# Spectral properties of an acoustic-elastic transmission eigenvalue problem
with applications

Huaian Diao  School of Mathematics, Jilin University, Changchun 130012, China
<EMAIL_ADDRESS><EMAIL_ADDRESS>,  Hongjie Li  Department of Mathematics,
The Chinese University of Hong Kong, Hong Kong SAR, China
<EMAIL_ADDRESS>,  Hongyu Liu  Department of Mathematics, City
University of Hong Kong, Kowloon, Hong Kong SAR, China
<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Jiexin Tang  School of
Mathematics, Northeast Normal University, Changchun 130024, China
<EMAIL_ADDRESS>
###### Abstract.

We are concerned with a coupled-physics spectral problem arising in the
coupled propagation of acoustic and elastic waves, which is referred to as the
acoustic-elastic transmission eigenvalue problem. There are two major
contributions in this work which are new to the literature. First, under a
mild condition on the medium parameters, we prove the existence of an
acoustic-elastic transmission eigenvalue. Second, we establish a geometric
rigidity result of the transmission eigenfunctions by showing that they tend
to localize on the boundary of the underlying domain. Moreover, we also
consider interesting implications of the obtained results to the effective
construction of metamaterials by using bubbly elastic structures and to the
inverse problem associated with the fluid-structure interaction.

Keywords: acoustic-elastic; coupled-physics; transmission eigenvalues;
transmission eigenfunctions; spectral geometry; boundary localization; bubbly
elastic medium; fluid-structure interaction

2010 Mathematics Subject Classification: 35C20, 35M10, 35M30, 35P25

##  1\. Introduction

###  1.1. Mathematical setup and summary of major findings

Initially focusing on the mathematics, but not the physics, we present the
mathematical setup of our study. Let $\Omega$ be a simply connected domain
with $\partial\Omega\in C^{1,s}$ for some $0<s<1$ in $\mathbb{R}^{N},N=2,3$.
The complement $\mathbb{R}^{N}\backslash\overline{\Omega}$ is connected.
Henceforth, we let $\nu\in\mathbb{S}^{N-1}$ signify the exterior unit normal
vector to $\partial\Omega$. Let $\rho_{b},\rho_{e}\in\mathbb{R}_{+}$ and
$\kappa\in\mathbb{R}_{+}$. Let $\tilde{\lambda},\tilde{\mu}$ be the Lamé
constants which satisfy the following strong convexity conditions

$\tilde{\mu}>0\ \ \mbox{and}\ \ N\tilde{\lambda}+2\tilde{\mu}>0.$ (1.1)

Let $\mathbf{u}(\mathbf{x})$, $\mathbf{x}\in\Omega$, be a
$\mathbb{C}^{N}$-valued function.

Define the Lamé operator $\mathcal{L}_{\tilde{\lambda},\tilde{\mu}}$ and the
traction operator $T_{\tilde{\nu}}$ respectively as follows:

$\begin{split}\mathcal{L}_{{\tilde{\lambda}},{\tilde{\mu}}}\mathbf{u}(\mathbf{x}):=&\tilde{\lambda}\Delta\mathbf{u}(\mathbf{x})+(\tilde{\lambda}+\tilde{\mu})\nabla(\nabla\cdot\mathbf{u}(\mathbf{x})),\quad\mathbf{x}\in\Omega,\\\
T_{\tilde{\nu}}{\mathbf{u}}(\mathbf{x}):=&\tilde{\lambda}(\nabla\cdot{\mathbf{u}}(\mathbf{x}))\nu(\mathbf{x})+2\tilde{\mu}(\nabla^{s}{\mathbf{u}}(\mathbf{x}))\nu(\mathbf{x}),\quad\mathbf{x}\in\partial\Omega,\end{split}$
(1.2)

where

$\nabla^{s}{\mathbf{u}}:=\frac{1}{2}(\nabla{\mathbf{u}}+\nabla{{\mathbf{u}}^{\top}}).$
(1.3)

Let $\omega\in\mathbb{R}_{+}$ and $v(\mathbf{x})$, $\mathbf{x}\in\Omega$, be a
$\mathbb{C}$-valued function. We are concerned with the following spectral
problem for $(\mathbf{u},v)\in H^{1}(\Omega)^{N}\times H^{1}(\Omega)$:

$\begin{cases}\mathcal{L}_{{\tilde{\lambda}},{\tilde{\mu}}}{\mathbf{u}}(\mathbf{x})+\omega^{2}\rho_{e}{\mathbf{u}}(\mathbf{x})=0&\mbox{in}\quad\Omega,\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\displaystyle{\nabla\cdot(\frac{1}{\rho_{b}}\nabla
v(\mathbf{x}))+\frac{\omega^{2}}{\kappa}v(\mathbf{x})=0}&\mbox{in}\quad\Omega,\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\displaystyle{{\mathbf{u}}(\mathbf{x})\cdot\nu-\frac{1}{\rho_{b}\omega^{2}}\nabla
v(\mathbf{x})\cdot\nu=0}&\mbox{on}\quad\partial\Omega,\vskip 3.0pt plus 1.0pt
minus 1.0pt\\\
T_{\tilde{\nu}}{\mathbf{u}}+v(\mathbf{x})\nu=0&\mbox{on}\quad\partial\Omega.\end{cases}$
(1.4)

In the physical setup, the first equation in (1.4) is known as the Lamé
equation which describes the propagation of elastic deformation, whereas the
second one is the Helmholtz equation which governs the acoustic wave
propagation. Hence, (1.4) is a coupled-physics spectral problem and shall be
referred to as the acoustic-elastic transmission eigenvalue problem in what
follows. In (1.4), $\kappa$ and $\rho_{b}$ describe the bulk modulus and
density of the acoustic medium. The physical parameters $\tilde{\lambda}$ and
$\tilde{\mu}$ characterize the compressional modulus and the shear modulus of
the elastic material, respectively. Furthermore, $\rho_{e}$ specifies the
density of the elastic medium.

It is clear that $(\mathbf{u},v)\equiv(\mathbf{0},0)$ is a pair of trivial
solutions to (1.4). If there exists a nontrivial pair of solutions to (1.4),
$\omega\in\mathbb{R}_{+}$ is called an AE (acoustic-elastic) transmission
eigenvalue and $(\mathbf{u},v)$ is the associated pair of transmission
eigenfunctions. It is particularly noted that if $v\equiv 0$, one has from
(1.4) that

$\begin{cases}\mathcal{L}_{{\tilde{\lambda}},{\tilde{\mu}}}{\mathbf{u}}(\mathbf{x})+\omega^{2}\rho_{e}{\mathbf{u}}(\mathbf{x})=0&\mbox{in}\quad\Omega,\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\displaystyle{{\mathbf{u}}(\mathbf{x})\cdot\nu=0,\quad
T_{\tilde{\nu}}{\mathbf{u}}=0}&\mbox{on}\quad\partial\Omega,\end{cases}$ (1.5)

which is the classical Jones eigenvalue problem. The Jones eigenvalue problem
arises in studying the fluid-structure interaction [23] and has been
extensively studied in the literature [20, 21, 34, 32]. It is known that under
certain conditions of the medium parameters
$\tilde{\lambda},\tilde{\mu},\rho_{e}$ as well as the domain $\Omega$, there
exist Jones eigenvalues [20]. Clearly, Jones eigenvalues to (1.4) are a
special subset of the AE transmission eigenvalues to (1.5). In this paper, we
show that in addition to the Jones eigenvalues, there exist AE transmission
eigenvalues to (1.4) under two generic scenarios. In the first scenario with
no geometric restriction on $\Omega$ but a minor condition on the medium
parameters, we show the existence of AE transmission eigenvalues. We employ
the layer potential theory and Gohberg-Sigal theory to establish the
aforementioned result. In the second scenario, if $\Omega$ is of a radial
shape and no restriction is imposed on the medium parameters, we show the
existence of infinitely many AE transmission eigenvalues. The derivation of
the second result employs Fourier series expansions and involves highly
technical and subtle calculations. The results are contained in Sections 2 and
3 in what follows.

In addition to the spectral properties of the transmission eigenvalues, we
further show that the transmission eigenfunctions tend to localize on
$\partial\Omega$ in the sense that their $L^{2}$-energies tend to concentrate
on $\partial\Omega$. In fact, we rigorously show the existence of a sequence
of transmission eigenfunctions $(\mathbf{u}_{m},v_{m})_{m\in\mathbb{N}}$
associated with $\omega_{m}\rightarrow\infty$ such that the aforementioned
boundary localization pattern occurs. Furthermore, it is intriguing to note
that depending on the configuration of the medium parameters, it may happen
that both $\mathbf{u}_{m}$ and $v_{m}$ are boundary-localized, which are
referred to as a pair of bi-localized eigen-modes; and it may also happen that
only $v_{m}$ is boundary-localized whereas $\mathbf{u}_{m}$ is not, which are
referred to as a pair of mono-localized eigen-modes. The results of this part
are rigorously proved in the radial case and numerically verified in the non-
radial case, which are contained in Section 3.

###  1.2. Physical relevance and implications

We discuss the physical background and motivation of our study and the
interesting implications that it may have. We first consider the fluid-
structure interaction problem which is described by the following PDE system
(cf. [23, 20, 32]):

$\begin{cases}\mathcal{L}_{{\tilde{\lambda}},{\tilde{\mu}}}{\mathbf{u}}(\mathbf{x})+\omega^{2}\rho_{e}{\mathbf{u}}(\mathbf{x})=0&\mbox{in}\quad\Omega,\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\displaystyle{\nabla\cdot(\frac{1}{\rho_{b}}\nabla
v^{t}(\mathbf{x}))+\frac{\omega^{2}}{\kappa}v^{t}(\mathbf{x})=0}&\mbox{in}\quad\mathbb{R}^{N}\backslash\overline{\Omega},\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\displaystyle{{\mathbf{u}}(\mathbf{x})\cdot\nu=\frac{1}{\rho_{b}\omega^{2}}\nabla
v^{t}(\mathbf{x})\cdot\nu,\
T_{\tilde{\nu}}{\mathbf{u}}=-v^{t}(\mathbf{x})\nu}&\mbox{on}\quad\partial\Omega,\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ v^{t}-v^{i}\ \ \mbox{satisfies the Sommerfeld
radiation condition},\end{cases}$ (1.6)

where $v^{i}$ is an entire solution to $(\Delta+\mathfrak{m}^{2})v^{i}=0$ in
$\mathbb{R}^{N}$ with $\mathfrak{m}:=\omega/c_{b}$ and
$c_{b}:=\sqrt{\kappa/\rho_{b}}$. The last condition in (1.6) signifies that
the scattered field $v^{s}:=v^{t}-v^{i}$ satisfies

$\lim_{r\rightarrow\infty}r^{(N-1)/2}(\partial_{r}v^{s}-\mathrm{i}\mathfrak{m}v^{s})=0,\
\ \ r:=|\mathbf{x}|.$ (1.7)

The well-posedness of the forward problem (1.6) is known. In the physical
setup, $(\Omega;\tilde{\lambda},\tilde{\mu},\rho_{e})$ signifies an elastic
body which is embedded in a fluid whose acoustic property is characterized by
$\kappa$ and $\rho_{b}$. $v^{i}$ denotes an incident acoustic wave with
$\omega\in\mathbb{R}_{+}$ being its angular frequency and its impingement on
the elastic body generates the scattering phenomenon where the elastic
response inside the elastic body is governed by the first equation in (1.6)
and the acoustic wave propagation outside $\Omega$ is governed by the second
equation in (1.6). The elastic and acoustic fields are coupled together via
the transmission conditions across $\partial\Omega$. An inverse problem of
practical importance in sonar technology is to identify the solid structure
$\Omega$ by knowledge of the acoustic field $v^{s}$ away from $\Omega$. It is
clear that if $v^{s}\equiv 0$, i.e. $v^{t}=v^{i}$ in
$\mathbb{R}^{N}\backslash\overline{\Omega}$, the solid body is invisible with
respect to the acoustic scanning by using $v^{i}$. In such a case, one can
directly verify that $v=v^{i}|_{\Omega}$ and $\mathbf{u}$ fulfils (1.4),
namely they form a pair of AE transmission eigenfunctions. That is, if
invisibility occurs, the scattering pattern, namely the perturbative wave
pattern, is trapped inside the solid body. Hence, in order to understand the
invisibility phenomenon, one needs to study the AE transmission eigenvalue
problem (1.4). In fact, by following a similar argument in [4] for the
acoustic transmission eigenvalue problem, one can show that if
$(v,\mathbf{u})$ is a pair of AE transmission eigenfunctions, then $v$ can be
extended by the Herglotz approximation to form an incident field whose
impingement on $\Omega$ generates a nearly-vanishing scattered field. Hence,
our results on the existence of AE transmission eigenvalues indicates that
(near) invisibility is not a sporadic phenomenon. Moreover, our results on the
boundary localization properties of the AE transmission eigenfunctions
characterize the quantitative behaviours of both the elastic and acoustic
fields when (near) invisibility occurs. It is also interesting to note the
mono-localized eigen-modes (i.e. $v$ is boundary-localized but $\mathbf{u}$ is
not) can be used for the design of one-way information transmission in that
the acoustic observable is void outside the solid, but the elastic observable
inside the solid is non-void. Finally, we would like to mention that the
spectral pattern of boundary localization was recently investigated in [8, 12]
for the acoustic transmission eigenfunctions and it has been used to produce a
super-resolution scheme for acoustic imaging. By following a similar spirit,
one can also use the spectral pattern discovered in the current article to
develop novel imaging scheme for the fluid-structure interaction problem. We
shall consider these and other developments in a forthcoming paper.

Next, we consider the elastodynamics in bubbly elastic media. For simplicity,
we consider the case with a single air bubble $(\Omega;\kappa,\rho_{e})$
embedded in an elastic medium
$(\mathbb{R}^{N}\backslash\overline{\Omega};\tilde{\lambda},\tilde{\mu},\rho_{b})$.
The linear elastic deformation is governed by the following PDE system (cf.
[30]):

$\begin{cases}\mathcal{L}_{{\tilde{\lambda}},{\tilde{\mu}}}{\mathbf{u}}^{t}(\mathbf{x})+\omega^{2}\rho_{e}{\mathbf{u}}^{t}(\mathbf{x})=0&\mbox{in}\quad\mathbb{R}^{N}\backslash\overline{\Omega},\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\displaystyle{\nabla\cdot(\frac{1}{\rho_{b}}\nabla
v(\mathbf{x}))+\frac{\omega^{2}}{\kappa}v(\mathbf{x})=0}&\mbox{in}\quad\Omega,\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\displaystyle{{\mathbf{u}}^{t}(\mathbf{x})\cdot\nu=\frac{1}{\rho_{b}\omega^{2}}\nabla
v(\mathbf{x})\cdot\nu,\
T_{\tilde{\nu}}{\mathbf{u}}^{t}=-v(\mathbf{x})\nu}&\mbox{on}\quad\partial\Omega,\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ \mathbf{u}^{t}-\mathbf{u}^{i}\ \
\mbox{satisfies the Kupradze radiation condition},\end{cases}$ (1.8)

where $\mathbf{u}^{i}$ is an entire solution to
$\mathcal{L}_{{\tilde{\lambda}},{\tilde{\mu}}}{\mathbf{u}}^{i}+\omega^{2}\rho_{e}{\mathbf{u}}^{i}=0$
in $\mathbb{R}^{N}$. The Kupradze radiation condition is given by decomposing
the elastic field $\mathbf{u}^{s}:=\mathbf{u}^{t}-\mathbf{u}^{i}$ into its
shear and compressional parts and requiring that each part fulfils the
Sommerfeld radiation condition (1.7); see e.g. [28, 29] for a more detailed
description. Clearly, one can consider the invisibility issue for (1.8) and
the identically vanishing of $\mathbf{u}^{s}$ leads again to the transmission
eigenvalue problem (1.4) with $\mathbf{u}=\mathbf{u}^{i}|_{\Omega}$. However,
we are more interested in understanding under what conditions such that when
$\mathbf{u}^{i}\equiv 0$, there exists a nontrivial solution to (1.8). In
fact, a systematic study was provided in [30] and it is shown that if
$\rho_{e}/\rho_{b}\gg 1$ and $\omega\ll 1$, the aforementioned resonance
phenomenon indeed may happen (at least asymptotically). This low-frequency
resonance phenomenon is referred to as the Minnaert resonance and forms the
fundamental basis for the effective realisation of elastic metamaterials by
bubble-elastic structures (cf. [24, 30, 38]). Intriguingly, the Minnaert
resonant mode possesses the same boundary-localization pattern as the AE
transmission eigenfunctions (cf. [19]), which defines the polarizability of
the nano-bubbles. Hence, we expect that the discovery in the current article
may produce interesting applications in effective construction of elastic
metamaterials, which is definitely worth our further study.

