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On the Dirichlet problem for the Beltrami equations in finitely connected domains

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Ukrainian Mathematical Journal Aims and scope

We establish a series of criteria for the existence of regular solutions of the Dirichlet problem for degenerate Beltrami equations in arbitrary Jordan domains. We also formulate the corresponding criteria for the existence of pseudoregular and multivalued solutions of the Dirichlet problem in the case of finitely connected domains.

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References

  1. B. V. Boyarskii, “Generalized solutions of a system of first-order elliptic differential equations with discontinuous coefficients,” Mat. Sb., 43(85), 451–503 (1957).

    MathSciNet  Google Scholar 

  2. I. N. Vekua, Generalized Analytic Functions [in Russian], Fizmatgiz, Moscow (1959).

    Google Scholar 

  3. Yu. Dybov, “On regular solutions of the Dirichlet problem for the Beltrami equations,” Complex Var. Elliptic Equat., 55, No. 12, 1099–1116 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  4. V. Ryazanov, U. Srebro, and E. Yakubov, “On strong solutions of the Beltrami equations,” Complex Var. Elliptic Equat., 55, No. 1–3, 219–236 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  5. V. Ryazanov, U. Srebro, and E. Yakubov, “To strong ring solutions of the Beltrami equations,” Uzbek. Math. J., 1, 127–137 (2009).

    MathSciNet  Google Scholar 

  6. T. V. Lomako, “On the extension of some generalizations of quasiconformal mappings to the boundary,” Ukr. Mat. Zh., 61, No. 10, 1329–1337 (2009); English translation: Ukr. Math. J., 61, No. 10, 1568–1577 (2009).

    Article  MathSciNet  Google Scholar 

  7. A. Hurwitz and R. Courant, Allgemeine Funktionentheorie und Elliptische Functionen, Springer, Berlin (1964).

    Book  Google Scholar 

  8. B. V. Boyarskii, V. Ya. Gutlyanskii, and V. I. Ryazanov, “General Beltrami equations and BMO,” Ukr. Mat. Vestn., 5, No. 3, 305–326 (2008).

    Google Scholar 

  9. B. Bojarski, V. Gutlyanskii, and V. Ryazanov, “On the Beltrami equations with two characteristics,” Complex Var. Ellipt. Equat., 54, No. 10, 935–950 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  10. B. Bojarski, V. Gutlyanskii, and V. Ryazanov, “On integral conditions for the general Beltrami equations,” Compl. Anal. Oper. Theory, 5, No. 3, 835–845 (2011).

    Article  MathSciNet  Google Scholar 

  11. S. L. Krushkal’ and R. Kühnay, Quasiconformal Mappings—New Methods and Applications [in Russian], Nauka, Novosibirsk (1984).

    Google Scholar 

  12. F. W. Gehring, “Rings and quasiconformal mappings in space,” Trans. Amer. Math. Soc., 103, 353–393 (1962).

    Article  MathSciNet  MATH  Google Scholar 

  13. V. Ryazanov, U. Srebro, and E. Yakubov, “On ring solutions of Beltrami equations,” J. Anal. Math., 96, 117–150 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  14. O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, Moduli in Modern Mapping Theory, Springer, New York (2009).

    MATH  Google Scholar 

  15. R. L. Wilder, Topology of Manifolds, American Mathematical Society, New York (1949).

    MATH  Google Scholar 

  16. E. S. Afanas’eva and V. I. Ryazanov, “Regular domains in the theory of mappings on Riemann manifolds,” in: Proc. of the Institute of Applied Mathematics and Mechanics, Ukrainian National Academy of Sciences [in Russian], 22 (2011), pp. 23–32.

  17. A. A. Ignat’ev and V. I. Ryazanov, “Finite mean oscillation in the theory of mappings,” Ukr. Mat. Vestn., 2, No. 3, 395–417 (2005).

    MathSciNet  MATH  Google Scholar 

  18. G. M. Goluzin, Geometric Theory of Functions of Complex Variable [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  19. V. Ryazanov, U. Srebro, and E. Yakubov, “Integral conditions in the mapping theory,” Ukr. Mat. Vestn., 7, No. 1, 524–535 (2010).

    MathSciNet  Google Scholar 

  20. S. Stoilov, Lectures on the Topological Principles of the Theory of Analytic Functions [in Russian], Nauka, Moscow (1964).

    Google Scholar 

  21. V. Ryazanov, U. Srebro, and E. Yakubov, “Integral conditions in the theory of the Beltrami equations,” Complex Var. Elliptic Equat., DOI: 10.1080/17476933.2010.534790 (2011).

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 7, pp. 932–944, July, 2012.

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Kovtonyuk, D.A., Petkov, I.V. & Ryazanov, V.I. On the Dirichlet problem for the Beltrami equations in finitely connected domains. Ukr Math J 64, 1064–1077 (2012). https://doi.org/10.1007/s11253-012-0699-9

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  • DOI: https://doi.org/10.1007/s11253-012-0699-9

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