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Homogenization of elliptic and parabolic equations with periodic coefficients in a bounded domain under the Neumann condition T. A. Suslina Saint Petersburg State University
Russian version:
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 2024, Volume 88, Issue 4, DOI: https://doi.org/10.4213/im9520 
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    IntroductionThe paper is concerned with homogenization theory of periodic differential operators, a topic which has been extensively studied. In the first place, we mention the books [1]–[4]. 0.1. The class of operatorsLet $\Gamma$ be a lattice in $\mathbb{R}^d$, and let $\Omega$ be the cell of $\Gamma$. For $\Gamma$-periodic functions, we use the notation $\psi^\varepsilon (\mathbf{x}):=\psi (\mathbf{x}/\varepsilon)$ and $\overline{\psi}:=|\Omega|^{-1} \int_\Omega \psi (\mathbf{x})\,d\mathbf{x}$. We also set $\mathbf{D} := - i \nabla$. Let $\mathcal{O}\subset\mathbb{R}^d$ be a bounded domain of class $C^{1,1}$. In $L_2(\mathcal{O};\mathbb{C}^n)$, we study a selfadjoint matrix strongly elliptic second-order differential operator $\mathcal{B}_{N,\varepsilon}$, $0<\varepsilon\leqslant 1$, with the Neumann boundary condition. The principal part $A_{N,\varepsilon}$ of the operator $\mathcal{B}_{N,\varepsilon}$ is given in a factorized form $A_{N,\varepsilon}=b(\mathbf{D})^*g^\varepsilon (\mathbf{x})b(\mathbf{D})$, where $b(\mathbf{D})$ is a matrix homogeneous first-order differential operator and $g(\mathbf{x})$ is a bounded and positive definite $\Gamma$-periodic matrix-valued function in $\mathbb{R}^d$. The homogenization problem for the operator $A_{N,\varepsilon}$ was studied in [5] and [6]. Now, we will study a more general class of selfadjoint differential operators $\mathcal{B}_{N,\varepsilon}$ involving lower order terms:
$$
\begin{equation}
\mathcal{B}_{N,\varepsilon}=b(\mathbf{D})^* g^\varepsilon b(\mathbf{D}) +\sum_{j=1}^d\bigl(a_j^\varepsilon (\mathbf{x})D_j +D_ja_j^\varepsilon(\mathbf{x})^*\bigr)+Q^\varepsilon (\mathbf{x}).
\end{equation}
\tag{0.1}
$$
Here, $a_j(\mathbf{x})$, $j=1,\dots,d$, and $Q(\mathbf{x})$ are $\Gamma$-periodic matrix-valued functions; in general, they are unbounded. The precise definition of the operator $\mathcal{B}_{N,\varepsilon}$ is given in terms of the corresponding quadratic form defined on the Sobolev space $H^1(\mathcal{O};\mathbb{C}^n)$. Assumptions are made ensuring that the operator $\mathcal{B}_{N,\varepsilon}$ is strongly elliptic. The coefficients of the operator (0.1) oscillate rapidly for small $\varepsilon$. A typical homogenization problem for the operator $\mathcal{B}_{N,\varepsilon}$ is to approximate the resolvent $(\mathcal{B}_{N,\varepsilon}-\zeta I)^{-1}$ or the generalized resolvent $(\mathcal{B}_{N,\varepsilon}-\zeta Q_0^\varepsilon)^{-1}$ as $\varepsilon\to 0$. Here, $Q_0(\mathbf{x})$ is a positive definite $\Gamma$-periodic matrix-valued function such that $Q_0$ and $Q_0^{-1}$ are bounded. An example of the operator of the form (0.1) is considered in § 11; this is the Schrödinger operator (11.1) with periodic rapidly oscillating coefficients (metric, magnetic potential and electric potential containing a singular term $\varepsilon^{-1} v^\varepsilon(\mathbf{x})$). 0.2. A survey of the results on the operator error estimatesIn a series of papers [7]–[9], Birman and Suslina developed an operator-theoretic (spectral) approach to homogenization problems. They studied the operator
$$
\begin{equation}
A_\varepsilon =b(\mathbf{D})^*g^\varepsilon (\mathbf{x})b(\mathbf{D})
\end{equation}
\tag{0.2}
$$
acting in $L_2(\mathbb{R}^d;\mathbb{C}^n)$. Here, $g(\mathbf{x})$ is a bounded and positive definite $\Gamma$-periodic $(m\times m)$-matrix-valued function in $\mathbb{R}^d$; $b(\mathbf{D})$ is a matrix differential operator of the form $b(\mathbf{D}) = \sum_{j=1}^d b_j D_j$, where $b_j$ are constant $(m \times n)$-matrices. It is assumed that $m \geqslant n$ and the symbol $b(\boldsymbol{\xi})$ has maximal rank. Many operators of mathematical physics can be written in the form (0.2). A simplest example is the acoustics operator $A_\varepsilon = - \operatorname{div}g^\varepsilon (\mathbf{x}) \nabla = \mathbf{D}^* g^\varepsilon (\mathbf{x}) \mathbf{D}$; in this case, we have $m=d$, $n=1$, and $b(\mathbf{D}) = \mathbf{D}$. The operator of elasticity theory also can be represented in the form (0.2). Another example arises when studying the Maxwell system; this is an operator of the form $A_\varepsilon = \operatorname{rot} \eta^\varepsilon(\mathbf{x}) \operatorname{rot} - \operatorname{\nabla} \nu^\varepsilon(\mathbf{x}) \operatorname{div}$ (for $d=3$). These and other examples are discussed in detail in [7]. In [7], it was shown that the resolvent $(A_\varepsilon +I)^{-1}$ converges in the operator norm on $L_2(\mathbb{R}^d;\mathbb{C}^n)$ to the resolvent of the effective operator $A^0=b(\mathbf{D})^*g^0b(\mathbf{D})$, as $\varepsilon\to 0$. Here, $g^0$ is a constant positive effective matrix. It was proved that
$$
\begin{equation}
\| (A_\varepsilon +I)^{-1}-(A^0 +I)^{-1}\|_{L_2(\mathbb{R}^d)\to L_2(\mathbb{R}^d)} \leqslant C\varepsilon.
\end{equation}
\tag{0.3}
$$
In [9], an approximation for the resolvent $(A_\varepsilon +I)^{-1}$ in the norm of operators acting from $L_2(\mathbb{R}^d;\mathbb{C}^n)$ to the Sobolev space $H^1(\mathbb{R}^d;\mathbb{C}^n)$ was obtained:
$$
\begin{equation}
\| (A_\varepsilon +I)^{-1}-(A^0+I)^{-1}-\varepsilon K(\varepsilon)\|_{L_2(\mathbb{R}^d)\to H^1(\mathbb{R}^d)}\leqslant C\varepsilon.
\end{equation}
\tag{0.4}
$$
In this approximation, the corrector $K(\varepsilon)$ is taken into account. The operator $K(\varepsilon)$ contains rapidly oscillating factors, and therefore, depends on $\varepsilon$. We have $\|\varepsilon K(\varepsilon)\|_{L_2\to H^1}=O(1)$. Estimates (0.3) and (0.4) are order-sharp. The constants in estimates are controlled explicitly in terms of the problem data. Such results are called operator error estimates in homogenization theory. The method of [7]–[9] is based on the scaling transformation, the Floquet–Bloch theory, and the analytic perturbation theory. Later, the spectral method was adapted by Suslina [10], [11] to the case of the operator
$$
\begin{equation}
\mathcal{B}_\varepsilon =A_\varepsilon +\sum_{j=1}^d \bigl( a_j^\varepsilon (\mathbf{x})D_j +D_j (a_j^\varepsilon (\mathbf{x}))^*\bigr) +Q^\varepsilon(\mathbf{x})
\end{equation}
\tag{0.5}
$$
acting in $L_2(\mathbb{R}^d;\mathbb{C}^n)$. It is convenient to fix a real-valued parameter $\lambda$ so that the operator $B_\varepsilon := \mathcal{B}_\varepsilon+\lambda Q_0^\varepsilon$ is positive definite. In [10], the following analogs of estimates (0.3) and (0.4) were obtained:
$$
\begin{equation}
\|B_\varepsilon^{-1}-(B^0)^{-1}\|_{L_2(\mathbb{R}^d)\to L_2(\mathbb{R}^d)} \leqslant C\varepsilon,
\end{equation}
\tag{0.6}
$$
$$
\begin{equation}
\|B_\varepsilon^{-1}-(B^0)^{-1}-\varepsilon \mathcal{K}(\varepsilon)\|_{L_2(\mathbb{R}^d)\to H^1(\mathbb{R}^d)} \leqslant C\varepsilon.
\end{equation}
\tag{0.7}
$$
Here, $B^0$ is the corresponding effective operator and $\mathcal{K}(\varepsilon)$ is the corresponding corrector. The spectral method was applied to parabolic systems by Suslina in [12]–[14]. In [12], [13], the principal term of approximation for the operator exponential $e^{-A_\varepsilon t}$, $t>0$, in the $(L_2 \to L_2)$-norm was found, and in [14], an approximation for this exponential in the $(L_2 \to H^1)$-norm with corrector taken into account was obtained:
$$
\begin{equation}
\bigl\| e^{-A_\varepsilon t}-e^{-A^0 t}\bigr\|_{L_2(\mathbb{R}^d)\to L_2(\mathbb{R}^d)} \leqslant C\varepsilon (t+\varepsilon^2)^{-1/2}, \qquad t \geqslant 0,
\end{equation}
\tag{0.8}
$$
$$
\begin{equation}
\bigl\| e^{-A_\varepsilon t}-e^{-A^0 t}-\varepsilon \mathcal{K}(t;\varepsilon)\bigr\|_{L_2(\mathbb{R}^d)\to H^1(\mathbb{R}^d)} \leqslant C\varepsilon (t^{-1/2}+t^{-1}), \qquad t \geqslant \varepsilon^2.
\end{equation}
\tag{0.9}
$$
The exponential of the operator $B_\varepsilon$ was studied by Meshkova in [15], where analogs of inequalities (0.8) and (0.9) were proved. A different approach to operator error estimates in homogenization theory was suggested by Zhikov and Pastukhova. In [16]–[18], estimates of the form (0.3), (0.4) were obtained for the acoustics operator and the operator of elasticity theory. The method, called by the authors the “modified first approximation method” or the “shift method”, was based on analysis of the first-order approximation to the solution and introduction of an additional parameter into the problem. In [17], [18], in addition to the problems in $\mathbb{R}^d$, homogenization problems in a bounded domain $\mathcal{O}\subset \mathbb{R}^d$ with the Dirichlet or Neumann boundary conditions were studied. The shift method was applied to parabolic equations in [19]. Further results of Zhikov, Pastukhova and their pupils are reflected in the survey [20]. In the presence of lower order terms, the homogenization problem for the operator (0.5) in $\mathbb{R}^d$ was studied by Borisov [21]. An expression for the effective operator $\mathcal{B}^0$ was found, and error estimates of the form (0.6), (0.7) were obtained. Moreover, it was assumed that the coefficients of the operator depend not only on the fast variable, but also on the slow variable. However, in [21], the coefficients of $\mathcal{B}_\varepsilon$ were assumed to be sufficiently smooth. We also mention the paper [22] by Senik, where the non-selfadjoint second-order strongly elliptic operator (involving lower order terms) on an infinite cylinder was studied. The coefficients are periodic along the cylinder and oscillate rapidly; estimates of the form (0.6), (0.7) are obtained. Operator error estimates for second-order elliptic equations (without lower order terms) in a bounded domain under Dirichlet or Neumann conditions were studied by many authors. Apparently, the first result is due to Moskow and Vogelius who obtained an estimate (see Corollary 2.2 in [23]), that can be written in operator terms:
$$
\begin{equation}
\| A_{D,\varepsilon}^{-1}-(A_D^0)^{-1}\|_{L_2(\mathcal{O})\to L_2(\mathcal{O})}\leqslant C\varepsilon.
\end{equation}
\tag{0.10}
$$
Here, the operator $A_{D,\varepsilon}$ acts in $L_2(\mathcal{O})$, where $\mathcal{O}\subset \mathbb{R}^2$, and is given by the expression $-\operatorname{div} g^\varepsilon (\mathbf{x})\nabla$ with the Dirichlet condition on $\partial\mathcal{O}$, and the matrix-valued function $g(\mathbf{x})$ is assumed to be infinitely smooth. In the case of the Neumann condition, a similar estimate was obtained in [24], Corollary 1. Also, in [24], an approximation was found with corrector for the inverse operator in the norm of operators acting from $L_2(\mathcal{O})$ to the Sobolev space $H^1(\mathcal{O})$, with error estimate of order $O(\sqrt{\varepsilon}\,)$. The order of this estimate is worse than in $\mathbb{R}^d$ because of the boundary influence. For arbitrary dimension, elliptic problems in a bounded domain with sufficiently smooth boundary were studied in [17] and [18]. The smoothness of the coefficients was not assumed. For the acoustics and elasticity operators with the Dirichlet or Neumann boundary conditions, approximation of the resolvent in the $(L_2\to H^1)$-norm with the corrector taken into account and with error of order $O(\sqrt{\varepsilon}\,)$ was obtained. As a consequence, estimate of the form (0.10), but with error of order $O(\sqrt{\varepsilon}\,)$ was deduced. Close results for the operator $-\operatorname{div} g^\varepsilon (\mathbf{x})\nabla$ in a bounded domain $\mathcal{O}\subset\mathbb{R}^d$ with the Dirichlet or Neumann conditions on $\partial\mathcal{O}$ were obtained by Griso [25], [26] with the help of the “unfolding” method. In [26], for the same operator, the order-sharp estimate (0.10) was obtained for the first time. For elliptic systems, similar results were independently obtained in [27]–[29]. Further advancements and a detailed survey can be found in [5], [6]. Operator error estimates for the second-order matrix elliptic operator (with lower order terms) in a bounded domain with the Dirichlet or Neumann conditions were found by Xu [30]–[32]. However, in those papers a rather restrictive condition of uniform ellipticity was imposed. Below in § 0.3 we will compare our results with those from [32]. We also mention the book [33] by Shen, the paper [34] and the references therein. Up to now, we have discussed the results on approximations for the resolvent at a fixed regular point. Approximation for the resolvent $(A_\varepsilon -\zeta I)^{-1}$ of the operator (0.2) with error estimates depending on $\varepsilon$ and the spectral parameter $\zeta\in\mathbb{C}\setminus\mathbb{R}_+$ was obtained by Suslina [6]. In that paper, the operators $A_{D,\varepsilon}$ and $A_{N,\varepsilon}$ of the form (0.2) were also studied. Approximations for the resolvents of these operators with two-parametric error estimates (with respect to $\varepsilon$ and $\zeta$) were obtained. Investigation of the two-parametric error estimates was stimulated by the study of homogenization of parabolic systems. Approximations for the exponential of the operators $A_{D,\varepsilon}$ and $A_{N,\varepsilon}$ were obtained by Meshkova and Suslina [35]:
$$
\begin{equation*}
\begin{alignedat}{2} \bigl\| e^{-A_{\flat,\varepsilon}t}-e^{-A_\flat^0 t}\bigr\|_{L_2(\mathcal{O}) \to L_2(\mathcal{O})} &\leqslant C\varepsilon (t+\varepsilon^2)^{-1/2}e^{-c t}, &\qquad t &\geqslant 0, \\ \bigl\| e^{-A_{\flat,\varepsilon}t}-e^{-A_\flat^0 t} -\varepsilon\mathcal{K}_\flat(t;\varepsilon)\bigr\|_{L_2(\mathcal{O})\to H^1(\mathcal{O})} &\leqslant C\varepsilon^{1/2}t^{-3/4}e^{-c t}, &\qquad t&\geqslant \varepsilon^2. \end{alignedat}
\end{equation*}
\notag
$$
Here, $\flat =D,N$. The method of [35] was based on the identity
$$
\begin{equation*}
e^{-A_{\flat,\varepsilon}t}=-\frac{1}{2\pi i}\int_\gamma e^{-\zeta t} (A_{\flat,\varepsilon}-\zeta I)^{-1}\,d\zeta,
\end{equation*}
\notag
$$
where $\gamma \subset \mathbb{C}$ is a positively oriented contour enclosing the spectrum of $A_{\flat,\varepsilon}$. This identity allows us to deduce approximations of the operator exponential $e^{-A_{\flat,\varepsilon}t}$ from the corresponding approximations of the resolvent $(A_{\flat,\varepsilon}-\zeta I)^{-1}$ with two-parametric error estimates (with respect to $\varepsilon$ and $\zeta$). The operator with coefficients periodic in spacial variables and in time was studied by Geng and Shen [36]. In [36], operator error estimates for the equation
$$
\begin{equation*}
\partial_t\mathbf{u}_\varepsilon(\mathbf{x},t)=-\operatorname{div}g (\varepsilon^{-1}\mathbf{x},\varepsilon^{-2}t) \nabla\mathbf{u}_\varepsilon(\mathbf{x},t)
\end{equation*}
\notag
$$
in a cylinder $\mathcal{O} \times (0,T)$ with the Dirichlet or Neumann boundary conditions were obtained; here, $\mathcal{O}$ is a bounded domain of class $C^{1,1}$. The present paper relies on the following two-parametric estimates for the operator $B_\varepsilon$ obtained in [37]:
$$
\begin{equation}
\bigl\|(B_\varepsilon -\zeta Q_0^\varepsilon)^{-1} - (B^0-\zeta\overline{Q_0}\,)^{-1}\bigr\|_{L_2(\mathbb{R}^d)\to L_2(\mathbb{R}^d)} \leqslant C(\phi)\varepsilon|\zeta|^{-1/2},
\end{equation}
\tag{0.11}
$$
$$
\begin{equation}
\bigl\|(B_\varepsilon -\zeta Q_0^\varepsilon )^{-1} - (B^0-\zeta\overline{Q_0}\,)^{-1}-\varepsilon K(\varepsilon;\zeta)\bigr\|_{L_2(\mathbb{R}^d)\to H^1(\mathbb{R}^d)} \leqslant C(\phi)\varepsilon.
\end{equation}
\tag{0.12}
$$
Here, $\phi =\operatorname{arg}\zeta\in (0,2\pi)$, $|\zeta|\geqslant 1$. The dependence of the constants in estimates on $\phi$ is traced. Estimates (0.11) and (0.12) are uniform with respect to $\phi$ in any domain of the form
$$
\begin{equation}
\{\zeta=|\zeta| e^{i\phi}\in\mathbb{C}\colon |\zeta|\geqslant 1, \, \phi_0\leqslant \phi\leqslant 2\pi-\phi_0\}
\end{equation}
\tag{0.13}
$$
with arbitrarily small $\phi_0 >0$. In [37], error estimates in the case $|\zeta|<1$, $\phi \in (0,2\pi)$, were also obtained. Note that in recent years, operator error estimates in various homogenization problems for differential operators have been extensively studied, and many meaningful results were obtained. A fairly detailed survey of the current state of this area can be found in the introduction to [38]. 0.3. Main resultsIn this paper, we study homogenization of elliptic and parabolic problems in a bounded domain under the Neumann condition for a strongly elliptic second-order operator of the form (0.1) (with inclusion of the lower order terms). As already noted, such problems were previously investigated in [5], [6], [35] for the operator $A_{N,\varepsilon}$ (without lower order terms). Before we formulate the results, it is convenient to turn to the positive definite operator $B_{N,\varepsilon}=\mathcal{B}_{N,\varepsilon}+\lambda Q_0^\varepsilon$ and choose an appropriate constant $\lambda$. Let $B_N^0$ be the corresponding effective operator. Our main results are the following estimates:
$$
\begin{equation}
\bigl\|(B_{N,\varepsilon}-\zeta Q_0^\varepsilon )^{-1}-(B_N^0-\zeta \overline{Q_0}\,)^{-1}\bigr\|_{L_2(\mathcal{O})\to L_2(\mathcal{O})} \leqslant C(\phi)\varepsilon|\zeta|^{-1/2},
\end{equation}
\tag{0.14}
$$
$$
\begin{equation}
\begin{split} &\bigl\|(B_{N,\varepsilon}-\zeta Q_0^\varepsilon )^{-1}-(B_N^0-\zeta \overline{Q_0}\,)^{-1}-\varepsilon K_N(\varepsilon;\zeta)\bigr\|_{L_2(\mathcal{O})\to H^1(\mathcal{O})} \\ &\qquad\leqslant C(\phi)(\varepsilon^{1/2}|\zeta|^{-1/4}+\varepsilon) \end{split}
\end{equation}
\tag{0.15}
$$
for $\zeta\in\mathbb{C}\setminus\mathbb{R}_+$, $|\zeta|\geqslant 1$, and sufficiently small $\varepsilon$. The constants $C(\phi)$ are controlled explicitly in terms of the problem data and the angle $\phi$. Estimates (0.14) and (0.15) are uniform with respect to $\phi$ in any domain (0.13) with arbitrarily small $\phi_0 >0$. For fixed $\zeta$, estimate (0.14) has sharp order $O(\varepsilon)$. The order of estimate (0.15) is worse than in $\mathbb{R}^d$ (cf. (0.7)) because of the boundary influence. In the general case, the corrector in (0.15) contains a smoothing operator. We distinguish the cases where a simpler corrector can be used. In addition to estimates for the generalized resolvent, we also find approximation in the $(L_2\to L_2)$-norm for the operator $g^\varepsilon b(\mathbf{D})(B_{N,\varepsilon}-\zeta Q_0^\varepsilon )^{-1}$ corresponding to the flux. We show that in a strictly interior subdomain $\mathcal{O}'$ of $\mathcal{O}$ it is possible to obtain an approximation of the operator $(B_{N,\varepsilon}-\zeta Q_0^\varepsilon )^{-1}$ in the $(L_2(\mathcal{O}) \to H^1(\mathcal{O}'))$-norm with an error estimate of sharp order $O(\varepsilon)$. We also find approximations for the operator $(B_{N,\varepsilon}-\zeta Q_0^\varepsilon)^{-1}$ valid in a larger domain of the spectral parameter $\zeta$; the corresponding error estimates have a different behaviour with respect to $\zeta$ (see § 9 for details). Two-parametric estimates (0.14) and (0.15) are applied to study the behaviour of the solution of the initial boundary value problem for a parabolic equation
$$
\begin{equation}
\begin{cases} Q_0^\varepsilon (\mathbf{x})\, \partial_t \mathbf{u}_\varepsilon (\mathbf{x},t)=- (B_{\varepsilon}\mathbf{u}_\varepsilon) (\mathbf{x},t),& \mathbf{x}\in\mathcal{O},\, t>0, \\ Q_0^\varepsilon (\mathbf{x})\mathbf{u}_\varepsilon (\mathbf{x},0)=\boldsymbol{\varphi}(\mathbf{x}), & \mathbf{x}\in\mathcal{O}, \end{cases}
\end{equation}
\tag{0.16}
$$
under the natural condition (the Neumann condition) on $\partial \mathcal{O} \times \mathbb{R}_+$. Here, $\boldsymbol{\varphi}\in L_2(\mathcal{O};\mathbb{C}^n)$. The solution is understood in the generalized sense as a solution from the “energy” class (cf. Chap. 3 in [39]). We show that, as $\varepsilon\to 0$, the solution $\mathbf{u}_\varepsilon (\,{\cdot}\,,t)$ converges in $L_2(\mathcal{O};\mathbb{C}^n)$ to the solution $\mathbf{u}_0(\,{\cdot}\,,t)$ of the effective problem
$$
\begin{equation*}
\begin{cases} \overline{Q_0}\, \partial_t \mathbf{u}_0 (\mathbf{x},t)=- (B^0\mathbf{u}_0) (\mathbf{x},t),& \mathbf{x}\in\mathcal{O},\, t>0, \\ \overline{Q_0}\mathbf{u}_0 (\mathbf{x},0) =\boldsymbol{\varphi}(\mathbf{x}),& \mathbf{x}\in\mathcal{O} \end{cases}
\end{equation*}
\notag
$$
with constant coefficients under the Neumann condition on $\partial \mathcal{O} \times \mathbb{R}_+$. We obtain the estimate
$$
\begin{equation}
\|\mathbf{u}_\varepsilon(\,{\cdot}\,,t)-\mathbf{u}_0(\,{\cdot}\,,t)\|_{L_2(\mathcal{O})} \leqslant C\varepsilon (t+\varepsilon^2)^{-1/2}e^{-c t}\| \boldsymbol{\varphi}\|_{L_2(\mathcal{O})},\qquad t\geqslant 0,
\end{equation}
\tag{0.17}
$$
for sufficiently small $\varepsilon$. For fixed time $t>0$, this estimate has sharp order $O(\varepsilon)$. Our second result for problem (0.16) is an approximation of the solution $\mathbf{u}_\varepsilon (\,{\cdot}\,,t)$ in the energy norm:
$$
\begin{equation}
\| \mathbf{u}_\varepsilon(\,{\cdot}\,,t) -\mathbf{v}_\varepsilon(\,{\cdot}\,,t)\|_{H^1(\mathcal{O})} \leqslant C(\varepsilon^{1/2}t^{-3/4}+\varepsilon t^{-1}) e^{-c t}\| \boldsymbol{\varphi}\|_{L_2(\mathcal{O})},\qquad t>0.
\end{equation}
\tag{0.18}
$$
Here, $\mathbf{v}_\varepsilon (\,{\cdot}\,,t)=\mathbf{u}_0(\,{\cdot}\,,t)+\varepsilon \mathcal{K}_N(t;\varepsilon )\boldsymbol{\varphi}(\,{\cdot}\,)$ is the first-order approximation of the solution $\mathbf{u}_\varepsilon (\,{\cdot}\,,t)$ and the operator $\mathcal{K}_N(t;\varepsilon)$ is a corrector. For fixed $t$, estimate (0.18) is of the order $O(\varepsilon^{1/2})$ because of the boundary influence. In general, the corrector contains the smoothing operator. We will distinguish conditions under which it is possible to use a simpler corrector without the smoothing operator. Along with estimate (0.18), we will find an approximation of the flux $g^\varepsilon b(\mathbf{D})\mathbf{u}_\varepsilon (\,{\cdot}\,,t)$ in the $L_2$-norm. In a strictly interior subdomain $\mathcal{O}'\subset \mathcal{O}$, we obtain an approximation of the solution $\mathbf{u}_\varepsilon(\,{\cdot}\,,t)$ in the norm on $H^1(\mathcal{O}'; \mathbb{C}^n)$ with error estimate of sharp order $O(\varepsilon)$. Estimates (0.17) and (0.18) can be rewritten in the uniform operator topology. In the simpler case, when $Q_0(\mathbf{x})=\mathbf{1}_n$, the results are as follows:
$$
\begin{equation*}
\begin{alignedat}{2} \bigl\| e^{-B_{N,\varepsilon}t}-e^{-B_N^0t}\bigr\|_{L_2(\mathcal{O})\to L_2(\mathcal{O})} &\leqslant C\varepsilon (t+\varepsilon^2)^{-1/2}e^{-c t}, &\qquad t &\geqslant 0, \\ \bigl\| e^{-B_{N,\varepsilon}t}-e^{-B_N^0t}-\varepsilon\mathcal{K}_N(t;\varepsilon) \bigr\|_{L_2(\mathcal{O})\to H^1(\mathcal{O})} &\leqslant C (\varepsilon^{1/2}t^{-3/4}+\varepsilon t^{-1}) e^{-c t}, &\qquad t &>0. \end{alignedat}
\end{equation*}
\notag
$$
Similar results for the operator $B_{D,\varepsilon}$ with the Dirichlet condition were obtained by Meshkova and Suslina; the paper [40] (see also [41]) was devoted to the elliptic Dirichlet problem, and [42], to the first initial boundary value problem for a parabolic equation. Note that the two-parametric error estimates for elliptic problems have already been applied to obtain operator error estimates not only for parabolic, but also for hyperbolic problems (see [43]). Let us compare our results for elliptic problems with the results of a close paper [32]. Let us list our advantages. First, we study a strongly elliptic operator of the form (0.1), and in [32] (as well as in [27], [30], [31]) a very restrictive condition of uniform ellipticity was imposed on the operator. Second, we include in consideration the lower order terms with unbounded coefficients (from appropriate $L_p(\Omega)$-classes), while in [32] the coefficients of the lower order terms were assumed to be bounded. Third, we get two-parametric error estimates (with respect to $\varepsilon$ and $\zeta$), while in [32] the estimates were one-parametric (in $\varepsilon$). On the other hand, in [32] some results were obtained in Lipschitz domains and non-selfadjoint operators (with selfadjoint principal part) were allowed. 0.4. MethodThe proofs of estimates (0.14), (0.15) rely on the method developed in [5], [6]. It is based on consideration of the associated problem in $\mathbb{R}^d$, using estimates (0.11), (0.12) (obtained in [37]), an introduction of the boundary layer correction term, and a careful analysis of this term. A significant technical role is played by the Steklov smoothing (borrowed from [18]) and estimates in the $\varepsilon$-vicinity of the boundary. We trace the dependence of estimates on the spectral parameter carefully. Additional technical difficulties (as compared with [6]) are related to the presence of the lower order terms with unbounded coefficients. First, the case of $\operatorname{Re}\zeta \leqslant 0$, $|\zeta|\geqslant 1$, is considered. We prove estimate (0.15) and then estimate (0.14), using inequality (0.15) and the duality arguments. Next, we transfer the already proved estimates from the point $\zeta$ in the left half-plane to the symmetric point in the right half-plane, using suitable identities for the resolvents. Approximations in a larger domain of the parameter $\zeta$ are deduced from the already proved estimates at the point $\zeta =-1$ and appropriate resolvent identities. 0.5. Plan of the paperThe paper consists of eleven sections. In § 1, we introduce the class of operators $B_\varepsilon$ acting in $L_2(\mathbb{R}^d;\mathbb{C}^n)$ and formulate the results about homogenization of the generalized resolvent $(B_\varepsilon -\zeta Q_0^\varepsilon)^{-1}$ obtained in [37]. In § 2, the class of operators $B_{N,\varepsilon}$ is described and the effective operator $B_N^0$ is defined. In § 3 we formulate the main results of the paper, introduce the boundary layer correction term, and obtain a theorem on approximation of the solution $\mathbf{u}_\varepsilon = (B_{N,\varepsilon} -\zeta Q_0^\varepsilon)^{-1} \mathbf{F}$ with this correction term taken into account. Section 4 contains the auxiliary material. In § 5, we estimate the correction term in the $H^1$-norm and find approximation (0.15) for the generalized resolvent in the $(L_2 \to H^1)$-norm in the case where $\operatorname{Re}\zeta \leqslant 0$. In § 6, we estimate the correction term in the $L_2$-norm and find approximation for the generalized resolvent in the $(L_2\to L_2)$-norm with estimate (0.14) in the case where $\operatorname{Re}\zeta \leqslant 0$. In § 7, the results are transferred to the point $\zeta$ in the right half-plane; the proof of the main results for the generalized resolvent is completed. In § 8, we distinguish conditions under which the smoothing operator in the corrector can be removed; special cases are considered; estimates in a strictly interior subdomain are obtained. Estimates valid in a wider domain of the spectral parameter are obtained in § 9. Section 10 is devoted to homogenization of the solutions of the second initial boundary value problem for a parabolic equation. An example of applications of the general results is considered in § 11. 0.6. NotationLet $\mathfrak{H}$ and $\mathfrak{H}_*$ be complex separable Hilbert spaces. The symbols $(\,{\cdot}\,,{\cdot}\,)_\mathfrak{H}$ and $\|\,{\cdot}\,\|_\mathfrak{H}$ stand for the inner product and the norm in $\mathfrak{H}$; the symbol $\|\,{\cdot}\,\|_{\mathfrak{H}\to\mathfrak{H}_*}$ denotes the norm of a linear continuous operator acting from $\mathfrak{H}$ to $\mathfrak{H}_*$. The symbols $\langle \,{\cdot}\,,{\cdot}\,\rangle$ and $|\,{\cdot}\,|$ stand for the inner product and the norm in $\mathbb{C}^n$, respectively; $\mathbf{1}_n$ is the unit $(n\times n)$-matrix. For an $(m\times n)$-matrix $a$, the symbol $|a|$ denotes the norm of $a$ viewed as a linear operator from $\mathbb{C}^n$ to $\mathbb{C}^m$. For $z\in\mathbb{C}$, we denote by $z^*$ the complex conjugate number. (This non-standard notation is employed because we write $\overline{g}$ for the mean value of a periodic function $g$.) We use the notation $\mathbf{x}=(x_1,\dots, x_d)\in\mathbb{R}^d$, $iD_j=\partial_j =\partial /\partial x_j$, $j=1,\dots,d$, $\mathbf{D}=-i\nabla=(D_1,\dots,D_d)$. The $L_p$-classes of $\mathbb{C}^n$-valued functions in a domain $\mathcal{O}\subset\mathbb{R}^d$ are denoted by $L_p(\mathcal{O};\mathbb{C}^n)$, $1\leqslant p\leqslant \infty$. The Sobolev spaces of $\mathbb{C}^n$-valued functions in a domain $\mathcal{O}\subset\mathbb{R}^d$ are denoted by $H^s(\mathcal{O};\mathbb{C}^n)$. Next, $H^1_0(\mathcal{O};\mathbb{C}^n)$ is the closure of $C_0^\infty(\mathcal{O};\mathbb{C}^n)$ in $H^1(\mathcal{O};\mathbb{C}^n)$. If $n=1$, we write simply $L_p(\mathcal{O})$, $H^s(\mathcal{O})$, etc., but sometimes we use this short notation also for spaces of vector-valued or matrix-valued functions. We denote $\mathbb{R}_+=[0,\infty)$. Different constants in estimates are denoted by $c$, $\mathfrak{c}$, $C$, $\mathcal{C}$, $\mathfrak{C}$, $\mathrm C$, $\beta$, $\gamma$ (possibly, with indices and marks). § 1. Homogenization problem for elliptic operator acting in $L_2(\mathbb{R}^d;\mathbb{C}^n)$In this section, we formulate the results about homogenization of elliptic systems in $\mathbb{R}^d$ obtained in [37]. 1.1. Lattices in $\mathbb{R}^d$Let $\Gamma \subset \mathbb{R}^d$ be a lattice generated by a basis $\mathbf{a}_1,\dots,\mathbf{a}_d \in \mathbb{R}^d$:
$$
\begin{equation*}
\Gamma = \biggl\{ \mathbf{a}\in \mathbb{R}^d \colon \mathbf{a}=\sum_{j=1}^d \nu_j \mathbf{a}_j, \, \nu_j\in \mathbb{Z} \biggr\},
\end{equation*}
\notag
$$
and let $\Omega$ be the elementary cell of $\Gamma$:
$$
\begin{equation*}
\Omega =\biggl\{ \mathbf{x}\in \mathbb{R}^d \colon \mathbf{x}=\sum_{j=1}^d \tau_j \mathbf{a}_j,\, -\frac{1}{2}<\tau_j<\frac{1}{2}\biggr\}.
\end{equation*}
\notag
$$
Let $|\Omega| $ be the volume of $\Omega$. The basis $\mathbf{b}_1,\dots,\mathbf{b}_d$ in $\mathbb{R}^d$ dual to the basis $\mathbf{a}_1,\dots,\mathbf{a}_d$ is defined by the relations $\langle \mathbf{b}_i,\mathbf{a}_j \rangle =2\pi \delta_{ij}$. This basis generates the lattice $\widetilde{\Gamma}$ dual to $\Gamma$. Denote
$$
\begin{equation*}
2 r_0 := \min_{0 \ne \mathbf{b} \in \widetilde{\Gamma}} |\mathbf{b}|, \qquad 2r_1 :=\operatorname{diam}\Omega.
\end{equation*}
\notag
$$
By $\widetilde{H}^1(\Omega)$ we denote the subspace of all functions in $H^1(\Omega)$ whose $\Gamma$-periodic extension to $\mathbb{R}^d$ belongs to $H^1_{\mathrm{loc}}(\mathbb{R}^d)$. If $f (\mathbf{x})$ is a $\Gamma$-periodic matrix-valued function in $\mathbb{R}^d$, we put $f^\varepsilon (\mathbf{x}):=f (\mathbf{x}/\varepsilon)$, $\varepsilon >0$;
$$
\begin{equation*}
\overline{f}:=|\Omega|^{-1}\int_\Omega f(\mathbf{x})\,d\mathbf{x}, \qquad \underline{f}:=\biggl(|\Omega|^{-1}\int_\Omega f(\mathbf{x})^{-1}\,d\mathbf{x} \biggr)^{-1}.
\end{equation*}
\notag
$$
Here, in the definition of $\overline{f}$ it is assumed that $f\,{\in}\, L_{1,\mathrm{loc}}(\mathbb{R}^d)$, and in the definition of $\underline{f}$ it is assumed that the matrix $f$ is square and non-degenerate, and $f^{-1}\in L_{1,\mathrm{loc}}(\mathbb{R}^d)$. By $[f^\varepsilon ]$ we denote the operator of multiplication by the matrix-valued function $f^\varepsilon (\mathbf{x})$. 1.2. The Steklov smoothingThe Steklov smoothing operator $S_\varepsilon^{(k)}$ acts in the space $L_2(\mathbb{R}^d;\mathbb{C}^k)$ (where $k\in\mathbb{N}$) and is given by
$$
\begin{equation}
(S_\varepsilon^{(k)} \mathbf{u})(\mathbf{x})=|\Omega|^{-1}\int_\Omega \mathbf{u}(\mathbf{x}-\varepsilon \mathbf{z})\,d\mathbf{z},\qquad \mathbf{u}\in L_2(\mathbb{R}^d;\mathbb{C}^k).
\end{equation}
\tag{1.1}
$$
We shall omit the index $(k)$ in the notation and write simply $S_\varepsilon$. Obviously, $S_\varepsilon \mathbf{D}^\alpha \mathbf{u}=\mathbf{D}^\alpha S_\varepsilon \mathbf{u}$ for $\mathbf{u}\in H^s(\mathbb{R}^d;\mathbb{C}^k)$ and any multiindex $\alpha$ such that $|\alpha| \leqslant s$. Note that
$$
\begin{equation}
\| S_\varepsilon \|_{H^l(\mathbb{R}^d)\to H^l(\mathbb{R}^d)}\leqslant 1,\qquad l\in \mathbb{Z}_+.
\end{equation}
\tag{1.2}
$$
We need the following properties of the operator $S_\varepsilon$ (see Lemmas 1.1 and 1.2 in [18] or Propositions 3.1 and 3.2 in [28]). Proposition 1.1. For any function $\mathbf{u}\in H^1(\mathbb{R}^d;\mathbb{C}^k)$, where $2r_1=\operatorname{diam}\Omega$. Proposition 1.2. Let $f$ be a $\Gamma$-periodic function in $\mathbb{R}^d$ such that $f\in L_2(\Omega)$. Then the operator $[f^\varepsilon ]S_\varepsilon $ is continuous in $L_2(\mathbb{R}^d)$, and 1.3. The class of operators $A_\varepsilon$In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, we consider the operator $A_\varepsilon$ formally given by the differential expression $A_\varepsilon {=}\,b(\mathbf{D})^*g^\varepsilon (\mathbf{x})b(\mathbf{D})$. Here, $g(\mathbf{x})$ is a $\Gamma$-periodic Hermitian $(m\times m)$-matrix-valued function (in general, with complex entries). It is assumed that $g(\mathbf{x})>0$ and $g,g^{-1}\in L_\infty (\mathbb{R}^d)$. The differential operator $b(\mathbf{D})$ is given by where $b_j$, $j=1,\dots,d$, are constant matrices of size $m\times n$ (in general, with complex entries). It is assumed that $m\geqslant n$ and the symbol $b(\boldsymbol{\xi})=\sum_{j=1}^d b_j\xi_j$ of the operator $b(\mathbf{D})$ has maximal rank:
$$
\begin{equation}
\operatorname{rank}b(\boldsymbol{\xi})=n,\qquad 0\neq \boldsymbol{\xi}\in\mathbb{R}^d.
\end{equation}
\tag{1.4}
$$
This condition is equivalent to the estimates
$$
\begin{equation}
\alpha_0\mathbf{1}_n \leqslant b(\boldsymbol{\theta})^*b(\boldsymbol{\theta}) \leqslant \alpha_1\mathbf{1}_n,\qquad \boldsymbol{\theta}\in \mathbb{S}^{d-1},\quad 0<\alpha_0\leqslant \alpha_1<\infty,
\end{equation}
\tag{1.5}
$$
with some constants $\alpha_0$ and $\alpha_1$. From (1.5) it follows that The precise definition of the operator $A_\varepsilon$ is given in terms of the quadratic form
$$
\begin{equation*}
\mathfrak{a}_\varepsilon [\mathbf{u},\mathbf{u}]=\int_{\mathbb{R}^d}\langle g^\varepsilon (\mathbf{x})b(\mathbf{D})\mathbf{u},b(\mathbf{D})\mathbf{u}\rangle \,d\mathbf{x},\qquad \mathbf{u}\in H^1(\mathbb{R}^d;\mathbb{C}^n).
\end{equation*}
\notag
$$
Under the above assumptions, this form is closed and non-negative. Using the Fourier transform and condition (1.5), it is easy to check that
$$
\begin{equation}
\alpha_0\| g^{-1}\|^{-1}_{L_\infty}\| \mathbf{D}\mathbf{u}\|^2_{L_2(\mathbb{R}^d)}\leqslant \mathfrak{a}_\varepsilon [\mathbf{u},\mathbf{u}]\leqslant \alpha_1\| g\|_{L_\infty}\| \mathbf{D}\mathbf{u}\|^2_{L_2(\mathbb{R}^d)},\qquad \mathbf{u}\in H^1(\mathbb{R}^d;\mathbb{C}^n).
\end{equation}
\tag{1.7}
$$
Let $c_1:=\alpha_0^{-1/2}\| g^{-1}\|^{1/2}_{L_\infty}$. Then the lower estimate (1.7) can be written as
$$
\begin{equation}
\|\mathbf{D}\mathbf{u}\|^2_{L_2(\mathbb{R}^d)}\leqslant c_1^2 \mathfrak{a}_\varepsilon [\mathbf{u},\mathbf{u}],\qquad \mathbf{u}\in H^1(\mathbb{R}^d;\mathbb{C}^n).
\end{equation}
\tag{1.8}
$$
1.4. The operator $B_\varepsilon$We will study a selfadjoint operator $B_\varepsilon$ whose principal part coincides with $A_\varepsilon$. To define the lower order terms, we will introduce $\Gamma$-periodic $(n\times n)$-matrix-valued functions $a_j$, $j=1,\dots,d$, (in general, with complex entries) such that
$$
\begin{equation}
a_j \in L_\rho (\Omega ), \qquad\rho =2 \ \text{ for }\ d=1,\quad\rho >d \ \text{ for } \ d\geqslant 2,\quad j=1,\dots,d.
\end{equation}
\tag{1.9}
$$
Next, let $Q$ and $Q_0$ be $\Gamma$-periodic Hermitian $(n\times n)$-matrix-valued functions (with complex entries) such that
$$
\begin{equation}
\begin{gathered} \, Q\in L_s(\Omega ),\qquad s=1 \ \text{ for }\ d=1,\quad s >\frac{d}2 \ \text{ for } \ d\geqslant 2, \\ Q_0(\mathbf{x})>0;\qquad Q_0,Q_0^{-1}\in L_\infty (\mathbb{R}^d). \nonumber \end{gathered}
\end{equation}
\tag{1.10}
$$
(Our assumptions on $Q$ correspond to Example 2.4 from [37].) In what follows, we assume that $0< \varepsilon \leqslant 1$. Consider the quadratic form
$$
\begin{equation}
\begin{aligned} \, \mathfrak{b}_\varepsilon [\mathbf{u},\mathbf{u}]&=\mathfrak{a}_\varepsilon [\mathbf{u},\mathbf{u}]+2\operatorname{Re}\sum_{j=1}^d(a_j^\varepsilon D_j \mathbf{u},\mathbf{u})_{L_2(\mathbb{R}^d)} \nonumber \\ &\qquad+(Q^\varepsilon \mathbf{u},\mathbf{u})_{L_2(\mathbb{R}^d)}+\lambda (Q_0^\varepsilon \mathbf{u},\mathbf{u})_{L_2(\mathbb{R}^d)},\qquad \mathbf{u}\in H^1(\mathbb{R}^d;\mathbb{C}^n). \end{aligned}
\end{equation}
\tag{1.11}
$$
Here, the constant $\lambda$ is chosen in such a way (see (1.16)) that the form $\mathfrak{b}_\varepsilon$ is non-negative. For convenience of further references, the following set of parameters is called the “initial data”:
$$
\begin{equation}
\begin{gathered} \, d,\ m,\ n,\ \rho,\ s; \ \alpha_0,\ \alpha_1,\ \|g\|_{L_\infty},\ \|g^{-1}\|_{L_\infty},\ \| a_j\|_{L_\rho (\Omega)},\quad j=1,\dots,d; \\ \| Q\|_{L_s(\Omega)};\ \| Q_0\|_{L_\infty},\ \| Q_0^{-1}\|_{L_\infty},\ \lambda;\ \text{the parameters of the lattice } \Gamma. \end{gathered}
\end{equation}
\tag{1.12}
$$
Let us check that the form $\mathfrak{b}_\varepsilon$ is closed. By the Hölder inequality and the Sobolev embedding theorem, it is easily seen (see [10], (5.11)–(5.14)) that, for any $\nu>0$, there exist constants $C_j(\nu)>0$ such that
$$
\begin{equation*}
\|a_j^*\mathbf{u}\|^2_{L_2(\mathbb{R}^d)}\leqslant \nu \|\mathbf{D}\mathbf{u}\|^2_{L_2(\mathbb{R}^d)}+ C_j(\nu)\|\mathbf{u}\|^2_{L_2(\mathbb{R}^d)}, \qquad \mathbf{u}\in H^1(\mathbb{R}^d;\mathbb{C}^n),\quad j=1,\dots,d.
\end{equation*}
\notag
$$
Using the change of variable $\mathbf{y}:=\varepsilon^{-1}\mathbf{x}$ and denoting $\mathbf{u}(\mathbf{x})=:\mathbf{v}(\mathbf{y})$, we deduce
$$
\begin{equation*}
\begin{aligned} \, \|(a_j^\varepsilon)^*\mathbf{u}\|^2_{L_2(\mathbb{R}^d)} &=\int_{\mathbb{R}^d} |a_j(\varepsilon^{-1}\mathbf{x})^*\mathbf{u}(\mathbf{x})|^2\,d\mathbf{x} =\varepsilon^d\int_{\mathbb{R}^d} |a_j(\mathbf{y})^*\mathbf{v}(\mathbf{y})|^2\,d\mathbf{y} \\ &\leqslant \varepsilon^d\nu \int_{\mathbb{R}^d} |\mathbf{D}_{\mathbf{y}}\mathbf{v}(\mathbf{y})|^2\,d\mathbf{y} +\varepsilon^d C_j(\nu)\int_{\mathbb{R}^d} |\mathbf{v}(\mathbf{y})|^2\,d\mathbf{y} \\ &\leqslant \nu \|\mathbf{D}\mathbf{u}\|^2_{L_2(\mathbb{R}^d)}{+}\, C_j(\nu)\|\mathbf{u}\|^2_{L_2(\mathbb{R}^d)},\qquad \mathbf{u}\,{\in}\,H^1(\mathbb{R}^d;\mathbb{C}^n), \quad 0 < \varepsilon \leqslant 1. \end{aligned}
\end{equation*}
\notag
$$
Hence by (1.7), for any $\nu >0$, there exists a constant $C(\nu)>0$ such that
$$
\begin{equation}
\sum_{j=1}^d \|(a_j^\varepsilon)^*\mathbf{u}\|^2_{L_2(\mathbb{R}^d)} \leqslant \nu \mathfrak{a}_\varepsilon [\mathbf{u},\mathbf{u}] +C(\nu)\|\mathbf{u}\|^2_{L_2(\mathbb{R}^d)}, \qquad\mathbf{u}\in H^1(\mathbb{R}^d;\mathbb{C}^n).
\end{equation}
\tag{1.13}
$$
If $\nu$ is fixed, then $C(\nu)$ depends only on $d$, $\rho$, $\alpha_0$, the norms $\|g^{-1}\|_{L_\infty}$, $\|a_j\|_{L_\rho (\Omega)}$, $j=1,\dots,d$, and the parameters of the lattice $\Gamma$. From (1.8) and (1.13) it follows that
$$
\begin{equation}
2\biggl|\operatorname{Re}\sum_{j=1}^d (D_j\mathbf{u},(a_j^\varepsilon)^*\mathbf{u})_{L_2(\mathbb{R}^d)}\biggr| \leqslant\frac{1}{4}\mathfrak{a}_\varepsilon [\mathbf{u},\mathbf{u}]+c_2\| \mathbf{u}\|^2_{L_2(\mathbb{R}^d)},\qquad \mathbf{u}\in H^1(\mathbb{R}^d;\mathbb{C}^n),
\end{equation}
\tag{1.14}
$$
where $c_2=8c_1^2C(\nu_0)$ with $\nu_0=2^{-6}\alpha_0\|g^{-1}\|^{-1}_{L_\infty}$. Next, by condition (1.10) on $Q$, for any $\nu >0$, there exists a constant $C_Q(\nu)>0$ such that
$$
\begin{equation}
|(Q^\varepsilon \mathbf{u},\mathbf{u})_{L_2(\mathbb{R}^d)}| \leqslant\nu \|\mathbf{D}\mathbf{u}\|^2_{L_2(\mathbb{R}^d)}+C_Q(\nu)\| \mathbf{u}\|^2_{L_2(\mathbb{R}^d)},\qquad \mathbf{u}\in H^1(\mathbb{R}^d;\mathbb{C}^n).
\end{equation}
\tag{1.15}
$$
For fixed $\nu$, the constant $C_Q(\nu)$ is controlled in terms of $d$, $s$, $\|Q\|_{L_s(\Omega)}$, and the parameters of the lattice $\Gamma$. We assume that the constant $\lambda$ in (1.11) satisfies
$$
\begin{equation}
\lambda \geqslant \lambda_* := (C_Q(\nu_*)+c_2)\| Q_0^{-1}\|_{L_\infty}\quad\text{for}\quad \nu_* =2^{-1}\alpha_0\| g^{-1}\|^{-1}_{L_\infty}.
\end{equation}
\tag{1.16}
$$
Combining (1.14), (1.15) with $\nu=\nu_*$, and (1.16), and taking (1.8) into account, we obtain the following lower estimate for the form (1.11):
$$
\begin{equation}
\mathfrak{b}_\varepsilon[\mathbf{u},\mathbf{u}]\geqslant \frac{1}{4}\, \mathfrak{a}_\varepsilon [\mathbf{u},\mathbf{u}]\geqslant c_* \|\mathbf{D}\mathbf{u}\|^2_{L_2(\mathbb{R}^d)},\qquad\mathbf{u}\in H^1(\mathbb{R}^d;\mathbb{C}^n);
\end{equation}
\tag{1.17}
$$
$$
\begin{equation}
c_*=\frac{1}{4}\, \alpha_0\| g^{-1}\|^{-1}_{L_\infty}.
\end{equation}
\tag{1.18}
$$
From (1.7), (1.14), and (1.15) with $\nu=1$ it follows that
$$
\begin{equation*}
\begin{gathered} \, \mathfrak{b}_\varepsilon [\mathbf{u},\mathbf{u}]\leqslant C_*\|\mathbf{D}\mathbf{u}\|^2_{L_2(\mathbb{R}^d)}+ c_3\|\mathbf{u}\|^2_{L_2(\mathbb{R}^d)},\qquad \mathbf{u}\in H^1(\mathbb{R}^d;\mathbb{C}^n), \\ C_*=\frac{5}{4}\, \alpha_1\|g\|_{L_\infty}+1,\qquad c_3=C_Q(1)+ \lambda\|Q_0\|_{L_\infty}+c_2. \end{gathered}
\end{equation*}
\notag
$$
So, the form $\mathfrak{b}_\varepsilon$ is closed and non-negative. The selfadjoint operator in $L_2(\mathbb{R}^d;\mathbb{C}^n)$ corresponding to this form is denoted by $B_\varepsilon$. Formally, we have
$$
\begin{equation}
B_\varepsilon = b(\mathbf{D})^* g^\varepsilon (\mathbf{x})b(\mathbf{D})+\sum_{j=1}^d \bigl(a_j^\varepsilon (\mathbf{x})D_j +D_j a_j^\varepsilon (\mathbf{x})^*\bigr) +Q^\varepsilon (\mathbf{x}) +\lambda Q_0^\varepsilon (\mathbf{x}).
