Global regularity in the Monge-Ampère obstacle problem

In this paper, we establish the global $W^{2,p}$ estimate for the Monge-Ampère obstacle problem: $(Du{)}_{\mathrm{♯}}f\chi {}_{\{u>\frac{1}{2}\mathrm{\mid }x{\mathrm{\mid }}^2\}}=g$, where $f$ and $g$ are positive continuous functions supported in disjoint bounded $C^2$ uniformly convex domains $\bar{\mathrm{\Omega }}$ and $\overline{{\mathrm{\Omega }}^{\ast }}$, respectively. Furthermore, we assume that ${\int }_{\mathrm{\Omega }}f\ge {\int }_{{\mathrm{\Omega }}^{\ast }}g$. The main result shows that $Du:\bar{U}\to \overline{{\mathrm{\Omega }}^{\ast }}$, where $U=\{u>\frac{1}{2}\mathrm{\mid }x{\mathrm{\mid }}^2\}$, is a $W^{1,p}$ diffeomorphism for any $p\in (1,\mathrm{\infty })$. Previously, it was only known to be a continuous homeomorphism according to Caffarelli and McCann \cite{CM}. It is worth noting that our result is sharp, as we can construct examples showing that even with the additional assumption of smooth densities, the optimal map $Du$ is not Lipschitz. This obstacle problem arises naturally in optimal partial transportation.