Riesz energy problems with external fields and related theory

TL;DR

This paper studies Riesz energy problems with general external fields on unbounded conductors, establishing conditions for equilibrium measures and extending classical potential theory results, including cases with signed external measures.

Contribution

It introduces new conditions for equilibrium measure existence under general external fields and extends classical theorems in Riesz potential theory.

Findings

Established sufficient conditions for equilibrium measures.

Analyzed cases with discrete external measures.

Extended classical theorems like de La Vallée-Poussin's in Riesz potentials.

Abstract

In this paper, we investigate Riesz energy problems on unbounded conductors in $\R^d$ in the presence of general external fields $Q$, not necessarily satisfying the growth condition $Q(x)\to\infty$ as $x\to\infty$ assumed in several previous studies. We provide sufficient conditions on $Q$ for the existence of an equilibrium measure and the compactness of its support. Particular attention is paid to the case of the hyperplanar conductor $\R^{d}$, embedded in $\R^{d+1}$, when the external field is created by the potential of a signed measure $\nu$ outside of $\R^{d}$. Simple cases where $\nu$ is a discrete measure are analyzed in detail. New theoretic results for Riesz potentials, in particular an extension of a classical theorem by de La Vall\'ee-Poussin, are established. These results are of independent interest.