On the basin of attraction for the free boundary free elastic flow

TL;DR

This paper investigates the basin of attraction for the free boundary free elastic flow, establishing a lower bound and providing numerical evidence that challenges previous conjectures about the maximum basin size.

Contribution

The authors prove a lower bound for the basin of attraction in elastic flow and present numerical evidence questioning the conjectured maximum basin size.

Findings

The basin of attraction extends at least to 1.9615π in Euler's scale-invariant energy.

The proof method cannot reach the conjectured level of 2π.

Numerical evidence suggests the conjecture of a 2π basin may be false.

Abstract

The free boundary free elastic flow is the steepest descent gradient flow for the elastic energy of curves meeting parallel lines perpendicularly. In this article we prove that the straight line has, measured in Euler's scale-invariant bending energy, a basin of attraction at least to the level $1.9615\, \pi$. We show that our method of proof cannot be pushed to the previously conjectured level $2\pi$, and in addition present numerical evidence that this conjecture may in fact be false.