Oscillatory integrals and periodic homogenization of Robin boundary value problems
Jun Geng Jinping Zhuge


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Oscillatory integrals and homogenization]Oscillatory integrals and periodic homogenization of Robin boundary value problems


Key words and phrase: Oscillatory integrals; Robin boundary condition; Homogenization.

J. Geng, and J. Zhuge]
Jun Geng     Jinping Zhuge

[2010]35B27, 42B20


In this paper, we consider a family of second-order elliptic systems subject to a periodically oscillating Robin boundary condition. We establish the qualitative homogenization theorem on any Lipschitz domains satisfying a non-resonance condition. We also use the quantitative estimates of oscillatory integrals to obtain the dimension-dependent convergence rates in L^2, assuming that the domain is smooth and strictly convex.
 

 


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    December 30, 2023
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§ INTRODUCTION




The primary purpose of this paper is to study a family of elliptic systems with oscillating Robin boundary condition in a bounded Lipschitz domain Ω⊂, d≥ 3, 

    {ℒ_ (u_)    = F       in    Ω,
    ∂ u_ε/∂ν_ε+b(x/ε)u_   =g       on    ∂Ω,
    .

where 

    ℒ_ε =- div[A(x/ε)∇] =-∂/∂ x_i[a_ij^αβ(x/ε)
    ∂/∂ x_j],     ε>0,

and u_ = (u_^1,u_^2, ⋯, u_^m) and 1≤α, β≤ m, 1≤ i,j ≤ d. The summation convention is used throughout this paper. We assume that the coefficients A = (a^αβ_ij) and b = (b^αβ) satisfy the following conditions

	
  * Ellipticity condition: there exists μ>0 such that
	
    μ|ξ|^2≤ a_ij^αβ(y)ξ_i^αξ_j^β≤μ^-1 |ξ|^2,
    	     for y∈, ξ=(ξ_i^α)∈ℝ^m× d,

	and
	
    μ|η|^2≤ b^αβ(y)η^αη^β≤μ^-1 |η|^2,
    	     for y∈, η=(η^α)∈.

	
  * Periodicity condition: for any y∈, z∈ℤ^d,
	
    A(y+z)=A(y),

	and
	
    b(y+z) = b(y).

	
  * Continuity: 
	
    The coefficient matrix  b  is continuous in .



The equation (<ref>) with m = 1 may be used to model, for example, the heat conductivity problems of heterogeneous materials surrounded by a certain fluid. On the surface of the material, a Robin boundary condition is needed due to the convection or phase transition. Actually, a general convective boundary condition is described as

    heat flux = b(T - T_liquid),     on ∂Ω,

where T is the temperature of the material surface, T_liquid is the temperature of the fluid and b is known as the heat transfer coefficient established by measurements (depending both on the material and liquid). Under the conditions (<ref>) – (<ref>) and assuming T_liquid is a constant and u_ = T -  T_liquid, we know the heat flux is given by -∂ u_/∂ν_ and b takes a form of b(x/), which is periodic. Therefore, the boundary condition (<ref>) may be rewritten as

    ∂ u_/∂ν_ + b(x/) u_ = 0.

If T_liquid is not a constant and could be measured near the material surface, then (<ref>) may be reduced to a nontrivial oscillating Robin boundary data that takes a form of g(x,x/) and g(x,y) is 1-periodic with respect to y. This case may also be handled by the method in this paper.

Recall that the variational form of equation (<ref>) may be written as

    ∫_Ω A(x/) ∇ u_·∇ϕ + ∫_∂Ω b(x/) u_·ϕ = ∫_∂Ω g ·ϕ + ∫_Ω F ·ϕ,

for any ϕ∈ H^1(Ω;). Then, the Lax-Milgram Theorem implies the existence and uniqueness of the weak solution of (<ref>), provided F∈ H^-1(Ω;) and g∈ H^-1/2(∂Ω;). Moreover, the sequence { u_: >0 } is uniformly bounded in H^1(Ω;). Now, we may ask a standard question in homogenization theory: as → 0, do the solutions u_ converge to some function u_0 in H^1(Ω;), while u_0 is the weak solution of the effective equation with a certain effective Robin boundary condition? It turns out that the answer to this question is quite different from the usual Dirichlet or Neumann boundary value problems, due to the oscillating factor of b(x/) on the boundary. We mention that the Dirichlet and Neumann problems have been well-studied in many literatures and we refer to a recent excellent monograph <cit.> by Z. Shen.

The purpose of this paper is two-fold: (i) prove the qualitative homogenization theorem under very weak assumptions on the coefficients and domain; (ii) obtain the L^2 convergence rates under sufficiently strong assumptions.



 §.§ Qualitative homogenization

Our method of proving the qualitative homogenization theorem is based on the direct asymptotic analysis of the variational equation (<ref>). In view of the second integral on the left-hand side of (<ref>), the key for homogenization to take place is the asymptotic behavior of the following oscillatory integral

    ∫_S f(x, λ x) dσ(x),     as λ→∞,

where S = ∂Ω, dσ is the surface measure on S (namely, (d-1)-dimensional Hausdoff measure) and f(x,y) is a 1-periodic function in y∈. The asymptotic behavior of the integral (<ref>) may be derived through the classical oscillatory integral theory if S is smooth and possesses certain geometric conditions. For example, if S is smooth and strictly convex and f is smooth, then (<ref>) converges with error O(λ^1-d/2) as λ→∞. Recently, S. Kim and K.-A. Lee showed in <cit.> that if S is a C^1 surface and satisfies a non-resonance condition (which is called irrational direction dense condition in <cit.>), then the integral (<ref>) converges qualitatively as λ→ 0. In this paper, we further weaken the regularity of S to any Lipschitz surfaces, which is sharp in the sense that dσ is well-defined and the following non-resonance condition makes sense. 




	Let S be a Lipschitz surface. We say S satisfies the non-resonance condition with respect to ℤ^d (or simply non-resonance condition), if
	
    σ( {x∈ S: n(x)  is well-defined and  n(x) ∈ℝℤ^d } ) = 0.

	Here n(x) is the unit normal to S at x. (A vector a∈ is called rational if a∈ℝℤ^d.)


The non-resonance condition in Definition <ref>, generalizing the notion in <cit.>, covers all the classes of domains people innovated to deal with PDEs subject to periodically oscillating boundary value, including strictly convex smooth domains <cit.>, finite-type domains <cit.>, polygonal domains <cit.> and more <cit.> (also see Remark <ref> for a possible extension for layered or directional materials). Under this non-resonance condition, we are able to prove Theorem <ref>, a (qualitative) version of the Weyl's equidistribution theorem, based on the fundamental ideas from <cit.> and <cit.>. Among many potential applications of the Weyl's equidistribution theorem in analysis and PDEs (see, e.g., Theorem <ref>), we prove the qualitative homogenization theorem for (<ref>).

	Let (A,b) satisfy (<ref>) - (<ref>). Assume Ω is a bounded Lipschitz domain satisfying the non-resonance condition with respect to ℤ^d. Suppose that u_ is the weak solution of (<ref>)
	
	
	
	
	
	
	
	
	with F∈ H^-1(Ω;) and g∈ H^-1/2(∂Ω;). Then, there exists some function u_0∈ H^1(Ω;) such that, as → 0,
	
    {
    	u_   → u_0          weakly in  H^1(Ω;)  and strongly in  L^2(Ω;), 
    
    	A_(x/)  ∇ u_   →A∇ u_0         weakly in  L^2(Ω;ℝ^m× d),
    	.
 
	where A is the homogenized coefficient matrix. Moreover, u_0 is the weak solution of
	
    {
    	- div(A∇ u_0)    = F          in Ω,
    ∂ u_0/∂ν_0+b̅ u_0     =g          on ∂Ω,
    	.

	where ∂/∂ν_0 = n·A∇ and b̅ = ∫_𝕋^d b(y)dy.


Theorem <ref> is a straightforward corollary of Theorem <ref> which is proved in a more general setting. Note that the effective heat transfer coefficient in (<ref>) is the average of b. This fact may be extended without any real difficulty to more general Robin boundary conditions, such as

    ∂ u_/∂ν_ + b(x,x/) u_ = g(x,x/),

where b(x,y) and g(x,y) are 1-periodic with respect to y. In this case, the homogenized boundary condition reads

    ∂ u_0/∂ν_0 + b̅(x) u_0 = g̅(x),

where b̅(x) = ∫_𝕋^d b(x,y)dy and g̅(x) = ∫_𝕋^d g(x,y)dy.






 §.§ Convergence rates

The second part of the paper is concerned with the convergence rates of u_ to u_0 in L^2(Ω;), which seems to be of more interest. The sharp convergence rates for the Dirichlet and Neumann problems of elliptic systems 

have been well-studied by a standard framework; see, e.g., <cit.>. In particular, it has been established for both the Dirichlet and Neumann problems that

    u_ - u_0_L^2(Ω)≤ Cu_0_H^2(Ω),

for any dimension d≥ 2 and any bounded C^1,1 domains. However, the convergence rates for the Robin problem (<ref>) is quite different due to the oscillating factor b(x/) on the boundary which essentially requires an additional strong geometric assumption on ∂Ω in order to carry out a quantitative analysis. In this paper, we will try to obtain the best possible convergence rates under the strongest geometric assumption that is commonly used in the quantitative analysis of the oscillating boundary layers <cit.>, namely, Ω is smooth and strictly convex (in the sense that all the principle curvatures of ∂Ω are positive).

The following is our main result.

	Let (A,b) satisfy (<ref>) – (<ref>). In addition, assume b ∈ C^∞(𝕋^d;ℝ^m× m) and Ω is a smooth and strictly convex domain. Let u_ and u_0 be the same as before and u_0 be sufficiently smooth. Then, for any σ∈ (0,1), there exists C>0 such that
	
    u_ - u_0_L^2(Ω)≤{   C^7/8 - σ( u_0_W^1,∞(Ω) + u_0_H^2(Ω))        if  d =3, 
       C^1-σ( u_0_W^1,∞(Ω) + u_0_H^2(Ω))         if  d = 4,
       C^3/4 + 3/4(d-1) - σu_0_W^2,d/2(Ω)       if  d ≥ 5.
    	.




It is crucial to point out that, in the above theorem, the convergence rate for d=4 is nearly sharp since σ>0 can be arbitrarily small; but the rates are possibly not nearly sharp for other dimensions, especially for d≥ 5. The intrinsic reasons for the appearance of the dimension-dependent exponents in (<ref>) will be explained later. Nevertheless, we provide an effective framework in this paper that allows further improvements. On the other hand, if we consider the estimates of u_ - u_0_L^p(Ω) with some p∈ (1,2) depending on d, then the almost sharp convergence rate O(^1-σ) may still be obtained for all dimensions; see Remark <ref>. Finally, we mention that, to be simple and more concentrated, we will not try to optimize the norms of u_0 on the right-hand side of (<ref>).

