# Unique determination by a single far-field measurement for an inverse
elastic problem

Huaian Diao  School of Mathematics and Key Laboratory of Symbolic Computation
and Knowledge Engineering of Ministry of Education, Jilin University,
Changchun, Jilin, China<EMAIL_ADDRESS><EMAIL_ADDRESS>,  Ruixiang Tang
School of Mathematics, Jilin University, Changchun, Jilin 130012, China.
<EMAIL_ADDRESS><EMAIL_ADDRESS>,  Hongyu Liu  Department of
Mathematics, City University of Hong Kong, Kowloon, Hong Kong SAR, China.
<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Jiexin Tang  The No. 3
Junior High School of Yantai Economic and Technological Development Zone, No.
1 Changzhou Street, Yantai Economic and Technological Development Zone,
Yantai, Shandong 265503, China<EMAIL_ADDRESS>
###### Abstract.

This paper is concerned with the unique identification of the shape of a
scatterer through a single far-field pattern in an inverse elastic medium
scattering problem with a generalized transmission boundary condition. The
uniqueness issue by a single far-field measurement is a challenging problem in
inverse scattering theory, which has a long and colorful history. In this
paper, we establish the uniqueness results by a single far-field measurement
under a generic scenario when dealing with underlying elastic scatterers
exhibiting polygonal-nest or polygonal-cell structures. Furthermore, for
polygonal-nest or polygonal-cell structure scatterers associated with density
and boundary impedance parameters as piecewise constants, we show that these
physical quantities can be uniquely determined simultaneously by a single far-
field measurement. The corresponding proof relies heavily on examining the
singular behaviour of a coupled PDE system near a corner in a microlocal
manner.

Keywords: inverse elastic scattering, generalized transmission boundary
condition, single far-field measurement, polygonal-nest and polygonal-cell
structure, corner singularity, unique identifiability

2020 Mathematics Subject Classification: 35R30; 74J20; 86A22

##  1\. Introduction

###  1.1. Mathematical setup

Initially, we focus on the mathematical model for our subsequent study. We
assume that an elastic scatterer $\Omega$ is a bounded Lipschitz domain in a
homogenous background $\mathbb{R}^{2}$ with the corresponding medium
configuration $\lambda$, $\mu$, $q\in L^{\infty}(\Omega)$ and $\eta\in
L^{\infty}(\partial\Omega)$, where $q$ and $\eta$ are real valued functions.
Indeed, $q$ characterizes the density of the scatterer $\Omega$. Here the Lamé
constants $\lambda$ and $\mu$ satisfy the following strong convexity condition

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

The homogenous background $\mathbb{R}^{2}$ is characterized by the Lamé
constants $\lambda$ and $\mu$. Let ${\mathbf{u}}^{i}$ be an incident elastic
wave field, which is an entire solution to the Lamé system as follows:

$\mathcal{L}{\mathbf{u}}^{i}+\omega^{2}{\mathbf{u}}^{i}={\mathbf{0}}\
\mbox{in}\ \mathbb{R}^{2},$ (1.2)

where $\mathcal{L}:=\mu\Delta+(\lambda+\mu)\nabla(\nabla\cdot)$ and
$\omega\in\mathbb{R}_{+}$ signifies the angular frequency of the time-harmonic
elastic wave propagation. Therefore, the resulting elastic scattering is
caused by the interaction between the incident field ${\mathbf{u}}^{i}$ and
the elastic scatterer $(\Omega;\lambda,\mu,q,\eta)$ associated with a
generalized transmission boundary condition. Throughout of this paper, the
compressional and shear wave numbers are defined by

$k_{p}:=\frac{\omega}{\sqrt{2\mu+\lambda}},\quad
k_{s}:=\frac{\omega}{\sqrt{\mu}}.$ (1.3)

Let $\bf curl$ and curl operators be defined in $\mathbb{R}^{2}$ as follows:

${\rm curl}~{}{\mathbf{u}}=\partial_{1}u_{2}-\partial_{2}u_{1},\quad{\bf
curl}~{}{u}=(\partial_{1}u,-\partial_{2}u)^{\top}.$ (1.4)

where $\mathbf{u}=(u_{1},u_{2})$ and $u$ are vector-valued and scalar
functions, respectively. For a vector-valued function ${\mathbf{u}}$, the
compressional and shear parts of ${\mathbf{u}}$ are defined by

${\mathbf{u}}_{p}:=-\frac{1}{k_{p}^{2}}\nabla(\nabla\cdot{\mathbf{u}}),\quad{\mathbf{u}}_{s}:=\frac{1}{k_{s}^{2}}{\bf
curl}\ \mathrm{curl}{\mathbf{u}}.$ (1.5)

In what follows, for a given vector-valued function ${\mathbf{u}}$, the
traction operator $T_{\nu}{\mathbf{u}}$ on $\partial\Omega$ is defined as

$T_{\nu}{\mathbf{u}}=\lambda(\nabla\cdot{\mathbf{u}})\nu+2\mu(\nabla^{s}{\mathbf{u}})\nu,\quad\nabla^{s}{\mathbf{u}}=\frac{1}{2}\left(\nabla{\mathbf{u}}+\nabla{{\mathbf{u}}^{\top}}\right),$
(1.6)

where $\nu$ is the exterior normal direction to $\partial\Omega$. The
aforementioned elastic scattering problem is governed by the Lamé system as
follows:

$\begin{cases}\mathcal{L}{\mathbf{u}}+\omega^{2}q{\mathbf{u}}={\mathbf{0}},&\mbox{in}\quad\Omega,\\\
\mathcal{L}{\mathbf{u}}+\omega^{2}{\mathbf{u}}={\mathbf{0}},&\mbox{in}\quad\mathbb{R}^{2}\backslash\overline{\Omega},\\\
{\mathbf{u^{+}}}={\mathbf{u^{-}}},\
T_{\nu}{\mathbf{u^{+}}}+\eta{\mathbf{u^{+}}}=T_{\nu}{\mathbf{u^{-}}},\quad&\mbox{on}\quad\partial\Omega,\\\
{\mathbf{u}}={\mathbf{u}}^{i}+{\mathbf{u}}^{sc},\quad&\mbox{in}\quad\mathbb{R}^{2},\end{cases}$
(1.7)

where ${\mathbf{u}}$ is the total elastic wave field and ${\mathbf{u}}^{sc}$
denotes the scattered elastic wave field satisfying the following Kupradze
radiation condition

$\lim_{r\to\infty}r^{\frac{1}{2}}\left(\frac{\partial{\mathbf{u}}_{\beta}^{sc}}{\partial
r}-{\rm i}k_{\beta}{\mathbf{u}}_{\beta}^{sc}\right)=0,\ r:=|{\mathbf{x}}|,\
\beta=p,s.$ (1.8)

Here the compressional wave ${\mathbf{u}}_{p}^{s}$ and shear wave
${\mathbf{u}}_{s}^{s}$ of ${\mathbf{u}}^{s}$ are defined as in (1.5).

It should be noted that the generalized transmission condition
$T_{\nu}{\mathbf{u^{+}}}+\eta{\mathbf{u^{+}}}=T_{\nu}{\mathbf{u^{-}}}$ in
(1.7) provides not only mathematical generalization but also physical
significance. In fact, the conductive boundary condition is the term used to
describe the similar transmission condition across the boundary in the context
of electromagnetic scattering [2]. This boundary condition arises when
effectively describing a thin layer of highly conducting coating [2]. In
(1.7), the generalized transmission boundary condition can be employed to
accurately depict a thin layer of highly lossy elastic coating. We shall not
elaborate on this point, however, as it is not the focus of this article. In
particular, when the parameter $\eta$ in (1.7) equals zero, the generalized
transmission boundary condition reduces to the classical transmission
condition across the boundary of $\Omega$. This boundary condition describes
the continuity of the elastic wave with respect to the Dirichlet and Neumann
data across the boundary of $\Omega$. Furthermore, if $\eta\equiv 0$, the
scattering model (1.7) is simplified to the conventional problem of scattering
in an elastic medium. In the following text, the parameter $\eta$ is referred
to as the boundary impedance parameter. The well-posedness of the system (1.7)
with $\eta\equiv 0$ can be found in [17, 18, 3]. To prove the existence and
uniqueness of a solution
$\mathbf{u}=\mathbf{u}^{-}\chi_{\Omega}+\mathbf{u}^{+}\chi_{\mathbb{R}^{2}\backslash\overline{\Omega}}\in
H_{\rm loc}^{1}(\mathbb{R}^{2})^{2}$ to (1.7), the variational argument is
employed in Section 2. The asymptotic expansions (see [17, 12]) for
${\mathbf{u}}^{{sc}}$ can be characterized by

$\begin{split}{\mathbf{u}}^{{sc}}({\mathbf{x}})=&\frac{{e}^{\mathrm{i}k_{p}r}}{{r}}u_{p}^{\infty}(\hat{{\mathbf{x}}})\hat{{\mathbf{x}}}+\frac{{e}^{\mathrm{i}k_{s}r}}{{r}}U_{s}^{\infty}(\hat{{\mathbf{x}}})+O\left(\frac{1}{r^{3/2}}\right),\quad
U_{s}^{\infty}(\hat{{\mathbf{x}}})\cdot\hat{{\mathbf{x}}}=0,\quad\forall{\mathbf{x}}\in\mathbb{S}^{1}\end{split}$
(1.9)

as $r=|{\mathbf{x}}|\rightarrow\infty$, where the scalar analytic function
$u_{p}^{\infty}$ and the vector-valued analytic function $U_{s}^{\infty}$
defined on the unit sphere
$\mathbb{S}^{1}:=\\{\hat{{\mathbf{x}}}\in\mathbb{R}^{2}\big{|}|\hat{{\mathbf{x}}}|=1\\}$
are referred to the far-field pattern of ${\mathbf{u}}_{p}^{{s}}$ and
${\mathbf{u}}_{p}^{{s}}$ respectively. Define the far-field pattern
${\mathbf{u}}_{t}^{\infty}$ of ${\mathbf{u}}^{\mathrm{s}}$ as

${\mathbf{u}}_{t}^{\infty}(\hat{{\mathbf{x}}}):=u_{p}^{\infty}(\hat{{\mathbf{x}}})\hat{{\mathbf{x}}}+U_{s}^{\infty}(\hat{{\mathbf{x}}}).$

It is obvious that we have the following one-to-one relationship
${u}_{p}^{\infty}(\hat{\mathbf{x}})={\mathbf{u}}_{t}^{\infty}(\hat{{\mathbf{x}}})\cdot\hat{{\mathbf{x}}}$
and
${U}_{s}^{\infty}(\hat{{\mathbf{x}}})\cdot\hat{{\mathbf{x}}}^{\perp}={\mathbf{u}}_{t}^{\infty}(\hat{{\mathbf{x}}})\cdot\hat{{\mathbf{x}}}^{\perp}$.
Moreover, the one-to-one correspondence between ${\mathbf{u}}_{t}^{\infty}$
and $\mathbf{u}^{\mathrm{s}c}$ is a result of the Rellich Theorem [18]. This
paper primarily focuses on solving the following inverse problem:

$\mathcal{F}(\Omega;\lambda,\mu,q,\eta)={\mathbf{u}}_{\beta}^{\infty}(\hat{\mathbf{x}};\mathbf{u}^{i}),\quad\beta=t,\
p,\mbox{ or }s,$ (1.10)

where $\mathcal{F}$ is implicitly defined by the scattering system (1.7). If
the a far-field measurement
${\mathbf{u}}_{t}^{\infty}\left(\hat{\mathbf{x}};{\mathbf{u}}^{i}\right)(\hat{\mathbf{x}}\in\mathbb{S}^{1})$
is provided in association with a fixed incident wave ${\mathbf{u}}^{i}$, it
is referred to as a single far-field measurement for the inverse problem
(1.10). Otherwise it is referred to as multiple measurements. Our focus is on
the geometrical inverse problem (1.10), which involves recovering $\Omega$
independently of $q$ and $\eta$ through a single far-field measurement. The
unique determination of the shape of a scatterer through a single far-field
measurement has a rich and intricate history in inverse scattering problems
[8, 9, 13]. Our focus lies specifically on scenarios where $\Omega$ exhibits
polygonal-nest or polygonal-cell structures, which will be elaborated further
in Section 3.

###  1.2. Connections to previous results and main findings

Determining the geometrical information of an underlying scatterer by
measurement is a fundamental problem in inverse scattering. It is referred to
as the unique identifiability of the scatterer shape. Since the cardinality of
${\mathbf{u}}_{t}^{\infty}\left(\hat{\mathbf{x}};{\mathbf{u}}^{i}\right)$
associated with a single fixed incident wave ${\mathbf{u}}^{i}$ is equal to
the cardinality of $\partial\Omega$, it is clear that the geometrical inverse
problem (1.10) is formally determined by a single far-field measurement. On
the contrary, the inverse problem of determining the physical parameters $q$
and $\eta$ associated with $\Omega$ through a single far-field measurement is
underdetermined, given that $q$ and $\eta$ are variable functions. The
uniqueness of identifying the position and shape of the scatterer from a
single far-field measurement was referred to as Schiffer’s problem in the
literature [8]. Up to now, the corresponding developments to Schiffer’s
problem have been made by imposing a-priori information on the scatterer
$\Omega$, such as the size or geometry of $\Omega$. Notably, Colton and
Sleeman [10] employed a spectral argument for the Laplacian operator to
demonstrate that a sound-soft obstacle can be uniquely identified through a
finite number of far-field measurements. Kirsch and Kress [21] established the
uniqueness of the sound-hard obstacle scenario through infinitely many far-
field patterns. Isakov [20] used infinitely many far-field patterns to
determine the shape of a penetrable inhomogeneous medium uniquely [20].
Recently, there has been an increasing interest in establishing the uniqueness
of a single far-field measurement for determining the shape of an impenetrable
or penetrable polyhedral scatterer using a single far-field pattern. See [1,
4, 5, 7, 22, 23, 24, 25, 6, 27, 15, 16] and the references cited therein for
further details. For the elastic medium scattering problem, specifically when
$\eta\equiv 0$ in (1.7), we note that [19, 17] provided the identifiable
result to the inverse problem (1.10) when multiple far-field measurements are
employed. Under general circumstances, a single far-field measurement is
capable of uniquely determining a polygonal elastic scatterer with the
generalized transmission boundary condition, as illustrated in the work of
[14], where the parameter $\eta$ for the corresponding generalized
transmission condition can also be simultaneously identified. When a scatterer
has the polygonal-nest or polygonal-cell structure, in the acoustic
scattering, it was shown that the uniqueness result for determining the shape
of the underlying scatterer by a single far-field measurement hold under
generic physical conditions (cf.[5]). If a scatterer has a thin layer with
high conductivity, a single far-field measurement under generic conditions
(cf. [6]) can uniquely determine the shape and location of the scatterer with
the polygon-nest or polygon-cell structure.

In this study, we aim to demonstrate the uniqueness results for the
identification of the shape and location of an elastic polygonal nest or
polygonal cell structure scatterer associated with (1.7) by a single far-field
measurement. In addition, a single far-field measurement can also determine
the physical parameters $q$ and $\eta$ in (1.7) if they are piecewise
constants. To achieve the desired result, we will study the coupled
transmission PDE system around the underlying corner and apply the complex
geometric optics (CGO) solution to investigate the singularity caused by a
corner in a microlocal way.

Finally, the remainder of the paper is structured as follows. In Section 2, we
examine the existence and uniqueness of the solution to the direct scatter
problem. In Section 3, we provide a number of necessary geometrical and
mathematical setups. In Section 4, we derive auxiliary lemmas to facilitate
subsequent analysis. In Section 5, we define admissible scatterers and prove
the unique identifiability results.

