Abstract: In this talk, we introduce some recent regularity results of the free boundary in optimal transportation with the quadratic cost. Assuming the densities $f, g$ are bounded away from zero and infinity, and supported on convex domains $\Omega, \Lambda$, respectively, our first result shows that the free boundaries are $C^{1,\alpha}$ inside the domains. Furthermore, when $f, g \in C^\alpha$, and $\partial\Omega, \partial\Lambda \in C^{1,1}$ are far apart, by adopting our recent results on boundary regularity of Monge-Ampere equations, 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.
