# Homogenisation of Laminated Metamaterials
and the
Inner Spectrum

Marcus Waurick

###### Abstract

We study homogenisation problems for divergence form equations with rapidly
sign-changing coefficients. With a focus on problems with piecewise constant,
scalar coefficients in a ($d$-dimensional) crosswalk type shape, we will
provide a limit procedure in order to understand potentially ill-posed and
non-coercive settings. Depending on the integral mean of the coefficient and
its inverse, the limits can either satisfy the usual homogenisation formula
for stratified media, be entirely degenerate or be a non-local differential
operator of 4th order. In order to mark the drastic change of nature, we
introduce the ‘inner spectrum’ for conductivities. We show that even though
$0$ is contained in the inner spectrum for all strictly positive periods, the
limit inner spectrum can be empty. Furthermore, even though the spectrum was
confined in a bounded set uniformly for all strictly positive periods and not
containing $0$, the limit inner spectrum might have $0$ as an essential
spectral point and accumulate at $\infty$ or even be the whole of
$\mathbb{C}$. This is in stark contrast to the classical situation, where it
is possible to derive upper and lower bounds in terms of the values assumed by
the coefficients in the pre-asymptotics. In passing, we also develop a theory
for Sturm–Liouville type operators with indefinite weights, reduce the
question on solvability of the associated Sturm–Liouville operator to
understanding zeros of a certain explicit polynomial and show that generic
real perturbations of piecewise constant coefficients lead to continuously
invertible Sturm–Liouville expressions.

Keywords: metamaterials, sign-changing coefficients, divergence form
equations, laminated materials, Sturm–Liouville problems, indefinite weights,
$G$-convergence, holomorphic $G$-convergence, $T$-coercivity, inner spectrum,
spectrum

MSC 2020: Primary 35B27; Secondary 35J25; 35P05; 35Q61; 35R25; 35B20; 35B34;
47A52; 47A53; 74Q05

###### Contents

  1. 1 Introduction
  2. 2 The main theorems
  3. 3 Abstract divergence-form operators
  4. 4 The one-dimensional problem – well-posedness
  5. 5 Sturm–Liouville problems with indefinite coefficients
  6. 6 The higher-dimensional problem – preliminaries
  7. 7 The higher dimensional problem – well-posedness
  8. 8 The inner spectrum in $d$ dimensions
  9. 9 Classical $G$-convergence
  10. 10 Holomorphic $G$-convergence
  11. 11 The one-dimensional problem – homogenisation
  12. 12 The higher-dimensional problem – homogenisation
  13. 13 Proofs
    1. 13.1 Results in Section 5
    2. 13.2 Results in Section 6
    3. 13.3 Results in Section 7
    4. 13.4 Results in Section 12
  14. 14 Conclusion

## Acknowledgments

The author is very grateful for the hospitality extended to him by the Erwin-
Schrödinger Institute. The workshop on “Spectral Theory of Differential
Operators in Quantum Theory” in November 2022 led to a wonderful scientific
atmosphere with valuable discussions and talks that have helped to improve the
presentation of the material at hand. The authors thanks Jonathan Stanfill for
careful reading. The support of the GrK2583/1 during an early stage of this
research is gratefully acknowledged.

##  1 Introduction

The aim of the present article is to understand divergence form problems with
piecewise constant possibly sign-changing coefficients and to address the
limit of period tending to $0$ for divergence form equations with periodic
coefficients of the said type. In more applied contexts sign-changing
coefficients are used to describe so-called ‘metamaterials’ and the
mathematical challenge lies in the fact that the problem considered is non-
coercive by nature. There is an abundance of literature concerning real-world
applications of metamaterials, we exemplarily refer to [46, 36, 15].

The rigorous limit of period tending to $0$ is classically treated in the area
of mathematical homogenisation, see, e.g., [23, 66]. This theory aims at
understanding microscopically heterogenous materials (e.g., periodic with
‘small’ period) by finding a suitable homogeneous (i.e., constant coefficient)
replacement sharing a more or less similar behaviour on a macroscopic level.
The homogenisation problem (in the context of coercive coefficients) with
coefficients depending on one variable only has been studied under the
umbrella term of ‘stratified media’ or ‘laminated materials’. Treating
coefficients being constant on slabs, we shall thus analyse homogenisation
theory for periodic laminated metamaterials.

In order to obtain a flavour of the type of problems we encounter here, we
recall one of the arguably ‘easiest’ non-trivial examples (see also [30] for a
(possibly) more elementary situation), namely that of a divergence form
equation on an open rectangle symmetric about the $y$-axis: On
$\Omega=(-1,1)\times(0,1)$ consider

$-\operatorname{div}a\operatorname{grad}u=f\in L_{2}(\Omega)$

subject to $u|_{\partial\Omega}=0$, where the conductivity $a$ satisfies

$a(x,y)=\operatorname{sgn}(x)=\begin{cases}1,&x\geqslant 0,\\\
-1,&x<0.\end{cases}$

This problem has been thoroughly analysed by [5] and it was shown that the
associated operator-realisation with homogeneous Dirichlet boundary conditions
on $L_{2}$ is self-adjoint, but not invertible; it has $0$ as an eigenvalue of
infinite multiplicity; thus implying that the corresponding operator possesses
non-trivial essential spectrum and has no compact resolvent. The operator is
studied by adding $\mathrm{i}\eta$, $\eta\in\mathbb{R}$ small, to $-1$ on
$x<0$, thus making the perturbed operator coercive (after suitable
multiplication with a complex scalar) for all $\eta\neq 0$ and then consider
the limit $\eta\to 0$.

A detailed analysis of this problem with the techniques developed in this
manuscript shows that _any_ (sufficiently small) number added to both sides of
the coefficients leads to a well-posed albeit non-coercive problem, see
Example 8.4 below.

In a series of papers, the well-posedness of sign-changing coefficients for
divergence form problems (also involving Maxwell type equations) has been
addressed mainly by the French school, particularly by Bonnet-Ben Dhia and
Ciarlet jr., and co-authors, see, [8] and, e.g., [11, 10, 9]; see, however,
also [14]. In [8], the authors introduce the concept of
$\mathbb{T}$-coercivity, which may be interpreted as coercivity after
multiplication with an operator and is equivalent to well-posedness of the
problem considered. The main task is in constructing a suitable topological
isomorphism $\mathbb{T}$, which may or may not be easy to accomplish in
applications, such that the standard bilinear form weighted with $\mathbb{T}$
becomes coercive. For given $f\in L_{2}(\Omega)$, in [8], the authors studied
(among other things) problems of finding $u\in H_{0}^{1}(\Omega)$ such that

$\operatorname{div}\varepsilon^{-1}\operatorname{grad}u=f\in L_{2}(\Omega)$

for $\Omega\subseteq\mathbb{R}^{d}$ open, bounded, $d=2$ or $d=3$, with
$\varepsilon,\varepsilon^{-1}\in L_{\infty}(\Omega;\mathbb{R})$. Here, they
additionally assume that $\Omega$ has two open, mutually disjoint (connected)
subdomains $\Omega_{+},\Omega_{-}$ with Lipschitz boundaries satisfying
$\overline{\Omega}=\overline{\Omega}_{+}\cup\overline{\Omega}_{-}$. The
coefficient $\varepsilon$ respectively satisfies on $\Omega_{+}$,
$\varepsilon\geqslant c_{+}>0$ and, on $\Omega_{-}$, $\varepsilon\leqslant-
c_{-}<0$ for some $c_{\pm}>0$. For the case

$\frac{-\sup_{\Omega_{-}}\varepsilon}{\sup_{\Omega_{+}}\varepsilon}>\kappa^{2}$

$\mathbb{T}$-coercivity and, hence, well-posedness for the problem in question
has been shown in [8]; here $\kappa$ satisfies

$\|v\|_{1/2,+}\leqslant\kappa\|v\|_{1/2,-}\quad(v\in
H^{1/2}(\Sigma);\Sigma=\overline{\Omega}_{+}\cap\overline{\Omega}_{-})$

and $\|v\|_{1/2,\pm}$ denotes the norm of the trace $v=u|_{\Sigma}$ for $u\in
H^{1}(\Omega_{\pm})$. With the more particular situation of piecewise
constant, laminated materials, we are able to provide a more precise well-
posedness condition, which in turn is even a characterisation of well-
posedness. Moreoever, since we allow for several sign-changes, $\Omega_{-}$
and $\Omega_{+}$ are not connected anymore; thus, the results presented here
paint a more detailed picture of a particular situation studied in [8, 10] and
they do complement the mentioned results in that we provide results for
disconnected $\Omega_{-}$ and $\Omega_{+}$.

In [35], K. Pankrashkin discusses whether or not the sign-indefinite Laplacian
given by

$-\operatorname{div}h\operatorname{grad}u=f\in L_{2}(\Omega)$

subject to homogeneous Dirichlet boundary conditions for $u$ with
$h\colon\Omega\to\\{-\mu,1\\}$ for some $\mu>0$ admits self-adjoint
realisations. In the paper, similar to the approach in [5], the notion of
boundary triplets are used. Similar to the discussion we develop in the
present article (see Section 5), the author analyses the indefinite Laplacian
using operator-valued Sturm–Liouville problems by a separation of variables
ansatz. We also mention [21], where the case $\mu=1$ has been addressed in
higher-dimensions again using the theory of boundary triples.

By means of representation theory, the thesis [53] treats the case of sign-
indefinite coefficients. We particularly note [53, Theorem 8.2.2 and Corollary
8.4.11] for results in the flavour of the general Theorem 3.1.

To proceed with the present approach, we need a well-posedness theorem for
divergence form problems other than the classical Lax–Milgram lemma. In this
manuscript, we recall one that _characterises_ well-posedness in terms of
invertibility of a projected variant of the conductivity matrix, see [54] and
Theorem 3.1 below. The core observation underlying [54] is from [37] and its
main consequence can be summarised as follows. Let
$\Omega\subseteq\mathbb{R}^{d}$ be open and bounded. Then for $a\in
L(L_{2}(\Omega)^{d})$ the Dirichlet-problem of finding $u\in
H_{0}^{1}(\Omega)$ such that for given $f\in H^{-1}(\Omega)$ we have

$-\operatorname{div}a\operatorname{grad}u=f$

is _equivalent_ to

$0\in\rho(\iota_{0}^{*}a\iota_{0}),$

where $\iota_{0}\colon g_{0}(\Omega)\hookrightarrow L_{2}(\Omega)^{d}$,
$g_{0}(\Omega)\coloneqq\operatorname{grad}[H_{0}^{1}(\Omega)]$, is the
canonical embedding. This observation gives rise to the definition of
$\sigma_{g_{0}(\Omega)}(a)$, the inner spectrum of $a$ with respect to this
Dirichlet problem, given by

$\sigma_{g_{0}(\Omega)}(a)=\\{\lambda\in\mathbb{C};(\iota_{0}^{*}(a-\lambda)\iota_{0})^{-1}\in
L(g_{0}(\Omega))\\}(=\sigma(\iota_{0}^{*}a\iota_{0})).$

In order to determine the (essential) spectrum for Maxwell’s equations the
relevance of the inner spectrum has been shown in [1, 7]; some examples
complementing the present setting are given there as well. Here, we will
explore the inner spectrum for piecewise constant sign-changing coefficients,
that is, for metamaterials. In [53, Section 8.3] the inner spectrum for the
one-dimensional case has been thoroughly discussed; see in particular [53,
Theorem 8.3.1]. We provide the concise arguments for this particular case in
Section 4. For higher-dimensional cases and scalar coefficients $a$ admitting
two values with $\Omega_{+}=[a>0]$ and $\Omega_{-}=[a<0]$ not necessarily
connected, the inner spectrum of $a$ has been computed in [53, Theorem 8.4.8].
The result is presented in terms of the spectrum of a certain combination of
the Dirichlet-to-Neumann maps associated to finding harmonic functions on
$\Omega_{+}$ and $\Omega_{-}$ on $H^{1/2}(\Gamma)$, where $\Gamma$ denotes the
interface of $\Omega_{+}$ and $\Omega_{-}$.

Using a non-classical variant of the homogenisation process (allowing for the
coercivity constant not to be uniformly bounded below or above in the
homogenisation parameter), people were able to rigorously derive constitutive
relations associated to metamaterials, see, e.g., [12, 13, 15, 31]. Even
though the coercivity constants of the problems considered were allowed to
degenerate as $\varepsilon\to 0$, the $\varepsilon$-parameter problem is
always coercive. Here, we adopt a different point of view in that the problem
class we start out with is neither assumed to be coercive nor well-posed.
Thus, we study highly oscillatory (periodic, laminated) metamaterials from the
outset.

For the coercive case of periodic laminated materials explicit formulas for
the homogenised coefficients are due to Tartar and Murat and are well-
established, see [33] or [23, Theorem 5.12]. As a consequence, see also
Examples 2.6 and 2.9, when the inner spectrum is concerned, the following
formulas can be shown for periodic, real, coefficients $\alpha_{\\#}$
depending on one variable only and assuming the values
$\alpha_{0},\ldots,\alpha_{r}\in\mathbb{R}\setminus\\{0\\}$ on slabs of equal
width. Denote by $\mathfrak{m}(\alpha_{\\#})$ the integral mean over the
period interval, which for simplicity we assume here to be $(0,1)$. Then if
$a_{n}$ is the multiplication operator on $L_{2}(0,1)^{d}$ induced by
$\alpha_{n}(x_{1},\ldots,x_{d})\coloneqq\alpha_{\\#}(nx_{1})$, $a_{n}$
$G$-converges (see below for the definition) to

$a_{\infty}=\begin{cases}\frac{1}{\mathfrak{m}(\alpha_{\\#}^{-1})},&d=1,\\\
\operatorname{diag}(\frac{1}{\mathfrak{m}(\alpha_{\\#}^{-1})},\mathfrak{m}(\alpha_{\\#}),\ldots,\mathfrak{m}(\alpha_{\\#}))\in\mathbb{R}^{d\times
d},&d\geqslant 2,\end{cases}$

and we have

$\begin{cases}\sigma_{g_{0}(\Omega)}(a_{\infty})=\big{\\{}\frac{1}{\mathfrak{m}(\alpha_{\\#}^{-1})}\big{\\}},&d=1,\\\
\sigma_{g_{0}(\Omega)}(a_{\infty})\subseteq{\operatorname{conv}}\\{\frac{1}{\mathfrak{m}(\alpha_{\\#}^{-1})},\mathfrak{m}(\alpha_{\\#})\\},&d\geqslant
2.\end{cases}$

We emphasise that, classically, only properties of the spectra of the (self-
adjoint) divergence form problem are considered, see, e.g., [66], and to the
best of the author’s knowledge the inner spectrum has not been addressed so
far. In any case, the mentioned formulas reflect the rather weak convergence
of the coefficients; see also [50, Lemma 6.7] for a related statement.
However, as the values of $\alpha_{\\#}$ (for piecewise constant coefficients)
belong to the inner spectrum (Theorem 2.1), the inner spectrum of the
homogenised coefficients still belongs to a suitable convex hull of the inner
spectrum of $a_{1}$ (note
$\mathfrak{m}(\alpha_{\\#})\in\operatorname{conv}\\{\alpha_{1},\ldots,\alpha_{r}\\}\subseteq\sigma_{g_{0}(\Omega)}(a_{1})$
and
$\mathfrak{m}(\alpha_{\\#}^{-1})\in\operatorname{conv}\\{1/\alpha_{1},\ldots,1/\alpha_{r}\\}\subseteq
1/[\sigma_{g_{0}(\Omega)}(a_{1})]$). Given the lack of coercivity and well-
posedness of the problems considered here, the corresponding inclusion
formulas do not hold in general.

In the series of papers [18, 19, 20], the homogenisation of sign-changing
coefficients has been addressed with a combination of the
$\mathbb{T}$-coercivity approach and the method of periodic unfolding. Even
though the coefficients are changing sign on a complicated domain, the
coefficients only assume two different values. Moreover, the setting is
arranged in a way that both the pre-asymptotic regime together with the
homogenised limit are well-posed allowing for standard arguments in
homogenisation theory.

This contrast the present situation, for example, take $\alpha$ to be the
$1$-periodic extension of
$\mathbbold{1}_{(0,1/4)\cup[3/4,1)}-2\mathbbold{1}_{[1/4,3/4)}$ and consider
$a_{n}$ as before. Then we will show and make precise the following statement

$a_{n}\text{ `$G$-converges to' }-2\Delta_{(0,1)}(-\Delta_{(0,1)^{d-1}}^{-1})$

or, written in terms of the corresponding divergence form operators,

$-\operatorname{div}a_{n}\operatorname{grad}\text{ `converges to'
}-2\Delta_{(0,1)}^{2}(-\Delta_{(0,1)^{d-1}}^{-1})-2\Delta_{(0,1)}$

Thus, in a generalised sense, the limit of the highly oscillatory sign-
changing problem is a 4th order pseudo-differential operator, where the
differential operator on the left-hand side is subject to homogeneous
Dirichlet boundary conditions and considered on $L_{2}((0,1)^{d})$ and
$-\Delta_{\Omega}$ denotes the Dirichlet–Laplacian as the operator realisation
in $L_{2}(\Omega)$ for an open and bounded subset
$\Omega\subseteq\mathbb{R}^{d}$. Depending on the dimension and the values of
$\mathfrak{m}(\alpha_{\\#})$ and $\mathfrak{m}(\alpha_{\\#}^{-1})$ we will
show that even though there exists a bounded set $\Sigma$ such that
$\bigcup_{n\in\mathbb{N}}\sigma_{g_{0}(\Omega)}(a_{n})\subseteq\Sigma$, the
corresponding generalised $G$-limit, $a_{\infty}$, may either satisfy the
above inclusions valid in the classical case or one of the following
alternatives

$\emptyset=\sigma_{g_{0}(\Omega)}(a_{\infty})\text{ even though
}0\in\bigcap_{n\in\mathbb{N}}\sigma_{g_{0}(\Omega)}(a_{n})\quad(d=1)$

or

$\sigma_{g_{0}(\Omega)}(a_{\infty})=\mathbb{C}\text{ or
}\sigma_{g_{0}(\Omega)}(a_{\infty})=\overline{\\{\lambda_{k};k\in\mathbb{N}\\}}\quad(d\geqslant
2)$

for some (explicit) sequence $(\lambda_{k})_{k}$ in $\mathbb{R}_{>0}$
accumulating at $0$ and $\infty$.

For making the $G$-convergence statement rigorous, we develop a new
homogenisation concept, namely that of _holomorphic $G$-convergence_. This
convergence generalises classical $G$-convergence of symmetric-matrix-valued
coefficients developed and initially studied by Spagnolo, see [47, 48]. The
key ingredients for the method to work are standard facts from operator theory
and complex analysis, we refer, e.g., to [32, 2]. We note in passing that
tools from analytic perturbation theory are of great importance in the context
of (quantitative) homogenisation theory, see in particular the seminal series
of papers by Birman and Suslina, see, e.g., [6].

The idea of holomorphic $G$-convergence is as follows: Revisiting the above
example, for a dense set of right-hand sides $f\in L_{2}(\Omega)^{d}$
($\operatorname{dom}(\Delta_{d})$ will be sufficient), $n\in\mathbb{N}$ and
suitable $\lambda\in\mathbb{C}$, we consider $u_{n}(\lambda)\in
H_{0}^{1}(\Omega)$ the solution of

$-\operatorname{div}(a_{n}-\lambda)\operatorname{grad}u_{n}(\lambda)=f.$

Then, it can be shown that there exists $\omega\subseteq\mathbb{C}$ open with
$0\in\overline{\omega}$ such that $\langle
v,u_{n}(\cdot)\rangle_{L_{2}(\Omega)}$ converges for all $v\in L_{2}(\Omega)$
in the compact open topology of holomorphic functions on $\omega$ to some
$\langle v,u(\cdot)\rangle_{L_{2}(\Omega)}$, which extends holomorphically to
$0$. Scalar holomorphy and vector-valued holomorphy being the same by
Dunford’s Theorem, see, e.g., [2, Proposition A.3], this defines an element
$u(0)\in H_{0}^{1}(\Omega)$, which satisfies

$\big{(}-2\Delta_{(0,1)}^{2}(-\Delta_{(0,1)^{d-1}}^{-1})-2\Delta_{(0,1)}\big{)}u(0)=f.$

The details of the definition of holomorphic $G$-convergence are provided in
Sections 2 and 10.

Homogenisation in an operator theoretic set-up has been addressed in various
articles predominantly for the time-dependent situation focussing on so-called
evolutionary equations, see, e.g., [34, 63, 62, 60, 59, 58, 57]. Problems in
an (almost) time-independent situation were treated in [64, 65] or [22, 24],
where in the latter references also quantitative results were established. In
the mentioned references, the core ideas recycled here to understand
homogenisation processes are the well-posedness theorems from [54] and [38,
39] (see also [45, Theorem 6.2.1]) in conjunction with certain stability
properties of operator-valued holomorphic functions under weak limits, see,
e.g., [62] or the Montel type Theorem 10.5 below.

