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Homogenization of monotone systemsof Hamilton-Jacobi equations

Published online by Cambridge University Press:  21 October 2008

Fabio Camilli ,
Olivier Ley  and
Paola Loreti
Fabio Camilli
Affiliation:
Dip. di Matematica Pura e Applicata, Univ. dell'Aquila, loc. Monteluco di Roio, 67040 l'Aquila, Italy. [email protected]
Olivier Ley
Affiliation:
Université François-Rabelais, Tours; Laboratoire de Mathématiques et Physique Théorique, CNRS UMR 6083, Fédération de Recherche Denis Poisson (FR 2964); Faculté des Sciences et Techniques, Parc de Grandmont, 37200 Tours, France. [email protected]
Paola Loreti
Affiliation:
Dip. di Metodi e Modelli Matematici per le Scienze Applicate, Facoltà di Ingegneria, Sapienza Università di Roma, via Scarpa 16, 00161 Roma, Italy. [email protected]
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Abstract

In this paper we study homogenization for a class of monotone systems of first-order time-dependent periodic Hamilton-Jacobi equations.We characterize the Hamiltonians of the limit problem by appropriate cell problems. Hence weshow the uniform convergence of the solution of the oscillating systems tothe boundeduniformly continuous solution of thehomogenized system.

Keywords

Systems of Hamilton-Jacobi equationsviscosity solutionshomogenization
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 16 , Issue 1 , January 2010 , pp. 58 - 76
Copyright
© EDP Sciences, SMAI, 2008

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References

Alvarez, O. and Bardi, M., Singular perturbations of nonlinear degenerate parabolic PDEs: a general convergence result. Arch. Ration. Mech. Anal. 170 (2003) 17–61. CrossRef
M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser Boston Inc., Boston, MA, USA (1997).
G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi. Springer-Verlag, Paris, France (1994).
Barles, G., Some homogenization results for non-coercive Hamilton-Jacobi equations. Calc. Var. Partial Differential Equations 30 (2007) 449–466. CrossRef
Barles, G., Biton, S. and Ley, O., A geometrical approach to the study of unbounded solutions of quasilinear parabolic equations. Arch. Ration. Mech. Anal. 162 (2002) 287–325. CrossRef
Camilli, F. and Loreti, P., Comparison results for a class of weakly coupled systems of eikonal equations. Hokkaido Math. J. 37 (2008) 349–362. CrossRef
Capuzzo-Dolcetta, I. and Ishii, H., On the rate of convergence in homogenization of Hamilton-Jacobi equations. Indiana Univ. Math. J. 50 (2001) 1113–1129.
Concordel, M.C., Periodic homogenization of Hamilton-Jacobi equations: additive eigenvalues and variational formula. Indiana Univ. Math. J. 45 (1996) 1095–1117.
Eizenberg, A. and Freidlin, M., On the Dirichlet problem for a class of second order PDE systems with small parameter. Stochastics Stochastics Rep. 33 (1990) 111–148. CrossRef
Engler, H. and Lenhart, S.M., Viscosity solutions for weakly coupled systems of Hamilton-Jacobi equations. Proc. London Math. Soc. (3) 63 (1991) 212–240. CrossRef
Evans, L.C., The perturbed test function method for viscosity solutions of nonlinear PDE. Proc. Roy. Soc. Edinburgh Sect. A 111 (1989) 359–375 CrossRef
Ishii, H., Perron's method for monotone systems of second-order elliptic partial differential equations. Differential Integral Equations 5 (1992) 1–24.
Ishii, H. and Koike, S., Remarks on elliptic singular perturbation problems. Appl. Math. Optim. 23 (1991) 1–15. CrossRef
Ishii, H. and Koike, S., Viscosity solutions for monotone systems of second-order elliptic PDEs. Comm. Partial Differential Equations 16 (1991) 1095–1128. CrossRef
Lions, P.-L. and Souganidis, P.E., Correctors for the homogenization of Hamilton-Jacobi equations in the stationary ergodic setting. Comm. Pure Appl. Math. 56 (2003) 1501–1524. CrossRef
P.-L. Lions, B. Papanicolaou and S.R.S. Varadhan, Homogenization of Hamilton-Jacobi equations. Preprint (1986).
Shimano, K., Homogenization and penalization of functional first-order PDE. NoDEA Nonlinear Differ. Equ. Appl. 13 (2006) 1–21. CrossRef