Optimal Transportation Networks

The transportation problem can be formalized as the problem of ?nding + ? the optimal paths to transport a measure ? onto a measure ? with the same mass. In contrast with the Monge-Kantorovich formalization, recent approaches model the branched structure of such supply networks by an energy functional whose essential feature is to favor wide roads. Given a ?ow ? in a tube or a road or a wire, the transportation cost per unit length ? is supposed to be proportional to ? with 0< 1. For the Monge- Kantorovich energy,? = 1 so that it is equivalent to have two roads with ?ow 1/2oralargeronewith?ow1. Ifinstead0< 1,aroadwith?ow? +? is 1 2 ? ? ? preferable to two individual roads ? and ? because (? +? ) 1? N N such structures can irrigate a whole bounded open set ofR . The aim of this set of lectures is to give a mathematical proof of s- eral existence, structure and regularity properties empirically observed in transportation networks. This will be done in a simple mathematical fra- work (measures on the set of paths) unifying several di?erent approaches and results due to Brancolini, Buttazzo, Devillanova, Maddalena, Pratelli, Santambrogio, Solimini, Stepanov, Xia and the authors.