Optimal matching problems with costs given by Finsler distances

Previous Article
An obstacle problem for Tug-of-War games
CPAA Home
This Issue
Next Article
Optimal reaction exponent for some qualitative properties of solutions to the $p$-heat equation

January 2015, 14(1): 229-244. doi: 10.3934/cpaa.2015.14.229

Optimal matching problems with costs given by Finsler distances

J. M. Mazón ^1, , Julio D. Rossi ^2, and J. Toledo ^1,

1.	Departament d'Anàlisi Matemàtica, U. de València, Valencia, Spain, Spain
2.	Departamento de Análisis Matemático, Universidad de Alicante, Ap 99, 03080, Alicante

Received January 2014 Revised April 2014 Published September 2014

Abstract
Related Papers

In this paper we deal with an optimal matching problem, that is, we want to transport two commodities (modeled by two measures that encode the spacial distribution of each commodity) to a given location, where they will match, minimizing the total transport cost that in our case is given by the sum of the two different Finsler distances that the two measures are transported. We perform a method to approximate the matching measure and the pair of Kantorovich potentials associated with this problem taking limit as $p\to \infty$ in a variational system of $p-$Laplacian type.

Keywords: MongeKantorovichs mass transport theory, $p$Laplacian systems., Optimal matching problem.

Mathematics Subject Classification: Primary: 49J20, 35J9.

Citation: J. M. Mazón, Julio D. Rossi, J. Toledo. Optimal matching problems with costs given by Finsler distances. Communications on Pure & Applied Analysis, 2015, 14 (1) : 229-244. doi: 10.3934/cpaa.2015.14.229

References:

[1]	L. Ambrosio, Lecture Notes on Optimal Transport Problems,, Mathematical aspects of evolving interfaces (Funchal, (2000), 1. doi: 10.1007/978-3-540-39189-0_1. Google Scholar
[2]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Universitext, (2011). Google Scholar
[3]	G. Bouchitté, G. Buttazzo and L. A. De Pascale, A p-Laplacian approximation for some mass optimization problems,, \emph{J. Optim. Theory Appl., 118 (2003), 1. doi: 10.1023/A:1024751022715. Google Scholar
[4]	G. Carlier, Duality and existence for a class of mass transportation problems and economic applications,, \emph{Adv. Math. Econ.}, 5 (2003), 1. doi: 10.1007/978-4-431-53979-7_1. Google Scholar
[5]	P-A. Chiappori, R. McCann and L. Nesheim, Hedoniic prices equilibria, stable matching, and optimal transport: equivalence, topolgy, and uniqueness,, \emph{Econ. Theory, 42 (2010), 317. doi: 10.1007/s00199-009-0455-z. Google Scholar
[6]	I. Ekeland, An optimal matching problem,, \emph{ESAIM COCV, 11 (2005), 57. doi: 10.1051/cocv:2004034. Google Scholar
[7]	I. Ekeland, Existence, uniqueness and efficiency of equilibrium in hedonic markets with multidimensional types,, \emph{Econ. Theory, 42 (2010), 275. doi: 10.1007/s00199-008-0427-8. Google Scholar
[8]	I. Ekeland, Notes on optimal transportation,, \emph{Econ. Theory, 42 (2010), 437. doi: 10.1007/s00199-008-0426-9. Google Scholar
[9]	I. Ekeland, J. J Hecckman and L. Nesheim, Identificacation and estimates of Hedonic models,, \emph{Journal of Political Economy, 112 (2004). Google Scholar
[10]	L. C. Evans, Partial differential equations and Monge-Kantorovich mass transfer,, \emph{Current Developments in Mathematics}, (1997), 65. Google Scholar
[11]	L. C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem,, \emph{Mem. Amer. Math. Soc.}, 137 (1999). doi: 10.1090/memo/0653. Google Scholar
[12]	K. Fan, Minimax theorems,, \emph{Pro. Nat. Acad. Sci., 39 (1953), 42. Google Scholar
[13]	M. Feldman and R. J. McCann, Monge's transport problem on a Riemannian manifold,, \emph{Trans. Amer. Math. Soc., 354 (2002), 1667. doi: 10.1090/S0002-9947-01-02930-0. Google Scholar
[14]	N. Igbida, J.M. Mazón, J. D. Rossi and J. J. Toledo, A Monge-Kantorovich mass transport problem for a discrete distance,, \emph{J. Funct. Anal., 260 (2011), 3494. doi: 10.1016/j.jfa.2011.02.023. Google Scholar
[15]	J. M. Mazón, J. D. Rossi and J. Toledo, An optimal transportation problem with a cost given by the Euclidean distance plus import/export taxes on the boundary,, \emph{Rev. Mat. Iberoam., 30 (2014), 277. doi: 10.4171/RMI/778. Google Scholar
[16]	J. M. Mazón, J. D. Rossi and J. Toledo, An optimal matching problem for the Euclidean distance,, \emph{SIAM J. Math. Anal., 46 (2014), 233. doi: 10.1137/120901465. Google Scholar
[17]	N. Igbida, J.M. Mazón, J. D. Rossi and J. Toledo, Mass transport problems for costs given by Finsler distances via $p$-Laplacian approximations,, Preprint., (). Google Scholar
[18]	C. Villani, Topics in Optimal Transportation,, Graduate Studies in Mathematics. Vol. 58, (2003). doi: 10.1007/b12016. Google Scholar
[19]	C. Villani, Optimal Transport. Old and New,, Grundlehren der MathematischenWissenschaften (Fundamental Principles of Mathematical Sciences), (2009). doi: 10.1007/978-3-540-71050-9. Google Scholar

