Regularity of elliptic systems in divergence form with directional homogenization

In this paper, we study regularity of solutions of elliptic systems in divergence form with directional homogenization. Here directional homogenization means that the coefficients of equations are rapidly oscillating only in some directions. We will investigate the different regularity of solutions on directions with homogenization and without homogenization. Actually, we obtain uniform interior \begin{document}$W^{1, p}$\end{document} estimates in all directions and uniform interior \begin{document}$C^{1, γ}$\end{document} estimates in the directions without homogenization.


1.
Introduction. In the present paper, we study the following elliptic systems in divergence form Here Ω is a domain in R n with n ≥ 2, f = (f i α ) (1 ≤ i ≤ n and 1 ≤ α ≤ N ) are suitable functions and x ∈ Ω is separated into two parts x = (x 1 , ..., x q ) and x = (x q+1 , ..., x n ) (1.2) for some integer 0 < q < n. Observe that the coefficients (a ij αβ ) of (1.1) are rapidly oscillating only in x as ε → 0 + . We call this directional homogenization and call x the homogenization direction. Our main target is to investigate the different regularity of solutions of (1.1) on x and on x .
(ii) Compared to the so called partial regularity results (cf. [9]), Theorem 1.1 and 1.2 reveal higher regularities. For example, as N = 1, under the same assumptions in x and L ∞ assumptions in x for a, by [9], we have (1.4) holds with 0 <γ < min{γ 0 , γ}, where γ 0 is the Hölder exponent appearing in the De Giorgi-Nash-Moser's estimate, which may be very small. That is, the homogenization results are better than the results that we only assume a is L ∞ in x . Actually, one of the target of studying homogenization problem is to discover this kind of hidden higher regularity.
(iii) If the homogenization direction is one-dimensional, from [7], better results can be obtained. (See also [12].) However, if the homogenization directions are multi-dimensional, the results in [7] are unadapted here.
As far as to the authors' knowledge scope, there is not too much literatures concerning the directional homogenization problems. In [13], Suslina considered the following elliptic equation in a strip Π = (0, a) × R: Here the coefficient matrix is a diagonal matrix. If g j ( j = 1, 2) are bounded, periodic in x 2 and Lipschitz in x 1 , it was proved that where u 0 solves the homogenized problem ε )u ε , a similar result was established in [5]. In Section 2, we will give the weak convergence of u ε to u 0 in H 1 .
We now briefly describe the ideas of the proofs of our main results. Since the behaviour of variables on the homogenization directions and the non-homogenization directions are essentially different, we deal with them by different methods. To obtain the regularity of u ε on the homogenization directions x , we use a three-step compactness method, which was demonstrated by Avellaneda and Lin in [1], while for the regularity on the non-homogenization directions x , the frozen coefficients method is exploited. Conclusion (i) in Theorem 1.1 and 1.2 will be used in the proofs of conclusion (ii) in Theorem 1.1 and 1.2 respectively.
The remaining sections are organized in the following way. In section 2, it is given the homogenized operator L 0 of L ε as ε → 0 + , whose properties will be used in the compactness method, and whose coefficients, different from the general homogenization, are not constant. In section 3, we prove Theorem 1.1 (i) and 1.2 (i), that is, the regularity of solutions of (1.1) in all directions. Theorem 1.1 (ii) and 1.2 (ii), that is, the higher regularity in directions without homogenization is derived in section 4.
Throughout the paper, we use standard notations. Before the end of this section, we list them in the following.
C : a positive constant independent of ε and the exact value may change from line to line.
Ω : a domain in R n .
2. Homogenization and weak convergence. In this section, we give the homogenized operator L 0 of L ε and a weak convergence result as ε → 0 + . Different from the general homogenization problems, here we will see the homogenized operator L 0 has non-constant coefficients. We first introduce χ = χ(x , y ) = χ j ιβ (x , y ), the so called the matrix of correctors for L ε , where x ∈ R q and y ∈ R n−q . Throughout this section, we confine the indices 1 ≤ i, j, ≤ n, q + 1 ≤ s, t ≤ n and 1 ≤ α, β, ι ≤ N . Actually, χ(x , y ) is obtained by the following way. For any fixed x , we solve χ(x , y ) as a periodic function of y by and We have the following convergent result as ε → 0 + .

