A new isoperimetric inequality for the elasticae

Abstract

For a smooth curve $\gamma$, we define its elastic energy as $E(\gamma)= \frac 12 \int_{\gamma} k^2 (s) ds$ where $k(s)$ is the curvature. The main purpose of the paper is to prove that among all smooth, simply connected, bounded open sets of prescribed area in $\mathbb{R}^2$, the disc has the boundary with the least elastic energy. In other words, for any bounded simply connected domain $\Omega$, the following isoperimetric inequality holds: $E^2(\partial \Omega)A(\Omega)\geq \pi ^3$. The analysis relies on the minimization of the elastic energy of drops enclosing a prescribed area, for which we give as well an analytic answer.