Partial regularity result for non-autonomous elliptic systems with general growth

Abstract

In this paper we prove a Hölder partial regularity result for weak solutions $u:\Omega \to {\mathbb{R}}^N$, $N\ge 2$, to non-autonomous elliptic systems with general growth of the type: \begin{equation*} -\rm{div}\, a(x, u, Du)= b(x, u, Du) \quad \mbox{ in } Ω. \end{equation*} The crucial point is that the operator $a$ satisfies very weak regularity properties and a general growth, while the inhomogeneity $b$ has a controllable growth.