Spectral stability of the curlcurl operator via uniform Gaffney inequalities on perturbed electromagnetic cavities

1.   Introduction

In this paper we study the spectral stability of the $curl curl$ operator on an electromagnetic cavity $\Omega$ in ${\mathbb{R}}^3$ upon perturbation of the shape of $\Omega$. The cavity $\Omega$ is a bounded connected open set (shortly, a bounded domain), the boundary of which is enough regular to guarantee the validity of the celebrated Gaffney inequality, that is

for all vector fields $u\in L^2(\Omega)^3$ with distributional $\operatorname{curl} u \in L^2(\Omega)^3$ and $\operatorname{div} u \in L^2(\Omega)$, and satisfying the so-called electric boundary conditions

Here $\nu$ denotes the unit outer normal to $\partial\Omega$ and $H^1(\Omega)$ is the standard Sobolev space of functions in $L^2(\Omega)$ with first order weak derivatives in $L^2(\Omega)$. It is classical that the Gaffney inequality holds for domains $\Omega$ with boundaries of class $C^{2}$, but the regularity can be relaxed in order to include boundaries of class $C^{1, \beta}$ with $\beta > 1/2$, see ^[22,38].

The eigenvalue problem under consideration is

and is immediately derived from the time-harmonic Maxwell's equations

where $E, H$ denote the spatial parts of the electric and the magnetic field respectively and $\omega > 0$ is the angular frequency. Indeed, taking the curl in the first equation of (1.3) and setting $\lambda = \omega^2$, one immediately obtains problem (1.2). Note that here the medium filling $\Omega$ is homogeneous and isotropic and for simplicity the corresponding electric permittivity $\varepsilon$ and magnetic permeability $\mu$ have been normalized by setting $\varepsilon = \mu = 1$. The boundary conditions are those of a perfect conductor, namely $\nu \times E = 0 \; {\rm{and }}\; H\cdot \nu = 0$. Thus, the vector field $u$ in (1.2) plays the role of the electric field $E$ (similarly, the magnetic field would satisfy the same equation but with the other boundary conditions $u\cdot \nu = 0$ and $\nu \times \operatorname{curl}\, u = 0$).

We observe that the study of electromagnetic cavities is quite important in applications, for example in designing cavity resonators or shielding structures for electronic circuits, see e.g., [24,Chp. 10]. We also refer to ^{[12,17,30,35,36,39]} for details and references concerning the mathematical theory of electromagnetism. See also ^{[13,14,15,16,31,32,37,41,42,43,44]}.

The spectrum of problem (1.2) is discrete and consists of a divergent sequence of positive eigenvalues $\lambda_n[\Omega]$ of finite multiplicity.

In this paper, we study the dependence of $\lambda_n[\Omega]$ and the corresponding eigenfunctions upon variation of $\Omega$. It seems to us that very little is known in the literature. The case of domain perturbations of the form $\Phi (\Omega)$ where $\Phi$ is a regular diffeomorphism from $\Omega$ to $\Phi (\Omega)$ are considered in ^[29] and ^[33] where differentiability results and Hadamard type formulas for shape derivatives are proved. We also quote the pioneering work ^[28] where the Hadamard formula was found on the base of heuristic computations. We note that shape derivatives are used in inverse electromagnetic scattering in ^[25,26,27].

The aim of the present paper is to prove spectral stability results under less stringent assumptions on the families of domain perturbations. For this purpose, we adopt the approach of ^[3] further developed in ^[2,18,19,20].

Given a fixed domain $\Omega$, we consider a family of domains $\Omega_{\epsilon}$, $\epsilon > 0$ converging to $\Omega$ as $\epsilon \to 0$. The convergence of $\Omega_{\epsilon}$ to $\Omega$ will be described by means of a fixed atlas ${\mathcal{A}}$, that is a finite collection of rotated parallelepipeds $V_j$, $j = 1, \dots, s$ covering the domains under consideration and such that if $V_j$ touches the boundaries of the domains then $\Omega\cap V_j$ and $\Omega_{\epsilon}\cap V_j$ are given by the subgraphs of two functions $g_j, g_{\epsilon, j}$ in two variables, say $\bar x = (x_1, x_2)$. Thus the convergence of $\Omega_{\epsilon}$ to $\Omega$ is understood in terms of the convergence of $g_{\epsilon, j}$ to $g_j$ as $\epsilon \to 0$.

It is not surprising that if $g_{\epsilon, j}$ converges uniformly to $g_j$ together with its first and second derivatives as $\epsilon \to 0$ (in which case one talks of $C^2$-convergence) then we have spectral stability of the $curl curl$ operator, which means that the eigenvalues and eigenfunctions of the problem in $\Omega_{\epsilon}$ converge to those in $\Omega$ as $\epsilon \to 0$. It is also not surprising that if $g_{\epsilon, j}$ converges uniformly to $g_j$ together with its first derivatives and

then we have spectral stability again. (These results are also immediate consequences of the results of the present paper.) The main question here is whether it is possible to relax condition (1.4). For example, if we assume that $g_{\epsilon, j}$ is of the form

where $\alpha > 0$ and $b_j$ is a fixed $C^{1, 1}$ function, condition $(1.4)$ is encoded by the inequality $\alpha \geq 2$. In this model case, the question is whether one can get spectral stability for $\alpha < 2$. Note that a profile of the form (1.5) is typical in the study of boundary homogenization problems and thin domains, see for example ^{[2,3,4,9,11,19,20,21]}.

This problem was solved for the biharmonic operator with intermediate boundary conditions (modelling an elastic hinged plate) in ^[3] where condition (1.4) is relaxed by introducing a suitable notion of weighted convergence which allows to prove spectral stability for $\alpha > 3/2$ in the model problem above. That condition is described here in (3.14). It is remarkable that the threshold $3/2$ is sharp since for $\alpha \le 3/2$ spectral stability does not occur for the problem discussed in ^[3] (in particular, it is proved in ^[3] that for $\alpha < 3/2$ a degeneration phenomenon occurs and for $\alpha = 3/2$ a strange term in the limit appears, as in many homogenization problems). An analogous trichotomy is found in ^[20] for the biharmonic operator subject to certain Steklov type boundary conditions.

In this paper, we prove that the relaxed convergence (3.14) guarantees the spectral stability of the $curl curl$ operator. Our result requires that the Gaffney inequality (1.1) holds for all domains $\Omega_{\epsilon}$ with a constant $C$ independent of $\epsilon$. Again, if one does not assume the validity of the uniform bound (1.4), then proving that a uniform Gaffney inequality holds is highly non-trivial. Here we manage to do this, by exploting the approach of [34,Ch. 14] based on the use of Sobolev multipliers and the notion of domains of class $\mathcal{M}^{3/2}_2(\delta)$. In particular, if we assume that

for all $\bar x, \bar y\in {\mathbb{R}}^2$, with $\beta \in]1/2, 1]$ and $M$ independent of $\epsilon$, and we also assume that the $\sup$-norms of functions $|\nabla g_{\epsilon, j } |$ are sufficienlty small, then our domains belong to the class $\mathcal{M}^{3/2}_2(\delta)$ with $\delta$ small enough. This allows to apply [34,Thm. 14.5.1] which guarantees the validity of a uniform $H^2$- a priori estimate for the Dirichlet Laplacian which, in turn, is equivalent to the uniform Gaffney inequality.

In conclusion, the convergence of the domains $\Omega_{\epsilon}$ to $\Omega$ in the sense of (3.14) combined with the validity of (1.6) and the smallness of the gradients of the profile functions $g_{\epsilon, j}$ guarantees the spectral stability of the $curl curl$ operator. Note that, in principle, since $\Omega$ is of class $C^1$ one may think of choosing from the very beginning an atlas which guarantees that the gradients of the profile functions are as small as required (indeed, it is enough to adapt the atlas to the tangent planes of a sufficiently big number of boundary points of $\Omega$). Then the convergence in the sense of (3.14) would imply the smallness of the gradients of the profile functions of $\Omega_{\epsilon}$ as well.

By setting $\beta = \alpha-1$, we deduce that a uniform Gaffney inequality holds for the example provided by (1.5) if $\alpha > 3/2$. Moreover, if $\alpha > 3/2$ then spectral stability occurs for the same example since in this case also the convergence (3.14) occurs.

The case $\alpha \le 3/2$ is more involved and we plan to address it in a forthcoming paper, see Remark 4. We note that if $\alpha < 3/2$ one cannot expect the validity of uniform Gaffney inequalities, in particular because the regularity assumptions $C^{1, \beta}$ for $\beta > 1/2$ is optimal for the validity of the Gaffney inequality itself, see ^[22,38].

One of the main tools used in this paper is a Piola-type transform which allows to pull back functions from $\Omega$ to $\Omega_{\epsilon}$ preserving the boundary conditions. In particular the transformation depends on $\epsilon$ and is constructed in such a way that for any fixed compact set $K$ contained in $\Omega\cap\Omega_{\epsilon}$, it does not modify the values of the vector fields on $K$ for $\epsilon$ sufficiently small. Our Piola transform is constructed by pasting together local Piola transforms defined in each local chart of the atlas and for this reason it is called here Atlas Piola transform. We believe that our construction has its own interest.

This paper is organized as follows. Section 2 is devoted to preliminaries and notation concerning the atlas classes, the functions spaces and the weak formulations of our problems. Section 3 is devoted to the construction of the Atlas Piola transform and to the proof of its main properties, see Theorem 2. In Section 4 we prove our main stability theorem, namely Theorem 4. Section 5 is devoted to the proof of uniform a priori estimates and uniform Gaffney inequalities - see Corollaries 1, 2 - and contains the corresponding applications to the spectral stability problems, see Theorems 8, 9.

