Rate of convergence for multiscale homogenization of HamiltonJacobi equations

Proceedings of the Royal Society of Edinburgh Section A: Mathematics

Volumn 139, Issue 3, 2009, Pages 519-539

  Marchi, Claudio ^a  

^a UNIVERSITY OF CALABRIA   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 67650713685     PISSN: 03082105     EISSN: 14737124     Source Type: Journal    
DOI: 10.1017/S0308210507000704     Document Type: Article

References (29)

1. G. Allaire and M. Briane. Multiscale convergence and reiterated homogenization. Proc. R. Soc. Edinb. A 126 (1996), 297-342.
2. O. Alvarez. Homogenization of Hamilton-Jacobi equations in perforated sets. J. Diff. Eqns 159 (1999), 543-577.
3. O. Alvarez and M. Bardi. Viscosity solutions methods for singular perturbations in deterministic and stochastic control. SIAM J. Control Optim. 40 (2001), 1159-1188.
4. O. Alvarez and M. Bardi. Singular perturbations of nonlinear degenerate parabolic PDEs: a general convergence result. Arch. Ration. Mech. Analysis 170 (2003), 17-61.
5. O. Alvarez and M. Bardi. Ergodicity, stabilization and singular perturbations for Bellman-Isaacs equations. Mem. Amer. Math. Soc. (In the press.)
6. O. Alvarez and H. Ishii. Hamilton-Jacobi equations with partial gradient and application to homogenization. Commun. PDEs 26 (2001), 983-1002. (Pubitemid 33731162)
7. O. Alvarez, M. Bardi and C. Marchi. Multiscale problems and homogenization for second-order Hamilton-Jacobi equations. J. Diff. Eqns 243 (2007), 349-387.
8. O. Alvarez, M. Bardi and C. Marchi. Multiscale singular perturbation and homogenization of optimal control problems. In Geometric control and nonsmooth analysis (ed. F. Ancona et al.), Series on Advances in Mathematics for Applied Sciences, vol. 76 (World Scientific, 2008).
9. M. Arisawa. Quasi-periodic homogenizations for second-order Hamilton-Jacobi-Bellman equations. Adv. Math. Sci. Appl. 11 (2001), 465-480.
10. Z. Artstein and V. Gaitsgory. The value function of singularly perturbed control systems. Appl. Math. Optim. 41 (2000), 425-445. (Pubitemid 30596843)
11. M. Avellaneda. Iterated homogenization, differential effective medium theory and applications. Commun. Pure Appl. Math. 40 (1987), 527-554.
12. M. Bardi and I. Capuzzo-Dolcetta. Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations (Birkḧauser, 1997).
13. A. Bensoussan, J.-L. Lions and G. Papanicolaou. Asymptotic analysis for periodic structures (Amsterdam: North-Holland, 1978).
14. I. Birindelli and J. Wigniolle. Homogenization of Hamilton-Jacobi equations in the Heisen-berg group. Commun. Pure Appl. Analysis 2 (2003), 461-479.
15. A. Braides and A. Defranceschi. Homogenizations of multiple integrals (Oxford: Clarendon, 1998).
16. I. Capuzzo-Dolcetta and H. Ishii. On the rate of convergence in homogenization of Hamil-ton-Jacobi equations. Indiana Univ. Math. J. 50 (2001), 1113-1129.
17. L. Evans. The perturbed test function method for viscosity solutions of nonlinear PDE. Proc. R. Soc. Edinb. A 111 (1989), 359-375.
18. L. Evans. Periodic homogenisation of certain fully nonlinear partial differential equations. Proc. R. Soc. Edinb. A 120 (1992), 245-265.
19. L. Evans and P. E. Souganidis. Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations. Indiana Univ. Math. J. 33 (1984), 773-797.
20. K. Horie and H. Ishii. Homogenization of Hamilton-Jacobi equations on domains with small scale periodic structure. Indiana Univ. Math. J. 47 (1998), 1011-1058.
21. H. Ishii. Almost periodic homogenization of Hamilton-Jacobi equations. In Proc. Equa-diff '99 Int. Conf. on Differential Equations (ed. B. Fiedler), vol. 1, pp. 600-605 (World Scientific, 2000).
22. J.-L. Lions, D. Lukkassen, L. E. Persson and P. Wall. Reiterated homogenization of nonlinear monotone operators. Chin. Annals Math. B22 (2001), 1-12.
23. P.-L. Lions and P. E. Souganidis. Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications. Annales Inst. H. Poincaŕe Analyse Non Lińeaire 22 (2005), 667-677.
24. P.-L. Lions and P. E. Souganidis. Homogenization of 'viscous' Hamilton-Jacobi equations in stationary ergodic media. Commun. PDEs 30 (2005), 335-375.
25. P.-L. Lions, G. Papanicolaou and S. R. S. Varadhan. Homogeneization of Hamilton-Jacobi equations. Preprint (1986).
26. D. Lukkassen. Reiterated homogenization of nonstandard Lagrangians. C. R. Acad. Sci. Paris Śer. I 332 (2001), 999-1004.
27. F. Rezakhanlou and J. E. Tarver. Homogenization for stochastic Hamilton-Jacobi equations. Arch. Ration. Mech. Analysis 151 (2000), 277-309.
28. P. E. Souganidis. Stochastic homogenization of Hamilton-Jacobi equations and some applications. Asymp. Analysis 20 (1999), 1-11.
29. B. Stroffolini. Homogenization of Hamilton-Jacobi equations in Carnot groups. ESAIM: Control Optim. Calc. Variations 13 (2007), 107-119.