A concept of weak Riesz energy with application to condensers with touching plates

TL;DR

This paper advances the study of weak Riesz energy problems for condensers with touching plates, establishing conditions for solutions and demonstrating the advantages of weak energy over standard energy in such configurations.

Contribution

It introduces a Gauss type problem involving weak energy with added constraints and external sources, and analyzes the existence, properties, and topology dependence of solutions.

Findings

Criteria for solution existence are established.

Weak energy provides advantages over standard energy in touching plate condensers.

Solutions depend continuously on the condenser and constraints.

Abstract

We proceed further with the study of minimum weak Riesz energy problems for condensers with touching plates, initiated jointly with Bent Fuglede (Potential Anal. 51 (2019), 197--217). Having now added to the analysis constraint and external source of energy, we obtain a Gauss type problem, but with weak energy involved. We establish sufficient and/or necessary conditions for the existence of solutions to the problem and describe their potentials. Treating the solution as a function of the condenser and the constraint, we prove its continuity relative to the vague topology and the topologies determined by the weak and standard energy norms. We show that the criteria for the solvability thus obtained fail in general once the problem is reformulated in the setting of standard energy, thereby justifying an advantage of weak energy when dealing with condensers with touching plates.