Analyticity of the solutions to degenerate Monge-Ampère equations

Abstract

This paper is devoted to study the following degenerate Monge-Ampère equation: \begin{eqnarray}\label{ab1} \begin{cases} \det D^2 u=Λ_q (-u)^q \quad \text{in}\quad Ω,\\ u=0 \quad\text{on}\quad \partialΩ\end{cases} \end{eqnarray} for some positive constant ${\Lambda }_q$. Suppose $\Omega \subset \subset {\mathbb{R}}^n$ is uniformly convex and analytic. Then the solution of the degenerate Monge-Ampère equation is analytic in $\bar{\Omega }$ provided $q\in {\mathbb{Z}}^{+}$.