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This article is cited in 1 scientific paper (total in 1 paper) Operator estimates for elliptic equations in multidimensional domains with strongly curved boundaries 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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    IntroductionThe issues of convergence and asymptotic behaviour of solutions of boundary value problems with rapidly oscillating boundaries were studied in a large number of works; see, for example, [1], Ch. V, § 7, [2], Ch. III, § 4, and [3]–[15], as well as the references there. The classical results in homogenization theory give a classification of homogenized problems depending on the geometry of oscillations and boundary conditions on boundaries of this type and describe the convergence of solutions of perturbed problems to solutions of homogenized ones. Convergence itself was proved, as a rule, for given right-hand sides of the equations and boundary conditions; it was established in the weak or strong topology of the space $L_2$ or $W_2^1$. In the language of relevant resolvents, for linear equations these results mean that the weak or strong resolvent convergence takes place. A much stronger result is uniform resolvent convergence, when the convergence of resolvents is established in the operator norm. Estimates of this type for problems in oscillating domains were obtained by several authors in some particular cases. The first estimate was considered in [2], Ch. III, § 4, for a scalar operator in a two-dimensional domain with Robin boundary condition on the oscillating boundary defined as the graph of a periodic function such that the period and amplitude of oscillations coincide and are a small parameter. The graph was understood as the dependence of the shift along the normal to the unperturbed boundary on the natural parameter. The convergence of resolvents acting from $L_2$ to $W_2^1$ was proved in the operator norm with suitable estimates for the rate of convergence; estimates of this type are usually called operator $L_2$- and $W_2^1$-estimates. In [16] and [17] the linear system of Stokes’ equations and Poisson’s equation were considered in a multidimensional domain with Dirichlet boundary condition on the oscillating boundary, which was assumed to be compact while oscillations were locally periodic. Operator $L_2$- and $W_2^1$-estimates were deduced. The paper [18] considered a general self-adjoint elliptic operator of the second order in a plane strip with rapidly periodically oscillating boundary such that the period and amplitude of oscillations were described by two independent small parameters. Boundary conditions of some classical type were prescribed on the oscillating boundary; possible homogenized problems were described depending on the geometry of oscillations, and operator $W_2^1$-estimates were derived. In [19] these studies were continued for a general plane domain with oscillating boundary defined as the graph of an arbitrary bounded function depending in an arbitrary way on a small parameter equal to the amplitude of oscillations. On such a boundary Dirichlet or Neumann boundary conditions were prescribed, which were preserved after homogenization. Operator $L_2$- and $W_2^1$-estimates were proved. The paper [20] considered a second-order elliptic operator of the general form in a two-dimensional domain with oscillating boundary defined again as the graph of an arbitrary bounded function. In that case the boundary was partitioned into two parts; Dirichlet boundary conditions were prescribed on one part and Neumann conditions on the other. After homogenization mixed boundary conditions are preserved on the limit boundary, which extends the domain of the homogenized operator beyond the space $W_2^2$. The main result in [20] is operator $L_2$- and $W_2^1$-estimates, to which the above extension of the domain of the homogenized operator contributes. In this paper we consider a boundary value problem for an elliptic semilinear system of equations of general form. The nonlinearity is localized in the free term and depends only on the unknown function. The growth of the nonlinearity with respect to the unknown function is at most of the first power. The equation is considered in an arbitrary domain that can be bounded or unbounded alike. One connected component of the boundary is arbitrarily curved. The main assumption is that the curved boundary lies in a thin layer along the unperturbed boundary. Dirichlet or Neumann boundary conditions are prescribed on the perturbed boundary. In the case of Dirichlet conditions no additional restrictions are imposed on the geometry of the curved boundary, which makes it possible to consider a very wide class of perturbations, including regular perturbations, classical oscillating boundaries, thin long spikes, fine perforation and so on. In Figures 1–3 we intentionally show the perturbed boundary on a large scale, to make clear the possible geometry of perturbations. In the case of Neumann conditions additional conditions are imposed on the geometry of the curved boundary, which are natural in the following sense: their significant violation eliminates the very possibility to perform the homogenization procedure. In the cases of both Dirichlet and Neumann conditions on the curved boundary operator $W_2^1$- and $L_2$-estimates are established; as concerns the $L_2$-estimates, the rate of convergence turns out to be twice as high. § 1. Statement of the problem and basic resultsLet $x=(x_1,\dots,x_n)$ be Cartesian coordinates in $\mathbb {R}^n$, $n\geqslant 2$, $\Omega\subset\mathbb{R}^n$ be some domain with nonempty boundary of class $C^2$ and $\Gamma_0$ be one or several connected components of this boundary. We assume that the surface $\Gamma_0$ is closed, has no self-intersections, is orientable, and the domain $\Omega$ lies on one side of $\Gamma_0$. We let $\nu$ denote the unit inward normal to $\Gamma_0$ relative $\Omega$, and we let $\tau$ denote the distance to a point measured along $\nu$. We assume that for some fixed $\tau_0>0$ local variables $(s,\tau)$ are well defined in the layer
$$
\begin{equation*}
\Pi_{\tau_0}:=\{x\in \Omega\colon \operatorname{dist}(x,\Gamma_0)<\tau_0\},
\end{equation*}
\notag
$$
where $s$ denotes some variables on the surface $\partial\Omega$, and the derivatives of $x$ with respect to $(s,\tau)$ and of $(s,\tau)$ with respect to $x$ are uniformly bounded in $\Pi_{\tau_0}$. Let $\varepsilon$ be a small positive parameter and $\Omega_\varepsilon$ be a subdomain of $\Omega$ obtained by curving the boundary component $\Gamma_0$ arbitrarily slightly. More precisely, our main assumption is as follows:
$$
\begin{equation}
\Omega_\varepsilon\subset \Omega\quad\text{and} \quad \Omega\setminus\Omega_\varepsilon\subseteq\Pi_\varepsilon.
\end{equation}
\tag{1.1}
$$
It immediately follows, in particular, that for sufficiently small $\varepsilon$ the component of the boundary of $\Omega_\varepsilon$ lying in $\Pi_\varepsilon$ is disjoint from components of $\partial\Omega$ other than $\Gamma_0$, that is,
$$
\begin{equation*}
\partial\Omega_\varepsilon\cap (\partial\Omega\setminus\Gamma_0)=\varnothing.
\end{equation*}
\notag
$$
We let $\mathbb{M}_m$ denote the set of square matrices of size $m\times m$, where $m\geqslant 1$; we assume that $\mathbb{M}_1=\mathbb{R}$ for $m=1$. The symbol $L_\infty(\Omega;\mathbb{M}_m)$ denotes the space of essentially bounded functions on $\Omega$ with values in $\mathbb{M}_m$. In other words, this space consists of matrix functions of size $m\times m$ each of whose components is an element of the space $L_\infty(\Omega)$. The norm on $L_\infty(\Omega;\mathbb{M}_m)$ is defined by
$$
\begin{equation*}
\|M\|_{L_\infty(\Omega;\mathbb{M}_m)}:=\operatorname*{ess\,sup}_{x\in\Omega} \|M(x)\|_{\mathbb{M}_m},
\end{equation*}
\notag
$$
where $\|\cdot\|_{\mathbb{M}_m}$ is the matrix norm on the space $\mathbb{M}_m$ defined by
$$
\begin{equation*}
\|M\|_{\mathbb{M}_m}:=\sup_{u\in \mathbb{C}^m} \frac{\|M U\|_{\mathbb{C}^m}} {\|U\|_{\mathbb{C}^m}}.
\end{equation*}
\notag
$$
In this paper we use similar Sobolev and Lebesgue spaces of matrix-valued functions and vector-valued functions, which are denoted in the standard way: for example, we use the notation $L_2(\Omega;\mathbb{M}_m)$ and $L_2(\Omega;\mathbb{C}^m)$. Let $A_{ij}=A_{ij}(x)$ and $A_j=A_j(x)$, $i,j=1,\dots,n$, be matrix-valued functions with values in $\mathbb{M}_m$ that are defined on $\Omega$ and satisfy the following conditions:
$$
\begin{equation}
\begin{gathered} \, A_{ij}\in W_\infty^1(\Omega;\mathbb{M}_m), \qquad A_j\in L_\infty(\Omega;\mathbb{M}_m), \\ \operatorname{Re} \sum_{i,j=1}^{n} (A_{ij}(x)U_j,U_i)_{\mathbb{C}^m}\geqslant c_0 \sum_{i=1}^{n} \|U_i\|_{\mathbb{C}^m}^2, \end{gathered}
\end{equation}
\tag{1.2}
$$
where the inequality holds for all $U_i\in \mathbb{C}^m$ and almost all $x\in\Omega$, with a positive constant $c_0$ independent of $x$ and $U_i$. We let $A_0=A_0(x,u)$ denote another matrix-valued function defined on $\Omega\times \mathbb{C}^m$ and satisfying the conditions
$$
\begin{equation}
A_0(x,0)=0 \quad\text{and} \quad \|A_0(x,u_1)-A_0(x,u_2)\|_{\mathbb{C}^m} \leqslant c_1 \|u_1-u_2\|_{\mathbb{C}^m}
\end{equation}
\tag{1.3}
$$
for almost all $x\in\Omega$ with a constant $c_1$ independent of $x$ and $u$. We assume additionally that the function $A_0(x,u(x))$ is measurable on $\Omega$ for any vector function $u\in L_2(\Omega;\mathbb{C}^m)$. The matrix-valued functions introduced above are assumed to have complex-valued components. We introduce the differential expression
$$
\begin{equation}
\widehat{\mathcal{H}}u=-\sum_{i,j=1}^{n} \frac{\partial\ }{\partial x_i} A_{ij}\,\frac{\partial u}{\partial x_j} + \sum_{j=1}^{n} A_j \,\frac{\partial u}{\partial x_j} + A_0(\,\cdot\,,u).
\end{equation}
\tag{1.4}
$$
In our paper we study boundary value problems for a system of semilinear elliptic equations of the form
$$
\begin{equation}
\widehat{\mathcal{H}}u_\varepsilon-\lambda u_\varepsilon=f \quad\text{in } \Omega_\varepsilon, \qquad u_\varepsilon=0 \quad\text{on } \Gamma:=\partial\Omega\setminus \Gamma_0
\end{equation}
\tag{1.5}
$$
with Dirichlet boundary conditions
$$
\begin{equation}
u_\varepsilon=0 \quad\text{on } \Gamma_\varepsilon:=\partial\Omega_\varepsilon\setminus \Gamma
\end{equation}
\tag{1.6}
$$
or Neumann boundary conditions
$$
\begin{equation}
\frac{\partial u_\varepsilon}{\partial\boldsymbol{\nu}_\varepsilon}=0 \quad\text{on } \Gamma_\varepsilon:=\partial\Omega_\varepsilon\setminus \Gamma, \qquad \frac{\partial\ }{\partial\boldsymbol{\nu}_\varepsilon}:=\sum_{i,j=1}^{n} \nu^i_\varepsilon A_{ij} \,\frac{\partial\ }{\partial x_j},
\end{equation}
\tag{1.7}
$$
where $\nu_\varepsilon=(\nu_\varepsilon^1,\dots,\nu_\varepsilon^n)$ is the unit inward to $\Gamma_\varepsilon$. The symbol $\lambda$ in (1.5) denotes a complex number to be chosen below, and $f$ is an arbitrary vector-valued function in the space $L_2(\Omega;\mathbb{M}_m)$. The aim of our study is to describe the behaviour of solutions of problems (1.5), (1.6) and (1.5), (1.7) as $\varepsilon\to+0$, namely, to find the form of the homogenized (limiting) problem and establish relevant operator estimates. To state the main results we need some auxiliary notation. We let $\mathring{W}_2^1(\Omega,\gamma)$ denote the subspace of the Sobolev space $W_2^1(\Omega)$ consisting of the functions with zero trace on the surface $\gamma$ in the domain $\Omega$. The analogous space of vector-valued functions is denoted by $\mathring{W}_2^1(\Omega,\gamma;\mathbb{C}^m)$. On $\mathring{W}_2^1(\Omega_\varepsilon,\Gamma;\mathbb{C}^m)$ we consider the nonlinear form
$$
\begin{equation*}
\begin{aligned} \, \mathfrak{h}^N_\varepsilon(u,v) &:=\sum_{i,j=1}^{n} \biggl(A_{ij}\,\frac{\partial u}{\partial x_j},\frac{\partial v}{\partial x_i}\biggr)_{L_2(\Omega_\varepsilon;\mathbb{C}^m)} + \sum_{j=1}^{n} \biggl(A_j\,\frac{\partial u}{\partial x_j},v\biggr)_{L_2(\Omega_\varepsilon;\mathbb{C}^m)} \\ &\qquad + \bigl(A_0(\,\cdot\,,u),v\bigr)_{L_2(\Omega_\varepsilon;\mathbb{C}^m)}. \end{aligned}
\end{equation*}
\notag
$$
Its restriction to the subspace $\mathring{W}_2^1(\Omega_\varepsilon,\partial\Omega_\varepsilon;\mathbb{C}^m)$ is denoted by $\mathfrak{h}_D^\varepsilon$. Solutions of boundary value problems under consideration are understood in the generalized sense. To be precise, a solution of problem (1.5), (1.6) is a vector-valued function in $W_2^1(\Omega_\varepsilon,\partial\Omega_\varepsilon;\mathbb{C}^m)$ satisfying the integral identity
$$
\begin{equation}
\mathfrak{h}^D_\varepsilon(u_\varepsilon,v)- \lambda (u_\varepsilon,v)_{L_2(\Omega_\varepsilon;\mathbb{C}^m)} =(f,v)_{L_2(\Omega_\varepsilon;\mathbb{C}^m)}
\end{equation}
\tag{1.8}
$$
for all $v\in \mathring{W}_2^1(\Omega_\varepsilon,\partial\Omega_\varepsilon;\mathbb{C}^m)$. A solution of problem (1.5), (1.7) is a vector-valued function in $W_2^1(\Omega_\varepsilon,\Gamma;\mathbb{C}^m)$ satisfying the integral identity
$$
\begin{equation}
\mathfrak{h}^N_\varepsilon(u_\varepsilon,v)- \lambda (u_\varepsilon,v)_{L_2(\Omega_\varepsilon;\mathbb{C}^m)} =(f,v)_{L_2(\Omega_\varepsilon;\mathbb{C}^m)}
\end{equation}
\tag{1.9}
$$
for all $v\in \mathring{W}_2^1(\Omega_\varepsilon,\Gamma;\mathbb{C}^m)$. We will show that the homogenized problem for (1.5), (1.6) is the Dirichlet problem
$$
\begin{equation}
\widehat{\mathcal{H}}u_0-\lambda u_0=f \quad\text{in } \Omega,
\end{equation}
\tag{1.10}
$$
In the case of the Neumann problem (1.5), (1.7) some additional restrictions must be imposed on the structure of the set $\Omega_\varepsilon$. More precisely, there must exists a cover of the set $\Omega\setminus\Omega_\varepsilon$ with certain properties. The condition that such a covering exists is briefly called condition (C) in what follows, and the covering itself is constructed in several steps as follows. We introduce the notation
$$
\begin{equation*}
\eta_0=\eta_0(\varepsilon):=\frac{\varepsilon}{3\sqrt{n}}\quad\text{and} \quad K_a:=\bigl\{x\in\mathbb{R}^n\colon x_i\in(0,a),\ i=1,\dots,n\bigr\}.
