Geodesic
A thresholding algorithm for Willmore-type flows via fourth-order linear parabolic equation
Katsuyuki Ishii1 orcid ; Yoshihito Kohsaka1 orcid ; Nobuhito Miyake2 orcid ; Koya Sakakibara3 orcid
1Kobe University, Kobe, Japan
2Kyushu University, Fukuoka, Japan
3Kanazawa University, Kanazawa, Japan; RIKEN, Wako, Japan
Interfaces and free boundaries, Tome 27 (2025) no. 2, pp. 235-283

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We propose a thresholding algorithm for Willmore-type flows in RN. 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 Ω0​. The main results of this paper demonstrate that the boundary ∂Ω(t) of the new set Ω(t), generated by our algorithm, is included in O(t)-neighborhood of ∂Ω0​ for small t>0 and that the normal velocity from ∂Ω0​ to ∂Ω(t) is nearly equal to the L2-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.
DOI : 10.4171/ifb/533
Classification : 53E40, 35K30, 35R37
Mots-clés : thresholding algorithm, Willmore-type energy, gradient flow

Katsuyuki Ishii  1   ; Yoshihito Kohsaka  1   ; Nobuhito Miyake  2   ; Koya Sakakibara  3

1 Kobe University, Kobe, Japan
2 Kyushu University, Fukuoka, Japan
3 Kanazawa University, Kanazawa, Japan; RIKEN, Wako, Japan 
Katsuyuki Ishii; Yoshihito Kohsaka; Nobuhito Miyake; Koya Sakakibara. A thresholding algorithm for Willmore-type flows via fourth-order linear parabolic equation. Interfaces and free boundaries, Tome 27 (2025) no. 2, pp. 235-283. doi: 10.4171/ifb/533
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     title = {A thresholding algorithm for {Willmore-type} flows via fourth-order linear parabolic equation},
     journal = {Interfaces and free boundaries},
     pages = {235--283},
     year = {2025},
     volume = {27},
     number = {2},
     doi = {10.4171/ifb/533},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ifb/533/}
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