L p estimates for parabolic equations in Reifenberg domains

Transcription

1 Journal of Functional Analysis 223 (2005) L p estimates for parabolic equations in Reifenberg domains Sun-Sig Byun a,, Lihe Wang b,c a Department of Mathematics, Seoul National University, Seoul 5 747, Korea b Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA c College of Sciences, Xian Jiaotong University, Xian 70049, China Received 3 April 2004; received in revised form October 2004; accepted 26 October 2004 Communicated by R.B. Melrose Available online 2 January 2005 Abstract The inhomogeneous Neumann problem for parabolic equations in divergence form is studied. An optimal regularity requirement on the domain for the L p -theory is investigated, assuming that the principal coefficients are supposed to be in the John Nirenberg space with small BMO semi-norms and that the domain is a Reifenberg flat domain Elsevier Inc. All rights reserved. MSC: primary 35R05; 35R35; secondary 35J5; 35J25 Keywords: Reifenberg flat domain; Maximal function; Vitali covering lemma. Introduction The study of parabolic equations has a very close relation to the study of elliptic equations. Recently, the authors [7,8] have investigated a suitable and minimal condition on the domain for the W,p regularity theory, <p<, for linear elliptic equations in divergence form. This work is an extension of results in [7,8] to the parabolic setting. This work was supported in part by NSF Grant # Corresponding author. addresses: byun@math.snu.ac.kr (S.-S. Byun), lwang@math.uiowa.edu (L. Wang) /$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:0.06/j.jfa

2 S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) We assume Ω to be an open, bounded subset of R n, and set Ω T = Ω (0,T] for some fixed time T>0. We will study the following initial/boundary value problem: u t div (A u) = div f in Ω T, (A u + f) ν = 0 on Ω [0,T], u = 0 on Ω t = 0}, (.) where f = ( f,...,f n) and A =a ij } are given, and ν(,t)is the outward pointing unit normal vector field along Ω for each fixed time 0 t T. We remark that the conormal derivative boundary condition in general is not defined, but its weak formulation is welldefined (see Definition.3, 3., 4., and 4.3). For the concept of weak solutions of PDE (.) we refer to the papers [3,28 3,33]. In this note we introduce an intrinsic Sobolev space W,p (see Definition 2.4) ( ) ( ) W,p (Ω T ) = L p 0,T; W,p (Ω) W,p 0,T; W,q (Ω), where <p,q< with p + q = and W,q (Ω) is the dual space of W,p 0 (Ω). Then we employ the method used in [43] to show the well-posedness in W,p (Ω T ) of the Neumann problem (.) with the estimate u W,p (Ω T ) C f L p (Ω T ) (.2) for some constant C independent of u and f. Throughout this paper the matrix A(x, t) =a ij (x, t)} of coefficients is supposed to be defined on R n R, as it follows by the papers [,22]. The main condition on the coefficients is that they are in the John Nirenberg space BMO (cf. [2]) of the functions of bounded mean oscillation with small BMO seminorms. We use the following definition. Definition. (Small BMO semi-norm condition). We say that the matrix A of coefficients is (δ,r)-vanishing if sup sup 0<r R (x,t) R n R K r K r (x,t) A(y, s) A Kr 2 (x,t) dy ds δ, where K r (x, t) = B r (x) (t r 2 /2,t+ r 2 /2] is a centered parabolic cube and A Kr (x,t) is the average of A over K r (x, t). We would like to point out that our condition on the coefficients weakens the condition in the papers (see eg.[2,4,,2,5,28,35,38 40]) that the matrix A of principal coefficients is in VMO space (cf. [37]). We also remark that both R and δ are fixed

3 46 S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) constant, and so, they can depend on each other. However by a scaling argument, δ is independent of R for R>. Our condition on the domain is that it is a Reifenberg domain. More precisely, we have the following definition. Definition.2 (Reifenberg flat domain condition). We say that a domain Ω is (δ,r)- Reifenberg flat if every x Ω and every r (0,R], there exists an (n ) dimensional plane L(x, r) such that r D[ Ω B r(x), L(x, r) B r (x)] δ, where D denotes the Hausdorff distance, namely, D[A, B] =supdist(a, B) : a A}+supdist(b, A) : b B}. We should point out that the previous definition is only significant for small δ > 0. A Reifenberg domain was introduced by Reifenberg in the paper [36] where the author proved that it is locally a topological disk. A typical example of Reifenberg flat domains is the well-known Van Koch snowflake. The Van Koch curve is a self-similar Jordan curve and a prototypical fractal set. We mention the very interesting paper [4] where the authors constructed a Reifenberg flat domain whose boundary has a Hausdorff dimension greater than n in R n+. Thus the domain considered in this work might have fractal dimension. Fractals are geometric shapes that are very complex and infinitely detailed. We can zoom in on a section and it will have just as much detail as the whole fractal. They are recursively defined and small sections of them are similar to large ones. They are found in real-world systems such as blood vessels, the internal structure of the lungs, graphs of stock market data, bacteria and fern growth, clouds, mountains and so on. We refer the reader to the papers [3,4,9,20,24 26,36,42] for further discussions of the notion of Reifenberg flat domains. The remarkable thing for Reifenberg flat domains is that they are W,p ( p ) extension domains. Thus extension theorem and Sobolev inequalities are available on a (δ,r)-reifenberg flat domain, which is very important to the W,p regularity theory for elliptic equations as well as W,p regularity theory for parabolic equations. For this property of a domain with nonsmooth boundary, we refer to [7,8,23]. We also remark that one might assume that R in both Definitions. and.2 is by a scaling argument. Through this paper we mean δ to be a small positive constant. The aim of this paper is to show that the estimate (.2) holds true under the conditions that A is (δ,r)-vanishing and Ω is (δ,r)-reifenberg flat. According to classical works when p = 2 (see [3,27,32]), as long as A is uniformly parabolic (see Definition 2.) and f L 2 (Ω T ), (.) has a unique (upto a constant) weak solution u, namely, u is a function in C 0 (0,T; L 2 (Ω)) L 2 (0,T; H (Ω))

4 S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) satisfying the integral identity uφt A u φ f φ } dx dt = u(x, 0)φ(x, 0) dx Ω T Ω for all φ C (Ω T ). Moreover, this solution belongs to W,2 = L 2 (0,T; H (Ω)) H /2 (0,T; L 2 (Ω)). Consequently, the estimate (.2) holds true under the conditions considered in this paper for the case that p = 2. We recall some of them in the next section. In this paper, we use the following definition of weak solutions. Definition.3. Let <p, q<, function u W,p (Ω T ) such that for all φ W,q (Ω T ). p + q =. Then a weak solution of (.) is a uφt A u φ f φ } dx dt = u(x, 0)φ(x, 0) dx Ω T Ω Remark.4. Under the condition in the above definition, Ω T u(x, t)φ(x,t)dx is continuous in t. For its proof, we refer to Chapter III in [27]. Remark.5. We remark that by an approximation argument, we can take φ in Definition.3 from the space C (Ω T ). In fact, we can approximate this competitor φ by the standard mollification in spatial variable and the Steklov average of φ in time variable, φ ε h (x, t) = h h Let us state the main result of this work. 0 φ ε (x, t + s)ds. Theorem.6. Let p be a real number with <p< and R>0. Then there is a small δ = δ(λ,p,n,r)>0 so that for all A with A uniformly parabolic (see Definition 2.) and (δ,r)-vanishing, for all Ω with Ω (δ,r)-reifenberg flat, and for all f with

