The anisotropic interior transmission eigenvalue problem with a conductive
boundary

Victor Hughes and Isaac Harris
Department of Mathematics, Purdue University, West Lafayette, IN 47907
Email<EMAIL_ADDRESS>and<EMAIL_ADDRESS>

Jiguang Sun
Department of Mathematics, Michigan Technological University, Houghton, MI
49931
Email<EMAIL_ADDRESS>

###### Abstract

In this paper, we study the transmission eigenvalue problem for an anisotropic
material with a conductive boundary. We prove that the transmission
eigenvalues for this problem exist and are at most a discrete set. We also
study the dependence of the transmission eigenvalues on the physical
parameters and prove that the first transmission eigenvalue is monotonic. We
then consider the limiting behavior of the transmission eigenvalues as the
conductive boundary parameter $\eta$ vanishes or goes to infinity in
magnitude. Finally, we provide some numerical examples on three different
domains to demonstrate our theoretical results.

Keywords: Transmission Eigenvalues; Inverse Scattering; Anisotropic Media

MSC: 35P25, 35J30, 65N30, 65N15

##  1 Introduction

In this paper, we study the transmission eigenvalue problem for an anisotropic
material with a conductive boundary. This can be seen as covering the boundary
of the scatterer with a thin highly conductive layer. This is modeled by
assuming that there is a jump across the boundary of the scatterer with
respect to the ‘normal’ derivatives of the total field. Transmission
eigenvalues are a relevant area of research due to the fact that they can be
used to retrieve information about the material properties of the scattering
object (see e.g. [7, 10, 17, 21]). It is well known that the transmission
eigenvalues can be recovered by the scattering data [8, 20, 29]. Therefore,
these eigenvalues can be used as a target signature, i.e., one can determine
defects in a material from the measured scattering data. The relevant inverse
problem is, given the eigenvalues, determine/estimate parameters in the
differential operator. We analyze the dependence of the transmission
eigenvalues on the conductivity parameter.

As we shall see, the eigenvalue problem is non-self-adjoint and non-linear.
The standard theory for eigenvalues for an elliptic operator does not apply.
This makes the study of the transmission eigenvalues interesting and
challenging mathematically. We refer to [3, 13, 32] on some recent work
connecting the transmission eigenvalues to non-scattering frequencies as well
as [12, 25, 26, 27, 35] for some numerical algorithms to compute the
eigenvalues. Here we study the discreteness, existence, and dependence on the
parameters for real transmission eigenvalues.

Now we introduce the scattering problem associated with the transmission
eigenvalues of interests. To this end, let $D\subset\mathbb{R}^{d}$ for $d=$ 2
or 3 be a simply connected open set with Lipschitz boundary $\partial D$ and
$\nu$ be the unit outward normal vector. The region $D$ denotes the scatterer
that we illuminate with an incident plane wave
$u^{i}\mathrel{\mathop{\mathchar 58\relax}}=\text{e}^{\text{i}kx\cdot\hat{y}}$
such that $k>0$ is the wave number and $\hat{y}\in\mathbb{S}^{d-1}$ denotes
the incident direction. Let $A(x)\in L^{\infty}(D,\mathbb{R}^{d\times d})$ be
a symmetric matrix valued function that is uniformly positive definite in $D$
satisfying

$0<A_{\text{min}}=\inf\limits_{x\in D}\inf\limits_{|\xi|=1}\overline{\xi}\cdot
A(x)\xi\quad\text{and}\quad A_{\text{max}}=\sup\limits_{x\in
D}\sup\limits_{|\xi|=1}\overline{\xi}\cdot A(x)\xi.$

We also assume that the refractive index $n(x)\in L^{\infty}(D)$ such that

$0<n_{\text{min}}\leq n(x)\leq n_{\text{max}}\quad\text{ a.e. in }\,\,D.$

Lastly, let the conductivity parameter $\eta\in L^{\infty}(\partial D)$
satisfy

$\eta_{\text{min}}\leq\eta(x)\leq\eta_{\text{max}}\quad\text{ a.e. on
}\,\,\partial D.$

The direct scattering problem for an anisotropic material with a conductive
boundary condition is formulated as follows: find $u^{s}\in
H^{1}_{loc}(\mathbb{R}^{d})$ such that

$\displaystyle\Delta u^{s}+k^{2}u^{s}=0\quad\text{in
$\mathbb{R}^{d}\setminus\overline{D}$}\quad\text{ and }\quad\nabla\cdot
A\nabla u+k^{2}nu=0\quad$ $\displaystyle\text{in $D$},$ (1)
$\displaystyle(u^{s}+u^{i})^{+}=u^{-}\quad\text{ and
}\quad\partial_{\nu}(u^{s}+u^{i})^{+}=\nu\cdot A\nabla u^{-}-\eta u\quad$
$\displaystyle\text{on $\partial D$},$ (2)

where $u=u^{s}+u^{i}$ denotes the total field. Here the superscripts $+$ and
$-$ demonstrate approaching the boundary from the outside and inside of $D$,
respectively. The scattered field satisfies the radiation condition

$\partial_{r}u^{s}-\text{i}ku^{s}=\mathcal{O}\Big{(}\frac{1}{r^{(d+1)/2}}\Big{)}\quad\text{
as }\quad r\to\infty$

uniformly with respect to $\hat{x}=\frac{x}{r}$ where $r=|x|$. Similar
arguments as in [5] lead to the well-posedness of the direct problem.

If there exists a general incident field satisfying the Helmholtz equation in
$\mathbb{R}^{d}$ that does not produce a scattered field on the exterior of
$D$, then, by (1)–(2), one has that $w=u^{s}+u^{i}$ and $v=u^{i}$ are in
$H^{1}(D)$ satisfying

$\displaystyle\Delta v+k^{2}v=0\quad$ $\displaystyle\text{ and
}\quad\nabla\cdot A\nabla w+k^{2}nw=0\quad\text{ in }D,$ (3) $\displaystyle
w=v\quad$ $\displaystyle\text{ and }\quad\nu\cdot A\nabla
w=\partial_{\nu}v+\eta v\quad\text{on $\partial D$.}$ (4)

The values $k\in\mathbb{C}$ such that there exists a nontrivial solution
$(w,v)\in H^{1}(D)\times H^{1}(D)$ to (3)–(4) are called transmission
eigenvalues.

The rest of the paper is organized as follows. First, we prove that the
transmission eigenvalues form an at most discrete set in the complex plane.
This is done by considering an equivalent variational formulation of the
problem and appealing to the analytic Fredholm theorem (see e.g. [9]). We then
prove the existence of infinitely many real transmission eigenvalues. It is
shown that the eigenvalues are monotone with respect to the coefficients. This
implies that the eigenvalues can be used to determine defects in the
scatterer. The limiting cases as $\eta\to 0$ as well as $\eta\to\pm\infty$ are
considered. In either case, the limiting value for the transmission eigenpairs
can be determined. Lastly, we provide some numerical validation for the
theoretical results.

##  2 Discreteness of Transmission Eigenvalues

In this section, we study the discreteness of the transmission eigenvalues
(TEVs) for an anisotropic scatterer with a conductive boundary condition. This
problem has been studied for the case when $A=I$ in [4, 20, 21]. Also, see
[19, 23] the study of this problem associated with an electromagnetic
scatterer. This result is useful in application since many qualitative
reconstruction methods such as the factorization method [6, 18, 22, 28] and
generalized linear sampling methods [1, 2, 9] fail to reconstruct the
scatterer $D$ from measured data if $k$ is a TEV. Here we study an equivalent
variational formulation of (3)–(4) to treat the case of $A\neq I$ and
$\eta\neq 0$, which is not covered in [4, 9, 16, 20, 21]. We refer the readers
to [7, 11, 14, 16, 21] for discreteness results of other TEV problems.

We begin with the case when $A_{\text{min}}-1>0$ and $\eta_{\text{max}}<0$.
Multiplying the second equation in (3) by $\overline{\phi}\in H^{1}(D)$ and
appealing to Green’s 1st theorem, we have that

$\int_{D}\nabla\overline{\phi}\cdot A\nabla
w-k^{2}nw\overline{\phi}\,\text{d}x\ =\int_{\partial D}(\nu\cdot A\nabla
w)\overline{\phi}\,\text{d}s.$

By applying the boundary conditions in (4) we have that

$0=\int_{D}\nabla\overline{\phi}\cdot A\nabla
w-k^{2}nw\overline{\phi}\,\text{d}x\ -\int_{\partial
D}\overline{\phi}\big{(}T_{k}w+\eta w\big{)}\,\text{d}s\quad\text{for
all}\,\,\phi\in H^{1}(D),$

where the Dirichlet-to-Neumann(DtN) map for the Helmholtz equation is defined
as

$T_{k}\mathrel{\mathop{\mathchar 58\relax}}H^{1/2}(\partial D)\longrightarrow
H^{-1/2}(\partial D)\quad\text{given by}\quad T_{k}f\mathrel{\mathop{\mathchar
58\relax}}=\partial_{\nu}v$

such that

$\Delta v+k^{2}v=0\quad\text{in }D\quad\text{and}\quad v=f\quad\text{on
}\partial D.$

Note that $T_{k}$ is well-defined for all complex values $k^{2}$ that are not
Dirichlet eigenvalues of the negative Laplacian in $D$. Moreover, $T_{k}$ as
an operator depends analytically on $k$ for all values for which it is well-
defined [24].

We now define the bounded sesquilinear form
$a_{k}(\cdot\,,\cdot)\mathrel{\mathop{\mathchar 58\relax}}H^{1}(D)\times
H^{1}(D)\mapsto\mathbb{C}$ such that

$a_{k}(w,\phi)\mathrel{\mathop{\mathchar
58\relax}}=\int_{D}\nabla\overline{\phi}\cdot A\nabla
w-k^{2}nw\overline{\phi}\,\text{d}x\ -\int_{\partial
D}\overline{\phi}\big{(}T_{k}w+\eta w\big{)}\,\text{d}s.$ (5)

It is clear that $k$ is a TEV if and only if there is a non-trivial $w$
satisfying $a_{k}(w,\phi)=0$ for all $\phi\in H^{1}(D)$, provided that $k^{2}$
is not a Dirichlet eigenvalue of the negative Laplacian in $D$. We denote the
set of Dirichlet eigenvalues as $\\{\lambda_{j}(D)\\}_{j=1}^{\infty}$.

To prove the discreteness, we appeal to the analytic Fredholm theorem. We
first show that the sesquilinear form associated with the transmission
eigenvalue problem is represented by a Fredholm operator with index zero that
depends on $k$ analytically in an open subset of the complex plane. To this
end, let $a_{k}(\cdot,\cdot)=b(\cdot,\cdot)+c_{k}(\cdot,\cdot)$, where

$b(w,\phi)=\int_{D}\nabla\overline{\phi}\cdot A\nabla
w+A_{\text{min}}w\overline{\phi}\,\text{d}x\ -\int_{\partial
D}\overline{\phi}T_{\text{i}}w\,\text{d}s.$ (6)

Here $T_{\text{i}}$ is the DtN mapping with $k=\text{i}$ and

$c_{k}(w,\phi)=-\int_{D}\big{(}k^{2}n+A_{\text{min}}\big{)}w\overline{\phi}\,\text{d}x\
-\int_{\partial
D}\overline{\phi}(T_{k}-T_{\text{i}})w+\eta\overline{\phi}w\,\text{d}s.$ (7)

In the next few lemmas, we show that $a_{k}(\cdot,\cdot)$, $b(\cdot,\cdot)$,
and $c_{k}(\cdot,\cdot)$ have analytical properties that allow us to use the
analytic Fredholm theorem.

######  Lemma 2.1.

Let $b(\cdot,\cdot)$ and $c_{k}(\cdot,\cdot)$ be given as in (6) and (7),
respectively. If we assume $A_{min}-1>0$ then $b(\cdot,\cdot)$ is coercive and
$c_{k}(\cdot,\cdot)$ is compact and analytic with respect to $k$ provided that
$k^{2}\in\mathbb{C}\setminus\\{\lambda_{j}(D)\\}_{j=1}^{\infty}$.

