Optimal matching problems with costs given by Finsler distances

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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
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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 and 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), 1-52, Lecture Notes in Math., 1812, Springer, Berlin, 2003. doi: 10.1007/978-3-540-39189-0_1.
[2]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.
[3]	G. Bouchitté, G. Buttazzo and L. A. De Pascale, A p-Laplacian approximation for some mass optimization problems, J. Optim. Theory Appl., 118 (2003), 1-25. doi: 10.1023/A:1024751022715.
[4]	G. Carlier, Duality and existence for a class of mass transportation problems and economic applications, Adv. Math. Econ., 5 (2003), 1-21. doi: 10.1007/978-4-431-53979-7_1.
[5]	P-A. Chiappori, R. McCann and L. Nesheim, Hedoniic prices equilibria, stable matching, and optimal transport: equivalence, topolgy, and uniqueness, Econ. Theory, 42 (2010), 317-354. doi: 10.1007/s00199-009-0455-z.
[6]	I. Ekeland, An optimal matching problem, ESAIM COCV, 11 (2005), 57-71. doi: 10.1051/cocv:2004034.
[7]	I. Ekeland, Existence, uniqueness and efficiency of equilibrium in hedonic markets with multidimensional types, Econ. Theory, 42 (2010), 275-315. doi: 10.1007/s00199-008-0427-8.
[8]	I. Ekeland, Notes on optimal transportation, Econ. Theory, 42 (2010), 437-459. doi: 10.1007/s00199-008-0426-9.
[9]	I. Ekeland, J. J Hecckman and L. Nesheim, Identificacation and estimates of Hedonic models, Journal of Political Economy, 112 (2004), S60-S109.
[10]	L. C. Evans, Partial differential equations and Monge-Kantorovich mass transfer, Current Developments in Mathematics, 1997 (Cambridge, MA), 65-126, Int. Press, Boston, MA, 1999.
[11]	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.
[12]	K. Fan, Minimax theorems, Pro. Nat. Acad. Sci., 39 (1953), 42-47.
[13]	M. Feldman and R. J. McCann, Monge's transport problem on a Riemannian manifold, Trans. Amer. Math. Soc., 354 (2002), 1667-1697. doi: 10.1090/S0002-9947-01-02930-0.
[14]	N. Igbida, J.M. Mazón, J. D. Rossi and J. J. Toledo, A Monge-Kantorovich mass transport problem for a discrete distance, J. Funct. Anal., 260 (2011), 3494-3534. doi: 10.1016/j.jfa.2011.02.023.
[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, Rev. Mat. Iberoam., 30 (2014), 277-308. doi: 10.4171/RMI/778.
[16]	J. M. Mazón, J. D. Rossi and J. Toledo, An optimal matching problem for the Euclidean distance, SIAM J. Math. Anal., 46 (2014), 233-255 doi: 10.1137/120901465.
[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.
[18]	C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics. Vol. 58, 2003. doi: 10.1007/b12016.
[19]	C. Villani, Optimal Transport. Old and New, Grundlehren der MathematischenWissenschaften (Fundamental Principles of Mathematical Sciences), vol. 338. Springer, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.

show all references

References:

[1]	L. Ambrosio, Lecture Notes on Optimal Transport Problems, Mathematical aspects of evolving interfaces (Funchal, 2000), 1-52, Lecture Notes in Math., 1812, Springer, Berlin, 2003. doi: 10.1007/978-3-540-39189-0_1.
[2]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.
[3]	G. Bouchitté, G. Buttazzo and L. A. De Pascale, A p-Laplacian approximation for some mass optimization problems, J. Optim. Theory Appl., 118 (2003), 1-25. doi: 10.1023/A:1024751022715.
[4]	G. Carlier, Duality and existence for a class of mass transportation problems and economic applications, Adv. Math. Econ., 5 (2003), 1-21. doi: 10.1007/978-4-431-53979-7_1.
[5]	P-A. Chiappori, R. McCann and L. Nesheim, Hedoniic prices equilibria, stable matching, and optimal transport: equivalence, topolgy, and uniqueness, Econ. Theory, 42 (2010), 317-354. doi: 10.1007/s00199-009-0455-z.
[6]	I. Ekeland, An optimal matching problem, ESAIM COCV, 11 (2005), 57-71. doi: 10.1051/cocv:2004034.
[7]	I. Ekeland, Existence, uniqueness and efficiency of equilibrium in hedonic markets with multidimensional types, Econ. Theory, 42 (2010), 275-315. doi: 10.1007/s00199-008-0427-8.
[8]	I. Ekeland, Notes on optimal transportation, Econ. Theory, 42 (2010), 437-459. doi: 10.1007/s00199-008-0426-9.
[9]	I. Ekeland, J. J Hecckman and L. Nesheim, Identificacation and estimates of Hedonic models, Journal of Political Economy, 112 (2004), S60-S109.
[10]	L. C. Evans, Partial differential equations and Monge-Kantorovich mass transfer, Current Developments in Mathematics, 1997 (Cambridge, MA), 65-126, Int. Press, Boston, MA, 1999.
[11]	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.
[12]	K. Fan, Minimax theorems, Pro. Nat. Acad. Sci., 39 (1953), 42-47.
[13]	M. Feldman and R. J. McCann, Monge's transport problem on a Riemannian manifold, Trans. Amer. Math. Soc., 354 (2002), 1667-1697. doi: 10.1090/S0002-9947-01-02930-0.
[14]	N. Igbida, J.M. Mazón, J. D. Rossi and J. J. Toledo, A Monge-Kantorovich mass transport problem for a discrete distance, J. Funct. Anal., 260 (2011), 3494-3534. doi: 10.1016/j.jfa.2011.02.023.
[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, Rev. Mat. Iberoam., 30 (2014), 277-308. doi: 10.4171/RMI/778.
[16]	J. M. Mazón, J. D. Rossi and J. Toledo, An optimal matching problem for the Euclidean distance, SIAM J. Math. Anal., 46 (2014), 233-255 doi: 10.1137/120901465.
[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.
[18]	C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics. Vol. 58, 2003. doi: 10.1007/b12016.
[19]	C. Villani, Optimal Transport. Old and New, Grundlehren der MathematischenWissenschaften (Fundamental Principles of Mathematical Sciences), vol. 338. Springer, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.

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