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Synchronized traffic plans and stability of optima

Published online by Cambridge University Press:  30 January 2008

Marc Bernot  and
Alessio Figalli
Marc Bernot
Affiliation:
UMPA, ENS Lyon, 46 Allée d'Italie, 69007 Lyon, France; mbernot@umpa.ens-lyon.fr
Alessio Figalli
Affiliation:
Scuola Normale Superiore, Piazza dei Cavalieri 7, 56100 Pisa, Italy; a.figalli@sns.it
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Abstract

The irrigation problem is the problem of finding an efficient way to transport a measure μ+onto a measure μ-. By efficient, we mean that a structure that achieves the transport (which, following [Bernot, Caselles and Morel, Publ. Mat.49 (2005) 417–451], we call traffic plan)is better if it carries the mass in a grouped way rather than in a separate way.This is formalized by considering costs functionals that favorize this property.The aim of this paper is to introduce a dynamical cost functional on traffic plans that we argue to be more realistic.The existence of minimizers is proved in two ways: in some cases, we can deduce it from a classical semicontinuity argument; the other cases are treated by studying the link between our cost and the one introduced in [Bernot, Caselles and Morel, Publ. Mat.49 (2005) 417–451].Finally, we discuss the stability of minimizers with respect to specific variations of the cost functional.

Keywords

Irrigation problemtraffic plansdynamical coststability
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 14 , Issue 4 , October 2008 , pp. 864 - 878
Copyright
© EDP Sciences, SMAI, 2008

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References

L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variations and free discontinuity problems, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press (2000).
M. Bernot, Irrigation and Optimal Transport. Ph.D. thesis, École Normale Supérieure de Cachan, France (2005). Available at http://www.umpa.ens-lyon.fr/ ~mbernot.
Bernot, M., Caselles, V. and Morel, J.-M., Traffic plans. Publ. Mat. 49 (2005) 417–451. CrossRef
M. Bernot, V. Caselles and J.-M. Morel, The structure of branched transportation networks. Calc. Var. Partial Differential Equations (online first). DOI: 10.1007/s00526-007-0139-0.
Brancolini, A., Buttazzo, G. and Santambrogio, F., Path functionals over Wasserstein spaces. J. EMS 8 (2006) 414–434.
W. D'Arcy Thompson, On Growth and Form. Cambridge University Press (1942).
R.M. Dudley, Real Analysis and Probability. Cambridge University Press (2002).
Gilbert, E.N., Minimum cost communication networks. Bell System Tech. J. 46 (1967) 2209–2227. CrossRef
Kantorovich, L., On the transfer of masses. Dokl. Acad. Nauk. USSR 37 (1942) 7–8.
Maddalena, F., Solimini, S. and Morel, J.M., A variational model of irrigation patterns. Interfaces and Free Boundaries 5 (2003) 391–416. CrossRef
G. Monge, Mémoire sur la théorie des déblais et de remblais. Histoire de l'Académie Royale des Sciences de Paris (1781) 666–704.
J.D. Murray, Mathematical Biology, Biomathematics Texts 19. Springer (1993).
A.M. Turing, The chemical basis of morphogenesis. Phil. Trans. Soc. Lond. B237 (1952) 37–72.
C. Villani, Topics in optimal transportation, Graduate Studies in Mathematics 58. American Mathematical Society, Providence, RI (2003).
Xia, Q., Optimal paths related to transport problems. Commun. Contemp. Math. 5 (2003) 251–279. CrossRef