# Asymptotic expansions of Stekloff eigenvalues for perturbations of
inhomogeneous media

Samuel Cogar Department of Mathematics, Rutgers University, Piscataway, NJ
08854<EMAIL_ADDRESS>
###### Abstract

Eigenvalues arising in scattering theory have been envisioned as a potential
source of target signatures in nondestructive testing of materials, whereby
perturbations of the eigenvalues computed for a penetrable medium would be
used to infer changes in its constitutive parameters relative to some
reference values. We consider a recently introduced modification of the class
of Stekloff eigenvalues, in which the inclusion of a smoothing operator
guarantees that infinitely many eigenvalues exist under minimal assumptions on
the medium, and we derive precise formulas that quantify the perturbation of a
simple eigenvalue in terms of the coefficients of a perturbed inhomogeneous
medium. These formulas rely on the theory of nonlinear eigenvalue
approximation and regularity results for elliptic boundary-value problems with
heterogeneous coefficients, the latter of which is shown to have a strong
influence on the sensitivity of the eigenvalues corresponding to an
anisotropic medium. A simple numerical example in two dimensions is used to
verify the estimates and suggest future directions of study.

Key words. inverse scattering, nondestructive testing, non-selfadjoint
eigenvalue problems, Laplace-Beltrami operator, nonlinear eigenvalue problems

AMS subject classifications. 35J25, 35P05, 35P25, 35R30

##  1 Introduction

A problem of interest in inverse scattering theory is to detect changes in a
penetrable medium from a knowledge of how waves are scattered from it, serving
as an example of nondestructive testing of materials. A recent approach is to
use certain classes of eigenvalues arising in scattering theory as target
signatures, in which perturbations of the eigenvalues may potentially be used
to infer changes in the medium relative to some reference configuration. This
approach was initially envisioned for the class of transmission eigenvalues
(cf. [6]); however, while this avenue of research has been fruitful from a
mathematical perspective, difficulties have arisen concerning their practical
implementation. In particular, measured scattering data may only be used to
detect real transmission eigenvalues, none of which exist for an absorbing
medium. Moreover, the detection of transmission eigenvalues requires the
collection of multifrequency data in a predetermined range, thereby providing
little freedom in the choice of interrogating frequency.

By following the same reasoning that led to the development of transmission
eigenvalues for a penetrable medium, a new class of eigenvalues was introduced
in [7] by considering a modification of the measured acoustic scattering data
obtained by subtracting from it the scattering data for an exterior impedance
problem dependent upon a parameter $\lambda$. This specific case resulted in
the class of Stekloff eigenvalues, and the appearance of the eigenparameter
$\lambda$ in the so-called auxiliary problem rather than the physical problem
removed many of the aforementioned restrictions that complicate the practical
application of transmission eigenvalues: eigenvalues anywhere in the complex
plane can be computed from measured scattering data, and a single
interrogation frequency may be chosen at will. Concerning the latter point,
the collection of scattering data for multiple choices of the frequency is
essentially replaced by the computation of auxiliary scattering data for
multiple choices of the eigenparameter $\lambda$.

From this approach was born various new eigenvalue problems (cf. [3, 10, 11,
12, 13, 14, 15, 16, 17, 19, 18, 21]), which share many qualities but often
require different techniques in the course of their analysis. Two properties
of these classes of eigenvalues are of special interest in a practical setting
but difficult to study: the existence of eigenvalues and their sensitivity to
changes in the medium to which they correspond. The former property is
necessary for eigenvalues to find use as a target signature. It is generally a
simple matter to establish for a nonabsorbing penetrable medium, as the
resulting eigenvalue problem is often self-adjoint; however, in the absorbing
case it has been shown only under stringent smoothness requirements on the
medium for some classes of eigenvalues (cf. [7, 15]) using Agmon’s theory of
non-selfadjoint boundary value problems (cf. [2]), but existence results are
not available otherwise. This issue was addressed in [13] by introducing a
smoothing operator in the construction of Stekloff eigenvalues, leading to the
class of $\delta$-Stekloff eigenvalues, for which it was shown that infinitely
many eigenvalues exist–even for an absorbing medium–provided that the degree
of smoothing is sufficiently high. It is for this modified class of Stekloff
eigenvalues (and consequently the standard class by choosing the smoothing
parameter $\delta=0$) that we address the second issue of sensitivity.
Knowledge of the relationship between the eigenvalues and changes in the
medium is of vital importance in the intended application of nondestructive
testing, as the perturbations of the eigenvalues are the only information that
is obtained when using eigenvalues as a target signature. Thus, precise
results in this direction may improve the applicability of using eigenvalues
as target signatures in a realistic setting.

While the sensitivity of eigenvalues was also discussed in [13], the main
effort was to provide stability estimates for a certain solution operator that
was related to the class of $\delta$-Stekloff eigenvalues, and the
perturbations of the eigenvalues themselves were not quantified. The present
aim is to improve upon this idea in three ways by reformulating the
$\delta$-Stekloff eigenvalue problem as a nonlinear eigenvalue problem and
using the asymptotic correction formula given in [23] (see [8, 9] for a
similar approach to transmission eigenvalues). First, we obtain an asymptotic
formula for the perturbation of a simple $\delta$-Stekloff eigenvalue due to a
change in the consitutitive parameters of the medium. Second, we use this
formula to arrive at upper bounds relative to the norm of the perturbed
coefficients. Third, we extend our study of eigenvalue perturbations to the
case of anisotropic media, which necessitates precise results on the
regularity of the medium and significantly increases the applicability of our
findings.

The remainder of our discussion is outlined as follows. As we will frequently
use a few key results throughout our investigation, Section 2 is devoted to
their statements and the basic notation on which they rely. In Section 3 we
introduce the $\delta$-Stekloff eigenvalue problem that we will consider and
provide some basic results from [13], including the aforementioned result that
infinitely many eigenvalues exist when the smoothing parameter $\delta$ is
sufficiently large relative to the dimension. We reformulate the
$\delta$-Stekloff eigenvalue problem as a nonlinear eigenvalue problem in
Section 4 and use the first-order correction formula from [23] to arrive at
our main result, which quantifies the sensitivity of the eigenvalues to
changes in the medium. In Section 5 we briefly consider the simpler case in
which the material is isotropic, allowing us to leverage higher regularity
results in order to improve the upper bound on the perturbations of the
eigenvalues (in the sense of achieving a lower $L^{p}$-norm of the perturbed
coefficients). After comparing the results of a simple numerical example in
Section 6 to the theoretical results of the preceding section, we conclude in
Section 7 with an overview of the impact and potential use of the main results
along with some unanswered questions.

##  2 Preliminaries and notation

We devote this section to introducing the notation and relevant results that
will be used throughout our investigation. We consider a domain
$B\subseteq\mathbb{R}^{d}$, $d=2,3$, which we assume to have connected
exterior $\mathbb{R}^{d}\setminus\overline{B}$ and smooth, simply-connected
boundary $\partial B$, the latter of which has a unit outward normal vector
denoted by $\nu$. On this domain we define the standard $L^{p}$-spaces for
$p\in[1,\infty)$ by

$L^{p}(B)=\left\\{f:B\to\mathbb{C}\;\middle|\;f\text{ is measurable and
}\int_{B}\left|f\right|^{p}\,dx<\infty\right\\}$

and equip each space with the norm

$\left\lVert
f\right\rVert_{L^{p}(B)}=\left(\int_{B}\left|f\right|^{p}\,dx\right)^{1/p}.$

In particular, we recall that $L^{2}(B)$ is a Hilbert space, and we denote its
inner product by $(\cdot,\cdot)_{B}$, which is defined as

$(f_{1},f_{2})_{B}=\int_{B}f_{1}\overline{f_{2}}\,dx\quad\forall
f_{1},f_{2}\in L^{2}(B).$

We define the space $L^{\infty}(B)$ by

$L^{\infty}(B)=\left\\{f:B\to\mathbb{C}\;\middle|\;f\text{ is measurable and
}\text{ess supp}(\left|f\right|)<\infty\right\\}$

and equip it with the norm

$\left\lVert f\right\rVert_{L^{\infty}(B)}=\text{ess supp}(\left|f\right|).$

For any positive integer $m>0$, we define the Sobolev space $H^{m}(B)$ in the
usual way (cf. [1, 5]) as

$H^{m}(B)=\\{f\in L^{2}(B)\mid D^{\alpha}f\in
L^{2}(B),\;\left|\alpha\right|\leq m\\},$

where $D^{\alpha}$ refers to the distributional derivative corresponding to
the multi-index $\alpha$. For $t\in(0,1)$, the fractional Sobolev space
$H^{t}(B)$ is defined by means of interpolation between $H^{1}(B)$ and
$L^{2}(B)$ (cf. [27]), and we define the space $H^{t+1}(B)$ as

$H^{t+1}(B)=\\{f\in H^{1}(B)\mid\nabla f\in(H^{t}(B))^{d}\\}$

with norm $\left\lVert f\right\rVert_{H^{t+1}(B)}^{2}=\left\lVert
f\right\rVert_{H^{1}(B)}^{2}+\left\lVert\nabla
f\right\rVert_{(H^{t}(B))^{d}}^{2}$ (cf. [4]). We denote by
$\tilde{H}^{-t}(B)$ the corresponding dual space of $H^{t}(B)$, defined in
terms of a Gelfand triple with pivot space $L^{2}(B)$.

If $X$ denotes any of these spaces, we use the notation $(X)^{d}$ and
$(X)^{d\times d}$ to represent spaces of $d$-dimensional vectors and $d\times
d$ matrices, respectively, whose entries lie in $X$, and we define the
respective norms by

$\left\lVert(u_{1},\dots,u_{d})^{T}\right\rVert_{(X)^{d}}=\sum_{i=1}^{d}\left\lVert
u_{i}\right\rVert_{X},\quad\left\lVert\left(\begin{array}[]{ccc}u_{11}&\cdots&u_{1d}\\\
\vdots&\ddots&\vdots\\\
u_{d1}&\cdots&u_{dd}\end{array}\right)\right\rVert_{(X)^{d\times
d}}=\sum_{i,j=1}^{d}\left\lVert u_{ij}\right\rVert_{X}.$

We also define the Sobolev space $H^{t}(\partial B)$ for $t>0$ as the trace
space of $H^{t+\frac{1}{2}}(B)$ with the image norm (cf. [1]), and we define
the dual space $H^{-t}(\partial B)$ with $L^{2}(\partial B)$ as the pivot
space, where the inner product $\left\langle\cdot,\cdot\right\rangle_{\partial
B}$ on $L^{2}(\partial B)$ defined by

$\left\langle f_{1},f_{2}\right\rangle_{\partial B}=\int_{\partial
B}f_{1}\overline{f_{2}}\,ds\quad\forall f_{1},f_{2}\in L^{2}(\partial B)$

will also be used to denote the duality pairing of $H^{-t}(\partial B)$ and
$H^{t}(\partial B)$.

We state a special case of the Sobolev embedding theorem (cf. [1]), recalling
that $B$ is assumed to be a smooth domain in $\mathbb{R}^{d}$.

######  Theorem 1.

(Sobolev Embedding Theorem) Suppose that $f\in H^{t}(B)$ for some $t>0$.

  1. (i)

If $2t>d$, then $f\in C_{b}(B)$, the space of bounded continuous functions on
$B$ equipped with the uniform norm, with the estimate

$\left\lVert f\right\rVert_{C_{b}(B)}\leq C_{t}\left\lVert
f\right\rVert_{H^{t}(B)}.$

  2. (ii)

If $2t=d$, then $f\in L^{p}(B)$ for every $p\in[2,\infty)$ with the estimate

$\left\lVert f\right\rVert_{L^{p}(B)}\leq C_{t,p}\left\lVert
f\right\rVert_{H^{t}(B)}.$

  3. (iii)

If $2t<d$, then $f\in L^{2d/(d-2t)}(B)$ with the estimate

$\left\lVert f\right\rVert_{L^{2d/(d-2t)}(B)}\leq C_{t}\left\lVert
f\right\rVert_{H^{t}(B)}.$

Each constant is independent of $f$.

We will also use this result for vector-valued functions, in which case we
apply it to each component individually. The following result from [5] is a
specific form of Hölder’s inequality involving the product of three functions
lying in suitable $L^{p}$-spaces. We will frequently apply the resulting
estimate with $q$ and $r$ determined by the Sobolev embedding theorem and $p$
determined by the condition in the theorem.

