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Boundary regularity for solutions of degenerate parabolic equations
Nonlinear Andyzir. Theory. Printed in Great Britain.
Merhods
& Appircmons.
Vol.
14.
No.
6. pp.
501-524.
1’9%.
0362-546X 90 53.00+ .CKl E 19w Pergamon Press plc
BOUNDARY REGULARITY FOR SOLUTIONS DEGENERATE PARABOLIC EQUATIONS GARY
OF
M. LIEBERMAN
Department of Mathematics, Iowa State University, Ames, IA 50011, U.S.A. (Received Key words and phrases:
1987; received for publication
9 September
21 April
1989)
Parabolic equations, degenerate equations, boundary value problems, ~1priori
estimates.
INTRODUCTION IN THIS paper, we consider the conormal derivative problem for equations with degeneracies of the same type as
m>
- u, + div(lDu I”Du) = 0,
-1.
(0.1)
By modifying the method of DiBenedetto and Friedman [3-51 as in [IS], we shall show that the spatial gradient of a solution of (0.1) with conormal boundary conditions is Hijlder continuous up to the lateral boundary of the domain. To be more specific, let Sz C IT?”be a domain with C’*O boundary for some /3 E (0, l] and inner unit normal y, fix T > 0, and set QT = SXIx (0, T) and SQ = 6’sZx (0, T). Then for constants II. > 0, A 1 2, Y, AZ, A, 2 0, m > - 1, and K L 0, we consider: a vector-valued function A defined on QT x R x R” satisfying (0.2a) (0.2b)
I&, ~,z,P) - ACV,f, w,P)I 5 forxandyinR,O
(1 +
bi)“+‘[Alx - yl” + AlIz - ~1’1
(0.2c)
A(1 +
(0.2d)
T,zandwinR,andpER”, IA(x, t, z,p) - A(x,s,
for x E X2, 0 < s < t < T, z E R, and
z,p)I
p E IR”;
I&K z, P)I for some function b E L4(QT) with
I
5
Ipl)m+‘lt
a scalar-valued
A,IPI~+~
q > n + m +
-
sI~‘(“‘+~)
function B with (0.2e)
+ b(X)
2 and
I/q bq
and a scalar-valued function I,Uwith
I
A2;
(0.20
(! QT >
Iw(X,z) - w(Y, w)l I ‘-I’(lx - yIB + It - sI”(~+‘)) 501
+ AlIz, - WI’
(0.2g)
502
G. M. LIEBERMAN
for X and Y = b,s) in SQ, and z and w in R. We are concerned boundary value problem in QT
u, = div .4(X, U, Du) + B(X, U, Du) A(X, U, DU) * y(x) + ly(X, u) = 0
on SQ
with solutions u of the
(0.3a) (0.3b)
with u E Lm(Qd+ Du E Lw(Q x (E, T))
(0.4a)
for any a E (0, T).
(0.4b)
1. Let u be a weak solution of (0.3) satisfying (0.4) and suppose conditions (0.2) are satisfied. If m 1 0 or if A, is sufficiently small, then there is a constant 0 E (0, p] determined only by p, A/A, m, q and n such that for any E E (0, T) and X and Y in !J x (E, T),
THEOREM
IDu(X) - Du(Y)l 5 C@,e,A,A,,&,&
m, ‘I’, )u],,n,q, T, 52)()x- ~1” + 11- ~j~‘(“‘+~)). (0. 5)
We leave the study of regularity near the initial surface to a sequel [17] because of the difference in method of proof and the weaker results in that case. The differing exponents with respect to x and t are based on the natural scaling properties of (0. l), which are more fully exploited here than in 141. The approach in [21] for degenerate systems also uses differing exponents, but not the ones here which seem to be more natural. The other new element here is the boundary estimation. In [4] DiBenedetto and Friedman (see also (211) prove interior gradient Holder estimates for systems under stronger structure conditions than ours; their method will give the interior estimates for single equations under our conditions. Also, they assume that m > max( - I, -4/(n + 2)) but not that u is bounded, and they prove their continuity estimate assuming a gradient bound; we prove the gradient bound from the continuity estimate. Our assumption that u have bounded gradient can be relaxed, but we shall not go into detail on this matter. The proof of theorem 1 proceeds in several steps. First, in Section 1, we prove some preliminary results for solutions of (0.3) under simpler, slightly different hypotheses. The Holder continuity of the gradient under these hypotheses is proved in Section 2, and the full strength of theorem 1 is demonstrated in Section 3 for nonnegative m. The few differences in proof needed for negative m are given in Section 4. Finally a crucial existence theorem is proved in Section 5. I. PRELIMINARIES
FOR
THE
SPECIAL
CASE
We begin by studying (0.3) when A depends only on p and some other dependencies are simplified. Suppose that F is a positive, continuous function on (0, co) and that there is a positive constant ci such that F(s) r ciF(4s)
SF(S) _( aF(o)
for 0 < s,
(l.la)
for 0 < s < ~7.
(l.lb)
(There is no loss of generality in using (1 .Ib) rather than the inequality SF(S) 4 c2cC(a), as in
Degenerate parabolic equations [ 151,
503
because the function f given by 1 f(s) = - sup &(a) s (IBS
satisfies (1.1) and F I
f I
c2 F.)
Now for x0 E IR”and positive constants R and M, define
B(x,,, R) = (x E II?“:Ix - x0( < Rj, B+(x,,R)
I(M,R;b)=
= (x~B(x~,R):x”>
(
R2 10-m,ro
9 >
0),
B”(xo, R) = {x E B(x,, R): x” = 01, Q+W,R;xo,4,) Q’W,
= B+(xo,N
R;xo,
to) = B’(xo,
CPQ+(M,R,x,,t,) Q*W,R;xo,to)
x I(M,R;fo), R)
x IW,
R; to),
= (X~i3Q+(M,R;x~,t,):t
< to),
= sQ+(M,R;xo,to)\Q’(M,R;xo,to).
is called the parabolic boundary of Q’ . In what follows, we generally suppress the arguments of the Qs and Bs when they are clear from the context; for technical reasons, we shall suppress the second argument of Q’ or B+ when it is 2R rather than R. We assume that @ is a constant. that A is a vector-valued C’(P) function with derivatives aAi/apj = au satisfying SQ’
a”CP)t*Tj
for all r E IR”
2 F(ipt)ki2
la%-45 ~(IPI),
(1.2a) (1.2b) (1.2c)
IA@)] 5 Al~lF(l~l)
for all p E IR”. (In many applications, condition (1.2~) is a consequence of (1.2b).) For to = 0 and fixed M, R, and x0 with x0”= 0, we now consider the boundary value problem V, = div A(Dv)
in Q’ ,
A”(Du)
+ Q, = 0 on Q”,
and we suppose that the solution u of (1.3) has weak second spatial derivatives in L’(Q’) satisfies the additional inequality max ]DiV] I M in Q’ ; isn
as in [4], this assumption and the use of the half-cylinder the keys to our arguments. As a final piece of notation, we set
(1.3) and (1.4)
Q’ rather than B+ x (- R2, 0) are
M’ = max sup I&V]. ken
LEMMA
Q+
1.1. For any h E (0, I), there is a positive constant fi = p(h) such that if M’ L hh4 and (1.5)
504
G. M. LIEBERMAN
for some k < n, then D,v
2 =
in Q’ (M, R).
8
(1.6)
Proof. Let q be a C’*‘(Q+) function vanishing on Q* and let g be a Lipschitz function on R. Define G by
G(a) =
O&s)ds, s0
set t, = - (2R)‘/F(M), and obey the summation convention only for the indices i and j. Now use D&g(Dkv)) as test function in the weak form of the equation for v and integrate by parts. For each r E (tt, 0), we thus infer
rlGG%v) + B+xb+
t,@DjkvDikvg’(Dk~) B+X(rl,T)
=
A’D,,Vg’Dit/ B+xul, 7)
+ A’gDiktl - rl,G. (1.7)
Now we choose g and q in this equation by setting, for q > n + 1, w=min[$,(F-Dkv)j,
g(Dbv)=@-“-’
+-(#[;-I],
V=p-n.
From (1.4), it follows that
C(h)
IAiDkkvl I &DjkvDi,v
in Q+,
+ -M2F(M)
and, because M’ L hM, we conclude that SUP t,
+ s sx(+(Wo2q-n s
F(M)Q+ ~D(wC;)~-“‘~~~I
Bf
C(h)q4M2F(M) s Q+ (W02q-n-2*
For K = (n + 2)/n, we infer from the Sobolev inequality in the form [ll, (1.6)] (see also [16, lemma 1.21) that x(2’?-n)
Q+(wo
5
C(h)
If we now set dx = (~/IQ’I)(M/w~)“‘~ rewritten as
44x
r$rM2
dx
(I,+
(~a~~-~-~)~.
dt on C = (wl > 01, then this inequality can be
( wo2qx cLZI C(h)q4” (~,(wc)%x)x IE and then an easy interation scheme (along with Holder’s inequality) gives sup (Wan+4 E
Degenerate
parabolic
505
equations
whence
Because w = 0 if D,v > M’/2,
it follows from (1 S) that 1C 1 I p/Q’ 1 and hence S.?
C(h)M’p”‘“+J’.
WCI
Therefore, if ZAis small enough, we have WCI M/8 in Q' ; (1.6) follows from this inequality because M’ L hM. N For our next lemma, we introduce some more notation. s(0) = (X E Q’ : Dkv(X) > (1 - @M’ ),
For 0 E (0, ‘/2), we write
S(8, 2) = (x E IR”:(x, t) E s(e)).
LEMMA1.2. For any h E (0, l), ,u E (0, l), 0 E (0, %I, if M’ 1 hkf, and
nQ’1
IW$j
>dQ+I,
(1.f>
then inf
Is(0, t,r)l 5
(1.9)
+$++I.
@/2<7<1
Proof.
Set t2 = ft,,
z=
(*/s(e,t)ldt= Is(e)n[B+ x(t,,t2)]l, I 11 J = inf IS@, t,r)l. p/Z
Then (1.8) implies that Z I IS@)/ I (1 - p)lQ’I. ZrJ[t,-
tl] =
On the other hand
( > 1
IQ'1 -$ .zm.
The desired result follows by combining these two inequalities. Lemma 1.2 is, except for a slight variation in notation, For our next lemma, we define for fi E (0, l),
4
[4, lemma 3.6).
l’(n+Z) ’
r(e)
= (XE s(e): 1x1 c VRJ.
Note that 0 < v c 2 and there is a constant p,,(n) such that P c p, implies v > 1.
(l.lOa) (l.lOb)
G. hl. LIEBERMAN
506
LEMMA 1.3. For any h E (0, l), p E (0, pug), if M’ 1 hMand (1.8) holds, then there is a positive integer r such that
sup IS’(2_‘/f, f)] 5 f ]B+(VR)j. &VI Proof.
(1.11)
Let E > 0, t’ E (@/2)t,, 0), and choose t” E (tl, (~/2)t,)
from lemma 1.2 so that
For r a positive integer at our disposal, we define the function Y by Y(aM”)
= log+
P ,U (a 1 + p)+ + 21-‘/f [
1’
Now, for c E Co’ ‘(B(2R)) with C;= 0 on aB(2R) to be further specified, we set
&Y(o)= 2WW’(d,
v(x, t> = [(x)2.
An argument similar to the one leading to (1.7) implies that r2(1 + Y)(Y
&J2 + B+
X(f’]
=
s
s
E+
!
B +
V’I’ ‘a”ojkvDJ
l2Y2 - 4 B+ (f”. rr)
B+ X(P)
I
‘)‘a’iDjkvDikU
X(P,P)
xp~C2Y2 +
Bf x(t”,t’)
c2(1 + Y)(Y’)‘a’jDjkVDikU
+ CF(M)
Y’IDd2. B+x(r”,r,)
If c is chosen so that [ = 1 in B(vR) and JD[l s C(v)/R in B(2R), it follows that Y2 5 IS@, t”)l[r - 1) log 212 + C(v)lB+ I(r - 1)
(1.12)
B(vR)x(f’)
)
+ E [(r - 1) log 212+ C(v)(r
q*
- 1) ]B+ I. I
We now send E to zero and conclude that
IO
i * [r - 1) log 21’ + C(v)@ - 1) IB+(vR)I 1 1 s B(vR)x(r)
Y2 1 [(r - 2)log212 [S/(2-‘p, t”)l.
Thus ]S’(2_‘/& t’)( I
“I:‘!
i21) + (%)‘(;)‘3
]B+(vR)I.
