# On new surface-localized transmission eigenmodes

Youjun Deng  School of Mathematics and Statistics, HNP-LAMA, Central South
University, Changsha, Hunan, China<EMAIL_ADDRESS><EMAIL_ADDRESS>,  Yan Jiang  Department of Mathematics, Jilin
University, Changchun, Jilin, China<EMAIL_ADDRESS>,  Hongyu
Liu  Department of Mathematics, City University of Hong Kong, Hong Kong SAR,
China<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Kai Zhang
Department of Mathematics, Jilin University, Changchun, Jilin, China.
<EMAIL_ADDRESS>
###### Abstract.

Consider the transmission eigenvalue problem

$(\Delta+k^{2}\mathbf{n}^{2})w=0,\ \ (\Delta+k^{2})v=0\ \ \mbox{in}\ \
\Omega;\quad w=v,\ \ \partial_{\nu}w=\partial_{\nu}v=0\ \ \mbox{on}\
\partial\Omega.$

It is shown in [12] that there exists a sequence of eigenfunctions
$(w_{m},v_{m})_{m\in\mathbb{N}}$ associated with $k_{m}\rightarrow\infty$ such
that either $\\{w_{m}\\}_{m\in\mathbb{N}}$ or $\\{v_{m}\\}_{m\in\mathbb{N}}$
are surface-localized, depending on $\mathbf{n}>1$ or $0<\mathbf{n}<1$. In
this paper, we discover a new type of surface-localized transmission
eigenmodes by constructing a sequence of transmission eigenfunctions
$(w_{m},v_{m})_{m\in\mathbb{N}}$ associated with $k_{m}\rightarrow\infty$ such
that both $\\{w_{m}\\}_{m\in\mathbb{N}}$ and $\\{v_{m}\\}_{m\in\mathbb{N}}$
are surface-localized, no matter $\mathbf{n}>1$ or $0<\mathbf{n}<1$. Though
our study is confined within the radial geometry, the construction is subtle
and technical.

Keywords: Transmission eigenfunctions, spectral geometry, surface
localization, wave concentration

2010 Mathematics Subject Classification: 35P25, 78A46 (primary); 35Q60, 78A05
(secondary).

##  1\. Introduction

###  1.1. Mathematical setup and discussion on the major finding

Let $\Omega$ be a bounded domain in $\mathbb{R}^{N}$, $N=2,3$, with a
connected complement $\mathbb{R}^{N}\backslash\overline{\Omega}$ and
$\mathbf{n}\in L^{\infty}(\Omega)$ be a positive function. Consider the
following transmission eigenvalue problem for $w\in H^{1}(\Omega)$ and $v\in
H^{1}(\Omega)$:

$\left\\{\begin{array}[]{ll}\Delta w+k^{2}\mathbf{n}^{2}w=0&\text{in}\
\Omega,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\ \Delta v+k^{2}v=0&\text{in}\
\Omega,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\ \displaystyle{w=v,\ \
\frac{\partial w}{\partial\nu}=\frac{\partial v}{\partial\nu}}&\text{on}\
\partial\Omega,\\\ \end{array}\right.$ (1.1)

where $\nu$ is the exterior unit normal vector to $\partial\Omega$. Clearly,
$w\equiv v\equiv 0$ are a pair of trivial solutions to (1.1). If there exists
a non-trivial pair of solutions $(w,v)$ to (1.1), $k\in\mathbb{R}_{+}$ is
called a transmission eigenvalue, and $w,v$ are the associated transmission
eigenfunctions. The transmission eigenvalue problem connects to the inverse
acoustic scattering theory in many aspects in delicate and mysterious manners,
from unique identifiability, reconstruction algorithm to invisibility
cloaking. Its study has a long and colourful history in the literature, and we
refer to [13, 11, 20] for historical accounts and surveys on the state-of-the-
arts developments.

In this paper, we are mainly concerned with the geometry of the transmission
eigenfunctions, which was initiated in [8] by showing that the transmission
eigenfunctions generically vanish around a corner. The discovery was inspired
by the corresponding study in the context of characterizing the wave
scattering from corner singularities [9]. The major difference is that the
weaker regularities of the transmission eigenfunctions than those in the
scattering problems require more technical and subtle treatments. The study
has received considerable attentions recently in the literature; see [2, 4, 5,
3, 6, 7, 10, 14, 15, 17, 16, 15, 23] and the references cited therein. The
results mentioned above are all of a local feature which are localized around
certain peculiar geometrical points on $\partial\Omega$. In [12], a global
geometric rigidity property was discovered for the transmission
eigenfunctions. In fact, it is shown that there exists a sequence of
eigenfunctions $(w_{m},v_{m})_{m\in\mathbb{N}}$ associated with
$k_{m}\rightarrow\infty$ such that either $\\{w_{m}\\}_{m\in\mathbb{N}}$ or
$\\{v_{m}\\}_{m\in\mathbb{N}}$ are surface-localized, depending on
$\mathbf{n}>1$ or $0<\mathbf{n}<1$. In fact, if $\mathbf{n}>1$,
$\\{v_{m}\\}_{m\in\mathbb{N}}$ are surface-localized, but
$\\{w_{m}\\}_{m\in\mathbb{N}}$ are not surface-localized; whereas if
$0<\mathbf{n}<1$, $\\{w_{m}\\}_{m\in\mathbb{N}}$ are surface-localized, but
$\\{v_{m}\\}_{m\in\mathbb{N}}$ are not surface-localized. Here, by surface-
localization, we mean the $L^{2}$-energy of the transmission eigenmode is
concentrated in a sufficiently small neighbourhood of $\partial\Omega$.
Moreover, two interesting applications of practical importance were generated
by using the surface-localization of the transmission eigenmodes in [12],
including a super-resolution wave scheme and a possible pseudo plasmon sensing
scheme.

In the current article, we show that there exists a new type of surface-
localised transmission eigenmodes which are different from those found in
[12]. In fact, we shall construct a sequence of transmission eigenfunctions
$(w_{m},v_{m})_{m\in\mathbb{N}}$ associated with $k_{m}\rightarrow\infty$ such
that both $\\{w_{m}\\}_{m\in\mathbb{N}}$ and $\\{v_{m}\\}_{m\in\mathbb{N}}$
are surface-localized, no matter $\mathbf{n}>1$ or $0<\mathbf{n}<1$. We shall
mainly consider our study for the case that $\Omega$ is radially symmetric and
$\mathbf{n}$ is a constant. Though the result is presented for a special
setup, it turns out that the construction is subtle and technical. We believe
the result holds for more general case, which shall be the subject of our
future study.

