A Priori Estimates For Hessian Equations On Riemannian Manifolds

The Hessian equation is a class of fully nonlinear partial differential equation which depends only on the eigenvalues of the Hessian of its solution in form. In this thesis we are mainly concerned with the a priori C2estimates for solutions to Hessian equations of elliptic and parabolic type on Riemannian manifolds. We also consider the regularity for a class of obstacle problem of Hessian equations. Our interests in studying such equations on Riemannian manifolds are from their applications to some geometric problems such as the Minkowski problem and its generalizations, the Alexandrov problem of prescribed curvature measure, the Weyl problem, the k-Yamabe problem and so on. Another moti-vation to study such equations comes from the problem of optimal transportation. The potential of an optimal transportation problem satisfies a Monge-Ampere type equation which is a special case of our equation.As is well known, in the study of fully nonlinear elliptic or parabolic equations, the a priori C2estimates are crucial to establishing the existence and regularity of solutions. Such estimates are also important in applications. For example, in this thesis, we study the regularity for solutions to an obstacle problem of Hessian equations by using these methods.We establish the a priori C2estimates for solutions to the Dirchlet problem of a class of elliptic Hessian equations on a compact Riemannian manifold with boundary under various conditions. The higher order estimates are established by using Evans-Krylov theory and Schauder theory. The existence of smooth solutions is proved by virtue of the method of continuity and degree theory based on these estimates.The a priori Cx,t2,1estimates are established for solutions to the first initial-boundary value problem for a class of Hessian equations of parabolic type on MT=M×(0,T](?) M×R, where M is a compact Riemannian manifold with boundary, under the assumption that there exists a strict subsolution which is used to construct a barrier function.Next, by considering an approximated problem and studying the level hypersurface of smooth convex functions, we prove the existence of C1,1viscosity solutions to the Dirichlet problem for an obstacle problem of Hessian equations on a compact Riemannian manifold with boundary under various conditions. Finally, we consider another obstacle problem for Hessian equations on Riemannian manifolds. The regularity for the greatest solution is studied. We prove that the greatest solution is also the solution to the obstacle problem above under various conditions.We shall construct some smooth or non-smooth subsolutions in some special cases because of the importance of subsolutions in our proof.