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On operator estimates for elliptic equations in two-dimensional domains with fast oscillating boundary and frequent alternation of boundary conditions D. I. Borisovab, R. R. Suleimanovc a Institute of Mathematics with Computing Centre, Ufa Federal Research Centre of the Russian Academy of Sciences, Ufa, Russia b Peoples' Friendship University of Russia, Moscow, Russia c Ufa University of Science and Technology, Ufa, Russia
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  IntroductionThis paper is dedicated to Arlen Mikhailovich Il’in, academician of the Russian Academy of Sciences, the world-famous scientist and founder of a scientific school in asymptotic analysis to which these authors belong. The first-named author is a scientific grandson of Il’in, and the other author is his scientific great-grandson. January 8, 2022 marked the 90th anniversary of Arlen Il’in’s birth, June 23, 2023 was ten years from the date of his death, and we would like to remember our teacher. A large number of works are devoted to the convergence of solutions to boundary value problems with fast oscillating boundaries and frequent alternation of boundary conditions: see, for example, [1]–[15]. Classical results classify the possible homogenized problems depending on the type and structure of perturbations and describe the convergence of solutions of perturbed problems to solutions of the homogenized ones. Convergence has been established in the weak and strong sense in the spaces $W_2^1$ and $L_2$ for fixed right-hand sides of the equation and boundary conditions. In the case of linear equations this weak or strong convergence of solutions is equivalent to the weak or strong resolvent convergence of the corresponding resolvents. If the convergence of resolvents can be established in the operator norm, then this is a significantly stronger result. An estimate of the rate of convergence of the norm of a resolvent difference is called an operator estimate. Equivalently, it can be understood as an estimate for the norm of the difference of solutions of the perturbed and homogenized problems which is uniform with respect to the norm of the right-hand side of the equation. Similar results for various problems with frequently alternating boundary conditions and fast oscillating boundaries were established in [2], Ch. III, § 4, and [15]–[28]. In [29] a linear elliptic equation was considered in a multidimensional bounded domain with aperiodic fast oscillating boundary on which the Dirichlet condition was set. The uniform resolvent convergence of this boundary-value problem to limiting problems with different types of boundary conditions — the Dirichlet problem and the third boundary value problem — depending on the frequency of oscillation of the boundary in the perturbed problem, was proved. The compact embedding of the space $W_2^1$ in $L_2$ was used substantially in the proof; operator estimates were not established. In [2], Ch. III, § 4, [27], [28] and [15] oscillating boundaries were defined as the graphs of periodic or locally periodic functions. In the first of these works a two-dimensional boundary value problem was considered for a scalar operator with a third boundary condition on the boundary of a domain with oscillating boundary defined as the graph of a periodic function whose period and amplitude of oscillations were equal to the same small parameter. In [27] and [28] a multidimensional problem was considered for the system of Stokes’ equations and Poisson’s equation with the Dirichlet boundary condition on the oscillating compact boundary defined in terms of a locally periodic function. In these cases $W_2^1$- and $L_2$-operator estimates were established. In [15] the problem for a general self-adjoint elliptic operator of the second order in a plane strip with fast and periodically oscillating boundary was consider, where the period and amplitude of the oscillations were determined by a pair of mutually independent small parameters. Depending on the relationship between these parameters and the geometry of oscillations, various homogenized problems arose in the limit. For each of them $W_2^1$-operator estimates were established. In [18] the general elliptic operator of the second order was considered in a plane domain with oscillating boundary, and the boundary was described by a bounded function depending arbitrarily on a small parameter, which characterizes the amplitude of oscillations. On the oscillating boundary the Dirichlet or Neumann boundary condition was set, which was preserved under the homogenization. For such a problem $W_2^1$- and $L_2$-operator estimates were established. In [17] a similar operator was considered in a two-dimensional domain with oscillating boundary described in the same way of as in [18]; however, this boundary was now divided into two parts, on one of which the Dirichlet condition was set, while the Neumann condition was set on the other. The homogenization resulted in the same mixed boundary conditions and in the extension of the domain of the homogenized operator beyond the space $W_2^2$. Operator estimates were obtained, and the change of boundary conditions influenced the rate of convergence. In [16] a system of semilinear elliptic equations was considered in a multidimensional domain. In a thin layer along the boundary, this boundary was curved in an arbitrary way, and the Dirichlet or Neumann condition was set on this curved boundary. In the case of the Neumann condition very weak and natural conditions were additionally imposed on the structure of the curvature, still allowing one to consider a very wide class of curvatures, including classical fast oscillating boundaries. When this thin layer was compressed and the curved boundary approached the unperturbed one, the original system of semilinear equations arose with same type of boundary conditions as in the perturbed problem. For this case $W_2^1$- and $L_2$-operator estimates were established. Some $W_2^1$-operator estimates for problems with frequent alternation were obtained in [23]–[25], where boundary-value problems for the Laplace operator were considered in a plane strip with periodic or almost periodic alternation of the boundary conditions on one component of the boundary. In [23] the case where homogenization replaces the alternation of boundary conditions by the Dirichlet the condition was considered, and in [24] the case where homogenization leads to the Neumann condition. The occurrence of one or another the homogenized problem depends on the ratio of the lengths of the Dirichlet and Neumann parts in the perturbed problem. In [25] the Schrödinger operator in a plane strip was considered; a frequent alternation of the third boundary condition and the Dirichlet condition was specified on one component of the boundary, and this alternation was aperiodic. Depending on the ratio of the lengths with the Dirichlet and third boundary condition, a homogenized problem with only the Dirichlet or only the third boundary condition arose in the limit; for each of these the uniform resolvent convergence of the perturbed problem to the corresponding homogenized problem was proved, and operator estimates were established in various norms. In [19]–[21] an elliptic operator in a multidimensional domain with frequent alternation of boundary conditions was considered in the case where the Dirichlet boundary value problem for the same operator arose after the homogenization. The uniform resolvent convergence of the perturbed problems to the homogenized ones was proved there, and estimates for the rate of convergence were obtained. In [26], for a second-order elliptic operator of general form the authors succeeded to omit completely the restrictions on the periodicity of alternation on the boundary of an arbitrary smooth plane domain. Order-sharp $W_2^1$- and $L_2$-operator estimates were established. In [22] a boundary-value problem for the Laplace operator was considered in a domain with a smooth boundary. On a small part of this boundary the Dirichlet condition was imposed, and the Steklov condition was set on the rest of the boundary. The behaviour of the original problem as the small parameter responsible for the size of the part carrying the Dirichlet condition tends to zero was investigated, and operator estimates were obtained. We also note that the above works on operator estimates in problems of boundary homogenization were motivated by a large number of publications on operator estimates for operators with fast oscillating coefficients: see [30], [31] and the extensive lists of references in these surveys. In the present paper we propose a model in which the rapid oscillation of the boundary is combined with the frequent alternation. Namely, we consider a two-dimensional boundary value problem for a weakly nonlinear second-order elliptic equation. An arbitrary connected component of the boundary of the domain is replaced by a fast oscillating boundary with a small amplitude of oscillations. The oscillating boundary is defined as the graph of an arbitrary smooth function. On this oscillating boundary the frequent alternation is specified, which is general and aperiodic, without any restrictions on its structure: the Dirichlet and Neumann conditions are alternately set on small parts of the boundary. We consider the case where, after the homogenization of such a problem, a boundary-value problem for the same equation, with the Dirichlet condition on the originally specified connected component of the boundary of the unperturbed domain, arises in the limit. For this to occur the parts of the boundary with the Dirichlet condition in the perturbed problem must be placed sufficiently close to one another, and their lengths must satisfy a certain relation. Our main result is $W_2^1$- and $L_2$-operator estimates for such a problem. We stress that the above constraint, distinguishing the homogenized Dirichlet problem, differs significantly from the analogous constraint in classical problems with frequently alternating boundary conditions. This unexpected effect is a nontrivial manifestation of the combination of fast oscillations of the boundary with the frequent alternation of boundary conditions. § 1. Problem statement and main resultsLet $\Omega \subseteq \mathbb{R}^2$ be a domain with boundary of class $C^1$, and let one of the connected components of the boundary belong to the class $C^2$. We denote this component by $\Gamma_0$ and assume that it is closed (and therefore bounded) or infinite, and that the domain $\Omega$ lies on one side of $\Gamma_0$. Let $s$ be a natural parameter on $\Gamma_0$ and $\nu$ be the unit inward normal to $\Gamma_0$ relative to $\Omega$. Let $\tau$ denote the distance measured along $\nu$. We assume that the component $\Gamma_0$ is regular, namely, that the variables ($\tau,s$) are well defined in some strip
$$
\begin{equation*}
\Pi_{\tau_0} := \{x\in\Omega\colon \operatorname{dist}(x,\Gamma_0)\leqslant\tau_0\},
\end{equation*}
\notag
$$
where $\tau_0$ is a positive number, and the first and second derivatives of the variables $x$ with respect to ($\tau, s$) and of ($\tau, s$) with respect to $x$ are uniformly bounded in $\Pi_{\tau_0}$. In this paper we consider a perturbation of $\Gamma_0$ described by
$$
\begin{equation*}
\Gamma_\varepsilon:= \{x\colon \tau=\varepsilon b_\varepsilon(s),\, s\in\Gamma_0\},
\end{equation*}
\notag
$$
where $\varepsilon$ is a small positive parameter and $b_\varepsilon(s)$ is an arbitrary nonnegative real function bounded above by one and belonging to the space $C^1(\Gamma_0)$. The curve $\Gamma_\varepsilon$ is a component of the boundary of the perturbed domain
$$
\begin{equation*}
\Omega_\varepsilon :=\Omega\setminus\{x \in \Omega\colon 0 \leqslant \tau \leqslant \varepsilon b_\varepsilon(s)\}.
