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

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For a finite collection A=(Aᵢ)ᵢ∈ ᵢ of locally closed sets in Rⁿ, n3, with the sign ±1 prescribed such that the oppositely charged plates are mutually disjoint, we consider the minimum energy problem relative to the α-Riesz kernel |x-y|α⁻ⁿ, α∈(0,2], over positive vector Radon measures μ=(μⁱ)ᵢ∈ ᵢ such that each μⁱ, i∈ I, is carried by Aᵢ and normalized by μⁱ(Aᵢ)=aᵢ∈(0,∞). 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 λξₐ=(λⁱₐ)ᵢ∈ ᵢ (also in the presence of an external field) if we restrict ourselves to μ with μⁱξⁱ, i∈ I, where the constraint ξ=(ξⁱ)ᵢ∈ ᵢ is properly chosen. We establish the sharpness of the sufficient conditions on the solvability thus obtained, provide descriptions of the weighted vector α-Riesz potentials of the solutions, single out their characteristic properties, and analyze the supports of the λⁱₐ, i∈ I. Our approach is based on the simultaneous use of the vague topology and an appropriate semimetric structure defined in terms of the α-Riesz energy on a set of vector measures associated with A, as well as on the establishment of an intimate relationship between the constrained minimum α-Riesz energy problem and a constrained minimum α-Green energy problem, suitably formulated. The results are illustrated by examples.