The Neumann problem for elliptic equations with multiscale coefficients: operator estimates for homogenization

We prove an $L^2$-estimate for the homogenization of an elliptic operator $A_\varepsilon$ in a domain $\Omega$ with a Neumann boundary condition on the boundary $\partial\Omega$. The coefficients of the operator $A_\varepsilon$ are rapidly oscillating over different groups of variables with periods of different orders of smallness as $\varepsilon\to 0$. We assume minimal regularity of the data, which makes it possible to impart to the result the meaning of an estimate in the operator $(L^2(\Omega)\to L^2(\Omega))$-norm for the difference of the resolvents of the original and homogenized problems. We also find an approximation to the resolvent of the original problem in the operator $(L^2(\Omega)\to H^1(\Omega))$-norm.
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