In the course of the manuscript, we also have occasion to study
Sturm–Liouville-type problems with sign-changing conductivities. In a slightly
different situation similar problems have been considered in [16, 17] (and the
references stated there). Focussing rather on the classical spectrum, in these
references however, the authors had as their key assumptions mere
differentiability (except for finitely many points) of the said conductivity
in order to allow for an analysis invoking an integrating factor. Moreover,
the boundary conditions were periodic and the derivative at some value needed
to be non-zero. This situation is entirely complementary to the equations we
consider in the present research in that we mostly consider piecewise constant
coefficients yielding vanishing of the classical derivative where ever it
exists.

We briefly comment on the structure of the article at hand. In Section 2 we
provide a concise presentation of the main results of the present manuscript.
Section 3 recalls the operator-theoretic perspective to divergence form
operators due to [54]. This section particularly contains a characterisation
result instrumental for the main results of the present article as it asserts
well-posedness of abstract second order divergence form equations
_irrespective_ of whether or not the corresponding conductivity is accretive,
thus, justifying the introduction of the inner spectrum. In Section 4 we apply
the well-posedness criterion to one-dimensional divergence form problems with
$L_{\infty}$-conductivity and show that the essential condition to solve this
type of equations is the inverse of the coefficient being bounded and this
inverse having non-vanishing integral mean. In this 1-d set-up, we also
compute the inner spectrum. A preparation for well-posedness results for sign-
changing coefficients of a corresponding higher-dimensional setting is
provided in Section 5 containing aspects of Sturm–Liouville theory.
Considering piecewise constant conductivities, only, we prove a result
characterising well-posedness involving _all_ values of the coefficients
differing from the criterion presented in [8]. We start to address the higher-
dimensional set-up in Section 6. Making use of the laminated structure of the
coefficients, we write the Laplacian with variable coefficients as sum of a
$1$-dimensional second order differential operator plus a $(d-1)$-dimensional
Laplacian with conductivity $1$ multiplied with a function not depending on
the $d-1$ variables contained in the derivatives of this Laplacian. Using the
spectral theorem for the $(d-1)$-dimensional operator, we are therefore in the
position to apply results from Section 5 developing a solution theory for the
higher-dimensional set-up in Section 7. Section 8 is devoted to describe the
inner spectrum in the higher-dimensional setting.

In Section 9 we recap the concept of $G$-convergence for (self-adjoint,
matrix-valued) coefficients and recall an operator-theoretic characterisation
for this convergence from [61]. Equipped with this knowledge, we proceed in
Section 10 to generalise $G$-convergence by introducing the concept of
_holomorphic $G$-convergence_. As a first application of this concept, we
provide a homogenisation theorem for general sign-changing coefficients in
Section 11. It turns out that the only requirement needed for a homogenisation
theorem is convergence in the $L_{\infty}(\Omega)$-weak-* topology of the
inverses of the conductivities. Holomorphic $G$-convergence requires mere
periodicity of the conductivity and the consideration of the usual limit of
period tending to $0$. The final application of the results and concepts
developed here is carried out in Section 12, where we consider the
homogenisation problem in the higher-dimensional case. The more lengthy proofs
are presented in Section 13. We conclude the article with some summarising
thoughts and future directions in Section 14.

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.

##  2 The main theorems

In this short section, we present the major results of the present article in
a rather concise form. For more details we refer to the subsequent sessions.
We focus on results concerning the inner spectrum and piecewise constant
coefficients. For the precise results and for Sturm–Liouville theory we refer
to the individual sections.

For $\alpha_{0},\ldots,\alpha_{r}\in\mathbb{C}$, we let $\alpha_{\\#}$ be the
$1$-periodic extension of
$\sum_{j=0}^{r}\alpha_{j}\mathbbold{1}_{(jh,(j+1)h]}$ on the whole of
$\mathbb{R}$. For all $d\geqslant 1$ and ${\Omega}\subseteq\mathbb{R}^{d}$
open, the multiplication operator on $L_{2}(\Omega)^{d}$ induced by
$(x_{1},\ldots,x_{d})\mapsto\alpha_{\\#}(x_{1})$ is denoted by $\alpha({\rm
m}_{1})$ (if $d=1$, we just write $\alpha({\rm m})$); correspondingly we will
write $\alpha(n{\rm m}_{1})$ for the respective multiplication operator
induced by $(x_{1},\ldots,x_{d})\mapsto\alpha_{\\#}(nx_{1})$. For a Lebesgue
integrable function $\beta$ defined on $(0,1)$, we denote by
$\mathfrak{m}(\beta)\coloneqq\int_{(0,1)}\beta$ its integral mean.

In the one-dimensional situation, the inner spectrum can be properly
understood; for this we recall one of the results of [53] for convenience.
Later we provide the proof of these in order to have a sample application of
Theorem 3.1.

######  Theorem 2.1 (see Theorem 4.3, see [53, Theorem 8.3.1] or [52]).

Let $\alpha_{0},\ldots,\alpha_{r}\in\mathbb{C}$. Then for
$V_{1}=\\{1\\}^{\bot}\subseteq L_{2}(0,1)$ we have

$\sigma_{V_{1}}(\alpha({\rm
m}))=\\{\lambda\in\mathbb{C};(\alpha_{\\#}-\lambda)^{-1}\in
L_{\infty},\mathfrak{m}((\alpha_{\\#}-\lambda)^{-1})=0\\}\cup\\{\alpha_{j};j\in\\{0,\ldots,r\\}\\}.$

With significantly more work, we then derive a condition involving checking
for zeros of a certain explicit polynomial that characterises well-posedness
of the higher-dimensional situation. To keep the presentation concise we just
mention an idea of the sort of theorems we derive.

######  Theorem 2.2 (see Corollary 7.6, Theorem 6.5, and Corollary 7.2).

Let $\alpha_{0},\ldots,\alpha_{r}\in\mathbb{R}\setminus\\{0\\}$,
$\widehat{\Omega}\subseteq\mathbb{R}^{d-1}$ open, bounded; put
$\Omega\coloneqq(0,1)\times\widehat{\Omega}$ and assume that
$\mathfrak{m}(\alpha_{\\#}^{-1})\neq 0$. If for all $t\in(0,1]$

$p_{\alpha}(t)=\begin{pmatrix}1&\alpha_{r}^{-1}t\end{pmatrix}\begin{pmatrix}1&\alpha_{r-1}^{-1}t\\\
\alpha_{r-1}t&1\end{pmatrix}\cdots\begin{pmatrix}1&\alpha_{1}^{-1}t\\\
\alpha_{1}t&1\end{pmatrix}\begin{pmatrix}t\\\ \alpha_{0}\end{pmatrix}\neq 0,$

then $-\Delta_{\alpha}\coloneqq-\overline{\operatorname{div}\alpha({\rm
m}_{1})\operatorname{grad}}$, where

$-\operatorname{div}\alpha({\rm m}_{1})\operatorname{grad}\colon\\{u\in
H_{0}^{1}(\Omega);\alpha({\rm m}_{1})\operatorname{grad}u\in
H(\operatorname{div},\Omega)\\}\subseteq L_{2}(\Omega)\to L_{2}(\Omega),$

is continuously invertible.

A more detailed account of the understanding highlighted in the previous
theorem helps us to deduce the following result on the inner spectrum for
divergence form problems with sign-changing coefficients.

We introduce for $\Omega\subseteq\mathbb{R}^{d}$ open and bounded, the set

$g_{0}(\Omega)\coloneqq\operatorname{grad}[H_{0}^{1}(\Omega)]\text{ and
}\iota_{0}\colon g_{0}(\Omega)\hookrightarrow L_{2}(\Omega)^{d}.$ (1)

Then, by the standard Poincaré inequality, we have $g_{0}(\Omega)\subseteq
L_{2}(\Omega)^{d}$ is a closed subspace and, thus, $\iota_{0}^{*}$ is well-
defined; $\iota_{0}^{*}q=q_{0}\in g_{0}(\Omega)$ where $q=q_{0}+q_{1}$ for
some $q_{1}\in g_{0}(\Omega)^{\bot}$.

######  Theorem 2.3 (see Theorem 8.2).

Let $\alpha_{0},\ldots,\alpha_{r}\in\mathbb{R}\setminus\\{0\\}$ and
$\widehat{\Omega}\subseteq\mathbb{R}^{d-1}$ open and bounded;
$\Omega\coloneqq(0,1)\times\widehat{\Omega}$, $A>\max_{j}|\alpha_{j}|$. Then,
there exists a countable nowhere dense set $\Sigma\subseteq[-A,A]$ such that

$\sigma_{g_{0}(\Omega)}(\alpha({\rm
m}_{1}))\setminus\sigma_{g_{0}(\Omega),\textnormal{c}}(\alpha({\rm
m}_{1}))=\Sigma\cup\\{\alpha_{j};j\in\\{0,\ldots,r\\}\\}.$

where $\sigma_{g_{0}(\Omega),\textnormal{c}}(\alpha({\rm m}_{1}))$ is the
inner continuous spectrum of $\alpha({\rm m}_{1})$; we have
$\sigma_{g_{0}(\Omega),\textnormal{c}}(\alpha({\rm m}_{1}))\subseteq[-A,A]$.

Equipped with the understanding of when sign-changing coefficients lead to
well-posed divergence form operators, we can now proceed to the understanding
of homogenisation of potentially ill-posed problems and present the respective
results. For this we introduce a new notion of convergence related to
homogenisation.

Let $(\tilde{a}_{n})_{n}$ in $L_{\infty}(\Omega)^{d\times d}$, identified with
a sequence in $L(L_{2}(\Omega)^{d})$. Define
$a_{n}\coloneqq\iota_{0}^{*}\tilde{a}_{n}\iota_{0}$ and assume there exists
$\omega\subseteq\bigcap_{n\in\mathbb{N}}\rho(a_{n})$ open in $\mathbb{C}$ with
$0\in\overline{\omega}$ such that $(a_{n}-\cdot)^{-1}$ is locally bounded on
$\omega$. Let $a\subseteq g_{0}(\Omega)\times g_{0}(\Omega)$ be a relation.
Then we say that $(a_{n})_{n}$ holomorphically $G$-converges (on $\omega$) to
$a$, $a_{n}\stackrel{{\scriptstyle\textnormal{hol}-G}}{{\to}}a$, if the set
$\mathcal{G}$ of $\varphi\in g_{0}(\Omega)$ satisfying the following two
conditions (a) and (b) is dense, where

  1. (a)

there exists $f_{\varphi}\colon\omega\to g_{0}(\Omega)$ holomorphic, such
that, for all $\psi\in g_{0}(\Omega)$,
$\langle\psi,(a_{n}-\cdot)^{-1}\varphi\rangle\to\langle\psi,f_{\varphi}(\cdot)\rangle$
in the compact open topology (i.e., uniform convergence on compact sets),

  2. (b)

$f_{\varphi}$ admits a holomorphic extension to $0$;

and

$a^{-1}=\\{(\varphi,\psi)\in\mathcal{G}\times
g_{0}(\Omega);\psi=f_{\varphi}(0)\\}.$

$(\tilde{a}_{n})_{n}$ in $L_{\infty}(\Omega)^{d\times d}$ holomorphically
$G$-converges (on $\omega$) to $\tilde{a}\subseteq L_{2}(\Omega)^{d}\times
L_{2}(\Omega)^{d}$,
$\tilde{a}_{n}\stackrel{{\scriptstyle\textnormal{hol}-G}}{{\to}}\tilde{a}$, if
$(a_{n})_{n}$ holomorphically $G$-converges on $\omega$ to $a\subseteq
g_{0}(\Omega)\times g_{0}(\Omega)$ and
$a^{-1}=(\iota_{0}^{*}\tilde{a}\iota_{0})^{-1}$.

A comparison of classical $G$-convergence111Classically $G$-convergence is
formulated for strictly positive definite coefficients, only; we use the same
notion for the non-coercive situation, see Section 9. and holomorphic
$G$-convergence reads as follows.

######  Theorem 2.4.

Let $(\tilde{a}_{n})_{n}$ in $L_{\infty}(\Omega)^{d\times d}$, $\tilde{a}\in
L_{\infty}(\Omega)^{d\times d}$;
$a_{n}\coloneqq\iota_{0}^{*}\tilde{a}_{n}\iota_{0}$,
$a\coloneqq\iota_{0}^{*}\tilde{a}\iota_{0}$. Assume
$0\in\bigcap_{n\in\mathbb{N}}\rho(a_{n})\cap\rho(a)$ with $(a_{n}^{-1})_{n}$
uniformly bounded. Consider the following assertions:

  1. (i)

$(a_{n})_{n}$ holomorphically $G$-converges to $a$;

  2. (ii)

every subsequence $(a_{\pi(n)})_{n}$ contains a subsequence holomorphically
$G$-converging to $a$;

  3. (iii)

$\tilde{a}_{n}$ $G$-converges to $\tilde{a}$, i.e., for all $f\in
H^{-1}(\Omega)$ and $u_{n}\in H_{0}^{1}(\Omega)$ such that
$-\operatorname{div}\tilde{a}_{n}\operatorname{grad}u_{n}=f$ we have

$u_{n}\rightharpoonup u\in H_{0}^{1}(\Omega)\text{ and
}-\operatorname{div}\tilde{a}\operatorname{grad}u=f$

Then (i)$\Rightarrow$(iii)$\Leftrightarrow$(ii).

We are now in the position to present our main homogenisation theorems on
(holomorphic) $G$-convergence for sign-changing coefficients. We start off
with a result the formulation of which does not require the notion of
holomorphic $G$-convergence.

######  Theorem 2.5 (see Theorem 11.1).

Let $(\alpha_{n})_{n}$ be a sequence in $L_{\infty}(0,1)$ such that
$(\alpha_{n}^{-1})_{n}$ converges in $\sigma(L_{\infty},L_{1})$ to some
$\beta\in L_{\infty}(0,1)$ with $\mathfrak{m}(\beta)\neq 0$. If
$\alpha_{\infty}\coloneqq\beta^{-1}\in L_{\infty}(0,1)$ then, for the
associated multiplication operators on $L_{2}(0,1)$,

$\alpha_{n}({\rm m})\stackrel{{\scriptstyle G}}{{\to}}\alpha_{\infty}({\rm
m})$

as $n\to\infty$.

######  Example 2.6.

Let $\alpha_{0},\ldots,\alpha_{r}\in\mathbb{R}_{>0}$. Then, by Theorem 2.1,

$\\{\alpha_{0},\ldots,\alpha_{r}\\}\subseteq\sigma_{g_{0}(0,1)}(\alpha(n{\rm
m})).$

By Theorem 2.5 (and Theorem 10.1), we obtain

$\alpha(n{\rm
m})\stackrel{{\scriptstyle\textnormal{$G$}}}{{\to}}\Big{(}\frac{1}{r+1}\big{(}\frac{1}{\alpha_{0}}+\cdots+\frac{1}{\alpha_{r}}\big{)}\Big{)}^{-1}=\mathfrak{m}(\alpha_{\\#}^{-1})^{-1}\eqqcolon\alpha_{\infty}.$

Thus, for all $n\in\mathbb{N}$,

$\emptyset\neq\big{(}1/[\sigma_{g_{0}(0,1)}(\alpha_{\infty})]\big{)}=\\{\mathfrak{m}(\alpha_{\\#}^{-1})\\}\subseteq\operatorname{conv}\big{(}1/[\sigma_{g_{0}(0,1)}(\alpha(n{\rm
m}))]\big{)},$

where $\operatorname{conv}$ denotes the convex hull.

The degenerate case reads as follows:

######  Theorem 2.7 (see Theorem 11.3).

Let $\alpha\in L_{\infty}(\mathbb{R};\mathbb{R})$ be $1$-periodic with
$\alpha^{-1}\in L_{\infty}(\mathbb{R})$ and assume
$\mathfrak{m}(\alpha^{-1})=0$; define $\alpha_{n}\coloneqq\alpha(n\cdot)$.
Then

$\alpha_{n}({\rm
m})\stackrel{{\scriptstyle\textnormal{hol-$G$}}}{{\to}}\alpha_{\infty}\coloneqq\\{0\\}\times
L_{2}(\Omega).$

In particular,

$\sigma_{g_{0}(\Omega)}(\alpha_{\infty})=\emptyset.$

Hence, note that by Theorem 2.1, $0\in\sigma_{g_{0}(\Omega)}(\alpha_{n}({\rm
m}))$ for all $n\in\mathbb{N}$; but still the limit has empty spectrum.

The higher-dimensional situation is more involved and provides – in the
degenerate case – a more complex picture concerning the limit behaviour of the
coefficients. Using some known results from homogenisation theory and the
following representation we can provide the behaviour of the inner spectrum in
the classical situation.

######  Lemma 2.8 (see Lemma 12.2).

Let $\Omega=(0,1)^{d}$ and $\gamma>0$; denote
$\Gamma\coloneqq\operatorname{diag}(\gamma,1,\ldots,1)$. Then

$\sigma_{g_{0}(\Omega)}(\Gamma)=\overline{\\{\Big{(}\frac{\gamma
k_{1}^{2}+\sum_{m=2}^{d}k_{m}^{2}}{\sum_{m=1}^{d}k_{m}^{2}}\Big{)};k\in\mathbb{N}_{>0}^{d}\\}}.$

######  Example 2.9.

Let $\alpha_{0},\ldots,\alpha_{r}\in\mathbb{R}_{>0}$. Then, classically (see
[33] or Theorem 12.1 below),

$\displaystyle\alpha(n{\rm m}_{1})$
$\displaystyle\stackrel{{\scriptstyle\textrm{$G$}}}{{\to}}\operatorname{diag}(1/(\mathfrak{m}(\alpha_{\\#}^{-1})),\mathfrak{m}(\alpha_{\\#}),\ldots,\mathfrak{m}(\alpha_{\\#}))$
$\displaystyle=\mathfrak{m}(\alpha_{\\#})\operatorname{diag}(\frac{1}{\mathfrak{m}(\alpha_{\\#}^{-1})\mathfrak{m}(\alpha_{\\#})},1,\ldots,1)\eqqcolon\alpha_{\infty}.$

Hence, by Lemma 2.8, we infer

$\displaystyle\sigma_{g_{0}(\Omega)}(\alpha_{\infty})$
$\displaystyle=\mathfrak{m}(\alpha_{\\#})\overline{\\{\Big{(}\frac{\frac{1}{\mathfrak{m}(\alpha_{\\#}^{-1})\mathfrak{m}(\alpha_{\\#})}k_{1}^{2}+\sum_{m=2}^{d}k_{m}^{2}}{\sum_{m=1}^{d}k_{m}^{2}}\Big{)};k\in\mathbb{N}_{>0}^{d}\\}}$
$\displaystyle\subseteq{\operatorname{conv}}\\{\frac{1}{\mathfrak{m}(\alpha_{\\#}^{-1})},\mathfrak{m}(\alpha_{\\#})\\}$

where both $\mathfrak{m}(\alpha_{\\#}^{-1})$ and $\mathfrak{m}(\alpha_{\\#})$
can be determined from $\sigma_{g_{0}(\Omega)}(\alpha(n{\rm
m}_{1}))\supseteq\\{\alpha_{0},\ldots,\alpha_{r}\\}$, for the last inclusion
see Theorem 2.3.

######  Theorem 2.10 (see Theorem 12.4 and Corollary 12.5).

Let $\alpha_{0},\ldots,\alpha_{r}\in\mathbb{R}\setminus\\{0\\}$. Then the
following statements hold

  1. (a)

If $\mathfrak{m}(\alpha_{\\#}^{-1})=0$, then there is
$\alpha_{\infty}^{-1}\subseteq L_{2}(\Omega)^{d}\times\\{0\\}$ densely (but
not everywhere) defined, such that

$\alpha(n{\rm
m}_{1})\stackrel{{\scriptstyle\textnormal{hol-}G}}{{\longrightarrow}}\alpha_{\infty}.$

We have $\sigma_{g_{0}(\Omega)}(a_{\infty})=\mathbb{C}$

  2. (b)

If $\mathfrak{m}(\alpha_{\\#})=0$ and $\mathfrak{m}(\alpha_{\\#}^{-1})\neq 0$,
then

$\alpha(n{\rm
m}_{1})\stackrel{{\scriptstyle\textnormal{hol-}G}}{{\longrightarrow}}-\frac{1}{\mathfrak{m}(\alpha_{\\#}^{-1})}\Delta_{(0,1)}(-\Delta_{(0,1)^{(d-1)}})^{-1},$

where $\Delta_{(0,1)}$ denotes the Dirichlet–Laplace operator on $L_{2}(0,1)$
tensorised with $d-1$ copies of the identity of $L_{2}(0,1)$; similarly
$\Delta_{(0,1)^{d-1}}$ is the identity on $L_{2}(0,1)$ tensorised with the
Dirichlet–Laplacian on $L_{2}((0,1)^{d-1})$. The limit inner spectrum is

$\frac{1}{\mathfrak{m}(\alpha_{\\#}^{-1})}\overline{\\{\frac{k_{1}^{2}}{k_{2}^{2}+\ldots+k_{d}^{2}};k_{1},\ldots,k_{d}\in\mathbb{N}_{>0}\\}}.$

  3. (c)

If $\mathfrak{m}(\alpha_{\\#})\mathfrak{m}(\alpha_{\\#}^{-1})\neq 0$, then

$\alpha(n{\rm
m}_{1})\stackrel{{\scriptstyle\textnormal{hol-}G}}{{\longrightarrow}}\operatorname{diag}(\mathfrak{m}(\alpha_{\\#}^{-1})^{-1},\mathfrak{m}(\alpha_{\\#}),\ldots,\mathfrak{m}(\alpha_{\\#}))$

If, in addition,
$\mathfrak{m}(\alpha_{\\#})\mathfrak{m}(\alpha_{\\#}^{-1})>0$, the limit inner
spectrum is

$\overline{\\{\frac{\mathfrak{m}(\alpha_{\\#}^{-1})\sum_{m=1}^{d}k_{m}^{2}}{k_{1}^{2}+\mathfrak{m}(\alpha_{\\#}^{-1})\mathfrak{m}(\alpha_{\\#})\sum_{m=2}^{d}k_{m}^{2}};k_{1},\ldots,k_{m}\in\mathbb{N}_{>0}\\}}.$

In the following we embark on the quest to prove the above statements. The
starting point for all of the developed theory is the well-posedness statement
from [54], which is dealt with next.

