Annales de l'Institut Henri Poincare (C) Non Linear Analysis Volume 22 Issue 6 2005 [Doi 10.1016_j.anihpc.2004.10.011] Frank Duzaar; Giuseppe Mingione -- Second Order Parabolic Systems,

Ann. I. H. Poincar AN 22 (2005) 705751www.elsevier.com/locate/anihpc

Second order parabolic systems, optimal regularity,and singular sets of solutions

Systmes paraboliques de la deuxime ordre, regularit optimale,et des ensembles singuliers des solutions

Frank Duzaar a,, Giuseppe Mingione b

a Mathematisches Institut der Friedrich-Alexander-Universitt, zu Nrnberg-Erlangen, Bismarckstr. 1 1/2, 91054 Erlangen, Germanyb Dipartimento di Matematica, Universit di Parma, Via DAzeglio 85/A, 43100 Parma, Italy

Received 14 September 2004; accepted 28 October 2004

Available online 10 May 2005

Abstract

We present a new, complete approach to the partial regularity of solutions to non-linear, second order parabolic systems ofthe form

ut divA(x, t, u,Du) = 0.In the first part we introduce the A-caloric approximation lemma, a parabolic analogue of the harmonic approximation lemmaof De Giorgi [Sem. Scuola Normale Superiore Pisa (19601961); Lectures in Math., ETH Zrich, Birkhuser, Basel, 1996] inthe version of Simon. This allows to prove optimal partial regularity results for solutions in an elementary way, under minimaland natural assumptions. In the second part we provide estimates for the parabolic Hausdorff dimension of the singular sets ofsolutions; the proof makes use of parabolic fractional Sobolev spaces. 2005 Elsevier SAS. All rights reserved.Rsum

Nous prsentons une nouvelle approche complte auprs de la regularit partielle des solutions des systmes paraboliques,non-linaires, de la deuxime ordre de la forme

ut divA(x, t, u,Du) = 0.Dans une premire partie nous introduisons le lemme dapproximation A-calorique, un analogue parabolique du lemme dap-proximation harmonique de De Giorgi [Sem. Scuola Normale Superiore Pisa (19601961) ; Lectures in Math., ETH Zrich,

* Corresponding author.E-mail addresses: [email protected] (F. Duzaar), [email protected] (G. Mingione).0294-1449/$ see front matter 2005 Elsevier SAS. All rights reserved.doi:10.1016/j.anihpc.2004.10.011

706 F. Duzaar, G. Mingione / Ann. I. H. Poincar AN 22 (2005) 705751

Birkhuser, Basel, 1996] daprs la version Simon. Cela permet de prouver des rsultats optimales de la regularit partielle pourdes solutions dune faon elementaire sous des hypothses minimales et naturelles. Dans la deuxime partie nous donnons desestimations des ensembles singuliers des solutions pour la dimension parabolique de Hausdorff ; la preuve se sert des espacesparaboliques fractionnaires de Sobolev. 2005 Elsevier SAS. All rights reserved.

Keywords: Partial regularity; Parabolic systems; Singular sets

MSC: 35K55; 35D10

1. Introduction and results

In this paper we are concerned with the study of regularity properties of solutions to non-linear, second-order,parabolic systems of the type

ut divA(x, t, u,Du) = 0, (x, t) (T ,0) QT , (1.1)where Rn is a bounded domain and T > 0; for precise notation we refer to the next section.

Our aim is twofold. First we want to explain a new method to prove partial regularity of solutions, that willavoid to assume additional structural assumptions on the system and a priori additional regularity on the solutions.With this new method it is no longer necessary to use various involved tools as Reverse-Hlder inequalities and(parabolic) Gehrings lemma. The method is based on an approximation result that we called the A-caloric approx-imation lemma, which is explained in Section 3, below. This is the parabolic analogue of the classical harmonicapproximation lemma of De Giorgi [7,41] and allows to approximate functions with solutions to parabolic systemswith constant coefficients in the same way as the classical harmonic approximation lemma does with harmonicfunctions. More precisely, in the case of the classical heat system we have (with B Rn denoting the unit ball andQ := B (1,0))

Lemma 1.1 (caloric approximation lemma). For every > 0 there exists (n, ) (0,1) with the following prop-erty: if u L2(1,0;W 1,2(B,RN)) with

Q(|u|2 + |Du|2)dz 1 is approximatively caloric in the sense that

Q

(ut Du D)dz sup

Q

|D| for all C0 (Q,RN)

then there exists a caloric function h L2(1,0;W 1,2(B,RN)), i.e. ht h = 0 in Q, such thatQ

(|h|2 + |Dh|2)dz 1 and Q

|u h|2 dz .

This lemma and its variant in Section 3 allow to prove partial regularity properties of solutions to non-linearparabolic systems by linearization arguments (see Sections 6 and 7) in a particular efficient and elementary way.In the elliptic setting the possibility of using the harmonic approximation lemma and its non-isotropic variant,the A-harmonic approximation lemma, was already exploited both in the setting of Geometric Measure Theoryto prove optimal regularity results for minimizing currents to elliptic integrands by Duzaar and Steffen [15] andto prove regularity results for non-linear elliptic systems by Duzaar and Grotowski [11]. In the last case such amethod allowed to get an elementary proof of the known regularity results for elliptic systems. These techniquesfind their origins in Simons approach, via De Giorgis harmonic approximation, to Allards regularity theorem and

to the regularity of harmonic maps [40,41]. Later this has been generalized to the degenerate elliptic problems bythe authors [12,13].

F. Duzaar, G. Mingione / Ann. I. H. Poincar AN 22 (2005) 705751 707

In the present paper the use of Lemma 1.1, combined with new ad hoc arguments, allow to prove optimalregularity results for solutions that was not possible to obtain before; it is actually our second aim to provide acomplete study of regularity properties of weak solutions: optimal partial regularity exponents and estimates forthe dimension of the singular sets. Indeed, the first regularity result of the paper is the following:

Theorem 1.2. Let u L2(T ,0;W 1,2(,RN)) be a weak solution to the system (1.1) under the assumptions(2.1)(2.3) and (2.5) and denote by Q0 the set of regular points of u in QT :

Q0 :={z QT : Du C,/2(A,RnN), A( QT ) is a neighborhood of z

}.

Then Q0 is an open subset with full measure and thereforeDu C,/2(Q0,RnN), |QT \Q0| = 0.

See Section 8 for the proof; of course by |S| we denote the usual Lebesgue measure of a set S Rn+1; moreover,C,/2(A) is the space of functions which are Hlder continuous with exponent with respect to space variable xand with exponent /2 with respect to the time variable t ; in other words they are Hlder continuous with exponent in the parabolic metric in Rn+1 given by

distp((x, t), (x0, t0)

) :=|x x0|2 + |t t0|, x, x0 Rn, t, t0 R. (1.2)As far as we are aware, Theorem 1.2 was not known under the general, minimal assumptions considered here.Partial regularity of solutions has been proven for quasi-linear systems [44,17,18,2,25], for non-linear systemswith special structure [32] or in low dimensions [37,22,23,34,36], and for non-linear systems only assuming thatsolutions were a priori more regular [47,29] i.e. bounded or even Hlder continuous; everywhere regularity ispossible only under very special (diagonal type) structures, as for instance in the case of the p-Laplacian system[9,33], otherwise it fails in general, as shown by counterexamples [45,42,21] and already in the case of ellipticsystems [8,20,46]. On the other hand, recently, non-a-priori regular solutions have been considered, but againassuming more regularity of A with respect to the coefficients (x, t) and, in particular, no dependence on thevariable u [38,1]; the methods of this last paper are suited for systems with growth conditions more general thanthe one treated here, but again, they are not suitable to treat the low regularity assumptions we consider. In anycase, the optimal regularity stating that the Hlder exponent of the spatial gradient is exactly the same one of thecoefficients was never achieved, even under extra assumptions, due to the different techniques available before thispaper. We stress here the fact that the use of the A-caloric approximation lemma, proved in Section 3, allows toget a completely elementary proof of Theorem 1.2, without the use of ReverseHlder inequalities; this will be animportant point in a forthcoming paper [14], where together with K. Steffen we are going to treat systems withsuper-linear growth. For such systems higher integrability of solutions has been recently proved by Kinnunen andLewis [24], and later refined by Misawa [32], but their proof does not yield a ReverseHlder inequality since thescaling for parabolic systems of the type in (1.1) with non-linear growth is non-isotropic; therefore the applicationto partial regularity seems to be not immediately possible, while a further variant of our method will allow to dealwith such systems too.

Theorem 1.2 immediately poses a natural problem. Let us call the set of non-regular points the singular set ofthe solution u:

:= QT \Q0.The natural question is now: how large can be? This question can be answered considering the so called parabolicHausdorff measure in Rn+1, that is the canonical Hausdorff measure constructed in Rn+1 with respect to theparabolic metric from (1.2). Here we shall consider its cylindrical variant, that leads to slightly stronger estimates.To be precise, let us denote, x0 Rn and t0 RB(x0,R) :={x Rn: |x x0|


Then we define for s [0, n+ 2] and F Rn+1

Ps (F ) := inf{

i=1Rsi : F

i=1

Q((xi, ti),Ri

), Ri

}, Ps(F ) := sup

>0Ps (F ).

The parabolic Hausdorff dimension is then usually defined according to

dimP (F ) := inf{s > 0: Ps(F ) = 0

}= sup{s > 0: Ps(F ) = }.Let us observe that due to the stretching in the time direction of the cubes, the limit dimension is n+2: dimP (F )n+ 2 for every F Rn+1, while Pn+2 is comparable to the Lebesgue measure in Rn+1.

