Advertisement

To the theory of mappings of the Sobolev class with the critical index

  • Elena S. Afanas’evaEmail author
  • Vladimir I. Ryazanov
  • Ruslan R. Salimov
Article
  • 1 Downloads

Abstract

It is established that any homeomorphism f of the Sobolev class \( {W}_{\mathrm{loc}}^{1,1} \) with outer dilatation \( {K}_O\left(x,f\right)\in {L}_{\mathrm{loc}}^{n-1} \) is the so-called lower Q-homeomorphism with Q(x) = KO(x, f) and also a ring Q-homeomorphism with \( Q(x)={K}_O^{n-1}\left(x,f\right) \). This allows us to apply the theory of boundary behavior of ring and lower Q-homeomorphisms. In particular, we have found the conditions imposed on the outer dilatation KO(x, f) and the boundaries of domains under which any homeomorphism of the Sobolev class \( {W}_{\mathrm{loc}}^{1,1} \) admits continuous or homeomorphic extensions to the boundary.

Keywords

Ring Q-homeomorphisms lower Q-homeomorphisms Sobolev classes mappings with finite distortion continuous extension homeomorphic extension moduli 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    E. S. Afanas’eva, V. I. Ryazanov, R. R. Salimov, “On mappings in the Orlicz–Sobolev classes on Riemannian manifolds,” J. Math. Sci., 181, No. 1, 1–17 (2012).MathSciNetCrossRefGoogle Scholar
  2. 2.
    C. Andreian Cazacu and V. Stanciu, “BMO-mappings in the plane,” in: Topics in Analysis and Its Applications, Kluwer, Dordrecht, 2004, pp. 11–30.Google Scholar
  3. 3.
    K. Astala, “A remark on quasiconformal mappings and BMO-functions,” Michigan Math. J., 80, 209–212 (1983).MathSciNetzbMATHGoogle Scholar
  4. 4.
    K. Astala and F. W. Gehring, “Injectivity, the BMO norm and the universal Teichm´’uller space,” J. Analyse Math., 46, 16–57 (1986).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    M. Csörnyei, S. Hencl, and J. Maly, “Homeomorphisms in the Sobolev space W 1,n-1,” J. Reine Angew. Math., 644, 221–235 (2010).MathSciNetzbMATHGoogle Scholar
  6. 6.
    H. Federer, Geometric Measure Theory, Springer, Berlin, 1996.CrossRefzbMATHGoogle Scholar
  7. 7.
    F. W. Gehring, Characteristic Properties of Quasidisks, Univ. de Montreal, Montreal, 1982.Google Scholar
  8. 8.
    F. W. Gehring and O. Lehto, “On the total differentiability of functions of a complex variable,” Ann. Acad. Sci. Fenn. Ser. A1. Math., 272, 3–8 (1959).Google Scholar
  9. 9.
    F. W. Gehring and O. Martio, “Quasiextremal distance domains and extension of quasiconformal mappings,” J. Anal. Math., 45, 181–206 (1985).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    V. Gutlyanskii, V. Ryazanov, U. Srebro, and E. Yakubov, The Beltrami Equation: A Geometric Approach, Springer, New York, 2012.Google Scholar
  11. 11.
    J. Heinonen, T. Kilpelainen, and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Clarendon Press, Oxford, 1993.Google Scholar
  12. 12.
    A. A. Ignat’ev and V. I. Ryazanov, “Finite mean iscillation in the theory of mappings,” Ukr. Math. Bull., 2, No. 3, 403–424 (2005).Google Scholar
  13. 13.
    A. A. Ignat’ev and V. I. Ryazanov, “To the theory of boundary behavior of spatial mappings,” Ukr. Math. Bull., 3, No. 2, 189–201 (2006).Google Scholar
  14. 14.
    T. Iwaniec and G. Martin, Geometrical Function Theory and Non-linear Analysis, Clarendon Press, Oxford, 2001.Google Scholar
  15. 15.
    T. Iwaniec and V. Šverák, “On mappings with integrable dilatation,” Proc. Amer. Math. Soc., 118, 181–188 (1993).Google Scholar
  16. 16.
    F. John and L. Nirenberg, “On functions of bounded mean oscillation,” Comm. Pure Appl. Math., 14, 415–426 (1961).Google Scholar
  17. 17.
    P. W. Jones, “Extension theorems for BMO,” Indiana Univ. Math. J., 29, 41–66 (1980).Google Scholar
  18. 18.
    D. Kovtonyuk and V. Ryazanov, “On the theory of boundaries of spatial domains,” Trudy IPMM NAN Ukr., 13, 110–120 (2006).Google Scholar
  19. 19.
    D. A. Kovtonyuk and V. I. Ryazanov, “On the theory of lower Q-homeomorphisms,” Ukr. Math. Bull., 5, No. 2, 157–181 (2008).Google Scholar
  20. 20.
    D. Kovtonyuk and V. Ryazanov, “On the boundary behavior of generalized quasi-isometries,” J. Anal. Math., 115, 103–119 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    D. Kovtonyuk and V. Ryazanov, “On the theory of mappings with finite area distortion,” J. Anal. Math., 104, 291–306 (2008).Google Scholar
  22. 22.
