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$ L^{p, q} $ estimates on the transport density
Laboratoire de Mathématiques d'Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay Cedex, France |
In this paper, we show a new regularity result on the transport density $ \sigma $ in the classical Monge-Kantorovich optimal mass transport problem between two measures, $ \mu $ and $ \nu $, having some summable densities, $ f^+ $ and $ f^- $. More precisely, we prove that the transport density $ \sigma $ belongs to $ L^{p,q}(\Omega) $ as soon as $ f^+,\,f^- \in L^{p,q}(\Omega) $.
References:
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L. Ambrosio, Lecture notes on optimal transport problems, in Mathematical Aspects of Evolving Interfaces, Lecture Notes in Mathematics (1812), Springer, New York, 2003, 1–52.
doi: 10.1007/978-3-540-39189-0_1. |
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M. Beckmann,
A continuous model of transportation, Econometrica, 20 (1952), 643-660.
doi: 10.2307/1907646. |
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R. E. Castillo and H. Rafeiro, An Introductory Course in Lebesgue Spaces, Springer International Publishing, 2016.
doi: 10.1007/978-3-319-30034-4. |
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L. De Pascale, L. C. Evans and A. Pratelli,
Integral estimates for transport densities, Bull. of the London Math. Soc., 36 (2004), 383-395.
doi: 10.1112/S0024609303003035. |
| [5] |
L. De Pascale and A. Pratelli,
Sharp summability for Monge transport density via interpolation, ESAIM Control Optim. Calc. Var., 10 (2004), 549-552.
doi: 10.1051/cocv:2004019. |
| [6] |
L. C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc., 137 (1999), no. 653.
doi: 10.1090/memo/0653. |
| [7] |
M. Feldman and R. McCann,
Uniqueness and transport density in Monge's mass transportation problem, Calc. Var. Par. Diff. Eq., 15 (2002), 81-113.
doi: 10.1007/s005260100119. |
| [8] |
L. Kantorovich,
On the transfer of masses, Dokl. Acad. Nauk. USSR, 37 (1942), 7-8.
|
| [9] |
G. Monge,
Mémoire sur la théorie des déblais et des remblais, Histoire de l'Académie Royale des Sciences de Paris, (1781), 666-704.
|
| [10] |
F. Santambrogio,
Absolute continuity and summability of transport densities: simpler proofs and new estimates, Calc. Var. Par. Diff. Eq., 36 (2009), 343-354.
doi: 10.1007/s00526-009-0231-8. |
| [11] |
F. Santambrogio, Optimal Transport for Applied Mathematicians, in Progress in Nonlinear Differential Equations and Their Applications, 87, Birkhäuser Basel (2015).
doi: 10.1007/978-3-319-20828-2. |
| [12] |
C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, Vol. 58, 2003.
doi: 10.1007/b12016. |
show all references
References:
| [1] |
L. Ambrosio, Lecture notes on optimal transport problems, in Mathematical Aspects of Evolving Interfaces, Lecture Notes in Mathematics (1812), Springer, New York, 2003, 1–52.
doi: 10.1007/978-3-540-39189-0_1. |
| [2] |
M. Beckmann,
A continuous model of transportation, Econometrica, 20 (1952), 643-660.
doi: 10.2307/1907646. |
| [3] |
R. E. Castillo and H. Rafeiro, An Introductory Course in Lebesgue Spaces, Springer International Publishing, 2016.
doi: 10.1007/978-3-319-30034-4. |
| [4] |
L. De Pascale, L. C. Evans and A. Pratelli,
Integral estimates for transport densities, Bull. of the London Math. Soc., 36 (2004), 383-395.
doi: 10.1112/S0024609303003035. |
| [5] |
L. De Pascale and A. Pratelli,
Sharp summability for Monge transport density via interpolation, ESAIM Control Optim. Calc. Var., 10 (2004), 549-552.
doi: 10.1051/cocv:2004019. |
| [6] |
L. C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc., 137 (1999), no. 653.
doi: 10.1090/memo/0653. |
| [7] |
M. Feldman and R. McCann,
Uniqueness and transport density in Monge's mass transportation problem, Calc. Var. Par. Diff. Eq., 15 (2002), 81-113.
doi: 10.1007/s005260100119. |
| [8] |
L. Kantorovich,
On the transfer of masses, Dokl. Acad. Nauk. USSR, 37 (1942), 7-8.
|
| [9] |
G. Monge,
Mémoire sur la théorie des déblais et des remblais, Histoire de l'Académie Royale des Sciences de Paris, (1781), 666-704.
|
| [10] |
F. Santambrogio,
Absolute continuity and summability of transport densities: simpler proofs and new estimates, Calc. Var. Par. Diff. Eq., 36 (2009), 343-354.
doi: 10.1007/s00526-009-0231-8. |
| [11] |
F. Santambrogio, Optimal Transport for Applied Mathematicians, in Progress in Nonlinear Differential Equations and Their Applications, 87, Birkhäuser Basel (2015).
doi: 10.1007/978-3-319-20828-2. |
| [12] |
C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, Vol. 58, 2003.
doi: 10.1007/b12016. |
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