# Symmetry of minimizers with a level surface parallel to the boundary

Giulio Ciraolo, Rolando Magnanini, Shigeru Sakaguchi

Risultato della ricerca: Article

16 Citazioni (Scopus)

### Abstract

We consider the functional $I_\Omega(v)=\int_\Omega [f(|Dv|)-v] dx$; 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 $W^{1,1}_0(\Omega)$ 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.
Lingua originale English 2789-2804 16 Journal of the European Mathematical Society 17 Published - 2015

### All Science Journal Classification (ASJC) codes

• Mathematics(all)
• Applied Mathematics