PERIODIC HOMOGENIZATION OF ELLIPTIC SYSTEMS WITH STRATIFIED STRUCTURE

. This paper concerns with the quantitative homogenization of second-order elliptic systems with periodic stratiﬁed structure in Lipschitz domains. Under the symmetry assumption on coeﬃcient matrix, the sharp O ( ε )-convergence rate in L p 0 (Ω) with p 0 = 2 d d − 1 is obtained based on detailed discussions on stratiﬁed functions. Without the symmetry assumption, an O ( ε σ )-convergence rate is also derived for some σ < 1 by the Meyers estimate. Based on this convergence rate, we establish the uniform interior Lipschitz es- timate. The uniform interior W 1 ,p and H¨older estimates are also obtained by the real variable method.

Under the assumptions above, one can verify that A(x, ρ(x)/ε) is measurable by the invertibility of the matrix ( ∂ρ l ∂xj ) (see the nondegeneracy of ρ in Section 2.1). Moreover, L ε is elliptic, which implies that problem (1) is well-posed.
The systems with stratified structure generally describes materials which cannot be generated by translating one elementary cell repeatedly. This can be found in many branches of biomechanics and engineering. For example, the human heart is formed by biological fibers nearly parallel to the cardiac wall with orientations varying continuously. More mathematical models arising from materials science includes multilayered spherical particles, cylindrical fibers, wavy multilayered composites, wavy radially multilayered composites and so on (see [9,10,37,36] for more details). The layers or the fibers composing these materials are characterized by the parametric equation ρ(x) = const. As an example, the wavy fibered composite where ρ(x) = (x 2 − H sin( 2π L x 1 ), 0, x 3 ) is composed by sin-shaped wavy fibers. Moreover, when ρ(x) = x 1 specifically, problem (1) is reduced to the directional homogenization, which has been studied widely (see [19,20,33,11,13] and references therein).
Periodic homogenization of elliptic systems with stratified structure was first introduced in [8]. There the authors considered two cases, where A(x, ρ(x)/ε) = A(x, x/ε) and A(x, ρ(x)/ε) = A(ρ(x)/ε) separately, and proved the qualitative results for the former case. The work on the latter case was developed in [9,10] together with some interesting biomechanics and engineering applications. Then the G-convergence for general stratified media was presented in [21], where abstract differential operators with n = 1 were discussed via explicit expressions. In [37], the analytical and computational homogenization of stratified elastic materials were studied, especially for the wavy multilayers and wavy fibered composites. Later on, in [36] the authors built a systematic theory of homogenization for heterogeneous elastoplastic composites, including the qualitative results, the computational algorithm and some numerical examples. For more works on related topics, we refer readers to [32] for approximating nonperiodic materials by locally periodic structures and to [15,14] for approximations by the solutions of heterogeneous multiscale method.
It is known that under assumptions (3)-(5), L ε is G-convergent to an operator L 0 , where L 0 is a heterogeneous elliptic operator with coefficient A defined in Section 2.2. Moreover, the weak limit of u ε in H 1 (Ω; R m ) as ε → 0 satisfies    L 0 u 0 = f in Ω, Our main goal is to establish the quantitative homogenization for problem (1), including the convergence rate of u ε to u 0 and the uniform interior regularity of u ε . The first result concerns with the optimal convergence rate of u ε to u 0 in L p0 (Ω) with p 0 = 2d d−1 .

Convergence rates in the periodic homogenization of elliptic equations (or systems)
− div(A(x/ε)∇u ε ) = f in Ω, (9) has been largely studied in recent years. We refer readers to [22,34,35,31,30] for the convergence rate in L 2 (Ω) and [29,30] for the convergence rate in L p0 (Ω). Theorem 1.1 extends the corresponding results for system (9) to the general stratified systems. Note that δ in (7) depends on the correctors and the flux correctors, thereby on the function ρ and the coefficient A. For system (9), δ is trivially bounded and, therefore, the convergence rate in L p0 (Ω) for system (9) can be deduced directly from Theorem 1.1. The next two theorems provide uniform interior Lipschitz and W 1,p estimates of u ε .
