# Nonlocal $H$-convergence for topologically nontrivial domains

Marcus Waurick111Institute for Applied Analysis, Faculty for Mathematics and
Computer Science, TU Bergakademie Freiberg, Freiberg, Germany, Email:
marcus.wauhugo<EMAIL_ADDRESS>
###### Abstract

The notion of nonlocal $H$-convergence is extended to domains with nontrivial
topology, that is, domains with non-vanishing harmonic Dirichlet and/or
Neumann fields. If the space of harmonic Dirichlet (or Neumann) fields is
infinite-dimensional, there is an abundance of choice of pairwise incomparable
topologies generalising the one for topologically trivial $\Omega$. It will be
demonstrated that if the domain satisfies the Maxwell’s compactness property
the corresponding natural version of the corresponding (generalised) nonlocal
$H$-convergence topology has no such ambiguity. Moreover, on multiplication
operators the nonlocal $H$-topology coincides with the one induced by (local)
$H$-convergence introduced by Murat and Tartar. The topology is used to obtain
nonlocal homogenisation results including convergence of the associated energy
for electrostatics. The derived techniques prove useful to deduce a new
compactness criterion relevant for nonlinear static Maxwell problems.

Keywords homogenisation, meta-materials, electrostatics, static Maxwell
problem, $G$-convergence, $H$-convergence, nonlocal $H$-convergence

MSC 2020 Primary 35B27; Secondary 35Q61; 74Q05

###### Contents

  1. 1 Introduction
  2. 2 Abstract divergence-form operators
  3. 3 The Helmholtz decomposition
  4. 4 Two formulations for electrostatics
  5. 5 Nonlocal $H$-convergence
  6. 6 Infinite-dimensional harmonic Dirichlet fields
  7. 7 Homogenisation theorems
  8. 8 Convergence of energy and the div-curl Lemma
  9. 9 A compactness criterion with moving coefficients
  10. 10 Relationship to local $H$-convergence
  11. 11 Conclusion

##  1 Introduction

Many physical phenomena such as the classical wave equation, Maxwell’s
equations or the heat equation consist of coefficients describing the specific
properties of the material the effects are considered in. These specifics are
also provided in so-called constitutive relations. Classically, these
relations are represented by (matrix-valued) functions depending on the
material point. With this it is possible to model a composite consisting of
two materials with, say, different thermal conductivities. In order to model
more sophisticated physical phenomena it is not enough to restrict oneself to
matrix-valued functions modelling the material’s properties. In fact, also
nonlocal effects need to be taken into account. This is particularly the case
for the description of interacting particles, where the McKean–Vlasov
equations are used, see, e.g., [2]. We refer to the references in [2] for a
multitude of applications and to [29, Example 2.7] for a linear version of
these equations.

The modelling of the electro-magnetic effects of metamaterials characterised
by nonlocal constitutive relations is provided for instance in [6, 3, 8].
Moreover, a bespoke homogenisation process has been provided in these
references to have a workable model at hand. The type of equations considered
are of the form

$\operatorname{curl}(1+\rho_{n}*)\operatorname{curl}E_{n}=f\in
L_{2}(\Omega)^{3}$

with $E_{n}$ subject to suitable boundary conditions and $(\rho_{n})_{n}$ a
sequence of integral kernels defined on $\Omega\times\Omega$.

These sources were the reason for the invention of the notion of _nonlocal
$H$-convergence_ in [29]. This notion describes the functional analytic
properties the homogenisation processes both local and nonlocal problems have
in common. In order to briefly describe the notion of both nonlocal and local
$H$-convergence, we let $\Omega\subseteq\mathbb{R}^{3}$ be open and bounded
with connected complement. Then consider a sequence of suitable matrix-valued
functions $(\varepsilon_{n})_{n}$. This sequence is said to (locally)
$H$-converge to some matrix-valued function $\varepsilon$, if for all $f\in
H^{-1}(\Omega)(=H_{0}^{1}(\Omega)^{\prime})$ the following properties are
satisfied. Consider the sequence of solutions $(u_{n})_{n}$ in
$H_{0}^{1}(\Omega)$ with

$-\operatorname{div}\varepsilon_{n}(x)\operatorname{grad}u_{n}=f$

in the distributional sense. Then $(u_{n})_{n}$ weakly converges to $u\in
H_{0}^{1}(\Omega)$ and $(\varepsilon_{n}\operatorname{grad}u_{n})_{n}$ weakly
converges to $\varepsilon\operatorname{grad}u\in L_{2}(\Omega)^{3}$, where $u$
satisfies

$-\operatorname{div}\varepsilon(x)\operatorname{grad}u_{n}=f.$

The set of suitable coefficients $\varepsilon$ can now be endowed with the
topology induced by (local) $H$-convergence. For uniformly bounded and equi-
coercive coefficients, this set becomes a metrisable compact Hausdorff space,
see [9, 24]. It can be shown that for the corresponding second order
$\operatorname{curl}$-problem a similar property holds, see also [10]. For
this, let $\operatorname{curl}$ be the distributional $\operatorname{curl}$
derivative realised as a closed, densely defined operator in
$L_{2}(\Omega)^{3}$ with maximal domain. The restriction of
$\operatorname{curl}$ to
$\operatorname{dom}(\operatorname{curl})\cap\operatorname{ker}(\operatorname{curl})^{\bot}$
considered as an operator from
$\operatorname{ker}(\operatorname{curl})^{\bot}$ to
$\operatorname{ran}(\operatorname{curl})$ (which is assumed to be closed) is
denoted by $\operatorname{curl_{\textnormal{red}}}$ (see below for the
details). We denote by $H^{1}(\operatorname{curl_{\textnormal{red}}})$ the
domain of $\operatorname{curl_{\textnormal{red}}}$ endowed with the graph
norm. Now, if $(\varepsilon_{n})_{n}$ locally $H$-converges to some
$\varepsilon$, then for all $g\in
H^{-1}(\operatorname{curl_{\textnormal{red}}})\coloneqq
H^{1}(\operatorname{curl_{\textnormal{red}}})^{\prime}$ and (uniquely
determined) $v_{n}\in H^{1}(\operatorname{curl_{\textnormal{red}}})$
satisfying

$\operatorname{curl_{\textnormal{red}}^{\diamond}}\varepsilon_{n}(x)^{-1}\operatorname{curl_{\textnormal{red}}}v_{n}=g,$

where $\operatorname{curl_{\textnormal{red}}^{\diamond}}$ is an appropriate
dual operator of $\operatorname{curl_{\textnormal{red}}}$ (see below for the
details), we obtain $v_{n}\to v$ weakly and
$\varepsilon_{n}^{-1}\operatorname{curl_{\textnormal{red}}}v_{n}\to\varepsilon^{-1}\operatorname{curl_{\textnormal{red}}}v$
weakly in the respective spaces
$H^{1}(\operatorname{curl_{\textnormal{red}}})$ and $L_{2}(\Omega)^{3}$ with
$v\in H^{1}(\operatorname{curl_{\textnormal{red}}})$ uniquely solving

$\operatorname{curl_{\textnormal{red}}^{\diamond}}\varepsilon(x)^{-1}\operatorname{curl_{\textnormal{red}}}v=g.$

In [29] this consequence of local $H$-convergence is used in order to
generalise the class of admissible coefficients to also include coefficients
of convolution type mentioned above. Indeed, basing on the fact that for
$\Omega$ with connected complement the space of harmonic Dirichlet fields,
that is,

$\mathcal{H}_{D}(\Omega)\coloneqq\operatorname{ker}(\operatorname{div})\cap\operatorname{ker}(\operatorname{curl}^{*})$

is $0$-dimensional, we have the decomposition

$L_{2}(\Omega)^{3}=\operatorname{ran}(\operatorname{\mathring{grad}})\oplus\operatorname{ran}(\operatorname{curl}).$

Thus, any bounded linear operator
$\varepsilon\in\mathcal{B}(L_{2}(\Omega)^{3})$ admits a representation as
$2$-by-$2$ matrix according to this decomposition. Then, in the present
situation of vanishing harmonic Dirichlet fields, a suitable sequence of
bounded linear operators $(\varepsilon_{n})_{n}$ is said to converge to
$\varepsilon$ in the nonlocal $H$-sense, if

$\displaystyle\begin{cases}\forall f\in
H^{-1}(\Omega)=H_{0}^{1}(\Omega)^{\prime}&\\\ \forall u_{n}\in
H_{0}^{1}(\Omega)&\\\
-\operatorname{div}\varepsilon_{n}\operatorname{grad}u_{n}=f&\end{cases}\Longrightarrow\begin{cases}u_{n}\to
u\in H_{0}^{1}(\Omega)&\\\
\varepsilon_{n}\operatorname{grad}u_{n}\to\varepsilon\operatorname{grad}u\in
L_{2}(\Omega)^{3}\end{cases}\text{ and }$ $\displaystyle\begin{cases}\forall
g\in H^{-1}(\operatorname{curl_{\textnormal{red}}})&\\\ \forall v_{n}\in
H^{1}(\operatorname{curl_{\textnormal{red}}})&\\\
\operatorname{curl_{\textnormal{red}}^{\diamond}}\varepsilon_{n}^{-1}\operatorname{curl}v_{n}=g&\end{cases}\Longrightarrow\begin{cases}v_{n}\to
v\in H^{1}(\operatorname{curl_{\textnormal{red}}})&\\\
\varepsilon_{n}^{-1}\operatorname{curl}v_{n}\to\varepsilon^{-1}\operatorname{curl}v\in
L_{2}(\Omega)^{3}&\end{cases}$

The introduction of the concept of nonlocal $H$-convergence leads then to a
well-defined operator topology, $\tau_{\textrm{nlH}}$, on (a subset of)
$\mathcal{B}(L_{2}(\Omega)^{3})$, which is weaker than both strong operator
and norm topology but incomparable to the weak operator topology. It can be
shown that $\tau_{\textrm{nlH}}$ is _not_ a linear topology. In any case, the
developed concept hinges on the vanishing of harmonic Dirichlet fields. It is
the aim of the present article to understand the situation of when the space
of harmonic Dirichlet fields is non-zero. Domains with existing non-zero
harmonic Dirichlet fields or harmonic Neumann fields
($\operatorname{ker}(\operatorname{grad}^{*})\cap\operatorname{ker}(\operatorname{curl})$)
are called topologically nontrivial.

For $0<\alpha<\beta$ we define the set

$\mathcal{M}(\alpha,\beta)\coloneqq\\{a\in\mathcal{B}(L_{2}(\Omega)^{3});\operatorname{Re}a\geqslant\alpha,\operatorname{Re}a^{-1}\geqslant
1/\beta\\}.,$

We say that $\tau$ is a nonlocal $H$-topology222We provide a slightly more
general setting in Section 5 below. on $\mathcal{M}(\alpha,\beta)$, if there
exists a closed subspace $\mathcal{H}_{0}\subseteq L_{2}(\Omega)^{3}$ such
that
$\operatorname{ran}(\operatorname{\mathring{grad}})\subseteq\mathcal{H}_{0}$
and
$\operatorname{ran}(\operatorname{curl})\subseteq\mathcal{H}_{0}^{\bot}\eqqcolon\mathcal{H}_{1}$
such that identifying any operator $a\in\mathcal{M}(\alpha,\beta)$ with
$(a_{jk})_{j,k}\in\mathcal{B}(\mathcal{H}_{0}\oplus\mathcal{H}_{1})$ the
topology $\tau$ is the initial topology induced by

$a\mapsto a_{00}^{-1},\ a\mapsto a_{00}^{-1}a_{01},\ a\mapsto
a_{10}a_{00}^{-1},\ a\mapsto a_{11}-a_{10}a_{00}^{-1}a_{01}$

with the target spaces all endowed with corresponding weak operator
topologies.

The first major result of this article reads as follows (part (a) turns out to
be rather trivial, whereas part (b) is the main part of the theorem):

######  Theorem A (consistence and uniqueness of nonlocal $H$-topologies).

(a) For all $\Omega\subseteq\mathbb{R}^{3}$ open with
$\operatorname{ran}(\operatorname{\mathring{grad}}),\operatorname{ran}(\operatorname{curl})\subseteq
L_{2}(\Omega)^{3}$ closed, define

$\mathfrak{T}\coloneqq\\{\tau;\tau\text{ nonlocal $H$-topology on
}\mathcal{M}(\alpha,\beta)\\}.$

Given $(a_{n})_{n}$, $a$ in $\mathcal{M}(\alpha,\beta)$ and if
$\mathcal{H}_{D}(\Omega)=\\{0\\}$, then for all $\tau_{\Omega}\in\mathfrak{T}$

$a_{n}\to a\text{ in }\tau_{\Omega}\iff a_{n}\to a\text{ in
}\tau_{\textnormal{nlH}}.$

(b) If $\dim(\mathcal{H}_{D}(\Omega))<\infty$, $\mathfrak{T}$ is a singleton.

We provide three applications of the Theorem A and the topology constructed,
next. For these applications, we use the following standing assumption.

Let $\Omega\subseteq\mathbb{R}^{3}$ open, bounded Lipschitz domain and
$\tau_{\Omega}$ be the unique topology on $\mathcal{M}(\alpha,\beta)$ as in
(b) of Theorem A.

The first application concerns the homogenisation problem associated to
electrostatics. The main challenge is to understand the relationship of the
coefficients to the projection of the modified harmonic Dirichlet field

$\mathcal{H}_{\varepsilon_{n}}\coloneqq\operatorname{ker}(\operatorname{div}\varepsilon_{n})\cap\operatorname{ker}(\operatorname{curl}^{*})$

In the course of this manuscript, we will define a (not necessarily
orthogonal) continuous projection

$\pi_{\varepsilon}\colon L_{2}(\Omega)^{3}\to\mathcal{H}_{\varepsilon}.$

The corresponding projection for $\varepsilon=\operatorname{id}$ is denoted by
$\pi_{D}$ (orthogonally) projecting onto $\mathcal{H}_{D}(\Omega)$. We shall
establish the following result:

######  Theorem B (nonlocal homogenisation theorem for electrostatics and
convergence of energy).

For all $f\in H^{-1}(\Omega)$, $g\in
H^{-1}(\operatorname{curl_{\textnormal{red}}})$ and
$x\in\mathcal{H}_{D}(\Omega)$ and $E_{n}\in L_{2}(\Omega)^{3}$ solving

$\operatorname{div}\varepsilon_{n}E_{n}=f,\
\operatorname{curl_{\textnormal{red}}^{\diamond}}E_{n}=g,\
\pi_{D}\pi_{\varepsilon_{n}}E_{n}=x,$

we obtain

$E_{n}\to E\text{ weakly in }L_{2}(\Omega)^{3},$

where $E\in L_{2}(\Omega)^{3}$ is the unique solution of

$\operatorname{div}\varepsilon E=f,\
\operatorname{curl_{\textnormal{red}}^{\diamond}}E=g,\
\pi_{D}\pi_{\varepsilon}E=x.$

Moreover, $\varepsilon_{n}E_{n}\to\varepsilon E$ weakly and

$\langle\varepsilon_{n}E_{n},E_{n}\rangle_{L_{2}}\to\langle\varepsilon
E,E\rangle_{L_{2}}.$

The actual result is more general and the solution theory for this kind of
system of equations is based on the solution theory developed in [25] pathing
the way for possible nonlinear generalisations. The main technique for the
solution theory relies on a Helmholtz type decomposition previously known for
self-adjoint and coercive coefficients.

As a second application, we show that the introduced nonlocal topology
coincides with the one of local $H$-convergence on the space of multiplication
operators

$M(\alpha,\beta;\Omega)\coloneqq\\{a\in\mathcal{M}(\alpha,\beta);a\in
L_{\infty}(\Omega)^{3\times 3}\\}.$

######  Theorem C (compatibility of nonlocal $H$-convergence with local
$H$-convergence).

For $(a_{n})_{n}$, $a$ in $M(\alpha,\beta;\Omega)$ the following conditions
are equivalent:

  1. (i)

$a_{n}\to a$ in the local $H$-sense;

  2. (ii)

$a_{n}\to a$ in $\tau_{\Omega}$.

In the third application, we derive a new compactness result generalising the
Picard–Weber–Weck compactness theorem in the variable coefficient case to
‘moving coefficients’. This is particularly relevant for nonlinear equations
with dielectricity depending on the electric field.

######  Theorem D (compactness criterion for moving coefficients).

Let $(\varepsilon_{n})_{n}$, $\varepsilon$ in $\mathcal{M}(\alpha,\beta)$.

Assume $\varepsilon_{n}\to\varepsilon$ in the weak operator topology and in
$\tau_{\Omega}$; and for all $E\in L_{2}(\Omega)^{3}$, assume
$\\{\operatorname{div}\varepsilon_{n}E;n\in\mathbb{N}\\}$ to be relatively
compact in $H^{-1}(\Omega)$.

Let $(E_{n})_{n}$ be a bounded sequence in
$\operatorname{dom}(\operatorname{curl}^{*})$ and assume
$\\{\operatorname{div}\varepsilon_{n}E_{n};n\in\mathbb{N}\\}$ to be bounded in
$L_{2}(\Omega)$. Then $(E_{n})_{n}$ contains an $L_{2}(\Omega)^{3}$-convergent
subsequence.

The methods developed here also work for a different set of boundary
conditions. Thus, analogous results also hold for the magneto-static
equations. In particular, a compactness statement with moving magnetic
permittivities also holds even though, we will not explicitly spell out the
result here. The whole theory developed here – similar to the notion of
nonlocal $H$-convergence in [29] – is based on the concept of (closed/compact)
Hilbert complexes. In order to keep the present manuscript as accessible as
possible we refrain from providing the general statements in the framework of
Hilbert complexes and rather refer to future research. Consequently, we shall
also address the particular cases for other established closed/compact Hilbert
complexes in a forthcoming publication.

Before we provide a brief overview of the arrangement of the contents of the
paper we provide a short literature review to relevant results regarding the
article at hand.

The idea of addressing possibly non-periodic homogenisation problems by
introducing an abstract notion of convergence goes back to Spagnolo (for
symmetric coefficients) in [22, 23] and, independently, to Murat and Tartar
(see, e.g., [9, 24]) (for general matrix-valued multiplication operators). The
generalisation of these concepts to homogenisation for nonlocal problems was
introduced in [29]. It has been applied to variational problems in the
situation of curl-div problems in [10]. In this source an abstract notion for
nonlocal $H$-convergence has been introduced, which is similar to the one to
be introduced below. However, the concepts have been applied to topologically
trivial domains only (or even more restrictive situations). We note that for
topologically trivial domains, the notion of nonlocal $H$-convergence has been
used to deduce (nonlocal) homogenisation theorems for time-dependent Maxwell’s
equations or the wave equation, see [29, 27].

For proving the above mentioned compactness statement, we will have the
occasion to use a particular version of the div-curl lemma. This div-curl
lemma used here is based on the abstract findings in [28]. These functional
analytic variants of the div-curl lemma can also be found in [11] with plenty
of concrete examples; see also [19]. We emphasise that in [11] one can also
find examples with variable coefficients. The idea of the abstract results can
be dated back to the notion of compensated compactness (see e.g. [24] and the
references therein).

Both for the Helmholtz decomposition as well as for the solution theory for
the electrostatic equations considered here, we refer (for the special case of
variable, self-adjoint and coercive coefficients) to the so-called ‘FA-
toolbox’ gathered by D. Pauly and presented for instance in [14]; see also
[17]. We note that also Friedrich’s type inequalities are accessible within
this framework. We exemplarily refer to some Banach space versions of the
Helmholtz decomposition in [4, 7, 21].

The basic compact embedding result for Maxwell’s equations is the
Picard–Weber–Weck selection theorem, see [16]. The core observation is that
compact embedding results of the type

$\operatorname{dom}(\operatorname{div})\cap\operatorname{dom}(\operatorname{curl}^{*})\hookrightarrow\hookrightarrow
L_{2}(\Omega)^{3}$ ($*$)

can be shown _without_ the validity of Gaffney’s inequality thus allowing to
prove this compactness result for more general $\Omega$ such as domains with
boundaries that are Lipschitz manifolds (i.e., weak Lipschitz domains). We
refer to [1] for generalisations to mixed boundary conditions. The proof in
[16] is based on Lipschitz transformations reducing the compactness statement
to whether or not the embedding

$\operatorname{dom}(\operatorname{div}\varepsilon)\cap\operatorname{dom}(\operatorname{curl}^{*})\hookrightarrow
L_{2}(\Omega)^{3}$

is compact for self-adjoint, coercive (matrix-valued) multiplication operators
$\varepsilon$ and $\Omega$ being the unit cube. This in turn can be shown
using weighted scalar products in Hilbert spaces and the classical compact
embedding result (that makes use of Gaffney’s inequality). We particularly
refer to the derivations in [13, Lemma 5.1 and Section 6.1] for these
techniques. In particular, one obtains

$\operatorname{dom}(\operatorname{div}\varepsilon)\cap\operatorname{dom}(\operatorname{curl}^{*})\hookrightarrow\hookrightarrow
L_{2}(\Omega)^{3}$

for self-adjoint, coercive $\varepsilon$ and bounded weak Lipschitz domains
$\Omega$. It appears that already for constant sequences
$(\varepsilon_{n})_{n}=(\varepsilon)_{n}$ for some nonselfadjoint, coercive
$\varepsilon$, the above Theorem D is new.