The proof is completed by choosing r so large that
C(W - 1) $(l (r_2)2
- (;)“2),
(=Yst.
n
Degenerate
parabolic
507
equations
Lemma 1.3 is essentially the same as [4, lemma 3.71 and [3, lemma 6.21. LEMMA1.4. For any h E (0, l), ,U E (0, p,,), if M’ 2 hM and (1.8) holds, then there is a positive constant q < 1 such that
in Q’
I&VI 5 VM’
M,$R (
Proof.
(1.13)
. >
Set w = (D,v - (1 - B)M’)+,t”
=ifl,
and let L:be a Czg ‘(Q+) function with &Y) = 0 if 1x1 > vR or t c t”. For q a positive constant at our disposal, we then use Dk(c$“+2)q-n w4-‘) as test function to see that, for any I’ E (t”, 0), 1
c 4 if+ X(I’} -s
(n+2)4-ng + (q - 1)
= ((n + 2)q - n)
4
~~p+m-ns B+X(f”,f’)
If we now take c so that 1~1 I 8F(M)/pR2,
s B+
ld.
(1.14)
and 0 5 [ 5 0, then
IDC) I 8/vR,
F(M)
([“+‘w)~ + F(M)
~D((~n+2w)q’2~-“‘2)~2 I Cq2T B+
X(f’}
(n+2)q-nWq-ZaijDjkUDikV
,$n+Z)q-n-l w4-‘a’jDjkvD;[ B+X(l”.f’)
(n + 2)q - n
+
c B+ X(f”.f’)
LJ+ X(f”.f’)
X(P,I’)
@r(
n+2)q-n-2
*
Setting Q = B+ x (t”, 0), C = (cw > 0), and S”(e) = S’(0) fl (t > t”], we see from the Sobolev inequality and iteration scheme that sup r”“W I CL
IQIs E
If, finally, c = 1 in B+(R)
x (Y/2,0),
wI
ce.nF.
it follows that in B+(R)
x
for Is”(@~/~QI small enough. This ratio is controlled as in [3, lemma 6.41. With C ’ = [w > 0, t > P’) and c2& replacing p+2)q-nWq-1, the proof of (1.14) yields ‘h
s s pw” +
B+ x(0)
and therefore
c2ai’DjkUDikv = 2
X’
c2ti + F(M) Bf x(0)
w[a”ojkVDJ
+
s I’
jDDku12c2 5 s E’
x$ s 1’
CF
n+d. s
G.
508
M. LIEBERMAN
Hence, for C ’ = 1X E C ‘: 1x1 < vR] and appropriate
1
lDD/plZ5 c y [
L”
[,
I2/Q+(M, vR)I.
We now choose 13= 2-jpM for various positive integers j and set 0 ’ c (j] = S”(2 -‘Z&
co
= co\co’-
l),
Z=
IDD,c/.
SIf’
I:(i. 0
To proceed, we use (3.5) of Chapter I1 in [IO] to infer that, for any r, C(n, v)R=+’
2 -j-‘/.lM’ I c (j, t)l I
JB+(VR)J - ICo’
- 1, t)l s &DDkUI.
If j 1 r@), the constant from lemma 1.3, then Z > C(U) and therefore,
because
’ 1C (j, t)/2 -j-‘PMR
dr = C(D) 2 -‘-‘PM
1C (J]IR
s f*
Z I C2-‘PM
[I C(j)IR-‘IQ+(M,
vR]]“~,
we see that
I C(j)15 WNIQ’W,
v&l I C ‘QIY~.
It
follows from this inequality that Is”
small by choosing 8
Lemmata 1.1 and 1.4 are easily combined to get our basic estimate for the tangential derivatives. To state this result more easily, we write Q(r) for Q+(M, r). THEOREM1.5. For any h E (0, I), if M’ r hM, then there are positive constants q(h) and 6,(h) such that either max osc Dkv I q max osc Dku ken
k
Q&r)
(1.15a)
Q(2r)
for all r E (0, R) or max sup l&U1 I q max sup ID,vl. k
(1.15b)
Note that we have used a simple variant of the usual Holder estimate [9, theorem V.1.11 or [19, theorem 4.21 in deriving (1.15a). From lemma 1.5, we see that for any h E (0, 1) there is a constant 6,(h) E (0, 1) such that max osc Dkv I hM. k
Q&R)
We now restrict our attention to Q&Z?). If ID,ul < M/4 in Q(S2R) and h < ?4, then max SUP IDiUl I SM. ian
(1.16)
Q(6zR)
Because we wish to prove a version of (1.15) with all the derivatives of v instead of just its tangential derivatives, we asume that (1.16) fails and look in Q((6,/2)R). Thus we may
Degenerate
parabolic
509
equations
assume that max osc Dkv I hM, k
sup ID,+ L
MI
(1.17a)
Q(ZR)
(1.17b)
‘/zM.
QGN
Our goal now is to prove analogs of the preceding lemmata under these assumptions with k replaced by n. LEMMA1.6. There are constants h, and p’ such that if h E (0, h,), and (1.17) and (1.18) hold, then (1.19) Proof. Let [ and g be as in lemma 1.1. If g’ (D,v) vanishes on Q”, the proof of (1.7) is easily modified to yield
s=?
CG(D,v) +
B+ x(1=?+
!
CdjDj,, VDi, vg ’ (D,v) ia+ x(r1.T)
A’D,,Vg’D,r
+ A’gDi,r
- rrG -
B+x(?I.r)
B+X(fl,r)
~D,,vD,k’
- @D,&.
(1.20)
Moreover if g = 0 on Q”, then (1.7) holds. In addition, for K = Ad%, ID,v(X)
- D,v(Y)I
I K Ill:; [IDkv(X)
- Dk(Y)j)
for x, Y in Q”.
(1.21)
Now we consider the two possibilities: D,v
2 Feverywhere
on Q”,
(1.22a)
D,v
< Fsomewhere
on Q”.
(1.22b)
If (1.22a) holds, then w from lemma 1.1 vanishes on Q”. The proof of lemma 1.1 applies without any restriction on h to show that ID,vl 2 3M/8 in Q’(R). If (1.22b) holds, say D,v(x) < M/2, then for any Yin Q”, D,,v( Y) I D&X)
+ K max osc D,v k
I r
+ KhM.
Q’
Hence if ho = 1/4K, D,v I 3M/4 on Q”. With this ho, we set w = min(M/8, ((7M/8) - D,v) + ) and note that 1~1 I C’F(M)M. The remainder of the proof follows lemma 1.1. n Because lemma 1.2 did not use any properties of Dkv other than its measurability, need to reprove it for D,v.
we do not
G. M. LIEBERMAN
510
For our next lemma, we recall the definition replaced by n.
of S(0), etc., from lemma 1.3 with k now
LEMMA1.7. For any P’ E (0, pO), there are positive constants hi, v1 < 1 and r such that if (1.17) holds and if (1.23) then sup IS(2--‘P’, f) fU?(v,R)I &‘/Z)rl 0, t’ E (0(‘/2)t,,
5 vJB+(V,R)I.
0), and choose t” E (t,, @‘/2)t,)
IS(j.i’, t”)I
I
so that
(l’_-p:;2+c)
p3+1.
For p, a positive integer, to be chosen, suppose that D,v > (1 - (~‘12~))Msomewhere If h _( (1 - @‘/2”))M, define Y as in lemma 1.3 with r replaced by p, and set kf*=
(1-$M,>
(1.24)
Y’,(a) = Y(min(a, M*j),
on Q”.
g(o) = 2Y,WY’;(o).
Because g(D,v) = 0 on Q”, the proof is completed in this case as in lemma 1.3 by taking p large enough and vi L v(,u’). On the other hand if D,u I (1 - (~‘/29)M everywhere on Q” for this p, then
Now the proof of lemma 1.3 gives v2 c 1 and r determined by p and ,u’ such that IS(2 -ICI’, t)l I (1 - v2)(BC(v2R)I. Clearly r 2 p, and hence (1.24) holds with v, = maxlv, v,).
n
LEMMA1.8. For any P’ E (0, l), there are constants h2, q, 6, in (0, 1) such that if 0 < h I h2 and (1.17) and (1.23) hold, then ID,,vl I VA4 in Q’(M,
6,R).
(1.25)
Proof. Suppose first that D,u I (1 - 16Kh)M everywhere on Q” and that h, 5 p”’/32K. By proceeding as in lemma 1.4 with ,uu/2replaced by 16Kh, we obtain (1.25). If DJJ > (1 - 16Kh)M somewhere on Q”, then the argument in lemma 1.7 shows that D,u 1 (1 - 18Kh)M everywhere on Q”. Now for B > 18Kh, we set
A(P) = A(P) - A(09 ***,O,M),
G = @ + A(0, . . . , 0, M)
and note that u, = div ii
in KJ,
A(Du) + 6 = 0 on &D.
Now set w = min(M4, (D,p - (1 - 2B)M)+ J,
511
Degenerate parabolic equations
and write W for the support of w. On W, it is easy to check that ]u’~]d CF(A4) and hence that ]A] 5 C&VU’(M); also the boundary condition implies that ]ri] I CBMF(M). Therefore we can repeat the proof of lemma 1.4 with all functions replaced by their barred counterparts and thus conclude (1.25) in this case because the constants will be independent of h. H We now choose first ,D’ from lemma 1.6, then h = min[h,, h, , h, ), and finally p from lemma 1.1; combining all the possibilities gives our basic result. THEOREM
1.9. There are constants rl and 6, in (0, 1) such that either max osc D;v 5 q max osc Diu isn Q(81r) ian Q(2r)
(1.26a)
for all r E (0, R) or max sup il;n
2. HOLDER
]D,ul 5 qA4.
(1.26b)
Q(61R)
CONTINUITY
IN THE
SPECIAL
CASE
We now show how to infer both a bound and a modulus of continuity estimate for the gradient from theorem 1.9 by using some of the ideas from [4, Section 21; however, there are some important differences to be noted here. First, our proof works directly in cylinders with the appropriate scaling for the equation and, second, we do not make the complicated adjustments in [4], so our proof applies regardless of the sign of the constant m. (We also point out that the proof of Holder continuity of the gradient given in [4] needs a minor adjustment in the time scaling of the cylinders Q. and Qr on page 7 of that work.) Our arguments simplify even further when F(r) = (K + T)~ for some K I 0 and m > -1. We also point out that some of the new elements here are also present in [21], but the author was unaware of the contents of that work while writing this one. To simplify notation, we write F* for the inverse to the function g given by g(s) = ?/F(l/s), we introduce the norm and metric 1x1 = max]]x], F*(t)],
P(X, Y) =
Ix - YI,
and, for X = (x,,, t,,) with 4 = 0, we introduce the scaled cylinders Q+(R) = (Y = 01,s): ]A’Q’(R)
= (y:
IX -
Y] < R,s < t,,y”
YI < R,s
< t,,y”
> 01,
= 01.
We suppose that F and A are as in the previous section and that v is a weak solution of the problem u, = div A@) in Q’ (R), A”(Du) + @ = 0 on Q’(R) (2.1) with bovnded gradient for some R > 0. Set MO = max sup ]Div] + f , i
K=
R R, = K’/2 -
R&f,,
Q+(R)
and note that K > 1. With 6, and q as in theorem 1.9 and q
2
‘A, set Rj = 6’Ro.
’
512
G. M. LIEBERMAN
Now observe that R2 R2 KF(M,,) - F( l/R)
R; PC_<_ F(M,) SO
that Q’(M,,
R,) C Q’(R),
and therefore MO b max
SUP
i
Furthermore,
JDiVl.
Q+Wo.Ro,
according to (1.1) and our choice of 6, we have
Ri
R$P
CIR: R: 1 4a2F(M,) L F(M,
F(M,) = F(M,/q) so that Q’(M,,
R,) C Q+(Mo, R,). Hence if max i
SUP
lDiul 5 rl”09
Q+Wo,Rd
then max
SUP
Q+WI
i
lDiU/
I: Ml*
.Rd
Continuing in this way, we see that as long as max i
IDi 5 ef-j-19
sup
Q’(Mj-t.Rj)
(2.2)
we have max i
IDiUl I Mj I M,
SUP
(2.3a)
Q’(Mj.Rj)
for any o I log,,, (l/q). We now denote by J the first integer j for which (2.2) fails; in the terminology of [4], Rj is the switching radius. It then follows from theorem 1.9 that for any j > J, max i
osc Q + WJ. Rj)
DiU I qmax i
osc
DiV
I
2qjeJrnax
Q + (Mr. Rj/o)
i
SUP
[DiVl I 2q’Mo 5 C $
Q + NJ, Rj)
OMo.