Uniform regularity estimates (in ε > 0) are another main issue in quantitative homogenization. It goes back to a series of papers of M. Avellaneda and F. Lin [4,5,6,7], where interior and boundary estimates for problem (9) under Dirichlet condition were established by a compactness method. The corresponding boundary estimates under Neumann condition were obtained by C. Kenig, F. Lin and Z. Shen in [23]. Recently, another scheme for large-scale uniform regularity estimates was formulated in [3] and further developed in [2,29]. It is based on convergence rates and is effective for both Lipschitz and Hölder estimates. Uniform W 1,p estimates in the homogenization of elliptic systems have also been studied largely (see e.g., [12,27,28,16,17,39]). Especially, an approach called the real variable method was initiated in [12] and further developed in [27,28]. It reduces W 1,p estimates to weak reverse Hölder inequalities, which can be derived from the large-scale Lipschitz or Hölder estimates.
It is worth noting that the constants C in the regularity estimates of Theorems 1.2 and 1.3 are dependent on the radius r or R of ball B obliquely. We call estimates like these local estimates. This is more or less sharp for the estimates in Theorems 1.2 and 1.3 since A is heterogeneous. For this reason, we could not establish Liouville theorems and global size estimates of fundamental solutions for system (1). However, the corresponding estimates for (9) are global and uniform, i.e., the constants C are independent of the radius (see e.g. [30]). In fact, Θ B R is trivially null in the case of (9). Moreover, condition (10) is also not needed for (9), which is used here to bound the quantities involving correctors and flux correctors (see Lemma 4.7).
The paper is organized as follows. In Section 2, we introduce some preliminaries, including the nondegeneracy of ρ, the properties of stratified functions of the form h(x, ρ(x)/ε), as well as the correctors. Afterwards, we establish the qualitative homogenization for L ε by the Div-Curl Lemma and an approximating argument. In Section 3, the sharp convergence rate in Theorem 1.1 is proved under the symmetry assumption by a general duality scheme (see e.g. [34,29,31]) together with further properties of stratified functions. The process is much complicated since the auxiliary functions are double-variable and the matrix of effective coefficients is variable. Then in Section 4, we derive the (uniform) local Lipschitz estimates in Theorem 1.2, based on a weaker L 2 -convergence rate obtained from the Meyers estimate without the symmetry assumption. The rescaling property of system (1) plays an essential role in these regularity estimates. The Lipschitz estimate is used in Section 5 to obtain the local W 1,p estimates in Theorem 1.3 via the real variable method. As a corollary, we also provide the Hölder and L ∞ estimates for u ε .
In this paper, we use the notation h ρ,ε (x) = h(x, ρ(x)/ε). Note that for h(x, y), For the sake of simplicity, we write Thus, We should point out that D i is commutative with D j as well as the partial derivatives with respect to y, but not commutative with the partial derivatives with respect to x. Throughout this paper, we will use C to denote positive constants which may depend on d, m, n, µ if unindicated. It should be understood that C may differ from each other even in the same line. For simplicity, we may omit the superscripts α, β if it's clear to understand. We also use the notation ffl E f := (1/|E|)´E f for the integral average of f over E.

2.
Preliminaries and qualitative results. In this section, we will present some useful lemmas and establish the qualitative homogenization for problem (1). Some technical facts on ρ are stated in Section 2.1, which are essential in our discussions. Then we introduce the correctors in Section 2.2 and prove the qualitative homogenization briefly in Section 2.3.
For fixed x 0 ∈ Ω, it follows from (5) that the rank of the d×n matrix ( ∂ρ l ∂xj (x 0 )) is n. Therefore, if n = d, ( ∂ρ l ∂xj (x 0 )) is nondegenerate and the inverse function theorem implies that ρ is a diffeomorphism in a neighbourhood U (x 0 ) of x 0 .
Proof. Note that we have assumed that d = n and ρ is a diffeomorphism locally.
