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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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