Homogenization of a parabolic Dirichlet problem by a method of Dahlberg

Consider the linear parabolic operator in divergence form $$\mathcal{H} u =\partial_t u(X,t)-\text{div}(A(X)\nabla u(X,t)).$$ We employ a method of Dahlberg to show that the Dirichlet problem for $\mathcal{H}$ in the upper half plane is well-posed for boundary data in $L^p$, for any elliptic matrix of coefficients $A$ which is periodic and satisfies a Dini-type condition. This result allows us to treat a homogenization problem for the equation $\partial_t u_\varepsilon(X,t)-\text{div}(A(X/\varepsilon)\nabla u_\varepsilon(X,t))$ in Lipschitz domains with $L^p$-boundary data.


Introduction, notation and main results
In this paper we are interested in the well-posedness of low regularity Dirichlet problems associated with the divergence type parabolic operator for a certain periodic matrix of coefficients A. That is, we would like to guarantee existence and uniqueness of solutions and continuous dependence on the boundary data, under minimal regularity assumptions on the coefficients and on the domain. For the upper half space we prove that the L p Dirichlet problem is well-posed if A is periodic in the λ-direction. This extends previous results for the upper half space, where it is assumed that A is either independent of λ, or that A is a perturbation of a matrix that is independent of λ. The theory developed for the upper half space allows us to study homogenization problems in bounded, time-independent Lipschitz domains.
We start by briefly putting these problems into context, mentioning just a few papers that precede this work. For the ordinary heat equation, in which case the matrix A is simply the identity matrix, Fabes and Rivière ( [7]) established the solvability in C 1 -cylinders. Later, Fabes and Salsa ( [8]) and Brown ([2]) extended the result to Lipschitz cylinders. For more involved time-varying domains, the situation has been analyzed by Lewis and Murray ([12]) and Hofmann and Lewis ([10]). The next step was to allow non-constant coefficients. Mitrea ([13]) studied the situation of A ∈ C ∞ ; Castro, Rodríguez-López and Staubach ( [4]) considered Hölder matrices and Nyström ([16]) the case of complex elliptic matrices, but independent of one of the spatial variables.
In all previous contexts, the matrices were time-independent. Allowing time-dependence is a very challenging problem, which has been understood very recently by Auscher, Moritz and Nyström ([1]), following a first order approach. They consider elliptic matrices depending on time and all spatial variables, which are certain perturbations of matrices independent of one single spatial direction (see [1,Section 2.15] for precise definitions).
It is also worth noting that in almost all the aforementioned papers, the analysis was carried out via the so called method of layer potentials, that we will not follow this time here. We consider the parabolic Dirichlet problem in Lipschitz cylinders for merely elliptic coefficients, depending on all spatial variables. However, we need to assume periodicity in one direction and a Dini-type condition in the same variable, as made precise below.
We show that if the coefficient matrix A is time-independent and periodic with period 1 in the spatial direction of the normal of the boundary, then the Dirichlet problem is solvable. Moreover, the estimates that we obtain for the solution are independent of the period of A. For periodic matrices A(X) and ε > 0, we can then obtain estimates that are uniform in ε for the solution u ε to the Dirichlet problem with coefficient matrix A(x/ε) with period ε. In particular, we prove that, as ε → 0, u ε converges to a limit functionū that solves the Dirichlet problem with a constant coefficient matrixĀ. A limit process of this type is called homogenization. For elliptic operators, these estimates were obtained by Kenig and Shen in [11]. In [11], the authors have two independent ways of proving the estimates. The first is an approximation argument that relies on certain integral identities, the second is through a potential theoretic method due to Dahlberg. For parabolic problems these integral estimates are not available and we rely instead on a parabolic version of the theorem by Dahlberg.
Let H denote the parabolic operator Hu := (∂ t + L)u, where is defined in R n+2 = {(X, t) = (x 1 , . . . , x n+1 , t) ∈ R n+1 × R}, n ≥ 1; and A = {A i,j (X, t)} n+1 i,j=1 is an (n + 1) × (n + 1) real and symmetric matrix which satisfies: • for certain 1 ≤ Λ < ∞, the uniform ellipticity condition • independence of the time variable t, • a Dini-type condition in the x n+1 variable In virtue of the hypothesis (1.2) and (1.4), the x n+1 direction is of special interest. Along this paper we call λ := x n+1 . Accordingly, ∇ := (∇ || , ∂ λ ) := (∂ x1 , . . . , ∂ xn , ∂ λ ). Depending on the situation, we refer to a point in R n+2 either as (X, t), X = (x, λ), or (x, t, λ), with an obvious abuse of notation. The latter is convenient when we consider the Dirichlet problem in the upper half space, where (x, t) denotes a point on the boundary. Our theorems are formulated in time-independent Lipschitz domains. By D we denote the domain which is an unbounded cylinder in time, whose spatial base is the region above the Lipschitz graph φ, i.e., φ satisfies |φ(x) − φ(y)| ≤ m|x − y|, x, y ∈ R n , for certain m > 0. The (lateral) boundary of D is given by We shall also consider bounded Lipschitz cylinders (1.6) where Ω is a bounded Lipschitz domain in R n+1 .
It will be assumed that Ω is a (m, r 0 ) domain in the following sense: For any X 0 ∈ ∂Ω, there exists a Lipschitz continuous function φ such that, after a rotation of the coordinates, one has X 0 = (x 0 , λ 0 ) and

