Local regularity result for an optimal transportation problem with rough measures in the plane

We investigate the properties of convex functions in R2 that satisfy a local inequality which generalizes the notion of sub-solution of Monge-Ampère equation for a Monge-Kantorovich prob- lem with quadratic cost between non-absolutely continuous measures. For each measure, we introduce a discrete scale so that the measure behaves as an absolutely continuous measure up to that scale. Our main theorem then proves that such convex functions cannot exhibit any flat part at a scale larger than the corresponding discrete scales on the measures. This, in turn, implies a C1 regularity result up to the discrete scale for the Legendre transform. Our result applies in particular to any Kantorovich potential associated to an optimal transportation prob- lem between two measures that are (possibly only locally) sums of uniformly distributed Dirac masses. The proof relies on novel explicit estimates directly based on the optimal transportation problem, instead of the Monge-Ampère equation.

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Work Title Local regularity result for an optimal transportation problem with rough measures in the plane
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Open Access
Creators
  1. P.-E. Jabin
  2. A. Mellet
  3. M. Molina-Fructuoso
Keyword
  1. Optimal transportation
  2. Singular measures
  3. Non-convex domains
License In Copyright (Rights Reserved)
Work Type Article
Publisher
  1. Journal of Functional Analysis
Publication Date April 9, 2021
Publisher Identifier (DOI)
  1. https://doi.org/10.1016/j.jfa.2021.109041
Deposited March 25, 2024

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Version 1
published

  • Created
  • Added LocalRegularityOptimalTransportationpdfa.pdf
  • Added Creator Pierre-Emmanuel Jabin
  • Added Creator Antoine Mellet
  • Added Creator Martin Molina Fructuoso
  • Published
  • Updated
  • Updated Work Title Show Changes
    Work Title
    • Optimal transportation in a discrete setting
    • Local regularity result for an optimal transportation problem with rough measures in the plane
  • Updated Keyword, Publisher Identifier (DOI), Publication Date Show Changes
    Keyword
    • Optimal transportation, Singular measures, Non-convex domains
    Publisher Identifier (DOI)
    • https://doi.org/10.1016/j.jfa.2021.109041
    Publication Date
    • 2021-05-01
    • 2021-04-09
  • Renamed Creator P.-E. Jabin Show Changes
    • Pierre-Emmanuel Jabin
    • P.-E. Jabin
  • Renamed Creator A. Mellet Show Changes
    • Antoine Mellet
    • A. Mellet
  • Renamed Creator M. Molina-Fructuoso Show Changes
    • Martin Molina Fructuoso
    • M. Molina-Fructuoso