###  1.3. Comments on our study

The AE transmission eigenvalue problem (1.4) was first considered in [25], and
it is proved that if the AE transmission eigenvalue exist, they form a
discrete set and can accumulate only at $\infty$. To our best knowledge, we
proved the first general result in the literature that under generic
scenarios, there indeed exist AE transmission eigenvalues with a mil
dcondition on the medium parameters. The boundary-localization of transmission
eigenfunctions was first discovered in [8] for the acoustic transmission
eigenvalue problem. It is further extended to the Maxwell system for
electromagnetic transmission eigenfunctions in [15], and to the Lamé system
for elastic transmission eigenfunctions [22]. It seems that the boundary-
localization phenomenon seems to be universal for different waves when
invisibility/non-scattering occurs. However, we would like to emphasize that
the different physics underlying this intriguing phenomenon leads to different
technical challenges and mathematical treatments, as well as different
applications. Indeed, we note that the AE transmission eigen-system (1.4)
consists of two PDEs with one vector-valued and the other one scalar-valued,
and moreover the eigenvalue $\omega$ is also coupled in the boundary
transmission conditions, which give rise to significant technical difficulties
compared to the studies in [8, 15]. The boundary-localization properties in
[8] lead to a super-resolution imaging scheme and generating the so-called
pseudo plasmon modes, and in [15] lead to an artificial mirage scheme. As
discussed earlier, the boundary-localization properties in the current article
may have the potential to be applied to the fluid-structure inverse problems
and the effective construction of elastic metamaterials. Finally, we would
like to mention that in all of the aforementioned works [8, 15, 22], the
boundary-localization properties were all rigorously verified for the radial
geometry and numerically verified for the general geometry. The only exception
is in [9] where the general geometry is considered and the boudnary-
localization is rigorously justified for the acoustic transmission
eigenfunctions via the theory of pseudo-differential operators and generalised
Wely’s law. However, due to the technical requirement, [9] actually studies
the so-called generalised transmission eigenfunctions and moreover the
boundary-localization properties are not so sharp and thorough compared to the
relevant results for the radial geometry. Hence, we shall follow a similar
spirt in the current study by focusing on treating the boundary-localization
for the radial geometry rigorously and the general geometry numerically. We
shall consider justifying the general geometry in a forthcoming paper.

##  2\. Existence of infinitely many transmission eigenvalues

In this section, we show that in a certain generic scenario there exist
transmission eigenvalues for the system (1.4).

###  2.1. Parameter configuration and nondimensionalization

To facilitate analyzing the coupled-physics system (1.4), we introduce the
following nondimensional parameters. Define

$\delta=\rho_{b}/\rho_{e},\quad\tau=\frac{c_{b}}{\tilde{c_{p}}}=\frac{\sqrt{\kappa/\rho_{b}}}{\sqrt{(\tilde{\lambda}+2\tilde{\mu})/\rho_{e}}},\quad
c_{b}=\sqrt{\kappa/\rho_{b}},\quad\tilde{c_{p}}=\sqrt{(\tilde{\lambda}+2\tilde{\mu})/\rho_{e}}.$
(2.1)

Let $l_{\Omega}$ be the average length of the domain $\Omega$ and we denote
the following parameters by

$\begin{split}{\mathbf{x}}^{\prime}=\frac{\mathbf{x}}{l_{\Omega}},\ \
k&=\frac{\omega}{c_{b}}l_{\Omega},\ \
{\mathbf{u}}^{\prime}=\frac{\mathbf{u}}{l_{\Omega}},\\\
\mu=\frac{\tilde{\mu}}{\tilde{\lambda}+2\tilde{\mu}},\ \
\lambda&=\frac{\tilde{\lambda}}{\tilde{\lambda}+2\tilde{\mu}},\ \
{v}^{\prime}=\frac{v}{\rho_{b}c_{b}^{2}}.\end{split}$ (2.2)

We would like to mention that the parameters defined in (2.1) and (2.2) are
all nondimensional. Through substituting these parameters into (1.4) and
dropping their primes, we obtain the following nondimensional coupled PDE
system (cf. [30]):

$\begin{cases}\mathcal{L}_{\lambda,\mu}{\mathbf{u}}(\mathbf{x})+k^{2}\tau^{2}{\mathbf{u}}(\mathbf{x})=0&\mbox{in}\quad\Omega,\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ \Delta
v(\mathbf{x})+k^{2}v(\mathbf{x})=0&\mbox{in}\quad\Omega,\vskip 3.0pt plus
1.0pt minus 1.0pt\\\ {\mathbf{u}}(\mathbf{x})\cdot\nu-\frac{1}{k^{2}}\nabla
v(\mathbf{x})\cdot\nu=0&\mbox{on}\quad\partial\Omega,\vskip 3.0pt plus 1.0pt
minus 1.0pt\\\
T_{\nu}{\mathbf{u}}+\delta\tau^{2}v(\mathbf{x})\nu=0&\mbox{on}\quad\partial\Omega.\end{cases}$
(2.3)

It is remarked that the system (2.3) is equivalent to the original system
(1.4). Indeed, one can obtain the system (1.4) from (2.3) by substituting the
parameters in (2.1) and (2.2) into the system (2.3). Consequently, in what
follows we focus on studying the system (2.3) instead of the system (1.4).

###  2.2. Layer potentials and integral reformulation

We shall rely on the layer potential theory to reformulate the transmission
eigenvalue problem (2.3) into an eigenvalue problem associated with a system
of integral equations. To that end, we first introduce the layer potential
operators for our subsequent use.

Let $G^{k}(\mathbf{x})$ be the fundamental solution of the operator
$\triangle+k^{2}$(cf. [27]), namely

$G^{k}(\mathbf{x})=\left\\{\begin{array}[]{ll}\displaystyle{-\frac{\mathrm{i}}{4}H_{0}^{(1)}(k|\mathbf{x}|)},&N=2,\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle{-\frac{e^{\mathrm{i}k|\mathbf{x}|}}{4\pi|\mathbf{x}|}},&N=3,\end{array}\right.$
(2.4)

where $H_{0}^{(1)}$ is the zeroth-order Hankel function of the first kind. The
single layer potential associated with the Helmholtz system is defined by

$S_{\partial\Omega}^{k}[\varphi](\mathbf{x})=\int_{\partial\Omega}G^{k}(\mathbf{x}-\mathbf{y})\varphi(\mathbf{y})ds(\mathbf{y})\quad\mathbf{x}\in\mathbb{R}^{N},$
(2.5)

with $\varphi(\mathbf{x})\in H^{-1/2}(\partial\Omega)$. Then the conormal
derivative of the single layer potential enjoys the jump formula

$\nabla
S_{\partial\Omega}^{k}[\varphi]\cdot\nu|_{\pm}(\mathbf{x})=\left(\pm\frac{1}{2}I+K_{\partial\Omega}^{k,*}\right)[\varphi](\mathbf{x})\quad\mathbf{x}\in\partial\Omega,$
(2.6)

where $I$ is an identity operator and

$K_{\partial\Omega}^{k,*}[\varphi](\mathbf{x})=\int_{\partial\Omega}\nabla_{\mathbf{x}}G^{k}(\mathbf{x}-\mathbf{y})\cdot\nu_{\mathbf{x}}\varphi(\mathbf{y})ds(\mathbf{y})\quad\mathbf{x}\in\partial\Omega,$

which is also known as the Neumann-Poincaré (N-P) operator associated with
Helmholtz system. Here and also in what follows, the subscript $\pm$ indicates
the limits from outside and inside $\Omega$, respectively.

For the Lamé system, the fundamental solution
$\mathbf{\Gamma}^{k}=(\Gamma^{k}_{i,j})_{i,j=1}^{N}$ of the operator
$\mathcal{L}_{{\lambda},{\mu}}+\rho_{e}k^{2}$ can be expressed as (cf.[30]):

$\mathbf{\Gamma}^{k}=\mathbf{\Gamma}^{k}_{s}+\mathbf{\Gamma}^{k}_{p},$ (2.7)

where

$\mathbf{\Gamma}^{k}_{p}=-\frac{1}{\rho_{e}k^{2}}\nabla\nabla
G^{{k}_{p}}\quad\mbox{and}\quad\mathbf{\Gamma}^{k}_{s}=\frac{1}{\rho_{e}k^{2}}({k}_{s}^{2}\mathbf{I}+\nabla\nabla)G^{{k}_{s}},$

with $\mathbf{I}$ denoting the $N\times N$ identity matrix and $G^{k}$ given
in (2.4). In the last equation ${k}_{s}$ and ${k}_{p}$ are defined by

$k_{p}:=\frac{k\sqrt{\rho_{e}}}{\sqrt{2\mu+\lambda}},\ \
k_{s}:=\frac{k\sqrt{\rho_{e}}}{\sqrt{\mu}}.$ (2.8)

When $k=0$, $\mathbf{\Gamma}^{0}(\mathbf{x})$ is defined as follows

$\mathbf{\Gamma}^{0}(\mathbf{x})=-\frac{\gamma_{1}}{4\pi}\frac{1}{|\mathbf{x}|}\mathbf{I}-\frac{\gamma_{2}}{4\pi}\frac{\mathbf{x}\mathbf{x}^{\top}}{|\mathbf{x}|^{3}},$
(2.9)

and

$\gamma_{1}=\frac{1}{2}\left(\frac{1}{\mu}+\frac{1}{2\mu+\lambda}\right)\quad\mbox{and}\quad\gamma_{2}=\frac{1}{2}\left(\frac{1}{\mu}-\frac{1}{2\mu+\lambda}\right).$
(2.10)

Then the single layer potential operator associated with the fundamental
solution $\mathbf{\Gamma}^{\omega}$ is defined by

$\mathbf{S}_{\partial\Omega}^{\omega}[\bm{\varphi}](\mathbf{x})=\int_{\partial\Omega}\mathbf{\Gamma}^{\omega}(\mathbf{x}-\mathbf{y})\bm{\varphi}(\mathbf{y})ds(\mathbf{y}),\quad\mathbf{x}\in\mathbb{R}^{N},$
(2.11)

for $\bm{\varphi}\in H^{-1/2}(\partial\Omega)^{N}$. On the boundary
$\partial\Omega$, the conormal derivative of the single layer potential
satisfies the following jump formula

$T_{\nu}\mathbf{S}_{\partial\Omega}^{\omega}[\bm{\varphi}]|_{\pm}(\mathbf{x})=\left(\pm\frac{1}{2}\mathbf{I}+\mathbf{K}_{\partial\Omega}^{\omega,*}\right)[\bm{\varphi}](\mathbf{x})\quad\mathbf{x}\in\partial\Omega,$
(2.12)

where

$\mathbf{K}_{\partial\Omega}^{\omega,*}[\bm{\varphi}](\mathbf{x})=\int_{\partial\Omega}T_{\nu_{\mathbf{x}}}\mathbf{\Gamma}^{\omega}(\mathbf{x}-\mathbf{y})\bm{\varphi}(\mathbf{y})ds(\mathbf{y}).$
(2.13)

The operator $\mathbf{K}_{\partial\Omega}^{\omega,*}$ is called the (N-P)
operator associated with the Lamé system.

With the above preparations, we next derive the integral reformulation of the
transmission eigenvalue problem (2.3). The main result is stated as follows.

######  Theorem 2.1.

Consider the following spectral problem associated with a system of integral
operator equations:

$\mathcal{A}(k,\delta)[\Phi](\mathbf{x})=0,\quad\mathbf{x}\in\partial\Omega,$
(2.14)

where

$\mathcal{A}(k,\delta):=\left(\begin{array}[]{cc}-\frac{I}{2}+K^{k,*}_{\partial\Omega}&-k^{2}\nu\cdot{\mathbf{S}}_{\partial\Omega}^{k\tau}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ \delta\tau^{2}\nu
S^{k}_{\partial\Omega}&-\frac{\mathbf{I}}{2}+{\mathbf{K}}^{k\tau,*}_{\partial\Omega}\\\
\end{array}\right),\quad\Phi:=\left(\begin{array}[]{c}\varphi_{b}\\\
\bm{\varphi}_{e}\\\ \end{array}\right),$ (2.15)

with the surface densities $\varphi_{b}\in
H^{-1/2}(\partial\Omega),\bm{\varphi}_{e}\in H^{-1/2}(\partial\Omega)^{N}$.
Then there exists a $k\in\mathbb{R}_{+}$ such that (2.14) possesses a
nontrivial solution $\Phi\in H^{-1/2}(\partial\Omega)\times
H^{-1/2}(\partial\Omega)^{N}$ if and only if $k$ is a transmission eigenvalue
to the system (2.3).

###### Proof.

We first prove the sufficiency part. If $k$ is a transmission eigenvalue to
the system (2.3), by the layer potential theory presented above, the solution
to the system (2.3) can be written as

$\left\\{\begin{array}[]{ll}v=S_{\partial\Omega}^{k}[\varphi_{b}](\mathbf{x}),&\mathbf{x}\in\Omega,\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\mathbf{u}=\mathbf{S}_{\partial\Omega}^{k\tau}[\bm{\varphi}_{e}](\mathbf{x}),&\mathbf{x}\in\Omega,\end{array}\right.$
(2.16)

where density functions $\varphi_{b}\in
H^{-1/2}(\partial\Omega),\bm{\varphi}_{e}\in H^{-1/2}(\partial\Omega)^{N}$.
Moreover, the density functions $\Phi=(\varphi_{b},\varphi_{e})$ solve the
equation (2.14). Indeed, the equation (2.14) follows from the third condition
and the fourth condition in the system (2.3), the fact $k\neq 0$ and the jump
formulas (2.6) as well as (2.12). Thus there exists a $k\in\mathbb{R}_{+}$
such that (2.14) possesses a nontrivial solution, which proves the sufficient
part.

Next, we show the necessity part. If there exists a $k\in\mathbb{R}_{+}$ such
that (2.14) possesses a nontrivial solution $\Phi=(\varphi_{b},\varphi_{e})\in
H^{-1/2}(\partial\Omega)\times H^{-1/2}(\partial\Omega)^{N}$, the functions
$(\mathbf{u},v)$ represented in (2.16) will solve the system (2.3). Indeed,
the functions $(\mathbf{u},v)$ represented in (2.16) satisfy the first and the
second condition in (2.3) by the potential theory. The third and the fourth
equations follow from $\Phi$ solving the equation (2.14). Thus all the
equations in (2.3) are fulfilled and this completes the proof.

∎

###  2.3. Existence of AE transmission eigenvalues

In this subsection, we show that there exist AE transmission eigenvalues to
the system (2.3). To that end, we firstly present some preliminaries for the
subsequent analysis.

######  Lemma 2.1.

The operators $S_{\partial\Omega}^{k}:H^{-1/2}(\partial\Omega)\rightarrow
H^{1/2}(\partial\Omega)$ and
$K_{\partial\Omega}^{k,*}:H^{-1/2}(\partial\Omega)\rightarrow
H^{-1/2}(\partial\Omega)$ defined in (2.5) and (2.6) enjoy the following
asymptotic expansion for $k\ll 1$ :

$S_{\partial\Omega}^{k}=S_{\partial\Omega}^{0}+kP,\qquad
K_{\partial\Omega}^{k,*}=K_{\partial\Omega}^{0,*}+k^{2}Q,$ (2.17)

where $S_{\partial\Omega}^{0}$ and $K_{\partial\Omega}^{0,*}$ are defined in
(2.5) and (2.6) with $k=0$, and the operators
$P\in\mathcal{L}(H^{-1/2}(\partial\Omega),H^{1/2}(\partial\Omega))$,
$Q\in\mathcal{L}(H^{-1/2}(\partial\Omega))$.

