On discontinuity of planar optimal transport maps
Consider two bounded domains $\Omega$ and $\Lambda$ in $\mathbb{R}^{2}$, and two sufficiently regular probability measures $\mu$ and $\nu$ supported on them. By Brenier's theorem, there exists a unique transportation map $T$ satisfying $T_\#\mu=\nu$ and minimizing the quadratic cost $\int_{\mathbb{R}^{n}}|T(x)-x|^{2}d\mu(x)$. Furthermore, by Caffarelli's regularity theory for the real Monge--Amp\`ere equations, if $\Lambda$ is convex, $T$ is continuous. We study the reverse problem, namely, when is $T$ discontinuous if $\Lambda$ fails to be convex? We prove a result guaranteeing the discontinuity of $T$ in terms of the geometries of $\Lambda$ and $\Omega$ in the two-dimensional case. The main idea is to use tools of convex analysis and the extrinsic geometry of $\partial\Lambda$ to distinguish between Brenier and Alexandrov weak solutions of the Monge--Amp\`ere equation. We also use this approach to give a new proof of a result due to Wolfson and Urbas. We conclude by revisiting an example of Caffarelli, giving a detailed study of a discontinuous map between two explicit domains, and determining precisely where the discontinuities occur.