Regularity of optimal transport maps [after Ma-Trudinger-Wang and Loeper]
Séminaire Bourbaki : volume 2008/2009 exposés 997-1011 - Avec table par noms d'auteurs de 1848/49 à 2008/09, Astérisque, no. 332 (2010), Exposé no. 1009, 28 p.
@incollection{AST_2010__332__341_0,
     author = {Figalli, Alessio},
     title = {Regularity of optimal transport maps [after {Ma-Trudinger-Wang} and {Loeper]}},
     booktitle = {S\'eminaire Bourbaki : volume 2008/2009 expos\'es 997-1011  - Avec table par noms d'auteurs de 1848/49 \`a 2008/09},
     series = {Ast\'erisque},
     note = {talk:1009},
     pages = {341--368},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {332},
     year = {2010},
     mrnumber = {2648684},
     zbl = {1211.49054},
     language = {en},
     url = {http://www.numdam.org/item/AST_2010__332__341_0/}
}
TY  - CHAP
AU  - Figalli, Alessio
TI  - Regularity of optimal transport maps [after Ma-Trudinger-Wang and Loeper]
BT  - Séminaire Bourbaki : volume 2008/2009 exposés 997-1011  - Avec table par noms d'auteurs de 1848/49 à 2008/09
AU  - Collectif
T3  - Astérisque
N1  - talk:1009
PY  - 2010
SP  - 341
EP  - 368
IS  - 332
PB  - Société mathématique de France
UR  - http://www.numdam.org/item/AST_2010__332__341_0/
LA  - en
ID  - AST_2010__332__341_0
ER  - 
%0 Book Section
%A Figalli, Alessio
%T Regularity of optimal transport maps [after Ma-Trudinger-Wang and Loeper]
%B Séminaire Bourbaki : volume 2008/2009 exposés 997-1011  - Avec table par noms d'auteurs de 1848/49 à 2008/09
%A Collectif
%S Astérisque
%Z talk:1009
%D 2010
%P 341-368
%N 332
%I Société mathématique de France
%U http://www.numdam.org/item/AST_2010__332__341_0/
%G en
%F AST_2010__332__341_0
Figalli, Alessio. Regularity of optimal transport maps [after Ma-Trudinger-Wang and Loeper], dans Séminaire Bourbaki : volume 2008/2009 exposés 997-1011  - Avec table par noms d'auteurs de 1848/49 à 2008/09, Astérisque, no. 332 (2010), Exposé no. 1009, 28 p. http://www.numdam.org/item/AST_2010__332__341_0/

[1] Y. Brenier - "Decomposition polaire et réarrangement monotone des champs de vecteurs", C.R. Acad. Sci. Paris Ser. I Math. 305 (1987), p. 805-808. | MR | Zbl

[2] Y. Brenier, "Polar factorization and monotone rearrangement of vector-valued functions", Comm. Pure Appl. Math. 44 (1991), p. 375-417. | DOI | MR | Zbl

[3] X. Cabré - "Nondivergent elliptic equations on manifolds with nonnegative curvature", Comm. Pure Appl. Math. 50 (1997), p. 623-665. | DOI | MR | Zbl

[4] L. A. Caffarelli - "A localization property of viscosity solutions to the Monge-Ampere equation and their strict convexity", Ann. of Math. 131 (1990), p. 129-134. | DOI | MR | Zbl

[5] L. A. Caffarelli, "Some regularity properties of solutions of Monge Ampere equation", Comm. Pure Appl. Math. 44 (1991), p. 965-969. | DOI | MR | Zbl

[6] L. A. Caffarelli, "The regularity of mappings with a convex potential", J. Amer. Math. Soc. 5 (1992), p. 99-104. | DOI | MR | Zbl

[7] L. A. Caffarelli, "Boundary regularity of maps with convex potentials. II", Ann. of Math. 144 (1996), p. 453-496. | DOI | MR | Zbl

[8] M. Castelpietra & L. Rifford - "Regularity properties of the distance function to conjugate and cut loci for viscosity solutions of Hamilton-Jacobi equations and applications in Riemannian geometry", to appear in ESAIM Control Optim. Calc. Var.. | EuDML | Numdam | MR | Zbl

[9] D. Cordero-Erausquin, R. J. Mccann & M. Schmuckenschläger - "A Riemannian interpolation inequality a la Borell, Brascamp and Lieb", Invent. Math. 146 (2001), p. 219-257. | DOI | MR | Zbl

[10] J. A. Cuesta & C. Matrán - "Notes on the Wasserstein metric in Hilbert spaces", Ann. Probab. 17 (1989), p. 1264-1276. | DOI | MR | Zbl