###  1.2. Connection to inverse scattering theory

In order to provide a physical background of our spectral study, we briefly
discuss the time-harmonic acoustic scattering due to an incident field $u^{i}$
and an inhomogeneous refractive medium $(\Omega,\mathbf{n})$. Here, $u^{i}$ is
an entire solution to homogeneous Helmholtz equation,

$\Delta u^{i}+k^{2}u^{i}=0\quad\text{in}\ \mathbb{R}^{N}.$ (1.2)

In the physical context, $k\in\mathbb{R}_{+}$ is the wavenumber and
$\mathbf{n}$ is the refractive index. For notational convenience, we extend
$\mathbf{n}$ to be $1$ outside $\Omega$. The forward scattering problem is
described by the following Helmholtz system

$\left\\{\begin{array}[]{ll}\Delta u+k^{2}\mathbf{n}^{2}(x)u=0&\text{in}\
\mathbb{R}^{N},\vskip 6.0pt plus 2.0pt minus 2.0pt\\\ u=u^{i}+u^{s}&\text{in}\
\mathbb{R}^{N},\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\displaystyle\lim\limits_{r\to\infty}r^{\frac{N-1}{2}}\left(\partial_{r}-\mathrm{i}k\right)u^{s}=0,&r=|x|,\end{array}\right.$
(1.3)

where $u$ and $u^{s}$ are respectively referred to as the total and scattered
fields. The last limit in (1.3) is known as the Sommerfeld radiation condition
which holds uniformly in the angular variable $\hat{x}:=x/|x|$ and
characterizes the outgoing nature of the scattered wave field $u^{s}$. The
well-posedness of the scattering system (1.3) is known [22], and there exists
a unique solution $u\in H_{loc}^{2}(\mathbb{R}^{d})$ which admits the
following asymptotic expansion:

$u^{s}(x;u^{i},(\Omega,\mathbf{n}))=\frac{e^{\mathrm{i}kr}}{r^{(N-1)/2}}u_{\infty}(\hat{x})+\mathcal{O}(r^{-(N+1)/2})\quad\mbox{as}\
r\rightarrow\infty.$

$u_{\infty}$ is known as the far-field pattern. Introduce an abstract operator
which sends the inhomogeneity $(\Omega,\mathbf{n})$ to its far-field pattern
under the probing of the incident field as follows:

$\mathcal{F}((\Omega,\mathbf{n}),u^{i})=u_{\infty}.$ (1.4)

An inverse problem of practical importance is to recover the refractive
inhomogeneity $(\Omega,\mathbf{n})$ by knowledge of the far-field measurement.
From the inverse problem point of view, it seems more practical for one to
characterize the range of $\mathcal{F}$, namely $\mathrm{Range}(\mathcal{F})$
associated with all possible $u^{i}$, which contains all the “visible”
information. However, a different perspective was proposed in [12] and in fact
one can achieve super-resolution reconstruction for the inverse problem (1.4)
if instead using $\mathrm{Ker}(\mathcal{F})$. Indeed, it is shown in [12] that
$\mathcal{K}_{\epsilon}(\mathcal{F})$ can be obtained by knowledge of
$\mathrm{Range}(\mathcal{F})$, where $\epsilon\in\mathbb{R}_{+}$ is
sufficiently small. Here, $u^{i}\in\mathcal{K}_{\epsilon}(\mathcal{F})$ means
that $\|\mathcal{F}((\Omega,\mathbf{n}),u^{i})\|\leq\epsilon$. It turns out
that the $\mathrm{Ker}_{\epsilon}(\mathcal{F})$ actually consists of the
Herglotz extensions of the transmission eigenfunctions in (1.1). Hence, the
surface-localized transmission eigenmodes carry the geometric information of
the underlying refractive inhomogeneity, which forms the basis of the super-
resolution imaging scheme in [12]. Therefore, our study not only unveils a new
spectral phenomenon of high theoretical value, but also is of significant
practical interest.

##  2\. Main results

Let us consider the transmission eigenvalue problem (1.1) with $\Omega$ being
a ball in $\mathbb{R}^{N}$, $N=2,3$, and $\mathbf{n}$ being a positive
constant. By scaling and translation if necessary, we can assume that $\Omega$
is the unit ball, namely $\Omega:=\\{x\in\mathbb{R}^{N};|x|<1\\}$. In what
follows, we set

$\Omega_{\tau}:=\\{x\in\mathbb{R}^{N};|x|<\tau\\},\quad\tau\in(0,1).$ (2.1)

######  Definition 2.1.

Consider a function $\psi\in L^{2}(\Omega)$. It is said to be surface-
localized if there exists $\tau_{0}\in(0,1)$, sufficiently close to $1$, such
that

$\frac{\|\psi\|_{L^{2}(\Omega_{\tau_{0}})}}{\|\psi\|_{L^{2}(\Omega)}}\ll 1.$
(2.2)

It is easy to see that if $\psi$ is surface-localized, then its $L^{2}$-energy
concentrate in a small neighbourhood of $\partial\Omega$, namely
$\Omega\backslash\Omega_{\tau_{0}}$. The qualitative asymptotic smallness in
(2.2) shall become more quantitatively definite in what follows. In fact, we
can prove

######  Theorem 2.1.

Consider the transmission eigenvalue problem (1.1) and assume that $\Omega$ is
the unit ball and $\mathbf{n}\neq 1$ is a positive constant. Then for any
given $\tau\in(0,1)$, there exists a sequence of transmission eigenfunctions
$\\{w_{m},v_{m}\\}_{m\in\mathbb{N}}$ associated to eigenvalues $k_{m}$ such
that

$k_{m}\rightarrow\infty\ \mbox{as}\ m\rightarrow\infty\ \ \mbox{and}\ \
\lim_{m\rightarrow\infty}\frac{\|\psi_{m}\|_{L^{2}(\Omega_{\tau})}}{\|\psi_{m}\|_{L^{2}(\Omega)}}=0,\
\ \psi_{m}=w_{m},v_{m}.$ (2.3)

######  Remark 2.1.

By (2.3), it is clear that both $\\{w_{m}\\}_{m\in\mathbb{N}}$ and
$\\{v_{m}\\}_{m\in\mathbb{N}}$ are surface-localized according to Definition
2.1. In our subsequent proof of Theorem 2.3, it can be seen that the
transmission eigenmode corresponding to higher mode number is more localized
around the surface. In other words, in (2.3), $\tau$ can be very close to $1$
provided $k_{m}$ is sufficiently large. It is interesting to point out that
$1-\tau$ is actually the localizing radius of the eigenmode, which defines the
super-resolution power of the wave imaging scheme proposed in [12].