The proof of Theorem <ref> follows a line as the usual Dirichlet or Neumann problems (see, e.g., <cit.>), but relies essentially on the quantitative estimates of certain oscillatory integrals. Let w_ = u_ - u_0 - χ(x/) S_(  η_∇ u_0) be the error of the first-order approximation of u_ (which is the same as Dirichlet or Neumann problems). Then a direct computation shows that

    ∫_Ω A_∇ w_·∇φ + ∫_∂Ω b_ w_·φ   = ∫_∂Ω (b^αβ - b^αβ(x/)) u_0^β·φ^α dσ
            + other familiar integrals over Ω,

where φ∈ H^1(Ω;) is a test function. The key to (<ref>) is the quantitative estimate of the oscillatory integral on ∂Ω in the following form

    ∫_∂Ω f(x/)·ϕ(x) dσ,

where f(y) is a 1-periodic vector-value function with zero mean. At first sight, one may expect to employ the oscillatory integral theory to analyze the quantitative behavior of (<ref>) as → 0, which is feasible if ϕ is smooth enough. However, in our application, ϕ = u_0 φ is only supposed to be in certain Sobolev spaces that a direct using of oscillatory integral theory will only give rough estimates. To establish the convergence rates as better as possible, in this paper, we will develop a “duality approach” to analyze (<ref>) via an auxiliary Neumann problem. Precisely, let v_ be the solution of the following Neumann problem

    {
    -div(A∇ v_)     =0       in    Ω,
    
    n·A∇ v_    =f(x/) - M_      on    ∂Ω.
    .

where A is the (constant) homogenized matrix and M_ is a constant vector such that the compatibility condition is satisfied. Note that (<ref>) may be formally viewed as a simplified version of the Robin problem (<ref>). By the integration by parts, it is not hard to see that the estimate of (<ref>) is reduced to the estimates of v_; see (<ref>). Unlike using the oscillatory integral theory to (<ref>) directly, our method completely get rid of the difficulty caused by the lower regularity of ϕ.


The Neumann problem (<ref>), of independent interest itself, has been studied in <cit.> by H. Aleksanyan, H. Shahgholian and P. Sjölin. In particular, they obtained the estimates of v__L^1(Ω) and ∇ v__L^1(Ω) for system (<ref>); see Theorem <ref>. To obtain better estimates of v__L^p(Ω) with general p>1, which are critical in our application, we establish a new estimate of v__L^∞(Ω) in this paper, namely,

    v__L^∞(Ω)≤ C^1/2.

This estimate is proved by using the integral representation for (<ref>) and taking advantage of the lower singularity of the Neumann function (compared to Poisson kernel). 

Then, combing these estimates, we conclude by interpolation that (see Theorem <ref>)

    v__L^2(Ω)≤{   C^7/8- σ       if  d =3, 
       C       if  d= 4,
       C^5/4 |ln|^1/2       if  d≥ 5.
    	.

Note that the estimates above are worse for lower dimensions due to the nature of oscillatory integrals with non-degenerate phases. In particular, we point out that the rate ^7/8-σ for d=3 follows naturally by an interpolation of ∇ v__L^1(Ω)≤ C^1-σ and (<ref>). This explains why the convergence rate for d=3 in Theorem <ref> cannot exceed ^7/8-σ.

On the other hand, the estimate of the integral on the right-hand side of (<ref>) definitely depends on the regularity of u_0 and φ. Under the assumption u_0∈ H^2(Ω;), we have

    |∫_∂Ω (b - b(x/) -M_) u_0 ·φ dσ| ≤{   C^2/3- σu_0_H^2(Ω)φ_H^1(Ω)     if  d =3, 
       C^1/2-σu_0_H^2(Ω)φ_H^1(Ω)     if  d = 4,
       C^1/4 + 3/4(d-1) - σu_0_H^2(Ω)φ_H^1(Ω)     if  d ≥ 5.
    .

See Lemma <ref> (ii) for details. The above estimates are worse (than ^1/2) for d≥ 5, because u_0∈ H^2(Ω;) actually implies worse integrability of u_0 and ∇ u_0 if d is larger; see Remark <ref>. Consequently, by a duality argument, the L^2 convergence rates may be raised to ^3/4 + 3/4(d-1) - σ for d≥ 5 as shown in Theorem <ref>.

Finally, we would like to emphasize that even though the estimates of v__L^1(Ω) and ∇ v__L^1(Ω) in Theorem <ref>, as well as (<ref>), are possibly optimal in view of the singularity of the Neumann function, it is still open whether (<ref>) and (<ref>) are optimal, which are very basic questions related to singular oscillatory integrals (see (<ref>)) and of independent interest. Clearly, through our framework, any further improvement of (<ref>) for d = 3 or (<ref>) for d≥ 5 will lead to an corresponding improvement for Theorem <ref>.



 §.§ Organization of the paper

We prove a Weyl's equidistribution theorem and Theorem <ref> in Section 2 and 3, respectively. In Section 4, we obtain the quantitative estimates of the auxiliary Neumann system (<ref>). In Section 5, we prove Theorem <ref>.




§ AN EQUIDISTRIBUTION THEOREM

This section is devoted to the proof of a (qualitative) version of the Weyl's equidistribution theorem under the non-resonance condition.

	Let S be a closed Lipschitz surface satisfing the non-resonance condition with respect to ℤ^d. Assume f(x,y) : S×𝕋^d ↦ℝ is 1-periodic and continuous in y for each x∈ S. Moreover, assume
	
    ∫_Sf(x,·)_C(𝕋^d) dσ(x) < ∞.

	Then
	
    lim_λ→∞∫_S f(x,λ x) dσ(x) = ∫_S∫_𝕋^d  f(x,y) dy dσ(x).



For a given unit vector ξ∈𝕊^d-1, denote by H_ξ : = {x∈: ξ· x = 0 } the hyperplane perpendicular to ξ.


	Let ξ∈𝕊^d-1 be irrational. If g ∈ C(𝕋^d) is a 1-periodic continuous function and f ∈ L^1(H_ξ). Then
	
    lim_λ→∞sup_y∈| ∫_H_ξ g(y+ λ x) f(x) dσ(x) - g∫_H_ξ f(x) dσ(x) | = 0.



	First of all, it follows from the classical ergodic property of quasi-periodic functions that
	
    lim_λ→∞∫_H_ξ g(λ x) f(x) dσ(x) = g∫_H_ξ f(x) dσ(x),

	for 1-periodic g∈ C^∞(𝕋^d) and f ∈ C_0^∞(H_ξ), where g = ∫_𝕋^d g(y)dy. Then the general case with g ∈ C(𝕋^d) and f ∈ L^1(H_ξ) follows by an approximation argument. Now, we prove (<ref>) by contradiction. If (<ref>) is not true, since g is 1-periodic, there exist δ>0, and sequences of y_k∈𝕋^d and λ_k ∈ℝ such that lim_k→∞λ_k = ∞ and
	
    | ∫_H_ξ g(y_k+ λ_k x) f(x) dσ(x) - g∫_H_ξ f(x) dσ(x) | > δ,   for any  k∈ℕ.

	Now, by the compactness of { g(y+·): y∈𝕋^d } in C(𝕋^d), we may choose a subsequence of {k_ℓ}⊂ℕ such that y_k_ℓ→ z ∈𝕋^d and g(y_k_ℓ+·) → g(z+·) in C(𝕋^d), as k_ℓ→∞. Observe that
	
    | ∫_H_ξ g(y_k_ℓ+ λ_k_ℓ x) f(x) dσ(x) - g∫_H_ξ f(x) dσ(x) | 
         ≤| ∫_H_ξ [ g(y_k_ℓ+ λ_k_ℓ x) - g(z+λ_k_ℓ x) ] f(x) dσ(x) | 
            + | ∫_H_ξ  g(z+λ_k_ℓ x) f(x) dσ(x) - g∫_H_ξ f(x) dσ(x) |.

	Clearly, the first integral in the right-hand side of (<ref>) is bounded by g(y_k_ℓ + ·) - g(z + ·)_C(𝕋^d)f_L^1(H_ξ), which tends to zero as k_ℓ→∞. Also, (<ref>) implies that the second integral in the right-hand side of (<ref>) tends to zero as k_ℓ→∞. It follows that
	
    | ∫_H_ξ g(y_k_ℓ+ λ_k_ℓ x) f(x) dσ(x) - g∫_H_ξ f(x) dσ(x) | → 0,   as  k_ℓ→∞.

	This contracts with (<ref>) and hence proves (<ref>).

	
Now, we may use Lemma <ref> to prove a simplified version of Theorem <ref>.

	Let S be a closed Lipschitz surface and satisfy the non-resonance condition with respect to ℤ^d. Then if g∈ C(𝕋^d) and f∈ L^1(S),
	
    lim_λ→∞∫_S g(λ x) f(x) dσ(x) = g∫_S f(x) dσ(x).




	Step 1: Reduction.
	First of all, we may assume g and f are smooth. The general case follows by an approximation argument. By a partition of unity, we may restrict ourself to a local coordinate system (parallel to the original coordinate system so that the non-resonance condition is still valid with respect to ℤ^d) such that for some x_0 = (x_0',x_0,d)∈ S,  B(x_0,R_0) ∩ S is the graph of
	
    x_d = ϕ(x')   for  x'∈ Q(x_0',R_0) ⊂ℝ^d-1,

	where Q(x'_0,R_0) is a (d-1)-dimensional cube parallel to coordinates and centered at x_0' with radius R_0. Put Q_0 = Q(x'_0,R_0). Without loss of generality, we assume |Q_0| = 1. Thus, it suffices to concentrate on
	
    ∫_{(x',ϕ(x')): x' ∈ Q_0  } g(λ x) f(x) dσ(x) 
        = ∫_Q_0 g(λ(x',ϕ(x'))) f(x',ϕ(x')) √(1+|∇ϕ(x')|^2 ) dx'

	We may reset f(x',ϕ(x')) √(1+|∇ϕ(x')|^2 ) to be f_0(x'), and assume again that f_0 is smooth. Then, we only need to show
	
    lim_λ→∞∫_Q_0 g(λ(x',ϕ(x'))) f_0(x') dx' = g∫_Q_0 f_0(x') dx'.

	Or, more specifically, we would like to show that for any given η>0, there exists λ_η >0 such that for any λ> λ_η,
	
    | ∫_Q_0 g(λ(x',ϕ(x'))) f_0(x') dx' - g∫_Q_0 f_0(x') dx'| ≤η.

	
	Step 2: Ruling out bad points.
	By our assumption, ϕ(x') is a Lipschitz function and thus ∇ϕ(x') is well-defined for almost every x'∈ Q_0. Let E ⊂ Q_0 be the set that ∇ϕ is not defined. Now, for x'∈ Q_0∖ E, define
	
    J(x',r) = sup_|y'-x'|≤ r|ϕ(y') - ϕ(x') - ∇ϕ(x') · (y'-x')|/|y'-x'|

	Since ϕ(x') is differentiable at x', then
	
    lim_r→ 0 J(x',r) = 0,   for any  x'∈ Q_0∖ E.

