Skip to main content
SpringerLink
Log in

Inequalities for eigenvalues of a clamped plate problem

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

In this paper we study eigenvalues of a clamped plate problem on compact domains in complete manifolds. For complete manifolds admitting special functions, we prove universal inequalities for eigenvalues of clamped plate problem independent of the domains of Payne–Pólya–Weinberger–Yang type. These manifolds include Hadamard manifolds with Ricci curvature bounded below, a class of warped product manifolds, the product of Euclidean spaces with any complete manifolds and manifolds admitting eigenmaps to a sphere. In the case of warped product manifolds, our result implies a universal inequality on hyperbolic space proved by Cheng–Yang. We also strengthen an inequality for eigenvalues of clamped plate problem on submanifolds in a Euclidean space obtained recently by Cheng, Ichikawa and Mametsuka.

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., Safalov, Yu. (eds.) Spectral Theory and Geometry (Edingurgh, 1998), London Mathematical Society Lecture Notes, vol. 273, pp. 95–139. Cambridge University Press, Cambridge (1999)

  2. Ashbaugh M.S.: Universal eigenvalue bounds of Payne-Pólya-Weinberger, Hile-Protter and H C Yang. Proc. India Acad. Sci. Math. Sci. 112, 3–30 (2002)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  4. 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  MathSciNet  Google Scholar 

  5. Ashbaugh M.S., Hermi L.: A unified approach to universal inequalities for eigenvalues of elliptic operators. Pac. J. Math. 217, 201–219 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. Ballmann W., Gromov M., Schroeder V.: Manifolds of Nonpositive Curvature. Birkhäuser, Boston (1985)

    MATH  Google Scholar 

  7. Chavel, I.: Eigenvalues in Riemannian geometry. Including a Chapter by Burton Randol. With an Appendix by Jozef Dodziuk. Pure and Applied Mathematics, vol. 115, xiv + 362 pp. Academic Press, Inc., Orlando (1984)

  8. Chen D., Cheng Q.M.: Extrinsic estimates for eigenvalues of the Laplace operator. J. Math. Soc. Jpn. 60, 325–339 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chen Z.C., Qian C.L.: Estimates for discrete spectrum of Laplacian operator with any order. J. China Univ. Sci. Technol. 20, 259–266 (1990)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  12. Cheng Q.M., Yang H.C.: Universal bounds for eigenvalues of a buckling problem. Commun. Math. Phys. 262(3), 663–675 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  14. Cheng, Q.M., Yang, H.C.: Universal inequalities for eigenvalues of a clamped plate problem on a hyperbolic space. Proc. Am. Math. Soc. (in press)

  15. Cheng Q.M., Ichikawa T., Mametsuka S.: Inequalities for eigenvalues of Laplacian with any order. Commun. Contemp. Math. 11, 639–655 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  16. Cheng Q.M., Ichikawa T., Mametsuka S.: Estimates for eigenvalues of a clamped plate problem on Riemannian manifolds. J. Math. Soc. Jpn. 62, 673–686 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  17. EI Soufi A., Harrell E.M., Ilias S.: Universal inequalities for the eigenvalues of Laplace and Schrödinger operators on submanifolds. Trans. Am. Math. Soc. 361, 2337–2350 (2009)

    Article  Google Scholar 

  18. Hardy G., Littlewood J.E., Pólya G.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1994)

    Google Scholar 

  19. Harrell E.M.: Some geometric bounds on eigenvalue gaps. Commun. Partial Differ. Equ. 18, 179–198 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  20. Harrell E.M.: Commutators, eigenvalue gaps, and mean curvature in the theory of Schrödinger operators. Commun. Partial Differ. Equ. 32, 401–413 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  21. Harrell E.M., Michel P.L.: Commutator bounds for eigenvalues with applications to spectral geometry. Commin. Partial Differ. Equ. 19, 2037–2055 (1994)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  23. Harrell E.M., Yolcu S.Y.: Eigenvalue inequalities for Klein-Gordon operators. J. Funct. Anal. 254, 3977–3995 (2009)

    Article  Google Scholar 

  24. Heintz E., Im Hof H.C.: Goemetry of horospheres. J. Diff. Geom. 12, 481–489 (1977)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  26. Hook S.M.: Domain independent upper bounds for eigenvalues of elliptic operator. Trans. Am. Math. Soc. 318, 615–642 (1990)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  28. Nash J.F.: The imbedding problem for Riemannian manifolds. Ann. Math. 63, 20–63 (1956)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  30. Sakai T.: On Riemannian manifolds admitting a function whose gradient is of constant norm. Kodai Math. J. 19, 39–51 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  31. Schoen, R., Yau, S.T.: Lectures on Differential Geometry. Conference Proceedings and Lecture Notes in Geometry and Topology, I, v + 235 pp. International Press, Cambridge (1994)

  32. Sun H.J., Cheng Q.M., Yang H.C.: Lower order eigenvalues of Dirichlet Laplacian. Manuscripta Math. 125, 139–156 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  33. Wang Q., Xia C.: Universal bounds for eigenvalues of the buckling problem on spherical domains. Commun. Math. Phys. 270, 759–775 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  34. Wang Q., Xia C.: Universal bounds for eigenvalues of the biharmonic operator on Riemannian manifolds. J. Funct. Anal. 245, 334–352 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  35. Wang Q., Xia C.: Universal bounds for eigenvalues of Schrödinger operator on Riemannian manifolds. Ann. Acad. Sci. Fenn. Math. 33, 319–336 (2008)

    MATH  MathSciNet  Google Scholar 

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

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

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática, Universidade de Brasília, Brasília, DF, 70910-900, Brazil

    Qiaoling Wang & Changyu Xia

Authors
  1. Qiaoling Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Changyu Xia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qiaoling Wang.

Additional information

Communicated by J. Jost.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, Q., Xia, C. Inequalities for eigenvalues of a clamped plate problem. Calc. Var. 40, 273–289 (2011). https://doi.org/10.1007/s00526-010-0340-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00526-010-0340-4

Mathematics Subject Classification (2000)

Search

Navigation