Skip to main content

On the compactness of classes of the solutions of the Dirichlet problem

Abstract

Some theorems concerning the compact classes of homeomorphisms with hydrodynamic normalization, which are solutions of the Beltrami equation and the characteristics of which are compactly supported and satisfy certain constraints of the theoretical-set type, have been proved. As a consequence, we obtained results on the compact classes of solutions of the corresponding Dirichlet problems considered in a certain Jordan domain.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    V. Ya. Gutlyanskii and V. I. Ryazanov, The Geometric and Topological Theory of Functions and Mappings [in Russian], Kiev, Naukova Dumka, 2011.

  2. 2.

    V. Gutlyanskii, V. Ryazanov, U. Srebro, and E. Yakubov, The Beltrami Equation: A Geometric Approach, Springer, New York, 2012.

    Book  Google Scholar 

  3. 3.

    V. Ya. Gutlyanskii and V. I. Ryazanov. Infinitesimal Geometry of Spatial Mappings, Akademperiodyka, Kiev, 2013.

  4. 4.

    V. Ya. Gutlyanskii, V. I. Ryazanov, and E. Yakubov. “The Beltrami equations and prime ends,” Ukr. Math. Bull., 12(1), 27–66 (2015); transl. in J. Math. Sci., 210(1), 22–51 (2015).

  5. 5.

    Yu. P. Dybov. “Compactness of classes of the solutions of the Dirichlet problem for the Beltrami equations,” Trudy IPMM NAS Ukrainy, 19, 81–89 (2009).

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Yu. P. Dybov. “On regular solutions of the Dirichlet problem for the Beltrami equations,” Complex Variables and Elliptic Equations, 55(12), 1099–1116 (2010).

    MathSciNet  Article  Google Scholar 

  7. 7.

    T. Lomako. “On the theory of convergence and compactness for Beltrami equations with constraints of set-theoretic type,” Ukrainian Mathematical Journal, 63(9), 1400–1414 (2012).

    MathSciNet  Article  Google Scholar 

  8. 8.

    J. Väisälä. Lectures on n-Dimensional Quasiconformal Mappings. Lecture Notes in Math., 229, Springer–Verlag, Berlin etc, 1971.

  9. 9.

    V. Ryazanov, U. Srebro, and E. Yakubov. “Finite mean oscillation and the Beltrami equation,” Israel Math. J., 153, 247–266 (2006).

    MathSciNet  Article  Google Scholar 

  10. 10.

    V. Ryazanov, R. Salimov, and E. Sevost’yanov. “On Convergence Analysis of Space Homeomorphisms,” Siberian Advances in Mathematics, 23(4), 263–293 (2013).

    MathSciNet  Article  Google Scholar 

  11. 11.

    F. W. Gehring and O. Martio. “Quasiextremal distance domains and extension of quasiconformal mappings,” J. d’Anal. Math., 24, 181–206 (1985).

    MathSciNet  Article  Google Scholar 

  12. 12.

    E. A. Sevost’yanov. “Equicontinuity of homeomorphisms with unbounded characteristic,” Siberian Advances in Mathematics, 23(2), 106–122 (2013).

    MathSciNet  Article  Google Scholar 

  13. 13.

    T. Lomako, R. Salimov, and E. Sevost’yanov. “On equicontinuity of solutions to the Beltrami equations,” Ann. Univ. Bucharest (math. series), V. LIX(2), 261–271 (2010).

  14. 14.

    O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov. Moduli in Modern Mapping Theory. Springer Science + Business Media, New York LLC, 2009.

    MATH  Google Scholar 

  15. 15.

    G. M. Goluzin. Geometric Theory of Complex-Variable Functions [in Russian], Fizmatgiz, Moscow, 1966.

    MATH  Google Scholar 

  16. 16.

    V. Ryazanov, U. Srebro, and E. Yakubov. “On convergence theory for Beltrami equations,” Ukr. Math. Bull., 5(4), 517–528 (2008).

    MathSciNet  Google Scholar 

  17. 17.

    S. Hencl and P. Koskela. “Regularity of the inverse of a planar Sobolev homeomorphism,” Arch. Ration. Mech. and Anal., 180(1), 75–95 (2006).

    MathSciNet  Article  Google Scholar 

  18. 18.

    S. P. Ponomarev. “N -1-property of mappings and Luzin’s condition (N),” Matematicheskie Zametki, 58, 411–418 (1995).

    MathSciNet  Google Scholar 

  19. 19.

    J. Maly and O. Martio. “Lusin’s condition N and mappings of the class \( {W}_{loc}^{1,n} \) ,” J. Reine Angew. Math., 458, 19–36 (1995).

  20. 20.

    H. Federer. Geometric Measure Theory. Springer-Verlag, Berlin, 1996.

    Book  Google Scholar 

  21. 21.

    L. V. Ahlfors. Lectures on Quasiconformal Mappings, D. Van Nosrtrand, Princeton, NJ, 1969.

    MATH  Google Scholar 

  22. 22.

    Yu. G. Reshetnyak, Spatial Nappings with Limited Distortion [in Russian], Nauka, Novosibirsk, 1982.

    Google Scholar 

  23. 23.

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

    Book  Google Scholar 

  24. 24.

    E. O. Sevost’yanov, S. O. Skvortsov, and O. P. Dovgopyatyi, “On nonhomeomorphic mappings with the inverse Poletsky inequality,” Ukr. Mat. Bull., 17(3), 414–436 (2020); transl. in J. Math. Sci., 252(4), 541–557 (2021).

  25. 25.

    S. Stoilov, Lectures on Topological Principles of the Theory of Analytic Functions [in Russian], Nauka, Moscow, 1964.

    Google Scholar 

  26. 26.

    E. A. Sevost’yanov. “Analog of the Montel theorem for mappings of the Sobolev class with finite distortion,” Ukrainian Mathematical Journal, 67(6), 938–947 (2015).

    MathSciNet  Article  Google Scholar 

  27. 27.

    D. A. Kovtonyuk, I. V. Petkov, V. I. Ryazanov, and R. R. Salimov, “The boundary behavior and the Dirichlet problem for the Beltrami equations,” Algebra i Analiz, 25(4), 101–124 (2013); transl. in St. Petersburg Math. J., 25(4), 587–603 (2013).

  28. 28.

    A. Hurwitz and R. Courant, Funktionentheorie, Springer, Berlin, 1966.

    MATH  Google Scholar 

  29. 29.

    V. I. Ryazanov, “On the accuracy of some convergence theorems,” Doklady Akademii Nauk SSSR, 315(2), 317–319 (1990).

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Oleksandr P. Dovhopiatyi.

Additional information

Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 18, No. 3, pp. 319–337, July–September, 2021.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Dovhopiatyi, O.P., Sevost’yanov, E.A. On the compactness of classes of the solutions of the Dirichlet problem. J Math Sci 259, 23–36 (2021). https://doi.org/10.1007/s10958-021-05598-7

Download citation

Keywords

  • Quasiconformal analysis
  • convergence theorems
  • compactness theorems
  • Beltrami equation