Global \(W^{2,p}\) estimates for the Monge-Ampère equation (Q2845432)

In this paper, the author answers the natural question of whether the interior estimates can be extended up to the boundary for solutions to the Monge-Ampère equation. The main result establishes a global \(W^{2,p}\)-estimate under natural assumptions on the domain and boundary data. The proof is based on the localization theorem, which asserts that if the boundary data has a quadratic growth near \([x_n=0]\) then each section of the solution at the origin is equivalent to a half-ellipsoid centered at the origin.