Thus, introducing
The lateral boundary of Ω T is denoted by ∂ L Ω T := ∂Ω × (0, T ) and the parabolic boundary is given by ∂ P Ω T := Ω × {0}. Note that for D as in (1.5), ∂ L D = ∂D and ∂ P D = ∅. On ∂ L Ω T and ∂D we define L p spaces with respect to the measure where σ is the surface measure on ∂Ω and {(x, φ(x)) : x ∈ R n }, respectively. We shall need to introduce some more notation that will be needed to state our main results. For (X, t) ∈ R n+1 × R, we define its parabolic norm ||(X, t)|| as the unique positive solution ρ of the equation Given (x 0 , t 0 ) ∈ R n+1 and η > 0, we define the cone and the standard parabolic cube centered at (x, t) ∈ R n+1 with side length (Q) = r > 0 by Similarly, we consider parabolic cubes Q in R n+2 centered at (X, t) as follows, It will also be useful to introduce the set For any function u defined in R n+2 + := {(x, t, λ) ∈ R n+2 : λ > 0}, we consider the following non-tangential maximal operator Having made such a choice of η we simply denote N (u) = N η (u). In all our estimates C denotes a constant that depends only upon the dimension n, the ellipticity constant Λ and possibly m, r 0 . Theorem 1.1. Suppose that A is a real and symmetric matrix satisfying (1.1) -(1.4) and D is an unbounded Lipschitz domain defined as in (1.5). Then, for certain 0 < δ < 1 and any f ∈ L p (∂D), 2 − δ < p < ∞, there exists a unique solution to the Dirichlet problem With Theorem 1.1 in place we are able to analyze a homogenization problem that we now describe. In addition to (1.1) and (1.2) we assume that (1.9) A(X + Z) = A(X), for all Z ∈ Z n+1 , and (1.10) That is, A is periodic with respect to the lattice Z n+1 and satisfies a Dini condition in all variables. For each ε > 0, consider the operator L ε given by We also need to introduceL,L u := −div(Ā∇u), where the matrixĀ is determined bȳ and the auxiliary function w α solves the problem  Let Ω T be as in (1.6). Then for any ε > 0 and f ∈ L p (∂ L Ω T ), 2 − δ < p < ∞, there exists a unique solution u ε to the Dirichlet problem Moreover, as ε → 0, u ε converges locally uniformly in Ω T toū, which is the unique solution to In the elliptic case, Theorem 1.1 and the first part of Theorem 1.2 ((1.11) and (1.12)) was proved by Kenig and Shen in [11]. In [11] the authors also treat the Neumann and regularity problems. The theory for the Neumann and regularity problems is based on the use of integral identities to estimate certain nontangential maximal functions. These integral identities are not available in the parabolic case and thus homogenization of Neumann and regularity problems remain an interesting and challenging open problem.
The main tools in our analysis are Harnack inequalities and the estimation of Green's function in terms of L-caloric measure and vice versa, see Section 2.1. The main difficulty in the parabolic setting is the time-lag that is present in these estimates. Our requiring that the matrix A is time independent and symmetric leads to spatial symmetry and time-invariance of Green's function, see (2.9). This becomes a key point in the proof of the parabolic version of Dahlbergs theorem in Section 2.3