5 48 S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) f L p (Ω T ; R n ), the Neumann problem (.) has a unique (up to a constant) weak solution u with the estimate u W,p (Ω T ) C( u L p (Ω T ) + f L p (Ω T )), where the constant Cis independent of u and f. Remark.7. We only consider the case that p > 2. Uniqueness upto a constant follows easily from the case p = 2. Then a duality argument ends the proof when <p<2. We will hereafter focus attention exclusively on the case that p>2. Our approach is very much influenced by [0,43]. In [0], the Calderón Zygmund decomposition was used. In this paper the Vitali covering lemma will be used as in [5 8,43]. Our basic tools in this approach are the Vitali covering lemma, the Hardy Littlewood maximal function and the use of compactness method. The remaining sections are organized in the following way. In Section 2, we give auxiliary notations, necessary function spaces, some definitions and some geometric analysis results. In Section 3, we discuss the interior regularity. A global regularity is derived for the Neumann problem of (.) in Section Some preliminary facts from real analysis 2.. Geometric notation () R n = n-dimensional real Euclidean space. (2) e i = (0,...,,...,0) = ith standard coordinate vector. (3) A typical point in R n R is (x, t) = (x,x n,t). (4) R n + =x Rn : x n > 0}. (5) B r =y R n : y <r} is an open ball on R n with center 0 and radius r>0, B r (x) = B r + x, B r + = B r R n +, B+ r (x) = B+ r + x, T r = B r x n = 0}, T r (x) = T r + x, and c B r + = B r R n + is the curved part of B+ r. (6) Ω r = Ω B r, Ω r (x) = Ω B r (x). (7) Ω is the boundary of Ω, w Ω r = Ω B r is the wiggled part of Ω r, and c Ω r = Ω r \ w Ω r is the covered part of Ω r. (8) Ω T = Ω (a, a + T ], a is some real number, is a cylinder, p Ω T = Ω [a,a + T ] Ω t = a} is its parabolic boundary. (9) Q r = B r ( r 2, 0] is a parabolic cube, Q r (x, t) = Q r + (x, t), p Q r = B r [ r 2, 0] B r r 2 } is its parabolic boundary, Q + r = B r + ( r2, 0], Q + r (x, t) = Q + r + (x, t), T r = T r [ r 2, 0], and T r (x, t) = T r + (x, t). (0) K r = B r ( r 2 /2,r 2 /2] is a centered parabolic cube, K r (x, t) = K r + (x, t). () Ω r = Ω r ( r 2, 0] and Ω r (x, t) = Ω r + (x, t). (2) Ω p r = Ω r [ r 2, 0] Ω r t = r 2 } is the parabolic boundary of Ω r, Ω w r = w Ω r [ r 2, 0] is the wiggled part of Ω p r, and Ω c r = Ω p r \ Ω w r is the covered part of Ω p r.

6 2.2. Matrix of coefficients S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) Definition 2.. We say that A is uniformly parabolic if there exists a positive constant Λ such that Λ ξ 2 A(x, t)ξ ξ Λ ξ 2, a.e. (x,t) R n R, ξ R n. () We write A =a ij } to mean an n n matrix with (i,j)th entry a ij. (2) A = ni,j= (A : A) = aij 2 and A = sup (y,s) A(y, s). (3) A is supposed to be uniformly parabolic. (4) A is supposed to be (δ,r)-vanishing (see Definition.). Remark 2.2. In this paper A is allowed to be nonsymmetric Notation for function () If u : Ω T R, we write u(x, t)((x, t) Ω T )). If f : Ω T R n, we write f(x, t) = (f (x,t),...,f n (x, t)). (2) f Kr = f(x,t)dx dt K r K r is the average of f over K r Notation for derivatives () u = (u x,...,u xn ) is the gradient of u with respect to spatial variable x. (2) div f = n i= (f i ) xi = (f i ) xi is the divergence of f = (f,f 2,...,f n ) Notation for estimates We employ the letter C to denote a universal constant depending usually on the dimension, uniform parabolicity, and the geometric quantities of p Ω T Function spaces () The space C0 (Ω T ) is the Banach space consisting of all infinitely differentiable functions which vanish near p Ω T. (2) The Sobolev space Wp,0 (Ω T ),<p<, is the Banach space consisting of all elements of L p (Ω T ) having a finite norm If p = 2, we usually write ) u W,0 p (Ω ( u T ) = p /p L p (Ω T ) + u p L p (Ω T ). H,0 (Ω T ) = W,0 2 (Ω T ).

7 50 S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) The letter H is used, since as we see H,0 (Ω T ) is a Hilbert space with scalar product (u, v) H,0 (Ω T ) = (uv + u v)dx dt. Ω T (3) The Sobolev space W, (Ω T ) is the Banach space consisting of all elements of L p (Ω T ) having a finite norm u W, (Ω T ) = ess sup Ω T u +ess sup u +ess sup u t. Ω T Ω T Remark 2.3. In present work we want to obtain L p estimates in W,p (Ω T ). For some technical reason, one can assume that a weak solution considered hereafter is defined on Ω R from the following classical argument: The solution u and the equation can be extended forward by taking f = 0 so that all properties in question are preserved. For backward extension one can use the zero extension of u. We introduce a certain non-isotropic Sobolev space whose members have weak derivatives of spatial order and time order /2 lying in the L 2 spaces. Definition 2.4. We say u W,p (Ω T ),<p<, ifu Wp,0 (Ω T ) and there exist functions F L p (Ω T ; R n ) and g L p (Ω T ) such that u t = div F g in Ω T in the sense that for all φ C 0 (Ω T ), uφt F φ gφ } dx dt = u(x, T )φ(x, T) dx, (2.) Ω T Ω where Ω T = Ω (0,T]. Furthermore, we define its norm by u,p W (Ω T ) = u Wp,0 (Ω T ) + inf ( ) } /p ( F p + g p )dxdt, Ω T where the infimum runs over all the functions satisfying (2.). Remark 2.5. Sometimes we write above definition as ( ) W,p (Ω T ) = L p 0,T; W,p (Ω) W,p (0,T; W,q (Ω)),

8 S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) where <p< and p + q =. In particular, for the classic case that p = 2 (see [3,27]), we have ( ) W,2 (Ω T ) = L 2 0,T; H (Ω) H /2 (0,T; L 2 (Ω)) L 2 (Ω T ), (2.2) that is, W,2 (Ω T ) is compact in L 2 (Ω T ) Preliminary tools We use the following standard arguments of measure theory. Lemma 2.6 (Caffarelli and Cabré [9]). Suppose that f is a nonnegative and measurable function in R n R. Suppose further that f has a compact support in a bounded subset E of R n R. Let θ > 0 and m> be constants. Then for 0 <p<, f L p (E) if and only if S := m kp (x, t) E : f(x,t)>θm k} < k and C S f p L p (E) C( E +S), where C>0 is a constant depending only on θ, m, and p. We use the Hardy Littlewood maximal function. Definition 2.7. Let f be a locally integrable function. Then (Mf )(x, t) = sup r>0 f(y,s) dy ds K r (x, t) K r (x,t) is called the Hardy Littlewood maximal function of f. We also use M E f = M ( χ E f ), if f is not defined outside E. We will dropthe index E if E is understood clearly in the context. The basic theorem for the Hardy Littlewood maximal function is the following. Theorem 2.8 (Stein [4]). If f L p (R n R) withp>, then Mf L p (R n R). Moreover, Mf L p C f L p. (2.3)

9 52 S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) If f L (R n R), then (x, t) R n R : (Mf )(x, t) > α} C α f dx dt. (2.4) Inequality (2.3) is called strong p p estimates and (2.4) is called weak estimates. The techniques involved in this paper are those of a general measure-theoretic flavor; indeed, one of the main tool is the Vitali covering Lemma. Lemma 2.9 (Mattila [34]). Let E be a measurable set. Suppose that a class of balls B α covers E: E α B α. Suppose the radius of B α is bounded from above. Then there exist a disjoint B αi } i= B α } α such that E i 5B αi, where 5B αi is the ball with 5 times the radius of B αi. Consequently, we have E 5 n i B αi. We use a variant of the Vitali covering lemma. Theorem 2.0. Let 0 < ε < and E F Ω := Ω R be two measurable sets. Assume that Ω is (δ, )-Reifenberg flat and Assume that the following property holds: E < ε K. (2.5) (x, t) Ω, r (0, ] with E K r (x, t) ε K r (x, t), K r (x, t) Ω F. (2.6) Then E ( ) 0 n+2 ε F. δ Proof. In view of (2.5), for a.e. (x, t) E, there is r = r (x,t) (0, ] such that E K r(x,t) (x, t) =ε K r(x,t) (2.7)

