Length-constrained, length-penalised and free elastic flows of planar curves inside cones

TL;DR

This paper investigates the long-term behavior of elastic flows of planar curves inside cones under various length constraints, proving convergence to specific shapes under certain conditions.

Contribution

It introduces new results on the exponential and self-similar convergence of elastic flows with length constraints inside cones, extending understanding of their asymptotic shapes.

Findings

Solutions converge exponentially to circular arcs for fixed or penalized length cases.

Free elastic flow solutions converge to expanding self-similar arcs.

Existence of smooth solutions is guaranteed under general initial conditions.

Abstract

We study families of smooth, embedded, regular planar curves $ \alpha : \left [-1,1 \right ]\times \left [0,T \right )\to \mathbb{R}^{2}$ with generalised Neumann boundary conditions inside cones, satisfying three variants of the fourth-order nonlinear $L^2$- gradient flow for the elastic energy: (1) elastic flow with a length penalisation, (2) elastic flow with fixed length and (3) the unconstrained or `free' elastic flow. Assuming neither end of the evolving curve reaches the cone tip, existence of smooth solutions for all time given quite general initial data is well known, but classification of limiting shapes is generally not known. For cone angles not too large and with suitable smallness conditions on the $L^2$-norm of the first arc length derivative of curvature of the initial curve, we prove in cases (1) and (2) smooth exponential convergence of solutions in the $C^\infty$-topology to particular circular arcs, while in case (3), we show smooth convergence to an expanding self-similar arc.