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.