Abstract

Some variational models have been recently introduced to the aim of modeling ramified structures, such as trees, rivers and so on. We introduce a general scheme in which the notion of "transport distance" is introduced starting from a general transport cost functional, through relaxation arguments. Then we apply this general framework to the irrigation cost, which is a particular cost functional depending on a parameter α between 0 and 1. We discuss the equivalence between this abstract approach and the above models

Author information

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Francesco Maddalena

Dip. di Matematica, Politecnico di Bari, Via Orabona 4, 70125 Bari, Italy

f.maddalena@poliba.it

Sergio Solimini

Dip. di Matematica, Politecnico di Bari, Via Orabona 4, 70125 Bari, Italy

solimini@poliba.it

References

  1. L. Ambrosio: Lecture notes on optimal transport problems, in: Mathematical Aspects of Evolving Interfaces (Madeira, 2000), P. Colli et al. (ed.), Lecture Notes in Mathematics 1812, Springer, Berlin (2003) 1–52.https://doi.org/10.1007/978-3-540-39189-0_1Google Scholar
  2. L. Ambrosio, N. Gigli, G. Savaré: Gradient Flows in Metric Spaces of Probability Measures, Birkhäuser, Basel (2005).Google Scholar
  3. M. Bernot, V. Caselles, J. M. Morel: Traffic plans, Publ. Mat., Barc. 49(2) (2005) 417–451.https://doi.org/10.5565/PUBLMAT_49205_09Google Scholar
  4. M. Bernot, V. Caselles, J. M. Morel: The structure of branched transportation networks, Calc. Var. Partial Differ. Equ. 32(3) (2008) 279–317.https://doi.org/10.1007/s00526-007-0139-0Google Scholar
  5. M. Bernot, V. Caselles, J. M. Morel: Branched Transportation Networks, Lecture Notes in Mathematics, Springer (2007).https://doi.org/10.1007/s00526-007-0139-0Google Scholar
  6. P. Billingsley: Convergence of Probability Measures, 2nd Ed., Wiley, Chichester (1999).https://doi.org/10.1002/9780470316962Google Scholar
  7. A. Brancolini, G. Buttazzzo: Optimal networks for mass transportation problems, ESAIM, Control Optim. Calc. Var. 11 (2005) 88–101.https://doi.org/10.1051/cocv:2004032Google Scholar
  8. A. Brancolini, G. Buttazzo, F. Santambrogio: Path functionals over Wasserstein spaces, J. Eur. Math. Soc. 8 (2006) 415–434.https://doi.org/10.4171/jems/61Google Scholar
  9. G. Devillanova, S. Solimini: Elementary properties of optimal irrigation patterns, Calc. Var. Partial Differ. Equ. 28 (2007) 317–349.https://doi.org/10.1007/s00526-006-0046-9Google Scholar
  10. G. Devillanova, S. Solimini: On the dimension of an irrigable measure, Rend. Semin. Mat. Univ. Padova 117 (2007) 1–49.Google Scholar
  11. R. M. Dudley: Real Analysis and Probability, Cambridge University Press, Cambridge (2002).https://doi.org/10.1017/CBO9780511755347Google Scholar
  12. L. C. Evans: Partial differential equations and Monge-Kantorovich mass transfer, in: Current Developments in Mathematics (Cambridge, 1997), R. Bott et al. (ed.), International Press, Boston (1999) 65–126.https://doi.org/10.4310/CDM.1997.v1997.n1.a2Google Scholar
  13. L. C. Evans, W. Gangbo: Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc. 653 (1999).https://doi.org/10.1090/memo/0653Google Scholar
  14. R. F. Gariepy, W. Ziemer: Modern Real Analysis, PWS, Boston (1995).Google Scholar
  15. E. N. Gilbert: Minimum cost communication networks, Bell Syst. Tech. J. 46 (1967) 2209–2227.https://doi.org/10.1002/j.1538-7305.1967.tb04250.xGoogle Scholar
  16. E. N. Gilbert, H. O. Pollak: Steiner minimal trees, SIAM J. Appl. Math. 16 (1968) 1–29.https://doi.org/10.1137/0116001Google Scholar
  17. L. Kantorovich: On the transfer of masses, Dokl. Akad. Nauk. USSR 37(7–8) (1942) 227–229.Google Scholar
  18. F. Maddalena, J. M. Morel, S. Solimini: A variational model of irrigation patterns, Interfaces Free Bound. 5(4) (2003) 391–415.https://doi.org/10.4171/ifb/85Google Scholar
  19. F. Maddalena, S. Solimini: Synchronous and asynchronous descriptions of irrigation problems, to appear.Google Scholar
  20. G. Monge: Mémoire sur la théorie des déblais et de remblais, Historie de l’Académie Royale des Sciences de Paris (1781) 666–704.Google Scholar
  21. J. M. Morel, F. Santambrogio: Comparison of distances between measures, Appl. Math. Lett. 20 (2007) 427–432.https://doi.org/10.1016/j.aml.2006.05.009Google Scholar
  22. Qinglan Xia: Optimal paths related to transport problems, Commun. Contemp. Math. 5(2) (2003) 251–279.https://doi.org/10.1142/S021919970300094XGoogle Scholar
  23. C. Villani: Topics in Mass Transportation, Graduate Studies in Mathematics 58, AMS, Providence (2003).https://doi.org/10.1090/gsm/058Google Scholar
  24. C. Villani: Optimal Transport. Old and New, Springer, Berlin (2009).https://doi.org/10.1007/978-3-540-71050-9Google Scholar

Suggested citation

F. Maddalena, S. Solimini. “Transport Distances and Irrigation Models.” Journal of Convex Analysis 16 (2009), No. 1, 121–152. https://doi.org/10.68381/jca16006

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Published by Heldermann Verlag, 2009. Rights now held by Banach Press LLC.