Skip to main content
Log in

Logarithmic Asymptotics of the Nonlinear Cauchy–Riemann–Beltrami Equation

  • Published:
Ukrainian Mathematical Journal Aims and scope

We study regular solutions of the nonlinear Cauchy–Riemann–Beltrami equation for the logarithmic asymptotics in terms of the lower limits and solve an extreme problem for the functional disk image area in a certain class of solutions to the nonlinear Cauchy–Riemann–Beltrami system.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

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

    Book  Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

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

  5. E. A. Sevost’yanov, “On quasilinear Beltrami-type equations with degeneration,” Math. Notes, 90, No. 3-4, 431–438 (2011).

    Article  MathSciNet  Google Scholar 

  6. E. A. Sevost’yanov, “Generalization of one Poletskii lemma to classes of space mappings,” Ukr. Mat. Zh., 61, No. 7, 969–975 (2009); English translation: Ukr. Math. J., 61, No. 7, 1151–1157 (2009).

  7. D. A. Kovtonyuk, R. R. Salimov, and E. A. Sevost’yanov, On the Mapping Theory for the Sobolev and Orlicz–Sobolev Classes [in Russian], Naukova Dumka, Kiev (2013).

  8. M. Cristea, “Local homeomorphisms having local ACLn inverses,” Complex Var. Elliptic Equat., 53, No. 1, 77–99 (2008).

    Article  MathSciNet  Google Scholar 

  9. M. Cristea, “Open, discrete mappings having local ACLn inverses,” Complex Var. Elliptic Equat., 55, No. 1-3, 61–90 (2010).

    Article  MathSciNet  Google Scholar 

  10. M. Cristea, “Local homeomorphisms satisfying generalized modular inequalities,” Complex Var. Elliptic Equat., 59, No. 2, 232–246 (2014).

    Article  MathSciNet  Google Scholar 

  11. M. Cristea, “Some properties of open, discrete generalized ring mappings,” Complex Var. Elliptic Equat., 61, No. 5, 623–643 (2016).

    Article  MathSciNet  Google Scholar 

  12. K. Astala, T. Iwaniec, and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton Univ. Press, Princeton (2009).

  13. C.-Y. Guo and M. Kar, “Quantitative uniqueness estimates for p-Laplace type equations in the plane,” Nonlin. Anal.: Theory, Meth. Appl., 143, 19–44 (2016).

  14. M. A. Lavrent’ev and B. V. Shabat, “Geometric properties of solutions of the nonlinear systems of partial differential equations,” Dokl. Akad. Nauk SSSR, 112, No. 5, 810–811 (1957).

    MathSciNet  Google Scholar 

  15. M. A. Lavrent’ev, “General problem of quasiconformal mappings of plane domains,” Mat. Sb., 21(63), No. 2, 285–320 (1947).

  16. M. A. Lavrent’ev, Variational Method in Boundary-Value Problems for Systems of Equations of the Elliptic Type [in Russian], Akad. Nauk SSSR, Moscow (1962).

  17. B. V. Shabat, “Geometric meaning of the notion of ellipticity,” Usp. Mat. Nauk, 12, No. 6 (78), 181–188 (1957).

  18. B. V. Shabat, “On the notion of derivative system according to M. A. Lavrent’ev,” Dokl. Akad. Nauk SSSR, 136, No. 6, 1298–1301 (1961).

    Google Scholar 

  19. R. Kuhnau, “Minimal surfaces and quasiconformal mappings in the mean,” in: Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, 7, No. 2, 104–131 (2010).

  20. S. L. Krushkal’ and R. Kühnay, Quasiconformal Mappings—New Methods and Applications [in Russian], Nauka, Novosibirsk (1984).

  21. T. Adamowicz, “On p-harmonic mappings in the plane,” Nonlin. Anal., 71, No. 1-2, 502–511 (2009).

    Article  MathSciNet  Google Scholar 

  22. G. Aronsson, “On certain p-harmonic functions in the plane,” Manuscripta Math., 61, No. 1, 79–101 (1988).

    Article  MathSciNet  Google Scholar 

  23. A. S. Romanov, “Capacity relations in a plane tetrahedron,” Sib. Mat. Zh., 49, No. 4, 886–897 (2008).

    Article  MathSciNet  Google Scholar 

  24. B. Bojarski and T. Iwaniec, “p-Harmonic equation and quasiregular mappings,” Banach Center Publ., 19, No. 1, 25–38 (1987).

