# Stable determination of an elastic medium scatterer by a single far-field
measurement and beyond

Zhengjian Bai  School of Mathematical Sciences, Xiamen University, Xiamen
361005, China<EMAIL_ADDRESS>,  Huaian Diao  School of Mathematics, Jilin
University, Changchun 130012, China<EMAIL_ADDRESS>,  Hongyu Liu  Department
of Mathematics, City University of Hong Kong, Kowloon, Hong Kong, China
<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Qingle Meng  School of
Mathematical Sciences, Xiamen University, Xiamen 361005, China
<EMAIL_ADDRESS>
###### Abstract.

We are concerned with the time-harmonic elastic scattering due to an
inhomogeneous elastic material inclusion located inside a uniformly
homogeneous isotropic medium. We establish a sharp stability estimate of
logarithmic type in determining the support of the elastic scatterer,
independent of its material content, by a single far-field measurement when
the support is a convex polyhedral domain in $\mathbb{R}^{n}$, $n=2,3$. Our
argument in establishing the stability result is localized around a corner of
the medium scatterer. This enables us to further establish a byproduct result
by proving that if a generic medium scatterer, not necessary to be a
polyhedral shape, possesses a corner, then there exists a positive lower bound
of the scattered far-field patterns. The latter result indicates that if an
elastic material object possesses a corner on its support, then it scatters
every incident wave stably and invisibility phenomenon does not occur.

Keywords: Inverse shape problem, elastic scattering, stability, single
measurement, corner scattering, invisibility

2020 Mathematics Subject Classification: 35Q60, 78A46, 35P25

##  1\. Introduction

###  1.1. Mathematical setup

We are mainly concerned with the time-harmonic elastic wave scattering due to
the impingement of an incident field on an inhomogeneous isotropic medium
scatterer as well as the associated inverse problem of determining the
scatterer from the corresponding far-field measurement. We first introduce the
mathematical formulation of our study.

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^{n}$ with a
connected complement $\mathbb{R}^{n}\backslash\overline{\Omega}$, $n=2,3$. In
the physical setup, $\Omega$ is the support of an inhomogeneous elastic
scatterer embedded in a uniformly homogeneous background space. The elastic
medium parameters are characterised by $\rho$ and $\lambda,\mu$, which are
respectively referred to as the density and the bulk moduli. It is assumed
that $\lambda$ and $\mu$ are real constants satisfying the strong convexity
conditions:

$\mu>0\quad\mbox{ and }\quad n\,\lambda+2\mu>0.$

It is also assumed that $\rho\in L^{\infty}(\mathbb{R}^{n})$ and
${\Omega=\mathrm{supp}(1-\rho)}$. That is, by normalisation, we assume that
the density of the homogeneous background space is $1$, whereas the scatterer
is characterised by the inhomogeneous density $\rho$ in $\Omega$.

To introduce the Lamé system that describes the elastic scattering, we let
$\omega\in\mathbb{R}_{+}$ signify the angular frequency of the time-harmonic
elastic wave. Denote

$\kappa_{\mathrm{s}}=\omega\sqrt{1/\mu}\mbox{ and
}\kappa_{\mathrm{p}}=\omega\sqrt{1/(\lambda+2\mu)}$

by the shear and compressional wave numbers, respectively. Let
$\mathbf{u}^{i}$ denote the time-harmonic plane incident wave of the following
form:

$\displaystyle\mathbf{u}^{i}=\alpha_{1}\,\mathbf{d}\,e^{\imath\kappa_{\mathrm{p}}\mathbf{x}\cdot\mathbf{d}}+\alpha_{2}\,\mathbf{d}^{\perp}e^{\imath\kappa_{\mathrm{s}}\mathbf{x}\cdot\mathbf{d}},\quad\alpha_{1},\alpha_{2}\in\mathbb{C},\quad|\alpha_{1}|+|\alpha_{2}|\neq
0,$ (1.1)

where $\mathbf{d}\in\mathbb{S}^{n-1}$ is called the incident direction, and
$\mathbf{d}^{\perp}$ is a unit vector orthogonal to $\mathbf{d}$. Due to the
interaction between the incident elastic wave $\mathbf{u}^{i}$ and the elastic
scatter $\Omega$, the scattered elastic wave $\mathbf{u}^{s}\in\mathbb{C}^{n}$
is generated. This induces the total elastic wave
$\mathbf{u}\in\mathbb{C}^{n}$, which is the superposition of $\mathbf{u}^{i}$
and $\mathbf{u}^{s}$, namely $\mathbf{u}:=\mathbf{u}^{i}+\mathbf{u}^{s}$, and
satisfies the Navier equation

$\Delta^{*}\mathbf{u}+\rho\,\omega^{2}\mathbf{u}={\bf
0},\quad\Delta^{*}\mathbf{u}:=\mu\Delta\mathbf{u}+(\lambda+\mu)\nabla(\nabla\cdot\mathbf{u}),\quad\mbox{in}\
\mathbb{R}^{n}\backslash\overline{\Omega}.$ (1.2)

Clearly, $\mathbf{u}^{i}$ is an entire solution to (1.2) with $\rho=1$. By the
Helmholtz decomposition, any solution $\mathbf{u}\in
H_{loc}^{2}(\mathbb{R}^{n})^{n}$ to (1.2) can be decomposed as follows:

$\mathbf{u}=\mathbf{u}_{\mathrm{p}}+\mathbf{u}_{\mathrm{s}},$ (1.3)

where $\mathbf{u}_{\mathrm{p}}$ and $\mathbf{u}_{\mathrm{s}}$ satisfy the
equations:

$\begin{cases}(\Delta+\kappa^{2}_{\mathrm{p}})\mathbf{u}_{\mathrm{p}}={\bf
0},&\quad\nabla\times\mathbf{u}_{\mathrm{p}}=\bf 0,\vskip 6.0pt plus 2.0pt
minus 2.0pt\\\ (\Delta+\kappa^{2}_{\mathrm{s}})\mathbf{u}_{\mathrm{s}}={\bf
0},&\quad\nabla\,\cdot\,\mathbf{u}_{\mathrm{s}}=0.\end{cases}$ (1.4)

The scattered wave $\mathbf{u}^{s}$ satisfies the Kurpradze radiation
condition

$\begin{cases}\lim_{|\mathbf{x}|\to\infty}|\mathbf{x}|^{(n-1)/2}(\frac{\partial{\mathbf{u}_{\mathrm{p}}^{s}}}{\partial{|\mathbf{x}|}}-\imath\kappa_{p}\mathbf{u}_{\mathrm{p}}^{s})=\bf
0,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\lim_{|\mathbf{x}|\to\infty}|\mathbf{x}|^{(n-1)/2}(\frac{\partial{\mathbf{u}_{\mathrm{s}}^{s}}}{\partial{|\mathbf{x}|}}-\imath\kappa_{s}\mathbf{u}_{\mathrm{s}}^{s})=\bf
0,\end{cases}$ (1.5)

and admits the following asymptotic expansion (cf. [31]):

$\displaystyle\mathbf{u}^{s}(\mathbf{x})=\dfrac{\exp(\imath\kappa_{\mathrm{p}}|\mathbf{x}|)}{|\mathbf{x}|^{(n-1)/2}}\mathbf{U}_{\mathrm{p}}(\hat{\mathbf{x}})+\dfrac{\exp(\imath\kappa_{\mathrm{s}}|\mathbf{x}|)}{|\mathbf{x}|^{(n-1)/2}}\mathbf{U}_{\mathrm{s}}(\hat{\mathbf{x}})+\mathcal{O}(|\mathbf{x}|^{-\frac{n+1}{2}})\quad\mbox{as}\quad|\mathbf{x}|\to\infty,$
(1.6)

which holds uniformly in all directions $\hat{\mathbf{x}}\in\mathbb{S}^{n-1}$,
where $\imath:=\sqrt{-1}$ is the imaginary unit. The vector fields
$\mathbf{U}_{\mathrm{p}}$ and $\mathbf{U}_{\mathrm{s}}$ are referred to as the
longitudinal and transversal far-field patterns, respectively, which
respectively characterize the asymptotic behaviors of the normal part
$\mathbf{u}^{s}_{\mathrm{p}}$ and of the tangential part
$\mathbf{u}^{s}_{\mathrm{s}}$ of $\mathbf{u}^{s}$. Let
$\mathbf{U}=(\mathbf{U}_{\mathrm{p}},\,\mathbf{U}_{\mathrm{s}})$ denote the
far-field pattern of $\mathbf{u}^{s}$, which is an element of the liner space
$L^{2}(\mathbb{S}^{n-1})^{n}\times L^{2}(\mathbb{S}^{n-1})^{n}$ equipped with
the following norm

$\|(\mathbf{w}_{1},\mathbf{w}_{2})\|^{2}_{L^{2}(\mathbb{S}^{n-1},\mathbb{C}^{n}\times\mathbb{C}^{n})}=\|\mathbf{w}_{1}\|^{2}_{L^{2}(\mathbb{S}^{n-1})^{n}}+\|\mathbf{w}_{2}\|^{2}_{L^{2}(\mathbb{S}^{n-1})^{n}},$

where $(\mathbf{w}_{1},\mathbf{w}_{2})\in L^{2}(\mathbb{S}^{n-1})^{n}\times
L^{2}(\mathbb{S}^{n-1})^{n}$.

The inverse problem of our study can be introduced as determining the support
of the inhomogeneous scattering object, namely $\Omega$, independent of its
material content $(\lambda,\mu,\rho)$, by knowledge of the far-field pattern
$\mathbf{U}(\hat{\mathbf{x}})$. By introducing an operator $\mathscr{F}$ which
is defined by the forward scattering system (1.1)–(1.6) and sends the
scatterer $(\Omega;\lambda,\mu,\rho)$ to the corresponding far-field pattern
associated with an incident field $\mathbf{u}^{i}$, the inverse problem can be
abstractly given as:

$\mathscr{F}(\Omega)=\mathbf{U}(\hat{\mathbf{x}};\mathbf{u}^{i}),\quad\hat{\mathbf{x}}\in\mathbb{S}^{n-1}.$
(1.7)

It is emphasized that in our study of (1.7), the material content of $\Omega$,
namely $(\lambda,\mu,\rho)$, is not required to be known in advance though it
belongs to a certain general a-priori class as shall be introduced in what
follows. Moreover, we shall consider the case that only a fixed
$\mathbf{u}^{i}$ is used, namely $\mathbf{u}^{i}$ is given in (1.1) with fixed
$\eta_{1},\eta_{2},\mathbf{d}$ and $\mathbf{d}^{\perp}$. In such a case,
$\mathbf{U}(\hat{\mathbf{x}};\mathbf{u}^{i})$ is said to be a single far-field
measurement. The inverse scattering problem with a single far-field
measurement constitutes a longstanding problem in the literature. Finally, we
remark that by direction verifications, the inverse problem (1.7) is nonlinear
and it is formally-determined with a single far-field measurement.

The second focus of our study is under what conditions,
$\mathbf{U}\equiv\mathbf{0}$. This is another aspect of the inverse scattering
problem (1.7) and in such a case, the underlying object
$(\Omega;\lambda,\mu,\rho)$ is invisible with respect to the far-field
measurement.

###  1.2. Statement of the main stability result and its implication to
invisibility

First, we introduce the a-priori class of elastic scatterers for our study.

######  Definition 1.1.

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^{n}$ with a
connected complement $\mathbb{R}^{n}\backslash\overline{\Omega}$, $n=2,3$. In
addition, $\Omega=\mathrm{supp}(1-\rho)\subset\overline{B_{R}}$, where $B_{R}$
signifies a central ball of radius $R$. We say $(\Omega;\rho)\in\mathcal{K}$
if the following conditions are satisfied:

  * (a)

$\Omega\subset\mathbb{R}^{2}$ is a convex polygon and the opening angle at
each vertex of $\Omega$ is in $(2\alpha_{m},2\alpha_{M})$, $\alpha_{m}>0$ and
$\alpha_{M}<\pi/2$;

  * (b)

$\Omega\subset\mathbb{R}^{3}$ is a convex polyhedron;

  * (c)

The distances of any vertex of $\Omega$ to its non-adjacent edges are at least
$l_{0}$, $0<l_{0}\leq 1$;

  * (d)

$\rho(\mathbf{x})$ is a uniformly $\theta$-Hölder continuous function in
$\overline{\Omega}$, $0<\theta\leq 1$. In addition,
$|\rho(\mathbf{x}_{0})-1|\geq\epsilon_{0}>0$ at any vertex $\mathbf{x}_{0}$ of
$\Omega$.

######  Definition 1.2.

We say $\rho$ is called an admissible density function if for a time-harmonic
plane incident wave $\mathbf{u}^{i}$ of the form (1.1), the forward scattering
system (1.1)–(1.5) admits a unique solution $\mathbf{u}\in
H_{loc}^{2}(\mathbb{R}^{n})^{n}$ such that the scattered wave
$\mathbf{u}^{s}=\mathbf{u}-\mathbf{u}^{i}$ is the radiating solution and
$\|\mathbf{u}^{s}\|_{H^{2}(B_{2R})^{n}}\leq\mathcal{N}$, where $\mathcal{N}$
is an a-priori positive constant and $\mathbf{u}(\mathbf{x})\neq 0$ in
$S_{\Omega}\cup B_{R}\backslash\overline{\Omega}$, where $S_{\Omega}$ denotes
the set of vertices of $\Omega$.

In what follows, $(\Omega,\rho)$ is said to be an admissible polyhedral
scatterer if it fulfils the conditions in Definitions 1.1 and 1.2. The
parameters $\\{R,\mathcal{N},l_{0},\epsilon_{0},\alpha_{m},\alpha_{M}\\}$ in
the above admissibility definitions are referred to as the a-priori
parameters. It is remarked that the admissibility requirements in Definitions
1.1 and 1.2 will be needed in our stability study and can be fulfilled in
certain general scenarios, which shall become clearer in our subsequent
discussion.

