Vol. 8, No. 8, 2015

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A junction condition by specified homogenization and application to traffic lights

Giulio Galise, Cyril Imbert and Régis Monneau

Vol. 8 (2015), No. 8, 1891–1929

Given a coercive Hamiltonian which is quasiconvex with respect to the gradient variable and periodic with respect to time and space, at least “far away from the origin”, we consider the solution of the Cauchy problem of the corresponding Hamilton–Jacobi equation posed on the real line. Compact perturbations of coercive periodic quasiconvex Hamiltonians enter into this framework, for example. We prove that the rescaled solution converges towards the solution of the expected effective Hamilton–Jacobi equation, but whose “flux” at the origin is “limited” in a sense made precise by Imbert and Monneau. In other words, the homogenization of such a Hamilton–Jacobi equation yields to supplement the expected homogenized Hamilton–Jacobi equation with a junction condition at the single discontinuous point of the effective Hamiltonian. We also illustrate possible applications of such a result by deriving, for a traffic flow problem, the effective flux limiter generated by the presence of a finite number of traffic lights on an ideal road. We also provide meaningful qualitative properties of the effective limiter.

Hamilton–Jacobi equations, quasiconvex Hamiltonians, homogenization, junction condition, flux-limited solution, viscosity solution
Mathematical Subject Classification 2010
Primary: 35B27, 49L25, 35F21
Received: 8 September 2014
Revised: 16 March 2015
Accepted: 11 May 2015
Published: 23 December 2015
Giulio Galise
Department of Mathematics
University of Salerno
Via Giovanni Paolo II, 132
I-84084 Fisciano (SA)
Cyril Imbert
CNRS, UMR 7580
CNRS, Université Paris-Est Créteil
61 avenue du Général de Gaulle
94010 Paris Créteil
Régis Monneau
Université Paris-Est
6–8 Avenue Blaise Pascal
Cité Descartes
F-77455 Champs-sur-Marne Marne-la-Vallée Cedex 2