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

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.