OPTIMIZATION PROBLEMS IN MASS TRANSPORTATION THEORY ...

where h is assumed convex and superlinear(i.e. h(t)/t → +∞ as t → +∞). The increasingand diverging function h(t)/t thenrepresents the unhappiness to live in an areawith population density t.Finally, there is a third term G(f − ) whichpenalizes sparse services. We force f − tobe a sum of Dirac masses and we considerG(f − ) a functional defined on measures,of the form studied by Bouchitté-Buttazzo(Nonlinear An. 1990, IHP 1992, IHP 1993):{ ∑G(f − ) = n g(a n) if f − = ∑ n a nδ xn+∞ otherwise,where g is concave and with infinite slopeat the origin. Every single term g(a n ) inthe sum represents here the cost for buildingand managing a service pole of dimensiona n , located at the point x n ∈ Ω.38