######  Theorem 2.

(Three-Term Hölder’s Inequality) If $f_{1}\in L^{p}(B)$, $f_{2}\in L^{q}(B)$,
and $f_{3}\in L^{r}(B)$ for $p,q,r\in[1,\infty]$ satisfying

$\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=1,$

then $f_{1}f_{2}f_{3}\in L^{1}(B)$ and

$\left\lVert f_{1}f_{2}f_{3}\right\rVert_{L^{1}(B)}\leq\left\lVert
f_{1}\right\rVert_{L^{p}(B)}\left\lVert
f_{2}\right\rVert_{L^{q}(B)}\left\lVert f_{3}\right\rVert_{L^{r}(B)}.$ (1)

We remark that the three-term Hölder’s inequality may be applied when
$f_{1}=\mathbf{F}_{1}$ is a matrix-valued function and $f_{2}=\mathbf{f}_{2}$
and $f_{3}=\mathbf{f}_{3}$ are vector-valued functions (with the integrand
replaced by $\mathbf{F}_{1}\mathbf{f}_{2}\cdot\mathbf{f}_{3}$). While the
exact implementation of the result in this case requires expanding the matrix
multiplication and evaluating the dot product in the integrand, the resulting
inequality is identical in form to (2), and consequently we use it without
further comment from this point onward.

Finally, we state the following interpolation inequality from [5], which can
be seen as a consequence of the standard Hölder’s inequality, to estimate the
remainder terms in our asymptotic formulas. Note that this statement is more
specific than that found in [5], covering only the case of a bounded domain.

######  Theorem 3.

(Hölder’s Interpolation Inequality) Assume that $f\in L^{q}(B)$ for some
$q\in[1,\infty]$ and that $p\in[1,q]$. If $r\in[p,q]$ and $\alpha\in[0,1]$
satisfy

$\frac{1}{r}=\frac{\alpha}{p}+\frac{1-\alpha}{q},$

then

$\left\lVert f\right\rVert_{L^{r}(B)}\leq\left\lVert
f\right\rVert_{L^{p}(B)}^{\alpha}\left\lVert
f\right\rVert_{L^{q}(B)}^{1-\alpha}.$ (2)

For both Theorems 2 and 3, we adopt the convention that $\frac{1}{\infty}=0$
in the case that one of the functions lies in $L^{\infty}(B)$.

##  3 The $\delta$-Stekloff eigenvalue problem

We recall that the domain $B$ introduced in the previous section has smooth,
simply-connected boundary $\partial B$. Before we present the eigenvalue
problem of interest from [13], we must introduce a smoothing operator that
will be necessary for certain existence results. Denoting by $\nabla_{\partial
B}$, $\textnormal{div}_{\partial B}$, $\textnormal{curl}_{\partial B}$, and
$\textbf{curl}_{\partial B}$ the surface gradient, surface divergence, scalar
surface curl, and vector surface curl, respectively, we define the nonnegative
Laplace-Beltrami operator as

$\Delta_{\partial B}:=-\textnormal{div}_{\partial B}\nabla_{\partial
B}=\textnormal{curl}_{\partial B}\textbf{curl}_{\partial B}.$

We refer to [22, 24] for details on these surface differential operators, and
we note that we have introduced a negative sign as in [20] in order to ensure
nonnegativity of $\Delta_{\partial B}$. We state the following results from
[26] on the spectral properties of this operator.

######  Proposition 4.

There exists an orthonormal basis $\\{Y_{m}\\}_{m=0}^{\infty}$ of
$L^{2}(\partial B)$ and a nondecreasing divergent sequence of nonnegative real
numbers $\\{\mu_{m}\\}_{m=0}^{\infty}$, counted according to multiplicity,
such that

$\Delta_{\partial B}Y_{m}=\mu_{m}Y_{m},\;m\geq 0.$

The first eigenvalue is $\mu_{0}=0$ with constant eigenfunction
$Y_{0}=\left|\partial B\right|^{-1/2}$, and $\mu_{m}>0$ for $m\geq 1$.

If $B$ is chosen to be a ball in $\mathbb{R}^{d}$, then the eigenfunctions
$\\{Y_{m}\\}$ are given by complex exponentials when $d=2$ and spherical
harmonics when $d=3$ (cf. [24]). By representing a given $\xi\in
L^{2}(\partial B)$ (or, more generally, any surface distribution on $\partial
B$) in the form

$\xi=\sum_{m=0}^{\infty}\xi_{m}Y_{m},\;\xi_{m}:=\left\langle\xi,Y_{m}\right\rangle_{\partial
B},$ (3)

it can be shown (cf. [24]) that the Sobolev space $H^{t}(\partial B)$, $t\geq
0$, may be characterized as

$H^{t}(\partial B)=\left\\{\xi\in L^{2}(\partial
B)\;\middle|\;\sum_{m=0}^{\infty}(1+\mu_{m})^{t}\left|\xi_{m}\right|^{2}<\infty\right\\}$

with equivalent norm

$\left\lVert\xi\right\rVert_{H^{t}(\partial
B)}^{2}:=\sum_{m=0}^{\infty}(1+\mu_{m})^{t}\left|\xi_{m}\right|^{2}.$

We use this characterization of Sobolev spaces on $\partial B$ to define the
_Bessel potential operator_ $S_{\delta}:=(I+\Delta_{\partial B})^{-\delta}$
for a real number $\delta$, which may be expressed as

$S_{\delta}\xi=\sum_{m=0}^{\infty}(1+\mu_{m})^{-\delta}\xi_{m}Y_{m}.$

We now proceed to introduce the $\delta$-Stekloff eigenvalue problem from
[13], which depends upon the constitutive parameters $A$ and $n$ of an
inhomogeneous medium that is supported in a bounded Lipschitz domain
$D\subseteq\mathbb{R}^{d}$ with connected exterior. The matrix-valued function
$A$ represents the anisotropic properties of the medium, and we assume that
$A(x)$ is Hermitian and positive-definite for each $x\in\mathbb{R}^{d}$. The
index of refraction $n$ is assumed to lie in $L^{\infty}(\mathbb{R}^{d})$ such
that $\textnormal{Re}(n)\geq\alpha>0$ and $\textnormal{Im}(n)\geq 0$ a.e. in
$D$. We assume that $D$ is compactly contained within $B$, implying that $A=I$
and $n=1$ in the open set $B\setminus\overline{D}$. We adopt one final
assumption to ensure regularity results that will be necessary later (cf.
[4]).

######  Assumption 5.

We assume that $B$ admits a partition $\\{\Omega_{m}\\}_{m=1}^{M}$ with
interface $\Sigma$ for which $A$ lies in the space

$W_{\Sigma}^{1,\infty}(B):=\left\\{\tilde{A}\in(L^{\infty}(B))^{d\times
d}\;\left|\;\begin{array}[]{cc}\nabla(\tilde{a}_{ij}|_{\Omega_{m}})\in(L^{\infty}(\Omega_{m}))^{d}\\\
1\leq i,j\leq d,\;1\leq m\leq M\end{array}\right.\right\\},$

where $\tilde{a}_{ij}$ denotes the corresponding entry of the matrix-valued
function $\tilde{A}$.

We equip the space $W_{\Sigma}^{1,\infty}(B)$ introduced in Assumption 5 with
the natural norm

$\left\lVert\tilde{A}\right\rVert_{W_{\Sigma}^{1,\infty}(B)}=\left\lVert\tilde{A}\right\rVert_{(L^{\infty}(B))^{d\times
d}}+\sum_{m=1}^{M}\sum_{i,j=1}^{d}\left\lVert\nabla(\tilde{a}_{ij}|_{\Omega_{m}})\right\rVert_{(L^{\infty}(\Omega_{m}))^{d}}.$

For a given $\delta\geq 0$, we consider the _$\delta$ -Stekloff eigenvalue
problem_ in which we seek $\lambda\in\mathbb{C}$ and a nonzero $u\in H^{1}(B)$
satisfying

$\displaystyle\nabla\cdot A\nabla u+k^{2}nu$ $\displaystyle=0\text{ in }B,$
(4a) $\displaystyle\frac{\partial u}{\partial\nu_{A}}+\lambda S_{\delta}u$
$\displaystyle=0\text{ on }\partial B,$ (4b)

where $k>0$ is a fixed wave number resulting from the chosen frequency of
incident wave and $\frac{\partial u}{\partial\nu_{A}}=\nu\cdot A\nabla u$ is
the conormal derivative. We refer to [13] for a derivation of (4) along with a
discussion of its relationship to the problem of acoustic scattering by an
inhomogeneous medium. We call a value of $\lambda$ for which (4) has a
nontrivial solution a _$\delta$ -Stekloff eigenvalue_, and we remark that with
$\delta=0$ we recover the standard class of Stekloff eigenvalues introduced in
[7] (see also [3]). This problem with $A=I$ was introduced and studied in
[13], where the primary aim was to use Lidski’s theorem (cf. [25]) to prove
that infinitely many eigenvalues exist even when $n$ is generally complex-
valued. Since we have assumed that $A=I$ and $n=1$ in
$B\setminus\overline{D}$, the same approach may be applied to the anisotropic
problem to obtain the following result.

######  Theorem 6.

Assume that there exists no nontrivial $w\in H^{1}(B)$ satisfying

$\displaystyle\nabla\cdot A\nabla w+k^{2}nw$ $\displaystyle=0\text{ in }B,$
(5a) $\displaystyle w$ $\displaystyle=0\text{ on }\partial B.$ (5b)

If $\delta>\frac{d}{2}-1$, then the $\delta$-Stekloff eigenvalues form an
infinite discrete set without finite accumulation point.

With the eigenvalue problem of interest formulated and basic existence results
in hand, we proceed in the next section to reformulate (4) as a nonlinear
eigenvalue problem in order to investigate the sensitivity of the
$\delta$-Stekloff eigenvalues to changes in $A$ and $n$. We remark that
Assumption 5 is essential to our investigation, as it is sufficient to obtain
higher regularity of solutions to (4) and related nonhomogeoneous problems
that will influence the sensitivity of the eigenvalues.

##  4 Perturbation estimates

We consider a sequence $\\{(A_{h},n_{h})\\}_{h\geq 0}$ of coefficient pairs
that satisfy the assumptions for $(A,n)$ given in Section 3, where in
particular Assumption 5 is satisfied by all $A_{h}$ with the same partition.
The pair $(A_{0},n_{0})$ corresponds to the reference medium, of which each
pair $(A_{h},n_{h})$ with $h>0$ is viewed as a perturbation. We also redefine
the inner product on $H^{1}(B)$ as

$(u,u^{\prime})_{H^{1}(B)}=(A_{0}\nabla u,\nabla
u^{\prime})_{B}+k^{2}(u,u^{\prime})_{B}\quad\forall u,u^{\prime}\in H^{1}(B)$

and consider the corresponding induced norm
$\left\lVert\cdot\right\rVert_{H^{1}(B)}$, which are equivalent to the
standard definitions due to our assumptions on $A_{0}$. We begin by
reformulating (4) with $(A,n)=(A_{h},n_{h})$ as a nonlinear eigenvalue
problem.

We define the operators $\mathbb{A}_{h},\mathbb{K}_{h},\mathbb{B}:H^{1}(B)\to
H^{1}(B)$ by means of the Riesz representation theorem such that

$\displaystyle(\mathbb{A}_{h}u,u^{\prime})_{H^{1}(B)}$
$\displaystyle=(A_{h}\nabla u,\nabla u^{\prime})_{B}+k^{2}(u,u^{\prime})_{B},$
$\displaystyle(\mathbb{K}_{h}u,u^{\prime})_{H^{1}(B)}$
$\displaystyle=-k^{2}((n_{h}+1)u,u^{\prime})_{B},$
$\displaystyle(\mathbb{B}u,u^{\prime})_{H^{1}(B)}$
$\displaystyle=-\left\langle S_{\delta}u,u^{\prime}\right\rangle_{\partial B}$

for all $u,u^{\prime}\in H^{1}(B)$, and we observe that (4) is equivalent to
finding $\lambda\in\mathbb{C}$ and a nonzero $u\in H^{1}(B)$ for which

$(\mathbb{A}_{h}+\mathbb{K}_{h})u=\lambda\mathbb{B}u.$ (6)

Note that each $\mathbb{A}_{h}$ is invertible and that, due to our
redefinition of the inner product on $H^{1}(B)$, $\mathbb{A}_{0}$ is equal to
the identity operator. By following the procedure outlined in [23, Section
5.1] for nonzero $\lambda$, this problem is in turn equivalent to the
nonlinear eigenvalue problem of finding $\lambda\in\mathbb{C}\setminus\\{0\\}$
and a nonzero $u\in H^{1}(B)$ such that

$\lambda T_{h}(\lambda)u=u,$ (7)

where the operator $T_{h}(\lambda):H^{1}(B)\to H^{1}(B)$ is defined by

$T_{h}(\lambda)=-\lambda^{-1}\mathbb{A}_{h}^{-1}\mathbb{K}_{h}+\mathbb{A}_{h}^{-1}\mathbb{B}.$
(8)

Writing the $\delta$-Stekloff eigenvalues in terms of a nonlinear eigenvalue
problem allows us to apply the asymptotic formulas of [23], namely the
following result (cf. [23, Corollary 4.1]), which we have adapted to the
specific case of operators on a Hilbert space.

