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.

@article{ASNSP_2003_5_2_4_631_0, 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}, publisher = {Scuola normale superiore}, volume = {Ser. 5, 2}, number = {4}, year = {2003}, pages = {631-678}, zbl = {1127.49031}, mrnumber = {2040639}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2003_5_2_4_631_0} }

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, Serie 5, Volume 2 (2003) no. 4, pp. 631-678. http://www.numdam.org/item/ASNSP_2003_5_2_4_631_0/

[1] “Functions of Bounded Variation and Free Discontinuity Problems”, Oxford mathematical monographs. Oxford University Press, Oxford, 2000. | MR 1857292 | Zbl 0957.49001

- - ,[2] L. Ambrosio - P. Tilli, ‘Selected Topics on “Analysis in Metric Spaces”, Quaderni della Scuola Normale Superiore, Pisa, 2000. | MR 2012736 | Zbl 1084.28500

[3] Mean curvature of a measure and related variational problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), 179-196. | Numdam | MR 1655514 | Zbl 1015.49015

- - ,[4] Characterization of optimal shapes and masses through Monge-Kantorovich equation, J. European Math. Soc. 3 (2001), 139-168. | MR 1831873 | Zbl 0982.49025

- ,[5] Optimal transportation problems with free Dirichlet regions, Progress Nonlinear Differential Equations Appl. 51 (2002), 41-65. | MR 2197837 | Zbl 1055.49029

- - ,[6] Mass transportation and urban planning problems, forthcoming.

- - - ,[7] On regularity of transport density in the Monge-Kantorovich problem, Preprint del Dipartimento di Matematica, Università di Pisa, 2001. | MR 2002147 | Zbl 1084.49036

- ,[8] Constructing optimal maps for Monge's transport problem as a limit of strictly convex costs, J. Amer. Math. Soc. 15 (2002), 1-206. | MR 1862796 | Zbl 1053.49032

- - ,[9] “Analysis of and on uniformly rectifiable sets”, Vol. 38 of Math. Surveys Monographs. Amer. Math. Soc., Providence, RI, 1993. | MR 1251061 | Zbl 0832.42008

- ,[10] “Topologie”, Vol. 1, Państwowe Wydawnictwo Naukowe, Warszawa, 1958, in French. | MR 90795 | Zbl 0078.14603

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