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Strong convergence to the homogenized limit of elliptic equations with random coefficients

Authors: Joseph G. Conlon and Thomas Spencer
Journal: Trans. Amer. Math. Soc. 366 (2014), 1257-1288
MSC (2010): Primary 81T08, 82B20, 35R60, 60J75
Published electronically: October 23, 2013
MathSciNet review: 3145731
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Abstract: Consider a discrete uniformly elliptic divergence form equation on the $ d$ dimensional lattice $ \mathbf {Z}^d$ with random coefficients. It has previously been shown that if the random environment is translational invariant, then the averaged Green's function, together with its first and second differences, are bounded by the corresponding quantities for the constant coefficient discrete elliptic equation. It has also been shown that if the random environment is ergodic, then solutions of the random equation converge under diffusive scaling to solutions of a homogenized elliptic PDE on $ \mathbf {R}^d$. In this paper point-wise estimates are obtained on the difference between the averaged Green's function and the homogenized Green's function for certain random environments which are strongly mixing.

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Additional Information

Joseph G. Conlon
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109

Thomas Spencer
Affiliation: School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540

Keywords: Euclidean field theory, PDE with random coefficients, homogenization
Received by editor(s): March 1, 2011
Received by editor(s) in revised form: November 23, 2011
Published electronically: October 23, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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