Calculus of Variations and Geometric Measure Theory

A. Diana

Elastic flow of curves with partial free boundary

created by diana1 on 28 Sep 2023
modified on 09 May 2024


Submitted Paper

Inserted: 28 sep 2023
Last Updated: 9 may 2024

Year: 2023


We consider a curve with boundary points free to move on a line in $\mathbb R^2$, which evolves by the $L^2$--gradient flow of the elastic energy, that is a linear combination of the Willmore and the length functional. For such planar evolution problem we study the short and long--time existence. Once we establish under which boundary conditions the PDE's system is well--posed (in our case the Navier boundary conditions), employing the Solonnikov theory for linear parabolic systems in H\"older space, we show that there exists a unique flow in a maximal time interval $[0,T)$. Then, using energy methods we prove that the maximal time is actually $T= + \infty$.