Since Ω is compact, there exist finite subsets U i ⊂ Ω, i = 1, . . . , N , such that ρ is a diffeomorphsim on each U i and where c i depends only on the continuity modulus of ∇ρ and ∇(ρ −1 ) L ∞ (Ω) , and C depends only on the Lipschitz character of Ω. Therefore, |h(y/ε)||detJρ −1 |dy. (20) Note that (18) implies that |detJρ| ≥ µ d 2 , which, together with the inverse function theorem, yields that Moreover, for each i, Let ε 0 = min i {c i }, which depends only on d, µ and the continuity modulus of ∇ρ. By combining (19)- (22) and the periodicity of h, we obtain for ε ≤ min where C depends only on d, µ and the Lipschitz character of Ω. The proof is completed.
Recall that D i h(x, y) = ∂ i ρ l (x)∂ y l h(x, y). Thus χ β j satisfies Denote Obviously, A is heterogeneous. We will show later that A is in fact the matrix of effective coefficients for A(x, ρ(x)/ε). By the equation of χ β j , we have the following estimates.
i) Then there exists p > 2, depending only on µ, such that for any x ∈ Ω, and for any x 1 , x 2 ∈ Ω, where C depends only on µ.
Proof. To prove estimates (26) and (27), one can apply the reverse Hölder inequality (see e.g. [18]) to the equations of χ β j , ∇ x χ β j , as well as the equation of , which is derived by taking differences. Similar argument also gives estimate (28). Finally, case (iii) follows from W 1,p estimates of elliptic systems with VMO coefficients.
Moreover, by replacing A with its adjoint A * , we can define χ * β j and A * similarly. Let per (Y ; R m ) and denote P γ k (y) = y k e γ , e γ = (0, . . . , 1, . . . , 0) with 1 at the γth position. Direct computations yields that where we have used the equation of χ β j in the second step. Therefore, it follows from the invertibility of the matrix ( ∂ρ l ∂xj ) (see the nondegeneracy of ρ in Section 2.1) that ( A) * = A * . Similarly, one can show that A satisfies the ellipticity condition.
Next we present more properties of A, which are useful in our research next. By the definition of A and Lemma 2.2, we have for any x 1 , x 2 ∈ Ω, where C depends only on µ. Thus, the conditions on A, such as VMO condition and Hölder continuity condition, can be reduced to the corresponding conditions on ∇ρ and A, which are involved in the estimates of u 0 . Furthermore, note that A is a C 0,1 -tensor defined on Ω. The lemma below extends A to the whole space. Proof. This is deduced from a result of [38]. It says that, if U ⊂ R n and f : U → R n is a Lipschitz function, then f can be extended onto R n preserving the Lipschitz property with the range contained in the closed convex hull of f (U ). Thus, by regarding Ω as a subset of R d 2 ×m 2 , we can extend A onto R d preserving the Lipschitz property. The boundedness property and the ellipticity condition follow from the fact that the value of A at each point outside Ω is a limit of convex combinations of the points in A(Ω).
where C depends only on d, µ, p, Ω. ii) If h ∈ L q (Ω; W 1,p per (Y )) with p > d and q ≥ 1, then for any ε > 0, where C depends only on d and p.
Proof. Let us prove (30) under the assumption that h(x, y) is smooth. The result for h ∈ W 1,p (Ω; L p per (Y )) follows from a standard density argument. By Sobolev imbedding theorem, for a.e. x ∈ Ω and any y ∈ R d , we have for p > d where C depends only on Ω and p. By setting y = ρ(x)/ε in (32) and taking L q -norm over Ω, we get where C depends on d, µ, Ω, p, and we have used Lemma 2.1 in the second inequality. This implies (30) directly. The proof of (ii) is similar and much easier. Instead of (32), one may start with the inequality which is insured by Sobolev imbedding theorem and the periodicity of h(x, ·).
Proof. First we prove (34) for h ∈ W 1,p (Ω; L p per (Y )). By considering h(x, y) − Y h(x, ·), we may assume that´Y h(x, ·) = 0 for any x ∈ Ω. Let u(x, ·) ∈ H 2 per (Y ) be a 1-periodic function such that Note that for any fixed x, by (5), L I y is a scalar elliptic operator with symmetric, constant coefficients.
Now we are prepared to consider the qualitative homogenization of problem (1).
Proof. We split the proof into two steps. In step 1, we focus on the case A(x, y) ∈ C 0,1 (Ω; C ∞ per (Y )).
