Multiscale problems and homogenization for second-order Hamilton-Jacobi equations

Journal of Differential Equations

Volumn 243, Issue 2, 2007, Pages 349-387

  Alvarez, Olivier ^a   Bardi, Martino ^b   Marchi, Claudio ^c  

^a UNIVERSITÉ DE ROUEN   (France)

^b UNIVERSITY OF PADOVA   (Italy)

^c UNIVERSITY OF CALABRIA   (Italy)

Author keywords

Bellman Isaacs equations; Ergodicity; Hamilton Jacobi equations; Homogenization; Hypoelliptic operators; Iterated homogenization; Multiscale problem; Nonlinear parabolic equations; Oscillating initial data; Singular perturbations; Stabilization; Viscosity solutions

Indexed keywords

EID: 36049037276     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2007.05.027     Document Type: Article

References (57)

1. Allaire G. Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992) 1482-1518
2. Allaire G., and Briane M. Multiscale convergence and reiterated homogenisation. Proc. Roy. Soc. Edinburgh Sect. A 126 (1996) 297-342
3. Alvarez O. Homogenization of Hamilton-Jacobi equations in perforated sets. J. Differential Equations 159 (1999) 543-577
4. Alvarez O., and Bardi M. Viscosity solutions methods for singular perturbations in deterministic and stochastic control. SIAM J. Control Optim. 40 (2001) 1159-1188
5. Alvarez O., and Bardi M. Singular perturbations of nonlinear degenerate parabolic PDEs: A general convergence result. Arch. Ration. Mech. Anal. 170 (2003) 17-61
6. Alvarez O., and Bardi M. Ergodic problems in differential games. In: Jorgensen S., Quincampoix M., and Vincent T.L. (Eds). Advances in Dynamic Game Theory. Ann. Internat. Soc. Dynam. Games vol. 9 (2007), Birkhäuser, Boston 131-152
7. Alvarez O., and Bardi M. Ergodicity, stabilization and singular perturbations for Bellman-Isaacs equations. Tech. report, University of Padova, 2007, available at. http://cpde.iac.rm.cnr.it/ricerca.php
8. O. Alvarez, M. Bardi, C. Marchi, Multiscale singular perturbations and homogenization of optimal control problems, in: F. Ancona, A. Bressan, P. Cannarsa, F.H. Clarke, P.R. Wolenski (Eds.), Geometric Control and Nonsmooth Analysis, in: Ser. Adv. Math. Appl. Sci., World Scientific Publishing, in press
9. Alvarez O., and Ishii H. Hamilton-Jacobi equations with partial gradient and application to homogenization. Comm. Partial Differential Equations 26 (2001) 983-1002
10. Arisawa M. Quasi-periodic homogenizations for second-order Hamilton-Jacobi-Bellman equations. Adv. Math. Sci. Appl. 11 (2001) 465-480
11. Arisawa M., and Giga Y. Anisotropic curvature flow in a very thin domain. Indiana Univ. Math. J. 52 (2003) 257-281
12. Arisawa M., and Lions P.-L. On ergodic stochastic control. Comm. Partial Differential Equations 23 (1998) 2187-2217
13. Artstein Z., and Gaitsgory V. The value function of singularly perturbed control systems. Appl. Math. Optim. 41 (2000) 425-445
14. Avellaneda M. Iterated homogenization, differential effective medium theory and applications. Comm. Pure Appl. Math. 40 (1987) 527-554
15. Bardi M. Homogenization of quasilinear elliptic equations with possibly superquadratic growth. In: Marino A., and Murthy M.K.V. (Eds). Nonlinear Variational Problems and Partial Differential Equations. Isola d'Elba, 1990. Pitman Res. Notes Math. vol. 320 (1995), Longman, Harlow 44-56
16. G. Barles, F. Da Lio, P.-L. Lions, P.E. Souganidis, Ergodic problems and periodic homogenization for fully nonlinear equations in half-space type domains with Neumann boundary conditions, preprint
17. Bensoussan A. Perturbation Methods in Optimal Control (1988), Wiley/Gauthiers-Villars, Chichester
18. Bensoussan A., Boccardo L., and Murat F. Homogenization of elliptic equations with principal part not in divergence form and Hamiltonian with quadratic growth. Comm. Pure Appl. Math. 39 (1986) 769-805
19. Bensoussan A., Lions J.-L., and Papanicolaou G. Asymptotic Analysis for Periodic Structures (1978), North-Holland, Amsterdam
20. Birindelli I., and Wigniolle J. Homogenization of Hamilton-Jacobi equations in the Heisenberg group. Comm. Pure Appl. Anal. 2 (2003) 461-479
21. Biroli M., Mosco U., and Tchou N. Homogenization for degenerate operators with periodical coefficients with respect to the Heisenberg group. C. R. Acad. Sci. Paris Sér. I Math. 322 (1996) 439-444
22. Biroli M., Mosco U., and Tchou N. Homogenization by the Heisenberg group. Adv. Math. Sci. Appl. 7 (1997) 809-831
23. Biroli M., Picard C., and Tchou N. Homogenization of the p-Laplacian associated with the Heisenberg group. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 22 (1998) 23-42
24. Brahim-Otsmane S., Francfort G.A., and Murat F. Correctors for the homogenization of the wave and heat equations. J. Math. Pures Appl. 71 (1992) 197-231
25. Braides A., and Defranceschi A. Homogenizations of Multiple Integrals (1998), Clarendon Press, Oxford
