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

Abstract

We propose a thresholding algorithm to 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 $\hat{I}{\text{c◯}}_0$. The main results of this paper demonstrate that the boundary $\partial \hat{I}\text{c◯}(t)$ of the new set $\hat{I}\text{c◯}(t)$, generated by our algorithm, is included in $O(t)$-neighborhood of $\partial \hat{I}{\text{c◯}}_0$ for small $t>0$ and that the normal velocity from $\partial \hat{I}{\text{c◯}}_0$ to $\partial \hat{I}\text{c◯}(t)$ 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.