In terms of layer potential methods, this paper is devoted to study the L 2 boundary value problems for nonhomogeneous elliptic operators with rapidly oscillating coefficients in a periodic setting. Under a low regularity assumption on the coefficients, we establish the solvability for Dirichlet, regular and Neumann problems in a bounded Lipschitz domain, as well as, the uniform nontangential maximal function estimates and square function estimates. The main difficulty is reflected in two aspects: (i) we can not treat the lower order terms as a compact perturbation to the leading term due to the low regularity assumption; (ii) the nonhomogeneous systems do not possess a scaling-invariant property in general. Although this work may be regarded as a follow-up to C. Kenig and Z. Shen's in [21], we make an effort to find a clear way of how to handle the nonhomogeneous operators by using the known results of the homogenous ones. Also, we mention that the periodicity condition plays a key role in the scaling-invariant estimates.

In this paper, we extend the uniform regularity estimates obtained by M. Avellaneda and F. Lin in [3,6] to the more general second order elliptic systems in divergence form {L ε , ε > 0}, with rapidly oscillating periodic coefficients. We establish not only sharp W 1,p estimates, Hölder estimates, Lipschitz estimates and non-tangential maximal function estimates for the Dirichlet problem on a bounded C 1,η domain, but also a sharp O(ε) convergence rate in H 1 0 (Ω) by virtue of the Dirichlet correctors. Moreover, we define the Green's matrix associated with L ε and obtain its decay estimates. We remark that the well known compactness methods are not employed here, instead we construct the transformations (1.11) to make full use of the results in [3,6].

In this paper, we mainly employed the idea of the previous paper [36] to study the sharp uniform W 1,p estimates with 1 < p ≤ ∞ for more general elliptic systems with the Neumann boundary condition on a bounded C 1,η domain, arising in homogenization theory. Based on the skills developed by Z. Shen in [29] and by T. Suslina in [33,34], we also established the L 2 convergence rates on a bounded C 1,1 domain and a Lipschitz domain, respectively. Here we found a "rough" version of the first order correctors (see (1.12)), which can unify the proof in [29] and [34]. It allows us to skip the corresponding convergence results on R d that are the preconditions in [33,34]. Our results can be regarded as an extension of [24] developed by C. Kenig, F. Lin, Z. Shen, as well as of [34] investigated by respectively. Then the result of [24, Theorem 1.2] can be directly applied to (D 2 ). Due to the same reason as explained in [36], we need to derivewhich follows from the decay estimates of Neumann matrixes defined in [24, pp.916] (see Theorem 4.2), as well as ∇Ψ ε,0 C 0,σ 1 (Ω) = O(ε −σ 1 ) and ∇u ε C 0,σ 1 (Ω) = O(ε −σ 2 ) as ε → 0, which are the main conclusions of Corollary 4.6 and Lemma 4.8, where 0 < σ 1 < σ 2 < 1 are independent of ε. The above two estimates together with (1.6) guarantee that the right hand side of (D 2 ) can be uniformly bounded by the given data in Theorem 1.2. Note that the right hand side of (D 2 ) which involves div(f ) is, as a matter of fact, more general than that in [24, Theorem 1.2]. We find a simple way inspired by [31] to derive the uniform Lipschitz estimate for the weak solution u ε to L ε (u ε ) = div(f ) in Ω and ∂u ε /∂ν ε = −n · f on ∂Ω. The key ingredient is to construct the auxiliary functions {v ε,k } d k=1 , which satisfy L ε (v ε,k ) = 0 in Ω, ∂v ε,k /∂ν ε = −n k I on ∂Ω

In this paper, we are interested in the periodic homogenization of quasilinear elliptic equations. We obtain error estimates O(ε 1/2 ) for a C 1,1 domain, and O(ε σ ) for a Lipschitz domain, in which σ ∈ (0, 1/2) is close to zero. Based upon the convergence rates, an interior Lipschitz estimate, as well as a boundary Hölder estimate can be developed at large scales without any smoothness assumption, and these will implies reverse Hölder estimates established for a C 1 domain. By a real method developed by Z.Shen [26], we consequently derive a global W 1,p estimate for 2 ≤ p < ∞. This work may be regarded as an extension of [5,24] to a nonlinear operator, and our results may be extended to the related Neumann boundary problems without any real difficulty.

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