Next, we sketch the contents of the sections of the present article. In
Section 2, we recall the core finding of [25] which forms the foundation of
the rationale to follow. The Helmholtz decomposition in the most general form
of this article is provided in Section 3. This decomposition is used to obtain
a solution theory for electrostatics, which is presented in Section 4. The
concept of nonlocal $H$-convergence is recalled in Section 5. In this section,
we will also argue that finite-dimensional harmonic Dirichlet/Neumann fields
will not have an effect on the nonlocal $H$-topology. In Section 6, we provide
an example for a domain with infinite-dimensional harmonic Dirichlet fields
eventually leading to the possibility of an infinite set of mutually
incomparable nonlocal $H$-topologies. The nonlocal homogenisation result for
electrostatic problems is provided in Section 7. Section 8 contains the
convergence of the corresponding energy. This result is applied in Section 9
to derive a proof of Theorem D, where we also provide an elementary example of
nonlocal coefficients. In Section 10, we will show that the well-known notion
of local $H$-convergence is equivalent to nonlocal $H$-convergence on
multiplication operators for any underlying $\Omega$ satisfying the electric
compactness property; that is, ($*$ ‣ 1).

We provide a conclusion of our findings in Section 11.

For linear operators the term ‘invertible’ will always be reserved for
surjective and injective transformations. Since we only consider closed
operators defined on Hilbert spaces here; invertible will thus always mean
‘continuously invertible’. $\mathbb{K}\in\\{\mathbb{R},\mathbb{C}\\}$, scalar
products are anti-linear in the first and linear in the second component.
Direct sums of vector subspaces are denoted by $\dot{+}$; if these sums happen
to be orthogonal, we als use $\oplus$.

##  2 Abstract divergence-form operators

The Helmholtz decomposition as much as the solution theory for the considered
electrostatic equations discussed here is based on [25, Theorem 3.1]. We
shortly set the stage for this detour. For a densely defined closed linear
operator
$C\colon\operatorname{dom}(C)\subseteq\mathcal{H}_{0}\to\mathcal{H}_{1}$
acting from the Hilbert spaces $\mathcal{H}_{0}$ into $\mathcal{H}_{1}$, we
define
$H^{1}(C)\coloneqq(\operatorname{dom}(C),\langle\cdot,\cdot\rangle_{C})$ with
$\langle\cdot,\cdot\rangle_{C}$ being the graph scalar product of $C$. We
define $C^{\diamond}\colon\mathcal{H}_{1}\to H^{-1}(C)\coloneqq(H^{1}(C))^{*}$
via

$C^{\diamond}\varphi\colon
H^{1}(C)\to\mathbb{K},u\mapsto\langle\varphi,Cu\rangle_{\mathcal{H}_{1}}.$

It is then not difficult to see that $C^{\diamond}$ extends $C^{*}$, where we
identify $\mathcal{H}_{1}$ with its dual via the unitary Riesz mapping.

For a Hilbert space $\mathcal{H}$ and a closed subspace
$\mathcal{V}\subseteq\mathcal{H}$, we define

$\iota_{\mathcal{V}}\colon\mathcal{V}\hookrightarrow\mathcal{H},x\mapsto x$

the canonical embedding. Note that then
$\iota_{\mathcal{V}}^{*}\colon\mathcal{H}\rightarrow\mathcal{V}$ acts as the
orthogonal projection onto $\mathcal{V}$ and
$\pi_{\mathcal{V}}\coloneqq\iota_{\mathcal{V}}\iota_{\mathcal{V}}^{*}$ is the
actual orthogonal projection; see [20, Lemma 11.3.3].

If $C\colon\operatorname{dom}(C)\subseteq\mathcal{H}_{0}\to\mathcal{H}_{1}$ is
closed and densely defined with closed range
$\mathcal{V}_{1}\coloneqq\operatorname{ran}(C)\subseteq\mathcal{H}_{1}$, the
closed range theorem yields that
$\mathcal{V}_{0}\coloneqq\operatorname{ran}(C^{*})\subseteq\mathcal{H}_{0}$ is
closed as well. In this case, we define $C_{{\rm
red}}\coloneqq\iota_{\mathcal{V}_{1}}^{*}C\iota_{\mathcal{V}_{0}}$, the
reduced operator.

######  Theorem 2.1 ([25, Theorem 3.1]).

Let $\mathcal{H}_{0}$, $\mathcal{H}_{1}$ be Hilbert spaces,
$a\in\mathcal{B}(\mathcal{H}_{1})$,
$C\colon\operatorname{dom}(C)\subseteq\mathcal{H}_{0}\to\mathcal{H}_{1}$
densely defined, closed with closed range
$\mathcal{V}_{1}\coloneqq\operatorname{ran}(C)\subseteq\mathcal{H}_{1}$. Then
the following conditions are equivalent:

  1. (i)

for all $f\in H^{-1}(C_{{\rm red}})$ there exists a unique $u\in H^{1}(C_{{\rm
red}})$ such that

$\langle aCu,C\varphi\rangle=f(\varphi)\quad(\varphi\in H^{1}(C_{{\rm
red}}));$

  2. (ii)

for all $f\in H^{-1}(C_{{\rm red}})$ there exists a unique $u\in H^{1}(C_{{\rm
red}})$ such that

$C_{{\rm
red}}^{\diamond}\iota_{\mathcal{V}_{1}}^{*}a\iota_{\mathcal{V}_{1}}C_{{\rm
red}}u=f;$

  3. (iii)

the operator
$\iota_{\mathcal{V}_{1}}^{*}a\iota_{\mathcal{V}_{1}}\in\mathcal{B}(\mathcal{V}_{1})$
is continuously invertible.

In either case, if $f\in H^{-1}(C_{{\rm red}})$ then

$u=(C_{{\rm red}})^{-1}(\iota_{V_{1}}^{*}a\iota_{V_{1}})^{-1}(C_{{\rm
red}}^{\diamond})^{-1}f.$

If $f\in\operatorname{ran}(C_{{\rm red}}^{*})$ then

$u=(C_{{\rm red}})^{-1}(\iota_{V_{1}}^{*}a\iota_{V_{1}})^{-1}(C_{{\rm
red}}^{*})^{-1}f$

and the operator

$D_{{\rm red},a}\colon\operatorname{dom}(D_{{\rm red},a})\subseteq V_{0}\to
V_{1},u\mapsto C_{{\rm red}}^{\diamond}aC_{{\rm red}}u$

with domain

$\operatorname{dom}(D_{{\rm red},a})=\\{u\in\operatorname{dom}(C_{{\rm
red}});aC_{{\rm red}}u\in\operatorname{dom}(C_{{\rm red}}^{*})\\}$

is densely defined, closed and continuously invertible. If
$\operatorname{dom}(C)\cap\operatorname{ker}(C)^{\bot}\hookrightarrow\mathcal{H}_{0}$
compactly, then $\operatorname{dom}(D_{{\rm red},a})$ has compact resolvent.

######  Remark 2.2.

We recall that the standard assumption for the coefficients
$a\in\mathcal{B}(\mathcal{H}_{1})$ for weak variational problems of the type
discussed in (a) is that $\operatorname{Re}a\geqslant c$ for some $c>0$ (i.e.,
$a$ is coercive) in the sense of positive definiteness. A moment’s thought
reveals that this condition yields
$\operatorname{Re}\iota_{\mathcal{V}_{1}}^{*}a\iota_{\mathcal{V}_{1}}\geqslant
c\operatorname{id}_{\mathcal{V}_{1}}$. In fact, the same holds for $a$ being a
maximal monotone relation with $\operatorname{dom}(a)=\mathcal{H}_{1}$, see
[18]. This has been exploited in [25] to provide a solution theory for
nonlinear divergence form problems.

##  3 The Helmholtz decomposition

It is the aim of this section to revisit the Helmholtz decomposition for
material coefficients satisfying the well-posedness criterion in Theorem 2.1.
For this, throughout this section, we let $\Omega\subseteq\mathbb{R}^{3}$ be
open. Next, we properly establish the needed vector analytical operators
throughout this manuscript. We use standard notation from the theory of
Sobolev spaces. Define $\operatorname{grad}\colon H^{1}(\Omega)\subseteq
L_{2}(\Omega)\to L_{2}(\Omega)^{3},\varphi\mapsto\nabla\varphi$ and
$\operatorname{\mathring{grad}}\coloneqq\operatorname{grad}|_{H_{0}^{1}(\Omega)}$.
Moreover, we put
$\operatorname{div}\coloneqq\operatorname{\mathring{grad}}^{*}$ and
$\operatorname{curl}\colon H^{1}(\operatorname{curl})\subseteq
L_{2}(\Omega)^{3}\to L_{2}(\Omega)^{3}$ endowed with their respective maximal
domains $H(\operatorname{div})$ and $H^{1}(\operatorname{curl})$. Note that
$\operatorname{curl}^{*}\eqqcolon\operatorname{\mathring{curl}}$ is the
completion of $C_{c}^{\infty}(\Omega)^{3}$ with respect to the graph norm of
$\operatorname{curl}$ thus realising the abstract implementation of vanishing
tangential component.

As a general assumption in this section, we shall ask $\Omega$ to be range
closed; that is, both

$\operatorname{ran}(\mathring{\operatorname{grad}})\subseteq
L_{2}(\Omega)^{3}\text{ and }\operatorname{ran}(\operatorname{curl})\subseteq
L_{2}(\Omega)^{3}\text{ are closed.}$

The classical Helmholtz decomposition (see e.g. [17]) asserts that we can
decompose any $L_{2}(\Omega)^{3}$-vector field $E$ with
$L_{2}(\Omega)^{3}$-orthogonal decomposition in the form

$E=\operatorname{\mathring{grad}}u+\operatorname{curl}H+x,$

where $u\in H_{0}^{1}(\Omega)$,
$H\in\operatorname{dom}(\operatorname{curl})\cap\operatorname{ker}(\operatorname{curl})^{\bot}$
and $x\in\mathcal{H}_{D}(\Omega)$ are uniquely determined. Here,

$\mathcal{H}_{D}(\Omega)\coloneqq\operatorname{ker}(\operatorname{div})\cap\operatorname{ker}(\mathring{\operatorname{curl}})$

denotes the space of harmonic Dirichlet fields. Tailored for static Maxwell
problems, the Helmholtz decomposition with variable coefficients looks similar
and still asserts a direct though in general not orthogonal decomposition. For
this, we call $\varepsilon\in\mathcal{B}(L_{2}(\Omega)^{3})$ admissible, if
the following three conditions hold:

  1. (a1)

$\varepsilon$ is invertible;

  2. (a2)

$\iota^{*}\varepsilon\iota$ is invertible, where
$\iota\colon\operatorname{ran}(\operatorname{\mathring{grad}})\hookrightarrow
L_{2}(\Omega)^{3}$;

  3. (a3)

$\kappa^{*}\varepsilon^{-1}\kappa$ is invertible, where
$\kappa\colon\operatorname{ran}(\operatorname{curl})\hookrightarrow
L_{2}(\Omega)^{3}$.

######  Remark 3.1.

(a) The assumption (a2) on $\varepsilon$ implies that Theorem 2.1 applies to
$C=\operatorname{\mathring{grad}}$ (then $C_{{\rm
red}}=\iota^{*}\operatorname{\mathring{grad}}$) in order that for all $f\in
H^{-1}(\iota^{*}\operatorname{\mathring{grad}})=H^{-1}(\Omega)$ the
variational problem

$\langle\varepsilon\operatorname{\mathring{grad}}u,\operatorname{\mathring{grad}}u\rangle=f(\varphi)\quad(\varphi\in
H_{0}^{1}(\Omega)=\operatorname{dom}(\operatorname{\mathring{grad}}))$

admits a unique solution $u\in H_{0}^{1}(\Omega)$. In the following we shall
refer to this variational problem also by employing the usual distributional
divergence $\operatorname{\mathring{grad}^{\diamond}}$ which extends the above
defined $\operatorname{div}=\operatorname{\mathring{grad}}^{*}$, reusing the
notion $\operatorname{div}$ and just writing

$-\operatorname{div}\varepsilon\operatorname{\mathring{grad}}u=f.$

(b) Quite similarly, the assumption (a1) in conjunction with (a3) enable the
usage of Theorem 2.1 for $C=\operatorname{curl}$. Recall that we assumed
$\Omega$ to be range closed, particularly implying that
$\operatorname{ran}(C)=\operatorname{ran}(\operatorname{curl})$ is closed. In
this case, we obtain $C_{{\rm red}}=\operatorname{curl_{\textnormal{red}}}$
and, thus,
$\operatorname{dom}(\operatorname{curl_{\textnormal{red}}})=H^{1}(\operatorname{curl})\cap\operatorname{ker}(\operatorname{curl})^{\bot}$.
Consequently, due to Theorem 2.1, for all $g\in
H^{-1}(\operatorname{curl_{\textnormal{red}}})$, there exists a unique $v\in
H^{1}(\operatorname{curl_{\textnormal{red}}})$ such that

$\langle\varepsilon^{-1}\operatorname{curl}v,\operatorname{curl}\psi\rangle=g(\psi)\quad(\psi\in
H^{1}(\operatorname{curl_{\textnormal{red}}})).$

We emphasise that the boundary condition is (weakly) imposed by this
variational formulation. Indeed, if $g\in L_{2}(\Omega)^{3}$, we would obtain
$\varepsilon^{-1}\operatorname{curl}v\in\operatorname{dom}(\operatorname{curl}^{*})=\operatorname{dom}(\operatorname{\mathring{curl}})$.
Hence, the tangential component of $\varepsilon^{-1}\operatorname{curl}v$
vanishes at the boundary (in a generalised sense). Similar to the notation in
(a) as a reminder of the boundary condition, we sometimes reuse the notation
for the adjoint of $\operatorname{curl}^{*}=\operatorname{\mathring{curl}}$ to
denote $\operatorname{{curl}^{\diamond}}$. It then depends on the context
whether we have applied $\operatorname{{curl}^{\diamond}}$ or
$\operatorname{curl}^{*}$.

The latter remark together with the following shorthands finally puts us in
the position to state and prove the generalised Helmholtz decomposition we are
aiming for in this section. For admissible $\varepsilon$, we define

$\mathcal{H}_{\varepsilon}\coloneqq\operatorname{ker}(\operatorname{div}\varepsilon)\cap\operatorname{ker}(\operatorname{\mathring{curl}})\text{
and
}\mathcal{H}^{\varepsilon}\coloneqq\operatorname{ker}(\operatorname{div})\cap\operatorname{ker}(\operatorname{\mathring{curl}}\varepsilon^{-1}).$

######  Theorem 3.2 (generalised Helmholtz decomposition).

Let $\Omega\subseteq\mathbb{R}^{3}$ open and range closed. Let
$\varepsilon\in\mathcal{B}(L_{2}(\Omega)^{3})$ be admissible. Then the
following decompositions are direct

$\displaystyle L_{2}(\Omega)^{3}$
$\displaystyle=\operatorname{ran}(\operatorname{\mathring{grad}})\dot{+}\varepsilon^{-1}\operatorname{ran}({\operatorname{curl}})\dot{+}\mathcal{H}_{\varepsilon}$
$\displaystyle=\varepsilon\operatorname{ran}(\operatorname{\mathring{grad}})\dot{+}\operatorname{ran}({\operatorname{curl}})\dot{+}\mathcal{H}^{\varepsilon}.$

Moreover,

$\varepsilon\colon\mathcal{H}_{\varepsilon}\to\mathcal{H}^{\varepsilon},x\mapsto\varepsilon
x$

is a bijection.

###### Proof.

Since the rationale for both decompositions is (almost) the same, we only show
the first decomposition. We show that this decomposition is direct first. Let
$E\in\operatorname{ran}(\mathring{\operatorname{grad}})\cap\varepsilon^{-1}\operatorname{ran}({\operatorname{curl}})$.
Then we find $u\in H_{0}^{1}(\Omega)$ and
$H\in\operatorname{dom}(\operatorname{curl})\cap\operatorname{ker}(\operatorname{curl})^{\bot}$
such that

$E=\operatorname{\mathring{grad}}u=\varepsilon^{-1}\operatorname{curl}H.$

Applying $\operatorname{div}\varepsilon$, we infer

$\operatorname{div}\varepsilon
E=\operatorname{div}\varepsilon\mathring{\operatorname{grad}}u=\operatorname{div}\operatorname{curl}H=0.$

Thus, $u=0$, by Theorem 2.1 and (a2) and, hence, $E=0$.

If
$E=\operatorname{ran}(\mathring{\operatorname{grad}})\cap\mathcal{H}_{\varepsilon}$,
then we find $u\in H_{0}^{1}(\Omega)$ and
$x\in\operatorname{ker}(\operatorname{div}\varepsilon)\cap\operatorname{ker}(\mathring{\operatorname{curl}})$
such that

$E={\operatorname{\mathring{grad}}}u=x.$

Again, applying $\operatorname{div}\varepsilon$, we obtain

$\operatorname{div}\varepsilon
E=\operatorname{div}\varepsilon{\operatorname{\mathring{grad}}}u=\operatorname{div}\varepsilon
x=0.$

Hence, by Theorem 2.1 and (a2), $u=0$ and, thus, $E=0$.

Finally, if
$E\in\operatorname{ran}(\operatorname{curl})\cap\mathcal{H}_{\varepsilon}$,
then we find
$H\in\operatorname{dom}(\operatorname{curl})\cap\operatorname{ker}(\operatorname{curl})^{\bot}$
and
$x\in\operatorname{ker}(\operatorname{div}\varepsilon)\cap\operatorname{ker}(\mathring{\operatorname{curl}})$
such that

$E=\varepsilon^{-1}\operatorname{curl}H=x.$

Applying $\operatorname{\mathring{curl}}$, we infer

$\operatorname{\mathring{curl}}E=\operatorname{\mathring{curl}}\varepsilon^{-1}\operatorname{curl}H=\operatorname{curl}x=0.$

Hence, by Theorem 2.1 and (a2), $H=0$, and, therefore, $E=0$.

Finally, let $E\in L_{2}(\Omega)^{3}$ we define $u\in H_{0}^{1}(\Omega)$ and
$H\in\operatorname{ran}(\operatorname{curl})\cap\operatorname{ker}(\operatorname{curl})^{\bot}$
satisfying

$\operatorname{div}\varepsilon{\operatorname{\mathring{grad}}}u=\operatorname{div}\varepsilon
E\text{ and
}\operatorname{\mathring{curl}}\varepsilon^{-1}\operatorname{curl}H=\operatorname{\mathring{curl}}\varepsilon^{-1}E.$

Then, it follows that

$x\coloneqq
E-\mathring{\operatorname{grad}}u-\varepsilon^{-1}\operatorname{curl}H\in\operatorname{ker}(\operatorname{div}\varepsilon)\cap\operatorname{ker}(\mathring{\operatorname{curl}})=\mathcal{H}_{\varepsilon}.$

The last statement in the theorem is an easy consequence of the definition of
$\mathcal{H}_{\varepsilon}$ and $\mathcal{H}^{\varepsilon}$. ∎

Theorem 3.2 asserts that the spaces
$\operatorname{ran}(\mathring{\operatorname{grad}})$,
$\varepsilon^{-1}\operatorname{ran}(\operatorname{curl})$ and
$\mathcal{H}_{\varepsilon}$ are complemented in $L_{2}(\Omega)^{3}$ and as
these spaces are closed the projection onto these spaces is continuous. For
$\varepsilon\operatorname{ran}(\mathring{\operatorname{grad}})$,
$\operatorname{ran}(\operatorname{curl})$ and $\mathcal{H}^{\varepsilon}$ a
similar observation holds.

Next, it is the aim to provide some connections of the continuous projections

$\pi_{\varepsilon}\colon\operatorname{ran}(\mathring{\operatorname{grad}})\dot{+}\varepsilon^{-1}\operatorname{ran}({\operatorname{curl}})\dot{+}\mathcal{H}_{\varepsilon}\to\mathcal{H}_{\varepsilon}$
(1)

and

$\pi^{\varepsilon}\colon\varepsilon\operatorname{ran}(\mathring{\operatorname{grad}})\dot{+}\operatorname{ran}({\operatorname{curl}})\dot{+}\mathcal{H}^{\varepsilon}\to\mathcal{H}^{\varepsilon}$
(2)

to the orthogonal projection

$\pi_{D}\colon
L_{2}(\Omega)^{3}\to\mathcal{H}_{D}(\Omega)\coloneqq\operatorname{ker}(\operatorname{div})\cap\operatorname{ker}(\mathring{\operatorname{curl}}).$

######  Corollary 3.3.

Let $\varepsilon\in\mathcal{B}(L_{2}(\Omega)^{3})$ be admissible. Then the
following statements hold:

(a) $\pi_{D}|_{\mathcal{H}_{\varepsilon}}$ is bijective with inverse given by
$\pi_{\varepsilon}|_{\mathcal{H}_{D}(\Omega)}$.