###### Proof.

For $k\ll 1$, the three dimensional fundamental solution given in (2.4)
satisfies the following asymptotic expansion

$\displaystyle{-\frac{e^{\mathrm{i}k|\mathbf{x}|}}{4\pi|\mathbf{x}|}}=\sum_{j=1}^{\infty}-\frac{{\mathrm{i}}}{4\pi}\frac{k^{j}({\mathrm{i}}|\mathbf{x}|)^{j-1}}{j!}.$
(2.18)

Thus, from the definition of the operator $S_{\partial\Omega}^{k}$ in (2.5),
one has that

$S_{\partial\Omega}^{k}=S_{\partial\Omega}^{0}+kP,$

where the operator $P$ is given by

$P[\varphi](\mathbf{x})=-\frac{{\mathrm{i}}}{4\pi}\sum_{j=1}^{\infty}\int_{\partial
D}\frac{({\mathrm{i}}k|\mathbf{x}-\mathbf{y}|)^{j-1}}{j!}\varphi(\mathbf{y})ds(\mathbf{y}),$

and is a bounded operator from $H^{-1/2}(\partial\Omega)$ to
$H^{1/2}(\partial\Omega)$ (cf. [29]).

For the N-P operator $K_{\partial\Omega}^{k,*}$, one has that

$K_{\partial\Omega}^{k,*}=K_{\partial\Omega}^{0,*}+k^{2}Q,$

where the operator $Q$ is written by

$Q[\varphi](\mathbf{x})=\sum_{j=2}^{\infty}-\frac{({\mathrm{i}}k)^{j}(j-1)}{4\pi
j!}\int_{\partial
D}|\mathbf{x}-\mathbf{y}|^{j-3}(\mathbf{x}-\mathbf{y})\cdot\nu_{\mathbf{x}}\varphi(\mathbf{y})ds(\mathbf{y}),$

and is a bounded operator from $H^{-1/2}(\partial\Omega)$ to
$H^{-1/2}(\partial\Omega)$ (cf. [30]). This completes the proof.

∎

######  Remark 2.1.

We only give the proof for the three dimensional case in Lemma 2.1. The two
dimensional situation can be proved similarly. The same principle is also
applied to Lemmas 2.2 and 2.4.

######  Lemma 2.2.

For the operators
$\mathbf{S}_{\partial\Omega}^{k}:H^{-1/2}(\partial\Omega)^{N}\rightarrow
H^{1/2}(\partial\Omega)^{N}$ and
$\mathbf{K}_{\partial\Omega}^{k,*}:H^{-1/2}(\partial\Omega)^{N}\rightarrow
H^{-1/2}(\partial\Omega)^{N}$ defined in (2.11) and (2.13), respectively, the
following asymptotic expansions hold:

$\mathbf{S}_{\partial\Omega}^{k}=\mathbf{S}_{\partial\Omega}^{0}+k\mathbf{P},\qquad\mathbf{K}_{\partial\Omega}^{k,*}=\mathbf{K}_{\partial\Omega}^{0,*}+k^{2}\mathbf{Q},$
(2.19)

where $\mathbf{S}_{\partial\Omega}^{0}$ and
$\mathbf{K}_{\partial\Omega}^{0,*}$ are defined in (2.11) and (2.13) with
$k=0$, and the operator
$\mathbf{P}\in\mathcal{L}\left(H^{-1/2}(\partial\Omega)^{N},H^{1/2}(\partial\Omega)^{N}\right)$
and $\mathbf{Q}\in\mathcal{L}\left(H^{-1/2}(\partial\Omega)^{N}\right)$.

###### Proof.

From the expression of the fundamental solution $\mathbf{\Gamma}^{\omega}$ in
(2.7) and the asymptotic expansion in (2.18), one has that

$\mathbf{\Gamma}^{k}(\mathbf{x})=\sum_{j=0}^{\infty}k^{j}\mathbf{\Gamma}_{j}(\mathbf{x}),$

where

$\begin{split}\mathbf{\Gamma}_{j}(\mathbf{x})=&-\frac{1}{4\pi}\frac{\mathrm{i}^{j}}{(j+2)j!}\left((j+1)\left(\frac{\rho_{e}}{\lambda+2\mu}\right)^{\frac{j+2}{2}}+\left(\frac{\rho_{e}}{\mu}\right)^{\frac{j+2}{2}}\right)|\mathbf{x}|^{j-1}\mathbf{I}\\\
&+\frac{1}{4\pi}\frac{\mathrm{i}^{j}(j-1)}{(j+2)j!}\left(\left(\frac{\rho_{e}}{\lambda+2\mu}\right)^{\frac{j+2}{2}}-\left(\frac{\rho_{e}}{\mu}\right)^{\frac{j+2}{2}}\right)|\mathbf{x}|^{j-3}\mathbf{x}\mathbf{x}^{T}.\end{split}$

Thus the single layer potential operator $\mathbf{S}_{\partial\Omega}^{k}$ can
be written as

$\mathbf{S}_{\partial\Omega}^{k}=\mathbf{S}_{\partial\Omega}^{0}+k\mathbf{P},$

where the operator $\mathbf{P}$ is given by

$\mathbf{P}[\bm{\varphi}](\mathbf{x})=\sum_{j=1}^{\infty}k^{j-1}\int_{\partial\Omega}\mathbf{\Gamma}_{j}(\mathbf{x}-\mathbf{y})\bm{\varphi}(\mathbf{y})ds(\mathbf{y}),$

and is bounded from $H^{-1/2}(\partial\Omega)^{3}$ to
$H^{1/2}(\partial\Omega)^{3}$ (cf. [14]).

For the N-P operator $\mathbf{K}_{\partial\Omega}^{k,*}$, one has that

$\mathbf{K}_{\partial\Omega}^{k,*}=\mathbf{K}_{\partial\Omega}^{0,*}+k^{2}\mathbf{Q}$

where the operator $\mathbf{Q}$ is written by

$\mathbf{Q}[\bm{\varphi}](\mathbf{x})=\sum_{j=2}^{\infty}k^{j-2}\int_{\partial\Omega}T_{\nu_{\mathbf{x}}}\mathbf{\Gamma}_{j}(\mathbf{x}-\mathbf{y})\bm{\varphi}(\mathbf{y})ds(\mathbf{y}),$

and is bounded operator from $H^{-1/2}(\partial\Omega)^{3}$ to
$H^{-1/2}(\partial\Omega)^{3}$ (cf. [14]). This completes the proof.

∎

######  Lemma 2.3.

The dimension of the kernel of the operator $-1/2+K_{\partial\Omega}^{0,*}$ is
$1$.

###### Proof.

Assume that $\varphi$ is the kernel of the operator
$-1/2+K_{\partial\Omega}^{0,*}$. Thus the function $u(\mathbf{x})$ expressed
by

$v(\mathbf{x})=S_{\partial\Omega}^{0}[\varphi],\quad\mathbf{x}\in\Omega,$
(2.20)

solves the following Neumann boundary value problem

$\begin{cases}\Delta v(\mathbf{x})=0&\mbox{in}\quad\Omega,\\\ \nabla
v(\mathbf{x})\cdot\nu=0&\mbox{on}\quad\partial\Omega.\end{cases}$ (2.21)

The dimension of the nontrivial solution to the system (2.21) is $1$. Indeed,
by the variational principle, the nontrivial solution to the equation (2.21)
is $v(\mathbf{x})=c$ with $c$ denoting an arbitrary constant.

Moreover, if $v(\mathbf{x})$ solves the system (2.21), the function
$v(\mathbf{x})$ can be written in the form of (2.20), which follows from that
the operator $S_{\partial\Omega}^{0}$ is invertible from
$H^{-1/2}(\partial\Omega)$ to $H^{1/2}(\partial\Omega)$. The density function
$\varphi$ will be the kernel of the operator $-1/2+K_{\partial\Omega}^{0,*}$
due to $v(\mathbf{x})$ satisfying the second equation in (2.21) and the jump
formula (2.6). The proof is complete. ∎

######  Lemma 2.4.

The kernel of the operator $-1/2+\mathbf{K}_{\partial\Omega}^{0,*}$ is
nontrivial. The dimension of the kernel is $3$ for $N=2$ and $6$ for $N=3$
(cf. [2]).

###### Proof.

Assume that $\bm{\varphi}$ is the kernel of the operator
$-1/2+\mathbf{K}_{\partial\Omega}^{0,*}$. Thus the function
$\mathbf{u}(\mathbf{x})$ expressed by

$\mathbf{u}(\mathbf{x})=\mathbf{S}_{\partial\Omega}^{0}[\bm{\varphi}],\quad\mathbf{x}\in\Omega,$
(2.22)

solves the following Neumann boundary value problem

$\begin{cases}\mathcal{L}_{\lambda,\mu}\mathbf{u}(\mathbf{x})=0&\mbox{in}\quad\Omega,\\\
T_{\nu}\mathbf{u}(\mathbf{x})=0&\mbox{on}\quad\partial\Omega.\end{cases}$
(2.23)

The dimension of the nontrivial solution to the system (2.23) is $6$ for $N=3$
(cf. [2]).

Moreover, if $\mathbf{u}(\mathbf{x})$ solves the system (2.23), the function
$\mathbf{u}(\mathbf{x})$ can be written in the form of (2.22), which follows
from that the operator $\mathbf{S}_{\partial\Omega}^{0}$ is invertible from
$H^{-1/2}(\partial\Omega)$ to $H^{1/2}(\partial\Omega)$. The density function
$\bm{\varphi}$ will be the kernel of the operator
$-1/2+\mathbf{K}_{\partial\Omega}^{0,*}$ due to $\mathbf{u}(\mathbf{x})$
satisfying the second equation in (2.23) and the jump formula (2.12). The
proof is complete. ∎

For our use in what follows, we give the definition for the characteristic
value [3].

######  Definition 2.1.

Denote by $B$ a Banach space. Let $\mathcal{U}_{z_{0}}$ be the set of all
operator-valued functions with values in $\mathcal{L}(B)$ which are
holomorphic in some neighborhood of $z_{0}$, except possibly at $z_{0}$. The
point $z_{0}$ is called a characteristic value of $A(z)\in\mathcal{U}_{z_{0}}$
if there exists a vector-valued function $\varphi(z)$ with values in $B$ such
that

  1. (1)

$\varphi(z)$ is holomorphic at $z_{0}$ and $\varphi(z_{0})\neq 0$,

  2. (2)

$A(z)\varphi(z)$ is holomorphic at $z_{0}$ and vanishes at this point.

Based on the definition above and Theorem 2.1, one can conclude the following
proposition.

######  Proposition 2.1.

The existence of the transmission eigenvalues for the system (2.3) is
equivalent to the existence of a characteristic value $k$ to the operator
$\mathcal{A}(k,\delta)$ defined in (2.14).

In the next theorem, we show the existence of characteristic values to the
operator $\mathcal{A}(k,\delta)$.

######  Theorem 2.2.

Assume that the parameters introduced in Subsection 2.1 satisfy

$k=o(1),\ \ \delta=o(1),\ \ \tau=\mathcal{O}(1),\ \ \mu=\mathcal{O}(1),\ \
\lambda=\mathcal{O}(1).$ (2.24)

Then, for any small $\delta$, there exists a characteristic value
$k^{*}(\delta)\ll 1$, which depends on $\delta$, to the operator-valued
analytic function $\mathcal{A}(k,\delta)$; that is, there exists a nontrivial
pair $(k^{*},\Phi^{*})$ such that $\mathcal{A}(k^{*},\delta)[\Phi^{*}]=0$.

######  Remark 2.2.

We would like to point out that in (2.2), for any given small $\delta$, we
only prove the existence of at least one AE transmission eigenvalue to (1.4),
though it is generically believed that there should be infinitely many
eigenvalues.

###### Proof of Theorem 2.2.

From the assumption in (2.24), the operator $\mathcal{A}(k,\delta)$ has the
following asymptotic expansion

$\mathcal{A}(k,\delta)=\mathcal{A}_{0}+\mathcal{B},$

where

$\mathcal{A}_{0}=\left(\begin{array}[]{cc}-\frac{I}{2}+K^{*}_{\partial\Omega}&0\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
0&-\frac{I}{2}+{\mathbf{K}}^{*}_{\partial\Omega}\\\ \end{array}\right),$

and $\mathcal{B}=\mathcal{O}(k^{2}+\delta)$ following from Lemmas 2.1 and 2.2.
Moreover, the kernel of the operator $\mathcal{A}_{0}$ is nontrivial. Indeed,
Lemmas 2.3 and 2.4 show that the dimension of the kernel for the operator
$\mathcal{A}_{0}$ is $3$ in two dimensions and $6$ in three dimensions. Thus
$0$ is a characteristic value of the operator $\mathcal{A}(k,0)$. Since the
operator $\mathcal{A}(k,\delta)$ is a Fredholm operator, one can find a curve
$\mathcal{C}$ in $\mathbb{C}$ that encloses the origin point $0$, such that
the operator $\mathcal{A}(k,0)$ is invertible for $k\in\mathcal{C}$. By the
Gohberg-Sigal theory [3], there exists a characteristic value
$k^{*}(\delta)\ll 1$, which is located in the region enclosed by the curve
$\mathcal{C}$, to the operator-valued analytic function
$\mathcal{A}(k,\delta)$. This completes the proof. ∎

######  Remark 2.3.

The assumption in (2.24) states that the size of the object $\Omega$ is
smaller compared with the wavelengths of all the waves in the system: the
acoustic wave and the compressional part as well as the shear part the elastic
wave. In such a case, the air bubble $(\Omega;\kappa,\rho_{e})$ is known as a
nano-bubble, which is widely used in the composite elastic medium theory (cf.
[33]).

Finally, we would like to mention that in the next section, we shall study the
boundary-localization of the AE transmission eigenfunctions, where as a
byproduct we show that within the radial geometry and no restriction on the
medium parameters, there are infinitely many AE transmission eigenvalues.

##  3\. Boundary localization of transmission eigenfunctions

In this section, we study the boundary-localization of the AE transmission
eigenfunctions in 2D and 3D. Let us first focus on the radial case, i.e. we
consider the acoustic-elastic transmission eigenvalue problem (2.3) with
$\Omega$ being a ball in $\mathbb{R}^{N}$, $N=2,3$. Since $\Delta$ and
$\mathcal{L}_{\lambda,\mu}$ are invariant under rigid motions and also noting
(2.2), we can assume that $\Omega$ is the unit ball, namely
$\Omega:=\\{\mathbf{x}\in\mathbb{R}^{N};|\mathbf{x}|<1\\}$. Define

$\Omega_{\epsilon}:=\\{\mathbf{x}\in\mathbb{R}^{N};|\mathbf{x}|<\epsilon\\},\
\epsilon\in(0,1).$ (3.1)

######  Definition 3.1.

Consider a function $\phi\in L^{2}(\Omega)$. It is said to be boundary-
localized if there exists $\epsilon\ll 1$ such that

$\frac{\|\phi\|_{L^{2}(\Omega_{\epsilon})}}{\|\phi\|_{L^{2}(\Omega)}}\ll 1.$
(3.2)

######  Remark 3.1.

It is clear that if $\phi\in L^{2}(\Omega)$ is boundary-localized, its
$L^{2}$-energy concentrate on a small neighbourhood of $\partial\Omega$,
namely $\Omega\backslash\Omega_{\epsilon}$. In what follows, we shall make the
asymptotic expressions in Definition 3.1 more rigorous. In fact, we shall
construct a sequence of eigenfunctions $\\{\phi_{m}\\}_{m\in\mathbb{N}}$ such
that for any given $\epsilon\in(0,1)$, one has

$\lim_{m\rightarrow\infty}\frac{\|\phi_{m}\|_{L^{2}(\Omega_{\epsilon})}}{\|\phi_{m}\|_{L^{2}(\Omega)}}=0.$
(3.3)

In such a case, we simply refer to $\\{\phi_{m}\\}$ as boundary-localized.

