W 1, p(·)-Regularity for Elliptic Equations with Measurable Coefficients in Nonsmooth Domains

Abstract

We establish global W 1, p(·)-estimates for second order elliptic equations in divergence form under the natural assumption that p(·) is log-Holder continuous. To this end, we assume that the coefficients are measurable in one variable and have small BMO semi-norms in the other variables and the boundary of the domain is Reifenberg flat. Our work is an optimal and natural extension of W 1,p -regularity for such equations with merely measurable coefficients beyond Lipschitz domains.