Next, we present the logarithmic stability estimate for the inverse problem
(1.7) on the shape determination of the elastic medium scatterer by a single
far-field pattern. Henceforth, we define the Hausdorff distance of two medium
scatterers $(\Omega;\rho)$ and $(\Omega^{\prime};\rho^{\prime})$ as follows,

$d_{\mathcal{H}}(\Omega,\Omega^{\prime})=\max\left\\{\sup_{\mathbf{x}\in\Omega}\mathrm{dist}(\mathbf{x},\,\Omega^{\prime}),\,\sup_{\mathbf{x}\in\Omega^{\prime}}\mathrm{dist}(\mathbf{x},\,\Omega)\right\\}.$
(1.8)

######  Theorem 1.1.

Let $(\Omega;\rho)$ and $(\Omega^{\prime};\rho^{\prime})$ be two admissible
polyhedral scatterers. Let $\mathbf{u}^{i}$ be a common time-harmonic plane
incident wave of the form (1.1). Assume that $\mathbf{U}$ and
$\mathbf{U}^{\prime}$ are the far-field patterns of the scattered waves
$\mathbf{u}^{s}$ and $\mathbf{u}^{\prime s}$ by the medium scatterers
$(\Omega;\rho)$ and $(\Omega^{\prime};\rho^{\prime})$, respectively. For
sufficiently small $\varepsilon\in\mathbb{R}_{+}$, if

$\|\mathbf{U}-\mathbf{U}^{\prime}\|_{L^{2}(\mathbb{S}^{n-1},\mathbb{C}^{n}\times\mathbb{C}^{n})}\leq\varepsilon,$

then

$d_{\mathcal{H}}(\Omega,\Omega^{\prime})\leq
C(\ln\ln(\mathcal{N}/\varepsilon))^{-\gamma},$

where $\mathcal{N}$ is given in Definition 1.2, and $C$ and $\gamma$ are
positive constants, depending only on the a-priori parameters involved in
Definitions 1.1 and 1.2 as well as the Lamé constants $\lambda,\mu$.

The argument in proving Theorem 1.1 is localized around a corner of the
underlying polyhedral scatterer. As an interesting byproduct, we can establish
another stability result where the medium scatterer is not necessarily
polyhedral as long as it possesses a corner. The full technical details of the
result will be given in Theorems 5.1 and 5.2, and we only provide a rough
summary in the following theorem.

######  Theorem 1.2.

Consider the scattering problem (1.1)–(1.5) associated with a general medium
scatterer $(\Omega,\rho)$, which is not necessarily polyhedral. Suppose that
$\partial\Omega$ possesses a corner as described in Theorem 5.2. Then it holds
that

$\|\mathbf{U}\|_{L^{2}(\mathbb{S}^{n-1},\mathbb{C}^{n}\times\mathbb{C}^{n})}\geq
C,$ (1.9)

where $C$ is a positive constant depending on a certain set of a-priori
parameters.

Theorem 1.2 indicates that for a general medium scatterer, if it possesses a
corner, then it generically scatters every incident field stably, i.e.
invisibility phenomenon cannot occur.

###  1.3. Connection to existing studies and discussion

Determining the shape of an inhomogeneous object by minimal/optimal scattering
measurements has been a longstanding problem in the literature with a long and
colorful history; see [15, 24, 30] for reviews and surveys. Recently, several
qualitative uniqueness results in determining convex polyhedral medium
scatterers were established in [25] for electrostatics, [11, 22, 18, 12, 6]
for acoustic scattering, [28, 8] for electromagnetic scattering and [20] for
elastic scattering. In [7, 27], the shape determination of medium scatterers
whose boundaries possess high-curvature parts was also considered for
electrostatics and acoustic scattering. In [5, 26], quantitative stability
estimates of double-logarithmic type were established in determining the
convex polyhedral shape of an acoustic medium scatterer.

Another issue of significant physical interest is the occurrence of
invisibility, namely the scattering pattern is identically zero. Generically,
it is believed that geometric singularities on the support of a generic medium
scatterer prevents the occurrence of the invisibility phenomenon. We refer to
[2, 3, 9, 5, 7, 8, 10, 11, 18, 24, 28, 29, 34, 35] for related studies on this
intriguing topic in different physical contexts. In [7], a geometrically
singular point (say, e.g. a corner point) on the boundary of a shape is
treated as with infinite curvature, and it is further shown that a medium
scatterer whose smooth shape possesses a sufficiently high curvature point can
also prevent the occurrence of invisibility. All of the aforementioned results
are qualitative and in [5] sharp estimates were established by showing that
there exist positive lower bounds of the scattering patterns, which quantify
the non-invisibility phenomena due to the presence of the shape singularities.
It is noted that the lower scattering bounds have been derived only for the
acoustic scattering.

Our study in this article extends the related studies in [5, 25, 26] for the
electrostatics and acoustic scattering to the elastic scattering which
possesses more complicated and technical nature, both in physics and
mathematics. It is pointed out that as remarked in [5], the double-logarithmic
stability estimates are generically optimal for inverse scattering problems.
Finally, we would like to mention in passing a closely related topic on the
geometric structures of transmission eigenfunctions which is beyond the scope
of this article [4, 18, 20, 13, 14, 16, 19, 23, 17].

The rest of the paper is organized as follows. In Section 2, we derive a
critical auxiliary result about the propagation of smallness from far field to
the boundary of the scatterer. Section 3 is devoted to a micro-local analysis
of the scattering solution around a corner. The full details of the proof of
Theorem 1.1 is presented in Section 4. In Section 5, we present the full
technical details of Theorem 1.2.

##  2\. Propagation of smallness from far-field to boundary

The main goal of this section is to show how the smallness from the far-field
pattern propagates to the boundary of the elastic medium scatterer, which is a
key ingredient in the stability proof of Theorem 1.1. Our argument follows the
general strategy developed in [5] for the acoustic scattering. Throughout the
rest of the paper, we let $\mathbf{u}$ and $\mathbf{u}^{\prime}$,
respectively, denote the total wave fields corresponding to the scatterers
$(\Omega;\rho)$ and $(\Omega^{\prime};\rho^{\prime})$ in Theorem 1.1.

###  2.1. Stability Estimates: from Far-field to Near-field

In this subsection, our aim is to estimate the difference of $\mathbf{u}$ and
$\mathbf{u}^{\prime}$ in $B_{2R}\backslash B_{R}$ close to the convex hull of
$\Omega$ and $\Omega^{\prime}$. To begin with, we need to generalise Theorem
4.1 in [33] and Proposition 5.2 in [5] to the elastic inhomogeneous medium
scatterer case.

######  Lemma 2.1.

Fix $t\in(1,2]$ and $\mathcal{T}>0$. Let $\mathbf{u}\in
H^{2}_{loc}(\mathbb{R}^{n})^{n}$ be a solution to the Navier equation (1.2) in
$\mathbb{R}^{n}\backslash\overline{B_{R}}$ with $\rho=1$, which satisfies the
Kupradze radiation condition. Let $\mathbf{U}$ be the far field pattern of
$\mathbf{u}$ with the norm
$\varepsilon=\|\mathbf{U}\|_{L^{2}(\mathbb{S}^{n-1},\mathbb{C}^{n}\times\mathbb{C}^{n})}$.
Assume that $\|\mathbf{u}_{\mathrm{\alpha}}\|_{L^{2}(B_{2R}\backslash
B_{R})^{n}}\leq\mathcal{T},\mathrm{\alpha}=\mathrm{p},\mathrm{s}$, where
$\mathbf{u}_{\mathrm{p}}$ and $\mathbf{u}_{\mathrm{s}}$ are the longitudinal
and the transversal parts of $\mathbf{u}$, respectively. Then there exists a
constant $C=C(\kappa_{\mathrm{p}},\kappa_{\mathrm{s}},\mathcal{T},n,R,t)>0$
such that the following stability estimate holds for
$\varepsilon\leq\frac{\mathcal{T}}{C}$:

$\|\mathbf{u}\|_{L^{2}(B_{2tR}\backslash B_{tR})^{n}}\leq
Ce^{-\hat{c}\sqrt{\ln\frac{\mathcal{T}}{\varepsilon}}},$ (2.1)

where $\kappa=\min\\{\kappa_{\mathrm{p}},\kappa_{\mathrm{s}}\\}$ and
$\hat{c}\leq\ln t\sqrt{\frac{e\kappa R}{2}}$.

###### Proof.

Let $u^{j}$ and $u^{j}_{\mathrm{\alpha}}$ denote the $j$-th components of
$\mathbf{u}$ and $\mathbf{u}_{\mathrm{\alpha}}$, respectively, where
$\mathrm{\alpha}=\mathrm{p},\mathrm{s}$. From (1.3) and (1.4), we obtain that
$u^{j}_{\mathrm{\alpha}}$ satisfies the Helmoltz equation

$(\Delta+\kappa^{2}_{\mathrm{\alpha}})u^{j}_{\mathrm{\alpha}}=0,\quad
j=1,2,\ldots,n.$

Then by [5, Proposition 5.2], we know that if
$\varepsilon\leq\frac{\mathcal{T}}{C_{\mathrm{\alpha}}}$, then

$\|u^{j}_{\mathrm{\alpha}}\|_{L^{2}(B_{2tR}\backslash B_{tR})}\leq
C_{\mathrm{\alpha}}\mathcal{T}e^{-\hat{c}\sqrt{\ln\frac{\mathcal{T}}{\varepsilon}}},$

where
$C_{\mathrm{\alpha}}=C_{\mathrm{\alpha}}(\kappa_{\mathrm{\alpha}},n,R,t)>0$,
$\mathrm{\alpha}=\mathrm{p},\mathrm{s}$, $\hat{c}\leq\ln t\sqrt{\frac{e\kappa
R}{2}}$ and $\kappa=\min\\{\kappa_{\mathrm{p}},\kappa_{\mathrm{s}}\\}$.

When
$\varepsilon\leq\frac{\mathcal{T}}{\sqrt{2n\,(C^{2}_{\mathrm{p}}+C^{2}_{\mathrm{s}})}}$,
we can derive

$\displaystyle\|\mathbf{u}\|_{L^{2}(B_{2tR}\backslash B_{tR})^{n}}$
$\displaystyle=\sqrt{\sum_{j=1}^{n}\|u^{j}\|^{2}_{L^{2}(B_{2tR}\backslash
B_{tR})}}\leq\sqrt{\sum_{\mathrm{\alpha}=\mathrm{p},\mathrm{s}}\sum_{j=1}^{n}\|u^{j}_{\mathrm{\alpha}}\|^{2}_{L^{2}(B_{2tR}\backslash
B_{tR})}}$
$\displaystyle\leq\sqrt{2n\,(C^{2}_{\mathrm{p}}+C^{2}_{\mathrm{s}})}\,e^{-\hat{c}\sqrt{\ln\frac{\mathcal{T}}{\varepsilon}}}.$

The proof is complete. ∎

With the help of elliptic interior regularity, we can derive the following
stability estimate of the near-fields in $B_{2R}\backslash\overline{B_{R}}$.

######  Proposition 2.1.

Fix $t\in(1,2]$ and $\mathcal{T}>0$. Assume that $\mathbf{u}\in
H^{2}_{loc}(\mathbb{R}^{n})^{n}$ satisfies the Navier equation
$(\Delta^{*}+\omega^{2})\mathbf{u}=\bf 0$ in
$\mathbb{R}^{n}\backslash\overline{B_{R}}$ and the Kupradze radiation
condition. Let $\mathbf{U}$ be the far field pattern of $\mathbf{u}$ with the
norm
$\varepsilon=\|\mathbf{U}\|_{L^{2}(\mathbb{S}^{n-1},\mathbb{C}^{n}\times\mathbb{C}^{n})}$.
Assume that $\|\mathbf{u}_{\mathrm{\alpha}}\|_{L^{2}(B_{2R}\backslash
B_{R})^{n}}\leq\mathcal{T},\mathrm{\alpha}=\mathrm{p},\mathrm{s}$, where
$\mathbf{u}_{\mathrm{p}}$ and $\mathbf{u}_{\mathrm{s}}$ are the longitudinal
and the transversal parts of $\mathbf{u}$, respectively. Let $K\subset
B_{2R}\backslash\overline{B_{tR}}$ be a domain. Then there exists a positive
constant $C=C(\kappa_{\mathrm{p}},\kappa_{\mathrm{s}},n,R,t)$ such that the
following stability estimate holds for $\varepsilon\leq\frac{\mathcal{T}}{C}$:

$\displaystyle\|\mathbf{u}\|_{H^{r}(K)^{n}}\leq
C\mathcal{T}e^{-\hat{c}\sqrt{\ln\frac{\mathcal{T}}{\varepsilon}}},$ (2.2)

where $r\in\mathbb{N}$ is a smoothness index,
$\kappa=\min\\{\kappa_{\mathrm{p}},\kappa_{\mathrm{s}}\\}$ and $\hat{c}\leq\ln
t\sqrt{\frac{e\kappa R}{2}}$.

###### Proof.

Note that $\mathbf{u}$ is the superposition of $\mathbf{u}_{\mathrm{p}}$ and
$\mathbf{u}_{\mathrm{s}}$, which are vector-valued weak solutions to the
Helmholtz equation with wave numbers $\kappa_{\mathrm{p}}$ and
$\kappa_{\mathrm{s}}$, respectively. We use $u^{j}_{\mathrm{\alpha}}$ to
denote the $j$-th component of $\mathbf{u}_{\mathrm{\alpha}}$, where
$\mathrm{\alpha}=\mathrm{p},\mathrm{s}$. Given two domains $\tilde{K}$ and
$K^{\prime}$ with $\tilde{K}\subset K^{\prime}\subset
B_{2R}\backslash\overline{B_{tR}}$ and
$\mathrm{dist}(\partial\tilde{K},\partial K^{\prime})>0$. For any
$s\in\mathbb{R}$ and the smooth cutoff function $\tilde{\phi}\in
C^{\infty}_{0}(K)$ with $\tilde{\phi}\equiv 1$ in $\tilde{K}$, it is not
difficult to prove the two subsequent properties:

$\|u^{j}_{\mathrm{\alpha}}\|_{H^{s+2}(\mathbb{R}^{n})}=\|(1+\kappa^{2}_{\mathrm{\alpha}})u^{j}_{\mathrm{\alpha}}-(\Delta+\kappa^{2}_{\mathrm{\alpha}})u^{j}_{\mathrm{\alpha}}\|_{H^{s}(\mathbb{R}^{n})}$

and

$\|(\Delta+\kappa^{2}_{\mathrm{\alpha}})(\tilde{\phi}\,u^{j}_{\mathrm{\alpha}})\|_{H^{s+2}(\mathbb{R}^{n})}=\|2\,\nabla\tilde{\phi}\cdot\nabla
u^{j}_{\mathrm{\alpha}}+u^{j}_{\mathrm{\alpha}}\Delta\phi\|_{H^{s}(\mathbb{R}^{n})}\leq
C_{\mathrm{\alpha}}\|u^{j}_{\mathrm{\alpha}}\|_{H^{s+1}(\Omega)}.$

