Minimum Riesz energy problems for a condenser with "touching plates"

TL;DR

This paper investigates minimum Riesz energy problems for condensers with touching plates, establishing conditions for solvability, properties of minimizers, and their uniqueness, with applications illustrated through examples.

Contribution

It provides new solvability criteria, characterizes minimizers via variational inequalities, and extends understanding of Riesz energy problems in complex condenser configurations.

Findings

Conditions for problem solvability are established.

Uniqueness of minimizers is proved.

Characterizations via variational inequalities are provided.

Abstract

Minimum Riesz energy problems in the presence of an external field are analyzed for a condenser with touching plates. We obtain sufficient and/or necessary conditions for the solvability of these problems in both the unconstrained and the constrained settings, investigate the properties of minimizers, and prove their uniqueness. Furthermore, characterization theorems in terms of variational inequalities for the weighted potentials are established. The results obtained are illustrated by several examples.