Free boundary problems in optimal transportation

In this talk, we introduce some recent regularity results of free boundary in
optimal transportation. Particularly for higher order regularity, when
densities are Hölder continuous and domains are $C^2$, uniformly convex, we obtain the free boundary is $C^{2,\alpha}$ smooth. We also consider another mode
case that the target consists of two disjoint convex sets, in which
singularities of optimal transport mapping arise. Under similar assumptions,
we show that the singular set of the optimal mapping is an $(n-1)$-dimensional
$C^{2,\alpha}$ regular sub-manifold of $\mathbb{R}^n$. These are based on a
series of joint work with Shibing Chen and Xu-Jia Wang.