Geodesic
Optimal transportation networks as free Dirichlet regions for the Monge-Kantorovich problem
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 4, pp. 631-678

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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 negligible when it 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.

Classification : 49Q10, 49Q15, 49N60, 90B10 
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     author = {Buttazzo, Giuseppe and Stepanov, Eugene},
     title = {Optimal transportation networks as free {Dirichlet} regions for the {Monge-Kantorovich} problem},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {631--678},
     publisher = {Scuola normale superiore},
     volume = {Ser. 5, 2},
     number = {4},
     year = {2003},
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     zbl = {1127.49031},
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Buttazzo, Giuseppe; Stepanov, Eugene. Optimal transportation networks as free Dirichlet regions for the Monge-Kantorovich problem. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 4, pp. 631-678. http://geodesic.mathdoc.fr/item/ASNSP_2003_5_2_4_631_0/