Higher regularity for minimizers of very degenerate convex integrals

Abstract

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 $u:\Omega \subset {\mathbb{R}}^n\to {\mathbb{R}}^N$, 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 $\mathcal{K}:\Omega \times {\mathbb{R}}^{nN}\to \mathbb{R}$ vanishing on $\{(x,\xi )\in \Omega \times {\mathbb{R}}^{nN}:|\xi {|}_{\gamma (x)}\le 1\}$.