10 and for all 0 <r (x,t) <r, S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) E K r(x,t) (x, t) < ε K r(x,t). (2.8) Since K r(x,t) (x, t) E : (x, t) E} is a covering of E, by the Vitali s covering lemma, there exists a disjoint K ri (x i,t i ) E : (x i,t i ) E} i= such that E i K 5ri (x i,t i ) and E 5 n+2 K ri. (2.9) Then it follows from (2.7) and (2.8) that E K 5ri (x i,t i ) < ε K 5ri (x i,t i =5 n+2 ε K ri (x i,t i =5 n+2 E K ri (x i,t i ). (2.0) Observe that δ is small enough and we will claim that sup 0<r sup (x,t) Ω ( ) K r (x, t) 2 n+2 K r (x, t) Ω. (2.) δ To do this, choose any r (0, ] and any x Ω. Ifdist(x, Ω) r, it follows from the fact B r (x) Ω. So suppose that dist(x, Ω) <r. Then there exists y Ω so that dist(x, Ω) = dist(x,y)<r. Since Ω is (δ, )-Reifenberg flat, without loss of generality, we may assume B r (x) x n > δ} B r (x) Ω B r (x) x n > δ} in some appropriate coordinate system in which y = 0. Then from the geometry and an easy observation, we see that ( ) K r (x, t) K r (x, t) Ω K r (x, t) 2 n+2 K r (x, t) x n > δ}, δ which shows (2.). Finally, by (2.9) (2.) and (2.6), we get ( E = B5ri (x i,t i ) E ) i i K 5ri (x i,t i ) E

11 54 S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) < ε i K 5ri (x i,t i ) = 5 n+2 ε i K ri (x i,t i ) ( ) 2 n+2 5 n+2 ε ( ) K ri (x i,t i Ω ) δ i ( ) 0 n+2 ( ) = ε Kri (x δ i,t i ) Ω ( ) 0 n+2 ε F, δ i which completes the proof. 3. Interior estimates This section will be devoted to obtain interior W,p estimates for 2 <p< concerning the following divergence form parabolic equation u t div(a u) = div f (3.) in a bounded parabolic cylinder in Ω T = Ω (a, a + T ] R n R, where a>0 is some real number. Our main condition is that the matrix A(x, t) of coefficients is (δ,r)-vanishing. In light of our scaling structure, we denote by Q R = B R ( R 2, 0] to mean the circular cylinder of radius R, height R 2, and topcenter point (0, 0). We will use the following definition. Definition 3.. We say that u L 2 ( R 2, 0; H0 (Q R) ) H ( R 2, 0; H (Q R ) ) is a weak solution of (3.) in Q R if uφt A u φ f φ } dx dt = u(x, 0)φ(x, 0)dx (3.2) Q R B R for all φ C 0 (Q R). Remark 3.2. We remark that our above definition is actually equivalent to the popular one in [6], that is, we say a function ( ) ( ) u L 2 R 2, 0; H0 (B R), with u t L 2 R 2, 0; H (B R )

12 S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) is a weak solution of (3.) if u t, φ + A u φ dx = f φ dx B R B R for each φ H 0 (B R) and a.e. time R 2 t 0, where, is the paring of H (B R ) and H 0 (B R), and H is the dual space of H 0. Let us state then main theorem of this section. Theorem 3.3. Assume that u is a weak solution of (3.) in Ω T and K r is a centered parabolic cube with K 7r Ω T. If (x, t) : M( u 2 )(x, t) > N 2} K r ε K r, then K r (x, t) : M( u 2 )(x, t) > } (x, t) : M( f 2 )(x, t) > δ 2 }. We start with the following standard energy estimate which will be used in the proof of Lemma 3.8. Lemma 3.4. Let u be a weak solution of the parabolic PDE (3.) in Q 2. Then we have u 2 dx dt C ( u 2 + f 2 )dxdt. Q Q 2 Proof. Assume first that u is smooth. Let η = η(x, t) be a smooth cut-off function, that is, 0 η, η = onq, and η = 0 near p Q 2. (3.3) Then one can replace the test function φ by η 2 u in Remark 3.2, that is, one can multiply the Eq. (3.) by η 2 u. Using the integration by parts formula over B 2,wefind u t (η 2 u) dx A u (η 2 u) dx = f (η 2 u) dx. B 2 B 2 B 2 Now we write the resulting expression as I + I 2 = I 3 + I 4, for I = η t B2 2 u 2 2 dx, I 2 = η 2 (A u u) dx, B 2

13 56 S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) I 3 = ηη t u 2 dx 2 B 2 ηu(a u η)dx, B 2 I 4 = f (η 2 u) dx. B 2 Estimate of I 2 : Estimate of I 3 : I 2 = η 2 (A u u) dx Λ η 2 u 2 dx. B 2 Ω 2 I 3 = ηη t u 2 dx 2 B 2 ηu(a u η)dx B 2 C( + /τ) u 2 dx + Cτ B 2 η 2 u 2 dx. B 2 Estimate of I 4 : I 4 = f (η 2 u) dx B 2 = ((f u)η 2 + 2(f η)ηu) dx B 2 τ η 2 u 2 dx + ( f 2 dx + C u 2 + f 2) dx B 2 4τ B 2 B 2 τ η 2 u 2 dx + C u 2 dx + C( + /τ) f 2 dx. B 2 B 2 B 2 We then combine the estimates I i ( i 4) to find η t B2 2 u 2 dx + Λ η 2 u 2 dx 2 B 2 I + I 2 = I 3 + I 4 C( + /τ) u 2 dx + Cτ B 2 η 2 u 2 dx B 2 +τ η 2 u 2 dx + C B u 2 dx + C( + /τ) B f 2 dx B 2 C( + /τ) ( u 2 + f 2 )dx+ Cτ B 2 η 2 u 2 dx. B 2

14 Taking τ so small, we see S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) t η B2 2 u 2 2 dx + η 2 u 2 dx C ( u 2 + f 2 )dx. B 2 B 2 Integrating with respect to time from 4 to 0 and noting (3.3), we see u 2 dx dt C ( u 2 + f 2 )dxdt. (3.4) Q Q 2 By approximation we find the inequality (3.4) holds with the smooth function u replaced by our weak solution. In fact we take sequences of smooth functions A k } and f k }. Then we can proceed with a sequence of corresponding smooth solutions u k }. We would like to point out that the intrinsic Sobolev space W,2 (Q R ) is compactly embedded in L 2 (Q R ), which is crucial in the following two lemmas. For this we refer to [3,27,33]. Lemma 3.5. Let u be a weak solution of (3.) in Q 2. Then we have u u Q 2 L 2 (Q ) C( u 2 L 2 (Q ) + f 2 L 2 ). (3.5) (Q ) Proof. We prove it by contradiction. If not, there exist A k } k=, u k} k= and f k} k= such that u k is a weak solution of in Q with (u k ) t div(a k u k ) = div f k (3.6) ) u k u kq ( u 2 L 2 (Q ) k k 2 L 2 (Q ) + f k 2 L 2. (Q ) We can normalize so that u k u kq L 2 (Q + ) =, to see and u k u kq ( u 2 W,2 C k 2 (Q ) L 2 (Q ) + u k 2 L 2 (Q ) + f k 2 L 2 (Q ) C( + /k) C ) u k 2 L 2 (Q ) + f k 2 L 2 (Q ) 0ask. (3.7) k

