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

TL;DR

This paper introduces a fourth-order regularization for the curvature flow of immersed plane curves with fixed boundaries, demonstrating convergence to the classical flow as the regularization parameter approaches zero.

Contribution

The work presents a novel elastica-type regularization for curvature flow with Dirichlet boundary conditions and proves smooth convergence to the unregularized flow before singularities.

Findings

Flow converges smoothly as regularization parameter tends to zero

Regularization maintains boundary conditions during evolution

Convergence holds up to the first singularity of the limit flow

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.