Higher order expansion at infinity of solutions for Monge–Ampère equations in the half space

Abstract

In this paper, we investigate the asymptotic behavior of viscosity solutions for Monge–Ampère equations in the half space with a Dirichlet boundary condition on the flat boundary. Via the Kelvin transform, we characterize the asymptotic remainders by a single function near the origin. Such a function is smooth in the neighborhood of the origin in even dimension, but only $C^{n-1,\alpha }$ $(0<\alpha <1)$ in odd dimension.