The Dirichlet Problem
We now turn to the proof of Theorem 1.1. Since D is globally defined by a Lipschitz graph, the situation of the proof may be reduced to the upper half space in a standard way, see for example [11, p. 905]. Thus, the goal of this section is to solve the Dirichlet problem for the operator H in the upper half space R n+2 + with given boundary data on ∂R n+2 Definition 2.1. We say that the Dirichlet problem for Hu = 0 in R n+2 + is solvable in L p if there exists 0 < δ < 1 such that for every 2 − δ < p < ∞ and every f ∈ C c (R n+1 ), the solution to the Dirichlet problem It can be shown that (2.1) has a unique solution by analyzing, for any k = 1, 2, . . ., the problems and define u := lim k→∞ u k which will solve (2.1). This allows us to define the L-caloric measure ω := ω Z,τ on R n+1 , which satisfies The caloric measure is a doubling measure, i.e.
see [6] for a proof. Assuming that dω and dxdt are mutually absolutely continuous, we define the kernel K(Z, τ ; x, t) with respect to the point (Z, τ ) ∈ R n+2 The solution to (2.1) may thus be represented as We recall that the solvability in L 2 of the Dirichlet problem in R n+2 + (in the sense of Definition 2.1) is equivalent to the reverse Hölder inequality for the kernel K (see Lemma 2.6 below): The reverse Hölder inequality is self improving in the sense that if (2.4) holds, then there exists α > 2 such that This is a consequence of Gehring's Lemma ([9, Lemma 3]), adapted to parabolic cubes. In turn, the reverse Hölder inequality is equivalent to the following condition (see Proposition 2.7 below): T2r(x0,t0) |u(x, t, λ)| 2 dxdtdλ, r > 0, . Shortly, we call (2.6) a local solvability condition when (2.6) holds for 0 < r ≤ 1. If (2.1) holds for H * = −∂ t + L instead of H = ∂ t + L, we say that u solves the adjoint Dirichlet problem. Analogously, we define the adjoint L-caloric measure ω * and the adjoint kernel K * (Z, τ ; y, s). It is easy to see that the adjoint Dirichlet problem is solvable if and only if the Dirichlet problem for H is solvable by considering the change of variables t → −t. This leads to analogous equivalent solvability conditions for the adjoint Dirichlet problem. For example, (2.6) holds for caloric functions if and only if it holds for adjoint caloric functions.
Our first step in the proof of Theorem 1.1 is to establish (2.6) for 0 < r < 1. This is achieved by localizing the operator and using the perturbation theory developed in [15]. Then we utilize an ingenious technique developed by Dahlberg to show that the periodicity of A implies that (2.6) also holds for all r > 1, see Theorem 2.12 below.
For Lipschitz cylinders Ω T = Ω × (0, T ), we say that the L p Dirichlet problem is solvable in Ω T if there exists 0 < δ < 1 such that for every 2 − δ < p < ∞ and for every f ∈ C c (∂ L Ω T ), there exists a solution to the Dirichlet problem and with the measure dσ(X, t), see (1.8), in place of dxdt.

