A thresholding algorithm to Willmore-type flows via fourth order linear parabolic equation
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866912461336084480 |
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| author | Ishii, Katsuyuki Kohsaka, Yoshihito Miyake, Nobuhito Sakakibara, Koya |
| author_facet | Ishii, Katsuyuki Kohsaka, Yoshihito Miyake, Nobuhito Sakakibara, Koya |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_13155 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A thresholding algorithm to Willmore-type flows via fourth order linear parabolic equation Ishii, Katsuyuki Kohsaka, Yoshihito Miyake, Nobuhito Sakakibara, Koya Analysis of PDEs Numerical Analysis 35K30, 35R37, 53E40 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. |
| title | A thresholding algorithm to Willmore-type flows via fourth order linear parabolic equation |
| topic | Analysis of PDEs Numerical Analysis 35K30, 35R37, 53E40 |
| url | https://arxiv.org/abs/2311.13155 |