Hölder continuity for solutions of certain degenerate parabolic systems

Nonlineor Analysis, Theory, Printed in Great Britain.

M&hods

& Applicalrons,

Vol.

18, No. 3, pp. 235-243,

1992. 0

HOLDER CONTINUITY CERTAIN DEGENERATE

0362-546X/92 $5.00+ .W 1992 Pergamon Press plc

FOR SOLUTIONS OF PARABOLIC SYSTEMS

HI JUN CHOE? Institute

for Mathematics

and Its Applications,

University

of Minnesota,

(Received 25 October 1990; received for publication Key words and phrases: Degenerate parabolic Poincart inequality, Campanato space.

system,

Holder

Minneapolis,

MN 55455, U.S.A

19 April 1991)

continuity,

weak

Harnack

inequality,

1. INTRODUCTION WE CONSIDER

the parabolic

system U, - div(lVulP-2

VU) + b(x, U, VU) = 0

(1)

where u = u(x, t); R” x R + RN, V = grad, and x varies in an open set n C RN. Naturally we assume 1 < p < 03. We assume that b(x, t, U, Q) E C’(R” x R X RN X MN" + RN) satisfies the following controllable growth condition: lb&,

1, U, Q)i + I&(x, f, ~9Q)I IQ1+ b,$,

t, ~3Q>Q:i 5 41 + iQiP-l)

for all x E R”, I E R, u E RN and Q E MN*, and for some c. Suppose that u is a solution to (1) which means u E C’[O, T; L2(Q + IR”)] 13 L’[O, T; W1*p(C2 + lRN)] and u satisfies - u$, + Ivz.+-~VU - V4 + b(z, u, Vu)+ dz = 0 .r ar for all 4 E C,“(n, -+ RN) where z = (x, t) and s2r = a x (0, T). The main result in this paper is the following theorem. THEOREM

1. Suppose

that u E L;O,,(Cl,), r,, > 42

- p)/p

then

for some CY> 0. Note that from the definition of the solution u E L&,(Cl,). Thus the requirement that r,, > 42 - p)/p becomes restrictive only when n(2 - p)/p 2 2, i.e. when p I 2n/(n + 2). When p L 2, DiBenedetto [5] proved that solutions for equations (1) with a natural growth condition on b are Holder continuous. Essentially his proof lies on the truncation idea of DeGiorgi and a scaling, and it seems inapplicable to systems. In the case 1 < p < 2 Ya-zhe

t Present Kyungbuk,

address: Department Republic of Korea.

of Mathematics,

Pohang

Institute

235

of Science and Technology,

P.O. Box 125, Pohang,

H. J. CHOE

236

and DiBenedetto

[IO] proved

Holder

continuity of solutions for equations. Also when [ll] proved Holder continuity for solutions of

p > 2n/(n + 2), Ya-zhe and DiBenedetto

parabolic systems up to the boundary. On the other hand DiBenedetto and Friedman [6] proved that Vu E C& for homogeneous systems when p > 2n/(n + 2). Independently Wiegner [9] proved that VU E CP,, when p 2 2. When 1 < p < 2, Choe [l] proved that VU E C$, for homogeneous systems. Here we prove an inequality of Poincare type. Once we have a Poincare inequality, a Campanato type growth estimate for u follows from the L” estimate of VU. Thus theorem 1 follows from the isomorphism theorem of Da Prato [4]. We assume U, u,, VU, V2u belong to a suitable Lq space. Justification for this appears in [6] when p 1 2 and in [l] when 1 < p < 2. 2. L- BOUND FOR Vu

In this section we prove that u is bounded by the following Moser iteration idea and we construct a weak Harnack inequality for 1~1.In this section we define a cylinder QR by QR = ((x, t); (x - x,1 < R, I, - RP < t I to] where (x0, to) E I?” x R is a generic point. Also we define AR = {t; to - Rp < t < to). A similar theorem appears in [l]. THEOREM 2. Suppose

that u E L;O,,(sZ,), then u E L~“,,(sZ,), where r, > n(2 - p)/p 1 < p < 2 and r,, = p when p 2 2. Moreover, we have that for all Q2R, c nr

when

for some constant c depending only on n, N and p, where rl = r,, - (N/p)(2 - p) when 1
we can assume r, 2 2. Let p < R and w be the standard cut-off w=l

I)/=0

in a neighborhood

in

Qp

of parabolic boundary of QR

Let (Ye= r, - 2. We apply u[u[~I+# as a test function to (1) where cy 2 cx,,2 0. So we have s’11p ~uJ~+~I,u~ dx + (a! + 1)2-p s

I c

s

JV(y/JuJ(ff+p)‘p)~p dz

Iu~~+~v/~-~[v/~(dz + c(a + 1)2-p

l@+PIVWIPdz I

s + c

(1 +

.i

IvuIY-lIula+llypdz.