Preliminaries.
We now recall some well known results that will be needed for the proof of Theorem 1.1. For the Lemmas 2.2-2.5 below we refer to [6] and the references therein. For a time-independent Lipschitz domain D (given either by (1.5) or (1.6)), we denote by G Green's function with respect to D, with the convention that G(X, t; Z, τ ) is Green's function with pole at (Z, τ ) ∈ D. Green's function G = G(·; Z, τ ), as a function of (X, t), satisfies Since the operator L is symmetric we have G(X, t; Z, τ ) = G(Z, t; X, τ ). Additionally, the timeindependence of A implies that G(X, t; Z, τ ) depends only on the time difference t − τ . To see this we note that if the function v(X, t) is L-caloric, then so is v(X, t + t 0 ). It follows that G(X, t + t 0 ; Z, τ + t 0 ) satisfies (2.7) and (2.8). Combining the symmetry in space and the timeinvariance we obtain We also recall the estimate We shall also consider the adjoint Green's function G * (X, t) with pole at (Z, τ ), given by which is adjoint L-caloric as a function of (X, t) for t < τ .
We say that u is locally Hölder continuous in a domain D if there exist constants C > 0 and 0 < α < 1 verifying for every parabolic cube Q := Q r ⊂ R n+2 such that 2 Q := Q 2r ⊂ D. Moreover, any u satisfying (2.11) also satisfies Moser's local estimate It is well known that if (2.11) or (2.12) hold for one single value of p, then they hold for all 1 ≤ p < ∞. We remark that (2.11)-(2.13) also holds for solutions to H * u = 0.
Let ω be the L-caloric measure of the domain R n+2 then there is a positive constant c such that Lemma 2.6. The reverse Hölder inequality holds if and only if the Dirichlet problem is solvable in L p , in the sense of Definition 2.1.

Local solvability.
In order to state the next lemma we shall need to introduce some notation.
Let Ω T be a Lipschitz cylinder as in (1.6) and let S = ∂Ω × (0, T ) be its lateral boundary. If (X, t) ∈ S we let Γ η (X, t) be a parabolic nontangential cone of opening η and vertex (X, t). We choose η so that for all (X, t) ∈ S, Γ η (X, t) ∩ S = {(X, t)} in an appropriate system of coordinates.
Our goal is to prove that the Dirichlet problem for H 1 is solvable in T 12 and thus satisfies the local solvability condition (2.6) in that domain (see Proposition 2.7). In particular, this gives us T2r |u(x, t, λ)| 2 dxdtdλ, for all 0 < r < 1, whenever H 1 u = 0 in T 4r and u(x, t, 0) = 0 on Q 4r . Notice that, when Hu = 0 in T 4r , then also H 1 u = 0 in T 4r and thus the local solvability condition for H follows from that of H 1 .

2.3.
Local solvability implies (2.6) for all r > 1. By localizing the operator H we were able to prove local solvability in the previous section. Now, using the periodicity of A we infer (2.6) for all r > 1. This proof is based on an unpublished work of Dahlberg, which is available in [11,Appendix]. We shall need the following Cacciopolli inequality in the proof.
Then, for (x, t, λ) ∈ Ω 2 × (0, 4R 2 ) such that λ ≥ R, we have Proof. By the periodicity of A, HQu = 0 in Ω 3 × (0, 8R 2 ). Thus, for (x, t, λ) ∈ Ω 2 × (0, 4R 2 ) such that λ ≥ R, (2.13) yields An application of the fundamental theorem of calculus, Hölder's inequality and Fubini's theorem leads to where in the last inequality we also applied Lemma 2.10. Proof. For the sake of simplicity we assume that (x 0 , t 0 ) = (0, 0) and write T r = T r (0, 0) and Q r = Q r (0, 0). We need to prove that for all u such that Hu = 0 in T 4r and u = 0 on Q 4r . If r ≤ 6 we may cover Q r by cubes Q 1/2 (x k , t k ) and apply the local solvability condition (2.6) for 0 < r ≤ 1 to each of them to prove (2.25). Assume r > 6 and that Hu = 0 in T 4r and u = 0 on Q 4r . We choose a covering {Q 1/2 (x k , t k )} k of Q r such that Q r ⊂ k Q 1/2 (x k , t k ) ⊂ Q r+1 and k χ Q 1/2 (x k ,t k ) ≤ C, where C is independent of r.
2.4. Solvability. As a consequence of Proposition 2.9, Theorem 2.12 and Proposition 2.7, we know that the reverse Hölder inequality (2.4) holds. Thus the following proposition follows now directly from Lemma 2.6.
Proposition 2.13. Suppose that A is a real and symmetric matrix satisfying (1.1) -(1.4). Let f ∈ C c (R n+1 ). Then, there exists 0 < δ < 1 (which depends only in the dimension n and the constants appearing in (1.1) and (2.6)) such that the solution to the classical Dirichlet problem verifies, for any 2 − δ < p < ∞,