0 ’
(2.3b)
Our next step is to use these inequalities to estimate the modulus of continuity of Du with respect to the metric p. To this end we consider separately continuity with respect to space and time. Because the spatial estimates are so simple, we only look at continuity with respect to time. To this end, we begin by proving certain estimates related to the function F. We set p = ‘/2 log,,(l/c,) and assume without loss of generality that 0 < p c ‘/o. By choosing the integer k so that 4’-’ I P < 4’ and then applying (1. la) k times and (1.1 b) once, we find that
Also it follows from (1 .l) that, for any w E (0, 1) and any s > 0,
(M2 -“F(l/s) F( 1/OS)
OS2
Degenerate
parabolic
513
equations
and hence, because KZa+” 5 K, (wK“/M,)~
< 16c~~R;
F(M,/oP)
- F(M,).
We now set X = (x0, to), Y = (x,,, t), and suppose that R;
7 = F*(lt
-
(2.4)
toI)
and consider three cases. First
R2
(2Sa)
F(Mo )
From (2.4) with w = l/16, we infer that
7 L
K“-‘R/16
and hence
]Du(X) - Du(Y)] I 2M, I CM,
;
OK(l-“)O
0 for any cr > 0. Next we suppose that R; F(Mo)
R:
Riz F(Mj)
(2Sb)
,
RJf< to - t s F(Mj_,)’
Then, by an argument similar to the one leading to (2.4), we have
By choosing o appropriately,
160
(WP/Mj)2
<
F(Mj/OP)
- (SV)~‘F(M.).J
we now infer that
R;
[(6rl)“/16]~-“‘Rj
7 2
and hence
Finally we suppose that R:
o
(2.5c)
FW,)
and choosej so that Ri’ F(M,)
Rj- , < to - t 5 F(MJ).
Similarly to our previous cases, we conclude that (Du(X)
- Dv(Y)I
I CM,
2 0
7 1
(63’~-1’2/16)Rj,
and therefore
“4cr3ti’4 I CMo(;)u’4k+4. 0
It follows that there are positive constants o and E with E < o such that, for X = (x0, to) and Yin Q’(R), IDu(X)
- Du( Y) I CM&-’
O.
(2.6)
513
G.
M.
LIEBERMAN
We are now ready to prove our estimates using an interpolation argument in the same spirit as [20]. Let r > 0 and t, = 0, set d(X) = y irrro p(X, Y) and d(X, Y) = min(d(X), d(Y)), and E introduce the weighted seminorms
14; = ;yp, dP4, I
I40 = Q”u,“,14,
r
[4r+o =
d(X,
sup
Y)‘+“~Du(X)
- Du(Y)I/fX
-
YI”.
X. YinQ+(r)
By an easy variant of [7, (6.86)], we therefore have [u]? 5 C(m, n, a)(lul,)“‘(l+u’([u]:+,
+ ]f.&)“(‘+?
(2.7)
Moreover, a simple modification of the proof of (2.6) shows that it remains valid for any X = (x, t) and Y = (JJ, s) in Q+(r) such that t 2 sand IX - Y] d !hd(X, Y) I R. By choosing X and Y so that [UK+, 5 2d(X, and R = %d(X,
Y)‘+“jDu(X)
Y), we see from (2.6) that [u]:,,
- Du(Y)I/jX
-
YI”,
I C([u]: + l)r+“-‘; then (2.7) gives
I@+, 5 c’h4cl+ 1)
(2.8)
for some constant C’ which depends also on ]ulO. Clearly the proof of (2.8) carries over to the case that F has the form F(r) = f(~ + r) for some function f also satisfying (1.1) and some K E (0, 1) if we define Q’, Q”, and F* with f replacing F. From (2.8) we now derive an estimate in terms of an integral norm of Du in the special case that F(r) = (K + T)~ for some constant m > - 1. For this purpose, another set of weighted norms is required. With IX - Y] = max(Ix - y I, It - sI”(~+~)) as per our remarks, we define q(X,P)=(Y=(y,s)EQ+(r):s
YI
for w E L’(q), and we introduce the Campanato
type seminorm l/(m+Z)
(WI X;R
sup
=
TX
O
s 4(X,
(
XeQ+(R)
Iw
-
iqx,
flp+2
>
v
for A > 0 and recall from [l] that there are positive constants C, and C2 for which tWh;R
1
cl X*
lWh;R
5
sup
YinQ+(R/Z)
c2 X*
sup YinQ+(R)
Iw(m- w(Y)I Ix-
Y]”
’
Iww> - w(Y)I Ix-
Y]”
for 1 E (n + m + 2, n + 2(m + 2)) and 0 = (A - (n + m + 2))/(m We now define ( w);Z = o~;lr (r - R)X’(mc2)(~)~;Rr
Ilwll, = (1
+ 2).
Iw,,+y(m+2J Q + VI
+ 1.
Degenerate parabolic equations It
515
therefore follows from (2.8) that @Ix* 5 C’(PJ,*
+ IINO)
for ,I = n + m + 2 + a(m + 2) and any,u E (n + m + 2,1). Because of the obvious interpolation inequality between the weighted seminorms, we have PJK,,
5 C’
ll~4lo9
(2.9)
which is the desired estimate. 3. PROOF
OF THEOREM
1 WHEN
m r0
We now prove theorem 1 when m L 0. Our proof is a simple variant of the perturbation argument of Giaquinta and Giusti [6] as modified to parabolic conormal problems in [14]. We begin with some lemmata related to the maximum principle. The first lemma gives a precise form of the maximum bound for solutions of parabolic differential inequalities. Although the basic result is certainly well-known, we know of no place where the precise dependence is written down, so we present it here although [16, lemma 2.11 gives the case q = 00. LEMMA
3.1. Suppose that u is a weak solution of the differential
inequalities
- cru, + div A(X, Du) L b in Q’(R), A”(X, Du) 2 0 on Q’(R),
(3.la)
u I 0 on Q*(R),
(3.lb)
and suppose that A satisfies the structure condition A(X,p)
‘p L jp/m+2 - d;+2
(3.2)
for some constant a, 2 0, that OL2 a,
(3.3)
b E L*(Q+)
(3.4)
for some positive constant a,, and that
for some q > 1 + n/(m + 2). Then u I C(a,, q, n, m) (u~R)‘+““~ + Rm+’ (R -(n+m+2) I,+ by*].
(3.5)
[ Proof. The proof is a simple modification of the Moser iteration method, used in Section 1. We include only enough detail to verify the specific form of the estimate, which will be important later. Let us set M
=
(a,R)l+m/2
bq)l/Q,
+ Rm+z(R-(n+“+2)
Q' m+2 K=l+-,
n
171, =
q2h (q-
- 1) l)K--q’
(1 - qh m2=(q-
l)K--q’
516
G. M. LIEBERMAN
and observe that m, > 0 > m, . For r > 0 at our disposal, we take as test function in the weak form of (3.1) the function (0 =
r+m2
v”’
M'
( > 1
_
_
v +’
In this way we see that max -Rm+2
( -
Vml”m2+l(l s
_
!!):”
+
I,+
Vml’-m,-l(l
_
~)3Du,“i2
B+ cRlx(t)
Cr2MR
-m-2+(n+m+2)/q [S
Q+
We now set
and note that
It follows from the Sobolev inequality (in the form [16, (1.8)] with K, = 0) and the iteration scheme of lemma 1.1 that v I Mmax
2, C R-“-m-2M-q I
j,+
vy’m’].
(3.6)
(
The inner integral is estimated by using the test function a,
=
(vq-1 - Mq-‘)+.
We now find, after an application of Young’s inequality, that vq I CMqR” + CR -m-2 1B+
Vq.
j
x(f)
B+x(-Rm+*,f)
Gronwall’s inequality yields
vq 5 CM’lR”+m+2. Q' The proof is completed by combining (3.6) and (3.7).
(3.7)
n
Our next lemma is a version of Gilding’s estimate [8] relating the time and space moduli of continuity of solutions of parabolic equations. LEMMA 3.2.Let u. > 1, p. - .u3, A, A, R be positive constants. Suppose that A is weakly differentiable with respect to p and that 44 @(X, Z, P)CiCj2 n(K +
2 PO,
IPI)“ICI’
(3.8) for all c E R”,
(3.9a)
Degenerate
parabolic
517
equations
(3.9b) (3.9c) (3.10) (3.11a) for some nonnegative function b E Lq with (3.11b) If u is a continuous
function with bounded spatial gradient which satisfies CYU~ = div A(X, U, Du) + B(X, u, Du) in Q’(R), on Q”W,
A” (X, U, DU) + q/(X, U) = 0 osc UIUo-
1,
SUP
Q+(R)
(3.12a) (3.12b)
IDUI5Lc3,
(3.12~)
Q+(R)
then osc
u 5 C(L A, cc,, m, n, u,)U
+ iu2 +
(3.13)
p3)R.
Q + (R/2)
Proof. Clearly it suffices to estimate the oscillation of u(xo, *) over an interval of length proportional to Rmc2. Fix t, E (to - R”‘+’ , to), and for c a constant to be chosen, set
p = [cA(to - f,)Ur/p0]“(m+2),
s =
sup
MO
I,
90 - Nxo, t,11,
and assume without loss of generality that s > 0 and p I ARAW. Define [Ix]] = (A’]x’ 12/64A2 + ]x”]~)“~, and introduce the sets N+ , No by replacing Ix - x0 I with [Ix - x0 II in the definition of Q’ and Q”, respectively. We now consider the function w(X) =
2,U,(m + 1) + CAS t - to + U~‘(m+%r’(m+r) [Ix - x0 112 + m+21 P P2 > PO
+ [ I”r(;+ By choosing c appropriately
l)]“‘m+‘)+4u~‘“id)sl/(m+)](p_i+*/d). we can make
- OIW,+ div A(x,, t, u(xo, t), Dw) I 0 t, MO, 0, Dw) + w(x0, 0 5 0
A?,,
in N+, on No,
and hence if 1 b”
=
#(x0,
f =‘M,
t, u(x”,
t),
ODU + (1 - a)Dv)da,
u, Du) - Nxo,
t, MO, 0, Du),
rl(3P
G. M.
518
LIEBERMAN
we find that u = u - u(x,,, tt ) - w is a weak solution of - CtU,+
Di(b'jDjU
+
b”jDjV + f” 2 0 on No,
f')
L
in NC,
b
v~Oon6N~W~.
We now apply lemma 3.1 with a, = C(1 + ,D~)and A4 = C[(l + P~)R]‘+~‘~; note that the use of N+ in place of Q’ does not affect the proof beyond a slight adjustment of constants. Therefore evaluation of (3.5) at x = x0 gives u(x,,
t) -
u(X,)
5
c(t
-
to)
+
f
+ p3p
+
C[(l
+ /f2 + /f3)p]i+m’2.
Taking the supremum over t and using the inequality &x0, t) - u&X,) I u. gives the desired result. W From this point the proof of theorem 1 is an easy combination of the proofs of the corresponding results in [14] and [15]. We fix a point X0 E SQ and use an ti2-t-@) flattening of the boundary to assume that u satisfies in Q’(2R),
ol(x)u, = div A(X, U, Du) + B(X, u, Du) A”(X, u, Du) + t//(X, u) = 0
(3.14a)
on Q0(2R)
(3.14b)
for some positive constant R and functions A, B, and IC/satisfying the hypotheses of theorem 1 and Q satisfying IDal I C(Y)‘-‘,
a zruo,
(3.15)
Ml3 5 C
for some nonnegative constant C(n, m, Q) and positive constant po. For M’ = max sup JDiUl i
Q’(2R)
(recall that Du is assumed bounded), lemma 3.2 implies that osc u I CM’R.
