Comptes Rendus
Partial differential equations
The five gradients inequality for non quadratic costs
Thibault Caillet1
1Institut Camille Jordan, Université Claude Bernard - Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne cedex
Comptes Rendus. Mathématique, Volume 361 (2023), pp. 715-721

We give a proof of the “five gradients inequality” of Optimal Transportation Theory for general costs of the form c(x,y)=h(x-y) where h is a C 1 strictly convex radially symmetric function.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.444

Thibault Caillet  1

1 Institut Camille Jordan, Université Claude Bernard - Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne cedex
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Thibault Caillet. The five gradients inequality for non quadratic costs. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 715-721. doi: 10.5802/crmath.444

[1] Martial Agueh Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory, Adv. Differ. Equ., Volume 10 (2005) no. 3, pp. 309-360 | MR | Zbl

[2] Luigi Ambrosio; Nicola Fusco; Diego Pallara Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, Oxford University Press, 2000

[3] Luigi Ambrosio; Nicola Gigli; Giuseppe Savaré Gradient flows in metric spaces and in the spaces of probability measures, Lectures in Mathematics, Birkhäuser, 2008

[4] Giuseppe Buttazzo; Guillaume Carlier; Maxime Laborde On the Wasserstein distance between mutually singular measures, Adv. Calc. Var., Volume 13 (2020) no. 2, pp. 141-154 | MR | Zbl | DOI

[5] Guido De Philippis; Alpár Richárd Mészáros; Filippo Santambrogio; Bozhidar Velichkov BV estimates in Optimal Transportation and Applications, Arch. Ration. Mech. Anal., Volume 212 (2020) no. 2, pp. 829-860 | MR | Zbl

[6] Simone Di Marino; Filippo Santambrogio JKO estimates in linear and non-linear Fokker-Planck equations, and Keller-Segel: L p and Sobolev bounds (2019) (to appear in Ann. Inst. H. Poincaré C, Anal. Non Linéaire) | arXiv

[7] Lawrence Craig Evans; Ronald F. Gariepy Measure Theory and Fine Properties of Functions. Revised Edition, Textbooks in Mathematics, Chapman & Hall/CRC, 2015 | DOI

[8] Grégoire Loeper On the regularity of solutions of optimal transportation problems, Acta Math., Volume 202 (2009) no. 2, pp. 241-283 | MR | Zbl | DOI

[9] Xi-Nan Ma; Neil S. Trudinger; Xu-Jia Wang Regularity of potential functions of the optimal transportation problem, Arch. Ration. Mech. Anal., Volume 117 (2005) no. 2, pp. 151-183 | MR | Zbl

[10] Felix Otto Doubly Degenerate Diffusion Equations as Steepest Descent (unpublished)

[11] Filippo Santambrogio Optimal Transport for Applied Mathematicians, Progress in Nonlinear Differential Equations and their Applications, 87, Birkhäuser, 2015 | DOI

[12] Cédric Villani Topics in Optimal Transportation, Graduate Studies in Mathematics, 58, American Mathematical Society, 2003

[13] Cédric Villani Optimal Transport, Old and New, Grundlehren der Mathematischen Wissenschaften, 338, Springer, 2009 | DOI

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