Theorem 2.1 (Weak convergence).
Let Ω be a bounded Lipschitz domain in R n . Let u ε ∈ W 1,2 (Ω) be a weak solution of

4)
where the coefficients of L ε satisfy (H 1 )-(H 3 ) and f ∈ L 2 (Ω). Then Proof. Suppose ã ij αβ (x, y) are elliptic and continuous in Ω × R n , periodic with respect to y. The following elliptic system are the so called non-uniformly oscillating coefficients homogenization.
From [3], one has that the solution u ε of (2.5) converges weakly in and χ j ιβ are solutions to the cell problems Observe that if ã ij αβ (x, y) do not depend on x and y , then System (2.5) is deduced to be System (2.4). And it is easy to see that in this case (2.1) and (2.6) are the same. Thus, our conclusion follows immediately.
Remark 2.2. L 0 given by (2.3) is called the homogenized operator of L ε as ε → 0 + . We claim that although the coefficient matrix of L 0 , â ij αβ (x ) , is not a constant matrix (depending only on x ), it satisfies the assumption (H 2 ) and (H 3 ). Actually, we have the following proofs.
Letx be a point other than x and χ(x , y ) satisfy From this and (2.1), we obtain Then, by W 1,2 estimates, we have where C depends only on n, N , λ and M . By this and Hölder's inequality, we get where C depends only on n, N , λ and M . This implies that â ij αβ (x ) defined in (2.2) satisfies (H 3 ).
Next, we prove that â ij αβ (x ) satisfies (H 2 ). Let 1 ≤ m ≤ n and 1 ≤ ρ ≤ N . Note that (2.1) is equivalent tô for any ϕ with the properties that´[ 0,1] n−q ϕ(y )dy = 0 and ϕ is periodic with respect to Z n−q , where P j β = y j (0, ..., 1, ..., 0) with 1 in the β th position and P j ιβ is the ι th component of P j β . (Here we use D ym P j ιβ = δ mj δ ιβ .) Observe (2.2) can be rewritten aŝ It follows that for any and a(x , y ) satisfies the condition (H 2 ), we obtain from (2.8) that This implies that â ij αβ (x ) satisfies (H 2 ). The observation that â ij αβ (x ) satisfies (H 2 ) and (H 3 ) will be used in Section 3 as applying the compactness method.
3. Regularity in all directions. In this section, we give the proofs of uniform interior W 1,p estimates of all variables with 2 ≤ p ≤ ∞ for solutions of (1.1), thus the result (i) in both Theorem 1.1 and 1.2.

RONG DONG, DONGSHENG LI AND LIHE WANG
Then we have where C depends only on n, N , M , p and ||Dη|| L ∞ . From (3.1), it follows that Since x 0 is fixed, (3.3) can be regarded as a general homogenization problem. By Theorem C in [2], we obtain that where C depends only on n, N , M , λ, γ and p. We choose R to be sufficiently small such that CR γ [a] x ,γ;B R(x 0 ) ≤ 1/2 and then where C depends only on n, N , M , λ, γ and p. By Sobolev imbedding inequality, for some constant C depending only on n, N , M , p and ||Dη|| L ∞ . Plug this and (3.2) into (3.4). Then we have where Caccioppoli's inequality is used to derive the last inequality and C depends only on n, N , M , λ, γ, p, ||Dη|| L ∞ and R. Now we choose η ≡ 1 in B R/2 (x 0 ) such that ||Dη|| L ∞ ≤ 2 R . Then the above inequality becomes where C depends only on n, N , M , λ, γ, p and R. After a standard covering, the estimate (1.3) is proved for the case 2 ≤ p ≤ 2 * = 2n/(n − 2).
To prove Theorem 1.2 (i), we apply the three-step compactness method introduced by Avellaneda and Lin in [1]. The first and second steps are given by the following two lemmas.
Proof. We prove (3.5) by contradiction. Suppose, to the contrary, that (3.5) is not true. There exists a sequence ε k → 0 + and sequences u ε k and f k so that for any 0 < θ < 1. Note that Theorem 1.1 (i) and the Sobolev imbedding theorem imply {||u ε k || C τ } is bounded. After extraction of a subsequence (same notation for subsequence ), there exist u 0 ∈ L ∞ (B 1 ) and f 0 ∈ C τ (B 1 ) so that Du ε k → Du 0 in L 2 (B θ ) weakly, with u 0 and f 0 satisfying From (3.7) and the Schauder estimates, we have u 0 ∈ C 1,τ loc (B 1 ) and sup for any 0 < θ ≤ 1 2 , where C depends only on n, N , M , λ, γ and τ . It follows that for any k large enough. Hence we can and we do choose θ > 0 to be small enough such that sup Fix this θ and since, by (2.1), as k is large enough. This contradicts with (3.6).
Proof. The proof is by induction on k. By Lemma 3.1, estimate (3.10) holds for k = 1 with b ε 1 = 0 and B ε 1 = (Du ε ) 0,θ . Suppose that (3.10) holds for some k such that ε/θ k < ε 0 . Let With the aid of (2.6), we have Rewriting this inequality in terms of u ε , we get for some constant C depending only on n and θ. We deduce from iterating the recursive formulae above that |b ε k+1 | + |B ε k+1 | ≤ CJ, where C depends only on n, N , M , λ, γ and τ .
Proof of Theorem 1.2 (i). We only consider the case ε < ε 0 , since the case ε ≥ ε 0 follows from the classical Schauder estimates. Suppose B 2 ⊂ Ω and to prove (1.5), it is suffice to prove where τ = min{γ, δ} and C depends only on n, N , M , λ, γ and δ. Let θ be given by Lemma 3.2 and k ∈ N be such that The assumed condition where C depends only on θ and J is given by (3.8). Rescaling the above inequality, we obtain by (3.9) that sup |x|<1/ε0 for some constant C depending only on n, N , M , λ, γ and δ.