2.   Preliminaries and notation

2.1.   Classes of open sets

In this paper we consider open sets $\Omega$ in ${\mathbb{R}}^N$, in particular in ${\mathbb{R}}^3$, with sufficiently regular boundaries. This means that $\Omega$ can be described in a neighborhood of any point of the boundary as the subgraph of a sufficiently regular function $g$ defined in a local system of orthogonal coordinates. The regularity of $\Omega$ depends on the regularity of the functions $g$. Since we aim at studying domain perturbation problems, following ^[8] and ^[20], we find convenient to use the notion of atlas, that is a collection ${\mathcal{A}}$ of rotated parallelepipeds $V_j$, $j = 1, \dots, s$, which cover $\Omega$ and such that if $V_j$ touches the boundary of $\Omega$ then $\Omega\cap V_j$ is a subgraph of a function $g_j$. The parallelepipeds will also be called local charts. More precisely, in Definition 1 below the atlas $\mathcal{A}$ is defined as $(\rho, s, s', \{V_j\}_{j = 1}^s, \{r_j\}^s_{j = 1})$ where $s$ is the total number of cuboids used to cover $\Omega$, $s'$ is the number of cuboids touching the boundary of $\Omega$, $V_j$ are the cuboids, $r_j$ are the rotations used to change variables in the representations of the local charts, and $\rho$ is a parameter controlling the minima and maxima of the functions $g_j$. Note that in this paper the atlas ${\mathcal{A}}$ will be often fixed, while the functions $g_j$, hence $\Omega$, will be perturbed.

Given a set $V\subset \mathbb{R}^N$ and a parameter $\rho > 0$, we write $V_\rho : = \{x \in V : d(x, \partial V) > \rho\}$.

Definition 1. Let $\rho > 0, s, s' \in \mathbb{N}, s' \leq s$ and $\{V_j\}_{j = 1}^s$ be a family of bounded open cuboids (i.e., rotations of rectangle parallelepipeds in $\mathbb{R}^N$) and $\{r_j\}^s_{j = 1}$ be a family of rotations in $\mathbb{R}^N$. We say that $\mathcal{A} = (\rho, s, s', \{V_j\}_{j = 1}^s, \{r_j\}^s_{j = 1}$) is an atlas in $\mathbb{R}^N$ with parameters $\rho, s, s', \{V_j\}_{j = 1}^s, \{r_j\}^s_{j = 1}$, briefly an atlas in $\mathbb{R}^N$.

A bounded domain $\Omega \subset \mathbb{R}^N$ is said to be of class $C^{k, \gamma}_M(\mathcal{A})$ with $k \in \mathbb{N} \cup \{0\}$, $\gamma \in [0, 1]$ and $M > 0$ if it satisfies the following conditions:

$({\rm{i}})$ $\Omega \subset \bigcup_{j = 1}^s (V_j)_\rho$ and $(V_j)_\rho \cap \Omega \neq \emptyset$;

$({\rm{ii}})$ $V_j \cap \partial\Omega \neq \emptyset$ for $j = 1, \dots, s'$ and $V_j \cap \partial \Omega = \emptyset$ for $s'+1 \leq j \leq s$;

$({\rm{iii}})$ for $j = 1, \dots, s$ we have

for $j = 1, \dots, s'$ we have

where $\bar{x} = (x_1, x_2)$,

and the functions $g_j \in C^{k, \gamma}(\overline{W_j})$ for any $j = 1, \dots, s'$. Moreover, for $j = 1, \dots, s'$

for all $\bar{x}\in \overline{W_j}$.

${\rm{(iv)}}$

for $j = 1, \dots, s'$.

We say that $\Omega$ is of class $C^{k, \gamma}(\mathcal{A})$ if it is of class $C^{k, \gamma}_M(\mathcal{A})$ for some $M > 0$; we say that $\Omega$ is of class $C^{k, \gamma}$ if it is of class $C^{k, \gamma}(\mathcal{A})$ for some atlas $\mathcal{A}$.

2.2.   Function spaces

In this section, we recall basic facts and notation for the function spaces that will be used in the following. We refer e.g., to [23,Ch. 2] for more details.

Here by $\Omega$ we denote a bounded domain - that is, a bounded connected open set - in ${\mathbb{R}}^3$. Since the differential problems under consideration are associated with self-adjoint operators, the space $L^2(\Omega)$ is understood here as a space of real-valued functions and is endowed with the scalar product $\int_{\Omega}u\cdot v \, dx$ defined for all vector fields $u, v\in L^2(\Omega)^3$. The space of vector fields $u\in L^2(\Omega)^3$ with distributional curl in $L^2(\Omega)^3$ is denoted by $H(\operatorname{curl}, \Omega)$ and is endowed with the norm defined by

for all $u\in H(\operatorname{curl}, \Omega)$. The closure in $H(\operatorname{curl}, \Omega)$ of the space of ${\mathcal{C}}^{\infty}$-functions with compact support in $\Omega$ is denoted by $H_{0}(\operatorname{curl}, \Omega)$. The following lemma characterizes the space $H_0(\operatorname{curl}, \Omega)$ and is analogous to the well-known characterization of the Sobolev space $H^1_0(\Omega)$ (see e.g., ^[6]). We include a short proof. Here by $v^0$ we denote the extension-by-zero of a vector field $v$, that is

Lemma 1. Let $\Omega$ be a bounded open set of class $C^{0, 1}$ and $u \in H(\operatorname{curl}, \Omega)$. Then $u \in H_0(\operatorname{curl}, \Omega)$ if and only if $u^0\in H(\operatorname{curl}, \mathbb{R}^3)$, in which case $\operatorname{curl}(u^0) = (\operatorname{curl}u)^0$.

Proof. Suppose that $u^0$ belongs to $H(\operatorname{curl}, \mathbb{R}^3)$. Thus, there exists $v \in (L^2(\mathbb{R}^3))^3$ such that

Since it holds in particular for all test functions $\varphi \in (C^\infty_c(\Omega))^3$, then necessarily $v = \operatorname{curl}u$ on $\Omega$. On the other hand, since we can take any $\varphi \in (C^\infty_c(\mathbb{R}^3 \setminus \bar{\Omega}))^3,$ we see that $v = 0$ outside $\Omega$. Hence we can rewrite (2.1) as follows

By [23,Lemma 2.4] it follows that $u \in H_0(\operatorname{curl}, \Omega)$. The converse implication is straighforward.

We note that if $\Omega$ is sufficiently regular, say of class $C^{0, 1}$, the space $H_0(\operatorname{curl}, \Omega)$ coincides with the set of square integrable vector fields in $\Omega$ whose curl is also square integrable, and such that their tangential trace at the boundary $\partial \Omega$ is zero (see [23,Thm. 2.12]). In particular, we have that

where $C^\infty(\bar{\Omega})$ denotes smooth compactly supported functions of $\mathbb{R}^3$ restricted to $\bar{\Omega}$.

We denote by $H(\operatorname{div}, \Omega)$ the space of vector fields $u\in L^2(\Omega)^3$ with distributional divergence in $L^2(\Omega)^3$, endowed with the norm defined by

for all $u\in H(\operatorname{div}, \Omega)$. Finally, we set

and we consider the space

endowed with the norm defined above, that is $||u||_{ X_{\rm{N}}(\Omega) }\! = \! ||u||_{ X(\Omega) }\!$ for all $u\in X_{\rm{N}}(\Omega)$. We also set $X_{\rm{N}}(\operatorname{div} 0, \Omega) : = \{u \in X_{\rm{N}}(\Omega) : \operatorname{div} u = 0 \, \, {\rm{in}}\ \Omega \}$.

Recall that $H^1(\Omega)$ is the standard Sobolev space of functions in $L^2(\Omega)$ with first order weak derivatives in $L^2(\Omega)$. The celebrated Gaffney inequality allows to prove that the space $X_{{\rm{N}}} (\Omega)$ is continuously embedded into the space $H^1(\Omega)^3$ provided $\Omega$ is sufficiently regular. Namely, we have the following result, see e.g., [23,Theorem 3.7].

Theorem 1. Let $\Omega$ is a bounded open set in ${\mathbb{R}}^3$ of class $C^{1, 1}$. Then $X_{\rm{N}}(\Omega)$ is continuously embedded into $H^1(\Omega)^3$, and there exists $C > 0$ such that the Gaffney inequality

holds for all $u\in X_{\rm{N}}(\Omega)$.

By the previous theorem it immediately follows that the space $X_{{\rm{N}}} (\Omega)$ is compactly embedded into $L^2(\Omega)^3$, since this is true for the space $H^1(\Omega)^3$.

As we shall see, the regularity assumptions on $\Omega$ in Theorem 1 can be relaxed since the inequality holds for domains of class $C^{1, \beta}$ with $\beta \in]1/2, 1]$, but some care is required, see Section 5.

2.3.   Weak formulations and resolvent operators

Since for our purposes we prefer to work in the space $X_{{\rm{N}}} (\Omega)$ rather than in the space $X_{{\rm{N}}}(\operatorname{div}\, 0, \Omega)$, following ^[14,15], we introduce a penalty term in the equation and we replace problem (1.2) by the problem

where $\tau$ is any fixed positive real number.

It is easy to see that problem (2.4) can be formulated in the weak sense as follows

in the unknowns $u\in X_{{\rm{N}}} (\Omega)$ and $\lambda \in {\mathbb{R}}$. Is obvious that the solutions of (1.2) are exactly the divergence free solutions of (2.5). (Moreover, the weak formulation of (1.2) can be obtained simply by replacing $X_{{\rm{N}}} (\Omega)$ by $X_{{\rm{N}}} (\operatorname{div} 0, \Omega)$ in (2.5).)

Problem (2.5) admits also solutions which are not divergence free and which are given by the gradients of the solutions to the Helmohltz equation with Dirichlet boundary conditions. Namely, $u = \nabla f$ where $f$ solves the following problem

with $\Lambda = \frac{\lambda}{\tau}$. In fact, we have the following result from ^[15].

Lemma 2. If $\Omega$ is a bounded domain in ${\mathbb{R}}^3$ of class ${\mathcal{C}}^{0, 1}$, then the set of all eigenpairs $(\lambda, u)$ of problem $(2.4)$ is the union of the set of all eigenpairs $(\lambda, u)$ of problem $(1.2)$ and the set of all eigenpairs of the form $(\tau \Lambda, \nabla f)$ where $(\Lambda, f)$ is an eigenpair of problem $(2.6)$.

Thus, we can directly study problem (2.5) rather than the original problem (1.2): this will always be understood in the following. In fact, studying the spectral stability of problem (2.5) is equivalent to studying the spectral stability of problem (1.2) because the spurious eigenpairs introduced by the penalty term are given by the eigenpairs of the Dirichlet Laplacian which are stable for our class of domain perturbations (see ^[3]).

