Asymptotic behavior at infinity of solutions of Monge-Ampere equations in half spaces

It is proved that any convex viscosity solution of det D-u(2) = 1 outside a bounded domain of the half space is asymptotic to a quadratic polynomial at infinity under reasonable assumptions, where the asymptotic rate is the same as the Poisson kernel of the half space. Consequently, it follows the Liouville type theorem on Monge-Ampere equation in the half space. Meanwhile, it is established the existence theorem for the Dirichlet problem with prescribed asymptotic behavior at infinity. (C) 2019 Elsevier Inc. All rights reserved.