##  3 Abstract divergence-form operators

The theory underlying the solution method for the differential equation
discussed here is the identification of divergence form problems as a
composition of three invertible mappings, see [54]. The main difference to the
traditional Lax–Milgram lemma approach is that the main result in [54]
_characterises_ well-posedness instead of providing a criterion for it. We
refer to [41] for a more detailed discussion of the Lax–Milgram lemma in
connection to accretive/coercive operators/relations. What is more, the
results in [54] provide a more explicit solution formula than the classical
approach. This solution formula will prove instrumental for our approach
proving our main results.

For a densely defined closed linear operator
$C\colon\operatorname{dom}(C)\subseteq H_{0}\to H_{1}$ acting from the Hilbert
spaces $H_{0}$ into $H_{1}$, we set
$H^{1}(C)\coloneqq(\operatorname{dom}(C),\langle\cdot,\cdot\rangle_{C})$ with
$\langle\cdot,\cdot\rangle_{C}$ being the graph scalar product. We define
$C^{\diamond}\colon H_{1}\to H^{-1}(C)\coloneqq(H^{1}(C))^{*}$ via

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

It is then not difficult to see that $C^{\diamond}$ extends $C^{*}$, where we
identify $H_{1}$ with its dual via the unitary Riesz mapping; for details we
refer to [45, Chapter 9].

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

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

the canonical embedding. Note that then $\iota_{V}^{*}\colon H\rightarrow V$
acts as the orthogonal projection onto $V$ and
$\pi_{V}\coloneqq\iota_{V}\iota_{V}^{*}$ is the actual orthogonal projection.

Now, throughout the rest of this section, let $H_{0}$, $H_{1}$ be Hilbert
spaces. We assume that $C\colon\operatorname{dom}(C)\subseteq H_{0}\to H_{1}$
is a closed, densely defined linear operator with closed range

$V_{1}\coloneqq\operatorname{ran}(C)\subseteq H_{1};$

the closed range theorem yields that

$V_{0}\coloneqq\operatorname{ran}(C^{*})\subseteq H_{0}$

is closed as well, see, e.g., [54, Corollary 2.5]. In this case, we define,
$C_{{\rm red}}\coloneqq\iota_{V_{1}}^{*}C\iota_{V_{0}}$, the reduced operator.
For $a\in L(H_{1})$ we also define the operator

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

with domain

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

The replacement for the Lax–Milgram lemma reads as follows.

######  Theorem 3.1 ([54, Theorem 3.1]).

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

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

  3. (iii)

the operator $\iota_{V_{1}}^{*}a\iota_{V_{1}}\in L(V_{1})$ is continuously
invertible.

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

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

and, if $f\in\operatorname{ran}(C_{{\rm red}}^{*})$,

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

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

Motivated by the observation that the abstract divergence form operator is
invertible if, and only if, $\iota_{V_{1}}^{*}a\iota_{V_{1}}$ is, we define
the inner spectrum of $a$ (with respect to $V_{1}$) via

$\sigma_{V_{1}}(a)\coloneqq\\{\lambda\in\mathbb{C};(\iota_{V_{1}}^{*}a\iota_{V_{1}}-\lambda\operatorname{id}_{V_{1}})^{-1}\in
L(V_{1})\\}=\sigma(\iota_{V_{1}}^{*}a\iota_{V_{1}}).$

The inner resolvent set of $a$ (with respect to $V_{1}$) is defined by

$\rho_{V_{1}}(a)\coloneqq\mathbb{C}\setminus\sigma_{V_{1}}(a)$

######  Remark 3.2.

The sets just introduced depend on $V_{1}$. In the following we only use
$V_{1}$ being the range of the gradient restricted to $H_{0}^{1}(\Omega)$
functions; that is,
$V_{1}=g_{0}(\Omega)=\operatorname{grad}[H_{0}^{1}(\Omega)]$.

Introducing the spectral parts inner point spectrum,
$\sigma_{V_{1},\textnormal{p}}(a)$, inner continuous spectrum,
$\sigma_{V_{1},\textnormal{c}}(a)$, and inner residual spectrum,
$\sigma_{V_{1},\textnormal{r}}(a)$, respectively given by

$\displaystyle\sigma_{V_{1},\textnormal{p}}(a)$
$\displaystyle=\\{\lambda\in\sigma_{V_{1}}(a);\iota_{V_{1}}^{*}a\iota_{V_{1}}-\lambda\text{
not one-to-one}\\}$ $\displaystyle\sigma_{V_{1},\textnormal{c}}(a)$
$\displaystyle=\\{\lambda\in\sigma_{V_{1}}(a);\iota_{V_{1}}^{*}a\iota_{V_{1}}-\lambda\text{
one-to-
one},\operatorname{ran}(\iota_{V_{1}}^{*}(a-\lambda)\iota_{V_{1}})\subseteq
V_{1}\text{ dense}\\}$ $\displaystyle\sigma_{V_{1},\textnormal{r}}(a)$
$\displaystyle=\\{\lambda\in\sigma_{V_{1}}(a);\iota_{V_{1}}^{*}a\iota_{V_{1}}-\lambda\text{
one-to-
one},\operatorname{ran}(\iota_{V_{1}}^{*}(a-\lambda)\iota_{V_{1}})\subseteq
V_{1}\text{ not dense}\\},$

we can use the operator $D_{{\rm red},a}$ in order to understand parts of the
inner spectrum. We use similar notation for the usual spectral parts.

######  Lemma 3.3.

Let $\lambda\in\mathbb{C}$. Then

$0\in\sigma_{\textnormal{p}}(D_{{\rm
red},a-\lambda})\iff\lambda\in\sigma_{V_{1},\textnormal{p}}(a).$

###### Proof.

If $0\notin\sigma_{\textnormal{p}}(D_{{\rm red},a})$, then

$C_{{\rm red}}^{*}\iota_{V_{1}}^{*}a\iota_{V_{1}}C_{{\rm red}}u=0;$

has only the trivial solution. Since $\operatorname{ker}(C_{{\rm
red}}^{*})=\operatorname{ker}(C_{{\rm red}}^{\diamond})$ (see, e.g., [45,
Proposition 9.2.2]), the equation

$C_{{\rm red}}^{\diamond}\iota_{V_{1}}^{*}a\iota_{V_{1}}C_{{\rm red}}u=0;$

has only the trivial solution either. As both $C_{{\rm red}}$ and $C_{{\rm
red}}^{\diamond}$ are topological isomorphisms (see [54]),
$\iota_{V_{1}}^{*}a\iota_{V_{1}}$ is one-to-one. Reverting the argument, we
infer that $\iota_{V_{1}}^{*}a\iota_{V_{1}}$ one-to-one implies
$0\neq\sigma_{\textnormal{p}}(D_{{\rm red},a})$. The assertion follows from

$0\in\sigma_{\textnormal{p}}(D_{{\rm red},a-\lambda})\iff
0\in\sigma_{V_{1},\textnormal{p}}(a-\lambda)\iff\lambda\in\sigma_{V_{1},\textnormal{p}}(a).\qed$

######  Theorem 3.4.

If $0\in\rho(D_{{\rm red},a})$, then
$0\in\rho_{V_{1}}(a)\cup\sigma_{V_{1},\textnormal{c}}(a)$. Moreover,

$\rho_{V_{1}}(a)\subseteq\\{\lambda\in\mathbb{C};0\in\rho(D_{{\rm
red},\alpha-\lambda})\\}\subseteq\rho_{V_{1}}(a)\cup\sigma_{V_{1},\textnormal{c}}(a),$

or, equivalently,

$\sigma_{V_{1},\textnormal{r}}(a)\cup\sigma_{V_{1},\textnormal{p}}(a)\subseteq\\{\lambda\in\mathbb{C};0\in\sigma(D_{{\rm
red},\alpha-\lambda})\\}\subseteq\sigma_{V_{1}}(a).$

###### Proof.

Assume $0\in\rho(D_{{\rm red},a})$. By Lemma 3.3,
$0\notin\sigma_{V_{1},\textnormal{p}}(a)$. Furthermore, if
$f\in\operatorname{ran}(C_{{\rm red}}^{*})=V_{0}$, then $D_{{\rm
red},a}^{-1}f\eqqcolon u$ is a solution of

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

In particular,

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

which shows that $\operatorname{dom}(C^{*})\cap
V_{1}=\operatorname{ran}((C_{{\rm
red}}^{*})^{-1})\subseteq\operatorname{ran}(\iota_{V_{1}}^{*}a\iota_{V_{1}})$
is dense in $V_{1}$ (see, e.g., [51, Lemma 4.4]). Thus,
$0\in\rho_{V_{1}}(a)\cup\sigma_{V_{1},\textnormal{c}}(a)$, as required. In
particular, the second inclusion is thus proved. The first one is a
consequence of Theorem 3.1. The remaining line of inclusions follows from the
previous one by computing complements. ∎

######  Example 3.5.

The inclusion $\\{\lambda\in\mathbb{C};0\in\rho(D_{{\rm
red},\alpha-\lambda})\\}\subseteq\rho_{V_{1}}(a)\cup\sigma_{V_{1},\textnormal{c}}(a)$
is optimal in the sense that $\sigma_{V_{1},\textnormal{c}}(a)$ cannot be
dropped. Indeed, take a Hilbert space $H$ and $a\in L(H)$ with
$0\in\sigma_{\textnormal{c}}(a)$. There exists
$T\colon\operatorname{ran}(a)\subseteq H\to H$ closed, onto and one-to-one. In
particular, $\operatorname{ran}(T)=H$ is closed and, thus, so is
$\operatorname{ran}(T^{*})\subseteq H$. Moreover, $T$ one-to-one leads to
$T^{*}$ mapping onto $H$. Consider $\mathcal{T}_{a}\coloneqq TaT^{*}$. Then
for all $f\in H_{0}$, $u\coloneqq(T^{*})^{-1}a^{-1}T^{-1}f$ is a solution of
$\mathcal{T}_{a}u=f$. Moreover, $T^{-1}$ is continuous and, as an inverse of a
bounded linear operator, $a^{-1}$ is closed. Thus, $a^{-1}T^{-1}$ is closed,
and, as it maps from $H$ to $H$, it is continuous by the closed graph theorem.
Finally, as $(T^{*})^{-1}=(T^{-1})^{*}$ is, too, continuous,
$\mathcal{T}_{a}^{-1}\in L(H)$. Hence, $0\in\rho(\mathcal{T}_{a})$. However,
$0\in\sigma_{\textnormal{c}}(a)=\sigma_{H,\textnormal{c}}(a)=\sigma_{\operatorname{ran}(T^{*}),\textnormal{c}}(a)$.

######  Remark 3.6.

(a) We recall that the standard assumption for the coefficients $a\in
L(H_{1})$ for weak variational problems of the type discussed in (i) is that
$\operatorname{Re}a\geqslant c$ for some $c>0$ in the sense of positive
definiteness. A moment’s reflection reveals that this condition yields
$\operatorname{Re}\iota_{V_{1}}^{*}a\iota_{V_{1}}\geqslant
c\operatorname{id}_{V_{1}}$ and, by the boundedness of
$\iota_{V_{1}}^{*}a\iota_{V_{1}}$, the invertibility of
$\iota_{V_{1}}^{*}a\iota_{V_{1}}$ follows; see, e.g., [45, Proposition
6.2.3(b)]. In fact, the same holds true for $a$ being a maximal monotone
relation with $\operatorname{dom}(a)=H_{1}$. The corresponding argument is,
however, somewhat more involved, see [44, Theorem 4]. This has been exploited
in [54] to provide a solution theory for nonlinear divergence form problems.

(b) Note that if $C$ is one-to-one (as it will always be the case in the
applications to follow), $D_{{\rm red},a}=D_{a}$, where
$D_{a}\colon\operatorname{dom}(D_{a})\subseteq H_{0}\to H_{0},u\mapsto
C^{*}aCu,$ and

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

where we used that $\operatorname{dom}(C)=\operatorname{dom}(C_{{\rm red}})$
as $C$ is one-to-one and that $aCu\in\operatorname{dom}(C^{*})$ if and only if
$aCu\in\operatorname{dom}(C_{{\rm red}}^{*})$. Also, by the injectivity of $C$
(together with the closed range of $C$), it follows that $C^{*}$ is onto;
i.e., $\operatorname{ran}(C^{*})=H_{0}$.

##  4 The one-dimensional problem – well-posedness

Having discussed the underlying solution method, we may consider the one-
dimensional case of sign-changing coefficients. In fact, we shall see that
this case is rather elementary to deal with and can be treated quite elegantly
with the results from [54]. Note that the results in the present section have
been obtained also in [53, Chapter 8]. Note that in [53] it suffices to have
$\alpha^{-1}\in L_{2}(\Omega)$ for the well-posedness Theorem 4.2. We recall
these results here to demonstrate that in one dimensions the results are
computable. To this extent, this is a property not shared by the higher-
dimensional case.

######  Setting 4.1.

Let $\Omega\subseteq\mathbb{R}$ be an open and bounded interval. Let
$\alpha\in L_{\infty}(\Omega)$. Then we define $X_{\alpha}$ acting as

$X_{\alpha}u=-(\alpha u^{\prime})^{\prime}.$

in the distributional sense and consider this operator as an operator from
$H_{0}^{1}(\Omega)$ into $H^{-1}(\Omega)$. The operator

$D_{\alpha}\colon\operatorname{dom}(D_{\alpha})\subseteq L_{2}(\Omega)\to
L_{2}(\Omega)$

is the $L_{2}(\Omega)$-realisation of $X_{\alpha}$ with maximal domain
$\operatorname{dom}(D_{\alpha})$ contained in $H_{0}^{1}(\Omega)$.

Under mild conditions on $\alpha$, the differential equation associated with
$X_{\alpha}$ turns out to be well-posed. For this, we introduce
$\mathfrak{m}_{\Omega}(\beta)\coloneqq\frac{1}{\lambda{(\Omega)}}\int_{\Omega}\beta$
for $\beta\in L_{1}(\Omega)$, $\lambda{(\Omega)}$ being the Lebesgue measure
of $\Omega$.

######  Theorem 4.2.

Assume Setting 4.1. In addition, assume $\alpha^{-1}\in L_{\infty}(\Omega)$
and $\mathfrak{m}_{\Omega}(\alpha^{-1})\neq 0$.

Then for all $f\in H^{-1}(\Omega)$ there exists a unique $u\in
H_{0}^{1}(\Omega)$ such that

$X_{\alpha}u=f\quad\text{(in the distributional sense)}.$

The problem is equivalent to finding $\varphi\in L_{2}(\Omega)$ given
$\psi\in\\{1\\}^{\bot}\subseteq L_{2}(\Omega)$ such that

$\alpha\varphi-\mathfrak{m}_{\Omega}(\alpha\varphi)=\psi;$

which can be found via

$\varphi=\alpha^{-1}\psi-\alpha^{-1}\frac{\mathfrak{m}_{\Omega}(\alpha^{-1}\psi)}{\mathfrak{m}_{\Omega}(\alpha^{-1})}.$

Moreover, $D_{\alpha}$, the $L_{2}(\Omega)$ realisation of $X_{\alpha}$ is
continuously invertible and has compact resolvent and
$D_{\alpha}^{*}=D_{\alpha^{*}}$.

The rationale follows similar lines to the slightly simpler situation in [41,
Example 1.10].

###### Proof of Theorem 4.2.

We apply Theorem 3.1 to $C=\mathring{\partial}\colon
H_{0}^{1}(\Omega)\subseteq L_{2}(\Omega)\to L_{2}(\Omega)$ and $a=\alpha({\rm
m})$, the multiplication operator induced by the multiplication by $\alpha$.
Then $C$ is densely defined, closed. $C$ has closed range by the Poincaré
inequality, see, e.g., [3]. Moreover,

$V_{1}=\operatorname{ran}(\mathring{\partial})=\operatorname{ker}(\partial)^{\bot}=\\{1\\}^{\bot}.$

We compute $\alpha_{1}\coloneqq\iota_{V_{1}}^{*}a\iota_{V_{1}}$ next. For
this, note that since $\operatorname{ker}(\partial)=\\{1\\}$ is spanned by the
unit vector $g\coloneqq\frac{1}{\lambda(\Omega)^{1/2}}$, for any $f\in
L_{2}(\Omega)$, we have

$\iota_{V_{1}}^{*}f=f-\frac{1}{\lambda(\Omega)^{1/2}}\langle
1,f\rangle\frac{1}{\lambda(\Omega)^{1/2}}1=f-\mathfrak{m}_{\Omega}(f).$

Next, let $\varphi,\psi\in\\{1\\}^{\bot}$ be such that

$\alpha_{1}\varphi=\psi.$

Then

$\psi=\alpha_{1}\varphi=\alpha\varphi-\mathfrak{m}_{\Omega}(\alpha\varphi)$

and so
$\alpha^{-1}\psi=\varphi-\alpha^{-1}\mathfrak{m}_{\Omega}(\alpha\varphi)$.
Hence,

$\mathfrak{m}_{\Omega}(\alpha^{-1}\psi)=\mathfrak{m}_{\Omega}(\varphi)-\mathfrak{m}_{\Omega}(\alpha^{-1})\mathfrak{m}_{\Omega}(\alpha\varphi)=-\mathfrak{m}_{\Omega}(\alpha^{-1})\mathfrak{m}_{\Omega}(\alpha\varphi),$

since $\varphi\in V_{1}$. Finally,

$\varphi=\alpha^{-1}\psi+\alpha^{-1}\mathfrak{m}_{\Omega}(\alpha\varphi)=\alpha^{-1}\psi-\alpha^{-1}\frac{\mathfrak{m}_{\Omega}(\alpha^{-1}\psi)}{\mathfrak{m}_{\Omega}(\alpha^{-1})},$

which leads to

$\alpha_{1}^{-1}\colon\psi\mapsto\alpha^{-1}\psi-\alpha^{-1}\frac{\mathfrak{m}_{\Omega}(\alpha^{-1}\psi)}{\mathfrak{m}_{\Omega}(\alpha^{-1})}.$

Since $\alpha_{1}^{-1}$ is continuous and everywhere defined, we infer the
invertibility of $X_{\alpha}$ and $D_{\alpha}$. The remaining assertions
follow from Theorem 3.1, together with
$H_{0}^{1}(\Omega)\hookrightarrow\hookrightarrow L_{2}(\Omega)$. ∎

A closer look into the proof of Theorem 4.2 – particularly for the case of
piecewise constant coefficients – reveals the inner spectrum with respect to
$g_{0}(\Omega)=\\{1\\}^{\bot}$. For illustrational purposes we only consider a
special case; for $\alpha_{0},\ldots,\alpha_{r}\in\mathbb{C}$ write
$\alpha({\rm m})$ for the multiplication operator induced by

$\alpha=\sum_{j=0}^{r}\alpha_{j}\mathbbold{1}_{(jh,(j+1)h]}$

on $L_{2}(0,1)$.

