200-Luminy conference on Quasilinear Elliptic and Parabolic Equations and Systems, Electronic Journal of Differential Equations, Conference 08, 2002, pp 2 3. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) A note on W,p estimates for quasilinear parabolic equations Ireneo Peral & Fernando Soria Abstract This work deals with the study of the W,p regularity for the solutions to parabolic equations in divergence form. An argument by perturbation based in real analysis is used. Introduction In this paper we study interior W,p estimates for solutions to quasilinear parabolic equations in divergence form, namely, solutions to the equation u t div a(x, t, u) = 0, x Ω, t > 0, (.) where a : Ω (0, ) N N. We assume that a(x, t, ξ) is a Caratheodory function (in the sense that it is measurable in (x, t) and continuous with respect to ξ for each x) and that satisfies the following conditions: (a) a(x, t, 0) = 0 (a2) a(x, t, ξ) a(x, t, η), (ξ η) γ ξ 2. (a3) a(x, t, η) Γ η, where γ and Γ are positive constants. We will work under the following hypotheses. (H) (eference operator.) For fixed a 0 satisfying (a) (a3), we consider the corresponding parabolic equation w t div(a 0 ( w)) = 0 (.2) If u is a weak solution to (.2) (see Definition 3.2 below) then there exists γ > 0 such that sup x u(x, t) 2 γ x u(x, t) 2 dx dt, (.3) Mathematics Subject Classifications: 35K0, 35K55, 42B25. Key words: semilinear parabolic equations, gradient estimates, Calderon-Zygmund theory. c 2002 Southwest Texas State University. Published October 22, 2002. I. Peral was supported by grant BFM200-083 from M.C.Y.T. Spain. F. Soria was supported by grant BFM200-089 from M.C.Y.T. 2
22 W,p estimates for quasilinear parabolic equations for all parabolic rectangles = { (x, t) : x i x 0 i < ρ, i =,... N, t 0 ρ 2 } < t < t 0 Ω, where = { (x, t) : x i x 0 i < ρ 2, i =,... N, t 0 ( ρ } 2 )2 < t < t 0. (H2) (Approximation property.) The vector field a(x, t, ξ) is close to a 0 (ξ) in the following sense: a(x, t, ξ) a 0 (ξ) ɛ ξ, (.4) uniformly in x and t. The simplest example for a 0 (ξ) is just a 0 (ξ) = ξ, which corresponds to the heat equation. A classical example for a is a(x, t, ξ) = A(x, t)ξ with A(x, t) a bounded N N matrix such that A(x, t) I N N ɛ. Even in the linear case, the results presented here seem to be new. We also point out that more general situations can be considered by the method that we develop. For simplicity, we restrict ourselves to the quadratic growth case. With these hypotheses we will be able to show some sort of parabolic Meyers type inequalities. More precisely we have the following main result. Theorem. Let a be a vector field satisfying (a) (a3). Assume that (H) holds. Given p > 2 there exists ɛ 0 > 0 such that if for some 0 < ɛ < ɛ 0 (H2) holds, then any weak solution to satisfies that x u L q loc, 2 < q < p. u t div(a(x, t, x u)) = 0, The idea is to estimate the level sets of x u and obtain the required growth of their measure to have the integrability property. To do that, we will use a form of the Calderón-Zygmund covering result. For the elliptic case see []. 2 Preliminary results In this section we present some tools that will be used along the paper. First of all we will prove the corresponding Calderón-Zygmund covering result and explain the properties of the maximal operator which naturally arises in the parabolic setting under study. To be systematic we will give a general Covering Lemma that includes the particular case that we will need.