(b) Let $E\in L_{2}(\Omega)^{3}$. Then

$\displaystyle\pi_{\varepsilon}\pi_{D}E$
$\displaystyle=(1-\mathring{\operatorname{grad}}(\operatorname{div}\varepsilon\mathring{\operatorname{grad}})^{-1}\operatorname{div}\varepsilon)\pi_{D}E\text{
and }$ $\displaystyle\pi_{D}\pi_{\varepsilon}E$
$\displaystyle=(1-\mathring{\operatorname{grad}}(\operatorname{div}\mathring{\operatorname{grad}})^{-1}\operatorname{div})\pi_{\varepsilon}E.$

###### Proof.

(a) Let $x_{\varepsilon}\in\mathcal{H}_{\varepsilon}$. If
$\pi_{D}x_{\varepsilon}=0$, then by Theorem 3.2 (for
$\varepsilon=\operatorname{id}$) we find $u\in H_{0}^{1}(\Omega)$, $v\in
H^{1}(\operatorname{curl_{\textnormal{red}}})$ with

$x_{\varepsilon}=\operatorname{\mathring{grad}}u+\operatorname{curl_{\textnormal{red}}}v.$

Applying $\operatorname{\mathring{curl}}$, we infer

$0=\operatorname{\mathring{curl}}x_{\varepsilon}=\operatorname{\mathring{curl}}\operatorname{curl_{\textnormal{red}}}v.$

Hence, $v=0$. The application of $\operatorname{div}\varepsilon$ yields

$0=\operatorname{div}\varepsilon
x_{\varepsilon}=\operatorname{div}\varepsilon\operatorname{\mathring{grad}}u.$

Hence, $u=0$. Thus, $\pi_{D}|_{\mathcal{H}_{\varepsilon}}$ is injective. Now,
let $x\in\mathcal{H}_{D}(\Omega)$. Again, by Theorem 3.2 applied in its full
generality, we obtain uniquely determined $u\in H_{0}^{1}(\Omega)$, $v\in
H^{1}(\operatorname{curl_{\textnormal{red}}})$ and
$x_{\varepsilon}\in\mathcal{H}_{\varepsilon}$ such that

$x=\operatorname{\mathring{grad}}u+\varepsilon^{-1}\operatorname{curl}v+x_{\varepsilon}.$

Then, applying $\operatorname{\mathring{curl}}$ yields

$0=\operatorname{\mathring{curl}}\varepsilon^{-1}\operatorname{curl}v=\operatorname{curl_{\textnormal{red}}^{\diamond}}\kappa^{*}\varepsilon^{-1}\kappa\operatorname{curl_{\textnormal{red}}}v;$

thus $v=0$. Hence,

$\pi_{D}(x_{\varepsilon})=\pi_{D}(x-\operatorname{\mathring{grad}}u)=\pi_{D}x=x.$

Computing the inverse works similarly.

(b) We only show the first equality, the second is proven similarly. We let
$\pi_{D}E=x$. Using the Helmholtz decomposition again, we obtain

$x=\pi_{D}E=\operatorname{\mathring{grad}}u_{\varepsilon}+\varepsilon^{-1}\operatorname{curl}v_{\varepsilon}+x_{\varepsilon}$

Applying $\operatorname{\mathring{curl}}$, we infer

$0=\operatorname{\mathring{curl}}x=\operatorname{\mathring{curl}}\varepsilon^{-1}\operatorname{curl}v_{\varepsilon}$

and, hence, $v_{\varepsilon}=0$ by (a3) and Theorem 2.1. Next,

$\operatorname{div}\varepsilon
x=\operatorname{div}\varepsilon\operatorname{\mathring{grad}}u_{\varepsilon}.$

Thus, by (a2) and Theorem 2.1,
$u_{\varepsilon}=(\operatorname{div}\varepsilon\operatorname{\mathring{grad}})^{-1}\operatorname{div}\varepsilon
x$ and so

$\displaystyle x$
$\displaystyle=\operatorname{\mathring{grad}}u_{\varepsilon}+\varepsilon^{-1}\operatorname{curl}v_{\varepsilon}+x_{\varepsilon}$
$\displaystyle=\operatorname{\mathring{grad}}(\operatorname{div}\varepsilon\operatorname{\mathring{grad}})^{-1}\operatorname{div}\varepsilon
x+x_{\varepsilon},$

which implies the desired equality. ∎

We shortly comment on the standing assumptions in this section.

######  Remark 3.4 (On range closed).

In [12, Remark 4.7] one can find conditions for the closedness of
$\operatorname{ran}(\operatorname{curl})$ (or, equivalently,
$\operatorname{ran}(\operatorname{\mathring{curl}})$). A sufficient condition
is for $\Omega$ to be bounded with boundary that is a Lipschitz manifold,
i.e., $\Omega$ is weak Lipschitz domain. One argues by contradiction invoking
Picard–Weber–Weck’s selection theorem (below). By Poincaré’s inequality, for
$\Omega$ contained in a slab, it follows that
$\operatorname{ran}(\operatorname{\mathring{grad}})$ is closed; see [20,
Proposition 11.3.1 and Corollary 11.3.2].

######  Theorem 3.5 (Picard–Weber–Weck selection theorem, [16]).

Let $\Omega\subseteq\mathbb{R}^{3}$ be a weak Lipschitz domain. Then the
embeddings

$\operatorname{dom}(\operatorname{div})\cap\operatorname{dom}(\mathring{\operatorname{curl}})\hookrightarrow
L_{2}(\Omega)^{3}\text{ and
}\operatorname{dom}(\mathring{\operatorname{div}})\cap\operatorname{dom}({\operatorname{curl}})\hookrightarrow
L_{2}(\Omega)^{3}$

are compact.

######  Remark 3.6.

If $\varepsilon\in\mathcal{B}(L_{2}(\Omega)^{3})$ satisfies
$\operatorname{Re}\varepsilon\geqslant c>0$, then $\varepsilon$ satisfies
(a1), (a2), and (a3), see [20, Proposition 6.2.3(b)].

######  Proposition 3.7.

Let $\varepsilon\in\mathcal{B}(L_{2}(\Omega)^{3})$ satisfy (a1). Assume that
$\mathcal{H}_{\textnormal{D}}(\Omega)=\\{0\\}$. Then (a2) $\iff$ (a3).

###### Proof.

The assertion follows from the formula stated in Lemma 3.8 below, which in
turn follows from a straightforward computation.∎

######  Lemma 3.8.

Let $\mathcal{L}_{0},\mathcal{L}_{1}$ be Hilbert spaces and
$a=(a_{jk})_{j,k\in\\{0,1\\}}\in\mathcal{B}(\mathcal{L}_{0}\oplus\mathcal{L}_{1})$
and assume $a_{00}$ is invertible. Then

$\begin{pmatrix}1&0\\\
-a_{10}a_{00}^{-1}&1\end{pmatrix}\begin{pmatrix}a_{00}&a_{01}\\\
a_{10}&a_{11}\end{pmatrix}\begin{pmatrix}1&-a_{00}^{-1}a_{01}\\\
0&1\end{pmatrix}=\begin{pmatrix}a_{00}&0\\\
0&a_{11}-a_{10}a_{00}^{-1}a_{01}\end{pmatrix}.$

Moreover, the following statements are equivalent:

  1. (i)

$a$ invertible;

  2. (ii)

$a_{S}\coloneqq a_{11}-a_{10}a_{00}^{-1}a_{01}$ invertible.

In either case, we have

$\begin{pmatrix}a_{00}&a_{01}\\\
a_{10}&a_{11}\end{pmatrix}^{-1}=\begin{pmatrix}a_{00}^{-1}+a_{00}^{-1}a_{01}a_{S}^{-1}a_{10}a_{00}^{-1}&-a_{00}^{-1}a_{01}a_{S}^{-1}\\\
-a_{S}^{-1}a_{10}a_{00}^{-1}&a_{S}^{-1}\end{pmatrix}.$

##  4 Two formulations for electrostatics

In this section, we present two entirely equivalent formulations of the
problem for electrostatics.

Similar to the previous section, we let $\Omega\subseteq\mathbb{R}^{3}$ be
open and range closed; that is, we assume that both

$\operatorname{ran}(\operatorname{\mathring{grad}})\subseteq
L_{2}(\Omega)^{3}\text{ and }\operatorname{ran}(\operatorname{curl})\subseteq
L_{2}(\Omega)^{3}\text{ are closed.}$

Moreover, throughout this section, we assume that
$\varepsilon\in\mathcal{B}(L_{2}(\Omega)^{3})$ is admissible; that is,
$\varepsilon$ satisfies the conditions (a1), (a2), and (a3) from the previous
section. The solution theory for the two problems to be defined next is based
on the Helmholtz decomposition presented in Theorem 3.2. Since we frequently
employ the generalised Helmholtz decomposition in the following, we will not
mark every instance of application of this theorem. We will also use the
continuous projections $\pi_{\varepsilon}$ and $\pi^{\varepsilon}$ introduced
in (1) and (2). Finally, we make use of the distribution spaces
$H^{-1}(\Omega)$ and $H^{-1}(\operatorname{curl_{\textnormal{red}}})$
mentioned in Remark 3.1.

######  Problem 4.1.

For given $f\in H^{-1}(\Omega)$, $g\in
H^{-1}(\operatorname{curl_{\textnormal{red}}})$, and
$x_{\varepsilon}\in\operatorname{ker}(\operatorname{div}\varepsilon)\cap\operatorname{ker}(\mathring{\operatorname{curl}})$
find

$E\in L_{2}(\Omega)^{3}$

such that

$\begin{cases}\operatorname{div}\varepsilon E=f,&\\\
\operatorname{\mathring{curl}}E=g,&\\\
\pi_{\varepsilon}E=x_{\varepsilon}.\end{cases}$
($\mathbb{P}_{\varepsilon}(f,g,x_{\varepsilon})$)

######  Problem 4.2.

For given $f\in H^{-1}(\Omega)$, $g\in
H^{-1}(\operatorname{curl_{\textnormal{red}}})$, and
$x^{\varepsilon}\in\operatorname{ker}(\operatorname{div})\cap\operatorname{ker}(\operatorname{\mathring{curl}}\varepsilon^{-1})$
find

$H\in L_{2}(\Omega)^{3}$

such that

$\begin{cases}\operatorname{div}H=f,&\\\
\operatorname{\mathring{curl}}\varepsilon^{-1}H=g,&\\\
\pi^{\varepsilon}H=x^{\varepsilon}.\end{cases}$
($\mathbb{P}^{\varepsilon}(f,g,x^{\varepsilon})$)

We establish well-posedness of either of the above problems:

######  Theorem 4.3.

Let $\Omega\subseteq\mathbb{R}^{3}$ be open and range closed and
$\varepsilon\in\mathcal{B}(L_{2}(\Omega)^{3})$ admissible. Let $f\in
H^{-1}(\Omega)$, $g\in H^{-1}(\operatorname{curl_{\textnormal{red}}})$, and
$x_{\varepsilon}\in\mathcal{H}_{\varepsilon}$ and
$x^{\varepsilon}\in\mathcal{H}^{\varepsilon}$.

(a) Problem 4.1 is uniquely solvable in $L_{2}(\Omega)^{3}$ with continuous
solution operator; the solution is given by

$E=\operatorname{\mathring{grad}}(\operatorname{div}\varepsilon\operatorname{\mathring{grad}})^{-1}f+\varepsilon^{-1}\operatorname{curl_{\textnormal{red}}}(\operatorname{curl_{\textnormal{red}}^{\diamond}}\varepsilon^{-1}\operatorname{curl_{\textnormal{red}}})^{-1}g+x_{\varepsilon}.$

If $f\in L_{2}(\Omega)$ and
$g\in\operatorname{ran}(\operatorname{\mathring{curl}})$, then
$E\in\operatorname{dom}(\operatorname{div}\varepsilon)\cap\operatorname{dom}(\operatorname{\mathring{curl}})$;
with corresponding continuous solution operator.

(b) Problem 4.2 is uniquely solvable in $L_{2}(\Omega)^{3}$ with continuous
solution operator; the solution is given by

$H=\varepsilon{\operatorname{\mathring{grad}}}(\operatorname{div}\varepsilon{\operatorname{\mathring{grad}}})^{-1}f+\operatorname{curl_{\textnormal{red}}}({\operatorname{curl_{\textnormal{red}}^{\diamond}}}\varepsilon^{-1}\operatorname{curl_{\textnormal{red}}})^{-1}g+x^{\varepsilon}.$

If $f\in L_{2}(\Omega)$ and
$g\in\operatorname{ran}(\operatorname{\mathring{curl}})$, then
$H\in\operatorname{dom}(\operatorname{div})\cap\operatorname{dom}({\operatorname{\mathring{curl}}}\varepsilon^{-1})$;
with corresponding continuous solution operator.

(c) The Problems 4.1 and 4.2 are equivalent in the sense that for every
solution $E$ solving ($\mathbb{P}_{\varepsilon}(f,g,x_{\varepsilon})$),
$H=\varepsilon E$ is a solution for
($\mathbb{P}^{\varepsilon}(f,g,x^{\varepsilon})$) with
$x^{\varepsilon}=\varepsilon x_{\varepsilon}$; similarly, if $H$ solves
($\mathbb{P}^{\varepsilon}(f,g,x^{\varepsilon})$), then $E=\varepsilon^{-1}H$
solves ($\mathbb{P}_{\varepsilon}(f,g,x_{\varepsilon})$) with
$x_{\varepsilon}=\varepsilon^{-1}x^{\varepsilon}$.

###### Proof.

(a) We decompose

$E=\operatorname{\mathring{grad}}u+\varepsilon^{-1}\operatorname{curl_{\textnormal{red}}}F+q,$

where $F\in\operatorname{dom}(\operatorname{curl_{\textnormal{red}}})$, $u\in
H_{0}^{1}(\Omega)$ and $q\in\mathcal{H}_{\varepsilon}$ are uniquely determined
by $E$. Next, applying $\operatorname{div}\varepsilon$ and
$\operatorname{\mathring{curl}}$, we infer

$f=\operatorname{div}\varepsilon
E=\operatorname{div}\varepsilon\operatorname{\mathring{grad}}u\text{ and
}g=\operatorname{\mathring{curl}}E=\operatorname{\mathring{curl}}\varepsilon^{-1}\operatorname{curl}F$

Since, clearly, $q=x_{\varepsilon}$, we deduce

$E=\operatorname{\mathring{grad}}(\operatorname{div}\varepsilon\operatorname{\mathring{grad}})^{-1}f+\varepsilon^{-1}\operatorname{curl_{\textnormal{red}}}(\operatorname{curl_{\textnormal{red}}^{\diamond}}\varepsilon^{-1}\operatorname{curl_{\textnormal{red}}})^{-1}F+x_{\varepsilon},$

which is easily seen to be continuous; see in particular Theorem 2.1.
Uniqueness follows from the uniqueness in the Helmholtz decomposition.

(b) For Problem 4.2, we use the other Helmholtz type decomposition:

$H=\varepsilon\operatorname{\mathring{grad}}u+\operatorname{curl}F+q,$

for $u\in H_{0}^{1}(\Omega)$,
$F\in\operatorname{dom}(\operatorname{curl_{\textnormal{red}}})$ and
$q\in\mathcal{H}^{\varepsilon}$ being uniquely determined by $H$.

Next, we apply $\operatorname{div}$ and
$\operatorname{\mathring{curl}}\varepsilon^{-1}$ to obtain

$f=\operatorname{div}\varepsilon\operatorname{\mathring{grad}}u\text{ and
}g=\operatorname{\mathring{curl}}\varepsilon^{-1}\operatorname{curl}F.$

Thus,

$H=\varepsilon\operatorname{\mathring{grad}}(\operatorname{div}\varepsilon\operatorname{\mathring{grad}})^{-1}f+\operatorname{curl_{\textnormal{red}}}(\operatorname{curl_{\textnormal{red}}^{\diamond}}\varepsilon^{-1}\operatorname{curl_{\textnormal{red}}})^{-1}g+x^{\varepsilon}.$

(c) Using the solution formula for $H$, we obtain

$\displaystyle H$
$\displaystyle=\varepsilon{\operatorname{\mathring{grad}}}(\operatorname{div}\varepsilon{\operatorname{\mathring{grad}}})^{-1}f+\operatorname{curl_{\textnormal{red}}}({\operatorname{curl_{\textnormal{red}}^{\diamond}}}\varepsilon^{-1}\operatorname{curl_{\textnormal{red}}})^{-1}F+x^{\varepsilon}$
$\displaystyle=\varepsilon({\operatorname{\mathring{grad}}}(\operatorname{div}\varepsilon{\operatorname{\mathring{grad}}})^{-1}f+\varepsilon^{-1}\operatorname{curl_{\textnormal{red}}}({\operatorname{curl_{\textnormal{red}}^{\diamond}}}\varepsilon^{-1}\operatorname{curl_{\textnormal{red}}})^{-1}F+\varepsilon^{-1}x^{\varepsilon})$
$\displaystyle=\varepsilon E.$

Finally, note that $\varepsilon$ bijects $\mathcal{H}_{\varepsilon}$ onto
$\mathcal{H}^{\varepsilon}$. ∎

######  Corollary 4.4.

Let $E\in L_{2}(\Omega)^{3}$, $x_{\varepsilon}\in\mathcal{H}_{\varepsilon}$.

Then

$\pi_{\varepsilon}E=x_{\varepsilon}\iff\pi^{\varepsilon}\varepsilon
E=\varepsilon x_{\varepsilon}.$

###### Proof.

It follows from Theorem 4.3 with $f=\operatorname{div}\varepsilon E\in
H^{-1}(\Omega)$ and $g=\operatorname{\mathring{curl}}E\in
H^{-1}(\operatorname{curl_{\textnormal{red}}})$

$\displaystyle E$
$\displaystyle=\operatorname{\mathring{grad}}(\operatorname{div}\varepsilon\operatorname{\mathring{grad}})^{-1}f+\varepsilon^{-1}\operatorname{curl_{\textnormal{red}}}(\operatorname{curl_{\textnormal{red}}^{\diamond}}\varepsilon^{-1}\operatorname{curl_{\textnormal{red}}})^{-1}g+x_{\varepsilon}$
$\displaystyle=\varepsilon^{-1}\big{(}\varepsilon{\operatorname{\mathring{grad}}}(\operatorname{div}\varepsilon{\operatorname{\mathring{grad}}})^{-1}f+\operatorname{curl_{\textnormal{red}}}(\operatorname{curl_{\textnormal{red}}^{\diamond}}\varepsilon^{-1}\operatorname{curl_{\textnormal{red}}})^{-1}g+\varepsilon
x_{\varepsilon}\big{)}$ $\displaystyle=\varepsilon^{-1}H.$

This equality together with the uniqueness statement in the Helmholtz
decomposition shows the assertion. ∎

######  Remark 4.5.

With the help of the Corollaries 3.3 and 4.4 it is possible to reformulate
Problems 4.1 and 4.2 in such a way that the data on
$\mathcal{H}_{\varepsilon}$ and $\mathcal{H}^{\varepsilon}$ becomes
independent of $\varepsilon$.

(a) Instead of Problem 4.1 one looks for $E\in L_{2}(\Omega)$ satisfying for
given $f\in H^{-1}(\Omega)$ and $g\in
H^{-1}(\operatorname{curl_{\textnormal{red}}})$ and
$x\in\mathcal{H}_{D}(\Omega)$ such that

$\operatorname{div}\varepsilon E=f,\ \operatorname{\mathring{curl}}E=g,\
\text{and}\ \pi_{D}\pi_{\varepsilon}E=x.$ (3)

By Corollary 3.3, the last equation is equivalent to

$\pi_{\varepsilon}E=\pi_{\varepsilon}\pi_{D}\pi_{\varepsilon}E=(1-\operatorname{\mathring{grad}}(\operatorname{div}\varepsilon\operatorname{\mathring{grad}})^{-1}\operatorname{div}\varepsilon)x\eqqcolon
x_{\varepsilon}.$

Hence, (3) is equivalent to Problem
($\mathbb{P}_{\varepsilon}(f,g,x_{\varepsilon})$).

