Calculus of Variations and Geometric Measure Theory

M. Bernot - A. Figalli

Synchronized traffic plans and stability of optima

created by figalli on 31 Aug 2007
modified on 08 Sep 2007


Accepted Paper

Inserted: 31 aug 2007
Last Updated: 8 sep 2007

Journal: ESAIM: COCV
Year: 2007


The irrigation problem is the problem of finding an efficient way to transport a measure $\mu^+$ onto a measure $\mu^-$. By efficient, we mean that a structure that achieves the transport (which 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 by Bernot, Caselles and Morel. Finally, we discuss the stability of minimizers with respect to specific variations of the cost functional.