Non-local Flows Of Closed-convex Plane Curves With Monotonicity Of Area And Length And Its Application

In this thesis,the non-local flows of the closed convex plane curve with the support function in the velocity function are studied,and the monotonicity of the area length of the evolution curve are given under the different values of the non-local term in the velocity.The evolution curve remains the initial closed convexity,and it converges to a finite circle in C∞ norm as t→∞.This result generalizes 2π/L(t)of Pan Shengliang in paper[3]to the interval[πL(t)/(σ+1)L2(t)-2π(2σ+1)A(t),T(t)/2A(t)](-1/2<σ<∞,L(t)A(t)represent the perimeter and area of the evolution curve,respectively),and the scope of the research is more extensive,and the results are richer.As an application of this flows,this thesis presents the strengthened forms of existing geometric inequalities on the plane,including the strengthened form of Ros’s theorem and generalized Chernoff’s inequality on the plane,and provides a new way to prove the inverse isoperimetric inequality conjecture,further,it is given that the equality holds of the geometric inequality of the strengthened form and the inverse isoperimetric inequality if and only if the closed convex plane curve is a geometric depiction of circle or a hexapolynomial curve.