Local regularity result for an optimal transportation problem with rough measures in the plane
| Autor: | Pierre Emmanuel Jabin, Antoine Mellet, M. Molina-Fructuoso |
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| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
Scale (ratio)
Plane (geometry) Dirac (video compression format) 010102 general mathematics Transportation theory Absolute continuity 01 natural sciences Measure (mathematics) Legendre transformation symbols.namesake Mathematics - Analysis of PDEs 0103 physical sciences symbols FOS: Mathematics Applied mathematics 010307 mathematical physics 0101 mathematics Convex function Analysis Mathematics Analysis of PDEs (math.AP) |
| Popis: | We investigate the properties of convex functions in R 2 that satisfy a local inequality which generalizes the notion of sub-solution of Monge-Ampere 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 C 1 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-Ampere equation. |
| Databáze: | OpenAIRE |
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