Neumann Problems For Hessian Equations And Geometric Applications

Hessian equations are important kind of fully nonlinear second order elliptic pari-tial differential equations. Boundary values problems of these equations are mainly di-vided into two categories:Dirichlet problems and Neumann problems. Dirichlet prob-lems have studied for a long time. Its existence and regularity results are well known in this area. While the Neumann problems are still unkown for many decades. In this paper, we mainly study Neumann boundary problems of Hessian equations. We com-pletely solve Neumann problems for Hessian equaions which is also a conjecture of Trudinger. The main contribution of this paper is that we construct a new auxiliary function and use maximum principle to prove second order derivative estimates on the boundary then existence results. Moreover, we use the solutions of these boundary problems to obtian a new proof of some geometric inequalities.