######  Remark 2.2.

It is sufficient for us to prove Theorem 2.1 only for the case $\mathbf{n}>1$.
In fact, let us suppose that Theorem 2.1 holds true for $\mathbf{n}>1$, and
instead consider the other case with $0<\mathbf{n}<1$. Set
$\tilde{k}=k\mathbf{n}$, $\tilde{\mathbf{n}}=\mathbf{n}^{-1}$, $\tilde{w}=v$
and $\tilde{v}=w$. Then (1.1) can be recast as

$\left\\{\begin{array}[]{ll}\Delta\tilde{w}+\tilde{k}^{2}\tilde{\mathbf{n}}^{2}\tilde{w}=0&\text{in}\
\Omega,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\Delta\tilde{v}+\tilde{k}^{2}\tilde{v}=0&\text{in}\ \Omega,\vskip 6.0pt plus
2.0pt minus 2.0pt\\\ \displaystyle{\tilde{w}=\tilde{v},\ \
\frac{\partial\tilde{w}}{\partial\nu}=\frac{\partial\tilde{v}}{\partial\nu}}&\text{on}\
\partial\Omega.\\\ \end{array}\right.$ (2.4)

Since $\tilde{\mathbf{n}}>1$, we readily have that there exist
$(v_{m},w_{m})=(\tilde{w}_{m},\tilde{v}_{m})$, $m\in\mathbb{N}$, associated to
$k_{m}=\tilde{k}_{m}/\mathbf{n}\rightarrow\infty$, which are surface-
localized. Hence, throughout the rest of the paper, we assume that
$\mathbf{n}>1$.

###  2.1. Two-dimensional result

In this subsection, we prove Theorem 2.1 in the two-dimensional case. Let
$x=(r\cos\theta,r\sin\theta)\in\mathbb{R}^{2}$ denote the polar coordinate. By
Fourier expansion, the solutions to (1.1) have the following series
expansions:

$w(x)=\sum_{m=0}^{\infty}\alpha_{m}J_{m}(k\mathbf{n}|x|)e^{\mathrm{i}m\theta},\quad
v(x)=\sum_{m=0}^{\infty}\beta_{m}J_{m}(k|x|)e^{\mathrm{i}m\theta},$ (2.5)

where $J_{m}$ is the $m$-th order Bessel function and
$\alpha_{m},\beta_{m}\in\mathbb{C}$ are the Fourier coefficients. Set

$w_{m}(x)=\alpha_{m}J_{m}(k\mathbf{n}|x|)e^{\mathrm{i}m\theta},\quad
v_{m}(x)=\beta_{m}J_{m}(k|x|)e^{\mathrm{i}m\theta}.$ (2.6)

In what follows, we shall construct the surface-localized transmission
eigenmodes of the form (2.6) to fulfil the requirement in Theorem 2.1. In
order to make $(w_{m},v_{m})$ in (2.6) transmission eigenfunctions of (1.1),
one has by using the two transmission conditions on $\partial\Omega$, together
with straightforward calculations that

$\beta_{m}=\frac{J_{m}(k\mathbf{n})}{J_{m}(k)}\alpha_{m},$ (2.7)

and $k$ must be a root of the following function

$f_{m}(k)=J_{m-1}(k)J_{m}(k\mathbf{n})-\mathbf{n}J_{m}(k)J_{m-1}(k\mathbf{n}),\
\ m\geq 1.$ (2.8)

Next, we prove the existence of transmission eigenvalues by finding roots of
$f_{m}$. In the sequel, we let $j_{m,s}$ denote the $s$-th positive root of
$J_{m}(t)$ (arranged according to the magnitude), and $j_{m,s}^{\prime}$
denote the $s$-th positive root of $J_{m}^{\prime}(t)$. Here, it is pointed
out that both $J_{m}(t)$ and $J_{m}^{\prime}(t)$ possess infinitely many
positive roots, accumulating only at $\infty$ (cf. [LiuZou]).

######  Lemma 2.1.

Let $\mathbf{n}>1$ and $s_{0}\in\mathbb{N}$ be fixed. Then there exists
$m_{0}(\mathbf{n},s_{0})\in\mathbb{N}$ such that when
$m>m_{0}(\mathbf{n},s_{0})$, one has

$\frac{j_{m,s_{0}}}{\mathbf{n}}\leq m.$ (2.9)

###### Proof.

According to the formula (1.2) in [24], we know

$m-\frac{a_{s}}{2^{1/3}}m^{1/3}<j_{m,s}<m-\frac{a_{s}}{2^{1/3}}m^{1/3}+\frac{3}{20}a_{s}^{2}\frac{2^{1/3}}{m^{1/3}},$

where $a_{s}$ is the $s$-th negative zero of the Airy function and has the
representation

$a_{s}=-\left[\frac{3\pi}{8}(4s-1)\right]^{2/3}(1+\sigma_{s}).$ (2.10)

Here, $\sigma_{s}$ in (2.10) can be estimated by

$0\leq\sigma_{s}\leq 0.130\left[\frac{3\pi}{8}(4s-1.051)\right]^{-2}.$ (2.11)

By combining the above estimates, one can show by straightforward calculations
that

$\frac{j_{m,s_{0}}}{\mathbf{n}}\leq m.$

The proof is complete. ∎

######  Lemma 2.2.

Let $\mathbf{n}>1$ and $s_{0}\in\mathbb{N}$ be fixed. Then there exists
$m_{0}(\mathbf{n},s_{0})\in\mathbb{N}$ such that when
$m>m_{0}(\mathbf{n},s_{0})$, the function $f_{m}(k)$ in (2.8) possesses at
least one zero point in $(\frac{j_{m,s_{0}}}{\mathbf{n}}$,
$\frac{j_{m,s_{0}+1}}{\mathbf{n}})$.