In order to study spectral stability problems, it is also convenient to recast the eigenvalue problems under consideration in the form of eigenvalue problems for compact self-adjoint operators and this can be done by passing to the analysis of the corresponding resolvent operators. A direct way of doing so, consists in defining the operator $T$ from $X_{{\rm{N}}} (\Omega)$ to its dual $(X_{{\rm{N}}} (\Omega))'$ by setting

for all $u, \varphi \in X_{{\rm{N}}} (\Omega)$, and considering the map $J$ from $L^2(\Omega)^3$ to $(X_{{\rm{N}}} (\Omega))'$ defined by

for all $u\in L^2(\Omega)^3$ and $\varphi \in X_{{\rm{N}}} (\Omega)$. By restricting $J$ to $X_{{\rm{N}}}(\Omega)$ (and denoting the restriction by the same symbol $J$), and using the Riesz Theorem it turns out that the operator $T+J$ is a homeomorphism from $X_{{\rm{N}}} (\Omega)$ to its dual. The inverse operator $(T+J)^{-1}$ will serve for our purposes as discussed above. In fact, the following theorem holds.

Lemma 3. If $\Omega$ is a bounded domain in ${\mathbb{R}}^3$ such that the embedding $\iota$ of $X_{{\rm{N}}} (\Omega)$ into $L^2(\Omega)^3$ is compact, then the operator $S_{\Omega}$ from $L^2(\Omega)^3$ to itself defined by

is a non-negative compact self-adjoint operator in $L^2(\Omega)^3$ whose eigenvalues $\mu$ are related to the eigenvalues $\lambda$ of problem $(2.5)$ by the equality $\mu = (\lambda +1)^{-1}$.

By the previous lemma and standard spectral theory it follows that the spectrum $\sigma (S_{\Omega})$ of $S_{\Omega}$ can be represented as $\sigma (S_{\Omega}) = \{0\} \cup \{\mu_n (\Omega) \}_{n\in {\mathbb{N}}}$, where $\mu_n(\Omega)$, $n\in {\mathbb{N}}$ is a decreasing sequence of positive eigenvalues of finite multiplicity, which converges to zero. Consequently, the eigenvalues of problem (2.5) can be represented by the sequence $\lambda_n(\Omega)$, $n\in {\mathbb{N}}$ defined by $\lambda_n(\Omega) = \mu_n^{-1}(\Omega)-1$. Moreover, the classical Min-Max Principle yields the following variational representation

3.   A Piola-type approximation of the identity

Given two domains $\Omega$ and $\tilde \Omega$ in ${\mathbb{R}}^3$ and a diffeomorphism $\Phi :\tilde\Omega \to \Omega$ of class $C^{1, 1}$, the standard way to pull-back vector fields from $X_{{\rm{N}}}(\Omega)$ to $X_{{\rm{N}}}(\tilde\Omega)$ consists in using the (covariant) Piola transform defined by

for all $v\in X_{{\rm{N}}}(\Omega)$, see e.g., ^[35]. In fact, it turns out that $v\in H_0(\operatorname{curl}, \Omega)$ if and only if $u\in H_0(\operatorname{curl}, \tilde \Omega)$, in which case we have

Note that for functions $u, v$ in $H^1$ we also have

and in this case $v \in X_{{\rm{N}}}(\Omega) \cap H^1(\Omega)^3$ if and only if $u \in X_{{\rm{N}}}(\tilde \Omega) \cap H^1(\tilde \Omega)^3$. See ^[33] for more details. Unfortunately, given two domains $\Omega$ and $\tilde \Omega$, in general it is not possible to define explicitly a diffeomorphism between $\Omega$ and $\tilde \Omega$ (even if it is known a priori that the two domains are diffeomorphic). Nevertheless, it is important for our purposes to define an operator which allows to pass from $X_{{\rm{N}}}(\Omega)$ to $X_{{\rm{N}}}(\tilde\Omega)$ as the Piola transform does. This can be done by assuming that $\Omega$ and $\tilde\Omega$ belong to the same atlas class and using a partition of unity in order to paste together Piola transforms defined locally, as described in the following. Note that the specific choice of local Piola transforms reflects our need for a transformation close to the identity.

Let $\mathcal{A}$ be a fixed atlas in ${\mathbb{R}}^3$ and let $\Omega, \tilde \Omega$ be two domains of class $C^{1, 1}(\mathcal{A})$. Let $g_j, \tilde g_j$ be the profile functions of $\Omega$ and $\tilde \Omega$ as in Definition 1. Assume that $k \in]0, +\infty[$ is such that

For any $j = 1, \dots, s'$ we set

and we define the map $h_{j}: r_j(\overline{\tilde \Omega \cap V_j}) \to \mathbb{R}$

and the map

Note that $\Phi_{j}$ coincides with the identity map on the set

Finally, if $s'+1\leq j\leq s$ we define $\Phi_{j}: r_j(\overline{V_j}) \to r_j(\overline{V_j})$ to be the identity map.

Observe that since $h_{j} \in C^{1, 1}(r_j(\overline{\tilde\Omega \cap V_j}))$, then $\Phi_{j}$ is of class $C^{1, 1}$, and so is the following map

An easy computation shows that if

for some constant $\alpha \in]0, 1[$ then

Let $\{\psi_j\}_{j = 1}^s$ be a $C^{\infty}$-partition of unity associated with the open cover $\{V_j\}_{j = 1}^s$ of the compact set $\overline{\cup_{j = 1}^s(V_{j})_{\rho}}$ that is $0\leq \psi_j \leq 1$, $\operatorname{supp}(\psi_j) \subset V_j$ for all $j = 1, \dots, s$, and $\sum_{j = 1}^s \psi_j \equiv 1$ in $\overline{\cup_{j = 1}^s(V_{j})_{\rho}}$, in particular also in $\overline{\Omega\cup\tilde\Omega}$. Note that this is a partition of unity is independent of $\Omega, \tilde \Omega$ in the atlas class under consideration.

Since for any $\varphi \in X_{{\rm{N}}}(\Omega)$ we have $\varphi = \sum_{j = 1}^s \varphi_j$ where $\varphi_j = \psi_j \varphi$, then it is natural to give the following definition (note that here we consider open sets of class $C^{1, 1}$ hence the spaces $X_{{\rm{N}}}$ are embedded into $H^1$).

Definition 2. Let ${\mathcal{A}}$ be an atlas in ${\mathbb{R}}^3$ and $\Omega, \tilde\Omega$ be two domains of class $C^{1, 1}({\mathcal{A}})$. Assume that $k > 0$ satisfies $(3.4)$, and $\{\psi_j\}_{j = 1}^s$ is a partition of unity as above. The Atlas Piola transform from $\Omega$ to $\tilde \Omega$, with parameters ${\mathcal{A}}$, $k$, and $\{\psi_j\}_{j = 1}^s$, is the map from $X_{{\rm{N}}}(\Omega)$ to $X_{{\rm{N}}}(\tilde\Omega)$ defined by

for all $\varphi \in X_{{\rm{N}}}(\Omega)$, where

for any $j = 1, \dots, s'$.

Note that $\mathcal{P}\varphi \in X_{{\rm{N}}}(\tilde{\Omega})$ because $(\varphi_j \circ \Psi_{j}) \operatorname{D}\Psi_{j} \in X_{{\rm{N}}}(\tilde{\Omega} \cap V_j)$ (observe that the support of $\varphi_j$ is compact in $V_j$), hence $\tilde{\varphi}_j \in X_{{\rm{N}}}(\tilde{\Omega})$.

This Atlas Piola transform will be used in this paper for a family $\Omega_\epsilon, \epsilon > 0$ of domains of class $C^{1, 1}(\mathcal{A})$, converging in some sense to a domain $\Omega$ of class $C^{1, 1}(\mathcal{A})$. In this case, $\Omega_{\epsilon}$ will play the role of the domain $\tilde \Omega$ and the corresponding transformation will allow us to pass from $X_{{\rm{N}}}(\Omega)$ to $X_{{\rm{N}}}(\Omega_{\epsilon})$.

Given a family of domains $\Omega_{\epsilon}$, $\epsilon > 0$, and a fixed domain $\Omega$, all of class $C^{1, 1}(\mathcal{A})$, we shall denote by $g_{\epsilon, j}$ and $g_j$ the corresponding profile functions (defined on $W_j$) of $\Omega_{\epsilon}$ and $\Omega$ respectively, as in Definition 1. Following ^[3,20], we use a notion of convergence for the open sets $\Omega_{\epsilon}$ to $\Omega$, which is expressed in terms of convergence of the the profile functions $g_{\epsilon, j}$ to $g_j$. Namely, we assume that for any $\epsilon > 0$ there exists $\kappa_\epsilon > 0$ such that for any $j\in \{1, \dots, s'\}$

Note that if every function $g_{\epsilon, j}$ converges to $g_j$ uniformly together with the first order derivatives and condition (1.4) is satisfied (in particular, if the second order derivatives of $g_{\epsilon, j}$ converge uniformly to those of $g_j$) then conditions (3.14) are fulfilled, see ^[3]. Note also that the exponent $3/2$ in (3.14) turns out to be optimal in the analysis of ^[3] and plays a crucial role for instance in proving inequality (3.41).

We now fix a partition of unity $\{ \psi_j\}_{j = 1}^s$ associated with the covering of cuboids of the atlas ${\mathcal{A}}$ as above, and independent of $\Omega_{\epsilon}$ and $\Omega$. We also choose $k = 6 \kappa_{\epsilon}$ and we denote by ${\mathcal{P}}_{\epsilon}$ the Atlas Piola transform from $\Omega$ to $\Omega_{\epsilon}$ (with parameters ${\mathcal{A}}$, $k$, $\{ \psi_j\}_{j = 1}^s$). Note that conditions (3.4), (3.10) (3.11) are satisfied with $\alpha = 1/2$ if $\epsilon$ is sufficiently small.

In the following, we shall denote by $\hat g_{\epsilon, j}$, $h_{\epsilon, j}$, $\Phi_{\epsilon, j}$, $K_{\epsilon, j}$, $\Psi_{\epsilon, j}$, $\tilde \varphi_{\epsilon, j}$ all quantities defined in (3.5), (3.6), (3.7), (3.8), (3.9), (3.13) respectively, with $\tilde \Omega = \Omega_{\epsilon}$ and $k = 6\kappa_{\epsilon}$.

