Classification and stability of penalized pinned elasticae

This paper considers critical points of the length-penalized elastic bending energy among planar curves whose endpoints are fixed. We classify all critical points with an explicit parametrization. The classification strongly depends on a special penalization parameter $\hat{\lambda}\simeq 0.70107$. Stability of all the critical points is also investigated, and again the threshold $\hat{\lambda}$ plays a decisive role. In addition, our explicit parametrization is applied to compare the energy of critical points, leading to uniqueness of minimal nontrivial critical points. As an application we obtain eventual embeddedness of elastic flows.