###### Proof.

According to the formula (9.5.2) in [1], we know

$m\leq
j_{m,1}^{\prime}<j_{m,1}<j_{m,2}^{\prime}<j_{m,2}<j_{m,3}^{\prime}<\cdots.$

By Lemma 2.1, one has $\frac{j_{m,s_{0}+1}}{\mathbf{n}}\leq m\leq
j_{m,1}^{\prime}$ for $m>m_{0}(\mathbf{n},s_{0}+1)$. Next we consider
$f_{m}(k)$ for
$k\in(\frac{j_{m,s_{0}}}{\mathbf{n}},\frac{j_{m,s_{0}+1}}{\mathbf{n}})$. We
have

$J_{m}(k)\geq 0,\quad k\in\left[0,j_{m,1}^{\prime}\right],$

and this implies $J_{m}(k)>0$ for
$k\in(\frac{j_{m,s_{0}}}{\mathbf{n}},\frac{j_{m,s_{0}+1}}{\mathbf{n}})$.
Following Lemma 2.1 in [21], we know that the positive zeros of $J_{m-1}$ are
interlaced with those of $J_{m}$, and hence

$J_{m-1}(j_{m,s_{0}})\cdot J_{m-1}(j_{m,s_{0}+1})<0.$

By using the above fact, one can show that

$\displaystyle f_{m}\left(\frac{j_{m,s_{0}}}{\mathbf{n}}\right)\cdot
f_{m}\left(\frac{j_{m,s_{0}+1}}{\mathbf{n}}\right)$ (2.12) $\displaystyle=$
$\displaystyle\,\mathbf{n}^{2}J_{m}\left(\frac{j_{m,s_{0}}}{\mathbf{n}}\right)\cdot
J_{m}\left(\frac{j_{m,s_{0}+1}}{\mathbf{n}}\right)\cdot
J_{m-1}(j_{m,s_{0}})\cdot J_{m-1}(j_{m,s_{0}+1})$ $\displaystyle\leq$
$\displaystyle\mathbf{n}^{2}(J_{m}(j^{\prime}_{m,1}))^{2}\cdot
J_{m-1}(j_{m,s_{0}})\cdot J_{m-1}(j_{m,s_{0}+1})$ $\displaystyle<$
$\displaystyle\,0.$

which readily implies by Rolle’s theorem that there exists at least one zero
point of $f(k)$ in $(\frac{j_{m,s_{0}}}{\mathbf{n}}$,
$\frac{j_{m,s_{0}+1}}{\mathbf{n}})$.

The proof is complete. ∎

Clearly, Lemma 2.2 proves the existence of transmission eigenvalues. In what
follows, for a fixed $s_{0}$, we let the transmission eigenvalue be denoted by

$k_{l_{m}}:=k_{m,s_{0}}\in\left(\frac{j_{m,s_{0}}}{\mathbf{n}},\frac{j_{m,s_{0}+1}}{\mathbf{n}}\right),\quad
m=m_{0}+1,m_{0}+2,m_{0}+3,\cdots,$ (2.13)

where $m_{0}=m_{0}(\mathbf{n},s_{0})$ is sufficiently large fulfilling the
requirements in Lemmas 2.1 and 2.2.

######  Lemma 2.3.

There exists constants $C$ amd $\gamma$ such that

$\frac{J^{\prime}_{m}(k_{l_{m}})}{J_{m}(k_{l_{m}})}\leqslant Cm^{\gamma}.$
(2.14)

###### Proof.

By Theorem 1 in [19], we have for $0<x<\sqrt{(m+1)(m+3)}$ that

$\begin{split}&\frac{J_{m}^{\prime}(x)}{J_{m}(x)}\\\
\leq&\frac{4x^{2}-12m-6+\sqrt{((2m+1)(2m+3)-4x^{2})^{3}+((2m+1)(2m+3))^{2}}}{2x((2m+1)(2m+5)-4x^{2})}.\end{split}$
(2.15)

The items in the numerator of the RHS of (2.15) can be estimated by

$\displaystyle 4x^{2}-12m-6\leqslant 4m^{2},$ (2.16)
$\displaystyle\sqrt{((2m+1)(2m+3)-4x^{2})^{3}+((2m+1)(2m+3))^{2}}\leqslant
4m^{3}.$

According to the formula (1.2) in [24], we further have

$m-\frac{a_{s}}{2^{1/3}}m^{1/3}<j_{m,s}<m-\frac{a_{s}}{2^{1/3}}m^{1/3}+\frac{3}{20}a_{s}^{2}\frac{2^{1/3}}{m^{1/3}}.$

Hence, it holds that

$\frac{m}{\mathbf{n}}\left(1+2(\frac{s_{0}}{m})^{\frac{2}{3}}\right)\leqslant\frac{j_{m,s_{0}}}{\mathbf{n}}<k_{l_{m}}<\frac{j_{m,s_{0}+1}}{\mathbf{n}}\leqslant\frac{m}{\mathbf{n}}\left(1+3(\frac{s_{0}+1}{m})^{\frac{2}{3}}+2(\frac{s_{0}+1}{m})^{\frac{4}{3}}\right).$

One thus has for sufficiently large $m$ that

$\frac{m}{\mathbf{n}}\leqslant
k_{l_{m}}\leqslant\frac{(\mathbf{n}+1)m}{2\mathbf{n}}.$

By substituting $x=k_{l_{m}}$ in (2.15), when $\mathbf{n}>1$

$2x((2m+1)(2m+5)-4x^{2})>\frac{2m}{\mathbf{n}}(4m^{2}+12m+5-4m^{2})>\frac{1}{\mathbf{n}}.$
(2.17)

Finally, by substituting (2.16) and (2.17) in (2.15), together with
straightforward calculations, we can arrive at (2.28).

The proof is complete. ∎

We are in a position to present the proof of Theorem 2.1, which shall be split
into two theorems as follows.

######  Theorem 2.2.

Consider the same setup as that in Theorem 2.1 in $\mathbb{R}^{2}$ and assume
that $\mathbf{n}>1$ and $\tau\in(0,1)$ is fixed. Let $(w_{m},v_{m})$ be the
pair of transmission eigenfunctions associated with $k_{l_{m}}$ in (2.13).
Then it holds that

$\lim\limits_{m\rightarrow\infty}\frac{\|v_{m}\|_{L^{2}(\Omega_{\tau})}}{\|v_{m}\|_{L^{2}(\Omega)}}=0.$
(2.18)

###### Proof.

Let $\beta_{m}=1$ in (2.7). Then one has

$\displaystyle\left\|v_{m}\right\|_{L^{2}(\Omega_{\tau})}^{2}$
$\displaystyle=\int_{\Omega_{\tau}}\left|J_{m}(k_{l_{m}}|x|)\right|^{2}\mathrm{d}x$
$\displaystyle=2\pi\int_{0}^{\tau}\left|J_{m}(k_{l_{m}}r)\right|^{2}r\mathrm{d}r$
$\displaystyle=2\pi\int_{0}^{\tau}rJ^{2}_{m}(k_{l_{m}}r)\mathrm{d}r.$

Similarly,

$\left\|v_{m}\right\|_{L^{2}(\Omega)}^{2}=2\pi\int_{0}^{1}rJ^{2}_{m}(k_{l_{m}}r)\mathrm{d}r\
\text{.}$

