2014/12/15 by Dorin Bucur, Bucur, Dorin, Antoine Henrot +1
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Optimization and Control (math.OC) #math.DG #math.OC
paper · pdf · doi:10.48550/arxiv.1412.4536
We have corrected the name of authors in the Metadata
arxiv created 2014/12/16 · arxiv updated 2014/12/17
For a smooth curve γ, we define its elastic energy as E(γ)= \frac 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.