A Class Of Nonlocal Flows Of Curvature On The Plane

There are three results in this paper.We first investigate a family of non-local closed convex log-type plane curvature flows.The flow takes the velocity function as ln k+α ln L/2π+(1-α)ln(?)(0≤α≤1),where k、A、L represent the curvature,enclosing area and perimeter of the curve respectively.Under this flow the evolution curve remains convex,and the circumference of the curve becomes smaller,the area enclosed by the curve becomes larger.The closed convex curve evenly converges to a fixed circle as time approaches infinity.A larger range of results from Pan Shengliang and Xing Qiaofang were included.The second result of this paper is to obtain a new class of curvature integral inequalities for the case of Wulff through the Green—Osher inequality.In addition,the third result of this paper is that there is a recurrence relationship between the high-order curvature integral inequality of k and the low-order case when the convex function takes F(x)=1/xn in the case of Wulff using the Green-Osher inequality.