Resolvent Convergence for Differential–Difference Operators with Small Variable Translations

Abstract

We consider general higher-order matrix elliptic differential–difference operators in arbitrary domains with small variable translations in lower-order terms. The operators are introduced by means of general higher-order quadratic forms on arbitrary domains. Each lower-order term depends on its own translation and all translations are governed by a small multi-dimensional parameter. The operators are considered either on the entire space or an arbitrary multi-dimensional domain with a regular boundary. The boundary conditions are also arbitrary and general and involve small variable translations. Our main results state that the considered operators converge in the norm resolvent sense to ones with zero translations in the best possible operator norm. Estimates for the convergence rates are established as well. We also prove the convergence of the spectra and pseudospectra.

1. Introduction

Differential–difference equations are those, in which a differentiation operator is composed with a translation operator. Due to the presence of the translation operator(s), the solutions to such equations and their corresponding various initial and boundary value problems exhibit interesting properties that do not appear in classical differential equations and problems. Nowadays, there are quite a lot of works, in which differential–difference equations are studied. Most of these works are on qualitative properties, such as the well-posedness of boundary and initial problems for differential–difference equations, solvability issues, and the behavior of solutions, in particular, the structure of singularities generated by the translation operator(s).

Many interesting qualitative results regarding elliptic differential–difference equations were obtained by A.L. Skubachevskii and scientists around him. Here, we first of all mention the book [1] and some recent papers [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17] (see also the references therein). A nice survey of previous results, including ones by other authors, as well as a survey of results on ordinary differential equations with delays are given in book [1] and paper [2]. The majority of the obtained results were devoted to the equations with the Dirichlet condition on the boundary. Some results regarding Neumann and Robin problems for differential–difference equations were obtained recently in [9,10,11,18,19]. Evolutionary differential–difference equations were studied quite intensively; see, for instance, papers [20,21,22,23,24,25,26,27], survey [28], book [29], and the references therein.

There is also a series of papers on nonlinear differential–difference equations, both elliptic and evolutionary; see [30,31,32,33,34,35,36] and the references therein.

Apart from translations, similar nonlocalities in differential equations can be produced by the operators of dilatation and contractions. The properties of solutions to various problems for such equations were studied in works [37,38,39,40,41,42]; see also book [43] and the many references therein.

The examples of applications of differential–difference equations to real-world applications were briefly mentioned in ([1], Chs. III, V).

One of the interesting and natural directions of studying differential–difference equations is the development of the perturbation theory, and a specific perturbation for such equations is the case of a small translation. For ordinary differential equations, this issue was addressed in papers [44,45,46,47,48]. It was shown that the solutions to the considered equations are approximated by those for zero delay, and, in some cases, these solutions can be represented by an appropriate power series in a small parameter governing the delay. We also mention the results obtained in ([43], Ch. 5, Sections 5.3 and 5.4) for elliptic operators with contractions and dilatations in domains with small perturbations of the boundaries. Under appropriate restrictions for the perturbation, it was proved that the eigenvalues of the considered perturbed operators converge to those of the operators in the limiting domains, and the estimates for the convergence rates were obtained.

To the best of the authors’ knowledge, a detailed analysis of differential–difference operators with small translations has not been conducted before, and there are no results, even for particular examples. The first, most natural question is: what are the limiting operators as the translations tend to zero in an appropriate sense, and in what sense is the convergence of the operators to be described? Of course, one should compare the resolvents of the perturbed and limiting operators, and the best possible result is to establish the norm resolvent convergence. In other words, it is preferable to prove the convergence of the resolvents of the operators with small translations to ones of the limiting operators in an appropriate operator norm. Apart from the convergence, it is interesting to establish estimates for the convergence rates, which are usually referred to in short as operator estimates. We stress that during the last 20 years such operator estimates have been actively studied and proved for the operators in the homogenization theory. A lot of results were established for operators with fast and (locally) periodically oscillating coefficients; see, for instance [49,50,51,52], and the reference therein. These results stimulated similar studies for operators with arbitrary fast oscillations in lower-order terms [53], for operators in perforated domains [54,55,56,57], and for operators from the boundary homogenization [58,59]; see also many references therein.

In the present work, we consider four classes of general even-order elliptic differential–difference operators with small variable translations in the lower-order terms. Each such term involves its own small translation governed by a multi-dimensional small parameter. Our main results state that as the translations go to zero, the considered operators converge to limiting ones with no translations. Such limiting operators turn out to be classical differential operators with no nonlocalities. Both perturbed and limiting operators are defined via appropriate sesquilinear forms, the coefficients of which are only essentially bounded. This allows us to consider very general classes of perturbed operators. The convergence is established in the norm resolvent sense, and, as the norm for the resolvent, we choose one for the operators acting from $L_2$ into $W_2^m$, where $2m$ is the order of the considered operators. This is the best possible operator norm for the resolvents since, due to the absence of extra smoothness conditions for the coefficients of the considered forms, we can just state that the domains of the associated operators are some subsets in $W_2^m$. Apart from the norm resolvent convergence, we also establish estimates for the convergence rates; that is, we in fact prove operator estimates for the considered differential–difference operators with small variable translations. Apart from the operator estimates, we also prove the convergence of the spectra of the perturbed operators to that of the unperturbed operators. The considered operators are usually non-self-adjoint, and, as is known, the non-self-adjoint operator $ϵ$-pseudospectra and $(p,ϵ)$-pseudospectra are important spectral characteristics [60,61,62]. In our work, we also establish the convergence of such pseudospectra.

Therefore, since the considered operators are usually non-self-adjoint, we also establish the convergence of the $ϵ$-pseudospectra.

2. Problems and Main Results

In this work, we consider four types of general even-order elliptic operators with small variable translations in lower-order terms. It is convenient to define each type of operator in a separate subsection and to formulate the corresponding main results. In the first subsection, we introduce some general notation.

2.1. Notation

Let $x=(x_1,\dots ,x_d)$ be Cartesian coordinates in ${\mathbb{R}}^d$, $d⩾1$. By ${\mathbb{M}}_n$, $n⩾1$, we denote the space of all square matrices of a size $n\times n$ with complex entries; this space is equipped with one of the possible equivalent norms ${|·|}_{{\mathbb{M}}_n}$. By $\mathrm{\Omega }\subset {\mathbb{R}}^d$, $\mathrm{\Omega }\ne {\mathbb{R}}^d$, we denote an arbitrary domain, the boundary of which has the smoothness $C^2$. In the case $d=1$, without loss of generality, we suppose that $\mathrm{\Omega }$ is either a bounded interval or a half-line.

The outward unit normal to the boundary $\partial \mathrm{\Omega }$ is denoted by $\nu$, and let $\tau$ be the distance to a point $x\in \mathrm{\Omega }$ measured along the normal $\nu$. If $d⩾2$, we additionally suppose that there exists a fixed positive ${\tau }_0$ and local variables s on $\partial \mathrm{\Omega }$ such that on the set

$${\mathrm{\Pi }}_{{\tau }_0}:=\left\{x\in \mathrm{\Omega }:\mathrm{dist}(x,\partial \mathrm{\Omega })⩽{\tau }_0\right\}$$

(1)

the variables $(\tau ,s)$ are well defined, and all derivatives of x in $(\tau ,s)$ and of $(\tau ,s)$ in x up to the second order are bounded uniformly on ${\mathrm{\Pi }}_{{\tau }_0}$. We shall show in Section 4 that this condition ensures the inequality

$${\parallel u\parallel }_{L_2(\partial \mathrm{\Omega };{\mathbb{C}}^n)}⩽C{\parallel u\parallel }_{W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)}$$

(2)

for all $u\in W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)$ with a constant C independent of u.

The symbol $L_{\infty }(\mathrm{\Omega };{\mathbb{M}}_n)$ stands for the space of matrix-valued functions defined on ${\mathbb{R}}^d$ with values in ${\mathbb{M}}_n$, each entry of which is an element of $L_{\infty }\left({\mathbb{R}}^d\right)$. The norm in this space is defined as

$${\parallel A\parallel }_{L_{\infty }({\mathbb{R}}^d;{\mathbb{M}}_n)}=\underset{x\in {\mathbb{R}}^d}{esssup}{\left|A\left(x\right)\right|}_{{\mathbb{M}}_n},$$

(3)

where esssup denotes the essential supremum. We shall also employ various Sobolev spaces of matrix-valued functions, for instance, $W_2^j(\mathrm{\Omega };{\mathbb{M}}_n)$. By ${\mathring{W}}_2^j(\mathrm{\Omega };{\mathbb{M}}_n)$, we denote the subspace of $W_2^j(\mathrm{\Omega };{\mathbb{M}}_n)$ consisting of the functions, the traces of which, and of all its normal derivatives up to the $(j-1)$th order on $\partial \mathrm{\Omega }$, are zero.

We shall consider operators of an order $2m$, where $m⩾1$ is a natural number. By ${\mathbb{Z}}_{+}$, we denote the set of nonnegative integers and $\alpha$, $\beta$ are multi-indices from ${\mathbb{Z}}_{+}^d$ such that $\left|\alpha \right|⩽m$, $\left|\beta \right|⩽m$. Let $\epsilon =({\epsilon }_1,\dots ,{\epsilon }_k)$, $k⩾1$, be a multi-dimensional small parameter, that is, ${\epsilon }_i\to 0$, $i=1,\dots ,k$. The symbol ${\parallel ·\parallel }_{X\to Y}$ stands for a norm of a bounded operator acting from a Banach space X into a Banach space Y.

Given a closed operator $\mathcal{A}$ on a Hilbert space H, by $\sigma \left(\mathcal{A}\right)$, we denote the spectrum of the operator, while ${\sigma }_{ϵ}\left(\mathcal{A}\right)$ and ${\sigma }_{p,ϵ}\left(\mathcal{A}\right)$ are, respectively, the $ϵ$-pseudospectrum and the $(p,ϵ)$-pseudospectrum:

$$\begin{array}{cc}\hfill {\sigma }_{ϵ}\left(\mathcal{A}\right):=& \left\{{\lambda \in \mathbb{C}:\parallel (\mathcal{A}-\lambda )}^{-1}{\parallel }_{H\to H}>{ϵ}^{-1}\right\},\hfill \\ \hfill {\sigma }_{p,ϵ}\left(\mathcal{A}\right):=& \left\{{\lambda \in \mathbb{C}:\parallel (\mathcal{A}-\lambda )}^{-2^p}{\parallel }_{H\to H}^{\frac{1}{2^p}}>{ϵ}^{-1}\right\},p\in {\mathbb{Z}}_{+},\hfill \end{array}$$

(4)

with the usual convention

$${\parallel (\mathcal{A}-\lambda )}^{-1}{\parallel }_{H\to H}={\parallel {(\mathcal{A}-\lambda )}^{-2^p}\parallel }_{H\to H}^{\frac{1}{2^p}}=+\infty \mathrm{for}\lambda \in \sigma \left(\mathcal{A}\right).$$

For a self-adjoint operator $\mathcal{A}$, by ${\mathcal{P}}_{[a,b]}\left(\mathcal{A}\right)$, we denote the spectral projection of the operator $\mathcal{A}$ corresponding to a segment $[a,b]$ on the real line with $-\infty <a<b<\infty$.

Let ${\mathrm{dist}}_{\mathrm{H}}(K_1,K_2)$ be the Hausdorff distance between sets $K_1$ and $K_2$ in the complex plane:

$${\mathrm{dist}}_{\mathrm{H}}(K_1,K_2):=max\left\{\underset{\lambda \in K_1}{sup}\underset{z\in K_2}{inf}|\lambda -z|,\underset{\lambda \in K_2}{sup}\underset{z\in K_1}{inf}|\lambda -z|\right\}.$$

(5)

2.2. Operators on Entire Space

Let

$$A_{\alpha \beta }=A_{\alpha \beta }\left(x\right),\left|\alpha \right|=\left|\beta \right|=m,$$

(6)

and

$$A_{\alpha \beta }^i=A_{\alpha \beta }^i\left(x\right),\left|\alpha \right|,\left|\beta \right|⩽m,\left|\alpha \right|+\left|\beta \right|⩽2m-1,i=1,\dots ,N_{\alpha \beta },$$

(7)

be two given families of functions defined on ${\mathbb{R}}^d$ and belonging to $L_{\infty }({\mathbb{R}}^d;{\mathbb{C}}^n)$; here, $N_{\alpha \beta }$ are some natural numbers. As $\left|\alpha \right|⩽m-1$ and $\left|\beta \right|=m$, we additionally suppose that $A_{\alpha \beta }^i\in W_{\infty }^1({\mathbb{R}}^d;{\mathbb{M}}_n)$. The matrices $A_{\alpha \beta }$ with $\left|\alpha \right|=\left|\beta \right|=m$ satisfy the following ellipticity condition:

$$Re\sum _{\left|\alpha \right|=\left|\beta \right|=m}{(A_{\alpha \beta }\left(x\right)z_{\alpha },z_{\beta })}_{{\mathbb{C}}^d}⩾c_0\sum _{\left|\alpha \right|=m}{\left|z_{\alpha }\right|}_{{\mathbb{C}}^d}^2,z_{\alpha }\in {\mathbb{C}}^d,x\in {\mathbb{R}}^d,$$

(8)

where $c_0$ is some fixed positive constant independent of x and $z_{\alpha }$.

We introduce a family of vector-valued functions ${\varphi }_{\alpha \beta }^i={\varphi }_{\alpha \beta }^i(x,\epsilon )$, defined for $x\in {\mathbb{R}}^d$ and containing a sufficiently small $\epsilon$ with values in ${\mathbb{C}}^d$, where the indices $\alpha$, $\beta$, and i range as in (7). We suppose that, for each $\epsilon$, the functions ${\varphi }_{\alpha \beta }^i(·,\epsilon )$ are elements of $W_{\infty }^1({\mathbb{R}}^d;{\mathbb{C}}^n)$, and the inequalities

$$|{\varphi }_{\alpha \beta }^i(x,\epsilon ){|}_{{\mathbb{C}}^n}+\sum _{j=1}^d|\frac{\partial {\varphi }_{\alpha \beta }^i}{\partial x_j}(x,\epsilon ){|}_{{\mathbb{C}}^n}⩽\mu \left(\epsilon \right)$$

(9)

hold, where $\mu \left(\epsilon \right)$ is positive function independent of x, $\alpha$, $\beta$, and i and $\mu \left(\epsilon \right)\to 0$ as $\epsilon \to 0$. By means of the functions ${\varphi }_{\alpha \beta }$, we introduce a family of operators on $L_2({\mathbb{R}}^d;{\mathbb{C}}^n)$ that acts by the rule

$${\mathcal{T}}_{\alpha \beta }^{\epsilon }:=\sum _{i=1}^{N_{\alpha \beta }}{\mathcal{T}}_{\alpha \beta }^{\epsilon ,i},\left({\mathcal{T}}_{\alpha \beta }^{\epsilon ,i}u\right)\left(x\right):=A_{\alpha \beta }^i\left(x\right)u\left(x+{\varphi }_{\alpha \beta }^i(x,\epsilon )\right).$$

(10)

The main object of this study, in this subsection, is a nonlocal operator

$${\mathcal{H}}_{\epsilon }={(-1)}^m\sum _{\left|\alpha \right|=\left|\beta \right|=m}\frac{{\partial }^{\alpha }}{\partial x^{\alpha }}{A}_{\alpha \beta }\frac{{\partial }^{\beta }}{\partial x^{\beta }}+\sum _{\begin{array}{c}\left|\alpha \right|,\left|\beta \right|⩽m\\ \left|\alpha \right|+\left|\beta \right|⩽2m-1\end{array}}\frac{{\partial }^{\alpha }}{\partial x^{\alpha }}{\mathcal{T}}_{\alpha \beta }^{\epsilon }\frac{{\partial }^{\beta }}{\partial x^{\beta }}$$

(11)

in $L_2({\mathbb{R}}^d;{\mathbb{C}}^n)$. Rigorously, we define it as follows. We first introduce a sesquilinear form

$${\mathfrak{h}}_{\epsilon }(u,v):=\sum _{\left|\alpha \right|=\left|\beta \right|=m}{\left(A_{\alpha \beta }\frac{{\partial }^{\beta }u}{\partial x^{\beta }},\frac{{\partial }^{\alpha }v}{\partial x^{\alpha }}\right)}_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}+\sum _{\begin{array}{c}\left|\alpha \right|,\left|\beta \right|⩽m\\ \left|\alpha \right|+\left|\beta \right|⩽2m-1\end{array}}{(-1)}^{\left|\alpha \right|}{\left({\mathcal{T}}_{\alpha \beta }^{\epsilon }\frac{{\partial }^{\beta }u}{\partial x^{\beta }},\frac{{\partial }^{\alpha }v}{\partial x^{\alpha }}\right)}_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}$$

(12)

in $L_2({\mathbb{R}}^d;{\mathbb{C}}^n)$ on the domain $\mathfrak{D}\left({\mathfrak{h}}_{\epsilon }\right):=W_2^m({\mathbb{R}}^d;{\mathbb{C}}^n)$. We shall show in Section 3 that this form is m-sectorial, and, by the first representation theorem ([63], Ch. 6, Section 2.1, Theorem 2.1), there exists a unique m-sectorial operator associated with this form. This operator is denoted precisely as ${\mathcal{H}}_{\epsilon }$.

Together with ${\mathcal{H}}_{\epsilon }$, we consider also an operator ${\mathcal{H}}_0$ in $L_2({\mathbb{R}}^d;{\mathbb{C}}^n)$, which is formally written as

$${\mathcal{H}}_0={(-1)}^m\sum _{\left|\alpha \right|=\left|\beta \right|=m}\frac{{\partial }^{\alpha }}{\partial x^{\alpha }}{A}_{\alpha \beta }\frac{{\partial }^{\beta }}{\partial x^{\beta }}+\sum _{\begin{array}{c}\left|\alpha \right|,\left|\beta \right|⩽m\\ \left|\alpha \right|+\left|\beta \right|⩽2m-1\end{array}}\sum _{i=1}^{N_{\alpha \beta }}\frac{{\partial }^{\alpha }}{\partial x^{\alpha }}{A}_{\alpha \beta }^i\frac{{\partial }^{\beta }}{\partial x^{\beta }}.$$

(13)

Rigorously, we define it as associated with a sesquilinear form

$$\begin{array}{cc}\hfill {\mathfrak{h}}_0(u,v):=& \sum _{\left|\alpha \right|=\left|\beta \right|=m}{\left(A_{\alpha \beta }\frac{{\partial }^{\beta }u}{\partial x^{\beta }},\frac{{\partial }^{\alpha }v}{\partial x^{\alpha }}\right)}_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}\hfill \\ \hfill & +\sum _{\begin{array}{c}\left|\alpha \right|,\left|\beta \right|⩽m\\ \left|\alpha \right|+\left|\beta \right|⩽2m-1\end{array}}\sum _{i=1}^{N_{\alpha \beta }}{(-1)}^{\left|\alpha \right|}{\left(A_{\alpha \beta }^i\frac{{\partial }^{\beta }u}{\partial x^{\beta }},\frac{{\partial }^{\alpha }v}{\partial x^{\alpha }}\right)}_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}\hfill \end{array}$$

(14)

in $L_2({\mathbb{R}}^d;{\mathbb{C}}^n)$ on the domain $\mathfrak{D}\left({\mathfrak{h}}_0\right):=W_2^m({\mathbb{R}}^d;{\mathbb{C}}^n)$. We shall show in Section 3 that this form is also m-sectorial, and, the first representation theorem ([63], Ch. 6, Section 2.1, Theorem 2.1), again ensures that the operator ${\mathcal{H}}_0$ is well defined and that it is m-sectorial.

Our main result in this subsection is the following theorem.

Theorem 1.

There exists ${\lambda }_0\in \mathbb{R}$ independent of ε such that all λ obeying $Re\lambda ⩽{\lambda }_0$ are in the resolvent sets of the operators ${\mathcal{H}}_{\epsilon }$ and ${\mathcal{H}}_0$. For $Re\lambda ⩽{\lambda }_0$, the estimate holds:

$$\parallel {({\mathcal{H}}_{\epsilon }-\lambda )}^{-1}-{({\mathcal{H}}_0-\lambda )}^{-1}{\parallel }_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)\to W_2^m({\mathbb{R}}^d;{\mathbb{C}}^n)}⩽C\mu \left(\epsilon \right),$$

(15)

where C is some constant independent of ε but dependent on λ.

2.3. Operators in Domains with Almost Identity Diffeomorphisms

In this subsection, we consider an elliptic operator involving compositions of derivatives with arbitrary, almost identity diffeomorphisms in a multi-dimensional domain $\mathrm{\Omega }$, for which we make the assumptions described in Section 2.1. We again introduce two families of matrix functions, as in (6) and (7), but, now, they are defined for $x\in \mathrm{\Omega }$ and belong to $L_{\infty }(\mathrm{\Omega };{\mathbb{M}}_n)$; for $\left|\alpha \right|⩽m-1$ and $\left|\beta \right|=m$, we suppose that $A_{\alpha \beta }^i\in W_{\infty }^1(\mathrm{\Omega };{\mathbb{M}}_n)$. Family (6) obeys ellipticity condition (8) uniformly in $x\in \mathrm{\Omega }$. We then introduce a family of functions ${\varphi }_{\alpha \beta }^i={\varphi }_{\alpha \beta }^i(x,\epsilon )$ on $\mathrm{\Omega }$ with values in ${\mathbb{C}}^d$, where the indices $\alpha$, $\beta$, and i range in accordance with (7). We suppose that, for each $\epsilon$, the functions ${\varphi }_{\alpha \beta }^i(·,\epsilon )$ belong to $W_{\infty }^1(\mathrm{\Omega };{\mathbb{C}}^d)$, the inequalities in (9) are satisfied, and, for each $\epsilon$, the mapping $x\mapsto x+{\varphi }_{\alpha \beta }^i(x,\epsilon )$ is a diffeomorphism of $\mathrm{\Omega }$ onto $\mathrm{\Omega }$. By means of the functions ${\varphi }_{\alpha \beta }^i$, we again introduce families of operators, ${\mathcal{T}}_{\alpha \beta }^{\epsilon }$ and ${\mathcal{T}}_{\alpha \beta }^{\epsilon ,i}$, on $L_2(\mathrm{\Omega };{\mathbb{C}}^n)$ via Formula (10).

