Continuity of weak solutions of elliptic partial differential equations

Abstract

The continuity of weak solutions of elliptic partial differential equations $$div \mathcal{A}(x,\nabla u) = 0$$ is considered under minimal structure assumptions. The main result guarantees the continuity at the pointx0 for weakly monotone weak solutions if the structure ofA is controlled in a sequence of annuli\(B(x_0 ,R_j )\backslash \bar B(x_0 ,r_j )\) with uniformly bounded ratioR j /r j such that lim j→∞ R j =0. As a consequence, we obtain a sufficient condition for the continuity of mappings of finite distortion.