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Keywords:
mass transport; Monge-Kantorovich problem; $p$-Laplacian equation
mass transport; Monge-Kantorovich problem; $p$-Laplacian equation
Summary:
We deal with an optimal matching problem, that is, we want to transport two measures to a given place (the target set) where they will match, minimizing the total transport cost that in our case is given by the sum of two different multiples of the Euclidean distance that each measure is transported. We show that such a problem has a solution with an optimal matching measure supported in the target set. This result can be proved by an approximation procedure using a $p$-Laplacian system. We prove that any optimal matching measure for this problem is supported on the boundary of the target set when the two multiples that affect the Euclidean distances involved in the cost are different. Moreover, we present simple examples showing uniqueness or non-uniqueness of the optimal measure.
References:
[1] Agueh, M., Carlier, G.: Barycenters in the Wasserstein space. SIAM J. Math. Anal. 43 (2011), 904-924. DOI 10.1137/100805741 | MR 2801182 | Zbl 1223.49045
[2] Ambrosio, L.: Lecture notes on optimal transport problems. Mathematical Aspects of Evolving Interfaces. Lectures given at the C.I.M.-C.I.M.E. joint Euro-summer school, Madeira, Funchal, Portugal Lecture Notes in Math. 1812 Springer, Berlin (2003), 1-52 P. Colli et al. \goodbreak. MR 2011032 | Zbl 1047.35001
[3] Carlier, G.: Duality and existence for a class of mass transportation problems and economic applications. S. Kusuoka et al. Advances in Mathematical Economics 5 Springer, Tokyo (2003), 1-21. DOI 10.1007/978-4-431-53979-7_1 | MR 2160899 | Zbl 1176.90409
[4] Carlier, G., Ekeland, I.: Matching for teams. Econ. Theory 42 (2010), 397-418. DOI 10.1007/s00199-008-0415-z | MR 2564442 | Zbl 1183.91112
[5] Chiappori, P.-A., McCann, R. J., Nesheim, L. P.: Hedonic price equilibria, stable matching, and optimal transport: Equivalence, topology, and uniqueness. Econ. Theory 42 (2010), 317-354. MR 2564439 | Zbl 1183.91056
[6] Ekeland, I.: 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 | MR 2564438 | Zbl 1203.91153
[7] Ekeland, I.: Notes on optimal transportation. Econ. Theory 42 (2010), 437-459. DOI 10.1007/s00199-008-0426-9 | MR 2564444 | Zbl 1185.90019
[8] Ekeland, I.: An optimal matching problem. ESAIM, Control Optim. Calc. Var. 11 (2005), 57-71. DOI 10.1051/cocv:2004034 | MR 2110613 | Zbl 1106.49054
[9] Ekeland, I., Heckman, J. J., Nesheim, L.: Identification and Estimates of Hedonic Models. Journal of Political Economy 112 (2004), S60--S109. DOI 10.1086/379947
[10] Evans, L. C., Gangbo, W.: Differential equations methods for the Monge-Kantorovich mass transfer problem. Mem. Am. Math. Soc. 137 (1999), 66. MR 1464149 | Zbl 0920.49004
[11] Igbida, N., Mazón, J. M., Rossi, J. D., Toledo, J.: 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 | MR 2781969 | Zbl 1225.49047
[12] Mazón, J. M., Rossi, J. D., Toledo, J.: An optimal matching problem for the Euclidean distance. SIAM J. Math. Anal. 46 (2014), 233-255. DOI 10.1137/120901465 | MR 3151384 | Zbl 1297.49006
[13] Mazón, J. M., Rossi, J. D., Toledo, J.: 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 | MR 3186940
[14] Pass, B.: Multi-marginal optimal transport and multi-agent matching problems: uniqueness and structure of solutions. Discrete Contin. Dyn. Syst. 34 (2014), 1623-1639. DOI 10.3934/dcds.2014.34.1623 | MR 3121634 | Zbl 1278.49054
[15] Pass, B.: Regularity properties of optimal transportation problems arising in hedonic pricing models. ESAIM, Control Optim. Calc. Var. 19 (2013), 668-678. DOI 10.1051/cocv/2012027 | MR 3092356 | Zbl 1271.91053
[16] Villani, C.: Optimal Transport. Old and New. Grundlehren der Mathematischen Wissenschaften 137 Springer, Berlin (2009). DOI 10.1007/978-3-540-71050-9_28 | MR 2459454 | Zbl 1156.53003
[17] Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics 58 American Mathematical Society, Providence (2003). MR 1964483 | Zbl 1106.90001
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