Synchronized traffic plans and stability of optima

¹ 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: 20 April 2007

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.