Date of Award
2024
Degree Type
Open Access Dissertation
Degree Name
Mathematics, PhD
Program
Institute of Mathematical Sciences
Advisor/Supervisor/Committee Chair
Chiu-Yen Kao
Dissertation or Thesis Committee Member
Marina Chugunova
Dissertation or Thesis Committee Member
Ali Nadim
Dissertation or Thesis Committee Member
Ali Nadim
Terms of Use & License Information
Rights Information
© 2024 Nathan Philip Schroeder
Keywords
Eigenvalue Optimization, Shape Optimization, Spectral Theory, Spherical Harmonics, Steklov
Subject Categories
Mathematics
Abstract
We consider Steklov eigenvalues on nearly spherical and nearly annular domains in d dimensions where d is any given positive integer. By using the Green-Beltrami identity for spherical harmonic functions, the derivatives of Steklov eigenvalues with respect to the domain perturbation parameter can be determined by the eigenvalues of a matrix involving the integral of the product of three spherical harmonic functions. By using the addition theorem for spherical harmonic functions, we determine conditions when the trace of this matrix becomes zero. These conditions can then be used to determine when spherical and annular regions are critical points while we optimize Steklov eigenvalues subject to a volume constraint. In addition, we develop numerical approaches based on particular solutions and show that numerical results in two and three dimensions are in agreement with our analytic results.
ISBN
9798382748214
Recommended Citation
Schroeder, Nathan Philip. (2024). Steklov Eigenvalue Problems on Nearly Spherical and Annular Domains. CGU Theses & Dissertations, 799. https://scholarship.claremont.edu/cgu_etd/799.