	Now, by the Egoroff's theorem, for a given δ>0 (to be determined), there exists a measurable subset F ⊂ Q_0, E⊂ F such that |F| < δ/2 and
	
    J(x',r)  converges to 0 uniformly for any  x'∈ Q_0∖ F.
	
	Define
	
    ω_1(r) = sup_x'∈ Q_0∖ F J(x',r).

	Then, ω_1(r) is a increasing function and
	
    lim_r→ 0ω_1(r) = 0.


	
	
	Next, we are going to use the non-resonance condition. Let n(x') be the unit vector normal vector of the graph x_d = ϕ(x') at (x',x_d), if there exists. Let
	
    G = {(x',x_d)∈ S: n(x')  is well-defined and  n(x') ∈ℝℤ^d }.

	Clearly, by the non-resonance condition with respect to ℤ^d, |G| = 0. Moreover, for x'∈ Q_0 ∖ G, we know
	
    n(x') = (∇ϕ(x'), -1)/√(1+|∇ϕ(x')|^2)  is irrational.

	Thus, we may define a good subset of Q_0,
	
    U = Q_0∖ (F ∪ G).

	
	Step 3: Decomposition.
	We construct a family of dyadic cubes in Q_0. Let {Q_k^j: j =1,2,3,⋯, 2^dk} be a collection of dyadic cubes at level k. Put
	
    P_k = { Q_k^j: U ∩ Q_k^j ≠∅}.

	
	Before we proceed, we claim that, there exists a decreasing step function ρ(λ), taking discrete values in { 2^-k: k∈ℕ}, such that ρ(λ) → 0 as λ→∞. Moreover,	
	
    lim_λ→∞λρ(λ) ω_1( √(d)ρ(λ) ) = 0   and  lim_λ→∞λρ(λ) = ∞.

	
	The existence of such ρ(λ) follows from a concrete construction. Actually, let
	
    ρ(λ):= sup{ 2^-k: k∈ℕ, λ 2^-k√(ω_1(√(d) 2^-k))≤ 1 }.

	It is not hard to see ρ(λ) is a well-defined decreasing step function, since r √(ω_1(√(d) r)), as a function of r, is increasing. Moreover, ρ(λ) → 0 as λ→∞. Now, note that (<ref>) implies
	
    λρ(λ) √(ω_1(√(d)ρ(λ)))≤ 1.

	It follows
	
    λρ(λ) ω_1( √(d)ρ(λ) ) ≤√(ω_1(√(d)ρ(λ)))→ 0,   as λ→∞.

	
	To see the second part of (<ref>), by the definition (<ref>), we know
	
    λ 2 ρ(λ) √(ω_1(√(d) 2 ρ(λ))) > 1.

	which yields
	
    λρ(λ) > 1/2 √(ω_1(√(d)2 ρ(λ)))→∞,   as λ→∞.

	This proves the claim as desired.
	
	Now, given any λ≥ 1, let ρ(λ) be the decreasing step function as above such that ρ(λ) = 2^-k(λ), where k(λ) ∈ℕ. For x'∈ U,
	
    Q(x',λ) = the dyadic cube in  P_k(λ) containing  x'.

	Note that ρ(λ) is the side length of Q(x',λ). Then, for each λ, U may be covered by a sequence of dyadic cubes Q(x',λ) with x'∈ U, at level k(λ).
	
	Step 4: Local approximation.
	With ρ(λ) and Q(x',λ) as constructed above, we claim that for any x'∈ U,
	
    lim_λ→ 0| _Q(x',λ) g(λ(y',ϕ(y'))) f_0(y') dy' - g_Q(x',λ) f_0(y') dy' | = 0 ,

	
	To prove (<ref>), we temporarily fix x' and λ and consider
	
    K(x',λ) := _Q(x',λ) g(λ(y',ϕ(y'))) f_0(y') dy'.

	By the Taylor's expansion at x', for y'∈ Q(x',λ)
	
    ϕ(y') = ϕ(x') + ∇ϕ(x')(y'-x') + R(y',x')

	where, by the fact x'∈ Q_0∖ F and the definition of ω_1, the remainder satisfies
	
    |R(y',x')| ≤ |y'-x'|ω_1(|y'-x'|).

	Thus, by the smoothness of g, we have
	
    |g(λ(y',ϕ(y'))) - g(λ (y', ϕ(x') + ∇ϕ(x')· (y'-x') ))  |    ≤  L_g|y'-x'| ω_1 (|y'-x'|) 
       ≤ L_g √(d)ρ(λ) ω_1( √(d)ρ(λ) ),

	where L_g = sup |∇ g|. Substituting (<ref>) and (<ref>) into (<ref>), we have
	
    |K(x',λ) - _Q(x',λ) g(λ(y',ϕ(x') + ∇ϕ(x')(y'-x'))) f_0(x') dy' | 
       ≤ L_g M_f √(d)λρ(λ) ω_1( √(d)ρ(λ) ) + M_g L_f √(d)ρ(λ),

	where M_f = sup |f_0|, M_g = sup |g| and L_f = sup|∇ f_0|. Observe that the above error is independent of x'. Thus, (<ref>) tends to zero as λ→∞ by the construction of ρ(λ) and the first part of (<ref>).
	
	Next, we consider
	
    I_1(x',λ)     := f_0(x') _Q(x',λ) g(λ(y',ϕ(x') + ∇ϕ(x')· (y'-x')))  dy' 
        = f_0(x') _Q(x',λ) g(λ((x',ϕ(x')) + ( y'-x', ∇ϕ(x')· (y'-x')))  dy'.

	By a change of variable w = (w',w_d) such that
	
    w' = y'-x'/ρ(λ)  and   w_d = ∇ϕ(x')· w',

	we see
	
    I_1(x',λ) = f_0(x') _Q(x',λ) g(λ (x',ϕ(x')) + λρ(λ) w)  dσ(w)

	where Q(x',λ), the image of Q(x',λ) under w, is some parallel polyhedron on the irrational hyperplane (perpendicular to the irrational direction (∇ϕ (x'), -1), since x'∈ U). The position of Q(x',λ) may depends on λ; but its shape and size, depending only on ∇ϕ(x'), are fixed. Moreover, |Q(x')| = (√(1+|∇ϕ(x')|^2))^-1. Since we have λρ(λ) →∞ as λ→∞, by Lemma <ref>,
	
    lim_λ→∞ |I_1(x',λ) - g f_0(x')| = 0.

	
	On the other hand, it is clear that
	
    lim_λ→ 0|g f_0(x') - g_Q(x',λ) f_0(y') dy' | = 0,

	by the smoothness of f_0.
	
	Finally, combining (<ref>), (<ref>) and (<ref>), we obtain (<ref>)
	
	Step 5: Completing the proof. Define
	
    T(x',λ) := | _Q(x',λ) g(λ(y',ϕ(y'))) f_0(y') dy' - g_Q(x',λ) f_0(y') dy' |

	It has been prove in Step 4 that
	
    lim_λ→∞ T(x',λ) = 0,   for any  x'∈ U.

	By the Egoroff's theorem, for a given δ>0, there exists H ⊂ U such that |H| ≤δ/2 and
	
    ω_2(λ): = sup_x'∈ V T(x',λ) → 0   as λ→ 0,

	where V = U∖ H. Note that |Q_0∖ V| ≤δ.
	
	Now given η>0, choose δ>0 small enough so that
	
    δ < η/4 M_f M_g.

	By (<ref>), we can choose λ_η large enough such that
	
    T(x',λ)< η/2     for any  x'∈ V  and λ>λ_η.

	For any given λ>λ_η, there exists a collection of Q(x',λ), x'∈ V such that
	
    V ⊂⋃_x'∈ V Q(x',λ).

	Therefore, we can find finite Q(x'_j,λ) with 1≤ j ≤ N_0 and x'_j∈ V, such that
	
    V ⊂ W:=⋃_1≤ j≤ N_0 Q(x'_j,λ)   and   Q(x'_i,λ)∩ Q(x'_j,λ) = ∅,  if  i≠ j.

	Obviously,
	
    |Q_0 ∖ W| ≤ |Q_0 ∖ V| ≤δ.

	
	It  follows that
	
    | ∫_Q_0 g(λ(x',ϕ(x'))) f_0(x') dx' - g∫_Q_0 f_0(x') dx'| 
       ≤| ∫_Q_0∖ W g(λ(x',ϕ(x'))) f_0(x') dx' - g∫_Q_0 ∖ W f_0(x') dx'| 
            + ∑_j=1^N_0| ∫_Q(x'_j,λ) g(λ(x',ϕ(x'))) f_0(x') dx' - g∫_Q(x'_j,λ) f_0(x') dx'| 
       ≤ 2 M_f M_g |Q_0∖ W| +  ∑_j=1^N_0|Q(x'_j,λ)| T(x'_j,λ)
       ≤ 2δ M_f M_g + η |W|/2
       ≤η,

	where we have used (<ref>), (<ref>) and (<ref>) in the third inequality and used (<ref>) in the last inequality. This finishes the proof of (<ref>).




	We use an approximation argument. Let ϕ∈ C_0^∞(B_1(0)) such that ∫ϕ =1 and let ϕ_δ(x) = δ^-dϕ(δ^-1 x) with 0<δ<1. Define
	
    f_δ(x,y) = ∫_ f(x,y-z) ϕ_δ(z) dz.

	Then, it is easy to see that
	
    f_δ(x,·)_C^m(𝕋^d)≤ C_m,δf(x,·)_C(𝕋^d),     for any  x∈ S, m∈ℕ,

	and
	
    |f_δ(x,y) - f(x,y)| ≤ Cω(x,δ),     for any  (x,y) ∈ S×𝕋^d,

	where ω(x,δ) is the modulo of the continuity of f(x,·), and C is independent of δ. By the fact ω(x,δ) → 0 as δ→ 0 for each x∈ S, and the dominant convergence theorem, one obtains
	
    lim_δ→ 0∫_Sω(x,δ) dσ(x) = 0.

	Consequently, in view of (<ref>), to see (<ref>), it suffices to show
	
    lim_λ→∞∫_S f_δ(x,λ x) dσ(x) = ∫_S∫_𝕋^d  f_δ(x,y) dy dσ(x).

	
	By Fourier series expansion of f_δ(x,y) in terms of y, we have
	
    f_δ(x,y) = ∑_k∈ℤ^d a_k(x) e^2π i k · y,     where  a_k(x) = ∫_𝕋^d f_δ(x,y) e^2π i k · y dy.

	In view of (<ref>), we have
	
    |a_k(x)| ≤ C_m,δ |k|^-mf(x,·)_C(𝕋^d),     for every  m∈ℕ.

	It follows that
	
    ∫_S f_δ(x,λ x) dσ(x) = ∑_|k| ≤ M∫_S a_k(x) e^2π i λ k· x dσ(x) + ∑_|k| > M∫_S a_k(x) e^2π i λ k· x dσ(x).