(3)

HGlder continuity

for solutions

of parabolic

231

systems

Using Young’s inequality on (3) we have syp

IUJ~+2l/ dx +

\V(&#a+P)‘P)IP dz i

s

5 (R f

p)”

QRIuIo1+2t,up-1 dz +

’ (R - P)”

dz + clQRI.

Iul”l+p

(4)

QR

We assume 2 5 p. By Holder’s inequality and Sobolev’s inequality on (4) we have Iuln+%

sup

+ .r

5 CR’ P)”

[.i

/~I”+~dz + c QR

1

@-Pvn

I4 (~+P)(n/@-P))

&

dt

IQRI

(5)

(R - P)“’

Hence using (5) and Holder’s inequality we have

5

[

s;p .r,

Iula+2 dx]p’n[

iAP (1,

dz +

juln+P QR

Iu((a+P)(N’(N-P)) dx)@-p”n

1

IQRI

dt]

l+(P/n)

(R -P)”

(6)

*

We define p, = (R,/2)(1 + 2-9, v = 0, 1,2, . . . , and Q, = Qp,. We now define 01,by %+1

=

( > 1 +f

ffo -- 0.

,,+$

Then we see that CL”= 2(0” - 1) where 19= 1 + (p/n). 8” lim V-rrn(Yy

Also note that

1 = 2’

We define 4, by 4, = then we can write (6) as

u~v+~dz, ! Q.

+ “+I I c”4,” + cy

(7)

where c depends only on n, N and p. Iterating (7) we prove theorem 2 when p L 2. Now we assume 1 < p < 2. From (4) we get s;p

B IUla+%l.x + I4 (~+P)w(n-P)) s AP[i B, P

s(R ’ P)P

Iula+2dz + c QR

IQRI

CR- P)"

1

(n-PI/n

&

dt

(8)

H. _I. CHOE

238

and

r

luI (ol+P)(l+(P/n)((U+2)/(n+P)))

&

JQ,

IBp Iu(a+2dxrn[ IA0 (.r,,u,(a+p)(“‘(“-P))dx)@-“” df]

I

[ syp

I

’tR-P)” 1

dz +

i QR

[

Defining ay, by

lula+2

a,+1 +

IQRI (R -P)” 2P

I+ (P/N 1 *

(9)

(Yg = r, - 2,

2 = a,0 + p + -, n

we can have 6 “+I 5 c@ + cy

(10)

for some constant c depending only on n, N and p. We note that lim Y’m

% + 2 B” = r, - i(2

Iterating (10) we prove theorem 2 when 1 < p < 2.

-p). I

We define a new cylinder S, by S, = BR(xo) x (to- R2, to). We denote ;1 = sups,/ul. Now we recall the following theorem due to Choe (see [ 1, theorem 41) which proves that VU E Ly”,, for 1 < p < 2. 3. Suppose S,,, c f& and 1 < p < 2. Then there exists a constant c depending only on 1, p, q,,, N and n such that

THEOREM

(11) where q. > (n/2)(2 - p) and q1 = 2/(2q, - n(2 - p)). We note that theorem 3 implies proposition P>-

3.1’ in [ll]. Suppose that 2n n+2

then we see P>5(2-PI

and we can take q,, = p and q1 = 2/(2p + n(p - 2)) in theorem 3. Consequently we have the following corollary.

239

Hiilder continuity for solutions of parabolic systems

COROLLARY. Suppose SzR, c OZT and (2n/(n + 2)) < p < 2. Then there exists a constant

c

depending only on n, N and p such that sup IVuJ 5 c([isR, S&/2

Ivul’dz~z’i2p+~ip-2”

+ 1).

