The Evolution Of Convex Plane Curves Driven By Anisotropic Curvature Flows

In this paper,we study the evolution of two types of curvature flows in the plane,which arise from the phase transition models.Firstly,we study the evolution of immersed locally convex plane curves driven by anisotropic flow with inner normal velocity V=1/αψ（θ）κ^αforα<0 orα>1,whereθ∈[0,2mπ]is tangential angle at the point on evolving curves.For-1≤α<0,we show the flow exists globally and the rescaled flow has a full-time convergence.Forα<-1 orα>1,we show only type-I singularity arises in the flow,and the rescaled flow has subsequential convergence,i.e.,for any time sequence,there is a time subsequence along which the rescaled curvature of evolving curves converges to a limit function?furthermore,if the anisotropic functionψand the initial curve both have some symmetric structure,the subsequential convergence could be refined to be full-time convergence.Secondly,we study an area-preserving anisotropic curvature flow with inner normal ve-locity V=（κ_σ-（（?）_γκ_σds）/（L（t）））,where κ_σ=(σ^′′（θ）+σ（θ）)κ（θ,t）is the anisotropic curvature.We show that for any convex closed curve,the evolving curves converge to the boundary of the Wulff shape defined by anisotropic functionσ（θ）as time goes to infinity.