\end{equation*}
\notag
$$
We consider a periodic lattice $\eta_0 \mathbb{Z}^n$ in the space $\mathbb{R}^n$, and let $M^0_\varepsilon$ denote the union of all cubes with side $\eta_0$ and vertices at points of this lattice that lie fully in the layer $\Pi_{2\varepsilon}\setminus\Pi_\varepsilon$, that is,
$$
\begin{equation*}
M^0_\varepsilon:=\bigcup_{z_0\in L^0_\varepsilon} \bigl(z_0+K_{\eta_0(\varepsilon)}\bigr), \ \ \text{and} \ \ L^0_\varepsilon:=\{z_0\in\eta_0(\varepsilon)\mathbb{Z}^n\colon z+K_{\eta_0(\varepsilon)}\subset\Pi_{2\varepsilon}\setminus\Pi_\varepsilon\}.
\end{equation*}
\notag
$$
By the choice of $\eta_0(\varepsilon)$ the length of the longest diagonal of each cube $z_0+K_{\eta_0(\varepsilon)}$ is ${\varepsilon}/{3}$; therefore, the interior of the set $M^0_\varepsilon$ is simply connected and lies fully in $\overline{\Pi_{2\varepsilon}}\setminus\Pi_\varepsilon$. The set $(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon)\setminus\overline{M^0_\varepsilon}$ turns out to be disconnected, and its part adjoining $\partial\Pi_\varepsilon\setminus\partial\Omega_\varepsilon$ is a subset of $\Pi_{\frac{4}{3}\varepsilon}$. Let $\eta_k=\eta_k(\varepsilon)$, $k=1,2,\dots$, be positive functions satisfying the estimates
$$
\begin{equation}
\eta_{k+1}(\varepsilon)\leqslant \eta_k(\varepsilon), \qquad k=0,1,2,\dots\,.
\end{equation}
\tag{1.12}
$$
We construct recursively a family of auxiliary sets $M^k_\varepsilon$, $k=1,2,\dots$, such that $M^0_\varepsilon$ is the initial set. Each $M^k_\varepsilon$ has the form
$$
\begin{equation*}
M^k_\varepsilon=\bigcup_{z\in L^k_\varepsilon} \bigl(z+S_z K_{\eta_k(\varepsilon)}\bigr),
\end{equation*}
\notag
$$
where $L^k_\varepsilon$ is an at most countable set of points in $\Omega_\varepsilon\cap \Pi_{\frac{4}{3}\varepsilon}$ and $S_z$ is some linear orthogonal transformation of $\mathbb{R}^n$. Assume that $M^k_\varepsilon$ and $L^k_\varepsilon$ have been constructed. Then the set $L^{k+1}_\varepsilon$ is chosen on the basis of the following conditions:
We assume that this procedure of constructing the sets $M^k_\varepsilon$ can be carried out until $k=N(\varepsilon)$, where $N(\varepsilon)$ is some finite number satisfying the condition
$$
\begin{equation}
\varepsilon\mu(\varepsilon)\to+0 \quad\text{as } \varepsilon\to+0, \quad\text{where } \mu(\varepsilon):=c_2^{N(\varepsilon)}\quad\text{and} \quad c_2:=\sqrt{(2^n+1)p}>1.
\end{equation}
\tag{1.13}
$$
We introduce the notation
$$
\begin{equation*}
M_\varepsilon:=\bigcup_{k=0}^{N(\varepsilon)} M^k_\varepsilon\quad\text{and} \quad L_\varepsilon:=\bigcup_{k=0}^{N(\varepsilon)} L^k_\varepsilon.
\end{equation*}
\notag
$$
In what follows we assume that the set $\Pi_{\frac{4}{3}\varepsilon}\cap\Omega_\varepsilon\setminus\overline{M_\varepsilon}$ can be covered by sets of the form $T^z_\varepsilon$, where $z$ ranges over some subset $L^\partial_\varepsilon \subseteq L_\varepsilon$ and the sets $T^z_\varepsilon$ themselves are defined as follows:
$$
\begin{equation}
T^z_\varepsilon:=\bigl\{x\in\mathbb{R}^n\colon 0\leqslant \rho< \eta_k(\varepsilon)\phi_z(\xi,\varepsilon),\ \xi\in\Upsilon^z_\varepsilon \bigr\}, \qquad z\in L^k_\varepsilon\cap L^\partial_\varepsilon.
\end{equation}
\tag{1.14}
$$
Here $\Upsilon^z_\varepsilon\subset z+\eta_k(\varepsilon) K_{\eta_k(\varepsilon)}$ is some oriented manifold of codimension $1$, possibly with boundary, on which a continuous field of normals $\nu^z_\varepsilon=\nu^z_\varepsilon(\xi)$, $\xi\in \Upsilon^z_\varepsilon$, is defined, and $\rho$ is the distance along this field of normals. We assume that there are positive functions $\rho_k(\varepsilon)$ such that the bijective diffeomorphism $x=\xi+\rho \nu^z_\varepsilon(\xi)$ is well defined on the set $\{(\rho,\xi)\colon 0<\rho<\rho_k(\varepsilon) \eta_k(\varepsilon),\ \xi\in \Upsilon^k_\varepsilon\}$, which performs a transition to the variables $x$, the derivatives of $x$ with respect to $(\xi,\rho)$ and of $(\xi,\rho)$ with respect to $x$ are bounded uniformly in $\varepsilon$, $z$ and the space variables on $\overline{T^z_\varepsilon}$, and we have
$$
\begin{equation}
0\leqslant \phi_z(\xi,\varepsilon)\leqslant \rho_k(\varepsilon), \quad \xi\in \Upsilon^z_\varepsilon, \quad\text{and} \quad 0<c_3\leqslant \rho_k(\varepsilon)\leqslant c_4,
\end{equation}
\tag{1.15}
$$
where $c_3$ and $c_4$ are fixed constants independent of $k$ and $\varepsilon$. We assume that each point $x$ in $\Pi_{\frac{4}{3}\varepsilon}\cap\Omega_\varepsilon\setminus\overline{M_\varepsilon}$ lies in finitely many domains $T^z_\varepsilon$, this finite number of domains is bounded uniformly in $x$ and $\varepsilon$, and
$$
\begin{equation*}
\bigl\{x\colon 0<\rho<c_3,\ \xi\in \Upsilon^z_\varepsilon\bigr\}\subset z+\eta_k(\varepsilon) K_{\eta_k(\varepsilon)}.
\end{equation*}
\notag
$$
We understand the above covering of the set $\Pi_{\frac{4}{3}\varepsilon}\cap\Omega_\varepsilon\setminus\overline{M_\varepsilon}$ in the sense of the equality
$$
\begin{equation*}
\biggl(M_\varepsilon\cup \bigcup_{z\in L_\varepsilon^\partial} T_\varepsilon^z \biggr)\cap \Pi_{\frac{4}{3}\varepsilon}=\Omega_\varepsilon\cap\Pi_{\frac{4}{3}\varepsilon}.
\end{equation*}
\notag
$$
We assume additionally that the smoothness of $\Gamma_\varepsilon$ is such that the Sobolev space $W_2^1(\Omega;\mathbb{C}^m)$ is a separable Hilbert space and the trace operator on $\Gamma_\varepsilon$ is bounded as an operator from $W_2^1(\Omega;\mathbb{C}^m)$ to $L_2(\Gamma_\varepsilon;\mathbb{C}^m)$; the norm of this operator can arbitrarily depend on $\varepsilon$. For example, it suffices to assume that $\Gamma_\varepsilon\in C^1$ or $\Gamma_\varepsilon$ consists of several continuously differentiable surfaces with appropriate conditions imposed on their boundaries. An example of a perturbed boundary in shown in Figure 3. The above covering in a neighbourhood of a small part of such a boundary is schematically shown in Figure 4. We will show that, when condition (C) mentioned above holds, the homogenized problem for the Neumann problem (1.5), (1.7) is a Neumann problem in $\Omega$, or, more precisely, a boundary value problem for equation (1.10) with Neumann boundary conditions:
$$
\begin{equation}
\frac{\partial u_0}{\partial\boldsymbol{\nu}}=0 \quad\text{on } \Gamma_0\quad\text{and}\quad \frac{\partial u}{\partial\boldsymbol{\nu}}:=\sum_{i,j=1}^{2} \nu^i A_{ij} \,\frac{\partial\ }{\partial x_j}, \quad \nu=(\nu^1,\dots,\nu^n).
\end{equation}
\tag{1.16}
$$
Solutions of the homogenized problems are also understood in the generalized sense. To be precise, we introduce the form
$$
\begin{equation*}
\begin{aligned} \, \mathfrak{h}_0^N(u,v) &:=\sum_{i,j=1}^{2} \biggl(A_{ij}\, \frac{\partial u}{\partial x_j},\frac{\partial v}{\partial x_i}\biggr)_{L_2(\Omega;\mathbb{C}^m)} + \sum_{j=1}^{2} \biggl(A_j\,\frac{\partial u}{\partial x_j},v\biggr)_{L_2(\Omega;\mathbb{C}^m)} \\ &\qquad + \bigl(A_0(\,\cdot\,,u),v\bigr)_{L_2(\Omega;\mathbb{C}^m)} \end{aligned}
\end{equation*}
\notag
$$
with domain $\mathfrak{D}(\mathfrak{h}_0^N):=\mathring{W}_2^1(\Omega,\Gamma;\mathbb{C}^m)$ in the space $L_2(\Omega;\mathbb{C}^m)$. A solution of the Neumann problem (1.10), (1.16) is a function $u\in\mathfrak{D}(\mathfrak{h}_0^N)$ satisfying the integral identity
$$
\begin{equation}
\mathfrak{h}_0(u_0,v)- \lambda (u_0,v)_{L_2(\Omega;\mathbb{C}^m)}=(f,v)_{L_2(\Omega;\mathbb{C}^m)},
\end{equation}
\tag{1.17}
$$
where $\mathfrak{h}_0:=\mathfrak{h}_0^N$, for all $v\in \mathfrak{D}(\mathfrak{h}_0^N)$. We let $\mathfrak{h}_0^D$ denote the restriction of $\mathfrak{h}_0^N$ to the subspace $\mathfrak{D}(\mathfrak{h}_0^D):=\mathring{W}_2^1(\Omega,\partial\Omega;\mathbb{C}^m)$. A solution of the Dirichlet problem (1.10), (1.11) is a function $u\in\mathfrak{D}(\mathfrak{h}_0^D)$ satisfying the integral identity (1.17), where $\mathfrak{h}_0:=\mathfrak{h}_0^D$, for all $v\in \mathfrak{D}(\mathfrak{h}_0^D)$. We now state the main results. Theorem 1.1. There exists $\lambda_0$ independent of $\varepsilon$ such that for $\operatorname{Re}\lambda \leqslant \lambda_0$ the Dirichlet problems (1.5), (1.6) and (1.10), (1.11) are uniquely solvable for all $f\in L_2(\Omega;\mathbb{C}^m)$ and where the constant $C$ is independent of $\varepsilon$ and $f$ but depends on $\lambda$. In addition, if then where the constant $C$ is independent of $\varepsilon$ and $f$ but depends on $\lambda$. Theorem 1.2. There exists $\lambda_0$ independent of $\varepsilon$ such that for $\operatorname{Re}\lambda \leqslant \lambda_0$ the Neumann problems (1.5), (1.7) and (1.10), (1.16) are uniquely solvable for all $f\in L_2(\Omega;\mathbb{C}^m)$, and We discuss the problems under consideration and the main results. The perturbed problems (1.5), (1.6), (1.7) are posed for a general equation of the second order with variable coefficients and a weakly nonlinear potential term; see the differential expression (1.4). This is a matrix differential expression; therefore, we actually deal with a system of weakly nonlinear elliptic equations. The perturbation is a small irregular perturbation of the component $\Gamma_0$ of the boundary of the domain $\Omega$, and the main assumption is that the amplitude of the perturbations is small. This is expressed by condition (1.1), which means that the perturbed boundary component $\Gamma_\varepsilon$ lies in a narrow layer of width $\varepsilon$ along the original unperturbed boundary component $\Gamma_0$. In the case of the Dirichlet problem (1.5), (1.6) no other restrictions are imposed on the boundary perturbation. It turns out in this case that the homogenized problem for this problem is the Dirichlet problem (1.10), (1.11), and the difference between the solutions of the perturbed and homogenized problems satisfies (1.18). An advantage of this estimate is that the dependence on the right-hand side $f$ is explicit, this is, in fact, an operator estimate for the weakly nonlinear elliptic system under consideration. If we consider the weaker $L_2$-norm of the difference between the solutions, then the rate of convergence increases twice; see (1.20). A condition under which this estimate holds is (1.19); it is essentially used in the proof and probably cannot be dropped without destroying estimate (1.20). We stress that condition (1.1) is very weak and holds for a wide class of boundary perturbations. It is obviously satisfied for weak regular deformations of the boundary. It also holds for classical rapidly oscillating boundaries, and there is no need to assume the periodicity of the oscillations in this case. The oscillations themselves do not necessarily have to be defined by graphs of the form $\tau=\varepsilon b(s,\varepsilon)$ with some bounded function $b$: perturbations not described by simple graphs of this form are also possible. In addition, perturbations changing the connectivity of the domain, like fine perforation along the boundary, are admissible. Various thin spikes of finite length positioned along the unperturbed boundary are another example. Figures 1–3 show examples of different admissible perturbations. In the case of the Neumann problem (1.5), (1.7) condition (1.1) is not sufficient for a meaningful convergence result. A reason is that the perturbed domain can contain a separate connected component of small size, for example, a ball of small radius. As a differential expression, it suffices to take $\widehat{\mathcal{H}}:=-\Delta+1$. Choosing a right-hand side $f$ concentrated on this component alone, we can easily verify that the limiting Neumann problem has a solution with small $W_2^1$-norm in this case. On the other hand the solution of the perturbed problem vanishes everywhere away from this small component, whereas on this component the solution has a finite $W_2^1$-norm, which is not small. Therefore, we need to assume about the Neumann problem that the perturbation does not change the connectivity of the domain. However, this is also not sufficient since the above small component can be connected with the rest of the perturbed domain by a very thin channel of finite length, and then we actually have the same phenomenon as in the case of an isolated small component. Thus, we have to impose additional restrictions on the geometry of the perturbation in the case of the Neumann problem. In our case these restrictions are formulated in condition (C). The meaning of this condition is that the part of the perturbed domain contained in the thin layer $\Pi_{\frac{4}{3}\varepsilon}$ can be covered in a prescribed way. More precisely, the first covering system consists of the cubes $K_{\eta_0(\varepsilon)}$ associated