An optimal matching problem for the Euclidean distance

To the memory of Vicent Caselles, an outstanding mathematician and friend. 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 the Euclidean distance that each measure is transported. We show that such a problem has a solution with matching measure concentrated on the boundary of the target set. Furthermore we perform a method to approximate the solution of the problem taking limit as p→ ∞ in a system of PDE’s of p−Laplacian type. Key words. Optimal matching problem, Monge-Kantorovich’s mass transport theory, p-Lapla-cian systems AMS subject classifications. 49J20, 35J92 1. Introduction. W