Skip to main content

Advertisement

SpringerLink
Log in

On a conjecture of Ashbaugh and Benguria about lower eigenvalues of the Neumann laplacian

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

In this paper, we prove an isoperimetric inequality for lower order eigenvalues of the free membrane problem on bounded domains in a Euclidean space or a hyperbolic space which strengthens the well-known Szegö–Weinberger inequality and supports a celebrated conjecture of Ashbaugh–Benguria.

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. Ashbaugh, M. S.: Open problems on eigenvalues of the Laplacian. Analytic and Geometric Inequalities and Applications. pp. 13–28 (1999)

  2. Ashbaugh, M.S., Benguria, R.D.: Universal bounds for the low eigenvalues of Neumann Laplacians in \(N\) dimensions. SIAM J. Math. Anal. 24, 557–570 (1993)

    Article  MATH  Google Scholar 

  3. Ashbaugh, M.S., Benguria, R.D.: Sharp upper bound to the first nonzero Neumann eigenvalue for bounded domains in spaces of constant curvature. J. Lond. Math. Soc. 52, 402–416 (1995)

    Article  MATH  Google Scholar 

  4. Brasco, L., De Philippis, G.: Spectral inequalities in quantitative form. Shape optimization and spectral theory, pp. 201–281, De Gruyter Open, Warsaw (2017)

  5. Brasco, L., Pratelli, A.: Sharp stability of some spectral inequalities. Geom. Funct. Anal. 22, 107–135 (2012)

    Article  MATH  Google Scholar 

  6. Chavel, I.: Eigenvalues in Riemannian Geometry. Academic, New York (1984)

    MATH  Google Scholar 

  7. Henrot, A.: Extremum problems for eigenvalues of elliptic operators, Birkhäuser Verlag, Basel, x + 202 pp. (2006). ISBN 978-3-76437705-2

  8. Hile, G.N., Xu, Z.: Inequalities for sums of the reciprocals of eigenvalues. J. Math. Anal. Appl. 180, 412–430 (1993)

    Article  MATH  Google Scholar 

  9. Nadirashvili, N.: Conformal maps and isoperimetric inequalities for eigenvalues of the Neumann problem. Proceedings of the Ashkelon Workshop on Complex Function Theory (1996), 197–201, Israel Math. Conf. Proc. 11, Bar-Ilan Univ., Ramat Gan (1997)

  10. Schoen, R., Yau, R., S. T.: Lectures on Differential Geometry. International Press, Cambridge, MA (2004)

  11. Szegö, G.: Inequalities for certain eigenvalues of a membrane of given area. J. Rational Mech. Anal. 3, 343–356 (1954)

    MATH  Google Scholar 

  12. Weinberger, H.F.: An isoperimetric inequality for the \(n\)-dimensional free membrane problem. J. Rational Mech. Anal. 5, 633–636 (1956)

    MATH  Google Scholar 

  13. Xia, C.: A universal bound for the low eigenvalues of Neumann Laplacians on compact domains in a Hadamard manifold. Monatsh. Math. 128, 165–171 (1999)

    Article  MATH  Google Scholar 

  14. Xu, Y.: The first nonzero eigenvalue of neumann problem on Riemannian manifolds. J. Geom. Anal. 5, 151–165 (1995)

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referee for the careful reading of the manuscript and the encouragements.

Author information

Authors and Affiliations

  1. Department of Mathematics, Southern University of Science and Technology, Shenzhen, Guandong, China

    Changyu Xia

  2. Departamento de Matemática, Universidade de Brasilia, Brasilia, DF, 70910-900, Brazil

    Qiaoling Wang

Authors
  1. Changyu Xia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qiaoling Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Changyu Xia.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

Communicated by Giga.

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

Xia, C., Wang, Q. On a conjecture of Ashbaugh and Benguria about lower eigenvalues of the Neumann laplacian. Math. Ann. 385, 863–879 (2023). https://doi.org/10.1007/s00208-021-02336-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-021-02336-x

Mathematics Subject Classification

Search

Navigation