Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-25T22:13:49.009Z Has data issue: false hasContentIssue false
  • English
  • Français

Estimates for sums and gaps of eigenvalues of Laplacians on measure spaces

Part of: Classical measure theory Spectral theory and eigenvalue problems Ordinary differential operators Numerical analysis: Ordinary differential equations Elliptic equations and systems

Published online by Cambridge University Press:  02 June 2020

Da-Wen Deng  and
Sze-Man Ngai
Da-Wen Deng
Affiliation:
Hunan Key Laboratory for Computational and Simulation in Science and Engineering, School of Mathematics and Computational Sciences, Xiangtan University, Hunan411105, People's Republic of China ([email protected]; [email protected])
Sze-Man Ngai
Affiliation:
Key Laboratory of High Performance Computing and Stochastic Information Processing (HPCSIP) (Ministry of Education of China), College of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan410081, People's Republic of China Department of Mathematical Sciences, Georgia Southern University, StatesboroGA30460-8093, USA ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

For Laplacians defined by measures on a bounded domain in ℝn, we prove analogues of the classical eigenvalue estimates for the standard Laplacian: lower bound of sums of eigenvalues by Li and Yau, and gaps of consecutive eigenvalues by Payne, Pólya and Weinberger. This work is motivated by the study of spectral gaps for Laplacians on fractals.

