Stability and convergence for the length-penalized elastic flow of curves with partial free boundary

Abstract

We investigate the asymptotic stability of the length-penalized elastic flow of curves with boundary points constrained to the $x$-axis in ${\mathbb{R}}^2$. The main tool in our analysis is the Lojasiewicz--Simon inequality, which is used to prove that the flow smoothly converges to an elastica.

There is a conceptual problem in the main result (Proposition 3.6). This issue will be fixed in the next version