SCHAUDER ESTIMATES FOR HIGHER-ORDER PARABOLIC SYSTEMS WITH TIME IRREGULAR COEFFICIENTS

SCHAUDER ESTIMATES FOR HIGHER-ORDER PARABOLIC SYSTEMS WITH TIME IRREGULAR COEFFICIENTS HONGJIE DONG AND HONG ZHANG Abstract. We prove Schauder estimates for solutions to both divergence and non-divergence type higher-order parabolic systems in the whole space and a half space. We also provide an existence result for the divergence type systems in a cylindrical domain. All coefficients are assumed to be only measurable in the time variable and Hölder continuous in the spatial variables. 1. Introduction This paper is devoted to Schauder estimates for divergence and nondivergence type higher-order parabolic systems. It is well known that the Schauder estimates play an important role in the existence and regularity theory for elliptic and parabolic equations and systems. The classical approach is to first study the fundamental solutions for equations with constant coefficients, then use the argument of freezing the coefficients for general equations with regular coefficients; see, for instance, [1, 8, 15, 18]. For systems it has become customary to use Campanato s technique first introduced in [4], the application of which to second-order elliptic systems was explained comprehensively in Giaquinta [9]; see also [17, 7] for second-order parabolic systems. For higher-order systems, we also refer the reader to [13, Chap. 3] and the references therein. The classical Schauder estimates were established under the assumption that the coefficients are regular in both space and time. In this paper, we consider the coefficients which are regular only with respect to spatial variables. This type of coefficient has been studied by several authors mostly for second-order equations; see, for instance, [2, 14, 10, 12, 16]. In [10, 11] Lieberman studied interior and boundary Schauder estimates for secondorder parabolic equations with time irregular coefficients using the maximum principle and a Campanato type approach. More recently, Boccia [3] considered higher-order non-divergence type parabolic systems in the whole space. Key words and phrases. higher-order systems, Schauder estimates. H. Dong was partially supported by the NSF under agreement DMS-1056737. H. Zhang was partially supported by the NSF under agreement DMS-1056737. 1

2 H. DONG AND H. ZHANG To present our results precisely, let Lu = A αβ D α D β u, Lu =, β m where m is a positive integer,, β m D α = D α 1 1 Dα d d, α = (α 1,..., α d, D α (A αβ D β u, and, for each α and β, A αβ = [A αβ ij ]n i,j=1 is an n n real matrix-valued function. Moreover, the leading coefficients satisfy the so-called Legendre Hadamard ellipticity condition (see (2.1, which is more general than the usual strong ellipticity condition. The functions used throughout this paper u = (u 1,..., u n tr, f = (f 1,..., f n tr, f α = (f 1 α,..., f n α tr are real vector-valued functions. The parabolic systems which we study are u t ( 1 m Lu = f, u t ( 1 m Lu = D α f α in the whole space, or in the half space or cylindrical domains with the Dirichlet boundary condition u = Du =... = D m 1 u = 0, where the first system is in the non-divergence form and the second system is in the divergence form. We assume that all the coefficients and data are Hölder continuous with respect to the spatial variables and merely measurable with respect to the time variable. For the non-divergence form systems, we prove that D 2m u is Hölder continuous with respect to both variables in the whole space, and in the half space all the 2mth order derivatives of u are Hölder continuous with respect to both variables up to the boundary with the exception of Dd 2m u, where x d is the normal direction of the boundary. For the divergence form systems, we prove that all the mth order derivatives are Hölder continuous with respect to both variables up to the boundary. We also prove an existence theorem for the divergence form systems in a cylindrical domain, provided that the boundary of the domain is sufficiently smooth. To our best knowledge, these results are new for higher-order systems and they extend the corresponding results found in Lieberman [10] for second-order scalar equations. In the special case of second-order parabolic systems, compared to [17] our conditions on the coefficients and data are more general. In particular, we do not require the data to vanish on the lateral boundary of the domain, i.e., the compatibility condition imposed in [17]. For the proof, we use some results in [5], in which the first author and Kim proved L p estimates for divergence and non-divergence type higherorder parabolic systems with time-irregular coefficients. Let us give the outline of the proofs. In the divergence case, the classical L 2 estimates and

an interior estimate obtained in [5] imply D m u (D m u Qr(X0 2 dx dt Q r(x 0 C( r R 2d2m Q R (X 0 SCHAUDER ESTIMATES 3 D m u (D m u QR (X 0 2 dx dt, r R, where Q r (X 0 (or Q R (X 0 is a parabolic cylinder with radius r (or R, respectively and center X 0, (D m u Qr(X0 (or (D m u QR (X 0, respectively is the average of D m u in Q r (X 0 (or Q R (X 0, respectively. For precise definitions, see Section 2, C is a constant independent of r, R, X 0, and u, and u is a solution of Here u t ( 1 m L 0 u = 0 in Q 2R. L 0 = α = β =m D α (A αβ (td β and X 0 Q R. The coefficients A αβ, which depend only on t, are called simple coefficients. For the boundary estimates, we combine the L p estimates established in [5] and the Sobolev embedding theorem to prove the Hölder continuity and obtain, for instance, the following mean oscillation type estimate for systems with simple coefficients, Dd m u (Dm d u Q r (X 0 2 dx dt Q r (X 0 C( r R 2γd2m Q R (X 0 D m d u (Dm d u Q R (X 0 2 dx dt, r R, where X 0 {x d = 0} Q R, and γ (0, 1 are arbitrary, Q r,r (X 0 = Q r,r (X 0 {x d > 0}, C is a constant independent of r, R, X 0, and u, and u satisfies u t ( 1 m L 0 u = 0 in Q 2R with the Dirichlet boundary condition u = D d u =... = D m 1 d u = 0 on {x d = 0}. We then use a perturbation argument to treat the general systems. Similar interior and boundary estimates for the non-divergence form systems can be established in the same fashion. Here the idea is that we can differentiate the system { ut ( 1 m L 0 u = 0 in Q 2R u = 0, D d u = 0,..., D m 1 d u = 0 on {x d = 0} Q 2R with respect to tangential direction x. This together with the Wp 1,2m estimates implies that D x D 2m 1 u is in some Hölder space C a 2m,a (Q R with a arbitrarily close to 1 from below, which yields the mean oscillation type estimates.

