A new isoperimetric inequality for the elasticae

For a smooth curve γ, we define its elastic energy as E(γ) = 12 ∫ γ k(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, 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(∂Ω)A(Ω) ≥ π. 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.