{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/86811"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/86811","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Continuity Properties and Variational Problems Involving the Determinant of the Hessian","abstract":"This thesis investigates continuity properties of the distributional determinant of the Hessian of a scalar function u in various function spaces. It is well known that when D is a bounded, open subset of n-dimensional Euclidean space Rn, the distributional determinant of the Hessian is weakly continuous in the Sobolev space of functions whose second derivatives are Lp functions on D, denoted by W(2,p)(D), for p greater than (nn/n+2) when n is greater than or equal to 3, and for p greater than or equal to 1 when n = 2. I show that it not strongly continuous in the norm topology in W(2,p)(D), when D is a subset of Rn for n greater than or equal to 3 and p is less than (nn/n+2) and similarly that it fails to be strongly continuous in W(1,p)(D), when D is a subset of R2 and p is less than 2. As my main result, I prove that when D is a subset of R3 then the map from u to Det(Hessian of u) is a continuous function from the intersection of BV2 and W(1, infty)(D), with a suitable topology, into the space of distributions. Here BV2 is the space of functions whose first derivatives are functions of bounded variation. This function space is naturally chosen to investigate a variational problem involving integral of the absolute value of det(Hessian of u).","abstract_html":"This thesis investigates continuity properties of the distributional determinant of the Hessian of a scalar function u in various function spaces. It is well known that when D is a bounded, open subset of n-dimensional Euclidean space Rn, the distributional determinant of the Hessian is weakly continuous in the Sobolev space of functions whose second derivatives are Lp functions on D, denoted by W(2,p)(D), for p greater than (nn/n+2) when n is greater than or equal to 3, and for p greater than or equal to 1 when n = 2. I show that it not strongly continuous in the norm topology in W(2,p)(D), when D is a subset of Rn for n greater than or equal to 3 and p is less than (nn/n+2) and similarly that it fails to be strongly continuous in W(1,p)(D), when D is a subset of R2 and p is less than 2. As my main result, I prove that when D is a subset of R3 then the map from u to Det(Hessian of u) is a continuous function from the intersection of BV2 and W(1, infty)(D), with a suitable topology, into the space of distributions. Here BV2 is the space of functions whose first derivatives are functions of bounded variation. This function space is naturally chosen to investigate a variational problem involving integral of the absolute value of det(Hessian of u).","abstract_has_math":false,"creators":["Jung, Nara"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Jerrard, Robert L."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-09-28T15:19:39Z","date_published":"2015-09-28T15:19:39Z","updated_at":"2026-07-22T22:26:28Z","subjects":["Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(MiAaPQ)AAI3086092"],"render_values":[{"text":"(MiAaPQ)AAI3086092","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/86811","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Jerrard, Robert L."]},{"key":"dc:creator","label":"Author","values":["Jung, Nara"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2015-09-28T15:19:39Z","10000-01-01","2003"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"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","label":"Dc Language","values":["eng"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/86811","(MiAaPQ)AAI3086092"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["This thesis investigates continuity properties of the distributional determinant of the Hessian of a scalar function u in various function spaces. It is well known that when D is a bounded, open subset of n-dimensional Euclidean space Rn, the distributional determinant of the Hessian is weakly continuous in the Sobolev space of functions whose second derivatives are Lp functions on D, denoted by W(2,p)(D), for p greater than (nn/n+2) when n is greater than or equal to 3, and for p greater than or equal to 1 when n = 2. I show that it not strongly continuous in the norm topology in W(2,p)(D), when D is a subset of Rn for n greater than or equal to 3 and p is less than (nn/n+2) and similarly that it fails to be strongly continuous in W(1,p)(D), when D is a subset of R2 and p is less than 2. As my main result, I prove that when D is a subset of R3 then the map from u to Det(Hessian of u) is a continuous function from the intersection of BV2 and W(1, infty)(D), with a suitable topology, into the space of distributions. Here BV2 is the space of functions whose first derivatives are functions of bounded variation. This function space is naturally chosen to investigate a variational problem involving integral of the absolute value of det(Hessian of u).","Made available in DSpace on 2015-09-28T15:19:39Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 3086092.pdf: 2172205 bytes, checksum: 9b1beb05a360e4796c9b8ad5eac88683 (MD5) Previous issue date: 2003","Embargo set by: Seth Robbins for item 88092 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","56 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2003."]},{"key":"dc:title","label":"Title","values":["Continuity Properties and Variational Problems Involving the Determinant of the Hessian"]}]}],"canonical_facts":{"dc:contributor":["Jerrard, Robert L."],"dc:creator":["Jung, Nara"],"dc:date":["2015-09-28T15:19:39Z","10000-01-01","2003"],"dc:description":["This thesis investigates continuity properties of the distributional determinant of the Hessian of a scalar function u in various function spaces. It is well known that when D is a bounded, open subset of n-dimensional Euclidean space Rn, the distributional determinant of the Hessian is weakly continuous in the Sobolev space of functions whose second derivatives are Lp functions on D, denoted by W(2,p)(D), for p greater than (nn/n+2) when n is greater than or equal to 3, and for p greater than or equal to 1 when n = 2. I show that it not strongly continuous in the norm topology in W(2,p)(D), when D is a subset of Rn for n greater than or equal to 3 and p is less than (nn/n+2) and similarly that it fails to be strongly continuous in W(1,p)(D), when D is a subset of R2 and p is less than 2. As my main result, I prove that when D is a subset of R3 then the map from u to Det(Hessian of u) is a continuous function from the intersection of BV2 and W(1, infty)(D), with a suitable topology, into the space of distributions. Here BV2 is the space of functions whose first derivatives are functions of bounded variation. 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