Repulsive knot energies and pseudodifferential calculus for O’Hara’s knot energy family E(α), α ∈ [2, 3)
✍ Scribed by Philipp Reiter
- Book ID
- 102496052
- Publisher
- John Wiley and Sons
- Year
- 2012
- Tongue
- English
- Weight
- 363 KB
- Volume
- 285
- Category
- Article
- ISSN
- 0025-584X
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✦ Synopsis
Abstract
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.