	Observe that the second term in the last equality is bounded by
	
    C_d+1,δ∑_|k|> M |k|^-(d+1)∫_Sf(x,·)_C(𝕋^d) dσ(x) ≤C_d+1,δ/M∫_Sf(x,·)_C(𝕋^d) dσ(x).

	On the other hand, by Theorem <ref>, we know
	
    lim_λ→∞∑_|k| ≤ M∫_S a_k(x) e^2π i λ k· x dσ(x) = ∫_S a_0(x) dσ(x) = ∫_S∫_𝕋^d  f_δ(x,y) dy dσ(x).

	As a result,
	
    | lim_λ→∞∫_S f_δ(x,λ x) dσ(x) - ∫_S∫_𝕋^d  f_δ(x,y) dy dσ(x) | ≤C_d+1,δ/M∫_Sf(x,·)_C(𝕋^d) dσ(x).

	Since M may be chosen arbitrarily large, the above estimate implies (<ref>) by letting M→∞. This ends the proof.



Observe that the convergence in (<ref>) may depend on the function f(x,y). However, one may have a fixed convergence rate, provided some compactness on the underlying function spaces. Let 

    L^1(S; C(𝕋^d) ) := { f(x,y):  ∫_Sf(x,·)_C(𝕋^d) dσ(x) < ∞}.





	Let X be a Banach space compactly embedded into L^1(S; C(𝕋^d) ), endowed with norm ·_X. Then there exists a rate function ω(λ), depending only on d,S and X, such that ω(λ) ↓ 0 as λ→∞, and
	
    | ∫_S f(x,λ x) dσ(x) - ∫_S∫_𝕋^d  f(x,y) dy dσ(x) | ≤ω(λ) f_X,

	for any f∈ X.


	Let B_X = {f∈ X: f_X ≤ 1  }. To prove (<ref>), it suffices to show
	
    ω(λ):= sup_f∈ B_X| ∫_S f(x,λ x) dσ(x) - ∫_S∫_𝕋^d  f(x,y) dy dσ(x) | → 0,   as λ→∞.
	
	
	The above statement may be proved by contradiction. Suppose (<ref>) is not true. Then there exist δ>0 and a sequence {λ_k: k∈ℕ} such that lim_k→∞λ_k = ∞ and ω(λ_k) > δ. This implies that there exists a sequence {f_k: k∈ℕ}⊂ B_X such that, for any k>0
	
    |  ∫_S f_k(x,λ_k x) dσ(x) - ∫_S∫_𝕋^d  f_k(x,y) dy dσ(x) | > δ/2.

	By the compactness of B_X ⊂ L^1(S; C(𝕋^d) ), we may choose a subsequence { k_ℓ}⊂ℕ and f_∞∈ L^1(S; C(𝕋^d) ) such that f_k_ℓ→ f_∞ in L^1(S; C(𝕋^d) ), as k_ℓ→∞. Now, observe that
	
    | ∫_S f_k_ℓ(x,λ_k_ℓ x) dσ(x) - ∫_S∫_𝕋^d  f_k_ℓ(x,y) dy dσ(x) | 
       ≤| ∫_S( f_k_ℓ(x,λ_k_ℓ x) - f_∞(x,λ_k_ℓ x)) dσ(x)  | 
            + | ∫_S f_∞(x,λ_k_ℓ x) dσ(x) -  ∫_S∫_𝕋^d  f_∞(x,y) dy dσ(x) | 
            + | ∫_S∫_𝕋^d  f_k_ℓ(x,y) dy dσ(x) - ∫_S∫_𝕋^d  f_∞(x,y) dy dσ(x) |
       ≤| ∫_S f_∞(x,λ_k_ℓ x) dσ(x) -  ∫_S∫_𝕋^d  f_∞(x,y) dy dσ(x) | + 2∫_Sf_k_ℓ(x,·) - f_∞(x,·)_C(𝕋^d) dσ(x).

	As k_ℓ→∞, the first term in the last inequality tends to zero by Theorem <ref>, while the second term tends to zero since { f_k_ℓ} converge to f_∞ in L^1(S; C(𝕋^d) ). This contradicts to (<ref>) and proves (<ref>).


As a straightforward application, we use Theorem <ref> to derive a homogenization theorem for harmonic functions with oscillating boundary data in Lipschitz domains satisfying the non-resonance condition.


	Let Ω be a Lipschitz domain satisfying the non-resonance condition with respect to ℤ^d. Assume  g(x,y) : ∂Ω×𝕋^d ↦ℝ is 1-periodic and continuous in y for each x∈∂Ω. Moreover, assume
	
    ∫_∂Ωg(x,·)_C(𝕋^d)^2 dσ(x) < ∞.

	Let u_ be the solution of
	
    {
    	- Δ u_    = 0       in    Ω,
    
    	u_(x)     =g(x,x/)       on    ∂Ω.
    	.

	Then, u_ converges to u_0 pointwise (and in L^2(Ω)), where u_0 is the solution of
	
    {
    	-Δ u_0     = 0       in    Ω,
    
    	u_0(x)     =g̅(x)       on    ∂Ω,
    	.

	where g̅(x) = ∫_𝕋^d g(x,y) dy.



	First of all, observe that g(x,x/) ∈ L^2(∂Ω). By the L^2 theory for elliptic equations in Lipschitz domains (see, e.g., <cit.> or <cit.>), the Dirichlet problem (<ref>) is solvable and
	
    𝒩(u_)_L^2(∂Ω)≤g(·,·/)_L^2(∂Ω)≤( ∫_∂Ωg(x,·)_C(𝕋^d)^2 dσ(x) )^1/2,

	where 𝒩(u_) is the non-tangential maximal function. Moreover, the harmonic measure ω^z(x), with z∈Ω, is absolutely continuous with respect to σ(x) and
	
    k(·,z):= dω^z/dσ∈ L^2(∂Ω,dσ),

	and the solution u_ may be represented by
	
    u_(z) = ∫_∂Ω g(x,x/) k(x,z) dσ(x),     for  z∈Ω.

	Let f_z(x,y) = g(x,y) k(x,z) and note that by (<ref>) and (<ref>),
	
    ∫_∂Ω f_z(x,·)_C(𝕋^d) dσ(x)     = ∫_∂Ω g(x,·)_C(𝕋^d) k(x,z) dσ(x) 
       ≤( ∫_∂Ω g(x,·)_C(𝕋^d)^2 dσ(x) )^1/2( ∫_∂Ω k^2(x,z) dσ(x) )^1/2
       ≤ C_z

	It follows from Theorem <ref> that
	
    lim_→ 0 u_(z)     = lim_→ 0∫_∂Ω f_z(x,x/) dσ(x) 
        = ∫_∂Ω∫_𝕋^d f_z(x,y) dy dσ(x) 
        = ∫_∂Ω( ∫_𝕋^d g(x,y) dy ) k(x,z) dσ(x) 
        = u_0(z),

	for each z∈Ω, where u_0 is exactly the solution of (<ref>). This gives the pointwise convergence of u_.
	
	Finally, we show that
	
    lim_→ 0u_ - u_0_L^2(Ω) = 0.

	Actually, by the maximal principle, we have
	
    -u̅(z) ≤ u_(z) ≤u̅(z),

	where u̅ is the solution of
	
    {
    	-Δu̅    = 0       in    Ω,
    u̅ (x)     =g(x,·)_C(𝕋^d)      on    ∂Ω.
    	.

	By (<ref>) and the L^2 solvability of (<ref>) in Lipschitz domains (see <cit.>), we know u̅_L^2(Ω) < ∞. Therefore, (<ref>) follows readily from the pointwise convergence (<ref>), uniform estimate (<ref>) and the dominant convergence theorem.







§ QUALITATIVE HOMOGENIZATION




In this section, we will apply Theorem <ref> to establish the qualitative homogenization theorem for the Robin boundary value problem (<ref>) in Lipschitz domains with a non-resonance condition. We begin with the definitions of correctors and the homogenized coefficient matrix <cit.>. For each 1≤ j≤ d, 1≤β≤ m, let χ = (χ_j^β) = (χ_j^1β,χ_j^2β,⋯,χ_j^mβ) denote the correctors for ℒ_, which are 1-periodic functions satisfying the cell problem

    {ℒ_1 (χ^β_j + P_j^β)    = 0      in 𝕋^d, 
    ∫_𝕋^dχ_j^β    = 0,
    .

where P_j^β(x) = x_je^β with e^β being βth Cartesian basis in . Recall that the homogenized matrix A = (â^αβ_ij) is defined by

    â^αβ_ij = ∫_𝕋^d[ a^αβ_ij + a^αγ_ik∂/∂ x_k (χ^γβ_j) ] dx,

and the homogenized operator is given by ℒ_0 = -div(A∇ ).




The following is the main theorem of this section.

	Let {A_ℓ, b_ℓ} be a sequence of coefficient matrices satisfying (<ref>) – (<ref>). Moreover, we assume {b_ℓ} are equicontinuous. Assume Ω is a bounded Lipschitz domain whose boundary satisfies the non-resonance condition with respect to ℤ^d, { F_ℓ}⊂ H^-1(Ω;) and { g_ℓ}⊂ H^1/2(∂Ω;). Suppose that { u_ℓ} are the weak solutions of
	
    {
    	- div(A_ℓ(x/_ℓ) ∇ u_ℓ)    = F_ℓ         in Ω,
    
    	n· A_ℓ(x/_ℓ)∇ u_ℓ+b_ℓ(x/_ℓ)u_ℓ    =g_ℓ         on ∂Ω,
    	.

	where _ℓ→ 0 as ℓ→∞, u_ℓ∈ H^1(Ω;). We further assume, as ℓ→∞,
	
    {
    	F_ℓ   → F          in  H^-1(Ω;),
    
    	g_ℓ   → g         in  H^-1/2(∂Ω;),
    
    	u_ℓ   ⇀ u          weakly in  H^1(Ω;), 
    A_ℓ   → A^0,      
    b_ℓ   → b^0,      .

	where A_ℓ denotes the effective coefficient matrix of A_ℓ and b_ℓ denotes the effective diffusive matrix of b_ℓ. Then
	
    A_ℓ(x/_ℓ)  ∇ u_ℓ⇀ A^0 ∇ u     weakly in  L^2(Ω;ℝ^m× d),

	Moreover, u is a weak solution of
	
    {
    	- div(A^0 ∇ u)    = F          in Ω,
    
    	n· A^0 ∇ u+b^0 u     =g          on ∂Ω.
    	.




