Isoperimetric inequalities for Schatten norms of Riesz potentials

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-Szegö inequalities.

arXiv comment14 pages; This is a revised version to replace previous ones