Calculus of Variations and Geometric Measure Theory

G. Buttazzo - E. Stepanov

Optimal transportation networks as free Dirichlet regions for the Monge-Kantorovich problem

created on 23 Jan 2003
modified on 24 Mar 2004


Published Paper

Inserted: 23 jan 2003
Last Updated: 24 mar 2004

Journal: Ann. Sc. Norm. Sup. Pisa Cl. Sci.
Volume: II
Number: 4
Pages: 631-678
Year: 2003


In the paper the problem of constructing an optimal urban transportation network in a city with given densities of population and of workplaces is studied. The network is modeled by a closed connected set of assigned length, while the optimality condition consists in minimizing the Monge-Kantorovich functional representing the total transportation cost. The cost of trasporting a unit mass between two points is assumed to be proportional to the distance between them when the transportation is carried out outside of the network, and constant when is is carried out along the network. The same problem can be also viewed as finding an optimal Dirichlet zone minimizing the Monge-Kantorovich cost of transporting the given two measures. The paper basically studies qualitative topological and geometrical properties of optimal networks. A mild regularity result for optimal networks is also provided.

Keywords: transportation network, Mass transportation, free boundary problem, urban planning problem