Journal of Mathematical Sciences

, Volume 173, Issue 4, pp 397–407 | Cite as

On integral conditions in the mapping theory

  • Vladimir Ryazanov
  • Uri Srebro
  • Eduard Yakubov


Interconnections between various integral conditions that play an important role in the theory of space mappings and in the theory of degenerate Beltrami equations in the plane are established.


Integral conditions mapping theory Beltrami equations Sobolev classes 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    L. Ahlfors, “On quasiconformal mappings,” J. Analyse Math., 3, 1–58 (1953/54).Google Scholar
  2. 2.
    K. Astala, T. Iwaniec, and G. J. Martin, Elliptic Differential Equations and Quasiconformal Mappings in the Plane, Princeton Univ. Press, Princeton, 2009.MATHGoogle Scholar
  3. 3.
    P. A. Biluta, “Extremal problems for mappings quasiconformal in the mean,” Sib. Mat. Zh., 6, 717–726 (1965).MathSciNetMATHGoogle Scholar
  4. 4.
    B. Bojarski, V. Gutlyanskii, and V. Ryazanov, “On the Beltrami equations with two characteristics,” Complex Var. Ellipt. Equa., 54, No. 10, 935–950 (2009).MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    M. A. Brakalova and J. A. Jenkins, “On solutions of the Beltrami equation. II,” Publ. de l’Inst. Math., 75(89), 3–8 (2004).MathSciNetCrossRefGoogle Scholar
  6. 6.
    Z. G. Chen, “μ(x)-homeomorphisms of the plane,” Michigan Math. J., 51, No. 3, 547–556 (2003).MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    V. Gutlyanskii, O. Martio, T. Sugawa, and M. Vuorinen, “On the degenerate Beltrami equation,” Trans. Amer. Math. Soc., 357, 875–900 (2005).MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    A. Golberg, “Homeomorphisms with finite mean dilatations,” Contemp. Math., 382, 177–186 (2005).MathSciNetGoogle Scholar
  9. 9.
    T. Iwaniec and G. Martin, “The Beltrami equation,” Memoirs of AMS, 191, 1–92 (2008).MathSciNetGoogle Scholar
  10. 10.
    V. I. Kruglikov, “Capacities of condensors and quasiconformal in the mean mappings in space,” Mat. Sb., 130, No. 2, 185–206 (1986).MathSciNetGoogle Scholar
  11. 11.
    S. L. Krushkal’, “On mappings quasiconformal in the mean,” Dokl. Akad. Nauk SSSR, 157, No. 3, 517–519 (1964).MathSciNetGoogle Scholar
  12. 12.
    S. L. Krushkal’ and R. Kühnau, Quasiconformal Mappings, New Methods, and Applications [in Russian], Nauka, Novosibirsk, 1984.Google Scholar
  13. 13.
    V. S. Kud’yavin, “Behavior of a class of mappings quasiconformal in the mean at an isolated singular point,” Dokl. Akad. Nauk SSSR, 277, No. 5, 1056–1058 (1984).MathSciNetGoogle Scholar
  14. 14.
    O. Lehto, “Homeomorphisms with a prescribed dilatation,” Lecture Notes in Math., 118, 58–73 (1968).MathSciNetCrossRefGoogle Scholar
  15. 15.
    O. Martio, V. Ryazanov, U. Srebro and E. Yakubov, Moduli in Modern Mapping Theory, Springer Monographs in Mathematics, Springer, New York, 2009.Google Scholar
  16. 16.
    V. M. Miklyukov and G. D. Suvorov, On existence and uniqueness of quasiconformal mappings with unbounded characteristics, in Investigations in the Theory of Functions of Complex Variables and Its Applications [in Russian], Kiev, Inst. Math. (1972).Google Scholar
  17. 17.
    M. Perovich, “Isolated singularity of the mean quasiconformal mappings,” Lect. Notes Math., 743, 212–214 (1979).CrossRefGoogle Scholar
  18. 18.
    I. N. Pesin, “Mappings quasiconformal in the mean,” Dokl. Akad. Nauk SSSR, 187, No. 4, 740–742 (1969).MathSciNetGoogle Scholar
  19. 19.
    V. I. Ryazanov, “On mappings quasiconformal in the mean,” Sib. Mat. Zh., 37, No. 2, 378–388 (1996).MathSciNetGoogle Scholar
  20. 20.
    V. Ryazanov, U. Srebro, and E. Yakubov, “Degenerate Beltrami equation and radial Q-homeomorphisms,” Reports Dept. Math. Helsinki, 369, 1–34 (2003).Google Scholar
  21. 21.
    V. Ryazanov, U. Srebro, and E. Yakubov, “On ring solutions of Beltrami equation,” J. d’Analyse Math., 96, 117–150 (2005).MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    V. Ryazanov, U. Srebro, and E. Yakubov, “The Beltrami equation and ring homeomorphisms,” Ukr. Math. Bull., 4, No. 1, 79–115 (2007).MathSciNetGoogle Scholar
  23. 23.
    V. Ryazanov, U. Srebro, and E. Yakubov, “On strong ring solutions of the Beltrami equations,” Complex Var. Ellipt. Equ., 55, No. 1–3, 219–236 (2010).MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    V. Ryazanov, U. Srebro, and E. Yakubov, “Integral conditions in the theory of the Beltrami equations,” arXiv: 1001.2821v1 [math.CV] Jan. 16, 2010, 1–27.Google Scholar
  25. 25.
    S. Saks, Theory of the Integral, Dover, New York, 1964.MATHGoogle Scholar
  26. 26.
    Yu. F. Strugov, “Compactness of the classes of mappings quasiconformal in the mean,” Dokl. Akad. Nauk SSSR, 243, No. 4, 859–861 (1978).MathSciNetGoogle Scholar
  27. 27.
    V. A. Zorich, “Admissible order of growth of the characteristic of quasiconformality in the Lavrent’ev theorem,” Dokl. Akad. Nauk SSSR, 181, 530–533 (1968).MathSciNetGoogle Scholar
  28. 28.
    V. A. Zorich, “Isolated singularities of mappings with bounded distortion,” Mat. Sb., 81, 634–638 (1970).MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  1. 1.Institute of Applied Mathematics and MechanicsNAS of UkraineDonetskUkraine
  2. 2.Technion — Israel Institute of TechnologyHaifaIsrael
  3. 3.Holon Institute of TechnologyHolonIsrael

Personalised recommendations