Step 2 is devoted to removing the smoothness condition.
For the left hand side, which, by virtue of Lemma 2.5 and the Div-Curl Lemma, implies that the l.h.s. of (38) converges to´Ω H γ k ϕ as ε → 0. On the other hand, thanks to Lemma 2.5, where we have used the equation of χ * γ k in the second step as well as Lemma 2.5 in the last step. Thus, by applying the Div-Curl Lemma and taking (39)-(40) into consideration, we conclude that the limit of the r.h.s. of (38) iŝ where we have also used the fact that A * = A * . Consequently, we obtain which, together with the arbitrariness of ϕ in C 1 0 (Ω), shows that H = A∇u 0 .
Step 2. For general A, we can find a sequence of A k (x, y) ∈ C 0,1 (Ω; C ∞ per (Y )) such that A k satisfies the same conditions as A and where p is to be determined and can be arbitrarily large. By Lemma 2.1, (41) tells us that For each ε, let v k,ε satisfy L A k ε (v k,ε ) = f in Ω and v k,ε = u ε on ∂Ω, where L A k ε = −div(A ρ,ε k (x)∇). Then, by a duality argument and the reverse Hölder inequality, there exist constants p and C, depending only on µ, m, Ω, such that, (see the proof of Theorem 1.4 in [40] for details). We can assume that p in (42) coincides with the one in (43). By virtue of Step 1, it's not hard to show that v k,ε v k,0 weakly in H 1 (Ω) as ε → 0, and by (42)-(43), where v k,0 is the unique solution of L A k 0 (v k,0 ) = f in Ω and v k,0 = u 0 on ∂Ω. Moreover, by using (26), one can verify that for any x, which together with the equations of v k,0 implies that any weak limit of v k,0 in H 1 (Ω) satisfies Therefore, it follows from (44) that v k,0 u 0 weakly in H 1 (Ω) and u 0 is the unique solution of (45). This completes the proof.
3.2. Convergence rates. Now we turn to the problem of convergence rate. We start with the equation of u ε − u 0 , where L ε u ε = f in Ω, ε ≥ 0. By calculation, where K ε (·) is a linear operator to be determined. Observing that informally where D k is given by (16) and For the last term, we obtain Next we introduce the flux correctors. For fixed x, consider the equation where L I y = D i D i and D i is defined by (16). Recall that for any fixed x, by (5), L I y is a scalar elliptic operator with symmetric, constant coefficients. Therefore, there exists a unique solution φ ij (x, ·) satisfying . We define the flux corrector as B kij (x, y) = D k φ ij (x, y) − D i φ kj (x, y) which is skew-symmetric. By (17), In view of equation (24), for any α, β, L I y [D k φ αβ kj (x, ·)] = D k B αβ kj (x, ·) = 0, which, together with the periodicity in y, implies that D k φ kj (x, ·) is constant in y. Therefore, D k D i φ kj (x, y) = 0 for any x, y. By taking this into (50), we obtain Thus, where we have used the skew-symmetry of B in the last equality. By taking (51) into (48), we finally get In the following, suppose η ε ∈ C ∞ 0 (Ω) is a cut-off function satisfying 0 ≤ η ε ≤ 1, |∇η ε | ≤ C/ε, η ε = 0 on Ω 3ε , η ε = 1 on Ω \ Ω 4ε . (53) Let Ω be a bounded Lipschitz domain in R d . Assume that A ∈ C 0,1 (Ω; L ∞ per (Y )) satisfies (3)-(4) and ρ ∈ C 1,1 (Ω; R n ) satisfies (5). Let w ε be given by (47) and K ε (u) = S 2 ε (u)η ε . Then for any ψ ∈ H 1 0 (Ω; R m ), where C depends only on µ and The first term in (56) can be bounded by For the second term, by using Lemma 3.1 and the definition of K ε , we have Similarly, the third term can be dominated by . Then we can obtain (54) by combining the estimates above. and where C depends only on µ and Ω.