4 H. DONG AND H. ZHANG The paper is organized as follows. In the next section we introduce some notation and state our main results. Section 3 is devoted to some preliminary lemmas. In Section 4 we make necessary preparations and in Section 5 prove our main result for the divergence type systems. Section 6 deals with the non-divergence type systems. 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 = (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 = {x R d : x x 0 < 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 R d, Q r (t 0, x 0 = Q r (t 0, x 0 O. We use the abbreviations, for example, B r if x 0 = 0 and Q r if (t 0, x 0 = (0, 0. The parabolic boundary of Q r (t 0, x 0 is defined to be p Q r (t 0, x 0 = [t 0, t 0 r 2m B r (x 0 {t = t 0 r 2m } B r (x 0. The parabolic boundary of Q r (t 0, x 0 is We denote p Q r (t 0, x 0 = ( p Q r (t 0, x 0 O (Q r (t 0, x 0 {x d = 0}. f, g Ω = Ω f tr g = n j=1 Ω f j g j. For a function f defined 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]. 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]. We also use 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,

SCHAUDER ESTIMATES 5 where a (0, 1]. The space corresponding to a,d is denoted by Ca (D. For a (0, 1], set f a 2m,a,D = f L (D [f] a 2m,a,D. The space corresponding to a 2m,a,D is denoted by C a 2m,a (D. For a (1, 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] is the fractional part of a. We use the following Sobolev space W 1,2m p ((S, T Ω = { u : u t, D α u L p ((S, T Ω, 0 α 2m }. We denote the average of f in D R d1 to be (f D = 1 f(t, x dx dt = f(t, x dx dt. D D 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, we impose the Legendre Hadamard ellipticity condition with a constant λ > 0 on the leading coefficients, i.e., A αβ ij ξ iξ j η α η β λ ξ 2 η 2m, (2.1 α = β =m where ξ R n, η R d, and η α = η α 1 1 ηα 2 2 ηα d d. Here we call Aαβ ij coefficients if α = β = m. Throughout this paper L 0 u = L 0 u = D α (A αβ (td β u, α = β =m D 2m the leading where A αβ are measurable in t and satisfy (2.1. We are ready to state the main results of the paper. The first result is about the Schauder estimate and solvability for divergence type higher-order systems in cylindrical domains. Theorem 2.1. Let a (0, 1 and T (0, ]. Assume f α C a for α = m and f α L for α < m. Suppose that the operator L satisfies the

6 H. DONG AND H. ZHANG Legendre Hadamard condition (2.1, and A αβ C a. function in R d1 and Ω C m,a. Then u t ( 1 m Lu = D α f α in (0, T Ω, Let g be a smooth u = g, Du = Dg,..., D m 1 u = D m 1 g on [0, T Ω, u = g on {0} Ω has a unique solution u such that u C am 2m,am ([0, min(t, k Ω for any k > 0, and it satisfies u am 2m,am,(0,T Ω C( u L2 ((0,T Ω F G, (2.2 where F = [f α ] a,(0,t Ω f α L ((0,T Ω, G = D α g a,(0,t Ω D α g L ((0,T Ω g t L ((0,T Ω, and C > 0 is a constant depending only on d, n, m, λ, K, A αβ a, Ω, and a. Moreover, for any constant k > 0, we have where u am 2m,am,(0,min(T,k Ω CeC 1k ( F k G k, (2.3 F k = [f α ] a,(0,min(t,k Ω f α L ((0,min(T,k Ω, G k = D α g a,(0,min(t,k Ω g t L ((0,min(T,k Ω, D α g L ((0,min(T,k Ω C > 0 is a constant depending only on d, n, m, λ, K, A αβ a, Ω, and a, and C 1 > 0 is a constant depending only on d, n, m, λ, and K. The next theorem is regarding the a priori interior and boundary Schauder estimates for the non-divergence type systems. Theorem 2.2. Let a (0, 1. Suppose that A αβ C a and f C a. Let u C loc (Rd1 be a solution to u t ( 1 m Lu = f, (2.4 where L satisfies (2.1. For any R 1, I. Interior case: if (2.4 holds in Q 4R, then there exists a constant C depending only on d, n, m, λ, K, A αβ a, R, and a such that u t a,q R D 2m u a 2m,a,Q C( f R a,q 4R u L2 (Q 4R ;

SCHAUDER ESTIMATES 7 II. Boundary case: if (2.4 holds in Q 4R with the homogeneous Dirichlet boundary condition on {x d = 0} Q 4R, then there exists a constant C depending only on d, n, m, λ, K, A αβ a, R, and a such that [D x D 2m 1 u] a C[f] C D γ u 2m,a,Q R a,q 4R L (Q. 4R γ 2m We note that the boundary estimate in Theorem 2.2 is optimal even for the heat equation, in the sense that near the boundary Dd 2 u might be discontinuous with respect to x if f has jump discontinuities in t. See, for instance, [19, 3] and [10, 16]. In order to estimate the Hölder semi-norm of Dd 2mu, we need to impose more regularity conditions on Aαβ and f. See Remark 6.3 below for a discussion. 3. Technical preparation In this section, we state several technical lemmas which are useful in our proofs. First we need the following version of Campanato s theorem. The proof can be found in [9] and [17]. Lemma 3.1. (i Let f L 2 (Q 2 and a (0, 1. Assume that f (f Qr(t0,x 0 2 dx dt A 2 r 2a, (t 0, x 0 Q 1, (3.1 Q r(t 0,x 0 and 0 < r 1. Then f C a 2m,a (Q 1 and [f] a 2m,a,Q CA, 1 with C = C(d. ii Let f L 2 (Q 2 and a (0, 1. Assume that (3.1 holds for r < x 0d. Moreover, f (f Q r (t 1,x 1 2 dx dt A 2 r 2a, (t 1, x 1 {x d = 0} Q 1, Q r (t 1,x 1 and 0 < r 1. Then f C a 2m,a (Q 1 and with C = C(d. [f] a 2m,a,Q 1 CA, The following lemma will be frequently used in this paper. The proof can be found in [10] and [17]. Lemma 3.2. Let Φ be a nonnegative, nondecreasing function on (0, r 0 ] such that Φ(ρ A( ρ r a Φ(r Br b for any 0 < ρ < r r 0, where 0 < b < a are fixed constants. Then Φ(r Cr b (r b 0 Φ(r 0 B, r (0, r 0 with a constant C = C(A, a, b.

8 H. DONG AND H. ZHANG Here is another technical lemma. Lemma 3.3. Assume that Ω is an open bounded domain in R d. Then there exists a function ξ C0 (Ω with a unit integral such that for any 0 < α m ξ(yy α dy = 0. Proof. Let M = m l=1 (ld 1 d 1 Ω. We look for ξ in the form ξ(y = a β ξ β (y, β m where β is an n-tuple multi-index and ξ β C0 (Ω, β m, are functions to be chosen. The problem is then reduced to the following linear system of {a β }: a β ξ β (yy α dy = 1 α =0, α m. β m Ω It suffices to choose ξ β such that the (M 1 (M 1 coefficient matrix { } A := ξ β (yy α dy Ω α, β m is invertible. To this end, we use a perturbation argument. It is easily seen that the matrix { } y β y α dy Ω α, β m is invertible. Since C0 (Ω is dense in L 2(Ω, for any ε > 0 and β m, there exists ξ β C0 (Ω such that ξ β y β L2 (Ω ε. By the Cauchy Schwarz inequality, ξ β (yy α dy y β y α dy Cε, Ω Ω where C is a positive constant independent of ε. By the continuity of matrix determinant, upon taking ε sufficiently small, A is still invertible. The lemma is proved. 4. Estimates for divergence type systems This section is devoted to the interior and boundary a priori estimates for the divergence type higher-order systems. 4.1. Interior estimates. The main result of this subsection is the following proposition. Proposition 4.1. Let a (0, 1. Assume that R 1 and u Cloc (Rd1 satisfying u t ( 1 m Lu = D α f α in Q 2R. (4.1

