Global $C^{1,\beta }$ and $W^{2,p}$ regularity for some singular Monge-Ampère equations

We establish global $C^{1,\beta }$ and $W^{2,p}$ regularity for singular Monge-Ampère equations of the form $$\det D^{2}u\sim {\text{dist}}^{-\alpha }(\cdot ,\mathrm{\partial }\mathrm{\Omega }),\alpha \in (0,1),$$ under suitable conditions on the boundary data and domains. Our results imply that the convex Aleksandrov solution to the singular Monge-Ampère equation $$\det D^{2}u=\mathrm{\mid }u{\mathrm{\mid }}^{-\alpha }\text{in}\mathrm{\Omega },u=0\text{in}\mathrm{\partial }\mathrm{\Omega },\alpha \in (0,1),$$ where $\mathrm{\Omega }$ is a $C^3$, bounded, and uniformly convex domain, is globally $C^{1,\beta }$ and belongs to $W^{2,p}$ for all $p<1\mathrm{/}\alpha$.