The Oblique Boundary Value Problem For Monge-ampère Equations

The study of the Monge-Ampere equations det D2u=f(x)in Ωis an fundamental problems in the field of nonlinear partial differential equations and geometric analysis,etc.The global regularity theory for convex solutions u to the Dirichlet problems and the nature boundary value problems for Monge-Ampere equations are well known,while only higher-order regularity results are available for the oblique problems.In this paper,we study the oblique problems for the Monge-Ampere equations det D2u=f(x)in Ω,Dβu=b(x)on ?Ω.Here,Ω? Rn is bounded convex domain,and β is a oblique vector field on ?Ω.We give a suitable definition of convex viscosity solution to the oblique problem and establish several theories.Which includes the comparison principle,the geometric interpretation of the oblique derivatives,the global Lipschitz regularity,the compactness theorem,and the existence of solutions to the Robin problem via the Perron method.Inspired by the Schauder estimates of Dirichlet problems,and the nature boundary value problems from optimal transport,we study the level sets with base point x0 ∈?Ω,We introduce the Good Shape properties for the level sets,and give the uniformly compactness for the local problems of normalization family at x0 under general conditions.We shows that the limit model of these local problems corresponds to the following Liouville problem below and and proving the corresponding Liouville theorem.Using the perturbation method,we obtain the global W2,1+ε,W2,p and Schauder estimate,and generate them to the case n≥3 under the quadratic growth assumption on boundary u(x)-u(x0)-▽u(x0)·(x-x0)≤C0|x-x0|2,?x ∈?Ω.