## Abstract

Consider two bounded domains ω and λ in 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 ∫_{R}_{n}/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 δλ 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.

Original language | English (US) |
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Pages (from-to) | 239-260 |

Number of pages | 22 |

Journal | Journal of Topology and Analysis |

Volume | 7 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1 2015 |

## All Science Journal Classification (ASJC) codes

- Analysis
- Geometry and Topology

## Keywords

- Monge-Ampère equation
- Optimal transportation
- singular solutions