In addition, we define two extra families of matrix-valued functions ${\gamma }_{\alpha \beta }^i={\gamma }_{\alpha \beta }^i\left(x\right)$, $B_{\alpha }^{jk}=B_{\alpha }^{jk}\left(x\right)$, $x\in \partial \mathrm{\Omega }$, and a family of vector-valued functions ${\psi }_{\alpha \beta }^i={\psi }_{\alpha \beta }^i(x,\epsilon )$, $x\in \mathrm{\Omega }$, for $\left|\alpha \right|,\left|\beta \right|⩽m-1$, $i=1,\dots ,K_{\alpha \beta }$, $j=1,\dots ,K$, $k=1,\dots ,K_j$, where $K_{\alpha \beta }$ and $K_j$ are some natural numbers, while K is a nonnegative integer. In the case that $K=0$, no functions $B_{\alpha }^{jk}$ are introduced.

We suppose that ${\gamma }_{\alpha \beta }^i,B_{\alpha }^{jk}\in L_{\infty }(\partial \mathrm{\Omega };{\mathbb{M}}_n)$, ${\psi }_{\alpha \beta }^i(·,\epsilon )\in W_{\infty }^1(\mathrm{\Omega };{\mathbb{C}}^d)$, the functions ${\psi }_{\alpha \beta }^i$ satisfy the inequalities in (9), and the mappings $x\mapsto x+{\psi }_{\alpha \beta }^i(x,\epsilon )$ are diffeomorphisms of $\mathrm{\Omega }$ onto $\mathrm{\Omega }$. As $\left|\alpha \right|=m-1$, we additionally assume that ${\gamma }_{\alpha \beta }^i\in W_{\infty }^1(\partial \mathrm{\Omega };{\mathbb{M}}_n)$. In the case that $d=1$, the functions ${\gamma }_{\alpha \beta }^i={\gamma }_{\alpha \beta }^i\left(x\right)$ and $B_{\alpha }^{jk}=B_{\alpha }^{jk}\left(x\right)$ are corresponding sets of constant matrices associated with each boundary point of the set $\mathrm{\Omega }$. For example, if $\mathrm{\Omega }$ is a bounded interval, then each ${\gamma }_{\alpha \beta }^i$ is a pair of matrices associated with a corresponding end of the interval $\mathrm{\Omega }$. The space $L_2(\partial \mathrm{\Omega };{\mathbb{C}}^n)$, then, is naturally identified with ${\mathbb{C}}^n\oplus {\mathbb{C}}^n$ if $\mathrm{\Omega }$ is a bounded interval, and it hence has two boundary points, or, with ${\mathbb{C}}^n$, if $\mathrm{\Omega }$ is the half-line, it has one boundary point.

We define operators acting from $W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)$ into $L_2(\partial \mathrm{\Omega };{\mathbb{C}}^n)$:

$${\mathcal{Q}}_{\alpha \beta }^{\epsilon }:=\sum _{i=1}^{K_{\alpha \beta }}{\mathcal{Q}}_{\alpha \beta }^{\epsilon ,i},\left({\mathcal{Q}}_{\alpha \beta }^{\epsilon ,i}u\right)\left(x\right):={\gamma }_{\alpha \beta }^i\left(x\right)u\left(x+{\psi }_{\alpha \beta }^i(x,\epsilon )\right).$$

(16)

As in the previous subsection, we introduce our main operator via an appropriate sesquilinear form. This form is defined as

$$\begin{array}{cc}\hfill {\mathfrak{h}}_{\epsilon }(u,v):=& \sum _{\left|\alpha \right|=\left|\beta \right|=m}{\left(A_{\alpha \beta }\frac{{\partial }^{\beta }u}{\partial x^{\beta }},\frac{{\partial }^{\alpha }v}{\partial x^{\alpha }}\right)}_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}+\sum _{\begin{array}{c}\left|\alpha \right|,\left|\beta \right|⩽m\\ \left|\alpha \right|+\left|\beta \right|⩽2m-1\end{array}}{(-1)}^{\left|\alpha \right|}{\left({\mathcal{T}}_{\alpha \beta }^{\epsilon }\frac{{\partial }^{\beta }u}{\partial x^{\beta }},\frac{{\partial }^{\alpha }v}{\partial x^{\alpha }}\right)}_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}\hfill \\ \hfill & +\sum _{\left|\alpha \right|,\left|\beta \right|⩽m-1}{\left({\mathcal{Q}}_{\alpha \beta }^{\epsilon }\frac{{\partial }^{\beta }u}{\partial x^{\beta }},\frac{{\partial }^{\alpha }v}{\partial x^{\alpha }}\right)}_{L_2(\partial \mathrm{\Omega };{\mathbb{C}}^n)}\hfill \end{array}$$

(17)

on a domain $\mathfrak{D}\left({\mathfrak{h}}_{\epsilon }\right)=\mathfrak{W}$, where $\mathfrak{W}$ is a subspace in $W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)$. This subspace $\mathfrak{W}$ is a set of functions from $W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)$ obeying the boundary conditions

$$\sum _{\left|\alpha \right|⩽m-1}\sum _{k=1}^{K_j}{B}_{\alpha }^{jk}\frac{{\partial }^{\alpha }u}{\partial x^{\alpha }}=0\mathrm{on}\partial \mathrm{\Omega },j=1,\dots ,K.$$

(18)

It is clear that ${\mathring{W}}_2^m(\mathrm{\Omega };{\mathbb{C}}^n)$ is a subspace of $\mathfrak{W}$, and, hence, the form ${\mathfrak{h}}_{\epsilon }$ is densely defined in $L_2(\mathrm{\Omega };{\mathbb{C}}^n)$. We shall show in Section 4 that this form is closed and sectorial, and, hence, by the first representation theorem ([63], Ch. 6, Section 2.1, Theorem 2.1), there exists a unique associated m-sectorial operator, which we denote by ${\mathcal{H}}_{\epsilon }$. This is precisely the main operator that we study in this subsection.

A formal definition of the operator ${\mathcal{H}}_{\epsilon }$ can be written through formal integration by parts in the form of ${\mathfrak{h}}_{\epsilon }$. This gives differential expression (11) for the operator ${\mathcal{H}}_{\epsilon }$. The issue of boundary operators describing the boundary conditions is much more delicate, and we shall discuss it later in Section 2.7.

The limiting operator is also introduced via an appropriate sesquilinear form:

$$\begin{array}{cc}\hfill {\mathfrak{h}}_0(u,v):=& \sum _{\left|\alpha \right|=\left|\beta \right|=m}{\left(A_{\alpha \beta }\frac{{\partial }^{\beta }u}{\partial x^{\beta }},\frac{{\partial }^{\alpha }v}{\partial x^{\alpha }}\right)}_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}\hfill \\ \hfill & +\sum _{\begin{array}{c}\left|\alpha \right|,\left|\beta \right|⩽m\\ \left|\alpha \right|+\left|\beta \right|⩽2m-1\end{array}}\sum _{i=1}^{N_{\alpha \beta }}{(-1)}^{\left|\alpha \right|}{\left(A_{\alpha \beta }^i\frac{{\partial }^{\beta }u}{\partial x^{\beta }},\frac{{\partial }^{\alpha }v}{\partial x^{\alpha }}\right)}_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}\hfill \\ \hfill & +\sum _{\left|\alpha \right|,\left|\beta \right|⩽m-1}\sum _{i=1}^{K_{\alpha \beta }}{\left({\gamma }_{\alpha \beta }^i\frac{{\partial }^{\beta }u}{\partial x^{\beta }},\frac{{\partial }^{\alpha }v}{\partial x^{\alpha }}\right)}_{L_2(\partial \mathrm{\Omega };{\mathbb{C}}^n)}\hfill \end{array}$$

(19)

on a domain $\mathfrak{D}\left({\mathfrak{h}}_0\right)=\mathfrak{W}$. We shall show in Section 4 that this form is sectorial and closed, and, by the first representation theorem ([63], Ch. 6, Section 2.1, Theorem 2.1), there exists a unique associated m-sectorial operator. This operator is denoted by ${\mathcal{H}}_0$. Its differential expression is (13), some of the boundary conditions are again given by (18), while other are to be obtained by an appropriate integration by parts in the form ${\mathfrak{h}}_0$; we shall discuss them in Section 2.7.

Now we are in position to formulate the main result of this subsection.

Theorem 2.

There exists ${\lambda }_0$ independent of ε such that all λ obeying $Re\lambda ⩽{\lambda }_0$ are in the resolvent sets of the operators ${\mathcal{H}}_{\epsilon }$ and ${\mathcal{H}}_0$. For $Re\lambda ⩽{\lambda }_0$ the estimate holds:

$$\parallel {({\mathcal{H}}_{\epsilon }-\lambda )}^{-1}-{({\mathcal{H}}_0-\lambda )}^{-1}{\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)\to W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)}⩽C\mu \left(\epsilon \right),$$

(20)

where C is some constant independent of ε but depending on λ.

2.4. Differential–Difference Operators with Small Variable Translations: Dirichlet Condition

In this subsection we consider differential–difference operators in a multi-dimensional domain $\mathrm{\Omega }$ with the Dirichlet condition. These operators are introduced similarly to ones in the previous subsection with the only main feature: now we do not suppose that these translations map the domain onto itself.

For the domain $\mathrm{\Omega }$, we make the assumptions described in Section 2.1. We introduce families (6) and (7) but for $x\in \mathrm{\Omega }$, which belong to $L_{\infty }(\mathrm{\Omega };{\mathbb{M}}_n)$ and for $\left|\alpha \right|⩽m-1$ and $\left|\beta \right|=m$, we again have $A_{\alpha \beta }^i\in W_{\infty }^1(\mathrm{\Omega };{\mathbb{M}}_n)$ and ellipticity condition (8) uniformly in $x\in \mathrm{\Omega }$. We also introduce a family of vector-valued functions ${\varphi }_{\alpha \beta }^i={\varphi }_{\alpha \beta }^i(x,\epsilon )$ on $\mathrm{\Omega }$ with values in ${\mathbb{C}}^d$, where the indices $\alpha$, $\beta$ and i range as in the previous section. We suppose that for each $\epsilon$ the functions ${\varphi }_{\alpha \beta }^i(·,\epsilon )$ are elements of $W_{\infty }^1(\mathrm{\Omega };{\mathbb{C}}^n)$ and the inequalities in (9) hold uniformly in $x\in \mathrm{\Omega }$, $\alpha$, $\beta$ and $\epsilon$. In contrast to the previous subsection, we do not suppose that the mapping $x\mapsto x+{\varphi }_{\alpha \beta }^i(x,\epsilon )$ is a diffeomorphism of $\mathrm{\Omega }$ onto $\mathrm{\Omega }$. Instead of this we suppose that the image of the domain $\mathrm{\Omega }$ under such mapping does not necessarily coincides with $\mathrm{\Omega }$.

By $\mathcal{L}$ we denote an operator of continuation by zero outside $\mathrm{\Omega }$, while $\mathcal{R}$ stands for an operator of restriction to $\mathrm{\Omega }$. Namely, the operator $\mathcal{L}$ acts from $L_2(\mathrm{\Omega };{\mathbb{C}}^n)$ into $L_2({\mathbb{R}}^d;{\mathbb{C}}^n)$ and it extends the function by zero outside $\mathrm{\Omega }$, while the operator $\mathcal{R}$ acts from $L_2({\mathbb{R}}^d;{\mathbb{C}}^n)$ into $L_2(\mathrm{\Omega };{\mathbb{C}}^n)$ and describes the restriction:

$$\mathcal{R}u:=u\mathrm{on}\mathrm{\Omega },\mathcal{L}u:=\left\{\begin{array}{cc}\hfill & u\mathrm{in}\mathrm{\Omega },\hfill \\ \hfill & 0\mathrm{outside}\mathrm{\Omega }.\hfill \end{array}\right.$$

(21)

Let ${\mathrm{\Omega }}_{\alpha \beta }^{\epsilon ,i}$ be the image of the domain $\mathrm{\Omega }$ under the mapping $x\mapsto x+{\varphi }_{\alpha \beta }^i(x,\epsilon )$. We define a family of auxiliary operators acting from $L_2({\mathrm{\Omega }}_{\alpha \beta }^{\epsilon ,i};{\mathbb{C}}^n)$ into $L_2(\mathrm{\Omega };{\mathbb{C}}^n)$:

$$\left({\mathcal{S}}_{\alpha \beta }^{\epsilon ,i}u\right)\left(x\right):=u\left(x+{\varphi }_{\alpha \beta }^i(x,\epsilon )\right),x\in \mathrm{\Omega }.$$

(22)

By means of these operators we introduce a family similar to (10):

$${\mathcal{T}}_{\alpha \beta }^{\epsilon }:=\sum _{i=1}^{N_{\alpha \beta }}{\mathcal{T}}_{\alpha \beta }^{\epsilon ,i},{\mathcal{T}}_{\alpha \beta }^{\epsilon ,i}u:=A_{\alpha \beta }^i\mathcal{R}{\mathcal{S}}_{\alpha \beta }^{\epsilon ,i}\mathcal{L}u.$$

(23)

The action of these operators is as follows. Given a function $u\in L_2(\mathrm{\Omega };{\mathbb{C}}^n)$, it is continued by zero outside $\mathrm{\Omega }$, this continuation is restricted on ${\mathrm{\Omega }}_{\alpha \beta }^{\epsilon ,i}$ and then it is mapped back to $\mathrm{\Omega }$ by means of the diffeomorphism $x+{\varphi }_{\alpha \beta }^i(x,\epsilon )$. To the resulting vector function, the matrix function $A_{\alpha \beta }^i$ is applied.

The main operator here is again denoted by ${\mathcal{H}}_{\epsilon }$ and formally this operator can be introduced as having differential expression (11) with the operators ${\mathcal{T}}_{\alpha \beta }^{\epsilon ,i}$ defined in (23) subject to the Dirichlet conditions:

$$\frac{{\partial }^{i}u}{\partial {\nu }^i}=0\mathrm{on}\partial \mathrm{\Omega },i=0,\dots ,m-1,$$

(24)

where $\nu$ is the outward unit normal to $\partial \mathrm{\Omega }$. Rigorously we again define it as associated with an appropriate sesquilinear form. This form reads as

$${\mathfrak{h}}_{\epsilon }(u,v):=\sum _{\left|\alpha \right|=\left|\beta \right|=m}{\left(A_{\alpha \beta }\frac{{\partial }^{\beta }u}{\partial x^{\beta }},\frac{{\partial }^{\alpha }v}{\partial x^{\alpha }}\right)}_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}+\sum _{\begin{array}{c}\left|\alpha \right|,\left|\beta \right|⩽m\\ \left|\alpha \right|+\left|\beta \right|⩽2m-1\end{array}}{(-1)}^{\left|\alpha \right|}{\left({\mathcal{T}}_{\alpha \beta }^{\epsilon }\frac{{\partial }^{\beta }u}{\partial x^{\beta }},\frac{{\partial }^{\alpha }v}{\partial x^{\alpha }}\right)}_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}$$

(25)

on a domain $\mathfrak{D}\left({\mathfrak{h}}_{\epsilon }\right)=\mathfrak{W}$, where $\mathfrak{W}$ is the closure of $C_0^{\infty }(\mathrm{\Omega };{\mathbb{C}}^n)$ in $W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)$. Such form is densely defined and we shall show in Section 5 that this form is closed and sectorial and hence, by the first representation theorem ([63], Ch. 6, Section 2.1, Theorem 2.1), there exists a uniquely associated m-sectorial operator, which is exactly ${\mathcal{H}}_{\epsilon }$.

The corresponding limiting operator ${\mathcal{H}}_0$ can be formally defined as having differential expression (13) in $\mathrm{\Omega }$, subject to the Dirichlet condition. Rigorously, it is defined as associated with the sesquilinear form

$$\begin{array}{cc}\hfill {\mathfrak{h}}_0(u,v):=& \sum _{\left|\alpha \right|=\left|\beta \right|=m}{\left(A_{\alpha \beta }\frac{{\partial }^{\beta }u}{\partial x^{\beta }},\frac{{\partial }^{\alpha }v}{\partial x^{\alpha }}\right)}_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}\hfill \\ \hfill & +\sum _{\begin{array}{c}\left|\alpha \right|,\left|\beta \right|⩽m\\ \left|\alpha \right|+\left|\beta \right|⩽2m-1\end{array}}\sum _{i=1}^{N_{\alpha \beta }}{(-1)}^{\left|\alpha \right|}{\left(A_{\alpha \beta }^i\frac{{\partial }^{\beta }u}{\partial x^{\beta }},\frac{{\partial }^{\alpha }v}{\partial x^{\alpha }}\right)}_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}\hfill \end{array}$$

(26)

in $L_2(\mathrm{\Omega };{\mathbb{C}}^n)$ on $\mathfrak{D}\left({\mathfrak{h}}_0\right):=\mathfrak{W}$. We shall show in Section 5 that this form is closed and sectorial and the first representation theorem, then, uniquely defines the m-sectorial operator ${\mathcal{H}}_0$.

Our main result in this subsection is as follows.

Theorem 3.

There exists ${\lambda }_0$ independent of ε such that all obeying λ are in the resolvent sets of the operators ${\mathcal{H}}_{\epsilon }$ and ${\mathcal{H}}_0$. For $Re\lambda ⩽{\lambda }_0$ the estimate holds:

$$\parallel {({\mathcal{H}}_{\epsilon }-\lambda )}^{-1}-{({\mathcal{H}}_0-\lambda )}^{-1}{\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)\to W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)}⩽C\mu \left(\epsilon \right),$$

(27)

where C is some constant independent of ε but dependent on λ.

2.5. Differential–Difference Operators with Small Variable Translations: General Case

In this subsection, we consider the differential–difference operators in a multi-dimensional domain, $\mathrm{\Omega }$, with a general boundary condition; for the domain, we make the assumptions from Section 2.1. We introduce families (6) and (7) for $x\in \mathrm{\Omega }$ and the families ${\varphi }_{\alpha \beta }^i$ with the same properties, and we define the operators $\mathcal{R}$, $\mathcal{L}$ by (21), as well as the operators ${\mathcal{S}}_{\alpha \beta }^{\epsilon ,i}$ and ${\mathcal{T}}_{\alpha \beta }^{\epsilon ,i}$ by (22) and (23). In addition, as in Section 2.3, we introduce two extra families of matrix-valued functions, ${\gamma }_{\alpha \beta }^i={\gamma }_{\alpha \beta }^i\left(x\right)$, $x\in \mathrm{\Omega }$, and $B_{\alpha }^{jk}=B_{\alpha }^{jk}\left(x\right)$, $x\in \partial \mathrm{\Omega }$, and a family of vector-valued functions ${\psi }_{\alpha \beta }^i={\psi }_{\alpha \beta }^i(x,\epsilon )$, $x\in \mathrm{\Omega }$, with the same properties as in Section 2.3, except for one: namely, now we do not suppose that the mappings $x\mapsto x+{\psi }_{\alpha \beta }^i(x,\epsilon )$ are diffeomorphisms of $\mathrm{\Omega }$ onto $\mathrm{\Omega }$. Instead of this, we suppose that the images of $\mathrm{\Omega }$ under such mappings are not necessarily $\mathrm{\Omega }$.

We define operators ${\mathcal{S}}_{\alpha \beta }^{\epsilon ,i}$ and ${\mathcal{T}}_{\alpha \beta }^{\epsilon ,i}$ as in (22) and (23). We introduce mappings

$${\mathcal{G}}_{\alpha \beta }^{\epsilon ,i}\left(x\right):=x+{\psi }_{\alpha \beta }^{\epsilon ,i}\left(x\right)$$

(28)

and, due to the assumed properties of ${\psi }_{\alpha \beta }^{\epsilon ,i}$, there exist inverse mappings ${\left({\mathcal{G}}_{\alpha \beta }^{\epsilon ,i}\right)}^{-1}$, which are of the form ${\left({\mathcal{G}}_{\alpha \beta }^{\epsilon ,i}\right)}^{-1}\left(y\right)=y+{\tilde{\psi }}_{\alpha \beta }^{\epsilon ,i}\left(y\right)$, where the functions ${\tilde{\psi }}_{\alpha \beta }^{\epsilon ,i}$ possess the same properties as ${\psi }_{\alpha \beta }^{\epsilon ,i}$. We then let

$${\mathrm{\Gamma }}_{\alpha \beta }^{\epsilon ,i}:=\partial \mathrm{\Omega }\cap \overline{{\mathrm{{\rm Y}}}_{\alpha \beta }^{\epsilon ,i}},{\mathrm{{\rm Y}}}_{\alpha \beta }^{\epsilon ,i}:={\left({\mathcal{G}}_{\alpha \beta }^{\epsilon ,i}\right)}^{-1}\left(\mathrm{\Omega }\cap {\mathcal{G}}_{\alpha \beta }^{\epsilon ,i}\left(\mathrm{\Omega }\right)\right).$$

The sets ${\mathrm{{\rm Y}}}_{\alpha \beta }^{\epsilon ,i}$ consist of all points in $\mathrm{\Omega }$ mapped by ${\mathcal{G}}_{\alpha \beta }^{\epsilon ,i}$ into $\mathrm{\Omega }$. Without loss of generality, we suppose that if $K_{\alpha \beta }\ne 0$, then at least one of corresponding functions ${\gamma }_{\alpha \beta }^i$ is not identically zero on ${\mathrm{\Gamma }}_{\alpha \beta }^{\epsilon ,i}$ for all $\epsilon$. For all given $\alpha$, $\beta$, the set of such indices i is denoted by ${\mathbb{I}}_{\alpha \beta }$. Provided ${\mathbb{I}}_{\alpha \beta }\ne \varnothing$, for at least one pair of indices $\alpha$, $\beta$, we make one more assumption:

$$\begin{array}{cc}\hfill & \underset{\left|\alpha \right|,\left|\beta \right|⩽m-1}{max}\underset{i\in {\mathbb{I}}_{\alpha \beta }}{max}{mes}_{d-1}\partial \mathrm{\Omega }\setminus {\mathrm{\Gamma }}_{\alpha \beta }^{\epsilon ,i}⩽\eta \left(\epsilon \right),d⩾2,\eta \left(\epsilon \right)\to 0,\mathrm{as}\epsilon \to 0,\hfill \\ \hfill & \mathrm{\Omega }\subseteq {\mathrm{{\rm Y}}}_{\alpha \beta }^{\epsilon ,i}\mathrm{for}\mathrm{all}\epsilon \mathrm{as}\left|\alpha \right|,\left|\beta \right|⩽m-1,i\in {\mathbb{I}}_{\alpha \beta },d=1,\hfill \end{array}$$

(29)

where ${mes}_{d-1}$ is the $(d-1)$-dimensional measure on $\partial \mathrm{\Omega }$ and $\eta =\eta \left(\epsilon \right)$ is some nonnegative function. We introduce the operators

$${\mathcal{Q}}_{\alpha \beta }^{\epsilon ,i}u:={\gamma }_{\alpha \beta }^i\mathcal{R}{\mathcal{E}}_{\alpha \beta }^{\epsilon ,i}\mathcal{L}u,\left({\mathcal{E}}_{\alpha \beta }^{\epsilon ,i}u\right)\left(x\right):=u\left({\mathcal{G}}_{\alpha \beta }^{\epsilon ,i}\left(x\right)\right),$$

(30)

acting from $W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)$ into $L_2(\mathrm{\Omega };{\mathbb{C}}^n)$ and $W_2^1({\mathrm{{\rm Y}}}_{\alpha \beta }^{\epsilon ,i};{\mathbb{C}}^n)$.

