Estimate Of Solutions To Mean Curvature Flow

The aim of this dissertation is to study the mean curvature flow with two different boundary conditions in Rⁿ.Using Maximum Principle,we obtain the longtime existence for f_z(x,z)?-?.The content of this paper is arranged as follows:In Section 1,we mainly introduce the history of "mean curvature flow" and the main conclusions of this paper.In Section 2,we list the preliminary knowledge,preparing for the proof in next sections.In Section 3,we give ut estimates and gradient estimates of the solution to the equation with Neumann boundary value condition in and obtain the long time existence.In Section 4,we obtain the long-time existence of the solutions to mean curvature flow with general capillary-type boundary condition in Rⁿ.The main results of this paper are as follows:Theorem 1 Assume that ?(?)Rⁿ(n? 2)is a bounded C³ domain.Let v be the inward unit normal vector to(?)?.For fixed T>0,suppose u?C²((?)×[0,T])?C³((?)×[0,T])is the solution of with |u|?M₀,where M₀>0.Assume f(x,u)?C¹((?)×[-M₀,M₀])and ?(x)?C³((?))are given functions which satisfy the following conditions:for some constants L₁,L₂ then for ??(0,1),there exists a unique C^2,?((?)×[0,?))solution to(1.1).Theorem 2 Assume that ?(?)Rⁿ is a strictly convex bounded C³ domain,n? 2,and u?C²((?)×[0,T])?C³(?×[0,T])is the solution of with |u|?M₀(M₀>0).Let ? be an inward unit normal vector to(?)?,q>0,v=(?).Suppose f(x,z)?C¹((?)×[-M₀,M₀])and ??C²((?))are given functions which satisfy the conditions(1.2),(1.3)and then for ??(0,1),there exists a unique C^2,?((?)×[0,?))solution to(1.5).