An Area-Preserving Flow for Closed Convex Plane Curves

Abstract

Motivated by Gage [On an area-preserving evolution equation for plane curves, in Nonlinear Problems in Geometry, ed. D. M. DeTurck, Contemporary Mathematics, Vol. 51 (American Mathematical Society, Providence, RI, 1986), pp. 51–62] and Ma–Cheng [A non-local area preserving curve flow, preprint (2009), arXiv:0907.1430v2, [math.DG]], in this paper, an area-preserving flow for convex plane curves is presented. This flow will decrease the perimeter of the evolving curve and make the curve more and more circular during the evolution process. And finally, as t goes to infinity, the limiting curve will be a finite circle in the C∞ metric.

A quasilinear parabolic problem with a source term and a nonlocal absorption

“…It can be inferred that u(x, t) ≥ e −CI0 u(x 1 (t), t) = aI −1 0 e −CI0 , ∀ (x, t) ∈ [0, a] × [0, ∞), and (17) follows immediately with C 1 = aI −1 0 e −CI0 . When p > 2 and a < π, similar as above, one can find that for every t > 0, there is an x 1 (t) ∈ [0, a], such that…”

“…Note that (17) illustrates that the first equation of problem (1) is uniformly parabolic. Joining (16) with Lemma 3.4, one can employ the classical theory of parabolic type equation to derive the regularity estimate (18), please refer to [14,15] for more details.…”

“…. Moreover, as long as u exists, it can be shown that u is positive (see Lemma 2.1) and its derivatives u x has regularity estimates ( [17]).…”

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].…”

Untitled

“…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].…”