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In this paper, we extend the uniform regularity estimates obtained by M.Avellanda and F. Lin in the paper of Compactness methods in the theory ofhomogenization (Comm. Pure Appl. Math. 40(1987), no.6, 803-847) to the moregeneral second order elliptic systems in divergences form, with rapidlyoscillating periodic coefficients. We establish not only sharp $W^{1,p}$estimates, Holder estimates, Lipschitz estimates and non-tangential maximalfunction estimates for the Dirichelt problem on a $C^{1,\eta}$ domain, but alsoa sharp $O(\varepsilon)$ convergence rate in $H_0^1(\Omega)$ by virtue of theDirichlet correctors. The well known compactness methods are not employed here,instead we construct the transformations to make full use of the correspondingresults developed by M. Avellanda and F. Lin.

Uniform Regularity Estimates in Homogenization Theory of Elliptic System with Lower Order terms