# Dislocations with corners in an elastic body with applications to fault
detection

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>,  Hongyu Liu
Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong
Kong SAR, China<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Qingle
Meng  Department of Mathematics, City University of Hong Kong, Kowloon Tong,
Hong Kong SAR, China<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

This paper focuses on an elastic dislocation problem that is motivated by
applications in the geophysical and seismological communities. In our model,
the displacement satisfies the Lamé system in a bounded domain with a mixed
homogeneous boundary condition. We also allow the occurrence of
discontinuities in both the displacement and traction fields on the fault
curve/surface. By the variational approach, we first prove the well-posedness
of the direct dislocation problem in a rather general setting with the Lamé
parameters being real-valued $L^{\infty}$ functions and satisfy the strong
convexity condition. Next, by considering the scenario that the Lamé
parameters are constant and the fault curve/surface possesses certain corner
singularities, we establish a local characterisation of the slip vectors at
the corner points over the dislocation curve/surface. In our study the
dislocation is geometrically rather general and may be open or closed. For
both cases, we establish the uniqueness results for the inverse problem of
determining the dislocation curve/surface and the slips.

Keywords: dislocations, elasticity, corners, slips, well-possdness, inverse
problem, uniqueness.

##  1\. Introduction

In this study, our focus lies on the phenomenon known as elastic dislocation.
An elastic dislocation refers to a surface or a crack within an elastic solid
across which there are discontinuities of the elastic displacement fields. It
may arise in various practical scenarios, such as a fault plane undergoing
slip for a limited duration or the sliding of faces in a crack. The modeling
and comprehension of interior elastic dislocations hold significant importance
in the geophysical and seismological communities. Specifically, the study can
finds important applications in monitoring, understanding, and mitigating
earthquakes and landslides. For further details and additional references on
this subject, we refer to [6, 8, 11, 13, 16, 17, 20] and the related
literature cited therein. Though the problem has been extensively and
intensively studied in the physical literature, there is only limited
theoretical understanding. In fact, the problem has only been investigated in
the recent papers [1, 3], where the elastic dislocation is modelled as an
open, oriented fault surface within an elastostatic system and there is
discontinuity in the displacement filed across such fault surface. The authors
proved that both the fault surface and the slip vector can be uniquely
determined by a single passive measurement of the elastostatic field on part
of the elastic solid boundary. Nevertheless, the technical argument developed
therein is of a global nature, which is based on carefully analyzing the
regularity of the solution of the direct problem around the fault surface and
inevitably requires technical restrictions, say e.g. one has to work within a
fixed coordinate frame. Moreover, the slip field is required to be compactly
supported within the fault surface or purely normal; see more details in [1,
3].

Motivated by the practical and theoretical studies mentioned above, we shall
propose and study the elastic dislocation problem in more sophisticated and
challenging setups. We confine ourselves to the elastic dislocation problem in
an extremely general form, which allows the occurrence of discontinuities in
both the displacement and traction fields across the fault curve/surface.
Moreover, the dislocation is geometrically rather general. In fact, the fault
curve/surface for describing the dislocation can be an open or closed
curve/surface in our study, whereas the previous ones in [1, 3] are only
closed.

In this paper, we investigate both the direct and inverse dislocation
problems. Our mathematical setup allows the presence of discontinuities in
both the displacement and traction fields across the fault curve/surface. The
direct problem is to determine the elastic displacements of the elastic
transmission problem with mixed homogeneous boundary conditions (see Problem
(2.6)), assuming that the dislocation curve/surface $\mathcal{S}$, the elastic
stiffness tensor $\mathbb{C}(\mathbf{x})$, and the slips $\mathbf{f}$ and
$\mathbf{g}$ over $\mathcal{S}$ are known. The inverse problem is to determine
the dislocation curve/surface $\mathcal{S}$ and the slip vectors $\mathbf{f}$
and $\mathbf{g}$ from the measurement of the displacement field. We focus on a
single measurement of displacement to study the inverse dislocation problems,
which means that we only need a pair of the Cauchy datum of the corresponding
elastic displacement to determine $\mathcal{S}$, $\mathbf{f}$ and
$\mathbf{g}$. Actually, various mature technologies, such as Synthetic
Aperture Radar (SAR) and Global Positioning System (GPS) arrays, can be used
to measure surface displacement measurements.

