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.

翻译：本文提出了一种用于$\mathbb{R}^N$中Willmore型流的阈值算法。该算法基于四阶线性抛物型偏微分方程初值问题解的渐近展开构造，其初始数据为紧集$\Omega_0$上的指示函数。本文主要结果表明：对于小参数$t>0$，由算法生成的新集合$\Omega(t)$的边界$\partial\Omega(t)$包含在$\partial\Omega_0$的$O(t)$邻域内，且从$\partial\Omega_0$到$\partial\Omega(t)$的法向速度近似等于Willmore型能量的$L^2$梯度。最后，通过应用本阈值算法，给出了由Willmore流控制的平面曲线数值算例。