A Liouville theorem for the Neumann problem of the Monge-Ampere equation

Abstract

In this paper, we study the Neumann problem of Monge-Amp\`ere equations in Semi-space. For two dimensional case, we prove that its viscosity convex solutions must be a quadratic polynomial. When the space dimension $n\geq 3$, we show that the conclusion still holds if either the boundary value is zero or the viscosity convex solutions restricted on some $n-2$ dimensional subspace is bounded from above by a quadratic function.