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Constrained minimum Riesz energy problems for a condenser with intersecting plates

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Abstract

We study the constrained minimum energy problem with an external field relative to the α-Riesz kernel x−yα−n of order α ∈ (0, n) for a generalized condenser A = (Ai)i∈I in ℝn, n ⩾ 3, whose oppositely charged plates intersect each other over a set of zero capacity. Conditions sufficient for the existence of minimizers are found, and their uniqueness and vague compactness are studied. Conditions obtained are shown to be sharp. We also analyze continuity of the minimizers in the vague and strong topologies when the condenser and the constraint both vary, describe the weighted equilibrium vector potentials, and single out their characteristic properties. Our arguments are based particularly on the simultaneous use of the vague topology and a suitable semimetric structure on a set of vector measures associated with A, and the establishment of completeness theorems for proper semimetric spaces. The results remain valid for the logarithmic kernel on ℝ2 and A with compact Ai, i ∈ I. The study is illustrated by several examples.

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References

  1. A. I. Aptekarev, Asymptotics of simultaneously orthogonal polynomials in the Angelesco case, Math. USSR Sb. 64 (1989), 57–84.

    MathSciNet  MATH  Google Scholar 

  2. A. I. Aptekarev and A. B. J. Kuijlaars, Hermite-Padé approximations and multiple orthogonal polynomial ensembles, Russ. Math. Surv. 66 (2011), 1133–1199.

    MATH  Google Scholar 

  3. B. Beckermann and A. Gryson, Extremal rational functions on symmetric discrete sets and superlinear convergence of the ADI method, Constr. Approx. 32 (2010), 393–428.

    MathSciNet  MATH  Google Scholar 

  4. B. Beckermann, V. Kalyagin, A. C. Matos and F. Wielonsky, Equilibrium problems for vector potentials with semidefinite interaction matrices and constrained masses, Constr. Approx. 37 (2013), 101–134.

    MathSciNet  MATH  Google Scholar 

  5. N. Bourbaki, Elements of Mathematics. General Topology. Chapters 1-4, Springer, Berlin, 1989.

    MATH  Google Scholar 

  6. N. Bourbaki, Elements of Mathematics. General Topology. Chapters 5-10, Springer, Berlin, 1989.

    MATH  Google Scholar 

  7. N. Bourbaki, Elements of Mathematics. Integration. Chapters 1-6, Springer, Berlin, 2004.

    MATH  Google Scholar 

  8. M. Brelot, On Topologies and Boundaries in Potential Theory, Springer, Berlin, 1971.

    MATH  Google Scholar 

  9. H. Cartan, Théorie du potentiel Newtonien: énergie, capacité, suites de potentiels, Bull. Soc. Math. France 73 (1945), 74–106.

    MathSciNet  MATH  Google Scholar 

  10. H. Cartan, Théorie générale du balayage en potentiel Newtonien, Ann. Univ. Grenoble 22 (1946), 221–280.

    MathSciNet  MATH  Google Scholar 

  11. J. Deny, Les potentiels d'énergie finie, Acta Math. 82 (1950), 107–183.

    MathSciNet  MATH  Google Scholar 

  12. J. Deny, Sur la définition de l'énergie en théorie du potentiel, Ann. Inst. Fourier Grenoble 2 (1950), 83–99.

    MathSciNet  MATH  Google Scholar 

  13. P. D. Dragnev, B. Fuglede, D. P. Hardin, E. B. Saff and N. Zorii, Minimum Riesz energy problems for a condenser with touching plates, Potential Anal. 44 (2016), 543–577.

    MathSciNet  MATH  Google Scholar 

  14. P. D. Dragnev, B. Fuglede, D. P. Hardin, E. B. Saff and N. Zorii, Condensers with touching plates and constrained minimum Riesz and Green energy problems, Constr. Approx. 50 (2019), 369–401.

    MathSciNet  MATH  Google Scholar 

  15. P. D. Dragnev and E. B. Saff, Constrained energy problems with applications to orthogonal polynomials of a discrete variable, J. Anal. Math. 72 (1997), 223–259.

    MathSciNet  MATH  Google Scholar 

  16. R. Edwards, Functional Analysis. Theory and Applications, Holt, Rinehart and Winston, New York, 1965.

    MATH  Google Scholar 

  17. B. Fuglede, On the theory of potentials in locally compact spaces, Acta Math. 103 (1960), 139–215.

    MathSciNet  MATH  Google Scholar 

  18. B. Fuglede, Capacity as a sublinear functional generalizing an integral, Mat. Fys. Medd. Dan. Vid. Selsk. 38 (1971).

  19. B. Fuglede, Symmetric function kernels and sweeping of measures, Analysis Math. 42 (2016), 225–259.

    MathSciNet  MATH  Google Scholar 

  20. B. Fuglede and N. Zorii, Green kernels associated with Riesz kernels, Ann. Acad. Sci. Fenn. Math. 43 (2018), 121–145.

    MathSciNet  MATH  Google Scholar 

  21. C. F. Gauss, Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnisse des Quadrats der Entfernung wirkenden Anziehungs- und Abstoßungs-Kräfte (1839), Werke, Ergänzungsreihe, Band V, Königlichen Gesellschaft der Wissenschaften, Göttingen, 1867, pp. 197-244.

    MATH  Google Scholar 

  22. A. A. Gonchar and E.A. Rakhmanov, On the equilibrium problem for vector potentials, Russ. Math. Surv. 40 (1985), 183–184.

    MathSciNet  MATH  Google Scholar 

  23. H. Harbrecht, W. L. Wendland and N. Zorii, Riesz minimal energy problems on Ck-1,k-manifolds, Math. Nachr. 287 (2014), 48–69.