Ireneo Peral & Fernando Soria 23 The rectangle collection For every k Z, let B k denote a collection of rectangles satisfying the following properties: i) All the rectangles of B k have the same side lengths. ii) Any two distinct rectangles of B k have disjoints interiors. iii) { B k } = N iv) If B k then = { B k } v) If δ k denotes the length of the diagonal of any B k then δ k 2δ k+ Observe that given B k there exists a unique rectangle in B k, its predecessor which we denote by so that. With these properties in hand we have the following observations. Lemma 2. Fix 0 B k0 and 0 < δ <. Assume that A 0 and that 0 < A < δ 0. Then, there exists a sequence { d } k=k 0 B k such that:. A \ d d = 0, 2. A d δ d, 3. If d for some B k, with 0, then A δ (in k=k 0 particular A δ ). Proof. The proof goes via the usual stopping-time argument. Observe that 0 does not satisfies 2) by hypothesis. We select all the rectangles in B k0+ for which 2) holds. If 0, B k0+ has not been selected we consider the partition of by rectangles of B k0+2 and we take those satisfying 2). In this way and by induction, we obtain a maximal sequence { d } satisfying 2) and 3). We call B A = d. To see why { d } satisfies ) too, we observe that if x 0 \ { : B A } then there exists a unique sequence { k (x)} so that x k (x) B k, k k 0. Moreover, since diam k (x) = δ k 0 as k, we obtain that from the Lebesgue differentiation theorem, A k (x) lim =, a.e. x A. k k (x) In particular this says that A d a.e. and, therefore, ) holds.
24 W,p estimates for quasilinear parabolic equations Observe that the sequence { d } in Lemma 2., is given by maximal rectangles in 0 for which property 2) holds. We will call this sequence the Calderón- Zygmund covering of A. Note that given a rectangle k in a Calderón-Zygmund covering of the set A, there exists a nested finite family of parabolic rectangles k 2 k r(k) k k, for which l k A δ l k. r(k) k We say that is the predecessor of k and for simplicity of notation we will r(k) write k k. We give some applications of Lemma 2. that are relevant for this work. Lemma 2.2 The maximal operator is of weak type (, ). Mf(x) = sup x f B k The proof uses the Vitali covering lemma and the observation that if, B = B k and with their interiors with nonempty intersection, then either or. Lemma 2.3 Fix 0 B and 0 < δ <. Given A B 0 with the properties, a) A < δ 0, b) If d is a rectangle in the Calderón-Zygmund decomposition of A (with respect to 0 and δ), then d B, we conclude that A δ B Proof. Consider the sequence {d } of predecessors of the C-Z covering of A with respect to 0 and δ. Select a subsequence {d l } with disjoint interiors so that dl = d, (a maximal subsequence). Then, A = l A d l δ l d l = δ d l δ B.
Ireneo Peral & Fernando Soria 25 3 Approximations esults Denote the open unit cube in N+ by Q 0 = {(x, t) N+ x i <, i =, 2,..., N, t < } and call Q + 0 = Q 0 { t > 0 } and Q 0 = Q 0 { t < 0 }. A parabolic rectangle is then the image of Q 0 through any transformation in N+ of the form φ α (x, t) = (x 0 + αx, t 0 + α 2 t), α > 0, (x 0, t 0 ) N+. Let us consider the following dyadic subdivision: Given a parabolic rectangle i) Divide into 2 equal parts each spatial side. ii) Divide into 2 2 equal parts the temporal side. We call this procedure a parabolic subdivision. It is obvious that with this procedure we obtain 2 N+2 new parabolic subrectangles. Associated to this subdivision we have the corresponding Calderón-Zygmund decomposition as in the previous section. Before we continue, some notation is in order. Given a parabolic rectangle = {(x, t) : x i x 0 i < ρ, i =,... N, t 0 ρ 2 < t < t 0 } we define the parabolic boundary of as p = {(x, t) : x i x 0 i ρ, t 0 ρ 2 = t} {(x, t) : x i x 0 i = ρ, t 0 ρ 2 < t < t 0 }. Example 3. In the case of the heat equation (the simplest model) we have the following result by Moser [4]. Let w be a measurable function in Q 0 such that (i) w(x, y) 0 (ii) w t, x w, w L 2 (Q 0 ) (iii) w t w 0 in D (Q 0 ). We call such w a positive subsolution of the heat equation. According to the classical results by Moser in [4] we have that there exists a positive constant γ such that if w is a positive subsolution to the heat equation then where sup w 2 (x, t) γ w 2 (x, t)dx dt, (3.) = { (x, t) : x i x 0 i < ρ, i =,... N, t 0 ρ 2 < t < t 0 } and = φ /2 (), that is, = { (x, t) : x i x 0 i < ρ 2, i =,... N, t 0 ( ρ 2 )2 < t < t 0 }.
