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Averaging of symmetric diffusion in random medium. (English. Russian original) Zbl 0614.60051
Sib. Math. J. 27, 603-613 (1986); translation from Sib. Mat. Zh. 27, No. 4(158), 167-180 (1986).
The accuracy of convergence is studied of the solutions $$u_{\epsilon}$$ of the elliptic equations depending on a random parameter $$L_{\epsilon}u_{\epsilon}=f$$, $$x\in Q\subseteq {\mathbb{R}}^ d$$, $$u_{\epsilon}=g$$, $$x\in \partial Q$$, $$\epsilon >0$$, to the solution of the averaging problem $$\bar LU=f$$, $$x\in Q$$, $$U=g$$, $$x\in \partial Q$$ where $$L_{\epsilon}u=(1/2)D_ i(a_{ij}(y,\omega)D_ ju)$$, $$D_ i=\partial /\partial x_ i$$, $$y=x/\epsilon$$, $$\bar Lu=(1/2)\bar a_{ij}D_ iD_ ju$$, $$\bar a{}_{ij}$$ constants, $$d\geq 3$$, and $$L_{\epsilon}$$ is approaching $$\bar L$$ using the corresponding resolvent operators.
The diffusion coefficients are supposed to be homogeneous, $$a_{ij}(y+z,\omega)=a_{ij}(y,\omega)$$, $$z\in {\mathbb{Z}}^ d$$, strictly elliptic, $$C_ 1| \xi |^ 2\leq a_{ij}(y,\omega)\xi_ i\xi_ j\leq C_ 2| \xi |^ 2$$, $$\xi \in {\mathbb{R}}^ d$$, and to satisfy an ergodic property with respect to the random parameter $$\omega\in \Omega$$. The main theorem states the following estimate $E| u_{\epsilon}-U|^ 2_{\infty}\leq \tilde C\epsilon^{\gamma},\quad \tilde C,\gamma >0$ where $$| v|_{\infty}=\sup \{| v(x)|$$, $$x\in Q\}$$.
Reviewer: C.Vârsan

MSC:
 60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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References:
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