Periodic Homogenization Of Elliptic Systems With Singular Perturbations

We consider the quantitative theory of periodic homogenization of elliptic systems with singular perturbations.Let ? be the bounded region in Rd(d?2),and consider the following elliptic systems.?(1)where elliptic operator L?=?2?2-div[A(x/?)?].(2)In this paper,we study the optimal convergence rate of the periodic homogenization problem on the bounded domain for the equation(1).Since the elliptic equation and its homogenization problem have different orders,we introduce the appropriate auxiliary function w?,and obtain H1(?)estimates of w?.We establish the sharp convergence rate in L2(?)by using the duality.Finally,we study the uniform regularity estimates of the equation(1).We first prove the large-scale interior Lipschitz estimation by the compact method,and the internal Lipschitz estimation of u? is derived by using the blow up argument.