\end{equation*}
\notag
$$
We introduce two ordered sets of numbers $a_j^\varepsilon$ and $b_j^\varepsilon$, $j \in\mathbb{I}_\varepsilon$, where $\mathbb{I}_\varepsilon \subseteq\mathbb{N}$ is a subset of the set of positive integers. The $a_j^\varepsilon$ and $b_j^\varepsilon$ are ordered in the sense of the inequality
$$
\begin{equation*}
a_i^\varepsilon<b_i^\varepsilon<a_j^\varepsilon<b_j^\varepsilon, \qquad i<j, \quad i,j \in\mathbb {I}_\varepsilon.
\end{equation*}
\notag
$$
We assume in addition that each $a_j^\varepsilon$ and each $b_j^\varepsilon$ is an admissible value of the natural parameter $s$ on $\Gamma_0$. Let $\Gamma_D^\varepsilon$ denote the following system of arcs on the curve $\Gamma_\varepsilon$:
$$
\begin{equation}
\Gamma_D^\varepsilon :=\{x\colon \tau=\varepsilon b_\varepsilon(s),\, \varepsilon a^\varepsilon_j < s < \varepsilon b^\varepsilon_j,\, j=0,\dots,N_\varepsilon\}.
\end{equation}
\tag{1.1}
$$
We denote the complement of this system to the full curve $\Gamma_\varepsilon$ by $\Gamma_N^\varepsilon$:
$$
\begin{equation*}
\Gamma_N^\varepsilon := \Gamma_\varepsilon \setminus \overline{\Gamma_D^\varepsilon}.
\end{equation*}
\notag
$$
A schematic example of the domain $\Omega_\varepsilon$ and the partition of its oscillating boundary into the parts $\Gamma_D^\varepsilon$ and $\Gamma_N^\varepsilon$ is shown in Figure 1. We consider the semilinear differential expression
$$
\begin{equation}
\widehat{\mathcal{H}}u=-\sum_{i,j=1}^{2} \frac{\partial}{\partial x_i} A_{ij}\,\frac{\partial u}{\partial x_j} + \sum_{j=1}^{2} A_j \, \frac{\partial u}{\partial x_j} + A_0(\,\cdot\,,u).
\end{equation}
\tag{1.2}
$$
Here the $A_{ij}=A_{ij}(x)$ are real-valued functions, $A_j=A_j(x)$ and $A_0=A_0(x,u)$ are complex-valued functions defined on $\Omega$ and $\Omega\times \mathbb{C}$, respectively, and we have
$$
\begin{equation}
\begin{gathered} \, A_{ij}\in W^1_\infty(\Omega), \qquad A_j\in L_\infty(\Omega), \\ \operatorname{Re} \sum_{i,j=1}^{2} A_{ij}(x)z_i \overline{z_j}\geqslant c_0|z|^2, \qquad x \in \Omega, \quad z=(z_1,z_2)\in \mathbb{C}^2, \\ A_0(x,0)=0\quad\text{and} \quad |A_0(x,u_1)-A_0(x,u_2)|\leqslant c_1 |u_1-u_2| \end{gathered}
\end{equation}
\tag{1.3}
$$
for almost all $x\in\Omega$ with a constant $c_1$ independent of $x$ and $u$. We assume in addition that for any $u \in L_2(\Omega)$ the function $A_0(x,u(x))$ is measurable on $\Omega$. The main object of our study is a boundary-value problem for a semilinear elliptic equation:
$$
\begin{equation}
\begin{gathered} \, \widehat{\mathcal{H}}u_\varepsilon-\lambda u_\varepsilon=f \quad\text{in } \Omega_\varepsilon, \qquad u=0 \quad\text{on } \Gamma:=\partial\Omega\setminus\Gamma_0, \\ u=0 \quad\text{on } \Gamma_D^\varepsilon , \qquad \frac{\partial u }{\partial \boldsymbol{\nu_\varepsilon}}=0 \quad\text{on } \Gamma_N^\varepsilon, \end{gathered}
\end{equation}
\tag{1.4}
$$
where the conormal derivative is given by
$$
\begin{equation}
\frac{\partial u}{\partial\boldsymbol{\nu_\varepsilon}}:=\sum_{i,j=1}^{2} \nu^i_\varepsilon A_{ij} \, \frac{\partial}{\partial x_j},
\end{equation}
\tag{1.5}
$$
and $\nu_\varepsilon=(\nu_\varepsilon^1,\nu_\varepsilon^2)$ is the inward unit normal to $\Gamma_\varepsilon$ relative to the domain $\Omega_\varepsilon$. In (1.4) $\lambda$ is a complex number and $f$ denotes an arbitrary function in $L_2(\Omega)$. The aim of this paper is to describe the behaviour of solutions of the boundary value problem (1.4) as $\varepsilon\to+0$, namely, to clarify the form of the homogenized problem and prove appropriate operator estimates. To formulate the main results it is necessary to introduce auxiliary notation first. Let $\mathring{W}_2^1(\Omega,\gamma)$ denote the set of functions with zero trace on the curve $\gamma$ in $\Omega$ that belong to the Sobolev space $W_2^1(\Omega)$. We define the following nonlinear form on the space $\mathring{W}_2^1(\Omega_\varepsilon,\Gamma\cup\Gamma_D^\varepsilon)$:
$$
\begin{equation*}
\mathfrak{h}_\varepsilon(u,v) :=\sum_{i,j=1}^{n} \biggl(A_{ij}\,\frac{\partial u}{\partial x_j}, \frac{\partial v}{\partial x_i}\biggr)_{L_2(\Omega_\varepsilon)} + \sum_{j=1}^{n} \biggl(A_j\,\frac{\partial u}{\partial x_j},v\biggr)_{L_2(\Omega_\varepsilon)} + \bigl(A_0(\,\cdot\,,u),v\bigr)_{L_2(\Omega_\varepsilon)}.
\end{equation*}
\notag
$$
We treat solutions of the boundary-value problem under consideration in a generalized sense. Namely, a solution of (1.4) is a function in $\mathring{W}_2^1(\Omega_\varepsilon,\Gamma\cup\Gamma_D^\varepsilon)$ satisfying the integral identity
$$
\begin{equation}
\mathfrak{h}_\varepsilon(u_\varepsilon,v)- \lambda (u_\varepsilon,v)_{L_2(\Omega_\varepsilon)}=(f,v)_{L_2(\Omega_\varepsilon)}
\end{equation}
\tag{1.6}
$$
for all $\mathring{W}_2^1(\Omega_\varepsilon,\Gamma\cup\Gamma_D^\varepsilon)$. In this paper we consider the case when the homogenized problem is the one with the Dirichlet condition on the boundary component $\Gamma_0$,
$$
\begin{equation}
\widehat{\mathcal{H}}u_0-\lambda u_0=f \quad\text{in } \Omega, \qquad u_0=0\quad\text{on } \partial\Omega.