with the lattice $\eta_0\mathbb{Z}^n$. The next systems are constructed recursively on the basis of conditions (A1) and (A2). Specifically, each new system of cubes has edges no longer than those of cubes in the previous system. Each new cube intersects some cube in the previous system, and this intersection should not be too small. Finally, the multiplicity of the covering by cubes of this type must be uniformly bounded. In finitely many steps a large part of the domain $\Pi_{\frac{4}{3}\varepsilon}\cap\Omega_\varepsilon$ is covered by such cubes. The rest of this domain is covered by the domains $T_\varepsilon^z$, parts of whose boundaries, described by equations $\rho=\eta_k(\varepsilon)\phi_z(\xi,\varepsilon)$, $\xi\in \Upsilon_\varepsilon^z$, lie on $\Gamma_\varepsilon$; see Figures 3 and 4. Note that methods of this type, which are based on coverings by domains of decreasing size, are used in various embedding theorems for Soboles spaces; see, for example, [21], Ch. 2, § 2.1.5, Theorem 1. Under the above conditions the homogenized problem is problem (1.10), (1.16), and an operator estimate holds again; see (1.21). The term $\varepsilon\mu(\varepsilon)$ in this estimate describes the contribution of the geometry of the perturbed boundary, and it is small because of assumption (1.13). If the difference between solutions of the perturbed and homogenized problems is considered in the $L_2$-norm, then the rate of convergence becomes better: see (1.22). The coefficient of $\|f\|_{L_2(\Omega;\mathbb{C}^m)}$ in this estimate has the double order of smallness in comparison with (1.21), whereas the second term has the same order. At the same time, the second term contains the norm of the function $f$ in $L_2(\Omega\setminus\Omega_\varepsilon;\mathbb{C}^m)$, and these are values of $f$ that do not participate in the perturbed problem and have no effect on the form of its solution. For example, we can assume that $f=0$ in $\Omega\setminus\Omega_\varepsilon$, and then there is no such term. On the other hand, if the function $f$ is originally defined in the whole domain, then it is convenient to work with this domain, and in this case the second term in (1.22) estimates the contribution of the solution of the homogenized problem that is generated by the restriction of the right-hand side to $\Omega\setminus\Omega_\varepsilon$. § 2. Auxiliary assertionsIn this section we prove a series of lemmas used below in the proofs of the main results. We start with the existence of a number $\lambda_0$ ensuring the solvability of the perturbed and homogenized problems under consideration. Lemma 2.1. There exists $\lambda_0$ independent of $\varepsilon$ such that for $\operatorname{Re}\lambda\leqslant \lambda_0$ the Dirichlet problems (1.5), (1.6) and (1.10), (1.11) and the Neumann problems (1.5), (1.7) and (1.10), (1.16) are uniquely solvable for all $f\in L_2(\Omega;\mathbb{C}^m)$, and for all $u,v\in\mathfrak{D}(\mathfrak{h}^\natural_\varepsilon)$, where $\natural\in\{D,N\}$. The solutions of the homogenized problems belong to the space $W_2^2(\Omega;\mathbb{C}^m)$ and satisfy where the constants $C_1$, $C_2$ and $C_3$ are independent of $f$, $\lambda$ and $\lambda_0$, and $C_3<-\lambda_0$. Proof. We use standard methods of the theory of monotone operators: see [22], Ch. VI, § 18.4, and [23], Ch. 1, § 2$^0$. Specifically, on the spaces $\mathfrak{D}(\mathfrak{h})$, where $\mathfrak{h}$ is the appropriate form out of $\mathfrak{h}^D_\varepsilon$, $\mathfrak{h}^N_\varepsilon$, $\mathfrak{h}_0^D$ and $\mathfrak{h}_0^N$, we introduce the operators that assign antilinear continuous functionals on these spaces to each function $u$ by the rule $u\mapsto \mathfrak{h}(u,\,\cdot\,)$. By general results ([22], Ch. VI, § 18.4, and [23], Ch. 1, § 2$^0$), to prove the unique solvability of the problems under consideration it suffices to verify that one of the following three properties holds:
Assume that a sequence $u_k\in\mathfrak{D}(\mathfrak{h})$ converges weakly to $u$ in $W_2^1(\Omega^\natural;\mathbb{C}^m)$, where $\Omega^\natural=\Omega$ or $\Omega^\natural=\Omega_\varepsilon$. Then for an arbitrary $v\in\mathfrak{D}(\mathfrak{h})$, as $k\to\infty$, we have
$$
\begin{equation}
\begin{gathered} \, \sum_{i,j=1}^{2} \biggl(A_{ij}\, \frac{\partial u_k}{\partial x_j}, \frac{\partial v}{\partial x_j}\biggr)_{L_2(\Omega;\mathbb{C}^m)} \to \sum_{i,j=1}^{2} \biggl(A_{ij}\, \frac{\partial u}{\partial x_j}, \frac{\partial v}{\partial x_j}\biggr)_{L_2(\Omega;\mathbb{C}^m)}, \\ \sum_{j=1}^{2} \biggl(A_j\,\frac{\partial u_k}{\partial x_j},v\biggr)_{L_2(\Omega;\mathbb{C}^m)} \to \sum_{j=1}^{2} \biggl(A_j\,\frac{\partial u}{\partial x_j},v\biggr)_{L_2(\Omega;\mathbb{C}^m)}, \\ \lambda(u_k,v)_{L_2(\Omega;\mathbb{C}^m)}\to \lambda(u,v)_{L_2(\Omega;\mathbb{C}^m)}. \end{gathered}
\end{equation}
\tag{2.3}
$$
Using the standard diagonal procedure, since the embedding of $W_2^1(Q;\mathbb{C}^m)$ in $L_2(Q;\mathbb{C}^m)$ is compact for the compact domain $Q$, we select a subsequence of $u_k$ that converges strongly to $u$ in $L_2(Q;\mathbb{C}^m)$ for all compact subdomains $Q\subset \Omega^\natural$. Since $v\in W_2^1(\Omega^\natural)$, for an arbitrary $\delta>0$ there exists a compact domain $Q_\delta\subset \Omega^\natural$ such that
$$
\begin{equation*}
\|v\|_{W_2^1(\Omega^\natural\setminus Q;\mathbb{C}^m)}\leqslant \delta;
\end{equation*}
\notag
$$
from the Lipschitz inequality in (1.3) for $u_2=0$ and the equality $A(\,\cdot\,,0)=0$ it follows that
$$
\begin{equation}
\begin{aligned} \, \notag &\bigl|\bigl(A_0(\,\cdot\,,u_k),v\bigr)_{L_2(\Omega^\natural\setminus Q_\delta;\mathbb{C}^m)} -\bigl(A_0(\,\cdot\,,u),v\bigr)_{L_2(\Omega^\natural\setminus Q_\delta;\mathbb{C}^m)}\bigr| \\ &\qquad \leqslant c_1\bigl(\|u_k\|_{L_2(\Omega^\natural\setminus Q_\delta;\mathbb{C}^m)}+\|u\|_{L_2(\Omega^\natural\setminus Q_\delta;\mathbb{C}^m)}\bigr) \|v\|_{L_2(\Omega^\natural\setminus Q_\delta;\mathbb{C}^m)} \leqslant C\delta, \end{aligned}
\end{equation}
\tag{2.4}
$$
where $C$ is a constant independent of $\delta$ and $k$. Since $u_k$ converges strongly to $u$ in $L_2(Q_\delta;\mathbb{C}^m)$, from the relations in (1.3) we derive similarly that
$$
\begin{equation*}
\begin{aligned} \, &\bigl|\bigl(A_0(\,\cdot\,,u_k),v\bigr)_{L_2(Q_\delta;\mathbb{C}^m)} -\bigl(A_0(\,\cdot\,,u),v\bigr)_{L_2(Q_\delta;\mathbb{C}^m)}\bigr| \\ &\qquad \leqslant c_1\|u_k-u\|_{L_2(Q_\delta;\mathbb{C}^m)} \|v\|_{L_2(Q_\delta;\mathbb{C}^m)}\to0 \end{aligned}
\end{equation*}
\notag
$$
as $k\to\infty$. Since $\delta$ is arbitrary, in view of (2.4) we infer that
$$
\begin{equation*}
\bigl(A_0(\,\cdot\,,u_k),v\bigr)_{L_2(\Omega^\natural;\mathbb{C}^m)} \to\bigl(A_0(\,\cdot\,,u),v\bigr)_{L_2(\Omega^\natural;\mathbb{C}^m)}, \qquad k\to\infty,
\end{equation*}
\notag
$$
which, in combination with (2.3), proves property (1).
It follows from conditions (1.2) and (1.3) and the Cauchy–Bunyakovsky–Schwarz inequality that
$$
\begin{equation*}
\begin{aligned} \, & \biggl|\sum_{j=1}^{n} \biggl(A_j\,\frac{\partial (u-v)}{\partial x_j},u-v\biggr)_{L_2(\Omega^\natural;\mathbb{C}^m)} + \bigl(A_0(\,\cdot\,,u)-A_0(\,\cdot\,,v),u-v\bigr)_{L_2(\Omega^\natural;\mathbb{C}^m)}\biggr| \\ &\qquad\leqslant \frac{c_0}{2} \|\nabla (u-v)\|_{L_2(\Omega^\natural)}^2 + C \|u-v\|_{L_2(\Omega^\natural)}^2 \end{aligned}
\end{equation*}
\notag
$$
for some constant $C$ independent of $u,v\in\mathfrak{D}(\mathfrak{h})$. In view of the ellipticity condition in (1.2) we see that
$$
\begin{equation*}
\begin{aligned} \, &\operatorname{Re} \bigl(\mathfrak{h}(u,u-v)-\mathfrak{h}(v,u-v)-\lambda\|u-v\|_{L_2(\Omega^\natural)}^2\bigr) \\ &\qquad \geqslant \frac{c_0}{2} \|\nabla (u-v)\|_{L_2(\Omega^\natural)}^2 -(\operatorname{Re}\lambda +C) \|u-v\|_{L_2(\Omega^\natural)}^2, \end{aligned}
\end{equation*}
\notag
$$
which yields that property (2) and estimate (2.1) hold for $\operatorname{Re}\lambda\leqslant \lambda_0\leqslant -C-{c_0}/{2}$. In addition, setting $v=0$ and taking account of the integral identity (1.17), we see that property (3) is valid and
$$
\begin{equation*}
\begin{aligned} \, &\frac{c_0}{2}\|\nabla u_0\|_{L_2(\Omega;\mathbb{C}^m)}^2+ (-\operatorname{Re}\lambda-C) \|u_0\|_{L_2(\Omega;\mathbb{C}^m)}^2 \\ &\qquad\leqslant \|f\|_{L_2(\Omega;\mathbb{C}^m)}\|u_0\|_{L_2(\Omega;\mathbb{C}^m)} \leqslant \frac{2}{c_0}\|f\|_{L_2(\Omega;\mathbb{C}^m)}^2+\frac{c_0}{2}\|u_0\|_{L_2(\Omega;\mathbb{C}^m)}^2 \end{aligned}
\end{equation*}
\notag
$$
for solutions of both problems (1.10), (1.11) and (1.10), (1.16). This fact implies the second estimate in (2.2) and the relation
$$
\begin{equation}
\|\nabla u_0\|_{L_2(\Omega;\mathbb{C}^m)}\leqslant \frac{2}{c_0} \|f\|_{L_2(\Omega;\mathbb{C}^m)}.
\end{equation}
\tag{2.5}
$$
Since solutions of problems (1.10), (1.11) and (1.10), (1.16) are in $W_2^1(\Omega;\mathbb{C}^m)$, we easily derive from (1.3) that the function $A_0(\,\cdot\,,u_0)$ is an element of $L_2(\Omega;\mathbb{C}^m)$ and
$$
\begin{equation*}
\|A_0(\,\cdot\,,u_0)\|_{L_2(\Omega;\mathbb{C}^m)}\leqslant c_1 \|u_0\|_{L_2(\Omega;\mathbb{C}^m)} \leqslant \frac{C}{|C_3+\operatorname{Re}\lambda|} \|f\|_{L_2(\Omega;\mathbb{C}^m)},
\end{equation*}
\notag
$$
where the constant $C$ is independent of $f$ and $\lambda$. Moving the function $A(\,\cdot\,,u_0)$ and the terms containing first derivatives to the right-hand side of (1.10), owing to standard theorems on improved smoothness of solutions of elliptic boundary value problems, the second inequality in (2.2) and (2.5), we arrive at estimates (2.2).
Lemma 2.1 is proved. We let $\chi=\chi(t)$ denote an infinitely differentiable cutoff function equal to one for $t<{1}/{3}$ and zero for $t>{2}/{3}$. The following lemma plays a key role in the proof of Theorem 1.1. Lemma 2.2. The following inequalities hold: Proof. Let $u\in \mathring{W}_2^1(\Omega_\varepsilon,\Gamma_\varepsilon;\mathbb{C}^m)$. We extend this function by zero to $\Omega\setminus\Omega_\varepsilon$ and denote the resulting extension by $u$ again. Since $u$ has zero trace on $\Gamma_\varepsilon$, this extension turns out to be an element of $\mathring{W}_2^1(\Omega,\Gamma_0;\mathbb{C}^m)$. By condition (1.1) and our assumptions on the regularity of the surface $\Gamma$, the variables $(s,\tau)$ are well defined on the set $\Omega\setminus\Omega_\varepsilon$; therefore,
$$
\begin{equation}
u(x)=\int_{0}^{\tau} \frac{\partial u}{\partial\tau}\,d\tau, \qquad x\in\Pi_{2\varepsilon}\cap\Omega_\varepsilon.
\end{equation}
\tag{2.9}
$$
From the uniform boundedness of derivatives of $x$ with respect to $(s,\tau)$ and the Cauchy–Bunyakovsky–Schwarz inequality it follows that
$$
\begin{equation}
|u(x)|^2\leqslant C \varepsilon\int_{0}^{\varepsilon}|\nabla u|^2\,d\tau, \qquad x\in\Pi_{2\varepsilon}\cap\Omega_\varepsilon,
\end{equation}
\tag{2.10}
$$
where the constant $C$ is independent of $u$, $x$ and $\varepsilon$. Integrating this inequality over $\Pi_{2\varepsilon}\cap\Omega_\varepsilon$ and taking account of the boundedness of the Jacobian of the change of variables $x\leftrightarrow(s,\tau)$ we obtain (2.6).
Let $u\in W_2^1(\Omega;\mathbb{C}^m)$. Then
$$
\begin{equation*}
u(x)=\int_{\tau}^{\tau_0} \frac{\partial\ }{\partial\tau}\chi\biggl(\frac{\tau}{\tau_0}\biggr) u \,d\tau, \qquad x\in\Pi_{2\varepsilon},
\end{equation*}
\notag
$$
which by the Cauchy–Bunyakovsky–Schwarz inequality yields
$$
\begin{equation*}
|u(x)|^2\leqslant C \int_{\tau}^{\tau_0} \bigl(|u|^2+|\nabla u|^2\bigr)\,d\tau, \qquad x\in\Pi_{2\varepsilon},
\end{equation*}
\notag
$$
where the constant $C$ is independent of $\varepsilon$, $u$ and $x$. Integrating this inequality over $\Pi_{2\varepsilon}$, we infer (2.8).
For functions $u\,{\in}\, W_2^2(\Omega;\mathbb{C}^m)\,{\cap}\, \mathring{W}_2^1(\Omega,\Gamma_0;\mathbb{C}^m)$, by the above calculations we have
$$
\begin{equation*}
\|u\|_{L_2(\Pi_{2\varepsilon};\mathbb{C}^m)}\leqslant C\varepsilon \|\nabla u\|_{L_2(\Pi_{2\varepsilon};\mathbb{C}^m)},
\end{equation*}
\notag
$$
where $C$ is a constant independent of $u$ and $\varepsilon$. Applying (2.8) to the right-hand side we arrive at (2.7).
Lemma 2.2 is proved. Proof. It is clearly sufficient to prove this lemma in the case of a scalar function $u\in W_2^1(K_1)$, that is, for $m=1$. We let $V$ denote the characteristic function of the cube $K_{{1}/{2}}$. We consider the operator $\mathcal{A}_\alpha:=-\Delta+a V$ with Neumann boundary conditions on the cube $K_1$. By the minimax principle the least eigenvalue of such an operator is expressed by the equality it is clear that this eigenvalue is strictly positive for all $a>0$. This equality implies that for all $u\in W_2^1(K_1)$.