Thus we obtain

$\displaystyle\|\mathbf{u}\|^{2}_{H^{s+2}(\tilde{K})^{n}}$
$\displaystyle=\|\mathbf{u}_{\mathrm{p}}+\mathbf{u}_{\mathrm{s}}\|^{2}_{H^{s+2}(\tilde{K})^{n}}\leq\sum_{\mathrm{\alpha}=\mathrm{p},\mathrm{s}}\sum^{n}_{j=1}\|u^{j}_{\mathrm{\alpha}}\|^{2}_{H^{s+2}(\tilde{K})}$
$\displaystyle\leq
2\sum_{\mathrm{\alpha}=\mathrm{p},\mathrm{s}}\sum^{n}_{j=1}\|\tilde{\phi}\,u^{j}_{\mathrm{\alpha}}\|^{2}_{H^{s+2}(\mathbb{R}^{n})}\leq\sum_{\mathrm{\alpha}=\mathrm{p},\mathrm{s}}\sum^{n}_{j=1}C^{2}_{\kappa_{\mathrm{\alpha}},\tilde{\phi}}\,\|u^{j}_{\mathrm{\alpha}}\|^{2}_{H^{s}(K^{\prime})}$
$\displaystyle\leq
C_{\kappa_{\mathrm{p}},\kappa_{\mathrm{s}},n,\tilde{\phi}}^{2}\,\|\mathbf{u}\|^{2}_{H^{s+1}(K^{\prime})^{n}},$

which implies that

$\|\mathbf{u}\|_{H^{s+2}(\tilde{K})^{n}}\leq
C_{\kappa_{\mathrm{p}},\kappa_{\mathrm{p}},n,\tilde{\phi}}\|\mathbf{u}\|_{H^{s+1}(K^{\prime})^{n}}.$
(2.3)

By fixing $r\in\mathbb{N}$, there exists a subdomain sequence
$\\{K_{j},\phi_{j}\\}^{r}_{j=0}$ such that

$\begin{cases}&K_{j}\subset K_{j-1},\,\,\mathrm{dist}(\partial K_{j},\partial
K_{j-1})>0,\\\ &\phi_{j}\in C_{0}^{\infty}(K_{j-1}),\quad\phi_{j}\equiv
1\quad\mathrm{in}\,\,K_{j},\\\
&K_{0}=B_{2R}\backslash\overline{B_{tR}},\quad\,K_{r}=K.\end{cases}$

By using (2.3) and Lemma 2.1 repeatedly, we have

$\displaystyle\|\mathbf{u}\|_{H^{r}(K)^{n}}$ $\displaystyle\leq
C_{\kappa_{\mathrm{p}},\kappa_{\mathrm{s}},\phi_{r}}\|\mathbf{u}\|_{H^{r-1}(K)^{n}}\leq\cdots\leq
C_{\kappa_{\mathrm{p}},\kappa_{\mathrm{s}},\phi_{r},\ldots,\phi_{1}}\|\mathbf{u}\|_{L^{2}(K)^{n}}$
$\displaystyle\leq
C_{\kappa_{\mathrm{p}},\kappa_{\mathrm{s}},\phi_{r},\ldots,\phi_{1}}\mathcal{T}e^{-\hat{c}\sqrt{\ln(\mathcal{T}/\varepsilon)}}.$

The proof is complete. ∎

###  2.2. Stability Estimates: from Near-field to Boundary

In this subsection, we establish the propagation of smallness from the near-
field to the boundary of $Q$ which is the convex hull of $\Omega$ and
$\Omega^{\prime}$. We are mainly interested in estimating the sum of
$|\mathbf{u}-\mathbf{u}^{\prime}|$ and
$|\nabla\mathbf{u}-\nabla\mathbf{u}^{\prime}|$ on $\partial Q$. To begin with,
we introduce the following three-spheres inequalities which may be found, for
instance, in [1].

######  Lemma 2.2.

[32] There exist positive constants $\hat{R}$, $C$ and $c$, $0<c<1$, depending
on $\kappa$ only. Let $0<r_{1}<r_{2}<r_{3}<\hat{R}$, and $u$ be a solution to
$\Delta u+\kappa^{2}u=0$ in $B_{r_{3}}$. For any $s\in(r_{2},r_{3})$, we have

$\|u\|_{L^{\infty}(B_{r_{2}})}\leq
C\big{(}1-\frac{r_{2}}{s}\big{)}^{-3/2}\|u\|^{1-\beta}_{L^{\infty}(B_{r_{3}})}\|u\|^{\beta}_{L^{\infty}(B_{r_{1}})},$

where $\beta$ satisfies the following inequality,

$\frac{c\log\frac{r_{3}}{s}}{\log\frac{r_{3}}{r_{1}}}\leq\beta\leq
1-\frac{c\log\frac{s}{r_{1}}}{\log\frac{r_{3}}{r_{1}}}.$

From now on, we fix $r\in\mathbb{R}_{+}$ and set $r_{1}=r$, $r_{2}=2r$,
$r_{3}=4r$, $s=2\sqrt{2}r$.

######  Lemma 2.3.

Let $V\subset\mathbb{R}^{n}$ be a bounded connected domain and $\Gamma\subset
V$ be a rectifiable curve with endpoints $\mathbf{x}$ and
$\mathbf{x}^{\prime}$. Fix positive constants $\mathcal{T}$ and $\hat{R}$ such
that $4r<\hat{R}$ and $\mathcal{T}\geq 1$. Let
$B(\Gamma,4r)=\bigcup_{\mathbf{y}\in\Gamma}B_{4r}(\mathbf{y})\subset V$.
Moreover, we assume that $\mathbf{u}\in L^{\infty}(V)^{n}$ satisfies

$\begin{cases}&(\Delta^{*}+\omega^{2})\mathbf{u}={\bf
0},\quad\|\mathbf{u}_{\mathrm{\alpha}}\|_{L^{\infty}(V)^{n}}\leq\mathcal{T},\,\mathrm{\alpha}=\mathrm{p},\mathrm{s},\\\
&\max\big{\\{}\|\mathbf{u}_{\mathrm{p}}\|_{L^{\infty}(B_{r}(\mathbf{x}))^{n}},\|\mathbf{u}_{\mathrm{s}}\|_{L^{\infty}(B_{r}(\mathbf{x}))^{n}}\big{\\}}\leq
1,\end{cases}$

where $\mathbf{u}_{\mathrm{p}}$ and $\mathbf{u}_{\mathrm{s}}$ represent the
longitudinal and the transversal parts of $\mathbf{u}$, respectively. Then the
following result holds

$\displaystyle\|\mathbf{u}_{\mathrm{\alpha}}\|_{L^{\infty}(B_{r}(\mathbf{x}^{\prime}))^{n}}\leq
C\mathcal{T}\|\mathbf{u}_{\mathrm{\alpha}}\|^{\beta^{\frac{d_{\Gamma}}{r}+1}}_{L^{\infty}(B_{r}(\mathbf{y}))^{n}},\quad\mathrm{\alpha}=\mathrm{p},\mathrm{s},$

where $d_{\Gamma}$ is the distance measured along $\Gamma$.

###### Proof.

Let $N=\lceil d_{\Gamma}/r\rceil$. There exists a sequence of balls, each of
radius $r$ and centred respectively at
$\mathbf{x}=\mathbf{x}_{1},\mathbf{x}_{2},\cdots,\mathbf{x}_{N+1}=\mathbf{x}^{\prime}$,
such that $\mathbf{x}_{k}\in\Gamma$ and $|\mathbf{x}_{k+1}-\mathbf{x}_{k}|\leq
d_{\Gamma}(\mathbf{x}_{k+1},\mathbf{x}_{k})\leq r$, where
$d_{\Gamma}(\mathbf{x}_{k+1},\mathbf{x}_{k})$ signifies the length along the
curve $\Gamma$ between the points $\mathbf{x}_{k}$ and $\mathbf{x}_{k+1}$.
Clearly, $B_{r}(\mathbf{x}_{k})\subset B_{2r}(\mathbf{x}_{k-1})$. Let
$u^{j}_{\mathrm{\alpha}}$ denote the $j$-th component of
$\mathbf{u}_{\mathrm{\alpha}}$, $\mathrm{\alpha}=\mathrm{p},\mathrm{s}$. Then
we have

$\displaystyle\|\mathbf{u}_{\mathrm{\alpha}}\|_{L^{\infty}(B_{r}(\mathbf{x}^{\prime}))^{n}}$
$\displaystyle\leq\sum^{n}_{j=1}\|u^{j}_{\mathrm{\alpha}}\|_{L^{\infty}(B_{r}(\mathbf{x}^{\prime}))}\leq\sum^{n}_{j=1}\Big{\\{}C_{1}\mathcal{T}^{1-\beta}\|u^{j}_{\mathrm{\alpha}}\|_{L^{\infty}(B_{r}(\mathbf{x}_{N}))}\Big{\\}}$
$\displaystyle\leq\cdots$
$\displaystyle\leq\sum^{n}_{j=1}\Big{\\{}C_{N}\mathcal{T}^{(1-\beta)(1+\beta+\cdots+\beta^{N-1})}\|u^{j}_{\mathrm{\alpha}}\|^{\beta^{N}}_{L^{\infty}(B_{r}(\mathbf{x}))}\Big{\\}}.$

Note that $1+\beta+\cdots+\beta^{N}\leq 1/(1-\beta)$ and $N\leq
d_{\Gamma}/r+1$, hence we have the following claim

$\displaystyle\|\mathbf{u}_{\mathrm{\alpha}}\|_{L^{\infty}(B_{r}(\mathbf{x}^{\prime}))^{n}}\leq
C\mathcal{T}\|\mathbf{u}_{\mathrm{\alpha}}\|^{\beta^{N}}_{L^{\infty}(B_{r}(\mathbf{x}))^{n}}\leq
C\mathcal{T}\|\mathbf{u}_{\mathrm{\alpha}}\|^{\beta^{\frac{d_{\gamma}}{r}+1}}_{L^{\infty}(B_{r}(\mathbf{x}))^{n}},\quad\mathrm{\alpha}=\mathrm{p},\mathrm{s}.$

The proof is complete. ∎

We now give the stability estimations of $\mathbf{u}_{\mathrm{p}}$ and
$\mathbf{u}_{\mathrm{s}}$ near the scatterer $(\Omega;\rho)$.

######  Proposition 2.2.

Given positive constants $\mathcal{T},\hat{R}$ and $\theta,\beta\in(0,1)$ such
that $\mathcal{T}\geq 1$. Let $Q\subset B_{R}$ be a convex polytope. Let
$\mathbf{u}\in H^{2}_{loc}(B_{2R})^{n}$ satisfy the Navier equation
$\Delta^{*}\mathbf{u}+\omega^{2}\mathbf{u}=\bf 0$ in
$B_{2R}\backslash\overline{Q}$. Moreover, we assume that

$\|\mathbf{u}_{\mathrm{\alpha}}\|_{C^{\theta}\big{(}\overline{B_{3/2R}}\big{)}^{n}}\leq\mathcal{T},\quad\|\mathbf{u}_{\mathrm{\alpha}}\|_{L^{\infty}(B_{7R/4}\backslash
B_{5R/4})^{n}}\leq\delta<1,\,\,\alpha=\mathrm{p},\mathrm{s},$

where $\mathbf{u}_{\mathrm{p}}$ and $\mathbf{u}_{\mathrm{s}}$ are the
longitudinal and the transversal parts of $\mathbf{u}$, respectively. Let
$r=\frac{9R|\ln\beta|}{4(1-\theta)\ln|\ln\delta|}$. If

$\delta<\Big{[}\exp\exp\Big{\\{}\frac{4\,r\ln|\ln\beta|}{\min\\{\hat{R},R/2\\}}\Big{\\}}\Big{]}^{-1},$
(2.4)

then

$\displaystyle\|\mathbf{u}_{\mathrm{\alpha}}\|_{L^{\infty}(Q^{\prime})^{n}}\leq
M(\ln|\ln\delta|)^{-\theta}\mathcal{T},\quad\mathrm{\alpha}=\mathrm{p},\mathrm{s}.$
(2.5)

Here $Q^{\prime}=\big{\\{}\mathbf{x}\in
B_{3/2R}\,\big{|}\,\mathrm{dist}(\mathbf{x},\partial Q)\leq 4\,r\big{\\}}$ and
$M=M(\omega,R,\theta,\beta)>1$.

###### Proof.

Let $Q_{1}=\big{\\{}\mathbf{x}\in
B_{2R}\,\big{|}\,\mathrm{dist}(\mathbf{x},\partial Q)<4\,r\big{\\}}$. It is
easy to reduce that $4\,r<\hat{R}$ and $2\,r<\frac{R}{4}$ from the upper bound
on $\delta$ and the expression of $r$. For any point $\mathbf{x}^{\prime}\in
B_{2R}\backslash\overline{Q_{1}}$, there always exists a ray from
$\mathbf{x}^{\prime}$ into $B_{2R}\backslash B_{5/4R}$. We consider a special
segment of the ray with endpoints $\mathbf{x}^{\prime}$ and $\mathbf{y}$,
where $\mathbf{y}\in B_{7/4R}\backslash B_{5/4R}$ and
$\mathrm{dist}(\mathbf{y},\partial B_{5/4R})=r$. One can see that
$\|\mathbf{u}_{\mathrm{\alpha}}\|_{L^{\infty}(B_{r}(\mathbf{y}))^{n}}\leq\delta,\mathrm{\alpha}=\mathrm{p},\mathrm{s}$.
The length of that segment is smaller than $\frac{5R}{2}+r$. By Lemma 2.3, we
have

$\displaystyle\|\mathbf{u}_{\mathrm{\alpha}}\|_{L^{\infty}(B_{r}(\mathbf{x}^{\prime}))^{n}}\leq
C\mathcal{T}\|\mathbf{u}_{\mathrm{\alpha}}\|^{\beta^{\frac{5R}{2\,r}+2}}_{L^{\infty}(B_{r}(\mathbf{y}))^{n}}\leq
C\mathcal{T}\delta^{\beta^{\frac{5R}{2r}+2}},\quad\mathrm{\alpha}=\mathrm{p},\mathrm{s}.$

Then

$\displaystyle\|\mathbf{u}_{\mathrm{\alpha}}\|_{L^{\infty}(B_{2R}\backslash
Q_{1})^{n}}\leq
C^{\prime}\mathcal{T}\delta^{\beta^{\frac{5R}{2r}+2}},\quad\mathrm{\alpha}=\mathrm{p},\mathrm{s}.$

For any point $\mathbf{x}^{\prime}\in Q_{1}$, there must be
$\mathbf{y}\in\partial Q$ such that $|\mathbf{x}^{\prime}-\mathbf{y}|\leq
4\,r$. By the convexity of $Q$, there exists
$\mathbf{x}\in\mathbb{R}^{n}\backslash Q$ such that
$\mathrm{dist}(\mathbf{x},Q)=|\mathbf{x}-\mathbf{y}|=4\,r$. The upper bound of
$\delta$ implies $4\,r\leq\frac{R}{2}$, and thus

$|\mathbf{x}|\leq|\mathbf{x}-\mathbf{y}|+|\mathbf{y}|\leq
4\,r+R\leq\frac{3R}{2}.$

Moreover,
$|\mathbf{x}^{\prime}-\mathbf{x}|\leq|\mathbf{x}^{\prime}-\mathbf{y}|+|\mathbf{y}-\mathbf{x}|\leq
8\,r$. From the $\theta$-Hölder continuity of $\mathbf{u}_{\mathrm{\alpha}}$
and proposition 5.7 in [5], we can obtain that

$\displaystyle\|\mathbf{u}_{\mathrm{\alpha}}\|_{L^{\infty}(Q_{1})^{n}}$
$\displaystyle\leq\sum^{n}_{j=1}\|u^{j}_{\mathrm{\alpha}}\|_{L^{\infty}(Q_{1})}\leq\sum^{n}_{j=1}\Big{(}\|u^{j}_{\mathrm{\alpha}}\|_{\mathbf{C}^{\theta}(B_{3/2R})}|\mathbf{x}-\mathbf{x}^{\prime}|^{\theta}+\|u^{j}_{\mathrm{\alpha}}\|_{L^{\infty}(Q_{1})}\Big{)}$
$\displaystyle\leq\sum^{n}_{j=1}C\mathcal{T}\Big{(}(8\,r)^{\theta}+\delta^{\beta^{\frac{5R}{2\,r}+2}}\Big{)}\leq\widetilde{C}\mathcal{T}\Big{(}(8\,r)^{\theta}+\delta^{\beta^{\frac{5R}{2\,r}+2}}\Big{)},$
(2.6)

where $\widetilde{C}$ is a positive constant depending on
$\kappa_{\mathrm{\alpha}}$ and $n$.

