Isoperimetric inequalities for Schatten norms of Riesz potentials

TL;DR

This paper proves that among all domains of a fixed measure, the ball maximizes certain Schatten p-norms of Riesz potential operators, leading to isoperimetric inequalities for eigenvalues of related nonlocal operators.

Contribution

It establishes new isoperimetric inequalities for Schatten norms of Riesz potentials and eigenvalues of polyharmonic operators, extending classical results to nonlocal boundary value problems.

Findings

Balls maximize Schatten p-norms of Riesz potentials among fixed measure domains.

Derived isoperimetric inequalities for eigenvalues of polyharmonic Newton potential operators.

Extended classical inequalities to nonlocal operators related to the poly-Laplacian.

Abstract

In this note we prove that the ball is a maximiser of some Schatten $p$-norms of the Riesz potential operators among all domains of a given measure in $\mathbb R^{d}$. In particular, the result is valid for the polyharmonic Newton potential operator, which is related to a nonlocal boundary value problem for the poly-Laplacian extending the one considered by M. Kac in the case of the Laplacian, so we obtain and isoperimetric inequalities for its eigenvalues as well, namely, analogues of Rayleigh-Faber-Krahn and Hong-Krahn-Szeg\"o inequalities.