A new isoperimetric inequality for the elasticae

الأصل · EN

For a smooth curve γ, we define its elastic energy as E(γ)= 12 ∫γ k² (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 R², 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: E²(∂ Ω)A(Ω)≥ π³. 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.