Interior Capacities of Condensers in Locally Compact Spaces

Abstract

The study deals with the theory of interior capacities of condensers in a locally compact space, a condenser being treated here as a finite collection of arbitrary sets with sign + 1 or − 1 prescribed such that the closures of oppositely signed sets are mutually disjoint. We are motivated by the known fact that, in the noncompact case, the main minimum-problem of the theory is in general unsolvable, and this occurs even under very natural assumptions (e.g., for the Newtonian, Green, or Riesz kernels in \(\mathbb R^n\) and closed condensers). Therefore it was particularly interesting to find statements of variational problems dual to the main minimum-problem (and hence providing new equivalent definitions to the capacity), but now always solvable (e.g., even for nonclosed condensers). For all positive definite kernels satisfying Fuglede’s condition of consistency between the strong and vague (= weak*) topologies, problems with the desired properties are posed and solved. Their solutions provide a natural generalization of the well-known notion of interior equilibrium measures associated with a set. We describe those solutions and the corresponding equilibrium constants, analyze their uniqueness and continuity, and point out their characteristic properties. Such results are new even for classical kernels in \(\mathbb R^n\), which is important in applications.