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Extremal Problems in Logarithmic Potential Theory

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Abstract

We pose and solve an extremal problem of logarithmic potential theory that is dual to the main minimum problem in the theory of interior capacities of condensers but, in contrast to the latter, it is solvable even in the case of a nonclosed condenser. Its solution is a natural generalization of the classical notion of interior equilibrium measure of a set. A condenser is treated as a finite collection of signed sets such that the closures of sets with opposite signs are pairwise disjoint. We also prove several assertions on the continuity of extremals.

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REFERENCES

  1. N. V. Zorii, “Extremal problems in the theory of capacities of condensers in locally compact spaces. I,” Ukr. Mat. Zh., 53, No. 2, 168–189 (2001).

    Google Scholar 

  2. N. V. Zorii, “Extremal problems in the theory of capacities of condensers in locally compact spaces. II,” Ukr. Mat. Zh., 53, No. 4, 466–488 (2001).

    Google Scholar 

  3. N. V. Zorii, “Extremal problems in the theory of capacities of condensers in locally compact spaces. III,” Ukr. Mat. Zh., 53, No. 6, 758–782 (2001).

    Google Scholar 

  4. N. S. Landkof, Foundations of Modern Potential Theory [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  5. A. A. Gonchar and E. A. Rakhmanov, “On the equilibrium problem for vector potentials,” Usp. Mat. Nauk, 40, Issue 4 (244), 155–156 (1985).

    Google Scholar 

  6. N. Bourbaki, Integration. Measures, Integration of Measures [Russian translation], Nauka, Moscow (1967).

    Google Scholar 

  7. B. Fuglede, “On the theory of potentials in locally compact spaces,” Acta Math., 103, No. 3–4, 139–215 (1960).

    Google Scholar 

  8. M. Ohtsuka, “On potentials in locally compact spaces,” J. Sci. Hiroshima Univ. Ser. A-1, 25, No. 2, 135–352 (1961).

    Google Scholar 

  9. J. L. Kelley, General Topology [Russian translation], Nauka, Moscow (1981).

    Google Scholar 

  10. R. E. Edwards, Functional Analysis. Theory and Applications [Russian translation], Mir, Moscow (1969).

    Google Scholar 

  11. T. Bagby, “The modulus of a plane condenser,” J. Math. Mech., 17, No. 4, 315–329 (1967).

    Google Scholar 

  12. P. M. Tamrazov, “On variational problems in the theory of logarithmic potentials,” in: Investigations in Potential Theory [in Russian], Preprint No. 80.25, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1980), pp. 3–13.

    Google Scholar 

  13. N. Bourbaki, General Topology. Main Structures [Russian translation], Nauka, Moscow (1968).

    Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev

    N. V. Zorii

  2. Shevchenko Kiev University, Kiev

    A. A. Latyshev

Authors
  1. N. V. Zorii
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  2. A. A. Latyshev
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Zorii, N.V., Latyshev, A.A. Extremal Problems in Logarithmic Potential Theory. Ukrainian Mathematical Journal 54, 1471–1491 (2002). https://doi.org/10.1023/A:1023415918753

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  • DOI: https://doi.org/10.1023/A:1023415918753

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