Curve diffusion and straightening flows on parallel lines

Abstract

In this paper, we study families of immersed curves $\gamma :(-1,1)\times [0,T)\to {\mathbb{R}}^2$ with free boundary supported on parallel lines $\{{\eta }_1,{\eta }_2\}:\mathbb{R}\to {\mathbb{R}}^2$ evolving by the curve diffusion flow and the curve straightening flow. The evolving curves are orthogonal to the boundary and satisfy a no-flux condition. We give estimates and monotonicity on the normalised oscillation of curvature, yielding global results for the flows.

39 pages, 3 figures