Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications

Lions, Pierre-Louis; Souganidis, Panagiotis E.

doi:10.1016/j.anihpc.2004.10.009

Lions, Pierre-Louis ; Souganidis, Panagiotis E.

Annales de l'I.H.P. Analyse non linéaire, Tome 22 (2005) no. 5, pp. 667-677.

MR Zbl | 5 citations dans Numdam

DOI : 10.1016/j.anihpc.2004.10.009

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     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},
     pages = {667--677},
     publisher = {Elsevier},
     volume = {22},
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     doi = {10.1016/j.anihpc.2004.10.009},
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     zbl = {02235973},
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     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2004.10.009/}
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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) no. 5, pp. 667-677. doi : 10.1016/j.anihpc.2004.10.009. http://www.numdam.org/articles/10.1016/j.anihpc.2004.10.009/

Bibliographie
Cité par

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

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

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

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

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

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

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

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

[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 | Zbl

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

[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 | Zbl

[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

[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 | Zbl

[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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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

[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 | Zbl

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

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