\end{equation}
\tag{1.19}
$$
1.5. The effective matrixThe effective operator for $A_\varepsilon$ is given by $A^0=b(\mathbf{D})^*g^0b(\mathbf{D})$. Here, $g^0$ is a constant effective $(m\times m)$-matrix defined in terms of the solution of an auxiliary cell problem. Suppose that a $\Gamma$-periodic $(n\times m)$-matrix-valued function $\Lambda (\mathbf{x})$ is the (weak) solution of the problem
$$
\begin{equation}
b(\mathbf{D})^*g(\mathbf{x})(b(\mathbf{D})\Lambda (\mathbf{x})+\mathbf{1}_m)=0,\qquad \int_{\Omega}\Lambda (\mathbf{x})\,d\mathbf{x}=0.
\end{equation}
\tag{1.20}
$$
Then the effective matrix is given by
$$
\begin{equation}
g^0=|\Omega|^{-1}\int_{\Omega} \widetilde{g}(\mathbf{x})\,d\mathbf{x},
\end{equation}
\tag{1.21}
$$
$$
\begin{equation}
\widetilde{g}(\mathbf{x}):= g(\mathbf{x})(b(\mathbf{D})\Lambda (\mathbf{x})+\mathbf{1}_m).
\end{equation}
\tag{1.22}
$$
It can be checked that the matrix $g^0$ is positive definite. Using (1.20), it is easy to check that
$$
\begin{equation}
\|g^{1/2} b(\mathbf{D})\Lambda\|_{L_2(\Omega)} \leqslant |\Omega|^{1/2} \|g\|_{L_\infty}^{1/2},
\end{equation}
\tag{1.23}
$$
$$
\begin{equation}
\|\Lambda\|_{L_2(\Omega)}\leqslant |\Omega|^{1/2}M_1,\qquad M_1=(2r_0)^{-1}\alpha_0^{-1/2}\|g\|^{1/2}_{L_\infty}\|g^{-1}\|^{1/2}_{L_\infty},
\end{equation}
\tag{1.24}
$$
$$
\begin{equation}
\|\mathbf{D}\Lambda\|_{L_2(\Omega)}\leqslant |\Omega|^{1/2}M_2,\qquad M_2= \alpha_0^{-1/2}\|g\|^{1/2}_{L_\infty}\|g^{-1}\|^{1/2}_{L_\infty}.
\end{equation}
\tag{1.25}
$$
The effective matrix satisfies estimates (1.26) known in homogenization theory as the Voigt–Reuss bracketing (see, for example, [7], Chap. 3, Theorem 1.5). Proposition 1.3. Let $g^0$ be the effective matrix (1.21). Then If $m=n$, then $g^0=\underline{g}$. From (1.26) it follows that
$$
\begin{equation}
|g^0| \leqslant \|g\|_{L_\infty},\qquad |(g^0)^{-1}| \leqslant \|g^{-1}\|_{L_\infty}.
\end{equation}
\tag{1.27}
$$
Now we distinguish the cases where one of the inequalities in (1.26) becomes an identity (see [7], Chap. 3, Propositions 1.6 and 1.7). Proposition 1.4. The identity $g^0=\overline{g}$ is equivalent to the relations where $\mathbf{g}_k(\mathbf{x})$, $k=1,\dots,m$, are the columns of the matrix $g(\mathbf{x})$. Proposition 1.5. The identity $g^0 =\underline{g}$ is equivalent to the representations where $\mathbf{l}_k(\mathbf{x})$, $k=1,\dots,m$, are the columns of the matrix $g(\mathbf{x})^{-1}$. 1.6. The effective operatorTo describe how the lower order terms of $B_\varepsilon$ are homogenized, consider a $\Gamma$-periodic $(n\times n)$-matrix-valued function $\widetilde{\Lambda}(\mathbf{x})$ which is the solution of the problem
$$
\begin{equation}
b(\mathbf{D})^*g(\mathbf{x})b(\mathbf{D})\widetilde{\Lambda }(\mathbf{x})+ \sum_{j=1}^dD_ja_j(\mathbf{x})^*=0,\qquad \int_{\Omega }\widetilde{\Lambda} (\mathbf{x})\,d\mathbf{x}=0.
\end{equation}
\tag{1.30}
$$
(This equation is understood in the weak sense.) It is easily seen that
$$
\begin{equation}
\|g^{1/2} b(\mathbf{D})\widetilde{\Lambda}\|_{L_2(\Omega)} \leqslant C_a \alpha_0^{-1/2}\|g^{-1}\|^{1/2}_{L_\infty},
\end{equation}
\tag{1.31}
$$
$$
\begin{equation}
\|\widetilde{\Lambda}\|_{L_2(\Omega)}\leqslant (2r_0)^{-1}C_a \alpha_0^{-1} \|g^{-1}\|_{L_\infty}=: \widetilde{M}_1 |\Omega|^{1/2},
\end{equation}
\tag{1.32}
$$
$$
\begin{equation}
\|\mathbf{D}\widetilde{\Lambda}\|_{L_2(\Omega)}\leqslant C_a \alpha_0^{-1} \|g^{-1}\|_{L_\infty} =: \widetilde{M}_2 |\Omega|^{1/2},
\end{equation}
\tag{1.33}
$$
where $C_a^2 :=\sum_{j=1}^d \int_\Omega |a_j(\mathbf{x})|^2\,d\mathbf{x}$. We define constant matrices $V$ and $W$ as follows:
$$
\begin{equation}
V =|\Omega|^{-1}\int_{\Omega}\bigl(b(\mathbf{D})\Lambda (\mathbf{x})\bigr)^*g(\mathbf{x}) \bigl(b(\mathbf{D})\widetilde{\Lambda}(\mathbf{x})\bigr)\,d\mathbf{x},
\end{equation}
\tag{1.34}
$$
$$
\begin{equation}
W =|\Omega|^{-1}\int_{\Omega} \bigl(b(\mathbf{D})\widetilde{\Lambda}(\mathbf{x})\bigr)^* g(\mathbf{x})\bigl(b(\mathbf{D})\widetilde{\Lambda}(\mathbf{x})\bigr)\,d\mathbf{x}.
\end{equation}
\tag{1.35}
$$
From (1.23), (1.31), (1.34), and (1.35) we have where $C_V= \alpha_0^{-1/2} \|g\|_{L_\infty}^{1/2} \|g^{-1}\|_{L_\infty}^{1/2} C_a |\Omega|^{-1/2}$ and $C_W = \alpha_0^{-1} \|g^{-1}\|_{L_\infty} C^2_a |\Omega|^{-1}$. The effective operator for the operator (1.19) is given by
$$
\begin{equation}
B^0 =b(\mathbf{D})^*g^0b(\mathbf{D})-b(\mathbf{D})^*V-V^*b(\mathbf{D}) +\sum _{j=1}^d (\overline{a_j+a_j^*})D_j-W+\overline{Q}+\lambda \overline{Q_0}.
\end{equation}
\tag{1.37}
$$
The operator $B^0$ is the elliptic second-order operator with constant coefficients with the symbol
$$
\begin{equation}
L(\boldsymbol{\xi})=b(\boldsymbol{\xi})^*g^0 b(\boldsymbol{\xi})-b(\boldsymbol{\xi})^*V-V^*b(\boldsymbol{\xi})+\sum_{j=1}^d (\overline{a_j+a_j^*})\xi_j -W +\overline{Q}+\lambda \overline{Q_0}.
\end{equation}
\tag{1.38}
$$
It is easily seen (see Lemma 1.6 in [40]) that the symbol (1.38) satisfies the estimates
$$
\begin{equation}
c_*|\boldsymbol{\xi}|^2 \mathbf{1}_n\leqslant L(\boldsymbol{\xi})\leqslant C_L(|\boldsymbol{\xi}|^2+1)\mathbf{1}_n, \qquad\boldsymbol{\xi}\in \mathbb{R}^d.
\end{equation}
\tag{1.39}
$$
Here, the constant $c_*$ is given by (1.18). The constant $C_L$ depends only on the initial data (1.12). This implies the following estimates for the quadratic form $\mathfrak{b}^0$ of the operator (1.37):
$$
\begin{equation*}
c_*\|\mathbf{D}\mathbf{u}\|^2_{L_2(\mathbb{R}^d)}\leqslant \mathfrak{b}^0[\mathbf{u},\mathbf{u}]\leqslant C_L\|\mathbf{u}\|^2_{H^1(\mathbb{R}^d)}, \qquad \mathbf{u}\in H^1(\mathbb{R}^d;\mathbb{C}^n).
\end{equation*}
\notag
$$
1.7. The results about approximation of the generalized resolventIn this subsection, we formulate the results proved in [37], Theorems 5.1, 5.2, and 5.4. Theorem 1.6 (see [37]). Suppose that the assumptions of §§ 1.3–1.6 are satisfied. Let $\zeta \in\mathbb{C}\setminus \mathbb{R}_+$, $\zeta =|\zeta| e^{i\phi}$, $\phi \in (0,2\pi)$, and $|\zeta| \geqslant 1$. We also set Next, we introduce a corrector
$$
\begin{equation}
K(\varepsilon;\zeta)=\bigl([\Lambda^\varepsilon] b(\mathbf{D}) +[\widetilde{\Lambda}^\varepsilon] \bigr)S_\varepsilon (B^0-\zeta \overline{Q_0}\,)^{-1},
\end{equation}
\tag{1.41}
$$
which is a bounded operator acting from $L_2(\mathbb{R}^d;\mathbb{C}^n)$ to $H^1(\mathbb{R}^d;\mathbb{C}^n)$. This can be easily checked by using Proposition 1.2 and since $\Lambda,\widetilde{\Lambda}\in \widetilde{H}^1(\Omega)$. Note that $\|\varepsilon K(\varepsilon;\zeta)\|_{L_2 \to H^1}=O(1)$ for small $\varepsilon$ and fixed $\zeta$. Denote
$$
\begin{equation}
G(\varepsilon;\zeta):=\bigl( \widetilde{g}^{\,\varepsilon} b(\mathbf{D}) +g^\varepsilon (b(\mathbf{D})\widetilde{\Lambda})^\varepsilon \bigr) S_\varepsilon (B^0-\zeta \overline{Q_0}\,)^{-1}.
\end{equation}
\tag{1.42}
$$
By Proposition 1.2 and since $\widetilde{g}, g (b(\mathbf{D})\widetilde{\Lambda}) \in L_2(\Omega)$, the operator (1.42) is bounded from $L_2(\mathbb{R}^d;\mathbb{C}^n)$ to $L_2(\mathbb{R}^d;\mathbb{C}^m)$, and $\| G(\varepsilon;\zeta)\|_{L_2\to L_2}=O(1)$. Theorem 1.7 (see [37]). Under the hypotheses of Theorem 1.6, let $K(\varepsilon;\zeta)$ and $G(\varepsilon;\zeta)$ be the operators defined by (1.41) and (1.42), respectively. Then, for $0<\varepsilon \leqslant 1$ and $|\zeta| \geqslant 1$, § 2. Statement of the problem. The effective operator2.1. CoercivityWe impose an additional restriction on the symbol $b(\boldsymbol{\xi})$ of the operator $b(\mathbf{D})$. Condition 2.1. Suppose that the matrix-valued function $b(\boldsymbol{\xi})= \sum_{l=1}^d b_l \xi_l$ satisfies Note that condition (2.1) is more restrictive than condition (1.4). The following assertion was obtained in the book [44] (see Theorem 7.8 in § 3.7; in this assertion, it suffices to assume that the boundary $\partial \mathcal{O}$ is Lipschitz). Proposition 2.2 (see [44]). Let $\mathcal{O}\subset \mathbb{R}^d$ be a bounded domain with Lipschitz boundary. Condition 2.1 is necessary and sufficient for the existence of constants $k_1>0$, $k_2 \geqslant 0$ such that the Gärding inequality holds: Remark 2.3. The constants $k_1$, $k_2$ depend on the matrix $b(\boldsymbol{\xi})$ and on the domain $\mathcal{O}$, but in the general case it is difficult to control them explicitly. However, they can often be found for specific operators. Therefore, in what follows we will refer to the dependence of other constants on $k_1$ and $k_2$. 2.2. The operator $A_{N,\varepsilon}$Let $\mathcal{O}\,{\subset}\, \mathbb{R}^d$ be a bounded domain of class $C^{1,1}$. In $L_2(\mathcal{O};\mathbb{C}^n)$, we consider the operator $A_{N,\varepsilon}$ formally given by the differential expressiion $b(\mathbf{D})^* g^\varepsilon(\mathbf{x}) b(\mathbf{D})$ with the Neumann boundary condition. Let us give the precise definition: $A_{N,\varepsilon}$ is a selfadjoint operator in $L_2(\mathcal{O};\mathbb{C}^n)$ generated by the quadratic form
$$
\begin{equation}
\mathfrak{a}_{N,\varepsilon}[\mathbf{u},\mathbf{u}] = \int_\mathcal{O} \langle g^\varepsilon(\mathbf{x}) b(\mathbf{D})\mathbf{u},b(\mathbf{D})\mathbf{u} \rangle \, d\mathbf{x}, \qquad \mathbf{u}\in H^1(\mathcal{O};\mathbb{C}^n).
\end{equation}
\tag{2.3}
$$
Relations (1.3) and (1.6) imply the upper estimate
$$
\begin{equation}
\mathfrak{a}_{N,\varepsilon}[\mathbf{u},\mathbf{u}] \leqslant d \alpha_1 \|g\|_{L_\infty} \|\mathbf{D}\mathbf{u}\|^2_{L_2(\mathcal{O})}, \qquad \mathbf{u}\in H^1(\mathcal{O};\mathbb{C}^n).
\end{equation}
\tag{2.4}
$$
The lower estimate follows from (2.2):
$$
\begin{equation}
\mathfrak{a}_{N,\varepsilon}[\mathbf{u},\mathbf{u}] \geqslant \|g^{-1}\|^{-1}_{L_\infty} \bigl(k_1 \|\mathbf{D}\mathbf{u}\|^2_{L_2(\mathcal{O})}- k_2 \|\mathbf{u}\|^2_{L_2(\mathcal{O})}\bigr), \qquad \mathbf{u}\in H^1(\mathcal{O};\mathbb{C}^n).
\end{equation}
\tag{2.5}
$$
By (2.3)–(2.5), the form $\mathfrak{a}_{N,\varepsilon }$ is closed and non-negative. 2.3. The operator $B_{N,\varepsilon}$Let $0< \varepsilon \leqslant 1$. Now, we consider a more general operator $B_{N,\varepsilon}$ adding lower order terms to $A_{N,\varepsilon}$. Formally, the operator $B_{N,\varepsilon}$ is given by the differential expression
$$
\begin{equation*}
b(\mathbf{D})^* g^\varepsilon (\mathbf{x})b(\mathbf{D})+\sum_{j=1}^d \bigl( a_j^\varepsilon(\mathbf{x})D_j +D_j a_j^\varepsilon (\mathbf{x})^* \bigr) +Q^\varepsilon (\mathbf{x})+\lambda Q_0^\varepsilon (\mathbf{x})
\end{equation*}
\notag
$$
with the Neumann boundary condition. The coefficients $g$, $a_j$, $Q$, $Q_0$ satisfy the assumptions of §§ 1.3, 1.4; the operator $b(\mathbf{D})$ of the form (1.3) is subject to Condition 2.1; the constant $\lambda >0$ will be fixed below. The precise definition of the operator $B_{N,\varepsilon}$ is given in terms of the quadratic form
$$
\begin{equation}
\begin{aligned} \, \mathfrak{b}_{N,\varepsilon}[\mathbf{u},\mathbf{u}] &= \mathfrak{a}_{N,\varepsilon}[\mathbf{u},\mathbf{u}] +2\operatorname{Re}\sum_{j=1}^d (D_j \mathbf{u},(a_j^\varepsilon)^*\mathbf{u})_{L_2(\mathcal{O})} \nonumber \\ &\qquad+(Q^\varepsilon \mathbf{u},\mathbf{u})_{L_2(\mathcal{O})}+\lambda (Q_0^\varepsilon \mathbf{u},\mathbf{u})_{L_2(\mathcal{O})},\qquad \mathbf{u}\in H^1(\mathcal{O};\mathbb{C}^n). \end{aligned}
\end{equation}
\tag{2.6}
$$
Let us estimate the form $\mathfrak{b}_{N,\varepsilon}$. We start with the analysis of the lower order terms. We have
$$
\begin{equation}
\biggl| 2\operatorname{Re}\sum_{j=1}^d (D_j \mathbf{u},(a_j^\varepsilon)^*\mathbf{u})_{L_2(\mathcal{O})}\biggr| \leqslant 2 \|\mathbf{D}\mathbf{u}\|_{L_2(\mathcal{O})} \biggl( \sum_{j=1}^d \int_\mathcal{O} |a_j^\varepsilon (\mathbf{x})|^2| \mathbf{u}(\mathbf{x})|^2\,d\mathbf{x}\biggr)^{1/2}.
\end{equation}
\tag{2.7}
$$
By the Hölder inequality,
$$
\begin{equation}
\int_\mathcal{O}| a_j^\varepsilon (\mathbf{x})|^2 |\mathbf{u}(\mathbf{x})|^2\,d\mathbf{x} \leqslant \biggl(\int_\mathcal{O} |a_j^\varepsilon(\mathbf{x})|^\rho\,d\mathbf{x}\biggr)^{2/\rho} \|\mathbf{u}\|_{L_q(\mathcal{O})}^2.
\end{equation}
\tag{2.8}
$$
Here, $\rho$ is as in condition (1.9), $q=\infty$ for $d=1$, $q=2\rho/(\rho -2)$ for $d\geqslant 2$. Let us cover the domain $\mathcal{O}$ by the union of the cells of the lattice $\varepsilon\Gamma$ that have a non-empty intersection with $\mathcal{O}$. By $N_\varepsilon$ we denote the number of the cells in this covering. Clearly, this union of cells is contained in the domain $\widetilde{\mathcal{O}}$, which is the $2r_1$-neighbourhood of $\mathcal{O}$, where $2r_1=\operatorname{diam}\Omega$. Therefore, we can estimate $N_\varepsilon$ from above: $N_\varepsilon\leqslant \mathfrak{c}_1\varepsilon^{-d}$, where $\mathfrak{c}_1$ depends only on the domain $\mathcal{O}$ and the parameters of the lattice $\Gamma$. We have
$$
\begin{equation}
\int_\mathcal{O} |a_j^\varepsilon (\mathbf{x})|^\rho \,d\mathbf{x}\leqslant \mathfrak{c}_1\varepsilon^{-d}\int_{\varepsilon\Omega} |a_j^\varepsilon (\mathbf{x})|^\rho \,d\mathbf{x} =\mathfrak{c}_1\|a_j\|^\rho_{L_\rho (\Omega)}.
\end{equation}
\tag{2.9}
$$
Now, (2.8) and (2.9) imply that
$$
\begin{equation}
\int_\mathcal{O}|a_j^\varepsilon (\mathbf{x})|^2 |\mathbf{u}(\mathbf{x})|^2\,d\mathbf{x} \leqslant \mathfrak{c}_1^{2/\rho}\|a_j\|^2_{L_\rho (\Omega)} \|\mathbf{u}\|^2_{L_q(\mathcal{O})}.
\end{equation}
\tag{2.10}
$$
By the compactness of the embedding $H^1(\mathcal{O};\mathbb{C}^n)\hookrightarrow L_q(\mathcal{O};\mathbb{C}^n)$, for any $\mu >0$ there exists a constant $\check{\mathcal{C}}(\mu)>0$ such that
$$
\begin{equation}
\|\mathbf{u}\|^2_{L_q(\mathcal{O})} \leqslant \mu \|\mathbf{D} \mathbf{u}\|^2_{L_2(\mathcal{O})} + \check{\mathcal{C}}(\mu) \|\mathbf{u}\|^2_{L_2(\mathcal{O})}, \qquad \mathbf{u} \in H^1(\mathcal{O};\mathbb{C}^n).
\end{equation}
\tag{2.11}
$$
For a fixed $\mu$, $\check{\mathcal{C}}(\mu)$ depends only on $d$, $\rho$, and the domain $\mathcal{O}$. From (2.7), (2.10), and (2.11) it follows that, for any $\nu>0$, there exists a constant $\mathcal{C}(\nu)$ such that
$$
\begin{equation}
\biggl| 2\operatorname{Re}\sum_{j=1}^d (D_j \mathbf{u},(a_j^\varepsilon)^*\mathbf{u})_{L_2(\mathcal{O})}\biggr| \leqslant \nu \|\mathbf{D}\mathbf{u}\|^2_{L_2(\mathcal{O})} + \mathcal{C}(\nu) \|\mathbf{u}\|^2_{L_2(\mathcal{O})}, \qquad \mathbf{u} \in H^1(\mathcal{O};\mathbb{C}^n).
\end{equation}
\tag{2.12}
$$
For a fixed $\nu$, $\mathcal{C}(\nu)$ depends only on $d$, $\rho$, $\|a_j\|_{L_\rho(\Omega)}$, $j=1,\dots,d$, the domain $\mathcal{O}$, and the parameters of the lattice $\Gamma$. Similarly, using condition (1.10) and the Hölder inequality, we obtain
$$
\begin{equation}
\int_\mathcal{O}| Q^\varepsilon (\mathbf{x})| | \mathbf{u}(\mathbf{x})|^2\,d\mathbf{x} \leqslant \mathfrak{c}_1^{1/s}\| Q \|_{L_s (\Omega)}\| \mathbf{u}\|^2_{L_{\check{q}}(\mathcal{O})},
\end{equation}
\tag{2.13}
$$
where $\check{q} = \infty$ for $d=1$, $\check{q}= 2s/(s-1)$ for $d\geqslant 2$. By the compactness of the embedding $H^1(\mathcal{O};\mathbb{C}^n)\hookrightarrow L_{\check{q}} (\mathcal{O};\mathbb{C}^n)$, it follows that, for any $\nu>0$, there exists a constant $\mathcal{C}_Q(\nu)>0$ such that
$$
\begin{equation}
|(Q^\varepsilon \mathbf{u},\mathbf{u})_{L_2(\mathcal{O})}| \leqslant \nu \|\mathbf{D}\mathbf{u}\|^2_{L_2(\mathcal{O})} + \mathcal{C}_Q(\nu) \|\mathbf{u}\|^2_{L_2(\mathcal{O})}, \qquad \mathbf{u} \in H^1(\mathcal{O};\mathbb{C}^n).
\end{equation}
\tag{2.14}
$$
For a fixed $\nu$, $\mathcal{C}_Q(\nu)$ depends only on $d$, $s$, $\|Q\|_{L_s(\Omega)}$, the domain $\mathcal{O}$, and the parameters of the lattice $\Gamma$. Now, relations (2.5), (2.6), (2.12), and (2.14) imply that
$$
\begin{equation}
\begin{aligned} \, &\mathfrak{b}_{N,\varepsilon }[\mathbf{u},\mathbf{u}] \geqslant \bigl(k_1 \|g^{-1}\|^{-1}_{L_\infty} - 2 \nu\bigr) \|\mathbf{D}\mathbf{u}\|^2_{L_2(\mathcal{O})} \nonumber \\ &\qquad+ \bigl(\lambda \| Q_0^{-1}\|^{-1}_{L_\infty} - k_2 \|g^{-1}\|^{-1}_{L_\infty} - \mathcal{C}(\nu) - \mathcal{C}_Q(\nu)\bigr) \|\mathbf{u}\|^2_{L_2(\mathcal{O})}, \qquad \mathbf{u}\in H^1(\mathcal{O};\mathbb{C}^n), \end{aligned}
\end{equation}
\tag{2.15}
$$
for any $\nu>0$. We choose $\nu$ equal to $\widehat{\nu}_0 := \frac{1}{4} k_1 \|g^{-1}\|^{-1}_{L_\infty}$ and impose the following restriction on the constant $\lambda$ in (2.6):
$$
\begin{equation}
\lambda \geqslant \lambda_0 := \|Q_0^{-1}\|_{L_\infty} \biggl(\biggl(\frac{k_1}2 + k_2\biggr) \|g^{-1}\|^{-1}_{L_\infty} + \mathcal{C}(\widehat{\nu}_0) + \mathcal{C}_Q(\widehat{\nu}_0)\biggr).
\end{equation}
\tag{2.16}
$$
Now (2.15) implies the lower estimate for the form (2.6):
$$
\begin{equation}
\mathfrak{b}_{N,\varepsilon}[\mathbf{u},\mathbf{u}] \geqslant c_4 \|\mathbf{u}\|^2_{H^1(\mathcal{O})}, \qquad \mathbf{u}\in H^1(\mathcal{O};\mathbb{C}^n); \quad c_4 := \frac{1}{2}\, k_1 \|g^{-1}\|^{-1}_{L_\infty}.
\end{equation}
\tag{2.17}
$$
The upper estimate for the form (2.6) follows from (2.4) and (2.12), (2.14) (with $\nu=1$):
$$
\begin{equation}
\begin{gathered} \, \mathfrak{b}_{N,\varepsilon}[\mathbf{u},\mathbf{u}] \leqslant c_5 \|\mathbf{u}\|^2_{H^1(\mathcal{O})}, \qquad \mathbf{u}\in H^1(\mathcal{O};\mathbb{C}^n); \\ c_5 := \max \{d \alpha_1 \|g\|_{L_\infty} + 2;\, \mathcal{C}(1) + \mathcal{C}_Q(1) + \lambda \|Q_0\|_{L_\infty} \}. \end{gathered}
\end{equation}
\tag{2.18}
$$
Inequalities (2.17) and (2.18) show that the form (2.6) is closed and positive definite. Let $B_{N,\varepsilon}$ be the selfadjoint operator in $L_2(\mathcal{O};\mathbb{C}^n)$ generated by this form. . We need to introduce an auxiliary operator $\widetilde{B}_{N,\varepsilon}$. We factorize the matrix $Q_0$: Let $\widetilde{B}_{N,\varepsilon}$ be a selfadjoint operator in $L_2(\mathcal{O};\mathbb{C}^n)$ generated by the quadratic form
$$
\begin{equation}
\begin{gathered} \, \widetilde{\mathfrak{b}}_{N,\varepsilon} [\mathbf{u},\mathbf{u}]:= \mathfrak{b}_{N,\varepsilon}[f^\varepsilon \mathbf{u},f^\varepsilon \mathbf{u}], \\ \operatorname{Dom}\widetilde{\mathfrak{b}}_{N,\varepsilon}=\{\mathbf{u}\in L_2(\mathcal{O};\mathbb{C}^n) \colon f^\varepsilon \mathbf{u}\in H^1(\mathcal{O};\mathbb{C}^n)\}. \end{gathered}
\end{equation}
\tag{2.20}
$$
We have $\widetilde{B}_{N,\varepsilon}=(f^\varepsilon )^*B_{N,\varepsilon}f^\varepsilon $. Note that
$$
\begin{equation}
(B_{N,\varepsilon}-\zeta Q_0^\varepsilon)^{-1}=f^\varepsilon (\widetilde{B}_{N,\varepsilon}-\zeta I)^{-1}(f^\varepsilon)^*.
\end{equation}
\tag{2.21}
$$
2.4. Estimates for the generalized resolvent $(B_{N,\varepsilon}-\zeta Q_0^\varepsilon)^{-1}$Our goal is to approximate the generalized resolvent $(B_{N,\varepsilon}-\zeta Q_0^\varepsilon )^{-1}$ and to prove two-parametric error estimates (with respect to $\varepsilon$ and $\zeta$). Assume that $\zeta \in \mathbb{C}\setminus\mathbb{R}_+$. In other words, we are interested in the behaviour of the solution
$$
\begin{equation*}
\mathbf{u}_\varepsilon =(B_{N,\varepsilon }-\zeta Q_0^\varepsilon )^{-1}\mathbf{F}, \qquad \mathbf{F}\in L_2(\mathcal{O};\mathbb{C}^n),
\end{equation*}
\notag
$$
of the Neumann problem for small $\varepsilon$. The solution $\mathbf{u}_\varepsilon \in H^1(\mathcal{O};\mathbb{C}^n)$ satisfies the identity
$$
\begin{equation}
\mathfrak{b}_{N,\varepsilon}[\mathbf{u}_\varepsilon,\boldsymbol{\eta}]-\zeta (Q_0^\varepsilon \mathbf{u}_\varepsilon,\boldsymbol{\eta})_{L_2(\mathcal{O})} =(\mathbf{F},\boldsymbol{\eta})_{L_2(\mathcal{O})},\qquad \boldsymbol{\eta}\in H^1(\mathcal{O};\mathbb{C}^n).
\end{equation}
\tag{2.22}
$$
Lemma 2.4. Let $\zeta \in \mathbb{C}\setminus \mathbb{R}_+$. For $0<\varepsilon \leqslant 1$, Proof. From (2.19), (2.21), and the inequality $\widetilde{B}_{N,\varepsilon}> 0$ it follows that
$$
\begin{equation*}
\begin{aligned} \, \|(B_{N,\varepsilon}-\zeta Q_0^\varepsilon)^{-1}\|_{L_2(\mathcal{O})\to L_2(\mathcal{O})} &\leqslant \|f\|^2_{L_\infty} \bigl\|(\widetilde{B}_{N,\varepsilon}-\zeta I)^{-1}\bigr\|_{L_2(\mathcal{O})\to L_2(\mathcal{O})} \\ &\leqslant (\operatorname{dist}\{\zeta;\mathbb{R}_+\})^{-1} \|Q_0^{-1}\|_{L_\infty} =c(\phi)|\zeta|^{-1} \|Q_0^{-1}\|_{L_\infty}, \end{aligned}
\end{equation*}
\notag
$$
which implies (2.23).
To check (2.24), substituting $\boldsymbol{\eta}= \mathbf{u}_\varepsilon$ in the identity (2.22) and applying estimate (2.17) and the already proved inequality (2.23), we obtain
$$
\begin{equation*}
c_4 \|\mathbf{u}_\varepsilon\|^2_{H^1 (\mathcal{O})} \leqslant |\zeta|^{-1} \bigl( c(\phi) \|Q_0^{-1}\|_{L_\infty}+c(\phi)^2\|Q_0\|_{L_\infty} \|Q_0^{-1}\|^2_{L_\infty}\bigr) \|\mathbf{F}\|_{L_2(\mathcal{O})}^2.
\end{equation*}
\notag
$$
This implies (2.24) and completes the proof. 2.5. The effective operatorIn $L_2(\mathcal{O};\mathbb{C}^n)$, we consider the quadratic form
$$
\begin{equation}
\begin{aligned} \, \mathfrak{b}_N^0[\mathbf{u},\mathbf{u}] &= \bigl(g^0b(\mathbf{D})\mathbf{u},b(\mathbf{D})\mathbf{u}\bigr)_{L_2(\mathcal{O})} +2\operatorname{Re}\sum_{j=1}^d (\overline{a_j}D_j\mathbf{u},\mathbf{u})_{L_2(\mathcal{O})} \nonumber \\ &\qquad -2\operatorname{Re}\bigl(V\mathbf{u},b(\mathbf{D})\mathbf{u}\bigr)_{L_2(\mathcal{O})} -(W\mathbf{u},\mathbf{u})_{L_2(\mathcal{O})} \nonumber \\ &\qquad+(\overline{Q}\mathbf{u},\mathbf{u})_{L_2(\mathcal{O})} +\lambda (\overline{Q_0}\mathbf{u},\mathbf{u})_{L_2(\mathcal{O})}, \qquad \mathbf{u}\in H^1(\mathcal{O};\mathbb{C}^n). \end{aligned}
\end{equation}
\tag{2.25}
$$
Let us estimate the form (2.25). We start with the lower order terms. Obviously,
$$
\begin{equation}
\begin{aligned} \, &\biggl| 2\operatorname{Re}\sum_{j=1}^d (\overline{a_j} D_j \mathbf{u},\mathbf{u})_{L_2(\mathcal{O})}\biggr| \leqslant 2 \|\mathbf{D}\mathbf{u}\|_{L_2(\mathcal{O})} \|\mathbf{u}\|_{L_2(\mathcal{O})} \biggl( \sum_{j=1}^d |\overline{a_j}|^2 \biggr)^{1/2} \nonumber \\ &\leqslant 2 C_a |\Omega|^{-1/2} \|\mathbf{D}\mathbf{u}\|_{L_2(\mathcal{O})}\| \mathbf{u}\|_{L_2(\mathcal{O})} \nonumber \\ &\leqslant \frac{1}{4}\, k_1 \|g^{-1}\|^{-1}_{L_\infty} \|\mathbf{D}\mathbf{u}\|_{L_2(\mathcal{O})}^2 + 4 k_1^{-1} \|g^{-1}\|_{L_\infty} C_a^2 |\Omega|^{-1} \|\mathbf{u}\|_{L_2(\mathcal{O})}^2, \qquad \mathbf{u}\in H^1(\mathcal{O};\mathbb{C}^n). \end{aligned}
\end{equation}
\tag{2.26}
$$
Next, from (1.36) it follows that
$$
\begin{equation}
\begin{aligned} \, &\bigl|2\operatorname{Re}\bigl(V \mathbf{u}, b(\mathbf{D}) \mathbf{u}\bigr)_{L_2(\mathcal{O})}\bigr| \leqslant 2 C_V \|b(\mathbf{D}) \mathbf{u}\|_{L_2(\mathcal{O})} \|\mathbf{u}\|_{L_2(\mathcal{O})} \nonumber \\ &\quad\leqslant \frac{1}{4}\, \|g^{-1}\|^{-1}_{L_\infty} \|b(\mathbf{D})\mathbf{u}\|^2_{L_2(\mathcal{O})} + 4 C_V^2 \|g^{-1}\|_{L_\infty} \|\mathbf{u}\|_{L_2(\mathcal{O})}^2, \qquad \mathbf{u}\in H^1(\mathcal{O};\mathbb{C}^n). \end{aligned}
\end{equation}
\tag{2.27}
$$
Finally, for $\mathbf{u}\in H^1(\mathcal{O};\mathbb{C}^n)$, we have
$$
\begin{equation}
|(W\mathbf{u},\mathbf{u})_{L_2(\mathcal{O})}| \leqslant C_W \|\mathbf{u}\|_{L_2(\mathcal{O})}^2,
\end{equation}
\tag{2.28}
$$
$$
\begin{equation}
|(\overline{Q} \mathbf{u},\mathbf{u})_{L_2(\mathcal{O})}| \leqslant |\overline{Q}\,| \|\mathbf{u}\|_{L_2(\mathcal{O})}^2 \leqslant |\Omega|^{-1} \|Q\|_{L_1(\Omega)} \|\mathbf{u}\|_{L_2(\mathcal{O})}^2,
\end{equation}
\tag{2.29}
$$
$$
\begin{equation}
\lambda(\overline{Q_0}\mathbf{u},\mathbf{u})_{L_2(\mathcal{O})} \geqslant \lambda \|Q_0^{-1}\|_{L_\infty}^{-1} \|\mathbf{u}\|_{L_2(\mathcal{O})}^2.
\end{equation}
\tag{2.30}
$$
Combining (2.26)–(2.30) and taking (1.27) into account, we obtain the following estimate for the form (2.25):
$$
\begin{equation*}
\begin{gathered} \, \begin{aligned} \, \mathfrak{b}_N^0[\mathbf{u},\mathbf{u}] &\geqslant \frac{3}{4}\, \|g^{-1}\|^{-1}_{L_\infty} \|b(\mathbf{D}) \mathbf{u}\|^2_{L_2(\mathcal{O})} - \frac{1}{4} \, k_1 \|g^{-1}\|^{-1}_{L_\infty} \|\mathbf{D}\mathbf{u}\|_{L_2(\mathcal{O})}^2 \\ &\qquad+ \bigl(\lambda \|Q_0^{-1}\|_{L_\infty}^{-1} - C_5\bigr) \|\mathbf{u}\|^2_{L_2(\mathcal{O})}, \qquad \mathbf{u}\in H^1(\mathcal{O};\mathbb{C}^n), \end{aligned} \\ C_5:= 4 k_1^{-1} \|g^{-1}\|_{L_\infty} C_a^2 |\Omega|^{-1} + 4 C_V^2 \|g^{-1}\|_{L_\infty} + |\Omega|^{-1} \|Q\|_{L_1(\Omega)} + C_W. \end{gathered}
\end{equation*}
\notag
$$
Now, we use (2.2) and impose one more restriction on $\lambda$:
$$
\begin{equation}
\lambda \geqslant \widehat{\lambda}:= \|Q_0^{-1}\|_{L_\infty} \biggl( \biggl( \frac{k_1}2 + \frac{3k_2}4\biggr) \|g^{-1}\|_{L_\infty}^{-1} + C_5 \biggr).
\end{equation}
\tag{2.31}
$$
As a result, we arrive at the estimate
$$
\begin{equation}
\mathfrak{b}_N^0[\mathbf{u},\mathbf{u}] \geqslant c_4 \|\mathbf{u}\|^2_{H^1(\mathcal{O})}, \qquad \mathbf{u}\in H^1(\mathcal{O};\mathbb{C}^n),
\end{equation}
\tag{2.32}
$$
where the constant $c_4$ is defined by (2.17). We proceed with the upper estimate for the form (2.25). By (1.27), (2.26), (2.27), and (2.29), and since $W \geqslant 0$, we deduce
$$
\begin{equation*}
\begin{gathered} \, \mathfrak{b}_N^0[\mathbf{u},\mathbf{u}] \leqslant 2 \|g\|_{L_\infty} \|b(\mathbf{D})\mathbf{u}\|^2_{L_2(\mathcal{O})}+ \|\mathbf{D}\mathbf{u}\|_{L_2(\mathcal{O})}^2 + C_6 \|\mathbf{u}\|^2_{L_2(\mathcal{O})}, \qquad \mathbf{u}\in H^1(\mathcal{O};\mathbb{C}^n), \\ C_6:= |\Omega|^{-1} (C_a^2 + \|Q\|_{L_1(\Omega)})+ C_V^2 \|g\|^{-1}_{L_\infty} + \lambda \|Q_0\|_{L_\infty}. \end{gathered}
\end{equation*}
\notag
$$
Together with (1.3) and (1.6), this yields
$$
\begin{equation}
\mathfrak{b}_N^0[\mathbf{u},\mathbf{u}] \leqslant c_6 \|\mathbf{u}\|^2_{H^1(\mathcal{O})}, \qquad \mathbf{u}\in H^1(\mathcal{O};\mathbb{C}^n),
\end{equation}
\tag{2.33}
$$
where $c_6 = \max \{ 2 d \alpha_1 \|g\|_{L_\infty} +1,\, C_6 \}$. Thus, by (2.32) and (2.33), the form (2.25) is closed and positive definite. A selfadjoint operator in $L_2(\mathcal{O};\mathbb{C}^n)$ corresponding to this form is denoted by $B_N^0$. Now, we finally fix the value of the parameter $\lambda$:
$$
\begin{equation}
\lambda = \max \{ \lambda_*, \lambda_0, \widehat{\lambda}\, \},
\end{equation}
\tag{2.34}
$$
where $\lambda_*$, $\lambda_0$, $\widehat{\lambda}$ are defined by (1.16), (2.16), and (2.31), respectively. This ensures that the operator $B_\varepsilon$ acting in $L_2(\mathbb{R}^d; \mathbb{C}^n)$ is non-negative and the operators $B_{N,\varepsilon}$ and $B_N^0$ acting in $L_2(\mathcal{O};\mathbb{C}^n)$ are positive definite. For the convenience of further references, we call the following set of parameters the “initial data”:
$$
\begin{equation}
\begin{gathered} \, d,\ m, \ n, \ \rho, \ s; \ \alpha_0, \ \alpha_1, \ \|g\|_{L_\infty}, \ \|g^{-1}\|_{L_\infty},\ \|a_j\|_{L_\rho (\Omega)},\qquad j=1,\dots,d; \\ \|Q\|_{L_s(\Omega)}, \ \|Q_0\|_{L_\infty}, \ \|Q_0^{-1}\|_{L_\infty}, \ k_1, \ k_2; \\ \text{the parameters of the lattice } \Gamma, \ \text{the domain } \mathcal{O}. \end{gathered}
\end{equation}
\tag{2.35}
$$
Clearly, the constant (2.34) and also the constants $c_4$, $c_5$, $c_6$ are controlled in terms of the initial data (2.35). Since $\partial\mathcal{O}\in C^{1,1}$, the operator $(B_N^0)^{-1}$ is a continuous mapping from $L_2(\mathcal{O};\mathbb{C}^n)$ to $H^2(\mathcal{O};\mathbb{C}^n)$. We have
$$
\begin{equation}
\|(B_N^0)^{-1}\|_{L_2(\mathcal{O})\to H^2(\mathcal{O})}\leqslant \widehat{c}.
\end{equation}
\tag{2.36}
$$
Here, the constant $\widehat{c}$ depends only on the initial data (2.35). To justify this fact, we refer to the theorems on the regularity of solutions of strongly elliptic systems (see [45], Chap. 4). Remark 2.5. Instead of the condition $\partial\mathcal{O}\in C^{1,1}$, one could impose the following implicit condition: a bounded domain $\mathcal{O}\subset \mathbb{R}^d$ with Lipschitz boundary is such that estimate (2.36) holds. The results of the paper remain valid for such domain. In the case of the scalar elliptic operators, wide sufficient conditions on $\partial \mathcal{O}$ ensuring (2.36) can be found in [46] and [47], Chap. 7 (in particular, it suffices that $\partial\mathcal{O}\in C^\alpha$, $\alpha >3/2$). We use the following factorization: $\overline{Q_0}=f_0^{-2}$. By (2.19),
$$
\begin{equation}
| f_0| \leqslant \| f\|_{L_\infty}=\| Q_0^{-1}\|^{1/2}_{L_\infty},\qquad | f_0^{-1}| \leqslant \| f^{-1}\|_{L_\infty}=\| Q_0\|^{1/2}_{L_\infty}.
\end{equation}
\tag{2.37}
$$
We need the auxiliary operator $\widetilde{B}_N^0=f_0 B_N^0 f_0$. Note that
$$
\begin{equation}
(B_N^0-\zeta\overline{Q_0}\,)^{-1}=f_0\bigl(\widetilde{B}_N^0-\zeta I\bigr)^{-1}f_0.
\end{equation}
\tag{2.38}
$$
The function
$$
\begin{equation*}
\mathbf{u}_0=(B_N^0-\zeta \overline{Q_0}\,)^{-1}\mathbf{F}, \qquad \mathbf{F} \in L_2(\mathcal{O};\mathbb{C}^n),
\end{equation*}
\notag
$$
is the solution of the homogenized Neumann problem. The solution $\mathbf{u}_0 \in H^1(\mathcal{O};\mathbb{C}^n)$ satisfies the identity
$$
\begin{equation}
\mathfrak{b}_{N}^0 [\mathbf{u}_0,\boldsymbol{\eta}]-\zeta (\overline{Q_0} \mathbf{u}_0,\boldsymbol{\eta})_{L_2(\mathcal{O})} =(\mathbf{F},\boldsymbol{\eta})_{L_2(\mathcal{O})},\qquad \boldsymbol{\eta}\in H^1(\mathcal{O};\mathbb{C}^n).
\end{equation}
\tag{2.39}
$$
Lemma 2.6. For $\zeta \in \mathbb{C}\setminus \mathbb{R}_+$, Proof. Estimates (2.40) and (2.41) can be checked by the same way as estimates of Lemma 2.4. Let us prove (2.42). Obviously,
$$
\begin{equation}
\begin{aligned} \, &\bigl\| (B_N^0-\zeta \overline{Q_0}\,)^{-1} \bigr\|_{L_2(\mathcal{O})\to H^2(\mathcal{O})} \nonumber \\ &\qquad\leqslant \| (B_N^0)^{-1}\|_{L_2(\mathcal{O})\to H^2(\mathcal{O})} \bigl\| B_N^0(B_N^0-\zeta \overline{Q_0}\,)^{-1} \bigr\|_{L_2(\mathcal{O})\to L_2(\mathcal{O})}. \end{aligned}
\end{equation}
\tag{2.43}
$$
By (2.38),
$$
\begin{equation*}
B_N^0(B_N^0-\zeta\overline{Q_0}\,)^{-1}=B_N^0f_0\bigl(\widetilde{B}_N^0-\zeta I\bigr)^{-1}f_0= f_0^{-1}\widetilde{B}_N^0 \bigl(\widetilde{B}_{N}^0-\zeta I\bigr)^{-1}f_0.
\end{equation*}
\notag
$$
Hence
$$
\begin{equation}
\begin{aligned} \, \bigl\| B_N^0(B_N^0-\zeta \overline{Q_0}\,)^{-1} \bigr\|_{L_2(\mathcal{O})\to L_2(\mathcal{O})} &\leqslant |f_0^{-1}|\, |f_0| \sup_{x\geqslant 0}\frac{x}{| x-\zeta|} \nonumber \\ &\leqslant \|Q_0\|_{L_\infty}^{1/2}\| Q_0^{-1}\|_{L_\infty}^{1/2}c(\phi), \end{aligned}
\end{equation}
\tag{2.44}
$$
where we have taken (2.37) into account. Now (2.42) follows from (2.36), (2.43), and (2.44). This completes the proof. § 3. Formulation of the results. Introduction of the boundary layer correction term3.1. The case $\zeta \,{\in}\, \mathbb{C}\setminus \mathbb{R}_+$, $| \zeta | \,{\geqslant}\, 1$Let $(\partial\mathcal{O})_{\varepsilon} \,{:=}\,\{ \mathbf{x}\,{\in}\, \mathbb{R}^d \colon \!\operatorname{dist}\{ \mathbf{x};\partial\mathcal{O}\} \,{<}\,\varepsilon \}$. We choose the numbers $\varepsilon_0,\varepsilon_1\in (0,1]$ according to the following condition. Condition 3.1. Let $\varepsilon_0\,{\in}\, (0,1]$ be such that the strip $(\partial\mathcal{O})_{\varepsilon_0}$ can be covered by a finite number of open sets admitting diffeomorphisms of class $C^{0,1}$ rectifying the boundary $\partial\mathcal{O}$. Denote $\varepsilon_1=\varepsilon_0 (1+r_1)^{-1}$, where $2r_1=\operatorname{diam}\Omega$. Clearly, $\varepsilon_1$ depends only on the domain $\mathcal{O}$ and the parameters of the lattice $\Gamma$. Note that Condition 3.1 would be provided only by the assumption that $\partial\mathcal{O}$ is Lipschitz. We have imposed a more restrictive condition $\partial\mathcal{O}\in C^{1,1}$ in order to ensure estimate (2.36). Let us formulate the main results of the paper. Theorem 3.2. Suppose that $\mathcal{O}\subset \mathbb{R}^d$ is a bounded domain of class $C^{1,1}$. Let $\zeta =| \zeta | e^{i\phi}\in \mathbb{C}\setminus \mathbb{R}_+$, $|\zeta| \geqslant 1$. Let $\mathbf{u}_\varepsilon=(B_{N,\varepsilon} -\zeta Q_0^\varepsilon)^{-1}\mathbf{F}$ and $\mathbf{u}_0=(B_N^0-\zeta \overline{Q_0}\,)^{-1}\mathbf{F}$, where $\mathbf{F} \in L_2(\mathcal{O};\mathbb{C}^n)$. Suppose that $\varepsilon_1$ satisfies Condition 3.1. Then, for $0<\varepsilon\leqslant \varepsilon_1$, In order to approximate the solution in the Sobolev space $H^1(\mathcal{O};\mathbb{C}^n)$, we introduce a corrector. For this, we fix a linear continuous extension operator
$$
\begin{equation}
P_\mathcal{O}: H^l(\mathcal{O};\mathbb{C}^n)\to H^l(\mathbb{R}^d;\mathbb{C}^n),\qquad l \in \mathbb{Z}_+.
\end{equation}
\tag{3.3}
$$
Such a “universal” extension operator exists for any bounded Lipschitz domain (see [48] or [49]). We have
$$
\begin{equation}
\| P_{\mathcal O} \|_{H^l({\mathcal O}) \to H^l({\mathbb R}^d)} \leqslant C_{\mathcal O}^{(l)}, \qquad l \in \mathbb{Z}_+,
\end{equation}
\tag{3.4}
$$
where the constant $C_{\mathcal O}^{(l)}$ depends only on $l$ and the domain ${\mathcal O}$. By $R_\mathcal{O}$ we denote the operator of restriction of functions in $\mathbb{R}^d$ to the domain $\mathcal{O}$. We put
$$
\begin{equation}
K_N(\varepsilon;\zeta) := R_\mathcal{O}\bigl([\Lambda^\varepsilon]b(\mathbf{D}) +[\widetilde{\Lambda}^\varepsilon ]\bigr)S_\varepsilon P_\mathcal{O} (B_N^0-\zeta \overline{Q_0}\,)^{-1}.
\end{equation}
\tag{3.5}
$$
We also need the operator
$$
\begin{equation}
G_N(\varepsilon;\zeta) := R_\mathcal{O} [\widetilde{g}^{\,\varepsilon}] S_\varepsilon b(\mathbf{D})P_\mathcal{O}(B_N^0-\zeta\overline{Q_0}\,)^{-1}+ R_\mathcal{O} \bigl[g^\varepsilon\bigl( b(\mathbf{D})\widetilde{\Lambda}\bigr)^\varepsilon\bigr] S_\varepsilon P_\mathcal{O} (B_N^0-\zeta \overline{Q_0}\,)^{-1}.
\end{equation}
\tag{3.6}
$$
Lemma 3.3. Suppose that the operators $K_N(\varepsilon;\zeta)$ and $G_N(\varepsilon;\zeta)$ are given by (3.5), (3.6), respectively. Then $K_N(\varepsilon;\zeta)$ is continuous from $L_2(\mathcal{O};\mathbb{C}^n)$ to $H^1(\mathcal{O};\mathbb{C}^n)$ and $G_N(\varepsilon;\zeta)$ is continuous from $L_2(\mathcal{O};\mathbb{C}^n)$ to $L_2(\mathcal{O};\mathbb{C}^m)$. For $\zeta\in\mathbb{C}\setminus\mathbb{R}_+$, $|\zeta|\geqslant 1$, and $0< \varepsilon \leqslant 1$, Proof. By Proposition 1.2 and relations (1.5), (1.24), (1.32), (3.4), we have
$$
\begin{equation*}
\begin{aligned} \, \varepsilon \|K_N(\varepsilon;\zeta)\|_{L_2(\mathcal{O}) \to L_2(\mathcal{O})} & \leqslant \varepsilon M_1 \alpha_1^{1/2} C_\mathcal{O}^{(1)} \bigl\| (B_N^0-\zeta \overline{Q_0}\,)^{-1} \bigr\|_{L_2(\mathcal{O}) \to H^1(\mathcal{O})} \\ &\qquad+ \varepsilon \widetilde{M}_1 C_\mathcal{O}^{(0)} \bigl\| (B_N^0-\zeta \overline{Q_0}\,)^{-1} \bigr\|_{L_2(\mathcal{O}) \to L_2(\mathcal{O})}. \end{aligned}
\end{equation*}
\notag
$$
Together with (2.40), (2.41), and the restriction $|\zeta|\geqslant 1$, this implies estimate (3.7) with the constant $C_K'= M_1 \alpha_1^{1/2} C_\mathcal{O}^{(1)} \mathcal{C}_1 + \widetilde{M}_1 C_\mathcal{O}^{(0)} \|Q_0^{-1}\|_{L_\infty}$.
