Multiscale stochastic homogenization of monotone operators

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March 2007, 2(1): 181-192. doi: 10.3934/nhm.2007.2.181

Multiscale stochastic homogenization of monotone operators

1.	Department of Computational Mathematics, Chalmers University, SE-412 96 Göteborg, Sweden

Received February 2006 Revised September 2006 Published December 2006

Abstract
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Multiscale stochastic homogenization is studied for divergence structure parabolic problems. More specifically we consider the asymptotic behaviour of a sequence of realizations of the form
$\frac{\partial u^\omega_\varepsilon}{\partial t}- $div$(a(T_1(\frac{x}{\varepsilon_1})\omega_1, T_2(\frac{x}{\varepsilon_2})\omega_2 ,t, D u^\omega_\varepsilon))=f.$
It is shown, under certain structure assumptions on the random map $a(\omega_1,\omega_2,t,\xi)$, that the sequence $\{u^\omega_\e}$ of solutions converges weakly in $ L^p(0,T;W^{1,p}_0(\Omega))$ to the solution $u$ of the homogenized problem $ \frac{\partial u}{\partial t} - $div$( b( t,D u )) = f$.

Keywords: homogenization, G-convergence, stochastic, parabolic., elliptic, multiscale.

Mathematics Subject Classification: 35B27, 35B4.

Citation: Nils Svanstedt. Multiscale stochastic homogenization of monotone operators. Networks & Heterogeneous Media, 2007, 2 (1) : 181-192. doi: 10.3934/nhm.2007.2.181

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