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Repulsive knot energies and pseudodifferential calculus for O’Hara’s knot energy family E (α) , α ∈ [2, 3)
Author(s) -
Reiter Philipp
Publication year - 2012
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201000090
Subject(s) - mathematics , knot (papermaking) , sobolev space , differentiable function , injective function , bilinear interpolation , smoothness , combinatorics , energy (signal processing) , mathematical analysis , calculus (dental) , statistics , chemical engineering , engineering , medicine , dentistry
We develop a precise analysis of J. O’Hara’s knot functionals E (α) , α ∈ [2, 3), that serve as self‐repulsive potentials on (knotted) closed curves. First we derive continuity of E (α) on injective and regular H 2 curves and then we establish Fréchet differentiability of E (α) and state several first variation formulae. Motivated by ideas of Z.‐X. He in his work on the specific functional E (2) , the so‐called Möbius Energy, we prove C ∞ ‐smoothness of critical points of the appropriately rescaled functionals \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\tilde{E}^{(\alpha )}= {\rm length}^{\alpha -2}E^{(\alpha )}$\end{document} by means of fractional Sobolev spaces on a periodic interval and bilinear Fourier multipliers.