Minimizing measures on condensers of infinitely many plates

TL;DR

This paper investigates the minimal energy problem for infinite-dimensional vector measures on condensers with infinitely many plates, establishing conditions for equilibrium measures and analyzing their properties in electrostatic models.

Contribution

It introduces new sufficient conditions for the existence and uniqueness of equilibrium measures in infinite condenser problems with positive definite kernels.

Findings

Existence of equilibrium measures under certain conditions

Uniqueness and compactness properties of these measures

Continuity results for measures in the electrostatic framework

Abstract

The study deals with a minimal energy problem over noncompact classes of infinite dimensional vector measures in a locally compact space. The components are positive measures (charges) satisfying certain normalizing assumptions and supported by given closed sets (plates) with the sign +1 or -1 prescribed such that oppositely signed sets are mutually disjoint, and the interaction matrix for the charges corresponds to an electrostatic interpretation of a condenser. For all positive definite kernels satisfying Fuglede's condition of consistency between the weak* and strong topologies, sufficient conditions for the existence of equilibrium measures are established and properties of their uniqueness, vague compactness, and continuity are studied.