The Gradient Estimation Of The Neumann Problem Of A Class Of Elliptic Equations

In the theoretical study of second-order elliptic partial differential equations,the existence of solutions to boundary value problems is one of the most important issues.At present,Neumann problem is an important research content of boundary value problems of second-order elliptic partial differential equations.Obtaining a priori estimation of the solutions of boundary value problems is the key to studying the existence of solutions of boundary value problems.This article mainly studies the gradient estimation of the Neumann problem of the following two equations:AndThe proof is divided into three cases according to the area where the point is located:Case 1:If x0 ?(?)?,|Du|(x0)is bounded by the Hopf lemma;Case 2:If x0 ?(?)??0??,it can be attributed to internal gradient estimation;Case 3:Ifx0 ? ??0,the principle of maximum value can be used to prove that|Du|(x0)is bounded.In the proof of case 3,by introducing a special frame,constructing an auxiliary function,and using the principle of extreme value,the gradient estimate of the equation solution at the near edge is obtained.Combining the results of the three cases,sum up the upper bound of |Du|(x0),and discuss the gradient estimation of the equation solution.