Skip to main content
SpringerLink
Log in

Space quasiconformal composition operators with applications to Neumann eigenvalues

  • Published:
Analysis and Mathematical Physics Aims and scope Submit manuscript

Abstract

In this article we obtain estimates of Neumann eigenvalues of p-Laplace operators in a large class of space domains satisfying quasihyperbolic boundary conditions. The suggested method is based on composition operators generated by quasiconformal mappings and their applications to Sobolev–Poincaré-inequalities. By using a sharp version of the reverse Hölder inequality we refine our estimates for quasi-balls, that is, images of balls under quasiconformal mappings of the whole space.

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. Ahlfors, L.: Lectures on Quasiconformal Mappings. D. Van Nostrand Co Inc, Toronto (1966)

    MATH  Google Scholar 

  2. Astala, K., Koskela, P.: Quasiconformal mappings and global integrability of the derivative. J. Anal. Math. 57, 203–220 (1991)

    Article  MathSciNet  Google Scholar 

  3. Astala, K.: Area distortion of quasiconformal mappings. Acta Math. 173, 37–60 (1994)

    Article  MathSciNet  Google Scholar 

  4. Bojarski, B., Iwaniec, T.: Analytic foundations of the theory of quasiconformal mappings in \({\mathbb{R}}^n\). Ann. Acad. Sci. Fenn. Ser. A. I. Math. 8, 257–324 (1983)

    Article  MathSciNet  Google Scholar 

  5. Brandolini, B., Chiacchio, F., Dryden, E.B., Langford, J.J.: Sharp Poincaré inequalities in a class of non-covex sets. J. Spectr. Theory 8, 1583–1615 (2018)

    Article  MathSciNet  Google Scholar 

  6. Edmunds, D.E., Hurri-Syrjänen, R.: Rellich’s theorem in irregular domains. Houst. J. Math. 30, 577–586 (2004)

    MathSciNet  MATH  Google Scholar 

  7. Esposito, L., Nitsch, C., Trombetti, C.: Best constants in Poincaré inequalities for convex domains. J. Convex Anal. 20, 253–264 (2013)

    MathSciNet  MATH  Google Scholar 

  8. Gehring, F.W.: The \(L^p\)-integrability of the partial derivatives of a quasiconformal mapping. Acta Math. 130, 265–277 (1973)

    Article  MathSciNet  Google Scholar 

  9. Gehring, F.W., Martio, O.: Lipschitz classes and quasiconformal mappings. Ann. Acad. Sci. Fenn. Ser. A I Math. 10, 203–219 (1985)

    Article  MathSciNet  Google Scholar 

  10. Gehring, F.W., Martin, G., Palka, B.: An Introduction to the Theory of Higher-Dimensional Quasiconformal Mappings. Mathematical Surveys and Monographs, vol. 216. American Mathematical Society, Providence (2017)

    MATH  Google Scholar 

  11. Gol’dshtein, V.: The degree of summability of generalized derivatives of quasiconformal homeomorphisms. Sib. Math. J. 22(6), 821–836 (1981)

    Article  MathSciNet  Google Scholar 

  12. Gol’dshtein, V., Gurov, L.: Applications of change of variables operators for exact embedding theorems. Integral Equ. Oper. Theory 19, 1–24 (1994)

    Article  MathSciNet  Google Scholar 

  13. Gol’dshtein, V., Hurri-Syrjänen, R., Ukhlov, A.: Space quasiconformal mappings and Neumann eigenvalues in fractal type domains. Georgian Math. J. 25, 221–233 (2018)

    Article  MathSciNet  Google Scholar 

  14. Gol’dshtein, V., Pchelintsev, V., Ukhlov, A.: Spectral estimates of the \(p\)-Laplace Neumann operator and Brennan’s conjecture. Boll. Unione Mat. Ital. 11, 245–264 (2018)

    Article  MathSciNet  Google Scholar 

  15. Gol’dshtein, V., Pchelintsev, V., Ukhlov, A.: Integral estimates of conformal derivatives and spectral properties of the Neumann–Laplacian. J. Math. Anal. Appl. 463, 19–39 (2018)

    Article  MathSciNet  Google Scholar 

  16. Gol’dshtein, V., Pchelintsev, V., Ukhlov, A.: On the first eigenvalue of the degenerate p-Laplace operator in non-convex domains. Integral Equ. Oper. Theory 90, 43 (2018)

    Article  MathSciNet  Google Scholar 

  17. Gol’dshtein, V., Pchelintsev, V., Ukhlov, A.: Spectral properties of the Neumann–Laplace operator in quasiconformal regular domains. Contemp. Math. 734, 129–144 (2019)

    Article  MathSciNet  Google Scholar 

  18. Gol’dshtein, V., Ukhlov, A.: Weighted Sobolev spaces and embedding theorems. Trans. Am. Math. Soc. 361, 3829–3850 (2009)

    Article  MathSciNet  Google Scholar 

  19. Gol’dshtein, V., Ukhlov, A.: About homeomorphisms that induce composition operators on Sobolev spaces. Complex Var. Elliptic Equ. 55, 833–845 (2010)

    Article  MathSciNet  Google Scholar 

  20. Gol’dshtein, V., Ukhlov, A.: On the first eigenvalues of free vibrating membranes in conformal regular domains. Arch. Ration. Mech. Anal. 221(2), 893–915 (2016)

