Gradient Estimation Of Neumann Boundary Value Problems For A Class Of Mean Curvature Equations

The boundary value problem of second-order elliptic partial differential equations has always been one of the most concerned problems,including dirichlet problem has been solved by Serrin,etc,but Neumann problem is still trying to solve the problem,especially the mean curvature equation of Neumann boundary problem of gradient estimation,the present study the results of the very few,Xu Jinju had studied Neumann boundary gradient estimates for mean curvature equation based on the bounded solutions.Using the research techniques of Spruck,Lieberman and Xu-Jia Wang,we obtain the existence of Neumann problems for this kind of mean curvature equation.The gradient estimation of the solution to Neumann boundary problem of mean curvature equation is solved by maximum principle.The paper is mainly composed of two parts:In the first part.We prof the gradient of a Laplace equation's Neumann boundary problem estimation by means of the maximum principle and Hopf lemma,which is divided into three situations according to the region of the point:Casel:u is the maximum value,which is reduced to the gradient interior estimation in(?)??0??.Case2:If u is the maximum,then|Du(x0)| is bounded according to Hopf lemma in(?)?.Case3:If u is the maximum,then we can prove boundedness by maximum principle in ??0.In the case of different premises,we use different methods obtaining the final conclusion.The second part is mainly by means of the curved surface maximum principle and the moving frame,The Neumann boundary problem of gradient estimates for we obtain a kind of mean curvature equation.This equation is discussed by the three kinds of circumstances.But it is different with above.we use moving frame on curved surface solving gradient estimates,which simplifies the calculation.