The elastica problem under area constraint

Abstract

Abstract. We show that the elastic energy E(γ) of a closed curve γ has a minimizer among all plane simple regular closed curves of given enclosed area A(γ), and that the minimum is attained for a circle. The proof is of a geometric nature and deforms parts of γ in a finite number of steps to construct some related convex sets with smaller energy.

Two dimensions are easier

“…Alternatively, one can prescribe the enclosed area of a closed curve without intersection points and ask for the shape with minimal elastic energy. That this shape is again a disk was recently shown in [5]. The proof uses the scale invariant functional J(γ) = κ 2 ds |Ω| 1/2 , where |Ω| is the area enclosed by γ and a reduction from simply connected sets Ω to convex ones.…”

Elastic energy of a convex body

“…This is an easy consequence of Gage's inequality together with the isoperimetric inequality: the disk is the unique minimizer of the elastic energy under a constraint of area among convex bodies. This result has been very recently extended to any (bounded) simply connected plane domains in two different papers by D. Bucur and the second author, in [4] and by V. Ferone, www.mn-journal.com B. Kawohl and C. Nitsch in [5]. It is noteworthy that these two papers use very different techniques (more analytic in the first one, more geometric in the second one).…”

Quantitative estimates for bending energies and applications to non-local variational problems

“…Our aim here is to show that actually the elastica energy W (E) itself controls the distance to balls. This is a quantitative version of [7,14] which could be of independent interest.…”

“…We first show that the energy of each connected component F of E can be strictly decreased by transforming it into a ball or an annulus. If F is simply connected, by [7,14] and the isoperimetric inequality…”

“…1 We choose to keep the factor 1 4 in dimension three to stick with the traditional notation.F λ,Q (E).(1.3)We start by considering the planar problem d = 2 and first focus on the uncharged case Q = 0. For λ = 0 no global minimizer exists in M(|B 1 |), but it has been recently shown in [7,14] that balls minimize the elastica energy under volume constraint in the class of simply connected sets. Our first result is a quantitative version of this fact in the spirit of the quantitative isoperimetric inequality [18,10].Theorem 1.1.…”