15 58 S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) In view of our compactness argument (see (2.2)), there exists a subsequence of u k u kq }, which we denote as u k u k } and a function u 0 W,2 (Q ) such that Moreover, we have u k u k u 0 in L 2 (Q ), u k u 0 in W,2 (Q ). (3.8) Now we want to claim that u 0 is a weak solution of u 0Q = 0, u 0 L 2 (Q ) =. (3.9) (u 0 ) t = 0 (3.0) in Q. To do this, choose any φ C0 (Q ). Then by (3.6), we get (uk u k )φ t A k u k φ f k φ } dx dt = (u k u k )(x, 0)φ(x, 0)dx. Q B Let k in the above identity to find u 0 φ t dx dt = u 0 (x, 0)φ(x, 0)dx Q B in view of (3.7) and (3.8), which shows (3.0). Then in light of (3.0) and (3.9), u 0 = 0, which is a contradiction. Once we get Lemma 3.5, we can obtain the following estimate by using PDE (3.) and our definition of an intrinsic Sobolev space W,2 (Q R ). Corollary 3.6. Let u be a weak solution of (3.) in Q 2. Then we have u u Q 2 W,2 (Q ) C( u 2 L 2 (Q ) + f 2 L 2 (Q ) ). Lemma 3.7. For any ε > 0, there is a δ = δ(ε) >0 such that for any weak solution u of the parabolic PDE (3.) in Q 4 with u 2 dx dt (3.) Q 4 Q 4 and Q 4 Q 4 ( f 2 + A A Q4 2 )dxdt δ 2, (3.2)

16 S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) there exists a weak solution v of v t div ( A Q4 v ) = 0 in Q 4 such that Q 4 (u u Q4 ) v 2 dx dt ε 2. (3.3) Proof. We prove this lemma by contradiction. If not, there exist ε 0 > 0, A k } k=, u k } k= and f k} k= such that in Q 4 and u k 2 dx dt, Q 4 Q 4 (u k ) t div(a k u k ) = div f k (3.4) ( f k 2 + A k A kq4 2 ) Q 4 Q 4 k 2. (3.5) But, Q 4 (u k u kq4 ) v k 2 dx dt > ε 2 0. (3.6) for any weak solution v k that solves (v k ) t div(a kq4 v k ) = 0inQ 4. (3.7) By (3.4), (3.5) and Corollary 3.6, u k u kq4 },2 k= is bounded in W (Q 4 ). In view of our compactness argument (see (2.2), it has a subsequence, which we still denote as u k u kq4 }, such that u k u kq4 u 0 in L 2 (Q 4 ), u k u kq4 u 0 in W,2 (Q 4 ) (3.8) for some u 0 W,2. Since A kq4 } k= is bounded in L, it has a subsequence, which we still denote as A kq4 } k=, such that A kq4 A 0 0ask (3.9) for some constant matrix A 0. But then, by (3.5), we have A k A 0 in L 2 (Q 4 ). (3.20)

17 60 S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) Now, we will show that u 0 is a weak solution of (u 0 ) t div(a 0 u 0 ) = 0 (3.2) in Q 4. To do this, choose any φ C0 (Q 4). From (3.5), we have (uk u kq4 )φ t A k u k φ f k φ } dx dt = (u k u kq4 )(x, 0)φ(x, 0)dx. Q 4 B 4 We recall (3.8), (3.9) and (3.5), to find upon passing to weak limits that u0 φ t u 0 φ } dx dt = u 0 (x, 0)φ(x, 0)dx, Q 4 B 4 which shows (3.2). Note that (u 0 ) t div ( A kq4 u 0 ) = (u0 ) t div ([ A kq4 A 0 ] u0 ) div (A0 u 0 ) = div ([ A kq4 A 0 ] u0 ) + (u0 ) t div (A 0 u 0 ) = div ([ A kq4 A 0 ] u0 ) in O 4, where we have used (3.2). Let h k be the weak solution of (hk ) t div(a kq4 h k ) = div([a kq4 A 0 ] u 0 ) in Q 4, h k = 0 on p Q 4. (3.22) Then u 0 h k is a weak solution of (u 0 h k ) t div ( A kq4 (u 0 h k ) ) = 0 (3.23) in Q 4. Furthermore, by (3.22), we get h k L 2 (Q 4 ) h k H, (Q 4 ) C (A kq4 A 0 ) u 0 L 2 (Q 4 ) C A kq4 A 0 u 0 L 2 (Q 4 ) C A kq4 A 0.

18 Thus, we have S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) (u k u kq4 ) (u 0 h k ) L 2 (Q 4 ) (u k u kq4 ) u 0 L 2 (Q 4 ) + h k L 2 (Q 4 ) (u k u kq4 ) u 0 L 2 (Q 4 ) +C A kq4 A 0. This estimate, (3.8) and (3.9) imply that (u k u kq4 ) (u 0 h k ) L 2 (Q 4 ) 0ask. But this is a contradiction to (3.6) by (3.23). Corollary 3.8. For any ε > 0, there is a small δ = δ(ε) >0 such that for any weak solution u of (3.) in Q 4 with u 2 dx dt, Q 4 Q 4 there exists a weak solution v of ( f 2 + A AQ5 2 ) dx dt δ 2, (3.24) Q 4 Q 4 v t div(a Q4 v) = 0 in Q 4 such that (u v) 2 L 2 (Q 2 ) ε2. Proof. In view of (3.24) and Lemma 3.7, for any α > 0, there is a small δ = δ(α) and a corresponding weak solution v of v t div(a Q4 v) = 0 in Q 4 such that provided that Q 4 (u u Q4 ) v 2 α 2 (3.25) (f 2 + A A Q4 2 )dxdt δ 2. Q 4 Q 4 First we see that w := (u u Q4 ) v is a weak solution of w t div(a w) = div [ f ( ) ] A A Q4 v (3.26)

19 62 S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) in Q 4.Now(3.26) and Lemma 3.4 imply that (u v) 2 L 2 (Q 2 ) C( (u u Q 4 ) v 2 L 2 (Q 3 ) + f ( A A Q4 ) v 2 L 2 (Q 3 ) ) C( (u u Q4 ) v 2 L 2 (Q 3 ) + f 2 L 2 (Q 3 ) + A A Q4 2 L 2 (Q 3 ) ) C( (u u Q4 ) v 2 L 2 (Q 4 ) + f 2 L 2 (Q 4 ) + A A Q4 2 L 2 (Q 4 ) ). Here we have used the interior W, finally regularity for v. This estimate and (3.25) imply (u v) 2 L 2 (Q 2 ) C(α2 + Q 4 δ 2 ) = ε 2 by taking α and δ satisfying the last identity above. This completes our proof. Lemma 3.9. There is a constant N so that for any ε > 0, there exists δ = δ(ε) >0 and if u is a weak solution of (3.) in Ω T = Ω (a, a + T ] Q 7 (0, 2) with Q (x, t) : M( u 2 ) } (x, t) : M( f 2 ) δ 2 } = (3.27) and A uniformly parabolic and (δ, 7)-vanishing, then (x, t) : M( u 2 )>N 2 } Q < ε Q. Proof. From condition (3.27), we see that there is a point (x 0,t 0 ) Q such that u 2, K r K r (x 0,t 0 ) Ω T Since Q 4 (0, 2) K 6 (x 0,t 0 ),by(3.28), we have f 2 dx dt K 6 Q 4 Q 4 (0,2) Q 4 K 6 Similarly, we see that f 2 δ 2, K r (x 0,t 0 ). (3.28) K r K r (x 0,t 0 ) Ω T u 2 dx dt Q 4 Q 4 (0,2) f 2 dx dt K 6 (x 0,t 0 ) ( ) 6 n+2 δ 2. (3.29) 4 ( ) 6 n+2. (3.30) 4

20 S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) In view of (3.29), (3.30) and from the assumption on A, we can apply Corollary 3.8 with u replaced by ( 4 6 ) n+2 u, f by ( 46 ) n+2 f and Q4 by Q 4 (0, 2), respectively, to find for any α > 0, there is a small δ = δ(α) and a corresponding weak solution v of v t div(a Q4 (0,2) v) = 0 (3.3) in Q 4 (0, 2) such that (u v) 2 dx dt α 2 (3.32) Q 2 (0,2) provided that ( f 2 + A A Q4 (0,2) 2) dx dt δ 2. Q 4 Q 4 (0,2) Then in view of (3.3), we can use the interior W, is a constant N 0 so that regularity of v to see that there Now we set N 2 := max4n 2 0, 2n+2 } and claim that sup v 2 }=N0 2. (3.33) Q 3 (0,2) (x, t) : M( u 2 )>N 2 } Q (x, t) : M Q4 (0,2)( (u v) 2 )>N 2 0 } Q. (3.34) To prove this, suppose that (x,t ) } (x, t) : M Q4 (0,2)( (u v) 2 ) N0 2 Q. (3.35) For r 2, K r (x,t ) Q 3 (0, 2) and by (3.33) and (3.35), we have u 2 dx dt 2 ( (u v) 2 + v 2 )dxdt 4N0 2 K r K r (x,t ) K r. Q 3 (0,2) For r>2, K r (x,t ) K 2r (x 0,t 0 ) and by (3.28), we have u 2 dx dt u 2 dx dt 2 n+2. K r K r (x,t ) Ω T K r K 2r (x 0,t 0 ) Ω T Thus (x,t ) (x, t) : M( u 2 ) N 2 } Q. (3.36) The claim (3.34) easily follows from (3.35) and (3.36).

