Averaging differential operators with almost periodic, rapidly oscillating coefficients

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.
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