Once again, estimates for the singular set have been obtained in very particular situations and when the systemin (1.1) shows a simpler structure: ut divA(Du) = 0. In this case it is possible to prove that dimP () n [5].Actually the problem of proving Hausdorff dimension estimates for systems including Hlder coefficients remainedopen for a long time already in the elliptic case: divA(x,u,Du) = 0 (see the open problems in [16], page 191). Ithas been finally settled in [31], where it is shown, among other things, that the Hausdorff dimension of solutions togeneral non-linear elliptic systems in Rn is always strictly less than n. Here we shall derive the parabolic analogueof this result, using a difference quotient technique in the setting of parabolic fractional Sobolev spaces; in any casewe treat systems with uniformly continuous coefficients, which is a quite standard assumption for partial regularity.Our first result in this direction is the following theorem, whose proof will be presented in Section 9:

Theorem 1.3. Let u L2(T ,0;W 1,2(,RN)) be a weak solution to the system (1.1) under the assumptions(2.1)(2.3) and (2.9) and denote by the singular set of u. Then there exists a number (,L/) > 0 suchthat

dimP () n+ 2 .

The dependence of upon the regularity of the coefficients and the ellipticity ratio L/ is critical in the sensethat

lim0 = 0 and limL/ = 0. (1.3)

The presence of the small but quantifiable number (see Remark 9.6 below), rather than a more consistentquantity, is due to the fact that the vector field A explicitly depends on the function u(x, t), which is a priori onlymeasurable; this yields a strong lack of smoothness for the function

(x, t) A(x, t, u(x, t), ),which prevents the singular set reduction. Indeed, when no dependence on u takes place, the previous result can besubstantially improved in the following, which extends previous elliptic results [30]:

Theorem 1.4. Let u L2(T ,0;W 1,2(,RN)) be a weak solution to the systemut divA(x, t,Du) = 0, (x, t) (T ,0) QT , (1.4)

under the assumptions (2.1)(2.3) and (2.9) and denote by the singular set of u. Then there exists a number (,L/) > 0 such that

dimP () n+ 2 2 . (1.5)

The last result, whose proof is in Section 10, shows that, independently of the ellipticity ratio L/, the singular

set dimension depends in a sensitive way on the regularity of the coefficients when considering systems of the typein (1.4). Note that the bound (1.5) is in some sense natural: in the differentiable case = 1, we find dimP () < n


which agrees (and actually improves) some known estimates for the case ut divA(Du) = 0 [5]. We remark thatTheorems 1.3 and 1.4 are the only results where we shall use in a crucial way the a priori higher integrabilityproperties of solutions via the parabolic Gehrings lemma [24].

Finally we point out that most of the previous results, and in particular Theorem 1.2, take place for more general,non-homogeneous systems of the type

ut divA(x, t, u,Du) = B(x, t, u,Du), (x, t) (T ,0) QT ,where the vector field b exhibits critical growth conditionsB(x, t, u,Du) L(1 + |Du|2),and the solution is assumed to be bounded with u satisfying a suitable smallness assumption of the type

2Lu 0. Here we specify the exact assumptions we are going to consider on the parabolic systems. Weshall distinguish the assumptions for partial regularity, Theorem 1.2, from the ones for the singular set estimates,Theorems 1.3, 1.4.

Assumptions for Theorem 1.2. We shall consider a vector field A :QT RN RnN RnN . If z = (x, t), u RNand p RnN we shall denote the coefficients by A(z,u,p) = A(x, t, u,p). We assume that the functions

(z, u,p) A(z,u,p), (z, u,p) Ap

(z,u,p)

are continuous in QT RN RnN and that the following growth and ellipticity conditions are satisfied:A(z,u,p) L(1 + |p|), (2.1)Ap (z,u,p) L, (2.2)

A

p(z,u,p)p p |p|2, (2.3)

for all z QT , u Rn and p, p RnN where > 0 and 1 L < ; actually, up to enlarging the constant L,(2.1) is a consequence of (2.2); we reported both of them for future convenience. Now we shall specify the regularityassumptions on A(x, t, u,p) with respect to the coefficients (z, u); we shall assume that the function

(z, u) A(z,u,p)1 + |p| (2.4)

is Hlder continuous with respect to the parabolic metric (1.2) with Hlder exponent (0,1), but not necessarilyuniformly Hlder continuous; namely we shall assume thatA(z,u,p)A(z0, u0,p) L(|u| + |u0|, |x x0| +|t t0| + |u u0|)(1 + |p|) (2.5)for any z = (x, t) and z0 = (x0, t0) in QT , u and u0 in Rn and for all p RnN where(y, s) := min{1, K(y)s}


and K : [0,) [1,) is a given non-decreasing function. Note that is concave in the second argument. Thisis the standard way to prescribe (non-uniform) Hlder continuity of the function in (2.4). We find it a bit difficultto handle, therefore, in many points of the paper, we shall useA(z,u,p)A(z0, u0,p)K(|u|)(|x x0| +|t t0| + |u u0|)(1 + |p|) (2.6)valid for any z = (x, t) and z0 = (x0, t0) in QT , u and u0 in Rn and for all p RnN where (0,1) andK : [0,) [L,) is a given non-decreasing function. We note that (2.6) is weaker than (2.5).

Finally we remark a trivial consequence of the continuity of A/p; this implies the existence of a function : [0,) [0,) [0,) with (t,0) = 0 for all t such that t (t, s) is nondecreasing for fixed s, s (t, s)2 is concave and nondecreasing for fixed t , and such thatAp (x, t, u,p) Ap (x0, t0, u0,p0)

(M, |x x0|2 + |t t0| + |u u0|2 + |p p0|2) (2.7)for any z = (x, t) and z0 = (x0, t0) in QT , any u, u0 in Rn and p, p0 RnN whenever |u| + |p| + |u u0| +|p p0|M .

Remark 2.1. In the case of systems of the type (1.4) both (2.5) and (2.6) must be replaced byA(z,p)A(z0,p) L(|x x0| +|t t0| )(1 + |p|), (2.8)while, in order to apply (2.7), we just need to require that |p| + |p p0|M .

Additional assumptions for Theorems 1.3 and 1.4. In order to prove Theorems 1.3 and 1.4 we shall be forcedto consider systems with uniformly Hlder continuous coefficients: we shall consider the following reinforcementof (2.6):A(z,u,p)A(z0, u0,p) L(|x x0| +|t t0| + |u u0|)(1 + |p|) (2.9)where : [0,) [0,1] is a continuous concave function such that

(s) s, s > 0.

Finally we recall that a weak solution to the system (1.1) is a function u L2(T ,0;W 1,2(,RN)) such thatQT

(ut A(z,u,Du)D

)dz = 0, for all C0 (QT ,RN). (2.10)

3. Preliminaries

If v is an integrable function in Q(z0, ) Q(z0) = B(x0) (t0 2, t0), z0 = (x0, t0), we will denote itsaverage by

(v)z0, :=

Q(z0)

v dz := 1nn+2

Q(z0)

v dz,

where n denotes the volume of the unit ball in Rn. We remark that in the following, when not crucial, the centerof the cylinder will be often unspecified e.g. Q(z0) Q; the same convention will be adopted for balls in Rn

thereby denoting B(x0, ) B(x0). Finally in the rest of the paper the symbol c will denote a positive, finiteconstant that may vary from line to line; the relevant dependencies will be specified.


For u L2(Q(z0),RN) we denote by z0, the unique affine function (in space) (z) = (x) minimizing

Q(z0)

|u |2 dz

amongst all affine functions a(z) = a(x) which are independent of t . To get an explicit formula for z0, we notethat such a unique minimum point exists and takes the form

z0,(x) = z0, + z0,(x x0),where z0, RnN . A straightforward computation yields that

Q(z0)

u a(x)dz =

Q(z0)

z0,(x) a(x)dz

for any affine function a(x) = + (x x0) with RN and RnN . This implies in particular that

z0, =

Q(z0)

udz = (u)z0, and z0, =n+ 2

2

Q(z0)

u (x x0)dz.

For convenience of the reader we recall from [26] the following:

Lemma 3.1. Let u L2(Q(z0),RN), 0 < < 1, and z0, respectively z0, the unique affine functions mini-mizing

Q(z0)|u |2 dz respectively

Q(z0)|u |2 dz. Then there holds

|z0, z0,|2 n(n+ 2)()2

Q(z0)

u (u)z0, z0,(x x0)2 dz.Moreover, if Du L2(Q(z0),RnN) we havez0, (Du)z0,2 n(n+ 2)2

Q(z0)

u (u)z0, (Du)z0,(x x0)2 dz.The following lemma is a (parabolic variant) of a well known measure theoretical result; the proof can be

obtained along the lines of [19], Chapter 3 and [30], Section 4.

Lemma 3.2. Let :B1 Bn R+ be a bounded, increasing set-function (here B1, Bn denote the families of Borelsubsets of (T ,0) and , respectively), which also satisfies

iN(QRi (zi)

)

( (T ,0)),

whenever {QRi (zi)}iN is a family of pairwise disjoint parabolic cylinders in (T ,0). If we letA :=

{z0 (T ,0): lim inf

0 s

(Q(z0)

)> 0

}, 0 < s n+ 2,

thendimP (A) s.


Now we recall the definition of the parabolic fractional Sobolev spaces for which we refer to [27]. We shallsay that a function u L2(QT ,Rk) belongs to the fractional Sobolev space W,;2(QT ,Rk), , (0,1), k N,provided

0T

|u(x, t) u(y, t)|2|x y|n+2 dx dy dt +

0T

0T

|u(x, t) u(x, s)|2|t s|1+2 dt ds dx =: [u],;QT < .

The local variant W,;2loc (QT ,Rk) can be defined in the usual way; it is also possible to define spacesW,;2(QT ,Rk) for higher values of and ; for this we refer to [27]. The following Poincar type inequalitycan then be obtained in a standard way (for instance imitating the proof of Proposition 3.1 from [10]):

Q(z0)

u(z) (u)z0,2 dz c(n)2 [u],/2;Q(z0), (0,1), (3.1)for any function u W,/2;2(Q,Rk).

