Interior Estimates For Hessian Equations On Riemannian Manifolds

The Hessian equation,especially its special case,the Monge-Ampère equation,is a class of fully nonlinear partial differential equations that deserves attention.It is not only valuable in the theory of partial differential equations,but also has many applications in geometric problems and optimal transportation problems.As we know,in the theory of partial differential equations,the a priori estimate is crucial for the study of the existence and regularity of the solution.This thesis focuses on the interior estimate of the second derivatives of the general Hessian type equations on Riemannian manifolds.In particular,we obtain a second-order interior estimate for admissible solutions in some special cases,with nearly optimal conditions.First of all,we introduce the development background of the Hessian equation,the current research status and the background knowledge of the related Riemannian manifold.Secondly,the related definitions of the k-Hessian equation and the property theorem of the solution are introduced as the theoretical basis.Thirdly,the MTW condition is introduced in Chapter 3.This condition is proposed by Ma,Trudinger and Wang when studying the regularity of the optimal transportation problem.It is widely used in the study of partial differential equation theory and optimal transportation problems.Finally,the second-order interior estimates for admissible solutions of the elliptic and parabolic k-Hessian equations is established under the MTW condition.