Geodesic
How to unify the total/local-length-constraints of the gradient flow for the bending energy of plane curves
Kybernetika, Tome 45 (2009) no. 4, pp. 615-624 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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The gradient flow of bending energy for plane curve is studied. The flow is considered under two kinds of constraints; one is under the area and total-length constraints; the other is under the area and local-length constraints. The fundamental results (the local existence and uniqueness) were obtained independently by Kurihara and the second author for the former one; by Okabe for the later one. For the former one the global existence was shown for any smooth initial curves, but the asymptotic behavior has not been studied. For the later one, the global existence was guaranteed for only curves with the rotation number one, and the behavior was well studied. It is desirable to compensate the results with each other. In this note, it is proposed how to unify the two flows.
The gradient flow of bending energy for plane curve is studied. The flow is considered under two kinds of constraints; one is under the area and total-length constraints; the other is under the area and local-length constraints. The fundamental results (the local existence and uniqueness) were obtained independently by Kurihara and the second author for the former one; by Okabe for the later one. For the former one the global existence was shown for any smooth initial curves, but the asymptotic behavior has not been studied. For the later one, the global existence was guaranteed for only curves with the rotation number one, and the behavior was well studied. It is desirable to compensate the results with each other. In this note, it is proposed how to unify the two flows.
Classification : 35K30, 35K55, 53A04, 53C44, 58J35
Keywords: gradient flow; bending energy; total-length constraint; local-length constraint 
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     title = {How to unify the total/local-length-constraints of the gradient flow for the bending energy of plane curves},
     journal = {Kybernetika},
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Miyamoto, Yuki; Nagasawa, Takeyuki; Suto, Fumito. How to unify the total/local-length-constraints of the gradient flow for the bending energy of plane curves. Kybernetika, Tome 45 (2009) no. 4, pp. 615-624. http://geodesic.mathdoc.fr/item/KYB_2009_45_4_a5/

[1] G. Dziuk, E. Kuwert, and R. Schätzle: Evolution of elastic curves in $\mathbb R^n$: existence and computation. SIAM J. Math. Anal. 33 (2002), 5, 1228–1245. | MR

[2] T. Kurihara and T. Nagasawa: On the gradient flow for a shape optimization problem of plane curves as a singular limit. Saitama J. Math. 24 (2006/2007), 43–75. | MR

[3] K. Mikula and D. Ševčovič: Evolution of plane curves driven by a nonlinear function of curvature and anisotropy. SIAM J. Appl. Math. 61 (2001), 5, 1473–1501. | MR

[4] K. Mikula and D. Ševčovič: A direct method for solving an anisotropic mean curvature flow of plane curves with an external force. Math. Methods Appl. Sci. 27 (2004), 13, 1545–1565. | MR

[5] K. Mikula and D. Ševčovič: Computational and qualitative aspects of evolution of curves driven by curvature and external force. Comput. Vis. Sci. 6 (2004), 4, 211–225. | MR

[6] K. Mikula and D. Ševčovič: Evolution of curves on a surface driven by the geodesic curvature and external force. Appl. Anal. 85 (2006), 4, 345–362. | MR

[7] Y. Miyamoto: Reformulation of Local-Constraint-Gradient Flow for Bending Energy of Plane Curves Applying the Fredholm Alternative (in Japanese). Master Thesis, Saitama University, 2009.

[8] S. Okabe: The motion of elastic planar closed curve under the area-preserving condition. Indiana Univ. Math. J. 56 (2007), 4, 1871–1912. | MR

[9] F. Suto: On the Global Existence for Local/Total-Constraint-Gradient Flows for the Bending Energy of Plane Curves (in Japanese). Master Thesis, Saitama University, 2009.