By standard theorems in the regular perturbation theory (see, for example, [24], Ch. VII, § 3.5) this eigenvalue is holomorphic in $a$ and coincides for $a=0$ with the least eigenvalue of the Laplacian with the Neumann boundary conditions in $K_1$, that is, $\lambda(a)=0$. Therefore, the first terms of the Taylor series for $\lambda(a)$ have the form where the constant $\lambda_1$ ensures the solvability of the boundary value problem
$$
\begin{equation}
-\Delta\psi_1+V=\lambda_1 \quad\text{in } K_1, \qquad \frac{\partial\psi_1}{\partial\nu}=0 \quad\text{on } \partial K_1;
\end{equation}
\tag{2.13}
$$
here $\nu$ is the normal to the boundary of $K_1$. Here we have also used the fact that an eigenfunction of the Laplace operator with Neumann boundary condition in $K_1$ can be taken identically equal to one in $K_1$. The solvability condition for problem (2.13) is obtained by the standard integration of the equation in this problem over $K_1$, which yields
$$
\begin{equation*}
\lambda_1=\frac{\operatorname{mes} K_{1/2}}{\operatorname{mes} K_1}=\frac{1}{2^n},
\end{equation*}
\notag
$$
where $\operatorname{mes}$ is the standard Lebesgue measure in $\mathbb{R}^n$. We now substitute this formula and the decomposition (2.12) into (2.11) and then choose $a>0$ to be sufficiently small to guarantee the inequality The assertion of the lemma follows from (2.11).
Lemma 2.3 is proved. Lemma 2.4. Assume that condition (C) holds. Then for all $u\in W_2^1(\Omega_\varepsilon,\partial\Pi_{\frac{3}{2}\varepsilon}\setminus\Gamma_0;\mathbb{C}^m)$, where the constant $C$ is independent of $\varepsilon$ and $u$. Proof. Let $u\in W_2^1(\Omega_\varepsilon,\partial\Pi_{\frac{3}{2}\varepsilon}\setminus\Gamma_0;\mathbb{C}^m)$. Then the following relations, similar to (2.9) and (2.10), are true:
$$
\begin{equation*}
u(x)=\int_{\frac{3}{2}\varepsilon}^{\tau} \frac{\partial u}{\partial\tau}\,d\tau\quad\text{and} \quad |u(x)|^2\leqslant C \varepsilon\int_{\varepsilon}^{\frac{3}{2}\varepsilon}|\nabla u|^2\,d\tau, \qquad x\in\Pi_{\frac{3}{2}\varepsilon}\setminus\Pi_\varepsilon.
\end{equation*}
\notag
$$
Integrating the inequality over $\Pi_{\frac{3}{2}\varepsilon}\setminus\Pi_\varepsilon$, we easily derive that
$$
\begin{equation}
\begin{gathered} \, \|u\|_{L_2(M_\varepsilon^0;\mathbb{C}^m)}^2\leqslant C\varepsilon^2 \|\nabla u\|_{W_2^1(\Pi_{\frac{3}{2}\varepsilon}\setminus\Pi_\varepsilon;\mathbb{C}^m)}^2, \\ \|u\|_{L_2(\Pi_{\frac{3}{2}\varepsilon}\setminus\Pi_\varepsilon;\mathbb{C}^m)}^2\leqslant C\varepsilon^2 \|\nabla u\|_{W_2^1(\Pi_{\frac{3}{2}\varepsilon}\setminus\Pi_\varepsilon;\mathbb{C}^m)}^2, \end{gathered}
\end{equation}
\tag{2.14}
$$
where the constant $C$ is independent of $u$ and $\varepsilon$.
Applying Lemma 2.3 to the function $v(y)=u(\eta y)$ for $u\,{\in}\, W_2^1(K_{\eta};\mathbb{C}^m)$, we obtain
$$
\begin{equation}
\|u\|_{L_2(K_\eta;\mathbb{C}^m)}^2\leqslant c_5 \eta^2 \|\nabla u\|_{L_2(K_\eta;\mathbb{C}^m)}^2 +(2^n+1)\|u\|_{L_2(K_{{\eta}/{2}};\mathbb{C}^m)}^2.
\end{equation}
\tag{2.15}
$$
Now consider a pair of intersecting cubes $z+S_z K_{\eta_{k+1}(\varepsilon)}$ and $\widetilde{z}+S_{\widetilde{z}} K_{\eta_k(\varepsilon)}$ from condition (A1). This condition makes it possible to apply estimate (2.15) to $z+ S_z K_{\eta_{k+1}(\varepsilon)}$:
$$
\begin{equation*}
\begin{aligned} \, \|u\|_{L_2(z+S_z K_{\eta_{k+1}(\varepsilon)};\mathbb{C}^m)}^2 &\leqslant c_5 \eta_{k+1}^2(\varepsilon) \|\nabla u\|_{L_2(z+S_z K_{\eta_{k+1}(\varepsilon)};\mathbb{C}^m)}^2 \\ &\qquad + (2^n+1) \|u\|_{L_2((z+S_z K_{\eta_{k+1}(\varepsilon)})\cap (\widetilde{z}+ S_{\widetilde{z}} K_{\eta_k(\varepsilon)});\mathbb{C}^m)}^2 \end{aligned}
\end{equation*}
\notag
$$
for all $u\in W_2^1(z+S_z K_{\eta_{k+1}(\varepsilon)};\mathbb{C}^m)$. In view of condition (A2) and (1.12), summing the estimate obtained for all $z\in L^{k+1}_\varepsilon$ yields another inequality:
$$
\begin{equation*}
\begin{aligned} \, \|u\|_{L_2(M^{k+1}_\varepsilon;\mathbb{C}^m)}^2 &\leqslant c_5p\eta_{k+1}^2(\varepsilon) \|\nabla u\|_{L_2(M^{k+1}_\varepsilon;\mathbb{C}^m)}^2 +c_2^2 \|u\|_{L_2(M^k_\varepsilon;\mathbb{C}^m)}^2 \\ &\leqslant c_5p\eta_1^2(\varepsilon) \|\nabla u\|_{L_2(M^{k+1}_\varepsilon;\mathbb{C}^m)}^2 +c_2^2 \|u\|_{L_2(M^k_\varepsilon;\mathbb{C}^m)}^2. \end{aligned}
\end{equation*}
\notag
$$
Applying this inequality by induction with respect to $k$ and taking account of condition (A2) we deduce the relation
$$
\begin{equation*}
\|u\|_{L_2(M^k_\varepsilon;\mathbb{C}^m)}^2\leqslant c_5p\eta_1^2(\varepsilon) \sum_{j=0}^{k}c_2^{2(k-j)}\|\nabla u\|_{L_2(M_\varepsilon^j;\mathbb{C}^m)}^2 +c_2^{2k} \|u\|_{L_2(M^0_\varepsilon;\mathbb{C}^m)}^2.
\end{equation*}
\notag
$$
We add the above estimates for $k=1,\dots, N(\varepsilon)$:
$$
\begin{equation*}
\begin{aligned} \, \|u\|_{L_2(M_\varepsilon;\mathbb{C}^m)}^2 &\leqslant c_5p\eta_1^2(\varepsilon) \sum_{k=1}^{N(\varepsilon)} \sum_{j=0}^{k}c_2^{2(k-j)}\|\nabla u\|_{L_2(M_\varepsilon^j;\mathbb{C}^m)}^2 + \sum_{k=1}^{N(\varepsilon)} c_2^{2k} \|u\|_{L_2(M^0_\varepsilon;\mathbb{C}^m)}^2 \\ &\leqslant c_5p\eta_1^2(\varepsilon) \sum_{j=0}^{N}\|\nabla u\|_{L_2(M_\varepsilon^j)}^2 \sum_{k=\max\{j,1\}}^{N(\varepsilon)} c_2^{2(k-j)} + \|u\|_{L_2(M_\varepsilon^0)}^2 \sum_{k=1}^{N(\varepsilon)} c_2^{2k} \\ &\leqslant \frac{c_5 c_6 p^2}{c_6-1}\bigl(\eta_1^2(\varepsilon) c_2^{2N(\varepsilon)} \|\nabla u\|_{L_2(M_\varepsilon^0)}^2 + \|u\|_{L_2(M_\varepsilon^0)}^2 \bigr). \end{aligned}
\end{equation*}
\notag
$$
By (2.14) and (1.12) for $k=0$,
$$
\begin{equation}
\|u\|_{L_2(M_\varepsilon;\mathbb{C}^m)}^2 \leqslant C \varepsilon^2 c_2^{2N(\varepsilon)} \|u\|_{W_2^1(\Omega_\varepsilon;\mathbb{C}^m)}^2,
\end{equation}
\tag{2.16}
$$
where $C$ is a constant independent of $u$ and $\varepsilon$.
Now we extend the above estimates to the sets $T^z_\varepsilon$ defined in (1.14). By the definition of the cutoff function $\chi$ and estimates (1.15) we have
$$
\begin{equation*}
u(x)=\int_{0}^{\rho} \frac{\partial\ }{\partial t} u(\xi+t\nu^z_\varepsilon(\xi)) \biggl(1-\chi\biggl(\frac{\rho}{c_3\eta_k(\varepsilon)}\biggr)\biggr)\,dt
\end{equation*}
\notag
$$
for all $x\in T^z_\varepsilon$ and $\rho>c_3$. From the Cauchy–Bunyakovsky–Schwarz inequality, the assumptions of boundedness of the derivatives of $x$ with respect to $(\xi,\rho)$ and $(\xi,\rho)$ with respect $x$ in condition (C) and estimates (1.15) we infer the inequality
$$
\begin{equation*}
\begin{aligned} \, |u(x)|^2 &\leqslant C\eta_k(\varepsilon) \int_{0}^{\eta_k(\varepsilon)\phi_z(\xi,\varepsilon)} |\nabla_x u(\xi+\rho\nu^z_\varepsilon(\xi))|^2\,d\rho \\ &\qquad + C\eta_k^{-1}(\varepsilon) \int_{0}^{\eta_k(\varepsilon)\phi_z(\xi,\varepsilon)} |u(\xi+\rho\nu^z_\varepsilon(\xi))|^2\,d\rho, \end{aligned}
\end{equation*}
\notag
$$
where the constant $C$ is independent of $u$, $\varepsilon$, $k$, $z$ and the space variables. Integrating this estimate over $T^z_\varepsilon$ and taking account again of the boundedness of the derivatives of $x$ with respect to $(\xi,\rho)$ and $(\xi,\rho)$ with respect to $x$, we arrive at the inequality
$$
\begin{equation*}
\|u\|_{L_2(T^z_\varepsilon)}^2\leqslant C \eta_k^2(\varepsilon) \|\nabla u\|_{L_2(T^z_\varepsilon)}^2 + C \|u\|_{L_2(T^z_\varepsilon\cap (z+K_{\eta_k(\varepsilon)}))}^2,
\end{equation*}
\notag
$$
where the constant $C$ is independent of $u$, $\varepsilon$, $z$ and $k$. Summing these estimates over $z\in L^\partial_\varepsilon$ and using (2.16) and (1.12), we conclude that
$$
\begin{equation*}
\begin{aligned} \, \|u\|_{L_2(\Pi_{\frac{4}{3}\varepsilon}\cap\Omega_\varepsilon \setminus\overline{M_\varepsilon})} &\leqslant C \varepsilon^2 \|\nabla u\|_{L_2(M_\varepsilon)}^2 + C \|u\|_{L_2(M_\varepsilon)}^2 \\ &\leqslant C\varepsilon^2 c_2^{2N(\varepsilon)} \|u\|_{W_2^1(\Pi_{\frac{3}{2}\varepsilon};\mathbb{C}^m)}^2, \end{aligned}
\end{equation*}
\notag
$$
where the constant $C$ is independent of $u$ and $\varepsilon$. This estimate and (2.16) imply the assertion of the lemma.
Lemma 2.4 is proved. § 3. Operator estimates for the Dirichlet problemIn this section we prove Theorem 1.1. The solvability of problems (1.5), (1.6) and (1.10), (1.11) was established in Lemma 2.1. Therefore, we assume in what follows that $f$ is an arbitrary function in $L_2(\Omega;\mathbb{C}^m)$ and the parameter $\lambda$ is chosen so that $\operatorname{Re}\lambda\leqslant \lambda_0$ and is fixed. Then problems (1.5), (1.6) and (1.10), (1.11) are uniquely solvable; let $u_\varepsilon$ and $u_0$ be their solutions. We prove the operator estimates (1.18) and (1.20) in separate subsections. 3.1. $W_2^1$-estimateIn this subsection we prove (1.18). We set
$$
\begin{equation*}
\chi_\varepsilon(x):= \begin{cases} 1-\chi\biggl(\dfrac{\tau}{3\varepsilon}\biggr) &\text{in } \Pi_{2\varepsilon}, \\ 1 & \text{in } \Omega\setminus\Pi_{2\varepsilon}, \end{cases}
\end{equation*}
\notag
$$
where $\chi$ is the cutoff function introduced before Lemma 2.2. The above function is clearly identically equal to zero on $\Pi_\varepsilon$ and infinitely differentiable in $\overline{\Omega}$ and satisfies the relations
$$
\begin{equation}
\operatorname{supp}\nabla\chi_\varepsilon\subseteq \overline{\Pi_{2\varepsilon}\setminus\Pi_\varepsilon}\quad\text{and}\quad |\nabla\chi_\varepsilon(x)|\leqslant C\varepsilon^{-1}, \quad x\in\Pi_{2\varepsilon}\setminus\Pi_\varepsilon,
\end{equation}
\tag{3.1}
$$
where $C$ is a constant independent of $x$ and $\varepsilon$. In view of the indicated properties, for an arbitrary $v\in \mathring{W}_2^1(\Omega_\varepsilon,\Gamma;\mathbb{C}^m)$ the function $\chi_\varepsilon v$ is an element in the space $\mathring{W}_2^1(\Omega_\varepsilon,\Gamma_\varepsilon;\mathbb{C}^m)$. Taking account of this fact, we conclude that the function $v_\varepsilon:=u_\varepsilon-\chi_\varepsilon u_0$ is in $\mathring{W}_2^1(\Omega_\varepsilon,\partial\Omega_\varepsilon;\mathbb{C}^m)$, while the function $\chi_\varepsilon v_\varepsilon$ extended by zero to $\Omega\setminus\Omega_\varepsilon$ is an element of $\mathring{W}_2^1(\Omega,\partial\Omega)$. Now we write the integral identity (1.8) with test function $v_\varepsilon$ and integral identity (1.17) for problem (1.10), (1.11) with test function $\chi_\varepsilon v_\varepsilon$:
$$
\begin{equation}
\mathfrak{h}^D_\varepsilon(u_\varepsilon, v_\varepsilon)- \lambda (u_\varepsilon, v_\varepsilon)_{L_2(\Omega_\varepsilon;\mathbb{C}^m)}=(f, v_\varepsilon)_{L_2(\Omega_\varepsilon;\mathbb{C}^m)},
\end{equation}
\tag{3.2}
$$
$$
\begin{equation}
\mathfrak{h}_0^D(u_0,\chi_\varepsilon v_\varepsilon)- \lambda (u_0,\chi_\varepsilon v_\varepsilon)_{L_2(\Omega;\mathbb{C}^m)}=(f, \chi_\varepsilon v_\varepsilon)_{L_2(\Omega;\mathbb{C}^m)}.
