Skip to main content
Log in

Functional Asymptotics of Solutions of the Nonlinear Cauchy–Riemann–Beltrami System

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

We prove a series of theorems on the asymptotic behavior of regular homeomorphic solutions of the nonlinear Cauchy–Riemann–Beltrami-type equation. Sufficient conditions for the logarithmic and exponential growth of regular solutions are obtained.

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).

    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).

    Google Scholar 

  4. U. Srebro and E. Yakubov, “The Beltrami equation,” in: Handbook in Complex Analysis: Geometric Function Theory, 2, Elseiver Sci. B.V., Amsterdam (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).

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

    Google Scholar 

  7. 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).

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

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

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

    Article  MathSciNet  Google Scholar 

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

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

    Google Scholar 

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

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

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  19. 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).

  20. 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 

  21. A. Golberg, R. Salimov, and M. Stefanchuk, “Asymptotic dilation of regular homeomorphisms,” Complex Anal. Oper. Theory, 13, No. 6, 2813–2827 (2019); https://doi.org/https://doi.org/10.1007/s11785-018-0833-2.

    Article  MathSciNet  Google Scholar 

  22. R. R. Salimov and M. V. Stefanchuk, “Logarithmic asymptotics of the nonlinear Cauchy–Riemann–Beltrami equation,” Ukr. Mat. Zh., 73, No. 3, 395–407 (2021); English translation: Ukr. Math. J., 73, No. 3, 463–478 (2021).

  23. R. R. Salimov and M. V. Stefanchuk, “On the local properties of solutions of the nonlinear Beltrami equation,” J. Math. Sci., 248, No. 2, 203–216 (2020).

    Article  MathSciNet  Google Scholar 

  24. R. Salimov and M. Stefanchuk, “Nonlinear Beltrami equation and asymptotics of its solution,” J. Math. Sci., 264, No. 4, 441–454 (2022).

    Article  MathSciNet  Google Scholar 

  25. A. Golberg and R. Salimov, “Nonlinear Beltrami equation,” Complex Var. Elliptic Equat., 65, No. 1, 6–21 (2019); https://doi.org/10.1080/17476933.2019.1631292.

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

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to M. V. Stefanchuk.

Additional information

Translated from Neliniini Kolyvannya, Vol. 25, No. 4, pp. 388–403, October–December, 2022.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Salimov, R.R., Stefanchuk, M.V. Functional Asymptotics of Solutions of the Nonlinear Cauchy–Riemann–Beltrami System. J Math Sci 277, 311–328 (2023). https://doi.org/10.1007/s10958-023-06835-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-023-06835-x

Navigation