A new isoperimetric inequality for the elasticae
Résumé
For a smooth curve γ, we define its elastic energy as E(γ) = 1 /2 \int 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.
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