######  Theorem 7.

Let $\\{T_{h}(\lambda)\\}_{h\geq 0}$ be a sequence of functions of $\lambda$
which are analytic in a region $U$ of the complex plane and have values in the
space of compact linear operators on a Hilbert space $\mathcal{H}$ such that
$T_{h}(\lambda)\to T_{0}(\lambda)$ in norm as $h\to 0$, uniformly for
$\lambda\in U$. Let $\lambda_{0}\in U$ be a nonzero simple nonlinear
eigenvalue for $h=0$, define $DT_{0}(\lambda_{0})$ as the derivative of
$T_{0}$ with respect to $\lambda$ evaluated at $\lambda_{0}$, and let $u_{0}$
be the normalized eigenfunction corresponding to $\lambda_{0}$. For
sufficiently small $h>0$, there exists a simple nonlinear eigenvalue
$\lambda_{h}$ of $T_{h}$ such that, if

$\lambda_{0}^{2}(DT_{0}(\lambda_{0})u_{0},u_{0})_{\mathcal{H}}\neq-1,$

then we have the formula

$\displaystyle\lambda_{h}=\lambda_{0}$
$\displaystyle+\frac{\lambda_{0}^{2}((T_{0}(\lambda_{0})-T_{h}(\lambda_{0}))u_{0},u_{0})_{\mathcal{H}}}{1+\lambda_{0}^{2}(DT_{0}(\lambda_{0})u_{0},u_{0})_{\mathcal{H}}}$
$\displaystyle+O\left(\sup_{\lambda\in
U}\left\lVert(T_{h}(\lambda)-T_{0}(\lambda))|_{R(E_{\lambda_{0}})}\right\rVert\left\lVert(T_{h}^{*}(\lambda)-T_{0}^{*}(\lambda))|_{R(E_{\lambda_{0}}^{*})}\right\rVert\right),$

where $E_{\lambda_{0}}$ is the spectral projection onto the one-dimensional
eigenspace corresponding to $\lambda_{0}$.

We prove that $\\{T_{h}(\lambda)\\}_{h\geq 0}$ as defined in (8) satisfies the
hypotheses of Theorem 7 and compute the corresponding asymptotic formula. We
begin with the following lemma, in which we state the basic properties of
$\\{T_{h}(\lambda)\\}$.

######  Lemma 8.

For each $h\geq 0$, the operator $T_{h}(\lambda)$ is compact for each
$\lambda\in\mathbb{C}\setminus\\{0\\}$, and the mapping $\lambda\mapsto
T_{h}(\lambda)$ is analytic in the domain
$U=\mathbb{C}\setminus\overline{B_{R}(0)}$ for every $R>0$.

Proof. We begin by showing that the operators $\mathbb{K}_{h}$ and
$\mathbb{B}$ are both compact. For every $u\in H^{1}(B)$ we see from the
assumption that $n_{h}\in L^{\infty}(B)$ and the Cauchy-Schwarz inequality
that

$\displaystyle\left\lVert\mathbb{K}_{h}u\right\rVert_{H^{1}(B)}$
$\displaystyle=\sup_{u^{\prime}\neq
0}\frac{\left|(\mathbb{K}_{h}u,u^{\prime})_{H^{1}(B)}\right|}{\left\lVert
u^{\prime}\right\rVert_{H^{1}(B)}}=k^{2}\sup_{u^{\prime}\neq
0}\frac{\left|((n_{h}+1)u,u^{\prime})_{B}\right|}{\left\lVert
u^{\prime}\right\rVert_{H^{1}(B)}}\leq C_{h}\left\lVert
u\right\rVert_{L^{2}(B)},$

and by another application of the Cauchy-Schwarz inequality and the trace
theorem we obtain

$\left\lVert\mathbb{B}u\right\rVert_{H^{1}(B)}=\sup_{u^{\prime}\neq
0}\frac{\left|(\mathbb{B}u,u^{\prime})_{H^{1}(B)}\right|}{\left\lVert
u^{\prime}\right\rVert_{H^{1}(B)}}=\sup_{u^{\prime}\neq
0}\frac{\left|\left\langle S_{\delta}u,u^{\prime}\right\rangle_{\partial
B}\right|}{\left\lVert u^{\prime}\right\rVert_{H^{1}(B)}}\leq C\left\lVert
S_{\delta}u\right\rVert_{\partial B}.$

Suppose that a sequence $\\{u_{j}\\}$ in $H^{1}(B)$ converges weakly to some
$u_{0}\in H^{1}(B)$, in which case the compact embedding of $H^{1}(B)$ into
$L^{2}(B)$ implies that $u_{j}\to u_{0}$ in $L^{2}(B)$ and the compact
embedding of $H^{1/2}(\partial B)$ into $L^{2}(\partial B)$ and boundedness of
$S_{\delta}$ imply that $S_{\delta}u_{j}\to S_{\delta}u_{0}$ in
$L^{2}(\partial B)$. It follows that

$\left\lVert\mathbb{K}_{h}(u_{j}-u_{0})\right\rVert_{H^{1}(B)}\leq
C_{h}\left\lVert u_{j}-u_{0}\right\rVert_{L^{2}(B)}\to 0$

and

$\left\lVert\mathbb{B}(u_{j}-u_{0})\right\rVert_{H^{1}(B)}\leq C\left\lVert
S_{\delta}u_{j}-S_{\delta}u_{0}\right\rVert_{\partial B}\to 0$

as $j\to\infty$. As a consequence, we observe that
$\mathbb{K}_{h}u_{j}\to\mathbb{K}_{h}u_{0}$ and
$\mathbb{B}u_{j}\to\mathbb{B}u_{0}$ in $H^{1}(B)$, and we conclude that
$\mathbb{K}_{h}$ and $\mathbb{B}$ are compact. Since composition with bounded
operators preserves compactness, we have shown that $T_{h}(\lambda)$ is
compact for each $\lambda\in\mathbb{C}\setminus\\{0\\}$. Analyticity of the
mapping $\lambda\mapsto T_{h}(\lambda)$ for $\lambda\in
U:=\mathbb{C}\setminus\overline{B_{R}(0)}$ is clear from the definition of
$T_{h}(\lambda)$. ∎

∎

Before we proceed to show norm convergence of $\\{T_{h}(\lambda)\\}$ to
$T_{0}(\lambda)$, we require a regularity result that depends strongly on
Assumption 5.

######  Proposition 9.

If $w_{h}\in H^{1}(B)$ satisfies

$\displaystyle\nabla\cdot A_{h}\nabla w_{h}-k^{2}w_{h}$ $\displaystyle=f\text{
in }B,$ (9a) $\displaystyle\frac{\partial w_{h}}{\partial\nu_{A_{h}}}$
$\displaystyle=\xi\text{ on }\partial B,$ (9b)

for given $f\in\tilde{H}^{s-1}(B)$ and $\xi\in H^{1/2}(\partial B)$, then
there exists $\tau_{h}\in\left(0,\frac{1}{2}\right)$ such that for every
$s\in[0,\tau_{h})$ we have $w_{h}\in H^{s+1}(B)$ with the estimate

$\left\lVert w_{h}\right\rVert_{H^{s+1}(B)}\leq C_{s,h}\left(\left\lVert
f\right\rVert_{\tilde{H}^{s-1}(B)}+\left\lVert\xi\right\rVert_{H^{1/2}(\partial
B)}\right).$ (10)

Moreover, if $A_{h}\to A_{0}$ in $W_{\Sigma}^{1,\infty}(B)$ as $h\to 0$, then
both $\tau_{h}$ and $C_{s,h}$ can be chosen independently of $h$ for
sufficiently small $h$.

Proof. We begin by defining a lifting function $\varphi\in H^{1}(B)$ as the
unique solution of

$\displaystyle\Delta\varphi-k^{2}\varphi$ $\displaystyle=0\text{ in }B,$
$\displaystyle\frac{\partial\varphi}{\partial\nu}$ $\displaystyle=\xi\text{ on
}\partial B,$

from which we observe that $w_{h}-\varphi\in H^{1}(B)$ satisfies

$\displaystyle\nabla\cdot A_{h}\nabla(w_{h}-\varphi)-k^{2}(w_{h}-\varphi)$
$\displaystyle=f+\nabla\cdot(I-A_{h})\nabla\varphi\text{ in }B,$
$\displaystyle\frac{\partial(w_{h}-\varphi)}{\partial\nu_{A_{h}}}$
$\displaystyle=0\text{ on }\partial B,$

with the boundary condition following from the assumption that $A_{h}=I$ in
the open set $B\setminus\overline{D}$. Due to smoothness of $\partial B$,
standard elliptic regularity estimates imply that $\varphi\in H^{2}(B)$ with
$\left\lVert\varphi\right\rVert_{H^{2}(B)}\leq
C\left\lVert\xi\right\rVert_{H^{1/2}(\partial B)}$ (cf. [1]). It follows that
$\nabla\varphi\in(H^{1}(B))^{d}$, which in turn implies that
$\nabla\varphi\in(H^{s}(B))^{d}$ for all $s\in[0,1]$. By Proposition 2.1 in
[4], from Assumption 5 it follows that $A_{h}$ is a multiplier of the space
$(H^{s}(B))^{d}$ for $s\in[0,\frac{1}{2})$, and we obtain
$(I-A_{h})\nabla\varphi\in(H^{s}(B))^{d}$ and hence
$\nabla\cdot(I-A_{h})\nabla\varphi\in\tilde{H}^{s-1}(B)$ for
$s\in[0,\frac{1}{2})$.

Since $f+\nabla\cdot(I-A_{h})\nabla\varphi\in\tilde{H}^{s-1}(B)$, we now apply
Theorem 3.1 in [4] to conclude that there exists
$\tau_{h}\in\left(0,\frac{1}{2}\right)$ dependent upon $A_{h}$, $B$, and the
partition $\\{\Omega_{m}\\}_{m=1}^{M}$ from Assumption 5 such that
$w_{h}-\varphi\in H^{s+1}(B)$ for $s\in[0,\tau_{h})$ with the estimate

$\left\lVert w_{h}-\varphi\right\rVert_{H^{s+1}(B)}\leq C_{s,h}\left\lVert
f+\nabla\cdot(I-A_{h})\nabla\varphi\right\rVert_{\tilde{H}^{s-1}(B)}.$

The construction of $\varphi$ and its subsequent elliptic regularity estimate
together imply (10). Finally, it can be seen from the proof of Theorem 3.1 in
[4] that convergence of $\\{A_{h}\\}$ to $A_{0}$ in $W_{\Sigma}^{1,\infty}(B)$
is sufficient to permit a choice of $\tau_{h}$ and $C_{s,h}$ that is
independent of $h$ when $h$ is sufficiently small. ∎

∎

######  Remark 10.

As a consequence of Proposition 9, we will from this point forward assume that
$A_{h}\to A_{0}$ in $W_{\Sigma}^{1,\infty}(B)$, allowing us to choose
$\tau=\tau_{h}$ independently of $h$ and fix an element $s$ of $(0,\tau)$.

The following estimate concerning the operator
$\mathbb{A}_{h}^{-1}-\mathbb{A}_{0}^{-1}$ will also be useful in many of the
following results.

######  Proposition 11.

For sufficiently small $h>0$ we have the estimate

$\left|((\mathbb{A}_{h}^{-1}-\mathbb{A}_{0}^{-1})u,v)_{H^{1}(B)}\right|\leq
C_{s}\left\lVert A_{h}-A_{0}\right\rVert_{(L^{d/s}(B))^{d\times d}}\left\lVert
u\right\rVert_{H^{1}(B)}\left\lVert v\right\rVert_{H^{s+1}(B)}$ (11)

for all $u\in H^{1}(B)$ and $v\in H^{s+1}(B)$.