26. Buckdahn R., Hu Y., and Peng S. Probabilistic approach to homogenization of viscosity solutions of parabolic PDEs. NoDEA Nonlinear Differential Equations Appl. 6 (1999) 395-411
27. Buckdahn R., and Ichihara N. Limit theorems for controlled backward SDEs and homogenization of Hamilton-Jacobi-Bellman equations. Appl. Math. Optim. 51 (2005) 1-33
28. Caffarelli L.A. A note on nonlinear homogenization. Comm. Pure Appl. Math. 52 (1999) 829-838
29. Caffarelli L.A., Souganidis P.E., and Wang L. Homogenization of fully nonlinear, uniformly elliptic and parabolic partial differential equations in stationary ergodic media. Comm. Pure Appl. Math. 58 (2005) 319-361
30. Crandall M.G., Ishii H., and Lions P.-L. User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 (1992) 1-67
31. Evans L. The perturbed test function method for viscosity solutions of nonlinear P.D.E. Proc. Roy. Soc. Edinburgh Sect. A 111 (1989) 359-375
32. Evans L. Periodic homogenisation of certain fully nonlinear partial differential equations. Proc. Roy. Soc. Edinburgh Sect. A 120 (1992) 245-265
33. Fleming W.H., and Souganidis P.E. On the existence of value functions of two-players, zero-sum stochastic differential games. Indiana Univ. Math. J. 38 (1989) 293-314
34. Franchi B., and Tesi M.C. Two-scale homogenization in the Heisenberg group. J. Math. Pures Appl. 81 (2002) 495-532
35. García-Azorero J., Gutiérrez C.E., and Peral I. Homogenization of quasilinear parabolic equations in periodic media. Comm. Partial Differential Equations 28 (2003) 1887-1910
36. Horie K., and Ishii H. Homogenization of Hamilton-Jacobi equations on domains with small scale periodic structure. Indiana Univ. Math. J. 47 (1998) 1011-1058
37. Ichihara K., and Kunita H. A classification of the second order degenerate elliptic operators and its probabilistic characterization. Z. Wahrsch. Verw. Geb. 30 (1974) 235-254
38. Ishii H. Almost periodic homogenization of Hamilton-Jacobi equations. In: Fiedler B. (Ed). International Conference on Differential Equations, Proceedings of the Conference, vol. 1. Equadiff '99, Singapore (2000), World Scientific 600-605
39. Ishii H., and Lions P.-L. Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Differential Equations 83 (1990) 26-78
40. Jensen R., and Lions P.-L. Some asymptotic problems in fully nonlinear elliptic equations and stochastic control. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11 (1984) 129-176
41. Jikov V.V., Kozlov S.M., and Oleinik O.A. Homogenization of Differential Operators and Integral Functionals (1994), Springer-Verlag, Berlin
42. Kabanov Y., and Pergamenshchikov S. Two-Scale Stochastic Systems. Asymptotic Analysis and Control (2003), Springer-Verlag, Berlin
43. Katsoulakis M.A. A representation formula and regularizing properties for viscosity solutions of second-order fully nonlinear degenerate parabolic equations. Nonlinear Anal. 24 (1995) 147-158
44. Kushner H.J. Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems (1990), Birkhäuser, Boston
45. Lions J.-L., Lukkassen D., Persson L.E., and Wall P. Reiterated homogenization of nonlinear monotone operators. Chinese Ann. Math. Ser. B 22 (2001) 1-12
46. P.-L. Lions, G. Papanicolaou, S.R.S. Varadhan, Homogenization of Hamilton-Jacobi equations, unpublished, 1986
47. Lions P.-L., and Souganidis P.E. Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications. Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005) 667-677
48. Lions P.-L., and Souganidis P.E. Homogenization of "viscous" Hamilton-Jacobi equations in stationary ergodic media. Comm. Partial Differential Equations 30 (2005) 335-375
49. Lukkassen D. Reiterated homogenization of nonstandard Lagrangians. C. R. Acad. Sci. Paris Sér. I Math. 332 (2001) 999-1004
50. Marchi C. Homogenization for fully nonlinear parabolic equations. Nonlinear Anal. 60 (2005) 411-428
51. Pardoux E. Homogenization of linear and semi-linear second order parabolic PDEs with periodic coefficients: A probabilistic approach. J. Funct. Anal. 167 (1999) 498-520
52. Rezakhanlou F., and Tarver J.E. Homogenization for stochastic Hamilton-Jacobi equations. Arch. Ration. Mech. Anal. 151 (2000) 277-309
53. Safonov M.V. On the classical solution of nonlinear elliptic equations of second-order. Math. USSR Izv. 33 (1989) 597-612 Engl. transl.:
54. Safonov M.V. On the classical solution of nonlinear elliptic equations of second-order. Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988)
55. Souganidis P.E. Stochastic homogenization of Hamilton-Jacobi equations and some applications. Asymptot. Anal. 20 (1999) 1-11
56. Swiech A. Another approach to the existence of value functions of stochastic differential games. J. Math. Anal. Appl. 204 (1996) 884-897
57. Tartar L. An introduction to the homogenization method in optimal design. Optimal Shape Design. Tróia, 1998 (2000), Springer-Verlag, Berlin 47-156