MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV ( that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint

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

Chun-Chi Lin and Hartmut Schwetlick


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.

MSC Codes:
35K55, 41A15, 53C44
fourth-order flow, elastic curve, knot point, nonlinear spline, curve fitting

Related publications

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

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