A viscosity equation for minimizers of a class of very degenerate elliptic functionals

Abstract

We consider the functional $$J(v) = \int_Ω[f(|\nabla v|) - v] dx,$$ where $Ω$ is a bounded domain and $f:[0,+\infty)\to \mathbb{R}$ is a convex function vanishing for $s\in [0,σ]$, with $σ>0$. We prove that a minimizer $u$ of $J$ satisfies an equation of the form $$\min(F(\nabla u, D^2 u), |\nabla u|-σ)=0$$ in the viscosity sense.