Proposition 3.3. Let u W,/2;2loc (QT ,Rk) for > 0 and let

A :={z0 QT : lim inf

0

Q(z0)

u (u)z0,2 dz > 0},B :=

{z0 QT : lim sup

0

(u)z0,= }.Then

dimP (A) n+ 2 2, dimP (B) n+ 2 2.

The proof can be obtained along the lines of the analogue, classical results for standard fractional Sobolev func-tions (see for instance [10,30] for elementary proofs). We briefly sketch the main arguments, confining ourselvesto the estimate for A. By a standard localization argument [30] we may suppose without loss of generality thatu W,/2;2(QT ,Rk); then by (3.1) we have that A S where

S :={z0 QT : lim inf

0 (n+2)+2 [u],/2;Q(z0) > 0

}.

Therefore, applying Lemma 3.2 we have dimP (A) dimP (S) n + 2 2; we observe that the application ofLemma 3.2 is possible since the set function

: I I

|u(x, t) u(y, t)|2|x y|n+2 dx dy dt +

I

I

|u(x, t) u(x, s)|2|t s|1+ dt ds dx,

with I (T ,0) and , Borel sets, clearly meets all the requirements of Lemma 3.2.Finally we conclude with the parabolic version of the well known relation between Nikolski spaces and Frac-

tional Sobolev spaces; the proof can be obtained by a straightforward adaptation of the standard elliptic result[30,10].

Proposition 3.4. Let u L2(QT ,Rk). Suppose that 2 2

Q

u(x, t + h) u(x, t) dx dt c1|h| , (0,1),


where Q := (T +d,d) and , for every h R such that |h|min{d,A}, where d (0, T /8), A> 0are positive constants. Then there exists a constant c1 c1(, , d,A, c1,uL2(QT )) > 0 such that

dT+d

dT+d

|u(x, t) u(x, s)|2|t s|1+2 dt ds dx c1, (0, ).

Suppose thatQ

u(x + hes, t) u(x, t)2 dx dt c2|h|2 , (0,1),for every |h|min{dist(, ),A}, s {1, . . . , n}, where {es}1sn is the standard basis of Rn. Then for everyO there exists a constant c2 c2(n, , ,A, c2,dist(, ),dist(O, ),uL2(QT )) > 0 such that

dT+d

O

O

|u(x, t) u(y, t)|2|x y|n+2 dx dy dt c2, (0, ).

4. A-caloric approximation

We recall that a strongly elliptic bilinear form A on RnN with ellipticity constant > 0 and upper bound > 0means that

|p|2 A(p, p), A(p, p)|p| |p| p, p RnN .We shall say that a function h L2(1,0;W 1,2(B,RN)) is A-caloric on Q if it satisfies

Q

(ht A(Dh,D)

)dz = 0 for all C0 (Q,RN).

Obviously, when A(p, p) |p|2 for every p RnN , then an A-caloric function is just a caloric functionht h = 0,

and therefore Lemma 1.1 is just a particular case of the following:

Lemma 4.1 (A-caloric approximation lemma). There exists a positive function (n,N,,, ) 1 with the fol-lowing property: Whenever A is a bilinear form on RnN which is strongly elliptic with ellipticity constant > 0and upper bound , is a positive number, and u L2(1,0;W 1,2(B,RN)) with

Q

(|u|2 + |Du|2)dz 1is approximatively A-caloric in the sense that

Q

(ut A(Du,D)

)dz sup

Q

|D| for all C0 (Q,RN)

then there exists an A-caloric function h such that (|h|2 + |Dh|2)dz 1 and |u h|2 dz .

Q Q


Proof. Were the assertion false, we could find > 0, a sequence (Ak) of bilinear forms on RnN , with uni-form ellipticity bound > 0 and uniform upper bound , and a sequence of functions (vk)kN with vk L2(1,0;W 1,2(B,RN)), such that

Q

(|vk|2 + |Dvk|2)dz 1 (4.1)and

Q

(vkt Ak(Dvk,D)

)dz 1k supQ |D| (4.2)

for all C10(B,RN) and k N, butQ

|vk h|2 dz > for all h Hk , (4.3)

where here

Hk ={f L2(1,0;W 1,2(B,RN)): f is an Ak-caloric function on Q,

Q

(|f |2 + |Df |2)dz 1}.Passing to a subsequence (also labeled with k) we obtain the existence of v L2(1,0;W 1,2(B,RN)) and A suchthat there holdsvk v weakly in L

2(Q,RN),Dvk Dv weakly in L2(Q,RnN),Ak A as bilinear forms on RnN .

(4.4)

Using the lower semicontinuity of v Q(|v|2 + |Dv|2)dz with respect to weak convergence in L2(1,0;

W 1,2(B,RN)) we obtainQ

(|v|2 + |Dv|2 dz) 1. (4.5)Moreover, for C0 (Q,RN) we have

Q

(vt A(Dv,D)

)dz

=Q

((v vk)t A(Dv Dvk,D)

)dz

Q

(AAk)(Dvk,D)dz+Q

(vkt Ak(Dvk,D)

)dz.

Passing to the limit k we see that the first term of the right-hand side converges to 0 due to (4.4); the sameholds for the second term in view of the uniform bound of Dvk in L2(Q,RnN) (see (4.1)) and the convergence ofthe Aks; the third term vanishes in the limit k via (4.2). This shows that the weak limit v is an A-caloricfunction on Q, i.e. ( ) NQ

vt A(Dv,D) dz = 0, C0 (Q,R ).


Now, in order to get compactness in L2(Q,RN), i.e. vk v in L2(Q,RN), we estimate the time derivatives of vk .For this we let C0 (Q,RN) and compute (z = (x, t) and Q = B (1,0)):

Q

vkt dz

0

1

B

Ak(Dvk,D)dx dt

+ 1k sup1t0D(, t)L(B) |Ak|

01

Dvk(, t)L2(B)D(, t)L2(B) dt + 1k sup1t0D(, t)L(B) |Ak|

( 01

Dvk(, t)2L2(B) dt)1/2( 0

1

D(, t)2L2(B) dt

)1/2+ 1

ksup

1t0

D(, t)L(B)

|Ak|0

1

D(, t)2L2(B) dt +

1k

sup1t0

DL(B). (4.6)

Here we have used in turn (4.2), the CauchySchwartz inequality and (4.1). Now, for 1 < s1 < s2 < 0 and > 0small enough we choose

(t) =

0, for 1 t s1 ,1(t s1 + ) for s1 t s1,

1 for s1 t s2, 1

(t s2 ) for s2 t s2 + ,

0 for s2 + t 1,and let (x, t) = (t)(x) for C0 (B,RN). Testing (4.6) with we obtain

B

(1

s1s1

vk(x, t)dt 1

s2+s2

vk(x, t)dt)

(x)dx

|Ak|( 0

1(t)

2 dt)1/2

DL2(B) +1kDL(B) sup

1t0(t)

(|Ak|

s2 s1 + 2 + 1

k

)DL(B).

By Sobolev-embedding

DL(B) c(n, )W,20 (B), >n+ 2

2,

we see thatB

(1

s1s1

vk(x, t)dt 1

s2+s2

vk(x, t)dt)

(x)dx( ) c(n, ) |Ak|s2 s1 + 2 + 1

k

W,20 (B)

.


Passing to the limit 0 we obtain for a.e. 1 < s1 < s2 < 0 B

(vk(, s2) vk(, s1)

) dx c(n, )(|Ak|s2 s1 + 1k)

W,20 (B)

for any C0 (B,RN). By density of C0 (B,RN) in W,20 (B,RN) the last inequality is also valid for any W,20 (B,RN). Taking the supremum over all W,20 (B,RN) with W,20 (B) 1 we infervk(, s2) vk(, s1)W,2(B,RN) c(,n)(|Ak|s2 s1 + 1k

). (4.7)

Interpolating L2(B,RN) between W 1,2(B,RN) and W,2(B,RN) it follows for > 0 that

h1

vk(, t + h) vk(, t)2L2(B) dt

h1

vk(, t + h) vk(, t)2W 1,2(B) dt + c()h

1

vk(, t + h) vk(, t)2W,2(B) dt 4

01

vk(, t)2W 1,2(B) dt + c()c2(|Ak|h+ 1k)2 4+ 2c()c2

(|Ak|2h+ 1

k2

). (4.8)

Here, we have used in the first line the interpolation inequality

w2L2(B) w2W 1,2(B) + c()w2W,2(B)

valid for w W 1,2(B,RN). Moreover, in the second-last line we have used the bound (4.7) for vk(, t + h) vk(, t)W,2(B) from above and the bound (4.1), i.e.

Q(|vk|2 + |Dvk|2)dz 1.

We are now in the position to show that

limh0

h1

vk(, t + h) vk(, t)2L2(B) dt = 0 uniformly in k. (4.9)In order to do this we recall that Ak A as k so that supk1 |Ak| a < . Using this in (4.8) we obtain

h1

vk(, t + h) vk(, t)2L2(B) dt 4+ 2c()c2(a2h+ 1k2).

For given > 0 we choose = 12 . This fixes and also c() = c( 112). Next we choose k0 N such that2c()c2

k2< 3 for any k k0. Then, for k = 1, . . . , k0 1 we choose h1 > 0 such that

h 2

1

vk(, t + h) vk(, t)L2(B) dt < 0 < h< h1, k = 1, . . . , k0 1.


Finally, we choose h2 > 0 such that 2c()c2a2h < 3 for any 0 < h < h2. Then, for any k N and 0 < h < h0 :=min(h1, h2) we have

h1

vk(, t + h) vk(, t)2L2(B) dt < which proves (4.9).

Since the sequence (vk)kN is also bounded in L2(1,0;W 1,2(B,RN)) we are able to apply Theorem 3 of [39]with the choice X = W 1,2(B,RN), B = L2(B,RN), F = (vk)kN to obtain a subsequence (vk)kN (again labeledby k) such that

vk v strongly in L2(Q,RN) = L2(1,0;L2(B,RN)).