###### Proof.

We first estimate the boundary term in the variational form. By Green’s 1st
theorem and the fact that $v=w$ on $\partial D$, it holds that

$I=\int_{\partial D}\overline{w}T_{\text{i}}w\,\text{d}s\ =\int_{\partial
D}\overline{v}\partial_{\nu}v\,\text{d}s\ =\int_{D}|\nabla
v|^{2}+|v|^{2}\,\text{d}x\ =||v||^{2}_{H^{1}(D)}.$

The Cauchy-Schwartz inequality implies that

$I=\int_{\partial D}\overline{w}\partial_{\nu}v\,\text{d}s\
=\int_{D}\nabla\overline{w}\cdot\nabla
v+\overline{w}v\,\text{d}x\leq||v||_{H^{1}(D)}||w||_{H^{1}(D)}=\sqrt{I}||w||_{H^{1}(D)}.$

Consequently, $I\leq||w||^{2}_{H^{1}(D)}$. Then $b(\cdot,\cdot)$ is coercive
since

$b(w,w)=\int_{D}\nabla\overline{w}\cdot A\nabla
w+A_{\text{min}}|w|^{2}\,\text{d}x\ -\int_{\partial
D}\overline{w}T_{\text{i}}w\,\text{d}s\geq\big{(}A_{\text{min}}-1\big{)}||w||^{2}_{H^{1}(D)}.$

By the compact embeddings of $H^{1}(D)$ into $L^{2}(D)$ and $H^{1/2}(\partial
D)$ into $L^{2}(\partial D)$ as well as the fact that $T_{k}-T_{\text{i}}$ is
compact(see for e.g. [24]), there exists a compact operator
$F_{k}\mathrel{\mathop{\mathchar 58\relax}}H^{1}(D)\to H^{1}(D)$ such that
$c_{k}(w,\phi)=(F_{k}w,\phi)_{H^{1}(D)}$ for all $w,\phi\in H^{1}(D)$ due to
the Riesz representation theorem. Furthermore, $T_{k}$ is analytic provided
that $k^{2}$ is not a Dirichlet eigenvalue of the Laplacian in $D$.
Consequently, $F_{k}$ is analytic with respect to
$k^{2}\in\mathbb{C}\setminus\\{\lambda_{j}(D)\\}_{j=1}^{\infty}$, proving the
claim. ∎

Since $b(\cdot,\cdot)$ is bounded and coercive, there exists an invertible
operator $B\mathrel{\mathop{\mathchar 58\relax}}H^{1}(D)\to H^{1}(D)$ such
that $b(w,\phi)=(Bw,\phi)_{H^{1}(D)}$ for all $w,\phi\in H^{1}(D)$. Hence the
operator $B+F_{k}$ represents the sesquilinear form $a_{k}(\cdot\,,\cdot)$
which is Fredholm and analytic with respect to
$k^{2}\in\mathbb{C}\setminus\\{\lambda_{j}(D)\\}_{j=1}^{\infty}$. Next we show
that $a_{k}(\cdot\,,\cdot)$ is injective.

######  Lemma 2.2.

Let $a_{k}(\cdot,\cdot)$ be given as in (5) with $A_{\text{min}}-1>0$ and
$\eta_{\text{max}}<0$ then $a_{0}(\cdot,\cdot)$ is coercive (i.e. injective).

###### Proof.

We first notice that for $k=0$

$a_{0}(w,w)=\int_{D}\nabla\overline{w}\cdot A\nabla w\,\text{d}x\
-\int_{\partial D}\overline{w}T_{0}w+\eta|w|^{2}\,\text{d}s.$

By the boundary condition $v=w$ on $\partial D$ we obtain that

$I=\int_{\partial D}\overline{w}T_{0}w\,\text{d}s\ =\int_{\partial
D}\overline{v}\partial_{\nu}v\,\text{d}s\ =\int_{D}|\nabla v|^{2}\,\text{d}x\
=||\nabla v||^{2}_{L^{2}(D)}.$

We conclude that $I\leq||\nabla w||^{2}_{L^{2}(D)}$. Therefore,
$a_{0}(\cdot,\cdot)$ is coercive (injective) due to

$a_{0}(w,w)\geq\big{(}A_{\text{min}}-1\big{)}||\nabla
w||^{2}_{L^{2}(D)}-\eta_{\text{max}}||w||^{2}_{L^{2}(\partial D)}.$

This proves coercivity by the equivalence of norms $||\cdot||^{2}_{H^{1}(D)}$
and $||\nabla\cdot||^{2}_{L^{2}(D)}+||\cdot||^{2}_{L^{2}(\partial D)}$ in
$H^{1}(D)$(see for e.g. [33]) as well as the fact that $\eta_{\text{max}}<0$.
∎

To continue, we now consider the case when $A_{\text{max}}-1<0$ and
$\eta_{\text{min}}>0$. Similar to the previous case, we study the equivalent
variational form for the TEV problem. Multiplying the first equation in (3) by
the conjugate of $\phi\in H^{1}(D)$ and appealing to Green’s 1st theorem we
have that

$\int_{D}\nabla\overline{\phi}\cdot\nabla v-k^{2}v\overline{\phi}\,\text{d}x\
=\int_{\partial D}\overline{\phi}\partial_{\nu}v\,\text{d}s.$

Define the DtN map

$S_{k}\mathrel{\mathop{\mathchar 58\relax}}H^{1/2}(\partial D)\longrightarrow
H^{-1/2}(\partial D)\quad\text{given by}\quad S_{k}f\mathrel{\mathop{\mathchar
58\relax}}=\nu\cdot A\nabla w$

for the boundary value problem

$\nabla A\nabla w+k^{2}nw=0\quad\text{in }D\quad\text{and}\quad
w=f\quad\text{on }\partial D.$

Provided that $k^{2}$’s are not Dirichlet eigenvalues of $-n^{-1}\nabla\cdot
A\nabla$ for $D$, the DtN mapping is a well-defined bounded linear operator.
Denote by $\\{\lambda_{j}(D,A,n)\\}_{j=1}^{\infty}$ the set of Dirichlet
eigenvalues for the operator $-n^{-1}\nabla\cdot A\nabla$ for the region $D$.
If $k^{2}\in\mathbb{C}\setminus\\{\lambda_{j}(D,A,n)\\}_{j=1}^{\infty}$ then
$S_{k}$ is analytic with respect to $k$.

We can now write $a_{k}(\cdot\,,\cdot)\mathrel{\mathop{\mathchar
58\relax}}H^{1}(D)\times H^{1}(D)\mapsto\mathbb{C}$ as

$a_{k}(v,\phi)\mathrel{\mathop{\mathchar
58\relax}}=\int_{D}\nabla\overline{\phi}\cdot\nabla
v-k^{2}v\overline{\phi}\,\text{d}x\ -\int_{\partial
D}\overline{\phi}\big{(}S_{k}v-\eta v\big{)}\,\text{d}s$ (8)

and decomposed it into $a_{k}(\cdot,\cdot)=b(\cdot,\cdot)+c_{k}(\cdot,\cdot)$,
where

$b(v,\phi)=\int_{D}\nabla\overline{\phi}\cdot\nabla
v+v\overline{\phi}\,\text{d}x\ -\int_{\partial
D}\overline{\phi}S_{0}v\,\text{d}s$ (9)

and

$c_{k}(v,\phi)=-\int_{D}\big{(}k^{2}+1\big{)}v\overline{\phi}\,\text{d}x\
-\int_{\partial D}\overline{\phi}(S_{k}-S_{0})v-\eta
v\overline{\phi}\,\text{d}s.$ (10)

Note that both $b(\cdot,\cdot)$ and $c_{k}(\cdot,\cdot)$ are bounded
sesquilinear forms on $H^{1}(D)\times H^{1}(D)$. In the next few lemmas, we
show that $a_{k}(\cdot,\cdot)$ can be represented by an operator that is
analytic with respect to
$k^{2}\in\mathbb{C}\setminus\\{\lambda_{j}(D,A,n)\\}_{j=1}^{\infty}$ and
Fredholm with index zero.

######  Lemma 2.3.

Let $b(\cdot,\cdot)$ and $c_{k}(\cdot,\cdot)$ be given as in (9) and (10),
respectively. If $A_{\text{max}}-1<0$ then $b(\cdot,\cdot)$ is coercive and,
provided that
$k^{2}\in\mathbb{C}\setminus\\{\lambda_{j}(D,A,n)\\}_{j=1}^{\infty}$,
$c_{k}(\cdot,\cdot)$ is compact and analytic with respect to $k$.

###### Proof.

Since $A$ is a symmetric real-valued positive definite matrix, there exists
$A^{1/2}$ such that $A=A^{1/2}\cdot A^{1/2}$. Using Green’s 1st theorem and
the boundary condition $v=w$ on $\partial D$, it holds that

$I=\int_{\partial D}\overline{v}S_{0}v\,\text{d}s\ =\int_{\partial
D}\overline{w}(\nu\cdot A\nabla w)\,\text{d}s=\big{(}\nabla w,A\nabla
w\big{)}_{L^{2}(D)}=||A^{1/2}\nabla w||^{2}_{L^{2}(D)}.$

Therefore,

$I=\int_{\partial D}\overline{v}(\nu\cdot A\nabla w)\,\text{d}s\
=\big{(}\nabla v,A\nabla w\big{)}_{L^{2}(D)}=\big{(}A^{1/2}\nabla
v,A^{1/2}\nabla w\big{)}_{L^{2}(D)}.$

It holds that

$I\leq||A^{1/2}\nabla v||_{L^{2}(D)}||A^{1/2}\nabla
w||_{L^{2}(D)}\leq\sqrt{A_{\text{max}}}||\nabla v||_{L^{2}(D)}||A^{1/2}\nabla
w||_{L^{2}(D)}.$

This implies that $I\leq A_{\text{max}}||\nabla v||^{2}_{L^{2}(D)}\leq
A_{\text{max}}||v||^{2}_{H^{1}(D)}$. Therefore, $b(\cdot,\cdot)$ is coercive
since

$b(v,v)=\int_{D}|\nabla v|^{2}+|v|^{2}\,\text{d}x\ -\int_{\partial
D}\overline{v}S_{0}v\,\text{d}s\
\geq\big{(}1-A_{\text{max}}\big{)}||v||^{2}_{H^{1}(D)}.$

By the compact embeddings of $H^{1}(D)$ into $L^{2}(D)$ and $H^{1/2}(\partial
D)$ into $L^{2}(\partial D)$ and the fact that $S_{k}-S_{0}$ is compact, there
exists a compact operator $F_{k}\mathrel{\mathop{\mathchar
58\relax}}H^{1}(D)\to H^{1}(D)$ such that
$c_{k}(v,\phi)=(F_{k}v,\phi)_{H^{1}(D)}$ for all $v,\phi\in H^{1}(D)$. We also
have that $S_{k}$ is analytic, and thus $c_{k}(\cdot,\cdot)$ is analytic for
all $k$ such that
$k^{2}\in\mathbb{C}\setminus\\{\lambda_{j}(D,A,n)\\}_{j=1}^{\infty}$. The
proof is complete. ∎

Consequently, there exist an invertible operator $B$ and a compact operator
$F_{k}$ which is Fredholm and analytic with respect to
$k^{2}\in\mathbb{C}\setminus\\{\lambda_{j}(D,A,n)\\}_{j=1}^{\infty}$
associating with the sesquilinear form $a_{k}(\cdot\,,\cdot)$. Now we prove
the injectivity of $a_{k}(\cdot\,,\cdot)$ for
$k^{2}\in\mathbb{C}\setminus\\{\lambda_{j}(D,A,n)\\}_{j=1}^{\infty}$.

######  Lemma 2.4.

Let $a_{k}(\cdot,\cdot)$ be given as in (8) with $A_{\text{max}}-1<0$ and
$\eta_{\text{min}}>0$. Then $a_{0}(\cdot,\cdot)$ is coercive (i.e. injective).

###### Proof.

Note that

$a_{0}(v,v)=\int_{D}|\nabla v|\,\text{d}x\ -\int_{\partial
D}\overline{v}S_{0}v-\eta|v|^{2}\,\text{d}s$

and $I\leq A_{\text{max}}||\nabla v||^{2}_{L^{2}(D)}$. Thus,
$a_{0}(\cdot,\cdot)$ is coercive since

$a_{0}(v,v)\geq\big{(}1-A_{\text{max}}\big{)}||\nabla
v||^{2}_{L^{2}(D)}+\eta_{\text{min}}||v||^{2}_{L^{2}(\partial D)}$

by the equivalence of $||\cdot||^{2}_{H^{1}(D)}$ and
$||\nabla\cdot||^{2}_{L^{2}(D)}+||\cdot||^{2}_{L^{2}(\partial D)}$ in
$H^{1}(D)$. ∎

With this, we have proven that $a_{k}(\cdot,\cdot)$ is analytic and Fredholm
for all complex-valued $k$ except for a discrete set on the real line. For
either case, we have that $a_{0}(\cdot,\cdot)$ is an injective sesquilinear
form. We now state the main theorem for this section.

