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

P. E. Jabin, A. Mellet, M. Molina-Fructuoso

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish (US)
Article number109041
JournalJournal of Functional Analysis
Volume281
Issue number2
DOIs
StatePublished - Jul 15 2021

All Science Journal Classification (ASJC) codes

  • Analysis

Fingerprint

Dive into the research topics of 'Local regularity result for an optimal transportation problem with rough measures in the plane'. Together they form a unique fingerprint.

Cite this