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On the theory of convergence and compactness for Beltrami equations

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Ukrainian Mathematical Journal Aims and scope

Theorems on convergence and compactness are proved for the classes of regular solutions of degenerate Beltrami equations with restrictions of integral type imposed on the dilatation.

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References

  1. Yu. G. Reshetnyak, Space Mappings with Bounded Distortion [in Russian], Nauka, Novosibirsk (1982).

    Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  4. 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 

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

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  7. W. Rudin, Function Theory in Polydiscs [Russian translation], Mir, Moscow (1974).

    Google Scholar 

  8. 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 

  9. S. Saks, Theory of the Integral, PWN, Warsaw (1937).

    Google Scholar 

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

    MathSciNet  Google Scholar 

  11. Yu. S. Kolomoitsev and V. I. Ryazanov, “Uniqueness of approximate solutions of the Beltrami equations,” in: Proceedings of the Institute of Applied Mathematics and Mechanics of the Ukrainian National Academy of Sciences,Vol. 19 (2009), pp. 116–124.

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

    Article  MathSciNet  MATH  Google Scholar 

  13. G. D. Suvorov, Families of Plane Topological Mappings [in Russian], Sibir. Otdelen. Akad. Nauk SSSR, Novosibirsk (1965).

    Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. K. Astala, T. Iwaniec, and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton University Press, Princeton (2009).

    MATH  Google Scholar 

  17. J. Väisälä, Lectures on n-Dimensional Quasiconformal Mappings, Springer, Berlin (1971).

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. M. A. Brakalova and J. A. Jenkins, “On solutions of the Beltrami equation,” J. Anal. Math., 76, 67–92 (1998).

    Article  MathSciNet  MATH  Google Scholar 

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 3, pp. 341–349, March, 2011.

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Lomako, T. On the theory of convergence and compactness for Beltrami equations. Ukr Math J 63, 393–402 (2011). https://doi.org/10.1007/s11253-011-0510-3

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  • DOI: https://doi.org/10.1007/s11253-011-0510-3

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