######  Theorem 2.1.

If $A_{\text{min}}-1>0$ and $\eta_{\text{max}}<0$ or $A_{\text{max}}-1<0$ and
$\eta_{\text{min}}>0$, then the set of transmission eigenvalues is at most
discrete.

###### Proof.

This is a consequence of the fact that TEV problem is given by

$0=a_{k}(w,\phi)=\left((B+F_{k})w,\phi\right)_{H^{1}(D)}\quad\text{ for
all}\,\,\phi\in H^{1}(D)$

where $B$ is invertible and $F_{k}$ is a compact operator and analytic for $k$
except for a discrete set. The result follows the analytic Fredholm theorem
since $a_{0}(\cdot,\cdot)$ is an injective sesquilinear form. ∎

##  3 Existence of Transmission Eigenvalues

In this section, we show that there exist infinitely many real TEVs. We shall
use a technique that was first introduced in [11] and used in [14, 22] to
prove the existence of TEVs associated with other scattering problems. Again,
we remark that this problem has not yet been studied when $A\neq I$. We shall
employ the assumptions that $A_{\text{min}}-1>0$, $n_{\text{max}}-1<0$, and
$\eta_{\text{max}}<0$ or $A_{\text{max}}-1<0$, $n_{\text{min}}-1>0$, and
$\eta_{\text{min}}>0$. Note that the discreteness needs no assumption on the
refractive index $n$. Once the existence is established we show that the first
TEV is monotone with respect to the coefficients. The result can be used to
study of the inverse spectral problem of estimating the coefficients from the
TEVs.

To show the existence of real transmission eigenvalues, we consider an
auxiliary problem for $u=w-v$ in $H^{1}_{0}(D)$. Note that, by (3)–(4), $u$
satisfies

$\displaystyle\nabla\cdot A\nabla u+k^{2}nu$
$\displaystyle=\nabla\cdot(I-A)\nabla v-k^{2}(n-1)v\quad\text{in $D$,}$ (11)
$\displaystyle\nu\cdot A\nabla u$ $\displaystyle=\nu\cdot(I-A)\nabla v+\eta
v\quad\text{on $\partial D$,}$ (12)

where $\Delta v+k^{2}v=0$ in $D$. Following [11], the above system (11)–(12)
can be viewed as a boundary value problem for $v\in H^{1}(D)$ given $u\in
H_{0}^{1}(D)$. We shall see that the system is well-posed for any
$k\in\mathbb{R}_{\geq 0}$. Once we have this we will need to enforce that $v$
obtained by solving (11)–(12) is a solution to the Helmholtz equation in $D$.

The variational formulation of (11)–(12) is to find $v$ such that

$\displaystyle\int_{D}\nabla\overline{\phi}\cdot(I-A)\nabla
v+k^{2}(n-1)v\overline{\phi}\,\text{d}x\ $ $\displaystyle+\int_{\partial
D}\eta v\overline{\phi}\,\text{d}s$
$\displaystyle=\int_{D}\nabla\overline{\phi}\cdot A\nabla
u-k^{2}nu\overline{\phi}\,\text{d}x.\ $ (13)

Using the assumption on the coefficients that either $A_{\text{min}}-1>0$,
$n_{\text{max}}-1<0$, and $\eta_{\text{max}}<0$ or $A_{\text{max}}-1<0$,
$n_{\text{min}}-1>0$, and $\eta_{\text{min}}>0$, it can be seen that the left
hand side is a coercive sesquilinear form for all $k\in\mathbb{R}_{\geq 0}$.
The right hand side is a bounded conjugate linear functional whose norm is
bounded by the norm of $u\in H^{1}_{0}(D)$. By applying to the Lax-Milgram
lemma we have that (13) is well-posed, i.e., for any given $u\in
H_{0}^{1}(D)$, there exists a unique solution $v_{u}\in H^{1}(D)$ satisfying
(11)–(12). The mapping $u\longmapsto v_{u}$ from $H_{0}^{1}(D)$ to $H^{1}(D)$
is a bounded linear operator. Now, we want the eigenfunction $v_{u}$ to
satisfy the Helmholtz equation in $D$ i.e.

$0=\int_{D}\nabla
v_{u}\cdot\nabla\overline{\psi}-k^{2}v_{u}\overline{\psi}\,\text{d}x\quad\text{for
all}\,\,\psi\in H_{0}^{1}(D).$

We define a bounded linear operator $\mathbb{L}_{k}\mathrel{\mathop{\mathchar
58\relax}}H_{0}^{1}(D)\to H_{0}^{1}(D)$ such that

$\big{(}\mathbb{L}_{k}u,\psi\big{)}_{H^{1}(D)}\mathrel{\mathop{\mathchar
58\relax}}=\int_{D}\nabla
v_{u}\cdot\nabla\overline{\psi}-k^{2}v_{u}\overline{\psi}\,\text{d}x\quad\text{for
all}\,\,\psi\in H_{0}^{1}(D).$ (14)

If there is a $k$ such that $\mathbb{L}_{k}u=0$ with $u\neq 0$ then $v_{u}$ is
a solution to the Helmholtz equation in $D$. Letting $w=u+v_{u}$ we have that
$(w,v_{u})\in H^{1}(D)\times H^{1}(D)$ satisfies (3)–(4) with eigenvalue $k$.
Conversely, if there is a TEV $k$ then $u=w-v$(nontrivial). In addition, $v$
satisfies (11)–(12) and is a solution for Helmholtz equation. From this, we
deduce that $k\in\mathbb{R}_{\geq 0}$ is a TEV if and only if $\mathbb{L}_{k}$
has a nontrivial null space. The existence of real TEVs is equivalent to
finding values when the null space of $\mathbb{L}_{k}$ is nontrivial.

######  Lemma 3.1.

The operator $\mathbb{L}_{k}$ defined by (14) is self-adjoint.

###### Proof.

We shall show that $\big{(}\mathbb{L}_{k}u,u\big{)}_{H^{1}(D)}$ is real-valued
for any $u\in H^{1}_{0}(D)$ which implies self-adjointness by Theorem 3:10-3
in [30]. Letting $\phi=u$ and $v=v_{u}$, (13) becomes

$\int_{D}\nabla\overline{u}\cdot(I-A)\nabla
v+k^{2}(n-1)v\overline{u}\,\text{d}x\ =\int_{D}\nabla\overline{u}\cdot A\nabla
u-k^{2}n|u|^{2}\,\text{d}x.$ (15)

Letting $\phi=v$, (13) becomes

$\displaystyle\int_{D}\nabla\overline{v}\cdot(I-A)\nabla
v+k^{2}(n-1)|v|^{2}\,\text{d}x\ $ $\displaystyle+\int_{\partial
D}\eta|v|^{2}\,\text{d}s$ $\displaystyle=\int_{D}\nabla\overline{v}\cdot
A\nabla u-k^{2}nu\overline{v}\,\text{d}x.$ (16)

Note that (14) can be written as

$\displaystyle\big{(}\mathbb{L}_{k}u,u\big{)}_{H^{1}(D)}=\int_{D}\nabla\overline{u}\cdot(I-A)\nabla
v$ $\displaystyle+k^{2}(n-1)v\overline{u}\,\text{d}x\ $
$\displaystyle+\int_{D}\nabla\overline{u}\cdot A\nabla
v-k^{2}nv\overline{u}\,\text{d}x.$

Using (15) and (16), we have that

$\displaystyle\big{(}\mathbb{L}_{k}u,u\big{)}_{H^{1}(D)}=\int_{D}\nabla\overline{u}\cdot
A\nabla u$ $\displaystyle-k^{2}n|u|^{2}\,\text{d}x\ $
$\displaystyle+\int_{D}\nabla\overline{v}\cdot(I-A)\nabla
v+k^{2}(n-1)|v|^{2}\,\text{d}x+\int_{\partial D}\eta|v|^{2}\,\text{d}s.$

From the fact that $A$ and $I-A$ are symmetric real-valued matrices, we can
infer that the above expression is real-valued for any $u\in H^{1}_{0}(D)$.
Therefore, $\mathbb{L}_{k}$ is self-adjoint. ∎

We now show that $\mathbb{L}_{0}$ is coercive if $A_{\text{max}}-1<0$ and
$\eta_{\text{min}}>0$ and $-\mathbb{L}_{0}$ is coercive if
$A_{\text{min}}-1>0$ and $\eta_{\text{max}}<0$. This is consistent with the
results in the previous section where a different variational argument is
used.

######  Lemma 3.2.

The operator $\mathbb{L}_{0}$ is coercive if $A_{\text{max}}-1<0$ and
$\eta_{\text{min}}>0$ and $-\mathbb{L}_{0}$ is coercive if
$A_{\text{min}}-1>0$ and $\eta_{\text{max}}<0$.

###### Proof.

We first consider $\mathbb{L}_{0}$. Recall in the proof of the previous lemma
that

$\displaystyle\big{(}\mathbb{L}_{k}u,u\big{)}_{H^{1}(D)}=\int_{D}\nabla\overline{u}\cdot
A\nabla u$ $\displaystyle-k^{2}n|u|^{2}\,\text{d}x$
$\displaystyle+\int_{D}\nabla\overline{v}\cdot(I-A)\nabla
v+k^{2}(n-1)|v|^{2}\,\text{d}x\ +\int_{\partial D}\eta|v|^{2}\,\text{d}s.$

It holds that

$\displaystyle\big{(}\mathbb{L}_{0}u,u\big{)}_{H^{1}(D)}$
$\displaystyle=\int_{D}\nabla\overline{u}\cdot A\nabla u\,\text{d}x\
+\int_{D}\nabla\overline{v}\cdot(I-A)\nabla v\,\text{d}x\ +\int_{\partial
D}\eta|v|^{2}\,\text{d}s\ $ $\displaystyle\geq A_{\text{min}}||\nabla
u||^{2}_{L^{2}(D)}+(1-A_{\text{max}})||\nabla
v||^{2}_{L^{2}(D)}+\eta_{\text{min}}||v||^{2}_{L^{2}(\partial D)}$
$\displaystyle\geq A_{\text{min}}||\nabla u||^{2}_{L^{2}(D)}.$

Thus, by the Poincaré inequality, $\mathbb{L}_{0}$ is coercive if
$A_{\text{max}}-1<0$ and $\eta_{\text{min}}>0$.

Next we show that $-\mathbb{L}_{0}$ is coercive. Letting $w=u+v$, (14) becomes

$\big{(}\mathbb{L}_{k}u,u\big{)}_{H^{1}(D)}=\int_{D}\nabla
w\cdot\nabla\overline{u}-k^{2}w\overline{u}\,\text{d}x\ -\int_{D}|\nabla
u|^{2}-k^{2}|u|^{2}\,\text{d}x.$ (17)

Taking $\phi=w$ in (13), we have that

$\int_{D}\nabla\overline{w}\cdot(I-A)\nabla
v+k^{2}(n-1)v\overline{w}\,\text{d}x\ +\int_{\partial D}\eta
v\overline{w}\,\text{d}s\ =\int_{D}\nabla\overline{w}\cdot A\nabla
u-k^{2}nu\overline{w}\,\text{d}x.$

It implies that

$\int_{D}\nabla\overline{w}\cdot\nabla v-k^{2}v\overline{w}\,\text{d}x\
+\int_{\partial D}\eta|w|^{2}\,\text{d}s\ =\int_{D}\nabla\overline{w}\cdot
A\nabla w-k^{2}n|w|^{2}\,\text{d}x,$

where we use that $w=u+v$ and $w=v$ on $\partial D$. Then adding
${\displaystyle\int_{D}\nabla\overline{w}\cdot\nabla
u-k^{2}u\overline{w}\,\text{d}x}$ to both sides of the above equation and
rearranging the terms, one obtains

$\displaystyle\int_{D}\nabla\overline{w}\cdot\nabla
u-k^{2}u\overline{w}\,\text{d}x\ =\int_{D}\nabla\overline{w}\cdot(I-A)\nabla
w+k^{2}(n-1)|w|^{2}\,\text{d}x+\int_{\partial D}\eta|w|^{2}\,\text{d}s.$ (18)

Using (18), we can rewrite (17) as

$\displaystyle\big{(}\mathbb{L}_{k}u,u\big{)}_{H^{1}(D)}=\int_{D}\nabla\overline{w}\cdot(I-A)\nabla
w$ $\displaystyle+k^{2}(n-1)|w|^{2}\,\text{d}x$ $\displaystyle+\int_{\partial
D}\eta|w|^{2}\,\text{d}s\ -\int_{D}|\nabla u|^{2}-k^{2}|u|^{2}\,\text{d}x.$

We let $k=0$ and obtain

$\displaystyle-\big{(}\mathbb{L}_{0}u,u\big{)}_{H^{1}(D)}$
$\displaystyle=\int_{D}\nabla\overline{w}\cdot(A-I)\nabla w\,\text{d}x\
-\int_{\partial D}\eta|w|^{2}\,\text{d}s\ +\int_{D}|\nabla u|^{2}\,\text{d}x$
$\displaystyle\geq||\nabla u||^{2}_{L^{2}(D)},$

where we have used the fact that $A_{\text{min}}-1>0$ and
$\eta_{\text{max}}<0$. Hence, by the Poincaré inequality, $-\mathbb{L}_{0}$ is
coercive. ∎

######  Lemma 3.3.

The operator $\mathbb{L}_{k}-\mathbb{L}_{0}$ is compact provided that either
$A_{max}-1<0$ and $\eta_{min}>0$ or $A_{min}-1>0$ and $\eta_{max}<0$.

