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).
Similar content being viewed by others
Data Availability
There is no data in the article
References
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)
Ashbaugh, M.S.: Isoperimetric and universal inequalities for eigenvalues. In Spectral theory and geometry (Edinburgh,: London Math, p. 273. Soc, Lecture Note Ser. (1998)
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)
Bauer, F., Hua, B., Jost, J.: The dual Cheeger constant and spectra of infinite graphs. Adv. Math. 251, 147–194 (2014)
Chavel, I.: Eigenvalues in Riemannian Geometry, volume 115 of Pure and Applied Mathematics. Academic Press Inc., Orlando (1984)
Chen, D., Cheng, Q.-M.: Extrinsic estimates for eigenvalues of the Laplace operator. J. Math. Soc. Jpn. 60(2), 325–339 (2008)
Chen, D., Zheng, T., Lu, M.: Eigenvalue estimates on domains in complete noncompact Riemannian manifolds. Pac. J. Math. 255(1), 41–54 (2012)
Chen, D., Zheng, T., Yang, H.C.: Estimates of the gaps between consecutive eigenvalues of Laplacian. Pac. J. Math. 282(2), 293–311 (2016)
Cheng, Q.-M., Yang, H.C.: Estimates on eigenvalues of Laplacian. Math. Ann. 331(2), 445–460 (2005)
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)
Cheng, Q.-M., Yang, H.C.: Bounds on eigenvalues of Dirichlet Laplacian. Math. Ann. 337(1), 159–175 (2007)
Cheng, Q.-M., Yang, H.C.: Estimates for eigenvalues on Riemannian manifolds. J. Differ. Equ. 247(8), 2270–2281 (2009)
Chung, F.R.K., Oden, K.: Weighted graph Laplacians and isoperimetric inequalities. Pac. J. Math. 192(2), 257–273 (2000)
Chung, F.R.K., Yau, S.-T.: A Harnack inequality for Dirichlet eigenvalues. J. Graph Theory 34(4), 247–257 (2000)
Coulhon, T., Grigor’yan, A.: Random walks on graphs with regular volume growth. Geom. Funct. Anal. 8(4), 656–701 (1998)
Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. I. Interscience Publishers Inc, New York (1953)
Dodziuk, J.: Difference equations, isoperimetric inequality and transience of certain random walks. Trans. Am. Math. Soc. 284(2), 787–794 (1984)
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)
Friedman, J.: Some geometric aspects of graphs and their eigenfunctions. Duke Math. J. 69(3), 487–525 (1993)
Grigor’yan, A.: Analysis on graphs. (2009). https://www.math.uni-bielefeld.de/~grigor/aglect.pdf
Harrell, E.M.: II. Some geometric bounds on eigenvalue gaps. Commun. Partial Differ. Equ. 18(1–2), 179–198 (1993)
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)
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)
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)
Hile, G.N., Protter, M.H.: Inequalities for eigenvalues of the Laplacian. Indiana Univ. Math. J. 29(4), 523–538 (1980)
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)
Li, P.: Eigenvalue estimates on homogeneous manifolds. Comment. Math. Helv. 55(3), 347–363 (1980)
Li, P., Yau, S.T.: On the Schrödinger equation and the eigenvalue problem. Commun. Math. Phys. 88(3), 309–318 (1983)
Payne, L.E., Polya, G., Weinberger, H.F.: On the ratio of consecutive eigenvalues. J. Math. Phys. 35, 289–298 (1956)
Polya, G.: On the eigenvalues of vibrating membranes. Proc. Lond. Math. Soc. 11(3), 419–433 (1961)
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
Sun, H., Cheng, Q.-M., Yang, H.C.: Lower order eigenvalues of Dirichlet Laplacian. Manuscripta Math. 125(2), 139–156 (2008)
Thompson, C.J.: On the ratio of consecutive eigenvalues in \(N\)-dimensions. Stud. Appl. Math. 48, 281–283 (1969)
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)
Yang, H.C.: An estimate of the difference between consecutive eigenvalues. Preprintn IC/91/60 of ICTP, Trieste (1991)
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)
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
Corresponding author
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.
About this article
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
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12220-023-01284-z