On a planar areapreserving curvature flow
Authors:
XiaoLi Chao, XiaoRan Ling and XiaoLiu Wang
Journal:
Proc. Amer. Math. Soc. 141 (2013), 17831789
MSC (2010):
Primary 53C44; Secondary 35B40, 35K59, 37B25
Published electronically:
September 19, 2012
MathSciNet review:
3020863
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Abstract 
References 
Similar Articles 
Additional Information
Abstract: A classical nonlocal curvature flow preserving the enclosed area is reinvestigated. The uniform upper bound and lower bound of curvature are established for the first time. As a result, a detailed proof is presented for the asymptotic behavior of the flow.
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 B. Andrews, Evolving convex curves, Calc. Var. Partial Differential Equations 7 (1998) 315371. MR 1660843 (99k:58038)
 [2]
 S. Angenent, Parabolic equations for curves on surfaces. I. Curves with integrable curvature, Ann. of Math.(2) 132 (1990) 451483; Parabolic equations for curves on surfaces II. Intersections, blowup and generalized solutions, Ann. of Math. (2) 133 (1991) 171215. MR 1078266 (91k:35102); MR 1087347 (92b:58039)
 [3]
 K.S. Chou, X.L. Wang, The curve shortening problem under Robin boundary condition, NoDea Nonlinear Differential Equations and Applications 19 (2012) 177194. MR 2902186
 [4]
 K.S. Chou, X.P. Zhu, The curve shortening problem. Chapman Hall/CRC, Boca Raton, FL, 2001. MR 1888641 (2003e:53088)
 [5]
 M. Gage, On an areapreserving evolution equation for plane curves, Nonlinear problems in geometry (Mobile, Ala., 1985), 5162, Contemp. Math., 51, Amer. Math. Soc., Providence, RI, 1986. MR 848933 (87g:53003)
 [6]
 M. Gage, R. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom. 23 (1986) 6996. MR 840401 (87m:53003)
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 M.A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom. 26 (1987) 285314. MR 906392 (89b:53005)
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 L.S. Jiang, S.L. Pan, On a nonlocal curve evolution problem in the plane, Comm. Anal. Geom. 16 (2008) 126. MR 2411467 (2009e:53083)
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 Y.C. Lin, D.H. Tsai, On a simple maximum principle technique applied to equations on the circle, J. Diff. Eq. 245 (2008) 377391. MR 2428003 (2010b:35252)
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 Y.C. Lin, D.H. Tsai, Nonlocal flow of convex plane curves and isoperimetric inequalities, Arxiv: 1005.0438v1, 2010.
 [11]
 Y.C. Lin, D.H. Tsai, On a general linear nonlocal curvature flow of convex plane curves, Arxiv: 1012.0114v1, 2010.
 [12]
 Y.C. Lin, D.H. Tsai, Application of Andrews and GreenOsher inequalities to nonlocal flow of convex plane curves, preprint, 2012.
 [13]
 L. Ma, L. Cheng, A nonlocal area preserving curve flow, Arxiv: 0907.1430v1, 2009.
 [14]
 L. Ma, A.Q. Zhu, On a length preserving curve flow, Monatshefte für Mathematik 165 (2012) 5778. MR 2886123
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 S.L. Pan, H. Zhang, On a curve expanding flow with a nonlocal term, Comm. Contemp. Math. 12 (2010) 815829. MR 2733199 (2012c:53102)
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 S.L. Pan, J.N. Yang, On a nonlocal perimeterpreserving curve evolution problem for convex plane curves, Manuscripta Math. 127 (2008) 469484. MR 2457190 (2010h:53099)
 [17]
 G. Sapiro, A. Tannenbaum, Area and length preserving geometric invariant scalespaces, Pattern Analysis and Machine Intelligence, IEEE Transactions 17 (1995) 6772.
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 X.L. Wang, The stability of fold circles in the curve shortening problem, Manuscripta Math. 134 (2011) 493511. MR 2765723 (2012d:53221)
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 X.L. Wang, W.F. Wo, On the stability of stationary line and grim reaper in planar curvature flow, Bulletin of the Australian Mathematical Society 83 (2011) 177188. MR 2784776 (2012c:35208)
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Additional Information
XiaoLi Chao
Affiliation:
Department of Mathematics, Southeast University, Nanjing, People’s Republic of China
XiaoRan Ling
Affiliation:
Department of Mathematics, Southeast University, Nanjing, People’s Republic of China
XiaoLiu Wang
Affiliation:
Department of Mathematics, Southeast University, Nanjing, People’s Republic of China
DOI:
http://dx.doi.org/10.1090/S000299392012117459
PII:
S 00029939(2012)117459
Keywords:
Mean curvature flow,
nonlocal flow,
asymptotic behavior
Received by editor(s):
August 24, 2011
Published electronically:
September 19, 2012
Additional Notes:
This work was supported by PRC Grants NSFC 11101078, 10971029 and 11171064 and the Natural Science Foundation of Jiangsu Province BK2011583
Communicated by:
Lei Ni
Article copyright:
© Copyright 2012
American Mathematical Society
