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 $\Omega_0$. 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.