| Sbornik: Mathematics |
|
|
|
|
|
This article is cited in 3 scientific papers (total in 3 papers) Asymptotics for problems in perforated domains with Robin nonlinear condition on the boundaries of cavities D. I. Borisova, A. I. Mukhametrakhimovaba a Institute of Mathematics with Computing Centre, Ufa Federal Research Centre of the Russian Academy of Sciences, Ufa, Russia b Bashkir State Pedagogical University n. a. M. Akmulla, Ufa, Russia
Russian version:
Matematicheskii Sbornik, 2022, Volume 213, Number 10, DOI: https://doi.org/10.4213/sm9739 
Bibliographic databases:
    § 1. IntroductionElliptic problems in domains finely perforated along a submanifold are a classical example of problems in the theory of boundary homogenization. They have been investigated by many authors; we mention only [1]–[11] and the monographs [12] and [13] for example. The aim of those investigations was to prove that solutions of such problems converge to solutions of certain homogenized problems, which differ from the original problems by replacing the perforation by a certain homogenized condition on the surface along which the perforation is concentrated. Convergence was mostly established in the norms of $L_2$ or $W_2^1$. From the operator standpoint, such results correspond to the presence of strong or weak resolution convergence. In some papers on boundary homogenization published recent years authors established a stronger, uniform resolution convergence in classical settings and proved so-called operator estimates, in which the difference between the solutions of the perturbed and homogenized problems was measured in terms of a small quantity times the $L_2$-norm of the right-hand side. Such results were obtained for operators with frequently alternating boundary conditions [14]–[22] and for problems in domains with fast oscillating boundary [19]. In [23] such results were established for a second-order elliptic operator of general form in a planar strip perforated by small holes along a fixed curve, where the first (Dirichlet) or third (Robin) boundary conditions were set arbitrarily on the boundaries of the holes and boundary conditions of different types on different holes were allowed. Apart from various convergence results there are many papers considering asymptotic expansions for the resolvents of such problems and asymptotic expansions of various spectral characteristics. Without aiming at or being able to list all of these papers, we only mention some of them, concerning asymptotic expansions for frequent alternation [9], [20]–[22], [24], related papers on concentrated masses [25], [26] and on fast oscillating boundary [27], and the recent paper [28] on a model with small rivets; see also the bibliography in these papers. In all these papers, on the basis of various combinations of methods of asymptotic analysis, authors constructed asymptotic expansions for solutions of some problems in the theory of boundary homogenization. In the majority of cases the distribution of oscillations was assumed to have a strictly periodic or a locally periodic structure, so that suitable ansätze could be put together. In this paper we consider an elliptic boundary value problem in a multidimensional domain with dense perforation finely arranged along a fixed hypersurface; a Robin nonlinear condition is set at the boundaries of the cavities. Let us take a deeper look at the preceding papers that served as a motivation for our work. In [29] we investigated the boundary value problem for a general second-order elliptic equation in a multidimensional domain that was aperiodically perforated by small cavities along a fixed submanifold. The Dirichlet condition or the Robin nonlinear condition was set on the boundaries of the cavities. Assumptions made about the sizes of the cavities, the distances between them, and the distribution of the cavities with Dirichlet boundary conditions ensured that we obtain a Dirichlet condition. We showed that the solution of the perturbed problem converges to the solution of the homogenized one in the $W^1_2$-norm uniformly with respect to the $L_2$-norm of the right-hand side, and we obtained a sharp in order estimate for the rate of convergence. In the case when the cavities were arranged strictly periodically along the given hypersurface and the Dirichlet conditions and Robin nonlinear condition were set periodically on the boundaries of the cavities, we constructed a complete asymptotic expansion for the solution of the perturbed problem. In [30] we also considered the model from [29], but there were no cavities with the Direchlet boundary conditions nor were there significant restrictions on the sizes of the cavities and the distances between them. Then either the submanifold disappears after homogenization or a boundary condition of the type of nonlinear delta interaction arises on it. We proved that the solution of the perturbed problem converges to the solution of the homogenized problem in the $W_2^1$-norm uniformly with respect to the right-hand side of the equation and established a sharp in order estimate for the rate of convergence. We also showed that when one estimates the difference between the solutions in the $L_2$-norm, then the rate of convergence is higher. Here we continue the investigations carried out in [29] and [30]. We consider a boundary value problem for a second-order elliptic equation with variable coefficients in a domain perforated periodically along a prescribed hypersurface. We set the Robin nonlinear boundary condition on the boundaries of cavities; the Direchlet boundary condition is not set. The distances between cavities are proportional to a small parameter $\varepsilon$, while their linear sizes are proportional to $\varepsilon\eta(\varepsilon)$, where $\eta=\eta(\varepsilon)$ is a fixed function taking values in the interval $[0,1]$. Unlike [29], we impose no restrictions on the behaviour of this function. Our main result is the construction of an asymptotic expansion for the solution of the perturbed problem. It is a combination of an inner and an outer expansion, which is constructed using the method of matched asymptotic expansions (see [31]) and the multiscale method (see [32]). The inner expansion is constructed in a neighbourhood of the hypersurface along which the perforation is arranged; it is used to take account of the microstructure of the perforation. The outer expansion approximates the solution outside a small neighbourhood of the perforated region. The outer and inner expansions are expansions in powers of $\varepsilon$, with coefficients depending on $\eta$. Their dependence on the parameter $\eta$ is significantly different from the case considered in [29]: now the coefficients are infinitely differentiable with respect to $\eta\in(0,1]$ and uniformly bounded as $\eta\to+0$. Thus we construct a two-parameter asymptotic expansion. The leading term is a solution of the homogenized problems stated in [30] for the perforation under consideration here. The structure of the paper is as follows. In § 2 we present the statement of the problem and state the main results. In § 3 we describe the general scheme of the formal construction of the solution of this problem: we state an ansatz in the form of an outer and an inner expansion, deduce boundary value problems for the coefficients and write out the matching conditions. Section 4 is devoted to the analysis of the solvability of a special case of the model problem for functions participating in the inner expansion, and in § 5 we obtain uniform estimates for these functions and their spatial derivatives. In § 6 we use these results to investigate the solvability of the general model problem for the functions involved in the inner expansion and obtain uniform estimates for the solution. In § 7 we show that the solutions of the model problems for functions involved in the inner and outer expansions are infinitely differentiable with respect to $\eta$. In § 8 we extend our general results to the coefficients of the formal asymptotic expansion constructed in § 3. In the final section, § 9, we justify these formal expansions rigorously; namely, we find estimates for the remainder terms in asymptotic expansions. § 2. Statements of the problem and the resultsLet $x=(x',x_n)$ and $x'=(x_1,\dots,x_{n-1})$ be Cartesian variables in $\mathbb{R}^n$ and $\mathbb{R}^{n-1}$, respectively, where $n\geqslant3$. Let $\Omega$ be an arbitrary domain with $C^2$-boundary in $\mathbb{R}^n$. We also assume that there exists $\tau>0$ such that
$$
\begin{equation*}
\Omega_{\tau}:=\{x\colon |x_n|<\tau\}\subseteq \Omega.
\end{equation*}
\notag
$$
Let $\varepsilon$ be a small positive parameter and $\eta=\eta(\varepsilon)$ be a function satisfying ${0<\eta(\varepsilon)\leqslant1}$. Set
$$
\begin{equation*}
\Pi:=\square\times\mathbb{R}, \qquad \square:=\bigl\{x\colon - b_i <x_i< b_i,\,i=1,\dots,n-1\bigr\},
\end{equation*}
\notag
$$
where the $b_i$ are some positive numbers. Let $\omega$ be a fixed bounded set with infinitely smooth boundary. We assume that for all $\eta\in(0,1]$ we have $\overline{\omega^\eta}\subset\Pi$ and $\omega^\eta:=\bigl\{x\colon \eta^{-1}x\in\omega\bigr\}$. Consider the sets
$$
\begin{equation*}
\begin{gathered} \, \omega_k^\varepsilon:=\bigl\{x\colon (x-\varepsilon M_k)\varepsilon^{-1}\eta^{-1}(\varepsilon)\in \omega\bigr\}\quad\text{and} \quad \theta^\varepsilon:=\bigcup_{k\in\mathbb{Z}^{n-1}}\omega_k^\varepsilon, \\ \text{where}\quad M_k:=(2b_1k_1,\dots,2b_{n-1}k_{n-1}), \qquad k:=(k_1,\dots,k_{n-1})\in\mathbb{Z}^{n-1}. \end{gathered}
\end{equation*}
\notag
$$
From $\Omega$ we remove the cavities $\omega_k^\varepsilon$, $k\in\mathbb{M}^\varepsilon$, and let $\Omega^\varepsilon$ denote the resulting domain: $\Omega^\varepsilon:=\Omega\setminus\overline{\theta^\varepsilon}$; see Figure 1. In $\Omega$ we consider functions $A_{ij}=A_{ij}(x)$, $A_i=A_i(x)$ and $A_0=A_0(x)$ satisfying the conditions
$$
\begin{equation*}
\begin{gathered} \, A_{ij}, A_i\in W_\infty^1(\Omega), \qquad A_0\in L_\infty(\Omega), \qquad A_{ij}=A_{ji}, \quad i,j=1,\dots,n, \\ \sum_{i,j=1}^n A_{ij}(x)z_i \overline{z_j}\geqslant c_0|z|^2, \qquad x\in\Omega, \quad z=(z_1\dots,z_n)\in\mathbb{C}^n, \end{gathered}
\end{equation*}
\notag
$$
where $c_0$ is a positive constant independent of $x$ and $z$. The functions $A_{ij}$ are real while the $A_j$ and $A_0$ are complex-valued. Let $a=a(x,u)$ denote a complex-valued function defined for $u \in \mathbb{C}$ and $|x_n|\leqslant \tau$ that is infinitely differentiable with respect to $x$, $u_{\mathrm{r}}:=\operatorname{Re} u$ and $u_{\mathrm{i}}:=\operatorname{Im} u$ and satisfies
$$
\begin{equation}
a(u,0)=0, \qquad \biggl|\frac{\partial a}{\partial u_{\mathrm{r}}}(x,u)\biggr|+\biggl|\frac{\partial a}{\partial u_{\mathrm{i}}}(x,u)\biggr|\leqslant a_1,
\end{equation}
\tag{2.1}
$$
$$
\begin{equation}
\biggl|\frac{\partial^{|\beta|}a}{\partial x^\beta}(x,u)\biggr|\leqslant a_{\beta,0}|u|\quad\text{and} \quad \biggl|\frac{\partial^{|\beta|+\gamma_1+\gamma_2}a}{\partial x^\beta \, \partial u_{\mathrm{r}}^{\gamma_1}\, \partial u_{\mathrm{i}}^{\gamma_2}} (x,u)\biggr|\leqslant a_{\beta,\gamma},
\end{equation}
\tag{2.2}
$$
where $\beta\in\mathbb{Z}_+^n$, $\gamma:=(\gamma_1,\gamma_2)\in\mathbb{Z}_+^2\setminus\{0\}$, and $a_1$ and $a_{\beta,\gamma}$ are some constants independent of $x$ and $u$. Let $f\in L_2(\Omega)$ be a function and $\lambda$ be a real number. In this paper we consider the following boundary value problem:
$$
\begin{equation}
\begin{gathered} \, (\mathcal{L}-\lambda) u_\varepsilon=f \quad\text{in } \Omega^\varepsilon, \\ u_\varepsilon=0 \quad\text{on } \partial\Omega, \qquad \frac{\partial u_\varepsilon}{\partial\mathrm{n}}+a(\cdot,u_\varepsilon)=0 \quad\text{on } \partial\theta^\varepsilon, \end{gathered}
\end{equation}
\tag{2.3}
$$
where the differential expression $\mathcal{L}$ and the conormal derivative are defined by
$$
\begin{equation*}
\mathcal{L}:= -\sum_{i,j=1}^n\frac{\partial}{\partial x_i}A_{ij}\frac{\partial}{\partial x_j}+\sum_{j=1}^n A_j\frac{\partial}{\partial x_j} +A_0\quad\text{and} \quad \frac{\partial}{\partial\mathrm{n}}=\sum_{i,j=1}^n A_{ij}\nu_i\frac{\partial}{\partial x_j},
\end{equation*}
\notag
$$
respectively, and the $\nu_i$ are the components of the unit inward normal $\nu$ to $\partial\theta^\varepsilon$. Our main aim is to construct an asymptotic expansion for the solution of this problem as $\varepsilon\to0$. Let $\mathring{W}_2^1(\Omega^\varepsilon,\partial\Omega)$ denote the subspace of functions in $W_2^1(\Omega^\varepsilon)$ that vanish on $\partial\Omega$. We understand the solution of (2.3) in the generalized sense: this is a function $u_\varepsilon$ in $W_2^1(\Omega^\varepsilon)$ that satisfies the integral identity
$$
\begin{equation*}
\mathfrak{h}_a(u_\varepsilon,v)-\lambda (u,v)_{L_2(\Omega^\varepsilon)}=(f,v)_{L_2(\Omega^\varepsilon)}
\end{equation*}
\notag
$$
for all $v_\varepsilon\in\mathring{W}_2^1(\Omega^\varepsilon,\partial\Omega)$, where
$$
\begin{equation*}
\mathfrak{h}_a(u,v) :=\mathfrak{h}_0(u,v)+(a(\cdot,u),v)_{L_2(\partial\theta^\varepsilon)}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\mathfrak{h}_0(u,v) :=\sum_{i,j=1}^n\biggl(A_{ij}\frac{\partial u}{\partial x_j},\frac{\partial v}{\partial x_i}\biggr)_{L_2(\Omega^\varepsilon)} +\sum_{j=1}^n\biggl( A_j\frac{\partial u}{\partial x_j},v\biggr)_{L_2(\Omega^\varepsilon)} +(A_0 u,v)_{L_2(\Omega^\varepsilon)}.
\end{equation*}
\notag
$$
Here the integral over the boundary $\partial\theta^\varepsilon$ of the cavities is treated in the sense of traces. It was proved in [30] that there exists $\lambda_0$ independent of $\varepsilon$ such that for $\lambda<\lambda_0$ problem (2.3) has a unique solution for all $f\in L_2(\Omega)$. According to [30], the solution of (2.3) converges to a generalized solution of the homogenized problem
$$
\begin{equation}
\begin{gathered} \, (\mathcal{L}-\lambda)u_0=f \quad\text{in } \Omega\setminus S, \qquad u_0=0 \quad \text{on } \partial\Omega, \\ [u_0]_0=0, \qquad\biggl[\frac{\partial u_0}{\partial \mathrm{n}}\biggr]_0 -\alpha a(x,u_0)=0 \quad\text{on } S, \end{gathered}
\end{equation}
\tag{2.4}
$$
where $[u]_0=u|_{x_n=+0}-u|_{x_n=-0}$ and $S:=\{x\colon x_n=0\}$. Here
$$
\begin{equation*}
\begin{aligned} \, \alpha:=0 & \ \text{ if }\ \eta(\varepsilon)\to0\text{ as } \varepsilon\to+0 \quad\text{or}\quad a\equiv 0, \\ \alpha:=\eta_0^{n-1}\frac{|\partial\omega|}{|\square|} &\ \text{ if }\ \eta(\varepsilon)\to\eta_0\text{ as } \varepsilon\to+0. \end{aligned}
\end{equation*}
\notag
$$
By [30] problem (2.4) is uniquely solvable for $\lambda < \lambda_0$. Note also that the boundary condition in (2.4) describes nonlinear delta interaction on the surface $S$. Here we construct an asymptotic representation for the solution of (2.3) under certain additional conditions. Namely, let for all $p\in\mathbb{N}$. We also assume that
$$
\begin{equation*}
A_{ij}=1, \qquad A_j=0\quad\text{and} \quad A_0=0 \quad \text{for } |x_n|\leqslant\tau.
\end{equation*}
\notag
$$
Consider the system of boundary value problems
$$
\begin{equation}
\begin{gathered} \, -\Delta_\xi v_m=f_m \quad\text{in } \mathbb{R}^n\setminus\theta_\eta, \qquad \frac{\partial v_m}{\partial \nu_\xi}=\psi_m \quad\text{on } \partial\theta_\eta, \\ f_0=0, \qquad \psi_0=0, \qquad \theta_\eta=\bigcup_{k\in\mathbb{Z}^{n-1}} \bigl\{\xi\colon \eta^{-1}(\xi-M_k)\in\omega\bigr\}, \nonumber \\ f_m:=\frac{ \xi_n^{m-2}}{(m-2)!} \,\frac{\partial^{m-2} f}{\partial x_n^{m-2}}(x',0) + 2\sum_{i=1}^{n-1}\frac{\partial^2 v_{m-1}}{\partial \xi_i\, \partial x_i}+(\Delta_{x'}+\lambda)v_{m-2}, \nonumber \\ \psi_m:=-\sum_{i=1}^{n-1}\frac{\partial v_{m-1}}{\partial x_i}\nu_i- T_{m-1}(v_1,\dots,v_{m-1}), \nonumber \end{gathered}
\end{equation}
\tag{2.6}
$$
where $\nu_\xi$ is the unit inward normal to $\theta_\eta$, the $\nu_i$ are the components of $\nu_\xi$, the functions $T_m$ appear as coefficients in the following asymptotic equality:
$$
\begin{equation}
a\biggl(x',\varepsilon\xi_n,\sum_{m=0}^{\infty} \varepsilon^j v_m\biggr)=T_0(x',v_0)+\sum_{m=1}^{\infty} \varepsilon^m T_m(x',v_1,\dots,v_m)
\end{equation}
\tag{2.7}
$$
and $T_0(x',v_0)=a(x',0,v_0)$. For any positive number $R$ set $\Pi_R:=\square\times(-R,R)$. Let $\chi=\chi(x_n)$ be an infinitely differentiable cut-off function equal to zero for $|x_n|<1$ and to one for $|x_n|>2$, and let
$$
\begin{equation*}
\chi^\varepsilon(x_n) = \begin{cases} \chi(x_n\varepsilon^{-1/2}), &|x_n|>\tau, \\ 0, &|x_n|\leqslant\tau. \end{cases}
\end{equation*}
\notag
$$
The following theorem is the main result of this paper. Theorem 2.1. The solution of problem (2.3) has the following asymptotic representation in the norm of $W_2^1(\Omega^\varepsilon)$: The function $u_0$ is a solution of (2.4) for $\alpha=\eta^{n-1}$, and the $u_m$ are the solutions of the problems
$$
\begin{equation}
(\mathcal{L}-\lambda) u_m=0 \quad\textit{in } \Omega\setminus S, \qquad u_m=0 \quad\textit{on } \partial\Omega, \quad m\geqslant1,
\end{equation}
\tag{2.10}
$$
with boundary conditions
$$
\begin{equation*}
[u_m]_0=U_{m,0}^+-U_{m,0}^-\quad\textit{and} \quad \biggl[\frac{\partial u_m}{\partial x_n}\biggr]_0=U_{m,1}^+-U_{m,1}^- \quad\textit{on } S.
\end{equation*}
\notag
$$
The estimate holds, where the constant $C$ is independent of $\varepsilon$ and $\eta$, but depends on $m$.Now we discuss our results briefly. The equation in (2.3) is a second-order linear elliptic equation such that the lower-order terms have complex coefficients. The strictly periodic perforation is arranged along the hyperplane $S$. On the boundary of the cavities $\theta^\varepsilon$ we set the Robin nonlinear condition described in terms of the function $a$. This is a complex-valued function, and the basic conditions (2.1) and (2.2) indicate that it has an at most linear growth in $u_\mathrm{r}$ and $u_\mathrm{i}$. As a function of the complex variable $u$, $a$ is not assumed to enjoy any analyticity. Note that even when $a$ is linear in $u$, in problem (2.3) we deal with a nonselfadjoint operator. The right-hand side of the equation in (2.3) has improved regularity: see (2.5). This regularity and the strict periodicity of the perforation are keys to constructing complete asymptotic expansions for the solution of (2.3). We construct an asymptotic expansion for the solution of (2.3) by combining the method of matched asymptotic expansions (see [31]) and multiscale method (see [32]), as a combination of an outer expansion with coefficients $u_m$ and an inner expansion with coefficients $v_m$. The same combination was used in [29]. However, here we, first, do not have cavities with the Dirichlet boundary condition and, second, impose no restrictions on the parameter $\eta$. This changes significantly the structure of the inner expansion in comparison to [29]. Namely, because of the Neumann condition on the boundaries of the cavities, in problems (2.6) for the functions involved in the inner expansion we obtain solvability conditions. As now the parameter $\eta$ can vary arbitrarily, we must track the dependence of the coefficients of both inner and outer expansion on arbitrary values of $\eta$ in the interval $[0,1]$. Our main result, Theorem 2.1, states that the coefficients of the inner and outer expansions are infinitely differentiable with respect to $\eta\in(0,1]$ and uniformly bounded in $\eta\in[0,1]$ and, moreover, for strong norms. This is significantly different from [29], where only estimates as $\eta\to+0$ for the norms of the analogous functions were found, which were enough to ensure that the expansion of the solution constructed in [29] was asymptotic. In our case the asymptotic expansion (2.8) is in fact two-parameter: we construct an expansion in powers of $\varepsilon$, while the coefficients are smooth in $\eta>0$ and uniformly bounded as $\eta\to+0$. In particular, fixing $\eta_0>0$, for $\eta\to\eta_0$ we can expand the coefficients of the inner and outer expansions in Taylor series, thus obtaining an asymptotic expansion in two parameters, $\varepsilon$ and $\eta-\eta_0$, for the solution $u_\varepsilon$. Investigations of the dependence on $\eta$ of the functions involved in the inner and outer expansions take most of the paper. Since the original problem involves a nonlinear boundary condition on the boundaries of the cavities, in the boundary conditions on $\partial\theta_\eta$ the right-hand sides depend in a nonlinear way on the preceding functions from the inner expansion. So, as in [29], this means that we need additional uniform bounds for the moduli of the functions $v_m$ and their derivatives. However, a much more delicate analysis is required now because we make no a priori assumptions about the behaviour of the parameter $\eta$. We prove that the functions $u_m$ and $v_m$ are infinitely differentiable using quite different techniques. For the $v_m$, this is a construction of a special diffeomorphism of the domain $\Pi$ and Schauder’s estimates. For the functions $u_m$, $m\geqslant 1$, the property of infinite differentiability with respect to $\eta\in(0,1]$ is easily seen to be inherited from a similar property of the right-hand sides in the problems for the $u_m$. Verifying the same property of the solution of (2.4) turns out quite unexpectedly to be a nontrivial task due to the nonlinear delta interaction on $S$. Eventually, we could solve this problem by adapting suitably some techniques and approaches used in [33] to analyse linear and nonlinear problems. § 3. Formal construction of an asymptotic expansionIn this section we combine the method of matched asymptotic expansions (see [31]) and multiscale method (see [32]) to construct a formal asymptotic expansion for the solution of boundary value problem (2.3). In a neighbourhood of the cavities $\theta^\varepsilon$ we introduce the rescaled variables $\xi=(\xi',\xi_n)=(x'\varepsilon^{-1},x_n\varepsilon^{-1})$. We seek an asymptotic expansion for the solution of boundary value problem (2.3) as a combination of the outer $u_\varepsilon^{\mathrm{ex}}$ and inner $u_\varepsilon^{\mathrm{in}}$ expansions:
$$
\begin{equation*}
u_\varepsilon(x,\xi,\eta)=\chi^\varepsilon(x_n) u_\varepsilon^{\mathrm{ex}}(x,\eta)+(1-\chi^\varepsilon(x_n))u_\varepsilon^{\mathrm{in}}(\xi,x',\eta).
\end{equation*}
\notag
$$
The outer and inner expansions are set to be as follows;
$$
\begin{equation}
u^{\mathrm{ex}}_\varepsilon(x,\eta) =u_0(x,\eta)+\sum_{m=1}^{\infty}\varepsilon^m u_m(x,\eta)
\end{equation}
\tag{3.1}
$$
and
$$
\begin{equation}
u^{\mathrm{in}}_\varepsilon(\xi,x',\eta) =v_0(\xi,x',\eta)+\sum_{m=1}^{\infty}\varepsilon^m v_m(\xi,x',\eta).
\end{equation}
\tag{3.2}
$$
The aim of the formal construction of an asymptotic expansion is to specify the coefficients of the above two expansions. We write out problems for the coefficients of the outer expansion. To do this we substitute (3.1) into (2.3) and equate the coefficients of like powers of $\varepsilon$. Then for $u_0$ we obtain the equation and boundary condition on $\partial\Omega$ presented in (2.4), while for the other functions $u_m$ these are presented in (2.10). Next we write out problems for the coefficients of the inner expansion. In view of theorems on embedding Sobolev spaces in spaces of functions with continuous derivatives, condition (2.5) means that $f$ is infinitely differentiable for $|x_n|<\tau_0$. We expand $f$ in a Taylor series for $x_n\to 0$, and then make the change of variable $x_n=\varepsilon\xi_n$:
$$
\begin{equation}
f(x)=\sum_{m=0}^{\infty}\frac{1}{m!}\,\frac{\partial^m f}{\partial x_n^m}(x',0)x_n^m =\sum_{m=0}^{\infty}\frac{\varepsilon^m}{m!}\,\frac{\partial^m f}{\partial x_n^m}(x',0)\xi_n^m.
\end{equation}
\tag{3.3}
$$
Expanding $a(x',\varepsilon\xi_n,u^{\mathrm{in}}_\varepsilon)$ in an asymptotic series in powers of $\varepsilon$ we obtain (2.7), where the $T_m$ are some fixed polynomials of $\operatorname{Re} v_1,\,\dots,\,\operatorname{Re} v_m$ and $\operatorname{Im} v_1,\dots,\operatorname{Im} v_m$ with coefficients infinitely differentiable with respect to $x'$ and $v_0$, such that for each monomial
$$
\begin{equation*}
C(x',v_0) (\operatorname{Re} v_1)^{p_1}(\operatorname{Im} v_1)^{q_1}(\operatorname{Re} v_2)^{p_2}(\operatorname{Im} v_2)^{q_2}\dotsb (\operatorname{Re} v_m)^{p_m}(\operatorname{Im} v_m)^{q_m}
\end{equation*}
\notag
$$
we have In particular,
$$
\begin{equation*}
T_1(v_1)=\frac{\partial a}{\partial u_\mathrm{r}}(x',0,v_0)\operatorname{Re} v_1+\frac{\partial a}{\partial u_\mathrm{i}}(x',0,v_0)\operatorname{Im} v_1+\frac{\partial a}{\partial x_n}a(x',0,v_0)\xi_n.
