The Properties Of The Solution To The Neumann Boundary Value Conditional Mean Curvature Equation

Firstly,this paper studies the estimation and convergence of solutions of curvature equation in R1.Then discussed the properties of the solutions of the mean curvature equation with specific conditions in Rn.The content of this paper is arranged as follows:In section 1,we mainly introduce the development background of "mean curvature" and the theoretical source of this paper.In section 2,we list the relevant symbols in this paper and Preliminary knowledge,prepare for the next chapter of the proof.In section 3,we give C0 and C1 estimates and ut estimates of the solution of the equation,Prepare for the proof of the next chapter,in this part,we use a special auxiliary function.In section 4,we discuss the longtime existence of solutions under Neumann boundary conditions.In section 5,we will use Schauder theory to study the existence of solutions of mean curvature equations under Neumann boundary conditions in Rn.The main results of this paper are as follows:Theorem 1 Let ?=[0,1],f are function defined on ?×R,u(x,t)is the solution of the following equation,where f(x,u)satisfies that(?)with k?0,and then for t ?(0,T)we have the estimate|Dxu(x,t)?C,(1.2)where C=C(T,k,|f|C0(?×R),|Dxf|C0(?)).Corollary 1 Under the same conditions as described in Theorem 1,the following equa-tion has a smooth solution u=u(x,t).Theorem 2 Let ?=[0,1],u(x,t)is the solution of the mean curvature flow equation as follows,where a,b is constant.Then u(x,t)will converge to ?t-?,where ? is a solution of equation(1,5)as follows,Theorem 3 Let ? is strictly convex bounded domain in Rn,where n?2,and the boundary is smooth.For any(?),there exists a unique solution of ??R and(?)for the following equation where v is an inward normal vector to(?),the solution w is unique up to a constant.