The asymptotic behavior of viscosity solutions of Monge–Ampère equations in half space

Abstract

In this paper we report the asymptotic behavior at infinity of convex viscosity solution of detD2u=1 outside a bounded domain of the upper half space. It is shown that if the solution is a quadratic polynomial plus a logarithmic function at the flat boundary, then it tends to a quadratic polynomial plus a “log” term at infinity, where the “log” term means that it can be controlled by logarithmic function. Meanwhile, more accurate asymptotic behaviors at infinity are acquired.