SCHAUDER ESTIMATES 9 Suppose that A αβ C a, f α C a if α = m, and f α L if α < m. Then there exists a constant C depending only on d, n, m, λ, K, A αβ a, R, and a such that where [D m u] a 2m,a,Q R C ( β m u 1 2 a 2m,Q R C ( β m D β u L (Q2R F, (4.2 D β u L (Q2R F, (4.3 F = [f α ] a,q 2R f α L (Q2R. For the proof of the proposition, we first consider homogeneous systems with the simple coefficients. Lemma 4.2. Assume that u C loc (Rd1 and satisfies u t ( 1 m L 0 u = 0 in Q 2R, (4.4 where R (0,. Then for any X 0 Q R and r < R, There is a constant C depending only on d, n, m, and λ such that u (u Qr(X0 2 dx dt Q r(x 0 C( r R 22md Q R (X 0 u (u QR (X 0 2 dx dt. (4.5 Proof. By scaling and translation of the coordinates, without loss of generality, we can assume that R = 1 and X 0 = (0, 0. First we consider the case r 1/4. Since Q r u (u Qr 2 dx dt Cr 22md sup Q r By Lemma 2 in [5], Hence, by scaling and (4.6, sup( Du u t C u L2 (Q 1. Q 1/4 Du 2 Cr 4m2md sup Q r u t 2. (4.6 Q r u (u Qr 2 dx dt Cr 22md u 2 L 2 (Q 1. Clearly, the inequality above holds true in the case when r [1/4, 1 as well. Since u (u Q1 also satisfies (4.4, upon substituting u by u (u Q1, the lemma is proved. Lemma 4.3. Let a (0, 1. Assume that r < R 1 and u Cloc (Rd1 satisfying u t ( 1 m L 0 u = D α f α in Q 2R,

10 H. DONG AND H. ZHANG f α C a if α = m, and f α L if α < m. Then for any X 0 Q R, there exists a constant C depending only on n, m, d, and λ such that D m u (D m u Qr(X0 2 dx dt Q r(x 0 C( r R 22md Q R (X 0 D m u (D m u QR (X 0 2 dx dt C [f α ] 2 a,q 2R R 2a2md C f α 2 L (Q 2R R22md. Proof. Let w be the weak solution of the following system w t ( 1 m L 0 w = D α f α in Q 2R, w = 0, Dw = 0,..., D m 1 w = 0 on p Q 2R. Multiply w to both sides of the system and integrate over Q 2R. Due to the homogeneous Dirichlet boundary condition of w, we can replace D α f α by D α (f α (t, x f α (t, 0 when α = m. By extending w = 0 outside Q 2R, we have A αβ D α wd β w dx dt = A αβ D α wd β w dx dt Q 2R O 0 λ D m w 2 dx dt = λ D m w 2 dx dt, (4.7 O 0 Q 2R where we use the Fourier transform and (2.1 in the inequality. By Young s inequality and (4.7, we get for any ε (0, 1, λ D m w 2 dx dt Q 2R ( 1 m( f α (t, x f α (t, 0 D α w dx dt Q 2R ( 1 α f α D α w dx dt Q 2R ε D m w 2 L 2 (Q 2R f α L (Q 2R C(ε f α (t, x f α (t, 0 2 L 2 (Q 2R Q 2R D α w dx dt. (4.8 Since w = 0, Dw = 0,..., D m 1 w = 0 on p Q 2R, for α < m and R 1 by the Poincaré inequality D α w 2 dx dt CR 2 D m w 2 dx dt, Q 2R Q 2R

SCHAUDER ESTIMATES 11 where C = C(d, m > 0. By the Cauchy Schwarz inequality, the inequality above, and Young s inequality, we can estimate the last term in (4.8 as follows: f α L (Q2R D α w dx dt Q 2R f α L (Q 2R Q 2R 1/2( Q 2R D α w 2 dx dt 1/2 C f α L (Q2R R Q 2R 1/2( 1/2 D m w 2 dx dt Q 2R C(ε f α 2 L (Q 2R R2d2m ε D m w 2 dx dt. Q 2R Choosing ε sufficiently small, we obtain D m w 2 dx dt Q 2R C f α (t, x f α (t, 0 2 L 2 (Q 2R C f α 2 L (Q 2R R2d2m C [f α ] 2 a,q 2R R 2ad2m C f α 2 L (Q 2R R2d2m. (4.9 We now temporarily assume that A αβ and f α are smooth functions. By the classical theory, we know that w is smooth. Consider v = u w, which is a smooth function as well, and satisfies v t ( 1 m L 0 v = 0 in Q 2R. Since D m v satisfies the same system as v, applying Lemma 4.2 to D m v, we have D m v (D m v Qr(X0 2 dx dt Q r(x 0 C( r R 2d2m Q R (X 0 By the triangle inequality, D m u (D m u Qr(X0 2 dx dt Q r(x 0 = C Q r(x 0 Q r(x 0 D m v (D m v QR (X 0 2 dx dt. (4.10 D m w (D m w Qr(X 0 D m v (D m v Qr(X 0 2 dx dt D m w 2 dx dt C Q r(x 0 D m v (D m v Qr(X 0 2 dx dt.

12 H. DONG AND H. ZHANG By (4.9, (4.10, and the inequality above, we get D m u (D m u Qr(X0 2 dx dt C Q r(x 0 C( r R 2d2m Q R (X 0 C D m w 2 dx dt Q 2R C( r R 2d2m Q R (X 0 Q r(x 0 D m v (D m v QR (X 0 2 dx dt D m u (D m u QR (X 0 2 dx dt D m w 2 dx dt C [f α ] 2 a,q 2R R 2ad2m C f α 2 L (Q 2R R2d2m C( r R 2d2m Q R (X 0 D m u (D m u QR (X 0 2 dx dt. Thus the lemma is proved under the additional smoothness assumption. By a standard argument of mollification, it is easily seen that we can remove the smoothness assumption. Applying Lemmas 3.2 and 4.3, we get the Campanato type estimate D m u (D m u Qr(X0 2 dx dt Q r(x 0 C( r R 2ad2m ( C Q R (X 0 D m u (D m u QR (X 0 2 dx dt [f α ] a,q 2 2R f α 2 L (Q 2R r 2ad2m for any r < R 1 and X 0 Q R. Then, 1 r 2ad2m D m u (D m u Qr(X0 2 dx dt Q r(x 0 C R 2ad2m D m u (D m u QR (X 0 2 dx dt CF 2 Q R (X 0 C R 2a2md D m u 2 dx dt CF 2. Q 2R Thanks to Lemma 3.1, we get [D m u] a 2m,a,Q R C ( R (2ad2m Q 2R D m u 2 dx dt F 2 1 2. (4.11 As for the coefficients which also depend on x, the estimates follow from the standard argument of freezing the coefficients. Specifically, we first consider the operator L that only has the highest-order terms. We fix a

