Asymptotic behavior at infinity of solutions of Monge-Ampère equations in half spaces (Q1986508)

The authors study the asymptotic behavior at infinity of convex solutions to the Monge-Ampère equation in the half space \[ \det D^2 u=f \text{ in } \mathbb R^n_+, \quad u=p \text{ on } \{x_n=0\},\] where \(n\geq 2\), \(p\) is a quadratic polynomial with \(D^2p>0\), and \(f\) satisfies, for some \(R_0>0\), \[ \text{supp}(f-1)\subset B_{R_0}^+ \text{ and } 0<\inf_{\mathbb R^n_+}f \leq \sup_{\mathbb R^n_+} f<\infty. \] They prove that, if \(u\) has quadratic growth at infinity, i.e., \[ \mu|x|^2 \leq u(x)\leq \mu^{-1}|x|^2 \text{ in } \mathbb R^n_+\setminus B_{R_0}^+ \] for some \(0<\mu\leq\frac{1}{2}\), then \(u\) is asymptotic to a quadratic polynomial at infinity, where the asymptotic rate is the same as the Poisson kernel of the half space. This result extends the result by \textit{L. Caffarelli} and \textit{Y. Li} [Commun. Pure Appl. Math. 56, No. 5, 549--583 (2003; Zbl 1236.35041)] to the half space.