Next, we have
$$
\begin{equation}
\begin{aligned} \, \varepsilon D_l K_N(\varepsilon;\zeta) &= (D_l \Lambda)^\varepsilon S_\varepsilon b(\mathbf{D}) P_\mathcal{O}(B_N^0-\zeta\overline{Q_0}\,)^{-1}+ ( D_l \widetilde{\Lambda})^\varepsilon S_\varepsilon P_\mathcal{O} (B_N^0-\zeta \overline{Q_0}\,)^{-1} \nonumber \\ &\qquad+ \varepsilon \Lambda^\varepsilon S_\varepsilon b(\mathbf{D}) D_l P_\mathcal{O}(B_N^0-\zeta\overline{Q_0}\,)^{-1}+ \varepsilon \widetilde{\Lambda}^\varepsilon S_\varepsilon D_l P_\mathcal{O} (B_N^0-\zeta \overline{Q_0}\,)^{-1}. \end{aligned}
\end{equation}
\tag{3.10}
$$
Combining this with Proposition 1.2 and relations (1.5), (1.24), (1.25), (1.32), (1.33), (3.4), we obtain
$$
\begin{equation*}
\begin{aligned} \, \varepsilon \| \mathbf{D} K_N(\varepsilon;\zeta)\|_{L_2(\mathcal{O}) \to L_2(\mathcal{O})} &\leqslant (M_2 \alpha_1^{1/2} + \varepsilon \widetilde{M}_1) C_\mathcal{O}^{(1)} \bigl\| (B_N^0-\zeta \overline{Q_0}\,)^{-1}\bigr\|_{L_2(\mathcal{O}) \to H^1(\mathcal{O})} \\ &\qquad+ \widetilde{M}_2 C_\mathcal{O}^{(0)} \bigl\| (B_N^0-\zeta \overline{Q_0}\,)^{-1} \bigr\|_{L_2(\mathcal{O}) \to L_2(\mathcal{O})} \\ &\qquad+\varepsilon {M}_1 \alpha_1^{1/2} C_\mathcal{O}^{(2)} \bigl\| (B_N^0-\zeta \overline{Q_0}\,)^{-1} \bigr\|_{L_2(\mathcal{O}) \to H^2(\mathcal{O})}. \end{aligned}
\end{equation*}
\notag
$$
Together with estimates (2.40)–(2.42), this yields inequality (3.8) with the constant $C_K''= \max\bigl\{ \widetilde{M}_1 C_\mathcal{O}^{(1)} \mathcal{C}_1 + M_1 \alpha_1^{1/2} C_\mathcal{O}^{(2)} \mathcal{C}_2;\, M_2 \alpha_1^{1/2} C_\mathcal{O}^{(1)} \mathcal{C}_1 + \widetilde{M}_2 C_\mathcal{O}^{(0)} \|Q_0^{-1}\|_{L_\infty} \bigr\}$.
Now, we estimate the operator (3.6) by using Proposition 1.2 and relations (1.5), (1.22), (1.23), (1.31), and (3.4):
$$
\begin{equation*}
\begin{aligned} \, \| G_N(\varepsilon;\zeta)\|_{L_2(\mathcal{O}) \to L_2(\mathcal{O})} &\leqslant C_G' \bigl\| (B_N^0-\zeta \overline{Q_0}\,)^{-1} \bigr\|_{L_2(\mathcal{O}) \to H^1(\mathcal{O})} \\ &\qquad+ C_G'' \bigl\| (B_N^0-\zeta \overline{Q_0}\,)^{-1} \bigr\|_{L_2(\mathcal{O}) \to L_2(\mathcal{O})}, \end{aligned}
\end{equation*}
\notag
$$
where $C_G' = 2 \|g\|_{L_\infty} \alpha_1^{1/2} C_\mathcal{O}^{(1)}$ and $C_G'' = |\Omega|^{-1/2} \alpha_0^{-1/2} C_a \|g\|^{1/2}_{L_\infty} \|g^{-1}\|^{1/2}_{L_\infty} C_\mathcal{O}^{(0)}$. Combining this with (2.40) and (2.41), we obtain estimate (3.9) with the constant $C_G = C_G' \mathcal{C}_1 + C_G'' \|Q_0^{-1}\|_{L_\infty}$. This completes the proof. Let $\widetilde{\mathbf{u}}_0=P_\mathcal{O}\mathbf{u}_0$ and let $\mathbf{v}_\varepsilon$ be the first-order approximation of the solution $\mathbf{u}_\varepsilon$:
$$
\begin{equation}
\widetilde{\mathbf{v}}_\varepsilon : =\widetilde{\mathbf{u}}_0+\varepsilon \Lambda^\varepsilon S_\varepsilon b(\mathbf{D})\widetilde{\mathbf{u}}_0 +\varepsilon\widetilde{\Lambda}^\varepsilon S_\varepsilon \widetilde{\mathbf{u}}_0,
\end{equation}
\tag{3.11}
$$
$$
\begin{equation}
\mathbf{v}_\varepsilon :=\widetilde{\mathbf{v}}_\varepsilon |_{\mathcal{O}}.
\end{equation}
\tag{3.12}
$$
In other words, $\mathbf{v}_\varepsilon = (B_N^0-\zeta \overline{Q_0}\,)^{-1}\mathbf{F} +\varepsilon K_N(\varepsilon;\zeta )\mathbf{F}$, where $K_N(\varepsilon;\zeta)$ is the operator (3.5). Theorem 3.4. Under the hypotheses of Theorem 3.2, let the matrix-valued functions $\Lambda (\mathbf{x})$ and $\widetilde{\Lambda}(\mathbf{x})$ be $\Gamma$-periodic solutions of problems (1.20) and (1.30), respectively. Let $S_\varepsilon $ be the Steklov smoothing operator (1.1), and let $P_\mathcal{O}$ be the extension operator (3.3). Suppose that $\widetilde{\mathbf{u}}_0=P_\mathcal{O}\mathbf{u}_0$ and $\mathbf{v}_\varepsilon$ is the function defined by (3.11), (3.12). Then, for $\zeta\in\mathbb{C}\setminus\mathbb{R}_+$, $|\zeta|\geqslant 1$, and $0<\varepsilon \leqslant \varepsilon_1$, Corollary 3.5. Under the assumptions of Theorem 3.4, for $\zeta\in\mathbb{C}\setminus\mathbb{R}_+$, $|\zeta|\geqslant 1$, and $0<\varepsilon \leqslant \varepsilon_1$, we have 3.2. The first part of the proof. Associated problem in $\mathbb{R}^d$By Lemma 2.6 and (3.4), for $\zeta \in \mathbb{C} \setminus \mathbb{R}_+$ we have
$$
\begin{equation}
\| \widetilde{\mathbf{u}}_0\|_{L_2(\mathbb{R}^d)} \leqslant C_7 c(\phi)| \zeta |^{-1}\| \mathbf{F}\|_{L_2(\mathcal{O})}; \qquad C_7 := C_\mathcal{O}^{(0)}\| Q_0^{-1}\|_{L_\infty},
\end{equation}
\tag{3.18}
$$
$$
\begin{equation}
\| \widetilde{\mathbf{u}}_0\|_{H^1(\mathbb{R}^d)} \leqslant C_8 c(\phi)| \zeta |^{-1/2}\| \mathbf{F}\|_{L_2(\mathcal{O})}; \qquad C_8 := C_\mathcal{O}^{(1)} \mathcal{C}_1,
\end{equation}
\tag{3.19}
$$
$$
\begin{equation}
\| \widetilde{\mathbf{u}}_0\|_{H^2(\mathbb{R}^d)} \leqslant C_9 c(\phi)\| \mathbf{F}\|_{L_2(\mathcal{O})}; \qquad C_9 := C_\mathcal{O}^{(2)}\mathcal{C}_2.
\end{equation}
\tag{3.20}
$$
Let
$$
\begin{equation}
\widetilde{\mathbf{F}}:=(B^0-\zeta \overline{Q_0}\,)\widetilde{\mathbf{u}}_0,
\end{equation}
\tag{3.21}
$$
where $B^0$ is the operator (1.37). Then $\widetilde{\mathbf{F}}\in L_2(\mathbb{R}^d;\mathbb{C}^n)$ and $\widetilde{\mathbf{F}}|_{\mathcal{O}}=\mathbf{F}$. By the upper estimate (1.39), (3.18), and (3.20),
$$
\begin{equation}
\begin{gathered} \, \| \widetilde{\mathbf{F}}\|_{L_2(\mathbb{R}^d)}\leqslant C_L \| \widetilde{\mathbf{u}}_0\|_{H^2(\mathbb{R}^d)}+|\zeta|\, | \overline{Q_0}|\, \|\widetilde{\mathbf{u}}_0\|_{L_2(\mathbb{R}^d)} \leqslant C_{10}c(\phi) \|\mathbf{F}\|_{L_2(\mathcal{O})}, \\ C_{10}:= C_L C_9 + C_7 \| Q_0\|_{L_\infty}. \nonumber \end{gathered}
\end{equation}
\tag{3.22}
$$
Let $\widetilde{\mathbf{u}}_\varepsilon \in H^1(\mathbb{R}^d;\mathbb{C}^n)$ be the solution of the following equation in $\mathbb{R}^d$:
$$
\begin{equation}
B_\varepsilon \widetilde{\mathbf{u}}_\varepsilon -\zeta Q_0^\varepsilon \widetilde{\mathbf{u}}_\varepsilon =\widetilde{\mathbf{F}},
\end{equation}
\tag{3.23}
$$
that is, $\widetilde{\mathbf{u}}_\varepsilon =(B_\varepsilon -\zeta Q_0^\varepsilon)^{-1}\widetilde{\mathbf{F}}$. Combining (3.21)–(3.23) and applying Theorems 1.6 and 1.7, for $0<\varepsilon \leqslant 1$ and $\zeta\in\mathbb{C}\setminus\mathbb{R}_+$, $| \zeta | \geqslant 1$, we obtain
$$
\begin{equation}
\| \widetilde{\mathbf{u}}_\varepsilon -\widetilde{\mathbf{u}}_0\|_{L_2(\mathbb{R}^d)} \leqslant C_1 C_{10} c(\phi)^3 \varepsilon | \zeta |^{-1/2}\| \mathbf{F}\|_{L_2(\mathcal{O})},
\end{equation}
\tag{3.24}
$$
$$
\begin{equation}
\| \widetilde{\mathbf{u}}_\varepsilon -\widetilde{\mathbf{v}}_\varepsilon \|_{L_2(\mathbb{R}^d)}\leqslant C_2 C_{10}c(\phi)^3\varepsilon | \zeta |^{-1/2}\| \mathbf{F}\|_{L_2(\mathcal{O})},
\end{equation}
\tag{3.25}
$$
$$
\begin{equation}
\| \mathbf{D}(\widetilde{\mathbf{u}}_\varepsilon -\widetilde{\mathbf{v}}_\varepsilon)\|_{L_2(\mathbb{R}^d)}\leqslant C_3 C_{10}c(\phi)^3\varepsilon\| \mathbf{F}\|_{L_2(\mathcal{O})},
\end{equation}
\tag{3.26}
$$
$$
\begin{equation}
\bigl\| g^\varepsilon b(\mathbf{D})\widetilde{\mathbf{u}}_\varepsilon - \widetilde{g}^{\,\varepsilon} S_\varepsilon b(\mathbf{D})\widetilde{\mathbf{u}}_0- g^\varepsilon \bigl(b(\mathbf{D}) \widetilde{\Lambda}\bigr)^\varepsilon S_\varepsilon \widetilde{\mathbf{u}}_0 \bigr\|_{L_2(\mathbb{R}^d)}\leqslant C_4 C_{10}c(\phi)^3\varepsilon\| \mathbf{F}\|_{L_2(\mathcal{O})}.
\end{equation}
\tag{3.27}
$$
Now, (3.25), (3.26), and the inequality $| \zeta |\geqslant 1$ imply that
$$
\begin{equation}
\| \widetilde{\mathbf{u}}_\varepsilon -\widetilde{\mathbf{v}}_\varepsilon \|_{H^1(\mathbb{R}^d)} \leqslant (C_2+C_3)C_{10}c(\phi)^3\varepsilon\| \mathbf{F}\|_{L_2(\mathcal{O})}.
\end{equation}
\tag{3.28}
$$
3.3. The second part of the proof. Introduction of the boundary layer correction termWe introduce a correction term $\mathbf{w}_\varepsilon \in H^1(\mathcal{O};\mathbb{C}^n)$ as a function satisfying the integral identity
$$
\begin{equation}
\mathfrak{b}_{N,\varepsilon} [\mathbf{w}_\varepsilon,\boldsymbol{\eta}]-\zeta (Q_0^\varepsilon \mathbf{w}_\varepsilon,\boldsymbol{\eta})_{L_2(\mathcal{O})}= \mathcal{I}_\varepsilon[\boldsymbol{\eta}], \qquad \boldsymbol{\eta}\in H^1(\mathcal{O};\mathbb{C}^n),
\end{equation}
\tag{3.29}
$$
where
$$
\begin{equation}
\begin{aligned} \, \mathcal{I}_\varepsilon[\boldsymbol{\eta}] &:= \bigl( \widetilde{g}^{\,\varepsilon} S_\varepsilon b(\mathbf{D})\widetilde{\mathbf{u}}_0 + g^\varepsilon (b(\mathbf{D}) \widetilde{\Lambda})^\varepsilon S_\varepsilon \widetilde{\mathbf{u}}_0, b(\mathbf{D})\boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})} \nonumber \\ &\qquad+ \sum_{j=1}^d \bigl( \bigl( D_j \mathbf{v}_\varepsilon, (a_j^\varepsilon)^* \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})} + \bigl( (a_j^\varepsilon)^* \mathbf{v}_\varepsilon, D_j \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})} \bigr) + (Q^\varepsilon \mathbf{v}_\varepsilon, \boldsymbol{\eta})_{L_2(\mathcal{O})} \nonumber \\ &\qquad +(\lambda - \zeta) (Q_0^\varepsilon \mathbf{u}_0, \boldsymbol{\eta})_{L_2(\mathcal{O})} - (\mathbf{F},\boldsymbol{\eta})_{L_2(\mathcal{O})}, \qquad \boldsymbol{\eta}\in H^1(\mathcal{O};\mathbb{C}^n). \end{aligned}
\end{equation}
\tag{3.30}
$$
It is easily seen that the functional (3.30) is antilinear and continuous in the space $H^1(\mathcal{O};\mathbb{C}^n)$. The continuity of the first term follows from Proposition 1.2 and the relations $\widetilde{g}, g(b(\mathbf{D}) \widetilde{\Lambda}) \in L_2(\Omega)$. Next, using (2.10) and the continuous embedding $H^1(\mathcal{O};\mathbb{C}^n)\hookrightarrow L_q(\mathcal{O};\mathbb{C}^n)$ (with the embedding constant $C(q;\mathcal{O})$), we obtain
$$
\begin{equation}
\biggl( \sum_{j=1}^d \| (a_j^\varepsilon)^* \boldsymbol{\eta} \|^2_{L_2(\mathcal{O})} \biggr)^{1/2} \leqslant C_{11} \| \boldsymbol{\eta} \|_{H^1(\mathcal{O})}, \qquad \boldsymbol{\eta}\in H^1(\mathcal{O};\mathbb{C}^n),
\end{equation}
\tag{3.31}
$$
where $C_{11}= \mathfrak{c}_1^{1 / \rho} C(q;\mathcal{O}) \bigl( \sum_{j=1}^d \|a_j\|^2_{L_\rho(\Omega)} \bigr)^{1/2}$. It follows that the second term in (3.30) is continuous in $H^1(\mathcal{O};\mathbb{C}^n)$. Similarly, using (2.13) and the continuity of the embedding $H^1(\mathcal{O};\mathbb{C}^n)\hookrightarrow L_{\check{q}} (\mathcal{O};\mathbb{C}^n)$ (with the embedding constant $C(\check{q};\mathcal{O})$), we obtain
$$
\begin{equation}
\| \,|Q^\varepsilon|^{1/2} \boldsymbol{\eta} \|_{L_2(\mathcal{O})} \leqslant C_{12} \|\boldsymbol{\eta} \|_{H^1(\mathcal{O})}, \qquad \boldsymbol{\eta}\in H^1(\mathcal{O};\mathbb{C}^n),
\end{equation}
\tag{3.32}
$$
where $C_{12}= \mathfrak{c}_1^{1/2s} C(\check{q};\mathcal{O}) \| Q\|^{1/2}_{L_s(\Omega)}$. Therefore, the third term in (3.30) is continuous in $H^1(\mathcal{O};\mathbb{C}^n)$. Obviously, the last two terms are continuous. It is checked in a standard way that the solution $\mathbf{w}_\varepsilon$ of problem (3.29) exists and is unique. The correction term $\mathbf{w}_\varepsilon$ of such type is often called a “boundary layer correction term”. Lemma 3.6. Suppose that $\zeta \in \mathbb{C} \setminus \mathbb{R}_+$ and $|\zeta| \geqslant 1$. Let $\mathbf{u}_\varepsilon=(B_{N,\varepsilon} -\zeta Q_0^\varepsilon)^{-1}\mathbf{F}$, where $\mathbf{F} \in L_2(\mathcal{O};\mathbb{C}^n)$. Let $\widetilde{\mathbf{u}}_\varepsilon$ be defined as in § 3.2. Suppose that $\mathbf{w}_\varepsilon \in H^1(\mathcal{O};\mathbb{C}^n)$ satisfies identity (3.29). Then Proof. We set $\mathbf{V}_\varepsilon :=\mathbf{u}_\varepsilon -\widetilde{\mathbf{u}}_\varepsilon+\mathbf{w}_\varepsilon$. In view of (2.22), (3.12), and (3.29), the function $\mathbf{V}_\varepsilon\in H^1(\mathcal{O};\mathbb{C}^n)$ satisfies the identity
$$
\begin{equation}
\begin{aligned} \, &\mathfrak{b}_{N,\varepsilon}[\mathbf{V}_\varepsilon,\boldsymbol{\eta}]-\zeta (Q_0^\varepsilon\mathbf{V}_\varepsilon,\boldsymbol{\eta})_{L_2(\mathcal{O})} \nonumber \\ &\ = - \bigl( g^\varepsilon b(\mathbf{D}) \widetilde{\mathbf{u}}_\varepsilon - \widetilde{g}^{\,\varepsilon} S_\varepsilon b(\mathbf{D})\widetilde{\mathbf{u}}_0 - {g}^\varepsilon (b(\mathbf{D}) \widetilde{\Lambda})^\varepsilon S_\varepsilon \widetilde{\mathbf{u}}_0, b(\mathbf{D})\boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})} \nonumber \\ &\ \qquad- \sum_{j=1}^d \bigl( \bigl( D_j (\widetilde{\mathbf{u}}_\varepsilon - \widetilde{\mathbf{v}}_\varepsilon), (a_j^\varepsilon)^* \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})} + \bigl( (a_j^\varepsilon)^* (\widetilde{\mathbf{u}}_\varepsilon - \widetilde{\mathbf{v}}_\varepsilon), D_j \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})} \bigr) \nonumber \\ &\ \qquad - \bigl( Q^\varepsilon (\widetilde{\mathbf{u}}_\varepsilon - \widetilde{\mathbf{v}}_\varepsilon), \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})} \,{-}\, (\lambda - \zeta) \bigl( Q_0^\varepsilon ( \widetilde{\mathbf{u}}_\varepsilon - \widetilde{\mathbf{u}}_0), \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})}, \,\quad \boldsymbol{\eta}\,{\in}\, H^1(\mathcal{O};\mathbb{C}^n). \end{aligned}
\end{equation}
\tag{3.35}
$$
We denote the right-hand side of (3.35) by $J_\varepsilon[\boldsymbol{\eta}]$. Using (3.24), (3.26)–(3.28), (3.31), and (3.32), we see that
$$
\begin{equation}
| J_\varepsilon[\boldsymbol{\eta}] | \leqslant C_{13} c(\phi)^3 \varepsilon \| \mathbf{F} \|_{L_2(\mathcal{O})} \bigl(\| \mathbf{D} \boldsymbol{\eta} \|_{L_2(\mathcal{O})} + |\zeta|^{1/2} \| \boldsymbol{\eta}\|_{L_2(\mathcal{O})}\bigr),
\end{equation}
\tag{3.36}
$$
where the constant $C_{13}$ depends only on the problem data (2.35).
Substituting $\boldsymbol{\eta} = \mathbf{V}_\varepsilon$ in the identity (3.35), taking the imaginary part, and applying (3.36), we have
$$
\begin{equation}
|{\operatorname{Im}\zeta}| (Q_0^\varepsilon\mathbf{V}_\varepsilon,\mathbf{V}_\varepsilon )_{L_2(\mathcal{O})} \leqslant C_{13} c(\phi)^3 \varepsilon \| \mathbf{F} \|_{L_2(\mathcal{O})} \bigl( \| \mathbf{D} \mathbf{V}_\varepsilon \|_{L_2(\mathcal{O})} + |\zeta|^{1/2} \| \mathbf{V}_\varepsilon \|_{L_2(\mathcal{O})}\bigr).
\end{equation}
\tag{3.37}
$$
If $\operatorname{Re}\zeta\geqslant 0$ (and then $\operatorname{Im}\zeta \neq 0$), we deduce
$$
\begin{equation}
\begin{aligned} \, (Q_0^\varepsilon\mathbf{V}_\varepsilon,\mathbf{V}_\varepsilon )_{L_2(\mathcal{O})} &\leqslant 2 C_{13}c(\phi)^4 \varepsilon |\zeta|^{-1} \| \mathbf{D} \mathbf{V}_\varepsilon \|_{L_2(\mathcal{O})} \| \mathbf{F}\|_{L_2(\mathcal{O})} \nonumber \\ &\qquad+ C_{13}^2 \| Q_0^{-1}\|_{L_\infty} c(\phi)^8 \varepsilon^2 |\zeta|^{-1} \| \mathbf{F}\|^2_{L_2(\mathcal{O})}, \qquad\operatorname{Re}\zeta\geqslant 0. \end{aligned}
\end{equation}
\tag{3.38}
$$
If $\operatorname{Re}\zeta <0$, we take the real part in the identity (3.35) with $\boldsymbol{\eta}=\mathbf{V}_\varepsilon$. Note that $c(\phi)=1$ for such $\zeta$. Taking (3.36) into account, we obtain
$$
\begin{equation}
|{\operatorname{Re}\zeta}| (Q_0^\varepsilon\mathbf{V}_\varepsilon,\mathbf{V}_\varepsilon )_{L_2(\mathcal{O})} \leqslant C_{13}\varepsilon \| \mathbf{F} \|_{L_2(\mathcal{O})} \bigl( \| \mathbf{D} \mathbf{V}_\varepsilon \|_{L_2(\mathcal{O})} + |\zeta|^{1/2} \| \mathbf{V}_\varepsilon \|_{L_2(\mathcal{O})}\bigr).
\end{equation}
\tag{3.39}
$$
Summing up (3.37) and (3.39), we arrive at the inequality
$$
\begin{equation*}
|\zeta| (Q_0^\varepsilon \mathbf{V}_\varepsilon,\mathbf{V}_\varepsilon )_{L_2(\mathcal{O})}\leqslant 2 C_{13}\varepsilon \| \mathbf{F} \|_{L_2(\mathcal{O})} \bigl( \| \mathbf{D} \mathbf{V}_\varepsilon \|_{L_2(\mathcal{O})} + |\zeta|^{1/2} \| \mathbf{V}_\varepsilon \|_{L_2(\mathcal{O})}\bigr).
\end{equation*}
\notag
$$
Hence
$$
\begin{equation*}
\begin{aligned} \, (Q_0^\varepsilon \mathbf{V}_\varepsilon,\mathbf{V}_\varepsilon )_{L_2(\mathcal{O})} &\leqslant 4 C_{13}\varepsilon |\zeta|^{-1} \| \mathbf{D} \mathbf{V}_\varepsilon \|_{L_2(\mathcal{O})} \| \mathbf{F}\|_{L_2(\mathcal{O})} \\ &\qquad +4 C_{13}^2 \|Q_0^{-1}\|_{L_\infty} \varepsilon^2 |\zeta|^{-1} \| \mathbf{F}\|^2_{L_2(\mathcal{O})},\qquad \operatorname{Re}\zeta <0. \end{aligned}
\end{equation*}
\notag
$$
Combining this with (3.38), we obtain the following inequality for all values of $\zeta$ under consideration:
$$
\begin{equation}
\begin{aligned} \, (Q_0^\varepsilon\mathbf{V}_\varepsilon,\mathbf{V}_\varepsilon )_{L_2(\mathcal{O})} &\leqslant 4C_{13}c(\phi)^4\varepsilon|\zeta|^{-1}\| \mathbf{D} \mathbf{V}_\varepsilon\|_{L_2 (\mathcal{O})}\| \mathbf{F}\|_{L_2(\mathcal{O})} \nonumber \\ &\qquad +4 C_{13}^2 \|Q_0^{-1}\|_{L_\infty} c(\phi)^8\varepsilon^2|\zeta|^{-1}\| \mathbf{F}\|^2_{L_2(\mathcal{O})}. \end{aligned}
\end{equation}
\tag{3.40}
$$
Now, from (3.35) with $\boldsymbol{\eta}=\mathbf{V}_\varepsilon$ and (3.36), we have
$$
\begin{equation*}
\begin{aligned} \, \mathfrak{b}_{N,\varepsilon}[\mathbf{V}_\varepsilon,\mathbf{V}_\varepsilon ] &\leqslant |J_\varepsilon [\mathbf{V}_\varepsilon ]| +|\zeta| (Q_0^\varepsilon\mathbf{V}_\varepsilon,\mathbf{V}_\varepsilon )_{L_2(\mathcal{O})} \\ &\leqslant C_{13} c(\phi)^3 \varepsilon \| \mathbf{F}\|_{L_2(\mathcal{O})} \| \mathbf{D} \mathbf{V}_\varepsilon\|_{L_2(\mathcal{O})} \\ &\qquad+C_{13}^2 \| Q_0^{-1}\|_{L_\infty} c(\phi)^6\varepsilon^2 \| \mathbf{F}\|^2_{L_2(\mathcal{O})} +2 |\zeta| (Q_0^\varepsilon \mathbf{V}_\varepsilon,\mathbf{V}_\varepsilon)_{L_2(\mathcal{O})}. \end{aligned}
\end{equation*}
\notag
$$
Combining this with (2.17) and (3.40), we obtain the required estimate (3.33) with the constant $\mathcal{C}_6^2= 81 C_{13}^2 c_4^{-2} + 18 C_{13}^2 c_4^{-1}\| Q_0^{-1}\|_{L_\infty}$. Relations (3.33) and (3.40) imply inequality (3.34) with the constant $\mathcal{C}_7^2= 4 C_{13} \mathcal{C}_6 \| Q_0^{-1}\|_{L_\infty}+ 4 C_{13}^2 \| Q_0^{-1}\|^2_{L_\infty}$. This completes the proof. Lemma 3.6 and estimate (3.28) imply the following theorem, which shows that, taking the correction term $\mathbf{w}_\varepsilon$ into account, we can approximate the solution $\mathbf{u}_\varepsilon$ by the function $\mathbf{v}_\varepsilon - \mathbf{w}_\varepsilon$ in the norm on $H^1(\mathcal{O};\mathbb{C}^n)$ with error of sharp order $O(\varepsilon)$. Theorem 3.7. Under the assumptions of Theorem 3.4, let $\mathbf{w}_\varepsilon{\kern1pt}{\in}{\kern1pt}H^1(\mathcal{O};\mathbb{C}^n)$ satisfy identity (3.29). Then, for $0<\varepsilon\leqslant 1$ and $\zeta \in \mathbb{C}\setminus \mathbb{R}_+$, $| \zeta | \geqslant 1$, The rest of the proof of Theorems 3.2 and 3.4 is as follows. We will first prove estimate (3.14) for $\operatorname{Re} \zeta \leqslant 0$. Then we will check (3.2) also for $\operatorname{Re} \zeta \leqslant 0$, using the already proved estimate (3.14) and the duality arguments. After that, we will complete the proofs of the theorems, relying on suitable identities for the resolvents that allow us to transfer the already proved estimates from the point $\zeta$ in the left half-plane to the symmetric point in the right half-plane. (Such technique was previously used in [40], § 10.) Сonclusions. 1) From (3.41) it follows that 2) From (3.24) and (3.34) it follows that
$$
\begin{equation}
\| \mathbf{u}_\varepsilon -\mathbf{u}_0 \|_{L_2(\mathcal{O})} \leqslant \mathcal{C}_9 \varepsilon |\zeta|^{-1/2} \| \mathbf{F}\|_{L_2(\mathcal{O})} + \| \mathbf{w}_\varepsilon \|_{L_2(\mathcal{O})}, \qquad \operatorname{Re} \zeta \leqslant 0, \quad |\zeta| \geqslant 1,
\end{equation}
\tag{3.43}
$$
where $\mathcal{C}_9= \mathcal{C}_7+ C_1 C_{10}$. So, the proof of Theorem 3.2 (for $\operatorname{Re} \zeta \leqslant 0$) is reduced to estimation of $\mathbf{w}_\varepsilon$ in $L_2(\mathcal{O};\mathbb{C}^n)$. § 4. Auxiliary results4.1. Estimates in the neighbourhood of the boundaryIn this subsection, we give auxiliary results related to estimates of integrals over the narrow neighbourhood of the boundary. Lemma 4.1. Let Condition 3.1 be met. We set $\Upsilon_\varepsilon =(\partial\mathcal{O})_\varepsilon\cap \mathcal{O}$. Then, for any function $u\in H^1(\mathcal{O})$, The constant $\beta$ depends only on the domain $\mathcal{O}$. Lemma 4.2. Let Condition 3.1 be met. Next, let $f(\mathbf{x})$ be a $\Gamma$-periodic function in $\mathbb{R}^d$ such that $f\in L_2(\Omega)$, and let $S_\varepsilon$ be the operator (1.1). Denote $\beta_*=\beta (1+r_1)$, where $2r_1=\operatorname{diam}\Omega$. Then, for $0<\varepsilon\leqslant\varepsilon_1$ and any function $\mathbf{u}\in H^1(\mathbb{R}^d;\mathbb{C}^k)$, Lemma 4.2 is similar to Lemma 2.6 from [18]. Lemmas 4.1 and 4.2 were established in [28], § 5, under the assumption $\partial \mathcal{O} \in C^1$, but the proofs can be transferred to the case of Condition 3.1. 4.2. Traditional lemma of homogenization theoryWe need the following version of the traditional lemma of homogenization theory (see, for example, [4], Chap. 1, § 1); the proof of this version of the lemma can be found in Lemma 3.1 of [5]. Lemma 4.3. Let $f_l(\mathbf{x})$, $l=1,\dots,d,$ be $\Gamma$-periodic $(n \times m)$-matrix-valued functions in $\mathbb{R}^d$ such that 4.3. Lemma on $h^\varepsilon - \overline{h}$Let us assume that $0 < \varepsilon \leqslant \varepsilon_0$. Let us fix a cut-off function $\theta_\varepsilon (\mathbf{x})$ in $\mathbb{R}^d$ such that
$$
\begin{equation}
\begin{gathered} \, \theta_\varepsilon \in C_0^\infty (\mathbb{R}^d),\qquad \operatorname{supp}\theta_\varepsilon \subset (\partial \mathcal{O})_\varepsilon,\qquad 0\leqslant \theta_\varepsilon (\mathbf{x})\leqslant 1, \\ \theta_\varepsilon (\mathbf{x})=1\quad \text{for}\quad \mathbf{x}\in \partial \mathcal{O};\qquad \varepsilon| \nabla \theta_\varepsilon (\mathbf{x})| \leqslant \kappa =\mathrm{Const}. \end{gathered}
\end{equation}
\tag{4.4}
$$
The constant $\kappa$ depends only on $d$ and the domain $\mathcal{O}$. Lemma 4.4. Let $p(\mathbf{x})$, $a(\mathbf{x})$ be $\Gamma$-periodic $(n\times n)$-matrix-valued functions in $\mathbb{R}^d$ such that $p \in L_2(\Omega)$ and $a \in L_\rho(\Omega)$, where $\rho=2$ for $d=1$, $\rho >d$ for $d\geqslant 2.$ Let $h(\mathbf{x}) := a(\mathbf{x}) p(\mathbf{x})$ and $\overline{h}=|\Omega|^{-1} \int_\Omega h(\mathbf{x})\,d\mathbf{x}$. Let $S_\varepsilon$ be the operator (1.1). Let $\widetilde{\mathbf{u}} \in H^1(\mathbb{R}^d;\mathbb{C}^n)$ and $\boldsymbol{\eta} \in H^1(\mathcal{O};\mathbb{C}^n)$. Denote $1^\circ$. For $0< \varepsilon \leqslant \varepsilon_1$,
$$
\begin{equation}
\begin{aligned} \, | \mathfrak{t}_\varepsilon[\widetilde{\mathbf{u}}, \boldsymbol{\eta}]| &\leqslant C' \varepsilon \|a\|_{L_\rho(\Omega)} \|p\|_{L_2(\Omega)} \bigl( \| \widetilde{\mathbf{u}}\|_{H^1(\mathbb{R}^d)} \| \boldsymbol{\eta}\|_{L_2(\mathcal{O})} + \| \widetilde{\mathbf{u}}\|_{L_2(\mathbb{R}^d)} \| \boldsymbol{\eta}\|_{H^1(\mathcal{O})}\bigr) \nonumber \\ &\qquad+ C'' \varepsilon^{1/2} \|p\|_{L_2(\Omega)} \| \widetilde{\mathbf{u}}\|_{H^1(\mathbb{R}^d)}^{1/2} \| \widetilde{\mathbf{u}}\|_{L_2(\mathbb{R}^d)}^{1/2} \| (a^\varepsilon)^* \boldsymbol{\eta}\|_{L_2(\Upsilon_\varepsilon)}. \end{aligned}
\end{equation}
\tag{4.5}
$$
Here, $C''$ depends only on the domain $\mathcal{O}$ and the lattice $\Gamma$, and $C'$ depends on the same parameters and on $\rho$. $2^\circ$. Under the additional assumption that $a(\mathbf{x})=1$ (that is, $h=p \in L_2(\Omega)$), for $0< \varepsilon \leqslant \varepsilon_1$, we have
$$
\begin{equation}
| \mathfrak t_\varepsilon[\widetilde{\mathbf{u}}, \boldsymbol{\eta}]| \leqslant C''' \varepsilon \| h\|_{L_2(\Omega)} \bigl( \| \widetilde{\mathbf{u}}\|_{H^1(\mathbb{R}^d)} \| \boldsymbol{\eta}\|_{L_2(\mathcal{O})} + \| \widetilde{\mathbf{u}}\|_{L_2(\mathbb{R}^d)} \| \boldsymbol{\eta}\|_{H^1(\mathcal{O})}\bigr).
\end{equation}
\tag{4.6}
$$
The constant $C'''$ depends only on the domain $\mathcal{O}$ and the lattice $\Gamma$. $3^\circ$. Under the additional assumption that $h \in L_\infty$, for $0< \varepsilon \leqslant \varepsilon_1$, we have
$$
\begin{equation*}
\begin{gathered} \, \bigl| \bigl( (h^\varepsilon -\overline{h}) \mathbf{u}, \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})} \bigr| \leqslant \widetilde{C}''' \varepsilon \| h\|_{L_\infty} \bigl( \| {\mathbf{u}}\|_{H^1(\mathcal{O})} \| \boldsymbol{\eta}\|_{L_2(\mathcal{O})} + \| {\mathbf{u}}\|_{L_2(\mathcal{O})} \| \boldsymbol{\eta}\|_{H^1(\mathcal{O})}\bigr) \\ \forall\, \mathbf{u}, \boldsymbol{\eta} \in H^1(\mathcal{O};\mathbb{C}^n). \end{gathered}
\end{equation*}
\notag
$$
The constant $\widetilde{C}'''$ depends only on the domain $\mathcal{O}$ and the lattice $\Gamma$. Proof. $1^\circ$. Let $\Phi(\mathbf{x})$ be a $\Gamma$-periodic (weak) solution of the problem
$$
\begin{equation*}
\Delta \Phi(\mathbf{x})= h(\mathbf{x}) - \overline{h}, \qquad \int_\Omega \Phi(\mathbf{x})\, d\mathbf{x}=0.
\end{equation*}
\notag
$$
Then the $(n\times n)$-matrix-valued function $\Phi \in \widetilde{H}^1(\Omega)$ satisfies the identity
$$
\begin{equation}
(\nabla \Phi {\mathbf C}, \nabla \boldsymbol{\psi})_{L_2(\Omega)} = - (p {\mathbf C}, a^* \boldsymbol{\psi})_{L_2(\Omega)} + (\overline{h} {\mathbf C}, \boldsymbol{\psi} )_{L_2(\Omega)} \quad \forall\, \boldsymbol{\psi} \in \widetilde{H}^1(\Omega;\mathbb{C}^n), \quad {\mathbf C} \in \mathbb{C}^n.
\end{equation}
\tag{4.7}
$$
Substituting $\boldsymbol{\psi} =\Phi {\mathbf C}$ in identity (4.7), using the estimate which follows from the Hölder inequality and embedding theorems, and the Poincaré inequality $\| \Phi \|_{L_2(\Omega)} \leqslant (2 r_0)^{-1} \| \nabla \Phi\|_{L_2(\Omega)}$, we obtain where the constant $\check{C}$ depends only on $\rho$ and the parameters of the lattice $\Gamma$.
We put $\varphi_j(\mathbf{x}):= \partial_j \Phi(\mathbf{x})$, $j=1,\dots,d$. Then $h(\mathbf{x}) - \overline{h} = \sum_{j=1}^d \partial_j \varphi_j(\mathbf{x})$ (this relation is understood in the sense of distributions). Hence $h^\varepsilon - \overline{h} = \varepsilon \sum_{j=1}^d \partial_j \varphi_j^\varepsilon$, and therefore,
$$
\begin{equation}
\mathfrak{t}_\varepsilon[\widetilde{\mathbf{u}}, \boldsymbol{\eta}] = \mathfrak{t}^{(1)}_\varepsilon[\widetilde{\mathbf{u}}, \boldsymbol{\eta}] - \mathfrak{t}_\varepsilon^{(2)}[\widetilde{\mathbf{u}}, \boldsymbol{\eta}],
\end{equation}
\tag{4.9}
$$
$$
\begin{equation}
\mathfrak{t}^{(1)}_\varepsilon[\widetilde{\mathbf{u}}, \boldsymbol{\eta}] = \varepsilon \sum_{j=1}^d \bigl( \partial_j (\varphi_j^\varepsilon S_\varepsilon \widetilde{\mathbf{u}}), \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})},
\end{equation}
\tag{4.10}
$$
$$
\begin{equation}
\mathfrak{t}^{(2)}_\varepsilon[\widetilde{\mathbf{u}}, \boldsymbol{\eta}] = \varepsilon \sum_{j=1}^d \bigl( \varphi_j^\varepsilon S_\varepsilon \partial_j\widetilde{\mathbf{u}}, \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})}.
\end{equation}
\tag{4.11}
$$
The term (4.11) is estimated with the help of Proposition 1.2 and (4.8):
$$
\begin{equation}
\begin{aligned} \, \bigl| \mathfrak{t}^{(2)}_\varepsilon[\widetilde{\mathbf{u}}, \boldsymbol{\eta}] \bigr| &\leqslant \varepsilon |\Omega|^{-1/2} \| \nabla \Phi\|_{L_2(\Omega)} \| \mathbf{D} \widetilde{\mathbf{u}}\|_{L_2(\mathbb{R}^d)} \| \boldsymbol{\eta}\|_{L_2(\mathcal{O})} \nonumber \\ &\leqslant \check{C} \varepsilon |\Omega|^{-1/2} \| p\|_{L_2(\Omega)} \| a\|_{L_\rho(\Omega)} \| \mathbf{D} \widetilde{\mathbf{u}}\|_{L_2(\mathbb{R}^d)} \| \boldsymbol{\eta}\|_{L_2(\mathcal{O})}. \end{aligned}
\end{equation}
\tag{4.12}
$$
Let $\theta_\varepsilon$ be a cut-off function satisfying conditions (4.4). The functional (4.10) can be represented as
$$
\begin{equation}
\mathfrak{t}^{(1)}_\varepsilon[\widetilde{\mathbf{u}}, \boldsymbol{\eta}] = \widetilde{\mathfrak{t}}^{\,(1)}_\varepsilon[\widetilde{\mathbf{u}}, \boldsymbol{\eta}] + \widehat{\mathfrak{t}}^{\,(1)}_\varepsilon[\widetilde{\mathbf{u}}, \boldsymbol{\eta}],
\end{equation}
\tag{4.13}
$$
$$
\begin{equation}
\widetilde{\mathfrak{t}}^{\,(1)}_\varepsilon[\widetilde{\mathbf{u}}, \boldsymbol{\eta}] = \varepsilon \sum_{j=1}^d \bigl( \partial_j ( \theta_\varepsilon \varphi_j^\varepsilon S_\varepsilon \widetilde{\mathbf{u}}), \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})},
\end{equation}
\tag{4.14}
$$
$$
\begin{equation}
\widehat{\mathfrak{t}}^{\,(1)}_\varepsilon[\widetilde{\mathbf{u}}, \boldsymbol{\eta}] = -\varepsilon \sum_{j=1}^d \bigl( (1- \theta_\varepsilon) \varphi_j^\varepsilon S_\varepsilon \widetilde{\mathbf{u}}, \partial_j \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})}.
\end{equation}
\tag{4.15}
$$
The term (4.15) is estimated via (4.4), Proposition 1.2, and (4.8) as follows:
$$
\begin{equation}
\begin{aligned} \, \bigl| \widehat{\mathfrak{t}}^{\,(1)}_\varepsilon[\widetilde{\mathbf{u}}, \boldsymbol{\eta}] \bigr| &\leqslant \varepsilon |\Omega|^{-1/2} \| \nabla \Phi\|_{L_2(\Omega)} \| \widetilde{\mathbf{u}}\|_{L_2(\mathbb{R}^d)} \| \mathbf{D} \boldsymbol{\eta}\|_{L_2(\mathcal{O})} \nonumber \\ &\leqslant \check{C} \varepsilon |\Omega|^{-1/2} \| p\|_{L_2(\Omega)} \| a\|_{L_\rho(\Omega)} \| \widetilde{\mathbf{u}}\|_{L_2(\mathbb{R}^d)} \| \mathbf{D} \boldsymbol{\eta}\|_{L_2(\mathcal{O})}. \end{aligned}
\end{equation}
\tag{4.16}
$$
Next, the term (4.14) can be represented as
$$
\begin{equation}
\begin{aligned} \, \widetilde{\mathfrak{t}}^{\,(1)}_\varepsilon [\widetilde{\mathbf{u}}, \boldsymbol{\eta}] &= \varepsilon \sum_{j=1}^d \bigl( ( \partial_j \theta_\varepsilon) \varphi_j^\varepsilon S_\varepsilon \widetilde{\mathbf{u}}, \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})} + \bigl( \theta_\varepsilon (h^\varepsilon - \overline{h}) S_\varepsilon \widetilde{\mathbf{u}}, \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})} \nonumber \\ &\qquad+ \varepsilon \sum_{j=1}^d (\theta_\varepsilon \varphi_j^\varepsilon S_\varepsilon \partial_j \widetilde{\mathbf{u}}, \boldsymbol{\eta})_{L_2(\mathcal{O})}. \end{aligned}
\end{equation}
\tag{4.17}
$$
Applying Lemmas 4.1, 4.2 and relations (4.4), (4.8), we estimate the first term on the right-hand side of (4.17) as follows:
$$
\begin{equation}
\begin{aligned} \, &\varepsilon \biggl| \sum_{j=1}^d \bigl( ( \partial_j \theta_\varepsilon) \varphi_j^\varepsilon S_\varepsilon \widetilde{\mathbf{u}}, \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})} \biggr| \leqslant \kappa \| (\nabla \Phi)^\varepsilon S_\varepsilon \widetilde{\mathbf{u}}\|_{L_2((\partial \mathcal{O})_\varepsilon)} \| \boldsymbol{\eta}\|_{L_2(\Upsilon_\varepsilon)} \nonumber \\ &\leqslant \check{C}' \varepsilon \|p \|_{L_2(\Omega)} \|a\|_{L_\rho(\Omega)} \| \widetilde{\mathbf{u}}\|^{1/2}_{H^1(\mathbb{R}^d)} \| \widetilde{\mathbf{u}}\|^{1/2}_{L_2(\mathbb{R}^d)} \| \boldsymbol{\eta}\|^{1/2}_{H^1(\mathcal{O})} \| \boldsymbol{\eta}\|^{1/2}_{L_2(\mathcal{O})} \nonumber \\ &\leqslant \frac{1}{2} \check{C}' \varepsilon \|p \|_{L_2(\Omega)} \|a\|_{L_\rho(\Omega)} \bigl( \| \widetilde{\mathbf{u}}\|_{H^1(\mathbb{R}^d)} \| \boldsymbol{\eta}\|_{L_2(\mathcal{O})} {+}{\kern1pt} \| \widetilde{\mathbf{u}}\|_{L_2(\mathbb{R}^d)} \| \boldsymbol{\eta}\|_{H^1(\mathcal{O})} \bigr), \,\quad 0 \,{<}\, \varepsilon \,{\leqslant}\, \varepsilon_1. \end{aligned}
\end{equation}
\tag{4.18}
$$
Here, $\check{C}' = \kappa (\beta \beta_*)^{1/2} |\Omega|^{-1/2} \check{C}$.
The second term on the right-hand side of (4.17) is estimated with the help of (4.4) and Lemmas 4.1 and 4.2. We have
$$
\begin{equation}
\begin{aligned} \, &\bigl| \bigl( \theta_\varepsilon (h^\varepsilon - \overline{h}) S_\varepsilon \widetilde{\mathbf{u}}, \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})} \bigr| \nonumber \\ &\ \leqslant \| p^\varepsilon S_\varepsilon \widetilde{\mathbf{u}}\|_{L_2((\partial \mathcal{O})_\varepsilon)} \| (a^\varepsilon)^* \boldsymbol{\eta}\|_{L_2(\Upsilon_\varepsilon)} + \| \overline{h} S_\varepsilon \widetilde{\mathbf{u}}\|_{L_2((\partial \mathcal{O})_\varepsilon)} \| \boldsymbol{\eta}\|_{L_2(\Upsilon_\varepsilon)} \nonumber \\ &\ \leqslant \beta_*^{1/2} |\Omega|^{-1/2} \varepsilon^{1/2} \|p\|_{L_2(\Omega)} \| \widetilde{\mathbf{u}}\|^{1/2}_{H^1(\mathbb{R}^d)} \| \widetilde{\mathbf{u}}\|^{1/2}_{L_2(\mathbb{R}^d)} \| (a^\varepsilon)^* \boldsymbol{\eta}\|_{L_2(\Upsilon_\varepsilon)} \nonumber \\ &\ \qquad+ (\beta \beta_*)^{1/2} |\Omega|^{-1} \varepsilon \|p\|_{L_2(\Omega)} \| a\|_{L_2(\Omega)} \| \widetilde{\mathbf{u}}\|^{1/2}_{H^1(\mathbb{R}^d)} \| \widetilde{\mathbf{u}}\|^{1/2}_{L_2(\mathbb{R}^d)} \| \boldsymbol{\eta}\|^{1/2}_{H^1(\mathcal{O})} \| \boldsymbol{\eta}\|^{1/2}_{L_2(\mathcal{O})} \nonumber \\ &\ \leqslant \beta_*^{1/2} |\Omega|^{-1/2} \varepsilon^{1/2} \|p\|_{L_2(\Omega)} \| \widetilde{\mathbf{u}}\|^{1/2}_{H^1(\mathbb{R}^d)} \| \widetilde{\mathbf{u}}\|^{1/2}_{L_2(\mathbb{R}^d)} \| (a^\varepsilon)^* \boldsymbol{\eta}\|_{L_2(\Upsilon_\varepsilon)} \nonumber \\ &\ \qquad+ \frac{1}{2}(\beta \beta_*)^{1/2} |\Omega|^{-1} \varepsilon \|p\|_{L_2(\Omega)} \| a\|_{L_2(\Omega)} \nonumber \\ &\ \qquad\qquad\times \bigl( \| \widetilde{\mathbf{u}}\|_{H^1(\mathbb{R}^d)} \| \boldsymbol{\eta}\|_{L_2(\mathcal{O})} +\| \widetilde{\mathbf{u}}\|_{L_2(\mathbb{R}^d)} \| \boldsymbol{\eta}\|_{H^1(\mathcal{O})} \bigr), \qquad 0 < \varepsilon \leqslant \varepsilon_1. \end{aligned}
\end{equation}
\tag{4.19}
$$
Finally, the third term on the right-hand side of (4.17) is estimated similarly to (4.12). We have
$$
\begin{equation}
\varepsilon \biggl| \sum_{j=1}^d (\theta_\varepsilon \varphi_j^\varepsilon S_\varepsilon \partial_j \widetilde{\mathbf{u}}, \boldsymbol{\eta})_{L_2(\mathcal{O})}\biggr| \leqslant \check{C} \varepsilon |\Omega|^{-1/2} \| p\|_{L_2(\Omega)} \| a\|_{L_\rho(\Omega)} \| \mathbf{D} \widetilde{\mathbf{u}}\|_{L_2(\mathbb{R}^d)} \| \boldsymbol{\eta}\|_{L_2(\mathcal{O})}.
\end{equation}
\tag{4.20}
$$
Now the required estimate (4.5) follows from (4.9), (4.12), (4.13), (4.16)–(4.20). $2^\circ$. Under the additional assumption that $a(\mathbf{x})=1$, we take into account that
$$
\begin{equation*}
\| \boldsymbol{\eta} \|_{L_2(\Upsilon_\varepsilon)} \leqslant \beta^{1/2} \varepsilon^{1/2} \| \boldsymbol{\eta} \|^{1/2}_{H^1(\mathcal{O})} \| \boldsymbol{\eta} \|^{1/2}_{L_2(\mathcal{O})},
\end{equation*}
\notag
$$
which follows from Lemma 4.1. Combining this with (4.5) (for $a(\mathbf{x})=1$), we arrive at (4.6).
$3^\circ$. Now, let $h\in L_\infty$. Let $\mathbf{u}, \boldsymbol{\eta} \in H^1(\mathcal{O};\mathbb{C}^n)$. We put $\widetilde{\mathbf{u}} = P_\mathcal{O} \mathbf{u} \in H^1(\mathbb{R}^d;\mathbb{C}^n)$. Obviously,
$$
\begin{equation*}
\bigl( (h^\varepsilon - \overline{h}) \mathbf{u}, \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})} = \mathfrak{t}_\varepsilon[\widetilde{\mathbf{u}}, \boldsymbol{\eta}] + \bigl( (h^\varepsilon - \overline{h}) (I - S_\varepsilon) \widetilde{\mathbf{u}}, \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})}.
\end{equation*}
\notag
$$
By the already proved assertion $2^\circ$, the first term on the right satisfies estimate (4.6). The second term is estimated by Proposition 1.1 as follows:
$$
\begin{equation*}
\bigl| \bigl( (h^\varepsilon - \overline{h}) (I - S_\varepsilon) \widetilde{\mathbf{u}}, \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})} \bigr| \leqslant 2 \|h\|_{L_\infty} r_1 \varepsilon \|\widetilde{\mathbf{u}}\|_{H^1(\mathbb{R}^d)} \| \boldsymbol{\eta}\|_{L_2(\mathcal{O})}.