26 W,p estimates for quasilinear parabolic equations We point out that if u is a solution to the heat equation then x u 2 is also a positive subsolution to the heat equation. Hence, by (2), sup x u(x, t) 2 γ x u(x, t) 2 dx dt, (3.2) where and are parabolic rectangles related as above. It is clear that hypothesis (H) extends the previous Example. Definition 3.2 We say that u L 2 (Q 0 ), with u t, x u L 2 (Q 0 ), is a weak solution to the equation u t div x (a(x, t, x u)) = 0 if we have uψ t dxdt + a(x, t, x u) ψ dx dt = 0, Q 0 Q 0 for all ψ in W,2 0, the completion of C 0 (Ω) with respect to the L 2 -norm of the function and its gradient. Lemma 3.3 Assume that (H),(H2) hold. equation Let u be a weak solution to the u t div x (A(x, t, x u) = 0, (3.3) such that for, a parabolic rectangle contained in Q 0, u satisfies If v is the solution to the problem x u 2 dx dt µ. v t div x (a 0 ( v)) = 0, v p = u, (x, t) then. x(u v) 2 dx dt µɛ 2 2. xv 2 dx dt µ( + ɛ) 2
Ireneo Peral & Fernando Soria 27 Proof. Write = Q (T, T 2 ). Observe that γ (u v) 2 dx dt a 0 ( u) a 0 ( v), (u v) dx dt u(x, T 2 ) v(x, T 2 ) 2 dx Q + a 0 ( u) a 0 ( v), (u v) dx dt = [(u v) t (div x a 0 ( u) div x a 0 ( v))] (u v)dx dt = (u t div x a 0 ( u))(u v)dx dt (v t div x a 0 ( v))(u v)dx dt = [u t div x (a(x, t, x u)](u v)dx dt + x (u v), (a 0 ( u) a(x, t, x u)) dx dt ɛ ( ) /2 ( /2, x (u v) 2 dx dt x u 2 dx dt) where we have used for the last inequality that u is a solution to equation (.), condition (H2) and Cauchy-Schwarz inequality. We thus obtain (). Now (2) is an easy consequence of () since, in fact, x v 2 dx dt ( = ( ( ( + ɛ)µ /2. ) /2 ) /2 x (u + (v u)) 2 dx dt ) /2 ( ) /2 [ x u 2 dx dt + x (v u)) 2 dx dt We will now introduce the (dyadic) parabolic maximal operator, defined for f L loc by Mf(x) = sup f(y) dy, x where the supremum is taken over all parabolic rectangles containing x N+. Taking into account that N+ is a homogeneous space with respect to the seminorm d(x, t) = max{ x i, t /2, i =, 2... N},
28 W,p estimates for quasilinear parabolic equations more precisely that φ 2 () 2 N+2, we have that M is of weak type (, ) as a consequence of the usual Besicovich covering lemma. The following result is the main ingredient in Theorem. below. Lemma 3.4 Assume that u is a weak solution to u t div x (a(x, t, x u)) = 0, in Q 0. Then, there exists a constant C > so that for 0 < δ < fixed, one can find ɛ 0 = ɛ 0 (δ) > 0 such that if (H2) holds with ɛ < ɛ 0, for all parabolic rectangle k in the Calderón-Zygmund δ-covering of then its predecessor satisfies In particular, we have {(x, t) N+ : M( x u 2 )(x, t) > Cµ}, k {(x, t) : M( x u 2 )(x, t) > µ}. {(x, t) N+ : M( x u 2 )(x, t) > Cµ} {(x, t) : M( x u 2 )(x, t) > µ}. Proof. The proof is similar to Lemma 3 in [] and we sketch it here for the sake of completeness. Since we look for a local result, we can assume that a parabolic dilation of k is contained in Q 0, to be more precise, let us assume, say, that Q = φ 4 ( k ) Q 0. We argue by contradiction. If k satisfies the hypothesis, namely, k {x : M( x u 2 ) > Cµ} > δ k and k does not satisfy the conclusion, there exists (x 0, t 0 ) k for which, x u 2 dx dt µ, for all parabolic rectangles with (x 0, t 0 ). We solve the problem Then, according to Lemma 3.3 we get:. Q Q xv 2 dx dt ( + ɛ) 2 µ 2. Q Q x(u v) 2 dx dt ɛ 2 µ v t div x (a 0 ( v)) = 0, (x, t) Q 0 v pq 0 = u.
