TY - JOUR

T1 - Local regularity result for an optimal transportation problem with rough measures in the plane

AU - Jabin, P. E.

AU - Mellet, A.

AU - Molina-Fructuoso, M.

N1 - Funding Information:
Partially supported by NSF DMS Grant 161453, 1908739, 2049020 and NSF Grant RNMS (Ki-Net) 1107444.Partially supported by NSF Grant NSF Grant DMS-1501067 and DMS-2009236.
Publisher Copyright:
© 2021 Elsevier Inc.

PY - 2021/7/15

Y1 - 2021/7/15

N2 - 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 problem 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 problem 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.

AB - 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 problem 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 problem 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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U2 - 10.1016/j.jfa.2021.109041

DO - 10.1016/j.jfa.2021.109041

M3 - Article

AN - SCOPUS:85104055948

VL - 281

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 2

M1 - 109041

ER -