A thresholding algorithm for Willmore-type flows via fourth-order linear parabolic equation

Abstract

We propose a thresholding algorithm for Willmore-type flows in ${\mathbb{R}}^N$. This algorithm is constructed based on the asymptotic expansion of the solution to the initial value problem for a fourth-order linear parabolic partial differential equation whose initial data is the indicator function on the compact set ${\Omega }_0$. The main results of this paper demonstrate that the boundary $\partial \Omega (t)$ of the new set $\Omega (t)$, generated by our algorithm, is included in $O(t)$-neighborhood of $\partial {\Omega }_0$ for small $t>0$ and that the normal velocity from $\partial {\Omega }_0$ to $\partial \Omega (t)$ is nearly equal to the $L^2$-gradient of Willmore-type energy for small $t>0$. Finally, numerical examples of planar curves governed by the Willmore flow are provided by using our thresholding algorithm.