Higher regularity for minimizers of very degenerate convex integrals

In this paper, we consider minimizers of integral functionals of the type \begin{equation*} $\mathcal{F}$(u):= \int_Ω\dfrac{1}{p} \bigl( |Du(x)|{γ(x)}-1\bigr)+^p \ $\mathrm{d}$x, \end{equation*} for $p>1$, where $u:\mathrm{\Omega }\subset {\mathbb{R}}^n\to {\mathbb{R}}^N$, with $N\ge 1$, is a possibly vector-valued function. Here, $\mathrm{\mid }\cdot {\mathrm{\mid }}_{\gamma }$ is the associated norm of a bounded, symmetric and coercive bilinear form on ${\mathbb{R}}^{nN}$. We prove that $\mathcal{K}(x,Du)$ is continuous in $\mathrm{\Omega }$, for any continuous function $\mathcal{K}:\mathrm{\Omega }\times {\mathbb{R}}^{nN}\to \mathbb{R}$ vanishing on $\{(x,\xi )\in \mathrm{\Omega }\times {\mathbb{R}}^{nN}:\mathrm{\mid }\xi {\mathrm{\mid }}_{\gamma (x)}\le 1\}$.