Extremal Problems in the Theory of Capacities of Condensers in Locally Compact Spaces. I

Abstract

The present paper is the first part of a work devoted to the development of the theory of κ-capacities of condensers in a locally compact space X; here, κ: X × X → (−∞, +∞] is a lower-semicontinuous function. Condensers are understood in a generalized sense. We investigate the corresponding problem on the minimum of energy on fairly general classes of normalized signed Radon measures. We describe potentials of minimal measures, establish their characteristic properties, and study the uniqueness problem. (The subsequent two parts of this work are devoted to the problem of existence of minimal measures in the noncompact case and to the development of the corresponding approaches and methods.) As an auxiliary result, we investigate the continuity of the mapping

$$\left( {x,{\mu }} \right) \mapsto \int {\kappa \left( {x,y} \right)} d{\mu }\left( y \right),\quad \left( {x,{\mu }} \right) \in X \times \mathfrak{M}^ + \left( X \right),$$

where $\mathfrak{M}^ +$ is the cone of positive measures in X equipped with the topology of vague convergence.