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

Abstract

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

To appear in Ann. Inst. Fourier (Grenoble)