# zbMATH — the first resource for mathematics

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)
Full Text:
##### References:
  S. M. Kozlov, ?Averaging of random operators,? Mat. Sb.,109, No. 2, 188-202 (1979).  V. V. Yurinskii, ?On averaging of elliptic boundary value problem with random coefficients,? Sib. Mat. Zh.,21, No. 3, 209-223 (1980).  V. V. Zhikov, S. M. Kozlov, O. A. Oleinik, and Ha Tien Ngoan, ?Averaging and G-convergence of elliptic operators,? Usp. Mat. Nauk,34, No. 5, 65-133 (1979). · Zbl 0445.35096  V. V. Yurinskii, ?On averaging of diffusion in a random medium,? Teor. Veroyatn. Primen.,29, No. 3, 607 (1984).  V. V. Yurinskii, ?On averaging of diffusion in random medium,? Tr. IM SO AN SSSR,5, 75-85 (1985).  I. P. Kornfel’d, Ya. G. Sinai, and S. V. Fomin, Ergodic Theory [in Russian], Nauka, Moscow (1980).  I. I. Gikhman and A. V. Skorokhod, Introduction to Theory of Random Processes [in Russian], Nauka, Moscow (1977). · Zbl 0429.60002  J. Neveu, Mathematical Foundations of the Calculus of Probability, Holden-Day (1965). · Zbl 0137.11301  D. G. Aronson, ?Nonnegative solutions of linear parabolic equations,? Ann. della Scuola Normale Superiore di Pisa, Ser. III,22, No. 4, 607-694 (1968). · Zbl 0182.13802  Yu. V. Kazachenko and M. I. Yadrenko, ?Local properties of sampling functions of random fields. I,? in: Theory of Probability and Mathematical Statistics [in Russian], No. 14, Vishcha Shkola, Kiev (1976), pp. 53-66.  O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Elliptic Equations, Academic Press (1968).  I. A. Ibragimov and Yu. V. Linnik, Independent and Stationarily Connected Quantities [in Russian], Nauka, Moscow (1965). · Zbl 0154.42201  S. M. Kozlov, ?Conductivity of two-dimensional random media,? Usp. Mat. Nauk,34, No. 4, 193-194 (1979).  O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc. (1968).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.