	Thanks to the qualitative homogenization for the operator ℒ_ (see <cit.>), our assumptions implies (<ref>) and the equation - div(A^0 ∇ u) = F in Ω. Thus, the key point is to verify the boundary condition. Note that the variational form of (<ref>) gives
	
    ∫_Ω A_ℓ(x/_ℓ) ∇ u_ℓ·∇ϕ + ∫_∂Ω b_ℓ(x/_ℓ) u_ℓ·ϕ = ∫_∂Ω g_ℓ·ϕ + ∫_Ω F_ℓ·ϕ,

	for any ϕ∈ C^∞_0(;ℝ^m). Clearly, by (<ref>), we have
	
    ∫_Ω A_ℓ(x/_ℓ) ∇ u_ℓ·∇ϕ→∫_Ω A^0 ∇ u ·∇ϕ,     as ℓ→∞.

	To verify the variational form of (<ref>), it suffices to show
	
    ∫_∂Ω b_ℓ(x/_ℓ) u_ℓ·ϕ→∫_∂Ω b^0 u ·ϕ.

	
	First of all, by the equicontinuity and periodicity of b_ℓ, as well as the convergence b_ℓ→ b^0, we know sup_ℓ |b_ℓ(y)| ≤ C.
	Since u_ℓ converges to u weakly in H^1(Ω;), u_ℓ converges to u strongly in H^s(Ω;) with 1/2<s<1. By the trace theorem,
	
    u_ℓ - u_L^2(∂Ω)→ 0,  as ℓ→∞.

	
	On the other hand, the Arzelà - Ascoli theorem implies that the set { b_ℓ(y)u(x)·ϕ(x) } is compact in L^1(S; C(𝕋^d) ). Hence, by Theorem <ref>,
	
    | ∫_∂Ω b_ℓ(x/_ℓ) u ·ϕ - ∫_∂Ωb_ℓ u ·ϕ| ≤ω(_ℓ),

	where ω(_ℓ) → 0 as ℓ→∞, and this rate depends at most on Ω, u·ϕ_H^1(Ω) and the modulus of equicontinuity of {b_ℓ}. Consequently,
	
    | ∫_∂Ω b_ℓ(x/_ℓ) u_·ϕ - ∫_∂Ω b^0 u ·ϕ| 
           ≤ Cu_ℓ - u_L^2(∂Ω)ϕ_L^2(∂Ω) + ω(_ℓ) + |b_ℓ - b^0| u_L^2(∂Ω)ϕ_L^2(∂Ω),

	where the right-hand side converges to 0 as ℓ→∞. This proves (<ref>) and hence
	
    ∫_Ω A^0 ∇ u ·∇ϕ + ∫_∂Ω b^0 u ·ϕ = ∫_∂Ω g·ϕ + ∫_Ω f·ϕ,

	which ends the proof.



	This is a simple corollary of Theorem <ref>.





Many materials in practice may have special microscopic structures beyond periodicity, such as layered materials (lamina) and directional materials (fiber, wood). These additional stronger structures may allows us to weaken the non-resonance condition that is indispensable for homogenization to take place on the boundary. Consider a material with a certain physical property,  described by a function B, in a fixed coordinate system. Assume that B depends only on k orthogonal directions p_1, p_2,⋯, p_k, and remains constant along the rest orthogonal directions p_k+1, ⋯, p_d. Let P = (p_1, p_2,⋯, p_k) be a d× k matrix. Then, the previous assumption on B is equivalent to the following structure equation

    B(x) = B(PP^T x),     for any  x∈.

In particular, the layered and directional materials mentioned above are corresponding to the special cases k = 1 and k=d-1, respectively. We identify the structure equation for these two interesting cases:

    Layered materials:      B(x) = B((p_1⊗ p_1) x);
    Directional materials:      B(x) = B((I - p_d⊗ p_d)x).

Now, for such a material with structure (<ref>), the periodicity condition for B will be imposed only on the linear subspace spanned by {p_1, p_2,⋯, p_k } (which is equivalent to PP^T). As a result, we may redefine a weaker non-resonance condition as follows.







	Let Γ⊂ PP^T be a periodic lattice. We say a closed Lipschitz surface S satisfies the non-resonance condition with respect to Γ if
	
    σ( {x∈ S: n(x)  is well-defined and  n(x) ∈ℝΓ} ) = 0.


Definition <ref> is weaker than Definition <ref> since we only need to verify (<ref>) for “much less" directions. For example, we only need to verify a single direction p_1 (see (<ref>)) for a given layered material. Therefore, by a similar argument as before, the homogenization theorem may be established on a larger class of domains for directional or layered materials. The details will be omitted.





§ AUXILIARY NEUMANN PROBLEMS


This section and next one will be devoted to the quantitative convergence rates. As we have noticed, the main difficulty in Robin boundary value problems is the analysis of the integral in a form of

    ∫_∂Ω f(x/)·ϕ(x) dσ(x),

where f(y) is a 1-periodic -valued function. In this paper, we will use a “duality approach” to analyze (<ref>) quantitatively. Precisely, let v_ be the solution of the following auxiliary Neumann problem

    {
    -div(A∇ v_)     =0       in    Ω,
    
    n·A∇ v_    =f(x/) - M_      on    ∂Ω.
    .

where A is the (constant) homogenized matrix and M_ is a constant vector such that the compatibility condition is satisfied, i.e.,

    ∫_∂Ω( f(x/) - M_)dσ(x) = 0.

Now, observing that by (<ref>) and the integration by parts, one has

    ∫_∂Ω f(x/)·ϕ(x) dσ(x)    = ∫_ΩA∇ v_(x) ·∇ϕ(x) dx + M_∫_∂Ωϕ(x)dσ(x)
        = - ∫_Ω v_(x) ·div(A^*∇ϕ(x)) dx + ∫_∂Ω v_(x) (n·A^* ∇ϕ(x))dσ(x) 
           + M_∫_∂Ωϕ(x)dσ(x).

Thus, the estimate of (<ref>) is effectively reduced to the estimates of v_ or ∇ v_.

In this section, we focus on the estimates of v_ given by (<ref>). Actually, (<ref>) has been studied in <cit.> and we include their results (adapted to our situation) in the following theorem.

	Let Ω be strictly convex and smooth. If v_ is the solution of (<ref>) satisfying ∫_∂Ω v_ dσ = 0, then
	
    v__L^p(Ω)≤{   C^1/p       if  d =3, 
       C^3/2p       if  d=4,
       C^2/p |ln|^1/p       if  d≥ 5.
    	.

	and
	
    ∇ v__L^p(Ω)≤ C_σ^1/p - σ ,      d≥ 3

	for any 1≤ p<∞ and σ>0.



	The proof of Theorem <ref> is based on the integral representation for the solution and the estimates of oscillatory integrals on the boundary.
	Precisely, the solution of (<ref>) may be given by
	
    v_(x) = ∫_∂Ω N_0(x,y)(f(y/) - M_) dσ(y).

	where N_0(x,y) is the Neumann function of -div(A∇) in Ω. Recall that the Neumann function N_0(x,y) = (N_0^αβ(x,y)) is a matrix such that <cit.>
	
    {   ℒ_0(N^β(·, y)) = e^βδ_y        in Ω,
       ∂/∂ν_0(N_0(·,y) ) = -e^β |∂Ω|^-1       on ∂Ω,
       ∫_∂Ω N_0(x,y) dσ(x) = 0   ,
    	.

	where N^β_0 = (N_0^1β, N_0^2β,⋯, N_0^mβ) and e^β = (0,⋯, 1,⋯, 0) is the βth Cartesian basis in . Moreover, since Ω is smooth, we have
	
    |∇_x^i ∇_y^j N_0(x,y)| ≤C_i,j/|x-y|^d-2+i+j.

	for any i,j≥ 0.
	

	If x is an interior point of Ω, the oscillatory integral theory for non-degenerate phase implies |v_(x)| ≤ C^d-1/2 (see <cit.>), which shows that the estimate in (<ref>) should be sharp for p = 1 and 3≤ d≤ 5. On the other hand, however, if p>1, the estimates in (<ref>) do not give the sharp convergence rates. In particular, (<ref>) and a Sobolev inequality lead to
	
    v__L^p(Ω)≤ C^1/p+1/d - σ,

	for any d/d-1≤ p≤∞ and any σ>0. Compared to (<ref>), this is a better convergence rate for lower dimensions or larger p. In the most interesting case with p = 2, combining (<ref>) (for d≥ 4) and (<ref>) (for d=3), we arrive at
	
    v__L^2(Ω)≤{   C^5/6-σ       if  d =3, 
       C^3/4       if  d=4,
       C |ln|^1/2       if  d≥ 5.
    	.



In the following theorem, we make a key improvement on the estimate of v_.


	Let Ω and v_ be the same as Theorem <ref>. Then, for d≥ 3,
	
    v__L^∞(Ω)≤ C^1/2.



	The proof follows the idea of <cit.> and relies on the integral representation (<ref>). By subtracting a constant, we may simply assume the mean of f is zero. Also, since f is 1-periodic and smooth, by the Fourier series expansion, we may first consider f(x) = g_k e^2π i k· x with k∈ℤ^d∖{ 0 } and f_k∈ is a constant vector. In this case
	
    M_ = f_k/|∂Ω|∫_∂Ω e^2π i^-1 k· x dσ(x).

	Since Ω is smooth and strictly convex, |M_| ≤ C|f_k|^d-1/2; see <cit.>. 
	
	Hence, (<ref>) gives
	
    v_(x)     = ∫_∂Ω N_0(x,y)f_k ( e^2π i ^-1 k· y - M_) dσ(y) 
        = ∫_∂Ω N_0(x,y)f_k e^2π i ^-1 k· y  dσ(y) + O(|f_k|^d-1/2).

	To obtain the pointwise estimate of the integral in (<ref>), it suffices to consider a typical case that x∈∂Ω. Fix such an x∈∂Ω. Let η be a smooth cut-off function such that η(y) = 1 for y∈ B_r_0(x) and η(y) = 0 for y∉ B_2r_0(x), where r_0>0 is an appropriately chosen radius (depending only on d and Ω) that ∂Ω may be localized near x. Moreover, |∇^ℓη| ≤ C_ℓ. To deal with the singularity of the Neumann function at x, we introduce another cut-off function to rule out a small neighborhood of x. Let θ_(y) = 0 in B_^1/2(x) and θ_ (y) = 1 in ∖ B_5^1/2(x), and |∇^ℓθ_| ≤ C_ℓ^-ℓ/2. Consequently, we have
	
    ∫_∂Ω N_0(x,y) f_k e^2π i ^-1 k· y  dσ(y)    = ∫_∂Ωη(y) θ_(y)  N_0(x,y) f_k e^2π i ^-1 k· y  dσ(y) 
            + ∫_∂Ω (1 - η(y))  N_0(x,y) f_k e^2π i ^-1 k· y  dσ(y) 
            + ∫_∂Ω (1 - θ_(y))  N_0(x,y) f_k e^2π i ^-1 k· y  dσ(y) 
        = R_1 + R_2 + R_3.