(3.16)
Q+(R)
We observe that u is also a solution of a boundary value problem of the form U, = div A(X, u, Du) + B(X, u, Du) A”(X,
u, Du) + v(X,
u) = 0
in Q’(2R), on Q0(2R)
(3.14a)’ (3.14b)’
with A and w satisfying the hypotheses of theorem 1 and B satisfying ]B(x, GP)] 5 A~]P]“‘+~ -+ b(X) + A,(1 + ]p()m+‘(~)S-l. (Note that the As and vvs in (3.14) and (3.14)’ are not the same!) If u is the solution of the boundary value problem V, = div A(X,, u(X,), Du)
in
Q'(R),
A”(&, Go), Du) + w(Xo, 4X0)) = 0 on Q’(R),
u = u on Q*(R)
(3.17)
Degenerate
parabolic
519
equations
with A and 9 from (3.14) (given in Section 5), using 9 = u - u in the integral forms of the equations for u and u gives n j B + CR)x {toI
9* +
=
j Q+(R)
m
D9
. LJW,,
* MKJ,
W,),
UGG),
Du)
-
Du)
-
AW,
A(&,
WG),
WI
u, Du)l
i Q+(R)
B w, u,
+
Du)rp
(3.18)
+
s Q+(R)
The left hand side of this equation is no smaller than n /I ID$Op+*. j Q+(R) To estimate the right hand side of (3.18), we first use a simple variant of lemma 3.1 to infer that sup ]U - VI I Cmin((1 + M’)R, 1). Q+(R)
On the other hand because of the modulus of continuity estimate of DiBenedetto conclude that
[2] for u, we
sup ]U - u] + osc u I min(C&P + s(M’R)“, C)
(3.19)
Q’(R)
Q+(R)
for all sufficiently small E and 6 (specifically any 6 < 1). The right hand side of (3.18) is then estimated almost exactly as in [14, Section 41. First
I
I-
for o
c /3/(m
j,+,,
+
[fw + A,(@
3
+ &(M’R)6)@]
s
A 4 Q+(R) ID91”+*
+ C-@,frR)(l+@(m+*)]R”
+ [Cp+“)(m+*)
1). The estimate of the integral involving B is slightly more complicated:
B9 5 jQ+,,, b9 + A, jQ+,R, 9lD@+* I (j,+,,
]D91(1 + ]D#‘+’
Q+(R)
b’)1’9( j,+,.,
99/(9-‘y9
+ CA2(1 + IW)~+‘(R
+ A~ j,+,,
+
[C&P
+
(l + ]DU]),+1(Y)8-19
CE(MfR~G](Ml)m+*Rn+m+z
+ IWR)R”+~+‘+“.
We now estmate the integral involving 9 via (3.19) to conclude that
c
JQ+W
B9
I
[C&(l+@(m+*)
+ ,-E(j$f’R)(l+‘d(‘“+*)]R”
G. M. LIEBERMAN
520
for o sufficiently small depending on q. Finally, we infer from Green’s theorem and Young’s inequality that
QO(R)lv(X, u) - VW,, ma)lp
&M’R]”
3 144
I
C[C&R +
I
C[C&A”+ &(M’R)6]o
Q'(R)
IDd
Q+(R) A I-
Q+cR,IDp]m+2
4
+ CE(M)R)(l+o)(m+2)]Rn.
[c$?('+'-+*)
+
We therefore obtain
J
ID(u
Q+(R)
- u)l”‘+’
(C,
5
+
E(Ml)(m+2)(l+o))Rn+(m+2)(l+a).
It now follows from (2.9) that [Dvl I CM’ in Q’(R/2) and the considerations of Section 2 imply that
(3.20)
and then the proof of [15, theorem l]
blT+o 5 C([u]f + 1). Just as before, this estimate proves theorem 1. 4. PROOF
OF THEOREM
1 WHEN
m < 0
The proof of theorem 1 for m < 0 is very similar to the proof in the previous section so we only indicate the significant changes that are made. We note first that lemmata 3.1 and 3.2 require only one change in this case: because now m c 0, the conclusion of lemma 3.2 becomes Qyc)
C[(l + ru2 + ,u~)R]~+~‘~.
u I
(4.1)
Also in our flattening of the boundary, we only use (3.15) and the last two inequalities in (3.16). Then we take u to be the solution of a(xo)u, = div A(&,
u(&),
Du)
in
Q’(R),
A’VG, uWd, Du) + u/(&, uCW) = 0 on Q”W, If we abbreviate S = B+(R) j$xo)~~ =
+ jQ+cR@
1
x (to), we obtain in place of (3.18)
* VW,,
uWo1, DuJ - AWo,
Do, * IAWo, u(Xo), W
Q+(R)
+
s
Q+(R)
u = u on Q*(R).
BW, u, Du)cp+
- A@‘, u, WI
s Q’(R)
unbox, WI
l~(x
U) - U/(X,, U(X,))IV +
s
sla(xo) - &P.
(4.2)
Degenerate
parabolic
521
equations
The left hand side of this equation is estimated from below by
and using Holder’s inequality to relate the second integral to 3
J
1Dpl”+?
Q+(R)
The first two integrals on the right hand side of (4.2) are estimated as before. For the last, we assume without loss of generality that r&Y,,) = 0, so that
s
[01(x()) -
a]ucpI
CY(X&*+
1%
s
CR*O
s i u* s CR”(IWR)~+*
for some C depending on the maxigum of u. By using the smallness of A, in place of the modulus of continuity estimate for u, which at present is not known for this situation, we again infer (3.20) and hence (0.5). 5. AN
EXISTENCE
THEOREM
We now show that the boundary value problem u, = div A(Du) in Q’(R),
A”(Du)
-t @ = 0 on Q’(R),
u = u on Q*(R)
(5.1)
is solvable in a suitable space for continuous u provided we make one additional hypothesis beyond those in Sections 1 and 2. When F has the form F(r) = (K + T)~, a simple variant of [9, theorem V.6.71 shows that (5.1) has a weak solution only for m > max[ 1, -4/(n + 2)). For this reason, and also to demonstrate the power of using our general Fs, we use a different approach which does not use the existence of weak solutions. We write H* for the space of uniformly continuous functions with finite norm IG+,
=
Ido+ [ulr+,,
and we seek solutions of (5.1) in this space. The idea is to approximate this problem by a family of problems which are known to have solutions and then take advantage of certain uniform estimates of the approximate solutions. To this end we now state our additional hypothesis: that there is a constant a! E (0, 1) for which s”F(.s) I a”F(a)
for 0 < s I 0.
(5.2)
Under this assumption (see [18, Section 5]), there is a locally Lipschitz function P on II?, such that c&s) 5 F(s) I P(s), SF’(S) s (1 - a)F(s) for all s > 0. For A, 1 2A/(l - 01)a constant to be further determined, we observe that
we now set A(p) = ~,$(lpl)p,
and
G. M. LIEBERMAN
522
Now we introduce a parameter A and define a function fx as follows: I A(s) =
1
ifsr1,
e2A(x-s)
Ik
if A < s I 21,
21e-‘*‘(31
- s)
if 1. < s I 3A,
0
ifs > 31.
If we now set
AA(P)= fA(lPIMP) + (1 - A(lPl))4P), we see that -i
f$ ti
Then, by choosing A sufficiently large (independent of A), we see that A, satisfies the hypotheses of Section 2 with ci and F independent of 1 and hence all the estimates of that section are valid for the approximating problems. Of course, we still do not know that these approximating problems have solutions, so we move to a further approximation. We set F,* = sup F(s) + A, O
F,(s) = min(F,*, F(s)) if IpI 5 4A if IpI > 4A.
It then follows from [12, theorem 31 (and [14, lemma 4.11) that u, = div A,(Du) in Q’(R),
A”(u) + @ = 0 on Q’(R),
u = u on Q*(R)
(5.1)’
has an H* solution for each i > 0, and the results of Section 2 give a uniform estimate on the H* norms of these solutions. To see that the solutions have a uniform limit as i. tends to infinity, we need a uniform modulus of continuity estimate. (The limit is in H* and satisfies (5.1) by the H* estimate.) If u is in C”, this estimate follows from the considerations in [13, Section 21 and the gradient bound from Section 2 at any point X0 in the closure of Q + unless x0” = 0 -6 > 0, $ = 0,
and
to = 0, and
1x01< R, 1x01 = R,
and
or to = 0,
(5.3a) or
(5.3b)
to > 0.
(5.3c)
In all cases, the modulus of continuity can be estimated by a simple variant of the estimates in [12]. In case (5.3a), we set w(x, t) = fl - (101 + &,)P + &2(t + Ix - x0/)* + U(X,) and note that -w, + divA,(Dw)
I Oin Q+,
A;(Dw) I @on Q”,
w > UonQ*
(5.4)
for suitable constants E, E, , and a2. It follows that w > u in Q ‘, which gives a one-sided modulus estimate; the estimate on the other side is proved similarly.
Degenerate
parabolic
523
equations
In case (5.3b), we see that W = t” + u(x,, 0) + Du(x,, 0) * (x - xg) + &,(t + lx-$ - J! I)’ + &22(t + Ix - x, I)2, for suitable positive E, ci, and e2, satisfies (5.4), which leads to a modulus of continuity estimate in this case. In case (5.3c), [12, lemma 21 provides a function w,(x) and positive constants E and C such that, for any matrix (a”) and vector p for which I I (a”) 5 N,
p” 1 1,
we have U’~,jWi I -IX
-
xo Ice2
IPI 5 & in B +,
/? *Dw, I -Ix - x0 I’-’ on B”, Ix - x0 I&I WI 5 clx - x0 I&,
IDW,~ s C~X - xOla-l in B+.
Then w(X) = It - to I2 + El w,(x) + u(X,) satisfies (5.4), so a modulus of continuity estimate follows in this The proof of existence (for fixed A) follows by approximating u tions and seeing that the corresponding us converge uniformly. If and u2 with Iu, - u2 I < E for some E > 0, the maximum principle and hence if the USconverge uniformly, so do the us.
case also. uniformly by smooth funcvi and u2 correspond to ui gives Ju, - u2 I 5 E in Q’,
REFERENCES I. DAPRATO G., Spazi Co’,*) (a, 6) e loro propierta, Annali Mat. pura appl. 69, 383-392 (1965). 2. DIBENEDETTO E., On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients, Annali Scu. norm sup. Pisa (4) 13, 487-535 (1986). 3. D~BENEDETTO E. & FRIEDMAN A., Regularity of solutions of nonlinear degenerate parabolic systems, J. Reine
Angew. Math. 349, 83-128 (1984). 4. DIBENEDETTO E. & FRIEDMAN A., Holder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Mafh. 357, l-22 (1985). 5. DIBENEDETTO E. & FRIEDMAN A., Addendum to “Holder estimates for nonlinear degenerate parabolic systems”, J. Reine Angew. Math. 363, 217-220 (1985). 6. GIAQUINTA M. & GIUSTI E., Global C”” -regularity for second order quasilinear elliptic equations in divergence form, J. Reine Angew. Math. 351, 55-65 (1984). 7. GILBARG D. & TRUDINGER N. S., Elliptic Partial Differential Equations of Second Order, 2nd edition. Springer, Berlin (1983). 8. GILDING B. H., Holder continuity of solutions of parabolic equations, J. London Mafh. Sot. 13, 103-106 (1976). 9. LADYZENSKAJA 0. A., SOLONNIKOV V. A. & URAL’CEVA N. N., Linear and Quasilinear Equations of Parabolic Type. Am. Math. Sot., Providence, R.I. (1968). 10. LADYZHENSKAYA 0. A. & URAL’TSEVA N. N., Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968).
11. LIEBERMAN G. M., The conormal derivative problem for elliptic equations of variational type, J. dvf. Eqns 49, 219-257 (1983). 12. LIEBERMAN G. M., Mixed boundary value problems for elliptic and parabolic differential equations of second order, J. Math. Analysis Applic. 113, 422-440 (1986). 13. L~EBERMANG. M., The first initial-boundary value problem for quasilinear second order parabolic equations, Annali Scu. norm. sup Piss (4) 13, 347-387 (1986). 14. L~EBERMANG. M., Hilder continuity of the gradient of solutions of uniformly parabolic equations with conormal boundary conditions, Annali mat. pura appl. 148, 77-99, 397-398 (1987). 15. LIEBERMAN G. M., Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Analysis 12, 1203-1219 (1988). 16. L~EBERMANG. M., The conormal derivative problem for non-uniformly parabolic equations, Indiana Univ. Math. J. 37, 23-72 (1988).
524
G. M. LIEBERMAN
17. LIEBERMAN G. M., Initial regularity for solutions of degenerate parabolic equations, Nonlinear Analysis 14, 525-536 (1990). 18. SPERNER, Jr E., Schauder’s existence theorem for cr-Dini continuous data, Ark. Mar. 19, 193-213 (1981). 19. TRUDINGER N. S., Pointwise estimates and quasilinear parabolic equations, Communs. pure appl. Math. 21, 205-226 (1968). 20. TRUDINGER N. S., On an interpolation inequality and its applications to nonlinear elliptic equations, Proc. Am. math. Sot. 95, 73-78 (1985). 21. WIEGNER M., On C,-regularity of the gradient of solutions of degenerate parabolic systems, Annali Mat. pura appl. 145, 385-406 (1986). Added in proof-If A is independent of have shown that u is Holder continuous.