\end{equation}
\tag{1.7}
$$
To this end we impose the following condition on the numbers $a_j^\varepsilon$ and $b_j^\varepsilon$. (A) There is a subset $\mathbb{J}_\varepsilon\subseteq \mathbb{I}_\varepsilon$ and a fixed positive constant $c_1$ independent of $\varepsilon$ such that
$$
\begin{equation}
\Gamma_0 \subseteq \bigcup_{j\in J_\varepsilon} \{x\colon s\in (\varepsilon a_j^\varepsilon-c_1 \varepsilon,\varepsilon b_j^\varepsilon+c_1\varepsilon) \}
\end{equation}
\tag{1.8}
$$
and
$$
\begin{equation}
\varepsilon^{1/2}\eta^{-1}(\varepsilon) \to +0, \qquad \eta(\varepsilon):=\min_{j\in\mathbb{J}_\varepsilon} |b_j^\varepsilon-a_j^\varepsilon|.
\end{equation}
\tag{1.9}
$$
Every point $x\in\Gamma_0$ occurs in a finite number of intervals $(\varepsilon a_j^\varepsilon-c_1 \varepsilon,\varepsilon b_j^\varepsilon + c_1\varepsilon)$, and this number is bounded uniformly with respect to $\varepsilon$ and $\Pi_\varepsilon$. Solutions of this homogenized problem are also understood in a generalized sense. Namely, we introduce a form in $L_2(\Omega)$ with domain $\mathfrak{D}(\mathfrak{h}_0):=\mathring{W}_2^1(\Omega,\partial\Omega)$:
$$
\begin{equation*}
\mathfrak{h}_0(u,v):=\sum_{i,j=1}^{2} \biggl(A_{ij} \, \frac{\partial u}{\partial x_j},\frac{\partial v}{\partial x_i}\biggr)_{L_2(\Omega)} + \sum_{j=1}^{2} \biggl(A_j \, \frac{\partial u}{\partial x_j},v\biggr)_{L_2(\Omega)} + \bigl(A_0(\,\cdot\,,u),v\bigr)_{L_2(\Omega)}.
\end{equation*}
\notag
$$
By a solution of problem (1.7) we mean a function $u\in\mathfrak{D}(\mathfrak{h}_0)$ satisfying the integral identity
$$
\begin{equation}
\mathfrak{h}_0(u_0,v)- \lambda (u_0,v)_{L_2(\Omega)}=(f,v)_{L_2(\Omega)}
\end{equation}
\tag{1.10}
$$
for all $v\in \mathfrak{D}(\mathfrak{h}_0)$. Let us formulate the main result of our paper. Theorem 1.1. There exists $\lambda_0$ independent of $\varepsilon$ such that for $\operatorname{Re}\lambda\leqslant \lambda_0$ problems (1.4) and (1.7) are uniquely solvable for all $f\in L_2(\Omega)$, and the bound holds for a constant $C$ independent of $\varepsilon$ and $f$ and depending on $\lambda$. In addition, if then the following bound holds: where the constant $C$ is independent of $\varepsilon$ and $f$ and depends on $\lambda$. We briefly discuss the main features of the problem in question and our main results. We consider the perturbed problem (1.4), (1.5) for a semilinear second-order elliptic equation with variable coefficients — see the differential expression (1.2). The nonlinearity in the differential expression in question is contained in the free term, and it is sufficiently weak, namely, the Lipschitz condition in (1.3) must be satisfied, together with the vanishing condition for the function zero. The perturbation is specified as a combination of two classical perturbations in homogenization theory, namely, the frequent alternation of the boundary conditions and an oscillating boundary. The oscillating boundary is described by an equation $\tau=\varepsilon b_\varepsilon(s)$, where $b_\varepsilon(s)$ is an arbitrary function in $C^1$; no periodicity is assumed for it, which is a significant difference from many known works on oscillating boundaries. The other perturbation, frequent alternation, is defined on the oscillating boundary, and we do not assume any periodicity for it either. The frequent alternation is defined by dividing the perturbed boundary $\Gamma_\varepsilon$ into a large number of small pieces on which the Dirichlet and Neumann conditions are alternately specified. Pieces with the Dirichlet condition on the perturbed boundary correspond to parts of the unperturbed boundary $\Gamma_0$ described in terms of the natural parameter; see (1.1). The choice of these parts is arbitrary, but condition (A) must be satisfied. Its idea is as follows. The pieces with the Dirichlet condition must be placed sufficiently close to one another to ensure that the embedding (1.8) holds. In this case, the lengths of the parts corresponding to values of the index $j\in \mathbb{J}_\varepsilon$ must be at least $\varepsilon\eta$, where the function $\eta$ satisfies (1.9). For example, $\eta(\varepsilon)$ in condition (1.9) can be a constant function, the function $\eta(\varepsilon)=\varepsilon^\alpha$, where $\alpha < 1/2$, the function $\eta(\varepsilon)=\varepsilon^{1/2}|{\ln\varepsilon}|$ and so on. These examples show that the lengths of parts of $\Gamma_0$ can be significantly less than $\varepsilon$. On the other hand, depending on the function $b_\varepsilon(s)$, which defines the oscillation of the curve $\Gamma_\varepsilon$ along the normal to $\Gamma_0$, parts of the curve $\Gamma_\varepsilon$ with the Dirichlet condition can have lengths of another order. If $b_\varepsilon(s)$ is constant or is independent of $\varepsilon$ on some parts of $\Gamma_0$, then the pieces of $\Gamma_\varepsilon$ with the Dirichlet condition corresponding to this part of $\Gamma_0$ have lengths of the same order as the corresponding parts of $\Gamma_0$. If $b_\varepsilon(s)$ oscillates strongly and the oscillations grow as $\varepsilon \to 0$, then the curvature of the corresponding part of the curve can increase unboundedly. Then the lengths of pieces with the Dirichlet condition can also grow and exceed $\varepsilon\eta$ significantly. A similar observation applies to parts of $\Gamma_\varepsilon$ with the Neumann condition. If condition (1.9) is satisfied, then the homogenization of the perturbed problem (1.4), (1.5) leads to the Dirichlet problem (1.7). Our main result is operator estimates in $W_2^1$ and $L_2$, that is, bounds for the $W_2^1$- and $L_2$-norms of the difference of solutions of the perturbed and homogenized problems that are uniform with respect to the $L_2$-norm of the right-hand side of $f$. For the $L_2$-bound the additional condition (1.12) must hold, which is typical and arose previously in the proof of $L_2$-estimates for other perturbations in boundary homogenization theory (see [16]–[18])). The rate of convergence in the $L_2$-bound is twice as high as in the $W_2^1$-bound, which agrees with the fact that in the first case the norm is stronger. A specific feature of the present paper is the significant difference between (1.9) and a similar condition in classical problems with frequent alternation, namely,
$$
\begin{equation*}
\varepsilon \ln\eta(\varepsilon) \to -0 \quad \text{as } \varepsilon\to+0.