##  2\. The direct problem

In this section, we will demonstrate the well-posedness of (1.7) to (1.8).
Since the incident wave ${\mathbf{u}}^{i}$ fulfills (1.2), by introducing

$\displaystyle{\mathbf{f}}:=\omega^{2}(1-q){\mathbf{u}}^{i},\quad{\mathbf{g}}:=-\eta{\mathbf{u}}^{i},$
(2.1)

the scattered wave filed ${\mathbf{u}}^{s}$ of (1.7) to (1.8) satisfies the
following problem

$\displaystyle\mathcal{L}{\mathbf{u}}^{s}+\omega^{2}q{\mathbf{u}}^{s}={\mathbf{f}},\quad$
$\displaystyle\mbox{in}\quad\Omega,$ (2.2)
$\displaystyle\mathcal{L}{\mathbf{u}}^{s}+\omega^{2}{\mathbf{u}}^{s}={\mathbf{0}},\quad$
$\displaystyle\mbox{in}\quad\mathbb{R}^{2}\backslash\overline{\Omega},$ (2.3)
$\displaystyle({\mathbf{u^{s}}})^{+}-({\mathbf{u^{s}}})^{-}={\mathbf{0}},\
T_{\nu}({\mathbf{u^{s}}})^{+}+\eta{\mathbf{(u^{s})^{+}}}-T_{\nu}({\mathbf{u}}^{s})^{-}={\mathbf{g}},\quad$
$\displaystyle\mbox{on}\quad\partial\Omega,$ (2.4)
$\displaystyle\lim_{r\to\infty}r^{\frac{1}{2}}\left(\frac{\partial{\mathbf{u}}_{\beta}^{s}}{\partial
r}-{\rm i}k_{\beta}{\mathbf{u}}_{\beta}^{s}\right)=0,\quad$ $\displaystyle
r:=|{\mathbf{x}}|,\ \beta=p,s,$ (2.5)

where $k_{\beta}$ and ${\mathbf{u}}^{s}_{\beta}$ are defined by (1.3) and
(1.5), respectively, $\beta=p,s$.

Define the elastic tensor $\mathcal{C}=(C_{ijkl})_{i,j,k,l=1}^{2}$ as

$C_{ijkl}=\lambda\delta_{ij}\delta_{kl}+\mu(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}),$

where $\delta_{ij}$ is the Kronecker symbol, $\lambda$ and $\mu$ are the Lamé
constants satisfying (1.1). Then it yields that

$\mathcal{L}{\mathbf{u}}=\nabla\cdot(\mathcal{C}:\nabla{\mathbf{u}}),\quad\mathcal{C}:\nabla{\mathbf{u}}=\left(\sum_{j,k,l=1}^{2}C_{ijkl}\partial_{l}u_{k}\right)_{i=1}^{2},\quad{\mathbf{u}}=(u_{i})_{i=1}^{2},$

where $\mathcal{L}$ is defined in (1.2). For ${\mathbf{f}}\in
L^{2}(\Omega)^{2}$ and ${\mathbf{g}}\in H^{-1/2}(\Omega)^{2}$, the solution
${\mathbf{u}}^{s}\in H^{1}_{\rm loc}(\mathbb{R}^{2})^{2}$ of (2.2) to (2.4) is
understood in the weak sence:

$\displaystyle\int_{\mathbb{R}^{2}\backslash\overline{\Omega}}\left((\mathcal{C}:\nabla{\mathbf{u}}^{s}):\nabla\overline{\bm{\varphi}}-\omega^{2}{\mathbf{u}}^{s}\cdot\overline{\bm{\varphi}}\right)\mathrm{d}\mathbf{x}+\int_{\Omega}\left((\mathcal{C}:\nabla{\mathbf{u}}^{s}):\nabla\overline{\bm{\varphi}}-\omega^{2}q{\mathbf{u}}^{s}\cdot\overline{\bm{\varphi}}\right)\mathrm{d}\mathbf{x}$
(2.6)
$\displaystyle\quad-\int_{\partial\Omega}\eta{\mathbf{u}}^{s}\cdot\overline{\bm{\varphi}}\mathrm{d}\sigma=-\int_{\Omega}{\mathbf{f}}\cdot\overline{\bm{\varphi}}\mathrm{d}x-\int_{\partial\Omega}{\mathbf{g}}\cdot\overline{\bm{\varphi}}\mathrm{d}\sigma$

for any test function $\bm{\varphi}\in H_{\rm loc}^{1}(\mathbb{R}^{2})^{2}$,
where $\bm{A}:\bm{B}=\sum_{i,j=1}^{2}a_{ij}b_{ij}$ for
$\bm{A}=(a_{ij})_{i,j=1}^{2}$ and $\bm{B}=(b_{ij})_{i,j=1}^{2}$.

Firstly, we demonstrates that there is at most one solution in $H_{\rm
loc}^{1}(\mathbb{R}^{2})^{2}$ for (2.2) to (2.5), or equivalently, a unique
solution exists in $H_{\rm loc}^{1}(\mathbb{R}^{2})^{2}$ for (2.6) and (2.5).
Prior to this, we revisit the Rellich’s type lemma in linear elasticity, which
is outlined in the following lemma. The subsequent lemma will be utilized to
substantiate Theorem 2.1.

######  Lemma 2.1.

[18] Let $B_{R}$ be an appropriate disk centered at origin with a radius
$R\in\mathbb{R}_{+}$, and assume that $\mathbf{u}^{s}$ is a radiating solution
to

$\mu\Delta\mathbf{u}^{s}+(\lambda+\mu)\nabla(\nabla\cdot\mathbf{u}^{s})+\omega^{2}\mathbf{u}^{s}={\bf
0},$

in $|\mathbf{x}|\geq R$, where $\mu$ and $\lambda$ satisfy the strong
convexity condition (1.1). If

$\displaystyle\Im\big{(}\int_{\partial
B_{R}}{T}_{\nu}\overline{\mathbf{u}^{s}}\cdot{\mathbf{u}^{s}}\mathrm{d}s\big{)}\geq
0,$ (2.7)

then $\mathbf{u}^{s}=\bf 0$ in $|\mathbf{x}|\geq R$.

######  Theorem 2.1.

For any ${\mathbf{f}}\in L^{2}(\Omega)^{2}$ and ${\mathbf{g}}\in
H^{-1/2}(\Omega)^{2}$, then there exists at most one solution
${\mathbf{u}}^{s}\in H^{1}_{\rm loc}(\mathbb{R}^{2})^{2}$ of (2.2) to (2.5).

###### Proof.

Let ${\mathbf{u}}^{s}_{1}$ and ${\mathbf{u}}^{s}_{2}$ be two solutions of
(2.2) to (2.5). Set ${\mathbf{w}}={\mathbf{u}}^{s}_{1}-{\mathbf{u}}^{s}_{2}$,
which fulfills (2.2) to (2.5) with ${\mathbf{f}}={\mathbf{g}}={\mathbf{0}}$.

Suppose $R$ is sufficient large such that $\overline{\Omega}\subset B_{R}$,
where $B_{R}$ is a disk centered at the origin with the radius $R$. Let
$\phi\in C^{\infty}(\mathbb{R}^{2})$ such that $\phi(\mathbf{x})\equiv 1$ for
$|\mathbf{x}|<R$ and $\phi(\mathbf{x})\equiv 0$ for $|\mathbf{x}|\geq R+1$.
Choosing $\bm{\varphi}=\phi{\mathbf{w}}$ in (2.6), we deduce that

$\displaystyle\int_{R\leq|\mathbf{x}|\leq
R+1}\left((\mathcal{C}:\nabla{\mathbf{w}}):\nabla\overline{\bm{\varphi}}-\omega^{2}{\mathbf{w}}\cdot\overline{\bm{\varphi}}\right)\mathrm{d}\mathbf{x}+\int_{B_{R}\backslash\overline{\Omega}}\left((\mathcal{C}:\nabla{\mathbf{w}}):\nabla\overline{{\mathbf{w}}}-\omega^{2}|{\mathbf{w}}|^{2}\right)\mathrm{d}\mathbf{x}$
$\displaystyle\quad+\int_{\Omega}\left((\mathcal{C}:\nabla{\mathbf{w}}):\nabla\overline{{\mathbf{w}}}-\omega^{2}q|{\mathbf{w}}|^{2}\right)\mathrm{d}\mathbf{x}-\int_{\partial\Omega}\eta|{\mathbf{w}}|^{2}\mathrm{d}\sigma=0.$

By applying the first Betti’s formula, it follows that

$\displaystyle-\int_{\partial
B_{R}}T_{\nu}{\mathbf{w}}\cdot\overline{{\mathbf{w}}}\mathrm{d}\sigma+\int_{B_{R}\backslash\overline{\Omega}}\left((\mathcal{C}:\nabla{\mathbf{w}}):\nabla\overline{{\mathbf{w}}}-\omega^{2}|{\mathbf{w}}|^{2}\right)\mathrm{d}\mathbf{x}$
$\displaystyle\quad+\int_{\Omega}\left((\mathcal{C}:\nabla{\mathbf{w}}):\nabla\overline{{\mathbf{w}}}-\omega^{2}q|{\mathbf{w}}|^{2}\right)\mathrm{d}\mathbf{x}-\int_{\partial\Omega}\eta|{\mathbf{w}}|^{2}\mathrm{d}\sigma=0.$

Due to (1.1), the elastic tensor $\mathcal{C}$ satisfies the uniform Legendre
ellipticity condition:

$c_{\min}\|\bm{A}\|\leq({\mathcal{C}}:\bm{A}):\overline{\bm{A}}\leq
c_{\max}\|\bm{A}\|,\quad\forall\bm{A}\in\mathbb{C}^{2\times 2}\quad\mbox{being
a symmetric matrix},$ (2.8)

where $\|\bm{A}\|$ is the Frobenius norm, $c_{\min}$ and $c_{\max}$ are two
positive constants. In view of (2.8), one has

$\Im\big{(}\int_{\partial
B_{R}}{T}_{\nu}\overline{{\mathbf{w}}}\cdot{{\mathbf{w}}}\mathrm{d}\sigma\big{)}\geq
0,$

which implies that ${\mathbf{w}}$ vanishes in $\mathbb{R}^{2}\backslash B_{R}$
by using Lemma 2.1. Due to the interior regularity of the elliptic PDE, it
follows that ${\mathbf{w}}$ is real analytic in
$\mathbb{R}^{2}\backslash\overline{\Omega}$. Subsequently, the unique
continuation principle establishes that ${\mathbf{w}}\equiv{\mathbf{0}}$ in
$\mathbb{R}^{2}\backslash\overline{\Omega}$. Utilizing the trace theorem and
the boundary condition of ${\mathbf{w}}$, it follows that
${\mathbf{w}}={\mathbf{0}}$ and $T_{\nu}{\mathbf{w}}={\mathbf{0}}$ on
$\partial\Omega$. According to Holmgren’s principle, it can be concluded that
${\mathbf{w}}\equiv{\mathbf{0}}$ in $\overline{\Omega}$.

The proof is complete. ∎

Next, we will examine the existence of (2.2) to (2.5) using variational
techniques. For this purpose, we will initially present an equivalent problem
of (2.2) to (2.5) in a bounded domain by adopting the Dirichlet-to-Neumann
operator (DtN).

######  Definition 2.1.

For any $\mathbf{h}\in H^{1/2}(\partial B_{R})^{2}$, where $B_{R}$ is a ball
centered at the origin with radius $R$ such that $\overline{\Omega}\subset
B_{R}$, suppose that $\mathbf{v}^{s}\in H^{1}_{\rm
loc}(\mathbb{R}^{2}\backslash\overline{B}_{R})^{2}$ is the unique radiating
solution to the boundary value problem

$\mathcal{L}\mathbf{v}^{s}+\omega^{2}\mathbf{v}^{s}=\mathbf{0}\quad\mbox{in}\quad\mathbb{R}^{2}\backslash\overline{B}_{R},\quad\mathbf{v}^{s}=\mathbf{h}\quad\mbox{on}\quad\partial
B_{R},$

then the DtN operator $\mathcal{T}$ from $H^{1/2}(\partial B_{R})^{2}$ to
$H^{-1/2}(\partial B_{R})^{2}$ is defined as follows

$\mathcal{T}\mathbf{h}:=T_{\nu}\mathbf{v}^{s}|_{\partial B_{R}}.$

The subsequent lemma outlines key properties of the DtN map $\mathcal{T}$,
which will be utilized to establish the existence of (2.2) to (2.5).

######  Lemma 2.2.

[3, Lemma 2.8] The operator $\mathcal{T}$ is bounded from $H^{1/2}(\partial
B_{R})^{2}$ to $H^{-1/2}(\partial B_{R})^{2}$. The operator $-\mathcal{T}$ can
be decomposed into the sum of a positive operator $\mathcal{T}_{1}$ and a
compact operator $\mathcal{T}_{2}$, that is,

$\Re(\int_{\partial
B_{R}}\mathcal{T}_{1}w\cdot\bar{w}\mathrm{~{}d}\sigma)\geqslant
c\|w\|_{H^{1/2}\left(\partial B_{R}\right)}^{2}$

for some constant $c>0$, such that
$\mathcal{T}_{2}=-\mathcal{T}-\mathcal{T}_{1}:H^{1/2}\left(\partial
B_{R}\right)\rightarrow H^{-1/2}\left(\partial B_{R}\right)$ is compact.

By the DtN operator, we may now rephrase (2.2)-(2.4) as an equivalent form in
a bounded domain. Specifically, given ${\mathbf{f}}\in L^{2}(\Omega)^{2}$ and
${\mathbf{g}}\in H^{-1/2}(\Omega)^{2}$, we want to find ${\mathbf{v}}\in
H^{1}\left(B_{R}\right)^{2}$ satisfying (2.2)-(2.4) and the boundary condition

$\mathcal{T}\mathbf{v}:=T_{\nu}\mathbf{v}\quad\text{on}\ \partial B_{R}.$
(2.9)

Therefore we can deduce that ${\mathbf{v}}\in H^{1}\left(B_{R}\right)^{2}$
solves

$a({\mathbf{v}},\bm{\varphi})=b(\bm{\varphi})\quad\forall\bm{\varphi}\in
H^{1}\left(B_{R}\right),$ (2.10)

where

$\displaystyle a({\mathbf{v}},\bm{\varphi})$
$\displaystyle=\int_{B_{R}\backslash\overline{\Omega}}\left((\mathcal{C}:\nabla{\mathbf{v}}):\nabla\overline{\bm{\varphi}}-\omega^{2}{\mathbf{v}}\cdot\overline{\bm{\varphi}}\right)\mathrm{d}\mathbf{x}-\int_{\partial
B_{R}}\mathcal{T}\mathbf{v}\cdot\overline{\bm{\varphi}}\mathrm{d}\sigma$
$\displaystyle+\int_{\Omega}\left((\mathcal{C}:\nabla{\mathbf{v}}):\nabla\overline{\bm{\varphi}}-\omega^{2}q{\mathbf{v}}\cdot\overline{\bm{\varphi}}\right)\mathrm{d}\mathbf{x}-\int_{\partial\Omega}\eta{\mathbf{v}}\cdot\overline{\bm{\varphi}}\mathrm{d}\sigma,$

and

$b(\bm{\varphi})=-\int_{\Omega}{\mathbf{f}}\cdot\overline{\bm{\varphi}}\mathrm{d}x-\int_{\partial\Omega}{\mathbf{g}}\cdot\overline{\bm{\varphi}}\mathrm{d}\sigma.$

We can show that a solution ${\mathbf{v}}$ to problem (2.2)-(2.4) and (2.9)
can be extended to a solution to the scattering problem (2.2)-(2.5).
Similarly, ${\mathbf{v}}$, which is is a solution to the scattering problem
(2.2)-(2.5), can be restricted to $B_{R}$ and solves (2.2)-(2.4) and (2.9).
Therefore, according to Theorem 2.1, the problem (2.2) to (2.4) and (2.9) can
have at most one solution. Additionally, we provide the proof of existence for
the solution to problem (2.2) to (2.5).

######  Theorem 2.2.

Let ${\mathbf{f}}\in L^{2}(\Omega)$ and ${\mathbf{g}}\in
H^{-1/2}(\partial\Omega)$. If $\eta\leq 0$ a.e. on $\partial\Omega$, then
problem (2.10) has a unique solution ${\mathbf{v}}\in
H^{1}\left(B_{R}\right)$. Furthermore,

$\|{\mathbf{v}}\|_{H^{1}\left(B_{R}\right)}\leqslant
C\left(\left\|{\mathbf{f}}\right\|_{L^{2}(\Omega)}+\|{\mathbf{g}}\|_{H^{-1/2}(\partial\Omega)}\right).$
(2.11)

with a positive constant $C$ independent of ${\mathbf{f}}$ and ${\mathbf{g}}$.

