CONORMAL PROBLEM OF HIGHER-ORDER PARABOLIC SYSTEMS WITH TIME IRREGULAR COEFFICIENTS

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1 CONORMAL PROBLEM OF HIGHER-ORDER PARABOLIC SYSTEMS WITH TIME IRREGULAR COEFFICIENTS HONGJIE DONG AND HONG ZHANG Abstract. The paper is a comprehensive study of L p and Schauder estimates for higher-order divergence type parabolic systems with discontinuous coefficients on a half space and cylindrical domains with the conormal derivative boundary conditions. For the L p estimates, we assume that the leading coefficients are only bounded and measurable in the t variable and have vanishing mean oscillations (VMO x) with respect to x. We also prove the Schauder estimates in two situations: the coefficients are Hölder continuous only in the x variable; the coefficients are Hölder continuous in the t variable as well on the lateral boundary. Contents 1. Introduction 1 2. Main results 6 3. Some auxiliary estimates L p estimate of Dd m u for systems with special coefficients Mean oscillation estimate for D x D m 1 u Proof of Theorem Schauder estimates for systems on a half space Proof of Theorem Acknowledgement 52 References Introduction This paper is devoted to the study of L p and Schauder estimates for higher-order divergence type parabolic systems with the conormal derivative boundary conditions. Many authors have studied the L p estimates for parabolic equations and systems with discontinuous coefficients. It is of particular interest not only 2010 Mathematics Subject Classification. 35K52, 35J58, 35B45, 35R05. Key words and phrases. higher-order systems, L p estimates, Schauder estimates. H. Dong was partially supported by the NSF under agreement DMS H. Zhang was partially supported by the NSF under agreement DMS

2 2 H. DONG AND H. ZHANG because of its various important applications in nonlinear equations and systems, but also due to its subtle link with the theory of stochastic processes. A good reference is [26]. In this paper, we expand the L p theory of higher-order parabolic systems to include a large class of discontinuous coefficients. For systems in the whole space and on a half space with the homogeneous Dirichlet boundary conditions, the first author and Kim [11] obtained the L p estimates with leading coefficients having vanishing mean oscillations with respect to spatial variables (VMO x ) under the Legendre Hadamard ellipticity condition (cf. (2.7)). Later in [12], they considered the conormal problem for higher-order elliptic systems with coefficients merely measurable in one direction and have small mean oscillation in orthogonal directions on each small ball, under the strong ellipticity condition (cf. (2.6)). The current paper can be viewed as a continuation of [11] and [12]. Compared to the strong ellipticity condition used in [12], the coefficients in this paper satisfy a Gårding type inequality (cf. (2.5)), which is weaker than the strong ellipticity condition. To present our results, we define the operator Lu = D α (A αβ D β u),, β m where m is a positive integer, A αβ are n n matrices, and D α = D α 1 1 Dα d The functions used throughout this paper u = (u 1,..., u n ) tr, f α = (f 1 α,..., f n α ) tr are complex vector-valued functions. For the L p estimates, the parabolic systems which we consider are of the form u t ( 1) m Lu λu = D α f α in (, T ) Ω (1.1) k=0 with the conormal derivative boundary conditions on (, T ) Ω (cf. (2.9)), where λ 0, T (, ], and Ω R d is either a half space or a C m 1,1 domain. We prove that if the leading coefficients A αβ (t, x) with α = β = m are VMO x and satisfy a Gårding type inequality (2.5), then the solution u of (1.1) with the conormal derivative boundary conditions satisfies the estimate m λ 1 k 2m D k u Lp((,T ) Ω) N 2m fα Lp((,T ) Ω), where N is a constant independent of u and λ is sufficiently large. To our best knowledge, this result is new in the higher-order case. We note that in the second-order case such result was proved in [9] by using the result in the whole space and the technique of odd/even extensions. However, such technique does not work for higher-order equations or systems. d.

3 CONORMAL PROBLEM 3 For the proof, in many references including, for example, [1], the L p estimates for second-order and higher-order systems with constant or continuous coefficients are obtained by relying on the exact representation of solutions and the Calderón Zygmund theorem. Another approach for such L p estimates is that of Campanato Stampachia using Stampachia s interpolation theorem (see [16]). Our proof is in the spirit of an approach introduced by Krylov [27, 28] to deal with the second-order elliptic and parabolic equations with VMO x coefficients in the whole space, which is well explained in his book [29]. Generally speaking, this approach consists of two steps. The first step is to establish mean oscillation estimates for systems with simple coefficients, i.e., the coefficients which depend only on t. The second step is to use a perturbation argument, which is well suited to the mean oscillation estimates, together with the Fefferman Stein theorem on sharp functions and the Hardy Littlewood theorem on maximal functions, in order to obtain the desired L p estimates. In our case, we estimate Dd mu and D x Dm 1 u separately. Here is an outline of our proof. We begin by considering special systems with simple coefficients as follows. Let u be a solution of u t ( 1) m L0 u = 0 in {x d > 0} with the conormal derivative boundary conditions on {x d = 0}, where d 1 L 0 u = Dj 2m u D ˆα (Aˆαˆα (t)d ˆα u) j=1 and ˆα = (0,..., 0, m). A key observation here is that the conormal derivative boundary conditions corresponding to the special systems are given by D m d u = = D2m 1 d u = 0 on {x d = 0}, which allows us to use some estimates for non-divergence type systems with the Dirichlet boundary conditions obtained in [11] to get 1/2, Dd m u (Dm d u) Q r (X 0 ) dx dt Nκ 1( Dd m u 2 dx dt) Q r (X 0 ) Q κr(x 0 ) where Q r (X 0 ) is a parabolic cylinder with radius r and center X 0 {x d = 0}, Q r (X 0 ) = Q r (X 0 ) {x d > 0}, (Dd mu) Q r (X 0 ) is the average of Dm d u in Q r (X 0 ), κ 32, and N is a constant independent of r, X 0, and u. The mean oscillation estimate above, together with the Fefferman Stein theorem and the Hardy Littlewood theorem, yields the L p estimate of Dd m u on a half space for the special systems. For general systems on a half space with simple coefficients, which is u t ( 1) m L 0 u λu = D α f α, (1.2)

4 4 H. DONG AND H. ZHANG with the conormal derivative boundary conditions on {x d = 0}, where L 0 u = D α (A αβ (t)d β u), α = β =m A αβ (t) are functions of t, and λ 0, we implement a scaling argument and the L p estimates of D m d u for the special systems to control the L p norm of D m d u by the L p norms of f α and D x D m 1 u. Then by the Sobolev embedding theorem and a bootstrap argument, we are able to obtain a local Hölder estimate of D m 1 u for the homogeneous system u t ( 1) m L 0 u λu = 0 in Q 2, (1.3) where Q 2 = Q 2 (0). Note that we can differentiate (1.3) with respect to x without affecting the system and boundary conditions, which enables us to estimate the mean oscillation of D x D m 1 u for (1.2). Finally, for coefficients depending on both x and t, we use the argument of freezing the coefficients to obtain the L p estimate of D x D m 1 u, and apply the same method for the operator with simple coefficients to bound the L p norm of Dd m u by those of f α and D x D m 1 u. Let us mention some other related results in the literature. In [7, 8], Chiarenza, Frasca, and Longo initiated the study of the Wp 2 estimates for second-order elliptic equations with VMO leading coefficients. Their proof is based on certain estimates of the Calderón Zygmund theorem and the Coifman Rochberg Weiss commutator theorem. We refer the reader to Bramanti and Cerutti [4], Bramanti, Cerutti, and Manfredini [5], Di Fazio [14], Maugeri, Palagachev, and Softova [31], Palagachev and Softova [32], Krylov [29], and the references therein. The second objective of this paper is to obtain the Schauder estimates for solutions to the system: u t ( 1) m Lu = D α f α in (0, T ) Ω (1.4) with the conormal derivative boundary conditions on (0, T ) Ω and the zero initial condition on {0} Ω, where Ω C m,a, for some a (0, 1). There is a vast literature on the Schauder estimates for parabolic and elliptic equations; see, for instance, [15, 22, 25]. The classical approaches are based on analyzing the fundamental solutions of the equations with a perturbation argument. On the other hand, when dealing with systems, it has become customary to use Campanato s technique which was first introduced in [6], and is well explained in [16, 34]. However, most of these results are obtained under the assumption that the coefficients are sufficiently regular in both t and x. In this paper, for systems on a half space, we estimate the Hölder norms of all the mth order spatial derivatives with the exception of Dd m u, when the coefficients are only measurable in the t variable. This type of coefficients has been studied by several authors mostly for second-order equations; see, for

