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

### Giulio Ciraolo

Università di Palermo, Italy### Rolando Magnanini

Università di Firenze, Italy### Shigeru Sakaguchi

Tohoku University, Sendai, Japan

## Abstract

We consider the functional

where $\Omega$ is a bounded domain and $f$ is a convex function. Under general assumptions on $f$, Crasta [Cr1] has shown that if $\mathcal I_{\Omega}$ 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.

## Cite this article

Giulio Ciraolo, Rolando Magnanini, Shigeru Sakaguchi, Symmetry of minimizers with a level surface parallel to the boundary. J. Eur. Math. Soc. 17 (2015), no. 11, pp. 2789–2804

DOI 10.4171/JEMS/571