\end{equation*}
\notag
$$
Substituting this expansion, (3.3) and (3.2) in (2.3) and collecting the coefficients of like powers of $\varepsilon$ we obtain problems (2.6) for the functions participating in the inner expansion. Now we match the inner and outer expansions. To do this we write the (asymptotic) Taylor series for the $u_m$ as $x_n\to\pm0$ and make the change of variable $x_n=\varepsilon\xi_n$:
$$
\begin{equation}
u_m(x,\eta)=\sum_{j=0}^{\infty}\frac{1}{j!}\,\frac{\partial^j u_m}{\partial x_n^j}(x',\pm0,\eta) x_n^j =\sum_{j=0}^{\infty} \frac{\varepsilon^j}{j!}\,\frac{\partial^j u_m}{\partial x_n^j}(x',\pm0,\eta) \xi_n^j.
\end{equation}
\tag{3.4}
$$
In accordance with the method of matched asymptotic expansions, these equalities mean that the functions $v_m$ have the following expansions as $\xi_n\to \pm\infty$:
$$
\begin{equation}
\begin{gathered} \, v_m(\xi,x',\eta)=P_m^\pm(x',\xi_n,\eta)+\frac{\partial u_{m-1}}{\partial x_n}(x',\pm 0,\eta)\xi_n+u_m(x',\pm 0,\eta)+o(1), \\ P_m^\pm:=\sum_{j=2}^{m}\frac{1}{j!}\,\frac{\partial^j u_{m-j}}{\partial x_n^j}(x',\pm 0,\eta)\xi_n^j. \nonumber \end{gathered}
\end{equation}
\tag{3.5}
$$
Boundary value problems (2.6), (3.5) have a $\square$-periodic structure with respect to $\xi'$, so we construct periodic solutions of these problems. To do this it suffices to replace (2.6), (3.5) by similar problems in $\Pi\setminus\overline{\omega^\eta}$ with periodic boundary conditions on the lateral faces of $\Pi$. After constructing suitable solutions of problems in $\Pi\setminus\overline{\omega^\eta}$, we find solutions of (2.6) and (3.5) by simple $\square$-periodic extension with respect to $\xi'$. Solutions of the above problems in $\Pi\setminus\overline{\omega^\eta}$ depend on $\eta$, so apart from the solvability of these problems we must also analyse the dependence of solutions on $\eta$. To do this, in the next sections we start by considering a model boundary value problem in $\Pi\setminus\overline{\omega^\eta}$, and then apply the results obtained to the problems for the $v_m$ stated above. This analysis is one of the most difficult steps in our work. § 4. Model problem for the coefficients of the inner expansionIn this section we consider a model boundary value problem in $\Pi\setminus\overline{\omega^\eta}$ for functions involved in the inner expansion. We investigate the solvability of the problem in question and establish some preliminary results on the dependence of its solution on the parameter $\eta$. 4.1. Stating the problemConsider the model boundary value problem
$$
\begin{equation}
-\Delta_\xi v=F \quad\text{in } \Pi\setminus\overline{\omega^\eta}, \qquad\frac{\partial v}{\partial\nu_\xi}=\phi \quad\text{on} \quad\partial\omega^\eta,
\end{equation}
\tag{4.1}
$$
$$
\begin{equation}
v|_{\xi_i=- b_i}=v|_{\xi_i= b_i}, \qquad\frac{\partial v}{\partial\xi_i}\bigg|_{\xi_i=- b_i}=\frac{\partial v}{\partial \xi_i}\bigg|_{\xi_i=b_i}, \quad i=1,\dots,n-1,
\end{equation}
\tag{4.2}
$$
where $F\in L_2(\Pi\setminus\omega^\eta)$ and $\phi\in L_2(\partial\omega^\eta)$ are some functions. We understand the solution to (4.1), (4.2) in the generalized sense. By a generalized solution of (4.1), (4.2) we mean a function $v$ that belongs to the space $W_2^1(\Pi_R\setminus\omega^\eta)$ for each $R>0$ and satisfies the integral identity
$$
\begin{equation*}
(\nabla_\xi v,\nabla_\xi w)_{L_2(\Pi_R\setminus\omega^\eta)}-(\phi,w )_{L_2(\partial\omega^\eta)}=(F,w)_{L_2(\Pi_R\setminus\omega^\eta)}
\end{equation*}
\notag
$$
for all functions $w\in C^2(\overline{\Pi\setminus\omega^\eta})$ satisfying periodic boundary conditions on the lateral faces of $\Pi$ and vanishing identically for $|\xi_n|>d>0$, where $d>0$ depends on the choice of $w$. We specify the behaviour of the solution of (4.1), (4.2) in what follows, as we investigate the solvability of the problem. 4.2. An operator equationIn this subsection we consider problem (4.1), (4.2) with compactly supported right-hand side $F$ and homogeneous boundary conditions on $\partial\omega^\eta$. We assume that $F$ vanishes outside $\Pi_{R_0}$ for some fixed $R_0>1$ such that $\overline{\omega^\eta}\subset \Pi_{R_0-1}$ for all $\eta\in (0,1]$. We look for a solution of this problem that is bounded at infinity, namely,
$$
\begin{equation}
v(\xi,\eta)=D_\pm(\eta)+o(1), \qquad\xi_n\to\pm\infty.
\end{equation}
\tag{4.3}
$$
Our immediate aim is to reduce (4.1)–(4.3) to an appropriate operator equation. The scheme of such a reduction repeats the scheme employed in [29], § 7.2, for problem (7.1)–(7.3), but it features some differences: in [29] the domain $ \Pi$ contains also a cavity with Dirichlet condition on the boundary. This ensures the unique solvability of the model problem for any right-hand sides of the equation and boundary conditions. There is no such cavity in our case, so that certain conditions for solvability arise. Furthermore, we analyze much more thoroughly the dependence of the solution on $\eta$ in comparison to [29]. Now we present briefly the general scheme of the argument in [29]; only the main necessary modifications will be discussed in detail. For an arbitrary $R>0$ set
$$
\begin{equation*}
\Pi_R^\pm:=\bigl\{\xi\colon \xi'\in\square,\,0<\pm\xi_n<R\bigr\}\quad\text{and} \quad \Pi^\eta:=\Pi_{R_0}\setminus\overline{\omega^\eta}.
\end{equation*}
\notag
$$
For an arbitrary function $g\in L_2(\Pi_{R_0})$ extended by zero to $\Pi\setminus\Pi_{R_0}$ we consider an auxiliary problem:
$$
\begin{equation*}
-\Delta_\xi V_1^\pm=g \quad\text{in } \Pi^\pm_0, \qquad V_1^\pm=0 \quad\text{on } \square\times\{0\},
\end{equation*}
\notag
$$
with periodic boundary conditions (4.2). In [29] we solved this problem by separating the variables:
$$
\begin{equation*}
\begin{gathered} \, V_1^\pm(\xi)=\sum_{k\in\mathbb{Z}^{n-1}} X_k^\pm(\xi_n)\exp(2\pi\mathrm{i} k_b\cdot\xi'), \\ X_k^\pm(\xi_n):= \int_{\Pi_{R_0}^\pm\setminus\Pi_{R_0-1}^\pm} J_k^\pm(\xi',\xi_n\mp (R_0-1),t)g(t)\, dt, \qquad k\ne0, \\ X_0^+(\xi_n):=\int_{\Pi_{R_0}^\pm\setminus\Pi_{R_0-1}^\pm}J_0^\pm(\xi_n\mp (R_0-1),t_n)g(t)\,dt, \\ J_0^+(\xi_n,t_n):=-\min\{\xi_n,t_n\}, \qquad J_0^-(\xi_n,t_n):=\max\{\xi_n,t_n\}, \\ \begin{aligned} \, J_k^\pm(\xi',\xi_n,t) &:=\frac{1}{4 \pi |k_b|}\bigl(\exp(\mp 2\pi |k_b|(\xi_n+t_n)) \\ &\qquad-\exp(-2\pi |k_b||\xi_n-t_n|)\bigr) \exp(2\pi\mathrm{i} k_b\cdot t'), \qquad k\ne0, \end{aligned} \\ k=(k_1,k_2,\dots,k_{n-1}), \qquad k_b:= \biggl(\frac{k_1}{2b_1},\frac{k_2}{2b_2},\dots, \frac{k_{n-1}}{2b_{n-1}}\biggr), \\ t=(t',t_n), \qquad t'=(t_1,t_2,\dots, t_{n-1}), \end{gathered}
\end{equation*}
\notag
$$
where ‘ $\cdot$ ’ denotes the inner product in $\mathbb{R}^{n-1}$. Since $g$ has compact support, for $\pm\xi_n>R_0$ the functions $X_0^\pm$ are constant:
$$
\begin{equation*}
X_0^\pm(\xi_n)\equiv D_\pm \quad\text{for } \pm\xi_n>R_0, \qquad D_\pm:= \int_{\Pi_{R_0}^\pm\setminus\Pi_{R_0-1}^\pm}|t_n| g(t)\,dt.
\end{equation*}
\notag
$$
It is also easy to see that for $|\xi_n|\geqslant R_0$
$$
\begin{equation*}
\begin{gathered} \, X_k^\pm(\xi)=D_k^\pm \exp(-2\pi |k_b|(\xi_n-R_0)), \\ \text{where } D_k^\pm:=-\frac{1}{2\pi |k_b|}\int_{\Pi_{R_0}^\pm\setminus\Pi_{R_0-1}^\pm} g(t)\exp(2\pi\mathrm{i} k_b\cdot t') \operatorname{sh} 2\pi |k_b| t_n \,dt, \end{gathered}
\end{equation*}
\notag
$$
and the constants $D_k^\pm$ satisfy
$$
\begin{equation*}
|D_k^\pm|\leqslant \frac{C \exp(2\pi |k_b| R_0)}{ |k_b|^{3/2}}\|g\|_{L_2(\Pi_{R_0}^\pm\setminus\Pi_{R_0-1}^\pm)},
\end{equation*}
\notag
$$
where the constant $C$ is independent of $k$ and $g$. Consider the function $V_1(\xi):=V_1^\pm(\xi)\Pi\cap\{\xi\colon \pm\xi_n>0\}$. By Lemma 7.1 in [29] it can be represented as $V_1=\mathcal{B}_1 g$, where $\mathcal{B}_1 $ is a bounded linear operator from $L_2(\Pi_{R_0})$ to $W_2^2(\Pi_R^+)\oplus W_2^2(\Pi_R^-)$ for each $R>0$. We also consider another auxiliary problem:
$$
\begin{equation*}
-\Delta_\xi V_2=g \quad\text{in } \Pi^\eta, \qquad V_2=V_1^\pm\quad\text{on } \square\times\{\pm R_0\}, \qquad \frac{\partial V_2}{\partial \nu_\xi}=0 \quad\text{on } \partial\omega^\eta,
\end{equation*}
\notag
$$
with periodic boundary conditions (4.2). Similarly to [29], § 7.2, we can show that this problem is uniquely solvable and $V_2$ has a representation $V_2=\mathcal{B}_2(\eta) g$, where $\mathcal{B}_2(\eta)$ is a linear bounded operator from $L_2(\Pi_{R_0})$ to $W_2^2(\Pi^\eta)$. We have the estimate
$$
\begin{equation}
\|V_2\|_{W_2^2(\Pi^\eta)}\leqslant C\|g\|_{L_2(\Pi_{R_0})},
\end{equation}
\tag{4.4}
$$
where the constant $C$ is independent of $V_2$ and $g$, but depends on $R_0$ and $\eta$. Let $\chi_1=\chi_1(\xi_n)$ be an even infinitely differentiable cut-off function that vanishes for $|\xi_n|<R_0-2/3$ and is equal to one for $|\xi_n|>R_0-1/3$. We construct a solution of (4.1), (4.2) in the following form:
$$
\begin{equation}
v(\xi)=(\mathcal{B}_3(\eta) g)(\xi)=\chi_1(\xi_n)V_1(\xi) +(1-\chi_1(\xi_n))V_2(\xi),
\end{equation}
\tag{4.5}
$$
where $\mathcal{B}_3(\eta)$ is a bounded linear operator from $L_2(\Pi_{R_0})$ to $W_2^2(\Pi^\eta)$ for all $R_0>0$. Similarly to how equation (7.12) was deduced in [29], we can show that $v$ solves the boundary value problem (4.1), (4.2) if $g$ solves the following equation: where $\mathcal{I}$ is the identity map, the operator $\mathcal{B}_4(\eta)$ is defined by
$$
\begin{equation}
\mathcal{B}_4(\eta) g=2 \frac{\partial(V_2-V_1)}{\partial\xi_n} \chi_1'+(V_2-V_1)\chi_1''
\end{equation}
\tag{4.7}
$$
and acts from $L_2(\Pi_{R_0})$ to $L_2(\Pi^\eta)$, and we extend $F$ by zero inside $\omega^\eta$. Note that by the definition (4.7) of $\mathcal{B}_4$ the image function is only distinct form zero in $\Pi_{R_0-1/3}\setminus \Pi_{R_0-2/3}$. For this reason, below we extend this image by zero inside $\omega^\eta$ and assume that $\mathcal{B}_4$ acts in the space $L_2(\Pi_{R_0})$. Also, since we have extended $F$ by zero inside $\omega^\eta$, solutions of (4.6) certainly vanish in $\omega^\eta$. The next lemma claims that the original boundary value problem (4.1)–(4.3) is equivalent to operator equation (4.6); we prove it similarly to Lemma 7.3 in [29]. Lemma 4.1. Equation (4.6) is equivalent to problem (4.1)–(4.3): for each solution $g$ of (4.6) problem (4.1)–(4.3) has a solution defined by (4.5), and for each solution $v$ of (4.1)–(4.3) equation (4.6) has a unique solution $g$ which is related to $v$ by (4.5). The proof of the next lemma is a literal reproduction of the proof of Lemma 7.3 in [29]. Lemma 4.2. $\mathcal{B}_4(\eta)$ is a compact linear operator in $L_2(\Pi_{R_0})$ for all $\eta\geqslant 0$. As $\mathcal{B}_4(\eta)$ is a compact operator, we can apply Fredholm’s alternative to (4.6). In particular, equation (4.6) is only solvable when its right-hand side is orthogonal to all linearly independent solutions of the corresponding homogeneous adjoint equation Lemma 4.3. Both the homogeneous equation (4.8) and the corresponding homogeneous equation (4.6) with $F=0$ have a unique nontrivial solution each. These solutions are given by Proof. Set $h_0=h_0(x)\equiv 1$ in $\Pi^\eta$. For each function $g\in L_2(\Pi^\eta)$ we have
$$
\begin{equation*}
((\mathcal{I}+\mathcal{B}_4(\eta))g,h_0)_{L_2(\Pi^\eta)}=-\int_{\Pi^\eta} \Delta_\xi \mathcal{B}_3(\eta)g\,d\xi=0.
\end{equation*}
\notag
$$
Therefore, $h_0\equiv 1$ is a solution of (4.8). We show that there are no other solutions.
By Fredholm’s alternative equation (4.6) with $F=0$ and the adjoint homogeneous equation (4.8) have the same number of linearly independent solutions. Equation (4.6) with $F=0$ is equivalent to boundary value problem (4.1)–(4.3) with homogeneous right-hand side and homogeneous boundary condition on $\partial\omega^\eta$. This problem has a unique solution, which is a constant. Hence equations (4.8) and (4.6) with $F=0$ have only one solution each. We have already found the solution of (4.8). In formula (4.5) the solution of (4.6) for $F=0$ corresponds to ${v\equiv 1}$. Repeating the arguments in the proof of Lemma 7.2 in [29] we can easily verify that this solution has the expression (4.9). The proof is complete. Let $\chi_2=\chi_2(\xi)$ be an infinitely differentiable function equal to one in a fixed neighbourhood of the cavity $\omega^\eta$ for all $\eta\in[0,1]$ and equal to zero outside some larger neighbourhood, which lies strictly inside $\Pi_{R_0-1}$. Consider the vector-valued function where $t$ is a positive real parameter. Clearly, this vector function is infinitely differentiable. It is a diffeomorphism of $\overline{\Pi}$ onto itself for $t\in[1-t_0,1+t_0]$, where $t_0$ is a fixed sufficiently small positive quantity. In a neighbourhood of the cavities $\omega^\eta$, the map $\Xi$ acts as a local dilation with coefficient $t$. Hence for any $\eta_1,\eta_2\in(0,1)$ such that $\eta_2\eta_1^{-1}\in[1-t_0,1+t_0]$ the map $\Xi(\eta_2\eta_1^{-1},\xi)$ takes the domain $\Pi\setminus\omega^{\eta_1}$ to $\Pi\setminus\omega^{\eta_2}$ and takes $\Pi^{\eta_1}$ to $\Pi^{\eta_2}$. Let $\Xi^{-1}(t,\xi)$ be the diffeomorphism inverse to $\Xi$.Now fix some $\eta_0\in(0,1]$ and introduce the new variables $\widetilde{\xi}=\Xi( \eta_0 \eta^{-1},\xi)$ in $\overline{\Pi}\setminus\omega^\eta$, where $\eta\in(0,1]$ is an arbitrary number such that $ \eta_0\eta^{-1}\in[1-t_0,1+t_0]$. By the properties of $\Xi$ the variables $\widetilde{\xi}$ range in $\Pi\setminus\omega^{\eta_0}$. Let $\Upsilon=\Upsilon(\xi,\eta)$ denote the Jacobian of this change of variables:
$$
\begin{equation}
\Upsilon(\xi,\eta):= {\det}^{-1} \biggl( \frac{\partial\Xi}{\partial\xi_1} ( \eta_0\eta^{-1},\xi) \ \dotsb \ \frac{\partial\Xi}{\partial\xi_n} ( \eta_0 \eta^{-1},\xi) \biggr).
\end{equation}
\tag{4.11}
$$
By the definition (4.10) of $\Xi$ the function $\Upsilon$ has the representation
$$
\begin{equation}
\Upsilon(\xi,\eta)=1+(\eta-\eta_0)\Upsilon_1(\widetilde{\xi},\eta),
\end{equation}
\tag{4.12}
$$
where $\Upsilon_1$ is infinitely differentiable with respect to $(\xi,\eta)\in \overline{\Pi}\setminus\omega^{\eta_0}\times[\eta_0-\delta(\eta_0), \eta_0+ \delta(\eta_0)]$ for some $\delta(\eta_0)>0$ and vanishes outside the support of $\chi_2$ for all $\eta$ under consideration. We have
$$
\begin{equation}
\Upsilon(\xi,\eta)=\frac{\eta^n}{\eta_0^n}\quad \text{on} \quad \{\xi\colon \chi_2(\xi)=1\}.
\end{equation}
\tag{4.13}
$$
Also note the obvious formula
$$
\begin{equation}
\Upsilon \Delta_\xi=\Delta_{\widetilde{\xi}}+(\eta-\eta_0)\mathcal{B}_5(\eta_0,\eta),
\end{equation}
\tag{4.14}
$$
where $\mathcal{B}_5$ is some differential operator of the second order whose coefficients have compact support and are distinct from zero only on the support of $\chi_2$. These coefficients are infinitely differentiable with respect to $(\widetilde{\xi},\eta)$, where $\widetilde{\xi}$ ranges in the support of $\chi_2$ and $\eta\in[\eta_0-\delta(\eta_0),\eta_0+\delta(\eta_0)]$, and they are uniformly bounded together with all derivatives with respect to the space variables and $\eta$. The operator $\mathcal{B}_5$ satisfies
$$
\begin{equation}
(\eta-\eta_0)\mathcal{B}_5(\eta_0,\eta)=\Upsilon \Delta_{\xi} \Upsilon^{-1}-\Delta_{\widetilde{\xi}}.
\end{equation}
\tag{4.15}
$$
The following lemma describes how $\mathcal{B}_4$ depends on $\eta$. Proof. For fixed function $g\in L_2(\Pi_{R_0})$ let $V_2^0$ denote the solution of the problem
$$
\begin{equation*}
-\Delta_\xi V_2^0=g \quad\text{in } \Pi_{R_0}, \qquad V_2^0=V_1^\pm \quad\text{on } \square\times\{\pm R_0\}
\end{equation*}
\notag
$$
with the periodic boundary conditions (4.2). We set it to be equal to $V_2$ for $\eta=0$. Then we can extend the operator $\mathcal{B}_4$ to $\eta=0$ using the same formula (4.7) as before. It also follows directly from this formula that for each pair $\eta_1,\eta_2\in[0,1]$ we have
$$
\begin{equation}
\bigl(\mathcal{B}_4(\eta_2)-\mathcal{B}_4(\eta_1)\bigr)=2\chi_1' \, \frac{\partial}{\partial \xi_n}(\widetilde{V}_{\eta_2}-\widetilde{V}_{\eta_1}) + ( \widetilde{V}_{\eta_2}-\widetilde{V}_{\eta_1})\chi_1'',
\end{equation}
\tag{4.16}
$$
where
$$
\begin{equation*}
\widetilde{V}_\eta(\xi):=V_2(\xi,\eta) - \chi_1(\xi_n) V_1(\xi,\eta).
\end{equation*}
\notag
$$
The function $\widetilde{V}_\eta$ is a solution of the equation
$$
\begin{equation}
\mathcal{B}_6(\eta) \widetilde{V}_\eta=\widetilde{g}, \qquad \widetilde{g}:=(1-\chi_1) g - 2\chi_1'\, \frac{\partial V_1}{\partial\xi_n} - \chi_1'' V_1,
\end{equation}
\tag{4.17}
$$
where $\mathcal{B}_6(\eta)$ is the operator $-\Delta_{\xi}$ in the domain $\Pi^\eta$ with Dirichlet boundary condition on $\square\times\{\pm R_0\}$, Neumann boundary condition on $\partial\omega^\eta$ and the periodic boundary conditions (4.2). Such an operator is selfadjoint and lower semibounded on its domain, which consists of the functions in $W_2^2(\Pi^\eta)$ satisfying the indicated boundary conditions. It follows directly from Theorem 1.1 and Lemma 2.1 in [34] that for sufficiently small $\eta$ we have
$$
\begin{equation*}
\|\widetilde{V}_\eta-\widetilde{V}_0\|_{W_2^1(\Pi_{R_0}\setminus \Pi_{R_0-2/3})}\leqslant C\eta \|\widetilde{g}\|_{L_2(\Pi_{R_0})}
\end{equation*}
\notag
$$
where the constant $C$ is independent of $\eta$ and $\widetilde{g}$. Hence it follows from (4.16), since $\mathcal{B}_1$ is bounded, that the operator $\mathcal{B}_4(\eta)$ is continuous at $\eta=0$.
Fix some $\eta_0\in(0,1]$ and take an arbitrary $\eta\in(0,1]$ sufficiently close to $\eta_0$. Then we go over to the variables $\widetilde{\xi}=\Xi(\eta_0\eta^{-1},\xi)$ in (4.17) taking (4.14) into account. We obtain the following equation for the $\widetilde{V}_\eta$ as a function of $\widetilde{\xi}$:
$$
\begin{equation*}
\bigl(\mathcal{B}_6(\eta_0)+(\eta-\eta_0)\mathcal{B}_5(\eta_0,\eta)\bigr) \widetilde{V}_\eta(\Xi^{-1}) =\Upsilon \widetilde{g}.
\end{equation*}
\notag
$$
Bearing in mind the properties of the coefficients of $\mathcal{B}_5$ described above we conclude immediately that this equation can be solved as follows:
$$
\begin{equation*}
\widetilde{V}_\eta (\Xi^{-1})=\bigl(\mathcal{B}_6(\eta_0)+(\eta-\eta_0)\mathcal{B}_5(\eta_0,\eta)\bigr)^{-1} \Upsilon \widetilde{g}.
\end{equation*}
\notag
$$
From this and (4.12), since the coefficients of $\mathcal{B}_5$ are uniformly bounded, it follows that
$$
\begin{equation*}
\|\widetilde{V}_\eta(\Xi^{-1})-\widetilde{V}_{\eta_0}\|_{W_2^2(\Pi^{\eta_0})} \leqslant C|\eta-\eta_0|\|\widetilde{g}\|_{L_2(\Pi_{R_0})},
\end{equation*}
\notag
$$
where the constant $C$ is independent of $\eta$ and $\widetilde{g}$. Taking the definition of $\Xi$ into account we obtain
$$
\begin{equation*}
\|\widetilde{V}_\eta-\widetilde{V}_{\eta_0}\|_{W_2^1(\Pi_{R_0}\setminus \Pi_{R_0-1})}\leqslant C|\eta-\eta_0|\|\widetilde{g}\|_{L_2(\Pi_{R_0})},
\end{equation*}
\notag
$$
where the constant $C$ is independent of $\eta$ and $\widetilde{g}$. This and (4.16) are sufficient to see the continuity of $\mathcal{B}_4(\eta)$ at $\eta_0$. The proof is complete. Since (4.8) has only a constant solution, equation (4.6) is solvable if Then by Banach’s inverse operator theorem there exists a bounded inverse operator $(\mathcal{I}+\mathcal{B}_4(\eta))^{-1}\colon \mathfrak{L}_*\to \mathfrak{L}$, where we set
$$
\begin{equation*}
\mathfrak{L}_*:=\{g\in L_2(\Pi_{R_0})\colon (g,h_0)_{L_2(\Pi^\eta)}=0,\ g=0\ \text{in }\omega^\eta\}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\mathfrak{L}:=\{g\in L_2(\Pi_{R_0})\colon (g,g_0)_{L_2(\Pi^\eta)}=0, \ g=0\ \text{in } \omega^\eta\}.
