Various concepts of Riesz energy of measures and application to condensers with touching plates

TL;DR

This paper studies weak Riesz energy minimization problems for generalized condensers with touching plates, providing conditions for solutions and linking them to Green energy problems, with applications to equilibrium measures.

Contribution

It introduces new concepts of weak Riesz energy for condensers with intersecting plates and establishes a connection to Green energy problems, advancing potential theory.

Findings

Established existence conditions for minimizers.

Described supports and potentials of solutions.

Linked weak Riesz energy problems to Green energy problems.

Abstract

We investigate minimum weak $\alpha$-Riesz energy problems with external fields in both the unconstrained and constrained settings for generalized condensers $(A_1,A_2)$ such that the closures of $A_1$ and $A_2$ in $\mathbb R^n$ are allowed to intersect one another. (Such problems with the standard $\alpha$-Riesz energy in place of the weak one would be unsolvable, which justifies the need for the concept of weak energy when dealing with condenser problems.) We obtain sufficient and/or necessary conditions for the existence of minimizers, provide descriptions of their supports and potentials, and single out their characteristic properties. To this end we have discovered an intimate relation between minimum weak $\alpha$-Riesz energy problems over signed measures associated with $(A_1,A_2)$ and minimum $\alpha$-Green energy problems over positive measures carried by $A_1$. Crucial for our analysis of the latter problems is the perfectness of the $\alpha$-Green kernel, established in our recent paper. As an application of the results obtained, we describe the support of the $\alpha$-Green equilibrium measure.