To obtain the desired contradiction we denote by wk :Q RN a solution to the following initial-Dirichlet problemand possessing the properties below; its existence can be deduced from standard existence arguments [27,28].wk C

([1,0];L2(B,RN))L2(1,0;W 1,20 (B,RN)),twk L2

(1,0;W1,2(B,RN)),wk(,1) = 0;

Q

(wkt Ak(Dwk,D)

)dz =

Q

(AAk)(Dv,D)dz C0 (Q,RN); (4.10)

12wk(, t)2L2(B) +

B(1,t)Ak(Dwk,Dwk)dz

=

B(1,t)(Ak A)(Dv,Dwk)dz for a.e. t (1,0).

Using the ellipticity of the bilinear forms Ak we see that the second term of the left-hand side of (4.10) is boundedfrom below by

B(1,t) |Dwk|2 dz. Moreover the right-hand side of (4.10) is estimated easily by the use of

CauchySchwarz inequality, the boundQ

|Dv|2 dz 1 from (4.5), and Youngs inequality

B(1,t)(Ak A)(Dv,Dwk)dz |AAk|

(Q

|Dv|2 dz)1/2(

B(1,t)|Dwk|2 dz

)1/2

2|AAk|2 + 2

B(1,t)

|Dwk|2 dz.

This implies in particular

12

B

wk(, t)2 dx + 2

B(1,t)|Dwk|2 dz 2

|Ak A|2 for a.e. t [1,0] and k N.

Taking the supremum over t (1,0) we arrive at1 2 2sup

t(1,0) 2B

wk(, t) dx + 2Q

|Dwk| dz 0 as k . (4.11)


Letting gk := v wk L2(1,0,W 1,2(B,RN)) we easily see that gk agrees with v on the parabolic boundarypQ of Q and satisfies

Q

gkt Ak(Dgk,D)dz = 0, C0 (Q,RN).

From (4.11) and the definition of gk we see thatQ

(|gk v|2 + |Dgk Dv|2)dz 0 as k ,which implies in particular that

Q

(|gk|2 + |Dgk|2)dz Q

(|v|2 + |Dv|2)dz 1 as k .Letting

bk := max{

1,Q

(|gk|2 + |Dgk|2)dz}, gk := gkbk

we see that bk 1 andQ(|gk|2 + |Dgk|2)dz 1 for any k N. Note that gk Hk . Furthermore,(

Q

|gk v|2 dz)1/2

(

Q

|gk v|2 dz)1/2

+(

1 1bk

)(Q

|gk|2 dz)1/2

0 as k ,

which yields the desired contradiction to (4.3). Remark 4.2. From (4.7) we infer

h1

vk(, t + h) vk(, t)W,2(B) dt c(ah+ 1k)

which yields, reasoning as aboveh

1

vk(, t + h) vk(, t)W,2(B) dt 0 uniformly in k N as h 0.Therefore we could have had directly applied Theorem 5 of [39] with the choice X = W 1,2(B,RN), B =L2(B,RN), Y = W,2(B,RN), F = (vk)kN, p = 2 to conclude that (vk)kN is relatively compact in L2(Q,RN)= L2(1,0;L2(B,RN)).

Lemma 4.3. There exists a positive function (n,N,,, ) 1 with the following property: Whenever A is abilinear form on RnN which is strongly elliptic with ellipticity constant > 0 and upper bound , is a positivenumber, and u L2(t0 2, t0;W 1,2(B(x0),RN)) with

2

2

2

Q(z0)

|u| dz+ Q(z0)

|Du| dz 1


is approximatively A-caloric in the sense that Q(z0)

(ut A(Du,D)

)dz sup

Q(z0)|D| for all C0

(Q(z0),RN

)then there exists h L2(t0 2, t0;W 1,2(B(x0),RN)) A-caloric on Q(z0) such that

2

Q(z0)

|h|2 dz+

Q(z0)

|Dh|2 dz 1 and 2

Q(z0)

|u h|2 dz . (4.12)

Proof. For a general parabolic cylinder Q(z0, ) = B(x0) (t0 2, t0) we can apply Lemma 4.1 (in a suitableaveraged version) to the rescaled function v(x, t) := 1u(x0 + x, t0 + 2t) defined on Q to obtain the existenceof an A-caloric H :Q RN satisfying (4.12) on Q Q. Rescaling via h(z) := H(1(x x0), 2(t t0))yields the desired result.

5. The Caccioppoli inequality

From the definitions given in Section 2 we setH(s) = K(s)(1 + s). (5.1)

Now we are ready to derive the following Caccioppoli inequality which slightly differs from the usual ones in thatit shows the correct dependence on H(M), a fact that will be needed later.

Lemma 5.1. Let u L2(T ,0;W 1,2(,RN)) be a weak solution to (1.1) under the assumptions (2.1)(2.3).Then, for any M > 0, any affine function (z) (x) independent of t and satisfying |(z0)| + |D|M , and anyQ(z0)QT with 1 we have

Q/2(z0)

|DuD|2 dz cCacc(

Q(z0)

u 2 dz+ 2

), (5.2)

where the constant cCacc depends only on , L and H(M).

Proof. The following calculations will be a bit sloppy. To proceed in a rigorous way, one should use a smoothingprocedure in time via a family of non-negative mollifying functions or via Steklov averages. Since this is a standardargument and yields only technical minor changes we shall proceed formally.

In (2.10) we take the test-function = 2 2(u ), where C10(B(x0)) is a cut-off function in space suchthat 0 1, 1 in B/2(x0) and |D| 41 while C1(R) is a cut-off function in time such that, with0 < < 2/4 being arbitrary

1, on(t0

2

4, t0 2

),

0, on (, t0 2) (t0,),0 1, on R,

t 0, on(t0

2

4,

),( 2) |t | 3

2, on t0 2, t0 4 .


Inserting in (2.10) we obtainQ(z0)

A(z,u,Du)D(u ) 22 dz

= 2

Q(z0)

A(z,u,Du) 2 (u )dz+

Q(z0)

ut dz. (5.3)

We further have

Q(z0)

A(z,u,D)D(u ) 22 dz = 2

Q(z0)

A(z,u,D) 2 (u )dz

Q(z0)

A(z,u,D)D dz

and

0 =

Q(z0)

A(z0, (z0),D)D dz = 0.

Adding the last and second-last equation to (5.3) and using also that t 0 we deduceQ(z0)

(A(z,u,Du)A(z,u,D))D(u ) 22 dz

= 2

Q(z0)

(A(z,u,Du)A(z,u,D)) 2 (u )dz

Q(z0)

(A(z,u,D)A(z, (z),D))D dz

Q(z0)

(A(z, (z),D

)A(z0, (z0),D))D dz+ Q(z0)

(u )t dz

=: I + II + III + IV. (5.4)Estimate for I : The Lipschitz bound |A(z,u,p)A(z,u, p)| L|p p| for p, p RnN and an application of

Youngs inequality yield (for 0


IV =

Q(z0)

(u )t dz =

Q(z0)

|u |22t 2 dz+ 12

Q(z0)

22t |u |2 dz

= 12

Q(z0)

|u |22t 2 dz =

Q(z0)

|u |22 t dz.

Taking into account that t 0 for t > t0 2/4 and that |t | 32 we infer

IV 3

Q(z0)

u 2 dz. (5.6)

Estimate for II: Using the assumption imposed for the modulus of continuity of (z, u) A(z,u,p), i.e. (2.6),we see that

|II|H(M)

Q(z0)

|u | |D|dz II1 + II2

where

II1 := H(M)

Q(z0)

|u | 22D(u )dz,II2 := 2H(M)

Q(z0)

|u | 2|||u |dz.

To estimate II1 we use Youngs inequality twice

II1

Q(z0)

2 2D(u )2 dz+ 1

H(M)22

Q(z0)

u 2 dz

Q(z0)

2D(u )2 dz+ 1

H(M)2

(Q(z0)2/(1) + Q(z0)

u 2 dz). (5.7)

Arguing similarly we obtain

II2 8H(M)(Q(z0)2/(1) +

Q(z0)

u 2 dz). (5.8)

Estimate for III: To estimate III we proceed as follows: Using (2.6) again we see thatIII 2H(M)1+

Q(z0)

|D|dz III1 + III2

where

III1 := 2H(M)1+

Q(z0)

22D(u )dz,

1+

2III2 := 4H(M) Q(z0)

|||u |dz.


Now, using Youngs inequality twice again, we have

III1 + III2

Q(z0)

2 2D(u )2 dz+ c

H(M)2(1+)

(Q(z0)2 + Q(z0)

u 2 dz). (5.9)

Combining (5.5), (5.7)(5.9), (5.6) with (5.4) and using the fact that 2/(1) 2 for 0 < 1 we arrive atQ(z0)

(A(z,u,Du)A(z,u,D))D(u ) 22 dz

3

Q(z0)

|DuD|2 22 dz+ c(L,H(M))

( Q(z0)

u 2 dz+ Q(z0)2).

We next estimate the integral on the left-hand side of the previous inequality using the ellipticity condition for thecoefficients A, which in turns implies strict monotonicity, i.e. (A(z,u,p) A(z,u, p))(p p) |p p|2 forany p, p RnN . We therefore obtain

( 3)

Q(z0)

|DuD|22 2 dz c(L,H(M))

( Q(z0)

u 2 dz+ Q(z0)2).

Choosing small enough, i.e. = min(1, /6), and taking into account that 1 for t [t0 2/4, t0 2], that 1 on B

2(x0) we infer that

t02t02/4

B/2(x0)

|DuD|2 dz c(,L,H(M))( Q(z0)

u 2 dz+ Q(z0)2).

Now, the desired Caccioppoli inequality follows by taking the limit 0. Remark 5.2. Remark 2.1 and a careful inspection of the previous proof, reveal that in the case of systems of thetype (1.4) the term II drops out while the estimate for III simplifies. Therefore the only condition we have to takeon (z) is that |D|M .