(12)

Now we consider the case p L 2. Again by following the Moser iteration we prove a weak Harnack inequality for VU. In this case we recall that Ya-zhe and DiBenedetto proved a similar theorem using a more complicated iteration (see [ 11, proposition 3.11). THEOREMS. Suppose SZRoc c&- and p 2 2. Then there exists a constant c depending only on n, N and p such that

(13)

Proof. Differentiating

(1) with respect to x7 we have

(UQ - (a$\vUIp-2U&~),0

+ bf;JX’ t, U, Vu)

+ b’,j(z, U, VU)Z& + b&*(X, t,

2.4, VU)Uh

where

a n f = 0,

(14)

and 6 is the Kronecker delta function. We introduce a new cut-off function I,Usuch that w=l y/=0

Taking 4 = U,yl~~ln~2, s;p

in a neighborhood

01 zz

in S, of parabolic boundary of S,

0 as a test function to (14) we get

I~uI~+~t/~dx+ - Iv(IvU[("+p)'2ty)12dz .i ?

Iv~l”+~t&l

s c

dz + c

s

Ivul”+PIvy/12dz + c

(1 + IVuip+u)v/2 dz.

(15)

1

By Holder’s inequality and Sobolev’s inequality we have

s

Iv4 (a+P)(l+(2/n)((n+2)/(nCp)))

&

5

c

lWv4

%

I&l CR

-

PI2

(a+Pw5#

1

&

1

1+(2/n) (16) *

H. J. CHOE

240

We set py = (1 + 2~‘)(R,/2), p = pV+i and R = p,. We define ,LI= 1 + (2/n). Also we define a, by 4 %+1

(110= 0.

+P=wv+P+;’

If we define CD(v) = is

ET.,

‘vu’oI”+p dzp

then (16) can be written as follows @(v + 1) I c”@(v)‘”+ cy

(17)

for some c depending only on n and p. We note that

So iterating (17), we prove theorem 4.

n

3. HOLDER

CONTINUITY

OF u

In this section we define & = (t,, - 2RP, to - Rp) and Qi = BR x A;. We introduce a cutoff function q E C,“(B,) such that ?I=1 in RR/2

Also we define

and

First we prove a lemma which is essential for a Poincare inequality for solutions of a degenerate parabolic system. LEMMA

1. Suppose Q2R C L&-, then u satisfies the following inequality: SUP teA,

ds I A> I

~2b(*,

t, -

uR,si2b

1 BR

12.4- Qt

(Vujp dz + c

CR” Q2R

I2 dZ + cRs2

I QR

for all R < R, where c depends only on n, N and p, and s1 = p and s2 = n + 2p s1 = p(p - 1) and s2 = n + p + p(p - 1)

when p 1 2 when 1 < p < 2.

(18)

241

Hlilder continuity for solutions of parabolic systems

Proof. Since u E C’[O, T, L2(Q)], there exists f(s) E AR for all s E A; such that

s

rfP(u(x,f) - uR,s(2dx= sup OZrZs s B, BR We take (u - ~~,~)~%t~,~l as a test function to (1) where X,,,f, = IR* F? is the characteristic function such that M,,,r,(s) = 1 for all s E [s, t] and Mrs,il(~) = 0 for all s $ [s, i]. Hence we have u, * (u - ~,,,W’W+~

IVU~~-~ Vu * V((u - u,,,)rlP)X,,,rldz

dz +

+

- uR,J@-‘X~~,~~ dz = 0.

b(x, u, vu)(u

(19)

Now we assume p E [2, 03). Using Young’s inequality and the structure condition on b we have s

Iu - %7,s12@(x, 0 dx +

s

I24- uR,s~2~p(x, s) dx + fj

c

I

B x [5tl IVulp#’ dz R 3

s x

B,

[s,?I

lu -

uR,s lp’lpdz

lVulPdz + cR~+~ (20) J BR x [s,TI for some c independent of R. Integrating (20) with respect to s from 1, - 2RP to to - RP, we have +C

ds s ai?

lu - ~R,s12~p(x, 0 ~ i BR dz + Iu - uR,t12qP

SC .i

Iu - u~,si2rlpdz

$

QR

B,

X

IS,

?I

+ cRp c jVulp dz + cR”+~~. J QZR

(21)

By the choice of i we have that for small E r a,

ds

Iu - ~R,s12~p(x, f) d-x

1BR

Iu -

SC

uR,,12qPdz + cRp

lVulPdz + cR”+~~. .iQZR

i

(22)