\end{equation}
\tag{3.3}
$$
By the definition of the function $\chi_\varepsilon$ and its properties (3.1) we can write the first term on the left-hand side of the second identity in the form
$$
\begin{equation}
\mathfrak{h}_0^D(u_0,\chi_\varepsilon v_\varepsilon)=\mathfrak{h}^D_\varepsilon(\chi_\varepsilon u_0, v_\varepsilon)+ \mathfrak{g}_\varepsilon(u_0,v_\varepsilon),
\end{equation}
\tag{3.4}
$$
where
$$
\begin{equation*}
\begin{aligned} \, &\mathfrak{g}_\varepsilon(u,v) :=\sum_{i,j=1}^{n} \biggl(A_{ij}\,\frac{\partial u}{\partial x_j},\frac{\partial \chi_\varepsilon}{\partial x_i}v\biggr)_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon;\mathbb{C}^m)} \\ &\qquad - \sum_{i,j=1}^{n} \biggl(A_{ij}\,\frac{\partial \chi_\varepsilon}{\partial x_j}u,\frac{\partial v}{\partial x_i}\biggr)_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon;\mathbb{C}^m)} -\sum_{j=1}^{2} \biggl(A_j\,\frac{\partial\chi_\varepsilon}{\partial x_j}u,v\biggr)_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon;\mathbb{C}^m)} \\ &\qquad-\bigl(A_0(\,\cdot\,,\chi_\varepsilon u)-\chi_\varepsilon A_0(\,\cdot\,,u), v\bigr)_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon;\mathbb{C}^m)}. \end{aligned}
\end{equation*}
\notag
$$
We substitute (3.4) into (3.3) and then subtract the resulting identity from (3.2):
$$
\begin{equation}
\begin{aligned} \, &\mathfrak{h}^D_\varepsilon(u_\varepsilon,v_\varepsilon)- \mathfrak{h}^D_\varepsilon(u_0\chi_\varepsilon,v_\varepsilon) -\lambda \|v_\varepsilon\|_{L_2(\Omega;\mathbb{C}^m)}^2 \nonumber \\ &\qquad= (f,(1-\chi_\varepsilon)v_\varepsilon)_{L_2(\Omega;\mathbb{C}^m)} +\mathfrak{g}_\varepsilon(u_0,v_\varepsilon). \end{aligned}
\end{equation}
\tag{3.5}
$$
The left-hand side of this identity satisfies (2.1), that is,
$$
\begin{equation}
\operatorname{Re} \bigl(\mathfrak{h}^D_\varepsilon(u_\varepsilon,v_\varepsilon)- \mathfrak{h}^D_\varepsilon(u_0\chi_\varepsilon,v_\varepsilon) -\lambda \|v_\varepsilon\|_{L_2(\Omega;\mathbb{C}^m)}^2 \bigr) \geqslant C \|v_\varepsilon\|_{W_2^1(\Omega_\varepsilon;\mathbb{C}^m)}^2.
\end{equation}
\tag{3.6}
$$
Throughout the rest of the section we let $C$ denote different constants independent of $x$, $\varepsilon$, $u_0$, $v_\varepsilon$ and $f$. In view of (3.6) our next step is to estimate the right-hand side of (3.5), which will make it possible to estimate the $W_2^1(\Omega_\varepsilon;\mathbb{C}^m)$-norm of $v_\varepsilon$. The first term on the right-hand side can be estimated on the basis of (2.6) for $u=v_\varepsilon$ and the definition of the function $\chi_\varepsilon$:
$$
\begin{equation}
\begin{aligned} \, \notag &\bigl|(f,(1-\chi_\varepsilon)v_\varepsilon)_{L_2(\Omega;\mathbb{C}^m)}\bigr| \\ \notag &\qquad =\bigl|(f,(1-\chi_\varepsilon)v_\varepsilon)_{L_2(\Pi_{2\varepsilon};\mathbb{C}^m)}\bigr| \leqslant \|f\|_{L_2(\Pi_{2\varepsilon};\mathbb{C}^m)} \|v_\varepsilon\|_{L_2(\Pi_{2\varepsilon};\mathbb{C}^m)} \\ &\qquad\leqslant C \varepsilon \|f\|_{L_2(\Pi_{2\varepsilon};\mathbb{C}^m)} \|\nabla v_\varepsilon\|_{L_2(\Pi_{2\varepsilon};\mathbb{C}^m)}. \end{aligned}
\end{equation}
\tag{3.7}
$$
Using (2.7) for $u=u_0$, (2.8) for $u=\nabla u_0$, (2.6) for $u=v_\varepsilon$ and the inequality for $\nabla\chi_\varepsilon$ in (3.1), we obtain estimates for the first three terms in $\mathfrak{g}_\varepsilon(u_0,v_\varepsilon)$:
$$
\begin{equation}
\begin{aligned} \, &\biggl|\sum_{i,j=1}^{n} \biggl(A_{ij}\,\frac{\partial u_0}{\partial x_j},\frac{\partial \chi_\varepsilon}{\partial x_i}v_\varepsilon\biggr)_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon;\mathbb{C}^m)}\biggr| \\ &\qquad\leqslant C \varepsilon^{-1}\|\nabla u_0\|_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon;\mathbb{C}^m)} \|v_\varepsilon\|_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon;\mathbb{C}^m)} \\ &\qquad\leqslant C \varepsilon^{1/2}\|u_0\|_{W_2^2(\Omega;\mathbb{C}^m)} \|\nabla v_\varepsilon\|_{L_2(\Omega_\varepsilon\cap\Pi_{2\varepsilon};\mathbb{C}^m)}, \\ &\biggl|\sum_{i,j=1}^{n} \biggl(A_{ij}\,\frac{\partial \chi_\varepsilon}{\partial x_j}u_0,\frac{\partial v_\varepsilon}{\partial x_i}\biggr)_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon;\mathbb{C}^m)}\biggr| \\ &\qquad\leqslant C \varepsilon^{-1}\|u_0\|_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon;\mathbb{C}^m)} \|\nabla v_\varepsilon\|_{L_2(\Pi_{2\varepsilon} \setminus\Pi_\varepsilon;\mathbb{C}^m)} \\ &\qquad\leqslant C \varepsilon^{1/2}\|u_0\|_{W_2^2(\Omega;\mathbb{C}^m)} \|\nabla v_\varepsilon\|_{L_2(\Omega_\varepsilon\cap\Pi_{2\varepsilon};\mathbb{C}^m)}, \\ &\biggl|\sum_{j=1}^{2} \biggl(A_j\,\frac{\partial\chi_\varepsilon}{\partial x_j}u,v\biggr)_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon;\mathbb{C}^m)}\biggr| \\ &\qquad\leqslant C \varepsilon^{-1} \|u_0\|_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon;\mathbb{C}^m)} \|v_\varepsilon\|_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon;\mathbb{C}^m)} \\ &\qquad\leqslant C \varepsilon \|u_0\|_{W_2^2(\Omega;\mathbb{C}^m)} \|\nabla v_\varepsilon\|_{L_2(\Omega_\varepsilon\cap\Pi_{2\varepsilon};\mathbb{C}^m)}. \end{aligned}
\end{equation}
\tag{3.8}
$$
It follows from condition (1.3) and the definition of $\chi_\varepsilon$ that
$$
\begin{equation}
\|A_0(\,\cdot\,,u)\|_{\mathbb{M}_m}\leqslant c_1 \|u\|_{\mathbb{C}^m}.
\end{equation}
\tag{3.9}
$$
This estimate, the definition of $\chi_\varepsilon$, inequality (2.7) for $u=u_0$ and (2.6) for $u=v_\varepsilon$ make it possible to estimate the remaining term in $\mathfrak{g}_\varepsilon(u_0,v_\varepsilon)$ in the following way:
$$
\begin{equation}
\begin{aligned} \, \notag &\bigl|\bigl(\chi_\varepsilon A_0(\,\cdot\,,u_0)-A_0(\,\cdot\,,\chi_\varepsilon u_0), v_\varepsilon\bigr)_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon;\mathbb{C}^m)}\bigr| \nonumber \\ &\qquad\leqslant 2c_1\|u_0\|_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon;\mathbb{C}^m)} \|v_\varepsilon\|_{L_2(\Omega\setminus\Omega_\varepsilon;\mathbb{C}^m)} \nonumber \\ &\qquad\leqslant C\varepsilon^2 \|u_0\|_{W_2^2(\Omega;\mathbb{C}^m)} \|\nabla v_\varepsilon\|_{L_2(\Omega_\varepsilon\cap\Pi_{2\varepsilon};\mathbb{C}^m)}. \end{aligned}
\end{equation}
\tag{3.10}
$$
This estimate, (3.8), (3.7), and (2.2) yield the final estimate for the right-hand side of (3.5):
$$
\begin{equation*}
\begin{aligned} \, &\bigl|(f,(1-\chi_\varepsilon)v_\varepsilon)_{L_2(\Omega;\mathbb{C}^m)} +\mathfrak{g}_\varepsilon(u_0,v_\varepsilon)\bigr| \\ &\qquad\leqslant C \varepsilon^{1/2} \bigl(\|f\|_{L_2(\Pi_{2\varepsilon};\mathbb{C}^m)} + \|u_0\|_{W_2^2(\Omega;\mathbb{C}^m)}\bigr) \|\nabla v_\varepsilon\|_{L_2(\Omega_\varepsilon\cap\Pi_{2\varepsilon};\mathbb{C}^m)} \\ &\qquad\leqslant C \varepsilon^{1/2} \|f\|_{L_2(\Omega;\mathbb{C}^m)}\|\nabla v_\varepsilon\|_{L_2(\Omega_\varepsilon\cap\Pi_{2\varepsilon};\mathbb{C}^m)}. \end{aligned}
\end{equation*}
\notag
$$
In view of (3.5) and (3.6) it now follows that
$$
\begin{equation}
\|v_\varepsilon\|_{W_2^1(\Omega_\varepsilon;\mathbb{C}^m)}\leqslant C \varepsilon^{1/2} \|f\|_{L_2(\Omega;\mathbb{C}^m)}.
\end{equation}
\tag{3.11}
$$
We also note that
$$
\begin{equation}
u_\varepsilon-u_0=v_\varepsilon-(1-\chi_\varepsilon)u_0.
\end{equation}
\tag{3.12}
$$
Using (2.7) for $u=u_0$, (2.8) for $u=\nabla u_0$, (2.2) and the estimate for $\nabla\chi_\varepsilon$ in (3.1) we verify that
$$
\begin{equation}
\begin{aligned} \, \notag \|(1-\chi_\varepsilon)u_0\|_{L_2(\Omega;\mathbb{C}^m)} &=\|(1-\chi_\varepsilon)u_0\|_{L_2(\Pi_{2\varepsilon};\mathbb{C}^m)} \\ &\leqslant C\varepsilon^{3/2}\|u_0\|_{W_2^2(\Omega;\mathbb{C}^m)}\leqslant C\varepsilon^{3/2}\|f\|_{L_2(\Omega;\mathbb{C}^m)}, \\ \notag \|\nabla(1-\chi_\varepsilon)u_0\|_{L_2(\Omega;\mathbb{C}^m)} &\leqslant \|(1-\chi_\varepsilon)\nabla u_0\|_{L_2(\Omega;\mathbb{C}^m)}+ \|u_0 \nabla \chi_\varepsilon\|_{L_2(\Omega;\mathbb{C}^m)} \\ &\leqslant C\varepsilon^{1/2}\|f\|_{L_2(\Omega;\mathbb{C}^m)}. \notag \end{aligned}
\end{equation}
\tag{3.13}
$$
Taking account of (3.12) and (3.11), we derive from these estimates that (1.18) holds. 3.2. $L_2$-estimateIn this subsection we prove estimate (1.20). With this aim in view, we follow the approach based on duality (see, for example, [25]–[29]) with a slight modification proposed in [19]. We have managed to extend this approach to the case of weakly nonlinear equations under consideration. We extend the function $v_\varepsilon$ by zero to $\Omega\setminus\Omega_\varepsilon$, keeping the same notation for the extension. Since the function $v_\varepsilon$ has zero trace on $\Gamma_\varepsilon$, this extension is an element of the space $\mathring{W}_2^1(\Omega,\partial\Omega;\mathbb{C}^m)$. We consider the auxiliary problem
$$
\begin{equation}
\widehat{\mathcal{L}} w-\overline{\lambda} w =v_\varepsilon \quad\text{in } \Omega, \qquad w=0 \quad\text{on }\partial\Omega,
\end{equation}
\tag{3.14}
$$
where
$$
\begin{equation*}
\widehat{\mathcal{L}}:= -\sum_{i,j=1}^{n} \frac{\partial\ }{\partial x_j} A_{ij}^*\,\frac{\partial\ }{\partial x_i} - \sum_{j=1}^{n} \frac{\partial\ }{\partial x_j}A_j^*.