6. Linearization

The next inequality will later allow us to apply the A-caloric approximation lemma.

Lemma 6.1. Let u L2(T ,0;W 1,2(,RN)) be a weak solution to (1.1) under the assumptions (2.1)(2.3)and (2.9). Then for any M > 0 we have

Q(z0)

((u )t A

p

(z0, (z0),D

)(DuD)D

)dz

cEu((M + 1,2)

2 +2 +2 +

)sup

Q(z0)|D|,

Nfor any Q(z0) QT and C0 (Q(z0),R ) with 1 and any affine function (z) = (x) independent oftime, satisfying |(z0)| + |D|M . Here cEu c(H(M),L) (the function H is defined in (5.1)). We write


2 2(z0, ,D) :=

Q(z0)

|DuD|2 dz,

2 2(z0, , ) :=

Q(z0)

u 2 dz.

Proof. Without loss of generality we can assume that supQ(z0) |D| 1. Using (2.10), the fact that

Q(z0)

A(z0, (z0),D

)D dz = 0,

and Q(z0)

t dz = 0 we deduce

Q(z0)

((u )t A

p

(z0, (z0),D

)(DuD)D

)dz = I + II + III,

where we have abbreviated

I :=

Q(z0)

(A(z0, (z0),Du

) Ap

(z0, (z0),D

)(DuD)D

)d dz,

II :=

Q(z0)

(A(z,u,Du)A(z, ,Du))D dz,

III :=

Q(z0)

(A(z, ,Du)A(z0, (z0),Du))D dz.

In turn we split the first integral as follows

I = 1|Q(z0)|S1

( )dz+ 1|Q(z0)|S2

( )dz =: IV + V

and

S1 := Q(z0){z: |DuD| 1}, S2 := Q(z0) {z: |DuD| > 1}.

We proceed estimating the two resulting pieces. As for IV we write

IV = 1|Q(z0)|S1

[ 10

[A

p

(z0, (z0),D+(DuD)

) Ap

(z0, (z0),D

)](DuD)D

]d dz.

We estimate IV using the modulus of continuity (, ) for (z, u,p) Ap

(z,u,p) from (2.7), the fact that s 2(t, s) is concave, and Jensens inequality (note that, by assumption, on S1 we have |(z0)|+ |D|+ |DuD|M + 1): ( 2)|IV|

Q(z0)

M + 1, |DuD| |DuD|dz


(

Q(z0)

2(M + 1, |DuD|2)dz)1/2(

Q(z0)

|DuD|2 dz)1/2

(M + 1,

Q(z0)

|DuD|2 dz)(

Q(z0)

|DuD|2 dz)1/2

.

To estimate V we preliminarily observe that, using Hlders inequality

|S2|S2

|DuD|dz|S2|Q(z0)( Q(z0)

|DuD|2 dz)1/2

,

and therefore|S2||Q(z0)|

(

Q(z0)

|DuD|2 dz)1/2

.

Using (2.1), (2.2) and the previous inequality we then conclude the estimate of V as follows

|V | 2L|Q(z0)|S2

1 + |D| + |DuD|dz 2L|Q(z0)|S2

1 +M + |DuD|dz

2L(M + 1) |S2||Q(z0)| + 2L|S2||Q(z0)|

(

Q(z0)

|DuD|2 dz)1/2

4L(M + 1)

Q(z0)

|DuD|2 dz.

Combining the estimates found for IV and V we have

|I | (M + 1,2)2 + 4L(M + 1)2.

For the remaining pieces, using the modulus of continuity (z, u) A(z,u,p) from (2.6), we deduce

|II|K((z0)+ |D|) Q(z0)

|u |(1 + |D| + |DuD|)dzH(M)

Q(z0)

u dz+H(M)

Q(z0)

u |DuD|dz

2H(M)[

Q(z0)

|DuD|2 dz+

Q(z0)

u 2 dz+ ].

Here we have used that 1 K(M) H(M) and the assumption that 1. Using again (2.6) and Youngs in-equality we estimate

( )( ) ( )|III| 2 K (z0) 1 + |D|

Q(z0)

1 + |D| + |DuD| dz


2H(M)2 + 2H(M)(

2 +

Q(z0)

|DuD|2 dz)

3H(M)2(

+

Q(z0)

|DuD|2 dz).

Combining the estimates just found for I , II and III, we obtain Q(z0)

((u )t A

p

(z0, (z0),D

)(DuD)D

)dz

c(H(M),L

)[(M + 1,2)

2 +2 +2 +

].

A simple scaling argument yields the result for general . Remark 6.2. The same observations on the affine function (z) in Remark 5.2 apply to the previous proof; there-fore, in the case of systems (1.4), once again we may assume only that |D|M .

The next lemma is a standard estimate for weak solutions to linear parabolic systems with constant coeffi-cients [4], Lemma 5.1.

Lemma 6.3. Let h L2(t0 2, t0;W 1,2(B(x0),RN)) be a weak solution in Q(z0) = B(x0) (t0 2, t0) ofthe following linear parabolic system with constant coefficients:

Q(z0)

(ht A(Dh,D)

)dz = 0, C0

(Q(z0),R

N),

where the coefficients A satisfyA(p,p) |p|2, A(p, p) L|p||p|,

for any p , p RnN . Then h is smooth in Q(z0) and there exists a constant cpa = cpa(n,N,L/) 1, such that (z0, ) cPa 2 (z0, ) 0 < < 1.

Here we write for 0 < (z0, ) = 1

2

Q (z0)

h (h)z0, (Dh)z0, (x x0)2 dz.

7. Regular points

In this section we consider a weak solution u of the nonlinear parabolic system (1.1) on a fixed sub-cylinderQ(z0)QT , under the assumptions described in Section 2. In the following we shall always consider 1.

Let M > 1 be given. We first want to apply Lemma 6.1 on Q/2(z0) tov := u ,

where (z) = (x) is an affine function independent of t satisfying |(z0)| + |D| M . We observe that 2 hasthe following property: u 2 n+4 u 2 n+42(z0, /2, ) =

Q/2(z0)

/2 dz 2 Q(z0)

dz = 2 2(z0, , ). (7.1)


From Caccioppolis inequality (5.2), we infer

2(z0, /2,D) cCacc(

Q(z0)

u 2 dz+ 2)= cCacc(2(z0, , )+ 2)= cCacc2(z0, , ), (7.2)

where we have abbreviated

2 2(z0, , ) := 2(z0, , )+ 2.From Lemma 6.1 we therefore get for any C0 (Q/2(z0),RN) (note also that (M + 1, cs) c(M + 1, s)for c 1, since s (M + 1, s) is concave)

Q/2(z0)

(vt A

p

(z0, (z0),D

)DvD

)dz

c1[(M + 1, 2(z0, , )

)2(z0, , )+ 2(z0, , )+

]sup

Q/2(z0)|D|, (7.3)

where c1 = c1(,L,,H(M)).For given > 0 (to be specified later) we let = (n,N,,L, ) (0,1] be the constant from Lemma 4.3. We

define

:= 4c12(z0, , )+ 22 and w := 1v = 1(u ).

Then, from (7.3) we deduce that for all C0 (Q/2(z0),RN) there holds

Q/2(z0)

(wt A

p

(z0, (z0),D

)DwD

)dz

14[(M + 1, 2(z0, , )

)+2(z0, , )+ ] supQ/2(z0)

|D|

[2(M + 1, 2(z0, , )

)+ 2(z0, , )+ 122]1/2

supQ/2(z0)

|D|. (7.4)

Moreover, we estimate using Caccioppolis inequality (7.2) and (7.1)(

2

)2

Q/2(z0)

|w|2 dz+

Q/2(z0)

|Dw|2 dz 2n+4 + cCacc

16c21 1, (7.5)

provided we have chosen c1 1 large enough. We further setA(p, p) := A

p

(z0, (z0),D

)(p, p) p, p RnN .

From (2.2) and (2.3) we see that the bilinear form A satisfies the following conditions:|p|2 A(p, p), A(p, p) L|p||p| p, p RnN ,

i.e. the bilinear form A fulfills the assumptions of Lemma 4.3. Therefore (7.4) and (7.5) allow us to applyLemma 4.3 to w, A on Q/2(z0). Assuming the smallness condition2(M + 1, 2(z0, , )

)+ 2(z0, , ) 122


the application of Lemma 4.3 yields the existence of h L2(t0 2/4, t0;W 1,2(B/2(x0),RN) solving the A-heatequation on Q/2(z0) and satisfying(

2

)2

Q/2(z0)

|h|2 dz+

Q/2(z0)

|Dh|2 dz 1 (7.6)

and (

2

)2

Q/2(z0)

|w h|2 dz . (7.7)

From Lemma 6.3 we recall that h satisfies for any 0


Now, we choose = n+6. Then (7.8) yields (z0, /2, z0,/2) 2c2 2

( (z0, , )+ 22

).

Given < < 1 we choose 0 < < 1 such that

21+2c2 2 2,

i.e. = (n,N,,L,,,H(M)). This also fixes the constants = (n,N,,L,,,H(M)) and = (n,N,,L,,,H(M)) (0,1]. Thus we have shown

Lemma 7.1. Given M > 0 and < < 1 there exist (0, 12 ) and (0,1] depending only on n, N , , L, , and H(M) such that if

2(M + 1, 2(z0, , z0,)

)+ 2(z0, , z0,) 122on Q(z0)QT for some 0 < 1 and such ifz0,(z0)+ |Dz0,|M, (7.9)then

2(z0, , z0,) 22(z0, , z0,)+ c32,where c3 := 1 + 2.

Remark 7.2. Keeping into account the content of Remarks 5.2 and 6.2 we have that in the case of systems of thetype (1.4) the condition in (7.9) can be relaxed to |Dz0,|M .

