The {\L}ojasiewicz-Simon gradient inequality for open elastic curves

TL;DR

This paper establishes a gradient inequality for the elastic energy of open curves with fixed endpoints, proving convergence of the flow to elastica, a critical point of the energy functional.

Contribution

It introduces the ojasiewicz-Simon gradient inequality for elastic energy of open curves with boundary conditions, enabling convergence analysis of the gradient flow.

Findings

Proved the ojasiewicz-Simon inequality for elastic energy.

Demonstrated convergence of the gradient flow to elastica.

Provided mathematical framework for analyzing elastic curves with boundary conditions.

Abstract

In this paper we consider the elastic energy for open curves in Euclidean space subject to clamped boundary conditions and obtain the \L ojasiewicz-Simon gradient inequality for this energy functional. Thanks to this inequality we can prove that a (suitably reparametrized) solution to the associated $L^2$-gradient flow converges for large time to an elastica, that is to a critical point of the functional.