\end{equation*}
\notag
$$
Condition (1.9) shows that in combining frequent alternation with a rapidly oscillating boundary a nontrivial effect arises, and the oscillating boundary begins to influence significantly the classification of homogenized problems in comparison to the analogous classical classification in the problems with frequent alternation. Although we do not prove that condition (1.9) is order sharp, there are good reasons to believe that this condition is essential and not due to the shortcomings of our techniques. This condition arises from Lemma 2.2, whose proof is based on the best possible bounds, which gives us ground to believe that (1.9) is either optimal or close to this. We assume that this optimality of the estimate can be shown by considering a suitable arrangement of the parts with the Dirichlet condition on the oscillating boundary. Such an arrangement is shown schematically in Figure 2. In this figure parts with the Dirichlet condition are shaded light gray, and parts with the Neumann condition are shaded dark gray. We expect that if condition (1.9) is violated in such a configuration and the parameter $\eta$ becomes much less than $\varepsilon^{1/2}$, then, in the limit, a homogenized problem with the Neumann condition on $\Gamma_0$ arises. We will devote a separate paper to this issue.  Figure 2.An example of an arrangement of pieces with the Dirichlet condition on an oscillating curve, for which, as expected, condition (1.9) is optimal. § 2. Auxiliary statementsHere we present two lemmas that are used below to prove the main theorems. We begin with the existence of a number $\lambda_0$ ensuring the solvability of the perturbed and homogenized problems. Lemma 2.1. There exists $\lambda_0$ independent of $\varepsilon$ such that for $\operatorname{Re}\lambda\leqslant \lambda_0$ problems (1.4), (1.7) are uniquely solvable for all $f\in L_2(\Omega)$ and the bounds hold for all $u,v\in\mathfrak{D}(\mathfrak{h}_\varepsilon)$. The solution of the homogenized problem belongs to the space $W_2^2(\Omega)$, and the following inequalities hold: where the constants $C_1$, $C_2$ and $C_3$ are independent of $f$, $\lambda$ and $\lambda_0$, and $C_3<-\lambda_0$. The proof of this lemma reproduces literally the proof of Lemma 2.1 in [17], and the type of boundary conditions is not important here. Lemma 2.2. The following bounds hold: Proof. The bounds (2.3) and (2.4) are proved in the same way as the bounds (2.8) and (2.7) in [17]. Consider the sets
$$
\begin{equation*}
S_j^\varepsilon :=\bigl\{ x\in\Omega \colon s\in (\varepsilon a_j^\varepsilon,\varepsilon b_j^\varepsilon),\, \tau \in (\varepsilon b_\varepsilon(s),2\varepsilon)\bigr\}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\Upsilon_j^\varepsilon :=\bigl\{x\colon s\in (\varepsilon a_j^\varepsilon-c_1 \varepsilon, \varepsilon b_j^\varepsilon+c_1\varepsilon),\, \tau \in (\varepsilon b_\varepsilon(s),2\varepsilon) \bigr\}.
\end{equation*}
\notag
$$
Let us choose an arbitrary function $u(x)$ in the space $\mathring{W}_2^1(\Omega_\varepsilon,\Gamma\cup\Gamma_D^\varepsilon)$. Then the following formula holds:
$$
\begin{equation}
u(x)=\int_{\varepsilon b_\varepsilon(s)}^{\tau} \frac{\partial u}{\partial\tau}(t,s)\,d\tau, \qquad x\in S_j^\varepsilon, \quad j=0,\dots,N_\varepsilon, \quad \tau \in (\varepsilon b_\varepsilon(s),\tau_0),
\end{equation}
\tag{2.6}
$$
where we set $u(x)=u(\tau,s)$ in the sense of the local variables $(\tau,s)$ in a neighbourhood of the boundary $\Gamma_0$. Using the Cauchy–Bunyakovskii–Schwarz inequality we obtain
$$
\begin{equation}
|u(x)|^2 \leqslant (\tau-\varepsilon b_\varepsilon(s)) \int_{\varepsilon b_\varepsilon(s)}^{\tau} \biggl|{\frac{\partial u}{\partial\tau}}\biggr|^2 \,d\tau \leqslant \tau \int_{\varepsilon b_\varepsilon(s)}^{\tau} \biggl|{\frac{\partial u}{\partial\tau}}\biggr|^2 \,d\tau.
\end{equation}
\tag{2.7}
$$
We integrate the resulting inequality over the set $S_j^\varepsilon$:
$$
\begin{equation}
\int_{S_j^\varepsilon} |u|^2 \,d x \leqslant \int_{S_j^\varepsilon} \,d x \, \tau \int_{\varepsilon b_\varepsilon(s)}^{\tau} \biggl|{\frac{\partial u}{\partial\tau}}\biggr|^2 \,d\tau \leqslant C \varepsilon^2 \int_{\varepsilon a_j^\varepsilon}^{\varepsilon b_j^\varepsilon} \,d s \int_{\varepsilon b_\varepsilon(s)}^{\tau} \biggl|{\frac{\partial u}{\partial\tau}}\biggr|^2 \,d\tau.
\end{equation}
\tag{2.8}
$$
Hence we derive the inequality Here and throughout what follows $C$ denotes various inessential constants independent of $\varepsilon$, $\eta$, $u$ and $j \in\mathbb {I}_\varepsilon$.
We set
$$
\begin{equation*}
Q_j^\varepsilon :=\bigl\{ x\in\Omega \colon s\in (\varepsilon a_j^\varepsilon-c_1 \varepsilon,\varepsilon b_j^\varepsilon+c_1\varepsilon), \tau \in (\varepsilon,2\varepsilon)\bigr\}, \qquad j \in\mathbb {I}_\varepsilon.
\end{equation*}
\notag
$$
Let us introduce an infinitely differentiable cutoff function $\chi_\varepsilon(x)$ that is equal to zero for
$$
\begin{equation*}
\biggl|s-\varepsilon \frac{a_j^\varepsilon+b_j^\varepsilon}{2}\biggr| < \varepsilon \frac{b_j^\varepsilon-a_j^\varepsilon}{3},
\end{equation*}
\notag
$$
to one for
$$
\begin{equation*}
\biggl|s-\varepsilon \frac{a_j^\varepsilon+b_j^\varepsilon}{2}\biggr| > 2 \varepsilon \frac{b_j^\varepsilon-a_j^\varepsilon}{3}
\end{equation*}
\notag
$$
and satisfies the relations
$$
\begin{equation}
\operatorname{supp} \nabla \chi_\varepsilon \subset Q_j^\varepsilon \cap S_j^\varepsilon\quad\text{and} \quad |\nabla \chi_\varepsilon(x)| \leqslant C\varepsilon^{-1}\eta^{-1}, \qquad x\in Q_j^\varepsilon \cap S_j^\varepsilon.
\end{equation}
\tag{2.10}
$$
Then the following equality holds for $x\in Q_j^\varepsilon \setminus S_j^\varepsilon$:
$$
\begin{equation}
u(x)=u(x)\chi_\varepsilon(x)= \int_{\varepsilon (a_j^\varepsilon+b_j^\varepsilon)/2}^{s} \frac{\partial}{\partial s} u(\tau,s)\chi_\varepsilon(x)\,d s.
\end{equation}
\tag{2.11}
$$
We use the Cauchy–Bunyakovskii–Shwarz inequality:
$$
\begin{equation*}
|u(x)|^2 \leqslant C\varepsilon \int_{\varepsilon (a_j^\varepsilon +b_j^\varepsilon)/2}^{s} (|\nabla u|^2 \chi_\varepsilon^2 + |u|^2|\nabla \chi_\varepsilon|^2)\,d s.
\end{equation*}
\notag
$$
We integrate this inequality over $Q_j^\varepsilon$ and take the properties of $\chi_\varepsilon(x)$ and relations (2.10) and (2.9) into account. Then we obtain
$$
\begin{equation}
\begin{aligned} \, \notag \int_{Q_j^\varepsilon}|u(x)|^2\,dx & \leqslant C\varepsilon^2 \int_{\varepsilon}^{2\varepsilon}\,d\tau \int_{\varepsilon a_j^\varepsilon -c_1 \varepsilon}^{\varepsilon b_j^\varepsilon+c_1\varepsilon}(|\nabla u|^2 + \varepsilon^{-2}\eta^{-2}|u|^2)\,d s \\ \notag &\leqslant C\varepsilon^2\|\nabla u\|_{L_2(Q_j^\varepsilon)}^2 +C\eta^{-2}\|\nabla u\|_{L_2(Q_j^\varepsilon \cap S_j^\varepsilon)}^2 \\ &\leqslant C\varepsilon^2\|\nabla u\|_{L_2(Q_j^\varepsilon)}^2 +C\varepsilon^2\eta^{-2}\|\nabla u\|_{L_2(S_j^\varepsilon)}^2. \end{aligned}
\end{equation}
\tag{2.12}
$$
Set $T_j^\varepsilon :=\Upsilon_j^\varepsilon \setminus S_j^\varepsilon$. We consider an infinitely differentiable cutoff function $\chi^\varepsilon(x)$ equal to one for $\tau \in (\varepsilon b_\varepsilon(s),\varepsilon)$, to zero for $\tau \geqslant 2\varepsilon$ and satisfying the conditions
$$
\begin{equation}
\operatorname{supp} \nabla \chi^\varepsilon \in T_j^\varepsilon \cap Q_j^\varepsilon\quad\text{and} \quad |\nabla \chi^\varepsilon| \leqslant C\varepsilon^{-1}, \qquad x\in T_j^\varepsilon \cap Q_j^\varepsilon.
