Revista Matemática Complutense

, Volume 31, Issue 2, pp 407–447 | Cite as

An optimal matching problem with constraints

  • J. M. Mazón
  • J. D. Rossi
  • J. Toledo


We deal with an optimal matching problem with constraints, that is, we want to transport two measures with the same total mass in \({\mathbb {R}}^N\) 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 \(p\rightarrow \infty \) 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.


Mass transport theory Matching problems p-Laplacian equation 

Mathematics Subject Classification

35J62 49J20 49J45 



The authors have been partially supported by the Spanish MINECO and FEDER, Project MTM2015-70227-P.


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© Universidad Complutense de Madrid 2018

Authors and Affiliations

  1. 1.Departament d’Anàlisi MatemàticaUniversidad de ValènciaValenciaSpain
  2. 2.Departamento de MatemáticasUniversidad de Buenos AiresBuenos AiresArgentina

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