A new isoperimetric inequality for the elasticae

Abstract

For a smooth curve $\gamma$, we define its elastic energy as $E(\gamma )=\frac{1}{2}{\int }_{\gamma }{k}^2(s)ds$ 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: $E^2(\partial \Omega )A(\Omega )\ge {\pi }^3$. 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.