Constrained Gauss variational problem for condensers with touching plates

TL;DR

This paper investigates a constrained energy minimization problem involving Riesz kernels for condensers with touching plates, establishing conditions for solvability through a novel metric space framework.

Contribution

It introduces a new metric structure on vector measures for condensers and proves completeness, advancing the understanding of energy problems with touching plates.

Findings

Derived sufficient conditions for problem solvability

Established a completeness theorem for the measure space

Provided a framework for analyzing condensers with touching plates

Abstract

We study a constrained minimum energy problem with an external field relative to the Riesz kernel of an arbitrary order for a generalized condenser with touching oppositely-charged plates. Conditions sufficient for the solvability of the problem are obtained. Our arguments are mainly based on the definition of an appropriate metric structure on a set of vector measures associated with a generalized condenser and the establishment of a completeness theorem for the corresponding metric space.