	Note that (1 - η(y))  N_0(x,y) has no singularity on the boundary and therefore, |R_2| ≤ C|f_k| ^d-1/2. To estimate R_3, note that 1-θ_ is supported in B_5^1/2(x) and thus
	
    |R_3| ≤ |f_k| ∫_∂Ω∩ B_5^1/2(x) |N_0(x,y)| dσ(y) ≤ C|f_k| ^1/2,

	where we have used (<ref>) with i=j=0 in the last inequality. Hence, it suffices to estimate R_1. To do so, we first transform the surface integral to the usual one in ℝ^d-1. Precisely, we assume z = Q^t(y-x) moves x∈∂Ω to the origin and transform the tangent plane at x to z_d = 0, where Q∈ℝ^d× d is an orthogonal matrix. As a result, ∂Ω∩ B(x,2r_0) is transformed to the local graph z_d = ϕ(z') which satisfies ϕ(0) = 0 and ∇ϕ(0) = 0. It follows that
	
    R_1    = ∫_{ |z'|< 2r_0}η(x+Qz) θ_(x+Qz) N_0(x,x+Qz) f_k e^2π i ^-1 k· (x+Qz)√(1+|∇ϕ(z')|^2) dz' 
        = K_x ∫_{ |z'|< 2r_0}η̃(z) θ̃_(z) N_0(z)f_k  e^2π i ^-1 Q^t k· z dz',

	where K_x = e^2π i ^-1 k· x, η̃(z) =η(x+Qz) √(1+|∇ϕ(z')|^2), θ̃_(z) = θ_(x+Qz) and N_0(z) = N_0(x,x+Qz). By our construction, one knows η̃(z) is a smooth cut-off function supported in B_2r_0(0); θ̃_(z) is a smooth cut-off function vanishing in B_^1/2(0) and |∇^ℓθ̃_| ≤ C^-ℓ/2. Moreover, (<ref>) implies
	
    |∇^ℓN_0(z)| ≤C/|z'|^d-2+ℓ,     ℓ≥ 0.

	
	Now, let n = (n',n_d) = Q^t k/|Q^t k|. Using z_d = ϕ(z'), we have
	
    R_1 = K_x ∫_{ |z'|< 2r_0}η̃(z) θ̃_(z) N_0(z)f_k  e^2π i ^-1|k| (n'· z' + n_d ϕ(z')) dz'.

	Let F(z') = n'· z' + n_d ϕ(z'). We need to discuss two cases separately.
	
	Case 1: |n'|>C_0|n_d| for some C_0>0. In this case, there exists n_j with some 1≤ j≤ d-1 so that |n_j|≥ |n'|/(d-1) > C_0|n_d|/(d-1). Thus, if C_0 is large enough,
	
    |∂/∂ z_j F(z')| = |n_j + n_d ∂/∂ z_jϕ(z') | ≥|n_j|/2≥1/4(d-1).

	for any z' with |z'|< 2r_0. Then, by an integration by parts, (<ref>) gives
	
    R_1  = -K_x ∫_{ |z'|< 2r_0}∂/∂ z_j[ η̃(z) θ̃_(z) N_0(z)f_k ( 2π i ^-1|k| ∂/∂ z_j F(z') )^-1] e^2π i ^-1|k| F(z')  dz'.

	It follows from (<ref>) and (<ref>) that
	
    |R_1|    ≤C|f_k| /|k|∫_{^1/2 < |z'| < 2r_0} |N_0(z)|dz' + C|f_k|^1/2/|k|∫_{^1/2 < |z'| < 5^1/2} |N_0(z)|dz' 
            + C|f_k|/|k|∫_{^1/2 < |z'| < 2r_0} |∇N_0(z)|dz' 
       ≤C|f_k|· |ln|/|k|,

	which gives a desired bound for the first case.
	
	Case 2: |n'|≤ C_0 |n_d| for some C_0. Note that this implies |n_d| ≥ 1/√(1+C_0^2). It turns out that ∇^2 F(z') = n_d ∇^2 ϕ(z') is non-degenerate in { |z'| < 2r_0 }. Therefore, there exists at most one point w' ∈{ |z'|<2r_0 } such that ∇ F(w') = 0. Without loss of generality, we may assume ∇^2 F(w') is diagonal with a minimum eigenvalue λ_0>0 depending only on the domain Ω (This can be done by making a rotation for z'). It follows that there exists c_0>0 such that if |z' - w'|<c_0,
	
    ∂^2 F(z')/∂z_j^2≥99/100λ_0,     ∂^2 F(z')/∂ z_i ∂ z_j ≤1/100dλ_0,     if  i≠ j.

	This implies that
	
    |∂ F(z')/∂ z_j| ≥ c|z_j|,

	if |z'|<2r_0 and z'-w'∈𝒞_j, where 𝒞_j is a cone defined by
	
    𝒞_j: = {z'∈ℝ^d-1: |z_j|> 1/2√(d-1) |z|  }.

	For each j = 1,2,⋯, d-1, there exists smooth ρ_j supported in 𝒞_j ∖ B_^1/2(0) such that
	
    ∑_j = 1^d-1ρ_j(z') = 1,    for any  z'∈ℝ^d-1∖ B_2 ^1/2(0),

	and |∇^ℓρ_j(z')|≤ C^-ℓ/2. Let ρ_0 = 1 - ∑_j=1^d-1ρ_j.
	
	To proceed, we write
	
    R_1 = ∑_j =0^d-1 K_x ∫_{ |z'|< 2r_0}ρ_j(z'-w') η̃(z) θ̃_(z) N_0(z)f_k  e^2π i ^-1|k| F(z') dz'.

	Let R_1^j be the integral above with j≥ 0. Since ρ_0(z'-w') is supported in B_2^1/2(w'), it is easy to see
	
    |R_1^0| ≤ C|f_k| ∫_B_2^1/2(w') |N_0(z)|dz' ≤ C|f_k| ^1/2

	For 1≤ j ≤ d-1, applying the integration by parts twice, one has
	
    R_1^j 
    	=    K_x (2π i ^-1|k| )^-2∫_{ |z'|< 2r_0} e^2π i ^-1|k| F(z')
           ×∂/∂ z_j[ ( ∂ F(z')/∂ z_j)^-1∂/∂ z_j[ ( ∂ F(z')/∂ z_j)^-1ρ_j(z'-w') η̃(z) θ̃_(z) N_0(z)f_k ]]   dz'

	It follows from (<ref>) and (<ref>) that
	
    |R_1^j|     ≤C|f_k| ^2/|k|^2∫_{^1/2≤ |z'|<2r_0, |z'-w'|≥^1/2}|N_0(z')|/|z'-w'|^4  dz' 
            + C|f_k| ^2/|k|^2∫_{^1/2≤ |z'|<2r_0, |z'-w'|≥^1/2}^-1/2 |N_0(z')| + |∇N_0(z')|/|z'-w'|^3 dz' 
            + C|f_k| ^2/|k|^2∫_{^1/2≤ |z'|<2r_0, |z'-w'|≥^1/2}^-1 |N_0(z')| + ^-1/2|∇N_0(z')| +  |∇^2 N_0(z')|/|z'-w'|^2 dz'

	By using the (<ref>), we see that each term above is bounded by C|k|^-2 |f_k| ^1/2. This gives a desired estimate of R_1 in the second case.
	
	Finally, combining the estimates for R_1, R_2 and R_3, as well as the estimate of M_, we obtain |v_(x)| ≤ C|f_k| ^1/2 in the particular situation that f(x) = f_k e^2π i k· x. For general smooth 1-periodic function f = ∑_k≠ 0 f_k e^2π i k· x, the previous estimate leads to
	
    |v_(x)| ≤ C^1/2∑_k≠ 0 |f_k| ≤ C^1/2,

	where the last inequality is valid because f is smooth.




	Let Ω and v_ be the same as Theorem <ref>. Then,
	
    v__L^p(Ω)≤{   C^min{1/2 + 3/4p, 1}- σ       if  d =3, 
       C^1/2 + 1/p       if  d=4,
       C^1/2 + 3/2p |ln|^1/p       if  d≥ 5,
    	.

	where 1≤ p≤∞ and σ>0 may be arbitrarily small (σ = 0 if p = 1 or ∞).


	If d=3, the estimate ∇ v__L^1(Ω)≤ C^1-σ and the Sobolev inequality implies that
	
    v__L^3/2(Ω)≤ C^1-σ.

	Thus, the same estimate holds for v__L^p(Ω) for any 1< p<3/2. If 3/2<p<∞, interpolating the estimate (<ref>) and (<ref>) gives
	
    v__L^p(Ω)≤ C^1/2+3/4p.

	
	For d = 4 or d≥ 5, an interpolation between (<ref>) with p = 1 and (<ref>) gives the desired estimates.




	For the special case p = 2, the above theorem gives
	
    v__L^2(Ω)≤{   C^7/8- σ       if  d =3, 
       C       if  d= 4,
       C^5/4|ln|^1/2       if  d≥ 5,
    	.

	which is a significant improvement of (<ref>). In this paper, the expected convergence rates for the Robin problem (<ref>) cannot be better than (<ref>) in any dimensions. 


The following estimate for the trace of v_ will also be useful.

	Let Ω and v_ be the same as Theorem <ref>. Then for any 1≤ p≤∞,
	
    v__L^p(∂Ω)≤{   C^1/2+1/2p-σ,      if     d = 3,4,
       C^min{1/2+1/p, 1} -σ,      if     d≥ 5.
    		.



	The lemma is a straightforward of Theorem <ref> and (<ref>). First of all, for any g ∈ C_0^∞(), one has
	
    g_L^1(∂Ω)≤ Cg_L^1(Ω) + C∇ g_L^1(Ω).

	Applying g = v_ to the above inequality and using Theorem <ref>, we have v__L^1(∂Ω)≤ C^1-σ. Then the general estimates of v__L^p(Ω) for d = 3 or 4 follows from an interpolation with (<ref>).
	
	
	To handle the case d≥ 5, we claim that for any q∈ (1,∞),
	
    g_L^2(∂Ω)≤ Cg_L^2(Ω) + Cg_L^q(Ω)^1/2∇ g_L^q'(Ω)^1/2.

	It turns out that the above trace estimate implies v__L^2(∂Ω)≤ C^1-σ for d≥ 5. Actually, applying (<ref>) to v_ and using Theorem <ref>, we obtain that
	
    v__L^2(∂Ω)   ≤ Cv__L^2(Ω) + Cv__L^q(Ω)^1/2∇ v__L^q'(Ω)^1/2
       ≤ C|ln|^1/2 + C ^1/q |ln|^1/2q^1/2q'
       ≤ C ^1/q.

	The desired estimate follows by choosing q sufficiently close to 1 such that 1/q = 1-σ. Finally, the estimate of v__L^p(∂Ω) for p>2 follows from the interpolation with (<ref>).







