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The length-preserving elastic flow with free boundary on hypersurfaces in $\mathbb{R}^n$

Anna Dall'Acqua, Manuel Schlierf

TL;DR

This work analyzes the length-preserving elastic flow of open curves with orthogonal free boundary on a hypersurface $M\subset\mathbb{R}^n$, a nonlocal, fourth-order geometric evolution posed in codimension $n-1$ with nonlinear boundary conditions. The authors develop an analytic reformulation with maximal $L^p$-regularity, prove local well-posedness and instantaneous smoothing, and then derive geometric interpolation estimates and Lagrange-multiplier bounds to obtain global existence and subconvergence to elastica. In a planar special case, they show full convergence and classify limits as circle arcs or figure-eight elastica, while in the general setting a uniform non-flatness hypothesis governs the asymptotics and enables partial convergence results. The results advance understanding of long-time behavior for length-preserving, higher-order geometric flows with free boundary on ambient hypersurfaces and provide a framework for analyzing convergence to elastica under complex boundary interactions.

Abstract

We study the length-preserving elastic flow of curves in arbitrary codimension with free boundary on hypersurfaces. This constrained gradient flow is given by a nonlocal evolution equation with nonlinear higher-order boundary conditions. We prove global existence and subconvergence to critical points. The proof strategy involves a careful treatment of short-time existence, uniqueness, and parabolic energy estimates.

The length-preserving elastic flow with free boundary on hypersurfaces in $\mathbb{R}^n$

TL;DR

This work analyzes the length-preserving elastic flow of open curves with orthogonal free boundary on a hypersurface , a nonlocal, fourth-order geometric evolution posed in codimension with nonlinear boundary conditions. The authors develop an analytic reformulation with maximal -regularity, prove local well-posedness and instantaneous smoothing, and then derive geometric interpolation estimates and Lagrange-multiplier bounds to obtain global existence and subconvergence to elastica. In a planar special case, they show full convergence and classify limits as circle arcs or figure-eight elastica, while in the general setting a uniform non-flatness hypothesis governs the asymptotics and enables partial convergence results. The results advance understanding of long-time behavior for length-preserving, higher-order geometric flows with free boundary on ambient hypersurfaces and provide a framework for analyzing convergence to elastica under complex boundary interactions.

Abstract

We study the length-preserving elastic flow of curves in arbitrary codimension with free boundary on hypersurfaces. This constrained gradient flow is given by a nonlocal evolution equation with nonlinear higher-order boundary conditions. We prove global existence and subconvergence to critical points. The proof strategy involves a careful treatment of short-time existence, uniqueness, and parabolic energy estimates.

Paper Structure

This paper contains 26 sections, 37 theorems, 234 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The elastic flow
  3. Free boundary elastic flows and main results
  4. Structure of the article
  5. Preliminaries
  6. Assumptions on and basic properties of the hypersurface $M$
  7. Evolution equations and natural boundary conditions
  8. Preliminaries for the tensor notation used in the energy estimates
  9. Higher-order identities following from the free boundary conditions
  10. Short-time existence and uniqueness
  11. Reformulation of the boundary conditions
  12. The analytic problem
  13. Existence and uniqueness of solutions to the analytic problem
  14. Instantaneous smoothing for the analytic problem
  15. Existence and regularity of reparametrizations: from analytic problem to equation with zero tangential speed and back
  16. ...and 11 more sections

Key Result

Theorem 1

Let $p>5$ and $\gamma_0\in W^{4(1-\frac{1}{p}),p}([-1,1],\mathbb{R}^n)$ be an immersion with $\mathcal{E}(\gamma_0)>0$, satisfying eq:intro-free-bcs. There exist $T>0$ and a family of immersions $\gamma\colon[0,T)\times[-1,1]\to\mathbb{R}^n$ with $\gamma\in C^\infty((0,T)\times [-1,1])$ such that $[ where $\lambda(\gamma)$ is given by eq:def-lambda and $\partial_t^\bot=\partial_t-\langle\partial_t

Figures (3)

  • Figure 1: Curve $\gamma$ with orthogonal free boundary on a round sphere $M\subseteq \mathbb{R}^3$ with unit normal $\xi$. Here $\langle \partial_s\gamma(\pm1),\xi(\gamma(\pm1))\rangle = \mp1$.
  • Figure 2: An initial datum (left) and the associated limit (right).
  • Figure 3: An example where a straight line $\bar{\gamma}$ of length $L_0$ satisfying \ref{['eq:intro-free-bcs']} with $\langle\partial_s\bar{\gamma}(\pm1),\xi(\bar{\gamma}(\pm1))\rangle=\mp1$ exists for exactly one length $L_0=L_0^*(M)>0$.

Theorems & Definitions (84)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Remark 1
  • Remark 2: Extension of functions defined on $M$
  • Remark 3: Extending the shape operator
  • Lemma 2
  • Definition 1
  • Remark 4: On the Lagrange multiplier
  • Remark 5
  • ...and 74 more