Global C^1, and W^2, p regularity for some singular Monge-Amp\`ere equations

We establish global C^1, and W^2, p regularity for singular Monge-Amp\`ere equations of the form \ D² u dist^- (, ), (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 \ D² u=|u|^- in, u=0 in, (0, 1), \ where is a C³, bounded, and uniformly convex domain, is globally C^1, and belongs to W^2, p for all p<1/.