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We consider degenerate Monge-Ampere equations of the type $$\det D^2 u= f\quad \{in $\Om$}, \quad \quad f \sim \, d_{\p \Om}^\alpha \quad \{near $\p\Om$,}$$ where $d_{\p \Om}$ represents the distance to the boundary of thedomain $\Om$ and $\alpha>0$ is a positive power. We obtain $C^2$ estimates atthe boundary under natural conditions on the boundary data and the right handside. Similar estimates in two dimensions were obtained by J.X. Hong, G. Huangand W. Wang.

A localization theorem and boundary regularity for a class of degenerate Monge Ampere equations