### Abstract

We consider the functional Iω.v/ D Z ω Tf .jDvj/ vU dx; where ω is a bounded domain and f is a convex function. Under general assumptions on f , Crasta [Cr1] has shown that if Iω admits a minimizer in W 1;1 0 .ω/ depending only on the distance from the boundary of ω, then ω 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.

Original language | English |
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Pages (from-to) | 2789-2804 |

Number of pages | 16 |

Journal | Journal of the European Mathematical Society |

Volume | 17 |

Issue number | 11 |

DOIs | |

Publication status | Published - 2015 |

### Keywords

- Minimizers of integral functionals
- Overdetermined problems

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Journal of the European Mathematical Society*,

*17*(11), 2789-2804. https://doi.org/10.4171/JEMS/571