Full $C^{1,\alpha }$-Regu1arity for Minimizers of Integral Functionals with $L$ log $L$-Growth

Abstract

We consider the integral functional with nearly-linear growth ${\int }_{\Omega }|Du|\log (1+|Du|)dx$ where $u:\Omega \subset {\mathbb{R}}^n\to {\mathbb{R}}^N(n\ge 2,N\ge 1)$ and we prove that any local minimizer $u$ has locally Hölder continuous gradient in the interior of $\Omega$ thus excluding the presence of singular sets in $\Omega$. This functional has recently been considered by several authors in connection with variational models for problems from the theory of plasticity with logarithmic hardening. We also give extensions of this result to more general cases.