{"id":{"repo_id":"temple","oai_identifier":"oai:scholarshare.temple.edu:20.500.12613/2452"},"canonical_url":"https://search.dev.ndltd.org/etd/temple/oai:scholarshare.temple.edu:20.500.12613/2452","repository":{"repo_id":"temple","name":"Temple University","base_url":"https://scholarshare.temple.edu/server/oai/request"},"display":{"title":"On Generalized Solutions to Some Problems in Electromagnetism and Geometric Optics","abstract":"The Maxwell equations of electromagnetism form the foundation of classical electromagnetism, and are of interest to mathematicians, physicists, and engineers alike. The first part of this thesis concerns boundary value problems for the anisotropic Maxwell equations in Lipschitz domains. In this case, the material parameters that arise in the Maxwell system are matrix valued functions. Using methods from functional analysis, global in time solutions to initial boundary value problems with general nonzero boundary data and nonzero current density are obtained, only assuming the material parameters are bounded and measurable. This problem is motivated by an electromagnetic inverse problem, similar to the classical Calder\\'on inverse problem in Electrical Impedance Tomography. The second part of this thesis deals with materials having negative refractive index. Materials which possess a negative refractive index were postulated by Veselago in 1968, and since 2001 physicists were able to construct these materials in the laboratory. The research on the behavior of these materials, called metamaterials, has been extremely active in recent years. We study here refraction problems in the setting of Negative Refractive Index Materials (NIMs). In particular, it is shown how to obtain weak solutions (defined similarly to Brenier solutions for the Monge-Amp\\`ere equation) to these problems, both in the near and the far field. The far field problem can be treated using Optimal Transport techniques; as such, a fully nonlinear PDE of Monge-Amp\\`ere type arises here.","abstract_html":"The Maxwell equations of electromagnetism form the foundation of classical electromagnetism, and are of interest to mathematicians, physicists, and engineers alike. The first part of this thesis concerns boundary value problems for the anisotropic Maxwell equations in Lipschitz domains. In this case, the material parameters that arise in the Maxwell system are matrix valued functions. Using methods from functional analysis, global in time solutions to initial boundary value problems with general nonzero boundary data and nonzero current density are obtained, only assuming the material parameters are bounded and measurable. This problem is motivated by an electromagnetic inverse problem, similar to the classical Calder\\'on inverse problem in Electrical Impedance Tomography. The second part of this thesis deals with materials having negative refractive index. Materials which possess a negative refractive index were postulated by Veselago in 1968, and since 2001 physicists were able to construct these materials in the laboratory. The research on the behavior of these materials, called metamaterials, has been extremely active in recent years. We study here refraction problems in the setting of Negative Refractive Index Materials (NIMs). In particular, it is shown how to obtain weak solutions (defined similarly to Brenier solutions for the Monge-Amp\\`ere equation) to these problems, both in the near and the far field. The far field problem can be treated using Optimal Transport techniques; as such, a fully nonlinear PDE of Monge-Amp\\`ere type arises here.","abstract_has_math":false,"creators":["Stachura, Eric Christopher"],"institution":"Temple University. Libraries","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Gutiérrez, Cristian E., 1950-"],"committee_chairs":[],"committee_members":["Berhanu, Shiferaw","Mendoza, Gerardo A.","Strain, Robert M."],"year":2016,"date_issued":"2016","date_published":"2016","updated_at":"2026-07-27T21:21:26Z","subjects":["Mathematics","Electromagnetics","Optics","Anisotropic Maxwell equations","Inverse problem","Maxwell semigroup","Monge-Ampere equation","Negative refraction","Weak solutions"],"languages":["eng"],"rights":["IN COPYRIGHT- This Rights Statement can be used for an Item that is in copyright. Using this statement implies that the organization making this Item available has determined that the Item is in copyright and either is the rights-holder, has obtained permission from the rights-holder(s) to make their Work(s) available, or makes the Item available under an exception or limitation to copyright (including Fair Use) that entitles it to make the Item available."],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["965642287"],"render_values":[{"text":"965642287","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/20.500.12613/2452","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Gutiérrez, Cristian E., 1950-"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Berhanu, Shiferaw","Mendoza, Gerardo A.","Strain, Robert M."]},{"key":"dc:creator","label":"Author","values":["Stachura, Eric Christopher"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2020-11-03T15:33:49Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2020-11-03T15:33:49Z"]},{"key":"dc:date.issued","label":"Date","values":["2016"]},{"key":"dc:publisher","label":"Institution","values":["Temple University. 