Steklov Eigenvalue Problems on Nearly Spherical and Annular Domains

<p>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.</p>

Name thesis:degree_name

Mathematics, PhD

Level thesis:degree_level

Open Access Dissertation

Discipline thesis:degree_discipline

Institute of Mathematical Sciences

Year dc:date.available

2024

Author dc:creator

• Schroeder, Nathan Philip

Contributors dc:contributor

• Marina Chugunova
• Ali Nadim

• Eigenvalue Optimization
• Shape Optimization
• Spectral Theory
• Spherical Harmonics
• Steklov
• Mathematics

Repository record dc:identifier

https://scholarship.claremont.edu/cgu_etd/799

OAI identifier oai:identifier

oai:scholarship.claremont.edu:cgu_etd-1821