Combining (2.4) with $|\ln\delta|>1$, we have

$\displaystyle
r^{\theta}=\frac{(9R|\ln\beta|)^{\theta}}{((4-4\theta)\ln|\ln\delta|)^{\theta}},\quad\quad\frac{5R}{2\,r}=\frac{10(1-\theta)\ln|\ln\delta|}{9|\ln\beta|},$
(2.7)

and

$\displaystyle\delta^{\beta^{\frac{5R}{2\,r}+2}}$
$\displaystyle=e^{-|\ln\delta|\beta^{\frac{5R}{2\,r}+2}}=e^{-\beta^{2}|\ln\delta|^{\frac{10\theta-1}{9}}}\leq
e^{-\beta^{2}|\ln\delta|^{\theta}}$
$\displaystyle\leq\beta^{-2}(|\ln\delta|)^{-\theta}\,\leq\,\beta^{-2}(\ln|\ln\delta|)^{-\theta}.$
(2.8)

Therefore, it yields that

$\displaystyle\|\mathbf{u}_{\mathrm{\alpha}}\|_{L^{\infty}(Q^{\prime})^{n}}\leq\|\mathbf{u}_{\mathrm{\alpha}}\|_{L^{\infty}(Q_{1})^{n}}\leq
M(\ln|\ln\delta|)^{-\theta}\mathcal{T},\quad\mathrm{\alpha}=\mathrm{p},\mathrm{s},$

where
$M=\widetilde{C}\big{[}2^{\theta}R^{\theta}|\ln\beta|^{\theta}(1-\theta)^{-\theta}+\beta^{-2}\big{]}$.

The proof is complete. ∎

The next proposition states the propagation of smallness from the near-field
to the boundary of $Q$, which is the convex hull of $\Omega$ and
$\Omega^{\prime}$.

######  Proposition 2.3.

Fix $n=2,3$ and the a-prior parameters $\mathcal{M},\mathcal{P}$ and
$\mathcal{N}$ greater than $1$. Let $(\Omega;\rho)$ and
$(\Omega^{\prime};\rho^{\prime})$ be two medium scatterers associated with two
admissible density functions $\rho$ and $\rho^{\prime}$. Let $\mathbf{u}^{i}$
be an incident wave with the form (1.1) and
$\|\mathbf{u}^{i}\|_{H^{2}(B_{2R})^{n}}\leq\mathcal{P}$. Assume that
$\mathbf{U}$ and $\mathbf{U}^{\prime}$ are the far field patterns of the
scattered waves $\mathbf{u}^{s}$ and $\mathbf{u}^{\prime s}$ by the medium
scatterers $(\Omega;\rho)$ and $(\Omega^{\prime};\rho^{\prime})$,
respectively, where $\mathbf{u}^{s},\mathbf{u}^{\prime s}$ can be bounded by
$\mathcal{N}>1$ in $H^{2}(B_{2R})^{n}$. Let $\varepsilon$ be a sufficiently
small constant. If

$\|\mathbf{U}-\mathbf{U}^{\prime}\|_{L^{2}(\mathbb{S}^{n-1},\mathbb{C}^{n}\times\mathbb{C}^{n})}\leq\varepsilon,$

then $\mathbf{u}-\mathbf{u}^{\prime}$ and
$\nabla\mathbf{u}-\nabla\mathbf{u}^{\prime}$ are continuous in $B_{R}$ and
moreover, we have

$\displaystyle\sup_{\partial
Q}\big{(}|\mathbf{u}-\mathbf{u}^{\prime}|+|\nabla\mathbf{u}-\nabla\mathbf{u}^{\prime}|\big{)}\leq
C\left(\ln\ln(\mathcal{N}/\varepsilon)\right)^{-1/2},$ (2.9)

where $Q$ is the convex hull of $\Omega$ and $\Omega^{\prime}$, and
$C=C(\kappa_{\mathrm{p}},\kappa_{\mathrm{s}},n,R,\mathcal{P},\mathcal{M},\mathcal{N})>0$.

###### Proof.

Denote
$\varepsilon_{0}=\|\mathbf{U}-\mathbf{U}^{\prime}\|_{L^{2}(\mathbb{S}^{n-1},\mathbb{C}^{n}\times\mathbb{C}^{n})}$
and
$\mathbf{v}=\mathbf{u}-\mathbf{u}^{\prime}=\mathbf{u}^{s}-\mathbf{u}^{\prime
s}$. One can easily see that $\|\mathbf{v}\|_{H^{2}(B_{2R})}\leq
2\mathcal{P}$. Let $\Omega=B_{7/4R}\backslash B_{5/4R}$. From Proposition 2.1
and the Sobolev embedding $H^{2}(\Omega)\rightarrow L^{\infty}(\Omega)$, there
exists a positive constant
$C_{1}=C_{1}(\kappa_{\mathrm{p}},\kappa_{\mathrm{s}},n,R)$ such that the
following estimates hold for sufficiently small $\varepsilon$,

$\displaystyle\|\mathbf{v}\|_{L^{\infty}(\Omega)^{n}}\leq\|\mathbf{v}\|_{H^{2}(\Omega)^{n}}\leq
C_{1}\mathcal{N}e^{-c_{0}\sqrt{\ln\frac{\mathcal{N}}{\varepsilon_{0}}}}\leq
C_{1}\mathcal{N}e^{-c_{0}\sqrt{\ln\frac{\mathcal{N}}{\varepsilon}}},$
$\displaystyle\|\nabla\mathbf{v}\|_{L^{\infty}(\Omega)^{n}}\leq\|\mathbf{v}\|_{H^{2}(\Omega)^{n}}\leq
C_{1}\mathcal{N}e^{-c_{0}\sqrt{\ln\frac{\mathcal{N}}{\varepsilon_{0}}}}\leq
C_{1}\mathcal{N}e^{-c_{0}\sqrt{\ln\frac{\mathcal{N}}{\varepsilon}}},$

where $\kappa=\min\\{\kappa_{\mathrm{p}},\kappa_{\mathrm{s}}\\}$ and
$c_{0}=\ln(\frac{5}{4})\sqrt{\frac{e\kappa R}{2}}$. According to the interior
regularity of the elliptic PDE (cf. [21]) and the Sobolev embedding, it is not
difficult to prove that

$\displaystyle\|\mathbf{u}_{\mathrm{\alpha}}\|_{C^{1,\frac{1}{2}}(\overline{B_{3/2R}})^{n}}\leq
C_{\mathrm{\alpha}}(1+\|\rho\|_{C^{\frac{1}{2}}}(B_{3/2R})^{n})\|\mathbf{u}_{\mathrm{\alpha}}\|_{H^{2}(B_{2R})^{n}}\leq
C_{\mathrm{\alpha}}(1+\mathcal{M})(\mathcal{P}+\mathcal{N}),$
$\displaystyle\|\mathbf{u}^{\prime}_{\mathrm{\alpha}}\|_{C^{1,\frac{1}{2}}(\overline{B_{3/2R}})^{n}}\leq
C_{\mathrm{\alpha}}(1+\|\rho\|_{C^{\frac{1}{2}}}(B_{3/2R})^{n})\|\mathbf{u}^{\prime}_{\mathrm{\alpha}}\|_{H^{2}(B_{2R})^{n}}\leq
C_{\mathrm{\alpha}}(1+\mathcal{M})(\mathcal{P}+\mathcal{N}),$

where $(\mathbf{u}_{\mathrm{s}},\mathbf{u}_{\mathrm{p}})$ and
$(\mathbf{u}^{\prime}_{\mathrm{s}},\mathbf{u}^{\prime}_{\mathrm{p}})$ are
transverse and longitudinal wave pairs of $\mathbf{u}$ and
$\mathbf{u}^{\prime}$, respectively. Thus $\mathbf{v}$ and
$\partial_{j}\mathbf{v}\in
C^{\frac{1}{2}}(\overline{B_{3/2R}})^{n},\,j=1,2,\ldots,n$. This shows that
$\mathbf{u}-\mathbf{u}^{\prime}$ and
$\nabla\mathbf{u}-\nabla\mathbf{u}^{\prime}$ are continuous in $B_{R}$.

Fix $\beta\in(0,1)$ and denote $Q^{\prime}=\big{\\{}\mathbf{x}\in
B_{3/2R}\big{|}\,\,\mathrm{dist}(\mathbf{x},\partial Q)\leq
4\cdot\frac{9R|\ln\beta|}{2\ln|\ln\delta|}\big{\\}}$. We choose
$\delta(\varepsilon)=C_{2}\,\mathcal{N}e^{-c_{0}\sqrt{\ln(\mathcal{N}/\varepsilon)}}$
satisfying (2.4). According to Proposition 2.2, we know that

$\displaystyle\|\mathbf{v}_{\mathrm{\alpha}}\|_{L^{\infty}(Q^{\prime})^{n}}$
$\displaystyle\leq
C_{\mathrm{\alpha}}(1+\mathcal{M})(2\mathcal{P}+\mathcal{N})(\ln|\ln\delta|)^{-\frac{1}{2}},\quad\mathrm{\alpha}=\mathrm{p},\mathrm{s}.$

Hence, one can deduce that

$\displaystyle\|\mathbf{v}\|_{L^{\infty}(Q^{\prime})^{n}}$
$\displaystyle\leq\max\\{C_{\mathrm{p}},C_{\mathrm{s}}\\}(1+\mathcal{M})(2\mathcal{P}+\mathcal{N})(\ln\ln|\delta|)^{-\frac{1}{2}}$
$\displaystyle\leq\max\\{C_{\mathrm{p}},C_{\mathrm{s}}\\}(1+\mathcal{M})(2\mathcal{P}+\mathcal{N})\Big{[}\ln\big{(}\frac{c_{0}}{2}\ln(\mathcal{N}/\varepsilon)\,\big{)}\Big{]}^{-\frac{1}{2}}$
$\displaystyle\leq\max\\{C_{\mathrm{p}},C_{\mathrm{s}}\\}(1+\mathcal{M})(2\mathcal{P}+\mathcal{N})\big{(}\ln\ln(\mathcal{N}/\varepsilon)^{\frac{1}{4}}\big{)}^{-\frac{1}{2}}$
$\displaystyle\leq
2\max\\{C_{\mathrm{p}},C_{\mathrm{s}}\\}(1+\mathcal{M})(2\mathcal{P}+\mathcal{N})\big{(}\ln\ln(\mathcal{N}/\varepsilon)\big{)}^{-\frac{1}{2}}.$
(2.10)

Similar to the upper bound of $\|\mathbf{v}\|_{L^{\infty}(Q^{\prime})^{n}}$ in
(2.2), we can obtain the upper bound of $\partial_{j}\mathbf{v}$ by following
a similar argument. Hence, we can prove (2.9).

The proof is complete. ∎

##  3\. Micro-local analysis of corner scattering

The present section is devoted to analyzing the quantitative behaviours of the
scattered field locally around a corner. We first present two auxiliary
lemmas.

######  Lemma 3.1.

[5, Lemma 8.2] Fix $n=2$. Let $Q$ be the convex hull of $\Omega$ and
$\Omega^{\prime}$, where $\Omega$ and $\Omega^{\prime}$ are two open bounded
convex polygons. If $\mathbf{x}_{0}$ is a vertex of $\Omega$ such that
$\mathrm{dist}(\mathbf{x}_{0},\Omega^{\prime})=\mathrm{dist}_{\mathcal{H}}(\Omega,\Omega^{\prime})$,
then $\mathbf{x}_{0}$ is also a vertex of $Q$. Let $\sigma$ denote the angle
of $K$ at $\mathbf{x}_{0}$, then the angle $\sigma^{\prime}$ of $Q$ at
$\mathbf{x}_{0}$ satisfies the inequality
$\sigma<\sigma^{\prime}\leq(\sigma+\pi)/2<\pi$.

######  Lemma 3.2.

[5, Lemma 8.3] Fix $n=3$. Let $Q$ be the convex hull of $\Omega$ and
$\Omega^{\prime}$, where $\Omega$ and $\Omega^{\prime}$ are open convex
polyhedral cones. If $\mathbf{x}_{0}$ is a vertex of $\Omega$ such that
$\mathrm{dist}(\mathbf{x}_{0},\Omega^{\prime})=\mathrm{dist}_{\mathcal{H}}(\Omega,\Omega^{\prime})$,
then $\mathbf{x}_{0}$ is also a vertex of $Q$. And $Q$ can fit inside such an
open spherical cone $\mathcal{A}$ that $\mathbf{x}_{0}$ is the vertex of
$\mathcal{A}$ whose open angle $\sigma$ at the vertex $\mathbf{x}_{0}$ is
smaller than $\pi$, where $\sigma$ does not depend on $\Omega$,
$\Omega^{\prime}$ and their locations.

Next, we establish an integral identity which follows from the Betti’s second
formula.

