Optimal transport of closed differential forms for convex costs

Bernard Dacorogna; Wilfrid Gangbo; Olivier Kneuss

doi:10.1016/j.crma.2015.09.015

Partial differential equations

Optimal transport of closed differential forms for convex costs
[Transport optimal des formes fermées pour des coûts convexes]

Présenté par : Haïm Brézis

Bernard Dacorogna ¹ ; Wilfrid Gangbo ² ; Olivier Kneuss ³

¹ Section de Mathématiques, EPFL, CH-1015 Lausanne, Switzerland
² School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA
³ Department of Mathematics, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil

Comptes Rendus. Mathématique, Volume 353 (2015) no. 12, pp. 1099-1104.

Résumé
Abstract

Soient $c : Λ^{k - 1} \to R_{+}$ une fonction convexe et $Ω \subset R^{n}$ un domaine borné. Soient $f_{0}$ et $f_{1}$ des k-formes fermées sur Ω satisfaisant des conditions de bord appropriées. Nous nous intéressons à la minimisation de $\int_{Ω} c (A) d x$ sur l'ensemble des $(k - 1)$ -formes A telles que $d A + f_{1} - f_{0} = 0$ , ainsi que sa relation à un problème de transport des formes symplectiques. La Section 3 sert d'étape intermédiaire vers la Section 4, qui est plus riche, car reliée à des problèmes variationnels avec une multitude de minimiseurs.

Let $c : Λ^{k - 1} \to R_{+}$ be convex and $Ω \subset R^{n}$ be a bounded domain. Let $f_{0}$ and $f_{1}$ be two closed k-forms on Ω satisfying appropriate boundary conditions. We discuss the minimization of $\int_{Ω} c (A) d x$ over a subset of $(k - 1)$ -forms A on Ω such that $d A + f_{1} - f_{0} = 0$ , and its connection with a transport of symplectic forms. Section 3 mainly serves as a step toward Section 4, which is richer, as it connects to variational problems with multiple minimizers.

Reçu le : 2015-08-04
Accepté le : 2015-09-18
Publié le : 2015-11-02

DOI : 10.1016/j.crma.2015.09.015

Affiliations des auteurs :

Bernard Dacorogna ¹ ; Wilfrid Gangbo ² ; Olivier Kneuss ³

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     author = {Bernard Dacorogna and Wilfrid Gangbo and Olivier Kneuss},
     title = {Optimal transport of closed differential forms for convex costs},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1099--1104},
     publisher = {Elsevier},
     volume = {353},
     number = {12},
     year = {2015},
     doi = {10.1016/j.crma.2015.09.015},
     language = {en},
}

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Bernard Dacorogna; Wilfrid Gangbo; Olivier Kneuss. Optimal transport of closed differential forms for convex costs. Comptes Rendus. Mathématique, Volume 353 (2015) no. 12, pp. 1099-1104. doi : 10.1016/j.crma.2015.09.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.09.015/

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