show all references

References:

[1]	L. Ambrosio, Lecture Notes on Optimal Transport Problems,, Mathematical aspects of evolving interfaces (Funchal, (2000), 1. doi: 10.1007/978-3-540-39189-0_1. Google Scholar
[2]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Universitext, (2011). Google Scholar
[3]	G. Bouchitté, G. Buttazzo and L. A. De Pascale, A p-Laplacian approximation for some mass optimization problems,, \emph{J. Optim. Theory Appl., 118 (2003), 1. doi: 10.1023/A:1024751022715. Google Scholar
[4]	G. Carlier, Duality and existence for a class of mass transportation problems and economic applications,, \emph{Adv. Math. Econ.}, 5 (2003), 1. doi: 10.1007/978-4-431-53979-7_1. Google Scholar
[5]	P-A. Chiappori, R. McCann and L. Nesheim, Hedoniic prices equilibria, stable matching, and optimal transport: equivalence, topolgy, and uniqueness,, \emph{Econ. Theory, 42 (2010), 317. doi: 10.1007/s00199-009-0455-z. Google Scholar
[6]	I. Ekeland, An optimal matching problem,, \emph{ESAIM COCV, 11 (2005), 57. doi: 10.1051/cocv:2004034. Google Scholar
[7]	I. Ekeland, Existence, uniqueness and efficiency of equilibrium in hedonic markets with multidimensional types,, \emph{Econ. Theory, 42 (2010), 275. doi: 10.1007/s00199-008-0427-8. Google Scholar
[8]	I. Ekeland, Notes on optimal transportation,, \emph{Econ. Theory, 42 (2010), 437. doi: 10.1007/s00199-008-0426-9. Google Scholar
[9]	I. Ekeland, J. J Hecckman and L. Nesheim, Identificacation and estimates of Hedonic models,, \emph{Journal of Political Economy, 112 (2004). Google Scholar
[10]	L. C. Evans, Partial differential equations and Monge-Kantorovich mass transfer,, \emph{Current Developments in Mathematics}, (1997), 65. Google Scholar
[11]	L. C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem,, \emph{Mem. Amer. Math. Soc.}, 137 (1999). doi: 10.1090/memo/0653. Google Scholar
[12]	K. Fan, Minimax theorems,, \emph{Pro. Nat. Acad. Sci., 39 (1953), 42. Google Scholar
[13]	M. Feldman and R. J. McCann, Monge's transport problem on a Riemannian manifold,, \emph{Trans. Amer. Math. Soc., 354 (2002), 1667. doi: 10.1090/S0002-9947-01-02930-0. Google Scholar
[14]	N. Igbida, J.M. Mazón, J. D. Rossi and J. J. Toledo, A Monge-Kantorovich mass transport problem for a discrete distance,, \emph{J. Funct. Anal., 260 (2011), 3494. doi: 10.1016/j.jfa.2011.02.023. Google Scholar
[15]	J. M. Mazón, J. D. Rossi and J. Toledo, An optimal transportation problem with a cost given by the Euclidean distance plus import/export taxes on the boundary,, \emph{Rev. Mat. Iberoam., 30 (2014), 277. doi: 10.4171/RMI/778. Google Scholar
[16]	J. M. Mazón, J. D. Rossi and J. Toledo, An optimal matching problem for the Euclidean distance,, \emph{SIAM J. Math. Anal., 46 (2014), 233. doi: 10.1137/120901465. Google Scholar
[17]	N. Igbida, J.M. Mazón, J. D. Rossi and J. Toledo, Mass transport problems for costs given by Finsler distances via $p$-Laplacian approximations,, Preprint., (). Google Scholar
[18]	C. Villani, Topics in Optimal Transportation,, Graduate Studies in Mathematics. Vol. 58, (2003). doi: 10.1007/b12016. Google Scholar
[19]	C. Villani, Optimal Transport. Old and New,, Grundlehren der MathematischenWissenschaften (Fundamental Principles of Mathematical Sciences), (2009). doi: 10.1007/978-3-540-71050-9. Google Scholar