###### Proof.

Using Lemma 2.2, for $a$ defineded in (2.10), we decompose $a$ as
$a=a_{1}+a_{2}$, where

$\displaystyle a_{1}({\mathbf{v}},\bm{\varphi})$
$\displaystyle=\int_{B_{R}}\left((\mathcal{C}:\nabla{\mathbf{v}}):\nabla\overline{\bm{\varphi}}+m{\mathbf{v}}\cdot\overline{\bm{\varphi}}\right)\mathrm{d}\mathbf{x}+\int_{\partial
B_{R}}\mathcal{T}_{1}\mathbf{v}\cdot\overline{\bm{\varphi}}\mathrm{d}\sigma-\eta\int_{\partial\Omega}{\mathbf{v}}\cdot\overline{\bm{\varphi}}\mathrm{d}\sigma,$

and

$a_{2}({\mathbf{v}},\bm{\varphi})=-\int_{B_{R}\backslash\overline{\Omega}}(m+\omega^{2}){\mathbf{v}}\cdot\overline{\bm{\varphi}}\mathrm{d}\mathbf{x}+\int_{\partial
B_{R}}\mathcal{T}_{2}\mathbf{v}\cdot\overline{\bm{\varphi}}\mathrm{d}\sigma-\int_{\Omega}(m+\omega^{2}q){\mathbf{v}}\cdot\overline{\bm{\varphi}}\mathrm{d}\mathbf{x},$
(2.12)

where $m$ is a sufficient large positive constant, $\mathcal{T}_{1}$ and
$\mathcal{T}_{2}$ are the operator defined in Lemma 2.2. According to the
boundedness of $\mathcal{T}_{1}$ and the trace theorem, we know that $a_{1}$
is bounded. Hence, using the Riesz representation theorem, it yields that

$a_{1}({\mathbf{v}},\bm{\varphi})=\left(A_{1}{\mathbf{v}},\bm{\varphi}\right)\text{
for all }\bm{\varphi}\in H^{1}\left(B_{R}\right).$

where $A_{1}:H^{1}\left(B_{R}\right)\rightarrow H^{1}\left(B_{R}\right)$ is a
bounded linear operator. By Korn inequality, the uniform Legendre ellipticity
condition of $\mathcal{C}$ and Lemma 2.2, for all ${\mathbf{v}}\in
H^{1}\left(B_{R}\right)$,

$\displaystyle\Re\left[a_{1}({\mathbf{v}},{\mathbf{v}})\right]$
$\displaystyle\geqslant\widetilde{C}\|{\mathbf{v}}\|_{H^{1}\left(B_{R}\right)}^{2}-\int_{\partial
B_{R}}\mathcal{T}_{1}{\mathbf{v}}\bar{{\mathbf{v}}}\mathrm{d}s-\eta\int_{\partial\Omega}{\mathbf{v}}{\mathbf{\bar{v}}}\mathrm{d}\sigma$
$\displaystyle\geqslant\widetilde{C}\|{\mathbf{v}}\|_{H^{1}\left(B_{R}\backslash\bar{\Omega}\right)}^{2}+c\|{\mathbf{v}}\|_{H^{\frac{1}{2}}\left(\partial
B_{R}\right)}^{2}$
$\displaystyle\geqslant\widetilde{C}\|{\mathbf{v}}\|_{H^{1}\left(B_{R}\right)}^{2},$

where $\widetilde{C}$ is a positive constant not depending on $\mathbf{v}$.
Hence $a_{1}$ is strictly coercive. From the Lax-Milgram theorem, we know that
the operator $A_{1}:H^{1}\left(B_{R}\right)\rightarrow
H^{1}\left(B_{R}\right)$ has a bounded inverse. Similarly, by the Riesz
representation theorem, there exists a bounded linear operator
$A_{2}:H^{1}\left(B_{R}\right)\rightarrow H^{1}\left(B_{R}\right)$ fulfilling

$a_{2}({\mathbf{v}},\bm{\varphi})=\left(A_{2}{\mathbf{v}},\bm{\varphi}\right)\quad\text{
for all }\bm{\varphi}\in H^{1}\left(B_{R}\right).$

By Rellich’s embedding theorem we know that the embedding of
$H^{1}\left(B_{R}\right)$ into $L^{2}\left(B_{R}\right))$ is compact. Since
$\mathcal{T}_{2}$ is compact, it yields that $A_{2}$ is compact. Using Theorem
2.1, we know that $A_{1}+A_{2}$ is of Fredholm type with index 0. According to
Riesz-Fredholm theory, we prove that (2.10) has an unique solution. The
estimate (2.11) holds by noting that $A_{1}$ is bounded and coercive, and
$A_{2}$ is compact.

The proof is complete. ∎

######  Remark 2.1.

Recall that ${\mathbf{f}}$ and ${\mathbf{g}}$ are defined in (2.1). By (2.11),
for the unique solution ${\mathbf{u}}^{sc}\in H^{1}_{\rm loc}(\mathbb{R}^{2})$
to (2.2) to (2.5), it yields that

$\displaystyle\|{\mathbf{u}}^{sc}\|_{H^{1}\left(B_{R}\right)}\leqslant
C\left(\omega^{2}\left\|(1-q){\mathbf{u}}^{i}\right\|_{L^{2}(\Omega)}+\|\eta{\mathbf{u}}^{i}\|_{H^{-1/2}(\partial\Omega)}\right)$
$\displaystyle=\left(\omega^{2}\|1-q\|_{L^{\infty}(\Omega)}+\|\eta\|_{L^{\infty}(\partial\Omega)}\right)\left\|{\mathbf{u}}^{i}\right\|_{L^{2}(\Omega)}.$

##  3\. Geometrical and mathematical setups for the inverse problem

In this section, we first introduce the geometric and mathematical setups for
subsequent analysis. The definition below gives the polygonal-nest and
polygonal-cell elastic scatterers. Following this, we will present the
corresponding generalized transmission conditions and inverse problem of these
two kinds of elastic layered media.

######  Definition 3.1.

$\Omega$ is considered to have a partition made up of a polygonal-nest
structure, if there are $\Pi_{\zeta},\zeta=1,2,\cdots,N,N\in\mathbb{N}$, where
$\\{\Pi_{\zeta}\\}_{\zeta=1}^{N}$ fulfills the following conditions:

  * 1.

Each $\Pi_{\zeta}$ is a convex polygon that is open, bounded and simply
connected.

  * 2.

$\Pi_{N}\Subset\Pi_{N-1}\Subset\cdots\Subset\Pi_{2}\Subset\Pi_{1}=\Omega$.

######  Definition 3.2.

$\Omega$ is considered to have a partition made up of a polygonal-cell
structure, if there are $\Pi_{\zeta},\zeta=1,2,\cdots,N,N\in\mathbb{N}$, where
$\\{\Pi_{\zeta}\\}_{\zeta=1}^{N}$ fulfills the following conditions:

  * 1.

Each $\Pi_{\zeta}$ is an open and bounded simply-connected polygon.

  * 2.

$\Pi_{\zeta}\subset\Omega$,
$\bigcup\nolimits_{\zeta=1}^{N}\overline{\Pi}_{\zeta}=\overline{\Omega}$ and
$\Pi_{\zeta}\cap\Pi_{\hat{\zeta}}=\emptyset$ if $\zeta\neq\hat{\zeta}$.

  * 3.

For each $\Pi_{\zeta}$, there is at least one vertex ${\mathbf{x}}_{c,\zeta}$
such that $\Gamma_{\zeta}^{\pm}\subset\partial\Omega$, where
$\Gamma_{\zeta}^{\pm}$ are the two adjacent edges of $\partial\Pi_{\zeta}$
associated with ${\mathbf{x}}_{c,\zeta}$;

Figure 1. Polygonal-nest geometry

Figure 2. Polygonal-cell geometry

######  Definition 3.3.

Suppose that $(\Omega;\lambda,\mu,q,\eta)$ is an elastic scatterer that
possesses a piecewise polygonal-nest structure, which fulfills the following
conditions:

  * 1.

$\Omega$ has a polygonal-nest partition as described in Definition 3.1.

  * 2.

Let $(\mathcal{H}_{\zeta};q_{\zeta},\eta_{\zeta})$ ($\zeta=1,2,\cdots,N$)
denote an elastic medium
$\mathcal{H}_{\zeta}:=\Pi_{\zeta}\backslash\overline{\Pi}_{\zeta+1}$ with
medium configurations $q_{\zeta}\in\mathbb{R}$ in $\mathcal{H}_{\zeta}$ and
$\eta_{\zeta}\in\mathbb{R}$ on $\partial\Pi_{\zeta}$, where each $\Pi_{\zeta}$
is an elastic scatterer and $\Pi_{\zeta+1}:=\emptyset$. Namely,
$q\big{|}_{\mathcal{H}_{\zeta}}=q_{\zeta}$ and
$\eta\big{|}_{\partial\Pi_{\zeta}}=\eta_{\zeta}$.

The schematic illustration of a scatter $\Omega$ with a polygonal-nest
structure is displayed in Figure 2. As per Definition 3.3, a polygonal-nest
elastic scatterer $(\Omega;\lambda,\mu,q,\eta)$ can be expressed as.

$(\Omega;\lambda,\mu,q,\eta)=\bigcup\limits_{\zeta=1}^{N}(\mathcal{H}_{\zeta};q_{\zeta},\eta_{\zeta}),$
(3.1)

where

$\Omega=\bigcup_{\zeta=1}^{N}\mathcal{H}_{\zeta},\quad
q=\sum_{\zeta=1}^{N}q_{\zeta}\chi_{\mathcal{H}_{\zeta}},\quad\eta=\sum_{\zeta=1}^{N}{\eta_{\zeta}}\chi_{\partial\Pi_{\zeta}}.$
(3.2)

Consider the elastic scattering problem (1.7) associated with a polygonal-nest
elastic scatterer as defined by (3.1). The generalized transmssion boundary
conditions are adjusted as follows:

${\mathbf{u}}_{\zeta}|_{\partial\Pi_{\zeta+1}}={\mathbf{u}}_{\zeta+1}|_{\partial\Pi_{\zeta+1}},\
(T_{\nu}{\mathbf{u}}_{\zeta}+\eta_{\zeta+1}{\mathbf{u}}_{\zeta})|_{\partial\Pi_{\zeta+1}}=T_{\nu}{\mathbf{u}}_{\zeta+1}|_{\partial\Pi_{\zeta+1}},$
(3.3)

where
${\mathbf{u}}_{\zeta}={\mathbf{u}}|_{\mathcal{H}_{\zeta}},\zeta=0,1,2,\cdots,N-1$
and $\mathcal{H}_{0}:=\mathbb{R}^{2}\backslash\overline{\Pi}_{\zeta}$. Using
the variational approach, the well-posedness of (1.7) associated with the
polygonal-nest elastic scatterer $(\Omega;\lambda,\mu,q,\eta)$ can be
demonstrated. As we focus on the corresponding inverse problem in this paper,
we assume that there is a unique solution $\mathbf{u}\in H_{\rm
loc}^{1}(\mathbb{R}^{2})^{2}$ of (1.7), given that
$(\Omega;\lambda,\mu,q,\eta)$ describes a polygonal-nest elastic scatterer.
According to (1.10), the inverse problem related to (1.7) associated with a
polygonal-nest elastic scatterer as described by (3.1) can be formulated by

$\mathcal{F}(\bigcup_{\zeta=1}^{N}(\mathcal{H}_{\zeta};q_{\zeta},\eta_{\zeta}))={\mathbf{u}}_{\beta}^{\infty}(\hat{{\mathbf{x}}};{\mathbf{u}}^{i}),\
\hat{{\mathbf{x}}}\in{\mathbb{S}^{1}},\quad\beta=t,p,\mbox{ or }s.$ (3.4)

In this paper, we aim to determine the configuration of $\mathcal{H}_{\zeta}$,
as well as the corresponding physical parameters $q_{\zeta}$ and
$\eta_{\zeta}$, using the information obtained from a single far field
measurement
${\mathbf{u}}_{\beta}^{\infty}(\hat{{\mathbf{x}}};{\mathbf{u}}^{i})$. This
implies that
${\mathbf{u}}_{\beta}^{\infty}(\hat{{\mathbf{x}}};{\mathbf{u}}^{i})$ is
measured at a fixed frequency $\omega$ and with a fixed incident wave
${\mathbf{u}}^{i}$.

In the subsequent discussion, we will examine the issue of an elastic
scattering concerning the problem (1.7) associated with an elastic scatterer
of a polygonal-cell structure. Firstly, we will give the definition of an
elastic scatterer of a piecewise polygonal-cell structure.

######  Definition 3.4.

The elastic scatterer $(\Omega;\lambda,\mu,q,\eta)$ is said to have a
piecewise polygonal-cell structure, if it fulfills the following conditions:

  * 1.

$\Omega$ has a partition with a polygonal-cell structure as described in
Definition 3.2.

  * 2.

Suppose that each elastic medium $(\Pi_{\zeta};q_{\zeta},\eta^{*})$ for
$\zeta=1,2,\cdots,N$ has the constant configuration parameters
$q_{\zeta}\in\mathbb{R}$ in $\Pi_{\zeta}$ and $\eta^{*}\in\mathbb{R}$ on
$\partial\Pi_{\zeta}$.

Similar to (3.1), according to Definition 3.4, an elastic scatterer
$(\Omega;\lambda,\mu,q,\eta)$ with a polygonal-cell structure can be expressed
as

$(\Omega;\lambda,\mu,q,\eta)=\bigcup_{\zeta=1}^{N}(\Pi_{\zeta};q_{\zeta},\eta^{*}),$
(3.5)

where

$\Omega=\bigcup_{\zeta=1}^{N}\Pi_{\zeta},\quad
q=\sum_{\zeta=1}^{N}q_{\zeta}\chi_{\Pi_{\zeta}},\quad\eta=\sum_{\zeta=1}^{N}\eta^{*}\chi_{\partial\Pi_{\zeta}}.$
(3.6)

For the elastic wave scattering defined in (1.7) related to an elastic
scatterer with a polygonal-cell structure as defined in (3.5), we define
${\mathbf{u}}_{\zeta}={\mathbf{u}}|_{\Pi_{\zeta}}$, where
$\zeta=0,1,2,\cdots,N-1$ and
$\Pi_{0}:=\mathbb{R}^{2}\backslash\overline{\bigcup}_{\zeta=1}^{N}\Pi_{\zeta}$.
It is clear to see that the boundary of the set $\partial\Pi_{0}$ is the same
as the boundary of the outer set $\partial\Omega$. This pertains to the
specific geometry of a scatterer made up of polygonal cells. Moving forward,
we will examine how interior transmission works across the internal
boundaries. Let us assume that $\Pi_{\zeta_{1}}$ and $\Pi_{\zeta_{2}}$ are
neighboring cells and that their intersection is defined as $\Gamma$, where
$\zeta_{2}>\zeta_{1}$. In light of (3.6), and through some suitable amendments
from (3.3), one can express the generalized transmission condition with
respect to a polygonal-cell elastic scatterer as

${\mathbf{u}}_{\zeta_{1}}|_{\partial\Pi_{\zeta_{2}}}={\mathbf{u}}_{\zeta_{2}}|_{\partial\Pi_{\zeta_{2}}},\
(T_{\nu}{\mathbf{u}}_{\zeta_{1}}+\eta^{*}{\mathbf{u}}_{\zeta_{1}})|_{\partial\Pi_{\zeta_{2}}}=T_{\nu}{\mathbf{u}}_{\zeta_{2}}|_{\partial\Pi_{\zeta_{2}}},$
(3.7)

where $\Pi_{\zeta_{1}}$ and $\Pi_{\zeta_{2}}$ are two adjacent cells, with
$\zeta_{2}>\zeta_{1}$ and
$\partial\Pi_{\zeta_{2}}=\Pi_{\zeta_{1}}\cap\Pi_{\zeta_{2}}$. The direct and
inverse problems of an elastic scatterer with a piecewise polygonal-cell
structure can be formulated in a similar manner to that of a piecewise
polygonal-nest structure. Additionally, the inverse problem of a piecewise
polygonal-cell elastic scatterer can be formulated as follows:

$\mathcal{F}(\bigcup_{\zeta=1}^{N}(\Pi_{\zeta};q_{\zeta},\eta^{*}))={\mathbf{u}}_{\beta}^{\infty}(\hat{{\mathbf{x}}};{\mathbf{u}}^{i}),\
\hat{{\mathbf{x}}}\in{\mathbb{S}^{1}},\quad\beta=t,p,\mbox{ or }s.$ (3.8)

##  4\. Auxiliary lemmas

In this section, we will establish various lemmas for subsequent analysis.
First, we introduce geometric notation for a planar corner, which will be
frequently used in our analysis.