We now want to iterate Lemma 7.1; in the following, for fixed z0 we shall denote z0, . For given M > 1(and < < 1) we determine = (2M), = (2M) and c3 = c3(2M) according to Lemma 7.1. Then we canfind 0(M) > 0 sufficiently small, such that

2(2M,20(M)

)+ 20(M) 122 (7.10)and

0(M)1

4(n+ 2)2 M2n+4(1 )2. (7.11)

Given this we can also find 0(M) (0,1] so small that, writing

c4(M) := c3(2M)2 2 ,

we have

c4(M)0(M)2 min

{2, 0(M),

14(n+ 2)2 M

2n+4(1 )2}. (7.12)

Now, suppose that the conditions

(i) |(z0)| + |D|M ,

(ii) 0(M),

(iii) 2() 0(M)


are satisfied on Q(z0)QT . Then, for j = 1,2,3, . . . we shall show

(I)j 2(j) 2j 2()+ c4(M)(j)2 ,(II)j |j(z0)| + |Dj| 2M .

Note first that (I)j combined with (ii), (iii) and (7.12) yields

(I)j 2(j) 20(M).

We now proceed by induction. We first consider the case j = 1. From (iii), (7.10) and the monotonicity of weinfer

2(M,2()

)+ 2() 2(2M,0(M))+ 20(M) 122.Moreover, we have 0(M) 1 and |(z0)| + |D| M . Therefore we can apply Lemma 7.1 to concludethat (I)1 holds. Furthermore, using Lemma 3.1, (iii) and (7.11) we deduce(z0)+ |D| (z0)+ |D| + (z0) (z0)+ |D D|

M + Q(z0)

(u (u)

)dz+ [n(n+ 2)()2

Q(z0)

|u |2 dz]1/2

M +[

Q(z0)

|u |2 dz]1/2

+[n(n+ 2)()2

Q(z0)

|u |2 dz]1/2

M +[22

n+42()

]1/2+[n(n+ 2)n+4

2()

]1/2M + 1 +

n(n+ 2)n+4

0(M)1/2

M + n+ 2n+4

0(M)1/2 2M,

i.e. (II)1 holds. We now assume (I)m and (II)m (and therefore (I)m) for m = 1, . . . , j 1. Then, (I)m, (II)mand (7.10) imply that we can apply Lemma 7.1 for m = 1, . . . , j 1. Recalling the definition of c4(M) we calculate

2(j) 2j2()+ c3(2M)(j)22

j1m=0

2()m 2j2()+ c3(2M)2 2 (

j)2

= 2j2()+ c4(M)(j)2,showing (I)j . To show (II)j we estimatej (z0)+ |Dj|

(z0)+ |D| + j

m=1

m(z0) m1(z0)+ jm=1

|Dm Dm1|

M +j

m=1

[

Qm(z0)

|u m1|2 dz]1/2

+j

m=1

[n(n+ 2)(m)2

Qm(z0)

|u m1|2 dz]1/2

1 + n(n+ 2) j n+ 2 j1M + n+4 m=1

2(m1)1/2 M +

n+4 m=02m2()+ c4(M)(m)2


M + (n+ 2)n+4

(2()

1 +c4(M)2

1 )M + M

2+ M

2= 2M.

Here we have used in turn Lemma 3.1, the definition of 2(m1), (I)m for m = 1, . . . , j 1, (7.11) and (7.12).The above reasoning proves the first assertion of the following

Lemma 7.3. For M > 1 and Q(z0)QT , suppose that the conditions

(i) |(z0)| + |D|M ,(ii) 0(M),

(iii) 2() 0(M),

are satisfied. Then for every j N we have2(

j) 2j 2()+ c4(M)(j)2 andj(z0)+ |Dj| 2M.

Moreover, the limit

z0 := limj(Du)z0,j /2

exists, and the estimate

Qr(z0)

|Du z0 |2 dz c[(

r

/2

)22()+ r2

]

is valid for 0 < r /2 for a constant c = c(n,N,,,L,,M).

Proof. Since |Dj| 2M we are in a position to apply Lemma 5.1. We obtain

2(j/2, (Du)j/2

)2

(j/2, (D)j

) cCacc(2M)2(j)

cCacc(2M)(2j 2()+ c4(M)(j)2

). (7.13)

We now consider 0 < r /2. We fix k N {0} with k+1/2 < r k/2. Then the previous estimate implies

2(r, (Du)r

)= Qr(z0)

Du (Du)r 2 dz n2 Q

k/2(z0)

Du (Du)k/22 dz n2cCacc(2M)

[2k2()+ c4(M)(k)2

] n2cCacc(2M)

[2

(r

/2

)22()+ c4(M)222r2

] 4n22cCacc(2M)

[(r

/2

)22()+

(c4(M)+ 1

)r2

] cdec(M)

[(r

/2

)22()+ r2

].Next, we show that ((Du)j /2)jN is a Cauchy sequence in RnN . For k > j we deduce


(Du)j /2 (Du)k/2 km=j+1

(Du)m/2 (Du)m1/2n2

k1m=j

[

Qm/2(z0)

Du (Du)m/22 dz]1/2

=n2

k1m=j

2(m/2)

n2cCacc(2M)

k1m=j

2m2()+ c4(M)(m)2

n2cCacc(2M)

(2()

1 j +

c4(M)2

1 j

).

This proves the claim. Therefore the limit z0 = limj(Du)j /2 RnN exists and from the previous estimatewe infer (taking the limit k )(Du)j /2 z0 c5(M)2j 2()+ (j)2.Combining this with (7.13) we arrive at

Qj /2(z0)

|Du z0 |2 dz 22(j/2)+ 2(Du)j /2 z0 2 c(M)(2j 2()+ (j)2).

For 0 < r /2 we find k N {0} with k+1/2 < r k/2. Then the previous estimate implies

Qr(z0)

|Du z0 |2 dz n2

Qk/2(z0)

|Du z0 |2 dz n2c(M)[2k2()+ (k)2

]

c(M)[(

r

/2

)22()+ r2

].

This proves the assertion of the lemma. Remark 7.4. Using Remark 7.2, a careful reading of the proof above yields that condition (i) in Lemma 7.3 can berelaxed to |D|M .

An immediate consequence of the previous lemma and of the isomorphism theorem of CampanatoDa Prato [6]is the first regularity result of the paper:

Theorem 7.5 (Description of regular points). Let u L2(T ,0;W 1,2(,RN)) be a weak solution to the system(1.1) under the assumptions (2.1)(2.3) and (2.6) and denote by the singular set of u (as explained in Section 1).Then

0 2,

where


0 :={z0 QT : lim inf

0 2

Q(z0)

u (u)z0, (Du)z0,(x x0)2 dz > 0},2 :=

{z0 QT : lim sup

0((u)z0,+ (Du)z0,)= }.

Remark 7.6. Remark 7.4 gives that for systems of the type (1.4) the set 2 above can be replaced by the smaller2 :=

{z0 QT : lim sup

0(Du)z0,= }.

Now it happens that while |2| = 0 and |2| = 0 by Lebesgue theory, we cannot a-priori assert the sameabout 0, since the function u is not assumed to be differentiable with respect to time. Therefore the previous resultsonly characterize the regular points but is not a partial regularity result in the sense that it does not immediatelyimply that | | = 0. For this, we still need another section.

8. Partial regularity

In this section we finally prove Theorem 1.2; this is a consequence of the following:

Theorem 8.1 (Almost everywhere regularity). Let u L2(T ,0;W 1,2(,RN)) be a weak solution to the system(1.1) under the assumptions (2.1)(2.3) and (2.5) and denote by the singular set of u; then

1 2,where 2 is as in Theorem 7.5 and


0

Q(z0)

Du (Du)z0,2 dz > 0} {z0 QT : lim inf

0

Q(z0)

u (u)z0,2 dz > 0}.Proof. We start taking a point z0 (x0, t0) QT such that

lim inf

0

Q(z0)

Du (Du)z0,2 dz = 0, lim inf

0

Q(z0)

u (u)z0,2 dz = 0 (8.1)and

sup

>0

(u)z0,+ sup

>0

(Du)z0,M < . (8.2)The proof is complete if we show that such points are a regular points.

Step 1: a comparison estimate. The main goal here is to achieve the estimates (8.5), (8.6) below. The followingargument can be justified by the use of Steklov-averages; we shall omit all the details, only proceeding formally.Consider the unique weak solution v L2(t0 42, t0;W 1,2(B2(x0),RN)) of the initial boundary value problem

Q2(z0)

(vt A

(z0, (u)z0,2,Dv

)D

)dz = 0 C0

(Q2(z0),R

N),v = u on B2(x0) {t0 42} B2(x0) (t0 42, t0).


Then the difference u v satisfiesQ2(z0)

((u v)t

(A(z,u,Du)A(z0, (u)z0,2,Dv))D)dz = 0,

for every C0 (Q2(z0),RN). We now choose := (t)(u v) with 1 for (, s), 0 on (s + ,),and (t) = (s + t)/ for s t s + , where [s, s + ] (t0 42, t0). Then we have

12

Q2(z0)

t(|u v|2)dz+ 1

2

Q2(z0)

|u v|2t dz

Q2(z0)

(A(z,u,Du)A(z0, (u)z0,2,Dv))(DuDv) dz = 0.

Letting 0 we easily obtain that for a.e. s (t0 42, t0)12u(, s) v(, s)2

L2(B2(x0))+

B2(x0)(t042,s)

(A(z0, (u)z0,2,Du

)A(z0, (u)z0,2,Dv))D(u v)dz=

B2(x0)(t042,s)

(A(z0, (u)z0,2,Du

)A(z,u,Du))D(u v)dz.The second term of the left-hand side of the previous equation can be estimated by the use of monotonicity, i.e.

