Precise estimates of the difference between the homogenized solution with its first corrector and the original One

The goal of this paper is to estimate the error done while approximating the solutions of an elliptic boundary problem with periodic structures by the two first asymptotic expansion terms in the case of non smooth data. After some preliminary technical results in Section 2, we obtain the convergence theorems in Section 3. The method used herein is inspired from [1] and from [2] where numerical scheme and differential equation solutions were studied with nonsmooth data Classically, weak convergence follows from the G - convergence as for instance in [3] or [4] and estimates for smooth data are obtained like for instance in [5] or [6]. We start in this paper by studying the case with some regularity assumptions on the data, in a second forthcoming paper we are studying the case with less regularity assumptions.