Studies On The Dirichlet Problem For Mixed Hessian Type Equation On Riemannian Manifolds

Hessian equations are an important class of fully nonlinear elliptic partial differential equations,and their Dirichlet problems have been studied extensively and deeply.In this article,we consider the Dirichlet problem for a class of mixed Hessian equations,and the existence of the solution to this problem is obtained under the assumption on the existence of a subsolution.The main body of this thesis is organized as follows.In Chapter 1,we briefly introduce the research background and the research status of the mixed Hessian equations,as well as the main contents and conclusions of this thesis;In Chapter 2,we give the fundamental knowledge related to the research of this paper,including the definition and properties of basic symmetric polynomials,as well as some formulas and notations;In Chapter 3,we obtain the C1 and C2 estimates for solutions to the mixed Hessian equation;In Chapter 4,combing Schauder theory and Evans-Krylov theorem,we use the continuity method to prove the existence of the admissible solutions.