Abstract
For any \(\alpha >0,\) we study \(k^{\alpha }\)-type length-preserving and area-preserving nonlocal flow of convex closed plane curves and show that these two types of flow evolve such curves into round circles in \(C^{\infty } \)-norm. Other relevant \(k^{\alpha }\)-type nonlocal flow is also discussed when \(\alpha \ge 1.\)
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Notes
Throughout this paper, “convex” always means “uniformly convex”, i.e., the curvature is strictly positive everywhere.
From now on, all closed curves are assumed to be simple unless otherwise stated.
We thank the referee for pointing out this important issue, which we had neglected at first.
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Acknowledgments
We thank the referee for his careful reading of our paper and useful comments and suggestions. We are very grateful to Professor Ben Andrews of the Australian National University. Without consulting him so often, it is unlikely for us to finish this paper. The first author is supported by the National Science Council of Taiwan with Grant Number 102-2115-M-007-012-MY3. The second author is supported by the National Natural Science Foundation of China 11101078, 11171064, and the Natural Science Foundation of Jiangsu Province BK20130596.
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Communicated by Y. Giga.
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Tsai, DH., Wang, XL. On length-preserving and area-preserving nonlocal flow of convex closed plane curves. Calc. Var. 54, 3603–3622 (2015). https://doi.org/10.1007/s00526-015-0915-1
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DOI: https://doi.org/10.1007/s00526-015-0915-1