The main operator is again introduced via an appropriate sesquilinear form, which reads as

$$\begin{array}{cc}\hfill {\mathfrak{h}}_{\epsilon }(u,v):=& \sum _{\left|\alpha \right|=\left|\beta \right|=m}{\left(A_{\alpha \beta }\frac{{\partial }^{\beta }u}{\partial x^{\beta }},\frac{{\partial }^{\alpha }v}{\partial x^{\alpha }}\right)}_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}\hfill \\ \hfill & +\sum _{\begin{array}{c}\left|\alpha \right|,\left|\beta \right|⩽m\\ \left|\alpha \right|+\left|\beta \right|⩽2m-1\end{array}}{(-1)}^{\left|\alpha \right|}{\left({\mathcal{T}}_{\alpha \beta }^{\epsilon }\frac{{\partial }^{\beta }u}{\partial x^{\beta }},\frac{{\partial }^{\alpha }v}{\partial x^{\alpha }}\right)}_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}\hfill \\ \hfill & +\sum _{\left|\alpha \right|,\left|\beta \right|⩽m-1}\sum _{i=1}^{K_{\alpha \beta }}{\left({\mathcal{Q}}_{\alpha \beta }^{\epsilon ,i}\frac{{\partial }^{\beta }u}{\partial x^{\beta }},\frac{{\partial }^{\alpha }v}{\partial x^{\alpha }}\right)}_{L_2({\mathrm{\Gamma }}_{\alpha \beta }^{\epsilon ,i};{\mathbb{C}}^n)}\hfill \end{array}$$

(31)

on a domain $\mathfrak{D}\left({\mathfrak{h}}_{\epsilon }\right)=\mathfrak{W}$, where $\mathfrak{W}$ is a subspace in $W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)$ determined by the boundary conditions in (18). The limiting operator is introduced via the sesquilinear form ${\mathfrak{h}}_0$, which is defined in (19) on the same domain $\mathfrak{D}\left({\mathfrak{h}}_0\right)=\mathfrak{W}$ that is determined by (18). We shall show that both forms ${\mathfrak{h}}_{\epsilon }$ and ${\mathfrak{h}}_0$ are closed and sectorial and that, by the first representation Theorem ([63], Ch. 6, Section 2.1, Theorem 2.1), there exist uniquely associated m-sectorial operators ${\mathcal{H}}_{\epsilon }$ and ${\mathcal{H}}_0$.

The main result is formulated in the following theorem.

Theorem 4.

There exists ${\lambda }_0$ independent of ε such that all λ obeying $Re\lambda ⩽{\lambda }_0$ are in the resolvent sets of the operators ${\mathcal{H}}_{\epsilon }$ and ${\mathcal{H}}_0$. For $Re\lambda ⩽{\lambda }_0$, the estimate holds:

$$\begin{array}{cccc}\hfill & \parallel {({\mathcal{H}}_{\epsilon }-\lambda )}^{-1}-{({\mathcal{H}}_0-\lambda )}^{-1}{\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)\to W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)}⩽C{\mu }^{\frac{1}{2}}\left(\epsilon \right),\hfill & \hfill & ifd=1,\hfill \\ \hfill & \parallel {({\mathcal{H}}_{\epsilon }-\lambda )}^{-1}-{({\mathcal{H}}_0-\lambda )}^{-1}{\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)\to W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)}⩽C\left({\mu }^{\frac{1}{2}}\left(\epsilon \right)+{\eta }^{\theta }\left(\epsilon \right)\right),\hfill & \hfill & ifd⩾2,\hfill \end{array}$$

(32)

where C is some constant independent of ε but dependent on λ and θ, while

$$\theta :=\left\{\begin{array}{cccc}\hfill & \mathit{arbitrary}\mathit{in}(0,1),\hfill & \hfill & d=2,\hfill \\ \hfill & \phantom{arbit}\frac{1}{d-1},\hfill & \hfill & d⩾3.\hfill \end{array}\right.$$

(33)

2.6. Spectral and Pseudospectral Convergence

In this subsection, we provide the results of the convergence of the spectra and pseudospectra of the operators introduced in the previous subsections. Let ${\mathcal{H}}_{\epsilon }$ be one of the perturbed operators introduced in Section 2.2, Section 2.3, Section 2.4 and Section 2.5, and let ${\mathcal{H}}_0$ be the corresponding limiting operator. In a case in which the operators are on the entire space, we let $\mathrm{\Omega }:={\mathbb{R}}^d$. Then, in accordance with Theorems 1–4, we have

$$\parallel {\mathcal{B}}_{\epsilon }{\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)\to W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)}⩽\zeta \left(\epsilon \right),{\mathcal{B}}_{\epsilon }:={({\mathcal{H}}_{\epsilon }-{\lambda }_0)}^{-1}-{({\mathcal{H}}_0-{\lambda }_0)}^{-1},$$

(34)

where $\zeta \left(\epsilon \right)$ corresponds to the right-hand sides in estimates (15), (20), (27), and (32).

Our first main result in this subsection is about the convergence of the spectra.

Theorem 5.

Let K be a non-empty compact set in the complex plane. Then

$$\sigma \left({\mathcal{H}}_{\epsilon }\right)\cap K\subseteq \overline{{\sigma }_{\varkappa \left(\epsilon \right)}\left({\mathcal{H}}_0\right)}\cap K,\sigma \left({\mathcal{H}}_0\right)\cap K\subseteq \overline{{\sigma }_{\varkappa \left(\epsilon \right)}\left({\mathcal{H}}_{\epsilon }\right)}\cap K,$$

(35)

where

$$\varkappa \left(\epsilon \right):=\frac{{\mathrm{\Lambda }}^2\left(K\right)\zeta \left(\epsilon \right)}{1-\mathrm{\Lambda }\left(K\right)\zeta \left(\epsilon \right)},\mathrm{\Lambda }\left(K\right):=\underset{\lambda \in K}{sup}|\lambda -{\lambda }_0|.$$

(36)

Our second main result describes the convergence of the $ϵ$-pseudospectra and $(p,ϵ)$-pseudospectra.

Theorem 6.

Let K be a compact set in the complex plane such that

$$\overline{{\sigma }_{ϵ}\left({\mathcal{H}}_0\right)}\cap K=\overline{{\sigma }_{ϵ}\left({\mathcal{H}}_0\right)\cap K}\ne \varnothing$$

(37)

or

$$\overline{{\sigma }_{p,ϵ}\left({\mathcal{H}}_0\right)}\cap K=\overline{{\sigma }_{p,ϵ}\left({\mathcal{H}}_0\right)\cap K}\ne \varnothing .$$

(38)

Then, in the case of condition (37), the convergence

$${\mathrm{dist}}_{\mathrm{H}}\left(\overline{{\sigma }_{ϵ}\left({\mathcal{H}}_0\right)}\cap K,\overline{{\sigma }_{ϵ}\left({\mathcal{H}}_{\epsilon }\right)}\cap K\right)\to 0,\epsilon \to 0,$$

(39)

holds, while in the case of condition (38), we have

$${\mathrm{dist}}_{\mathrm{H}}\left(\overline{{\sigma }_{p,ϵ}\left({\mathcal{H}}_0\right)}\cap K,\overline{{\sigma }_{p,ϵ}\left({\mathcal{H}}_{\epsilon }\right)}\cap K\right)\to 0,\epsilon \to 0.$$

(40)

Additionally, the third main result treats the case of the self-adjoint operators ${\mathcal{H}}_{\epsilon }$ and ${\mathcal{H}}_0$.

Theorem 7.

Let both operators ${\mathcal{H}}_0$ and ${\mathcal{H}}_{\epsilon }$ be self-adjoint and $K=[a,b]$ with some finite $a<b$. Then,

$${\mathrm{dist}}_{\mathrm{H}}\left(\overline{\sigma \left({\mathcal{H}}_0\right)}\cap K,\overline{\sigma \left({\mathcal{H}}_{\epsilon }\right)}\cap K\right)⩽\varkappa \left(\epsilon \right)$$

(41)

with the function $\varkappa \left(\epsilon \right)$ defined in (36). If a and b are in the resolvent set of ${\mathcal{H}}_0$, then they are also in the resolvent set of ${\mathcal{H}}_{\epsilon }$ for a sufficiently small ε and for the corresponding spectral projections that a convergence holds:

$$\parallel {\mathcal{P}}_{[a,b]}\left({\mathcal{H}}_{\epsilon }\right)-{\mathcal{P}}_{[a,b]}\left({\mathcal{H}}_0\right){\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)\to L_2(\mathrm{\Omega };{\mathbb{C}}^n)}\to 0,\epsilon \to 0.$$

(42)

2.7. Discussion

Here, we discuss the main features of the considered operators and the main advantages of our primary results. We consider four types of operators, and, in all four cases, these are general even-order operators with variable coefficients and small variable translations. The leading terms of the operators are classical in the sense that they involve no translations, and the latter are present only in the lower-order terms. The operators are introduced via the appropriate sesquilinear forms, and the coefficients in these forms satisfy the minimal smoothness conditions; namely, most of these coefficients are just essentially bounded. The assumption that the coefficients $A_{\alpha \beta }^i$ for $\left|\alpha \right|⩽m-1$ and $\left|\beta \right|=m$ should possess essentially bounded first derivatives is not technical; it is essentially employed in the proofs, and it seems to be unavoidable.

The main property of the translations is that they are small in the sense of condition (9), and, in fact, this is the only restriction that we impose. This allows us to consider general variable translations, which can arbitrarily depend on a multidimensional parameter, $\epsilon$. The considered operators involve a translation in each lower-order term, and each such term can involve several translations (see definitions (10) and (23) of the operators ${\mathcal{T}}_{\alpha \beta }^{\epsilon }$ and definition (16) of the operators ${\mathcal{Q}}_{\alpha \beta }^{\epsilon }$). Such an admissible structure of the lower-order terms describes the very wide classes of the differential–difference operators, to which our results apply.

The operators considered in Section 2.3 and Section 2.5 are subject to general boundary conditions on the boundary, and this issue is to be discussed separately. These boundary conditions are introduced in terms of two main objects: the domain $\mathfrak{W}$ of these forms and the boundary terms in forms (17) and (31), expressed as scalar products in $L_2(\partial \mathrm{\Omega };{\mathbb{C}}^n)$. The domain $\mathfrak{W}$ is determined by boundary conditions (18), and the total number K of these conditions as well as their coefficients can be arbitrary. However, we do not claim that all such conditions are independent and form a minimal (in an appropriate sense) set. For instance, if the number K is too big, it can turn out that conditions (18) are satisfied only for the functions from ${\mathring{W}}_2^m(\mathrm{\Omega };{\mathbb{C}}^n)$. However, as we show in Section 4, the set $\mathfrak{W}$ is a subspace of $W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)$ no matter how many or what conditions are prescribed in (18), and exactly this fact ensures the closedness of forms (17) and (31). Moreover, the coefficients $B_{\alpha }^{jk}$ can vanish on some subsets of $\partial \mathrm{\Omega }$, and, in this way, it is possible to introduce arbitrary mixed conditions, for instance, the Dirichlet condition on some subset of $\partial \mathrm{\Omega }$ and the Neumann condition on another.

Boundary terms in forms (17) and (31) are a natural way of prescribing the Robin condition. In our case, they involve small translations described by ${\psi }_{\alpha \beta }^i(x,\epsilon )$, and this makes these conditions nonlocal. Formal writing of the corresponding boundary operators can be obtained through the formal integration by parts, under which we move all derivatives from the function v to the function u in (17) and (31). However, once the order m is high, namely, as $m⩾2$, such an integration by parts can be performed in various ways, each producing its own boundary terms. For instance, the integration by parts in the integral $\underset{\mathrm{\Omega }}{\int }\frac{{\partial }^{2}u}{\partial x_1\partial x_2}\overline{v}dx$ can be performed in two ways. Namely, we can first move the derivative in $x_1$ to v, and then the derivative in $x_2$ goes, or we can interchange this order. By integrating in these two ways, we obtain two different sums for the boundary integrals, which, of course, should coincide, since the integrals over $\mathrm{\Omega }$ in these integrations are the same. One more gentle point is that even if a way of integration by parts is fixed, we finally obtain quite a lot of scalar products in $L_2(\partial \mathrm{\Omega };{\mathbb{C}}^n)$ of the function v and all its derivatives up to mth order with various differential expressions of u. However, we should not suppose that each differential expression of u should vanish. In each obtained scalar product the corresponding derivative of v should be rewritten in the local variables $(\tau ,s)$ mentioned in Section 2.1. Then the derivatives with respect to the variable s should be moved to the differential expressions of u through an appropriate integration by parts over $\partial \mathrm{\Omega }$. Finally, after all of these quite technically complicated integrations, we obtain the scalar products in $L_2(\partial \mathrm{\Omega };{\mathbb{C}}^n)$ of v and its derivatives in $\tau$ of up to the mth order with some new differential expressions of u. Exactly these expressions should be equated to zero.

Apart from inequality (9), we make no other conditions for the translations, but, depending on their structures, we distinguish different cases presented in Section 2.2, Section 2.3, Section 2.4 and Section 2.5. In the first case, we consider the operators in the entire space. Then, for any translation ${\varphi }_{\alpha \beta }^i(x,\epsilon )$, the mapping $x\mapsto x+{\varphi }_{\alpha \beta }^i(x,\epsilon )$ is a diffeomorphism of ${\mathbb{R}}^d$ onto itself. In Section 2.3, we consider operators in some domain, $\mathrm{\Omega }$, but we still suppose that these mappings are diffeomorphisms of $\mathrm{\Omega }$ onto itself. The situation in which this condition is violated is treated in Section 2.4 and Section 2.5. In all cases, we establish that the considered operators converge in the norm resolvent sense to the limiting operators corresponding to zero translations; we choose that the operator norm for these operators is from $L_2$ into $W_2^m$. This is the best possible norm for the considered classes of the operators since the coefficients $A_{\alpha \beta }$ and $A_{\alpha \beta }^i$ are only essentially bounded, and this is why, according to the first representation theorem, we can just say that the domains of the considered operators are subsets of $W_2^m$ and that no additional smoothness improvements are possible. One of the advantages of our main results is that we can prove the norm resolvent convergence in the best possible norm under the minimal smoothness conditions for the coefficients. In addition, inequalities (15), (20), (27), and (32) also provide estimates for the convergence rates. As these inequalities show, the convergence rate depends on the structure of the considered operators. According to (15) and (20), if the mappings $x\mapsto x+{\varphi }_{\alpha \beta }^i(x,\epsilon )$ are diffeomorphisms of the domain where the operator is considered onto itself, then the convergence rate is $\mu \left(\epsilon \right)$. The same is true when the mentioned mappings are violated, provided the operator is subject to the Dirichlet condition. If the mappings $x\mapsto x+{\varphi }_{\alpha \beta }^i(x,\epsilon )$ change the domain $\mathrm{\Omega }$, and, on the boundary, we impose general conditions, then the convergence rate becomes worse (see inequality (32)). Moreover, here, the convergence rate is specific to the one-dimensional case.

Apart from the resolvent convergence and corresponding operator estimates, we also establish the spectral and pseudospectral convergence. Theorem 5 states that the spectra of the perturbed operators are located in specific neighborhoods of the spectra of the limiting operators, and vice versa; the limiting spectra are located in specific neighborhoods of the perturbed ones, as shown in (35). It should be stressed that the mentioned neighborhoods are the $ϵ$-pseudospectra defined in (4), and their shapes can be quite complicated for general non-self-adjoint operators [62]. However, this follows from the definition of the $ϵ$-pseudospectrum that, for an arbitrary closed operator, $\mathcal{A}$, and a compact set in the complex plane, we have ([62], Ch. I, Sect. 2, Equation (2.5))

$$\sigma \left(\mathcal{A}\right)=\bigcup _{ϵ>0}{\sigma }_{ϵ}\left(\mathcal{A}\right).$$

In view of this identity and the fact that $\varkappa \left(\epsilon \right)\to 0$ as $\epsilon \to 0$, the first inclusion in (35) means that the spectrum of the perturbed operator ${\mathcal{H}}_{\epsilon }$, located in the compact set K, converges to the spectrum of the limiting operator ${\mathcal{H}}_0$, located in the same compact set. At the same time, the convergence rate highly depends on the behavior of the resolvent ${({\mathcal{H}}_0-{\lambda }_0)}^{-1}$ near the limiting spectrum, and this behavior can be very complicated once the operator ${\mathcal{H}}_0$ is non-self-adjoint. If both operators ${\mathcal{H}}_0$ and ${\mathcal{H}}_{\epsilon }$ are self-adjoint, then the behavior of their resolvents is described by classical results in the spectral theory, which imply the convergence of the perturbed spectrum to the limiting one in the sense of the Hausdorff distance; see (41). In addition, in this case, the standard behavior of the spectral projectors holds; see (42).

For general non-self-adjoint operators, other important (pseudo)spectral characteristics are the $ϵ$-pseudospectrum and $(p,ϵ)$-pseudospectrum, which are defined in (4). The natural question of how these sets of operators ${\mathcal{H}}_{\epsilon }$ behave as $\epsilon$ goes to zero is answered in Theorem 6. Their convergence is established in the sense of the Hausdorff distance, cf. (39) and (40), based on quite recent general results from [60]; see also [64].

3. Convergence for Operators in Entire Space

In this section, we prove Theorem 1. We begin with the existence of the number ${\lambda }_0$ described in the formulation of the theorem.