5 CONORMAL PROBLEM 5 instance, [17, 20, 24, 30]. Lieberman [20] studied the interior and boundary Schauder estimates for second-order parabolic equations with time irregular coefficients. In the proof, he used the Campanato type approach and the maximum principle, the latter of which no longer works for systems or higher-order equations. Here we implement the L p estimates obtained in the first part, with a bootstrap argument, to obtain a local Hölder regularity for systems with simple coefficients, which yields the following mean oscillation estimate: D x D m 1 u (D x D m 1 u) Q Q r 2 dx dt r N( r R )2γ2md D x D m 1 u (D x D m 1 u) Q 2 dx dt, R Q R where γ (0, 1), 0 < r < R <, and u is a solution of u t ( 1) m L 0 u = 0 in Q R with the conormal derivative boundary conditions on {x d = 0} Q R. We then prove that if the coefficients A αβ and f α in (1.4) are Hölder continuous in the spatial variables x, then D x D m 1 u is Hölder continuous in both t and x. In contrast to the Dirichlet boundary condition case in [13], to estimate Dd m u, more regularity assumptions on the coefficients and data are necessary. In fact, this is not surprising by considering the second-order equation u t D 11 u D 2 (a(t)d 2 u) = D 2 f(t) in (0, T ) R 2 with the conormal derivative boundary condition. Clearly, the corresponding boundary condition is given by D 2 u = f(t)/a(t) on {x 2 = 0}, which implies that D 2 u is not necessarily continuous on {x 2 = 0}. In this paper, besides the case when the coefficients and data are Hölder continuous in x, we also consider the case when the coefficients and data are also Hölder continuous in t but only on the lateral boundary of the cylindrical domain. We show that under this stronger assumption all the mth order derivatives of u are Hölder continuous in both t and x. For the proof, it is sufficient to estimate Dd m u. To this end, we consider a system with special coefficients as in the L p case, and then use a scaling argument to bound the Hölder semi-norm of Dd mu by f α and D x D m 1 u. Combining with the estimate of D x D m 1 u, which we obtained under the assumption that coefficients and data are just measurable in the t variable, we are able to estimate D m u. For linear systems, our Schauder estimate extends the results in Lieberman [19], in which the author considered second-order quasilinear parabolic equations with the conormal derivative boundary condition. Schauder estimates for higher-order non-divergence type parabolic systems in the whole space, with the coefficients measurable in t and Hölder continuous in x, was

6 6 H. DONG AND H. ZHANG considered recently in Boccia [33]. With the same class of coefficients as in [33], in [13] the authors obtained Schauder estimates for both divergence and non-divergence type higher-order parabolic systems on a half space with the Dirichlet boundary conditions. For more results about the conormal derivative problems, we refer the reader to Lieberman [18, 21] and his new book [23]. The paper is organized as follows. In the next section, we introduce some notation and state our main results. The remaining part of the article can be divided into two parts. In the first part, we treat the L p estimates. Section 3 provides some necessary preparations and Section 4 deals with the L p estimate of Dd m u for systems with special coefficients. Section 5 and Section 6 are devoted to the proof of our main result of the L p estimates (Theorem 2.2). The second part is about the Schauder estimates. In Section 7, we establish necessary lemmas and prove the Schauder estimates near a flat boundary with coefficients Hölder continuous only with respect to x, or also Hölder continuous in t on the boundary where the conormal boundary conditions are given. Finally in Section 8, we prove the Schauder estimates in cylindrical domains when the coefficients are not only Hölder continuous with respect to x but also Hölder continuous in t on the lateral boundary. 2. Main results We first introduce some notation used throughout the paper. A point in R d is denoted by x = (x 1,..., x d ), also by x = (x, x d ), where x R d 1. A point in R d1 = R R d = {(t, x) : t R, x R d } is denoted by X = (t, x). For T (, ], set O T = (, T ) R d, O T = (, T ) Rd, where R d := {x = (x 1,..., x d ) R d : x d > 0}. In particular, when T =, we use O := R R d. Denote B r (x 0 ) = {y R d : x 0 y < r}, Q r (t 0, x 0 ) = (t 0 r 2m, t 0 ) B r (x 0 ), B r (x 0 ) = B r (x 0 ) {x d > 0}, Q r (t 0, x 0 ) = Q r (t 0, x 0 ) O. We use the abbreviation, for instance, Q r to denote the parabolic cylinder centered at (0, 0). We denote n f, g Ω = f tr ḡ = f j g j. Ω For a function f on D R d1, we set { } f(t, x) f(s, y) [f] a,b,d := sup t s a : (t, x), (s, y) D, (t, x) (s, y), x y b where a, b (0, 1]. For 0 < a 1, let j=1 f a 2m,a,D = f L (D) [f] a 2m,a,D. Ω

7 CONORMAL PROBLEM 7 The space corresponding to a 2m,a,D is denoted by C a 2m,a (D). For 1 < a < 2m not an integer, we define f a 2m,a,D = f L (D) [D α f]a α,{a},d, α <a where {a} = a [a]. The Hölder semi-norm with respect to t is denoted by { } f(t, x) f(s, x) f a,d := sup t s a : (t, x), (s, x) D, t s, where a (0, 1]. Define f t a,d := f L (D) f a,d and denote the space corresponding to t a,d by Ca t (D). We also define the Hölder semi-norm with respect to x { } f(t, x) f(t, y) [f] a,d := sup x y a : (t, x), (t, y) D, x y, and denote f a,d = f L (D) [f] a,d, where a (0, 1]. For a nonnegative integer m and a (0, 1], we define In particular, 2m f ma,d = f L (D) [D m f] a,d. f m,d = f L (D) D m f L (D). The space corresponding to a,d (or ma,d ) is denoted by Ca (D) (or C ma (D), respectively). We denote the average of f in D to be (f) D = 1 D D f(t, x) dx dt = D f(t, x) dx dt. Sometimes we take the average only with respect to x. For instance, (f) BR (x 0 )(t) = f(t, x) dx. B R (x 0 ) Throughout this paper, we assume that all the coefficients are measurable and bounded: A αβ K. In addition, in the main theorems below we assume that the leading coefficients satisfy a Gårding type inequality: for fixed X R d1 and any u W m 2 (Rd ), R R d α = β =m D α ū tr (y)a αβ (X)D β u(y) dy δ R d d Dj m u(y) 2 dy, j=1 (2.5)

8 8 H. DONG AND H. ZHANG where δ > 0 is a constant. Clearly, this condition is weaker than the strong ellipticity condition used, for instance, in [12]: for any ξ = (ξ α ), ξ α C n, R ( ξtr α A αβ ) ξ β δ ξ α 2. (2.6) α = β =m We will show in Lemma 3.1 that (2.5) is stronger than the Legendre Hadamard ellipticity condition used, for instance, in [11, 13]: for any η R d and ξ C n, R ( A αβ ij ξ i ξ j η α η β) δ ξ 2 η 2m, (2.7) α = β =m where α = (α 1,..., α d ) and η α = η α 1 1 ηα 2 2 ηα d d. Here we call Aαβ the leading coefficients if α = β = m. All the other coefficients are called lower-order coefficients. When A αβ only depend on t, we assume that (2.5) holds with A αβ = A αβ (t) for any t R. In order to state and prove our results in Sobolev spaces, in addition to the well-known L p and Wp k spaces, we introduce the following function spaces. If Ω = R d, let equipped with the norm Here H m p ((S, T ) R d ) = (1 ) m 2 W 1,2m p ((S, T ) R d ) u H m p ((S,T ) R d ) = (1 ) m 2 u W 1,2m p ((S,T ) R d ). W 1,2m p ((S, T ) R d ) = {u : u t, D α u L p ((S, T ) R d ), 0 α 2m} equipped with its natural norm. Notice that if we set then H m p ((S, T ) R d ) = (1 ) m 2 Lp ((S, T ) R d ), f H m p ((S,T ) R d ) = (1 ) m 2 f Lp((S,T ) R d ), u H m p ((S,T ) R d ) = u t H m p ((S,T ) R d ) D α u Lp((S,T ) R d ). For a general domain Ω R d, we set { H m p ((S, T ) Ω) = f : f = } D α f α, f α L p ((S, T ) Ω), { f H m p ((S,T ) Ω) = inf f α Lp((S,T ) Ω) : f = D α f α },

