A regularity theory for variational integrals with $L\ln L$-Growth

Abstract.

In the present paper we study regularity for local minimizers of the convex variational integral \[ J(u) = \int\limits_{\Omega} |\nabla u|\ln(1+|\nabla u|)\,dx \] defined on certain classes of vector–valued functions $u:{\Bbb R}^n\supset\Omega\rightarrow{\Bbb R}^N$. The underlying energy spaces are natural from the point of view of existence theory. We then show that local minimizers are of class $C^1$ apart from a closed singular set with vanishing Lebesgue measure, provided $n\leq 4$. For twodimensional problems we obtain smoothness in the interior of $\Omega$.