Several Existence Results Of Hessian Equations

Hessian equations are important classes of fully nonlinear partial differential equa-tions,and they arise widely in differential geometry,complex geometry and theory of convex bodies.In this paper,we mainly consider several existence results for Hessian equations.By establishing a priori estimates,we use the method of continuity to prove the existence of smooth solutions for Hessian equations.In the first part,we consider the existence problem for the ?-Hessian equation on compact Hermitian manifolds.According to the method of continuity and Evans-Krylov theory,the solvability can be reduced to prove C2 a prior estimates.We obtain the C0 estimate by some basic inequlities of the ?-Hessian operator and the induction argument.By choosing a suitable auxiliary function,we can show the complex Hessian estimate of Hou-Ma-Wu type.Based on this estimate,the gradient estimate follows similarly as the Kahler case.In the second part,we consider the Neumann problem for elliptic Hessian quotient equatons,which are natural generalization of ?-Hessian equations.We firstly derive the gradient estimates and the second order derivative estimate under general structural conditions.As an application,we prove that there exists a unique smooth solution solv-ing the Neumann problem for a class of Hessian quotient equations on smooth bounded convex and strictly(?-1)-convex domain in Rn.In the third part,we consider the Neumann problem for parabolic ?-Hessian equa-tions and Hessian quotient equations and the related translating solution problem.Un-der some natural structural conditions,we can establish ut estimates and C0 estimates which are both independent of the time.Modifying the proof in the elliptic case,we then derive the gradient estimates and the second order derivative esimate.Then,we can show the longtime existence and convergence.Moreover,the solution converges to that of the related eliptic equation.At last,we consider the translating solution prob-lem for the ?-Hessian equation.The key step is to derive the gradient estimate which is independent of the time and the C0 norm of the solution.