Let $(r,\theta)$ be the polar coordinates in $\mathbb{R}^{2}$. Namely, for
${\mathbf{x}}=(x_{1},x_{2})^{\top}$, the polar coordinate of ${\mathbf{x}}$ is
given by ${\mathbf{x}}=(r\cos\theta,r\sin\theta)^{\top}$. Denote an open
sector $W\subseteq{\mathbb{R}^{2}}$ and its boundary $\Gamma^{\pm}$ as
follows:

$\displaystyle W$
$\displaystyle=\\{{\mathbf{x}}\in\mathbb{R}^{2}|{\mathbf{x}}\neq{\mathbf{0}},\theta_{m}<\mathrm{arg}(x_{1}+{{\mathrm{i}}}x_{2})<\theta_{M}\\},$
(4.1) $\displaystyle\Gamma^{+}$
$\displaystyle=\\{{\mathbf{x}}\in\mathbb{R}^{2}|{\mathbf{x}}\neq{\mathbf{0}},\mathrm{arg}(x_{1}+{{\mathrm{i}}}x_{2})=\theta_{M}\\},$
$\displaystyle\Gamma^{-}$
$\displaystyle=\\{{\mathbf{x}}\in\mathbb{R}^{2}|{\mathbf{x}}\neq{\mathbf{0}},\mathrm{arg}(x_{1}+{{\mathrm{i}}}x_{2})=\theta_{m}\\},$

where $-\pi<\theta_{m}<\theta_{M}<\pi$. Here $\theta_{M}-\theta_{m}$ is the
opening angle of $W$. Let $B_{h}$ is an open disk centered at ${\mathbf{0}}$
with radius $h\in\mathbb{R}_{+}$ in the remaining of the paper, set

$S_{h}=W\cap B_{h},\ \Gamma_{h}^{\pm}=\Gamma^{\pm}\cap B_{h},\
\overline{S_{h}}=\overline{W}\cap B_{h},\ \Lambda_{h}=S_{h}\cap\partial
B_{h}.$ (4.2)

Since $\mathcal{L}$ is invariant under rigid motion, it is worth noting that
the sector $S_{h}$ represents a region in the vicinity of a corner vertex of a
polygonal elastic scatterer for the remainder of the paper. Consequently, we
can assume that $h\in\mathbb{R}_{+}$ is sufficiently small so that $S_{h}$ is
completely contained in the elastic scatterer. Additionally,
$\Gamma_{h}^{\pm}$ lie completely on the two edges associated with the elastic
scatterer. To establish unique identifiability results, we will use the
following complex geometric optics (CGO) solution in the following analysis.

######  Lemma 4.1.

Let the convex sectorial corner $S_{h}$ be defined by (4.2) satisfying the
opening angle $\theta_{M}-\theta_{m}\in(0,\pi)$. Then there exist unit vectors
$\mathbf{d}\in\mathbb{S}^{1}$ and $\mathbf{d}^{\perp}\in\mathbb{S}^{1}$ such
that

$-1\leq\mathbf{d}\cdot\hat{\mathbf{x}}\leq-\delta_{W}<0\quad\mbox{for
all}\quad\mathbf{x}\in
S_{h},\quad\mbox{and}\quad\mathbf{d}\cdot\mathbf{d}^{\perp}=0,$ (4.3)

where $\hat{\mathbf{x}}=\mathbf{x}/|\mathbf{x}|$ and $\delta_{W}$ is a
positive constant only depending on $W$. Denote

${\mathbf{u}}_{0}({\mathbf{x}})=\left(\begin{array}[]{l}1\\\
{{\mathrm{i}}}\end{array}\right)\exp(\bm{\rho}\cdot\mathbf{x})\buildrel\Delta\over{=}\left(\begin{array}[]{l}{\hat{u}_{1}}({\mathbf{x}})\\\
{\hat{u}_{2}}({\mathbf{x}})\end{array}\right),\quad\bm{\rho}=s\left(\mathbf{d}+{\mathrm{i}}\mathbf{d}^{\perp}\right),\quad\mathrm{i}=\sqrt{-1},$
(4.4)

where $s\in\mathbb{R}_{+}$. Then

$\mathcal{L}\mathbf{u}_{0}=0\quad\mbox{in}\quad\mathbb{R}^{2}.$ (4.5)

Recall that $W$ is defined in (4.1). Let

$\displaystyle\mathbf{d}=(\cos\theta_{d},\sin\theta_{d})^{\top}\quad\mbox{and}\quad\mathbf{d}^{\perp}=(-\sin\theta_{d},\cos\theta_{d})^{\top}$
(4.6)

satisfy (4.3). One has

$\int_{W}\hat{u}_{1}(s{\mathbf{x}})\mathrm{d}{\mathbf{x}}=0.5{\mathrm{i}}e^{2{\mathrm{i}}\theta_{d}}(e^{-2{\mathrm{i}}\theta_{M}}-e^{-2{\mathrm{i}}\theta_{m}})s^{-2}$
(4.7)

and

$\int_{W}|\hat{u}_{j}({\mathbf{x}})||{\mathbf{x}}|^{\alpha}\mathrm{d}{\mathbf{x}}\leq\frac{(\theta_{M}-\theta_{m})\Gamma(\alpha+2)}{\delta_{W}^{\alpha+2}}s^{-\alpha-2},\quad
j=1,2,$ (4.8)

as $s\rightarrow+\infty$. Moreover, for $\alpha\geq 0$ and $j\in\\{1,2\\}$,
the following integrals can be estimated

and

$\int_{W\backslash
S_{h}}|\hat{u}_{j}({\mathbf{x}})||\mathbf{x}|^{\alpha}\mathrm{d}{\mathbf{x}}\leq\frac{2(\theta_{M}-\theta_{m})}{\delta_{W}}s^{-1}e^{-\delta_{W}sh/2}$
(4.9)

as $s\rightarrow+\infty$. The following boundary integrals on
$\Gamma_{h}^{\pm}$ satisfy

$\displaystyle\int_{\Gamma_{h}^{+}}\hat{u}_{1}({\mathbf{x}})\mathrm{d}\sigma$
$\displaystyle=-e^{{\mathrm{i}}(\theta_{d}-\theta_{M})}s^{-1}-{\sf
R({\theta_{M}})},\quad\int_{\Gamma_{h}^{-}}\hat{u}_{1}({\mathbf{x}})\mathrm{d}\sigma=-e^{{\mathrm{i}}(\theta_{d}-\theta_{m})}s^{-1}-{\sf
R({\theta_{m}})},$ (4.10)

where

${\sf
R(\theta_{M})}=\int_{h}^{+\infty}e^{sr\exp({\mathrm{i}}(\theta_{M}-\theta_{d}))}\mathrm{d}r,\quad{\sf
R(\theta_{m})}=\int_{h}^{+\infty}e^{sr\exp({\mathrm{i}}(\theta_{m}-\theta_{d}))}\mathrm{d}r$

satisfying $|{\sf R(\theta_{M})}|\leq 2/\delta_{W}e^{-h\delta_{W}/2}$ and
$|{\sf R(\theta_{m})}|\leq 2/\delta_{W}e^{-h\delta_{W}/2}$.

###### Proof.

Since $S_{h}$ is convex, it can be direct to see that (4.3) can be fulfilled.
By the definition of $\bm{\rho}$ given in (4.3), one has
$\bm{\rho}\cdot\bm{\rho}=0$, which implies that

$\displaystyle\Delta\hat{u}_{1}(\mathbf{x})=0\quad\mbox{in}\quad\mathbb{R}^{2}.$
(4.11)

By the definition of $\mathbf{u}_{0}$, we know that
$\hat{u}_{2}={\mathrm{i}}\hat{u}_{1}$. Hence by directly calculations it
yields that

$\displaystyle\nabla(\nabla\cdot\mathbf{u}_{0})=(\rho_{1}\hat{u}_{1}+\rho_{2}\hat{u}_{2})\bm{\rho}=\mathbf{0}\quad\mbox{in}\quad\mathbb{R}^{2},\quad\bm{\rho}=s\exp(-{\mathrm{i}}\theta_{d})\begin{bmatrix}1\\\
{\mathrm{i}}\end{bmatrix}=\begin{bmatrix}\rho_{1}\\\ \rho_{2}\end{bmatrix},$

which implies that $\mathcal{L}u_{0}=\mathbf{0}$ in $\mathbb{R}^{2}$ by noting
(4.11) and the definition of the Lamé operator $\mathcal{L}$.

For any given $\alpha\in\mathbb{R}_{+}\cup\\{0\\}$ and
$\varepsilon\in\mathbb{R}_{+}$, if $\Re(\gamma)\gg 1$ with
$\gamma\in\mathbb{C}$, using Laplace transform, one can obtain that

$\int_{0}^{+\infty}r^{\alpha}e^{-\gamma
r}\mathrm{d}\mathrm{r}=\frac{\Gamma(\alpha+1)}{\gamma^{\alpha+1}},\quad\left|\int_{\varepsilon}^{\infty}r^{\alpha}e^{-\gamma
r}\mathrm{d}\mathrm{r}\right|\leq\frac{2}{\Re{\gamma}}e^{-\varepsilon\Re{\gamma}/2},$
(4.12)

where $\Gamma$ is the Gamma function. Therefore, according to (4.12), for
(4.7) we can deduce that

$\displaystyle\int_{W}\hat{u}_{1}({\mathbf{x}})\mathrm{d}{\mathbf{x}}$
$\displaystyle=\int_{0}^{\infty}\int_{\theta_{m}}^{\theta_{M}}\exp(sr\exp({\mathrm{i}}(\theta-\theta_{d})))r\mathrm{d}r\mathrm{d}\theta=\int_{\theta_{m}}^{\theta_{M}}\frac{\Gamma(2)}{s^{2}\exp(2{\mathrm{i}}(\theta-\theta_{d}))}\mathrm{d}\theta$
$\displaystyle=\frac{{\mathrm{i}}e^{2{\mathrm{i}}\theta_{d}}(e^{-2{\mathrm{i}}\theta_{M}}-e^{-2{\mathrm{i}}\theta_{m}})}{2s^{2}}.$

Using the similar argument for (4.7), we can derive (4.8). By virtue of (4.12)
and polar coordinate transformation we can obtain (4.9). Analogously we adopt
the polar transformation and utilize (4.12) to prove (4.10). ∎

######  Lemma 4.2.

Let ${\Lambda_{h}}$ be given in (4.1) and ${{\mathbf{u}}}_{0}({{\mathbf{x}}})$
be defined in (4.4). Recall that ${\delta_{W}}>0$ is given in Lemma 4.1. It
yields that

$\displaystyle\|{{\mathbf{u}}_{0}}\|_{H^{1}{{(\Lambda_{h})}^{2}}}$
$\displaystyle\leq\sqrt{2h(1+2s^{2})}\sqrt{\theta_{M}-\theta_{m}}e^{-sh\delta_{W}},$
(4.13a) $\displaystyle\|T_{\nu}{\mathbf{u}}_{0}\|_{L^{2}{(\Lambda_{h})}^{2}}$
$\displaystyle\leq 2\mu s\sqrt{3(\theta_{M}-\theta_{m})}e^{-sh\delta_{W}},$
(4.13b)

as $s\rightarrow+\infty$.

###### Proof.

By virtue of (4.4), there holds

$\|\hat{u}_{1}(s{\mathbf{u}})\|_{L^{2}(\Lambda_{h})^{2}}\leq\sqrt{h}e^{-sh\delta_{W}}\sqrt{\theta_{M}-\theta_{m}}.$
(4.14)

Using (4.3), it yields that

$\|\nabla\hat{u}_{1}(s{\mathbf{x}})\|_{L^{2}(\Lambda_{h})^{2}}\leq\sqrt{2}|\bm{\rho}|\left(\int_{\theta_{m}}^{\theta_{M}}he^{2sh\mathbf{d}\cdot\mathbf{\hat{x}}}\mathrm{d}\theta\right)^{1/2}\leq
s\sqrt{2h}e^{-sh\delta_{W}}\sqrt{\theta_{M}-\theta_{m}},$ (4.15)

as $s\rightarrow+\infty$. Furthermore, noting
$\hat{u}_{2}({\mathbf{x}})={{\mathrm{i}}}\hat{u}_{1}({\mathbf{x}})$, and
combing (4.14) with (4.15), one has (4.13a).

Using $\hat{u}_{2}(s{\mathbf{x}})={{\mathrm{i}}}\hat{u}_{1}(s{\mathbf{x}})$
again, we have $\nabla\cdot{\mathbf{u}}_{0}=0$. Then, combing (1.6), we can
deduce that

$T_{\nu}{\mathbf{u}}_{0}=\mu\left[\begin{array}[]{cc}2\frac{\partial\hat{u}_{1}}{\partial
x_{1}}&\frac{\partial\hat{u}_{1}}{\partial
x_{2}}+\frac{\partial\hat{u}_{2}}{\partial x_{1}}\\\
\frac{\partial\hat{u}_{2}}{\partial x_{1}}+\frac{\partial\hat{u}_{1}}{\partial
x_{2}}&2\frac{\partial\hat{u}_{2}}{\partial x_{2}}\end{array}\right]\cdot\nu.$

Therefore, we can conclude that

$\displaystyle\left\|T_{\nu}{\mathbf{u}}_{0}\right\|_{L^{2}(\Lambda_{h})^{2}}^{2}$
$\displaystyle=\mu^{2}\int_{\Lambda_{h}}{\left|2\frac{\partial\hat{u}_{1}}{\partial
x_{1}}\nu_{1}+\frac{\partial\hat{u}_{1}}{\partial
x_{2}}\nu_{2}+\frac{\partial\hat{u}_{2}}{\partial
x_{1}}\nu_{2}\right|}^{2}\mathrm{d}\sigma+\mu^{2}\int_{\Lambda_{h}}{\left|\frac{\partial\hat{u}_{2}}{\partial
x_{1}}\nu_{1}+\frac{\partial\hat{u}_{1}}{\partial
x_{2}}\nu_{1}+2\frac{\partial\hat{u}_{2}}{\partial
x_{2}}\nu_{2}\right|}^{2}\mathrm{d}\sigma$ $\displaystyle\leq
6\mu^{2}\int_{\Lambda_{h}}{\left|\frac{\partial\hat{u}_{1}}{\partial
x_{1}}\right|}^{2}\mathrm{d}\sigma\leq
12s^{2}\mu^{2}e^{-2sh\delta_{W}}(\theta_{M}-\theta_{m}).$

The proof is complete. ∎

In the subsequent lemma, with appropriate regularity around the corner on the
solution of a PDE transmission problem (4.16), we shall show that the
corresponding solution must vanish at the corner.

