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

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 ${\mathrm{\Omega }}_0$. The main results of this paper demonstrate that the boundary $\mathrm{\partial }\mathrm{\Omega }(t)$ of the new set $\mathrm{\Omega }(t)$, generated by our algorithm, is included in $O(t)$-neighborhood of $\mathrm{\partial }{\mathrm{\Omega }}_0$ for small $t>0$ and that the normal velocity from $\mathrm{\partial }{\mathrm{\Omega }}_0$ to $\mathrm{\partial }\mathrm{\Omega }(t)$ 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.