A new estimate in optimal mass transport
Authors:
G. Bouchitté, C. Jimenez and M. Rajesh
Journal:
Proc. Amer. Math. Soc. 135 (2007), 3525-3535
MSC (2000):
Primary 39B62, 46N10, 49Q20
DOI:
https://doi.org/10.1090/S0002-9939-07-08877-6
Published electronically:
July 3, 2007
MathSciNet review:
2336567
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: Let be a bounded Lipschitz regular open subset of
and let
be two probablity measures on
. It is well known that if
is absolutely continuous, then there exists, for every
, a unique transport map
pushing forward
on
and which realizes the Monge-Kantorovich distance
. In this paper, we establish an
bound for the displacement map
which depends only on
, on the shape of
and on the essential infimum of the density
.
- 1. Luis A. Caffarelli, Interior 𝑊^{2,𝑝} estimates for solutions of the Monge-Ampère equation, Ann. of Math. (2) 131 (1990), no. 1, 135–150. MR 1038360, https://doi.org/10.2307/1971510
- 2. Luis A. Caffarelli, Some regularity properties of solutions of Monge Ampère equation, Comm. Pure Appl. Math. 44 (1991), no. 8-9, 965–969. MR 1127042, https://doi.org/10.1002/cpa.3160440809
- 3. Luigi Ambrosio, Lecture notes on optimal transport problems, Mathematical aspects of evolving interfaces (Funchal, 2000) Lecture Notes in Math., vol. 1812, Springer, Berlin, 2003, pp. 1–52. MR 2011032, https://doi.org/10.1007/978-3-540-39189-0_1
- 4. Luigi Ambrosio and Aldo Pratelli, Existence and stability results in the 𝐿¹ theory of optimal transportation, Optimal transportation and applications (Martina Franca, 2001) Lecture Notes in Math., vol. 1813, Springer, Berlin, 2003, pp. 123–160. MR 2006307, https://doi.org/10.1007/978-3-540-44857-0_5
- 5. Guy Bouchitté, Chloé Jimenez, and Mahadevan Rajesh, Asymptotique d’un problème de positionnement optimal, C. R. Math. Acad. Sci. Paris 335 (2002), no. 10, 853–858 (French, with English and French summaries). MR 1947712, https://doi.org/10.1016/S1631-073X(02)02575-X
- 6. Luis A. Caffarelli, Mikhail Feldman, and Robert J. McCann, Constructing optimal maps for Monge’s transport problem as a limit of strictly convex costs, J. Amer. Math. Soc. 15 (2002), no. 1, 1–26. MR 1862796, https://doi.org/10.1090/S0894-0347-01-00376-9
- 7. Lawrence C. Evans, Partial differential equations and Monge-Kantorovich mass transfer, Current developments in mathematics, 1997 (Cambridge, MA), Int. Press, Boston, MA, 1999, pp. 65–126. MR 1698853
- 8. L. C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc. 137 (1999), no. 653, viii+66. MR 1464149, https://doi.org/10.1090/memo/0653
- 9. Wilfrid Gangbo and Robert J. McCann, The geometry of optimal transportation, Acta Math. 177 (1996), no. 2, 113–161. MR 1440931, https://doi.org/10.1007/BF02392620
- 10. L. V. Kantorovich, On the transfer of masses, Dokl. Akad. Nauk. SSSR 37 227-229 (1942).
- 11. G. Monge, Mémoire sur la theorie des deblais et des remblais, Histoire de l'Académie Royale des Sciences, Paris (1781).
- 12. V. N. Sudakov, Geometric problems in the theory of infinite-dimensional probability distributions, Proc. Steklov Inst. Math. 2 (1979), i–v, 1–178. Cover to cover translation of Trudy Mat. Inst. Steklov 141 (1976). MR 530375
- 13. Cédric Villani, Topics in optimal transportation, Graduate Studies in Mathematics, vol. 58, American Mathematical Society, Providence, RI, 2003. MR 1964483
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Additional Information
G. Bouchitté
Affiliation:
UFR Sciences, Université du Sud-Toulon-Var, BP20132, 83957 La Garde Cedex, France
Email:
bouchitte@univ-tln.fr
C. Jimenez
Affiliation:
UFR Sciences, Université du Sud-Toulon-Var, BP20132, 83957 La Garde Cedex, France
Email:
c.jimenez@sns.it
M. Rajesh
Affiliation:
Departemento de Matematica, Facultad de Ciencias Fisicas y Matematicas, Universidad de Concepcion, Casilla 160-C. Concepcion, Chile
Email:
rmahadevan@udec.cl
DOI:
https://doi.org/10.1090/S0002-9939-07-08877-6
Keywords:
Wasserstein distance,
optimal transport map,
uniform estimates
Received by editor(s):
January 9, 2006
Received by editor(s) in revised form:
June 23, 2006
Published electronically:
July 3, 2007
Communicated by:
David Preiss
Article copyright:
© Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.