9 CONORMAL PROBLEM 9 and H m p ((S, T ) Ω) = {u : u t H m p ((S, T ) Ω), D α u L p ((S, T ) Ω), 0 α m}, u H m p ((S,T ) Ω) = u t H m p ((S,T ) Ω) D α u Lp((S,T ) Ω). Let T (, ] and Q = {Q r (t, x) O T : (t, x) O T, r (0, )}. For a function g defined on O T, we denote its (parabolic) maximal and sharp functions, respectively, by Mg(t, x) = g(s, y) dy ds, Then g # (t, x) = sup Q Q:(t,x) Q sup Q Q:(t,x) Q Q Q g(s, y) (g) Q dy ds. g Lp(O T ) N g# Lp(O T ), Mg L p(o T ) N g L p(o T ), if g L p, where 1 < p < and N = N(d, p). As is well known, the first inequality above is due to the Fefferman Stein theorem on sharp functions and the second one is the Hardy Littlewood maximal function theorem. Now we state our regularity assumption on the leading coefficients for the L p estimates. Let t osc x (A αβ, Q r (t, x)) = t r 2m B r (x) A αβ (s, y) A αβ (s, z) dz dy ds, B r (x) which is the mean oscillation in the spatial variables. Then we set A # R = sup sup sup (t,x) O r R α = β =m osc x (A αβ, Q r (t, x)). We impose on the leading coefficients a small mean oscillation condition with a parameter ρ > 0, which is specified later. Assumption 2.1. (ρ). There is a constant R 0 (0, 1] such that A # R 0 ρ. We are now ready to present our main results. The first one is about the L p estimates. For any S < T and Ω R d, we say that u H m p ((S, T ) Ω) is a solution to u t ( 1) m Lu λu = D α f α in (S, T ) Ω (2.8)

10 10 H. DONG AND H. ZHANG with the conormal derivative boundary conditions on (S, T ) Ω if the equality u, φ t (S,T ) Ω ( 1) m α A αβ D β u, D α φ (S,T ) Ω, β m λ u, φ (S,T ) Ω = ( 1) α f α, D α φ (S,T ) Ω (2.9) holds for any φ C 0 k=0 ((S, T ) Ω). Theorem 2.2. Let p (1, ), Ω C m 1,1, T (, ], and f α L p ((, T ) Ω) for α m. The operator L satisfies (2.5) for any X (, T ] Ω. Then there exist constants ρ = ρ(d, n, m, δ, p, K), and λ 0 = λ 0 (d, n, m, δ, p, K, R 0 ) such that, under Assumption 2.1 (ρ), for any u Hp m ((, T ) Ω) satisfying (2.8) with the conormal derivative boundary conditions on (, T ) Ω, we have m λ 1 k 2m D k u Lp((,T ) Ω) N 2m fα Lp((,T ) Ω) (2.10) provided that λ λ 0, where N depends only on d, n, m, δ, p, K, and Ω. Moreover, if λ > λ 0, there exists a unique solution u H m p ((, T ) Ω) of (2.8) with the conormal derivative boundary conditions on (, T ) Ω. We say u H2 m ((0, T ) Ω) is a weak solution to u t ( 1) m Lu = D α f α in (0, T ) Ω with the conormal derivative boundary conditions on (0, T ) Ω and the zero initial condition on {0} Ω if (2.9) with S = λ = 0 holds for any φ C ([0, T ) Ω). The following result is regarding the Schauder estimates near a flat boundary with coefficients and data only measurable in the t variable. Theorem 2.3. Assume that a (0, 1), u Cloc m (O 0 ) satisfying u t ( 1) m Lu = D α f α in Q 2R with the conormal derivative boundary conditions on {x d = 0} Q 2R, f α C a (Q 2R ) if α = m, f α L (Q 2R ) if α < m, and Aαβ C a (Q 2R ). The operator L satisfies (2.5) for any X Q 2R. Then for any R 1, there exists a constant N depending only on d, n, m, δ, K, a, A αβ and a,q 2R, R such that [D x D m 1 u] a 2m,a,Q R ( N D α u L (Q 2R ) [f α ] f a,q α 2R L (Q ). 2R ) α <m

11 CONORMAL PROBLEM 11 Our last result is regarding the Schauder estimates in cylindrical domains with more regularity assumptions on the coefficients and data. Theorem 2.4. Assume that a (0, 1), Ω C m,a, T (0, ), f α C a ([0, T ) Ω) C a 2m t ([0, T ) Ω), f α = 0 on {0} Ω if α = m, and f α L ((0, T ) Ω) if α < m. The operator L satisfies (2.5) for any X [0, T ] Ω, and A αβ C a ([0, T ) Ω) C a 2m t ([0, T ) Ω). Then the equation u t ( 1) m Lu = D α f α in (0, T ) Ω with the conormal derivative boundary conditions on (0, T ) Ω and the zero initial condition on {0} Ω has a unique solution u C am 2m,am ([0, T ) Ω). Moreover, there exists a constant N depending only on d, n, m, δ, K, A αβ a,(0,t ) Ω, Aαβ a 2m,(0,T ) Ω, Ω, and a such that, where G := u am 2m,am,(0,T ) Ω N( u L2 ((0,T ) Ω) G ), (2.11) ([f α ] a,(0,t ) Ω f α a 2m,(0,T ) Ω ) f α L ((0,T ) Ω). It is worth noting that in view of the example given at the end of the introduction, without the compatibility condition that f α = 0 on {0} Ω for α = m, in general D m u is not Hölder continuous near {t = 0} Ω. This is in contrast to the Dirichlet case, where such condition is not needed; see, for instance, [13]. 3. Some auxiliary estimates In this section, we consider operators without lower-order terms. Denote L 0 u = D α (A αβ D β u), α = β =m where A αβ = A αβ (t). Let C0 (O T ) be the collection of infinitely differentiable functions defined on O T vanishing for large (t, x). We prove (2.5) implies (2.7) in the following lemma. Recall that when the coefficients only depends on t, we assume that (2.5) holds with A αβ = A αβ (t) for any t R. Lemma 3.1. Under the condition (2.5), (2.7) holds with a possibly different δ > 0. Proof. By shifting the coordinates in the x d -direction, it is easy to see that (2.5) implies d R D α ū tr A αβ D β u dx δ Dj m u 2 dx (3.12) R d R d α = β =m j=1

12 12 H. DONG AND H. ZHANG for any u C0 (Rd ), and thus for any u W2 m(rd ) since C0 (Rd ) is dense in W2 m(rd ). We claim that (3.12) is equivalent to (2.7) with a possibly different δ > 0. Indeed, by taking the Fourier transform on both sides above and using Parseval s identity, (3.12) is equivalent to ( ) d (iη) α û tr A αβ (iη) β û dη δ η j 2m û 2 dη. (3.13) α = β =m R d R R d j=1 Clearly, (2.7) implies to (3.13). Conversely, let φ C0 (Rd ) and ξ C n. Choosing û(η) = ξφ(η) in (3.13), we get ( R ( η α η β ξtr A αβ ξ ) d δ η j 2m ξ 2) φ 2 dη 0. R d α = β =m By the arbitrariness of φ, we have for any ξ C n and η R d, R ( η α η β ξtr A αβ ξ ) d δ η j 2m ξ 2, which implies that α = β =m α = β =m j=1 j=1 R ( η α η β ξtr A αβ ξ ) δ N η 2m ξ 2, where N > 0 depends on d and m. The proof is completed. We begin with the following L 2 estimate for parabolic operators in the divergence form with measurable coefficients satisfying (2.5). Theorem 3.2. There exists a constant N = N(d, m, n, δ) such that under the condition (2.5), for any λ 0 and T (, ], α 1 λ 2m D α u L2 (O T ) N 2m fα L2 (O T ) (3.14) provided that u H2 m(o T ) satisfies u t ( 1) m L 0 u λu = D α f α (3.15) in O T with the conormal derivative boundary conditions on {x d = 0}, and f α L 2 (O T ), α m. Furthermore, for any λ > 0 and f α L 2 (O T ), there exists a unique solution u H2 m(o T ) to the system (3.15) with the conormal derivative boundary conditions on {x d = 0}. Proof. We assume λ > 0. If λ = 0 the inequality (3.14) holds trivially or we obtain D α u L2 (O T ) N f α L2 (O T ) if f α = 0 for α < m

