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On Hölder continuity-in-time of the optimal transport map towards measures along a curve

Part of: Manifolds Classical measure theory

Published online by Cambridge University Press:  31 March 2011

Nicola Gigli
Nicola Gigli
Affiliation:
UFR Mathématiques et Informatique, Université de Bordeaux, 351 cours de la Libération, 33405 Talence cedex, France
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Abstract

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We discuss the problem of the regularity-in-time of the map t ↦ Tt ∊ Lp(ℝd, ℝd; σ), where Tt is a transport map (optimal or not) from a reference measure σ to a measure μt which lies along an absolutely continuous curve t ↦ μt in the space (). We prove that in most cases such a map is no more than 1/p-Hölder continuous.

Keywords

optimal transportHölder continuityregularity-in-time

MSC classification

Secondary: 49Q20: Variational problems in a geometric measure-theoretic setting 28A33: Spaces of measures, convergence of measures
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 54 , Issue 2 , June 2011 , pp. 401 - 409
Copyright
Copyright © Edinburgh Mathematical Society 2011

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