Global gradient estimates in directional homogenization

: In this research, we investigate a higher regularity result in periodic directional homogenization for divergence-form elliptic systems with discontinuous coe ﬃ cients in a bounded nonsmooth domain. The coe ﬃ cients are assumed to have small bounded mean oscillation (BMO) seminorms and the domain has the δ -Reifenberg property. Under these assumptions we derive global uniform Calder´on-Zygmund estimates by proving that the gradient of the weak solution is as integrable as the given nonhomogeneous term.


Introduction
In this study, we consider elliptic systems in directional homogenization of the following form: for 1 ≤ α, β ≤ n and 1 ≤ i, j ≤ m with m ≥ 1, where the nonhomogeneous term F = { f i α } is given by a matrix-valued function.Here, Ω is a bounded domain in R n with n ≥ 2 and 0 < ≤ 1. Especially, in order to treat directional homogenization we define the coefficients A = {A αβ, i j } for 0 < ≤ 1 from A = {A αβ i j }, A αβ i j : R n → R, to be as follows: A αβ i j (x) = A αβ,1 i j (x) and A αβ, i j (x , x ) = A αβ i j x , x where x = (x , x ), x = (x 1 , • • • , x l ) ∈ R l and x = (x l+1 , • • • , x n ) ∈ R n−l with 0 ≤ l ≤ n.In addition, we assume the following periodicity condition on A αβ i j (x) : 3) The coefficients are assumed to have uniform ellipticity and uniform boundedness.In other words, we assume that there exist positive constants ν and L such that for every matrix ξ ∈ R mn and for almost every x ∈ R n .We note that since if l = n, we do not need to treat homogenization and if l = 0, our problem is periodic homogenization, throughout this research in order to consider directional homogenzition we assume that 1 ≤ l ≤ n − 1.
In this paper, we consider the weak solution u = (u 1 , • • • , u m ) ∈ H 1 0 (Ω, R m ) to (1.1) which satisfies Here we note that if F ∈ L 2 (Ω, R mn ), the weak solution u ∈ H 1 0 (Ω, R m ) exists and satisfies the estimate where the constant c does not depend on , by the Lax-Milgram lemma.Now, we introduce some basic facts for directional homogenization; see for details in [2,16].The matrix of correctors χ = χ i j α (x , x ) , with 1 ≤ i, j ≤ m and l + 1 ≤ α ≤ n, is the weak solution to the following cell problem: which satisfies the following estimate : Then the linear elliptic system given by is the homogenized problem of (1.1), whose weak solution u 0 of (1.10) is the weak limit of the weak solution u in H 1 0 (Ω, R m ) as → 0. Regularity theories for elliptic equations in homogenization are widely studied for partial differential equations; see [1, 3-5, 11, 15-18, 20, 23, 27] and the references therein.Among these, under the settings, our goal is to obtain global uniform Calderón-Zygmund estimates, that is, we would like to prove that if F ∈ L p (Ω, R mn ), then the L p norm of Du is controlled by the L p norm of F and is independent of .The authors proved the Calderón-Zygmund theory for (1.1) under the condition of periodic homogenization in [3].Recently, the authors of [15,16] gave several different interior regularity results for directional homogenization.Given these viewpoints, here, we consider the global estimates in directional homogenization.
For homogenization problems, we want to derive estimates which are independent of 0 < ≤ 1.Since our desired result includes the case = 1 for (1.1) when there is no homogenization, our research relies on the conditions that the L p regularity theory for the gradient is established; see [6-9, 14, 22].Thus, we prove the global Calderón-Zygmund theory for (1.1) subject to periodic directional homogenization under the conditions of Definitions 2.2 and 2.3 described in Section 2. In fact, in view of the condition for the coefficients of the regularity theory for (1.1) with = 1 in this literature, we may consider some weaker assuptions than Definition 2.2 but in that case, we can only obtain some local results instead of the global regularity; see Remark 2.8.
