On continuity of minimizers for certain quadratic growth functionals

Abstract

In this paper we treat the regularity problem for minimizers u(x): Ω ⊂ \R^m → \R^n of quadratic growth functionals ∈t_Ω A(x, u, Du)dx. About the dependence on the variable x we assume only that A(•, 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 ≤ 4.