Riesz equilibrium on a ball in the external field of a point charge

TL;DR

This paper studies Riesz energy minimization on a ball influenced by a point charge, analyzing equilibrium measures for both attractive and repulsive cases, and extends methods to Coulomb and logarithmic energies across dimensions.

Contribution

It introduces a modified balayage method to determine equilibrium measures under external fields, addressing mass loss phenomena in Riesz potential theory.

Findings

Support of equilibrium measure determined in 1D case

Modified balayage handles mass loss in Riesz balayage

Results extend to Coulomb and logarithmic energies in various dimensions

Abstract

We investigate the Riesz energy minimization problem on a $d$-dimensional ball in the presence of an external field created by a point charge above the ball in $\R^{d+1}$, $d\geq1$. Both cases of an attractive charge and a repulsive charge are considered. The notion of a signed equilibrium measure is one of the main tools in the present study. For the case of a positive (repulsive) charge, the determination of the support of the equilibrium measure is a nontrivial question. We solve it in the one-dimensional case by making use of iterated balayage, a method already applied in logarithmic potential theory. Here we use a modified version of it, in order to handle the phenomenon of mass loss, characteristic of the Riesz balayage of positive measures. Moreover, we also consider minimization of Coulomb energy on the ball in dimension $d\geq2$, and of logarithmic energy on the segment in dimension 1. Different techniques are used for these two cases.