OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

We recall that, in the case Ω compact,W p (Ω) is the space of all Borel probabilitymeasures µ on Ω equipped with the p-Wasserstein distance( ∫w p (µ 1 , µ 2 ) = infΩ×Ω) 1/pc(x, y) p λ(dx, dy)where the infimum is taken on all transportplans λ between µ 1 and µ 2 , that is onall probability measures λ on Ω × Ω whosemarginals π # 1 λ and π# 2 λ coincide with µ 1and µ 2 respectively.In order to define the functionalJ (γ) =∫ 10J(γ(t))|γ ′ |(t) dtit remains to fix the “coefficient” J.Wetake a l.s.c. functional on the space of measures,of the kind considered by Bouchittéand Buttazzo (Nonlinear Anal. 1990, Ann.IHP 1992, Ann. IHP 1993):54