Uniform Lipschitz estimates in stochastic homogenization
Scott Armstrong1
1 Ceremade (UMR CNRS 7534) Université Paris-Dauphine Place du Maréchal de Lattre de Tassigny 75775 Paris Cedex 16, France
Journées équations aux dérivées partielles (2014), article no. 1, 11 p.

We review some recent results in quantitative stochastic homogenization for divergence-form, quasilinear elliptic equations. In particular, we are interested in obtaining L ∞ -type bounds on the gradient of solutions and thus giving a demonstration of the principle that solutions of equations with random coefficients have much better regularity (with overwhelming probability) than a general equation with non-constant coefficients.

DOI: 10.5802/jedp.104
Classification: 35B27, 60H25, 35J20, 35J62
Keywords: Stochastic homogenization, Lipschitz regularity, error estimate
Scott Armstrong 1

1 Ceremade (UMR CNRS 7534) Université Paris-Dauphine Place du Maréchal de Lattre de Tassigny 75775 Paris Cedex 16, France 
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Scott Armstrong. Uniform Lipschitz estimates in stochastic homogenization. Journées équations aux dérivées partielles (2014), article  no. 1, 11 p. doi : 10.5802/jedp.104. https://proceedings.centre-mersenne.org/articles/10.5802/jedp.104/

[1] S. N. Armstrong; J.-C. Mourrat Lipschitz regularity for elliptic equations with random coefficients (Preprint)

[2] S. N. Armstrong; Z. Shen Lipschitz estimates in almost-periodic homogenization (Preprint, arXiv:1409.2094)

[3] S. N. Armstrong; C. K. Smart Quantitative stochastic homogenization of convex integral functionals (Preprint, arXiv:1406.0996)

[4] M. Avellaneda; F.-H. Lin Compactness methods in the theory of homogenization, Comm. Pure Appl. Math., Volume 40 (1987) no. 6, pp. 803-847 | DOI | MR | Zbl

[5] M. Avellaneda; F.-H. Lin L p bounds on singular integrals in homogenization, Comm. Pure Appl. Math., Volume 44 (1991) no. 8-9, pp. 897-910 | DOI | MR | Zbl

[6] G. Dal Maso; L. Modica Nonlinear stochastic homogenization, Ann. Mat. Pura Appl. (4), Volume 144 (1986), pp. 347-389 | DOI | MR | Zbl

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

[8] A. Gloria; S. Neukamm; F. Otto A regularity theory for random elliptic operators, Preprint, arXiv:1409.2678

[9] A. Gloria; F. Otto An optimal variance estimate in stochastic homogenization of discrete elliptic equations, Ann. Probab., Volume 39 (2011) no. 3, pp. 779-856 | DOI | MR | Zbl

[10] A. Gloria; F. Otto An optimal error estimate in stochastic homogenization of discrete elliptic equations, Ann. Appl. Probab., Volume 22 (2012) no. 1, pp. 1-28 | DOI | MR

[11] A. Gloria; F. Otto Quantitative results on the corrector equation in stochastic homogenization, Preprint

[12] S. M. Kozlov The averaging of random operators, Mat. Sb. (N.S.), Volume 109(151) (1979) no. 2, p. 188-202, 327 | MR | Zbl

[13] A Naddaf; T. Spencer Estimates on the variance of some homogenization problems, 1998, Unpublished preprint

[14] G. C. Papanicolaou; S. R. S. Varadhan Boundary value problems with rapidly oscillating random coefficients, Random fields, Vol. I, II (Esztergom, 1979) (Colloq. Math. Soc. János Bolyai), Volume 27, North-Holland, Amsterdam, 1981, pp. 835-873 | MR | Zbl

[15] V. V. Yurinskiĭ Averaging of symmetric diffusion in a random medium, Sibirsk. Mat. Zh., Volume 27 (1986) no. 4, p. 167-180, 215 | MR | Zbl

[16] V. V. Yurinskiĭ Homogenization error estimates for random elliptic operators, Mathematics of random media (Blacksburg, VA, 1989) (Lectures in Appl. Math.), Volume 27, Amer. Math. Soc., Providence, RI, 1991, pp. 285-291 | MR | Zbl

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