Eigenvalues and Eigenfunctions of One-Dimensional Fractal Laplacians

We study the eigenvalues and eigenfunctions of one-dimensional weighted fractal Laplacians. These Laplacians are defined by self-similar measures with overlaps. We first prove the existence of eigenvalues and eigenfunctions. We then set up a framework for one-dimensional measures to discretize the e...

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Bibliographic Details
Published in:Journal of nonlinear mathematical physics Vol. 30; no. 3; pp. 996 - 1010
Main Authors: Tang, Wei, Guo, Jia
Format: Journal Article
Language:English
Published: Dordrecht Springer Netherlands 01.09.2023
Springer Nature B.V
Subjects:
Convergence
Eigenvalues
Eigenvectors
Finite element analysis
Finite element method
Fractals
Mathematical Physics
Mathematics
Mathematics and Statistics
Research Article
Self-similarity
Secondary: 65L60
Fractal
Primary: 28A80
Self-similar measure with overlaps
Eigenfunction
34L16
Eigenvalue
Laplacian
ISSN:1776-0852, 1402-9251, 1776-0852
Online Access:Get full text
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Summary:We study the eigenvalues and eigenfunctions of one-dimensional weighted fractal Laplacians. These Laplacians are defined by self-similar measures with overlaps. We first prove the existence of eigenvalues and eigenfunctions. We then set up a framework for one-dimensional measures to discretize the equation defining the eigenvalues and eigenfunctions, and obtain numerical approximations of the eigenvalue and eigenfunction by using the finite element method. Finally, we show that the numerical eigenvalues and eigenfunctions converge to the actual ones and obtain the rate of convergence.
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ISSN:1776-0852
1402-9251
1776-0852
DOI:10.1007/s44198-023-00113-9