######  Proposition 3.1.

Let $\omega>0$ and $\rho(\mathbf{x})\in C^{\theta}(S)$, where
$S\subset\mathbb{R}^{n}$ is a bounded Lipschitz domain and $\theta\in(0,1)$.
If $\mathbf{u},\mathbf{u}^{\prime}$ and $\mathbf{u}_{0}\in H^{2}(S)^{n}$
satisfy the Navier equations

$\displaystyle\left\\{\begin{array}[]{ll}\Delta^{*}\mathbf{u}\,+\rho\,\omega^{2}\mathbf{u}=\bf
0,\\\ \Delta^{*}\mathbf{u}^{\prime}+~{}\omega^{2}\mathbf{u}^{\prime}=\bf 0,\\\
\Delta^{*}\mathbf{u}_{0}+\omega^{2}\mathbf{u}_{0}=\bf 0\end{array}\right.$

in $S$, then

$\displaystyle\omega^{2}\int_{S}(\rho-1)\,\mathbf{u}\cdot\mathbf{u}_{0}\,\mathrm{d}\mathbf{x}=\int_{\partial
S}\mathbf{u}_{0}\cdot
T_{\hat{\nu}}(\mathbf{u}^{i}-\mathbf{u})-(\mathbf{u}^{i}-\mathbf{u})\cdot
T_{\hat{\nu}}\,(\mathbf{u}_{0})\,\mathrm{d\sigma}.$ (3.1)

Here, $\hat{\nu}$ denotes the outward unit normal to the boundary of $S$ and
the the conormal derivative $T_{\hat{\nu}}(\mathbf{u})$ with $u^{j}$ being the
$j$-th component of $\mathbf{u}$ is defined by

$\displaystyle T_{\hat{\nu}}(\mathbf{u})$
$\displaystyle=\left\\{\begin{array}[]{lr}2\mu\,\partial_{\hat{\nu}}\mathbf{u}+\lambda\,\hat{\nu}(\nabla\cdot\mathbf{u})+\mu\,(\partial_{2}u^{1}-\partial_{1}u^{2})\hat{\nu}^{\perp},&\quad
n=2,\\\\[10.0pt]
2\mu\,\partial_{\hat{\nu}}\mathbf{u}+\lambda\,\hat{\nu}\,(\nabla\cdot\mathbf{u})+\mu\,\hat{\nu}\times(\nabla\times\mathbf{u}),&\quad
n=3.\end{array}\right.$ (3.4)

The following complex geometric optics (CGO) solution $\mathbf{u}_{0}$ is
introduced in [31]:

$\mathbf{u}_{0}(\mathbf{x})=e^{\xi\cdot(\mathbf{x}-\mathbf{x}_{0})}\eta,$
(3.5)

where

$\displaystyle\xi=\tau\mathbf{p}+\imath\sqrt{\kappa^{2}_{\mathrm{s}}+\tau^{2}}\,\mathbf{p}^{\perp},\,\eta=\mathbf{p}^{\perp}-\imath\sqrt{1+\kappa^{2}_{\mathrm{s}}/\tau^{2}}\,\mathbf{p},$
$\displaystyle\tau>\kappa_{\mathrm{s}},\quad\mathbf{p}\cdot\mathbf{p}^{\perp}=0,\quad\mathbf{p}^{\perp},\,\mathbf{p}\in\mathbb{S}^{n-1}.$
(3.6)

It is directly verified that

$\Delta^{*}\mathbf{u}_{0}+\,\omega^{2}\mathbf{u}_{0}={\bf 0}\ \mbox{ in }\
\mathbb{R}^{n}.$ (3.7)

Note that

$\displaystyle|\xi|=\sqrt{2\tau^{2}+\kappa^{2}_{\mathrm{s}}}\quad\mbox{and}\quad|\eta|=\sqrt{2+\kappa^{2}_{\mathrm{s}}/\tau^{2}}\leq\sqrt{3}.$
(3.8)

In Proposition 3.2 we shall present asymptotic analysis of the volume integral
of the CGO solution $\mathbf{u}_{0}$ over a polyhedral cone with respect to
$\tau$ as $\tau\rightarrow+\infty$. Before proving that, we now introduce a
special cone pair $(\mathcal{A},\mathcal{C})$ as follows.

######  Definition 3.1.

Let $n\in\\{2,3\\}$, $0<\alpha_{m}<\alpha_{M}<\frac{\pi}{2}$, $\omega>0$.
Assume that $\mathcal{A},\mathcal{C}\subset\mathbb{R}^{n}$ are, respectively,
an open polyhedral cone and an open spherical cone. We say
$(\mathcal{A},\mathcal{C})\in\mathcal{F}(\alpha_{m},\alpha_{M},n)$ if the
following conditions are fulfilled:

  * (a)

$\mathcal{A}\subset\mathbb{R}^{n}$ is a convex polyhedral cone;

  * (b)

$\mathcal{A}$, $\mathcal{C}$ both take $\mathbf{x}_{0}\in\mathbb{R}^{n}$ as
the apex and $\mathcal{A}\subset\mathcal{C}$;

  * (c)

the opening angle of $\mathcal{C}$ at the apex $\mathbf{x}_{0}$ is no more
than $\pi$;

  * (d)

the opening angle of $\mathcal{A}\subset\mathbb{R}^{2}$ at the apex
$\mathbf{x}_{0}$ is in $(2\alpha_{m},2\alpha_{M})$.

######  Proposition 3.2.

Fix $n\in\\{2,3\\}$, $0<2\alpha_{m}<2\alpha_{M}<\pi$, $\omega>0$. Let
$\mathbf{u}$ be the solution to (1.2) with $\Omega=\mathrm{supp}(\rho-1)$. In
addition, $\Omega$ takes $\mathbf{x}_{0}$ as its apex and
$\mathbf{u}(\mathbf{x}_{0})$ is a nontrivial constant complex vector. Let
$\mathcal{A}$ and $\mathcal{C}$ be an open polyhedral cone and an open
spherical cone, respectively, they both take $\mathbf{x}_{0}$ as their vertex.
If $(\mathcal{A},\mathcal{C})\in\mathcal{F}(\alpha_{m},\alpha_{M},n)$, then
there exists a vector $\xi$ defined by (3) such that

$\displaystyle\mathbf{p}\cdot(\mathbf{x}-\mathbf{x}_{0})\leq-\delta_{0}\big{|}\mathbf{x}-\mathbf{x}_{0}\big{|}\mbox{
for any }\,\mathbf{x}\in\mathcal{A},$ (3.9)

where $\delta_{0}$ is a positive constant only depending on $\mathbf{p}$,
$\Omega$ and $\mathcal{A}$. Moreover, we have

$\displaystyle\Big{|}\int_{\mathcal{A}}e^{\xi\cdot(\mathbf{x}-\mathbf{x}_{0})}\mathrm{d}x\Big{|}\geq
C\,\tau^{-n},$ (3.10)

and

$\displaystyle\big{|}\mathbf{u}(\mathbf{x}_{0})\cdot\eta\big{|}\geq C_{0}>0,$
(3.11)

where $\tau=\big{|}\Re\xi\big{|}$, $\eta$ is defined in (3),
$\mathbf{p}^{\perp}\in\mathbb{S}^{n-1}$ satisfies
$\mathbf{p}\cdot\mathbf{p}^{\perp}=0$, and $C_{0}$ is a positive constant only
depending on $\Re\mathbf{u}(\mathbf{x}_{0})$, $\Im\mathbf{u}(\mathbf{x}_{0})$,
$\mathbf{p}$ and $\mathbf{p}^{\perp}$. Here $\Re\mathbf{u}(\mathbf{x}_{0})$
and $\Im\mathbf{u}(\mathbf{x}_{0})$ stand for the real and imaginary parts of
$\mathbf{u}(\mathbf{x}_{0})$, respectively.

###### Proof.

The above conclusions can be obtained directly from [5, Lemma 6.3] and [8,
Lemma 2.2] except that we need to prove the inequality (3.11). Note that

$\displaystyle\mathbf{u}(\mathbf{x}_{0})\cdot\eta=$
$\displaystyle\,\,\imath\big{(}\Im\mathbf{u}(\mathbf{x}_{0})\cdot\mathbf{p}^{\perp}-\sqrt{1+\kappa^{2}_{\mathrm{s}}/\tau^{2}}\,\Re\mathbf{u}(\mathbf{x}_{0})\cdot\mathbf{p}\big{)}$
$\displaystyle+\Re\mathbf{u}(\mathbf{x}_{0})\cdot\mathbf{p}^{\perp}+\sqrt{1+\kappa^{2}_{\mathrm{s}}/\tau^{2}}\,\Im\mathbf{u}(\mathbf{x}_{0})\cdot\mathbf{p}.$
(3.12)

Since $\mathbf{u}(\mathbf{x}_{0})$ is a nontrivial constant complex vector,
without loss of generality, we assume that
$\Im\mathbf{u}(\mathbf{x}_{0})\neq\bf 0$. For simplicity, we only consider
three-dimensional scenarios and the two-dimensional conclusion can be
similarly proved. In fact, once $\mathbf{p}$ is fixed, $\mathbf{p}^{\perp}$
belongs to $\Pi$, where $\Pi$ represents the plane defined by
$\Pi=\\{\mathbf{x}\in\mathbb{R}^{3}~{}|~{}\mathbf{x}\cdot\mathbf{p}=0\\}$. We
can easily prove (3.11) by virtue of choosing a special unit vector
$\mathbf{p}^{\perp}$ as follows:

  * (I)

Case 1: $\Im\mathbf{u}(\mathbf{x}_{0})\cdot\mathbf{p}<0$. In order to prove
(3.11), we consider the lower bound for the real part of
$\mathbf{u}(\mathbf{x}_{0})\cdot\eta.$

Let us distinguish two seperate situations. If
$\Re\mathbf{u}(\mathbf{x}_{0})\in\Pi$ (including
$\Re\mathbf{u}(\mathbf{x}_{0})\mathrel{/\mkern-5.0mu/}\Pi$), we can take
$\mathbf{p}^{\perp}\in\Pi$ such that
$\angle(\mathbf{p}^{\perp},\Re\mathbf{u}(\mathbf{x}_{0}))\in(\pi/2,\pi)$,
where $\angle(\mathbf{p}^{\perp},\Re\mathbf{u}(\mathbf{x}_{0}))$ is the angle
between $\Re\mathbf{u}(\mathbf{x}_{0})$ and $\mathbf{p}^{\perp}$. If
$\Re\mathbf{u}(\mathbf{x}_{0})\notin\Pi$, it is clear that there must exist a
unit vector $\mathbf{p}^{\perp}\in\Pi$ such that
$\angle(\mathbf{p}^{\perp},\Re\mathbf{u}(\mathbf{x}_{0}))\in(\pi/2,\pi).$
Hence, we always can choose the unit vector $\mathbf{p}^{\perp}\in\Pi$
fulfilling the following inequality,

$\Re\mathbf{u}(\mathbf{x}_{0})\cdot\mathbf{p}^{\perp}<0.$

It can be directly seen that the signs of
$\Re\mathbf{u}(\mathbf{x}_{0})\cdot\mathbf{p}^{\perp}$ and
$\Im\mathbf{u}(\mathbf{x}_{0})\cdot\mathbf{p}$ are the same. Therefore, we
have

$\displaystyle\big{|}\mathbf{u}(\mathbf{x}_{0})\cdot\eta\big{|}$
$\displaystyle\geq\big{|}\Re\mathbf{u}(\mathbf{x}_{0})\cdot\mathbf{p}^{\perp}+\sqrt{1+\kappa^{2}_{\mathrm{s}}/\tau^{2}}\,\Im\mathbf{u}(\mathbf{x}_{0})\cdot\mathbf{p}\big{|}$
$\displaystyle\geq\big{|}\Re\mathbf{u}(\mathbf{x}_{0})\cdot\mathbf{p}^{\perp}+\Im\mathbf{u}(\mathbf{x}_{0})\cdot\mathbf{p}\big{|}:=C_{0}>0.$
(3.13)

  * (II)

Case 2: $\Im\mathbf{u}(\mathbf{x}_{0})\cdot\mathbf{p}=0$. In order to prove
(3.11), we consider the lower bound for the imaginary part of
$\mathbf{u}(\mathbf{x}_{0})\cdot\eta.$

Let us investigate the following two situations. If
$\Re\mathbf{u}(\mathbf{x}_{0})\in\Pi$ (including
$\Re\mathbf{u}(\mathbf{x}_{0})\mathrel{/\mkern-5.0mu/}\Pi$), then
$\Re\mathbf{u}(\mathbf{x}_{0})\cdot\mathbf{p}=0$. Thus we can take
$\mathbf{p}^{\perp}\in\Pi$ such that
$\angle(\mathbf{p}^{\perp},\Re\mathbf{u}(\mathbf{x}_{0}))\neq\pi/2$. If
$\Re\mathbf{u}(\mathbf{x}_{0})\notin\Pi$, let us suppose that
$\Re\mathbf{u}(\mathbf{x}_{0})\cdot\mathbf{p}>0$, we can choose a unit vector
$\mathbf{p}^{\perp}$ belonging to $\Pi$ such that
$\angle(\mathbf{p}^{\perp},\Im\mathbf{u}(\mathbf{x}_{0}))\in(\pi/2,\pi)$,
which implies ${\sf
sign}(\Im\mathbf{u}(\mathbf{x}_{0})\cdot\mathbf{p}^{\perp})=-{\sf
sign}(\Re\mathbf{u}(\mathbf{x}_{0})\cdot\mathbf{p})$. Here ${\sf sign}(a)$ is
the sign of a real number $a$. Therefore, we obtain

$\displaystyle\big{|}\mathbf{u}(\mathbf{x}_{0})\cdot\eta\big{|}$
$\displaystyle\geq\big{|}\Im\mathbf{u}(\mathbf{x}_{0})\cdot\mathbf{p}^{\perp}-\sqrt{1+\kappa^{2}_{\mathrm{s}}/\tau^{2}}\,\Re\mathbf{u}(\mathbf{x}_{0})\cdot\mathbf{p}\big{|}$
$\displaystyle\geq\big{|}\Im\mathbf{u}(\mathbf{x}_{0})\cdot\mathbf{p}^{\perp}-\Re\mathbf{u}(\mathbf{x}_{0})\cdot\mathbf{p}\big{|}:=C_{0}>0.$

  * (III)

Case 3: If $\Im\mathbf{u}(\mathbf{x}_{0})\cdot\mathbf{p}>0$. Similar to Case
1, in order to prove (3.11), we can consider the lower bound for the real part
of $\mathbf{u}(\mathbf{x}_{0})\cdot\eta.$

Similar to the above two cases, let us consider the following two situations.
If $\Re\mathbf{u}(\mathbf{x}_{0})\in\Pi$ (including
$\Re\mathbf{u}(\mathbf{x}_{0})\mathrel{/\mkern-5.0mu/}\Pi$), we can take
$\mathbf{p}^{\perp}\in\Pi$ such that
$\angle(\mathbf{p}^{\perp},\Re\mathbf{u}(\mathbf{x}_{0}))\in[0,\pi/2)$. If
$\Re\mathbf{u}(\mathbf{x}_{0})\notin\Pi$, there must exist a unit vector
$\mathbf{p}^{\perp}\in\Pi$ such
$\angle(\mathbf{p}^{\perp},\Re\mathbf{u}(\mathbf{x}_{0}))\in[0,\pi/2)$. Hence,
for the above two situations, we can always choose a unit vector
$\mathbf{p}^{\perp}$ fulfilling the following inequality as
$\mathbf{p}^{\perp}$,

$\Re\mathbf{u}(\mathbf{x}_{0})\cdot\mathbf{p}^{\perp}\geq 0.$

By the same deduction for Case 1, we can also obtain the inequality ((I)).