######  Lemma 4.3.

Recall that $S_{h}$ is defined in (4.1). Suppose that
$({\mathbf{v}},{\mathbf{w}})\in H^{2}(S_{h})^{2}\times H^{1}(S_{h})^{2}$ are a
pair of solutions to the following PDE system:

$\begin{cases}\mathcal{L}{\mathbf{w}}+{\omega}^{2}q{\mathbf{w}}={\mathbf{0}},&\mbox{in}\quad
S_{h},\\\
\mathcal{L}{\mathbf{v}}+{\omega}^{2}{\mathbf{v}}={\mathbf{0}},&\mbox{in}\quad
S_{h},\\\
{\mathbf{w}}={\mathbf{v}},T_{\nu}{\mathbf{v}}+\eta{\mathbf{v}}=T_{\nu}{\mathbf{w}},&\mbox{on}\quad\Gamma_{h}^{\pm},\end{cases}$
(4.16)

where $\nu\in\mathbb{S}^{1}$ signifies the exterior unit normal vector to
$\Gamma_{h}^{\pm}$, ${\omega}\in\mathbb{R}_{+}$,
$q\in\mathbb{R}\backslash\\{0\\}$, $\eta$ is a real valued function and
$\eta({\mathbf{x}})\in C^{\alpha}(\overline{\Gamma_{h}^{\pm}})$ for
$0<\alpha<1$. Assumed that either $\eta\equiv 0$ on $\Gamma_{h}^{\pm}$ or
$\eta({\mathbf{0}})\neq 0$ when $\eta$ is a variable function on
$\Gamma_{h}^{\pm}$ . If the following conditions are fulfilled:

(a) the function ${\mathbf{w}}$ is Hölder continuous in $\overline{S_{h}}$
where $\alpha\in(0,1)$, namely

${\mathbf{w}}\in C^{\alpha}(\overline{S_{h}})^{2},$ (4.17)

(b) the angles $\theta_{M}$ and $\theta_{m}$ of $S_{h}$ satisfy

$-\pi<\theta_{m}<\theta_{M}<\pi,\ \theta_{M}-\theta_{m}\neq\pi,$ (4.18)

one has

${\mathbf{v(0)}}={\mathbf{w(0)}}={\mathbf{0}}.$ (4.19)

###### Proof.

Let us consider the following two cases. For the case that
$\eta({\mathbf{x}})\in C^{\alpha}(\overline{\Gamma_{h}^{\pm}})$ for
$0<\alpha<1$ satisfying $\eta({\mathbf{0}})\neq 0$, by virtue of [14, Theorem
2.3], we can obtain

${\mathbf{v(0)}}={\mathbf{0}}.$ (4.20)

For another case that $\eta\equiv 0$ on $\Gamma_{h}^{\pm}$, using [14, Remark
2.3], it yields (4.20). By using the boundary conditions, it indicates that
${\mathbf{v(0)}}={\mathbf{w(0)}}={\mathbf{0}}$.

The proof is complete. ∎

For a PDE transmission problem (4.21), the subsequent lemma states that the
underlying solution is Hölder continuous at the corner, which will help to
prove the main unique identifiability results in the next section.

######  Lemma 4.4.

[14, Lemma4.2] For $h\in\mathbb{R}_{+}$, let $S_{2h}=W\cap B_{2h}$ and
$\Gamma_{2h}^{\pm}=\partial S_{2h}\backslash\partial B_{2h}$, where $W$ is the
infinite sector defined in (4.1) with the opening angle
$\theta_{W}\in(0,\pi)$. Suppose that ${\mathbf{u}}\in H^{1}(B_{2h})^{2}$
satisfies the following PDE system:

$\begin{cases}\mathcal{L}{\mathbf{u}}^{-}+{\omega}^{2}q{\mathbf{u}}^{-}={\mathbf{0}},&\mbox{in}\quad
S_{2h},\\\
\mathcal{L}{\mathbf{u}}^{+}+{\omega}^{2}{\mathbf{u}}^{+}={\mathbf{0}},&\mbox{in}\quad
B_{2h}\backslash\overline{S_{2h}},\\\
{\mathbf{u}}^{+}={\mathbf{u}}^{-},&\mbox{on}\quad\Gamma_{2h}^{\pm},\end{cases}$
(4.21)

where ${\mathbf{u}}^{-}={\mathbf{u}}|_{S_{2h}}$ and
${\mathbf{u}}^{+}={\mathbf{u}}|_{B_{2h}\backslash\overline{S}_{2h}}$, $\omega$
is a positive constant and $q\in L^{\infty}(S_{2h})$. Assume that
${\mathbf{u}}^{+}$ is real analytic in $B_{2h}\backslash\overline{S_{2h}}$.
There exists $\alpha\in(0,1)$ such that ${\mathbf{u}}^{-}\in
C^{\alpha}(\overline{S_{h}})^{2}$.

Lemma 4.5 constitutes a key ingredient for the proof of Theorems 5.2 to 5.3,
where we shall prove that the density and boundary parameters in (4.22)
coincide with their counterparts under the generic assumption. To analyze the
resulting transmission PDE system (4.25) near a corner, we rely upon the CGO
solution presented in Lemma 4.1.

######  Lemma 4.5.

Let $S_{h}$, $\Gamma_{h}^{\pm}$ and $B_{h}$ be defined in (4.1). Consider the
PDE system below:

$\begin{cases}\mathcal{L}{\mathbf{u}}_{1}^{-}+{\omega}^{2}q_{1}{\mathbf{u}}_{1}^{-}={\mathbf{0}},&\mbox{in}\quad
S_{h},\\\
\mathcal{L}{\mathbf{u}}_{1}^{+}+{\omega}^{2}q_{1}^{+}{\mathbf{u}}_{1}^{+}={\mathbf{0}},&\mbox{in}\quad
B_{h}\backslash\overline{S_{h}},\\\
\mathcal{L}{\mathbf{u}}_{2}^{-}+{\omega}^{2}q_{2}{\mathbf{u}}_{2}^{-}={\mathbf{0}},&\mbox{in}\quad
S_{h},\\\
\mathcal{L}{\mathbf{u}}_{2}^{+}+{\omega}^{2}q_{2}^{+}{\mathbf{u}}_{2}^{+}={\mathbf{0}},&\mbox{in}\quad
B_{h}\backslash\overline{S_{h}},\\\
{\mathbf{u}}_{1}^{+}={\mathbf{u}}_{1}^{-},T_{\nu}{\mathbf{u}}_{1}^{+}+\eta_{1}{\mathbf{u}}_{1}^{+}=T_{\nu}{\mathbf{u}}_{1}^{-},&\mbox{on}\quad\Gamma_{h}^{\pm},\\\
{\mathbf{u}}_{2}^{+}={\mathbf{u}}_{2}^{-},T_{\nu}{\mathbf{u}}_{2}^{+}+\eta_{2}{\mathbf{u}}_{2}^{+}=T_{\nu}{\mathbf{u}}_{2}^{-},&\mbox{on}\quad\Gamma_{h}^{\pm},\end{cases}$
(4.22)

where $q_{j}$ and $q_{j}^{+}$ ($j=1,2$) are nonzero real constants in $S_{h}$
and $B_{h}\backslash\overline{S_{h}}$, respectively. Moreover, $\eta_{j}$
($j=1,2$) are constants on $\Gamma_{h}^{\pm}$. Assume that
${{\mathbf{u}}_{j}}={{\mathbf{u}}_{j}^{-}}\chi_{S_{h}}+{{\mathbf{u}}_{j}^{+}}\chi_{B_{h}\backslash\overline{S_{h}}}\in
H^{1}(B_{h})^{2}$, $j=1,2$, are solutions to (4.22) satisfying

${\mathbf{u}}_{1}^{+}={\mathbf{u}}_{2}^{+}\ \mbox{in}\
B_{h}\backslash\overline{S_{h}},\quad\mbox{and}\quad{\mathbf{u}}_{2}({\mathbf{0}})\neq{\mathbf{0}}.$
(4.23)

One has $\eta_{1}=\eta_{2}$ and $q_{1}=q_{2}$. Moreover, we have

${\mathbf{u}}_{1}^{-}={\mathbf{u}}_{2}^{-}\quad\mbox{in}\quad S_{h}.$ (4.24)

###### Proof.

In what follows, the proof of this lemma shall be divided into three parts.

Part 1. First, we shall prove that $\eta_{1}=\eta_{2}$.

By Lemma 4.4 , we can directly derive that ${\mathbf{u}}_{1}^{-}\in
C^{\alpha}(\overline{S_{h}})^{2}$ and ${\mathbf{u}}_{2}^{-}\in
C^{\alpha}(\overline{S_{h}})^{2}$.

Using (4.23), by the trace theorem, we know that
${\mathbf{u}}_{1}^{+}={\mathbf{u}}_{2}^{+}$ on $\Gamma_{h}^{\pm}$, which can
be used to obtain that
$T_{\nu}{\mathbf{u}}_{1}^{+}=T_{\nu}{\mathbf{u}}_{2}^{+}$ on
$\Gamma_{h}^{\pm}$. Furthermore we can derive that
${\mathbf{u}}_{1}^{-}={\mathbf{u}}_{1}^{+}={\mathbf{u}}_{2}^{+}={\mathbf{u}}_{2}^{-}$
on $\Gamma_{h}^{\pm}$, by the generalized transmission boundary conditions in
(4.22).

Define ${\mathbf{v}}:={\mathbf{u}}_{1}^{-}-{\mathbf{u}}_{2}^{-}$, by using
(4.22), we have

$\begin{cases}\mathcal{L}{\mathbf{v}}+{\omega}^{2}q_{1}{\mathbf{v}}={\omega}^{2}(q_{2}-q_{1}){{\mathbf{u}}_{2}^{-}},&\mbox{in}\quad
S_{h},\\\ {\mathbf{v}}={\mathbf{0}},\
T_{\nu}{\mathbf{v}}=(\eta_{1}-\eta_{2}){\mathbf{u}}_{2}^{-},&\mbox{on}\quad\Gamma_{h}^{\pm}.\end{cases}$
(4.25)

Recall that the CGO solution ${\mathbf{u}}_{0}$ is defined in (4.4), where
$\mathbf{d}$ and $\mathbf{d}^{\perp}$ are given by (4.6). Since
${\mathbf{u}}_{1}^{-},{\mathbf{u}}_{2}^{-}\in H^{1}(B_{h})^{2}$, from (4.22),
we obviously have
$\mathcal{L}{{\mathbf{u}}_{1}^{-}},\mathcal{L}{{\mathbf{u}}_{2}^{-}}\in
H^{1}(B_{h})^{2}$, where $S_{h}$ is the bounded Lipschitz domain. By the fact
that $\mathcal{L}{\mathbf{u}}_{0}={\mathbf{0}}$ in $S_{h}$ and
${\mathbf{v=0}}$ on $\Gamma_{h}^{\pm}$, we can obtain the integral identity
from the second Green identity (cf. [11, Lemm 3.4], [14, Lemma 2.5] and [26,
Theorem 4.4]) and (4.25) as follows:

$\int_{S_{h}}{\mathbf{u}}_{0}({\mathbf{x}})\cdot\mathcal{L}{\mathbf{v}}dx=\int_{S_{h}}({\omega}^{2}(q_{2}-q_{1}){\mathbf{u}}_{2}^{-}-{\omega}^{2}q_{1}{\mathbf{v}})\cdot{\mathbf{u}}_{0}({\mathbf{x}})\mathrm{d}{\mathbf{x}},$

and

$\displaystyle\int_{S_{h}}{\mathbf{u}}_{0}({\mathbf{x}})\cdot\mathcal{L}{\mathbf{v}}\mathrm{d}{\mathbf{x}}=\int_{\partial
S_{h}}[T_{\nu}{\mathbf{v(x)}}\cdot{\mathbf{u}}_{0}({\mathbf{x}})-T_{\nu}{\mathbf{u}}_{0}({\mathbf{x}})\cdot{\mathbf{v(x)}}]\mathrm{d}\sigma$
$\displaystyle=$
$\displaystyle\int_{\Gamma_{h}^{\pm}}(\eta_{1}-\eta_{2}){\mathbf{u}}_{2}^{-}({\mathbf{x}})\cdot{\mathbf{u}}_{0}({\mathbf{x}})\mathrm{d}\sigma+\int_{\Lambda_{h}}[T_{\nu}{\mathbf{v(x)}}\cdot{{\mathbf{u}}}_{0}({{\mathbf{x}}})-T_{\nu}{\mathbf{u}}_{0}({\mathbf{x}})\cdot{\mathbf{v(x)}}]\mathrm{d}\sigma.$

By rearranging terms, it can be rewritten as

$\displaystyle\int_{S_{h}}{\omega}^{2}(q_{2}-q_{1}){{\mathbf{u}}_{2}^{-}}\cdot{\mathbf{u}}_{0}({\mathbf{x}})\mathrm{d}{\mathbf{x}}$
(4.26) $\displaystyle=$
$\displaystyle\int_{S_{h}}{\omega}^{2}q_{1}{\mathbf{v}}\cdot{\mathbf{u}}_{0}({\mathbf{x}})\mathrm{d}{\mathbf{x}}+\int_{\Gamma_{h}^{\pm}}(\eta_{1}-\eta_{2}){\mathbf{u}}_{2}^{-}({\mathbf{x}})\cdot{\mathbf{u}}_{0}({\mathbf{x}})\mathrm{d}\sigma$
$\displaystyle+\int_{\Lambda_{h}}[T_{\nu}{\mathbf{v(x)}}\cdot{\mathbf{u}}_{0}({\mathbf{x}})-T_{\nu}{\mathbf{u}}_{0}({\mathbf{x}})\cdot{\mathbf{v(x)}}]\mathrm{d}\sigma.$

Denote

$I_{1}:=\int_{\Lambda_{h}}[T_{\nu}{\mathbf{v(x)}}\cdot{\mathbf{u}}_{0}({\mathbf{x}})-T_{\nu}{\mathbf{u}}_{0}({\mathbf{x}})\cdot{\mathbf{v(x)}}]\mathrm{d}\sigma.$
(4.27)

Using the Hölder inequality, Lemma 4.2 and the trace theorem, one has

$\displaystyle|I_{1}|$
$\displaystyle\leq\|T_{\nu}{\mathbf{v}}\|_{H^{-1/2}(\Lambda_{h})^{2}}\|{\mathbf{u}}_{0}\|_{H^{1/2(\Lambda_{h})^{2}}}+\|T_{\nu}{\mathbf{u}}_{0}\|_{{L^{2}}(\Lambda_{h})^{2}}^{2}\|{\mathbf{v}}\|_{L^{2}(\Lambda_{h})^{2}}$
(4.28)
$\displaystyle\leq(\|{\mathbf{u}}_{0}\|_{H^{1}(\Lambda_{h})^{2}}+\|T_{\nu}{\mathbf{u}}_{0}\|_{L^{2}(\Lambda_{h})^{2}})\|{\mathbf{v}}\|_{H^{1}(S_{h})^{2}}\leq
Cse^{-sh\delta_{W}}$

as $s\to+\infty$, where $\delta_{W}$ is defined in Lemma 4.1 and $C$ is a
positive constant, which does not depend on $s$.