\end{equation*}
\notag
$$
Since the operator $\mathcal{B}_4(\eta)$ is continuous as shown in Lemma 4.4, the inverse operator $(\mathcal{I}+\mathcal{B}_4(\eta))^{-1}$ is uniformly bounded for $\eta\in[0,1]$. The solution of operator equation (4.6) has the following form:
$$
\begin{equation}
g=\widehat{g}+c g_0, \qquad \widehat{g}:=(\mathcal{I}+\mathcal{B}_4(\eta))^{-1}F,
\end{equation}
\tag{4.18}
$$
where $c$ is an arbitrary constant and the function $g_0$ is defined in (4.9). We have
$$
\begin{equation}
\|\widehat{g}\|_{L_2(\Pi^\eta)}\leqslant C\|F\|_{L_2(\Pi^\eta)},
\end{equation}
\tag{4.19}
$$
where the constant $C$ is independent of $F$. By (4.18) the solution of the boundary value problem (4.1), (4.2) with homogeneous boundary conditions on $\partial\omega^\eta$ has the following form: where $\widehat{v}$ is a solution of problem (4.1), (4.2) with homogeneous boundary conditions on $\partial\omega^\eta$ that corresponds to the solution $\widehat{g}$ of operator equation (4.6) in the sense described by Lemma 4.1, and $c$ is the same constant as in (4.18). The function $\widehat{v}$ has the following asymptotic behaviour at infinity:
$$
\begin{equation*}
\widehat{v}(\xi,\eta)=\widehat{D}_\pm(\eta)+o(1), \qquad \xi_n\to\pm\infty,
\end{equation*}
\notag
$$
where the constants $\widehat{D}_\pm$ are defined by
$$
\begin{equation}
\widehat{D}_\pm= \int_{\Pi_{R_0}^\pm\setminus\Pi_{R_0-1}^\pm}|t_n| \widehat{g}(t)\,dt.
\end{equation}
\tag{4.20}
$$
For $|\xi_n|>R_0$ the function $\widehat{v}$ has the following form:
$$
\begin{equation*}
\begin{gathered} \, \widehat{v}(\xi,\eta)=\widehat{D}_\pm(\eta)+ \sum_{k\in\mathbb{Z}^{n-1}}\widehat{D}_k^\pm(\eta) \exp(-2\pi |k_b||\xi_n|) \exp(2\pi\mathrm{i} k_b\cdot\xi'), \qquad \pm \xi_n> R_0, \\ \widehat{D}_k^\pm:=-\frac{1}{2\pi |k_b|} \int_{\Pi_{R_0}^\pm\setminus\Pi_{R_0-1}^\pm} \widehat{g}(t) \exp(2\pi\mathrm{i} k_b\cdot t') \operatorname{sh} 2\pi |k_b| t_n \,dt. \end{gathered}
\end{equation*}
\notag
$$
By this equality and (4.19) we have
$$
\begin{equation*}
|\widehat{D}_k^\pm|\leqslant \frac{C \exp(2\pi |k_b| R_0)}{ |k_b|^{3/2}}\|F\|_{L_2(\Pi^\eta)},
\end{equation*}
\notag
$$
where the constant $C$ is independent of $k$, $F$ and $\eta$. Since the operator $\mathcal{B}_1$ is bounded, it follows from (4.5) and inequalities (4.4) and (4.19) that
$$
\begin{equation*}
\|\widehat{v}\|_{W_2^1(\Pi^\eta)}\leqslant C\|F\|_{L_2(\Pi^\eta)},
\end{equation*}
\notag
$$
where the constant $C$ is independent of $F$ and $\eta$. Now we add the constant $(\widehat{D}_++\widehat{D}_-)/2$ to the solution $\widehat{v}$ of problem (4.1), (4.2) with homogeneous boundary condition on $\partial\omega^\eta$. We denote the result by $\widetilde{v}$. Then $\widetilde{v}$ has the following asymptotic behaviour at infinity:
$$
\begin{equation*}
\widetilde{v}(\xi,\eta)=\widetilde{D}_\pm(\eta)+o(1), \qquad \xi_n\to\pm\infty, \qquad\widetilde{D}_\pm=\widehat{D}_\pm-\frac{1}{2}(\widehat{D}_++\widehat{D}_-).
\end{equation*}
\notag
$$
The constants $\widetilde{D}_\pm$ satisfy $\widetilde{D}_++\widetilde{D}_-=0$. From (4.20) and (4.19) we obtain
$$
\begin{equation*}
|\widehat{D}_++\widehat{D}_-|\leqslant C\|F\|_{L_2(\Pi^\eta)},
\end{equation*}
\notag
$$
where the constant $C$ is independent of $F$. Now from (4.5) and inequalities (4.4) and (4.19), since $\mathcal{B}_1$ is a bounded operator, we obtain
$$
\begin{equation*}
\|\widetilde{v}\|_{W_2^1(\Pi^\eta)}\leqslant C\|F\|_{L_2(\Pi^\eta)},
\end{equation*}
\notag
$$
where the constant $C$ is independent of $F$ and $\eta$. Thus we have proved the following. Lemma 4.5. Let $F\in L_2(\Pi^\eta)$ be a function vanishing outside $\Pi_{R_0}$ such that Then boundary value problem (4.1), (4.2) with homogeneous boundary conditions on $\partial\omega^\eta$ is solvable. This problem with homogeneous boundary condition on $\partial\omega^\eta$ has a unique solution that has the asymptotic behaviour (4.3) as $\xi_n\to\pm\infty$, where the constants $D_\pm$ satisfy $D_++D_-=0$. The general solution of problem (4.1), (4.2) with homogeneous boundary condition on $\partial\omega^\eta$ is different from this solution by an arbitrary constant. For $|\xi_n|>R_0$ the function $v$ has the form The estimates hold, where the constant $C$ is independent of $F$ and $\eta$.4.3. The solvability of the model problemIn this subsection we investigate the solvability of model problem (4.1), (4.2). Lemma 4.6. Let $F\in L_2(\Pi^\eta)$ be a function vanishing for $\Pi_{R_0}$. Then the boundary value problem (4.1), (4.2) is solvable if and only if For $|\xi_n|>R_0$ the function $v$ has the form (4.21). The following inequalities hold: and where the constant $C$ is independent of $F$, $\phi$ and $\eta$. Proof. Consider the problem Making the substitution $\widetilde{\xi}:=\xi\eta^{-1}$ we rewrite it as follows: where $\widetilde{u}(\widetilde{\xi}):=u(\widetilde{\xi}\eta)$ and $\widetilde{\phi}(\widetilde{\xi}):=\eta \phi(\widetilde{\xi}\eta)$.
The Green’s function of (4.25) has the form
$$
\begin{equation}
G(\widetilde{\xi},y)=\frac{1}{\sigma_n(n-2)|\widetilde{\xi}-y|^{n-2}}+G_1(\widetilde{\xi},y),
\end{equation}
\tag{4.26}
$$
where the function $G_1$ belongs to the space $C^1(\mathbb{R}^n)$ with respect to $\widetilde{\xi}$, satisfies the equation $\Delta_{\widetilde{\xi}} G_1(\widetilde{\xi}, y)=0$, and $G_1(\widetilde{\xi},y)= O( |\widetilde{\xi}\,{-}\,y|^{-n+1})$ as $\xi_n\to\infty$. The function $G$ has the following properties:
$$
\begin{equation}
G(\widetilde{\xi},y)=G(y, \widetilde{\xi}),\quad \widetilde{\xi},y\in\mathbb{R}^n\setminus\overline{\omega},\quad\text{and} \quad \frac{\partial G}{\partial \nu_{\widetilde{\xi}}}(\widetilde{\xi},y)=0, \quad \widetilde{\xi}\in\partial\omega, \quad y\in\mathbb{R}^n\setminus\overline{\omega}.
\end{equation}
\tag{4.27}
$$
We can represent the solution of (4.25) as
$$
\begin{equation}
\widetilde{u}(\widetilde{\xi})=\int_{\partial\omega} G(\widetilde{\xi},y)\widetilde{\phi}(y)\,ds.
\end{equation}
\tag{4.28}
$$
By [35], Ch. I, § 1.6, the function $\widetilde{u}$ belongs to $C^1(\mathbb{R}^n)$.
We seek the solution of (4.1), (4.2) in the form $v=\chi_2 u+\widetilde{v}$. The function $\widetilde{v}$ is a solution of the problem
$$
\begin{equation}
\Delta_{\xi} \widetilde{v}=\widetilde{F} \quad\text{in } \Pi^\eta, \qquad \frac{\partial \widetilde{v}}{\partial \nu_{\xi}}=0 \quad\text{on } \partial \omega^{\eta}, \qquad \widetilde{F}:=F -2 \sum_{i=1}^n \frac{\partial u}{\partial \xi_i}\, \frac{\partial \chi_2}{\partial \xi_i} - u \Delta_{\xi}\chi_2,
\end{equation}
\tag{4.29}
$$
with periodic boundary conditions (4.2). By the properties of $\chi_2$ and $G$ we have
$$
\begin{equation}
\|\widetilde{F}\|_{L_2(\Pi^\eta)}\leqslant C \bigl(\|F\|_{L_2(\Pi^\eta)} +\eta^{n-1}\|\phi\|_{L_2(\partial\omega^\eta)}\bigr),
\end{equation}
\tag{4.30}
$$
where the constant $C$ is independent of $F$, $\phi$ and $\eta$.
By Lemma 4.5 problem (4.29), (4.2) is solvable of $\displaystyle\int _{\Pi^\eta} \widetilde{F}\,d\xi=0$. Now we integrate by parts as follows:
$$
\begin{equation*}
\int_{\Pi^\eta}\widetilde{F}\,d\xi=\int_{\Pi^\eta} (F-\Delta_{\xi}(\chi_2 u))\, d\xi=\int_{\Pi^\eta}F\,d\xi +\int_{\partial\omega^\eta} \phi \, ds.
\end{equation*}
\notag
$$
Hence condition (4.22) ensures the solvability of (4.29), (4.2). When it is satisfied, this problem has a unique solution $\widetilde{v}$ with asymptotic behaviour (4.3) as $\xi_n\to\pm\infty$, where the constants $D_\pm$ satisfy $D_++D_-=0$. For $|\xi_n|>R_0$ this solution has the form (4.21), and we have
$$
\begin{equation*}
\sup_{k\in\mathbb{Z}^{n-1}} \exp(-2\pi |k_b|)|D_k^\pm(\eta)|\leqslant C\|\widetilde{F}\|_{L_2(\Pi^\eta)}\quad\text{and} \quad \|\widetilde{v}\|_{W_2^1(\Pi^\eta)}\leqslant C\|\widetilde{F}\|_{L_2(\Pi^\eta)},
\end{equation*}
\notag
$$
where the constants $C$ are independent of $F$, $\phi$ and $\eta$. Returning to the function $v$ and taking (4.30) into account we obtain the required result. The lemma is proved. § 5. Estimates for the maximum of the solution of the model problem and for its derivativesOur aim in this section is to give estimates for the maximum modulus of the solutions of (4.1), (4.2) and for the moduli of its derivatives. We assume that $F$ and $\phi$ are functions in $C^{\vartheta}(\overline{\Pi^\eta})$ and $C^{1+\vartheta}(\partial\omega^\eta)$, respectively, where $\vartheta\in(0,1)$ is a fixed number. Then by Schauder’s estimates (see [33], Ch. III, §§ 2 and 3) model boundary value problem (4.1), (4.2) is solvable and its solution belongs to $C^{2+\vartheta}(\overline{\Pi})$. The classical Schauder estimates do not allow us to describe the dependence of the norm of this solution in $C^{2+\vartheta}$ on the parameter $\eta$ for small values of this parameter. The approach described in [29], § 7.5, gives only very crude estimates, particularly for derivatives. So in this section we establish more refined estimates than those in [29], § 7.5, where we used other techniques. 5.1. An estimate for the maximum of the solutionIn this subsection we estimate the maximum modulus of the solution of (4.1), (4.2). Lemma 5.1. The unique solution of problem (4.1), (4.2) that exists by Lemma 4.6 satisfies the inequality where the constant $C$ is independent of the functions $F$ and $\phi$ and the parameter $\eta$. Proof. Generally speaking, the proof uses the pattern of the proof of (7.49) in [29] in the case when $\eta<\eta_0$.
The function $\widetilde{v}=(1-\chi_1)v$ (recall that we defined the cut-off function $\chi_1$ before (4.5)) is a solution of the problem
$$
\begin{equation*}
\begin{gathered} \, -\Delta_\xi \widetilde{v}=G \quad\text{in } \Pi_{2R_0}\setminus\overline{\omega^\eta}, \qquad \frac{\partial \widetilde{v}}{\partial\nu_\xi}=\phi \quad\text{on } \partial\omega^\eta, \\ \widetilde{v}=0 \quad\text{on } \square\times\{-2R_0,\, 2R_0\}, \\ G=(1-\chi_1)F + 2\nabla_\xi\chi_1\cdot \nabla_\xi v + v\Delta_\xi \chi_1, \end{gathered}
\end{equation*}
\notag
$$
with periodic boundary conditions (4.2).
Let $\widetilde{\xi}=(\widetilde{\xi}_1,\dots,\widetilde{\xi}_n)$ be Cartesian variables in $\mathbb{R}^n$. Consider the outer boundary value problems
$$
\begin{equation*}
\begin{gathered} \, -\Delta_{\widetilde{\xi}} Y_i=0 \quad\text{in } \mathbb{R}^n\setminus\overline{\omega}, \qquad i=0,1,2, \\ \frac{\partial Y_0}{\partial\nu_{\widetilde{\xi}}}=1, \qquad \frac{\partial Y_1}{\partial\nu_{\widetilde{\xi}}}=\frac{\partial \widetilde{\xi}_n}{\partial\nu_{\widetilde{\xi}}}, \qquad \frac{\partial Y_2}{\partial\nu_{\widetilde{\xi}}}=\widetilde{\xi}_n\, \frac{\partial \widetilde{\xi}_n}{\partial\nu_{\widetilde{\xi}}} \quad\text{on } \partial\omega, \end{gathered}
\end{equation*}
\notag
$$
where $\nu_{\widetilde{\xi}}$ is the unit inward normal to $\partial\omega$. These problems have classical solutions in the space $C^\infty(\mathbb{R}^n\setminus\omega)\cap C^{2+\vartheta}(\overline{B_R(0)})$, where $R$ is a sufficiently large fixed number, which have the following asymptotic behaviour at infinity:
$$
\begin{equation*}
Y_i(\widetilde{\xi})=O(|\widetilde{\xi}|^{-n+2}), \qquad \widetilde{\xi}\to\infty, \quad i=0,1,2.
\end{equation*}
\notag
$$
The functions $Y_i$, $i=0,1,2$, have the estimates
$$
\begin{equation}
\bigl|\chi_2(\xi) Y_i ( \xi \eta^{-1}) \bigr|\leqslant C\quad\text{and} \quad \bigl|\Delta_\xi\eta \chi_2(\xi)Y_i (\xi \eta^{-1})\bigr| \leqslant C\eta^{n-1},
\end{equation}
\tag{5.2}
$$
where $C$ is a constant independent of $\xi$ and $\eta$, and $\chi_2$ is the cut-off function introduced after Lemma 4.3.
Consider the functions
$$
\begin{equation*}
Y_3(\xi,\eta):=(\xi_n^2-4R_0^2)-\chi_2(\xi) \bigl(2\eta^2 Y_2( \xi \eta^{-1}) +2\eta \xi_n Y_1( \xi \eta^{-1})\bigr)
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
Y_4(\xi,\eta):=\eta \chi_2(\xi) Y_0 ( \xi \eta^{-1}).
\end{equation*}
\notag
$$
By the definitions of the $Y_i$, $i=0,1,2$, and (5.2) the functions $Y_3$ and $Y_4$ satisfy
$$
\begin{equation*}
-\Delta_\xi Y_3\geqslant 1 \quad\text{in } \Pi^\eta, \qquad \frac{\partial Y_3}{\partial\nu_\xi}=0 \quad\text{on } \partial\omega^\eta,
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
|\Delta_\xi Y_4|\leqslant C\eta^{n-1} \quad\text{in } \Pi^\eta, \qquad \frac{\partial Y_4}{\partial\nu_\xi}=1 \quad\text{on } \partial\omega^\eta,
\end{equation*}
\notag
$$
with periodic boundary conditions (4.2), where $C$ is a constant independent of $\xi$ and $\eta$. Consider the functions
$$
\begin{equation*}
u\pm 2\|\phi\|_{C(\overline{\Pi^\eta})} Y_4 + 2\bigl(\|G\|_{C(\overline{\Pi^\eta})} +C\eta^n\|\phi\|_{C(\overline{\Pi^\eta})}\bigr) Y_3.
\end{equation*}
\notag
$$
Similarly to how we deduced (7.53) and (7.54) in [29], we can show that
$$
\begin{equation*}
|u|\leqslant 2\|\phi\|_{C(\overline{\Pi^\eta})} Y_4 + 2\bigl(\|G\|_{C(\Pi^\eta)} +C\eta^n\|\phi\|_{C(\overline{\Pi^\eta})}\bigr) Y_3.
\end{equation*}
\notag
$$
From this, (5.2) and the definition of $Y_3$ and $Y_4$ we obtain
$$
\begin{equation*}
\|\widetilde{v}\|_{C(\overline{\Pi^\eta})}\leqslant C \bigl(\|G\|_{C(\overline{\Pi^\eta})}+\eta\|\phi\|_{C(\partial\omega^\eta)}\bigr),
\end{equation*}
\notag
$$
where the constant $C$ is independent of $F$, $\phi$ and $\eta$. This is sufficient to establish (5.1). The proof is complete. 5.2. An estimate for the maximum of the derivatives of the solutionIn this subsection we obtain estimates for the maximum modulus of the derivatives of the unique solution of (4.1), (4.2) that exists by Lemma 4.6. Set $\breve{v}(\xi):=(1-\chi_2(\xi))v(\xi)$, where we recall that $\chi_2$ is the cut-off function in the proof of Lemma 4.6. We extend $F$ and $\breve{v}$ by zero inside the cavities $\omega^\eta$. By the definition of $\chi_2$ the function $\breve{v}$ vanishes in a neighbourhood of $\omega^\eta$, so the extension of $\breve{v}$ by zero inside $\omega^\eta$ does not affect its smoothness properties and we have $\breve{v}\in C^{2+\vartheta}(\overline{\Pi})$. The function $\breve{v}$ solves the problem
$$
\begin{equation}
-\Delta_{\xi}\breve{v}=F_1 \quad \text{in } \Pi, \qquad F_1:=(1-\chi_2)F+2\sum_{i=1}^{n}\frac{\partial \chi_2}{\partial \xi_i}\,\frac{\partial v}{\partial \xi_i}+v\Delta_{\xi}\chi_2,
\end{equation}
\tag{5.3}
$$
with periodic boundary conditions (4.2), where we assume that $F_1$ is extended by zero inside $\omega^\eta$. The function $F_1$ vanishes in a neighbourhood of $\omega^\eta$ and belongs to $C^\vartheta({\overline{\Pi}})$. Note that the problem (5.3), (4.2) is considered in the whole domain $\Pi$ without the cavity. For this problem we can construct the Green’s function explicitly. Namely, using the ideas from [24], § 3.1, we can prove the following lemma. Lemma 5.2. The Green’s function of (5.3), (4.2) is given by Since the solution of (5.3), (4.2) belongs to $C^{2+\vartheta}(\overline{\Pi})$, it has the representation
$$
\begin{equation}
\breve{v}(\xi)=\int_{\Pi_{R_0}} G_{\Pi}(\xi-y) F_1(y)\,dy.
\end{equation}
\tag{5.6}
$$
Lemma 5.3. The inequality holds, where the constant $C$ is independent of the functions $v$ and $F$ and the parameter $\eta$. Proof. By (5.6) and the definition of $F_1$ in (5.3) we have
$$
\begin{equation*}
\begin{aligned} \, \breve{v}(\xi)&\,{=}\int_{\Pi_{R_0}}\!(1\,{-}\,\chi_2(y))(y)F(y)G_{\Pi}(\xi\,{-}\,y)\,dy \,{+}\, 2\sum_{i=1}^n \int_{\Pi_{R_0}} \frac{\partial v}{\partial y_i}(y) \frac{\partial \chi_2}{\partial y_i}(y) G_{\Pi}(\xi\,{-}\,y)\,dy \\ &\qquad +\int_{\Pi_{R_0}} v(y) G_{\Pi}(\xi-y)\Delta_y \chi_2(y)\,dy. \end{aligned}
\end{equation*}
\notag
$$
We integrate by parts in the second integral on the right:
$$
\begin{equation*}
\begin{aligned} \, \breve{v}(\xi)&=\int_{\Pi_{R_0}}\bigl((1-\chi_2(y))F(y)-v(y)\Delta_y \chi_2(y)\bigr) G_{\Pi}(\xi-y)\,dy \\ &\qquad +2\sum_{i=1}^n \int_{\Pi_{R_0}} v(y) \frac{\partial \chi_2}{\partial y_i}(y) \, \frac{\partial G_{\Pi}}{\partial \xi_i}(\xi-y)\,dy. \end{aligned}
\end{equation*}
\notag
$$
Using formula (1.24) from [33], Ch. III, § 1, we find that
$$
\begin{equation}
\begin{aligned} \, \notag \frac{\partial\breve{v}}{\partial\xi_j}(\xi) &=\int_{\Pi_{R_0}}\bigl((1-\chi_2(y))F(y)- v(y)\Delta_y \chi_2(y)\bigr)\,\frac{\partial G_{\Pi}}{\partial\xi_j}(\xi-y)\,dy \\ \notag &\qquad-2\lim_{\rho\to+0} \int_{\Pi\cap\{y\colon |\xi-y|\geqslant\rho\}}v(y) \sum_{i=1}^n\frac{\partial \chi_2}{\partial y_i}(y)\, \frac{\partial^2 G_{\Pi}}{\partial \xi_i\, \partial\xi_j}(\xi-y)\,dy \\ &\qquad-\frac{2\delta_{ij}}{n} v(\xi)\, \frac{\partial\chi_2}{\partial\xi_i}(\xi), \end{aligned}
\end{equation}
\tag{5.8}
$$
where $j=1,\dots,n$ and $\delta_{ij}$ is the Kronecker delta. Since $\widetilde{G}_\Pi$ is infinitely differentiable in $\overline{\Pi}$, it follows directly from the formula for $G_\Pi$ in (5.4) and the asymptotic expression (5.5) that all the derivatives of $\widetilde{G}_\Pi$ are uniformly bounded in $\overline{\Pi}$. Taking this and the fact that $G_\Pi$ is $\square$-periodic in $\xi'$ into account we estimate the first integral on the right-hand side of the above equality:
$$
\begin{equation}
\begin{aligned} \, \notag & \biggl|\int_{\Pi_{R_0}}\bigl((1-\chi_2(y))F(y)- v(y)\Delta_y \chi_2(y)\bigr)\, \frac{\partial G_{\Pi}}{\partial\xi_j}(\xi-y)\,dy \biggr| \\ \notag &\qquad \leqslant C \Bigl(\max_{\overline{\Pi_{R_0}}}|F| +\max_{\overline{\Pi_{R_0}}}|v|\Bigr) \int_{\Pi_{R_0}}\biggl( \frac{1}{|\xi-y|^{n-1}} +\biggl|\frac{\partial\widetilde{G}_\Pi}{\partial\xi_j}(\xi-y)\biggr|\biggr)\,dy, \\ &\qquad \leqslant C \Bigl(\max_{\overline{\Pi_{R_0}}}|F| +\max_{\overline{\Pi_{R_0}}}|v|\Bigr) \int_{\Pi_{R_0}}\biggl( \frac{1}{|\xi-y|^{n-1}} + 1 \biggr)\,dy, \end{aligned}
\end{equation}
\tag{5.9}
$$
where $C$ is a constant independent of $F$ and $\xi$. Let $\xi\notin\Pi_{2R_0}$. Then $|\xi-y|\geqslant R_0>1$, which immediately implies that
For $\xi\in\Pi_{2R_0}$ and $y\in\Pi_{R_0}$ we have
$$
\begin{equation*}
|\xi-y|<\rho_1, \qquad \rho_1:=3R_0+2\biggl(\sum_{i=1}^{n-1}b_i^2\biggr)^{1/2}.
\end{equation*}
\notag
$$
This leads to the estimate
$$
\begin{equation*}
\int_{\Pi_{R_0}} \frac{1}{|\xi-y|^{n-2}} \,dy < \int_{B_{\rho_1}(0)} \frac{dt}{|t|^{n-1}} \leqslant \rho_1^n |B_1(0)|.
\end{equation*}
\notag
$$
Using this and (5.10) we can continue the estimates in (5.9):
$$
\begin{equation}
\biggl|\int_{\Pi_{R_0}}\bigl((1-\chi_2(y))F(y)- v(y)\Delta_y \chi_2(y)\bigr) \, \frac{\partial G_{\Pi}}{\partial\xi_j}(\xi-y)\,dy \biggr| \leqslant C \Bigl(\max_{\overline{\Pi_{R_0}}}|F| +\max_{\overline{\Pi_{R_0}}}|v|\Bigr),
\end{equation}
\tag{5.11}
$$
where the constant $C$ is independent of $\xi$, $y$, $j$ and $F$.