Proof of (7) in Theorem 1.1. Note that By (59) and (60), it is easy to see that where q 0 = 2d d+1 . To bound I 2 , we use (58) to obtain where we have also used the estimates (61) and (65) for v 0 . Again, by (58), where η ε is defined as (53). Note that by Lemma 3.1, where we have used estimate (61) for v 0 in the second inequality. Similarly,

Thus,
In view of the estimates of I 1 , I 2 , I 3 and (70), we obtain ˆΩ which, by duality, gives the desired result. The proof is completed.
Similar to Lemma 2.2, we have the following lemma for φ (see (49)).
i) Then for any x ∈ Ω, and for any x 1 , x 2 ∈ Ω, where p is given by case (i) in Lemma 2.2 and C depends only on µ.
Proof. By the definition of B(x, y) and Lemma 2.2, one can verify that, for any and for any Then Lemma 3.7 can be derived in the same way as Lemma 2.2. We omit the details. : Ω × R n → R is 1-periodic in y and ρ ∈ C 1 (Ω; R n ) satisfies (5). Let ε 0 be given in Lemma 2.1.
where C depends only on d, µ, k and the Lipschitz character of Ω. ii) If h ∈ W 1,∞ (Ω; L p per (Y )) with p > d, then for any 2 ≤ q ≤ p and ε ≤ ε 0 , where C depends only on d, µ, p and the Lipschitz character of Ω. iii) If h ∈ L ∞ (Ω; W 1,p per (Y )) with p > d, then for any q ≥ 1 and ε > 0, where C depends only on d and p.
Proof. Thanks to Sobolev imbedding theorem, for a.e. x ∈ Ω ∩ B(y, 2ε), where C depends only on k and the Lipschitz character of Ω. By taking the L qaverage of (80) over Ω ∩ B(y, 2ε), we obtain Ω∩B(y,2ε) where we have used Lemma 2.1 and C depends only on d, µ, k and the Lipschitz character of Ω. By taking the supremum of y over Ω, we get case (i). Cases (ii) and (iii) are obtained by similar ideas and the inequalities as (32) and (33) respectively. We omit the details.
Remark 2. The additional conditions on A and ρ can be replaced by the uniformly VMO condition of A without changing (8). To do this, one need to make use of (iii) in Lemmas 2.2, 3.7 and (ii)-(iii) in Lemma 3.8.

4.
Interior Lipschitz estimates. This section is devoted to the interior Lipschitz estimates for u ε .
We first study the rescaling and translation properties for the problem L ε (u ε ) = f in Ω. By setting v ε (x) = u ε (rx + x 0 ), we have where F (x) = r 2 f (rx + x 0 ) and Ω r,x0 = {(x − x 0 )/r : x ∈ Ω}. Denote by L A,ρ ε the elliptic operator associated to A, ρ and ε, that is, Then v ε satisfies where A r,x0 (x, y) = A(rx + x 0 , y), ρ r,x0 (x) = ρ(rx + x 0 )/r. Note that ρ r,x0 satisfies condition (5) with the same µ. For system (81), we can define the correctors χ r,x0 j as (23) and the homogenized matrix A r,x0 as (25) with A, ρ replaced by A r,x0 , ρ r,x0 respectively. It's easy to verify that χ r,x0 (x, y) = χ(rx + x 0 , y) and A r,x0 (x) = A(rx + x 0 ). Moreover, one can also define the associated flux corrector B r,x0 and function φ r,x0 which satisfy B r,x0 (x, y) = B(rx + x 0 , y) and φ r,x0 (x, y) = φ(rx + x 0 , y). Now we turn to the Lipschitz estimates for u ε by using a scheme of regularity estimates of large scale (see e.g. [3,2,29]). It is based on the O(ε σ )-convergence rate of u ε to u 0 . Recall that the convergence rate in Section 3 was obtained under the assumption that A = A * . Thus, to avoid the symmetry assumption in this section, we shall establish another O(ε σ )-convergence rate.
Proof. First of all, observing that we obtain that the third term in (56) can be dominated by , where we have used Hölder's inequality and Lemma 3.1. Besides, we have Thus, similar to Lemma 3.4, it follows that By setting ψ = w ε in (84) and applying (68), we obtain (82).