Ireneo Peral & Fernando Soria 29 Then the restricted maximal operator, M ( x u 2 )(x, t) = sup x u 2 dyds, x, φ 2( k ) satisfies M( x u 2 )(x, t) max{m ( x u 2 )(x, t), 4 N+2 µ}. Consider C = max{4 N+2, 4( + ɛ) 2 }. Then {(x, t) k : M( x u 2 ) > Cµ} {(x, t) k : M ( x u 2 ) > C µ 2 } + {(x, t) k : M ( x (u v) 2 ) > C µ 2 } {(x, t) k : M ( x (u v) 2 ) > C µ 2 }. By the (,) weak type estimate for the maximal operator we conclude {x k : M( x u 2 ) > Cµ} {x k : M ( x (u v) 2 ) > C µ 2 } A 2 x (u v) 2 dx dt Cµ k A 2 Cµ ɛ2 k. Taking ɛ > 0 so that A 2 Cµ ɛ2 < δ we reach a contradiction. 4 Proof of the main result As a consequence of the approximation Lemma 3.3 and the behavior of the level sets of the maximal operator described in Lemma 3.4 we can formulate the following regularity result. Theorem 4. Assume that (H) holds. Given p > 2 there exists ɛ 0 > 0 such that if for some 0 < ɛ < ɛ 0, (H2) holds, then any weak solution to satisfies that x u L q loc, 2 < q < p. u t div(a(x, t) x u) = 0, Proof. For s > 0, call ω(s) = {(x, t) : M( x u 2 (x, t) > s}, the distribution function of the maximal operator. Take δ (0, ) in such a way that C q/2 δ <, where C is as in Lemma 3.4. Now, there exists ɛ 0, such that if 0 < ɛ < ɛ 0 and (H2) holds, then Lemmas 3.4 and 2.3 imply ω(cµ 0 ) δω(µ 0 ).
30 W,p estimates for quasilinear parabolic equations Hence by recurrence ω(c k µ 0 ) δ k ω(µ 0 ). (4.) Now x u q L if, in particular, M( x u 2 ) L q/2, and this is equivalent to the convergence of the series C k(q/2) ω(c k µ 0 ). k= But, C q/2 δ < and from estimate (7) we obtain C k(q/2) ω(c k µ 0 ) (C q/2 δ) k ω(µ 0 ) < k= k= Corollary 4.2 Assume A(x, t) a N N matrix which is continuous in Ω and such that A(x, t)ξ, ξ γ ξ 2. Then if u is a weak solution to we have u W,p loc u t div x (A(x, t) x u) = 0, for all < p <. Proof. As reference equation we take the heat equation. The hypothesis (H2) is obtained easily by an orthogonal change of variables in N and our assumptions on the continuity of A. By the same method we are able to get estimates for equations that are close to the p-laplacian. For estimates to u t div( u p 2 u) see [2]. eferences [] L. A. Caffarelli and I. Peral, On W,p Estimates for Elliptic Equations in Divergence Form, Comm. Pure App. Math., 5, 998, -2. [2] E. Di Benedetto Degenerate Parabolic Equations, Springer-Verlag, 993 [3] G. M. Lieberman, Second Order Parabolic Differential Equations. World Scientific, 996. [4] J. Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 7, 964, 0 34. [5] O. A. Ladyzhenskaya, V.A. Solonnikov, N.N. Ural tseva, Linear and Quasilinear Equations of Parabolic Type Translations of Mathematics Monographs, A.M.S. 968.
Ireneo Peral & Fernando Soria 3 [6] P. Tolksdorff, egularity for a more general class of quasilinear elliptic equations J. of Diff. Equations 5 (984), 26-50 Ireneo Peral (e-mail: ireneo.peral@uam.es) Fernando Soria (e-mail: fernando.soria@uam.es) Departamento de Mátematicas U.A.M. 28049 Madrid, Spain