(A(z0, (u)z0,2,Du)A(z0, (u)z0,2,Dv))D(u v) |DuDv|2. We therefore obtain12u(, t) v(, t)2

L2(B2(x0))+

B2(x0)(t042,t)

|DuDv|2 dz

B2(x0)(t042,s)

(A(z0, (u)z0,2,Du

)A(z,u,Du))D(u v)dz = I.To estimate the right-hand side we use (2.5) which easily yieldsA(z0, (u)z0,2,Du)A(z,u,Du) L(2(u)z0,2+ u (u)z0,2,4 + u (u)z0,2)(1 + |Du|).Using the previous estimate, Youngs inequality and the fact that 1, we have

|I | 2

B2(x0)(t042,s)

|DuDv|2 dz

+ 2L2

Q2(z0)

(2(u)z0,2+ u (u)z0,2,4 + u (u)z0,2)(1 + |Du|2)dz.

Absorbing the term 2B2(x0)(t042,s) |DuDv|2 dz on the left-hand side we arrive at

12u(, t) v(, t)2

L2(B2(x0))+

2

B2(x0)(t042,t)

|DuDv|2 dz

4L2 ( )( 2) 4L2

Q2(z0)

2(u)z0,2 + u (u)z0,2 ,4 + u (u)z0,2 1 + |Du| dz =: II. (8.3)


We shall provide an estimate for II. We denote

:=

Q (z0)

u (u)z0, dz, > 0.If we let At At := {z Q2(z0): |u (u)z0,2| t} then

|At | 1t

Q2(z0)

u (u)z0,2dz |Q2|t 2. (8.4)We now split II

II =At

(. . .)dz+

Q2(z0)\At(. . .)dz =: III + IV,

and estimate III and IV . We have, using that 1, (8.4) and (8.2)

III At

(1 + |Du|2)dz 2

Q2(z0)

Du (Du)z0,22 dz+ (1 + 2(Du)z0,22)|At | 2

Q2(z0)

Du (Du)z0,22 dz+ 2(1 + 2(Du)z0,22) |Q2|t 2 2

Q2(z0)

Du (Du)z0,22 dz+ 2(1 + 2M2) |Q2|t 2.From the definition of (Section 2) we have

IV 4K(2M + t)( + t)

Q2(z0)

(1 + |Du|2)dz.

We now choose the parameter t carefully, i.e.

t := 2.Connecting the previous estimates for II, III and IV to (8.3), we easily have the estimate we were interested in; thatis

Q2(z0)

|DuDv|2 dz+ supt042tt0

2

B2(z0)

u(x, t) v(x, t)2 dx c(,L)K

(2M + 2

)(

+ 2

) Q2(z0)

(1 + |Du|2)dz

+

Q2(z0)

Du (Du)z0,22 dz+ 2(1 +M2) c

[K(2M + 2

)(

+ 2

) + 1] Q2(z0)

Du (Du)z0,22 dz

+ c[K(2M + 2 )( + 2 ) + 2 ](1 +M2) =: S(). (8.5)


In particular, we see that

2

Q2(z0)

u(z) v(z)2 dz supt042tt0

2

B2(z0)

u(x, t) v(x, t)2 dx S(). (8.6)We observe that as a consequence of (8.1) and (8.2) we have that

lim inf

0 S() = 0. (8.7)

Step 2: A Poincar type inequality. Our aim is to derive (8.8). Let us definev := v (Dv)z0,2(x x0).

Therefore v solvesQ2(z0)

(vt A(Dv)D

)dz = 0 C0

(Q2(z0),R

N),

where A(p) := A(z0, (u)z0,2, (Dv)z0,2 + p) for every p RnN . By (2.2) and (2.3) we haveAp (p) L, Ap (p)(p, p) |p|2, p, p RnN .

The last properties allow us to apply the relevant regularity theory for systems without coefficients of the typeut div A(Du) = 0: from [5], Theorem 3.1 we conclude that v W 1,2(t0 2, t0;W 1,2(B(x0), RN)) and that

2

Q(z0)

|tv|2 dz = 2

Q(z0)

|t v|2 dz c(,L)

Q2(z0)

|Dv|2 dz = c(,L)

Q2(z0)

Dv (Dv)z0,22 dz c(,L)

[

Q2(z0)

|DuDv|2 dz+

Q2(z0)

Du (Du)z0,22 dz].In view of the previous estimate, using Poincars inequality for v and (8.5) we find

2

Q(z0)

v (v)z0, (Dv)z0,(x x0)2 dz c[ Q(z0)

Dv (Dv)z0,2 dz+ 2 Q(z0)

|tv|2 dz]

c[

Q2(z0)

|DuDv|2 dz+

Q2(z0)

Du (Du)z0,22 dz] c[S()+ Q2(z0)

Du (Du)z0,22 dz],where c = c(n,,L). Finally, by comparison, we get the Poincar inequality for u via (8.5) and the previousestimate

2

Q(0)

u (u)z0, (Du)z0,(x x0)2 dz c

[

2

Q(z0)

|u v|2 dz+

Q(z0)

|DuDv|2 dz+ 2

Q(z0)

v (v)z0, (Dv)z0,(x x0)2 dz]

c[S()+

Q2(z0)

Du (Du)z0,22 dz] (8.8)

for a constant c = c(n,,L).


Step 3: Conclusion. From the previous estimate and (8.7) the assertion readily follows. Indeed if z0 QTsatisfies (8.1) and (8.2) then we have

lim inf

0

2

Q(0)

u (u)z0, (Du)z0(x x0)2 dz = 0therefore z0 is a regular point in view of Theorem 7.5. Theorem 8.2 (Almost everywhere regularity for simpler systems). Let u L2(T ,0;W 1,2(,RN)) be a weaksolution to the system (1.4) under the assumptions (2.1)(2.3) and (2.8) and denote by the singular set of u; then

1 2,where 2 is as in Remark 7.6 and


0

Q(z0)

Du (Du)z0,2 dz > 0}.Proof. The proof is based on a simple re-reading of Theorem 8.1; indeed it suffices to start with a point z0 (x0, t0) QT such that

lim inf

0

Q(z0)

Du (Du)z0,2 dz = 0 and sup

>0

(Du)z0,M < . (8.9)Since the vector field A does not depend on u the estimates in Step 1 simplify, especially those for II. In par-ticular, in the definition of S() in (8.5) we can take 0, therefore (8.9) suffices to have S() 0, which isthe fundamental information to use the estimates (8.5) and (8.6). The remaining Steps 2 and 3 do not need anyadjustment.

9. Proof of Theorem 1.3

For f L1(QT ) and 0 < h < T we recall the definition of Steklov averages: fh of f are defined for allT < t < 0 by

fh(x, t) :=

1h

t+ht

f (x, s)ds, t (T ,h),

0, t > h,respectively

fh(x, t) :=

1h

tth

f (x, s)ds, t (T + h,0),

0, t < T + h.For the properties of the Steklov averages we refer for instance to [9,27]. The following relations will be particularlyuseful in the sequel: For a.e. (x, t) (T ,h) and for a.e. (x, t) (T + h,0), we havefh

t= 1

h

(f (x, t + h) f (x, t)), fh

t= 1

h

(f (x, t) f (x, t h)).


Finally, for |h| > 0 and t (T |h|,|h|) we defineh(f ) hf (hf )(x, t) := f (x, t + h) f (x, t).

The following lemma asserts a well-known property of the time derivative that we restate in the way it is requiredlater.

Lemma 9.1 (Fractional time derivative of u). Let u L2(T ,0;W 1,2(,RN)) be a weak solution of the nonlinearparabolic system (1.1) under the only assumption (2.1), (t0, t1) (T ,0), and C0 () a cut-off function withspt . Then, whenever 0 < |h| < 1/2 min{|t1|, T |t0|,1} the following estimate holds:

t1t0

2u(x, t + h) u(x, t)2 dx dt c(2L + 2L)|h|

QT

(1 + |Du|2)dz,

where c = c(L).

Proof. For brevity we shall only give the proof for the case h > 0, the other one being the same, using uh insteadof uh. Using Steklov averages to formulate (1.1) we have that for a.e. t (t0, t1) and h as above

(t (uh) +

[A(x, t, u,Du)

]hD

)dz = 0, W 1,20 (,RN).

Using the fact that ht (uh) = hu we get

(hu

h + [A(x, t, u,Du)]

hD

)dz = 0.

We use the test function = 2hu and integrate with respect to t over the interval (t0, t1), obtainingt1

t0

|hu|2h

2 dx dt = t1

t0

[A(x, t, u,Du)

]h

[2 hu+ 2Dhu

]dx dt =: I + II. (9.1)

The first term is estimated using (2.1) and Youngs inequality

|I | 12

t1t0

|hu|2h

2 dx dt + ht1

t0

[A(x, t, u,Du)]h

2||2 dx dt 1

2

t1t0

|hu|2h

2 dx dt +L22Lht1

t0

[(1 + |Du|)2]

hdx dt

12

t1t0

|hu|2h

2 dx dt +L22Lht1+ht0

(1 + |Du|)2 dx dt

1t1 |hu|2 2 2 2 ( )2

2t0

h dx dt +L Lh

QT

1 + |Du| dz.


Here we have used in the second-last line the elementary estimatet1

t0

|fh|2 dx dt t1+ht0

|f |2 dx dt

where f L2(QT ), (t0, t1) (T ,0) and 0 < h< t1. The second term is estimated similarly

|II|( t1

t0

2[a(x, t, u,Du)]

h

2 dx dt)1/2( t1t0

2|Dhu|2 dx dt)1/2

2L( t1

t0

2[(

1 + |Du|)2]h

dx dt)1/2( t1+h

t0

2|Du|2 dx dt)1/2

2L2Lt1+ht0

(1 + |Du|)2 dx dt 2L2L

QT

(1 + |Du|)2 dz,

while in the second line we have used the elementary estimatet1

t0

|hf |2 dx dt 2t1+ht0

|f |2 dx dt.

Combining the estimates for I and II with (9.1) we finally conclude witht1

t0

|hu|2h

2 dx dt c(L)(2L + 2L)h

QT

(1 + |Du|)2 dz.