(b) Instead of Problem 4.2 one looks for $H\in L_{2}(\Omega)$ satisfying for
given $f\in H^{-1}(\Omega)$ and $g\in
H^{-1}(\operatorname{curl_{\textnormal{red}}})$ and
$x\in\mathcal{H}_{D}(\Omega)$ such that

$\operatorname{div}H=f,\ \operatorname{\mathring{curl}}\varepsilon^{-1}H=g,\
\text{and}\ \pi_{D}\pi_{\varepsilon}\varepsilon^{-1}H=x.$ (4)

We again reformulate $\pi_{D}\pi_{\varepsilon}\varepsilon^{-1}H=x$. We obtain,
by Corollary 3.3, that

$\pi_{\varepsilon}\varepsilon^{-1}H=\pi_{\varepsilon}\pi_{D}\pi_{\varepsilon}\varepsilon^{-1}H=(1-\operatorname{\mathring{grad}}(\operatorname{div}\varepsilon\operatorname{\mathring{grad}})^{-1}\operatorname{div}\varepsilon)x.$

Hence, by Corollary 4.4,

$\pi^{\varepsilon}H=\varepsilon\pi_{\varepsilon}\varepsilon^{-1}H=\varepsilon(1-\operatorname{\mathring{grad}}(\operatorname{div}\varepsilon\operatorname{\mathring{grad}})^{-1}\operatorname{div}\varepsilon)x\eqqcolon
x^{\varepsilon}.$

Thus, (4) is equivalent to Problem
($\mathbb{P}^{\varepsilon}(f,g,x^{\varepsilon})$).

In the next section we recall the core properties of nonlocal $H$-convergence
viewed at from an operator-theoretic perspective.

##  5 Nonlocal $H$-convergence

For nonlocal $H$-convergence – even though coming from sequences/complexes of
closed operators in Hilbert spaces – the core concept really can be put into
general perspective of operators on general Hilbert spaces. One upshot of [29]
is that the process of homogenisation of linear (operator) coefficients can be
framed in terms of an operator topology, which is incomparable to the weak
operator topology and strictly weaker than the strong operator and the norm
topology. At the end of this section, we provide a proof for Theorem A.

Let $\mathcal{H}$ be a Hilbert space, $\mathcal{H}_{0}\subseteq\mathcal{H}$ a
closed subspace and $\mathcal{H}_{1}\coloneqq\mathcal{H}_{0}^{\bot}$. Using
the canonical embeddings
$\iota_{0}\colon\mathcal{H}_{0}\hookrightarrow\mathcal{H}$ and
$\iota_{1}\colon\mathcal{H}_{1}\hookrightarrow\mathcal{H}$, we define the set

$\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{1})\coloneqq\\{a\in\mathcal{B}(\mathcal{H});a_{00}^{-1}\in\mathcal{B}(\mathcal{H}_{0}),a^{-1}\in\mathcal{B}(\mathcal{H})\\},$

where we abbreviated $a_{jk}\coloneqq\iota_{j}^{*}a\iota_{k}$ for all
$j,k\in\\{0,1\\}$. The topology for nonlocal $H$-convergence (with respect to
$(\mathcal{H}_{0},\mathcal{H}_{1})$) (or Schur topology or nonlocal
$H$-topology), $\tau(\mathcal{H}_{0},\mathcal{H}_{1})$, is the initial
topology induced by the mappings

$\displaystyle\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{1})\ni a$
$\displaystyle\mapsto
a_{00}^{-1}\in\mathcal{B}_{\textnormal{w}}(\mathcal{H}_{0})$
$\displaystyle\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{1})\ni a$
$\displaystyle\mapsto
a_{00}^{-1}a_{01}\in\mathcal{B}_{\textnormal{w}}(\mathcal{H}_{1},\mathcal{H}_{0})$
$\displaystyle\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{1})\ni a$
$\displaystyle\mapsto
a_{10}a_{00}^{-1}\in\mathcal{B}_{\textnormal{w}}(\mathcal{H}_{0},\mathcal{H}_{1})$
$\displaystyle\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{1})\ni a$
$\displaystyle\mapsto
a_{11}-a_{10}a_{00}^{-1}a_{01}\in\mathcal{B}_{\textnormal{w}}(\mathcal{H}_{1}).$

Here, the index w indicates that the weak operator topology is being used for
the operator spaces mentioned.

In order to get a feeling for the topology just introduced, we provide some
first examples next.

######  Example 5.1.

(a) If $\mathcal{H}_{0}=\\{0\\}$, then $a_{11}=a$. In this case
$\tau(\\{0\\},\mathcal{H})$ coincides with the weak operator topology on
$\mathcal{M}(\\{0\\},\mathcal{H})=\mathcal{GL}(\mathcal{H})$, the set of
invertible operators.

(b) If $\mathcal{H}_{0}=\mathcal{H}$, then $\mathcal{H}_{1}=\\{0\\}$. In this
case $\tau(\mathcal{H},\\{0\\})$ coincides with the weak operator topology for
the inverses on the set
$\mathcal{M}(\mathcal{H},\\{0\\})=\mathcal{GL}(\mathcal{H})$ of invertible
linear operators on $\mathcal{H}$.

For topologically trivial $\Omega$, the Schur topology with respect to
$(\operatorname{ran}(\operatorname{\mathring{grad}}),\operatorname{ran}(\operatorname{curl}))$
provides a Hausdorff topology on
$\mathcal{M}(\operatorname{ran}(\operatorname{\mathring{grad}}),\operatorname{ran}(\operatorname{curl}))$
generalising local $H$-convergence on $L_{\infty}(\Omega)$-valued matrices.

######  Example 5.2 (classical nonlocal $H$-convergence).

Let $\Omega\subseteq\mathbb{R}^{3}$ be bounded, range closed and assume that
$\mathcal{H}_{D}(\Omega)=\\{0\\}$; that is, (see [15, 13]) $\Omega$ has
continuous boundary and connected complement. Then Theorem 3.2 for
$\varepsilon=\operatorname{id}$ reads

$L_{2}(\Omega)^{3}=\operatorname{ran}(\operatorname{\mathring{grad}})\oplus\operatorname{ran}(\operatorname{curl}).$

Thus, for $\mathcal{H}=L_{2}(\Omega)^{3}$, we can choose

$\mathcal{H}_{0}=\operatorname{ran}(\operatorname{\mathring{grad}})\text{ and
}\mathcal{H}_{1}=\operatorname{ran}(\operatorname{curl}).$

The aim of this section is to abstractly understand what happens if the
assumption $\mathcal{H}_{D}(\Omega)=\\{0\\}$ is violated in Example 5.2. In
other terms, what is the correct333‘Correct’ in the sense that it is
consistent with convergence in the local $H$-sense for multiplication
operators.nonlocal $H$-topology, if

$L_{2}(\Omega)^{3}=\operatorname{ran}(\operatorname{\mathring{grad}})\oplus\operatorname{ran}(\operatorname{curl})\oplus\mathcal{H}_{D}(\Omega)\neq\operatorname{ran}(\operatorname{\mathring{grad}})\oplus\operatorname{ran}(\operatorname{curl})??$

It will turn out that there is not so much ambiguity if
$\mathcal{H}_{D}(\Omega)$ is finite-dimensional. However, if
$\mathcal{H}_{D}(\Omega)$ is infinite-dimensional, many mutually incomparable
examples can be constructed. We provide the abstract rationale for this in the
next example. The construction of a range closed $\Omega$ with infinite-
dimensional space of harmonic Dirichlet fields will be carried out in Section
6, see particularly Example 6.5.

######  Example 5.3.

(a) Let $\mathcal{H}_{0},\mathcal{H}_{1},\mathcal{H}_{2}$ be three Hilbert
spaces; $\mathcal{H}_{1}$ infinite-dimensional. Consider a sequence of
invertible bounded linear operators $(b_{n})_{n}$ on
$\mathcal{B}(\mathcal{H}_{1})$ with $(b_{n}^{-1})_{n}$ bounded on
$\mathcal{B}(\mathcal{H}_{1})$ such that

$b_{n}\to b\in\mathcal{B}_{\textnormal{w}}(\mathcal{H}_{1})$

but $(b_{n}^{-1})_{n}$ is not convergent in
$\mathcal{B}_{\textnormal{w}}(\mathcal{H}_{1})$; such sequences exist as
$\mathcal{H}_{1}$ infinite-dimensional: Use [20, Remark 13.2.5 and Proposition
13.2.1 (c)] to find two sequences of $(f_{n})_{n},(g_{n})_{n}$ of strictly
positive functions in $L_{\infty}(\mathbb{R})$ such that the associated
multiplication operators $M_{f_{n}}$ and $M_{g_{n}}$ on $L_{2}(\mathbb{R})$
satisfy $M_{f_{n}},M_{g_{n}}\to\operatorname{id}_{L_{2}(\mathbb{R})}$ in the
weak operator topology and
$M_{f_{n}}^{-1}=M_{f_{n}^{-1}}\to\alpha\operatorname{id}_{L_{2}(\mathbb{R})}$
as well as
$M_{g_{n}}^{-1}=M_{g_{n}^{-1}}\to\beta\operatorname{id}_{L_{2}(\mathbb{R})}$
in the weak operator topology for some scalars $\alpha\neq\beta$; the desired
sequence $b_{n}$ is then given by $b_{n}\coloneqq M_{f_{n/2}}$ for $n$ even
and $b_{n}\coloneqq M_{g_{(n-1)/2}}$ for $n$ odd.

In this setting, the operator sequence

$(c_{n})_{n}\coloneqq
n\mapsto\begin{pmatrix}\operatorname{id}_{\mathcal{H}_{0}}&0&0\\\ 0&b_{n}&0\\\
0&0&1_{\mathcal{H}_{2}}\end{pmatrix}\in\mathcal{M}\coloneqq\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{1}\oplus\mathcal{H}_{2})\cap\mathcal{M}(\mathcal{H}_{0}\oplus\mathcal{H}_{1},\mathcal{H}_{2})$

converges in
$\tau_{0}\coloneqq\tau(\mathcal{H}_{0},\mathcal{H}_{1}\oplus\mathcal{H}_{2})$;
it does not converge in
$\tau_{1}\coloneqq\tau(\mathcal{H}_{0}\oplus\mathcal{H}_{1},\mathcal{H}_{2})$.
On the other hand, $(c_{n}^{-1})_{n}$ is also a sequence in $\mathcal{M}$,
converging in $\tau_{1}$ and not converging in $\tau_{0}$. Thus, $\tau_{0}$
and $\tau_{1}$ are incomparable topologies.

(b) Let $\mathcal{H}$ be a separable, infinite-dimensional Hilbert space;
$\mathcal{H}_{0}$, $\mathcal{H}_{2}$ Hilbert spaces. Let
$(\varphi_{n})_{n\in\mathbb{N}}$ be an orthonormal basis of $\mathcal{H}$ and
let $(N_{n})_{n\geqslant 1}$ be a sequence of pairwise disjoint, infinite
subsets of $\mathbb{N}$ such that $\bigcup_{n}N_{n}=\mathbb{N}$. Defining
$\check{\mathcal{H}}_{n}\coloneqq\overline{\operatorname{lin}}\\{\varphi_{k};k\in
N_{\ell},1\leqslant\ell\leqslant n\\}$ and
$\widehat{\mathcal{H}}_{n}\coloneqq{\check{\mathcal{H}}}_{n}^{\bot}\subseteq\mathcal{H}$,
we put

$\tau_{n}\coloneqq\tau(\mathcal{H}_{0}\oplus\check{\mathcal{H}}_{n},\widehat{\mathcal{H}}_{n}\oplus\mathcal{H}_{2}).$

Then, by (a), for all $n,n^{\prime}\in\mathbb{N}$ with $n\neq n^{\prime}$,
$\tau_{n}$ and $\tau_{n^{\prime}}$ are incomparable.

For finite-dimensional $\mathcal{H}_{D}(\Omega)$, the, in fact, infinite
amount of possible pairwise incomparable topologies does not exist, if we
restrict the set of topologies slightly in that we only consider appropriately
bounded sequences/nets. This restriction, however, is irrelevant for
applications. Before we go into details, we need to mention another structural
property of nonlocal $H$-convergence. For this, note that the Examples 5.1 (a)
and (b) suggest a symmetry in inversion and simultaneous change of
$\mathcal{H}_{0}$ and its orthogonal complement. This symmetry holds in the
following general sense.

######  Lemma 5.4.

Let $\mathcal{H}$ be a Hilbert space and $\mathcal{H}_{0}\subseteq\mathcal{H}$
be a closed subspace. Then

$(\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{1}),\tau(\mathcal{H}_{0},\mathcal{H}_{1}))\ni
a\mapsto
a^{-1}\in(\mathcal{M}(\mathcal{H}_{1},\mathcal{H}_{0}),\tau(\mathcal{H}_{1},\mathcal{H}_{0}))$

is a homeomorphism. More precisely, we have for all
$a\in\mathcal{B}(\mathcal{H})$

$a\in\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{1})\iff
a^{-1}\in\mathcal{M}(\mathcal{H}_{1},\mathcal{H}_{0}).$

In either case, the following equalities hold

$\displaystyle[a^{-1}]_{11}^{-1}$
$\displaystyle=a_{11}-a_{10}a_{00}^{-1}a_{01}$
$\displaystyle[a^{-1}]_{11}^{-1}[a^{-1}]_{10}$
$\displaystyle=-a_{10}a_{00}^{-1}$
$\displaystyle[a^{-1}]_{01}[a^{-1}]_{11}^{-1}$
$\displaystyle=-a_{10}a_{00}^{-1}$
$\displaystyle[a^{-1}]_{00}-[a^{-1}]_{01}[a^{-1}]_{11}^{-1}[a^{-1}]_{10}$
$\displaystyle=a_{00}^{-1}.$

###### Proof.

We show that

$\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{1})=[\mathcal{M}(\mathcal{H}_{1},\mathcal{H}_{0})]^{-1}$

and in passing that

$\displaystyle[a^{-1}]_{11}^{-1}$
$\displaystyle=a_{11}-a_{10}a_{00}^{-1}a_{01}$
$\displaystyle[a^{-1}]_{11}^{-1}[a^{-1}]_{10}$
$\displaystyle=-a_{10}a_{00}^{-1}$
$\displaystyle[a^{-1}]_{01}[a^{-1}]_{11}^{-1}$
$\displaystyle=-a_{10}a_{00}^{-1}$
$\displaystyle[a^{-1}]_{00}-[a^{-1}]_{01}[a^{-1}]_{11}^{-1}[a^{-1}]_{10}$
$\displaystyle=a_{00}^{-1}.$

These formulas together with the definition of the topology establish all the
remaining claims. Now, let $a\in\mathcal{B}(\mathcal{H})$ be continuously
invertible.

Firstly, let us assume that
$a\in\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{1})$. Then
$a_{00}^{-1}\in\mathcal{B}(\mathcal{H}_{0})$. By Lemma 3.8 it follows that
$a_{S}\coloneqq a_{11}-a_{10}a_{00}^{-1}a_{01}$ is also invertible. The
formula for the inverse of $a$ in Lemma 3.8 then shows the desired equalities
as well as $a^{-1}\in\mathcal{M}(\mathcal{H}_{1},\mathcal{H}_{0})$. On the
other hand, if $a^{-1}\in\mathcal{M}(\mathcal{H}_{1},\mathcal{H}_{0})$, then
$[a^{-1}]_{11}$ is invertible. As $a^{-1}$ is invertible, by Lemma 3.8
interchanging the roles of $a$ and $a^{-1}$ and $\mathcal{H}_{0}$ and
$\mathcal{H}_{1}$ respectively, we obtain that
$[a^{-1}]_{S}\coloneqq[a^{-1}]_{00}-[a^{-1}]_{01}[a^{-1}]_{11}^{-1}[a^{-1}]_{10}$
is invertible. Moreover, we infer $[a^{-1}]_{S}^{-1}=a_{00}$ from the
concluding formula in Lemma 3.8. In particular, $a_{00}$ is invertible. Thus,
$a\in\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{1})$, which concludes the proof.
∎

Since $\tau(\mathcal{H}_{0},\mathcal{H}_{1})$ is not a locally convex
topology, the notion of boundedness is a priori unclear. In the following it
will be a desired property for the considered weakly convergent sequences to
be bounded. With this in mind, we call a subset
$\mathcal{A}\subseteq\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{1})$
$\tau(\mathcal{H}_{0},\mathcal{H}_{1})$-bounded, if all of the following sets

$\displaystyle\\{a\in\mathcal{A};a_{00}^{-1}\\}$
$\displaystyle\subseteq\mathcal{B}(\mathcal{H}_{0})$
$\displaystyle\\{a\in\mathcal{A};a_{00}^{-1}a_{01}\\}$
$\displaystyle\subseteq\mathcal{B}(\mathcal{H}_{1},\mathcal{H}_{0})$
$\displaystyle\\{a\in\mathcal{A};a_{10}a_{00}^{-1}\\}$
$\displaystyle\subseteq\mathcal{B}(\mathcal{H}_{0},\mathcal{H}_{1})$
$\displaystyle\\{a\in\mathcal{A};a_{11}-a_{10}a_{00}^{-1}a_{01}\\}$
$\displaystyle\subseteq\mathcal{B}(\mathcal{H}_{1})$

are bounded in the respective norm topologies. The number

$\textnormal{b}(\mathcal{A};\tau(\mathcal{H}_{0},\mathcal{H}_{1}))\coloneqq\sup_{a\in\mathcal{A}}\max\\{\|a_{00}^{-1}\|,\|a_{00}^{-1}a_{01}\|,\|a_{10}a_{00}^{-1}\|,\|a_{11}-a_{10}a_{00}^{-1}a_{01}\|\\}$

is called the $\tau(\mathcal{H}_{0},\mathcal{H}_{1})$-bound of $\mathcal{A}$.
We denote the set of operators in
$\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{1})$ with a
$\tau(\mathcal{H}_{0},\mathcal{H}_{1})$-bound less than some $\lambda\geqslant
0$ by $B_{\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{1})}(\lambda)$. For the
comparison of topologies for different decompositions of $\mathcal{H}$, we
need to restrict our attention to bounded subsets of
$\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{1})$.

Along the same lines in [29, Theorem 5.1], one obtains a metrisability result
in the separable case.

######  Theorem 5.5.

If both $\mathcal{H}_{0}$ and $\mathcal{H}_{1}$ are separable, then for all
$\lambda\geqslant 0$,
$(B_{\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{1})}(\lambda),\tau(\mathcal{H}_{0},\mathcal{H}_{1}))$
is metrisable.

To keep the exposition rather lean and to avoid too many technical assumptions
in the respective propositions, we introduce
$\tau_{\textnormal{b}}(\mathcal{H}_{0},\mathcal{H}_{1})$, the  topology of
bounded nonlocal $H$-convergence, which is the inductive topology on
$\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{1})$ in order that for all
$\lambda>0$ the mappings

$\big{(}B_{\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{1})}(\lambda),\tau(\mathcal{H}_{0},\mathcal{H}_{1})\cap
B_{\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{1})}(\lambda)\big{)}\hookrightarrow\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{1})$

are continuous with the understanding that
$\tau(\mathcal{H}_{0},\mathcal{H}_{1})\cap
B_{\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{1})}(\lambda)$ denotes the
relative topology on
$B_{\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{1})}(\lambda)$ induced by
$\tau(\mathcal{H}_{0},\mathcal{H}_{1})$.

######  Remark 5.6.

A similar statement to the one for the inversion of operators in Lemma 5.4 is
also true for $\tau_{\textnormal{b}}$. In fact we have
$B_{\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{1})}(\lambda)=B_{[\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{1})]^{-1}}(\lambda)\coloneqq
B_{\mathcal{M}(\mathcal{H}_{1},\mathcal{H}_{0})}(\lambda)$ for all
$\lambda\geqslant 0$. In consequence,

$(\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{1}),\tau_{\textnormal{b}}(\mathcal{H}_{0},\mathcal{H}_{1}))\ni
a\mapsto
a^{-1}\in(\mathcal{M}(\mathcal{H}_{1},\mathcal{H}_{0}),\tau_{\textnormal{b}}(\mathcal{H}_{1},\mathcal{H}_{0}))$

is a homeomorphism. Indeed, this statement is a straightforward consequence of
the formulas presented in Lemma 5.4.

######  Proposition 5.7.

Let $Y$ be a topological space and
$g\colon\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{1})\to Y$. Then the following
conditions are equivalent:

  1. (i)

$g$ is continuous;

  2. (ii)

for all $\lambda\geqslant 0$,
$g_{\lambda}\colon\big{(}B_{\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{1})}(\lambda),\tau(\mathcal{H}_{0},\mathcal{H}_{1})\cap
B_{\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{1})}(\lambda)\big{)}\to Y,a\mapsto
g(a)$ is continuous;

  3. (iii)

for all $\tau(\mathcal{H}_{0},\mathcal{H}_{1})$-bounded convergent nets
$(a^{(\iota)})_{\iota}$ to some
$a\in\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{1})$, one has
$g(a^{(\iota)})\stackrel{{\scriptstyle\iota}}{{\to}}g(a)$.