\end{equation}
\tag{2.13}
$$
For $x\in T_j^\varepsilon$ the following equality holds:
$$
\begin{equation}
u(x)=u(x)\chi^\varepsilon(x)= \int_{\varepsilon}^{\tau} \frac{\partial}{\partial \tau} u(x)\chi^\varepsilon(x)\,d\tau.
\end{equation}
\tag{2.14}
$$
Using this equality, similarly to (2.11) and (2.12), applying (2.13) and (2.14) we obtain
$$
\begin{equation*}
\|u\|_{L_2(T_j^\varepsilon)}^2 \leqslant C \varepsilon^2 \|\nabla u\|_{L_2(T_j^\varepsilon \cup Q_j^\varepsilon)}^2 + C \|u\|_{L_2(T_j^\varepsilon\cap Q_j^\varepsilon)}^2.
\end{equation*}
\notag
$$
Hence from (2.12) we arrive at the following bound:
$$
\begin{equation*}
\|u\|_{L_2(T_j^\varepsilon)}^2 \leqslant C \varepsilon^2 \|\nabla u\|_{L_2(T_j^\varepsilon \cup Q_j^\varepsilon)}^2 + C \varepsilon^{2} \eta^{-2} \|\nabla u\|_{L_2(S_j^\varepsilon)}^2,
\end{equation*}
\notag
$$
from which we derive that
$$
\begin{equation*}
\|u\|_{L_2(\Upsilon_j^\varepsilon)}^2 \leqslant C \varepsilon^{2} \eta^{-2} \|\nabla u\|_{L_2(\Upsilon_j^\varepsilon)}^2.
\end{equation*}
\notag
$$
By the last inequality and condition (A) the following chain of relations holds:
$$
\begin{equation*}
\begin{aligned} \, \|u\|_{L_2(\Pi_{2\varepsilon}\cap \Omega_\varepsilon)}^2 &= \int_{\Pi_{2\varepsilon}\cap \Omega_\varepsilon} |u(x)|^2\,d x \leqslant \sum _{j} \int_{\Upsilon_j^\varepsilon} |u(x)|^2\,d x \\ &\leqslant C \varepsilon^{2} \eta^{-2} \sum _{j} \|\nabla u\|_{L_2(\Upsilon_j^\varepsilon)}^2 \leqslant C \varepsilon^{2} \eta^{-2} \|\nabla u\|_{L_2(\Omega_\varepsilon)}^2; \end{aligned}
\end{equation*}
\notag
$$
hence (2.5) follows.
This completes the proof of Lemma 2.2. § 3. Operator estimatesIn this section we present the proof of Theorem 1.1. In what follows we assume that $f$ is an arbitrary function in $L_2(\Omega)$, and the parameter $\lambda$ satisfies the condition $\operatorname{Re}\lambda\leqslant \lambda_0$ and is fixed. We denote the corresponding unique solutions of problems (1.4) and (1.7) by $u_\varepsilon$ and $u_0$, respectively. We present the proof of the operator estimates (1.11) and (1.13) in two separate sections. 3.1. The $W_2^1$-estimateIn the proof of the $W_2^1$-estimate we follow the approach of [15], [16] and [18]. Consider the function
$$
\begin{equation*}
\chi_\varepsilon(x):= \begin{cases} 1-\chi\biggl(\dfrac{\tau}{3\varepsilon}\biggr) &\text{in }\Pi_{2\varepsilon}, \\ 1 &\text{in }\Omega\setminus\Pi_{2\varepsilon}, \end{cases}
\end{equation*}
\notag
$$
where $\chi=\chi(t)$ is an infinitely differentiable cutoff function equal to one for $t < 1/3$ and to zero for $t > 2/3$. The function $\chi_\varepsilon(x)$ obviously vanishes on $\Pi_\varepsilon$, is infinitely differentiable on $\overline \Omega$ and satisfies the relations
$$
\begin{equation}
\operatorname{supp}\nabla\chi_\varepsilon\subseteq \overline{\Pi_{2\varepsilon}\setminus\Pi_\varepsilon}, \qquad |\nabla\chi_\varepsilon(x)|\leqslant C\varepsilon^{-1}, \qquad x\in\Pi_{2\varepsilon}\setminus\Pi_\varepsilon,
\end{equation}
\tag{3.1}
$$
where $C$ is some constant independent of $x$ and $\varepsilon$. Taking into account the indicated properties, we see that for an arbitrary $v\in \mathring{W}_2^1(\Omega_\varepsilon,\Gamma)$ the function $\chi_\varepsilon v$ belongs to $\mathring{W}_2^1(\Omega_\varepsilon,\Gamma_\varepsilon)$. Then the function $v_\varepsilon:=u_\varepsilon-\chi_\varepsilon u_0$ is an element of $\mathring{W}_2^1(\Omega_\varepsilon,\Gamma\cup\Gamma_D^\varepsilon)$, and the function $\chi_\varepsilon v_\varepsilon$, extended by zero to $\Omega\setminus\Omega_\varepsilon$, is an element of $\mathring{W}_2^1(\Omega,\partial\Omega)$. We write out the integral identity (1.6) and identity (1.10) for problem (1.7), using $v_\varepsilon$ and $\chi_\varepsilon v_\varepsilon$ as test functions, respectively:
$$
\begin{equation}
\mathfrak{h}_\varepsilon(u_\varepsilon, v_\varepsilon)- \lambda (u_\varepsilon, v_\varepsilon)_{L_2(\Omega_\varepsilon)} =(f, v_\varepsilon)_{L_2(\Omega_\varepsilon)}
\end{equation}
\tag{3.2}
$$
and
$$
\begin{equation}
\mathfrak{h}_0(u_0,\chi_\varepsilon v_\varepsilon)- \lambda (u_0,\chi_\varepsilon v_\varepsilon)_{L_2(\Omega)} =(f, \chi_\varepsilon v_\varepsilon)_{L_2(\Omega)}.
\end{equation}
\tag{3.3}
$$
Taking the definition and properties (3.1) of $\chi_\varepsilon$ into account, we can rewrite the first term on the left-hand side of the second identity in the form
$$
\begin{equation}
\mathfrak{h}_0(u_0,\chi_\varepsilon v_\varepsilon) =\mathfrak{h}_\varepsilon(\chi_\varepsilon u_0, v_\varepsilon) + \mathfrak{g}_\varepsilon(u_0,v_\varepsilon),
\end{equation}
\tag{3.4}
$$
where
$$
\begin{equation*}
\begin{aligned} \, \mathfrak{g}_\varepsilon(u,v) &:=\sum_{i,j=1}^{n} \biggl(A_{ij}\,\frac{\partial u}{\partial x_j}, \frac{\partial \chi_\varepsilon}{\partial x_i}v\biggr)_{L_2(\Pi_{2\varepsilon} \setminus\Pi_\varepsilon)} - \sum_{i,j=1}^{n} \biggl(A_{ij}\,\frac{\partial \chi_\varepsilon}{\partial x_j}u, \frac{\partial v}{\partial x_i}\biggr)_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon)} \\ &\quad -\sum_{j=1}^{2} \biggl(A_j\,\frac{\partial\chi_\varepsilon}{\partial x_j}u,v\biggr)_{L_2(\Pi_{2 \varepsilon}\setminus\Pi_\varepsilon)} -\bigl(A_0(\,\cdot\,,\chi_\varepsilon u) -\chi_\varepsilon A_0(\,\cdot\,,u), v\bigr)_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon)}. \end{aligned}
\end{equation*}
\notag
$$
Substituting (3.4) into (3.3) and then subtracting the resulting identity from (3.2), we derive that
$$
\begin{equation}
\mathfrak{h}_\varepsilon(u_\varepsilon,v_\varepsilon)- \mathfrak{h}_\varepsilon(u_0\chi_\varepsilon,v_\varepsilon) -\lambda \|v_\varepsilon\|_{L_2(\Omega)}^2 = (f,(1-\chi_\varepsilon)v_\varepsilon)_{L_2(\Omega)}+\mathfrak{g}_\varepsilon(u_0,v_\varepsilon).
