Isoperimetric inequalities for Schatten norms of Riesz potentials

2015/06/21 by Rozenblum, Grigori, Ruzhansky, Michael, Suragan, Durvudkhan
#35P99 #35S15 #47G40 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1506.06355

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.

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