Keywords

Eigenvalue estimateFractalMeasureLaplacian

MSC classification

Primary: 35P15: Estimation of eigenvalues, upper and lower bounds 28A80: Fractals 35J05: Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation
Secondary: 34L16: Numerical approximation of eigenvalues and of other parts of the spectrum 65L15: Eigenvalue problems 65L60: Finite elements, Rayleigh-Ritz, Galerkin and collocation methods
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 2 , April 2021 , pp. 842 - 861
Check for updates
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Andrews, U., Bonik, G., Chen, J. P., Martin, R. W. and Teplyaev, A.. Wave equation on one-dimensional fractals with spectral decimation and the complex dynamics of polynomials. J. Fourier Anal. Appl. 23 (2017), 994–1027.CrossRefGoogle Scholar
2Atkinson, F. V.. Discrete and continuous boundary problems. Mathematics in Science and Engineering, vol. 8 (New York-London: Academic Press, 1964).Google Scholar
3Bird, E. J., Ngai, S.-M. and Teplyaev, A.. Fractal Laplacians on the unit interval. Ann. Sci. Math. Québec 27 (2003), 135–168.Google Scholar
4Chan, J. F.-C., Ngai, S.-M. and Teplyaev, A.. One-dimensional wave equations defined by fractal Laplacians. J. Anal. Math. 127 (2015), 219–246.CrossRefGoogle Scholar
5Chavel, I.. Eigenvalues in Riemannian geometry (Orlando, FL: Academic Press Inc., 1984).Google Scholar
6Chen, J. and Ngai, S.-M.. Eigenvalues and eigenfunctions of one-dimensional fractal Laplacians defined by iterated function systems with overlaps. J. Math. Anal. Appl. 364 (2010), 222–241.CrossRefGoogle Scholar
7Davies, E. B.. Spectral theory and differential operators. Cambridge Studies in Advanced Mathematics, vol. 42 (Cambridge: Cambridge University Press, 1995).CrossRefGoogle Scholar
8Deng, D.-W. and Ngai, S.-M.. Eigenvalue estimates for Laplacians on measure spaces. J. Funct. Anal. 268 (2015), 2231–2260.CrossRefGoogle Scholar
9Drenning, S. and Strichartz, R. S.. Spectral decimation on Hambly's homogeneous hierarchical gaskets. Illinois J. Math. 53 (2009), 915–937.CrossRefGoogle Scholar
10Feller, W.. On second order differential operators. Ann. Math. (2) 61 (1955), 90–105.CrossRefGoogle Scholar
11Feller, W.. Generalized second order differential operators and their lateral conditions. Illinois J. Math. 1 (1957), 459–504.CrossRefGoogle Scholar
12Freiberg, U.. Analytical properties of measure geometric Krein-Feller-operators on the real line. Math. Nachr. 260 (2003), 34–47.CrossRefGoogle Scholar
13Freiberg, U.. Dirichlet forms on fractal subsets of the real line. Real Anal. Exch. 30 (2004/05), 589–603.CrossRefGoogle Scholar
14Freiberg, U.. Spectral asymptotics of generalized measure geometric Laplacians on Cantor like sets. Forum Math. 17 (2005), 87–104.CrossRefGoogle Scholar
15Freiberg, U. and Löbus, J.-U.. Zeros of eigenfunctions of a class of generalized second order differential operators on the Cantor set. Math. Nachr. 265 (2004), 3–14.CrossRefGoogle Scholar
16Freiberg, U. and Zähle, M.. Harmonic calculus on fractals—a measure geometric approach, I. Potential Anal. 16 (2002), 265–277.CrossRefGoogle Scholar
17Fujita, T.. A fractional dimension, self-similarity and a generalized diffusion operator. In Probabilistic methods in mathematical physics (Katata/Kyoto, 1985), pp. 83–90 (Boston, MA: Academic Press, 1987).Google Scholar
18Gu, Q., Hu, J. and Ngai, S.-M.. Two-sided sub-Gaussian estimates of heat kernels on intervals for self-similar measures with overlaps. Commun. Pure Appl. Anal. 19 (2020), 641–676.CrossRefGoogle Scholar
19Hare, K.-E. and Zhou, D.. Gaps in the ratios of the spectra of Laplacians on fractals. Fractals 17 (2009), 523–535.CrossRefGoogle Scholar
20Hu, J., Lau, K.-S. and Ngai, S.-M.. Laplace operators related to self-similar measures on ℝd. J. Funct. Anal. 239 (2006), 542–565.CrossRefGoogle Scholar
21Kac, I. S. and Kreĭn, M. G.. Criteria for the discreteness of the spectrum of a singular string. Izv. Vyss. Ucebn. Zaved. Mat. 1958 (1958), 136–153.Google Scholar
22Kac, I. S. and Kreĭn, M. G.. On the spectral functions of the string. Am. Math. Soc. Transl. (2) 103 (1974), 19–102.Google Scholar
23Kröger, P.. Estimates for sums of eigenvalues of the Laplacian. J. Funct. Anal. 126 (1994), 217–227.CrossRefGoogle Scholar
24Li, P. and Yau, S.-T.. On the Schrödinger equation and the eigenvalue problem. Commun. Math. Phys. 88 (1983), 309–318.CrossRefGoogle Scholar
25McKean, H. P. and Ray, D. B.. Spectral distribution of a differential operator. Duke Math. J. 29 (1962), 281–292.CrossRefGoogle Scholar
26Naimark, K. and Solomyak, M.. The eigenvalue behaviour for the boundary value problems related to self-similar measures on ℝd. Math. Res. Lett. 2 (1995), 279–298.CrossRefGoogle Scholar
27Ngai, S.-M.. Spectral asymptotics of Laplacians associated with one-dimensional iterated function systems with overlaps. Can. J. Math. 63 (2011), 648–688.CrossRefGoogle Scholar
28Ngai, S.-M. and Tang, W.. Eigenvalue asymptotics and Bohr's formula for fractal Schrödinger operators. Pacific J. Math. 300 (2019), 83–119.CrossRefGoogle Scholar
29Ngai, S.-M., Tang, W. and Xie, Y.. Spectral asymptotics of one-dimensional fractal Laplacians in the absence of second-order identities. Discrete Cont. Dyn. Syst. 38 (2018), 1849–1887.CrossRefGoogle Scholar
30Ngai, S.-M., Tang, W. and Xie, Y.. Wave propagation speed on fractals. J. Fourier Anal. Appl. 26 (2020), 31.CrossRefGoogle Scholar
31Payne, L. E., Pólya, G. and Weinberger, H. F.. On the ratio of consecutive eigenvalues. Stud. Appl. Math. 35 (1956), 289–298.Google Scholar
32Pinasco, J. P. and Scarola, C.. Eigenvalue bounds and spectral asymptotics for fractal Laplacians. J. Fractal Geom. 6 (2019), 109–126.CrossRefGoogle Scholar
33Schoen, R. and Yau, S.-T.. Lectures on differential geometry. Conference Proceedings and Lecture Notes in Geometry and Topology, vol. 1 (Cambridge: International Press, 1994).Google Scholar
34Solomyak, M. and Verbitsky, E.. On a spectral problem related to self-similar measures. Bull. London Math. Soc. 27 (1995), 242–248.CrossRefGoogle Scholar
35Strichartz, R. S.. Estimates for sums of eigenvalues for domains in homogeneous spaces. J. Funct. Anal. 137 (1996), 152–190.CrossRefGoogle Scholar
36Strichartz, R. S.. Laplacians on fractals with spectral gaps have nicer Fourier series. Math. Res. Lett. 12 (2005), 269–274.CrossRefGoogle Scholar
37Zähle, M.. Harmonic calculus on fractals—a measure geometric approach, II. Trans. Am. Math. Soc. 357 (2005), 3407–3423.CrossRefGoogle Scholar
38Zhou, D.. Spectral analysis of Laplacians on the Vicsek set. Pacific J. Math. 241 (2009), 369–398.CrossRefGoogle Scholar