Proof. For given $u\in H^{1}(B)$ and $v\in H^{s+1}(B)$, we define
$v_{h}=\mathbb{A}_{h}^{-1}u$, and we observe that

$\left|((\mathbb{A}_{h}^{-1}-\mathbb{A}_{0}^{-1})u,v)_{H^{1}(B)}\right|=\left|(v_{h}-v_{0},v)_{H^{1}(B)}\right|=\left|((A_{h}-A_{0})\nabla
v_{h},\nabla v)_{B}\right|.$

Since $\nabla v\in(H^{s}(B))^{d}$, the Sobolev embedding theorem implies that
$\nabla v\in(L^{2d/(d-2s)}(B))^{d}$ for $d=2,3$, in which case we may apply
the three-term Hölder’s inequality with $p=\frac{d}{s}$, $q=2$, and
$r=\frac{2d}{d-2s}$ to obtain

$\displaystyle\left|((A_{h}-A_{0})\nabla v_{h},\nabla v)_{B}\right|$
$\displaystyle\hskip 40.00006pt\leq\left\lVert
A_{h}-A_{0}\right\rVert_{(L^{d/s}(B))^{d\times d}}\left\lVert\nabla
v_{h}\right\rVert_{(L^{2}(B))^{d}}\left\lVert\nabla
v\right\rVert_{(L^{2d/(d-2s)}(B))^{d}}$ $\displaystyle\hskip 40.00006pt\leq
C_{s}\left\lVert A_{h}-A_{0}\right\rVert_{(L^{d/s}(B))^{d\times d}}\left\lVert
v_{h}\right\rVert_{H^{1}(B)}\left\lVert v\right\rVert_{H^{s+1}(B)}.$

Finally, we have

$\displaystyle\left\lVert v_{h}\right\rVert_{H^{1}(B)}^{2}$
$\displaystyle=(A_{0}\nabla v_{h},\nabla v_{h})_{B}+k^{2}(v_{h},v_{h})_{B}$
$\displaystyle=-((A_{h}-A_{0})\nabla v_{h},\nabla
v_{h})_{B}+(u,v_{h})_{H^{1}(B)}$ $\displaystyle\leq\left\lVert
A_{h}-A_{0}\right\rVert_{(L^{\infty}(B))^{d\times d}}\left\lVert
v_{h}\right\rVert_{H^{1}(B)}^{2}+\left\lVert
u\right\rVert_{H^{1}(B)}\left\lVert v_{h}\right\rVert_{H^{1}(B)},$

where we applied the three-term Hölder’s inequality with $p=\infty$ and
$q=r=2$ to obtain the last inequality. Convergence of $\\{A_{h}\\}$ to $A_{0}$
in $(L^{\infty}(B))^{d\times d}$ implies that $\left\lVert
v_{h}\right\rVert_{H^{1}(B)}\leq C\left\lVert u\right\rVert_{H^{1}(B)}$ for
sufficiently small $h$, from which we obtain (11). ∎

∎

######  Remark 12.

Since $\mathbb{A}_{h}^{-1}-\mathbb{A}_{0}^{-1}$ is a self-adjoint operator,
the estimate (11) also holds with the roles of $u$ and $v$ interchanged on the
right-hand side, provided that each lies in the appropriate space.

We now estimate the norm $\left\lVert
T_{h}(\lambda)-T_{0}(\lambda)\right\rVert$ for $\lambda\in U$.

######  Lemma 13.

If $A_{h}\to A_{0}$ in $W_{\Sigma}^{1,\infty}(B)$ and $n_{h}\to n_{0}$ in
$L^{2}(B)$ as $h\to 0$, then the sequence $\\{T_{h}(\lambda)\\}$ satisfies the
norm estimate

$\left\lVert T_{h}(\lambda)-T_{0}(\lambda)\right\rVert\leq
C_{s}\left(\left\lVert A_{h}-A_{0}\right\rVert_{(L^{d/s}(B))^{d\times
d}}+\left\lVert n_{h}-n_{0}\right\rVert_{L^{p_{0}}(B)}\right),$ (12)

where for each $\epsilon>0$ we have

$p_{0}=\left\\{\begin{array}[]{cl}1+\epsilon&\text{ if }d=2,\\\
\frac{3}{2}&\text{ if }d=3,\end{array}\right.$ (13)

and the constant $C_{s}$ is independent of $h$ and $\lambda\in
U=\mathbb{C}\setminus\overline{B_{R}(0)}$ but depends on $\epsilon$ whenever
$d=2$. As a result, we have $T_{h}(\lambda)\to T_{0}(\lambda)$ in norm as
$h\to 0$, uniformly for $\lambda\in U$.

Proof. Let $\lambda\in U$. We first observe that we may write

$T_{h}(\lambda)-T_{0}(\lambda)=-(\mathbb{A}_{h}^{-1}-\mathbb{A}_{0}^{-1})(\lambda^{-1}\mathbb{K}_{h}-\mathbb{B})-\lambda^{-1}\mathbb{A}_{0}^{-1}(\mathbb{K}_{h}-\mathbb{K}_{0}),$

immediately providing the estimate

$\displaystyle\left|((T_{h}(\lambda)-T_{0}(\lambda))u,u^{\prime})_{H^{1}(B)}\right|$
$\displaystyle\leq\left|((\mathbb{A}_{h}^{-1}-\mathbb{A}_{0}^{-1})(\lambda^{-1}\mathbb{K}_{h}-\mathbb{B})u,u^{\prime})_{H^{1}(B)}\right|$
$\displaystyle\hskip
40.00006pt+\left|\lambda\right|^{-1}\left|(\mathbb{A}_{0}^{-1}(\mathbb{K}_{h}-\mathbb{K}_{0})u,u^{\prime})_{H^{1}(B)}\right|$

for all $u,u^{\prime}\in H^{1}(B)$. In order to estimate the first term, we
notice that $(\lambda^{-1}\mathbb{K}_{h}-\mathbb{B})u$ satisfies (9) with
$f=\lambda^{-1}k^{2}(n_{h}+1)u\in L^{2}(B)$ and $\xi=S_{\delta}u\in
H^{1/2}(\partial B)$, from which Proposition 9 implies that
$(\lambda^{-1}\mathbb{K}_{h}-\mathbb{B})u\in H^{s+1}(B)$ with the estimate

$\displaystyle\left\lVert(\lambda^{-1}\mathbb{K}_{h}-\mathbb{B})u\right\rVert_{H^{s+1}(B)}$
$\displaystyle\leq
C_{s}\left(\left\lVert\lambda^{-1}k^{2}(n_{h}+1)u\right\rVert_{\tilde{H}^{s-1}(B)}+\left\lVert
S_{\delta}u\right\rVert_{H^{1/2}(\partial B)}\right)$ $\displaystyle\leq
C_{s}\left(\left\lVert n_{h}u\right\rVert_{\tilde{H}^{s-1}(B)}+\left\lVert
u\right\rVert_{H^{1}(B)}\right)$

with $C_{s}$ independent of $\lambda\in U$. We apply a duality argument and
the three-term Hölder’s inequality to estimate $\left\lVert
n_{h}u\right\rVert_{\tilde{H}^{s-1}(B)}$. By definition of the dual norm
$\left\lVert\cdot\right\rVert_{\tilde{H}^{s-1}(B)}$ we have

$\left\lVert
n_{h}u\right\rVert_{\tilde{H}^{s-1}(B)}=\sup_{\left\lVert\psi\right\rVert_{H^{1-s}(B)}=1}\left|(n_{h}u,\psi)_{B}\right|,$

where we interpret the duality pairing in terms of the Gelfand triple
$H^{1-s}(B)\subseteq L^{2}(B)\subseteq\tilde{H}^{s-1}(B)$. Since $u\in
H^{1}(B)$ and $\psi\in H^{1-s}(B)$, the Sobolev embedding theorem implies that
$u\in L^{q}(B)$ for all $q\in[2,\infty)$ and $\psi\in L^{2/s}(B)$ when $d=2$
and that $u\in L^{6}(B)$ and $\psi\in L^{6/(1+2s)}(B)$ when $d=3$, each with a
continuous embedding. For $d=2$, applying the three-term Hölder’s inequality
with $p=\frac{3}{2-s}$, $q=\frac{6}{2-s}$, and $r=\frac{2}{s}$ yields

$\displaystyle\left|(n_{h}u,\psi)_{\tilde{H}^{s-1}(B)}\right|$
$\displaystyle\leq\left\lVert n_{h}\right\rVert_{L^{3/(2-s)}(B)}\left\lVert
u\right\rVert_{L^{6/(2-s)}(B)}\left\lVert\psi\right\rVert_{L^{2/s}(B)}$
$\displaystyle\leq C_{s}\left\lVert
n_{h}\right\rVert_{L^{3/(2-s)}(B)}\left\lVert
u\right\rVert_{H^{1}(B)}\left\lVert\psi\right\rVert_{H^{1-s}(B)}.$

For $d=3$, we apply the three-term Hölder’s inequality with $p=\frac{3}{2-s}$,
$q=6$, and $r=\frac{6}{1+2s}$ to obtain

$\displaystyle\left|(n_{h}u,\psi)_{\tilde{H}^{s-1}(B)}\right|$
$\displaystyle\leq\left\lVert n_{h}\right\rVert_{L^{3/(2-s)}(B)}\left\lVert
u\right\rVert_{L^{6}(B)}\left\lVert\psi\right\rVert_{L^{6/(1+2s)}(B)}$
$\displaystyle\leq C_{s}\left\lVert
n_{h}\right\rVert_{L^{3/(2-s)}(B)}\left\lVert
u\right\rVert_{H^{1}(B)}\left\lVert\psi\right\rVert_{H^{1-s}(B)}.$

Thus, in either case we have $\left\lVert
n_{h}u\right\rVert_{\tilde{H}^{s-1}(B)}\leq C_{s}\left\lVert
n_{h}\right\rVert_{L^{3/(2-s)}(B)}\left\lVert u\right\rVert_{H^{1}(B)}$.
Convergence of $\\{n_{h}\\}$ to $n_{0}$ in $L^{2}(B)$ implies uniform
boundedness of $\\{n_{h}\\}$ in the weaker $L^{3/(2-s)}(B)$-norm, and hence
there exists a constant $C_{s}$ independent of $h$ for which

$\left\lVert(\lambda^{-1}\mathbb{K}_{h}-\mathbb{B})u\right\rVert_{H^{s+1}(B)}\leq
C_{s}\left\lVert u\right\rVert_{H^{1}(B)}.$

We now proceed to the main estimate. Applying the estimate from Proposition 11
(with the functions reversed as in Remark 12) yields

$\displaystyle\left|((\mathbb{A}_{h}^{-1}-\mathbb{A}_{0}^{-1})(\lambda^{-1}\mathbb{K}_{h}-\mathbb{B})u,u^{\prime})_{H^{1}(B)}\right|$
$\displaystyle\hskip 40.00006pt\leq C_{s}\left\lVert
A_{h}-A_{0}\right\rVert_{(L^{d/s}(B))^{d\times
d}}\left\lVert(\lambda^{-1}\mathbb{K}_{h}-\mathbb{B})u\right\rVert_{H^{s+1}(B)}\left\lVert
u^{\prime}\right\rVert_{H^{1}(B)}$ $\displaystyle\hskip 40.00006pt\leq
C_{s}\left\lVert A_{h}-A_{0}\right\rVert_{(L^{d/s}(B))^{d\times d}}\left\lVert
u\right\rVert_{H^{1}(B)}\left\lVert u^{\prime}\right\rVert_{H^{1}(B)}.$

For the second term we recall that $\mathbb{A}_{0}$ is equal to the identity
operator, which allows us to write

$\left|(\mathbb{A}_{0}^{-1}(\mathbb{K}_{h}-\mathbb{K}_{0})u,u^{\prime})_{H^{1}(B)}\right|=k^{2}\left|((n_{h}-n_{0})u,u^{\prime})\right|_{B}.$

Since $u,u^{\prime}\in H^{1}(B)$, the Sobolev embedding theorem implies that
$u,u^{\prime}\in L^{q}(B)$, where $q\in[2,\infty)$ for $d=2$ and $q=6$ for
$d=3$. In the case $d=2$, we apply the three-term Hölder’s inequality for a
given $\epsilon>0$ with $p=1+\epsilon$ and
$q=r=\frac{2(1+\epsilon)}{\epsilon}$, and in the case $d=3$ we apply the
inequality with $p=\frac{3}{2}$ and $q=r=6$ to obtain

$\displaystyle\left|((n_{h}-n_{0})u,u^{\prime})_{B}\right|$
$\displaystyle\leq\left\lVert
n_{h}-n_{0}\right\rVert_{L^{p_{0}}(B)}\left\lVert
u\right\rVert_{L^{q}(B)}\left\lVert u^{\prime}\right\rVert_{L^{r}(B)}$
$\displaystyle\leq C\left\lVert
n_{h}-n_{0}\right\rVert_{L^{p_{0}}(B)}\left\lVert
u\right\rVert_{H^{1}(B)}\left\lVert u^{\prime}\right\rVert_{H^{1}(B)},$

where $p_{0}$ is given by (13). Combining these estimates and noting that all
constants are independent of $h$ and $\lambda\in U$, the estimate (12) follows
from the definition of the operator norm. Moreover, convergence of
$\\{A_{h}\\}$ to $A_{0}$ and of $\\{n_{h}\\}$ to $n_{0}$ in the spaces
$W_{\Sigma}^{1,\infty}(B)$ and $L^{2}(B)$, respectively, implies that the
right-hand side of (12) converges to $0$ as $h\to 0$, proving the final
assertion. ∎

∎

With the results of Lemmas 8 and 13 in hand, we proceed to apply Theorem 7 to
the sequence $\\{T_{h}(\lambda)\\}$.

