Gradient regularity for widely degenerate elliptic partial differential equations

Abstract

In this paper, we investigate the regularity of weak solutions $u:\hat{I}\text{c◯}\to \mathbb{R}$ to elliptic equations of the type \begin{equation*} \mathrm{div}\, \nabla \mathcal{F}(x,Du) = f\qquad\text{in $\hat{I}\text{c◯}$}, \end{equation*} whose ellipticity degenerates in a fixed bounded and convex set $E\subset {\mathbb{R}}^n$ with $0\in \mathrm{Int}E$. Here, $\hat{I}\text{c◯}\subset {\mathbb{R}}^n$ denotes a bounded domain, and $\mathcal{F}:\hat{I}\text{c◯}\times {\mathbb{R}}^n\to {\mathbb{R}}_{\ge 0}$ is a function with the properties: for any $x\in \hat{I}\text{c◯}$, the mapping $\hat{I}¾\mapsto \mathcal{F}(x,\hat{I}¾)$ is regular outside $E$ and vanishes entirely within this set. Additionally, we assume $f\in L^{n+\ddot{I}ƒ}(\hat{I}\text{c◯})$ for some $\ddot{I}ƒ>0$, representing an arbitrary datum. Our main result establishes the regularity \begin{equation*} \mathcal{K}(Du)\in C^0(Ω) \end{equation*} for any continuous function $\mathcal{K}\in C^0({\mathbb{R}}^n)$ vanishing on $E$.