First, we are going to confirm that both forms ${\mathfrak{h}}_{\epsilon }$ and ${\mathfrak{h}}_0$ are closed and sectorial. Estimate (8) and the boundedness of the coefficients $A_{\alpha \beta }$ imply:

$$\begin{array}{cc}\hfill & Re\sum _{\left|\alpha \right|=\left|\beta \right|=m}{\left(A_{\alpha \beta }\frac{{\partial }^{\beta }u}{\partial x^{\beta }},\frac{{\partial }^{\alpha }u}{\partial x^{\alpha }}\right)}_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}⩾c_0\sum _{\left|\alpha \right|=m}\parallel \frac{{\partial }^{\alpha }u}{\partial x^{\alpha }}{\parallel }_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}^2,\hfill \\ \hfill & Re\sum _{\left|\alpha \right|=\left|\beta \right|=m}{\left(A_{\alpha \beta }\frac{{\partial }^{\beta }u}{\partial x^{\beta }},\frac{{\partial }^{\alpha }u}{\partial x^{\alpha }}\right)}_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}⩽C\sum _{\left|\alpha \right|=m}\parallel \frac{{\partial }^{\alpha }u}{\partial x^{\alpha }}{\parallel }_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}^2,\hfill \end{array}$$

(43)

where C is a fixed constant independent of u. Let $\varphi =\varphi (x,\epsilon )$ be a vector-valued function defined for $x\in {\mathbb{R}}^d$ and a sufficiently small $\epsilon$, taking values in ${\mathbb{C}}^d$ and obeying an estimate similar to (9):

$$|\varphi (x,\epsilon ){|}_{{\mathbb{C}}^n}+\sum _{j=1}^d|\frac{\partial \varphi }{\partial x_j}(x,\epsilon ){|}_{{\mathbb{C}}^n}⩽\mu \left(\epsilon \right).$$

(44)

In terms of this function, in the space $L_2({\mathbb{R}}^d;{\mathbb{C}}^n)$, we introduce an operator similar to ${\mathcal{T}}_{\alpha \beta }^{\epsilon ,i}$:

$$\left({\mathcal{T}}^{\epsilon }u\right)\left(x\right):=A\left(x\right)u(x+\varphi (x,\epsilon )),$$

(45)

where $A\in L_{\infty }({\mathbb{R}}^d;{\mathbb{M}}_n)$ is some matrix-valued function. Then, changing the variables $y=x+\varphi (x,\epsilon )$ in the following integrals, for an arbitrary $u\in L_2({\mathbb{R}}^d;{\mathbb{C}}^n)$, we obtain:

$$\parallel {\mathcal{T}}^{\epsilon }{u\parallel }_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}^2⩽{\parallel A\parallel }_{L_{\infty }({\mathbb{R}}^d;{\mathbb{M}}_n)}^2\underset{{\mathbb{R}}^d}{\int }{\left|u\left(y\right)\right|}_{{\mathbb{C}}^n}^2{J}_1(y,\epsilon )dy,$$

(46)

where $J_1(y,\epsilon )$ is the Jacobian associated with the above change:

$$J_1(y,\epsilon )={|det(E_d+{\nabla }_x\varphi (x,\epsilon ))|}^{-1},$$

and $E_d$ is the unit matrix of a size $d\times d$. It follows from (9) that

$$|J_1(y,\epsilon )-1|⩽C\mu \left(\epsilon \right),$$

(47)

where C is an absolute constant independent of y and $\epsilon$. This estimate applied with ${\mathcal{T}}^{\epsilon }={\mathcal{T}}_{\alpha \beta }^{\epsilon ,i}$ and (46) yield that

$$\parallel {\mathcal{T}}_{\alpha \beta }^{\epsilon }{u\parallel }_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}^2⩽C{\parallel u\parallel }_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}^2,$$

(48)

where a constant C is independent of $\epsilon$, u, $\alpha$, $\beta$, and i. Using this estimate, the Cauchy–Schwarz inequality, the uniform boundedness of the coefficients $A_{\alpha \beta }$, and standard interpolation inequalities, we estimate the lower-order terms in the form ${\mathfrak{h}}_{\epsilon }$:

$$\begin{array}{cc}\hfill |\sum _{\begin{array}{c}\left|\alpha \right|,\left|\beta \right|⩽m\\ \left|\alpha \right|+\left|\beta \right|⩽2m-1\end{array}}{(-1)}^{\left|\alpha \right|}& {\left(A_{\alpha \beta }{\mathcal{T}}_{\alpha \beta }^{\epsilon }\frac{{\partial }^{\beta }u}{\partial x^{\beta }},\frac{{\partial }^{\alpha }v}{\partial x^{\alpha }}\right)}_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}|\hfill \\ \hfill ⩽& C\sum _{\begin{array}{c}\left|\alpha \right|,\left|\beta \right|⩽m\\ \left|\alpha \right|+\left|\beta \right|⩽2m-1\end{array}}\parallel {\mathcal{T}}_{\alpha \beta }^{\epsilon }\frac{{\partial }^{\beta }u}{\partial x^{\beta }}{\parallel }_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}\parallel \frac{{\partial }^{\alpha }u}{\partial x^{\alpha }}{\parallel }_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}\hfill \\ \hfill ⩽& C\sum _{\begin{array}{c}\left|\alpha \right|,\left|\beta \right|⩽m\\ \left|\alpha \right|+\left|\beta \right|⩽2m-1\end{array}}\parallel \frac{{\partial }^{\beta }u}{\partial x^{\beta }}{\parallel }_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}\parallel \frac{{\partial }^{\alpha }u}{\partial x^{\alpha }}{\parallel }_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}\hfill \\ \hfill ⩽& \frac{c_0}{3}\sum _{\left|\alpha \right|=m}\parallel \frac{{\partial }^{\alpha }u}{\partial x^{\alpha }}{\parallel }_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}^2+C{\parallel u\parallel }_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}^2,\hfill \end{array}$$

(49)

where C are some positive constants independent of u and $\epsilon$. It follows from these estimates and from (43) that

$$\begin{array}{cc}\hfill & Re{\mathfrak{h}}_{\epsilon }(u,u)⩾\frac{2c_0}{3}\sum _{\left|\alpha \right|=m}\parallel \frac{{\partial }^{\alpha }u}{\partial x^{\alpha }}{\parallel }_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}^2-c_1{\parallel u\parallel }_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}^2,\hfill \\ \hfill & |Im{\mathfrak{h}}_{\epsilon }(u,u)|⩽c_2\sum _{\left|\alpha \right|=m}\parallel \frac{{\partial }^{\alpha }u}{\partial x^{\alpha }}{\parallel }_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}^2+c_3{\parallel u\parallel }_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}^2,\hfill \end{array}$$

(50)

where $c_1$, $c_2$, and $c_3$ are some positive constants independent of $\epsilon$ and u. Using the former of the obtained estimates, it is easy to confirm that the form ${\mathfrak{h}}_{\epsilon }$ is closed, while both these estimates imply that numerical range of the form ${\mathfrak{h}}_{\epsilon }$ is contained in the sector

$$\left\{z\in \mathbb{C}:|Imz|⩽\frac{3c_2}{2c_0}Rez+\frac{3c_1{c}_2}{2c_0}+c_3\right\}.$$

(51)

Hence, the form ${\mathfrak{h}}_{\epsilon }$ is also sectorial. This allows us to apply the first representation theorem as it was described in Section 2.2 and to define the operator ${\mathcal{H}}_{\epsilon }$. In particular, the spectrum of this operator is contained in sector (51). The form ${\mathfrak{h}}_0$ is a particular case of the form ${\mathfrak{h}}_{\epsilon }$ corresponding to the case $\epsilon =0$, and, hence, this form is also sectorial and closed. It defines the operator ${\mathcal{H}}_0$, and the spectrum of the latter is contained in sector (51). This proves the existence of the number ${\lambda }_0$ in the formulation of the theorem; in particular, it is sufficient to take

$${\lambda }_0<-c_1-\frac{2c_0{c}_3}{3c_2}.$$

We also observe that as $Re\lambda ⩽{\lambda }_0$, it follows from the first estimate in (50) and standard interpolation inequalities that

$$\begin{array}{cc}\hfill Re\left({\mathfrak{h}}_{\epsilon }(u,u)-\lambda {\parallel u\parallel }_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}^2\right)⩾& \frac{2c_0}{3}\sum _{\left|\alpha \right|=m}\parallel \frac{{\partial }^{\alpha }u}{\partial x^{\alpha }}{\parallel }_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}^2+\frac{2c_0{c}_3}{3c_2}{c}_1{\parallel u\parallel }_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}^2\hfill \\ \hfill ⩾& c_4{\parallel u\parallel }_{W_2^m({\mathbb{R}}^d;{\mathbb{C}}^n)}^2,\hfill \end{array}$$

(52)

where $c_4$ is some fixed positive constant independent of u and $\epsilon$. The same estimate is also true for the form ${\mathfrak{h}}_0$.

We proceed to proving the resolvent convergence. The proof is based on the following auxiliary lemma.

Lemma 1.

For all $u\in W_2^1({\mathbb{R}}^d;{\mathbb{C}}^n)$, the estimate

$$\parallel {\mathcal{T}}^{\epsilon }{u-Au\parallel }_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}⩽C\mu \left(\epsilon \right){\parallel u\parallel }_{W_2^1({\mathbb{R}}^d;{\mathbb{C}}^n)}$$

(53)

holds, where C is some constant independent of u and ε.

Proof. 

We choose an arbitrary $u\in W_2^1({\mathbb{R}}^d;{\mathbb{C}}^n)$ and, for almost each $x\in {\mathbb{R}}^d$, we have:

$$u(x+\varphi (x,\epsilon ))-u\left(x\right)=\underset{0}{\overset{1}{\int }}\frac{d}{dt}u(x+t\varphi (x,\epsilon ))dt=\sum _{j=1}^d\underset{0}{\overset{1}{\int }}\frac{\partial u}{\partial x_j}(x+t\varphi (x,\epsilon )){\varphi }_j(x,\epsilon )dt,$$

(54)

where ${\varphi }_j$ are the components of $\varphi$, that is, $\varphi =({\varphi }_1,\dots ,{\varphi }_d)$. Then, by the Cauchy–Schwarz inequality, we obtain:

$$\begin{array}{cc}\hfill |u(x+\varphi (x,\epsilon ))-u\left(x\right){|}_{{\mathbb{C}}^n}^2⩽& \sum _{j=1}^d\underset{0}{\overset{1}{\int }}|\frac{\partial u}{\partial x_j}(x+t\varphi (x,\epsilon )){|}_{{\mathbb{C}}^n}^{2}dt\underset{0}{\overset{1}{\int }}{\left|\varphi (x,\epsilon )\right|}_{{\mathbb{C}}^n}^{2}dt\hfill \\ \hfill ⩽& {\mu }^2\left(\epsilon \right)\sum _{j=1}^d\underset{0}{\overset{1}{\int }}|\frac{\partial u}{\partial x_j}(x+t\varphi (x,\epsilon )){|}_{{\mathbb{C}}^n}^{2}dt.\hfill \end{array}$$

(55)

We integrate the obtained inequality over $x\in {\mathbb{R}}^d$ and make the change in variables $y=x+t\varphi (x,\epsilon )$:

$$\begin{array}{cc}\hfill \underset{{\mathbb{R}}^d}{\int }|u(x+\varphi (x,\epsilon ))-u\left(x\right){|}_{{\mathbb{C}}^n}^{2}dx⩽& {\mu }^2\left(\epsilon \right)\sum _{j=1}^d\underset{0}{\overset{1}{\int }}dt\underset{{\mathbb{R}}^d}{\int }|\frac{\partial u}{\partial x_j}(x+t\varphi (x,\epsilon )){|}_{{\mathbb{C}}^n}^{2}dx\hfill \\ \hfill =& {\mu }^2\left(\epsilon \right)\sum _{j=1}^d\underset{0}{\overset{1}{\int }}dt\underset{{\mathbb{R}}^d}{\int }|\frac{\partial u}{\partial x_j}\left(y\right){|}_{{\mathbb{C}}^n}^2{J}_t(y,\epsilon )dy,\hfill \end{array}$$

(56)

where $J_t$ is the Jacobian associated with the completed change in variables:

$$J_t(y,\epsilon )={|det(E_d+t{\nabla }_x\varphi (x,\epsilon ))|}^{-1}.$$

(57)

Inequality (9) implies that the Jacobian $J_t$ also satisfies estimate (47) with some constant C independent of y, t, and $\epsilon$. Substituting this estimate into (56), we obtain

$${\parallel u(·+\varphi (·,\epsilon ))-u\parallel }_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}⩽C\mu \left(\epsilon \right){\parallel u\parallel }_{W_2^1({\mathbb{R}}^d;{\mathbb{C}}^n)},$$

where C is some constant independent of $\epsilon$ and u. Hence,

$$\parallel {\mathcal{T}}^{\epsilon }{u-Au\parallel }_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}⩽{C\mu \left(\epsilon \right)\parallel A\parallel }_{L_{\infty }({\mathbb{R}}^d;{\mathbb{M}}_n)}{\parallel u\parallel }_{W_2^1({\mathbb{R}}^d;{\mathbb{C}}^n)}$$

and this implies (53). The proof is complete. □

We fix $\lambda \in \mathbb{C}$, obeying the inequality $Re\lambda ⩽{\lambda }_0$, and choose an arbitrary $f\in L_2({\mathbb{R}}^d;{\mathbb{C}}^n)$. Then, we denote

$$u_{\epsilon }:={({\mathcal{H}}_{\epsilon }-\lambda )}^{-1}f,u_0:={({\mathcal{H}}_0-\lambda )}^{-1}f.$$

(58)

These functions satisfy the identities

$$\begin{array}{cc}\hfill & {\mathfrak{h}}_{\epsilon }(u_{\epsilon },v)-\lambda {(u_{\epsilon },v)}_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}={(f,v)}_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)},\hfill \\ \hfill & {\mathfrak{h}}_0(u_0,v)-\lambda {(u_0,v)}_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}={(f,v)}_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}\hfill \end{array}$$

(59)

for all $v\in W_2^m({\mathbb{R}}^d;{\mathbb{C}}^n)$. Then, by letting $v=v_{\epsilon }:=u_{\epsilon }-u_0$ and calculating the difference of the above identities, we immediately obtain:

$${\mathfrak{h}}_{\epsilon }(v_{\epsilon },v_{\epsilon })-\lambda {\parallel v_{\epsilon }\parallel }_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}^2={\mathfrak{h}}_0(u_0,v_{\epsilon })-{\mathfrak{h}}_{\epsilon }(u_0,v_{\epsilon }).$$

(60)

Taking the real part of this identity and using (52), we find:

$$c_4\parallel v_{\epsilon }{\parallel }_{W_2^m({\mathbb{R}}^d;{\mathbb{C}}^n)}^2⩽|{\mathfrak{h}}_{\epsilon }(u_0,v_{\epsilon })-{\mathfrak{h}}_0(u_0,v_{\epsilon })|.$$

(61)

It follows from definitions (12) and (14) of the forms ${\mathfrak{h}}_{\epsilon }$ and ${\mathfrak{h}}_0$ that

$$\begin{array}{cc}\hfill {\mathfrak{h}}_0(u_0,v_{\epsilon })-{\mathfrak{h}}_{\epsilon }(u_0,v_{\epsilon })=& -\sum _{\begin{array}{c}\left|\alpha \right|,\left|\beta \right|⩽m\\ \left|\alpha \right|+\left|\beta \right|⩽2m-1\end{array}}\sum _{i=1}^{N_{\alpha \beta }}{(-1)}^{\left|\alpha \right|}{\left(({\mathcal{T}}_{\alpha \beta }^{\epsilon ,i}-A_{\alpha \beta }^i)\frac{{\partial }^{\beta }{u}_0}{\partial x^{\beta }},\frac{{\partial }^{\alpha }{v}_{\epsilon }}{\partial x^{\alpha }}\right)}_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}\hfill \\ \hfill =& S_1+S_2,\hfill \end{array}$$

(62)

where

$$\begin{array}{cc}\hfill & S_1:=-\sum _{\begin{array}{c}\left|\alpha \right|⩽m-1\\ \left|\beta \right|⩽m\end{array}}\sum _{i=1}^{N_{\alpha \beta }}{(-1)}^{\left|\alpha \right|}{\left(({\mathcal{T}}_{\alpha \beta }^{\epsilon ,i}-A_{\alpha \beta }^i)\frac{{\partial }^{\beta }{u}_0}{\partial x^{\beta }},\frac{{\partial }^{\alpha }{v}_{\epsilon }}{\partial x^{\alpha }}\right)}_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)},\hfill \\ \hfill & S_2:=-\sum _{\begin{array}{c}\left|\alpha \right|=m\\ \left|\beta \right|⩽m-1\end{array}}\sum _{i=1}^{N_{\alpha \beta }}{(-1)}^{\left|\alpha \right|}{\left(({\mathcal{T}}_{\alpha \beta }^{\epsilon ,i}-A_{\alpha \beta }^i)\frac{{\partial }^{\beta }{u}_0}{\partial x^{\beta }},\frac{{\partial }^{\alpha }{v}_{\epsilon }}{\partial x^{\alpha }}\right)}_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}.\hfill \end{array}$$

We estimate the first sum on the right-hand side of the obtained formula by using Lemma 1 with ${\mathcal{T}}^{\epsilon }={\mathcal{T}}_{\alpha \beta }^{\epsilon }$:

$$|S_1|⩽C\mu \left(\epsilon \right)\parallel u_0{\parallel }_{W_2^m({\mathbb{R}}^d;{\mathbb{C}}^n)}{\parallel v_{\epsilon }\parallel }_{W_2^m({\mathbb{R}}^d;{\mathbb{C}}^n)},$$

(63)

where C is some constant independent of $\epsilon$, $u_0$, and $v_{\epsilon }$.

In order to estimate the sum $S_2$, we substitute the definition of the operator ${\mathcal{T}}_{\alpha \beta }^{\epsilon }$ from (10), and then, in each obtained term, we make the change in variables $y=x+{\varphi }_{\alpha \beta }^i(x,\epsilon )$:

$${\left(({\mathcal{T}}_{\alpha \beta }^{\epsilon ,i}-A_{\alpha \beta }^i)\frac{{\partial }^{\beta }{u}_0}{\partial x^{\beta }},\frac{{\partial }^{\alpha }{v}_{\epsilon }}{\partial x^{\alpha }}\right)}_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}={\left(\frac{{\partial }^{\beta }{u}_0}{\partial x^{\beta }},{\left({\mathcal{T}}_{\alpha \beta }^{\epsilon ,i}-A_{\alpha \beta }^i\right)}^{*}\frac{{\partial }^{\alpha }{v}_{\epsilon }}{\partial x^{\alpha }}\right)}_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)},$$

(64)

where

$$\left({\left({\mathcal{T}}_{\alpha \beta }^{\epsilon ,i}\right)}^{*}u\right)\left(y\right)=J_1(y,\epsilon )A_{\alpha \beta }^{*}\left(x(y,\epsilon )\right)u\left(x(y,\epsilon )\right),$$

and $x(y,\epsilon )$ is the inverse change in the variables. It is clear that the inverse change reads as $x=y+{\tilde{\varphi }}_{\alpha \beta }^i(x,\epsilon )$, where the function ${\tilde{\varphi }}_{\alpha \beta }^i$ possesses the same properties as the function ${\varphi }_{\alpha \beta }^i$. Hence, since $A_{\alpha \beta }\in W_{\infty }^1({\mathbb{R}}^d;{\mathbb{M}}_n)$, it is clear that the operator ${\left({\mathcal{T}}_{\alpha \beta }^{\epsilon ,i}\right)}^{*}$ is of the same structure as ${\mathcal{T}}_{\alpha \beta }^{\epsilon ,i}$. Therefore, Lemma 1 can be also applied to the functions ${\left({\mathcal{T}}_{\alpha \beta }^{\epsilon ,i}-A_{\alpha \beta }^i\right)}^{*}u$ while maintaining estimate (53). Applying this estimate to Formula (64) and using inequality (47), we obtain:

$$\left|{\left(\frac{{\partial }^{\beta }{u}_0}{\partial x^{\beta }},{\left({\mathcal{T}}_{\alpha \beta }^{\epsilon ,i}-A_{\alpha \beta }\right)}^{*}\frac{{\partial }^{\alpha }{v}_{\epsilon }}{\partial x^{\alpha }}\right)}_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}\right|⩽C\mu \left(\epsilon \right)\parallel u_0{\parallel }_{W_2^m({\mathbb{R}}^d;{\mathbb{C}}^n)}{\parallel v_{\epsilon }\parallel }_{W_2^m({\mathbb{R}}^d;{\mathbb{C}}^n)},$$

(65)

where C is a constant independent of $\epsilon$, $\alpha$, $\beta$, i, $u_0$, and $v_{\epsilon }$. Summing up the obtained estimate over i, $\alpha$, and $\beta$, we arrive at a similar inequality for $S_2$:

$$|S_2|⩽C\mu \left(\epsilon \right)\parallel u_0{\parallel }_{W_2^m({\mathbb{R}}^d;{\mathbb{C}}^n)}{\parallel v_{\epsilon }\parallel }_{W_2^m({\mathbb{R}}^d;{\mathbb{C}}^n)},$$

(66)

where a constant C is independent of $\epsilon$, $u_0$, and $v_{\epsilon }$. This inequality alongisde (63) and (62) allow us to estimate the right-hand side of (61), and this gives:

$$\parallel v_{\epsilon }{\parallel }_{W_2^m({\mathbb{R}}^d;{\mathbb{C}}^n)}⩽C\mu \left(\epsilon \right){\parallel u_0\parallel }_{W_2^m({\mathbb{R}}^d;{\mathbb{C}}^n)},$$

(67)

where a constant C is independent of $\epsilon$, $u_0$, and $v_{\epsilon }$. It also follows from the definition of the function $u_0$, identity (59), and estimate (52) for $\epsilon =0$ that

$$\parallel u_0{\parallel }_{W_2^m({\mathbb{R}}^d;{\mathbb{C}}^n)}⩽C{\parallel f\parallel }_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}$$

with a constant C independent of f. This estimate and (67) yield

$$\parallel u_{\epsilon }-u_0{\parallel }_{W_2^m({\mathbb{R}}^d;{\mathbb{C}}^n)}=\parallel v_{\epsilon }{\parallel }_{W_2^m({\mathbb{R}}^d;{\mathbb{C}}^n)}⩽C\mu \left(\epsilon \right){\parallel f\parallel }_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}$$

with a constant C independent of f and $\epsilon$. The obtained inequality is equivalent to (15). The proof of Theorem 1 is complete.

4. Convergence for Operators with Almost Identity Diffeomorphisms

In this section, we prove Theorem 2. We begin by checking the sectoriality and closedness of the forms ${\mathfrak{h}}_{\epsilon }$ and ${\mathfrak{h}}_0$. Ellipticity condition (8) implies estimates similar to (43):

$$\begin{array}{cc}\hfill & Re\sum _{\left|\alpha \right|=\left|\beta \right|=m}{\left(A_{\alpha \beta }\frac{{\partial }^{\beta }u}{\partial x^{\beta }},\frac{{\partial }^{\alpha }u}{\partial x^{\alpha }}\right)}_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}⩾c_0\sum _{\left|\alpha \right|=m}\parallel \frac{{\partial }^{\alpha }u}{\partial x^{\alpha }}{\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}^2,\hfill \\ \hfill & Re\sum _{\left|\alpha \right|=\left|\beta \right|=m}{\left(A_{\alpha \beta }\frac{{\partial }^{\beta }u}{\partial x^{\beta }},\frac{{\partial }^{\alpha }u}{\partial x^{\alpha }}\right)}_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}⩽C\sum _{\left|\alpha \right|=m}\parallel \frac{{\partial }^{\alpha }u}{\partial x^{\alpha }}{\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}^2,\hfill \end{array}$$

(68)

where C is a fixed constant independent of u.

Let $\varphi =\varphi (x,\epsilon )$ be a vector function defined for $x\in \mathrm{\Omega }$ and a sufficiently small $\epsilon$, taking values in ${\mathbb{C}}^d$ and obeying estimate (44). We again introduce the operator ${\mathcal{T}}^{\epsilon }$ by Formula (45), but, this time, $A\in L_{\infty }(\mathrm{\Omega };{\mathbb{M}}_n)$, and the operator ${\mathcal{T}}^{\epsilon }$ is considered in $L_2(\mathrm{\Omega };{\mathbb{C}}^n)$. Proceeding then as in (46), we easily prove an estimate similar to (48):

$$\parallel {\mathcal{T}}^{\epsilon }{u\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}^2⩽C{\parallel u\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}^2,$$

(69)

which then implies an inequality similar to (49):

$$\begin{array}{cc}\hfill \left|\sum _{\begin{array}{c}\left|\alpha \right|,\left|\beta \right|⩽m\\ \left|\alpha \right|+\left|\beta \right|⩽2m-1\end{array}}{(-1)}^{\left|\alpha \right|}{\left(A_{\alpha \beta }{\mathcal{T}}_{\alpha \beta }^{\epsilon }\frac{{\partial }^{\beta }u}{\partial x^{\beta }},\frac{{\partial }^{\alpha }v}{\partial x^{\alpha }}\right)}_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}\right|⩽& \frac{c_0}{3}\sum _{\left|\alpha \right|=m}\parallel \frac{{\partial }^{\alpha }u}{\partial x^{\alpha }}{\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}^2+c_5{\parallel u\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}^2,\hfill \end{array}$$

(70)

where $c_5$ is some positive constant independent of u and $\epsilon$.