######  Theorem 14.

Suppose that $A_{h}\to A_{0}$ in $W_{\Sigma}^{1,\infty}(B)$ and that $n_{h}\to
n_{0}$ in $L^{2}(B)$ as $h\to 0$, and suppose that there exist no nontrivial
solutions of (5) for $(A,n)=(A_{0},n_{0})$. Let $\lambda_{0}$ be a nonzero
simple $\delta$-Stekloff eigenvalue for $(A,n)=(A_{0},n_{0})$ with
$H^{1}(B)$-normalized eigenfunction $u_{0}$, choose $R>0$ such that
$R<\left|\lambda_{0}\right|$, and let
$U=\mathbb{C}\setminus\overline{B_{R}(0)}$. For sufficiently small $h>0$ there
exists a simple $\delta$-Stekloff eigenvalue $\lambda_{h}$ for
$(A,n)=(A_{h},n_{h})$ that satisfies the formula

$\displaystyle\begin{split}\lambda_{h}=\lambda_{0}&+\frac{-((A_{h}-A_{0})\nabla
u_{0},\nabla u_{0})_{B}+k^{2}((n_{h}-n_{0})u_{0},u_{0})_{B}}{\left\langle
S_{\delta}u_{0},u_{0}\right\rangle_{\partial B}}\\\ &\hskip
30.00005pt+O\left(\left\lVert A_{h}-A_{0}\right\rVert_{(L^{d/s}(B))^{d\times
d}}^{2}+\left\lVert
n_{h}-n_{0}\right\rVert_{L^{p^{\prime}}(B)}^{2}\right),\end{split}$ (14)

where the exponent $p^{\prime}$ is given by

$p^{\prime}=\left\\{\begin{array}[]{cl}1+\epsilon&\text{ if }d=2,\\\
\frac{3}{2+s}&\text{ if }d=3,\end{array}\right.$ (15)

for a given $\epsilon>0$.

Proof. We begin with the observation that Lemmas 8 and 13 verify that the
hypotheses of Theorem 7 are satisfied by the sequence $\\{T_{h}(\lambda)\\}$
for $U=\mathbb{C}\setminus\overline{B_{R}(0)}$ with
$R<\left|\lambda_{0}\right|$. Moreover, from the assumption that $u_{0}$ is an
$H^{1}(B)$-normalized eigenfunction corresponding to $\lambda_{0}$, we obtain

$\displaystyle 1+\lambda_{0}^{2}(DT_{0}(\lambda_{0})u_{0},u_{0})_{H^{1}(B)}$
$\displaystyle=1+\lambda_{0}^{2}(\lambda_{0}^{-2}\mathbb{A}_{0}^{-1}\mathbb{K}_{0}u_{0},u_{0})_{H^{1}(B)}$
$\displaystyle=1-(\lambda_{0}T_{0}(\lambda_{0})u_{0},u_{0})_{H^{1}(B)}+\lambda_{0}(\mathbb{B}u_{0},u_{0})_{H^{1}(B)}$
$\displaystyle=1-(u_{0},u_{0})_{H^{1}(B)}-\lambda_{0}\left\langle
S_{\delta}u_{0},u_{0}\right\rangle_{\partial B}$
$\displaystyle=-\lambda_{0}\left\langle
S_{\delta}u_{0},u_{0}\right\rangle_{\partial B}.$

Since $\lambda_{0}\neq 0$ and $u_{0}\neq 0$ by assumption (for otherwise
$u_{0}$ would be a nontrivial solution of (5) for $(A,n)=(A_{0},n_{0})$), it
follows that $\lambda_{0}\left\langle
S_{\delta}u_{0},u_{0}\right\rangle_{\partial B}\neq 0$, and by Theorem 7 there
exists a simple $\delta$-Stekloff eigenvalue $\lambda_{h}$ for
$(A,n)=(A_{h},n_{h})$ such that

$\displaystyle\lambda_{h}=\lambda_{0}$
$\displaystyle+\frac{\lambda_{0}^{2}((T_{0}(\lambda_{0})-T_{h}(\lambda_{0}))u_{0},u_{0})_{H^{1}(B)}}{1+\lambda_{0}^{2}(DT_{0}(\lambda_{0})u_{0},u_{0})_{H^{1}(B)}}$
$\displaystyle+O\left(\sup_{\lambda\in
U}\left\lVert(T_{h}(\lambda)-T_{0}(\lambda))|_{R(E_{\lambda_{0}})}\right\rVert\left\lVert(T_{h}^{*}(\lambda)-T_{0}^{*}(\lambda))|_{R(E_{\lambda_{0}}^{*})}\right\rVert\right).$

We first estimate the remainder term

$\mathcal{R}_{h}=\sup_{\lambda\in
U}\left\lVert(T_{h}(\lambda)-T_{0}(\lambda))|_{R(E_{\lambda_{0}})}\right\rVert\left\lVert(T_{h}^{*}(\lambda)-T_{0}^{*}(\lambda))|_{R(E_{\lambda_{0}}^{*})}\right\rVert.$

For the leftmost norm, we may begin as in the proof of Lemma 13 with the
estimate

$\displaystyle\left|((T_{h}(\lambda)-T_{0}(\lambda))u,u^{\prime})_{H^{1}(B)}\right|$
$\displaystyle\leq\left|((\mathbb{A}_{h}^{-1}-\mathbb{A}_{0}^{-1})(\lambda^{-1}\mathbb{K}_{h}-\mathbb{B})u,u^{\prime})_{H^{1}(B)}\right|$
$\displaystyle\hskip
40.00006pt+\left|\lambda\right|^{-1}\left|(\mathbb{A}_{0}^{-1}(\mathbb{K}_{h}-\mathbb{K}_{0})u,u^{\prime})_{H^{1}(B)}\right|$

for $u\in R(E_{\lambda_{0}})$ and $u^{\prime}\in H^{1}(B)$. The first term may
be estimated as in that proof to obtain

$\displaystyle\left|((\mathbb{A}_{h}^{-1}-\mathbb{A}_{0}^{-1})(\lambda^{-1}\mathbb{K}_{h}-\mathbb{B})u,u^{\prime})_{H^{1}(B)}\right|$
$\displaystyle\hskip 40.00006pt\leq C_{s}\left\lVert
A_{h}-A_{0}\right\rVert_{(L^{d/s}(B))^{d\times d}}\left\lVert
u\right\rVert_{H^{1}(B)}\left\lVert u^{\prime}\right\rVert_{H^{1}(B)}.$

We may proceed as before to estimate the second term, but Proposition 9 now
implies that $u\in H^{s+1}(B)$ with the estimate

$\left\lVert u\right\rVert_{H^{s+1}(B)}\leq C_{s}\left\lVert
u\right\rVert_{H^{1}(B)},$ (16)

and we may take advantage of this higher regularity. For $d=2$ the Sobolev
embedding theorem implies that $u\in C_{b}(B)$ and $u^{\prime}\in L^{r}(B)$
for every $r\in[2,\infty)$, and applying the three-term Hölder’s inequality
with $p=\frac{r}{r-1}$, $q=\infty$, and any $r\in[2,\infty)$ yields

$\displaystyle\left|\lambda^{-1}(\mathbb{A}_{0}^{-1}(\mathbb{K}_{h}-\mathbb{K}_{0})u,u^{\prime})_{H^{1}(B)}\right|$
$\displaystyle\hskip
60.00009pt=k^{2}\left|\lambda\right|^{-1}\left|((n_{h}-n_{0})u,u^{\prime})_{B}\right|$
$\displaystyle\hskip 60.00009pt\leq k^{2}R^{-1}\left\lVert
n_{h}-n_{0}\right\rVert_{L^{r/(r-1)}(B)}\left\lVert
u\right\rVert_{C_{b}(B)}\left\lVert u^{\prime}\right\rVert_{L^{r}(B)}$
$\displaystyle\hskip 60.00009pt\leq C_{s,\epsilon}\left\lVert
n_{h}-n_{0}\right\rVert_{L^{1+\epsilon}(B)}\left\lVert
u\right\rVert_{H^{1}(B)}\left\lVert u^{\prime}\right\rVert_{H^{1}(B)},$

where in the last inequality we chose $r=\frac{1+\epsilon}{\epsilon}$ for a
given $\epsilon>0$ and applied the Sobolev embedding estimates along with
(16). For $d=3$ we have $u\in L^{6/(1-2s)}(B)$ and $u^{\prime}\in L^{6}(B)$,
and by applying the three-term Hölder’s inequality with $p=\frac{3}{2+s}$,
$q=\frac{6}{1-2s}$, and $r=6$ we obtain

$\displaystyle\left|\lambda^{-1}(\mathbb{A}_{0}^{-1}(\mathbb{K}_{h}-\mathbb{K}_{0})u,u^{\prime})_{H^{1}(B)}\right|$
$\displaystyle\hskip
60.00009pt=k^{2}\left|\lambda\right|^{-1}\left|((n_{h}-n_{0})u,u^{\prime})_{B}\right|$
$\displaystyle\hskip 60.00009pt\leq k^{2}R^{-1}\left\lVert
n_{h}-n_{0}\right\rVert_{L^{3/(2+s)}(B)}\left\lVert
u\right\rVert_{L^{6/(1-2s)}(B)}\left\lVert u^{\prime}\right\rVert_{L^{6}(B)}$
$\displaystyle\hskip 60.00009pt\leq C_{s,\epsilon}\left\lVert
n_{h}-n_{0}\right\rVert_{L^{3/(2+s)}(B)}\left\lVert
u\right\rVert_{H^{1}(B)}\left\lVert u^{\prime}\right\rVert_{H^{1}(B)},$

By combining these estimates along with the definition of the operator norm,
we arrive at the estimate

$\displaystyle\begin{split}&\left\lVert(T_{h}(\lambda)-T_{0}(\lambda))|_{R(E_{\lambda_{0}})}\right\rVert\\\
&\hskip 20.00003pt\leq C_{s}\biggr{(}\left\lVert
A_{h}-A_{0}\right\rVert_{(L^{d/s}(B))^{d\times d}}+\left\lVert
n_{h}-n_{0}\right\rVert_{L^{p^{\prime}}(B)}\biggr{)},\end{split}$ (17)

where $p^{\prime}$ is defined as in (15). The constant $C_{s}$ is uniform with
respect to $h$ and $\lambda$, but it depends on $\epsilon$ in the case $d=2$.
We now estimate the rightmost norm in the remainder term $\mathcal{R}_{h}$,
which is in terms of the sequence $\\{T_{h}^{*}(\lambda)\\}$ of adjoint
operators. Since the operators $\mathbb{A}_{h}$ and $\mathbb{B}$ are self-
adjoint, we may write the operator $T_{h}^{*}(\lambda)-T_{0}^{*}(\lambda)$ in
the form

$T_{h}^{*}(\lambda)-T_{0}^{*}(\lambda)=-(\overline{\lambda}^{-1}\mathbb{K}_{h}^{*}-\mathbb{B})(\mathbb{A}_{h}^{-1}-\mathbb{A}_{0}^{-1})-\overline{\lambda}^{-1}(\mathbb{K}_{h}^{*}-\mathbb{K}_{0}^{*})\mathbb{A}_{0}^{-1}.$

For the first term, we observe that

$\displaystyle\left|((\overline{\lambda}^{-1}\mathbb{K}_{h}^{*}-\mathbb{B})(\mathbb{A}_{h}^{-1}-\mathbb{A}_{0}^{-1})u^{*},u^{\prime})_{H^{1}(B)}\right|$
$\displaystyle\hskip
80.00012pt=\left|((\mathbb{A}_{h}^{-1}-\mathbb{A}_{0}^{-1})u^{*},(\lambda^{-1}\mathbb{K}_{h}-\mathbb{B})u^{\prime})_{H^{1}(B)}\right|$

for $u^{*}\in R(E_{\lambda_{0}})$ and $u^{\prime}\in H^{1}(B)$. Recalling from
the proof Lemma 13 that $(\lambda^{-1}\mathbb{K}_{h}-\mathbb{B})$ maps
$H^{1}(B)$ continuously into $H^{s+1}(B)$, uniformly for sufficiently small
$h$, we apply Proposition 11 to obtain