13 CONORMAL PROBLEM 13 using the inequality (3.14) for λ > 0 and letting λ 0. If we have proved the inequality (3.14), then due to the fact that u t = ( 1) m D α (A αβ D β u) λu D α f α, we obtain α = β =m u H m 2 (O T ) N f α L2 (O T ), where N = N(d, n, m, δ, λ, K). Then using this estimate, the method of continuity, and the unique solvability of system with coefficients A αβ = δ αβ I n n (for instance, see [15, Chapter 10], as the systems is decoupled when A αβ = δ αβ I n n ) we prove the second assertion of the theorem. Hence, it is clear that we only need to prove the inequality (3.14). Moreover, by a density argument it is obvious that we can assume that u C0 (O T ). By multiplying ū tr to both sides of (3.15) and integrating over (, T ] Ω, we have u, u t O D α u, A αβ D β u T O λ u, u T O T α = β =m = ( 1) α D α u, f α O. By (2.5), we get m δ Dj m u 2 dx dt O T j=1 From [11, Proposition 1], we know that Moreover, R u, u t O T D m u L2 (O T ) N = 1 2 O T T α = β =m R D α u, A αβ D β u O. T d Dj m u L2 (O ). T j=1 t u 2 (t, x) dx dt = 1 u(t, x) 2 dx 0. 2 R d Hence for any ε > 0, by Young s inequality, δ D m u 2 dx dt λ u 2 dx dt N ( 1) α R D α u, f α O O T ε λ m α m O T O T D α u 2 dx dt Nε 1 λ m α m O T T f α 2 dx dt. To finish the proof, it suffices to use the interpolation inequalities and choose ε sufficiently small depending on δ, d, m, and n.

14 14 H. DONG AND H. ZHANG By Theorem 3.2 and adapting the proofs of [10, Lemmas 3.2 and 7.2] to the conormal case, we have the following local L 2 estimates. Lemma 3.3. Let 0 < r < R <. Assume that u C loc (O 0 ) satisfies u t ( 1) m L 0 u = 0 in Q R with the conormal derivative boundary conditions on {x d = 0} Q R. Then under the condition (2.5), there exists a constant N = N(d, m, n, δ, K) such that for j = 1,..., m, D j u L2 (Q r ) N(R r) j u L2 (Q R ). The technical lemma below is useful in our proof. Note that we do not require any ellipticity condition for this lemma. Lemma 3.4. Let R (0, ) and f α L (Q 2R ). Assume that u C loc (O 0 ) satisfies ut ( 1) m L0u = D α f α in Q 2R with the conormal derivative boundary conditions on {x d = 0} Q 2R. Let P (x) be a vector-valued polynomial of order m 1 and satisfy (D α P (x)) Q = (D α u) R Q (3.16) R for α < m. Let v = u P (x). Then there exists a constant N = N(d, n, m, K) such that D α v L2 (Q R ) NR m α D m u L2 (Q R ) N β m where α < m. If f α = 0 for any α, then it holds that where α < β m. R 3md/2 α β f β L (Q R ), D α v L2 (Q R ) NR β α D β v L2 (Q R ), Proof. By using a scaling argument, we only need to consider the case when R = 1. Choose ξ(y) C0 (B 1 ) with a unit integral and set 0 g α (t) = D α v(t, y)ξ(y) dy, c α = g α (t) dt B 1 for α < m. By Hölder s inequality and the Poincaré inequality, we have D α v(t, x) g α (t) 2 dx = (D α v(t, x) D α v(t, y))ξ(y) dy 2 dx B 1 N B 1 B 1 B 1 B 1 D α v(t, x) D α v(t, y) 2 dy dx N 1 B 1 D α 1 v(t, y) 2 dy.

15 CONORMAL PROBLEM 15 = 0, by the triangle inequality and the Poincaré inequal- Because (D α v) Q 1 ity, D α v L2 (Q 1 ) Dα v c α L2 (Q 1 ) D α v g α (t) L2 (Q 1 ) g α(t) c α L2 (Q 1 ) N D α 1 v L2 (Q 1 ) N tg α L2 ( 1,0). (3.17) Since v satisfies the same equation as u, by the definition of g α, t g α (t) = ξ(y)d α t v(t, y) dy = B 1 B 1 ( 1) m1 ξ(y)d α L 0 v(t, y) dy β m B 1 ξ(y)d α D β f β dy. For the first term on the right-hand side of the equality above, we leave α 1 derivatives on v and move all the others to ξ. For the second term, we move all the derivatives to ξ. Therefore, by the Cauchy Schwarz inequality, t g α (t) N D α 1 v dy N f α L (Q 1 ) B 1 N D α 1 v(t, ) L2 (B 1 ) N f α L (Q 1 ). (3.18) Combining (3.18) and (3.17), we prove the desired estimates for R = 1 by induction. 4. L p estimate of Dd m u for systems with special coefficients In this section, we consider the special operator: L 0 = D m d 1 I n n D ˆα (Aˆαˆα (t)d ˆα ), where ˆα = (0,..., 0, m), d 1 Dd 1 m = Dj 2m, j=1 and Aˆαˆα (t) is only measurable in t and satisfies Aˆαˆα δi n n. Note that L 0 satisfies (2.5) so that Theorem 3.2 is applicable. We have the following observation. Lemma 4.1. Assume that u C loc (O 0 ) satisfies u t ( 1) m L0 u = 0 (4.19) in Q 4 with the conormal derivative boundary conditions on {x d = 0} Q 4. Then the boundary conditions are given by D m d u = = D2m 1 d u = 0 on {x d = 0} Q 4.

16 16 H. DONG AND H. ZHANG Proof. By the weak formulation of the system, u, φ t Q 4 d 1 j=1 D m j u, D m j φ Q 4 Aˆαˆα (t)d m d u, Dm d φ Q 4 = 0 (4.20) for any φ C 0 (Q 4). Since only the boundary conditions on {x d = 0} are considered, we integrate by parts and boundary terms only appear in the last term of the equation above. Let us denote Σ := {x d = 0} Q 4. First, we integrate by parts once and get Aˆαˆα (t)dd m u, Dm d φ Q 4 = d φ dx dt Aˆαˆα (t)d m1 d u, D m 1 d φ Q. Aˆαˆα (t)dd m udm 1 Σ We continue this process by integrating by parts the second term on the right-hand side of the equality above. By induction, it is easily seen that Aˆαˆα (t)dd m u, Dm d φ Q 4 m = ( 1) j Aˆαˆα D mj 1 d u D m j d φ dx dt ( 1) m Aˆαˆα (t)dd 2m u, φ Q. 4 j=1 Σ Plug the equality above into (4.20). Since φ is an arbitrary smooth function, we get m ( 1) j Aˆαˆα D mj 1 d u D m j d φ dx dt = 0, and thus j=1 Σ Aˆαˆα D mj 1 d u = 0 on Σ for j = 1, 2,..., m. Since Aˆαˆα is positive definite, we obtain Dd m u =... = D2m 1 d u = 0 on Σ. The lemma is proved. We state a conclusion in [11, Remark 6] and notice that by Lemma 3.1 L 0 satisfies the Legendre Hadamard condition (2.7). Lemma 4.2. Let R (0, ) be a constant. Assume that u C loc (O ) satisfies u t ( 1) m L0 u = 0 in Q 2R, u = = D m 1 d u = 0 on Q 2R {x d = 0}. Then for any r (0, R), there exists a constant N depending only on d, n, m, δ, K, r, R, and a such that 4 where 0 < a < 2m. u a 2m,a,Q r N u L 2 (Q R ),