§ CONVERGENCE RATES

This section is devoted to the convergence rates for the Robin problem (<ref>) under perfect conditions. Precisely, we assume that b ∈ C^∞(𝕋^d;ℝ^m× m) and Ω is a smooth and strictly convex domain. Since we do not impose any regularity on the matrix A, as usual, we need a smoothing operator to deal with this situation. Let ϕ∈ C^∞_0(B_1(0)) such that ∫ϕ = 1. Define ϕ_(x) = ^-dϕ(^-1 x) and

    S_(f)(x) = ∫_ϕ_(y) f(x-y) dy.

Many useful properties of operator S_ may be found in, e.g., <cit.>.

Assume u_ and u_0 are the solutions of (<ref>) and the corresponding homogenized system (<ref>). First of all, we construct the first-order expansion and establish the convergence rate in H^1. Let η_ be a cut-off function such that η_ = 1 in {x∈Ω: dist(x,∂Ω) > 2}, η = 0 in {x∈Ω: dist(x,∂Ω) < } and |∇η_| ≤ C^-1. Define the error of the first-order approximation by

    w_ = u_ - u_0 - χ(x/) S_(  η_∇ u_0) .

By the variational equations for u_ and u_0, for any φ∈ H^1(Ω;), one has
	
    ∫_Ω A_∇ u_·∇φ + ∫_∂Ω b_ u_·φ = ∫_ΩA∇ u_0 ·∇φ + ∫_∂Ωb u_0 ·φ,

	where A_(x) = A(x/). Then, a straightforward computation shows that w_ satisfies
	
    ∫_Ω A_∇ w_·∇φ + ∫_∂Ω b_ w_·φ     = ∫_Ω (A - A_ - A_∇χ(x/)) S_ (η_∇ u_0) ·∇φ
            + ∫_Ω (A - A_) (∇ u_0 -  S_( η_∇ u_0 ))·∇φ
            + ∫_Ω A_χ(x/) ∇ S_ (η_∇ u_0) ·∇φ
            + ∫_∂Ω (b - b(x/)) u_0 ·φ
        = I_1 + I_2 + I_3 + I_4.

	The estimates for I_1, I_2 and I_3 follow from the same argument as Dirichlet or Neumann problems; see, e.g., <cit.>. Indeed, one can prove that
	
    |I_1| + |I_2| + |I_3| ≤ C^1/2u_0_H^2(Ω)( ^1/2∇φ_L^2(Ω) + ∇φ_L^2(Ω_2)),

	where Ω_t = { x∈Ω: dist(x,∂Ω)<t }.
	
	The main difficulty caused by the Robin boundary condition is the estimate of I_4. To handle this term, we write
	
    I_4 = ∫_∂Ω (b - b(x/) - M_) u_0 ·φ +  ∫_∂Ω M_ u_0 ·φ,

	where M_∈ℝ^m× m is a constant matrix given by
	
    M_ = _∂Ω (b - b(x/) ) dσ(x).

	Since Ω is strictly convex and smooth, we may employ the classical results in oscillatory integral theory (with non-degenerate phases) to obtain
	
    |M_| ≤ C^d-1/2,

	where the constant C depends only on b and Ω. It follows that
	
    | ∫_∂Ω M_ u_0 ·φ| ≤ Cu_0_L^2(∂Ω)φ_L^2(∂Ω).

	for d≥ 3.
	
	Thus, it is sufficient to consider the first integral of (<ref>), namely,
	
    I_osc = ∫_∂Ω (b - b(x/) - M_) u_0 ·φ.

	As we have mentioned, this can be done by a “duality approach" via a Neumann problem. For each β with 1≤β≤ m, let b^β = (b^1β, b^2β, ⋯, b^mβ). Let v_^β be the solution of
	
    {
    	-div(A∇ v_^β)     =0       in    Ω,
    
    	n·A∇ v_^β    =b^β - b^β(x/) - M_^β      on    ∂Ω.
    	.

	where b̅^β and M_^β are the βth column of b̅ and M_. Under the assumption that Ω is strictly convex and smooth, all the estimates in the previous section are valid for v_ = (v_^1,v_^2,⋯, v_^m).
	
	
		Let φ∈ H^1(Ω;).
		
		(i) If u_0∈ W^1,d(Ω;), then
		
    |I_osc| ≤ C^1/2 - σu_0_W^1,d(Ω)φ_H^1(Ω).

		
		(ii) If u_0∈ H^2(Ω;), then
		
    |I_osc| ≤{   C^2/3- σu_0_H^2(Ω)φ_H^1(Ω)       if  d =3, 
       C^1/2-σu_0_H^2(Ω)φ_H^1(Ω)       if  d = 4,
       C^1/4 + 3/4(d-1) - σu_0_H^2(Ω)φ_H^1(Ω)       if  d ≥ 5.
    		.

	

	
	
(i) By the construction of v_ in (<ref>) and the integration by parts, one has
	
    |I_osc|     = | ∫_∂Ω (n·A∇ v_) u_0 ·φ| 
         = | ∫_Ω∇ v_∇ u_0  ·φ + ∫_Ω∇ v_ u_0 ·∇φ|
       ≤ C(  ∇ v__L^2(Ω)∇ u_0_L^d(Ω) + ∇ v__L^2+σ(Ω) u_0_L^4/σ+2(Ω)) φ_H^1(Ω)
       ≤ C^1/2 - σu_0_W^1,d(Ω)φ_H^1(Ω),

	where we have used the Sobolev inequality and (<ref>) with appropriate σ in the last two inequalities.
	
	(ii) If u_0∈ H^2(Ω;), the argument in (i) only gives |I_osc| ≤ C^2/d-σu_0_H^2(Ω)φ_H^1(Ω), which gives the desired estimates for d = 3,4. In the following, we use a more careful argument to improve this estimate for d≥ 5. We first assume φ∈ H^1(∂Ω;). Then the Sobolev inequality implies u_0 φ∈ W^1,p_0(∂Ω;), where 1/p_0 = 1-3/2(d-1), and
	
    u_0 φ_W^1,p_0(∂Ω)≤ Cu_0_H^3/2(∂Ω)φ_H^1(∂Ω).

	
	Now, we construct Φ = (Φ^β) as the solution of
	
    {
    	-div(A^* ∇Φ^β)     =0       in    Ω,
    Φ^β    =u_0^βφ      on    ∂Ω.
    	.

	By the regularity theory of (<ref>) with W^1,p Dirichlet boundary value in smooth domains, we have
	
    ∇Φ_L^p_0(∂Ω)≤𝒩(∇Φ)_L^p_0(∂Ω)≤ C∇_tan (u_0 φ)_L^p_0(∂Ω).

	where 𝒩(∇Φ) is the non-tangential maximal function defined by
	
    𝒩(f)(Q) = sup{ |f(x)|: x∈Ω, |x-Q|< (1+α) dist(x,∂Ω) },

	where α>0 is a fixed constant. Particularly, (<ref>) implies that the Dirichlet-to-Neumann map Φ^β↦ n·A^* ∇Φ^β is bounded from W^1,p_0(∂Ω;) to L^p_0(∂Ω;). Also, it follows from the Green's second identity that
	
    I_osc = ∫_∂Ω (n·A∇ v_) ·Φ = ∫_∂Ω  v_· (n·A^* ∇Φ).

	As a result, if d≥ 5, (<ref>) and (<ref>) imply that
	
    |I_osc|    ≤ Cv__L^p_0'(∂Ω)∇Φ_L^p_0(∂Ω)
       ≤ C^1/2 + 1/p_0' -σu_0_H^3/2(∂Ω)φ_H^1(∂Ω)
       ≤ C^1/2 + 3/2(d-1) -σu_0_H^3/2(∂Ω)φ_H^1(∂Ω),

	where we have used (<ref>) and the fact that 1/p_0' = 3/2(d-1).
	
	Clearly, note that
	
    |I_osc| ≤ C u_0_L^2(∂Ω)φ_L^2(∂Ω).

	Thus, an interpolation between (<ref>) and (<ref>) gives
	
    |I_osc| ≤ C^1/4 + 3/4(d-1)u_0_H^3/2(∂Ω)φ_H^1/2(∂Ω)≤ C^1/4 + 3/4(d-1)u_0_H^2(Ω)φ_H^1(Ω).

	This ends the proof.



	Let u_ and u_0 be the same as before. Then
	
	(i) If u_0∈ H^2(Ω;),
	
    w__H^1(Ω)≤{   C^1/2u_0_H^2(Ω)       if  d =3, 
       C^1/2-σu_0_H^2(Ω)       if  d = 4,
       C^1/4 + 3/4(d-1) - σu_0_H^2(Ω)       if  d ≥ 5.
    	.

	
	(ii) If d≥ 5 and u_0∈ H^2 ∩ W^1,d(Ω;),
	
    w__H^1(Ω)≤ C^1/2 -σ( u_0_W^1,d(Ω) + u_0_H^2(Ω)).



	(i) In view of (<ref>), (<ref>), (<ref>) and Lemma <ref> (ii), one has
		
    | ∫_Ω A_∇ w_·∇φ + ∫_∂Ω b_ w_·φ|    ≤ Cu_0_H^2(Ω)( φ_H^1(Ω) + ^1/2∇φ_L^2(Ω_2)) 
            + {   C^2/3- σu_0_H^2(Ω)φ_H^1(Ω)       if  d =3, 
       C^1/2-σu_0_H^2(Ω)φ_H^1(Ω)       if  d = 4,
       C^1/4 + 3/4(d-1) - σu_0_H^2(Ω)φ_H^1(Ω)       if  d ≥ 5.
    			.

	Now, choosing φ = w_ in the above inequality and using (<ref>) and (<ref>), we obtain
	
    ∇ w__L^2(Ω)^2 + w__L^2(∂Ω)^2 ≤{   C^1/2u_0_H^2(Ω)w__H^1(Ω)       if  d =3, 
       C^1/2-σu_0_H^2(Ω)w__H^1(Ω)       if  d = 4,
       C^1/4 + 3/4(d-1) - σu_0_H^2(Ω)w__H^1(Ω)       if  d ≥ 5.
    	.

	The desired estimate follows from a simple observation w__H^1(Ω)≃∇ w__L^2(Ω) + w__L^2(∂Ω) and the Hölder inequality.


	(ii) The estimate (<ref>) follows from Lemma <ref> (i) and a similar argument.



	Lemma <ref> and Theorem <ref> shows that the estimate of I_osc and the convergence rates of w_ are very sensitive to the regularity of u_0. For instance, u_0∈ H^2(Ω;) implies that u_0∈ W^1,2d/d-2(Ω;), which gives worse integrability as d increases (d≥ 5) and hence causes lower convergence rates. On the other hand, u_0 ∈ W^1,d(Ω;) is a very natural condition to guarantee good convergence rates for d≥ 5.



		Assume d≥ 5. Let φ∈ H^2(Ω;) and u_0∈ W^2,d/2(Ω;).
	Then	
	
    | ∫_Ω A_∇ w_·∇φ + ∫_∂Ω b_ w_·φ| ≤ C^1-σu_0_W^2,d/2(Ω)φ_H^2(Ω),

	for any σ>0.



