Interior capacities of condensers in locally compact spaces

TL;DR

This paper investigates the interior capacities of condensers in locally compact spaces, addressing solvability issues of variational problems and introducing dual problems that are always solvable, thus generalizing classical capacity concepts.

Contribution

It formulates and solves dual variational problems for interior capacities, providing new definitions and properties of interior capacitary distributions in a broad setting.

Findings

Dual problems are always solvable under certain kernel conditions.

Solutions are characterized by uniqueness and continuity properties.

The work generalizes classical notions of capacitary distributions.

Abstract

The study is motivated by the known fact that, in the noncompact case, the main minimum-problem of the theory of interior capacities of condensers in a locally compact space is in general unsolvable, and this occurs even under very natural assumptions (e.g., for the Newton, Green, or Riesz kernels in an Euclidean space and closed condensers). Therefore it was particularly interesting to find statements of variational problems dual to the main minimum-problem (and hence providing some new equivalent definitions of the capacity), but always solvable (e.g., even for nonclosed condensers). For all positive definite kernels satisfying B. Fuglede's condition of consistency between the strong and vague topologies, problems with the desired properties are posed and solved. Their solutions provide a natural generalization of the well-known notion of interior capacitary distributions associated with a set. We give a description of those solutions, establish statements on their uniqueness and continuity, and point out their characteristic properties. A condenser is treated as a finite collection of arbitrary sets with sing +1 or -1 prescribed, such that the closures of opposite-signed sets are mutually disjoint.