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