By noting (2.2), it is remarked that the p-wavenumber and s–wavenumber
associated with the Lamé operator in (2.3) satisfy that

$\begin{split}k_{p}&=\frac{k\tau}{\sqrt{\lambda+2\mu}}=k\tau,\quad
k_{s}=\frac{k\tau}{\sqrt{\mu}},\end{split}$ (3.4)

where $k,\lambda,\mu$ and $\tau$ are the parameters in (2.3).  It is remarked
that in (3.4) we have used the relation $\sqrt{\lambda+2\mu}=1$ that follows
from (2.2). Moreover, it is noted that the wave speed in elastic media is
larger than that in fluid media, which means that $\tau<1$ from (2.1). Thus in
what follows, we shall confine our study on the case $\tau\in(0,1)$ for
physical relevance.

In the sequel, we let $m\in\mathbb{N}$ be a positive integer,
$J_{m}(|\mathbf{x}|)$ be the first kind Bessel function of order $m$, and
$J^{\prime}_{m}(|\mathbf{x}|)$ be the derivative of $J_{m}(|\mathbf{x}|)$.
Furthermore, we let $j_{m,s}$ denote the $s$-th positive zero of
$J_{m}(|\mathbf{x}|)$, in the mean while, $j^{\prime}_{m,s}$ denote the $s$-th
positive zero of $J^{\prime}_{m}(|\mathbf{x}|)$. From [1], one has

$\begin{split}&m\leq
j^{\prime}_{m,1}<j_{m,1}<j^{\prime}_{m,2}<j_{m,2}<j^{\prime}_{m,3}<j_{m,3}<\ldots,\
J_{m}(|x|)=\frac{(|x|/2)^{m}}{\Gamma(m+1)}\prod_{s=1}^{\infty}\left(1-\frac{|x|^{2}}{j^{2}_{m,s}}\right).\end{split}$
(3.5)

In order to prove the result of boundary-localization, we assume that the
order $m$ of the Bessel function $J_{m}(x)$ is sufficiently large.

###  3.1. Two-dimensional results

In this subsection, we mainly prove the results of boundary-localization in
$\mathbb{R}^{2}$. Let
$\mathbf{x}:=(x_{1},x_{2})=(r\cos\theta,r\sin\theta)\in\mathbb{R}^{2}$ denote
the polar coordinate. Using Fourier expansion, the acoustic-elastic
transmission eigenfunctions to (2.3) associated with the transmission
eigenvalues $k\in\mathbb{R}_{+}$ have series expansions as follow (cf.[37, 5,
10, 16, 17]):

$\begin{split}\mathbf{u}=&\sum_{m=0}^{+\infty}\Biggl{\\{}\alpha_{m}\left(k_{p}J^{\prime}_{m}(k_{p}|\mathbf{x}|)e^{\mathrm{i}m\theta}\cdot\hat{r}+\frac{\mathrm{i}m}{|\mathbf{x}|}J_{m}(k_{p}|\mathbf{x}|)e^{\mathrm{i}m\theta}\cdot\hat{\theta}\right)\\\
&+\gamma_{m}\left(\frac{\mathrm{i}m}{r}J_{m}(k_{s}|\mathbf{x}|)e^{\mathrm{i}m\theta}\cdot\hat{r}-k_{s}J^{\prime}_{m}(k_{s}|\mathbf{x}|)e^{\mathrm{i}m\theta}\cdot\hat{\theta}\right)\Biggr{\\}},\\\
v=&{\sum_{m=0}^{+\infty}\beta_{m}e^{\mathrm{i}m\theta}J_{m}(k|\mathbf{x}|),}\end{split}$
(3.6)

where $\alpha_{m}$, $\beta_{m}$ and $\gamma_{m}$are complex constants, $k_{p}$
is given in (3.4), and

$\hat{r}:=\left(\begin{array}[]{c}\cos\theta\\\ \sin\theta\end{array}\right)\
\quad\mbox{and}\quad\hat{\theta}:=\left(\begin{array}[]{c}-\sin\theta\\\
\cos\theta\end{array}\right).$ (3.7)

Next, we show that there exists a sequence of discrete transmission
eigenvalues of (2.3). Moreover, we prove that infinity is the only
accumulation point of this sequence.

######  Lemma 3.1.

Consider (2.3), where $\Omega\in\mathbb{R}^{2}$ is the unit disk and the
parameter $\tau$ is chosen such that $\tau<j_{m,1}/j_{m,2}<1$. Let
$\\{k_{m,\ell}~{}|~{}\ell=1,2,\ldots\\}$ be the subset of AE transmission
eigenvalues $k$ of (2.3), where $m$ is the order of the Bessel functions and
$\ell$ denotes the $\ell$-th eigenvalue for a fixed $m$. Then there exists a
subsequence of $\\{k_{m,\ell}\\}$, denoted by $\\{k_{m,s(m)}\\}$, such that
for $m$ sufficiently large, it holds that

$k_{m,s}\in\left(j_{m,1},\ j_{m,2}\right),$ (3.8)

Furthermore, it yields that

$\frac{k_{m,s}}{m}\to 1,\quad\mbox{as}\quad m\to+\infty.$ (3.9)

and

$k_{m,s}=m\left(1+C_{0}m^{-\frac{2}{3}}+o(m^{-\frac{2}{3}})\right),$ (3.10)

where $C_{0}$ is a positive constant not depending on $m$.

###### Proof.

Let $k$ be an AE transmission eigenvalue of (2.3). Recall that $k_{p}$ is
defined in (3.4). For a fixed $m\in\mathbb{N}_{+}$, using Fourier expansion,
it is not difficult to see that

$\begin{split}\mathbf{u}_{m}(\mathbf{x})&=\alpha_{m}k_{p}J^{\prime}_{m}(k_{p}|\mathbf{x}|)e^{\mathrm{i}m\theta}\cdot\hat{r}+\alpha_{m}\frac{\mathrm{i}m}{|\mathbf{x}|}J_{m}(k_{p}|\mathbf{x}|)e^{\mathrm{i}m\theta}\cdot\hat{\theta},\
v_{m}(\mathbf{x})=\beta_{m}J_{m}(k|\mathbf{x}|)e^{\mathrm{i}m\theta},\end{split}$
(3.11)

where $\alpha_{m}$ and $\beta_{m}$ are nonzero constants, is a pair solution
to
$\mathcal{L}_{\lambda,\mu}{\mathbf{u}}(\mathbf{x})+k^{2}\tau^{2}{\mathbf{u}}(\mathbf{x})=\mathbf{0}$
and $\Delta v(\mathbf{x})+k^{2}v(\mathbf{x})=0$ in $\Omega$. Hence, by the
first transmission condition of (2.3) on $\partial\Omega$, one can arrive at
that

${\mathbf{u}}_{m}(\mathbf{x})\cdot\nu-\frac{1}{k^{2}}\nabla
v_{m}(\mathbf{x})\cdot\nu={\mathbf{u}}_{m}\cdot\hat{r}-\frac{1}{k^{2}}\frac{\partial
v_{m}}{\partial
r}=\alpha_{m}k_{p}J^{\prime}_{m}(k_{p}r)e^{\mathrm{i}m\theta}-\frac{1}{k}\beta_{m}J^{\prime}_{m}(kr)e^{\mathrm{i}m\theta}=0.$

It is ready to know that

$\beta_{m}=\alpha_{m}k_{p}k\frac{J^{\prime}_{m}(k_{p})}{J^{\prime}_{m}(k)}.$
(3.12)

it is obvious that $\alpha_{m}$ and $\beta_{m}$ are not zero by contradiction.
By direct calculation, we can obtain that

$\nabla\cdot{\mathbf{u}}_{m}=\alpha_{m}k_{p}^{2}J^{\prime\prime}_{m}(k_{p}r)e^{\mathrm{i}m\theta}+\alpha_{m}\frac{k_{p}}{r}J^{\prime}_{m}(k_{p}r)e^{\mathrm{i}m\theta}-\alpha_{m}\frac{m^{2}}{r^{2}}J_{m}(k_{p}r)e^{\mathrm{i}m\theta},$
(3.13)

and

$\begin{split}\frac{\partial u_{m,1}}{\partial
x_{2}}=&\alpha_{m}k_{p}^{2}J^{\prime\prime}_{m}(k_{p}r)e^{\mathrm{i}m\theta}\cos\theta\sin\theta+\alpha_{m}\frac{{\mathrm{i}}m}{r^{2}}J_{m}(k_{p}r)e^{\mathrm{i}m\theta}\sin^{2}\theta-\alpha_{m}\frac{{\mathrm{i}}m}{r}k_{p}J^{\prime}_{m}(k_{p}r)\\\
&\times
e^{\mathrm{i}m\theta}\sin^{2}\theta+\alpha_{m}\frac{{\mathrm{i}}m}{r}k_{p}J^{\prime}_{m}(k_{p}r)e^{\mathrm{i}m\theta}\cos^{2}\theta-\alpha_{m}\frac{k_{p}}{r}J^{\prime}_{m}(k_{p}r)e^{\mathrm{i}m\theta}\sin\theta\cos\theta\\\
&+\alpha_{m}\frac{m^{2}}{r^{2}}J_{m}(k_{p}r)e^{\mathrm{i}m\theta}\sin\theta\cos\theta-\alpha_{m}\frac{{\mathrm{i}}m}{r^{2}}J_{m}(k_{p}r)e^{\mathrm{i}m\theta}\cos^{2}\theta=\frac{\partial
u_{m,2}}{\partial x_{1}}.\end{split}$ (3.14)

By using the second transmission condition of (2.3) on $\partial\Omega$, one
has

$\begin{split}&\left(\lambda(\nabla\cdot{\mathbf{u}}_{m})I_{2}+2\mu)\nabla^{s}{\mathbf{u}}_{m}+\delta\tau^{2}v_{m}\right)\cdot\hat{r}\\\
=&\lambda\left(\nabla\cdot{\mathbf{u}}_{m}\right)\left(\begin{array}[]{c}\cos\theta\\\
\sin\theta\end{array}\right)+2\mu\left(\begin{array}[]{c}\frac{\partial
u_{m,1}}{\partial x_{1}}\cos\theta+\frac{\partial u_{m,1}}{\partial
x_{2}}\sin\theta\\\ \frac{\partial u_{m,2}}{\partial
x_{1}}\cos\theta+\frac{\partial u_{m,2}}{\partial
x_{2}}\sin\theta\end{array}\right)+\delta\tau^{2}v_{m}\left(\begin{array}[]{c}\cos\theta\\\
\sin\theta\end{array}\right)=0.\end{split}$ (3.15)

Comparing the coefficients of the functions $\sin\theta$ and $\cos\theta$ in
(3.15) and combining equations (3.11) to (3.14), there hold

$J_{m}(k_{p})=k_{p}J^{\prime}_{m}(k_{p}),$ (3.16)

and

$-k_{p}^{2}J_{m}(k_{p})J^{\prime}_{m}(k)-2\mu
k_{p}J^{\prime}_{m}(k_{p})J^{\prime}_{m}(k)+2\mu
m^{2}J_{m}(k_{p})J^{\prime}_{m}(k)+\delta\tau^{2}kk_{p}J^{\prime}_{m}(k_{p})J_{m}(k)=0.$
(3.17)

Now, we want to find a $k$ that satisfies both (3.16) and (3.17). Instead of
directly solving the two equations, we first substitute (3.16) into (3.17),
one has

$\left[-k_{p}^{2}J^{\prime}_{m}(k)+2\mu m^{2}J^{\prime}_{m}(k)-2\mu
J^{\prime}_{m}(k)+\delta\tau^{2}kJ_{m}(k)\right]J_{m}(k_{p})=0.$ (3.18)

Apparently, if $k$ solves the equations (3.16) and (3.17), the equation (3.18)
holds. On the other hand, if $k$ is the root of the equations (3.16) and
(3.18), $k$ solves the equation (3.17). Thus, in the rest of the proof, we
shall confine ourselves on finding the root $k$ of the equation (3.18). Then
we will show that the root $k$ also solves (3.16) in the asymptotic sense and
please refer to Remark 3.2.

By utilizing
$J^{\prime}_{m}(|\mathbf{x}|)=J_{m-1}(|\mathbf{x}|)-\frac{m}{|\mathbf{x}|}J_{m}(|\mathbf{x}|)$
and (3.4), the root $k$ of (3.18) coincides with that of the following
function:

$\begin{split}f(k):=&\left[-k^{2}\tau^{2}+2\mu(m^{2}-1)\right]J_{m-1}(k)J_{m}(\tau
k)\\\
&+\Big{[}k\tau^{2}m-\frac{2\mu(m^{2}-1)m}{k}+\delta\tau^{2}k\Big{]}J_{m}(k)J_{m}(\tau
k).\end{split}$ (3.19)

Next, we compute the following quantity:

$\begin{split}f_{m}(j_{m,1})f_{m}(j_{m,2})=&\left[-j_{m,1}^{2}\tau^{2}+2\mu(m^{2}-1)\right]\left[-j_{m,2}^{2}\tau^{2}+2\mu(m^{2}-1)\right]\\\
&\times J_{m-1}(j_{m,1})J_{m-1}(j_{m,2})J_{m}(\tau j_{m,1})J_{m}(\tau
j_{m,2})\\\ \end{split}$ (3.20)

Suppose that $a_{s}$ is the $s$th negative zero of the Airy function. From
[35], we have $a_{s}=-[3\pi/8(4s-1)]^{2/3}(1+\Upsilon_{s})$, where
$\Upsilon_{s}\in[0,\,0.13(3\pi/8(4s-1.051))^{-1}]$. The positive root of the
Bessel function can have the following sharp lower and upper bound. Indeed,
from [36] one knows that

$m\left(1-\frac{a_{s}}{(2m^{2})^{1/3}}\right)<j_{m,s}<m\left(1-\frac{a_{s}}{(2m^{2})^{1/3}}+\frac{3a_{s}^{2}}{20}\left(\frac{2}{m^{4}}\right)^{1/3}\right),$
(3.21)

where $j_{m,s}$ is the $s$th positive root of $J_{m}(|\mathbf{x}|)$. Hence,
when $m$ is sufficiently large, there hold that

$\begin{split}j_{m,1}&=m\Big{(}1+C_{1}m^{-2/3}+o(m^{-2/3})\Big{)},\\\
j_{m,2}&=m\Big{(}1+C_{2}m^{-2/3}+o(m^{-2/3})\Big{)},\end{split}$ (3.22)

where $C_{i},i=1,2$ is a positive constant not depending on $m$. By virtue of
(3.22), when $m$ is sufficient large, we have

$\begin{split}&\left[-j_{m,1}^{2}\tau^{2}+2\mu(m^{2}-1)\right]\left[-j_{m,2}^{2}\tau^{2}+2\mu(m^{2}-1)\right]\\\
&=m^{4}\,\Pi_{i=1}^{2}\left[\tau^{2}-2\mu+\tau^{2}C_{i}m^{-2/3}+o(m^{-2/3})+2\mu
m^{-2}\right]>0\end{split}$ (3.23)

for any positive constants $\tau$ and $\mu$. Moreover, from the choice of
$\tau j_{m,2}<j_{m,1}$, the following inequality holds

$J_{m}(\tau j_{m,1})J_{m}(\tau j_{m,2})>0.$ (3.24)

Since the positive zeros of $J_{m-1}$ are interlaced with those of $J_{m}$, we
have

$J_{m-1}(j_{m,1})J_{m-1}(j_{m,2})<0.$

By using the above facts, we derive that

$\begin{split}f_{m}(j_{m,1})f_{m}(j_{m,2})<0,\end{split}$ (3.25)

which implies that $k_{m,s}\in\left(j_{m,1},j_{m,2}\right)$.

