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

Giuseppe Buttazzo; Eugene Stepanov

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2003)

  • Volume: 2, Issue: 4, page 631-678
  • ISSN: 0391-173X

Abstract

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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.

How to cite

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Buttazzo, Giuseppe, and Stepanov, Eugene. "Optimal transportation networks as free Dirichlet regions for the Monge-Kantorovich problem." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 2.4 (2003): 631-678. <http://eudml.org/doc/84515>.

@article{Buttazzo2003,
abstract = {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.},
author = {Buttazzo, Giuseppe, Stepanov, Eugene},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {4},
pages = {631-678},
publisher = {Scuola normale superiore},
title = {Optimal transportation networks as free Dirichlet regions for the Monge-Kantorovich problem},
url = {http://eudml.org/doc/84515},
volume = {2},
year = {2003},
}

TY - JOUR
AU - Buttazzo, Giuseppe
AU - Stepanov, Eugene
TI - Optimal transportation networks as free Dirichlet regions for the Monge-Kantorovich problem
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2003
PB - Scuola normale superiore
VL - 2
IS - 4
SP - 631
EP - 678
AB - 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.
LA - eng
UR - http://eudml.org/doc/84515
ER -

References

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  1. [1] L. Ambrosio – N. Fusco – D. Pallara, “Functions of Bounded Variation and Free Discontinuity Problems”, Oxford mathematical monographs. Oxford University Press, Oxford, 2000. Zbl0957.49001MR1857292
  2. [2] L. Ambrosio – P. Tilli, ‘Selected Topics on “Analysis in Metric Spaces”, Quaderni della Scuola Normale Superiore, Pisa, 2000. Zbl1084.28500MR2012736
  3. [3] G. Bouchitté – G. Buttazzo – I. Fragalà, Mean curvature of a measure and related variational problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), 179–196. Zbl1015.49015MR1655514
  4. [4] G. Buttazzo – G. Bouchitté, Characterization of optimal shapes and masses through Monge-Kantorovich equation, J. European Math. Soc. 3 (2001), 139–168. Zbl0982.49025MR1831873
  5. [5] G. Buttazzo – E. Oudet – E. Stepanov, Optimal transportation problems with free Dirichlet regions, Progress Nonlinear Differential Equations Appl. 51 (2002), 41–65. Zbl1055.49029MR2197837
  6. [6] G. Buttazzo – A. Pratelli – S. Solimini – E. Stepanov, Mass transportation and urban planning problems, forthcoming. 
  7. [7] G. Buttazzo – E. Stepanov, On regularity of transport density in the Monge-Kantorovich problem, Preprint del Dipartimento di Matematica, Università di Pisa, 2001. Zbl1084.49036MR2002147
  8. [8] L. Caffarelli – M. Feldman – R. J. McCann, Constructing optimal maps for Monge’s transport problem as a limit of strictly convex costs, J. Amer. Math. Soc. 15 (2002), 1–206. Zbl1053.49032MR1862796
  9. [9] G. David – S. Semmes, “Analysis of and on uniformly rectifiable sets”, Vol. 38 of Math. Surveys Monographs. Amer. Math. Soc., Providence, RI, 1993. Zbl0832.42008MR1251061
  10. [10] C. Kuratowski, “Topologie”, Vol. 1, Państwowe Wydawnictwo Naukowe, Warszawa, 1958, in French. Zbl0078.14603MR90795

Citations in EuDML Documents

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  1. Guillaume Carlier, Filippo Santambrogio, A variational model for urban planning with traffic congestion
  2. Guillaume Carlier, Filippo Santambrogio, A variational model for urban planning with traffic congestion
  3. Alessio Brancolini, Giuseppe Buttazzo, Optimal networks for mass transportation problems
  4. Alessio Brancolini, Giuseppe Buttazzo, Filippo Santambrogio, Eugene Stepanov, Long-term planning versus short-term planning in the asymptotical location problem
  5. Alessio Brancolini, Giuseppe Buttazzo, Filippo Santambrogio, Eugene Stepanov, Long-term planning short-term planning in the asymptotical location problem
  6. Alessio Brancolini, Giuseppe Buttazzo, Optimal networks for mass transportation problems
  7. Antoine Lemenant, Edoardo Mainini, On convex sets that minimize the average distance

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