  In this paper, we obtain some regularities of the free boundary in optimal
transportation with the quadratic cost. Our first result is about the
$C^{1,\alpha}$ regularity of the free boundary for optimal partial transport
between convex domains for densities $f, g$ bounded from below and above. When
$f, g \in C^\alpha$, and $\partial\Omega, \partial\Omega^*\in C^{1,1}$ are far
apart, by adopting our recent results on boundary regularity of Monge-Amp\`ere
equations \cite{CLW1}, our second result shows that the free boundaries are
$C^{2,\alpha}$. As an application, in the last we also obtain these
regularities of the free boundary in an optimal transport problem with two
separate targets.
