On A Non-local Perimeter-preserving Curve Evolution Problem For Convex Plane Curves

As one of the main topics of modern mathematics, " geometry flow " is to study the deformation of geometric object with its applications by tools of geometry and analysis. Since the 1980s, it has always been one of the hot research in the field of geometric analysis, we try to study some relate problems on the background of above researches.The aim of this thesis is to investigate a non-local curve evolution problem in the plane. Let X(u,t) : [a,b]×[0,∞)→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) is the length of the curve at time t and N = N(u, t) the unit inward pointing normal vector along the curve.In this thesis, it is proved that a closed convex plane curve will preserve convexity, and its perimeter,but enlarge the area it bounds during the evolution process. As the time t goes to infinity,it makes the evolving curve more and more circular during the evolution process. And the final shape of the evolving curve will be a circle in the C^∞metric.