$\displaystyle\left|((\overline{\lambda}^{-1}\mathbb{K}_{h}^{*}-\mathbb{B})(\mathbb{A}_{h}^{-1}-\mathbb{A}_{0}^{-1})u^{*},u^{\prime})_{H^{1}(B)}\right|$
$\displaystyle\hskip 60.00009pt\leq C_{s}\left\lVert
A_{h}-A_{0}\right\rVert_{(L^{d/s}(B))^{d\times d}}\left\lVert
u^{*}\right\rVert_{H^{1}(B)}\left\lVert u^{\prime}\right\rVert_{H^{1}(B)}.$

For the second term, we note that $u^{*}\in R(E_{\lambda_{0}}^{*})$ satisfies
(4) with $(A,n)=(A_{0},\overline{n_{0}})$ and
$\lambda=\overline{\lambda_{0}}$, from which Proposition 9 implies that
$u^{*}\in H^{s+1}(B)$ with the estimate $\left\lVert
u^{*}\right\rVert_{H^{s+1}(B)}\leq C_{s}\left\lVert
u^{*}\right\rVert_{H^{1}(B)}$. Observing that

$((\mathbb{K}_{h}^{*}-\mathbb{K}_{0}^{*})\mathbb{A}_{0}^{-1}u^{*},u^{\prime})_{H^{1}(B)}=-k^{2}((\overline{n_{h}}-\overline{n_{0}})u^{*},u^{\prime})_{H^{1}(B)},$

we may apply the three-term Hölder’s inequality to the second term just as we
did when estimating the leftmost term in $\mathcal{R}_{h}$ to obtain

$\left|(\overline{\lambda}^{-1}(\mathbb{K}_{h}^{*}-\mathbb{K}_{0}^{*})\mathbb{A}_{0}^{-1}u^{*},u^{\prime})_{H^{1}(B)}\right|\leq
C_{s}\left\lVert n_{h}-n_{0}\right\rVert_{L^{p^{\prime}}(B)}\left\lVert
u^{*}\right\rVert_{H^{1}(B)}\left\lVert u^{\prime}\right\rVert_{H^{1}(B)}.$

By combining these estimates, the definition of the operator norm yields

$\displaystyle\begin{split}&\left\lVert(T_{h}^{*}(\lambda)-T_{0}^{*}(\lambda))|_{R(E_{\lambda_{0}}^{*})}\right\rVert\\\
&\hskip 20.00003pt\leq C_{s}\biggr{(}\left\lVert
A_{h}-A_{0}\right\rVert_{(L^{d/s}(B))^{d\times d}}+\left\lVert
n_{h}-n_{0}\right\rVert_{L^{p^{\prime}}(B)}\biggr{)},\end{split}$ (18)

where $p^{\prime}$ is defined as in (15), and the constant $C_{s}$ is uniform
with respect to $h$ and $\lambda$ but depends on $\epsilon$ in the case $d=2$.
Noticing that the right-hand sides of (LABEL:Th1) and (LABEL:Th2) coincide, we
arrive at the result

$\mathcal{R}_{h}=O\left(\left\lVert
A_{h}-A_{0}\right\rVert_{(L^{d/s}(B))^{d\times d}}^{2}+\left\lVert
n_{h}-n_{0}\right\rVert_{L^{p^{\prime}}(B)}^{2}\right).$ (19)

In order to evaluate the correction term, we write the numerator in the form

$\displaystyle((T_{0}(\lambda_{0})-T_{h}(\lambda_{0}))u_{0},u_{0})_{H^{1}(B)}$
$\displaystyle\hskip
40.00006pt=-((\mathbb{A}_{h}-\mathbb{A}_{0})(\lambda_{0}^{-1}\mathbb{K}_{0}-\mathbb{B})u_{0},u_{0})_{H^{1}(B)}$
(20a) $\displaystyle\hskip
60.00009pt+\lambda_{0}^{-1}((\mathbb{K}_{h}-\mathbb{K}_{0})u_{0},u_{0})_{H^{1}(B)}$
(20b) $\displaystyle\hskip
60.00009pt-((\mathbb{A}_{h}^{-1}-\mathbb{A}_{0}^{-1})(\mathbb{A}_{h}-\mathbb{A}_{0})(\lambda_{0}^{-1}\mathbb{K}_{0}-\mathbb{B})u_{0},u_{0})_{H^{1}(B)}$
(20c) $\displaystyle\hskip
60.00009pt+\lambda_{0}^{-1}((\mathbb{A}_{h}^{-1}-\mathbb{A}_{0}^{-1})(\mathbb{K}_{h}-\mathbb{K}_{0})u_{0},u_{0})_{H^{1}(B)},$
(20d)

where we have used the identity

$\mathbb{A}_{h}^{-1}-\mathbb{A}_{0}^{-1}=-\mathbb{A}_{0}^{-1}(\mathbb{A}_{h}-\mathbb{A}_{0})\mathbb{A}_{0}^{-1}-(\mathbb{A}_{h}^{-1}-\mathbb{A}_{0}^{-1})(\mathbb{A}_{h}-\mathbb{A}_{0})\mathbb{A}_{0}^{-1}$

to force all terms involving $\mathbb{A}_{h}^{-1}-\mathbb{A}_{0}^{-1}$ to be
sufficiently small that they may be absorbed into the remainder term, as we
shall show. We note that we also used the fact that $\mathbb{A}_{0}^{-1}$ is
the identity operator to ignore it anytime it is composed with another
operator; however, we have left it in place when appearing in a difference of
operators in order to clearly see the influence of the perturbation. We first
use Proposition 11 to show that (20c) and (20d) may be absorbed into the
remainder term $\mathcal{R}_{h}$. Indeed, since $u_{0}$ is an eigenfunction
corresponding to $\lambda_{0}$, we have
$(\lambda_{0}^{-1}\mathbb{K}_{0}-\mathbb{B})u_{0}=-\lambda_{0}^{-1}u_{0}$ and
$u_{0}\in H^{s+1}(B)$ by Proposition 9. Thus, Proposition 11 asserts that

$\displaystyle\left|((\mathbb{A}_{h}^{-1}-\mathbb{A}_{0}^{-1})(\mathbb{A}_{h}-\mathbb{A}_{0})(\lambda_{0}^{-1}\mathbb{K}_{0}-\mathbb{B})u_{0},u_{0})_{H^{1}(B)}\right|$
$\displaystyle\hskip
60.00009pt=\left|\lambda_{0}\right|^{-1}\left|((\mathbb{A}_{h}^{-1}-\mathbb{A}_{0}^{-1})(\mathbb{A}_{h}-\mathbb{A}_{0})u_{0},u_{0})_{H^{1}(B)}\right|$
$\displaystyle\hskip 60.00009pt\leq C_{s}\left\lVert
A_{h}-A_{0}\right\rVert_{(L^{d/s}(B))^{d\times
d}}\left\lVert(\mathbb{A}_{h}-\mathbb{A}_{0})u_{0}\right\rVert_{H^{1}(B)}\left\lVert
u_{0}\right\rVert_{H^{s+1}(B)}.$

We observe that $\left\lVert u_{0}\right\rVert_{H^{s+1}(B)}\leq
C_{s}\left\lVert u_{0}\right\rVert_{H^{1}(B)}=C_{s}$ by Proposition 9 and that

$\displaystyle\left\lVert(\mathbb{A}_{h}-\mathbb{A}_{0})u_{0}\right\rVert_{H^{1}(B)}$
$\displaystyle=\sup_{\left\lVert\psi\right\rVert_{H^{1}(B)}=1}\left|((A_{h}-A_{0})\nabla
u_{0},\nabla\psi)_{B}\right|$ $\displaystyle\leq C_{s}\left\lVert
A_{h}-A_{0}\right\rVert_{(L^{d/s}(B))^{d\times d}},$

where we have applied the three-term Hölder’s inequality with $p=\frac{d}{s}$,
$q=\frac{2d}{d-2s}$, and $r=2$ in a similar manner to the proof of Proposition
11. It follows that (20c) satisfies the estimate

$\left|((\mathbb{A}_{h}^{-1}-\mathbb{A}_{0}^{-1})(\mathbb{A}_{h}-\mathbb{A}_{0})(\lambda_{0}^{-1}\mathbb{K}_{0}-\mathbb{B})u_{0},u_{0})_{H^{1}(B)}\right|\leq
C_{s}\left\lVert A_{h}-A_{0}\right\rVert_{(L^{d/s}(B))^{d\times d}}^{2}$

and may be absorbed into the remainder term $\mathcal{R}_{h}$. The estimate of
(20d) follows similarly, as from Proposition 11 we have

$\displaystyle\left|\lambda_{0}^{-1}((\mathbb{A}_{h}^{-1}-\mathbb{A}_{0}^{-1})(\mathbb{K}_{h}-\mathbb{K}_{0})u_{0},u_{0})_{H^{1}(B)}\right|$
$\displaystyle\hskip 40.00006pt\leq C_{s}\left\lVert
A_{h}-A_{0}\right\rVert_{(L^{d/s}(B))^{d\times
d}}\left\lVert(\mathbb{K}_{h}-\mathbb{K}_{0})u_{0}\right\rVert_{H^{1}(B)}\left\lVert
u_{0}\right\rVert_{H^{s+1}(B)}.$

As with the previous estimate, the bound $\left\lVert
u_{0}\right\rVert_{H^{s+1}(B)}\leq C_{s}$ implies that

$\displaystyle\left\lVert(\mathbb{K}_{h}-\mathbb{K}_{0})u_{0}\right\rVert_{H^{1}(B)}$
$\displaystyle=\sup_{\left\lVert\psi\right\rVert_{H^{1}(B)}=1}\left|((n_{h}-n_{0})u_{0},\psi)_{H^{1}(B)}\right|$
$\displaystyle\leq C_{s}\left\lVert
n_{h}-n_{0}\right\rVert_{L^{p^{\prime}}(B)},$

where we have applied the three-term Hölder’s inequality as we did in the
estimate of the remainder $\mathcal{R}_{h}$. From the arithmetic-geometric
mean inequality we obtain

$\displaystyle\left|\lambda_{0}^{-1}((\mathbb{A}_{h}^{-1}-\mathbb{A}_{0}^{-1})(\mathbb{K}_{h}-\mathbb{K}_{0})u_{0},u_{0})_{H^{1}(B)}\right|$
$\displaystyle\hskip 60.00009pt\leq C_{s}\left\lVert
A_{h}-A_{0}\right\rVert_{(L^{d/s}(B))^{d\times d}}\left\lVert
n_{h}-n_{0}\right\rVert_{L^{p^{\prime}}(B)}$ $\displaystyle\hskip
60.00009pt\leq C_{s}\biggr{(}\left\lVert
A_{h}-A_{0}\right\rVert_{(L^{d/s}(B))^{d\times d}}^{2}+\left\lVert
n_{h}-n_{0}\right\rVert_{L^{p^{\prime}}(B)}^{2}\biggr{)},$

and we conclude that (20d) may also be absorbed into the remainder term
$\mathcal{R}_{h}$. Finally, we compute the correction term from the remaining
terms (20a) and (20b). Again noting that
$(\lambda_{0}^{-1}\mathbb{K}_{0}-\mathbb{B})u_{0}=-\lambda_{0}^{-1}u_{0}$, we
may write (20a) as

$\displaystyle-((\mathbb{A}_{h}-\mathbb{A}_{0})(\lambda_{0}^{-1}\mathbb{K}_{0}-\mathbb{B})u_{0},u_{0})_{H^{1}(B)}$
$\displaystyle=\lambda_{0}^{-1}((\mathbb{A}_{h}-\mathbb{A}_{0})u_{0},u_{0})_{H^{1}(B)}$
$\displaystyle=\lambda_{0}^{-1}((A_{h}-A_{0})\nabla u_{0},\nabla u_{0})_{B},$

and we may write (20b) in the form

$\lambda_{0}^{-1}((\mathbb{K}_{h}-\mathbb{K}_{0})u_{0},u_{0})_{H^{1}(B)}=-\lambda_{0}^{-1}k^{2}((n_{h}-n_{0})u_{0},u_{0})_{B}.$

Thus, we arrive at the correction formula given by (14). ∎

∎

In the following corollary, we use (14) to obtain an upper bound on
perturbations of the eigenvalues.

