{"id":{"repo_id":"aachen","oai_identifier":"oai:publications.rwth-aachen.de:51124"},"canonical_url":"https://search.dev.ndltd.org/etd/aachen/oai:publications.rwth-aachen.de:51124","repository":{"repo_id":"aachen","name":"RWTH Aachen University","base_url":"https://publications.rwth-aachen.de/oai2d"},"display":{"title":"Repulsive knot energies and pseudodifferential calculus : rigorous analysis and regularity theory for O'Hara's knot energy family E (alpha), alpha in [2,3)","abstract":"In this thesis, we consider J. O'Hara's knot functionals E^(alpha), $alphain[2,3)$, proving Fréchet differentiability and $C^infty$ regularity of critical points. Using some ideas of Z.-X. He and filling major gaps in his investigation of the Möbius Energy E^(2), we furnish a rigorous proof of an even more general statement. We start with proving continuity of E^(alpha) on injective and regular H^2 curves, moreover we establish Fréchet differentiability of E^(alpha). Among other things, the proof draws on the fact that reparametrization of a sequence of curves to arc-length preserves H^2 convergence. Additionally, we derive several formulae of the first variation. In the second part, we consider the rescaled functional $ilde E = ext{length}^{alpha-2}E$ establishing a bootstrap argument, which gives $C^infty$ regularity for critical points in $H^alphacap H^{2,3}$ being injective and parametrized by arc-length. The major technique is to introduce fractional Sobolev spaces on a periodic interval and to study bilinear Fourier multipliers.","abstract_html":"In this thesis, we consider J. O&#x27;Hara&#x27;s knot functionals E^(alpha), $alphain[2,3)$, proving Fréchet differentiability and <span class=\"etd-inline-math\">C<sup>i</sup>nfty</span> regularity of critical points. Using some ideas of Z.-X. He and filling major gaps in his investigation of the Möbius Energy E^(2), we furnish a rigorous proof of an even more general statement. We start with proving continuity of E^(alpha) on injective and regular H^2 curves, moreover we establish Fréchet differentiability of E^(alpha). Among other things, the proof draws on the fact that reparametrization of a sequence of curves to arc-length preserves H^2 convergence. Additionally, we derive several formulae of the first variation. In the second part, we consider the rescaled functional <span class=\"etd-inline-math\">ilde E = ext{length}<sup>alpha-2</sup>E</span> establishing a bootstrap argument, which gives <span class=\"etd-inline-math\">C<sup>i</sup>nfty</span> regularity for critical points in <span class=\"etd-inline-math\">H<sup>a</sup>lphacap H<sup>2,3</sup></span> being injective and parametrized by arc-length. The major technique is to introduce fractional Sobolev spaces on a periodic interval and to study bilinear Fourier multipliers.","abstract_has_math":true,"creators":["Reiter, Philipp"],"institution":"Publikationsserver der RWTH Aachen University","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":["von der Mosel, Heiko"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2009,"date_issued":"2009","date_published":"2009","updated_at":"2026-07-30T19:40:33Z","subjects":["info:eu-repo/classification/ddc/510","Knoten <Mathematik>","Fourier-Reihe","Harmonische Analyse","Mathematik","Knotenenergie","bilinearer Fouriermultiplikator","Regularitätstheorie","Möbius-Energie","knot energy","bilinear Fourier multiplier","regularity theory","Möbius energy"],"languages":["eng"],"rights":["info:eu-repo/semantics/openAccess"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-113439%22"],"render_values":[{"text":"https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-113439%22","href":"https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-113439%22","code":true}]}]},"links":{"outbound_url":"https://publications.rwth-aachen.de/record/51124","outbound_label":"Repository record","outbound_source":"dc:identifier"},"source_record":{"url":"https://publications.rwth-aachen.de/oai2d?verb=GetRecord&metadataPrefix=oai_dc&identifier=oai%3Apublications.rwth-aachen.de%3A51124","prefix":"oai_dc"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["von der Mosel, Heiko"]},{"key":"dc:creator","label":"Author","values":["Reiter, Philipp"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:coverage","label":"Dc Coverage","values":["DE"]},{"key":"dc:date","label":"Dc Date","values":["2009"]},{"key":"dc:publisher","label":"Institution","values":["Publikationsserver der RWTH Aachen University"]},{"key":"dc:relation","label":"Dc