Constrained minimum Riesz energy problems for a condenser with intersecting plates

TL;DR

This paper investigates the existence, uniqueness, and properties of minimizers for constrained minimum Riesz energy problems involving intersecting plates in a generalized condenser, with applications to potential theory.

Contribution

It establishes sharp conditions for the existence and uniqueness of minimizers, analyzes their continuity, and develops a framework using vague topology and semimetric spaces for vector measures.

Findings

Existence and uniqueness conditions for minimizers are established.

Continuity properties of minimizers are analyzed under varying conditions.

Results extend to logarithmic kernels and compact condensers.

Abstract

We study the constrained minimum energy problem with an external field relative to the $\alpha$-Riesz kernel $|x-y|^{\alpha-n}$ of order $\alpha\in(0,n)$ for a generalized condenser $\mathbf A=(A_i)_{i\in I}$ in $\mathbb R^n$, $n\geqslant 3$, whose oppositely charged plates intersect each other over a set of zero capacity. Conditions sufficient for the existence of minimizers are found, and their uniqueness and vague compactness are studied. Conditions obtained are shown to be sharp. We also analyze continuity of the minimizers in the vague and strong topologies when the condenser and the constraint both vary, describe the weighted equilibrium vector potentials, and single out their characteristic properties. Our arguments are based particularly on the simultaneous use of the vague topology and a suitable semimetric structure on a set of vector measures associated with $\mathbf A$, and the establishment of completeness theorems for proper semimetric spaces. The results remain valid for the logarithmic kernel on $\mathbb R^2$ and $\mathbf A$ with compact $A_i$, $i\in I$. The study is illustrated by several examples.