  In this paper, we establish the global $W^{2,p}$ estimate for the
Monge-Amp\`ere obstacle problem:
$(Du)_{\sharp}f\chi{_{\{u>\frac{1}{2}|x|^2\}}}=g$, where $f$ and $g$ are
positive continuous functions supported in disjoint bounded $C^2$ uniformly
convex domains $\overline{\Omega}$ and $\overline{\Omega^*}$, respectively.
Furthermore, we assume that $\int_{\Omega}f\geq \int_{\Omega^*}g$. The main
result shows that $Du:\overline U\rightarrow\overline{\Omega^*}$, 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.
