\(W^{2,1}\) regularity for solutions of the Monge-Ampère equation (Q1949227)

Let \(\Omega\subset{\mathbb R}^n\) be a bounded convex domain and assume that \(0<\lambda\leq f\leq\Lambda\). This paper deals with the study of solutions in the sense of Alexandrov of the Monge-Ampère equation \[ \text{det}\,D^2u=f\quad\text{in}\;\Omega \] satisfying \(u=0\) on \(\partial\Omega\). The main result of this paper establishes that if \(u\) is continuous and strictly convex then \(u\in W^{2,1}_{\text{loc}}(\Omega)\). The key idea of the proof is to first establish higher integrability \textit{a priori} estimates for \(D^2u\), namely \(D^2u\in L\log^kL\) for all positive numbers \(k\).