\end{equation}
\tag{3.5}
$$
For the left-hand side of the above identity inequality (2.1) holds:
$$
\begin{equation}
\operatorname{Re} \bigl(\mathfrak{h}_\varepsilon(u_\varepsilon,v_\varepsilon)- \mathfrak{h}_\varepsilon(u_0\chi_\varepsilon,v_\varepsilon) -\lambda \|v_\varepsilon\|_{L_2(\Omega)}^2 \bigr) \geqslant C \|v_\varepsilon\|_{W_2^1(\Omega_\varepsilon)}^2.
\end{equation}
\tag{3.6}
$$
Here and throughout the rest of the section, $C$ denotes various constants that do not depend on $x$, $\varepsilon$, $u_0$, $v_\varepsilon$ and $f$. The next step is to estimate the right-hand side of (3.5). Using inequality (2.5) for $u=v_\varepsilon$ and the definition of $\chi_\varepsilon$, we can estimate the first term on the right-hand side:
$$
\begin{equation}
\begin{aligned} \, \notag \bigl|(f,(1-\chi_\varepsilon)v_\varepsilon)_{L_2(\Omega)}\bigr| &=\bigl|(f,(1-\chi_\varepsilon)v_\varepsilon)_{L_2(\Pi_{2\varepsilon})}\bigr| \leqslant \|f\|_{L_2(\Pi_{2\varepsilon})} \|v_\varepsilon\|_{L_2(\Pi_{2\varepsilon})} \\ &\leqslant C \varepsilon\eta^{-1} \|f\|_{L_2(\Omega)} \|\nabla v_\varepsilon\|_{L_2(\Omega_\varepsilon)}. \end{aligned}
\end{equation}
\tag{3.7}
$$
We apply (2.4) for $u=u_0$, (2.3) for $u=\nabla u_0$, (2.5) for $u=v_\varepsilon$ and the inequality for $\nabla \chi_\varepsilon$ from (3.1) to the first three terms in $\mathfrak{g}_\varepsilon(u_0,v_\varepsilon)$:
$$
\begin{equation}
\begin{aligned} \, \biggl|\sum_{i,j=1}^{n} \biggl(A_{ij}\,\frac{\partial u_0}{\partial x_j}, \frac{\partial \chi_\varepsilon}{\partial x_i}v_\varepsilon\biggr)_{L_2(\Pi_{2\varepsilon} \setminus\Pi_\varepsilon)}\biggr| &\leqslant C \varepsilon^{-1}\|\nabla u_0\|_{L_2(\Pi_{2\varepsilon} \setminus\Pi_\varepsilon)} \|v_\varepsilon\|_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon)} \\ &\leqslant C \varepsilon^{1/2}\eta^{-1}\|u_0\|_{W_2^2(\Omega)} \|\nabla v_\varepsilon\|_{L_2(\Omega_\varepsilon)}, \\ \biggl|\sum_{i,j=1}^{n} \biggl(A_{ij}\,\frac{\partial \chi_\varepsilon}{\partial x_j}u_0, \frac{\partial v_\varepsilon}{\partial x_i}\biggr)_{L_2(\Pi_{2\varepsilon} \setminus\Pi_\varepsilon)}\biggr| &\leqslant C \varepsilon^{-1}\|u_0\|_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon)} \|\nabla v_\varepsilon\|_{L_2(\Pi_{2\varepsilon} \setminus\Pi_\varepsilon)} \\ &\leqslant C \varepsilon^{1/2}\|u_0\|_{W_2^2(\Omega)} \|\nabla v_\varepsilon\|_{L_2(\Omega_\varepsilon)}, \\ \biggl|\sum_{j=1}^{2} \biggl(A_j\,\frac{\partial\chi_\varepsilon}{\partial x_j}u_0, v_\varepsilon\biggr)_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon)}\biggr| &\leqslant C \varepsilon^{-1} \|u_0\|_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon)} \|v_\varepsilon\|_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon)} \\ &\leqslant C \varepsilon^{3/2}\eta^{-1} \|u_0\|_{W_2^2(\Omega)} \|\nabla v_\varepsilon\|_{L_2(\Omega_\varepsilon)}. \end{aligned}
\end{equation}
\tag{3.8}
$$
Using condition (1.3) and the definition of the function $\chi_\varepsilon$ we immediately derive the bound Using it, the definition of the function $\chi_\varepsilon$, inequality (2.4) for $u = u_0$, and (2.5) for $v=v_\varepsilon$ we can estimate the remaining term in the function $\mathfrak{g}_\varepsilon(u_0,v_\varepsilon)$:
$$
\begin{equation}
\begin{aligned} \, \notag \bigl|\bigl(\chi_\varepsilon A_0(\,\cdot\,,u_0)-A_0(\,\cdot\,,\chi_\varepsilon u_0), v_\varepsilon\bigr)_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon)}\bigr| &\leqslant 2c_1\|u_0\|_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon)} \|v_\varepsilon\|_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon)} \\ &\leqslant C\varepsilon^{5/2}\eta^{-1} \|u_0\|_{W_2^2(\Omega)} \|\nabla v_\varepsilon\|_{L_2(\Omega_\varepsilon)}. \end{aligned}
\end{equation}
\tag{3.10}
$$
Applying this bound and (3.8), (3.7) and (2.2) to the right-hand side of equality (3.5) we obtain
$$
\begin{equation*}
\begin{aligned} \, &\bigl|(f,(1-\chi_\varepsilon)v_\varepsilon)_{L_2(\Omega)} +\mathfrak{g}_\varepsilon(u_0,v_\varepsilon)\bigr| \\ &\qquad \leqslant C \varepsilon^{1/2} \eta^{-1} \bigl(\|f\|_{L_2(\Omega)} + \|u_0\|_{W_2^2(\Omega)}\bigr) \|\nabla v_\varepsilon\|_{L_2(\Omega_\varepsilon)} \\ &\qquad \leqslant C \varepsilon^{1/2} \eta^{-1} \|f\|_{L_2(\Omega)}\|v_\varepsilon\|_{W_2^1(\Omega_\varepsilon)}. \end{aligned}
\end{equation*}
\notag
$$
This inequality, in combination with (3.5) and (3.6), yields
$$
\begin{equation}
\|v_\varepsilon\|_{W_2^1(\Omega_\varepsilon)}\leqslant C \varepsilon^{1/2} \eta^{-1} \|f\|_{L_2(\Omega)}.
\end{equation}
\tag{3.11}
$$
The following equality holds:
$$
\begin{equation}
u_\varepsilon-u_0=v_\varepsilon-(1-\chi_\varepsilon)u_0.
