Document ESAIM: COCV
DOI: 10.1051/cocv:2008012
Synchronized traffic plans and stability of optima
Marc Bernot1 and Alessio Figalli21 UMPA, ENS Lyon, 46 Allée d'Italie, 69007 Lyon, France; mbernot@umpa.ens-lyon.fr
2 Scuola Normale Superiore, Piazza dei Cavalieri 7, 56100 Pisa, Italy; a.figalli@sns.it
(Received April 20, 2007. Published online January 30, 2008)
Abstract
The irrigation problem is the problem of finding an efficient way to transport a measure 
onto a measure 
. 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.
Mathematics Subject Classification. 49Q20, 90B10, 90B06, 90B20
Key words: Irrigation problem, traffic plans, dynamical cost, stability
© EDP Sciences, SMAI 2008