Comptes Rendus
no. 12
Partial differential equations
Optimal transport of closed differential forms for convex costs

Presented by: Haïm Brézis

Bernard Dacorogna1; Wilfrid Gangbo2; Olivier Kneuss3
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.

Let c:Λk−1→R+ be convex and Ω⊂Rn be a bounded domain. Let f0 and f1 be two closed k-forms on Ω satisfying appropriate boundary conditions. We discuss the minimization of ∫Ωc(A)dx over a subset of (k−1)-forms A on Ω such that dA+f1−f0=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.

Soient c:Λk−1→R+ une fonction convexe et Ω⊂Rn un domaine borné. Soient f0 et f1 des k-formes fermées sur Ω satisfaisant des conditions de bord appropriées. Nous nous intéressons à la minimisation de ∫Ωc(A)dx sur l'ensemble des (k−1)-formes A telles que dA+f1−f0=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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2015.09.015

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 
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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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