Due to ${\mathbf{u}}_{j}^{-}\in C^{\alpha}(\overline{S_{h}})^{2}$ ($j=1,2$),
we have ${\mathbf{v}}\in C^{\alpha}(\overline{S_{h}})^{2}$ for
$\alpha\in(0,1)$. Then ${\mathbf{v}}$ has the expansion:

${\mathbf{v(x)}}={\mathbf{v(0)}}+\delta{\mathbf{v(x)}},\quad|\delta{\mathbf{v(x)}}|\leq{\|{\mathbf{v}}\|}_{C^{\alpha}(\overline{S_{h}})}|{\mathbf{x}}|^{\alpha},$
(4.29)

where ${\mathbf{v(0)=0}}$ due to ${\mathbf{v=0}}$ on $\Gamma_{h}^{\pm}$. Using
(4.29) we can show that

$\int_{S_{h}}{\omega}^{2}q_{1}{\mathbf{v}}\cdot{\mathbf{u}}_{0}({\mathbf{x}})d{\mathbf{x}}={\omega}^{2}q_{1}\int_{S_{h}}\delta{\mathbf{v(x)}}\cdot{\mathbf{u}}_{0}({\mathbf{x}})\mathrm{d}{\mathbf{x}}.$
(4.30)

Denote

$I_{2}:=\int_{S_{h}}\delta{\mathbf{v(x)}}\cdot{\mathbf{u}}_{0}({\mathbf{x}})\mathrm{d}{\mathbf{x}}.$
(4.31)

Based on (4.7), it is readily to know that

$\displaystyle|I_{2}|$
$\displaystyle\leq\int_{S_{h}}|\delta{\mathbf{v(x)}}||{\mathbf{u}}_{0}({\mathbf{x}})|\mathrm{d}{\mathbf{x}}\leq\sqrt{2}{\|{\mathbf{v}}\|}_{C^{\alpha}(\overline{S_{h}})}\int_{W}|\hat{u}_{1}({\mathbf{x}})||{\mathbf{x}}|^{\alpha}\mathrm{d}{\mathbf{x}}$
(4.32)
$\displaystyle\leq\sqrt{2}{\|{\mathbf{v}}\|}_{C^{\alpha}(\overline{S_{h}})}\frac{(\theta_{M}-\theta_{m})\Gamma(\alpha+2)}{\delta_{W}^{\alpha+2}}s^{-\alpha-2},$

as $s\rightarrow+\infty$. Denote

$I_{3}^{\pm}:=\int_{\Gamma_{h}^{\pm}}(\eta_{1}-\eta_{2}){\mathbf{u}}_{2}^{-}({\mathbf{x}})\cdot{\mathbf{u}}_{0}({\mathbf{x}})\mathrm{d}\sigma.$
(4.33)

In view of ${{\mathbf{u}}_{2}^{-}}\in C^{\alpha}(\overline{S_{h}})^{2}$ for
$\alpha\in(0,1)$, it has a expansion:

${\mathbf{u}}_{2}^{-}({\mathbf{x}})={\mathbf{u}}_{2}^{-}({\mathbf{0}})+\delta{\mathbf{u}}_{2}^{-}({\mathbf{x}}),\quad|\delta{\mathbf{u}}_{2}^{-}(x)|\leq{\|{\mathbf{u}}_{2}^{-}\|}_{C^{\alpha}(\overline{S_{h}})}|{\mathbf{x}}|^{\alpha}.$
(4.34)

Substituting (4.34) into $I_{3}^{+}$, there holds that

$\displaystyle I_{3}^{+}$
$\displaystyle=(\eta_{1}-\eta_{2})\int_{\Gamma_{h}^{+}}({\mathbf{u}}_{2}^{-}({\mathbf{0}})+\delta{\mathbf{u}}_{2}^{-}({\mathbf{x}}))\cdot{\mathbf{u}}_{0}({\mathbf{x}})\mathrm{d}\sigma$
(4.35)
$\displaystyle=(\eta_{1}-\eta_{2}){\mathbf{u}}_{2}^{-}({\mathbf{0}})\cdot\int_{\Gamma_{h}^{+}}{\mathbf{u}}_{0}({\mathbf{x}})\mathrm{d}\sigma+(\eta_{1}-\eta_{2})\int_{\Gamma_{h}^{+}}\delta{\mathbf{u}}_{2}^{-}({\mathbf{x}})\cdot{\mathbf{u}}_{0}({\mathbf{x}})\mathrm{d}\sigma.$

Let

$I_{31}^{+}=\int_{\Gamma_{h}^{+}}{\mathbf{u}}_{0}({\mathbf{x}})\mathrm{d}\sigma,\quad
I_{32}^{+}=\int_{\Gamma_{h}^{+}}\delta{\mathbf{u}}_{2}^{-}({\mathbf{x}})\cdot{\mathbf{u}}_{0}({\mathbf{x}})\mathrm{d}\sigma.$
(4.36)

Utilizing (4.10), we have

$\displaystyle I_{31}^{+}$ $\displaystyle=\left(\begin{array}[]{c}1\\\
{\mathrm{i}}\end{array}\right)\int_{0}^{h}e^{sr\exp{{\mathrm{i}}(\theta_{M}-\theta_{d})}}\mathrm{d}r=\left({\begin{array}[]{*{20
}{c}}1\\\
{\mathrm{i}}\end{array}}\right)\left(-e^{{\mathrm{i}}(\theta_{d}-\theta_{M})}s^{-1}-{\sf
R({\theta_{M}})}\right),$ (4.41)

where ${\sf R({\theta_{M}})}$ is defined in (4.10). By (4.34) and (4.12), it
can be deduced that

$|I_{32}^{+}|\leq\sqrt{2}{\|{\mathbf{u}}_{2}^{-}\|}_{C^{\alpha}(\overline{S_{h}})}\int_{0}^{h}r^{\alpha}e^{sr\cos(\theta_{M}-\theta_{d})}dr=O(s^{-\alpha-1}),$
(4.42)

as $s\rightarrow+\infty$. Utilizing the similar argument for $I_{3}^{+}$, one
can obtain that

$\displaystyle I_{3}^{-}$
$\displaystyle=(\eta_{1}-\eta_{2})\int_{\Gamma_{h}^{-}}({\mathbf{u}}_{2}^{-}({\mathbf{0}})+\delta{\mathbf{u}}_{2}^{-}({\mathbf{0}}))\cdot{\mathbf{u}}_{0}({\mathbf{x}})\mathrm{d}\sigma$
(4.43) $\displaystyle:$
$\displaystyle=(\eta_{1}-\eta_{2}){\mathbf{u}}_{2}^{-}({\mathbf{0}})\cdot
I_{31}^{-}+(\eta_{1}-\eta_{2})I_{32}^{-},$

where

$\displaystyle I_{31}^{-}$ $\displaystyle=\left(\begin{array}[]{c}1\\\
{\mathrm{i}}\end{array}\right)\int_{0}^{h}e^{sr\exp{{\mathrm{i}}(\theta_{m}-\theta_{d})}}\mathrm{d}r=\left({\begin{array}[]{*{20
}{c}}1\\\
{\mathrm{i}}\end{array}}\right)\left(-e^{{\mathrm{i}}(\theta_{d}-\theta_{m})}s^{-1}-{\sf
R({\theta_{m}})}\right),$ (4.48)

and

$|I_{32}^{-}|\leq\sqrt{2}{\|{\mathbf{u}}_{2}^{-}\|}_{C^{\alpha}(\overline{S_{h}})}\int_{0}^{h}r^{\alpha}e^{-sr\cos(\theta_{m}-\theta_{d})}\mathrm{d}r=O(s^{-\alpha-1}),$
(4.49)

as $s\rightarrow+\infty$. Here ${\sf R({\theta_{m}})}$ is defined in (4.10).
Furthermore, using (4.34), there holds that

$\displaystyle\int_{S_{h}}{\omega}^{2}(q_{2}-q_{1}){\mathbf{u}}_{2}^{-}\cdot{\mathbf{u}}_{0}({\mathbf{x}})\mathrm{d}{\mathbf{x}}={\omega}^{2}(q_{2}-q_{1})\int_{S_{h}}\left({\mathbf{u}}_{2}^{-}({\mathbf{0}})+\delta{\mathbf{u}}_{2}^{-}({\mathbf{0}})\right)\cdot{\mathbf{u}}_{0}({\mathbf{x}})\mathrm{d}{\mathbf{x}}$
$\displaystyle=$
$\displaystyle{\omega}^{2}(q_{2}-q_{1}){\mathbf{u}}_{2}^{-}({\mathbf{0}})\cdot\int_{S_{h}}{\mathbf{u}}_{0}({\mathbf{x}})\mathrm{d}{\mathbf{x}}+{\omega}^{2}(q_{2}-q_{1})\int_{S_{h}}\delta{\mathbf{u}}_{2}^{-}({\mathbf{0}})\cdot{\mathbf{u}}_{0}({\mathbf{x}})\mathrm{d}{\mathbf{x}}$
$\displaystyle=$
$\displaystyle{\omega}^{2}(q_{2}-q_{1}){\mathbf{u}}_{2}^{-}({\mathbf{0}})\cdot\int_{W}{\mathbf{u}}_{0}({\mathbf{x}})\mathrm{d}{\mathbf{x}}-{\omega}^{2}(q_{2}-q_{1}){\mathbf{u}}_{2}^{-}({\mathbf{0}})\cdot\int_{W\backslash{S_{h}}}{\mathbf{u}}_{0}({\mathbf{x}})\mathrm{d}{\mathbf{x}}$
$\displaystyle+{\omega}^{2}(q_{2}-q_{1})\int_{S_{h}}\delta{\mathbf{u}}_{2}^{-}({\mathbf{0}})\cdot{\mathbf{u}}_{0}({\mathbf{x}})\mathrm{d}{\mathbf{x}}.$
(4.50)

Define

$I_{4}:={\mathbf{u}}_{2}^{-}({\mathbf{0}})\cdot\int_{W\backslash{S_{h}}}{\mathbf{u}}_{0}({\mathbf{x}})\mathrm{d}{\mathbf{x}},\quad
I_{5}:=\int_{S_{h}}\delta{\mathbf{u}}_{2}^{-}({\mathbf{0}})\cdot{\mathbf{u}}_{0}({\mathbf{x}})\mathrm{d}{\mathbf{x}}.$
(4.51)

According to (4.9), it yields that

$|I_{4}|\leq
2\sqrt{2}|{\mathbf{u}}_{2}^{-}({\mathbf{0}})|\frac{\theta_{M}-\theta_{m}}{\delta_{W}}s^{-1}e^{-\delta_{W}sh/2},$
(4.52)

as $s\rightarrow+\infty$. Utilizing the analogous argument for $I_{2}$, it
readily to know that

$|I_{5}|\leq\sqrt{2}{\|{\mathbf{u}}_{2}^{-}\|}_{C^{\alpha}(\overline{S_{h}})}\frac{(\theta_{M}-\theta_{m})\Gamma(\alpha+2)}{\delta_{W}^{\alpha+2}}s^{-\alpha-2}=O(s^{-\alpha-2}),$
(4.53)

as $s\rightarrow+\infty$, where $\alpha\in(0,1).$ Hence, from the integral
identity (LABEL:eq:inter_indentity) we can deduce that

$\displaystyle{\omega}^{2}(q_{2}-q_{1}){\mathbf{u}}_{2}^{-}({\mathbf{0}})\cdot\int_{W}{\mathbf{u}}_{0}({\mathbf{x}})\mathrm{d}{\mathbf{x}}$
(4.54) $\displaystyle=$ $\displaystyle
I_{1}+{\omega}^{2}q_{1}I_{2}+I_{3}^{\pm}+{\omega}^{2}(q_{2}-q_{1})I_{4}+{\omega}^{2}(q_{1}-q_{2})I_{5}.$

Substituting (4.7), (4.35) and (4.43) into (LABEL:eq:itg_idt1), after
rearranging terms, there holds that

$\displaystyle(\eta_{2}-\eta_{1})e^{{\mathrm{i}}\theta_{d}}\left(e^{-{\mathrm{i}}\theta_{m}}+e^{-{\mathrm{i}}\theta_{M}}\right)s^{-1}{\mathbf{u}}_{2}^{-}({\mathbf{0}})\cdot\left({\begin{array}[]{*{20}{c}}1\\\
{\mathrm{i}}\end{array}}\right)$ (4.57) $\displaystyle=$
$\displaystyle-0.5{\mathrm{i}}{\omega}^{2}(q_{1}-q_{2})e^{2{\mathrm{i}}\theta_{d}}(e^{-2{\mathrm{i}}\theta_{M}}-e^{-2{\mathrm{i}}\theta_{m}})s^{-2}{\mathbf{u}}_{2}^{-}({\mathbf{0}})\cdot\left({\begin{array}[]{*{20}{c}}1\\\
{\mathrm{i}}\end{array}}\right)$ (4.60)
$\displaystyle-(\eta_{2}-\eta_{1})\left({\sf R({\theta_{M}})}+{\sf
R({\theta_{m}})}\right){\mathbf{u}}_{2}^{-}({\mathbf{0}})\cdot\left({\begin{array}[]{*{20}{c}}1\\\
{\mathrm{i}}\end{array}}\right)+I_{1}+{\omega}^{2}q_{1}I_{2}+(\eta_{1}-\eta_{2})I_{32}^{+}$
(4.63)
$\displaystyle+(\eta_{1}-\eta_{2})I_{32}^{-}+{\omega}^{2}(q_{2}-q_{1}){\mathbf{u}}_{2}^{-}({\mathbf{0}})I_{4}+{\omega}^{2}(q_{1}-q_{2})I_{5}.$
(4.64)

Multiplying $s$ on the both sides of (4.57), combining with (4.28), (4.32),
(4.42), (4.48), (4.49), (4.52) and (4.53), taking $s\to+\infty$, it yields
that

$(\eta_{2}-\eta_{1})e^{{\mathrm{i}}\theta_{d}}\left(e^{-{\mathrm{i}}\theta_{m}}+e^{-{\mathrm{i}}\theta_{M}}\right){\mathbf{u}}_{2}^{-}({\mathbf{0}})\cdot\left({\begin{array}[]{*{20}{c}}1\\\
{\mathrm{i}}\end{array}}\right)=0.$ (4.65)

By noting that $-\pi<\theta_{m}<\theta_{M}<\pi$ and
$\theta_{m}-\theta_{M}\neq\pi$, one can claim that
$e^{-{\mathrm{i}}\theta_{m}}+e^{-{\mathrm{i}}\theta_{M}}\neq 0$. So we know
$\mu(\theta_{M})^{-2}+\mu(\theta_{m})^{-2}\neq 0$. Due to the fact that the
physical parameters in (4.9) is real valued, without loss of generality, we
assume that the solutions $\mathbf{u}_{j}$ ($j=1,2$) are real valued by
considering the real and imaginary part of (4.9) respectively. Since $\eta$ is
a real valued function and
${\mathbf{u}}_{2}^{-}({\mathbf{0}})\neq{\mathbf{0}}$, one has

$\eta_{1}=\eta_{2},$ (4.66)

which completes the proof of Part 1.

Part 2. We prove $q_{1}=q_{2}$ in this part.

Substituting (4.66) into (4.57), there holds that

$\displaystyle
0.5{\mathrm{i}}{\omega}^{2}(q_{1}-q_{2})e^{2{\mathrm{i}}\theta_{d}}(e^{-2{\mathrm{i}}\theta_{M}}-e^{-2{\mathrm{i}}\theta_{m}})s^{-2}{\mathbf{u}}_{2}^{-}({\mathbf{0}})\cdot\left({\begin{array}[]{*{20}{c}}1\\\
{\mathrm{i}}\end{array}}\right)$ (4.69) $\displaystyle=$ $\displaystyle
I_{1}+{\omega}^{2}q_{1}I_{2}+{\omega}^{2}(q_{2}-q_{1}){\mathbf{u}}_{2}^{-}({\mathbf{0}})I_{4}+{\omega}^{2}(q_{1}-q_{2})I_{5}.$
(4.70)

Combining with (4.28), (4.32), (4.52) and (4.53), multiplying $s^{2}$ on the
both sides of (4.69) and taking $s\to+\infty$, we can derive that

$(q_{2}-q_{1}){\mathbf{u}}_{2}^{-}({\mathbf{0}})(e^{-2{{\mathrm{i}}}\theta_{M}}-e^{-2{{\mathrm{i}}}\theta_{m}})\cdot\left(\begin{array}[]{*{20}{c}}1\\\
{\mathrm{i}}\end{array}\right)=0.$ (4.71)

Due to $-\pi<\theta_{m}<\theta_{M}<\pi$, it yields that
$e^{-2\theta_{M}{{\mathrm{i}}}}-e^{-2\theta_{m}{{\mathrm{i}}}}\neq 0$. By
virtue of (4.23), it is easy to derive that
${\mathbf{u}}_{1}^{-}({\mathbf{0}})={\mathbf{u}}_{2}^{-}({\mathbf{0}})\neq{\mathbf{0}}$.
Therefore, by virtue of (4.71) and $q_{j}$ being a real valued constant, we
can deduce that $q_{1}=q_{2}$.

Part 3. In this part, we prove (4.24). In fact, utilizing (4.69), (4.23), the
generalized impedance boundary conditions in (4.22) and the trace theorem, it
indicates that

${\mathbf{u}}_{1}^{-}={\mathbf{u}}_{1}^{+}={\mathbf{u}}_{2}^{+}={\mathbf{u}}_{2}^{-},\
T_{\nu}{\mathbf{u}}_{1}^{+}=T_{\nu}{\mathbf{u}}_{2}^{+},\
T_{\nu}{\mathbf{u}}_{1}^{-}=T_{\nu}{\mathbf{u}}_{2}^{-},\ \mbox{on}\
\Gamma_{h}^{\pm},$ (4.72)

Moreover, by using $q_{1}=q_{2}$ and (4.72), we can deduce that

$\begin{cases}\mathcal{L}{\mathbf{v}}+{\omega}^{2}q_{1}{\mathbf{v}}={\mathbf{0}},&\mbox{in}\quad{S_{h}},\\\
{\mathbf{v}}=T_{\nu}{\mathbf{v}}={\mathbf{0}},&\mbox{on}\quad\Gamma_{h}^{\pm},\end{cases}$
(4.73)

where ${\mathbf{v}}:={\mathbf{u}}_{1}^{-}-{\mathbf{u}}_{2}^{-}$. By Holmgren’s
uniqueness principle, we derive that
${\mathbf{u}}_{1}^{-}={\mathbf{u}}_{2}^{-}\ \mbox{in}\ S_{h}$.