To prove our result, we use a perturbation argument based on localization which includes scaling, translation and rotation.In fact, even though in (1.1) with (1.2) the direction of homogenization is globally fixed so that the direction is changed under rotation, the condition of the coefficients in Definition 2.2 below is invariant under rotation.This makes our method suitable for directional homogenization.Also, for directional homogenization, when we solve our problem, there are some differences between the x -direction which is not involved with homogenization and the xdirection for homogenization.The former gives macroscopic properties in the x -direction and the latter represents microscopic oscillation in the x -direction.With this observation, we can apply estimates from the case without homogenization; see [6,7] for x and results corresponding to periodic homogenization in [3] for x in our proof; see details in Lemma 3.3.This paper is organized as follows.In Section 2, we introduce some notations and definitions and announce our result as Theorem 2.6.In Section 3, we show our key lemma, Lemma 3.3, and then finally give the proof of the result.

Assumptions and main result
We start this section with some notations and definitions.If the center is the origin, we denote B r (0) by B r .Similarly, for y = (y , y ) ∈ R l × R n−l , an open ball in R l with the center y with radius r > 0 is defined to be an open ball in R n−l with the center y with radius r > 0 is defined to be and if the center is the origin, y = (0 , 0 ), then we denote B r (0 ) ⊂ R l by B r and B r (0 ) ⊂ R n−l by B r .
(2) The integral average of g ∈ L 1 (U) over the bounded domain U in R n is denoted by and we denote the integral average over U ⊂ R n−l by ( For our global regularity result we assume that the coefficient A enjoys the small bounded mean oscillation (BMO) condition which is a generalization of the vanishing mean oscillation (VMO) condition.The following is the precise definition that is to be used throughout this paper.
Additionally, we consider the domain Ω is a Reifenberg domain; see [26], which is an extension of Lipschitz domains with small Lipschitz constants.The definition is as follows: Definition 2.3.Let Ω be a bounded domain in R n .We say that Ω is (δ, R)-Reifenberg flat if for every x ∈ ∂Ω and every r ∈ (0, R], there exists a coordinate system {y 1 , . . ., y n } that is dependent on r and x so that x = 0 in this coordinate system and Similar to Definition 2.2, we can define and Remark 2.4.We can see that A αβ i j being (δ, R)-vanishing is equivalent to the condition that A αβ i j is (δ, R)-vanishing with respect to x and x in the following sense.From direct computations by using the properties of averages, if A αβ i j is (δ, R)-vanishing with respect to x and x , then -vanishing with respect to x and x where c From this equivalence, we consider the (δ, R)-vanishing condition instead of the (δ, R)-vanishing conditions with respect to x and x throughout this paper since the (δ, R)-vanishing condition has a rotational invariant property.
Remark 2.5.We give some comments on Definitions 2.2 and 2.3.First of all, by the scaling invariant property of our problem (1.1), the value of R in the definitions of both coefficients and domains can be 1 or any other constants greater than 1.For this reason R ≥ 1 is to be selected for our purpose.In addition, the constant δ to be determined is also invariant under this scaling.
For (2.1), with Ω being a Reifenberg flat domain, it is known that δ ≤ δ * for some constant δ * = δ * (n).We note that δ * = δ * (n) < 2 −n−1 ≤ 1 8 for n ≥ 2; see [28].From this, we can assume that δ < 1 8 throughout this paper.Moreover, even though the Reifenberg flatness condition given by (2.1) does not mean any smoothness on the boundary, this gives the following measure density condition: for every y ∈ ∂Ω and r ∈ (0, R].This will be used in our L 2 approach since (2.4) implies the p-capacity condition with p = 2; see [25, Section 2.2.3], which makes us apply the higher integrability result in [21] to our method.Now let us state the global estimate of this paper.
Remark 2.8.In view of the regularity results [6,9,14,22], just for interior estimates or local boundary estimates we can give weaker conditions than those of Theorem 2.6.The weaker conditions are that A αβ i j is (δ, R)-vanishing of codimension 1 in [4] for interior cases and (A αβ i j , Ω) is (δ, R)-vanishing of codimension 1 in [6,19] for boundary cases.Both conditions allow A αβ i j to be merely measurable for one variable, while they have small BMO seminorms for the other variables in some appropriate coordinates.Second, Ω is to be a (δ, R) Reifenberg flat domain.