(Archs ration. Mech. Analysis 103, 319-245; 1988) A, in theorem 1 need not be restricted in this case.
t, Chen and DiBenedetto Hence the constant
Merhods
& Appircmons.
Vol.
14.
No.
6. pp.
501-524.
1’9%.
0362-546X 90 53.00+ .CKl E 19w Pergamon Press plc
BOUNDARY REGULARITY FOR SOLUTIONS DEGENERATE PARABOLIC EQUATIONS GARY
OF
M. LIEBERMAN
Department of Mathematics, Iowa State University, Ames, IA 50011, U.S.A. (Received Key words and phrases:
1987; received for publication
9 September
21 April
1989)
Parabolic equations, degenerate equations, boundary value problems, ~1priori
estimates.
INTRODUCTION IN THIS paper, we consider the conormal derivative problem for equations with degeneracies of the same type as
m>
- u, + div(lDu I”Du) = 0,
-1.
(0.1)
By modifying the method of DiBenedetto and Friedman [3-51 as in [IS], we shall show that the spatial gradient of a solution of (0.1) with conormal boundary conditions is Hijlder continuous up to the lateral boundary of the domain. To be more specific, let Sz C IT?”be a domain with C’*O boundary for some /3 E (0, l] and inner unit normal y, fix T > 0, and set QT = SXIx (0, T) and SQ = 6’sZx (0, T). Then for constants II. > 0, A 1 2, Y, AZ, A, 2 0, m > - 1, and K L 0, we consider: a vector-valued function A defined on QT x R x R” satisfying (0.2a) (0.2b)
I&, ~,z,P) - ACV,f, w,P)I 5 forxandyinR,O
(1 +
bi)“+‘[Alx - yl” + AlIz - ~1’1
(0.2c)
A(1 +
(0.2d)
T,zandwinR,andpER”, IA(x, t, z,p) - A(x,s,
for x E X2, 0 < s < t < T, z E R, and
z,p)I
p E IR”;
I&K z, P)I for some function b E L4(QT) with
I
5
Ipl)m+‘lt
a scalar-valued
A,IPI~+~
q > n + m +
-
sI~‘(“‘+~)
function B with (0.2e)
+ b(X)
2 and
I/q bq
and a scalar-valued function I,Uwith
I
A2;
(0.20
(! QT >
Iw(X,z) - w(Y, w)l I ‘-I’(lx - yIB + It - sI”(~+‘)) 501
+ AlIz, - WI’
(0.2g)
502
G. M. LIEBERMAN
for X and Y = b,s) in SQ, and z and w in R. We are concerned boundary value problem in QT
u, = div .4(X, U, Du) + B(X, U, Du) A(X, U, DU) * y(x) + ly(X, u) = 0
on SQ
with solutions u of the
(0.3a) (0.3b)
with u E Lm(Qd+ Du E Lw(Q x (E, T))
(0.4a)
for any a E (0, T).
(0.4b)
1. Let u be a weak solution of (0.3) satisfying (0.4) and suppose conditions (0.2) are satisfied. If m 1 0 or if A, is sufficiently small, then there is a constant 0 E (0, p] determined only by p, A/A, m, q and n such that for any E E (0, T) and X and Y in !J x (E, T),
THEOREM
IDu(X) - Du(Y)l 5 C@,e,A,A,,&,&
m, ‘I’, )u],,n,q, T, 52)()x- ~1” + 11- ~j~‘(“‘+~)). (0. 5)
We leave the study of regularity near the initial surface to a sequel [17] because of the difference in method of proof and the weaker results in that case. The differing exponents with respect to x and t are based on the natural scaling properties of (0. l), which are more fully exploited here than in 141. The approach in [21] for degenerate systems also uses differing exponents, but not the ones here which seem to be more natural. The other new element here is the boundary estimation. In [4] DiBenedetto and Friedman (see also (211) prove interior gradient Holder estimates for systems under stronger structure conditions than ours; their method will give the interior estimates for single equations under our conditions. Also, they assume that m > max( - I, -4/(n + 2)) but not that u is bounded, and they prove their continuity estimate assuming a gradient bound; we prove the gradient bound from the continuity estimate. Our assumption that u have bounded gradient can be relaxed, but we shall not go into detail on this matter. The proof of theorem 1 proceeds in several steps. First, in Section 1, we prove some preliminary results for solutions of (0.3) under simpler, slightly different hypotheses. The Holder continuity of the gradient under these hypotheses is proved in Section 2, and the full strength of theorem 1 is demonstrated in Section 3 for nonnegative m. The few differences in proof needed for negative m are given in Section 4. Finally a crucial existence theorem is proved in Section 5. I. PRELIMINARIES
FOR
THE
SPECIAL
CASE
We begin by studying (0.3) when A depends only on p and some other dependencies are simplified. Suppose that F is a positive, continuous function on (0, co) and that there is a positive constant ci such that F(s) r ciF(4s)
SF(S) _( aF(o)
for 0 < s,
(l.la)
for 0 < s < ~7.
(l.lb)
(There is no loss of generality in using (1 .Ib) rather than the inequality SF(S) 4 c2cC(a), as in
Degenerate parabolic equations [ 151,
503
because the function f given by 1 f(s) = - sup &(a) s (IBS
satisfies (1.1) and F I
f I
c2 F.)
Now for x0 E IR”and positive constants R and M, define
B(x,,, R) = (x E II?“:Ix - x0( < Rj, B+(x,,R)
I(M,R;b)=
= (x~B(x~,R):x”>
(
R2 10-m,ro
9 >
0),
B”(xo, R) = {x E B(x,, R): x” = 01, Q+W,R;xo,4,) Q’W,
= B+(xo,N
R;xo,
to) = B’(xo,
CPQ+(M,R,x,,t,) Q*W,R;xo,to)
x I(M,R;fo), R)
x IW,
R; to),
= (X~i3Q+(M,R;x~,t,):t
< to),
= sQ+(M,R;xo,to)\Q’(M,R;xo,to).
is called the parabolic boundary of Q’ . In what follows, we generally suppress the arguments of the Qs and Bs when they are clear from the context; for technical reasons, we shall suppress the second argument of Q’ or B+ when it is 2R rather than R. We assume that @ is a constant. that A is a vector-valued C’(P) function with derivatives aAi/apj = au satisfying SQ’
a”CP)t*Tj
for all r E IR”
2 F(ipt)ki2
la%-45 ~(IPI),
(1.2a) (1.2b) (1.2c)
IA@)] 5 Al~lF(l~l)
for all p E IR”. (In many applications, condition (1.2~) is a consequence of (1.2b).) For to = 0 and fixed M, R, and x0 with x0”= 0, we now consider the boundary value problem V, = div A(Dv)
in Q’ ,
A”(Du)
+ Q, = 0 on Q”,
and we suppose that the solution u of (1.3) has weak second spatial derivatives in L’(Q’) satisfies the additional inequality max ]DiV] I M in Q’ ; isn
as in [4], this assumption and the use of the half-cylinder the keys to our arguments. As a final piece of notation, we set
(1.3) and (1.4)
Q’ rather than B+ x (- R2, 0) are
M’ = max sup I&V]. ken
LEMMA
Q+
1.1. For any h E (0, I), there is a positive constant fi = p(h) such that if M’ L hh4 and (1.5)
504
G. M. LIEBERMAN
for some k < n, then D,v
2 =
in Q’ (M, R).
8
(1.6)
Proof. Let q be a C’*‘(Q+) function vanishing on Q* and let g be a Lipschitz function on R. Define G by
G(a) =
O&s)ds, s0
set t, = - (2R)‘/F(M), and obey the summation convention only for the indices i and j. Now use D&g(Dkv)) as test function in the weak form of the equation for v and integrate by parts. For each r E (tt, 0), we thus infer
rlGG%v) + B+xb+
t,@DjkvDikvg’(Dk~) B+X(rl,T)
=
A’D,,Vg’Dit/ B+xul, 7)
+ A’gDiktl - rl,G. (1.7)
Now we choose g and q in this equation by setting, for q > n + 1, w=min[$,(F-Dkv)j,
g(Dbv)=@-“-’
+-(#[;-I],
V=p-n.
From (1.4), it follows that
C(h)
IAiDkkvl I &DjkvDi,v
in Q+,
+ -M2F(M)
and, because M’ L hM, we conclude that SUP t,
+ s sx(+(Wo2q-n s
F(M)Q+ ~D(wC;)~-“‘~~~I
Bf
C(h)q4M2F(M) s Q+ (W02q-n-2*
For K = (n + 2)/n, we infer from the Sobolev inequality in the form [ll, (1.6)] (see also [16, lemma 1.21) that x(2’?-n)
Q+(wo
5
C(h)
If we now set dx = (~/IQ’I)(M/w~)“‘~ rewritten as
44x
r$rM2
dx
(I,+
(~a~~-~-~)~.
dt on C = (wl > 01, then this inequality can be
( wo2qx cLZI C(h)q4” (~,(wc)%x)x IE and then an easy interation scheme (along with Holder’s inequality) gives sup (Wan+4 E
Degenerate
parabolic
505
equations
whence
Because w = 0 if D,v > M’/2,
it follows from (1 S) that 1C 1 I p/Q’ 1 and hence S.?
C(h)M’p”‘“+J’.
WCI
Therefore, if ZAis small enough, we have WCI M/8 in Q' ; (1.6) follows from this inequality because M’ L hM. N For our next lemma, we introduce some more notation. s(0) = (X E Q’ : Dkv(X) > (1 - @M’ ),
For 0 E (0, ‘/2), we write
S(8, 2) = (x E IR”:(x, t) E s(e)).
LEMMA1.2. For any h E (0, l), ,u E (0, l), 0 E (0, %I, if M’ 1 hkf, and
nQ’1
IW$j
>dQ+I,
(1.f>
then inf
Is(0, t,r)l 5
(1.9)
+$++I.
@/2<7<1
Proof.
Set t2 = ft,,
z=
(*/s(e,t)ldt= Is(e)n[B+ x(t,,t2)]l, I 11 J = inf IS@, t,r)l. p/Z
Then (1.8) implies that Z I IS@)/ I (1 - p)lQ’I. ZrJ[t,-
tl] =
On the other hand
( > 1
IQ'1 -$ .zm.
The desired result follows by combining these two inequalities. Lemma 1.2 is, except for a slight variation in notation, For our next lemma, we define for fi E (0, l),
4
[4, lemma 3.6).
l’(n+Z) ’
r(e)
= (XE s(e): 1x1 c VRJ.
Note that 0 < v c 2 and there is a constant p,,(n) such that P c p, implies v > 1.
(l.lOa) (l.lOb)
G. hl. LIEBERMAN
506
LEMMA 1.3. For any h E (0, l), p E (0, pug), if M’ 1 hMand (1.8) holds, then there is a positive integer r such that
sup IS’(2_‘/f, f)] 5 f ]B+(VR)j. &VI Proof.
(1.11)
Let E > 0, t’ E (@/2)t,, 0), and choose t” E (tl, (~/2)t,)
from lemma 1.2 so that
For r a positive integer at our disposal, we define the function Y by Y(aM”)
= log+
P ,U (a 1 + p)+ + 21-‘/f [
1’
Now, for c E Co’ ‘(B(2R)) with C;= 0 on aB(2R) to be further specified, we set
&Y(o)= 2WW’(d,
v(x, t> = [(x)2.
An argument similar to the one leading to (1.7) implies that r2(1 + Y)(Y
&J2 + B+
X(f’]
=
s
s
E+
!
B +
V’I’ ‘a”ojkvDJ
l2Y2 - 4 B+ (f”. rr)
B+ X(P)
I
‘)‘a’iDjkvDikU
X(P,P)
xp~C2Y2 +
Bf x(t”,t’)
c2(1 + Y)(Y’)‘a’jDjkVDikU
+ CF(M)
Y’IDd2. B+x(r”,r,)
If c is chosen so that [ = 1 in B(vR) and JD[l s C(v)/R in B(2R), it follows that Y2 5 IS@, t”)l[r - 1) log 212 + C(v)lB+ I(r - 1)
(1.12)
B(vR)x(f’)
)
+ E [(r - 1) log 212+ C(v)(r
q*
- 1) ]B+ I. I
We now send E to zero and conclude that
IO
i * [r - 1) log 21’ + C(v)@ - 1) IB+(vR)I 1 1 s B(vR)x(r)
Y2 1 [(r - 2)log212 [S/(2-‘p, t”)l.
Thus ]S’(2_‘/& t’)( I
“I:‘!
i21) + (%)‘(;)‘3
]B+(vR)I.
