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Published February 2003 | Submitted
Journal Article Open

Approximation of the effective conductivity of ergodic media by periodization


This paper is concerned with the approximation of the effective conductivity σ(A, μ) associated to an elliptic operator ∇_xA(x,η)∇_x where for xЄℝ^d,d ≥ 1, A(x,η) is a bounded elliptic random symmetric d×d matrix and η takes value in an ergodic probability space (X, μ). Writing A^N (x, η) the periodization of A(x, η) on the torus T^d_N of dimension d and side N we prove that for μ-almost all η lim ^(N→+∞) σ(A^N, η) = σ(A,μ) We extend this result to non-symmetric operators ∇_x (a+E(x, η))∇_x corresponding to diffusions in ergodic divergence free flows (a is d×d elliptic symmetric matrix and E(x, η) an ergodic skew-symmetric matrix); and to discrete operators corresponding to random walks on ℤ^d with ergodic jump rates. The core of our result is to show that the ergodic Weyl decomposition associated to L^2(X,μ) can almost surely be approximated by periodic Weyl decompositions with increasing periods, implying that semi-continuous variational formulae associated to L^2(X,μ) can almost surely be approximated by variational formulae minimizing on periodic potential and solenoidal functions.

Additional Information

© 2003 Springer-Verlag. Received: 10 January 2002. Revised version: 12 August 2002. Published online: 14 November 2002. Part of this work was supported by the Aly Kaufman fellowship. The author would like to thank Dmitry Ioffe for his hospitality during his stay at the Technion, for suggesting this problem and for stimulating and helpful discussions. Thanks are also due to the referee for many useful comments.

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