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November 2019 Constrained minimum Riesz and Green energy problems for vector measures associated with a generalized condenser
Bent Fuglede, Natalia Zorii
Hiroshima Math. J. 49(3): 399-437 (November 2019). DOI: 10.32917/hmj/1573787036

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.

Citation

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Bent Fuglede. Natalia Zorii. "Constrained minimum Riesz and Green energy problems for vector measures associated with a generalized condenser." Hiroshima Math. J. 49 (3) 399 - 437, November 2019. https://doi.org/10.32917/hmj/1573787036

Information

Received: 28 December 2017; Revised: 25 May 2019; Published: November 2019
First available in Project Euclid: 15 November 2019

zbMATH: 07180035
MathSciNet: MR4031738
Digital Object Identifier: 10.32917/hmj/1573787036

Subjects:
Primary: 31C15

Rights: Copyright © 2019 Hiroshima University, Mathematics Program

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Vol.49 • No. 3 • November 2019
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