\end{equation}
\tag{3.12}
$$
Inequality (2.4) for $u = u_0$, (2.3) for $u=\nabla u_0$, (2.2) and the bounds for $\nabla \chi_\varepsilon$ in (3.1) lead to the following estimates:
$$
\begin{equation}
\begin{aligned} \, \|(1-\chi_\varepsilon)u_0\|_{L_2(\Omega)} &=\|(1-\chi_\varepsilon)u_0\|_{L_2(\Pi_{2\varepsilon})} \leqslant C\varepsilon^{3/2}\|u_0\|_{W_2^2(\Omega)} \leqslant C\varepsilon^{3/2}\|f\|_{L_2(\Omega)}, \\ \|\nabla(1-\chi_\varepsilon)u_0\|_{L_2(\Omega)} &\leqslant \|(1-\chi_\varepsilon)\nabla u_0\|_{L_2(\Omega)}+ \|u_0 \nabla \chi_\varepsilon\|_{L_2(\Omega)} \leqslant C\varepsilon^{1/2}\|f\|_{L_2(\Omega)}. \nonumber \end{aligned}
\end{equation}
\tag{3.13}
$$
Hence (3.11) and (3.12) imply the bound (1.11). 3.2. The $L_2$-estimateIn proving the $L_2$-estimate we follow the approach using duality — see, for example, [32]–[36] — with a slight modification proposed in [18]. Using this approach, we are able to consider the case of weakly nonlinear equations. We extend the function $v_\varepsilon$ introduced above by zero to the domain $\Omega\setminus\Omega_\varepsilon$ and keep the notation for this extension. Consider the boundary value problem
$$
\begin{equation}
\begin{gathered} \, \widehat{\mathcal{L}} w-\overline{\lambda} w =v_\varepsilon \quad\text{in } \Omega, \qquad w=0 \quad\text{on } \partial\Omega, \\ \notag \widehat{\mathcal{L}}:= -\sum_{i,j=1}^{n} \frac{\partial}{\partial x_j} \overline{A_{ij}}\,\frac{\partial }{\partial x_i} - \sum_{j=1}^{n} \frac{\partial}{\partial x_j}\overline{A_j}. \end{gathered}
\end{equation}
\tag{3.14}
$$
This problem is similar in structure to (1.7), and therefore the statements in Lemma 2.1 are valid for it. We choose a number $\lambda_0$ so as to ensure the solvability of both (1.7) and (3.14). Bounds similar to (2.2) hold:
$$
\begin{equation}
\|w\|_{W_2^2(\Omega)}\leqslant C_1\|v_\varepsilon\|_{L_2(\Omega_\varepsilon)}\quad\text{and} \quad \|w\|_{L_2(\Omega)}\leqslant \frac{C_2}{|\operatorname{Re}\lambda+C_3|} \|v_\varepsilon\|_{L_2(\Omega_\varepsilon)},
\end{equation}
\tag{3.15}
$$
where the constants $C_1$, $C_2$ and $C_3$ are independent of $v_\varepsilon$ and $\lambda$, and $C_3<-\lambda_0$. We write down the integral identity corresponding to problem (3.14), taking $\chi_\varepsilon v_\varepsilon$ as a test function:
$$
\begin{equation*}
\begin{aligned} \, &\sum_{i,j=1}^{n} \biggl(\frac{\partial \chi_\varepsilon v_\varepsilon}{\partial x_j}, \overline{A_{ij}}\,\frac{\partial w}{\partial x_i}\biggr)_{L_2(\Omega)}- \sum_{j=1}^{n} \biggl(\chi_\varepsilon v_\varepsilon,\frac{\partial}{\partial x_j} \overline{A_j} w\biggr)_{L_2(\Omega)} \\ &\qquad\qquad - \lambda (\chi_\varepsilon v_\varepsilon, w)_{L_2(\Omega)} =(\chi_\varepsilon v_\varepsilon,v_\varepsilon)_{L_2(\Omega)}. \end{aligned}
\end{equation*}
\notag
$$
Integrating by parts once and taking into account that the function $v_\varepsilon$ is equal to zero in $\Omega\setminus\Omega_\varepsilon$, we obtain
$$
\begin{equation*}
\begin{aligned} \, (\chi_\varepsilon v_\varepsilon,v_\varepsilon)_{L_2(\Omega)} &=\sum_{i,j=1}^{n} \biggl(A_{ij}\,\frac{\partial \chi_\varepsilon v_\varepsilon}{\partial x_j},\frac{\partial w}{\partial x_i}\biggr)_{L_2(\Omega_\varepsilon)} +\sum_{j=1}^{n} \biggl(A_j \,\frac{\partial \chi_\varepsilon v_\varepsilon}{\partial x_j}, w\biggr)_{L_2(\Omega_\varepsilon)} \\ &\qquad- \lambda (\chi_\varepsilon v_\varepsilon,w)_{L_2(\Omega_\varepsilon)} \\ &=\sum_{i,j=1}^{n} \biggl(A_{ij}\,\frac{\partial \chi_\varepsilon}{\partial x_j}v_\varepsilon, \frac{\partial w}{\partial x_i}\biggr)_{L_2(\Omega_\varepsilon)} -\sum_{i,j=1}^{n} \biggl(A_{ij}\,\frac{\partial v_\varepsilon}{\partial x_j}, \frac{\partial \chi_\varepsilon}{\partial x_i}w\biggr)_{L_2(\Omega_\varepsilon)} \\ &\qquad +\sum_{j=1}^{n} \biggl(A_j \,\frac{\partial \chi_\varepsilon}{\partial x_j}v_\varepsilon, w\biggr)_{L_2(\Omega_\varepsilon)} + \sum_{i,j=1}^{n} \biggl(A_{ij}\,\frac{\partial v_\varepsilon}{\partial x_j}, \frac{\partial \chi_\varepsilon w}{\partial x_i}\biggr)_{L_2(\Omega_\varepsilon)} \\ &\qquad +\sum_{j=1}^{n} \biggl(A_j\, \frac{\partial v_\varepsilon}{\partial x_j}, \chi_\varepsilon w\biggr)_{L_2(\Omega_\varepsilon)} - \lambda ( v_\varepsilon,\chi_\varepsilon w)_{L_2(\Omega_\varepsilon)}. \end{aligned}
\end{equation*}
\notag
$$
This equality, the definition of the form $\mathfrak{h}_\varepsilon$, and the properties of the cutoff function $\chi_\varepsilon$ give:
$$
\begin{equation}
\begin{aligned} \, \notag (\chi_\varepsilon v_\varepsilon,v_\varepsilon)_{L_2(\Omega)} &=\mathfrak{h}_\varepsilon(u_\varepsilon,\chi_\varepsilon w) - \mathfrak{h}_\varepsilon(\chi_\varepsilon u_0,\chi_\varepsilon w) -\lambda ( v_\varepsilon,\chi_\varepsilon w)_{L_2(\Omega_\varepsilon)} \\ \notag &\qquad+ \sum_{i,j=1}^{n} \biggl(A_{ij}\,\frac{\partial \chi_\varepsilon}{\partial x_j}v_\varepsilon, \frac{\partial w}{\partial x_i}\biggr)_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon)} \\ \notag &\qquad - \sum_{i,j=1}^{n} \biggl(A_{ij}\,\frac{\partial v_\varepsilon}{\partial x_j}, \frac{\partial \chi_\varepsilon}{\partial x_i} w\biggr)_{L_2(\Pi_{2\varepsilon} \setminus\Pi_\varepsilon)} \\ \notag &\qquad +\sum_{j=1}^{n} \biggl(A_j\, \frac{\partial \chi_\varepsilon}{\partial x_j}v_\varepsilon, w\biggr)_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon)} \\ &\qquad - \bigl(A_0(\,\cdot\,, u_\varepsilon)-A_0(\,\cdot\,, \chi_\varepsilon u_0), \chi_\varepsilon w\bigr)_{L_2(\Omega_\varepsilon)}. \end{aligned}
\end{equation}
\tag{3.16}
$$
Now we write integral identities as in the derivation of (3.5), but use now $\chi_\varepsilon w$ and $\chi_\varepsilon^2 w$, rather than $v_\varepsilon$ and $\chi_\varepsilon v_\varepsilon$, as test functions. Then we obtain
$$
\begin{equation*}
