Some results on radial symmetry in partial differential equations

Research output: Journal PublicationArticlepeer-review

2 Citations (Scopus)

Abstract

In this paper we will discuss three different problems which share the same conclusions. In the first one we revisit the well known Faber-Krahn inequality for the principal eigenvalue of the p-Laplace operator with zero homogeneous Dirichlet boundary conditions. Motivated by Chatelain, Choulli, and Henrot, 1996, we show in case the equality holds in the Faber-Krahn inequality, the domain of interest must be a ball. In the second problem we consider a generalization of the well known torsion problem and accordingly define a quantity that we name the p-torsional rigidity of the domain of interest. We maximize this quantity relative to a set of domains having the same volume, and prove that the optimal domain is a ball. The last problem is very similar in spirit to the second one. We consider a Hamilton-Jacobi boundary value problem, and define a quantity to be maximized relative to a set of domains having fixed volume. Again, we prove that the optimal domain is a ball. The main tools in our analysis are the method of domain derivatives, an appropriate generalized version of the Pohozaev identity, and the classical symmetrization techniques.

Original languageEnglish
Pages (from-to)241-255
Number of pages15
JournalNew York Journal of Mathematics
Volume20
Publication statusPublished - 2014

Keywords

  • Domain derivative
  • Equality case
  • Faber-krahn inequality
  • Hamilton-jacobi system
  • Maximization
  • P-laplace
  • Pohozaev identity
  • Principal eigenvalue
  • Volume constraint

ASJC Scopus subject areas

  • Mathematics (all)

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