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A fourth-order regularization of the curvature flow of immersed plane curves with Dirichlet boundary conditions

Giovanni Bellettini, Virginia Lorenzini, Matteo Novaga, Riccardo Scala

TL;DR

A fourth-order elastica-type regularization is introduced for the curvature flow of planar curves with Dirichlet boundaries, via $F_\varepsilon(\gamma)=\int_0^{\ell(\gamma)}(1+\varepsilon \kappa_\gamma^2)\,ds$, and the gradient flow is shown to converge to the curvature flow with Dirichlet data as $\varepsilon\to0^+$ on any interval before the first singularity. The authors develop $\varepsilon$-uniform estimates for length, energy, curvature and all spatial derivatives, together with bounds on the tangential velocity, enabling compactness and the passage to the limit. They prove convergence of the reparameterized flows $\Upsilon_\varepsilon$ to the limit flow $\Upsilon$ in $C_{\mathrm{loc}}^\infty((0, T_{\mathrm{sing}})\times[0,1])$, and under extra endpoint regularity, obtain $C_{\mathrm{loc}}^\infty([0, T_{\mathrm{sing}})\times[0,1])$ convergence. The analysis also shows that the convergence time reaches the first singularity, providing a rigorous route to generalized solutions via regularization and clarifying the role of boundary terms and tangential velocity in the Dirichlet setting.

Abstract

We consider a fourth-order regularization of the curvature flow for an immersed plane curve with fixed boundary, using an elastica-type functional depending on a small positive parameter $\varepsilon$. We show that the approximating flow smoothly converges, as $\varepsilon \to 0^+$, to the curvature flow of the curve with Dirichlet boundary conditions for all times before the first singularity of the limit flow.

A fourth-order regularization of the curvature flow of immersed plane curves with Dirichlet boundary conditions

TL;DR

A fourth-order elastica-type regularization is introduced for the curvature flow of planar curves with Dirichlet boundaries, via , and the gradient flow is shown to converge to the curvature flow with Dirichlet data as on any interval before the first singularity. The authors develop -uniform estimates for length, energy, curvature and all spatial derivatives, together with bounds on the tangential velocity, enabling compactness and the passage to the limit. They prove convergence of the reparameterized flows to the limit flow in , and under extra endpoint regularity, obtain convergence. The analysis also shows that the convergence time reaches the first singularity, providing a rigorous route to generalized solutions via regularization and clarifying the role of boundary terms and tangential velocity in the Dirichlet setting.

Abstract

We consider a fourth-order regularization of the curvature flow for an immersed plane curve with fixed boundary, using an elastica-type functional depending on a small positive parameter . We show that the approximating flow smoothly converges, as , to the curvature flow of the curve with Dirichlet boundary conditions for all times before the first singularity of the limit flow.
Paper Structure (8 sections, 16 theorems, 120 equations)

This paper contains 8 sections, 16 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Estimates of geometric quantities
  4. epsilon-uniform estimates of length, energy and normal velocity
  5. epsilon-uniform estimate of norm
  6. epsilon-uniform estimates of higher derivatives
  7. epsilon-uniform estimate of tangential velocity
  8. Proof of Theorem \ref{['teo:asymptotic_convergence']}

Key Result

Theorem 1.1

Let $\overline \gamma$ be a planar immersed regular curve satisfying eq:inidatmcf, eq:inidatmcf_curvature, and let $\gamma$ be the solution to eq:initial_bdry_value_prb_intro in $[0, T_{\mathrm{sing}})$. For any $\varepsilon\in(0,1]$ denote by $\gamma_\varepsilon$ the solution to eq:eps_dir_motion_e

Theorems & Definitions (29)

  • Theorem 1.1: Asymptotic convergence
  • Lemma 3.1: Evolution of geometric quantities
  • proof
  • Lemma 3.2: Energy dissipation equality
  • proof
  • Remark 3.3
  • Corollary 3.4: $L^2$-estimate of the normal velocity
  • proof
  • Proposition 3.5
  • Proposition 3.5
  • ...and 19 more