On continuity of minimizers for certain quadratic growth functionals

Abstract

In this paper we treat the regularity problem for minimizers $u\left(x\right):\mathrm{\Omega }\subset {\mathbf{R}}^m\to {\mathbf{R}}^n$ of quadratic growth functionals ${\int }_{\mathrm{\Omega }}A(x,u,Du)dx$. About the dependence on the variable $x$ we assume only that $A(\cdot ,u,p)$ is in the class $VMO$ as a function of $x$. Namely, we do not assume the continuity of $A(x,u,p)$ with respect to $x$. We will prove a partial regularity result for the case $m\le 4$.