The proof is complete. ∎

A critical asymptotic estimate with respect to $\Re\xi$ for the volume
integral associated with the CGO solution $\mathbf{u}_{0}$ and the total wave
field $\mathbf{u}$ near a vertex of $\Omega$ is presented in the following
proposition, which is the key ingredient to derive the stability result in
Theorem 1.1.

######  Proposition 3.3.

Let $\mathbf{u}$ and $\mathbf{u}^{\prime}$ be the solutions to (1.2) with
$\Omega,\Omega^{\prime},\rho,\rho^{\prime}$ satisfying the assumptions in
Theorem 1.1. Let $\mathbf{x}_{0}$ be a vertex of $\Omega$ such that
$\mathrm{dist}(\mathbf{x}_{0},\Omega^{\prime})=\mathrm{dist}_{\mathcal{H}}(\Omega,\Omega^{\prime})$.
Fix $\omega,\tau>0$ and $h\in(0,1)$ such that
$B_{h}(\mathbf{x}_{0})\cap\Omega^{\prime}=\emptyset$. Denote
$S_{h}=B_{h}(\mathbf{x}_{0})\cap\Omega$ and $D_{h}=B_{h}(\mathbf{x}_{0})\cap
Q$, where $Q$ is the convex hull of $\Omega$ and $\Omega^{\prime}$. Let
$\mathbf{u}(\mathbf{x}_{0})$ be a nontrivial constant complex vector. Assume
that $\mathcal{A}$ and $\mathcal{B}$ represent the polyhedral cone generated
by $K$ and $Q$ at the vertex $\mathbf{x}_{0}$, respectively. Let $\rho\in
C^{\theta_{1}}(S_{h})$ and $\mathbf{u}\in C^{\theta_{2}}(S_{h})^{n}$,
$\theta_{j}\in(0,1)$, $j=1,2$, which yield that

$\displaystyle\rho(\mathbf{x})=1+\rho(\mathbf{x}_{0})+\rho_{1}(\mathbf{x}),\,\,\,|\rho_{1}(\mathbf{x})|\leq\mathcal{M}|\mathbf{x}-\mathbf{x}_{0}|^{\theta_{1}},\,\,\theta_{1}\in(0,1),$
(3.14)
$\displaystyle\mathbf{u}(\mathbf{x})=\mathbf{u}(\mathbf{x}_{0})+\mathbf{u}_{1}(\mathbf{x}),\quad\quad|\mathbf{u}_{1}(\mathbf{x})|\leq\mathcal{R}|\mathbf{x}-\mathbf{x}_{0}|^{\theta_{2}},\,\,\theta_{2}\in(0,1),$
(3.15)

where $\mathcal{M}=\|\rho_{1}\|_{C^{\theta_{1}}(\Omega)}$ and
$\mathcal{R}=\|\mathbf{u}_{1}\|_{C^{\theta_{2}}(\Omega)}$.

Then we have the integral identity

$\displaystyle(\rho(\mathbf{x}_{0})-1)\int_{\mathcal{A}}e^{\xi\cdot(\mathbf{x}-\mathbf{x}_{0})}\eta\cdot\mathbf{u}(\mathbf{x}_{0})\mathrm{d}x$
$\displaystyle=\frac{1}{\omega^{2}}\int_{\partial D_{h}}\mathbf{u}_{0}\cdot
T_{\hat{\nu}}(\mathbf{u}^{i}-\mathbf{u})-(\mathbf{u}^{i}-\mathbf{u})T_{\hat{\nu}}(\mathbf{u}_{0})\mathrm{d\sigma}$
$\displaystyle\quad+(\rho(\mathbf{x}_{0})-1)\int_{\mathcal{A}\backslash
S_{h}}e^{\xi\cdot(\mathbf{x}-\mathbf{x}_{0})}\eta\cdot\mathbf{u}(\mathbf{x}_{0})\mathrm{d}x$
$\displaystyle\quad-\int_{S_{h}}e^{\xi\cdot(\mathbf{x}-\mathbf{x}_{0})}(\rho_{1}+1)\eta\cdot\mathbf{u}(\mathbf{x}_{0})\mathrm{d}x$
$\displaystyle\quad-\int_{S_{h}}e^{\xi\cdot(\mathbf{x}-\mathbf{x}_{0})}(\rho-1)\eta\cdot\mathbf{u}_{1}\mathrm{d}x,$
(3.16)

and the estimate

$\displaystyle
C\big{|}(\rho(\mathbf{x}_{0})-1)\int_{\mathcal{A}}e^{\xi\cdot(\mathbf{x}-\mathbf{x}_{0})}\eta\cdot\mathbf{u}(\mathbf{x}_{0})\mathrm{dx}\big{|}$
$\displaystyle\leq|\Re\xi|^{-n}e^{-\delta_{0}|\Re\xi|h/2}+|\Re\xi|^{-n-\min\\{\theta_{1},\theta_{2}\\}}$
$\displaystyle+h^{n-1}\sup_{\partial\mathcal{B}\cap
B_{h}(\mathbf{x}_{0})}\big{\\{}|\nabla\mathbf{u}-\nabla\mathbf{u}^{\prime}|+|\mathbf{u}-\mathbf{u}^{\prime}|\big{\\}}$
$\displaystyle+h^{n-1}e^{-\delta_{0}|\Re\xi|h}\big{\\{}\|\mathbf{u}_{\mathrm{s}}\|_{H^{2}(B_{2R})^{n}}+\|\mathbf{u}_{\mathrm{p}}\|_{H^{2}(B_{2R})^{n}}$
$\displaystyle+\|\mathbf{u}^{\prime}_{\mathrm{p}}\|_{H^{2}(B_{2R})^{n}}+\|\mathbf{u}^{\prime}_{\mathrm{s}}\|_{H^{2}(B_{2R})^{n}}\big{\\}},$
(3.17)

where $\delta_{0}$ coincides with the one in (3.9) and $C$ is a positive
constant depending on the a-priori parameters.

###### Proof.

According to (3.5) and (3.7), the integral identity (3.3) follows directly
from (3.1) by utilizing (3.14) and (3.15).

To estimate each term in the right-hand side of (3.3), let us recall the
incomplete Gamma functions $\gamma(\cdot,\cdot)$ and $\Gamma(\cdot,\cdot)$
(cf. [5]) which are defined by

$\displaystyle\gamma:\mathbb{R}_{+}\times\mathbb{R}_{+}\rightarrow\mathbb{R}:\,\,(s,x)\mapsto\gamma(s,x),\quad\Gamma:\mathbb{R}_{+}\times\mathbb{R}_{+}\rightarrow\mathbb{R}:\,(s,x)\mapsto\Gamma(s,x),$

where

$\displaystyle\gamma(s,x)=\int_{0}^{x}e^{-t}t^{s-1}\mathrm{dt},\quad\Gamma(s,x)=\int_{x}^{\infty}t^{s-1}e^{-t}\mathrm{dt}.$
(3.18)

From the fact $e^{-t}\leq e^{-t/2}e^{-x/2}$ and a variable substitution
$t^{\prime}=t/2$, it is easy to get the estimation:

$\displaystyle\gamma(s,x)\leq\Gamma(s)\leq\lceil
s-1\rceil,\quad\Gamma(s,x)\leq 2^{s}\Gamma(s)e^{-x/2},$ (3.19)

where $\lceil s\rceil$ the largest integer satisfying $\lceil s\rceil\leq s$,
and $\Gamma(s)$ is the complete Gamma function of $s$ bounded by $\lceil
s\rceil!$.

To estimate the first four terms in the right-hand side of (3.3), in the
following we derive some important asymptotic inequalities with respect to the
parameter $|\Re\xi|$ in the CGO solution $\mathbf{u}_{0}$. By using the polar
coordinate transformation, as well as (3.9), (3.18) and (3.19), we can obtain

$\displaystyle\Big{|}\int_{\mathcal{A}\backslash
S_{h}}e^{\xi\cdot(\mathbf{x}-\mathbf{x}_{0})}\mathrm{dx}\Big{|}$
$\displaystyle\leq\Big{|}\int_{\mathcal{A}\backslash
S_{h}}e^{-\delta_{0}|\Re\xi||\mathbf{x}-\mathbf{x}_{0}|}\mathrm{dx}\Big{|}\leq\sigma(\mathbb{S}^{n-1})\int^{\infty}_{h}e^{-\delta_{0}|\Re\xi|r}r^{n-1}\mathrm{dr}$
$\displaystyle\leq\sigma(\mathbb{S}^{n-1})\int_{h\delta_{0}|\Re\xi|r}^{\infty}e^{-t}t^{n-1}(\delta_{0}|\Re\xi|)^{-n}\mathrm{dt}$
$\displaystyle=\sigma(\mathbb{S}^{n-1})(\delta_{0}|\Re\xi|)^{-n}\Gamma(n,\delta_{0}|\Re\xi|h)$
$\displaystyle\leq
C_{\delta_{0},n}|\Re\xi|^{-n}e^{-\delta_{0}|\Re\xi|\frac{h}{2}}$ (3.20)

as $|\Re\xi|\rightarrow+\infty$. Similarly, we have

$\displaystyle\Big{|}\int_{S_{h}}e^{\xi\cdot(\mathbf{x}-\mathbf{x}_{0})}\mathrm{dx}\Big{|}$
$\displaystyle\leq\Big{|}\int_{S_{h}}e^{-\delta_{0}|\Re\xi||\mathbf{x}-\mathbf{x}_{0}|}\mathrm{dx}\leq\sigma(\mathbb{S}^{n-1})\int^{h}_{0}e^{-\delta_{0}|\Re\xi|r}r^{n-1}\mathrm{dr}$
$\displaystyle\leq\sigma(\mathbb{S}^{n-1})\int^{h\delta_{0}|\Re\xi|r}_{0}e^{-t}t^{n-1}(\delta_{0}|\Re\xi|)^{-n}\mathrm{dt}$
$\displaystyle\leq\sigma(\mathbb{S}^{n-1})(\delta_{0}|\Re\xi|)^{-n}\gamma(n,\delta_{0}|\Re\xi|h)$
$\displaystyle\leq C^{\prime}_{\delta_{0},n}|\Re\xi|^{-n}$ (3.21)

as $|\Re\xi|\rightarrow+\infty$, where $\sigma(\mathbb{S}^{n-1})$ is the
measure of $\mathbb{S}^{n-1}$. For scalar functions $f$ and $g$, which admit
$|f|\leq A|\mathbf{x}-\mathbf{x}_{0}|^{B}$ with $A$ and $B$ being positive
constants, and $g\in L^{q}(S_{h})$ with $1/p+1/q=1$. Combining the Hölder
inequality, (3.9), (3.18) with (3.19), we obtain

$\displaystyle\Big{|}\int_{S_{h}}e^{\xi\cdot(\mathbf{x}-\mathbf{x}_{0})}f\,g\mathrm{dx}\Big{|}$
$\displaystyle\leq
A\Big{(}\int_{S_{h}}e^{\Re\xi\cdot(\mathbf{x}-\mathbf{x}_{0})p}|\mathbf{x}-\mathbf{x}_{0}|^{Bp}\mathrm{dx}\Big{)}^{\frac{1}{p}}\,\|g\|_{L^{q}(S_{h})}$
$\displaystyle\leq
A\,\sigma(\mathbb{S}^{n-1})^{\frac{1}{p}}(\delta_{0}p|\Re\xi|)^{-B-n/p}\big{(}\gamma(Bp+n,\,\delta_{0}p|\Re\xi|h)\big{)}^{\frac{1}{p}}\,\|g\|_{L^{q}(S_{h})}$
$\displaystyle\leq
C_{A,B,\delta_{0},p}|\Re\xi|^{-B-n/p}\,\|g\|_{L^{q}(S_{h})}$ (3.22)

as $|\Re\xi|\rightarrow+\infty$. By similar arguments, we can derive the
following estimations for vector-type functions $\mathbf{f}$ and $\mathbf{g}$
satisfying $|\mathbf{f}|\leq A|\mathbf{x}-\mathbf{x}_{0}|^{B}$ with $A$ and
$B$ being positive constants and $\mathbf{g}\in L^{q}(S_{h})^{n}$ with
$1/p+1/q=1$,

$\displaystyle\Big{|}\int_{S_{h}}e^{\xi\cdot(\mathbf{x}-\mathbf{x}_{0})}\,\mathbf{f}\cdot\mathbf{g}\,\mathrm{dx}\Big{|}$
$\displaystyle\leq
A\Big{(}\int_{S_{h}}e^{\Re\xi\cdot(\mathbf{x}-\mathbf{x}_{0})p}|\mathbf{x}-\mathbf{x}_{0}|^{Bp}\mathrm{dx}\Big{)}^{\frac{1}{p}}\,\,\|\mathbf{g}\|_{L^{q}(S_{h})^{n}}$
$\displaystyle\leq
A\,\sigma(\mathbb{S}^{n-1})^{\frac{1}{p}}(\delta_{0}p|\Re\xi|)^{-B-n/p}\gamma(Bp+n,\,\delta_{0}p|\Re\xi|h)^{\frac{1}{p}}\,\,\|\mathbf{g}\|_{L^{q}(S_{h})^{n}}$
$\displaystyle\leq
C^{\prime}_{A,B,\delta_{0},p}|\Re\xi|^{-B-n/p}\,\|\mathbf{g}\|_{L^{q}(S_{h})^{n}}$
(3.23)

as $|\Re\xi|\rightarrow+\infty$.

