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

TL;DR

This paper studies the stability and convergence of a length-penalized elastic flow of curves with boundary points on the x-axis, proving smooth convergence to elastica using the Lojasiewicz--Simon inequality.

Contribution

It introduces a stability analysis for the elastic flow with boundary constraints and applies the Lojasiewicz--Simon inequality to establish convergence to elastica.

Findings

Flow converges smoothly to elastica

Lojasiewicz--Simon inequality is effective for stability analysis

Boundary constraints are incorporated into the flow analysis

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.