Riesz energy problems with external fields and related theory

Abstract

In this paper, we investigate Riesz energy problems on unbounded conductors in ${\mathbb{R}}^d$ 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 ${\mathbb{R}}^d$, embedded in ${\mathbb{R}}^{d+1}$, when the external field is created by the potential of a signed measure $\nu$ outside of ${\mathbb{R}}^d$. 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 Vallée-Poussin, are established. These results are of independent interest.