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

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

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])
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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
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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