The integral under the sign of limit on the right-hand side of (5.8) has the representation
$$
\begin{equation}
\begin{aligned} \, \notag & \sum_{i=1}^n \int_{\Pi\cap\{y\colon |\xi-y|\geqslant\rho\}}v(y)\, \frac{\partial \chi_2}{\partial y_i}(y)\, \frac{\partial^2 G_{\Pi}}{\partial \xi_i\, \partial\xi_j}(\xi,y)\,dy \\ \notag &\qquad =\sum_{i=1}^n \int_{\Pi\cap\{y\colon \rho\leqslant|\xi-y|\leqslant \rho_2\}}v(y)\, \frac{\partial \chi_2}{\partial y_i}(y)\, \frac{\partial^2 G_{\Pi}}{\partial \xi_i\, \partial\xi_j}(\xi,y)\,dy \\ &\qquad\qquad +\sum_{i=1}^n \int_{\Pi\cap\{y\colon |\xi-y|\geqslant \rho_2\}}v(y)\, \frac{\partial \chi_2}{\partial y_i}(y)\, \frac{\partial^2 G_{\Pi}}{\partial \xi_i\, \partial\xi_j}(\xi,y)\,dy, \end{aligned}
\end{equation}
\tag{5.12}
$$
where $\rho_2$ is a sufficiently small positive number independent of $\xi$ and $y$. We estimate the second term on the right-hand side of this equality:
$$
\begin{equation}
\biggl|\sum_{i=1}^n \int_{\Pi\cap\{y\colon |\xi-y|\geqslant \rho_2\}}v(y)\, \frac{\partial \chi_2}{\partial y_i}(y)\, \frac{\partial^2 G_{\Pi}}{\partial \xi_i\, \partial\xi_j}(\xi,y)\, dy\biggr|\leqslant \frac{C}{\rho_2^n}\max_{\overline{\Pi_{R_0}}}|v|,
\end{equation}
\tag{5.13}
$$
where the constant $C$ is independent of $\xi$, $y$ and $j$.
Similarly to the proof of inequality (1.27) in [33], Ch. III, § 1, we verify that
$$
\begin{equation}
\biggl| \int_{\Pi\cap\{y\colon \rho\leqslant|\xi-y|\leqslant \rho_3\}}\frac{\partial^2 G_{\Pi}}{\partial \xi_i\, \partial\xi_j}(\xi,y)\,dy\biggr|\leqslant C,
\end{equation}
\tag{5.14}
$$
where $C$ is independent of $\xi$, $y$ and $\rho_2$. For $|\xi-y|<\rho_2$ we have
$$
\begin{equation*}
|v(y)-v(\xi)|\leqslant C |\xi-y|\max_{\overline{\Pi_{R_0}}}|\nabla v|,
\end{equation*}
\notag
$$
where $C$ is independent of $v$, $\xi$ and $y$. Now, by the last inequality and (5.14) we have
$$
\begin{equation}
\begin{aligned} \, \notag &\biggl|\sum_{i=1}^n \int_{\Pi\cap\{y\colon \rho\leqslant|\xi-y|\leqslant \rho_2\}} v(y)\, \frac{\partial \chi_2}{\partial y_i}(y)\, \frac{\partial^2 G_{\Pi}}{\partial \xi_i\, \partial\xi_j}(\xi,y)\,dy\biggr| \\ \notag &\qquad=\biggl|v(\xi)\sum_{i=1}^n \int_{\Pi\cap\{y\colon \rho\leqslant|\xi-y|\leqslant \rho_2\}} \frac{\partial \chi_3}{\partial y_i}(y)\, \frac{\partial^2 G_{\Pi}}{\partial \xi_i\, \partial\xi_j}(\xi,y)\,dy\biggr| \\ \notag &\qquad\qquad +\biggl|\sum_{i=1}^n \int_{\Pi\cap\{y\colon \rho\leqslant|\xi-y|\leqslant \rho_2\}}(v(y)-v(\xi))\,\frac{\partial \chi_2}{\partial y_i}(y)\, \frac{\partial^2 G_{\Pi}}{\partial \xi_i\, \partial\xi_j}(\xi,y)\,dy\biggr| \\ &\qquad \leqslant C \Bigl( \max_{\overline{\Pi_{R_0}}}|v| + \rho_2\max_{\overline{\Pi_{R_0}}}|\nabla v| \Bigr), \end{aligned}
\end{equation}
\tag{5.15}
$$
where the constants $C$ and $C_5$ are independent of $\rho_2$, $v$, $j$, $\xi$ and $y$.
By (5.8), (5.11)–(5.13) and (5.15) we have
$$
\begin{equation*}
\max_{\overline{\Pi_{R_0}}}|\nabla\breve{v}|\leqslant C \Bigl(\max_{\overline{\Pi_{R_0}}}|F| +\max_{\overline{\Pi_{R_0}}}|v|\Bigr) +C\rho_2\max_{\overline{\Pi_{R_0}}}|\nabla v|,
\end{equation*}
\notag
$$
where the constant $C$ is independent of $F$, $\rho_2$, $v$ and $\xi$. Carrying over the last term to the left-hand side and taking sufficiently small $\rho_2$ we arrive at (5.7). Lemma 5.3 is proved. Set $\check{v}(\xi):=v(\xi)\chi_3(\xi)$, where $\chi_3=\chi_3(\xi)$ is an infinitely smooth cut-off function that is equal to one on the support of $1-\chi_2$ and to zero outside a neighbourhood of this support that lies strictly inside $\Pi$. The function $\check{v}$ is a solution of the problem
$$
\begin{equation}
\begin{gathered} \, -\Delta_{\xi} \check{v}=F_2 \quad \text{in } \mathbb{R}^n\setminus\overline{\omega^\eta}, \qquad \frac{\partial \check{v}}{\partial \nu_{\xi}}=\phi \quad\text{on } \partial \omega^{\eta}, \\ F_2:=\chi_3 F - 2 \sum_{i=1}^n \frac{\partial v}{\partial \xi_i}\,\frac{\partial \chi_3}{\partial \xi_i} - v\Delta_{\xi}\chi_3. \nonumber \end{gathered}
\end{equation}
\tag{5.16}
$$
Lemma 5.4. The inequality holds, where the constant $C$ is independent of $F$, $v$, $\phi$ and $\eta$. Proof. In (5.16) we make the substitution $\widetilde{\xi}:=\xi\eta^{-1}$. Then it takes the following form:
$$
\begin{equation}
\begin{aligned} \, -\Delta_{\widetilde{\xi}} \widetilde{v}=\widetilde{F}_2 \quad \text{in } \mathbb{R}^n\setminus\overline{\omega}, \qquad \frac{\partial \widetilde{v}}{\partial \nu_{\widetilde{\xi}}}=\widetilde{\phi} \quad\text{on } \partial \omega, \end{aligned}
\end{equation}
\tag{5.18}
$$
where $\widetilde{v}(\widetilde{\xi}):=\check{v}(\widetilde{\xi}\eta)$, $\widetilde{F}_2(\widetilde{\xi}):=\eta^2 F_2(\widetilde{\xi}\eta)$ and $\widetilde{\phi}(\widetilde{\xi}):=\eta \phi(\widetilde{\xi}\eta)$. Note that $\widetilde{v}$ has a compact support by definition.
We represent the solution of (5.18) as follows:
$$
\begin{equation}
\begin{gathered} \, \widetilde{v}(\widetilde{\xi})= \widetilde{v}_1(\widetilde{\xi}) + \widetilde{v}_2(\widetilde{\xi}), \\ \widetilde{v}_1(\widetilde{\xi}):=\int_{\mathbb{R}^n\setminus\omega} G(\widetilde{\xi},y)\widetilde{F}_2(y)\,dy\quad\text{and} \quad \widetilde{v}_2(\widetilde{\xi}):= \int_{\partial\omega} G(\widetilde{\xi},y)\widetilde{\phi}(y)\,ds, \end{gathered}
\end{equation}
\tag{5.19}
$$
where $G$ is the Green’s function of (5.18), which has the form (4.26). By [35], Ch. I, § 1.6, the function $\widetilde{v}$ belongs to $C^1(\mathbb{R}^n)$.
We set
$$
\begin{equation*}
\widetilde{v}_1(\widetilde{\xi}):=\int_{\mathbb{R}^n\setminus\omega} G(\widetilde{\xi},y)\widetilde{F}_2(y)\,dy\quad\text{and} \quad \widetilde{v}_2(\widetilde{\xi}):=\int_{\partial\omega} G(\widetilde{\xi},y)\widetilde{\phi}(y)\,ds
\end{equation*}
\notag
$$
and estimate the derivatives of $\widetilde{v}_1$. From (4.26) we obtain
$$
\begin{equation}
\biggl|\frac{\partial \widetilde{v}_1}{\partial \widetilde{\xi}_i}(\widetilde{\xi})\biggr|\leqslant C\max_{B^\eta}|\widetilde{F}_2| \biggl(\biggl|\int_{B^\eta}\frac{\partial}{\partial\widetilde{\xi}_i} \,\frac{1}{|\widetilde{\xi}-y|^{n-2}}\,dy\biggr| +\biggl|\int_{B^\eta}\frac{\partial G_1}{\partial\widetilde{\xi}_i}(\widetilde{\xi}, y)\,dy\biggr|\biggr),
\end{equation}
\tag{5.20}
$$
where $B^\eta:=B_{R_2\eta^{-1}}(0)\setminus\omega$ and $C$ is a constant independent of $\widetilde{F}_2$, $\widetilde{\xi}$ and $y$. In the first integral on the right-hand side of (5.20) we make the change of variables $z=\widetilde{\xi}-y$ and then introduce the spherical variables. As a result, we obtain
$$
\begin{equation*}
\biggl|\int_{B^\eta}\frac{\partial}{\partial\widetilde{\xi}_i}\, \frac{1}{|\widetilde{\xi}-y|^{n-2}}\,dy\biggr|\leqslant \biggl|\int_{\mathbb{S}^{n-1}}\,d\varphi \int_{0}^{\eta^{-1}}\,dr\biggr|\leqslant C\eta^{-1},
\end{equation*}
\notag
$$
where $\mathbb{S}^{n-1}$ is the unit sphere $\mathbb{R}^n$ and $C$ is a constant independent of $\eta$, $\widetilde{\xi}$, $y$ and $i$. By the properties of $G_1$ we have
$$
\begin{equation*}
\biggl|\int_{B^\eta}\frac{\partial G_1}{\partial \widetilde{\xi_i}}(\widetilde{\xi},y)\,dy\biggr|\leqslant C,
\end{equation*}
\notag
$$
where $C$ is independent of $\widetilde{\xi}$, $y$ and $i$. It follows from the last two inequalities and (5.20) that
$$
\begin{equation}
\biggl|\frac{\partial \widetilde{v}_1}{\partial \widetilde{\xi}_i}(\widetilde{\xi})\biggr|\leqslant C\eta^{-1}\max_{B^\eta}|\widetilde{F}_2|, \qquad \widetilde{\xi}\in\mathbb{R}^n\setminus\omega,
\end{equation}
\tag{5.21}
$$
where the constant $C$ is independent of $\eta$, $\widetilde{\xi}$, $y$ and $i$.
Let $\widetilde{\xi}\notin\partial\omega$. Then
$$
\begin{equation*}
\frac{\partial\widetilde{v}_2}{\partial\widetilde{\xi}_i}(\widetilde{\xi})=\int_{\partial\omega} \frac{\partial G}{\partial \widetilde{\xi}_i}(\widetilde{\xi},y)\widetilde{\phi}(y)\,ds.
\end{equation*}
\notag
$$
We introduce smooth local coordinates $s=(s_1,\dots,s_{n-1})$ on $\partial\omega$. The Jacobian $J=J(s)$ of the change of variables from $y$ to $s$ is a $C^1$-function, which is uniformly bounded together with all derivatives. By (4.27) we have
$$
\begin{equation*}
\begin{aligned} \, \int_{\partial\omega} \widetilde{\phi}\, \frac{\partial\widetilde{G}}{\partial\widetilde{\xi}_i}\,ds &=\int_{\partial\omega} \widetilde{\phi}\, \frac{\partial\widetilde{G}}{\partial y_i}\,ds \\ &=\int_{\partial\omega} J\phi\biggl(\frac{\partial s_1}{\partial y_i}\,\frac{\partial G}{\partial s_1}+\dots+\frac{\partial s_{n-1}}{\partial y_i}\,\frac{\partial G}{\partial s_{n-1}}\biggr) \,ds, \qquad \widetilde{\xi}\notin\partial\omega. \end{aligned}
\end{equation*}
\notag
$$
We integrate by parts:
$$
\begin{equation*}
\begin{aligned} \, &\int_{\partial\omega} J\phi\biggl(\frac{\partial s_1}{\partial y_i}\,\frac{\partial G}{\partial s_1}+\dots+\frac{\partial s_{n-1}}{\partial y_i}\,\frac{\partial G}{\partial s_{n-1}}\biggr)\,ds \\ &\qquad =-\int_{\partial\omega}G \biggl(\frac{\partial}{\partial s_1}\,\frac{\partial s_1}{\partial y_i}\, J\phi+\dots+\frac{\partial}{\partial s_{n-1}}\,\frac{\partial s_{n-1}}{\partial y_i}\, J\phi\biggr)\,ds, \qquad \widetilde{\xi}\notin\partial\omega. \end{aligned}
\end{equation*}
\notag
$$
Hence, as $J$ is bounded, we obtain
$$
\begin{equation}
\biggl|\frac{\partial\widetilde{v}_2}{\partial\widetilde{\xi}_i}(\widetilde{\xi})\biggr| \leqslant C\biggl(\max_{\partial \omega}|\widetilde{\phi}| +\max_{\partial \omega}\biggl|\frac{\partial \widetilde{\phi}}{\partial y_i}\biggr|\biggr), \qquad \widetilde{\xi}\notin\partial\omega,
\end{equation}
\tag{5.22}
$$
where the constant $C$ is independent of $\widetilde{\phi}$, $i$, $\widetilde{\xi}$ and $y$.
Let $\widetilde{\xi}_0\in\partial\omega$. Taking the limit as $\widetilde{\xi}\to\widetilde{\xi}_0$ in this inequality, since ${\partial\widetilde{v}_2}/{\partial\xi_i}$ is continuous, we see that (5.22) holds for all $\widetilde{\xi}\in \mathbb{R}^n\setminus\omega$. Returning to the variables $\xi$ and using (5.19), (5.21) and (5.22) we obtain
$$
\begin{equation*}
\max_{\overline{\Pi^\eta}}|\nabla \breve{v} |\leqslant C\Bigl( \max_{\overline{\Pi^\eta}} |F_2|+\max_{\partial \omega}|\phi|+\max_{\partial \omega}|\nabla \phi|\Bigr),
\end{equation*}
\notag
$$
where the constant $C$ is independent of $F_2$, $\phi$, $\xi$ and $y$. Inequality (5.17) is a consequence of this estimate and Lemmas 5.1 and 5.3. The proof is complete. Lemmas 5.1, 5.3 and 5.4 yield the following result. Lemma 5.5. The unique solution of problem (4.1), (4.2) that exists by Lemma 4.6 satisfies where the constant $C$ is independent of $\eta$, $F$ and $\phi$. § 6. Solvability of the model problemsHere we apply the results of the previous section to investigate the solvability of model problem (4.1), (4.2) with right-hand side of the general form and to describe how the solution depends on the parameter $\eta$. We also consider similar questions for the model problem stated for the coefficients of the outer expression: we present the statement of this problem in this section. Let $\mathfrak{P}$ be the space of functions $f=f(\xi)$ on $\overline{\Pi}\setminus\omega^\eta$ such that for $|\xi_n|>R_0$ the Fourier series of $f$ has the following form:
$$
\begin{equation}
f(\xi)= \sum_{k\in\mathbb{Z}^{n-1}} T_k^\pm(\xi_n)\exp(-2\pi |k_b| |\xi_n|)\exp(2\pi\mathrm{i} k_b\cdot\xi'),
\end{equation}
\tag{6.1}
$$
where the $T_k^\pm(\xi_n)$ are polynomials of degree bounded uniformly in $k$. We assume that the coefficients of these polynomials satisfy
$$
\begin{equation*}
\mathfrak{p}(f):= \sup_{k\in\mathbb{Z}^{n-1}} \exp(-2\pi |k_b| R_0)|\!|\!| T_k^+|\!|\!| + \sup_{k\in\mathbb{Z}^{n-1}} \exp(-2\pi |k_b| R_0)|\!|\!| T_k^-|\!|\!| < \infty.
\end{equation*}
\notag
$$
Here, given a polynomial $L$, we let $|\!|\!| L|\!|\!|$ denote the maximum absolute value of its coefficients. Lemma 6.1. Let Proof. The problems
$$
\begin{equation*}
\begin{aligned} \, &-\Delta_\xi v_F^\pm=F_0 + \sum_{j=1}^{n-1}\frac{\partial F_j}{\partial\xi_j} \quad\text{in } \Pi_{R_0-1}^\pm \end{aligned}
\end{equation*}
\notag
$$
with periodic boundary conditions (4.2) were solved in [29], § 7.6, using the method of separation of variables:
$$
\begin{equation*}
v_F^\pm(\xi)=\sum_{k\in\mathbb{Z}^{n-1}}\widetilde{Q}^\pm_k(\xi_n) \exp(2\pi\mathrm{i} k_b\cdot\xi'),
\end{equation*}
\notag
$$
where the functions $\widetilde{Q}_k^\pm$ are determined from the equations
$$
\begin{equation*}
-\frac{\partial^2\widetilde{Q}^\pm_k}{\partial\xi_n^2} \pm 4\pi| k_b| \, \frac{\partial \widetilde{Q}^\pm_k}{\partial\xi_n}=\widetilde{T}_{k,0}^\pm+ \pi \mathrm{i}\sum_{j=1}^{n-1}\frac{k_j}{b_j}\widetilde{T}_{k,j},
\end{equation*}
\notag
$$
and the functions $\widetilde{T}_{k,j}^\pm=\widetilde{T}_{k,j}^\pm(\xi_n)$ are defined as the coefficients of the Fourier expansions of the functions $F_j$:
$$
\begin{equation*}
F_j(\xi)=\sum_{k\in\mathbb{Z}^{n-1}} \widetilde{T}_{k,j}^\pm(\xi_n) \exp(2\pi\mathrm{i} k_b\cdot\xi'), \qquad \widetilde{T}_{k,j}^\pm(\xi_n)=T_{k,j}^\pm(\xi_n) \quad\text{for } \pm\xi_n>R_0.
\end{equation*}
\notag
$$
Similarly to the proof of Lemma 3.1 in [36], it is easy to show that we have $v_F^\pm\in W_2^2(\Pi_R^\pm\setminus \Pi_{R_0-1}^\pm)$ for each $R>R_0$ and that
$$
\begin{equation}
\begin{aligned} \, \notag \|v_F^\pm\|_{W_2^2(\Pi_R^\pm\setminus \Pi_{R_0-1}^\pm)} &\leqslant C(R) \biggl(\|F_0\|_{L_2(\Pi_{R_0}^\pm\setminus \Pi_{R_0-1}^\pm)} + \sum_{j=1}^{n-1}\|\nabla F_j\|_{L_2(\Pi_{R_0}^\pm\setminus \Pi_{R_0-1}^\pm)} \\ &\qquad\qquad +\sum_{j=0}^{n-1}\mathfrak{p}(F_j)\biggr), \end{aligned}
\end{equation}
\tag{6.4}
$$
where $C(R)$ is a constant independent of the functions $F_j$, $j=0,\dots,n-1$.
We seek the solution of (4.1), (4.2) in the form
$$
\begin{equation*}
v=\widetilde{v}+ \chi_1 v_F, \qquad v_F(\xi):=v_F^\pm(\xi) \quad\text{for } \pm \xi_n>R_0-1,
\end{equation*}
\notag
$$
where the cut-off function $\chi_1$ was defined before (4.5). The function $\widetilde{v}$ is a solution of the problem
$$
\begin{equation}
\begin{gathered} \, -\Delta_\xi\widetilde{v}=\widetilde{F} \quad\text{in } \Pi_R\setminus\omega^\eta, \qquad \frac{\partial\widetilde{v}}{\partial \nu_\xi}=0 \quad\text{on } \partial\omega^\eta, \\ \widetilde{F}:=F\chi_1 -2 \frac{\partial v_F}{\partial \xi_n} \chi_1'- v_F\chi_1', \end{gathered}
\end{equation}
\tag{6.5}
$$
with the periodic boundary conditions (4.2).
Using (4.30) and (6.4) we deduce the inequality
$$
\begin{equation*}
\begin{aligned} \, \|\widetilde{F}\|_{L_2(\Pi_{R_0})} &\leqslant C \biggl(\eta^{n-1}\|\phi\|_{L_2(\partial\omega^\eta)}+ \|F_0\|_{L_2(\Pi_{R_0}^\pm\setminus \Pi_{R_0-1}^\pm)} \\ &\qquad\qquad+ \sum_{j=1}^{n-1}\|\nabla F_j\|_{L_2(\Pi_{R_0}^\pm\setminus \Pi_{R_0-1}^\pm)} +\sum_{j=0}^{n-1}\mathfrak{p}(F_j) \biggr), \end{aligned}
\end{equation*}
\notag
$$
where the constant $C$ is independent of $\eta$, $\phi$ and the $F_j$, $j=0,\dots,n-1$.
We verify the condition of solvability from Lemma 4.5 for problem (6.5), (4.2). To do this we integrate by parts:
$$
\begin{equation*}
\int_{\Pi^\eta}\widetilde{F}\,d\xi+\int_{\partial\omega^\eta}\phi \,ds=\int_{\Pi^\eta}(F-\Delta_{\xi} \chi_1 v_F)\,d\xi =\int_{\Pi^\eta} Fd\,\xi+\int_{\partial\omega^\eta}\phi \,ds=0.
\end{equation*}
\notag
$$
Hence it follows from Lemma 4.5 that problem (6.5), (4.2) has a unique solution with asymptotic behaviour (4.3) as $\xi_n\to\pm\infty$, where the constants $D_\pm$ satisfy $D_++D_-=0$. For $ |\xi_n|>R_0$ this solution has a representation (4.21), where the coefficients $\widehat{D}_k^{\pm}(\eta)$ have the estimates
$$
\begin{equation}
\begin{aligned} \, \notag \sup_{k\in\mathbb{Z}^{n-1}}\exp(-2\pi |k_b|)|\widehat{D}_k^\pm(\eta)| &\leqslant \frac{C}{|k_b|}\biggl(\eta^{n-1}\|\phi\|_{C(\partial\omega^\eta)} +\|F_0\|_{C(\overline{\Pi^\eta})} \\ &\qquad\qquad+ \sum_{j=1}^{n-1} \|\nabla_\xi F_j\|_{C(\overline{\Pi^\eta})} +\sum_{j=0}^{n-1}\mathfrak{p}(F_j)\biggr) \end{aligned}
\end{equation}
\tag{6.6}
$$
with $C$ independent of $\eta$, $\phi$ and the $F_j$, $j=0,\dots,n-1$. Hence problem (4.1), (4.2) is also solvable and has a unique solution with asymptotic behaviour (4.3) as ${\xi_n\!\to\!\pm\infty}$, where the constants $D_\pm$ satisfy $D_++D_-=0$. For $|\xi_n|>R_0$ this solution has the form (6.1) where $T_k^\pm=Q_k^\pm$ and $Q_k^\pm(\xi,\eta):=\widetilde{Q}_k^{\pm}(\xi_n)+\widehat{D}_k^{\pm}(\eta)$. For this solution inequality (6.2) is a direct consequence of (6.6).
Let $\chi_4=\chi_4(\xi_n)$ be an infinitely smooth cut-off function equal to one for ${|\xi_n|<R_0+4/3}$ and to zero for $|\xi_n|>R_0+5/3$. The function $\chi_4 v$ is a solution of the problem
$$
\begin{equation}
\begin{gathered} \, -\Delta_\xi\chi_4 v=\chi_4 F-2\nabla_\xi\chi_4\cdot \nabla_\xi v - v\Delta_\xi \chi_4 \quad\text{in } \Pi_{R_0+2}\setminus\overline{\omega^\eta}, \\ \frac{\partial\chi_4 v}{\partial \nu_\xi}=\phi \quad\text{on } \partial\omega^\eta, \qquad \chi_4v =0 \quad\text{on } \square\times\{\pm (R_0+2)\}, \end{gathered}
\end{equation}
\tag{6.7}
$$
with the periodic boundary conditions (4.2). Applying Lemma 5.5 to this problem and taking (6.2) into account we obtain (6.3). Lemma 6.1 is proved. § 7. Infinite differentiability with respect to $\eta$In this section we investigate the smoothness of the solution of (4.1), (4.2) that exists by Lemma 4.6 with respect to the parameter $\eta\in(0,1]$; we also investigate the smoothness in $\eta\in[0,1]$ of the solution of a pair of model problems in $\Omega\setminus S$ for coefficients of the outer expansion. 7.1. The smoothness of the solution of the model problem for coefficients of the inner expansionIn this subsection we investigate the smoothness of the solution of (4.1), (4.2) with respect to $\eta$. Throughout the subsection we assume that $F$ vanishes outside $\Pi_{R_0}$ and is infinitely differentiable, as also is $\phi$: namely, $F\in C^\infty(\overline{\Pi_{R_0}}\setminus\omega^\eta)$ and $\phi\in C^\infty(\partial\omega^\eta)$. Taking account of the behaviour of the solution of (4.1), (4.2) at infinity, which we described in Lemma 6.1, the solution is infinitely differentiable with respect to $\xi$ in $\overline{\Pi}\setminus\Pi_{R_0+1}$. The function $\widetilde{v}=v\chi_4$ solves problem (6.7) with periodic boundary conditions on the lateral faces. The right-hand side of the equation in this problem is obviously infinitely smooth in $\overline{\Pi_{R_0+2}}\setminus\omega^\eta$, so by Schauder’s standard estimates it is straightforward to see that $\widetilde{v}\in C^\infty(\overline{\Pi_{R_0+2}}\setminus\omega^\eta)$. As $v$ is infinitely differentiable in $\overline{\Pi}\setminus \Pi_{R_0+1}$, it follows directly that $v\in C^\infty(\overline{\Pi}\setminus\omega^\eta)$. Also note the obvious estimates
$$
\begin{equation}
\|v\|_{C^{k+2+1/2}(\overline{\Pi}\setminus\omega^\eta)} \leqslant C_k(\eta) \bigl(\|F\|_{C^{k+1/2}(\overline{\Pi_{R_0}}\setminus\omega^\eta)} +\|\phi\|_{C^{k+1+1/2}(\partial\omega^\eta)}\bigr),
\end{equation}
\tag{7.1}
$$
where the $C_k(\eta)$ are constants independent of $F$ and $\phi$ and the parameter $\eta$ is strictly positive. Throughout what follows we assume that for all $\eta\in(0,1]$ the function $\phi$ in the boundary condition on $\partial\omega^\eta$ in (4.1) is the trace of a smooth function $\Phi=\Phi(\xi,\eta)$ in $\overline{\Pi}\setminus\omega^\eta$ that vanishes outside $\{\xi\colon \chi_2(\xi)=1\}$. Our main result in this subsection is as follows. Lemma 7.1. Let $\Phi=\Phi(\xi,\eta)$ and $F=F(\xi,\eta)$ be functions in $\overline{\Pi}\setminus\omega^\eta$ that are infinitely differentiable with respect to $\xi$ for each $\eta\in(0,1]$, let $F$ vanish outside $\Pi_{R_0}$ and $\Phi$ vanish outside $\{\xi\colon \chi_2(\xi)=1\}$ for all $\eta\in(0,1]$, and let The rest of this section is devoted to the proof of the lemma. By Lemma 6.1 the second equality in (7.2) ensures that (4.1), (4.2) is solvable for each $\eta\in(0,1]$. We have proved above that the solution is infinitely differentiable with respect to $\xi$ and established estimates (7.1). Fix some $\eta_0\in(0,1]$ and $k\in\mathbb{N}$, and let $\eta\in[\eta_0-\delta_k(\eta_0), \eta_0+\delta_k(\eta_0)]$. Consider the boundary value problem
$$
\begin{equation*}
\Delta_\xi v_\phi=0 \quad\text{in } \mathbb{R}^d\setminus\overline{\omega^\eta}, \qquad \frac{\partial v_\phi}{\partial\nu_\xi}=\Phi(\xi,\eta) \quad\text{on } \partial\omega^\eta.
