/2dx~O[(p/R)"+1l+X(B)] .

1

IDu- (Du).12th

+HoCJUBI + I CDU)BI +cPCR)2)R"+8, 1

(1.8)

where x(R) =cu( IUB f + I CDu)Bl +cPCR) 2", c/>CR)). wet, s) is inoreasing in t for fixed s, and w-+O uniformly with respect to tE [-M, M] when 8~O; HCt) is inoreasing in t, Proof. It is sufficient to prove (18) for p
Wo

E Q and set

J:

A~= A~.i(",o, 'Y-l

UB, (Du)B/4+ t (.Du(",) - (Du)B/4) )dt.

Let 'lJEHlnL"~(BB/4' BN) be the solution of the Diriohlet problem:

182

ACTA MATHEMATICA SCIENTIA

Vol. 10

Da,{Att.Dsvi) =0 on B~/4 { v-uE H&(BB/4' RN) then from the theory of linear elliptic systems (see [2J, Chap. III), we have for R/4 IDv- (Dv)pI2dx~O(p/R)"+JJ JDv- (DV)R/41 2dx.

J

~

p<

B~

/,-·1

Set w=v-u, then wEH5,1'nLn2-'Y (BB/4' RN) and

t"

IDu- (DU)pjlldx
(19)

Note that w is a weak solution of the following elliptio system

J

Aft.Ds~Da,¢'dili=JB [A~-Aff] [Ds1i -

BB"

Il"

+Jo

BBl,

.

(DsUi)R/4]Da,¢'dili

[Af(ilio, UR, DU(ili»)-At(x. U(ili), Du(ili»)]Da,¢'dili

V¢EH~nL";=~ (B R/4,

+ tSI' B,(x, U, DU)¢'dili,

RN).

Putting ep=w into the above equations we deduce

J

BB'

IDwI2dx~OJB ~IA~-AffI2IDu- (DU)R/4f 2dx

+0 +0

J

R •

IAf(xo,

UR, DU(aJ») - Af(x, u(z), Du(x)) 12dx

tSI' jB(ili, U, Du) Ilwldili. RRI'

(20)

Now let's estimate the right hand side of (20). Without loss of generality we assume pE (2, 2,,), then O<~= (2,,-p) /(p-'Y) <2(1'-1)/(2-'Y),we get

tRl< IDu I

'Y

~(J ~ (J

BBl.

Iw Idili<

IDu I' dai)" "1/'[(" J IDu I

fJ

B RI •

(tSI. IDu III da:r II(t.,. Iw /1I1
\J BBl,

B RI•

1'

dx)"I/ [ ( BR1I.

IDw IJ dx)(J

~ORn+(1'-2)/(1'-1)F(R/4) +1. J '±

B ll t

BRII.

Iw I"PIll dili )2/"J1-"I/1'

I win ;=~ ax)

8(2-'Y) n()'-l)

R

2-8 •

!=.!. '}'-1 ] 1-')",

l-Vw1 2 ax.

Similar to (8) and (9) we have

o

t.,.

(!u! ... + I b(ili) /) /1LJ!dili

~ORn(f

JBBI.

«ou-» (

Iwlll·dili)1/2* [(( 1

BRII.

JBBII.

lulll.... dili)1/2*

+(f

IDWI2dx)2' [F(RI4)+R~a,'+R-a,.]

+1.J

~ORn+2aF2(R/4)

4

BRII.

IDwI 2 da1.

fb.J2*d.~)1/2*]

BBl.'

WEAK SOLUTIONS FOR NONLIN. ELLIPT. SYS.

No.2

Thus for some 8>0 we have

o

· JB./t

.

IB(x,

U,

Du) 1·lwlttxoe;;;;O.R"+8pll(R/4) +

1 · "2J

B."

183

/DwI 2da:.

By Propositton 1, Proposition 2 and the standard Giaquinta technique (see [2], Chap. VI) we have the following estimates for the first and second terms on the right hand side of (20):

fB ~ IA~(xo,

'UB,

Du(x)) - Af(x, u, Du(x))

R"

~H1(luBI"+'1 (DU)BI

f

B R"

1

+ep(R)I)R-+.,

~IAft: -A~tl~·IDu- (DU)altI 2dai .

oe;;;;o {X(R) where H1(t),

t.,.

