Equilibrium problems for infinite dimensional vector potentials with external fields

TL;DR

This paper investigates the existence and properties of equilibrium measures in infinite-dimensional vector potential problems with external fields, extending classical electrostatic models to more complex, noncompact settings.

Contribution

It establishes sufficient conditions for equilibrium measure existence, analyzes their uniqueness, and studies continuity and variational properties in an infinite-dimensional context.

Findings

Existence of equilibrium measures under certain conditions

Uniqueness and compactness properties of solutions

Continuity of equilibrium constants and potentials

Abstract

The study deals with a minimal energy problem in the presence of an external field over noncompact classes of vector measures of infinite dimension 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. We also obtain variational inequalities for the weighted equilibrium potentials, single out their characteristic properties, and analyze continuity of the equilibrium constants.