Well-posedness Of Solutions To Some Classes Of Nonlinear PDEs Involving Infinity-Laplacian

Infinity Laplacian equations are very important topics that deal with calculus of variations, functional analysis, differential geometry and quasi-linear partial differential equations etc. L∞minimal problem is the source of infinity Laplacian equations. The infinity Laplacian equa-tions have been widely applied in game theory, shape deformation, optimal transport, image processing, elastic mechanics, physics and other fields.The classical form of infinity Laplacian equations is a quasi-linear and highly degenerate partial differential equation:During the last twenty years infinity Laplacian equations have been a subject of intensive studies. This thesis mainly studies four aspects as follows:First, existence and uniqueness of viscosity solutions for Dirichlet problem of some elliptic inhomogeneous infinity Laplacian equation are discussed, and then the properties of solutions near an isolated singularity and the priori estimates of the smooth solutions for some elliptic infinity Laplacian equation are stud-ied; Second, the uniqueness of the positive solutions for the initial-boundary value problem of a homogeneous parabolic infinity Laplacian equation is discussed, a kind of solutions of special form is obtained by super-geometric functions and the asymptotic behavior is also discussed; Third, existence and uniqueness of viscosity solutions for the initial-boundary value problem of a parabolic inhomogeneous infinity Laplacian equation are discussed by the method of reg-ularized equations and standard perturbation theory of viscosity solutions respectively; Finally, the uniqueness of viscosity solutions for the initial-boundary value problem of a parabolic in-homogeneous normalized p-Laplace equation and the asymptotic mean value formula of the solutions in the viscosity sense are studied.Some classes of equations involving infinity Laplace operator are studied in this paper. The main results achieved include:(1) We prove the existence of viscosity solutions for Dirichlet problem of an elliptic inho-mogeneous infinity Laplacian equation by the classical Perron’s method. Then we prove some properties of the solutions near an isolated singularity. Finally, we obtain the priori estimates of the smooth solutions for some elliptic infinity Laplacian equation.(2) We prove the uniqueness of the positive solutions for the initial-boundary value prob-lem of a homogeneous parabolic infinity Laplacian equation. Then we give some special solu-tions by super-geometric functions. Finally we give the asymptotic behavior of the solutions.(3) We prove the comparison principle and uniqueness of viscosity solutions for the initial-boundary value problem of a parabolic inhomogeneous infinity Laplacian equation. Then we prove the existence of the solutions by the uniform estimates.(4) We prove the comparison principle of viscosity solutions for the initial-boundary value problem of a parabolic inhomogeneous normalized p-Laplace equation. Then we give the asymptotic mean value formula of the solutions in the viscosity sense.In this thesis, we mainly use the methods including Perron’s method, barrier functions, a uniform estimate, separation of variables and perturbation method.