Global C1,β and W2, p regularity for some singular Monge-Amp\`ere equations

We establish global C1,β and W2, p regularity for singular Monge-Amp\`ere equations of the form \[ D^2 u ~ dist-α(·,∂Ω), α∈ (0, 1),\] under suitable conditions on the boundary data and domains. Our results imply that the convex Aleksandrov solution to the singular Monge-Amp\`ere equation \[ D^2 u=|u|-α inΩ, u=0 in ∂Ω, α∈ (0, 1),\] where Ω is a C³, bounded, and uniformly convex domain, is globally C1,β and belongs to W2, p for all p<1/α.