###### Proof.

Assume that there exists a sequence $u^{j}\rightharpoonup 0$ in
$H^{1}_{0}(D)$. By (13), there exist sequences $v_{k}^{j}\rightharpoonup 0$
and $v_{0}^{j}\rightharpoonup 0$ in $H^{1}(D)$. These sequences correspond to
the solutions of (13) for each given $k$. We now define
$(\mathbb{L}_{k}-\mathbb{L}_{0})u^{j}$ in terms of $v_{k}^{j}$ and
$v_{0}^{j}$. To this end, using the variational form (13), we have that, for
all $\phi\in H^{1}(D)$,

$\displaystyle\int_{D}\nabla\overline{\phi}\cdot(I-A)\nabla
v_{k}^{j}+k^{2}(n-1)v_{k}^{j}\overline{\phi}\,\text{d}x\ $
$\displaystyle+\int_{\partial D}\eta v_{k}^{j}\overline{\phi}\,\text{d}s$
$\displaystyle=\int_{D}\nabla\overline{\phi}\cdot A\nabla
u^{j}-k^{2}nu^{j}\overline{\phi}\,\text{d}x\ $

and

$\int_{D}\nabla\overline{\phi}\cdot(I-A)\nabla v_{0}^{j}\,\text{d}x\
+\int_{\partial D}\eta v_{0}^{j}\overline{\phi}\,\text{d}s\
=\int_{D}\nabla\overline{\phi}\cdot A\nabla u^{j}\,\text{d}x.$

We let $\phi=v_{k}^{j}-v_{0}^{j}$ and subtract the equations to obtain

$\displaystyle\int_{D}\nabla\overline{(v_{k}^{j}-v_{0}^{j})}\cdot(I-A)\nabla(v_{k}^{j}-v_{0}^{j})\,\text{d}x\
$ $\displaystyle+\int_{\partial D}\eta|v_{k}^{j}-v_{0}^{j}|^{2}\,\text{d}s$
$\displaystyle=\int_{D}-k^{2}nu^{j}\overline{(v_{k}^{j}-v_{0}^{j})}-k^{2}(n-1)v_{k}^{j}\overline{(v_{k}^{j}-v_{0}^{j})}\,\text{d}x.$
(19)

The assumption that $A_{\text{max}}-1<0$ and $\eta_{\text{min}}>0$ or
$A_{\text{min}}-1>0$ and $\eta_{\text{max}}<0$ implies that the left-hand side
of (19) is equivalent to $||v_{k}^{j}-v_{0}^{j}||^{2}_{H^{1}(D)}$. This
implies that

$||v_{k}^{j}-v_{0}^{j}||_{H^{1}(D)}\leq
C\left(||v_{k}^{j}||_{L^{2}(D)}+||u^{j}||_{L^{2}(D)}\right),$

where $C$ is independent of $j$. By the compact embedding of $H^{1}(D)$ into
$L^{2}(D)$ it holds that

$||v_{k}^{j}-v_{0}^{j}||_{H^{1}(D)}\longrightarrow 0\quad\text{as }\quad
j\to\infty.$

By the definition of $\mathbb{L}_{k}$ in (14) we obtain that, for all $\psi\in
H_{0}^{1}(D)$,

$\displaystyle\big{(}(\mathbb{L}_{k}-\mathbb{L}_{0})u^{j},\psi\big{)}_{H^{1}(D)}=\int_{D}\nabla(v_{k}^{j}-v_{0}^{j})\cdot\nabla\overline{\psi}-k^{2}v_{k}^{j}\overline{\psi}\,\text{d}x.$

Letting $\psi=(\mathbb{L}_{k}-\mathbb{L}_{0})u^{j}$ and using the Cauchy-
Schwartz inequality, we have that

$||(\mathbb{L}_{k}-\mathbb{L}_{0})u^{j}||_{H^{1}(D)}\leq\left(||v_{k}^{j}-v_{0}^{j}||_{H^{1}(D)}+k^{2}||v_{k}^{j}||_{L^{2}(D)}\right)\longrightarrow
0\quad\text{as }\quad j\to\infty.$

Thus, $(\mathbb{L}_{k}-\mathbb{L}_{0})u^{j}$ strongly converges to $0$ in
$H^{1}(D)$, proving the compactness. ∎

From (13) and (14) we see that $\mathbb{L}_{k}$ is continuous with with
respect to $k\in\mathbb{R}_{\geq 0}$. Next we prove that the operator
$\pm\mathbb{L}_{k}$ is positive for certain $k$’s.

######  Lemma 3.4.

Let $\lambda_{1}(D)$ be the first Dirichlet eigenvalue of $-\Delta$ in $D$ and
$k^{2}$ be a real transmission eigenvalue. We have the following:

  1. 1.

If $A_{\text{min}}-1>0$, $n_{\text{max}}-1<0$, and $\eta_{\text{max}}<0$, then
$-\mathbb{L}_{k}$ is positive for $k^{2}<\lambda_{1}(D)$.

  2. 2.

If $A_{\text{max}}-1<0$, $n_{\text{min}}-1>0$, and $\eta_{\text{min}}>0$, then
$\mathbb{L}_{k}$ is positive for
$k^{2}<\frac{A_{\text{min}}}{n_{\text{max}}}\lambda_{1}(D)$.

###### Proof.

1\. Let $A_{\text{min}}-1>0$, $n_{\text{max}}-1<0$, and $\eta_{\text{max}}<0$.
From the proof of Lemma 3.2,

$\displaystyle-\big{(}\mathbb{L}_{k}u,u\big{)}_{H^{1}(D)}=-\int_{D}\nabla\overline{w}\cdot(I-A)\nabla
w$ $\displaystyle+k^{2}(n-1)|w|^{2}\,\text{d}x$ $\displaystyle-\int_{\partial
D}\eta|w|^{2}\,\text{d}s\ +\int_{D}|\nabla u|^{2}-k^{2}|u|^{2}\,\text{d}x\ .$

Using the Poincaré inequality
$||u||^{2}_{L^{2}(D)}\leq\frac{1}{\lambda_{1}(D)}||\nabla
u||^{2}_{L^{2}(D)},u\in H_{0}^{1}(D),$ we obtain that

$-\big{(}\mathbb{L}_{k}u,u\big{)}_{H^{1}(D)}\geq\int_{D}|\nabla
u|^{2}-k^{2}|u|^{2}\,\text{d}x\
\geq\left(1-\frac{k^{2}}{\lambda_{1}(D)}\right)||\nabla u||^{2}_{L^{2}(D)}.$

Hence $-\mathbb{L}_{k}$ is positive for values of $k$ such that
$k^{2}<\lambda_{1}(D)$.

2\. Let $A_{\text{max}}-1<0$, $n_{\text{min}}-1>0$, and $\eta_{\text{min}}>0$.
From Lemma 3.1,

$\displaystyle\big{(}\mathbb{L}_{k}u,u\big{)}_{H^{1}(D)}=\int_{D}\nabla\overline{u}\cdot
A\nabla u$ $\displaystyle-k^{2}n|u|^{2}\,\text{d}x$
$\displaystyle+\int_{D}\nabla\overline{v}\cdot(I-A)\nabla
v+k^{2}(n-1)|v|^{2}\,\text{d}x\ +\int_{\partial D}\eta|v|^{2}\,\text{d}s.$

Again, using the Poincaré Inequality, we have that

$\displaystyle\big{(}\mathbb{L}_{k}u,u\big{)}_{H^{1}(D)}$
$\displaystyle\geq\int_{D}\nabla\overline{u}\cdot A\nabla
u-k^{2}n|u|^{2}\,\text{d}x\ \geq\int_{D}A_{\text{min}}|\nabla
u|^{2}-k^{2}n_{\text{max}}|u|^{2}\,\text{d}x$
$\displaystyle\geq\left(A_{\text{min}}-n_{\text{max}}\frac{k^{2}}{\lambda_{1}(D)}\right)||\nabla
u||^{2}_{L^{2}(D)}.$

Thus $\mathbb{L}_{k}$ is positive for values of $k$ such that
$k^{2}<\frac{A_{\text{min}}}{n_{\text{max}}}\lambda_{1}(D)$. ∎

From the above lemma, all real TEVs must satisfy either Faber–Krahn type
inequality

$k^{2}\geq\lambda_{1}(D)\quad\text{provided that}\quad
A_{\text{min}}-1>0,\,\,\,n_{\text{max}}-1<0,\,\,\,\eta_{\text{max}}<0$

or

$k^{2}\geq\frac{A_{\text{min}}}{n_{\text{max}}}\lambda_{1}(D)\quad\text{
provided that}\quad
A_{\text{max}}-1<0,\,\,\,n_{\text{min}}-1>0,\,\,\,\eta_{\text{min}}>0.$

It is consistent with the case when $\eta=0$.

######  Lemma 3.5.

There exists a $\tau>0$ such that $-\mathbb{L}_{\tau}$ with
$A_{\text{min}}-1>0$, $n_{\text{max}}-1<0$, and $\eta_{\text{max}}<0$ or
$\mathbb{L}_{\tau}$ with $A_{\text{max}}-1<0$, $n_{\text{min}}-1>0$, and
$\eta_{\text{min}}>0$ is non-positive on some $N$–dimensional subspace of
$H_{0}^{1}(D)$ for any $N\in\mathbb{N}$.

###### Proof.

We construct a finite-dimensional subspace of $H_{0}^{1}(D)$ where
$-\mathbb{L}_{\tau}$ is non-positive (the other case is similar) by
considering small disjoint balls $B_{\epsilon}\subset D$. Assume that
$A_{\text{min}}-1>0$, $n_{\text{max}}-1<0$, and $\eta_{\text{max}}<0$.
Consider the ball $B_{\epsilon}$ of radius $\epsilon>0$ such that
$B_{\epsilon}\subset D$. Using Theorems 2.5 and 2.6 from [11] there exist TEVs
for

$\displaystyle\Delta v_{\epsilon}+\tau^{2}v_{\epsilon}=0\quad$
$\displaystyle\text{ and }\quad\nabla\cdot A_{\text{min}}\nabla
w_{\epsilon}+\tau^{2}n_{\text{max}}w_{\epsilon}=0\quad\text{in
$B_{\epsilon}$},$ (20) $\displaystyle w_{\epsilon}=v_{\epsilon}\quad$
$\displaystyle\text{ and }\quad\nu\cdot A_{\text{min}}\nabla
w_{\epsilon}=\partial_{\nu}v_{\epsilon}\quad\text{on $\partial
B_{\epsilon}$.}$ (21)

Let $u_{\epsilon}=w_{\epsilon}-v_{\epsilon}$ associated with the auxiliary
eigenvalue $\tau$. Similar calculation as in Lemma 3.2 leads to

$0=\int_{B_{\epsilon}}|\nabla
u_{\epsilon}|^{2}-\tau^{2}|u_{\epsilon}|^{2}\,\text{d}x+\int_{B_{\epsilon}}\nabla\overline{w_{\epsilon}}\cdot(A_{\text{min}}-1)\nabla
w_{\epsilon}-\tau^{2}(n_{\text{max}}-1)|w_{\epsilon}|^{2}\,\text{d}x.$ (22)

