In this paper, we investigate the regularity of weak solutions \(u:\Omega \rightarrow {\mathbb {R}}\) to elliptic equations of the type \(\begin{aligned} \textrm{div}\, \nabla {\mathcal {F}}(x,Du) = f\qquad \text {in }\Omega , \end{aligned}\) whose ellipticity degenerates in a fixed bounded and convex set \(E\subset {\mathbb {R}}^n\) with \(0\in \textrm{Int}\, E\) . Here, \(\Omega \subset {\mathbb {R}}^n\) denotes a bounded domain, and \({\mathcal {F}}:\Omega \times {\mathbb {R}}^n \rightarrow {\mathbb {R}}_{\ge 0}\) 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 \(\begin{aligned} {\mathcal {K}}(Du)\in C^0(\Omega ) \end{aligned}\) for any continuous function \({\mathcal {K}}\in C^0({\mathbb {R}}^n)\) vanishing on E.
