Partial regularity for elliptic systems with critical growth

Abstract

We prove a partial Hölder regularity result for weak solutions $u:\Omega\to\mathbb{R}^N$, $N\geq 2$, to non-autonomous elliptic systems with general growth of the type $$ \-\operatorname{div} a(x, u, Du) = b(x, u, Du)\quad\text{in }\Omega. $$ The crucial point is that the operator $a$ satisfies very weak regularity properties and general growth, while for the inhomogeneity $b$ we allow a critical growth condition.