Homogenization and vanishing viscosity in fully nonlinear elliptic equations: Rate of convergence estimates

Advanced Nonlinear Studies

Volumn 11, Issue 2, 2011, Pages 405-428

  Camilli, Fabio ^a   Cesaroni, Annalisa ^b   Marchi, Claudio ^b  

^a UNIVERSITY OF L'AQUILA   (Italy)

^b UNIVERSITY OF PADOVA   (Italy)

Author keywords

Fully nonlinear elliptic equations; Homogenization; Rate of convergence; Vanishing viscosity; Viscosity solutions

Indexed keywords

EID: 80051989204     PISSN: 15361365     EISSN: None     Source Type: Journal    
DOI: 10.1515/ans-2011-0210     Document Type: Article

References (23)

1. Y. Achdou, F. Camilli, I. Capuzzo Dolcetta: Homogenization of Hamilton-Jacobi equations: Numerical Methods, M3AS Math. Models Methods Appl. Sci. 18 (2008), no. 7, 1115-1143.
2. O. Alvarez, M. Bardi: Viscosity solutions methods for singular perturbations in deterministic and stochastic control, Siam J. Control Optim. 40 (2001), no. 4, 1159-1188.
3. O. Alvarez, M. Bardi: Singular perturbations of nonlinear degenerate parabolic PDEs: a general convergence result, Arch. Ration. Mech. Anal. 170 (2003), no. 1, 17-61. (Pubitemid 37364563)
4. O. Alvarez, M. Bardi: Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equations, Memoirs of the Amer. Math. Soc. 204 (2010), no. 960.
5. M. Arisawa, P.-L. Lions: On Ergodic Stochastic Control, Comm. Partial Differential Equations 23 (1998), no. 11-12, 2187-2217. (Pubitemid 128361780)
6. G. Barles: Solutions de Viscosité des Équations de Hamilton-Jacobi, Springer Verlag, Berlin, 1994.
7. M. Bardi, I. Capuzzo Dolcetta: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkäuser, Boston, 1997.
8. A. Bensoussan, J.L.Lions, G.Papanicolau, Asymptotic Analysis for Periodic Structures, North-Holland Publishing Co., Amsterdam-New York, 1978.
9. F. Camilli, C. Marchi: Rate of convergence in periodic homogeneization of fully nonlinear uniformly elliptic PDEs, Nonlinearity 22 (2009), no. 6, 1481-1498.
10. I. Capuzzo Dolcetta, H. Ishii: On the rate of convergence in homogenization of Hamilton-Jacobi equations, Indiana Univ. Math. J. 50 (2001), no. 3, 1113-1129.
11. M. G. Crandall, H. Ishii, P.-L. Lions: User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1-67.
12. A.-L. Dalibard: Homogenization of a quasilinear parabolic equation with vanishing viscosity, J. Math. Pures Appl. 86 (2006), no. 2, 133-154. (Pubitemid 44159220)
13. L. Evans: Periodic homogenisation of certain fully nonlinear partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A 120 (1992), no. 3-4, 245-265.
14. D. Gilbarg, N.S. Trudinger: Elliptic Partial Differential Equations of Second Order, Classics in Mathematics. Springer-Verlag, Berlin, 2001 (2nd edition).
15. K. Horie, H. Ishii: Simultaneous effects of homogenization and vanishing viscosity in fully nonlinear elliptic equations, Funkcial. Ekvac. 46 (2003), no. 1, 63-88.
16. S. M. Kozlov, A.L.Piatnitski: Degeneration of effective diffusion in the presence of periodic potential. Ann. Inst. H. Poincar Probab. Statist. 32 (1996), no. 5, 571-587.
17. P.-L. Lions: Existence results for first-order Hamilton-Jacobi equations, Ricerche Mat. 32 (1983), no. 1, 1-23.
18. P.-L. Lions, G. Papanicolaou, S.R.S. Varadhan: Homogenization of Hamilton-Jacobi Equations, unpublished, 1986.
19. P.-L. Lions, P.E. Souganidis: Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), no. 5, 667-677. (Pubitemid 41328788)
20. C. Marchi: Rate of convergence for multiscale homogenization of Hamilton-Jacobi equations, Proc. Roy. Soc. Edinburgh Sect. A 139 (2009), no. 3, 519-539.
21. A.Oberman, R.Takei, A.Vladimirsky: Homogenization of Metric Hamilton-Jacobi Equations, Multiscale Model. Simul. 8 (2009), no. 1, 269-295.
22. M. V. Safonov: Classical solution of second-order nonlinear elliptic equations, translation in Math. USSR-Izv. 33 (1989), no. 3, 597-612.
23. N.S. Trudinger: Comparison principles and pointwise estimates for viscosity solutions of nonlinear elliptic equations, Rev. Mat. Iberoamericana 4 (1988), no. 3-4, 453-468.