Higher regularity for minimizers of very degenerate convex integrals

In this paper, we consider minimizers of integral functionals of the type $F\left(u\right)≔{\int }_{\Omega }\frac{1}{p}{\left({\left|Du\left(x\right)\right|}_{\gamma \left(x\right)}-1\right)}_{+}^{p}dx,$for $p>1$, where $u:\Omega \subset R^n\to R^N$, with $N\ge 1$, is a possibly vector-valued function. Here, ${\left|\cdot \right|}_{\gamma }$ is the associated norm of a bounded, symmetric and coercive bilinear form on $R^{nN}$. We prove that $K(x,Du)$ is continuous in $\Omega$, for any continuous function $K:\Omega \times R^{nN}\to R$ vanishing on $\left\{(x,\xi )\in \Omega \times R^{nN}:{\left|\xi \right|}_{\gamma \left(x\right)}\le 1\right\}$.