The free boundary free elastic flow is the steepest descent gradient flow for the elastic energy of curves meeting two parallel lines in the euclidean plane perpendicularly. Straight lines connecting these orthogonally are obviously equilibria and absolute minimisers of the elastic energy.
It is proved that these have, measured in Euler's scale-invariant bending energy, a basin of attraction at least to the level $1.9615\, \pi$. It was originally conjectured that this level should be $2\pi$, which is the energy of a half period of Euler's elastica, which is unstable and (up to scaling, translation and rotation) the only nontrivial equilibrium.
The method of proof cannot be pushed to the previously conjectured level, and numerical simulations give evidence that this conjecture may in fact be false.
The core of the argument is an interpolation inequality which looks purely analytical. However, the necessary conditions on possibly optimal functions turn out to be the elastica equations with respect to the euclidean, spherical or hyperbolic metric, respectively.
This talk reports on joint work with Klaus Deckelnick, Robert Nürnberg (numerical analysis & experiments), and Glen and Valentina-Mira Wheeler (geometric analysis).