Averaging on a background of vanishing viscosity

Elliptic equations of the form
$$\begin{gather*} \biggl(\mu a_{ij}\biggl (\frac x\varepsilon\biggr)\frac\partial{\partial x_i} \frac\partial{\partial x_j}+\varepsilon^{-1}v_i\biggl (\frac x\varepsilon\biggr) \frac\partial{\partial x_i}\biggr)u^{\mu,\varepsilon}(x)=0, \\ u^{\mu,\varepsilon}\big|_{\partial\Omega}=\varphi(x) \end{gather*}$$
with periodic coefficients are considered; $\mu$ and $\varepsilon$ are small parameters. For potential fields $v(y)$ and constants $a_{ij}=\delta_{ij}$, the asymptotic behavior as $\mu\to 0$ of the coefficients of the averaged operator (which is customarily also called the effective diffusion) is studied. It is shown that as $\mu\to 0$ the effective diffusion $\sigma(\mu)=\sigma_{ij}(\mu)$ decays exponentially, and the limit $\lim\limits_{\mu\to 0}\mu\ln\sigma(\mu)$ is found. Sufficient conditions are found for the existence of a limit operator as $\mu$ and $\varepsilon$ tend to 0 simultaneously. The structure of this operator depends on the symmetry reserve of the coefficients $a_{ij}(y)$ and $v_i(y)$; in particular, it may decompose into independent operators in subspaces of lower dimension.