Elliptic Equations in Divergence Form with Partially BMO Coefficients

Abstract

The solvability in Sobolev spaces is proved for divergence form second order elliptic equations in the whole space, a half space, and a bounded Lipschitz domain. For equations in the whole space or a half space, the leading coefficients aij are assumed to be only measurable in one direction and have locally small BMO semi-norms in the other directions. For equations in a bounded domain, additionally we assume that aij have small BMO semi-norms in a neighborhood of the boundary of the domain. We give a unified approach of both the Dirichlet boundary problem and the conormal derivative problem. We also investigate elliptic equations in Sobolev spaces with mixed norms under the same assumptions on the coefficients.