The Obstacle Problem For A Class Of Hessian Equations On Riemannian Manifolds

The obstacle problem for Hessian equations has important applications in differential geometry and the problem originated in the study of hypersurface with an upper(lower) obstacle on Euclidean space under certain conditions.This dissertation studies the existence and regularity of solutions for a class of Hessian equations on Riemann manifolds. In order to study the existence and regularity of the solution to the obstacle problem, we use the penalty function to convert it into singular perturbation equation. Then through solving the singularly perturbation equation, we can get the existence and regularity of the allowable solution to the obstacle problem for Hessian equations. For singular perturbation equations, a priori estimate can guarantee the existence and the regularity for the allowable solution, thus a priori estimate for the allowable solution becomes very important to the obstacle problem of Hessian equations.Firstly, we use the principle of maximum to get the priori0 C estimate of allowable solutions on a n-dimensional compact Riemannian manifold with boundary.Secondly, we use the principle of maximum to get a prior1 C estimate on the boundary with the conclusion of a priori0 C estimates; By selecting the appropriate test function, we can obtain a priori C1 estimate inside the manifold. Thus we get the prior1 C estimate to the obstacle problem of Hessian equations.Finally, we construct the proper gate function by using the distance function to the boundary, and then we get the priori C2 estimates of allowable solutions on the manifold boundary. By selecting the appropriate test function, we can obtain a priori2 C estimate inside the manifold with the principle of maximum. Thus we get the prior2 C estimate to the obstacle problem of Hessian equations.