2018/02/19 by Fuglede, Bent, Zorii, Natalia
#31C15 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.1802.07171
For a finite collection \mathbf A=(Ai)i∈ I of locally closed sets in \mathbb Rn, n\geqslant3, 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 \boldsymbolμ=(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 \boldsymbolλ\boldsymbolξ\mathbf A=(λi\mathbf A)i∈ I (also in the presence of an external field) if we restrict ourselves to \boldsymbolμ with μi\leqslantξi, i∈ I, where the constraint \boldsymbolξ=(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 λi\mathbf A, 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 \mathbf 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.