21 64 S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) Now (3.34), parabolic weak estimates (see Theorem 2.8) and (3.32) finally yield: (x, t) : M( u 2 )>N 2 } Q (x, t) : M Q4 (0,2)( (u v) 2 )>N0 2 } Q < C N0 2 (u v) 2 dx dt Q 2 (0,2) C N 2 0 α 2 = ε Q, by taking α (and δ) satisfying the last equality above. This completes our proof. Remark 3.0. By a scaling argument we can substitute Q R (0, ε+) for Q 7 (0, 2) in Lemma 3.2 as long as R>. Now we are set to prove the main theorem of this section, Theorem 3.3. Proof. The proof comes directly from Lemma 3.9 and a scaling argument. We end this section by stating an interior regularity theory whose proof is coming from the global regularity theory addressed in coming section with u replaced by u for an appropriately chosen cut-off function. Theorem 3.. Let p be a real number with <p< and let R>0. There is a small δ = δ(λ,p,n,r) > 0 so that for all A with A uniformly parabolic and (δ,r)- vanishing, and for all f with f L p (Q 2 ), if u is a weak solution of the parabolic PDE (3.) in Ω T Q 2, then u belongs to W,p (Q ) with the estimate u,p W (Q ) C ( ) u L p (Q 2 ) + f L p (Q 2 ), where the constant Cis independent of u and f. 4. The Neumann problem in Reifenberg domains In this section we study an optimal regularity requirement on Ω for W,p estimates. The boundary condition considered in this paper is that Ω is (δ,r)-reifenberg flat. Let us start with the following classical theory. Definition 4.. We say that u L 2 (0,T; H (Ω)) H (0,T; H (Ω)) is a weak solution of (.) if uφt A u φ f φ } dx dt = u(x, 0)φ(x, 0) dx (4.) Ω T Ω for all φ C (Ω T ).

22 S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) Theorem 4.2 (Baiocchi [3], Ladyzhenskaya et al. [27]). There exists a unique (up to a constant) weak solution of the parabolic PDE (.). Now we would like to refer to some geometric notations given in Section 2 and point out that our weak solution is supposed to be defined on Ω R. For our purpose we localize our interest into the case B R x n > δ} Ω R B + R with δ small enough. Then we consider a weak solution of u t div(a u) = div f in Ω R, (A u + f) ν = 0 on w Ω R (4.2) and a corresponding weak solution to vt div ( à v ) = 0 in Q + R à v ν = 0on T R, (4.3) where à is a constant matrix with A Q + R à sufficiently small. We will henceforth write for any R>0, A( Ω R ) =φ C (Q R ) : φ = 0on B R [ R 2, 0] Ω R t = R 2 } to denote this class of admissible functions φ. Note that we have the Dirichlet data on and the Neumann data on c B + R [ R2, 0] Ω R t = R 2 } w Ω R. We will also use the notation A(Q + R ) to mean the collection of all elements in C (Q R ) satisfying φ = 0on B R x n > 0} [ R 2, 0] B + R t = R2 }. Let us introduce the following definition. Definition 4.3. We say that u L 2 ( R 2, 0; H (Ω R ) ) H ( R 2, 0; H (Ω R ) ) is a weak solution of (4.2) if for all φ A( Ω R ), uφt A u φ f φ } dx dt = u(x, 0)φ(x, 0)dx. Ω R Ω R

23 66 S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) We say that v L 2 ( R 2, 0; H (B R + )) H ( R 2, 0; H (B R + )) is a weak solution of (4.3) if for all φ A(Q + R ), vφt à v φ } dx dt = v(x, 0)φ(x, 0) dx. Q + R Theorem 4.4. Let u be a weak solution of the parabolic PDE (4.2) in Ω. Then we have ( ) u u Ω 2 L 2 ( Ω C u 2 ) L 2 ( Ω ) + f 2 L 2 ( Ω. ) B + R Proof. We prove it by contradiction. If not, there exist A k } k=, u k} k= and f k} k= such that u k is a weak solution of (u k ) t div(a k u k ) = div f k in Ω, (A k u k + f k ) ν = 0 on Ω (4.4) w with ( ) u k u k Ω 2 L 2 ( Ω k u k 2 ) L 2 ( Ω + f k 2 ) L 2 ( Ω. ) We can normalize so that u k u k Ω L 2 ( Ω =, and we have ) u k u k Ω 2 W,2 ( Ω ) C( u k u k Ω 2 L 2 ( Ω ) + u k 2 L 2 ( Ω ) + f k 2 L 2 ( Ω ) ) C( + /k) C and u k 2 L 2 ( Ω + f k 2 ) L 2 ( Ω /k 0ask. (4.5) ) Then in view of our compactness argument (see (2.2)), there exists a subsequence of u k u k Ω }, which we denote as u k u k }, and a function u 0 W,2 ( Ω ) such that Moreover, we have u k u k u 0 in L 2 ( Ω ), u k u 0 in W,2 ( Ω ). (4.6) u 0 Ω = 0, u 0 L 2 ( Ω =. (4.7) ) Now we want to claim that u 0 is a weak solution of in Ω with the corresponding Neumann boundary condition. (u 0 ) t = 0 (4.8)

24 S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) To do this, choose any φ A( Ω ). Then by (4.4), we get (uk u k )φ t A k u k φ f k φ } dx dt = (u k u k )(x, 0)φ(x, 0)dx. Ω Ω Note (4.5) and (4.6) and take k in the identity above to discover u 0 φ t dx dt = u 0 (x, 0)φ(x, 0)dx, Ω Ω = 0, which is a contra- which shows (4.8). Then in light of (4.8) and (4.7), u 0 diction. Lemma 4.5. Let u be a weak solution of the parabolic PDE (4.2) in Ω. Then we have u u Ω 2 W,2 ( Ω ) C( u 2 L 2 ( Ω ) + f 2 L 2 ( Ω ) ). Proof. We turn to PDE (4.2) to invoke Definition 2.4. Then we apply Theorem 4.4 to get the following estimates: u u Ω 2 W,2 ( Ω u u ) Ω 2 L 2 ( Ω ) + u 2 L 2 ( Ω + (A u + ) f) 2 L 2 ( Ω ) ( ) C u 2 L 2 ( Ω 2 ) + f 2 L 2 ( Ω. 2 ) The following lemma is so-called compactness method. Lemma 4.6. For any ε > 0, there exists a small δ = δ(ε) >0 such that for any weak solution u of (4.2) in Ω 4 with and u 2 dx dt, Q 4 Ω 4 B 4 x n > δ} Ω 4 B + 4 (4.9) ( f 2 + A A Q 4 Ω 2 Ω 4 )dxdt δ 2, (4.0) 4 there exist a constant matrix à with A Ω Ã 4 ε and a corresponding weak solution v of (4.3) in Q + 4 such that (u u Q + ) v 2 dx dt ε 2. (4.) 4 Q + 4