Then we can prove the following theorem. We note that in the proof, some technical issues related to pasting together functions defined in different charts are treated in the spirit of the arguments used in ^[20] for the Sobolev spaces $H^2(\Omega)$.

Theorem 2. Let $\Omega_\epsilon, \epsilon > 0$, and $\Omega$ be bounded domains of class $C^{1, 1}(\mathcal{A})$. Assume that $\Omega_{\epsilon}$ converges to $\Omega$ as $\epsilon\to 0$ in the sense of $(3.14)$. Let ${\mathcal{P}}_{\epsilon}$ be the Atlas Piola transform from $\Omega$ to $\Omega_{\epsilon}$ defined for $\epsilon$ sufficiently small as above. Then the following statements hold:

$({\rm{i}})$ for any $\epsilon > 0$ the function ${\mathcal P}_\epsilon$ maps $X_{{\rm{N}}}(\Omega)$ to $X_{{\rm{N}}}(\Omega_\epsilon)$ with continuity;

$({\rm{ii}})$ for any compact set ${\mathcal{K}}$ contained in $\Omega$ there exists $\epsilon_{\mathcal{K}} > 0$ such that

for all $\epsilon \in]0, \epsilon_{\mathcal{K}}[$ and $\varphi \in X_{{\rm{N}}}(\Omega)$;

$({\rm{iii}})$ the limit

holds for all $\varphi \in X_{{\rm{N}}}(\Omega)$;

$({\rm{iv}})$ the limit

holds for all $\varphi \in X_{{\rm{N}}}(\Omega)$.

Proof. Let $\varphi\in X_{{\rm{N}}}(\Omega)$ be fixed. Note that $\Omega$ is of class $C^{1, 1}$ hence the Gaffney inequality holds and $\varphi\in H^1(\Omega)^3$. Moreover, $\varphi_j\in X_{{\rm{N}}}(\Omega)$ for all $j = 1, \dots, s'$ hence $\tilde \varphi_{\epsilon, j }$ belongs to $X_{{\rm{N}}}(\Omega_{\epsilon })$ for all $j = 1, \dots, s'$. It follows that ${\mathcal P}_\epsilon \varphi \in X_{{\rm{N}}}(\Omega_\epsilon)$. The continuity of the operator follows by standard calculus, the Gaffney inequality and formulas (3.2), (3.3). Thus, statement (ⅰ) holds.

For any fixed compact set ${\mathcal{K}}$ contained in $\Omega$, since $\hat g_{\epsilon, j}$ converges uniformly to $g_j$, we have

for all $j = 1, \dots, s'$ and $\epsilon$ sufficiently small; this, combined with the fact that $\Phi_{\epsilon, j}$ coincides with the identity on $K_{\epsilon, j }$, it follows that $\tilde \varphi_{\epsilon, j} = \varphi_j$ on ${\mathcal{K}}$ for all $\epsilon$ sufficiently small and (3.15) follows.

We now prove statement (ⅲ). We have to prove the following limiting relations:

We begin by proving (3.18). To see this, it just suffices to show that

and

for any $j, h \in \{1, \dots, s'\}$ and $i\in \{s'+1, \dots, s\}$. We will only show (3.21), since the computations to prove (3.22) are similar. We will first see that

Notice that for any $j \in \{1, \dots, s'\}$ we have $|{(\Omega \cap V_j) \setminus r_j^{-1}(K_{\epsilon, j})}|\to 0$ as $\epsilon$ goes to 0. Moreover, if $w\in \mathbb{R}^3$ is a vector, then $|{w \operatorname{D}\Psi_{\epsilon, j}}| = |{w \operatorname{D}\Phi_{\epsilon, j}}| \leq C |{w}|$, since

and the first derivatives of $h_{\epsilon, j}$ are all bounded due to the hypothesis on the functions $g_{\epsilon, j}$ (see also (3.37)). Note that here and in what follows, by $c$ we denote a constant independent of $\epsilon$ which may vary from line to line. Then by using also (3.11), we have

By (3.23) we deduce that

Indeed, since $\Psi_{\epsilon, j}$ is the identity on $r_j^{-1}(K_{\epsilon, j}) \subset \Omega \cap \Omega_{\epsilon}$, using (3.23) yields

Observe now that

It is obvious that

Here and in the following we will make use of the identity

Observe that by (3.23), (3.24) we get

and

Hence, by formula (3.27), we see that the second term of the sum in (3.25) vanishes as $\epsilon$ goes to zero, hence we deduce the validity of (3.21) from (3.25) and (3.26).

We now prove (3.19). Again, we need to check that

for any $j, h \in \{1, \dots, s'\}$. Note that

and recall that $\Psi_{\epsilon, j} = r_j^{-1} \circ \Phi_{\epsilon, j} \circ r_j$. Moreover, by (3.2) we have

so that

Then, with computations analogous to those performed above, we get

It is also obvious that

and thus

By using the same argument above, formula (3.27) together with the new identities (3.31)–(3.33), we obtain (3.30).

Finally, we prove (3.20). To do so, we need to prove that

for any $j, h \in \{1, \dots, s'\}$. Here and in the rest of the proof, the vectors under consideration will be represented as follows: $\varphi_j = (\varphi_j^1, \varphi_j^2, \varphi_j^3)$ and $\Psi_{\epsilon, j} = (\Psi_{\epsilon, j}^1, \Psi_{\epsilon, j}^2, \Psi_{\epsilon, j}^3)$.

Since $\varphi \in X_{{\rm{N}}}(\Omega)$ and the Gaffney inequality holds on $\Omega$, it follows that $\varphi \in H^1(\Omega)^3$. Thus, recalling that $\tilde{\varphi}_{\epsilon, j}(x) = (\varphi_j \circ \Psi_{\epsilon, j}(x)) \operatorname{D}\Psi_{\epsilon, j}(x)$ for all $x \in \Omega_\epsilon \cap V_j$, it is possible to apply the chain rule and obtain that

where the terms in the first sum are called of type A and the others are called terms of type B.

Recall that $h_{\epsilon, j}$ are the functions in (3.6) used to define the diffeomorphisms $\Phi_{\epsilon, j}$. We observe that by the Leibniz rule we have

hence by standard calculus (note that the denominator in the previous formula is the constant $k = 6\kappa_{\epsilon}$) and (3.14) we get

Therefore

for all $\epsilon > 0$ sufficiently small. It follows by the definitions of $\Psi_{\epsilon, j}, \Phi_{\epsilon, j}$, by (3.37) and part $(iii)$ of condition (3.14), that for all $m, i = 1, 2, 3$

We claim that

To prove that, we analyse first the terms of type A in (3.35). By changing variables in integrals we get:

We now consider the terms of type B. By setting $\eta_j(z) : = \varphi_j(r^{-1}_j(z))$ and recalling (3.38) we have that

Here we have used the following one dimensional embedding estimate for Sobolev functions (see e.g., Burenkov ^[7]):

for all $f \in H^1(a, b)$, where the constant $c = c(d)$ is uniformly bounded for $|{b-a}| > d$. We conclude that (3.39) holds.

Using (3.39), the fact that $\Psi_{\epsilon, j}$ in $r^{-1}_j(K_{\epsilon, j})$ coincides with the identity and that $|{(\Omega \cap V_j) \setminus r^{-1}_j(K_{\epsilon, j})}| \to 0$ as $\epsilon$ goes to 0, we deduce that

and

With (3.39), (3.42) and (3.43) in mind, in order to prove (3.34), it suffices to reproduce the same argument used before starting from (3.25) combined with formula (3.27). We omit the details. Thus statement (ⅲ) is proved.

The proof of statement (ⅳ) follows by the same considerations above. First of all, for any $j = 1, \dots, s'$ the function $\tilde \varphi_{\epsilon, j}$ coincides with $\varphi_j$ on $r_j^{-1}(K_{\epsilon, j})$. Thus ${\mathcal P}_{\epsilon}\varphi = \varphi$ on $(\cup_{j = 1, \dots, s'} r_j^{-1}(K_{\epsilon, j})) \cup (\cup_{j = s'+1, \dots, s} V_j)$. It follows that

This combined with by the limiting relations (3.23), (3.31) and (3.39) yields the validity of statement (ⅳ).

4.   Spectral stability

Let $\Omega_\epsilon$, $\epsilon > 0$, and $\Omega$ be bounded domains in $\mathbb{R}^3$ of class $C^{1, 1}(\mathcal{A})$. For simplicity, it is convenient to set $\Omega_0 = \Omega$. In this section, we prove that if $\Omega_{\epsilon}$ converges to $\Omega$ as $\epsilon\to 0$ in the sense of (3.14), and a uniform Gaffney inequality holds on the domains $\Omega_{\epsilon}$ then we have spectral stability for the $\operatorname{curl} \operatorname{curl}$ operator defined on the domains $\Omega_{\epsilon}$ with respect to the reference domain $\Omega$. By uniform Gaffney inequality, we mean that the spaces $X_{{\rm{N}}}(\Omega_{\epsilon})$ are embedded into $H^1(\Omega_{\epsilon})^3$ and there exists a positive constant $C$ independent of $\epsilon$ such that

for all $u\in X_{{\rm{N}}}(\Omega_{\epsilon})$ and $\epsilon > 0$. (Note that by Theorem 1, for every $\epsilon > 0$ there exists a positive constant $C_{\epsilon}$, possibly depending on $\epsilon$, such that (4.1) holds, but here we need a constant independent of $\epsilon$).

To do so, for any $\epsilon \geq 0$, we denote by $S_{\epsilon}$ the operator $S_{\Omega_{\epsilon}}$ from $L^2(\Omega_{\epsilon})$ to itself defined in Lemma 3. Recall that if $f_\epsilon\in L^2(\Omega_\epsilon)^3$ is the datum of the following Poisson problem

then the unique solution $v_{\epsilon }\in X_{{\rm{N}}}(\Omega_{\epsilon })$ is precisely $S_{\epsilon}f_{\epsilon }$, that is $v_{\epsilon } = S_{\epsilon}f_{\epsilon }$. Recall that $\tau$ is a fixed positive constant (one could normalise it by setting $\tau = 1$ but we prefer to keep it as it is also with reference to other papers where it is important to have the possibility to use different values of $\tau$, see for example [33,Remark 2.13]). In this section we prove that $S_{\epsilon}$ compactly converges to $S_{0}$ as $\epsilon \to 0$, and this implies spectra stability. This has to be understood in the following sense.

