Minimum Riesz energy problems with external fields

TL;DR

This paper advances the theory of minimum Riesz energy problems with external fields by establishing existence conditions, characterizations, and support descriptions of minimizers on general sets using a novel topological approach.

Contribution

It introduces new necessary and sufficient conditions for minimizer existence and provides comprehensive characterizations and support descriptions, improving the theoretical framework significantly.

Findings

Established existence criteria for minimizers under general external fields.

Provided new characterizations of the Riesz energy minimizers.

Analyzed the continuity of minimizers and their associated constants.

Abstract

The paper deals with minimum energy problems in the presence of external fields with respect to the Riesz kernels $|x-y|^{\alpha-n}$, $0<\alpha<n$, on $\mathbb R^n$, $n\geqslant2$. For quite a general (not necessarily lower semicontinuous) external field $f$, we obtain necessary and/or sufficient conditions for the existence of $\lambda_{A,f}$ minimizing the Gauss functional \[\int|x-y|^{\alpha-n}\,d(\mu\otimes\mu)(x,y)+2\int f\,d\mu\] over all positive Radon measures $\mu$ with $\mu(\mathbb R^n)=1$, concentrated on quite a general (not necessarily closed) $A\subset\mathbb R^n$. We also provide various alternative characterizations of the minimizer $\lambda_{A,f}$, analyze the continuity of both $\lambda_{A,f}$ and the modified Robin constant for monotone families of sets, and give a description of the support of $\lambda_{A,f}$. The significant improvement of the theory in question thereby achieved is due to a new approach based on the close interaction between the strong and the vague topologies, as well as on the theory of inner balayage, developed recently by the author.