November  2019, 18(6): 3001-3009. doi: 10.3934/cpaa.2019134

$ L^{p, q} $ estimates on the transport density

Laboratoire de Mathématiques d'Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay Cedex, France

Received  September 2018 Revised  March 2019 Published  May 2019

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) $.

Citation: Samer Dweik. $ L^{p, q} $ estimates on the transport density. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3001-3009. doi: 10.3934/cpaa.2019134
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. Google Scholar

[2]

M. Beckmann, A continuous model of transportation, Econometrica, 20 (1952), 643-660. doi: 10.2307/1907646. Google Scholar

[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. Google Scholar

[4]

L. De PascaleL. C. Evans and A. Pratelli, Integral estimates for transport densities, Bull. of the London Math. Soc., 36 (2004), 383-395. doi: 10.1112/S0024609303003035. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

L. Kantorovich, On the transfer of masses, Dokl. Acad. Nauk. USSR, 37 (1942), 7-8. Google Scholar

[9]

G. Monge, Mémoire sur la théorie des déblais et des remblais, Histoire de l'Académie Royale des Sciences de Paris, (1781), 666-704. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, Vol. 58, 2003. doi: 10.1007/b12016. Google Scholar

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. Google Scholar

[2]

M. Beckmann, A continuous model of transportation, Econometrica, 20 (1952), 643-660. doi: 10.2307/1907646. Google Scholar

[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. Google Scholar

[4]

L. De PascaleL. C. Evans and A. Pratelli, Integral estimates for transport densities, Bull. of the London Math. Soc., 36 (2004), 383-395. doi: 10.1112/S0024609303003035. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

L. Kantorovich, On the transfer of masses, Dokl. Acad. Nauk. USSR, 37 (1942), 7-8. Google Scholar

[9]

G. Monge, Mémoire sur la théorie des déblais et des remblais, Histoire de l'Académie Royale des Sciences de Paris, (1781), 666-704. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, Vol. 58, 2003. doi: 10.1007/b12016. Google Scholar

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