Partial regularity for optimal transport maps
De Philippis, Guido1 ; Figalli, Alessio2
1 Scuola Normale Superiore p.za dei Cavalieri 7 56126 Pisa Italy
2 Department of Mathematics, The University of Texas at Austin 1 University Station C1200 78712 Austin TX USA
Publications Mathématiques de l'IHÉS, Tome 121 (2015), pp. 81-112.

We prove that, for general cost functions on Rn, or for the cost d2/2 on a Riemannian manifold, optimal transport maps between smooth densities are always smooth outside a closed singular set of measure zero.

DOI : 10.1007/s10240-014-0064-7
Mots-clés : Riemannian Manifold, Partial Regularity, Optimal Transport, Optimal Transportation, Smooth Density
De Philippis, Guido 1 ; Figalli, Alessio 2

1 Scuola Normale Superiore p.za dei Cavalieri 7 56126 Pisa Italy
2 Department of Mathematics, The University of Texas at Austin 1 University Station C1200 78712 Austin TX USA 
@article{PMIHES_2015__121__81_0,
     author = {De Philippis, Guido and Figalli, Alessio},
     title = {Partial regularity for optimal transport maps},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {81--112},
     publisher = {Springer Berlin Heidelberg},
     address = {Berlin/Heidelberg},
     volume = {121},
     year = {2015},
     doi = {10.1007/s10240-014-0064-7},
     language = {en},
     url = {http://www.numdam.org/articles/10.1007/s10240-014-0064-7/}
}
TY  - JOUR
AU  - De Philippis, Guido
AU  - Figalli, Alessio
TI  - Partial regularity for optimal transport maps
JO  - Publications Mathématiques de l'IHÉS
PY  - 2015
SP  - 81
EP  - 112
VL  - 121
PB  - Springer Berlin Heidelberg
PP  - Berlin/Heidelberg
UR  - http://www.numdam.org/articles/10.1007/s10240-014-0064-7/
DO  - 10.1007/s10240-014-0064-7
LA  - en
ID  - PMIHES_2015__121__81_0
ER  - 
%0 Journal Article
%A De Philippis, Guido
%A Figalli, Alessio
%T Partial regularity for optimal transport maps
%J Publications Mathématiques de l'IHÉS
%D 2015
%P 81-112
%V 121
%I Springer Berlin Heidelberg
%C Berlin/Heidelberg
%U http://www.numdam.org/articles/10.1007/s10240-014-0064-7/
%R 10.1007/s10240-014-0064-7
%G en
%F PMIHES_2015__121__81_0
De Philippis, Guido; Figalli, Alessio. Partial regularity for optimal transport maps. Publications Mathématiques de l'IHÉS, Tome 121 (2015), pp. 81-112. doi : 10.1007/s10240-014-0064-7. http://www.numdam.org/articles/10.1007/s10240-014-0064-7/

[1.] Ambrosio, L.; Gigli, N.; Savaré, G. Gradient Flows in Metric Spaces and in the Space of Probability Measures (2008) | Zbl

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

[3.] Caffarelli, L. A. A localization property of viscosity solutions to the Monge-Ampère equation and their strict convexity, Ann. Math. (2), Volume 131 (1990), pp. 129-134 | DOI | MR | Zbl

[4.] Caffarelli, L. A. Some regularity properties of solutions of Monge Ampère equation, Commun. Pure Appl. Math., Volume 44 (1991), pp. 965-969 | DOI | MR | Zbl

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

[6.] Caffarelli, L. A. Interior W2,p estimates for solutions of the Monge-Ampère equation, Ann. Math. (2), Volume 131 (1990), pp. 135-150 | DOI | MR | Zbl

[7.] L. A. Caffarelli, M. M. Gonzáles and T. Nguyen, A perturbation argument for a Monge-Ampère type equation arising in optimal transportation. Preprint (2011).

[8.] Caffarelli, L. A.; Li, Y. Y. A Liouville theorem for solutions of the Monge-Ampère equation with periodic data, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 21 (2004), pp. 97-120 | Numdam | MR | Zbl

[9.] Cordero-Erausquin, D.; McCann, R. J.; Schmuckenschläger, M. A Riemannian interpolation inequality à la Borell, Brascamp and Lieb, Invent. Math., Volume 146 (2001), pp. 219-257 | DOI | MR | Zbl

[10.] Delanoë, P.; Ge, Y. Regularity of optimal transportation maps on compact, locally nearly spherical, manifolds, J. Reine Angew. Math., Volume 646 (2010), pp. 65-115 | MR | Zbl

[11.] Delanoë, P.; Rouvière, F. Positively curved Riemannian locally symmetric spaces are positively squared distance curved, Can. J. Math., Volume 65 (2013), pp. 757-767 | DOI | Zbl

[12.] Evans, L. C.; Gariepy, R. F. Measure Theory and Fine Properties of Functions (1992) | Zbl

[13.] Fathi, A.; Figalli, A. Optimal transportation on non-compact manifolds, Isr. J. Math., Volume 175 (2010), pp. 1-59 | DOI | MR | Zbl

[14.] Figalli, A. Existence, uniqueness, and regularity of optimal transport maps, SIAM J. Math. Anal., Volume 39 (2007), pp. 126-137 | DOI | MR | Zbl

[15.] A. Figalli, Regularity of optimal transport maps [after Ma-Trudinger-Wang and Loeper]. (English summary) Séminaire Bourbaki. Volume 2008/2009. Exposés 997–1011. Astérisque No. 332 (2010), Exp. No. 1009, ix, 341–368.

