Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications
Lions, Pierre-Louis ; Souganidis, Panagiotis E.
Annales de l'I.H.P. Analyse non linéaire, Tome 22 (2005), p. 667-677 / Harvested from Numdam
@article{AIHPC_2005__22_5_667_0,
     author = {Lions, Pierre-Louis and Souganidis, Panagiotis E.},
     title = {Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {22},
     year = {2005},
     pages = {667-677},
     doi = {10.1016/j.anihpc.2004.10.009},
     mrnumber = {2171996},
     zbl = {02235973},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2005__22_5_667_0}
}
Lions, Pierre-Louis; Souganidis, Panagiotis E. Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications. Annales de l'I.H.P. Analyse non linéaire, Tome 22 (2005) pp. 667-677. doi : 10.1016/j.anihpc.2004.10.009. http://gdmltest.u-ga.fr/item/AIHPC_2005__22_5_667_0/

[1] Arisawa M., Quasi-periodic homogenizations for second-order Hamilton-Jacobi-Bellman equations, Adv. Math. Sci. Appl. 11 (2001) 465-480. | MR 1842387 | Zbl 1014.49018

[2] Arisawa M., Lions P.-L., On ergodic stochastic control, Comm. Partial Differential Equations 23 (1998) 2187-2217. | MR 1662180 | Zbl 01247980

[3] Barles G., A weak Bernstein method for fully nonlinear elliptic equations, Differential Integral Equations 4 (1991) 241-262. | MR 1081182 | Zbl 0733.35014

[4] Bensoussan A., Blakenship G., Controlled diffusions in a random medium, Stochastics 24 (1988) 87-120. | MR 972975 | Zbl 0666.93131

[5] Bhattacharya K., Cracium B., Homogenization of a Hamilton-Jacobi equation associated with the geometric motion of an interface, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003) 773-805. | MR 2006202 | Zbl 1043.35028

[6] Bourgeat A., Piatniski A., Approximations of effective coefficients in stochastic homogenization, Ann. Inst. H. Poincaré Probab. Statist. 40 (2004) 153-165. | Numdam | MR 2044813 | Zbl 1058.35023

[7] Cabre X., Caffarelli L.A., Fully nonlinear elliptic partial differential equations, Amer. Math. Soc., 1997.

[8] Caffarelli L.A., A note on nonlinear homogenization, Comm. Pure Appl. Math. 52 (1999) 829-838. | MR 1682808 | Zbl 0933.35022

[9] L.A. Caffarelli, P.-L., Lions, P.E. Souganidis, in preparation.

[10] Caffarelli L.A., Souganidis P.E., Wang L., Stochastic homogenization for fully nonlinear, second-order partial differential equations, Comm. Pure Appl. Math. LVII (2005) 319-361. | MR 2116617 | Zbl 1063.35025

[11] Castell F., Homogenization of random semilinear PDEs, Probab. Theory Related Fields 121 (2001) 492-524. | MR 1872426 | Zbl 0989.35022

[12] Cocordel M., Periodic homogenization of Hamilton-Jacobi equations: additive eigenvalue and variational formulas, Indiana Univ. Math. J. 45 (1996) 1095-1117. | MR 1444479 | Zbl 0871.49025

[13] Crandall M.G., Ishii H., Lions P.-L., User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. 27 (1992) 1-67. | MR 1118699 | Zbl 0755.35015

[14] Dal Maso G., Modica L., Nonlinear stochastic homogenization and ergodic theory, J. Reine Angew. Math. 368 (1986) 28-42. | MR 850613 | Zbl 0582.60034

[15] N. Dirr, A. Yip, personal communication.

[16] E W., A class of homogenization problems in the calculus of variations, Comm. Pure Appl. Math. XLIV (1991) 733-759. | MR 1115092 | Zbl 0773.49007

[17] Evans L.C., Periodic homogenization of certain fully nonlinear partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A 120 (1992) 245-265. | MR 1159184 | Zbl 0796.35011

[18] Evans L.C., The perturbed test function method for viscosity solutions of nonlinear pde, Proc. Roy. Soc. Edinburgh Sect. A 111 (1989) 359-375. | MR 1007533 | Zbl 0679.35001

[19] Freidlin M., On factorization of a non-negative definite matrix, Probab. Theory Appl. 13 (1968) 375-378, (in Russian). | Zbl 0169.20603

