The Dirichlet Problem For The K-hessian Equations

In differential geometry, complex analysis and applied sciences there is a kind of fully nonlinear partial differential equation, named Monge-Ampere equation which rose in many geometry problems. The k-Hessian equation discussed in this paper is an ex-tension of the Monge-Ampere equation. Furthermore, the k-Hessian equation itself has been playing an important role in differential geometry and some applied sciences. Thus it is of great worth to study the k-Hessian equation. The k-Hessian equation is a kind of very complicated fully nonlinear partial differential equation when k≥2. It is a hard job to study the equation which need a wide knowledge, such as geometry, algebra, analysis, partial differential equations and so on. The purpose of this paper is to introduce some skills and conclusions in studying the Dirichlet problem for the k-Hessian equation.In chapter 1 we first introduce the background and definition of the k-Hessian equation and some corresponding concepts. In chapter 2, we further introduce the k-admissible functions and the ellipticity and concavity of the k-Hessian operators under the k-admissible functions. Chapter 3 contains the existence and regularity of the clas-sical solution of the Dirichlet problem for k-Hessian equation. k-Hessian measures and weak solutions appear in chapter 4 where we also introduce the existence of weak so-lutions as well as the advance of the regularity of non-smooth k-admissible functions. The k-Hessian capacity and uniqueness of the weak solutions of the k-Hessian equation form the final chapter where we also make a further study of the uniqueness of the weak solutions.As a new research field, the k-Hessian equation, still consists of lots of problems such as the regularity and uniqueness of the weak solutions and the existence of the clas-sical solutions under some weaker conditions.