Author(s)

P. Daniele, G. Idone & A. Maugeri

Abstract

A continuum model of the transportation network with elastic demand is presented. The equilibrium conditions are expressed in terms of a Variational Inequality, for which an existence theorem and a computational procedure are provided. 1 Introduction In the papers [5], [7], the authors consider a continuum model of transportation network and characterize the equilibrium conditions by means of the following Variational Inequality: Find u ∈ K such that
Ω c (x,u (x ))(v (x ) −u (x ))dx ≥0 ∀v ∈ K (1) where K =  v ∈E (Ω,R 2 ):s (x ) ≤v (x ) ≤z (x )a.e. in Ω, div v (x )=t (x )a.e. in Ω,v (x ) | ∂Ω=ϕ (x )  Ωis a simply connected bounded domain in R 2 of generic point x =(x 1 ,x 2 ), with a Lipschitz boundary ∂Ω, v (x )=(v 1 (x ),v 2 (x ))represents the unknown flow at each point x ∈Ωand the components v 1 (x ),v 2 (x )are the traffic density through a neighbourhood of x in the directions of the increasing axes x 1 and x 2 . s (x )=(s 1 (x ),s 2 (x )), z (x )=(z 1 (x ),z 2 (x ))are the capacity constraints and it is assumed that s (x ),z (x ) ∈E (Ω,R 2 ), 0 ≤s (x )Keywords