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Using this statement implies that the organization making this Item available has determined that the Item is in copyright and either is the rights-holder, has obtained permission from the rights-holder(s) to make their Work(s) available, or makes the Item available under an exception or limitation to copyright (including Fair Use) that entitles it to make the Item available."]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["965642287"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/20.500.12613/2452"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The Maxwell equations of electromagnetism form the foundation of classical electromagnetism, and are of interest to mathematicians, physicists, and engineers alike. The first part of this thesis concerns boundary value problems for the anisotropic Maxwell equations in Lipschitz domains. In this case, the material parameters that arise in the Maxwell system are matrix valued functions. Using methods from functional analysis, global in time solutions to initial boundary value problems with general nonzero boundary data and nonzero current density are obtained, only assuming the material parameters are bounded and measurable. This problem is motivated by an electromagnetic inverse problem, similar to the classical Calder\\'on inverse problem in Electrical Impedance Tomography. The second part of this thesis deals with materials having negative refractive index. Materials which possess a negative refractive index were postulated by Veselago in 1968, and since 2001 physicists were able to construct these materials in the laboratory. The research on the behavior of these materials, called metamaterials, has been extremely active in recent years. We study here refraction problems in the setting of Negative Refractive Index Materials (NIMs). In particular, it is shown how to obtain weak solutions (defined similarly to Brenier solutions for the Monge-Amp\\`ere equation) to these problems, both in the near and the far field. The far field problem can be treated using Optimal Transport techniques; as such, a fully nonlinear PDE of Monge-Amp\\`ere type arises here."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph.D."]},{"key":"dc:title","label":"Title","values":["On Generalized Solutions to Some Problems in Electromagnetism and Geometric Optics"]}]}],"canonical_facts":{"dc:contributor.advisor":["Gutiérrez, Cristian E., 1950-"],"dc:contributor.committeemember":["Berhanu, Shiferaw","Mendoza, Gerardo A.","Strain, Robert M."],"dc:creator":["Stachura, Eric Christopher"],"dc:date.accessioned":["2020-11-03T15:33:49Z"],"dc:date.available":["2020-11-03T15:33:49Z"],"dc:date.issued":["2016"],"dc:description.abstract":["The Maxwell equations of electromagnetism form the foundation of classical electromagnetism, and are of interest to mathematicians, physicists, and engineers alike. The first part of this thesis concerns boundary value problems for the anisotropic Maxwell equations in Lipschitz domains. In this case, the material parameters that arise in the Maxwell system are matrix valued functions. Using methods from functional analysis, global in time solutions to initial boundary value problems with general nonzero boundary data and nonzero current density are obtained, only assuming the material parameters are bounded and measurable. This problem is motivated by an electromagnetic inverse problem, similar to the classical Calder\\'on inverse problem in Electrical Impedance Tomography. The second part of this thesis deals with materials having negative refractive index. Materials which possess a negative refractive index were postulated by Veselago in 1968, and since 2001 physicists were able to construct these materials in the laboratory. The research on the behavior of these materials, called metamaterials, has been extremely active in recent years. We study here refraction problems in the setting of Negative Refractive Index Materials (NIMs). In particular, it is shown how to obtain weak solutions (defined similarly to Brenier solutions for the Monge-Amp\\`ere equation) to these problems, both in the near and the far field. The far field problem can be treated using Optimal Transport techniques; as such, a fully nonlinear PDE of Monge-Amp\\`ere type arises here."],"dc:description.degree":["Ph.D."],"dc:identifier.other":["965642287"],"dc:identifier.uri":["http://hdl.handle.net/20.500.12613/2452"],"dc:language.iso":["eng"],"dc:publisher":["Temple University. Libraries"],"dc:rights":["IN COPYRIGHT- This Rights Statement can be used for an Item that is in copyright. Using this statement implies that the organization making this Item available has determined that the Item is in copyright and either is the rights-holder, has obtained permission from the rights-holder(s) to make their Work(s) available, or makes the Item available under an exception or limitation to copyright (including Fair Use) that entitles it to make the Item available."],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:subject":["Mathematics","Electromagnetics","Optics","Anisotropic Maxwell equations","Inverse problem","Maxwell semigroup","Monge-Ampere equation","Negative refraction","Weak solutions"],"dc:title":["On Generalized Solutions to Some Problems in Electromagnetism and Geometric Optics"],"dc:type":["Text"]},"updated_at":"2026-07-27T21:21:26Z"}