The proof is completed by choosing r so large that
C(W - 1) $(l (r_2)2
- (;)“2),
(=Yst.
n
Degenerate
parabolic
507
equations
Lemma 1.3 is essentially the same as [4, lemma 3.71 and [3, lemma 6.21. LEMMA1.4. For any h E (0, l), ,U E (0, p,,), if M’ 2 hM and (1.8) holds, then there is a positive constant q < 1 such that
in Q’
I&VI 5 VM’
M,$R (
Proof.
(1.13)
. >
Set w = (D,v - (1 - B)M’)+,t”
=ifl,
and let L:be a Czg ‘(Q+) function with &Y) = 0 if 1x1 > vR or t c t”. For q a positive constant at our disposal, we then use Dk(c$“+2)q-n w4-‘) as test function to see that, for any I’ E (t”, 0), 1
c 4 if+ X(I’} -s
(n+2)4-ng + (q - 1)
= ((n + 2)q - n)
4
~~p+m-ns B+X(f”,f’)
If we now take c so that 1~1 I 8F(M)/pR2,
s B+
ld.
(1.14)
and 0 5 [ 5 0, then
IDC) I 8/vR,
F(M)
([“+‘w)~ + F(M)
~D((~n+2w)q’2~-“‘2)~2 I Cq2T B+
X(f’}
(n+2)q-nWq-ZaijDjkUDikV
,$n+Z)q-n-l w4-‘a’jDjkvD;[ B+X(l”.f’)
(n + 2)q - n
+
c B+ X(f”.f’)
LJ+ X(f”.f’)
X(P,I’)
@r(
n+2)q-n-2
*
Setting Q = B+ x (t”, 0), C = (cw > 0), and S”(e) = S’(0) fl (t > t”], we see from the Sobolev inequality and iteration scheme that sup r”“W I CL
IQIs E
If, finally, c = 1 in B+(R)
x (Y/2,0),
wI
ce.nF.
it follows that in B+(R)
x
for Is”(@~/~QI small enough. This ratio is controlled as in [3, lemma 6.41. With C ’ = [w > 0, t > P’) and c2& replacing p+2)q-nWq-1, the proof of (1.14) yields ‘h
s s pw” +
B+ x(0)
and therefore
c2ai’DjkUDikv = 2
X’
c2ti + F(M) Bf x(0)
w[a”ojkVDJ
+
s I’
jDDku12c2 5 s E’
x$ s 1’
CF
n+d. s
G.
508
M. LIEBERMAN
Hence, for C ’ = 1X E C ‘: 1x1 < vR] and appropriate
1
lDD/plZ5 c y [
L”
[,
I2/Q+(M, vR)I.
We now choose 13= 2-jpM for various positive integers j and set 0 ’ c (j] = S”(2 -‘Z&
co
= co\co’-
l),
Z=
IDD,c/.
SIf’
I:(i. 0
To proceed, we use (3.5) of Chapter I1 in [IO] to infer that, for any r, C(n, v)R=+’
2 -j-‘/.lM’ I c (j, t)l I
JB+(VR)J - ICo’
- 1, t)l s &DDkUI.
If j 1 r@), the constant from lemma 1.3, then Z > C(U) and therefore,
because
’ 1C (j, t)/2 -j-‘PMR
dr = C(D) 2 -‘-‘PM
1C (J]IR
s f*
Z I C2-‘PM
[I C(j)IR-‘IQ+(M,
vR]]“~,
we see that
I C(j)15 WNIQ’W,
v&l I C ‘QIY~.
It
follows from this inequality that Is”
small by choosing 8
Lemmata 1.1 and 1.4 are easily combined to get our basic estimate for the tangential derivatives. To state this result more easily, we write Q(r) for Q+(M, r). THEOREM1.5. For any h E (0, I), if M’ r hM, then there are positive constants q(h) and 6,(h) such that either max osc Dkv I q max osc Dku ken
k
Q&r)
(1.15a)
Q(2r)
for all r E (0, R) or max sup l&U1 I q max sup ID,vl. k
(1.15b)
Note that we have used a simple variant of the usual Holder estimate [9, theorem V.1.11 or [19, theorem 4.21 in deriving (1.15a). From lemma 1.5, we see that for any h E (0, 1) there is a constant 6,(h) E (0, 1) such that max osc Dkv I hM. k
Q&R)
We now restrict our attention to Q&Z?). If ID,ul < M/4 in Q(S2R) and h < ?4, then max SUP IDiUl I SM. ian
(1.16)
Q(6zR)
Because we wish to prove a version of (1.15) with all the derivatives of v instead of just its tangential derivatives, we asume that (1.16) fails and look in Q((6,/2)R). Thus we may
Degenerate
parabolic
509
equations
assume that max osc Dkv I hM, k
sup ID,+ L
MI
(1.17a)
Q(ZR)
(1.17b)
‘/zM.
QGN
Our goal now is to prove analogs of the preceding lemmata under these assumptions with k replaced by n. LEMMA1.6. There are constants h, and p’ such that if h E (0, h,), and (1.17) and (1.18) hold, then (1.19) Proof. Let [ and g be as in lemma 1.1. If g’ (D,v) vanishes on Q”, the proof of (1.7) is easily modified to yield
s=?
CG(D,v) +
B+ x(1=?+
!
CdjDj,, VDi, vg ’ (D,v) ia+ x(r1.T)
A’D,,Vg’D,r
+ A’gDi,r
- rrG -
B+x(?I.r)
B+X(fl,r)
~D,,vD,k’
- @D,&.
(1.20)
Moreover if g = 0 on Q”, then (1.7) holds. In addition, for K = Ad%, ID,v(X)
- D,v(Y)I
I K Ill:; [IDkv(X)
- Dk(Y)j)
for x, Y in Q”.
(1.21)
Now we consider the two possibilities: D,v
2 Feverywhere
on Q”,
(1.22a)
D,v
< Fsomewhere
on Q”.
(1.22b)
If (1.22a) holds, then w from lemma 1.1 vanishes on Q”. The proof of lemma 1.1 applies without any restriction on h to show that ID,vl 2 3M/8 in Q’(R). If (1.22b) holds, say D,v(x) < M/2, then for any Yin Q”, D,,v( Y) I D&X)
+ K max osc D,v k
I r
+ KhM.
Q’
Hence if ho = 1/4K, D,v I 3M/4 on Q”. With this ho, we set w = min(M/8, ((7M/8) - D,v) + ) and note that 1~1 I C’F(M)M. The remainder of the proof follows lemma 1.1. n Because lemma 1.2 did not use any properties of Dkv other than its measurability, need to reprove it for D,v.
we do not
G. M. LIEBERMAN
510
For our next lemma, we recall the definition replaced by n.
of S(0), etc., from lemma 1.3 with k now
LEMMA1.7. For any P’ E (0, pO), there are positive constants hi, v1 < 1 and r such that if (1.17) holds and if (1.23) then sup IS(2--‘P’, f) fU?(v,R)I &‘/Z)rl
5 vJB+(V,R)I.
0), and choose t” E (t,, @‘/2)t,)
IS(j.i’, t”)I
I
so that
(l’_-p:;2+c)
p3+1.
For p, a positive integer, to be chosen, suppose that D,v > (1 - (~‘12~))Msomewhere If h _( (1 - @‘/2”))M, define Y as in lemma 1.3 with r replaced by p, and set kf*=
(1-$M,>
(1.24)
Y’,(a) = Y(min(a, M*j),
on Q”.
g(o) = 2Y,WY’;(o).
Because g(D,v) = 0 on Q”, the proof is completed in this case as in lemma 1.3 by taking p large enough and vi L v(,u’). On the other hand if D,u I (1 - (~‘/29)M everywhere on Q” for this p, then
Now the proof of lemma 1.3 gives v2 c 1 and r determined by p and ,u’ such that IS(2 -ICI’, t)l I (1 - v2)(BC(v2R)I. Clearly r 2 p, and hence (1.24) holds with v, = maxlv, v,).
n
LEMMA1.8. For any P’ E (0, l), there are constants h2, q, 6, in (0, 1) such that if 0 < h I h2 and (1.17) and (1.23) hold, then ID,,vl I VA4 in Q’(M,
6,R).
(1.25)
Proof. Suppose first that D,u I (1 - 16Kh)M everywhere on Q” and that h, 5 p”’/32K. By proceeding as in lemma 1.4 with ,uu/2replaced by 16Kh, we obtain (1.25). If DJJ > (1 - 16Kh)M somewhere on Q”, then the argument in lemma 1.7 shows that D,u 1 (1 - 18Kh)M everywhere on Q”. Now for B > 18Kh, we set
A(P) = A(P) - A(09 ***,O,M),
G = @ + A(0, . . . , 0, M)
and note that u, = div ii
in KJ,
A(Du) + 6 = 0 on &D.
Now set w = min(M4, (D,p - (1 - 2B)M)+ J,
511
Degenerate parabolic equations
and write W for the support of w. On W, it is easy to check that ]u’~]d CF(A4) and hence that ]A] 5 C&VU’(M); also the boundary condition implies that ]ri] I CBMF(M). Therefore we can repeat the proof of lemma 1.4 with all functions replaced by their barred counterparts and thus conclude (1.25) in this case because the constants will be independent of h. H We now choose first ,D’ from lemma 1.6, then h = min[h,, h, , h, ), and finally p from lemma 1.1; combining all the possibilities gives our basic result. THEOREM
1.9. There are constants rl and 6, in (0, 1) such that either max osc D;v 5 q max osc Diu isn Q(81r) ian Q(2r)
(1.26a)
for all r E (0, R) or max sup il;n
2. HOLDER
]D,ul 5 qA4.
(1.26b)
Q(61R)
CONTINUITY
IN THE
SPECIAL
CASE
We now show how to infer both a bound and a modulus of continuity estimate for the gradient from theorem 1.9 by using some of the ideas from [4, Section 21; however, there are some important differences to be noted here. First, our proof works directly in cylinders with the appropriate scaling for the equation and, second, we do not make the complicated adjustments in [4], so our proof applies regardless of the sign of the constant m. (We also point out that the proof of Holder continuity of the gradient given in [4] needs a minor adjustment in the time scaling of the cylinders Q. and Qr on page 7 of that work.) Our arguments simplify even further when F(r) = (K + T)~ for some K I 0 and m > -1. We also point out that some of the new elements here are also present in [21], but the author was unaware of the contents of that work while writing this one. To simplify notation, we write F* for the inverse to the function g given by g(s) = ?/F(l/s), we introduce the norm and metric 1x1 = max]]x], F*(t)],
P(X, Y) =
Ix - YI,
and, for X = (x,,, t,,) with 4 = 0, we introduce the scaled cylinders Q+(R) = (Y = 01,s): ]A’Q’(R)
= (y:
IX -
Y] < R,s < t,,y”
YI < R,s
< t,,y”
> 01,
= 01.
We suppose that F and A are as in the previous section and that v is a weak solution of the problem u, = div A@) in Q’ (R), A”(Du) + @ = 0 on Q’(R) (2.1) with bovnded gradient for some R > 0. Set MO = max sup ]Div] + f , i
K=
R R, = K’/2 -
R&f,,
Q+(R)
and note that K > 1. With 6, and q as in theorem 1.9 and q
2
‘A, set Rj = 6’Ro.
’
512
G. M. LIEBERMAN
Now observe that R2 R2 KF(M,,) - F( l/R)
R; PC_<_ F(M,) SO
that Q’(M,,
R,) C Q’(R),
and therefore MO b max
SUP
i
Furthermore,
JDiVl.
Q+Wo.Ro,
according to (1.1) and our choice of 6, we have
Ri
R$P
CIR: R: 1 4a2F(M,) L F(M,
F(M,) = F(M,/q) so that Q’(M,,
R,) C Q+(Mo, R,). Hence if max i
SUP
lDiul 5 rl”09
Q+Wo,Rd
then max
SUP
Q+WI
i
lDiU/
I: Ml*
.Rd
Continuing in this way, we see that as long as max i
IDi 5 ef-j-19
sup
Q’(Mj-t.Rj)
(2.2)
we have max i
IDiUl I Mj I M,
SUP
(2.3a)
Q’(Mj.Rj)
for any o I log,,, (l/q). We now denote by J the first integer j for which (2.2) fails; in the terminology of [4], Rj is the switching radius. It then follows from theorem 1.9 that for any j > J, max i
osc Q + WJ. Rj)
DiU I qmax i
osc
DiV
I
2qjeJrnax
Q + (Mr. Rj/o)
i
SUP
[DiVl I 2q’Mo 5 C $
Q + NJ, Rj)
OMo.
