Skip to main content
SpringerLink
Log in

On the local properties of solutions of the nonlinear Beltrami equation

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

Abstract

A power estimate of the area of the image of a disk for regular homeomorphisms possessing the Luzin N-property is obtained in terms of the p-angular dilation for p > 2. The result generalizes the known estimate by M.A. Lavrent’ev. A number of theorems on the asymptotic behavior of regular homeomorphic solutions of the nonlinear Beltrami equation are proved, and an extreme analog of the Ikoma–Schwartz lemma is formulated.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    Book  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  4. V. Gutlyanskii, V. Ryazanov, U. Srebro, and E. Yakubov, “On recent advances in the degenerate Beltrami equations,” Ukr. Mat. Visn., 7(4), 467-515 (2010).

    MATH  Google Scholar 

  5. U. Srebro and E. Yakubov, “The Beltrami equation,” in: Handbook in Complex Analysis: Geometric function theory, Vol. 2, Elsevier, Amsterdam, 2005, pp. 555-597.

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

    Article  MathSciNet  MATH  Google Scholar 

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

  8. M. A. Lavrent’ev and B. V. Shabat, “Geometric properties of solutions of nonlinear systems of partial differential equations,” Dokl. AN SSSR, 112(5), 810-811 (1957).

    Google Scholar 

  9. M. A. Lavrent’ev, “A general problem of the theory of quasiconformal mappings of planar domains,” Matem. Sb., 21(63)(2), 285-320.

  10. M. A. Lavrent’ev, The Variational Method in Boundary-Value Problems for Systems of Equations of the Elliptic Type [in Russian], Izd. AN SSSR, Moscow, 1962.

  11. B. V. Shabat, “The geometric meaning of the notion of ellipticity,” Uspekhi Mat. Nauk, 12(6)(78), 181-188 (1957).

  12. B. V. Shabat, “To the notion of a derivative system in the meaning by M. A. Lavrent’ev,” Dokl. AN SSSR, 136(6), 1298-1301 (1961).

    Google Scholar 

  13. R. Kuhnau, “Minimal surfaces and quasiconformal mappings in the mean,” Zbir. Prats. Inst. Mat. NANU, 7(2), 104–131 (2010).

    MathSciNet  MATH  Google Scholar 

  14. S. L. Krushkal’ and R. Kuhnau, Quasiconformal Mappings – New Methods and Applications [in Russian], Nauka, Moscow, 1984.

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. A. S. Romanov, “Capacity relations in a plane four-side figure,” Siber. Math. J., 49(4), 709-717 (2008).

    Article  Google Scholar 

  18. B. Bojarski and T. Iwaniec, “p-harmonic equation and quasiregular mappings,” Banach Center Publ., 19(1), 25-38 (1987).

    Article  MathSciNet  MATH  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 Lin., 34(6), 1543-1559 (2017).

    Article  MathSciNet  MATH  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  MATH  Google Scholar 

  21. A. Golberg, R. Salimov, and M. Stefanchuk, “Asymptotic dilation of regular homeomorphisms,” Compl. Anal. Oper. Theory, 13(6), 2813–2827 (2019).

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Google Scholar 

  24. T. Iwaniec, P. Koskela, and J. Onninen, “Mappings of finite distortion: Compactness,” Ann. Acad. Sci. Fenn. Ser. A1 Math., 27(2), 391-417 (2002).

    MathSciNet  MATH  Google Scholar 

  25. S. K. Vodop’yanov and A. D. Ukhlov, “Operators of superposition in Sobolev spaces,” Izv. Vyssh. Ucheb. Zav. Mat., No. 10, 11-33 (2002).

    MATH  Google Scholar 

  26. S. K. Vodop’yanov, “Mappings with finite co-distortion and the Sobolev classes of functions,” Dokl. RAN, 440(3), 301-305 (2011).

    MathSciNet  Google Scholar 

  27. S. K. Vodop’yanov, “On the regularity of mappings inverse to Sobolev ones,” Matem. Sb., 203(10), 3-32 (2012).

    Article  Google Scholar 

  28. S. K. Vodop’yanov, “On the differentiability of mappings of the Sobolev class \( {W}_{n-1}^1 \) with conditions on the distortion function,” Sib. Mat. Zh., 59(6), 1240-1267 (2018).

