Uniform Lipschitz estimates in stochastic homogenization
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 : https://doi.org/10.5802/jedp.104
Classification:  35B27,  60H25,  35J20,  35J62
Keywords: Stochastic homogenization, Lipschitz regularity, error estimate
@article{JEDP_2014____A1_0,
     author = {Armstrong, Scott},
     title = {Uniform Lipschitz estimates in stochastic homogenization},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2014},
     doi = {10.5802/jedp.104},
     language = {en},
     url = {http://www.numdam.org/item/JEDP_2014____A1_0}
}
Armstrong, Scott. Uniform Lipschitz estimates in stochastic homogenization. Journées équations aux dérivées partielles (2014), article  no. 1, 11 p. doi : 10.5802/jedp.104. http://www.numdam.org/item/JEDP_2014____A1_0/

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

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

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

[4] Avellaneda, M.; Lin, F.-H. Compactness methods in the theory of homogenization, Comm. Pure Appl. Math., Tome 40 (1987) no. 6, pp. 803-847 | Article | MR 910954 | Zbl 0632.35018

[5] Avellaneda, M.; Lin, F.-H. L p bounds on singular integrals in homogenization, Comm. Pure Appl. Math., Tome 44 (1991) no. 8-9, pp. 897-910 | Article | MR 1127038 | Zbl 0761.42008

[6] Dal Maso, G.; Modica, L. Nonlinear stochastic homogenization, Ann. Mat. Pura Appl. (4), Tome 144 (1986), pp. 347-389 | Article | MR 870884 | Zbl 0607.49010

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

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

[9] Gloria, A.; Otto, F. An optimal variance estimate in stochastic homogenization of discrete elliptic equations, Ann. Probab., Tome 39 (2011) no. 3, pp. 779-856 | Article | MR 2789576 | Zbl 1215.35025

[10] Gloria, A.; Otto, F. An optimal error estimate in stochastic homogenization of discrete elliptic equations, Ann. Appl. Probab., Tome 22 (2012) no. 1, pp. 1-28 | Article | MR 2932541

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

[12] Kozlov, S. M. The averaging of random operators, Mat. Sb. (N.S.), Tome 109(151) (1979) no. 2, p. 188-202, 327 | MR 542557 | Zbl 0415.60059

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

[14] Papanicolaou, G. C.; Varadhan, S. R. S. Boundary value problems with rapidly oscillating random coefficients, Random fields, Vol. I, II (Esztergom, 1979), North-Holland, Amsterdam (Colloq. Math. Soc. János Bolyai) Tome 27 (1981), pp. 835-873 | MR 712714 | Zbl 0499.60059

[15] Yurinskiĭ, V. V. Averaging of symmetric diffusion in a random medium, Sibirsk. Mat. Zh., Tome 27 (1986) no. 4, p. 167-180, 215 | MR 867870 | Zbl 0614.60051

[16] Yurinskiĭ, V. V. Homogenization error estimates for random elliptic operators, Mathematics of random media (Blacksburg, VA, 1989), Amer. Math. Soc., Providence, RI (Lectures in Appl. Math.) Tome 27 (1991), pp. 285-291 | MR 1117252 | Zbl 0729.60060