Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5,
56127 Pisa, Italy.
We revisit the duality theorem for multimarginal optimal transportation problems. In particular, we focus on the Coulomb cost. We use a discrete approximation to prove equality of the extremal values and some careful estimates of the approximating sequence to prove existence of maximizers for the dual problem (Kantorovich’s potentials). Finally we observe that the same strategy can be applied to a more general class of costs and that a classical results on the topic cannot be applied here.
Mathematics Subject Classification: 49J45 / 49N15 / 49K30
Key words: Multimarginal optimal transportation / Monge−Kantorovich problem / duality theory / Coulomb cost
The research of the author is part of the project 2010A2TFX2 Calcolo delle Variazioni financed by the Italian Ministry of Research.
The research of the author is partially financed by the “Fondi di ricerca di ateneo” of the University of Pisa.
© EDP Sciences, SMAI 2015