y B R, and define Then L 0y = u t ( 1 m L 0y u = where f α = f α ( 1 m( β =m f α = f α if α < m. SCHAUDER ESTIMATES 13 α = β =m D α (A αβ (t, yd β. D α f α ( 1 m (L L 0y u = (A αβ (t, y A αβ (t, xd β u D α fα, if α = m, Applying the method in proving Lemma 4.3, as (4.9 we obtain ( D m w 2 L 2 (Q 2R C [A αβ ] 2 a D m u 2 L (Q 2R R2a2md α = β =m Following the proof of (4.11, we reach f α 2 L (Q 2R R22md [f α ] 2 a,q 2R R 2a2md. [D m u] a 2m,a,Q R C ( R (2ad2m Q 2R D m u 2 dx dt α = β =m [A αβ ] 2 a D m u 2 L (Q 2R F 2 1 2. (4.12 For systems with lower-order terms, we move all the lower-order terms to the right-hand side to get u t ( 1 m L h u = D α( f α ( 1 m1 A αβ D β u β <m D α( f α ( 1 m1 β m A αβ D β u, where L h u denotes the sum of the highest-order terms. It suffices to substitute ˆf α = f α ( 1 m1 β Aαβ D β u into the estimate (4.12 and notice that when α = m [A αβ D β u] a,q 2R β <m C β <m ( A αβ L [D β u] a,q 2R [A αβ ] a D β u L (Q2R. An easy calculation then completes the proof of (4.2.

14 H. DONG AND H. ZHANG It remains to prove (4.3. We estimate u(t, x 0 u(s, x 0, where (t, x 0, (s, x 0 Q R. Let ρ = t s 1 2m and η(x = 1 ξ( x ρ d ρ, where ξ(x is the function in Lemma 3.3 with Ω = B 1. We define u(t, y = u(t, y T m,x0 u(t, y u(t, x 0, where T m,x0 u(t, y is the Taylor expansion of u(t, y in y at x 0 up to mth order. By the triangle inequality, u(t, x 0 u(s, x 0 u(t, x 0 η(y x 0 u(t, y dy B ρ(x 0 u(s, x 0 η(y x 0 u(s, y dy B ρ(x 0 η(y x 0 u(s, y dy η(y x 0 u(t, y dy. (4.13 B ρ(x 0 B ρ(x 0 The first two terms on the right-hand side of (4.13 can be estimated in a similar fashion: noting that B ρ η(x dx = 1, Since u(t, x 0 = B ρ(x 0 B ρ(x 0 η(y x 0 u(t, y dy η(y x 0 (u(t, x 0 u(t, y dy. u(t, y u(t, x 0 = u(t, y T m,x0 u(t, y C[D m u] a,q 2R y x 0 ma, we get u(t, x 0 B ρ(x 0 η(y x 0 u(t, y dy C[D m u] a,q 2R ρ ma. (4.14 For the last term on the right-hand side of (4.13, by the definition of η, η(y x 0 u(s, y dy = η(y x 0 u(s, y dy. Therefore, = = B ρ(x 0 B ρ(x 0 B ρ(x 0 t s η(y x 0 u(t, y dy B ρ(x 0 B ρ(x 0 η(y x 0 u(t, y dy B ρ(x 0 η(y x 0 u(s, y dy B ρ(x 0 η(y x 0 u t (τ, y dy dτ. η(y x 0 u(s, y dy

Because u satisfies (4.1, t η(y x 0 u t (τ, y dy dτ s = B ρ(x 0 t s B ρ(x 0 SCHAUDER ESTIMATES 15 η(y x 0 ( ( 1 m1, β m D α (A αβ D β u(τ, y D α f α dy dτ. (4.15 Since η(y x 0 has compact support in B ρ (x 0, we use f α (f α Bρ(x 0 to substitute f α when α = m. Moreover, for α = m, D α (A αβ D β u = D α( (A αβ (A αβ Bρ(x 0 D β u D α( (A αβ Bρ(x 0 (D β u (D β u Bρ(x 0. We plug these into (4.15 and integrate by parts. It follows easily that t η(y x 0 u t (τ, y dy dτ s B ρ(x 0, β m Q ρ(x 0, β m C β m C ([A αβ ] a D β u L (Q 2R [D β u] a,q2r A αβ L ρ a D m η(y x 0 dy dt A αβ L D β u L (Q 2R f α L (Q 2R Q ρ(x 0 D m η L [f α ] a,q 2R Q ρ ρ a Q ρ(x 0 D α η(y x 0 dy dt D β u a,q 2R ρ ma C [f α ] a,q 2R ρ ma ( f α L (Q 2R β m D α η(y x 0 dy dt A αβ L D β u L (Q2R ρ m1, where we use the fact that D α η L Cρ d α for any α m. Hence, combining with (4.14, we reach u(t, x 0 u(s, x 0 Cρ ma( D β u a,q 2R F, β m which, together with (4.2 and the interpolation inequality, implies (4.3. This completes the proof of the proposition.

16 H. DONG AND H. ZHANG 4.2. Boundary estimates. As in the previous subsection, we first consider systems with the simple coefficients without lower-order terms: u t ( 1 m L 0 u = 0 in Q 2R, (4.16 u = 0, D d u = 0,..., D m 1 d u = 0 on {x d = 0} Q 2R. (4.17 Thanks to the L p estimates in a half space obtained in [5], we are able to derive a local Wp 1,2m estimate for solution of (4.16 (4.17 by the method in Lemma 1 in [5]. Namely, u W 1,2m p (Q R C(R, p u L p(q 2R, (4.18 where C(R, p is a constant. Now we are ready to prove the following lemma. Lemma 4.4. Assume that 0 < r R <, γ (0, 1, and u Cloc (Rd1 satisfying (4.16 (4.17. Then there exists a constant C depending only on d, n, m, λ, K, and γ, such that u 2 dx dt C( r R d4m u 2 dx dt, (4.19 Q r (X 0 Q r (X 0 Q r (X 0 Q R (X 0 D l d u 2 dx dt C( r R 2(m ld2m D m d u (Dm d u Q r (X 0 2 dx dt C( r R 2γd2m Q R (X 0 where X 0 {x d = 0} Q R and 0 < l < m. Q R (X 0 D l d u 2 dx dt, (4.20 D m d u (Dm d u Q R (X 0 2 dx dt, (4.21 Proof. By scaling and translation of the coordinates, without loss of generality, we can assume R = 1 and X 0 = (0, 0. From (4.18, for any r 1 [1, 2, there exists a constant C = C(r 1 so that By the Sobolev embedding theorem, u W 1,2m 2 (Q r 1 C u L 2 (Q 2. u Lp(Q r 1 C u W 1,2m 2 (Q r 1, where 1/p > 1/2 1/(d 1. Using (4.18 again, we get u W 1,2m p (Q r 2 C u L p(q r 1 C u L 2 (Q 2, where r 2 < r 1 and C = C(r 1, r 2. Using a standard argument of bootstrap with a sequence of shrinking cylinders Q rl, for any p > 2 there is a C depending on p such that u W 1,2m p (Q 1 C u L 2 (Q 2. (4.22

