On A Non-local Curve Expanding Evolution Problem In The Plane

The aim of this paper is to investigate a new curve evolution problem in the plane. Let X(u, t) : [a, b]×[0,∞)â†'R² be a family of closed planar curves with X(u,0) = X₀(u) being a positively oriented, closed, strictly convex curve. Consider the following evolution problem:where k = k(u, t) is the signed curvature of the evolving curve, L=L(t) and A=A{t) are respectively the length of the curve and the area it bounds at time t and TV = N(u, t) the unit inward pointing normal vector along the curve.This curve evolution problem is a curve expanding evolution problem which will increase both the length of the evolving curve and the area it bounds and make the evolving curve more and more circular during the evolution process. And the limiting shape of the evolving curve will be a finite circle (i.e., a circle with finite radius) as the time t goes to infinity.