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On the error of averaging symmetric elliptic systems. (English. Russian original) Zbl 0702.35254
Math. USSR, Izv. 35, No. 1, 183-201 (1990); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 53, No. 4, 851-867 (1989).
The paper deals with differential operators with random coefficients depending on a small parameter \[ (L\cdot \epsilon u)^{(\alpha)}(x)\equiv \sum^{d}_{i,j=1}\sum^{m}_{\beta =1}(\partial /\partial x_ i)(a_{ij}^{(\alpha \beta)}(x/\epsilon,w)(\partial /\partial x_ j)u^{(\beta)}(x)). \] It is shown, that there is G-convergence for these operators where \(\epsilon\to 0\) and some restrictions are imposed on the coefficients, that is, the resolvents of the averaging operators of these operators converge to the resolvents of the “averaging” operators of the same kind possessing the non-random coefficients.
The paper is dedicated to averaging errors. The coefficients of differential operators are supposed to satisfy the conditions of symmetry and ellipticity. The values of the coefficients at distant points are weakly dependent.
For obtaining averaging error estimates one investigates sequential approximations of some iterative process applying here the implicit integral representations.
Reviewer: I.N.Molčanov

MSC:
35R60 PDEs with randomness, stochastic partial differential equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35J45 Systems of elliptic equations, general (MSC2000)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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