Skip to main content

Advertisement

SpringerLink
Log in

Payne–Polya–Weinberger, Hile–Protter and Yang’s Inequalities for Dirichlet Laplace Eigenvalues on Integer Lattices

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

In this paper, we prove some analogues of Payne–Polya–Weinberger, Hile–Protter and Yang’s inequalities for Dirichlet (discrete) Laplace eigenvalues on any subset in the integer lattice \({{\mathbb {Z}}}^n\). This partially answers a question posed by Chung and Oden (Pac J Math 192(2):257–273, 2000).

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.

Similar content being viewed by others

Data Availability

There is no data in the article

References

  1. Ashbaugh, M.S., Benguria, R.D.: Bounds for ratios of the first, second, and third membrane eigenvalues. In Nonlinear problems in applied mathematics, pp. 30–42. SIAM, Philadelphia, PA (1996)

  2. Ashbaugh, M.S.: Isoperimetric and universal inequalities for eigenvalues. In Spectral theory and geometry (Edinburgh,: London Math, p. 273. Soc, Lecture Note Ser. (1998)

  3. Ashbaugh, M.S.: The universal eigenvalue bounds of Payne–Pólya–Weinberger, Hile–Protter, and H. C. Yang. Proc. Indian Acad. Sci. Math. Sci. 112(1), 3–30 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bauer, F., Hua, B., Jost, J.: The dual Cheeger constant and spectra of infinite graphs. Adv. Math. 251, 147–194 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chavel, I.: Eigenvalues in Riemannian Geometry, volume 115 of Pure and Applied Mathematics. Academic Press Inc., Orlando (1984)

    Google Scholar 

  6. Chen, D., Cheng, Q.-M.: Extrinsic estimates for eigenvalues of the Laplace operator. J. Math. Soc. Jpn. 60(2), 325–339 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, D., Zheng, T., Lu, M.: Eigenvalue estimates on domains in complete noncompact Riemannian manifolds. Pac. J. Math. 255(1), 41–54 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, D., Zheng, T., Yang, H.C.: Estimates of the gaps between consecutive eigenvalues of Laplacian. Pac. J. Math. 282(2), 293–311 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cheng, Q.-M., Yang, H.C.: Estimates on eigenvalues of Laplacian. Math. Ann. 331(2), 445–460 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. 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(2), 545–561 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cheng, Q.-M., Yang, H.C.: Bounds on eigenvalues of Dirichlet Laplacian. Math. Ann. 337(1), 159–175 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cheng, Q.-M., Yang, H.C.: Estimates for eigenvalues on Riemannian manifolds. J. Differ. Equ. 247(8), 2270–2281 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chung, F.R.K., Oden, K.: Weighted graph Laplacians and isoperimetric inequalities. Pac. J. Math. 192(2), 257–273 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chung, F.R.K., Yau, S.-T.: A Harnack inequality for Dirichlet eigenvalues. J. Graph Theory 34(4), 247–257 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Coulhon, T., Grigor’yan, A.: Random walks on graphs with regular volume growth. Geom. Funct. Anal. 8(4), 656–701 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. I. Interscience Publishers Inc, New York (1953)

    MATH  Google Scholar 

  17. Dodziuk, J.: Difference equations, isoperimetric inequality and transience of certain random walks. Trans. Am. Math. Soc. 284(2), 787–794 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  18. El Soufi, A., Harrell, E.M., II., Saïd, I.: Universal inequalities for the eigenvalues of Laplace and Schrödinger operators on submanifolds. Trans. Am. Math. Soc. 361(5), 2337–2350 (2009)

    Article  MATH  Google Scholar 

  19. Friedman, J.: Some geometric aspects of graphs and their eigenfunctions. Duke Math. J. 69(3), 487–525 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  20. Grigor’yan, A.: Analysis on graphs. (2009). https://www.math.uni-bielefeld.de/~grigor/aglect.pdf

  21. Harrell, E.M.: II. Some geometric bounds on eigenvalue gaps. Commun. Partial Differ. Equ. 18(1–2), 179–198 (1993)

    Article  MATH  Google Scholar 

  22. Harrell, E.M., Commutators, I.I.: Eigenvalue gaps, and mean curvature in the theory of Schrödinger operators. Commun. Partial Differ. Equ. 32(1–3), 401–413 (2007)

    Article  MATH  Google Scholar 

  23. Harrell, E.M., II., Michel, P.L.: Commutator bounds for eigenvalues, with applications to spectral geometry. Commun. Partial Differ. Equ. 19(11–12), 2037–2055 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  24. Harrell, E.M., II., Stubbe, J.: On trace identities and universal eigenvalue estimates for some partial differential operators. Trans. Am. Math. Soc. 349(5), 1797–1809 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hile, G.N., Protter, M.H.: Inequalities for eigenvalues of the Laplacian. Indiana Univ. Math. J. 29(4), 523–538 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  26. Leung, P.F.: On the consecutive eigenvalues of the Laplacian of a compact minimal submanifold in a sphere. J. Austral. Math. Soc. Ser. A 50(3), 409–416 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  27. Li, P.: Eigenvalue estimates on homogeneous manifolds. Comment. Math. Helv. 55(3), 347–363 (1980)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  30. Polya, G.: On the eigenvalues of vibrating membranes. Proc. Lond. Math. Soc. 11(3), 419–433 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  31. Schoen, R., Yau, S.-T.: Lectures on differential geometry. Conference Proceedings and Lecture Notes in Geometry and Topology, I. International Press, Cambridge, MA, (1994). Lecture notes prepared by W.Y. Ding, K.C. Chang [G.Q. Zhang], J.Q. Zhong and Y.C. Xu, Translated from the Chinese by Ding and S.Y. Cheng, Preface translated from the Chinese by K. Tso

  32. Sun, H., Cheng, Q.-M., Yang, H.C.: Lower order eigenvalues of Dirichlet Laplacian. Manuscripta Math. 125(2), 139–156 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  34. Weyl, H.: Das asymptotische verteilungsgesetz der eigenwerte linearer partieller differentialgleichungen (mit einer anwendung auf die theorie der hohlraumstrahlung). Math. Ann. 71(4), 441–479 (1912)

    Article  MathSciNet  MATH  Google Scholar 

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

  36. Yang, P.C., Yau, S.-T.: Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7(1), 55–63 (1980)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

B. Hua is supported by NSFC, Grant Nos. 11831004 and 11401106. Y. Lin is supported by NSFC, Grant No. 12071245. Y. Su is supported by NSF of Fujian Province through Grant Nos. 2021J01615, 2017J01556, 2016J01013.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, LMNS, Fudan University, Shanghai, 200433, China

    Bobo Hua

  2. Yau Mathematical Sciences Center and Department of Mathematics, Tsinghua University, Beijing, 100084, China

    Yong Lin

  3. School of Mathematics and Statistics, Fuzhou University, Fuzhou, 350116, China

    Yanhui Su

Authors
  1. Bobo Hua
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yong Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yanhui Su
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bobo Hua.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hua, B., Lin, Y. & Su, Y. Payne–Polya–Weinberger, Hile–Protter and Yang’s Inequalities for Dirichlet Laplace Eigenvalues on Integer Lattices. J Geom Anal 33, 217 (2023). https://doi.org/10.1007/s12220-023-01284-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12220-023-01284-z

Keywords

Mathematics Subject Classification

Search

Navigation