By the Sobolev embedding theorem, we have SCHAUDER ESTIMATES 17 Wp 1,2m (Q 1 C a 2m,a (Q 1, u a 2m,a,Q 1 C u W 1,2m p (Q 1, (4.23 where a = 2m (d 2m/p. Since p < can be arbitrarily large, a can be arbitrarily close to 2m from below. Let us first prove (4.19. We only need to consider r (0, 1/2], otherwise it is obvious. By the Poincaré inequality and (4.23 with some a > m, u 2 dx dt Cr 2m D Q d m u 2 L (Q r Q r Cr d4m Dd m u 2 L (Q 1/2 r Cr d4m u 2 Wp 1,2m (Q 1/2 Crd4m u 2 dx dt. Q 1 The last inequality is due to the L p estimate in (4.22 with 1/2 and 1 in place of 1 and 2, respectively. Next, we prove (4.20 and only consider the case r (0, 1/2] as in the proof of (4.19. By the Poincaré inequality and (4.23 with some a > m, we have Dd l u 2 dx dt r 2(m l Dd m u 2 dx dt Q r Q r Cr 2(m ld2m sup Dd m u 2 Cr 2(m ld2m u 2 Q Wp 1,2m (Q 1/2 1/2 Cr 2(m ld2m u 2 dx dt Cr 2(m ld2m Dd l u 2 dx dt. Q 1 Finally for (4.21, let v = u xm d m! (Dd mu Q so that v also satisfies the same 1 system as u. Moreover, Dd mv = Dm d u (Dm d u Q. Then, by (4.23, (4.22, 1 and the Poincaré inequality, for a = m γ (m, m 1 and r (0, 1/2], D Q d m v (Dm d v Q 2 dx dt Cr 2(a md2m v 2 a r 2m,a,Q r r Cr 2γd2m v 2 dx dt Cr 2γd2m Dd m v 2 dx dt. The lemma is proved. Q 1 Similar to Lemma 4.3, we consider u t ( 1 m L 0 u = and the boundary condition (4.17. Q 1 Q 1 D α f α in Q 2R (4.24 Lemma 4.5. Let a (0, 1. Assume that u Cloc (Rd1 satisfying (4.24 and (4.17, f α C a for α = m, and f α L for α < m. Then for any

18 H. DONG AND H. ZHANG 0 < r < R 1, γ (0, 1, and X 0 {x d = 0} Q R, there exists a constant C depending only on d, m, n, λ, K, and γ such that Dx m u 2 dx dt Q r (X 0 C( r R 4md C Q r (X 0 Q R (X 0 Dx m u 2 dx dt C [f α ] 2 R 2a2md a,q 2R f α 2 L (Q 2R R22md, (4.25 D m d u (Dm d u Q r (X 0 2 dx dt C( r R 2γd2m C [f α ] 2 a,q 2R Q R (X 0 and D Q d l Dm l x u 2 dx dt r (X 0 C( r R 2(m ld2m C [f α ] 2 where 0 < l < m. a,q 2R D m d u (Dm d u Q R (X 0 2 dx dt R 2a2md C Q R (X 0 Dd l Dm l x u 2 dx dt R 2a2md C f α 2 L (Q 2R R22md, (4.26 f α 2 L (Q 2R R22md, (4.27 Proof. We follow the proof of Lemma 4.3. Let w be the weak solution of the following system w t ( 1 m L 0 w = D α f α in Q 2R, w = 0, Dw = 0,..., D m 1 w = 0 on p Q 2R. Then the Poincaré inequality and the method in the proof of Lemma 4.3 yield D m w 2 dx dt Q 2R C [f α ] 2 a,q 2R R 2a2md C f α 2 L (Q 2R R22md. We again use the mollification argument as in Lemma 4.3 so that w is smooth. Let v = u w. Then v satisfies all the conditions in Lemma 4.4, and so does D m x v. Thus we obtain (4.19 with D m x v in place of u.

SCHAUDER ESTIMATES 19 Combining the estimates of v and w, we obtain (4.25. The proofs of (4.26 and (4.27 are similar. The lemma is proved Now we are ready to prove the boundary Hölder estimates similar to the interior case. From (4.25, (4.26, and (4.27, taking γ > a and using Lemma 3.2, we get that for any X 0 {x d = 0} Q R and r (0, R, where Q r (X 0 m 1 Dx m u 2 Dd m u (Dm d u Q r (X 0 2 l=1 D l d Dm l x u 2 dx dt I, I := C (R (2a2md D m u 2 dx dt F 2 r 2a2md, Q 2R F = [f α ] f a,q α 2R L (Q. 2R This estimate, together with the interior estimates and Lemma 3.1, implies that [D m u] a C (R (2a2md D m u 2 dx dt F 2 1 2. 2m,a,Q R Q 2R Similar to the proof of (4.2, we can apply the argument of freezing the coefficients to deal with the general operator L with lower-order terms. Here we just state the conclusion as the following proposition. Proposition 4.6. Assume that R 1 and u Cloc (Rd1 satisfying u t ( 1 m Lu = D α f α in Q 2R, u = 0, D d u = 0,..., D m 1 d u = 0 on {x d = 0} Q 2R. Suppose that L and f α satisfy the conditions in Theorem 2.1. Then there exists a constant C depending only on d, n, m, λ, K, A αβ a, R, and a such that u 1 2 a [D m u] 2m,Q a C R 2m,a,Q R ( β m 5. Proof of Theorem 2.1 D β u L (Q 2R F. Before proving the main theorem, let us show a technical lemma. Lemma 5.1. Assume that a (0, 1, u C loc (Rd1, Ω is an open set in R d, Ω C 2, and T (0, ]. Then for any ε > 0 sufficiently small, we have u L ((0,T Ω ε[u] a 2m,a,(0,T Ω C(ε u L2 ((0,T Ω, where C(ε is a constant depending on ε.