25 68 S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) Proof. If not, there exist ε 0 > 0, A k } k=, u k} k=, and Ω k } 4 such that u k is a k= weak solution of with (u k ) t div(a k u k ) = div f k in Ω k 4, (A k u k + f k ) ν = 0 on Ω k w 4 (4.2) B 4 x n > /k} Ω k 4 B+ 4 (4.3) and u Q 4 Ω k k 2 dx dt, 4 Q 4 Ω 4( f k k 2 + A k A k Ω k 2 )dxdt /k 2. (4.4) 4 But, Q + 4 (u k u k Q + 4 ) v 2 dx dt > ε 2 0 (4.5) for any constant matrix à with A k Ω k à ε 0 and for any corresponding weak 4 solution v of (4.3) in Q + 4. In view of Lemma 4.5 and (4.4), u k u Ω },2 4 k= is uniformly bounded in W (Q + 4 ). So there exists a subsequence, which we denote as u k u k }, such that u k u k u 0 in W,2 (Q + 4 ) and u k u k u 0 in L 2 (Q + 4 ) (4.6) } for some u 0 in W,2 (Q + 4 A ). Since k Q + is uniformly bounded in 4 k= L, it has a subsequence, which we denote as } A k, such that A k A 0 0ask. (4.7) But then, by (4.7) and (4.4), we have A k A 0 in L 2 (Q + 4 ) (4.8) for some constant matrix A 0. Now we will show that u 0 itself is a weak solution of (u 0 ) t div(a 0 u 0 ) = 0 (4.9)

26 S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) in Q + 4 with the corresponding Neumann boundary condition. To do this, fix any φ A(Q + 4 ). Then by (4.2), we have (uk u k )φ t A k u k φ } dx dt + (uk u k )φ t A k u k φ } dx dt Q + 4 = = = Ω k 4 Ω k 4 Q (uk u k )φ t A k u k φ } dx dt f k φ dx dt + f k φ dx dt + B + 4 Ω k 4 Ω k 4 \Q+ 4 (u k u k )(x, 0)φ(x, 0)dx Ω k 4 \Q+ 4 f k φ dx dt (u k u k )(x, 0)φ(x, 0)dx+ We write the resulting equality above as Ω k 4 \B+ 4 (u k u k )(x, 0)φ(x, 0)dx. where First note that I = I 2 + I 3 + I 4 + I 5 + I 6 + I 7, I = (uk u k )φ t A k u k φ } dx dt, Q + 4 I 2 = f k φ dx dt, Q + 4 I 3 = Ω k 4 \Q+ 4 (u k u k ) φ t dx dt, I 4 = A Ω k k u k φ dx dt, 4 \Q+ 4 I 5 = f k φ dx dt, I 6 = I 7 = Ω k 4 \Q+ 4 B + 4 Ω k 4 \B+ 4 (u k u k )(x, 0)φ(x, 0)dx, (u k u k )(x, 0)φ(x, 0)dx. for some constant C independent of k. u k u k W,2 ( Ω k C, f k 4 ) L 2 ( Ω k C (4.20) 4 )

27 70 S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) Estimate of I 3 : By Cauchy s inequality and (4.20), we have I 3 = (u k u k ) φ t dx dt C Ω k 4 \Q+ 4 Ω k 4 \Q+ 4 u k u k 2 dx dt Ω k 4 \Q+ 4 dx dt C/ k. Estimate of I 4 : By Cauchy s inequality and (4.20), we have I 4 = A Ω k k u k φ dx dt 4 \Q+ 4 C u Ω k k dx dt 4 \Q+ 4 C u k 2 dx dt dx dt C/ k. Ω k 4 \Q+ 4 Ω k 4 \Q+ 4 Estimate of I 5 : By Cauchy s inequality and (4.20), we have I 5 = f Ω k k φ dx dt 4 \Q+ 4 C f k dx dt C C/ k. Ω k 4 \Q+ 4 Ω k 4 \Q+ 4 f k 2 dx dt Ω k 4 \Q+ 4 dx dt Estimate of I 7 : Cauchy s inequality, (4.20) and (4.3), we have I 7 = (u k u k )(x, 0)φ(x, 0)dxdx C Ω k 4 \B+ 4 Ω k 4 \B+ 4 (u k u k )(x, 0) 2 dx dt Ω k 4 \B+ 4 dx dt C/ k.

28 S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) We finally combine estimates I i s, to discover (uk u k )φ t A k u k φ f k φ } dx dt Q + 4 = B + 4 (u k u k )(x, 0)φ(x, 0)dx+ O(/ k) as k. Then in view of this identity, (4.6), (4.8) and (4.4), we have Q + 4 u0 φ t A 0 u 0 φ } dx dt = B + 4 u 0 (x, 0)φ(x, 0)dx, which says (4.9). Finally, we get a contradiction to (4.5) by taking à = A 0, v = u 0 and k large enough. Corollary 4.7. For any ε > 0, there exists a small δ = δ(ε) >0 such that for any weak solution u of (4.2) in Ω 4 with B 4 x n > δ} Ω 4 B + 4 (4.2) and u 2 dx dt, Q 4 Ω 4 ( f 2 + Q 4 Ω 4 ) 2 A A Ω 4 δ 2, (4.22) there exist a constant matrix à with solution v of (4.3) in Q + 4 such that A Ω Ã 4 ε and a corresponding weak Q + 2 (u v δ ) 2 dx dt ε 2, (4.23) where v δ (x, t) = v(x + δe n,t) for (x, t), that is, the function v translated a distance δ in the e n direction. Proof. By Lemma 4.6, (4.2) and (4.22), for any α > 0, there exist δ = δ(α), a constant matrix à with A Ω Ã 4 α, and a corresponding weak solution v of (4.3) in Q + 4 such that u v 2 dx dt α 2 (4.24) Q + 4

29 72 S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) provided that ( f 2 ) 2 + A A Q 4 Ω dx dt + Ω 4 D( w Ω 4,T 4 ) δ 2, 4 where D denotes the Hausdorff distance. Now set w := u v δ and select a standard smooth cut-off function η = η(x, t) satisfying 0 η, η = onq 2, and η = 0 near p Q 3. (4.25) Now in light of approximation argument, we may assume that w is smooth and η 2 w A( ). Then by the definition of our weak solutions, we have ( ( ) ( )} u η Ω3 2 w )t A u η 2 w f η 2 w dx dt ( ) = u(x, 0) η 2 w (x, 0) dx (4.26) Ω 3 and Q + 3 ( ) ( ) v η 2 w à v η 2 w dx dt = t B 3 + ( ) v(x, 0) η 2 w (x, 0) dx. (4.27) By elementary computation using (4.26) and (4.27), we have ( ) ( ) w η 2 w dx dt A w η 2 w dx dt t = ( f + (A Ã) v Ω3 δ) ( ) η 2 w dx dt + ( ) + (v δ v)(x, 0) η 2 w (x, 0)dx+ B 3 + Q + 3 \Q + 3 ( ) (v δ v) η 2 w t à (vδ v) ( ) w(x, 0) Ω 3 η 2 w (x, 0)dx ( ) v δ (x, 0) η 2 w (x, 0)dx Ω 3 \B + 3 ( η 2 w ( ) ( )} v δ η 2 w t à vδ η 2 w dx dt. Then using integration by parts formula in t and the identity t )} dx dt η Ω3 2 w 2 ( ) } w 2 dx = t η 2 w + w 2 ηη t dx Ω 3

30 S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) for a.e. 9 t 0, we can write the resulting expression as I + I 2 = I 3 + I 4 + I 5 + I 6 + I 7 + I 8 + I 9 + I 0, where 0 } I = η 9 t Ω3 2 w 2 2 dx dt, I 2 = A w wdxdt, I 3 = 2ηw A w η dx dt, } I 4 = 2ηw f η + η Ω3 2 f w dx dt, } I 5 = 2ηw(A Ã) v Ω3 δ η + η 2 (A Ã) v δ w dx dt, ( ) I 6 = (v δ v)(x, 0) η 2 w (x, 0)dx, B 3 + ( ) I 7 = v δ (x, 0) η 2 w (x, 0)dx, Ω 3 \B 3 + I 8 = w 2 ηη t dx dt, ( ) ( )} I 9 = v δ η 2 w \Q + t à vδ η 2 w dx dt, 3 ( ) ( )} I 0 = (v δ v) η 2 w Q + t à (vδ v) η 2 w dx dt. 3 Estimate of I : I = 0 9 t } η Ω3 2 w 2 2 dx dt 0. Estimate of I 2 : Uniformly parabolicity condition implies that I 2 = η 2 A w wdxdt Λ η 2 w 2 dx dt.

