Repulsive knot energies and pseudodifferential calculus : rigorous analysis and regularity theory for O'Hara's knot energy family E (alpha), alpha in [2,3)

In this thesis, we consider J. O'Hara's knot functionals E^(alpha), $alphain[2,3)$, proving Fréchet differentiability and Cⁱnfty 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-2E establishing a bootstrap argument, which gives Cⁱnfty 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.

Grantor dc:publisher

Publikationsserver der RWTH Aachen University

Year dc:date

2009

Author dc:creator

• Reiter, Philipp

Contributors dc:contributor

• von der Mosel, Heiko

• 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

Statement dc:rights

• info:eu-repo/semantics/openAccess

Language dc:language

eng

Identifier

https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-113439%22

OAI record oai:identifier

https://publications.rwth-aachen.de/oai2d?verb=GetRecord&metadataPrefix=oai_dc&identifier=oai%3Apublications.rwth-aachen.de%3A51124