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_\Omega \dfrac{1}{p} \bigl( |Du(x)|_{\gamma(x)}-1\bigr)_+^p \ \mathrm{d}x, \end{equation*} for $p >1$, where $u : \Omega \subset \mathbb{R}^n \to \mathbb{R}^N$, with $N \ge 1$, is a possibly vector-valued function. Here, $| \cdot |_\gamma$ 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 $\Omega$, for any continuous function $\mathcal{K}: \Omega \times \mathbb{R}^{nN} \rightarrow \mathbb{R}$ vanishing on $\bigl\{ (x,\xi ) \in \Omega \times \mathbb{R}^{nN} : |\xi|_{\gamma(x)} \le 1 \bigr\}$.