    MathSciNet  Google Scholar 

  29. J. Kauhanen, P. Koskela, and J. Maly, “Mappings of finite distortion: discreteness and openness,” Arch. Ration. Mech. Anal., 160(2), 135-151 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  30. J. Kauhanen, P. Koskela, and J. Maly, “Mappings of finite distortion: condition N,” Michigan Math. J., 49(1), 169-181 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  31. J. Kauhanen, P. Koskela, and J. Maly, “Mappings of finite distortion: sharp Orlicz-conditions,” Rev. Mat. Iberoamer., 19(3), 857-872 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  32. 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 

  33. D. A. Kovtonyuk, V. I. Ryazanov, R. R. Salimov, and E. A. Sevost’yanov, “To the theory of Orlicz–Sobolev classes,” Alg. Analiz, 25(6), 50-102 (2013).

    MathSciNet  MATH  Google Scholar 

  34. D. A. Kovtonyuk, V. I. Ryazanov, R. R. Salimov, and E. A. Sevost’yanov, “Boundary behavior of Orlicz–Sobolev classes,” Mat. Zamet., 95(4), 564-576 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  35. D. A. Kovtonyuk, I. V. Petkov, V. I. Ryazanov, and R. R. Salimov, “Boundary behavior and the Dirichlet problem for Beltrami equations,” Alg. Analiz, 25(4), 101-124 (2013).

    MathSciNet  MATH  Google Scholar 

  36. R. R. Salimov, “Metric properties of Orlicz–Sobolev classes,” Ukr. Mat. Visn., 13(1), 129-141 (2016).

    Google Scholar 

  37. R. R. Salimov, “On the finite Lipschitz property of Orlicz–Sobolev classes,” Vladikavkaz. Mat. Zh., 17(1), 64-77 (2015).

    MathSciNet  Google Scholar 

  38. B. Bojarski, V. Gutlyanskii, O. Martio, and V. Ryazanov, Infinitesimal Geometry of Quasiconformal and Bi-Lipschitz Mappings in the Plane, E.M.S., Warsaw–Donetsk–Helsinki, 2013.

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  41. C. Andreian Cazacu, “Influence of the orientation of the characteristic ellipses on the properties of the quasiconformal mappings,” in: Proc. Rom. Finn. Sem., Publ. House Acad. Soc. Rep. Rom., Bucharest, 1971, pp. 65-85.

  42. M. A. Brakalova and J. A. Jenkins, “On solutions of the Beltrami equation,” J. Analyse Math., 76, 67-92 (1998).

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  45. V. Gutlyanskii and A. Golberg, “Rings and Lipschitz continuity of quasiconformal mappings,” in: Analysis and Mathematical Physics, Birkh\( {}^{\overset{\prime }{,}} \)auser, Basel, 2009, pp.187-192.

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  48. V. V. Bilet, B. A. Klishchuk, and R. R. Salimov, “Estimates of the area of the image of a disk for Sobolev classes,” Zbir. Prats. Inst. Mat. NANU, 14(1), 39-45 (2017).

    MATH  Google Scholar 

  49. T. V. Lomako and R. R. Salimov, “To the theory of extreme problems,” Zbir. Prats. Inst. Mat. NANU, 7(2), 264-269 (2010).

    MATH  Google Scholar 

  50. R. R. Salimov, “Lower bounds for a p-modulus and mappings of a Sobolev class,” Alg. Analiz, 26(6), 143-171 (2014).

    Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Ruslan R. Salimov & Mariia V. Stefanchuk

Authors
  1. Ruslan R. Salimov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mariia V. Stefanchuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ruslan R. Salimov.

Additional information

Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 17, No. 1, pp. 77–95 January–March, 2020.

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. On the local properties of solutions of the nonlinear Beltrami equation. J Math Sci 248, 203–216 (2020). https://doi.org/10.1007/s10958-020-04870-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-020-04870-6

Keywords

Search

Navigation