Monge-Amp(?)re Type Equations With Neumann Boundary Conditions On Riemannian Manifolds

The Monge-Amp(?)re type equation is a very important type of fully nonlinear partial differential equation.It originates from the problem of optimal transporta-tion,and is also widely used in affine geometry,geometric optics,conformal geom-etry and other problems.In ?,the general form of Monge-Amp(?)re type equation is det[D2u-A(x,u,Du)]=B(x,u,Du),when A?0,the Monge-Amp(?)re type equa-tion degenerates into the classic Monge-Amp(?)re equation.The Monge-Amp(?)re equa-tion was discovered by the French mathematicians G.Monge(1784)and A.M.Amp(?)re(1820).Such equations are widely used in geometry(e.g.Minkowski problem,e-quidistant embedding problem);physics(e.g.Mass transportation problem,transonic problem).In recent years,there have been more systematic research results on the Dirichlet boundary value problem of elliptic Monge-Amp(?)re type equations in Euclidean space.For its Neumann boundary value problem,the global regularity of classical solutions was obtained by the continuity method by Lions-Trudinger-Urbas[17].Compared with the Dirichlet problem,the estimation of the second derivative of the boundary of the Neumann problem is more complicated.In fact,Dirichlet boundary conditions provide all the information in the tangential direction,while Neumann boundary value condi-tions only give the derivative information in the normal direction.On the one hand,how to convert the derivative of the tangential direction to the normal direction derivative es-timation becomes difficult when the boundary of the region is not strictly convex.On the other hand,the construction of the auxiliary function is more complicated when the method is estimated.Therefore,the Neumann boundary value problem is very different from the Dirichlet boundary value problem.Combined with its application background,the research of this kind of problem has theoretical value and practical significance.In this paper,we consider the global regularity for Monge-Amp(?)re type equations with the Neumann boundary conditions on Riemannian manifolds,and extend the main conclusions in the Euclidean flat space to curved spaces.This article is mainly divided into four parts,and its structure is as follows:In the first part,we give the research background,research status and the main content of this paper.In the second part,some preliminary conclusions are given,such as the comparison theorem for Neumann problem(1.1)and(1.2),maximum modulus and gradient estimation.In the third part,we get the global C2 estimates and the boundary estimates,then the main theorem 1.1 is proved.Finally,in the fourth part,we completed the proof of theorem 1.2.