\end{equation*}
\notag
$$
It remains to take (3.4) into account. This completes the proof. 4.4. The properties of the matrix-valued functions $\Lambda$ and $\widetilde{\Lambda}$The following result was proved in [28], Corollary 2.4. Lemma 4.5. Let the $\Gamma$-periodic solution $\Lambda (\mathbf{x})$ of problem (1.20) be bounded: $\Lambda\in L_\infty$. Then, for any function $u\in H^1(\mathbb{R}^d)$ and $\varepsilon >0$, The constants $\beta_1$ and $\beta_2$ depend on $m$, $d$, $\alpha_0$, $\alpha_1$, $\| g\|_{L_\infty}$, and $\| g^{-1}\|_{L_\infty}$. The following result can be proved by using the Hölder inequality and embedding theorems (cf. [37], Lemma 3.5). Lemma 4.6. Let $f(\mathbf{x})$ be a $\Gamma$-periodic function in $\mathbb{R}^d$ such that Then, for $0<\varepsilon\leqslant 1$ the operator $[f^\varepsilon ]$ is continuous from $H^1(\mathbb{R}^d)$ to $L_2(\mathbb{R}^d)$, and where $C(\widehat{q};\Omega)$ is the norm of the embedding operator $H^1(\Omega)\hookrightarrow L_{\widehat{q}}(\Omega)$. Here, $\widehat{q}=\infty$ for $d=1$ and $\widehat{q}=2p(p-2)^{-1}$ for $d\geqslant 2$. The following result was obtained in [37], Corollary 3.6. Lemma 4.7. Suppose that the $\Gamma$-periodic solution $\widetilde{\Lambda}(\mathbf{x})$ of problem (1.30) satisfies condition (4.21). Then, for any $u\in H^2(\mathbb{R}^d)$ and $0<\varepsilon\leqslant 1$, § 5. Estimate for the correction term $\mathbf{w}_\varepsilon$ in $H^1(\mathcal{O})$ for $\operatorname{Re} \zeta \leqslant 0$5.1. The case $\operatorname{Re} \zeta \leqslant 0$. The estimate for the functional $\mathcal{I}_\varepsilon[\boldsymbol{\eta}]$Using (2.25) and (2.39), we represent the functional (3.30) as
$$
\begin{equation}
\mathcal{I}_\varepsilon[\boldsymbol{\eta}] = \sum_{k=1}^5 \mathcal{I}^{(k)}_\varepsilon [\boldsymbol{\eta}], \qquad \boldsymbol{\eta}\in H^1(\mathcal{O};\mathbb{C}^n),
\end{equation}
\tag{5.1}
$$
where
$$
\begin{equation}
\begin{aligned} \, \mathcal{I}^{(1)}_\varepsilon [\boldsymbol{\eta}] &= \bigl( \widetilde{g}^{\,\varepsilon} S_\varepsilon b(\mathbf{D})\widetilde{\mathbf{u}}_0, b(\mathbf{D})\boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})} - \bigl( {g}^0 b(\mathbf{D}) {\mathbf{u}}_0, b(\mathbf{D})\boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})}, \end{aligned}
\end{equation}
\tag{5.2}
$$
$$
\begin{equation}
\begin{aligned} \, \mathcal{I}^{(2)}_\varepsilon [\boldsymbol{\eta}] &= \bigl( g^\varepsilon (b(\mathbf{D}) \widetilde{\Lambda})^\varepsilon S_\varepsilon \widetilde{\mathbf{u}}_0, b(\mathbf{D})\boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})} + \sum_{ l=1}^d \bigl( (a_l^\varepsilon)^* {\mathbf{v}}_\varepsilon, D_l \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})} \nonumber \\ &\qquad- \sum_{l=1}^d \bigl( \overline{a_l^*} {\mathbf{u}}_0, D_l \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})} + \bigl(V \mathbf{u}_0, b(\mathbf{D})\boldsymbol{\eta}\bigr)_{L_2(\mathcal{O})}, \end{aligned}
\end{equation}
\tag{5.3}
$$
$$
\begin{equation}
\begin{aligned} \, \mathcal{I}^{(3)}_\varepsilon [\boldsymbol{\eta}] &= \sum_{l=1}^d \bigl( D_l {\mathbf{v}}_\varepsilon, (a_l^\varepsilon)^* \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})} - \sum_{l=1}^d (\overline{a_l} D_l {\mathbf{u}}_0, \boldsymbol{\eta})_{L_2(\mathcal{O})} \nonumber \\ &\qquad + \bigl( b(\mathbf{D}) \mathbf{u}_0, V \boldsymbol{\eta}\bigr)_{L_2(\mathcal{O})} + (W \mathbf{u}_0,\boldsymbol{\eta})_{L_2(\mathcal{O})}, \end{aligned}
\end{equation}
\tag{5.4}
$$
$$
\begin{equation}
\mathcal{I}^{(4)}_\varepsilon [\boldsymbol{\eta}] = \bigl(Q^\varepsilon {\mathbf{v}}_\varepsilon - \overline{Q} {\mathbf{u}}_0, \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})},
\end{equation}
\tag{5.5}
$$
$$
\begin{equation}
\mathcal{I}^{(5)}_\varepsilon [\boldsymbol{\eta}] = (\lambda - \zeta) \bigl((Q_0^\varepsilon - \overline{Q_0}\,) {\mathbf{u}}_0, \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})}.
\end{equation}
\tag{5.6}
$$
Note that for $\operatorname{Re} \zeta \leqslant 0$ we have $c(\phi)=1$. The following assertion can be checked similarly to Lemma 11.1 in [6]. Lemma 5.1. Let $\operatorname{Re} \zeta \leqslant 0$ and $|\zeta| \geqslant 1$. Suppose that the number $\varepsilon_1$ is chosen according to Condition 3.1. For $0< \varepsilon \leqslant \varepsilon_1$ the functional (5.2) satisfies Proof. We represent the functional (5.2) as
$$
\begin{equation}
\mathcal{I}^{(1)}_\varepsilon [\boldsymbol{\eta}] \,= \widetilde{\mathcal{I}}^{(1)}_\varepsilon [\boldsymbol{\eta}]+ \widehat{\mathcal{I}}^{(1)}_\varepsilon [\boldsymbol{\eta}],
\end{equation}
\tag{5.8}
$$
The term (5.9) is estimated with the help of Proposition 1.1 and relations (1.3), (1.5), (1.6), and (3.20) as follows:
$$
\begin{equation}
\bigl| \widetilde{\mathcal{I}}^{(1)}_\varepsilon [\boldsymbol{\eta}] \bigr| \leqslant \|g\|_{L_\infty} r_1 \varepsilon \| \mathbf{D} b(\mathbf{D})\widetilde{\mathbf{u}}_0\|_{L_2(\mathbb{R}^d)} \| b(\mathbf{D})\boldsymbol{\eta} \|_{L_2(\mathcal{O})} \leqslant \gamma_1^{(1)} \varepsilon \| \mathbf{F} \|_{L_2(\mathcal{O})} \| \mathbf{D} \boldsymbol{\eta} \|_{L_2(\mathcal{O})},
\end{equation}
\tag{5.11}
$$
where $\gamma_1^{(1)} = \|g\|_{L_\infty} r_1 \alpha_1 d^{1/2}C_9$.
Using (1.3), we represent the term (5.10) as
$$
\begin{equation}
\widehat{\mathcal{I}}^{(1)}_\varepsilon [\boldsymbol{\eta}] = \sum_{l=1}^d \bigl( f_l^\varepsilon S_\varepsilon b(\mathbf{D})\widetilde{\mathbf{u}}_0, D_l \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})},
\end{equation}
\tag{5.12}
$$
where $f_l(\mathbf{x}):= b_l^* (\widetilde{g}(\mathbf{x}) - g^0)$, $l=1,\dots, d$. By (1.6) and (1.21)–(1.23), we have
$$
\begin{equation}
\| f_l \|_{L_2(\Omega)}\leqslant \alpha_1^{1/2} \| g (b(\mathbf{D}) \Lambda + \mathbf{1}) \|_{L_2(\Omega)} \leqslant |\Omega|^{1/2} \mathfrak{C}, \qquad l=1,\dots,d,
\end{equation}
\tag{5.13}
$$
where $\mathfrak{C}:= 2 \alpha_1^{1/2} \| g\|_{L_\infty}$. By (1.20)–(1.22), the functions $f_l$, $l=1,\dots,d,$ satisfy the assumptions of Lemma 4.3, whence there exist $\Gamma$-periodic matrix-valued functions $M_{lj}(\mathbf{x})$ in $\mathbb{R}^d$, $l,j=1,\dots,d$, satisfying conditions (4.1)–(4.3). From (4.3) and (5.13) it follows that
$$
\begin{equation}
\| M_{lj} \|_{L_2(\Omega)}\leqslant r_0^{-1} |\Omega|^{1/2} \mathfrak{C}, \qquad l,j=1,\dots,d.
\end{equation}
\tag{5.14}
$$
By (4.2), we have $f_l^\varepsilon(\mathbf{x}) = \varepsilon \sum_{j=1}^d \partial_j M_{lj}^\varepsilon(\mathbf{x})$, $l=1,\dots,d$. Together with (5.12), this yields
$$
\begin{equation}
\widehat{\mathcal{I}}^{(1)}_\varepsilon [\boldsymbol{\eta}] \,= \mathfrak{A}_\varepsilon'[\boldsymbol{\eta}] + \mathfrak{A}_\varepsilon''[\boldsymbol{\eta}],
\end{equation}
\tag{5.15}
$$
$$
\begin{equation}
\mathfrak{A}_\varepsilon'[\boldsymbol{\eta}] := \varepsilon \sum_{l,j=1}^d \bigl( \partial_j (M_{lj}^\varepsilon S_\varepsilon b(\mathbf{D})\widetilde{\mathbf{u}}_0), D_l \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})},
\end{equation}
\tag{5.16}
$$
$$
\begin{equation}
\mathfrak{A}_\varepsilon''[\boldsymbol{\eta}] := - \varepsilon \sum_{l,j=1}^d \bigl( M_{lj}^\varepsilon S_\varepsilon b(\mathbf{D}) \partial_j \widetilde{\mathbf{u}}_0, D_l \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})}.
\end{equation}
\tag{5.17}
$$
Using Proposition 1.2 and relations (1.5), (3.20), (5.14), we estimate the term (5.17):
$$
\begin{equation}
\begin{aligned} \, |\mathfrak{A}_\varepsilon''[\boldsymbol{\eta}]| &\leqslant \varepsilon \sum_{l,j=1}^d |\Omega|^{-1/2} \| M_{lj} \|_{L_2(\Omega)} \| b(\mathbf{D}) \partial_j \widetilde{\mathbf{u}}_0 \|_{L_2(\mathbb{R}^d)} \| D_l \boldsymbol{\eta} \|_{L_2(\mathcal{O})} \nonumber \\ &\leqslant \gamma_1^{(2)} \varepsilon \| \mathbf{F} \|_{L_2(\mathcal{O})} \| \mathbf{D} \boldsymbol{\eta} \|_{L_2(\mathcal{O})}, \end{aligned}
\end{equation}
\tag{5.18}
$$
where $\gamma_1^{(2)} = d \alpha_1^{1/2} r_0^{-1} \mathfrak{C} C_9$.
Now, consider the term (5.16). Let $\theta_\varepsilon$ be a cut-off function in $\mathbb{R}^d$ satisfying conditions (4.4). We have
$$
\begin{equation*}
\sum_{j,l=1}^d \bigl( \partial_j ( (1-\theta_\varepsilon) M_{lj}^\varepsilon S_\varepsilon b(\mathbf{D}) \widetilde{\mathbf{u}}_0), D_l \boldsymbol{\eta}\bigr)_{L_2(\mathcal{O})} =0, \qquad \boldsymbol{\eta} \in H^1(\mathcal{O};\mathbb{C}^n),
\end{equation*}
\notag
$$
which can be checked by integration by parts in view of $M_{lj} = - M_{jl}$ (here, we first assume that $\boldsymbol{\eta} \in H^2(\mathcal{O};\mathbb{C}^n)$, and then we close the result by continuity). Therefore,
$$
\begin{equation*}
\mathfrak{A}_\varepsilon' [\boldsymbol{\eta}] = \varepsilon \sum_{j,l=1}^d \bigl( \partial_j ( \theta_\varepsilon M_{lj}^\varepsilon S_\varepsilon b(\mathbf{D})\widetilde{\mathbf{u}}_0), D_l \boldsymbol{\eta}\bigr)_{L_2(\mathcal{O})} = \sum_{l=1}^d \bigl( \boldsymbol{\psi}_l(\varepsilon), D_l \boldsymbol{\eta}\bigr)_{L_2(\mathcal{O})},
\end{equation*}
\notag
$$
where $\boldsymbol{\psi}_l(\varepsilon) := \varepsilon \sum_{j =1}^d \partial_j (\theta_\varepsilon M_{lj}^\varepsilon S_\varepsilon b(\mathbf{D}) \widetilde{\mathbf{u}}_0)$, $l=1,\dots,d$. We have
$$
\begin{equation*}
\boldsymbol{\psi}_l(\varepsilon) = \varepsilon \theta_\varepsilon \sum_{j =1}^d M_{lj}^\varepsilon S_\varepsilon b(\mathbf{D}) \partial_j \widetilde{\mathbf{u}}_0 + \varepsilon \sum_{j =1}^d (\partial_j \theta_\varepsilon) M_{lj}^\varepsilon S_\varepsilon b(\mathbf{D}) \widetilde{\mathbf{u}}_0 + \theta_\varepsilon f_l^\varepsilon S_\varepsilon b(\mathbf{D})\widetilde{\mathbf{u}}_0.
\end{equation*}
\notag
$$
Let $\boldsymbol{\psi}_l^{(1)}(\varepsilon)$, $\boldsymbol{\psi}_l^{(2)}(\varepsilon)$, $\boldsymbol{\psi}_l^{(3)}(\varepsilon)$ the consecutive terms on the right-hand side. By (4.4), we have
$$
\begin{equation}
\begin{aligned} \, |\mathfrak{A}_\varepsilon' [\boldsymbol{\eta}]| &\leqslant \sum_{l=1}^d \bigl\| \boldsymbol{\psi}_l^{(1)}(\varepsilon) \bigr\|_{L_2(\mathbb{R}^d)} \| D_l \boldsymbol{\eta} \|_{L_2(\mathcal{O})} \nonumber \\ &\qquad + \sum_{l=1}^d \bigl( \bigl\| \boldsymbol{\psi}_l^{(2)}(\varepsilon)\bigr\|_{L_2(\mathbb{R}^d)}+ \bigl\| \boldsymbol{\psi}_l^{(3)}(\varepsilon) \bigr\|_{L_2(\mathbb{R}^d)}\bigr) \| D_l \boldsymbol{\eta} \|_{L_2(\Upsilon_\varepsilon)}. \end{aligned}
\end{equation}
\tag{5.19}
$$
To estimate $\boldsymbol{\psi}_l^{(1)}(\varepsilon)$, we apply Proposition 1.2 and relations (1.5), (3.20), (4.4), (5.14). We have
$$
\begin{equation}
\begin{aligned} \, \bigl\| \boldsymbol{\psi}_l^{(1)}(\varepsilon) \bigr\|_{L_2(\mathbb{R}^d)} &\leqslant \varepsilon \sum_{j=1}^d \| M_{lj}^\varepsilon S_\varepsilon b(\mathbf{D}) \partial_j \widetilde{\mathbf{u}}_0 \|_{L_2(\mathbb{R}^d)} \leqslant ( d \alpha_1)^{1/2} r_0^{-1} \mathfrak{C} \varepsilon \| \widetilde{\mathbf{u}}_0 \|_{H^2(\mathbb{R}^d)} \nonumber \\ &\leqslant C_{14} \varepsilon \| \mathbf{F} \|_{L_2(\mathcal{O})}, \qquad l=1,\dots,d, \end{aligned}
\end{equation}
\tag{5.20}
$$
where $C_{14} = (d \alpha_1)^{1/2} r_0^{-1} \mathfrak{C} C_9$. From (4.4), (5.14), and Lemma 4.2 it follows that, for $0< \varepsilon \leqslant \varepsilon_1$,
$$
\begin{equation*}
\begin{aligned} \, &\bigl\| \boldsymbol{\psi}_l^{(2)}(\varepsilon) \bigr\|_{L_2(\mathbb{R}^d)} \leqslant \kappa \biggl( \sum_{j=1}^d \| M_{lj}^\varepsilon S_\varepsilon b(\mathbf{D}) \widetilde{\mathbf{u}}_0 \|^2_{L_2((\partial \mathcal{O})_\varepsilon)} \biggr)^{1/2} \\ &\qquad\leqslant \varepsilon^{1/2} \kappa (d\beta_*)^{1/2} r_0^{-1}\mathfrak{C} \| b(\mathbf{D})\widetilde{\mathbf{u}}_0 \|^{1/2}_{H^1(\mathbb{R}^d)} \| b(\mathbf{D})\widetilde{\mathbf{u}}_0 \|^{1/2}_{L_2(\mathbb{R}^d)}, \qquad l=1,\dots,d. \end{aligned}
\end{equation*}
\notag
$$
Combining this with (1.5), (3.19), and (3.20), we obtain
$$
\begin{equation}
\bigl\| \boldsymbol{\psi}_l^{(2)}(\varepsilon) \bigr\|_{L_2(\mathbb{R}^d)} \leqslant C_{15} \varepsilon^{1/2} |\zeta|^{-1/4} \| \mathbf{F} \|_{L_2(\mathcal{O})}, \qquad 0< \varepsilon \leqslant \varepsilon_1,\quad l=1,\dots,d,
\end{equation}
\tag{5.21}
$$
where $C_{15} = \kappa (d \beta_* \alpha_1)^{1/2} r_0^{-1} \mathfrak{C} (C_8 C_9)^{1/2}$. By Lemma 4.2 and relations (1.5), (3.19), (3.20), (4.4), and (5.13), we arrive at the estimate
$$
\begin{equation}
\begin{aligned} \, \bigl\| \boldsymbol{\psi}_l^{(3)}(\varepsilon) \bigr\|_{L_2(\mathbb{R}^d)} &\leqslant \| f_l^\varepsilon S_\varepsilon b(\mathbf{D}) \widetilde{\mathbf{u}}_0 \|_{L_2((\partial \mathcal{O})_\varepsilon)} \nonumber \\ &\leqslant \varepsilon^{1/2} \beta_*^{1/2} \mathfrak{C} \| b(\mathbf{D})\widetilde{\mathbf{u}}_0 \|^{1/2}_{H^1(\mathbb{R}^d)} \| b(\mathbf{D})\widetilde{\mathbf{u}}_0 \|^{1/2}_{L_2(\mathbb{R}^d)} \nonumber \\ &\leqslant C_{16} \varepsilon^{1/2} |\zeta|^{-1/4} \| \mathbf{F} \|_{L_2(\mathcal{O})}, \qquad 0< \varepsilon \leqslant \varepsilon_1,\quad l=1,\dots,d, \end{aligned}
\end{equation}
\tag{5.22}
$$
where $C_{16} = (\beta_* \alpha_1)^{1/2} \mathfrak{C} (C_8 C_9)^{1/2}$.
As a result, from (5.19)–(5.22) we have
$$
\begin{equation*}
\begin{aligned} \, |\mathfrak{A}_\varepsilon' [\boldsymbol{\eta}]| &\leqslant \gamma_1^{(3)} \varepsilon \| \mathbf{F} \|_{L_2(\mathcal{O})} \| \mathbf{D} \boldsymbol{\eta} \|_{L_2(\mathcal{O})} \\ &\qquad+ \gamma_2 \varepsilon^{1/2} |\zeta|^{-1/4} \| \mathbf{F} \|_{L_2(\mathcal{O})} \| \mathbf{D} \boldsymbol{\eta} \|_{L_2(\Upsilon_\varepsilon)}, \qquad 0 < \varepsilon \leqslant \varepsilon_1, \end{aligned}
\end{equation*}
\notag
$$
where $\gamma_1^{(3)} = C_{14} d^{1/2}$ and $\gamma_2 = (C_{15}+ C_{16}) d^{1/2}$. Together with (5.8), (5.11), (5.15), and (5.18), this yields the required estimate (5.7) with the constant $\gamma_1 = \gamma_1^{(1)} + \gamma_1^{(2)} + \gamma_1^{(3)}$. This completes the proof. Lemma 5.2. Let $\operatorname{Re} \zeta \leqslant 0$ and $|\zeta| \geqslant 1$, and let the number $\varepsilon_1$ satisfy Condition 3.1. Then, for $0< \varepsilon \leqslant \varepsilon_1$, the functional (5.3) satisfies the estimate Proof. Using (3.11) and (3.12), we represent the functional (5.3) as
$$
\begin{equation}
\mathcal{I}^{(2)}_\varepsilon [\boldsymbol{\eta}] = \mathcal{T}_\varepsilon^{(1)}[\boldsymbol{\eta}] + \mathcal{T}_\varepsilon^{(2)}[\boldsymbol{\eta}] + \mathcal{T}_\varepsilon^{(3)}[\boldsymbol{\eta}],
\end{equation}
\tag{5.24}
$$
where
$$
\begin{equation}
\begin{aligned} \, \mathcal{T}_\varepsilon^{(1)}[\boldsymbol{\eta}] &:= \varepsilon \sum_{l=1}^d \bigl( (a_l^\varepsilon)^* ( \Lambda^\varepsilon S_\varepsilon b(\mathbf{D}) \widetilde{\mathbf{u}}_0 + \widetilde{\Lambda}^\varepsilon S_\varepsilon \widetilde{\mathbf{u}}_0), D_l \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})}, \end{aligned}
\end{equation}
\tag{5.25}
$$
$$
\begin{equation}
\begin{aligned} \, \mathcal{T}_\varepsilon^{(2)}[\boldsymbol{\eta}] &:= \sum_{l=1}^d \bigl( (a_l^\varepsilon - \overline{a_l})^* (I- S_\varepsilon) \widetilde{\mathbf{u}}_0, D_l \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})} + \bigl( V (I- S_\varepsilon) \widetilde{\mathbf{u}}_0, b(\mathbf{D}) \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})}, \end{aligned}
\end{equation}
\tag{5.26}
$$
$$
\begin{equation}
\begin{aligned} \, \mathcal{T}_\varepsilon^{(3)}[\boldsymbol{\eta}] &:= \bigl( g^\varepsilon ( b(\mathbf{D}) \widetilde{\Lambda} )^\varepsilon S_\varepsilon \widetilde{\mathbf{u}}_0, b(\mathbf{D}) \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})} \nonumber \\ &\qquad+ \sum_{l=1}^d \bigl( (a_l^\varepsilon - \overline{a_l})^* S_\varepsilon \widetilde{\mathbf{u}}_0, D_l \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})} + \bigl( V S_\varepsilon \widetilde{\mathbf{u}}_0, b(\mathbf{D}) \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})}. \end{aligned}
\end{equation}
\tag{5.27}
$$
We first estimate the term (5.25). By the Hölder inequality and since the embedding $H^1(\Omega)\hookrightarrow L_q(\Omega)$ is continuous (with the embedding constant $C(q;\Omega)$), we have
$$
\begin{equation}
\begin{aligned} \, \| a_l^* \Lambda \|_{L_2(\Omega)} &\leqslant C(q;\Omega) \| a_l \|_{L_\rho(\Omega)} \| \Lambda \|_{H^1(\Omega)} \nonumber \\ &\leqslant C(q;\Omega) \| a_l \|_{L_\rho(\Omega)} (M_1 + M_2) |\Omega|^{1/2} =: C_{17,l} |\Omega|^{1/2}, \end{aligned}
\end{equation}
\tag{5.28}
$$
where we have used estimates (1.24) and (1.25). Similarly, taking (1.32) and (1.33) into account, we obtain
$$
\begin{equation}
\| a_l^* \widetilde{\Lambda} \|_{L_2(\Omega)} \leqslant C(q;\Omega) \| a_l \|_{L_\rho(\Omega)} (\widetilde{M}_1 + \widetilde{M}_2) |\Omega|^{1/2} =: C_{18,l} |\Omega|^{1/2}.
\end{equation}
\tag{5.29}
$$
The functional (5.25) is estimated via Proposition 1.2 and inequalities (5.28), (5.29) as follows:
$$
\begin{equation*}
\bigl| \mathcal{T}_\varepsilon^{(1)}[\boldsymbol{\eta}] \bigr| \leqslant \varepsilon \sum_{l=1}^d \bigl( C_{17,l} \| b(\mathbf{D}) \widetilde{\mathbf{u}}_0\|_{L_2(\mathbb{R}^d)} + C_{18,l} \| \widetilde{\mathbf{u}}_0\|_{L_2(\mathbb{R}^d)} \bigr) \| D_l \boldsymbol{\eta} \|_{L_2(\mathcal{O})}.
\end{equation*}
\notag
$$
Combining this with (1.5), (3.18), (3.19) and taking into account that $|\zeta| \geqslant 1$, we obtain the estimate
$$
\begin{equation}
\bigl| \mathcal{T}_\varepsilon^{(1)}[\boldsymbol{\eta}] \bigr| \leqslant \gamma_3^{(1)} \varepsilon \| \mathbf{F} \|_{L_2(\mathcal{O})} \| \mathbf{D} \boldsymbol{\eta} \|_{L_2(\mathcal{O})},
\end{equation}
\tag{5.30}
$$
where $\gamma_3^{(1)} = \bigl(\sum_{l=1}^d ( C_{17,l} C_8 \alpha_1^{1/2} + C_{18,l} C_7)^2\bigr)^{1/2}$.
Now, we consider the term (5.26). Similarly to (3.31),
$$
\begin{equation}
\begin{aligned} \, &\biggl( \sum_{l=1}^d \| (a_l^\varepsilon - \overline{a_l})^* (I- S_\varepsilon) \widetilde{\mathbf{u}}_0 \|^2_{L_2(\mathcal{O})} \biggr)^{1/2} \leqslant 2 C_{11}\| (I- S_\varepsilon) \widetilde{\mathbf{u}}_0\|_{H^1(\mathbb{R}^d)} \nonumber \\ &\qquad\leqslant 2 C_{11} r_1 \varepsilon \|\widetilde{\mathbf{u}}_0\|_{H^2(\mathbb{R}^d)} \leqslant 2 C_9 C_{11} r_1 \varepsilon \| \mathbf{F} \|_{L_2(\mathcal{O})}, \end{aligned}
\end{equation}
\tag{5.31}
$$
where we have applied Proposition 1.1 and (3.20). Next, from Proposition 1.1, estimates (1.36), (3.19), and the restriction $|\zeta| \geqslant 1$ it follows that
$$
\begin{equation}
\| V (I- S_\varepsilon) \widetilde{\mathbf{u}}_0\|_{L_2(\mathbb{R}^d)} \leqslant C_{V} r_1 \varepsilon \|\widetilde{\mathbf{u}}_0\|_{H^1(\mathbb{R}^d)} \leqslant C_8 C_{V} r_1 \varepsilon \| \mathbf{F} \|_{L_2(\mathcal{O})}.
\end{equation}
\tag{5.32}
$$
From (1.3), (1.6), (5.26), (5.31), and (5.32) we have
$$
\begin{equation}
\bigl| \mathcal{T}_\varepsilon^{(2)}[\boldsymbol{\eta}] \bigr| \leqslant \gamma_3^{(2)} \varepsilon \| \mathbf{F} \|_{L_2(\mathcal{O})} \| \mathbf{D} \boldsymbol{\eta} \|_{L_2(\mathcal{O})},
\end{equation}
\tag{5.33}
$$
where $\gamma_3^{(2)} = 2 C_9 C_{11}r_1 + C_8 C_V r_1 (d \alpha_1)^{1/2}$.
Using (1.3), we represent the functional (5.27) as
$$
\begin{equation}
\mathcal{T}_\varepsilon^{(3)}[\boldsymbol{\eta}] = \sum_{l=1}^d \bigl( \check{f}_l^\varepsilon S_\varepsilon \widetilde{\mathbf{u}}_0, D_l \boldsymbol{\eta}\bigr)_{L_2(\mathcal{O})},
\end{equation}
\tag{5.34}
$$
where
$$
\begin{equation}
\check{f}_l(\mathbf{x}) := b_l^* g(\mathbf{x}) b(\mathbf{D}) \widetilde{\Lambda}(\mathbf{x}) + a^*_l(\mathbf{x}) - \overline{a_l^*} + b_l^* V, \qquad l=1,\dots,d.
\end{equation}
\tag{5.35}
$$
Obviously, $\check{f}_l(\mathbf{x})$ are $\Gamma$-periodic $(n \times n)$-matrix-valued functions such that $\check{f}_l \in L_2(\Omega)$ and, by equation (1.30), we have $\sum_{l=1}^d D_l \check{f}_l(\mathbf{x})=0$. Note that relations (1.20) and (1.34) imply that $V = - \overline{g (b(\mathbf{D}) \widetilde{\Lambda})}$. Therefore, $\int_\Omega \check{f}_l(\mathbf{x})\,d\mathbf{x} =0$, $l=1,\dots,d$. Thus, the assumptions of Lemma 4.3 are satisfied. By this lemma, there exist $\Gamma$-periodic $(n \times n)$-matrix-valued functions $\check{M}_{lj}(\mathbf{x})$, $l,j=1,\dots,d$, such that $\check{M}_{lj} \in \widetilde{H}^1(\Omega)$, $\int_\Omega \check{M}_{lj}(\mathbf{x})\,d\mathbf{x}=0$,
$$
\begin{equation}
\begin{gathered} \, \check{M}_{lj}(\mathbf{x}) = - \check{M}_{jl}(\mathbf{x}),\qquad l,j=1,\dots,d, \\ \check{f}_l(\mathbf{x}) = \sum_{j=1}^d \partial_j \check{M}_{lj}(\mathbf{x}), \qquad l=1,\dots,d, \\ \| \check{M}_{lj} \|_{L_2(\Omega)} \leqslant (2r_0)^{-1} \bigl( \| \check{f}_l \|_{L_2(\Omega)} + \| \check{f}_j \|_{L_2(\Omega)}\bigr). \end{gathered}
\end{equation}
\tag{5.36}
$$
Hence $\check{f}_l^\varepsilon(\mathbf{x}) = \varepsilon \sum_{j=1}^d \partial_j \check{M}^\varepsilon_{lj}(\mathbf{x})$, $l=1,\dots,d$. Together with (5.34), this yields
$$
\begin{equation}
\mathcal{T}_\varepsilon^{(3)}[\boldsymbol{\eta}] \,= \widetilde{\mathcal{T}}_\varepsilon^{(3)}[\boldsymbol{\eta}] + \widehat{\mathcal{T}}_\varepsilon^{(3)}[\boldsymbol{\eta}],
\end{equation}
\tag{5.37}
$$
$$
\begin{equation}
\widetilde{\mathcal{T}}_\varepsilon^{(3)}[\boldsymbol{\eta}] := \varepsilon \sum_{j,l=1}^d \bigl( \partial_j ( \check{M}_{lj}^\varepsilon S_\varepsilon \widetilde{\mathbf{u}}_0), D_l \boldsymbol{\eta}\bigr)_{L_2(\mathcal{O})},
\end{equation}
\tag{5.38}
$$
$$
\begin{equation}
\widehat{\mathcal{T}}_\varepsilon^{(3)}[\boldsymbol{\eta}] := - \varepsilon \sum_{j,l=1}^d \bigl( \check{M}_{lj}^\varepsilon S_\varepsilon \partial_j \widetilde{\mathbf{u}}_0, D_l \boldsymbol{\eta}\bigr)_{L_2(\mathcal{O})}.
\end{equation}
\tag{5.39}
$$
$$
\begin{equation}
\| \check{f}_l \|_{L_2(\Omega)} \leqslant \alpha_1^{1/2} \|g\|^{1/2}_{L_\infty} \| g^{1/2}b(\mathbf{D}) \widetilde{\Lambda} \|_{L_2(\Omega)} + \| a_l \|_{L_2(\Omega)} \leqslant C_{19,l}, \qquad l=1,\dots,d,
\end{equation}
\tag{5.40}
$$
where $C_{19,l} = \alpha_0^{-1/2} \alpha_1^{1/2} \|g\|^{1/2}_{L_\infty} \|g^{-1}\|^{1/2}_{L_\infty} C_a + \|a_l\|_{L_2(\Omega)}$. Combining this with (5.36), we see that
$$
\begin{equation}
\| \check{M}_{lj} \|_{L_2(\Omega)} \leqslant (2r_0)^{-1} (C_{19,l} + C_{19,j}), \qquad l,j=1,\dots,d.
\end{equation}
\tag{5.41}
$$
The term (5.39) is estimated by using Proposition 1.2, relations (3.19), (5.41), and the restriction $|\zeta| \geqslant 1$:
$$
\begin{equation}
\bigl| \widehat{\mathcal{T}}_\varepsilon^{(3)}[\boldsymbol{\eta}] \bigr| \leqslant \gamma_3^{(3)} \varepsilon \| \mathbf{F} \|_{L_2(\mathcal{O})} \| \mathbf{D} \boldsymbol{\eta}\|_{L_2(\mathcal{O})},
\end{equation}
\tag{5.42}
$$
where $\gamma_3^{(3)} = d^{1/2} r_0^{-1} |\Omega|^{-1/2} C_8 \bigl(\sum_{l=1}^d C^2_{19,l}\bigr)^{1/2}$.
It remains to consider the term (5.38). Let $\theta_\varepsilon (\mathbf{x})$ be a cut-off function in $\mathbb{R}^d$ satisfying conditions (4.4). We have
$$
\begin{equation*}
\sum_{j,l=1}^d \bigl( \partial_j ( (1-\theta_\varepsilon) \check{M}_{lj}^\varepsilon S_\varepsilon \widetilde{\mathbf{u}}_0), D_l \boldsymbol{\eta}\bigr)_{L_2(\mathcal{O})} =0, \qquad \boldsymbol{\eta} \in H^1(\mathcal{O};\mathbb{C}^n),
\end{equation*}
\notag
$$
which can be checked by integration by parts with the help of the equalities $\check{M}_{lj} = - \check{M}_{jl}$ (for a proof, we first assume that $\boldsymbol{\eta} \in H^2(\mathcal{O};\mathbb{C}^n)$, and then we close the result by continuity). Hence
$$
\begin{equation*}
\widetilde{\mathcal{T}}_\varepsilon^{(3)}[\boldsymbol{\eta}] = \varepsilon \sum_{j,l=1}^d \bigl( \partial_j ( \theta_\varepsilon \check{M}_{lj}^\varepsilon S_\varepsilon \widetilde{\mathbf{u}}_0), D_l \boldsymbol{\eta}\bigr)_{L_2(\mathcal{O})} = \sum_{l=1}^d \bigl( \check{\boldsymbol{\psi}}_l(\varepsilon), D_l \boldsymbol{\eta}\bigr)_{L_2(\mathcal{O})},
\end{equation*}
\notag
$$
where $\check{\boldsymbol{\psi}}_l(\varepsilon) := \varepsilon \sum_{j =1}^d \partial_j (\theta_\varepsilon \check{M}_{lj}^\varepsilon S_\varepsilon \widetilde{\mathbf{u}}_0)$, $l=1,\dots,d$. We have
$$
\begin{equation*}
\check{\boldsymbol{\psi}}_l(\varepsilon) = \varepsilon \theta_\varepsilon \sum_{j =1}^d \check{M}_{lj}^\varepsilon S_\varepsilon \partial_j \widetilde{\mathbf{u}}_0 + \varepsilon \sum_{j =1}^d (\partial_j \theta_\varepsilon) \check{M}_{lj}^\varepsilon S_\varepsilon \widetilde{\mathbf{u}}_0 + \theta_\varepsilon \check{f}_l^\varepsilon S_\varepsilon \widetilde{\mathbf{u}}_0.
\end{equation*}
\notag
$$
Let $\check{\boldsymbol{\psi}}_l^{(1)}(\varepsilon)$, $\check{\boldsymbol{\psi}}_l^{(2)}(\varepsilon)$, $\check{\boldsymbol{\psi}}_l^{(3)}(\varepsilon)$ be the consecutive summands on the right-hand side. From (4.4) we have
$$
\begin{equation}
\begin{aligned} \, \bigl| \widetilde{\mathcal{T}}_\varepsilon^{(3)}[\boldsymbol{\eta}]\bigr| &\leqslant \sum_{l=1}^d \bigl\| \check{\boldsymbol{\psi}}_l^{(1)}(\varepsilon) \bigr\|_{L_2(\mathbb{R}^d)} \| D_l \boldsymbol{\eta} \|_{L_2(\mathcal{O})} \nonumber \\ &\qquad+\sum_{l=1}^d \bigl( \bigl\| \check{\boldsymbol{\psi}}_l^{(2)}(\varepsilon) \bigr\|_{L_2(\mathbb{R}^d)}+ \bigl\| \check{\boldsymbol{\psi}}_l^{(3)}(\varepsilon) \bigr\|_{L_2(\mathbb{R}^d)}\bigr) \| D_l \boldsymbol{\eta} \|_{L_2(\Upsilon_\varepsilon)}. \end{aligned}
\end{equation}
\tag{5.43}
$$
To estimate $\check{\boldsymbol{\psi}}_l^{(1)}(\varepsilon)$, we apply Proposition 1.2, relations (3.19), (5.41), and take into account the restriction $|\zeta| \geqslant 1$. As a result, we have
$$
\begin{equation}
\bigl\| \check{\boldsymbol{\psi}}_l^{(1)}(\varepsilon) \bigr\|_{L_2(\mathbb{R}^d)} \leqslant \varepsilon \sum_{j=1}^d \| \check{M}_{lj}^\varepsilon S_\varepsilon \partial_j \widetilde{\mathbf{u}}_0 \|_{L_2(\mathbb{R}^d)} \leqslant C_{20,l} \varepsilon \| \mathbf{F} \|_{L_2(\mathcal{O})}, \qquad l=1,\dots,d,
\end{equation}
\tag{5.44}
$$
where $C_{20,l} = C_8 (2r_0)^{-1} |\Omega|^{-1/2} \bigl( \sum_{j=1}^d (C_{19,l} + C_{19,j})^2 \bigr)^{1/2}$. By (4.4) and Lemma 4.2, for $0< \varepsilon \leqslant \varepsilon_1$, we have
$$
\begin{equation*}
\begin{aligned} \, \bigl\| \check{\boldsymbol{\psi}}_l^{(2)}(\varepsilon) \bigr\|_{L_2(\mathbb{R}^d)} &\leqslant \kappa \biggl( \sum_{j=1}^d \| \check{M}_{lj}^\varepsilon S_\varepsilon \widetilde{\mathbf{u}}_0 \|^2_{L_2((\partial \mathcal{O})_\varepsilon)} \biggr)^{1/2} \\ &\leqslant \varepsilon^{1/2} \kappa \beta_*^{1/2} |\Omega|^{-1/2} \biggl( \sum_{j=1}^d \| \check{M}_{lj}\|^2_{L_2(\Omega)} \biggr)^{1/2} \| \widetilde{\mathbf{u}}_0 \|^{1/2}_{H^1(\mathbb{R}^d)} \| \widetilde{\mathbf{u}}_0 \|^{1/2}_{L_2(\mathbb{R}^d)}, \\ &\qquad \qquad 0< \varepsilon \leqslant \varepsilon_1,\quad l=1,\dots,d. \end{aligned}
\end{equation*}
\notag
$$
Combining this with (3.18), (3.19), (5.41), and the restriction $|\zeta| \geqslant 1$, we obtain
$$
\begin{equation}
\bigl\| \check{\boldsymbol{\psi}}_l^{(2)}(\varepsilon) \bigr\|_{L_2(\mathbb{R}^d)} \leqslant C_{21,l} \varepsilon^{1/2} |\zeta|^{-1/4} \| \mathbf{F} \|_{L_2(\mathcal{O})}, \qquad l=1,\dots,d,
\end{equation}
\tag{5.45}
$$
where $C_{21,l} = \kappa \beta_*^{1/2} (2 r_0)^{-1} |\Omega|^{-1/2} (C_7 C_8)^{1/2}\bigl( \sum_{j=1}^d (C_{19,l} + C_{19,j})^2 \bigr)^{1/2}$. Similarly, using (4.4) and Lemma 4.2, and then (3.18), (3.19), and (5.40), we arrive at the estimate
$$
\begin{equation}
\begin{aligned} \, \bigl\| \check{\boldsymbol{\psi}}_l^{(3)}(\varepsilon) \bigr\|_{L_2(\mathbb{R}^d)} &\leqslant \| \check{f}_l^\varepsilon S_\varepsilon \widetilde{\mathbf{u}}_0 \|_{L_2((\partial \mathcal{O})_\varepsilon)} \nonumber \\ &\leqslant \varepsilon^{1/2} \beta_*^{1/2} |\Omega|^{-1/2}\| \check{f}_l\|_{L_2(\Omega)} \| \widetilde{\mathbf{u}}_0 \|^{1/2}_{H^1(\mathbb{R}^d)} \| \widetilde{\mathbf{u}}_0 \|^{1/2}_{L_2(\mathbb{R}^d)} \nonumber \\ &\leqslant C_{22,l} \varepsilon^{1/2} |\zeta|^{-1/4} \| \mathbf{F} \|_{L_2(\mathcal{O})}, \qquad 0< \varepsilon \leqslant \varepsilon_1,\quad l=1,\dots,d, \end{aligned}
\end{equation}
\tag{5.46}
$$
where $C_{22,l} = \beta_*^{1/2} |\Omega|^{-1/2} (C_7 C_8)^{1/2} C_{19,l}$.
Relations (5.43)–(5.46) imply that
$$
\begin{equation}
\begin{aligned} \, \bigl| \widetilde{\mathcal{T}}_\varepsilon^{(3)}[\boldsymbol{\eta}] \bigr| &\leqslant \gamma_3^{(4)} \varepsilon \| \mathbf{F} \|_{L_2(\mathcal{O})} \| \mathbf{D} \boldsymbol{\eta} \|_{L_2(\mathcal{O})} \nonumber \\ &\qquad+ \gamma_4 \varepsilon^{1/2} |\zeta|^{-1/4} \| \mathbf{F} \|_{L_2(\mathcal{O})} \| \mathbf{D} \boldsymbol{\eta} \|_{L_2(\Upsilon_\varepsilon)}, \qquad 0 < \varepsilon \leqslant \varepsilon_1, \end{aligned}
\end{equation}
\tag{5.47}
$$
where $\gamma_3^{(4)} = \bigl( \sum_{l=1}^d C_{20,l}^2\bigr)^{1/2}$ and $\gamma_4 = \bigl( \sum_{l=1}^d (C_{21,l} + C_{22,l})^2\bigr)^{1/2}$.
As a result, relations (5.24), (5.30), (5.33), (5.37), (5.42), and (5.47) yield the required estimate (5.23) with the constant $\gamma_3 = \gamma_3^{(1)} + \gamma_3^{(2)} + \gamma_3^{(3)} + \gamma_3^{(4)}$. This completes the proof. Lemma 5.3. Let $\operatorname{Re} \zeta \leqslant 0$, $|\zeta| \geqslant 1$, and let the number $\varepsilon_1$ satisfy Condition 3.1. Then, for $0< \varepsilon \leqslant \varepsilon_1$, the functional (5.4) satisfies the estimate Proof. By (3.11) and (3.12), the functional (5.4) can be represented as
$$
\begin{equation}
\mathcal{I}^{(3)}_\varepsilon [\boldsymbol{\eta}] = \sum_{k=1}^5 \mathcal{J}^{(k)}_\varepsilon [\boldsymbol{\eta}],
\end{equation}
\tag{5.49}
$$
$$
\begin{equation}
\begin{aligned} \, \mathcal{J}^{(1)}_\varepsilon [\boldsymbol{\eta}] &:= \sum_{l=1}^d \bigl( (I - S_\varepsilon) D_l \widetilde{\mathbf{u}}_0, (a_l^\varepsilon - \overline{a_l})^* \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})} \nonumber \\ &\qquad+ \bigl( V^* (I - S_\varepsilon) b(\mathbf{D}) \widetilde{\mathbf{u}}_0, \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})} + \bigl(W (I - S_\varepsilon) \widetilde{\mathbf{u}}_0, \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})}, \end{aligned}
\end{equation}
\tag{5.50}
$$
$$
\begin{equation}
\mathcal{J}^{(2)}_\varepsilon [\boldsymbol{\eta}] := \varepsilon \sum_{l=1}^d \bigl( a_l^\varepsilon (\Lambda^\varepsilon S_\varepsilon b(\mathbf{D}) D_l \widetilde{\mathbf{u}}_0 + \widetilde{\Lambda}^\varepsilon S_\varepsilon D_l \widetilde{\mathbf{u}}_0), \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})},
\end{equation}
\tag{5.51}
$$
$$
\begin{equation}
\mathcal{J}^{(3)}_\varepsilon [\boldsymbol{\eta}] := \sum_{l=1}^d \bigl( (a_l^\varepsilon - \overline{a_l}) S_\varepsilon D_l \widetilde{\mathbf{u}}_0, \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})},
\end{equation}
\tag{5.52}
$$
$$
\begin{equation}
\mathcal{J}^{(4)}_\varepsilon [\boldsymbol{\eta}] := \sum_{l=1}^d \bigl( a_l^\varepsilon (D_l \Lambda)^\varepsilon S_\varepsilon b(\mathbf{D}) \widetilde{\mathbf{u}}_0, \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})} + \bigl( V^* S_\varepsilon b(\mathbf{D}) \widetilde{\mathbf{u}}_0, \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})},
\end{equation}
\tag{5.53}
$$
$$
\begin{equation}
\mathcal{J}^{(5)}_\varepsilon [\boldsymbol{\eta}] := \sum_{l=1}^d \bigl( a_l^\varepsilon (D_l \widetilde{\Lambda})^\varepsilon S_\varepsilon \widetilde{\mathbf{u}}_0, \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})} + \bigl( W S_\varepsilon \widetilde{\mathbf{u}}_0, \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})}.
\end{equation}
\tag{5.54}
$$
Similarly to (3.31),
$$
\begin{equation}
\biggl( \sum_{l=1}^d \| (a_l^\varepsilon - \overline{a_l})^* \boldsymbol{\eta} \|^2_{L_2(\mathcal{O})} \biggr)^{1/2} \leqslant 2 C_{11} \| \boldsymbol{\eta} \|_{H^1(\mathcal{O})}.
\end{equation}
\tag{5.55}
$$
By Proposition 1.1, relations (1.5), (3.19), (3.20), and the restriction $|\zeta| \geqslant 1$, we have
$$
\begin{equation}
\| (I - S_\varepsilon) \mathbf{D} \widetilde{\mathbf{u}}_0\|_{L_2(\mathbb{R}^d)} \leqslant r_1 \varepsilon \| \widetilde{\mathbf{u}}_0\|_{H^2(\mathbb{R}^d)} \leqslant C_9 r_1 \varepsilon \| \mathbf{F}\|_{L_2(\mathcal{O})},
\end{equation}
\tag{5.56}
$$
$$
\begin{equation}
\| (I - S_\varepsilon) b(\mathbf{D}) \widetilde{\mathbf{u}}_0\|_{L_2(\mathbb{R}^d)} \leqslant r_1 \varepsilon \| b(\mathbf{D}) \widetilde{\mathbf{u}}_0\|_{H^1(\mathbb{R}^d)} \leqslant C_9 r_1 \alpha_1^{1/2} \varepsilon \| \mathbf{F}\|_{L_2(\mathcal{O})},
\end{equation}
\tag{5.57}
$$
$$
\begin{equation}
\| (I - S_\varepsilon) \widetilde{\mathbf{u}}_0\|_{L_2(\mathbb{R}^d)} \leqslant r_1 \varepsilon \| \widetilde{\mathbf{u}}_0\|_{H^1(\mathbb{R}^d)} \leqslant C_8 r_1 \varepsilon \| \mathbf{F}\|_{L_2(\mathcal{O})}.
\end{equation}
\tag{5.58}
$$
Combining (1.36) and (5.55)–(5.58), we obtain the following estimate for the functional (5.50):
$$
\begin{equation}
\bigl| \mathcal{J}^{(1)}_\varepsilon [\boldsymbol{\eta}] \bigr| \leqslant \gamma_5^{(1)} \varepsilon \| \mathbf{F} \|_{L_2(\mathcal{O})} \| \boldsymbol{\eta}\|_{H^1(\mathcal{O})},
\end{equation}
\tag{5.59}
$$
where $\gamma_5^{(1)} = r_1 (2 C_9 C_{11} + C_V C_9 \alpha_1^{1/2} + C_W C_8)$.
Now, consider the term (5.51). Similarly to (5.28) and (5.29),
$$
\begin{equation}
\| a_l \Lambda \|_{L_2(\Omega)} \leqslant C_{17,l} |\Omega|^{1/2},
\end{equation}
\tag{5.60}
$$
$$
\begin{equation}
\| a_l \widetilde{\Lambda} \|_{L_2(\Omega)} \leqslant C_{18,l} |\Omega|^{1/2}.
\end{equation}
\tag{5.61}
$$
By Proposition 1.2, relations (1.5), (3.19), (3.20), (5.60), (5.61), and the restriction $|\zeta| \geqslant 1$, we have
$$
\begin{equation}
\bigl| \mathcal{J}^{(2)}_\varepsilon [\boldsymbol{\eta}] \bigr| \leqslant \gamma_5^{(2)} \varepsilon \| \mathbf{F} \|_{L_2(\mathcal{O})} \| \boldsymbol{\eta}\|_{L_2(\mathcal{O})},
\end{equation}
\tag{5.62}
$$
where $\gamma_5^{(2)} = \alpha_1^{1/2} C_9 \bigl( \sum_{l=1}^d C_{17,l}^2\bigr)^{1/2} + C_8 \bigl( \sum_{l=1}^d C_{18,l}^2\bigr)^{1/2}$.
To estimate the functional (5.52), we apply Lemma 4.4($2^\circ$) with $a(\mathbf{x})=1$ and $p(\mathbf{x})= a_l(\mathbf{x})$. Recall the notation $C_a = \bigl( \sum_{l=1}^d \|a_l\|^2_{L_2(\Omega)}\bigr)^{1/2}$. We obtain
$$
\begin{equation*}
\bigl| \mathcal{J}^{(3)}_\varepsilon [\boldsymbol{\eta}] \bigr| \leqslant 2 C''' C_a \varepsilon \| \mathbf{D} \widetilde{\mathbf{u}}_0 \|_{H^1(\mathbb{R}^d)} \| \boldsymbol{\eta}\|_{H^1(\mathcal{O})},\qquad 0 < \varepsilon \leqslant \varepsilon_1.
\end{equation*}
\notag
$$
Together with (3.20), this implies that
$$
\begin{equation}
\bigl| \mathcal{J}^{(3)}_\varepsilon [\boldsymbol{\eta}] \bigr| \leqslant \gamma_5^{(3)} \varepsilon \|\mathbf{F} \|_{L_2(\mathcal{O})} \| \boldsymbol{\eta}\|_{H^1(\mathcal{O})},
\end{equation}
\tag{5.63}
$$
where $\gamma_5^{(3)} = 2 C''' C_a C_9$.
Let us now consider the term (5.53). From (1.30) and (1.34) we have $V^* = - \sum_{l=1}^d \overline{a_l (D_l \Lambda)}$. Therefore, the term (5.53) can be written as
$$
\begin{equation*}
\mathcal{J}^{(4)}_\varepsilon [\boldsymbol{\eta}] = \sum_{l=1}^d \bigl( \bigl(a_l^\varepsilon (D_l \Lambda)^\varepsilon - \overline{a_l (D_l \Lambda)}\,\bigr) S_\varepsilon b(\mathbf{D}) \widetilde{\mathbf{u}}_0, \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})}.
\end{equation*}
\notag
$$
We next apply Lemma 4.4($1^\circ$) with $a(\mathbf{x})=a_l(\mathbf{x})$ and $p(\mathbf{x})= D_l \Lambda(\mathbf{x})$. Setting $\widetilde{C}^2_a := \sum_{l=1}^d \|a_l \|_{L_\rho(\Omega)}^2$, we have
$$
\begin{equation*}
\begin{aligned} \, &\bigl| \mathcal{J}^{(4)}_\varepsilon [\boldsymbol{\eta}] \bigr| \leqslant 2 C' \widetilde{C}_a \varepsilon \| \mathbf{D} \Lambda\|_{L_2(\Omega)} \| b(\mathbf{D}) \widetilde{\mathbf{u}}_0\|_{H^1(\mathbb{R}^d)} \| \boldsymbol{\eta}\|_{H^1(\mathcal{O})} \\ &\qquad + C'' \varepsilon^{1/2}\| \mathbf{D} \Lambda\|_{L_2(\Omega)} \| b(\mathbf{D}) \widetilde{\mathbf{u}}_0\|^{1/2}_{H^1(\mathbb{R}^d)} \| b(\mathbf{D}) \widetilde{\mathbf{u}}_0\|^{1/2}_{L_2(\mathbb{R}^d)} \biggl( \sum_{l=1}^d \| (a_l^\varepsilon)^* \boldsymbol{\eta} \|^2_{L_2(\Upsilon_\varepsilon)} \biggr)^{1/2}. \end{aligned}
\end{equation*}
\notag
$$
Combining this with (1.5), (1.25), (3.19), (3.20), we obtain
$$
\begin{equation}
\begin{aligned} \, &\bigl| \mathcal{J}^{(4)}_\varepsilon [\boldsymbol{\eta}] \bigr| \leqslant \gamma_5^{(4)} \varepsilon \| \mathbf{F} \|_{L_2(\mathcal{O})} \| \boldsymbol{\eta}\|_{H^1(\mathcal{O})} \nonumber \\ &\qquad+ \gamma_6^{(1)} \varepsilon^{1/2} |\zeta|^{-1/4} \| \mathbf{F} \|_{L_2(\mathcal{O})} \biggl( \sum_{l=1}^d \| (a_l^\varepsilon)^* \boldsymbol{\eta} \|^2_{L_2(\Upsilon_\varepsilon)} \biggr)^{1/2}, \qquad 0< \varepsilon \leqslant \varepsilon_1, \end{aligned}
\end{equation}
\tag{5.64}
$$
where $\gamma_5^{(4)}= 2 C' \widetilde{C}_a |\Omega|^{1/2} M_2 \alpha_1^{1/2} C_9$, $\gamma_6^{(1)}= C'' |\Omega|^{1/2} M_2 \alpha_1^{1/2} (C_8C_9)^{1/2}$.
