A Priori Estimates Of Solutions Of A Kind Of Hessian Equations

The Hessian equations of elliptic and parabolic type are kinds of fully nonlinear partial differential equations that hard to solve.We now focus on its a priori C2 estimates and regularity.In this thesis,we want to consider the regularity of a wide range of parabolic Hessian equations defined in a compact Riemannian manifold of n dimension to obtain the regularity of solutions of the Dirichlet problem.Based on lots of former study by others,we establish some estimates of a kind of approximation problems.As we all know,a priori estimates of second order is crucial important to us in establishing the existence and regularity of solutions.In the elliptic type,if we establish a seconde order estimates for an admissible solution,under the fundamental structure conditions we immediately get a higher smoothness,the existence of smooth solutions and higher estimates by continuous methods,Evans-Krylov theorem and Schauder theorem.Because of the similarity of elliptic and parabolic type,we consider parabolic problems in the same way though its degeneration.In a priori estimates of approximation problem of second order parabolic equations of this paper,we use an normal assumption to overcome it.In this thesis,we firstly introduce some background and research about Hessian equations.we give some necessary knowledge and an inequality.Besides,we establish gradient estimates about the approximation problem under three different assumptions.Then,we establish C2 global estimates.After that,we give a part of proof of boundary estimates.In the last of this thesis,a conclusion is given.Important tool in this proof is the assumption of strict subsolution.