Sbornik: Mathematics, 2022, Volume 213, Issue 10, Pages 1318–1371 DOI: https://doi.org/10.4213/sm9739e (Mi sm9739)

DOI: https://doi.org/10.4213/sm9739e

Funding agency	Grant number
Russian Science Foundation	20-11-19995
This research was supported by the Russian Science Foundation under grant no. 20-11-19995, https://rscf.ru/en/project/20-11-19995/.

Russian version:
Matematicheskii Sbornik, 2022, Volume 213, Number 10, Pages 3–59
DOI: https://doi.org/10.4213/sm9739

Bibliographic databases:

§ 1. Introduction

Elliptic 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 results

Let $x=(x',x_n)$ 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

Let $\varepsilon$ be a small positive parameter and $\eta=\eta(\varepsilon)$ be a function satisfying ${0<\eta(\varepsilon)\leqslant1}$. Set

Consider the sets

Zoom inZoom out

Figure 1.Schematic image of a part of the domain $\Omega^\varepsilon$.

In $\Omega$ we consider functions $A_{ij}=A_{ij}(x)$, $A_i=A_i(x)$ and $A_0=A_0(x)$ satisfying the conditions

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:

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

According to [30], the solution of (2.3) converges to a generalized solution of the homogenized problem

Here we construct an asymptotic representation for the solution of (2.3) under certain additional conditions. Namely, let

Consider the system of boundary value problems

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

The following theorem is the main result of this paper.

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 expansion

In 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:

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 $x_n=\varepsilon\xi_n$:

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 $x_n=\varepsilon\xi_n$:

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 expansion

In 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 problem

Consider the model boundary value problem

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 equation

In 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,

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

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:

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:

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].

The proof of the next lemma is a literal reproduction of the proof of Lemma 7.3 in [29].

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

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

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:

By the definition (4.10) of $\Xi$ the function $\Upsilon$ has the representation

Also note the obvious formula

The following lemma describes how $\mathcal{B}_4$ depends on $\eta$.

Since (4.8) has only a constant solution, equation (4.6) is solvable if

The solution of operator equation (4.6) has the following form:

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:

4.3. The solvability of the model problem

In this subsection we investigate the solvability of model problem (4.1), (4.2).

§ 5. Estimates for the maximum of the solution of the model problem and for its derivatives

Our 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 solution

In this subsection we estimate the maximum modulus of the solution of (4.1), (4.2).

5.2. An estimate for the maximum of the derivatives of the solution

In 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

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.

Since the solution of (5.3), (4.2) belongs to $C^{2+\vartheta}(\overline{\Pi})$, it has the representation

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

Lemmas 5.1, 5.3 and 5.4 yield the following result.

§ 6. Solvability of the model problems

Here 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:

§ 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 expansion

In 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

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.

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

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

Consider the function 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:

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

For an arbitrary function $f\in\mathfrak{C}^k$ consider the boundary value problem

Since $\Upsilon F_1$ is an element of $\mathfrak{C}^k$ and (7.9) holds, it follows from (7.8) that

7.2. The smoothness of the solution with respect to the parameter of the problem in the case of coefficients of the outer expansion

In 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 $\Omega_r^\pm:=\{x\colon 0<\pm x_n<r\}$.

The next result is a direct consequence of this lemma.

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:

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,

Set

Using this lemma in combination with standard theorems on improved smoothness for linear problems we can establish the following key result.

§ 8. Properties of the coefficients of the inner and outer expansions

In 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

To find a solution of problem (2.6), (3.5) for $m=1$, we consider the auxiliary problems

Now consider the boundary value problems

Now the solution of (2.6), (3.5) for $m=1$ can be found explicitly:

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.

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 asymptotics

Set

Set $\widehat{u}_{\varepsilon,N}:=u_{\varepsilon,N}-u_\varepsilon$. This is a solution of the problem

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Citation in format AMSBIB

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