31 74 S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) Estimate of I 3 : The condition on A and Cauchy s inequality with τ imply that I 3 = 2ηw A w η dx dt A L 2 η w w η dx dt Cτ η 2 w 2 dx dt + C w η 2 dx dt. τ Estimate of I 4 : The Cauchy s inequality with τ and (4.25) imply that } I 4 = 2ηwf η + η Ω3 2 f w dx dt } 2 η f w η +η Ω3 2 f w dx dt η 2 f 2 + η 2 w 2 + 4τ } Ω η2 f 2 + τη 2 w 2 dx dt 3 ( + /4τ) f 2 dx dt + w η 2 dx dt + τ η 2 w 2 dx dt. Estimate of I 5 : } I 5 = 2ηw(A Ã) v Ω3 δ η + η 2 (A Ã) v δ w dx dt v δ L ( ) } 2 A Ã w η +η Ω3 2 A Ã w dx dt C ( + /4τ) A Ã 2 dx dt + w η 2 dx dt + τ η 2 w 2 dx dt. Estimate of I 6 : ( ) I 6 = (v δ v)(x, 0) η 2 w (x, 0)dx B 3 + ) (v δ v)(x, 0) L 2 (B (η 3 + ) 2 w (x, 0) L 2 (B + 3 ) C (v δ v)(x, 0) L 2 (B + 3 ) Cδ.

32 S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) Estimate of I 7 : I 7 = C C Ω 3 \B + 3 Ω 3 \B + 3 ( ) v δ (x, 0) η 2 w (x, 0)dx ηw(x, 0) dx Ω 3 ηw 2 dx Ω 3 \B + 3 dx Cδ. Estimate of I 8 : Hölder inequality and Sobolev inequality imply that I 8 = ηw 2 η t dx dt C ηw 2 dx dt C ηw 2 dx dt + C ηw 2 dx dt \Q + 3 Q } = C ηw 2 dx dt + C ηw 2 dx dt 9 Ω 3 \B 3 + Q C 9 ( Ω 3 \B + 3 ηw) 2 dx ) n 2 2 ( Ω 3 \B + 3 ) 2 n dx dt +C ηw 2 dx dt Q + 3 Cδ + 2 η(u v) 2 + η(v v δ ) 2} dx dt Q + 3 ( C δ + α 2). ) Estimate of I 9 : Note that v δ W ( Ω, 3 \ Q + 3 and A L. Then use integration of parts formula in t and Sobolev inequality along with Hölder s inequality to get the following estimates: I 9 = \Q + 3 = \Q + 3 Cδ. v δ ( η 2 w ) t à vδ ( )} η 2 w dx dt ( v δ) ( ) ( )} η 2 w à v δ η 2 w dx dt + t Ω 3 \B + 3 ( ) v δ η 2 w (x, 0)dx

33 76 S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) Estimate of I 0 : First use integration of parts formula in t and note that translation is continuous in the L 2 -norm. Then like Estimate of I 9, we see the following estimates: I 0 = Q + 3 = + Cδ. ( ) ( )} (v δ v) η 2 w t à (vδ v) η 2 w dx dt Q + 3 B + 3 )( ) ( ) ( )} ((v t ) δ v t η 2 w + à v δ v η 2 w dx dt ( ) ( ) v δ v (x, 0) η 2 w (x, 0)dx Note w = u v δ and w = (u v) + (v v δ ) in Q + 3. Then w η 2 dx dt = w η 2 dx dt + (u v) η + (v v δ ) η 2 dx dt \Q + 3 Q + 3 ( 0 ) n 2 ( w η 2 n 2 n n ) 2 n dx dx dt 9 Ω 3 \B + 3 Ω 3 \B + 3 +C u v 2 dx dt + C v v δ dx dt Q + 3 Q + 3 ( C δ + α 2), where we have used Hölder s inequality, Sobolev inequality and the fact that B 4 x n > δ } Ω 4 B + 4 and that translation is continuous in L 2 -norm. Thus, we have ( w η 2 dx dt C δ + α 2). (4.28) We finally combine estimates of I i ( i 0) and (4.28), to obtain Λ η 2 w 2 dx dt I + I 2 = I 3 + I 4 + I 5 + I 6 + I 7 + I 8 + I 9 + I 0 Cτ η 2 w 2 dx dt + C u v 2 dx dt Q + 3

34 S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) C ( + /τ) f Ω3 2 + A Ã 2} dx dt + Cδ ( Cτ η 2 w 2 dx dt + C ( + /τ) δ + α 2). Now take τ small enough to find ( η 2 w 2 dx dt C δ + α 2). Then, this estimate finally implies ( (u v δ ) 2 dx dt η 2 w 2 dx dt C δ + α 2) = ε 2, Ω 2 by taking α and δ satisfying the last identity above. This completes our proof. It suffices to consider only the estimates on the lateral boundary. The zero extension of the solution can lead to the estimates on the bottom and corner of the boundary. Lemma 4.8. There is a constant N > 0 so that for any ε > 0, there is δ = δ(ε) >0 with A uniformly parabolic and (δ, 7)-vanishing, and if u is a weak solution of (.) in Ω T = Ω (a, a + T ] Q 7 (0, 2) with and B 7 x n > δ} Ω 7 B + 7 (4.29) Ω (x, t) : M( u 2 ) } (x, t) : M( f 2 ) δ 2 } =, (4.30) then (x, t) : M( u 2 )>N 2 } Ω < ε Ω. (4.3) Proof. From condition (4.30), we see that there is a point (x 0,t 0 ) Ω such that u 2 dx dt, K r K r (x 0,t 0 ) Ω T f 2 dx dt δ 2 (4.32) K r K r (x 0,t 0 ) Ω T for all r>0. First we note the condition (4.29) with δ sufficiently small. From the fact that (x 0,t 0 ) Ω and via an easy geometry, we observe that Ω 4 (0, 2) = (Ω B 4 (0)) ( 4, 2] K 6 (x 0,t 0 ) Ω T.

35 78 S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) This gives f 2 dx dt K 6 f 2 dx dt Q 4 Ω 4 (0,2) Q 4 K 6 K 6 (x 0,t 0 ) Ω T Similarly, we see that u 2 dx dt Q 4 Ω 4 (0,2) ( ) 6 n+2 δ 2. (4.33) 4 ( ) 6 n+2. (4.34) 4 Then in view of (4.29), (4.33) and (4.34), we can employ Corollary 4.7 with u replaced with ( ) 4 n+2, ( 6 f with 64 ) n+2 f, and Q4 with Q 4 (0, 2), respectively, to find that for any α > 0, there exist a small δ = δ(α) >0, a constant matrix à with A Ω 4 (0,2) à α, and a corresponding weak solution v of (4.3) in Q + 4 (0, 2) such that (u v δ ) 2 dx dt α 2 (4.35) Ω 2 (0,2) provided that ( f 2 + A A Q 4 Ω Ω 4 (0,2) 2) dx dt + D( w Ω 4,T 4 ) δ 2, 4 (0,2) where D denotes the Hausdorff distance and v δ (x, t) = v(x + δe n,t) for (x, t). Then, we can use the local estimates v δ 2 dx dt C K 4 Ω 4 (0,2) to see that there is a constant N 0 so that sup v δ 2 }=N0 2. (4.36) (0,2) Now set N 2 := max 4N 2 0, 2n+2} and we show M( u 2 )>N 2 } Ω M Ω 4 (0,2) ( (u vδ ) 2 )>N 2 0 } Ω. (4.37) To show this, suppose that (x,t ) (x, t) Ω : M Ω 4 (0,2) ( (u vδ ) 2 )(x, t) N0 2 }. (4.38)

