Uniformly nondegenerate elliptic equations with partially BMO coefficients in nonsmooth domains

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.