Constrained minimum Riesz and Green energy problems for vector measures associated with a generalized condenser

Abstract

For a finite collection $\mathbf{A}=(A_i)_{i \in I}$ of locally closed sets in $\mathbb R ^n$, $n \geqslant 3$, with the sign $s_i=\pm 1$ prescribed such that the oppositely charged plates are mutually disjoint, we consider the minimum energy problem relative to the $\alpha$-Riesz kernel $|x − y|^{\alpha-n}, \alpha \in (0,2]$, over positive vector Radon measures $\mathbf{μ}=(\mu^i)_{i \in I}$ such that each $\mu^i, i \in I$, is carried by $A_i$ and normalized by $\mu^1(A_i)=a_i \in (0, \infty)$, while the interaction between $\mu^i, i \in I$ , is determined by the matrix $(s_is_j)_{i,j\in I^\bullet}$. We show that, though the closures of oppositely charged plates may intersect each other even in a set of nonzero capacity, this problem has a solution $\mathbf{λ}_\mathbf{A}^\xi=(\lambda_\mathbf{A}^i)_{i \in I}$ (also in the presence of an external field) if we restrict ourselves to $\mathbf{μ}$ with $\mu^i\leqslant\xi^i, i \in I$, where the constraint $\mathbf{ξ}=(\xi^i)_{i \in I}$ is properly chosen. We establish the sharpness of the suffcient conditions on the solvability thus obtained, provide descriptions of the weighted vector $\alpha$-Riesz potentials of the solutions, single out their characteristic properties, and analyze the supports of the $\lambda_\mathbf{A}^i, i \in I$. Our approach is based on the simultaneous use of the vague topology and an appropriate semimetric structure defined in terms of the $\alpha$-Riesz energy on a set of vector measures associated with $\mathbf{A}$, as well as on the establishment of an intimate relationship between the constrained minimum $\alpha$-Riesz energy problem and a constrained minimum $\alpha$-Green energy problem, suitably formulated. The results are illustrated by examples.