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 point x₀ for weakly monotone weak solutions if the structure of A is controlled in a sequence of annuli $B(x_0 ,R_j )\backslash \bar B(x_0 ,r_j )$ with uniformly bounded ratio R_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.