Set $f(r)=rJ_{m}^{2}(k_{l_{m}}r)$. By straightforward calculations, we have

$\displaystyle f^{\prime}(r)$ $\displaystyle=$ $\displaystyle
J_{m}^{2}(k_{l_{m}}r)+2k_{l_{m}}rJ_{m}(k_{l_{m}}r)J_{m}^{\prime}(k_{l_{m}}r),$
$\displaystyle f^{\prime\prime}(r)$ $\displaystyle=$ $\displaystyle
4k_{l_{m}}J_{m}(k_{l_{m}}r)J_{m}^{\prime}(k_{l_{m}}r)+2k_{l_{m}}^{2}rJ_{m}^{{}^{\prime}2}(k_{l_{m}}r)+2k_{l_{m}}^{2}rJ_{m}(k_{l_{m}}r)J_{m}^{\prime}(k_{l_{m}}r)$
$\displaystyle=$ $\displaystyle
4k_{l_{m}}J_{m}(k_{l_{m}}r)J_{m}^{\prime}(k_{l_{m}}r)+2k_{l_{m}}^{2}rJ_{m}^{{}^{\prime}2}(k_{l_{m}}r)$
$\displaystyle+$ $\displaystyle
2k_{l_{m}}^{2}rJ_{m}(k_{l_{m}}r)\left((\frac{m^{2}}{k_{l_{m}}^{2}r^{2}}-1)J_{m}(k_{l_{m}}r)-\frac{1}{k_{l_{m}}r}J_{m}^{\prime}(k_{l_{m}}r)\right)$
$\displaystyle=$ $\displaystyle
2k_{l_{m}}J_{m}(k_{l_{m}}r)J_{m}^{\prime}(k_{l_{m}}r)+2k_{l_{m}}^{2}rJ_{m}^{{}^{\prime}2}(k_{l_{m}}r)+2k_{l_{m}}^{2}rJ_{m}(k_{l_{m}}r)\left(\frac{m^{2}}{k_{l_{m}}^{2}r^{2}}-1\right)J_{m}(k_{l_{m}}r).$

Noting that $k_{l_{m}}r\leqslant m$, one clearly has

$\frac{m^{2}}{k_{l_{m}}^{2}r^{2}}-1\geq 0,\ f^{{}^{\prime\prime}}(r)\geq
0,\quad r\in[0,1].$ (2.19)

Hence $f(r)$ is a convex function on $[0,1]$. Therefore, $\int_{0}^{1}f(r)dr$
is bigger than the area of the triangle under the tangent of $f(1)$ (see 1 for
a schematic illustration).

Figure 1.  Schematic illustration of $\int_{0}^{1}f(r)dr$ which is bigger than
the area of the triangle under the tangent of $f(1)$.

The hypotenuse of the aforementioned triangle is

$p=(J^{2}_{m}(k_{l_{m}})+2k_{l_{m}}J_{m}(k_{l_{m}})J^{\prime}_{m}(k_{l_{m}}))(r-1)+J^{2}_{m}(k_{l_{m}}),$

and the lengths of its base and height are respectively
$l:=\frac{J_{m}(k_{l_{m}})}{J_{m}(k_{l_{m}})+2k_{l_{m}}J^{\prime}_{m}(k_{l_{m}})}$
and $h:=J^{2}_{m}(k_{l_{m}})$. Hence, the area of the triangle is

$\displaystyle S_{\triangle}=\frac{1}{2}lh$
$\displaystyle=\frac{\frac{1}{2}J^{3}_{m}(k_{l_{m}})}{J_{m}(k_{l_{m}})+2k_{l_{m}}J^{\prime}_{m}(k_{l_{m}})}$
$\displaystyle\leqslant\int_{0}^{1}rJ^{2}_{m}(k_{l_{m}}r)\mathrm{d}r.$

Since $f$ is monotonically increasing, one has
$\int_{0}^{\tau}f(r)\mathrm{d}r\leqslant\tau f(\tau)$,

$\int_{0}^{\tau}rJ^{2}_{m}(k_{l_{m}}r)\mathrm{d}r\leqslant\tau^{2}J^{2}_{m}(k_{l_{m}}\tau).$

Therefore, it holds that

$\displaystyle\frac{\|v_{m}\|^{2}_{L^{2}(\Omega_{\tau})}}{\|v_{m}\|^{2}_{L^{2}(\Omega)}}$
$\displaystyle=\frac{\int_{0}^{\tau}rJ^{2}_{m}(k_{l_{m}}r)\mathrm{d}r}{\int_{0}^{1}rJ^{2}_{m}(k_{l_{m}}r)\mathrm{d}r}$
$\displaystyle\leqslant\frac{\tau^{2}J^{2}_{m}(k_{l_{m}}\tau)}{\frac{\frac{1}{2}J^{3}_{m}(k_{l_{m}})}{J_{m}(k_{l_{m}})+2k_{l_{m}}J^{\prime}_{m}(k_{l_{m}})}}$
$\displaystyle\leqslant
2\tau^{2}\left(\frac{J_{m}(k_{l_{m}}\tau)}{J_{m}(k_{l_{m}})}\right)^{2}\left(1+2k_{l_{m}}\frac{J^{\prime}_{m}(k_{l_{m}})}{J_{m}(k_{l_{m}})}\right).$

By virtue of Lemma 2.3, it further holds that

$\displaystyle\frac{\|v_{m}\|^{2}_{L^{2}(\Omega_{\tau})}}{\|v_{m}\|^{2}_{L^{2}(\Omega)}}$
$\displaystyle\leqslant
2\tau^{2}\left(\frac{J_{m}(k_{l_{m}}\tau)}{J_{m}(k_{l_{m}})}\right)^{2}\left(1+2k_{l_{m}}\frac{J^{\prime}_{m}(k_{l_{m}})}{J_{m}(k_{l_{m}})}\right)$
(2.20) $\displaystyle\leqslant
36\mathbf{n}m^{4}\tau^{2}\left(\frac{J_{m}(k_{l_{m}}\tau)}{J_{m}(k_{l_{m}})}\right)^{2}\
\text{.}$

Next, $\frac{J_{m}(k_{l_{m}}\tau)}{J_{m}(k_{l_{m}})}$ in (2.20) can be
estimated by the Carlini formula (see formula (30.1) in [18]):

