Estamos trabajando para incorporar este artículo al repositorio
Consulte el artículo en la página del editor
Consulte la política de Acceso Abierto del editor


We deal with an optimal matching problem with constraints, that is, we want to transport two measures with the same total mass in RN to a given place (the target set), where they will match and in which we have constraints on the amount of matter we can take to points in the target set. This transport has to be done optimally, minimizing the total transport cost, that in our case is given by the sum of the Euclidean distances that each measure is transported. Here we show that such a problem has a solution. First, we solve the problem using mass transport arguments and next we perform a method to approximate the solution of the problem taking limit as p→ ∞ in a p-Laplacian type variational problem. In the particular case in which the target set is contained in a hypersurface, we deal with an optimal transport problem through a membrane, that is, we want to transport two measures which are located in different locations separated by a membrane (the hypersurface) which only let through a predetermined amount of matter. © 2018, Universidad Complutense de Madrid.


Documento: Artículo
Título:An optimal matching problem with constraints
Autor:Mazón, J.M.; Rossi, J.D.; Toledo, J.
Filiación:Departament d’Anàlisi Matemàtica, Universidad de València, Valencia, Spain
Departamento de Matemáticas, Universidad de Buenos Aires, Buenos Aires, Argentina
Palabras clave:Mass transport theory; Matching problems; p-Laplacian equation
Página de inicio:407
Página de fin:447
Título revista:Revista Matematica Complutense
Título revista abreviado:Rev. Mat. Complutense


  • Ambrosio, L., Lecture notes on optimal transport problems (1812) Mathematical Aspects of Evolving Interfaces (Funchal, 2000), Lecture Notes in Mathematics, pp. 1-52. , Springer, Berlin
  • Barrett, J.W., Prigozhin, L., Partial L1 Monge–Kantorovich problem: variational formulation and numerical approximation (2009) Interfaces Free Bound., 11, pp. 201-238
  • Brezis, H., (2011) Functional Analysis. Sobolev Spaces and Partial Differential Equations. Universitext, , Springer, New York
  • Carlier, G., Duality and existence for a class of mass transportation problems and economic applications (2003) Adv. Math. Econ., 5, pp. 1-21
  • Carlier, G., Ekeland, I., Matching for teams (2010) Econ. Theory, 42, pp. 397-418
  • Champion, T., De Pascale, L., The Monge problem in Rd (2011) Duke Math. J., 157, pp. 551-572
  • Chiappori, P.-A., McCann, R., Nesheim, L., Hedonic prices equilibria, stable matching, and optimal transport: equivalence, topology, and uniqueness (2010) Econ. Theory, 42, pp. 317-354
  • Ekeland, I., An optimal matching problem (2005) ESAIM COCV, 11, pp. 57-71
  • Ekeland, I., Existence, uniqueness and efficiency of equilibrium in hedonic markets with multidimensional types (2010) Econ. Theory, 42 (2), pp. 275-315
  • Ekeland, I., Notes on optimal transportation (2010) Econ. Theory, 42 (2), pp. 437-459
  • Ekeland, I., Hecckman, J.J., Nesheim, L., Identification and estimates of hedonic models (2004) J. Polit. Econ., 112, pp. S60-S109
  • Evans, L.C., (1998) Partial Differential Equations. Graduate Studies Mathematics, 19. , American Mathematical Society, Providence
  • Evans, L.C., Partial differential equations and Monge–Kantorovich mass transfer (1999) Current Developments in Mathematics, 1997, pp. 65-126. , (Cambridge, MA), International Press, Boston
  • Evans, L.C., Gangbo, W., Differential equations methods for the Monge–Kantorovich mass transfer problem (1999) Mem. Am. Math. Soc., 137, p. 653
  • Fan, K., Minimax theorems (1953) Proc. Natl. Acad. Sci., 39, pp. 42-47
  • Igbida, N., Mazón, J.M., Rossi, J.D., Toledo, J.J., A Monge–Kantorovich mass transport problem for a discrete distance (2011) J. Funct. Anal., 260, pp. 3494-3534
  • Kantorovich, L.V., On the tranfer of masses (1942) Dokl. Nauk. SSSR, 37, pp. 227-229
  • Mazón, J.M., Rossi, J.D., Toledo, J.J., An optimal transportation problem with a cost given by the euclidean distance plus import/export taxes on the boundary (2014) Rev. Mat. Iberoam., 30 (1), pp. 277-308
  • Mazón, J.M., Rossi, J.D., Toledo, J.J., An optimal matching problem for the Euclidean distance (2014) SIAM J. Math. Anal., 46, pp. 233-255
  • Mazón, J.M., Rossi, J.D., Toledo, J.J., Optimal matching problems with costs given by Finsler distances (2015) Commun. Pure Appl. Anal., 14, pp. 229-244
  • Mazón, J.M., Rossi, J.D., Toledo, J.J., Optimal mass transport on metric graphs (2015) SIAM J. Optim., 25, pp. 1609-1632
  • Villani, C., (2003) Topics in Optimal Transportation. Graduate Studies in Mathematics, 58. , AMS, Providence
  • Villani, C., (2009) Optimal Transport. Old and New. Grundlehren der MathematischenWissenschaften (Fundamental Principles of Mathematical Sciences), 338. , Springer, Berlin


---------- APA ----------
Mazón, J.M., Rossi, J.D. & Toledo, J. (2018) . An optimal matching problem with constraints. Revista Matematica Complutense, 31(2), 407-447.
---------- CHICAGO ----------
Mazón, J.M., Rossi, J.D., Toledo, J. "An optimal matching problem with constraints" . Revista Matematica Complutense 31, no. 2 (2018) : 407-447.
---------- MLA ----------
Mazón, J.M., Rossi, J.D., Toledo, J. "An optimal matching problem with constraints" . Revista Matematica Complutense, vol. 31, no. 2, 2018, pp. 407-447.
---------- VANCOUVER ----------
Mazón, J.M., Rossi, J.D., Toledo, J. An optimal matching problem with constraints. Rev. Mat. Complutense. 2018;31(2):407-447.