On discontinuity of planar optimal transport maps

Consider two bounded domains Omega and Lambda in 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 integral(Rn) vertical bar T(x)-x vertical bar(2)d mu(x). Furthermore, by Caffarelli's regularity theory for the real Monge-Ampere equation, 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 derivative Lambda to distinguish between Brenier and Alexandrov weak solutions of the Monge-Ampere 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.