On discontinuity of planar optimal transport maps

Abstract

Consider two bounded domains Ω and Λ in ℝ2, 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 ∫ℝn ∣T(x) - x∣2 dμ(x). Furthermore, by Caffarelli's regularity theory for the real Monge–Ampère equation, 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 ∂Λ 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.