Isoperimetric inequalities for the logarithmic potential operator

Michael Ruzhansky, Durvudkhan Suragan

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21 Citations (Scopus)

Abstract

In this paper we prove that the disc is a maximiser of the Schatten p-norm of the logarithmic potential operator among all domains of a given measure in R2, for all even integers 2≤p<∞. We also show that the equilateral triangle has the largest Schatten p-norm among all triangles of a given area. For the logarithmic potential operator on bounded open or triangular domains, we also obtain analogies of the Rayleigh-Faber-Krahn or Pólya inequalities, respectively. The logarithmic potential operator can be related to a nonlocal boundary value problem for the Laplacian, so we obtain isoperimetric inequalities for its eigenvalues as well.

Original languageEnglish
Pages (from-to)1676-1689
Number of pages14
JournalJournal of Mathematical Analysis and Applications
Volume434
Issue number2
DOIs
Publication statusPublished - Jan 1 2016

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Keywords

  • Characteristic numbers
  • Isoperimetric inequality
  • Logarithmic potential
  • Pólya inequality
  • Rayleigh-Faber-Krahn inequality
  • Schatten class

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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