If, in addition, $\mathcal{H}$ is separable, then the following condition is
also equivalent to (i), (ii) and (iii):

  1. (iv)

for all $\tau(\mathcal{H}_{0},\mathcal{H}_{1})$-convergent sequences
$(a^{(n)})_{n}$ to some $a\in\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{1})$,
$g(a^{(n)})\to g(a)$ as $n\to\infty$.

###### Proof.

The conditions (i) and (ii) being equivalent follow from the general fact for
final topologies, see e.g. [26, Theorem 10.1] in the context of topological
vectors spaces. (ii) and (iii) are a reformulation of one another using the
fact that also $a\in
B_{\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{1})}(\lambda)$ if $a_{\iota}\in
B_{\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{1})}(\lambda)$ for all $\iota$ and
some $\lambda\geqslant 0$.

If $\mathcal{H}$ is separable, then
$\big{(}B_{\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{1})}(\lambda),\tau(\mathcal{H}_{0},\mathcal{H}_{1})\cap
B_{\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{1})}(\lambda)\big{)}$ is
metrisable Theorem 5.5. Hence, continuity of $g_{\lambda}$ is characterised by
sequential continuity; thus (iv) implies (ii). If, on the other hand, (ii)
holds then for any $\tau(\mathcal{H}_{0},\mathcal{H}_{1})$-convergent sequence
$(a^{(n)})_{n}$ sequence with limit $a$ we find $\lambda\geqslant 0$ such that
$a^{(n)},a\in B_{\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{1})}(\lambda)$
employing the uniform boundedness principle. Thus, (iii) holds due to
continuity of $g_{\lambda}$. ∎

######  Theorem 5.8.

Let $\mathcal{H}$ be a Hilbert space and
$\mathcal{H}_{0}\subseteq\mathcal{K}_{0}\subseteq\mathcal{H}$ be closed
subspaces. Assume that $\mathcal{K}_{0}\ominus\mathcal{H}_{0}$ is finite-
dimensional. Then
$\tau_{\textnormal{b}}(\mathcal{H}_{0},\mathcal{H}_{0}^{\bot})=\tau_{\textnormal{b}}(\mathcal{K}_{0},\mathcal{K}_{0}^{\bot})$
on
$\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{0}^{\bot})\cap\mathcal{M}(\mathcal{K}_{0},\mathcal{K}_{0}^{\bot})$.

###### Proof.

We put $\mathcal{H}_{1}\coloneqq\mathcal{H}_{0}^{\bot}$ and
$\mathcal{K}_{1}\coloneqq\mathcal{K}_{0}^{\bot}$. It suffices to show that
with
$\mathcal{M}\coloneqq\mathcal{M}(\mathcal{H}_{0},\mathcal{H}_{1})\cap\mathcal{M}(\mathcal{K}_{0},\mathcal{K}_{1})$
the inclusion

$(\mathcal{M},\tau_{\textnormal{b}}(\mathcal{H}_{0},\mathcal{H}_{1}))\hookrightarrow(\mathcal{M},\tau_{\textnormal{b}}(\mathcal{K}_{0},\mathcal{K}_{1}))$
($*$)

is continuous. Indeed, the other inclusion is then a consequence of this
statement in conjunction with Remark 5.6 and Lemma 5.4.

For the continuity of the above inclusion, we denote
$\mathcal{L}_{0}\coloneqq\mathcal{H}_{0},\mathcal{L}_{1}\coloneqq\mathcal{K}_{0}\ominus\mathcal{H}_{0}$
and $\mathcal{L}_{2}\coloneqq\mathcal{K}_{0}^{\bot}$. Then
$\mathcal{H}=\mathcal{L}_{0}\oplus\mathcal{L}_{1}\oplus\mathcal{L}_{2}$. For
every $a\in\mathcal{B}(\mathcal{H})$ we use block operator notation and
suggestive indices to write $a$ as an operator in
$\mathcal{B}(\mathcal{L}_{0}\oplus\mathcal{L}_{1}\oplus\mathcal{L}_{2})$:

$\mathcal{B}(\mathcal{H})\ni a\sim\begin{pmatrix}a_{00}&a_{01}&a_{02}\\\
a_{10}&a_{11}&a_{12}\\\
a_{20}&a_{21}&a_{22}\end{pmatrix}\in\mathcal{B}(\mathcal{L}_{0}\oplus\mathcal{L}_{1}\oplus\mathcal{L}_{2}).$

With this identification, $\tau(\mathcal{H}_{0},\mathcal{H}_{1})$ is induced
by the mappings

$\displaystyle a$ $\displaystyle\mapsto a_{00}^{-1}$ $\displaystyle a$
$\displaystyle\mapsto\begin{pmatrix}a_{10}\\\
a_{20}\end{pmatrix}a_{00}^{-1}=\begin{pmatrix}a_{10}a_{00}^{-1}\\\
a_{20}a_{00}^{-1}\end{pmatrix}$ $\displaystyle a$ $\displaystyle\mapsto
a_{00}^{-1}\begin{pmatrix}a_{01}&a_{02}\end{pmatrix}=\begin{pmatrix}a_{00}^{-1}a_{01}&a_{00}^{-1}a_{02}\end{pmatrix}$
$\displaystyle a$ $\displaystyle\mapsto\begin{pmatrix}a_{11}&a_{12}\\\
a_{21}&a_{22}\end{pmatrix}-\begin{pmatrix}a_{10}\\\
a_{20}\end{pmatrix}a_{00}^{-1}\begin{pmatrix}a_{01}&a_{02}\end{pmatrix}=\begin{pmatrix}a_{11}-a_{10}a_{00}^{-1}a_{01}&a_{12}-a_{10}a_{00}^{-1}a_{02}\\\
a_{21}-a_{20}a_{00}^{-1}a_{01}&a_{22}-a_{20}a_{00}^{-1}a_{02}\end{pmatrix}.$

$\tau(\mathcal{K}_{0},\mathcal{K}_{1})$ is induced by

$\displaystyle a$ $\displaystyle\mapsto\begin{pmatrix}a_{00}&a_{01}\\\
a_{01}&a_{11}\end{pmatrix}^{-1}$ $\displaystyle a$
$\displaystyle\mapsto\begin{pmatrix}a_{20}&a_{21}\end{pmatrix}\begin{pmatrix}a_{00}&a_{01}\\\
a_{01}&a_{11}\end{pmatrix}^{-1}$ $\displaystyle a$
$\displaystyle\mapsto\begin{pmatrix}a_{00}&a_{01}\\\
a_{01}&a_{11}\end{pmatrix}^{-1}\begin{pmatrix}a_{02}\\\ a_{12}\end{pmatrix}$
$\displaystyle a$ $\displaystyle\mapsto
a_{22}-\begin{pmatrix}a_{20}&a_{21}\end{pmatrix}\begin{pmatrix}a_{00}&a_{01}\\\
a_{01}&a_{11}\end{pmatrix}^{-1}\begin{pmatrix}a_{02}\\\ a_{12}\end{pmatrix}$

Using the formula for the inverse in Lemma 3.8 and $a_{S}\coloneqq
a_{11}-a_{10}a_{00}^{-1}a_{01}$ , we obtain

$\displaystyle\begin{pmatrix}a_{00}&a_{01}\\\
a_{10}&a_{11}\end{pmatrix}^{-1}=\left(\begin{pmatrix}1&0\\\
a_{10}a_{00}^{-1}&1\end{pmatrix}\begin{pmatrix}a_{00}&0\\\
0&a_{S}\end{pmatrix}\begin{pmatrix}1&a_{00}^{-1}a_{01}\\\
0&1\end{pmatrix}\right)^{-1}$
$\displaystyle=\begin{pmatrix}1&-a_{00}^{-1}a_{01}\\\
0&1\end{pmatrix}\begin{pmatrix}a_{00}^{-1}&0\\\
0&a_{S}^{-1}\end{pmatrix}\begin{pmatrix}1&0\\\
-a_{10}a_{00}^{-1}&1\end{pmatrix}$
$\displaystyle=\begin{pmatrix}a_{00}^{-1}+a_{00}^{-1}a_{01}a_{S}^{-1}a_{10}a_{00}^{-1}&-a_{00}^{-1}a_{01}a_{S}^{-1}\\\
-a_{S}^{-1}a_{10}a_{00}^{-1}&a_{S}^{-1}\end{pmatrix}$

and, thus, $\tau(\mathcal{K}_{0},\mathcal{K}_{1})$ is induced by

$\displaystyle a$
$\displaystyle\mapsto\begin{pmatrix}a_{00}^{-1}+a_{00}^{-1}a_{01}a_{S}^{-1}a_{10}a_{00}^{-1}&-a_{00}^{-1}a_{01}a_{S}^{-1}\\\
-a_{S}^{-1}a_{10}a_{00}^{-1}&a_{S}^{-1}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}a_{00}^{-1}+a_{00}^{-1}a_{01}\times\operatorname{id}_{1}\times
a_{S}^{-1}\times\operatorname{id}_{1}\times
a_{10}a_{00}^{-1}&-a_{00}^{-1}a_{01}\times\operatorname{id}_{1}\times
a_{S}^{-1}\\\ -a_{S}^{-1}\times\operatorname{id}_{1}\times
a_{10}a_{00}^{-1}&a_{S}^{-1}\end{pmatrix}$ $\displaystyle a$
$\displaystyle\mapsto\begin{pmatrix}a_{20}a_{00}^{-1}+a_{20}a_{00}^{-1}a_{01}a_{S}^{-1}a_{10}a_{00}^{-1}+a_{21}(-a_{S}^{-1}a_{10}a_{00}^{-1})&-a_{20}a_{00}^{-1}a_{01}a_{S}^{-1}+a_{21}a_{S}^{-1}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}a_{20}a_{00}^{-1}+(a_{20}a_{00}^{-1}a_{01}-a_{21})\times\operatorname{id}_{1}\times
a_{S}^{-1}\times\operatorname{id}_{1}\times
a_{10}a_{00}^{-1}&(-a_{20}a_{00}^{-1}a_{01}+a_{21})\times\operatorname{id}_{1}\times
a_{S}^{-1}\end{pmatrix}$ $\displaystyle a$
$\displaystyle\mapsto\begin{pmatrix}a_{00}^{-1}a_{02}+a_{00}^{-1}a_{01}a_{S}^{-1}a_{10}a_{00}^{-1}a_{02}-a_{00}^{-1}a_{01}a_{S}^{-1}a_{12}\\\
-a_{S}^{-1}a_{10}a_{00}^{-1}a_{02}+a_{S}^{-1}a_{12}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}a_{00}^{-1}a_{02}+a_{00}^{-1}a_{01}\times\operatorname{id}_{1}\times
a_{S}^{-1}\times\operatorname{id}_{1}\times(a_{10}a_{00}^{-1}a_{02}-a_{12})\\\
a_{S}^{-1}\times\operatorname{id}_{1}\times(-a_{10}a_{00}^{-1}a_{02}+a_{12})\end{pmatrix}$
$\displaystyle a$ $\displaystyle\mapsto
a_{22}-\begin{pmatrix}a_{20}&a_{21}\end{pmatrix}\begin{pmatrix}a_{00}^{-1}+a_{00}^{-1}a_{01}a_{S}^{-1}a_{10}a_{00}^{-1}&-a_{00}^{-1}a_{01}a_{S}^{-1}\\\
-a_{S}^{-1}a_{10}a_{00}^{-1}&a_{S}^{-1}\end{pmatrix}\begin{pmatrix}a_{02}\\\
a_{12}\end{pmatrix}$
$\displaystyle=(a_{22}-a_{20}a_{00}^{-1}a_{02})-a_{20}a_{00}^{-1}a_{01}\times\operatorname{id}_{1}\times
a_{S}^{-1}\times\operatorname{id}_{1}\times(a_{10}a_{00}^{-1}a_{02}-a_{12})$
$\displaystyle\quad\quad-
a_{21}a_{S}^{-1}\times\operatorname{id}_{1}\times(-a_{10}a_{00}^{-1}a_{02}+a_{12})$
$\displaystyle=(a_{22}-a_{20}a_{00}^{-1}a_{02})+(a_{21}-a_{20}a_{00}^{-1}a_{01})\times\operatorname{id}_{1}\times
a_{S}^{-1}\times\operatorname{id}_{1}\times(a_{10}a_{00}^{-1}a_{02}-a_{12}),$

where $\operatorname{id}_{1}$ denotes the identity on $\mathcal{L}_{1}$. Since
$\mathcal{L}_{1}$ is finite-dimensional, $\operatorname{id}_{1}$ is a compact
operator. Moreover, in finite dimensions, the weak operator topology coincides
with the norm topology. Thus, inversion is a continuous process for invertible
operators on $\mathcal{L}_{1}$ in the weak operator topology. In order to show
continuity of ($*$ ‣ 5), it suffices to use the net characterisation in
Proposition 5.7 (iii), in combination with the result on weakly converging
products in Lemma 5.9 below. Note that the above expressions of the mappings
inducing the topology $\tau(\mathcal{K}_{0},\mathcal{K}_{1})$ are written as
sums and products of mappings inducing the topology on
$\tau(\mathcal{H}_{0},\mathcal{H}_{1})$ and the compact operator
$\operatorname{id}_{1}$. This eventually shows the assertion. ∎

######  Lemma 5.9.

Let $n\in\mathbb{N}$, $\mathcal{H}_{0},\ldots,\mathcal{H}_{n+1}$ Hilbert
spaces, $(A_{\iota}^{(\ell)})_{\iota}$, $\ell\in\\{0,\ldots,n\\}$ bounded nets
in $\mathcal{B}(\mathcal{H}_{\ell},\mathcal{H}_{\ell+1})$ respectively
converging in the weak operator topology to some
$A^{(\ell)}\in\mathcal{B}(\mathcal{H}_{\ell},\mathcal{H}_{\ell+1})$. For
$\ell\in\\{1,\ldots,n\\}$ let $T_{\ell}\in\mathcal{B}(\mathcal{H}_{\ell})$ be
compact. Then

$A_{\iota}^{(n)}T_{n}A_{\iota}^{(n-1)}T_{n-1}\cdots
A_{\iota}^{(1)}T_{1}A_{\iota}^{(0)}\to A^{(n)}T_{n}A^{(n-1)}T_{n-1}\cdots
A^{(1)}T_{1}A^{(0)}$

in the weak operator topology of
$\mathcal{B}(\mathcal{H}_{0},\mathcal{H}_{n+1})$.

###### Proof.

We prove the claim by induction on $n$. For $n=0$, there is nothing to show.
For the inductive step, we recall that compact operators map weakly convergent
sequences to norm convergent sequences. Thus, assuming the claim is true for
some $n-1$, we let $x\in\mathcal{H}_{0}$ and $y\in\mathcal{H}_{n+1}$. Then, by
the induction hypothesis, $z_{\iota}\coloneqq A_{\iota}^{(n-1)}T_{n-1}\cdots
A_{\iota}^{(1)}T_{1}A_{\iota}^{(0)}x$ weakly converges to $z\coloneqq
A^{(n-1)}T_{n-1}\cdots A^{(1)}T_{1}A^{(0)}x$ as $\iota\to\infty$. The aim is
to show that $A_{\iota}^{(n)}T_{n}z_{\iota}$ weakly converges to
$A^{(n)}T_{n}z$ as $\iota\to\infty$. As $T_{n}$ is compact and
$(z_{\iota})_{\iota}$ is bounded, we obtain $T_{n}z_{\iota}\to T_{n}z$ in
norm, as $\iota\to\infty$. Since computing the adjoint is continuous in the
weak operator topology, we infer for $\iota$:

$\langle A_{\iota}^{(n)}T_{n}z_{\iota},y\rangle_{\mathcal{H}_{n+1}}=\langle
T_{n}z_{\iota},(A_{\iota}^{(n)})^{*}y\rangle_{\mathcal{H}_{n}}\to\langle
T_{n}z,(A^{(n)})^{*}y\rangle_{\mathcal{H}_{n}}=\langle
A^{(n)}T_{n}z,y\rangle_{\mathcal{H}_{n+1}}$

as $\iota\to\infty$. This completes the proof. ∎

###### Proof of Theorem A.

At first note that for all $\Omega\subseteq\mathbb{R}^{3}$ open and range
closed and $0<\alpha<\beta$, the set $\mathcal{M}(\alpha,\beta)$ is a
$\tau(\mathcal{H},\mathcal{H}^{\bot})$-bounded subset for all closed subspaces
$\mathcal{H}\subseteq L_{2}(\Omega)^{3}$.

Next, we prove statement (a). Let $\Omega\subseteq\mathbb{R}^{3}$ open and
range closed and let $\mathcal{H}_{0}\subseteq L_{2}(\Omega)^{3}$ be a closed
subspace with the properties

$\operatorname{ran}(\operatorname{\mathring{grad}})\subseteq\mathcal{H}_{0}\text{
and }\operatorname{ran}(\operatorname{curl})\subseteq\mathcal{H}_{0}^{\bot}.$

Then, any $\mathcal{H}_{0}$ chosen in such a way defines an element in
$\mathfrak{T}$. Note that $\tau_{\Omega}$ is uniquely determined, if
$\mathcal{H}_{D}(\Omega)=\\{0\\}$ since then necessarily
$\mathcal{H}_{0}=\operatorname{ran}(\operatorname{\mathring{grad}})$ and
$\tau_{\Omega}$ is the classical nonlocal $H$-convergence topology on the
$\tau_{\Omega}$-bounded set $\mathcal{M}(\alpha,\beta)$, see also Example 5.2;
proving (a).

(b) The uniqueness statement follows from Theorem 5.8. Indeed, let
$\mathcal{H}_{0}$ be as above. Then
$\mathcal{H}_{0}\ominus\operatorname{ran}(\operatorname{\mathring{grad}})$ is
finite-dimensional since
$\mathcal{H}_{0}^{\bot}\supseteq\operatorname{ran}(\operatorname{curl})$ and
$\mathcal{H}_{D}(\Omega)$ is finite-dimensional. Thus, by Theorem 5.8, we
obtain

$\tau_{\textnormal{b}}(\operatorname{ran}(\operatorname{\mathring{grad}}),\operatorname{ran}(\operatorname{\mathring{grad}})^{\bot})=\tau_{\textnormal{b}}(\mathcal{H}_{0},\mathcal{H}_{0}^{\bot}),$

which yields the assertion in conjunction with the observation from the
beginning of this proof that $\mathcal{M}(\alpha,\beta)$ is both
$\tau(\operatorname{ran}(\operatorname{\mathring{grad}}),\operatorname{ran}(\operatorname{\mathring{grad}})^{\bot})$-bounded
and $\tau(\mathcal{H}_{0},\mathcal{H}_{0}^{\bot})$-bounded. ∎

##  6 Infinite-dimensional harmonic Dirichlet fields

In this section we aim to construct a range closed, open
$\Omega\subseteq\mathbb{R}^{3}$ such that $\mathcal{H}_{D}(\Omega)$ is
infinite-dimensional. This will then provide a particular example proving that
there exists infinitely many pairwise incomparable topologies all providing in
some sense an extension of the classical nonlocal $H$-convergence topology:

######  Example 6.1.

(a) Let $\Omega\subseteq\mathbb{R}^{3}$ be open and range closed with
$\mathcal{H}_{D}(\Omega)$ infinite-dimensional (to be constructed below).
Consider the topologies $(\tau_{k})_{k}$ in Example 5.3 (b) with
$\operatorname{ran}(\operatorname{\mathring{grad}})=\mathcal{H}_{0}$,
$\operatorname{ran}(\operatorname{curl})=\mathcal{H}_{2}$ and
$\mathcal{H}_{D}(\Omega)=\mathcal{H}$. Let $(a_{n})_{n}$ be an admissible
sequence in $\mathcal{B}(L_{2}(\Omega)^{3})$. Assume all $a_{n}$ have the
following common block structure

$\begin{pmatrix}*&0&*\\\ 0&*&0\\\
*&0&*\end{pmatrix}\in\mathcal{B}(\operatorname{ran}(\operatorname{\mathring{grad}})\oplus\mathcal{H}_{D}(\Omega)\oplus\operatorname{ran}(\operatorname{curl})),$

where $*$ symbolises potential non-zero entries. Let $f\in H^{-1}(\Omega)$ and
$g\in H^{-1}(\operatorname{curl_{\textnormal{red}}})$ and $u_{n}\in
H_{0}^{1}(\Omega)$ and $v_{n}\in
H^{1}(\operatorname{curl_{\textnormal{red}}})$ satisfying

$-\operatorname{div}a_{n}\operatorname{\mathring{grad}}u_{n}=f,\quad\operatorname{\mathring{curl}}a_{n}^{-1}\operatorname{curl}v_{n}=g.$

Now, let $a_{n}\to a$ in $\tau_{k}$ for some admissible $a$ and $k\geqslant
1$. Then using Theorem 2.1 it is not difficult to see that
$u_{n},a_{n}\operatorname{\mathring{grad}}u_{n},v_{n},a_{n}^{-1}\operatorname{\mathring{curl}}v_{n}$
all weakly converge to the respective weak limits
$u,a\operatorname{\mathring{grad}}u,v,a_{n}^{-1}\operatorname{\mathring{curl}}v$
with $u,v$ being uniquely determined by

$-\operatorname{div}a\operatorname{\mathring{grad}}u=f,\quad\operatorname{\mathring{curl}}a^{-1}\operatorname{curl}v=g.$

However, by Example 5.3 (b) there are choices for $(a_{n})_{n}$, where $a$ is
different for different $k$.

