  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.
