Skip to main content
SpringerLink
Log in

Lower order eigenvalues of Dirichlet Laplacian

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

In this paper, we investigate an eigenvalue problem for the Dirichlet Laplacian on a domain in an n-dimensional compact Riemannian manifold. First we give a general inequality for eigenvalues. As one of its applications, we study eigenvalues of the Laplacian on a domain in an n-dimensional complex projective space, on a compact complex submanifold in complex projective space and on the unit sphere. By making use of the orthogonalization of Gram–Schmidt (QR-factorization theorem), we construct trial functions. By means of these trial functions, estimates for lower order eigenvalues are obtained.

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.: Isoperimetric and universal inequalities for eigenvalues. In: Davies, E.B., Yu Safalov (eds.) spectral theory and geometry (Edinburgh, 1998), London Math. Soc. Lecture Notes, vol. 273. Cambridge University Press, Cambridge, pp. 95–139 (1999)

  2. Ashbaugh M.S. and Benguria R.D. (1991). Proof of the Payne-Pólya-Weinberger conjecture. Bull. Am. Math. Soc. 25: 19–29

    Article  MATH  MathSciNet  Google Scholar 

  3. Ashbaugh M.S. and Benguria R.D. (1992). A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions. Ann. Math. 135: 601–628

    Article  MathSciNet  Google Scholar 

  4. Ashbaugh M.S. and Benguria R.D. (1992). A second proof of the Payne-Pólya-Weinberger conjecture. Commun. Math. Phys. 147: 181–190

    Article  MATH  MathSciNet  Google Scholar 

  5. Ashbaugh M.S. and Benguria R.D. (1993). More bounds on eigenvalue ratios for Dirichlet Laplacians in n dimensions. SIAM J. Math. Anal. 24: 1622–1651

    Article  MATH  MathSciNet  Google Scholar 

  6. Brands J.J.A.M. (1964). Bounds for the ratios of the first three membrane eigenvalues. Arch. Ration. Mech. Anal. 16: 265–268

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  8. Cheng Q.-M. and Yang H.C. (2005). Estimates on eigenvalues of Laplacian. Math. Ann. 331: 445–460

    Article  MATH  MathSciNet  Google Scholar 

  9. Cheng Q.-M. and Yang H.C. (2006). Inequalities for eigenvalues of a clamped plate problem. Trans. Am. Math. Soc. 358: 2625–2635

    Article  MATH  MathSciNet  Google Scholar 

  10. Cheng Q.-M. and Yang H.C. (2006). Inequalities for eigenvalues of Laplacian on domains and compact complex hypersurfaces in complex projective spaces. J. Math. Soc. Jpn. 58: 545–561

    Article  MATH  MathSciNet  Google Scholar 

  11. Chiti G. (1983). A bound for the ratio of the first two eigenvalues of a membrane. SIAM J. Math. Anal. 14: 1163–1167

    Article  MATH  MathSciNet  Google Scholar 

  12. Vries H.L. (1967). On the upper bound for the ratio of the first two membrane eigenvalues. Z. Naturforschung 22: 152–153

    MATH  Google Scholar 

  13. Harrell E.M. II (1993). Some geometric bounds on eigenvalue gaps. Comm. Part. Diff. Equ. 18: 179–198

    Article  MATH  MathSciNet  Google Scholar 

  14. Michel P.L. and Harrell E.M. II (1994). Commutator bounds for eigenvalues with applications to spectral geometry. Comm. Part. Diff. Equ. 19: 2037–2055

    Article  MATH  MathSciNet  Google Scholar 

  15. Stubbe J. and Harrell E.M. II (1997). On trace identities and universal eigenvalue estimates for some partial differential operators. Trans. Am. Math. Soc. 349: 1797–1809

    Article  MATH  MathSciNet  Google Scholar 

  16. Hile G.N. and Protter M.H. (1980). Inequalities for eigenvalues of the Laplacian. Indiana Univ. Math. J. 29: 523–538

    Article  MATH  MathSciNet  Google Scholar 

  17. Leung P.-F. (1991). On the consecutive eigenvalues of the Laplacain of a compact minimal submanifold in a sphere. J. Aust. Math. Soc. 50: 409–426

    Article  MATH  Google Scholar 

  18. Li P. (1980). Eigenvalue estimates on homogeneous manifolds. Comment. Math. Helv. 55: 347–363

    Article  MATH  MathSciNet  Google Scholar 

  19. Levitin M. and Parnovski L. (2002). Commutators, spectral trace identities and universal estimates for eigenvalues. J. Funct. Anal. 192: 425–445

    Article  MATH  MathSciNet  Google Scholar 

  20. Marcellini P. (1980). Bounds for the third membrane eigenvalue. J. Diff. Equ. 37: 438–443

    Article  MATH  MathSciNet  Google Scholar 

  21. Payne L.E., Polya G. and Weinberger H.F. (1955). Sur le quotient de deux fréquences propres consécutives. Comptes Rendus Acad. Sci. Paris 241: 917–919

    MATH  MathSciNet  Google Scholar 

  22. Payne L.E., Polya G. and Weinberger H.F. (1956). On the ratio of consecutive eigenvalues. J. Math. Phys. 35: 289–298

    MathSciNet  MATH  Google Scholar 

  23. Protter M.H. (1987). Can one hear the shape of a drum?. SIAM Rev. 29: 185–197

    Article  MATH  MathSciNet  Google Scholar 

  24. Thompson C.J. (1969). On the ratio of consecutive eigenvalues in n-dimensions. Stud. Appl. Math. 48: 281–283

    MATH  Google Scholar 

  25. Yang, H.C.: An estimate of the difference between consecutive eigenvalues, preprint IC/91/60 of ICTP, Trieste (1991)

  26. Yang P.C. and Yau S.T. (1980). Eigenvalues of the Laplacian of compact Riemannian surfaces and minimal submanifolds. Ann. Scuola Norm. Sup. Pisa CI. Sci. 7: 55–63

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, College of Science, Nanjing University of Science and Technology, Nanjing, 210094, People’s Republic of China

    Hejun Sun

  2. Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga, 840-8502, Japan

    Qing-Ming Cheng

  3. Academy of Mathematics and Systematical Sciences, CAS, Beijing, 100080, China

    Hongcang Yang

Authors
  1. Hejun Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qing-Ming Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hongcang Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qing-Ming Cheng.

Additional information

Qing-Ming Cheng research was partially supported by a Grant-in-Aid for Scientific Research from JSPS.

Hejun Sun and Hongcang Yang research were partially supported by NSF of China.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sun, H., Cheng, QM. & Yang, H. Lower order eigenvalues of Dirichlet Laplacian. manuscripta math. 125, 139–156 (2008). https://doi.org/10.1007/s00229-007-0136-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-007-0136-9

Mathematics Subject Classification (2001)

Search

Navigation