Gradient regularity for widely degenerate elliptic partial differential equations

In this paper, we investigate the regularity of weak solutions  u:Ω → R to elliptic equations of the type [Formula: see text] whose ellipticity degenerates in a fixed bounded and convex set  E ⊂ Rn with  0 ∈ Int E . Here,  Ω ⊂ Rn denotes a bounded domain, and  F:Ω × Rn → R≥0 is a function with the properties: for any x ∈ Ω , the mapping  ξ ↦ F(x, ξ) is regular outside E and vanishes entirely within this set. Additionally, we assume  f ∈ Ln+σ(Ω) for some  σ > 0 , representing an arbitrary datum. Our main result establishes the regularity [Formula: see text] for any continuous function  K ∈ C0(Rn) vanishing on E.