Using the variational approach, we first prove the well-posedness of the
direct dislocation problem in a rather general setting with the Lamé
parameters being real-valued $L^{\infty}$ functions and satisfy the strong
convexity condition, where the fault curve/surface can be open or closed. For
the inverse dislocation problems, we consider the scenario that the Lamé
parameters are constant and the fault curve/surface possesses certain corner
singularities. We first establish a local characterisation of the slip vectors
at the corner points over the dislocation curve/surface by analyzing the
singularity formation of the elastic field locally around certain abnormal
points on the fault surface in a microlocal way. In fact we utilize the so-
called CGO (complex geometric optic) solutions for the underlying elastic
system to achieve the corresponding characterisation of the slip vectors at
the corner points, where subtle and delicate analysis is developed in our
study. For both cases that the dislocation may be open or closed, we establish
the uniqueness results for the inverse problem of determining the dislocation
curve/surface and the slips.

The paper is structured as follows. In Section 2, we introduce the
mathematical setup of our study and establish the well-posedness of the
dislocation problem. In Section 3, we propose some admissible assumptions
about $(\mathcal{S};\mathbf{f},\mathbf{g})$ for the scenario that
$\mathcal{S}$ may be closed or open. For both cases, assuming that Lamé
parameters are constants and the fault curve/surface posses has corner
singularities, we establish a local characterization of the slip vectors at
the corner points over the dislocation curve/surface. Furthermore, we also
establish global uniqueness results for the inverse dislocation problem for
determining the dislocation curve/surface $\mathcal{S}$ and the slip vectors
$\mathbf{f}$ and $\mathbf{g}$ in these two cases with additional geometrical
assumption about the dislocation curve/surface. In Section 4, we derive
several local results for the slip vectors $\mathbf{f}$ and $\mathbf{g}$ at
the corner points along $\mathcal{S}$. Section 5 is devoted to proving the
uniqueness of the inverse problem presented in Section 3.

##  2\. Mathematical setup and the direct problem

In this section, we pay our attention to the mathematical setup of the
dislocation problem and the study of the well-posedness of the direct problem.

###  2.1. Mathematical setup

We first introduce a geometric and mathematical setup for our study; see Fig.
1 for a schematic illustration in 2D.

Figure 1. Schematic illustration of an elastic dislocation, where
$\mathcal{S}_{i}\ (i=1,\ldots,4)$ are fault curves and
$\partial\Omega=\Sigma_{D}\cup\Sigma_{N}$.

Let $\lambda(\mathbf{x})$ and $\mu(\mathbf{x})$,
$\mathbf{x}=(x_{j})_{j=1}^{n}\in\Omega$, be real-valued $L^{\infty}$
functions, which are referred to as the Lamé parameters of the elastic solid
$\Omega$. We define
$\mathbb{C}(\mathbf{x})=(C_{ijkl}(\mathbf{x}))_{i,j,k,l=1}^{n}$,
$\mathbf{x}\in\Omega$, as a four-rank tensor given by:

$\mathbb{C}(\mathbf{x}):=\lambda(\mathbf{x})\mathbf{I}\otimes\mathbf{I}+2\mu(\mathbf{x})\mathbb{I},\
\ \text{where}\ \
C_{ijkl}(\mathbf{x})=\lambda(\mathbf{x})\delta_{ij}\delta_{kl}+\mu(\mathbf{x})(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}),$
(2.1)

Here, $\mathbf{I}$ and $\mathbb{I}$ respectively represent the identity
operators of the second- and fourth-rank tensors, and $\delta_{ij}$ is the
Kronecker delta. We assume that the Lamé parameters $\lambda(\mathbf{x})$ and
$\mu(\mathbf{x})$ satisfy the strong convexity conditions:

$\mu(\mathbf{x})>0,\quad
2\mu(\mathbf{x})+n\lambda(\mathbf{x})>0,\quad\forall\mathbf{x}\in\Omega,$
(2.2)

These conditions ensure the uniform strong convexity of the elastic stiffness
tensor $\mathbb{C}(\mathbf{x})$. For $\mathbf{A}=(a_{ij})_{i,j=1}^{n}$, we
define the operation “$:$” as:

$\mathbb{C}:\mathbf{A}=\big{(}C_{ijkl}:\mathbf{A}\big{)}_{ij}=(\Pi)_{ij},\quad\text{where}\quad\Pi_{ij}:=\sum_{k,l=1}^{n}C_{ijkl}a_{kl}.$
(2.3)