I~ dz

H~(t)

t.

IDu- (Du)B/2dx+.R'*eHJl(luBI

+ I (Du)BI +ep(R)i)},

are increasing functions. So from (20) we have

/DwI 2dxoe;;;;Ox(R)

t.

IDu- (Du)B/Jldai+H(luB/

Hence the results follows from

Definition 2. Let p~1,

linear space of functions

(1~).

A~O, Q(zo,

uE Y(D) s.t,

sup

X.ED

p-Af

O
+ I (Duhl +ep(R)J).R"+e.

p) =zQn Bp(xo), by y,A(Q) we denote the

IU(ai)-Ueo,pl'aai<+oo.

D(".,p)

Lemma 3. Let D be of type (A), Le, for any aioEQ, p
where

inff

2 1=={QioED: lim R-+O+ 22

={iVo E .o:

S

B.

IDu(x) -

(Du)BI 2dw> 0} ,

~ p( /uBI + I (DU)BI) =+oo}.

Proof. By Lemma 2 and the standard Giaquinta technique we have: suppose,

Qio

Et 2 1 U2 2 ,

"then

eav,

(21) Since ep(R) and IUBI + I (DU)BI are continuous functions of xo, if (21) holds for a point xoE D, then (21) holds for every x on a ball B-y(xo), I.e, DuE L2,n+8(B,,!(xo) , BrlN). Thus from Lemma 3, DuE OO,a(B"!(xo) , Rn.), so there exists an open Qos.t. Du EOO,a(Qo, RrtM) with.Q\Doc21U2~, and meas(D\Qo) =0 follows from the Lebesgue ep(p)~O(p/Ro)·,

'I'heorem,

Vp
184

ACTA MATHEMATICA SCIENTIA

Vol. 10

§ 4. Arguments on the Exponent n-')'-1

-')' 2--

Let us oonsider the fol low.lng Dirichlet problem

Llu=O!Du r { u=O

where

O=(~=~

t

1+ :

in B.R(O)

<,,<2,

n~3

on oBR(O)

(~=I +2-n), then both U= lail ~=r

-R

~-i

and U=O are

the weak solutions of the above Diriohlet problem whioh Violates the local unique-

ness prinoiple for the Diriohlet problem. Note that uE R 1 nY, 'v'p
H1

nirr», 1'-1

2-')'

so under natural growth condition 1+-<,,<2, the admissible class

n

n

')'-1

rb

for the weak solutions must at least be H1 L 2-')' to except the case of the above example. In this sense, the exponent can not be improve anymore.

Note that 1(,,/) =n~=~ (1+

:

<,,/<2) is a strictly Increasing and continuous

function with inf f (,,) = 2*, which is the oritical exponent in the embedding Jll~ Y, while limi(,,) =00; under controllable growth condition, it is only required ')'~2

that uE H1 while under quadratic growth condition uE H 1 just as a link between Hi and H1 L",

n

nL", bhus L n~ appears ~-1

References [ 1]

Giaquinta, M., G. Modica, Almost everywhere regularity results for solutions of nonlinear elliptic systems, Manu. Math., 28 (1979), 109--158. [ 2 ] Giaquinta,M.,M'llltiple Integrals in the Calculus of Variations and Nonlinea'J· Elliptic Systems,Princeton University Press, .1983. [ 3] Ladyzhenskaya, O. A., N. N. Ural'tseva, Linea'!" and Quarilinear Elliptic Equations, Nauka, Noscow, 196~; English Translation,Academic Press, New York, 1968; Second Russian Edition, Nauka, Moscow, 1973. [ 4 ] Giaquinta, M., G. Modica, Regularity results for some classes of higher order nonlinear elliptic systems, J. fur Reine una, Angew. Math., 311, 312 (1979), 145-109. [5] Adams, R. A., Sobole» Sf!