Eigenvalues and Eigenfunctions of One-Dimensional Fractal Laplacians Defined by Iterated Function Systems with Overlaps
Document Type
Article
Publication Date
4-1-2010
Publication Title
Journal of Mathematical Analysis and Applications
DOI
10.1016/j.jmaa.2009.10.009
ISSN
0022-247X
Abstract
Under the assumption that a self-similar measure defined by a one-dimensional iterated function system with overlaps satisfies a family of second-order self-similar identities introduced by Strichartz et al., we obtain a method to discretize the equation defining the eigenvalues and eigenfunctions of the corresponding fractal Laplacian. This allows us to obtain numerical solutions by using the finite element method. We also prove that the numerical eigenvalues and eigenfunctions converge to the true ones, and obtain estimates for the rates of convergence. We apply this scheme to the fractal Laplacians defined by the well-known infinite Bernoulli convolution associated with the golden ratio and the 3-fold convolution of the Cantor measure. The iterated function systems defining these measures do not satisfy the open set condition or the post-critically finite condition; we use second-order self-similar identities to analyze the measures.
Recommended Citation
Chen, Jie, Sze-Man Ngai.
2010.
"Eigenvalues and Eigenfunctions of One-Dimensional Fractal Laplacians Defined by Iterated Function Systems with Overlaps."
Journal of Mathematical Analysis and Applications, 364 (1): 222-241.
doi: 10.1016/j.jmaa.2009.10.009
https://digitalcommons.georgiasouthern.edu/math-sci-facpubs/123
