Solutions Of Curvature Flow Equation With Neumann Boundary Conditions

In this paper,we first study the curvature flow equation with Neumann boundary conditions on[0,1],we will obtain the estimate of solutions and prove convergence.Then we will prove the existence and uniqueness of solutions of the mean curvature equation with Neumann boundary value when the domain extended to a bounded domain in Rn.The section arrangement is as follows:In section 1,this part mainly makes a brief introduction to the historical development of the problem studied in this paper.In section 2,we summarize relevant symbols and the required knowledge.In section 3,we first derive ut estimate,according to the mean value theorem,we can get C0 estimate,and then we obtain C1 estimate with the help of a auxiliary function.In section 4,we mainly prove the convergence of solutions.In section 5,according to Schauder's theorem and Maximum principle,we will prove the existence and uniqueness of solutions of mean curvature equations with Neumann boundary conditions on a strictly convex bounded domain.The main conclusions of this paper are as follows:Theorem 1 Assume ?=[0,1],??R,f is a smooth function on ?×R.u(x,t)is a solution to this problem,where f(x,?)satisfies(?)?-? with ??0.then for t?[0,T],we have|ux(x,t)|?C,(1.2)where C=C(T,?,?,|f|C0(?×R),|Dxf|C0(?)).Proposition 1 The conditions are same as theorem 1,then parabolic problem has a smooth solution u=u(x,t).Proposition 2 Assume ?=[0,1],??R.then there exists a unique ??R and w?C?(?)satisfying what's more,the solution w is unique up to a constant.Theorem 2 Let ?=[0,1],??5.u(x,t)is a solution of this problem,then we can obtain where w is a suitable solution to(1.4).Theorem 3 Assume ? is a bounded and strictly convex domain in Rn(n?2),(?)? is smooth,?>0,f?C(?)satisfies then for ?(x)?C3(?),there exists a unique ??R and w?C2,?(?)satisfying where v is an inward unit normal vector to(?)?,??(0,1),and the solution w is unique up to a constant.