It remains to consider the term (5.54). From (1.30) and (1.35) we have the representation $W = - \sum_{l=1}^d \overline{a_l (D_l \widetilde{\Lambda})}$. Therefore, the term (5.54) can be written as
$$
\begin{equation*}
\mathcal{J}^{(5)}_\varepsilon [\boldsymbol{\eta}] = \sum_{l=1}^d \bigl( (a_l^\varepsilon (D_l \widetilde{\Lambda})^\varepsilon - \overline{a_l (D_l \widetilde{\Lambda})}) S_\varepsilon \widetilde{\mathbf{u}}_0, \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})}.
\end{equation*}
\notag
$$
An appeal to Lemma 4.4($1^\circ$) with $a(\mathbf{x})=a_l(\mathbf{x})$ and $p(\mathbf{x})= D_l \widetilde{\Lambda}(\mathbf{x})$ verifies that
$$
\begin{equation*}
\begin{aligned} \, &\bigl| \mathcal{J}^{(5)}_\varepsilon [\boldsymbol{\eta}] \bigr| \leqslant 2 C' \widetilde{C}_a \varepsilon \| \mathbf{D} \widetilde{\Lambda} \|_{L_2(\Omega)} \| \widetilde{\mathbf{u}}_0\|_{H^1(\mathbb{R}^d)} \| \boldsymbol{\eta}\|_{H^1(\mathcal{O})} \\ &+ C'' \varepsilon^{1/2}\| \mathbf{D} \widetilde{\Lambda} \|_{L_2(\Omega)} \| \widetilde{\mathbf{u}}_0\|^{1/2}_{H^1(\mathbb{R}^d)} \| \widetilde{\mathbf{u}}_0\|^{1/2}_{L_2(\mathbb{R}^d)} \biggl( \sum_{l=1}^d \| (a_l^\varepsilon)^* \boldsymbol{\eta} \|^2_{L_2(\Upsilon_\varepsilon)} \biggr)^{1/2}, \,\quad 0\,{<}\, \varepsilon \,{\leqslant}\, \varepsilon_1. \end{aligned}
\end{equation*}
\notag
$$
Together with (1.33), (3.18), (3.19), and the restriction $|\zeta| \geqslant 1$, this implies
$$
\begin{equation}
\begin{aligned} \, &\bigl| \mathcal{J}^{(5)}_\varepsilon [\boldsymbol{\eta}] \bigr| \leqslant \gamma_5^{(5)} \varepsilon \| \mathbf{F} \|_{L_2(\mathcal{O})} \| \boldsymbol{\eta}\|_{H^1(\mathcal{O})} \nonumber \\ &\qquad+ \gamma_6^{(2)} \varepsilon^{1/2} |\zeta|^{-1/4} \| \mathbf{F} \|_{L_2(\mathcal{O})} \biggl( \sum_{l=1}^d \| (a_l^\varepsilon)^* \boldsymbol{\eta} \|^2_{L_2(\Upsilon_\varepsilon)} \biggr)^{1/2}, \qquad 0< \varepsilon \leqslant \varepsilon_1, \end{aligned}
\end{equation}
\tag{5.65}
$$
where $\gamma_5^{(5)}= 2 |\Omega|^{1/2} C' \widetilde{C}_a \widetilde{M}_2 C_8$, $\gamma_6^{(2)}= |\Omega|^{1/2} C'' \widetilde{M}_2 (C_7C_8)^{1/2}$.
As a result, combining (5.49), (5.59), (5.62)–(5.65), we arrive at the required estimate (5.48) with the constants $\gamma_5 = \gamma_5^{(1)} +\gamma_5^{(2)} + \gamma_5^{(3)} + \gamma_5^{(4)}+ \gamma_5^{(5)}$, $\gamma_6= \gamma_6^{(1)}+ \gamma_6^{(2)}$. This completes the proof. Lemma 5.4. Let $\operatorname{Re} \zeta \leqslant 0$, $|\zeta| \geqslant 1$, and let the number $\varepsilon_1$ satisfy Condition 3.1. Then, for $0< \varepsilon \leqslant \varepsilon_1$, the functional (5.5) satisfies Proof. By (3.11) and (3.12), the functional (5.5) can be represented as
$$
\begin{equation}
\mathcal{I}^{(4)}_\varepsilon [\boldsymbol{\eta}] \,= \Sigma^{(1)}_\varepsilon [\boldsymbol{\eta}]+ \Sigma^{(2)}_\varepsilon [\boldsymbol{\eta}] + \Sigma^{(3)}_\varepsilon [\boldsymbol{\eta}],
\end{equation}
\tag{5.67}
$$
$$
\begin{equation}
\Sigma^{(1)}_\varepsilon [\boldsymbol{\eta}] := \bigl( (Q^\varepsilon - \overline{Q})(I - S_\varepsilon) \widetilde{\mathbf{u}}_0, \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})},
\end{equation}
\tag{5.68}
$$
$$
\begin{equation}
\Sigma^{(2)}_\varepsilon [\boldsymbol{\eta}] := \varepsilon \bigl( Q^\varepsilon (\Lambda^\varepsilon S_\varepsilon b(\mathbf{D}) \widetilde{\mathbf{u}}_0 + \widetilde{\Lambda}^\varepsilon S_\varepsilon \widetilde{\mathbf{u}}_0), \boldsymbol{\eta} \bigr)_{L_2(\mathcal{O})},
\end{equation}
\tag{5.69}
$$
Similarly to (3.32),
$$
\begin{equation}
\bigl\| |Q^\varepsilon - \overline{Q}\,|^{1/2} \boldsymbol{\eta} \bigr\|^2_{L_2(\mathcal{O})} \leqslant 2 C_{12} \| \boldsymbol{\eta} \|_{H^1(\mathcal{O})}, \qquad \boldsymbol{\eta} \in H^1(\mathcal{O};\mathbb{C}^n).
\end{equation}
\tag{5.71}
$$
Using Proposition 1.1 and relations (3.20), (5.71), we estimate the term (5.68):
$$
\begin{equation}
\begin{aligned} \, \bigl| \Sigma^{(1)}_\varepsilon [\boldsymbol{\eta}] \bigr| &\leqslant \bigl\| |Q^\varepsilon - \overline{Q}\,|^{1/2} (I - S_\varepsilon) \widetilde{\mathbf{u}}_0\bigr\|_{L_2(\mathcal{O})} \bigl\| |Q^\varepsilon - \overline{Q}\,|^{1/2} \boldsymbol{\eta} \bigr\|_{L_2(\mathcal{O})} \nonumber \\ &\leqslant 4 C_{12}^2 \|(I - S_\varepsilon) \widetilde{\mathbf{u}}_0\|_{H^1(\mathbb{R}^d)} \| \boldsymbol{\eta} \|_{H^1(\mathcal{O})} \leqslant \gamma_7^{(1)} \varepsilon \| \mathbf{F}\|_{L_2(\mathcal{O})}\| \boldsymbol{\eta} \|_{H^1(\mathcal{O})}, \end{aligned}
\end{equation}
\tag{5.72}
$$
where $\gamma_7^{(1)}= 4 C_9 C_{12}^2 r_1$.
Now, we consider the term (5.69). Using the Hölder inequality, the embedding theorem, and employing (1.24), (1.25), (1.32), and (1.33), we have
$$
\begin{equation}
\begin{aligned} \, \| |Q|^{1/2} \Lambda \|_{L_2(\Omega)} &\leqslant C(\check{q};\Omega) \| Q\|_{L_s(\Omega)}^{1/2} \| \Lambda\|_{H^1(\Omega)} \nonumber \\ &\leqslant C(\check{q};\Omega) \| Q\|_{L_s(\Omega)}^{1/2} (M_1+M_2) |\Omega|^{1/2} =: C_{23} |\Omega|^{1/2}, \end{aligned}
\end{equation}
\tag{5.73}
$$
$$
\begin{equation}
\begin{aligned} \, \| |Q|^{1/2} \widetilde{\Lambda} \|_{L_2(\Omega)} &\leqslant C(\check{q};\Omega) \| Q\|_{L_s(\Omega)}^{1/2} \| \widetilde{\Lambda} \|_{H^1(\Omega)} \nonumber \\ &\leqslant C(\check{q};\Omega) \| Q\|_{L_s(\Omega)}^{1/2} (\widetilde{M}_1+ \widetilde{M}_2) |\Omega|^{1/2} =: C_{24} |\Omega|^{1/2}. \end{aligned}
\end{equation}
\tag{5.74}
$$
Using Proposition 1.2, relations (1.5), (3.18), (3.19), (3.32), (5.73), (5.74), and the restriction $|\zeta| \geqslant 1$, we obtain
$$
\begin{equation}
\begin{aligned} \, \bigl| \Sigma^{(2)}_\varepsilon [\boldsymbol{\eta}] \bigr| &\leqslant \varepsilon \bigl\| |Q^\varepsilon|^{1/2} \bigl(\Lambda^\varepsilon S_\varepsilon b(\mathbf{D}) \widetilde{\mathbf{u}}_0 + \widetilde{\Lambda}^\varepsilon S_\varepsilon \widetilde{\mathbf{u}}_0\bigr)\bigr\|_{L_2(\mathcal{O})} \| |Q^\varepsilon|^{1/2}\boldsymbol{\eta} \|_{L_2(\mathcal{O})} \nonumber \\ &\leqslant \varepsilon \bigl( C_{23} \|b(\mathbf{D}) \widetilde{\mathbf{u}}_0\|_{L_2(\mathbb{R}^d)} + C_{24} \| \widetilde{\mathbf{u}}_0\|_{L_2(\mathbb{R}^d)} \bigr) C_{12} \| \boldsymbol{\eta} \|_{H^1(\mathcal{O})} \nonumber \\ &\leqslant \gamma_7^{(2)} \varepsilon \| \mathbf{F}\|_{L_2(\mathcal{O})}\| \boldsymbol{\eta} \|_{H^1(\mathcal{O})}, \end{aligned}
\end{equation}
\tag{5.75}
$$
where $\gamma_7^{(2)}= (C_8 C_{23} \alpha_1^{1/2} + C_7 C_{24}) C_{12}$.
Finally, to estimate the term (5.70), we apply Lemma 4.4($1^\circ$) with $a(\mathbf{x}) = |Q(\mathbf{x})|^{1/2}$, $p(\mathbf{x}) = |Q(\mathbf{x})|^{1/2}V(\mathbf{x})$ (it is assumed that $Q(\mathbf{x})= |Q(\mathbf{x})| V(\mathbf{x})$ is the polar decomposition of the matrix $Q(\mathbf{x})$) and $\rho =2s$. Now, for $0< \varepsilon \leqslant \varepsilon_1$,
$$
\begin{equation*}
\begin{aligned} \, \bigl| \Sigma^{(3)}_\varepsilon [\boldsymbol{\eta}] \bigr | &\leqslant 2 C' \varepsilon \|Q\|_{L_s(\Omega)}^{1/2} \| Q\|^{1/2}_{L_1(\Omega)} \| \widetilde{\mathbf{u}}_0\|_{H^1(\mathbb{R}^d)} \| \boldsymbol{\eta} \|_{H^1(\mathcal{O})} \\ &\qquad+ C'' \varepsilon^{1/2} \|Q\|^{1/2}_{L_1(\Omega)} \| \widetilde{\mathbf{u}}_0\|_{H^1(\mathbb{R}^d)}^{1/2} \| \widetilde{\mathbf{u}}_0\|^{1/2}_{L_2(\mathbb{R}^d)} \| | Q^\varepsilon |^{1/2} \boldsymbol{\eta} \|_{L_2(\Upsilon_\varepsilon)}. \end{aligned}
\end{equation*}
\notag
$$
Combining this with (3.18), (3.19), and since $|\zeta| \geqslant 1$, we arrive at the estimate
$$
\begin{equation}
\bigl| \Sigma^{(3)}_\varepsilon [\boldsymbol{\eta}] \bigr | \leqslant \gamma_7^{(3)} \varepsilon \| \mathbf{F}\|_{L_2(\mathcal{O})}\| \boldsymbol{\eta} \|_{H^1(\mathcal{O})} + \gamma_8 \varepsilon^{1/2} |\zeta|^{-1/4} \| \mathbf{F}\|_{L_2(\mathcal{O})} \| | Q^\varepsilon |^{1/2} \boldsymbol{\eta} \|_{L_2(\Upsilon_\varepsilon)}
\end{equation}
\tag{5.76}
$$
for $0< \varepsilon \leqslant \varepsilon_1$, where $\gamma_7^{(3)} = 2 {C}' C_8 \|Q\|^{1/2}_{L_s(\Omega)} \|Q\|^{1/2}_{L_1(\Omega)}$, $\gamma_8= {C}'' (C_7C_8)^{1/2} \|Q\|^{1/2}_{L_1(\Omega)}$.
Finally, the required estimate (5.66) with the constant $\gamma_7 = \gamma_7^{(1)}+ \gamma_7^{(2)} + \gamma_7^{(3)}$ follows from (5.67), (5.72), (5.75), and (5.76). This completes the proof. Lemma 5.5. Let $\operatorname{Re} \zeta \leqslant 0$, $|\zeta| \geqslant 1$, and let the number $\varepsilon_1$ satisfy Condition 3.1. Then, for $0< \varepsilon \leqslant \varepsilon_1$, the functional (5.6) satisfies Proof. By assertion $(3^\circ)$ of Lemma 4.4, for $0< \varepsilon \leqslant \varepsilon_1$, we have
$$
\begin{equation*}
\bigl| \mathcal{I}^{(5)}_\varepsilon [\boldsymbol{\eta}] \bigr| \leqslant |\lambda - \zeta| \widetilde{C}''' \|Q_0\|_{L_\infty} \varepsilon \bigl( \| \mathbf{u}_0\|_{H^1(\mathcal{O})} \| \boldsymbol{\eta} \|_{L_2(\mathcal{O})} + \| \mathbf{u}_0\|_{L_2(\mathcal{O})} \| \boldsymbol{\eta} \|_{H^1(\mathcal{O})} \bigr).
\end{equation*}
\notag
$$
Together with estimates (2.40) and (2.41), this implies the required inequality (5.77) with the constants $\gamma_9 = \lambda \widetilde{C}''' \|Q_0\|_{L_\infty} (\mathcal{C}_1 + \|Q_0^{-1}\|_{L_\infty}) + \widetilde{C}'''\| Q_0\|_{L_\infty} \|Q_0^{-1}\|_{L_\infty}$, $\gamma_{10} = \widetilde{C}''' \|Q_0\|_{L_\infty} \mathcal{C}_1$. This completes the proof. Let us summarize the results. From (5.1), (5.7), (5.23), (5.48), (5.66), and (5.77) it follows that, for $\operatorname{Re} \zeta \leqslant 0$, $|\zeta|\geqslant 1$, and $0< \varepsilon \leqslant \varepsilon_1$,
$$
\begin{equation}
\begin{aligned} \, |\mathcal{I}_\varepsilon [\boldsymbol{\eta}]| &\leqslant \gamma_{0}' \varepsilon \| \mathbf{F} \|_{L_2(\mathcal{O})} \| \boldsymbol{\eta} \|_{H^1(\mathcal{O})} + \gamma_{10} \varepsilon |\zeta|^{1/2} \| \mathbf{F} \|_{L_2(\mathcal{O})} \| \boldsymbol{\eta} \|_{L_2(\mathcal{O})} \nonumber \\ &\qquad+ \gamma_{0}'' \varepsilon^{1/2} |\zeta|^{-1/4} \| \mathbf{F} \|_{L_2(\mathcal{O})} \biggl(\| \mathbf{D} \boldsymbol{\eta}\|_{L_2(\Upsilon_\varepsilon)} \nonumber \\ &\qquad+ \biggl( \sum_{l=1}^d \| (a_l^\varepsilon)^* \boldsymbol{\eta} \|^2_{L_2(\Upsilon_\varepsilon)}\biggr)^{1/2} + \| |Q^\varepsilon|^{1/2} \boldsymbol{\eta}\|_{L_2(\Upsilon_\varepsilon)} \biggr) \end{aligned}
\end{equation}
\tag{5.78}
$$
with the constants $\gamma_0' = \gamma_1 + \gamma_3 + \gamma_5 +\gamma_7 + \gamma_9$, $\gamma_0'' = \max\{\gamma_2 + \gamma_4, \gamma_6,\gamma_8\}$. We will need estimate (5.78) in the next section to prove Theorem 3.2. However, estimate (3.14) can be obtained from the following more rough estimate:
$$
\begin{equation}
\begin{aligned} \, | \mathcal{I}_\varepsilon [\boldsymbol{\eta}]| &\leqslant \gamma_{0} (\varepsilon + \varepsilon^{1/2} |\zeta|^{-1/4}) \| \mathbf{F} \|_{L_2(\mathcal{O})} \| \boldsymbol{\eta} \|_{H^1(\mathcal{O})} \nonumber \\ &\qquad+ \gamma_{10} \varepsilon |\zeta|^{1/2} \| \mathbf{F} \|_{L_2(\mathcal{O})} \| \boldsymbol{\eta} \|_{L_2(\mathcal{O})}, \qquad 0 < \varepsilon \leqslant \varepsilon_1. \end{aligned}
\end{equation}
\tag{5.79}
$$
Estimate (5.79) follows from (5.78), the obvious estimates $\| \mathbf{D} \boldsymbol{\eta}\|_{L_2(\Upsilon_\varepsilon)} \leqslant \| \boldsymbol{\eta}\|_{H^1(\mathcal{O})}$, $\| (a_l^\varepsilon)^* \boldsymbol{\eta} \|_{L_2(\Upsilon_\varepsilon)} \leqslant \| (a_l^\varepsilon)^* \boldsymbol{\eta} \|_{L_2(\mathcal{O})}$, $\| |Q^\varepsilon|^{1/2} \boldsymbol{\eta}\|_{L_2(\Upsilon_\varepsilon)} \leqslant \| |Q^\varepsilon|^{1/2} \boldsymbol{\eta}\|_{L_2(\mathcal{O})}$, and (3.31), (3.32). The constant $\gamma_0 = \max\{ \gamma_0',\, \gamma_0'' (1+ C_{11} + C_{12})\}$ depends only on the problem data (2.35). 5.2. Completion of the proof of estimate (3.14) for $\operatorname{Re} \zeta \leqslant 0$ Lemma 5.6. Let $\operatorname{Re} \zeta \leqslant 0$ and $|\zeta| \geqslant 1$. Suppose that $\mathbf{w}_\varepsilon \in H^1(\mathcal{O};\mathbb{C}^n)$ satisfies identity (3.29), and the number $\varepsilon_1$ satisfies Condition 3.1. Then, for $0< \varepsilon \leqslant \varepsilon_1$, Proof. We substitute $\boldsymbol{\eta} = \mathbf{w}_\varepsilon$ in identity (3.29) and take the imaginary part of the obtained relation. By (5.79), for $0< \varepsilon \leqslant \varepsilon_1$, we have
$$
\begin{equation}
\begin{aligned} \, |{\operatorname{Im} \zeta}| (Q_0^\varepsilon \mathbf{w}_\varepsilon, \mathbf{w}_\varepsilon)_{L_2(\mathcal{O})} &\leqslant \gamma_0 (\varepsilon + \varepsilon^{1/2}|\zeta|^{-1/4}) \| \mathbf{F} \|_{L_2(\mathcal{O})}\| \mathbf{w}_\varepsilon\|_{H^1(\mathcal{O})} \nonumber \\ &\qquad + \gamma_{10} \varepsilon |\zeta|^{1/2} \| \mathbf{F} \|_{L_2(\mathcal{O})} \| \mathbf{w}_\varepsilon \|_{L_2(\mathcal{O})}. \end{aligned}
\end{equation}
\tag{5.81}
$$
Now, taking the real part of the obtained relation and using the restriction $\operatorname{Re} \zeta \leqslant 0$, for $0< \varepsilon \leqslant \varepsilon_1$ we obtain
$$
\begin{equation}
\begin{aligned} \, |{\operatorname{Re} \zeta}| (Q_0^\varepsilon \mathbf{w}_\varepsilon, \mathbf{w}_\varepsilon)_{L_2(\mathcal{O})} &\leqslant \gamma_0 (\varepsilon + \varepsilon^{1/2}|\zeta|^{-1/4}) \| \mathbf{F} \|_{L_2(\mathcal{O})}\| \mathbf{w}_\varepsilon\|_{H^1(\mathcal{O})} \nonumber \\ &\qquad+ \gamma_{10} \varepsilon |\zeta|^{1/2} \| \mathbf{F} \|_{L_2(\mathcal{O})} \| \mathbf{w}_\varepsilon \|_{L_2(\mathcal{O})}. \end{aligned}
\end{equation}
\tag{5.82}
$$
Summing up (5.81) and (5.82), we have
$$
\begin{equation*}
\begin{aligned} \, |\zeta| (Q_0^\varepsilon \mathbf{w}_\varepsilon, \mathbf{w}_\varepsilon)_{L_2(\mathcal{O})} &\leqslant 2 \gamma_0 (\varepsilon + \varepsilon^{1/2}|\zeta|^{-1/4}) \| \mathbf{F} \|_{L_2(\mathcal{O})}\| \mathbf{w}_\varepsilon\|_{H^1(\mathcal{O})} \\ &\qquad+ 2 \gamma_{10} \varepsilon |\zeta|^{1/2} \| \mathbf{F} \|_{L_2(\mathcal{O})} \| \mathbf{w}_\varepsilon \|_{L_2(\mathcal{O})}. \end{aligned}
\end{equation*}
\notag
$$
Hence
$$
\begin{equation}
\begin{aligned} \, (Q_0^\varepsilon \mathbf{w}_\varepsilon, \mathbf{w}_\varepsilon)_{L_2(\mathcal{O})} &\leqslant 4\gamma_0 |\zeta|^{-1} (\varepsilon + \varepsilon^{1/2}|\zeta|^{-1/4}) \| \mathbf{F} \|_{L_2(\mathcal{O})}\| \mathbf{w}_\varepsilon\|_{H^1(\mathcal{O})} \nonumber \\ &\qquad + 4 \gamma_{10}^2 \|Q_0^{-1}\|_{L_\infty} |\zeta|^{-1} \varepsilon^2 \| \mathbf{F} \|_{L_2(\mathcal{O})}^2, \qquad 0 < \varepsilon \leqslant \varepsilon_1. \end{aligned}
\end{equation}
\tag{5.83}
$$
For $\operatorname{Re} \zeta \leqslant 0$, from (3.29) with $\boldsymbol{\eta} = \mathbf{w}_\varepsilon$, (5.79), and (5.83), we have
$$
\begin{equation}
\begin{aligned} \, \mathfrak{b}_{N,\varepsilon} [\mathbf{w}_\varepsilon, \mathbf{w}_\varepsilon] &\leqslant 2 \gamma_0 (\varepsilon + \varepsilon^{1/2}|\zeta|^{-1/4}) \| \mathbf{F} \|_{L_2(\mathcal{O})}\| \mathbf{w}_\varepsilon\|_{H^1(\mathcal{O})} \nonumber \\ &\qquad + 2 \gamma_{10}^2 \|Q_0^{-1}\|_{L_\infty} \varepsilon^2 \| \mathbf{F} \|_{L_2(\mathcal{O})}^2, \qquad 0 < \varepsilon \leqslant \varepsilon_1. \end{aligned}
\end{equation}
\tag{5.84}
$$
Together with (2.17), this implies the required estimate (5.80) with This completes the proof. Completion of the proof of estimate (3.14) for $\operatorname{Re} \zeta \leqslant 0$Inequalities (3.42) and (5.80) yield estimate (3.13) in the case $\operatorname{Re} \zeta \leqslant 0$, $|\zeta|\geqslant 1$:
$$
\begin{equation*}
\| \mathbf{u}_\varepsilon - \mathbf{v}_\varepsilon \|_{H^1(\mathcal{O})} \leqslant \bigl( \mathcal{C}_{10} \varepsilon^{1/2} |\zeta|^{-1/4} + (\mathcal{C}_8 + \mathcal{C}_{10}) \varepsilon \bigr) \| \mathbf{F} \|_{L_2(\mathcal{O})}, \qquad 0< \varepsilon \leqslant \varepsilon_1.
\end{equation*}
\notag
$$
In operator terms, we have proved that
$$
\begin{equation}
\begin{aligned} \, &\bigl\|(B_{N,\varepsilon} - \zeta Q_0^\varepsilon )^{-1} - (B_{N}^0 - \zeta \overline{Q_0})^{-1} - \varepsilon K_N(\varepsilon;\zeta) \bigr\|_{L_2(\mathcal{O})\to H^1(\mathcal{O})} \nonumber \\ &\qquad\leqslant \mathcal{C}_{10} \varepsilon^{1/2} |\zeta|^{-1/4} +(\mathcal{C}_8 + \mathcal{C}_{10}) \varepsilon, \qquad \operatorname{Re} \zeta \leqslant 0,\quad |\zeta|\geqslant 1,\quad 0< \varepsilon \leqslant \varepsilon_1. \end{aligned}
\end{equation}
\tag{5.85}
$$
§ 6. Estimate for the correction term $\mathbf{w}_\varepsilon$ in $L_2(\mathcal{O})$. Proof of Theorem 3.2 for $\operatorname{Re} \zeta \leqslant 0$6.1. Estimate for $\mathbf{w}_\varepsilon$ in $L_2(\mathcal{O})$ Lemma 6.1. Let $\operatorname{Re} \zeta \leqslant 0$ and $|\zeta| \geqslant 1$. Suppose that $\mathbf{w}_\varepsilon \in H^1(\mathcal{O};\mathbb{C}^n)$ satisfies identity (3.29), and the number $\varepsilon_1$ satisfies Condition 3.1. Then, for $0< \varepsilon \leqslant \varepsilon_1$, Proof. We substitute $\boldsymbol{\eta}_\varepsilon = (B_{N,\varepsilon} -\zeta^* Q_0^\varepsilon)^{-1} \boldsymbol{\Phi}$ with $\boldsymbol{\Phi} \in L_2(\mathcal{O};\mathbb{C}^n)$ as a test function in identity (3.29). Now the left-hand side of (3.29) coincides with $(\mathbf{w}_\varepsilon, \boldsymbol{\Phi})_{L_2(\mathcal{O})}$, and the identity takes the form
$$
\begin{equation}
(\mathbf{w}_\varepsilon, \boldsymbol{\Phi})_{L_2(\mathcal{O})} = \mathcal{I}_\varepsilon[\boldsymbol{\eta}_\varepsilon].
\end{equation}
\tag{6.2}
$$
To approximate the function $\boldsymbol{\eta}_\varepsilon$, we apply the already proved inequality (5.85). We put $\boldsymbol{\eta}_0 = ( B_N^0 - \zeta^* \overline{Q_0}\,)^{-1} \boldsymbol{\Phi}$, $\widetilde{\boldsymbol{\eta}}_0 = P_\mathcal{O} \boldsymbol{\eta}_0$. The function $\boldsymbol{\eta}_\varepsilon$ is approximated by the function $\boldsymbol{\rho}_\varepsilon := \boldsymbol{\eta}_0 + \boldsymbol{\upsilon}_ \varepsilon$, where $\boldsymbol{\upsilon}_ \varepsilon:= \varepsilon \Lambda^\varepsilon S_\varepsilon b(\mathbf{D}) \widetilde{\boldsymbol{\eta}}_0 + \varepsilon \widetilde{\Lambda}^\varepsilon S_\varepsilon \widetilde{\boldsymbol{\eta}}_0$. From (5.85) it follows that
$$
\begin{equation}
\| \boldsymbol{\eta}_\varepsilon - {\boldsymbol{\rho}}_\varepsilon \|_{H^1(\mathcal{O})} \leqslant \bigl(\mathcal{C}_{10} \varepsilon^{1/2} |\zeta|^{-1/4} + (\mathcal{C}_8+ \mathcal{C}_{10}) \varepsilon \bigr) \| \boldsymbol{\Phi} \|_{L_2(\mathcal{O})}, \qquad 0 < \varepsilon \leqslant \varepsilon_1.
\end{equation}
\tag{6.3}
$$
We will also need the following estimate which follows from (2.23), (2.40), and (3.7):
$$
\begin{equation}
\| \boldsymbol{\eta}_\varepsilon - {\boldsymbol{\rho}}_\varepsilon \|_{L_2(\mathcal{O})} \leqslant (2 \|Q_0^{-1}\|_{L_\infty} |\zeta|^{-1} + C_K' \varepsilon |\zeta|^{-1/2}) \|\boldsymbol{\Phi} \|_{L_2(\mathcal{O})}, \qquad 0 < \varepsilon \leqslant 1.
\end{equation}
\tag{6.4}
$$
We rewrite identity (6.2) as
$$
\begin{equation}
(\mathbf{w}_\varepsilon, \boldsymbol{\Phi})_{L_2(\mathcal{O})} = \mathcal{I}_\varepsilon[\boldsymbol{\eta}_\varepsilon - \boldsymbol{\rho}_\varepsilon ] + \mathcal{I}_\varepsilon[\boldsymbol{\eta}_0] + \mathcal{I}_\varepsilon[\boldsymbol{\upsilon}_ \varepsilon ].
\end{equation}
\tag{6.5}
$$
By (5.79), (6.3), and (6.4), for $0< \varepsilon \leqslant \varepsilon_1$, we have
$$
\begin{equation}
\begin{aligned} \, |\mathcal{I}_\varepsilon [\boldsymbol{\eta}_\varepsilon - \boldsymbol{\rho}_\varepsilon]| &\leqslant \gamma_{0} (\varepsilon + \varepsilon^{1/2} |\zeta|^{-1/4}) \| \mathbf{F} \|_{L_2(\mathcal{O})} \| \boldsymbol{\eta}_\varepsilon - {\boldsymbol{\rho}}_\varepsilon \|_{H^1(\mathcal{O})} \nonumber \\ &\qquad+ \gamma_{10} \varepsilon |\zeta|^{1/2} \| \mathbf{F} \|_{L_2(\mathcal{O})} \| \boldsymbol{\eta}_\varepsilon - {\boldsymbol{\rho}}_\varepsilon \|_{L_2(\mathcal{O})} \nonumber \\ &\leqslant \gamma_{0} (\varepsilon + \varepsilon^{1/2} |\zeta|^{-1/4}) \bigl(\mathcal{C}_{10} \varepsilon^{1/2} |\zeta|^{-1/4} + (\mathcal{C}_8 +\mathcal{C}_{10}) \varepsilon \bigr) \| \mathbf{F} \|_{L_2(\mathcal{O})} \| \boldsymbol{\Phi} \|_{L_2(\mathcal{O})} \nonumber \\ &\qquad+ \gamma_{10} (2 \|Q_0^{-1}\|_{L_\infty} \varepsilon |\zeta|^{-1/2} + C_K'\varepsilon^2) \| \mathbf{F} \|_{L_2(\mathcal{O})} \| \boldsymbol{\Phi} \|_{L_2(\mathcal{O})} \nonumber \\ &\leqslant \mathcal{C}_{11}^{(1)} (\varepsilon |\zeta|^{-1/2} + \varepsilon^2) \| \mathbf{F} \|_{L_2(\mathcal{O})} \| \boldsymbol{\Phi} \|_{L_2(\mathcal{O})}, \end{aligned}
\end{equation}
\tag{6.6}
$$
where $\mathcal{C}_{11}^{(1)} = 2 \gamma_0 (\mathcal{C}_8 +\mathcal{C}_{10}) + \gamma_{10} \max \{2 \|Q_0^{-1}\|_{L_\infty};C_K'\}$.
It is clear that
$$
\begin{equation}
\mathcal{I}_\varepsilon[\boldsymbol{\eta}_0] = \mathcal{I}_\varepsilon[S_\varepsilon \widetilde{\boldsymbol{\eta}}_0] + \mathcal{I}_\varepsilon[(I - S_\varepsilon) \widetilde{\boldsymbol{\eta}}_0].
\end{equation}
\tag{6.7}
$$
By Proposition 1.1 and estimate (3.20) (for $\widetilde{\boldsymbol{\eta}}_0$), we have
$$
\begin{equation}
\| (I - S_\varepsilon) \widetilde{\boldsymbol{\eta}}_0\|_{H^1(\mathbb{R}^d)} \leqslant \varepsilon r_1 \| \widetilde{\boldsymbol{\eta}}_0 \|_{H^2(\mathbb{R}^d)} \leqslant C_9 r_1 \varepsilon \| \boldsymbol{\Phi} \|_{L_2(\mathcal{O})}.
\end{equation}
\tag{6.8}
$$
The second term in (6.7) is estimated via (1.2), (3.18) (for $\widetilde{\boldsymbol{\eta}}_0$), (5.79), and (6.8) as follows:
$$
\begin{equation}
\begin{aligned} \, |\mathcal{I}_\varepsilon [(I - S_\varepsilon)\widetilde{\boldsymbol{\eta}}_0]| &\leqslant \gamma_0 (\varepsilon + \varepsilon^{1/2} |\zeta|^{-1/4}) \| \mathbf{F} \|_{L_2(\mathcal{O})} \| (I - S_\varepsilon) \widetilde{\boldsymbol{\eta}}_0\|_{H^1(\mathbb{R}^d)} \nonumber \\ &\qquad+ 2 \gamma_{10} \varepsilon |\zeta|^{1/2} \| \mathbf{F} \|_{L_2(\mathcal{O})} \| \widetilde{\boldsymbol{\eta}}_0\|_{L_2(\mathbb{R}^d)} \nonumber \\ &\leqslant C_9 r_1 \gamma_{0} \varepsilon (\varepsilon + \varepsilon^{1/2} |\zeta|^{-1/4}) \| \mathbf{F} \|_{L_2(\mathcal{O})} \| \boldsymbol{\Phi} \|_{L_2(\mathcal{O})} \nonumber \\ &\qquad+ 2 C_7 \gamma_{10} \varepsilon |\zeta|^{-1/2} \| \mathbf{F} \|_{L_2(\mathcal{O})} \| \boldsymbol{\Phi} \|_{L_2(\mathcal{O})} \nonumber \\ &\leqslant \mathcal{C}_{11}^{(2)} (\varepsilon |\zeta|^{-1/2} + \varepsilon^{2}) \| \mathbf{F} \|_{L_2(\mathcal{O})} \| \boldsymbol{\Phi} \|_{L_2(\mathcal{O})}, \end{aligned}
\end{equation}
\tag{6.9}
$$
where $\mathcal{C}_{11}^{(2)} \,{=}\, 2 C_9 r_1 \gamma_0 \,{+}\, 2 C_7 \gamma_{10}$. To estimate the first term in (6.7), we apply estimate (5.78) for $0\,{<}\, \varepsilon \,{\leqslant}\, \varepsilon_1$. As a result, we have
$$
\begin{equation}
\begin{aligned} \, &| \mathcal{I}_\varepsilon [S_\varepsilon \widetilde{\boldsymbol{\eta}}_0 ]| \leqslant \gamma_{0}' \varepsilon \| \mathbf{F} \|_{L_2(\mathcal{O})} \| S_\varepsilon \widetilde{\boldsymbol{\eta}}_0 \|_{H^1(\mathcal{O})} + \gamma_{10} \varepsilon |\zeta|^{1/2} \| \mathbf{F} \|_{L_2(\mathcal{O})} \| S_\varepsilon \widetilde{\boldsymbol{\eta}}_0\|_{L_2(\mathcal{O})} \nonumber \\ &\quad+ \gamma_{0}'' \varepsilon^{1/2} |\zeta|^{-1/4} \| \mathbf{F} \|_{L_2(\mathcal{O})} \| \mathbf{D} S_\varepsilon \widetilde{\boldsymbol{\eta}}_0\|_{L_2(\Upsilon_\varepsilon)} \nonumber \\ &\quad+ \gamma_{0}'' \varepsilon^{1/2} |\zeta|^{-1/4} \| \mathbf{F} \|_{L_2(\mathcal{O})} \biggl( \biggl( \sum_{l=1}^d \| (a_l^\varepsilon)^* S_\varepsilon \widetilde{\boldsymbol{\eta}}_0 \|^2_{L_2(\Upsilon_\varepsilon)}\biggr)^{1/2} + \| |Q^\varepsilon|^{1/2} S_\varepsilon \widetilde{\boldsymbol{\eta}}_0\|_{L_2(\Upsilon_\varepsilon)} \biggr). \end{aligned}
\end{equation}
\tag{6.10}
$$
By (1.2) and estimates (3.18), (3.19) (for $\widetilde{\boldsymbol{\eta}}_0$), we have
$$
\begin{equation}
\| S_\varepsilon \widetilde{\boldsymbol{\eta}}_0 \|_{L_2(\mathcal{O})} \leqslant \| S_\varepsilon \widetilde{\boldsymbol{\eta}}_0 \|_{L_2(\mathbb{R}^d)} \leqslant \| \widetilde{\boldsymbol{\eta}}_0 \|_{L_2(\mathbb{R}^d)} \leqslant C_7 |\zeta|^{-1}\| \boldsymbol{\Phi} \|_{L_2(\mathcal{O})},
\end{equation}
\tag{6.11}
$$
$$
\begin{equation}
\| S_\varepsilon \widetilde{\boldsymbol{\eta}}_0 \|_{H^1(\mathcal{O})} \leqslant \| S_\varepsilon \widetilde{\boldsymbol{\eta}}_0 \|_{H^1(\mathbb{R}^d)} \leqslant \| \widetilde{\boldsymbol{\eta}}_0 \|_{H^1(\mathbb{R}^d)} \leqslant C_8 |\zeta|^{-1/2}\| \boldsymbol{\Phi} \|_{L_2(\mathcal{O})}.
\end{equation}
\tag{6.12}
$$
Applying Lemma 4.1, (1.2), and estimates (3.19), (3.20) (for $\widetilde{\boldsymbol{\eta}}_0$), for $0< \varepsilon \leqslant \varepsilon_1$, we obtain
$$
\begin{equation}
\begin{aligned} \, \| \mathbf{D} S_\varepsilon \widetilde{\boldsymbol{\eta}}_0\|_{L_2(\Upsilon_\varepsilon)} &\leqslant \beta^{1/2} \varepsilon^{1/2} \| \mathbf{D} S_\varepsilon \widetilde{\boldsymbol{\eta}}_0 \|_{H^1(\mathcal{O})}^{1/2} \| \mathbf{D} S_\varepsilon \widetilde{\boldsymbol{\eta}}_0 \|_{L_2(\mathcal{O})}^{1/2} \nonumber \\ &\leqslant \beta^{1/2} \varepsilon^{1/2} \| \widetilde{\boldsymbol{\eta}}_0 \|_{H^2(\mathbb{R}^d)}^{1/2} \| \widetilde{\boldsymbol{\eta}}_0 \|_{H^1(\mathbb{R}^d)}^{1/2} \leqslant C_{25} \varepsilon^{1/2} |\zeta|^{-1/4} \| \boldsymbol{\Phi} \|_{L_2(\mathcal{O})}, \end{aligned}
\end{equation}
\tag{6.13}
$$
where $C_{25} = (\beta C_8 C_9)^{1/2}$. Next, by Lemma 4.2 and estimates (3.18), (3.19) (for $\widetilde{\boldsymbol{\eta}}_0$), taking into account the restriction $|\zeta| \geqslant 1$, for $0< \varepsilon \leqslant \varepsilon_1$, we obtain
$$
\begin{equation}
\begin{aligned} \, \biggl( \sum_{l=1}^d \| (a_l^\varepsilon)^* S_\varepsilon \widetilde{\boldsymbol{\eta}}_0 \|^2_{L_2(\Upsilon_\varepsilon)}\biggr)^{1/2} &\leqslant \beta_*^{1/2} \varepsilon^{1/2}|\Omega|^{-1/2} C_a \| \widetilde{\boldsymbol{\eta}}_0 \|_{H^1(\mathbb{R}^d)}^{1/2} \| \widetilde{\boldsymbol{\eta}}_0 \|_{L_2(\mathbb{R}^d)}^{1/2} \nonumber \\ &\leqslant C_{26} \varepsilon^{1/2} |\zeta|^{-1/4} \| \boldsymbol{\Phi} \|_{L_2(\mathcal{O})}, \end{aligned}
\end{equation}
\tag{6.14}
$$
where $ C_{26} = (\beta_* C_7 C_8)^{1/2} |\Omega|^{-1/2} C_a$. Similarly, for $0< \varepsilon \leqslant \varepsilon_1$, we have
$$
\begin{equation}
\begin{aligned} \, \bigl\| |Q^\varepsilon|^{1/2} S_\varepsilon \widetilde{\boldsymbol{\eta}}_0 \bigr\|_{L_2(\Upsilon_\varepsilon)} &\leqslant \beta_*^{1/2} \varepsilon^{1/2}|\Omega|^{-1/2} \|Q\|^{1/2}_{L_1(\Omega)} \| \widetilde{\boldsymbol{\eta}}_0 \|_{H^1(\mathbb{R}^d)}^{1/2}\| \widetilde{\boldsymbol{\eta}}_0 \|_{L_2(\mathbb{R}^d)}^{1/2} \nonumber \\ &\leqslant C_{27} \varepsilon^{1/2} |\zeta|^{-1/4} \| \boldsymbol{\Phi} \|_{L_2(\mathcal{O})}, \end{aligned}
\end{equation}
\tag{6.15}
$$
where $C_{27}=(\beta_* C_7 C_8)^{1/2} |\Omega|^{-1/2} \|Q\|^{1/2}_{L_1(\Omega)}$.
As a result, from (6.10)–(6.15) it follows that, for $0< \varepsilon \leqslant \varepsilon_1$,
$$
\begin{equation}
|\mathcal{I}_\varepsilon [S_\varepsilon \widetilde{\boldsymbol{\eta}}_0]| \leqslant \mathcal{C}_{11}^{(3)} \varepsilon |\zeta|^{-1/2} \| \mathbf{F} \|_{L_2(\mathcal{O})} \| \boldsymbol{\Phi} \|_{L_2(\mathcal{O})}
\end{equation}
\tag{6.16}
$$
with the constant $\mathcal{C}_{11}^{(3)} = \gamma_0' C_8 + \gamma_{10}C_7 + \gamma_0''(C_{25} + C_{26} + C_{27})$.
It remains to estimate the third term in (6.5). By (5.78), for $0< \varepsilon \leqslant \varepsilon_1$, we have
$$
\begin{equation}
\begin{aligned} \, &|\mathcal{I}_\varepsilon [\boldsymbol{\upsilon}_ \varepsilon]| \leqslant \gamma_{0}' \varepsilon \| \mathbf{F} \|_{L_2(\mathcal{O})} \| \boldsymbol{\upsilon}_ \varepsilon \|_{H^1(\mathcal{O})} + \gamma_{10} \varepsilon |\zeta|^{1/2} \| \mathbf{F} \|_{L_2(\mathcal{O})} \| \boldsymbol{\upsilon}_ \varepsilon \|_{L_2(\mathcal{O})} \nonumber \\ &\ +\gamma_{0}'' \varepsilon^{1/2} |\zeta|^{-1/4} \| \mathbf{F} \|_{L_2(\mathcal{O})} \| \mathbf{D} \boldsymbol{\upsilon}_ \varepsilon \|_{L_2(\Upsilon_\varepsilon)} \nonumber \\ &\ + \gamma_{0}'' \varepsilon^{1/2} |\zeta|^{-1/4} \| \mathbf{F} \|_{L_2(\mathcal{O})} \biggl( \biggl( \sum_{l=1}^d \| (a_l^\varepsilon)^* \boldsymbol{\upsilon}_ \varepsilon \|^2_{L_2(\Upsilon_\varepsilon)}\biggr)^{1/2} + \bigl\| |Q^\varepsilon|^{1/2} \boldsymbol{\upsilon}_ \varepsilon \bigr\|_{L_2(\Upsilon_\varepsilon)} \biggr). \end{aligned}
\end{equation}
\tag{6.17}
$$
Applying Lemma 3.3, we estimate the function $\boldsymbol{\upsilon}_ \varepsilon= \varepsilon K_N(\varepsilon;\zeta) \boldsymbol{\Phi}$ as follows:
$$
\begin{equation}
\| \boldsymbol{\upsilon}_ \varepsilon \|_{L_2(\mathcal{O})} \leqslant C_K' \varepsilon |\zeta|^{-1/2} \| \boldsymbol{\Phi} \|_{L_2(\mathcal{O})},
\end{equation}
\tag{6.18}
$$
$$
\begin{equation}
\| \boldsymbol{\upsilon}_ \varepsilon \|_{H^1(\mathcal{O})} \leqslant (C_K'+C_K'') \varepsilon \| \boldsymbol{\Phi} \|_{L_2(\mathcal{O})} + C_{K}'' |\zeta|^{-1/2} \| \boldsymbol{\Phi} \|_{L_2(\mathcal{O})}.
\end{equation}
\tag{6.19}
$$
Next, by (3.10),
$$
\begin{equation*}
\begin{aligned} \, \| D_l \boldsymbol{\upsilon}_ \varepsilon \|_{L_2(\Upsilon_\varepsilon)} &\leqslant \varepsilon \| \Lambda^\varepsilon S_\varepsilon b(\mathbf{D}) D_l \widetilde{\boldsymbol{\eta}}_0 \|_{L_2(\mathbb{R}^d)} + \varepsilon \| \widetilde{\Lambda}^\varepsilon S_\varepsilon D_l\widetilde{\boldsymbol{\eta}}_0 \|_{L_2(\mathbb{R}^d)} \\ &\qquad + \| (D_l \Lambda)^\varepsilon S_\varepsilon b(\mathbf{D}) \widetilde{\boldsymbol{\eta}}_0 \|_{L_2(\Upsilon_\varepsilon)} + \| (D_l \widetilde{\Lambda})^\varepsilon S_\varepsilon \widetilde{\boldsymbol{\eta}}_0 \|_{L_2(\Upsilon_\varepsilon)}. \end{aligned}
\end{equation*}
\notag
$$
Combining this with Proposition 1.2, Lemma 4.2, and employing relations (1.5), (1.24), (1.25), (1.32), (1.33), estimates (3.18)–(3.20) (for $\widetilde{\boldsymbol{\eta}}_0$), and the restriction $|\zeta| \geqslant 1$, for $0< \varepsilon \leqslant \varepsilon_1$, we obtain
$$
\begin{equation}
\begin{aligned} \, \| \mathbf{D} \boldsymbol{\upsilon}_ \varepsilon \|_{L_2(\Upsilon_\varepsilon)} &\leqslant \varepsilon M_1 \alpha_1^{1/2} \| \widetilde{\boldsymbol{\eta}}_0 \|_{H^2(\mathbb{R}^d)} + \varepsilon \widetilde{M}_1 \| \widetilde{\boldsymbol{\eta}}_0 \|_{H^1(\mathbb{R}^d)} \nonumber \\ &\qquad+ \varepsilon^{1/2} \beta_*^{1/2} {M}_2 \alpha_1^{1/2} \| \widetilde{\boldsymbol{\eta}}_0 \|^{1/2}_{H^2(\mathbb{R}^d)} \| \widetilde{\boldsymbol{\eta}}_0 \|^{1/2}_{H^1(\mathbb{R}^d)} \nonumber \\ &\qquad + \varepsilon^{1/2} \beta_*^{1/2} \widetilde{M}_2 \| \widetilde{\boldsymbol{\eta}}_0 \|^{1/2}_{H^1(\mathbb{R}^d)} \| \widetilde{\boldsymbol{\eta}}_0 \|^{1/2}_{L_2(\mathbb{R}^d)} \nonumber \\ &\leqslant C_{28} \varepsilon \| \boldsymbol{\Phi} \|_{L_2(\mathcal{O})} + C_{29} \varepsilon^{1/2} |\zeta|^{-1/4} \| \boldsymbol{\Phi} \|_{L_2(\mathcal{O})}. \end{aligned}
\end{equation}
\tag{6.20}
$$
Here, $C_{28}= M_1 \alpha_1^{1/2} C_9 + \widetilde{M}_1 C_8$, $C_{29} = \beta_*^{1/2} \bigl( M_2 \alpha_1^{1/2} (C_8 C_9)^{1/2} + \widetilde{M}_2 (C_7 C_8)^{1/2} \bigr)$.
Obviously,
$$
\begin{equation*}
\| (a_l^\varepsilon)^* \boldsymbol{\upsilon}_ \varepsilon \|_{L_2(\Upsilon_\varepsilon)} \leqslant \varepsilon \| (a_l^\varepsilon)^* \Lambda^\varepsilon S_\varepsilon b(\mathbf{D}) \widetilde{\boldsymbol{\eta}}_0 \|_{L_2(\mathbb{R}^d)} + \varepsilon \| (a_l^\varepsilon)^* \widetilde{\Lambda}^\varepsilon S_\varepsilon \widetilde{\boldsymbol{\eta}}_0 \|_{L_2(\mathbb{R}^d)}.
\end{equation*}
\notag
$$
Applying Proposition 1.2, relations (1.5), (5.28), (5.29), estimates (3.18), (3.19) (for $\widetilde{\boldsymbol{\eta}}_0$), and the restriction $|\zeta| \geqslant 1$, we obtain
$$
\begin{equation}
\begin{aligned} \, \biggl( \sum_{l=1}^d \| (a_l^\varepsilon)^* \boldsymbol{\upsilon}_ \varepsilon \|^2_{L_2(\Upsilon_\varepsilon)}\biggr)^{1/2} &\leqslant \varepsilon \biggl(\sum_{l=1}^d C_{17,l}^2\biggr)^{1/2} \alpha_1^{1/2} \| \widetilde{\boldsymbol{\eta}}_0 \|_{H^1(\mathbb{R}^d)} \nonumber \\ &\qquad+ \varepsilon \biggl(\sum_{l=1}^d C_{18,l}^2\biggr)^{1/2} \| \widetilde{\boldsymbol{\eta}}_0 \|_{L_2(\mathbb{R}^d)} \nonumber \\ &\leqslant C_{30} \varepsilon \| \boldsymbol{\Phi} \|_{L_2(\mathcal{O})}, \end{aligned}
\end{equation}
\tag{6.21}
$$
where $C_{30}= \bigl(\sum_{l=1}^d C_{17,l}^2\bigr)^{1/2} \alpha_1^{1/2} C_8 +\bigl(\sum_{l=1}^d C_{18,l}^2\bigr)^{1/2} C_7$. Similarly, by (5.73) and (5.74),
$$
\begin{equation}
\begin{aligned} \, \bigl\| |Q^\varepsilon|^{1/2} \boldsymbol{\upsilon}_ \varepsilon \bigr\|_{L_2(\Upsilon_\varepsilon)} &\leqslant \varepsilon \bigl\| |Q^\varepsilon|^{1/2} \Lambda^\varepsilon S_\varepsilon b(\mathbf{D}) \widetilde{\boldsymbol{\eta}}_0 \bigr\|_{L_2(\mathbb{R}^d)} + \varepsilon \bigl\| |Q^\varepsilon|^{1/2} \widetilde{\Lambda}^\varepsilon S_\varepsilon \widetilde{\boldsymbol{\eta}}_0 \bigr\|_{L_2(\mathbb{R}^d)} \nonumber \\ &\leqslant C_{23} \alpha_1^{1/2} \varepsilon \| \widetilde{\boldsymbol{\eta}}_0 \|_{H^1(\mathbb{R}^d)} + C_{24} \varepsilon \| \widetilde{\boldsymbol{\eta}}_0 \|_{L_2(\mathbb{R}^d)} \leqslant C_{31} \varepsilon \| \boldsymbol{\Phi} \|_{L_2(\mathcal{O})}, \end{aligned}
\end{equation}
\tag{6.22}
$$
where $C_{31} = C_{23} \alpha_1^{1/2} C_8 + C_{24} C_7$.
As a result, relations (6.17)–(6.22) imply that
$$
\begin{equation}
|\mathcal{I}_\varepsilon [ \boldsymbol{\upsilon}_ \varepsilon]| \leqslant \mathcal{C}_{11}^{(4)} (\varepsilon |\zeta|^{-1/2} + \varepsilon^2)\| \mathbf{F} \|_{L_2(\mathcal{O})} \| \boldsymbol{\Phi} \|_{L_2(\mathcal{O})}, \qquad 0< \varepsilon \leqslant \varepsilon_1,
\end{equation}
\tag{6.23}
$$
with the constant $\mathcal{C}_{11}^{(4)} = \gamma_0' (C_{K}'+ C_{K}'') +\gamma_{10} C_K' + \gamma_0''(C_{28}+ C_{29} + C_{30} + C_{31})$.
