# A new isoperimetric inequality for the elasticae

### Dorin Bucur

Université de Savoie, Le-Bourget-du-Lac, France### Antoine Henrot

Université de Lorraine, Vandoeuvre-lès-Nancy, France

## Abstract

For a smooth curve $γ$, we define its elastic energy as $E(γ)=21 ∫_{γ}k_{2}(s)ds$ 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 $R_{2}$, 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: $E_{2}(∂Ω)A(Ω)≥π_{3}$. 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.

## Cite this article

Dorin Bucur, Antoine Henrot, A new isoperimetric inequality for the elasticae. J. Eur. Math. Soc. 19 (2017), no. 11, pp. 3355–3376

DOI 10.4171/JEMS/740