By virtue of (3.14) and (3.15), from (3)-(3), we can deduce that

$\displaystyle R_{1}:=\Big{|}\int_{\mathcal{A}\backslash
S_{h}}e^{\xi\cdot(\mathbf{x}-\mathbf{x}_{0})}\eta\cdot\mathbf{u}(\mathbf{x}_{0})\mathrm{dx}\Big{|}\leq
C_{\delta_{0},n}|\mathbf{u}(\mathbf{x}_{0})||\Re\xi|^{-n}e^{-\delta_{0}|\Re\xi|\frac{h}{2}},$
(3.24a) $\displaystyle
R_{2}:=\Big{|}\int_{S_{h}}e^{\xi\cdot(\mathbf{x}-\mathbf{x}_{0})}\eta\cdot\mathbf{u}(\mathbf{x}_{0})\mathrm{dx}\Big{|}\leq
C^{\prime}_{\delta_{0},n}|\mathbf{u}(\mathbf{x}_{0})||\Re\xi|^{-n},$ (3.24b)
$\displaystyle
R_{3}:=\Big{|}\int_{S_{h}}e^{\xi\cdot(\mathbf{x}-\mathbf{x}_{0})}(\rho_{1}+1)\,\eta\cdot\mathbf{u}(\mathbf{x}_{0})\mathrm{dx}\Big{|}\leq
C_{\mathcal{R},n,\delta_{0}}|\mathbf{u}(\mathbf{x}_{0})|\big{|}\Re\xi\big{|}^{-n-\theta_{1}},$
(3.24c) $\displaystyle
R_{4}:=\Big{|}\int_{S_{h}}e^{\xi\cdot(\mathbf{x}-\mathbf{x}_{0})}(\rho-1)\eta\cdot\mathbf{u}_{1}\mathrm{dx}\Big{|}\leq
C_{\mathcal{M},\mathcal{R},n,\delta_{0}}|\Re\xi|^{-n-\theta_{2}}.$ (3.24d)

For the boundary integral term in the right-hand side of (3.3), we divide the
integration area $\partial D_{h}$ into two parts $\partial\mathcal{B}\cap
B_{h}(\mathbf{x}_{0})$ and ${\mathcal{S}}:=\partial
S_{h}(\mathbf{x}_{0})\backslash(\partial\mathcal{B}\cap
B_{h}(\mathbf{x}_{0}))$. For the boundary integral over
$\partial\mathcal{B}\cap B_{h}(\mathbf{x}_{0})$, the Cauchy-Schwarz inequality
implies that

$\displaystyle I_{1}:$ $\displaystyle=\Big{|}\int_{\partial\mathcal{B}\cap
B_{h}(\mathbf{x}_{0})}\mathbf{u}_{0}\cdot
T_{\hat{\nu}}(\mathbf{u}^{\prime}-\mathbf{u})-(\mathbf{u}^{\prime}-\mathbf{u})T_{\hat{\nu}}(\mathbf{u}_{0})\mathrm{d\sigma}\Big{|}$
$\displaystyle\leq\sqrt{\sigma(\partial\mathcal{B}\cap
B_{h}(\mathbf{x}_{0}))}\,\,\Big{(}\int_{\partial\mathcal{B}\cap
B_{h}(\mathbf{x}_{0})}\big{|}\mathbf{u}_{0}\cdot
T_{\hat{\nu}}(\mathbf{u}^{\prime}-\mathbf{u})-(\mathbf{u}^{\prime}-\mathbf{u})T_{\hat{\nu}}(\mathbf{u}_{0})\big{|}^{2}\mathrm{d\sigma}\Big{)}^{1/2}$
$\displaystyle\leq\sqrt{\sigma(\partial\mathcal{B}\cap
B_{h}(\mathbf{x}_{0}))}\,\Big{[}\int_{\partial\mathcal{B}\cap
B_{h}(\mathbf{x}_{0})}\big{(}|\mathbf{u}_{0}|+|T_{\hat{\nu}}(\mathbf{u}_{0})|\big{)}^{2}\mathrm{d\sigma}\,\Big{]}^{1/2}\,T_{1},$
(3.25)

where $T_{1}=\sup\limits_{\partial\mathcal{B}\cap
B_{h}(\mathbf{x}_{0})}\big{\\{}|T_{\hat{\nu}}(\mathbf{u}^{\prime}-\mathbf{u})|,|\mathbf{u}-\mathbf{u}^{\prime}|\big{\\}}.$
By virtue of (3.4) and (3.5), one can directly derive that

$\displaystyle T_{\hat{\nu}}(\mathbf{u}_{0})=\mu
e^{\xi\cdot(\mathbf{x}-\mathbf{x}_{0})}\big{[}(\xi\cdot\hat{\nu})\,\eta+(\eta\cdot\hat{\nu})\,\xi\big{]}\quad\mbox{and}\quad\big{|}T_{\hat{\nu}}(\mathbf{u}-\mathbf{u}^{\prime})\big{|}\leq
C_{\lambda,\mu}|\nabla(\mathbf{u}-\mathbf{u}^{\prime})|,$

where $\nu$ is the outward unit normal vector to $\partial\mathcal{B}\cap
B_{h}(\mathbf{x}_{0})$ and $C_{\lambda,\mu}$ is a positive constant depending
on $\lambda$ and $\mu$. From (3), we obtain that

$\displaystyle\quad\quad\quad\quad I_{1}$
$\displaystyle\leq\sqrt{\sigma(\partial\mathcal{B}\cap
B_{h}(\mathbf{x}_{0}))}\,\Big{[}\int_{\partial\mathcal{B}\cap
B_{h}(\mathbf{x}_{0})}\Big{(}|e^{\xi\cdot(\mathbf{x}-\mathbf{x}_{0})}|+\big{|}\mu
e^{\xi\cdot(\mathbf{x}-\mathbf{x}_{0})}(\xi\cdot\hat{\nu})\eta$
$\displaystyle\hskip
142.26378pt+(\eta\cdot\hat{\nu})\xi\big{|}\big{)}^{2}\mathrm{d\sigma}\Big{]}^{1/2}\,T_{1}$
$\displaystyle\leq\sqrt{\sigma(\partial\mathcal{B}\cap
B_{h}(\mathbf{x}_{0}))}\,\Big{[}\int_{\partial\mathcal{B}\cap
B_{h}(\mathbf{x}_{0})}\big{(}|\eta|+\mu|\eta(\xi\cdot\hat{\nu})+\xi(\eta\cdot\hat{\nu})|\big{)}^{2}\mathrm{d\sigma}\Big{]}^{1/2}\,T_{1}$
$\displaystyle\leq
C^{\prime}_{\lambda,\mu}\,\,\,\sigma(\partial\mathcal{B}\cap
B_{h}(\mathbf{x}_{0}))\,\sup\limits_{\partial\mathcal{B}\cap
B_{h}(\mathbf{x}_{0})}\big{\\{}|\nabla(\mathbf{u}-\mathbf{u}^{\prime})|,|\mathbf{u}-\mathbf{u}^{\prime}|\big{\\}},$
(3.26)

where $C^{\prime}_{\lambda,\mu}$ is a positive constant depending on $\lambda$
and $\mu$. There exist finite half-spaces $\big{\\{}H_{j}\big{\\}}^{N}_{j=1}$
through $\mathbf{x}_{0}$ with a codimension of $1$ such that

$\partial\mathcal{B}\cap
B_{h}(\mathbf{x}_{0})\subset\bigcap\limits^{N}_{j=1}\big{(}H_{j}\cap
B_{h}(\mathbf{x}_{0})\big{)}.$

Hence, we have

$\sigma(\partial\mathcal{B}\cap B_{h}(\mathbf{x}_{0}))\leq
C_{N,n}\sigma(\partial H_{j}\cap B_{h}(\mathbf{x}_{0}))\leq
C_{N,n}\,\,h^{n-1},$ (3.27)

where $C_{N,n}$ is a positive constant depending only $n$ and $N$.
Substituting (3.27) into (3), one has

$I_{1}\leq Ch^{n-1}\sup\limits_{\partial\mathcal{B}\cap
B_{h}(\mathbf{x}_{0})}\big{\\{}|\nabla(\mathbf{u}-\mathbf{u}^{\prime})|,|\mathbf{u}-\mathbf{u}^{\prime}|\big{\\}},$
(3.28)

where $C$ is a a-priori positive constant depending on $\lambda$, $\mu$, $N$
and $n$.

For the boundary integral over ${\mathcal{S}}=\partial
S_{h}(\mathbf{x}_{0})\backslash(\partial\mathcal{B}\cap
B_{h}(\mathbf{x}_{0}))$, by the Cauchy-Schwarz inequality and the fact that
$\sigma(\mathcal{B}\cap S_{h}(\mathbf{x}_{0}))\leq Ch^{n-1}$, where $C$ is a
positive constant depending on $n$ and $\pi$, one has

$\displaystyle\quad\quad\quad\quad I_{2}:$
$\displaystyle=\Big{|}\int_{{\mathcal{S}}}\mathbf{u}_{0}\,T_{\hat{\nu}}(\mathbf{u}-\mathbf{u}^{\prime})-(\mathbf{u}-\mathbf{u}^{\prime})\,T_{\hat{\nu}}(\mathbf{u}_{0})\mathrm{d\sigma}\Big{|}$
$\displaystyle\leq\sqrt{\sigma({\mathcal{S}})}\,\,\big{(}\int_{{\mathcal{S}}}\big{|}\mathbf{u}_{0}\cdot
T_{\hat{\nu}}(\mathbf{u}^{\prime}-\mathbf{u})-(\mathbf{u}^{\prime}-\mathbf{u})T_{\hat{\nu}}(\mathbf{u}_{0})\big{|}^{2}\mathrm{d\sigma}\,\big{)}^{1/2}\,\,T_{2}$
$\displaystyle\leq\sqrt{\sigma({\mathcal{S}})}\,\,\big{(}\int_{{\mathcal{S}}}(\big{|}\mathbf{u}_{0}|+|T_{\hat{\nu}}(\mathbf{u}_{0})|\big{)}^{2}\mathrm{d\sigma}\,\big{)}^{1/2}\,\,T_{3}$
$\displaystyle\leq
C^{\prime}_{\lambda,\mu,n}\,\,\sigma({\mathcal{S}})\,\,e^{-\delta_{0}|\Re\xi|h}\,\,T_{3}$
$\displaystyle\leq
C^{\prime}_{\lambda,\mu,n}\,\,h^{n-1}\,e^{-\delta_{0}|\Re\xi|h}\,\,T_{4},$
(3.29)

where

$\displaystyle T_{2}$
$\displaystyle=\sup\limits_{{\mathcal{S}}}|\nabla(\mathbf{u}-\mathbf{u}^{\prime})|+\sup\limits_{{\mathcal{S}}}|\mathbf{u}-\mathbf{u}^{\prime}|,$
$\displaystyle T_{3}$
$\displaystyle=\|\mathbf{u}_{\mathrm{s}}-\mathbf{u}^{\prime}_{\mathrm{s}}\|_{C^{1}(\overline{B_{h}(\mathbf{x}_{0})})^{n}}+\big{\|}\mathbf{u}_{\mathrm{p}}-\mathbf{u}^{\prime}_{\mathrm{p}}\|_{C^{1}(\overline{B_{h}(\mathbf{x}_{0})})^{n}},$
$\displaystyle T_{4}$
$\displaystyle=\|\mathbf{u}_{\mathrm{s}}\|_{H^{2}(B_{2R})^{n}}+\|\mathbf{u}_{\mathrm{p}}\|_{H^{2}(B_{2R})^{n}}+\|\mathbf{u}^{\prime}_{\mathrm{s}}\|_{H^{2}(B_{2R})^{n}}+\|\mathbf{u}^{\prime}_{\mathrm{p}}\|_{H^{2}(B_{2R})^{n}}.$

Finally, it is easy to see that

$C\big{|}(\rho(\mathbf{x}_{0})-1)\int_{\mathcal{A}}e^{\xi\cdot(\mathbf{x}-\mathbf{x}_{0})}\eta\cdot\mathbf{u}(\mathbf{x}_{0})\mathrm{dx}\big{|}\leq
R_{1}+R_{2}+R_{3}+I_{1}+I_{2},$ (3.30)

where
$C=C\big{(}\kappa_{\mathrm{p}},\kappa_{\mathrm{s}},\mathcal{M},\mathcal{R},\theta_{1},\theta_{2},n,\delta_{0}\big{)}>0$.
Substituting (3.24a) to (3.24d), (3.28) and (3) into (3.30), we prove (3.3). ∎

##  4\. Proof of Theorem 1.1

In this section, we shall give the proof of Theorem 1.1.

###### Proof of Theorem 1.1.

Let $\mathbf{x}_{0}$ be a vertex of $\Omega$ such that
$\mathrm{dist}(\mathbf{x}_{0},\Omega^{\prime})=\mathrm{dist}_{\mathcal{H}}(\Omega,\Omega^{\prime})$
and $Q$ be the convex hull of $\Omega$ and $\Omega^{\prime}$. From Lemma 3.1
and Lemma 3.2, we know $\mathbf{x}_{0}$ must be a vertex of $Q$. Let
$\mathcal{B}$ be the open polyhedral cone generated by $Q$ at
$\mathbf{x}_{0}$. Then there must be an open spherical cone $\mathcal{D}$
whose opening angle at $\mathbf{x}_{0}$ is in
$(2\alpha_{M},\,\pi)\subset(2\alpha_{m},\,\pi)$. Let $\mathcal{A}$ denote the
open polyhedral cone generated by $\Omega$ at $\mathbf{x}_{0}$ and
$\mathcal{C}$ be an open spherical cone such that
$(\mathcal{A},\mathcal{C})\in\mathcal{F}(\alpha_{m},\alpha_{M},n)$ (cf.
Definition 3.1). Denote
$\hbar=\mathrm{dist}(\mathbf{x}_{0},\Omega^{\prime})<1$ and choose
$h\leq\min\\{l_{0},\hbar\\}$ such that

$\begin{cases}\Delta^{*}\mathbf{u}^{\prime}+\omega^{2}\mathbf{u}^{\prime}={\bf
0}\quad\mbox{in}\quad S_{h}\subset D_{h},\\\
S_{h}\cap\Omega^{\prime}=D_{h}\cap\Omega^{\prime}=\emptyset,\end{cases}$

where $S_{h}=\Omega\cap B_{h}(\mathbf{x}_{0})=\mathcal{A}\cap
B_{h}(\mathbf{x}_{0})$ and $D_{h}=Q\cap B_{h}(\mathbf{x}_{0})=\mathcal{B}\cap
B_{h}(\mathbf{x}_{0})$. From the proof of Proposition 2.3, we can derive that

$\|\mathbf{u}_{\mathrm{\alpha}}\|_{H^{2}(B_{2R})^{n}},\,\|\mathbf{u}^{\prime}_{\mathrm{\alpha}}\|_{H^{2}(B_{2R})^{n}}\leq
C_{\kappa_{\mathrm{p}},\kappa_{\mathrm{s}},R,\mathcal{S}},\quad\mathrm{\alpha}=\mathrm{p},\mathrm{s}.$

Thus

$\mathbf{u}\in H^{2}(B_{2R})^{n}\,\,\mbox{and }\,\,\mathbf{u}^{\prime}\in
H^{2}(B_{2R})^{n}.$

The a-priori admissibility assumptions of $\rho$ and the local
$H^{2}$-regularity of $\mathbf{u}$ near $S_{h}$ imply the following splittings

