THE GROWTH OF SOLUTIONS OF MONGE-AMPERE EQUATIONS IN HALF SPACES AND ITS APPLICATION

We consider the growth of the convex viscosity solution of the Monge-Ampere equation det D(2)u = 1 outside a bounded domain of the upper half space. We show that if u is a convex quadratic polynomial on the boundary {x(n) = 0} and there exists some epsilon > 0 such that u = O(|x|(3-epsilon)) at infinity, then u = O(|x|(2)) at infinity. As an application, we improve the asymptotic result at infinity for viscosity solutions of Monge-Ampere equations in half spaces of Jia, Li and Li ['Asymptotic behavior at infinity of solutions of Monge-Ampere equations in half spaces', J. Differential Equations 269(1) (2020), 326-348].