A minimising movement scheme for the $p$-elastic energy of curves

TL;DR

This paper establishes short-time existence for the negative gradient flow of the p-elastic energy of curves using a minimising movement scheme, addressing reparametrisation invariance through normal graph representations.

Contribution

It introduces a novel approach to handle degeneracy due to reparametrisation invariance by representing curves as approximate normal graphs, enabling rigorous existence results.

Findings

Proves short-time existence for the flow.

Provides a lower bound on the solution's lifetime.

Handles degeneracy via normal graph parametrization.

Abstract

We prove short-time existence for the negative $L^2$-gradient flow of the $p$-elastic energy of curves via a minimising movement scheme. In order to account for the degeneracy caused by the energy's invariance under curve reparametrisations, we write the evolving curves as approximate normal graphs over a fixed smooth curve. This enables us to establish short-time existence and give a lower bound on the solution's lifetime that depends only on the $W^{2,p}$-Sobolev norm of the initial data.