On discontinuity of planar optimal transport maps

Consider two bounded domains $Ω$ and $Λ$ in $\mathbb{R}^{2}$, and two sufficiently regular probability measures $μ$ and $ν$ supported on them. By Brenier's theorem, there exists a unique transportation map $T$ satisfying $T_\#μ=ν$ and minimizing the quadratic cost $\int_{\mathbb{R}^{n}}|T(x)-x|^{2}dμ(x)$. Furthermore, by Caffarelli's regularity theory for the real Monge--Ampère equations, if $Λ$ is convex, $T$ is continuous. We study the reverse problem, namely, when is $T$ discontinuous if $Λ$ fails to be convex? We prove a result guaranteeing the discontinuity of $T$ in terms of the geometries of $Λ$ and $Ω$ in the two-dimensional case. The main idea is to use tools of convex analysis and the extrinsic geometry of $\partialΛ$ to distinguish between Brenier and Alexandrov weak solutions of the Monge--Ampère 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.

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