    Article  MathSciNet  Google Scholar 

  25. K. Astala, A. Clop, D. Faraco, J. Jääskeläinen, and A. Koski, “Nonlinear Beltrami operators. Schauder estimates and bounds for the Jacobian,” Ann. Inst. H. Poincaré, Anal. Non Linéaire, 34, No. 6, 1543–1559 (2017).

  26. M. Carozza, F. Giannetti, A. Passarelli di Napoli, C. Sbordone, and R. Schiattarella, “Bi-Sobolev mappings and Kp-distortions in the plane,” J. Math. Anal. Appl., 457, No. 2, 1232–1246 (2018).

    Article  MathSciNet  Google Scholar 

  27. A. Golberg, R. Salimov, and M. Stefanchuk, “Asymptotic dilation of regular homeomorphisms,” Complex Anal. Oper. Theory, 13, No. 6, 2813–2827 (2019).

    Article  MathSciNet  Google Scholar 

  28. R. R. Salimov and M. V. Stefanchuk, “On the local properties of solutions of the nonlinear Beltrami equation,” J. Math. Sci., 248, 203–216 (2020).

    Article  Google Scholar 

  29. E. A. Sevost’yanov and R. R. Salimov, “On the Väisälä-type inequality for the angular dilatation of mappings and some its applications,” Ukr. Mat. Visn., 12, No. 4, 511–538 (2015).

    Google Scholar 

  30. M. Cristea, “On Poleckii-type modular inequalities,” Complex Var. Elliptic Equat.; https://doi.org/10.1080/17476933.2020.1783660.

  31. A. Golberg and R. Salimov, “Nonlinear Beltrami equation,” Complex Var. Elliptic Equat., 65, No. 1, 6–21 (2019).

    Article  MathSciNet  Google Scholar 

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

  33. B. Bojarski, V. Gutlyanskii, O. Martio, and V. Ryazanov, Infinitesimal Geometry of Quasiconformal and Bi-Lipschitz Mappings in the Plane, European Mathematical Society, Zürich (2013).

  34. E. Reich and H. Walczak, “On the behavior of quasiconformal mappings at a point,” Trans. Amer. Math. Soc., 117, 338–351 (1965).

    Article  MathSciNet  Google Scholar 

  35. A. Schatz, “On the local behavior of homeomorphic solutions of Beltrami equation,” Duke Math. J., 35, 289–306 (1968).

    Article  MathSciNet  Google Scholar 

  36. C. Andreian Cazacu, “Influence of the orientation of the characteristic ellipses on the properties of the quasiconformal mappings,” Proc. Rom. Finn. Sem., Romania (1969), Publ. House Acad. Soc. Rep. Rom., Bucharest (1971), pp. 65–85.

  37. M. A. Brakalova and J. A. Jenkins, “On solutions of the Beltrami equation,” J. Anal. Math., 76, 67–92 (1998).

    Article  MathSciNet  Google Scholar 

  38. V. Gutlyanskii and T. Sugawa, “On Lipschitz continuity of quasiconformal mappings,” Rep. Univ. Jyväskylä Dep. Math. Stat., 83, 91–108 (2001).

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  40. V. Gutlyanskii and A. Golberg, “Rings and Lipschitz continuity of quasiconformal mappings,” in: Analysis and Mathematical Physics, Trends in Mathematics, Birkhäuser, Basel (2009), pp. 187–192.

  41. V. Gutlyanskii, O. Martio, T. Sugawa, and M. Vuorinen, “On the degenerate Beltrami equation,” Trans. Amer. Math. Soc., 357, 875–900 (2005).

    Article  MathSciNet  Google Scholar 

  42. V. Ryazanov, R. Salimov, U. Srebro, and. Yakubov, “On boundary value problems for the Beltrami equations,” Contemp. Math., 591, 211–242 (2013).

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

    MathSciNet  MATH  Google Scholar 

  44. K. Ikoma, “On the distortion and correspondence under quasiconformal mappings in space,” Nagoya Math. J., 25, 175–203 (1965).

    Article  MathSciNet  Google Scholar 

  45. S. Saks, Theory of the Integral, Państwowe Wydawnictwo Naukowe, Warsaw (1937).

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to M. V. Stefanchuk.

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal,Vol. 73, No.3, pp.395–407, March, 2021. Ukrainian DOI: 10.37863/umzh.v73i3.6403.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Salimov, R.R., Stefanchuk, M.V. Logarithmic Asymptotics of the Nonlinear Cauchy–Riemann–Beltrami Equation. Ukr Math J 73, 463–478 (2021). https://doi.org/10.1007/s11253-021-01936-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-021-01936-9

Navigation