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

Abstract

We complete the construction of the theory of interior capacities of condensers in locally compact spaces begun in the previous two parts of the work. A condenser is understood as an ordered finite collection of sets each of which is marked with the sign + or − so that the closures of sets with opposite signs are mutually disjoint. The theory developed here is rich in content for arbitrary (not necessarily compact or closed) condensers. We obtain sufficient and (or) necessary conditions for the solvability of the main minimum problem of the theory of capacities of condensers and show that, under fairly general assumptions, these conditions form a criterion. For the main minimum problem (generally speaking, unsolvable even for a closed condenser), we pose and solve dual problems that are always solvable (even in the case of a nonclosed condenser). For all extremal problems indicated, we describe the potentials of minimal measures and investigate properties of extremals. As an auxiliary result, we solve the well-known problem of the existence of a condenser measure. The theory developed here includes (as special cases) the main results of the theory of capacities of condensers in \(\mathbb{R}^n\) , n ≥ 2, with respect to the classical kernels.