To proceed further and estimate the boundary terms in the form ${\mathfrak{h}}_{\epsilon }$, we need an additional lemma.

Lemma 2.

For each fixed $\delta >0$, there exists a constant $C\left(\delta \right)>0$ independent of ε such that for all $u\in W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)$ the estimate holds:

$${\parallel u(·+\varphi (·,\epsilon ))\parallel }_{L_2(\partial \mathrm{\Omega };{\mathbb{C}}^n)}⩽{\delta \parallel \nabla u\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}+C\left(\delta \right){\parallel u\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}.$$

(71)

Proof. 

Let $\chi =\chi \left(\tau \right)$ be an infinitely differentiable function such that $\chi \left(\tau \right)=0$ as $\left|\tau \right|>\frac{{\tau }_0}{2}$ and $\chi \left(\tau \right)=\tau$ as $\left|\tau \right|<\frac{{\tau }_0}{3}$. Since the variable $\tau$ is well defined in ${\mathrm{\Pi }}_{{\tau }_0}$, we regard the function $\chi \left(\tau \right)$ as a function of the variables x; we recall that the set ${\mathrm{\Pi }}_{{\tau }_0}$ was defined in (1). Due to our assumptions about the boundary $\partial \mathrm{\Omega }$, the functions ${\mathrm{\Delta }}_x\chi \left(\tau \right)$ and ${\nabla }_x\chi \left(\tau \right)$ are uniformly bounded in ${\mathrm{\Pi }}_{{\tau }_0}$, where ${\mathrm{\Delta }}_x$ is the Laplace operator in the variables x and ${\nabla }_x$ is the gradient. Then, integrating by parts, for an arbitrary scalar function $u\in W_2^1\left(\mathrm{\Omega }\right)$, we have:

$$\begin{array}{cc}\hfill \underset{{\mathrm{\Pi }}_{{\tau }_0}}{\int }{\left|u\right|}^2{\mathrm{\Delta }}_x\chi \left(\tau \right)dx=& \underset{\partial \mathrm{\Omega }}{\int }{\left|u\right|}^2\frac{\partial \chi }{\partial \tau }ds-\underset{\mathrm{\Omega }}{\int }{\nabla }_x\chi \left(\tau \right)·\nabla {\left|u\left(x\right)\right|}^{2}dx\hfill \\ \hfill =& \underset{\partial \mathrm{\Omega }}{\int }{\left|u\right|}^{2}ds-\underset{{\mathrm{\Pi }}_{{\tau }_0}}{\int }{\nabla }_x\chi \left(\tau \right)·\nabla {\left|u\left(x\right)\right|}^{2}dx\hfill \end{array}$$

(72)

and, due to the aforementioned boundedness and the Cauchy–Schwarz inequality, this implies the estimates:

$${\parallel u\parallel }_{L_2(\partial \mathrm{\Omega })}^2⩽C\left({\parallel u\parallel }_{L_2\left(\mathrm{\Omega }\right)}^2+{\parallel u\parallel }_{L_2\left(\mathrm{\Omega }\right)}{\parallel \nabla u\parallel }_{L_2\left(\mathrm{\Omega }\right)}\right)⩽{\delta \parallel \nabla u\parallel }_{L_2\left(\mathrm{\Omega }\right)}^2+C\left(\delta \right){\parallel u\parallel }_{L_2\left(\mathrm{\Omega }\right)}^2,$$

where $\delta >0$ is arbitrary and fixed and $C\left(\delta \right)>0$ is positive and independent of u. We then apply this estimate to the function $u(x+\varphi (x,\epsilon \left)\right)$ and use estimate (69) for the operator ${\mathcal{T}}^{\epsilon }$, defined in (45), with A being the identity matrix:

$${\parallel u(·+\varphi (·,\epsilon ))\parallel }_{L_2(\partial \mathrm{\Omega })}^2⩽C\left({\parallel u\parallel }_{L_2\left(\mathrm{\Omega }\right)}^2+{\parallel u\parallel }_{L_2\left(\mathrm{\Omega }\right)}{\parallel \nabla u\parallel }_{L_2\left(\mathrm{\Omega }\right)}\right)⩽{\delta \parallel \nabla u\parallel }_{L_2\left(\mathrm{\Omega }\right)}^2+C\left(\delta \right){\parallel u\parallel }_{L_2\left(\mathrm{\Omega }\right)}^2.$$

This proves (71) and completes the proof. □

Applying the proven lemma to all terms in the third sum in definition (17) of the form ${\mathfrak{h}}_{\epsilon }$ with a sufficiently small fixed $\delta >0$, we obtain:

$$\left|\sum _{\left|\alpha \right|,\left|\beta \right|⩽m-1}{\left({\mathcal{Q}}_{\alpha \beta }^{\epsilon }\frac{{\partial }^{\beta }u}{\partial x^{\beta }},\frac{{\partial }^{\alpha }v}{\partial x^{\alpha }}\right)}_{L_2(\partial \mathrm{\Omega };{\mathbb{C}}^n)}\right|⩽\frac{c_0}{3}\sum _{\left|\alpha \right|=m}\parallel \frac{{\partial }^{\alpha }u}{\partial x^{\alpha }}{\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}^2+C{\parallel u\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}^2,$$

(73)

where C is some positive constant independent of u and $\epsilon$. This estimate and (68) and (70) yield

$$\begin{array}{cc}\hfill & Re{\mathfrak{h}}_{\epsilon }(u,u)⩾\frac{c_0}{3}\sum _{\left|\alpha \right|=m}\parallel \frac{{\partial }^{\alpha }u}{\partial x^{\alpha }}{\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}^2-c_6{\parallel u\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}^2,\hfill \\ \hfill & |Im{\mathfrak{h}}_{\epsilon }(u,u)|⩽c_7\sum _{\left|\alpha \right|=m}\parallel \frac{{\partial }^{\alpha }u}{\partial x^{\alpha }}{\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}^2+c_8{\parallel u\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}^2,\hfill \end{array}$$

(74)

where $c_6$, $c_7$, and $c_8$ are some positive constants independent of $\epsilon$ and u. Proceeding, then, as in the previous section, we see that both forms ${\mathfrak{h}}_{\epsilon }$ and ${\mathfrak{h}}_0$ are sectorial; their numerical ranges are contained in some fixed sector independent of $\epsilon$ and

$$\begin{array}{cc}\hfill & Re\left({\mathfrak{h}}_{\epsilon }(u,u)-\lambda {\parallel u\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}^2\right)⩾c_9{\parallel u\parallel }_{W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)}^2,\hfill \\ \hfill & Re\left({\mathfrak{h}}_0(u,u)-\lambda {\parallel u\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}^2\right)⩾c_9{\parallel u\parallel }_{W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)}^2,\hfill \end{array}$$

(75)

where $c_9$ is a fixed positive constant independent of u and $\epsilon$ and $Re\lambda ⩽{\lambda }_0$ for some fixed ${\lambda }_0$. These estimates also imply that the forms ${\mathfrak{h}}_{\epsilon }$ and ${\mathfrak{h}}_0$ are closed. Indeed, if $u_p\in \mathfrak{D}\left({\mathfrak{h}}_{\epsilon }\right)$ is an arbitrary sequence such that $u_p\to u_{*}$ in $L_2(\mathrm{\Omega };{\mathbb{C}}^n)$ and ${\mathfrak{h}}_{\epsilon }(u_p-u_q,u_p-u_q)\to 0$ as $p,q\to \infty$, the above estimates yield that $u_p$ converges to $u_{*}$ in $W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)$. Then, Lemma 2 allows us to pass to the limit as $p\to \infty$ in boundary conditions (18) for $u_p$, and we obtain the same conditions for $u_{*}$. Hence, the form ${\mathfrak{h}}_{\epsilon }$ is closed. The proof for the form ${\mathfrak{h}}_0$ is similar. Thus, by the first representation theorem, the operators ${\mathcal{H}}_{\epsilon }$ and ${\mathcal{H}}_0$ are well defined.

We proceed to the proof of the convergence. We shall need an analog of Lemma 1.

Lemma 3.

For all $u\in W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)$, the estimate

$$\parallel {\mathcal{T}}^{\epsilon }{u-Au\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}⩽C\mu \left(\epsilon \right){\parallel u\parallel }_{W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)}$$

(76)

holds, where C is some constant independent of u and ε.

Proof. 

We continue u outside $\mathrm{\Omega }$, namely, we let:

$$u\left(x\right):=\left\{\begin{array}{cccc}\hfill u(-\tau ,& s\left)\chi \right(\tau )\hfill & \hfill & \mathrm{in}{\mathrm{\Pi }}_{{\tau }_0}\setminus \mathrm{\Omega },\hfill \\ \hfill & 0\hfill & \hfill & \mathrm{as}\mathrm{dist}(x,\partial \mathrm{\Omega })>\frac{2{\tau }_0}{3},x\notin \mathrm{\Omega },\hfill \end{array}\right.$$

(77)

where $\chi =\chi \left(\tau \right)$ is the cut-off function introduced in the proof of Lemma 2. The continued function is an element of the space $W_2^1({\mathbb{R}}^d;{\mathbb{C}}^n)$ and

$${\parallel u\parallel }_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}⩽{C\parallel u\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}{,\parallel u\parallel }_{W_2^1({\mathbb{R}}^d;{\mathbb{C}}^n)}⩽C{\parallel u\parallel }_{W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)},$$

(78)

where C is a constant independent of u. In the same way as in (77), we continue the vector function $\varphi$, the continuation is an element of $W_{\infty }^1({\mathbb{R}}^d;{\mathbb{C}}^n)$, and

$${\parallel \varphi (·,\epsilon )\parallel }_{L_{\infty }({\mathbb{R}}^d;{\mathbb{C}}^d)}⩽{C\parallel \varphi (·,\epsilon )\parallel }_{L_{\infty }(\mathrm{\Omega };{\mathbb{C}}^d)}{,\parallel \varphi (·,\epsilon )\parallel }_{W_{\infty }^1({\mathbb{R}}^d;{\mathbb{C}}^d)}⩽C{\parallel \varphi (·,\epsilon )\parallel }_{W_{\infty }^1(\mathrm{\Omega };{\mathbb{C}}^d)},$$

(79)

where a constant C is independent of u and $\epsilon$. The function A is continued by zero outside $\mathrm{\Omega }$. Then, by Lemma 1, we have:

$$\parallel {\mathcal{T}}^{\epsilon }{u-Au\parallel }_{L_2({\mathbb{R}}^d;{\mathbb{C}}^n)}⩽C\parallel {\mathcal{T}}^{\epsilon }{u-Au\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}⩽C\mu \left(\epsilon \right){\parallel u\parallel }_{W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)},$$

where C is a positive constant independent of $\epsilon$ and u. The proof is complete. □

We proceed to the proof of the convergence. We choose an arbitrary $f\in L_2(\mathrm{\Omega };{\mathbb{C}}^n)$ and $\lambda \in \mathbb{C}$ such that $Re\lambda ⩽{\lambda }_0$, and we define the functions $u_{\epsilon }$ and $u_0$ by Formulas (58) with the operators ${\mathcal{H}}_{\epsilon }$ and ${\mathcal{H}}_0$ from the current section. Since both functions $u_{\epsilon }$ and $u_0$ belong to the same domain $\mathfrak{D}\left({\mathfrak{h}}_{\epsilon }\right)=\mathfrak{D}\left({\mathfrak{h}}_0\right)$, the difference $v_{\epsilon }:=u_{\epsilon }-u_0$ is also in this domain. Proceeding as in (59) and (60), we immediately obtain an identity similar to (60) and (62):

$${\mathfrak{h}}_{\epsilon }(v_{\epsilon },v_{\epsilon })-\lambda {\parallel v_{\epsilon }\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}^2={\mathfrak{h}}_0(u_0,v_{\epsilon })-{\mathfrak{h}}_{\epsilon }(u_0,v_{\epsilon })=S_1+S_2+S_3+S_4,$$

(80)

where

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill S_1:=& -\sum _{\begin{array}{c}\left|\alpha \right|⩽m-1\\ \left|\beta \right|⩽m\end{array}}\sum _{i=1}^{N_{\alpha \beta }}{(-1)}^{\left|\alpha \right|}{\left(({\mathcal{T}}_{\alpha \beta }^{\epsilon ,i}-A_{\alpha \beta }^i)\frac{{\partial }^{\beta }{u}_0}{\partial x^{\beta }},\frac{{\partial }^{\alpha }{v}_{\epsilon }}{\partial x^{\alpha }}\right)}_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)},\hfill \\ \hfill S_2:=& -\sum _{\begin{array}{c}\left|\alpha \right|=m\\ \left|\beta \right|⩽m-1\end{array}}\sum _{i=1}^{N_{\alpha \beta }}{(-1)}^{\left|\alpha \right|}{\left(({\mathcal{T}}_{\alpha \beta }^{\epsilon ,i}-A_{\alpha \beta }^i)\frac{{\partial }^{\beta }{u}_0}{\partial x^{\beta }},\frac{{\partial }^{\alpha }{v}_{\epsilon }}{\partial x^{\alpha }}\right)}_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)},\hfill \end{array}\hfill \\ \hfill & \begin{array}{cc}\hfill S_3:=& -\sum _{\begin{array}{c}\left|\alpha \right|⩽m-2\\ \left|\beta \right|⩽m-1\end{array}}\sum _{i=1}^{K_{\alpha \beta }}{\left(({\mathcal{Q}}_{\alpha \beta }^{\epsilon ,i}-{\gamma }_{\alpha \beta }^i)\frac{{\partial }^{\beta }{u}_0}{\partial x^{\beta }},\frac{{\partial }^{\alpha }{v}_{\epsilon }}{\partial x^{\alpha }}\right)}_{L_2(\partial \mathrm{\Omega };{\mathbb{C}}^n)},\hfill \\ \hfill S_4:=& -\sum _{\begin{array}{c}\left|\alpha \right|=m-1\\ \left|\beta \right|⩽m-1\end{array}}\sum _{i=1}^{K_{\alpha \beta }}{\left(({\mathcal{Q}}_{\alpha \beta }^{\epsilon ,i}-{\gamma }_{\alpha \beta }^i)\frac{{\partial }^{\beta }{u}_0}{\partial x^{\beta }},\frac{{\partial }^{\alpha }{v}_{\epsilon }}{\partial x^{\alpha }}\right)}_{L_2(\partial \mathrm{\Omega };{\mathbb{C}}^n)}.\hfill \end{array}\hfill \end{array}$$

(81)

Proceeding as in (63)–(66) and employing Lemma 3 instead of Lemma 1, we obtain similar estimates:

$$|S_1|⩽C\mu \left(\epsilon \right)\parallel u_0{\parallel }_{W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)}\parallel v_{\epsilon }{\parallel }_{W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)},|S_2|⩽C\mu \left(\epsilon \right)\parallel u_0{\parallel }_{W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)}{\parallel v_{\epsilon }\parallel }_{W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)},$$

(82)

where C is a constant independent of $u_0$, $v_{\epsilon }$, and $\epsilon$.

We estimate the term $S_3$ by the Cauchy–Schwartz inequality and Lemma 2 applied to the derivatives of $v_{\epsilon }$ with $\varphi =0$ and to the derivatives of $u_0$ with $\varphi ={\varphi }_{\alpha \beta }^{\epsilon ,i}$:

$$|S_3|⩽C\parallel v_{\epsilon }{\parallel }_{W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)}\sum _{\begin{array}{c}\left|\alpha \right|⩽m-2\\ \left|\beta \right|⩽m-1\end{array}}\sum _{i=1}^{K_{\alpha \beta }}\parallel \frac{{\partial }^{\alpha }{u}_0}{\partial x^{\alpha }}(·+{\varphi }_{\alpha \beta }^i(·,\epsilon ))-\frac{{\partial }^{\alpha }{u}_0}{\partial x^{\alpha }}{\parallel }_{W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)}.$$

Then, we apply Lemma 3 with $A=0$ to the functions $\frac{{\partial }^{\alpha }{u}_0}{\partial x^{\alpha }}$ and its first derivatives and we obtain

$$|S_3|⩽C\mu \left(\epsilon \right)\parallel u_0{\parallel }_{W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)}{\parallel v_{\epsilon }\parallel }_{W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)},$$

(83)

where C is a constant independent of $\epsilon$, $u_0$, and $v_{\epsilon }$.

In order to estimate the term $S_4$ on the right-hand side of (80), we need the following lemma.

Lemma 4.

Let $\gamma \in W_{\infty }^1(\partial \mathrm{\Omega };{\mathbb{C}}^n)$. Then, for all $u,v\in W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)$, the estimate

$$\left|{\left(\gamma u(·+\varphi (·,\epsilon \left)\right)-\gamma u,v\right)}_{L_2(\partial \mathrm{\Omega };{\mathbb{C}}^n)}\right|⩽{C\mu \left(\epsilon \right)\parallel u\parallel }_{W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)}{\parallel v\parallel }_{W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)}$$

(84)

holds, where C is a constant independent of ε, u, and v.

Proof. 

It is sufficient to prove estimate (84) for the scalar functions $u,v\in W_2^1\left(\mathrm{\Omega }\right)$ and $\gamma \in W_{\infty }^1(\mathrm{\Omega };{\mathbb{C}}^n)$. We first continue the function A from $\partial \mathrm{\Omega }$ inside and outside $\mathrm{\Omega }$ by the rule $\gamma \left(x\right)=\gamma \left(x_0\right)\chi \left(\tau \right)$, where $\chi$ is the cut-off function from the proof of Lemma 2 and $x_0$ is the point on $\partial \mathrm{\Omega }$ such that $x=x_0+\tau \nu \left(x_0\right)$; it is clear that $\gamma \in W_{\infty }^1\left(\mathrm{\Omega }\right)$. Then, we rewrite the considered scalar product as

$${\left(\gamma u(·+\varphi (·,\epsilon \left)\right)-\gamma u,v\right)}_{L_2(\partial \mathrm{\Omega })}={\left(u(·+\varphi (·,\epsilon ))-u,\overline{\gamma }v\right)}_{L_2(\partial \mathrm{\Omega })}$$

(85)

and, redenoting $\overline{\gamma }v$ by v, we see that it is sufficient to consider only the case $\gamma =1$.

We denote $w\left(x\right):=u(x+\varphi (x,\epsilon \left)\right)-u\left(x\right)$ and integrate by parts, as in (72):

$$\begin{array}{cc}\hfill {(w,v)}_{L_2(\partial \mathrm{\Omega })}=& \underset{{\mathrm{\Pi }}_{{\tau }_0}}{\int }w\overline{v}\mathrm{\Delta }\chi dx+\underset{{\mathrm{\Pi }}_{{\tau }_0}}{\int }{\nabla }_x\chi ·{\nabla }_{x}w\overline{v}dx\hfill \\ \hfill =& \underset{{\mathrm{\Pi }}_{{\tau }_0}}{\int }w\overline{v}\mathrm{\Delta }\chi dx+\underset{{\mathrm{\Pi }}_{{\tau }_0}}{\int }w\nabla \chi ·\nabla \overline{v}dx+\underset{{\mathrm{\Pi }}_{{\tau }_0}}{\int }\overline{v}\nabla \chi ·\nabla wdx\hfill \end{array}$$

(86)

According to Lemma 3 with $A=1$ applied to u, we immediately obtain:

$$\left|\underset{{\mathrm{\Pi }}_{{\tau }_0}}{\int }w\overline{v}\mathrm{\Delta }\chi dx\right|+|\underset{{\mathrm{\Pi }}_{{\tau }_0}}{\int }w\nabla \chi ·\nabla \overline{v}dx|⩽{C\mu \left(\epsilon \right)\parallel u\parallel }_{W_2^1\left(\mathrm{\Omega }\right)}{\parallel v\parallel }_{W_2^1\left(\mathrm{\Omega }\right)},$$

(87)

where C is a constant independent of $\epsilon$, u, and v.