\end{equation*}
\notag
$$
The dilation $\widehat{\xi}=\eta_0\eta^{-1}\xi$ takes it to a similar problem:
$$
\begin{equation}
\Delta_{\widehat{\xi}} v_\phi=0 \quad\text{in}\ \mathbb{R}^d\setminus\overline{\omega^{\eta_0}}, \qquad \frac{\partial v_\phi}{\partial\nu_{\widehat{\xi}}}=\eta_0\eta^{-1} \Phi\bigl(\eta_0{\eta}^{-1}\xi,\eta\bigr) \quad\text{on } \partial\omega^{\eta_0}.
\end{equation}
\tag{7.5}
$$
By the definition of $\Phi$ and $\Xi$, on the support of $\Phi$ the diffeomorphism $\Xi$ is a mere dilation with coefficient $t-1$, so that
$$
\begin{equation*}
\Phi(\eta_0\eta^{-1} \xi,\eta) =\Phi(\Xi( \eta_0\eta^{-1},\xi),\eta)
\end{equation*}
\notag
$$
and this function is an element of the space from the first membership relation in (7.3). Making a further dilation, with coefficient $\eta_0$, we reduce (7.5) to problem (4.25), whose solution is given by a convolution with a fixed Green’s function; see (4.28). As a result, we conclude that $v_\phi=v_\phi(\xi,\eta)$ is infinitely differentiable with respect to $\xi$ in $\mathbb{R}^n\setminus\omega^\eta$ for each $\eta\in(0,1]$, while $v_\phi(\xi,\eta)\chi_2(\xi)$, viewed as a function in $\Pi\setminus\omega^\eta$, satisfies (7.4). We represent the solution of (4.1), (4.2) in the form $v=v_\phi\chi_2+v_F$. Then for $v_F$ we obtain the boundary value problem
$$
\begin{equation}
\begin{gathered} \, -\Delta v_F=F_1 \quad \text{in } \Pi\setminus\overline{\omega^\eta}, \qquad \frac{\partial v_F}{\partial\nu_\xi}=0 \quad \text{on } \partial\omega^\eta, \\ F_1:=F+2\sum_{j=1}^{n} \frac{\partial v_\phi}{\partial \xi_j} \, \frac{\partial\chi_2}{\partial\xi_j} +v_\phi \Delta_\xi \chi_2, \end{gathered}
\end{equation}
\tag{7.6}
$$
with periodic boundary conditions on the lateral faces. By the properties of $v_\phi$ established above, the function $F_1$ has the same properties as $F$, except that now the first equality in (7.2) must be replaced by
$$
\begin{equation}
\int_{\Pi\setminus\omega^\eta} F_1(\xi,\eta)\,d\xi=0, \qquad \eta\in(0,1].
\end{equation}
\tag{7.7}
$$
Consider the function space
$$
\begin{equation*}
\mathfrak{C}^k:=\biggl\{f\in C^{k+1/2}(\overline{\Pi}\setminus\omega^{\eta_0})\colon \int_{\Pi^{\eta_0}} f(\xi)\,d\xi=0, \ f=0 \text{ outside } \Pi_{R_0}\biggr\}
\end{equation*}
\notag
$$
with the norm of $C^{k+1/2}(\overline{\Pi^{\eta_0}})$. This norm makes of $\mathfrak{C}^k$ a Banach space. In $\overline{\Pi}\setminus\omega^\eta$ we introduce the new variables $\widetilde{\xi}=\Xi( \eta_0\eta^{-1},\xi)$. By the properties of $\Xi$ the variables $\widetilde{\xi}$ range over $\Pi\setminus\overline{\omega^{\eta_0}}$. Recall that the Jacobian $\Upsilon=\Upsilon(\xi,\eta)$ of this change of variables was introduced in (4.11); it satisfies (4.12) and (4.13). Taking these relations into account, as the functions $F_1$ and $\Upsilon$ are smooth, in the integral (7.7) we can go over to $\widetilde{\xi}$. Then we obtain directly that $\Upsilon F_1$, as a function of $\widetilde{\xi}$, belongs to $\mathfrak{C}^k$. Next we go over to the variables $\widetilde{\xi}$ in (7.6) and consider the new unknown function defined by $\widetilde{v}_F(\widetilde{\xi},\eta):=v_F(\xi,\eta)\Upsilon(\xi,\eta)$. Then in view of (4.12)–(4.15) we obtain the following problem:
$$
\begin{equation}
-(\Delta_{\widetilde{\xi}}+ (\eta-\eta_0)\mathcal{B}_5(\eta_0,\eta)) \widetilde{v}_F=\Upsilon F_1 \quad \text{in } \Pi\setminus\overline{\omega^{\eta_0}}, \qquad \frac{\partial v_F}{\partial\nu_{\widetilde{\xi}}}=0 \quad \text{on } \partial\omega^{\eta_0},
\end{equation}
\tag{7.8}
$$
with periodic boundary conditions on the lateral part of the boundary. By the properties of the operator $\mathcal{B}_5$ described after (4.14), this operator is bounded uniformly in $\eta\in[\eta_0-\delta_k(\eta_0), \eta_0+\delta_k(\eta_0)]$ as an operator from $C^{k+2+1/2}(\overline{\Pi_{R_0}}\setminus\omega^{\eta_0})$ to $C^{k+1/2}(\overline{\Pi_{R_0}}\setminus\omega^{\eta_0})$. Let $v\in C^{k+1/2}(\overline{\Pi}\setminus\omega^{\eta_0})$ be any function satisfying the Neumann boundary condition on $\partial\omega^{\eta_0}$. Then taking the definition of $\Xi$ and (4.13) into account, integrating by parts we obtain
$$
\begin{equation}
\begin{aligned} \, \notag &\int_{\Pi\setminus\omega^{\eta_0}} (\eta-\eta_0)\mathcal{B}_5(\eta_0,\eta) v\,d\widetilde{\xi}=\int_{\Pi_{R_0}\setminus\omega^{\eta_0}} (\Upsilon \Delta_{\xi} \Upsilon^{-1}-\Delta_{\widetilde{\xi}})v\,d\widetilde{\xi} \\ &\ =\int_{\Pi_{R_0}\setminus\omega^{\eta}} \Delta_{\xi} \Upsilon^{-1} v\,d\xi - \int_{\Pi_{R_0}\setminus\omega^{\eta_0}} \Delta_{\widetilde{\xi}}v\,d\widetilde{\xi} =\frac{\eta_0^n}{\eta^n}\int_{\partial\omega^\eta} \frac{\partial v}{\partial\nu_\xi}\,ds-\int_{\partial\omega^{\eta_0}} \frac{\partial v}{\partial\nu_{\widetilde{\xi}}}\,ds=0. \end{aligned}
\end{equation}
\tag{7.9}
$$
For an arbitrary function $f\in\mathfrak{C}^k$ consider the boundary value problem
$$
\begin{equation}
-\Delta_{\widetilde{\xi}} v=f \quad \text{in } \Pi\setminus\omega^{\eta_0}, \qquad \frac{\partial v}{\partial\nu_{\widetilde{\xi}}}=0 \quad \text{on } \partial\omega^{\eta_0},
\end{equation}
\tag{7.10}
$$
with periodic boundary conditions on the lateral part of the boundary. By Lemma 6.1 this problem is solvable, and by the above its solution belongs to $\mathfrak{C}^{k+2}$. Let $\mathcal{B}_7\colon \mathfrak{C}^k\to C^{k+2+1/2}(\overline{\Pi}\setminus\omega^{\eta_0}) $ be the linear operator taking functions $f$ to the solutions of (7.10). It is bounded by (7.1). Since $\Upsilon F_1$ is an element of $\mathfrak{C}^k$ and (7.9) holds, it follows from (7.8) that
$$
\begin{equation}
\widetilde{v}_F=\mathcal{B}_7 f, \qquad f=\Upsilon F_1+(\eta-\eta_0)\mathcal{B}_5(\eta_0,\eta) \widetilde{v}_F\in \mathfrak{C}^k.
\end{equation}
\tag{7.11}
$$
Substituting this into (7.8) we arrive at the operator equation
$$
\begin{equation*}
\bigl(\mathcal{I}-(\eta-\eta_0)\mathcal{B}_5(\eta_0,\eta) \mathcal{B}_7\bigr)g=\Upsilon F_1
\end{equation*}
\notag
$$
in $\mathfrak{C}^k$, where $\mathcal{I}$ is the identity operator. As $\mathcal{B}_7$ is bounded, the norm of $(\eta- \eta_0)\times \mathcal{B}_5(\eta_0,\eta)\mathcal{B}_7$ is small if $\eta$ is sufficiently close to $\eta_0$. This ensures that the inverse operator $\bigl(\mathcal{I}-(\eta- \eta_0)\mathcal{B}_5(\eta_0,\eta)\mathcal{B}_7\bigr)^{-1}$ exists, so we can define the function $g$, and then $\widetilde{v}_F$, using (7.11):
$$
\begin{equation*}
\widetilde{v}_F\mathcal{B}_7=\bigl(\mathcal{I}-(\eta-\eta_0)\mathcal{B}_5(\eta_0,\eta) \mathcal{B}_7\bigr)^{-1} \Upsilon F_1.
\end{equation*}
\notag
$$
Since the coefficients of $\mathcal{B}_5(\eta_0,\eta)$ are infinitely smooth with respect to $\eta$ and the space variables, we easily deduce from this that
$$
\begin{equation*}
\widetilde{v}_F\in C^{k+1/2}\bigl(\overline{\Pi}\setminus\omega^{\eta_0} \times[\eta_0-\widetilde{\delta}_k(\eta_0),\eta_0+\widetilde{\delta}_k(\eta_0)]\bigr).
\end{equation*}
\notag
$$
Now we return to the function $v$. Taking the properties of $v_\phi$ just established into account we arrive at the conclusion of Lemma 7.1. The proof is complete. 7.2. The smoothness of the solution with respect to the parameter of the problem in the case of coefficients of the outer expansionIn this subsection we consider the smoothness of the solutions of two model problems for the outer expansion. Set $\Omega^\pm:=\{x\colon \pm x_n>0\}\cap \Omega$ and $\Omega_r^\pm:=\{x\colon 0<\pm x_n<r\}$. Lemma 7.2. There exists $\lambda_0$ such that for $\lambda<\lambda_0$ the problem Proof. Let $\chi_5=\chi_5(x_n)$ be a function vanishing for $x_n\geqslant{2\tau}/{3}$ and $x_n<0$ and equal to one for $0<x_n\leqslant{\tau}/{3}$. We seek a solution of (7.12) in the form The function $\widetilde{u}$ solves the problem where $\widetilde{f}=f-(\mathcal{L} -\lambda)(\chi_5(x_n) \phi_1(x')+x_n \chi_5(x_n)\phi_2(x'))$ for $x_n>0$ and $\widetilde{f}=f$ for ${x_n<0}$.
Let $\mathcal{H}$ denote the operator in $L_2(\Omega)$ with differential expression $\mathcal{L}$ and the Dirichlet boundary condition on $\partial\Omega$. The corresponding sesquilinear form is equal to $\mathfrak{h}_0$ considered on the space $\mathring{W}_2^1(\Omega,\partial\Omega)$. This operator is $m$-sectorial, so its spectrum lies in a sector of complex plane with vertex on the real axis which is symmetric about the positive direction of this axis. Now we choose $\lambda_0$ that lies to the left of the vertex of this sector. The resolvent $(\mathcal{H}-\lambda)^{-1}$ is defined for $\lambda<\lambda_0$, so that problem (7.15) is solvable. Hence (7.12) is also solvable. Clearly, the operator $(\mathcal{H}-\lambda)$ is bounded from $\mathring{W}_2^2(\Omega,\partial\Omega)$ to $L_2(\Omega)$. Since it has an inverse operator $(\mathcal{H}-\lambda)^{-1}$ by the above, we conclude immediately from Banach’s inverse operator theorem that $(\mathcal{H}-\lambda)^{-1}\colon L_2(\Omega)\to\mathring{W}_2^2(\Omega,\partial\Omega)$ is bounded. This means that
$$
\begin{equation}
\|\widetilde{u}\|_{W_2^2(\Omega)}\leqslant C\|\widetilde{f}\|_{L_2(\Omega)}.
\end{equation}
\tag{7.16}
$$
Here and throughout the proof we denote by $C$ various insignificant constants which are independent of $\widetilde{f}$, $\widetilde{u}$ and $x\in\Omega$, as well as of the subscript $k$ introduced below.
We prove (7.13). In $\mathbb{R}^{n-1}$ we consider a partition of unity by means of infinitely differentiable functions $\zeta_k=\zeta_k(x')$ with compact supports, $k\in\mathbb{N}$, such that
$$
\begin{equation*}
0\leqslant \zeta_k\leqslant 1, \qquad \biggl|\frac{\partial \zeta_k}{\partial x_i}\biggr|+\biggl|\frac{\partial^2 \zeta_k}{\partial x_i\, \partial x_j}\biggr|\leqslant C\quad\text{and} \quad \sum_{k=1}^\infty \zeta_k =1.
\end{equation*}
\notag
$$
We also assume that the support of each $\zeta_k$ can be put in a fixed bounded set $Q$ independent of $k$ by means of a translation. In addition, at each point $x\in\Omega$ only finitely many supports of the $\zeta_k$ intersect; the multiplicity of intersection is bounded by a constant independent of $x$ and $k$. Set $\Omega^{(k)}:=\operatorname{supp}\zeta_k \times \mathbb{R}$ and $\Omega_\tau^{(k)}:=\operatorname{supp}\zeta_k \times (-\tau,\tau)$.
We represent the function $\widetilde{u}$ in the following form:
$$
\begin{equation}
\widetilde{u}(x)=\sum_{k=1}^\infty u_k(x), \quad \text{where } u_k(x)=\widetilde{u}(x)\zeta_k(x').
\end{equation}
\tag{7.17}
$$
Each function $u_k$ is a solution of the problem
$$
\begin{equation*}
\begin{gathered} \, (\mathcal{L} -\lambda) u_k=\zeta_k \widetilde{f}+\widetilde{F}_k \quad\text{for } x\in\Omega_\tau^{(k)}, \qquad u_k=0 \quad\text{for } x\in\partial\Omega_\tau^{(k)}, \\ \widetilde{F}_k=\widetilde{u}\sum_{i=1}^{n}\sum_{j=1}^{n-1} \frac{\partial}{\partial x_i}\, A_{ij}\, \frac{\partial \zeta_k}{\partial x_j}+2\sum_{i=1}^{n}\sum_{j=1}^{n-1} A_{ij}\, \frac{\partial \widetilde{u}}{\partial x_i}\, \frac{\partial\zeta_k}{\partial x_j} -\sum_{j=1}^{n-1} A_j \widetilde{u}\, \frac{\partial \zeta_k}{\partial x_j}. \end{gathered}
\end{equation*}
\notag
$$
As $f$ and $\widetilde{u}$ are smooth, by results on improved smoothness ([37], Ch. 4, § 2) we have the a priori estimates
$$
\begin{equation}
\|u_k\|^2_{W_2^p(\Omega_{\tau_0}^{(k)})}\leqslant C \bigl(\|\zeta_k \widetilde{f}\|^2_{W_2^{p-2}(\Omega_{\tau_1}^{(k)})} +\|\widetilde{F}_k\|^2_{W_2^{p-2}(\Omega_{\tau_1}^{(k)})} +\|u_k\|^2_{L_2(\Omega^{(k)})}\bigr),
\end{equation}
\tag{7.18}
$$
where $p\geqslant 2$, $p\in\mathbb{N}$ and $\tau_0<\tau_1<\tau$.
Using the definition of the $\zeta_k$ and the inequality $0\leqslant\zeta_k\leqslant1$, for any function $w\in L_2(\Omega)$ and any $\tau$ we obtain
$$
\begin{equation*}
\sum_{k=1}^\infty \|\zeta_k w\|_{L_2(\Omega_\tau^{(k)})}^2\leqslant C \sum_{k=1}^\infty \int_{\Omega_\tau^{(k)}} \zeta_k^2 |w|^2\,dx=\|w\|_{L_2(\Omega_\tau)}^2.
\end{equation*}
\notag
$$
By the last inequality, the postulated uniform estimates for derivatives of the $\zeta_k$, and (7.16) we have
$$
\begin{equation*}
\sum_{k=1}^\infty \|u_k\|_{L_2(\Omega^{(k)})}^2\leqslant C \|\widetilde{f}\|_{L_2(\Omega)}^2
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\sum_{k=1}^\infty \|\zeta_k \widetilde{f}\|^2_{W_2^{p-2}(\Omega_{\tau_1}^{(k)})}+\sum_{k=1}^\infty \| \widetilde{F}_k\|^2_{W_2^{p-2}(\Omega_{\tau_1}^{(k)})}\leqslant C \bigl( \|\widetilde{f}\|_{W_2^{p-2}(\Omega_\tau)}^2+\|\widetilde{u}\|_{W_2^{p-1}(\Omega)}^2\bigr).
\end{equation*}
\notag
$$
It follows from these two inequalities, (7.17) and (7.18) that
$$
\begin{equation*}
\|\widetilde{u}\|_{W_2^p(\Omega_{\tau_0})}^2\leqslant \sum_{k=1}^\infty \|u_k\|_{W_2^p(\Omega_{\tau_0}^{(k)})}^2\leqslant C \bigl(\|\widetilde{f}\|_{W_2^{p-2}(\Omega_{\tau_1})}^2 +\|\widetilde{u}\|_{W_2^{p-1}(\Omega_{\tau_1})}^2\bigr).
\end{equation*}
\notag
$$
Using this inequality recursively for $p\geqslant2$ and an appropriate monotone sequence $\tau'\to\tau_0$, taking (7.16) into account we obtain
$$
\begin{equation*}
\|\widetilde{u}\|_{W_2^p(\Omega_{\tau_0})}\leqslant C\bigl( \|\widetilde{f}\|_{W_2^{p-2}(\Omega_{\tau_1})} + \|\widetilde{f}\|_{L_2(\Omega)}\bigr)
\end{equation*}
\notag
$$
for any $\tau_0$ and $\tau_1$, $\tau_0<\tau_1<\tau$. Inequality (7.13) follows from this and (7.14). Lemma 7.2 is proved. The next result is a direct consequence of this lemma. Lemma 7.3. Let $\lambda<\lambda_0$, where $\lambda_0$ is from Lemma 7.2, and let $f=f(x,\eta)$, $\phi_1=\phi_1(x',\eta)$ and $\phi_2=\phi_2(x',\eta)$ be elements of the spaces $L_2(\Omega)\cap (W_2^p(\Omega_{\tau_1}^+)\oplus W_2^p(\Omega_{\tau_1}^-))$ and $ W_2^p(S)$ for all $p\in\mathbb{N}$ and some $\tau_1$, $0<\tau_1<\tau$, that are infinitely differentiable with respect to $\eta\in(0,1]$ and bounded uniformly in $\eta\in[0,1]$ in the norms of these spaces. Then the solution of (7.12) belongs to $\mathring{W}_2^1(\Omega)$ and $W_2^p(\Omega_{\tau_0}^+)\oplus W_2^p(\Omega_{\tau_0}^-)$ for all $p\in\mathbb{N}$ and $0<\tau_0<\tau_1$, and it is infinitely differentiable with respect to $\eta\in(0,1]$ and bounded uniformly in $\eta\in[0,1]$ in the norms of these spaces. In fact, the linear operator assigning the solution of (7.12) to a triple $(f,\phi_1,\phi_2)$ is bounded from $L_2(\Omega)\cap (W_2^p(\Omega_{\tau_1}^+)\oplus W_2^p(\Omega_{\tau_1}^-))\times W_2^{p+2}(S) \times W_2^{p+2}(S)$ to $(W_2^1(\Omega^+)\oplus W_2^1(\Omega^-))\cap (W_2^{p+2}(\Omega_{\tau_0}^+)\oplus W_2^{p+2}(\Omega_{\tau_0}^-))$ by Lemma 7.2. Therefore, if its arguments are infinitely differentiable and uniformly bounded with respect to a parameter, then the result of its action also has these properties. The rest of this section is concerned with the smoothness of the solution of (2.4) with respect to the space variables and the parameter $\alpha$. First we consider an auxiliary problem, the properties of whose solutions will play a key part in what follows:
$$
\begin{equation}
\begin{gathered} \, (-\Delta-\lambda)u=0 \quad\text{in } \Omega_{\tau_0}\setminus S, \qquad u=0 \quad\text{on } \partial\Omega_{\tau_0}\setminus S, \\ [u]_0=0, \qquad \biggl[\frac{\partial u}{\partial x_n}\biggr]_0+\mathrm{a}(x,u)=h \quad\text{on } S, \end{gathered}
\end{equation}
\tag{7.19}
$$
where $\tau_0<\tau$ is a fixed number, $\mathrm{a}$ is a complex function with the same domain of definition as $a$ and satisfying (2.1), and $h=h(x')$ is a fixed function, on which we assume at this point that $h\in L_2(S)$. Note straight away that there exists a fixed number $\lambda_0$ depending on $\mathrm{a}$, but independent of $h$ such that problem (7.19) is uniquely solvable in $W_2^1(\Omega_{\tau_0})$ for each function $h\in L_2(S)$. This can be verified similarly to how the solvability of problem (2.3) was proved in [30], § 4, Lemma 8, for an arbitrary function $\mathrm{a}(x,u)$ satisfying (2.1). Throughout what follows we consider $\lambda<\lambda_0$. It follows from the statement of problem (7.19) that its solution is even in $x_n$ and, by standard results on the interior smoothness of solutions of linear elliptic problems,
$$
\begin{equation}
u\in W_2^p(\Omega_{\tau_0}\setminus\Omega_{\tau_1}) \quad\text{for any } \tau_1<\tau_0\quad\text{and}\quad p\in\mathbb{N}.
\end{equation}
\tag{7.20}
$$
Set
$$
\begin{equation*}
u_\mathrm{r}:=\operatorname{Re} u, \ \ u_\mathrm{i}:=\operatorname{Im} u, \ \ h_\mathrm{r}:=\operatorname{Re} h,\ \ h_\mathrm{i}:=\operatorname{Im} h, \ \ a_\mathrm{r}:=\operatorname{Re} \mathrm{a}\ \ \ \text{and}\ \ \ a_\mathrm{i}:=\operatorname{Im} \mathrm{a}.
\end{equation*}
\notag
$$
By (2.1) the function $\mathrm{a}$ satisfies
$$
\begin{equation}
\begin{aligned} \, \notag |a_\mathrm{r}(x,u)u_\mathrm{r} + a_\mathrm{i}(x,u) u_\mathrm{i}| & \leqslant |u_r|^2 \sup_{x,u} \biggl|\frac{\partial a_\mathrm{r}}{\partial u_\mathrm{r}}\biggr| + |u_i|^2 \sup_{x,u} \biggl|\frac{\partial a_\mathrm{i}}{\partial u_\mathrm{r}}\biggr| \\ \notag &\qquad+ |u_r||u_i| \biggl( \sup_{x,u} \biggl|\frac{\partial a_\mathrm{r}}{\partial u_\mathrm{i}}\biggr|+ \sup_{x,u} \biggl|\frac{\partial a_\mathrm{i}}{\partial u_\mathrm{r}}\biggr|\biggr) \\ &\leqslant a_1 (|u_\mathrm{r}| +|u_\mathrm{i}|)^2\leqslant 2a_1 (|u_\mathrm{r}|^2 +|u_\mathrm{i}|^2). \end{aligned}
\end{equation}
\tag{7.21}
$$
Lemma 7.4. Let $h\in L_2(S)\cap L_\infty(S)$. Then there exists an absolute constant $\widetilde{\lambda}_0$ independent of $\mathrm{a}$ and $h$ such that for $\lambda<\widetilde{\lambda}_0$ the generalized solution of (7.19) belongs necessarily to $L_\infty(\Omega_{\tau_0})\cap C^\vartheta(\overline{\Omega_{\tau_0}})$, where $\vartheta$ is some absolute constant independent of $h$ and $\mathrm{a}$. Moreover, Proof. In the proof we use techniques from [33], Ch. VII, § 2. In that section of [33] problems with Dirichlet boundary condition for a linear elliptic system are considered. In fact, we reproduce some calculations from this monograph taking account of the nonlinear boundary condition currently present and estimating terms arising because of this boundary condition.
