Approximation and Comparison for Motion by Mean Curvature with Intersection Points

Abstract-Consider the motion of a curve in the plane under its mean curvature. It is a very interesting problem to investigate what happens when there are intersection points on the curve at which the mean curvature is singular. In this paper, we study this issue numerically by solving the Allen-Cahn equation and the nonlocal evolution equation with Kac potential. The Allen-Cahn equation is discretized by a monotone scheme, and the nonlocal evolution equation with Kac potential is discretized by the spectral method. Several curves with intersection points under motion by mean curvature are studied. From a simple analysis and our numerical results, we find that which direction to split of the curve at the intersection point depends on the angle of the curve at the point, i.e., it splits in horizontal direction when the angle o/ > 7r/2, in vertical direction when LY < 7r/2, and in either direction when 01 = n/2. @ 2003 Elsevier Ltd. All rights reserved.