Sala Seminari (Dip. Matematica).
The optimal transport problem consists in minimizing the total energy of the displacement among all the (vector-valued) functions having prescribed image measure. In this talk we are interested in a particular case of cost functions: c is given by h(y-x) where h is convex, has a bounded domain and is bounded on this domain. The fact that it takes infinite value makes, in particular, complicated the existence of solutions for the dual problem. The strategy consists in introducing a “time-parameter” t>0 and in studying the re-scaled problem, with cost h((y-x)/t). We show the continuity of the total cost with respect to t, and the existence of Kantorovich potentials for “supercritical” time. This is a joint work with A.Pratelli and F.Zeisler (Erlangen).