A non-local area preserving curve flow

Length-preserving evolution of non-simple symmetric plane curves

“…To obtain the regularity estimates, one may refer to the classical theory of parabolic equations (see or ). Some references can also be found in . Here, we apply the maximum principle to do this, and the main idea is from .…”

On a length preserving curve flow

“…It is interesting to study curve flow which preserve some geometry quantity, such as the area of the region bounded by the curve. For this, one may see [13] for a recent study. The main result of this paper is the following theorem.…”

Evolving convex curves to constant-width ones by a perimeter-preserving flow

“…Such problems arise in many fields, such as image processing [Cao 2003], phase transitions [Gurtin 1993], etc. In fact, the above evolution problem has been studied extensively, for example, for the popular curve-shortening flow [Gage 1984;Gage and Hamilton 1986;Grayson 1987], the area-preserving flows [Gage 1986;Mao et al 2013;Ma and Cheng 2014], the perimeter-preserving flows [Pan and Yang 2008;Ma and Zhu 2012] and in other related research [Angenent 1991;Chow and Tsai 1996;Andrews 1998;Urbas 1999;Chao et al 2013]. One can find more background material in the book [Chou and Zhu 2001].…”

“…If a smooth simple closed curve evolves under the curve shortening flow then it converges to a round point (see [Gage 1984;Gage and Hamilton 1986;Grayson 1987]). In the cases of nonlocal flows for convex curves, the limiting curves are finite circles (see [Gage 1986;Jiang and Pan 2008;Pan and Zhang 2010;Ma and Cheng 2014]). Forming a striking contrast to these researches, although in the present case the evolving curve keeps its convexity and becomes more and more circular, the limiting curve of the flow is only of constant width rather than being a circle.…”