Isoperimetric inequalities for the logarithmic potential operator

Michael Ruzhansky; Durvudkhan Suragan

doi:10.1016/j.jmaa.2015.07.041

Isoperimetric inequalities for the logarithmic potential operator

Michael Ruzhansky
, Durvudkhan Suragan

Department of Mathematics

Imperial College London
Al Farabi Kazakh National University

Research output: Contribution to journal › Article › peer-review

28 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 language	English
Pages (from-to)	1676-1689
Number of pages	14
Journal	Journal of Mathematical Analysis and Applications
Volume	434
Issue number	2
DOIs	https://doi.org/10.1016/j.jmaa.2015.07.041
Publication status	Published - Jan 1 2016

Funding

The authors were supported in parts by the EPSRC grant EP/K039407/1 and by the Leverhulme Trust Grant RPG-2014-02 , as well as by the MESRK grant 5127/GF4 .

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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10.1016/j.jmaa.2015.07.041

Cite this

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title = "Isoperimetric inequalities for the logarithmic potential operator",

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 P{\'o}lya 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.",

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N2 - 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.

AB - 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.

KW - Characteristic numbers

KW - Isoperimetric inequality

KW - Logarithmic potential

KW - Pólya inequality

KW - Rayleigh-Faber-Krahn inequality

KW - Schatten class

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Isoperimetric inequalities for the logarithmic potential operator

Abstract

Funding

Keywords

ASJC Scopus subject areas

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