JournalsjemsVol. 19, No. 11pp. 3355–3376

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
A new isoperimetric inequality for the elasticae cover
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Abstract

For a smooth curve γ\gamma, we define its elastic energy as E(γ)=12γk2(s)dsE(\gamma)= \frac {1}{2} \int_{\gamma} k^2 (s) ds where k(s)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 R2\mathbb R^2, the disc has the boundary with the least elastic energy. In other words, for any bounded simply connected domain Ω\Omega, the following isoperimetric inequality holds: E2(Ω)A(Ω)π3E^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.

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