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

We consider degenerate Monge–Ampere equations of the typedetD2u=fin Ω,f∼d∂Ωαnear ∂Ω, where d∂Ωd∂Ω represents the distance to the boundary of the domain Ω   and α>0α>0 is a positive power. We obtain C2C2 estimates at the boundary under natural conditions on the boundary data and the right-hand side. Similar estimates in two dimensions were obtained by J.X. Hong, G. Huang and W. Wang in [3].