General theory of balayage on locally compact spaces. Applications to weighted minimum energy problems

TL;DR

This paper develops a comprehensive theory of balayage for Radon measures on locally compact spaces, extending classical potential theory and enabling advanced solutions to weighted minimum energy problems.

Contribution

It introduces new methods for balayage of general Radon measures onto broad sets, extending existing theories beyond finite energy measures and classical Newtonian potentials.

Findings

Established existence and uniqueness of balayage measures.

Provided alternative characterizations of balayage.

Enhanced solutions to the Gauss variational problem.

Abstract

Under suitable requirements on a kernel on a locally compact space, we develop a theory of inner (outer) balayage of quite general Radon measures $\omega$ (not necessarily of finite energy) onto quite general sets (not necessarily closed). We prove the existence and the uniqueness of inner (outer) swept measures, analyze their properties, and provide a number of alternative characterizations. In spite of being in agreement with Cartan's theory of Newtonian balayage, the results obtained require essentially new methods and approaches, since in the case in question, useful specific features of Newtonian potentials may fail to hold. The theory thereby established extends considerably that by Fuglede (Anal. Math., 2016) and that by the author (Anal. Math., 2022), these two dealing with $\omega$ of finite energy. Such a generalization enables us to improve substantially our recent results on the Gauss variational problem (Constr. Approx., 2024), by strengthening their formulations and/or by extending the area of their validity. This study covers many interesting kernels in classical and modern potential theory, which also looks promising for other applications.