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On the theory of convergence and compactness for Beltrami equations with constraints of set-theoretic type

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Ukrainian Mathematical Journal Aims and scope

We prove theorems on convergence and compactness for the classes of regular solutions of degenerate Beltrami equations with constraints of the set-theoretic type imposed on the complex coefficient and construct variations for these classes.

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References

  1. O. Lehto and K. Virtanen, Quasiconformal Mappings in the Plane, Springer, New York (1973).

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. L. V. Ahlfors, Lectures on Quasiconformal Mappings [Russian translation], Mir, Moscow (1969).

    Google Scholar 

  4. M. Schiffer and G. Schober, “Representation of fundamental solutions for generalized Cauchy–Riemann equations by quasiconformal mappings,” Ann. Acad. Sci. Fenn., Ser. A1, 2, 501–531 (1976).

    MathSciNet  MATH  Google Scholar 

  5. G. David, “Solutions de l’equation de Beltrami avec ∥μ =1,” Ann. Acad. Sci. Fenn., Ser. A1. Math., 13, 25–70 (1988).

    MATH  Google Scholar 

  6. V. I. Ryazanov, Topological Aspects of the Theory of Quasiconformal Mappings [in Russian], Doctoral-Degree Thesis (Physics and Mathematics), Donetsk (1993).

  7. V. Ryazanov, U. Srebro, and E. Yakubov, “The Beltrami equation and ring homeomorphisms,” Ukr. Mat. Vestn., 4, No. 1, 79–115 (2007).

    MathSciNet  Google Scholar 

  8. R. Salimov, “On regular homeomorphisms in the plane,” Ann. Acad. Sci. Fenn. Math., 35, 285–289 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  9. V. I. Ryazanov and E. A. Sevost’yanov, “Equicontinuous classes of ring Q-homeomorphisms,” Sib. Mat. Zh., 48, No. 6, 1361–1376 (2007).

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  11. S. Hencl and P. Koskela, “Regularity of the inverse of a planar Sobolev homeomorphism,” Arch. Ration. Mech. Anal., 180, No. 1, 75–95 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  12. V. Ryazanov, U. Srebro, and E. Yakubov, “On convergence theory for Beltrami equations,” Ukr. Mat. Vestn., 5, No. 4, 524–535 (2008).

    MathSciNet  Google Scholar 

  13. V. I. Ryazanov, “Quasiconformal mappings with restrictions in measure,” Ukr. Mat. Zh., 45, No. 7, 1009–1019 (1993); English translation: Ukr. Math. J., 45, No. 7, 1121–1133 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  14. V. I. Ryazanov, “On the improvement of the Strebel convergence theorem,” Izv. Ros. Akad. Nauk, Ser. Mat., 56, No. 3, 636–653 (1992).

    Google Scholar 

  15. K. Strebel, “Ein Konvergensatz für Folgen quasikonformer Abbildungen,” Comm. Math. Helv., 44, No. 4, 469–475 (1969).

    Article  MathSciNet  MATH  Google Scholar 

  16. K. Kuratowski, Topology [Russian translation], Vol. 1, Mir, Moscow (1966).

    Google Scholar 

  17. I. I. Privalov, Introduction to the Theory of Functions of Complex Variable [in Russian], Nauka, Moscow (1977).

    Google Scholar 

  18. G. D. Suvorov, Families of Plane Topological Mappings [in Russian], Nauka, Novosibirsk (1965).

    Google Scholar 

  19. B. V. Shabat, Introduction to Complex Analysis [in Russian], Vol. 2, Nauka, Moscow (1976).

    Google Scholar 

  20. Iu. S. Kolomoitsev and V. I. Ryazanov, “Uniqueness of approximate solutions of the Beltrami equations,” in: Proc. of the Institute of Applied Mathematics and Mechanics, Ukrainian National Academy of Sciences [in Russian], 19 (2009), pp. 116–124.

  21. S. P. Ponomarev, “The N −1-property of mappings and the Luzin (N)-condition,” Mat. Zametki, 58, No. 3, 411–418 (1995).

    MathSciNet  Google Scholar 

  22. S. Saks, Theory of the Integral [Russian translation], Inostr. Literatura, Moscow (1949).

    Google Scholar 

  23. J. Väisälä, Lectures on n-Dimensional Quasiconformal Mappings, Lect. Notes Math., 299, Springer, Berlin (1971).

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  25. V. Ryazanov, U. Srebro, and E. Yakubov, “On integral conditions in the mapping theory,” Ukr. Mat. Vestn., 7, No. 1, 73–87 (2010).

    MathSciNet  Google Scholar 

  26. V. Ryazanov and E. Sevostyanov, “Equicontinuity of mappings quasiconformal in the mean,” Ann. Acad. Sci. Fenn. Math., 36 231–244 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  27. V. Ya. Gutlyanskii, “On the method of variations for univalent analytic functions with quasiconformal extension,” Sib. Mat. Zh., 21, No. 2, 61–78 (1980).

    MathSciNet  Google Scholar 

  28. A. Ukhlov, “Mappings generating imbeddings of Sobolev spaces,” Sib. Mat. Zh., 34, No. 1, 165–171 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  29. S. K. Vodop’yanov and A. Ukhlov, “Sobolev spaces and (P, Q)-quasiconformal mappings of the Carnot groups,” Sib. Mat. Zh., 39, No. 4, 665–682 (1998).

    Article  MathSciNet  Google Scholar 

  30. V. M. Gol’dshtein and Yu. G. Reshetnyak, Introduction to the Theory of Functions with Generalized Derivatives and Quasiconformal Mappings [in Russian], Nauka, Moscow (1983).

    Google Scholar 

  31. H. Federer, Geometric Measure Theory [Russian translation], Nauka, Moscow (1987).

    Google Scholar 

  32. T. V. Lomako and V. I. Ryazanov, “Theory of the variational method for Beltrami equations,” in: Proc. of the Institute of Applied Mathematics and Mechanics, Ukrainian National Academy of Sciences [in Russian], 22 (2011), pp. 1–10.

  33. D. Menchoff, “Sur les differentielles totales des fonctions univalentes,” Math. Ann., 105, 75–85 (1931).

    Article  MathSciNet  Google Scholar 

  34. Yu. Yu. Trokhimchuk, Removable Singularities of Analytic Functions [in Russian], Naukova Dumka, Kiev (1992).

    Google Scholar 

  35. M. D. Kirszbraun, “Uber die zusammenziehende und Lipschitzsche Transformationen,” Fund. Math. J., 22, 77–108 (1934).

    Google Scholar 

  36. E. J. McShane, “Extension of range of functions,” Bull. Amer. Math. Soc., 40, 837–842 (1934).

    Article  MathSciNet  Google Scholar 

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 9, pp. 1227–1240, September, 2011.

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Lomako, T.V. On the theory of convergence and compactness for Beltrami equations with constraints of set-theoretic type. Ukr Math J 63, 1400–1414 (2012). https://doi.org/10.1007/s11253-012-0587-3

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  • DOI: https://doi.org/10.1007/s11253-012-0587-3

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