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

We consider the functional J(v) = integral Omega [f(vertical bar del v vertical bar) - v]dx, where Omega is a bounded domain and f : [0,+ infinity) -> R is a convex function vanishing for s is an element of [0, sigma], with sigma > 0. We prove that a minimizer u of J satisfies an equation of the form min(F(del u, D(2)u), vertical bar del u vertical bar - sigma) = 0 in the viscosity sense.