On optimal matching measures for matching problems related to the Euclidean distance

José Manuel Mazón; Julio Daniel Rossi; Julián Toledo

Mathematica Bohemica (2014)

  • Volume: 139, Issue: 4, page 553-566
  • ISSN: 0862-7959

Abstract

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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.

How to cite

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Mazón, José Manuel, Rossi, Julio Daniel, and Toledo, Julián. "On optimal matching measures for matching problems related to the Euclidean distance." Mathematica Bohemica 139.4 (2014): 553-566. <http://eudml.org/doc/269836>.

@article{Mazón2014,
abstract = {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.},
author = {Mazón, José Manuel, Rossi, Julio Daniel, Toledo, Julián},
journal = {Mathematica Bohemica},
keywords = {mass transport; Monge-Kantorovich problem; $p$-Laplacian equation; mass transport; Monge-Kantorovich problem; -Laplacian equation},
language = {eng},
number = {4},
pages = {553-566},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On optimal matching measures for matching problems related to the Euclidean distance},
url = {http://eudml.org/doc/269836},
volume = {139},
year = {2014},
}

TY - JOUR
AU - Mazón, José Manuel
AU - Rossi, Julio Daniel
AU - Toledo, Julián
TI - On optimal matching measures for matching problems related to the Euclidean distance
JO - Mathematica Bohemica
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 139
IS - 4
SP - 553
EP - 566
AB - 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.
LA - eng
KW - mass transport; Monge-Kantorovich problem; $p$-Laplacian equation; mass transport; Monge-Kantorovich problem; -Laplacian equation
UR - http://eudml.org/doc/269836
ER -

References

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