Constrained energy problems with external fields for infinite dimensional vector measures

TL;DR

This paper investigates a constrained energy minimization problem for infinite dimensional vector measures with external fields, establishing existence, uniqueness, and continuity of solutions under broad conditions, with implications for classical kernels.

Contribution

It introduces new sufficient conditions for the existence and uniqueness of minimizers in constrained energy problems with external fields for infinite dimensional vector measures.

Findings

Existence of minimizers under broad conditions.

Uniqueness and weak* compactness of solutions.

Continuity properties when constraints and plates vary.

Abstract

We consider a constrained minimal energy problem with an external field over noncompact classes of infinite dimensional vector measures on a locally compact space. The components are positive measures (charges) that are constrained from above, satisfy some normalizing assumptions, and are supported by given closed sets (plates) with the sign +1 or -1 prescribed such that the oppositely signed plates 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 minimizers are established and their uniqueness and weak* compactness are studied. Examples illustrating the sharpness of the sufficient conditions are provided. We also analyze continuity properties of minimizers in both the weak* and strong topologies when the constraints and the plates are varied simultaneously. The results are new even for classical kernels in an Euclidean space, which is important in applications.