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

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.