20 H. DONG AND H. ZHANG Proof. Since Ω C 2, there exists a r 0 > 0 such that for any X 0 (0, T Ω and r (0, r 0, we can choose a cylinder Q r (s, y Ω so that X 0 Q r (s, y and u(x 0 u(x 0 u(t, x dx dt u(t, x dx dt Q r(s,y Cr a [u] a 2m,a Cr d2m 2 u L2. Therefore, u L ε[u] a 2m,a C(ε u L2. The lemma is proved. Next, we show a global a priori estimate. Q r(s,y Proposition 5.2. Let L, Ω, and f α satisfy the conditions in Theorem 2.1. Assume that u C ((0, T Ω and satisfies the following system u t ( 1 m Lu = D α f α in (0, T Ω, u = 0, Du = 0,..., D m 1 (5.1 u = 0 on [0, T Ω, u = 0 on {0} Ω. Then (2.2 holds with G = 0. Proof. Using the standard arguments of partition of the unity and flattening the boundary, we combine the interior and boundary estimates to get ( u am 2m,(0,T Ω [Dm u] a 2m,a,(0,T Ω C D β u L ((0,T Ω F. β m By the interpolation inequalities in Hölder spaces, for instance, see [15, Section 8.8], u am 2m,(0,T Ω [Dm u] a 2m,a,(0,T Ω C ( u L ((0,T Ω F. Applying Lemma 5.1 and the interpolation inequalities again, we get u L ((0,T Ω C(ε u L2 ((0,T Ω ε[d m u] a 2m,a,(0,T Ω ε u am Upon taking ε sufficiently small, we arrive at (2.2. proved.,(0,t Ω. 2m The proposition is In order to implement the method of continuity, we need the right-hand side of (2.2 to be independent of u and this leads us to consider the following system for T <, u t ( 1 m Lu κu = D α f α in (0, T Ω, u = 0, Du = 0,..., D m 1 (5.2 u = 0 on [0, T Ω, u = 0 on {0} Ω,

SCHAUDER ESTIMATES 21 where κ is a large constant to be specified later. We rewrite the system as u t ( 1 m L h u κu = D α f α ( 1 m (L h Lu, (5.3 where L h u is the sum of the highest-order terms. Then multiply u to both sides of (5.3 and integrate over (0, T Ω. Thus λ D m u 2 L 2 ((0,T Ω κ u 2 L 2 ((0,T Ω ( 1 α f α D α u dx dt, β m, β <m (0,T Ω ( 1 m α 1 ( 1 m α 1 (0,T Ω (0,T Ω A αβ D β ud α u dx dt A αβ D β ud α u dx dt. (5.4 Take a point x 0 Ω. Due to the homogeneous Dirichlet boundary condition, if α = m, the factor f α in the first integral on the right-hand side above can be replaced by f α (t, x f α (t, x 0. We use the Cauchy Schwarz inequality, Young s inequality, and the interpolation inequality to bound the right-hand side by ( C(n, m, d, λ, K, ε f α f α (t, x 0 1 2 dx dt (0,T Ω (0,T Ω u 2 dx dt ε D m u 2 L 2 ((0,T Ω. After taking ε sufficiently small to absorb the term ε D m u 2 L 2 ((0,T Ω to the left-hand side of (5.4 and choosing κ sufficiently large depending on n, m, d, λ, K, and ɛ, we reach u L2 ((0,T Ω C T F, (5.5 where C depends only on d, n, m, λ, K, and Ω. Combining (5.5 with (2.2, we get the following lemma. Lemma 5.3. Assume that L, Ω, and f α satisfy all the conditions in Theorem 2.1 and u C ((0, T Ω satisfies (5.2 with a sufficiently large constant κ depending only on n, m, d, λ, and K. Then u am,am,(0,t Ω CF, (5.6 2m where C depends only on d, m, n, λ, K, A αβ a, Ω, and a. Now we are ready to prove Theorem 2.1. Proof of Theorem 2.1. By considering u g instead of u, without loss of generality, we can assume g = 0. For T =, in general there is no

22 H. DONG AND H. ZHANG 2m,ma ([0, Ω solution (need a reference. We only consider the case T <. It remains to prove the solvability. Let κ be the constant from the calculation of (5.7previous lemma. Since u satisfies (5.1, the function v := e κt u satisfies v t ( 1 m Lv κv = e κt D α f α. C am By (5.5, we have which implies v L2 ((0,T Ω C T F, u L2 ((0,T Ω C T e κt F Ce 2κT F. We still use κ to denote 2κ. The inequality above combined with (2.2 yields u am 2m,am CeκT F. (5.7 We then reduce the problem to the solvability of v. By Lemma 5.3, (5.6 holds for v. Next we consider the following equation u t ( 1 m( slu (1 s m I n n u = D α f α with the same initial and boundary conditions, where the parameter s [0, 1]. It is well known that when s = 0 there is a unique solution in C am 2m,am ([0, T Ω. Then by the method of continuity and the a priori estimate (5.7, we find a solution when s = 1, which gives u. Finally, (2.3 follows from (5.6 with v in place of u. The theorem is proved. Remark 5.4. Actually the condition f α L for α < m can be relaxed. With slight modification in our proof, f α can be in some Morrey spaces. 6. Estimate for non-divergence type systems In this section, we deal with the non-divergence type systems. 6.1. Interior estimates. First, we consider the interior estimates for the non-divergence form system u t ( 1 m L 0 u = f in Q 2R. For this system, the interior estimates are simple consequences of the corresponding estimates for the divergence form systems. Indeed, we differentiate the system m times with respect to x to get D m u t ( 1 m L 0 D m u = D m f. Thanks to the estimates for the divergence type systems, by (4.11, [D 2m u] a 2m,a,Q R C( D 2m u L (Q 2R [f] a,q 2R. Next we deal with the general non-divergence form systems with lower-order terms and coefficients depending on both t and x. The coefficients are all

SCHAUDER ESTIMATES 23 in C a. Following exactly the same idea as in handling the divergence form systems, we first consider L = α = β =m Aαβ D α D β. Fix y B R and let L 0y = A αβ (t, yd α D β. α = β =m We rewrite the systems as follows u t ( 1 m L 0y u = f ( 1 m (L 0y Lu. Set ζ to be an infinitely differentiable function in R d1 such that ζ = 1 in Q 2R, ζ = 0 outside ( (4R 2m, (4R 2m B 4R. (6.1 From Theorem 2 and Remark 1 of [5], for T = (4R 2m, there is a unique solution w W 1,2m 2 to the following system w t ( 1 m L 0y w = ζ ( f (f B4R (t ζ( 1 m (L 0y Lu in ( T, 0 R d with the zero initial condition at t = T, and w satisfies D 2m w 2 L 2 (Q 4R C f (f B 4R (t 2 L 2 (Q 4R C (L 0y Lu 2 L 2 (Q 4R CR 2a2md [f] 2 a,q 4R CR 2a2md [A αβ ] 2 a D 2m u 2 L (Q 4R. α = β =m We apply the mollification argument so that w is smooth. Let v = u w which is also a smooth function. Moreover, since ζ = 1 in Q 2R, D 2m v satisfies (4.4. Therefore, (4.5 holds for D 2m v. Combining the estimate of w and D 2m v, similar to the proof of Proposition 4.1, we get [D 2m u] a 2m,a,Q R C( D2m u L (Q4R [f] a,q 4R. (6.2 For the operator L with the lower-order terms, we rewrite the systems as follows u t ( 1 m L h u = f ( 1 m1 A αβ D α D β u, where L h = α = β =m Aαβ D α D β. Let f = f ( 1 m1 An easy calculation shows that [ f] ( a,q 4R C [f] a,q 4R α β <2m α β <2m [A αβ ] a D α D β u L (Q4R. α β <2m A αβ D α D β u. ( A αβ L [D α D β u] a,q 4R From (6.2 with f in place of f, the following estimate holds ( [D 2m u] a 2m,a,Q C D γ u R L (Q4R [f] a,q 4R, γ 2m

