In this paper, we investigate the regularity of weak solutions $u\colon\Omega\to\mathbb{ R}$ to elliptic equations of the type \begin{ equation*} \mathrm{ div}\, \nabla \mathcal{ F}(x,Du) = f\qquad\text{ in $\Omega$}, \end{ equation*} whose ellipticity degenerates in a fixed bounded and convex set $E\subset\mathbb{ R}^n$ with $0\in \mathrm{ Int}\, E$. Here, $\Omega\subset\mathbb{ R}^n$ denotes a bounded domain, and $\mathcal{ F} \colon \Omega\times\mathbb{ R}^n \to\mathbb{ R}_{ \geq 0}$ is a function with the properties: for any $x\in\Omega$, the mapping $\xi\mapsto \mathcal{ F}(x,\xi)$ is regular outside $E$ and vanishes entirely within this set. Additionally, we assume $f\in L^{ n+\sigma}(\Omega)$ for some $\sigma > 0$, representing an arbitrary datum. Our main result establishes the regularity \begin{ equation*} \mathcal{ K}(Du)\in C^0(\Omega) \end{ equation*} for any continuous function $\mathcal{ K}\in C^0(\mathbb{ R}^n)$ vanishing on $E$.
Categories: math.AP