Analyticity of the solutions to degenerate Monge-Amp\`ere equations

Abstract

This paper is devoted to study the following degenerate Monge-Amp\`ere equation: \begin{eqnarray}\label{ab1} \begin{cases} \det D^2 u=\Lambda_q (-u)^q \quad \text{in}\quad \Omega,\\ u=0 \quad\text{on}\quad \partial\Omega \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-Amp\`ere equation is analytic in $\bar\Omega$ provided $q\in \mathbb Z^+$.