0 ’
(2.3b)
Our next step is to use these inequalities to estimate the modulus of continuity of Du with respect to the metric p. To this end we consider separately continuity with respect to space and time. Because the spatial estimates are so simple, we only look at continuity with respect to time. To this end, we begin by proving certain estimates related to the function F. We set p = ‘/2 log,,(l/c,) and assume without loss of generality that 0 < p c ‘/o. By choosing the integer k so that 4’-’ I P < 4’ and then applying (1. la) k times and (1.1 b) once, we find that
Also it follows from (1 .l) that, for any w E (0, 1) and any s > 0,
(M2 -“F(l/s) F( 1/OS)
OS2
Degenerate
parabolic
513
equations
and hence, because KZa+” 5 K, (wK“/M,)~
< 16c~~R;
F(M,/oP)
- F(M,).
We now set X = (x0, to), Y = (x,,, t), and suppose that R;
7 = F*(lt
-
(2.4)
toI)
and consider three cases. First
R2
(2Sa)
F(Mo )
From (2.4) with w = l/16, we infer that
7 L
K“-‘R/16
and hence
]Du(X) - Du(Y)] I 2M, I CM,
;
OK(l-“)O
0 for any cr > 0. Next we suppose that R; F(Mo)
R:
Riz F(Mj)
(2Sb)
,
RJf< to - t s F(Mj_,)’
Then, by an argument similar to the one leading to (2.4), we have
By choosing o appropriately,
160
(WP/Mj)2
<
F(Mj/OP)
- (SV)~‘F(M.).J
we now infer that
R;
[(6rl)“/16]~-“‘Rj
7 2
and hence
Finally we suppose that R:
o
(2.5c)
FW,)
and choosej so that Ri’ F(M,)
Rj- , < to - t 5 F(MJ).
Similarly to our previous cases, we conclude that (Du(X)
- Dv(Y)I
I CM,
2 0
7 1
(63’~-1’2/16)Rj,
and therefore
“4cr3ti’4 I CMo(;)u’4k+4. 0
It follows that there are positive constants o and E with E < o such that, for X = (x0, to) and Yin Q’(R), IDu(X)
- Du( Y) I CM&-’
O.
(2.6)
513
G.
M.
LIEBERMAN
We are now ready to prove our estimates using an interpolation argument in the same spirit as [20]. Let r > 0 and t, = 0, set d(X) = y irrro p(X, Y) and d(X, Y) = min(d(X), d(Y)), and E introduce the weighted seminorms
14; = ;yp, dP4, I
I40 = Q”u,“,14,
r
[4r+o =
d(X,
sup
Y)‘+“~Du(X)
- Du(Y)I/fX
-
YI”.
X. YinQ+(r)
By an easy variant of [7, (6.86)], we therefore have [u]? 5 C(m, n, a)(lul,)“‘(l+u’([u]:+,
+ ]f.&)“(‘+?
(2.7)
Moreover, a simple modification of the proof of (2.6) shows that it remains valid for any X = (x, t) and Y = (JJ, s) in Q+(r) such that t 2 sand IX - Y] d !hd(X, Y) I R. By choosing X and Y so that [UK+, 5 2d(X, and R = %d(X,
Y)‘+“jDu(X)
Y), we see from (2.6) that [u]:,,
- Du(Y)I/jX
-
YI”,
I C([u]: + l)r+“-‘; then (2.7) gives
I@+, 5 c’h4cl+ 1)
(2.8)
for some constant C’ which depends also on ]ulO. Clearly the proof of (2.8) carries over to the case that F has the form F(r) = f(~ + r) for some function f also satisfying (1.1) and some K E (0, 1) if we define Q’, Q”, and F* with f replacing F. From (2.8) we now derive an estimate in terms of an integral norm of Du in the special case that F(r) = (K + T)~ for some constant m > - 1. For this purpose, another set of weighted norms is required. With IX - Y] = max(Ix - y I, It - sI”(~+~)) as per our remarks, we define q(X,P)=(Y=(y,s)EQ+(r):s
YI
for w E L’(q), and we introduce the Campanato
type seminorm l/(m+Z)
(WI X;R
sup
=
TX
O
s 4(X,
(
XeQ+(R)
Iw
-
iqx,
flp+2
>
v
for A > 0 and recall from [l] that there are positive constants C, and C2 for which tWh;R
1
cl X*
lWh;R
5
sup
YinQ+(R/Z)
c2 X*
sup YinQ+(R)
Iw(m- w(Y)I Ix-
Y]”
’
Iww> - w(Y)I Ix-
Y]”
for 1 E (n + m + 2, n + 2(m + 2)) and 0 = (A - (n + m + 2))/(m We now define ( w);Z = o~;lr (r - R)X’(mc2)(~)~;Rr
Ilwll, = (1
+ 2).
Iw,,+y(m+2J Q + VI
+ 1.
Degenerate parabolic equations It
515
therefore follows from (2.8) that @Ix* 5 C’(PJ,*
+ IINO)
for ,I = n + m + 2 + a(m + 2) and any,u E (n + m + 2,1). Because of the obvious interpolation inequality between the weighted seminorms, we have PJK,,
5 C’
ll~4lo9
(2.9)
which is the desired estimate. 3. PROOF
OF THEOREM
1 WHEN
m r0
We now prove theorem 1 when m L 0. Our proof is a simple variant of the perturbation argument of Giaquinta and Giusti [6] as modified to parabolic conormal problems in [14]. We begin with some lemmata related to the maximum principle. The first lemma gives a precise form of the maximum bound for solutions of parabolic differential inequalities. Although the basic result is certainly well-known, we know of no place where the precise dependence is written down, so we present it here although [16, lemma 2.11 gives the case q = 00. LEMMA
3.1. Suppose that u is a weak solution of the differential
inequalities
- cru, + div A(X, Du) L b in Q’(R), A”(X, Du) 2 0 on Q’(R),
(3.la)
u I 0 on Q*(R),
(3.lb)
and suppose that A satisfies the structure condition A(X,p)
‘p L jp/m+2 - d;+2
(3.2)
for some constant a, 2 0, that OL2 a,
(3.3)
b E L*(Q+)
(3.4)
for some positive constant a,, and that
for some q > 1 + n/(m + 2). Then u I C(a,, q, n, m) (u~R)‘+““~ + Rm+’ (R -(n+m+2) I,+ by*].
(3.5)
[ Proof. The proof is a simple modification of the Moser iteration method, used in Section 1. We include only enough detail to verify the specific form of the estimate, which will be important later. Let us set M
=
(a,R)l+m/2
bq)l/Q,
+ Rm+z(R-(n+“+2)
Q' m+2 K=l+-,
n
171, =
q2h (q-
- 1) l)K--q’
(1 - qh m2=(q-
l)K--q’
516
G. M. LIEBERMAN
and observe that m, > 0 > m, . For r > 0 at our disposal, we take as test function in the weak form of (3.1) the function (0 =
r+m2
v”’
M'
( > 1
_
_
v +’
In this way we see that max -Rm+2
( -
Vml”m2+l(l s
_
!!):”
+
I,+
Vml’-m,-l(l
_
~)3Du,“i2
B+ cRlx(t)
Cr2MR
-m-2+(n+m+2)/q [S
Q+
We now set
and note that
It follows from the Sobolev inequality (in the form [16, (1.8)] with K, = 0) and the iteration scheme of lemma 1.1 that v I Mmax
2, C R-“-m-2M-q I
j,+
vy’m’].
(3.6)
(
The inner integral is estimated by using the test function a,
=
(vq-1 - Mq-‘)+.
We now find, after an application of Young’s inequality, that vq I CMqR” + CR -m-2 1B+
Vq.
j
x(f)
B+x(-Rm+*,f)
Gronwall’s inequality yields
vq 5 CM’lR”+m+2. Q' The proof is completed by combining (3.6) and (3.7).
(3.7)
n
Our next lemma is a version of Gilding’s estimate [8] relating the time and space moduli of continuity of solutions of parabolic equations. LEMMA 3.2.Let u. > 1, p. - .u3, A, A, R be positive constants. Suppose that A is weakly differentiable with respect to p and that 44 @(X, Z, P)CiCj2 n(K +
2 PO,
IPI)“ICI’
(3.8) for all c E R”,
(3.9a)
Degenerate
parabolic
517
equations
(3.9b) (3.9c) (3.10) (3.11a) for some nonnegative function b E Lq with (3.11b) If u is a continuous
function with bounded spatial gradient which satisfies CYU~ = div A(X, U, Du) + B(X, u, Du) in Q’(R), on Q”W,
A” (X, U, DU) + q/(X, U) = 0 osc UIUo-
1,
SUP
Q+(R)
(3.12a) (3.12b)
IDUI5Lc3,
(3.12~)
Q+(R)
then osc
u 5 C(L A, cc,, m, n, u,)U
+ iu2 +
(3.13)
p3)R.
Q + (R/2)
Proof. Clearly it suffices to estimate the oscillation of u(xo, *) over an interval of length proportional to Rmc2. Fix t, E (to - R”‘+’ , to), and for c a constant to be chosen, set
p = [cA(to - f,)Ur/p0]“(m+2),
s =
sup
MO
I,
90 - Nxo, t,11,
and assume without loss of generality that s > 0 and p I ARAW. Define [Ix]] = (A’]x’ 12/64A2 + ]x”]~)“~, and introduce the sets N+ , No by replacing Ix - x0 I with [Ix - x0 II in the definition of Q’ and Q”, respectively. We now consider the function w(X) =
2,U,(m + 1) + CAS t - to + U~‘(m+%r’(m+r) [Ix - x0 112 + m+21 P P2 > PO
+ [ I”r(;+ By choosing c appropriately
l)]“‘m+‘)+4u~‘“id)sl/(m+)](p_i+*/d). we can make
- OIW,+ div A(x,, t, u(xo, t), Dw) I 0 t, MO, 0, Dw) + w(x0, 0 5 0
A?,,
in N+, on No,
and hence if 1 b”
=
#(x0,
f =‘M,
t, u(x”,
t),
ODU + (1 - a)Dv)da,
u, Du) - Nxo,
t, MO, 0, Du),
rl(3P
G. M.
518
LIEBERMAN
we find that u = u - u(x,,, tt ) - w is a weak solution of - CtU,+
Di(b'jDjU
+
b”jDjV + f” 2 0 on No,
f')
L
in NC,
b
v~Oon6N~W~.
We now apply lemma 3.1 with a, = C(1 + ,D~)and A4 = C[(l + P~)R]‘+~‘~; note that the use of N+ in place of Q’ does not affect the proof beyond a slight adjustment of constants. Therefore evaluation of (3.5) at x = x0 gives u(x,,
t) -
u(X,)
5
c(t
-
to)
+
f
+ p3p
+
C[(l
+ /f2 + /f3)p]i+m’2.
Taking the supremum over t and using the inequality &x0, t) - u&X,) I u. gives the desired result. W From this point the proof of theorem 1 is an easy combination of the proofs of the corresponding results in [14] and [15]. We fix a point X0 E SQ and use an ti2-t-@) flattening of the boundary to assume that u satisfies in Q’(2R),
ol(x)u, = div A(X, U, Du) + B(X, u, Du) A”(X, u, Du) + t//(X, u) = 0
(3.14a)
on Q0(2R)
(3.14b)
for some positive constant R and functions A, B, and IC/satisfying the hypotheses of theorem 1 and Q satisfying IDal I C(Y)‘-‘,
a zruo,
(3.15)
Ml3 5 C
for some nonnegative constant C(n, m, Q) and positive constant po. For M’ = max sup JDiUl i
Q’(2R)
(recall that Du is assumed bounded), lemma 3.2 implies that osc u I CM’R.