Since $u_{\epsilon}\in H_{0}^{1}(B_{\epsilon})$, we can take an extension by
zero to the whole domain $D$ and denote this function as $u\in H_{0}^{1}(D)$.
Since $A_{\text{min}}-1>0$, $n_{\text{max}}-1<0$, and $\eta_{\text{max}}<0$,
we can construct a nontrivial $v\in H^{1}(D)$ that solves (13) with
coefficients $A$,$n$, and $\eta$ in $D$ where $w=u+v$. Adding

$\int_{D}\nabla
u\cdot\nabla\overline{\phi}-\tau^{2}u\overline{\phi}\,\text{d}x$

to both sides of (13) and rearranging the terms, we obtain that, for any
$\phi\in H^{1}_{0}(D)$,

$\displaystyle\int_{D}\nabla\overline{\phi}\cdot(A-I)\nabla
w-\tau^{2}(n-1)w\overline{\phi}\,\text{d}x\ -\int_{\partial D}\eta
w\overline{\phi}\,\text{d}s$ $\displaystyle=$ $\displaystyle-\int_{D}\nabla
u\cdot\nabla\overline{\phi}-\tau^{2}u\overline{\phi}\,\text{d}x$
$\displaystyle=$ $\displaystyle-\int_{B_{\epsilon}}\nabla
u_{\epsilon}\cdot\nabla\overline{\phi}-\tau^{2}u_{\epsilon}\overline{\phi}\,\text{d}x$
$\displaystyle=$
$\displaystyle\int_{B_{\epsilon}}\nabla\overline{\phi}\cdot(A_{\text{min}}-1)\nabla
w_{\epsilon}-\tau^{2}(n_{\text{max}}-1)w_{\epsilon}\overline{\phi}\,\text{d}x.$
(23)

The last line is obtained by replacing $w_{\epsilon}$ by $u_{\epsilon}$ in
(20)–(21). Since $A_{\text{min}}-1>0$, $n_{\text{max}}-1<0$, the right-hand
side of (23) is an inner product on $H^{1}(B_{\epsilon})$. Let $\phi=w$ in
(23) to obtain

$\displaystyle\int_{D}\nabla\overline{w}\cdot(A-I)\nabla w$
$\displaystyle-\tau^{2}(n-1)|w|^{2}\,\text{d}x-\int_{\partial
D}\eta|w|^{2}\,\text{d}s$
$\displaystyle=\int_{B_{\epsilon}}\nabla\overline{w}\cdot(A_{\text{min}}-1)\nabla
w_{\epsilon}-\tau^{2}(n_{\text{max}}-1)w_{\epsilon}\overline{w}\,\text{d}x.$

Using the Cauchy-Schwartz inequality and $\eta_{\text{max}}<0$, we obtain that

$\displaystyle\int_{D}\nabla\overline{w}\cdot(A-I)\nabla w$
$\displaystyle-\tau^{2}(n-1)|w|^{2}\,\text{d}x-\int_{\partial
D}\eta|w|^{2}\,\text{d}s$
$\displaystyle\leq\int_{B_{\epsilon}}\nabla\overline{w_{\epsilon}}\cdot(A_{\text{min}}-1)\nabla
w_{\epsilon}-\tau^{2}(n_{\text{max}}-1)|w_{\epsilon}|^{2}\,\text{d}x.$ (24)

From the proof of Lemma 3.2,

$\displaystyle-\big{(}\mathbb{L}_{\tau}u,u\big{)}_{H^{1}(D)}=\int_{D}\nabla\overline{w}\cdot(A-I)\nabla
w$ $\displaystyle-\tau^{2}(n-1)|w|^{2}\,\text{d}x-\int_{\partial
D}\eta|w|^{2}\,\text{d}s$ $\displaystyle+\int_{D}|\nabla
u|^{2}-\tau^{2}|u|^{2}\,\text{d}x.$

By the construction of $u$ and using (23)-(24) we have that

$\displaystyle-\big{(}\mathbb{L}_{\tau}u,u\big{)}_{H^{1}(D)}$
$\displaystyle=\int_{D}\nabla\overline{w}\cdot(A-I)\nabla
w-\tau^{2}(n-1)|w|^{2}\,\text{d}x-\int_{\partial D}\eta|w|^{2}\,\text{d}s$
$\displaystyle\hskip 72.26999pt+\int_{B_{\epsilon}}|\nabla
u_{\epsilon}|^{2}-\tau^{2}|u_{\epsilon}|^{2}\,\text{d}x\ $
$\displaystyle\leq\int_{B_{\epsilon}}\nabla\overline{w_{\epsilon}}\cdot(A_{\text{min}}-1)\nabla
w_{\epsilon}-\tau^{2}(n_{\text{max}}-1)|w_{\epsilon}|^{2}\,\text{d}x\ $
$\displaystyle\hskip 72.26999pt+\int_{B_{\epsilon}}|\nabla
u_{\epsilon}|^{2}-\tau^{2}|u_{\epsilon}|^{2}\,\text{d}x=0.$

Thus the operator $-\mathbb{L}_{\tau}$ is non-positive on this one–dimensional
subspace.

To construct an $N$–dimensional subspace for any $N\in\mathbb{N}$ we let
$B_{j}$ be the ball centered at $x_{j}\in D$ with radius $\epsilon>0$. Let
$\epsilon$ be sufficiently small such that $B_{i}\cap B_{j}=\emptyset$ for all
$i\neq j$. Then, consider (20)–(21) in each $B_{j}$ and let $u_{j}\in
H^{1}_{0}(D)$ be the difference of the eigenfunctions extended to $D$ by zero
with eigenvalue $\tau$. Note that
$\text{span}\\{{u}_{j}\mathrel{\mathop{\mathchar 58\relax}}\,j=1,\cdots,N\\}$
is an $N$–dimensional subspace of $H^{1}_{0}(D)$ due to the disjoint supports.
Hence $-\mathbb{L}_{\tau}$ is non-positive on this $N$–dimensional subspace of
$H^{1}_{0}(D)$. The case for $\mathbb{L}_{\tau}$ can be handled similarly. ∎

The following theorem states that there exists infinitely many transmission
eigenvalues.

######  Theorem 3.1.

Assume that either $A_{\text{min}}-1>0$, $n_{\text{max}}-1<0$, and
$\eta_{\text{max}}<0$, or $A_{\text{max}}-1<0$, $n_{\text{min}}-1>0$, and
$\eta_{\text{min}}>0$. Then there exists infinitely many real transmission
eigenvalues $k>0$.

###### Proof.

The theorem is a direct consequence of the lemmas proved in this section and
Theorem 2.6 of [11] (see also [9]). ∎

##  4 Dependence on the parameters

In this section, we study how the TEVs depend on the material parameters. It
has been shown in previous works [7, 10, 21] that the TEVs can be used to
estimate or determine the material parameters associated with the scattering
problem. This falls under the category of inverse spectral problems where one
wishes to infer information about the physical model from the knowledge of an
associated eigenvalue problem. We begin by proving the monotonicity of the
first (i.e. smallest positive) TEV with respect to the coefficients. Then we
consider limiting problems as $\eta\to 0$ and $\eta\to\pm\infty.$ Here, the
limits are understood in the sense that there are sequences of
$\eta_{j},j\in\mathbb{N}$ such that

$\|\eta_{j}\|_{L^{\infty}(\partial D)}\longrightarrow 0\quad\text{or
}\quad\inf\limits_{\partial D}|\eta_{j}|\longrightarrow\infty\quad\text{as
}\quad j\to\infty.$

See e.g. [14, 20, 21, 31] for other studies of limiting values for some TEV
problems.

###  4.1 Monotonicity Results

We consider the monotonicity of the first TEV, denoted by $k_{1}$, with
respect to the parameters $A,n$ and $\eta$. Note that $k_{1}>0$ is the
smallest value in $\mathbb{R}_{\geq 0}$ where the function $f(k)=\min_{u\neq
0}(\mathbb{L}_{k}u,u)_{H^{1}(D)}$ is non-positive (or non-negative) by Lemma
3.4 and the continuity of the mapping $k\mapsto\mathbb{L}_{k}$ from
$\mathbb{R}_{\geq 0}$ to the set of bounded linear operators from
$H^{1}_{0}(D)$ into itself. In our analysis, we say that $A_{1}\leq A_{2}$,
for matrix valued functions $A_{1}$ and $A_{2}$, if $A_{2}-A_{1}$ is semi-
positive definite.

######  Theorem 4.1.

Assume that $A_{j}-I>0$, $n_{j}-1<0$, and $\eta_{j}<0$ for $j=1,2$.

  1. (1)

If $n_{1}\leq n_{2}$ and $\eta_{1}\leq\eta_{2}$ then
$k_{1}(n_{1},\eta_{1})\leq k_{1}(n_{2},\eta_{2})$.

  2. (2)

If $A_{1}\leq A_{2}$ then $k_{1}(A_{2})\leq k_{1}(A_{1})$.

###### Proof.

We shall prove (1). Claim (2) can be proved in a similar way. If $n_{1}\leq
n_{2}$, then $-(n_{1}-1)\geq-(n_{2}-1)>0$. Let $\tau=k_{1}(A,n_{2},\eta_{2})$
and assume that $A>I$ with $\eta_{2}<0$, and $n_{2}-1<0$. Letting $v_{\tau}$
and $w_{\tau}$ denote the transmission eigenfunctions corresponding to the
transmission eigenvalue $\tau$. From the proof of Lemma 3.2 we have that

$\displaystyle 0$ $\displaystyle=\int_{D}|\nabla
u_{\tau}|^{2}-\tau^{2}|u_{\tau}|^{2}\,\text{d}x+\int_{D}\nabla\overline{w_{\tau}}\cdot(A-I)\nabla
w_{\tau}-\tau^{2}(n_{2}-1)|w_{\tau}|^{2}\,\text{d}x\ -\int_{\partial
D}\eta_{2}|w_{\tau}|^{2}\,\text{d}s$

where $u_{\tau}=w_{\tau}-v_{\tau}$.

By the well-posedness of (13), we have the existence of $v\in H^{1}(D)$ with
$u=u_{\tau}$, $n=n_{1}$, $\eta=\eta_{1}$ and $k=\tau$ where $w=v+u_{\tau}$.
From (13) along with similar calculations as in Lemma 3.5, we have

$\displaystyle\int_{D}\nabla\overline{\phi}\cdot(A-I)\nabla w$
$\displaystyle-\tau^{2}(n_{1}-1)w\overline{\phi}\,\text{d}x-\int_{\partial
D}\eta_{1}w\overline{\phi}\,\text{d}s$ $\displaystyle=-\int_{D}\nabla
u_{\tau}\cdot\nabla\overline{\phi}-\tau^{2}u_{\tau}\overline{\phi}\,\text{d}x$
$\displaystyle=\int_{D}\nabla\overline{\phi}\cdot(A-I)\nabla
w_{\tau}-\tau^{2}(n_{2}-1)w_{\tau}\overline{\phi}\,\text{d}x-\int_{\partial
D}\eta_{2}w_{\tau}\overline{\phi}\,\text{d}s.$

The right-hand side of the above equality is an inner-product on $H^{1}(D)$
due to the assumptions on the coefficients. Letting $\phi=w$, it holds that

$\displaystyle\int_{D}\nabla\overline{w}\cdot(A-I)\nabla w$
$\displaystyle-\tau^{2}(n_{1}-1)|w|^{2}\,\text{d}x-\int_{\partial
D}\eta_{1}|w|^{2}\,\text{d}s$
$\displaystyle=\int_{D}\nabla\overline{w}\cdot(A-I)\nabla
w_{\tau}-\tau^{2}(n_{2}-1)w_{\tau}\overline{w}\,\text{d}x-\int_{\partial
D}\eta_{2}w_{\tau}\overline{w}\,\text{d}s.$

We employ the Cauchy-Schwartz inequality to obtain

$\displaystyle\int_{D}\nabla\overline{w}\cdot(A-I)\nabla w$
$\displaystyle-\tau^{2}(n_{1}-1)|w|^{2}\,\text{d}x-\int_{\partial
D}\eta_{1}|w|^{2}\,\text{d}s$
$\displaystyle\leq\int_{D}\nabla\overline{w_{\tau}}\cdot(A-I)\nabla
w_{\tau}-\tau^{2}(n_{2}-1)|w_{\tau}|^{2}\,\text{d}x-\int_{\partial
D}\eta_{2}|w_{\tau}|^{2}\,\text{d}s.$

We have used that $-(n_{1}-1)\geq-(n_{2}-1)>0$ as well as
$-\eta_{1}\geq-\eta_{2}>0$.