2.5.
Uniqueness. Moving forward to the proof of Theorem 1.1, we start by showing that a solution to (2.28) where f ∈ L p (R n+1 ) and p > 1, is unique. The proof relies on the following lemma.
Proof of Theorem 1.1. Let f ∈ L p (R n+1 ), with 2 − δ < p < ∞; where 0 < δ < 1 was determined in Proposition 2.13. We can take functions Then, for each k ∈ N, call u k the solution provided in Proposition 2.13 with boundary data f k , which satisfies the estimate We also have that and from here we infer that there exists a function u such that u k −→ u, k → ∞, uniformly on compact sets of R n+2 + . Moreover, standard arguments guarantee that u is a weak solution of the Dirichlet problem For the fact that u = f n.t on R n+1 we refer to [6]. On the other hand, the uniqueness is a consequence of Proposition 2.15, since the kernel K(Z, τ ; y, s) ∈ L p (R n+1 ), for all (Z, τ ) = (z, σ, τ ) ∈ R n+2 + . Indeed, by duality, Here the supremum was taken over all g ∈ C c (R n+1 ) such that g L p (R n+1 ) ≤ 1; v g is the solution to the Dirichlet problem with boundary data g and in the third inequality we used (2.12).

Homogenization
We divide the proof of Theorem 1.2 in three steps.

3.2.
Proof of (1.11) and (1.12) for Ω T . We are going to prove that the kernel K ε associated to the caloric measure ω ε for ∂ t + L ε on ∂ L Ω T satisfies the reverse Hölder inequality.

3.3.
Proof of (1.13). We now turn to the homogenization result. Since the domain Ω T is bounded, the L p norm of u ε in Ω T can be estimated by the L p norm of its non tangential maximal function: u ε L p (Ω T ) ≤ C(diam(Ω T )) N (u ε ) L p (∂ L Ω T ) ≤ C(diam(Ω T )) f L p (∂ L Ω T ) .
We shall also need to extract a convergent subsequence of the Kernel K ε . If It thus follows by duality that K ε (x, t, λ; ·, ·) L q (∂ L Ω T ) is bounded uniformly in ε for (x, t, λ) as in (3.7), where q is the conjugate exponent of p. This clearly implies that K ε L q (B R (X0)×(t1,t2)×∂ L Ω T ) is bounded uniformly in ε. Thus, for a subsequence, K ε −→K, as ε → 0, weakly in L q (B R (X 0 ) × (t 1 , t 2 ) × ∂ L Ω T ).
Since this holds for any set of the type B R (X 0 ) × (t 1 , t 2 ) that is compactly contained in Ω T , we conclude that for a certain subsequence of {ε} ε>0 , u ε −→ū, weakly in W loc (Ω T ), It remains to prove thatK is indeed the kernel associated to ∂ t +L. That is, we need to show thatū = f n.t. on ∂ L Ω T . Assume that f is smooth. Then by the De Giorgi-Moser-Nash estimate (2.13), u ε is uniformly continuous up to the boundary, with estimates uniform in ε. Thus, u ε converges uniformly toū in any neighborhood N of the boundary, for a subsequence, andū = f on N ∩ ∂D. Since ∂ tū +Lū = 0 in Ω T we see that u(x, t, λ) = ∂ L Ω TK (x, t, λ; Y, s)f (Y, s)dσ(Y )ds solves the Dirichlet problem (1.13) when f is smooth. Since smooth functions are dense in L 2 , this proves thatK is the kernel associated to ∂ t +L.
Finally, taking into account that all convergent subsequences have the same unique limitū, we conclude that u ε converges locally uniformly, and locally weakly in W(Ω T ), to the solutionū of (1.13).