On a planar area-preserving curvature flow

Authors:
Xiao-Li Chao, Xiao-Ran Ling and Xiao-Liu Wang

Journal:
Proc. Amer. Math. Soc. **141** (2013), 1783-1789

MSC (2010):
Primary 53C44; Secondary 35B40, 35K59, 37B25

Published electronically:
September 19, 2012

MathSciNet review:
3020863

Full-text 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**, 10.1007/s005260050111**[2]**Sigurd Angenent,*Parabolic equations for curves on surfaces. I. Curves with 𝑝-integrable curvature*, Ann. of Math. (2)**132**(1990), no. 3, 451–483. MR**1078266**, 10.2307/1971426

Sigurd Angenent,*Parabolic equations for curves on surfaces. II. Intersections, blow-up and generalized solutions*, Ann. of Math. (2)**133**(1991), no. 1, 171–215. MR**1087347**, 10.2307/2944327**[3]**Kai-Seng Chou and Xiao-Liu Wang,*The curve shortening problem under Robin boundary condition*, NoDEA Nonlinear Differential Equations Appl.**19**(2012), no. 2, 177–194. MR**2902186**, 10.1007/s00030-011-0123-4**[4]**Kai-Seng Chou and Xi-Ping Zhu,*The curve shortening problem*, Chapman & Hall/CRC, Boca Raton, FL, 2001. MR**1888641****[5]**Michael Gage,*On an area-preserving 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**, 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****[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****[8]**Lishang Jiang and Shengliang Pan,*On a non-local curve evolution problem in the plane*, Comm. Anal. Geom.**16**(2008), no. 1, 1–26. MR**2411467****[9]**Yu-Chu Lin and Dong-Ho Tsai,*On a simple maximum principle technique applied to equations on the circle*, J. Differential Equations**245**(2008), no. 2, 377–391. MR**2428003**, 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 Green-Osher inequalities to nonlocal flow of convex plane curves, preprint, 2012.**[13]**L. Ma, L. Cheng, A non-local 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/s00605-011-0302-8**[15]**Shengliang Pan and Hong Zhang,*On a curve expanding flow with a non-local term*, Commun. Contemp. Math.**12**(2010), no. 5, 815–829. MR**2733199**, 10.1142/S0219199710003981**[16]**Shengliang Pan and Juanna Yang,*On a non-local perimeter-preserving curve evolution problem for convex plane curves*, Manuscripta Math.**127**(2008), no. 4, 469–484. MR**2457190**, 10.1007/s00229-008-0211-x**[17]**G. Sapiro, A. Tannenbaum, Area and length preserving geometric invariant scale-spaces, Pattern Analysis and Machine Intelligence, IEEE Transactions 17 (1995) 67-72.**[18]**Xiao-Liu Wang,*The stability of 𝑚-fold circles in the curve shortening problem*, Manuscripta Math.**134**(2011), no. 3-4, 493–511. MR**2765723**, 10.1007/s00229-010-0410-0**[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**, 10.1017/S0004972710001942

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

**Xiao-Li Chao**

Affiliation:
Department of Mathematics, Southeast University, Nanjing, People’s Republic of China

**Xiao-Ran Ling**

Affiliation:
Department of Mathematics, Southeast University, Nanjing, People’s Republic of China

**Xiao-Liu Wang**

Affiliation:
Department of Mathematics, Southeast University, Nanjing, People’s Republic of China

DOI:
https://doi.org/10.1090/S0002-9939-2012-11745-9

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