(3.16)
Q+(R)
We observe that u is also a solution of a boundary value problem of the form U, = div A(X, u, Du) + B(X, u, Du) A”(X,
u, Du) + v(X,
u) = 0
in Q’(2R), on Q0(2R)
(3.14a)’ (3.14b)’
with A and w satisfying the hypotheses of theorem 1 and B satisfying ]B(x, GP)] 5 A~]P]“‘+~ -+ b(X) + A,(1 + ]p()m+‘(~)S-l. (Note that the As and vvs in (3.14) and (3.14)’ are not the same!) If u is the solution of the boundary value problem V, = div A(X,, u(X,), Du)
in
Q'(R),
A”(&, Go), Du) + w(Xo, 4X0)) = 0 on Q’(R),
u = u on Q*(R)
(3.17)
Degenerate
parabolic
519
equations
with A and 9 from (3.14) (given in Section 5), using 9 = u - u in the integral forms of the equations for u and u gives n j B + CR)x {toI
9* +
=
j Q+(R)
m
D9
. LJW,,
* MKJ,
W,),
UGG),
Du)
-
Du)
-
AW,
A(&,
WG),
WI
u, Du)l
i Q+(R)
B w, u,
+
Du)rp
(3.18)
+
s Q+(R)
The left hand side of this equation is no smaller than n /I ID$Op+*. j Q+(R) To estimate the right hand side of (3.18), we first use a simple variant of lemma 3.1 to infer that sup ]U - VI I Cmin((1 + M’)R, 1). Q+(R)
On the other hand because of the modulus of continuity estimate of DiBenedetto conclude that
[2] for u, we
sup ]U - u] + osc u I min(C&P + s(M’R)“, C)
(3.19)
Q’(R)
Q+(R)
for all sufficiently small E and 6 (specifically any 6 < 1). The right hand side of (3.18) is then estimated almost exactly as in [14, Section 41. First
I
I-
for o
c /3/(m
j,+,,
+
[fw + A,(@
3
+ &(M’R)6)@]
s
A 4 Q+(R) ID91”+*
+ C-@,frR)(l+@(m+*)]R”
+ [Cp+“)(m+*)
1). The estimate of the integral involving B is slightly more complicated:
B9 5 jQ+,,, b9 + A, jQ+,R, 9lD@+* I (j,+,,
]D91(1 + ]D#‘+’
Q+(R)
b’)1’9( j,+,.,
99/(9-‘y9
+ CA2(1 + IW)~+‘(R
+ A~ j,+,,
+
[C&P
+
(l + ]DU]),+1(Y)8-19
CE(MfR~G](Ml)m+*Rn+m+z
+ IWR)R”+~+‘+“.
We now estmate the integral involving 9 via (3.19) to conclude that
c
JQ+W
B9
I
[C&(l+@(m+*)
+ ,-E(j$f’R)(l+‘d(‘“+*)]R”
G. M. LIEBERMAN
520
for o sufficiently small depending on q. Finally, we infer from Green’s theorem and Young’s inequality that
QO(R)lv(X, u) - VW,, ma)lp
&M’R]”
3 144
I
C[C&R +
I
C[C&A”+ &(M’R)6]o
Q'(R)
IDd
Q+(R) A I-
Q+cR,IDp]m+2
4
+ CE(M)R)(l+o)(m+2)]Rn.
[c$?('+'-+*)
+
We therefore obtain
J
ID(u
Q+(R)
- u)l”‘+’
(C,
5
+
E(Ml)(m+2)(l+o))Rn+(m+2)(l+a).
It now follows from (2.9) that [Dvl I CM’ in Q’(R/2) and the considerations of Section 2 imply that
(3.20)
and then the proof of [15, theorem l]
blT+o 5 C([u]f + 1). Just as before, this estimate proves theorem 1. 4. PROOF
OF THEOREM
1 WHEN
m < 0
The proof of theorem 1 for m < 0 is very similar to the proof in the previous section so we only indicate the significant changes that are made. We note first that lemmata 3.1 and 3.2 require only one change in this case: because now m c 0, the conclusion of lemma 3.2 becomes Qyc)
C[(l + ru2 + ,u~)R]~+~‘~.
u I
(4.1)
Also in our flattening of the boundary, we only use (3.15) and the last two inequalities in (3.16). Then we take u to be the solution of a(xo)u, = div A(&,
u(&),
Du)
in
Q’(R),
A’VG, uWd, Du) + u/(&, uCW) = 0 on Q”W, If we abbreviate S = B+(R) j$xo)~~ =
+ jQ+cR@
1
x (to), we obtain in place of (3.18)
* VW,,
uWo1, DuJ - AWo,
Do, * IAWo, u(Xo), W
Q+(R)
+
s
Q+(R)
u = u on Q*(R).
BW, u, Du)cp+
- A@‘, u, WI
s Q’(R)
unbox, WI
l~(x
U) - U/(X,, U(X,))IV +
s
sla(xo) - &P.
(4.2)
Degenerate
parabolic
521
equations
The left hand side of this equation is estimated from below by
and using Holder’s inequality to relate the second integral to 3
J
1Dpl”+?
Q+(R)
The first two integrals on the right hand side of (4.2) are estimated as before. For the last, we assume without loss of generality that r&Y,,) = 0, so that
s
[01(x()) -
a]ucpI
CY(X&*+
1%
s
CR*O
s i u* s CR”(IWR)~+*
for some C depending on the maxigum of u. By using the smallness of A, in place of the modulus of continuity estimate for u, which at present is not known for this situation, we again infer (3.20) and hence (0.5). 5. AN
EXISTENCE
THEOREM
We now show that the boundary value problem u, = div A(Du) in Q’(R),
A”(Du)
-t @ = 0 on Q’(R),
u = u on Q*(R)
(5.1)
is solvable in a suitable space for continuous u provided we make one additional hypothesis beyond those in Sections 1 and 2. When F has the form F(r) = (K + T)~, a simple variant of [9, theorem V.6.71 shows that (5.1) has a weak solution only for m > max[ 1, -4/(n + 2)). For this reason, and also to demonstrate the power of using our general Fs, we use a different approach which does not use the existence of weak solutions. We write H* for the space of uniformly continuous functions with finite norm IG+,
=
Ido+ [ulr+,,
and we seek solutions of (5.1) in this space. The idea is to approximate this problem by a family of problems which are known to have solutions and then take advantage of certain uniform estimates of the approximate solutions. To this end we now state our additional hypothesis: that there is a constant a! E (0, 1) for which s”F(.s) I a”F(a)
for 0 < s I 0.
(5.2)
Under this assumption (see [18, Section 5]), there is a locally Lipschitz function P on II?, such that c&s) 5 F(s) I P(s), SF’(S) s (1 - a)F(s) for all s > 0. For A, 1 2A/(l - 01)a constant to be further determined, we observe that
we now set A(p) = ~,$(lpl)p,
and
G. M. LIEBERMAN
522
Now we introduce a parameter A and define a function fx as follows: I A(s) =
1
ifsr1,
e2A(x-s)
Ik
if A < s I 21,
21e-‘*‘(31
- s)
if 1. < s I 3A,
0
ifs > 31.
If we now set
AA(P)= fA(lPIMP) + (1 - A(lPl))4P), we see that -i
f$ ti
Then, by choosing A sufficiently large (independent of A), we see that A, satisfies the hypotheses of Section 2 with ci and F independent of 1 and hence all the estimates of that section are valid for the approximating problems. Of course, we still do not know that these approximating problems have solutions, so we move to a further approximation. We set F,* = sup F(s) + A, O
F,(s) = min(F,*, F(s)) if IpI 5 4A if IpI > 4A.
It then follows from [12, theorem 31 (and [14, lemma 4.11) that u, = div A,(Du) in Q’(R),
A”(u) + @ = 0 on Q’(R),
u = u on Q*(R)
(5.1)’
has an H* solution for each i > 0, and the results of Section 2 give a uniform estimate on the H* norms of these solutions. To see that the solutions have a uniform limit as i. tends to infinity, we need a uniform modulus of continuity estimate. (The limit is in H* and satisfies (5.1) by the H* estimate.) If u is in C”, this estimate follows from the considerations in [13, Section 21 and the gradient bound from Section 2 at any point X0 in the closure of Q + unless x0” = 0 -6 > 0, $ = 0,
and
to = 0, and
1x01< R, 1x01 = R,
and
or to = 0,
(5.3a) or
(5.3b)
to > 0.
(5.3c)
In all cases, the modulus of continuity can be estimated by a simple variant of the estimates in [12]. In case (5.3a), we set w(x, t) = fl - (101 + &,)P + &2(t + Ix - x0/)* + U(X,) and note that -w, + divA,(Dw)
I Oin Q+,
A;(Dw) I @on Q”,
w > UonQ*
(5.4)
for suitable constants E, E, , and a2. It follows that w > u in Q ‘, which gives a one-sided modulus estimate; the estimate on the other side is proved similarly.
Degenerate
parabolic
523
equations
In case (5.3b), we see that W = t” + u(x,, 0) + Du(x,, 0) * (x - xg) + &,(t + lx-$ - J! I)’ + &22(t + Ix - x, I)2, for suitable positive E, ci, and e2, satisfies (5.4), which leads to a modulus of continuity estimate in this case. In case (5.3c), [12, lemma 21 provides a function w,(x) and positive constants E and C such that, for any matrix (a”) and vector p for which I I (a”) 5 N,
p” 1 1,
we have U’~,jWi I -IX
-
xo Ice2
IPI 5 & in B +,
/? *Dw, I -Ix - x0 I’-’ on B”, Ix - x0 I&I WI 5 clx - x0 I&,
IDW,~ s C~X - xOla-l in B+.
Then w(X) = It - to I2 + El w,(x) + u(X,) satisfies (5.4), so a modulus of continuity estimate follows in this The proof of existence (for fixed A) follows by approximating u tions and seeing that the corresponding us converge uniformly. If and u2 with Iu, - u2 I < E for some E > 0, the maximum principle and hence if the USconverge uniformly, so do the us.
case also. uniformly by smooth funcvi and u2 correspond to ui gives Ju, - u2 I 5 E in Q’,
REFERENCES I. DAPRATO G., Spazi Co’,*) (a, 6) e loro propierta, Annali Mat. pura appl. 69, 383-392 (1965). 2. DIBENEDETTO E., On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients, Annali Scu. norm sup. Pisa (4) 13, 487-535 (1986). 3. D~BENEDETTO E. & FRIEDMAN A., Regularity of solutions of nonlinear degenerate parabolic systems, J. Reine
Angew. Math. 349, 83-128 (1984). 4. DIBENEDETTO E. & FRIEDMAN A., Holder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Mafh. 357, l-22 (1985). 5. DIBENEDETTO E. & FRIEDMAN A., Addendum to “Holder estimates for nonlinear degenerate parabolic systems”, J. Reine Angew. Math. 363, 217-220 (1985). 6. GIAQUINTA M. & GIUSTI E., Global C”” -regularity for second order quasilinear elliptic equations in divergence form, J. Reine Angew. Math. 351, 55-65 (1984). 7. GILBARG D. & TRUDINGER N. S., Elliptic Partial Differential Equations of Second Order, 2nd edition. Springer, Berlin (1983). 8. GILDING B. H., Holder continuity of solutions of parabolic equations, J. London Mafh. Sot. 13, 103-106 (1976). 9. LADYZENSKAJA 0. A., SOLONNIKOV V. A. & URAL’CEVA N. N., Linear and Quasilinear Equations of Parabolic Type. Am. Math. Sot., Providence, R.I. (1968). 10. LADYZHENSKAYA 0. A. & URAL’TSEVA N. N., Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968).
11. LIEBERMAN G. M., The conormal derivative problem for elliptic equations of variational type, J. dvf. Eqns 49, 219-257 (1983). 12. LIEBERMAN G. M., Mixed boundary value problems for elliptic and parabolic differential equations of second order, J. Math. Analysis Applic. 113, 422-440 (1986). 13. L~EBERMANG. M., The first initial-boundary value problem for quasilinear second order parabolic equations, Annali Scu. norm. sup Piss (4) 13, 347-387 (1986). 14. L~EBERMANG. M., Hilder continuity of the gradient of solutions of uniformly parabolic equations with conormal boundary conditions, Annali mat. pura appl. 148, 77-99, 397-398 (1987). 15. LIEBERMAN G. M., Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Analysis 12, 1203-1219 (1988). 16. L~EBERMANG. M., The conormal derivative problem for non-uniformly parabolic equations, Indiana Univ. Math. J. 37, 23-72 (1988).
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G. M. LIEBERMAN
17. LIEBERMAN G. M., Initial regularity for solutions of degenerate parabolic equations, Nonlinear Analysis 14, 525-536 (1990). 18. SPERNER, Jr E., Schauder’s existence theorem for cr-Dini continuous data, Ark. Mar. 19, 193-213 (1981). 19. TRUDINGER N. S., Pointwise estimates and quasilinear parabolic equations, Communs. pure appl. Math. 21, 205-226 (1968). 20. TRUDINGER N. S., On an interpolation inequality and its applications to nonlinear elliptic equations, Proc. Am. math. Sot. 95, 73-78 (1985). 21. WIEGNER M., On C,-regularity of the gradient of solutions of degenerate parabolic systems, Annali Mat. pura appl. 145, 385-406 (1986). Added in proof-If A is independent of have shown that u is Holder continuous.
(Archs ration. Mech. Analysis 103, 319-245; 1988) A, in theorem 1 need not be restricted in this case.
t, Chen and DiBenedetto Hence the constant