Geodesic
Averaging differential operators with almost periodic, rapidly oscillating coefficients
Sbornik. Mathematics, Tome 35 (1979) no. 4, pp. 481-498 Cet article a éte moissonné depuis la source Math-Net.Ru

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The Dirichlet problem $$ D_ia_{ij}(x\varepsilon^{-1})D_ju_\varepsilon(x)=f(x)\quad\text{in}\quad\Omega,\qquad u_\varepsilon(x)|_{\partial\Omega}=f_1(x), $$ containing a small parameter $\varepsilon$ is considered, where the coefficients $a_{ij}(y)$ are almost periodic functions in the sense of Besicovitch. An averaged equation having constant coefficients is contracted, and the convergence of $u_\varepsilon(x)$ to the solution $u_0(x)$ of the averaged equation is proved. An estimate of the remainder $\sup_{x\in\Omega}|u_\varepsilon(x)-u_0(x)|\leqslant C\varepsilon$ is obtained under the condition that there are no anomalous commensurable frequences in the spectrum of the coefficients. For the problem in the whole space a complete asymptotic expansion in powers of $\varepsilon$ is constructed. Bibliography: 12 titles. 
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     author = {S. M. Kozlov},
     title = {Averaging differential operators with almost periodic, rapidly oscillating coefficients},
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     year = {1979},
     volume = {35},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1979_35_4_a2/}
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S. M. Kozlov. Averaging differential operators with almost periodic, rapidly oscillating coefficients. Sbornik. Mathematics, Tome 35 (1979) no. 4, pp. 481-498. http://geodesic.mathdoc.fr/item/SM_1979_35_4_a2/

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