Uniform regularity estimates in homogenization theory of elliptic system with lower order terms

In this paper, we extend the uniform regularity estimates obtained by M. Avellaneda and F. Lin in [3] and [6] to the more general second order elliptic systems in divergence form {Lε,ε>0}{Lε,ε>0}, with rapidly oscillating periodic coefficients. We establish not only sharp W1,pW1,p estimates, Hölder estimates, Lipschitz estimates and non-tangential maximal function estimates for the Dirichlet problem on a bounded C1,ηC1,η domain, but also a sharp O(ε)O(ε) convergence rate in H01(Ω) by virtue of the Dirichlet correctors. Moreover, we define the Green's matrix associated with LεLε 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] and [6].