So we have lemma 2 when; 1 2. In the case p E (1,2) we estimate the second term of (19) as follows

s

IvuI~-~vu - V((u - u,,,)rp)Xt,,~, dz =

Ivu~~~~X,~,~~ dz + p

~v~~~-~u~~~&i - z~k,,)~~-‘X~,,~~dz

(23)

H. J.

242

and using the fact that p(p - 1) < p and p/(p

CHOE

- 1) > 2, we have

~vu~~-~z&~~,(u~ - u;,,)~~-%~~,~J dz

IvuJp-‘Iv~~qp-llu - u,JX’~~,~~ dz

I

-$

I

.i

(u - uR,s(p’(p-l)rfXls,il

dz + &

s

(vulp’p-l’ dz

c

~R,s12rP~[s,I] c-k+ RP(2-P) ___

lvulp dz + cR~+~(~-~). B, x IS,0

(24)

Applying Young’s inequality and the structure condition of b on (19) and combining (19), (23) and (24) we have

Iu -

12qp(x,i) dz 5

uR,s

G,,I~~?~%FJ~~

B/Z + &n+P(P1)

+

+ &

B,xIs,il IVulpdz .i

(25)

c

BR

Integrating l
(25) with respect to s and using the choice of 7 we prove lemma 2 when n

Now we prove a Poincare inequality. THEOREM4. Suppose Q2R c C&, we then have

lu - u~,~)’dz

I cR2

QR/Z

(VU(2dz + CR”’

.iQ2R

/VU/~dz + cRsz,

.iQZR

(26)

where c is independent of R. Proof. Using lemma 2 we have Iu QR/~

- #R/212

dz I c

ju - uR,$ 12rlpdz 3 QR

C I-

I c

UR,si2~pdz

s QR

sup ds Iu - ~R,sI~~~(X,f, b tE& 1A’R 1BR

IC i Qi 5

Iu -

ds i A,

RP

cR2

dz + Iu - uR,f12qp (v~(~dz + CR”’

s QZR

CR”’

i QZR

Ivu(pdz + cRs2

jvulp& s QZR

+ Rs2

(27)

Holder

continuity

for solutions

of parabolic

243

systems

where we used a PoincarC type inequality for x variables only as follows

r BR

Iu - u~,~ 12qpdx I CR*

Jvu1*dx.

n

i BR

Proof of theorem 1. Since Vu is bounded, we have from theorem 4 IU - z+\*dz 5 cRflfp+*

when

p E

[2, a)

when

p E

(1,2)

.i QR < -

where c is independent theorem 1. n

&“+P+P(P-

1)

of R. Hence by the isomorphism theorem of Da Prato [4] we prove

Acknowledgement-I wish to thank the University of Minnesota.

the Institute

for Mathematics

and its Applications

for hospitality

and support

at

REFERENCES Communs 1. CHOE H. J., Holder regularity for the gradient of solutions of certain singular parabolic equations, partial difJ Eqns (to appear). 2. CHOE H. J., On the regularity of parabolic equations and obstacle problems with quadratic growth nonlinearities, J. d$x Eqns (submitted). elliptic partial differential equations and 3. CHOE H. J., A regularity theory for a more general class of quasilinear obstacle problems, Archs ration. Mech. 114, 383-394 (1991). Anna/i Mat. pura appl. 69, 383-392 (1965). 4. DA PRATO G., Spazi L@*“(a, s) e lora proprieta, 5. DIBENEDETTO E., On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients, Annali Scu. norm sup. Pisa 13, 487-535 (1986). 6. DIBENEDETTO E. & FRIEDMAN A., Holder estimates for nonlinear degenerate parabolic systems, J. reine angew. Math. 351, l-22 (1985). equations of parabolic 1. LADYZHENSKAYA0. A., SOLONNIKOVV. A. & URAL’TZEVA N. N., Linear and quasilinear type. Providence, RI (1968). regularity for solutions of degenerate parabolic equations, Nonlinear Analysis (to 8. LIEBERMAN G., Boundary appear). of the gradient of solutions of degenerate parabolic systems, Anna/i Mat. pura 9. WIEGNER M., On C*-regularity appl. 145, 385-405 (1986). 10. YA-ZHE C. & DIBENEDETTO E., On the local behaviour of solutions of singular parabolic equations, Archs ration. Mech. 103, 319-345 (1988). estimates for solutions of nonlinear degenerate parabolic systems, 11. YA-ZHE C. & DIBENEDETTO E., Boundary J. reine angew. Math. 395, 102-131 (1989).