    D. Kovtonyuk, I. Petkov, and V. Ryazanov, “On the boundary behaviour of solutions to the Beltrami equations,” Complex Var. Ellipt. Equ., 58, No. 5, 647–663 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    D. A. Kovtonyuk, I. V. Petkov, V. I. Ryazanov, and R. R. Salimov, “The boundary behavior and the Dirichlet problem for Beltrami equations,” St.-Petersburg Math. J., 25, No. 4, 587–603 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    D. A. Kovtonyuk, V. I. Ryazanov, R. R. Salimov, and E. A. Sevost’yanov, “To the theory of Orlicz–Sobolev classes,” St.-Petersburg Math. J., 25, No. 6, 929–963 (2014).Google Scholar
  25. 25.
    D. Kovtonyuk, V. Ryazanov, R. Salimov, and E. Sevost’yanov, “On mappings in the Orlicz–Sobolev classes,” Ann. Univ. Bucharest (Math. Ser.), 3 (LXI), 67–78 (2012).Google Scholar
  26. 26.
    D. A. Kovtonyuk, R. R. Salimov, and E. A. Sevost’yanov, To the Theory of Mappings of Sobolev and Orlicz–Sobolev Classes [in Russian], edited by V.I. Ryazanov, Naukova Dumka, Kiev, 2013.Google Scholar
  27. 27.
    K. Kuratowski, Topology, Academic Press, New York, 1968.Google Scholar
  28. 28.
    O. Lehto and K. Virtanen, Quasiconformal Mappings in the Plane, Springer, New York, 1973.Google Scholar
  29. 29.
    T. V. Lomako, “On the extension of some generalizations of quasiconformal mappings to the boundary,” Ukr. Mat. Zh., 61, No. 10, 1329–1337 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    J. Maly and O. Martio, “Lusin’s condition (N) and mappings of the class W 1,n,” J. Reine Angew. Math., 485, 19–36 (1995).Google Scholar
  31. 31.
    D. Menchoff, “Sur les differencelles totales des fonctions univalentes,” Math. Ann., 105, 75–85 (1931).Google Scholar
  32. 32.
    O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, Moduli in Modern Mapping Theory, Springer, New York, 2009.Google Scholar
  33. 33.
    O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, “On Q-homeomorphisms,” Ann. Acad. Sci. Fenn. Ser. A1. Math., 30, 49–69 (2005).Google Scholar
  34. 34.
    O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, “Q-homeomorphisms,” Contemp. Math., 364, 193–203 (2004).Google Scholar
  35. 35.
    V. G. Maz’ya, S. L. Sobolev’s Spaces [in Russian], Leningrad State Univ., Leningrad, 1985.Google Scholar
  36. 36.
    R. Nakki, “Boundary behavior of quasiconformal mappings in n-space,” Ann. Acad. Sci. Fenn. Ser. A1 Math., 484, 1–50 (1970).Google Scholar
  37. 37.
    S. P. Ponomarev, “On the N-property of homeomorphisms of the class \( {W}_p^1 \),” Sibir. Mat. Zh., 28, No. 2, 140–148 (1987).Google Scholar
  38. 38.
    T. Rado and P.V. Reichelderfer, Continuous Transformations in Analysis, Springer, Berlin, 1955.Google Scholar
  39. 39.
    H. M. Reimann and T. Rychener, “Functions of bounded mean oscillation and quasiconformal mappings,” Comment. Math. Helv., 49, 260–276 (1974).MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Yu. G. Reshetnyak, Spatial Mappings with Bounded Distortion [in Russian], Nauka, Novosibirsk, 1982.Google Scholar
  41. 41.
    V. I. Ryazanov and R. R. Salimov, “Weakly flat spaces and boundaries in the theory of mappings,” Ukr. Mat. Visn., 4, No. 2, 199–234 (2007).MathSciNetGoogle Scholar
  42. 42.
    V. I. Ryazanov and E. A. Sevost’yanov, “Equicontinuous classes of ring Q-homeomorphisms,” Siberian Math. J., 48, No. 6, 1093–1105 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    V. Ryazanov, U. Srebro, and E. Yakubov, “Integral conditions in the mapping theory,” J. Math. Sci., 173, No. 4, 397–407 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    V. Ryazanov, U. Srebro, and E. Yakubov, “On ring solutions of Beltrami equation,” J. Anal. Math., 96, 117–150 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    S. Saks, Theory of the Integral, Dover, New York, 1964.Google Scholar
  46. 46.
    J. Serrin, “On the differentiability of functions of several variables,” Arch. Rat. Mech. Anal., 7, 359–372 (1961).MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    V. Tengvall, “Differentiability in the Sobolev space W 1,n-1,” Calc. Var. Part. Diff. Equa., 51, Nos. 1–2, 381–399 (2014).Google Scholar
  48. 48.
    J. Väisälä, Lectures on n-Dimensional Quasiconformal Mappings, Springer, Berlin, 1971.CrossRefzbMATHGoogle Scholar
  49. 49.
    J. Väisälä, “On quasiconformal mappings in space,” Ann. Acad. Sci. Fenn. Ser. A1 Math., 298, 1–36 (1961).MathSciNetzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  • Elena S. Afanas’eva
    • 1
    Email author
  • Vladimir I. Ryazanov
    • 1
  • Ruslan R. Salimov
    • 2
  1. 1.Institute of Applied Mathematics and Mechanics of the NAS of UkraineSlavyanskUkraine
  2. 2.Institute of Mathematics of the NAS of UkraineKievUkraine

Personalised recommendations