We rewrite the third term on the right-hand side of (86) as

$$\begin{array}{cc}\hfill \underset{{\mathrm{\Pi }}_{{\tau }_0}}{\int }\overline{v}\nabla \chi ·\nabla wdx=& \underset{{\mathrm{\Pi }}_{{\tau }_0}}{\int }\overline{v}\left(x\right)\nabla \chi \left(\tau \right)·(E_d+\Phi (x,\epsilon ))(\nabla u)(x+\varphi (x,\epsilon ))dx\hfill \\ \hfill & -\underset{{\mathrm{\Pi }}_{{\tau }_0}}{\int }\overline{v}\left(x\right)\nabla \chi \left(\tau \right)·(\nabla u)\left(x\right)dx,\hfill \end{array}$$

(88)

where, we recall, $E_d$ is the unit $d\times d$ matrix, while $\Phi (x,\epsilon )$ is the matrix with the entries $\frac{\partial {\varphi }_j}{\partial x_i}(x,\epsilon )$, $\varphi =({\varphi }_1,\dots ,{\varphi }_d)$, and, due to (9), it satisfies the estimate

$${\parallel \Phi (·,\epsilon )\parallel }_{L_{\infty }(\mathrm{\Omega };{\mathbb{M}}_d)}⩽C\mu \left(\epsilon \right),$$

(89)

where C is a constant independent of $\epsilon$ and x. In the first integral, we make the change in the variables $y=x+\varphi (x,\epsilon )$. It is clear that the inverse change is of the form $x=y+\tilde{\varphi }(y,\epsilon )$, where the function $\tilde{\varphi }$ possesses the same properties as $\varphi$. Then, in view of (89), the fact that $\chi \left(\tau \right)$ vanishes for $\tau >\frac{{\tau }_0}{2}$ and with the Jacobian taken into account, after the change, the first integral becomes

$$\begin{array}{cc}\hfill \underset{{\mathrm{\Pi }}_{{\tau }_0}}{\int }\overline{v}\left(x\right)\nabla \chi \left(\tau \right)·(E+\Phi (x,\epsilon ))(\nabla u)(x+\varphi (x,\epsilon ))dx=& \underset{{\mathrm{\Pi }}_{{\tau }_0}}{\int }\overline{v}(y+\tilde{\varphi }(y,\epsilon ))\nabla \chi \left({\tau }_y\right)(\nabla u)\left(y\right)dy\hfill \\ \hfill & +\underset{{\mathrm{\Pi }}_{{\tau }_0}}{\int }\overline{v}(y+\tilde{\varphi }(y,\epsilon ))\mathrm{\Psi }(y,\epsilon )·\nabla u\left(y\right)dy,\hfill \end{array}$$

(90)

where ${\tau }_y$ is introduced in the same way as $\tau$ but is associated with y, while $\mathrm{\Psi }\in L_{\infty }(\mathrm{\Omega };{\mathbb{C}}^n)$ is some function such that

$${\parallel \mathrm{\Psi }(·,\epsilon )\parallel }_{L_{\infty }(\mathrm{\Omega };{\mathbb{M}}_n)}⩽C\mu \left(\epsilon \right),$$

(91)

C is some constant independent of $\epsilon$. Then, Formula (88) is rewritten as

$$\begin{array}{cc}\hfill \underset{{\mathrm{\Pi }}_{{\tau }_0}}{\int }\overline{v}\nabla \chi ·\nabla wdx=& \underset{{\mathrm{\Pi }}_{{\tau }_0}}{\int }\left(\overline{v}(y+\tilde{\varphi }(y,\epsilon ))-\overline{v}\left(y\right)\right)\nabla \chi \left(\tau \right)·\nabla w\left(y\right)dy\hfill \\ \hfill & +\underset{{\mathrm{\Pi }}_{{\tau }_0}}{\int }\overline{v}(y+\tilde{\varphi }(y,\epsilon ))\mathrm{\Psi }(y,\epsilon )·\nabla u\left(y\right)dy.\hfill \end{array}$$

(92)

By then applying Lemma 3 with $A=1$ to the function $\overline{v}$ and using (91), we obtain:

$$|\underset{{\mathrm{\Pi }}_{{\tau }_0}}{\int }\overline{v}\nabla \chi ·\nabla wdx|⩽{C\mu \left(\epsilon \right)\parallel u\parallel }_{W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)}{\parallel v\parallel }_{W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)},$$

where C is a constant independent of $\epsilon$, u, and v. This estimate and (86) and (87) lead us to the statement of the lemma. The proof is complete. □

Applying this lemma to each term in $S_4$, we obtain:

$$|S_4|⩽C\mu \left(\epsilon \right)\parallel u_0{\parallel }_{W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)}{\parallel v_{\epsilon }\parallel }_{W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)},$$

(93)

where a constant C is independent of $\epsilon$, $u_0$ and $v_{\epsilon }$.

The second estimate in (75) and the definition of ${\mathcal{H}}_0$ imply the inequality

$$\parallel u_0{\parallel }_{W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)}⩽C{\parallel f\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}$$

(94)

with a constant C independent of f. The first estimate in (75) ensures a lower bound for the left-hand side of identity (80), similar to (61), which together with (82), (83), (93), and (94) yields

$$\parallel v_{\epsilon }{\parallel }_{W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)}⩽C\mu \left(\epsilon \right)\parallel u_0{\parallel }_{W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)}⩽C\mu \left(\epsilon \right){\parallel f\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)},$$

where C is a constant independent of $\epsilon$ and f. The obtained inequality implies (20) and completes the proof of Theorem 2.

5. Convergence for Operators with Small Translations: Dirichlet Condition

In this section, we prove Theorem 3. We continue all functions in $\mathfrak{W}$ by zero outside $\mathrm{\Omega }$, and, due to the definition of $\mathfrak{W}$, all such continuations belong to $W_2^2({\mathbb{R}}^d;{\mathbb{C}}^n)$ and satisfy an obvious identity

$${\parallel u\parallel }_{W_2^j(\mathrm{\Omega };{\mathbb{C}}^n)}={\parallel u\parallel }_{W_2^j({\mathbb{R}}^d;{\mathbb{C}}^n)},j=0,\dots ,m.$$

(95)

For all $u\in \mathfrak{W}$, we also have the inequalities in (68).

Let $\varphi =\varphi (x,\epsilon )$ be a vector function defined for $x\in \mathrm{\Omega }$ and a sufficiently small $\epsilon$ with values in ${\mathbb{C}}^d$ and obeying estimate (44). By ${\mathcal{F}}^{\epsilon }$, we denote a mapping on $\mathrm{\Omega }$ acting according to the rule that ${\mathcal{F}}^{\epsilon }\left(x\right):=x+\varphi (x,\epsilon )$. The image of the domain $\mathrm{\Omega }$ under this mapping is not supposed to coincide with $\mathrm{\Omega }$. Then, the inverse mapping ${\left({\mathcal{F}}^{\epsilon }\right)}^{-1}$ obviously reads as ${\left({\mathcal{F}}^{\epsilon }\right)}^{-1}y=y+\tilde{\varphi }(y,\epsilon )$, where the function $\tilde{\varphi }$ has the same properties as $\varphi$. Given $A\in L_{\infty }(\mathrm{\Omega };{\mathbb{C}}^n)$, we let

$$\begin{array}{cc}\hfill & {\mathcal{T}}^{\epsilon }u:=A\mathcal{R}{\mathcal{S}}^{\epsilon }\mathcal{L}u,u\in L_2(\mathrm{\Omega };{\mathbb{C}}^n),\hfill \\ \hfill & \left({\mathcal{S}}^{\epsilon }v\right)\left(x\right):=v\left({\mathcal{F}}^{\epsilon }\left(x\right)\right),x\in \mathrm{\Omega },v\in L_2({\mathcal{F}}^{\epsilon }\left(\mathrm{\Omega }\right);{\mathbb{C}}^n),\hfill \end{array}$$

(96)

where the operator ${\mathcal{T}}^{\epsilon }u$ is treated as acting in $L_2(\mathrm{\Omega };{\mathbb{C}}^n)$. Making the change in the variables $y={\mathcal{F}}^{\epsilon }\left(x\right)$ and using the boundedness of the matrix function A, we obtain the following chain of inequalities:

$$\begin{array}{cc}\hfill \parallel {\mathcal{T}}^{\epsilon }{u\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}^2⩽& C\parallel {\mathcal{S}}^{\epsilon }{\mathcal{L}u\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}^2⩽C{\parallel \mathcal{L}u\parallel }_{L_2({\mathcal{F}}^{\epsilon }\left(\mathrm{\Omega }\right);{\mathbb{C}}^n)}^2\hfill \\ \hfill =& {C\parallel u\parallel }_{L_2(\mathrm{\Omega }\cap {\mathcal{F}}^{\epsilon }\left(\mathrm{\Omega }\right);{\mathbb{C}}^n)}^2⩽C{\parallel u\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}^2,\hfill \end{array}$$

(97)

where C is a constant independent of u and $\epsilon$. This leads us to estimate (70) for all $u\in \mathfrak{W}$ with the operators ${\mathcal{T}}_{\alpha \beta }^{\epsilon ,i}$ defined in (23). By estimates (68), we then arrive at (74) and (75). Proceeding as in the previous section after (75), we see that both forms ${\mathfrak{h}}_{\epsilon }$ and ${\mathfrak{h}}_0$ are sectorial and closed, and their numerical ranges are located in a fixed cone independent of $\epsilon$. Hence, there exists a number ${\lambda }_0$ such that the complex half-plane $Re\lambda ⩽{\lambda }_0$ is in the resolvent sets of both operators ${\mathcal{H}}_{\epsilon }$ and ${\mathcal{H}}_0$.

The convergence of the resolvent ${({\mathcal{H}}_{\epsilon }-\lambda )}^{-1}$ to ${({\mathcal{H}}_0-\lambda )}^{-1}$ follows the lines of the proof of Theorem 1 in Section 3. We choose an arbitrary $f\in L_2(\mathrm{\Omega };{\mathbb{C}}^n)$ and continue it by zero outside $\mathrm{\Omega }$. Then, we define the functions $u_{\epsilon }$ and $u_0$ by Formulas (58) with the operators ${\mathcal{H}}_{\epsilon }$ and ${\mathcal{H}}_0$ from the current section, and we also let $v_{\epsilon }:=u_{\epsilon }-u_0$. As in the previous sections, for the function $v_{\epsilon }$, we obtain the identity

$${\mathfrak{h}}_{\epsilon }(v_{\epsilon },v_{\epsilon })-\lambda {\parallel v_{\epsilon }\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}^2={\mathfrak{h}}_0(u_0,v_{\epsilon })-{\mathfrak{h}}_{\epsilon }(u_0,v_{\epsilon })=S_1+S_2,$$

(98)

where $S_1$ and $S_2$ are given by Formula (81) with all of the involved functions defined in this subsection.

To estimate the quantities $S_1$ and $S_2$, we need two lemmas.

Lemma 5.

For each $u\in {\mathring{W}}_2^1(\mathrm{\Omega };{\mathbb{C}}^n)$, the estimate holds:

$${\parallel u\parallel }_{L_2({\mathrm{\Pi }}_{\mu \left(\epsilon \right)};{\mathbb{C}}^n)}⩽C\mu \left(\epsilon \right){\parallel u\parallel }_{W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)},$$

where C is a constant independent of μ and u and the set ${\mathrm{\Pi }}_{\mu \left(\epsilon \right)}$ is defined by (1) with ${\tau }_0$ replaced by $\mu \left(\epsilon \right)$.

Proof. 

It is sufficient to consider the case of a scalar function $u\in {\mathring{W}}_2^1\left(\mathrm{\Omega }\right)$. Due to our assumptions on the boundary $\partial \mathrm{\Omega }$, the function ${(\tau -\mu )}^2$ is well defined on ${\mathrm{\Pi }}_{\mu \left(\epsilon \right)}$. It is clear that

$${\nabla }_x{(\tau -\mu )}^2=0\mathrm{as}\tau =\mu ,\left|{\nabla }_x{(\tau -\mu )}^2\right|⩽C\mu \left(\epsilon \right),x\in {\mathrm{\Pi }}_{\mu \left(\epsilon \right)},$$

(99)

where C is a constant independent of x and $\mu$, and

$${\mathrm{\Delta }}_x{(\tau -\mu )}^2=2{|\nabla \tau |}^2+2(\tau -\mu ){\mathrm{\Delta }}_x\tau .$$

(100)

It follows from the definition of $\tau$ that

$$1=\frac{d\tau }{d\tau }=\sum _{j=1}^d\frac{\partial x_j}{\partial \tau }\frac{\partial \tau }{\partial x_j}=\nu ·{\nabla }_x\tau ,$$

where the normal $\nu$ is taken at a point $P\in \partial \mathrm{\Omega }$ so that $P+\tau \nu =x$. Hence,

$$|\nabla \tau |⩾\left|\nu \right||\nu ·\nabla \tau |=1$$

and then identity (100) and the boundedness of the second derivatives of $\tau$ in x yield:

$${\mathrm{\Delta }}_x{(\tau -\mu )}^2⩾2+C(\tau -\mu )⩾1,x\in {\mathrm{\Pi }}_{\mu \left(\epsilon \right)},$$

for a sufficiently small $\tau$. Using this inequality, (99), and the fact that u has a zero trace on $\partial \mathrm{\Omega }$, we integrate by parts as follows:

$$\begin{array}{cc}\hfill {\parallel u\parallel }_{L_2\left({\mathrm{\Pi }}_{\mu \left(\epsilon \right)}\right)}^2⩽& \underset{{\mathrm{\Pi }}_{\mu \left(\epsilon \right)}}{\int }{\left|u\left(x\right)\right|}^2{\mathrm{\Delta }}_x{(\tau -\mu \left(\epsilon \right))}^{2}dx=-\underset{{\mathrm{\Pi }}_{\mu \left(\epsilon \right)}}{\int }{\nabla }_x{(\tau -\mu \left(\epsilon \right))}^2·{\nabla }_x{\left|u\left(x\right)\right|}^{2}dx\hfill \\ \hfill ⩽& C\mu \left(\epsilon \right)\underset{{\mathrm{\Pi }}_{\mu \left(\epsilon \right)}}{\int }\left|u\left(x\right)\right||{\nabla }_x{u\left(x\right)|dx⩽C\mu \left(\epsilon \right)\parallel u\parallel }_{L_2\left({\mathrm{\Pi }}_{\mu \left(\epsilon \right)}\right)}{\parallel \nabla u\parallel }_{L_2\left({\mathrm{\Pi }}_{\mu \left(\epsilon \right)}\right)}.\hfill \end{array}$$

(101)

This inequality implies the statement of the lemma. The proof is complete. □

Lemma 6.

For all $u\in {\mathring{W}}_2^1(\mathrm{\Omega };{\mathbb{C}}^n)$, estimate (76) holds with the operator ${\mathcal{T}}^{\epsilon }$ defined in (96).

Proof. 

For an arbitrary $t\in [0,1]$ we introduce an auxiliary mapping on $\mathrm{\Omega }$ by the rule ${\mathcal{F}}_t^{\epsilon }\left(x\right):=x+t\varphi (x,\epsilon )$. It is clear that the inverse mapping is well defined and that ${\left({\mathcal{F}}_t^{\epsilon }\right)}^{-1}\left(y\right)=y+\widehat{\varphi }(y,t,\epsilon )$, where the function $\widehat{\varphi }(y,t,\epsilon )$ has the same properties as $\varphi$, and an analog of estimate (44) is uniform in t. We then introduce an auxiliary domain:

$${\mathrm{\Omega }}^{\epsilon }:=\bigcap _{t\in [0,1]}{\left({\mathcal{F}}_t^{\epsilon }\right)}^{-1}(\mathrm{\Omega }\cap {\mathcal{F}}_t^{\epsilon }\left(\mathrm{\Omega }\right))\subseteq \mathrm{\Omega }.$$

The main property of the introduced domain is

$$x\in {\mathrm{\Omega }}^{\epsilon }\Rightarrow {\mathcal{F}}_t^{\epsilon }\left(x\right)\in \mathrm{\Omega }\cap {\mathcal{F}}_t^{\epsilon }\left(\mathrm{\Omega }\right)\subseteq \mathrm{\Omega }\mathrm{for}\mathrm{all}t\in [0,1].$$

(102)

In other words, the domain ${\mathrm{\Omega }}^{\epsilon }$ consists of all points in $\mathrm{\Omega }$ such that they are mapped by ${\mathcal{F}}_t^{\epsilon }$ inside ${\mathrm{\Omega }}^{\epsilon }$ for all $t\in [0,1]$.

Using the domain ${\mathrm{\Omega }}^{\epsilon }$, we represent the needed norm as

$$\parallel {\mathcal{T}}^{\epsilon }{u-Au\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}^2=\parallel {\mathcal{T}}^{\epsilon }{u-Au\parallel }_{L_2({\mathrm{\Omega }}^{\epsilon };{\mathbb{C}}^n)}^2+{\parallel {\mathcal{T}}^{\epsilon }u-Au\parallel }_{L_2(\mathrm{\Omega }\setminus {\mathrm{\Omega }}^{\epsilon };{\mathbb{C}}^n)}^2.$$

(103)

The first term on the right-hand side of this formula is estimated similarly to the proof of Lemma 1. Namely, we begin with inequality (55) and integrate it over ${\mathrm{\Omega }}^{\epsilon }$:

$$\underset{{\mathrm{\Omega }}^{\epsilon }}{\int }|u(x+\varphi (x,\epsilon ))-u\left(x\right){|}_{{\mathbb{C}}^n}^{2}dx⩽{\mu }^2\left(\epsilon \right)\sum _{j=1}^d\underset{0}{\overset{1}{\int }}dt\underset{{\mathrm{\Omega }}^{\epsilon }}{\int }|\frac{\partial u}{\partial x_j}(x+t\varphi (x,\epsilon )){|}_{{\mathbb{C}}^n}^{2}dx.$$

Since the points of the domain ${\mathrm{\Omega }}^{\epsilon }$ are mapped into $\mathrm{\Omega }$ (see (102)), we can make the change in the variables $y=x+t\varphi (x,\epsilon )$ in the above inequality:

$$\underset{{\mathrm{\Omega }}^{\epsilon }}{\int }|u(x+\varphi (x,\epsilon ))-u\left(x\right){|}_{{\mathbb{C}}^n}^{2}dx⩽{\mu }^2\left(\epsilon \right)\sum _{j=1}^d\underset{0}{\overset{1}{\int }}dt\underset{{\mathcal{F}}_t^{\epsilon }\left({\mathrm{\Omega }}^{\epsilon }\right)}{\int }|\frac{\partial u}{\partial x_j}\left(y\right){|}_{{\mathbb{C}}^n}^2{J}_t(y,\epsilon )dy,$$

where $J_t$ is the Jacobian defined by Formula (57). This Jacobian is bounded uniformly in $\epsilon$, t, and spatial variables. Hence, since according to (102) we have ${\mathcal{F}}_t^{\epsilon }\left({\mathrm{\Omega }}^{\epsilon }\right)\subseteq \mathrm{\Omega }$, we can continue estimating as

$$\begin{array}{cc}\hfill \underset{{\mathrm{\Omega }}^{\epsilon }}{\int }|u(x+\varphi (x,\epsilon ))-u\left(x\right){|}_{{\mathbb{C}}^n}^{2}dx⩽& C{\mu }^2\left(\epsilon \right)\sum _{j=1}^d\underset{0}{\overset{1}{\int }}dt\underset{{\mathcal{F}}_t^{\epsilon }\left({\mathrm{\Omega }}^{\epsilon }\right)}{\int }|\frac{\partial u}{\partial x_j}\left(y\right){|}_{{\mathbb{C}}^n}^{2}dy\hfill \\ \hfill ⩽& C{\mu }^2\left(\epsilon \right)\sum _{j=1}^d\underset{0}{\overset{1}{\int }}dt\underset{\mathrm{\Omega }}{\int }|\frac{\partial u}{\partial x_j}\left(y\right){|}_{{\mathbb{C}}^n}^{2}dy\hfill \\ \hfill ⩽& C{\mu }^2\left(\epsilon \right)\sum _{j=1}^d\underset{\mathrm{\Omega }}{\int }|\frac{\partial u}{\partial x_j}\left(y\right){|}_{{\mathbb{C}}^n}^{2}dy.\hfill \end{array}$$

(104)

We proceed to estimating the second term on the right-hand side of (103). It follows from the definition of the domain ${\mathrm{\Omega }}^{\epsilon }$ that the set $\mathrm{\Omega }\setminus {\mathrm{\Omega }}^{\epsilon }$ consists of points $x\in \mathrm{\Omega }$ such that there exists $t\in [0,1]$, for which ${\mathcal{F}}_t^{\epsilon }\left(x\right)\notin \mathrm{\Omega }$. At the same time, the definition of ${\mathcal{F}}_t^{\epsilon }$ and inequality (44) yield that $|{\mathcal{F}}_t^{\epsilon }{\left(x\right)-x|}_{{\mathbb{C}}^d}⩽\mu \left(\epsilon \right)$. Hence, if $x\in \mathrm{\Omega }$ and $\mathrm{dist}(x,\partial \mathrm{\Omega })>\mu \left(\epsilon \right)$, such a point x is surely in ${\mathrm{\Omega }}^{\epsilon }$ and does not belong to $\mathrm{\Omega }\setminus {\mathrm{\Omega }}^{\epsilon }$. Therefore, the set $\mathrm{\Omega }\setminus {\mathrm{\Omega }}^{\epsilon }$ satisfies $\mathrm{\Omega }\setminus {\mathrm{\Omega }}^{\epsilon }\subseteq \overline{{\mathrm{\Pi }}_{\mu \left(\epsilon \right)}}$. Applying now Lemma 5, we then obtain:

$${\parallel u\parallel }_{L_2(\mathrm{\Omega }\setminus {\mathrm{\Omega }}^{\epsilon };{\mathbb{C}}^n)}⩽{\parallel u\parallel }_{L_2({\mathrm{\Pi }}_{\mu \left(\epsilon \right)};{\mathbb{C}}^n)}⩽C\mu \left(\epsilon \right){\parallel u\parallel }_{W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)}$$

(105)

with a constant C that is independent of $\mu$, $\epsilon$, and u. This inequality and (104) imply the statement of the lemma. The proof is complete. □

We observe that the function $u_0$ satisfies estimate (94). Using this estimate and the proven lemma, we obtain:

$$|S_1|⩽C\mu \left(\epsilon \right)\parallel u_0{\parallel }_{W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)}\parallel v_{\epsilon }{\parallel }_{W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)}⩽{C\mu \left(\epsilon \right)\parallel f\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}{\parallel v_{\epsilon }\parallel }_{W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)},$$

(106)

where C is a constant independent of $\mu$, $\epsilon$, $u_0$, $v_{\epsilon }$, and f. In order to estimate the sum $S_2$, we substitute the definition of the operator ${\mathcal{T}}_{\alpha \beta }^{\epsilon }$ from (23), and, then, in each obtained term, we have a chain of identities:

$$\begin{array}{cc}\hfill {({\mathcal{T}}_{\alpha \beta }^{\epsilon ,i}{u}_0,v_{\epsilon })}_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}=& {(A_{\alpha \beta }^i\mathcal{R}{\mathcal{S}}_{\alpha \beta }^{\epsilon ,i}\mathcal{L}{u}_0,v_{\epsilon })}_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}={\left(\mathcal{R}{\mathcal{S}}_{\alpha \beta }^{\epsilon ,i}\mathcal{L}{u}_0,{\left(A_{\alpha \beta }^i\right)}^{*}{v}_{\epsilon }\right)}_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}\hfill \\ \hfill =& {\left({\mathcal{S}}_{\alpha \beta }^{\epsilon ,i}\mathcal{L}{u}_0,\mathcal{L}{\left(A_{\alpha \beta }^i\right)}^{*}{v}_{\epsilon }\right)}_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}={\left(\mathcal{L}{u}_0,{\left({\mathcal{S}}_{\alpha \beta }^{\epsilon ,i}\right)}^{*}\mathcal{L}{\left(A_{\alpha \beta }^i\right)}^{*}{v}_{\epsilon }\right)}_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}\hfill \\ \hfill =& {\left(u_0,\mathcal{R}{\left({\mathcal{S}}_{\alpha \beta }^{\epsilon ,i}\right)}^{*}\mathcal{L}{\left(A_{\alpha \beta }^i\right)}^{*}{v}_{\epsilon }\right)}_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)},\hfill \end{array}$$

(107)

where

$$\left({\left({\mathcal{S}}_{\alpha \beta }^{(\epsilon ,i)}\right)}^{*}v\right)\left(y\right)=v(y+\tilde{\varphi }(y,\epsilon ))J_1(y,\epsilon ),$$

and $J_1$ is the Jacobian defined by (57) with $\varphi ={\varphi }_{\alpha \beta }^i$. Hence,

$${\left(({\mathcal{T}}_{\alpha \beta }^{\epsilon ,i}-A_{\alpha \beta }^i)u_0,v_{\epsilon }\right)}_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}={\left(u_0,\mathcal{R}{\left({\mathcal{S}}_{\alpha \beta }^{\epsilon ,i}\right)}^{*}\mathcal{L}{\left(A_{\alpha \beta }^i\right)}^{*}{v}_{\epsilon }-{\left(A_{\alpha \beta }^i\right)}^{*}{v}_{\epsilon }\right)}_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}$$

(108)

and, by applying Lemma 6 to ${\left(A_{\alpha \beta }^i\right)}^{*}{v}_{\epsilon }$, we obtain:

$$|{\left(({\mathcal{T}}_{\alpha \beta }^{\epsilon ,i}-A_{\alpha \beta }^i)u_0,v_{\epsilon }\right)}_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}|⩽C\mu \left(\epsilon \right)\parallel u_0{\parallel }_{W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)}{\parallel v_{\epsilon }\parallel }_{W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)},$$

(109)

where C is some constant independent of $\mu$, $\epsilon$, $\alpha$, $\beta$, i, $u_0$, and $v_{\epsilon }$. This estimate, (94), and the definition of $S_2$ yield

$$|S_2|⩽C\mu \left(\epsilon \right)\parallel u_0{\parallel }_{W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)}\parallel v_{\epsilon }{\parallel }_{W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)}⩽{C\mu \left(\epsilon \right)\parallel f\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}{\parallel v_{\epsilon }\parallel }_{W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)},$$

(110)

where C is a constant independent of $\mu$, $\epsilon$, $u_0$, $v_{\epsilon }$, and f. This inequality and (106) allow us to estimate the right-hand side of (98). Then, by taking the real part of this identity and using the first inequality in (75), we obtain:

$$\parallel v_{\epsilon }{\parallel }_{W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)}⩽C\mu \left(\epsilon \right){\parallel f\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)},$$

where C is a constant independent of $\epsilon$ and f. This estimate proves (27).