\end{equation*}
\notag
$$
This problem is of the same type as (1.10), (1.11); thus, the relevant assertions in Lemma 2.1 are valid for it. Without loss of generality we assume that $\lambda_0$ is such that problem (3.14) is also solvable. By (2.2) we have the analogous estimates
$$
\begin{equation}
\begin{gathered} \, \|w\|_{W_2^2(\Omega;\mathbb{C}^m)}\leqslant C_1\|v_\varepsilon\|_{L_2(\Omega_\varepsilon;\mathbb{C}^m)}, \\ \|w\|_{L_2(\Omega;\mathbb{C}^m)}\leqslant \frac{C_2}{|\operatorname{Re}\lambda+C_3|}\|v_\varepsilon\|_{L_2(\Omega_\varepsilon;\mathbb{C}^m)}, \end{gathered}
\end{equation}
\tag{3.15}
$$
where the constants $C_1$, $C_2$ and $C_3$ are independent of $v_\varepsilon$ and $\lambda$, and $C_3<-\lambda_0$. The integral identity for problem (3.14) with test function $\chi_\varepsilon v_\varepsilon$ has the form
$$
\begin{equation*}
\begin{aligned} \, &\sum_{i,j=1}^{n} \biggl(\frac{\partial \chi_\varepsilon v_\varepsilon}{\partial x_j},A_{ij}^*\,\frac{\partial w}{\partial x_i}\biggr)_{L_2(\Omega;\mathbb{C}^m)} - \sum_{j=1}^{n} \biggl(\chi_\varepsilon v_\varepsilon,\frac{\partial\ }{\partial x_j} A_j^* w\biggr)_{L_2(\Omega;\mathbb{C}^m)} \\ &\qquad\qquad - \lambda (\chi_\varepsilon v_\varepsilon, w)_{L_2(\Omega;\mathbb{C}^m)}=(\chi_\varepsilon v_\varepsilon,v_\varepsilon)_{L_2(\Omega;\mathbb{C}^m)}. \end{aligned}
\end{equation*}
\notag
$$
Using the equality $v_\varepsilon=0$ in $\Omega\setminus\Omega_\varepsilon$ and integrating by parts once in the second term on the left-hand side of this integral identity, we obtain
$$
\begin{equation*}
\begin{aligned} \, &(\chi_\varepsilon v_\varepsilon,v_\varepsilon)_{L_2(\Omega;\mathbb{C}^m)} \\ &\qquad =\sum_{i,j=1}^{n} \biggl(A_{ij}\,\frac{\partial \chi_\varepsilon v_\varepsilon}{\partial x_j},\frac{\partial w}{\partial x_i}\biggr)_{L_2(\Omega_\varepsilon;\mathbb{C}^m)} +\sum_{j=1}^{n} \biggl(A_j \,\frac{\partial \chi_\varepsilon v_\varepsilon}{\partial x_j}, w\biggr)_{L_2(\Omega_\varepsilon;\mathbb{C}^m)} \\ &\qquad\qquad - \lambda (\chi_\varepsilon v_\varepsilon,w)_{L_2(\Omega_\varepsilon;\mathbb{C}^m)} \\ &\qquad=\sum_{i,j=1}^{n} \biggl(A_{ij}\,\frac{\partial \chi_\varepsilon}{\partial x_j}v_\varepsilon,\frac{\partial w}{\partial x_i}\biggr)_{L_2(\Omega_\varepsilon;\mathbb{C}^m)} -\sum_{i,j=1}^{n} \biggl(A_{ij}\,\frac{\partial v_\varepsilon}{\partial x_j},\frac{\partial \chi_\varepsilon}{\partial x_i}w\biggr)_{L_2(\Omega_\varepsilon;\mathbb{C}^m)} \\ &\qquad\qquad +\sum_{j=1}^{n} \biggl(A_j\, \frac{\partial \chi_\varepsilon}{\partial x_j}v_\varepsilon, w\biggr)_{L_2(\Omega_\varepsilon;\mathbb{C}^m)} +\sum_{i,j=1}^{n} \biggl(A_{ij}\,\frac{\partial v_\varepsilon}{\partial x_j},\frac{\partial \chi_\varepsilon w}{\partial x_i}\biggr)_{L_2(\Omega_\varepsilon;\mathbb{C}^m)} \\ &\qquad\qquad +\sum_{j=1}^{n} \biggl(A_j \,\frac{\partial v_\varepsilon}{\partial x_j}, \chi_\varepsilon w\biggr)_{L_2(\Omega_\varepsilon;\mathbb{C}^m)} - \lambda ( v_\varepsilon,\chi_\varepsilon w)_{L_2(\Omega_\varepsilon;\mathbb{C}^m)}. \end{aligned}
\end{equation*}
\notag
$$
In view of the definitions of the form $\mathfrak{h}^D_\varepsilon$ and the cutoff function $\chi_\varepsilon$ it follows that
$$
\begin{equation}
\begin{aligned} \, \notag &(\chi_\varepsilon v_\varepsilon,v_\varepsilon)_{L_2(\Omega;\mathbb{C}^m)} \\ \notag &\ =\mathfrak{h}^D_\varepsilon(u_\varepsilon,\chi_\varepsilon w) - \mathfrak{h}^D_\varepsilon(\chi_\varepsilon u_0,\chi_\varepsilon w) -\lambda ( v_\varepsilon,\chi_\varepsilon w)_{L_2(\Omega_\varepsilon;\mathbb{C}^m)} \\ \notag &\ \quad +\sum_{i,j=1}^{n} \biggl(A_{ij}\,\frac{\partial \chi_\varepsilon}{\partial x_j}v_\varepsilon,\frac{\partial w}{\partial x_i}\biggr)_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon;\mathbb{C}^m)} \\ \notag &\ \quad - \sum_{i,j=1}^{n} \biggl(A_{ij}\,\frac{\partial v_\varepsilon}{\partial x_j},\frac{\partial \chi_\varepsilon}{\partial x_i} w\biggr)_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon;\mathbb{C}^m)} +\sum_{j=1}^{n} \biggl(A_j \,\frac{\partial \chi_\varepsilon}{\partial x_j}v_\varepsilon, w\biggr)_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon;\mathbb{C}^m)} \\ &\ \quad - \bigl(A_0(\,\cdot\,, u_\varepsilon)-A_0(\,\cdot\,, \chi_\varepsilon u_0),\chi_\varepsilon w\bigr)_{L_2(\Omega_\varepsilon;\mathbb{C}^m)}. \end{aligned}
\end{equation}
\tag{3.16}
$$
As in the above derivation of (3.5), we write the integral identity (1.8) with test function $\chi_\varepsilon w$ and the integral identity (1.17) for problem (1.10), (1.11) with test function $\chi_\varepsilon^2 w$. Then we use formula (3.4) with $v_\varepsilon$ replaced by $\chi_\varepsilon w$ and then subtract the resulting identities one from the other. We arrive at the relation
$$
\begin{equation*}
\begin{aligned} \, &\mathfrak{h}^D_\varepsilon(u_\varepsilon,\chi_\varepsilon w)-\mathfrak{h}^D_\varepsilon(\chi_\varepsilon u_0,\chi_\varepsilon w)-\lambda(v_\varepsilon,\chi_\varepsilon w)_{L_2(\Omega_\varepsilon;\mathbb{C}^m)} \\ &\qquad=\bigl(f,\chi_\varepsilon(1-\chi_\varepsilon)w\bigr)_{L_2(\Pi_{2\varepsilon};\mathbb{C}^m)} + \mathfrak{g}_\varepsilon(u_0,\chi_\varepsilon w). \end{aligned}
\end{equation*}
\notag
$$
Taking account of (3.16) we derive from it the equality
$$
\begin{equation}
\begin{aligned} \, \notag &\|v_\varepsilon\|_{L_2(\Omega;\mathbb{C}^m)}^2+ \bigl(A_0(\,\cdot\,,u_\varepsilon)- A_0(\,\cdot\,,\chi_\varepsilon u_0), \chi_\varepsilon w\bigr)_{L_2(\Omega_\varepsilon;\mathbb{C}^m)} \\ &\qquad =\bigl((1-\chi_\varepsilon) v_\varepsilon,v_\varepsilon\bigr)_{L_2(\Omega_\varepsilon\cap\Pi_{2\varepsilon};\mathbb{C}^m)} +\bigl(f,\chi_\varepsilon(1-\chi_\varepsilon)w\bigr)_{L_2(\Pi_{2\varepsilon};\mathbb{C}^m)} + \mathfrak{r}_\varepsilon^{1}+\mathfrak{r}_\varepsilon^{2}, \end{aligned}
\end{equation}
\tag{3.17}
$$
where
$$
\begin{equation*}
\begin{aligned} \, \mathfrak{r}_\varepsilon^{1} &:=\sum_{i,j=1}^{n} \biggl(A_{ij}\,\frac{\partial \chi_\varepsilon}{\partial x_j}v_\varepsilon,\frac{\partial w}{\partial x_i}\biggr)_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon;\mathbb{C}^m)} - \sum_{i,j=1}^{n} \biggl(A_{ij}\,\frac{\partial v_\varepsilon}{\partial x_j},\frac{\partial \chi_\varepsilon}{\partial x_i} w\biggr)_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon;\mathbb{C}^m)} \\ &\qquad +\sum_{j=1}^{n} \biggl(A_j\, \frac{\partial \chi_\varepsilon}{\partial x_j}v_\varepsilon, w\biggr)_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon;\mathbb{C}^m)} \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, \mathfrak{r}_\varepsilon^{2} &:=\sum_{i,j=1}^{n} \biggl(A_{ij}\,\frac{\partial u_0}{\partial x_j},\frac{\partial \chi_\varepsilon}{\partial x_i}\chi_\varepsilon w\biggr)_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon;\mathbb{C}^m)} \\ &\qquad - \sum_{i,j=1}^{n} \biggl(A_{ij}\,\frac{\partial \chi_\varepsilon}{\partial x_j}u_0,\frac{\partial \chi_\varepsilon w}{\partial x_i}\biggr)_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon;\mathbb{C}^m)} \\ &\qquad -\sum_{j=1}^{2} \biggl(A_j\,\frac{\partial\chi_\varepsilon}{\partial x_j}u_0,\chi_\varepsilon w\biggr)_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon;\mathbb{C}^m)} \\ &\qquad -\bigl(A_0(\,\cdot\,,\chi_\varepsilon u_0)-\chi_\varepsilon A_0(\,\cdot\,,u_0), \chi_\varepsilon w\bigr)_{L_2(\Pi_{2\varepsilon}\setminus\Pi_\varepsilon;\mathbb{C}^m)}. \end{aligned}
\end{equation*}
\notag
$$
We estimate the right-hand side of the above equality. We infer from (2.7) for $u=w$ and the first inequality in (3.15) that
$$
\begin{equation}
\begin{aligned} \, \notag \bigl|\bigl(f,\chi_\varepsilon(1-\chi_\varepsilon)w\bigr)_{L_2(\Pi_{2\varepsilon}; \mathbb{C}^m)}\bigr| &\leqslant C\varepsilon^{3/2} \|f\|_{L_2(\Pi_{2\varepsilon};\mathbb{C}^m)} \|w\|_{W_2^2(\Omega;\mathbb{C}^m)} \\ &\leqslant C\varepsilon^{3/2} \|f\|_{L_2(\Pi_{2\varepsilon};\mathbb{C}^m)} \|v_\varepsilon\|_{W_2^2(\Omega_\varepsilon;\mathbb{C}^m)}. \end{aligned}
\end{equation}
\tag{3.18}
$$
Owing to (2.6) for $u=v_\varepsilon$ and (3.11), we have
$$
\begin{equation}
\begin{aligned} \, \notag &\bigl|\bigl((\chi_\varepsilon-1)v_\varepsilon, v_\varepsilon\bigr)_{L_2(\Omega_\varepsilon\cap\Pi_{2\varepsilon};\mathbb{C}^m)}\bigr| \\ \notag &\qquad\leqslant \|v_\varepsilon\|_{L_2(\Omega_\varepsilon\cap\Pi_{2\varepsilon};\mathbb{C}^m)} \|v_\varepsilon\|_{L_2(\Omega_\varepsilon;\mathbb{C}^m)} \leqslant C\varepsilon\|\nabla v_\varepsilon\|_{L_2(\Omega_\varepsilon\cap\Pi_{2\varepsilon};\mathbb{C}^m)} \|v_\varepsilon\|_{L_2(\Omega_\varepsilon;\mathbb{C}^m)} \\ &\qquad \leqslant C\varepsilon^{3/2}\|f\|_{L_2(\Omega;\mathbb{C}^m)}\|v_\varepsilon \|_{L_2(\Omega_\varepsilon;\mathbb{C}^m)}. \end{aligned}
\end{equation}
\tag{3.19}
$$
The function $\mathfrak{r}_\varepsilon^{1}$ is estimated on the basis of (2.6) for $u=v_\varepsilon$, (2.7) for $u=w$, (2.8) for $u={\partial w}/{\partial x_j}$ and relations (3.1), (3.15), and (3.11):
$$
\begin{equation}
|\mathfrak{r}_\varepsilon^1|\leqslant C\varepsilon^{1/2}\|v_\varepsilon\|_{W_2^1(\Omega_\varepsilon;\mathbb{C}^m)} \|w\|_{W_2^2(\Omega;\mathbb{C}^m)}\leqslant C \varepsilon \|f\|_{L_2(\Omega;\mathbb{C}^m)} \|v_\varepsilon\|_{L_2(\Omega_\varepsilon;\mathbb{C}^m)}.
\end{equation}
\tag{3.20}
$$
The first three terms in $\mathfrak{r}_\varepsilon^{2}$ are estimated similarly to (3.8), where it is necessary to use (2.7) for $u=w$ instead of (2.6) for $u=v_\varepsilon$ and write the derivative ${\partial \chi_\varepsilon w}/{\partial x_i}$ explicitly. The fourth term is estimated on the basis of (3.10) with $v_\varepsilon$ replaced by $\chi_\varepsilon w$. The resulting estimate is
$$
\begin{equation*}
|\mathfrak{r}_\varepsilon^2|\leqslant C\varepsilon \|u_0\|_{W_2^2(\Omega;\mathbb{C}^m)}\|w\|_{W_2^2(\Omega;\mathbb{C}^m)}\leqslant C \varepsilon \|f\|_{L_2(\Omega;\mathbb{C}^m)} \|v_\varepsilon\|_{L_2(\Omega_\varepsilon;\mathbb{C}^m)}.
\end{equation*}
\notag
$$
Taking account of (3.20), (3.19), (3.18), (3.15) and (2.2) we deduce the final estimate for the right-hand side of (3.17):
$$
\begin{equation}
\begin{aligned} \, \notag &\bigl|\bigl((1-\chi_\varepsilon) v_\varepsilon,v_\varepsilon\bigr)_{L_2(\Omega;\mathbb{C}^m)} +\bigl(f,\chi_\varepsilon(1-\chi_\varepsilon)w\bigr)_{L_2(\Pi_{2\varepsilon};\mathbb{C}^m)} + \mathfrak{r}_\varepsilon^{1}+\mathfrak{r}_\varepsilon^{2}\bigr| \\ &\qquad \leqslant C\varepsilon\|f\|_{L_2(\Omega;\mathbb{C}^m)} \|v_\varepsilon\|_{L_2(\Omega_\varepsilon;\mathbb{C}^m)}. \end{aligned}
\end{equation}
\tag{3.21}
$$
Now we estimate the second term on the left-hand side of (3.17) on the basis of the Lipschitz inequality in (1.3) and the second inequality in (3.15):
$$
\begin{equation*}
\begin{aligned} \, &\bigl|\bigl( A_0(\,\cdot\,,u_\varepsilon)-A_0(\,\cdot\,,\chi_\varepsilon u_0), \chi_\varepsilon w\bigr)_{L_2(\Omega_\varepsilon;\mathbb{C}^m)}\bigr| \\ &\qquad \leqslant c_1\|v_\varepsilon\|_{L_2(\Omega_\varepsilon;\mathbb{C}^m)} \|w\|_{L_2(\Omega_\varepsilon;\mathbb{C}^m)} \leqslant \frac{c_1 C_2}{|\operatorname{Re}\lambda+C_3|} \|v_\varepsilon\|_{L_2(\Omega_\varepsilon;\mathbb{C}^m)}^2. \end{aligned}
\end{equation*}
\notag
$$
Choosing $\lambda_0$ to be negative and sufficiently small so that $\operatorname{Re}\lambda$ is sufficiently large in absolute value, for the left-hand side of (3.17) we obtain
$$
\begin{equation*}
\bigl|\|v_\varepsilon\|_{L_2(\Omega_\varepsilon;\mathbb{C}^m)}^2 +\bigl( A_0(\,\cdot\,,u_\varepsilon)-A_0(\,\cdot\,,\chi_\varepsilon u_0), \chi_\varepsilon w\bigr)_{L_2(\Omega_\varepsilon;\mathbb{C}^m)}\bigr| \geqslant \frac{1}{2}\|v_\varepsilon\|_{L_2(\Omega_\varepsilon;\mathbb{C}^m)}^2.
\end{equation*}
\notag
$$
In view of (3.21) and (3.17) it follows that
$$
\begin{equation*}
\|v_\varepsilon\|_{L_2(\Omega_\varepsilon;\mathbb{C}^m)}\leqslant C \varepsilon \|f\|_{L_2(\Omega;\mathbb{C}^m)}.
\end{equation*}
\notag
$$
Using now (3.12) and estimate (3.13) we arrive at (1.20). Theorem 1.1 is proved. § 4. Operator estimates for the Neumann problemIn this section we prove Theorem 1.2. The solvability of problems (1.5), (1.7) and (1.10), (1.16) was already proved in Lemma 2.1. We assume that $f$ is an arbitrary function in $L_2(\Omega;\mathbb{C}^m)$, the parameter $\lambda$ is fixed so that $\operatorname{Re}\lambda\leqslant \lambda_0$ and $u_\varepsilon$ and $u_0$ are the corresponding unique solutions of problems (1.5), (1.7) and (1.10), (1.16), respectively. We introduce the notation $v_\varepsilon:=u_\varepsilon-u_0$. The estimates (1.21) and (1.22) are proved in the following two subsections. 4.1. $W_2^1$-estimateWe take the inner product in $L_2(\Omega_\varepsilon;\mathbb{C}^m)$ of the equation for $u_0$ and $v_\varepsilon$ and integrate by parts:
$$
\begin{equation}
\mathfrak{h}^N_\varepsilon(u_0,v_\varepsilon) -\lambda(u_0,v_\varepsilon)_{L_2(\Omega_\varepsilon;\mathbb{C}^m)}+\biggl(\frac{\partial u_0}{\partial\boldsymbol{\nu}_\varepsilon}, v_\varepsilon\biggr)_{L_2(\Gamma_\varepsilon;\mathbb{C}^m)}= (f,v_\varepsilon)_{L_2(\Omega_\varepsilon;\mathbb{C}^m)}.
