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On a Vӓisӓlӓ-type inequality for the angular dilatation of mappings and some of its applications

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Abstract

For a subtype of mappings with finite distortion f : D → D′, D, D′ ⊂ ℝn; n ≥ 2; which admit the existence of branch points, a modular inequality playing an essential role in the study of various problems of planar and spatial mappings is established. As an application, the problem of removal of isolated singularities of open discrete mappings with finite length distortion is investigated.

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References

  1. C. Andreian Cazacu, “On the length-area dilatation,” Complex Var. Theory Appl., 50, No. 7–11, 765–776 (2005).

    MathSciNet  MATH  Google Scholar 

  2. C. J. Bishop, V. Ya. Gutlyanskii, O. Martio, and M. Vuorinen, “On conformal dilatation in space,” Intern. J. Math. and Math. Sci., 22, 1397–1420 (2003).

  3. M. Cristea, “Local homeomorphisms having local ACLn inverses,” Compl. Var. and Ellipt. Equ., 53, No. 1, 77–99 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  4. H. Federer, Geometric Measure Theory, Springer, Berlin, 1996.

    Book  MATH  Google Scholar 

  5. A. Golberg and V. Gutlyanskii, “On Lipschitz continuity of quasiconformal mappings in space,” J. Anal. Math., 109, 233–251 (2009).

    Article  MathSciNet  MATH  Google Scholar 

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

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

  8. T. Iwaniec and G. Martin, Geometrical Function Theory and Non-Linear Analysis, Oxford, Clarendon Press, 2001.

    Google Scholar 

  9. A. Ignat'ev and V. Ryazanov, “A finite mean oscillation in the theory of mappings,” Ukr. Mat. Vest., 2, No. 3, 395–417 (2005).

    MathSciNet  MATH  Google Scholar 

  10. F. John and L. Nirenberg, “On functions of bounded mean oscillation,” Comm. Pure Appl. Math., 14, 415–426 (1961).

    Article  MathSciNet  MATH  Google Scholar 

  11. P. Koskela and J. Onninen, “Mappings of finite distortion: Capacity and modulus inequalities,” J. Reine Angew. Math., 599, 1–26 (2006).

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  13. O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, “Mappings with finite length distortion,” J. d'Anal. Math., 93, 215–236 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  14. S. P. Ponomarev, “N –1 property of mappings and the Luzin (N) condition,” Matem. Zam., 58, 411–418 (1995).

  15. Yu. G. Reshetnyak, Spatial Mappings with Bounded Distortion [in Russian], Nauka, Novosibirsk, 1982.

    MATH  Google Scholar 

  16. S. Rickman, Quasiregular Mappings, Berlin, Springer, 1993.

    Book  MATH  Google Scholar 

  17. V. Ryazanov, U. Srebro, and E. Yakubov, “Plane mappings with dilatation dominated by functions of bounded mean mean oscillation,” Sibir. Adv. in Math., 11, No. 2, 94–130 (2001).

    MathSciNet  MATH  Google Scholar 

  18. V. Ryazanov, U. Srebro, and E. Yakubov, “On ring solutions of Beltrami equations,” J. d'Anal. Math., 96, 117–150 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  19. S. Saks, Theory of the Integral, Dover, New York, 1964.

    MATH  Google Scholar 

  20. R. R. Salimov and E. A. Sevost'yanov, “The theory of ring Q-mappings in the geometrical theory of functions,” Matem. Sb., 201, No. 6, 131–158 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  21. E. A. Sevost'yanov, “The Vӓisӓlӓ inequality for mappings with finite length distortion,” Complex Var. Ellip. Equ., 55, No. 1–3, 91–101 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  22. E. A. Sevost'yanov, “On the local behavior of mappings with unbounded characteristic of the quasiconfor-mity,” Sibir. Mat. Zh., 53, No. 3, 648–662 (2012).

    MathSciNet  Google Scholar 

  23. E. A. Sevost'yanov, “The theory of moduli and capacities and normal families of mappings admitting a branching,” Ukr. Mat. Vest., 4, No. 4, 582–604 (2007).

  24. E. A. Sevost'yanov, “A generalization of a lemma by E.A. Poletskii on classes of space mappings,” Ukr. Mat. Zh., 61, No. 7, 969–975 (2009).

    MathSciNet  MATH  Google Scholar 

  25. E. A. Sevost'yanov and R. R. Salimov, “On internal dilatations of mappings with unbounded characteristic,” Ukr. Mat. Vest., 8, No. 1, 129–143 (2011).

    MathSciNet  MATH  Google Scholar 

  26. J. Vӓisӓlӓ, Lectures on n–Dimensional Quasiconformal Mappings, Springer, Berlin, 1971.

    Google Scholar 

  27. J. Vӓisӓlӓ, “Modulus and capacity inequalities for quasiregular mappings,” Ann. Acad. Sci. Fenn. Ser. A 1 Math., 509, 1–14 (1972).

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Author information

Authors and Affiliations

  1. Institute of Mathematics of the NAS of Ukraine, Kiev, Ukraine

    Ruslan R. Salimov

  2. I. Franko Zhitomir State University, Zhitomir, Ukraine

    Evgeny A. Sevost’yanov

Authors
  1. Evgeny A. Sevost’yanov
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  2. Ruslan R. Salimov
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Corresponding author

Correspondence to Evgeny A. Sevost’yanov.

Additional information

Presented by V. Ya. Gutlyanskii

Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 12, No. 4, pp. 511–538, September–December, 2015.

Translated from Russian by V. V. Kukhtin

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Sevost’yanov, E.A., Salimov, R.R. On a Vӓisӓlӓ-type inequality for the angular dilatation of mappings and some of its applications. J Math Sci 218, 69–88 (2016). https://doi.org/10.1007/s10958-016-3011-y

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  • DOI: https://doi.org/10.1007/s10958-016-3011-y

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