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$L^2$-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.