6. Convergence for Operators with Small Translations: General Case

Here, we prove Theorem 4. As in the previous subsections, first, we are going to confirm that the forms ${\mathfrak{h}}_0$ and ${\mathfrak{h}}_{\epsilon }$ are sectorial and closed. For the form ${\mathfrak{h}}_0$, this was completed in the proof of Theorem 3 in Section 4. Let us prove the same for the form ${\mathfrak{h}}_{\epsilon }$.

As in the previous subsection, we introduce the vector function $\varphi$, the corresponding mapping ${\mathcal{F}}^{\epsilon }$, and the operators ${\mathcal{T}}^{\epsilon }$ and ${\mathcal{S}}^{\epsilon }$ from (96). The operator ${\mathcal{T}}^{\epsilon }$ satisfies estimate (97). By Lemma 2, we obtain:

$$\begin{array}{cc}\hfill \sum _{\left|\alpha \right|,\left|\beta \right|⩽m-1}\sum _{i=1}^{K_{\alpha \beta }}\left|{\left({\mathcal{Q}}_{\alpha \beta }^{\epsilon ,i}\frac{{\partial }^{\beta }u}{\partial x^{\beta }},\frac{{\partial }^{\alpha }v}{\partial x^{\alpha }}\right)}_{L_2({\mathrm{\Gamma }}_{\alpha \beta }^{\epsilon ,i};{\mathbb{C}}^n)}\right|⩽& \parallel {\mathcal{Q}}_{\alpha \beta }^{\epsilon ,i}\frac{{\partial }^{\beta }u}{\partial x^{\beta }}{\parallel }_{L_2({\mathrm{\Gamma }}_{\alpha \beta }^{\epsilon ,i};{\mathbb{C}}^n)}\parallel \frac{{\partial }^{\alpha }v}{\partial x^{\alpha }}{\parallel }_{L_2({\mathrm{\Gamma }}_{\alpha \beta }^{\epsilon ,i};{\mathbb{C}}^n)}\hfill \\ \hfill ⩽& C\parallel {\mathcal{E}}_{\alpha \beta }^{\epsilon ,i}\mathcal{L}\frac{{\partial }^{\beta }u}{\partial x^{\beta }}{\parallel }_{L_2({\mathrm{\Gamma }}_{\alpha \beta }^{\epsilon ,i};{\mathbb{C}}^n)}\parallel \frac{{\partial }^{\alpha }v}{\partial x^{\alpha }}{\parallel }_{W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)}\hfill \\ \hfill ⩽& C\parallel \frac{{\partial }^{\beta }u}{\partial x^{\beta }}\left({\mathcal{G}}_{\alpha \beta }^{\epsilon ,i}\right){\parallel }_{W_2^1({\mathrm{{\rm Y}}}_{\alpha \beta }^{\epsilon ,i};{\mathbb{C}}^n)}\parallel \frac{{\partial }^{\alpha }v}{\partial x^{\alpha }}{\parallel }_{W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)}\hfill \\ \hfill ⩽& C\parallel \frac{{\partial }^{\beta }u}{\partial x^{\beta }}{\parallel }_{W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)}\parallel \frac{{\partial }^{\alpha }v}{\partial x^{\alpha }}{\parallel }_{W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)},\hfill \end{array}$$

where the mappings ${\mathcal{G}}_{\alpha \beta }^{\epsilon ,i}$, ${\mathcal{Q}}_{\alpha \beta }^{\epsilon ,i}$, and ${\mathcal{E}}_{\alpha \beta }^{\epsilon ,i}$ are defined in (28) and (30) and C is some constant independent of $\epsilon$, $\alpha$, $\beta$, i, u, and v. Using these estimates and (97), we obtain (74) and (75) and then proceed as in Section 4 after (75). Then, we see that the form ${\mathfrak{h}}_{\epsilon }$ is sectorial and closed; its numerical range is located in a fixed cone independent of $\epsilon$. Hence, there exists a number ${\lambda }_0$ such that the complex half-plane $Re\lambda ⩽{\lambda }_0$ is in the resolvent sets of both operators ${\mathcal{H}}_{\epsilon }$ and ${\mathcal{H}}_0$.

We proceed to prove the convergence. For an arbitrary $f\in L_2(\mathrm{\Omega };{\mathbb{C}}^n)$, we define functions $u_{\epsilon }$ and $u_0$ by Formulas (58) with the operators ${\mathcal{H}}_{\epsilon }$ and ${\mathcal{H}}_0$ from the current section, and we let $v_{\epsilon }:=u_{\epsilon }-u_0$. Then, as in Section 4, for the function $v_{\epsilon }$, we obtain the identity

$${\mathfrak{h}}_{\epsilon }(v_{\epsilon },v_{\epsilon })-\lambda {\parallel v_{\epsilon }\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}^2={\mathfrak{h}}_0(u_0,v_{\epsilon })-{\mathfrak{h}}_{\epsilon }(u_0,v_{\epsilon })=S_1+S_2+S_3+S_4+S_5,$$

(111)

where the quantities $S_1$, $S_2$ are given by Formulas (81) with all involved functions defined in this subsection, while

$$\begin{array}{cc}\hfill S_3:=& -\sum _{\begin{array}{c}\left|\alpha \right|⩽m-2\\ \left|\beta \right|⩽m-1\end{array}}\sum _{i=1}^{K_{\alpha \beta }}{\left(({\mathcal{Q}}_{\alpha \beta }^{\epsilon ,i}-{\gamma }_{\alpha \beta }^i)\frac{{\partial }^{\beta }{u}_0}{\partial x^{\beta }},\frac{{\partial }^{\alpha }{v}_{\epsilon }}{\partial x^{\alpha }}\right)}_{L_2({\mathrm{\Gamma }}_{\alpha \beta }^{\epsilon ,i};{\mathbb{C}}^n)},\hfill \\ \hfill S_4:=& -\sum _{\begin{array}{c}\left|\alpha \right|=m-1\\ \left|\beta \right|⩽m-1\end{array}}\sum _{i=1}^{K_{\alpha \beta }}{\left(({\mathcal{Q}}_{\alpha \beta }^{\epsilon ,i}-{\gamma }_{\alpha \beta }^i)\frac{{\partial }^{\beta }{u}_0}{\partial x^{\beta }},\frac{{\partial }^{\alpha }{v}_{\epsilon }}{\partial x^{\alpha }}\right)}_{L_2({\mathrm{\Gamma }}_{\alpha \beta }^{\epsilon ,i};{\mathbb{C}}^n)},\hfill \\ \hfill S_5:=& \sum _{\begin{array}{c}\left|\alpha \right|⩽m-1\\ \left|\beta \right|⩽m-1\end{array}}\sum _{i=1}^{K_{\alpha \beta }}{\left({\gamma }_{\alpha \beta }^i\frac{{\partial }^{\beta }{u}_0}{\partial x^{\beta }},\frac{{\partial }^{\alpha }{v}_{\epsilon }}{\partial x^{\alpha }}\right)}_{L_2(\partial \mathrm{\Omega }\setminus {\mathrm{\Gamma }}_{\alpha \beta }^{\epsilon ,i};{\mathbb{C}}^n)}.\hfill \end{array}$$

Due to assumption (29), in the case that $d=1$, the sets $\partial \mathrm{\Omega }\setminus {\mathrm{\Gamma }}_{\alpha \beta }^{\epsilon ,i}$ are empty, and $S_5$ vanishes.

In order to estimate the terms $S_5$ as $d⩾2$, we shall employ two statements similar to Lemmas 5 and 6.

Lemma 7.

For each $u\in W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)$, the estimate holds:

$${\parallel u\parallel }_{L_2({\mathrm{\Pi }}_{\mu \left(\epsilon \right)};{\mathbb{C}}^n)}⩽C{\mu }^{\frac{1}{2}}\left(\epsilon \right){\parallel u\parallel }_{W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)},$$

where C is a constant independent of μ and u.

Proof. 

It is sufficient to consider the case of a scalar function u. We integrate by parts as in (101) and use Lemma 2 with $\varphi =0$ and (99):

$$\begin{array}{cc}\hfill {\parallel u\parallel }_{L_2\left({\mathrm{\Pi }}_{\mu \left(\epsilon \right)}\right)}^2⩽& \underset{{\mathrm{\Pi }}_{\mu \left(\epsilon \right)}}{\int }{\left|u\right|}^2{\mathrm{\Delta }}_x{(\tau -\mu )}^{2}dx=-2\mu \left(\epsilon \right)\underset{\partial \mathrm{\Omega }}{\int }{\left|u\right|}^{2}ds-\underset{{\mathrm{\Pi }}_{\mu \left(\epsilon \right)}}{\int }{\nabla }_x{(\tau -\mu )}^2·{\nabla }_x{\left|u\right|}^{2}dx\hfill \\ \hfill ⩽& {C\mu \parallel u\parallel }_{W_2^1\left(\mathrm{\Omega }\right)}^2+{C\mu \parallel u\parallel }_{L_2\left({\mathrm{\Pi }}_{\mu \left(\epsilon \right)}\right)}{\parallel \nabla u\parallel }_{L_2\left({\mathrm{\Pi }}_{\mu \left(\epsilon \right)}\right)}⩽\frac{1}{2}{\parallel u\parallel }_{L_2\left({\mathrm{\Pi }}_{\mu \left(\epsilon \right)}\right)}^2+C\mu {\parallel u\parallel }_{W_2^1\left(\mathrm{\Omega }\right)}^2,\hfill \end{array}$$

where C is some constant independent of $\epsilon$ and u. The obtained inequalities imply the needed estimate. The proof is complete. □

Lemma 8.

For all $u\in W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)$, the estimate

$$\parallel {\mathcal{T}}^{\epsilon }{u-Au\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}⩽C{\mu }^{\frac{1}{2}}\left(\epsilon \right){\parallel u\parallel }_{W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)}$$

(112)

holds, where C is some constant independent of u and ε.

Proof. 

We proceed as in the proof of Lemma 6 and obtain identity (103) and estimate (104). The second term on the right-hand side of (103) is estimated similarly to (105), but now we use Lemma 7 instead of Lemma 8:

$${\parallel u\parallel }_{L_2(\mathrm{\Omega }\setminus {\mathrm{\Omega }}^{\epsilon };{\mathbb{C}}^n)}⩽{\parallel u\parallel }_{L_2({\mathrm{\Pi }}_{\mu \left(\epsilon \right)};{\mathbb{C}}^n)}⩽C{\mu }^{\frac{1}{2}}\left(\epsilon \right){\parallel u\parallel }_{W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)}$$

where the constant C is independent of $\mu$, $\epsilon$, and u. This inequality and (104) imply the statement of the lemma and complete the proof. □

The function $u_0$ satisfies estimate (94). Using this estimate, Lemmas 7 and 8 and the arguments presented in (106)–(110), we obtain:

$$|S_1|+|S_2|⩽C{\mu }^{\frac{1}{2}}{\left(\epsilon \right)\parallel f\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}{\parallel v_{\epsilon }\parallel }_{W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)},$$

(113)

where C is a constant independent of $\epsilon$, f, $u_0$, and $v_{\epsilon }$. Employing Lemma 2 with $\varphi =0$, Lemma 8, and estimate (94), we obtain:

$$\begin{array}{cc}\hfill |S_3|⩽& C\sum _{\begin{array}{c}\left|\alpha \right|⩽m-2\\ \left|\beta \right|⩽m-1\end{array}}\sum _{i=1}^{K_{\alpha \beta }}\parallel (\mathcal{R}{\mathcal{E}}_{\alpha \beta }^{\epsilon ,i}\mathcal{L}-1)\frac{{\partial }^{\beta }{u}_0}{\partial x^{\beta }}{\parallel }_{L_2({\mathrm{\Gamma }}_{\alpha \beta }^{\epsilon ,i};{\mathbb{C}}^n)}\parallel \frac{{\partial }^{\alpha }{v}_{\epsilon }}{\partial x^{\alpha }}{\parallel }_{L_2({\mathrm{\Gamma }}_{\alpha \beta }^{\epsilon ,i};{\mathbb{C}}^n)}\hfill \\ \hfill ⩽& C\parallel v_{\epsilon }{\parallel }_{W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)}\sum _{\begin{array}{c}\left|\alpha \right|⩽m-2\\ \left|\beta \right|⩽m-1\end{array}}\sum _{i=1}^{K_{\alpha \beta }}\parallel (\mathcal{R}{\mathcal{E}}_{\alpha \beta }^{\epsilon ,i}\mathcal{L}-1)\frac{{\partial }^{\beta }{u}_0}{\partial x^{\beta }}{\parallel }_{W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)}\hfill \\ \hfill ⩽& C{\mu }^{\frac{1}{2}}\left(\epsilon \right)\parallel v_{\epsilon }{\parallel }_{W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)}\parallel u_0{\parallel }_{W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)}⩽C{\mu }^{\frac{1}{2}}{\left(\epsilon \right)\parallel f\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}{\parallel v_{\epsilon }\parallel }_{W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)}.\hfill \end{array}$$

In order to estimate the terms $S_3$, $S_4$, we need an auxiliary notation and one more lemma. We introduce the mappings ${\mathcal{G}}^{\epsilon }\left(x\right):=x+\varphi (x,\epsilon )$, and, by the assumed properties of $\varphi$, there exists an inverse mapping ${\left({\mathcal{G}}^{\epsilon }\right)}^{-1}$ of the form ${\left({\mathcal{G}}^{\epsilon }\right)}^{-1}\left(y\right)=y+\tilde{\psi }(y,\epsilon )$, where the function $\tilde{\psi }(y,\epsilon )$ possesses the same properties as $\psi$. We let

$$\begin{array}{cc}\hfill & {\mathrm{\Gamma }}^{\epsilon }:=\partial \mathrm{\Omega }\cap \overline{{\mathrm{{\rm Y}}}^{\epsilon }},\phantom{{\mathrm{{\rm Y}}}^{\epsilon }}{\mathrm{{\rm Y}}}^{\epsilon }:={\left({\mathcal{G}}^{\epsilon }\right)}^{-1}\left(\mathrm{\Omega }\cap {\mathcal{G}}^{\epsilon }\left(\mathrm{\Omega }\right)\right),\hfill \\ \hfill & {\mathcal{Q}}^{\epsilon }u:=\gamma \mathcal{R}{\mathcal{E}}^{\epsilon }\mathcal{L}u,\left({\mathcal{E}}^{\epsilon }u\right)\left(x\right):=u\left({\mathcal{G}}^{\epsilon }\left(x\right)\right),x\in \partial \mathrm{\Omega },\hfill \end{array}$$

where $\gamma \in L_{\infty }(\partial \mathrm{\Omega };{\mathbb{M}}_n)$ is some matrix-valued function, and the operator is regarded as acting from $W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)$ into $L_2({\mathrm{\Gamma }}^{\epsilon };{\mathbb{C}}^n)$. We also assume that

$${mes}_{d-1}\partial \mathrm{\Omega }\setminus {\mathrm{\Gamma }}^{\epsilon }⩽\eta \left(\epsilon \right),d⩾2,\mathrm{\Omega }\subseteq {\mathrm{{\rm Y}}}^{\epsilon },d=1.$$

(114)

Lemma 9.

Let $d⩾2$. For all $u,v\in W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)$, the estimate

$$\left|{(\gamma u,v)}_{L_2(\partial \mathrm{\Omega }\setminus {\mathrm{\Gamma }}^{\epsilon };{\mathbb{C}}^n)}\right|⩽C{\eta }^{\theta }{\left(\epsilon \right)\parallel u\parallel }_{W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)}{\parallel v\parallel }_{W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)}$$

holds, where C is some constant independent of ε and u.

Proof. 

We have:

$$\left|{(\gamma u,v)}_{L_2(\partial \mathrm{\Omega }\setminus {\mathrm{\Gamma }}^{\epsilon };{\mathbb{C}}^n)}\right|⩽{C\parallel u\parallel }_{L_2(\partial \mathrm{\Omega }\setminus {\mathrm{\Gamma }}^{\epsilon };{\mathbb{C}}^n)}{\parallel v\parallel }_{L_2(\partial \mathrm{\Omega }\setminus {\mathrm{\Gamma }}^{\epsilon };{\mathbb{C}}^n)}.$$

(115)

Letting $q:=2\frac{d-1}{d-2}$ for $d⩾3$ and choosing an arbitrary $q>2$ for $d=2$, by ([65], Ch. V, Section 5.22, Theorem) and the Hölder inequality, we obtain the estimates:

$$\begin{array}{c}\hfill {\parallel u\parallel }_{L_2(\partial \mathrm{\Omega }\setminus {\mathrm{\Gamma }}^{\epsilon };{\mathbb{C}}^n)}^2⩽{\left({mes}_{d-1}\partial \mathrm{\Omega }\setminus {\mathrm{\Gamma }}^{\epsilon }\right)}^{\frac{q-2}{q}}{\left(\underset{\partial \mathrm{\Omega }\setminus {\mathrm{\Gamma }}^{\epsilon }}{\int }{\left|u\right|}_{{\mathbb{C}}^n}^{q}ds\right)}^{\frac{2}{q}}⩽C{\eta }^{\frac{q-2}{q}}\left(\epsilon \right){\parallel u\parallel }_{W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)},\end{array}$$

where C is some constant independent of $\epsilon$ and u. These estimates are applied to u and v, and (115) implies the statement of the lemma. The proof is complete. □

Lemma 10.

For all $u,v\in W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)$, the estimates

$$\begin{array}{cccc}\hfill & \left|{\left(({\mathcal{Q}}^{\epsilon }-\gamma )u,v\right)}_{L_2({\mathrm{\Gamma }}^{\epsilon };{\mathbb{C}}^n)}\right|⩽C\left({\eta }^{\theta }\left(\epsilon \right)+{\mu }^{\frac{1}{2}}\left(\epsilon \right)\right){\parallel u\parallel }_{W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)}{\parallel v\parallel }_{W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)},\hfill & \hfill & d⩾2,\hfill \end{array}$$

(116)

$$\begin{array}{cccc}\hfill & \left|{\left(({\mathcal{Q}}^{\epsilon }-\gamma )u,v\right)}_{L_2({\mathrm{\Gamma }}^{\epsilon };{\mathbb{C}}^n)}\right|⩽C{\mu }^{\frac{1}{2}}{\left(\epsilon \right)\parallel u\parallel }_{W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)}{\parallel v\parallel }_{W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)},\hfill & \hfill & d=1,\hfill \end{array}$$

(117)

hold, where C is some constant independent of ε and u.

Proof. 

