A Study On The Viscosity Solutions Of The Equation With A Three Times Infinity Laplacian

In recent years,the research on infinite Laplacian equation has made a lot of achievements.This kind of equations originated from the study of L?variational problem.The infinity Laplacian equations have been widely used in game theory,shape deformation,optimal transport,image processing,elastic mechanics,physics and other important research fields.The discussion on the existence,uniqueness and interior estimate of viscosity solutions of various infinity Laplace equations attractes the attention of many scholars.The main contents are given as follows:In the first chapter,we introduce the research background and development trend of parabolic equation with infinite Laplace operator,analyze and summarize the main work and innovation points of this paper.In the second chapter,we study the Dirichlet initial boundary value problem for a class of third-order homogeneous parabolic equation,prove the comparison principle of the viscosity solution by Jensen method,and obtain the existence of the positive viscosity solution by using asymptotic behavior and a uniform estimate.In the third chapter,we discuss the uniqueness and asymptotic behavior of the positive viscosity solution for the Dirichlet initial boundary value problem with 3-homogeneous parabolic infinite Laplace inhomogeneous equation,and also infer the existence of the nonnegative viscosity solution.In the fourth chapter,we study a class of inhomogeneous infinity Laplacian equation,prove the comparison principle of the positive viscosity solution of the initial boundary value problem of the equation,obtain the uniqueness of the solution,and obtain the existence result of the viscosity solution of the equation by the uniform estimates.In the fifth chapter,we study a kind of?-biased infinity Laplacian equation.By Bernstein's method and smooth solution,we establish Lipschitz's estimate of the viscosity solution of the equation.In the last chapter,we summarize the research contents of this paper and give some problems which could be studied in the future.