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.
Mots clés : systems of Hamilton-Jacobi equations, viscosity solutions, homogenization
@article{COCV_2010__16_1_58_0,
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},
number = {1},
year = {2010},
doi = {10.1051/cocv:2008061},
mrnumber = {2598088},
zbl = {1187.35008},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv:2008061/}
}
TY - JOUR AU - Camilli, Fabio AU - Ley, Olivier AU - Loreti, Paola TI - Homogenization of monotone systems of Hamilton-Jacobi equations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2010 SP - 58 EP - 76 VL - 16 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2008061/ DO - 10.1051/cocv:2008061 LA - en ID - COCV_2010__16_1_58_0 ER -
%0 Journal Article %A Camilli, Fabio %A Ley, Olivier %A Loreti, Paola %T Homogenization of monotone systems of Hamilton-Jacobi equations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2010 %P 58-76 %V 16 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2008061/ %R 10.1051/cocv:2008061 %G en %F COCV_2010__16_1_58_0
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] and , Singular perturbations of nonlinear degenerate parabolic PDEs: a general convergence result. Arch. Ration. Mech. Anal. 170 (2003) 17-61. | Zbl
[2] and , Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser Boston Inc., Boston, MA, USA (1997). | Zbl
[3] , Solutions de viscosité des équations de Hamilton-Jacobi. Springer-Verlag, Paris, France (1994). | Zbl
[4] , Some homogenization results for non-coercive Hamilton-Jacobi equations. Calc. Var. Partial Differential Equations 30 (2007) 449-466. | Zbl
[5] , and , A geometrical approach to the study of unbounded solutions of quasilinear parabolic equations. Arch. Ration. Mech. Anal. 162 (2002) 287-325. | Zbl
[6] and , Comparison results for a class of weakly coupled systems of eikonal equations. Hokkaido Math. J. 37 (2008) 349-362.
[7] and , On the rate of convergence in homogenization of Hamilton-Jacobi equations. Indiana Univ. Math. J. 50 (2001) 1113-1129.
[8] , Periodic homogenization of Hamilton-Jacobi equations: additive eigenvalues and variational formula. Indiana Univ. Math. J. 45 (1996) 1095-1117. | Zbl
[9] and , On the Dirichlet problem for a class of second order PDE systems with small parameter. Stochastics Stochastics Rep. 33 (1990) 111-148. | Zbl
[10] and , Viscosity solutions for weakly coupled systems of Hamilton-Jacobi equations. Proc. London Math. Soc. (3) 63 (1991) 212-240. | Zbl
[11] , The perturbed test function method for viscosity solutions of nonlinear PDE. Proc. Roy. Soc. Edinburgh Sect. A 111 (1989) 359-375 | Zbl
[12] , Perron's method for monotone systems of second-order elliptic partial differential equations. Differential Integral Equations 5 (1992) 1-24. | Zbl
[13] and , Remarks on elliptic singular perturbation problems. Appl. Math. Optim. 23 (1991) 1-15. | Zbl
[14] and , Viscosity solutions for monotone systems of second-order elliptic PDEs. Comm. Partial Differential Equations 16 (1991) 1095-1128. | Zbl
[15] and , Correctors for the homogenization of Hamilton-Jacobi equations in the stationary ergodic setting. Comm. Pure Appl. Math. 56 (2003) 1501-1524. | Zbl
[16] , and , Homogenization of Hamilton-Jacobi equations. Preprint (1986).
[17] , Homogenization and penalization of functional first-order PDE. NoDEA Nonlinear Differ. Equ. Appl. 13 (2006) 1-21. | Zbl
Cité par Sources :
