Asymptotic analysis of second-order boundary layer correctors

In this article we extend the ideas presented in Onofrei and Vernescu [Asymptotic Anal. 54 (2007), pp. 103-123] and introduce suitable second-order boundary layer correctors, to study the H-1-norm error estimate for the classical problem of homogenization, i.e. {-del.(A(x/epsilon)del u(epsilon)(x)) = f in Omega, u(epsilon) = 0 on partial derivative Omega. Previous second-order boundary layer results assume either smooth enough coefficients (which is equivalent to assuming smooth enough correctors chi(j), chi(ij) is an element of W-1,W-infinity), or smooth homogenized solution u(0), to obtain an estimate of order O(epsilon(3/2)). For this we use some ideas related to the periodic unfolding method proposed by Cioranescu et al. [C. R. Acad. Sci. Paris, Ser. I 335 (2002), pp. 99-104]. We prove that in two dimensions, for non-smooth coefficients and general data, one obtains an estimate of order O(epsilon(3/2)). In three dimensions the same estimate is obtained assuming chi(j), chi(ij) is an element of W-1,W-p, with p>3.