L²-flow of elastic curves with knot points and clamped ends

Chun-Chi Lin and Hartmut Schwetlick

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Submission date: 22. Aug. 2012 (revised version: October 2012)
Pages: 23
Bibtex
MSC-Numbers: 35K55, 41A15, 53C44
Keywords and phrases: fourth-order flow, elastic curve, knot point, nonlinear spline, curve fitting
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Abstract:
In this paper we investigate the L²-flow of elastic non-closed curves in n-dimensional Euclidean spaces with knot points and two clamped ends. The L²-flow corresponds to a fourth-order parabolic equation on each piece of curve between two successive knot points with certain dynamic interior boundary conditions at these interior knot points. For solutions of the L²-flow, we prove that they are not only piecewise C^∞-smooth but also globally C¹-smooth at each fixed time t if the initial curves are both piecewise C^∞-smooth and globally C¹-smooth. Moreover, the asymptotic limit curves are piecewise C^∞-smooth but globally C²-smooth. To the best of the authors’ knowledge, our parabolic PDE approach provides a new method in the literature for the curve fitting problem, instead of variational methods.