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

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 $\epsilon$. We show that the approximating flow smoothly converges, as $\epsilon \to 0^{+}$, to the curvature flow of the curve with Dirichlet boundary conditions for all times before the first singularity of the limit flow.