{"id":{"repo_id":"rice","oai_identifier":"oai:repository.rice.edu:1911/19224"},"canonical_url":"https://search.dev.ndltd.org/etd/rice/oai:repository.rice.edu:1911/19224","repository":{"repo_id":"rice","name":"Rice University","base_url":"https://repository.rice.edu/server/oai/request"},"display":{"title":"Optimal design problems for quasidisks and partially clamped drums: Existence, symmetrization, and numerical methods","abstract":"It is shown that the class of quasidisks in the complex plane, with fixed quasicircle constant and area, is compact in both the Hausdorff metric and in the sense of Caratheodory convergence. Compactness for chord-arc domains with fixed chord-arc constant and area is shown as a result of the quasidisk compactness. Compactness is used to show that each eigenvalue of the Laplacian, subject to Dirichlet boundary conditions, attains its extrema over each of these classes. The design problem of extremizing the fundamental tone of a drum fastened only on a fraction of the boundary is considered. The special case of minimizing the fundamental frequency of a circular drum is solved using symmetrization. The gradient of the tone of the drum with respect to the design is considered and approximated appropriately. This approximate gradient is then used to compute examples numerically.","abstract_html":"It is shown that the class of quasidisks in the complex plane, with fixed quasicircle constant and area, is compact in both the Hausdorff metric and in the sense of Caratheodory convergence. Compactness for chord-arc domains with fixed chord-arc constant and area is shown as a result of the quasidisk compactness. Compactness is used to show that each eigenvalue of the Laplacian, subject to Dirichlet boundary conditions, attains its extrema over each of these classes. The design problem of extremizing the fundamental tone of a drum fastened only on a fraction of the boundary is considered. The special case of minimizing the fundamental frequency of a circular drum is solved using symmetrization. The gradient of the tone of the drum with respect to the design is considered and approximated appropriately. This approximate gradient is then used to compute examples numerically.","abstract_has_math":false,"creators":["Uhlig, Paul Xavier"],"institution":"Rice University","degree_name":"Doctor of Philosophy","degree_level":"Doctoral","degree_discipline":"Engineering","degree_department":null,"school":null,"contributors":[],"advisors":["Cox, Steven J."],"committee_chairs":[],"committee_members":[],"year":1997,"date_issued":"1997","date_published":"1997","updated_at":"2026-07-24T04:10:37Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder."],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/1911/19224","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Cox, Steven J."]},{"key":"dc:creator","label":"Author","values":["Uhlig, Paul Xavier"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2009-06-04T08:36:36Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2009-06-04T08:36:36Z"]},{"key":"dc:date.issued","label":"Date","values":["1997"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Engineering"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Rice University"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/1911/19224"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["It is shown that the class of quasidisks in the complex plane, with fixed quasicircle constant and area, is compact in both the Hausdorff metric and in the sense of Caratheodory convergence. Compactness for chord-arc domains with fixed chord-arc constant and area is shown as a result of the quasidisk compactness. Compactness is used to show that each eigenvalue of the Laplacian, subject to Dirichlet boundary conditions, attains its extrema over each of these classes. The design problem of extremizing the fundamental tone of a drum fastened only on a fraction of the boundary is considered. The special case of minimizing the fundamental frequency of a circular drum is solved using symmetrization. The gradient of the tone of the drum with respect to the design is considered and approximated appropriately. This approximate gradient is then used to compute examples numerically."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Optimal design problems for quasidisks and partially clamped drums: Existence, symmetrization, and numerical methods"]}]}],"canonical_facts":{"dc:contributor.advisor":["Cox, Steven J."],"dc:creator":["Uhlig, Paul Xavier"],"dc:date.accessioned":["2009-06-04T08:36:36Z"],"dc:date.available":["2009-06-04T08:36:36Z"],"dc:date.issued":["1997"],"dc:description.abstract":["It is shown that the class of quasidisks in the complex plane, with fixed quasicircle constant and area, is compact in both the Hausdorff metric and in the sense of Caratheodory convergence. Compactness for chord-arc domains with fixed chord-arc constant and area is shown as a result of the quasidisk compactness. Compactness is used to show that each eigenvalue of the Laplacian, subject to Dirichlet boundary conditions, attains its extrema over each of these classes. The design problem of extremizing the fundamental tone of a drum fastened only on a fraction of the boundary is considered. The special case of minimizing the fundamental frequency of a circular drum is solved using symmetrization. The gradient of the tone of the drum with respect to the design is considered and approximated appropriately. This approximate gradient is then used to compute examples numerically."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["https://hdl.handle.net/1911/19224"],"dc:language.iso":["eng"],"dc:rights":["Copyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder."],"dc:subject":["Mathematics"],"dc:title":["Optimal design problems for quasidisks and partially clamped drums: Existence, symmetrization, and numerical methods"],"dc:type":["Thesis"],"thesis:degree_discipline":["Engineering"],"thesis:degree_level":["Doctoral"],"thesis:degree_name":["Doctor of Philosophy"],"thesis:institution_name":["Rice University"]},"updated_at":"2026-07-24T04:10:37Z"}