######  Theorem 4.3.

Let $\alpha_{0},\ldots,\alpha_{r}\in\mathbb{C}$, $h\coloneqq 1/(r+1)$. Then
for $V_{1}=\\{1\\}^{\bot}\subseteq L_{2}(0,1)$ we have

$\sigma_{V_{1}}(\alpha({\rm
m}))=\\{\lambda\in\mathbb{C};(\alpha-\lambda)^{-1}\in
L_{\infty},\mathfrak{m}((\alpha-\lambda)^{-1})=0\\}\cup\\{\alpha_{j};j\in\\{0,\ldots,r\\}\\}.$

###### Proof.

We apply Theorem 3.1 and, thus, need to look for exactly those
$\lambda\in\mathbb{C}$ such that the operator
$a_{\lambda}\coloneqq\iota_{V_{1}}^{*}(\alpha({\rm m})-\lambda)\iota_{V_{1}}$
is not invertible. For this let $\varphi,\psi\in V_{1}$ and consider the
equation

$a_{\lambda}\varphi=\psi,$

which reads

$\psi=(\alpha-\lambda)\varphi-\mathfrak{m}_{(0,1)}((\alpha-\lambda)\varphi).$
(2)

Now, if $\alpha-\lambda=0$ on some subinterval, $\psi$ is constant on this
subinterval. Thus, $a_{\lambda}$ is not onto. Hence,

$\\{\alpha_{j};j\in\\{0,\ldots,r\\}\\}\subseteq\sigma_{V_{1}}(a).$

Next, consider the case $(\alpha-\lambda)^{-1}\in L_{\infty}$ (which
complements the other as $\alpha$ is piecewise constant). If
$\mathfrak{m}((\alpha-\lambda)^{-1})\neq 0$, Theorem 4.2 applies and
$\lambda\notin\sigma_{V_{1}}(\alpha({\rm m}))$. On the other hand, if
$\mathfrak{m}((\alpha-\lambda)^{-1})=0$, then (2) leads to

$(\alpha-\lambda)^{-1}\psi=\varphi-(\alpha-\lambda)^{-1}\mathfrak{m}((\alpha-\lambda)\varphi)$

showing that

$\mathfrak{m}((\alpha-\lambda)^{-1}\psi)=0.$

Since $\mathfrak{m}((\alpha-\lambda)^{-1})=0$, we infer
$(\alpha-\lambda)^{-1}$ is not constant. Thus,
$\psi\in\\{1,(\alpha-\lambda)^{-1}\\}^{\bot}\neq\\{1\\}^{\bot}$. Hence, we
again deduce that $a_{\lambda}$ is not onto. ∎

######  Remark 4.4.

The precise detection of the set

$\\{\lambda\in\mathbb{C};(\alpha-\lambda)^{-1}\in
L_{\infty},\mathfrak{m}((\alpha-\lambda)^{-1})=0\\}=\big{\\{}\lambda\in\mathbb{C};\sum_{j=0}^{r}\frac{1}{\alpha_{j}-\lambda}=0\big{\\}}$

is a classical albeit nontrivial task, see, e.g., [56]; see also [26] for a
related question.

Next we address the solution theory for problems that are not well-posed and
consider small perturbations of these. In the higher-dimensional situation we
need to revisit this result. Since, for these problems, we will consider
piecewise constant coefficients only, the condition $\alpha^{-1}\in
L_{\infty}(\Omega)$ is automatically satisfied. If, for general $\alpha\in
L_{\infty}(\Omega)$, its inverse is contained in $L_{\infty}(\Omega)$ with
vanishing integral mean, the spectral point $0$ of the inner spectrum is
isolated. Again, let $\alpha({\rm m})$ be the multiplication operator on
$L_{2}(\Omega)$ induced by $\alpha$.

######  Corollary 4.5.

Assume Setting 4.1. In addition, assume $\alpha^{-1}\in
L_{\infty}(\Omega;\mathbb{R})$ and $\mathfrak{m}_{\Omega}(\alpha^{-1})=0$.

Then there exists $\varepsilon_{0}>0$ such that

$B_{\mathbb{C}}(0,\varepsilon_{0})\setminus\\{0\\}\subseteq\rho_{g_{0}(\Omega)}(\alpha({\rm
m})).$

######  Remark 4.6.

Since $\alpha$ is real,
$\mathbb{C}\setminus\mathbb{R}\subseteq\rho_{g_{0}(\Omega)}(\alpha({\rm m}))$
by the classical Lax–Milgram lemma. Indeed, for
$\lambda\in\mathbb{C}\setminus\mathbb{R}$, we find $\gamma\in\mathbb{C}$ such
that $\operatorname{Re}\gamma(\alpha-\lambda)\geqslant c$ for some $c>0$.

###### Proof of Corollary 4.5.

By Theorem 4.2, it suffices to show the existence of $\varepsilon_{0}>0$ such
that for all $\lambda\in B(0,\varepsilon_{0})\setminus\\{0\\}$ we have
$\mathfrak{m}_{\Omega}((\alpha-\lambda)^{-1})\neq 0$. For this, we find $c>0$
such that $|\alpha^{-1}(x)|\leqslant c$ for a.e. $x\in\Omega$. Thus, for all
$\lambda\in(-\tfrac{1}{2c},\tfrac{1}{2c})$, we obtain for a.e. $x\in\Omega$:

$|\alpha(x)-\lambda|=|\tfrac{1}{c}(c\alpha(x)-c\lambda)|\geqslant\tfrac{1}{c}(|c\alpha(x)|-\tfrac{1}{2})\geqslant\tfrac{1}{2c},$

thus proving $(\alpha-\lambda)^{-1}\in L_{\infty}(\Omega)$. Similar arguments
show that
$B_{\mathbb{C}}(0,\varepsilon_{0})\ni\lambda\mapsto(\alpha-\lambda)^{-1}\in
L_{\infty}(\Omega)$ is holomorphic (in fact it follows almost literally the
same lines as the proof of the holomorphicity of the resolvent map for linear
operators). In consequence, $F\colon
B_{\mathbb{C}}(0,\varepsilon_{0})\ni\lambda\mapsto\mathfrak{m}_{\Omega}((\alpha+\lambda)^{-1})$
is, too, holomorphic. Since, for $\lambda=i\delta$, $\delta\in\mathbb{R}$, we
obtain

$\Im\int_{\Omega}(\alpha(x)-i\delta)^{-1}\mathrm{d}x=\int_{\Omega}-\delta(\alpha(x)^{2}+\delta^{2})^{-1}\mathrm{d}x\neq
0.$

Hence, $F\neq 0$. Thus, the identity theorem yields that the zeros of $F$
cannot accumulate around $0$, implying the assertion. ∎

One might hope that the corresponding result for higher-dimensional cases with
laminated media is a somewhat straightforward generalisation of the one-
dimensional case. This is far from correct as the projection onto the range of
the gradient is not one-co-dimensional but infinite-co-dimensional instead.
That is why we need a lot more work to treat this case.

##  5 Sturm–Liouville problems with indefinite coefficients

The analysis of higher-dimensional stratified media requires a more detailed
picture of the structure of the operator $X_{\alpha}$ in Setting 4.1. In
passing, we shall provide some insight into the spectral theory of
Sturm–Liouville operators with indefinite coefficients. The focus here is on
invertibility rather than computing the whole (classical) spectrum even though
the one yields the other.

In the following, we address the solvability of the resolvent type problem
associated with the operator

$D_{\alpha}+\beta=(u\mapsto-(\alpha u^{\prime})^{\prime}+\beta u)$ (SLP)

in $L_{2}(0,1)$ for piecewise constant coefficients $\alpha$ and $\beta$ with
$\operatorname{dom}(D_{\alpha})\subseteq H_{0}^{1}(0,1).$

However, using the result from the previous section a first statement is
rather immediate from Fredholm theory for general coefficients. We will use
$\mathfrak{m}_{\Omega}$ for $\Omega=(0,1)$ only and, hence, just write
$\mathfrak{m}$ to denote the integral mean on $(0,1)$.

######  Proposition 5.1.

Let $\alpha,\alpha^{-1},\beta\in L_{\infty}(0,1)$. If
$\mathfrak{m}(\alpha^{-1})\neq 0$, then the following conditions are
equivalent:

  1. (i)

$D_{\alpha}+\beta$ is continuously invertible in $L_{2}(0,1)$.

  2. (ii)

$D_{\alpha}+\beta$ is one-to-one.

###### Proof.

(i)$\Rightarrow$(ii): Obvious.

(ii)$\Rightarrow$(i): Theorem 4.2 implies that $D_{\alpha}$ is continuously
invertible and has compact resolvent. In consequence, the multiplication with
$\beta$ is a $D_{\alpha}$-compact perturbation of $D_{\alpha}$. Thus,
$\operatorname{ind}(D_{\alpha}+\beta)=\operatorname{ind}(D_{\alpha})=\\{0\\}$,
see for instance [29, Theorem 3.6] (and references therein) for the notion of
unbounded Fredholm operators and their index. Hence, $D_{\alpha}+\beta$ is
onto if and only if it is one-to-one; yielding (i). ∎

We shall characterise the injectivity for a more particular situation in the
results to come. The setting is as follows.

######  Setting 5.2.

Let $\alpha_{0},\ldots,\alpha_{r}\in\mathbb{C}\setminus\\{0\\}$ and
$\beta_{0},\ldots,\beta_{r}\in\mathbb{C}$ and define for $h\coloneqq 1/(r+1)$

$\alpha=\sum_{j=0}^{r}\alpha_{j}\mathbbold{1}_{(jh,(j+1)h]}\text{ and
}\beta=\sum_{j=0}^{r}\beta_{j}\mathbbold{1}_{(jh,(j+1)h]}.$

Let $D_{\alpha}$ be as in Setting 4.1 for $\Omega=(0,1)$.

In order to properly study the intricacies of this situation, we introduce the
following notion, which is related to the monodromy matrix from Floquet
theory. However, the considered boundary conditions are different. In order
not to confused the following with the notion from Floquet theory, we coined a
different notion.

Let $h>0,\tau\in\mathbb{K}$, $A\in\mathbb{K}^{2\times 2}$. We call $A$
transition matrix with respect to $(h,\tau)$, if for all $v\in\mathbb{K}^{2}$
and $u$ being the solution of

$\begin{cases}u^{\prime\prime}=\tau u,&\text{ on }(0,h)\\\
\begin{pmatrix}u(0)\\\ u^{\prime}(0)\end{pmatrix}=v&\end{cases}$

we have

$Av=\begin{pmatrix}u(h)\\\ u^{\prime}(h)\end{pmatrix}.$

By the solution theory for linear ODEs, transition matrices are uniquely
determined from $(h,\tau)$. Next, we provide the necessary examples needed
here. Most prominently we shall require of case (b) in the subsequent
sections.

######  Example 5.3.

(a) Let $\tau=0$. Then $u(x)=1$ and $v(x)=x$ are linearly independent
solutions of $u^{\prime\prime}=0$. Since $u^{\prime}=0$ and $v(0)=0$ we get

$A=\begin{pmatrix}1&h\\\ 0&1\end{pmatrix}$

is the transition matrix with respect to $(h,0)$.

(b) Let $\tau=\mu^{2}>0$, $\mu>0$. Then $u(x)=\sinh(\mu x)$ and
$v(x)=\cosh(\mu x)$ are linearly independent solutions of
$u^{\prime\prime}=\tau u$. Then

$A=\begin{pmatrix}\cosh(\mu h)&\tfrac{1}{\mu}\sinh(\mu h)\\\ \mu\sinh(\mu
h)&\cosh(\mu h)\end{pmatrix}$

is the transition matrix w.r.t. $(h,\mu^{2})$,

(c) Let $\tau=-\mu^{2}<0$, $\mu>0$. Then $u(x)=\sin(\mu x)$ and $v(x)=\cos(\mu
x)$ are linearly independent solutions of $u^{\prime\prime}=\tau u$. Then

$A=\begin{pmatrix}\cos(\mu h)&\frac{1}{\mu}\sin(\mu h)\\\ -\mu\sin(\mu
h)&\cos(\mu h)\end{pmatrix}$

is the transition matrix w.r.t. $(h,-\mu^{2})$.

The main result of this section and at the same time key for understanding the
inner spectra considered in the forthcoming sections is as follows.

######  Theorem 5.4.

Assume Setting 5.2. If $\mathfrak{m}(\alpha^{-1})\neq 0$, then the following
conditions are equivalent:

  1. (i)

$D_{\alpha}+\beta$ is continuously invertible;

  2. (ii)

$D_{\alpha}+\beta$ is one-to-one;

  3. (iii)

$\begin{pmatrix}a_{00}^{(r)}&\alpha_{r}^{-1}a_{01}^{(r)}\end{pmatrix}\tilde{A}_{\alpha_{r-1}}\cdots\tilde{A}_{\alpha_{1}}\begin{pmatrix}a_{01}^{(0)}\\\
\alpha_{0}a_{11}^{(0)}\end{pmatrix}\neq 0$,

where $\tilde{A}_{\alpha_{j}}=\begin{pmatrix}1&0\\\
0&\alpha_{j}\end{pmatrix}A^{(j)}\begin{pmatrix}1&0\\\
0&\alpha_{j}^{-1}\end{pmatrix}$ and $A^{(j)}=(a_{kl}^{(j)})_{k,l\in\\{0,1\\}}$
is the transition matrix with respect to
$(\frac{1}{1+r},\tfrac{\beta_{j}}{\alpha_{j}})$, $j\in\\{1,\ldots,r\\}$.

The proof of Theorem 5.4 is rather technical, though elementary, and postponed
to Section 13. In the Setting 5.2, we call

$p_{\alpha,\beta}\coloneqq\begin{pmatrix}a_{00}^{(r)}&\alpha_{r}^{-1}a_{01}^{(r)}\end{pmatrix}\tilde{A}_{\alpha_{r-1}}\cdots\tilde{A}_{\alpha_{1}}\begin{pmatrix}a_{01}^{(0)}\\\
\alpha_{0}a_{11}^{(0)}\end{pmatrix}$

used in Theorem 5.4 the characteristic function of $(\alpha,\beta)$.

We conclude this section with an example, which in a way reproduces the
results developed earlier.

######  Example 5.5.

We consider the case $\beta=0$ and recall $h=1/(r+1)$. Then for
$j\in\\{0,\ldots,r\\}$,

$A^{(j)}=\begin{pmatrix}1&h\\\ 0&1\end{pmatrix}$

and, thus, for $j\in\\{1,\ldots,r-1\\}$

$\tilde{A}_{\alpha_{j}}=\begin{pmatrix}1&0\\\
0&\alpha_{j}\end{pmatrix}\begin{pmatrix}1&h\\\
0&1\end{pmatrix}\begin{pmatrix}1&0\\\
0&\alpha_{j}^{-1}\end{pmatrix}=\begin{pmatrix}1&h\\\
0&\alpha_{j}\end{pmatrix}\begin{pmatrix}1&0\\\
0&\alpha_{j}^{-1}\end{pmatrix}=\begin{pmatrix}1&h\alpha_{j}^{-1}\\\
0&1\end{pmatrix}.$

Hence,

$\displaystyle p_{\alpha,0}$
$\displaystyle=\begin{pmatrix}a_{00}^{(r)}&\alpha_{r}^{-1}a_{01}^{(r)}\end{pmatrix}\tilde{A}_{\alpha_{r-1}}\cdots\tilde{A}_{\alpha_{1}}\begin{pmatrix}a_{01}^{(0)}\\\
\alpha_{0}a_{11}^{(0)}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}1&\alpha_{r}^{-1}h\end{pmatrix}\begin{pmatrix}1&h\sum_{j=1}^{r-1}\alpha_{j}^{-1}\\\
0&1\end{pmatrix}\begin{pmatrix}h\alpha_{0}^{-1}\\\
1\end{pmatrix}\alpha_{0}=h\alpha_{0}\sum_{j=0}^{r}\alpha_{j}^{-1}=\alpha_{0}\mathfrak{m}(\alpha^{-1}).$

##  6 The higher-dimensional problem – preliminaries

In this section, we address the application of the results in the previous
section to coefficients modelling stratified media. More precisely, we
consider the $d$-dimensional open set
$\Omega\coloneqq(0,1)\times\widehat{\Omega}$ for some
$\widehat{\Omega}\subseteq\mathbb{R}^{d-1}$ open and bounded and address
solvability of

$-\Delta_{\alpha}u\coloneqq-\overline{\operatorname{div}\alpha({\rm
m}_{1})\operatorname{grad}}\,u=f\in L_{2}(\Omega)$

and $u\in H_{0}^{1}(\Omega)$, where $\alpha\in L_{\infty}(0,1)$ and
$\alpha({\rm m}_{1})$ denotes the multiplication operator on
$L_{2}(\Omega)^{d}$ induced by $(x_{1},\ldots,x_{d})\mapsto\alpha(x_{1})$. The
closure is taken as an operator acting in $L_{2}(\Omega)$. The tensor product
structure of $\Omega$ and the conductivity allows to reduce the problem to a
Sturm–Liouville type situation with unbounded potential. Before we go into the
details of this observation, we recall $g_{0}(\Omega)$ and the corresponding
$\iota_{0}\colon g_{0}(\Omega)\hookrightarrow L_{2}(\Omega)^{d}$ from (1) and
apply our results on the inner spectrum from Section 3 to the present
situation.

######  Theorem 6.1.

Let $\alpha\in L_{\infty}(0,1)$, $\lambda\in\mathbb{C}$. Then

  1. (a)

$0\in\sigma(\Delta_{\alpha-\lambda})\Rightarrow\lambda\in\sigma_{g_{0}(\Omega)}(\alpha({\rm
m}_{1}))$;

  2. (b)

$0\in\rho(\Delta_{\alpha-\lambda})\Rightarrow\lambda\in\rho_{g_{0}(\Omega)}(\alpha({\rm
m}_{1}))\cup\sigma_{g_{0}(\Omega),\textnormal{c}}(\alpha({\rm m}_{1}))$.

###### Proof.

The statements (a) and (b) are immediate applications of Theorem 3.4. (For the
second implication prove the contraposition and use that both point spectrum
as well as residual spectrum are stable under closure processes.) ∎

######  Remark 6.2.

The statement in Theorem 6.1 is not optimal. However, I still conjecture that
at least for piecewise constant $\alpha$, the statement in (a) is, in fact, an
equivalence. Note that for the case $d=1$, $\alpha({\rm m}_{1})$ has no inner
continuous spectrum, so that equivalence holds.

Concerning the inner spectrum, we provide an elementary observation next. If
$\alpha=0$ on some nontrivial subinterval of $(0,1)$, then $-\Delta_{\alpha}$
is not one-to-one: Indeed, choose $\widehat{\varphi}\in
C_{c}^{\infty}(\widehat{\Omega})\setminus\\{0\\}$ and a non-zero
$\check{\varphi}\in C_{c}^{\infty}(0,1)$ with $\check{\varphi}\alpha=0$. Then
$\varphi(x_{1},\ldots,x_{n})=\check{\varphi}(x_{1})\widehat{\varphi}(x_{2},\ldots,x_{d})$
satisfies

$-\Delta_{\alpha}\varphi=0.$

Hence, as a consequence, we deduce the following result:

######  Corollary 6.3.

In the present situation assume $\alpha$ takes the values
$\\{\alpha_{0},\ldots,\alpha_{r}\\}\subseteq\mathbb{C}$ on nondegenerate
subintervals of $(0,1)$.

Then

$\\{\alpha_{0},\ldots,\alpha_{r}\\}\subseteq\sigma_{g_{0}(\Omega)}(\alpha({\rm
m}_{1})).$

We conclude this section by providing the relation of Sturm–Liouville problems
to the Laplace equation with stratified media. For this, we call
$\Lambda\subseteq\mathbb{C}$ discrete, if

$\inf\\{|\lambda-\mu|;\lambda,\mu\in\Lambda,\lambda\neq\mu\\}>0.$

We call a sequence $(\lambda_{k})_{k}$ in $\mathbb{C}$ discrete, if
$\\{\lambda_{k};k\in\mathbb{N}\\}\subseteq\mathbb{C}$ is discrete.

######  Remark 6.4.

Note that for a discrete sequence $(\lambda_{k})_{k}$, the set of finite
accumulation values is

$\\{\lambda;\lambda_{k}=\lambda\text{ for infinitely many }k\in\mathbb{N}\\}.$

As a consequence, for any discrete sequence $(\lambda_{k})_{k}$ in
$\mathbb{R}_{>0}$, we obtain

$\inf_{k\in\mathbb{N}}|\lambda_{k}|=\min_{k\in\mathbb{N}}|\lambda_{k}|>0.$

The announced result on the unitary equivalence of $\Delta_{\alpha}$ is stated
next. Note that, as a consequence, since we provide results relative to _any_
discrete sequence in the following, we cover all geometries of
$\widehat{\Omega}$ as long as it is open and bounded.

######  Theorem 6.5.

Assume $\alpha,\alpha^{-1}\in L_{\infty}(0,1)$ with
$\mathfrak{m}(\alpha^{-1})\neq 0$. Recall $D_{{\rm red},\alpha}=D_{\alpha}$
from Setting 4.1. Then $-\Delta_{\alpha}$ is self-adjoint and unitarily
equivalent to the diagonal operator

$(D_{{\rm red},\alpha}+\alpha\lambda_{k})_{k\in\mathbb{N}}$

on $L_{2}(0,1;\ell_{2}(\mathbb{N}))$ for a discrete sequence
$(\lambda_{k})_{k}$ in $\mathbb{R}_{>0}$.

We provide the proof in Section 13.

