We consider the problem of minimizing the bending or elastic energy among Jordan curves confined in a given open set Ω. We prove existence, regularity and some structural properties of minimizers. In particular, when Ω is convex we show that a minimizer is necessarily a convex curve. We also provide an example of a minimizer with self-intersections.
IntroductionThe study of curvature-based energies for curves started with Bernoulli and Euler's works on elastic rods and thin beams, see [36] for a historical overview.Considering a regular curve γ with curvature κ, its bending energy -also called Bernoulli-Euler's elastic energy -is given bySuch energy is not only fundamental in the context of mechanics, it is also important in image processing and computer vision, see for instance the applications to amodal completion, image inpainting, image denoising, or shape smoothing in [33,32,2,18,10,12,13,14,30,39,38,9,40].Curves which are critical points or minimizers of the bending energy are called elasticae, and the study of their properties have motivated many contributions, see a few references below. Free minimization is not well-posed, since the energy of a circle with radius R vanishes as R → +∞, and the only curves with zero bending energy are straight lines. Therefore, additional constraints are needed in order to have non-trivial minimizers. Frequently the length of the curve is prescribed (see for instance [5,37]) but other constraints can also be used, as in the non-exhaustive following list:• prescribed enclosed area for closed curves in the plane [3,11,20];• open curves in the plane with clamped ends [15,21,24];• confined closed curves, in a phase-field approximation context [16,17];We recall a few notions about planar regular curves. Let Ω be a bounded and Lipschitz open set of R 2 .