We start by verifying the membership relations
$$
\begin{equation}
|u|^2,\, |u||\nabla u|,\, |u||\nabla |u||\in L_2(\Omega_{\tau_0}).
\end{equation}
\tag{7.23}
$$
We choose some nonnegative real locally integrable function $\psi$ with locally integrable first derivatives such that $\psi(x) u(x)$ is an element of $W_2^1(\Omega_{\tau_0})$ and has zero trace on $\partial\Omega_{\tau_0}\setminus S$. Taking the real and imaginary parts of the equations and boundary conditions in (7.19) we obtain a system of boundary problems for the pair of functions $u_\mathrm{r}$, $u_\mathrm{i}$. For the problem for $u_\mathrm{r}$ we write an integral identity with test function $u_{\mathrm{r}} \psi$, and for the problem for $u_\mathrm{i}$ we write an integral identity with test function $u_{\mathrm{i}} \psi$. Adding these identities, after elementary transformations we obtain
$$
\begin{equation}
\begin{aligned} \, \notag &\int_{\Omega_{\tau_0}} \biggl(\psi|\nabla u|^2+ \frac{1}{2} \nabla |u|^2\cdot \nabla \psi - \lambda|u|^2\psi\biggr)\,dx= \int_{S} (a_\mathrm{r} u_\mathrm{r} + a_\mathrm{i} u_\mathrm{i}) \psi\,dx' \\ &\qquad\qquad- \int_{S} (h_\mathrm{r} u_\mathrm{r} + h_\mathrm{i} u_\mathrm{i}) \psi\,dx'. \end{aligned}
\end{equation}
\tag{7.24}
$$
For an arbitrary fixed $r>0$ set
$$
\begin{equation}
\psi(x):=\min\{\mathrm{u}(x),r\}, \quad\text{where } \mathrm{u}(x):=|u(x)|^2.
\end{equation}
\tag{7.25}
$$
Using (7.21) we can estimate the first integral on the right-hand side of (7.24) as follows:
$$
\begin{equation}
\biggl|\int_{S} (a_\mathrm{r} u_\mathrm{r} + a_\mathrm{i} u_\mathrm{i}) \psi\,dx'\biggr| \leqslant 2a_1 \int_{S} \mathrm{u} \psi \,dx'= a_1\biggl|\int_{\Omega_{\tau_0}^+} \frac{\partial}{\partial x_n}\mathrm{u} \psi \,dx-\int_{\Omega_{\tau_0}^-} \frac{\partial}{\partial x_n}\mathrm{u} \psi \,dx\biggr|.
\end{equation}
\tag{7.26}
$$
It follows from the definition (7.25) of $\psi$ that
$$
\begin{equation*}
\biggl|\frac{\partial}{\partial x_n} \mathrm{u} \psi\biggr|= 4|u| \psi \biggl|\frac{\partial |u|}{\partial x_n}\biggr|\leqslant 4|u| \psi |\nabla u| \quad\text{for } |\mathrm{u}|\leqslant r
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\biggl|\frac{\partial}{\partial x_n} \mathrm{u} \psi\biggr|= 2 r\psi|u| \biggl|\frac{\partial |u|}{\partial x_n}\biggr|\leqslant 2\psi|u| \,|\nabla u| \quad \text{for } |\mathrm{u}|>r.
\end{equation*}
\notag
$$
Hence we can extend (7.26) as follows, using Cauchy’s inequality:
$$
\begin{equation}
\begin{aligned} \, \notag \biggl|\int_{S} (a_\mathrm{r} u_\mathrm{r} + a_\mathrm{i} u_\mathrm{i}) \psi\,dx'\biggr| & \leqslant 4a_1 \int_{\Omega_{\tau_0}} \psi |u||\nabla u| \,dx' \\ &\leqslant \int_{\Omega_{\tau_0}} \biggl(\frac{1}{4} \psi|\nabla u|^2 +16a_1^2 |u|^2\biggr)\,dx. \end{aligned}
\end{equation}
\tag{7.27}
$$
In a similar way we estimate the second integral on the right-hand side of (7.24):
$$
\begin{equation}
\begin{aligned} \, \notag &\biggl| \int_{S} (h_\mathrm{r} u_\mathrm{r} + h_\mathrm{i} u_\mathrm{i}) \psi\,dx'\biggr| \leqslant \int_{S} |h| |u| \psi\,dx'\leqslant \|h\|_{L_\infty(S)} \int_{S} |u|\psi\,dx' \\ \notag &\qquad\leqslant \frac{1}{2} \|h\|_{L_\infty(S)}\biggl|\int_{\Omega_{\tau_0}} \operatorname{sgn} x_n \, \frac{\partial}{\partial x_n} |u|\psi\, dx\biggr| \leqslant \frac{3}{2}\|h\|_{L_\infty(S)} \int_{\Omega_{\tau_0}} |\nabla u|\psi\, dx \\ \notag &\qquad\leqslant \frac{3}{2}\|h\|_{L_\infty(S)} \biggl(\int_{\Omega_{\tau_0}} |\nabla u|^2\psi\,dx\biggr)^{1/2}\biggl(\int_{\Omega_{\tau_0}} \psi\,dx\biggr)^{1/2} \\ &\qquad\leqslant \frac{1}{4} \int_{\Omega_{\tau_0}} |\nabla u|^2\psi\,dx + \frac{9}{4} \|h\|_{L_\infty(S)}^2 \|u\|_{L_2(\Omega_{\tau_0})}^2. \end{aligned}
\end{equation}
\tag{7.28}
$$
Substituting this and (7.27) into (7.24), assuming that $\lambda < \lambda_0 <-1/2$ without loss of generality, we obtain
$$
\begin{equation*}
\int_{\Omega_{\tau_0}} ( \psi|\nabla u|^2+ |\nabla \psi|^2 + |u|^2\psi)\,dx \leqslant \frac{9}{2} \|h\|_{L_\infty(S)}^2 \|u\|_{L_2(\Omega_{\tau_0})}^2 +32 a_1^2 \|u\|_{L_2(\Omega_{\tau_0})}^2.
\end{equation*}
\notag
$$
Taking here the limit as $r\to+\infty$ we see that (7.23) holds and
$$
\begin{equation*}
\int_{\Omega_{\tau_0}} (|u|^2|\nabla u|^2+ |u|^4)\,dx \leqslant \frac{9}{2} \|h\|_{L_\infty(S)}^2 \|u\|_{L_2(\Omega_{\tau_0})}^2+32 a_1^2 \|u\|_{L_2(\Omega_{\tau_0})}^2.
\end{equation*}
\notag
$$
In view of the obvious inequality $|\nabla |u||\leqslant |\nabla u|$ we arrive at (7.23).
We turn to the boundedness of the function. Assume that $U$ is extended by zero outside $\Omega_{\tau_0}$. Again, we fix some $r > 0$ and an arbitrary ball $B_\rho(x_0)$, where $x_0\in \Omega_{\tau_0}$ and ${\tau_0}/{2}\leqslant \rho\leqslant \tau_0$. Let $\chi_6=\chi_6(x)$ denote an infinitely smooth cut-off function taking values in the closed interval $[0,1]$, equal to one in the ball $B_{{\rho}/{2}}(x_0)$, to zero outside $B_\rho(x_0)$ and satisfying $|\nabla \chi_6|\leqslant C\rho^{-1}$ in $B_\rho(x_0)\setminus B_{{\rho}/{2}}(x_0)$ for sone constant $C$ independent of $x$, $\rho$ and $x_0$. We also set $E_{r,\rho}:=\{x\colon \mathrm{u}(x)> r\}\cap B_\rho(x_0)$. Let $\psi:=\max\{2(\mathrm{u}(x)-r)\chi_6^2,0\}$ in (7.24). Then the left-hand side of this equality has the following lower estimate:
$$
\begin{equation}
\begin{aligned} \, \notag &\int_{\Omega_{\tau_0}} \biggl(\psi|\nabla u|^2+ \frac{1}{2} \nabla |u|^2\cdot \nabla \psi - \lambda|u|^2\psi\biggr)\,dx \\ \notag &\qquad= \int_{E_{r,\rho}} \bigl(2(\mathrm{u}-r)|\nabla u|^2\chi_6^2+ |\nabla \mathrm{u}|^2\chi_6^2 + (\mathrm{u}-r)\chi_6 \nabla \mathrm{u} \cdot \nabla \chi_6 +|\lambda| (\mathrm{u}-r)\mathrm{u} \chi_6^2\bigr)\,dx \\ &\qquad\geqslant \int_{E_{r,\rho}} \biggl(2(\mathrm{u}-r)|\nabla u|^2\chi_6^2+ \frac{3}{4}|\nabla \mathrm{u}|^2\chi_6^2 - (\mathrm{u}-r)^2|\nabla\chi_6|^2 + |\lambda| (\mathrm{u}-r)^2 \chi_6^2\biggr)\,dx. \end{aligned}
\end{equation}
\tag{7.29}
$$
We estimate the integrals on the right-hand side of (7.24) similarly to (7.26), (7.27) and (7.28). The estimate of the first integral is as follows:
$$
\begin{equation}
\begin{aligned} \, \notag &\biggl|\int_{S} (a_\mathrm{r} u_\mathrm{r} + a_\mathrm{i} u_\mathrm{i}) \psi\,dx'\biggr| \leqslant 2a_1 \int_{E_{r,\rho}} \biggl|\frac{\partial}{\partial x_n}(\mathrm{u}-r)\mathrm{u}\chi_6^2 \biggr|\,dx \\ \notag &\qquad\leqslant 2a_1 \int_{E_{r,\rho}} \bigl( 2 (\mathrm{u}-r)\mathrm{u} \chi_6|\nabla \chi_6|+(2\mathrm{u}-r)|\nabla \mathrm{u}|\chi_6^2 \bigr)\,dx \\ \notag &\qquad\leqslant \frac{1}{4} \int_{E_{r,\rho}} |\nabla \mathrm{u}|^2\chi_6^2\,dx + C \int_{E_{r,\rho}} \bigl((\mathrm{u}-r)^2(\chi_6^2+|\nabla \chi_6|^2) + \mathrm{u}^2\chi_6^2\bigr)\,dx \\ &\qquad\leqslant \frac{1}{4} \int_{E_{r,\rho}} |\nabla \mathrm{u}|^2\chi_6^2\,dx + C \int_{E_{r,\rho}} (\mathrm{u}-r)^2(\chi_6^2+|\nabla \chi_6|^2)\,dx + C r^2 \operatorname{mes} E_{r,\rho}, \end{aligned}
\end{equation}
\tag{7.30}
$$
where we denote by $C$ absolute constants independent of $\rho$, $x_0$, $r$, $\mathrm{u}$, $u$ and $\chi_6$. The estimate of the second integral is as follows:
$$
\begin{equation*}
\begin{aligned} \, &\biggl|\int_{S} (h_\mathrm{r} u_\mathrm{r} + h_\mathrm{i} u_\mathrm{i}) \psi\,dx'\biggr| \leqslant \int_{S} |h| |u| \psi\,dx' \leqslant \|h\|_{L_\infty(S)} \int_{E_{r,\rho}} \biggl|\frac{\partial}{\partial x_n} |u|(\mathrm{u}-r)\chi_6^2\biggr| \,dx \\ &\qquad\leqslant \int_{E_{r,\rho}} \biggl(|\nabla u|^2 (\mathrm{u}-r)+\frac{1}{4}|\nabla \mathrm{u}|^2\biggr)\chi_6^2\,dx + \int_{E_{r,\rho}} |\mathrm{u}-r|^2 |\nabla \chi_6|^2 \,dx \\ &\qquad \qquad + C \int_{E_{r,\rho}}\bigl((\|h\|_{L_\infty(S)}^2+1)\mathrm{u} + \|h\|_{L_\infty(S)}^2(\mathrm{u}-r)\bigr) \chi_6^2\,dx \\ &\qquad\leqslant \int_{E_{r,\rho}} \biggl(|\nabla u|^2 (\mathrm{u}-r)+\frac{1}{4}|\nabla \mathrm{u}|^2\biggr)\chi_6^2\,dx \\ &\qquad\qquad+ \int_{E_{r,\rho}}\bigl((\mathrm{u}-r)^2 (|\nabla \chi_6|^2+|\chi_6|^2)+ \mathrm{u}^2 \chi_6^2\bigr) \,dx + C (\|h\|_{L_\infty(S)}^4+1) \operatorname{mes} E_{r,\rho} \\ &\qquad\leqslant \int_{E_{r,\rho}} \biggl((\mathrm{u}-r)|\nabla u|^2 +\frac{1}{4}|\nabla \mathrm{u}|^2\biggr)\chi_6^2\,dx + 3\int_{E_{r,\rho}} (\mathrm{u}-r)^2 \bigl(|\nabla \chi_6|^2+|\chi_6|^2\bigr)\,dx \\ &\qquad\qquad+ C (r^2+\|h\|_{L_\infty(S)}^4+1) \operatorname{mes} E_{r,\rho}, \end{aligned}
\end{equation*}
\notag
$$
where we denote by $C$ absolute constants independent of $r$, $\rho$, $x_0$, $u$, $\mathrm{u}$ and $h$. Substituting this estimate, (7.30) and (7.29) into (7.24), bearing in mind that
$$
\begin{equation*}
E_{r,\rho}\subseteq B_\rho(x_0) \quad\text{for } \rho\leqslant \tau_0\quad\text{and} \quad \operatorname{mes} E_{r,\rho}\leqslant \rho^n \operatorname{mes} B_1(0)\leqslant \tau_0^n \operatorname{mes} B_1(0),
\end{equation*}
\notag
$$
and then assuming that $\lambda_0$ has a modulus greater than a suitable absolute constant we obtain
$$
\begin{equation}
\begin{aligned} \, \notag &\int_{E_{r,\rho}}\bigl((\mathrm{u}-r)|\nabla u|^2\chi_6^2+ |\nabla \mathrm{u}|^2\chi_6^2+ (\mathrm{u}-r)^2 \chi_6^2 \bigr)\,dx \leqslant C \int_{E_{r,\rho}} (\mathrm{u}-r)^2 |\nabla \chi_6|^2 \,dx \\ &\qquad\qquad+ C (r^2+\|h\|_{L_\infty(S)}^4+1) \operatorname{mes}^{1-{1}/{n}} E_{r,\rho}, \end{aligned}
\end{equation}
\tag{7.31}
$$
where we denote by $C$ absolute constants independent of $r$, $\rho$, $x_0$, $u$, $\mathrm{u}$ and $h$. This estimate means that, for some fixed constants $c_1$ and $c_2$, $\mathrm{u}$ is in the class $\mathfrak{A}_2(B_{\tau_0}(x_0)\cap \Omega_{\tau_0},c_1,2,2,{1}/{n},c_2)$, introduced in [33], Ch. II, § 5, so that by Theorem 5.3 in [33], Ch. II, it also belongs to $L_\infty(\Omega_{\tau_0})$.
Inequality (7.31) means that $\mathrm{u}\in \mathfrak{B}_2(\Omega_{\tau_0},\|\mathrm{u}\|_{L_\infty(\Omega_{\tau_0})}, c_3,c_4,1,{1}/(2n))$, where the function class was defined in [33], Ch. II, § 6. Hence by Theorem 6.1 in [33], Ch. II, § 6, $\mathrm{u}$ is an element of the Hölder space $C^{2\vartheta}(\overline{\Omega_{\tau_0}})$ for some absolute constant $\vartheta$ independent of $h$ and $\mathrm{a}$. Now we view (7.19) as a system for the functions $(u_\mathrm{r},u_\mathrm{i})$ and reproduce the arguments in [33], Ch. VII, § 3, for this system, similarly to the above calculations for $\mathrm{u}$. In fact, we need an estimate of the form (7.31) for the functions $\pm u_\mathrm{r}-r$ and $\pm u_\mathrm{i}-r$, given a priori that $u\in L_\infty(\Omega_{\tau_0})$. Finally, these estimates enable us to use Theorems 8.2, 8.3 and Lemma 8.4 in [33], Ch. II, § 8, and conclude that $u\in C^\vartheta(\overline{\Omega_{\tau_0}})$ for some fixed exponent $\vartheta$. Lemma 7.4 is proved. Using this lemma in combination with standard theorems on improved smoothness for linear problems we can establish the following key result. Lemma 7.5. Let $\lambda<\min\{\lambda_0,\widetilde{\lambda}_0\}$, and let $h\in W_2^p(S)$ for all $p\in\mathbb{N}$. Then the solution of (7.19) is an element of $W_2^p(\Omega_{\tau_0}^-)\oplus W_2^p(\Omega_{\tau_0}^+)$ for all $p\in \mathbb{N}$ and some $\tau_0<\tau$. The norms of the solution $u$ in these spaces have estimates depending only on the constants $a_1$ and $a_{\beta,\gamma}$ in (2.1) and (2.2), the norm $\|u\|_{W_2^1(\Omega_{\tau_0})}$ and the norms $\|h\|_{W_2^p(S)}$. Proof. By the second inequality for the function $\mathrm{a}$ in (2.1), $\mathrm{a}(x,u(x))$ is an element of $W_2^1(\Omega_{\tau_0})$. Its trace is a term on the left-hand side of the boundary condition in (7.19), and by Theorem 25.1 in [38], § 25, there exists a function in $W_2^2(\Omega_{\tau_0}^+)\oplus W_2^2(\Omega_{\tau_0}^-)$ that satisfies the boundary conditions in problem (7.19). Hence from the standard theorems on improved smoothness for linear boundary value problems we conclude directly that $u\in W_2^2(\Omega_{\tau_0}^+)\oplus W_2^2(\Omega_{\tau_0}^-)$. Furthermore, using Lemma 7.4 we can claim that $u$ is also an element of $L_\infty(\Omega_{\tau_0})\cap C^\vartheta(\overline{\Omega_{\tau_0}})$.
Bearing in mind the function class of $u$ we differentiate the equation and boundary conditions in (7.19) with respect to the $x_j$, $j=1,\dots,n-1$. Then we see that the functions $u_j:={\partial u}/{\partial x_j}$, $j=1,\dots,n-1$, are generalized solutions of the boundary value problem
$$
\begin{equation}
\begin{gathered} \, (-\Delta-\lambda)u_j=0 \quad\text{in } \Omega_{\tau_0}\setminus S, \qquad u_j=0 \quad\text{on } \partial\Omega_{\tau_0}\setminus S, \\ [u_j]_0=0, \qquad \biggl[\frac{\partial u_j}{\partial x_n}\biggr]_0+\mathrm{a}_j(x,u_j)=h_j \quad\text{on } S, \end{gathered}
\end{equation}
\tag{7.32}
$$
where we set
$$
\begin{equation*}
\mathrm{a}_j(x,u_j):=\frac{\partial a}{\partial u_\mathrm{r}}(x,u(x))\operatorname{Re} u_j + \mathrm{i} \, \frac{\partial a}{\partial u_\mathrm{i}}(x,u(x))\operatorname{Im} u_j
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
h_i(x):=\frac{\partial a}{\partial x_i}(x,u(x))+\frac{\partial h}{\partial x_i}(x).
\end{equation*}
\notag
$$
From (2.1), (2.2), the relation $u\in L_\infty(\Omega_{\tau_0})$ and the supposed smoothness of $h$ it is easy to see that $\mathrm{a}_j$ also satisfies (2.1). Hence we can apply Lemma 7.4 to (7.32) and conclude that ${\partial u}/{\partial x_j}=u_j\in L_\infty(\Omega_{\tau_0})\cap C^\vartheta(\overline{\Omega_{\tau_0}})$.
Bearing in mind that $u$ is even in $x_n$, in view of (7.20) the function $u_n:={\partial u}/{\partial x_n}$ solves the same equation in (7.19), is odd in $x_n$, its trace on $\partial\Omega\setminus S$ is an element of $W_2^p(\partial\Omega\setminus S)$ for all $p\in\mathbb{N}$, and the boundary condition $u_n\big|_{x_n=\pm 0}=\pm a(x,u(x))\big|_{x_n=0}$ is satisfied. From the assumptions on $a$ and the properties of $u$ that we have established, we see that $\pm a(x,u(x))\big|_{x_n=0}$ are elements of the space $L_\infty(S)\cap C^\vartheta(S)$ for some $\vartheta$ and are the traces of some functions $a(x,u(x))$ in $W_2^1(\Omega_{\tau_0}^\pm)$. So we can apply Theorem 13.1 in [33], Ch. III, § 13, and Theorem 14.1 in [33], Ch. III, § 13, to the function $u_n$, which show that $u_n\in L_\infty(\Omega_{\tau_0}^\pm)\cap C^\vartheta(\overline{\Omega_{\tau_0}})$. Now, since $u,{\partial u}/{\partial x_j}\in L_\infty(\Omega_{\tau_0}^\pm)\cap C^\vartheta(\overline{\Omega_{\tau_0}})$, $j=1,\dots,n$, it is easy to verify, taking the supposed smoothness of $a$ and conditions (2.1) and (2.2) into account, that the function $a(x,u(x))$ is an element of $W_2^2(\Omega_{\tau_0}^\pm)$. Hence, similarly to the beginning of the proof, the function $u$ is an element of $W_2^3(\Omega_{\tau_0}^+)\oplus W_2^3(\Omega_{\tau_0}^-)$. Thus, we can differentiate (7.19) twice with respect to space variables and show, as a result, that the second derivatives of $u$ belong to $L_\infty(\Omega_{\tau_0}^\pm)\cap C^\vartheta(\overline{\Omega_{\tau_0}})$. Repeating this procedure recursively, we arrive at the required result. The lemma is proved. Lemma 7.6. There exists $\lambda_0$ such that for $\lambda<\lambda_0$ the problem Proof. Let $\chi_7=\chi_7(x_n)$ be an infinitely differentiable cut-off function equal to one for $|x_n|<{\tau_0}/{3}$ and to zero for $|x_n|>{2\tau_0}/{3}$. The substitution $u=\widetilde{u}+\frac{1}{2}\chi_7(x_n) |x_n|h(x')$ reduces problem (7.33), (7.34) to the special case of it for $h=0$ and a certain new right-hand side of the equation, where this right-hand side also satisfies (2.5). The solvability of (7.33), (7.34) for $h=0$ was shown in [30], § 4, Lemma 8, for any function $a(x,u)$; the value of $\lambda_0$ in this case is actually determined by $a_1$ in (2.1). Hence in the problem under consideration we can choose a common constant $a_1$ for all $\alpha\in[0,\alpha_*]$ and then use Lemma 8 in [30], § 4.
We prove that the solution is smooth in the space variables. Take an arbitrary $\alpha_0\in[0,\alpha_*]$, and let $u_0$ denote the corresponding solution of problem (7.33), (7.34) for some fixed function $f$. Then $\widetilde{u}_0:=u_0\chi_7$ is a solution of the boundary value problem
$$
\begin{equation}
\begin{gathered} \, (-\Delta-\lambda)\widetilde{u}_0=\widetilde{f} \quad\text{in } \Omega_{\tau_0}, \qquad \widetilde{u}_0=0 \quad\text{on } \partial\Omega_{\tau_0}, \\ [\widetilde{u}_0]_0=0, \qquad \biggl[\frac{\partial\widetilde{u}_0}{\partial x_n}\biggr]_0+a(x,\widetilde{u}_0)=g\quad\text{on } S, \end{gathered}
\end{equation}
\tag{7.36}
$$
where $\widetilde{f}:=\chi_7 f -2\chi_7'\,{\partial u_0}/{\partial x_n} - \chi_7'' u_0$ is a function in $W_2^p(\Omega_{\tau_0})$ for all $p\in\mathbb{N}$.
Let $u_f$ be the solution of the equation in (7.36) with Dirichlet boundary condition on $\partial\Omega_{\tau_0}\cup S$. This problem is solvable for $\lambda<0$, so we assume without loss of generality that $\lambda_0<0$. By standard smoothness improving theorems for solutions of linear elliptic problems we have $u_f\in W_2^p(\Omega_{\tau_0}^\pm)$ for all $p\in\mathbb{N}$. Set $u_a:=\widetilde{u}_0-u_f$. Then for $u_a$ we obtain problem (7.19), where $h=g+[{\partial u_f}/{\partial x_n}]_0$. By the properties of $u_f$ we have $h\in W_2^p(S)$ for all $p\in\mathbb{N}$. Now we conclude from Lemma 7.5 that $u_a\in W_2^p(\Omega_{\tau_0}^+)\oplus W_2^p(\Omega_{\tau_0}^-)$ for all $p\in\mathbb{N}$. Returning to the solution of (2.4) and bearing in mind that $u_f$ is smooth, we arrive at the required result. The lemma is proved. Lemma 7.7. Under the assumptions of Lemma 7.6 the solution of (2.4) is infinitely differentiable with respect to $\alpha\in[0,\alpha_*]$ in the norms of $W_2^1(\Omega)$ and $W_2^p(\Omega_{\tau_0}^+)\cap W_2^p(\Omega_{\tau_0}^-)$ for all $p\in\mathbb{N}$. Proof. We choose a sufficiently small $\epsilon$ and let $u_\epsilon$ denote the solution of (2.4) for $\alpha=\alpha_0+\epsilon\in[0,\alpha_*]$. The difference quotient $w_\epsilon:=(u_\epsilon-u_0)\epsilon^{-1}$ solves the boundary value problem
$$
\begin{equation*}
\begin{gathered} \, \biggl(-\sum_{i,j=1}^n\frac{\partial}{\partial x_i}\, A_{ij}\, \frac{\partial }{\partial x_j}+\sum_{j=1}^n A_j\, \frac{\partial}{\partial x_j} +A_0-\lambda \biggr)w_\epsilon=0 \quad\text{in } \Omega, \qquad w_\epsilon=0 \quad \text{on } \partial\Omega, \\ [w_\epsilon]_0=0, \qquad \biggl[\frac{\partial w_\epsilon}{\partial \mathrm{n}}\biggr]_0+a_\epsilon(x,w_\epsilon) +a(x,u_0(x))=0 \quad\text{on } S, \end{gathered}
\end{equation*}
\notag
$$
where we set
$$
\begin{equation*}
a_\epsilon(x,w):= ( \alpha_0+\epsilon) \epsilon^{-1}\bigl(a(x,u_0(x)+\epsilon w)-a(x,u_0(x))\bigr)\big|_{x_n=0}.
