Partial differential equations
Optimal transport of closed differential forms for convex costs
Comptes Rendus. Mathématique, Volume 353 (2015) no. 12, pp. 1099-1104.

Let c:Λk1R+ be convex and ΩRn 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 (k1)-forms A on Ω such that dA+f1f0=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:Λk1R+ une fonction convexe et ΩRn 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 (k1)-formes A telles que dA+f1f0=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
Dacorogna, Bernard 1; Gangbo, Wilfrid 2; Kneuss, Olivier 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
@article{CRMATH_2015__353_12_1099_0,
     author = {Dacorogna, Bernard and Gangbo, Wilfrid and Kneuss, Olivier},
     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},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2015.09.015/}
}
TY  - JOUR
AU  - Dacorogna, Bernard
AU  - Gangbo, Wilfrid
AU  - Kneuss, Olivier
TI  - Optimal transport of closed differential forms for convex costs
JO  - Comptes Rendus. Mathématique
PY  - 2015
SP  - 1099
EP  - 1104
VL  - 353
IS  - 12
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2015.09.015/
DO  - 10.1016/j.crma.2015.09.015
LA  - en
ID  - CRMATH_2015__353_12_1099_0
ER  - 
%0 Journal Article
%A Dacorogna, Bernard
%A Gangbo, Wilfrid
%A Kneuss, Olivier
%T Optimal transport of closed differential forms for convex costs
%J Comptes Rendus. Mathématique
%D 2015
%P 1099-1104
%V 353
%N 12
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2015.09.015/
%R 10.1016/j.crma.2015.09.015
%G en
%F CRMATH_2015__353_12_1099_0
Dacorogna, Bernard; Gangbo, Wilfrid; Kneuss, Olivier. 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. http://www.numdam.org/articles/10.1016/j.crma.2015.09.015/

[1] Ambrosio, L.; Gigli, N.; Savaré, G. Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics, ETH Verlag/Birkhäuser, Zürich/Basel, Switzerland, 2005

[2] Brenier, Y. (Lecture Notes in Mathematics), Volume vol. 1813, Springer (2003), pp. 91-122

[3] Csato, G.; Dacorogna, B.; Kneuss, O. The Pullback Equation for Differential Forms, Birkhaüser, 2012

[4] Dacorogna, B. Direct Methods in the Calculus of Variations, Springer-Verlag, 2007

[5] B. Dacorogna, W. Gangbo, O. Kneuss, Optimal transport of closed differential forms, preprint.

[6] B. Dacorogna, W. Gangbo, O. Kneuss, Symplectic decomposition, Darboux theorem and ellipticity, preprint.

[7] Evans, L.C.; Gangbo, W. Differential equations methods for the Monge–Kantorovich mass transfer problem, Mem. Amr. Math. Soc., Volume 137 (1999) no. 653 (66 p)

[8] Uhlenbeck, K. Regularity for a class of non-linear elliptic system, Acta Math., Volume 138 (1977), pp. 219-240

Cited by Sources: