Uniform Lipschitz estimates in stochastic homogenization

Scott Armstrong[1]

  • [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)

  • page 1-11
  • ISSN: 0752-0360

Abstract

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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.

How to cite

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Armstrong, Scott. "Uniform Lipschitz estimates in stochastic homogenization." Journées Équations aux dérivées partielles (2014): 1-11. <http://eudml.org/doc/275471>.

@article{Armstrong2014,
abstract = {We review some recent results in quantitative stochastic homogenization for divergence-form, quasilinear elliptic equations. In particular, we are interested in obtaining $L^\infty $-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.},
affiliation = {Ceremade (UMR CNRS 7534) Université Paris-Dauphine Place du Maréchal de Lattre de Tassigny 75775 Paris Cedex 16, France},
author = {Armstrong, Scott},
journal = {Journées Équations aux dérivées partielles},
keywords = {Stochastic homogenization; Lipschitz regularity; error estimate},
language = {eng},
pages = {1-11},
publisher = {Groupement de recherche 2434 du CNRS},
title = {Uniform Lipschitz estimates in stochastic homogenization},
url = {http://eudml.org/doc/275471},
year = {2014},
}

TY - JOUR
AU - Armstrong, Scott
TI - Uniform Lipschitz estimates in stochastic homogenization
JO - Journées Équations aux dérivées partielles
PY - 2014
PB - Groupement de recherche 2434 du CNRS
SP - 1
EP - 11
AB - We review some recent results in quantitative stochastic homogenization for divergence-form, quasilinear elliptic equations. In particular, we are interested in obtaining $L^\infty $-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.
LA - eng
KW - Stochastic homogenization; Lipschitz regularity; error estimate
UR - http://eudml.org/doc/275471
ER -

References

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  13. A Naddaf, T. Spencer, Estimates on the variance of some homogenization problems, (1998, Unpublished preprint) Zbl0871.35010
  14. G. C. Papanicolaou, S. R. S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, Random fields, Vol. I, II (Esztergom, 1979) 27 (1981), 835-873, North-Holland, Amsterdam Zbl0499.60059MR712714
  15. V. V. Yurinskiĭ, Averaging of symmetric diffusion in a random medium, Sibirsk. Mat. Zh. 27 (1986), 167-180, 215 Zbl0614.60051MR867870
  16. V. V. Yurinskiĭ, Homogenization error estimates for random elliptic operators, Mathematics of random media (Blacksburg, VA, 1989) 27 (1991), 285-291, Amer. Math. Soc., Providence, RI Zbl0729.60060MR1117252

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