Gradient Estimation Of Newman Boundary Value Problem For Hessian Equation

In the theoretical study of second order elliptic partial differential equations,the study of the existence of solutions to boundary value problems is one of the most important problems,and the boundary value problems are mainly Newman boundary value problems and Dirichlet boundary value problems,in which the boundary value problems are Newman boundary value problem and Dirichlet boundary value problem.The Hessian equation is a class of completely nonlinear second order elliptic partial differential equations with an important position.The Hessian equation with Dirichlet boundary value problem has been fully studied.The results related to the existence and regularity of the solutions have also been recognized in the domain and have been fully applied.For the Newman problem of the Hessian equation,there is a certain basis for the estimation of the boundary gradient.This paper is based on this basis.On the basis of constructing auxiliary functions,applying the maximum principle and Hopf Lemma,and limiting the boundary conditions of the equation,the gradient estimation of the solution of the slightly complex Hessian equation is carried out,and the conditions satisfied by the equation are divided into three cases.(1)reaching the extremum x0 ?(?)?,(2)reaching the extremum x0 ???0,(3)reaching the extremum X0 ?(?)??0??,Discuss three kinds of situations separately,for the first case,we will use Hopf lemma to prove the auxiliary function L(x);In the second case,for a sufficiently small constant ?0>0,we will use the maximum principle to prove and obtain upper bound of L(x);For the third case,we will use the theorem of intra-gradient estimation to prove and obtain the upper bound of |Du|(x),so that we can get a better result,upper bound ofL(x).Proof in the above three cases,we got a function,upper bound of L(x),from this,we obtain the gradient estimate of the solution.