$\displaystyle\left|\frac{J_{m}(k_{l_{m}}\tau)}{J_{m}(k_{l_{m}})}\right|$
$\displaystyle=\left|\frac{\frac{(m(\frac{k_{l_{m}}\tau}{m}))^{m}\exp\left\\{m\sqrt{1-(\frac{k_{l_{m}}\tau}{m})^{2}}\right\\}\exp(-V(m,\frac{k_{l_{m}}\tau}{m}))}{e^{m}\Gamma(m+1)(1-(\frac{k_{l_{m}}\tau}{m})^{2})^{1/4}\left\\{1+\sqrt{1-(\frac{k_{l_{m}}\tau}{m})^{2}}\right\\}^{m}}}{\frac{(m(\frac{k_{l_{m}}}{m}))^{m}\exp\left\\{m\sqrt{1-(\frac{k_{l_{m}}}{m})^{2}}\right\\}\exp(-V(m,\frac{k_{l_{m}}}{m}))}{e^{m}\Gamma(m+1)(1-(\frac{k_{l_{m}}}{m})^{2})^{1/4}\left\\{1+\sqrt{1-(\frac{k_{l_{m}}}{m})^{2}}\right\\}^{m}}}\right|$
(2.21)
$\displaystyle=\left|\left(\frac{1-(\frac{k_{l_{m}}}{m})^{2}}{1-(\frac{k_{l_{m}}\tau}{m})^{2}}\right)^{\frac{1}{4}}\times\left(\frac{\frac{(\frac{k_{l_{m}}\tau}{m})e^{\sqrt{1-(\frac{k_{l_{m}}\tau}{m})^{2}}}}{1+\sqrt{1-(\frac{k_{l_{m}}\tau}{m})^{2}}}}{\frac{(\frac{k_{l_{m}}}{m})e^{\sqrt{1-(\frac{k_{l_{m}}}{m})^{2}}}}{1+\sqrt{1-(\frac{k_{l_{m}}}{m})^{2}}}}\right)^{m}\times
e^{V(m,\frac{k_{l_{m}}}{m})-V(m,\frac{k_{l_{m}}\tau}{m})}\right|$
$\displaystyle:=I_{1}\times I_{2}\times I_{3}.$

Next, we estimate the terms $I_{i},i=1,2,3$ in (2.21). Since

$\frac{1}{\mathbf{n}}<\frac{k_{l_{m}}}{m}<\frac{1+\mathbf{n}}{2\mathbf{n}}\ ,$

we first have

$I_{1}:=\left|\left(\frac{1-(\frac{k_{l_{m}}}{m})^{2}}{1-(\frac{k_{l_{m}}\tau}{m})^{2}}\right)^{\frac{1}{4}}\right|<\frac{1}{1-\frac{\mathbf{n}^{2}+2\mathbf{n}+1}{4\mathbf{n}^{2}}}<\frac{1}{\mathbf{n}-1}\
\text{.}$ (2.22)

For $I_{3}$, since the limit of $\frac{k_{l_{m}}}{m}$ is strictly smaller than
$1$, one has

$e^{V(m,\frac{k_{l_{m}}}{m})}=1+o(1),\quad m\rightarrow\infty,$

and hence

$I_{3}:=\big{|}e^{V(m,\frac{k_{l_{m}}}{m})-V(m,\frac{k_{l_{m}}\tau}{m})}\big{|}=\frac{1+o(1)}{1+o(1)}.$

Therefore there exists a sufficiently large $m_{0}$ such that when $m>m_{0}$

$I_{3}=|e^{V(m,\frac{k_{l_{m}}}{m})-V(m,\frac{k_{l_{m}}\tau}{m})}|<2.$ (2.23)

For $I_{2}$, we introduce the auxiliary function $\varphi$:

$\varphi(x)=\frac{xe^{\sqrt{1-x^{2}}}}{1+\sqrt{1-x^{2}}}.$

Since
$\varphi^{\prime}(x)=\frac{(1-x^{2}+\sqrt{1-x^{2}})e^{\sqrt{1-x^{2}}}}{(1+\sqrt{1-x^{2}})^{2}}>0$,
$x\in(0,1)$, $\varphi$ is monotonically increasing. Noting that

$\lim\limits_{m\rightarrow\infty}\frac{k_{l_{m}}}{m}-\frac{k_{l_{m}}\tau}{m}=\frac{(1-\tau)}{\mathbf{n}}>0,$

there exists $\delta(\tau,\mathbf{n})>0$ such that

$\frac{\varphi(\frac{k_{l_{m}}}{m})}{\varphi(\frac{k_{l_{m}}\tau}{m})}<1-\delta(\tau,\mathbf{n}).$
(2.24)

Combining (2.22), (2.23) and (2.24), together with straightforward
calculations, one can show that

$\frac{\left\|v_{m}\right\|_{L^{2}(\Omega_{\tau})}^{2}}{\left\|v_{m}\right\|_{L^{2}(\Omega)}^{2}}\leqslant
144\frac{\mathbf{n}}{(\mathbf{n}-1)^{2}}m^{4}\tau^{2}(1-\delta(\tau,\mathbf{n}))^{2m}.$

That is

$\lim\limits_{m\rightarrow\infty}\frac{\left\|v_{m}\right\|_{L^{2}(\Omega_{\tau})}^{2}}{\left\|v_{m}\right\|_{L^{2}(\Omega)}^{2}}=0.$

The proof is complete. ∎

######  Theorem 2.3.

Consider the same setup as that in Theorem 2.2. The corresponding transmission
eigenfunctions $\\{w_{m}\\}_{m\in\mathbb{N}}$ are also surface-localized in
the sense that

$\lim_{m\rightarrow\infty}\frac{\left\|w_{m}\right\|_{L^{2}(\Omega_{\tau})}}{\left\|w_{m}\right\|_{L^{2}(\Omega)}}=0.$
(2.25)

###### Proof.

Since $w_{m}(x)=J_{m}(\mathbf{n}k_{l_{m}}x)$, one has

$\displaystyle\mathbf{n}k_{l_{m}}$
$\displaystyle>\mathbf{n}\frac{j_{m,s_{0}}}{\mathbf{n}}>m\Big{(}1+2\big{(}\frac{s_{0}+1}{m}\big{)}^{\frac{2}{3}}\Big{)}.$

Next, we show that for $m$ sufficiently large, one has
$\mathbf{n}k_{l_{m}}\tau<j^{\prime}_{m,1}$. Indeed, it can be deduced that

$\displaystyle\frac{m}{\mathbf{n}k_{l_{m}}}<\frac{j^{\prime}_{m,1}}{\mathbf{n}k_{l_{m}}}<\frac{m(1+3(\frac{1}{m})^{\frac{2}{3}}+2(\frac{1}{m})^{\frac{4}{3}})}{m(1+2(\frac{s_{0}+1}{m})^{\frac{2}{3}})},$
(2.26)

where we make use of the following fact

$\displaystyle\frac{m}{j_{m,s_{0}+1}}=\frac{m}{\mathbf{n}\cdot\frac{j_{m,s_{0}+1}}{\mathbf{n}}}<\frac{m}{\mathbf{n}k_{l_{m}}}<\frac{m}{\mathbf{n}\cdot\frac{j_{m,s_{0}}}{\mathbf{n}}}=\frac{m}{j_{m,s_{0}}}.$

