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Stochastic homogenization and random lattices

Lions, Pierre-Louis; Le Bris, Claude; Blanc, Xavier (2007), Stochastic homogenization and random lattices, Journal de mathématiques pures et appliquées, 88, 1, p. 34-63. http://dx.doi.org/10.1016/j.matpur.2007.04.006

Type
Article accepté pour publication ou publié
Date
2007-04
Journal name
Journal de mathématiques pures et appliquées
Volume
88
Number
1
Publisher
Elsevier
Pages
34-63
Publication identifier
http://dx.doi.org/10.1016/j.matpur.2007.04.006
Metadata
Show full item record
Author(s)
Lions, Pierre-Louis
Le Bris, Claude
Blanc, Xavier cc
Abstract (FR)
We present some variants of stochastic homogenization theory for scalar elliptic equations of the form−div[A( x ε ,ω)∇u(x,ω)]= f . These variants basically consist in defining stochastic coefficients A( x ε ,ω) from stochastic deformations (using random diffeomorphisms) of the periodic setting, as announced in [X. Blanc, C. Le Bris, P.-L. Lions, Une variante de la théorie de l’homogénéisation stochastique des opérateurs elliptiques (A variant of stochastic homogenization theory for elliptic operators), C. R. Acad. Sci. Sér. I 343 (2006) 717–727]. The settings we define are not covered by the existing theories. We also clarify the relation between this type of questions and our construction, performed in [X. Blanc, C. Le Bris, P.-L. Lions, A definition of the ground state energy for systems composed of infinitely many particles, Commun. Partial Differential Equations 28 (1–2) (2003) 439–475; X. Blanc, C. Le Bris, P.-L. Lions, The energy of some microscopic stochastic lattices, Arch. Rat. Mech. Anal. 184 (2) (2007) 303–339], of the energy of, both deterministic and stochastic, microscopic infinite sets of points in interaction.
Abstract (EN)
We present some variants of stochastic homogenization theory for scalar elliptic equations of the form -div[ A(x/ε,ω) ∇ u] = f. These variants basically consist in defining stochastic coefficients A(x/ε,ω) from stochastic deformations (using random diffeormorphisms) of the periodic setting, as announced in [4]. The settings we define are not covered by the existing theories. We also clarify the relation between this type of questions and our construction, performed in [3,5], of the energy of, both deterministic and stochastic, microscopic infinite sets of points in interaction.
Subjects / Keywords
Thermodynamique; Equation différentielle; Equation dérivée partielle; Differential equation; Homogenization; Elliptic operator

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