[16.] Figalli, A. Regularity properties of optimal maps between nonconvex domains in the plane, Commun. Partial Differ. Equ., Volume 35 (2010), pp. 465-479 | DOI | MR | Zbl

[17.] Figalli, A.; Gigli, N. Local semiconvexity of Kantorovich potentials on non-compact manifolds, ESAIM Control Optim. Calc. Var., Volume 17 (2011), pp. 648-653 | DOI | Numdam | MR | Zbl

[18.] Figalli, A.; Kim, Y. H. Partial regularity of Brenier solutions of the Monge-Ampère equation, Discrete Contin. Dyn. Syst., Volume 28 (2010), pp. 559-565 | DOI | MR | Zbl

[19.] Figalli, A.; Kim, Y. H.; McCann, R. J. Hölder continuity and injectivity of optimal maps, Arch. Ration. Mech. Anal., Volume 209 (2013), pp. 747-795 | DOI | MR | Zbl

[20.] Figalli, A.; Kim, Y. H.; McCann, R. J. Regularity of optimal transport maps on multiple products of spheres, J. Eur. Math. Soc. (JEMS), Volume 5 (2013), pp. 1131-1166 | DOI | MR

[21.] Figalli, A.; Loeper, G. C1 regularity of solutions of the Monge-Ampère equation for optimal transport in dimension two, Calc. Var. Partial Differ. Equ., Volume 35 (2009), pp. 537-550 | DOI | MR | Zbl

[22.] Figalli, A.; Rifford, L. Continuity of optimal transport maps and convexity of injectivity domains on small deformations of S2, Commun. Pure Appl. Math., Volume 62 (2009), pp. 1670-1706 | DOI | MR | Zbl

[23.] Figalli, A.; Rifford, L.; Villani, C. On the Ma-Trudinger-Wang curvature on surfaces, Calc. Var. Partial Differ. Equ., Volume 39 (2010), pp. 307-332 | DOI | MR | Zbl

[24.] Figalli, A.; Rifford, L.; Villani, C. Necessary and sufficient conditions for continuity of optimal transport maps on Riemannian manifolds, Tohoku Math. J. (2), Volume 63 (2011), pp. 855-876 | DOI | MR | Zbl

[25.] Figalli, A.; Rifford, L.; Villani, C. Nearly round spheres look convex, Am. J. Math., Volume 134 (2012), pp. 109-139 | DOI | MR | Zbl

[26.] Gutierrez, C. The Monge-Ampére Equation (2001) | DOI | Zbl

[27.] Jian, H.-Y.; Wang, X.-J. Continuity estimates for the Monge-Ampère equation, SIAM J. Math. Anal., Volume 39 (2007), pp. 608-626 | DOI | MR | Zbl

[28.] Y.-H. Kim, Counterexamples to continuity of optimal transport maps on positively curved Riemannian manifolds, Int. Math. Res. Not. IMRN 2008, Art. ID rnn120, 15 pp.

[29.] Kim, Y.-H.; McCann, R. J. Towards the smoothness of optimal maps on Riemannian submersions and Riemannian products (of round spheres in particular), J. Reine Angew. Math., Volume 664 (2012), pp. 1-27 | DOI | MR | Zbl

[30.] Liu, J.; Trudinger, N. S.; Wang, X.-J. Interior C2,α regularity for potential functions in optimal transportation, Commun. Partial Differ. Equ., Volume 35 (2010), pp. 165-184 | DOI | MR | Zbl

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

[32.] Loeper, G. Regularity of optimal maps on the sphere: The quadratic cost and the reflector antenna, Arch. Ration. Mech. Anal., Volume 199 (2011), pp. 269-289 | DOI | MR | Zbl

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

[34.] McCann, R. J. Polar factorization of maps on Riemannian manifolds, Geom. Funct. Anal., Volume 11 (2001), pp. 589-608 | DOI | MR | Zbl

[35.] Trudinger, N. S.; Wang, X.-J. On the second boundary value problem for Monge-Ampère type equations and optimal transportation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), Volume 8 (2009), pp. 143-174 | Numdam | MR | Zbl

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

[37.] Villani, C. Optimal Transport. Old and New (2009) | DOI | Zbl

Cité par Sources :