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MiS Preprint
54/2012

$L^2$-flow of elastic curves with knot points and clamped ends

Chun-Chi Lin and Hartmut Schwetlick

Abstract

In this paper we investigate the $L^2$-flow of elastic non-closed curves in $n$-dimensional Euclidean spaces with knot points and two clamped ends. The $L^2$-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 $L^2$-flow, we prove that they are not only piecewise $C^\infty$-smooth but also globally $C^1$-smooth at each fixed time $t$ if the initial curves are both piecewise $C^\infty$-smooth and globally $C^1$-smooth. Moreover, the asymptotic limit curves are piecewise $C^\infty$-smooth but globally $C^2$-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.

Received:
Aug 22, 2012
Published:
Sep 7, 2012
MSC Codes:
35K55, 41A15, 53C44
Keywords:
fourth-order flow, elastic curve, knot point, nonlinear spline, curve fitting

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Preprint
2012 Repository Open Access
Hartmut R. Schwetlick and Chun-Chi Lin

\({L^2}\)-flow of elastic curves with knot points and clamped ends