\end{equation}
\tag{4.1}
$$
We subtract this identity from (1.9) for $v=v_\varepsilon$ and obtain
$$
\begin{equation}
\mathfrak{h}^N_\varepsilon(u_\varepsilon,v_\varepsilon) -\mathfrak{h}^N_\varepsilon(u_0,v_\varepsilon) -\lambda\|v_\varepsilon\|_{L_2(\Omega_\varepsilon;\mathbb{C}^m)}^2=\biggl(\frac{\partial u_0}{\partial\boldsymbol{\nu}_\varepsilon}, v_\varepsilon\biggr)_{L_2(\Gamma_\varepsilon;\mathbb{C}^m)}.
\end{equation}
\tag{4.2}
$$
We introduce the auxiliary function
$$
\begin{equation*}
\widetilde{v}_\varepsilon(s,\tau):= \begin{cases} v_\varepsilon(s,\tau)&\text{in } \Pi_{3\varepsilon}\setminus \Pi_{\frac{3}{2}\varepsilon}, \\ v_\varepsilon(s,3-\tau) & \text{in } x\in\Pi_{\frac{3}{2}\varepsilon}. \end{cases}
\end{equation*}
\notag
$$
Clearly, it is an element of $W_2^1(\Pi_{3\varepsilon};\mathbb{C}^m)$, its trace on the surface $\bigl\{x\colon \tau=(3/2)\varepsilon\bigr\}$ coincides with the trace of $v_\varepsilon$, and by the assumed regularity of the surface $\Gamma_0$ and Lemma 2.4 we have
$$
\begin{equation}
\begin{gathered} \, \|\widetilde{v}_\varepsilon\|_{L_2(\Pi_{3\varepsilon};\mathbb{C}^m)}\leqslant C\|v_\varepsilon\|_{L_2(\Pi_{3\varepsilon}\setminus\Pi_{\frac{3}{2}\varepsilon};\mathbb{C}^m)}, \\ \|\widetilde{v}_\varepsilon\|_{W_2^1(\Pi_{3\varepsilon};\mathbb{C}^m)}\leqslant C\|v_\varepsilon\|_{W_2^1(\Pi_{3\varepsilon}\setminus\Pi_{\frac{3}{2}\varepsilon};\mathbb{C}^m)}, \end{gathered}
\end{equation}
\tag{4.3}
$$
where the constant $C$ is independent of $v_\varepsilon$. Also note that, repeating the derivation of (2.8) in the proof of Lemma 2.2, we can easily verify that
$$
\begin{equation}
\|v_\varepsilon\|_{L_2(\Pi_{3\varepsilon}\setminus\Pi_{\frac{3}{2}\varepsilon}; \mathbb{C}^m)}\leqslant C\varepsilon^{1/2} \|v_\varepsilon\|_{W_2^1(\Omega_\varepsilon;\mathbb{C}^m)}.
\end{equation}
\tag{4.4}
$$
Now we consider the inner products in $L_2(\Omega_\varepsilon\cap\Pi_{\frac{3}{2}\varepsilon};\mathbb{C}^m)$ and $L_2(\Pi_{\frac{3}{2}\varepsilon};\mathbb{C}^m)$ of the equation for $u_0$ with $v_\varepsilon$ and $\widetilde{v}_\varepsilon$, respectively, integrate by parts and subtract the resulting relations one from the other. As a result, we have
$$
\begin{equation}
\biggl(\frac{\partial u_0}{\partial\boldsymbol{\nu}_\varepsilon},v_\varepsilon\biggr)_{L_2(\Gamma_\varepsilon;\mathbb{C}^m)} =\mathfrak{q}_\varepsilon^1 + \mathfrak{q}_\varepsilon^2 -(f,v_\varepsilon)_{L_2(\Omega_\varepsilon\cap\Pi_{\frac{3}{2}\varepsilon};\mathbb{C}^m)}+ (f,\widetilde{v}_\varepsilon)_{L_2(\Pi_{\frac{3}{2}\varepsilon};\mathbb{C}^m)},
\end{equation}
\tag{4.5}
$$
where
$$
\begin{equation*}
\begin{aligned} \, &\mathfrak{q}_\varepsilon^1:=\sum_{i,j=1}^{n} \biggl(A_{ij}\, \frac{\partial u_0}{\partial x_j},\frac{\partial v_\varepsilon}{\partial x_i}\biggr)_{L_2(\Omega_\varepsilon\cap\Pi_{\frac{3}{2}\varepsilon};\mathbb{C}^m)} + \sum_{j=1}^{n} \biggl(A_j\,\frac{\partial u_0}{\partial x_j},v_\varepsilon\biggr)_{L_2(\Omega_\varepsilon\cap\Pi_{\frac{3}{2}\varepsilon};\mathbb{C}^m)} \\ &\qquad\quad-\sum_{i,j=1}^{n} \biggl(A_{ij} \, \frac{\partial u_0}{\partial x_j},\frac{\partial \widetilde{v}_\varepsilon}{\partial x_j}\biggr)_{L_2(\Pi_{\frac{3}{2}\varepsilon};\mathbb{C}^m)} - \sum_{j=1}^{n} \biggl(A_j\, \frac{\partial u_0}{\partial x_j},\widetilde{v}_\varepsilon\biggr)_{L_2(\Pi_{\frac{3}{2}\varepsilon};\mathbb{C}^m)} \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\mathfrak{q}_\varepsilon^2:= \bigl(A_0(\,\cdot\,,u_0)-\lambda u_0,v_\varepsilon\bigr)_{L_2(\Omega_\varepsilon\cap\Pi_{\frac{3}{2}\varepsilon};\mathbb{C}^m)} -\bigl(A_0(\,\cdot\,,u_0)-\lambda u_0,\widetilde{v}_\varepsilon\bigr)_{L_2(\Pi_{\frac{3}{2}\varepsilon};\mathbb{C}^m)}.
\end{equation*}
\notag
$$
Note that
$$
\begin{equation}
\begin{aligned} \, &(f,\widetilde{v}_\varepsilon)_{L_2(\Pi_{\frac{3}{2}\varepsilon};\mathbb{C}^m)} -(f,v_\varepsilon)_{L_2(\Omega_\varepsilon\cap\Pi_{\frac{3}{2}\varepsilon};\mathbb{C}^m)} \\ &\qquad = (f,\widetilde{v}_\varepsilon -v_\varepsilon)_{L_2(\Omega_\varepsilon\cap\Pi_{\frac{3}{2}\varepsilon};\mathbb{C}^m)} + (f,\widetilde{v}_\varepsilon)_{L_2(\Pi_{\frac{3}{2}\varepsilon} \setminus\Omega_\varepsilon;\mathbb{C}^m)}, \\ &\mathfrak{q}_\varepsilon^2=\bigl(A_0(\,\cdot\,,u_0)-\lambda u_0,v_\varepsilon-\widetilde{v}_\varepsilon\bigr)_{L_2 (\Omega_\varepsilon\cap\Pi_{\frac{3}{2}\varepsilon};\mathbb{C}^m)} \\ &\qquad\quad-\bigl(A_0(\,\cdot\,,u_0)-\lambda u_0, \widetilde{v}_\varepsilon\bigr)_{L_2(\Pi_{\frac{3}{2}\varepsilon} \setminus\Omega_\varepsilon;\mathbb{C}^m)}. \end{aligned}
\end{equation}
\tag{4.6}
$$
The first equality in (4.6) and equality (4.5) make it possible to rewrite identity (4.2) as
$$
\begin{equation}
\begin{aligned} \, \notag &\mathfrak{h}^N_\varepsilon(u_\varepsilon,v_\varepsilon) -\mathfrak{h}^N_\varepsilon(u_0,v_\varepsilon) -\lambda\|v_\varepsilon\|_{L_2(\Omega_\varepsilon;\mathbb{C}^m)}^2 \\ &\qquad = \mathfrak{q}_\varepsilon^1+ \mathfrak{q}_\varepsilon^2 + (f,\widetilde{v}_\varepsilon -v_\varepsilon)_{L_2(\Omega_\varepsilon\cap\Pi_{\frac{3}{2}\varepsilon};\mathbb{C}^m)} + (f,\widetilde{v}_\varepsilon)_{L_2(\Pi_{\frac{3}{2}\varepsilon} \Omega_\varepsilon;\mathbb{C}^m)}. \end{aligned}
\end{equation}
\tag{4.7}
$$
We estimate the right-hand side of this relation. Note that the function $\widetilde{v}_\varepsilon-v_\varepsilon$ has zero trace on $\partial\Pi_{\frac{3}{2}\varepsilon}\setminus\Gamma_0$. Hence we can apply to it Lemma 2.4 for $u=\widetilde{v}_\varepsilon -v_\varepsilon$, which, in view of (4.3), yields
$$
\begin{equation}
\begin{aligned} \, \notag \bigl|(f,\widetilde{v}_\varepsilon -v_\varepsilon)_{L_2(\Omega_\varepsilon\cap\Pi_{\frac{3}{2}\varepsilon};\mathbb{C}^m)}\bigr| &\leqslant C\varepsilon\mu(\varepsilon) \|f\|_{L_2(\Omega_\varepsilon)} \|\widetilde{v}_\varepsilon -v_\varepsilon\|_{W_2^1(\Pi_{2\varepsilon}\cap\Omega_\varepsilon;\mathbb{C}^m)} \\ &\leqslant C\varepsilon\mu(\varepsilon) \|f\|_{L_2(\Omega_\varepsilon)} \|v_\varepsilon\|_{W_2^1(\Omega_\varepsilon;\mathbb{C}^m)}; \end{aligned}
\end{equation}
\tag{4.8}
$$
here and in the proof that follows the symbol $C$ denotes different constants independent of $\varepsilon$, $u_0$, $v_\varepsilon$ and $f$. We estimate the third term on the right-hand side of (4.7) using (4.3) and (4.4):
$$
\begin{equation}
\bigl| (f,\widetilde{v}_\varepsilon)_{L_2(\Pi_{\frac{3}{2}\varepsilon} \setminus\Omega_\varepsilon;\mathbb{C}^m)}\bigr| \leqslant C\varepsilon^{1/2} \|f\|_{L_2(\Omega\setminus\Omega_\varepsilon;\mathbb{C}^m)} \|v_\varepsilon\|_{W_2^1(\Omega_\varepsilon;\mathbb{C}^m)}.
\end{equation}
\tag{4.9}
$$
Similarly, in view of (3.9) and the first inequality in (2.2), we can infer an estimate for $\mathfrak{q}_\varepsilon^2$:
$$
\begin{equation}
\begin{aligned} \, \notag |\mathfrak{q}_\varepsilon^2| &\leqslant C (\varepsilon^{3/2}\mu+\varepsilon) \|u_0\|_{W_2^1(\Omega;\mathbb{C}^m)} \|v_\varepsilon\|_{W_2^1(\Omega_\varepsilon;\mathbb{C}^m)} \\ &\leqslant C (\varepsilon^{3/2}\mu+\varepsilon) \|f\|_{L_2(\Omega;\mathbb{C}^m)} \|v_\varepsilon\|_{W_2^1(\Omega_\varepsilon;\mathbb{C}^m)}. \end{aligned}
\end{equation}
\tag{4.10}
$$
We estimate the function $\mathfrak{q}_\varepsilon^1$ using (2.8) for $u=u_0$ and $u={\partial u_0}/{\partial x_j}$, (4.3), and (2.2):
$$
\begin{equation*}
\begin{aligned} \, |\mathfrak{q}_\varepsilon^1| &\leqslant C \varepsilon^{1/2} \|u_0\|_{W_2^2(\Omega;\mathbb{C}^m)}\| v_\varepsilon\|_{W_2^1(\Omega_\varepsilon\cap\Pi_{\frac{3}{2}\varepsilon};\mathbb{C}^m)} \\ &\leqslant C \varepsilon^{1/2} \|f\|_{L_2(\Omega;\mathbb{C}^m)}\|v_\varepsilon\|_{W_2^1(\Omega_\varepsilon\cap \Pi_{\frac{3}{2}\varepsilon};\mathbb{C}^m)}. \end{aligned}
\end{equation*}
\notag
$$
Taking account of (4.10), (4.9) and (4.8), we infer the final estimate for the right-hand side of (4.7):
$$
\begin{equation*}
\begin{aligned} \, &\bigl|\mathfrak{q}_\varepsilon^1+ \mathfrak{q}_\varepsilon^2 + (f,\widetilde{v}_\varepsilon -v_\varepsilon)_{L_2(\Omega_\varepsilon\cap\Pi_{\frac{3}{2}\varepsilon};\mathbb{C}^m)} + (f,\widetilde{v}_\varepsilon)_{L_2(\Pi_{\frac{3}{2}\varepsilon} \setminus\Omega_\varepsilon;\mathbb{C}^m)}\bigr| \\ &\qquad\leqslant C(\varepsilon^{1/2}+\varepsilon\mu) \|f\|_{L_2(\Omega;\mathbb{C}^m)}\|v_\varepsilon\|_{W_2^1(\Omega_\varepsilon \cap\Pi_{\frac{3}{2}\varepsilon};\mathbb{C}^m)}. \end{aligned}
\end{equation*}
\notag
$$
In view of inequality (2.1) for the left-hand side of (4.7), we obtain (1.21). 4.2. $L_2$-estimateWe establish (1.22) by following the scheme used in the proof of (1.20). Again, we extend the function $v_\varepsilon$ by zero to $\Omega\setminus\Omega_\varepsilon$ and keep the same notation for the extension. We emphasize that the extended function $v_\varepsilon$ is an element only of $L_2(\Omega;\mathbb{C}^m)$ but, generally speaking, not of $W_2^1(\Omega;\mathbb{C}^m)$, since the trace of the original function on $\Gamma_\varepsilon$ is not necessarily zero. In $L_2(\Omega;\mathbb{C}^m)$ we introduce the sesquilinear form
$$
\begin{equation*}
\begin{aligned} \, \mathfrak{l}^N(u,v) &:=\sum_{i,j=1}^{n} \biggl(\frac{\partial u}{\partial x_i},A_{ij}\,\frac{\partial v}{\partial x_j}\biggr)_{L_2(\Omega;\mathbb{C}^m)}-\sum_{j=1}^{n} \biggl(\frac{\partial\ }{\partial x_j} A_j^*u,v\biggr)_{L_2(\Omega;\mathbb{C}^m)} \\ &\qquad-\sum_{j=1}^{n} (\nu^j A_j^*u,v)_{L_2(\Gamma_0;\mathbb{C}^m)} \\ &=\sum_{i,j=1}^{n} \biggl(A_{ij}^*\,\frac{\partial u}{\partial x_i},\frac{\partial v}{\partial x_j}\biggr)_{L_2(\Omega;\mathbb{C}^m)}+\sum_{j=1}^{n} \biggl(u,A_j\,\frac{\partial v}{\partial x_j}\biggr)_{L_2(\Omega;\mathbb{C}^m)} \end{aligned}
\end{equation*}
\notag
$$
with domain $\mathring{W}_2^1(\Omega,\Gamma;\mathbb{C}^m)$. Similarly to the proof of Lemma 2.1, we can easily verify that this form is sectorial and closed; hence by the first representation theorem ([24], Ch. VI, § 21) it is associated with an $m$-sectorial operator, which we denote by $\mathcal{L}^N$. Without loss of generality we assume in what follows that $\lambda_0$ is chosen so that the half-plane $\operatorname{Re}\lambda\leqslant \lambda_0$ lies in the resolvent set of this operator. We introduce the notation $w:=(\mathcal{L}^N-\overline{\lambda})^{-1}v_\varepsilon$. This function is obviously a solution of the boundary value problem
