Gradient Estimation Of Solutions For The Second Boundary Value Problem Of Mean Curvature Equation

The second boundary value problem of mean curvature equation is one of the important boundary value problems in partial differential equations,and the priori estimation of the solution is the key to study the existence of the solution.In this paper,we study the gradient estimation of the solutions of the mean curvature equation as follows,#12 here the second boundary value is mainly emphasized,and the first boundary value has been obtained.To some extent,we generalize xu jinju's gradient estimation of the solution of this equation when f only depends on x and u.By using the skills of Xu jinju and Simon,an effective auxiliary function is constructed,and restrict the conditions of f to obtain the conditions of f.The proof divide into three cases:Case 1:if x0?(?)?,we can prove|Du|(x0)is bounded by the Hopf lemma;Case 2:if x0?(?)??0??,it can be reduced to internal gradient estimation;Case 3:if x0???0,we can prove |Du|(x0)is bounded by the maximum principle.Combining the results of the three cases,the upper bound of |Du| is obtained,and the gradient estimation of the solution of the equation in question is given.In particular,we give the range of ? for a class of common second boundary value problems of mean curvature equation with gradient estimation as a verification of the general result:(?)...