	As before, we still have (<ref>) and (<ref>), which yields
	
    |I_1| + |I_2| + |I_3| ≤ Cu_0_H^2(Ω)φ_H^2(Ω),   for any φ∈ H^2(Ω;).

	Therefore, the main difficulty is to estimate
	
    I_osc = ∫_∂Ω (b^αβ - b^αβ(x/) - M_^αβ) u_0^β·φ^α.

	By the equations of v_^β in (<ref>) and the integration by parts, we have
	
    I_osc    = ∫_∂Ω n·A^αγ∇ v_^γβ u_0^β·φ^α
        = ∫_Ω∇ v_^γβ·A^*γα∇( u_0^βφ^α) 
        = - ∫_Ω v_^γβ·div( A^*γα∇ (u_0^βφ^α) ) + ∫_∂Ω v_^γβ (n·A^*γα∇ u_0^β) φ^α + ∫_∂Ω v_^γβ  u_0^β (n·A^*γα∇φ^α) 
        = J_1 + J_2 + J_3.

	
	To handle J_1, by (<ref>) and the Sobolev inequality, we have
	
    |J_1|    ≤ Cv__L^2+σ(Ω)u_0_W^2,d/2(Ω)φ_H^2(Ω)
       ≤ C^1-σu_0_W^2,d/2(Ω)φ_H^2(Ω),

	provided d≥ 4.
	On the other hand, if d≥ 5, (<ref>) and the trace theorem implies
	
    |J_2| + |J_3|    ≤ Cv__L^2+σ(∂Ω)( ∇ u_0 φ_W^1,2-σ_1(Ω) +  u_0 ∇φ_W^1,2-σ_1(Ω)) 
       ≤ C^1-σu_0_W^2,d/2(Ω)φ_H^2(Ω),

	where σ,σ_1 are small and 1/2+σ + 1/2-σ_1 = 1.
	This finishes the proof by combining (<ref>), (<ref>) and (<ref>).




	Assume d= 3 or 4. Let u_0∈ H^2∩ W^1,∞(Ω;). Let φ∈ H^2(Ω;) be the weak solution of
			
    {
    			- div(A^* ∇φ)     = G       in    Ω,
    
    			n·A^*∇φ + b̅^* φ    =0       on    ∂Ω.
    			.

	Then	
	
    | ∫_Ω A_∇ w_·∇φ + ∫_∂Ω b_ w_·φ| ≤{   C^7/8-σ(u_0_W^1,∞(Ω) + ∇^2 u_0_L^2(Ω)) φ_H^2(Ω),
       C^1-σ(u_0_W^1,∞(Ω) + ∇^2 u_0_L^2(Ω)) φ_H^2(Ω),
    	 .

	for any σ>0.



	The proof follows a line of Lemma <ref>. Note that (<ref>) still holds if d =4. If d = 3, it is replaced by
	
    |J_1| ≤ C^7/8 - σu_0_H^2(Ω)φ_H^2(Ω),

	thanks to (<ref>).
	
	The estimates for J_2 and J_3 will be sightly different since we only have a better rate for v__L^1(∂Ω) if d = 3 or 4, instead of v__L^2(∂Ω). Precisely, if d = 3, the Sobolev embedding theorem implies that φ_L^∞(Ω)≤ Cφ_H^2(Ω). Thus, (<ref>) implies
	
    |J_2| ≤ Cv__L^1(∂Ω)∇ u_0_L^∞(∂Ω)φ_L^∞(∂Ω)≤ C^1-σ∇ u_0_L^∞(∂Ω)φ_H^2(Ω).

	For J_3, we take advantage of the boundary condition of φ in (<ref>) and obtain
	
    J_3 = -∫_∂Ω v_^γβ  u_0^β (b̅^* φ)^γ,

	which yields
	
    |J_3| ≤ C^1-σ u_0_L^∞(∂Ω)φ_H^2(Ω).

	
	If d=4, φ is not bounded. Nevertheless, we still have
	
    φ_L^q(∂Ω)≤ Cφ_H^2(Ω),     for any  q<∞.

	Consequently,
	
    |J_2| + |J_3|    ≤ Cv__L^q'(∂Ω)u_0_W^1,∞(∂Ω)φ_L^q(∂Ω)
       ≤ C^1-σu_0_W^1,∞(∂Ω)φ_H^2(Ω),

	where in the last inequality, we have chosen q sufficiently large (hence, q' is sufficiently close to 1). Note that v__L^q'≤ C^1-σ if q' is close enough to 1, due to (<ref>). These complete the proof.



Now, we are in a position to prove Theorem <ref>.


	The proof is based on a duality argument. For a given function G∈ L^2(Ω;), let h_ be the solution of
	
    {
    	-div( A^*(x/)∇ h_)     = G       in    Ω,
    
    	n· A^*(x/) ∇ h_ + b^*(x/) h_    =0       on    ∂Ω.
    	.

	Let h_0 be the corresponding homogenized equation, namely,
	
    {
    	-div( A^* ∇ h_0)     = G       in    Ω,
    
    	n·A^* ∇ h_0 + b̅^* h_0     =0       on    ∂Ω.
    	.

	Since Ω is smooth, the H^2 regularity theory for (<ref>) implies h_0∈ H^2(Ω;) and h_0_H^2(Ω)≤ CG_L^2(Ω). It follows from Theorem <ref> (i) that
	
    h_ - h_0 - χ(x/) S_( η_∇ h_0) _H^1(Ω)≤{   C^1/2h_0_H^2(Ω)       if  d =3, 
       C^1/2-σh_0_H^2(Ω)       if  d = 4,
       C^1/4 + 3/4(d-1) - σh_0_H^2(Ω)       if  d ≥ 5.
    	.

	
	Now, let w_ be defined as (<ref>) and consider
	
    ∫_Ω w_· G    = ∫_Ω A_∇ w_·∇ h_ - ∫_∂Ω w_· (n· A^*(x/) ∇ h_) dσ
        = ∫_Ω A_∇ w_·∇ ( h_ - h_0 - χ(x/) S_( η_∇ h_0) ) 
            + ∫_∂Ω  w_· b^*(x/) (h_ - h_0) ·
            + ∫_Ω A_∇ w_·∇ h_0 + ∫_∂Ω b(x/) w_· h_0 dσ
            + ∫_Ω A_∇ w_·∇ (χ(x/) S_( η_∇ h_0) ) 
       = K_1 + K_2 + (K_3+K_4) + K_5.

	
	To estimate K_1, we use Theorem <ref>  and (<ref>) to obtain
	
    |K_1|    ≤w__H^1(Ω)h_ - h_0 - χ(x/) S_( η_∇ h_0) _H^1(Ω)
       ≤{   Cu_0_H^2(Ω)h_0_H^2(Ω)       if  d =3, 
       C^1-σu_0_H^2(Ω)h_0_H^2(Ω)       if  d = 4,
       C^3/4 + 3/4(d-1) - σ( u_0_W^1,d(Ω) + u_0_H^2(Ω)) h_0_H^2(Ω)       if  d ≥ 5.
    	.

	To estimate K_2, note that χ(x/) S_( η_∇ h_0) vanishes on the boundary. Then, K_2 has the same bound as K_1, by the trace theorem.

	Next, using Lemma <ref> and Lemma <ref>, we bound K_3+K_4 by
	
    |K_3 + K_4|    = | ∫_Ω A_∇ w_·∇ h_0 + ∫_∂Ω b(x/) w_· h_0 dσ| 
       ≤{   C^7/8 - σ( u_0_W^1,∞(Ω) + u_0_H^2(Ω)) h_0_H^2(Ω)       if  d =3, 
       C^1-σ( u_0_W^1,∞(Ω) + u_0_H^2(Ω)) h_0_H^2(Ω)       if  d = 4,
       C^1 - σu_0_W^2,d/2(Ω)h_0_H^2(Ω)       if  d ≥ 5.
    	.

	
	Finally, to estimate K_5, we use the fact that χ(x/) S_( η_∇ h_0) vanishes on a boundary layer with thickness 2, if we choose the cut-off function η_ appropriately. In view of (<ref>) and (<ref>), we have
	
    |K_5|     = | ∫_Ω A_∇ w_·∇ (χ(x/) S_( η_∇ h_0) ) + ∫_∂Ω b(x/) w_·χ(x/) S_( η_∇ h_0) dσ| 
       ≤ C^1/2u_0_H^2(Ω)( ^1/2∇ (χ(x/) S_( η_∇ h_0))_L^2(Ω) + ∇ (χ(x/) S_( η_∇ h_0))_L^2(Ω_2)) 
        = Cu_0_H^2(Ω)∇ (χ(x/) S_( η_∇ h_0))_L^2(Ω)
       ≤ Cu_0_H^2(Ω)h_0_H^2(Ω),

	where in the last inequality, we have used a property of S_; see <cit.> or <cit.>.
	
	As a result, we arrive at
	
    | ∫_Ω w_· G | ≤{   C^7/8 - σ( u_0_W^1,∞(Ω) + u_0_H^2(Ω)) h_0_H^2(Ω)       if  d =3, 
       C^1-σ( u_0_W^1,∞(Ω) + u_0_H^2(Ω)) h_0_H^2(Ω)       if  d = 4,
       C^3/4 + 3/4(d-1) - σu_0_W^2,d/2(Ω)h_0_H^2(Ω)       if  d ≥ 5.
    	.

	Recall that h_0_H^2(Ω)≤ CG_L^2(Ω). Then, by duality, we see that w__L^2(Ω) is bounded by the right-hand side of (<ref>), which yields the desired estimate (<ref>), since χ(x/) S_( η_∇ u_0) is clearly bounded by Cu_0_H^1(Ω).



	In this paper, we focus only on the estimate of u_ - u_0_L^2(Ω). However, the sharp convergence rates may be obtained if we consider u_ - u_0_L^p(Ω) with some p∈(1,2). For instance, if d = 3, (<ref>) implies that v__L^3/2(Ω)≤ C^1-σ. This leads to the improvement for the estimate of J_1 in Lemma <ref> and eventually yields 
	
    u_ - u_0_L^3/2(Ω)≤ C^1-σ ( u_0_W^1,∞(Ω) + u_0_H^2(Ω) ).

	Similar results are also true for d≥ 5.




















































































Acknowledgment.
Jun Geng was Supported in part by the NNSF of China (no.11571152) and Fundamental
Research Funds for the Central Universities (lzujbky-2017-161). Jinping Zhuge is supported in part by National Science Foundation grant DMS-1600520. Both of the authors would like to thank Professor
Zhongwei Shen for helpful discussions and suggestions.

abbrv







Jun Geng

School of Mathematics and Statistics,
Lanzhou University,
Lanzhou, P.R. China

E-mail:gengjun@lzu.edu.cn







Jinping Zhuge

Department of Mathematics,
University of Kentucky,
Lexington, Kentucky, 40506,
USA.

E-mail: jinping.zhuge@uky.edu







