Existence and convergence of the length-preserving elastic flow of clamped curves

Abstract

We study the evolution of curves with fixed length and clamped boundary conditions moving by the negative $L^2$-gradient flow of the elastic energy. For any initial curve lying merely in the energy space we show existence and parabolic smoothing of the solution. Applying previous results on long time existence and proving a constrained Lojasiewicz-Simon gradient inequality we furthermore show convergence to a critical point as time tends to infinity.

32 pages, final version. To appear in J. Evol. Equ

References in corpus (5)

• Convergence of elastic flows of curves into manifolds
• The Lojasiewicz-Simon inequality for the elastic flow
• The volume-preserving Willmore flow
• The Willmore flow with prescribed isoperimetric ratio
• Asymptotic convergence of evolving hypersurfaces

Cited by in corpus (2)

• Li-Yau type inequality for curves in any codimension
• Dynamics of Elastic Wires: Preserving Area without Nonlocality