Skip to main content
SpringerLink
Log in

On the boundary behavior of solutions of the Beltrami equations

  • Published:
Ukrainian Mathematical Journal Aims and scope

We show that any homeomorphic solution of the Beltrami equation v from the Sobolev class W 1,1loc is a so-called lower Q-homeomorphism with Q(z) = K μ(z), where K μ(z) is the dilatation ratio of this equation. On this basis, we develop the theory of boundary behavior and removing of singularities of these solutions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  3. V. Gutlyanskii, V. Ryazanov, U. Srebro, and E. Yakubov, “On recent advances in the degenerate Beltrami equations,” Ukr. Math. Bull., 7, No. 4, 467–515 (2010).

    Google Scholar 

  4. U. Srebro and E. Yakubov, “The Beltrami equation,” in: Handbook in Complex Analysis: Geometric Function Theory, Vol. 2 (2005), pp. 555–597.

  5. B. Fuglede, “Extremal length and functional completion,” Acta Math., 98, 171–219 (1957).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. D. Kovtonyuk and V. Ryazanov, “On the theory of lower Q-homeomorphisms,” Ukr. Mat. Vestn., 5, No. 2, 159–184 (2008).

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  10. D. Kovtonyuk and V. Ryazanov, “On the theory of boundaries of domains in the space,” in: Proc. of the Institute of Applied Mathematics and Mechanics, Ukrainian National Academy of Sciences [in Russian], 13 (2006), pp. 110–120.

  11. O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, “Q-homeomorphisms,” Contemp. Math., 364, 193–203 (2004).

    Article  MathSciNet  Google Scholar 

  12. O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, “On Q-homeomorphisms,” Ann. Acad. Sci. Fenn., 30, 49–69 (2005).

    MathSciNet  MATH  Google Scholar 

  13. J. Väisälä, Lectures on n-Dimensional Quasiconformal Mappings, Springer, New York (1971).

    MATH  Google Scholar 

  14. R. Näkki, “Boundary behavior of quasiconformal mappings in n-space,” Ann. Acad. Sci. Fenn., Ser. A1. Math., 484, 1–50 (1970).

    Google Scholar 

  15. F. W. Gehring and O. Martio, “Quasiextremal distance domains and extension of quasiconformal mappings,” J. Anal. Math., 45, 181–206 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  16. O. Martio and J. Sarvas, “Injectivity theorems in plane and space,” Ann. Acad. Sci. Fenn., Ser. A1. Math., 4, 384–401 (1978/1979).

    Google Scholar 

  17. J. Väisälä, “On the null-sets for extremal distances,” Ann. Acad. Sci. Fenn., Ser. A1. Math., 322, 1–12 (1962).

    Google Scholar 

  18. W. Hurewicz and H. Wallman, Dimension Theory, Princeton University Press, Princeton (1948).

    MATH  Google Scholar 

  19. H. Federer, Geometric Measure Theory, Springer, Berlin (1969).

    MATH  Google Scholar 

  20. F. W. Gehring and O. Lehto, “On the total differentiability of functions of a complex variable,” Ann. Acad. Sci. Fenn., Ser. A1. Math., 272, 1–9 (1959).

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  22. D. Kovtonyuk and V. Ryazanov, “On the theory of mappings with finite area distortion,” J. Anal. Math., 104, 291–306 (2008).

    Article  MathSciNet  Google Scholar 

  23. V. Maz’ya, Sobolev Classes, Springer, Berlin (1985).

  24. A. A. Ignat’ev and V. I. Ryazanov, “Finite mean oscillation in the mapping theory,” Ukr. Math. Bull., 2, No. 3, 403–424 (2005).

    MathSciNet  Google Scholar 

  25. V. Ryazanov, U. Srebro, and E. Yakubov, “Integral conditions in the mapping theory,” Ukr. Math. Bull., 7, No. 1, 73–87 (2010).

    Google Scholar 

  26. N. Bourbaki, Functions of a Real Variable, Springer, Berlin (2004).

    Book  MATH  Google Scholar 

  27. Yu. G. Reshetnyak, Space Mappings with Bounded Distortion, American Mathematical Society, Providence, RI (1989).

Download references

Author information

Authors and Affiliations

  1. Institute of Applied Mathematics and Mechanics, Ukrainian National Academy of Sciences, Donetsk, Ukraine

    D. A. Kovtonyuk, I. V. Petkov & V. I. Ryazanov

Authors
  1. D. A. Kovtonyuk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. V. Petkov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. V. I. Ryazanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 8, pp. 1078–1091, August, 2011.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kovtonyuk, D.A., Petkov, I.V. & Ryazanov, V.I. On the boundary behavior of solutions of the Beltrami equations. Ukr Math J 63, 1241–1255 (2012). https://doi.org/10.1007/s11253-012-0575-7

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-012-0575-7

Keywords

Search

Navigation