4.
Higher regularity in directions without homogenization. This section is devoted to Hölder continuity of derivatives of variables without homogenization.
Observe here the derivatives are taken in directions without homogenization. However, the Hölder continuity is in all variables. The conclusions (ii) in Theorem 1.1 and 1.2 shall be proved respectively. First, we recall a result that characterizes Hölder continuity by Campanato spaces (cf. Theorem 1.2 in [10]).

Lemma 4.1.
Let Ω be a bounded Lipschitz domain in R n . For each γ, 0 < γ < 1, there exist positive constants C 1 and C 2 , which depend only on γ, n, N and the geometry of Ω such that, for all u ∈ C γ (Ω), we have Next, we introduce a technical lemma (cf. Lemma 2.1 in [10]).

Lemma 4.2. Let φ(t) be a nonnegative and nondecreasing function on
Then there exists a constant 0 depending only on A, α and β so that if < 0 , for all 0 < r ≤ R 0 , we have where C depends only on A, α and β. Now, we are in a position to prove Theorem 1.1 (ii) and 1.2 (ii).
Notice that the a ij αβ (x 0 , x ε ) and f i α (x 0 , x ) in (4.1) are independent of x . We differentiate the first equation in (4.1) with respect to x and obtain that Take any µ ∈ (γ, 1). By Hölder estimate (cf. Lemma 9 in [1]), we obtain where C depends only on n, N , M , λ andγ. It follows that where C depends only on n, N , M , λ andγ. Next, we estimate´B R (x0) |Dw ε | 2 . In fact, from (4.2), where C depends only on λ and n. Apply Theorem 1.1 (i) to u ε with p = n/(γ−γ) > n and then , where C depends only on n, N , M , λ, γ,γ and d. Recall d = dist(Ω , ∂Ω). This and the above inequality implŷ where C depends only on n, N , M , λ, γ,γ and d. (4.5) In Lemma 4.2, we set α = n + 2µ, β = n + 2γ and Here φ(r) is nondecreasing. Actually, since for any function h,´Ω |h − c| 2 takes minimum as c = ffl Ω h, we have for any 0 < r 1 ≤ r 2 . Then it follows from Lemma 4.2, (4.5) and (1.3) that where C depends only on n, N , M , λ, γ,γ and d. From Lemma 4.1, we have Theorem 1.1 (ii) holds.
Putting (4.6) into (4.3), we havê (4.7) By Lemma 4.2, where we set α = n + 2µ, β = n + 2γ and we deduce from (4.7) and (1.3) that , where C depends only on n, N , M , λ, γ, δ and d. By Lemma 4.1, we have the conclusion (ii) in Theorem 1.2. Remark 4.3. In Theorem 1.1, if a ij αβ are independent of x and f ∈ C γ x (Ω), we can prove that D x u ε ∈ C γ (Ω ) and [D x u ε ] γ;Ω ≤ C u ε L 2 (Ω) + |f | x ,γ;Ω , (4.8) where C depends only on n, N , M , λ, γ, Ω and Ω. The proof is similar to that of Theorem 1.1 (ii). Note that here D x u ε is Hölder continuous in all variables. We point out that, in [8], Dong and Kim showed that if N = 1, the coefficients are independent of x , L ∞ in x and the data are Hölder continuous in x , then for any weak solution u of divergence form scalar elliptic equation, D x u is Hölder continuous only in x . However, to obtain (4.8), the periodicity in x is essential and in [8], no periodicity is assumed.