Soient une fonction convexe et un domaine borné. Soient et des k-formes fermées sur Ω satisfaisant des conditions de bord appropriées. Nous nous intéressons à la minimisation de sur l'ensemble des -formes A telles que , 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 be convex and be a bounded domain. Let and be two closed k-forms on Ω satisfying appropriate boundary conditions. We discuss the minimization of over a subset of -forms A on Ω such that , 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.
Accepté le :
Publié le :
@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, Tome 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] Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics, ETH Verlag/Birkhäuser, Zürich/Basel, Switzerland, 2005
[2] (Lecture Notes in Mathematics), Volume vol. 1813, Springer (2003), pp. 91-122
[3] The Pullback Equation for Differential Forms, Birkhaüser, 2012
[4] 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] Differential equations methods for the Monge–Kantorovich mass transfer problem, Mem. Amr. Math. Soc., Volume 137 (1999) no. 653 (66 p)
[8] Regularity for a class of non-linear elliptic system, Acta Math., Volume 138 (1977), pp. 219-240
Cité par Sources :