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

Natalia Zorii

TL;DR

The work develops a comprehensive, kernel-based balayage theory on locally compact spaces that treats general Radon measures, including infinite-energy cases, and general (not necessarily closed) sets. It establishes existence, uniqueness, and robust characterizations of inner and outer swept measures, with a novel approach beyond classical Newtonian potential theory. The results yield a rigorous framework for weighted minimum-energy problems, including solvability criteria, equilibrium constants, and convergence properties under set approximations, with concrete implications for Gauss variational problems and a broad class of kernels (e.g., Riesz, Green). The combination of variational duality, capacity theory, and strong-vague convergence arguments enhances the applicability of balayage to modern potential theory and related applications.

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 $ω$ (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 $ω$ 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.

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

TL;DR

The work develops a comprehensive, kernel-based balayage theory on locally compact spaces that treats general Radon measures, including infinite-energy cases, and general (not necessarily closed) sets. It establishes existence, uniqueness, and robust characterizations of inner and outer swept measures, with a novel approach beyond classical Newtonian potential theory. The results yield a rigorous framework for weighted minimum-energy problems, including solvability criteria, equilibrium constants, and convergence properties under set approximations, with concrete implications for Gauss variational problems and a broad class of kernels (e.g., Riesz, Green). The combination of variational duality, capacity theory, and strong-vague convergence arguments enhances the applicability of balayage to modern potential theory and related applications.

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 (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 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.
Paper Structure (26 sections, 26 theorems, 149 equations)

This paper contains 26 sections, 26 theorems, 149 equations.

Table of Contents

  1. Introduction and main results
  2. Statement of the problem and general conventions
  3. Inner and outer balayage for $\omega\in\mathfrak M^+$
  4. Proofs of Theorems \ref{['th1']} and \ref{["th1'"]}
  5. Proof of Theorem \ref{['th1']}
  6. Remark
  7. Proof of Theorem \ref{["th1'"]}
  8. Further properties of inner and outer swept measures
  9. Applications of balayage to minimum energy problems
  10. On the solvability of problem (\ref{['G']})
  11. On the inner $f$-weighted equilibrium constant for a set $A$
  12. Convergence theorems for $\lambda_{A,f}$ and $c_{A,f}$
  13. Alternative characterizations of the solution $\lambda_{A,f}$
  14. On the support of $\lambda_{A,f}$
  15. Proofs
  16. ...and 11 more sections

Key Result

Theorem 1.3

There is one and the same measure $\omega^A$, called the inner balayage of $\omega$ onto $A$, that is uniquely characterized by any one of the following (i)--(vi).

Theorems & Definitions (50)

  • Remark 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Lemma 2.1
  • proof
  • ...and 40 more