[11] P. Delanoë & Y. Ge - "Regularity of optimal transportation maps on compact, locally nearly spherical, manifolds", to appear in J. reine angew. Math.. | MR | Zbl

[12] A. Figalli & L. Rifford - "Continuity of optimal transport maps and convexity of injectivity domains on small deformations of 𝕊 2 ", Comm. Pure Appl. Math. 62 (2009), p. 1670-1706. | DOI | MR | Zbl

[13] A. Figalli, L. Rifford & - C. Villani "Nearly round spheres look convex", preprint, 2009. | MR | Zbl

[14] A. Figalli, "Necessary and sufficient conditions for continuity of optimal transport maps on Riemannian manifolds", in preparation. | DOI | Zbl

[15] A. Figalli, "On the Ma-Trudinger-Wang curvature tensor on surfaces", to appear in Calc. Var. Partial Differential Equations. | MR

[16] A. Figalli & C. Villani - "Optimal transport and curvature", Lecture notes, http://cvgmt.sns.it/cgi/get.cgi/papers/figvil08/0Tcurv-6.pdf. | DOI | MR | Zbl

[17] S. Gallot, D. Hulin & J. Lafontaine- Riemannian geometry, third ed., Universitext, Springer, 2004. | DOI | MR

[18] J.-I. Itoh & M. Tanaka - "The Lipschitz continuity of the distance function to the cut locus", Trans. Amer. Math. Soc. 353 (2001), p. 21-40. | DOI | MR | Zbl

[19] Y.-H. Kim - "Counterexamples to continuity of optimal transport maps on positively curved Riemannian manifolds", Int. Math. Res. Not. 2008 (2008), art. ID rnnl20, 15. | MR | Zbl

[20] Y.-H. Kim & R. J. Mccann - "Continuity, curvature, and the general covariance of optimal transportation", to appear in J. Eur. Math. Soc. | EuDML | MR | Zbl

[21] Y.-H. Kim & R. J. Mccann, "Towards the smoothness of optimal maps on Riemannian submersions and Riemannian products (of round spheres in particular)", preprint, to appear in J. reine angew. Math. | MR | Zbl

[22] Y. Li & L. Nirenberg - "The distance function to the boundary, Finsler geometry, and the singular set of viscosity solutions of some Hamilton-Jacobi equations", Comm. Pure Appl. Math. 58 (2005), p. 85-146. | DOI | MR | Zbl

[23] J. Liu - "Hölder regularity of optimal mappings in optimal transportation", Calc. Var. Partial Differential Equations 34 (2009), p. 435-451. | DOI | MR | Zbl

[24] G. Loeper - "On the regularity of solutions of optimal transportation problems", Acta Math. 202 (2009), p. 241-283. | DOI | MR | Zbl

[25] G. Loeper, "Regularity of optimal maps on the sphere: The quadratic cost and the reflector antenna", to appear in Arch. Ration. Mech. Anal. | MR | Zbl

[26] G. Loeper & C. Villani - "Regularity of optimal transport in curved geometry: the nonfocal case", preprint, to appear in Duke Math. J. | MR | Zbl

[27] X.-N. Ma, N. S. Trudinger & X.-J. Wang - "Regularity of potential functions of the optimal transportation problem", Arch. Ration. Mech. Anal. 177 (2005), p. 151-183. | DOI | MR | Zbl

[28] R. J. Mccann - "Polar factorization of maps on Riemannian manifolds", Geom. Fund. Anal. 11 (2001), p. 589-608. | DOI | MR | Zbl

[29] G. Monge - "Mémoire sur la théorie des déblais et des remblais", Hist. de l'Acad. des Sciences de Paris (1781), p. 666-704.

[30] S. T. Rachev & L. Rüschendorf - Mass transportation problems. Vol. I, Probability and its Applications (New York), Springer, 1998. | MR | Zbl

[31] N. S. Trudinger & X.-J. Wang - "On strict convexity and continuous differentiability of potential functions in optimal transportation", Arch. Ration. Mech. Anal. 192 (2009), p. 403-418. | DOI | MR | Zbl

[32] N. S. Trudinger & X.-J. Wang, "On the second boundary value problem for Monge-Ampere type equations and optimal transportation", Ann. Sc. Norm. Super. Pisa Cl. Sci. 8 (2009), p. 143-174. | Numdam | MR | Zbl

[33] J. Urbas - "On the second boundary value problem for equations of Monge-Ampère type", J. reine angew. Math. 487 (1997), p. 115-124. | EuDML | MR | Zbl

[34] C. Villani - Optimal transport, old and new, Grundl. Math. Wiss., vol. 338, Springer, 2009. | MR | Zbl

[35] X.-J. Wang - "On the design of a reflector antenna", Inverse Problems 12 (1996), p. 351-375. | DOI | MR | Zbl