Siberian Mathematical Journal

, Volume 53, Issue 6, pp 1061–1074 | Cite as

On the theory of degenerate alternating beltrami equations

  • A. N. KondrashovEmail author


The article focuses on the equation A(z)f z (z) + B(z)f ż(z) = 0. We aim at the study of the interrelation between the solutions to this equation and the solutions to the appropriate classical Beltrami equation.


degenerate Beltrami equation alternating Beltrami equation folds 


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  1. 1.
    Vekua I. N., Generalized Analytic Functions, Pergamon Press, Oxford, London, New York, and Paris (1962).zbMATHGoogle Scholar
  2. 2.
    Belinskiĭ P. P., General Properties of Quasiconformal Mappings [in Russian], Nauka, Novosibirsk (1974).Google Scholar
  3. 3.
    Lavrentieff M., “Sur une classe de representation continues,” Mat. Sb., 42, No. 4, 407–424 (1935). (Also see Lavrent’ev M. A., “On a class of continuous mappings,” in: Lavrent’ev M. A., Selected Works. Mathematics and Mechanics [in Russian], Nauka, Moscow, 1990, pp. 219–237.)Google Scholar
  4. 4.
    Volkovyskiĭ L. I., “Some problems of the theory of quasiconformal mappings,” in: Some Problems of Mathematics and Mechanics (to the seventieth birthday of M. A. Lavrent’ev) [in Russian], Novosibirsk, 1970, pp. 128–134.Google Scholar
  5. 5.
    Srebro U. and Yakubov E., “Branched folded maps and alternating Beltrami equations,” J. Anal. Math., 70, 65–90 (1996).MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Srebro U. and Yakubov E., “Uniformization of maps with folds,” Israel Math. Conf. Proc., 11, 229–232 (1997).MathSciNetGoogle Scholar
  7. 7.
    Trokhimchuk Yu. Yu., Continuous Mappings and Conditions of Monogeneity, Isr. Program for Scientific Translations, Jerusalem (1964).zbMATHGoogle Scholar
  8. 8.
    Yakubov È. Kh., “On solutions of Beltrami’s equation with degeneration,” Soviet Math., Dokl., 19, 1515–1516 (1978).zbMATHGoogle Scholar
  9. 9.
    Maz’ya V. G., Sobolev Spaces [in Russian], Leningrad Univ., Leningrad (1985).Google Scholar
  10. 10.
    Miklyukov V. M., “Isothermic coordinates on singular surfaces,” Sb.: Math., 195, No. 1, 65–83 (2004).MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Martio O. and Miklyukov V. M., “On existence and uniqueness of degenerate Beltrami equations,” Complex Variables, 49, 647–656 (2004).MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Goldstein V. M. and Reshetnyak Yu. G., Quasiconformal Mappings and Sobolev Spaces, Kluwer, Dordrecht (1983).Google Scholar
  13. 13.
    Srebro U. and Yakubov E., “The Beltrami equation,” in: Handbook of Complex Analysis, Geometry Function Theory. Vol. 2, Elsevier, Amsterdam, 2005, pp. 555–597.Google Scholar
  14. 14.
    Miklyukov V. M., Functions of Weighted Sobolev Classes, Anisotropic Metric, and Degenerate Quasiconformal Mappings, Izdat. Volgograd Univ., Volgograd (2010).Google Scholar
  15. 15.
    Srebro U. and Yakubov E., “μ-Homeomorphisms,” Contemp. Math., 211, 473–479 (1997).MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lavrent’ev M. A. and Shabat B. V., Methods of the Theory of Functions of Complex Variables [in Russian], Fizmatgiz, Moscow (1958).Google Scholar
  17. 17.
    Lehto O. and Virtanen K. I., Quasiconformal Mappings in the Plane, Springer-Verlag, New York, Heidelberg, and Berlin (1973).zbMATHCrossRefGoogle Scholar
  18. 18.
    Montel P., Normal Families of Analytic Functions [Russian translation], ONTI NKTP, Moscow and Leningrad (1936).Google Scholar
  19. 19.
    Goluzin G. M., Geometric Theory of Functions of a Complex Variable [in Russian], Nauka, Moscow (1966).Google Scholar
  20. 20.
    Zorich V. A., “A theorem of M. A. Lavrent’ev on quasiconformal space maps,” Math. USSR-Sb., 3, No. 3, 389–403 (1967).CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  1. 1.Institute of Mathematics and Computer Science Volgograd State UniversityVolgogradRussia

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