Solutions of elliptic equations with a level surface parallel to the boundary: Stability of the radial configuration

Giulio Ciraolo, Rolando Magnanini, Shigeru Sakaguchi

Research output: Contribution to journalArticle

7 Citations (Scopus)

Abstract

A positive solution of a homogeneous Dirichlet boundary value problem or initial-value problems for certain elliptic or parabolic equations must be radially symmetric and monotone in the radial direction if just one of its level surfaces is parallel to the boundary of the domain. Here, for the elliptic case, we prove the stability counterpart of that result. We show that if the solution is almost constant on a surface at a fixed distance from the boundary, then the domain is almost radially symmetric, in the sense that is contained in and contains two concentric balls (Formula presented.) and (Formula presented.) , with the difference re-ri (linearly) controlled by a suitable norm of the deviation of the solution from a constant. The proof relies on and elaborates arguments developed by Aftalion, Busca, and Reichel.

Original languageEnglish
Pages (from-to)337-353
Number of pages17
JournalJournal d'Analyse Mathematique
Volume128
Issue number1
DOIs
Publication statusPublished - 2016 Feb 1

ASJC Scopus subject areas

  • Analysis
  • Mathematics(all)

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