######  Corollary 15.

Assume the hypotheses of Theorem 14, and define the exponent
$p^{\prime\prime}$ by

$p^{\prime\prime}=\left\\{\begin{array}[]{cl}1&\text{ if }d=2,\\\
\frac{3}{2(s+1)}&\text{ if }d=3.\end{array}\right.$ (21)

For sufficiently small $h>0$ we have the bound

$\left|\lambda_{h}-\lambda_{0}\right|\leq C_{s}\biggr{(}\left\lVert
A_{h}-A_{0}\right\rVert_{(L^{d/2s}(B))^{d\times d}}+\left\lVert
n_{h}-n_{0}\right\rVert_{L^{p^{\prime\prime}}(B)}\biggr{)}.$ (22)

Proof. We first direct our attention to estimating the correction term in
(14). Since $\nabla u_{0}\in H^{s+1}(B)$ with the estimate $\left\lVert\nabla
u_{0}\right\rVert_{(H^{s}(B))^{d}}\leq C_{s}$, implying by the Sobolev
embedding theorem that $\nabla u_{0}\in(L^{2d/(d-2s)}(B))^{d}$, we may apply
the three-term Hölder’s inequality with $p=\frac{d}{2s}$ and
$q=r=\frac{2d}{d-2s}$ to obtain

$\left|(A_{h}-A_{0})\nabla u_{0},\nabla u_{0})_{B}\right|\leq C_{s}\left\lVert
A_{h}-A_{0}\right\rVert_{(L^{d/2s}(B))^{d\times d}}.$

For the second term, the Sobolev embedding theorem implies that $u_{0}\in
C_{b}(B)$ if $d=2$ and $u_{0}\in L^{6/(1-2s)}(B)$ if $d=3$. For $d=2$, we use
the observation that $u_{0}$ is uniformly bounded in $B$ to obtain

$\left|((n_{h}-n_{0})u_{0},u_{0})_{B}\right|\leq\left\lVert
n_{h}-n_{0}\right\rVert_{L^{1}(B)}\left\lVert
u_{0}\right\rVert_{C_{b}(B)}^{2}\leq C_{s}\left\lVert
n_{h}-n_{0}\right\rVert_{L^{1}(B)}.$

For $d=3$, the three-term Hölder’s inequality with $p=\frac{3}{2(s+1)}$ and
$q=r=\frac{6}{1-2s}$ yields

$\displaystyle\left|((n_{h}-n_{0})u_{0},u_{0})_{B}\right|$
$\displaystyle\leq\left\lVert
n_{h}-n_{0}\right\rVert_{L^{3/2(s+1)}(B)}\left\lVert
u_{0}\right\rVert_{L^{6/(1-2s)}(B)}^{2}$ $\displaystyle\leq C_{s}\left\lVert
n_{h}-n_{0}\right\rVert_{L^{3/2(s+1)}(B)}.$

Finally, we observe that the remainder term may be absorbed into the upper
bound for the correction term provided that

$\displaystyle\left\lVert A_{h}-A_{0}\right\rVert_{(L^{d/s}(B))^{d\times
d}}^{2}$ $\displaystyle\leq C\left\lVert
A_{h}-A_{0}\right\rVert_{(L^{d/2s}(B))^{d\times d}},$ (23a)
$\displaystyle\left\lVert n_{h}-n_{0}\right\rVert_{L^{p^{\prime}}(B)}^{2}$
$\displaystyle\leq C\left\lVert
n_{h}-n_{0}\right\rVert_{L^{p^{\prime\prime}}(B)},$ (23b)

for sufficiently small $h$. We use Hölder’s interpolation inequality (Theorem
3) to show that both (23a) and (23b) hold. Beginning with (23a), we first note
that

$\left\lVert A_{h}-A_{0}\right\rVert_{(L^{d/s}(B))^{d\times
d}}^{2}=\left(\sum_{i,j=1}^{d}\left\lVert(a_{h})_{ij}-(a_{0})_{ij}\right\rVert_{L^{d/s}(B)}\right)^{2},$

where $(a_{h})_{ij}$ represents the $i,j$ entry of the matrix $A_{h}$. We
apply Hölder’s interpolation inequality with $p=\frac{d}{s}$, $q=\infty$,
$r=\frac{d}{2s}$, and $\alpha=\frac{1}{2}$ to each term in the sum to obtain

$\displaystyle\left\lVert A_{h}-A_{0}\right\rVert_{(L^{d/s}(B))^{d\times
d}}^{2}$ $\displaystyle\hskip
40.00006pt\leq\left(\sum_{i,j=1}^{d}\left\lVert(a_{h})_{ij}-(a_{0})_{ij}\right\rVert_{L^{d/2s}(B)}^{1/2}\left\lVert(a_{h})_{ij}-(a_{0})_{ij}\right\rVert_{L^{\infty}(B)}^{1/2}\right)^{2},$

and an application of the Cauchy-Schwarz inequality yields

$\displaystyle\left\lVert A_{h}-A_{0}\right\rVert_{(L^{d/s}(B))^{d\times
d}}^{2}$ $\displaystyle\hskip
40.00006pt\leq\left(\sum_{i,j=1}^{d}\left\lVert(a_{h})_{ij}-(a_{0})_{ij}\right\rVert_{L^{d/2s}(B)}\right)\left(\sum_{i,j=1}^{d}\left\lVert(a_{h})_{ij}-(a_{0})_{ij}\right\rVert_{L^{\infty}(B)}\right)$
$\displaystyle\hskip 40.00006pt=\left\lVert
A_{h}-A_{0}\right\rVert_{(L^{d/2s}(B))^{d\times d}}\left\lVert
A_{h}-A_{0}\right\rVert_{(L^{\infty}(B))^{d\times d}}.$

Observing that $\left\lVert A_{h}-A_{0}\right\rVert_{(L^{\infty}(B))^{d\times
d}}$ is uniformly bounded as $h\to 0$ due to convergence of $\\{A_{h}\\}$ to
$A_{0}$ in $W_{\Sigma}^{1,\infty}(B)$, it follows that (23a) holds. We now
apply the same reasoning to establish (23b), and we begin by considering the
exponent $q=\frac{p^{\prime}p^{\prime\prime}}{2p^{\prime\prime}-p^{\prime}}$,
which may be explicitly written as

$q=\left\\{\begin{array}[]{cl}\frac{1+\epsilon}{1-\epsilon}&\text{ if }d=2,\\\
\frac{3}{2}&\text{ if }d=3.\end{array}\right.$

Noting that
$p^{\prime\prime}<p^{\prime}<\frac{p^{\prime}p^{\prime\prime}}{2p^{\prime\prime}-p^{\prime}}$
(for either choice of the dimension $d$), we apply Hölder’s interpolation
inequality with $p=p^{\prime\prime}$,
$q=\frac{p^{\prime}p^{\prime\prime}}{2p^{\prime\prime}-p^{\prime}}$,
$r=p^{\prime}$, and $\alpha=\frac{1}{2}$ to obtain

$\left\lVert n_{h}-n_{0}\right\rVert_{L^{p^{\prime}}(B)}^{2}\leq\left\lVert
n_{h}-n_{0}\right\rVert_{L^{p^{\prime\prime}}(B)}\left\lVert
n_{h}-n_{0}\right\rVert_{L^{q}(B)}.$

Convergence of $\\{n_{h}\\}$ to $n_{0}$ in $L^{2}(B)$ implies uniform
boundedness of $n_{h}-n_{0}$ in the weaker $L^{q}(B)$-norm as $h\to 0$, from
which we conclude that (23b) is satisfied. As a final note, we may absorb the
denominator of the correction term into the constant since it is independent
of $h$, providing the estimate (29). ∎

∎

We have considered perturbations that only require convergence of the
coefficients in certain spaces, but of course we may restrict to specific
types of perturbations (such as those corresponding to $L^{\infty}(B)$-bounded
diametrically small perturbations studied in [8, 9]) in order to gain more
information about the sensitivity of the eigenvalues in those cases. We will
provide such an example in a computational setting in Section 6 by examining
the effect of introducing a circular void to the medium.

##  5 The isotropic case

While our main interest has concerned the effects of perturbations of an
anisotropic medium on the set of $\delta$-Stekloff eigenvalues, it is worth
investigating the isotropic case (in which $A=I$) to obtain improved bounds
that result from the higher regularity that is available in this case. Thus,
we assume for the remainder of this section that $A_{h}=A_{0}=I$. The
following result is an analogue of Proposition 9 and follows directly from
standard elliptic regularity estimates (cf. [1]).

######  Proposition 16.

If $w\in H^{1}(B)$ satisfies

$\displaystyle\Delta w-k^{2}w$ $\displaystyle=f\text{ in }B,$ (24a)
$\displaystyle\frac{\partial w}{\partial\nu}$ $\displaystyle=\xi\text{ on
}\partial B,$ (24b)

for given $f\in L^{2}(B)$ and $\xi\in H^{1/2}(\partial B)$, then $w_{h}\in
H^{2}(B)$ with the estimate

$\left\lVert w\right\rVert_{H^{2}(B)}\leq C\left(\left\lVert
f\right\rVert_{L^{2}(B)}+\left\lVert\xi\right\rVert_{H^{1/2}(\partial
B)}\right).$ (25)

We continue with the following analogue of Lemma 13, which asserts the norm
convergence of $\\{T_{h}(\lambda)\\}$ to $T_{0}(\lambda)$, independently of
$\lambda\in U$, with an improved assumption on the convergence of
$\\{n_{h}\\}$. We remark that this improvement is a result of no longer
needing to uniformly bound the norm $\left\lVert
n_{h}u\right\rVert_{\tilde{H}^{s-1}(B)}$, which required convergence of
$\\{n_{h}\\}$ in $L^{2}(B)$ (more specifically, in $L^{3/(2-s)}(B)$).
Otherwise, the proof is identical to the estimate of the second term in the
proof of Lemma 13.

######  Lemma 17.

If $n_{h}\to n_{0}$ in $L^{3/2}(B)$ as $h\to 0$, then the sequence
$\\{T_{h}(\lambda)\\}$ satisfies the norm estimate

$\left\lVert T_{h}(\lambda)-T_{0}(\lambda)\right\rVert\leq C\left\lVert
n_{h}-n_{0}\right\rVert_{L^{p_{0}}(B)},$ (26)

where for each $\epsilon>0$ we have

$p_{0}=\left\\{\begin{array}[]{cl}1+\epsilon&\text{ if }d=2,\\\
\frac{3}{2}&\text{ if }d=3,\end{array}\right.$

and the constant $C$ is independent of $h$ and $\lambda\in
U=\mathbb{C}\setminus\overline{B_{R}(0)}$ but depends on $\epsilon$ whenever
$d=2$. As a result, we have $T_{h}(\lambda)\to T_{0}(\lambda)$ in norm as
$h\to 0$, uniformly for $\lambda\in U$.

With these initial results in hand, we may state the improved version of
Theorem 14 in the isotropic case, noting that for $d=3$ the new exponent
$p^{\prime}$ may be obtained formally by setting $s=\frac{1}{2}$ in (15).

######  Theorem 18.

Suppose that $n_{h}\to n_{0}$ in $L^{3/2}(B)$ as $h\to 0$, and suppose that
there exist no nontrivial solutions of (5) for $(A,n)=(I,n_{0})$. Let
$\lambda_{0}$ be a nonzero simple $\delta$-Stekloff eigenvalue for
$(A,n)=(I,n_{0})$ with $H^{1}(B)$-normalized eigenfunction $u_{0}$, choose
$R>0$ such that $R<\left|\lambda_{0}\right|$, and let
$U=\mathbb{C}\setminus\overline{B_{R}(0)}$. For sufficiently small $h>0$ there
exists a simple $\delta$-Stekloff eigenvalue for $(A,n)=(I,n_{h})$ that
satisfies the formula

$\lambda_{h}=\lambda_{0}+\frac{k^{2}((n_{h}-n_{0})u_{0},u_{0})_{B}}{\left\langle
S_{\delta}u_{0},u_{0}\right\rangle_{\partial B}}+O\left(\left\lVert
n_{h}-n_{0}\right\rVert_{L^{p^{\prime}}(B)}^{2}\right),$ (27)

where the exponent $p^{\prime}$ is given by

$p^{\prime}=\left\\{\begin{array}[]{cl}1+\epsilon&\text{ if }d=2,\\\
\frac{6}{5}&\text{ if }d=3,\end{array}\right.$ (28)

for a given $\epsilon>0$.

