A Study On Some Properties Of Monge-Ampère Type Equations

This thesis investigates some properties of Monge-Ampère type equations,one is the symmetry of solutions to a class of Monge-Ampère type equations,and the other is the boundary H (?)lder estimate for a class of nonlinear singular elliptic equations which includes Monge-Amp?ere type equations as a special case.We first consider the symmetry of solutions to a class of Monge-Ampère type equations from a few geometric problems.Under a proper structure assumptions,we use a new transform to analyze the asymptotic behavior of the solutions near the infinity.By this and a moving plane method,we prove the radially symmetry of the convex solutions.We then study the Dirichlet problem of a class of nonlinear elliptic equations which includes Monge-Amp?ere equation,K-Hessian equation and the usual linear elliptic equation.This problem becomes singular near the boundary of the domain.By carefully constructing sub-solutions and analysis techniques,we obtain the boundary H (?)lder estimate for the convex solution to the problem when the domain satisfies exterior sphere condition.