The proof is complete. ∎

######  Remark 4.1.

To obtain uniquely identifiable outcomes for an elastic scatterer with
piecewise polygonal-nest or polygonal-cell structures associated with the
scattering problem (1.7) through a single far-field measurement, by
contradiction argument, we need to study the singular behaviour of the
underlying total wave field around a corner of the underlying elastic
scatterer, which can be recast as a PDE system (4.22). Therefore, the crucial
role of Lemma 4.5 in proving Theorem 5.2 and Theorem 5.3 cannot be overlooked.

##  5\. Main uniqueness results

In this section, we will use a single far-field measurement to demonstrate the
local and global uniqueness result for determining the shape of an elastic
scatterer with a polygonal-nest or polygonal-cell configuration under a
generic physical scenario. Additionally, a single measurement enables the
unique identification of both the piecewise constant density and boundary
impedance parameter. Before establishing the uniqueness results of the inverse
problems (1.10), we shall introduce the admissibility conditions for an
elastic scatterer with polygonal-nest or polygonal-cell structures, which will
be used in the rest of this paper.

######  Definition 5.1.

Let $(\Omega;q,\eta)$ be a polygonal-nest or polygonal-cell elastic scatterer
$(\Omega;q,\eta)$ as described by Definitions 3.3 and Definition 3.4,
respectively. The scatterer is said to be admissible if the following
conditions is met:

  * 1.

Considering a polygonal-nest elastic scatterer $(\Omega;\lambda,\mu,q,\eta)$
associated with the polygonal-nest partition
$\\{\Pi_{\zeta}\\}_{\zeta=1}^{N}$,
$\forall{\mathbf{x_{c}}}\in\partial{\Pi_{\zeta}}$,
${\mathbf{u}}({\mathbf{x_{c}}})\neq{\mathbf{0}}$, where ${\mathbf{u}}$ is the
solution to (1.7) corresponding to $(\Omega;\lambda,\mu,q,\eta)$.

  * 2.

Considering a polygonal-cell elastic scatterer $(\Omega;\lambda,\mu,q,\eta)$,
${\mathbf{u}}({\mathbf{x_{c}}})\neq{\mathbf{0}}$,
$\forall{\mathbf{x_{c}}}\in\partial\Omega$, where where ${\mathbf{u}}$ is the
solution to (1.7) corresponding to $(\Omega;q,\eta)$.

The non-void admissible condition described in Definition 5.1 was applied in
[14] and can be fulfilled in various generic situations. For instance, when
the angular frequency $\omega$ is considerably smaller than the diameter of
the scatterer $\Omega$, i.e.,

$\displaystyle\omega\cdot{\rm diam}(\Omega)\ll 1.$ (5.1)

Therefore, as the scatterer $\Omega$ is significantly smaller than the
operating wavelength, it can be assumed that the total wave field is primarily
due to the incident wave ${\mathbf{u}}^{i}$, while the influence of the
corresponding scattering wave ${\mathbf{u}}^{sc}$ is negligible from a
physical standpoint. For the incident wave ${\mathbf{u}}^{i}$, we can choose
either a compressional plane wave
$\mathbf{d}\exp(\mathrm{i}k_{p}\mathbf{x}\cdot\mathbf{d})$ or a shear plane
wave $\mathbf{d}^{\perp}\exp(\mathrm{i}k_{s}\mathbf{x}\cdot\mathbf{d})$, where
${\mathbf{d}}$ and ${\mathbf{d}}^{\perp}$ are both elements of
$\mathbb{S}^{1}$ and ${\mathbf{d}}\cdot{\mathbf{d}}^{\perp}=0$, it is evident
that ${\mathbf{u}}^{i}$ is non-vanishing throughout. Consequently, when (5.1)
is fulfilled, ${\mathbf{u}}={\mathbf{u}}^{i}+{\mathbf{u}}^{sc}$ cannot vanish
everywhere. Indeed, we can rigorously demonstrate from Remark 2.1 that if
(5.1) is satisfied and the generalized transmission boundary parameter $\eta$
is sufficiently small, the energy of the scattered wave ${\mathbf{u}}^{sc}$ of
(1.7) is negligible. This ensures that the total wave field will not vanish
throughout.

In Theorem 5.1, we consider an elastic scatterer $\Omega$ that is in the form
of a polygonal-nest or polygonal-cell. We aim to present a local uniqueness
result for the shape determination of $\Omega$. If two scatterers,
$\Omega_{1}$ and $\Omega_{2}$, have identical far field measurements under a
fixed incident wave, we will prove that the difference between these two
scatterers cannot contain a corner. The proof will rely on the contraction,
Rellich theorem and Lemma 4.3. By observing the convexity of a scatterer with
a polygonal-nest structure, we can further prove a global determination of the
boundary of $\mathbb{R}^{2}\backslash\Omega$ by a single measurement.

######  Theorem 5.1.

Consider the elastic scattering problem (1.7) associated with two admissible
polygonal-nest or polygonal-cell elastic scatterers
$(\Omega_{j};\lambda,\mu,q_{j},\eta_{j})$ ($j=1,2$) in $\mathbb{R}^{2}$.
Recall that
${\mathbf{u}}_{\beta}^{j,\infty}(\hat{{\mathbf{x}}};{{\mathbf{u}}^{i}})$ is
the far-field pattern associated with the incident field ${{\mathbf{u}}^{i}}$
and the scatterers $(\Omega_{j};\lambda,\mu,q_{j},\eta_{j})$, $\beta=t,p\ or\
s$. Assume that

${\mathbf{u}}_{\beta}^{1,\infty}(\hat{{\mathbf{x}}};{{\mathbf{u}}^{i}})={\mathbf{u}}_{\beta}^{2,\infty}(\hat{{\mathbf{x}}};{{\mathbf{u}}^{i}}),\quad\forall\hat{{\mathbf{x}}}\in\mathbb{S}^{1},$
(5.2)

where ${\mathbf{u}}^{i}$ is a fixed incident wave. Then

$\Omega_{1}\Delta\Omega_{2}:=(\Omega_{1}\backslash\Omega_{2})\cup(\Omega_{2}\backslash\Omega_{1}).$
(5.3)

cannot contain a sectorial corner. Moreover, suppose that $\Omega_{1}$ and
$\Omega_{2}$ are two admissible polygonal-nest elastic scatterers, we can
conclude that

$\partial\Omega_{1}=\partial\Omega_{2}.$ (5.4)

###### Proof.

The first conclusion of this theorem is proved by contradiction. In the
following, we suppose that there is a corner contained within
$\Omega_{1}\Delta\Omega_{2}$. For a polygonal-nest or a polygonal-cell elastic
scatterer, the generalized transmission conditions are introduced in (3.3) and
(3.7), respectively. It is evident that these two generalized transmission
conditions for polygonal-cell and polygonal-nest elastic scatterer are
coherent. Therefore, it is only necessary to prove the case for the polygonal-
nest structure, and the polygonal-cell case can be demonstrated in a similar
way.

Following the notation introduced in (3.1), let two polygonal-nest elastic
scatterers be written as

$(\Omega_{j};\lambda,\mu,q_{j},\eta_{j})=\bigcup\limits_{\zeta=1}^{N_{j}}(\mathcal{H}_{\zeta,j};q_{\zeta,j},\eta_{\zeta,j}),$
(5.5)

where

$\Omega_{j}=\bigcup_{\zeta=1}^{N_{j}}\mathcal{H}_{\zeta,j},\
q=\sum_{\zeta=1}^{N_{j}}q_{\zeta,j}\chi_{\mathcal{H}_{\zeta,j}},\
\eta=\sum_{\zeta=1}^{N_{j}}{\eta_{\zeta,j}}\chi_{\partial\Pi_{\zeta,j}}.$
(5.6)

Assume that $\Omega_{1}\Delta\Omega_{2}$ possesses a corner. Since
$\mathcal{L}$ is invariant under rigid motion, without loss of generality, we
may suppose that the vertex $O$ of the corner $\Omega_{2}\cap W$ satisfies
that $O\in\partial\Omega_{2}$ and $O\notin\partial\overline{\Omega}_{1}$,
where $O$ is the origin. Moreover we assume that the underlying corner
satisfies that $S_{h}\subseteq\Pi_{1,2}$, where $\Pi_{1,2}$ is defined in
(5.6).

Due to
${\mathbf{u}}_{\beta}^{1,\infty}(\hat{{\mathbf{x}}};{{\mathbf{u}}^{i}})={\mathbf{u}}_{\beta}^{2,\infty}(\hat{{\mathbf{x}}};{{\mathbf{u}}^{i}}),\
\forall\hat{{\mathbf{x}}}\in\mathbb{S}^{1}$, it is easy to deduce that
${{\mathbf{u}}}_{1}^{sc}={{\mathbf{u}}}_{2}^{sc}$ in
$\mathbb{R}^{2}\backslash(\overline{\Omega}_{1}\cup\overline{\Omega}_{2})$ by
making use of Rellich Theorem [18]. Furthermore, it yields that

${\mathbf{u}}_{1}({\mathbf{x}})={\mathbf{u}}_{2}({\mathbf{x}}),\
\forall{\mathbf{x}}\in\mathbb{R}^{2}\backslash(\overline{\Omega}_{1}\cup\overline{\Omega}_{2}).$
(5.7)

With the help of the generalized transmission condition and the notations in
(4.2), one has that

${\mathbf{u}}_{2}^{-}={\mathbf{u}}_{2}^{+}={\mathbf{u}}_{1}^{+},\quad
T_{\nu}{\mathbf{u}}_{2}^{-}=T_{\nu}{\mathbf{u}}_{2}^{+}+\eta_{1,2}{\mathbf{u}}_{2}^{+}=T_{\nu}{\mathbf{u}}_{1}^{+}+\eta_{1,2}{\mathbf{u}}_{1}^{+},\quad\mbox{on}\
\Gamma_{h}^{\pm},$ (5.8)

where the superscripts $(\cdot)^{-},(\cdot)^{+}$ stand for the limits taken
from $\Pi_{1,2}\Subset\Omega_{2}$ and
$\mathbb{R}^{2}\backslash\overline{\Omega}_{2}$ respectively. Furthermore, we
may assume that the neighbourhood $B_{h}(O)$ is sufficiently small such that

$\begin{cases}\mathcal{L}{\mathbf{u}}_{1}^{+}+{\omega}^{2}{\mathbf{u}}_{1}^{+}={\mathbf{0}},\
&\mbox{in}\quad B_{h},\\\
\mathcal{L}{\mathbf{u}}_{2}^{-}+{\omega}^{2}q_{1,2}{\mathbf{u}}_{2}^{-}={\mathbf{0}},\
&\mbox{in}\quad S_{h}\Subset\Pi_{2,1},\end{cases}$

where $q_{1,2}$ is defined in (5.6). It is obvious that
${\mathbf{u}}_{1}^{+}\in H^{2}(S_{h})^{2}$ and ${\mathbf{u}}_{2}^{-}\in
H^{1}(S_{h})^{2}$. From Lemma 4.4, one has ${\mathbf{u}}_{2}^{-}\in
C^{\alpha}(\overline{S_{h}})^{2}$. Applying Lemma 4.3 and utilizing the
continuity of ${\mathbf{u_{1}}}$ at the vertex $O$, one has
${\mathbf{u}}_{1}({\mathbf{0}})={\mathbf{0}}$, which contradicts to the
admissibility condition in Definition 5.1. Hence we obtain that there is no
corner contained within $\Omega_{1}\Delta\Omega_{2}$.

Finally, we prove (5.4) using contradiction. We assume that
$\partial\Omega_{1}\neq\partial\Omega_{2}$. From Definition 3.1, it is evident
that $\Omega_{1}$ and $\Omega_{2}$ are convex. Therefore, we claim that the
boundary of
$\mathbb{R}^{2}\backslash(\overline{\Omega}_{1}\cup\overline{\Omega}_{2})$ has
a corner, which contradicts the first conclusion of this theorem.

The proof is complete. ∎

For the polygonal-cell elastic scatterer $\Omega$, in Theorem 5.2, we will
demonstrate that the physical parameters of density and boundary impedance
parameter of $\Omega$, which are piecewise constant, can be determined with a
single far field measurement, assuming a-prior knowledge of the shape and
partition of $\Omega$. Indeed, we will use the transmission PDE system and
Lemma 4.5 to prove the global recovery result for piecewise constant physical
parameters.

######  Theorem 5.2.

Consider the elastic scattering problems (1.7) associated with two admissible
polygonal-cell elastic scatterers $(\Omega;\lambda,\mu,q_{j},\eta_{j})$ in
$\mathbb{R}^{2}$ ($j=1,2$). Suppose that the physical parameter $q_{j}$ and
$\eta_{j}$ associated with $\Omega_{j}$ has a common polygonal-cell partition
$\\{\Pi_{\zeta}\\}_{\zeta=1}^{N}$ described by Definition 3.2. Let $q_{j}$ and
$\eta_{j}$ be defined as follows:

$q_{j}=\sum_{\zeta=1}^{N}q_{\zeta,j}\chi_{\Pi_{\zeta}},\
\eta_{j}=\sum_{\zeta=1}^{N}\eta_{j}^{*}\chi_{\partial\Pi_{\zeta}}.$ (5.9)

Let ${\mathbf{u}}_{\beta}^{j,\infty}(\hat{{\mathbf{x}}};{{\mathbf{u}}^{i}})$
be the far-field pattern associated with the incident field ${\mathbf{u}}^{i}$
and the scatterer $(\Omega;q_{j},\eta_{j})$ where $\beta=t,p\ \mbox{or}\ s$.
Assume that

${\mathbf{u}}_{\beta}^{1,\infty}(\hat{{\mathbf{x}}};{{\mathbf{u}}^{i}})={\mathbf{u}}_{\beta}^{2,\infty}(\hat{{\mathbf{x}}};{{\mathbf{u}}^{i}}),\quad\forall\hat{{\mathbf{x}}}\in\mathbb{S}^{1},$
(5.10)

where $\mathbf{u}^{i}$ is a fixed incident wave. Then one has $q_{1}=q_{2}$
and $\eta_{1}=\eta_{2}$.

###### Proof.

According to the statement of this theorem, the shape of $\Omega$ and its
corresponding polygonal-cell partitions $\\{\Pi_{\zeta}\\}_{\zeta=1}^{N}$ are
already known in advance. We establish the uniqueness in determining the
physical parameters of density and boundary impedance parameter by
contradiction. Let us assume that $\eta_{1}^{*}\neq\eta_{2}^{*}$, or if there
exists an index $1\leq\zeta_{0}\leq N$ such that $q_{\zeta_{0},1}\neq
q_{\zeta_{0},2}$. From Definition 3.2, there exists a vertex
${\mathbf{x_{c}}}$ in $\Pi_{\zeta_{0}}$ which is formed by two adjacent edges
$\Gamma^{\pm}$ of $\Omega$. Without loss of generality, it can be assumed that
${\mathbf{x_{c}}}$ coincides with the origin. Let $h>0$ be sufficiently small
so that $S_{h}=\Omega\cap{B_{h}}({\mathbf{0}})$, where
$\Gamma^{\pm}=\partial\Omega\cap{B_{h}}({\mathbf{0}})$ satisfying
${\Gamma^{\pm}}\subset\partial{\mathbf{G}}$. Here,
${\mathbf{G}}={\mathbb{R}^{2}}\backslash\overline{\Omega}$. Under this
geometrical setup, we assume that $S_{h}$ and $\Gamma_{h}^{\pm}$ are same as
(4.1); see Figure 3 for illustration.

