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Harmonic functions on Riemannian manifolds with ends

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Abstract

We study the problem of solvability of some boundary value problems on noncompact Riemannian manifolds with ends. We obtain the conditions for existence and uniqueness of solutions to the problems as well as the conditions for the fulfillment of Liouville-type theorems for harmonic functions on the manifolds.

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References

  1. Grigor’yan A., “Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds,” Bull. Amer. Math. Soc., 36, No. 2, 135–249 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  2. Yau S. T., “Nonlinear analysis in geometry,” Enseign. Math. (2), 33, 109–158 (1987).

    MATH  MathSciNet  Google Scholar 

  3. Miklyukov V. M., “Some parabolicity and hyperbolicity criteria for boundary sets of surfaces,” Izv. Math., 60, No. 4, 763–809 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  4. Li P. and Tam L. F., “Harmonic functions and the structure of complete manifolds,” J. Differential Geom., 35, No. 2, 359–383 (1992).

    MATH  MathSciNet  Google Scholar 

  5. Grigor’yan A. A., “The set of positive solutions of the Laplace-Beltrami equation on Riemannian manifolds of special form,” Soviet Math., 31, No. 2, 48–60 (1987).

    MATH  Google Scholar 

  6. Losev A. G., “Some Liouville theorems on Riemannian manifolds of a special type,” Soviet Math., 35, No. 12, 15–23 (1991).

    MATH  MathSciNet  Google Scholar 

  7. Korol’kov S. A. and Losev A. G., “On the set of positive solutions of the Laplace-Beltrami equation on model manifolds,” Vestnik VolGU Ser. 1: Mat. Fiz., No. 8, 48–61 (2003–2004).

  8. Colding T. H. and Minicozzi II V. P., “Harmonic functions with polynomial growth,” J. Differential Geom., 46, No. 1, 1–77 (1997).

    MATH  MathSciNet  Google Scholar 

  9. Li P., “Harmonic functions of polynomial growth,” Math. Res. Lect., 4, 35–44 (1997).

    MATH  Google Scholar 

  10. Losev A. G. and Mazepa E. A., “Bounded solutions of the Schrödinger equation on noncompact Riemannian manifolds of a special form,” Dokl. Akad. Nauk, 367, No. 2, 166–167 (1999).

    MATH  MathSciNet  Google Scholar 

  11. Mazepa E. A., “Boundary value problems for the stationary Schrödinger equation on Riemannian manifolds,” Siberian Math. J., 43, No. 3, 473–479 (2002).

    Article  MathSciNet  Google Scholar 

  12. Nakai M., “On the Evans potential,” Proc. Japan Acad., 38, 624–629 (1962).

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Volgograd State University, Volgograd, Russia

    S. A. Korol’kov

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  1. S. A. Korol’kov
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Correspondence to S. A. Korol’kov.

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Original Russian Text Copyright © 2008 Korol’kov S. A.

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Volgograd. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 49, No. 6, pp. 1319–1332, November–December, 2008.

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Korol’kov, S.A. Harmonic functions on Riemannian manifolds with ends. Sib Math J 49, 1051–1061 (2008). https://doi.org/10.1007/s11202-008-0101-1

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  • DOI: https://doi.org/10.1007/s11202-008-0101-1

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