Let $\mathbf{u}(\mathbf{x})=(u_{j}(\mathbf{x}))_{j=1}^{n}$,
$\mathbf{x}\in\Omega$, with each $u_{j}(\mathbf{x})$ being a complex-valued
function. We introduce the Lamé operator $\mathcal{L}$ as:

$\displaystyle\mathcal{L}\mathbf{u}:=\nabla\cdot(\mathbb{C}:\nabla\mathbf{u})=\mu\Delta\mathbf{u}+(\lambda+\mu)\nabla(\nabla\cdot\mathbf{u}),$
(2.4)

where $\nabla\mathbf{u}:=(\partial_{j}u_{i})_{i,j=1}^{n-1}$,
$\nabla\cdot\mathbf{u}:=\sum^{n}_{j=1}\partial_{j}u_{j}$, and
$\partial_{j}u_{j}=\partial u_{j}/\partial x_{j}$. Furthermore, let
$\bm{\nu}\in\mathbb{S}^{n-1}$ be the unit normal vector, and we define the
traction operator as:

$\displaystyle\mathcal{T}_{\bm{\nu}}(\mathbf{u})=\bm{\nu}\cdot(\mathbb{C}:\nabla\mathbf{u}).$
(2.5)

Let $\mathcal{S}\subset\Omega$ be an oriented Lipschitz curve/surface, which
can be open or closed. We define $\mathcal{S}^{\pm}$ as the two sides of
$\mathcal{S}$, with $\mathcal{S}^{+}$ representing the side where $\bm{\nu}$
points outward. We denote the jump of a function or tensor field $\mathbf{p}$
across $\mathcal{S}$ as
$[\mathbf{p}]_{\mathcal{S}}:=\mathbf{p}|_{\mathcal{S}}^{+}-\mathbf{p}|_{\mathcal{S}}^{-}$,
where $\mathbf{p}|_{\mathcal{S}}^{\pm}$ represent the non-tangential limits of
$\mathbf{p}$ on $\mathcal{S}^{\pm}$, respectively. The elastic dislocation
problem that we consider allows the occurrence of discontinuities in both the
displacement and traction fields, denoted by $\mathbf{f}$ and $\mathbf{g}$,
respectively.

###  2.2. Mathematical model

In this paper, our main focus is on the following elastostatic system for
$\mathbf{u}\in H^{1}(\Omega\backslash\overline{\mathcal{S}})^{n}$:

$\displaystyle\begin{cases}\mathcal{L}\mathbf{u}(\mathbf{x})=\mathbf{0},&\mathbf{x}\in\Omega\backslash\overline{\mathcal{S}},\\\
\mathcal{T}_{\bm{\nu}}\mathbf{u}\big{|}_{\Sigma_{N}}=\mathbf{0},&\mathbf{u}\big{|}_{\Sigma_{D}}=\mathbf{0},\\\
[\mathbf{u}]_{\mathcal{S}}=\mathbf{f},&[\mathcal{T}_{\bm{\nu}}\mathbf{u}]_{\mathcal{S}}=\mathbf{g}.\end{cases}$
(2.6)

Here, $\partial\Omega=\Sigma_{D}\cup\Sigma_{N}$ represents a Lipschitz
partition of $\partial\Omega$. To investigate both the direct and inverse
dislocation problems, we introduce some relevant function spaces for the slips
$\mathbf{f}$ and $\mathbf{g}$. Noting that the function spaces differ
depending on whether $\mathcal{S}$ is closed or open.

Class 1: When $\mathcal{S}$ is closed, we consider slips $\mathbf{f}$ and
$\mathbf{g}$ that satisfy

$\mathbf{f}\in
H^{\frac{1}{2}}(\mathcal{S})^{n}\quad\text{and}\quad\mathbf{g}\in
H^{-\frac{1}{2}}(\mathcal{S})^{n}.$

Class 2: When $\mathcal{S}$ is open, we assume that $\mathbf{f}$ and
$\mathbf{g}$ belong to appropriate weighted spaces with a favorable extension
property on such a curve/surface (see, for example, [14, 15, 19, 7]).
Following [15, 19, 7], we introduce the following space:

$\displaystyle H_{00}^{\frac{1}{2}}(\mathcal{S})^{n}$
$\displaystyle:=\Big{\\{}\mathbf{u}\in
H_{0}^{\frac{1}{2}}(\mathcal{S})^{n};\,\,\varrho^{-\frac{1}{2}}\mathbf{u}\in
L^{2}(\mathcal{S})^{n}\Big{\\}}$

associated with the following norms

$\displaystyle\|\mathbf{f}\|_{H_{00}^{\frac{1}{2}}(\mathcal{S})^{n}}$
$\displaystyle:=\|\mathbf{f}\|_{H^{\frac{1}{2}}(\mathcal{S})^{n}}+\|\varrho^{-\frac{1}{2}}\mathbf{f}\|_{L^{2}(\mathcal{S})^{n}}$
for $\displaystyle\mathbf{f}\,\in H_{00}^{\frac{1}{2}}(\mathcal{S})^{n}.$

Here, $H_{0}^{\frac{1}{2}}(\mathcal{S})^{n}$ is the closure of the space of
smooth functions with compact support in $\mathcal{S}$ with respect to the
$H^{\frac{1}{2}}(\mathcal{S})^{n}$ norm. In addition, $\mathbf{g}$ belongs to
the space $H^{-\frac{1}{2}}_{0}(\mathcal{S})^{n}$ which is the dual space of
$H_{00}^{\frac{1}{2}}(\mathcal{S})^{n}$. Let $\varrho\in
C^{\infty}(\overline{\mathcal{S}})$ denote a weight function which possesses
the certain properties:

  * (1)

$\varrho$ has the same order as the distance to the boundary, that is to say

$\lim\limits_{\mathbf{x}\rightarrow\mathbf{x}_{0}}\frac{\varrho(\mathbf{x})}{\mathrm{d}(\mathbf{x},\partial\mathcal{S})}=d\neq
0,\,\forall\,\mathbf{x}_{0}\,\in\partial\mathcal{S},$

  * (2)

$\varrho(\mathbf{x})$ is positive in $\mathcal{S}$ and $\varrho$ vanishes on
$\partial\mathcal{S}$.

We want to note that when $\mathcal{S}$ is a curve in the 2D case,
$\partial\mathcal{S}$ corresponds to the set of endpoints of $\mathcal{S}$.
Similarly, when $\mathcal{S}$ is a surface in the 3D case,
$\partial\mathcal{S}$ corresponds to the boundary curve of $\mathcal{S}$.

Assume that $\mathcal{S}$ is open. Let $\mathcal{S}$ be extended to a closed
Lipschitz curve/surface $\Gamma=\overline{\mathcal{S}}\cup\Gamma_{0}$
satisfying $\Gamma\cap\partial\Omega=\emptyset$, where $\Gamma_{0}$ is a curve
or a surface linking with the boundary $\partial\mathcal{S}$ satisfying
$\Gamma_{0}\cap(\mathcal{S}\backslash\partial\mathcal{S})=\emptyset$. Hence,
the Lipschitz domain $\Omega$ can be partitioned into two connected subdomains
$\Omega_{1}$ and $\Omega^{c}_{1}=\Omega\backslash\overline{\Omega}_{1}$, where
$\partial\Omega_{1}=\Gamma$ and
$\partial\Omega^{c}_{1}=\Gamma\cup\partial\Omega$. Let $\mathbf{f}$ be
continuously extended to $\widetilde{\mathbf{f}}\in H^{1/2}(\Gamma)^{n}$ by
zero on $\Gamma\backslash\mathcal{S}$ and $\mathbf{g}$ be continuously
extended to $\widetilde{\mathbf{g}}\in H^{-\frac{1}{2}}(\Gamma)^{n}$ by zero
on $\Gamma\backslash\mathcal{S}$. That is to say

$\widetilde{\mathbf{f}}=\begin{cases}\mathbf{f},\quad\mathbf{x}\in\mathcal{S},\\\
{\bf
0},\quad\mathbf{x}\in\Gamma\backslash\overline{\mathcal{S}}\end{cases}\quad\mbox{and}\qquad\widetilde{\mathbf{g}}=\begin{cases}\mathbf{g},\quad\mathbf{x}\in\mathcal{S},\\\
{\bf 0},\quad\mathbf{x}\in\Gamma\backslash\overline{\mathcal{S}}.\end{cases}$
(2.7)

In particular, when $\mathcal{S}$ is closed, $\Gamma=\mathcal{S}$,
$\widetilde{\mathbf{f}}=\mathbf{f}$ and $\widetilde{\mathbf{g}}=\mathbf{g}$.
Regardless of whether $\mathcal{S}$ is open or closed, we use

