Elastic flow of curves with partial free boundary

2023/10/23 by Diana, Antonia · 1 citation
#35A01 #35G31 #53E40 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2310.14738

Abstract

We consider a curve with boundary points free to move on a line in \mathbb R², which evolves by the L²--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 Hölder 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= + ∞.

Cited by

• Obstacle problems for the elastic flow and related topics

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• The elastic flow of curves in the hyperbolic plane
• The length-preserving elastic flow with free boundary on hypersurfaces in ℝⁿ
• A gradient flow for the p-elastic energy defined on closed planar curves
• The p-elastic flow for planar closed curves with constant parametrization
• Stability and convergence for the length-penalized elastic flow of curves with partial free boundary