##  7 The higher dimensional problem – well-posedness

We turn back to piecewise constant coefficients. For this we recall that for
$\alpha_{0},\ldots,\alpha_{r}\in\mathbb{R}\setminus\\{0\\}$ and $h\coloneqq
1/(r+1)$ we defined

$\alpha=\sum_{j=0}^{r}\alpha_{j}\mathbbold{1}_{(jh,(j+1)h]}.$

Let $D_{\alpha}$ be as in Setting 4.1 for $\Omega=(0,1)$. We address criteria
that ensure that for a given discrete sequence $(\lambda_{k})_{k}$ in
$\mathbb{R}_{>0}$, the diagonal operator

$S_{\alpha}\coloneqq((D_{\alpha}+\alpha\lambda_{k})^{-1})_{k\in\mathbb{N}}$

acting in $L_{2}(0,1;\ell_{2}(\mathbb{N}))$ is both well-defined and bounded.
For this particular situation, we rephrase Theorem 5.4. Before doing so, we
introduce a function with which the respective formulation is less cumbersome.
Define

$\tilde{q}(m_{0},\ldots,m_{r})\coloneqq\\\
\begin{pmatrix}1&\alpha_{r}^{-1}m_{r}^{-1}t_{r}\end{pmatrix}\begin{pmatrix}1&\alpha_{r-1}^{-1}m_{r-1}^{-1}t_{r-1}\\\
\alpha_{r-1}m_{r-1}t_{r-1}&1\end{pmatrix}\cdots\begin{pmatrix}1&\alpha_{1}^{-1}m_{1}^{-1}t_{1}\\\
\alpha_{1}m_{1}t_{1}&1\end{pmatrix}\begin{pmatrix}m_{0}^{-1}t_{0}\\\
\alpha_{0}\end{pmatrix}$

with $t_{j}=\tanh(m_{j}h)$, $j\in\\{0,\ldots,r\\}$, and
$m_{0},\ldots,m_{r}>0$. We present a reformulation of Theorem 5.4. Recall
$\mathfrak{m}(\beta)=\int_{0}^{1}\beta$ for $\beta\in L_{1}(0,1)$.

######  Theorem 7.1.

Let $\alpha_{0},\ldots,\alpha_{r}\in\mathbb{R}\setminus\\{0\\}$ with
$\mathfrak{m}(\alpha^{-1})\neq 0$. Let $(\lambda_{k})_{k}$ in
$\mathbb{R}_{>0}$ be discrete and choose
$C>1/(\min_{k\in\mathbb{N}}\lambda_{k}\min_{j}|\alpha_{j}|)$.

Then the following conditions are equivalent:

  1. (i)

for all $k\in\mathbb{N}$, the operator $D_{\alpha}+\alpha\lambda_{k}$ is
continuously invertible with $\|(D_{\alpha}+\alpha\lambda_{k})^{-1}\|\leqslant
C$;

  2. (ii)

$(-1/C,1/C)\subseteq\rho(D_{\alpha}+\alpha\lambda_{k})$

  3. (iii)

for all $k\in\mathbb{N}$ and $\delta\in[-1,1]$,
$\tilde{q}(m_{0},\ldots,m_{r})\neq 0$ where $t_{j}=\tanh(m_{j}h)$ and
$m_{j}=\sqrt{\lambda_{k}+\tfrac{\delta}{\alpha_{j}C}}$

The proof is carried out in Section 13. An elementary consequence of Theorem
7.1 is the following.

######  Corollary 7.2.

Assume the situation of Theorem 7.1. The following conditions are equivalent:

  1. (i)

$S_{\alpha}$ is well-defined and bounded;

  2. (ii)

there exists $\delta_{0}>0$ such that for all $d\in[-\delta_{0},\delta_{0}]$
and $k\in\mathbb{N}$, $\tilde{q}(m_{0},\ldots,m_{r})\neq 0$, where
$t_{j}=\tanh(m_{j}h)$ and
$m_{j}=\sqrt{\lambda_{k}+\operatorname{sgn}(\alpha_{j})d}$.

###### Proof.

(ii)$\Rightarrow$(i): This is a consequence of Theorem 7.1
(iii)$\Rightarrow$(i).

(i)$\Rightarrow$(ii): Since $S_{\alpha}$ is a diagonal operator, it is well-
defined and bounded, if, and only if, $0\in\rho(D_{\alpha}+\alpha\lambda_{k})$
for all $k\in\mathbb{N}$ and there exists $C\geqslant 0$ such that
$\sup_{k\in\mathbb{N}}\|(D_{\alpha}+\alpha\lambda_{k})^{-1}\|\leqslant C$;
hence, for $\delta\in[-1,1]$ the non-zero condition holds with

$m_{j}^{2}=\lambda_{k}+\tfrac{\delta}{\alpha_{j}C}=\lambda_{k}+\operatorname{sgn}(\alpha_{j})\tfrac{\delta}{|\alpha_{j}|C}\quad(k\in\mathbb{N},j\in\\{0,\ldots,r\\}),$

which eventually leads to the assertion. ∎

Condition (ii) in the latter characterisation is rather difficult to apply for
particular settings. Therefore, we require a more handy criterion ensuring
(ii) (and thus (i)) to hold.

Let $\alpha_{0},\ldots,\alpha_{r}\in\mathbb{R}\setminus\\{0\\}$ and
$(\lambda_{k})_{k}$ a discrete sequence in $\mathbb{R}_{>0}$. We say that
$\alpha_{0},\ldots,\alpha_{r}$ and $(\lambda_{k})_{k}$ satisfy the
$\tilde{q}$-criterion, if there exists $\delta_{0}>0$ such that for all
$d\in[-\delta_{0},\delta_{0}]$ and $k\in\mathbb{N}$

$\tilde{q}(m_{0},\ldots,m_{r})\neq 0,$ (3)

where $t_{j}=\tanh(m_{j}h)$ and
$m_{j}=\sqrt{\lambda_{k}+\operatorname{sgn}(\alpha_{j})d}$.

The satisfaction of the $\tilde{q}$-criterion works as follows. There is a
local part asking for a non-zero condition to be satisfied for $d=0$ and an
asymptotic condition that warrants the existence of $\delta_{0}>0$ _uniformly_
in $k\in\mathbb{N}$.

Recall the characteristic function $p_{\alpha,\beta}$ from Theorem 5.4 and
define for $\mu_{0}>0$

$M_{\mu_{0}}\coloneqq[\mu_{0},\infty)\times\mu_{0}[-\tfrac{1}{2},\tfrac{1}{2}]^{r+1}\text{
and }S(\mu,x)\coloneqq((\mu+x_{j}))_{j\in\\{0,\ldots,r\\}}\quad((\mu,x)\in
M_{\mu_{0}})$

Moreover, we set

$\displaystyle q$ $\displaystyle\colon
M_{\mu_{0}}\to\mathbb{R},(\mu,x)\mapsto\tilde{q}(S(\mu,x))$ $\displaystyle w$
$\displaystyle\colon
M_{\mu_{0}}\to\mathbb{R},(\mu,x)\mapsto\tilde{r}(S(\mu,x)),$

where

$\displaystyle\tilde{w}(m_{0},\ldots,m_{r})\coloneqq\begin{pmatrix}1&\alpha_{r}^{-1}m_{r}^{-1}\end{pmatrix}\begin{pmatrix}1&\alpha_{r-1}^{-1}m_{r-1}^{-1}\\\
\alpha_{r-1}m_{r-1}&1\end{pmatrix}\cdots\begin{pmatrix}1&\alpha_{1}^{-1}m_{1}^{-1}\\\
\alpha_{1}m_{1}&1\end{pmatrix}\begin{pmatrix}m_{0}^{-1}\\\
\alpha_{0}\end{pmatrix}$

for $m_{0},\ldots,m_{r}>0$.

######  Proposition 7.3.

(a) Let $m_{0},\ldots,m_{r}>0$, let $\alpha$ be composed of $\alpha_{j}$ and
$\beta$ be as in Setting 5.2 with $\beta_{j}=m_{j}^{2}\alpha_{j}$. Then

$p_{\alpha,\beta}=\big{(}\prod_{j=0}^{r}\cosh(m_{j}h)\big{)}\tilde{q}(m_{0},\ldots,m_{r}).$

(b) Let $\mu_{0}>0$. Then

$\lim_{\mu\to\infty}\mu
w(\mu,x)=\mu_{0}w(\mu_{0},0)=\prod_{j=0}^{r-1}(1+\alpha_{j+1}^{-1}\alpha_{j})\eqqcolon\chi(\alpha)$

uniformly for $x\in\mu_{0}[-\tfrac{1}{2},\tfrac{1}{2}]^{r+1}$.

(c) Let $\mu_{0}>0$. Then, as $\mu\to\infty$,
$q(\mu,\cdot),w(\mu,\cdot)\in\mathcal{O}(\tfrac{1}{\mu})$. Moreover,

$\lim_{\mu\to\infty}\|\mu(q(\mu,\cdot)-w(\mu,\cdot))\|_{\infty}=0.$

(d) For all $\mu_{0}>0$, the function

$\widehat{q}\colon M_{\mu_{0}}\ni(\mu,x)\mapsto\mu q(\mu,x)$

is uniformly continuous.

The proof can be found in Section 13. For the characterisation of the
$\tilde{q}$-criterion, we introduce another function:

$\displaystyle
p_{\alpha}(t)\coloneqq\begin{pmatrix}1&\alpha_{r}^{-1}t\end{pmatrix}\begin{pmatrix}1&\alpha_{r-1}^{-1}t\\\
\alpha_{r-1}t&1\end{pmatrix}\cdots\begin{pmatrix}1&\alpha_{1}^{-1}t\\\
\alpha_{1}t&1\end{pmatrix}\begin{pmatrix}t\\\ \alpha_{0}\end{pmatrix}.$

A first consequence of Proposition 7.3 is a condition guaranteeing the
$\tilde{q}$-criterion for $\alpha_{0},\ldots,\alpha_{r}$ and
$(\lambda_{k})_{k\geqslant k_{0}}$ for some $k_{0}\geqslant 0$ making use of
$\chi(\alpha)\coloneqq\prod_{j=0}^{r-1}(1+\alpha_{j+1}^{-1}\alpha_{j}),$
introduced in Proposition 7.3 (b).

######  Theorem 7.4.

Let $\alpha_{0},\ldots,\alpha_{r}\in\mathbb{R}\setminus\\{0\\}$,
$(\mu_{k})_{k}$ a discrete sequence in $\mathbb{R}_{>0}$ with
$\mu_{k}\to\infty$ as $k\to\infty$. If $\chi(\alpha)\neq 0$, then there exists
$k_{0}\in\mathbb{N}$ and $\delta>0$ such that $q(\mu_{k},x)\neq 0$ for all
$k\geqslant k_{0}$ and $x\in[-\delta,\delta]^{r+1}$.

###### Proof.

By Proposition 7.3 (b) and (c) and assumption, we get
$c\coloneqq\lim_{\mu\to\infty}\mu\tilde{q}(\mu,\ldots,\mu)\neq 0$. Hence, we
find $k_{0}\in\mathbb{N}$ such that, for all $k\geqslant k_{0}$,
$|\mu_{k}\tilde{q}(\mu_{k},\ldots,\mu_{k})|\geqslant c/2$. By uniform
continuity of $\widehat{q}$ from Proposition 7.3 (d), we find $\delta>0$ such
that for all $x\in[-\delta,\delta]^{r+1}$ and $k\in\mathbb{N}$, we have

$\mu_{k}|q(\mu_{k},x)|\geqslant\tfrac{c}{4};$

yielding the desired assertion. ∎

We are now in the position to summarise our findings concerning the
$\tilde{q}$-criterion, which we prove in Section 13.

######  Theorem 7.5.

Let $\alpha_{0},\ldots,\alpha_{r}\in\mathbb{R}\setminus\\{0\\}$,
$(\lambda_{k})_{k}$ a discrete sequence in $\mathbb{R}_{>0}$,
$\mu_{k}\coloneqq\sqrt{\lambda_{k}}$, with $\lambda_{k}\to\infty$ as
$k\to\infty$. Consider the following conditions:

  1. (i)

$\alpha_{0},\ldots,\alpha_{r}$ and $(\lambda_{k})_{k}$ satisfy the
$\tilde{q}$-criterion;

  2. (ii)

for all $k\in\mathbb{N}$, $q(\mu_{k},0)\neq 0$;

  3. (iii)

for all $k\in\mathbb{N}$, with $t_{k}\coloneqq\tanh(\mu_{k}h)$,
$p_{\alpha}(t_{k})\neq 0$;

  4. (iv)

for all $k\in\mathbb{N}$, $q(\mu_{k},0)\neq 0$ and $\chi(\alpha)\neq 0$;

  5. (v)

for all $k\in\mathbb{N}$, with $t_{k}\coloneqq\tanh(\mu_{k}h)$,
$p_{\alpha}(t_{k})\neq 0$ and $\chi(\alpha)\neq 0$.

Then
(v)$\Leftrightarrow$(iv)$\Rightarrow$(i)$\Rightarrow$(ii)$\Leftrightarrow$(iii).

Finally, we come to a rather handy sufficient condition for the satisfaction
of the $\tilde{q}$-criterion.

######  Corollary 7.6.

Let $\alpha_{0},\ldots,\alpha_{r}\in\mathbb{R}\setminus\\{0\\}$,
$(\lambda_{k})_{k}$ in $\mathbb{R}_{>0}$ discrete; $h=1/({r+1})$,
$0<t_{0}\leqslant\tanh(\min_{k\in\mathbb{N}}\sqrt{\lambda_{k}}h)$. If for all
$t\in[t_{0},1]$, $p_{\alpha}(t)\neq 0,$ then $\alpha_{0},\ldots,\alpha_{r}$
and $(\lambda_{k})_{k}$ satisfy the $\tilde{q}$-criterion.

###### Proof.

The statement follows from Theorem 7.5 since $p_{\alpha}(1)=\chi(\alpha)$. ∎

##  8 The inner spectrum in $d$ dimensions

In this section we want to apply the well-posedness criterion from the
previous section and want to obtain some information on the inner spectrum of
$\alpha({\rm m}_{1})$ given as multiplication operator induced by the function

$(x_{1},\ldots,x_{d})\mapsto\alpha(x_{1})=\sum_{j=0}^{r}\alpha_{j}\mathbbold{1}_{[j/{r+1},(j+1)/(r+1))}(x_{1}),$

where $\alpha_{0},\ldots,\alpha_{r}\in\mathbb{R}\setminus\\{0\\}$ for the case
$\Omega=(0,1)\times\widehat{\Omega}$,
$\widehat{\Omega}\subseteq\mathbb{R}^{d-1}$ open and bounded. For one, the
inner spectrum is real and, two, bounded above and below by
$\max_{j}|\alpha_{j}|$ (both statements follow from classical Lax–Milgram). We
use the results obtained in the previous sections to deduce that it contains
the actual values of $\alpha$ (see Corollary 6.3) and that the discrete part
of the spectrum is countable. The latter follows from the representation of
the inner spectrum as zeros of certain rational functions. Recall for $t>0$

$p_{\alpha}(t)=\begin{pmatrix}1&\alpha_{r}^{-1}t\end{pmatrix}\begin{pmatrix}1&\alpha_{r-1}^{-1}t\\\
\alpha_{r-1}t&1\end{pmatrix}\cdots\begin{pmatrix}1&\alpha_{1}^{-1}t\\\
\alpha_{1}t&1\end{pmatrix}\begin{pmatrix}t\\\ \alpha_{0}\end{pmatrix}.$

Corollary 7.6 confirms that if $p_{\alpha}$ has no zeros for $t\in(0,1]$, then
$\alpha_{0},\ldots,\alpha_{r}$ and any discrete $(\lambda_{k})_{k}$ in
$\mathbb{R}_{>0}$ satisfy the $\tilde{q}$-criterion and thus the problem of
finding $u\in H_{0}^{1}(\Omega)$ such that for given $f\in L_{2}(\Omega)$ and
$\alpha$ piecewise constant $\alpha_{j}$ on slabs of width $1/(r+1)$ we have

$-\Delta_{\alpha}=-\overline{\operatorname{div}\alpha({\rm
m}_{1})\operatorname{grad}}\,u=f$

admits a unique solution $u$ continuously depending on $f$.

For the next statement, we recall that in a metric space $X$, a set
$U\subseteq X$ is called nowhere dense, if
$\operatorname{int}(\overline{U})=\emptyset$. As a consequence, if
$X=[-A,A]\subseteq\mathbb{R}$ for some $A>0$ and $\overline{U}$ is countable,
$U$ is nowhere dense.

######  Theorem 8.1.

Let $\alpha_{0},\ldots,\alpha_{r}\in\mathbb{R}\setminus\\{0\\}$,
$A>\max_{j\in\\{0,\ldots,r\\}}|\alpha_{j}|$. Then for all $(\lambda_{k})_{k}$
in $\mathbb{R}_{>0}$ discrete, the set

$W\coloneqq\\{s\in[-A,A];\text{$\alpha_{0}-s,\ldots,\alpha_{r}-s$ and
$(\lambda_{k})_{k}$ satisfy the
$\tilde{q}$-criterion}\\}\setminus\\{\alpha_{0},\ldots,\alpha_{r}\\}$

as a subset of $[-A,A]$ is the complement of a nowhere dense, countable set.

###### Proof.

We define for all $t>0$ (using the obvious meaning for $\alpha-s$)

$[-A,A]\setminus\\{\alpha_{0},\ldots,\alpha_{r}\\}\ni s\mapsto
f_{t}(s)\coloneqq p_{\alpha-s}(t).$

Let $(\lambda_{k})_{k}$ be a discrete sequence in $\mathbb{R}_{>0}$,
$\mu_{k}\coloneqq\sqrt{\lambda_{k}}$, $h=1/(1+r)$, and choose any
$0<t_{0}\leqslant\tanh(\min_{k}\mu_{k}h)$. With
$T\coloneqq\\{\tanh(\mu_{k}h);k\in\mathbb{N}\\}\cup\\{1\\}$ consider the set

$Z\coloneqq\\{(t,s)\in[t_{0},1]\times\big{(}[-A,A]\setminus\\{\alpha_{0},\ldots,\alpha_{r}\\}\big{)};\exists
t\in T:f_{t}(s)=0\\}=\bigcup_{t\in T}[f_{t}=0].$

Then $Z$ is countable as countable union of zeros of rational functions. Next,
the set

$U\coloneqq\\{s\in[-A,A];\exists t\in[t_{0},1]\colon(t,s)\in Z\\}$

is, too, countable. Indeed, if $U$ was uncountable, due to the countability of
$Z$, we find $t^{*}\in[t_{0},1]$ such that
$U^{*}=\\{s\in[-A,A];f_{t^{*}}(s)=0\\}$ is uncountable, which contradicts the
fact that $f_{t^{*}}$, being a non-constant rational function
$(f_{t^{*}}(A)\neq 0)$, has only finitely many zeros.

Define $\tilde{U}\coloneqq U\cup\\{\alpha_{0},\ldots,\alpha_{r}\\}$. We show
that $\tilde{U}$ is closed. For this, let $(s_{n})_{n}$ be a sequence in
$\tilde{U}$ converging to some $s^{*}\in[-A,A]$. If
$s^{*}\in\\{\alpha_{0},\ldots,\alpha_{r}\\}$, we are done. Otherwise, without
restriction, we may assume the existence of $\varepsilon>0$ such that
$(s_{n})_{n}$ is a sequence in
$B\coloneqq[-A,A]\setminus\big{(}\cup_{j=0}^{r}(\alpha_{j}-\varepsilon,\alpha_{j}+\varepsilon)\big{)}$;
as a consequence $s^{*}\in B$. Then, for all $n\in\mathbb{N}$ we find
$t(s_{n})\in T$ such that $f_{t(s_{n})}(s_{n})=0$. Since $(\mu_{k})_{k}$ is
discrete possibly accumulating at $\infty$, we infer that
$T\subseteq[t_{0},1]$ is closed; hence compact. Consequently, we find a
subsequence $(t(s_{\pi(n)}))_{n}$ converging to some $t^{*}\in T$. By
continuity of $B\times T\ni(s,t)\mapsto f_{t}(s)$, $f_{t^{*}}(s^{*})=0$ and,
therefore, $s^{*}\in U$. Thus, $\tilde{U}$ is countable and closed.

Finally, note that, for all $k\in\mathbb{N}$ and $s\in
V\coloneqq[-A,A]\setminus\tilde{U}$, $f_{t_{k}}(s)\neq 0$ and $f_{1}(s)\neq 0$
where $t_{k}=\tanh(\mu_{k}h)$; hence $V\subseteq W$, by Theorem 7.5. As
$\tilde{U}$ is both countable and closed, the assertion follows. ∎

We are now in the position to describe the inner spectrum for laminated
metamaterials of the form discussed in this manuscript; we recall
$g_{0}(\Omega)=\operatorname{grad}[H_{0}^{1}(\Omega)]$.

