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Duality for the $ L^{\infty}$ optimal transport problem


Authors: E. N. Barron, M. Bocea and R. R. Jensen
Journal: Trans. Amer. Math. Soc. 369 (2017), 3289-3323
MSC (2010): Primary 35F21, 49J35, 49J45, 49K30, 49L25, 49Q20
DOI: https://doi.org/10.1090/tran/6759
Published electronically: September 15, 2016
MathSciNet review: 3605972
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Abstract: We derive the dual of the relaxed Monge-Kantorovich optimal mass transport problem in $ L^{\infty }$ in which one seeks to minimize
$ \mu $- $ \mathrm {ess\,sup}_{(x,y)\in \mathbb{R}^N \times \mathbb{R}^N} c(x,y)$ over Borel probability measures $ \mu $ with given marginals $ P_0, P_1.$ Several formulations of the dual problem are obtained using various techniques including quasiconvex duality. We also consider weighted optimal transport in $ L^{\infty }$ and we identify the form of the dual in the Lagrangian cost setting for both integral and essential supremum costs along trajectories. Finally, we prove a duality formula that relates a maximization problem which arises naturally in the $ L^{\infty }$ calculus of variations with a family of optimal partial transport problems.


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Additional Information

E. N. Barron
Affiliation: Department of Mathematics and Statistics, Loyola University Chicago, 1032 W. Sheridan Road, Chicago, Illinois 60660
Email: ebarron@luc.edu

M. Bocea
Affiliation: Department of Mathematics and Statistics, Loyola University Chicago, 1032 W. Sheridan Road, Chicago, Illinois 60660
Email: mbocea@luc.edu

R. R. Jensen
Affiliation: Department of Mathematics and Statistics, Loyola University Chicago, 1032 W. Sheridan Road, Chicago, Illinois 60660
Email: rjensen@luc.edu

DOI: https://doi.org/10.1090/tran/6759
Received by editor(s): October 1, 2014
Received by editor(s) in revised form: April 30, 2015
Published electronically: September 15, 2016
Additional Notes: The research of the first and third authors was partially supported by the National Science Foundation under Grants No. DMS-1008602 and DMS-1515871
The second author’s research was partially supported by the National Science Foundation under Grants No. DMS-1156393 and DMS-1515871
Article copyright: © Copyright 2016 American Mathematical Society