\begin{aligned} \, &\mathfrak{h}_\varepsilon(u_\varepsilon,\chi_\varepsilon w) -\mathfrak{h}_\varepsilon(\chi_\varepsilon u_0,\chi_\varepsilon w) -\lambda(v_\varepsilon,\chi_\varepsilon w)_{L_2(\Omega_\varepsilon;\mathbb{C}^m)} \\ &\qquad=\bigl(f,\chi_\varepsilon(1-\chi_\varepsilon)w\bigr)_{L_2(\Pi_{2\varepsilon};\mathbb{C}^m)} + \mathfrak{g}_\varepsilon(u_0,\chi_\varepsilon w). \end{aligned}
\end{equation*}
\notag
$$
From this equality and (3.16) we derive the equality
$$
\begin{equation}
\begin{aligned} \, \notag &\|v_\varepsilon\|_{L_2(\Omega)}^2+ \bigl(A_0(\,\cdot\,,u_\varepsilon)- A_0(\,\cdot\,,\chi_\varepsilon u_0), \chi_\varepsilon w\bigr)_{L_2(\Omega_\varepsilon)} \\ \notag &\qquad= \bigl((1-\chi_\varepsilon) v_\varepsilon,v_\varepsilon\bigr)_{L_2(\Omega_\varepsilon \cap\Pi_{2\varepsilon})} +\bigl(f,\chi_\varepsilon(1-\chi_\varepsilon)w\bigr)_{L_2(\Pi_{2\varepsilon})} \\ &\qquad\qquad + \mathfrak{r}_\varepsilon+\mathfrak{g}_\varepsilon(u_0,\chi_\varepsilon w), \end{aligned}
\end{equation}
\tag{3.17}
$$
where we use the notation
$$
\begin{equation*}
\begin{aligned} \, \mathfrak{r}_\varepsilon &:= \sum_{i,j=1}^{n} \biggl(A_{ij}\,\frac{\partial \chi_\varepsilon}{\partial x_j}v_\varepsilon, \frac{\partial w}{\partial x_i}\biggr)_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon)} - \sum_{i,j=1}^{n} \biggl(A_{ij}\,\frac{\partial v_\varepsilon}{\partial x_j}, \frac{\partial \chi_\varepsilon}{\partial x_i} w\biggr)_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon)} \\ &\qquad +\sum_{j=1}^{n} \biggl(A_j \,\frac{\partial \chi_\varepsilon}{\partial x_j}v_\varepsilon, w\biggr)_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon)}. \end{aligned}
\end{equation*}
\notag
$$
We go over to the right-hand side of the resulting equality. Using (2.4) for $u = w$ and the first inequality in (3.15) we immediately obtain
$$
\begin{equation}
\begin{aligned} \, \notag \bigl|\bigl(f,\chi_\varepsilon(1-\chi_\varepsilon)w\bigr)_{L_2(\Pi_{2\varepsilon})}\bigr| &\leqslant C\varepsilon^{3/2} \|f\|_{L_2(\Pi_{2\varepsilon})} \|w\|_{W_2^2(\Omega)} \\ &\leqslant C\varepsilon^{3/2} \|f\|_{L_2(\Omega)} \|v_\varepsilon\|_{L_2(\Omega_\varepsilon)}. \end{aligned}
\end{equation}
\tag{3.18}
$$
It follows directly from (2.5) and (3.11) that
$$
\begin{equation}
\begin{aligned} \, \notag \bigl|\bigl((\chi_\varepsilon-1) v_\varepsilon,v_\varepsilon)_{L_2(\Omega_\varepsilon \cap\Pi_{2\varepsilon}\bigr)}\bigr| &\leqslant \|v_\varepsilon\|_{L_2(\Omega_\varepsilon\cap\Pi_{2\varepsilon})} \|v_\varepsilon\|_{L_2(\Omega_\varepsilon)} \\ \notag &\leqslant C\varepsilon \eta^{-1} \|\nabla v_\varepsilon\|_{L_2(\Omega_\varepsilon)}\|v_\varepsilon\|_{L_2(\Omega_\varepsilon)} \\ &\leqslant C\varepsilon^{3/2} \eta^{-2}\|f\|_{L_2(\Omega)} \|v_\varepsilon\|_{L_2(\Omega_\varepsilon)}. \end{aligned}
\end{equation}
\tag{3.19}
$$
By (2.3) for $u=v_\varepsilon$, (2.4) for $u=w$, (2.5) for $u=v_\varepsilon$, (3.1), (3.11) and (3.15) the following bound holds for $\mathfrak{r}_\varepsilon$:
$$
\begin{equation}
|\mathfrak{r}_\varepsilon|\leqslant C\varepsilon^{1/2} \eta^{-1} \|v_\varepsilon\|_{W_2^1(\Omega_\varepsilon)}\|w\|_{W_2^2(\Omega)} \leqslant C \varepsilon\eta^{-2} \|f\|_{L_2(\Omega)} \|v_\varepsilon\|_{L_2(\Omega_\varepsilon)}.
\end{equation}
\tag{3.20}
$$
The first three terms in the expression $\mathfrak{g}_\varepsilon(u_0,\chi_\varepsilon w)$ are estimated similarly to (3.8), using (2.4) for $u = w$ instead of (2.5) for $u=v_\varepsilon$ and writing out the derivative ${\partial \chi_\varepsilon w}/{\partial x_i}$. The fourth term can be estimated using (3.10), by replacing $v_\varepsilon$ by $\chi_\varepsilon w$. As a result, we obtain
$$
\begin{equation*}
|\mathfrak{g}_\varepsilon(u_0,\chi_\varepsilon w)|\leqslant C\varepsilon \|u_0\|_{W_2^2(\Omega)}\|w\|_{W_2^2(\Omega)}\leqslant C \varepsilon \|f\|_{L_2(\Omega)} \|v_\varepsilon\|_{L_2(\Omega_\varepsilon)}.
\end{equation*}
\notag
$$
Hence from (3.18)–(3.20), (3.15) and (2.2) we obtain the final estimate for the right-hand side of (3.17),
$$
\begin{equation}
\begin{aligned} \, \notag &\bigl|\bigl((1-\chi_\varepsilon) v_\varepsilon,v_\varepsilon\bigr)_{L_2(\Omega)} +\bigl(f,\chi_\varepsilon(1-\chi_\varepsilon)w\bigr)_{L_2(\Pi_{2\varepsilon})} + \mathfrak{r}_\varepsilon+\mathfrak{g}_\varepsilon(u_0,\chi_\varepsilon w)\bigr| \\ &\qquad \leqslant C\varepsilon \eta^{-2} \|f\|_{L_2(\Omega)} \|v_\varepsilon\|_{L_2(\Omega_\varepsilon)}. \end{aligned}
\end{equation}
\tag{3.21}
$$
Using the Lipschitz inequality in (1.2) and the second inequality in (3.15), we can estimate the second term on the left-hand side of (3.17):
$$
\begin{equation*}
\begin{aligned} \, \bigl|\bigl( A_0(\,\cdot\,,u_\varepsilon)-A_0(\,\cdot\,,\chi_\varepsilon u_0), \chi_\varepsilon w\bigr)_{L_2(\Omega_\varepsilon)}\bigr| &\leqslant c_1\|v_\varepsilon\|_{L_2(\Omega_\varepsilon)} \|w\|_{L_2(\Omega)} \\ &\leqslant \frac{C_2}{|\operatorname{Re}\lambda+C_3|} \|v_\varepsilon\|_{L_2(\Omega_\varepsilon)}^2. \end{aligned}
\end{equation*}
\notag
$$
Hence, for negative $\lambda_0$ with sufficiently large absolute value the left-hand side of (3.17) satisfies
$$
\begin{equation*}
\bigl|\|v_\varepsilon\|_{L_2(\Omega_\varepsilon}^2 +\bigl( A_0(\,\cdot\,,u_\varepsilon)-A_0(\,\cdot\,,\chi_\varepsilon u_0), \chi_\varepsilon w\bigr)_{L_2(\Omega_\varepsilon)}\bigr| \geqslant \frac{1}{2}\|v_\varepsilon\|_{L_2(\Omega_\varepsilon)}^2.
\end{equation*}
\notag
$$
By the last inequality, (3.21), and (3.17) we have
$$
\begin{equation*}
\|v_\varepsilon\|_{L_2(\Omega_\varepsilon)}\leqslant C \varepsilon \eta^{-2} \|f\|_{L_2(\Omega)}.
\end{equation*}
\notag
$$
Now using (3.12) and the first inequality in (3.13) we obtain the final bound (1.12). This completes the proof of Theorem 1.1.
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