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Multimarginal Optimal Transport Maps for One–dimensional Repulsive Costs

Published online by Cambridge University Press:  20 November 2018

Maria Colombo ,
Luigi De Pascale  and
Simone Di Marino
Maria Colombo
Affiliation:
Scuola Normale Superiore, 56126 Pisa, Italy. e-mail: maria.colombo@sns.itsimone.dimarino@sns.it
Luigi De Pascale
Affiliation:
Dipartimento di Matematica, Universitá di Pisa, Pisa, Italy. e-mail: depascal@dm.unipi.it
Simone Di Marino
Affiliation:
Scuola Normale Superiore, 56126 Pisa, Italy. e-mail: maria.colombo@sns.itsimone.dimarino@sns.it
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Abstract

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We study a multimarginal optimal transportation problem in one dimension. For a symmetric, repulsive cost function, we show that, given a minimizing transport plan, its symmetrization is induced by a cyclical map, and that the symmetric optimal plan is unique. The class of costs that we consider includes, in particular, the Coulomb cost, whose optimal transport problem is strictly related to the strong interaction limit of Density Functional Theory. In this last setting, our result justifies some qualitative properties of the potentials observed in numerical experiments.

Keywords

49Q2049K30Monge–Kantorovich problemoptimal transport problemcyclical monotonicity
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 67 , Issue 2 , April 2015 , pp. 350 - 368
Copyright
Copyright © Canadian Mathematical Society 2015

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