We denote by $E = \{ E_\epsilon \}_{\epsilon > 0}$ the family of the extension-by-zero operators $E_\epsilon:L^2(\Omega)^3 \to L^2(\Omega_\epsilon)^3$ defined by

for all $\varphi \in L^2(\Omega)^3$. Note that under our assumptions we have that for all $\varphi \in L^2(\Omega)^3$

since $|{\Omega \setminus (\Omega_\epsilon \cap \Omega)}| \to 0$ as $\epsilon$ goes to 0. We recall the following definition from ^[40], see also ^[1] and ^[10].

Definition 3. Let $u_{\epsilon}\in L^2(\Omega_{\epsilon })$, for $\epsilon > 0$, be a family of functions. We say that $u_{\epsilon}$ $E$-converges to $u_0\in L^2(\Omega)$ as $\epsilon \to 0$ and we write $u_{\epsilon}\xrightarrow{E} u_0$ if

We also say that $S_{\epsilon }$ $E$-converges to $S_0$ as $\epsilon \to 0$ and we write $S_{\epsilon}\xrightarrow{EE} S_0$ if for any family of functions $f_{\epsilon }\in L^2(\Omega_{\epsilon })$, we have

Finally, we say that $S_{\epsilon }$ $E$-compact converges to $S_0$ as $\epsilon \to 0$ and we write $S_{\epsilon}\xrightarrow{C} S_0$ if $S_{\epsilon}\xrightarrow{EE} S_0$ and for any family of functions $f_{\epsilon }\in L^2(\Omega_{\epsilon })$, with $\| f_{\epsilon } \|_{L^2(\Omega_{\epsilon})} = 1$ and any sequence of positive numbers $\epsilon_n$ with $\epsilon_n\to 0$, there exists a subsequence $\epsilon_{n_k}$ and $u\in L^2(\Omega)$ such that $S_{\epsilon_{n_k}}f_{n_k} \xrightarrow{E} u$.

The following theorem from [40,Thm. 6.3] holds.

Theorem 3. If $S_{\epsilon}\xrightarrow{C} S_0$ then the eigenvalues of the operator $S_{\epsilon}$ converge to the eigenvalues of the operator $S_0$, and the eigenfunctions of the operator $S_{\epsilon}$ $E$-converge to the eigenfunctions of the operator $S_{0}$ as $\epsilon \to 0$.

If we denote by $\mu_n(\epsilon)$, $n\in \mathbb{N}$ the sequence of eigenvalues of $S_{\epsilon}$ and by $u_n(\epsilon)$, $n\in \mathbb{N}$ a corresponding orthonormal sequence of eigenfunctions, then the stability of eigenvalues and eigenfunctions stated above has to be interpreted in the following sense:

$({\rm{i}})$ $\mu_n(\epsilon)\to \mu_n(0)$ as $\epsilon \to 0$.

$({\rm{ii}})$ For any sequence $\epsilon_k$, $k\in \mathbb{N}$, converging to zero there exists an orthonormal sequence of eigenfunctions $u_n(0)$, $n\in \mathbb{N}$ in $L^2(\Omega)^3$ such that, possibly passing to a subsequence of $\epsilon_k$, $u_n(\epsilon_k) \xrightarrow{E} u_n(0)$.

$({\rm{iii}})$ Given $m$ eigenvalues $\mu_n(0), \dots, \mu_{n+m-1}(0)$ with $\mu_n(0)\ne \mu_{n-1}(0)$ and $\mu_{n+m-1}(0)\ne \mu_{n+m}(0)$ and corresponding orthonormal eigenfunctions $u_n(0), \dots, u_{n+m-1}(0)$, there exist $m$ orthonormal generalized eigenfunctions (i.e., linear combinations of eigenfunctions) $v_n(\epsilon), \dots, v_{n+m-1}(\epsilon)$ associated with $\mu_n(\epsilon), \dots, \mu_{n+m-1}(\epsilon)$ such that $v_{n+i}(\epsilon) \xrightarrow{E} u_{n+i}(0)$ for all $i = 0, 1, \dots, m-1$.

Recall that $\mu$ is an eigenvalue of $S_\epsilon$ if and only if $\lambda = \mu^{-1}$ is an eigenvalue of the problem

and that the corresponding eigenfunctions are the same. Note that the eigenvalues of (4.4) differ from those of (2.4) just by a translation. Thus, studying the stability of eigenvalues and eigenfunctions of the problem (4.4) or (2.4), is equivalent to studying the spectral stability of the family of operators $S_{\epsilon}$. To do so, we recall that the weak formulation of problem (4.2) reads as follows: find $v_\epsilon \in X_{{\rm{N}}}(\Omega_\epsilon)$ such that

for all $\eta \in X_{{\rm{N}}}(\Omega_\epsilon)$.

Suppose that for some $C > 0$ we have that $||{f_\epsilon}||_{L^2(\Omega_\epsilon)^3} \leq C$ for every $\epsilon > 0$. Then, setting $\eta = v_\epsilon$ in (4.5) and observing that $\int_{\Omega_\epsilon} f_\epsilon \cdot v_\epsilon \, dx \leq \frac{1}{2} \int_{\Omega_\epsilon}|{f_\epsilon}|^2 \, dx + \frac{1}{2} \int_{\Omega_\epsilon} |{v_\epsilon}|^2 \, dx$, we get

This in turn implies that for all $\epsilon > 0$

In order to prove the $E$-convergence of the operators $S_{\epsilon}$, it is necessary to consider the limit of functions $v_{\epsilon}$. We note that if $\Omega \subset \Omega_{\epsilon}$ for all $\epsilon > 0$ then it would suffice to consider the restriction of $v_{\epsilon}$ to $\Omega$ and pass to the weak limit in $\Omega$. Otherwise, it is convenient to extend functions $v_{\epsilon }$ to the whole of ${\mathbb{R}}^3$. To do so, we observe that by the uniform Gaffney inequality combined with inequality (4.6) it follows that $\| v_{\epsilon }\|_{H^1(\Omega_{\epsilon})^3}$ is uniformly bounded. Moreover, the domains $\Omega_{\epsilon}$ belong to the same Lipschitz class $C^{0, 1}_M({\mathcal{A}})$ for some $M > 0$ hence the functions $v_{\epsilon }$ can be extended to the whole of ${\mathbb{R}}^3$ with a uniformly bounded norm, see e.g., ^[7]. Thus, in the sequel we shall directly make the following assumption:

Thus the family $\{v_\epsilon\rvert_{ \Omega } \}_{\epsilon > 0}$ is bounded in $H(\operatorname{curl}; \Omega) \cap H(\operatorname{div}; \Omega)$ and we can extract a sequence $\{v_{\epsilon_n}\rvert_\Omega\}_{n \in \mathbb{N}}$, with $\epsilon_n \to 0$ as $n$ goes to $\infty$, such that

for some $v \in H(\operatorname{curl}; \Omega) \cap H(\operatorname{div}; \Omega)$. It turns out that $v$ preserves the boundary conditions as the following lemma clarifies.

Lemma 4. Assume that for some $\epsilon_n > 0$ with $\epsilon_n\to 0$, there exists $v\in H(\operatorname{curl}, \Omega) \cap H(\operatorname{div}, \Omega)$ such that $\{v_{\epsilon_n}\rvert_\Omega\}_{n \in \mathbb{N}}$ weakly converges to $v$ in $H(\operatorname{curl}, \Omega) \cap H(\operatorname{div}, \Omega)$. Then $v \in X_{{\rm{N}}}(\Omega)$.

Proof. To prove that $v \in X_{{\rm{N}}}(\Omega)$ we just need to make sure that $v\in H_0(\operatorname{curl}, \Omega)$. Since $v_{\epsilon_n} \in H_0(\operatorname{curl}, \Omega_{\epsilon_n})$ for all $n \in \mathbb{N}$, by Lemma 1 we know that the extension-by-zero $v_{\epsilon_n}^0$ of $v_{\epsilon_n}$ belongs to $H(\operatorname{curl}, \mathbb{R}^3)$ for all $n \in \mathbb{N}$. By the reflexivity of $H(\operatorname{curl}, \mathbb{R}^3)$ and the boundedness of the sequence $\{v^0_{\epsilon_n}\}_{n \in \mathbb{N}}$, we deduce that possibly passing to a subsequence, there exists a function $\tilde{v} \in H(\operatorname{curl}, \mathbb{R}^3)$ such that $v^0_{\epsilon_n} \rightharpoonup \tilde{v}$ weakly in $H(\operatorname{curl}, \mathbb{R}^3)$ as $n$ goes to $\infty$. It suffices to show that $\tilde{v} = v^0$. Since $v^0_{\epsilon_n}$ is equal to zero outside of $\Omega_{\epsilon_n}$, it is clear that $\tilde{v} = 0$ a.e. in $\mathbb{R}^3 \setminus \Omega$. Moreover, since $v_{\epsilon_n}\rvert_\Omega$ weakly converges to both $v, \tilde v$ in $H(\operatorname{curl}, \Omega)$, we have that $v = \tilde v$ a.e. in $\Omega$. Thus the extension by zero of $v$ to the whole of $\mathbb{R}^3$ is precisely $\tilde v$ and belongs to $H(\operatorname{curl}, {\mathbb{R}^3})$. Using Lemma 1 again, we see that $v \in H_0(\operatorname{curl}, \Omega)$.

Lemma 5. Assume that condition $(3.14)$ and the uniform Gaffney inequality $(4.1)$ hold. For any $\epsilon > 0$ let $f_\epsilon \in L^2(\Omega_\epsilon)^3$. Suppose that $\sup_{\epsilon > 0}||{f_\epsilon}||_{L^2(\Omega_\epsilon)^3} \ne \infty$ and that the extension-by-zero of the functions $f_\epsilon$ converge weakly in $L^2(\Omega)^3$ to some function $f \in L^2(\Omega)^3$ as $\epsilon \to 0$. For all $\epsilon > 0$, let $v_\epsilon : = S_\epsilon f_\epsilon$ the (unique) weak solution in $X_{{\rm{N}}}(\Omega_\epsilon)$ of $(4.5)$ with datum $f_\epsilon$. Assume $(4.7)$ and suppose that $v_\epsilon \rightharpoonup v$ weakly in $H(\operatorname{curl}, \Omega) \cap H(\operatorname{div}, \Omega)$ to some $v\in H(\operatorname{curl}, \Omega) \cap H(\operatorname{div}, \Omega)$. Then $v = S_0 f$.