    MathSciNet  MATH  Google Scholar 

  24. A. Hardy and A. B. J. Kuijlaars, Weakly admissible vector equilibrium problems, J. Approx. Theory 164 (2012), 854–868.

    MathSciNet  MATH  Google Scholar 

  25. J. L. Kelley, General Topology, Princeton, New York, 1957.

    MATH  Google Scholar 

  26. A. B. J. Kuijlaars, Multiple orthogonal polynomials in random matrix theory, in Proceedings of the International Congress of Mathematicians, Vol. III, Hindustan Book Agency, New Delhi, 2010, pp. 1417–1432.

    Google Scholar 

  27. N. S. Landkof, Foundations of Modern Potential Theory, Springer, Berlin, 1972.

    MATH  Google Scholar 

  28. V. G. Lysov and D. N. Tulyakov, On a vector potential-theory equilibrium problem with the Angelesco matrix, Proc. Steklov Inst. Math. 298 (2017), 170–200.

    MathSciNet  MATH  Google Scholar 

  29. E. H. Moore and H. L. Smith, A general theory of limits, Amer. J. Math. 44 (1922), 102–121.

    MathSciNet  MATH  Google Scholar 

  30. G. Of, W. L. Wendland and N. Zorii, On the numerical solution of minimal energy problems, Complex Var. Elliptic Equ. 55 (2010), 991–1012.

    MathSciNet  MATH  Google Scholar 

  31. M. Ohtsuka, Sur un espace complet de mesures positives dans la théorie du potentiel, Proc. Japan Acad. 32 (1956), 311–313.

    MathSciNet  MATH  Google Scholar 

  32. M. Ohtsuka, On potentials in locally compact spaces, J. Sci. Hiroshima Univ. Ser. A-1 25 (1961), 135–352.

    MathSciNet  MATH  Google Scholar 

  33. E.A. Rakhmanov, Equilibrium measure and the distribution of zeros of extremal polynomials of a discrete variable, Sb. Math. 187 (1996), 1213–1228.

    MathSciNet  MATH  Google Scholar 

  34. M. Riesz, Intégrales de Riemann-Liouville et potentiels, Acta Szeged 9 (1938), 1–42.

    MATH  Google Scholar 

  35. M.L. Yattselev, Strong asymptotics of Hermite-Padé approximants for Angelesco systems, Can. J. Math. 68 (2016), 1159–1201.

    MATH  Google Scholar 

  36. N. Zorii, An extremal problem of the minimum of energy for space condensers, Ukrain. Math. J. 38 (1986), 365–369.

    Google Scholar 

  37. N. Zorii, A problem of minimum energy for space condensers and Riesz kernels, Ukrain. Math. J. 41 (1989), 29–36.

    MathSciNet  Google Scholar 

  38. N. Zorii, A noncompact variational problem in Riesz potential theory. I; II, Ukrain. Math. J. 47 (1995), 1541–1553; 48 (1996), 671-682.

    MathSciNet  Google Scholar 

  39. N. Zorii, Equilibrium potentials with external fields, Ukrain. Math. J. 55 (2003), 1423–1444.

    MathSciNet  Google Scholar 

  40. N. Zorii, Interior capacities of condensers in locally compact spaces, Potential Anal. 35 (2011), 103–143.

    MathSciNet  MATH  Google Scholar 

  41. N. Zorii, Constrained energy problems with external fields for vector measures, Math. Nachr. 285 (2012), 1144–1165.

    MathSciNet  MATH  Google Scholar 

  42. N. Zorii, Equilibrium problems for infinite dimensional vector potentials with external fields, Potential Anal. 38 (2013), 397–432.

    MathSciNet  MATH  Google Scholar 

  43. N. Zorii, Necessary and sufficient conditions for the solvability of the Gauss variational problem for infinite dimensional vector measures, Potential Anal. 41 (2014), 81–115.

    MathSciNet  MATH  Google Scholar 

  44. N. Zorii, Constrained Gauss variational problem for condensers with touching plates, Bull. Soc. Sci. Lettres Łódź Sér. Rech. Déform. 65 (2015), 85–99.

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

We express our sincere gratitude to the anonymous referee for valuable suggestions, helping us in improving the exposition of the paper.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Purdue University Fort Wayne, Fort Wayne, IN, 46805, USA

    Peter D. Dragnev

  2. Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100, Copenhagen, Denmark

    Bent Fuglede

  3. Center for Constructive Approximation Department of Mathematics, Vanderbilt University, Nashville, TN, 37240, USA

    Doug P. Hardin & Edward B. Saff

  4. Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivska 3, 01601, Kyiv-4, Ukraine

    Natalia Zorii

Authors
  1. Peter D. Dragnev
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  2. Bent Fuglede
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  3. Doug P. Hardin
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  4. Edward B. Saff
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  5. Natalia Zorii
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Corresponding author

Correspondence to Natalia Zorii.

Additional information

The research of the first author was supported, in part, by a Simons Foundation grant no. 282207.

The research of the third and the fourth authors was supported, in part, by the U.S. National Science Foundation under grants DMS-1516400.

The research of the fifth author was supported, in part, by the Scholar-in-Residence program at PUFW and by the Department of Mathematical Sciences of the University of Copenhagen.

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Dragnev, P.D., Fuglede, B., Hardin, D.P. et al. Constrained minimum Riesz energy problems for a condenser with intersecting plates. JAMA 140, 117–159 (2020). https://doi.org/10.1007/s11854-020-0091-x

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  • DOI: https://doi.org/10.1007/s11854-020-0091-x

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