W1,p estimates in homogenization of elliptic systems with measurable coefficients in nonsmooth domains

Abstract

In this study, we establish uniform $W^{1,p}$ estimates for weak solutions in homogenization of elliptic systems in divergence-form with measurable coefficients in nonsmooth domains. We consider first an interior regularity and then we study boundary value problems, a Dirichlet problem and a conormal derivative problem. Our main purpose is to find an answer for minimal requirements on the coefficients and the boundary condition of the domains to ensure that Calder\'{o}n-Zygmund theory holds in a homogenization problem.