Homogenization of monotone systems of Hamilton-Jacobi equations
ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 1, pp. 58-76.

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 we show the uniform convergence of the solution of the oscillating systems to the bounded uniformly continuous solution of the homogenized system.

DOI : 10.1051/cocv:2008061
Classification : 35B27, 49L25, 35K45
Mots-clés : systems of Hamilton-Jacobi equations, viscosity solutions, homogenization 
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     author = {Camilli, Fabio and Ley, Olivier and Loreti, Paola},
     title = {Homogenization of monotone systems of {Hamilton-Jacobi} equations},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {58--76},
     publisher = {EDP-Sciences},
     volume = {16},
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     year = {2010},
     doi = {10.1051/cocv:2008061},
     mrnumber = {2598088},
     zbl = {1187.35008},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2008061/}
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Camilli, Fabio; Ley, Olivier; Loreti, Paola. Homogenization of monotone systems of Hamilton-Jacobi equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 1, pp. 58-76. doi : 10.1051/cocv:2008061. http://www.numdam.org/articles/10.1051/cocv:2008061/

[1] O. Alvarez and M. Bardi, Singular perturbations of nonlinear degenerate parabolic PDEs: a general convergence result. Arch. Ration. Mech. Anal. 170 (2003) 17-61. | Zbl

[2] M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser Boston Inc., Boston, MA, USA (1997). | Zbl

[3] G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi. Springer-Verlag, Paris, France (1994). | Zbl

[4] G. Barles, Some homogenization results for non-coercive Hamilton-Jacobi equations. Calc. Var. Partial Differential Equations 30 (2007) 449-466. | Zbl

[5] G. Barles, S. Biton and O. Ley, A geometrical approach to the study of unbounded solutions of quasilinear parabolic equations. Arch. Ration. Mech. Anal. 162 (2002) 287-325. | Zbl

[6] F. Camilli and P. Loreti, Comparison results for a class of weakly coupled systems of eikonal equations. Hokkaido Math. J. 37 (2008) 349-362.

[7] I. Capuzzo-Dolcetta and H. Ishii, On the rate of convergence in homogenization of Hamilton-Jacobi equations. Indiana Univ. Math. J. 50 (2001) 1113-1129.

[8] M.C. Concordel, Periodic homogenization of Hamilton-Jacobi equations: additive eigenvalues and variational formula. Indiana Univ. Math. J. 45 (1996) 1095-1117. | Zbl

[9] A. Eizenberg and M. Freidlin, On the Dirichlet problem for a class of second order PDE systems with small parameter. Stochastics Stochastics Rep. 33 (1990) 111-148. | Zbl

[10] H. Engler and S.M. Lenhart, Viscosity solutions for weakly coupled systems of Hamilton-Jacobi equations. Proc. London Math. Soc. (3) 63 (1991) 212-240. | Zbl

[11] L.C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE. Proc. Roy. Soc. Edinburgh Sect. A 111 (1989) 359-375 | Zbl

[12] H. Ishii, Perron's method for monotone systems of second-order elliptic partial differential equations. Differential Integral Equations 5 (1992) 1-24. | Zbl

[13] H. Ishii and S. Koike, Remarks on elliptic singular perturbation problems. Appl. Math. Optim. 23 (1991) 1-15. | Zbl

[14] H. Ishii and S. Koike, Viscosity solutions for monotone systems of second-order elliptic PDEs. Comm. Partial Differential Equations 16 (1991) 1095-1128. | Zbl

[15] P.-L. Lions and P.E. Souganidis, Correctors for the homogenization of Hamilton-Jacobi equations in the stationary ergodic setting. Comm. Pure Appl. Math. 56 (2003) 1501-1524. | Zbl

[16] P.-L. Lions, B. Papanicolaou and S.R.S. Varadhan, Homogenization of Hamilton-Jacobi equations. Preprint (1986).

[17] K. Shimano, Homogenization and penalization of functional first-order PDE. NoDEA Nonlinear Differ. Equ. Appl. 13 (2006) 1-21. | Zbl

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