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.
","abstract_html":"<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>","abstract_has_math":false,"creators":["Schroeder, Nathan Philip"],"institution":null,"degree_name":"Mathematics, PhD","degree_level":"Open Access Dissertation","degree_discipline":"Institute of Mathematical Sciences","degree_department":null,"school":null,"contributors":["Marina Chugunova","Ali Nadim"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2024,"date_issued":"2024-01-01T08:00:00Z","date_published":"2024-01-01T08:00:00Z","updated_at":"2026-07-24T01:40:49Z","subjects":["Eigenvalue Optimization","Shape Optimization","Spectral Theory","Spherical Harmonics","Steklov","Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholarship.claremont.edu/cgu_etd/799","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Marina Chugunova","Ali Nadim"]},{"key":"dc:creator","label":"Author","values":["Schroeder, Nathan Philip"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2024-06-01T07:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Institute of Mathematical Sciences"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Open Access Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Mathematics, PhD"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Eigenvalue Optimization","Shape Optimization","Spectral Theory","Spherical Harmonics","Steklov","Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholarship.claremont.edu/cgu_etd/799"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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.
"]},{"key":"dc:title","label":"Title","values":["Steklov Eigenvalue Problems on Nearly Spherical and Annular Domains"]}]}],"canonical_facts":{"dc:contributor":["Marina Chugunova","Ali Nadim"],"dc:creator":["Schroeder, Nathan Philip"],"dc:date.available":["2024-06-01T07:00:00Z"],"dc:description.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.
"],"dc:identifier":["https://scholarship.claremont.edu/cgu_etd/799"],"dc:subject":["Eigenvalue Optimization","Shape Optimization","Spectral Theory","Spherical Harmonics","Steklov","Mathematics"],"dc:title":["Steklov Eigenvalue Problems on Nearly Spherical and Annular Domains"],"thesis:degree_discipline":["Institute of Mathematical Sciences"],"thesis:degree_level":["Open Access Dissertation"],"thesis:degree_name":["Mathematics, PhD"]},"updated_at":"2026-07-24T01:40:49Z"}