[20] Ishii H., Almost periodic homogenization of Hamilton-Jacobi equations, in: Int. Conf. on Diff. Eqs., vol. 1, Berlin 1999, World Scientific, River Edge, NJ, 2000, pp. 600-605. | MR 1870203 | Zbl 0969.35018

[21] Ishii H., Lions P.-L., Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, JDE 83 (1990) 26-78. | MR 1031377 | Zbl 0708.35031

[22] Jikov V.V., Kozlov S.M., Oleinik O.A., Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, 1991. | MR 1329546 | Zbl 0838.35001

[23] H. Kosynga, F. Rezankhanlou, S.R.S. Varadhan, Stochastic homogenization of Hamilton-Jacobi-Bellman equations, Preprint.

[24] Kozlov S.M., The method of averaging and walks in inhomogeneous environments, Russian Math. Surveys 40 (1985) 73-145. | MR 786087 | Zbl 0615.60063

[25] Lions P.-L., Resolution de problemes elliptic quasilineaires, Arch. Rational Mech. Anal. 74 (1980) 335-353. | MR 588033 | Zbl 0449.35036

[26] Lions P.-L., Souganidis P.E., Correctors for the homogenization of Hamilton-Jacobi equations in a stationary ergodic setting, Comm. Pure Appl. Math. LVI (2003) 1501-1524. | MR 1988897 | Zbl 1050.35012

[27] Lions P.-L., Souganidis P.E., Homogenization of “viscous” Hamilton-Jacobi equations in stationary ergodic media, Comm. Partial Differential Equations 30 (2005) 335-375. | Zbl 1065.35047

[28] P.-L. Lions, P.E. Souganidis, in preparation.

[29] P.-L. Lions, G. Papanicolaou, S.R.S. Varadhan, Homogenization of Hamilton-Jacobi equations, Preprint.

[30] Majda A., Souganidis P.E., Large scale front dynamics for turbulent reaction-diffusion equations with separated velocity scales, Nonlinearity 7 (1994) 1-30. | MR 1260130 | Zbl 0839.76093

[31] S. Müller, private communication.

[32] Oleinik A., Alcuni visultati sulle equazioni lineari e quasilineri ellitico-paraboliche a derivate parziali del second ordine, Rend. Classe Sci. Fis. Mat., Nat. Acad. Naz. Lincei, Sci. 8 40 (1966) 774-784. | Zbl 0173.12906

[33] Papanicolaou G., Varadhan S.R.S., Boundary value problems with rapidly oscillating random coefficients, in: Fritz J., Lebaritz J.L., Szasz D. (Eds.), Rigorous Results in Statistical Mechanics and Quantum Field Theory, Proc. Colloq. on Random Fields, Colloquia Mathematica Societ. Janos Bolyai, vol. 10, 1979, pp. 835-873. | MR 712714 | Zbl 0499.60059

[34] Papanicolaou G., Varadhan S.R.S., Diffusion with random coefficients, in: Krishnaiah P.R. (Ed.), Essays in Statistics and Probability, North-Holland, 1981. | MR 659505 | Zbl 0486.60076

[35] Rezankhanlou F., Tarver J., Homogenization for stochastic Hamilton-Jacobi equations, Arch. Rational Mech. Anal. 151 (2000) 277-309. | MR 1756906 | Zbl 0954.35022

[36] Serrin J., The problem of Dirichlet of quasilinear elliptic differential equations with many independent variables, Philos. Trans. Roy. Soc. London Ser. A 264 (1969) 413-469. | MR 282058 | Zbl 0181.38003

[37] Souganidis P.E., Front propagation: Theory and applications, in: CIME Course on “Viscosity Solutions and their Applications”, Lecture Notes in Math., vol. 1660, Springer, 1997. | Zbl 0882.35016

[38] Souganidis P.E., Recent developments in the theory of front propagation and its applications, in: Sabiclussi G. (Ed.), Modern Methods in Scientific Computing and Applications, NATO Science Ser. II, vol. 75, Kluwer Academic, 2002. | MR 2004361 | Zbl 1052.35097

[39] Souganidis P.E., Stochastic homogenization of Hamilton-Jacobi equations and some applications, Asymptotic Anal. 20 (1999) 1-11. | MR 1697831 | Zbl 0935.35008