Proof. First of all, we note that by Lemma 4, $v\in X_{{\rm{N}}}(\Omega)$. Define for $u, w \in H(\operatorname{curl}, \Omega_\epsilon) \cap H(\operatorname{div}, \Omega_\epsilon)$

which is equivalent to the scalar product for the space $H(\operatorname{curl}, \Omega_\epsilon) \cap H(\operatorname{div}, \Omega_\epsilon)$. The square of the induced norm will be denoted with $Q_{\Omega_\epsilon}(\cdot)$. Note that since $v_\epsilon$ is the solution with datum $f_\epsilon$, then we have that

for all $\eta \in X_{{\rm{N}}}(\Omega_\epsilon)$.

Let $\varphi$ be any function in $X_{{\rm{N}}}(\Omega)$ and let ${\mathcal{P}}_{\epsilon }\varphi$ the Atlas Piola trasform of $\varphi$. Since ${\mathcal{P}}_{\epsilon }\varphi \in X_{{\rm{N}}}(\Omega_{\epsilon})$, we deduce tha

for all $\epsilon > 0$. We now show that

and

In order to prove the first limit, it suffices to prove that

for any $j = 1, \dots, s'$, where $\tilde{\varphi}_{\epsilon, j}$ is defined in (3.13) (with $\tilde \Omega$ replaced by $\Omega_{\epsilon}$), since it is obvious that

for any $j = s'+1, \dots, s$. We have that

Obviously

since the extension-by-zero of $f_\epsilon$ weakly converge to $f$ in $L^2(\Omega)^3$, $\sup_{\epsilon > 0} ||{f_\epsilon}||_{L^2(\Omega_\epsilon)^3} < \infty$ and $|{(\Omega \cap V_j) \setminus r_j^{-1}(K_{\epsilon, j})}|$ goes to 0 as $\epsilon \to 0$. Meanwhile

by (3.23) and the hypothesis that $\sup_{\epsilon > 0}||{f_\epsilon}||_{L^2(\Omega_\epsilon)^3} \ne \infty$. From (4.14), (4.15) and (4.16) we immediately deduce (4.12). Hence we have proved (4.10).

Let us now focus on (4.11). We write

By the weak convergence of $v_{\epsilon }$ to $v$ we have that

By the Cauchy-Schwarz inequality and Theorem 2, (ⅳ) we get that

Similarly,

Finally,

since by (3.23), (3.31) and (3.39) it follows that $Q_{\Omega_\epsilon \setminus \Omega }({\mathcal{P}}_{\epsilon }\varphi)\to 0$ as $\epsilon \to 0$. By combining (4.18)–(4.21), we deduce that limit (4.11) holds.

In conclusion, by using the limiting relations (4.10) and (4.11) in equation (4.9) we conclude that

which means that $v$ is the solution in $X_{{\rm{N}}}(\Omega)$ of the given problem with datum $f\in L^2(\Omega)$, as required.

Remark 1. A careful inspection of the proof of Lemma 5 reveals that the uniform Gaffney inequality has been used only to prove the limiting relations $(4.18)$–$(4.20)$ since the functions $v_{\epsilon}$ are required here to be defined on $\Omega$ and to have uniformly bounded norms. This problem does not occur if $\Omega \subset \Omega_{\epsilon}$ in which case only the Gaffney inequality in $\Omega$ is necessary. However, the uniform Gaffney inequality will be used in an essential way in the following statements also in the particular case $\Omega \subset \Omega_{\epsilon}$

In the next lemma we prove that $S_\epsilon$ $E$-converges to $S_0$ as $\epsilon \to 0$.

Lemma 6. Assume that condition $(3.14)$ and the uniform Gaffney inequality $(4.1)$ hold. Let $f_\epsilon \in L^2(\Omega_\epsilon)^3, \epsilon > 0$ be such that $f_\epsilon \xrightarrow[\epsilon \to 0]{E} f \in L^2(\Omega)^3$ for some function $f \in L^2(\Omega)^3$. Set $v_\epsilon : = S_\epsilon f_\epsilon$ and $v: = S_0 f$. Then $v_\epsilon \xrightarrow[\epsilon \to 0]{E} v$, hence $S_\epsilon \xrightarrow[\epsilon \to 0]{EE}S_0.$

Proof. Since $f_\epsilon \xrightarrow[\epsilon \to 0]{E} f$, then $||{f_\epsilon}||_{L^2(\Omega_\epsilon)^3} \leq C$ for all $\epsilon > 0$ sufficiently small and consequently $||{v_\epsilon}||_{ X_{{\rm{N}}}(\Omega_\epsilon)}$ is uniformly bounded with respect to $\epsilon$, as shown in $(4.6)$. By the uniform Gaffney inequality it follows that also $||{v_\epsilon}||_{H^1(\Omega_\epsilon)^3}$ is uniformly bounded. In particular

because $|{\Omega_\epsilon \setminus \Omega}|\to 0$ as $\epsilon$ goes to 0. This can be proved using the same argument used for (3.41) as follows:

Hence to prove that $v_\epsilon \xrightarrow[\epsilon \to 0]{E} v$ we just have to show that

Recall that $\{v_\epsilon\rvert_\Omega \}\subset H^1(\Omega)^3$ is bounded in $H^1$-norm. Select now a sequence $\{v_{\epsilon_n}\}_{n \in \mathbb{N}}$ from the family. By the compact embedding of $H^1(\Omega)^3$ into $L^2(\Omega)^3$ we have that, up to choosing a subsequence, $v_{\epsilon_n}\rvert_\Omega \to v^*$ strongly in $L^2(\Omega)^3$ and $v_{\epsilon_n}\rvert_\Omega \rightharpoonup v^*$ weakly in $H^1(\Omega)^3$ for some $v^* \in H^1(\Omega)^3$. By Lemma 5 we have that $v^* = S_0 f = v \in X_{{\rm{N}}}(\Omega)$. This shows that for any extracted sequence of the family $\{v_\epsilon\rvert_\Omega - v\}_{\epsilon > 0}$, there exist a subsequence such that $||{v_{\epsilon_{n_k}}\rvert_\Omega - v}||_{L^2(\Omega)^3} \xrightarrow[k \to \infty]{}0$. Thus we can conclude that $||{v_\epsilon\rvert_\Omega - v}||_{L^2(\Omega)^3} \xrightarrow[\epsilon \to 0]{}0$, which is exactly (4.22).

Remark 2. The hypothesis of Lemma 6 can be weakened to only require that $||{f_\epsilon}||_{L^2(\Omega_\epsilon)^3}$ are uniformly bounded and that the extenstion-by-zero of $f_\epsilon$ (restricted to $\Omega$) weakly converges to $f$ in $L^2(\Omega)^3$ as $\epsilon$ goes to 0, which is a weaker assumption than $f_\epsilon \xrightarrow[\epsilon \to 0]{E}f$.

Finally we can state and prove the main theorem

Theorem 4. Let $\mathcal{A}$ be an atlas in $\mathbb{R}^3$ and $\{\Omega_\epsilon\}_{\epsilon > 0}$ be a family of bounded domains of class $C^{1, 1}(\mathcal{A})$ converging to a bounded domain $\Omega$ of class $C^{1, 1}(\mathcal{A})$ as $\epsilon\to 0$, in the sense that condition $(3.14)$ holds. Suppose that the uniform Gaffney inequality $(4.1)$ holds.Then $S_\epsilon \xrightarrow[]{C}S_0$ as $\epsilon \to 0$. In particular, spectral stability occurs: the eigenvalues of the operator $S_{\epsilon}$ converge to the eigenvalues of the operator $S_0$, and the eigenfunctions of the operator $S_{\epsilon}$ $E$-converge to the eigenfunctions of the operator $S_{0}$ as $\epsilon \to 0$.

Proof. By Lemma 6 we have that $S_\epsilon \xrightarrow[\epsilon \to 0]{EE}S$. Now, suppose that we are given a family of data $\{f_\epsilon\}_{\epsilon > 0}$ such that $||{f_\epsilon}||_{L^2(\Omega_\epsilon)^3} \leq 1$ for all $\epsilon > 0$, and extract a sequence $\{f_{\epsilon_n}\}_{n \in \mathbb{N}}$ from it. We have to show that we can always find a subsequence $\epsilon_{n_k}\to 0$ and a function $v \in L^2(\Omega)^3$ such that

Possibly passing to a subsequence, we can find a function $f$ to which the restriction to $\Omega$ of the extension-by-zero of $\{f_{\epsilon_n}\}_{n \in \mathbb{N}}$ weakly converge in $L^2(\Omega)^3$. Setting $v: = S_0f$, we can apply Lemma 6 and Remark 2 to find out that (4.23) holds.

Finally, the spectral stability is a consequence of the compact convergence of compact operators as stated in Theorem 3.

5.   Uniform Gaffney Inequalities and applications to families of oscillating boundaries

In this section we give sufficient conditions that guarantee the validity of a uniform Gaffney inequality of the type (4.1) for a family of Lipschitz domains $\Omega_{\epsilon}$, $\epsilon > 0$, belonging to the same class $C^{0, 1}_M({\mathcal{A}})$. To do so, we exploit a known relation between Gaffney inequalities and a priori estimates for the Dirichlet Laplacian that we formulate in our setting. We note that one of the two implications (namely, the validity of the Gaffney inequality implies the validity of the a priori estimate) is quite standard. The other one is a bit more involved, hence, for the sake of completeness, we include a proof.

Theorem 5. Let $\Omega$ be a bounded domain in ${\mathbb{R}}^3$ of class $C^{0, 1}_M({\mathcal{A}})$. Then the Gaffney inequality $(2.3)$ holds for all $u \in X_{{\rm{N}}}(\Omega)$ and a constant $C > 0$ independent of $u$ if and only if (the weak, variational) solutions $\varphi \in H^1_0(\Omega)$ to the Poisson problem

satisfy the a priori estimate

for all $f\in L^2(\Omega)$ and a constant $\tilde C > 0$ independent of $f$. In particular, the constants $C$ and $\tilde{C}$ depend on each other, $M$ and ${\mathcal{A}}$.