We shall now provide an abstract detour to construct the $\Omega$ needed in
Example 6.1. Throughout, let $\mathcal{H}_{0},\mathcal{H}_{1},\mathcal{H}_{2}$
be Hilbert spaces. For a densely defined, closed linear operator

$A\colon\operatorname{dom}(A)\subseteq\mathcal{H}_{0}\to\mathcal{H}_{1},$

we define

$\gamma(A)\coloneqq\inf_{\varphi\in\operatorname{dom}(A)\cap\operatorname{ker}(A)^{\bot}}\frac{\|A\varphi\|_{\mathcal{H}_{0}}}{\|\varphi\|_{\mathcal{H}_{1}}},$

the minimum modulus of $A$. The number $\gamma(A)$ is decisive when it comes
to decide whether or not the range of $A$ is closed.

######  Theorem 6.2 ([5, Theorem IV.1.6]).

Let $A\colon\operatorname{dom}(A)\subseteq\mathcal{H}_{0}\to\mathcal{H}_{1}$
be a closed linear operator. Then the following conditions are equivalent:

  1. (i)

$\operatorname{ran}(A)\subseteq\mathcal{H}_{1}$ closed;

  2. (ii)

$\gamma(A)>0$.

The next observation provides an estimate for the minimum modulus if $A$ is
given as an orthogonal sum of many operators.

######  Proposition 6.3.

For $n\in\mathbb{N}$, let
$A^{(n)}\colon\operatorname{dom}(A^{(n)})\subseteq\mathcal{H}_{0}\to\mathcal{H}_{1}$
be densely defined, closed and linear. We define

$\mathcal{A}\coloneqq\bigoplus_{n\in\mathbb{N}}A^{(n)}\colon\operatorname{dom}(\mathcal{A})\subseteq\ell_{2}(\mathbb{N};\mathcal{H}_{0})\to\ell_{2}(\mathbb{N};\mathcal{H}_{1}),\quad(\varphi_{n})_{n}\mapsto(A^{(n)}\varphi_{n})_{n}$

where

$\operatorname{dom}(\mathcal{A})=\\{(\varphi_{n})_{n}\in\ell_{2}(\mathbb{N};\mathcal{H}_{0});\varphi_{n}\in\operatorname{dom}(A^{(n)})\text{
and }(A^{(n)}\varphi_{n})_{n}\in\ell_{2}(\mathbb{N};\mathcal{H}_{1})\\}.$

Then $\mathcal{A}$ is densely defined, closed, linear operator with
$\mathcal{A}^{*}=\bigoplus_{n\in\mathbb{N}}(A^{(n)})^{*}$ and

$\operatorname{ker}(\mathcal{A})=\bigoplus_{n\in\mathbb{N}}\operatorname{ker}(A^{(n)}).$

Moreover, if $\gamma(A^{(n)})>0$, then

$\inf_{n\in\mathbb{N}}\gamma(A^{(n)})\leqslant\gamma(\mathcal{A}).$

###### Proof.

Most of the statements being elementary and following from corresponding
statements for (operator-valued, abstract) multiplication operators, we only
prove the last inequality. Let $\varphi=(\varphi_{n})_{n}$ in
$\operatorname{dom}(\mathcal{A})\cap\operatorname{ker}(\mathcal{A})^{\bot}$.
By the formula for the kernel of $\mathcal{A}$ we obtain, for all
$n\in\mathbb{N}$,
$\varphi_{n}\in\operatorname{dom}(A^{(n)})\cap\operatorname{ker}(A^{(n)})^{\bot}$.
Let $\alpha>1$. Then, for $n\in\mathbb{N}$,

$\frac{\gamma(A^{(n)})}{\alpha}\leqslant\frac{\|A^{(n)}\varphi_{n}\|_{\mathcal{H}_{1}}}{\|\varphi_{n}\|_{\mathcal{H}_{0}}};\text{
and, thus,
}\big{(}\frac{\gamma(A^{(n)})}{\alpha}\big{)}^{2}\|\varphi_{n}\|_{\mathcal{H}_{0}}^{2}\leqslant\|A^{(n)}\varphi_{n}\|_{\mathcal{H}_{1}}^{2}.$

Hence, summing over $n\in\mathbb{N}$, we infer

$\frac{1}{\alpha^{2}}\inf_{n\in\mathbb{N}}\gamma(A^{(n)})^{2}\|\varphi\|^{2}_{\ell_{2}(\mathbb{N};\mathcal{H}_{0})}\leqslant\|\mathcal{A}\varphi\|_{\ell_{2}(\mathbb{N};\mathcal{H}_{1})}^{2}.$

Therefore,

$\frac{1}{\alpha}\inf_{n\in\mathbb{N}}\gamma(A^{(n)})\leqslant\gamma(\mathcal{A}).$

Letting $\alpha\to 1$, we deduce the asserted inequality. ∎

######  Theorem 6.4.

For $n\in\mathbb{N}$, $j\in\\{0,1\\}$, let
$A_{j}^{(n)}\colon\operatorname{dom}(A_{j}^{(n)})\subseteq\mathcal{H}_{j}\to\mathcal{H}_{j+1}$
be densely defined, closed linear operators satisfying
$\operatorname{ran}(A_{0}^{(n)})\subseteq\operatorname{ker}(A_{1}^{(n)})$.
Then for $\mathcal{A}_{j}\coloneqq\bigoplus_{n\in\mathbb{N}}A_{j}^{(n)}$,
$j\in\\{0,1\\}$ we have
$\operatorname{ran}(\mathcal{A}_{0})\subseteq\operatorname{ker}(\mathcal{A}_{1})$
and

$\operatorname{ker}(\mathcal{A}_{1})\cap\operatorname{ker}(\mathcal{A}_{0}^{*})=\bigoplus_{n\in\mathbb{N}}\operatorname{ker}(A_{1}^{(n)})\cap\operatorname{ker}((A_{0}^{(n)})^{*})\subseteq\ell_{2}(\mathbb{N};\mathcal{H}_{1}).$

Moreover, if for $j\in\\{0,1\\}$

$\inf_{n\in\mathbb{N}}\gamma(A_{j}^{(n)})>0,$

then $\operatorname{ran}(\mathcal{A}_{0}),\operatorname{ran}(\mathcal{A}_{1})$
are closed.

###### Proof.

The statements are almost direct consequences of Proposition 6.3. The
closedness of the ranges follows from Theorem 6.2 in conjunction with the
inequality asserted in Proposition 6.3.∎

We are now in the position to provide an example for the desired range closed
domain $\Omega$ with $\mathcal{H}_{D}(\Omega)$ infinite-dimensional.

######  Example 6.5.

Let $\Omega\subseteq\mathbb{R}^{3}$ be open, bounded and range closed with
$\mathcal{H}_{D}(\Omega)\neq\\{0\\}$444In view of the Picard–Weber–Weck
selection theorem together with [16, Theorem 1] or [13] $\Omega$ being smooth,
bounded with disconnected complement is a possible choice.. Then by Theorem
6.4, $\mathbb{N}$ identical copies of the operators
$\operatorname{\mathring{grad}}$ and $\operatorname{\mathring{curl}}$ yield 2
operators $\mathcal{A}_{0}$, $\mathcal{A}_{1}$ with closed range and infinite-
dimensional
$\operatorname{ker}(\mathcal{A}_{1})\cap\operatorname{ker}(\mathcal{A}_{0}^{*})$.
We note that this orthogonal sum, by translation invariance of the derivative
(and the locality of the operators involved), can be realised as a single set
$\widehat{\Omega}\subseteq\mathbb{R}^{3}$, which is a grid of copies of
$\Omega$ that are arranged in such a way they do not intersect.

##  7 Homogenisation theorems

In this section, we combine the results concerning well-posedness of
electrostatic equations and on the topology of nonlocal $H$-convergence in the
case where it can be guaranteed that $\mathcal{H}_{D}(\Omega)$ is finite-
dimensional, only. Here, we let $\Omega\subseteq\mathbb{R}^{3}$ open.
Furthermore, we shall assume in most of the statements in this section that
$\Omega$ satisfies the electric compactness property; that is,

$\operatorname{dom}(\operatorname{div})\cap\operatorname{dom}(\operatorname{\mathring{curl}})\hookrightarrow\hookrightarrow
L_{2}(\Omega)^{3}.$

By a contradiction argument, we infer that
$\operatorname{ran}(\operatorname{\mathring{curl}})$ and – by the closed range
theorem – $\operatorname{ran}(\operatorname{curl})$ are closed. Furthermore,
it follows that
$\mathcal{H}_{D}(\Omega)=\operatorname{ker}(\operatorname{div})\cap\operatorname{ker}(\operatorname{\mathring{curl}})$
is finite-dimensional. Moreover, we remark here that any $\Omega$ satisfying
the electric compactness property is necessarily bounded. Hence,
$\operatorname{dom}(\operatorname{\mathring{grad}})\hookrightarrow\hookrightarrow
L_{2}(\Omega)$ by the Rellich–Kondrachov theorem. Thus, both
$\operatorname{ran}(\operatorname{\mathring{grad}})$ and
$\operatorname{ran}(\operatorname{\mathring{grad}}^{*})=\operatorname{ran}(\operatorname{div})$
are closed. Consequently, $\Omega$ is range closed. We remark that the Maxwell
compactness property is stronger than the electric compactness property. Thus,
by the Picard–Weber–Weck selection theorem, for instance bounded weak
Lipschitz domains $\Omega$ satisfy the electric compactness property.

Next, we provide an explicit reformulation of Theorem 5.8. As short hand, we
will use

$\tau_{\textnormal{bH}}\coloneqq\tau_{\textnormal{b}}(\operatorname{ran}(\mathring{\operatorname{grad}}),\operatorname{ran}(\mathring{\operatorname{grad}})^{\bot})$
(5)

######  Theorem 7.1.

Let $\Omega\subseteq\mathbb{R}^{3}$ open satisfying the electric compactness
property. Then

$\tau_{\textnormal{bH}}=\tau_{\textnormal{b}}(\operatorname{ran}(\operatorname{curl})^{\bot},\operatorname{ran}(\operatorname{curl}))$

on admissible operators in $\mathcal{B}(L_{2}(\Omega)^{3})$. In particular,
for admissible sequences $(\varepsilon_{n})_{n}$ and an admissible
$\varepsilon$ the following conditions are equivalent:

  1. (i)

for all $f\in H^{-1}(\Omega)$ and $u_{n}\in H_{0}^{1}(\Omega)$ satisfying

$-\operatorname{div}\varepsilon_{n}\operatorname{\mathring{grad}}u_{n}=f,$

we obtain $u_{n}\rightharpoonup u$ in $H_{0}^{1}(\Omega)$ and
$\varepsilon_{n,00}^{-1}\varepsilon_{n,01}$,
$\varepsilon_{n,10}\varepsilon_{n,00}^{-1}$, and
$\varepsilon_{n,11}-\varepsilon_{n,10}\varepsilon_{n,00}^{-1}\varepsilon_{n,01}$
respectively converging in the weak operator topology to
$\varepsilon_{00}^{-1}\varepsilon_{01}$,
$\varepsilon_{10}\varepsilon_{00}^{-1}$, and
$\varepsilon_{11}-\varepsilon_{10}\varepsilon_{00}^{-1}\varepsilon_{01}$,
where we used the decomposition
$L_{2}(\Omega)^{3}=\operatorname{ran}(\operatorname{\mathring{grad}})\oplus\operatorname{ker}(\operatorname{div})$.

  2. (ii)

for all $g\in H^{-1}(\operatorname{curl_{\textnormal{red}}})$ and
$v_{n}\in\operatorname{dom}(\operatorname{curl_{\textnormal{red}}})$
satisfying

$\operatorname{curl_{\textnormal{red}}^{\diamond}}\varepsilon_{n}^{-1}\operatorname{curl_{\textnormal{red}}}v_{n}=g,$

we obtain $v_{n}\rightharpoonup v$ in
$H^{1}(\operatorname{curl_{\textnormal{red}}})$ and $\varepsilon_{n,00}^{-1}$,
$\varepsilon_{n,00}^{-1}\varepsilon_{n,01}$, and
$\varepsilon_{n,10}\varepsilon_{n,00}^{-1}$, respectively converging in the
weak operator topology to $\varepsilon_{00}^{-1}$,
$\varepsilon_{00}^{-1}\varepsilon_{01}$, and
$\varepsilon_{10}\varepsilon_{00}^{-1}$, where we used the decomposition
$L_{2}(\Omega)^{3}=\operatorname{ker}(\operatorname{curl_{\textnormal{red}}})\oplus\operatorname{ran}(\operatorname{curl})$.

###### Proof.

The assertion follows from Theorem 5.8 together with the consequence of the
electric compactness property in that
$\mathcal{H}_{\textnormal{D}}(\Omega)=\operatorname{ran}(\operatorname{\mathring{grad}})^{\bot}\cap\operatorname{ran}(\operatorname{curl_{\textnormal{red}}})^{\bot}=\operatorname{ker}(\operatorname{div})\cap\operatorname{ran}(\operatorname{\mathring{curl}})$
is finite-dimensional. For the equivalence stated note that this is just a
reformulation of the equality of the topologies. ∎

######  Remark 7.2.

If $\Omega$ is range closed (not necessarily having finite-dimensional
$\mathcal{H}_{D}(\Omega)$), then the convergence described in (i) of Theorem
7.1 is equivalent to convergence in $\tau_{\textnormal{bH}}$.

Particularly useful for the next section is the following consequence of the
previous theorem.

######  Corollary 7.3.

Let $\Omega\subseteq\mathbb{R}^{3}$ open satisfying the electric compactness
property. Assume that a sequence of admissible operators
$(\varepsilon_{n})_{n}$ converges to some admissible $\varepsilon$ in
$\tau_{\textnormal{bH}}$. Then the following conditions are satisfied:

  1. (a)

Let $(f_{n})_{n}$ in $H^{-1}(\Omega)$, $f\in H^{-1}(\Omega)$. Assume for
$n\in\mathbb{N}$ that $u_{n}\in H_{0}^{1}(\Omega)$ satisfies

$-\operatorname{div}\varepsilon_{n}\mathring{\operatorname{grad}}u_{n}=f_{n}$

If $f_{n}\to f$ in $H^{-1}(\Omega)$, we obtain $u_{n}\rightharpoonup u$ weakly
in $H_{0}^{1}(\Omega)$, and
$\varepsilon_{n}\operatorname{\mathring{grad}}u_{n}\rightharpoonup\varepsilon\operatorname{\mathring{grad}}u$
weakly in $L_{2}(\Omega)^{3}$.

  2. (b)

Let $(g_{n})_{n}$ in $H^{-1}(\operatorname{curl_{\textnormal{red}}})$, $g\in
H^{-1}(\operatorname{curl_{\textnormal{red}}})$. Assume for $n\in\mathbb{N}$
that $v_{n}\in\operatorname{dom}(\operatorname{curl}_{{\rm red}})$ satisfies

$\operatorname{curl_{\textnormal{red}}^{\diamond}}\varepsilon_{n}^{-1}\operatorname{curl_{\textnormal{red}}}v_{n}=g_{n}.$

Then $v_{n}\rightharpoonup v$ in
$H^{1}(\operatorname{curl_{\textnormal{red}}})$,
$\varepsilon_{n}^{-1}\operatorname{curl}v_{n}\rightharpoonup\varepsilon^{-1}\operatorname{curl}v$
in $L_{2}(\Omega)^{3}$.

  3. (c)

$\operatorname{\mathring{grad}}(\operatorname{div}\varepsilon_{n}\operatorname{\mathring{grad}})^{-1}\operatorname{div}\varepsilon_{n}\to\operatorname{\mathring{grad}}(\operatorname{div}\varepsilon\operatorname{\mathring{grad}})^{-1}\operatorname{div}\varepsilon$
in the weak operator topology.

  4. (d)

$\varepsilon_{n}(1-\operatorname{\mathring{grad}}(\operatorname{div}\varepsilon_{n}\operatorname{\mathring{grad}})^{-1}\operatorname{div}\varepsilon_{n})\to\varepsilon(1-\operatorname{\mathring{grad}}(\operatorname{div}\varepsilon\operatorname{\mathring{grad}})^{-1}\operatorname{div}\varepsilon)$
in the weak operator topology.

###### Proof.

For (a) and (b), in view of Theorem 7.1, it suffices to show convergence of
the respective fluxes (Note that the weak convergence of the solutions
$(u_{n})_{n}$ and $(v_{n})_{n}$ follow from the reformulation in Theorem 2.1
(b) and the assumed convergence of the operators involved in the weak operator
topology). By the symmetry of the argument (see Remark 5.6), it moreover
suffices to consider (a). For this, note that

$u_{n}=-(\operatorname{div}\varepsilon_{n}\mathring{\operatorname{grad}})^{-1}f=-(\iota_{0}^{*}\mathring{\operatorname{grad}})^{-1}\varepsilon_{n,00}^{-1}(\operatorname{div}\iota_{0})^{-1}f$

and so

$\displaystyle\varepsilon_{n}\mathring{\operatorname{grad}}u_{n}$
$\displaystyle=-\varepsilon_{n}\iota_{0}\iota_{0}^{*}\mathring{\operatorname{grad}}(\iota_{0}^{*}\mathring{\operatorname{grad}})^{-1}\varepsilon_{n,00}^{-1}(\operatorname{div}\iota_{0})^{-1}f$
$\displaystyle=-\begin{pmatrix}\iota_{0}&\iota_{1}\end{pmatrix}\begin{pmatrix}\iota_{0}^{*}\\\
\iota_{1}^{*}\end{pmatrix}\varepsilon_{n}\begin{pmatrix}\iota_{0}&\iota_{1}\end{pmatrix}\begin{pmatrix}\iota_{0}^{*}\\\
\iota_{1}^{*}\end{pmatrix}\iota_{0}\iota_{0}^{*}\mathring{\operatorname{grad}}(\iota_{0}^{*}\mathring{\operatorname{grad}})^{-1}\varepsilon_{n,00}^{-1}(\operatorname{div}\iota_{0})^{-1}f$
$\displaystyle=-\begin{pmatrix}\iota_{0}&\iota_{1}\end{pmatrix}\begin{pmatrix}\varepsilon_{n,00}&\varepsilon_{n,01}\\\
\varepsilon_{n,10}&\varepsilon_{n,11}\end{pmatrix}\begin{pmatrix}\iota_{0}^{*}\iota_{0}\varepsilon_{n,00}^{-1}(\operatorname{div}\iota_{0})^{-1}f\\\
0\end{pmatrix}$
$\displaystyle=-\begin{pmatrix}\iota_{0}&\iota_{1}\end{pmatrix}\begin{pmatrix}\varepsilon_{n,00}&\varepsilon_{n,01}\\\
\varepsilon_{n,10}&\varepsilon_{n,11}\end{pmatrix}\begin{pmatrix}\iota_{0}^{*}\iota_{0}\varepsilon_{n,00}^{-1}(\operatorname{div}\iota_{0})^{-1}f\\\
0\end{pmatrix}=\iota_{1}\varepsilon_{n,10}\varepsilon_{n,00}^{-1}(\operatorname{div}\iota_{0})^{-1}f,$

so the convergence of the flux follows from the convergence of
$\varepsilon_{n,10}\varepsilon_{n,00}^{-1}$ in the weak operator topology.

