Isoperimetric inequalities for some integral operators arising in potential theory

TL;DR

This paper reviews isoperimetric inequalities for logarithmic and Newton potential operators, highlighting their use in deriving bounds for spectral invariants of these operators on arbitrary domains.

Contribution

It provides a review of previous results on isoperimetric inequalities for potential operators and demonstrates their application in bounding spectral invariants.

Findings

Explicit examples of bounds for spectral invariants.

Reinforcement of the importance of geometric extremum problems.

Connections between potential theory and spectral analysis.

Abstract

In this paper we review our previous isoperimetric results for the logarithmic potential and Newton potential operators. The main reason why the results are useful, beyond the intrinsic interest of geometric extremum problems, is that they produce a priori bounds for spectral invariants of operators on arbitrary domains. We demonstrate these in explicit examples.