Now, relations (6.5)–(6.7), (6.9), (6.16), and (6.23) yield
$$
\begin{equation*}
|(\mathbf{w}_\varepsilon, \boldsymbol{\Phi})_{L_2(\mathcal{O})}| \leqslant \mathcal{C}_{11} (\varepsilon |\zeta|^{-1/2} + \varepsilon^2) \| \mathbf{F} \|_{L_2(\mathcal{O})} \| \boldsymbol{\Phi} \|_{L_2(\mathcal{O})}, \qquad 0< \varepsilon \leqslant \varepsilon_1,
\end{equation*}
\notag
$$
for any $\boldsymbol{\Phi} \in L_2(\mathcal{O};\mathbb{C}^n)$. Here, $\mathcal{C}_{11} = \mathcal{C}_{11}^{(1)} + \mathcal{C}_{11}^{(2)} + \mathcal{C}_{11}^{(3)} + \mathcal{C}_{11}^{(4)}$. This is equivalent to the required estimate (6.1). This completes the proof. 6.2. Completion of the proof of Theorem 3.2 for $\operatorname{Re} \zeta \leqslant 0$For $\operatorname{Re} \zeta \leqslant 0$, $|\zeta|\geqslant 1$, from (3.43) and (6.1) it follows that
$$
\begin{equation*}
\| \mathbf{u}_\varepsilon - \mathbf{u}_0 \|_{L_2(\mathcal{O})} \leqslant (\mathcal{C}_9 + \mathcal{C}_{11}) (\varepsilon |\zeta|^{-1/2} + \varepsilon^2)\| \mathbf{F} \|_{L_2(\mathcal{O})}, \qquad 0< \varepsilon \leqslant \varepsilon_1.
\end{equation*}
\notag
$$
For $|\zeta| \leqslant \varepsilon^{-2}$ we have $\varepsilon |\zeta|^{-1/2} + \varepsilon^2 \leqslant 2 \varepsilon |\zeta|^{-1/2}$. For $|\zeta| > \varepsilon^{-2}$, using estimates (2.23), (2.40) and noting that $ |\zeta|^{-1} < \varepsilon |\zeta|^{-1/2}$, we have
$$
\begin{equation*}
\| \mathbf{u}_\varepsilon - \mathbf{u}_0 \|_{L_2(\mathcal{O})} \leqslant 2 \| Q_0^{-1}\|_{L_\infty} |\zeta|^{-1} \| \mathbf{F} \|_{L_2(\mathcal{O})} \leqslant 2 \| Q_0^{-1}\|_{L_\infty} \varepsilon |\zeta|^{-1/2} \| \mathbf{F} \|_{L_2(\mathcal{O})}.
\end{equation*}
\notag
$$
As a result, we arrive at estimate (3.1) with the constant $\mathcal{C}'_3 = \max \{2 (\mathcal{C}_9 + \mathcal{C}_{11}); 2 \| Q_0^{-1}\|_{L_\infty} \}$. So, in operator terms, we have obtained the following estimate:
$$
\begin{equation}
\begin{gathered} \, \bigl\| (B_{N,\varepsilon} - \zeta Q_0^\varepsilon )^{-1} - (B_N^0 - \zeta \overline{Q_0}\,)^{-1} \bigr\|_{L_2(\mathcal{O})\to L_2(\mathcal{O})} \leqslant \mathcal{C}'_3 \varepsilon |\zeta|^{-1/2}, \\ \operatorname{Re} \zeta \leqslant 0,\quad |\zeta|\geqslant 1, \qquad 0<\varepsilon \leqslant \varepsilon_1. \end{gathered}
\end{equation}
\tag{6.24}
$$
§ 7. Completion of the proof of Theorems 3.2 and 3.47.1. The case $\operatorname{Re} \zeta > 0$. Completion of the proof of Theorem 3.2Now, let $\zeta \in \mathbb{C} \setminus \mathbb{R}_+$, $\operatorname{Re} \zeta > 0$, $|\zeta| \geqslant 1$. Setting $\widehat{\zeta} = - \operatorname{Re} \zeta+ i \operatorname{Im} \zeta$, we have $| \widehat{\zeta}| = |\zeta|$. Let us write down the already proved estimate (6.24) at the point $\widehat{\zeta}$ as follows:
$$
\begin{equation}
\bigl\| (B_{N,\varepsilon }- \widehat{\zeta} Q_0^\varepsilon )^{-1}-(B_N^0- \widehat{\zeta}\, \overline{Q_0}\,)^{-1} \bigr\|_{L_2(\mathcal{O})\to L_2(\mathcal{O})}\leqslant \mathcal{C}'_3 \varepsilon |\zeta|^{-1/2},\qquad 0<\varepsilon \leqslant \varepsilon_1.
\end{equation}
\tag{7.1}
$$
It is easily seen that
$$
\begin{equation}
\begin{aligned} \, &(B_{N,\varepsilon }-\zeta Q_0^\varepsilon )^{-1} - (B_N^0-\zeta \overline{Q_0}\,)^{-1} = (B_{N,\varepsilon} - \zeta Q_0^\varepsilon )^{-1}(B_{N,\varepsilon} - \widehat{\zeta} Q_0^\varepsilon) \nonumber \\ &\qquad\qquad\times\bigl( (B_{N,\varepsilon} - \widehat{\zeta}Q_0^\varepsilon )^{-1} - (B_N^0 - \widehat{\zeta}\, \overline{Q_0}\,)^{-1} \bigr) (B_N^0 - \widehat{\zeta}\, \overline{Q_0}\,)(B_N^0 -\zeta \overline{Q_0}\,)^{-1} \nonumber \\ &\qquad+(\zeta - \widehat{\zeta})(B_{N,\varepsilon}-\zeta Q_0^\varepsilon)^{-1} (Q_0^\varepsilon -\overline{Q_0}\,)(B_N^0-\zeta \overline{Q_0}\,)^{-1}. \end{aligned}
\end{equation}
\tag{7.2}
$$
Denote the terms on the right-hand side of (7.2) by $\mathfrak{T}_1(\varepsilon;\zeta)$ and $\mathfrak{T}_2(\varepsilon;\zeta)$. By (2.21),
$$
\begin{equation}
\begin{aligned} \, &\bigl\| (B_{N,\varepsilon}-\zeta Q_0^\varepsilon )^{-1}(B_{N,\varepsilon} - \widehat{\zeta} Q_0^\varepsilon) \bigr\|_{L_2(\mathcal{O}) \to L_2(\mathcal{O})} \nonumber \\ &\qquad= \bigl\| f^\varepsilon (\widetilde{B}_{N,\varepsilon}-\zeta I)^{-1} (\widetilde{B}_{N,\varepsilon} - \widehat{\zeta} I) (f^\varepsilon)^{-1} \bigr\|_{L_2(\mathcal{O}) \to L_2(\mathcal{O})} \nonumber \\ &\qquad\leqslant \| f\|_{L_\infty} \| f^{-1}\|_{L_\infty} \bigl\| (\widetilde{B}_{N,\varepsilon}-\zeta I)^{-1}(\widetilde{B}_{N,\varepsilon}- \widehat{\zeta} I)\bigr\|_{L_2(\mathcal{O})\to L_2(\mathcal{O})} \nonumber \\ &\qquad\leqslant \|f\|_{L_\infty} \|f^{-1}\|_{L_\infty} \sup_{x \geqslant 0} \frac{|x- \widehat{\zeta}|}{|x- {\zeta}|}. \end{aligned}
\end{equation}
\tag{7.3}
$$
Similarly, taking (2.37) and (2.38) into account, we obtain
$$
\begin{equation}
\bigl\| (B_N^0 - \widehat{\zeta}\, \overline{Q_0}\,)(B_N^0-\zeta \overline{Q_0}\,)^{-1} \bigr\|_{L_2(\mathcal{O}) \to L_2(\mathcal{O})} \leqslant \|f\|_{L_\infty} \|f^{-1}\|_{L_\infty} \sup_{x \geqslant 0} \frac{|x- \widehat{\zeta}|}{|x- {\zeta}|}.
\end{equation}
\tag{7.4}
$$
A calculation shows that
$$
\begin{equation}
\sup_{x \geqslant 0} \frac{|x- \widehat{\zeta}|}{|x- {\zeta}|} \leqslant 2 c(\phi).
\end{equation}
\tag{7.5}
$$
Relations (7.1) and (7.3)–(7.5) imply that
$$
\begin{equation}
\| \mathfrak{T}_1(\varepsilon;\zeta) \|_{L_2(\mathcal{O}) \to L_2(\mathcal{O})} \leqslant \mathcal{C}_3^{(1)} c(\phi)^2 \varepsilon |\zeta|^{-1/2}, \qquad 0< \varepsilon \leqslant \varepsilon_1,
\end{equation}
\tag{7.6}
$$
where $\mathcal{C}_3^{(1)} = 4 \mathcal{C}_3' \|f\|^2_{L_\infty} \|f^{-1}\|^2_{L_\infty}$. Now we consider the second term on the right-hand side of (7.2). Let $\boldsymbol{\Phi}_1, \boldsymbol{\Phi}_2 \in L_2(\mathcal{O};\mathbb{C}^n)$. By assertion $(3^\circ)$ of Lemma 4.4, for $0< \varepsilon \leqslant \varepsilon_1$, we have
$$
\begin{equation*}
\begin{aligned} \, &\bigl| \bigl( (Q_0^\varepsilon -\overline{Q_0}\,) (B_{N}^0 -\zeta \overline{Q_0}\, )^{-1} \boldsymbol{\Phi}_1, (B_{N,\varepsilon}-\zeta^* Q_0^\varepsilon )^{-1} \boldsymbol{\Phi}_2 \bigr)_{L_2(\mathcal{O})}\bigr| \\ &\qquad\leqslant \widetilde{C}''' \varepsilon \| Q_0\|_{L_\infty} \bigl( \|(B_{N}^0 -\zeta \overline{Q_0}\, )^{-1}\boldsymbol{\Phi}_1\|_{H^1(\mathcal{O})} \|(B_{N,\varepsilon}-\zeta^* Q_0^\varepsilon )^{-1} \boldsymbol{\Phi}_2\|_{L_2(\mathcal{O})} \\ &\qquad\qquad+ \|(B_{N}^0 -\zeta \overline{Q_0}\, )^{-1}\boldsymbol{\Phi}_1 \|_{L_2(\mathcal{O})} \|(B_{N,\varepsilon}-\zeta^* Q_0^\varepsilon )^{-1} \boldsymbol{\Phi}_2 \|_{H^1(\mathcal{O})} \bigr). \end{aligned}
\end{equation*}
\notag
$$
Combining this with (2.23), (2.24), (2.40), (2.41), and using the identity $\zeta - \widehat{\zeta} = 2 \operatorname{Re} \zeta$, we arrive at the estimate
$$
\begin{equation}
\| \mathfrak{T}_2(\varepsilon;\zeta) \|_{L_2(\mathcal{O}) \to L_2(\mathcal{O})} \leqslant \mathcal{C}_3^{(2)} c(\phi)^2 \varepsilon |\zeta|^{-1/2}, \qquad 0< \varepsilon \leqslant \varepsilon_1,
\end{equation}
\tag{7.7}
$$
where $\mathcal{C}_3^{(2)} = 4 \widetilde{C}''' \mathcal{C}_1 \|Q_0\|_{L_\infty} \|Q_0^{-1}\|_{L_\infty}$. As a result, relations (7.2), (7.6), and (7.7) imply the required estimate (3.2) with the constant $\mathcal{C}_3 = \mathcal{C}_3^{(1)} + \mathcal{C}_3^{(2)}$ in the case $\zeta \in \mathbb{C} \setminus \mathbb{R}_+$, $\operatorname{Re} \zeta > 0$, $|\zeta| \geqslant 1$. In view of the already proved estimate (6.24), this completes the proof of Theorem 3.2. 7.2. The case $\operatorname{Re} \zeta > 0$. Proof of estimate (3.14)Let us write down the already proved estimate (5.85) at the point $\widehat{\zeta}$ as follows:
$$
\begin{equation}
\begin{aligned} \, &\bigl\|(B_{N,\varepsilon}-\widehat{\zeta} Q_0^\varepsilon )^{-1} - (B_N^0-\widehat{\zeta}\, \overline{Q_0}\,)^{-1}-\varepsilon K_N (\varepsilon;\widehat{\zeta}) \bigr\|_{L_2(\mathcal{O})\to H^1(\mathcal{O})} \nonumber \\ &\qquad\leqslant \mathcal{C}_{10} \varepsilon^{1/2}| \zeta |^{-1/4}+ (\mathcal{C}_8+\mathcal{C}_{10}) \varepsilon,\qquad 0< \varepsilon \leqslant \varepsilon_1. \end{aligned}
\end{equation}
\tag{7.8}
$$
By using (7.2) and the obvious identity
$$
\begin{equation*}
K_N(\varepsilon;{\zeta}) = K_N(\varepsilon;\widehat{\zeta}) (B_N^0- \widehat{\zeta}\, \overline{Q_0}\,)(B_N^0-\zeta \overline{Q_0}\,)^{-1},
\end{equation*}
\notag
$$
it is easy to check that
$$
\begin{equation}
\begin{aligned} \, &(B_{N,\varepsilon}-\zeta Q_0^\varepsilon )^{-1} - (B_N^0-\zeta \overline{Q_0}\,)^{-1}-\varepsilon K_N(\varepsilon;\zeta) \nonumber \\ &\quad=\bigl((B_{N,\varepsilon}- \widehat{\zeta} Q_0^\varepsilon)^{-1} - (B_N^0 - \widehat{\zeta}\, \overline{Q_0}\,)^{-1}-\varepsilon K_N(\varepsilon;\widehat{\zeta}) \bigr) (B_N^0- \widehat{\zeta}\, \overline{Q_0}\,)(B_N^0-\zeta \overline{Q_0}\,)^{-1} \nonumber \\ &\quad\qquad+ (\zeta - \widehat{\zeta})(B_{N,\varepsilon}-\zeta Q_0^\varepsilon)^{-1} Q_0^\varepsilon \bigl((B_{N,\varepsilon} - \widehat{\zeta} Q_0^\varepsilon)^{-1} - (B_N^0- \widehat{\zeta}\, \overline{Q_0}\,)^{-1}\bigr) \nonumber \\ &\quad\qquad\qquad\times (B_N^0 - \widehat{\zeta}\,\overline{Q_0}\,)(B_N^0-\zeta \overline{Q_0}\,)^{-1} \nonumber \\ &\quad\qquad+(\zeta - \widehat{\zeta})(B_{N,\varepsilon }-\zeta Q_0^\varepsilon)^{-1} (Q_0^\varepsilon -\overline{Q_0}\,)(B_N^0-\zeta \overline{Q_0}\,)^{-1}. \end{aligned}
\end{equation}
\tag{7.9}
$$
Let $\mathfrak{L}_1(\varepsilon;\zeta)$, $\mathfrak{L}_2(\varepsilon;\zeta)$, and $\mathfrak{L}_3(\varepsilon;\zeta)$ be the consecutive summands on the right-hand side of (7.9). Note that $\mathfrak{L}_3(\varepsilon;\zeta) = \mathfrak{T}_2(\varepsilon;\zeta)$. From (7.4), (7.5), and (7.8) it follows that
$$
\begin{equation}
\| \mathfrak{L}_1(\varepsilon;\zeta) \|_{L_2(\mathcal{O}) \to H^1(\mathcal{O})} \leqslant \mathcal{C}_4 c(\phi) \varepsilon^{1/2} |\zeta|^{-1/4} + \mathcal{C}_5^{(1)} c(\phi) \varepsilon, \qquad 0< \varepsilon \leqslant \varepsilon_1,
\end{equation}
\tag{7.10}
$$
where $\mathcal{C}_4 = 2 \mathcal{C}_{10} \|f\|_{L_\infty} \|f^{-1}\|_{L_\infty}$, $\mathcal{C}_5^{(1)} = 2 (\mathcal{C}_8 + \mathcal{C}_{10}) \|f\|_{L_\infty} \|f^{-1}\|_{L_\infty}$. The second term on the right-hand side of (7.9) is estimated as
$$
\begin{equation*}
\begin{aligned} \, \| \mathfrak{L}_2(\varepsilon;\zeta)\|_{L_2(\mathcal{O})\to H^1(\mathcal{O})} &\leqslant 2 (\operatorname{Re}\zeta) \|(B_{N,\varepsilon}-\zeta Q_0^\varepsilon )^{-1} \|_{L_2(\mathcal{O})\to H^1(\mathcal{O})} \| Q_0 \|_{L_\infty} \\ &\qquad\times \bigl\| (B_{N,\varepsilon}- \widehat{\zeta} Q_0^\varepsilon)^{-1} - (B_N^0- \widehat{\zeta}\,\overline{Q_0}\,)^{-1} \bigr\|_{L_2(\mathcal{O}) \to L_2(\mathcal{O})} \\ &\qquad\times \bigl\| (B_N^0 - \widehat{\zeta}\,\overline{Q_0}\,)(B_N^0-\zeta \overline{Q_0}\,)^{-1} \bigr\|_{L_2(\mathcal{O}) \to L_2(\mathcal{O})}. \end{aligned}
\end{equation*}
\notag
$$
Combining this inequality with (2.24), (7.1), (7.4), and (7.5), we obtain
$$
\begin{equation}
\| \mathcal{L}_2(\varepsilon;\zeta) \|_{L_2(\mathcal{O}) \to H^1(\mathcal{O})} \leqslant \mathcal{C}_5^{(2)} c(\phi)^2 \varepsilon, \qquad 0 < \varepsilon \leqslant \varepsilon_1,
\end{equation}
\tag{7.11}
$$
where $\mathcal{C}_5^{(2)} = 4 \mathcal{C}'_3 \mathcal{C}_1 \| f\|_{L_\infty} \|f^{-1}\|^3_{L_\infty}$. Let us now consider the term $\mathfrak{L}_3(\varepsilon;\zeta)$. From (2.17), (2.20), and (2.21) we have
$$
\begin{equation}
\begin{aligned} \, &\| \mathfrak{L}_3(\varepsilon;\zeta)\|_{L_2(\mathcal{O}) \to H^1(\mathcal{O})} \leqslant c_4^{-1/2} \bigl\| B_{N,\varepsilon}^{1/2} \mathfrak{L}_3(\varepsilon;\zeta) \bigr\|_{L_2(\mathcal{O}) \to L_2(\mathcal{O})} \nonumber \\ &\quad= c_4^{-1/2} \bigl\| \widetilde{B}_{N,\varepsilon}^{1/2} (f^\varepsilon)^{-1}\mathfrak{L}_3(\varepsilon;\zeta) \bigr\|_{L_2(\mathcal{O}) \to L_2(\mathcal{O})} \nonumber \\ &\quad= 2 c_4^{-1/2} (\operatorname{Re}\zeta) \bigl\| \widetilde{B}_{N,\varepsilon}^{1/2} (\widetilde{B}_{N,\varepsilon}{-}\,\zeta I)^{-1} (f^\varepsilon)^* (Q_0^\varepsilon - \overline{Q_0}\,) (B_N^0 - \zeta \overline{Q_0}\,)^{-1} \bigr\|_{L_2(\mathcal{O}) \to L_2(\mathcal{O})}. \end{aligned}
\end{equation}
\tag{7.12}
$$
Let $\boldsymbol{\Phi}_1, \boldsymbol{\Phi}_2 \in L_2(\mathcal{O};\mathbb{C}^n)$. By assertion $(3^\circ)$ of Lemma 4.4, for $0< \varepsilon \leqslant \varepsilon_1$,
$$
\begin{equation}
\begin{aligned} \, &\bigl| \bigl( (Q_0^\varepsilon -\overline{Q_0}\,) (B_{N}^0 -\zeta \overline{Q_0}\, )^{-1} \boldsymbol{\Phi}_1, f^\varepsilon\widetilde{B}_{N,\varepsilon}^{1/2}(\widetilde{B}_{N,\varepsilon}-\zeta^* I )^{-1} \boldsymbol{\Phi}_2 \bigr)_{L_2(\mathcal{O})}\bigr| \nonumber \\ &\qquad\leqslant \widetilde{C}''' \varepsilon \| Q_0\|_{L_\infty} \bigl( \bigl\|(B_{N}^0 -\zeta \overline{Q_0}\, )^{-1}\boldsymbol{\Phi}_1 \bigr\|_{H^1(\mathcal{O})}\bigl\| f^\varepsilon\widetilde{B}_{N,\varepsilon}^{1/2}(\widetilde{B}_{N,\varepsilon}-\zeta^* I )^{-1} \boldsymbol{\Phi}_2 \bigr\|_{L_2(\mathcal{O})} \nonumber \\ &\qquad\qquad+ \|(B_{N}^0 -\zeta \overline{Q_0}\, )^{-1}\boldsymbol{\Phi}_1 \|_{L_2(\mathcal{O})} \bigl\| f^\varepsilon\widetilde{B}_{N,\varepsilon}^{1/2}(\widetilde{B}_{N,\varepsilon}-\zeta^* I )^{-1} \boldsymbol{\Phi}_2\bigr\|_{H^1(\mathcal{O})} \bigr). \end{aligned}
\end{equation}
\tag{7.13}
$$
Obviously,
$$
\begin{equation}
\begin{aligned} \, \bigl\| f^\varepsilon\widetilde{B}_{N,\varepsilon}^{1/2}(\widetilde{B}_{N,\varepsilon}-\zeta^* I )^{-1} \boldsymbol{\Phi}_2 \bigr\|_{L_2(\mathcal{O})} &\leqslant \|f\|_{L_\infty} \sup_{x \geqslant 0} \frac{x^{1/2}}{|x - \zeta^*|}\| \boldsymbol{\Phi}_2\|_{L_2(\mathcal{O})} \nonumber \\ &\leqslant \|f\|_{L_\infty} c(\phi) |\zeta|^{-1/2} \|\boldsymbol{\Phi}_2\|_{L_2(\mathcal{O})}. \end{aligned}
\end{equation}
\tag{7.14}
$$
According to (2.17) and (2.20),
$$
\begin{equation}
\begin{aligned} \, &\bigl\| f^\varepsilon\widetilde{B}_{N,\varepsilon}^{1/2}(\widetilde{B}_{N,\varepsilon}-\zeta^* I )^{-1} \boldsymbol{\Phi}_2 \bigr\|_{H^1(\mathcal{O})} \leqslant c_4^{-1/2} \bigl\|B_{N,\varepsilon}^{1/2} f^\varepsilon\widetilde{B}_{N,\varepsilon}^{1/2} (\widetilde{B}_{N,\varepsilon}-\zeta^* I )^{-1} \boldsymbol{\Phi}_2 \bigr\|_{L_2(\mathcal{O})} \nonumber \\ &\qquad= c_4^{-1/2} \bigl\| \widetilde{B}_{N,\varepsilon}(\widetilde{B}_{N,\varepsilon}-\zeta^* I )^{-1} \boldsymbol{\Phi}_2 \bigr\|_{L_2(\mathcal{O})} \leqslant c_4^{-1/2} \sup_{x \geqslant 0} x |x- \zeta^*|^{-1} \|\boldsymbol{\Phi}_2 \|_{L_2(\mathcal{O})} \nonumber \\ &\qquad\leqslant c_4^{-1/2} c(\phi) \|\boldsymbol{\Phi}_2 \|_{L_2(\mathcal{O})}. \end{aligned}
\end{equation}
\tag{7.15}
$$
From (2.40), (2.41), and (7.12)–(7.15) we have the estimate
$$
\begin{equation}
\| \mathfrak{L}_3(\varepsilon;\zeta) \|_{L_2(\mathcal{O}) \to H^1(\mathcal{O})} \leqslant \mathcal{C}_5^{(3)} c(\phi)^2 \varepsilon, \qquad 0< \varepsilon \leqslant \varepsilon_1,
\end{equation}
\tag{7.16}
$$
where $\mathcal{C}_5^{(3)} = 2 c_4^{-1/2} \widetilde{C}''' \|Q_0\|_{L_\infty}(\mathcal{C}_1 \|f\|_{L_\infty}+ c_4^{-1/2} \|Q_0^{-1}\|_{L_\infty})$. As a result, combining (7.9)–(7.11) and (7.16), we obtain estimate (3.14) with the constant $\mathcal{C}_5 = \mathcal{C}_5^{(1)}+ \mathcal{C}_5^{(2)} + \mathcal{C}_5^{(3)}$ in the case $\zeta \in \mathbb{C} \setminus \mathbb{R}_+$, $\operatorname{Re} \zeta > 0$, $|\zeta| \geqslant 1$. By (5.85), this completes the proof of estimate (3.14). 7.3. Proof of estimate (3.15)From (3.13) and (1.3), (1.6), we have, for $0 < \varepsilon \leqslant \varepsilon_1$,
$$
\begin{equation}
\| \mathbf{p}_\varepsilon - g^\varepsilon b(\mathbf{D}) \mathbf{v}_\varepsilon \|_{L_2(\mathcal{O})} \leqslant \| g\|_{L_\infty} (d \alpha_1)^{1/2} \bigl( \mathcal{C}_4 c(\phi) \varepsilon^{1/2}|\zeta|^{-1/4}+ \mathcal{C}_5c(\phi)^2 \varepsilon \bigr) \| \mathbf{F} \|_{L_2(\mathcal{O})}.
\end{equation}
\tag{7.17}
$$
By (3.11) and (3.12),
$$
\begin{equation}
\begin{aligned} \, g^\varepsilon b(\mathbf{D}) \mathbf{v}_\varepsilon &= g^\varepsilon b(\mathbf{D}) \mathbf{u}_0 + g^\varepsilon (b(\mathbf{D})\Lambda)^\varepsilon S_\varepsilon b(\mathbf{D}) \widetilde{\mathbf{u}}_0 + g^\varepsilon (b(\mathbf{D}) \widetilde{\Lambda})^\varepsilon S_\varepsilon \widetilde{\mathbf{u}}_0 \nonumber \\ &\qquad + \varepsilon \sum_{l=1}^d g^\varepsilon b_l (\Lambda^\varepsilon S_\varepsilon b(\mathbf{D}) D_l \widetilde{\mathbf{u}}_0 + \widetilde{\Lambda}^\varepsilon S_\varepsilon D_l \widetilde{\mathbf{u}}_0). \end{aligned}
\end{equation}
\tag{7.18}
$$
The fourth summand on the right of (7.18) is estimated via Proposition 1.2 and relations (1.5), (1.6), (1.24), (1.32), (3.19), (3.20) as follows:
$$
\begin{equation}
\begin{aligned} \, &\biggl\| \varepsilon \sum_{l=1}^d g^\varepsilon b_l (\Lambda^\varepsilon S_\varepsilon b(\mathbf{D}) D_l \widetilde{\mathbf{u}}_0 + \widetilde{\Lambda}^\varepsilon S_\varepsilon D_l \widetilde{\mathbf{u}}_0) \biggr\|_{L_2(\mathcal{O})} \nonumber \\ &\qquad\leqslant\varepsilon (d \alpha_1)^{1/2}\|g\|_{L_\infty} (M_1 \alpha_1^{1/2} \| \widetilde{\mathbf{u}}_0 \|_{H^2(\mathbb{R}^d)} + \widetilde{M}_1 \| \widetilde{\mathbf{u}}_0 \|_{H^1(\mathbb{R}^d)}) \nonumber \\ &\qquad \leqslant C_{32} c(\phi) \varepsilon \| \mathbf{F} \|_{L_2(\mathcal{O})}, \end{aligned}
\end{equation}
\tag{7.19}
$$
where $C_{32}= (d \alpha_1)^{1/2}\|g\|_{L_\infty}(M_1 \alpha_1^{1/2} C_9 + \widetilde{M}_1 C_8)$. Next, by Proposition 1.1 and relations (1.5), (3.20), we have
$$
\begin{equation}
\| g^\varepsilon b(\mathbf{D}) (I \,{-}\, S_\varepsilon) \widetilde{\mathbf{u}}_0 \|_{L_2(\mathbb{R}^d)} \leqslant r_1 \alpha_1^{1/2} \|g\|_{L_\infty} \varepsilon \| \widetilde{\mathbf{u}}_0 \|_{H^2(\mathbb{R}^d)}\leqslant C_{33} c(\phi) \varepsilon \| \mathbf{F} \|_{L_2(\mathcal{O})},
\end{equation}
\tag{7.20}
$$
where $C_{33}= r_1 \alpha_1^{1/2} \|g\|_{L_\infty} C_9$. As a result, combining (1.22) and (7.17)–(7.20), we arrive at the required estimate (3.15) with the constants $\widetilde{\mathcal{C}}_4 = \mathcal{C}_4 \|g\|_{L_\infty} (d \alpha_1)^{1/2}$, $\widetilde{\mathcal{C}}_5 = \mathcal{C}_5 \|g\|_{L_\infty} (d \alpha_1)^{1/2} + C_{32} + C_{33}$. This completes the proof of Theorem 3.4. 7.4. Proof of Corollary 3.5By (1.3), (1.6), (2.24), and (3.9), we have
$$
\begin{equation*}
\begin{aligned} \, &\| g^\varepsilon b(\mathbf{D}) (B_{N,\varepsilon} - \zeta Q_0^\varepsilon)^{-1} - G_N(\varepsilon;\zeta)\|_{L_2(\mathcal{O}) \to L_2(\mathcal{O})} \\ &\qquad \leqslant \| g \|_{L_\infty} (d \alpha_1)^{1/2} \| (B_{N,\varepsilon} - \zeta Q_0^\varepsilon)^{-1}\|_{L_2(\mathcal{O}) \to H^1 (\mathcal{O})} + \| G_N(\varepsilon;\zeta)\|_{L_2(\mathcal{O}) \to L_2(\mathcal{O})} \\ &\qquad\leqslant C_{34} c(\phi) |\zeta|^{-1/2}, \end{aligned}
\end{equation*}
\notag
$$
where $C_{34} = \|g\|_{L_\infty} (d \alpha_1)^{1/2} \mathcal{C}_1 + C_G$. Together with (3.16) this implies that, for $\zeta \in \mathbb{C}\setminus \mathbb{R}_+$, $|\zeta|\geqslant 1$, and $0 < \varepsilon \leqslant \varepsilon_1$,
$$
\begin{equation*}
\begin{aligned} \, &\| g^\varepsilon b(\mathbf{D}) (B_{N,\varepsilon} - \zeta Q_0^\varepsilon)^{-1} - G_N(\varepsilon;\zeta)\|_{L_2(\mathcal{O}) \to L_2(\mathcal{O})} \\ &\qquad\leqslant \min \bigl\{ C_{34} c(\phi) |\zeta|^{-1/2};\, \widetilde{\mathcal{C}}_4 c(\phi) \varepsilon^{1/2} |\zeta|^{-1/4} + \widetilde{\mathcal{C}}_5 c(\phi)^2 \varepsilon\bigr\} \\ &\qquad\leqslant \widetilde{\mathcal{C}}_4 c(\phi) \varepsilon^{1/2} |\zeta|^{-1/4} + \min \bigl\{ C_{34} c(\phi) |\zeta|^{-1/2};\, \widetilde{\mathcal{C}}_5 c(\phi)^2 \varepsilon\bigr\} \\ &\qquad\leqslant \widetilde{\mathcal{C}}_4 c(\phi) \varepsilon^{1/2} |\zeta|^{-1/4} + (C_{34} \widetilde{\mathcal{C}}_5)^{1/2}c(\phi)^{3/2} \varepsilon^{1/2} |\zeta|^{-1/4}. \end{aligned}
\end{equation*}
\notag
$$
This yields the required estimate (3.17) with the constant $\widetilde{\mathcal{C}}'_4 = \widetilde{\mathcal{C}}_4 + (C_{34} \widetilde{\mathcal{C}}_5)^{1/2}$.
§ 8. Removal of the smoothing operator. Special cases. Estimates in a strictly interior subdomain8.1. Removal of the operator $S_\varepsilon$ in the correctorLet us show that, under some additional assumptions about the solutions of the auxiliary problems, the smoothing operator $S_\varepsilon$ in the corrector can be removed (replaced by the identity operator keeping the same order of error). Condition 8.1. The $\Gamma$-periodic solution $\Lambda (\mathbf{x})$ of problem (1.20) is bounded, that is, $\Lambda\in L_\infty (\mathbb{R}^d)$. Condition 8.2. The $\Gamma$-periodic solution $\widetilde{\Lambda}(\mathbf{x})$ of problem (1.30) is such that Some cases where Conditions 8.1 and 8.2 hold automatically were distinguished in [9], Lemma 8.7, and [10], Proposition 8.11, respectively. Proposition 8.3 (see [9]). Suppose that at least one of the following assumptions is satisfied: $1^\circ)$ $d\leqslant 2$; $2^\circ)$ the dimension $d$ is arbitrary, and the operator $A_\varepsilon$ is of the form $A_\varepsilon =\mathbf{D}^* g^\varepsilon (\mathbf{x})\mathbf{D}$, where $g(\mathbf{x})$ is a symmetric matrix with real entries; $3^\circ)$ the dimension $d$ is arbitrary, and $g^0=\underline{g}$, that is, (1.29) holds. Then Condition 8.1 is fulfilled. Proposition 8.4 (see [10]). Suppose that at least one of the following assumptions is satisfied: $1^\circ)$ $d\leqslant 4$; $2^\circ)$ the dimension $d$ is arbitrary, and the operator $A_\varepsilon$ is of the form $A_\varepsilon =\mathbf{D}^* g^\varepsilon (\mathbf{x})\mathbf{D}$, where $g(\mathbf{x})$ is a symmetric matrix with real entries. Then Condition 8.2 is fulfilled. Remark 8.5. If $A_\varepsilon =\mathbf{D}^* g^\varepsilon (\mathbf{x})\mathbf{D}$, where $g(\mathbf{x})$ is a symmetric matrix with real entries, then it follows from [50], Chap. III, Theorem 13.1, that $\Lambda \in L_\infty$ and $\widetilde{\Lambda} \in L_\infty$. Hence Conditions 8.1 and 8.2 are fulfilled. Moreover, the norm $\|\Lambda\|_{L_\infty}$ does not exceed a constant depending on $d$, $\| g\|_{L_\infty}$, $\| g^{-1}\|_{L_\infty}$, and $\Omega$, while the norm $\| \widetilde{\Lambda}\|_{L_\infty}$ is estimated in terms of $d$, $\rho$, $\| g\|_{L_\infty}$, $\| g^{-1}\|_{L_\infty}$, $\| a_j\|_{L_\rho (\Omega)}$, $j=1,\dots,d$, and $\Omega$. We put
$$
\begin{equation}
K_N^0(\varepsilon;\zeta ) := \bigl( \Lambda^\varepsilon b(\mathbf{D}) + \widetilde{\Lambda}^\varepsilon \bigr) (B_N^0-\zeta \overline{Q_0}\,)^{-1},
\end{equation}
\tag{8.1}
$$
$$
\begin{equation}
G_N^0(\varepsilon;\zeta ) := \widetilde{g}^{\,\varepsilon} b(\mathbf{D})(B_N^0-\zeta\overline{Q_0}\,)^{-1} +g^\varepsilon \bigl(b(\mathbf{D})\widetilde{\Lambda}\bigr)^\varepsilon (B_N^0-\zeta\overline{Q_0}\,)^{-1}.
\end{equation}
\tag{8.2}
$$
If Conditions 8.1 and 8.2 are satisfied, then the operator (8.1) is continuous from $L_2(\mathcal{O};\mathbb{C}^n)$ to $H^1(\mathcal{O};\mathbb{C}^n)$, and the operator (8.2) is continuous from $L_2(\mathcal{O};\mathbb{C}^n)$ to $L_2(\mathcal{O};\mathbb{C}^m)$. This can be easily checked by using Lemmas 4.5–4.7. Our goal in this subsection is to prove the following theorem. Theorem 8.6. Under the hypotheses of Theorem 3.4, let Conditions 8.1 and 8.2 be met. Let $K_N^0(\varepsilon;\zeta )$ and $G_N^0(\varepsilon;\zeta )$ be the operators (8.1) and (8.2). Then, for $\zeta \in \mathbb{C}\setminus \mathbb{R}_+$, $|\zeta|\geqslant 1$, and $0<\varepsilon\leqslant\varepsilon_1$, To prove Theorem 8.6, we need the following assertions (see Lemmas 7.7 and 7.8 in [40]). Lemma 8.7. Suppose that Condition 8.1 is satisfied. Let $S_\varepsilon$ be the Steklov smoothing operator (1.1). Then, for $0<\varepsilon\leqslant 1$, The constant $\mathfrak{C}_\Lambda$ depends only on $m$, $d$, $\alpha_0$, $\alpha_1$, $\| g\|_{L_\infty}$, $\| g^{-1}\|_{L_\infty}$, the parameters of the lattice $\Gamma$, and the norm $\| \Lambda\|_{L_\infty}$. Lemma 8.8. Suppose that Condition 8.2 is satisfied. Let $S_\varepsilon$ be the smoothing operator (1.1). Then, for $0<\varepsilon\leqslant 1$, The constant $\mathfrak{C}_{\widetilde{\Lambda}}$ depends only on $n$, $d$, $\alpha_0$, $\alpha_1$, $\rho$, $\| g\|_{L_\infty}$, $\| g^{-1}\|_{L_\infty}$, the norms $\| a_j\|_{L_\rho (\Omega)}$, $j=1,\dots,d$, $p$, $\| \widetilde{\Lambda}\|_{L_p(\Omega)}$, and the parameters of the lattice $\Gamma$. It is easy to check Lemma 8.7 using Lemma 4.5, and Lemma 8.8 is proved with the help of Lemmas 4.6 and 4.7. 8.2. Proof of Theorem 8.6Combining (2.42), (3.4), and Lemma 8.7, we have, under Condition 8.1,
$$
\begin{equation}
\varepsilon \bigl\| \Lambda^\varepsilon (S_\varepsilon -I) b(\mathbf{D})P_\mathcal{O}(B_N^0-\zeta\overline{Q_0}\,)^{-1} \bigr\|_{L_2(\mathcal{O})\to H^1(\mathcal{O})} \leqslant \mathfrak{C}_\Lambda C_\mathcal{O}^{(2)}\mathcal{C}_2 c(\phi) \varepsilon.
\end{equation}
\tag{8.5}
$$
Similarly, employing (2.42), (3.4) and Lemma 8.8, we have, under Condition 8.2,
$$
\begin{equation}
\varepsilon\bigl\| \widetilde{\Lambda}^\varepsilon (S_\varepsilon -I) P_\mathcal{O}(B_N^0-\zeta\overline{Q_0}\,)^{-1}\bigr\|_{L_2(\mathcal{O})\to H^1(\mathcal{O})} \leqslant \mathfrak{C}_{\widetilde{\Lambda}}C_\mathcal{O}^{(2)}\mathcal{C}_2c(\phi) \varepsilon.
\end{equation}
\tag{8.6}
$$
From (3.14), (8.5), and (8.6) we have estimate (8.3) with the constant $\mathcal{C}_{12} = \mathcal{C}_5+ (\mathfrak{C}_{\Lambda} + \mathfrak{C}_{\widetilde{\Lambda}}) C_\mathcal{O}^{(2)} \mathcal{C}_2$. It remains to check (8.4). From (8.3), in view of (1.3) and (1.6), we have
$$
\begin{equation}
\begin{aligned} \, &\!\bigl\| g^\varepsilon b(\mathbf{D}) (B_{N,\varepsilon}{-}\,\zeta Q_0^\varepsilon )^{-1} {-}\, g^\varepsilon b(\mathbf{D}) \bigl(I+\varepsilon \bigl(\Lambda^\varepsilon b(\mathbf{D})+\widetilde{\Lambda}^\varepsilon \bigr)\bigr) (B_N^0-\zeta\overline{Q_0}\,)^{-1}\bigr\|_{L_2(\mathcal{O})\to L_2(\mathcal{O})} \nonumber \\ &\quad\leqslant (d\alpha_1)^{1/2}\| g\|_{L_\infty} \bigl(\mathcal{C}_4 c(\phi ) \varepsilon^{1/2}|\zeta|^{-1/4}+\mathcal{C}_{12}c(\phi)^2 \varepsilon\bigr) \end{aligned}
\end{equation}
\tag{8.7}
$$
for $0< \varepsilon \leqslant \varepsilon_1$. We have
$$
\begin{equation}
\begin{aligned} \, &\varepsilon g^\varepsilon b(\mathbf{D}) \bigl(\Lambda^\varepsilon b(\mathbf{D})+\widetilde{\Lambda}^\varepsilon \bigr)(B_N^0-\zeta\overline{Q_0}\,)^{-1} \nonumber \\ &\qquad= g^\varepsilon \bigl(b(\mathbf{D})\Lambda\bigr)^\varepsilon b(\mathbf{D})(B_N^0-\zeta \overline{Q_0}\,)^{-1} + g^\varepsilon \bigl( b(\mathbf{D})\widetilde{\Lambda}\bigr)^\varepsilon (B_N^0-\zeta\overline{Q_0}\,)^{-1} \nonumber \\ &\qquad\qquad+ \varepsilon\sum_{l=1}^d g^\varepsilon b_l \bigl(\Lambda^\varepsilon b(\mathbf{D}) + \widetilde{\Lambda}^\varepsilon \bigr) D_l(B_N^0-\zeta\overline{Q_0}\,)^{-1}. \end{aligned}
\end{equation}
\tag{8.8}
$$
From Condition 8.1 and (1.3), (1.6), and (2.42) it follows that
$$
\begin{equation}
\begin{aligned} \, &\varepsilon \sum_{l=1}^d \bigl\| g^\varepsilon b_l \Lambda^\varepsilon b(\mathbf{D})D_l(B_N^0-\zeta\overline{Q_0}\,)^{-1}\bigr\|_{L_2(\mathcal{O})\to L_2(\mathcal{O})} \nonumber \\ &\qquad\leqslant \varepsilon\alpha_1 d\| g\|_{L_\infty}\| \Lambda\|_{L_\infty}\bigl\| (B_N^0-\zeta\overline{Q_0}\,)^{-1} \bigr\|_{L_2(\mathcal{O})\to H^2(\mathcal{O})} \leqslant C_{35} c(\phi) \varepsilon, \end{aligned}
\end{equation}
\tag{8.9}
$$
where $C_{35} = d \alpha_1 \| g\|_{L_\infty}\| \Lambda\|_{L_\infty} \mathcal{C}_2$. Next, from (1.6), (2.42), (3.4), Condition 8.2, and Lemma 4.6, we have
$$
\begin{equation}
\begin{aligned} \, &\varepsilon\sum_{l=1}^d \bigl\| g^\varepsilon b_l \widetilde{\Lambda}^\varepsilon D_l (B_N^0-\zeta\overline{Q_0}\,)^{-1}\bigr\|_{L_2(\mathcal{O})\to L_2(\mathcal{O})} \nonumber \\ &\qquad\leqslant \varepsilon \alpha_1^{1/2}\| g\|_{L_\infty} \sum_{l=1}^d \bigl\| \widetilde{\Lambda}^\varepsilon P_\mathcal{O} D_l (B_N^0-\zeta\overline{Q_0}\,)^{-1}\bigr\|_{L_2(\mathcal{O})\to L_2(\mathbb{R}^d)} \nonumber \\ &\qquad\leqslant \varepsilon \alpha_1^{1/2}\| g\|_{L_\infty} C(\widehat{q};\Omega) \| \widetilde{\Lambda}\|_{L_p(\Omega)} \sum_{l=1}^d \bigl\| P_\mathcal{O}{D_l}(B_N^0-\zeta\overline{Q_0}\,)^{-1} \bigr\|_{L_2(\mathcal{O})\to H^1(\mathbb{R}^d)} \nonumber \\ &\qquad\leqslant \varepsilon (d\alpha_1)^{1/2}\| g\|_{L_\infty}\| \widetilde{\Lambda}\|_{L_p(\Omega)}C(\widehat{q}; \Omega)C_\mathcal{O}^{(1)} \bigl \| (B_N^0-\zeta\overline{Q_0}\,)^{-1} \bigr \|_{L_2(\mathcal{O})\to H^2(\mathcal{O})} \nonumber \\ &\qquad\leqslant C_{36} c(\phi) \varepsilon, \end{aligned}
\end{equation}
\tag{8.10}
$$
where $C_{36} = (d\alpha_1)^{1/2}\| g\|_{L_\infty}\| \widetilde{\Lambda}\|_{L_p(\Omega)}C(\widehat{q};\Omega)C_\mathcal{O}^{(1)}\mathcal{C}_2$. As a result, relations (8.7)–(8.10) together with (1.22) and (8.2) imply the required estimate (8.4) with the constant $\widetilde{\mathcal{C}}_{12}= (d\alpha_1)^{1/2}\| g\|_{L_\infty}\mathcal{C}_{12} + C_{35}+C_{36}$. This completes the proof. Remark 8.9. If only Condition 8.1 (respectively, Condition 8.2) is satisfied, then the smoothing operator $S_\varepsilon$ can be removed only in the term of the corrector containing $\Lambda^\varepsilon$ (respectively, $\widetilde{\Lambda}^\varepsilon$). 8.3. The case of vanishing correctorSuppose that $g^0=\overline{g}$, that is, relations (1.28) are satisfied. Then the $\Gamma$-periodic solution of problem (1.20) vanishes: $\Lambda (\mathbf{x})=0$. Suppose also that Then the $\Gamma$-periodic solution of problem (1.30) also vanishes: $\widetilde{\Lambda}(\mathbf{x})=0$. Therefore, in the case under consideration, the operator (3.5) is equal to zero, relation (3.14) simplifies, and Theorem 3.4 implies the following result.Proposition 8.10. Let (1.28) and (8.11) hold. Then, for $0<\varepsilon\leqslant \varepsilon_1$ and $\zeta\in\mathbb{C}\setminus\mathbb{R}_+$, $|\zeta|\geqslant 1$, 8.4. Special caseSuppose that $g^0=\underline{g}$, that is, relations (1.29) are satisfied. Then, by Proposition 8.3($3^\circ$), Condition 8.1 is met. According to Remark 3.5 in [8], the matrix-valued function (1.22) is constant and coincides with $g^0$, that is, $\widetilde{g}(\mathbf{x})=g^0=\underline{g}$. In addition, assume that (8.11) is fulfilled. Then $\widetilde{\Lambda}(\mathbf{x})=0$, and Theorem 8.6 implies the following result. Proposition 8.11. Let (1.29) and (8.11) hold. Then, for $0<\varepsilon\leqslant\varepsilon_1$ and $\zeta\in\mathbb{C}\setminus\mathbb{R}_+$, $|\zeta|\geqslant 1$, 8.5. Estimates in a strictly interior subdomainLet $\mathcal{O}'$ be a strictly interior subdomain of the domain $\mathcal{O}$. Using Theorem 3.2 and the results for homogenization problem in $\mathbb{R}^d$, it is easy to approximate the generalized resolvent $(B_{N,\varepsilon}-\zeta Q_0^\varepsilon )^{-1}$ in the $(L_2(\mathcal{O})\to H^1(\mathcal{O}'))$-norm with an error estimate of sharp order $O(\varepsilon)$. Theorem 8.12. Under the hypotheses of Theorem 3.4, let $\mathcal{O}'$ be a strictly interior subdomain of the domain $\mathcal{O}$. We set $\delta :=\operatorname{dist}\{ \mathcal{O}';\partial \mathcal{O}\}$. Then, for $\zeta\in\mathbb{C}\setminus\mathbb{R}_+$, $|\zeta|\geqslant 1$, and $0<\varepsilon\leqslant\varepsilon_1$, Proof. Let us start with the case $\operatorname{Re} \zeta \leqslant 0$, $|\zeta| \geqslant 1$.
We fix a smooth cut-off function $\chi(\mathbf{x})$ such that
$$
\begin{equation}
\chi \in C_0^\infty(\mathcal{O}); \quad 0\leqslant \chi(\mathbf{x}) \leqslant 1; \quad \chi(\mathbf{x})=1\ \text{ for } \ \mathbf{x} \in \mathcal{O}'; \quad |\nabla \chi(\mathbf{x})| \leqslant \varkappa \delta^{-1}.
\end{equation}
\tag{8.14}
$$
The constant $\varkappa$ depends only on the dimension $d$ and the domain $\mathcal{O}$.
Let $\mathbf{u}_\varepsilon = (B_{N,\varepsilon} - \zeta Q_0^\varepsilon)^{-1} \mathbf{F}$ and $\widetilde{\mathbf{u}}_\varepsilon = (B_{\varepsilon} - \zeta Q_0^\varepsilon)^{-1} \widetilde{\mathbf{F}}$. Then
$$
\begin{equation}
\mathfrak{b}_{N,\varepsilon} [\mathbf{u}_\varepsilon - \widetilde{\mathbf{u}}_\varepsilon, \boldsymbol{\eta}] - \zeta (Q_0^\varepsilon (\mathbf{u}_\varepsilon - \widetilde{\mathbf{u}}_\varepsilon), \boldsymbol{\eta})_{L_2(\mathcal{O})}=0, \qquad \boldsymbol{\eta} \in H^1_0 (\mathcal{O};\mathbb{C}^n).
\end{equation}
\tag{8.15}
$$
We substitute $\boldsymbol{\eta} = \chi^2 (\mathbf{u}_\varepsilon - \widetilde{\mathbf{u}}_\varepsilon)$ in (8.15) and denote
$$
\begin{equation*}
\mathfrak{U}(\varepsilon) = \mathfrak{b}_{N,\varepsilon} [\chi(\mathbf{u}_\varepsilon - \widetilde{\mathbf{u}}_\varepsilon),\, \chi(\mathbf{u}_\varepsilon - \widetilde{\mathbf{u}}_\varepsilon)].
\end{equation*}
\notag
$$
The corresponding identity can be written as
$$
\begin{equation}
\begin{aligned} \, &\mathfrak{U}(\varepsilon)- \zeta \bigl(Q_0^\varepsilon \chi (\mathbf{u}_\varepsilon - \widetilde{\mathbf{u}}_\varepsilon), \chi (\mathbf{u}_\varepsilon - \widetilde{\mathbf{u}}_\varepsilon)\bigr)_{L_2(\mathcal{O})} \nonumber \\ &\qquad= 2 i \operatorname{Im} \bigl(g^\varepsilon \mathbf{z}_\varepsilon, b(\mathbf{D}) \chi(\mathbf{u}_\varepsilon - \widetilde{\mathbf{u}}_\varepsilon)\bigr)_{L_2(\mathcal{O})} + (g^\varepsilon \mathbf{z}_\varepsilon, \mathbf{z}_\varepsilon )_{L_2(\mathcal{O})} \nonumber \\ &\qquad\qquad+ 2 i \operatorname{Im} \sum_{j=1}^d \bigl( (D_j \chi) (\mathbf{u}_\varepsilon - \widetilde{\mathbf{u}}_\varepsilon), (a_j^\varepsilon)^* \chi (\mathbf{u}_\varepsilon - \widetilde{\mathbf{u}}_\varepsilon)\bigr)_{L_2(\mathcal{O})}, \end{aligned}
\end{equation}
\tag{8.16}
$$
where $\mathbf{z}_\varepsilon := \sum_{l=1}^d b_l (D_l \chi) (\mathbf{u}_\varepsilon - \widetilde{\mathbf{u}}_\varepsilon)$. By (1.6) and (8.14),
$$
\begin{equation}
\| \mathbf{z}_\varepsilon\|_{L_2(\mathcal{O})} \leqslant (d \alpha_1)^{1/2} \varkappa \delta^{-1} \| \mathbf{u}_\varepsilon - \widetilde{\mathbf{u}}_\varepsilon \|_{L_2(\mathcal{O})}.
\end{equation}
\tag{8.17}
$$
We take the real part of relation (8.16):
$$
\begin{equation*}
\mathfrak{U}(\varepsilon)- (\operatorname{Re} \zeta) \bigl(Q_0^\varepsilon \chi (\mathbf{u}_\varepsilon - \widetilde{\mathbf{u}}_\varepsilon), \chi (\mathbf{u}_\varepsilon - \widetilde{\mathbf{u}}_\varepsilon)\bigr)_{L_2(\mathcal{O})} = (g^\varepsilon \mathbf{z}_\varepsilon, \mathbf{z}_\varepsilon )_{L_2(\mathcal{O})}.
\end{equation*}
\notag
$$
Since $\operatorname{Re} \zeta \leqslant 0$, both terms on the left are non-negative, whence
$$
\begin{equation}
\mathfrak{U}(\varepsilon) \leqslant (g^\varepsilon \mathbf{z}_\varepsilon, \mathbf{z}_\varepsilon )_{L_2(\mathcal{O})} \leqslant \|g\|_{L_\infty} \| \mathbf{z}_\varepsilon\|^2_{L_2(\mathcal{O})} \leqslant \|g\|_{L_\infty} d \alpha_1 \varkappa^2 \delta^{-2} \| \mathbf{u}_\varepsilon - \widetilde{\mathbf{u}}_\varepsilon \|^2_{L_2(\mathcal{O})}.
