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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

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
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References:
[1]

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.  Google Scholar

[2]

Universitext, Springer, New York, 2011.  Google Scholar

[3]

J. Optim. Theory Appl., 118 (2003), 1-25. doi: 10.1023/A:1024751022715.  Google Scholar

[4]

Adv. Math. Econ., 5 (2003), 1-21. doi: 10.1007/978-4-431-53979-7_1.  Google Scholar

[5]

Econ. Theory, 42 (2010), 317-354. doi: 10.1007/s00199-009-0455-z.  Google Scholar

[6]

ESAIM COCV, 11 (2005), 57-71. doi: 10.1051/cocv:2004034.  Google Scholar

[7]

Econ. Theory, 42 (2010), 275-315. doi: 10.1007/s00199-008-0427-8.  Google Scholar

[8]

Econ. Theory, 42 (2010), 437-459. doi: 10.1007/s00199-008-0426-9.  Google Scholar

[9]

Journal of Political Economy, 112 (2004), S60-S109. Google Scholar

[10]

Current Developments in Mathematics, 1997 (Cambridge, MA), 65-126, Int. Press, Boston, MA, 1999.  Google Scholar

[11]

Mem. Amer. Math. Soc., 137 (1999), no. 653. doi: 10.1090/memo/0653.  Google Scholar

[12]

Pro. Nat. Acad. Sci., 39 (1953), 42-47.  Google Scholar

[13]

Trans. Amer. Math. Soc., 354 (2002), 1667-1697. doi: 10.1090/S0002-9947-01-02930-0.  Google Scholar

[14]

J. Funct. Anal., 260 (2011), 3494-3534. doi: 10.1016/j.jfa.2011.02.023.  Google Scholar

[15]

Rev. Mat. Iberoam., 30 (2014), 277-308. doi: 10.4171/RMI/778.  Google Scholar

[16]

SIAM J. Math. Anal., 46 (2014), 233-255 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]

Graduate Studies in Mathematics. Vol. 58, 2003. doi: 10.1007/b12016.  Google Scholar

[19]

Grundlehren der MathematischenWissenschaften (Fundamental Principles of Mathematical Sciences), vol. 338. Springer, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.  Google Scholar

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