Advertisement

On the Local Behavior of Sobolev Classes on Two-Dimensional Riemannian Manifolds

  • E. A. Sevost’yanovEmail author
Article

We study open discrete maps of two-dimensional Riemannian manifolds from the Sobolev class. For these mappings, we establish the lower estimates of distortions of the moduli of families of the curves. As a consequence, we establish the equicontinuity of Sobolev classes at the interior points of the domain.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    E. A. Sevost’yanov and S. A. Skvortsov, “On the local behavior of Orlicz–Sobolev classes,” Ukr. Mat. Vestn., 13, No. 4, 543–569 (2016).MathSciNetGoogle Scholar
  2. 2.
    E. S. Afanas’eva, V. I. Ryazanov, and R. R. Salimov, “On mappings in the Orlicz–Sobolev classes on Riemannian manifolds,” Ukr. Mat. Vestn., 8, No. 3, 319–342 (2011).Google Scholar
  3. 3.
    D. A. Kovtonyuk, R. R. Salimov, and E. A. Sevost’yanov, On the Theory of Mappings of Sobolev and Orlicz–Sobolev Classes [in Russian], Naukova Dumka, Kiev (2013).zbMATHGoogle Scholar
  4. 4.
    R. R. Salimov, “Lower estimates for the p-module and mappings of the Sobolev classes,” Algebra Analiz, 26, No. 6, 143–171 (2014).Google Scholar
  5. 5.
    V. Ryazanov and S. Volkov, “On the boundary behavior of mappings in the class \( {W}_{\mathrm{loc}}^{1,1} \) on Riemann surfaces,” Complex Anal. Oper. Theory, 11, 1503–1520 (2017).Google Scholar
  6. 6.
    V. M. Miklyukov, Conformal Mapping of an Irregular Surface and Its Applications [in Russian], Volgograd University, Volgograd (2005).Google Scholar
  7. 7.
    T. Rado and P. V. Reichelderfer, Continuous Transformations in Analysis, Springer, Berlin (1955).CrossRefGoogle Scholar
  8. 8.
    O. Lehto and O. Virtanen, Quasiconformal Mappings in the Plane, Springer, New York (1973).CrossRefGoogle Scholar
  9. 9.
    A. Ignat’ev and V. Ryazanov, “Finite mean oscillation in the mapping theory,” Ukr. Mat. Vestn., 2, No. 3, 395–417 (2005).MathSciNetzbMATHGoogle Scholar
  10. 10.
    O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, Moduli in Modern Mapping Theory, Springer Science + Business Media, New York (2009).zbMATHGoogle Scholar
  11. 11.
    H. Federer, Geometric Measure Theory, Springer, New York (1969).zbMATHGoogle Scholar
  12. 12.
    J. Väisälä, Lectures on n-Dimensional Quasiconformal Mappings, Springer, Berlin (1971).CrossRefGoogle Scholar
  13. 13.
    J. M. Lee, Riemannian Manifolds: an Introduction to Curvature, Springer, New York (1997).CrossRefGoogle Scholar
  14. 14.
    D. P. Il’yutko and E. A. Sevost’yanov, “On the open discrete mappings with unbounded characteristic on Riemannian manifolds,” Mat. Sb., 207, No. 4, 65–112 (2016).MathSciNetCrossRefGoogle Scholar
  15. 15.
    V. G. Maz’ya, Sobolev Spaces [in Russian], Leningrad Univ., Leningrad (1985).CrossRefGoogle Scholar
  16. 16.
    T. Lomako, R. Salimov, and E. Sevost’yanov, “On equicontinuity of solutions to the Beltrami equations,” Ann. Univ. Bucharest. Math. Ser., 59, No. 2, 263–274 (2010).MathSciNetzbMATHGoogle Scholar
  17. 17.
    E. A. Sevost’yanov, “On the local behavior of open discrete mappings from the Orlicz–Sobolev classes,” Ukr. Math. Zh., 68, No. 9, 1259–1272 (2016); English translation: Ukr. Math. J., 68, No. 9, 1447–1465 (2017).Google Scholar
  18. 18.
    K. Kuratowski, Topology, Vol. 2, Academic Press, New York (1968).Google Scholar
  19. 19.
    B. Fuglede, “Extremal length and functional completion,” Acta Math., 98, 171–219 (1957).MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.I. Franko Zhitomir State UniversityZhitomirUkraine

Personalised recommendations