(c) We compute using the decomposition
$L_{2}(\Omega)^{3}=\operatorname{ran}(\mathring{\operatorname{grad}})\oplus\operatorname{ker}(\operatorname{div})$,

$\displaystyle\mathring{\operatorname{grad}}(\operatorname{div}\varepsilon_{n}\mathring{\operatorname{grad}})^{-1}\operatorname{div}\varepsilon_{n}$
$\displaystyle=\iota_{0}\iota_{0}^{*}\mathring{\operatorname{grad}}(\iota_{0}^{*}\mathring{\operatorname{grad}})^{-1}\varepsilon_{n,00}^{-1}(\operatorname{div}\iota_{0})^{-1}\operatorname{div}\iota_{0}\iota_{0}^{*}\varepsilon_{n}$
$\displaystyle=\iota_{0}\varepsilon_{n,00}^{-1}\iota_{0}^{*}\begin{pmatrix}\iota_{0}&\iota_{1}\end{pmatrix}\begin{pmatrix}\varepsilon_{n,00}&\varepsilon_{n,01}\\\
\varepsilon_{n,10}&\varepsilon_{n,11}\end{pmatrix}\begin{pmatrix}\iota_{0}^{*}\\\
\iota_{1}^{*}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\iota_{0}\varepsilon_{n,00}^{-1}&0\end{pmatrix}\begin{pmatrix}\varepsilon_{n,00}\iota_{0}^{*}+\varepsilon_{n,01}\iota_{1}^{*}\\\
\varepsilon_{n,10}\iota_{0}^{*}+\varepsilon_{n,11}\iota_{1}^{*}\end{pmatrix}$
$\displaystyle=\iota_{0}\varepsilon_{n,00}^{-1}(\varepsilon_{n,00}\iota_{0}^{*}+\varepsilon_{n,01}\iota_{1}^{*})$
$\displaystyle=\iota_{0}\varepsilon_{n,00}^{-1}\varepsilon_{n,00}\iota_{0}^{*}+\iota_{0}\varepsilon_{n,00}^{-1}\varepsilon_{n,01}\iota_{1}^{*}.$

Hence, the statement follows from the convergence statement in (i) of Theorem
7.1.

(d) With the same decomposition as in (c), we compute

$\displaystyle\varepsilon_{n}(1-\operatorname{\mathring{grad}}(\operatorname{div}\varepsilon_{n}\operatorname{\mathring{grad}})^{-1}\operatorname{div}\varepsilon_{n})$
$\displaystyle=\varepsilon_{n}(1-\iota_{0}\iota_{0}^{*}+\iota_{0}\varepsilon_{n,00}^{-1}\varepsilon_{n,01}\iota_{1}^{*})$
$\displaystyle=\begin{pmatrix}\iota_{0}&\iota_{1}\end{pmatrix}\begin{pmatrix}\varepsilon_{n,00}&\varepsilon_{n,01}\\\
\varepsilon_{n,10}&\varepsilon_{n,11}\end{pmatrix}\begin{pmatrix}\iota_{0}^{*}\\\
\iota_{1}^{*}\end{pmatrix}(\iota_{1}\iota_{1}^{*}-\iota_{0}\varepsilon_{n,00}^{-1}\varepsilon_{n,01}\iota_{1}^{*})$
$\displaystyle=\begin{pmatrix}\iota_{0}&\iota_{1}\end{pmatrix}\begin{pmatrix}\varepsilon_{n,00}&\varepsilon_{n,01}\\\
\varepsilon_{n,10}&\varepsilon_{n,11}\end{pmatrix}\begin{pmatrix}-\varepsilon_{n,00}^{-1}\varepsilon_{n,01}\\\
1\end{pmatrix}\iota_{1}^{*}$
$\displaystyle=\begin{pmatrix}\iota_{0}&\iota_{1}\end{pmatrix}\begin{pmatrix}0\\\
\varepsilon_{n,11}-\varepsilon_{n,10}\varepsilon_{n,00}^{-1}\varepsilon_{n,01}\end{pmatrix}\iota_{1}^{*}=\iota_{1}(\varepsilon_{n,11}-\varepsilon_{n,10}\varepsilon_{n,00}^{-1}\varepsilon_{n,01})\iota_{1}^{*}.$

The latter operator converges in the weak operator topology by Theorem 7.1(i).
∎

######  Remark 7.4.

Note that (a), (c) and (d) are true even in the case where
$\mathcal{H}_{D}(\Omega)$ is infinite-dimensional; that is, range closed
$\Omega$ is sufficient.

We now turn to the discussion of the homogenisation problem for electrostatic
problems as discussed in Section 4. We particularly refer to Remark 4.5 for
the equivalent formulations of
($\mathbb{P}_{\varepsilon}(f,g,x_{\varepsilon})$) and
($\mathbb{P}^{\varepsilon}(f,g,x^{\varepsilon})$) used in the following
theorem.

######  Theorem 7.5.

Let $\Omega\subseteq\mathbb{R}^{3}$ be open satisfying the electric
compactness property. Let $(\varepsilon_{n})_{n}$ in
$\mathcal{B}(L_{2}(\Omega)^{3})$ and
$\varepsilon\in\mathcal{B}(L_{2}(\Omega)^{3})$. Assume $\varepsilon$ and, for
all $n\in\mathbb{N}$, $\varepsilon_{n}$ are admissible. Furthermore, assume
$\varepsilon_{n}\to\varepsilon$ in $\tau_{\textnormal{bH}}$.

Let $f\in H^{-1}(\Omega)$, $g\in H^{-1}(\operatorname{curl}_{{\rm red}})$,
$x\in\mathcal{H}_{D}(\Omega)$ and $E_{n}\in L_{2}(\Omega)^{3}$ satisfying

$\begin{cases}\operatorname{div}\varepsilon_{n}E_{n}=f&\\\
\operatorname{curl}E_{n}=g&\\\
\pi_{D}\pi_{\varepsilon_{n}}E_{n}=x.&\end{cases}$

Then $E_{n}\to E$ and $\varepsilon_{n}E_{n}\to\varepsilon E$ weakly in
$L_{2}(\Omega)^{3}$ and $E$ satisfies

$\begin{cases}\operatorname{div}\varepsilon E=f&\\\ \operatorname{curl}E=g&\\\
\pi_{D}\pi_{\varepsilon}E=x.&\end{cases}$

###### Proof.

We use the solution formulas provided in Theorem 4.3 with

$\displaystyle x_{\varepsilon_{n}}$
$\displaystyle=(1-\operatorname{\mathring{grad}}(\operatorname{div}\varepsilon_{n}\operatorname{\mathring{grad}})^{-1}\operatorname{div}\varepsilon_{n})x,$
$\displaystyle x^{\varepsilon_{n}}$
$\displaystyle=\varepsilon_{n}(1-\operatorname{\mathring{grad}}(\operatorname{div}\varepsilon_{n}\operatorname{\mathring{grad}})^{-1}\operatorname{div}\varepsilon_{n})x$

(see also Remark 4.5) to obtain

$E_{n}=\operatorname{\mathring{grad}}(\operatorname{div}\varepsilon_{n}\operatorname{\mathring{grad}})^{-1}f\\\
+\varepsilon_{n}^{-1}\operatorname{curl_{\textnormal{red}}}(\operatorname{curl_{\textnormal{red}}^{\diamond}}\varepsilon_{n}^{-1}\operatorname{curl_{\textnormal{red}}})^{-1}g+(1-\operatorname{\mathring{grad}}(\operatorname{div}\varepsilon_{n}\operatorname{\mathring{grad}})^{-1}\operatorname{div}\varepsilon_{n})x$

and

$H_{n}=\varepsilon_{n}E_{n}=\varepsilon_{n}\operatorname{\mathring{grad}}(\operatorname{div}\varepsilon_{n}\operatorname{\mathring{grad}})^{-1}f\\\
+\operatorname{curl_{\textnormal{red}}}(\operatorname{curl_{\textnormal{red}}^{\diamond}}\varepsilon_{n}^{-1}\operatorname{curl_{\textnormal{red}}})^{-1}g+\varepsilon_{n}(1-\operatorname{\mathring{grad}}(\operatorname{div}\varepsilon_{n}\operatorname{\mathring{grad}})^{-1}\operatorname{div}\varepsilon_{n})x.$

Here, using Corollary 7.3, we may now let $n\to\infty$ to obtain the desired
result. ∎

######  Remark 7.6.

An analogous statement to Theorem 7.5 holds, if we replace $f$, $g$ and $x$ by
(strongly) convergent sequences $(f_{n})_{n}$, $(g_{n})_{n}$ and
$(x_{n})_{n}$.

##  8 Convergence of energy and the div-curl Lemma

Similarly to the previous section, we shall assume that
$\Omega\subseteq\mathbb{R}^{3}$ satisfies the electric compactness property.
We also recall the abbreviation

$\tau_{\textnormal{bH}}=\tau_{\textnormal{b}}(\operatorname{ran}(\mathring{\operatorname{grad}}),\operatorname{ran}(\mathring{\operatorname{grad}})^{\bot}).$

At the end of this section, we provide a proof for Theorem B.

In order to obtain convergence of the energy
$\langle\varepsilon_{n}E_{n},E_{n}\rangle$ for $E_{n}$ solving the system for
electrostatics, we need the following concrete version of the abstract div-
curl lemma from [28, Theorem 2.6].

######  Theorem 8.1.

Let $\Omega\subseteq\mathbb{R}^{3}$ open satisfying the electric compactness
property. Assume that $(r_{n})_{n},(q_{n})_{n}$ are weakly convergent
sequences in $L_{2}(\Omega)^{3}$ with respective limits $r$ and $q$. If both
$\\{\operatorname{div}r_{n};n\in\mathbb{N}\\}\subseteq H^{-1}(\Omega)$ and
$\\{\operatorname{{curl}^{\diamond}}q_{n};n\in\mathbb{N}\\}\subseteq
H^{-1}(\operatorname{curl})$ are relatively compact, then

$\lim_{n\to\infty}\langle r_{n},q_{n}\rangle_{L_{2}(\Omega)^{3}}=\langle
r,q\rangle_{L_{2}(\Omega)^{3}}.$

###### Proof.

The claim follows from [28, Theorem 2.6] for the special case
$A_{0}=\operatorname{\mathring{grad}}$ and
$A_{1}=\operatorname{\mathring{curl}}$ (where we used the notation there). The
assumptions in [28, Theorem 2.6] are that $\operatorname{ran}(A_{0})$ and
$\operatorname{ran}(A_{1})$ are closed and that
$\operatorname{ker}(A_{0}^{*})\cap\operatorname{ker}(A_{1})$ is finite-
dimensional. These assumptions are – as it is argued just after the definition
of the electric compactness property – consequences of this compactness
property. ∎

We state the main result of this section next.

######  Theorem 8.2 (Convergence of the energy).

Let $(\varepsilon_{n})_{n}$ be admissible and
$\tau_{\textnormal{bH}}$-convergent to some admissible $\varepsilon$. If
$f_{n}\to f$ and $g_{n}\to g$ strongly in $H^{-1}(\Omega)$ and
$H^{-1}(\operatorname{curl}_{\textnormal{red}})$, $x_{n}\to x$ in
$\mathcal{H}_{D}(\Omega)$. Then, given $E_{n}$ satisfies

$\operatorname{div}\varepsilon_{n}E_{n}=f_{n}\quad\operatorname{\mathring{curl}}E_{n}=g_{n},\pi_{D}\pi_{\varepsilon_{n}}E_{n}=x_{n}$

we obtain that both $E_{n}\to E$ and $\varepsilon_{n}E_{n}\to\varepsilon E$
weakly in $L_{2}(\Omega)^{3}$, where $E$ satisfies

$\operatorname{div}\varepsilon
E=f\quad\operatorname{\mathring{curl}}E=g,\pi_{D}\pi_{\varepsilon}=x.$

Moreover,

$\langle\varepsilon_{n}E_{n},E_{n}\rangle\to\langle\varepsilon E,E\rangle$

###### Proof.

The weak convergence of $E_{n}$ and $\varepsilon_{n}E_{n}$ to $E$ and
$\varepsilon E$ follows from Theorem 7.5 in conjunction with Remark 7.6.

The convergence of the scalar products follows from Theorem 8.1. Indeed, for
$q_{n}=E_{n}$ and $r_{n}=\varepsilon_{n}E_{n}$ the assumptions of Theorem 8.1
are satisfied proving the assertion. ∎

###### Proof of Theorem B.

This is a direct consequence of Theorem 7.5 together with Theorem 8.2. ∎

##  9 A compactness criterion with moving coefficients

In this section, we present a compactness criterion and provide a proof of
Theorem D. For this, let $\Omega\subseteq\mathbb{R}^{3}$ be open. Consider a
sequence of admissible coefficients $(\varepsilon_{n})_{n}$ in
$\mathcal{B}(L_{2}(\Omega)^{3})$. When is it true that a bounded
$L_{2}$-sequence of vector fields $(E_{n})_{n}$ satisfying the conditions

$\operatorname{div}\varepsilon_{n}E_{n},\operatorname{\mathring{curl}}E_{n}\text{
bounded in $L_{2}$}$

actually contains an $L_{2}$-convergent subsequence? If
$\varepsilon_{n}=\varepsilon_{0}$ for some selfadjoint, strictly positive
definite $\varepsilon_{0}$, then standard Hilbert space techniques confirm
that

$\operatorname{dom}(\operatorname{div}\varepsilon_{0})\cap\operatorname{dom}(\operatorname{\mathring{curl}})\hookrightarrow\hookrightarrow
L_{2}(\Omega)^{3},$

if $\Omega$ satisfies the electric compactness property, see e.g. [13]. Thus,
the compactness result hinges on $\Omega$, only. In the case of ‘moving’
coefficients, that is, non-constant $(\varepsilon_{n})_{n}$, some additional
assumptions on the limit behaviour are needed. The corresponding result reads
as follows, which by the Picard–Weber–Weck selection theorem is more general
than Theorem D from the introduction.

######  Theorem 9.1.

Let $\Omega\subseteq\mathbb{R}^{3}$ open satisfying the electric compactness
property. Let $(\varepsilon_{n})_{n\geqslant 1}$ be bounded sequence in
$\mathcal{B}(L_{2}(\Omega)^{3})$,
$\varepsilon_{0}\in\mathcal{B}(L_{2}(\Omega)^{3})$. Assume there exists $c>0$
such that

$\operatorname{Re}\varepsilon_{n}\geqslant c\quad(n\in\mathbb{N}).$

and assume that for all $E\in L_{2}(\Omega)^{3}$,
$(\operatorname{div}\varepsilon_{n}E)_{n}$ is relatively compact in
$H^{-1}(\Omega)$.

Assume $\varepsilon_{n}\to\varepsilon_{0}$ as $n\to\infty$ in both the weak
operator topology and $\tau_{\textnormal{bH}}$.

Let $(E_{n})_{n}$ bounded in $L_{2}(\Omega)^{3}$ such that

$(\operatorname{div}\varepsilon_{n}E_{n})_{n}\text{ and
}(\operatorname{\mathring{curl}}E_{n})_{n}$

are bounded in $L_{2}(\Omega)$ and $L_{2}(\Omega)^{3}$.

Then there exists an $L_{2}(\Omega)^{3}$-strongly convergent subsequence of
$(E_{n})_{n}$.

###### Proof of Theorem 9.1 and, thus, of Theorem D.

Without loss of generality, we may assume that $E_{n}\to E$ weakly in
$L_{2}(\Omega)^{3}$ for some $E\in L_{2}(\Omega)^{3}$. By Remark 3.6,
$\varepsilon_{n}$ is admissible for all $n\in\mathbb{N}$. Since
$\varepsilon_{n}\to\varepsilon_{0}$ in the weak operator topology, we obtain
$\operatorname{Re}\varepsilon\geqslant c$ and, thus, $\varepsilon_{0}$ is,
too, admissible. We define
$\tilde{f}_{n}\coloneqq\operatorname{div}\varepsilon_{n}E_{n}$ and
$\tilde{g}_{n}\coloneqq\mathring{\operatorname{curl}}E_{n}$. Next, we put

$f_{n}\coloneqq\tilde{f}_{n}-\operatorname{div}\varepsilon_{n}E;\text{ and
}g_{n}\coloneqq\tilde{g}_{n}-\mathring{\operatorname{curl}}E.$

Then, by assumption for another subsequence, $f_{n}\to
f\coloneqq\tilde{f}-\operatorname{div}\varepsilon E$ and $g_{n}\to
g\coloneqq\tilde{g}-\mathring{\operatorname{curl}}E$ strongly in
$H^{-1}(\Omega)$ and $H^{-1}(\operatorname{curl}_{\text{red}})$, respectively.
Hence, as $\varepsilon_{n}\to\varepsilon_{0}$ in $\tau_{\textnormal{bH}}$, we
obtain by Theorem 8.2, $E_{n}\to E$ and $\varepsilon_{n}E_{n}\to\varepsilon E$
weakly in $L_{2}(\Omega)^{3}$. Moreover, by Theorem 8.2 again, we deduce

$\displaystyle c\langle E_{n}-E,E_{n}-E\rangle$
$\displaystyle\leqslant\operatorname{Re}\langle
E_{n}-E,\varepsilon_{n}(E_{n}-E)\rangle\to 0.$

The claim follows. ∎

######  Corollary 9.2.

Let $\Omega\subseteq\mathbb{R}^{3}$ open satisfying the electric compactness
property. Let $\varepsilon\in\mathcal{B}(L_{2}(\Omega)^{3})$ with

$\operatorname{Re}\varepsilon\geqslant c$

for some $c>0$. Then

$\operatorname{dom}(\operatorname{div}\varepsilon)\cap\operatorname{dom}(\operatorname{\mathring{curl}})\hookrightarrow\hookrightarrow
L_{2}(\Omega)^{3}.$

###### Proof.

The assertion follows trivially from Theorem 9.1 for the constant sequence
$\varepsilon_{n}=\varepsilon$ with limit $\varepsilon_{0}=\varepsilon$. ∎

######  Remark 9.3.

If $\varepsilon_{n}\to\varepsilon_{0}$ in the strong operator topology
additionally satisfies the divergence condition, then the assumptions in
Theorem 9.1 are met.

The rest of this section is devoted to discuss an example. More precisely, we
shall establish the following result.

######  Theorem 9.4.

Let $\rho\in L_{\infty}(\mathbb{R}^{3})\cap L_{1}(\mathbb{R}^{3})$ and assume
that $\rho_{n}\coloneqq\rho(n\cdot)\to\tilde{\rho}$ in
$\sigma(L_{\infty}(\mathbb{R}^{3}),L_{1}(\mathbb{R}^{3}))$ for some
$\tilde{\rho}\in L_{\infty}(\mathbb{R}^{3})$. If $\|\rho\|_{L_{1}}<1$, then
$(\varepsilon_{n})_{n\geqslant 1}$ in $\mathcal{B}(L_{2}(\Omega))$ given by

$\varepsilon_{n}\varphi\coloneqq(1+\rho_{n}*)\varphi\coloneqq\varphi+(x\mapsto\int_{\Omega}\rho_{n}(x-y)\varphi(y)\mathrm{d}y)$

satisfies the assumptions of Theorem 9.1 with
$\varepsilon_{0}=1+\tilde{\rho}*$.

We prove Theorem 9.4 in several lemmas.

######  Lemma 9.5.

Let $\rho\in L_{1}(\mathbb{R}^{3}),E\in L_{2}(\Omega)^{3}$;
$\rho_{n}\coloneqq\rho(n\cdot)$. Then
$\\{((1+\rho_{n}*)E);n\in\mathbb{N},n\geqslant 1\\}\subseteq
L_{2}(\Omega)^{3}$ is relatively compact.

###### Proof.

We prove the claim in two steps.

First step: Assume additionally that $\rho\in W^{1}_{1}(\mathbb{R}^{3})$. In
this case, for $j\in\\{1,2,3\\}$, we note
$\partial_{j}\rho_{n}=n(\partial_{j}\rho)(n\cdot)$ and

$\|\partial_{j}\rho_{n}\|_{L_{1}(\Omega)}=\int_{\Omega}n|\partial_{j}\rho(nx)|\mathrm{d}x=\int_{n\Omega}|\partial_{j}\rho(x)|\mathrm{d}x\leqslant\|\partial_{j}\rho\|_{L_{1}(\mathbb{R}^{3})}.$

Hence, $(\partial_{j}\rho_{n}*E)_{n\in\mathbb{N},j\in\\{1,2,3\\}}$ is
uniformly bounded in $L_{2}(\Omega)^{3}$. In particular, $(\rho_{n}*E)_{n}$ is
uniformly bounded in $H^{1}(\Omega)^{3}$ and, by Rellich’s selection theorem,
$\\{\rho_{n}*E;n\in\mathbb{N},n\geqslant 1\\}$ is relatively compact in
$L_{2}(\Omega)^{3}$, proving the assertion for the first step.