Finally, using (3.22) one has (3.9) and (3.10). The proof is complete. ∎

######  Remark 3.2.

Here we remark that if $k_{m,s}$ solves (3.18), then $k_{m,s}$ solves the
equation (3.16) in the asymptotic sense. Indeed, if $k_{m,s}$ is a root of
(3.18), for sufficiently large $m$ and $\tau<j_{m,1}/j_{m,2}<1$, using
(3.8),(3.9) and the following asymptotic expansion (cf.[26])

$J_{m}(k_{m,s}r)=\frac{z^{m}e^{m\sqrt{1-z^{2}}}}{(2\pi
m)^{1/2}(1-z^{2})^{1/4}(1+\sqrt{1-z^{2}})^{m}}\left(1+o(1)\right),\
z=\frac{k_{m,s}r}{m},\ 0<r<1,$ (3.26)

we derive that

$J_{m}(k_{p})-k_{p}J^{\prime}_{m}(k_{p})=Cm^{\frac{1}{2}}\left(\frac{\tau
e^{\sqrt{1-\tau^{2}}}}{1+\sqrt{1-\tau^{2}}}\right)^{m-1}\rightarrow
0\quad\mbox{as}\quad m\rightarrow\infty,$ (3.27)

where C is a constant. In the derivation of the last inequality, we have used
the fact $0<\frac{\tau e^{\sqrt{1-\tau^{2}}}}{1+\sqrt{1-\tau^{2}}}<1$ due to
$\tau<1$ . Thus, $k_{m,s}$ solves (3.16) in the asymptotic sense.

Next, we shall prove that the AE transmission eigenfunction $v_{m}$ associated
with transmission eigenvalue $k_{m,s}$ given by Lemma 3.1 is boundary-
localized.

######  Theorem 3.1.

Consider the AE transmission eigenvalue problem (2.3). Let $\Omega$ be the
unit disk in $\mathbb{R}^{2}$ and $\Omega_{\epsilon}$ be given in (3.1). For
any fixed $\epsilon\in(0,1)$, there exist transmission eigenfunctions $v_{m}$,
$m\in\mathbb{N}$, associated with the eigenvalues $k_{m,s}$ of (2.3) described
in Lemma 3.1 such that

$\lim_{m\to\infty}\frac{\|v_{m}\|_{L^{2}(\Omega_{\epsilon})}}{\|v_{m}\|_{L^{2}(\Omega)}}=0.$
(3.28)

###### Proof.

Let the acoustic-elastic transmission eigenfunction $v_{m}$ associated with
the transmission eigenvalue $k_{m,s}$ be given in the form (3.11), where
$k_{m,s}$ fulfils (3.8). Hence by direct calculations, it yields that

$\begin{split}\|v_{m}\|^{2}_{L^{2}(\Omega_{\epsilon})}&=|\beta_{m}|^{2}\int_{\Omega_{\epsilon}}|J_{m}(k_{m,s}|\mathbf{x}|)|^{2}d\mathbf{x}=2\pi|\beta_{m}|^{2}\int_{0}^{\epsilon}rJ^{2}_{m}(k_{m,s}r)dr,\\\
\|v_{m}\|^{2}_{L^{2}(\Omega)}&=|\beta_{m}|^{2}\int_{\Omega}|J_{m}(k_{m,s}|\mathbf{x}|)|^{2}d\mathbf{x}=2\pi|\beta_{m}|^{2}\int_{0}^{1}rJ^{2}_{m}(k_{m,s}r)dr.\end{split}$
(3.29)

Recall that $j^{\prime}_{m,1}$ is the first zero of
$J^{\prime}_{m}(|\mathbf{x}|)$. According to (3.5) and (3.21), it readily
yields that

$j^{\prime}_{m,1}=m(1+O(m^{-2/3}))\quad\mbox{when}\quad m\rightarrow\infty.$
(3.30)

Due to (3.10) and (3.30), for sufficiently large $m$, one can claim that

$k_{m,s}r<j^{\prime}_{m,1},\quad r\in(0,1).$

Denote $H_{m}(t)=J_{m}(k_{m,s}r)$, where $r\in(0,1)$ and
$t=k_{m,s}r\in(0,j^{\prime}_{m,1})$. By virtue of (3.5), $J_{m}(t)\geq 0$ for
$t\in[0,j^{\prime}_{m,1}]$ and $j^{\prime}_{m,1}$ is the first maximum point
of $J_{m}(t)$. Hence $H_{m}(t)$ is monotonically increasing in $(0,1)$, which
implies that $J^{\prime}_{m}(k_{m,s}r)>0$ for $0<r<1$.

Hence, using (3.26) and (3.29), we can obtain the asymptotic upper bound for
the integral as follows

$\begin{split}&\int_{0}^{\epsilon}rJ^{2}_{m}(k_{m,s}r)dr\leqslant\epsilon^{2}J^{2}_{m}(k_{m,s}\epsilon)\\\
&=\frac{\epsilon^{2}}{2\pi
m\left(1-(\frac{k_{m,s}\epsilon}{m})^{2}\right)^{1/2}}\left(\frac{k_{m,s}\epsilon}{m}\frac{e^{\sqrt{1-(\frac{k_{m,s}\epsilon}{m})^{2}}}}{1+\sqrt{1-(\frac{k_{m,s}\epsilon}{m})^{2}}}\right)^{2m}\left(1+o(1)\right)^{2},\end{split}$
(3.31)

when $m$ is sufficient large. Furthermore, in view of (3.30), we denote

$\Xi=\frac{j^{\prime}_{m,1}}{k_{m,s}}.$ (3.32)

One has $\epsilon<\Xi<1$ when $m$ is sufficiently large. Hence, it yields that
$m<j^{\prime}_{m,1}=k_{m,s}r_{1}$. Using (3.10), (3.30) and the following
asymptotic formula:

$J_{m}(|\mathbf{x}|)=\sqrt{\frac{2}{\pi\sqrt{|\mathbf{x}|^{2}-m^{2}}}}\cos\left(\sqrt{|\mathbf{x}|^{2}-m^{2}}-\frac{m\pi}{2}+m\arcsin(\frac{m}{|\mathbf{x}|})-\frac{\pi}{4}\right)\left(1+o(1)\right),$
(3.33)

for sufficient large $m$, we can deduce that

$\begin{split}\int_{0}^{1}rJ^{2}_{m}(k_{m,s}r)dr\geqslant&\int_{\Xi}^{1}rJ^{2}_{m}(k_{m,s}r)dr=\frac{2(1+o(1))^{2}}{\pi}\int_{\Xi}^{1}\frac{r\cos^{2}x_{m}}{\left(k^{2}_{m,s}r^{2}-m^{2}\right)^{1/2}}dr\\\
\sim&\frac{1}{\pi}\int_{\Xi}^{1}\frac{r}{\left(k^{2}_{m,s}r^{2}-m^{2}\right)^{1/2}}dr\\\
\sim&\frac{1}{\pi
k^{2}_{m,s}}\Big{(}\sqrt{k_{m,s}^{2}-m^{2}}-\sqrt{j_{m,1}^{\prime
2}-m^{2}}\Big{)}.\end{split}$ (3.34)

where $x_{m}=\sqrt{{(\tau
k_{m,s}r)}^{2}-m^{2}}-\frac{m\pi}{2}+m\arcsin(\frac{m}{\tau
k_{m,s}r})-\frac{\pi}{4}$.

Combining (LABEL:vmeps1) and (3.34), there holds

$\begin{split}\frac{\|v_{m}\|^{2}_{L^{2}(\Omega_{\epsilon})}}{\|v_{m}\|^{2}_{L^{2}(\Omega)}}\leqslant&Cm^{\frac{1}{3}}\left(1-\epsilon^{2}\right)^{-\frac{1}{2}}\left(\frac{\epsilon
e^{\sqrt{1-\epsilon^{2}}}}{1+\sqrt{1-\epsilon^{2}}}\right)^{2m}\end{split}$
(3.35)

where $C$ is a constant. When $m$ is sufficiently large, let $m\to\infty$ one
has (3.28).

The proof is complete. ∎

######  Theorem 3.2.

Consider (2.3), where $\Omega$ is the unit disk in $\mathbb{R}^{2}$ and
$\Omega_{\epsilon}$ is given in (3.1). If the physical parameter $\tau$ in
(2.3) fulfilling $\tau\in(0,1)$, then for any $\epsilon\in(0,1)$, there exists
eigenfunctions $\mathbf{u}_{m}$, $m\in\mathbb{N}$, associated with
transmission eigenvalues $k_{m,s}$ described in Lemma 3.1 such that

$\lim_{m\to\infty}\frac{\|\mathbf{u}_{m}\|_{L^{2}(\Omega_{\epsilon})}}{\|\mathbf{u}_{m}\|_{L^{2}(\Omega)}}=0.$
(3.36)

###### Proof.

Suppose that the AE transmission eigenfunction ${\mathbf{u}}_{m}$ associated
with the transmission eigenvalue $k_{m,s}$ is given in the form (3.11), where
$k_{m,s}$ satisfies (3.8). By direct calculations and (3.4), it yields that

$\displaystyle\|\mathbf{u}_{m}\|_{L^{2}(\Omega)}$
$\displaystyle=2\pi|\alpha_{m}|^{2}\int_{0}^{1}\left(k_{p}^{2}rJ_{m}^{\prime
2}(k_{p}r)+\frac{m^{2}}{r}J_{m}^{2}(k_{p}r)\right)dr$ (3.37a)
$\displaystyle=2\pi|\alpha_{m}|^{2}\int_{0}^{1}\left(\tau^{2}k^{2}_{m,s}rJ_{m}^{\prime
2}(\tau k_{m,s}r)+\frac{m^{2}}{r}J_{m}^{2}(\tau k_{m,s}r)\right)dr,$
$\displaystyle\|\mathbf{u}_{m}\|_{L^{2}(\Omega_{\epsilon})}$
$\displaystyle=2\pi|\alpha_{m}|^{2}\int_{0}^{\epsilon}\left(\tau^{2}k^{2}_{m,s}rJ_{m}^{\prime
2}(\tau k_{m,s}r)+\frac{m^{2}}{r}J_{m}^{2}(\tau k_{m,s}r)\right)dr.$ (3.37b)

From [35], one has

$\displaystyle|J^{\prime}_{m}(mx)|$
$\displaystyle\leqslant\frac{\left(1+x^{2}\right)^{1/4}}{x\left(2\pi
m\right)^{1/2}}\frac{x^{m}e^{m\sqrt{1-x^{2}}}}{\left(1+\sqrt{1-x^{2}}\right)^{m}},\quad\mbox{for}\quad
m>0\quad\mbox{and}\quad 0<x\leqslant 1,$ (3.38a) $\displaystyle|J_{m}(mx)|$
$\displaystyle\leqslant\frac{x^{m}e^{m\sqrt{1-x^{2}}}}{\left(1+\sqrt{1-x^{2}}\right)^{m}},\quad\mbox{for}\quad
m\geqslant 0\quad\mbox{and}\quad 0<x\leqslant 1.$ (3.38b)

Under the assumption $\tau\leqslant 1$, by virtue of (3.10), if $m$ is
sufficient large, it yields that

$\frac{\tau k_{m,s}r}{m}<1,\quad\forall r\in[0,\epsilon].$

Therefore, due to (3.38a) and (3.38b), after tedious calculations we can
derive that

$\displaystyle\int_{0}^{\epsilon}\left(\tau^{2}k^{2}_{m,s}rJ_{m}^{\prime
2}(\tau k_{m,s}r)+\frac{m^{2}}{r}J_{m}^{2}(\tau k_{m,s}r)\right)dr$ (3.39)
$\displaystyle\leqslant$
$\displaystyle\int_{0}^{\epsilon}\left(\frac{\sqrt{1+(\frac{\tau
k_{m,s}r}{m})^{2}}}{2\pi}+m\right)\left(\frac{\tau
k_{m,s}e^{\sqrt{1-(\frac{\tau k_{m,s}r}{m})^{2}}}}{1+\sqrt{1-(\frac{\tau
k_{m,s}r}{m})^{2}}}\right)\left(\frac{\tau
k_{m,s}r}{m}\frac{e^{\sqrt{1-(\frac{\tau
k_{m,s}r}{m})^{2}}}}{1+\sqrt{1-(\frac{\tau
k_{m,s}r}{m})^{2}}}\right)^{2m-1}dr$ $\displaystyle=$
$\displaystyle\epsilon\cdot\left(\frac{\sqrt{1+(\frac{\tau
k_{m,s}\eta_{1}}{m})^{2}}}{2\pi}+m\right)\left(\frac{\tau
k_{m,s}e^{\sqrt{1-(\frac{\tau
k_{m,s}\eta_{1}}{m})^{2}}}}{1+\sqrt{1-(\frac{\tau
k_{m,s}\eta_{1}}{m})^{2}}}\right)\left(\frac{\tau
k_{m,s}\eta_{1}}{m}\frac{e^{\sqrt{1-(\frac{\tau
k_{m,s}\eta_{1}}{m})^{2}}}}{1+\sqrt{1-(\frac{\tau
k_{m,s}\eta_{1}}{m})^{2}}}\right)^{2m-1}$

where $\eta_{1}\in[0,\epsilon]$.

When $\tau<1$, recall that $\Xi$ is given in (3.32), using (3.26) we can
obtain the following inequality

$\begin{split}&\int_{0}^{1}\left(\tau^{2}k^{2}_{m,s}rJ_{m}^{\prime 2}(\tau
k_{m,s}r)+\frac{m^{2}}{r}J_{m}^{2}(\tau k_{m,s}r)\right)dr\\\
\geqslant&\int_{\Xi}^{1}\frac{m^{2}}{r}\frac{1}{2\pi m\sqrt{1-(\frac{\tau
k_{m,s}r}{m})^{2}}}\left(\frac{\tau k_{m,s}r}{m}\frac{e^{\sqrt{1-(\frac{\tau
k_{m,s}r}{m})^{2}}}}{1+\sqrt{1-(\frac{\tau
k_{m,s}r}{m})^{2}}}\right)^{2m}dr\cdot(1+o(1))^{2}\\\
\geqslant&\int_{\Xi}^{1}\frac{1}{2\pi\sqrt{1-(\frac{\tau
k_{m,s}r}{m})^{2}}}\left(\frac{\tau k_{m,s}e^{\sqrt{1-(\frac{\tau
k_{m,s}r}{m})^{2}}}}{1+\sqrt{1-(\frac{\tau
k_{m,s}r}{m})^{2}}}\right)\left(\frac{\tau
k_{m,s}r}{m}\frac{e^{\sqrt{1-(\frac{\tau
k_{m,s}r}{m})^{2}}}}{1+\sqrt{1-(\frac{\tau
k_{m,s}r}{m})^{2}}}\right)^{2m-1}dr\\\ =&\frac{1-\Xi}{2\pi\sqrt{1-(\frac{\tau
k_{m,s}\eta_{2}}{m})^{2}}}\left(\frac{\tau k_{m,s}e^{\sqrt{1-(\frac{\tau
k_{m,s}\eta_{2}}{m})^{2}}}}{1+\sqrt{1-(\frac{\tau
k_{m,s}\eta_{2}}{m})^{2}}}\right)\left(\frac{\tau
k_{m,s}\eta_{2}}{m}\frac{e^{\sqrt{1-(\frac{\tau
k_{m,s}\eta_{2}}{m})^{2}}}}{1+\sqrt{1-(\frac{\tau
k_{m,s}\eta_{2}}{m})^{2}}}\right)^{2m-1},\end{split}$ (3.40)

where $\eta_{2}\in[r_{1},1]$ and $\eta_{2}\to 1$ as $m\to\infty$.

