Symmetry of minimizers with a level surface parallel to the boundary

We consider the functional I-Omega(v) = integral(Omega) [f(vertical bar Dv vertical bar) - v]dx, where Omega is a bounded domain and f is a convex function. Under general assumptions on f, Crasta [Cr1] has shown that if I-Omega admits a minimizer in W-0(1,1)(Omega) depending only on the distance from the boundary of Omega, then Omega must be a ball. With some restrictions on f, we prove that spherical symmetry can be obtained only by assuming that the minimizer has one level surface parallel to the boundary (i.e. it has only a level surface in common with the distance). We then discuss how these results extend to more general settings, in particular to functionals that are not differentiable and to solutions of fully nonlinear elliptic and parabolic equations.