Some Research On Geometric Partial Differential Equations

This thesis contains two parts,three chapters.The first part consists of the sec-ond and third chapters,which investigate two approaches to do the apriori estimates in second order elliptic equations——subsolution method and P-function method,re-spectively.In the second chapter,taking the Dirichlet problem for the quaternionic Monge-Ampere equation for example,we derive apriori estimates for the solution on a general domain through the subsolution method,which gives us an existence theorem.Therefore,we can answer the fourth question proposed by S.Alesker in[4].The third chapter is mainly about the Dirichlet problem for the constant mean curvature equation in 3-dimensional Minkowski space.Based on the uniqueness of the critical point,we can improve the gradient estimate in[14]by the P-function method,and get a height es-timate for the solutions.At last,in the second part(i.e.the fourth chapter),we focus on the relationship between the immersed hypersurfaces in hyperbolic space and the con-formal metrics over the domains in sphere,and generalize the global correspondence in the[17]to the 2-dimension.Moreover,as applications,we obtain a Liouville type theorem and a new stronger Bernstein type theorem.