Now, we introduce the auxiliary function

$\displaystyle\phi(x)=\frac{xe^{\sqrt{1-x^{2}}}}{1+\sqrt{1-x^{2}}},$ (3.41)

which is monotonically increasing for $x\in(0,1)$. Noting that
$\lim_{m\to\infty}\left(\frac{\tau k_{m,s}\eta_{2}}{m}-\frac{\tau
k_{m,s}\eta_{1}}{m}\right)=\tau(1-\eta_{1})>0$. Therefore there exists
$\delta(\eta_{1})\in(0,1)$ such that

$0<\frac{\phi\left(\frac{\tau k_{m,s}\eta_{1}}{m}\right)}{\phi\left(\frac{\tau
k_{m,s}\eta_{2}}{m}\right)}<1-\delta(\eta_{1})$ (3.42)

for sufficiently large $m$. Combining (3.37a), (3.37b), (3.39) with
(LABEL:int2), there holds

$\frac{\|\mathbf{u}_{m}\|_{L^{2}(\Omega_{\epsilon})}}{\|\mathbf{u}_{m}\|_{L^{2}(\Omega)}}\leqslant
C\epsilon m^{\frac{5}{3}}(1-\delta(\eta_{1}))^{2m-1},$ (3.43)

where $C$ is a constant. Finally, letting $m\to\infty$ one has (3.36).

The proof is complete. ∎

###  3.2. Three-dimensional results

In this subsection, we mainly prove the results of boundary-localization of
transmission eigenfunctions to (2.3) in $\mathbb{R}^{3}$. Using Fourier
expansion, the acoustic-elastic transmission eigenfunctions to (2.3)
associated with the transmission eigenvalues $k\in\mathbb{R}_{+}$ have series
expansions as follow (cf.[11, 6, 7, 10]):

$\displaystyle\mathbf{u}=\sum_{m=0}^{\infty}\sum_{l=-m}^{m}$
$\displaystyle\Big{[}\alpha_{m}^{l}j^{\prime}_{m}(k_{p}|\mathbf{x}|)e^{\mathrm{i}m\varphi}P^{|l|}_{m}(\cos\theta)\cdot\hat{r}+\alpha_{m}^{l}\frac{j_{m}(k_{p}|\mathbf{x}|)}{k_{p}|\mathbf{x}|}e^{\mathrm{i}m\varphi}\frac{dP^{|l|}_{m}(\cos\theta)}{d\theta}\cdot\hat{\theta}$
$\displaystyle+\alpha_{m}^{l}\frac{j_{m}(k_{p}|\mathbf{x}|)}{k_{p}|\mathbf{x}|}\frac{\mathrm{i}m}{\sin\theta}e^{\mathrm{i}m\varphi}P^{|l|}_{m}(\cos\theta)\cdot\hat{\varphi}\Big{]}$
$\displaystyle+\sum_{m=1}^{\infty}\sum_{l=-m}^{m}$
$\displaystyle\Big{[}\delta_{m}^{l}j_{m}(k_{s}|\mathbf{x}|)\frac{\mathrm{i}m}{\sin\theta}e^{\mathrm{i}m\varphi}P^{|l|}_{m}(\cos\theta)\cdot\hat{\theta}-\delta_{m}^{l}j_{m}(k_{s}|\mathbf{x}|)e^{\mathrm{i}m\varphi}\frac{dP^{|l|}_{m}(\cos\theta)}{d\theta}\cdot\hat{\varphi}\Big{]}$
$\displaystyle+\sum_{m=1}^{\infty}\sum_{l=-m}^{m}$
$\displaystyle\Big{[}\gamma_{m}^{l}\frac{j_{m}(k_{s}|\mathbf{x}|)}{k_{s}|\mathbf{x}|}e^{\mathrm{i}m\varphi}P^{|l|}_{m}(\cos\theta)\cdot\hat{r}$
$\displaystyle+\gamma_{m}^{l}\Big{(}j^{\prime}(k_{s}|\mathbf{x}|)+\frac{j_{m}(k_{s}|\mathbf{x}|)}{k_{s}|\mathbf{x}|}\Big{)}\Big{(}e^{\mathrm{i}m\varphi}\frac{dP^{|l|}_{m}(\cos\theta)}{d\theta}\cdot\hat{\theta}+\frac{\mathrm{i}m}{\sin\theta}e^{\mathrm{i}m\varphi}P^{|l|}_{m}(\cos\theta)\cdot\hat{\varphi}\Big{)}\Big{]},$
$\displaystyle v=\sum_{m=0}^{\infty}\sum_{l=-m}^{m}$
$\displaystyle\mathrm{i}^{m}\beta_{m}^{l}j_{m}(k|\mathbf{x}|)Y^{l}_{m}(\hat{r})=\sum_{m=0}^{\infty}\sum_{l=-m}^{m}\sqrt{\dfrac{2n+1}{4\pi}\dfrac{(m-|l|)!}{(m+|l|)!}}\mathrm{i}^{m}\beta_{m}^{l}j_{m}(k|\mathbf{x}|)e^{\mathrm{i}m\varphi}P^{|l|}_{m}(\cos\theta),$
(3.44)

where $\alpha_{m}^{l}$, $\beta_{m}^{l}$, $\delta_{m}^{l}$ and $\gamma_{m}^{l}$
are complex constants,

$\hat{r}:=\left(\begin{array}[]{c}\sin\theta\cos\varphi\\\
\sin\theta\sin\varphi\\\
\cos\theta\end{array}\right),\quad\hat{\theta}:=\left(\begin{array}[]{c}\cos\theta\cos\varphi\\\
\cos\theta\sin\varphi\\\
-\sin\theta\end{array}\right),\quad\hat{\varphi}:=\left(\begin{array}[]{c}-\sin\varphi\\\
\cos\varphi\\\ 0\end{array}\right),$ (3.45)

and $P^{|l|}_{m}$ is the associated Legendre function of degree $m$ and order
$l$. The spherical Bessel function can be characterized by

$j_{m}(|\mathbf{x}|)=\sqrt{\frac{\pi}{2|\mathbf{x}|}}J_{m+1/2}(|\mathbf{x}|).$
(3.46)

Similar to Lemma 3.1, in the following lemma we prove that there exists a
sequence of discrete transmission eigenvalues of (2.3), where infinity is the
only accumulation point of this sequence.

######  Lemma 3.2.

Consider the acoustic-elastic transmission eigenvalue problem (2.3). Let
$\Omega\subset\mathbb{R}^{3}$ be the unit ball and $\tau$ is chosen such that
$\tau<j_{m+\frac{1}{2},1}/j_{m+\frac{1}{2},2}<1$. Let
$k_{m,\ell},\ell=1,2,\ldots,$ be the transmission eigenvalues of (2.3), where
$m$ is the order of the spherical Bessel function and $\ell$ denotes the
$\ell$th eigenvalue for a fixed $m$. Then there exists a subsequence of
$\\{k_{m,\ell}\\}$, denoted by $\\{k_{m,s(m)}\\}$, such that for $m$
sufficiently large, it holds that

$k_{m,s}\in\left(j_{m+\frac{1}{2},1},\ j_{m+\frac{1}{2},2}\right),$ (3.47)

Furthermore, it yields that

$\frac{k_{m,s}}{m+\frac{1}{2}}\to 1,\quad\mbox{as}\quad m\to+\infty.$ (3.48)

More specifically,

$k_{m,s}=\left(m+\frac{1}{2}\right)\left(1+C_{0}\left(m+\frac{1}{2}\right)^{-\frac{2}{3}}+o((m+\frac{1}{2})^{-\frac{2}{3}})\right),$
(3.49)

where $C_{0}$ is a constant independent of $m$.

###### Proof.

Let $m\in\mathbb{N}_{+}$ be fixed. For any nonzero constants $\alpha_{m}^{l}$
and $\beta_{m}^{l}$, it can be verified that

$\begin{split}\mathbf{u}_{m}(\mathbf{x})=&\alpha_{m}^{l}j^{\prime}_{m}(k_{p}|\mathbf{x}|)e^{\mathrm{i}m\varphi}P^{|l|}_{m}(\cos\theta)\cdot\hat{r}+\alpha_{m}^{l}\frac{j_{m}(k_{p}|\mathbf{x}|)}{k_{p}|\mathbf{x}|}e^{\mathrm{i}m\varphi}\frac{dP^{|l|}_{m}(\cos\theta)}{d\theta}\cdot\hat{\theta}\\\
&+\alpha_{m}^{l}\frac{j_{m}(k_{p}|\mathbf{x}|)}{k_{p}|\mathbf{x}|}\frac{\mathrm{i}m}{\sin\theta}e^{\mathrm{i}m\varphi}P^{|l|}_{m}(\cos\theta)\cdot\hat{\varphi},\\\
v_{m}(\mathbf{x})=&\beta_{m}^{l}j_{m}(k|\mathbf{x}|)e^{\mathrm{i}m\varphi}P^{|l|}_{m}(\cos\theta),\end{split}$
(3.50)

are solutions to
$\mathcal{L}_{\lambda,\mu}{\mathbf{u}}(\mathbf{x})+k^{2}\tau^{2}{\mathbf{u}}(\mathbf{x})=\mathbf{0}$
and $\Delta v(\mathbf{x})+k^{2}v(\mathbf{x})=0$ in $\Omega$. Using the
transmission conditions of (2.3) on $\partial\Omega$, we have

$\begin{split}{\mathbf{u}}_{m}(\mathbf{x})\cdot\nu-\frac{1}{k^{2}}\nabla
v_{m}(\mathbf{x})\cdot\nu&={\mathbf{u}}_{m}\cdot\hat{r}-\frac{1}{k^{2}}\frac{\partial
v_{m}}{\partial r}\\\
&=\alpha_{m}^{l}j^{\prime}_{m}(k_{p}r)e^{\mathrm{i}m\theta}P^{|l|}_{m}-\frac{1}{k}\beta_{m}^{l}j^{\prime}_{m}(kr)e^{\mathrm{i}m\theta}P^{|l|}_{m}=0,\end{split}$
(3.51)

where $r=|\mathbf{x}|$. Since $r=1$, it yields that

$\beta_{m}^{l}=\alpha_{m}^{l}k\frac{j^{\prime}_{m}(k_{p})}{j^{\prime}_{m}(k)}.$
(3.52)

it is obvious that $\alpha_{m}^{l}$ and $\beta_{m}^{l}$ are not zero by
contradiction. From direct calculations, we can obtain that

$\displaystyle\nabla\cdot{\mathbf{u}}_{m}=$
$\displaystyle\alpha_{m}^{l}k_{p}j^{\prime\prime}_{m}(k_{p}r)e^{\mathrm{i}m\varphi}P^{|l|}_{m}+2\alpha_{m}^{l}\frac{j^{\prime}_{m}(r)}{k_{p}r}e^{\mathrm{i}m\varphi}P^{|l|}_{m}+\alpha_{m}^{l}\frac{j_{m}(k_{p}r)}{k_{p}r^{2}}e^{\mathrm{i}m\varphi}\frac{d^{2}P^{|l|}_{m}}{d\theta^{2}}$
$\displaystyle+\alpha_{m}^{l}\frac{j_{m}(k_{p}r)}{k_{p}r^{2}}\frac{\cos\theta}{\sin\theta}e^{\mathrm{i}m\varphi}\frac{dP^{|l|}_{m}}{d\theta}-\alpha_{m}^{l}\frac{j_{m}(k_{p}r)}{k_{p}r^{2}}\frac{m^{2}}{\sin^{2}\theta}e^{\mathrm{i}m\varphi}P^{|l|}_{m},$
(3.53) $\displaystyle\nabla^{s}{\mathbf{u}}_{m}\cdot\hat{r}=$
$\displaystyle\frac{1}{2}\left(\nabla{\mathbf{u}}_{m}+\nabla{\mathbf{u}}_{m}^{t}\right)\cdot\hat{r}=\alpha_{m}^{l}k_{p}j^{\prime\prime}_{m}(k_{p}r)e^{\mathrm{i}m\varphi}P^{|l|}_{m}\cdot\hat{r}$
(3.54)
$\displaystyle+\Big{(}\alpha_{m}^{l}\frac{j^{\prime}_{m}(k_{p}r)}{k_{p}r}e^{\mathrm{i}m\varphi}\frac{dP^{|l|}_{m}}{d\theta}-\alpha_{m}^{l}\frac{j_{m}(k_{p}r)}{k_{p}r^{2}}e^{\mathrm{i}m\varphi}\frac{dP^{|l|}_{m}}{d\theta}\Big{)}\cdot\hat{\theta}$
$\displaystyle+\Big{(}\alpha_{m}^{l}\frac{\mathrm{i}m}{r\sin\theta}j^{\prime}_{m}{k_{p}r}e^{\mathrm{i}m\varphi}P^{|l|}_{m}-\alpha_{m}^{l}\frac{j_{m}(k_{p}r)}{k_{p}r^{2}}\frac{\mathrm{i}m}{r\sin\theta}e^{\mathrm{i}m\varphi}P^{|l|}_{m}\Big{)}\cdot\hat{\varphi}.$

By using the transmission conditions of (2.3) on $\partial\Omega$, and
combining with (3.50), (3.53) and (3.54), we can deduce that

$\displaystyle\left(\lambda(\nabla\cdot{\mathbf{u}}_{m})I_{3}+2\mu\nabla^{s}{\mathbf{u}}_{m}+\delta\tau^{2}v_{m}I_{3}\right)\cdot\hat{r}$
$\displaystyle=$
$\displaystyle\lambda\Big{[}\alpha_{m}^{l}k_{p}j^{\prime\prime}_{m}(k_{p})e^{\mathrm{i}m\varphi}P^{|l|}_{m}+2\alpha_{m}^{l}j^{\prime}_{m}(k_{p})e^{\mathrm{i}m\varphi}P^{|l|}_{m}+\alpha_{m}^{l}\frac{j_{m}(k_{p})}{k_{p}}e^{\mathrm{i}m\varphi}\frac{d^{2}P^{|l|}_{m}}{d\theta^{2}}+\alpha_{m}^{l}\frac{j_{m}(k_{p})}{k_{p}}\frac{\cos\theta}{\sin\theta}e^{\mathrm{i}m\varphi}$
$\displaystyle\times\frac{dP^{|l|}_{m}}{d\theta}-\alpha_{m}^{l}\frac{j_{m}(k_{p})}{k_{p}}\frac{m^{2}}{\sin^{2}\theta}e^{\mathrm{i}m\varphi}P^{|l|}_{m}\Big{]}\cdot\hat{r}+2\mu\alpha_{m}^{l}k_{p}j^{\prime\prime}_{m}(k_{p}r)e^{\mathrm{i}m\varphi}P^{|l|}_{m}\cdot\hat{r}+2\mu\Big{[}\alpha_{m}^{l}j^{\prime}_{m}(k_{p})e^{\mathrm{i}m\varphi}$
$\displaystyle\times\frac{dP^{|l|}_{m}}{d\theta}-\alpha_{m}^{l}\frac{j_{m}(k_{p})}{k_{p}}e^{\mathrm{i}m\varphi}\frac{dP^{|l|}_{m}}{d\theta}\Big{]}\cdot\hat{\theta}+2\mu\Big{[}\alpha_{m}^{l}\frac{\mathrm{i}m}{\sin\theta}j^{\prime}_{m}({k_{p}})e^{\mathrm{i}m\varphi}P^{|l|}_{m}-\alpha_{m}^{l}\frac{j_{m}(k_{p})}{k_{p}}\frac{\mathrm{i}m}{\sin\theta}e^{\mathrm{i}m\varphi}$
$\displaystyle\times
P^{|l|}_{m}\Big{]}\cdot\hat{\varphi}+\delta\tau^{2}\beta_{m}^{l}j_{m}(k)e^{\mathrm{i}m\varphi}P^{|l|}_{m}\cdot\hat{r}=\mathbf{0}.$
(3.55)

Using the linear independence of $\hat{r}$, $\hat{\theta}$ and
$\hat{\varphi}$, in view of the coefficient of $\hat{\varphi}$ and (3.52), we
obtain that

$j_{m}(k_{p})=k_{p}j^{\prime}_{m}(k_{p}).$ (3.56)

Similarly, according to the coefficient of $\hat{r}$ in (3.55), by using
(3.4), (3.46),(3.52), (3.56) and the following property (cf.[1])

$\displaystyle
j^{\prime}_{m}(|\mathbf{x}|)+j_{m+1}(|\mathbf{x}|)-\frac{m}{|\mathbf{x}|}j_{m}(|\mathbf{x}|)$
$\displaystyle=0,$ (3.57a)
$\displaystyle|\mathbf{x}|^{2}j^{\prime\prime}_{m}(|\mathbf{x}|)+2|\mathbf{x}|j^{\prime}_{m}(|\mathbf{x}|)+\left[|\mathbf{x}|^{2}-m(m+1)\right]j_{m}(|\mathbf{x}|)$
$\displaystyle=0,$ (3.57b)
$\displaystyle\frac{d^{2}P_{m}^{|l|}}{d\theta^{2}}+\frac{\cos\theta}{\sin\theta}\frac{dP_{m}^{|l|}}{\theta}+\Big{[}m(m+1)-\frac{m^{2}}{\sin^{2}\theta}\Big{]}P_{m}^{|l|}$
$\displaystyle=0,$ (3.57c)

after some calculations, we have

$-4\mu
k_{p}j^{\prime}_{m}(k_{p})j^{\prime}_{m}(k)-k_{p}^{2}j_{m}(k_{p})j^{\prime}_{m}(k)+2\mu
m(m+1)j_{m}(k_{p})j^{\prime}_{m}(k)+\delta\tau^{2}kj^{\prime}_{m}(k_{p})j_{m}(k)=0$
(3.58)

Now, we want to find a $k$ that satisfies both (3.56) and (3.58). Then, we
substitute (3.56) into (3.58), one has

$\left[-4\mu-k_{p}^{2}+2\mu
m(m+1)\right]j_{m}(k_{p})j^{\prime}_{m}(k)+\frac{\delta\tau^{2}k}{k_{p}}j_{m}(k_{p})j_{m}(k)=0$
(3.59)

Apparently, if $k$ solves the equations (3.56) and (3.58), the equation (3.59)
holds. On the other hand, if $k$ is the root of the equations (3.56) and
(3.59), $k$ solves the equation (3.58). Thus, in the rest of the proof, we
shall confine ourselves on finding the root $k$ of the equation (3.59). Then
we will show that the root $k$ solves (3.56) in the asymptotic sense and
please refer to Remark 3.3.