\end{equation}
\tag{8.18}
$$
We have taken (8.17) into account.
Next, we extend the function $\chi (\mathbf{u}_\varepsilon - \widetilde{\mathbf{u}}_\varepsilon)$ by zero to $\mathbb{R}^d \setminus \mathcal{O}$ and note that $\mathfrak{U}(\varepsilon) =\mathfrak{b}_{\varepsilon} [\chi(\mathbf{u}_\varepsilon - \widetilde{\mathbf{u}}_\varepsilon), \chi(\mathbf{u}_\varepsilon - \widetilde{\mathbf{u}}_\varepsilon)]$. Now from (1.17) and (8.18) we have
$$
\begin{equation}
\| \mathbf{D} \chi (\mathbf{u}_\varepsilon - \widetilde{\mathbf{u}}_\varepsilon) \|_{L_2(\mathcal{O})} \leqslant C_{37} \delta^{-1} \| \mathbf{u}_\varepsilon - \widetilde{\mathbf{u}}_\varepsilon \|_{L_2(\mathcal{O})},
\end{equation}
\tag{8.19}
$$
where $C_{37} = c_*^{-1/2} (d \alpha_1)^{1/2} \varkappa \|g\|_{L_\infty}^{1/2}$.
Estimates (3.1) and (3.24) show that
$$
\begin{equation}
\| \mathbf{u}_\varepsilon - \widetilde{\mathbf{u}}_\varepsilon \|_{L_2(\mathcal{O})} \leqslant \mathcal{C}_{14}^{(1)} \varepsilon |\zeta|^{-1/2} \| \mathbf{F} \|_{L_2(\mathcal{O})},\qquad 0< \varepsilon \leqslant \varepsilon_1,
\end{equation}
\tag{8.20}
$$
where $\mathcal{C}_{14}^{(1)}= \mathcal{C}_3 + C_1C_{10}$. By (8.19) and (8.20),
$$
\begin{equation}
\| \mathbf{D} \chi (\mathbf{u}_\varepsilon - \widetilde{\mathbf{u}}_\varepsilon) \|_{L_2(\mathcal{O})} \leqslant \mathcal{C}_{13}' \delta^{-1} \varepsilon |\zeta|^{-1/2} \| \mathbf{F} \|_{L_2(\mathcal{O})}, \qquad 0< \varepsilon \leqslant \varepsilon_1,
\end{equation}
\tag{8.21}
$$
where $\mathcal{C}_{13}' = C_{37} \mathcal{C}_{14}^{(1)}$. From (8.20) and (8.21) it follows that
$$
\begin{equation}
\| \mathbf{u}_\varepsilon - \widetilde{\mathbf{u}}_\varepsilon \|_{H^1(\mathcal{O}')} \leqslant (\mathcal{C}_{13}' \delta^{-1} +\mathcal{C}_{14}^{(1)}) \varepsilon |\zeta|^{-1/2} \| \mathbf{F} \|_{L_2(\mathcal{O})}, \qquad 0< \varepsilon \leqslant \varepsilon_1.
\end{equation}
\tag{8.22}
$$
By (3.28),
$$
\begin{equation}
\| \widetilde{\mathbf{u}}_\varepsilon - {\mathbf{v}}_\varepsilon \|_{H^1(\mathcal{O}')} \leqslant \| \widetilde{\mathbf{u}}_\varepsilon - \widetilde{\mathbf{v}}_\varepsilon \|_{H^1(\mathbb{R}^d)} \leqslant \mathcal{C}_{14}^{(2)} \varepsilon \| \mathbf{F} \|_{L_2(\mathcal{O})},
\end{equation}
\tag{8.23}
$$
where $\mathcal{C}_{14}^{(2)}= (C_2 + C_3) C_{10}$. As a result, relations (8.22) and (8.23) imply that
$$
\begin{equation}
\begin{gathered} \, \| {\mathbf{u}}_\varepsilon - {\mathbf{v}}_\varepsilon \|_{H^1(\mathcal{O}')} \leqslant (\mathcal{C}_{13}' |\zeta|^{-1/2}\delta^{-1} + \mathcal{C}_{14}') \varepsilon \| \mathbf{F} \|_{L_2(\mathcal{O})}, \\ \operatorname{Re}\zeta \leqslant 0, \quad |\zeta|\geqslant 1, \quad 0< \varepsilon \leqslant \varepsilon_1, \end{gathered}
\end{equation}
\tag{8.24}
$$
where $ \mathcal{C}_{14}' = \mathcal{C}_{14}^{(1)}+ \mathcal{C}_{14}^{(2)}$.
Now we consider $\zeta \in \mathbb{C} \setminus \mathbb{R}_+$, $\operatorname{Re}\zeta > 0$, $|\zeta| \geqslant 1$. Let $\widehat{\zeta} = - \operatorname{Re}\zeta + i \operatorname{Im}\zeta$. We use the identity (7.9). For the second and third terms on the right-hand side of (7.9) we apply the previous estimates (7.11) and (7.16). Let us estimate the first term on the right-hand side of (7.9). We have
$$
\begin{equation}
\begin{aligned} \, &\| \mathfrak{L}_1(\varepsilon; \zeta) \|_{L_2(\mathcal{O}) \to H^1(\mathcal{O}')} \nonumber \\ &\qquad \leqslant \bigl\| (B_{N,\varepsilon}- \widehat{\zeta} Q_0^\varepsilon)^{-1} - (B_N^0 - \widehat{\zeta}\,\overline{Q_0}\,)^{-1}-\varepsilon K_N(\varepsilon;\widehat{\zeta}) \bigr\|_{L_2(\mathcal{O}) \to H^1(\mathcal{O}')} \nonumber \\ &\qquad\qquad \times \bigl\| (B_N^0- \widehat{\zeta}\,\overline{Q_0}\,)(B_N^0-\zeta \overline{Q_0}\,)^{-1} \bigr\|_{L_2(\mathcal{O}) \to L_2(\mathcal{O})}. \end{aligned}
\end{equation}
\tag{8.25}
$$
From the already proved estimate (8.24) at the point $\widehat{\zeta}$ it follows that the first factor on the right satisfies
$$
\begin{equation}
\begin{aligned} \, &\bigl\| (B_{N,\varepsilon}- \widehat{\zeta} Q_0^\varepsilon)^{-1} - (B_N^0 - \widehat{\zeta}\,\overline{Q_0}\,)^{-1}-\varepsilon K_N(\varepsilon;\widehat{\zeta}) \bigr\|_{L_2(\mathcal{O}) \to H^1(\mathcal{O}')} \nonumber \\ &\qquad\leqslant (\mathcal{C}_{13}' |\zeta|^{-1/2}\delta^{-1} + \mathcal{C}_{14}') \varepsilon \end{aligned}
\end{equation}
\tag{8.26}
$$
for $0< \varepsilon \leqslant \varepsilon_1$. Combining (7.4), (7.5), (8.25), and (8.26), we arrive at the estimate
$$
\begin{equation}
\begin{aligned} \, &\| \mathfrak{L}_1(\varepsilon; \zeta) \|_{L_2(\mathcal{O}) \to H^1(\mathcal{O}')} \nonumber \\ &\qquad\leqslant 2 c(\phi) \| f\|_{L_\infty} \| f^{-1}\|_{L_\infty} (\mathcal{C}_{13}' |\zeta|^{-1/2}\delta^{-1} + \mathcal{C}_{14}') \varepsilon, \qquad 0< \varepsilon \leqslant \varepsilon_1. \end{aligned}
\end{equation}
\tag{8.27}
$$
Now the required estimate (8.12) with the constants $\mathcal{C}_{13} = 2 \| f\|_{L_\infty} \| f^{-1}\|_{L_\infty} \mathcal{C}_{13}'$ and $\mathcal{C}_{14} = 2 \| f\|_{L_\infty} \| f^{-1}\|_{L_\infty} \mathcal{C}_{14}' + \mathcal{C}_5^{(2)} + \mathcal{C}_5^{(3)}$ follows from (7.9), (7.11), (7.16), and (8.27).
Inequality (8.13) is deduced from (8.12) with the help of (7.18)–(7.20). This completes the proof. Under some additional assumptions about the solutions of the auxiliary problems, it is possible to remove the smoothing operator $S_\varepsilon$ in the corrector. Theorem 8.13. Under the hypotheses of Theorem 8.12, let Conditions 8.1 and 8.2 be met, and let $K_N^0(\varepsilon;\zeta)$ and $G_N^0(\varepsilon;\zeta)$ be the operators given by (8.1) and (8.2). Then, for $0<\varepsilon\leqslant\varepsilon_1$ and $\zeta\in\mathbb{C}\setminus\mathbb{R}_+$, $|\zeta| \geqslant 1$, Proof. Inequality (8.28) follows from (8.5), (8.6), and (8.12).
By (8.28),
$$
\begin{equation*}
\begin{aligned} \, &\bigl\| g^\varepsilon b(\mathbf{D})(B_{N,\varepsilon}-\zeta Q_0^\varepsilon)^{-1} {-}\,g^\varepsilon b(\mathbf{D})\bigl(I+\varepsilon\Lambda^\varepsilon b(\mathbf{D})+\varepsilon\widetilde{\Lambda}^\varepsilon\bigr)(B_D^0{-}\, \zeta\overline{Q_0}\,)^{-1} \bigr\|_{L_2(\mathcal{O})\to L_2(\mathcal{O}')} \\ &\qquad\leqslant (d\alpha_1)^{1/2}\| g\|_{L_\infty}(\mathcal{C}_{13} c(\phi) |\zeta|^{-1/2}\delta^{-1}+\mathcal{C}_{15}c(\phi)^2 ) \varepsilon. \end{aligned}
\end{equation*}
\notag
$$
Together with (8.8)–(8.10), this yields estimate (8.29). This completes the proof. § 9. “Another” approximation of the generalized resolventIn theorems from § 3 and § 8 it was assumed that $\zeta\in\mathbb{C}\setminus\mathbb{R}_+$ and $|\zeta| \geqslant 1$. In the present section, we obtain the results valid in a wider range of the spectral parameter. The material of this section is similar to § 9 from [40]. 9.1. The general case Theorem 9.1. Let $\mathcal{O}\subset\mathbb{R}^d$ be a bounded domain of class $C^{1,1}$. Let $c_\flat\geqslant 0$ be a common lower bound for the operators $\widetilde{B}_{N,\varepsilon}=(f^\varepsilon )^*B_{N,\varepsilon}f^\varepsilon $ and $\widetilde{B}_N^0=f_0B_N^0f_0$. Suppose that $\zeta\in\mathbb{C}\setminus [c_\flat,\infty)$. Let $\psi =\operatorname{arg}(\zeta -c_\flat)$, $0<\psi <2\pi$. We denote Remark 9.2. 1) The expression $c(\psi)^2| \zeta - c_\flat |^{-2}$ in (9.1) is inverse of the square of the distance from $\zeta$ to $[c_\flat,\infty)$. 2) By (2.17), (2.19), (2.32), and (2.37), one can take $c_\flat$ equal to $c_4 \|f^{-1}\|_{L_\infty}^{-2} = \frac{1}{2} k_1 \| g^{-1}\|^{-1}_{L_\infty} \| Q_0\|_{L_\infty}^{-1}$. 3) It is easy to estimate $c_\flat$ from above. From the variational arguments and estimates (2.18) and (2.33) it follows that $c_\flat \leqslant \min\{c_5,c_6\} \|Q_0^{-1}\|_{L_\infty}$. Therefore, the number $c_\flat$ is controlled in terms of the problem data (2.35). Remark 9.3. Estimates (9.2)–(9.7) are useful for bounded $|\zeta|$ and small $\varepsilon \rho_\flat(\zeta)$. In this case, $\varepsilon^{1/2} \rho_\flat(\zeta)^{1/2}+ \varepsilon | 1+ \zeta |^{1/2} \rho_ \flat (\zeta )$ is majorated by $\varepsilon^{1/2} \rho_\flat(\zeta)^{1/2}$. For large $|\zeta|$, estimates from Theorems 3.2 and 3.4 are preferable. We start with the following two lemmas similar to Lemmas 9.4 and 9.5 from [40]. Lemma 9.4. Under the assumptions of Theorem 9.1, for $0< \varepsilon \leqslant 1$ and $\zeta \in \mathbb{C} \setminus [c_\flat,\infty)$, Proof. Under our assumptions, the spectrum of the operator $\widetilde{B}_{N,\varepsilon}$ is contained in $[c_\flat,\infty)$. Therefore, $\|(\widetilde{B}_{N,\varepsilon} - \zeta I)^{-1}\|_{L_2(\mathcal{O}) \to L_2(\mathcal{O})} \leqslant c(\psi) |\zeta - c_\flat|^{-1}$. Together with (2.21), this implies (9.8).
Next, from (2.20) and (2.21) it follows that
$$
\begin{equation*}
\begin{aligned} \, \bigl\| B_{N,\varepsilon}^{1/2} (B_{N,\varepsilon}-\zeta Q_0^\varepsilon )^{-1} \bigr\|_{L_2\to L_2} &= \bigl\| \widetilde{B}_{N,\varepsilon}^{1/2} \bigl(\widetilde{B}_{N,\varepsilon}-\zeta I \bigr)^{-1} (f^\varepsilon)^* \bigr\|_{L_2\to L_2} \\ &\leqslant \| f \|_{L_\infty} \sup_{x \geqslant c_\flat} \frac{x^{1/2}}{|x-\zeta|}. \end{aligned}
\end{equation*}
\notag
$$
A calculation shows that
$$
\begin{equation*}
\sup _{x\geqslant c_\flat}\frac{x}{| x-\zeta |^2}\leqslant \begin{cases} (c_\flat+1) c(\psi)^2 |\zeta - c_\flat|^{-2}, &|\zeta - c_\flat| < 1, \\ (c_\flat+1) c(\psi)^2 |\zeta - c_\flat|^{-1}, &|\zeta - c_\flat| \geqslant 1. \end{cases}
\end{equation*}
\notag
$$
Note that $|\zeta|+1 \leqslant 2 + c_\flat$ for $|\zeta - c_\flat| < 1$ and $(|\zeta|+1) |\zeta - c_\flat|^{-1} \leqslant 2 + c_\flat$ for $|\zeta - c_\flat| \geqslant 1$. Hence
$$
\begin{equation*}
\begin{aligned} \, &(|\zeta| + 1)^{1/2} \bigl\| B_{N,\varepsilon}^{1/2} (B_{N,\varepsilon}-\zeta Q_0^\varepsilon)^{-1} \bigr\|_{L_2(\mathcal{O})\to L_2(\mathcal{O})} \\ &\qquad\leqslant \| f \|_{L_\infty} (c_\flat+1)^{1/2} (c_\flat+2)^{1/2} \rho_\flat(\zeta)^{1/2}. \end{aligned}
\end{equation*}
\notag
$$
Together with (2.17) this implies estimate (9.9) with
$$
\begin{equation*}
\mathfrak{C}_4=c_4^{-1/2} \| f \|_{L_\infty} (c_\flat+1)^{1/2} (c_\flat+2)^{1/2}.
\end{equation*}
\notag
$$
Estimates (9.10) and (9.11) are proved similarly to (9.8) and (9.9), with the help of (2.32), (2.37), and (2.38). It remains to check (9.12). By (2.36)–(2.38),
$$
\begin{equation}
\begin{aligned} \, &\| (B_{N}^0 - \zeta \overline{Q_0}\, )^{-1}\|_{L_2(\mathcal{O}) \to H^2(\mathcal{O})} \nonumber \\ &\qquad\leqslant \| (B_N^0)^{-1}\|_{L_2(\mathcal{O}) \to H^2(\mathcal{O})} \| B_N^0 (B_{N}^0 - \zeta \overline{Q_0}\, )^{-1}\|_{L_2(\mathcal{O}) \to L_2(\mathcal{O})} \nonumber \\ &\qquad\leqslant \widehat{c}\, \|f\|_{L_\infty} \| f^{-1}\|_{L_\infty} \sup_{x \geqslant c_\flat} x |x-\zeta|^{-1} \leqslant \widehat{c}\, \|f\|_{L_\infty} \| f^{-1}\|_{L_\infty} \sup_{x \geqslant c_\flat} (x+1) |x-\zeta|^{-1}. \end{aligned}
\end{equation}
\tag{9.13}
$$
A calculation shows that
$$
\begin{equation}
\sup _{x\geqslant c_\flat}\frac{(x+1)^2}{| x-\zeta |^2}\leqslant (c_\flat+2)^2 \rho_\flat(\zeta), \qquad \zeta \in \mathbb{C} \setminus [c_\flat,\infty).
\end{equation}
\tag{9.14}
$$
Relations (9.13) and (9.14) imply (9.12) with the constant This completes the proof. Lemma 9.5. Under the assumptions of Theorem 9.1, for $0< \varepsilon \leqslant 1$ and $\zeta \in \mathbb{C} \setminus [c_\flat,\infty)$, Proof. Combining Proposition 1.2 and relations (1.5), (1.24), (1.32), (3.4), (3.5), we obtain
$$
\begin{equation*}
\begin{aligned} \, \| K_{N} (\varepsilon;\zeta) \|_{L_2(\mathcal{O}) \to L_2(\mathcal{O})} &\leqslant \bigl\| \bigl( \Lambda^\varepsilon S_\varepsilon b(\mathbf{D}) + \widetilde{\Lambda}^\varepsilon S_\varepsilon \bigr) P_\mathcal{O} (B_{N}^0 - \zeta \overline{Q_0}\, )^{-1} \bigr\|_{L_2(\mathcal{O}) \to L_2(\mathbb{R}^d)} \\ &\leqslant M_1 \alpha_1^{1/2} C_\mathcal{O}^{(1)} \| (B_{N}^0 - \zeta \overline{Q_0}\, )^{-1} \|_{L_2(\mathcal{O}) \to H^1(\mathcal{O})} \\ &\qquad +\widetilde{M}_1 C_\mathcal{O}^{(0)} \| (B_{N}^0 - \zeta \overline{Q_0}\, )^{-1} \|_{L_2(\mathcal{O}) \to L_2(\mathcal{O})}. \end{aligned}
\end{equation*}
\notag
$$
Together with (9.11), this yields estimate (9.15) with the constant
Next, by (3.5),
$$
\begin{equation*}
\begin{aligned} \, &\varepsilon \|\mathbf{D} K_{N} (\varepsilon;\zeta) \|_{L_2(\mathcal{O}) \to L_2(\mathcal{O})} \\ &\qquad\leqslant \bigl\| \bigl( (\mathbf{D}\Lambda)^\varepsilon S_\varepsilon b(\mathbf{D}) + (\mathbf{D}\widetilde{\Lambda})^\varepsilon S_\varepsilon \bigr) P_\mathcal{O} (B_{N}^0 - \zeta \overline{Q_0}\, )^{-1} \bigr\|_{L_2(\mathcal{O}) \to L_2(\mathbb{R}^d)} \\ &\qquad\qquad+ \varepsilon \bigl\| \bigl( \Lambda^\varepsilon S_\varepsilon b(\mathbf{D}) + \widetilde{\Lambda}^\varepsilon S_\varepsilon\bigr) \mathbf{D} P_\mathcal{O} (B_{N}^0 - \zeta \overline{Q_0}\, )^{-1} \bigr\|_{L_2(\mathcal{O}) \to L_2(\mathbb{R}^d)}. \end{aligned}
\end{equation*}
\notag
$$
Applying Proposition 1.2 and (1.5), (1.24), (1.25), (1.32), (1.33), (3.4), we obtain
$$
\begin{equation*}
\begin{aligned} \, &\varepsilon \|\mathbf{D} K_{N} (\varepsilon;\zeta) \|_{L_2(\mathcal{O}) \to L_2(\mathcal{O})} \\ &\qquad\leqslant \bigl( M_2 \alpha_1^{1/2} + \widetilde{M}_2 + \varepsilon \widetilde{M}_1\bigr) C_\mathcal{O}^{(1)} \| (B_{N}^0 - \zeta \overline{Q_0}\, )^{-1}\|_{L_2(\mathcal{O}) \to H^1(\mathcal{O})} \\ &\qquad\qquad+ \varepsilon M_1 \alpha_1^{1/2} C_\mathcal{O}^{(2)} \| (B_{N}^0 - \zeta \overline{Q_0}\, )^{-1}\|_{L_2(\mathcal{O}) \to H^2(\mathcal{O})}. \end{aligned}
\end{equation*}
\notag
$$
Together with (9.11) and (9.12), this yields
$$
\begin{equation}
\varepsilon \|\mathbf{D} K_{N} (\varepsilon;\zeta) \|_{L_2(\mathcal{O}) \to L_2(\mathcal{O})}\leqslant \mathfrak{C}_7' (1+ |\zeta|)^{-1/2} \rho_\flat (\zeta)^{1/2} + \mathfrak{C}_7'' \varepsilon \rho_\flat (\zeta)^{1/2},
\end{equation}
\tag{9.17}
$$
where $\mathfrak{C}_7' = \bigl( M_2 \alpha_1^{1/2} + \widetilde{M}_2 + \widetilde{M}_1\bigr) C_\mathcal{O}^{(1)} \mathfrak{C}_4$ and $\mathfrak{C}_7'' = M_1 \alpha_1^{1/2} C_\mathcal{O}^{(2)} \mathfrak{C}_5$.
As a result, estimates (9.15) and (9.17) imply the required inequality (9.16) with the constant $\mathfrak{C}_7 = \max \{ \mathfrak{C}_7'; \mathfrak{C}_6 + \mathfrak{C}_7''\}$. This completes the proof. 9.2. Proof of Theorem 9.1We start with the proof of estimate (9.5). Inequality (6.24) at the point $\zeta =-1$ reads as:
$$
\begin{equation}
\bigl\| (B_{N,\varepsilon }+ Q_0^\varepsilon )^{-1}-(B_N^0+ \overline{Q_0}\,)^{-1} \bigr\|_{L_2(\mathcal{O})\to L_2(\mathcal{O})}\leqslant \mathcal{C}_3' \varepsilon,\qquad 0<\varepsilon \leqslant \varepsilon_1.
\end{equation}
\tag{9.18}
$$
Similarly to (7.2),
$$
\begin{equation}
\begin{aligned} \, &(B_{N,\varepsilon }-\zeta Q_0^\varepsilon )^{-1} - (B_N^0-\zeta \overline{Q_0}\,)^{-1} = (B_{N,\varepsilon}-\zeta Q_0^\varepsilon )^{-1}(B_{N,\varepsilon}+Q_0^\varepsilon) \nonumber \\ &\qquad\qquad\times\bigl( (B_{N,\varepsilon}+Q_0^\varepsilon )^{-1} - (B_N^0+\overline{Q_0}\,)^{-1} \bigr) (B_N^0+\overline{Q_0}\,)(B_N^0-\zeta \overline{Q_0}\,)^{-1} \nonumber \\ &\qquad+(1+\zeta)(B_{N,\varepsilon}-\zeta Q_0^\varepsilon )^{-1}(Q_0^\varepsilon -\overline{Q_0}\,)(B_N^0-\zeta \overline{Q_0}\,)^{-1}. \end{aligned}
\end{equation}
\tag{9.19}
$$
Let $\mathfrak{J}_1(\varepsilon;\zeta)$ and $\mathfrak{J}_2(\varepsilon;\zeta)$ be the terms on the right-hand side of (9.19). Similarly to (7.3) and (7.4),
$$
\begin{equation}
\|(B_{N,\varepsilon}-\zeta Q_0^\varepsilon )^{-1}(B_{N,\varepsilon}+Q_0^\varepsilon) \|_{L_2(\mathcal{O})\to L_2(\mathcal{O})} \leqslant \| f\|_{L_\infty}\| f^{-1}\|_{L_\infty} \sup_{x\geqslant c_\flat}\frac{(x+1)}{| x-\zeta |},
\end{equation}
\tag{9.20}
$$
$$
\begin{equation}
\| (B_N^0+\overline{Q_0}\,)(B_N^0-\zeta \overline{Q_0}\,)^{-1}\|_{L_2(\mathcal{O})\to L_2(\mathcal{O})} \leqslant \| f\|_{L_\infty}\| f^{-1}\|_{L_\infty}\sup_{x\geqslant c_\flat}\frac{(x+1)}{| x-\zeta |}.
\end{equation}
\tag{9.21}
$$
Relations (9.14), (9.18), (9.20), (9.21) imply the estimate
$$
\begin{equation}
\| \mathfrak{J}_1(\varepsilon;\zeta)\|_{L_2(\mathcal{O})\to L_2(\mathcal{O})} \leqslant \mathfrak{C}_8 \varepsilon \rho_\flat (\zeta), \qquad 0 < \varepsilon \leqslant \varepsilon_1,
\end{equation}
\tag{9.22}
$$
where $\mathfrak{C}_8 = \mathcal{C}'_3 \| f\|^2_{L_\infty}\| f^{-1}\|^2_{L_\infty}(c_\flat +2)^2$. Let us estimate the norm of the operator $\mathfrak{J}_2(\varepsilon;\zeta)$. Let $\boldsymbol{\Phi}_1, \boldsymbol{\Phi}_2 \in L_2(\mathcal{O};\mathbb{C}^n)$. By assertion $3^\circ$ of Lemma 4.4, for $0< \varepsilon \leqslant \varepsilon_1$, we have
$$
\begin{equation*}
\begin{aligned} \, &\bigl| \bigl( (Q_0^\varepsilon -\overline{Q_0}\,) (B_{N}^0 -\zeta \overline{Q_0}\, )^{-1} \boldsymbol{\Phi}_1, (B_{N,\varepsilon}-\zeta^*Q_0^\varepsilon )^{-1} \boldsymbol{\Phi}_2 \bigr)_{L_2(\mathcal{O})}\bigr| \\ &\ \leqslant 2 \widetilde{C}''' \varepsilon \| Q_0\|_{L_\infty} \|(B_{N}^0 -\zeta \overline{Q_0}\, )^{-1} \boldsymbol{\Phi}_1 \|_{H^1(\mathcal{O})} \| (B_{N,\varepsilon}-\zeta^*Q_0^\varepsilon )^{-1} \boldsymbol{\Phi}_2 \|_{H^1(\mathcal{O})} \\ &\ \leqslant 2 \widetilde{C}''' \varepsilon \| Q_0\|_{L_\infty} \mathfrak{C}_4^2 (1+ |\zeta|)^{-1} \rho_\flat(\zeta) \| \boldsymbol{\Phi}_1 \|_{L_2(\mathcal{O})} \| \boldsymbol{\Phi}_2 \|_{L_2(\mathcal{O})},\,\quad \boldsymbol{\Phi}_1, \boldsymbol{\Phi}_2 \in L_2(\mathcal{O};\mathbb{C}^n), \end{aligned}
\end{equation*}
\notag
$$
where we have taken estimates (9.9) and (9.11) into account. Hence
$$
\begin{equation}
\| \mathfrak{J}_2(\varepsilon;\zeta)\|_{L_2(\mathcal{O})\to L_2(\mathcal{O})} \leqslant \mathfrak{C}_9 \varepsilon \rho_\flat (\zeta), \qquad 0 < \varepsilon \leqslant \varepsilon_1,
\end{equation}
\tag{9.23}
$$
where $\mathfrak{C}_9 = 2 \widetilde{C}''' \| Q_0\|_{L_\infty} \mathfrak{C}_4^2$. As a result, relations (9.19), (9.22), and (9.23) imply (9.5) with the constant $\mathfrak{C}_1 = \mathfrak{C}_8 + \mathfrak{C}_9$. Now, we check estimate (9.6). We write estimate (5.85) at the point $\zeta =-1$:
$$
\begin{equation}
\begin{aligned} \, &\bigl\| (B_{N,\varepsilon}+Q_0^\varepsilon )^{-1}-(B_N^0+\overline{Q_0}\,)^{-1}-\varepsilon K_N(\varepsilon;-1) \bigr\|_{L_2(\mathcal{O})\to H^1(\mathcal{O})} \nonumber \\ &\qquad\leqslant (2 \mathcal{C}_{10} + \mathcal{C}_8)\varepsilon^{1/2},\qquad 0<\varepsilon \leqslant \varepsilon_1. \end{aligned}
\end{equation}
\tag{9.24}
$$
Similarly to (7.9),
$$
\begin{equation}
\begin{aligned} \, &(B_{N,\varepsilon}-\zeta Q_0^\varepsilon )^{-1} - (B_N^0-\zeta \overline{Q_0}\,)^{-1}-\varepsilon K_N(\varepsilon;\zeta) \nonumber \\ &\ =\bigl((B_{N,\varepsilon}+Q_0^\varepsilon)^{-1} - (B_N^0+\overline{Q_0}\,)^{-1}-\varepsilon K_N(\varepsilon;-1)\bigr) (B_N^0+\overline{Q_0}\,)(B_N^0-\zeta \overline{Q_0}\,)^{-1} \nonumber \\ &\ \qquad +(1+\zeta)(B_{N,\varepsilon}-\zeta Q_0^\varepsilon )^{-1}Q_0^\varepsilon \bigl((B_{N,\varepsilon}+Q_0^\varepsilon)^{-1} - (B_N^0+\overline{Q_0}\,)^{-1}\bigr) \nonumber \\ &\ \qquad \qquad \times(B_N^0+\overline{Q_0}\,)(B_N^0-\zeta \overline{Q_0}\,)^{-1} \nonumber \\ &\ \qquad +(1+\zeta )(B_{N,\varepsilon }-\zeta Q_0^\varepsilon )^{-1}(Q_0^\varepsilon -\overline{Q_0}\,)(B_N^0-\zeta \overline{Q_0}\,)^{-1}. \end{aligned}
\end{equation}
\tag{9.25}
$$
Let $\mathcal{L}_1(\varepsilon;\zeta)$, $\mathcal{L}_2(\varepsilon;\zeta)$, and $\mathcal{L}_3(\varepsilon;\zeta)$ be the consecutive terms on the right-hand side of (9.25). Note that $\mathcal{L}_3(\varepsilon;\zeta) = {\mathfrak J}_2(\varepsilon;\zeta)$. Obviously,
$$
\begin{equation*}
\begin{aligned} \, &\| \mathcal{L}_1(\varepsilon;\zeta) \|_{L_2(\mathcal{O}) \to H^1(\mathcal{O})} \\ &\qquad\leqslant \bigl\|(B_{N,\varepsilon}+Q_0^\varepsilon)^{-1} - (B_N^0+\overline{Q_0}\,)^{-1}-\varepsilon K_N(\varepsilon;-1) \bigr\|_{L_2(\mathcal{O})\to H^1(\mathcal{O})} \\ &\qquad\qquad\times \bigl\| (B_N^0+\overline{Q_0}\,)(B_N^0-\zeta \overline{Q_0}\,)^{-1} \bigr\|_{L_2(\mathcal{O}) \to L_2(\mathcal{O})}. \end{aligned}
\end{equation*}
\notag
$$
Together with (9.14), (9.21), and (9.24), this implies
$$
\begin{equation}
\| \mathcal{L}_1(\varepsilon;\zeta)\|_{L_2(\mathcal{O})\to H^1(\mathcal{O})} \leqslant \mathfrak{C}_{10} \varepsilon^{1/2} \rho_\flat (\zeta)^{1/2}, \qquad 0 < \varepsilon \leqslant \varepsilon_1,
\end{equation}
\tag{9.26}
$$
where $\mathfrak{C}_{10} = \| f\|_{L_\infty} \|f^{-1}\|_{L_\infty} (c_\flat+2) (2 \mathcal{C}_{10} + \mathcal{C}_8)$. Let us estimate the second term on the right-hand side of (9.25):
$$
\begin{equation*}
\begin{aligned} \, \| \mathcal{L}_2(\varepsilon;\zeta)\|_{L_2(\mathcal{O})\to H^1(\mathcal{O})} &\leqslant |1+\zeta|\, \|(B_{N,\varepsilon}-\zeta Q_0^\varepsilon )^{-1} \|_{L_2(\mathcal{O})\to H^1(\mathcal{O})} \| Q_0 \|_{L_\infty} \\ &\qquad\times \bigl\| (B_{N,\varepsilon}+Q_0^\varepsilon)^{-1} - (B_N^0+\overline{Q_0}\,)^{-1} \bigr\|_{L_2(\mathcal{O}) \to L_2(\mathcal{O})} \\ &\qquad\times\bigl\| (B_N^0+\overline{Q_0}\,)(B_N^0-\zeta \overline{Q_0}\,)^{-1} \bigr\|_{L_2(\mathcal{O}) \to L_2(\mathcal{O})}. \end{aligned}
\end{equation*}
\notag
$$
Combining this inequality and relations (9.9), (9.14), (9.18), and (9.21), we obtain
$$
\begin{equation}
\| \mathcal{L}_2(\varepsilon;\zeta) \|_{L_2(\mathcal{O}) \to H^1(\mathcal{O})} \leqslant \mathfrak{C}_{11} \varepsilon |1+\zeta|^{1/2} \rho_\flat (\zeta), \qquad 0 < \varepsilon \leqslant \varepsilon_1,
\end{equation}
\tag{9.27}
$$
where $\mathfrak{C}_{11} = \mathcal{C}'_3 \| f\|_{L_\infty} \|f^{-1}\|^3_{L_\infty} (c_\flat+2) \mathfrak{C}_4$. It remains to estimate $\mathcal{L}_3(\varepsilon;\zeta)$. Similarly to (7.12),
$$
\begin{equation}
\begin{aligned} \, &\| \mathcal{L}_3(\varepsilon;\zeta)\|_{L_2(\mathcal{O}) \to H^1(\mathcal{O})} \leqslant c_4^{-1/2} \bigl\| \widetilde{B}_{N,\varepsilon}^{1/2} (f^\varepsilon)^{-1} \mathcal{L}_3(\varepsilon;\zeta) \bigr \|_{L_2(\mathcal{O}) \to L_2(\mathcal{O})} \nonumber \\ &\ = c_4^{-1/2} |1+\zeta| \bigl\| \widetilde{B}_{N,\varepsilon}^{1/2} (\widetilde{B}_{N,\varepsilon}-\zeta I)^{-1} (f^\varepsilon)^* (Q_0^\varepsilon - \overline{Q_0}\,) (B_N^0 - \zeta \overline{Q_0}\,)^{-1} \bigr\|_{L_2(\mathcal{O}) \to L_2(\mathcal{O})}. \end{aligned}
\end{equation}
\tag{9.28}
$$
Let $\boldsymbol{\Phi}_1, \boldsymbol{\Phi}_2 \in L_2(\mathcal{O};\mathbb{C}^n)$. By assertion $3^\circ$ of Lemma 4.4, for $0< \varepsilon \leqslant \varepsilon_1$,
$$
\begin{equation}
\begin{aligned} \, &\bigl| \bigl( (Q_0^\varepsilon -\overline{Q_0}\,) (B_{N}^0 -\zeta \overline{Q_0}\, )^{-1} \boldsymbol{\Phi}_1, f^\varepsilon\widetilde{B}_{N,\varepsilon}^{1/2} \bigl(\widetilde{B}_{N,\varepsilon}-\zeta^* I \bigr)^{-1} \boldsymbol{\Phi}_2 \bigr)_{L_2(\mathcal{O})}\bigr| \nonumber \\ &\qquad\leqslant 2 \widetilde{C}''' \varepsilon \| Q_0\|_{L_\infty} \|(B_{N}^0 -\zeta \overline{Q_0}\, )^{-1}\boldsymbol{\Phi}_1\|_{H^1(\mathcal{O})} \bigl\| f^\varepsilon\widetilde{B}_{N,\varepsilon}^{1/2}\bigl(\widetilde{B}_{N,\varepsilon}-\zeta^* I\bigr)^{-1} \boldsymbol{\Phi}_2 \bigr\|_{H^1(\mathcal{O})}. \end{aligned}
\end{equation}
\tag{9.29}
$$
Similarly to (7.15), taking (9.14) into account, we obtain
$$
\begin{equation*}
\begin{aligned} \, \bigl\| f^\varepsilon\widetilde{B}_{N,\varepsilon}^{1/2} \bigl(\widetilde{B}_{N,\varepsilon}-\zeta^* I \bigr)^{-1} \boldsymbol{\Phi}_2 \bigr\|_{H^1(\mathcal{O})} &\leqslant c_4^{-1/2} \sup_{x \geqslant c_\flat} x |x- \zeta^*|^{-1} \|\boldsymbol{\Phi}_2 \|_{L_2(\mathcal{O})} \\ &\leqslant c_4^{-1/2} (c_\flat +2) \rho_\flat(\zeta)^{1/2} \|\boldsymbol{\Phi}_2 \|_{L_2(\mathcal{O})}. \end{aligned}
\end{equation*}
\notag
$$
Combining this with (9.11) and (9.29), we have
$$
\begin{equation*}
\begin{aligned} \, &\bigl| \bigl( (Q_0^\varepsilon -\overline{Q_0}\,) (B_{N}^0 -\zeta \overline{Q_0}\, )^{-1} \boldsymbol{\Phi}_1, f^\varepsilon\widetilde{B}_{N,\varepsilon}^{1/2} (\widetilde{B}_{N,\varepsilon}-\zeta^* I )^{-1} \boldsymbol{\Phi}_2 \bigr)_{L_2(\mathcal{O})}\bigr| \\ &\qquad\leqslant \widetilde{\mathfrak{C}}_{12} \varepsilon (1+ |\zeta|)^{-1/2} \rho_\flat(\zeta) \| \boldsymbol{\Phi}_1 \|_{L_2(\mathcal{O})} \|\boldsymbol{\Phi}_2 \|_{L_2(\mathcal{O})}, \qquad \boldsymbol{\Phi}_1, \boldsymbol{\Phi}_2 \in L_2(\mathcal{O};\mathbb{C}^n), \end{aligned}
\end{equation*}
\notag
$$
where $\widetilde{\mathfrak{C}}_{12} = 2 \widetilde{C}''' c_4^{-1/2} (c_\flat +2) \| Q_0\|_{L_\infty} \mathfrak{C}_4$. Together with (9.28), this implies
$$
\begin{equation}
\| \mathcal{L}_3(\varepsilon;\zeta)\|_{L_2(\mathcal{O}) \to H^1(\mathcal{O})} \leqslant \mathfrak{C}_{12} \varepsilon |1+ \zeta|^{1/2} \rho_\flat(\zeta), \qquad 0< \varepsilon \leqslant \varepsilon_1, \quad \mathfrak{C}_{12} = c_4^{-1/2} \widetilde{\mathfrak{C}}_{12}.
\end{equation}
\tag{9.30}
$$
Comparing (9.25)–(9.27) and (9.30), we arrive at the required estimate (9.6) with the constants $\mathfrak{C}_2= \mathfrak{C}_{10}$ and $\mathfrak{C}_3 = \mathfrak{C}_{11}+ \mathfrak{C}_{12}$. It remains to check (9.4). From (9.3), taking (1.3) and (1.6) into account, we find that, for $0< \varepsilon \leqslant \varepsilon_1$,
$$
\begin{equation}
\begin{aligned} \, &\| \mathbf{p}_\varepsilon - g^\varepsilon b(\mathbf{D})\mathbf{v}_\varepsilon\|_{L_2(\mathcal{O})} \nonumber \\ &\qquad \leqslant (d\alpha_1)^{1/2}\| g\|_{L_\infty}\bigl(\mathfrak{C}_2 \varepsilon^{1/2}\rho_\flat(\zeta)^{1/2} +\mathfrak{C}_3 \varepsilon | 1+\zeta |^{1/2} \rho_\flat (\zeta ) \bigr) \| \mathbf{F}\|_{L_2(\mathcal{O})}. \end{aligned}
\end{equation}
\tag{9.31}
$$
Next, by analogy with (7.18)–(7.20), using (9.11) and (9.12), we obtain
$$
\begin{equation}
\bigl\| g^\varepsilon b(\mathbf{D})\mathbf{v}_\varepsilon - \widetilde{g}^{\,\varepsilon} S_\varepsilon b(\mathbf{D})\widetilde{\mathbf{u}}_0 - g^\varepsilon \bigl(b(\mathbf{D})\widetilde{\Lambda}\bigr)^\varepsilon S_\varepsilon\widetilde{\mathbf{u}}_0 \bigr\|_{L_2(\mathcal{O})} \leqslant \mathfrak{C}_{13} \varepsilon \rho_\flat(\zeta)^{1/2} \|\mathbf{F}\|_{L_2(\mathcal{O})},
\end{equation}
\tag{9.32}
$$
where $\mathfrak{C}_{13}$ depends only on the problem data (2.35). Relations (9.31) and (9.32) imply estimate (9.4). This completes the proof. Corollary 9.6. Under the assumptions of Theorem 9.1, for $\zeta \in \mathbb{C} \setminus [c_\flat,\infty)$ and $0< \varepsilon \leqslant \varepsilon_1$, Proof. By (9.9), (9.11), and (9.16),
$$
\begin{equation}
\begin{aligned} \, &\bigl\| (B_{N,\varepsilon} - \zeta Q_0^\varepsilon)^{-1} - (B_N^0 - \zeta \overline{Q_0}\,)^{-1} - \varepsilon K_N(\varepsilon;\zeta) \bigr\|_{L_2(\mathcal{O}) \to H^1(\mathcal{O})} \nonumber \\ &\qquad\leqslant \mathfrak{C}_{15} \bigl( \varepsilon + (1+ |\zeta|)^{-1/2}\bigr) \rho_\flat(\zeta)^{1/2}, \qquad 0< \varepsilon \leqslant 1, \end{aligned}
\end{equation}
\tag{9.35}
$$
where $\mathfrak{C}_{15} = 2 \mathfrak{C}_4 + \mathfrak{C}_7$. For $|1+\zeta|^{1/2} \rho_\flat(\zeta)^{1/4} \leqslant \varepsilon^{-1/2}$ we use (9.6) and note that $\varepsilon |1+\zeta|^{1/2} \rho_\flat(\zeta) \leqslant \varepsilon^{1/2} \rho_\flat(\zeta)^{3/4}$. For $|1+\zeta|^{1/2} \rho_\flat(\zeta)^{1/4} > \varepsilon^{-1/2}$ we apply (9.35) and note that $(1+ |\zeta|)^{-1/2} \rho_\flat(\zeta)^{1/2} < \varepsilon^{1/2} \rho_\flat(\zeta)^{3/4}$. As a result, we obtain estimate (9.33) with the constant $\mathfrak{C}_{14}=\max \{ \mathfrak{C}_2+ \mathfrak{C}_3; 2 \mathfrak{C}_{15}\}$.
Combining relations (9.32), (9.33) and taking (1.3), (1.6) into account, we arrive at estimate (9.34) with the constant $\widetilde{\mathfrak{C}}_{14} = (d \alpha_1)^{1/2} \|g\|_{L_\infty} \mathfrak{C}_{14} + \mathfrak{C}_{13}$. This completes the proof. 9.3. Removal of the smoothing operator Theorem 9.7. Under the hypotheses of Theorem 9.1, let Conditions 8.1 and 8.2 be met, and let $K_N^0(\varepsilon;\zeta)$ and $G_N^0(\varepsilon;\zeta)$ be the operators defined by (8.1), (8.2). Then, for $0<\varepsilon\leqslant \varepsilon_1$ and $\zeta\in\mathbb{C}\setminus[c_\flat,\infty)$, Proof. Inequality (9.36) follows from (9.6) with the help of Lemmas 8.7 and 8.8 and relations (3.4), (9.12).
Similarly to (8.7)–(8.10), taking (9.12) into account, we deduce estimate (9.37) from (9.36). This completes the proof. 9.4. Special casesThe following assertions are proved by analogy with Propositions 8.10 and 8.11. Proposition 9.8. Under the hypotheses of Theorem 9.1, let (1.28) and (8.11) hold. Then, for $\zeta\in\mathbb{C}\setminus [c_\flat,\infty)$ and $0<\varepsilon\leqslant \varepsilon_1$, Proposition 9.9. Under the hypotheses of Theorem 9.1, let (1.29) and (8.11) hold. Then, for $\zeta\in\mathbb{C}\setminus [c_\flat,\infty)$ and $0<\varepsilon\leqslant \varepsilon_1$, 9.5. Estimates in a strictly interior subdomain Theorem 9.10. Under the hypotheses of Theorem 9.1, let $\mathcal{O}'$ be a strictly interior subdomain of the domain $\mathcal{O}$, and let $\delta := \operatorname{dist} \{\mathcal{O}'; \partial \mathcal{O}\}$. Then, for $\zeta \in \mathbb{C} \setminus [c_\flat,\infty)$ and $0 < \varepsilon \leqslant \varepsilon_1$, Proof. We write inequality (8.12) at the point $\zeta=-1$:
$$
\begin{equation}
\begin{aligned} \, &\bigl\| (B_{N,\varepsilon} + Q_0^\varepsilon )^{-1} - (B_N^0 + \overline{Q_0}\,)^{-1} -\varepsilon K_N(\varepsilon;-1 ) \bigr\|_{L_2(\mathcal{O})\to H^1(\mathcal{O}')} \nonumber \\ &\qquad\leqslant (\mathcal{C}_{13} \delta^{-1} + \mathcal{C}_{14}) \varepsilon, \qquad 0< \varepsilon \leqslant \varepsilon_1. \end{aligned}
\end{equation}
\tag{9.40}
$$
We apply identity (9.25). The first term on the right satisfies
$$
\begin{equation*}
\begin{aligned} \, &\| \mathcal{L}_1(\varepsilon;\zeta) \|_{L_2(\mathcal{O}) \to H^1(\mathcal{O}')} \\ &\qquad\leqslant \bigl\| (B_{N,\varepsilon}+Q_0^\varepsilon)^{-1} - (B_N^0+\overline{Q_0}\,)^{-1}-\varepsilon K_N(\varepsilon;-1) \bigr\|_{L_2(\mathcal{O})\to H^1(\mathcal{O}')} \\ &\qquad\qquad\times \bigl\| (B_N^0+\overline{Q_0}\,)(B_N^0-\zeta \overline{Q_0}\,)^{-1} \bigr\|_{L_2(\mathcal{O}) \to L_2(\mathcal{O})}. \end{aligned}
\end{equation*}
\notag
$$
Together with (9.14), (9.21), and (9.40), this implies
$$
\begin{equation}
\| \mathcal{L}_1(\varepsilon;\zeta) \|_{L_2(\mathcal{O}) \to H^1(\mathcal{O}')} \leqslant (\mathfrak{C}_{17} \delta^{-1} + \mathfrak{C}_{18}') \varepsilon \rho_ \flat (\zeta )^{1/2}, \qquad 0 < \varepsilon \leqslant \varepsilon_1,
\end{equation}
\tag{9.41}
$$
where $\mathfrak{C}_{17} = \mathcal{C}_{13} \|f\|_{L_\infty} \|f^{-1}\|_{L_\infty} (c_\flat +2)$, $\mathfrak{C}_{18}' = \mathcal{C}_{14} \|f\|_{L_\infty} \|f^{-1}\|_{L_\infty} (c_\flat +2)$. For the second and the third terms on the right-hand side of (9.25), we use inequalities (9.27) and (9.30). As a result, we arrive at estimate (9.38) with the constant $\mathfrak{C}_{18} = \mathfrak{C}_3 + \mathfrak{C}_{18}'$.
By analogy with (7.17)–(7.20), taking (9.12) into account, we deduce (9.39) from (9.38). This completes the proof. Under some additional assumptions about the solutions of the auxiliary problems, the following assertion is true. Theorem 9.11. Under the hypotheses of Theorem 9.10, let Conditions 8.1 and 8.2 be met, and let $K_N^0(\varepsilon;\zeta)$ and $G_N^0(\varepsilon;\zeta)$ be the operators defined by (8.1) and (8.2), respectively. Then, for $\zeta \in \mathbb{C} \setminus [c_\flat,\infty)$ and $0 < \varepsilon \leqslant \varepsilon_1$, Proof. Combining (9.38), Lemmas 8.7, 8.8, and relations (3.4), (9.12), we obtain estimate (9.42).
By analogy with (8.7)–(8.10), taking (9.12) into account, we deduce inequality (9.43) from (9.42). This completes the proof. § 10. Homogenization of solutions of the second initial boundary value problem for parabolic equationsIn this section, we study homogenization of solutions of the second initial boundary value problem for a parabolic equation (that is, a problem with the Neumann condition). The results are deduced from approximations of the generalized resolvent with two-parametric error estimates by integrating the resolvent along the contour. The proofs of the main results of this section are completely similar to those from [42], where the first initial boundary value problem (with the Dirichlet condition) was considered. Therefore, we will limit ourselves to formulations and brief comments, omitting the details. 10.1. Statement of the problemConsider a (weak) solution $\mathbf{u}_\varepsilon(\mathbf{x},t)$ of the problem
$$
\begin{equation}
\begin{cases} Q_0^\varepsilon(\mathbf{x}) \, \dfrac{\partial \mathbf{u}_\varepsilon(\mathbf{x},t)}{\partial t} = - (B_\varepsilon \mathbf{u}_\varepsilon)(\mathbf{x},t), &\mathbf{x} \in \mathcal{O},\ t>0, \\ Q_0^\varepsilon(\mathbf{x}) \mathbf{u}_\varepsilon(\mathbf{x},0) = \boldsymbol{\varphi}(\mathbf{x}), &\mathbf{x}\in \mathcal{O}, \end{cases}
\end{equation}
\tag{10.1}
$$
under the natural condition (Neumann condition) on $\partial \mathcal{O} \times \mathbb{R}_+$. Here, $B_\varepsilon$ is the differential expression (1.19), whose coefficients satisfy the assumptions of § 1. It is assumed that $\boldsymbol \varphi \in L_2(\mathcal{O};\mathbb{C}^n)$. Using (2.19), it is easily seen that
$$
\begin{equation*}
\mathbf{u}_\varepsilon(\,{\cdot}\,,t) = f^\varepsilon e^{- \widetilde{B}_{N,\varepsilon} t} (f^\varepsilon)^* \boldsymbol{\varphi}.
\end{equation*}
\notag
$$
We are interested in the behaviour of the solution $\mathbf{u}_\varepsilon(\mathbf{x},t)$ for small $\varepsilon$, that is, in the behaviour of the sandwiched operator exponential $f^\varepsilon e^{-\widetilde{B}_{N,\varepsilon} t} (f^\varepsilon)^*$. The corresponding effective problem is
$$
\begin{equation}
\begin{cases} \overline{Q_0} \, \dfrac{\partial \mathbf{u}_0(\mathbf{x},t)}{\partial t} = - (B^0 \mathbf{u}_0)(\mathbf{x},t), &\mathbf{x} \in \mathcal{O},\ t>0, \\ \overline{Q_0} \mathbf{u}_0(\mathbf{x},0) = \boldsymbol \varphi(\mathbf{x}), &\mathbf{x}\in \mathcal{O}, \end{cases}
\end{equation}
\tag{10.2}
$$
with the Neumann condition on $\partial \mathcal{O} \times \mathbb{R}_+$. The solution of the effective problem is represented as
$$
\begin{equation*}
\mathbf{u}_0(\,{\cdot}\,,t) = f_0 e^{- \widetilde{B}_{N}^0 t} f_0 \boldsymbol{\varphi}.
\end{equation*}
\notag
$$
As in § 9, let $c_\flat > 0$ be a common lower bound of the operators $\widetilde{B}_{N,\varepsilon}=(f^\varepsilon )^*B_{N,\varepsilon}f^\varepsilon $ and $\widetilde{B}_N^0=f_0B_N^0f_0$. According to Remark 9.2, we can fix $c_\flat$ as follows:
$$
\begin{equation}
c_\flat = c_4 \|f^{-1}\|_{L_\infty}^{-2} = \frac{1}{2}\, k_1 \| g^{-1}\|^{-1}_{L_\infty} \| Q_0\|_{L_\infty}^{-1}.