17 CONORMAL PROBLEM 17 As a consequence of Lemma 4.2, we get the following Hölder estimate of D m d u. Lemma 4.3. Let λ 0 be a constant. Assume that u C loc (O ) satisfies u t ( 1) m L0 u λu = 0 (4.21) in Q 2 with the conormal derivative boundary conditions on {x d = 0} Q 2. Then there exists N = N(d, m, n, δ, K) such that [D m d u] 1 2m,1,Q 1 N D m d u L 2 (Q 2 ). Proof. For the case when λ = 0, as noted in Lemma 4.1, the conormal derivative boundary conditions for (4.19) are given by D m d u = = D2m 1 d u = 0 on {x d = 0}. We differentiate (4.19) m times with respect to x d and let v = Dd m u. Then we arrive at v t ( 1) m L0 v = 0 in Q 2, v = D d v = = D m 1 d v = 0 on {x d = 0} Q 2. By Lemma 4.2 with a = 1, [v] 1 2m,1,Q 1 N v L2 (Q 2 ). Since v = Dd m u, we prove the case when λ = 0. For the case when λ > 0, we apply an idea of S. Agmon, the details of which can be found in Corollary 5.5. Lemma 4.4. Let r (0, ), κ [32, ), λ > 0, and X 0 = (t 0, x 0 ) O. Assume that u C loc (O ) satisfies (4.21) in Q κr(x 0 ) with the conormal derivative boundary conditions on {x d = 0} Q κr (X 0 ). Then we have ( D m d u (Dm d u) Q r (X 0 ) ) Q r (X 0 ) Nκ 1 ( D m d u 2 ) 1 2 Q κr(x 0 ), where N = N(d, n, m, δ, K). Proof. By scaling, it suffices to prove the inequality for r = 8/κ (0, 1/4]. We consider the following two cases. Case 1: the last coordinate of x 0 [0, 1). In this case, by denoting Y 0 = (t 0, x 0, 0), we have Q r (X 0 ) Q 2 (Y 0 ) Q 4 (Y 0 ) Q κr (X 0 ). After applying Lemma 4.3 to u with a scaling and translation of the coordinates, we obtain ( D m d u (Dm d u) Q r (X 0 ) ) Q r (X 0 ) Nr[Dm d u] 1 2m,1,Q 2 (Y 0) Nκ 1 ( D m d u 2 ) 1 2 Q 4 (Y 0) Nκ 1 ( D m d u 2 ) 1 2 Q κr(x 0 ).

18 18 H. DONG AND H. ZHANG Case 2: the last coordinate of x 0 1. This case is indeed an interior case. From Lemmas 2 and 3 in [11], we have [v] 1 2m,1,Q 1 N v L 2 (Q 4 ), (4.22) where v smooth is a solution of Taking v = Dd m u, we get v t ( 1) m L0 v λv = 0 in Q 4. ( D m d u (Dm d u) Q r(x 0 ) ) Qr(X 0 ) Nr[D m d u] 1 2m,1,Q 1/4(X 0 ) Nr D m d u L 2 (Q 1 (X 0 )) Nκ 1 ( D m d u 2 ) 1 2 Q κr(x 0 ), where we use (4.22) with a scaling and translation of the coordinates in the second inequality. Hence we prove the lemma. Now we are ready to establish a mean oscillation estimate of D m d u for systems with special coefficients on a half space. Theorem 4.5. Let r (0, ), T (, ], κ [64, ), λ > 0, X 0 = (t 0, x 0 ) O T, and f α L 2,loc (O T ), where α m. Assume that u H2,loc m (O T ) satisfies u t ( 1) m L0 u λu = D α f α (4.23) in Q 2κr (X 0) with the conormal derivative boundary conditions on {x d = 0} Q 2κr (X 0 ). Then we have ( D m d u (Dd m u) Q r (X 0 ) ) Q r (X 0 ) Nκ 1 ( D m d u 2 ) 1 2 Q κr(x 0 ) Nκm d 2 where N = N(d, m, n, δ, K). 2m 1 2 ( fα 2 ) 1 2 Q κr(x 0 ), (4.24) Proof. Choose two smooth functions ζ and ζ 1 defined on R d1 such that and ζ = 1 on Q κr/2 (X 0 ), ζ = 0 outside (t 0 (κr) 2m, t 0 (κr) 2m ) B κr (x 0 ). ζ 1 = 1 on Q κr (X 0 ), ζ 1 = 0 outside (t 0 (2κr) 2m, t 0 (2κr) 2m ) B 2κr (x 0 ). Define ũ = ζ 1 u. We claim that ũ satisfies the following equation in O t 0 ũ t ( 1) m L0 ũ λũ = D α F α (4.25)

19 CONORMAL PROBLEM 19 with the conormal derivative boundary conditions on {x d = 0}, where F α is the linear combination of terms like f α ζ 1, uζ 1t, D k ζ 1 f α, Aˆαˆα D m ud k ζ 1, D m ud k ζ 1, In order to show (4.25), we compute Aˆαˆα D m k ud k ζ 1, D m k ud k ζ 1, k 1. m 1 ũ, φ t O D m t0 j ũ, Dj m φ O Aˆαˆα D m t0 d ũ, Dm d φ O t (4.26) 0 j=1 for any φ C0 ((, t 0) R d ). Notice that m 1 D m (ũ) = D m (uζ 1 ) = D m uζ 1 C(d, m, j)d j ud m j ζ 1, ζ 1 D m φ = D m (ζ 1 φ) j=0 m C (d, m, j)d j ζ 1 D m j φ, j=1 and ζ 1 φ vanishes outside Q 2κr (X 0 ). Upon plugging the two equalities above into (4.26) and using the fact that u satisfies (4.23) in Q 2κr (X 0) with the conormal derivative boundary conditions on {x d = 0} Q 2κr (X 0 ), where we take ζ 1 φ as a test function, we obtain (4.25). On the other hand, by our assumption that u H2,loc m (O T ) and f α L 2,loc (O T ), it is easy to see that F α L 2 (O T ). Now we consider the values of u and f α in Q κr (X 0 ). Because ũ = u and f α = F α in Q κr (X 0 ), after extending f α to be zero when t > T, without loss of generality we can assume that f α L 2 (O ). By a standard approximation argument, we can assume Aˆαˆα and f α are smooth in O and (4.23) holds in O. Indeed, let Aˆαˆα ε be the mollification of Aˆαˆα in the t variable and f αε be the mollification of f α (we may further extend f α to be 0 when x d < 0). Therefore, f αε f α in L 2 (O ) and A αβ ε A αβ (a.e) as ε 0. Let u ε be the unique weak solution to the system u ε t ( 1) m L0ε u ε λv ε = D α f αε in O with the conormal derivative boundary conditions on {x d = 0}. By Theorem 3.2 and the dominated convergence theorem, u ε converges to u in H2 m(o t 0 ) as ε 0. Obviously, if (4.24) holds for u ε, then it holds for u as well. Thus, it is sufficient to assume that f α and Aˆαˆα are smooth and u satisfies (4.23) in O, which implies u Cloc (O ) (see, for instance, [1]). By Theorem 3.2, for any λ > 0, there exists a unique solution w H2 m(o ), which is also smooth, to the equation w t ( 1) m L0 w λw = D α (ζf α )

