A gradient flow for the p-elastic energy defined on closed planar curves

Abstract

We study the evolution of closed inextensible planar curves under a second order flow that decreases the p-elastic energy. A short time existence result for p ∈ (1, ∞) is obtained via a minimizing movements method. For p = 2, that is in the case of the classic elastic energy, long-time existence is retrieved.

A Second Order Gradient Flow of p-Elastic Planar Networks

“…Proof of Lemma 3.8. This is a straight-forward adaptation of [34,Lemma 3.13]. Notice first that, from (3.6), (3.5), and Theorem 3.4 it follows that |∂ s θ j i,n | p−2 ∂ s θ j i,n H 1 (I j ) ≤ C(1 + 3 r=1 V r i,n L 2 (I j ) ) (A11) for j = 1, 2, 3 and for all i = 1, .…”

“…Here we consider the evolution of the network Γ via a second order gradient flow first introduced by Y. Wen in [37] (see also [22,34,21]). More precisely we will consider the L 2 -gradient flow of the energy…”

A Survey of the Elastic Flow of Curves and Networks

“…By expressing the L 2 corresponding gradient flow by means of θ one get another geometric evolution equation. This is a second order gradient flow and it has been first considered by [54] and then further investigated by [30,31,42,45,55].…”

The $p$-elastic flow for planar closed curves with constant parametrization

“…It is significant to extend studies on the bending energy to those on the p-elastic energy with p = 2. Indeed, recently the p-elastic energy has attracted interest (e.g., [1,5,14,16,27,32,35,37]). The purpose of this paper is to construct and study the L 2 -gradient flow for the p-elastic energy for initial curves in the energy space.…”