As a consequence of the previous lemma we have

Corollary 9.2. Let u L2(T ,0;W 1,2(,RN)) be a weak solution of the nonlinear parabolic system (1.1) underthe only assumption (2.1), (t0, t1) (T ,0), and . Then, whenever 0 < |h| < 1/2 min(|t1|, T |t0|,1) thefollowing estimate holds:

t1t0

u(x, t + h) u(x, t)2 dx dt c|h| QT

(1 + |Du|2)dz, (9.2)

where c = c(L,dist(, )).

In the following we state a preliminary estimate for the space time derivative of a weak solution u of thenonlinear parabolic system (1.1) which we are going to use as a starting point; to be more precise we will estimate|hDu|. Let us fix T < t < t1 < 0, arbitrarily. In the following we shall always take

|h| 11000

min{|t1|, T + t0}, || 11000 min{|t1|, T + t0} (9.3)

and t (t0, t1) where t0 := (T + t )/2 < t . We shall fix a cut-off function in time C((T ,0)) such that

(t) = 0 on (T , t0), (t) = 1 for t > t, 0, | | 4

T + t0 . (9.4)


Moreover we choose a cut-off function in space C0 () such that |spt| > 0. We start once again from theSteklov-average formulation of (1.1)

(t (u) +

[A(x, t, u,Du)

]D

)dz = 0, W 1,20 (,RN). (9.5)

Writing the previous equation at levels t , t + h we get

(t (hu) + h

[A(x, t, u,Du)

]D

)dz = 0. (9.6)

Here we have used the identity h(tu) = t (hu). In (9.6) we choose the test function(x, t) := 2(t)2(x)(hu)(x, t).

It is easy to see that is admissible in (9.6) for a.e. t (t0, t1), i.e. that (, t) W 1,20 (,RN). Therefore taking(, t) at each level t and integrating over (t0, t1) with respect to t we obtain

t1t0

22t (hu)(hu)dx dt = t1

t0

2h[A(x, t, u,Du)

]

[2 (hu)+ 2D(hu)

]dx dt.

We are now in a position to treat the left-hand side in a standard way. Recalling that (t) = 0 for T < t t0 wecan rewrite the left-hand side of the previous identity in the form

t1t0

22t (hu)(hu)dx dt =t1

T

22t (hu)(hu)dx dt

= 12

2(t1)2|hu|2(, t1)dx

t1T

2|hu|2 dx dt.

Substituting this above and letting 0 we obtain for a.e. t1 as above

12

2(t1)2|hu|2(, t1)dx

t1T

2|hu|2 dx dt

+t1

t0

2h[A(x, t, u,Du)

][2 (hu)+ 2D(hu)

]dx dt = 0. (9.7)

In the sequel the contribution of the first integral appearing in (9.7) will be ignored. We shall use the followingdecomposition of h(A):

h[A(, , u(, ),Du(, ))](x, t)

= A(x, t + h,u(x, t + h),Du(x, t + h))A(x, t, u(x, t),Du(x, t))= A(x, t + h,u(x, t + h),Du(x, t + h))A(x, t + h,u(x, t + h),Du(x, t))

+A(x, t + h,u(x, t + h),Du(x, t))A(x, t + h,u(x, t),Du(x, t))+A(x, t + h,u(x, t),Du(x, t))A(x, t, u(x, t),Du(x, t))=:A(h)+B(h)+ C(h).


With such a notation (9.7) turns tot1

t0

22[A(h)+B(h)+ C(h)]D(hu)dx dt

2t1

t0

2[A(h)+B(h)+ C(h)] (hu)dx dt + t1

T

2|hu|2 dx dt. (9.8)

We proceed estimating the various pieces arising in (9.8).

Estimates for integrals involving A(h). We write

A(h)(x, t) =1

0

A

p

(x, t + h,u(x, t + h),Du(x, t)+ sh(Du)(x, t)

)dsh(Du)(x, t) =: A(h)h(Du)(x, t).

Using the ellipticity of A, i.e. (2.3) we see thatA(h), h(Du)= A(h)h(Du) h(Du) |hDu|2.Therefore we obtain

t1t0

22A(h)D(hu)dx dt t1

t0

22|hDu|2 dx dt.

From (2.2) we find that |A(h)| L so that

2t1

t0

2A(h)|hu|dx dt 2L t1

t0

2|||hDu||hu|dx dt

t1

t0

22|hDu|2 dx dt + L2

t1t0

spt

2||2|hu|2 dx dt

t1

t0

22|hDu|2 dx dt + L2

2L

t1t0

spt

2|hu|2 dx dt

t1

t0

22|hDu|2 dx dt + c2L|h|QT

(1 + |Du|)2 dz,

where c = c( 1 ,L,dist(spt, )). In the last line we used Corollary 9.2.

Estimates for integrals involving C(h). Using (2.9), we have |C(h)| L|h|/2(1 + |Du|) and sot1

2 2 /2 t1 2 2( )

t0

C(h)D(hu)dx dt L|h|t0

1 + |Du| |hDu|dx dt


t1

t0

22|hDu|2 dx dt + L2

|h|

t1t0

22(1 + |Du|)2 dx dt

t1

t0

22|hDu|2 dx dt + c|h|QT

(1 + |Du|)2 dz,

where c = c( 1 ,L). Using again Corollary 9.2 we gain

2t1

t0

2C(h)|||hu|dx dt 2

( t1t0

2C(h)2 dx dt)1/2( t1

t0

22|hu|2 dx dt)1/2

L|h|/2( QT

(1 + |Du|)2 dz)1/2( t1

t0

spt

|hu|2 dx dt)1/2

c|h|(1+)/2QT

(1 + |Du|)2 dz,

where now c = c(L,dist(spt, )).

Estimates for integrals involving B(h). Here we shall use the fact that there exists (L/) > 0 such that|Du| L2(1+)loc (QT ). (9.9)

More precisely, see also Theorem 10.1 below: For every open subset O QT there exists a constant c c(dist(O, QT )) such that

O|Du|2(1+) dz c

( QT

(|Du|2 + 1)dz)1+ . (9.10)This is a Gehrings type higher integrability result [17,35,24] and we note that in general

limL/ = 0. (9.11)

Using that |B(h)| L(|hu|)(1 + |Du|), Youngs inequality once again and keeping into account that 1 wehave

t1t0

22B(h)|hDu|dx dt t1

t0

22|hDu|2 dx dt + 1

t1t0

22B(h)2 dx dt

t1

t0

22|hDu|2 dx dt + L2

t1t0

spt

(|hu|)(1 + |Du|)2 dx dt. (9.12)

It remains to estimate the second integral appearing on the right-hand side. Using Hlders inequality, the higher

integrability of Du twice with (9.10), Jensens inequality and once again the estimate for the fractional time deriv-ative of u we obtain


t1t0

spt

(|hu|)(1 + |Du|)2 dx dt

| spt|(t1 t0)(

spt(t0,t1)(|hu|) 1+ dz) 1+ (

spt(t0,t1)

(1 + |Du|)2(1+) dz) 11+

| spt|(t1 t0)(

spt(t0,t1)|hu|dz

) 1+ (

spt(t0,t1)

(1 + |Du|)2(1+) dz) 11+

| spt|(t1 t0)(

spt(t0,t1)|hu|2 dz

) 2(1+)(

spt(t0,t1)

(1 + |Du|)2(1+) dz) 11+

c( QT

(1 + |Du|)2 dz) 2(1+)(

QT

(1 + |Du|)2 dz)|h| 2(1+)

c(,, |QT |,dist(spt, ),DuL2(QT )

)|h| 2(1+) .In the second last line we also used (9.10). For the remaining term we have

t1t0

2B(h)|||hu|dx dt

L( t1

t0

spt

2B(h)2dx dt)1/2( t1

t0

spt

|hu|2 dx dt)1/2

.

Now, the first integral appearing in the right-hand side of the previous estimate can be treated as the third integralin (9.12), while the second integral can be estimated via the estimate of the fractional time derivative for u fromCorollary 9.2. We finally arrive at

t1t0

2B(h)|||hu|dx dt c|h| 12 + 4(1+) c|h| 2(1+) ,

where c = c(L,,, |QT |,dist(spt, ),DuL2(QT ),L).We now turn our attention to the remaining term from the left-hand side of (9.8). Recalling that 0 4

T+t0this last term is estimated again via Corollary 9.2. We obtain

tt0

spt

2|hu|2 dx dt 4T + t0

t1t0

|hu|2 dx dt c|h|QT

(1 + |Du|)2 dz,

where c = c(L,dist(spt, ), t0).Connecting the previous estimates and taking in terms of suitably small we finally arrive at

t1 2 2 2 ( 1+ ) t0

|hDu| dx dt c1 |h| + |h| + |h| 2 + c2|h| 2(1+) c1|h| + c2|h| 2(1+) , (9.13)


wherec1 = c

(,L, |QT |,dist(spt, ), t0,L,DuL2(QT )

) (9.14)and

c2 = c(,L,,, |QT |,dist(spt, ),L,DuL2(QT )

). (9.15)

Remark 9.3. In the case of systems of the type (1.4), the above estimate holds with c2 = 0. Indeed the last term in(9.13) is due to the presence of B(h).

Letting

:= 1 + (9.16)

and using the fact that , , , t0, t1 considered above were arbitrary, we can therefore conclude, by Proposition 3.4

Lemma 9.4. Let u L2(T ,0;W 1,2(,RN)) be a weak solution of the nonlinear parabolic system (1.1) underthe assumptions (2.1)(2.3) and (2.9). Then for any , (t0, t1) (T ,0) we have in the special case ofcoefficients A A(x, t,Du) that

t1t0

t1t0

|Du(x, t)Du(x, s)|2|t s|1+2 dt ds dx < ,

Documents

Annales de l'Institut Henri Poincare (C) Non Linear Analysis Volume 22 Issue 6 2005 [Doi 10.1016_j.anihpc.2004.10.011] Frank Duzaar; Giuseppe Mingione -- Second Order Parabolic Systems,