Free boundary problems in optimal transportation

Abstract

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 model 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.

The Kantorovich Initiative