Global regularity in the Monge-Ampère obstacle problem

Abstract

In this paper, we establish the global $W^{2,p}$ estimate for the Monge-Ampère obstacle problem: $(Du{)}_{♯}f\chi {}_{\{u>\frac{1}{2}|x{|}^2\}}=g$, where $f$ and $g$ are positive continuous functions supported in disjoint bounded $C^2$ uniformly convex domains $\overline{\Omega }$ and $\overline{{\Omega }^{\ast }}$, respectively. Furthermore, we assume that ${\int }_{\Omega }f\ge {\int }_{{\Omega }^{\ast }}g$. The main result shows that $Du:\overline{U}\to \overline{{\Omega }^{\ast }}$, where $U=\{u>\frac{1}{2}|x{|}^2\}$, is a $W^{1,p}$ diffeomorphism for any $p\in (1,\infty )$. 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.