Gradient regularity for widely degenerate elliptic partial differential equations

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