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A new $ L^\infty$ estimate in optimal mass transport

Authors: G. Bouchitté, C. Jimenez and M. Rajesh
Journal: Proc. Amer. Math. Soc. 135 (2007), 3525-3535
MSC (2000): Primary 39B62, 46N10, 49Q20
Published electronically: July 3, 2007
MathSciNet review: 2336567
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Abstract: Let $ \Omega$ be a bounded Lipschitz regular open subset of $ \mathbb{R}^d$ and let $ \mu,\nu$ be two probablity measures on $ \overline{\Omega}$. It is well known that if $ \mu=f\, dx$ is absolutely continuous, then there exists, for every $ p>1$, a unique transport map $ T_p$ pushing forward $ \mu$ on $ \nu$ and which realizes the Monge-Kantorovich distance $ W_p(\mu,\nu)$. In this paper, we establish an $ L^\infty$ bound for the displacement map $ T_p x-x$ which depends only on $ p$, on the shape of $ \Omega$ and on the essential infimum of the density $ f$.

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Additional Information

G. Bouchitté
Affiliation: UFR Sciences, Université du Sud-Toulon-Var, BP20132, 83957 La Garde Cedex, France

C. Jimenez
Affiliation: UFR Sciences, Université du Sud-Toulon-Var, BP20132, 83957 La Garde Cedex, France

M. Rajesh
Affiliation: Departemento de Matematica, Facultad de Ciencias Fisicas y Matematicas, Universidad de Concepcion, Casilla 160-C. Concepcion, Chile

Keywords: Wasserstein distance, optimal transport map, uniform estimates
Received by editor(s): January 9, 2006
Received by editor(s) in revised form: June 23, 2006
Published electronically: July 3, 2007
Communicated by: David Preiss
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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