American Institute of Mathematical Sciences

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

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

Samer Dweik

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) $.

Keywords: Monge-Kantorovich problem, transport density, Lorentz regularity.
Mathematics Subject Classification: 35B65, 46N10, 49N60.
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 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.  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 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.  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

[1]

Giuseppe Buttazzo, Eugene Stepanov. Transport density in Monge-Kantorovich problems with Dirichlet conditions. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 607-628. doi: 10.3934/dcds.2005.12.607
[2]

Abbas Moameni. Invariance properties of the Monge-Kantorovich mass transport problem. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2653-2671. doi: 10.3934/dcds.2016.36.2653
[3]

Jesus Garcia Azorero, Juan J. Manfredi, I. Peral, Julio D. Rossi. Limits for Monge-Kantorovich mass transport problems. Communications on Pure & Applied Analysis, 2008, 7 (4) : 853-865. doi: 10.3934/cpaa.2008.7.853
[4]

Zuo Quan Xu, Jia-An Yan. A note on the Monge-Kantorovich problem in the plane. Communications on Pure & Applied Analysis, 2015, 14 (2) : 517-525. doi: 10.3934/cpaa.2015.14.517
[5]

Yupeng Li, Wuchen Li, Guo Cao. Image segmentation via $ L_1 $ Monge-Kantorovich problem. Inverse Problems & Imaging, 2019, 13 (4) : 805-826. doi: 10.3934/ipi.2019037
[6]

Nassif Ghoussoub, Bernard Maurey. Remarks on multi-marginal symmetric Monge-Kantorovich problems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1465-1480. doi: 10.3934/dcds.2014.34.1465
[7]

Paolo Maremonti. A remark on the Stokes problem in Lorentz spaces. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1323-1342. doi: 10.3934/dcdss.2013.6.1323
[8]

Qi-Rui Li, Xu-Jia Wang. Regularity of the homogeneous Monge-Ampère equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 6069-6084. doi: 10.3934/dcds.2015.35.6069
[9]

Alessio Figalli, Young-Heon Kim. Partial regularity of Brenier solutions of the Monge-Ampère equation. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 559-565. doi: 10.3934/dcds.2010.28.559
[10]

Dario Cordero-Erausquin, Alessio Figalli. Regularity of monotone transport maps between unbounded domains. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7101-7112. doi: 10.3934/dcds.2019297
[11]

Sun-Sig Byun, Yumi Cho. Lorentz-Morrey regularity for nonlinear elliptic problems with irregular obstacles over Reifenberg flat domains. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4791-4804. doi: 10.3934/dcds.2015.35.4791
[12]

Bingkang Huang, Lusheng Wang, Qinghua Xiao. Global nonlinear stability of rarefaction waves for compressible Navier-Stokes equations with temperature and density dependent transport coefficients. Kinetic & Related Models, 2016, 9 (3) : 469-514. doi: 10.3934/krm.2016004
[13]

Jingang Xiong, Jiguang Bao. The obstacle problem for Monge-Ampère type equations in non-convex domains. Communications on Pure & Applied Analysis, 2011, 10 (1) : 59-68. doi: 10.3934/cpaa.2011.10.59
[14]

Shouchuan Hu, Haiyan Wang. Convex solutions of boundary value problem arising from Monge-Ampère equations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 705-720. doi: 10.3934/dcds.2006.16.705
[15]

Gyula Csató. On the isoperimetric problem with perimeter density $r^p$. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2729-2749. doi: 10.3934/cpaa.2018129
[16]

Ping Chen, Daoyuan Fang, Ting Zhang. Free boundary problem for compressible flows with density--dependent viscosity coefficients. Communications on Pure & Applied Analysis, 2011, 10 (2) : 459-478. doi: 10.3934/cpaa.2011.10.459
[17]

Günther J. Wirsching. On the problem of positive predecessor density in $3n+1$ dynamics. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 771-787. doi: 10.3934/dcds.2003.9.771
[18]

Jishan Fan, Tohru Ozawa. A regularity criterion for 3D density-dependent MHD system with zero viscosity. Conference Publications, 2015, 2015 (special) : 395-399. doi: 10.3934/proc.2015.0395
[19]

John C. Schotland, Vadim A. Markel. Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation. Inverse Problems & Imaging, 2007, 1 (1) : 181-188. doi: 10.3934/ipi.2007.1.181
[20]

Christian Léonard. A survey of the Schrödinger problem and some of its connections with optimal transport. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1533-1574. doi: 10.3934/dcds.2014.34.1533

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (72)
  • HTML views (187)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences