MATHEMATICA BOHEMICA, Vol. 139, No. 4, pp. 553-566, 2014

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

José Manuel Mazón, Julio Daniel Rossi, Julián Toledo

José Manuel Mazón, Departament d'Anàlisi Matemàtica, Facultat d'Economia, Universitat de València, Avda. Doctor Moliner 50, 46100 Burjassot, València, Spain, e-mail: mazon@uv.es; Julio Daniel Rossi, Departamento de Análisis Matemático, Universidad de Alicante, Ap. Correos 99, 03080 Alicante, Spain, on leave from Depto. de Matematica, FCEyN UBA, Ciudad Universitaria, Pab 1 (1428), Buenos Aires, Argentina, e-mail: julio.rossi@ua.es; Julián Toledo, Departament d'Anàlisi Matemàtica, Facultat d'Economia, Universitat de València, Avda. Doctor Moliner 50, 46100 Burjassot, València, Spain, e-mail: toledojj@uv.es

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.

Keywords: mass transport; Monge-Kantorovich problem; $p$-Laplacian equation

Classification (MSC 2010): 49J20, 49J45, 45G10


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