The Research On DIRICHLET Problem And Obstacle Problem For K-HESSIAN Equation

This paper describes the research results and the latest progress of the Dirichlet problem and the obstacle problem for the k-Hessian equations. For the k-Hessian equation of Dirichlet problem and the obstacle problem, we focused on two aspects of the work, first we give a proof for the existence and uniqueness of the viscosity solution in the sense of viscosity for the equation and for the obstacle problem of the k-Hessian equation, the existence of the viscosity sub solution has been proved.This article includes two parts. In the first part, we introduce the k-Hessian equations of Dirichlet problems of development and research status. Respectively, from the most basic of classical solution, and later developed the theory of the weak solution and the theory of viscosity solutions has been studied, described the proof of the existence, uniqueness and regularity of classical solutions. In the weak solution theory, the comparison principle, combined with the nature of Dirichlet boundary conditions for the k-Hessian equation, the proof of the theorem has been given, which is used in the second part of the article. The theorem is very useful. In the viscosity solution theory part, the proof of the existence and uniqueness of the viscosity solution for a special class of k-Hessian equations has been given.The second part of the article was described the obstacle problem of the k-Hessian equation. First we introduced the absolute barrier function of the nonlinear second order elliptic partial differential equations, and then argued the general obstacle problem and in convex domain for the obstacle problem of the k-Hessian equation and the establishment of the existence of the greatest vicosity solution in a special non-convex regions of the obstacle problem for the k-Hessian equation and the C^1,1 regularity of the viscosity solutions. And finally by the existence of a special weak solution of the Dirichlet problems for the k-Hessian equation, we have a series of good conclusions of the obstacle problem for the k-Hessian equation.