Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Fabian Rupp, Adrian Spener

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.

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 -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.

Paper Structure

This paper contains 17 sections, 31 theorems, 137 equations.

Table of Contents

  1. Introduction and main results
  2. Short time existence
  3. On the Lagrange multiplier
  4. From the geometric problem to a quasilinear PDE
  5. Linearization of the analytic problem
  6. Contraction estimates
  7. The fixed point argument
  8. Parabolic smoothing
  9. Long time behavior and the proof of Theorem 1.2
  10. Long-time existence after reparametrization
  11. The length-preserving elastic flow as a gradient flow on a Hilbert manifold
  12. The constrained Lojasiewicz--Simon gradient inequality
  13. Convergence
  14. Explicit formulas in coordinates
  15. Function spaces
  16. ...and 2 more sections

Key Result

Theorem 1

Let $f_0\in W^{2,2}(I;\mathbb{R}^d)$ be immersed, let ${p_0, p_1\in \mathbb{R}^d}$ and $\tau_0, \tau_1 \in \mathbb{S}^{d-1}$ satisfy eq:BCCompatibility and eq:f_0assumption. Then, there exists $T>0$ and a solution ${f\in W^{1,2}(0,T;L^2(I,\mathbb{R}^d))\cap L^2(0,T;W^{4,2}(I;\mathbb{R}^d))}$ of eq:E

Theorems & Definitions (67)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Definition 1
  • Definition 2
  • Remark 1
  • Proposition 2.1
  • ...and 57 more