######  Theorem 8.2.

Let $\alpha_{0},\ldots,\alpha_{r}\in\mathbb{R}\setminus\\{0\\}$ and
$\widehat{\Omega}\subseteq\mathbb{R}^{d-1}$ open and bounded;
$\Omega\coloneqq(0,1)\times\widehat{\Omega}$, $A>\max_{j}|\alpha_{j}|$. Then

$\sigma_{g_{0}(\Omega),\textnormal{c}}(\alpha({\rm m}_{1}))\subseteq[-A,A]$

and there exists a countable and nowhere dense set
$\\{\alpha_{0},\ldots,\alpha_{r}\\}\subseteq\Sigma\subseteq[-A,A]$ such that

$\sigma_{g_{0}(\Omega)}(\alpha({\rm
m}_{1}))\setminus\sigma_{g_{0}(\Omega),\textnormal{c}}(\alpha({\rm
m}_{1}))=\Sigma.$

###### Proof.

Since $\alpha_{j}\in\mathbb{R}$, for any $s\in\mathbb{C}\setminus\mathbb{R}$,
$\gamma(\alpha({\rm m}_{1})-s)$ is coercive for a suitable $\gamma$, thus
showing that $\sigma_{g_{0}(\Omega)}(\alpha({\rm m}_{1}))\subseteq\mathbb{R}$.
Similarly, it follows that $\sigma_{g_{0}(\Omega)}(\alpha({\rm
m}_{1}))\subseteq[-A,A]$. By Theorem 6.5 there exists a discrete sequence
$(\lambda_{k})_{k}$ in $\mathbb{R}_{>0}$ such that $-\Delta_{\alpha}$ is
unitarily equivalent to $(D_{\alpha}+\alpha\lambda_{k})_{k}$. From Theorem 8.1
it follows that

$W=\\{s\in[-A,A];\text{$\alpha_{0}-s,\ldots,\alpha_{r}-s$ and
$(\lambda_{k})_{k}$ satisfy the
$\tilde{q}$-criterion}\\}\setminus\\{\alpha_{0},\ldots,\alpha_{r}\\}$

is the complement of a closed and countable set. Note that
$\mathfrak{m}((\alpha-s)^{-1})=0$ only for finitely many
$s_{1},\ldots,s_{k}\in\mathbb{R}$; recall $\rho_{g_{0}(\Omega)}(\alpha({\rm
m}_{1}))=\mathbb{C}\setminus\sigma_{g_{0}(\Omega)}(\alpha({\rm m}_{1}))$. By
Corollary 7.2 and Theorem 6.1 (b), we obtain that

$W\setminus\\{s_{1},\ldots,s_{k}\\}\subseteq\big{(}\rho_{g_{0}(\Omega)}(\alpha({\rm
m}_{1}))\cap[-A,A]\big{)}\cup\sigma_{g_{0}(\Omega),\textnormal{c}}(\alpha({\rm
m}_{1}))$

Since, by Theorem 8.1, the left-hand side is the complement of a countable and
nowhere dense set, so is the right-hand side. Therefore,
$\sigma_{g_{0}(\Omega)}(\alpha({\rm
m}_{1}))\setminus\sigma_{g_{0}(\Omega),\textnormal{c}}(\alpha({\rm
m}_{1}))\subseteq[-A,A]$ is countable and nowhere dense. The inclusion of
$\\{\alpha_{0},\ldots,\alpha_{r}\\}$ in the spectrum is shown in Corollary
6.3. ∎

In the next couple of examples we explicitly compute $p_{\alpha+s}(t)$ for the
sign-changing coefficients $\alpha_{j}=(-1)^{j}$.

######  Example 8.3.

Let $r\in\mathbb{N}$ odd, $(\alpha_{j})_{j}=((-1)^{j})_{j}$. Then we obtain
for $s\neq\pm 1$

$\displaystyle p_{\alpha+s}(t)$
$\displaystyle=\begin{pmatrix}1&(-1+s)^{-1}t\end{pmatrix}\begin{pmatrix}1&(1+s)^{-1}t\\\
(1+s)t&1\end{pmatrix}\cdots\begin{pmatrix}1&(-1+s)^{-1}t\\\
(-1+s)t&1\end{pmatrix}\begin{pmatrix}t\\\ 1+s\end{pmatrix}$
$\displaystyle=\begin{pmatrix}1&\frac{1}{-1+s}t\end{pmatrix}\begin{pmatrix}1+\frac{-1+s}{1+s}t^{2}&(\frac{1}{1+s}+\frac{1}{-1+s})t\\\
2st&1+\frac{1+s}{-1+s}t^{2}\end{pmatrix}^{(r-1)/2}\begin{pmatrix}t\\\
1+s\end{pmatrix}$

In particular, if $s=0$, we obtain

$\displaystyle p_{\alpha}(t)$
$\displaystyle=\begin{pmatrix}1&-t\end{pmatrix}\begin{pmatrix}1-t^{2}&0\\\
0&1-t^{2}\end{pmatrix}^{(r-1)/2}\begin{pmatrix}t\\\ 1\end{pmatrix}$
$\displaystyle=(1-t^{2})^{(r-1)/2}(t-t)$ $\displaystyle=0,$

which confirms the irregular situation one is confronted with in [5].

Continuing the latter example, we shall now address whether the set $W$ in
Theorem 8.1 is the entire interval (except $s=0$). It turns out that the
answer depends on the number of slabs in the crosswalk.

######  Example 8.4.

Let $r\in\mathbb{N}$ odd, $(\alpha_{j})_{j}=((-1)^{j})_{j}$. Then we obtain
for $s\in(-1,1)$

$\displaystyle p_{\alpha+s}(t)$
$\displaystyle=\begin{pmatrix}1&\frac{1}{-1+s}t\end{pmatrix}\begin{pmatrix}1+\frac{-1+s}{1+s}t^{2}&(\frac{1}{1+s}+\frac{1}{-1+s})t\\\
2st&1+\frac{1+s}{-1+s}t^{2}\end{pmatrix}^{(r-1)/2}\begin{pmatrix}t\\\
1+s\end{pmatrix}$
$\displaystyle=\begin{pmatrix}1&\frac{1}{-1+s}t\end{pmatrix}\Big{(}1_{2}+\begin{pmatrix}\frac{-1+s}{1+s}t^{2}&(\frac{1}{1+s}+\frac{1}{-1+s})t\\\
2st&\frac{1+s}{-1+s}t^{2}\end{pmatrix}\Big{)}^{(r-1)/2}\begin{pmatrix}t\\\
1+s\end{pmatrix}$
$\displaystyle=\sum_{\ell=0}^{(r-1)/2}\left[\begin{matrix}(r-1)/2\\\
\ell\end{matrix}\right]\begin{pmatrix}1&\frac{1}{-1+s}t\end{pmatrix}\begin{pmatrix}\frac{-1+s}{1+s}t^{2}&(\frac{1}{1+s}+\frac{1}{-1+s})t\\\
2st&\frac{1+s}{-1+s}t^{2}\end{pmatrix}^{\ell}\begin{pmatrix}t\\\
1+s\end{pmatrix},$

where $\left[\begin{matrix}(r-1)/2\\\ \ell\end{matrix}\right]$ denotes the
binomial coefficent. Using

$\displaystyle\begin{pmatrix}\frac{-1+s}{1+s}t^{2}&(\frac{1}{1+s}+\frac{1}{-1+s})t\\\
2st&\frac{1+s}{-1+s}t^{2}\end{pmatrix}$
$\displaystyle=\frac{t}{s^{2}-1}\begin{pmatrix}(-1+s)^{2}t&(-1+s+1+s)\\\
2s(s^{2}-1)&(1+s)^{2}t\end{pmatrix}$
$\displaystyle=\frac{t}{s^{2}-1}\begin{pmatrix}(-1+s)^{2}t&2s\\\
2s(s^{2}-1)&(1+s)^{2}t\end{pmatrix},$

we obtain

$\displaystyle p_{\alpha+s}(t)$
$\displaystyle=\begin{pmatrix}1&\frac{1}{-1+s}t\end{pmatrix}\begin{pmatrix}1+\frac{-1+s}{1+s}t^{2}&(\frac{1}{1+s}+\frac{1}{-1+s})t\\\
2st&1+\frac{1+s}{-1+s}t^{2}\end{pmatrix}^{(r-1)/2}\begin{pmatrix}t\\\
1+s\end{pmatrix}$
$\displaystyle=\sum_{\ell=0}^{(r-1)/2}\left[\begin{matrix}(r-1)/2\\\
\ell\end{matrix}\right]\big{(}\frac{t}{s^{2}-1}\big{)}^{\ell}\begin{pmatrix}1&\frac{1}{-1+s}t\end{pmatrix}\begin{pmatrix}(-1+s)^{2}t&2s\\\
2s(s^{2}-1)&(1+s)^{2}t\end{pmatrix}^{\ell}\begin{pmatrix}t\\\
1+s\end{pmatrix}.$

In the simplest case, $r=1$, we obtain

$p_{\alpha+s}(t)=\begin{pmatrix}1&\frac{1}{-1+s}t\end{pmatrix}\begin{pmatrix}t\\\
1+s\end{pmatrix}=t(1+\frac{1+s}{-1+s}),$

which, for all $s\neq 0$, leads to the applicability of Corollary 7.6 and,
particularly, to the satisfaction of the $\tilde{q}$-criterion for all
discrete $(\lambda_{k})_{k}$ in $\mathbb{R}_{>0}$.

For $r=3$, however, we see that

$\displaystyle p_{\alpha+s}(t)$
$\displaystyle=\begin{pmatrix}1&\frac{1}{-1+s}t\end{pmatrix}\begin{pmatrix}t\\\
1+s\end{pmatrix}+\frac{t}{s^{2}-1}\begin{pmatrix}1&\frac{1}{-1+s}t\end{pmatrix}\begin{pmatrix}(-1+s)^{2}t&2s\\\
2s(s^{2}-1)&(1+s)^{2}t\end{pmatrix}\begin{pmatrix}t\\\ 1+s\end{pmatrix}$
$\displaystyle=t(1+\frac{1+s}{-1+s})+\frac{t}{s^{2}-1}\begin{pmatrix}(-1+s)^{2}t+2st(1+s)&2s+2st\frac{(1+s)^{2}}{-1+s}t\end{pmatrix}\begin{pmatrix}t\\\
1+s\end{pmatrix}$
$\displaystyle=\frac{4st(s^{2}-1+(s^{2}+1)t^{2})}{(-1+s)^{2}(1+s)}.$

For $s\neq 0$ and $|s|<1$, we obtain

$z(s)\coloneqq\sqrt{\frac{1-s^{2}}{1+s^{2}}}$

as the only strictly positive zero of $p_{\alpha+s}$ in $(0,1)$. In
particular, since for $s\to 1$, we get $z(s)\to 0$ and, for $s\to 0$, $z(s)\to
1$, we deduce that for all $(\mu_{k})_{k}$ in $\mathbb{R}_{>0}$ discrete with
$\mu_{k}\to\infty$ and $\varepsilon_{0}>0$, we find
$s_{0}\in[-\varepsilon_{0},\varepsilon_{0}]\setminus\\{0\\}$ so that
$z(s_{0})=\tanh(\mu_{k}h)$ for some $k\in\mathbb{N}$. In particular, using
Theorem 7.5, we find a sequence $(s_{\ell})_{\ell}$ in
$[-\varepsilon_{0},\varepsilon_{0}]\setminus\\{0\\}$ such that the
$\tilde{q}$-criterion is violated for $\alpha+s_{\ell}$ and
$(\lambda_{k})_{k}$. Hence, $\Sigma$ from Theorem 8.2 accumulates at $0$
showing it to be in the essential spectrum. In particular, in contrast to
$r=1$, $0$ is _not_ an isolated inner spectral point.

The previous example shows that the statement in Theorem 8.1 is optimal, that
is, in the case $\mathfrak{m}(\alpha^{-1})=0$, one cannot hope for that $0$ is
isolated in $\sigma_{g_{0}(\Omega)}(\alpha({\rm m}_{1}))$. On the other hand,
unfortunately, the information on the inner continuous spectrum is rather
coarse, which emphasises the importance of the conjecture mentioned in Remark
6.2.

In order to study highly oscillatory sign-changing coefficients, we provide
some elements from operator theory next.

##  9 Classical $G$-convergence

We recall the concept of $G$-convergence, see [47, 48]. We let
$\Omega\subseteq\mathbb{R}^{d}$ open and bounded, and recall $\iota_{0}\colon
g_{0}(\Omega)\hookrightarrow L_{2}(\Omega)^{d}$ from (1). We introduce the set

$M(\Omega)\coloneqq\\{\tilde{a}\in L_{\infty}(\Omega)^{d\times
d};0\in\rho(\iota_{0}^{*}\tilde{a}\iota_{0})\\},$

where we identify $\tilde{a}\in M(\Omega)$ with the corresponding
multiplication operator on $L_{2}(\Omega)^{d}$. The restriction of $M(\Omega)$
to symmetric $L_{\infty}$-matrix valued functions will be denoted by
$M_{\operatorname{sym}}(\Omega)$.

By Theorem 3.1, if $\tilde{a}\in M(\Omega)$, for any given $f\in
H^{-1}(\Omega)$, the divergence form problem

$-\operatorname{div}\tilde{a}\operatorname{grad}u=f$

is uniquely solvable for $u\in H_{0}^{1}(\Omega)$. Now, if
$(\tilde{a}_{n})_{n}$ in $M(\Omega)$ and $\tilde{a}\in M(\Omega)$, we say that
$(\tilde{a}_{n})_{n}$ $G$-converges to $\tilde{a}$, if for all $f\in
H^{-1}(\Omega)$ and $u_{n}\in H_{0}^{1}(\Omega)$ with

$-\operatorname{div}\tilde{a}_{n}\operatorname{grad}u_{n}=f$

implies $u_{n}\to u$ weakly in $H_{0}^{1}(\Omega)$ and

$-\operatorname{div}\tilde{a}\operatorname{grad}u=f.$

######  Remark 9.1.

The concept of $G$-convergence is originally introduced for sequences
contained in the set

$M_{\operatorname{sym}}(\alpha,\beta,\Omega)\coloneqq\\{\tilde{a}\in
L_{\infty}(\Omega)^{d\times
d};\tilde{a}(x)=\tilde{a}(x)^{*}\geqslant\alpha,\|\tilde{a}\|\leqslant\beta\\},$

where $0<\alpha<\beta$. It can be shown that the $G$-limit is unique in
$M_{\operatorname{sym}}(\alpha,\beta,\Omega)$. We address uniqueness in
$M_{\operatorname{sym}}(\Omega)$ after the following characterisation of
$G$-convergence.

The basic operator theoretic description for $G$-convergence observed in [61]
and implicitly contained in [50] reads as follows, see also [51]. Again, it
uses the representation of the solution for divergence form problems provided
in Theorem 3.1. We will provide a slightly more general result in Theorem 9.3
below as it merely requires a closer inspection of the proof of the main
result in [61].

######  Theorem 9.2 ([61]).

Let $0<\alpha<\beta$ and $(\tilde{a}_{n})_{n}$ in
$M_{\operatorname{sym}}(\alpha,\beta,\Omega)$, $\tilde{a}\in
M_{\operatorname{sym}}(\alpha,\beta,\Omega)$. Then the following conditions
are equivalent:

  1. (i)

$(\tilde{a}_{n})_{n}$ $G$-converges to $\tilde{a}$;

  2. (ii)

$(\iota_{0}^{*}\tilde{a}_{n}\iota_{0})^{-1}\to(\iota_{0}^{*}\tilde{a}\iota_{0})^{-1}$
in the weak operator topology.

A (straightforward) consequence of the proof of Theorem 9.2 is the following
statement, which does not require coercivity conditions (nor the symmetry).

######  Theorem 9.3.

Let $(\tilde{a}_{n})_{n}$ in $M(\Omega)$, $\tilde{a}\in M(\Omega)$. Then the
following conditions are equivalent:

  1. (i)

$(\tilde{a}_{n})_{n}$ $G$-converges to $\tilde{a}$;

  2. (ii)

$(\iota_{0}^{*}\tilde{a}_{n}\iota_{0})^{-1}\to(\iota_{0}^{*}\tilde{a}\iota_{0})^{-1}$
in the weak operator topology.

###### Proof.

Let $f\in H^{-1}(\Omega)$. For $n\in\mathbb{N}$, by Theorem 3.1,

$u_{n}=(\iota_{0}^{*}\operatorname{grad})^{-1}(\iota_{0}^{*}\tilde{a}_{n}\iota_{0})^{-1}(\operatorname{div}\iota_{0})^{-1}f.$

Since $(\iota_{0}^{*}\operatorname{grad})^{-1}$ is a topological isomorphism,
$u_{n}\to u$ weakly in $H_{0}^{1}(\Omega)$ with

$u=(\iota_{0}^{*}\operatorname{grad})^{-1}(\iota_{0}^{*}\tilde{a}_{n}\iota_{0})^{-1}(\operatorname{div}\iota_{0})^{-1}f,$

if, and only if,
$(\iota_{0}^{*}\tilde{a}_{n}\iota_{0})^{-1}(\operatorname{div}\iota_{0})^{-1}f\to(\iota_{0}^{*}\tilde{a}_{n}\iota_{0})^{-1}(\operatorname{div}\iota_{0})^{-1}f$
weakly. So, if the former convergence holds for all $f$, the latter holds for
all
$q\in(\operatorname{div}\iota_{0})^{-1}[H^{-1}(\Omega)]=\operatorname{ran}(\iota_{0})=g_{0}(\Omega)$.
The assertion follows. ∎

######  Remark 9.4.

In the situation of Theorem 9.3 consider the $L_{2}(\Omega)$ realisations
$D_{\tilde{a}_{n}}$ of the divergence form operators
$-\operatorname{div}\tilde{a}_{n}\operatorname{grad}$ initially considered as
operator from $H_{0}^{1}(\Omega)$ to $H^{-1}(\Omega)$, see also Theorem 3.1.
Then, if $\tilde{a}_{n}$ $G$-converges to $\tilde{a}$, then
$D_{\tilde{a}_{n}}^{-1}\to D_{\tilde{a}}^{-1}$ in operator norm.

Indeed, this follows using a similar technique as in the proof of [51, Theorem
4.2]. For this we observe that both $\iota_{0}^{*}\operatorname{grad}$ and
$(\iota_{0}^{*}\operatorname{grad})^{*}=-\operatorname{div}\iota_{0}$ have
compact resolvent. In consequence, for all $(f_{n})_{n}$ in $L_{2}(\Omega)$
weakly convergent to some $f$, we obtain, using the representation in Theorem
3.1,

$u_{n}\coloneqq(\iota_{0}^{*}\operatorname{grad})^{-1}(\iota_{0}^{*}\tilde{a}_{n}\iota_{0})^{-1}(-\operatorname{div}\iota_{0})^{-1}f_{n}\to
u\coloneqq(\iota_{0}^{*}\operatorname{grad})^{-1}(\iota_{0}^{*}\tilde{a}\iota_{0})^{-1}(-\operatorname{div}\iota_{0})^{-1}f$

strongly in $L_{2}(\Omega)$ (recall that compact operators map weakly
convergent sequences to strongly convergent sequences). Now, assuming that
$D_{\tilde{a}_{n}}^{-1}$ does not converge to $D_{\tilde{a}}^{-1}$ in operator
norm, we find a sequence of unit vectors $(f_{n})_{n}$ in $L_{2}(\Omega)$ such
that

$\|D_{\tilde{a}_{n}}^{-1}f_{n}-D_{\tilde{a}}^{-1}f_{n}\|\geqslant\varepsilon,$

for some $\varepsilon>0$. Re-using the index $n$ for a suitable weakly
convergent subsequence of $(f_{n})_{n}$ and using that $D_{\tilde{a}}^{-1}$ is
a compact operator, we obtain a contradiction, as we have shown in the
argument above, that we also have that $D_{\tilde{a}_{n}}^{-1}f_{n}\to
D_{\tilde{a}}^{-1}f$ strongly as $n\to\infty$.

######  Remark 9.5.

Note that $\tilde{a}\in M(\Omega)$ if and only if $\gamma\tilde{a}\in
M(\Omega)$ for all $\gamma\in\mathbb{C}$, $\gamma\neq 0$. Thus, if

$W(\tilde{a})\coloneqq\\{\langle\tilde{a}\varphi,\varphi\rangle,\varphi\in
L_{2}(\Omega)^{d},\|\varphi\|=1\\},$

the numerical range of $\tilde{a}$, is contained in a half space of
$\mathbb{C}$ and $0\notin\overline{W(\tilde{a})}$, then we find
$\gamma\in\mathbb{C}\setminus\\{0\\}$ such that
$\operatorname{Re}\gamma\tilde{a}\geqslant c$ for some $c>0$. Next, let
$(\tilde{a}_{n})_{n}$ be bounded in $M(\Omega)\subseteq
L_{\infty}(\Omega)^{d\times d}$ with $\bigcup_{n}W(\tilde{a}_{n})$ being
contained in a half space and $0\notin\overline{\bigcup_{n}W(\tilde{a}_{n})}$.
Then we find $\gamma$ and $c>0$ such that for all $n\in\mathbb{N}$,
$\operatorname{Re}\gamma\tilde{a}_{n}\geqslant c$. If $(\tilde{a}_{n})_{n}$
$G$-converges to some $\tilde{a}\in M(\Omega)$, then, by the characterisation
in Theorem 9.3, $(\gamma\tilde{a}_{n})_{n}$ $G$-converges to
$\gamma\tilde{a}$. In particular, it follows that
$\operatorname{Re}\iota_{0}^{*}\gamma\tilde{a}\iota_{0}\geqslant\tilde{c}$ for
some $\tilde{c}>0$; see also [50, Proof of Theorem 6.5].

