高階與分數階的幾何變分問題

In this project we mainly focus on the investigation of the $L^2$-flow of elastic non-closed curves in $n$-dimensional Euclidean spaces with knot points and two clamped ends. For the case of fractional operator, we leave it to the future project. 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. As an application, the $L^2$-flow of non-closed elastic curves in this project provides a new approach for the curve fitting problem.