TY - JOUR

T1 - On discontinuity of planar optimal transport maps

AU - Chodosh, Otis

AU - Jain, Vishesh

AU - Lindsey, Michael

AU - Panchev, Lyuboslav

AU - Rubinstein, Yanir A.

N1 - Funding Information:
This research was supported by the Stanford University SURIM (Stanford Undergraduate Research Institute in Mathematics) and VPUE, and NSF grants DGE-1147470, DMS-1206284. Y.A.R. was also supported by a Sloan Research Fellowship. The authors are grateful to the referee for a very careful reading and numerous corrections, and in particular for Proposition 4.

PY - 2015/6/1

Y1 - 2015/6/1

N2 - 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/2dμ(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.

AB - 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/2dμ(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.

KW - Monge-Ampère equation

KW - Optimal transportation

KW - singular solutions

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UR - http://www.scopus.com/inward/citedby.url?scp=84951782828&partnerID=8YFLogxK

U2 - 10.1142/S1793525315500089

DO - 10.1142/S1793525315500089

M3 - Article

AN - SCOPUS:84951782828

VL - 7

SP - 239

EP - 260

JO - Journal of Topology and Analysis

JF - Journal of Topology and Analysis

SN - 1793-5253

IS - 2

ER -