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arXiv · 2311.13155

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

Katsuyuki Ishii·Yoshihito Kohsaka·Nobuhito Miyake·Koya Sakakibara
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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 $Ω_0$. The main results of this paper demonstrate that the boundary $\partialΩ(t)$ of the new set $Ω(t)$, generated by our algorithm, is included in $O(t)$-neighborhood of $\partialΩ_0$ for small $t>0$ and that the normal velocity from $ \partialΩ_0 $ to $ \partialΩ(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.

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Katsuyuki Ishii, Yoshihito Kohsaka, Nobuhito Miyake, Koya Sakakibara. 2024-11-30. A thresholding algorithm to Willmore-type flows via fourth order linear parabolic equation. https://doi.org/10.4171/ifb%2F533

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