Proof. Assume that the a priori estimate (5.2) holds. We set $H^1_{\rm{N}}(\Omega): = X_{{\rm{N}}}(\Omega)\cap H^1(\Omega)^3$ and $E(\Omega) = \{ \nabla \varphi:\ \varphi \in H^1_0(\Omega), \ \Delta \varphi \in L^2(\Omega)\}$. By [5,Thm. 4.1] there exists two linear operators $P: X_{{\rm{N}}}(\Omega)\to H^1_{\rm{N}}(\Omega)$ and $Q : X_{{\rm{N}}}(\Omega)\to E(\Omega)$ such that $u = Pu+Qu$ for all $u\in X_{{\rm{N}}}(\Omega)$ and such that

for all $u\in X_{{\rm{N}}}(\Omega)$, for some positive constant $C_{BS}$. A careful inspection of the proof of [5,Thm. 4.1] reveals that $C_{BS}$ depends only on $M$, ${\mathcal{A}}$. By definition, $Qu = \nabla \varphi$ with $\varphi\in H^1_0(\Omega)$ and $\Delta \varphi\in L^2(\Omega)$. Since we have assumed that (5.2) holds, then

Thus, since $\| Qu \|_{H^1(\Omega)^3}$ is obviously controlled by $\| \varphi\|_{H^2(\Omega)}$ we deduce that

for all $u\in X_{{\rm{N}}}(\Omega)$, and (2.3) is proved.

Viceversa, assume that (2.3) holds and let $\varphi$ be a solution to (5.1). Since $\nabla \varphi \in X_{{\rm{N}}}(\Omega)$, by (2.3) it follows that

where we have used [20,Lemma 1] and $c_{\mathcal{P}}$ denotes the usual Poincaré constant. This, combined with the Poincaré's inequality, immediately implies (5.2).

Example 1. Let $\Omega$ be a bounded domain in ${\mathbb{R}}^N$ of class $C^1$ such that around a boundary point (identified here with the origin) is described by the subgraph $x_N < g(\bar x)$ of the $C^1$ function defined by

It is proved in [34,14.6.1] that for this domain the a priori estimate $(5.2)$ does not hold. Thus, by Theorem 5 it follows that not even the Gaffney inequality holds for this domain for $N = 3$.

Theorem 5 highlights the importance of proving the a priori estimate (5.2) and getting information on the constant $\tilde C$. We do this by following the approach of Maz'ya and Shaposhnikova ^[34] and using the notion of domains $\Omega$ with boundaries $\partial \Omega$ of class $\mathcal{M}^{3/2}_2 (\delta)$. We re-formulate the definition in Maz'ya and Shaposhnikova [34,§ 14.3.1] by using the atlas classes. Here we can treat the general case of domains in ${\mathbb{R}}^N$ with $N\geq 2$.

Note that in this section, following ^[34] we find it convenient to assume directly that the functions $g_j$ describing the boundary of $\Omega$ as in Definition 1 are extended to the whole of ${\mathbb{R}}^{N-1}$ and belong to the corresponding function spaces defined on ${\mathbb{R}}^{N-1}$.

Definition 4. Let ${\mathcal{A}}$ be an atlas in ${\mathbb{R}}^N$ and $\delta > 0$. We say that a bounded domain $\Omega$ in ${\mathbb{R}}^N$ is of class $\mathcal{M}^{3/2}_2 (\delta, {\mathcal{A}})$ if $\Omega$ is of class $C^{0, 1}({\mathcal{A}})$ and the gradients $\nabla g_j$ of the functions $g_j$ describing the boundary of $\Omega$ as in Definition 1 belong to the space $MW^{1/2 }_2({\mathbb{R}}^{N-1})$ of Sobolev multipliers with

for all $j = 1, \dots s'$. We say that a bounded domain $\Omega$ in ${\mathbb{R}}^N$ is of class $\mathcal{M}^{3/2}_2 (\delta)$ if it is of class $\mathcal{M}^{3/2}_2 (\delta, {\mathcal{A}})$ for some atlas ${\mathcal{A}}$.

Recall that $MW^{1/2 }_2({\mathbb{R}}^{N-1}) = \{f\in W^{1/2}_{2, loc}({\mathbb{R}}^{N-1}):\ f\varphi \in W^{1/2}_2({\mathbb{R}}^{N-1})\ \; {\rm{for}}\; {\rm{all}}\; \ \varphi \in W^{1/2}_2({\mathbb{R}}^{N-1})\}$ and that $\| f\|_{MW^{1/2}_2({\mathbb{R}}^{N-1}) } = \sup \{ \| f\varphi \|_{ W^{1/2}_2({\mathbb{R}}^{N-1}) }:\| \varphi \|_{ W^{1/2}_2({\mathbb{R}}^{N-1}) } = 1\}$, where $W^{1/2}_2({\mathbb{R}}^{N-1})$ denotes the standard Sobolev space with fractional order of smoothness $1/2$ and index of summability $2$ (for simplicity, in (5.3) we use the the same symbol for the norm of a vector field).

Remark 3. Note that by [34,Thm. 4.1.1] there exists $c > 0$ depending only on $N$ such that the functions $g_j$ in Definition 4 satisfy the estimate $\| \nabla g_j \|_{L^\infty(\mathbb{R}^{N-1})} \le c \| \nabla g_j\|_{MW^{1/2}_2({\mathbb{R}}^{N-1}) }\le c\delta$, see also [34,Thm. 14.6.4]. Thus, if $\Omega$ is of class $\mathcal{M}^{3/2}_2 (\delta, {\mathcal{A}})$ then it is also of class $C^{0, 1}_M({\mathcal{A}})$ with $M = c\delta$.

The following theorem is a reformulation of [34,Thm. 14.5.1]

Theorem 6. Let ${\mathcal{A}}$ be an atlas in ${\mathbb{R}}^N$. If $\Omega$ is a bounded domain of class $\mathcal{M}^{3/2}_2 (\delta, {\mathcal{A}})$ for some $\delta$ sufficiently small (depending only on $N$) then the a priori estimate $(5.2)$ holds for some constant $\tilde C$ depending only on $N$ and ${\mathcal{A}}$.

By [34,Corollaries. 14.6.1,14.6.2] it is possible to prove the following theorem based on the condition (5.5) from [34,(14.6.9)]. Here, given an atlas $\mathcal{A} = (\rho, s, s', \{V_j\}_{j = 1}^s, \{r_j\}^s_{j = 1})$, by a refinement of $\mathcal{A}$ we mean an atlas of the type $\widetilde{\mathcal{A}} = (\tilde \rho, \tilde s, \tilde s', \{\tilde V_j\}_{j = 1}^{\tilde s}, \{\tilde r_j\}^{\tilde s}_{j = 1})$ where $\tilde \rho \le \rho$, $s\le \tilde s$, $s'\le \tilde s'$, $\cup_{j = 1}^{\tilde s}{\tilde V_j} = \cup_{j = 1}^{\ s}V_j$, $\{\tilde r_j\}^{\tilde s}_{j = 1}\subset \{ r_j\}^{ s}_{j = 1}$, which can be thought as an atlas constructed from $\mathcal{A}$ by replacing each cuboid $V_{j} = r_j(W_{j}\times]a_{N, j}, b_{N, j}[)$ by a finite number of cuboids of the form $\widetilde V_{j, l} = r_j(\widetilde W_{j, l}\times]a_{N, j}, b_{N, j}[)$, $l = 1, \dots m_j$, where $W_j = \cup_{l = 1}^{m_j} \widetilde W_{j, l}$.

Theorem 7. Let ${\mathcal{A}}$ be an atlas in ${\mathbb{R}}^N$ and let $\Omega$ be a bounded domain of class $C^{0, 1}_M({\mathcal{A}})$.Let $\omega$ be a (non-decreasing) modulus of continuity for the gradients $\nabla g_{j}$ of the functions $g_j$ describing the boundary of $\Omega$, that is

for all $\bar x, \bar y\in {\mathbb{R}}^{N-1}$. Assume that there exists $D > 0$ such that the function $\omega$ satisfies the following condition^*

^*Here only the integrability at zero really matters and one could consider integrals defined in a neigborhhood of zero.

Then there exists $C > 0$ depending only on $N$, ${\mathcal{A}}$, $D$ such that if $M\le C\delta$ then, possibly replacing the atlas ${\mathcal{A}}$ with a refinement of ${\mathcal{A}}$, $\Omega$ is of class $\mathcal{M}^{3/2}_2 (\delta, {\mathcal{A}})$.

Proof. We begin with the case $N\geq 3$. By [34,Cor. 14.6.1] there exists $c > 0$ depending only on $N$ such that if $x = (\bar x, g_j(\bar x))\in \partial \Omega$ is any point of the boundary represented in local charts by a profile function $g_j$ and the following inequality

is satisfied, then, possibly replacing the atlas ${\mathcal{A}}$ with a refinement of its, $\Omega$ is of class $\mathcal{M}^{3/2}_2 (\delta, {\mathcal{A}})$. Here $|E|$ denotes the $N-1$-dimensional Lebesgue measure of $E$,

and $B_{\rho}(\bar x)$ the ball in ${\mathbb{R}}^{N-1}$ of radius $\rho$ and centre $\bar x$. We refer to [34,§ 14.7.2] for the local characterization of the boundaries of domains of class $\mathcal{M}^{3/2}_2 (\delta, {\mathcal{A}})$.

We have

Here $\sigma_m$ denotes the $m$-dimensional measure of the $m$-dimensional unit sphere. Thus

hence

provided $\rho$ is sufficiently small. Thus, inequality (5.6) follows if we assume directly that $\| \nabla g_j\|_{L^{\infty }(B_{\rho}) } \le c\delta$.

In the case $N = 2$, by [34,Cor. 14.6.1] it suffices to replace (5.6) by the following inequality

and use the same argument as above.