The $G$-limit fails to be unique if one keeps the definition as is and
considers non-symmetric operators $\tilde{a}_{n}$. However, if the
coefficients are symmetric, that is, belong to the set

$M_{\operatorname{sym}}(\Omega)=\\{\tilde{a}\in
M(\Omega);\tilde{a}(x)=\tilde{a}(x)^{*}\;\text{a.e.~{}}x\in\Omega\\},$

the limit is unique:

######  Corollary 9.6 (Uniqueness of $G$-limit).

Let $(\tilde{a}_{n})_{n}$ in $M_{\operatorname{sym}}(\Omega)$ $G$-converge to
$\tilde{a}$ and $\tilde{b}$ in $M_{\operatorname{sym}}(\Omega)$. Then
$\tilde{a}=\tilde{b}$.

###### Proof.

We follow the argument in [51, p. 201]; see also [50]. By Theorem 9.3 it
follows that
$(\iota_{0}^{*}\tilde{a}\iota_{0})^{-1}=(\iota_{0}^{*}\tilde{b}\iota_{0})^{-1}$
and, thus, $\iota_{0}^{*}\tilde{a}\iota_{0}=\iota_{0}^{*}\tilde{b}\iota_{0}$.
By linearity, it suffices to show for $\tilde{a}\in
M_{\operatorname{sym}}(\Omega)$ that $\iota_{0}^{*}\tilde{a}\iota_{0}=0$
implies $\tilde{a}=0$. For this, take $\tau\in C_{c}^{\infty}(\Omega)$ and
$\xi\in\mathbb{R}^{d}$. Then define for $\lambda\geqslant 0$,
$u_{\lambda}(x)\coloneqq\tau(x){\rm e}^{\mathrm{i}\lambda x\xi}$, it follows
that $u_{\lambda}\in H_{0}^{1}(\Omega)$ and, by assumption,

$\displaystyle
0=\langle\tilde{a}\operatorname{grad}u_{\lambda},\operatorname{grad}u_{\lambda}\rangle=\langle\tilde{a}(\operatorname{grad}\tau+\mathrm{i}\lambda\xi\tau),(\operatorname{grad}\tau+\mathrm{i}\lambda\xi\tau)\rangle.$

Dividing by $\lambda^{2}$ and letting $\lambda\to\infty$, we deduce

$0=\int_{\Omega}|\tau(x)|^{2}\langle\tilde{a}(x)\xi,\xi\rangle\mathrm{d}x$

This leads to $\langle\tilde{a}(x)\xi,\xi\rangle=0$ for a.e. $x\in\Omega$ and,
by the separability of $\mathbb{R}^{d}$ and $\tilde{a}=\tilde{a}^{*}$, we
infer $\tilde{a}=0$. ∎

######  Remark 9.7.

The uniqueness statement can still be obtained for $H$-convergence replacing
$G$-convergence, see [33]. We shall not discuss this here since for our
purposes the above uniqueness statement is sufficient.

##  10 Holomorphic $G$-convergence

This section is devoted to generalise the concept of $G$-convergence in order
to deal with operators that are not necessarily invertible.

In order to provide the broader context of the results to come, we present an
example highlighting the intricacies of the weak operator topology. The
example is based on the following well-known result.

######  Theorem 10.1 (see, e.g., [23] or [45, Theorem 13.2.4 and Remark
13.2.5]).

Let $\Omega\subseteq\mathbb{R}^{d}$ open, $f\colon\mathbb{R}^{d}\to\mathbb{C}$
$1$-periodic (i.e., for all $k\in\mathbb{Z}^{d}$, $f(\cdot+k)=f$), bounded,
measurable. Then

$f(n\cdot)\to\int_{[0,1)^{d}}f(x)dx$

as $n\to\infty$ in $\sigma(L_{\infty}(\Omega),L_{1}(\Omega))$, the
weak*-topology of $L_{\infty}(\Omega)$.

The following remark recalls a well-known fact about the relationship between
the weak*-topology of $L_{\infty}(\Omega)$ and the weak operator topology on
$L(L_{2}(\Omega))$.

######  Remark 10.2.

For a bounded, measurable function $f\colon\Omega\to\mathbb{C}$ we denote the
associated multiplication operator by

$f({\rm m})\colon L_{2}(\Omega)\to L_{2}(\Omega),\varphi\mapsto[x\mapsto
f(x)\varphi(x)].$

Then it is easy to see that the mapping

$(L_{\infty}(\Omega),\sigma(L_{\infty}(\Omega),L_{1}(\Omega)))\ni f\mapsto
f({\rm m})\in(L(L_{2}(\Omega)),\tau_{\textnormal{w}})$

is a homeomorphism onto its image, see also [45, Proposition 13.2.1]. Indeed,
this follows from the fact that any $L_{1}$-function can be written as a
product of two $L_{2}$-functions and that, by the Cauchy–Schwarz inequality,
the product of any two $L_{2}$-functions is in $L_{1}$.

We come to the announced example showing that convergence of inverses in the
weak operator does not imply convergence of the respective resolvents.

######  Example 10.3.

(a) Let
$f=\frac{1}{2}\mathbbold{1}_{(0,1/2]}+\frac{1}{3}\mathbbold{1}_{(1/2,1]}$.
Periodically extend $f$ to whole of $\mathbb{R}$. Define $f_{n}\coloneqq
f(n\cdot).$ Then for all $\lambda$, sufficiently small, we have

$(f_{n}({\rm
m})+\lambda)^{-1}\to\int_{0}^{1}\frac{1}{f(x)+\lambda}dx=\frac{1}{2}(\frac{1}{1/2+\lambda}+\frac{1}{1/3+\lambda}).$

Note that for $\lambda=0$, we have

$f_{n}({\rm m})^{-1}\to\frac{5}{2}.$

(b) Define

$g_{n}\coloneqq\begin{cases}f_{n}({\rm m}),&n\text{ odd},\\\
\frac{2}{5},&n\text{ even}.\end{cases}$

Then $g_{n}^{-1}$ converges in the weak operator topology to $\frac{5}{2}$.
However, for $\lambda\neq 0$ we have

$\frac{1}{2}(\frac{1}{1/2+\lambda}+\frac{1}{1/3+\lambda})\neq(\frac{2}{5}+\lambda)^{-1}.$

Even though, we have just seen that the convergence of the resolvents is not
necessarily implied by the convergence of the inverses, we shall provide an
elementary characterisation next. This statement provides a link between
resolvents and inverses; which itself is a consequence of Cauchy’s integral
formula.

######  Proposition 10.4.

Let $(a_{n})_{n}$ in $L(H)$, $b\in L(H)$. Then the following conditions are
equivalent:

  1. (i)

$a_{n}\to b$ in the weak operator topology;

  2. (ii)

for all $n\in\mathbb{N}$ there exists
$f_{n}\colon\operatorname{dom}(f_{n})\subseteq\mathbb{C}\to L(H)$ holomorphic
in a neighbourhood of $0$ with $f_{n}(0)=a_{n}$ and

$\int_{\gamma_{n}}f_{n}(\lambda)\frac{1}{\lambda}d\lambda\to\int_{\gamma_{\infty}}f_{\infty}(\lambda)\frac{1}{\lambda}d\lambda$

in the weak operator topology, where
$f_{\infty}\colon\operatorname{dom}(f_{\infty})\subseteq\mathbb{C}\to L(H)$
holomorphic around $0$ with $f_{\infty}(0)=b$ and $\gamma_{n}$ is a closed,
piecewise continuously differentiable path in $\operatorname{dom}(f_{n})$ with
$\operatorname{ind}_{0}\gamma_{n}=1$ for all $n\in\mathbb{N}\cup\\{\infty\\}$.

###### Proof.

(i)$\Rightarrow$(ii) Choose $f_{n}(\lambda)\coloneqq
a_{n}$,$f_{\infty}(\lambda)\coloneqq b$, and any paths $\gamma_{n}$,
$\gamma_{\infty}$ with the described properties in $\mathbb{C}$.

(ii)$\Rightarrow$(i) By Cauchy’s integral formula,
$\int_{\gamma_{n}}f_{n}(\lambda)\frac{1}{\lambda}d\lambda=a_{n}$ and
$\int_{\gamma_{\infty}}f_{\infty}(\lambda)\frac{1}{\lambda}d\lambda=b$. ∎

This observation provides a way of generalising convergence in the weak
operator topology. The major application, we have in mind, of the above is
$f_{n}(\lambda)=(a_{n}-\lambda)^{-1}$ for $a_{n}$ invertible. In the above
theorem, if we had $(a_{n}-\lambda)^{-1}\to(a-\lambda)^{-1}$ in the weak
operator topology for all $\lambda$ in a sufficiently small neighbourhood of
$0$, we immediately get that $a_{n}^{-1}\to a^{-1}$ in the weak operator
topology. The above example however shows that the weak operator topology
limit of resolvents is not necessarily a resolvent again. Moreover, we are
interested in cases for $a_{n}$ not necessarily invertible. Thus, the
functions $f_{n}$ we are interested in have a singularity at $0$. Even worse,
the singularities are not isolated at $0$ (uniformly in $n$).

For a Hilbert spaces $H_{1},H_{2}$ and an open $\omega\subseteq\mathbb{C}$, we
introduce the set

$\mathcal{H}(\omega;L(H_{1},H_{2}))\coloneqq\\{f\colon\omega\to
L(H_{1},H_{2});f\text{ holomorphic}\\}$

of operator-valued holomorphic functions. Let $(f_{n})_{n}$ in
$\mathcal{H}(\omega;L(H_{1},H_{2}))$. We call $(f_{n})_{n}$ locally bounded or
normal, if for all $K\subseteq\omega$ compact,
$\sup_{n\in\mathbb{N}}\|f_{n}\|_{\infty,K}<\infty$, where
$\|f\|_{\infty,K}\coloneqq\sup_{\lambda\in K}\|f(\lambda)\|_{L(H_{1},H_{2})}$
for all $f\in\mathcal{H}(\omega;L(H_{1},H_{2}))$.

$(f_{n})_{n}$ in $\mathcal{H}(\omega;L(H_{1},H_{2}))$ is said to converge in
the compact open weak operator topology to some $f$,
$f_{n}\stackrel{{\scriptstyle\textnormal{co-}\tau_{\textnormal{w}}}}{{\to}}f$
or
$f=\textnormal{co-}\tau_{\textnormal{w}}\textnormal{-}\lim_{n\to\infty}f_{n}$,
if for all $\varphi,\psi\in H$

$\langle\varphi,f_{n}(\cdot)\psi\rangle\to\langle\varphi,f(\cdot)\psi\rangle$

in the compact open topology of uniform convergence on compact sets. If
$H_{2}=\mathbb{C}$, then $L(H_{1})=H_{1}$, by the Riesz representation
theorem, and we say compact open weak topology instead of compact open weak
operator topology and use $\stackrel{{\scriptstyle\textnormal{co-w}}}{{\to}}$
and $\textnormal{co-w-}\lim$ as suitable symbols.

With the help of Tikhonov’s theorem in conjunction with Montel’s theorem as
well as using Riesz representation and Dunford’s theorem characterising
vector-valued holomorphic functions, one can show Montel’s theorem also for
the compact open weak operator topology. For general Hilbert spaces, it is
only possible to provide the existence of an accumulation point. If $H$ is
separable, the full statement of the scalar version of Montel’s theorem can be
recovered:

######  Theorem 10.5 (Montel’s theorem, [58, Theorem 3.4], or [57]).

Let $H_{1},H_{2}$ be a separable Hilbert spaces, $\omega\subseteq\mathbb{C}$
open. $(f_{n})_{n}$ a normal family of holomorphic functions from $\omega$ to
$L(H_{1},H_{2})$. Then there exists a subsequence $(f_{\pi(n)})_{n}$ and
$f\colon\omega\to L(H_{1},H_{2})$ with
$\textnormal{co-}\tau_{\textnormal{w}}\textnormal{-}\lim_{n\to\infty}f_{\pi(n)}=f$.

###### Proof.

In [58, Theorem 3.4] it has been shown that uniformly bounded sequences admit
a convergent subsequence in the compact open weak operator topology. Using a
standard exhaustion of $\omega$ with open sets being compactly contained in
$\omega$, the statement follows upon applying a standard diagonal procedure. ∎

The abstract concepts now help to generalise convergence in the weak operator
topology of inverses of bounded linear operators. For this let $H$ be a
separable Hilbert space and $(a_{n})_{n}$ a sequence in $L(H)$ and assume
there exists $\omega\subseteq\bigcap_{n\in\mathbb{N}}\rho(a_{n})$ open in
$\mathbb{C}$ with $0\in\overline{\omega}$ such that $(a_{n}-\cdot)^{-1}$ is
locally bounded on $\omega$. Let $a\subseteq H\times H$ be a relation. Then we
say that $(a_{n})_{n}$ holomorphically $G$-converges (on $\omega$) to $a$, if
the set $\mathcal{G}$ of $\varphi\in H$ satisfying the following two
conditions (a) and (b) is dense in $H$, where

  1. (a)

there exists $f_{\varphi}\colon\omega\to H$ holomorphic, such that
$(a_{n}-\cdot)^{-1}\varphi\to f_{\varphi}$ in the compact open weak topology,

  2. (b)

$f_{\varphi}$ admits a holomorphic extension to $0$;

and

$a^{-1}=\\{(\varphi,\psi)\in\mathcal{G}\times H;\psi=f_{\varphi}(0)\\}.$

######  Remark 10.6.

(a) By the identity theorem, for $\varphi\in\mathcal{G}$, the holomorphic
extension of $f_{\varphi}$ in a neighbourhood of zero is uniquely determined
from its values on $\omega$ since $0\in\overline{\omega}$. Thus, the
holomorphic $G$-limit is unique.

(b) The holomorphic $G$-limit may depend on $\omega$. If, however,

$\widehat{\omega}\coloneqq\\{\lambda\in\bigcap_{n\in\mathbb{N}}\rho(a_{n});(a_{n}-\lambda)^{-1}\text{
bounded}\\}$

consists of only one connected component and $(a_{n})_{n}$ is bounded, then
the limit is independent of the choice of $\omega$. This property will
_always_ be satisfied in the applications considered in the present
manuscript.

Let us prove the just mentioned uniqueness statement. Let $(a_{n})_{n}$
holomorphically $G$-converge on $\omega_{1}$ to $a$ and on $\omega_{2}$ to
$b$. Denote $\mathcal{G}_{a}\coloneqq\operatorname{dom}(a^{-1})$ and
$\mathcal{G}_{b}\coloneqq\operatorname{dom}(b^{-1})$. By definition,
$(a_{n}-\cdot)^{-1}\varphi\eqqcolon
f_{n,\varphi}\stackrel{{\scriptstyle\textnormal{co-w}}}{{\to}}f_{\varphi}^{(a)}$
on $\omega_{1}$ and $(a_{n}-\cdot)^{-1}\psi\eqqcolon
f_{n,\psi}\stackrel{{\scriptstyle\textnormal{co-w}}}{{\to}}f_{\psi}^{(b)}$ on
$\omega_{2}$ for all $\varphi\in\mathcal{G}_{a}$ and $\psi\in\mathcal{G}_{b}$.
By the Vitali Convergence Theorem ([49, Theorem 6.2.8]) (carried out weakly),
for all $\varphi\in\mathcal{G}_{a}$,
$(a_{n}-\cdot)^{-1}\varphi=f_{n,\varphi}\stackrel{{\scriptstyle\textnormal{co-w}}}{{\to}}f_{\varphi}$
on the whole of $\widehat{\omega}$ for some
$f_{\varphi}\colon\widehat{\omega}\to H$ holomorphiccally extending
$f_{\varphi}^{(a)}$. In particular,
$f_{n,\varphi}\stackrel{{\scriptstyle\textnormal{co-w}}}{{\to}}f_{\varphi}$ on
$\omega_{2}$. By the identity theorem, it follows that
$f_{\varphi}=f_{\varphi}^{(b)}$ and $\varphi\in\mathcal{G}_{b}$ as well as
$(\varphi,f_{\varphi}(0))\in b^{-1}$. Thus, $a^{-1}\subseteq b^{-1}$. By
symmetry, $a=b$.

(c) Given the knowledge of $a_{n}^{-1}\to a^{-1}$ in the weak operator
topology does not imply holomorphic $G$-convergence of $(a_{n})_{n}$ to $a$ as
Example 10.3(b) confirms. It is however possible to still obtain a suitable
equivalence statement as Theorem 10.8 below shows.

(d) If $(a_{n})_{n}$ holomorphically $G$-converges to $a$ with
$\mathcal{G}_{a}=H$ and if there exists $\varepsilon>0$ such that
$f_{\varphi}$ is holomorphic on $B(0,\varepsilon)$ for all $\varphi\in H$.
Then, by Dunford’s theorem, $f_{n}=(a_{n}-\cdot)^{-1}$ converges in the
compact open weak operator topology on $B(0,\varepsilon)$.

(e) If $\widehat{\omega}$ from (b) satisfes
$B(0,\varepsilon)\subseteq\widehat{\omega}$ for some $\varepsilon>0$ and if
$(a_{n})_{n}$ holomorphically $G$-converges to $a$, then as
$(a_{n}-\cdot)^{-1}$ is a normal family on $B(0,\varepsilon)$, the density of
$\mathcal{G}_{a}$ yields $\mathcal{G}_{a}=H$. Hence, by (d),
$(a_{n}-\cdot)^{-1}$ is convergent in the compact open weak operator topology.

######  Lemma 10.7.

Let $(a_{n})_{n}$ in $L(H)$ invertible with $(a_{n}^{-1})_{n}$ bounded. Then
there exists $\varepsilon>0$ and $C>0$ such that
$B(0,\varepsilon)\subseteq\bigcap_{n\in\mathbb{N}}\rho(a_{n})$ and

$f_{n}\colon B(0,\varepsilon)\ni\lambda\to(a_{n}-\lambda)^{-1}$

defines a normal family of holomorphic functions with
$\sup_{n}\|f_{n}\|_{\infty,B(0,\varepsilon)}\leqslant C$.

###### Proof.

Let $C\coloneqq 2\sup_{n}\|a_{n}^{-1}\|$, $\varepsilon\coloneqq 1/(2C)$. Then
for $\lambda\in B(0,\varepsilon)$ we deduce that
$(a_{n}-\lambda)=a_{n}(1-a_{n}^{-1}\lambda)$ is invertible using the Neumann
series and that

$\|(a_{n}-\lambda)^{-1}\|=\|(1-a_{n}^{-1}\lambda)^{-1}a_{n}^{-1}\|\leqslant\frac{C}{2}\sum_{k=0}^{\infty}\|a_{n}^{-k}\lambda^{k}\|\leqslant\frac{C}{2}\frac{1}{1-1/2}=C.\qed$

######  Theorem 10.8.

Let $(a_{n})_{n}$ in $L(H)$ invertible with $(a_{n}^{-1})_{n}$ bounded, $a\in
L(H)$ invertible, $H$ separable. Consider the following assertions:

  1. (i)

$(a_{n})_{n}$ holomorphically $G$-converges to $a$;

  2. (ii)

every subsequence $(a_{\pi(n)})_{n}$ contains a subsequence holomorphically
$G$-converging to $a$;

  3. (iii)

$a_{n}^{-1}\to a^{-1}$ in the weak operator topology.

Then (i)$\Rightarrow$(iii)$\Leftrightarrow$(ii).

###### Proof.

(i)$\Rightarrow$(iii) By Lemma 10.7 $((a_{n}-\cdot)^{-1})_{n}$ is a normal
family on $B(0,\varepsilon)$ for some $\varepsilon>0$ Hence, by Remark 10.6
(e) in conjunction with (i), $((a_{n}-\cdot)^{-1})_{n}$ is convergent in the
compact open weak operator topology to some $g$. Thus, $a_{n}^{-1}\to a^{-1}$
in the weak operator topology by Proposition 10.4 (ii) applied to
$f_{n}=(a_{n}-\cdot)^{-1}$, $f_{\infty}=g$ and $\gamma_{n}=\gamma_{\infty}$ be
the circle of radius $\varepsilon/2$ around $0$.

(iii)$\Rightarrow$(ii) By Lemma 10.7, $f_{n}=(a_{\pi(n)}-\cdot)^{-1}$ is a
normal family on $B(0,\varepsilon)$. By the separability of $H$ and Theorem
10.5, we may choose a subsequence $(f_{\tilde{\pi}(n)})_{n}$ converging in the
compact open weak operator topology to some holomorphic $f\colon
B(0,\varepsilon)\to L(H)$. In particular, we have

$f(0)=\tau_{\textnormal{w}}\text{-}\lim_{n\to\infty}f_{\tilde{\pi}(n)}(0)=\tau_{\textnormal{w}}\text{-}\lim_{n\to\infty}a_{\tilde{\pi}(n)}^{-1}=a^{-1},$

which shows the assertion.

