Condensers with Touching Plates and Constrained Minimum Riesz and Green Energy Problems

  • P. D. Dragnev
  • B. Fuglede
  • D. P. Hardin
  • E. B. Saff
  • N. ZoriiEmail author


We study minimum energy problems relative to the \(\alpha \)-Riesz kernel \(|x-y|^{\alpha -n}\), \(\alpha \in (0,2]\), over signed Radon measures \(\mu \) on \({\mathbb {R}}^n\), \(n\geqslant 3\), associated with a generalized condenser \((A_1,A_2)\), where \(A_1\) is a relatively closed subset of a domain D and \(A_2={\mathbb {R}}^n\setminus D\). We show that although \(A_2\cap {\mathrm {Cl}}_{{\mathbb {R}}^n}A_1\) may have nonzero capacity, this minimum energy problem is uniquely solvable (even in the presence of an external field) if we restrict ourselves to \(\mu \) with \(\mu ^+\leqslant \xi \), where a constraint \(\xi \) is properly chosen. We establish the sharpness of the sufficient conditions on the solvability thus obtained, provide descriptions of the weighted \(\alpha \)-Riesz potentials of the solutions, single out their characteristic properties, and analyze their supports. The approach developed is mainly based on the establishment of an intimate relationship between the constrained minimum \(\alpha \)-Riesz energy problem over signed measures associated with \((A_1,A_2)\) and the constrained minimum \(\alpha \)-Green energy problem over positive measures carried by \(A_1\). The results are illustrated by examples.


Constrained minimum energy problems \(\alpha \)-Riesz kernels \(\alpha \)-Green kernels External fields Condensers with touching plates 

Mathematics Subject Classification




The authors express their sincere gratitude to the anonymous referees for valuable suggestions, helping us improve the exposition of the paper.


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Authors and Affiliations

  • P. D. Dragnev
    • 1
  • B. Fuglede
    • 2
  • D. P. Hardin
    • 3
  • E. B. Saff
    • 3
  • N. Zorii
    • 4
    Email author
  1. 1.Department of Mathematical SciencesPurdue University Fort WayneFort WayneUSA
  2. 2.Department of Mathematical SciencesUniversity of CopenhagenCopenhagenDenmark
  3. 3.Department of Mathematics, Center for Constructive ApproximationVanderbilt UniversityNashvilleUSA
  4. 4.Institute of Mathematics of National Academy of Sciences of UkraineKiev-4Ukraine

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