20 20 H. DONG AND H. ZHANG in O with the conormal derivative boundary conditions on {x d = 0}. Let v := u w C loc (O ). Then v satisfies v t ( 1) m L0 v λv = 0 in Q κr/2 (X 0) with the conormal derivative boundary conditions on {x d = 0} Q κr/2 (X 0 ). By applying Lemma 4.4 to v (note that κ/2 32), we have ( D m d v (D m d v) Q r (X 0 ) ) Q r (X 0 ) Nκ 1 ( D m d v 2 ) 1 2 Q κr/2 (X 0). (4.27) By Theorem 3.2 with T = t 0, we get α 1 λ 2m D α w L2 (O t ) N 0 In particular, ( D m w 2 ) 1 2 Q r (X 0 ) Nκmd/2 2m ζfα L2 (O t 0 ). (4.28) 2m 1 2 ( fα 2 ) 1 2 Q κr(x 0 ). (4.29) Now let us prove (4.24). By the triangle inequality, ( D m d u (D m d u) Q r (X 0 ) ) Q r (X 0 ) 2( Dm d u c ) Q r (X 0 ) for any constant c. By taking c = (Dd mv) Q r (X 0 ), we have ( D m d u (Dd m u) Q r (X 0 ) ) Q r (X 0 ) 2( Dm d u (Dm d v) Q r (X 0 ) ) Q r (X 0 ). By using the triangle inequality and the Cauchy Schwarz inequality, the right-hand side of the inequality above can be bounded by N ( D m d v (Dm d v) Q r (X 0 ) ) Q r (X 0 ) N( Dm w 2 ) 1 2 Q r (X 0 ). By using (4.27) and (4.29), the quantity above is less than Nκ 1 ( Dd m v 2 ) 1 2 Q κr/2 (X 0) Nκmd/2 2m 1 2 ( fα 2 ) 1 2. (4.30) Q κr(x 0 ) Finally, from (4.28) we know that ( D m d w 2 ) 1 2 Q κr/2 (X 0) N 2m 1 2 ( fα 2 ) 1 2 Q κr(x 0 ). Combining the inequality above and the fact that v = u w, we see that the first term of (4.30) is less than the right-hand side of (4.24). The theorem is proved. Now we are ready to prove an L p estimate of Dd m u for the system with special coefficients.

21 CONORMAL PROBLEM 21 Theorem 4.6. Let p [2, ), λ 0, T (, ], and f α L p (O T ) for α m. Then for any u Hp m (O T ) satisfying u t ( 1) m L0 u λu = D α f α in O T with the conormal derivative boundary conditions on {x d = 0}, we have λ 1 2 D m d u Lp(O T ) N 2m fα Lp(O ), T where N = N(d, m, n, p, δ, K). Proof. First we suppose that p (2, ). From Theorem 4.5, we deduce that (D m d u)# (X 0 ) Nκ 1 (M(D m d u)2 (X 0 )) 1 2 Nκ m d 2 2m 1 2 (M(fα ) 2 (X 0 )) 1 2 for any κ 64 and X 0 O T. This, together with the Fefferman Stein theorem and the Hardy Littlewood maximal function theorem, yields D m d u L p(o T ) N (Dm d u)# Lp(O T ) Nκ 1 (M(D m d u)2 ) 1 2 Lp(O T ) Nκm d 2 Nκ 1 D m d u L p(o T ) Nκm d 2 2m 1 2 (M(fα ) 2 ) 1 2 Lp(O T ) 2m 1 2 fα Lp(O T ). Now we choose κ sufficiently large such that the first term on the righthand side of the inequality above is absorbed in the left-hand side. Then we obtain the desired estimate. The case when p = 2 follows from Theorem 3.2. The theorem is proved. 5. Mean oscillation estimate for D x D m 1 u In this section, we prove a mean oscillation estimate for D x D m 1 u. The following lemma shows that Dd mu L p can be bounded by D x D m 1 u Lp and f α Lp for systems with simple coefficients. Lemma 5.1. Let T (, ] and p [2, ). Assume that u Hp m (O T ) and u t ( 1) m L 0 u λu = D α f α with the conormal derivative boundary conditions on {x d = 0}, where λ 0 and f α L p (O T ). Then under the condition Aˆαˆα δi n n, there exists a constant N, depending only on d, m, n, δ, K, and p, such that λ 1 2 D m u Lp(O T ) N 2m fα Lp(O T ) Nλ1 2 Dx D m 1 u Lp(O ). T

22 22 H. DONG AND H. ZHANG Especially, if λ = 0 and f α = 0 when α < m, then D m u Lp(O T ) N D x Dm 1 u Lp(O T ) N f α Lp(O T ). Proof. The case when λ = 0 follows by letting λ 0 after the estimate with λ > 0 is proved. We use a scaling argument. Let v(t, x, x d ) = u(µ 2m t, x, µ 1 x d ) with a sufficiently large constant µ to be chosen later. Then v satisfies in O µ 2m T, v t ( 1) m = α = β =m µ α d 2m D α fα µ α dβ d 2m D α (Ãαβ D β v)µ 2m λv with the conormal derivative boundary conditions on {x d = 0}, where à αβ (t) = A αβ (µ 2m t), fα (t, x, x d ) = f α (µ 2m t, x, µ 1 x d ). We leave the term ( 1) m D ˆα (Èαˆα D ˆα v) on the left-hand side and move all the other spatial derivatives to the right-hand side and add ( 1) m D m d 1 v to both sides of the system, v t ( 1) m( D ˆα (Aˆαˆα D ˆα v) D m d 1 v) µ 2m λv = where ˆf α = µ α d 2m fα for α < m, D α ˆfα ( 1) m D m d 1 v, ˆf α = µ α d 2m fα ( 1) m1 µ α dβ d 2m à αβ (t)d β v for α = m but α ˆα, and β =m ˆfˆα = µ m fˆα β ˆα β =m Then we implement Theorem 4.6 to get ( 1) m1 µ β d m Èαβ (t)d β v. µ m λ 1 2 D m v Lp(O µ 2m ) T N (µ 2m λ) α 2m ˆfα Lp(O µ 2m ) Nµ m λ 1 2 Dx D m 1 v Lp(O T µ 2m ) T

23 N CONORMAL PROBLEM 23 (µ 2m λ) α 2m µ α d 2m f α Lp(O µ 2m T ) Nµ m λ 1 2 ( α ˆα β =m µ α dβ d 2m D β v Lp(O µ 2m T ) µ βd m D β v Lp(O )) Nµ m λ 1 2 Dx D m 1 v Lp(O ). µ β ˆα 2m T µ 2m T β =m Let µ be sufficiently large such that N ( α ˆα β =m µ αdβd 2m β ˆα β =m Then we fix this µ and obtain µ m λ 1 2 D m v Lp(O µ 2m T ) N µ β d m ) 1/2. (µ 2m λ) α 2m µ α d 2m f α Lp(O µ 2m T ) Nµ m λ 1 2 Dx D m 1 v Lp(O µ 2m T ). After returning to u and f α, we complete the proof of the lemma. We localize Lemma 5.1 to get Lemma 5.2 following the proof of [11, Lemma 1]. Lemma 5.2. Let 0 < r < R < and p [2, ). Assume u H m p (Q R ) and u t ( 1) m L 0 u = 0 in Q R with the conormal derivative boundary conditions on {x d = 0} Q R. Then under the condition Aˆαˆα δi n n, there exists a constant N depending only on d, m, n, δ, r, R, p, and K such that D m u Lp(Q r ) N( D x Dm 1 u Lp(Q R ) u L p(q R )). (5.31) Next, we state a parabolic type Sobolev embedding theorem (cf. [2, pp. 187] and [3, pp. 72]). Lemma 5.3. Let r (0, ) and 1 q p <. Assume that 1 q 1 p 1 d 2m. Let ζ C0 be such that ζ = 1 in Q r. Then for any function u such that uζ Hq m (O 0 ), we have u, Dm 1 u L p (Q r ) and u Lp(Q r ) Dm 1 u Lp(Q r ) N uζ H m q (O 0 ), where N = N(d, m, r, p, q). Moreover, for any q > d 2m, we have u γ 2m,γ,Q r N u H m q (Q r ), where γ = 1 (d 2m)/q.