(ii)$\Rightarrow$(iii) This follows from the implication (i)$\Rightarrow$(ii)
in conjunction with a subsequence principle. ∎

###### Proof of Theorem 2.4.

It suffices to apply Theorem 10.8 and to note that, by Theorem 9.2, (iii) is
equivalent to
$(\iota_{0}^{*}\tilde{a}_{n}\iota_{0})^{-1}\to(\iota_{0}^{*}\tilde{a}\iota_{0})^{-1}$
in the weak operator topology. ∎

##  11 The one-dimensional problem – homogenisation

We will apply the abstract findings from the previous section to obtain
homogenisation results for divergence form problems with possibly rapidly
sign-changing coefficients. As for the analysis concerning well-posedness, we
start with the regular case. The regular case will then provide us with the
necessary tools to understand the irregular case, where the well-posedness
criteria are violated. This violation, however, happens to be only in a very
controlled way so that holomorphic $G$-convergence can be shown.

Define $\Omega\coloneqq(0,1)$, as usual let $\iota_{0}\colon
g_{0}(\Omega)\hookrightarrow L_{2}(\Omega)$ be the canonical embedding, where
$g_{0}(\Omega)=\partial[H_{0}^{1}(\Omega)]$ and recall
$\mathfrak{m}(\beta)=\int_{0}^{1}\beta$.

######  Theorem 11.1.

Assume Setting 4.1. Let $(\alpha_{n})_{n}$ be a sequence in
$L_{\infty}(\Omega)$ such that $(\alpha_{n}^{-1})_{n}$ converges in
$\sigma(L_{\infty}(\Omega),L_{1}(\Omega))$ to some $\beta\in
L_{\infty}(\Omega)$ with $\mathfrak{m}(\beta)\neq 0$. Then

$(\iota_{0}^{*}\alpha_{n}\iota_{0})^{-1}\to\big{[}\psi\mapsto\beta\psi-\beta\frac{\mathfrak{m}(\beta\psi)}{\mathfrak{m}(\beta)}\big{]}.$

in the weak operator topology of $L(g_{0}(\Omega))$. If, in addition,
$\alpha_{\infty}\coloneqq\beta^{-1}\in L_{\infty}(\Omega)$ then
$\alpha_{n}\stackrel{{\scriptstyle G}}{{\to}}\alpha_{\infty}$.

###### Proof.

The form of the solution operator in Theorem 4.2 shows that

$(\iota_{0}^{*}\alpha_{n}\iota_{0})^{-1}=\big{[}\psi\mapsto\alpha_{n}\psi-\alpha_{n}\frac{\mathfrak{m}(\alpha_{n}\psi)}{\mathfrak{m}_{\Omega}(\alpha_{n})}\big{]}.$

Given the assumed convergence in $L_{\infty}$-weak*, we deduce the
corresponding limit (note that in particular
${\mathfrak{m}(\alpha_{n}\psi)}\to{\mathfrak{m}(\beta\psi)}$ as $n\to\infty$
since by the boundedness of $\Omega$, $L_{2}(\Omega)\hookrightarrow
L_{1}(\Omega)$. The second assertion follows from Theorem 9.3 (the additional
assumptions on $\beta$ guarantee that $\beta^{-1}\in
M_{\operatorname{sym}}(\Omega)$). ∎

######  Remark 11.2.

The treatment in Theorem 11.1 provides a complete description of the
homogenisation problem for coefficients $\alpha$ satisfying
$\alpha,\alpha^{-1}\in L_{\infty}(\Omega)$ under the additional requirement
that $\mathfrak{m}(\alpha^{-1})\neq 0$.

Let us turn to the degenerate case.

######  Theorem 11.3.

Let $\alpha\in L_{\infty}(\mathbb{R};\mathbb{R})$ be $1$-periodic with
$\alpha^{-1}\in L_{\infty}(\mathbb{R})$ and assume that for
$\Omega\coloneqq(0,1)$, $\mathfrak{m}(\alpha^{-1})=0$; define
$\alpha_{n}\coloneqq\alpha(n\cdot)$. Then for all
$\lambda>\|\alpha\|_{\infty}$ or $\lambda\in\mathbb{C}\setminus\mathbb{R}$,

$(\alpha_{n}-\lambda)^{-1}\to\mathfrak{m}((\alpha-\lambda)^{-1})$

in $\sigma(L_{\infty}(\Omega),L_{1}(\Omega))$. Moreover,

$\iota_{0}^{*}\alpha_{n}\iota_{0}\stackrel{{\scriptstyle\textnormal{hol-}G}}{{\to}}0^{-1}=\\{0\\}\times
g_{0}(\Omega),$

or, defining, $\alpha_{\infty}\coloneqq\\{0\\}\times L_{2}(\Omega)$, we have

$\alpha_{n}\stackrel{{\scriptstyle\textnormal{hol-}G}}{{\to}}\alpha_{\infty}$

and $\sigma_{g_{0}(\Omega)}(\alpha_{\infty})=\emptyset$.

###### Proof.

For the first statement it suffices to realise that $\alpha$ is $1$-periodic
and, thus, so is $(\alpha-\lambda)^{-1}$. Hence, the first assertion follows
from Theorem 10.1. Next Theorem 11.1 implies for $\lambda>\|\alpha\|_{\infty}$
or $\lambda\in\mathbb{C}\setminus\mathbb{R}$, that

$\displaystyle(\iota_{0}^{*}(\alpha_{n}-\lambda)\iota_{0})^{-1}$
$\displaystyle\to\big{[}\psi\mapsto\mathfrak{m}((\alpha-\lambda)^{-1})\psi-\mathfrak{m}((\alpha-\lambda)^{-1})\frac{\mathfrak{m}(\mathfrak{m}((\alpha-\lambda)^{-1})\psi)}{\mathfrak{m}(\mathfrak{m}((\alpha-\lambda)^{-1}))}\big{]}$
$\displaystyle=\big{[}\psi\mapsto\mathfrak{m}((\alpha-\lambda)^{-1})\psi-\mathfrak{m}((\alpha-\lambda)^{-1})\mathfrak{m}(\psi)\big{]}.$

The right-hand side defines a holomorphic function $f$, which holomorphically
extends to $0$ by $0$; which shows that
$\iota_{0}^{*}\alpha_{n}\iota_{0}\text{ holomorphically $G$-converges to }0$.
∎

##  12 The higher-dimensional problem – homogenisation

In the higher-dimensional homogenisation problem, we focus on periodic
coefficients right away. We recall here the well-known theorem on the
homogenisation problem for periodic stratified media (or, equivalently,
laminated materials) due to [33]. We only consider the case of scalar
coefficients. For $\Omega\subseteq\mathbb{R}^{d}$, we call $\tilde{a}\in
L_{\infty}(\Omega)^{d\times d}$ laminated, if there exists
$\alpha\colon\mathbb{R}\to\mathbb{R}$ such that
$\tilde{a}(x)=\operatorname{diag}(\alpha(x_{1}),\ldots,\alpha(x_{1}))$.

######  Theorem 12.1 (see [33] or [23, Theorem 5.12]).

Let $\tilde{a}\in L_{\infty}(\mathbb{R}^{d})^{d\times d}$ $1$-periodic,
laminated with $\tilde{a}(x)\geqslant c$ for a.e. $x\in\mathbb{R}^{d}$ and
some $c>0$. Define $\tilde{a}_{n}(x)\coloneqq\tilde{a}(nx)$. Then for all
bounded, open $\Omega\subseteq\mathbb{R}^{d}$, $(\tilde{a}_{n})_{n}$
$G$-converges to

$\operatorname{diag}(\mathfrak{m}(\alpha^{-1})^{-1},\mathfrak{m}(\alpha),\ldots,\mathfrak{m}(\alpha)).$

For the degenerate case, we focus on piecewise constant coefficients, that is,
laminated materials with $\alpha$ being piecewise constant as in Setting 5.2.
Before we provide the homogenisation theorem, we need a technical lemma, the
proof of which is postponed to Section 13. For the rest of this section, we
let $\Omega\coloneqq(0,1)^{d}$.

######  Lemma 12.2.

There exists a unitary operator $U\colon\ell_{2}(\mathbb{N}_{>0}^{d})\to
g_{0}(\Omega)$ such that for all $\gamma>0$ and
$\Gamma\coloneqq\operatorname{diag}(\gamma,1,\ldots,1)$ we have

$U^{*}(\iota_{0}^{*}\Gamma\iota_{0})^{-1}U=\left[(x_{k})_{k\in\mathbb{N}_{>0}^{d}}\mapsto\Big{(}\frac{\sum_{m=1}^{d}k_{m}^{2}}{\gamma
k_{1}^{2}+\sum_{m=2}^{d}k_{m}^{2}}x_{k}\Big{)}_{k\in\mathbb{N}_{>0}^{d}}\right]\in
L(\ell_{2}(\mathbb{N}_{>0}^{d}))$

To compute the inner spectrum for operators of the type $\Gamma$ is now an
easy task:

######  Theorem 12.3.

Let $\gamma>0$ and $\Gamma\coloneqq\operatorname{diag}(\gamma,1,\ldots,1)$.
Then

$\sigma_{g_{0}(\Omega)}(\Gamma)=\overline{\big{\\{}\tfrac{\gamma
k_{1}^{2}+\sum_{m=2}^{d}k_{m}^{2}}{\sum_{m=1}^{d}k_{m}^{2}};k_{1},\ldots,k_{m}\in\mathbb{N}_{>0}\big{\\}}}$

###### Proof.

The statement follows from the unitary equivalence stated in Lemma 12.2 and
standard results on the spectrum of multiplication operators, see, e.g., [45,
Theorem 2.4.7]. ∎

We present the homogenisation theorem in this situation next. For this, we let
$\alpha_{0},\ldots,\alpha_{n}\in\mathbb{R}\setminus\\{0\\}$ and define
$\alpha$ to be the $1$-periodic extension to the whole of $\mathbb{R}^{d}$ of
the function

$(x_{1},\ldots,x_{d})\mapsto\sum_{j=0}^{r}\alpha_{j}\mathbbold{1}_{[j/{r+1},(j+1)/(r+1))}(x_{1}).$

######  Theorem 12.4.

Define $\alpha_{n}\coloneqq\alpha(n\cdot)$. Then for all
$\lambda>\|a\|_{\infty}$ or $\lambda\in\mathbb{C}\setminus\mathbb{R}$, we have

$(\alpha_{n}({\rm m}_{1})-\lambda)\textnormal{ $G$-converges to
}\operatorname{diag}(\mathfrak{m}((\alpha-\lambda)^{-1})^{-1},\mathfrak{m}(\alpha-\lambda),\ldots,\mathfrak{m}(\alpha-\lambda)).$

Moreover, the following holomorphic $G$-convergence statements hold:

  1. (a)

If $\mathfrak{m}(\alpha^{-1})=0$, then

$\iota_{0}^{*}\alpha_{n}({\rm
m}_{1})\iota_{0}\stackrel{{\scriptstyle\textnormal{hol-}G}}{{\longrightarrow}}\mathbf{0}^{-1}\subseteq
g_{0}(\Omega)\times g_{0}(\Omega),$

where $\mathbf{0}$ is a densely defined, proper restriction of the relation
$g_{0}(\Omega)\times\\{0\\}$.

  2. (b)

If $\mathfrak{m}(\alpha)=0$ and $\mathfrak{m}(\alpha^{-1})\neq 0$, then

$\iota_{0}^{*}\alpha_{n}({\rm
m}_{1})\iota_{0}\stackrel{{\scriptstyle\textnormal{hol-}G}}{{\longrightarrow}}\frac{1}{\mathfrak{m}(\alpha^{-1})}\Delta_{(0,1)}(\Delta_{(0,1)^{(d-1)}})^{-1}\subseteq
g_{0}(\Omega)\times g_{0}(\Omega),$

where $\Delta_{(0,1)}$ denotes the Dirichlet–Laplace operator on $L_{2}(0,1)$
tensorised with $d-1$ copies of the identity of $L_{2}(0,1)$; similarly
$\Delta_{(0,1)^{d-1}}$ is the identity on $L_{2}(0,1)$ tensorised with the
Dirichlet–Laplacian on $L_{2}((0,1)^{d-1})$.

  3. (c)

If $\mathfrak{m}(\alpha)\neq 0$ and $\mathfrak{m}(\alpha^{-1})\neq 0$, then

$\alpha_{n}({\rm
m}_{1})\stackrel{{\scriptstyle\textnormal{hol-}G}}{{\longrightarrow}}\operatorname{diag}(\mathfrak{m}((\alpha)^{-1})^{-1},\mathfrak{m}(\alpha),\ldots,\mathfrak{m}(\alpha))$

###### Proof.

The $G$-convergence result is a special case of Theorem 12.1. We have

$\operatorname{diag}(\mathfrak{m}((\alpha-\lambda)^{-1})^{-1},\mathfrak{m}(\alpha-\lambda),\ldots,\mathfrak{m}(\alpha-\lambda))\\\
=\mathfrak{m}(\alpha-\lambda)\operatorname{diag}(\frac{\mathfrak{m}((\alpha-\lambda)^{-1})^{-1}}{\mathfrak{m}(\alpha-\lambda)},1,\ldots,1)\eqqcolon\mathfrak{m}(\alpha-\lambda)\Gamma_{\lambda}$

By Lemma 12.2 we find a unitary operator independent of
$\lambda>\|\alpha\|_{\infty}$ such that

$\displaystyle f(\lambda)\coloneqq
U^{*}(\iota_{0}^{*}\mathfrak{m}(\alpha-\lambda)\Gamma_{\lambda}\iota_{0})^{-1}U$
$\displaystyle=\frac{1}{\mathfrak{m}(\alpha-\lambda)}\left(\frac{\sum_{m=1}^{d}k_{m}^{2}}{\frac{\mathfrak{m}((\alpha-\lambda)^{-1})^{-1}}{\mathfrak{m}(\alpha-\lambda)}k_{1}^{2}+\sum_{m=2}^{d}k_{m}^{2}}\right)_{k\in\mathbb{N}^{d}_{>0}}$
$\displaystyle=\left(\frac{\sum_{m=1}^{d}k_{m}^{2}}{\frac{1}{\mathfrak{m}((\alpha-\lambda)^{-1})}k_{1}^{2}+\mathfrak{m}(\alpha-\lambda)\sum_{m=2}^{d}k_{m}^{2}}\right)_{k\in\mathbb{N}^{d}_{>0}}$
$\displaystyle=\left(\frac{\mathfrak{m}((\alpha-\lambda)^{-1})\sum_{m=1}^{d}k_{m}^{2}}{k_{1}^{2}+\mathfrak{m}((\alpha-\lambda)^{-1})\mathfrak{m}(\alpha-\lambda)\sum_{m=2}^{d}k_{m}^{2}}\right)_{k\in\mathbb{N}^{d}_{>0}}$

where we identified the multiplication operator in the right-hand side with
the function it is multiplying with.

The expression on the right-hand side yields the following convergences
eventually implying the statements in (a), (b), and (c) of the present
theorem. By the identity theorem for holomorphic functions, it suffices to
treat the real limits only. The connectedness of the considered $\omega$ in
the definition of holomorphic $G$-convergence follows from the fact that both
$\mathbb{C}\setminus\mathbb{R}$ and all $\lambda\in\mathbb{R}$, $|\lambda|$
large enough belong to the inner resolvent of $\alpha_{n}({\rm m}_{1})$, see
also Theorem 8.2.

If $\mathfrak{m}(\alpha^{-1})=0$, then, as $\lambda\to 0$ for all canonical
basis vectors $\varphi$

$f_{\varphi}(\lambda)\to 0.$

Note that the convergence does not hold for all $\varphi$.

If $\mathfrak{m}(\alpha)=0$ and $\mathfrak{m}(\alpha^{-1})\neq 0$, then, as
$\lambda\to 0$, for all $\varphi$ such that
$U\varphi\in\operatorname{dom}(\Delta_{(0,1)^{(d-1)}}(-\Delta_{(0,1)})^{-1})$

$Uf(\lambda)U^{*}U\varphi\to-\mathfrak{m}(\alpha^{-1})\Delta_{(0,1)^{(d-1)}}(-\Delta_{(0,1)})^{-1}U\varphi,$

If $\mathfrak{m}(\alpha)\neq 0$ and $\mathfrak{m}(\alpha^{-1})\neq 0$, then,
as $\lambda\to 0$,

$Uf(\lambda)U^{*}\to(\iota_{0}^{*}(\mathfrak{m}(\alpha)\Gamma_{0})\iota_{0})^{-1}$

and with

$\operatorname{diag}(\mathfrak{m}((\alpha)^{-1})^{-1},\mathfrak{m}(\alpha),\ldots,\mathfrak{m}(\alpha))=\mathfrak{m}(\alpha)\Gamma_{0}$

the assertion follows. ∎

We briefly describe the inner spectra of the respective $G$-limits. The
statement in (a) follows from the fact that the coefficient is not defined
everywhere; the equations in (b) and (c) follows from the representation of
$f(\lambda)$ in the proof of Theorem 12.4 and the identity theorem.

######  Corollary 12.5.

In the situation of Theorem 12.4, the inner spectra of the limits in the
respective cases (a), (b), and (c) are

  1. (a)

$\sigma_{g_{0}(\Omega)}(\textnormal{hol-}G\text{-}\lim_{n\to\infty}\alpha_{n}({\rm
m}_{1}))=\mathbb{C}$;

  2. (b)

$\sigma_{g_{0}(\Omega)}(\textnormal{hol-}G\text{-}\lim_{n\to\infty}\alpha_{n}({\rm
m}_{1}))=\frac{1}{\mathfrak{m}(\alpha^{-1})}\overline{\\{\frac{k_{1}^{2}}{k_{2}^{2}+\ldots+k_{d}^{2}};k_{1},\ldots,k_{d}\in\mathbb{N}_{>0}\\}}$;

  3. (c)

if, in addition, $\mathfrak{m}(\alpha)\mathfrak{m}(\alpha^{-1})>0$,

$\sigma_{g_{0}(\Omega)}(\textnormal{hol-}G\text{-}\lim_{n\to\infty}\alpha_{n}({\rm
m}_{1}))=\overline{\\{\frac{\mathfrak{m}(\alpha^{-1})\sum_{m=1}^{d}k_{m}^{2}}{k_{1}^{2}+\mathfrak{m}(\alpha^{-1})\mathfrak{m}(\alpha)\sum_{m=2}^{d}k_{m}^{2}};k_{1},\ldots,k_{m}\in\mathbb{N}_{>0}\\}}$

We provide examples for the corresponding cases.

######  Example 12.6.

(a) $n=1$, $\alpha_{j}=(-1)^{j}$, $j\in\\{0,1\\}$, leading to
$\mathfrak{m}(\alpha^{-1})=0$; for all $d\geqslant 2$.

(b) $n=2$, $\alpha_{0}=1$, $\alpha_{1}=-2$, $\alpha_{2}=1$;
$\mathfrak{m}(\alpha)=\tfrac{1}{3}(1-2+1)=0$,
$\mathfrak{m}(\alpha^{-1})=(1-\tfrac{1}{2}+1)/3=\tfrac{1}{2}\neq 0$, for all
$d\geqslant 2$.

(c) $n=2$, $\alpha_{0}=1$, $\alpha_{1}=-1$, $\alpha_{2}=1$;
$\mathfrak{m}(\alpha)=\mathfrak{m}(\alpha^{-1})=\tfrac{1}{3}$, $d=2$.

##  13 Proofs

###  13.1 Results in Section 5

We start with some (standard) auxiliary material helping to prove Theorem 5.4.

######  Remark 13.1.

(a) Let $\Omega_{-},\Omega_{+}\subseteq\mathbb{R}$ disjoint, open intervals
such that $\Omega\coloneqq\Omega_{-}\cup\\{\gamma\\}\cup\Omega_{+}$ is
connected for some $\gamma\in\mathbb{R}$. Then for $\varphi\in
H^{1}(\Omega_{-})\cap H^{1}(\Omega_{+})$ the following conditions are
equivalent:

  1. (i)

$\varphi$ extends to a function in $H^{1}(\Omega)$;

  2. (ii)

$\varphi$ extends to a continuous function on $\Omega$.

The proof can be carried out by a suitable application of integration by parts
((ii)$\Rightarrow$(i)) and the Sobolev embedding theorem
((i)$\Rightarrow$(ii)).

(b) Let $M\in\mathbb{K}^{N\times N}$ be a block operator matrix

$M=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}$

with quadratic matrices $A$ and $D$ and matrices of appropriate size $B$ and
$C$. If $A$ is invertible, then

$\det M=\det A\det(D-CA^{-1}B).$

The proof is not difficult and uses the elementary representation

$M=\begin{pmatrix}1&0\\\ CA^{-1}&1\end{pmatrix}\begin{pmatrix}A&0\\\
0&D-CA^{-1}B\end{pmatrix}\begin{pmatrix}1&A^{-1}B\\\ 0&1\end{pmatrix}.$

###### Proof of Theorem 5.4.

(i)$\Leftrightarrow$(ii): Proposition 5.1.

The remainder of the proof is concerned with establishing the equivalence
between (ii) and (iii). For this, let $u\in H_{0}^{1}(\Omega)$ be a solution
of

$D_{\alpha}u=-(\alpha u^{\prime})^{\prime}=-\beta u.$