no. G1
Équations aux dérivées partielles
Une interprétation variationnelle de la relativité générale dans le vide en termes de transport optimal
Brenier, Yann1 
1 CNRS, Département de Mathématiques et Applications, École Normale Supérieure, Université PSL, 45 rue d’Ulm 75005 Paris, France
Comptes Rendus. Mathématique, Tome 360 (2022) no. G1, pp. 25-33.

On revoit les équations d’Einstein de la relativité générale dans le vide comme équations d’optimalité d’une sorte de problème de transport optimal quadratique généralisé.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.275
Brenier, Yann 1

1 CNRS, Département de Mathématiques et Applications, École Normale Supérieure, Université PSL, 45 rue d’Ulm 75005 Paris, France 
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Brenier, Yann. Une interprétation variationnelle de la relativité générale dans le vide en termes de transport optimal. Comptes Rendus. Mathématique, Tome 360 (2022) no. G1, pp. 25-33. doi : 10.5802/crmath.275. http://archive.numdam.org/articles/10.5802/crmath.275/

[1] Ambrosio, Luigi; Gigli, Nicola; Savaré, Giuseppe Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics, ETH Zürich, Birkhäuser, 2008 | Zbl

[2] Benamou, Jean-David; Brenier, Yann A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem, Numer. Math., Volume 84 (2000) no. 3, pp. 375-393 | DOI | MR | Zbl

[3] Berman, Robert J.; Boucksom, Sébastien; Guedj, Vincent; Zeriahi, Ahmed A variational approach to complex Monge–Ampère equations, Publ. Math., Inst. Hautes Étud. Sci., Volume 117 (2013), pp. 179-245 | DOI | Numdam | Zbl

[4] Berndtsson, Bo; Cordero-Erausquin, Dario; Klartag, Bo’az; Rubinstein, Yanir A. Complex Legendre duality, Am. J. Math., Volume 142 (2020) no. 1, pp. 323-339 | DOI | MR | Zbl

[5] Brenier, Yann Extended Monge-Kantorovich theory, Optimal transportation and applications. Lectures given at the C.I.M.E. summer school, Martina Franca, Italy, September 2–8, 2001 (Caffarelli, Luis A. et al., eds.) (Lecture Notes in Mathematics), Volume 1813, Springer, 2003, pp. 91-121 | MR | Zbl

[6] Brenier, Yann Hydrodynamic structure of the augmented Born–Infeld equations, Arch. Ration. Mech. Anal., Volume 172 (2004) no. 1, pp. 65-91 | DOI | MR | Zbl

[7] Brenier, Yann; Vorotnikov, Dmitry On optimal transport of matrix-valued measures, SIAM J. Math. Anal., Volume 52 (2020) no. 3, pp. 2849-2873 | DOI | MR | Zbl

[8] Brézis, Haïm Analyse fonctionnelle appliquée, Masson, 1982

[9] Brézis, Haïm; Coron, Jean-Michel; Lieb, Elliott H. Harmonic maps with defects, Commun. Math. Phys., Volume 107 (1986), pp. 649-705 | DOI | MR | Zbl

[10] Carlen, Eric A.; Maas, Jan Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance, J. Funct. Anal., Volume 273 (2017) no. 5, pp. 1810-1869 | DOI | MR | Zbl

[11] Chen, Yongxin; Gangbo, Wilfrid; Georgiou, Tryphon T.; Tannenbaum, Allen On the matrix Monge–Kantorovich problem, Eur. J. Appl. Math., Volume 31 (2020) no. 4, pp. 574-600 | DOI | MR

[12] Christodoulou, Demetrios; Klainerman, Sergiu The global nonlinear stability of the Minkowski space, Princeton Mathematical Series, 41, Princeton University Press, 1993 | Zbl

[13] Frisch, Uriel; Matarrese, Sabino; Mohayaee, Roya; Sobolevski, Andrei A reconstruction of the initial conditions of the Universe by optimal mass transportation, Nature, Volume 417 (2002), pp. 260-262 | DOI

[14] Guionnet, Alice First order asymptotics of matrix integrals ; a rigorous approach towards the understanding of matrix models, Commun. Math. Phys., Volume 244 (2004) no. 3, pp. 527-569 | DOI | MR | Zbl

[15] Huneau, Cécile; Luk, Jonathan Trilinear compensated compactness and Burnett’s conjecture in general relativity (2019) (https://arxiv.org/abs/1907.10743)

[16] John, Lott; Villani, Cédric Ricci curvature for metric-measure spaces via optimal transport, Ann. Math., Volume 169 (2009) no. 3, pp. 903-991 | DOI | MR | Zbl

[17] Lavenant, Hugo Harmonic mappings valued in the Wasserstein space, J. Funct. Anal., Volume 277 (2019) no. 3, pp. 688-785 | DOI | MR | Zbl

[18] McCann, Robert J. Displacement convexity of Boltzmann’s entropy characterizes the strong energy condition from general relativity, Camb. J. Math., Volume 8 (2020) no. 3, pp. 609-681 | DOI | MR | Zbl

[19] Mondino, Andrea; Suhr, Stefan An optimal transport formulation of the Einstein equations of general relativity (2018) (https://arxiv.org/abs/1810.13309v1)

[20] Otto, Felix; Westdickenberg, Michael Eulerian calculus for the contraction in the Wasserstein distance, SIAM J. Math. Anal., Volume 37 (2005) no. 4, pp. 1227-1255 | DOI | MR | Zbl

[21] Peyré, Gabriel; Chizat, Lénaïc; Vialard, François-Xavier; Solomon, Justin Quantum entropic regularization of matrix-valued optimal transport, Eur. J. Appl. Math., Volume 30 (2019) no. 6, pp. 1079-1102 | DOI | MR | Zbl

[22] Rachev, S.; Rûschendorf, L. Mass Transportation Problems. Vol. 1 : Theory. Vol. 2 : Applications, Springer Series in Statistics, Springer, 1998 | Zbl

[23] Santambrogio, Filippo Optimal transport for applied mathematicians, Progress in Nonlinear Differential Equations and their Applications, 87, Birkhäuser/Springer, 2015 | DOI | Zbl

[24] Sturm, Karl-Théodor On the geometry of metric measure spaces, Acta Math., Volume 196 (2006) no. 1, pp. 65-177 | DOI | MR | Zbl

[25] Villani, Cédric Topics in optimal transportation, Graduate Studies in Mathematics, 58, American Mathematical Society, 2003 | Zbl

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