Condensers with touching plates and constrained minimum Riesz and Green energy problems

TL;DR

This paper investigates constrained minimum energy problems involving touching plates and Riesz and Green energies, establishing conditions for unique solvability and analyzing the properties and supports of solutions.

Contribution

It introduces a new approach linking Riesz and Green energy problems, providing solvability conditions and detailed descriptions of solutions in complex condenser configurations.

Findings

Unique solvability under certain constraints despite nonzero capacity intersections

Characterization of solutions' potentials and supports

Illustrative examples demonstrating theoretical results

Abstract

We study minimum energy problems relative to the $\alpha$-Riesz kernel $|x-y|^{\alpha-n}$, $\alpha\in(0,2]$, over signed Radon measures $\mu$ on $\mathbb R^n$, $n\geqslant3$, associated with a generalized condenser $(A_1,A_2)$, where $A_1$ is a relatively closed subset of a domain $D$ and $A_2=\mathbb R^n\setminus D$. We show that, though $A_2\cap\mathrm{Cl}_{\mathbb R^n}A_1$ may have nonzero capacity, this minimum energy problem is uniquely solvable (even in the presence of an external field) if we restrict ourselves to $\mu$ with $\mu^+\leqslant\xi$, where a constraint $\xi$ is properly chosen. We establish the sharpness of the sufficient conditions on the solvability thus obtained, provide descriptions of the weighted $\alpha$-Riesz potentials of the solutions, single out their characteristic properties, and analyze their supports. The approach developed is mainly based on the establishment of an intimate relationship between the constrained minimum $\alpha$-Riesz energy problem over signed measures associated with $(A_1,A_2)$ and the constrained minimum $\alpha$-Green energy problem over positive measures carried by $A_1$. The results are illustrated by examples.