\end{equation*}
\notag
$$
Note that by the inequality $0\leqslant \alpha_0+\epsilon \leqslant \alpha_*$ and Hadamard’s lemma, $a_\epsilon$ satisfies (2.1) and (2.2) and, moreover, with the same constants as $a$. By Lemma 7.5 and the result on embedding $W_2^p$ in $C^{p-[{n}/{2}]-1}$ the trace of $u_0$ on $S$ belongs to $W_2^p(S)$ for all $p\in\mathbb{N}$ and is infinitely differentiable and uniformly bounded together with all derivatives. So we can use Lemma 7.6 and obtain the following estimate, which holds uniformly in $\epsilon$:
$$
\begin{equation}
\begin{gathered} \, \|w_\epsilon\|_{W_2^1(\Omega)}\leqslant C, \qquad \|w_\epsilon\|_{W_2^p(\Omega_{\tau_0}^+)} + \|w_\epsilon\|_{W_2^p(\Omega_{\tau_0}^-)} \leqslant C, \\ \|w_\epsilon\|_{W_\infty^p(\Omega_{\tau_0}^+)} + \|w_\epsilon\|_{W_\infty^p(\Omega_{\tau_0}^-)} \leqslant C, \qquad \max_{\overline{\Omega_{\tau_0}^+}} |\partial^\beta w_\epsilon| + \max_{\overline{\Omega_{\tau_0}^-}} |\partial^\beta w_\epsilon| \leqslant C, \end{gathered}
\end{equation}
\tag{7.37}
$$
where $p\in\mathbb{N}$, $\beta\in\mathbb{Z}_+^n$, and the constants $C$ are independent of $\epsilon$.
Now consider another problem:
$$
\begin{equation}
\begin{gathered} \, \biggl(-\sum_{i,j=1}^n\frac{\partial}{\partial x_i}\, A_{ij}\, \frac{\partial }{\partial x_j}+\sum_{j=1}^n A_j\, \frac{\partial}{\partial x_j} +A_0-\lambda \biggr)w_0=0 \quad\text{in } \Omega, \qquad w_0=0 \quad \text{on } \partial\Omega, \\ [w_0]_0=0, \qquad \biggl[\frac{\partial w_0}{\partial \mathrm{n}}\biggr]_0 + a_0(x,w_0) +a(x,u_0(x)) =0 \quad\text{on } S, \end{gathered}
\end{equation}
\tag{7.38}
$$
where we set
$$
\begin{equation*}
a_0(x,w ):= \alpha_0 \biggl( \frac{\partial a}{\partial u_\mathrm{r}}(x,u_0) \operatorname{Re} w + \mathrm{i}\, \frac{\partial a}{\partial u_\mathrm{i}}(x,u_0) \operatorname{Im} w \biggr).
\end{equation*}
\notag
$$
It is clear that the function $a_0$ satisfies conditions (2.1) and (2.2) with the same constants $a_1$ and $a_{\beta,\gamma}$ as the function $a$. Hence, for the same choice of $\lambda_0$ we obtain estimates for $w_0$ which are similar to (7.37): it is sufficient to replace $w_\epsilon$ by $w_0$.
Now set $\Theta_\epsilon:=w_\epsilon-w_0$. This is a solution of the problem
$$
\begin{equation}
\begin{gathered} \, \biggl(-\sum_{i,j=1}^n\frac{\partial}{\partial x_i}\, A_{ij}\, \frac{\partial }{\partial x_j}+\sum_{j=1}^n A_j\, \frac{\partial}{\partial x_j} +A_0-\lambda \biggr)\Theta_\epsilon=0 \quad\text{in } \Omega, \qquad \Theta_\epsilon=0\quad \text{on } \partial\Omega, \\ [\Theta_\epsilon]_0=0, \qquad \biggl[\frac{\partial \Theta_\epsilon}{\partial \mathrm{n}}\biggr]_0 + a_\epsilon(x,w_0+\Theta_\epsilon)=a_0(x,w_0) \quad\text{on } S. \end{gathered}
\end{equation}
\tag{7.39}
$$
It is easy to verify that
$$
\begin{equation*}
\begin{gathered} \, a_\epsilon(x,w_0+\Theta_\epsilon)-a_0(x,w_0)=a^\epsilon(x,\Theta_\epsilon) + h^\epsilon(x), \\ a^\epsilon(x,\Theta):=(\alpha_0+\epsilon) \epsilon^{-1} \bigl(a(x,u_0(x)+\epsilon w_0(x)+ \epsilon \Theta)- a(x,u_0(x)+\epsilon w_0(x))\bigr), \\ \begin{split} h^\epsilon(x)&:=(\alpha_0+\epsilon) \epsilon^{-1} \bigl(a(x,u_0(x)+\epsilon w_0(x)) - a(x,u_0(x))\bigr) \\ &\qquad -\alpha_0 \biggl( \frac{\partial a}{\partial u_\mathrm{r}}(x,u_0) \operatorname{Re} w_0(x) + \mathrm{i} \, \frac{\partial a}{\partial u_\mathrm{i}}(x,u_0) \operatorname{Im} w_0(x) \biggr). \end{split} \end{gathered}
\end{equation*}
\notag
$$
Taking the smoothness of $u_0$ and $v_0$ and estimates of type (7.37) for these functions into account we conclude straight away that $h^\epsilon \in W_2^p(S)\cap W_\infty^p(S)\cap C^\infty(S)$ for all ${p\in\mathbb{N}}$, and we have the estimates
$$
\begin{equation}
\max_{S} |\partial^\beta h^\epsilon|\leqslant C\epsilon\quad\text{and} \quad \|h^\epsilon\|_{W_2^p(S)} \leqslant C\epsilon, \qquad p\in\mathbb{N}, \quad \beta\in\mathbb{Z}_+^n,
\end{equation}
\tag{7.40}
$$
for some constants $C$ independent of $\epsilon$. The function $a^\epsilon$ satisfies (2.1) and (2.2) with the same constants as the function $a$. Using this, (7.40) and (7.35) we conclude that
$$
\begin{equation*}
\|\Theta_\epsilon\|_{W_2^1(\Omega)} + \|\Theta_\epsilon\|_{W_2^2 (\Omega_{\tau_0}^+)} + \|\Theta_\epsilon\|_{W_2^2 (\Omega_{\tau_0}^-)}\leqslant C\epsilon,
\end{equation*}
\notag
$$
where the constant $C$ is independent of $\epsilon$. Now we differentiate (7.39) with respect to the space variables and write out the relevant boundary value problems for the derivatives of $\Theta_\epsilon$. Then on the basis of (7.40) and (7.35) we easily verify using induction that
$$
\begin{equation*}
\|\Theta_\epsilon\|_{W_2^p (\Omega_{\tau_0}^+)} + \|\Theta_\epsilon\|_{W_2^p (\Omega_{\tau_0}^-)}\leqslant C\epsilon,
\end{equation*}
\notag
$$
where the constant $C$ is independent of $\epsilon$. These estimates mean that $w_\epsilon$ converges to $w_0$ as $\epsilon\to+0$ in the norms of $W_2^1(\Omega)$ and $W_2^p (\Omega_{\tau_0}^+) \oplus W_2^p (\Omega_{\tau_0}^-)$ for all $p\in\mathbb{N}$. Hence the solution of (2.4) is differentiable with respect to $\alpha$ in these spaces, and its derivative is the solution $w_0$ of problem (7.38). This problem is of the same type as the original problem (2.4), and the fact that its solution is differentiable with respect to $\alpha$ is similarly proved. Repeating this procedure recursively we arrive at the result of Lemma 7.7. § 8. Properties of the coefficients of the inner and outer expansionsIn this section we prove that the problems for the coefficients of the outer and inner expansions are solvable and investigate their properties. We start with problem (2.6), (3.5) for $m=0$. This is a homogeneous problem, which has a unique solution bounded at infinity by Lemma 6.1, namely, a constant. In view of the asymptotic expressions (3.5) this means that
$$
\begin{equation}
u_0(x',+0,\eta)=u_0(x',-0,\eta)\quad\text{and} \quad v_0(\xi, x', \eta)\equiv u_0(x',0,\eta).
\end{equation}
\tag{8.1}
$$
To find a solution of problem (2.6), (3.5) for $m=1$, we consider the auxiliary problems
$$
\begin{equation}
\Delta_\xi Z_\pm=0 \quad\text{in } \Pi\setminus\overline{\omega^\eta}, \qquad \frac{\partial Z_-}{\partial\nu_\xi}=0, \qquad \frac{\partial Z_+}{\partial\nu_\xi}=-\frac{|\square|}{|\partial\omega|} \quad\text{on } \partial\omega^\eta
\end{equation}
\tag{8.2}
$$
with the periodic boundary conditions (4.2) and the following behaviour at infinity:
$$
\begin{equation}
Z_-(\xi)=\frac{1}{2}\xi_n+O(1)\quad\text{and} \quad Z_+(\xi)=\frac{\eta^{n-1}}{2}|\xi_n|+O(1), \qquad \xi_n\to\pm\infty.
\end{equation}
\tag{8.3}
$$
We seek the solution of the problems in (8.2), (8.3) in the following form
$$
\begin{equation*}
Z_-(\xi)=\widetilde{Z}_-(\xi)+\frac{1}{2}\chi_1(\xi_n)\xi_n, \qquad Z_+(\xi)=\widetilde{v}_+(\xi)+\frac{\eta^{n-1}}{2}\chi_1(\xi_n)|\xi_n|.
\end{equation*}
\notag
$$
Then we obtain the following problems with periodic boundary conditions (4.2) for the functions $\widetilde{v}_\pm$:
$$
\begin{equation}
\begin{gathered} \, \Delta_\xi \widetilde{Z}_-=-\frac{1}{2}\Delta_\xi(\chi_1 \xi_n) \quad\text{in } \Pi\setminus\overline{\omega^\eta}, \qquad\frac{\partial\widetilde{Z}_-}{\partial \nu_\xi}=0 \quad\text{on } \partial\omega^\eta, \\ \Delta_\xi \widetilde{Z}_+=-\frac{\eta^{n-1}}{2}\Delta_\xi(\chi_1 |\xi_n|) \quad\text{in } \Pi\setminus\overline{\omega^\eta}, \qquad \frac{\partial\widetilde{Z}_+}{\partial \nu_\xi}=-\frac{|\square|}{|\partial\omega|} \quad\text{on } \partial\omega^\eta. \end{gathered}
\end{equation}
\tag{8.4}
$$
We verify that the solvability condition in Lemma 6.1 holds for these problems. To do this we integrate by parts:
$$
\begin{equation*}
\begin{aligned} \, &\int_{\partial\omega^\eta} \frac{|\square|}{|\partial\omega^\eta|}\,ds -\frac{\eta^{n-1}}{2}\int_{\Pi\setminus\omega^\eta}\Delta_\xi (|\xi_n|\chi_1) \,d\xi \\ &\qquad =|\square|\eta^{n-1} -\frac{1}{2}\lim_{R\to+\infty}\int_{|\square|}\frac{\partial |\xi_n|}{\partial \xi_n}\bigg|_{\xi_n=-R}^{\xi_n=R}\,d\xi=0. \end{aligned}
\end{equation*}
\notag
$$
For the problem with respect to $Z_-$ the solvability condition is verified in a similar way. Then it follows immediately from Lemmas 6.1 and 7.1 that problems (8.4), and therefore also (8.2), (8.3), are solvable, and for each $\eta\in(0,1]$ their solutions are infinitely differentiable in $\overline{\Pi}\setminus\omega^\eta$. For $|\xi_n|>R_0$ the functions $Z_\pm$ have the form
$$
\begin{equation}
\begin{split} Z_+(\xi,\eta) &=\frac{\eta^{n-1}}{2}|\xi_n| \\ &\qquad +\sum_{k\in\mathbb{Z}^{n-1}} D^{+,\pm}_k(\eta) \exp(-2\pi |k_b| |\xi_n|)\exp(2\pi\mathrm{i} k_b\cdot\xi'),\qquad \pm\xi_n>R_0, \end{split}
\end{equation}
\tag{8.5}
$$
and
$$
\begin{equation}
Z_-(\xi,\eta)=\frac{1}{2}\xi_n +\sum_{k\in\mathbb{Z}^{n-1}} D_k^{-,\pm}(\eta) \exp(-2\pi |k_b| |\xi_n|)\exp(2\pi\mathrm{i} k_b\cdot\xi'), \qquad \pm\xi_n> R_0,
\end{equation}
\tag{8.6}
$$
where $D$ and $D_k^{-,\pm}$ are infinitely differentiable functions with respect to ${\eta\!\in\!(0,1]}$. The following estimates hold:
$$
\begin{equation*}
\sup_{k\in\mathbb{Z}^{n-1}} \exp(-2\pi |k_b| R_0)|D^{+,\pm}_k(\eta)| + \sup_{k\in\mathbb{Z}^{n-1}} \exp(-2\pi |k_b| R_0)|D^{-,\pm}_k(\eta)| \leqslant C
\end{equation*}
\notag
$$
and where the constant $C$ is independent of $\eta$. For each $k\in\mathbb{N}$ and $\eta_0\in(0,1]$ there exists $\delta_k(\eta_0)$ such that
$$
\begin{equation}
Z_\pm\bigl(\Xi^{-1}(\eta_0\eta^{-1},\xi),\eta\bigr) \in C^{k+1/2}\bigl(\overline{\Pi}\setminus\omega^{\eta_0} \times[\eta_0-\delta_k(\eta_0),\eta_0+\delta_k(\eta_0)]\bigr).
\end{equation}
\tag{8.7}
$$
Now consider the boundary value problems
$$
\begin{equation*}
\Delta_\xi \varphi_{1j}=0 \quad\text{in } \Pi\setminus\omega^\eta, \qquad \frac{\partial \varphi_{1j}}{\partial \nu_\xi}=\nu_j \quad\text{on } \partial\omega^\eta
\end{equation*}
\notag
$$
with periodic boundary conditions on the lateral part of the boundary. The solvability conditions in Lemma 6.1 hold for these problems because
$$
\begin{equation*}
0=\int_{\omega^\eta}\Delta_\xi \xi_j \,d\xi=\int_{\partial \omega^\eta} \frac{\partial \xi_j}{\partial \nu}\, ds= \int_{\partial\omega^\eta}\nu_j\,ds.
\end{equation*}
\notag
$$
By Lemmas 6.1 and 7.1 the functions $\varphi_{1j}$ have the form
$$
\begin{equation}
\varphi_{1j}(\xi,\eta)=\sum_{k\in\mathbb{Z}^{n-1}}D^\pm_{1j,k}(\eta) \exp(-2\pi |k_b| |\xi_n|)\exp(2\pi\mathrm{i} k_b\cdot\xi'), \qquad \pm\xi_n> R_0,
\end{equation}
\tag{8.8}
$$
where the $D_{1j,k}^\pm=D_{1j,k}^\pm(\eta)\in C^\infty(0,1]$ are some functions. We have the estimates
$$
\begin{equation*}
\sup_{k\in\mathbb{Z}^{n-1}} e^{-2\pi |k_b| R_0}|D_{1j,k}^\pm| \leqslant C\quad\text{and} \quad \|\varphi_{1j}\|_{C^1(\overline{\Pi^\eta})}\leqslant C,
\end{equation*}
\notag
$$
where the constant $C$ is independent of $\eta$. The functions $\varphi_{1j}$ satisfy membership relations similar to (8.7). Now the solution of (2.6), (3.5) for $m=1$ can be found explicitly:
$$
\begin{equation}
\begin{aligned} \, \notag v_1(\xi, x', \eta) &=\biggl(\frac{\partial u_0}{\partial x_n}(x',+0,\eta)+\frac{\partial u_0}{\partial x_n}(x',-0,\eta) \biggr) Z_-(\xi,\eta) \\ \notag &\qquad+\eta^{-n+1} \biggl( \frac{\partial u_0}{\partial x_n}(x',+0,\eta)-\frac{\partial u_0}{\partial x_n}(x',-0,\eta) \biggr)Z_+(\xi,\eta) \\ &\qquad- \sum_{j=1}^{n-1}\frac{\partial u_0}{\partial x_j}(x',0,\eta)\, \varphi_{1j}(\xi,\eta)+v_1^{(0)}(x',\eta), \end{aligned}
\end{equation}
\tag{8.9}
$$
where $v_1^{(0)}$ is a function to be determined below. The above function satisfies the equation in (2.6) and periodic conditions on the lateral part of the boundary, and it exhibits the behaviour (3.5) for $m=1$. Since the boundary condition in (2.6) for $m=1$ must hold for $v_1$, taking the formula for $T_0(x',v_0)$, equalities (8.1) and the boundary conditions for $Z_\pm$ in (8.2) into account we arrive at the following boundary conditions for $u_0$:
$$
\begin{equation*}
[u_0]_0=0\quad\text{and} \quad \biggl[\frac{\partial u_0}{\partial x_n}\biggr]_0-\frac{\eta^{n-1}|\partial\omega|}{|\square|}a(x',0,u_0(x',0,\eta))=0,
\end{equation*}
\notag
$$
where the first equality follows from the first equality in (8.1). Using the second boundary condition we can rewrite (8.9) as follows:
$$
\begin{equation*}
\begin{aligned} \, &v_1(\xi, x', \eta) =\biggl(\frac{\partial u_0}{\partial x_n}(x',+0,\eta)+\frac{\partial u_0}{\partial x_n}(x',-0,\eta) \biggr) Z_-(\xi,\eta) \\ &\quad+\frac{|\partial\omega|}{|\square|}a(x',0,u_0(x',0,\eta))Z_+(\xi,\eta) - \sum_{j=1}^{n-1}\frac{\partial u_0}{\partial x_i}(x',0,\eta)\, \varphi_{1j}(\xi,\eta)+ v_1^{(0)}(x',\eta). \end{aligned}
\end{equation*}
\notag
$$
Comparing the third term in the asymptotic expression (3.5) for $v_1$ with the analogous term obtained from the last formula, and (8.5), (8.6) and (8.8) we arrive at the following boundary conditions for $u_1$:
$$
\begin{equation}
\begin{aligned} \, \nonumber u_1(x',\pm 0,\eta) &=\biggl(\frac{\partial u_0}{\partial x_n}(x',+0,\eta)+\frac{\partial u_0}{\partial x_n}(x',-0,\eta) \biggr)D_0^{-,\pm}(\eta) \\ \nonumber &\qquad + \frac{|\partial\omega|}{|\square|}a(x',0,u_0(x',0,\eta)) D_0^{+,\pm}(\eta) \\ \nonumber &\qquad- \sum_{j=1}^{n-1}\frac{\partial u_0}{\partial x_i}(x',0,\eta)\, D_{1j,0}^\pm(\eta) + v_1^{(0)}(x',\eta), \\ \nonumber [u_1]_0 &=\biggl(\frac{\partial u_0}{\partial x_n}(x',+0,\eta)+\frac{\partial u_0}{\partial x_n}(x',-0,\eta) \biggr)\bigl(D_0^{-,+}(\eta)-D_0^{-,-}(\eta)\bigr) \\ \nonumber &\qquad + \frac{|\partial\omega|}{|\square|}a(x',0,u_0(x',0,\eta)) \bigl(D_0^{+,+}(\eta)-D_0^{+,-}(\eta)\bigr) \\ &\qquad- \sum_{j=1}^{n-1}\frac{\partial u_0}{\partial x_j}(x',0,\eta)\bigl(D_{1j,0}^+(\eta)-D_{1j,0}^-(\eta)\bigr). \end{aligned}
\end{equation}
\tag{8.10}
$$
Note that by Lemma 7.7 the function $\partial u_0/\partial x_n(\,\cdot\,,\pm 0,\eta)$ is infinitely differentiable with respect to $\eta\in[0,1]$ in the norms of the spaces $ W_2^p(S)$, $p\in\mathbb{N}$. To find the jump of the derivative $[{\partial u_1}/{\partial x_n}]_0$ we must analyse the solvability of the problem for $v_2$. Once we have found this jump, we can on the one hand construct the function $u_1$ by solving the relevant boundary problem for it, and on the other hand we can solve the problem for $v_2$. Repeating this procedure recursively, we can find all functions in the outer and inner expansions. This algorithm and its results are stated as the following lemma. Lemma 8.1. Problems (2.6) and (3.5) are solvable. Their solutions have representations (2.9). For $|\xi_n|>R_0$ the functions $v_{mj}$ can be represented as follows: where the $K_{mj}^\pm$ are polynomials of degree at most $m$ such that $K_{mj}^\pm(0)=0$, the $Q_{mjk}^\pm$ are polynomials in $\xi_n$ of degree at most $m-1$ with coefficients depending on $\eta$, and the $D_{mj}^\pm$ are some functions. The functions $v_m^{(0)}$ and $\varphi_{mj}$ belong to all spaces $W_2^p(S)$, $p\in\mathbb{N}$, are infinitely differentiable with respect to $\eta\in(0,1]$ and are uniformly bounded in $\eta\in[0,1]$ in the norms of these spaces. The functions $v_{mj}$ are infinitely differentiable in $\overline{\Pi}\setminus\omega^\eta$ for each $\eta\in(0,1]$ and infinitely differentiable with respect to $\eta\in(0,1]$ in the sense of (7.4). The estimates
$$
\begin{equation}
\mathfrak{p}(v_{mj})+\|v_{mj}\|_{C^1(\overline{\Pi^\eta})}\leqslant C
\end{equation}
\tag{8.12}
$$
hold, where the constants $C$ are independent of $\eta\in[0,1]$ and $j$, but depend on $m$. On the plane $S$ the functions $u_m$ satisfy the boundary conditions
$$
\begin{equation}
[u_m]_0 =\sum_{j=1}^{N_m}\varphi_{mj}(x',\eta) (D_{mj}^+(\eta)-D_{mj}^-(\eta)),
\end{equation}
\tag{8.13}
$$
$$
\begin{equation}
\nonumber \biggl[\frac{\partial u_m}{\partial x_n}\biggr]_0 =-\frac{1}{|\square|}\int_{\partial\omega^\eta} \psi_{m+1}\,ds -\frac{1}{|\square|}\lim_{R\to+\infty}\biggl( \int_{\Pi_R\setminus \omega^\eta} f_{m+1}\,d\xi
\end{equation}
\notag
$$
$$
\begin{equation}
\qquad+\int_{\square\times\{R\}}\frac{\partial P_{m+1}^+}{\partial \xi_n}\,d\xi' -\int_{\square\times\{-R\}}\frac{\partial P_{m+1}^-}{\partial \xi_n}\,d\xi'\biggr).
\end{equation}
\tag{8.14}
$$
Problems (2.10) with these boundary conditions are uniquely solvable. The functions $u_m$ belong to the spaces $W_2^1(\Omega)$ and $W_2^p(\Omega_{\tau_0}^+)\oplus W_2^p(\Omega_{\tau_0}^-)$ for all $p\in\mathbb{N}$ and $\tau_0<\tau$, are infinitely differentiable with respect to $\eta\in(0,1]$ and are uniformly bounded in ${\eta\in[0,1]}$ in the norms of these spaces. Proof. The proof goes by induction. The base case $m=1$ was considered above: we constructed the functions $u_0$ and $v_1$, and for $u_1$ we obtained boundary condition (8.10).
Assume that we have constructed solutions of (2.6) and (3.5) up to some $m$, solutions of (2.10) up to $m-1$ and have found the boundary condition (8.13) for $[u_m]_0$. The function $v_m$ has the form (2.9), involving the function $v_m^{(0)}$, which we do not know yet. To find $v_m^{(0)}$ it suffices to specify $u_m$. Then, using (3.5) we can find $v_m^{(0)}$ by the formula
$$
\begin{equation}
v_m^{(0)}(x',\eta)=u_m(x',+0,\eta)-\sum_{j=1}^{N_m} \varphi_{mj}(x',\eta)D_{mj}^+(\eta).
