Estimation Of The Gradient Of The Solution Of A Quasi-linear Equation

In partial differential equations,it is very important to study the estimation of solutions,and the prior estimation of solutions is the key to study the properties of solutions.In this paper,Bernstein's method is used to study and obtain the following equations.(?)(1)(?)(2)(?)(3) gradient estimation of the solution.When equation(1)is an elliptic equation,the internal gradient estimation of the solution of the equation is obtained by Bernstein method.Equation(2)is a parabolic equation.The gradient estimation of the solution of Neumann problem of curvature equation is obtained by differential method and Bernstein method.Equation(3)is an average curvature flow equation.By selecting a proper test function,the internal gradient estimation of the solution of the average curvature flow equation is obtained by using the maximum principle.The full text is divided into four parts.The first part briefly introduces the research background of equations(1),(2)and(3).In the second part,the prior estimation of the solution of the equation is discussed.Generally,the estimation of the solution is discussed first.Taking a class of linear parabolic operators as an example,the estimation of the solution is given,and then the gradient estimation of the solution is made.In the third part,the internal gradient estimation of the solution of equation(1)is given.In the fourth part,the gradient estimation of the solution of equation(2)Neumann problem and the internal gradient estimation of the solution of equation(3)are established.