Skip to main content
Log in

Lower bounds for sums of eigenvalues of the buckling problem

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

This paper studies the lower bound of eigenvalues of the buckling problem with the Dirichlet boundary condition. We prove some Berezin–Li–Yau type inequalities with additional term for the buckling problem on a bounded domain in the Euclidean space.

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Ashbaugh, M. S.: On universal inequalities for the low eigenvalues of the buckling problem. Partial differential equations and inverse problems, 13–31, Contemp. Math., 362, Amer. Math. Soc., Providence, RI, (2004)

  2. Ashbaugh, M.S., Bucur, D.: On the isoperimetric inequality for the buckling of a clamped plate. Z. Angew. Math. Phys. 54(5), 756–770 (2003)

    Article  MathSciNet  Google Scholar 

  3. Ashbaugh, M. S., Laugesen, R. S.: Fundamental tones and buckling loads of clamped plates. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 23 (2), 383–402 (1996)

  4. Barnes, D.C.: Extremal problems for eigenvalues with applications to buckling, vibration and sloshing. SIAM J. Math. Anal. 16(2), 341–357 (1985)

    Article  MathSciNet  Google Scholar 

  5. Berezin, F.A.: Covariant and contravariant symbols of operators. Izv. Akad. Nauk SSSR Ser. Mat. 36, 1134–1167 (1972)

    MathSciNet  Google Scholar 

  6. Cheng, Q.M., Qi, X., Wang, Q., Xia, C.: Inequalities for eigenvalues of the buckling problem of arbitrary order. Ann. Mat. Pura Appl. 197(1), 211–232 (2018)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Cheng, Q.M., Yang, H.: Universal bounds for eigenvalues of a buckling problem II. Trans. Amer. Math. Soc. 364(11), 6139–6158 (2012)

    Article  MathSciNet  Google Scholar 

  9. Cianchi, A., Fusco, N.: Functions of bounded variation and rearrangements. Arch. Ration. Mech. Anal. 165(1), 1–40 (2002)

    Article  MathSciNet  Google Scholar 

  10. Conway, H.D., Leissa, A.W.: A method for investigating certain eigenvalue problems of the buckling and vibration of plates. J. Appl. Mech. 2, 557–558 (1960)

    Article  MathSciNet  Google Scholar 

  11. Du, F., Mao, J., Wang, Q., Wu, C.: Eigenvalue inequalities for the buckling problem of the drifting Laplacian on Ricci solitons. J. Differential Equations 260(7), 5533–5564 (2016)

    Article  MathSciNet  Google Scholar 

  12. Friedlander, L.: Remarks on the membrane and buckling eigenvalues for planar domains. Mosc. Math. J. 4(2), 369–375 (2004)

    Article  MathSciNet  Google Scholar 

  13. Ghidaglia, J.M., Marion, M., Temam, R.: Generalization of the Sobolev-Lieb-Thirring inequalities and applications to the dimension of attractors. Differential Integral Equations 1(1), 1–21 (1988)

    MathSciNet  MATH  Google Scholar 

  14. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Springer-Verlag, Cham (2001)

    Book  Google Scholar 

  15. Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1952)

    MATH  Google Scholar 

  16. Ilyin, A.A.: Lower bounds for the spectrum of the Laplace and Stokes operators. Discrete Contin. Dyn. Syst. 28(1), 131–146 (2010)

    Article  MathSciNet  Google Scholar 

  17. Kawohl, B., Levine, H.A., Velte, W.: Buckling eigenvalues for a clamped plate embedded in an elastic medium and related questions. SIAM J. Math. Anal. 24(2), 327–340 (1993)

    Article  MathSciNet  Google Scholar 

  18. Krbec, M., Kufner, A., Opic, B., Rákosník, J.: Nonlinear analysis, function spaces and applications, vol. 5. Prometheus Publishing House, Prague (1994)

    MATH  Google Scholar 

  19. Levine, H.A., Protter, M.H.: Unrestricted lower bounds for eigenvalues for classes of elliptic equations and systems of equations with applications to problems in elasticity. Math. Methods Appl. Sci. 7(2), 210–222 (1985)

    Article  MathSciNet  Google Scholar 

  20. Laptev, A., Weidl, T.: Recent results on Lieb–Thirring inequalities. Journées “Équations aux Dérivées Partielles" (La Chapelle sur Erdre, 2000), Exp. No. XX, 14 pp., Univ. Nantes, Nantes, (2000)

  21. Li, P., Yau, S.T.: On the Schrödinger equation and the eigenvalue problem. Comm. Math. Phys. 88(3), 309–318 (1983)

    Article  MathSciNet  Google Scholar 

  22. Lord Rayleigh, J.W.S.B.: The theory of sound, 2nd edn. Dover Publications, New York, N.Y. (1945)

    MATH  Google Scholar 

  23. Melas, A.D.: A lower bound for sums of eigenvalues of the Laplacian. Proc. Amer. Math. Soc. 131(2), 631–636 (2003)

    Article  MathSciNet  Google Scholar 

  24. Pólya, G., Szegö, G.: Isoperimetric Inequalities in Mathematical Physics. Princeton University Press, Princeton, N. J, Annals of Mathematics Studies (1951)

  25. Rother, W.: New bounds for the first eigenvalue of an elliptic equation occurring in the buckling problem for the plate. Appl. Anal. 31(1–2), 57–61 (1988)

    Article  MathSciNet  Google Scholar 

  26. Talenti, G.: Inequalities in rearrangement invariant function spaces. Nonlinear Anal, Funct Spaces Appl 5, 177–230 (1994)

    MathSciNet  MATH  Google Scholar 

  27. Wang, Q., Xia, C.: Inequalities for eigenvalues of the buckling problem of arbitrary order on bounded domains of \( {\mathbf{M}} \times {\mathbf{R}} \). Math. Nachr. 292(4), 922–930 (2019)

    Article  MathSciNet  Google Scholar 

  28. Yolcu, S.Y.: An improvement to a Berezin-Li-Yau type inequality. Proc. Amer. Math. Soc. 138(11), 4059–4066 (2010)

    Article  MathSciNet  Google Scholar 

  29. Yolcu, S.Y., Yolcu, T.: Multidimensional lower bounds for the eigenvalues of Stokes and Dirichlet Laplacian operators. J. Math. Phys. 53(4), 043508 (2012)

    Article  MathSciNet  Google Scholar 

  30. Yolcu, S.Y., Yolcu, T.: Estimates for the sums of eigenvalues of the fractional Laplacian on a bounded domain. Commun. Contemp. Math. 15(3), 1250048 (2013)

    Article  MathSciNet  Google Scholar 

  31. Yolcu, S.Y., Yolcu, T.: Estimates on the eigenvalues of the clamped plate problem on domains in Euclidean spaces. J. Math. Phys. 54(4), 043515 (2013)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author appreciates the anonymous reviewer(s) for their valuable suggestions and constructive comments to improving the paper. This work was partially supported by the National Natural Science Foundation of China under the grants 11801443.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Pan Liu.

Additional information

Communicated by Adrian Constantin.

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

Liu, P. Lower bounds for sums of eigenvalues of the buckling problem. Monatsh Math 198, 591–618 (2022). https://doi.org/10.1007/s00605-022-01695-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-022-01695-0

Keywords

Mathematics Subject Classification

Navigation