A New Proof For The Regularity Of Monge-Ampère Type Equations

In this paper,we mainly study the Monge-Ampère equation:det D2u = f(x,u,▽u),in Ω,where Ω(?)Rn is a bounded domain,u:Ω→R is a convex function,f:Ω×R×Rn→R+ is a smooth function.The main idea of the paper is that we use the Green function to proof the interior regularity of solutions to the Monge-Ampère equation.The proof also applies to the complex Monge-Ampère equation and the k-Hessian equation.In the section 1,we introduce the background of the problem and present the main result of the paper.In the section 2,we assume that sup |D2u(x)|≤A.By establishing the estimate of Green function G and uξξ.we obtain the mean value inequality for uξξ.Then,we prove that D2uk are uniformly continuous for any sequence of smooth,convex solutions uk.As a result,we have u ∈ C2,α(Ω).In the section 3,for the complex Monge-Ampère equation and the k-Hessian equation,we assume that Δu≤C.Then,we use the idea in the section 2 to prove the result.