Denote by $\mathbb{L}_{\tau}$ the operator with coefficients $A,n_{1}$ and
$\eta_{1}$. It holds that

$\displaystyle-\big{(}\mathbb{L}_{\tau}u_{\tau},u_{\tau}\big{)}_{H^{1}(D)}$
$\displaystyle=\int_{D}\nabla\overline{w}\cdot(A-I)\nabla
w-\tau^{2}(n_{1}-1)|w|^{2}\,dx-\int_{\partial D}\eta_{1}|w|^{2}\,ds$
$\displaystyle\hskip 72.26999pt+\int_{D}|\nabla
u_{\tau}|^{2}-\tau^{2}|u_{\tau}|^{2}\,dx\ $
$\displaystyle\leq\int_{D}\nabla\overline{w_{\tau}}\cdot(A-I)\nabla
w_{\tau}-\tau^{2}(n_{2}-1)|w_{\tau}|^{2}\,dx-\int_{\partial
D}\eta_{2}|w_{\tau}|^{2}\,ds\ $ $\displaystyle\hskip
72.26999pt+\int_{D}|\nabla u_{\tau}|^{2}-\tau^{2}|u_{\tau}|^{2}\,dx\ =0.$

This implies that $-\mathbb{L}_{\tau}$ associated with $A,n_{1}$ and,
$\eta_{1}$ is nonpositive on this one-dimensional subspace spanned by
$u_{\tau}$. Then the continuous dependence of $k\mapsto\mathbb{L}_{k}$ implies
the existence of a TEV corresponding to $A,n_{1}$ and $\eta_{1}$ in $(0,\tau]$
with $\tau=k_{1}(A,n_{2},\eta_{2})$. The proof is complete. ∎

Similar monotonicity results hold for $A_{\text{max}}-1<0$,
$n_{\text{min}}-1>0$, and $\eta_{\text{min}}>0$.

######  Theorem 4.2.

Assume that $A_{j}-I<0$, $n_{j}-1>0$, and $\eta_{j}>0$ for $j=1,2$.

  1. 1.

If $n_{1}\leq n_{2}$ and $\eta_{1}\leq\eta_{2}$ then we have that
$k_{1}(n_{2},\eta_{2})\leq k_{1}(n_{1},\eta_{1})$.

  2. 2.

If $A_{1}\leq A_{2}$ then we have that $k_{1}(A_{1})\leq k_{1}(A_{2})$.

where $k_{1}$ corresponds to the first transmission eigenvalue.

###### Proof.

The proof is similar to that of Theorem 4.1 and thus omitted. ∎

###  4.2 Convergence of the TEVs as $\eta\longrightarrow 0$

Now we give a more in depth study of the TEVs on the dependence on $\eta$. In
particular, we are interested in the limiting behavior as $\eta\to 0$. Let
$k_{\eta}$ correspond to a TEV for $\eta\neq 0$ and assume that it forms a
bounded set. Then $k_{\eta}$ has a limit point, denoted by $k_{0}$, as
$\eta\to 0$. By Lemma 3.4 and Theorems 4.1 and 4.2, $k_{\eta}$ converges as
$\eta\to 0$ monotonically.

Define the Hilbert space

$X(D)\mathrel{\mathop{\mathchar 58\relax}}=\left\\{(\phi_{1},\phi_{2})\in
H^{1}(D)\times H^{1}(D)\,\,\mathrel{\mathop{\mathchar
58\relax}}\,\,\phi_{1}=\phi_{2}\,\,\text{on}\,\,\partial D\right\\}$

with the standard norm. Assume that the non-trivial eigenfunctions
$(w_{\eta},v_{\eta})\in X(D)$ satisfying (3)–(4) are normalized in the
$L^{2}(D)\times L^{2}(D)$ norm such that

$||w_{\eta}||^{2}_{L^{2}(D)}+||v_{\eta}||^{2}_{L^{2}(D)}=1$

for any $\eta\neq 0$. We show that the eigenfunctions are bounded in $X(D)$ as
$\eta\to 0$.

######  Lemma 4.1.

Assume that either $A_{\text{min}}-1>0$ or $A_{\text{max}}-1<0$ and that
$k_{\eta}$ is bounded then $(w_{\eta},v_{\eta})\in X(D)$ is bounded as
$\eta\to 0$.

###### Proof.

We assume that $(w_{\eta},v_{\eta})$ satisfies (3)–(4) and let
$(\phi_{1},\phi_{2})\in X(D)$. Then, by Green’s first theorem, we have that

$\displaystyle\int_{D}A\nabla
w_{\eta}\cdot\nabla\overline{\phi_{1}}-k_{\eta}^{2}nw_{\eta}\overline{\phi_{1}}\,\text{d}x\
$ $\displaystyle=\int_{\partial D}\overline{\phi_{1}}(\nu\cdot A\nabla
w_{\eta})\,\text{d}s,\quad\text{and}$ (25) $\displaystyle\int_{D}\nabla
v_{\eta}\cdot\nabla\overline{\phi_{2}}-k_{\eta}^{2}v_{\eta}\overline{\phi_{2}}\,\text{d}x$
$\displaystyle=\int_{\partial
D}\overline{\phi_{2}}\partial_{\nu}v_{\eta}\,\text{d}s$ (26)

Subtracting the equations (25)–(26) and using the boundary conditions (4) we
get that

$\int_{D}A\nabla w_{\eta}\cdot\nabla\overline{\phi_{1}}-\nabla
v_{\eta}\cdot\nabla\overline{\phi_{2}}-k_{\eta}^{2}nw_{\eta}\overline{\phi_{1}}+k_{\eta}^{2}v_{\eta}\overline{\phi_{2}}\,\text{d}x=\int_{\partial
D}\eta v_{\eta}\overline{\phi_{2}}\,\text{d}s.$

(1) Consider the case when $A_{\text{min}}-1>0$. Write the above variational
formulation as

$a\left((w_{\eta},v_{\eta});(\phi_{1},\phi_{2})\right)=b\left((w_{\eta},v_{\eta});(\phi_{1},\phi_{2})\right)\quad\text{for
any}\quad(\phi_{1},\phi_{2})\in X(D)$

where

$a\left((w_{\eta},v_{\eta});(\phi_{1},\phi_{2})\right)=\int_{D}A\nabla
w_{\eta}\cdot\nabla\overline{\phi_{1}}+A_{\text{min}}w_{\eta}\overline{\phi_{1}}-\nabla
v_{\eta}\cdot\nabla\overline{\phi_{2}}-v_{\eta}\overline{\phi_{2}}\,\text{d}x\
$

and

$b\left((w_{\eta},v_{\eta});(\phi_{1},\phi_{2})\right)=\int_{D}(nk_{\eta}^{2}+A_{\text{min}})w_{\eta}\overline{\phi_{1}}-(k_{\eta}^{2}+1)v_{\eta}\overline{\phi_{2}}\,\text{d}x\
+\int_{\partial D}\eta v_{\eta}\overline{\phi_{2}}\,\text{d}s.$

We first show the boundedness. To this end, if $\phi_{1}=w_{\eta}$ and
$\phi_{2}=-v_{\eta}+2w_{\eta}$, then $(\phi_{1},\phi_{2})\in X(D)$. Hence

$\displaystyle\big{|}a\left((w_{\eta},v_{\eta});(w_{\eta},-v_{\eta}+2w_{\eta})\right)\big{|}$
$\displaystyle\geq\int_{D}A\nabla
w_{\eta}\cdot\nabla\overline{w_{\eta}}+A_{\text{min}}|w_{\eta}|^{2}+|\nabla
v_{\eta}|^{2}+|v_{\eta}|^{2}\,\text{d}x\ $ $\displaystyle\hskip
72.26999pt-2\Bigg{|}\int_{D}\nabla
v_{\eta}\cdot\nabla\overline{w_{\eta}}+v_{\eta}\overline{w_{\eta}}\,\text{d}x\
\Bigg{|}$
$\displaystyle\geq\Big{(}1-\frac{1}{\delta}\Big{)}||v_{\eta}||^{2}_{H^{1}(D)}+\Big{(}A_{\text{min}}-\delta\Big{)}||w_{\eta}||^{2}_{H^{1}(D)},$

where the last line uses Young’s inequality. So for any
$\delta\in(1,A_{\text{min}})$ we have that

$\big{|}a\left((w_{\eta},v_{\eta});(w_{\eta},-v_{\eta}+2w_{\eta})\right)\big{|}\geq\alpha\left(||v_{\eta}||^{2}_{H^{1}(D)}+||w_{\eta}||^{2}_{H^{1}(D)}\right),$

where $\alpha>0$ is independent of $\eta$. For $b(\cdot,\cdot)$, it holds that

$\displaystyle\big{|}b\left((w_{\eta},v_{\eta});(w_{\eta},-v_{\eta}+2w_{\eta})\right)\big{|}$
$\displaystyle\leq
C_{1}\Big{[}||v_{\eta}||^{2}_{L^{2}(D)}+||w_{\eta}||^{2}_{L^{2}(D)}\Big{]}+\big{|}(\eta
v_{\eta},v_{\eta})_{L^{2}(\partial D)}\big{|}$ $\displaystyle\leq
C_{1}+C_{2}\|\eta\|_{L^{\infty}(\partial
D)}\left(||v_{\eta}||^{2}_{H^{1}(D)}+||w_{\eta}||^{2}_{H^{1}(D)}\right),$

where the constants $C_{j}>0$ for $j=1,2$ are independent of $\eta$. Note that
we have used the normalization of the eigenfunctions and the Trace Theorem. He
it holds that

$\alpha\left(||v_{\eta}||^{2}_{H^{1}(D)}+||w_{\eta}||^{2}_{H^{1}(D)}\right)\leq
C_{1}+C_{2}\|\eta\|_{L^{\infty}(\partial
D)}\left(||v_{\eta}||^{2}_{H^{1}(D)}+||w_{\eta}||^{2}_{H^{1}(D)}\right),$

which implies that

$||w_{\eta}||^{2}_{H^{1}(D)}+||v_{\eta}||^{2}_{H^{1}(D)}\leq\frac{C_{1}}{\alpha-
C_{2}\|\eta\|_{L^{\infty}(\partial D)}}\quad\text{for}\quad\eta\ll 1.$

We conclude that $||w_{\eta}||^{2}_{H^{1}(D)}+||v_{\eta}||^{2}_{H^{1}(D)}$ is
bounded as $\eta\to 0$.

(2) For $A_{\text{max}}-1<0$, we rewrite (25)–(26) as

$\displaystyle\int_{D}A\nabla
w_{\eta}\cdot\nabla\overline{\phi_{1}}+A_{\text{max}}w_{\eta}\overline{\phi_{1}}$
$\displaystyle-\nabla
v_{\eta}\cdot\nabla\overline{\phi_{2}}-v_{\eta}\overline{\phi_{2}}\,\text{d}x$
$\displaystyle=\int_{D}(nk_{\eta}^{2}+A_{\text{max}})w_{\eta}\overline{\phi_{1}}-(k_{\eta}^{2}+1)v_{\eta}\overline{\phi_{2}}\,\text{d}x\
+\int_{\partial D}\eta v_{\eta}\overline{\phi_{2}}\,\text{d}s.$

Then the rest of the proof is similar to the previous case by taking
$\phi_{1}=w_{\eta}-2v_{\eta}$ and $\phi_{2}=-v_{\eta}$ such that
$(\phi_{1},\phi_{2})\in X(D)$. ∎

We now have that
$\left\\{k_{\eta};(w_{\eta},v_{\eta})\right\\}\in\mathbb{R}_{\geq 0}\times
X(D)$ is bounded and thus there exist
$\left\\{k_{0};(w_{0},v_{0})\right\\}\in\mathbb{R}_{\geq 0}\times X(D)$ such
that, up to a subsequence,

$k_{\eta}\longrightarrow k_{0}\quad\text{ as }\quad\eta\to 0$

and

$w_{\eta}\rightharpoonup w_{0}\hskip 5.69046pt\text{ and }\hskip
5.69046ptv_{\eta}\rightharpoonup v_{0}\hskip 5.69046pt\text{ in $H^{1}(D)$
}\quad\text{as $\eta\to 0$. }$

This implies that $(v_{0},w_{0})$ is nontrivial by appealing to the weak
convergence and the compact embedding of $H^{1}(D)$ into $L^{2}(D)$. Moreover,
$||w_{0}||^{2}_{L^{2}(D)}+||v_{0}||^{2}_{L^{2}(D)}=1.$