    Article  MathSciNet  Google Scholar 

  21. Gol’dshtein, V., Ukhlov, A.: The spectral estimates for the Neumann–Laplace operator in space domains. Adv. Math. 315, 166–193 (2017)

    Article  MathSciNet  Google Scholar 

  22. Gol’dshtein, V., Ukhlov, A.: Composition operators on Sobolev spaces and Neumann eigenvalues. Complex Anal. Oper. Theory 13, 2781–2798 (2019)

    Article  MathSciNet  Google Scholar 

  23. Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford Mathematical Monographs. Oxford University Press, Oxford (1993)

    MATH  Google Scholar 

  24. Hurri, R.: Poincaré domains in \({\mathbb{R}}^n\). Ann. Acad. Sci. Fenn., Ser. A, I. Math. Dissertationes 71, 1–42 (1988)

  25. Hurri-Syrjänen, R., Marola, N., Vähäkangas, A.V.: Poincaré inequalities in quasihyperbolic boundary condition domains. Manuscr. Math. 148, 99–118 (2015)

    Article  Google Scholar 

  26. Hurri-Syrjänen, R., Staples, S.G.: Quasiconformal maps and Poincaré domains. Rocky Mount. J. Math. 26, 1395–1423 (1996)

    Article  Google Scholar 

  27. Iwaniec, T.: The Gehring Lemma, Quasiconformal Mappings and Analysis, vol. 1998, pp. 181–204. Springer, Ann Arbor (1995)

    Google Scholar 

  28. Iwaniec, T., Nolder, C.: Hardy–Littlewood for quasiregular mappings in certain domains in \({\mathbb{R}}^n\). Ann. Acad. Sci. Fenn. Ser. A. I. Math 10, 267–282 (1985)

    Article  MathSciNet  Google Scholar 

  29. Koskela, P., Onninen, J., Tyson, J.T.: Quasihyperbolic boundary conditions and capacity: Poincaré domains. Math. Ann. 323, 811–830 (2002)

    Article  MathSciNet  Google Scholar 

  30. Maz’ya, V.: Sobolev Spaces: With Applications to Elliptic Partial Differential equations. Springer, Berlin (2010)

    Google Scholar 

  31. Rasila, A.: Introduction to quasiconformal mappings in \(n\)-space. In: Proceedings of the International Workshop on Quasiconformal Mappings and their Applications (2006)

  32. Stein, E.M.: Harmonic Analysis, Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993)

    MATH  Google Scholar 

  33. Ukhlov, A.: On mappings, which induce embeddings of Sobolev spaces. Sib. Math. J. 34, 185–192 (1993)

    Google Scholar 

  34. Vodop’yanov, S.K., Gol’dstein, V.M., Reshetnyak, YuG: On geometric properties of functions with generalized first derivatives. Usp. Mat. Nauk 34, 17–65 (1979)

    MathSciNet  Google Scholar 

  35. Vodop’yanov, S.K., Ukhlov, A.D.: Superposition operators in Sobolev spaces. Izvest. VUZ 46(4), 11–33 (2002). (in Russian)

    MathSciNet  MATH  Google Scholar 

  36. Vuorinen, M.: Conformal Geometry and Quasiregular Mappings. Lecture Notes in Mathematics, vol. 1319. Springer, Berlin (1988)

    Book  Google Scholar 

  37. Ziemer, W.P.: Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation. Graduate Texts in Mathematics, vol. 120. Springer, New York (1989)

    Book  Google Scholar 

Download references

Acknowledgements

The first author was supported by the United States-Israel Binational Science Foundation (BSF Grant No. 2014055). The second author, whose visit to the Ben-Gurion University of Negev was supported by a grant from the Finnish Academy of Science and Letters, Vilho, Yrjö and Kalle Väisälä Foundation, is grateful for the hospitality given by the Department of Mathematics of the Ben-Gurion University of the Negev. The third author was supported by RSF Grant No. 20-71-00037 (Results of Sect. 3).

Author information

Authors and Affiliations

  1. Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, Beer Sheva, 8410501, Israel

    Vladimir Gol’dshtein & Alexander Ukhlov

  2. Department of Mathematics and Statistics, University of Helsinki, Gustaf Hällströmin katu 2 b, 00014, Helsinki, Finland

    Ritva Hurri-Syrjänen

  3. Division for Mathematics and Computer Sciences, Tomsk Polytechnic University, Lenin Ave. 30, Tomsk, Russia, 634050

    Valerii Pchelintsev

  4. Department of Mathematical Analysis and Theory of Functions, Tomsk State University, Lenin Ave. 36, Tomsk, Russia, 634050

    Valerii Pchelintsev

Authors
  1. Vladimir Gol’dshtein
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ritva Hurri-Syrjänen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Valerii Pchelintsev
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Alexander Ukhlov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Ukhlov.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gol’dshtein, V., Hurri-Syrjänen, R., Pchelintsev, V. et al. Space quasiconformal composition operators with applications to Neumann eigenvalues. Anal.Math.Phys. 10, 78 (2020). https://doi.org/10.1007/s13324-020-00420-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13324-020-00420-0

Keywords

Mathematics Subject Classification

Search

Navigation