Constrained energy problems with external fields

TL;DR

This paper investigates a minimal energy problem involving external fields and constraints on measures supported on noncompact sets, establishing existence, continuity, and potential properties of solutions under general conditions.

Contribution

It introduces new existence and characterization results for constrained energy minimization problems with external fields, applicable even to classical kernels.

Findings

Existence of a minimizing measure under broad conditions

Continuity properties of the measure with respect to set and constraint variations

Characterization of the measure's weighted potential

Abstract

Given a positive definite kernel in a locally compact space, we study a minimal energy problem in the presence of an external field over the class of all nonnegative Radon measures that are supported by a given closed noncompact set, satisfy certain normalizing assumptions, and do not exceed a fixed measure serving as a constraint. Under general assumptions, we establish the existence of a minimizing measure and analyze its continuity properties in the weak* and strong topologies when the set and the constraint are both varied. We also give a description of the weighted potential of a minimizing measure and single out its characteristic properties. Such results are mostly new even for classical kernels, which is important in applications.