Proof. Observing that w ε ∈ H 1 0 (Ω), and by Lemma 3.1 where C depends only on µ and Ω. Note that by the Meyers estimate [25,18], there exists p > 2, depending only on µ and the Lipschitz character of Ω, such that where C depends only on µ and Ω. For the first term on the r.h.s. of (86), Moreover, by the interior H 2 estimates for elliptic systems, B(y,c δ(y)) where δ(y) := dist(y, ∂Ω), c is a very small constant and C depends only on µ.
Proof. Without loss of generality, we assume that b = 0. We first consider the case r = 1 and x 0 = 0. By Caccioppoli's inequality, where C depends only on µ. According to the co-area formula, there exists t ∈ [5/4, 3/2] such that Now let u 0 be the solution to L 0 u 0 = f in B t , u 0 = u ε on ∂B t . By Lemma 4.2, where C depends only on µ and d. This, together with (92), implies that For general r and where L A r,x 0 = −div( A r,x0 ∇),δ r,x0 B2 is given by (83) with χ and φ replaced by χ r,x0 and φ r,x0 respectively, and  Proof. First we assume that x 0 = 0 and r = 1. By choosing P (x) = u 0 (0)+∇u 0 (0)· x, we have |f | q 1/q for any P ∈ P, where α = 1 − d q , C depends only on µ, A C α (B1) and we have used the C 1,α estimates for elliptic systems with C α coefficients in the second inequality. By setting θ ∈ (0, 1/8) small enough, we obtain the desired result. Note that we also used the fact that For general r > 0 and x 0 ∈ R d , one can do rescaling as usual. Then the same result holds with θ depending on ∇ A r,x0 This completes the proof.
for some q > d and ε ∈ (0, 1/4). For 0 < r ≤ 1, define for any b ∈ R m , where we have used Lemma 4.3 in the last step. This is exactly what we desired.
The following lemma was proved in [29].
Proof. By applying Lemmas 2.2, 3.7 and 3.8, we have for any q ≥ 1 where C depends only on d, µ and (t). Thus, we havē where C depends only on d, µ and (t). This completes the proof.
Proof of Theorem 1.2. We first consider the case R = 1 and x 0 = 0. Since the case r ∈ [1/4, 1/2] is trivial, we also assume that ε ∈ (0, 1/4). Let H(r), Φ(r) be defined as in Lemma 4.5 and ω(t) = t σ which satisfies (97). Denote by P r the polynomial achieving the infimum in (94), i.e., P r satisfies H(r) = 1 r Br |u ε − P r | 2 1/2 + r 2 Br |f | q 1/q , and let h(r) = |∇P r |. Note that ∇P r is a constant matrix. For t ∈ [r, 2r], it is evident that H(t) ≤ CH(2r). Note that for t, s ∈ [r, 2r], We obtain that max for any r ∈ [ε, 1/2], where in the last step we have used the observation that The desired estimate for the case R = 1 and x 0 = 0 now follows from (99) by Caccioppoli's inequality. The general case can be reduced to this special case by rescaling and translation. We omit the details here.
5. W 1,p estimates. In this section, we establish the uniform interior W 1,p estimates under the additional assumption that A satisfies the uniformly VMO condition.
The next theorem is a real variable argument formulated in [12,27,28].
where C 1 , C 2 > 0 and 0 < c 0 < 1. Then F ∈ L p (B 0 ) and where C depends only on d, C 1 , C 2 , c 0 , p and q.
Clearly, F ≤ F B +R B on 2B. And (102) follows from the standard energy estimates. Moreover, by using Lemma 5.1, which is exactly (101). Thanks to Theorem 5.2, we obtain (15). This completes the proof.
The interior W 1,p estimate in Theorem 1.3 gives the following interior Hölder and L ∞ estimates by Sobolev imbedding. Corollary 2. Suppose the assumptions of Theorem 1.3 hold. Let u ε ∈ H 1 (2B; R m ) be a weak solution to where B = B(x 0 , r), and f 1 , f 2 ∈ L p (2B; R m ) for some p > d. Then for any x, y ∈ B |u ε (x) − u ε (y)| ≤ C |x − y| r where C depends only on p, (t) in (10) and Θ 2B in (12).