Symmetry of minimizers with a level surface parallel to the boundary

Giulio Ciraolo, Rolando Magnanini, Shigeru Sakaguchi

Research output: Contribution to journalArticle

16 Citations (Scopus)

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 languageEnglish
Pages (from-to)2789-2804
Number of pages16
JournalJournal of the European Mathematical Society
Volume17
Issue number11
DOIs
Publication statusPublished - 2015

Keywords

  • Minimizers of integral functionals
  • Overdetermined problems

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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