By using (3.46), we find that the transmission eigenvalues $k$’s to (2.3) are
positive zeros of the following function:

$\begin{split}f_{m+\frac{1}{2}}(k):=&\Big{[}\frac{4\mu}{k\sqrt{\tau}}+\tau^{\frac{3}{2}}-\frac{2\mu
m(m+1)}{\sqrt{\tau}k}\Big{]}J_{m+\frac{3}{2}}(k)J_{m+\frac{1}{2}}(\tau k)\\\
&+\Big{[}-\frac{4\mu m}{k^{2}\sqrt{\tau}}-m\tau^{\frac{3}{2}}-\frac{2\mu
m^{2}(m+1)}{\sqrt{\tau}k^{2}}+\delta\tau^{\frac{3}{2}}\Big{]}J_{m+\frac{1}{2}}(k)J_{m+\frac{1}{2}}(\tau
k).\end{split}$ (3.60)

Nest, we compute the following quantity:

$\begin{split}&f_{m+\frac{1}{2}}(j_{m+\frac{1}{2},1})f_{m+\frac{1}{2}}(j_{m+\frac{1}{2},2})\\\
=&\Big{[}\frac{4\mu}{j_{m+\frac{1}{2},1}\sqrt{\tau}}+\tau^{\frac{3}{2}}-\frac{2\mu
m(m+1)}{\sqrt{\tau}j_{m+\frac{1}{2},1}}\Big{]}\Big{[}\frac{4\mu}{j_{m+\frac{1}{2},2}\sqrt{\tau}}+\tau^{\frac{3}{2}}-\frac{2\mu
m(m+1)}{\sqrt{\tau}j_{m+\frac{1}{2},2}}\Big{]}\\\ &\times
J_{m+\frac{1}{2}}(\tau j_{m+\frac{1}{2},1})J_{m+\frac{1}{2}}(\tau
j_{m+\frac{1}{2},2})J_{m+\frac{3}{2}}(j_{m+\frac{1}{2},1})J_{m+\frac{3}{2}}(j_{m+\frac{1}{2},2})\\\
\end{split}$ (3.61)

Nest, using (3.21) when $m$ is sufficiently large, there hold that

$\begin{split}j_{m+\frac{1}{2},1}&=\left(m+\frac{1}{2}\right)\Big{(}1+C_{1}(m+\frac{1}{2})^{-2/3}+o((m+\frac{1}{2})^{-2/3})\Big{)},\\\
j_{m+\frac{1}{2},2}&=\left(m+\frac{1}{2}\right)\Big{(}1+C_{2}(m+\frac{1}{2})^{-2/3}+o((m+\frac{1}{2})^{-2/3})\Big{)},\end{split}$
(3.62)

where $C_{i},i=1,2$ is a positive constant not depending on $m$. By virtue of
(3.62), when $m$ is sufficient large, we have

$\begin{split}&\Big{[}\frac{4\mu}{j_{m+\frac{1}{2},1}\sqrt{\tau}}+\tau^{\frac{3}{2}}-\frac{2\mu
m(m+1)}{\sqrt{\tau}j_{m+\frac{1}{2},1}}\Big{]}\Big{[}\frac{4\mu}{j_{m+\frac{1}{2},2}\sqrt{\tau}}+\tau^{\frac{3}{2}}-\frac{2\mu
m(m+1)}{\sqrt{\tau}j_{m+\frac{1}{2},2}}\Big{]}\\\
&=m\,\Pi_{i=1}^{2}\left[\frac{2\mu}{\sqrt{\tau}}(1-2m^{-2})(1-C_{i}(m+\frac{1}{2})^{-2/3}-o((m+\frac{1}{2})^{-2/3}))-\tau^{3/2}m^{-1}\right]>0\end{split}$
(3.63)

for any positive constants $\tau$ and $\mu$.

Moreover, from the choice of $\tau j_{m+\frac{1}{2},2}<j_{m+\frac{1}{2},1}$,
the following inequality holds

$J_{m+\frac{1}{2}}(\tau j_{m+\frac{1}{2},1})J_{m+\frac{1}{2}}(\tau
j_{m+\frac{1}{2},2})>0.$ (3.64)

Since the positive zeros of $J_{m+\frac{3}{2}}$ are interlaced with those of
$J_{m+\frac{1}{2}}$, we have

$J_{m+\frac{3}{2}}(j_{m+\frac{1}{2},1})J_{m+\frac{3}{2}}(j_{m+\frac{1}{2},2})<0.$

By using the above fact, we derive that

$\begin{split}f_{m+\frac{1}{2}}(j_{m+\frac{1}{2},1})f_{m+\frac{1}{2}}(j_{m+\frac{1}{2},2})<0,\end{split}$
(3.65)

which implies that
$k_{m,s}\in\left(j_{m+\frac{1}{2},1},j_{m+\frac{1}{2},2}\right)$.

Finally, by using (3.62) one has (3.48) and (3.49).

The proof is complete. ∎

######  Remark 3.3.

Here we remark that if $k_{m,s}$ is a root of (3.59), the root $k_{m,s}$
solves the equation (3.56) in the asymptotic sense. Indeed, if $k_{m,s}$ is a
root of (3.59), for sufficiently large $m$ and
$\tau<j_{m+\frac{1}{2},1}/j_{m+\frac{1}{2},2}<1$, using (3.26), (3.46), (3.48)
and (3.49) we derive that

$j_{m}(k_{p})-k_{p}j^{\prime}_{m}(k_{p})=C\left(\frac{\tau
e^{\sqrt{1-\tau^{2}}}}{1+\sqrt{1-\tau^{2}}}\right)^{m-\frac{1}{2}}\rightarrow
0\quad\mbox{as}\quad m\rightarrow\infty,$ (3.66)

where C is a constant. In the derivation of the last inequality, we have used
the fact $0<\frac{\tau e^{\sqrt{1-\tau^{2}}}}{1+\sqrt{1-\tau^{2}}}<1$ due to
$\tau<1$ . Thus, $k_{m,s}$ satisfies (3.56) in the asymptotic sense.

We are in a position to consider the boundary-localization patterns of the AE
transmission eigenfunctions associated with the eigenvalues determined in
Lemma 3.2.

######  Theorem 3.3.

Consider (2.3) and let $\Omega$ be the unit ball in $\mathbb{R}^{3}$ and
$\Omega_{\epsilon}$ be given in (3.1). For any $\epsilon\in(0,1)$, there exist
transmission eigenfunctions $v_{m}$, $m\in\mathbb{N}$, associated with the
transmission eigenvalues $k_{m,s}$ described in Lemma 3.2 such that

$\lim_{m\to\infty}\frac{\|v_{m}\|_{L^{2}(\Omega_{\epsilon})}}{\|v_{m}\|_{L^{2}(\Omega)}}=0.$
(3.67)

###### Proof.

Recall that the transmission eigenvalue $k_{m,s}$ fulfils (3.49). Let $v_{m}$
be given by (3.50). One has

$\displaystyle\|v_{m}\|^{2}_{L^{2}(\Omega_{\epsilon})}$
$\displaystyle=|\beta_{m}^{l}|^{2}\int_{\Omega_{\epsilon}}|j_{m}(k_{m,s}|\mathbf{x}|)|^{2}d\mathbf{x}=\frac{2\pi|\beta_{m}^{l}|^{2}}{k_{m,s}}\int_{0}^{\epsilon}rJ^{2}_{m+\frac{1}{2}}(k_{m,s}r)dr,$
(3.68a) $\displaystyle\|v_{m}\|^{2}_{L^{2}(\Omega)}$
$\displaystyle=|\beta_{m}^{l}|^{2}\int_{\Omega}|j_{m}(k_{m,s}|\mathbf{x}|)|^{2}d\mathbf{x}=\frac{2\pi|\beta_{m}^{l}|^{2}}{k_{m,s}}\int_{0}^{1}rJ^{2}_{m+\frac{1}{2}}(k_{m,s}r)dr.$
(3.68b)

From the proof of Theorem 3.1, we have
$J^{\prime}_{m+\frac{1}{2}}(k_{m,s}r)>0$ for $0<r<1$. Hence, by using
(3.26),we can derive the following asymptotic bound:

$\begin{split}&\int_{0}^{\epsilon}rJ^{2}_{m+\frac{1}{2}}(k_{m,s}r)dr\leqslant\epsilon^{2}J^{2}_{m+\frac{1}{2}}(k_{m,s}\epsilon)\\\
=&\frac{\epsilon^{2}}{2\pi(m+\frac{1}{2})\left(1-(\frac{k_{m,s}\epsilon}{m+\frac{1}{2}})^{2}\right)^{1/2}}\left(\frac{k_{m,s}\epsilon}{m+\frac{1}{2}}\frac{e^{\sqrt{1-(\frac{k_{m,s}\epsilon}{m+\frac{1}{2}})^{2}}}}{1+\sqrt{1-(\frac{k_{m,s}\epsilon}{m+\frac{1}{2}})^{2}}}\right)^{2(m+\frac{1}{2})}\left(1+o(1)\right)^{2}\end{split}$
(3.69)

as $m\rightarrow\infty$. For sufficiently large $m$, one can choose $\Xi_{2}$
by

$\Xi_{2}=\frac{j^{\prime}_{m+\frac{1}{2},1}}{k_{m,s}},$ (3.70)

where $\epsilon<\Xi_{2}<1$. From (3.30), we have
$j^{\prime}_{m+\frac{1}{2},1}=(m+\frac{1}{2})\left(1+{O}((m+\frac{1}{2})^{-2/3})\right)$.
Hence, one has $m+\frac{1}{2}<j^{\prime}_{m+\frac{1}{2},1}=k_{m,s}\Xi_{2}$. It
yields that

$\displaystyle\int_{0}^{1}rJ^{2}_{m+\frac{1}{2}}(k_{m,s}r)dr\geqslant$
$\displaystyle\int_{\Xi_{2}}^{1}rJ^{2}_{m+\frac{1}{2}}(k_{m,s}r)dr=\frac{2(1+o(1))^{2}}{\pi}\int_{\Xi_{2}}^{1}\frac{r\cos^{2}x_{m+\frac{1}{2}}}{\left(k^{2}_{m,s}r^{2}-(m+\frac{1}{2})^{2}\right)^{1/2}}dr$
$\displaystyle\sim$
$\displaystyle\frac{1}{\pi}\int_{\Xi_{2}}^{1}\frac{r}{\left(k^{2}_{m,s}r^{2}-(m+\frac{1}{2})^{2}\right)^{1/2}}dr$
$\displaystyle\sim$ $\displaystyle\frac{1}{\pi
k^{2}_{m,s}}\Big{(}\sqrt{k_{m,s}^{2}-(m+\frac{1}{2})^{2}}-\sqrt{j_{m+\frac{1}{2},1}^{\prime
2}-(m+\frac{1}{2})^{2}}\Big{)}.$ (3.71)

where $x_{m+\frac{1}{2}}=\sqrt{{(\tau
k_{m,s}r)}^{2}-(m+\frac{1}{2})^{2}}-\frac{(m+\frac{1}{2})\pi}{2}+(m+\frac{1}{2})\arcsin(\frac{m+\frac{1}{2}}{\tau
k_{m,s}r})-\frac{\pi}{4}$.

Therefore, combining (3.68a), (3.68b), (LABEL:3d:vmeps1) and (3.2), there
holds

$\begin{split}\frac{\|v_{m}\|^{2}_{L^{2}(\Omega_{\epsilon})}}{\|v_{m}\|^{2}_{L^{2}(\Omega)}}\leqslant&C(m+\frac{1}{2})^{\frac{1}{3}}\left(1-\epsilon^{2}\right)^{-\frac{1}{2}}\left(\frac{\epsilon
e^{\sqrt{1-\epsilon^{2}}}}{1+\sqrt{1-\epsilon^{2}}}\right)^{2(m+\frac{1}{2})}\end{split}$
(3.72)

where $C$ is a constant. By letting $m\to\infty$ one has (3.67) ∎

######  Theorem 3.4.

Consider (2.3) and let $\Omega$ be the unit ball in $\mathbb{R}^{3}$ and
$\Omega_{\epsilon}$ be given in (3.1). Let the physical parameter $\tau$ in
(2.3) satisfy $\tau\in(0,1)$. For any $\epsilon\in(0,1)$, there exists
transmission eigenfunctions $\mathbf{u}_{m}$, $m\in\mathbb{N}$, associated
with the transmission eigenvalues $k_{m,s}$ described in Lemma 3.2 such that

$\lim_{m\to\infty}\frac{\|\mathbf{u}_{m}\|_{L^{2}(\Omega_{\epsilon})}}{\|\mathbf{u}_{m}\|_{L^{2}(\Omega)}}=0.$
(3.73)

###### Proof.

Suppose that the transmission eigenvalue $k_{m,s}$ fulfils (3.49). Let the
transmission eigenfunction $\mathbf{u}_{m}$ of (2.3) be associated with
$k_{m,s}$ in (3.50). By using (3.46), (3.57a) and the following properties
(cf. [1]):

