A minimising movement scheme for the p-elastic energy of curves
Simon Blatt, Christopher Hopper, Nicole Vorderobermeier
Abstract
Abstract We prove short-time existence for the negative $$L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>L</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math> -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 $$W^{2,p}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>W</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>p</mml:mi></mml:mrow></mml:msup></mml:math> -Sobolev norm of the initial data.