Isoperimetric inequalities for the logarithmic potential operator

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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 2p<. 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.

Keywords

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

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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.