Proof. From Lemma 17 we may apply Theorem 7 as we did in the proof of Theorem
14 for the anisotropic case. The correction term in the present case is
obtained by setting $A_{h}=A_{0}=I$ in (14), and as a consequence we need only
bound the remainder term. For any $\lambda\in U$ we observe that for all $u\in
R(E_{\lambda_{0}})$ and $u^{\prime}\in H^{1}(B)$ we have

$\left|((T_{h}(\lambda)-T_{0}(\lambda))u,u^{\prime})_{B}\right|\leq
k^{2}\left|\lambda\right|^{-1}\left|((n_{h}-n_{0})u,u^{\prime})_{B}\right|.$

By Proposition 16 we have $u\in H^{2}(B)$ with $\left\lVert
u\right\rVert_{H^{2}(B)}\leq C\left\lVert u\right\rVert_{H^{1}(B)}$, and the
Sobolev embedding theorem implies that $u\in C_{b}(B)$ for $d=2,3$. Thus, we
may apply the three-term Hölder’s inequality with $p=p^{\prime}$, $q=\infty$,
and $r\in[2,\infty)$ when $d=2$ and $r=6$ when $d=3$ to obtain

$\left|((T_{h}(\lambda)-T_{0}(\lambda))u,u^{\prime})_{B}\right|\leq
C\left\lVert n_{h}-n_{0}\right\rVert_{L^{p^{\prime}}(B)}\left\lVert
u\right\rVert_{H^{1}(B)}\left\lVert u^{\prime}\right\rVert_{H^{1}(B)}.$

The definition of the operator norm yields

$\left\lVert(T_{h}(\lambda)-T_{0}(\lambda))|_{R(E_{\lambda_{0}})}\right\rVert\leq
C\left\lVert n_{h}-n_{0}\right\rVert_{L^{p^{\prime}}(B)}$

with $C$ independent of $\lambda\in U$. By the same reasoning as in the proof
of Theorem 14, the adjoint operators restricted to $R(E_{\lambda_{0}}^{*})$
satisfy the same bound, and the result follows. ∎

∎

######  Remark 19.

In [7], perturbations of Stekloff eigenvalues (i.e., $\delta=0$) corresponding
to an isotropic medium with $n$ real-valued were shown to satisfy the
approximate formula (recast in the notation of the present work)

$\lambda_{h}-\lambda_{0}\approx\frac{k^{2}((n_{h}-n_{0})u_{0},u_{0})_{B}}{\left\langle
u_{0},u_{0}\right\rangle_{\partial B}},$

where second-order terms have been ignored. We remark that (27) provides a
rigorous form of this approximation with a remainder term that is also valid
for generally complex-valued $n$. The same approach may be used to rigorously
justify a similar approximate formula in [15] for the related class of
modified transmission eigenvalues.

We end this section by presenting the analogue of Corollary 15 in the
isotropic case, which is an improvement in the case $d=3$ and may be obtained
formally by setting $s=\frac{1}{2}$ in (21).

######  Corollary 20.

Under the hypotheses of Theorem 18, for sufficiently small $h>0$ we have the
bound

$\left|\lambda_{h}-\lambda_{0}\right|\leq C\left\lVert
n_{h}-n_{0}\right\rVert_{L^{1}(B)}.$ (29)

Proof. We observe that

$\left|((T_{h}(\lambda_{0})-T_{0}(\lambda_{0}))u_{0},u^{\prime})_{B}\right|\leq
k^{2}\left|\lambda_{0}\right|^{-1}\left|((n_{h}-n_{0})u_{0},u_{0})_{B}\right|.$

As in the proof of Theorem 18, from $u_{0}\in H^{2}(B)$ the Sobolev embedding
theorem implies that $u_{0}\in C_{b}(B)$ for $d=2,3$ with continuous
embedding. Noting that $u_{0}$ now appears in both components of the inner
product, we apply the three-term Hölder’s inequality with $p=1$ and
$q=r=\infty$ to obtain

$\left|((T_{h}(\lambda_{0})-T_{0}(\lambda_{0}))u_{0},u^{\prime})_{B}\right|\leq
C\left\lVert n_{h}-n_{0}\right\rVert_{L^{1}(B)}.$

All that remains is to verify that the remainder term may be absorbed into the
right-hand side of this estimate, as we did in the proof of Corollary 15. We
begin with the observation that the exponent
$q=\frac{p^{\prime}}{2-p^{\prime}}$ may be written explicitly as

$q=\left\\{\begin{array}[]{cl}\frac{1+\epsilon}{1-\epsilon}&\text{ if }d=2,\\\
\frac{3}{2}&\text{ if }d=3.\end{array}\right.$

Noting that $1<p^{\prime}<q$, we apply Hölder’s interpolation inequality with
$p=1$, $q=\frac{p^{\prime}}{2-p^{\prime}}$, $r=p^{\prime}$, and
$\alpha=\frac{1}{2}$ to obtain

$\left\lVert n_{h}-n_{0}\right\rVert_{L^{p^{\prime}}(B)}^{2}\leq\left\lVert
n_{h}-n_{0}\right\rVert_{L^{1}(B)}\left\lVert
n_{h}-n_{0}\right\rVert_{L^{q}(B)}.$

Convergence of $\\{n_{h}\\}$ to $n_{0}$ in $L^{3/2}(B)$ and hence also in the
(possibly) weaker $L^{q}(B)$-norm implies that $\left\lVert
n_{h}-n_{0}\right\rVert_{L^{q}(B)}$ is uniformly bounded as $h\to 0$. Thus, we
conclude that (29) holds. ∎

∎

##  6 Numerical examples

We now investigate the sensitivity of $\delta$-Stekloff eigenvalues
corresponding to an isotropic medium through an example in two dimensions. We
refer to [13] for a description of how $\delta$-Stekloff eigenvalues and the
corresponding eigenfunctions may be computed using the finite element method.
We choose a medium such that $n_{0}=4$ in an L-shaped domain $D$ (depicted in
Figure 1) given by removing the square $[0.1,1.1]\times[-1.1,-0.1]$ from the
square $[-0.9,1.1]\times[-1.1,0.9]$ and $n_{0}=1$ elsewhere, and we choose the
auxiliary domain $B$ to be the disk of radius $1.5$ centered at the origin,
which guarantees that the closure of $D$ is contained within $B$. Finally, we
choose $k=1$ and $\delta=\frac{1}{2}$.

For the perturbed medium, we introduce a circular void of radius $h>0$
centered at $(x_{c},y_{c})=(0.1,0.4)$ that we denote by
$B_{h}:=B_{h}(x_{c},y_{c})$, in which case we may write $n_{h}$ as

$n_{h}=\left\\{\begin{array}[]{cl}1&\text{ in }B_{h},\\\ n_{0}&\text{
elsewhere}.\end{array}\right.$

We note that $\text{supp}(n_{h}-n_{0})=B_{h}$ and $\left\lVert
n_{h}-n_{0}\right\rVert_{L^{\infty}(B)}=\left|n_{0}-1\right|=3$, from which we
obtain

$\left\lVert n_{h}-n_{0}\right\rVert_{L^{3/2}(B)}\leq\left\lVert
n_{h}-n_{0}\right\rVert_{L^{\infty}(B)}\left(\int_{B_{h}}\,dx\right)^{2/3}=3\pi
h^{4/3}.$

In particular, we observe that $n_{h}\to n_{0}$ in $L^{3/2}(B)$, and Theorem
18 asserts that for sufficiently small $h$ we have the eigenvalue bound

$\left|\lambda_{h}-\lambda_{0}\right|\leq C\left\lVert
n_{h}-n_{0}\right\rVert_{L^{1}(B)}\leq Ch^{2}.$ (30)

We recall from (27) that $\lambda_{h}-\lambda_{0}$ is also related to the
magnitude of the eigenfunction $u_{0}$ in a neighborhood of the support of
$n_{h}-n_{0}$, motivating us to investigate the eigenvalue bound for two
eigenvalues whose respective eigenfunctions have different magnitudes near
$B_{h}$. The normalized eigenfunctions for two such eigenvalues are shown in
Figure 1, where we have depicted the boundary of the region $B_{h}$ for
$h=0.1$ as a white circle. The corresponding shifts in the two eigenvalues
when the radius $h$ is reduced logarithmically from $h=0.1$ to $h=0.01$ are
shown in Figure 2.

(a) $\lambda_{0}=-4.25$

(b) $\lambda_{0}=-2.16$

Figure 1: Plots of the magnitudes of two eigenfunctions corresponding to
$\delta$-Stekloff eigenvalues for an L-shaped domain. The eigenfunction
corresponding to $\lambda=-4.25$ is much larger than that corresponding to
$\lambda=-2.16$ in a neighborhood of $B_{h}$, which is outlined in white for
$h=0.1$. Figure 2: Magnitudes of the shifts of the eigenvalues
$\lambda_{0}=-4.25$ (blue circles) and $\lambda_{0}=-2.16$ (red squares). The
black dashed lines represent $O(h^{2})$-convergence.

As expected from the observation that the eigenfunction for
$\lambda_{0}=-4.25$ has significantly higher magnitude near $B_{h}$ than the
eigenfunction for $\lambda_{0}=-2.16$, we see in Figure 2 that the former
eigenvalue shifts to a greater degree due to changes in $n$. Moreover, by
comparing with the black dashed lines indicating $O(h^{2})$-convergence, we
confirm the bound (30); however, for smaller values of $h$ it appears that
convergence might be even faster than $O(h^{2})$, possibly influenced by the
shape of the eigenfunctions in that region. This observation suggests that it
might be of interest to study the structure of the eigenfunctions in greater
detail, such as the location of nodal lines, the number of connected
components, and where the eigenfunctions are concentrated. Notably, an
eigenvalue whose eigenfunction is concentrated in the center of the domain $B$
(as seen in Figure 1(a)) should display high sensitivity for most localized
perturbations of $n$, and it would be advantageous to be able to identify such
eigenvalues in a practical setting.

##  7 Conclusion

By reformulating the $\delta$-Stekloff eigenvalue problem as a nonlinear
eigenvalue problem, we have derived precise first-order asymptotic correction
formulas for perturbations of the eigenvalues due to changes in the
coefficients $A$ and $n$ of the medium. In particular, we found that the
regularity of $A$ has a strong effect on the rate of convergence of the
eigenvalues as $A_{h}\to A_{0}$ and $n_{h}\to n_{0}$ in appropriate norms,
with the general effect of higher regularity increasing this rate. We also
investigated–both theoretically and with a simple numerical example–the
sensitivity of the eigenvalues in the isotropic case, which allowed us to
leverage classical elliptic regularity estimates to obtain improved bounds.
While the numerical example was restricted to perturbations of a piecewise-
homogeneous isotropic medium, it serves as an initial step toward the study of
more complicated examples involving a possibly heterogeneous absorbing medium.

A possible application of the asymptotic formulas we have obtained is to go
beyond simply detecting the presence of a flaw in a material and attempt to
localize it, but the author is not aware of any results in this direction,
even among a restricted class of perturbations. If we ignore the remainder
term in the asymptotic formula for an isotropic medium, we are presented with
the inverse problem of determining the support of the perturbation
$n_{h}-n_{0}$ from a knowledge of $\lambda_{h}-\lambda_{0}$ for multiple
eigenvalues and possibly their eigenfunctions. A constructive solution to this
problem would be of immense value to the applicability of eigenvalues as
target signatures in nondestructive testing.

Other interesting questions remain unanswered, one of which concerns the
effect of the smoothing parameter $\delta$ on the sensitivity of the
eigenvalues. This relationship is not clear from the asymptotic formulas that
we have derived, as the eigenfunction $u_{0}$ also depends on $\delta$. A
further modification of the $\delta$-Stekloff eigenvalue problem was
investigated in [13], in which the lowest order Fourier coefficient in the
explicit representation of $S_{\delta}$ was increased by an amplification
factor $\sigma$, and a series of numerical examples showed an improvement in
sensitivity of one of the eigenvalues. It would be of interest to examine this
observation in the theoretical context that we have developed in the preceding
sections.

Finally, the class of $\delta$-Stekloff eigenvalues was extended to the case
of electromagnetic scattering in [14]. Similar to [13], stability results were
obtained for the eigenvalues but no quantitative formulas were derived. Given
the similar structure in this case to the present problem, it may be possible
to apply the same techniques to arrive at asymptotic formulas for the
eigenvalues; however, the more complicated solvability requirements for
electromagnetic problems (namely the strong dependence of compactness results
on the coefficients) leaves it unclear how to reformulate the eigenvalue
problem as a nonlinear eigenvalue problem that satisfies the hypotheses of
Theorem 7.

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