24 H. DONG AND H. ZHANG where C depends only on d, m, n, λ, K, A αβ a, R, and a. Because it follows immediately that u t = ( 1 m1 Lu f, u t a,q R [D 2m u] a 2m,a,Q R C ( γ 2m D γ u L (Q 4R f a,q 4R. Implementing a standard interpolation argument, for instance, see [15, Section 8.8] and Lemma 5.1 of [9], we are able to prove the first part of Theorem 2.2. 6.2. Boundary estimates. In the non-divergence case, in order to prove the boundary Schauder estimates, better estimates than these in Lemma 4.4 are necessary. For (4.21, we actually can estimate more normal derivatives up to 2m 1-th order. In order to show this, let us prove the following lemma. A similar result can be found in Lemma 3.3 of [6]. Lemma 6.1. Assume that u Cloc (Rd1 satisfying (4.16 (4.17. Let Q(x be a vector-valued polynomial of order m 1 and P (x = x m d Q(x. Suppose that (D k Dd m P (x Q = (D k Dd m u(t, x R Q R, where 0 k m 1. Let v = u P (x, then there exists a constant C depending only on d, m, n, and K such that for any 0 k m 1. D k D m d v L 2 (Q R CRm 1 k D m 1 D m d v L 2 (Q R Proof. Without loss of generality, let us assume R = 1. Choose ξ(y C0 (B 1 with a unit integral. Then let 0 g k (t = ξ(yd k Dd m v(t, y dy, c k = g k (tdt. B 1 By the Poincaré inequality and Hölder s inequality, the following estimate holds D k Dd m v(t, x g k(t 2 dx B 1 = B 1 C C B 1 B 1 B 1 B 1 1 (D k D m d v(t, x Dk D m d v(t, yξ(y dy 2 dx (D k D m d v(t, x Dk D m d v(t, y 2 dy dx D k1 D m d v(t, y 2 dy.

SCHAUDER ESTIMATES 25 = 0, by the triangle inequality and the Poincaré in- Because (D k Dd mv Q 1 equality, we have D k D m d v L 2 (Q 1 Dk D m d v c k L2 (Q 1 D k D m d v g k(t L2 (Q 1 g k(t c k L2 (Q 1 C D k1 D m d v L 2 (Q 1 C tg k L2 ( 1,0. (6.3 Since v satisfies the same system as u, by the definition of g k and integrating by parts t g k (t = ξ(yd k Dd m tv(t, y dy B 1 = ( 1 mk1 B 1 D k ξ(yl 0 Dd m v(t, y dy. We integrate by parts again leaving m 1 derivatives on Dd m v and moving all the others onto ξ to get t g k (t C D m 1 Dd m v dy. Therefore, B 1 t g k (t 2 C D m 1 Dd m v(t, 2 L 2 (B 1. (6.4 Combining (6.3 and (6.4, we prove the lemma by induction. Next, we prove an estimate for D 2m 1 u. Lemma 6.2. Assume that u Cloc (Rd1 and satisfies (4.16 (4.17. Then for any 0 < r R < and γ (0, 1, there exists a constant C depending only on d, n, m, λ, and γ such that for any X 0 {x d = 0} Q R D 2m 1 u (D 2m 1 u Q r (X 0 2 dx dt Q r (X 0 C( r R 2γd2m Q R (X 0 D 2m 1 u (D 2m 1 u Q R (X 0 2 dx dt. Proof. By scaling and translation of the coordinates, without loss of generality, we can assume R = 1 and X 0 = (0, 0. Moreover, we only need to consider the case r (0, 1/2] for the same reason as in the proof of Lemmas 4.2 and 4.4. Recall that in Lemma 4.4 we have u a 2m,a,Q 1 C u W 1,2m p (Q 1 C u L 2 (Q 2, where a < 2m can be arbitrarily close to 2m. Set a = γ 2m 1 (2m 1, 2m. Applying the inequality above with 1/2 and 1 in place of 1

26 H. DONG AND H. ZHANG and 2, and the Poincaré inequality, we have D 2m 1 u (D 2m 1 u Q r 2 dx dt Q r Cr 2γd2m u 2 a 2m,a,Q r Cr2γd2m u 2 W 1,2m p (Q 1/2 Cr 2γd2m u 2 L 2 (Q 1 Cr2γd2m Dd m u 2 L 2 (Q 1. (6.5 Let P and v be the functions in Lemma 6.1. It is easily seen that for a given function u, such P and thus v exist and are unique. Note that v also satisfies (4.16 (4.17. Then the inequality (6.5 holds with v in place of u. Since P (x is a polynomial of degree 2m 1, using Lemma 6.1, we have D 2m 1 u (D 2m 1 u Q Q r 2 dx dt r = D 2m 1 v (D 2m 1 v Q Q r 2 dx dt r Cr 2γd2m Dd m v 2 dx dt Cr 2γd2m D m 1 Dd m v 2 dx dt = Cr 2γd2m Cr 2γd2m Q 1 Q 1 Q 1 Q 1 D m 1 Dd m u (Dm 1 Dd m u Q 2 dx dt 1 D 2m 1 u (D 2m 1 u Q 2 dx dt. 1 The lemma is proved. Let us turn to the non-homogeneous systems. Consider a smooth solution u Cloc (Rd1 of { ut ( 1 m L 0 u = f in Q 4R, u = 0, D d u = 0,..., D m 1 d u = 0 on Q 4R {x d = 0}. We shall show the estimates of all derivatives up to order 2m with the exception of Dd 2mu. Let ζ be the function in (6.1. For T = (4R2m, consider the following system ( w t ( 1 m L 0 w = ζ f (f B (t in ( T, 0 R d, 4R w = 0, D d w = 0,..., D m 1 d w = 0 on ( T, 0 {x d = 0}, w = 0 on {t = T }. We know from Theorem 4 and Remark 1 of [5] that the system above has a unique W 1,2m 2 -solution which satisfies the following estimate D 2m w 2 L 2 (Q 4R C f (f B (t 2 4R L 2 (Q 4R CR2a2md [f] 2. (6.6 a,q 4R

