The Neumann Boundary Value Problem For Hessian Type Equations

The Hessian type equations have important theoretical significance and application in differential geometry,complex analysis,fully nonlinear partial differential equation theory.If A = 0,the Hessian type equations degenerate into the k-Hessian equations.The existence,uniqueness and regularity of solutions are important properties of the Hessian type equations.In addition,the further study of the equations can also enrich fully nonlinear partial differential equation theory.In this paper,for the Neumann boundary value problem of the Hessian type e-quations,by using the property of function at the maximum value point,we obtain the inner gradient estimates of the Hessian type equations.Then by constructing differen-t auxiliary functions,we get the global estimates of the Hessian type equations with Neumann boundary value problem according to boundary gradient estimates,near the boundary gradient estimates and inner gradient estimates.In addition,we give two kinds of proof for the gradient estimates with the oblique boundary value problems of the Laplace equations by constructing different auxiliary functions.This dissertation includes four sections.In section 1,we introduce the background and the main results of Laplace equations and the Hessian type equations.In section 2,we give some notations.In section 3,the gradient estimates of Laplace equations with oblique boundary value problem have been proved.In section 4,the gradient estimates of the Hessian type equations with Neumann boundary value problem have been obtained.