Fully Degenerate Monge Amp\'ere Equations

In this paper, we consider the following nonlinear eigenvalue problem for the Monge-Amp\'ere equation: find a non-negative weakly convex classical solution $f$ satisfying {equation*} {cases} \det D^2 f=f^p \quad &\text{in $\Omega$} f=\vp \quad &text{on $\partial\Omega$} {cases} {equation*} for a strictly convex smooth domain $\Omega\subset\R^2$ and $0<p<2$. When $\{f=0\}$ contains a convex domain, we find a classical solution which is smooth on $\bar{\{f>0\}}$ and whose free boundary $\partial\{f=0\}$ is also smooth.