In fact, for the interior case, our argument in this paper can be applied to the interior version for Theorem 2.6 when the coefficients A αβ i j are (δ, R)-vanishing of codimension 1. Especially, for the interior case with l = n − 1, in view of [6,12] the interior estimate corresponding to Theorem 2.6 is obtained without considering homogenization since we can regard the direction of homogenization as just a measurable direction.
On the other hand, for the boundary case we cannot consider in general (A αβ i j , Ω) to be (δ, R)vanishing of codimension 1 because of consistency between the coefficients of homogenization and the domain even though the global regularity result holds under this condition when there is no homogenization; see [6].From this observation, with the same idea in this research, we can obtain a local boundary estimate at the point x 0 ∈ ∂Ω whose normal direction is only related to x in the sense of Definition 2.3 under the condition that (A αβ i j , Ω) is (δ, R)-vanishing of codimension 1.

Global gradient estimate
To establish our global gradient estimate, we first introduce some tools for the proof of Theorem 2.6.Our method is based on the Hardy-Littlewood maximal function.
First, let us recall the Hardy-Littlewood maximal function and its basic properties.If we suppose that g is a locally integrable function on R n , then the Hardy-Littlewood maximal function is given by |g(y)|dy.
If g is defined only on a bounded subset of R n , then we define where g is the zero extension of g from the bounded set to R n .This maximal function satisfies the conditions of the weak 1-1 estimate and the strong p-p estimates.Also, we define the restricted maximal function where 1 U is the characteristic function of U ⊂ R n .
Our goal in this article is to show the L p integrability of Du .For this, we would like to use a sum of certain estimates for super-level sets.The next lemma gives a relation between the integration and summation of super-level sets.Lemma 3.1.[10] Assume that g is a nonnegative, measurable function defined on the bounded domain Ω ⊂ R n , and let θ > 0 and λ > 1 be constants.Then for 0 < q < ∞, we have The positive constant c depends only on θ, λ and q.
The following lemma is the Vitali-type covering lemma for our proof.Here, we note that because of the scaling invariant property of δ and R in Definition 2.3 for Reifenberg flat domains, we only need to consider R = 1 in the next lemma.Lemma 3.2.[7,29] Assume that C and D are measurable sets with C ⊂ D ⊂ Ω and Ω being (δ, 1)-Reifenberg flat.Also, assume that there exists a small η > 0 such that and that for each x ∈ Ω and r The next one is the main lemma in our argument.This shows the second condition of Lemma 3.2 under our settings.In the following argument, we would like to refer [13,24] to help the readers.
