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
Fulltext PDF
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.
 [1]
Ben
Andrews, Evolving convex curves, Calc. Var. Partial
Differential Equations 7 (1998), no. 4,
315–371. MR 1660843
(99k:58038), 10.1007/s005260050111
 [2]
Sigurd
Angenent, Parabolic equations for curves on surfaces. I. Curves
with 𝑝integrable curvature, Ann. of Math. (2)
132 (1990), no. 3, 451–483. MR 1078266
(91k:35102), 10.2307/1971426
Sigurd
Angenent, Parabolic equations for curves on surfaces. II.
Intersections, blowup and generalized solutions, Ann. of Math. (2)
133 (1991), no. 1, 171–215. MR 1087347
(92b:58039), 10.2307/2944327
 [3]
KaiSeng
Chou and XiaoLiu
Wang, The curve shortening problem under Robin boundary
condition, NoDEA Nonlinear Differential Equations Appl.
19 (2012), no. 2, 177–194. MR
2902186, 10.1007/s0003001101234
 [4]
KaiSeng
Chou and XiPing
Zhu, The curve shortening problem, Chapman & Hall/CRC,
Boca Raton, FL, 2001. MR 1888641
(2003e:53088)
 [5]
Michael
Gage, On an areapreserving evolution equation for plane
curves, Nonlinear problems in geometry (Mobile, Ala., 1985) Contemp.
Math., vol. 51, Amer. Math. Soc., Providence, RI, 1986,
pp. 51–62. MR 848933
(87g:53003), 10.1090/conm/051/848933
 [6]
M.
Gage and R.
S. Hamilton, The heat equation shrinking convex plane curves,
J. Differential Geom. 23 (1986), no. 1, 69–96.
MR 840401
(87m:53003)
 [7]
Matthew
A. Grayson, The heat equation shrinks embedded plane curves to
round points, J. Differential Geom. 26 (1987),
no. 2, 285–314. MR 906392
(89b:53005)
 [8]
Lishang
Jiang and Shengliang
Pan, On a nonlocal curve evolution problem in the plane,
Comm. Anal. Geom. 16 (2008), no. 1, 1–26. MR 2411467
(2009e:53083)
 [9]
YuChu
Lin and DongHo
Tsai, On a simple maximum principle technique applied to equations
on the circle, J. Differential Equations 245 (2008),
no. 2, 377–391. MR 2428003
(2010b:35252), 10.1016/j.jde.2008.04.007
 [10]
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]
Li
Ma and Anqiang
Zhu, On a length preserving curve flow, Monatsh. Math.
165 (2012), no. 1, 57–78. MR
2886123, 10.1007/s0060501103028
 [15]
Shengliang
Pan and Hong
Zhang, On a curve expanding flow with a nonlocal term,
Commun. Contemp. Math. 12 (2010), no. 5,
815–829. MR 2733199
(2012c:53102), 10.1142/S0219199710003981
 [16]
Shengliang
Pan and Juanna
Yang, On a nonlocal perimeterpreserving curve evolution problem
for convex plane curves, Manuscripta Math. 127
(2008), no. 4, 469–484. MR 2457190
(2010h:53099), 10.1007/s002290080211x
 [17]
G. Sapiro, A. Tannenbaum, Area and length preserving geometric invariant scalespaces, Pattern Analysis and Machine Intelligence, IEEE Transactions 17 (1995) 6772.
 [18]
XiaoLiu
Wang, The stability of 𝑚fold circles in the curve
shortening problem, Manuscripta Math. 134 (2011),
no. 34, 493–511. MR 2765723
(2012d:53221), 10.1007/s0022901004100
 [19]
Xiaoliu
Wang and Weifeng
Wo, On the stability of stationary line and grim reaper in planar
curvature flow, Bull. Aust. Math. Soc. 83 (2011),
no. 2, 177–188. MR 2784776
(2012c:35208), 10.1017/S0004972710001942
 [1]
 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)
 [7]
 M.A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom. 26 (1987) 285314. MR 906392 (89b:53005)
 [8]
 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)
 [9]
 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)
 [10]
 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
 [15]
 S.L. Pan, H. Zhang, On a curve expanding flow with a nonlocal term, Comm. Contemp. Math. 12 (2010) 815829. MR 2733199 (2012c:53102)
 [16]
 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.
 [18]
 X.L. Wang, The stability of fold circles in the curve shortening problem, Manuscripta Math. 134 (2011) 493511. MR 2765723 (2012d:53221)
 [19]
 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)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2010):
53C44,
35B40,
35K59,
37B25
Retrieve articles in all journals
with MSC (2010):
53C44,
35B40,
35K59,
37B25
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
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