Since

$\displaystyle\lim\limits_{m\rightarrow\infty}\frac{m}{\mathbf{n}k_{l_{m}}}=1,$
(2.27)
$\displaystyle\lim\limits_{m\rightarrow\infty}\frac{m(1+3(\frac{1}{m})^{\frac{2}{3}}+2(\frac{1}{m})^{\frac{4}{3}})}{m(1+2(\frac{s_{0}+1}{m})^{\frac{2}{3}})}=1,$

we thus have from (2.26) that

$\lim\limits_{m\rightarrow\infty}\frac{j^{\prime}_{m,1}}{\mathbf{n}k_{l_{m}}}=1.$

Hence, for $\varepsilon=\frac{1}{2}(1-\tau)$, there exists a
$m_{0}\in\mathbb{N}$ such that when $m>m_{0}$, we have

$\frac{j^{\prime}_{m,1}}{\mathbf{n}k_{l_{m}}}>1-\varepsilon>\frac{1}{2}(1+\tau)>\tau.$

That is, $\mathbf{n}k_{l_{m}}\tau<j^{\prime}_{m,1}$. Next, by using the
Carlini formula again, we have

$\displaystyle\frac{\left\|w_{m}\right\|_{L^{2}(\Omega_{\tau})}^{2}}{\left\|w_{m}\right\|_{L^{2}(\Omega)}^{2}}$
$\displaystyle=\frac{\int_{0}^{\tau}rJ_{m}^{2}(\mathbf{n}k_{l_{m}}r)\mathrm{d}r}{\int_{0}^{1}rJ_{m}^{2}(\mathbf{n}k_{l_{m}}r)\mathrm{d}r}$
$\displaystyle\leqslant\frac{\int_{0}^{\tau}rJ_{m}^{2}(\mathbf{n}k_{l_{m}}r)\mathrm{d}r}{\int_{0}^{\frac{j^{\prime}_{m,1}}{\mathbf{n}k_{l_{m}}}}rJ_{m}^{2}(\mathbf{n}k_{l_{m}}r)\mathrm{d}r}.$

The rest of the proof is similar to that of Theorem 2.2, and by
straightforward calculations one can show that

$\frac{\left\|w_{m}\right\|_{L^{2}(\Omega_{\tau})}^{2}}{\left\|w_{m}\right\|_{L^{2}(\Omega)}^{2}}\leqslant
36\mathbf{n}m^{4}\tau^{2}\left(\frac{J_{m}(\mathbf{n}k_{l_{m}}\tau)}{J_{m}(j^{\prime}_{m,1})}\right)^{2},$

which readily implies (2.25).

The proof is complete. ∎

###  2.2. Three-dimensional result

The proof of Theorem 2.1 in three dimensions is similar to the two-dimensional
case in Theorems 2.2 and 2.3. We only sketch the necessary modifications in
what follows.

######  Theorem 2.4.

Consider the same setup as that in Theorem 2.1 in $\mathbb{R}^{3}$ and assume
that $\mathbf{n}>1$ and $\tau\in(0,1)$ is fixed. Then (2.3) holds true.

###### Proof.

By Fourier expansion, the solutions to (1.1) in $\mathbb{R}^{3}$ have the
following series expansions:

$\displaystyle w(x)$
$\displaystyle=\sum_{m=0}^{\infty}\sum_{l=-m}^{m}\alpha_{m}^{l}j_{m}(k\mathbf{n}|x|)Y_{m}^{l}(\hat{x}),$
(2.28) $\displaystyle v(x)$
$\displaystyle=\sum_{m=0}^{\infty}\sum_{l=-m}^{m}\beta_{m}^{l}j_{m}(k|x|)Y_{m}^{l}(\hat{x}),$

where $\hat{x}:=x/|x|$, $Y_{m}^{l}$ is the Spherical harmonic function of
order $m$ and degree $l$, and

$j_{m}(|x|)=\sqrt{\frac{\pi}{2|x|}}J_{m+1/2}(|x|),$ (2.29)

is known as the _spherical Bessel function_. In what follows, we shall look
for surface-localized transmission eigenfunctions of the following form:

$\displaystyle w_{l,m}(x)$
$\displaystyle=\alpha_{m}j_{m}(k\mathbf{n}|x|)Y_{m}^{l}(\hat{x}),$ (2.30)
$\displaystyle v_{l,m}(x)$
$\displaystyle=\beta_{m}j_{m}(k|x|)Y_{m}^{l}(\hat{x}),$

where $\alpha_{m}$ and $\beta_{m}$ are constants. By using the two
transmission conditions on $\partial\Omega$, together with straightforward
calculations, one can show that

$\alpha_{m}=1,\quad\beta_{m}=\mathbf{n}^{-\frac{1}{2}}\frac{J_{m+\frac{1}{2}}(k\mathbf{n})}{J_{m+\frac{1}{2}}(k)}\alpha_{m},$

and $k$ should be a root of the following function

$f_{m+\frac{1}{2}}(k)=J_{m-\frac{1}{2}}(k)J_{m+\frac{1}{2}}(k\mathbf{n})-\mathbf{n}J_{m+\frac{1}{2}}(k)J_{m-\frac{1}{2}}(k\mathbf{n}).$
(2.31)

Next, we construct the desired transmission eigenvalues by showing that
$f_{m+\frac{1}{2}}(k)$ has least one zero point in
$\big{(}\frac{j_{m+\frac{1}{2},s_{0}}}{\mathbf{n}},\frac{j_{m+\frac{1}{2},s_{0}+1}}{\mathbf{n}}\big{)}$.
Spherical Bessel function satisfies the same estimate in Lemma 2.2. Then for
any fixed $s_{0}$, there exists a sufficiently large
$m_{0}(\mathbf{n},s_{0})\in\mathbb{N}_{+}$ such that when
$m>m_{0}(\mathbf{n},s_{0})$, we have

$\frac{j_{m+\frac{1}{2},s_{0}+1}}{\mathbf{n}}\leq m+\frac{1}{2}.$

On the other hand, the spherical Bessel functions possess the following
property [21]:

$m+\frac{1}{2}\leq
j^{\prime}_{m+\frac{1}{2},1}<j_{m+\frac{1}{2},1}<j^{\prime}_{m+\frac{1}{2},2}<j_{m+\frac{1}{2},2}<j^{\prime}_{m+\frac{1}{2},3}<\cdots.$

