Repulsive knot energies and pseudodifferential calculus [Elektronische Ressource] : regorous analysis and regularity theory for O'Hara's knot energy family E_1hn(_1hnα_1hn), α _e63 (2,3) [E (alpha), alpha epsilon (2,3)] / vorgelegt von Philipp Reiter

RepulsiveKnotEnergiesandPseudodifferentialCalculusRigorousAnalysisandRegularityTheory(α)forO’Hara’sKnotEnergyFamily E ,α∈ [2,3)VonderFakultätfürMathematik,InformatikundNaturwissenschaftenderRWTHAachenUniversityzurErlangungdesakademischenGradeseinesDoktorsderNaturwissenschaftengenehmigteDissertationvorgelegtvonDipl. Math. PhilippReiterausHannoverBerichter UniversitätsprofessorDr. HeikovonderMoselUnivDr. JosefBemelmansTagdermündlichenPrüfung 13. März2009DieseDissertationistaufdenInternetseitenderHochschulbibliothekonlineverfügbar.Abstract(α)In this thesis, we consider J. O’H’s knot functionals E , α ∈ [2,3), proving∞FréchetdifferentiabilityandC regularityofcriticalpoints.Usingsomeideasof Z. X.H andfillingmajorgapsinhisinvestigationoftheMöbius(2)Energy E ,wefurnisharigorousproofofanevenmoregeneralstatement.(α) 2We start with proving continuity of E on injective and regular H curves, moreover(α)we establish Fréchet differentiability of E . Among other things, the proof draws2on the fact that reparametrization of a sequence of curves to arc length preserves Hconvergence. Additionally,wederiveseveralformulaeofthefirstvariation.α−2(α) (α)˜In the second part, we consider the rescaled functional E = length E establish ∞ α 2,3ing a bootstrap argument, which gives C regularity for critical points in H ∩ Hbeinginjectiveandparametrizedbyarc length.