$\displaystyle\rho(\mathbf{x})=1+\rho(\mathbf{x}_{0})+\rho_{1}(\mathbf{x}),\quad|\rho_{1}(\mathbf{x})|\leq\|\rho_{1}\|_{C^{\theta_{1}}(K\cap
B_{h}(\mathbf{x}_{0}))}|\mathbf{x}-\mathbf{x}_{0}|^{\theta_{1}},\,\,\theta_{1}\in(0,1),$
$\displaystyle\mathbf{u}(\mathbf{x})=\mathbf{u}(\mathbf{x}_{0})+\mathbf{u}_{1}(\mathbf{x}),\,\,\quad\quad|\mathbf{u}_{1}(\mathbf{x})|\leq\|\mathbf{u}_{1}\|_{C^{\theta_{2}}(K\cap
B_{h}(\mathbf{x}_{0}))^{n}}|\mathbf{x}-\mathbf{x}_{0}|^{\theta_{2}},\,\,\theta_{2}\in(0,1),$

where we use the Sobolev embedding theorem from $H^{2}$ to $C^{\theta_{2}}$.
So the conditions of Proposition 3.3 are all satisfied. We absorb all the
constants into the left-hand side, which depend only on the a-priori
parameters, then (3.3) reduces to

$\displaystyle
C\big{|}(\rho(\mathbf{x}_{0})-1)\int_{\mathcal{A}}e^{\xi\cdot(\mathbf{x}-\mathbf{x}_{0})}\eta\cdot\mathbf{u}(\mathbf{x}_{0})\mathrm{dx}\big{|}$
$\displaystyle\leq|\Re\xi|^{-n}e^{-\delta_{0}|\Re\xi|h/2}+|\Re\xi|^{-n-\min\\{\theta_{1},\theta_{2}\\}}$
$\displaystyle+h^{n-1}\sup_{\partial\mathcal{B}\cap
B_{h}(\mathbf{x}_{0})}\Big{(}|\mathbf{u}-\mathbf{u}^{\prime}|+|\nabla\mathbf{u}-\nabla\mathbf{u}^{\prime}|\Big{)}^{-1/2}$
$\displaystyle+h^{n-1}e^{-\delta_{0}|\Re\xi|h}.$ (4.1)

From Proposition 2.3 and the fact that $\partial
Q\supset\partial\mathcal{B}\cap B_{h}(\mathbf{x}_{0})$, we have

$\displaystyle\sup_{\partial\mathcal{B}\cap
B_{h}(\mathbf{x}_{0})}\Big{(}|\mathbf{u}-\mathbf{u}^{\prime}|+|\nabla\mathbf{u}-\nabla\mathbf{u}^{\prime}|\Big{)}\leq
C^{\prime}\left(\ln\ln(\mathcal{N}/\varepsilon)\right)^{-1/2}.$

Let
$\small\delta(\varepsilon)=\left(\ln\ln(\mathcal{N}/\varepsilon)\right)^{-1/2}$,
where $C^{\prime}$ depends only on those a-priori parameters.
On the other hand, note that
$(\mathcal{A},\mathcal{C})\in\mathcal{F}(\alpha_{m},\alpha_{M},n)$. We can
apply Proposition 3.2 to obtain an estimate of the term on the left-hand side
of (3.3) as follows

$\Big{|}\int_{\mathcal{A}}e^{\xi(\tau)\cdot(\mathbf{x}-\mathbf{x}_{0})}\,\eta\cdot\mathbf{u}(\mathbf{x}_{0})\mathrm{d}x\Big{|}\geq
C^{\prime}_{\mathcal{A}}\tau^{-n},$ (4.2)

where
$C^{\prime}_{\mathcal{A}}=C^{\prime}_{\mathcal{A}}(\mathcal{A},\mathbf{p},\mathbf{u}(\mathbf{x}_{0}))>0$.

From Definition 1.1, we have $|\rho(\mathbf{x}_{0})-1|\neq 0$. Combining (4)
with (4.2), we have

$\displaystyle C_{1}\big{|}(\rho(\mathbf{x}_{0})-1)\big{|}$ $\displaystyle\leq
e^{-\delta_{0}|\Re\xi|h/2}+\tau^{-\min\\{\theta_{1},\theta_{2}\\}}+h^{n-1}\tau^{n}\delta(\varepsilon)+h^{n-1}\tau^{n}e^{-\delta_{0}|\Re\xi|h},$

where $C_{1}$ is a positive constant depending only on those a- priori
parameters and $\mathbf{u}(\mathbf{x}_{0})$. Using the fact that
$e^{-\mathbf{x}}\leq\frac{1}{\mathbf{x}}$ and
$e^{-\mathbf{x}}\leq\frac{(n+4)!}{\mathbf{x}^{n+4}}$ for all $\mathbf{x}>\bf
0$, we get

$\displaystyle C_{2}\big{|}(\rho(\mathbf{x}_{0})-1)\big{|}$
$\displaystyle\leq\tau^{-1}h^{-1}+\tau^{-\min\\{\theta_{1},\theta_{2}\\}}+h^{n-1}\tau^{n+1}\delta(\varepsilon)+h^{-5}\tau^{-3}$
$\displaystyle\leq
C_{0}(\tau^{-m}h^{-5}+\tau^{n+1}\delta(\varepsilon)h^{n-1})$
$\displaystyle\leq
C_{0}h^{n-1}[\tau^{-m}h^{-n-4}+\tau^{n+1}\delta(\varepsilon)].$

Here, $m=\min\\{1,\theta_{1},\theta_{2}\\}$. Note that $\tau^{4}>1$,
$h^{-n}>1$ and $0<h^{n-1}<1$. Dividing the both sides of the above inequality
by $h^{n-1}$, we obtain the following inequality:

$C_{3}\big{|}(\rho(\mathbf{x}_{0})-1)\big{|}\leq\tau^{-m}h^{-n-5}+\tau^{n+5}\delta(\varepsilon).$
(4.3)

To determinate a minimum modulo constants of the right-hand side of (4.3). Set

$\tau=\tau_{e}=\Big{[}\frac{1}{h^{n+5}\delta(\varepsilon)}\Big{]}^{\frac{1}{m+n+1}},$
(4.4)

we have

$C_{4}\big{|}(\rho(\mathbf{x}_{0})-1)\big{|}\leq
2\,\delta(\varepsilon)^{\frac{m}{m+n+5}}\,h^{-\frac{(n+5)^{2}}{m+n+5}}.$ (4.5)

Note that
$\tau_{e}\geq\tau_{e}\,h^{\frac{n+5}{m+n+5}}=\delta(\varepsilon)^{-\frac{1}{m+n+5}}\geq\tau_{0}$
and $h\leq\hbar\leq l_{0}$ for sufficiently small $\varepsilon$. From (4.5),
we have

$\displaystyle\hbar$ $\displaystyle\leq
C_{5}\,\delta(\varepsilon)^{\frac{m}{m+n+5}}\big{|}\rho(\mathbf{x}_{0})\big{|}^{-\frac{m}{(n+5)^{2}}-\frac{1}{n+5}}\leq
C_{5}\,\delta(\varepsilon)^{\frac{m}{m+n+5}}\big{|}\epsilon_{0}+1\big{|}^{-\frac{m}{(n+5)^{2}}-\frac{1}{n+5}}$
$\displaystyle\leq
C_{6}\,\big{(}\ln\ln(\mathcal{N}/\varepsilon)\big{)}^{-\frac{m}{2(n+5)^{2}}},$

and the claim readily follows.

The proof is complete. ∎

##  5\. Related results about invisibility

The present section is devoted to the technical details of Theorem 1.2.

######  Theorem 5.1.

Let $(\Omega;\rho)$ be an admissible polyhedral scatterer. The plane incident
wave $\mathbf{u}^{i}$ of the form (1.1) is a nontrivial entire solution to
(1.2) with $\rho=1$. Let $\mathbf{U}$ be the far-field pattern of the
scattered wave $\mathbf{u}^{s}$ by medium scatterer $(\Omega;\rho)$. Then
there exists a positive constant $C$ only depending on the a-priori parameters
such that the following estimate holds:

$\displaystyle\|\mathbf{U}\|_{L^{2}(\mathbb{S}^{n-1},\mathbb{C}^{n}\times\mathbb{C}^{n})}\geq
C>0.$ (5.1)

In fact, $\Omega\subset\mathbb{R}^{n}$, $n=2,3$, can be relaxed to a general
bounded Lipschitz domain, which admits a convex polygonal/polyhedral point on
its boundary, namely, there exits $\mathbf{x}_{c}\in\partial\Omega$ such that
$\Omega\cap B_{l}(\mathbf{x}_{c})\subset\mathbb{R}^{2}$ is a plane sector for
$l_{0}>l>0$ and $\Omega\cap B_{l}(\mathbf{x}_{c})\subset\mathbb{R}^{3}$ is a
polyhedral cone with a spherical bottom surface for $l_{0}>l>0$. The general
result is stated as follows.

######  Theorem 5.2.

Let $\Omega\subset\mathbb{R}^{n}$, $n=2,3$, be a bounded Lipschitz domain,
with a convex polygonal/polyhedral point $\mathbf{x}_{c}\in\partial\Omega$.
Let $\rho(\mathbf{x})$ be a uniformly $\theta$-Hölder continuous function in
$\Omega$, $0<\theta\leq 1$. The plane incident wave $\mathbf{u}^{i}$ of the
form (1.1) is a nontrivial entire solution to (1.2) with $\rho=1$. Let
$\mathbf{U}$ be the far-field pattern of the scattered wave $\mathbf{u}^{s}$
by medium scatterer $(\Omega;\rho)$. Suppose that the following conditions are
fulfilled:

  * (1)

$\Omega=\mathrm{supp}(1-\rho)\subset\overline{B_{R}}$ for some $R>l>0$;

  * (2)

$\|\mathbf{u}^{s}\|_{H^{2}(B_{R})^{n}}\leq\mathcal{N}$ for $\mathcal{N}>0$;

  * (3)

$|\rho(\mathbf{x}_{c})-1|\geq\epsilon_{0}>0$ and
$\|\rho\|_{C^{\theta}(\Omega)}\leq\mathcal{M}$ for some $\mathcal{M}>0$;

  * (4)

For $n=2$, $0<2\alpha_{m}\leq\alpha\leq 2\alpha_{M}\leq\pi$, where $\alpha$
signifies the opening angle at $\mathbf{x}_{c}$.

Then the following estimate holds:

$\displaystyle\|\mathbf{U}\|_{L^{2}(\mathbb{S}^{n-1},\mathbb{C}^{n}\times\mathbb{C}^{n})}\geq
C>0,$ (5.2)

where $C$ is a positive constant, which depends on the a-priori parameters
$\theta,l,\mathcal{N},\mathcal{M},\epsilon_{0}$, and $R$.

###### Proof.

We follow similar arguments in the proof of Theorem 1.1 with some necessary
modifications. We take $\Omega^{\prime}=\emptyset$ and $\rho^{\prime}\equiv 1$
in $\mathbb{R}^{n}$. Clearly, the incident wave $\mathbf{u}^{i}$ of the form
(1.1) is an entire solution to
$\Delta^{*}\mathbf{u}^{\prime}+\omega^{2}\mathbf{u}^{\prime}=\bf 0$, which
yields $\mathbf{u}^{\prime s}\equiv\bf 0$ and $\mathbf{U}^{\prime}\equiv\bf
0$. Denote
$\varepsilon=\|\mathbf{U}\|_{L^{2}(\mathbb{S}^{n-1},\mathbb{C}^{n}\times\mathbb{C}^{n})}$.
In fact, we only consider the effect caused by $\mathbf{x}_{c}$ in its micro-
local area $\Omega\cap B_{h}(\mathbf{x}_{c})$, and therefore we can reduce the
corresponding reasoning in the proof Theorem 1.1. Here we take $h=l$. After a
series of derivations similar to Theorem 1.1, we can obtain the following
inequality

$\displaystyle
C_{1}\,\big{|}(\rho(\mathbf{x}_{c})-1)\big{|}\leq\tau^{-m}\,l^{-5}+\tau^{n+1}\delta(\varepsilon)\,l^{n-1},$
(5.3)

where
$\delta(\varepsilon)=\left(\ln\ln(\mathcal{N}/\varepsilon)\right)^{-1/2}$ and
$C_{1}>0$ does not depend on $l$.

Set

$\tau=\tau_{e}=\Big{[}\frac{1}{l^{n+5}\delta(\varepsilon)}\Big{]}^{\frac{1}{m+n+1}},$

we have

$\displaystyle C_{2}\big{|}(\rho(\mathbf{x}_{c})-1)\big{|}\leq
2\,\delta(\varepsilon)^{\frac{m}{m+n+5}}\,l^{-\frac{(n+5)^{2}}{m+n+5}}.$ (5.4)

It is straightforward to verify that for sufficient small $\varepsilon$, one
has

$\tau_{e}\geq\tau_{e}\,l^{\frac{n+5}{m+n+5}}=\frac{C_{3}}{\delta(\varepsilon)^{m+n+5}}=C_{3}(\ln\ln(\mathcal{N}/\varepsilon))^{\frac{(n+5)^{2}}{2(m+n+5)}}\geq\tau_{0},$

Using $\tau=\tau_{e}$ in (5.4), $|\rho(\mathbf{x}_{c})-1|\geq\epsilon_{0}>0$
and $\delta(\varepsilon)=\left(\ln\ln(\mathcal{N}/\varepsilon)\right)^{-1/2}$,

$\displaystyle\varepsilon$
$\displaystyle\geq\mathcal{N}\Big{[}\exp\exp\big{\\{}C_{3}\,l^{-\frac{(n+5)^{2}}{m}}|(\rho(\mathbf{x}_{c})-1)|^{-\frac{m+n+5}{m}}\big{\\}}\Big{]}^{-1}$
$\displaystyle\geq\mathcal{N}\Big{[}\exp\exp\big{\\{}C_{3}\,l^{-\frac{(n+5)^{2}}{m}}\epsilon_{0}^{-\frac{m+n+5}{m}}\big{\\}}\Big{]}^{-1}.$
(5.5)

That is,

$\displaystyle\|\mathbf{U}\|_{L^{2}(\mathbb{S}^{n-1},\mathbb{C}^{n}\times\mathbb{C}^{n})}\geq
C>0,$

where
$C=\mathcal{N}\Big{[}\exp\exp\big{\\{}C_{3}\,l^{-\frac{(n+5)^{2}}{m}}\epsilon_{0}^{-\frac{m+n+5}{m}}\big{\\}}\Big{]}^{-1}$.

The proof is complete. ∎

## Acknowledgement

The research of Z Bai was partially supported by the National Natural Science
Foundation of China under grant 11671337 and the Natural Science Foundation of
Fujian Province of China under grant 2021J01033. The work of H Diao was
supported in part by the startup fund from Jilin University. The work of H Liu
was supported by the startup fund from City University of Hong Kong and the
Hong Kong RGC General Research Fund (projects 12301420, 11300821 and
12301218), and the NSFC/RGC Joint Research Fund (project N_CityU101/21).

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