As in the proof of Lemma 4, we first continue the function $\gamma$ inside and outside $\mathrm{\Omega }$, and, arguing as in (85), we see that it is sufficient to consider the case of the scalar functions u and v and $\gamma \equiv 1$. We then continue the functions u, v, and $\varphi$ outside $\mathrm{\Omega }$ by Formula (77), keeping the same notations for the continuations, and rewrite the studied scalar product as

$${(({\mathcal{E}}^{\epsilon }-1)u,v)}_{L_2\left({\mathrm{\Gamma }}^{\epsilon }\right)}={\left(({\mathcal{E}}^{\epsilon }-1)u,v\right)}_{L_2(\partial \mathrm{\Omega })}-{\left(({\mathcal{E}}^{\epsilon }-1)u,v\right)}_{L_2(\partial \mathrm{\Omega }\setminus {\mathrm{\Gamma }}^{\epsilon })}.$$

(118)

Due to condition (114) in the case that $d=1$, the second scalar product on the right-hand side of the above identity vanishes, while, in the case $d⩾2$, it can be estimated by means of Lemma 9 and inequalities (78):

$$\begin{array}{cc}\hfill \left|{\left(({\mathcal{E}}^{\epsilon }-1)u,v\right)}_{L_2(\partial \mathrm{\Omega }\setminus {\mathrm{\Gamma }}^{\epsilon })}\right|⩽& C\left(\left|{({\mathcal{E}}^{\epsilon }u,v)}_{L_2(\partial \mathrm{\Omega }\setminus {\mathrm{\Gamma }}^{\epsilon })}\right|+\left|{(u,v)}_{L_2(\partial \mathrm{\Omega }\setminus {\mathrm{\Gamma }}^{\epsilon })}\right|\right)\hfill \\ \hfill ⩽& C{\eta }^{\theta }\left(\epsilon \right)\left({\parallel u\parallel }_{W_2^1\left(\mathrm{\Omega }\right)}+{\parallel {\mathcal{E}}^{\epsilon }u\parallel }_{W_2^1\left(\mathrm{\Omega }\right)}\right){\parallel v\parallel }_{W_2^1\left(\mathrm{\Omega }\right)}\hfill \\ \hfill ⩽& C{\eta }^{\theta }{\left(\epsilon \right)\parallel u\parallel }_{W_2^1\left(\mathrm{\Omega }\right)}{\parallel v\parallel }_{W_2^1\left(\mathrm{\Omega }\right)},\hfill \end{array}$$

(119)

where C is some constant independent of $\epsilon$, u, and v.

In order to estimate the first term on the right-hand side of (118), we denote $w:=({\mathcal{E}}^{\epsilon }-1)u$ and integrate by parts as in (86). Then, using Lemma 8 instead of Lemma 3, we obtain an estimate similar to (87):

$$\left|\underset{\mathrm{\Omega }}{\int }w\overline{v}\mathrm{\Delta }\chi dx\right|+|\underset{\mathrm{\Omega }}{\int }w\nabla \chi ·\nabla \overline{v}dx|⩽C{\mu }^{\frac{1}{2}}{\left(\epsilon \right)\parallel u\parallel }_{W_2^1\left(\mathrm{\Omega }\right)}{\parallel v\parallel }_{W_2^1\left(\mathrm{\Omega }\right)},$$

(120)

where C is a constant independent of $\epsilon$, u, and v.

In view of the above continuation of the function $\varphi$, the inverse mapping ${\left({\mathcal{G}}^{\epsilon }\right)}^{-1}$ is well defined on $\mathrm{\Omega }$. Then, we rewrite the third term in the right-hand side of (86) as

$$\begin{array}{cc}\hfill \underset{\mathrm{\Omega }}{\int }\overline{v}\left(x\right)\nabla \chi \left(\tau \right)& ·(E+\Phi (x,\epsilon \left)\right)(\nabla u)(x+\varphi (x,\epsilon \left)\right)dx\hfill \\ \hfill =& \underset{{\left({\mathcal{G}}^{\epsilon }\right)}^{-1}\left(\mathrm{\Omega }\right)}{\int }\overline{v}\left(x\right)\nabla \chi \left(\tau \right)·(E+\Phi (x,\epsilon ))(\nabla u)(x+\varphi (x,\epsilon ))dx\hfill \\ \hfill & +\underset{\mathrm{\Omega }\setminus {\left({\mathcal{G}}^{\epsilon }\right)}^{-1}\left(\mathrm{\Omega }\right)}{\int }\overline{v}\left(x\right)\nabla \chi \left(\tau \right)·(E+\Phi (x,\epsilon ))(\nabla u)(x+\varphi (x,\epsilon ))dx\hfill \\ \hfill & -\underset{{\left({\mathcal{G}}^{\epsilon }\right)}^{-1}\left(\mathrm{\Omega }\right)\setminus \mathrm{\Omega }}{\int }\overline{v}\left(x\right)\nabla \chi \left(\tau \right)·(E+\Phi (x,\epsilon ))(\nabla u)(x+\varphi (x,\epsilon ))dx.\hfill \end{array}$$

(121)

Then, arguing as before in (105) and using Lemma 7 and (78), we immediately obtain:

$$\begin{array}{cc}\hfill & |\underset{\mathrm{\Omega }\setminus {\left({\mathcal{G}}^{\epsilon }\right)}^{-1}\left(\mathrm{\Omega }\right)}{\int }\overline{v}\left(x\right)\nabla \chi \left(\tau \right)·(E+\Phi (x,\epsilon ))(\nabla u)(x+\varphi (x,\epsilon ))dx|\hfill \\ \hfill & +|\underset{{\left({\mathcal{G}}^{\epsilon }\right)}^{-1}\left(\mathrm{\Omega }\right)\setminus \mathrm{\Omega }}{\int }\overline{v}\left(x\right)\nabla \chi \left(\tau \right)·(E+\Phi (x,\epsilon ))(\nabla u)(x+\varphi (x,\epsilon ))dx|\hfill \\ \hfill & ⩽C{\mu }^{\frac{1}{2}}{\left(\epsilon \right)\parallel u\parallel }_{W_2^1\left(\mathrm{\Omega }\right)}{\parallel v\parallel }_{W_2^1\left(\mathrm{\Omega }\right)}{\parallel u\parallel }_{W_2^1\left(\mathrm{\Omega }\right)},\hfill \end{array}$$

(122)

where C is a constant independent of $\epsilon$, u, and v.

In the first integral on the right-hand side of (121), we make the change in the variable $y={\mathcal{G}}^{\epsilon }\left(x\right)$. Then, we proceed as in (90):

$$\begin{array}{cc}\hfill \underset{{\left({\mathcal{G}}^{\epsilon }\right)}^{-1}\left(\mathrm{\Omega }\right)}{\int }\overline{v}\left(x\right)\nabla \chi \left(\tau \right)& ·(E+\Phi (x,\epsilon ))(\nabla u)(x+\varphi (x,\epsilon ))dx-\underset{\mathrm{\Omega }}{\int }\overline{v}\left(x\right)\nabla \chi \left(\tau \right)·(\nabla u)\left(x\right)dx\hfill \\ \hfill & =\underset{\mathrm{\Omega }}{\int }\overline{v}\left({\left({\mathcal{G}}^{\epsilon }\right)}^{-1}\left(y\right)\right)\nabla \chi \left({\tau }_y\right)(\nabla u)\left(y\right)dy+\underset{\mathrm{\Omega }}{\int }\overline{v}\left({\left({\mathcal{G}}^{\epsilon }\right)}^{-1}\left(y\right)\right)\mathrm{\Psi }(y,\epsilon )·\nabla u\left(y\right)dy,\hfill \end{array}$$

where a function $\mathrm{\Psi }\in L_{\infty }(\mathrm{\Omega };{\mathbb{C}}^n)$ still satisfies estimate (91). By Lemma 3 with $A=1$ applied to v and by (78), we then obtain:

$$\begin{array}{cc}\hfill |\underset{{\left({\mathcal{G}}^{\epsilon }\right)}^{-1}\left(\mathrm{\Omega }\right)}{\int }\overline{v}\left(x\right)\nabla \chi \left(\tau \right)·(E+\Phi (x,\epsilon ))(\nabla u)(x+\varphi (x,\epsilon ))& dx-\underset{\mathrm{\Omega }}{\int }\overline{v}\left(x\right)\nabla \chi \left(\tau \right)·(\nabla u)\left(x\right)dx|\hfill \\ \hfill & ⩽C{\mu }^{\frac{1}{2}}{\left(\epsilon \right)\parallel u\parallel }_{W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)}{\parallel v\parallel }_{W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)}\hfill \end{array}$$

with a constant C independent of u and v. These estimates, (86), (120), (121), and (122), imply the statement of the lemma. The proof is complete. □

Lemma 9 and (94) allow us to estimate $S_5$ in the case that $d⩾2$:

$$|S_5|⩽C{\eta }^{\theta }\left(\epsilon \right)\parallel u_0{\parallel }_{W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)}\parallel v_{\epsilon }{\parallel }_{W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)}⩽C{\eta }^{\theta }{\left(\epsilon \right)\parallel f\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}{\parallel v_{\epsilon }\parallel }_{W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)}.$$

(123)

Lemma 10 and (94) imply an estimate for $S_4$:

$$\begin{array}{cc}\hfill |S_4|⩽& C\left({\eta }^{\theta }\left(\epsilon \right)+{\mu }^{\frac{1}{2}}\left(\epsilon \right)\right)\parallel u_0{\parallel }_{W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)}{\parallel v_{\epsilon }\parallel }_{W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)}\hfill \\ \hfill ⩽& C\left({\eta }^{\theta }\left(\epsilon \right)+{\mu }^{\frac{1}{2}}\left(\epsilon \right)\right){\parallel f\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}{\parallel v_{\epsilon }\parallel }_{W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)},\hfill & \hfill & d⩾2,\hfill \\ \hfill |S_4|⩽& C{\mu }^{\frac{1}{2}}\left(\epsilon \right)\parallel u_0{\parallel }_{W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)}{\parallel v_{\epsilon }\parallel }_{W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)}\hfill \\ \hfill ⩽& C{\mu }^{\frac{1}{2}}{\left(\epsilon \right)\parallel f\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}{\parallel v_{\epsilon }\parallel }_{W_2^1(\mathrm{\Omega };{\mathbb{C}}^n)},\hfill & \hfill & d=1.\hfill \end{array}$$

(124)

In the above estimates, C is some fixed constant independent of $\epsilon$, $u_0$, $v_{\epsilon }$, and f.

When we take the real part of identity (111) and use estimates (75), (113), (123), and (124), we obtain

$$\begin{array}{cc}\hfill & \parallel v_{\epsilon }{\parallel }_{W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)}⩽C\left({\eta }^{\theta }\left(\epsilon \right)+{\mu }^{\frac{1}{2}}\left(\epsilon \right)\right){\parallel f\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)},d⩾2,\hfill \\ \hfill & \parallel v_{\epsilon }{\parallel }_{W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)}⩽C{\mu }^{\frac{1}{2}}\left(\epsilon \right){\parallel f\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)},\phantom{\left({\eta }^{\theta }\left(\epsilon \right)+\right)}d=1,\hfill \end{array}$$

with a constant C independent of $\epsilon$ and f. This completes the proof of Theorem 4.

7. Convergence of Spectra and Pseudospectra

In this section, we prove Theorems 5–7.

Proof of Theorem 5. 

We choose and fix a compact set $K\subset \mathbb{C}$, and, for $\lambda \in K\setminus \sigma \left({\mathcal{H}}_0\right)$, we consider the resolvent equation for the operator ${\mathcal{H}}_{\epsilon }$:

$$({\mathcal{H}}_{\epsilon }-\lambda )u=f.$$

According to Theorems 1–4, the resolvent ${({\mathcal{H}}_{\epsilon }-{\lambda }_0)}^{-1}$ is well defined. We apply this resolvent to the above equation and employ the definition of the operator ${\mathcal{B}}_{\epsilon }$ from (34):

$$u-(\lambda -{\lambda }_0){({\mathcal{H}}_{\epsilon }-{\lambda }_0)}^{-1}u=\left(\mathcal{I}-(\lambda -{\lambda }_0){({\mathcal{H}}_0-{\lambda }_0)}^{-1}\right)u-(\lambda -{\lambda }_0){\mathcal{B}}_{\epsilon }u={({\mathcal{H}}_{\epsilon }-{\lambda }_0)}^{-1}f.$$

(125)

Since

$$\begin{array}{cc}\hfill & \mathcal{I}-(\lambda -{\lambda }_0){({\mathcal{H}}_0-{\lambda }_0)}^{-1}=({\mathcal{H}}_0-\lambda ){({\mathcal{H}}_0-{\lambda }_0)}^{-1},\hfill \\ \hfill & {\left(\mathcal{I}-(\lambda -{\lambda }_0){({\mathcal{H}}_0-{\lambda }_0)}^{-1}\right)}^{-1}=({\mathcal{H}}_0-{\lambda }_0){({\mathcal{H}}_0-\lambda )}^{-1}=\mathcal{I}-(\lambda -{\lambda }_0){({\mathcal{H}}_0-\lambda )}^{-1}:=\mathcal{U}\left(\lambda \right),\hfill \end{array}$$

by applying the latter operator to Equation (125), we find:

$$\left(\mathcal{I}-(\lambda -{\lambda }_0)\mathcal{U}\left(\lambda \right){\mathcal{B}}_{\epsilon }\right)u=\mathcal{U}\left(\lambda \right){({\mathcal{H}}_{\epsilon }-{\lambda }_0)}^{-1}f.$$

(126)

We can inverse the operator on the left-hand side of the obtained equation provided

$$\parallel (\lambda -{\lambda }_0)\mathcal{U}\left(\lambda \right){\mathcal{B}}_{\epsilon }{\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)\to L_2(\mathrm{\Omega };{\mathbb{C}}^n)}⩽\left(\mathrm{\Lambda }\left(K\right)+{\mathrm{\Lambda }}^2\left(K\right){\parallel {({\mathcal{H}}_0-\lambda )}^{-1}\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)\to L_2(\mathrm{\Omega };{\mathbb{C}}^n)}\right)\zeta \left(\epsilon \right)<1.$$

(127)

Under such a condition, we can solve Equation (126) and find the resolvent ${({\mathcal{H}}_{\epsilon }-\lambda )}^{-1}$:

$${({\mathcal{H}}_{\epsilon }-\lambda )}^{-1}f=u={\left(\mathcal{I}-(\lambda -{\lambda }_0)\mathcal{U}\left(\lambda \right){\mathcal{B}}_{\epsilon }\right)}^{-1}\mathcal{U}\left(\lambda \right){({\mathcal{H}}_{\epsilon }-{\lambda }_0)}^{-1}f.$$

Hence, the set of $\lambda$ violating condition (127) includes the spectrum of ${\mathcal{H}}_{\epsilon }$ located in K, that is,

$$\sigma \left({\mathcal{H}}_{\epsilon }\right)\cap K\subseteq \left\{\lambda \in K:\left(\mathrm{\Lambda }\left(K\right)+{\mathrm{\Lambda }}^2\left(K\right){\parallel {({\mathcal{H}}_0-\lambda )}^{-1}\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)\to L_2(\mathrm{\Omega };{\mathbb{C}}^n)}\right)\zeta \left(\epsilon \right)⩾1\right\}$$

(128)

The inequality in the definition of the latter set on the right-hand side can be equivalently rewritten as

$$\parallel {({\mathcal{H}}_0-\lambda )}^{-1}{\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)\to L_2(\mathrm{\Omega };{\mathbb{C}}^n)}⩾{\varkappa }^{-1}\left(\epsilon \right)$$

and, in view of the definition of the $ϵ$-pseudospectrum in (34), inclusion (128) becomes the first inclusion in (35). The second inclusion in (35) can be established by literally reproducing the above argument with the interchanging ${\mathcal{H}}_0$ and ${\mathcal{H}}_{\epsilon }$. The proof is complete. □

Proof of Theorems 6. 

Here, we apply general Theorems 2.1 and 2.5 from [60] to our operators ${\mathcal{H}}_{\epsilon }$ and ${\mathcal{H}}_0$ in order to establish convergences (39) and (40). First, we need to show that the assumptions of the mentioned theorems from [60] are obeyed. The first assumptions of these theorems are condition (37) and condition (38) (for [60], Theorem 2.5). Other conditions are as follows:

• The operator ${\mathcal{H}}_{\epsilon }$ converges to ${\mathcal{H}}_0$ in the generalized sense;
• The function $\lambda \mapsto \parallel {({\mathcal{H}}_0-\lambda )}^{-1}{\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)\to L_2(\mathrm{\Omega };{\mathbb{C}}^n)}$ is non-constant on any open subset in the resolvent set of ${\mathcal{H}}_0$ (for [60], Theorem 2.1);
• The function $\lambda \mapsto \parallel {({\mathcal{H}}_0-\lambda )}^{-2^p}{\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)\to L_2(\mathrm{\Omega };{\mathbb{C}}^n)}$ is non-constant on any open subset in the resolvent set of ${\mathcal{H}}_0$ for each $p\in \mathbb{N}$ (for [60], Theorem 2.5).

We recall that the definition of convergence in the generalized sense can be found in ([63] Ch. IV, Section 2.6), and, according to ([63], Ch. IV, Section 2.6, Theorem 2.23), it is equivalent to the norm resolvent convergence of ${\mathcal{H}}_{\epsilon }$ to ${\mathcal{H}}_0$, which was established in Theorems 1–4. This ensures condition 1.

In order to confirm the other two conditions, we first use the fact that Hilbert spaces are uniformly convex spaces [61]. Then, we use Theorems 3.2 and 3.4 from [60], in accordance with which it is sufficient to show that the norms $\parallel {({\mathcal{H}}_0-\lambda )}^{-1}{\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)\to L_2(\mathrm{\Omega };{\mathbb{C}}^n)}$ and $\parallel {({\mathcal{H}}_0-\lambda )}^{-2^p}{\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)\to L_2(\mathrm{\Omega };{\mathbb{C}}^n)}$ decay as $\lambda \to -\infty$ along the real line. Let us show the latter fact. Given an arbitrary $f\in L_2(\mathrm{\Omega };{\mathbb{C}}^n)$, for $\lambda <{\lambda }_0$, we denote $u:={({\mathcal{H}}_0-\lambda )}^{-1}f$, and, by estimates (52) and (75) with $\lambda ={\lambda }_0$, we obtain:

$$\begin{array}{cc}\hfill c_{10}{\parallel u\parallel }_{W_2^m(\mathrm{\Omega };{\mathbb{C}}^n)}^2⩽& Re\left({\mathfrak{h}}_0(u,u)-{\lambda }_0{\parallel u\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}^2\right)\hfill \\ \hfill =& Re\left({\mathfrak{h}}_0(u,u)-\lambda {\parallel u\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}^2\right)+(\lambda -{\lambda }_0){\parallel u\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}^2\hfill \\ \hfill =& {(f,u)}_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}+(\lambda -{\lambda }_0){\parallel u\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}^2\hfill \\ \hfill ⩽& {\parallel f\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}{\parallel u\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}-|\lambda -{\lambda }_0{|\parallel u\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}^2,\hfill \end{array}$$

where $c_{10}$ is some fixed positive constant independent of u and $\lambda$. The obtained inequality implies:

$$\parallel {({\mathcal{H}}_0-\lambda )}^{-1}{f\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}={\parallel u\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}⩽\frac{1}{|\lambda -{\lambda }_0|}{\parallel f\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)}$$

and, therefore,

$$\parallel {({\mathcal{H}}_0-\lambda )}^{-2^p}{\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)\to L_2(\mathrm{\Omega };{\mathbb{C}}^n)}^{\frac{1}{2^p}}⩽{\parallel {({\mathcal{H}}_0-\lambda )}^{-1}\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)\to L_2(\mathrm{\Omega };{\mathbb{C}}^n)}⩽\frac{1}{|\lambda -{\lambda }_0|},\lambda <{\lambda }_0.$$

This gives the desired decay of the norms $\parallel {({\mathcal{H}}_0-\lambda )}^{-2^p}{\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)\to L_2(\mathrm{\Omega };{\mathbb{C}}^n)}$, and, by ([60], Theorems 3.2 and 3.4), we confirm conditions 2 and 3. Applying now ([60], Theorems 2.1 and 2.5), we obtain convergences (39) and (40). The proof is complete. □

Proof of Theorem 7. 

Since the operators ${\mathcal{H}}_0$ and ${\mathcal{H}}_{\epsilon }$ are self-adjoint, the norms of their resolvents satisfy the well-known identity

$${\parallel (\mathcal{H}-\lambda )}^{-1}{\parallel }_{L_2(\mathrm{\Omega };{\mathbb{C}}^n)\to L_2(\mathrm{\Omega };{\mathbb{C}}^n)}=\frac{1}{\mathrm{dist}(\lambda ,\sigma (\mathcal{H}\left)\right)},\mathcal{H}={\mathcal{H}}_0\mathrm{or}\mathcal{H}={\mathcal{H}}_{\epsilon },$$

and, together with the definition of the $ϵ$-pseudospectrum in (4), this yields:

$${\sigma }_{ϵ}\left(\mathcal{H}\right)=\left\{\lambda \in \mathbb{C}:\mathrm{dist}(\lambda ,\sigma (\mathcal{H}\left)\right)<ϵ\right\},\mathcal{H}={\mathcal{H}}_0\mathrm{or}\mathcal{H}={\mathcal{H}}_{\epsilon }.$$

(129)

The above identity with $\mathcal{H}={\mathcal{H}}_0$ and the first inclusion in (35) imply the inequality:

$$\underset{\lambda \in \sigma \left({\mathcal{H}}_{\epsilon }\right)}{sup}\underset{z\in \sigma \left({\mathcal{H}}_0\right)}{inf}|\lambda -z|⩽\underset{\lambda \in \sigma \left({\mathcal{H}}_{\epsilon }\right)}{sup}\varkappa \left(\epsilon \right)=\varkappa \left(\epsilon \right),$$

while identity (129) with $\mathcal{H}={\mathcal{H}}_{\epsilon }$ and the second inclusion in (35), in the same way, yields

$$\underset{\lambda \in \sigma \left({\mathcal{H}}_0\right)}{sup}\underset{z\in \sigma \left({\mathcal{H}}_{\epsilon }\right)}{inf}|\lambda -z|⩽\underset{\lambda \in \sigma \left({\mathcal{H}}_0\right)}{sup}\varkappa \left(\epsilon \right)=\varkappa \left(\epsilon \right).$$

The two obtained inequalities and definition (5) of the Hausdorff distance prove (41). The convergence of the spectral projections claimed in (42) is an immediate implication of the standard theorem on the norm resolvent convergence for self-adjoint operators; see, for instance, ([66], Ch. VIII, Section 7, Theore VIII.23). The proof is complete. □