[1]	Pengwen Chen, Changfeng Gui. Alpha divergences based mass transport models for image matching problems. Inverse Problems & Imaging, 2011, 5 (3) : 551-590. doi: 10.3934/ipi.2011.5.551
[2]	Yangang Chen, Justin W. L. Wan. Numerical method for image registration model based on optimal mass transport. Inverse Problems & Imaging, 2018, 12 (2) : 401-432. doi: 10.3934/ipi.2018018
[3]	Brendan Pass. Multi-marginal optimal transport and multi-agent matching problems: Uniqueness and structure of solutions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1623-1639. doi: 10.3934/dcds.2014.34.1623
[4]	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
[5]	Yansheng Zhong, Yongqing Li. On a p-Laplacian eigenvalue problem with supercritical exponent. Communications on Pure & Applied Analysis, 2019, 18 (1) : 227-236. doi: 10.3934/cpaa.2019012
[6]	Francesca Colasuonno, Benedetta Noris. A p-Laplacian supercritical Neumann problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3025-3057. doi: 10.3934/dcds.2017130
[7]	Giuseppina Barletta, Roberto Livrea, Nikolaos S. Papageorgiou. A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1075-1086. doi: 10.3934/cpaa.2014.13.1075
[8]	Po-Chun Huang, Shin-Hwa Wang, Tzung-Shin Yeh. Classification of bifurcation diagrams of a $P$-Laplacian nonpositone problem. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2297-2318. doi: 10.3934/cpaa.2013.12.2297
[9]	Hugo Beirão da Veiga, Francesca Crispo. On the global regularity for nonlinear systems of the $p$-Laplacian type. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1173-1191. doi: 10.3934/dcdss.2013.6.1173
[10]	David Kinderlehrer, Adrian Tudorascu. Transport via mass transportation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (2) : 311-338. doi: 10.3934/dcdsb.2006.6.311
[11]	Danilo Coelho, David Pérez-Castrillo. On Marilda Sotomayor's extraordinary contribution to matching theory. Journal of Dynamics & Games, 2015, 2 (3&4) : 201-206. doi: 10.3934/jdg.2015001
[12]	Patrizia Pucci, Mingqi Xiang, Binlin Zhang. A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4035-4051. doi: 10.3934/dcds.2017171
[13]	Marek Galewski, Renata Wieteska. Multiple periodic solutions to a discrete $p^{(k)}$ - Laplacian problem. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2535-2547. doi: 10.3934/dcdsb.2014.19.2535
[14]	Rossella Bartolo, Anna Maria Candela, Addolorata Salvatore. Infinitely many radial solutions of a non--homogeneous $p$--Laplacian problem. Conference Publications, 2013, 2013 (special) : 51-59. doi: 10.3934/proc.2013.2013.51
[15]	Everaldo S. de Medeiros, Jianfu Yang. Asymptotic behavior of solutions to a perturbed p-Laplacian problem with Neumann condition. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 595-606. doi: 10.3934/dcds.2005.12.595
[16]	Julián Fernández Bonder, Leandro M. Del Pezzo. An optimization problem for the first eigenvalue of the $p-$Laplacian plus a potential. Communications on Pure & Applied Analysis, 2006, 5 (4) : 675-690. doi: 10.3934/cpaa.2006.5.675
[17]	Agnid Banerjee, Nicola Garofalo. On the Dirichlet boundary value problem for the normalized $p$-laplacian evolution. Communications on Pure & Applied Analysis, 2015, 14 (1) : 1-21. doi: 10.3934/cpaa.2015.14.1
[18]	Yuxiang Zhang, Shiwang Ma. Some existence results on periodic and subharmonic solutions of ordinary $P$-Laplacian systems. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 251-260. doi: 10.3934/dcdsb.2009.12.251
[19]	Wenbin Liu, Zhaosheng Feng. Periodic solutions for $p$-Laplacian systems of Liénard-type. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1393-1400. doi: 10.3934/cpaa.2011.10.1393
[20]	Maya Chhetri, D. D. Hai, R. Shivaji. On positive solutions for classes of p-Laplacian semipositone systems. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 1063-1071. doi: 10.3934/dcds.2003.9.1063

2018 Impact Factor: 0.925

American Institute of Mathematical Sciences