24 24 H. DONG AND H. ZHANG In the next lemma, we obtain a Hölder estimate of D m 1 u. Lemma 5.4. Assume that u C loc (O 0 ) satisfies u t ( 1) m L 0 u = 0 in Q 2 with the conormal derivative boundary conditions on {x d = 0} Q 2. Then under the condition (2.5), for any γ (0, 1) there exists a constant N = N(d, m, n, δ, γ, K) such that D m 1 u γ 2m,γ,Q 1 N u L2 (Q 2 ). Proof. Due to Lemma 3.3 and the definition of H m p, we have u H m 2 (Q r 1 ) N(d, r 1) u L2 (Q 2 ), where 1 < r 1 < 2. From Lemma 5.3, we know that there is a p 1 satisfying p 1 1 d2m such that u Lp1 (Q r 1 ) Dm 1 u Lp1 (Q r 1 ) N(d, r 1, r 1, p 1 ) u H m 2 (Q r 1 ) N(d, r 1, r 1, p 1 ) u L2 (Q 2 ), (5.32) where 1 < r 1 < r 1. Since D x u satisfies the same system and boundary conditions as u, with slight modifications of the argument above and Lemma 3.3 we get D x D m 1 u Lp1 (Q r 1 ) N(d, r 1, r 1, p 1 ) u L2 (Q 2 ). Choose r 2 (1, r 1 ). From Lemma 3.1, (5.31), and (5.32), we have D m u Lp1 (Q r 2 ) N( D x D m 1 u Lp1 (Q r 1 ) u L p1 (Q )) N u L2 r 1 (Q ), 2 which, combining with the fact that u satisfies the system, implies u H m p1 (Q r 2 ) N u L 2 (Q 2 ). By induction, we can choose an increasing sequence p 1, p 2,..., such that 1 1 p i p i1 1 d 2m, and a sequence of decreasing domains Q r 1 Q r 2, such that u H m pi (Q r 2i ) N u L 2 (Q 2 ). It is obvious that for any γ (0, 1), in finite steps we can always take p n = (d 2m)/(1 γ) and Q r 2n Q 1. Finally applying the last assertion of Lemma 5.3 for p > 2m d, we prove the lemma. Corollary 5.5. Let λ 0 and γ (0, 1). Assume that u C loc (O 0 ) satisfies u t ( 1) m L 0 u λu = 0

25 CONORMAL PROBLEM 25 in Q 2 with the conormal derivative boundary conditions on {x d = 0} Q 2. Then under the condition (2.5), there exists a constant N depending only on d, m, n, δ, γ, and K such that [D x D m 1 u] γ 2m,γ,Q 1 λ 1 2 [u] γ 2m,γ,Q 1 N m k=0 λ 1 2 k 2m D k u L2 (Q 2 ). (5.33) Proof. First let λ = 0. The inequality in the corollary becomes [D x D m 1 u] γ N D m u 2m,γ,Q 1 L2 (Q ). 2 We differentiate the system with respect to x and apply Lemma 5.4 to D x u to obtain [D x D m 1 u] γ 2m,γ,Q 1 N D x u L2 (Q 2 ). (5.34) Let P (x) be a vector-valued polynomial of order m 1 and satisfies (3.16) for any α < m. It is easily seen that such P exists and is unique. Define v = u P (x), which satisfies the same system and the boundary conditions as u. By (5.34) with v in place of u and Lemma 3.4, we get [D x D m 1 u] γ = [D 2m,γ,Q x D m 1 v] γ 1 2m,γ,Q 1 N D x v L2 (Q 2 ) N Dm u L2 (Q ). 2 In order to handle the case when λ > 0, we implement an argument originally due to S. Agmon. Specifically, let Note that ζ(y) = cos(λ 1 2m y) sin(λ 1 2m y). ( 1) m D 2m y ζ(y) = λζ(y), ζ(0) = 1, D m ζ(0) = λ 1 2. Denote (t, z) = (t, y, x) to be a point in R d2 and set û(t, z) = u(t, x)ζ(y), ˆQ r = ( r 2m, 0) {z : z < r, z d1 > 0, z R d1 }. Obviously, û satisfies û t ( 1) m L 0 û ( 1) m D 2m y û = 0 in ˆQ 2 with the conormal derivative boundary conditions on {z d1 = 0} ˆQ 2. Note that our new operator above satisfies (2.5). Upon applying the lemma with λ = 0 to û, we find [D z D m 1 û] γ 2m,γ, ˆQ N D m û 1 L2 ( ˆQ 2 ), (5.35) where z = (z 1,..., z d ). Since, for example λ 1 2 [u] γ 2m,γ,Q 1 [Dy m û] γ 2m,γ, ˆQ, 1 we only need to bound the right-hand side of (5.35) by the right-hand side of (5.33). This can be done easily, since D m û is a linear combination of terms like λ 1 2 k 2m cos(λ 1 2m y)d k x u(t, x), λ 1 2 k 2m sin(λ 1 2m y)d k x u(t, x), k = 0,..., m.

26 26 H. DONG AND H. ZHANG The corollary is proved. In the next lemma, we obtain a mean oscillation estimate of D x D m 1 u for homogeneous systems. Lemma 5.6. Let r (0, ), κ [32, ), γ (0, 1), and X 0 = (t 0, x 0 ) O 0. Assume that u C loc (O 0 ) satisfies u t ( 1) m L 0 u λu = 0 in Q κr(x 0 ) with the conormal derivative boundary conditions on {x d = 0} Q κr (X 0 ). Then under the condition (2.5), we have ( Dx D m 1 u (D x D m 1 u) Q r (X 0 ) ) Q r (X 0 ) λ ( 1 2 u (u)q r (X 0 ) ) Q r (X 0 ) Nκ γ m k=0 λ 1 2 k 2m ( D k u 2 ) 1 2 Q κr(x 0 ), where N = N(d, n, m, δ, γ, K). Proof. Using Corollary 5.5, the proof is exactly the same as that of Lemma 4.4 and thus is omitted. In the following proposition, we obtain a mean oscillation estimate of D x D m 1 u for systems with simple coefficients. Proposition 5.7. Let r (0, ), T (, ], κ [64, ), λ > 0, γ (0, 1), and X 0 = (t 0, x 0 ) O T. Assume that u Hm 2,loc (O T ) satisfies u t ( 1) m L 0 u λu = D α f α in Q 2κr (X 0) with the conormal derivative boundary conditions on {x d = 0} Q 2κr (X 0 ), where f α L 2,loc (O T ), α m. Then under the condition (2.5), we have ( Dx D m 1 u (D x D m 1 u) Q r (X 0 ) ) Q r (X 0 ) λ ( 1 2 u (u)q r (X 0 ) ) Q r (X 0 ) Nκ γ m k=0 λ 1 2 k 2m ( D k u 2 ) 1 2 Q κr(x 0 ) Nκm d 2 where N = N(d, m, n, δ, γ, K). 2m 1 2 ( fα 2 ) 1 2 Q κr(x 0 ), Proof. Similar to the approximation argument in Theorem 4.5, we assume that f α and Aˆαˆα are smooth and the system holds in O T. Let ζ be the function defined at the beginning of the proof of Theorem 4.5. By Theorem 3.2, there exists a unique solution w H2 m(o T ) of w t ( 1) m L 0 w λw = D α (ζf α )

27 CONORMAL PROBLEM 27 in O T with the conormal derivative boundary conditions on {x d = 0}. By the classical theory, w is smooth in O T. Moreover, we have α 1 λ 2m D α w L2 (O t ) N 2m ζfα L2 (O 0 t ), 0 from which, we get and ( D m w 2 ) 1 2 Q r (X 0 ) λ 1 2 ( w 2 ) 1 2 Q r (X 0 ) Nκm d 2 m k=0 λ 1 2 k 2m ( D k w 2 ) 1 2 Q κr(x 0 ) N 2m 1 2 ( fα 2 ) 1 2 Q κr(x 0 ), 2m 1 2 ( fα 2 ) 1 2 Q κr(x 0 ). Let v := u w, which is also smooth and satisfies v t ( 1) m L 0 v λv = D α ((1 ζ)f α ) in O T with the conormal derivative boundary conditions on {x d = 0}. Hence in Q κr/2 (X 0) v t ( 1) m L 0 v λv = 0. Applying Lemma 5.6 to v, we get ( Dx D m 1 v (D x D m 1 v) Q r (X 0 ) ) Q r (X 0 ) λ ( 1 2 v (v)q r (X 0 ) ) Q r (X 0 ) Nκ γ m k=0 λ 1 2 k 2m ( D k v 2 ) 1 2 Q κr/2 (X 0). With all the preparations above and following the proof of Theorem 4.5, by the triangle inequality and the Cauchy Schwarz inequality we have ( Dx D m 1 u (D x D m 1 u) Q r (X 0 ) ) Q r (X 0 ) λ ( 1 2 u (u)q r (X 0 ) ) Q r (X 0 ) N ( D x D m 1 v (D x D m 1 v) Q r (X 0 ) ) Q r (X 0 ) ( Nλ 1 2 v (v)q r (X 0 ) ) Q r (X 0 ) N( Dm w 2 ) 1 2 Q r (X 0 ) Nλ1 2 ( w 2 ) 1 2 Nκ γ Nκ γ m k=0 m k=0 λ 1 2 k 2m ( D k v 2 ) 1 2 Q κr(x 0 ) Nκm d 2 λ 1 2 k 2m ( D k u 2 ) 1 2 Q κr(x 0 ) Nκm d 2 Therefore, the proposition is proved. Q r (X 0 ) 2m 1 2 ( fα 2 ) 1 2 Q κr(x 0 ) 2m 1 2 ( fα 2 ) 1 2 Q κr(x 0 ). Next, we consider the case that A αβ are functions of both x and t and use the argument of freezing the coefficients to obtain:

