Isoperimetric inequalities for the logarithmic potential operator

TL;DR

This paper establishes isoperimetric inequalities for the Schatten p-norms of the logarithmic potential operator, showing that disks and equilateral triangles maximize these norms among domains of fixed measure or area.

Contribution

It proves that the disk maximizes the Schatten p-norm of the logarithmic potential operator among all domains of fixed measure, and the equilateral triangle among all triangles of fixed area, extending classical isoperimetric results.

Findings

Disks maximize Schatten p-norms among fixed measure domains.

Equilateral triangles maximize Schatten p-norms among fixed area triangles.

Analogues of Rayleigh-Faber-Krahn 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 $\mathbb R^{2}$, for all even integers $2\leq p<\infty$. 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.