Consider the function $f_{m+\frac{1}{2}}(k)$ in
$(\frac{j_{m+\frac{1}{2},s_{0}}}{\mathbf{n}},\frac{j_{m+\frac{1}{2},s_{0}+1}}{\mathbf{n}})$.
By the monotonicity of the Bessel function, we have

$J_{m+\frac{1}{2}}(k)\geq 0,\quad
k\in\left[0,j^{\prime}_{m+\frac{1}{2},1}\right].$

According to the formula (9.5.2) in [1], we know

$\displaystyle
f_{m+\frac{1}{2}}\left(\frac{j_{m+\frac{1}{2},s_{0}}}{\mathbf{n}}\right)\cdot
f_{m+\frac{1}{2}}\left(\frac{j_{m+\frac{1}{2},s_{0}+1}}{\mathbf{n}}\right)$
$\displaystyle=$
$\displaystyle\,\mathbf{n}^{2}J_{m+\frac{1}{2}}\left(\frac{j_{m+\frac{1}{2},s_{0}}}{\mathbf{n}}\right)\cdot
J_{m+\frac{1}{2}}\left(\frac{j_{m+\frac{1}{2},s_{0}+1}}{\mathbf{n}}\right)\cdot
J_{m-\frac{1}{2}}(j_{m+\frac{1}{2},s_{0}})\cdot
J_{m-\frac{1}{2}}(j_{m+\frac{1}{2},s_{0}+1})$ $\displaystyle\leq$
$\displaystyle\,\mathbf{n}^{2}(J_{m+\frac{1}{2}}(j_{m+\frac{1}{2},1}^{\prime}))^{2}\cdot
J_{m-\frac{1}{2}}(j_{m+\frac{1}{2},s_{0}})\cdot
J_{m-\frac{1}{2}}(j_{m+\frac{1}{2},s_{0}+1})$ $\displaystyle<$
$\displaystyle\,0.$

Therefore by Rolle’s theorem, $f_{m+\frac{1}{2}}(k)$ has at least one zero
point in
$\left(\frac{j_{m+\frac{1}{2},s_{0}}}{\mathbf{n}},\frac{j_{m+\frac{1}{2},s_{0}+1}}{\mathbf{n}}\right)$.

For any fixed $s_{0}\in\mathbb{N}$, we denote the aforementioned zero point in
$\left(\frac{j_{m+\frac{1}{2},s_{0}}}{\mathbf{n}},\frac{j_{m+\frac{1}{2},s_{0}+1}}{\mathbf{n}}\right)$
as $k_{l_{m}}$ (comparing to (2.13) in the two-dimensional case). Let
$(w_{m},v_{m})$ be the transmission eigenfunctions associated to the
eigenvalue $k_{l_{m}}$. Next we prove that $\\{v_{m}\\}_{m\in\mathbb{N}}$ are
surface-localized on $\partial\Omega$. By straightforward calculations, one
has

$\displaystyle\left\|v_{m}\right\|_{L^{2}(\Omega_{\tau})}^{2}=\int_{\Omega_{\tau}}\left|j_{m}(k_{l_{m}}|x|)\right|^{2}\mathrm{d}x$
$\displaystyle=$ $\displaystyle
2\pi\int_{0}^{\tau}\left|j_{m}(k_{l_{m}}r)\right|^{2}r^{2}\mathrm{d}r=2\pi\int_{0}^{\tau}r^{2}j_{m}^{2}(k_{l_{m}}r)\mathrm{d}r$
$\displaystyle=$ $\displaystyle
2\pi\int_{0}^{\tau}r^{2}\frac{\pi}{2r}J_{m+\frac{1}{2}}^{2}(k_{l_{m}}r)\mathrm{d}r=\pi^{2}\int_{0}^{\tau}rJ_{m+\frac{1}{2}}^{2}(k_{l_{m}}r)\mathrm{d}r.$

Hence, it holds that

$\frac{\left\|v_{m}\right\|_{L^{2}(\Omega_{\tau})}^{2}}{\left\|v_{m}\right\|_{L^{2}(\Omega)}^{2}}=\frac{\int_{0}^{\tau}rJ_{m+\frac{1}{2}}^{2}(k_{l_{m}}r)\mathrm{d}r}{\int_{0}^{1}rJ_{m+\frac{1}{2}}^{2}(k_{l_{m}}r)\mathrm{d}r}.$
(2.32)

Consider the function $\zeta(r)=rJ_{m+\frac{1}{2}}^{2}(k_{l_{m}}r)$. It can be
shown that $\zeta$ is convex in $(0,1)$. Using this fact, one can further
estimate that

$\displaystyle\frac{\left\|v_{m}\right\|_{L^{2}(\Omega_{\tau})}^{2}}{\left\|v_{m}\right\|_{L^{2}(\Omega)}^{2}}$
$\displaystyle=\frac{\int_{0}^{\tau}rJ_{m+\frac{1}{2}}^{2}(k_{l_{m}}r)\mathrm{d}r}{\int_{0}^{1}rJ_{m+\frac{1}{2}}^{2}(k_{l_{m}}r)\mathrm{d}r}$
$\displaystyle\leqslant
2\tau^{2}\left(\frac{J_{m+\frac{1}{2}}(k_{l_{m}}\tau)}{J_{m+\frac{1}{2}}(k_{l_{m}})}\right)^{2}\left(1+2k_{l_{m}}\frac{J_{m+\frac{1}{2}}^{\prime}(k_{l_{m}})}{J_{m+\frac{1}{2}}(k_{l_{m}})}\right).$

Similar to Lemma 2.3, one can show that there exists constants $C$ and
$\gamma$ such that

$\frac{k_{l_{m}}J^{\prime}_{m+\frac{1}{2}}(k_{l_{m}})}{J_{m+\frac{1}{2}}(k_{l_{m}})}<Cm^{\gamma}.$

Finally, by following a similar argument to the two-dimensional case and
combining the above estimates, together with the use of the Carlini formula,
one can show that

$\lim\limits_{m\rightarrow\infty}\frac{\left\|v_{m}\right\|_{L^{2}(\Omega_{\tau})}}{\left\|v_{m}\right\|_{L^{2}(\Omega)}}=0.$
(2.33)

By following a completely similar argument to that of Theorem 2.3 in the two-
dimensional case, one can show that (2.33) also holds for $w_{m}$.

The proof is complete. ∎

## Acknowledgement

The work of Y. Deng was supported by NSF grant of China No. 11971487 and NSF
grant of Hunan No. 2020JJ2038. The work of H Liu was supported by a startup
fund from City University of Hong Kong and the Hong Kong RGC grants (projects
12302018, 12302919 and 12301420). The work of K. Zhang was supported by the
NSF grant of China No. 11871245.

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