We now show that $\left\\{k_{0};(w_{0},v_{0})\right\\}$ satisfies (3)–(4).
Note that, by the weak convergence of the eigenfunctions, for any $\phi\in
H^{1}(D)$,

$\int_{\partial D}\overline{\phi}\Big{[}\nu\cdot A\nabla
w_{\eta}-\partial_{\nu}v_{\eta}\Big{]}\,\text{d}s\longrightarrow\int_{\partial
D}\overline{\phi}\Big{[}\nu\cdot A\nabla
w_{0}-\partial_{\nu}v_{0}\Big{]}\,\text{d}s\quad\text{ as }\quad\eta\to 0.$

Hence

$\displaystyle\Bigg{|}\int_{\partial D}\overline{\phi}\Big{[}\nu\cdot A\nabla
w_{\eta}-\partial_{\nu}v_{\eta}\Big{]}\,\text{d}s\ \Bigg{|}$
$\displaystyle=\Bigg{|}\int_{\partial D}\eta
v_{\eta}\overline{\phi}\,\text{d}s\Bigg{|}$
$\displaystyle\leq\|\eta\|_{L^{\infty}(\partial
D)}||v_{\eta}||_{H^{1/2}(\partial D)}||\phi||_{H^{1/2}(\partial D)}$
$\displaystyle\leq C\|\eta\|_{L^{\infty}(\partial
D)}||\phi||_{H^{1/2}(\partial D)},$

where we have used the boundedness of the eigenfunctions. This implies that

$\|\nu\cdot A\nabla w_{\eta}-\partial_{\nu}v_{\eta}\|_{H^{-1/2}(\partial
D)}\longrightarrow 0\quad\text{ as }\quad\eta\to 0$

and we can conclude that $\nu\cdot A\nabla w_{0}=\partial_{\nu}v_{0}$ on
$\partial D$. Due to the fact that the eigenfunctions $(w_{\eta},v_{\eta})\in
X(D)$ satisfy (3)–(4) and the weak convergence as $\eta\to 0$, it holds that

$\nabla\cdot A\nabla w_{0}+k_{0}^{2}nw_{0}=0\quad\text{and}\quad\Delta
v_{0}+k_{0}^{2}v_{0}=0\quad\text{ in $D$}.$

Consequently, $(w_{0},v_{0})\in X(D)$ is a non-trivial solution to (3)–(4)
with eigenvalue $k_{0}$. We thus have shown the main result of this section.

######  Theorem 4.3.

Assume that either $A_{\text{min}}-1>0$ or $A_{\text{max}}-1<0$ and that
$k_{\eta}$ is bounded then for $k_{\eta}\to k_{0}$ as $\eta\to 0$ it holds
that $(w_{\eta},v_{\eta})\rightharpoonup(w_{0},v_{0})$ in $X(D)$ as $\eta\to
0$ such that

$\displaystyle\Delta v_{0}+k_{0}^{2}v_{0}=0\quad$ $\displaystyle\text{ and
}\quad\nabla\cdot A\nabla w_{0}+k_{0}^{2}nw_{0}=0\quad\text{ in }D,$
$\displaystyle w_{0}=v_{0}\quad$ $\displaystyle\text{ and }\quad\nu\cdot
A\nabla w_{0}=\partial_{\nu}v_{0}\quad\text{on $\partial D$}$

with $(w_{0},v_{0})$ being nontrivial.

###  4.3 Convergence of the TEVs as $\eta\longrightarrow\pm\infty$

In this section, we consider the case when $\eta\to\pm\infty$. One will see
that there is an interesting bifurcation for this limit. We shall make
additional assumptions that the boundary $\partial D$ is of class
$\mathscr{C}^{2}$ or is polygonal with no reentrant corners and
$A\in\mathscr{C}^{1}(\overline{D},\mathbb{R}^{d\times d})$ so that we can use
the standard elliptic regularity [33]. Let $k_{\eta}$ denote the TEV with
respect to $\eta$. We assume that $k_{\eta}$ is bounded as $\eta\to\pm\infty$
and therefore has a limit point. Again, by Lemma 3.4 as well as Theorems 4.1
and 4.2, in some cases, $k_{\eta}$ has a limit point as $\eta\to\pm\infty$
monotonically.

Assume that the eigenfunctions $(w_{\eta},v_{\eta})$ satisfying (3)–(4) are
normalized in $X(D)$ such that

$||v_{\eta}||^{2}_{H^{1}(D)}+||w_{\eta}||^{2}_{H^{1}(D)}=1$

for any $|\eta|$ finite. We have that
$\left\\{k_{\eta};(w_{\eta},v_{\eta})\right\\}\in\mathbb{R}_{\geq 0}\times
X(D)$ is bounded and thus there exists
$\left\\{k_{\infty};(w_{\infty},v_{\infty})\right\\}\in\mathbb{R}_{\geq
0}\times X(D)$ such that, up to a subsequence,

$k_{\eta}\longrightarrow k_{\infty}\quad\text{ as }\quad\eta\to\pm\infty$

and

$w_{\eta}\rightharpoonup w_{\infty}\hskip 5.69046pt\text{ and }\hskip
5.69046ptv_{\eta}\rightharpoonup v_{\infty}\hskip 5.69046pt\text{ in
$H^{1}(D)$ }\quad\text{as $\eta\to\pm\infty$. }$

As in section 4.2, the weak limit $(w_{\infty},v_{\infty})\in X(D)$ satisfies

$\Delta v_{\infty}+k_{\infty}^{2}v_{\infty}=0\quad\text{ and }\quad\nabla\cdot
A\nabla w_{\infty}+k_{\infty}^{2}nw_{\infty}=0\quad\text{in $D$}$

by the weak convergence and Green’s first theorem.

To determine the limiting boundary value problem, we first show that the weak
limits have zero trace on $\partial D$. By the conductivity condition in (4),
we have that

$\int_{\partial D}\overline{\phi}\Big{[}\nu\cdot A\nabla
w_{\eta}-\partial_{\nu}v_{\eta}\Big{]}\,\text{d}s\ =\int_{\partial D}\eta
v_{\eta}\overline{\phi}\,\text{d}s\quad\text{ for all }\quad\phi\in H^{1}(D).$

Letting $\phi={v_{\eta}}$, the following estimate holds

$\inf\limits_{\partial D}|\eta|\int_{\partial
D}|v_{\eta}|^{2}\,\text{d}s\leq\left|\int_{\partial
D}\overline{v_{\eta}}\left[\nu\cdot A\nabla
w_{\eta}-\partial_{\nu}v_{\eta}\right]\,\text{d}s\ \right|,$

which implies that

$\|v_{\eta}\|^{2}_{L^{2}(\partial D)}\leq\frac{1}{\inf\limits_{\partial
D}|\eta|}||\nu\cdot A\nabla
w_{\eta}-\partial_{\nu}v_{\eta}||_{H^{-1/2}(\partial
D)}||v_{\eta}||_{H^{1/2}(\partial D)}.$

By the trace theorems and the normalization of the transmission
eigenfunctions, we have that there exist $C_{j}>0$, $j=1,2$, independent of
$\eta$ such that

$||\nu\cdot A\nabla w_{\eta}-\partial_{\nu}v_{\eta}||_{H^{-1/2}(\partial
D)}\leq C_{1}\quad\text{ and }\quad||v_{\eta}||_{H^{1/2}(\partial D)}\leq
C_{2}.$

Therefore, we obtain that

$||v_{\eta}||^{2}_{L^{2}(\partial D)}\leq\frac{C}{\inf\limits_{\partial
D}|\eta|}\longrightarrow 0\quad\text{ as }\quad\eta\longrightarrow\pm\infty.$

By the compact embedding of $H^{1/2}(\partial D)$ into $L^{2}(\partial D)$ we
have that $v_{\infty}=0$ and $w_{\infty}=0$ on $\partial D$ since
$(w_{\infty},v_{\infty})\in X(D)$.

Unlike the previous case, due to the normalization in $X(D)$, we can not
conclude that the eigenfunctions are nontrivial yet. To prove this, we need to
make an extra assumption on the boundary values for the eigenfunctions. In our
investigation, we are unable to avoid this technicality just as in [14]. With
this in mind, we now prove a lemma to show that the corresponding limits
$(w_{\infty},v_{\infty})$ are nontrivial.

######  Lemma 4.2.

If $\partial D$ is of class $\mathscr{C}^{2}$ or is polygonal with no
reentrant corners, $A\in\mathscr{C}^{1}(\overline{D},\mathbb{R}^{d\times d})$,
$v_{\eta}$ on $\partial D$ is bounded in $H^{3/2}(\partial D)$ and that
$k_{\eta}$ is bounded, then $v_{\eta}$ and $w_{\eta}$ are bounded in
$H^{2}(D)$.

###### Proof.

Recall that the eigenfunctions $v_{\eta}$ and $w_{\eta}$ satisfy

$\Delta v_{\eta}+k_{\eta}^{2}v_{\eta}=0\quad\text{ or }\quad\nabla\cdot
A\nabla w_{\eta}+k_{\eta}^{2}nw_{\eta}=0\quad\text{in $D$}.$

Since $v_{\eta},w_{\eta}\in H^{1}(D)\subset L^{2}(D)$ and $n\in
L^{\infty}(D)$, we have

$\Delta v_{\eta}=-k_{\eta}^{2}v_{\eta}\in L^{2}(D)\quad\text{ and
}\quad\nabla\cdot A\nabla w_{\eta}=-k_{\eta}^{2}nw_{\eta}\in L^{2}(D).$

Due to the regularity of the boundary and coefficient matrix, standard
elliptic regularity [33] gives that $v_{\eta},w_{\eta}\in H^{2}(D)$ such that

$||v_{\eta}||_{H^{2}(D)}\leq
C_{1}\Big{[}||v_{\eta}||_{L^{2}(D)}+k_{\eta}^{2}||v_{\eta}||_{L^{2}(D)}+||v_{\eta}||_{H^{3/2}(\partial
D)}\Big{]},$ $||w_{\eta}||_{H^{2}(D)}\leq
C_{2}\Big{[}||w_{\eta}||_{L^{2}(D)}+k_{\eta}^{2}n_{\text{max}}||w_{\eta}||_{L^{2}(D)}+||v_{\eta}||_{H^{3/2}(\partial
D)}\Big{]},$

where we have used the fact that $w_{\eta}=v_{\eta}$ on $\partial D$. Note
that the constants $C_{j}>0$ for $j=1,2$ are independent of $\eta$. Since
$v_{\eta}$ and $w_{\eta}$ are bounded in $H^{1}(D)$ and $k_{\eta}$ is bounded,
$||v_{\eta}||_{H^{2}(D)}$ and $||w_{\eta}||_{H^{2}(D)}$ are bounded. ∎

The compact embedding of $H^{2}(D)$ into $H^{1}(D)$ implies that

$||v_{\infty}||^{2}_{H^{1}(D)}+||w_{\infty}||^{2}_{H^{1}(D)}=1.$

Hence the limiting functions are nontrivial. Also, note that Lemma 4.2 still
holds provided that $\partial_{\nu}v_{\eta}$ on $\partial D$ is bounded in
$H^{1/2}(\partial D)$. Put the results together from this subsection, we have
proved the following theorem.

######  Theorem 4.4.

If $\partial D$ is class $\mathscr{C}^{2}$ or is polygonal with no reentrant
corners as well as $A\in\mathscr{C}^{1}(\overline{D},\mathbb{R}^{d\times d})$
with $v_{\eta}$ on $\partial D$ is bounded in $H^{3/2}(\partial D)$ and that
$k_{\eta}$ is bounded then for $k_{\eta}\to k_{\infty}$ as $\eta\to\pm\infty$
we have that $(w_{\eta},v_{\eta})\to(w_{\infty},v_{\infty})$ as
$\eta\to\pm\infty$ in $X(D)$ such that

$\Delta v_{\infty}+k_{\infty}^{2}v_{\infty}=0\quad\text{ and }\quad\nabla\cdot
A\nabla w_{\infty}+k_{\infty}^{2}nw_{\infty}=0\quad\text{in $D$}$

where $(w_{\infty},v_{\infty})\in H^{1}_{0}(D)\times H^{1}_{0}(D)$ nontrivial.