28 28 H. DONG AND H. ZHANG Lemma 5.8. Let L be the operator in Theorem 2.2 and T (, ]. Suppose the lower-order coefficients of L are all zero. Let ξ, ν (1, ) satisfying 1/ξ 1/ν = 1, γ (0, 1), and λ, R (0, ). Assume that u H2,loc m (O T ) vanishes outside Q R (X 1), where X 1 O, and u t ( 1) m Lu λu = D α f α with the conormal derivative boundary conditions on {x d = 0}, where f α L 2,loc (O T ). Suppose that (2.5) holds for any X O T. Then there exists a constant N = N(d, m, n, δ, ξ, K, γ) such that for any r (0, ), κ 64, and X 0 O T, we have ( Dx D m 1 u (D x D m 1 u) Q r (X 0 ) ) Q r (X 0 ) λ ( 1 2 u (u)q r (X 0 ) ) Q r (X 0 ) ( Nκ m d 2 2m 1 2 ( fα 2 ) 1 2 Q κr(x 0 ) (A# R ) 1 2ν ( D m u 2ξ 1 ) 2ξ ) Q κr(x 0 ) Nκ γ m k=0 λ 1 2 k 2m ( D k u 2 ) 1 2 Q κr(x 0 ). Proof. Throughout the proof, we assume that Q R (X 1) Q κr(x 0 ). Otherwise, the conclusion holds trivially. Fix a y R d and set L y u = D α (A αβ (t, y)d β u(t, x)). Then we have where f α = f α ( 1) m f α = f α α = β =m u t ( 1) m L y u λu = D α fα, (A αβ (t, y) A αβ (t, x))d β u when α = m, β =m otherwise. It follows from Proposition 5.7 that ( Dx D m 1 u (D x D m 1 u) Q r (X 0 ) ) Q r (X 0 ) λ ( 1 2 u (u)q r (X 0 ) ) Q r (X 0 ) m Nκ γ Note that k=0 λ 1 2 k 2m ( D k u 2 ) 1 2 Q κr(x 0 ) Nκm d 2 Q κr(x 0 ) f α 2 dx dt N Q κr(x 0 ) 2m 1 2 ( fα 2 ) 1 2 Q κr(x 0 ). (5.36) f α 2 dxdt NI y, (5.37)

29 where, for α = m, I y = Q κr(x 0 ) CONORMAL PROBLEM 29 (A αβ (t, y) A αβ (t, x))d β u 2 dx dt. Denote B to be B κr(x 0 ) if κr < R, or to be B R (x 1) otherwise; denote Q in the same fashion. Now we take the average of I y with respect to y in B. Since u vanishes outside Q R (X 1), by Hölder s inequality we get I y dy = (A αβ (t, y) A αβ (t, x))d β u 2 dx dt dy B B Q κr(x 0 ) Q R (X 1) B ( Q A αβ (t, y) A αβ (t, x) 2ν dx dt ) 1 ν dy ( D m u 2ξ dx dt ) 1 ξ, Q κr(x 0 ) Q R (X 1) where, by the boundedness of A αβ, Hölder s inequality as well as the definition of osc x and A # R, the integral over B in the last term above is estimated as follows ( A αβ (t, y) A αβ (t, x) 2ν dx dt) 1 ν dy B Q N( A αβ (t, y) A αβ (t, x) dx dt dy) 1 ν B Q N( Q osc x (A αβ, Q )) 1 ν N((κr) 2md A # R ) 1 ν. This together with (5.36) and (5.37) completes the proof of the lemma. 6. Proof of Theorem 2.2 Proof of Theorem 2.2. Due to the method of continuity, it suffices to prove the a priori estimate. First we note that the interior estimates are obtained in [11, Theorem 1]. With the standard arguments of partition of unity and flattening the boundary, it suffices to consider the case when Ω = R d. We may assume that all the lower-order coefficients are zero. Indeed, if we get the a priori estimate without the lower-order terms, for general systems, we can move the lower-order terms to the right-hand side and apply the interpolation inequality. Taking λ large enough, we then obtain the estimate for general systems. Case 1: p (2, ). First we suppose u Hp m (O T ) and vanishes on O T \ Q R 0 (X 1 ) for some X 1 = (t 1, x 1 ) O, where Q R 0 (X 1 ) = {(µ 2m t, x, µ 1 x d ) : (t, x) Q R 0 (X 1 )}, µ 1 is a parameter which will be determined later, and R 0 is the constant in Assumption 2.1 (ρ). Then it follows Q R 0 (X 1 ) Q R 0 ( X 1 ), where X 1 = (µ 2m t 1, x 1, µ 1 x 1d ). Choose ξ and ν (1, ) such that 2ξ < p and 1/ν

30 30 H. DONG AND H. ZHANG 1/ξ = 1, and fix γ (0, 1). Under these assumptions, from Lemma 5.8 we easily deduce (D x D m 1 u) # (X 0 ) λ 1 2 u # (X 0 ) Nκ γ m Nκ m d 2 ( k=0 λ 1 2 k 2m (M(D k u) 2 (X 0 )) 1 2 ) 2m 1 2 (M(fα ) 2 (X 0 )) ρ 2ν (M(D m u) 2ξ (X 0 )) 1 2ξ for any κ 64 and X 0 O T. This, together with the Fefferman Stein theorem and the Hardy Littlewood maximal function theorem, yields D x D m 1 u Lp λ 1 2 u Lp N( (D x D m 1 u) # Lp λ 1 2 u # Lp ) Nκ γ m k=0 Nκ m ( d 2 Nκ m d 2 λ 1 2 k 2m (M(D k u) 2 ) 1 2 Lp 2m 1 2 (M(fα ) 2 ) 1 2 Lp ρ 1 2ν (M(D m u) 2ξ ) 1 2ξ Lp ) 2m 1 2 fα Lp N(κ γ κ m d 2 ρ 1 2ν ) m k=0 λ 1 2 k 2m D k u Lp (6.38) for any κ 64, where L p = L p (O T ). Now we use the arguments of freezing the coefficients and scaling to get the estimate of Dd m u. As in Lemma 5.1, let Then v satisfies v(t, x, x d ) = u(µ 2m t, x, µ 1 x d ), à αβ (t, x, x d ) = A αβ (µ 2m t, x, µ 1 x d ), f α (t, x, x d ) = f α (µ 2m t, x, µ 1 x d ). v t ( 1) m Dd m (Èαˆα Dd m v) ( 1)m λµ 2m v = µ α d 2m D α f α α = β =m (α,β) ( ˆα, ˆα) µ α dβ d 2m D α (Ãαβ D β v) in O µ 2m T with the conormal derivative boundary conditions on {x d = 0}. We fix (t, y) O and move all the spacial derivatives to the right-hand side of the equation, then add ( 1) m (Dd 1 m v Dm d (Èαˆα (t, y)dd m v)) to both sides of the equation so that v t ( 1) m (Dd m (Èαˆα (t, y)dd m v) Dm d 1 v) λµ 2m v = D α ˆfα ( 1) m Dd m ( ( Èαˆα (t, y) Èαˆα (t, x))dd m v) ( 1) m Dd 1 m v,