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 $u : Ω\subset \mathbb{R}^n \to \mathbb{R}^N$, with $N \ge 1$, is a possibly vector-valued function. Here, $| \cdot |_γ$ 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 $Ω$, for any continuous function $\mathcal{K}: Ω\times \mathbb{R}^{nN} \rightarrow \mathbb{R}$ vanishing on $\bigl\{ (x,ξ) \in Ω\times \mathbb{R}^{nN} : |ξ|_{γ(x)} \le 1 \bigr\}$.

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