\end{equation}
\tag{8.15}
$$
At this point, in the problem for $u_m$ we know the right-hand side in terms of the functions $u_j$ for $ j\leqslant m-1$, the boundary condition on $\partial\Omega$ and the jump of the function itself on $S$: see (8.13). In view of Lemma 7.2 it is sufficient to determine the jump of the normal derivative of $u_m$ on $S$, and then we can uniquely solve the resulting problem for $u_m$. We find this jump by analysing the solvability of the problem for $v_{m+1}$. The functions $f$ and $u_j$, $j\leqslant m-1$, are elements of the spaces $W_2^1(\Omega)$ and $W_2^p(\Omega_{\tau_0}^+)\oplus W_2^p(\Omega_{\tau_0}^-)$ for all $p\in\mathbb{N}$ and $0<\tau_0<\tau_1$. Hence we conclude from the standard theorems on the embeddings of Sobolev spaces in spaces of continuously differentiable functions that the functions $u_j$, $j\leqslant m-1$, are infinitely differentiable in $\overline{\Omega_{\tau_0}^+}$ and $\overline{\Omega_{\tau_0}^-}$. Therefore, (3.4) holds, as well as a similar formula for $f$. We plug (3.4) and (3.3) into the equations in (2.4) and (2.10) by taking account of our assumptions on the coefficients $A_{ij}$ and $A_j$ for small $x_n$. Then we obtain the equalities
$$
\begin{equation}
\begin{gathered} \, -\frac{\partial u_0}{\partial x_n}(x',\pm0)=f(x',0), \\ -(\Delta_{x'}+\lambda)\, \frac{\partial^j u_0}{\partial x_n^j}(x',\pm0)-\frac{\partial^{j+2} u_0}{\partial x_n^{j+2}}(x',\pm 0)=\frac{\partial^j f}{\partial x_n^j}(x',0), \qquad j\geqslant 1, \\ -(\Delta_{x'}+\lambda)\, \frac{\partial^j u_p}{\partial x_n^j}(x',\pm0)-\frac{\partial^{j+2} u_p}{\partial x_n^{j+2}}(x',\pm 0)=0, \qquad j\geqslant 0. \end{gathered}
\end{equation}
\tag{8.16}
$$
We seek the solution of problem (2.6), (3.5) for $v_{m+1}$ in the form
$$
\begin{equation*}
v_{m+1}=\widetilde{v}_{m+1}+\biggl(\frac{\partial u_m}{\partial x_n}(x',+0,\eta)\, \xi_n+\frac{\partial u_m}{\partial x_n}(x',-0,\eta)\, \xi_n+P_{m+1}^++P_{m+1}^-\biggr)\chi_1,
\end{equation*}
\notag
$$
where $\chi_1$ is the cut-off function introduced before (4.5). Then we obtain the following problem for $\widetilde{v}_{m+1}$:
$$
\begin{equation}
\begin{gathered} \, -\Delta_\xi \widetilde{v}_{m+1}=\widetilde{F}_{m+1} \quad \text{in } \Pi\setminus\overline{\omega^\eta}, \qquad \frac{\partial \widetilde{v}_{m+1}}{\partial \nu_\xi}=\psi_{m+1} \quad \text{on } \partial\omega^\eta, \\ \widetilde{F}_{m+1}:=f_{m+1}+\Delta_\xi \biggl(\frac{\partial u_m}{\partial x_n}(x',+0,\eta)\, \xi_n+\frac{\partial u_m}{\partial x_n}(x',-0,\eta)\, \xi_n+P_{m+1}^++P_{m+1}^-\biggr)\chi_1, \nonumber \end{gathered}
\end{equation}
\tag{8.17}
$$
with the periodic boundary conditions (4.2).
We analyse the behaviour of $\widetilde{F}_{m+1}$ as $\xi_n\to\pm\infty$. To do this, in place of $v_{m-1}$ and $v_m$ we plug their asymptotic expressions as $\xi_n\to\pm\infty$ into $\widetilde{F}_{m+1}$. Taking (8.16) into account we obtain
$$
\begin{equation*}
\begin{aligned} \, \widetilde{F}_{m+1} &=\frac{\xi_n^{m-1}}{(m-1)!}\biggl(\frac{\partial^{m-1}f}{\partial x_n^{m-1}}(x',0)+(\Delta_{x'}+\lambda)\, \frac{\partial^{m-1}u_0} {\partial x_n^{m-1}}(x',\pm0,\eta) \\ &\qquad+\frac{\partial^{m+1}u_0}{\partial x_n^{m+1}}(x',\pm0,\eta)\biggr) +(\Delta_{x'}+\lambda)\sum_{j=0}^{m-2} \frac{\xi_n^j}{j!}\, \frac{\partial^j u_{m-1-j}}{\partial x_n^j}(x',\pm0,\eta) \\ &\qquad+\sum_{j=2}^{m} \frac{\xi_n^{j-2}}{(j-2)!}\, \frac{\partial^{j-2} u_{m+1-j}}{\partial x_n^{j-2}}(x',\pm0,\eta) +o(1)=o(1), \qquad \xi_n\to\pm\infty, \end{aligned}
\end{equation*}
\notag
$$
where $o(1)$ denotes the terms decreasing exponentially as $\xi_n\to\pm\infty$. By Lemma 6.1 problem (8.17) is solvable if
$$
\begin{equation}
\begin{aligned} \, \notag & \int_{\partial\omega^\eta} \psi_{m+1}\,ds+\int_{\Pi\setminus \omega^\eta} \Delta_\xi\biggl( \frac{\partial u_m}{\partial x_n}(x',+0,\eta)\, \xi_n+\frac{\partial u_m}{\partial x_n}(x',-0,\eta)\, \xi_n\biggr)\chi_1\,d\xi \\ &\qquad\qquad+\int_{\Pi\setminus \omega^\eta} \bigl( f_{m+1}+\Delta_\xi( P_{m+1}^+ + P_{m+1}^-)\chi_1 \bigr)\,d\xi=0. \end{aligned}
\end{equation}
\tag{8.18}
$$
We integrate by parts:
$$
\begin{equation*}
\begin{aligned} \, &\int_{\Pi\setminus \omega^\eta} \Delta_\xi\biggl( \frac{\partial u_m}{\partial x_n}(x',+0,\eta)\, \xi_n+\frac{\partial u_m}{\partial x_n}(x',-0,\eta)\, \xi_n\biggr)\chi_1\,d\xi \\ &\qquad=\lim_{R\to+\infty} \int_{\Pi_R\setminus \omega^\eta } \Delta_\xi\biggl( \frac{\partial u_m}{\partial x_n}(x',+0,\eta)\, \xi_n+\frac{\partial u_m}{\partial x_n}(x',-0,\eta)\, \xi_n\biggr)\chi_1\,d\xi \\ &\qquad=|\square|\biggl(\frac{\partial u_m}{\partial x_n}(x',+0,\eta)-\frac{\partial u_m}{\partial x_n}(x',-0,\eta)\biggr). \end{aligned}
\end{equation*}
\notag
$$
In a similar way we represent the third integral in (8.18) as a limit and integrate by parts. Then this solvability condition takes the following form:
$$
\begin{equation*}
\begin{aligned} \, &\int_{\partial\omega^\eta} \psi_{m+1}\,ds+|\square|\biggl(\frac{\partial u_m}{\partial x_n}(x',+0,\eta)-\frac{\partial u_m}{\partial x_n}(x',-0,\eta)\biggr) \\ &\qquad+\lim_{R\to+\infty}\biggl( \int_{\Pi_R\setminus \omega^\eta }f_{m+1}\,d\xi +\int_{\square\times\{R\}}\frac{\partial P_{m+1}^+}{\partial \xi_n}\,d\xi'-\int_{\square\times\{-R\}}\frac{\partial P_{m+1}^-}{\partial \xi_n}\,d\xi'\biggr)=0. \end{aligned}
\end{equation*}
\notag
$$
Note that, as all integrals in the original equality (8.18) are convergent, a finite limit exists in the resulting equality. Hence we can write down boundary condition (8.14) for $u_m$. From the inductive assumption on the smoothness of the $u_j$ for $j\leqslant m$, and the smoothness of the $v_j$ for $j\leqslant m$ in the space variables and by formula (2.9) for the $v_j$, $j\leqslant m$, it is straightforward that the right-hand side of (8.14), as a function of $x'$, belongs to all spaces $W_2^p(S)$, $p\in\mathbb{N}$. It follows immediately from the inductive assumption on the boundedness of the $v_j$, $j\leqslant m$, for ${\eta\in[0,1]}$ that the right-hand side of (8.14) is bounded uniformly in $\eta\in[0,1]$ in the norms of $W_2^p(S)$ for all ${p\in\mathbb{N}}$. Bearing in mind the inductive assumption that the functions $v_j$, $j\leqslant m$, are infinitely differentiable with respect to $\eta\in(0,1]$ and the corresponding precise expression of this property (7.4), for each $\eta_0\in(0,1]$ we make the change of variables $\xi\mapsto \Xi^{-1}(\eta_0\eta^{-1},\xi)$ in the integral over $\partial\omega^\eta$ on the right-hand side of (8.14). Then we conclude directly from the inductive assumption that the functions $u_j$ and $v_{mj}$, $j\leqslant m$, are infinitely differentiable with respect to $\eta\in(0,1]$ and from representations (2.9) that the integral over $\partial\omega^\eta$ on the right-hand side of (8.14) is infinitely differentiable with respect to $\eta\in(0,1]$ in the norms of the spaces $W_2^p(S)$, $p\in\mathbb{N}$. Now we write out an analogue of the representation (2.9) for the functions $f_{m+1}$ and use (8.16). Then we see that for $\pm \xi_n>R_0$ the function $f_{m+1}$ has the form
$$
\begin{equation*}
\begin{gathered} \, f_{m+1}(\xi,\eta)=-\frac{\partial^2\ }{\partial\xi_n^2} \bigl(P_{m+1}^+(\xi_n,\eta) + P_{m+1}^-(\xi_n,\eta)\bigr) + \widetilde{f}_{m+1}, \\ \widetilde{f}_{m+1}:=\sum_{k\in\mathbb{Z}^{n-1}} F_{(m+1)jk}^\pm(\xi_n,\eta)\exp(-2\pi |k_b|\,|\xi_n|) \exp(2\pi\mathrm{i} k_b\cdot\xi'), \end{gathered}
\end{equation*}
\notag
$$
where the $F_{mjk}^\pm$ are polynomials in $\xi_n$ of degree at most $m-1$ whose coefficients are infinitely differentiable with respect to $\eta\in(0,1]$, and where the quantity $\mathfrak{p}(f_{m+1})$ is bounded uniformly in $\eta\in[0,1]$. This representation enables us to write the limit in (8.14) as follows:
$$
\begin{equation*}
\begin{aligned} \, &\lim_{R\to+\infty} \biggl( \int_{\Pi_R\setminus \omega^\eta }f_{m+1}\,d\xi +\int_{\square\times\{R\}}\frac{\partial P_{m+1}^+}{\partial \xi_n}\,d\xi'-\int_{\square\times\{-R\}}\frac{\partial P_{m+1}^-}{\partial \xi_n}\,d\xi'\biggr) \\ &\qquad= \int_{\Pi_{2R_0}\setminus\omega^\eta} f_{m+1} \,d\xi + \int_{\Pi\setminus \Pi_{2R_0}} \widetilde{f}_{m+1}\,d\xi \\ &\qquad\qquad-\int_{\square\times\{2R_0\}}\frac{\partial P_{m+1}^+}{\partial \xi_n}\,d\xi' + \int_{\square\times\{-2R_0\}}\frac{\partial P_{m+1}^-}{\partial \xi_n}\,d\xi'. \end{aligned}
\end{equation*}
\notag
$$
Now it follows from the inductive assumption that the functions $v_{mj}$ and $u_j$, $j\leqslant m$, are infinitely differentiable with respect to $\eta\in(0,1]$ and from the formula obtained that the limit in (8.14) is infinitely differentiable with respect to $\eta\in(0,1]$ in the norms of the spaces $W_2^p(S)$, $p\in\mathbb{N}$.
In view of the above the boundary condition (8.14) which we have obtained completes the problem for $u_m$, and this problem is solvable by Lemma 7.2. The function $v_m^{(0)}$ is determined by (8.15). By the inductive assumption on the properties of the $u_j$ for $j\leqslant m$ and Lemma 7.3, the function $u_{m+1}$ is an element of the spaces $W_2^1(\Omega)$ and $W_2^p(\Omega_{\tau_0}^+)\oplus W_2^p(\Omega_{\tau_0}^-)$ for all $\tau_0<\tau$, and it is uniformly bounded for $\eta\in[0,1]$ and uniformly differentiable with respect to $\eta\in(0,1)$ in the norms of these spaces. By Lemma 6.1 condition (8.14) ensures the solvability of problem (8.17). Since the variables $(x',\eta)$ and $(\xi,\eta)$ are separated in the solution of problems (2.6), (3.5) for the functions $v_{m-1}$ and $v_m$, this separation of variables also occurs in $f_{m+1}$, $\psi_{m+1}$ and $\widetilde{F}_{m+1}$. Hence we also have such separation of variables in the solution of problem (2.6), (3.5) for $\widetilde{v}_{m+1}$:
$$
\begin{equation*}
\widetilde{v}_{m+1}(\xi,x',\eta)= \sum_{j=1}^{\widetilde{N}_m} \varphi_{m+1j}(x',\eta)\widetilde{v}_{m+1j}(\xi,\eta),
\end{equation*}
\notag
$$
where the $\widetilde{N}_m$ are some integers and the $ \varphi_{m+1j}$ are functions belonging to all spaces $W_2^p(S)$, $p\in\mathbb{N}$, bounded uniformly in $\eta\in[0,1]$ and infinitely differentiable with respect to $\eta\in(0,1]$ in the norms of these spaces. By the inductive assumption about the functions $v_{ij}$, $i\leqslant m$, and Lemma 7.1 the functions $v_{mj}$ are infinitely differentiable with respect to $\xi$ in $\overline{\Pi}\setminus\omega^\eta$ for each $\eta\in(0,1]$ and infinitely differentiable with respect to $\eta\in(0,1]$ in the sense of (7.4), and we have estimates (8.12). Now returning to $v_{m+1}$ we conclude that problem (2.6), (3.5) for it is also solvable. For $v_{m+1}$ we have representations (2.9) and (8.11) and estimates (8.12), where the functions involved in these relations have all properties required in the statement of the lemma. Now, writing an analogue of representation (8.11) for $v_m$ and comparing it with the asymptotic expressions (3.5) we obtain (8.13) with $m$ replaced by $m+1$, which completes the prof of the step of induction. Lemma 8.1 is proved. Note that it follows directly from this lemma that the functions $u_m$ and $v_m$ have all properties mentioned in the statement of Theorem 2.1. § 9. Justifying the asymptoticsSet
$$
\begin{equation*}
\begin{gathered} \, u_{\varepsilon,N}(x,\eta):= \chi^\varepsilon(x_n)u_{\varepsilon,N}^{\mathrm{ex}}(x,\eta) +(1-\chi^\varepsilon(x_n))u_{\varepsilon,N}^{\mathrm{in}}(x\varepsilon^{-1},x',\eta), \\ u_{\varepsilon,N}^{\mathrm{ex}}(x,\eta):=u_0(x)+\sum_{m=1}^N \varepsilon^m u_m(x,\eta) \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
u_{\varepsilon,N}^{\mathrm{in}}(\xi,x',\eta):=v_0(x',\xi,\eta) +\sum_{m=1}^N\varepsilon^m v_m(x',\xi,\eta),
\end{equation*}
\notag
$$
where $N\geqslant 3$ is a positive integer. Lemma 9.1. The function $u_{\varepsilon,N}$ is a solution of the problem Proof. The homogeneous Dirichlet boundary condition for the function $u_{\varepsilon,N}$ on $\partial\Omega$ follows directly from the same condition in problems (2.4) and (2.10) for the coefficients of the outer expansion.
Set
$$
\begin{equation*}
\phi_{\varepsilon,N}(x):= \frac{\partial u_{\varepsilon,N}^{\mathrm{in}}}{\partial \mathrm{n}}(x)+a(x,u_{\varepsilon,N}^{\mathrm{in}}(x))\quad\text{on } \partial\theta^\varepsilon.
\end{equation*}
\notag
$$
Note that by Lemma 8.1 the functions $v_j(x\varepsilon^{-1},x',\eta)$ satisfy estimates on $\partial\theta^\varepsilon$ which hold uniformly in $\varepsilon$, $\eta$ and $x$: where the constants $C$ are independent of $x$, $\varepsilon$ and $\eta$, and the $V_j(x')$ are functions belonging to all spaces $W_2^p(S)$, $p\in\mathbb{N}$. This, the fact that $a$ is smooth and (2.2) ensure Taylor’s formula for $x\in\partial\theta^\varepsilon$:
$$
\begin{equation*}
\begin{aligned} \, a(x,u^{\mathrm{in}}_\varepsilon(x)) &=a(x',\varepsilon\xi_n,u^{\mathrm{in}}_\varepsilon) \\ &= T_0(x',v_0(x',\xi,\eta))+\sum_{m=1}^{N-1} \varepsilon^m T_m\bigl(x',v_1(x',\xi,\eta),\dots,v_m(x',\xi,\eta)\bigr) \\ &\qquad+ \varepsilon^{N} T_{N,\varepsilon}\bigl(x',\xi_n,v_1(x',\xi,\eta),\dots,v_N(x',\xi,\eta)\bigr), \end{aligned}
\end{equation*}
\notag
$$
where $T_{N,\varepsilon}$ is a function satisfying the uniform estimate
$$
\begin{equation*}
\bigl| T_{N,\varepsilon}(x',\xi_n,v_1(x',\xi,\eta), \dots,v_N(x',\xi,\eta)) \bigr|\leqslant C \widetilde{T}_{N}(x'),
\end{equation*}
\notag
$$
where $C$ is independent of $\varepsilon$, $\eta$, $x$ and $\xi$, and $\widetilde{T}_N$ is a function belonging to all spaces $W_2^p(S)$, $p\in\mathbb{N}$. Now taking account of the boundary condition on $\partial\theta_\eta$ (see (2.6)) and the properties of the $v_j$ established in Lemma 8.1 we immediately obtain the estimate for $\phi_{\varepsilon,N}$ in (9.1).
Set $f_{\varepsilon,N}:= (\mathcal{L} -\lambda)u_{\varepsilon,N}-f$. Taking the equations in (2.4), (2.10) and (2.6) into account we verify directly that
$$
\begin{equation*}
\begin{gathered} \, f_{\varepsilon,N}=f_{\varepsilon,N}^{(1)}+f_{\varepsilon,N}^{(2)}+f_{\varepsilon,N}^{(3)}, \\ f_{\varepsilon,N}^{(1)}:=(\chi^\varepsilon(x_n)-1)\biggl(f(x)-\sum_{j=1}^{N-2}\frac{x_n^j}{j!}\, \frac{\partial^j f}{\partial x_n^j}(x',0)\biggr), \\ f_{\varepsilon,N}^{(2)}=\varepsilon^{N-1}(\chi^\varepsilon(x_n)-1)\biggl(\lambda (v_{N-1}+\varepsilon v_N)+2\sum_{j=1}^{n-1}\frac{\partial^2 v_N}{\partial\xi_j\, \partial x_j} \biggr) \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
f_{\varepsilon,N}^{(3)}:=-2(\chi^\varepsilon)'\frac{\partial}{\partial x_n} (u_{\varepsilon,N}^{\mathrm{ex}}-u_{\varepsilon,N}^{\mathrm{in}})-(u_{\varepsilon,N}^{\mathrm{ex}}-u_{\varepsilon,N}^{\mathrm{in}})(\chi^\varepsilon)''.
\end{equation*}
\notag
$$
As $f$ is smooth, we can estimate $f_{\varepsilon,N}^{(1)}$ using Taylor’s formula with remainder in the Lagrange form:
$$
\begin{equation}
\|f_{\varepsilon,N}^{(1)}\|_{L_2(\Omega^\varepsilon)}\leqslant C\varepsilon^{N/2-1/4},
\end{equation}
\tag{9.2}
$$
where $C$ is independent of $\varepsilon$.
The function $f_{\varepsilon,N}^{(2)}$ is nonzero only for $|x_n|\leqslant 2\varepsilon^{1/2}$, which in terms of the variables $\xi$ corresponds to the layer $\{\xi\colon |\xi_n|<2\varepsilon^{-1/2}\}$. Lemma 8.1 provides the necessary estimates for the functions $v_{mj}$, $\varphi_{mj}$, $v_m^{(0)}$, $K_{mj}^\pm$, $D_{mj}^\pm$ and $Q_{mjk}^\pm$, $m=N-1,N$, and their derivatives, so that we can estimate $f_{\varepsilon,N}^{(2)}$:
$$
\begin{equation}
\|f_{\varepsilon,N}^{(2)}\|_{L_2(\Omega^\varepsilon)}\leqslant C\varepsilon^{N/2-1/4}.
\end{equation}
\tag{9.3}
$$
By the definition of $\chi^\varepsilon$ the function $f_{\varepsilon,N}^{(3)}$ is nonzero only for $\varepsilon^{1/2}<|x_n|<2\varepsilon^{1/2}$. Hence to estimate $f_{\varepsilon,N}^{(3)}$ we must take matching conditions (3.5) into account, which ensure the required smallness of the difference $u_{\varepsilon,N}^{\mathrm{ex}}-u_{\varepsilon,N}^{\mathrm{in}}$, and also of the smoothness of the functions $v_j$ and $u_j$ and estimates for them established in Lemma 8.1. As a result, we have
$$
\begin{equation*}
\|f_{\varepsilon,N}^{(3)}\|_{L_2(\Omega^\varepsilon)}\leqslant C \varepsilon^{N/2-1/2}.
\end{equation*}
\notag
$$
Hence from (9.2) and (9.3) we obtain the first estimate in (9.1).
We establish (2.11) similarly to the above estimates for the functions $f_{\varepsilon,N}^{(i)}$. The proof is complete. Set $\widehat{u}_{\varepsilon,N}:=u_{\varepsilon,N}-u_\varepsilon$. This is a solution of the problem
$$
\begin{equation*}
\begin{gathered} \, (\mathcal{L}-\lambda)\widehat{u}_\varepsilon=f_{\varepsilon,N} \quad\text{in } \Omega^\varepsilon, \qquad \widehat{u}_{\varepsilon,N}=0 \quad\text{on } \partial\Omega, \\ \frac{\partial\widehat{u}_{\varepsilon,N}}{\partial N}+a(\,\cdot\,,u_{\varepsilon,N})-a(\,\cdot\,,u_\varepsilon)=\phi_{\varepsilon,N}\quad\text{on } \partial\theta^\varepsilon. \end{gathered}
\end{equation*}
\notag
$$
Its solution satisfies the integral identity
$$
\begin{equation}
\begin{aligned} \, \notag &\mathfrak{h}_0(\widehat{u}_{\varepsilon,N},\widehat{u}_{\varepsilon,N}) + (a(u_{\varepsilon,N}) -a(u_\varepsilon),\widehat{u}_{\varepsilon,N})_{L_2(\partial \theta^\varepsilon)} \\ &\qquad=(f_{\varepsilon,N},\widehat{u}_{\varepsilon,N})_{L_2(\Omega^\varepsilon)} +(\phi_{\varepsilon,N},\widehat{u}_{\varepsilon,N})_{L_2(\partial\theta^\varepsilon)}. \end{aligned}
\end{equation}
\tag{9.4}
$$
By [29], Lemma 3.4, for each function $u\in W_2^1(\Omega^\varepsilon)$ with zero trace on $\partial\Omega$ we have
$$
\begin{equation}
\|u\|_{L_2(\partial\theta^\varepsilon)}^2\leqslant (c\varepsilon+\delta)\|\nabla u\|_{L_2(\Omega^\varepsilon)}^2 + C(\delta) \|u\|_{L_2(\Omega^\varepsilon)}^2,
\end{equation}
\tag{9.5}
$$
where $\delta $ is an arbitrary fixed positive number, $c$ is a constant independent of $u$, $\varepsilon$ and $\delta$, and the constant $C(\delta)$ is independent of $\varepsilon$ and $u$. Note also an obvious inequality, which holds by (2.1):
$$
\begin{equation*}
|\mathfrak{h}_0(u,u)|\geqslant \frac{3c_0}{4} \|\nabla u\|_{L_2(\Omega^\varepsilon)}^2 - C\|u\|_{L_2(\Omega^\varepsilon)}^2,
\end{equation*}
\notag
$$
where the constant $C$ is independent of $\varepsilon$ and $u$. Using this inequality, (2.1) and (9.5) we can estimate the left-hand side of (9.4) for $\lambda<\lambda_0$, provided that $\lambda_0$ is negative and has a sufficiently large modulus:
$$
\begin{equation*}
\begin{aligned} \, &\bigl|\mathfrak{h}_0(\widehat{u}_{\varepsilon,N},\widehat{u}_{\varepsilon,N}) + (a(u_{\varepsilon,N}) -a(u_\varepsilon),\widehat{u}_{\varepsilon,N})_{L_2(\partial \theta^\varepsilon)}-\lambda\|\widehat{u}_{\varepsilon,N}\|_{L_2(\Omega^\varepsilon)}^2\bigr| \\ &\qquad\geqslant \bigl|\mathfrak{h}_0(\widehat{u}_{\varepsilon,N},\widehat{u}_{\varepsilon,N}) -\lambda\|\widehat{u}_{\varepsilon,N}\|_{L_2(\Omega^\varepsilon)}^2\bigr|-2|a_1| \|\widehat{u}_{\varepsilon,N}\|_{L_2(\theta^\varepsilon)}^2 \geqslant \frac{c_0}{2} \|\widehat{u}_{\varepsilon,N}\|_{W_2^1(\Omega^\varepsilon)}^2, \end{aligned}
\end{equation*}
\notag
$$
and we can estimate the right-hand side of (9.4):
$$
\begin{equation*}
\begin{aligned} \, &\bigl| (f,\widehat{u}_{\varepsilon,N})_{L_2(\Omega^\varepsilon)} +(\phi_{\varepsilon,N},\widehat{u}_{\varepsilon,N})_{L_2(\partial\theta^\varepsilon)} \bigr| \\ &\qquad\leqslant C\bigl(\|f_{\varepsilon,N}\|_{L_2(\Omega^\varepsilon)} + \|\phi_{\varepsilon,N}\|_{L_2(\partial\theta^\varepsilon)}\bigr) \|\widehat{u}_{\varepsilon,N}\|_{W_2^1(\Omega^\varepsilon)}, \end{aligned}
\end{equation*}
\notag
$$
where the constant $C$ is independent of $\varepsilon$. Now it follows from the above estimates and (9.4) that
$$
\begin{equation*}
\|\widehat{u}_{\varepsilon,N}\|_{W_2^1(\Omega^\varepsilon)} = \|u_{\varepsilon,N}-u_\varepsilon\|_{W_2^1(\Omega^\varepsilon)}\leqslant C\varepsilon^{N/2-1/4}.
\end{equation*}
\notag
$$
In this estimate we replace $N$ by $N+2$ and take estimates (2.11) for $m=N+1$ and $ N+2$ into account. Then we obtain (2.8). Theorem 2.1 is proved.
Citation in format AMSBIB
\Bibitem{BorMuk22} |
|
|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|
Contact us: |
Terms of Use
|
Registration to the website |
Logotypes |
|