Inner Riesz balayage in minimum energy problems with external fields

TL;DR

This paper investigates the existence and properties of minimum energy measures in Riesz potential problems with external fields, providing new conditions for existence and characterizations of solutions.

Contribution

It establishes new necessary and sufficient conditions for the existence of solutions in Riesz energy problems with external fields, improving recent results.

Findings

Existence of solutions depends on inner capacity and balayage measure finiteness.

Provides alternative characterizations of the solution measure.

Analyzes the support of the solution measure.

Abstract

For the Riesz kernel $\kappa_\alpha(x,y):=|x-y|^{\alpha-n}$ on $\mathbb R^n$, where $n\geqslant2$, $\alpha\in(0,2]$, and $\alpha<n$, we consider the problem of minimizing the Gauss functional \[\int\kappa_\alpha(x,y)\,d(\mu\otimes\mu)(x,y)+2\int f\,d\mu,\quad\text{where $f:=-\int\kappa_\alpha(\cdot,y)\,d\omega(y)$},\] $\omega$ being a given positive (Radon) measure on $\mathbb R^n$, and $\mu$ ranging over all positive measures of finite energy, concentrated on $A\subset\mathbb R^n$ and having unit total mass. We prove that if $A$ is a quasiclosed set of nonzero inner capacity $c_*(A)$, and if the inner balayage $\omega^A$ of $\omega$ onto $A$ is of finite energy, then the solution $\lambda_{A,f}$ to the problem in question exists if and only if either $c_*(A)<\infty$, or $\omega^A(\mathbb R^n)\geqslant1$. Despite its simple form, this result improves substantially some of the latest ones, e.g. those by Dragnev et al. (Constr. Approx., 2023) as well as those by the author (J. Math. Anal. Appl., 2023). We also provide alternative characterizations of $\lambda_{A,f}$, and analyze its support.