On The Dirichlet Problem For Special Lagrangian Curvature Potential Equations

In recent years,like many models for geometry and physics problems,the Existence and Uniqueness of a solution are the most important research topics in nonlinear elliptic equa-tions.In this thesis,we study the properties of solution to the second order fully nonlinear elliptic equations,using the special Lagrangian curvature potential equations.Specifically,we study the Dirichlet Problem for the Lagrange curvature potential equation in the re-al bounded strictly domain,Using the properties of special Lagrangian curvature potential operator and convex domain to construct an effective auxiliary function to obtain a priori estimate,then we see the Existence and Uniqueness of real Lagrangian curvature potential equations with Dirichlet boundary in the real domain by Laray-shaucder Degree theorem.The specific contents are as follows:The first chapter briefly introduces the topic selection back ground,current research status,main work and results of this thesis.The second chapter gives some properties of nonlinear elliptic equation,some definitions and properties of Lagrange curvature potential operator.The third chapter,first we prove that C~0 estimates by using the maximum principle,then we construct relevant auxiliary function to proveC~1estimates andC~2estimates.The key difficulty of this thesis is to proveC~2estimates.The fourth chapter,On the basis of the third chapter,we establish the existence and uniqueness of solutions by Laray-Shaucder degree theorem.The fifth chapter,we summarize and prospect this thesis.