Uniformly nondegenerate elliptic equations with partially BMO coefficients in nonsmooth domains

Abstract

A global estimate in weighted Lorentz–Sobolev spaces is obtained for the weak solutions to divergence form uniformly nondegenerate elliptic equations over a bounded nonsmooth domain. Here, the leading coefficients are assumed to be merely measurable in one variable and have small BMO semi-norms in the remaining variables under the assumption that one variable direction is perpendicular to the boundary points which is close to the boundary, while a geometric assumption on the boundary is a locally bounded Reifenberg flatness. In addition, we also investigate regularities in Lorentz–Morrey, Morrey, and Hölder spaces for elliptic equations under the same assumptions on the leading coefficients and the boundary of domain.