Gradient regularity for widely degenerate elliptic partial differential equations

In this paper, we investigate the regularity of weak solutions $u\text{\hspace{0.17em}⁣}:\mathrm{\Omega }\to \mathbb{R}$ to elliptic equations of the type \begin{equation*} $\mathrm{d}\mathrm{i}\mathrm{v}$, \nabla $\mathcal{F}$(x,Du) = f\qquad$\text{in }\mathrm{\Omega }$, \end{equation*} whose ellipticity degenerates in a fixed bounded and convex set $E\subset {\mathbb{R}}^n$ with $0\in \mathrm{I}\mathrm{n}\mathrm{t}E$. Here, $\mathrm{\Omega }\subset {\mathbb{R}}^n$ denotes a bounded domain, and $\mathcal{F}\text{\hspace{0.17em}⁣}:\mathrm{\Omega }\times {\mathbb{R}}^n\to {\mathbb{R}}_{\ge 0}$ is a function with the properties: for any $x\in \mathrm{\Omega }$, the mapping $\xi \mapsto \mathcal{F}(x,\xi )$ is regular outside $E$ and vanishes entirely within this set. Additionally, we assume $f\in L^{n+\sigma }(\mathrm{\Omega })$ for some $\sigma >0$, 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 $E$.