The Regularity of a Class of Degenerate Elliptic Monge-Ampere Equations

Abstract

In the present paper the regularity of solutions to Dirichlet problem of degenerate elliptic Monge-Ampère equations is studied. Let Ω⊂R^2 be smooth and convex. Suppose that u∈C^2(Ω) is a solution to the following problem: det(u_{ij}) = K(x) f (x,u,Du) in Ω with u = 0 on ∂Ω. Then u∈C^∞(\bar{Ω}) provided that f (x,u,p) is smooth and positive in \bar{Ω}×R×R^2, K > 0 in Ω and near ∂Ω, K=d^m\tilde{K}, where d is the distance to ∂Ω, m some integer bigger than 1 and \tilde{K} smooth and positive on \bar{Ω}.