Second step: We show that $(\rho_{n}*E)_{n}$ is totally bounded in
$L_{2}(\Omega)^{3}$. For this, let $\varepsilon>0$. Then we find $\eta\in
C_{c}^{\infty}(\Omega)$ such that
$\|\rho-\eta*\rho\|_{L_{1}}\leqslant\varepsilon$. By the first step, we find a
finite set $N\subseteq\mathbb{N}$ such that

$\bigcup_{n\in
N}B((\eta*\rho)_{n}*E,\varepsilon)\supseteq\\{(\eta*\rho)_{n}*E;n\in\mathbb{N}\\}.$

Thus, with Young’s inequality again, for $n\geqslant 1$,

$\|\rho_{n}*E-(\eta*\rho)_{n}*E\|_{L_{2}}\leqslant\frac{1}{n}\|\rho-\eta*\eta\|_{L_{1}}\|E\|_{L_{2}}\leqslant\varepsilon\|E\|_{L_{2}},$

the assertion follows. ∎

######  Lemma 9.6.

Let $(\kappa_{n})_{n}$ bounded in $L_{\infty}(\mathbb{R}^{3})\cap
L_{1}(\mathbb{R}^{3})$ and $\kappa\in L_{\infty}(\mathbb{R}^{n})$. Assume that
$\kappa_{n}\to\kappa$ in
$\sigma(L_{\infty}(\mathbb{R}^{3}),L_{1}(\mathbb{R}^{3}))$. Then
$\kappa_{n}*\to\kappa*$ in the weak operator topology of
$\mathcal{B}(L_{2}(\Omega)^{3})$. If, additionally, $\kappa_{n}=\rho(n\cdot)$
for some $\rho\in L_{1}(\mathbb{R}^{3})$, then $\kappa_{n}*\to\kappa*$ in the
strong operator topology.

###### Proof.

By Young’s inequality, $(\kappa_{n}*)_{n}$ considered as an operator sequence
in $\mathcal{B}(L_{2}(\mathbb{R}^{3}))$ is uniformly bounded. First we show
convergence in the weak operator topology. For this, by the observed
boundedness, it suffices to prove convergence of expressions of the type
$\langle\kappa_{n}*\psi,\varphi\rangle$ for $\psi,\varphi\in
C_{c}^{\infty}(\Omega)$, only. The latter, however, is a consequence of
Fubini’s and the dominated convergence theorem. Moreover, in the particular
case $\kappa_{n}=\rho(n\cdot)$ by Lemma 9.5, $(\kappa_{n}*E)_{n}$ relatively
compact in $L_{2}(\Omega)^{3}$. This together with weak convergence proves
strong operator topology convergence. ∎

###### Proof of Theorem 9.4.

By Lemma 9.5, it follows that $((1+\rho_{n}*)E)_{n}$ is relatively compact in
$L_{2}(\Omega)^{3}$. Hence, by the continuity of $\operatorname{div}\colon
L_{2}(\Omega)^{3}\to H^{-1}(\Omega)$, it follows that
$(\operatorname{div}(1+\rho_{n}*)E)_{n}$ is relatively compact in
$H^{-1}(\Omega)$. Moroever, we obtain

$\operatorname{Re}(1+\rho_{n}*)\geqslant 1-\sup_{n}\|\rho_{n}\|_{L_{1}}>0.$

Finally, as the strong operator topology of bounded sequences is stronger both
than the weak operator topology and $\tau_{\textnormal{bH}}$, the assertion
follows (see also Remark 9.3). ∎

Being formulated for potentially topologically nontrivial domains, convergence
in $\tau_{\textnormal{bH}}$ generalises the concept of nonlocal
$H$-convergence for bounded sequences as introduced in [29]. Since this
generalisation consists in the consideration of other domains, the theorem
stating the equivalence of local $H$-convergence and nonlocal $H$-convergence
as presented in [29, Theorem 5.11] or Theorem 10.3 below does not apply
anymore. However, in the next section, we show that local $H$-convergence is
equivalent to convergence in $\tau_{\textnormal{bH}}$ for multiplication
operators confirming the ‘correctness’ of our generalisation, see also Section
5.

##  10 Relationship to local $H$-convergence

Finally, we position convergence in $\tau_{\textnormal{bH}}$ in the context of
local $H$-convergence; that is, we provide a proof of Theorem C. As before, we
let throughout $\Omega\subseteq\mathbb{R}^{3}$ open satisfying the electric
compactness property and

$\tau_{\textnormal{bH}}=\tau_{\textnormal{b}}(\operatorname{ran}(\mathring{\operatorname{grad}}),\operatorname{ran}(\mathring{\operatorname{grad}})^{\bot}).$

Moreover, we define for $0<\alpha<\beta$

$M(\alpha,\beta;\Omega)\coloneqq\\{a\in L_{\infty}(\Omega)^{3\times
3};\operatorname{Re}a\geqslant\alpha,\operatorname{Re}a^{-1}\geqslant
1/\beta\\}.$

Recall that a sequence $(a_{n})_{n}$ in $M(\alpha,\beta;\Omega)$ locally
$H$-converges or converges in the local $H$-sense to some $a\in
M(\alpha,\beta;\Omega)$, if for all $f\in H^{-1}(\Omega)$ and $u_{n}\in
H_{0}^{1}(\Omega)$ satisfying

$-\operatorname{div}a_{n}\operatorname{\mathring{grad}}u_{n}=f,$

one obtains $u_{n}\to u$ weakly and
$a_{n}\operatorname{\mathring{grad}}u_{n}\to a\operatorname{\mathring{grad}}u$
weakly in $H_{0}^{1}(\Omega)$ and $L_{2}(\Omega)^{3}$, respectively. The main
result of this section reads as follows.

######  Theorem 10.1.

Let $(\varepsilon_{n})_{n}$ in $M(\alpha,\beta;\Omega)$ and $\varepsilon\in
M(\alpha,\beta;\Omega)$. Assume that $\varepsilon_{n}\to\varepsilon$ in the
local $H$-sense. Then $\varepsilon_{n}\to\varepsilon$ in
$\tau_{\textnormal{bH}}$.

The proof of this theorem needs some preparations. First, we recall a
characterisation of nonlocal $H$-convergence for topologically trivial domains
from [29]. Even though the proof in [29] is provided in a slightly less
general setting for the class of operator sequences considered, the proof for
admissible operators is (almost) literally the same. The required
modifications are obvious.

######  Theorem 10.2 ([29, Theorem 6.2; see also Theorem 6.5]).

Let $\Omega_{\textnormal{t}}\subseteq\mathbb{R}^{3}$ be open satisfying the
electric compactness property and
$\mathcal{H}_{D}(\Omega_{\textnormal{t}})=\\{0\\}$. Let $(a_{n})_{n}$ be
bounded in $\mathcal{B}(L_{2}(\Omega_{\textnormal{t}})^{3})$ and admissible,
$a\in\mathcal{B}(L_{2}(\Omega_{\textnormal{t}})^{3})$ admissible. Then the
following statements are equivalent:

  1. (i)

$(a_{n})_{n}$ converges to $a$ in
$\tau(\operatorname{ran}(\operatorname{\mathring{grad}},{\Omega_{\textnormal{t}}}),\operatorname{ran}(\operatorname{curl},{\Omega_{\textnormal{t}}}))$

  2. (ii)

for all $(q_{n})_{n}$ in $L_{2}(\Omega_{\textnormal{t}})^{3}$ weakly
convergent to some $q$ in $L_{2}(\Omega_{\textnormal{t}})^{3}$ and
$\kappa\colon\mathbb{N}\to\mathbb{N}$ strictly monotone we have: Given the
conditions

    1. (a)

$(\operatorname{div}(a_{\kappa(n)}q_{n}))_{n}$ is relatively compact in
$H^{-1}(\Omega_{\textnormal{t}})$,

    2. (b)

$(\operatorname{\Circ{\operatorname{curl}}}(q_{n}))_{n}$ is relatively compact
in $H^{-1}(\operatorname{curl_{\textnormal{red}}},\Omega_{\textnormal{t}})$,

then $a_{\kappa(n)}q_{n}\rightharpoonup aq$ as $n\to\infty$.

We recall the equivalence theorem of local and nonlocal $H$-convergence from
[29] for topologically trivial domains.

######  Theorem 10.3 ([29, Theorem 5.11]).

Let $\Omega_{\textnormal{t}}\subseteq\mathbb{R}^{3}$ be open satisfying the
electric compactness property and
$\mathcal{H}_{D}(\Omega_{\textnormal{t}})=\\{0\\}$. Let $(a_{n})_{n}$ be a
sequence in $M(\alpha,\beta;\Omega_{\textnormal{t}})$, $a\in
M(\alpha,\beta;\Omega_{\textnormal{t}})$. Then the following statements are
equivalent:

  1. (i)

$(a_{n})_{n}$ converges to $a$ in
$\tau(\operatorname{ran}(\operatorname{\mathring{grad}},{\Omega_{\textnormal{t}}}),\operatorname{ran}(\operatorname{curl},{\Omega_{\textnormal{t}}}))$

  2. (ii)

$(a_{n})_{n}$ locally $H$-converges to $a$.

Next, we establish the main result of this section.

###### Proof of Theorem 10.1.

We will show the property described in Theorem 7.1 (ii). For the operators in
question, we use the orthogonal decomposition induced by
$\operatorname{ran}(\operatorname{curl})^{\bot}\oplus\operatorname{ran}(\operatorname{curl})$;
the induced operators are denoted by indices as in Theorem 7.1 (ii). Before we
turn to the $\operatorname{curl}$-problem, we show that
$\varepsilon_{n,00}^{-1}$ converges in the weak operator topology to
$\varepsilon_{00}^{-1}$. For this, one notes that the operator

$G\coloneqq\begin{pmatrix}\operatorname{\mathring{grad}}_{\Omega}&\iota_{D}\end{pmatrix}\colon
H^{1}_{0}(\Omega)\oplus\mathcal{H}_{D}(\Omega)\subseteq
L_{2}(\Omega)\oplus\mathcal{H}_{D}(\Omega)\to
L_{2}(\Omega)^{3},(u,x)\mapsto\operatorname{\mathring{grad}}u+x$

is densely defined and closed. Moreover,

$\operatorname{ran}(G)=\operatorname{ran}(\operatorname{\mathring{grad}},\Omega)+\mathcal{H}_{D}(\Omega)(=\operatorname{ran}(\operatorname{curl})^{\bot})$

is closed as the sum is orthogonal and both
$\operatorname{ran}(\operatorname{\mathring{grad}},\Omega)$ and
$\mathcal{H}_{D}(\Omega)$ are closed. Next, $G$ is one-to-one. Consequently,
for $f\in L_{2}(\Omega)$ and $y\in\mathcal{H}_{D}(\Omega)$, we find uniquely
determined $u_{n}\in H_{0}^{1}(\Omega)$ and $x_{n}\in\mathcal{H}_{D}(\Omega)$
such that

$G^{*}\varepsilon_{n}G\begin{pmatrix}u_{n}\\\
x_{n}\end{pmatrix}=\begin{pmatrix}f\\\ y\end{pmatrix}.$

It follows that $U_{n}\coloneqq\begin{pmatrix}u_{n}\\\ x_{n}\end{pmatrix}$
forms a bounded sequence in $\operatorname{dom}(G)$. Thus, there exists a
weakly convergent subsequence $(U_{\kappa(n)})_{n}$ with limit $U$. We need to
uniquely identify $U$. For this, consider $q_{n}\coloneqq GU_{\kappa(n)}$ and
let $B\subseteq\Omega$ be an open ball with $\overline{B}\subseteq\Omega$.
Since local $H$-convergence on $\Omega$ implies the same on $B$, we infer
(that the restriction to $L_{2}(B)^{3}$ of) $(\varepsilon_{n})_{n}$ locally
$H$-converges $\varepsilon$. As $B^{\textnormal{c}}$ is connected,
$\mathcal{H}_{D}(B)=\\{0\\}$ (see [15, 13]). Hence, applying the Theorems 10.2
and 10.3 to $a_{n}=\varepsilon_{n}$ and $\Omega_{\textnormal{t}}=B$, we are in
the position to use condition (ii) in Theorem 10.2. For this, we let
$\varphi\in C_{c}^{\infty}(\Omega)$, $\operatorname{spt}\varphi\subseteq B$.
Then

$\displaystyle\operatorname{div}\varepsilon_{\kappa(n)}\varphi q_{n}$
$\displaystyle=\operatorname{div}\varphi\varepsilon_{\kappa(n)}q_{n}$
$\displaystyle=\langle\operatorname{grad}\varphi,\varepsilon_{\kappa(n)}q_{n}\rangle_{\mathbb{R}^{3}}+\varphi\operatorname{div}\varepsilon_{\kappa(n)}q_{n}$
$\displaystyle=\langle\operatorname{grad}\varphi,\varepsilon_{\kappa(n)}q_{n}\rangle_{\mathbb{R}^{3}}+f;$

thus, bounded in $L_{2}(B)$. Also,

$\operatorname{\mathring{curl}}\varphi q_{n}=\operatorname{grad}\varphi\times
q_{n},$

forming a bounded sequence in $L_{2}(B)^{3}.$ Since $B$ particularly satisfies
the electric compactness property by the Picard–Weber–Weck selection theorem
(Theorem 3.5), the embeddings $L_{2}(B)\hookrightarrow H^{-1}(B)$ and
$\operatorname{ran}(\operatorname{\mathring{curl}},B)\hookrightarrow
H^{-1}(\operatorname{curl_{\textnormal{red}}},B)$ are compact. Hence, by
condition (ii) in Theorem 10.2, we obtain, using $q\coloneqq GU$,

$\varepsilon_{\kappa(n)}\varphi q_{n}\to\varepsilon\varphi
q=\varepsilon\varphi U$

weakly in $L_{2}(B)$. As both $B$ and $\varphi$ were arbitrary, we infer that

$\varepsilon_{\kappa(n)}q_{n}\to\varepsilon q=\varepsilon GU$

weakly in $L_{2}(\Omega)$. Weak closedness of $G^{*}$ implies

$(f,y)=G^{*}\varepsilon_{\kappa(n)}GU_{\kappa(n)}=G^{*}\varepsilon_{\kappa(n)}q_{n}\to
G^{*}\varepsilon GU,$

which determines $U$ uniquely. Hence, $U_{n}\to U$ weakly. Repeating the
argument replacing $\kappa(n)$ by $n$, we deduce that also
$\varepsilon_{n}GU_{n}\to\varepsilon GU$ weakly. Since $(f,y)$ are arbitrary,
we thus obtain both

$\varepsilon_{n,00}^{-1}\to\varepsilon_{00}^{-1}$

and

$\varepsilon_{n,10}\varepsilon_{n,00}^{-1}\to\varepsilon_{10}\varepsilon_{00}^{-1}$

in the respective weak operator topologies.

Next, we turn to the $\operatorname{curl}$-problem. For this, we let
$g\in\operatorname{ran}(\operatorname{\mathring{curl}},\Omega)$. Then we find
uniquely determined
$v_{n}\in\operatorname{dom}(\operatorname{curl_{\textnormal{red}}})$ such that

$\operatorname{curl_{\textnormal{red}}^{\diamond}}\varepsilon_{n}^{-1}\operatorname{curl_{\textnormal{red}}}v_{n}=g.$

Since $\varepsilon_{n}\in M(\alpha,\beta;\Omega)$, we deduce that
$(v_{n})_{n}$ forms a bounded sequence in
$\operatorname{dom}(\operatorname{curl_{\textnormal{red}}})$. Thus, there is a
weakly convergent subsequence
$(r_{n})_{n}\coloneqq(\varepsilon_{\kappa(n)}^{-1}\operatorname{curl_{\textnormal{red}}}v_{\kappa(n)})_{n}$
of $(\varepsilon_{n}^{-1}\operatorname{curl_{\textnormal{red}}}v_{n})_{n}$ in
$L_{2}(\Omega)^{3}$; denote by $r$ its limit. Again, let $B$ be an open ball
strictly contained in $\Omega$ and $\varphi$ a smooth function supported on
$B$. Then appealing to condition (ii) in Theorem 10.2 with $q_{n}=\varphi
r_{n}$, we compute

$\operatorname{div}\varepsilon_{\kappa(n)}\varphi
r_{n}=\langle\operatorname{grad}\varphi,\varepsilon_{\kappa(n)}r_{n}\rangle_{\mathbb{R}^{3}}$

and

$\operatorname{\mathring{curl}}\varphi r_{n}=\operatorname{grad}\varphi\times
r_{n}+g.$

Hence, both (a) and (b) in the just mentioned condition (ii) are met with the
help of Theorem 3.5. By local $H$-convergence on $B$, we deduce

$\varepsilon_{\kappa(n)}\varphi r_{n}\to\varepsilon\varphi r.$

weakly. Thus,

$\operatorname{curl_{\textnormal{red}}}v_{\kappa(n)}=\varepsilon_{\kappa(n)}r_{n}\to\varepsilon
r$

weakly. The closedness of
$\operatorname{ran}(\operatorname{curl_{\textnormal{red}}})$ yields the
existence of some
$v\in\operatorname{dom}(\operatorname{curl_{\textnormal{red}}})$ such that
$\varepsilon r=\operatorname{curl_{\textnormal{red}}}v$. Since, by the (weak)
closedness of $\operatorname{\mathring{curl}}$,

$g=\operatorname{\mathring{curl}}r_{n}\to\operatorname{\mathring{curl}}r=\operatorname{\mathring{curl}}\varepsilon^{-1}\operatorname{curl_{\textnormal{red}}}v,$

$v$ is uniquely determined, which, in turn, uniquely determines $r$. Hence,

$\varepsilon_{n}^{-1}\operatorname{curl_{\textnormal{red}}}v_{n}\to
r=\varepsilon^{-1}\operatorname{curl_{\textnormal{red}}}v$

weakly. Repeating the argument with $n$ instead of $\kappa(n)$, we also obtain

$\operatorname{curl_{\textnormal{red}}}v_{n}=\varepsilon_{n}\varepsilon_{n}^{-1}\operatorname{curl_{\textnormal{red}}}v_{n}\to\varepsilon
r=\varepsilon\varepsilon^{-1}\operatorname{curl_{\textnormal{red}}}v=\operatorname{curl_{\textnormal{red}}}v.$

weakly. Thus, $v_{n}\to v$ weakly in
$\operatorname{dom}(\operatorname{curl_{\textnormal{red}}})$ as
$\operatorname{curl_{\textnormal{red}}}$ is continuously invertible. Since $g$
was arbitrary, the weak convergence
$\varepsilon_{n}^{-1}\operatorname{curl_{\textnormal{red}}}v_{n}\to\varepsilon^{-1}\operatorname{curl_{\textnormal{red}}}v$
implies
$\varepsilon_{n,00}^{-1}\varepsilon_{n,01}\to\varepsilon_{00}^{-1}\varepsilon_{01}$
in the weak operator topology. The weak convergence $v_{n}\to v$ is the last
required convergence in Theorem 7.1 (ii). ∎

We conclude this section stating that the local $H$-convergence topology,
$\tau_{\textnormal{H}}$, on $M(\alpha,\beta;\Omega)$ thus coincides with the
induced topology of $\tau_{\textnormal{bH}}$ on $M(\alpha,\beta;\Omega)$.

######  Corollary 10.4.

We have

$(M(\alpha,\beta;\Omega),\tau_{\textnormal{H}})=(M(\alpha,\beta;\Omega),\tau_{\textnormal{bH}}).$

###### Proof.

The fundamental theorem of local $H$-convergence states that
$(M(\alpha,\beta;\Omega),\tau_{\textnormal{H}})$ is a compact metric space.
$(M(\alpha,\beta;\Omega),\tau_{\textnormal{bH}})$ is a Hausdorff topological
space. By Theorem 10.1,
$\operatorname{id}\colon(M(\alpha,\beta;\Omega),\tau_{\textnormal{H}})\to(M(\alpha,\beta;\Omega),\tau_{\textnormal{bH}})$
is continuous. Hence,
$(M(\alpha,\beta;\Omega),\tau_{\textnormal{H}})=(M(\alpha,\beta;\Omega),\tau_{\textnormal{bH}})$.
∎

###### Proof of Theorem C.

This is Corollary 10.4. ∎

##  11 Conclusion

In this article we have introduced a nonlocal $H$-convergence topology for
$\Omega$ not necessarily satisfying $\mathcal{H}_{D}(\Omega)=\\{0\\}$. For the
case of trivial $\mathcal{H}_{D}(\Omega)$, this topological space coincides
with the topology of nonlocal $H$-convergence introduced in [29]. Moreover,
for local coefficients, the newly introduced topology coincides with the
topology of local $H$-convergence as introduced by Murat and Tartar in
particular if $\dim(\mathcal{H}_{D}(\Omega))<\infty$.

We have analysed electrostatic equations with coefficients converging in the
newly introduced topology thus obtaining new, nonlocal homogenisation results.
These results furthermore led to new compactness results akin to the
Picard–Weber–Weck selection theorem with moving coefficients. This type of
compactness has potential impact on nonlinear static Maxwell problems, which
will be addressed in future publications. Also, the presented notion can be
generalised to different boundary conditions as well as other differential
operators using the notion of compact Hilbert complexes of linear operators.
This, too, will be subject of a future text on this topic.

## Acknowledgements

The author wishes to thank Paula Ostmann, Helga Ostmann, Neewa Nopsi and
Andreas Buchinger for useful discussions helping to significantly improve the
paper.

## References

  * [1] S. Bauer, D. Pauly, and M. Schomburg  The Maxwell Compactness Property in Bounded Weak Lipschitz Domains with Mixed Boundary Conditions,  SIAM Journal on Mathematical Analysis, 48(4), 2912–2943, 2016