A new isoperimetric inequality for the elasticae

For a smooth curve $γ$, we define its elastic energy as $E(γ)= \frac 12 \int_γ k^2 (s) ds$ 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 $Ω$, the following isoperimetric inequality holds: $E^2(\partial Ω)A(Ω)\geq π^3$. 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.

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