|
Monge's transport problem on a Riemannian manifold
Author(s):
Mikhail
Feldman;
Robert
J.
McCann
Journal:
Trans. Amer. Math. Soc.
354
(2002),
1667-1697.
MSC (2000):
Primary 49Q20, 28A50
Posted:
December 4, 2001
Retrieve article in:
PDF DVI PostScript
This article is available free of charge
Abstract |
References |
Similar articles |
Additional information
Abstract:
Monge's problem refers to the classical problem of optimally transporting mass: given Borel probability measures  , find the measure-preserving map  between them which minimizes the average distance transported. Set on a complete, connected, Riemannian manifold  -- and assuming absolute continuity of  -- an optimal map will be shown to exist. Aspects of its uniqueness are also established.
References:
-
- 1.
- L. Ambrosio.
Lecture notes on optimal transport problems.
Preprint. - 2.
- L. Caffarelli.
Allocation maps with general cost functions.
In P. Marcellini et al., editor, Partial Differential Equations and Applications, number 177 in Lecture Notes in Pure and Appl. Math., pages 29-35. Dekker, New York, 1996. MR 97f:99055 - 3.
- L. Caffarelli, M. Feldman, R.J. McCann.
Constructing optimal maps for Monge's transport problem as a limit of strictly convex costs.
J. Amer. Math. Soc. 15 (2002), 1-26. - 4.
- J. Cheeger, D.G. Ebin. Comparison Theorems in Riemannian Geometry. North-Holland Publishing Co., Amsterdam-Oxford, 1975. MR 56:16538
- 5.
- M.P. Do Carmo. Riemannian geometry. Birkhäuser Boston, Inc., Boston, MA, 1992. MR 92i:53001
- 6.
- L.C. Evans.
Partial differential equations and Monge-Kantorovich mass transfer.
In R. Bott et al., editor, Current Developments in Mathematics, pages 65-128. International Press, Cambridge, 1999. MR 2000e:49001 - 7.
- L.C. Evans and W. Gangbo.
Differential equations methods for the Monge-Kantorovich mass transfer problem.
Mem. Amer. Math. Soc., 137:1-66, 1999. MR 99g:35132 - 8.
- L.C. Evans, R.F. Gariepy.
Measure theory and fine properties of functions.
CRC Press, 1992. - 9.
- H. Federer.
Geometric Measure Theory.
Springer-Verlag, New York, 1969. MR 41:1976 - 10.
- M. Feldman.
Variational evolution problems and nonlocal geometric motion.
Arch. Rat. Mech. Anal., 146:221-274, 1999. MR 2000h:35066 - 11.
- M. Feldman, R.J. McCann.
Uniqueness and transport density in Monge's mass transportation problem.
To appear in Calc. Var. Partial Differential Equations. - 12.
- L. Kantorovitch.
On the translocation of masses.
C.R. (Doklady) Acad. Sci. URSS (N.S.), 37:199-201, 1942. MR 5:174d - 13.
- W. Klingenberg Riemannian geometry. de Gruyter Studies in Mathematics, 1. Walter de Gruyter & Co., Berlin-New York, 1982. 396 pp. MR 84j:53001
- 14.
- R.J. McCann.
Polar factorization of maps on Riemannian manifolds.
Geom. Funct. Anal. 11 (2001), 589-608. - 15.
- G. Monge.
Mémoire sur la théorie des déblais et de remblais.
Histoire de l'Académie Royale des Sciences de Paris, avec les Mémoires de Mathématique et de Physique pour la même année, pages 666-704, 1781. - 16.
- S.T. Rachev and L. Rüschendorf.
Mass Transportation Problems.
Probab. Appl. Springer-Verlag, New York, 1998. MR 99k:28006; MR 99k:28007 - 17.
- M. Spivak
A comprehensive introduction to differential geometry.
Vol. I. Second edition. Publish or Perish, Inc., Wilmington, Del., 1979. xiv+668 pp MR 82g:53003a - 18.
- V.N. Sudakov.
Geometric problems in the theory of infinite-dimensional probability distributions.
Proc. Steklov Inst. Math., 141, 1979. MR 80e:60052 - 19.
- N.S. Trudinger, X.-J. Wang.
On the Monge mass transfer problem.
Calc. Var. Partial Differential Equations, 13 (2001), no. 1, 19-31. - 20.
- C. Villani.
Topics in mass transportation.
In preparation.
Similar Articles:
Retrieve articles in Transactions of the American Mathematical Society
with MSC
(2000):
49Q20, 28A50
Retrieve articles in all Journals with MSC
(2000):
49Q20, 28A50
Additional Information:
Mikhail
Feldman
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Email:
feldman@math.wisc.edu
Robert
J.
McCann
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3
Email:
mccann@math.toronto.edu
DOI:
10.1090/S0002-9947-01-02930-0
PII:
S 0002-9947(01)02930-0
Keywords:
Monge-Kantorovich mass transportation,
Riemannian manifold,
optimal map,
dual problem
Received by editor(s):
March 30, 2001
Posted:
December 4, 2001
Additional Notes:
The authors gratefully acknowledge the support of grants DMS 0096090 [MF] and 0074037 [MF and RJM] of the U.S. National Science Foundation, and grant 217006-99 [RJM] of the Natural Sciences and Engineering Research Council of Canada.
Copyright of article:
Copyright
2001,
by the authors
|
Comments: webmaster@ams.org