Variational analysis of inextensible elastic curves

Abstract

We minimize elastic energies on framed curves which penalize both curvature and torsion. We also discuss critical points using the infinite dimensional version of the Lagrange multipliers’ method. Finally, some examples arising from the applications are discussed and also numerical experiments are presented.

Elastic membranes spanning deformable curves

“…Using the framed curve approach (introduced by Gonzalez et al [35] and developed in ref. [25] for elastic curves), in this section we are able to consider a more general situation. Let 𝑝 > 1 and let (𝒕|𝒏|𝒃) ∈ 𝑊 1,𝑝 ((0, 2𝜋); 𝑆𝑂(3)), where 𝑆𝑂(3) is the set of all 3 × 3 rotation matrices (namely, {𝒕, 𝒏, 𝒃} is an orthonormal positively oriented basis in ℝ 3 ).…”

“…In this paper we aim to allow for self-intersections, which are physically evident, and to treat a general dependence on 𝜅, 𝜏. A first result in this direction is our contribution [25] where we derived the existence of minimizers of a functional of the following type…”

Geometric invariants of non-smooth framed curves

“…one needs to introduce a weak notion of curvature and torsion. For instance, in [5,1], we used the framed curve approach. Precisely, we considered functionals of the following type…”

“…We refer to [7,8], where they derived a corrected version of the Sadowsky functional within the theory of elastic ribbons and to [10,2,3,4], where the authors investigated equilibrium shapes of a system in which a closed flexible filament is spanned by a liquid film. Other results can be found in [5,1], where the framed curve approach has been employed to derive first-order necessary conditions for minimizers.…”

Elastic flow of curves with partial free boundary

“…Still later, Born, in his Thesis of 1906, was the first to publish figures of elasticae plotted numerically. However, in the last decades, many authors contribute to their classification, for instance we refer to Langer and Singer [23,24], Linnér [27], Djondjorov et al [17] and to Langer and Singer [25], Bevilacqua, Lussardi and Marzocchi [9], the same authors with Ballarin [8], for the case of functional which depends both on the curvature and the torsion of the curve. More recently, Miura and Yoshizawa in a series of papers [34,33,35], give a complete classification of both clamped and pinned p-elasticae.…”