We review some recent results in quantitative stochastic homogenization for divergence-form, quasilinear elliptic equations. In particular, we are interested in obtaining -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.
Keywords: Stochastic homogenization, Lipschitz regularity, error estimate
Scott Armstrong 1
Scott Armstrong. Uniform Lipschitz estimates in stochastic homogenization. Journées équations aux dérivées partielles (2014), article no. 1, 11 p.. doi: 10.5802/jedp.104
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author = {Scott Armstrong},
title = {Uniform {Lipschitz} estimates in stochastic homogenization},
booktitle = {},
series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
eid = {1},
pages = {1--11},
year = {2014},
publisher = {Groupement de recherche 2434 du CNRS},
doi = {10.5802/jedp.104},
language = {en},
url = {https://proceedings.centre-mersenne.org/articles/10.5802/jedp.104/}
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TY - JOUR AU - Scott Armstrong TI - Uniform Lipschitz estimates in stochastic homogenization JO - Journées équations aux dérivées partielles PY - 2014 SP - 1 EP - 11 PB - Groupement de recherche 2434 du CNRS UR - https://proceedings.centre-mersenne.org/articles/10.5802/jedp.104/ DO - 10.5802/jedp.104 LA - en ID - JEDP_2014____A1_0 ER -
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[1] Lipschitz regularity for elliptic equations with random coefficients (Preprint)
[2] Lipschitz estimates in almost-periodic homogenization (Preprint, arXiv:1409.2094)
[3] Quantitative stochastic homogenization of convex integral functionals (Preprint, arXiv:1406.0996)
[4] Compactness methods in the theory of homogenization, Comm. Pure Appl. Math., Volume 40 (1987) no. 6, pp. 803-847 | DOI | Zbl | MR
[5] bounds on singular integrals in homogenization, Comm. Pure Appl. Math., Volume 44 (1991) no. 8-9, pp. 897-910 | DOI | Zbl | MR
[6] Nonlinear stochastic homogenization, Ann. Mat. Pura Appl. (4), Volume 144 (1986), pp. 347-389 | DOI | Zbl | MR
[7] Nonlinear stochastic homogenization and ergodic theory, J. Reine Angew. Math., Volume 368 (1986), pp. 28-42 | Zbl | MR
[8] A regularity theory for random elliptic operators, Preprint, arXiv:1409.2678
[9] An optimal variance estimate in stochastic homogenization of discrete elliptic equations, Ann. Probab., Volume 39 (2011) no. 3, pp. 779-856 | DOI | Zbl | MR
[10] 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] Quantitative results on the corrector equation in stochastic homogenization, Preprint
[12] The averaging of random operators, Mat. Sb. (N.S.), Volume 109(151) (1979) no. 2, p. 188-202, 327 | Zbl | MR
[13] Estimates on the variance of some homogenization problems, 1998, Unpublished preprint
[14] 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 | Zbl | MR
[15] Averaging of symmetric diffusion in a random medium, Sibirsk. Mat. Zh., Volume 27 (1986) no. 4, p. 167-180, 215 | Zbl | MR
[16] 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 | Zbl | MR
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