$$
\begin{equation}
\begin{gathered} \, \widehat{\mathcal{L}} w-\overline{\lambda} w =v_\varepsilon \quad\text{in } \Omega, \qquad w=0 \quad\text{on } \Gamma, \\ \sum_{i,j=1}^{n} \nu^j A_{ij}^*\, \frac{\partial w}{\partial x_i} +\sum_{j=1}^{n}\nu^j A_j^* w=0\quad\text{on}\quad\partial\Gamma_0. \end{gathered}
\end{equation}
\tag{4.11}
$$
Using the theorems on improved smoothness, we can verify directly estimates (3.15) for the solution of this problem. We take the inner product in $L_2(\Omega_\varepsilon\setminus\Pi_{\frac{3}{2}\varepsilon};\mathbb{C}^m)$ of the equation in (4.11) with $v_\varepsilon$ and integrate by parts:
$$
\begin{equation}
\begin{aligned} \, \notag \|v_\varepsilon\|_{L_2(\Omega_\varepsilon\setminus\Pi_{\frac{3}{2}\varepsilon};\mathbb{C}^m)}^2 &=\sum_{i,j=1}^{n} \biggl(A_{ij}\,\frac{\partial v_\varepsilon}{\partial x_j},\frac{\partial w}{\partial x_i}\biggr)_{L_2(\Omega_\varepsilon\setminus\Pi_{\frac{3}{2}\varepsilon};\mathbb{C}^m)} \\ \notag &\qquad+\sum_{j=1}^{n} \biggl(A_j\,\frac{\partial v_\varepsilon}{\partial x_j},w \biggr)_{L_2(\Omega_\varepsilon\setminus\Pi_{\frac{3}{2}\varepsilon};\mathbb{C}^m)} -\lambda(v_\varepsilon,w)_{L_2(\Omega_\varepsilon\setminus\Pi_{\frac{3}{2}\varepsilon};\mathbb{C}^m)} \\ &\qquad+ \biggl(v_\varepsilon,\sum_{i,j=1}^{n} \nu_\varepsilon^j A_{ij}^* \,\frac{\partial w}{\partial x_i} +\sum_{j=1}^{n} \nu^j_\varepsilon A_j w\biggr)_{L_2(\partial\Pi_{\frac{3}{2}\varepsilon}\setminus\partial\Omega;\mathbb{C}^m)}. \end{aligned}
\end{equation}
\tag{4.12}
$$
We take the inner product in $L_2(\Pi_{\frac{3}{2}\varepsilon};\mathbb{C}^m)$ of the same equation with $\widetilde{v}_\varepsilon$ and integrate by parts again:
$$
\begin{equation}
\begin{aligned} \, \notag (\widetilde{v}_\varepsilon,v_\varepsilon)_{L_2(\Pi_{\frac{3}{2}\varepsilon};\mathbb{C}^m)} &= \sum_{i,j=1}^{n} \biggl(A_{ij}\,\frac{\partial \widetilde{v}_\varepsilon}{\partial x_j},\frac{\partial w}{\partial x_i}\biggr)_{L_2(\Pi_{\frac{3}{2}\varepsilon};\mathbb{C}^m)}+\sum_{j=1}^{n} \biggl(A_j\,\frac{\partial \widetilde{v}_\varepsilon}{\partial x_j},w \biggr)_{L_2(\Pi_{\frac{3}{2}\varepsilon};\mathbb{C}^m)} \\ \notag &\qquad-\biggl(\widetilde{v}_\varepsilon,\sum_{i,j=1}^{n} \nu_\varepsilon^j A_{ij}^* \,\frac{\partial w}{\partial x_i} +\sum_{j=1}^{n} \nu^j_\varepsilon A_j w\biggr)_{L_2(\partial\Pi_{\frac{3}{2}\varepsilon}\setminus\partial\Omega;\mathbb{C}^m)} \\ &\qquad- \lambda(\widetilde{v}_\varepsilon,w)_{L_2(\Pi_{\frac{3}{2}\varepsilon};\mathbb{C}^m)}. \end{aligned}
\end{equation}
\tag{4.13}
$$
In view of the relation $v_\varepsilon=\widetilde{v}_\varepsilon$ on $\partial\Pi_{\frac{3}{2}\varepsilon}\setminus\Gamma_0$ and formulae similar to (4.6), adding together the equalities obtained yields
$$
\begin{equation*}
\|v_\varepsilon\|_{L_2(\Omega_\varepsilon;\mathbb{C}^m)}^2=\mathfrak{t}_\varepsilon^1+\mathfrak{t}_\varepsilon^2,
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\begin{aligned} \, \mathfrak{t}_\varepsilon^1 &:=(v_\varepsilon-\widetilde{v}_\varepsilon,v_\varepsilon)_{L_2(\Pi_{\frac{3}{2} \varepsilon}\cap\Omega_\varepsilon;\mathbb{C}^m)} -\lambda (\widetilde{v}_\varepsilon-v_\varepsilon,w)_{L_2(\Pi_{\frac{3}{2}\varepsilon} \cap\Omega_\varepsilon;\mathbb{C}^m)} \\ &\qquad-\lambda (\widetilde{v}_\varepsilon,w)_{L_2(\Pi_{\frac{3}{2}\varepsilon} \setminus\Omega_\varepsilon;\mathbb{C}^m)} +\sum_{i,j=1}^{n} \biggl(A_{ij}\,\frac{\partial\ }{\partial x_j}(\widetilde{v}_\varepsilon-v_\varepsilon),\frac{\partial w}{\partial x_i}\biggr)_{L_2(\Pi_{\frac{3}{2}\varepsilon}\cap\Omega_\varepsilon;\mathbb{C}^m)} \\ &\qquad+\sum_{j=1}^{n} \biggl(A_j\,\frac{\partial\ }{\partial x_j}(\widetilde{v}_\varepsilon-v_\varepsilon),w \biggr)_{L_2(\Pi_{\frac{3}{2}\varepsilon}\cap\Omega_\varepsilon;\mathbb{C}^m)} \\ &\qquad+\sum_{i,j=1}^{n} \biggl(A_{ij}\,\frac{\partial\widetilde{v}_\varepsilon}{\partial x_j},\frac{\partial w}{\partial x_i}\biggr)_{L_2(\Pi_{\frac{3}{2}\varepsilon}\setminus\Omega_\varepsilon; \mathbb{C}^m)}+\sum_{j=1}^{n} \biggl(A_j\,\frac{\partial\widetilde{v}_\varepsilon}{\partial x_j},w \biggr)_{L_2(\Pi_{\frac{3}{2}\varepsilon}\setminus\Omega_\varepsilon;\mathbb{C}^m)} \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, \mathfrak{t}_\varepsilon^2 &:=\sum_{i,j=1}^{n} \biggl(A_{ij}\,\frac{\partial v_\varepsilon}{\partial x_j},\frac{\partial w}{\partial x_i}\biggr)_{L_2(\Omega_\varepsilon;\mathbb{C}^m)}+\sum_{j=1}^{n} \biggl(A_j\,\frac{\partial v_\varepsilon}{\partial x_j},w \biggr)_{L_2(\Omega_\varepsilon;\mathbb{C}^m)} \\ &\qquad-\lambda(v_\varepsilon,w)_{L_2( \Omega_\varepsilon;\mathbb{C}^m)}. \end{aligned}
\end{equation*}
\notag
$$
Identity (4.1) clearly holds for $v_\varepsilon$ replaced by $w$. Subtracting it from (1.9) for ${v=w}$ we obtain an analogue of (4.2):
$$
\begin{equation}
\begin{aligned} \, \notag \biggl(\frac{\partial u_0}{\partial\boldsymbol{\nu}_\varepsilon},w\biggr)_{L_2(\Gamma_\varepsilon;\mathbb{C}^m)} &= \mathfrak{h}^N_\varepsilon(u_\varepsilon,w)-\mathfrak{h}^N_\varepsilon(u_0,w) -\lambda(v_\varepsilon,w)_{L_2(\Omega_\varepsilon;\mathbb{C}^m)} \\ &=\mathfrak{t}_\varepsilon^2-\bigl(A_0(\,\cdot\,,u_\varepsilon) -A_0(\,\cdot\,,u_0),w\bigr)_{L_2(\Omega_\varepsilon;\mathbb{C}^m)}. \end{aligned}
\end{equation}
\tag{4.14}
$$
We also take the inner products in $L_2(\Omega_\varepsilon\cap\Pi_{\frac{3}{2}\varepsilon};\mathbb{C}^m)$ and $L_2(\Pi_{\frac{3}{2}\varepsilon};\mathbb{C}^m)$ of the equation for $u_0$ with $w$, integrate by parts and subtract the resulting relations one from the other. This procedure gives us an analogue of (4.5):
$$
\begin{equation*}
\biggl(\frac{\partial u_0}{\partial\boldsymbol{\nu}_\varepsilon},w\biggr)_{L_2(\Gamma_\varepsilon;\mathbb{C}^m)} =\mathfrak{t}_\varepsilon^3-(f,w)_{L_2(\Omega\setminus\Omega_\varepsilon;\mathbb{C}^m)},
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\begin{aligned} \, \mathfrak{t}_\varepsilon^3 &:=\sum_{i,j=1}^{n} \biggl(A_{ij}\, \frac{\partial u_0}{\partial x_j},\frac{\partial w}{\partial x_j}\biggr)_{L_2(\Omega\setminus\Omega_\varepsilon;\mathbb{C}^m)} + \sum_{j=1}^{n} \biggl(A_j\,\frac{\partial u_0}{\partial x_j},w\biggr)_{L_2(\Omega\setminus\Omega_\varepsilon;\mathbb{C}^m)} \\ &\qquad + \bigl(A_0(\,\cdot\,,u_0)-\lambda u_0,w\bigr)_{L_2(\Omega\setminus\Omega_\varepsilon;\mathbb{C}^m)}. \end{aligned}
\end{equation*}
\notag
$$
It follows from this formula, (4.12), the equality $v_\varepsilon=\widetilde{v}_\varepsilon$ on $\Gamma_\varepsilon$, and the sum of (4.13) and (4.14) that
$$
\begin{equation}
\begin{aligned} \, &\|v_\varepsilon\|_{L_2(\Omega_\varepsilon;\mathbb{C}^m)}^2 - \bigl(A(\,\cdot\,,u_\varepsilon)-A(\,\cdot\,,u_0),w\bigr)_{L_2(\Omega_\varepsilon;\mathbb{C}^m)} \nonumber \\ &\qquad= \mathfrak{t}_\varepsilon^1+\mathfrak{t}_\varepsilon^3-(f,w)_{L_2( \Omega\setminus\Omega_\varepsilon;\mathbb{C}^m)}. \end{aligned}
\end{equation}
\tag{4.15}
$$
We estimate the function $\mathfrak{t}_\varepsilon^1$ using (2.8) for $u=w$, $u={\partial w}/{\partial x_i}$, $u=u_0$ and $u={\partial u_0}/{\partial x_i}$, Lemma 2.4 for $u=\widetilde{v}_\varepsilon-v_\varepsilon$, and inequalities (4.3), (4.4), (1.21) and (3.15):
$$
\begin{equation}
\begin{aligned} \, \notag |\mathfrak{t}_\varepsilon^1| &\leqslant C \|v_\varepsilon\|_{W_2^1(\Omega_\varepsilon;\mathbb{C}^m)} \bigl(\varepsilon\mu\|v_\varepsilon\|_{L_2(\Omega_\varepsilon;\mathbb{C}^m)} + \varepsilon^{1/2} \|w\|_{W_2^2(\Omega;\mathbb{C}^m)} \bigr) \\ &\leqslant C (\varepsilon^2\mu^2+\varepsilon) \|f\|_{L_2(\Omega;\mathbb{C}^m)} \|v_\varepsilon\|_{L_2(\Omega_\varepsilon;\mathbb{C}^m)}. \end{aligned}
\end{equation}
\tag{4.16}
$$
The function $\mathfrak{t}_\varepsilon^3$ is estimated on the base of (2.8) for $u=w$, $u={\partial w}/{\partial x_i}$, $u=u_0$ and $u={\partial u_0}/{\partial x_i}$ and inequalities (3.9), (3.15) and (2.2):
$$
\begin{equation}
|\mathfrak{t}_\varepsilon^3|\leqslant C\varepsilon\|u_0\|_{W_2^2(\Omega;\mathbb{C}^m)} \|w\|_{W_2^2(\Omega;\mathbb{C}^m)} \leqslant C \varepsilon \|f\|_{L_2(\Omega;\mathbb{C}^m)} \|v_\varepsilon\|_{L_2(\Omega_\varepsilon;\mathbb{C}^m)}.
\end{equation}
\tag{4.17}
$$
The third term on the right-hand side of (4.15) is estimated on the base of (2.8) for $u=w$ and (3.15):
$$
\begin{equation}
\begin{aligned} \, \notag \bigl|(f,w)_{L_2(\Omega\setminus\Omega_\varepsilon;\mathbb{C}^m)}\bigr| &\leqslant C \varepsilon^{1/2} \|f\|_{L_2(\Omega\setminus\Omega_\varepsilon;\mathbb{C}^m)}\|w\|_{W_2^2(\Omega;\mathbb{C}^m)} \\ &\leqslant C \varepsilon^{1/2} \|f\|_{L_2(\Omega\setminus\Omega_\varepsilon;\mathbb{C}^m)} \|v_\varepsilon\|_{L_2(\Omega_\varepsilon;\mathbb{C}^m)}. \end{aligned}
\end{equation}
\tag{4.18}
$$
We estimate the second term on the left-hand side of (4.15) using the Lipschitz inequality in (1.3) and the second inequality in (3.15), which implies the estimate
$$
\begin{equation*}
\bigl|\bigl(A(\,\cdot\,,u_\varepsilon)-A(\,\cdot\,,u_0),w\bigr)_{L_2(\Omega_\varepsilon;\mathbb{C}^m)}\bigr| \leqslant \frac{C}{|{\operatorname{Re}\lambda+C_3}|}\|v_\varepsilon\|_{L_2(\Omega_\varepsilon;\mathbb{C}^m)}^2.
\end{equation*}
\notag
$$
Now we choose $\lambda_0$ to guarantee that
$$
\begin{equation*}
\frac{C}{|{\operatorname{Re}\lambda+C_3}|}<\frac{1}{2}.
\end{equation*}
\notag
$$
Then the left-hand side of (4.15) satisfies the inequality
$$
\begin{equation*}
\bigl|\|v_\varepsilon\|_{L_2(\Omega_\varepsilon;\mathbb{C}^m)}^2 - \bigl(A(\,\cdot\,,u_\varepsilon)-A(\,\cdot\,,u_0),w\bigr)_{L_2(\Omega_\varepsilon;\mathbb{C}^m)}\bigr|\geqslant \frac{1}{2} \|v_\varepsilon\|_{L_2(\Omega_\varepsilon;\mathbb{C}^m)}^2.
\end{equation*}
\notag
$$
It follows from this estimate and (4.16)–(4.18) that (1.22) is valid. Theorem 1.2 is proved.
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\Bibitem{BorSul25} |
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