Skip to main content
Log in

Nonlinear Beltrami equation and asymptotics of its solution

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

Abstract

We continue to study regular homeomorphic solutions to the nonlinear Beltrami equation introduced in [24]. Estimates of the Schwarz Lemma type have been obtained using the length-area method.

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. Gutlyanskiĭ, V. Ryazanov, U. Srebro, and E. Yakubov, The Beltrami equation. A geometric approach. Developments in Mathematics, 26. Springer, New York (2012).

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

  3. V. Gutlyanskiĭ, V. Ryazanov, U. Srebro, and E. Yakubov, “On recent advances in the Beltrami equations,” Ukr. Mat. Visn., 7 (4), 467–515 (2010); transl. in J. Math. Sci., 175 (4), 413–449 (2011).

  4. U. Srebro and E. Yakubov, Beltrami equation. Handbook of complex analysis: geometric function theory. Vol. 2, 555–597, Elsevier Sci. B. V., Amsterdam (2005).

  5. E. A. Sevost’yanov, “On quasilinear Beltrami-type equations with degeneration,” Mat. Zametki, 90 (3), 445–453; transl. in Math. Notes, 90 (3–4), 431–438 (2011).

  6. K. Astala, T. Iwaniec, and G. Martin, Elliptic partial differential equations and quasiconformal mappings in the plane. Princeton Mathematical Series, 48. Princeton University Press, Princeton, NJ (2009).

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

    Article  MathSciNet  Google Scholar 

  8. M. A. Lavrent’ev and B. V. Šabat, “Geometrical properties of solutions of non-linear systems of partial differential equations”, Dokl. Akad. Nauk SSSR (N.S.), 112, 810–811 (1957).

    MathSciNet  MATH  Google Scholar 

  9. M. A. Lavrent’ev, “A general problem of the theory of quasi-conformal representation of plane regions,” Mat. Sbornik N.S., 21 (63), 285–320 (1947).

    MathSciNet  Google Scholar 

  10. M. A. Lavrent’ev, The variational method in boundary-value problems for systems of equations of elliptic type. Izdat. Akad. Nauk SSSR, Moscow (1962).

  11. B. V. Šabat, “Geometric interpretation of the concept of ellipticity,” Uspehi Mat. Nauk, 12 (6), 181–188 (1957).

    MathSciNet  Google Scholar 

  12. Šabat, B.V. “On the notion of derivative system according to M. A. Lavrent’ev,” Dokl. Akad. Nauk SSSR, 136, 1298–1301; transl. in Soviet Math. Dokl., 2, 202–205 (1961).

  13. R. Kühnau, “Minimal surfaces and quasiconformal mappings in the mean,” Trans. of Institute of Mathematics, National Academy of Sciences of Ukraine, 7 (2), 104–131 (2010).

    MATH  Google Scholar 

  14. S. L. Kruschkal and R. Kühnau, Quasikonforme Abbildungen-neue Methoden und Anwendungen. German, Quasiconformal mappings-new methods and applications With English, French and Russian summaries. Teubner-Texte zurMathematik (Teubner Texts inMathematics), 54. BSB B. G. Teubner Verlagsgesellschaft, Leipzig (1983).

  15. T. Adamowicz, “On p-harmonic mappings in the plane,” Nonlinear Anal., 71 (1–2), 502–511 (2009).

    Article  MathSciNet  Google Scholar 

  16. G. Aronsson, “On certain p-harmonic functions in the plane,” Manuscripta Math., 61 (1), 79–101 (1988).

    Article  MathSciNet  Google Scholar 

  17. A. S. Romanov, “Capacity relations in a planar quadrilateral,” Sibirsk. Mat. Zh., 49 (4), 886–897; transl. in Sib. Math. J., 49 (4), 709–717 (2008).

  18. B. Bojarski and T. Iwaniec, p-harmonic equation and quasiregular mappings. Partial differential equations (Warsaw, 1984), 25–38, Banach Center Publ., 19, PWN, Warsaw (1987).

  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 (6), 1543–1559 (2017).

    Article  MathSciNet  Google Scholar 

  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 (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 (6), 2813–2827 (2019).

    Article  MathSciNet  Google Scholar 

  22. R. R. Salimov and M. V. Stefanchuk, “On the local properties of solutions of the nonlinear Beltrami equation,” Ukr. Math. Bull., 17 (1), 77–95; transl. in J. Math. Sci., 248 (2), 203–216 (2020).

  23. R. R. Salimov and M. V. Stefanchuk, “Logarithmic asymptotics of the nonlinear Cauchy-Riemann-Beltrami equation,” Ukr. Math. J., 73, 463–478 (2021).

    Article  MathSciNet  Google Scholar 

  24. A. Golberg and R. Salimov, “Nonlinear Beltrami equation,” Complex Var. Elliptic Equ., 65 (1), 6–21 (2020).

    Article  MathSciNet  Google Scholar 

  25. O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane. Second edition. Translated from the German by K. W. Lucas. Die Grundlehren der mathematischen Wissenschaften, Band 126. Springer-Verlag, New York-Heidelberg.(1973).

  26. B. Bojarski, V. Gutlyanskiĭ, O. Martio, and V. Ryazanov, Infinitesimal geometry of quasiconformal and bi-Lipschitz mappings in the plane. EMS Tracts in Mathematics, 19. European Mathematical Society (EMS), Zürich (2013).

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

    Article  MathSciNet  Google Scholar 

  28. S. Saks, Theory of the Integral. Second revised edition. English translation by L. C. Young. With two additional notes by Stefan Banach Dover Publications, Inc., New York (1964).

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ruslan Salimov.

Additional information

Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 19, No. 2, pp. 237–253, April–June, 2022.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Salimov, R., Stefanchuk, M. Nonlinear Beltrami equation and asymptotics of its solution. J Math Sci 264, 441–454 (2022). https://doi.org/10.1007/s10958-022-06010-8

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-022-06010-8

Keywords

Navigation