\end{equation}
\tag{10.3}
$$
The following simple assertion can be checked similarly to Lemma 2.1 from [42]. Lemma 10.1. For $0<\varepsilon\leqslant 1$, 10.2. Approximation of the solution in $L_2(\mathcal{O};\mathbb{C}^n)$The following result is deduced from Theorems 3.2 and 9.1 by analogy with the proof of Theorem 2.2 from [42]. Theorem 10.2. Let $\mathbf{u}_\varepsilon(\mathbf{x},t)$ be the solution of problem (10.1) with the Neumann condition on $\partial \mathcal{O} \times \mathbb{R}_+$. Let $\mathbf{u}_0(\mathbf{x},t)$ be the solution of problem (10.2) with the Neumann condition on $\partial \mathcal{O} \times \mathbb{R}_+$. Let $\varepsilon_1$ satisfy Condition 3.1. Then, for $0< \varepsilon \leqslant \varepsilon_1$, Proof. We have (see, for example, [51], Chap. IX, § 1.6)
$$
\begin{equation}
e^{-\widetilde{B}_{N,\varepsilon} t} = - \frac{1}{2\pi i} \int_\gamma e^{-\zeta t} (\widetilde{B}_{N,\varepsilon} - \zeta I)^{-1} \,d\zeta,\qquad t>0.
\end{equation}
\tag{10.8}
$$
Here, $\gamma\subset\mathbb{C}$ is a contour enveloping the spectrum of $\widetilde{B}_{N,\varepsilon}$ in the positive direction. The exponential of the operator $\widetilde{B}_N^0$ admits a similar representation. Since the constant (10.3) is a common lower bound of the operators $\widetilde{B}_{N,\varepsilon}$ and $\widetilde{B}_N^0$, it is convenient to choose the integration contour as follows:
$$
\begin{equation*}
\begin{aligned} \, \gamma &=\biggl\{\zeta \in\mathbb{C} \colon \operatorname{Im}\zeta \geqslant 0,\, \operatorname{Re}\zeta =\operatorname{Im} \zeta +\frac{c_\flat}2\biggr\} \\ &\qquad\cup \biggl\{ \zeta \in \mathbb{C} \colon \operatorname\zeta \leqslant 0,\, \operatorname{Re} \zeta =-\operatorname{Im} \zeta +\frac{c_\flat}2\biggr\}. \end{aligned}
\end{equation*}
\notag
$$
Multiplying (10.8) by $f^\varepsilon$ from the left and by $(f^\varepsilon)^*$ from the right and taking (2.21) into account, we obtain the following representation:
$$
\begin{equation*}
f^\varepsilon e^{-\widetilde{B}_{N,\varepsilon}t}(f^\varepsilon)^*=-\frac{1}{2\pi i}\int_\gamma e^{-\zeta t}(B_{N,\varepsilon}-\zeta Q_0^\varepsilon)^{-1}\,d\zeta,\qquad t>0.
\end{equation*}
\notag
$$
Similarly,
$$
\begin{equation*}
f_0 e^{-\widetilde{B}_N^0t}f_0=-\frac{1}{2\pi i}\int_\gamma e^{-\zeta t}(B_N^0-\zeta \overline{Q_0}\,)^{-1}\,d\zeta,\qquad t>0.
\end{equation*}
\notag
$$
Hence
$$
\begin{equation}
\begin{aligned} \, &f^\varepsilon e^{-\widetilde{B}_{N,\varepsilon}t}(f^\varepsilon)^*-f_0 e^{-\widetilde{B}_N^0t}f_0 \nonumber \\ &\qquad =-\frac{1}{2\pi i}\int_\gamma e^{-\zeta t} \bigl((B_{N,\varepsilon}-\zeta Q_0^\varepsilon)^{-1} - (B_N^0-\zeta \overline{Q_0}\,)^{-1}\bigr)\,d\zeta. \end{aligned}
\end{equation}
\tag{10.9}
$$
We set $\check{c}:= \max \{1;\, \sqrt{5} c_\flat/2\}$. Applying Theorem 9.1 for $\zeta \in \gamma$, $|\zeta| \leqslant \check{c}$, and Theorem 3.2 for $\zeta \in \gamma$, $|\zeta| > \check{c}$, it is easy to check that
$$
\begin{equation}
\bigl\| (B_{N,\varepsilon}-\zeta Q_0^\varepsilon)^{-1}-(B_N^0-\zeta \overline{Q_0}\,)^{-1} \bigr\|_{L_2(\mathcal{O})\to L_2(\mathcal{O})}\leqslant \mathrm{C}_1'| \zeta |^{-1/2}\varepsilon,\qquad \zeta \in \gamma,
\end{equation}
\tag{10.10}
$$
where the constant $\mathrm{C}_1'$ depends only on the initial data (2.35). By (10.9) and (10.10) it follows that
$$
\begin{equation}
\bigl\| f^\varepsilon e^{-\widetilde{B}_{N,\varepsilon}t}(f^\varepsilon)^*-f_0 e^{-\widetilde{B}_N^0t}f_0 \bigr\|_{L_2(\mathcal{O})\to L_2(\mathcal{O})}\leqslant \mathrm{C}_1'' \varepsilon t^{-1/2} e^{-c_\flat t/2}, \qquad \mathrm{C}_1'' = 2 \pi^{-1/2} \mathrm{C}_1'.
\end{equation}
\tag{10.11}
$$
Using (10.11) for $t \geqslant \varepsilon^2$ and the elementary estimate
$$
\begin{equation*}
\bigl\| f^\varepsilon e^{-\widetilde{B}_{N,\varepsilon}t}(f^\varepsilon)^*-f_0 e^{-\widetilde{B}_N^0t}f_0 \bigr\|_{L_2(\mathcal{O})\to L_2(\mathcal{O})}\leqslant 2 \|f\|_{L_\infty}^2 e^{-c_\flat t}
\end{equation*}
\notag
$$
for $0\leqslant t < \varepsilon^2$, it is easy to deduce the required inequality (10.7). This completes the proof. 10.3. Approximation of the solution in $H^1(\mathcal{O};\mathbb{C}^n)$Consider the corrector
$$
\begin{equation}
\mathcal{K}_N(t;\varepsilon):=R_\mathcal{O}\bigl([\Lambda^\varepsilon]S_\varepsilon b(\mathbf{D})+[\widetilde{\Lambda}^\varepsilon ]S_\varepsilon\bigr) P_\mathcal{O}f_0 e^{-\widetilde{B}_N^0t}f_0.
\end{equation}
\tag{10.12}
$$
We will also need the operator
$$
\begin{equation}
\mathcal{G}_N(t;\varepsilon):=\widetilde{g}^{\,\varepsilon} S_\varepsilon b(\mathbf{D})P_\mathcal{O}f_0e^{-\widetilde{B}_N^0 t}f_0 +g^\varepsilon \bigl(b(\mathbf{D})\widetilde{\Lambda}\bigr)^\varepsilon S_\varepsilon P_\mathcal{O}f_0 e^{-\widetilde{B}_N^0t}f_0.
\end{equation}
\tag{10.13}
$$
For $t>0$, operator (10.12) is continuous from $L_2(\mathcal{O};\mathbb{C}^n)$ to $H^1(\mathcal{O};\mathbb{C}^n)$. Indeed, by (10.6), for $t>0$ the operator $f_0e^{-\widetilde{B}_N^0t}f_0$ is continuous from $L_2(\mathcal{O};\mathbb{C}^n)$ to $H^2(\mathcal{O};\mathbb{C}^n)$. Hence the operator $b(\mathbf{D})P_\mathcal{O}f_0e^{-\widetilde{B}_N^0 t}f_0$ is continuous from $L_2(\mathcal{O};\mathbb{C}^n)$ to $H^1(\mathbb{R}^d;\mathbb{C}^m)$, and the operator $P_\mathcal{O}f_0 e^{-\widetilde{B}_N^0 t}f_0$ is continuous from $L_2(\mathcal{O};\mathbb{C}^n)$ to $H^1(\mathbb{R}^d;\mathbb{C}^n)$. It remains to take into account the continuity of the operators $[\Lambda^\varepsilon ]S_\varepsilon\colon H^1(\mathbb{R}^d;\mathbb{C}^m)\to H^1(\mathbb{R}^d;\mathbb{C}^n)$ and $[\widetilde{\Lambda}^\varepsilon ]S_\varepsilon \colon H^1(\mathbb{R}^d;\mathbb{C}^n) \to H^1(\mathbb{R}^d;\mathbb{C}^n)$, which follows from Proposition 1.2 and since $\Lambda,\widetilde{\Lambda}\in \widetilde{H}^1(\Omega)$. Similarly, one can check that operator (10.13) is continuous from $L_2(\mathcal{O};\mathbb{C}^n)$ to $L_2(\mathcal{O};\mathbb{C}^m)$. We put $\widetilde{\mathbf{u}}_0(\,{\cdot}\,,t):=P_\mathcal{O}\mathbf{u}_0(\,{\cdot}\,,t)$. By $\mathbf{v}_\varepsilon(\mathbf{x},t)$ we denote the first-order approximation to the solution $\mathbf{u}_\varepsilon(\mathbf{x},t)$ of problem (10.1):
$$
\begin{equation}
\begin{gathered} \, \widetilde{\mathbf{v}}_\varepsilon (\,{\cdot}\,,t) =\widetilde{\mathbf{u}}_0(\,{\cdot}\,,t)+\varepsilon \Lambda^\varepsilon S_\varepsilon b(\mathbf{D})\widetilde{\mathbf{u}}_0(\,{\cdot}\,,t)+\varepsilon\widetilde{\Lambda}^\varepsilon S_\varepsilon \widetilde{\mathbf{u}}_0(\,{\cdot}\,,t), \\ \mathbf{v}_\varepsilon(\,{\cdot}\,,t) :=\widetilde{\mathbf{v}}_\varepsilon (\,{\cdot}\,,t) |_{\mathcal{O}}, \end{gathered}
\end{equation}
\tag{10.14}
$$
that is, $\mathbf{v}_\varepsilon (\,{\cdot}\,,t) = f_0 e^{-\widetilde{B}_N^0t}f_0\boldsymbol{\varphi}(\,{\cdot}\,)+\varepsilon \mathcal{K}_N(t;\varepsilon )\boldsymbol{\varphi}(\,{\cdot}\,)$. Theorem 10.3. Under the hypotheses of Theorem 10.2, let function $\mathbf{v}_\varepsilon(\mathbf{x},t)$ be defined by (10.14), and let $\mathbf{p}_\varepsilon(\mathbf{x},t):= g^\varepsilon(\mathbf{x}) b(\mathbf{D}) \mathbf{u}_\varepsilon(\mathbf{x},t)$. Suppose that $\mathcal{K}_N(t;\varepsilon)$ and $\mathcal{G}_N(t;\varepsilon)$ are the operators defined by (10.12) and (10.13), respectively. Then, for $0<\varepsilon\leqslant\varepsilon_1$ and $t>0$, Proof. Similarly to (10.9),
$$
\begin{equation}
\begin{aligned} \, &f^\varepsilon e^{-\widetilde{B}_{N,\varepsilon}t}(f^\varepsilon)^*-f_0 e^{-\widetilde{B}_N^0t}f_0-\varepsilon \mathcal{K}_N(t;\varepsilon) \nonumber \\ &\qquad =-\frac{1}{2 \pi i}\int_\gamma e^{-\zeta t}\bigl( (B_{N,\varepsilon}-\zeta Q_0^\varepsilon)^{-1}-(B_N^0-\zeta \overline{Q_0}\,)^{-1}-\varepsilon K_N(\varepsilon;\zeta )\bigr) \,d\zeta, \end{aligned}
\end{equation}
\tag{10.17}
$$
$$
\begin{equation}
\begin{aligned} \, &g^\varepsilon b(\mathbf{D})f^\varepsilon e^{-\widetilde{B}_{N,\varepsilon}t}(f^\varepsilon)^*-\mathcal{G}_N(t;\varepsilon) \nonumber \\ &\qquad= -\frac{1}{2\pi i}\int_\gamma e^{-\zeta t}\bigl(g^\varepsilon b(\mathbf{D})(B_{N,\varepsilon}-\zeta Q_0^\varepsilon)^{-1} -G_N(\varepsilon;\zeta)\bigr)\,d\zeta. \end{aligned}
\end{equation}
\tag{10.18}
$$
Here, the operators $K_N(\varepsilon;\zeta )$ and $G_N(\varepsilon;\zeta )$ are given by (3.5) and (3.6), respectively.
Using (10.17), applying Theorem 9.1 for $\zeta \in \gamma$, $|\zeta| \leqslant \check{c}$, and employing Theorem 3.4 for $\zeta \in \gamma$, $|\zeta| > \check{c}$, it is easy to deduce estimate (10.15). Similarly, applying identity (10.18), using (9.7) for $\zeta \in \gamma$, $|\zeta| \leqslant \check{c}$, and invoking (3.17) for $\zeta \in \gamma$, $|\zeta| > \check{c}$, we arrive at (10.16). This completes the proof. Remark 10.4. Let $\lambda_1^0$ be the first eigenvalue of the operator $B_N^0$, and let $\sigma>0$ be an arbitrarily small number. Obviously, the number $\lambda_1^0 \| Q_0\|^{-1}_{L_\infty}$ is a lower bound of the operator $\widetilde{B}_N^0$. Due to the resolvent convergence of $B_{N,\varepsilon}$ to $B_N^0$, for sufficiently small $\varepsilon_\circ$, the number $\lambda_1^0 \| Q_0\|^{-1}_{L_\infty} - \sigma /2$ is a common lower bound of the operators $\widetilde{B}_{N,\varepsilon}$ for all $0< \varepsilon \leqslant \varepsilon_\circ$. Therefore, it is possible to shift the integration contour $\gamma$ so that it intersects the real axis at the point $\mathfrak{c}_\circ :=\lambda_1^0 \| Q_0\|^{-1}_{L_\infty} - \sigma$ instead of $c_\flat/2$. In this way, we obtain estimates of the form (10.7), (10.15), (10.16) with $e^{-c_\flat t/2}$ replaced by $e^{-\mathfrak{c}_\circ t}$ on the right-hand sides. At the same time, the constants in estimates will depend on $\sigma$. 10.4. Removal of the smoothing operator $S_\varepsilon$ in the correctorIt is possible to remove the smoothing operator in the corrector if the solutions of the auxiliary problems are subject to some additional conditions. The following result is checked similarly to Theorem 10.3 with the help of Theorems 8.6 and 9.7. Theorem 10.5. Under the hypotheses of Theorem 10.3, let Conditions 8.1 and 8.2 be met. We set Remark 10.6. If only Condition 8.1 (respectively, Condition 8.2) is satisfied, then the smoothing operator $S_\varepsilon$ can be removed in the corrector term containing $\Lambda^\varepsilon$ (respectively, $\widetilde{\Lambda}^\varepsilon$). It is also possible to remove the smoothing operator $S_\varepsilon$ in the corrector by strengthening the assumption about the boundary smoothness. Consider the case $d\geqslant 3$, because for $d\leqslant 2$ Theorem 10.5 applies (see Propositions 8.3 and 8.4). Theorem 10.7. Under the hypotheses of Theorem 10.2, let $d\geqslant 3$, and let $\mathcal{O}\subset \mathbb{R}^d$ be a bounded domain of class $C^{d/2,1}$ for even $d$ and of class $C^{(d+1)/2,1}$ for odd $d$. Suppose that $\mathcal{K}_N^0(t;\varepsilon)$ and $\mathcal{G}_N^0(t;\varepsilon)$ are the operators given by (10.19), (10.20), respectively. Then, for $t>0$ and $0<\varepsilon\leqslant\varepsilon_1$, Theorem 10.7 relies on the following lemma, which is similar to Lemma 2.8 in [42]. Lemma 10.8. Let $k \geqslant 2$ be an integer. Let $\mathcal{O}\subset\mathbb{R}^d$ be a bounded domain of class $C^{k-1,1}$. Then, for $t>0$, the operator $e^{-\widetilde{B}_N^0t}$ is continuous from $L_2(\mathcal{O};\mathbb{C}^n)$ to $H^q(\mathcal{O};\mathbb{C}^n)$, $0\leqslant q\leqslant k$, and The proofs of Lemma 10.8 and Theorem 10.7 are completely analogous to the case of the first initial boundary value problem considered in [42], § 2.7 and § 7. Clearly, it is convenient to apply Theorem 10.7 when $t$ is separated from zero. For small values of $t$, the order of the factor $(\varepsilon^{1/2}t^{-3/4}+\varepsilon t^{-d/4-1/2})$ grows with dimension. This is compensation for the removal of the smoothing operator. 10.5. Special casesLet us highlight special cases. Suppose that $g^0=\overline{g}$, that is, relations (1.28) are fulfilled. In addition, suppose that condition (8.11) is fulfilled. Then the $\Gamma$-periodic solutions of problems (1.20) and (1.30) are equal to zero: $\Lambda (\mathbf{x})=0$ and $\widetilde{\Lambda}(\mathbf{x})=0$. Theorem 10.3 leads to the following result. Proposition 10.9. Let (1.28) and (8.11) be satisfied and let the assumptions of Theorem 10.2 be met. Then, for $t>0$ and $0<\varepsilon\leqslant \varepsilon_1$, Now, assume that $g^0=\underline{g}$, that is, (1.29) holds. Hence Condition 8.1 is fulfilled by assertion $3^\circ$ of Proposition 8.3. In this case, $\widetilde{g}(\mathbf{x})=g^0=\underline{g}$. In addition, assume that (8.11) holds. Then $\widetilde{\Lambda}(\mathbf{x})=0$, and from Theorem 10.3, using Proposition 1.1, it is easy to deduce the following result (cf. the proof of Proposition 2.13 in [42]). Proposition 10.10. Let (1.29) and (8.11) be satisfied and let the assumptions of Theorem 10.2 be met. Then, for $0<\varepsilon\leqslant\varepsilon_1$ and $t>0$, 10.6. Estimates in a strictly interior subdomainIt is easy to deduce the following result by applying Theorems 8.12, 9.10 and identities (10.17), (10.18). Theorem 10.11. Under the hypotheses of Theorem 10.3, let $\mathcal{O}'$ be a strictly interior subdomain of the domain $\mathcal{O}$ and $\delta = \operatorname{dist} \{\mathcal{O}'; \partial \mathcal{O} \}$. Then, for $0<\varepsilon\leqslant\varepsilon_1$ and $t>0$, The following result is proved by using Theorems 8.13, 9.11 and identities (10.17), (10.18). Theorem 10.12. Under the assumptions of Theorem 10.11, let Conditions 8.1 and 8.2 be met. Let $\mathcal{K}_N^0(t;\varepsilon)$ and $\mathcal{G}_N^0(t;\varepsilon)$ be the operators defined by (10.19) and (10.20), respectively. Then, for $t>0$ and $0<\varepsilon\leqslant \varepsilon_1$, Note that it is possible to remove the smoothing operator $S_\varepsilon$ in the corrector in estimates from Theorem 10.11 without additional assumptions on the matrix-valued functions $\Lambda$ and $\widetilde{\Lambda}$. Consider the case $d\geqslant 3$ (otherwise, by Propositions 8.3 and 8.4, Theorem 10.12 applies). For $t>0$, the operator $e^{-\widetilde{B}_N^0t}$ is continuous from $L_2(\mathcal{O};\mathbb{C}^n)$ to $H^2(\mathcal{O};\mathbb{C}^n)$, and estimate (10.6) holds. In addition, we have the property of “increasing smoothness” inside the domain: for $t>0$, the operator $e^{-\widetilde{B}_N^0t}$ is continuous from $L_2(\mathcal{O};\mathbb{C}^n)$ to $H^l (\mathcal{O}';\mathbb{C}^n)$ for any integer $l \geqslant 3$, and
$$
\begin{equation}
\begin{gathered} \, \bigl\| e^{-\widetilde{B}_N^0t}\bigr\|_{L_2(\mathcal{O})\to H^l (\mathcal{O}')} \leqslant\mathrm{C}'_l t^{-1/2}(\delta^{-2}+t^{-1})^{(l -1)/2}e^{-c_\flat t/2}, \\ t>0,\quad l \in\mathbb{N},\quad l \geqslant 3. \end{gathered}
\end{equation}
\tag{10.21}
$$
The constant $\mathrm{C}'_l$ depends on $l$ and the initial data (2.35) (cf. estimate (2.45) from [42] and the comments therein). Using (10.21), as well as the properties of the matrix-valued functions $\Lambda$, $\widetilde{\Lambda}$ and the operator $S_\varepsilon$, we deduce the following result from Theorem 10.11. The proof is completely analogous to the case of the first initial boundary value problem (see [42], §§ 2.10 and 8). Theorem 10.13. Under the hypotheses of Theorem 10.11, let $d\geqslant 3$, and let $\mathcal{K}_N^0(t;\varepsilon)$ and $\mathcal{G}_N^0(t;\varepsilon)$ be the operators defined by (10.19) and (10.20), respectively. Denote 10.7. Homogenization of the solutions of the second initial boundary value problem for a non-homogeneous equationNow we consider the solution $\mathbf{u}_\varepsilon(\mathbf{x},t)$ of the problem
$$
\begin{equation}
\begin{cases} Q_0^\varepsilon(\mathbf{x}) \, \dfrac{\partial \mathbf{u}_\varepsilon(\mathbf{x},t)}{\partial t} = - (B_\varepsilon \mathbf{u}_\varepsilon)(\mathbf{x},t) + \mathbf{F}(\mathbf{x},t), &\mathbf{x} \in \mathcal{O},\ 0< t< T, \\ Q_0^\varepsilon(\mathbf{x}) \mathbf{u}_\varepsilon(\mathbf{x},0) = \boldsymbol{\varphi}(\mathbf{x}), & \mathbf{x}\in \mathcal{O}, \end{cases}
\end{equation}
\tag{10.22}
$$
under the Neumann condition on $\partial \mathcal{O} \times (0,T)$. It is assumed that $0< T \leqslant \infty$ and $\mathbf{F} \in \mathfrak{H}_r(T):= L_r((0,T); L_2(\mathcal{O};\mathbb{C}^n))$ for some $1\leqslant r \leqslant \infty$; $\boldsymbol{\varphi} \in L_2(\mathcal{O};\mathbb{C}^n)$. Then
$$
\begin{equation}
\mathbf{u}_\varepsilon(\,{\cdot}\,,t) = f^\varepsilon e^{- \widetilde{B}_{N,\varepsilon} t} (f^\varepsilon)^* \boldsymbol{\varphi} + \int_0^t f^\varepsilon e^{-\widetilde{B}_{N,\varepsilon} (t-\widetilde{t}\,)} (f^\varepsilon)^* \mathbf{F}(\,{\cdot}\,, \widetilde{t}\,) \, d\widetilde{t}.
\end{equation}
\tag{10.23}
$$
The corresponding effective problem takes the form
$$
\begin{equation}
\begin{cases} \overline{Q_0} \, \dfrac{\partial \mathbf{u}_0(\mathbf{x},t)}{\partial t} = - (B^0 \mathbf{u}_0)(\mathbf{x},t) + \mathbf{F}(\mathbf{x},t), &\mathbf{x} \in \mathcal{O},\ 0< t< T, \\ \overline{Q_0} \mathbf{u}_0(\mathbf{x},0) = \boldsymbol{\varphi}(\mathbf{x}), &\mathbf{x}\in \mathcal{O}, \end{cases}
\end{equation}
\tag{10.24}
$$
with the Neumann condition on $\partial \mathcal{O} \times (0,T)$. The solution of the effective problem is represented as
$$
\begin{equation}
\mathbf{u}_0(\,{\cdot}\,,t) = f_0 e^{- \widetilde{B}_{N}^0 t} f_0 \boldsymbol{\varphi} + \int_0^t f_0 e^{- \widetilde{B}_{N}^0 (t-\widetilde{t}\,)} f_0 \mathbf{F}(\,{\cdot}\,, \widetilde{t}\,) \, d\widetilde{t}.
\end{equation}
\tag{10.25}
$$
Subtracting (10.25) from (10.23) and using Theorem 10.2, we see that
$$
\begin{equation*}
\begin{aligned} \, &\| \mathbf{u}_\varepsilon(\,{\cdot}\,,t) - \mathbf{u}_0(\,{\cdot}\,,t)\|_{L_2(\mathcal{O})} \leqslant \mathrm{C}_1 \varepsilon (t+\varepsilon^2)^{-1/2} e^{-c_\flat t/2} \| \boldsymbol{\varphi}\|_{L_2(\mathcal{O})} \\ &\qquad+ \mathrm{C}_1 \varepsilon \int_0^t e^{-c_\flat (t-\widetilde{t}\,)/2} (\varepsilon^2 +t - \widetilde{t})^{-1/2} \| \mathbf{F}(\,{\cdot}\,, \widetilde{t}\,) \|_{L_2(\mathcal{O})} \, d\widetilde{t}. \end{aligned}
\end{equation*}
\notag
$$
Estimating the integral term for $1< r \leqslant \infty$, we obtain the following result (cf. the proof of Theorem 5.1 from [35]). Theorem 10.14. Let $\boldsymbol{\varphi} \in L_2(\mathcal{O};\mathbb{C}^n)$ and $\mathbf{F} \in \mathfrak{H}_r(T)$ with some $1< r \leqslant \infty$, where $0< T \leqslant \infty$, let $\mathbf{u}_\varepsilon(\mathbf{x},t)$ be the solution of problem (10.22) with the Neumann condition on $\partial \mathcal{O} \times (0,T)$, and let $\mathbf{u}_0(\mathbf{x},t)$ be the solution of problem (10.24) with the Neumann condition on $\partial \mathcal{O} \times (0,T)$. Suppose that the number $\varepsilon_1$ satisfies Condition 3.1. Then, for $0< \varepsilon \leqslant \varepsilon_1$ and $0< t <T$, By analogy with the proof of Theorem 5.2 from [35], it is easy to deduce the estimate for the norm of the difference $\mathbf{u}_\varepsilon - \mathbf{u}_0$ in the class $\mathfrak{H}_r(T)$ from Theorem 10.2. Theorem 10.15. Let $\boldsymbol{\varphi} \in L_2(\mathcal{O};\mathbb{C}^n)$ and $\mathbf{F} \in \mathfrak{H}_r(T)$ with some $1 \leqslant r < \infty$, where $0< T \leqslant \infty$, let $\mathbf{u}_\varepsilon(\mathbf{x},t)$ be the solution of problem (10.22) with the Neumann condition on $\partial \mathcal{O} \times (0,T)$, and let $\mathbf{u}_0(\mathbf{x},t)$ be the solution of problem (10.24) with the Neumann condition on $\partial \mathcal{O} \times (0,T)$. Suppose that the number $\varepsilon_1$ satisfies Condition 3.1. Then, for $0< \varepsilon \leqslant \varepsilon_1$, Remark 10.16. In the case $\boldsymbol{\varphi}=0$ and $\mathbf{F} \in \mathfrak{H}_\infty(T)$, Theorem 10.14 implies that Now, using Theorem 10.3, we obtain approximation of the solution of problem (10.22) in $H^1(\mathcal{O};\mathbb{C}^n)$. Difficulties arise when considering the integral term in (10.23) due to the singularity of the right-hand side of estimate (10.15) for small $t$. Assuming that $t \geqslant \varepsilon^2$, we divide the integration interval in (10.23) into two parts: $(0, t-\varepsilon^2)$ and $(t-\varepsilon^2,t)$. On the interval $(0, t-\varepsilon^2)$, we apply (10.15), and on $(t-\varepsilon^2,t)$, we use the estimate
$$
\begin{equation*}
\bigl\| f^\varepsilon e^{-\widetilde{B}_{N,\varepsilon}t} (f^\varepsilon)^* - f_0 e^{-\widetilde{B}_{N}^0t} f_0 \bigr\|_{L_2(\mathcal{O})\to H^1(\mathcal{O})} \leqslant 2 c_4^{-1/2} \|f\|_{L_\infty} t^{-1/2}e^{-c_\flat t/2},\qquad t>0,
\end{equation*}
\notag
$$
which follows from (10.4) and (10.5). This leads to the following result (cf. the proof of Theorem 5.4 from [35]). Setting
$$
\begin{equation*}
\mathbf{u}_{0,\varepsilon}(\,{\cdot}\,,t) = f_0 e^{- \widetilde{B}_{N}^0 t} f_0 \boldsymbol{\varphi} + \int_0^{t-\varepsilon^2} f_0 e^{- \widetilde{B}_{N}^0 (t-\widetilde{t}\,)} f_0 \mathbf{F}(\,{\cdot}\,, \widetilde{t}\,) \, d\widetilde{t},
\end{equation*}
\notag
$$
we have $\mathbf{u}_{0,\varepsilon}(\,{\cdot}\,,t) = f_0 e^{- \widetilde{B}_{N}^0 \varepsilon^2} f_0^{-1} \mathbf{u}_0(\,{\cdot}\,, t-\varepsilon^2)$. Let $\widetilde{\mathbf{u}}_{0,\varepsilon}(\,{\cdot}\,,t) = P_\mathcal{O} \mathbf{u}_{0,\varepsilon}(\,{\cdot}\,,t)$, where $P_\mathcal{O}$ is the extension operator (3.3). As a first-order approximation to the solution of problem (10.22), we take
$$
\begin{equation}
\mathbf{v}_{\varepsilon}(\,{\cdot}\,,t) = \mathbf{u}_0(\,{\cdot}\,,t) + \varepsilon \Lambda^\varepsilon S_\varepsilon b(\mathbf{D}) \widetilde{\mathbf{u}}_{0,\varepsilon}(\,{\cdot}\,,t) + \varepsilon \widetilde{\Lambda}^\varepsilon S_\varepsilon \widetilde{\mathbf{u}}_{0,\varepsilon}(\,{\cdot}\,,t).
\end{equation}
\tag{10.27}
$$
An approximation for the flux $\mathbf{p}_\varepsilon(\,{\cdot}\,,t) = g^\varepsilon b(\mathbf{D}) \mathbf{u}_\varepsilon(\,{\cdot}\,,t)$ is given by
$$
\begin{equation}
\mathbf{q}_{\varepsilon}(\,{\cdot}\,,t) = \widetilde{g}^{\,\varepsilon} S_\varepsilon b(\mathbf{D}) \widetilde{\mathbf{u}}_{0,\varepsilon}(\,{\cdot}\,,t) + g^\varepsilon (b(\mathbf{D}) \widetilde{\Lambda})^\varepsilon S_\varepsilon \widetilde{\mathbf{u}}_{0,\varepsilon}(\,{\cdot}\,,t).
\end{equation}
\tag{10.28}
$$
Theorem 10.17. Let $\boldsymbol{\varphi} \in L_2(\mathcal{O};\mathbb{C}^n)$ and $\mathbf{F} \in \mathfrak{H}_r(T)$ for some $2< r \leqslant \infty$, where $0< T \leqslant \infty$. Let $\mathbf{u}_\varepsilon(\mathbf{x},t)$ be the solution of problem (10.22) with the Neumann condition on $\partial \mathcal{O} \times (0,T)$. Let $\mathbf{u}_0(\mathbf{x},t)$ be the solution of problem (10.24) with the Neumann condition on $\partial \mathcal{O} \times (0,T)$. Let $\mathbf{p}_\varepsilon(\,{\cdot}\,,t) = g^\varepsilon b(\mathbf{D}) \mathbf{u}_\varepsilon(\,{\cdot}\,,t)$. Suppose that $\mathbf{v}_{\varepsilon}$ and $\mathbf{q}_\varepsilon$ are the functions defined by (10.27), (10.28), respectively. Suppose that the number $\varepsilon_1$ satisfies Condition 3.1. Then, for $0< \varepsilon \leqslant \varepsilon_1$ and $\varepsilon^2 \leqslant t < T$, Since the right-hand side of (10.16) has a smaller singularity for small $t$ than the right-hand side of (10.15), for $r>4$ it is possible to approximate the flux $\mathbf{p}_\varepsilon(\,{\cdot}\,,t)$ by the function
$$
\begin{equation}
\mathbf{q}^0_{\varepsilon}(\,{\cdot}\,,t) = \widetilde{g}^{\,\varepsilon} S_\varepsilon b(\mathbf{D}) \widetilde{\mathbf{u}}_{0}(\,{\cdot}\,,t) + g^\varepsilon \bigl(b(\mathbf{D}) \widetilde{\Lambda}\bigr)^\varepsilon S_\varepsilon \widetilde{\mathbf{u}}_{0}(\,{\cdot}\,,t).
\end{equation}
\tag{10.29}
$$
The following assertion is proved similarly to the proof of Proposition 3.5 from [42]. Proposition 10.18. Let $\boldsymbol{\varphi} \in L_2(\mathcal{O};\mathbb{C}^n)$ and $\mathbf{F} \in \mathfrak{H}_r(T)$ with some $4< r \leqslant \infty$, where $0< T \leqslant \infty$. Let $\mathbf{u}_\varepsilon(\mathbf{x},t)$ be the solution of problem (10.22) with the Neumann condition on $\partial \mathcal{O} \times (0,T)$. Let $\mathbf{p}_\varepsilon(\,{\cdot}\,,t) = g^\varepsilon b(\mathbf{D}) \mathbf{u}_\varepsilon(\,{\cdot}\,,t)$. Suppose that $\mathbf{q}^0_\varepsilon$ is the function defined by (10.29). Suppose that the number $\varepsilon_1$ satisfies Condition 3.1. Then, for $0< \varepsilon \leqslant \varepsilon_1$ and $0 < t < T$, Under some additional assumptions on the solutions of the auxiliary problems, it is easy to deduce the following result from Theorem 10.5. Theorem 10.19. Under the assumptions of Theorem 10.12, let Conditions 8.1 and 8.2 be met. We set It is easy to deduce approximations of the solution and the flux in a strictly interior subdomain from Theorem 10.11. Theorem 10.20. Under the hypotheses of Theorem 10.17, let $\mathcal{O}'$ be a strictly interior subdomain of the domain $\mathcal{O}$. Let $\delta = \operatorname{dist} \{\mathcal{O}'; \partial \mathcal{O} \}$. Then, for $0< \varepsilon \leqslant \varepsilon_1$ and $\varepsilon^2 \leqslant t < T$, Finally, under some additional assumptions on the solutions of the auxiliary problems, it is easy to derive the following result from Theorem 10.12. Theorem 10.21. Under the assumptions of Theorem 10.19, let Conditions 8.1 and 8.2 be met. Let $\check{\mathbf{v}}_{\varepsilon}$ and $\check{\mathbf{q}}_{\varepsilon}$ be the functions defined by (10.30) and (10.31), respectively. Then, for $0< \varepsilon \leqslant \varepsilon_1$ and $\varepsilon^2 \leqslant t < T$, § 11. An example of applying the general resultsFor elliptic systems in the entire space $\mathbb{R}^d$, the example in question was studied in [10] and [37]. For elliptic and parabolic systems in a bounded domain under the Dirichlet condition, this example was considered in [40] and [42], respectively. 11.1. Scalar elliptic operator with a singular potential of order $\varepsilon^{-1}$Let $n=1$, $m=d$, $b(\mathbf{D})=\mathbf{D}$, and let $g(\mathbf{x})$ be a $\Gamma$-periodic symmetric $(d\times d)$-matrix-valued function with real entries such that $g(\mathbf{x})>0$ and $g,g^{-1}\in L_\infty$. Condition (1.5) is valid with $\alpha_0=\alpha_1 =1$. Obviously, Condition 2.1 is fulfilled with $k_1=1$, $k_2=0$. We have $b(\mathbf{D})^*g^\varepsilon (\mathbf{x}) b(\mathbf{D})=-\operatorname{div}g^\varepsilon (\mathbf{x})\nabla$. Next, let $\mathbf{A}(\mathbf{x})=\operatorname{col}\{ A_1(\mathbf{x}),\dots,A_d(\mathbf{x})\}$, where $A_j(\mathbf{x})$, $j=1,\dots,d$, are $\Gamma$-periodic real-valued functions such that
$$
\begin{equation*}
A_j\in L_\rho (\Omega),\qquad\rho=2\ \text{ for }\ d=1,\qquad \rho >d\ \text{ for } \ d\geqslant 2;\qquad j=1,\dots,d.
\end{equation*}
\notag
$$
Let $v(\mathbf{x})$ and $\mathcal{V}(\mathbf{x})$ be real-valued $\Gamma$-periodic functions such that
$$
\begin{equation*}
v,\mathcal{V}\in L_s(\Omega),\qquad\int_\Omega v(\mathbf{x})\,d\mathbf{x}=0,\qquad s=1\ \text{ for }\ d=1,\qquad s>\frac{d}2\ \text{ for }\ d\geqslant 2.
\end{equation*}
\notag
$$
In $L_2(\mathcal{O})$, we consider the operator $\mathfrak{B}_{N,\varepsilon}$ formally given by the differential expression
$$
\begin{equation}
\mathfrak{B}_{\varepsilon}=(\mathbf{D}-\mathbf{A}^\varepsilon(\mathbf{x}))^*g^\varepsilon (\mathbf{x})(\mathbf{D}-\mathbf{A}^\varepsilon (\mathbf{x}))+\varepsilon^{-1}v^\varepsilon (\mathbf{x})+\mathcal{V}^\varepsilon (\mathbf{x})
\end{equation}
\tag{11.1}
$$
under the natural condition (Neumann condition) on $\partial\mathcal{O}$. This operator can be interpreted as the Schrödinger operator with rapidly oscillating metric $g^\varepsilon$, magnetic potential $\mathbf{A}^\varepsilon$ and electric potential $\varepsilon^{-1}v^\varepsilon +\mathcal{V}^\varepsilon$, involving a singular term $\varepsilon^{-1}v^\varepsilon$. The precise definition of the operator $\mathfrak{B}_{N,\varepsilon}$ is given in terms of the quadratic form
$$
\begin{equation*}
\mathfrak{b}_{N,\varepsilon}[u,u]=\int_\mathcal{O} \bigl( \langle g^\varepsilon (\mathbf{D}-\mathbf{A}^\varepsilon )u,(\mathbf{D}-\mathbf{A}^\varepsilon)u\rangle +(\varepsilon^{-1}v^\varepsilon+\mathcal{V}^\varepsilon)| u|^2 \bigr)\,d\mathbf{x},\qquad u\in H^1(\mathcal{O}).
\end{equation*}
\notag
$$
It is easily seen (see [10], § 13.1) that expression (11.1) can be rewritten as
$$
\begin{equation*}
\mathfrak{B}_{\varepsilon}=\mathbf{D}^*g^\varepsilon (\mathbf{x})\mathbf{D} +\sum_{j=1}^d\bigl(a_j^\varepsilon (\mathbf{x})D_j+D_j(a_j^\varepsilon (\mathbf{x}))^*\bigr) +Q^\varepsilon (\mathbf{x}).
\end{equation*}
\notag
$$
Here, $Q(\mathbf{x})$ is a real-valued function given by
$$
\begin{equation}
Q(\mathbf{x})=\mathcal{V}(\mathbf{x})+\langle g(\mathbf{x})\mathbf{A}(\mathbf{x}),\mathbf{A}(\mathbf{x})\rangle.
\end{equation}
\tag{11.2}
$$
The complex-valued functions $a_j(\mathbf{x})$ are defined by
$$
\begin{equation}
a_j(\mathbf{x})=-\eta_j(\mathbf{x})+i\xi_j(\mathbf{x}),\qquad j=1,\dots, d.
\end{equation}
\tag{11.3}
$$
Here, $\eta_j(\mathbf{x})$ are the components of the vector-valued function $\boldsymbol{\eta}(\mathbf{x})=g(\mathbf{x})\mathbf{A}(\mathbf{x})$, and the functions $\xi_j(\mathbf{x})$ are given by $\xi_j (\mathbf{x})=-\partial_j \Phi (\mathbf{x})$, where $\Phi (\mathbf{x})$ is the $\Gamma$-periodic solution of the problem $\Delta \Phi(\mathbf{x})=v(\mathbf{x})$, $\int_\Omega \Phi(\mathbf{x})\,d\mathbf{x}=0$. We have
$$
\begin{equation}
v(\mathbf{x})=-\sum_{j=1}^d\partial_j \xi_j(\mathbf{x}).
\end{equation}
\tag{11.4}
$$
It is easy to check that the functions (11.3) satisfy condition (1.9) with a suitable exponent $\rho '$ depending on $\rho$ and $s$, and the norms $\|a_j\|_{L_{\rho '}(\Omega)}$ are controlled in terms of $\| g\|_{L_\infty}$, $\|\mathbf{A}\|_{L_\rho (\Omega)}$, $\| v\|_{L_s(\Omega)}$, and the parameters of the lattice $\Gamma$. The function (11.2) satisfies condition (1.10) with a suitable exponent $s'=\min \{ s;\rho/2\}$. Let $Q_0(\mathbf{x})$ be a positive definite bounded $\Gamma$-periodic function. We consider a positive definite operator $\mathcal{B}_{N,\varepsilon}:=\mathfrak{B}_{N,\varepsilon}+\lambda Q_0^\varepsilon$. Here, the constant $\lambda$ is chosen from condition (2.34) for the operator with the coefficients $g$, $a_j$, $j=1,\dots,d$, $Q$, and $Q_0$ defined above. The operator $\mathcal{B}_{N,\varepsilon}$ can be written as
$$
\begin{equation*}
\mathcal{B}_{N,\varepsilon}=(\mathbf{D}-\mathbf{A}^\varepsilon (\mathbf{x}))^* g^\varepsilon (\mathbf{x})(\mathbf{D}-\mathbf{A}^\varepsilon (\mathbf{x})) +\varepsilon^{-1} v^\varepsilon (\mathbf{x})+\mathcal{V}^\varepsilon (\mathbf{x})+\lambda Q_0^\varepsilon (\mathbf{x}).
\end{equation*}
\notag
$$
In the case under consideration, the initial data (2.35) are reduced to the set
$$
\begin{equation}
\begin{gathered} \, d,\ \rho,\ s;\ \| g\|_{L_\infty},\ \| g^{-1}\|_{L_\infty}, \ \| \mathbf{A}\|_{L_\rho (\Omega)}, \ \| v\|_{L_s(\Omega)}, \ \| \mathcal{V}\|_{L_s(\Omega)}, \\ \| Q_0\|_{L_\infty}, \ \| Q_0^{-1}\|_{L_\infty}; \ \text{the parameters of the lattice }\Gamma,\ \text{the domain }\mathcal{O}. \end{gathered}
\end{equation}
\tag{11.5}
$$
Let us describe the effective operator. In the case under consideration, the $\Gamma$-periodic solution of problem (1.20) is the row-matrix $\Lambda (\mathbf{x})=i\Psi (\mathbf{x})$, $\Psi (\mathbf{x})=(\psi_1(\mathbf{x}),\dots,\psi_d(\mathbf{x}))$, where $\psi_j \in\widetilde{H}^1(\Omega)$ is the solution of the problem
$$
\begin{equation*}
\operatorname{div}g(\mathbf{x})(\nabla \psi_j (\mathbf{x})+\mathbf{e}_j)=0,\qquad \int_\Omega \psi_j(\mathbf{x})\,d\mathbf{x}=0.
\end{equation*}
\notag
$$
Here, $\mathbf{e}_j$, $j=1,\dots,d$, is the standard basis for $\mathbb{R}^d$. Clearly, the functions $\psi_j(\mathbf{x})$ are real-valued, and the entries of the matrix $\Lambda(\mathbf{x})$ are purely imaginary. According to (1.22), the columns of the $(d\times d)$-matrix-valued function $\widetilde{g}(\mathbf{x})$ are given by $g(\mathbf{x})(\nabla \psi_j (\mathbf{x})+\mathbf{e}_j)$, $j=1,\dots,d$. The effective matrix is defined according to (1.21): $g^0=|\Omega|^{-1}\int_\Omega\widetilde{g}(\mathbf{x})\,d\mathbf{x}$. Clearly, $\widetilde{g}(\mathbf{x})$ and $g^0$ have real entries. According to (11.3) and (11.4), the periodic solution of problem (1.30) is represented as $\widetilde{\Lambda}(\mathbf{x}) =\widetilde{\Lambda}_1(\mathbf{x})+i\widetilde{\Lambda}_2(\mathbf{x})$, where the real-valued $\Gamma$-periodic functions $\widetilde{\Lambda}_1(\mathbf{x})$ and $\widetilde{\Lambda}_2(\mathbf{x})$ are the solutions of the problems
$$
\begin{equation*}
\begin{gathered} \, -\operatorname{div} g(\mathbf{x})\nabla \widetilde{\Lambda}_1(\mathbf{x})+v(\mathbf{x})=0, \qquad\int_\Omega\widetilde{\Lambda}_1(\mathbf{x})\,d\mathbf{x}=0, \\ -\operatorname{div} g(\mathbf{x})\nabla \widetilde{\Lambda}_2(\mathbf{x})+\operatorname{div} g(\mathbf{x})\mathbf{A}(\mathbf{x})=0, \qquad\int_\Omega\widetilde{\Lambda}_2(\mathbf{x})\,d\mathbf{x}=0. \end{gathered}
\end{equation*}
\notag
$$
The column $V$ (see (1.34)) has the form $V=V_1+iV_2$, where $V_1$, $V_2$ are the columns with real entries defined by
$$
\begin{equation*}
\begin{aligned} \, V_1 &=|\Omega|^{-1}\int_\Omega (\nabla\Psi (\mathbf{x}))^t g(\mathbf{x})\nabla\widetilde{\Lambda}_2(\mathbf{x}) \,d\mathbf{x}, \\ V_2 &=-|\Omega|^{-1}\int_\Omega (\nabla\Psi (\mathbf{x}))^t g(\mathbf{x})\nabla \widetilde{\Lambda}_1(\mathbf{x}) \,d\mathbf{x}. \end{aligned}
\end{equation*}
\notag
$$
According to (1.35), the constant $W$ is given by
$$
\begin{equation*}
W=|\Omega|^{-1}\int_\Omega \bigl(\langle g(\mathbf{x})\nabla \widetilde{\Lambda}_1(\mathbf{x}),\nabla\widetilde{\Lambda}_1(\mathbf{x})\rangle +\langle g(\mathbf{x})\nabla\widetilde{\Lambda}_2(\mathbf{x}), \nabla\widetilde{\Lambda}_2(\mathbf{x})\rangle\bigr)\,d\mathbf{x}.
\end{equation*}
\notag
$$
The effective operator ${\mathcal B}_N^0$ for $\mathcal{B}_{N,\varepsilon}$ is given by
$$
\begin{equation*}
\mathcal{B}^0 u=-\operatorname{div} g^0\nabla u +2i\langle\nabla u, V_1+\overline{\boldsymbol{\eta}}\rangle +(-W+\overline{Q}+\lambda \overline{Q_0}\,)u
\end{equation*}
\notag
$$
under the Neumann boundary condition. This expression can be represented as
$$
\begin{equation*}
\mathcal{B}^0=(\mathbf{D}-\mathbf{A}^0)^*g^0(\mathbf{D}-\mathbf{A}^0)+\mathcal{V}^0+\lambda \overline{Q_0},
\end{equation*}
\notag
$$
where $\mathbf{A}^0=(g^0)^{-1}(V_1+\overline{g\mathbf{A}})$ and $\mathcal{V}^0=\overline{\mathcal{V}}+\overline{\langle g\mathbf{A},\mathbf{A}\rangle}-\langle g^0\mathbf{A}^0,\mathbf{A}^0\rangle -W$. 11.2. Elliptic resultsAccording to Remark 8.5, in the case under consideration Conditions 8.1 and 8.2 are fulfilled, and the norms $\| \Lambda\|_{L_\infty}$ and $\|\widetilde{\Lambda}\|_{L_\infty}$ are estimated in terms of the problem data (11.5). Therefore, the corrector (8.1) without the smoothing operator can be used:
$$
\begin{equation*}
K_N^0(\varepsilon;\zeta):=\bigl(\Lambda^\varepsilon\mathbf{D} +\widetilde{\Lambda}^\varepsilon\bigr)(\mathcal{B}_N^0-\zeta\overline{Q_0}\,)^{-1} =\bigl(\Psi^\varepsilon\nabla +\widetilde{\Lambda}^\varepsilon\bigr)(\mathcal{B}_N^0-\zeta\overline{Q_0}\,)^{-1}.
\end{equation*}
\notag
$$
The operator (8.2) can be written as
$$
\begin{equation*}
G_N^0(\varepsilon;\zeta )= -i \bigl(\widetilde{g}^{\,\varepsilon} \nabla(\mathcal{B}_N^0-\zeta\overline{Q_0}\,)^{-1} +g^\varepsilon (\nabla\widetilde{\Lambda})^\varepsilon (\mathcal{B}_N^0-\zeta \overline{Q_0}\,)^{-1} \bigr).
\end{equation*}
\notag
$$
Applying Theorems 3.2 and 8.6, we obtain the following result. Proposition 11.1. Suppose that the assumptions of § 11.1 are satisfied. Let $\zeta\in\mathbb{C}\setminus \mathbb{R}_+$, $\zeta =| \zeta | e^{i\phi}$, $0<\phi<2\pi$, $|\zeta| \geqslant 1$. Suppose that the number $\varepsilon_1$ satisfies Condition 3.1. Then, for $0<\varepsilon\leqslant\varepsilon_1$, “Another” approximation of the operator $(\mathcal{B}_{N,\varepsilon}-\zeta Q_0^\varepsilon )^{-1}$ follows from Theorems 9.1 and 9.7. Proposition 11.2. Suppose that the assumptions of § 11.1 are satisfied. Let $\zeta \in \mathbb{C}\setminus [c_\flat,\infty)$, where $c_\flat = \frac{1}{2}\|g^{-1}\|^{-1}_{L_\infty} \| Q_0\|^{-1}_{L_\infty}$. Let $\varrho_\flat (\zeta )$ be the value (9.1). Suppose that the number $\varepsilon_1$ satisfies Condition 3.1. Then, for $0<\varepsilon\leqslant \varepsilon_1$, We can also apply Theorems 8.13 and 9.11 on approximations of the operator $(\mathcal{B}_{N,\varepsilon }-\zeta Q_0^\varepsilon )^{-1}$ in a strictly interior subdomain. 11.3. Parabolic resultsLet us briefly discuss a parabolic problem. Let $u_\varepsilon(\mathbf{x},t)$ be the solution of the initial boundary value problem
$$
\begin{equation}
\begin{cases} Q_0^\varepsilon(\mathbf{x}) \dfrac{\partial u_\varepsilon(\mathbf{x},t)}{\partial t} = - (\mathbf{D}-\mathbf{A}^\varepsilon(\mathbf{x}))^*g^\varepsilon (\mathbf{x})(\mathbf{D}-\mathbf{A}^\varepsilon (\mathbf{x}))u_\varepsilon(\mathbf{x},t) \\ \quad {-}\bigl(\varepsilon^{-1}v^\varepsilon (\mathbf{x})+\mathcal{V}^\varepsilon (\mathbf{x}) + \lambda Q_0^\varepsilon(\mathbf{x}) \bigr) u_\varepsilon(\mathbf{x},t), \quad \mathbf{x} \in \mathcal{O},\ 0< t< T, \\ Q_0^\varepsilon(\mathbf{x}) u_\varepsilon(\mathbf{x},0) = \varphi(\mathbf{x}), \mathbf{x}\in \mathcal{O}, \end{cases}
\end{equation}
\tag{11.6}
$$
with the Neumann condition on $\partial \mathcal{O} \times \mathbb{R}_+$. Here, $\varphi \in L_2(\mathcal{O})$. The homogenized problem takes the form
$$
\begin{equation}
\begin{cases} \overline{Q_0}\, \dfrac{\partial u_0(\mathbf{x},t)}{\partial t} = -(\mathbf{D}-\mathbf{A}^0)^*g^0(\mathbf{D}-\mathbf{A}^0) u_0(\mathbf{x},t) \\ \quad {-}\bigl(\mathcal{V}^0 + \lambda \overline{Q_0}\, \bigr) u_0(\mathbf{x},t), \quad \mathbf{x} \in \mathcal{O},\ 0< t< T, \\ \overline{Q_0} u_0(\mathbf{x},0) = \varphi(\mathbf{x}), \mathbf{x}\in \mathcal{O}. \end{cases}
\end{equation}
\tag{11.7}
$$
Applying Theorems 10.2 and 10.5, we obtain the following assertion. Proposition 11.3. Suppose that the assumptions of § 11.1 are satisfied. Let $u_\varepsilon(\mathbf{x},t)$ be the solution of problem (11.6), and let $u_0(\mathbf{x},t)$ be the solution of the homogenized problem (11.7). Suppose that the number $\varepsilon_1$ satisfies Condition 3.1. Then, for $0<\varepsilon\leqslant\varepsilon_1$ and $t>0$, We can also apply Theorem 10.12 and obtain approximation of the solution $u_\varepsilon(\,{\cdot}\,,t)$ in $H^1(\mathcal{O}')$. It is also possible to consider the initial boundary value problem for the non-homogeneous equation (the analog of (11.6) with the additional term $F(\mathbf{x},t)$ on the right-hand side of the equation) and apply Theorems 10.14, 10.15, 10.19, and 10.21.
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