SCHAUDER ESTIMATES 27 We again apply the mollification argument so that w is smooth. Observe that v := u w satisfies { vt ( 1 m L 0 v = (1 ζf ζ(f B (t in Q 4R 2R, v = 0, D d v = 0,..., D m 1 d v = 0 on {x d = 0} Q 2R. Since ζ = 1 in Q 2R, ṽ := D x v satisfies { ṽt ( 1 m L 0 ṽ = 0 in Q 2R, ṽ = 0, D d ṽ = 0,..., D m 1 d ṽ = 0 on {x d = 0} Q 2R. (6.7 Thanks to Lemma 6.2, for any X 0 {x d = 0} Q R and r (0, R, we get D x D 2m 1 v (D x D 2m 1 v Q r (X 0 2 dx dt Q r (X 0 C( r R 2γd2m Q R (X 0 D x D 2m 1 v (D x D 2m 1 v Q R (X 0 2 dx dt. (6.8 Therefore, by the triangle inequality, D x D 2m 1 u (D x D 2m 1 u Q r (X 0 2 dx dt Q r (X 0 C( r R 2γd2m C Q 2R Q R (X 0 D 2m w 2 dx dt. D x D 2m 1 u (D x D 2m 1 u Q R (X 0 2 dx dt Combining with (6.6, we obtain D x D 2m 1 u (D x D 2m 1 u Q r (X 0 2 dx dt Q r (X 0 C( r R 2γd2m Q R (X 0 CR 2ad2m [f] 2. a,q 4R D x D 2m 1 u (D x D 2m 1 u Q R (X 0 2 dx dt Taking γ > a, from Lemmas 3.2 and 3.1, and the corresponding interior estimates, we obtain the estimate for the Hölder semi-norm, ( [D x D 2m 1 u] a C D 2m u 2m,a,Q R L2 (Q 4R [f]. a,q 4R Let us turn to the case that the coefficients are functions of both t and x. The method of freezing the coefficients is implemented in this case. We first consider the case when L = α = β =m Aαβ D α D β. Fix y B R and rewrite the system as follows u t ( 1 m L 0y u = f ( 1 m (L 0y Lu.

28 H. DONG AND H. ZHANG We use the same ζ as in (6.1 and let T = (4R 2m. By Theorem 4 and Remark 1 of [5], there is a unique solution w W 1,2m 2 to the system w t ( 1 m L 0y w = ζ ( f (f B (t ζ( 1 m (L 0y Lu 4R in ( T, 0 R d with the Dirichlet boundary conditions w = 0, D d w = 0,..., D m 1 d w = 0 on ( T, 0 {x d = 0} and the zero initial condition at t = T. Moreover, D 2m w 2 L 2 (Q 4R C f (f B (t 2 4R L 2 (Q 4R C (L 0y Lu 2 L 2 (Q 4R CR 2a2md [f] 2 CR 2a2md [A αβ ] 2 a,q a D 2m u 2 4R L (Q. 4R α = β =m Let v = u w. Then ṽ := D x v satisfies (6.7, which implies (6.8 holds for v. Consequently, taking γ > a, we get the following inequality ( [D x D 2m 1 u] a C [f] D 2m u 2m,a,Q R a,q 4R L (Q. 4R (6.9 If the operator L has lower-order terms, we can move all the lower-order terms to the right-hand side regarded as a part of f. Hence, we proved the second part of Theorem 2.2, i.e., ( [D x D 2m 1 u] a C [f] D γ u 2m,a,Q R a,q 4R L (Q. 4R γ 2m The proof of Theorem 2.2 is completed. Remark 6.3. We need more regularity assumptions to get the estimate of [Dd 2mu] a. For example, if assuming A αβ, f C a 2m,a,Q 2m,a, then for R 1, R by using a similar method one can show that ( [Dd 2m u] a [u t ] 2m,a,Q a C f R 2m,a,Q a u R 2m,a,Q 4R L2 (Q. 4R (6.10 We give a sketched proof. First, let us assume that the coefficients are constants, and u is smooth and satisfies (4.16 (4.17. In this case, we can differentiate the equation with respect to t which means u t satisfies the same equation. From Lemma 4.4, we know that for any r < R 1 and X 0 {x d = 0} Q R, there exists a constant C such that u t 2 dx dt C( r R d4m u t 2 dx dt. (6.11 Q r (X 0 Q R (X 0 We are now ready to show (6.10. For simplicity we only consider L which consists of highest-order terms. We use the same ζ as in (6.1, T = (4R 2m, and fix X 0 Q 1,2m R. Let w be the W2 -solution of the system w t ( 1 m L X0 w = ζ ( f (f Q (t ζ( 1 m (L X0 Lu 4R in ( T, 0 R d with the Dirichlet boundary conditions w = 0, D d w = 0,..., D m 1 d w = 0 on ( T, 0 {x d = 0} and the zero initial condition at

t = T, where SCHAUDER ESTIMATES 29 L X0 = A αβ (X 0 D α D β. Let v = u w, which satisfies { vt ( 1 m L X0 v = (f Q 4R in Q 2R, v = 0, D d v = 0,..., D m 1 d v = 0 on {x d = 0} Q 2R. Then (6.11 holds with v in place of u. By the triangle inequality, for any X 1 {x d = 0} Q R and r (0, R, u t 2 dx dt Q r (X 1 C( r R d4m Q R (X 1 By Theorem 4 and Remark 1 of [5], w t 2 L 2 (Q 4R u t 2 dx dt C Q R (X 1 w t 2 dx dt. (6.12 C f (f Q 4R 2 L 2 (Q 4R (L X 0 Lu 2 L 2 (Q 4R CR 2md2a( [f] 2 a [A αβ ] 2 2m,a,Q a 4R 2m,a D2m u 2 L (Q 4R. (6.13 α = β =m Note that under the assumption that A αβ, f C a/2m,a, for the interior estimates, by the classical Schauder theory it is easy to see that u t C a/2m,a. We combine (6.12, (6.13, and the corresponding interior estimates to obtain a Hölder estimate, [u t ] a 2m,a,Q R ( C [f] a [A αβ ] 2m,a,Q a 4R 2m,a D 2m u L (Q 4R u t L2 (Q 4R. α = β =m (6.14 It remains to estimate Dd 2m u. Let γ = (0, 0,..., 0, m, A γγ D γ D γ u = ( 1 m1 u t A αβ D α D β u ( 1 m f. (α,β (γ,γ Since A γγ is positive definite, it follows easily that [Dd 2m C u] a ( [u t ] a 2m,a,Q R 2m,a,Q R α (0,...,0,2m α =2m D α u a [f] 2m,a,Q a R 2m,a,Q R Plugging (6.9 and (6.14 into the inequality above and using the interpolation inequalities, we immediately prove (6.10..

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