Outer Approximation of the Spectrum of a Fractal Laplacian
Date of Award
2007
Document Type
Thesis
Degree Name
Bachelors
Department
Natural Sciences
First Advisor
McDonald, Patrick
Keywords
Fractal, Laplacian, Partial Differential Equations (PDE)
Area of Concentration
Mathematics
Abstract
The Neumann spectrum of a Laplacian on a fractal K in the plane is approximated as a renormalized limit of the Neumann spectra of the standard Laplacian on a sequence of domains which approximate K from the outside. Eigenvalues and eigenfunctions are approximated numerically for lower portions of the spectrum. A speculative description of the spectrum is presented for the Sierpinski carpet (SC) where existence, but not uniqueness, of a self-similar Laplacian is known, and also for the octagasket, where even the existence of a self-similar Laplacian is not known. New results about the structure of the spectrum involving "miniaturization" of eigenfunctions are presented.
Recommended Citation
Goff, Stacey, "Outer Approximation of the Spectrum of a Fractal Laplacian" (2007). Theses & ETDs. 3787.
https://digitalcommons.ncf.edu/theses_etds/3787