Semi-discrete linear curvature flows of curves with boundary

Abstract In [1], Chow and Glickenstein considered a second order linear semi-discrete analogue to the curve shortening flow for closed curves formed by joining an ordered set of points in the plane or in higher dimensional Euclidean space. In this article we consider similar flows with boundary conditions. Given distinct endpoints, results include characterisation of self-similar scaling solutions, and existence and uniqueness for solutions that evolve by pure translation in any given fixed direction and solutions that evolve by pure rotation with any fixed angular velocity. More generally, given any initial piecewise-linear curve with distinct boundary points, we provide a representation formula for the unique solution of the flow. Next we consider a semi-discrete analogue to the Yau problem of flowing one curve to another by curvature flow, here with boundaries. We show that any piecewise linear curve with distinct boundary points may flow to any other under the corresponding nonhomogeneous curvature flow; the solution is not unique unless the precise boundary point trajectories are prescribed. In the last section we consider two further variants of our semi-discrete curvature flow, where we allow anisotropy and vertices of varying mobility.