) is the weak solution to (1.1).Then there exists a universal constant η = η(ν, L, m, n, p) so that a small δ = δ(ν, L, m, n, p) > 0 is selected such that if A αβ i j is (δ, 336)-vanishing, Ω is (δ, 336)-Reifenberg flat, and for all y ∈ Ω and every where 80 7 then the following holds: Proof.We prove this by contradiction.We assume that (3.5) holds but (3.7) is false.Then there is a point for all r > 0. Under these conditions, there are two cases that we need to consider.One is B 14ρ (y) ⊂ Ω, which is an interior case, and the other is B 14ρ (y) Ω, which is a boundary case.Since the proof for the interior case is eventually the same as that of the boundary case, here, we prove this lemma for the boundary case.Now we consider the case that B 14ρ (y) Ω.Then we assume that since Ω is (δ, 336)-Reifenberg flat there exists an appropriate coordinate system, after suitable rotation and translation, so that Then from (3.8) we see that |F| 2 dx ≤ 2 n δ 2 . (3.12) Now we consider the following rescaled maps: Ω 336ρ .Then we see that ũ is a weak solution to the following: and Ã is (δ, 12)-vanishing.Under these settings, it suffices to consider the following case: This is because the (δ, 12)-vanishing condition, which is a small BMO condition, is invariant under rotation for the coefficients, even though the direction of homogenization for our problem (1.1) is changed under rotation.Next, we let w ∈ H 1 ( Ω11 , R m ) be the weak solution to the following: a standard L 2 estimate follows from (3.18) and (3.16) for some positive constant c = c(ν, L, m, n).In addition, since our domain satisfies the measure density condition which implies the p-capacity condition with p = 2 according to Remark 2.5, from (3.16) and (3.17) there exists positive constants σ 1 and c = c(ν, L, m, n) such that For the sake of our perturbation argument, by using the notation given by (2.2), we now let h ∈ H 1 ( Ω5 , R m ) be the weak solution to From this, (1.4), Remark 2.4 and (3.20), we compute for some constant c = c(ν, L, m, n) and hence Now we note that since our desired δ has an upper bound by Remark 2.5 we have from (3.23) that such h satisfies 1 for some constant c = c(ν, L, m, n) and similar to w for some constant c = c(ν, L, m, n).Thus according to [18,Lemma 3.4] for any κ > 0 there exists a small δ = δ(ν, L, m, n), which depends only on the given structure conditions, such that there exists a weak solution ṽ ∈ H for some constant c = c(ν, L, m, n) and (3.27)Moreover, since Ãαβ, z , we can extend our coefficient Ãαβ, (z ) for the z -variable to the z-variable, that is, Ãαβ, z .For this reason, we can apply the result [3,27] for periodic homogenization to (3.25); then, we obtain for any 2 < q < ∞, that there exists δ = δ(ν, L, m, n, q) such that with the estimate .
for some constant c = c(ν, L, m, n, q) independent of .Especially, by taking q = p + 1 we have that there exists δ = δ(ν, L, m, n, p) so that where the constant c = c(ν, L, m, n, p) is independent of .Considering u as the zero extension outside of the domain Ω, we assert that if For this, we denote z 1 = y 1 28ρ and let z 0 ∈ {z ∈ Ω1 : If r > 2, since z 1 ∈ B r (z 0 ) ⊂ B 2r (z 1 ), we obtain the following from (3.8) and (3.13) Thus, we prove (3.29) by showing that z 0 ∈ {z ∈ Ω1 : M(|Dũ | 2 ) ≤ N 2 1 }.Now, we let Ṽ be the zero extension of ṽ from B + 4 to B 4 , and we let where N 1 ≥ 1 is to be determined.Then from (3.29) we compute the following: For I 1 , we use (3.19); then, for some constant c = c(ν, L, m, n).Applying (3.23), we see that for some constant c = c(ν, L, m, n).
We now suppose that and where the constant δ 0 = δ 0 (ν, L, m, n, p) is to be determined.Then, we want to show first that To prove (3.37), we write By the weak 1-1 estimate and the L 2 -estimate, we see that if F L 2 (Ω) ≤ δ 2 , then we can take For any δ ≤ δ 0 , this verifies the first condition of Lemma 3.2.Moreover, the second condition of Lemma 3.2 follows from Lemma 3.3.Thus we apply Lemma 3.2 for such δ 0 to see that Then S 1 is estimated as follows: Also, S 2 is computed as follows: This completes the proof.

Conclusions
The proof of this paper is based on a perturbation argument in the main lemma, Lemma 3.3.Under this argument for directional homegenization, in the proof of Lemma 3.3 we can compare the original coefficients Ãαβ, i j (z , z ) with Ãαβ, i j B 5 (z ) by using the small BMO condition to derive (3.23) and then we can apply the result in [3] to (3.25) since we can consider (3.25) as homogenization for whole variables.For these reasons, we can derive Theorem 2.6 for directional homogenization.

Use of AI tools declaration
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Notations 2 . 1 . ( 1 )
An open ball in R n with a center y with radius r > 0 is defined to be B r (y) = {x ∈ R n : |x − y| < r}.