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.
DOI : 10.1051/m2an/2015035
Mots-clés : Multimarginal optimal transportation, Monge−Kantorovich problem, duality theory, Coulomb cost
@article{M2AN_2015__49_6_1643_0,
author = {De Pascale, Luigi},
title = {Optimal transport with {Coulomb} cost. {Approximation} and duality},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1643--1657},
publisher = {EDP-Sciences},
volume = {49},
number = {6},
year = {2015},
doi = {10.1051/m2an/2015035},
zbl = {1330.49048},
mrnumber = {3423269},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2015035/}
}
TY - JOUR AU - De Pascale, Luigi TI - Optimal transport with Coulomb cost. Approximation and duality JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2015 SP - 1643 EP - 1657 VL - 49 IS - 6 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2015035/ DO - 10.1051/m2an/2015035 LA - en ID - M2AN_2015__49_6_1643_0 ER -
%0 Journal Article %A De Pascale, Luigi %T Optimal transport with Coulomb cost. Approximation and duality %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2015 %P 1643-1657 %V 49 %N 6 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2015035/ %R 10.1051/m2an/2015035 %G en %F M2AN_2015__49_6_1643_0
De Pascale, Luigi. Optimal transport with Coulomb cost. Approximation and duality. ESAIM: Mathematical Modelling and Numerical Analysis , Optimal Transport, Tome 49 (2015) no. 6, pp. 1643-1657. doi : 10.1051/m2an/2015035. http://archive.numdam.org/articles/10.1051/m2an/2015035/
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