Relation","values":["info:eu-repo/semantics/altIdentifier/urn/urn:nbn:de:hbz:82-opus-28484"]},{"key":"dc:type","label":"Dc Type","values":["info:eu-repo/semantics/doctoralThesis","info:eu-repo/semantics/publishedVersion"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["info:eu-repo/classification/ddc/510","Knoten <Mathematik>","Fourier-Reihe","Harmonische Analyse","Mathematik","Knotenenergie","bilinearer Fouriermultiplikator","Regularitätstheorie","Möbius-Energie","knot energy","bilinear Fourier multiplier","regularity theory","Möbius energy"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["info:eu-repo/semantics/openAccess"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://publications.rwth-aachen.de/record/51124","https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-113439%22"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In this thesis, we consider J. O'Hara's knot functionals E^(alpha), $alphain[2,3)$, proving Fréchet differentiability and $C^infty$ regularity of critical points. Using some ideas of Z.-X. He and filling major gaps in his investigation of the Möbius Energy E^(2), we furnish a rigorous proof of an even more general statement. We start with proving continuity of E^(alpha) on injective and regular H^2 curves, moreover we establish Fréchet differentiability of E^(alpha). Among other things, the proof draws on the fact that reparametrization of a sequence of curves to arc-length preserves H^2 convergence. Additionally, we derive several formulae of the first variation. In the second part, we consider the rescaled functional $ilde E = ext{length}^{alpha-2}E$ establishing a bootstrap argument, which gives $C^infty$ regularity for critical points in $H^alphacap H^{2,3}$ being injective and parametrized by arc-length. The major technique is to introduce fractional Sobolev spaces on a periodic interval and to study bilinear Fourier multipliers."]},{"key":"dc:source","label":"Dc Source","values":["Aachen : Publikationsserver der RWTH Aachen University 86 S. (2009). = Aachen, Techn. Hochsch., Diss., 2009"]},{"key":"dc:title","label":"Title","values":["Repulsive knot energies and pseudodifferential calculus : rigorous analysis and regularity theory for O'Hara's knot energy family E (alpha), alpha in [2,3)"]}]}],"canonical_facts":{"dc:contributor":["von der Mosel, Heiko"],"dc:coverage":["DE"],"dc:creator":["Reiter, Philipp"],"dc:date":["2009"],"dc:description":["In this thesis, we consider J. O'Hara's knot functionals E^(alpha), $alphain[2,3)$, proving Fréchet differentiability and $C^infty$ regularity of critical points. Using some ideas of Z.-X. He and filling major gaps in his investigation of the Möbius Energy E^(2), we furnish a rigorous proof of an even more general statement. We start with proving continuity of E^(alpha) on injective and regular H^2 curves, moreover we establish Fréchet differentiability of E^(alpha). Among other things, the proof draws on the fact that reparametrization of a sequence of curves to arc-length preserves H^2 convergence. Additionally, we derive several formulae of the first variation. In the second part, we consider the rescaled functional $ilde E = ext{length}^{alpha-2}E$ establishing a bootstrap argument, which gives $C^infty$ regularity for critical points in $H^alphacap H^{2,3}$ being injective and parametrized by arc-length. The major technique is to introduce fractional Sobolev spaces on a periodic interval and to study bilinear Fourier multipliers."],"dc:identifier":["https://publications.rwth-aachen.de/record/51124","https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-113439%22"],"dc:language":["eng"],"dc:publisher":["Publikationsserver der RWTH Aachen University"],"dc:relation":["info:eu-repo/semantics/altIdentifier/urn/urn:nbn:de:hbz:82-opus-28484"],"dc:rights":["info:eu-repo/semantics/openAccess"],"dc:source":["Aachen : Publikationsserver der RWTH Aachen University 86 S. (2009). = Aachen, Techn. Hochsch., Diss., 2009"],"dc:subject":["info:eu-repo/classification/ddc/510","Knoten <Mathematik>","Fourier-Reihe","Harmonische Analyse","Mathematik","Knotenenergie","bilinearer Fouriermultiplikator","Regularitätstheorie","Möbius-Energie","knot energy","bilinear Fourier multiplier","regularity theory","Möbius energy"],"dc:title":["Repulsive knot energies and pseudodifferential calculus : rigorous analysis and regularity theory for O'Hara's knot energy family E (alpha), alpha in [2,3)"],"dc:type":["info:eu-repo/semantics/doctoralThesis","info:eu-repo/semantics/publishedVersion"]},"updated_at":"2026-07-30T19:40:33Z"}