36 S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) For r 2, K r (x,t ) Ω T (0, 2) so by (4.38) and (4.36), we have u 2 dx dt 2 ( (u v δ ) 2 + v δ 2 )dxdt K r K r (x,t ) Ω T K r K r (x,t ) Ω T 4N 2 0. For r>2, K r (x,t ) K 2r (x 0,t 0 ) so by (4.32), we have u K r Kr 2 dx dt 2n+2 u 2 dx dt 2 n+2. (x,t) ΩT K 2r K 2r (x 0,t 0 ) Ω T Consequently we find (x,t ) (x, t) Ω : M( u 2 ) N 2 }. (4.39) Thus (4.38) and (4.39) imply (4.37). Using (4.37), weak estimate and (4.35), we finally find: M( u 2 )>N 2 } Ω M Ω 4 (0,2) ( (u vδ ) 2 )>N0 2 } Ω < C N0 2 (u v δ ) 2 dx dt Ω 2 (0,2) C N 2 0 α 2 ε Ω, by taking α (and δ) satisfying the last inequality above. This finishes our proof. The next lemma follows immediately from Lemma 4.8 and a scaling argument. Lemma 4.9. There is a constant N > 0 so that for any ε,r >0, there exists a small δ = δ(ε) >0 with A uniformly parabolic and (δ, 7r)-vanishing, if u is a weak solution of (.) in Ω T = Ω (a, a + T ] Q 7r (0, 2r 2 ) with B 7r x n > δr} B 7r Ω B + 7r and K r (x, t) Ω T : M( u 2 ) } (x, t) : M( f 2 ) δ 2 } =,

37 80 S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) then (x, t) Ω T : M( u 2 )>N 2 } K r < ε K r. For the following theorem it is convenient to use K r for a parabolic cube centered at a point in Ω T. Theorem 4.0. There is a constant N > 0 so that for any ε, r > 0, there exists a small δ = δ(ε) >0 with A uniformly parabolic and (δ, 63)-vanishing, andifuisa weak solution of (.) in Ω T = Ω (a, a + T ] with Ω (δ, 63)-Reifenberg flat, and if the following property holds: (x, t) : M( u 2 )>N 2 } K r ε K r, (4.40) then K r Ω T (x, t) : M( u 2 )>} (x, t) : M( f 2 )>δ 2 }. (4.4) Proof. We argue by contradiction. If K r satisfies (4.40) and (4.4) is false, then there exists (x 0,t 0 ) Ω T K r such that u 2 dx dt, K r Ω T K r (x 0,t 0 ) f 2 dx dt δ 2 K r Ω T K r (x 0,t 0 ) for all r > 0. If B 7r Ω =, this is an interior estimate (see Theorem 3.3). So suppose that B 7r Ω =. Now observe that B 7r B 9r (x 0 ), and choose some y = (y,y n ) B 7r (x) Ω. AsΩ is (δ, 63r)-Reifenberg flat, we have in some appropriate coordinate system Ω x n > 63δr} Ω B 63r (0) B + 63r (0). Now one can apply Lemma 4.9 to the cube K 9r when ε is replaced by (x, t) : M( u 2 )>N 2 } K r (x, t) : M( u 2 )>N 2 } K 9r < ε 9 n+2 K 9r = ε K r, ε 9 n+2, to obtain which is contradiction to (4.40). Now we take N, ε, and the corresponding δ given in Theorem 4.0. The following lemma shows that the decay estimates on the size of distribution functions of maximal function M( u 2 ).

38 S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) Lemma 4.. Suppose that u is a weak solution of (.) in Ω T with A uniformly parabolic and (δ, 63)-vanishing and Ω (δ, 63)-Reifenberg flat. Assume that (x, t) Ω T : M( u 2 )>N 2 } < ε K. (4.42) Let k be a positive integer and set ε := (x, t) : M( u 2 )>N 2k } k ( 0 δ) n+2 ε. Then we have i= ε i (x, t) : M( f 2 )>δ 2 N 2(k i) } +ε k (x, t) Ω T : M( u 2 )>}. Proof. We want to prove this lemma by induction on k. For the case k =, set E :=(x, t) Ω T : M( u 2 )>N 2 } and F :=(x, t) Ω T : M( f 2 )>δ 2 } (x, t) Ω T : M( u 2 )>}. Since Ω is (δ, 63)-Reifenberg flat, Ω is (δ, )-Reifenberg flat. Then it follows from (4.42), Theorem 4.0 and Theorem 2.0 that E ε F and so (4.43) is true for k =. Assume now that (4.43) is true for some positive integer k. Let us define u = u N and corresponding f = f N. Then, u is a weak solution of (.) and satisfies (x, t) Ω T : M( u 2 )>N 2 } < ε K. Now it follows from induction hypothesis and simple computations that (x, t) Ω T : M( u 2 )>N 2(k+) } = (x, t) Ω T : M( u 2 )>N 2k } k i= ε i (x, t) Ω T : M( f 2 )>δ 2 N 2(k i) } +ε k (x, t) Ω T : M( u 2 )>}

39 82 S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) k+ = ε i (x, t) Ω T : M( f 2 )>δ 2 N 2(k+ i) } i= +ε k+ (x, t) Ω T : M( u 2 )>}. These estimates in turn complete the induction on k. Remark 4.2. We remark that one might assume the number 63 in the previous corollary to be R(> ) by scaling the given equation, while δ is scaling invariant. Finally we are set to prove the main result of this paper, Theorem.6. Proof. Now we wish to prove Theorem.6 in its full generality. In the case that p = 2, it is classical. The case that <p<2 will be easily recovered from the case that p>2 by a duality. So suppose that p>2. Multiplying (.) by a small constant depending on f L p (Ω T ) and u L 2 (Ω T ), one can assume and f L p (Ω T ) is small enough (4.43) (x, t) Ω T : M( u 2 )>N 2 } K < ε K. (4.44) Since f L p (Ω T ), it follows from strong p p estimates of the maximal functions that M( f 2 ) L p/2 (Ω T ). Then in view of Lemma 2.6, there is a constant C depending only on δ, p and N such that k=0 N pk (x, t) Ω T : M( f 2 )>δ 2 N 2k } C M( f 2 ) p/2 L p/2 (Ω T ). Now this estimate, strong p p estimates, and (4.43) imply k=0 N pk (x, t) Ω T : M( f 2 )>δ 2 N 2k }. (4.45) We are intended to claim that M( u 2 ) L p/2 by using Lemma 2.6 when f 2 = M( u 2 ) and m = N 2. Without loss of generality we may assume that u has a

40 S.-S. Byun, L. Wang / Journal of Functional Analysis 223 (2005) compact support. In fact one can take an appropriate cut-off function η(t) so that u has a compact support in a bounded domain. Let us compute k=0 = N pk (x, t) Ω T : M( u 2 )>N 2k } k= N pk ( k i= ε i (x, t) Ω T : M( f 2 )>δ 2 N 2(k i) } +ε k (x, t) Ω T : M( u 2 )>} ( ) (N p ε ) i N p(k i) (x, t) Ω T : M( u 2 )>δ 2 N 2(k i) } i= + C k=i (N p ε ) k (x, t) Ω T : M( u 2 )>} k= (N p ε ) k k= < +, where we used Lemma 4., (4.45) and selected ε so that N p ε <. This selection is possible since N is a universal constant depending on the dimension and parabolicity and we can take ε, and the corresponding δ > 0, so ε. In view of Lemma 2.6, we deduce M( u 2 ) L p 2 (Ω T ), and thus u L p (Ω T ), with the estimate ) u p L p (Ω T ) ( Ω C T + f p L p (Ω T ). ) Since f p L p (Ω T ), we have u p L p (Ω T ) C. Now utilizing linear property in this estimate, we deduce u p L p (Ω T ) C f p L p (Ω T ). (4.46) Recalling our definition of W,p (Ω T ) and using (4.46), we finally obtain u p W,p (Ω T ) u p L p (Ω T ) + u p L p (Ω T ) + A u + f p L p (Ω T ) ( ) C u p L p (Ω T ) + f p L p (Ω T ).