The above theorem states that the TEVs limit points as $\eta\to\pm\infty$ are
either the Dirichlet eigenvalues for $-\Delta$ or $-n^{-1}\nabla\cdot
A\nabla$. The analysis in this section works for any TEV with a limit point as
$\eta\to\pm\infty$. Therefore, we conclude that the complex TEVs (provided
they exist) limit points as $\eta\to\pm\infty$ all lie on the real line. To
summarize, we have shown the discreteness, existence, and dependence on the
parameters for TEVs.

##  5 Numerical Validation

In this section, we provide numerical examples that demonstrate the
theoretical results in the previous sections. We shall focus on the dependence
of the TEVs with respect to the coefficients $A,n,$ and $\eta$. From Section
4.1, the first TEV is monotone with respect to the coefficients. Also, in
sections 4.2 and 4.3 we established the limiting behavior of the TEVs as
$\eta\longrightarrow 0$ and $\eta\longrightarrow\pm\infty$. The following
numerical examples demonstrate the theory.

Figure 1:  The unit circle, unit square and L–Shaped domains

We consider three domains in $\mathbb{R}^{2}$: the unit disk, the unit square,
and an L–shaped domain (see Figure 1). For simplicity, we shall assume that
$A=aI$ where $a$ is constant and $I$ is the $2\times 2$ identity matrix.

###  5.1 Unit Disk

For the unit disk, by separation of variables, $k$ is a TEV if and only if
$d_{m}(k)=0$ for some $m\in\mathbb{N}$, where

$d_{m}(k)\mathrel{\mathop{\mathchar
58\relax}}=\text{det}\begin{pmatrix}J_{m}\Big{(}\sqrt{\frac{n}{a}}k\Big{)}&-J_{m}(k)\\\
k\sqrt{na}J^{\prime}_{m}\Big{(}\sqrt{\frac{n}{a}}k\Big{)}&-\Big{[}kJ^{\prime}_{m}(k)+\eta
J_{m}(k)\Big{]}\end{pmatrix}.$ (27)

where $n$ and $\eta$ are constant and $J_{m}$ denotes the Bessel function of
the first kind of order $m$.

We consider the case when $\eta\to 0$. By theorem 3.1 there exist infinitely
many real eigenvalues. Theorem 4.3 gives that the limit point as $\eta\to 0$
is the TEV with $\eta=0$, i.e. the TEVs are continuous at $\eta=0$. To verify
the convergence we compute the first root of $d_{0}(k)$ given by (27). We
remark that the first root of $d_{0}(k)$ in general is be the first TEV. The
estimated rate of convergence (EOC) is defined as

$\text{EOC}=\frac{1}{\log(2)}\log\Bigg{(}\frac{|k_{\eta_{p}}-k_{0}|}{|k_{\eta_{p+1}}-k_{0}|}\Bigg{)}\quad\text{with
$\eta_{p}=\pm\frac{1}{2^{p}}$ for $p=0,1,2,3,...$}$

The numerical results are given in Table 1.

$\eta$ | $k_{\eta}$ | EOC | $\eta$ | $k_{\eta}$ | EOC
---|---|---|---|---|---
1 | 1.6010 | N/A | -1 | 5.8032 | N/A
1/2 | 1.7353 | 1.7003 | -1/2 | 5.8143 | 0.9223
1/4 | 1.7697 | 1.2386 | -1/4 | 5.8203 | 0.9542
1/8 | 1.7832 | 1.1004 | -1/8 | 5.8234 | 0.9556
1/16 | 1.7893 | 1.0498 | -1/16 | 5.8250 | 0.9569
1/32 | 1.7922 | 1.0255 | -1/32 | 5.8258 | 0.9175
1/64 | 1.7936 | 1 | -1/64 | 5.8262 | 0.8480
1/128 | 1.7943 | 1 | -1/128 | 5.8264 | 0.7370
1/256 | 1.7946 | 1 | -1/256 | 5.8265 | 0.5850
1/512 | 1.7948 | 1 | -1/512 | 5.8266 | 1
Table 1: Convergence with respect to $\eta\to 0$ for the unit disk ($A=0.4I$
and $n=3$ for $\eta>0$; $A=3I$ and $n=0.7$ for $\eta<0$).

Note that the limit for $A=0.4I$ and $n=3$ is $k_{0}=1.7950$ and the limit for
$A=3I$ and $n=0.7$ is $k_{0}=5.8267$. The values $k_{0}$’s are given by the
first root of $d_{0}(k;\eta=0)$. The numerical results in Table 1 verify the
convergence in the previous sections and indicate that the EOC is of first
order, i.e., $|k_{\eta}-k_{0}|=\mathcal{O}(|\eta|)$ as $\eta\to 0$.

For $\eta\to\pm\infty$, from Theorem 4.4 the limiting value $k_{\infty}$ is
such that either

$\Delta v_{\infty}+k_{\infty}^{2}v_{\infty}=0\quad\text{ or }\quad\nabla\cdot
A\nabla w_{\infty}+k_{\infty}^{2}nw_{\infty}=0\quad\text{in $D$},$

where $v_{\infty}=w_{\infty}=0$ on $\partial D$. The limiting values are given
by

$k_{\infty}=\sqrt{\lambda_{j}(D)}\quad\text{ or }\quad
k_{\infty}=\sqrt{\lambda_{j}(D)\frac{a}{n}}\quad\text{ for some
$j\in\mathbb{N}$},$

where $\lambda_{j}(D)$ is a Dirichlet eigenvalues of the negative Laplacian in
$D$. For the unit disk, $\sqrt{\lambda_{1}(D)}=2.4048$. For $a=0.4$ and $n=3$,
$k_{\infty}=2.4048$ or $k_{\infty}=0.8781$ are possible limit points. The EOC
is defined by

$\text{EOC}=\frac{1}{\log(2)}\log\Bigg{(}\frac{|k_{\eta_{p}}-k_{\infty}|}{|k_{\eta_{p+1}}-k_{\infty}|}\Bigg{)}\quad\text{
with $\eta_{p}=\pm{2^{p}}$ for $p=0,1,2,3,...$}.$

$\eta$ | $k_{\eta}$ | EOC | $\eta$ | $k_{\eta}$ | EOC
---|---|---|---|---|---
1 | 1.6010 | N/A | -1 | 5.8032 | N/A
2 | 1.1415 | 1.4565 | -2 | 5.7839 | 0.00822
4 | 0.9828 | 1.3310 | -4 | 5.7537 | 0.01295
8 | 0.9258 | 1.1342 | -8 | 5.7124 | 0.0179
16 | 0.9010 | 1.0586 | -16 | 2.5944 | 4.1248
32 | 0.8893 | 1.0318 | -32 | 2.4881 | 1.1866
64 | 0.8837 | 1 | -64 | 2.4443 | 1.0765
128 | 0.8809 | 1 | -128 | 2.4241 | 1.0333
256 | 0.8795 | 1 | -256 | 2.4143 | 1.0266
512 | 0.8788 | 1 | -512 | 2.4095 | 1.0153
Table 2: Convergence with respect to $\eta\to\pm\infty$ for the unit disk.
($A=0.4I$ and $n=3$ for $\eta>0$ and $A=3I$ and $n=0.7$ for $\eta<0$).

The results are shown in Table 2, which verifies the convergence in the
previous sections. Similarly, it seems that EOC is first order i.e.
$|k_{\eta}-k_{\infty}|=\mathcal{O}\left(1/|\eta|\right)$ as
$\eta\to\pm\infty$. Note that by Table 2 we see that in the limit $k_{\eta}$
can tend to either a Dirichlet eigenvalues of the negative Laplacian or a
Dirichlet eigenvalues for $-n^{-1}\nabla\cdot A\nabla$ in $D$.

###  5.2 Validation for the Unit Square and L–Shaped Domain

In this section, we consider two Lipschitz domains, for which a continuous
finite element method is used [15, 25, 34]. The two domains are respectively
given by

$(-1/2,1/2)\times(-1/2,1/2)\quad\text{and}\quad(-1/2,1/2)\times(-1/2,1/2)\setminus[0,1/2]\times[0,1/2].$

Again, for simplicity, $A=aI$ where $a$ is constant and $n$ and $\eta$ are
constant.

$a$ | $k_{1}(a)$ | $a$ | $k_{1}(a)$
---|---|---|---
$1.5$ | $18.3370$ | $0.3$ | $3.3113$
$2$ | $14.1549$ | $0.4$ | $4.2377$
$2.5$ | $11.8953$ | $0.5$ | $5.4361$
$3$ | $10.7253$ | $0.6$ | $6.5176$
$3.5$ | $10.2221$ | $0.7$ | $7.6965$
$4$ | $9.9292$ | $0.8$ | $9.2953$
Table 3: Monotonicity with respect to $A=aI$ for the unit square. Here, we
take $n=0.75$ and $\eta=-2$ for $a>1$ where as $n=2$ and $\eta=2$ for $a<1$.

In Table 3 we see the monotonicity for the unit square with respect to $A=aI$.
The monotonicity with respect to $n$ for the L-shaped domain can be found in
Table 4. For both cases Theorems 4.1 and 4.2 are validated.

$n$ | $k_{1}(n)$ | $n$ | $k_{1}(n)$
---|---|---|---
$1.5$ | $12.9868$ | $0.3$ | $10.5159$
$2$ | $9.0795$ | $0.4$ | $11.0014$
$2.5$ | $7.2272$ | $0.5$ | $11.5661$
$3$ | $6.0632$ | $0.6$ | $12.2386$
$3.5$ | $5.3701$ | $0.7$ | $12.9390$
$4$ | $4.8812$ | $0.8$ | $13.5370$
Table 4: Monotonicity with respect to $n$ for the L–Shaped domain. Here, we
take $A=0.7I$ and $\eta=2$ for $n>1$ where as $A=3I$ and $\eta=-2$ for $n<1$.

Lastly, consider the case when $\eta\to\pm\infty$ for the L-shaped domain.
Note that this domain is not covered by Theorem 4.4 due to the reentrant
corner. The calculations here are to see if the smoothness of the boundary in
Theorem 4.4 is an essential assumption or technical one that can possibly be
removed. The results are shown in Table 5.

$\eta$ | $k_{\eta}$ | $\eta$ | $k_{\eta}$
---|---|---|---
$1$ | $3.9995$ | $-1$ | $13.0503$
$2$ | $3.8281$ | $-2$ | $12.9390$
$4$ | $3.4828$ | $-4$ | $12.6623$
$8$ | $2.7805$ | $-8$ | $11.7693$
$16$ | $2.4898$ | $-16$ | $10.1467$
$32$ | $2.3732$ | $-32$ | $7.2242$
$64$ | $2.3196$ | $-64$ | $6.6044$
$128$ | $2.2936$ | $-128$ | $6.3938$
$256$ | $2.2807$ | $-256$ | $6.3000$
$512$ | $2.2742$ | $-512$ | $6.2549$
Table 5: Convergence with respect to $\eta\to\pm\infty$ for the L-shaped
domain ($A=0.4I$ and $n=3$ for $\eta>0$ and $A=3I$ and $n=0.7$ for $\eta<0$).

From Table 5, the first TEV does indeed seem to have a limit as
$\eta\to\pm\infty$. The limiting value is given by
$k_{\infty}=\sqrt{\lambda_{1}(D)}$ or
$k_{\infty}=\sqrt{\lambda_{1}(D)\frac{a}{n}}$ where $\lambda_{1}(D)\approx
38.5674$. This is consistent with $k_{\infty}\approx 6.2103$ or $2.2677$ (i.e.
for the case $a=0.4$ and $n=3$). Therefore, we conjecture that Theorem 4.4 is
valid for polygonal domains with reentrant corners. Also, with this we see the
‘bifurcation’ numerically for the limiting case as $\eta\to\pm\infty$ since we
have seen for multiple examples that both possible limiting values are valid.

##  6 Conclusions

We study the scattering problem for an anisotropic material with a conductive
boundary. We prove that there exists infinitely transmission eigenvalues and
that they are at most a discrete set. We then study the dependence of the
transmission eigenvalues on the physical parameters and proved that the first
transmission eigenvalue is monotone with respect to each of these parameters.
We also prove that the transmission eigenvalues converge as the conductive
boundary parameter $\eta$ goes to $0$ or $\infty$ in magnitude. Lastly, we
provide some numerical examples for three different shapes, validating our
theoretical results for monotonicity and convergence. For the L-shaped domain,
the convergence result as $\eta\longrightarrow\pm\infty$ is consistent with
the theory, though it isn’t covered under our assumptions.

Acknowledgments: The research of I. Harris and V. Hughes is partially
supported by the NSF DMS Grant 2107891.

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