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%0 Thesis
%A Reiter, Philipp
%T Repulsive knot energies and pseudodifferential calculus : rigorous analysis and regularity theory for O'Hara's knot energy family E (alpha), alpha in [2,3)
%C Aachen
%I Publikationsserver der RWTH Aachen University
%M RWTH-CONV-113439
%P 86 S.
%D 2009
%Z Zusammenfassung in engl. und dt. Sprache
%Z Aachen, Techn. Hochsch., Diss., 2009
%X In this thesis, we consider J. O'Hara's knot functionals E^(alpha), alphain[2,3), proving Fréchet differentiability and C<sup>i</sup>nfty 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 = extlength<sup>alpha−2</sup>E establishing a bootstrap argument, which gives C<sup>i</sup>nfty regularity for critical points in H<sup>a</sup>lphacap H<sup>2,3</sup> 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.
%K Knoten <Mathematik> (SWD)
%K Fourier-Reihe (SWD)
%K Harmonische Analyse (SWD)
%F PUB:(DE-HGF)11
%9 Dissertation / PhD Thesis
%U https://publications.rwth-aachen.de/record/51124