On a planar area-preserving curvature flow

Chao, Xiao-Li; Ling, Xiao-Ran; Wang, Xiao-Liu

doi:10.1090/S0002-9939-2012-11745-9

On a planar area-preserving curvature flow
HTML articles powered by AMS MathViewer

by Xiao-Li Chao, Xiao-Ran Ling and Xiao-Liu Wang

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

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

Published electronically: September 19, 2012

PDF | Request permission

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.

References

Ben Andrews, Evolving convex curves, Calc. Var. Partial Differential Equations 7 (1998), no. 4, 315–371. MR 1660843, DOI 10.1007/s005260050111
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, DOI 10.2307/1971426
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, DOI 10.1007/s00030-011-0123-4
Kai-Seng Chou and Xi-Ping Zhu, The curve shortening problem, Chapman & Hall/CRC, Boca Raton, FL, 2001. MR 1888641, DOI 10.1201/9781420035704
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, DOI 10.1090/conm/051/848933
M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom. 23 (1986), no. 1, 69–96. MR 840401, DOI 10.4310/jdg/1214439902
Matthew A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom. 26 (1987), no. 2, 285–314. MR 906392
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, DOI 10.4310/CAG.2008.v16.n1.a1
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, DOI 10.1016/j.jde.2008.04.007
Y.C. Lin, D.H. Tsai, Nonlocal flow of convex plane curves and isoperimetric inequalities, Arxiv: 1005.0438v1, 2010.
Y.C. Lin, D.H. Tsai, On a general linear nonlocal curvature flow of convex plane curves, Arxiv: 1012.0114v1, 2010.
Y.C. Lin, D.H. Tsai, Application of Andrews and Green-Osher inequalities to nonlocal flow of convex plane curves, preprint, 2012.
L. Ma, L. Cheng, A non-local area preserving curve flow, Arxiv: 0907.1430v1, 2009.
Li Ma and Anqiang Zhu, On a length preserving curve flow, Monatsh. Math. 165 (2012), no. 1, 57–78. MR 2886123, DOI 10.1007/s00605-011-0302-8
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, DOI 10.1142/S0219199710003981
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, DOI 10.1007/s00229-008-0211-x
G. Sapiro, A. Tannenbaum, Area and length preserving geometric invariant scale-spaces, Pattern Analysis and Machine Intelligence, IEEE Transactions 17 (1995) 67–72.
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, DOI 10.1007/s00229-010-0410-0
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, DOI 10.1017/S0004972710001942