Recall that ${\mathbf{u}}_{1}$ and ${\mathbf{u}}_{2}$ be the total wave fields
associated with the admissible polygonal-cell elastic scatterers
$(\Omega_{j};q_{j},{\eta_{j}}),j=1,2$. Due to
${\mathbf{u}}_{\beta}^{1,\infty}(\hat{{\mathbf{x}}};{\mathbf{u}}^{i})={\mathbf{u}}_{\beta}^{2,\infty}(\hat{{\mathbf{x}}};{\mathbf{u}}^{i}),\forall\hat{{\mathbf{x}}}\in\mathbb{S}^{1}$,
with the help of Rellich Theorem [18] and the unique continuation principle,
it yields that

${\mathbf{u}}_{1}^{+}={\mathbf{u}}_{2}^{+}\quad\mbox{in}\quad\
B_{h}\backslash\overline{S_{h}}\subset{\mathbf{G}}.$ (5.11)

Furthermore, by using the generalized impedance transmission boundary
conditions in (1.7), one has

$\begin{cases}{\mathbf{u}}_{1}^{+}={\mathbf{u}}_{1}^{-},\
T_{\nu}{\mathbf{u}}_{1}^{+}+\eta_{1}^{*}{\mathbf{u}}_{1}^{+}=T_{\nu}{\mathbf{u}}_{1}^{-},\quad\mbox{on}\quad\Gamma_{h}^{\pm},\\\
{\mathbf{u}}_{2}^{+}={\mathbf{u}}_{2}^{-},\
T_{\nu}{\mathbf{u}}_{2}^{+}+\eta_{2}^{*}{\mathbf{u}}_{2}^{+}=T_{\nu}{\mathbf{u}}_{2}^{-},\quad\mbox{on}\quad\Gamma_{h}^{\pm}.\end{cases}$
(5.12)

Then, it indicates that the following system are satisfied:

$\begin{cases}\mathcal{L}{\mathbf{u}}_{1}^{-}+{\omega}^{2}q_{\zeta_{0},1}{\mathbf{u}}_{1}^{-}={\mathbf{0}},\
&\quad\mbox{in}\quad S_{h},\\\
\mathcal{L}{\mathbf{u}}_{1}^{+}+{\omega}^{2}{\mathbf{u}}_{1}^{+}={\mathbf{0}},\
&\quad\mbox{in}\quad B_{h}\backslash\overline{S_{h}},\\\
\mathcal{L}{\mathbf{u}}_{2}^{-}+{\omega}^{2}q_{\zeta_{0},2}{\mathbf{u}}_{2}^{-}={\mathbf{0}},\
&\quad\mbox{in}\quad S_{h},\\\
\mathcal{L}{\mathbf{u}}_{2}^{+}+{\omega}^{2}{\mathbf{u}}_{2}^{+}={\mathbf{0}},\
&\quad\mbox{in}\quad{B_{h}}\backslash\overline{S_{h}}.\end{cases}$ (5.13)

Since ${\mathbf{u}}_{1}^{+},{\mathbf{u}}_{2}^{+}\in
H^{2}(B_{h}\backslash\overline{S_{h}})^{2}$ and $\Omega$ is an admissible
polygonal-cell elastic scatterer, we can deduce that
${\mathbf{u}}_{1}^{+}({\mathbf{0}})={\mathbf{u}}_{2}^{+}({\mathbf{0}})\neq{\mathbf{0}}$.
By virtue of (5.11), (5.12), (5.13) and Lemma 4.5, one has

$q_{\zeta_{0},1}=q_{\zeta_{0},2}\quad\mbox{and}\quad\eta_{1}^{*}=\eta_{2}^{*},$

where we get the contradiction.

The proof is complete. ∎

Figure 3. The schematic illustration of the geometrical setup in the proof of
Theorem 5.2.

In the following theorem, we consider the global uniqueness result for the
identification of the shape of a polygonal-nest elastic scatterer $\Omega$ by
a single far-field measurement. Specifically, the outer boundary, internal
polygonal-nest pattern, and the corresponding constant physical parameters of
the density and boundary impedance parameter can be uniquely determined by a
single measurement. Compared to Theorem 5.2, the polygonal-nest structure
eliminates the a-prior knowledge of the shape and polygonal-nest partition of
$\Omega$.

######  Theorem 5.3.

Consider the elastic scattering problems (1.7) associated with two admissible
polygonal-nest elastic scatterers $(\Omega;q_{j},\eta_{j})$ in
$\mathbb{R}^{2}$ ($j=1,2$), where the associated polygonal-nest partitions
$\\{\Pi_{\zeta,1}\\}_{\zeta=1}^{N_{1}}$ and
$\\{\Pi_{\zeta,2}\\}_{\zeta=1}^{N_{2}}$ are described by Definition 3.1 with
$\Omega_{1}=\bigcup_{\zeta=1}^{N_{1}}{\mathcal{H}_{\zeta,1}}$ and
${\Omega_{2}}=\bigcup\nolimits_{\zeta=1}^{N_{2}}\mathcal{H}_{\zeta,2}$. Here
$\mathcal{H}_{\zeta,j}=\Pi_{\zeta,j}\backslash\overline{\Pi}_{\zeta+1,j},\zeta=1,\cdots,N_{j}$.
For $j=1,2$, the physical parameters $q_{j}$ and $\eta_{j}$ can be
characterized as follows:

$q_{j}=\sum_{\zeta=1}^{N_{j}}q_{\zeta,j}\chi_{\mathcal{H}_{\zeta,j}},\
\eta_{j}=\sum_{\zeta=1}^{N_{j}}\eta_{\zeta,j}\chi_{\partial\Pi_{\zeta,j}}.$
(5.14)

Let ${\mathbf{u}}_{\beta}^{j,\infty}(\hat{{\mathbf{x}}};{{\mathbf{u}}^{i}})$
be the far-field pattern associated with the incident field ${\mathbf{u}}^{i}$
and the scatterer $(\Omega;q_{j},\eta_{j})$ where $\beta=t,p\ \mbox{or}\ s$.
Assume that

${\mathbf{u}}_{\beta}^{1,\infty}(\hat{{\mathbf{x}}};{{\mathbf{u}}^{i}})={\mathbf{u}}_{\beta}^{2,\infty}(\hat{{\mathbf{x}}};{{\mathbf{u}}^{i}}),\
\hat{{\mathbf{x}}}\in\mathbb{S}^{1},$ (5.15)

where $\mathbf{u}^{i}$ is a fixed incident wave. Then $N_{1}=N_{2}=N,\
\partial\Pi_{\zeta,1}=\partial\Pi_{\zeta,2}$ for $\zeta=1,\cdots,N$,
$q_{1}=q_{2}$ and $\eta_{1}=\eta_{2}$.

Figure 4. The schematic illustration of the geometrical setup in the proof of
Theorem 5.3

###### Proof.

We will prove this theorem by mathematical induction. The subsequent proof
will be divided into three steps.

Step 1. When $\zeta=1$, using the admissible condition, by Theorem 5.1, we
have $\partial\Omega_{1}=\partial\Omega_{2}$, which indicates that
$\partial\Pi_{1,1}=\partial\Pi_{1,2}$. Let ${\mathbf{u}}_{j}$ be the total
wave fields associated with the admissible polygonal-nest elastic scatterers
$(\Omega_{j};q_{j},\eta_{j}),j=1,2$. Since
$\partial\Omega_{1}=\partial\Omega_{2}$ and $\mathcal{L}$ is invariant under
rigid motion, let $S_{h}$ be a sectoral corner with the apex $\mathbf{0}$ and
$h\in\mathbb{R}_{+}$ such that $S_{h}\Subset\Omega_{1}$, where $S_{h}$ is
defined in (4.2) with two adjacent edges
$\Gamma_{h}^{\pm}\in\partial\Omega_{1}$. Due to
${\mathbf{u}}_{\beta}^{1,\infty}(\hat{{\mathbf{x}}};{{\mathbf{u}}^{i}})={\mathbf{u}}_{\beta}^{2,\infty}(\hat{{\mathbf{x}}};{{\mathbf{u}}^{i}}),$
$\forall\hat{{\mathbf{x}}}\in\mathbb{S}^{1}$, with the help of Rellich Theorem
[18] and the unique continuation principle, it yields that

${\mathbf{u}}_{1}={\mathbf{u}}_{2}\quad\mbox{in}\quad
B_{h}\backslash\overline{S_{h}}\subset{\mathbf{G}},\quad{\mathbf{G}}=\mathbb{R}^{2}\backslash\overline{\Omega_{1}}.$
(5.16)

Using the generalized impedance transmission boundary conditions in (1.7), one
can claim that

$\begin{cases}{\mathbf{u}}_{1}^{+}={\mathbf{u}}_{1,1}^{-},\quad
T_{\nu}{\mathbf{u}}_{1}^{+}+\eta_{1,1}{\mathbf{u}}_{1}^{+}=T_{\nu}{\mathbf{u}}_{1,1}^{-},\quad\mbox{on}\quad\Gamma_{h}^{\pm},\\\
{\mathbf{u}}_{2}^{+}={\mathbf{u}}_{1,2}^{-},\quad
T_{\nu}{\mathbf{u}}_{2}^{+}+\eta_{1,2}{\mathbf{u}}_{2}^{+}=T_{\nu}{\mathbf{u}}_{1,2}^{-},\quad\mbox{on}\quad\Gamma_{h}^{\pm},\end{cases}$
(5.17)

where
${\mathbf{u}}_{j}^{+}={\mathbf{u}}_{j}\big{|}_{\mathbb{R}^{2}\backslash\Omega_{1}}$
and ${\mathbf{u}}_{1,j}^{-}={\mathbf{u}}_{j}\big{|}_{\mathcal{H}_{1,j}}$.
Therefore one has

$\begin{cases}\mathcal{L}{\mathbf{u}}_{1,1}^{-}+{\omega}^{2}q_{1,1}{\mathbf{u}}_{1,1}^{-}={\mathbf{0}},\quad&\mbox{in}\quad
S_{h},\\\
\mathcal{L}{\mathbf{u}}_{1}^{+}+{\omega}^{2}{\mathbf{u}}_{1}^{+}={\mathbf{0}},\quad&\mbox{in}\quad
B_{h}\backslash\overline{S_{h}},\\\
\mathcal{L}{\mathbf{u}}_{1,2}^{-}+{\omega}^{2}q_{1,2}{\mathbf{u}}_{1,2}^{-}={\mathbf{0}},\quad&\mbox{in}\quad
S_{h},\\\
\mathcal{L}{\mathbf{u}}_{2}^{+}+{\omega}^{2}{\mathbf{u}}_{2}^{+}={\mathbf{0}},\quad&\mbox{in}\quad
B_{h}\backslash\overline{S_{h}}.\end{cases}$ (5.18)

Using the fact that $\Omega$ is an admissible polygonal-nest elastic
scatterer, one has
${\mathbf{u}}_{1}({\mathbf{0}})={\mathbf{u}}_{2}({\mathbf{0}})\neq{\mathbf{0}}$.
Due to ${\mathbf{u}}_{1},{\mathbf{u}}_{2}\in
H^{2}(B_{h}\backslash\overline{S_{h}})^{2}$, in view of (5.16), (5.17) and
(5.18), with the help of Lemma 4.5, it indicates that

$q_{1,1}=q_{1,2}\quad\mbox{and}\quad\eta_{1,1}=\eta_{1,2}.$

Furthermore, by virtue of (4.24) in Lemma 4.5, one has

$\displaystyle{\mathbf{u}}_{1,1}^{-}={\mathbf{u}}_{1,2}^{-}\quad\mbox{in}\quad
S_{h}.$ (5.19)

Step 2. Assume that there exits an index $n\in\mathbb{N}\backslash\\{1\\}$
such that

$\partial\Pi_{\zeta,1}=\partial\Pi_{\zeta,2},\quad
q_{\zeta,1}=q_{\zeta,2},\quad\eta_{\zeta,1}=\eta_{\zeta,2},\quad\zeta=2,\ldots,N-1,$
(5.20)

where $\Pi_{\zeta,j},j=1,2$, are described in the statement of this theorem.
According to Lemma 4.5, by making use of
${\mathbf{u}}_{\beta}^{1,\infty}(\hat{{\mathbf{x}}};{{\mathbf{u}}^{i}})={\mathbf{u}}_{\beta}^{2,\infty}(\hat{{\mathbf{x}}};{{\mathbf{u}}^{i}})$,
we can recursively obtain that

${\mathbf{u}}_{\zeta,1}={\mathbf{u}}_{\zeta,2}\quad\mbox{in}\quad{\mathcal{H}_{\zeta}}=\Pi_{\zeta}\backslash\overline{\Pi_{\zeta+1}},\
\zeta=1,2,\cdots,N-1,$ (5.21)

where ${\mathbf{u}}_{\zeta,1}={\mathbf{u}}_{1}{|_{\mathcal{H}_{\zeta,1}}}$ and
${\mathbf{u}}_{\zeta,2}={\mathbf{u}}_{2}|_{\mathcal{H}_{\zeta,1}}$. From
(5.21), adopting the similar argument in proving Theorem 5.1, by Lemma 4.2, we
can prove that $\partial\Pi_{N,1}=\partial\Pi_{N,2}$.

Step 3. In the following we prove that

$q_{N,1}=q_{N,2},\quad\eta_{N,1}=\eta_{N,2}.$ (5.22)

Let ${\mathbf{x}}_{c}\in\Pi_{N,1}$. Without loss of generality, we may assume
that ${\mathbf{x_{c}}}={\mathbf{0}}$. For sufficiently small
$h\in\mathbb{R}_{+}$, we assume that
$S_{h}\Subset\mathcal{H}_{N,1}=\Pi_{N,1}\backslash\overline{\Pi_{N+1,1}}$. The
schematic illustration is displayed in Figure 4. Hence, there holds that

$\begin{cases}\mathcal{L}{\mathbf{u}}_{N,1}^{-}+{\omega}^{2}q_{N,1}{\mathbf{u}}_{N,1}^{-}={\mathbf{0}},&\mbox{in}\quad
S_{h},\\\
\mathcal{L}{\mathbf{u}}_{N-1,1}^{+}+{\omega}^{2}q_{N-1,1}{\mathbf{u}}_{N-1,1}^{+}={\mathbf{0}},&\mbox{in}\quad
B_{h}\backslash\overline{S_{h}},\\\
\mathcal{L}{\mathbf{u}}_{N,2}^{-}+{\omega}^{2}q_{N,2}{\mathbf{u}}_{N,2}^{-}={\mathbf{0}},&\mbox{in}\quad
S_{h},\\\
\mathcal{L}{\mathbf{u}}_{N-1,2}^{+}+{\omega}^{2}q_{N-1,2}{\mathbf{u}}_{N-1,2}^{+}={\mathbf{0}},&\mbox{in}\quad
B_{h}\backslash\overline{S_{h}},\\\
{\mathbf{u}}_{N,1}^{-}={\mathbf{u}}_{N-1,1}^{+},\
T_{\nu}{\mathbf{u}}_{N-1,1}^{+}+\eta_{N,1}{\mathbf{u}}_{N-1,1}^{+}=T_{\nu}{\mathbf{u}}_{N,1}^{-},&\mbox{on}\quad\Gamma_{h}^{\pm},\\\
{\mathbf{u}}_{n,2}^{-}={\mathbf{u}}_{N-1,2}^{+},\
T_{\nu}{\mathbf{u}}_{N-1,2}^{+}+\eta_{N,1}{\mathbf{u}}_{N-1,2}^{+}=T_{\nu}{\mathbf{u}}_{N,2}^{-},&\mbox{on}\quad\Gamma_{h}^{\pm}.\end{cases}$
(5.23)

According to (5.21), one has
${\mathbf{u}}_{n-1,1}^{+}={\mathbf{u}}_{n-1,2}^{+}$. In view of (5.23), with
the help of Lemma 4.5 and the admissible condition, one has (5.22).

Finally, it can be proved by contradiction that $N_{1}=N_{2}$. Without loss of
generality, let us assume that $N_{1}>N_{2}$. Therefore, $\Pi_{N_{2},2}$
encompasses a corner of $\Pi_{N_{1}+1,1}$. Using Lemma 4.3, we deduce that the
total wave field becomes zero at the vertex of this corner. This contradicts
the admissibility of the polygonal-nest elastic scatterer.

The proof is complete. ∎

## Acknowledgment

The work of H. Diao is supported by National Natural Science Foundation of
China (No. 12371422), the Fundamental Research Funds for the Central
Universities, JLU (No. 93Z172023Z01) and the National Key R&D Program of China
(No. 2020YFA0714102). The work of H. Liu was supported by the Hong Kong RGC
General Research Funds (projects 11311122, 11300821 and 12301420), NSF/RGC
Joint Research Fund (project N_CityU101/21) and the ANR/RGC Joint Research
Fund (project A_CityU203/19).

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