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)
MSC-Numbers: 35K55, 41A15, 53C44
Keywords and phrases: fourth-order flow, elastic curve, knot point, nonlinear spline, curve fitting
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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.