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# Multimarginal Optimal Transport Maps for $1$-dimensional Repulsive Costs

Published:2014-05-22

• Maria Colombo,
Scuola Normale Superiore, 56126 Pisa, Italy
• Luigi De Pascale,
Dipartimento di Matematica, Universitá di Pisa, Pisa, Italy
• Simone Di Marino,
Scuola Normale Superiore, 56126 Pisa, Italy
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## Abstract

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: Monge-Kantorovich problem, optimal transport problem, cyclical monotonicity
 MSC Classifications: 49Q20 - Variational problems in a geometric measure-theoretic setting 49K30 - Optimal solutions belonging to restricted classes

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