$\displaystyle\frac{dP_{m}^{|l|}(\cos\theta)}{d\theta}$
$\displaystyle=\frac{1}{2}\left[(m+|l|)(m-|l|+1)P_{m}^{|l|-1}(\cos\theta)-P_{m}^{|l|+1}(\cos\theta)\right],$
(3.74) $\displaystyle\frac{|m|}{\sin\theta}P_{m}^{|l|}(\cos\theta){d\theta}$
$\displaystyle=-\frac{1}{2}\left[P_{m-1}^{|l|+1}(\cos\theta)+(m+|l|-1)(m+|l|)P_{m-1}^{|l|-1}(\cos\theta)\right],$
(3.75) $\displaystyle\int_{-1}^{1}P_{m}^{l}(x)P_{m}^{k}(x)$
$\displaystyle=\begin{cases}0,&l\neq k\\\
\frac{(m+|l|)!}{(m-|l|)!}\frac{2}{2m+1},&l=k\end{cases},$ (3.76)

and similar to (LABEL:3d:vmeps1), one can show that

$\displaystyle\|{\mathbf{u}}_{m}\|^{2}_{L^{2}(\Omega_{\epsilon})}$
$\displaystyle=\int_{\Omega_{\epsilon}}\Bigg{(}|\alpha_{m}^{l}|^{2}|j^{\prime}_{m}(k_{p}|{\mathbf{x}}|)|^{2}|P_{m}^{|l|}|^{2}+|\alpha_{m}^{l}|^{2}\frac{|j_{m}(k_{p}|{\mathbf{x}}|)|^{2}}{k_{p}^{2}|{\mathbf{x}}|^{2}}(\frac{dP_{m}^{|l|}}{d\theta})^{2}$
$\displaystyle\quad+|\alpha_{m}^{l}|^{2}\frac{|j_{m}(k_{p}|{\mathbf{x}}|)|^{2}}{k_{p}^{2}|{\mathbf{x}}|^{2}}\frac{m^{2}}{\sin^{2}\theta}|P_{m}^{|l|}|^{2}\Bigg{)}d{\mathbf{x}}$
$\displaystyle\leqslant$
$\displaystyle\frac{|\alpha_{m}^{l}|^{2}}{k_{m,s}}\frac{(m+|l|)!}{(m-|l|)!(m+\frac{1}{2})}\frac{\epsilon}{2\pi(m+\frac{1}{2})\sqrt{1-(\frac{\tau
k_{m,s}\eta_{3}}{m+\frac{1}{2}})^{2}}}$ (3.77)
$\displaystyle\times\Biggl{[}\frac{\eta_{3}}{\tau}\Biggr{(}\frac{\tau
k_{m,s}\eta_{3}}{m+\frac{1}{2}}\frac{e^{\sqrt{1-(\frac{\tau
k_{m,s}\eta_{3}}{m+\frac{1}{2}})^{2}}}}{1+\sqrt{1-(\frac{\tau
k_{m,s}\eta_{3}}{m+\frac{1}{2}}})^{2}}\Biggr{)}^{3}$
$\displaystyle+\frac{Cm^{2}}{\tau^{3}k_{m,s}^{2}}\Biggr{(}\frac{\tau
k_{m,s}}{m+\frac{1}{2}}\frac{e^{\sqrt{1-(\frac{\tau
k_{m,s}\eta_{3}}{m+\frac{1}{2}})^{2}}}}{1+\sqrt{1-(\frac{\tau
k_{m,s}\eta_{3}}{m+\frac{1}{2}}})^{2}}\Biggr{)}+\frac{Cm^{2}\eta_{3}}{\tau^{3}k_{m,s}^{2}}$
$\displaystyle\times\Biggr{(}\frac{\tau
k_{m,s}\eta_{3}}{m+\frac{1}{2}}\frac{e^{\sqrt{1-(\frac{\tau
k_{m,s}\eta_{3}}{m+\frac{1}{2}})^{2}}}}{1+\sqrt{1-(\frac{\tau
k_{m,s}\eta_{3}}{m+\frac{1}{2}}})^{2}}\Biggr{)}\Biggr{]}\Biggr{(}\frac{\tau
k_{m,s}\eta_{3}}{m+\frac{1}{2}}\frac{e^{\sqrt{1-(\frac{\tau
k_{m,s}\eta_{3}}{m+\frac{1}{2}})^{2}}}}{1+\sqrt{1-(\frac{\tau
k_{m,s}\eta_{3}}{m+\frac{1}{2}}})^{2}}\Biggr{)}^{2m}(1+o(1))^{2},$

where $\eta_{3}\in[0,\epsilon]$.

When $\tau<1$. Recall that $\Xi_{2}$ is given in (3.70). Adopting a similar
argument for deriving (3.2), one has

$\displaystyle\|{\mathbf{u}}_{m}\|^{2}_{L^{2}(\Omega)}$
$\displaystyle\geqslant\frac{|\alpha_{m}^{l}|^{2}}{k_{m,s}}\frac{(m+|l|)!}{(m-|l|)!(m+\frac{1}{2})}\frac{1-\Xi_{2}}{2\pi(m+\frac{1}{2})\sqrt{1-(\frac{\tau
k_{m,s}\eta_{4}}{m+\frac{1}{2}})^{2}}}$ (3.78)
$\displaystyle\times\Biggl{[}\frac{\eta_{4}}{\tau}\Biggr{(}\frac{\tau
k_{m,s}\eta_{4}}{m+\frac{1}{2}}\frac{e^{\sqrt{1-(\frac{\tau
k_{m,s}\eta_{4}}{m+\frac{1}{2}})^{2}}}}{1+\sqrt{1-(\frac{\tau
k_{m,s}\eta_{4}}{m+\frac{1}{2}}})^{2}}\Biggr{)}^{3}+\frac{Cm^{2}}{\tau^{3}k_{m,s}^{2}}\Biggr{(}\frac{\tau
k_{m,s}}{m+\frac{1}{2}}\frac{e^{\sqrt{1-(\frac{\tau
k_{m,s}\eta_{4}}{m+\frac{1}{2}})^{2}}}}{1+\sqrt{1-(\frac{\tau
k_{m,s}\eta_{4}}{m+\frac{1}{2}}})^{2}}\Biggr{)}$
$\displaystyle+\frac{Cm^{2}\eta_{4}}{\tau^{3}k_{m,s}^{2}}\Biggr{(}\frac{\tau
k_{m,s}\eta_{4}}{m+\frac{1}{2}}\frac{e^{\sqrt{1-(\frac{\tau
k_{m,s}\eta_{4}}{m+\frac{1}{2}})^{2}}}}{1+\sqrt{1-(\frac{\tau
k_{m,s}\eta_{4}}{m+\frac{1}{2}}})^{2}}\Biggr{)}-\frac{2m}{\tau^{2}k^{2}_{m,s}}$
$\displaystyle\times\Biggr{(}\frac{\tau
k_{m,s}\eta_{4}}{m+\frac{1}{2}}\frac{e^{\sqrt{1-(\frac{\tau
k_{m,s}\eta_{4}}{m+\frac{1}{2}})^{2}}}}{1+\sqrt{1-(\frac{\tau
k_{m,s}\eta_{4}}{m+\frac{1}{2}}})^{2}}\Biggr{)}^{2}\Biggr{]}\Biggr{(}\frac{\tau
k_{m,s}\eta_{4}}{m+\frac{1}{2}}\frac{e^{\sqrt{1-(\frac{\tau
k_{m,s}\eta_{4}}{m+\frac{1}{2}})^{2}}}}{1+\sqrt{1-(\frac{\tau
k_{m,s}\eta_{4}}{m+\frac{1}{2}}})^{2}}\Biggr{)}^{2m}(1+o(1))^{2},$

where $\eta_{4}\in[\Xi_{2},1]$ and $\eta_{4}\to 1$ as $m\to\infty$.
Furthermore, by using a similar argument for (3.43), there exists
$0<\delta(\eta_{3})<1$ such that

$\frac{\|\mathbf{u}_{m}\|_{L^{2}(\Omega_{\epsilon})}}{\|\mathbf{u}_{m}\|_{L^{2}(\Omega)}}\leqslant
C(m+\frac{1}{2})^{\frac{2}{3}}(1-\delta(\eta_{3}))^{2m},$ (3.79)

where $C$ is positive constant independent of $m$. Therefore, we can prove
(3.73) by letting $m\rightarrow\infty$ in (3.79).

The proof is complete. ∎

###  3.3. Numerical results

We have conducted extensive numerical experiments to verify that the boundary-
localization phenomenon of the AE transmission eigenfunctions holds for
general domains of $\Omega$: radial/non-radial, smooth/non-smooth and
convex/non-convex.  Here, we apply the finite element method for the numerical
experiments. The system (1.4) is first reformulated as a variational problem
and the boundary conditions are the third and the forth conditions in (1.4).
The mesh is chosen to be the triangular and the piecewise linear basis
functions are used in the computations. In what follows, we present a few
representative examples for illustration; see Fig. 1 and 2.

Figure 1. (a) & (b): $|\mathbf{u}|$ to (1.4) associated with $\rho_{b}=20$,
$\kappa=\rho_{e}=\tilde{\lambda}=\tilde{\mu}=1$ and $w^{2}=43.27,201.52$,
respectively; (c) & (d): $v$ to (1.4) associated with $\rho_{e}=10$,
$\kappa=\rho_{b}=\tilde{\lambda}=\tilde{\mu}=1$ and $w^{2}=93.71,101.35$,
respectively.

Figure 2. (a) & (b): $v$ to (1.4) associated with $\rho_{e}=10$,
$\kappa=\rho_{b}=\tilde{\lambda}=\tilde{\mu}=1$ and $w^{2}=15.0140,5.9875$,
respectively.

## Acknowledgment

The work of H. Diao is supported by a startup fund from National Key R&D
Program of China (No. 2020YFA0714102) and NSFC/RGC Joint Research Grant No.
12161160314. The work of H. Li was supported by Direct Grant for Research,
CUHK (project 4053518). The work of H. Liu is supported by the Hong Kong RGC
General Research Funds (projects 11311122, 11300821 and 12301420), the
NSFC/RGC Joint Research Fund (project N_CityU101/21), and the ANR/RGC Joint
Research Grant, A_CityU203/19.

## References

  * [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1965.
  * [2] H. Ammari, E. Bretin, J. Garnier, H. Kang, H. Lee and A. Wahab, Mathematical Methods in Elasticity Imaging, Princeton University Press, 2015.
  * [3] H. Ammari, H. Kang, H. Lee, Layer Potential Techniques in Spectral Analysis, Mathematical Surveys and Monographs, vol.153, Amer. Math. Soc., Rhode Island, 2009.
  * [4] E. Blåsten and H. Liu, On vanishing near corners of transmission eigenfunctions, J. Funct. Anal., 273 (2017), no. 11, 3616–3632.
  * [5] X. Cao, H. Diao, H. Liu and J. Zou, On nodal and generalized singular structures of Laplacian eigenfunctions and applications, J. Math. Pures Appl., 143 (2020), 116–161.
  * [6] X. Cao, H. Diao, H. Liu and J. Zou, On novel geometric structures of Laplacian eigenfunctions in $\mathbb{R}^{3}$ and applications to inverse problems , SIAM J. Math. Anal., 53 (2021), no. 2, 1263–1294.
  * [7] X. Cao, H. Diao, H. Liu and J. Zou, Two single-measurement uniqueness results for inverse scattering problems within polyhedral geometries, Inverse Probl. Imaging, doi: 10.3934/ipi.2022023, 2022.
  * [8] Y. T. Chow, Y. Deng, Y. He, H. Liu and X. Wang, boundary-localized transmission eigenstates, super-resolution imaging, and pseudo surface plasmon modes. SIAM J. Imaging Sciences, 14 (2021), 946–975.
  * [9] Y. T. Chow, Y. Deng, H. Liu and M. Sunkula, Surface concentration of transmission eigenfunctions, arXiv:2109.14361, 2021.
  * [10] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 4th. Ed., Springer, New York, 2019.
  * [11] G. Dassios and Z. Rigou, _Elastic Herglotz functions_ , SIAM J. Appl. Math., 55 (1995), 1345–1361.
  * [12] Y. Deng, Y. Jiang, H. Liu and K. Zhang, On new surface-localized transmission eigenmodes, Inverse Probl. Imaging, 16 (2022), no. 3, 595–611.
  * [13] Y. Deng, H. Li and H. Liu, Spectral properties of Neumann-Poincaré operator and anomalous localized resonance in elasticity beyond quasi-static limit, Journal of Elasticity, 140 (2020), 213 – 242.
  * [14] Y. Deng, H. Li and H. Liu, Analysis of surface polariton resonance for nanoparticles in elastic system, SIAM J. Math. Anal., 52 (2020), 1786–1805.
  * [15] Y. Deng, H. Liu, X. Wang and W. Wu, On geometrical properties of electromagnetic transmission eigenfunctions and artificial mirage, SIAM J. Appl. Math., 82 (2022), no. 1, 1–24.
  * [16] H. Diao, H. Liu and L. Wang, On generalized Holmgren’s principle to the Lamé operator with applications to inverse elastic problems, Cal. Var. Partial Differential Equations, 59 (2020), 179.
  * [17] H. Diao, H. Liu, and L. Wang, Further results on generalized Holmgren’s principle to the Lamé operator and applications, J. Differential Equations, 309 (2022), 841–882.
  * [18] H. Diao, H. Liu and B. Sun, On a local geometric property of the generalized elastic transmission eigenfunctions and application, Inverse Problems, 37 (2021), no. 10, Paper No. 105015, 36 pp.
  * [19] H. Diao, H. Liu and R. Tang, Boundary localization of Minneart resonances in bubbly elastic media, preprint, 2022.
  * [20] S. Domínguez, N. Nigam and J. Sun, Revisiting the Jones eigenproblem in fluid-structure interaction, SIAM J. Appl. Math., 79 (2019), no. 6, 2385–2408.
  * [21] T. Hargé, Valeurs propres d’un corps élastique, C. R. Acad. Sci. Paris Sér. I Math., 311 (1990), 857–859.
  * [22] Y. Jiang, H. Liu, J. Zhang and K. Zhang, Spectral patterns of elastic transmission eigenfunctions: boundary localization, surface resonance and stress concentration, preprint, 2022.
  * [23] D. S. Jones, Low-frequency scattering by a body in lubricated contact, Quart. J. Mech. Appl. Math., 36 (1983), pp. 111–138.
  * [24] P. Juliette, D. Benjamin and L. Valentin, Resonant acoustic propagation and negative density in liquid foams, Phys. Rev. Lett., 14 (2014), 148307.
  * [25] A. Kirsch, A. Ruiz, The Factorization Method for an inverse fluid-solid interaction scattering problem, Inverse Problems and Imaging, 6 (2012), 681–695.
  * [26] B. G. Korenev, Bessel Functions and Their Applications, Chapman & Hall/CRC, Boca Raton, FL, 2002\.
  * [27] H. Li and H. Liu, On anomalous localized resonance and plasmonic cloaking beyond the quasistatic limit, Proceedings of the Royal Society A, 474: 20180165.
  * [28] H. Li, Recent progress on the mathematical study of anomalous localized resonance in elasticity, Electronic Research Archive, 28 (2020), 1257–1272.
  * [29] H. Li, J. Li and H. Liu, On novel elastic structures inducing polariton resonances with finite frequencies and cloaking due to anomalous localized resonance, J. Math. Pures Appl., 120 (2018), 195–219.
  * [30] H. Li, H. Liu and J. Zou, Minnaert resonances for bubbles in soft elastic materials, SIAM J. Appl. Math., 82 (2022), 119–141.
  * [31] H. Liu, On local and global structures of transmission eigenfunctions and beyond, J. Inverse Ill-Posed Probl., 30 (2022), no. 2, 287–305.
  * [32] C. J. Luke and P. A. Martin, Fluid-solid interaction: Acoustic scattering by a smooth elastic obstacle, SIAM J. Appl. Math., 55 (1995), 904–922.
  * [33] V. Leroy, A. Strybulevych, M. Lanoy, F. Lemoult, A. Tourin and J.H. Page, _Superabsorption of acoustic waves with bubble metascreens_ , Phys. Rev. B, 91 (2015), 020301.
  * [34] D. Natroshvili, G. Sadunishvili and I. Sigua, Some remarks concerning Jones eigenfrequencies and Jones modes, Georgian Math. J., 12 (2005), 337–348.
  * [35] F. Olver, D. Lozier, R. Boisvert and C. Clark, The NIST Handbook of Mathematical Functions, Cambridge Univ. Press, New York, 2010.
  * [36] C. K. Qu and R. Wong, “Best possible” upper and lower bounds for the zeros of the Bessel function $J_{\nu}(x)$, Trans. Amer. Math. Soc., 351 (2008), 2833–2859.
  * [37] V. Sevroglou and G. Pelekanos, _Two-dimensional elastic Herglotz functions and their application in inverse scattering_ , Journal of Elasticity, 68 (2002), 123–144.
  * [38] J. M. Solano-Altamirano, J. D. Malcolm and S. Goldman, Gas bubble dynamics in soft materials, Soft Matter, 11 (2015), 202–210.