By combining Theorems 6 and 7, we deduce the validity of the following result

Corollary 1. Under the same assumptions of Theorem 7, there exists $\tilde C > 0$ depending only on $N$, ${\mathcal{A}}$, $D$ such that if $M < \tilde C^{-1}$ then the a priori estimate $(5.2)$ holds.

Finally, by Theorem 5 and Corollary 1 we deduce the following result ensuring the validity of uniform Gaffney inequality that can be used in our spectral stability results.

Corollary 2. Under the same assumptions of Theorem 7 with $N = 3$, there exists $C > 0$ depending only on ${\mathcal{A}}$ and $D$ such that if $M < C^{-1}$ then the Gaffney inequality $(2.3)$.

5.1.   Applications to families of oscillating boundaries

It is clear that in order to apply Theorem 7 and Corollaries 1, 2, it suffices to assume that the gradients $\nabla g_j$ of the functions $g_j$ describing the boundary of a domain $\Omega$ as in Definition 1 are of class $C^{0, \beta}$ with $\beta \in]1/2, 1]$, that is

for some positive constant $K$ and all $\bar x, \bar y\in W_j$, and that the functions $g_j$ have sufficiently small Lipschitz constants. As we have already mentioned, in principle, the second condition is not a big obstruction to the application of these results, since for a domain of class $C^1$ one can find a sufficiently refined atlas, adapted to the tangent planes of a finite number of boundary points, such that the $C^1$ norms, hence the Lipschitz constants, of the profile functions $g_j$ are arbitrarily close to zero. Thus, we can apply our results to uniform classes of domains of class $C^{1, \beta}$ since condition (5.5) would be satisfied exactly because $\beta > 1/2$ (as we have said, here what matters is the behaviour of the modulus of continuity $\omega (t)$ for $t$ close to zero and one can assume directly that $\omega(t)$ is constant for $t$ big enough).

Thus, we can prove the following result. Note that here the domains $\Omega_\epsilon$ are assumed to be of class $C^{1, 1}$ and that they belong to the uniform class $C^{1, \beta}_K(\mathcal{A})$ with $K > 0$ fixed, which in particular implies the validity of (5.10) for all functions $g_{\epsilon, j}$ and all $\epsilon > 0$. (Recall that the operators $S_{\epsilon}$ are defined in the beginning of Section 4.)

Theorem 8. Let $\mathcal{A}$ be an atlas in $\mathbb{R}^3$ and $\{\Omega_\epsilon\}_{\epsilon > 0}$ be a family of bounded domains of class $C^{1, 1}(\mathcal{A})$ converging to a bounded domain $\Omega$ of class $C^{1, 1}(\mathcal{A})$ as $\epsilon\to 0$, in the sense that condition $(3.14)$ holds. Suppose that $\Omega$ is of class $C^{0, 1}_M(\mathcal{A})$ with $M$ small enough as in Corollary 2. Suppose also that all domains $\Omega_{\epsilon}$ are of class $C^{1, \beta}_K(\mathcal{A})$ with the same parameters $\beta \in]1/2, 1]$ and $K > 0$. Then the uniform Gaffney inequality $(4.1)$ holds provided $\epsilon$ is small enough. Moreover, $S_\epsilon \xrightarrow[]{C}S$ as $\epsilon \to 0$. In particular, spectral stability occurs: the eigenvalues of the operator $S_{\epsilon}$ converge to the eigenvalues of the operator $S_{0}$, and the eigenfunctions of the operator $S_{\epsilon}$ $E$-converge to the eigenfunctions of the operator $S_{0}$ as $\epsilon \to 0$.

Proof. Since $\Omega_\epsilon$ converges to $\Omega$ as $\epsilon \to 0$ in the sense that condition (3.14) holds, it follows that the gradients of the functions $g_{\epsilon, j}$ describing the boundary of $\Omega_{\epsilon }$ converge uniformly to the gradients of the functions $g_{ j}$ describing the boundary of $\Omega$. Thus, $\Omega_{\epsilon}$ is of class $C^{0, 1}_M(\mathcal{A})$ provided $\epsilon$ is small enough. By the discussion above, Corollary 2 is applicable and the uniform Gaffney inequality (4.1) holds provided $\epsilon$ is small enough. Then the last part of the statement follows by Theorem 4.

A prototype for the classes of domains under discussion is given by domains designed by profile functions often used in homogenization theory, in particular in the study of thin domains. Namely, assume that one of the profile functions $g_{\epsilon, j}$, call it $g_\epsilon$, is of the form

for some function $b$ of class $C^{1, 1}(\mathbb{R}^2)$ and $\alpha > 0$, and assume that the gradient of $b$ is bounded. If $\omega_{\nabla b}$ is a (non-decreasing) modulus of continuity of $\nabla b$, then we have

hence the function $\omega$ to be considered in (5.4) is given by $\omega(t) = \epsilon^{\alpha-1}\omega_{\nabla b}(t/\epsilon)$. Observe that

Moreover, since $b$ is assumed to be of class $C^{1, 1}$, we have that $\omega_{\nabla b}(t)\le ct$ for $t$ in a neighborhhood of zero. Thus, if $\alpha \geq 3/2$ and $\epsilon_0$ is any fixed positive constant, it follows that that

Since the gradient of $g_\epsilon$ is arbitrarily close to zero for $\epsilon$ sufficiently small, we have that Theorem 7 and Corollaries 1, 2 are applicable and the Gaffney inequality (4.1) holds for all $\epsilon$ sufficiently small, with a constant $C > 0$ independent of $\epsilon$. The same arguments can be applied to families of profile functions of the type

where $b$ is as above and $\psi$ is a fixed $C^{1, 1}$ function with bounded gradient. Thus, we can state the following stability result concerning a local perturbation of a domain $\Omega$.

Theorem 9. Let $W$ be a bounded open rectangle in ${\mathbb{R}}^2$, $b\in C^{1, 1}(\mathbb{R}^{2})$ with bounded gradient, $b\ge 0$, and $\psi \in C^{1, 1}_c(W)$, $\alpha > 3/2$. Assume that $\Omega$ and $\Omega_{\epsilon}$, $\epsilon > 0$ are domains of class $C^{1, 1}$ in ${\mathbb{R}}^3$ satisfying the following condition:

$(i)$ $\Omega \cap (W\times]-1, 1[) = \{(\bar x, x_3)\in {\mathbb{R}}^3:\ \bar x\in W, \ -1 < x_3 < 0 \}$;

$(ii)$ $\Omega_{\epsilon} \cap (W\times]-1, 1[) = \{(\bar x, x_3)\in {\mathbb{R}}^3:\ \bar x\in W, \ -1 < x_3 < \epsilon^{\alpha}b(\bar x/\epsilon)\psi(\bar x) \}$ where $b\in C^{1, 1}(\mathbb{R}^{2})$ has bounded gradient, and $\psi \in C^{1, 1}_c(W)$;

$(iii)$ $\Omega \setminus (W \times]-1, 1[) = \Omega_{\epsilon } \setminus (W \times]-1, 1[)$;

Then the family $\{\Omega_{\epsilon} \}_{\epsilon > 0}$ converges to $\Omega$ in the sense that condition $(3.14)$ holds. Moreover, the uniform Gaffney inequality $(4.1)$ holds and $S_\epsilon \xrightarrow[]{C}S$ as $\epsilon \to 0$. In particular, spectral stability occurs: the eigenvalues of the operator $S_{\epsilon}$ converge to the eigenvalues of the operator $S_0$, and the eigenfunctions of the operator $S_{\epsilon}$ $E$-converge to the eigenfunctions of the operator $S_{0}$ as $\epsilon \to 0$.

Proof. By assumptions, the domains $\Omega$ and $\Omega_{\epsilon}$ belong to the same atlas class $C^{1, 1}({\mathcal{A}})$ for a suitable atlas ${\mathcal{A}}$, and $W\times]-1, 1[$ is one of the local charts of ${\mathcal{A}}$. In particular, the profile functions describing the boundaries of $\Omega$ and $\Omega_{\epsilon}$ in that chart are given by $g(\bar x) = 0$ and $g_{\epsilon} = \epsilon^{\alpha}b(\bar x/\epsilon)\psi(\bar x)$ for all $\bar x\in W$.

As in the proof of [3,Thm. 7.4], if $\tilde \alpha \in]3/2, \alpha[$ is fixed then one can easily check that conditions (3.14) are satisfied with $k_{\epsilon} = \epsilon^{2\tilde \alpha /3}$. By (5.2) and the discussion above, it follows that the Gaffney inequality (2.3) holds with a constant $C$ independent of $\epsilon$, provided $\epsilon$ is sufficiently small. To complete the proof it suffices to apply Theorem 4.

Remark 4. It is clear that condition $(5.13)$ is satisfied also in the case $\alpha = 3/2$. Thus the uniform Gaffney inequality $(4.1)$ holds also in the case $\alpha = 3/2$ in Theorem 9. However, in this case the convergence of $\Omega_{\epsilon}$ to $\Omega$ in the sense of $(3.14)$ is not guaranteed hence we cannot directly deduce that we have spectral stability. Thus, another method has to be used in the analysis of the stability problem for $\alpha = 3/2$. For example, in the case of non-constant periodic functions $b$ one could use the unfolding method as in ^[11], adopted also in ^[2,3,19,20]: in those papers, for $\alpha = 3/2$ we have spectral instability in the sense that the limiting problem differs from the given problem in $\Omega$ by a strange term appearing in the boundary conditions (as often happens in homogenization problems). We plan to discuss the details of this problem for the $curl curl$ operator in a forthcoming paper, but we can already mention that a preliminary formal analysis would indicate that no strange limit appears in the limiting problem for $\alpha = 3/2$. On the other hand, at the moment we are not able to formulate any conjecture for the case $\alpha < 3/2$ although, on the base of the results of ^[11] concerning the Navier-Stokes system, a degeneration phenomenon (to Dirichlet boundary conditions) could not be excluded.

Acknowledgments

The authors would like to thank the referee for the careful reading of the paper and many useful suggestions. The authors are also very thankful to Dr. Francesco Ferraresso and Prof. Ioannis G. Stratis for useful discussions and references. The authors acknowledge financial support from the research project BIRD191739/19 "Sensitivity analysis of partial differential equations in the mathematical theory of electromagnetism" of the University of Padova. The authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

Conflict of interest

The authors declare no conflicts of interest in this paper.