On A Non-local Curve Shortening Flow In The Plane

The aim of this paper is to investigate a non-local evolution problem for convex plane curves. Let X(u, t):[a, b]×[0,∞)→R2 be a family of closed planer curves with X(u,0)=X0(u) being a closed, strictly convex curve. Consider the following problem:This flow will decrease the perimeter of the evolving curve and the area it bounds and make the evolving curve more and more circular during the evolution process. And finally, as the time t goes to infinity, the limiting curve will be a finite circle in the "C∞" metric.