$\displaystyle\Omega_{1}:={\mathrm{enclose}}(\mathcal{S})$ (2.8)

to denote the aforementioned domain satisfying $\partial\Omega_{1}=\Gamma$.
Similarly, when $\mathcal{S}$ is closed, let the domain
${\mathrm{enclose}}(\mathcal{S})$ satisfy that
$\partial\left(\mathrm{enclose}(\mathcal{S})\right)=\mathcal{S}.$

We now are in the position to show the existence of a unique weak solution
$\mathbf{u}\in
H^{1}_{\Sigma_{D},\Sigma_{N}}(\Omega\backslash\overline{\mathcal{S}})^{n}$ to
Problem (2.6) corresponding with the boundary data on $\Sigma_{D}$ and
$\Sigma_{N}$.

######  Theorem 2.1.

There exists a unique solution $\mathbf{u}\in
H_{\Sigma_{D},\Sigma_{N}}^{1}(\Omega\backslash\overline{\mathcal{S}})^{n}$ to
Problem (2.6).

###### Proof.

In the following we first prove the well-posedness of Problem (2.6) for the
case that $\mathcal{S}$ is open. When $\mathcal{S}$ is closed, the
corresponding proof can be obtained in a similar way. We shall use the
variational technique (cf.[12]) to verify that there exists a unique solution
to this problem. From [1, Lemma 3.2] and [2, Remark 3.3], Problem (2.6) can be
recast equivalently as the following PDE system for $\mathbf{u}_{1}\in
H^{1}(\Omega_{1})^{n}$ and $\mathbf{u}_{2}\in H^{1}(\Omega^{c}_{1})^{n}$ such
that

$\displaystyle\begin{cases}\mathcal{L}\mathbf{u}_{1}(\mathbf{x})=\mathbf{0},\quad\mathbf{x}\in\Omega_{1},\\\
\mathcal{L}\mathbf{u}_{2}(\mathbf{x})=\mathbf{0},\quad\mathbf{x}\in\Omega^{c}_{1},\\\
\mathcal{T}_{\bm{\nu}}\mathbf{u}_{2}\big{|}_{\Sigma_{N}}=\mathbf{0},\,\mathbf{u}_{2}\big{|}_{\Sigma_{D}}=\mathbf{0},\\\
\mathbf{u}_{2}\big{|}_{\Gamma}-\mathbf{u}_{1}\big{|}_{\Gamma}=\widetilde{\mathbf{f}},\\\
\mathcal{T}_{\bm{\nu}}\mathbf{u}_{2}\big{|}_{\Gamma}-\mathcal{T}_{\bm{\nu}}\mathbf{u}_{1}\big{|}_{\Gamma}=\widetilde{\mathbf{g}},\end{cases}$
(2.9)

where $\widetilde{\mathbf{f}}\in H^{1/2}(\Gamma)^{n}$ and
$\widetilde{\mathbf{g}}\in H^{-1/2}(\Gamma)^{n}$ defined by (2.7) are given.
Let $\mathbf{u}_{\,\widetilde{\mathbf{f}}}$ be the unique solution to the
following Dirichlet boundary value problem

$\mathcal{L}\,\mathbf{u}_{\,\widetilde{\mathbf{f}}}=\mathbf{0}\quad\mbox{in}\quad\Omega^{c}_{1},\quad\mathbf{u}_{\,\widetilde{\mathbf{f}}}=\widetilde{\mathbf{f}}\quad\mbox{on}\quad\Gamma,\quad\mathbf{u}_{\,\widetilde{\mathbf{f}}}={\bf
0}\quad\mbox{on}\quad{\partial\Omega}.$

Let us next consider an equivalent variational formulation of Problem (2.9):
Find $\mathbf{w}\in H_{\Sigma_{D},\Sigma_{N}}^{1}(\Omega)^{n}$ satisfying

$\displaystyle\int_{\Omega_{1}}(\mathbb{C}:\nabla\mathbf{w}):\nabla\overline{\bm{\phi}}\,\mathrm{d}\mathrm{\mathbf{x}}+\int_{\Omega^{c}_{1}}(\mathbb{C}:\nabla\mathbf{w}):\nabla\overline{\bm{\phi}}\,\mathrm{d}\mathrm{\mathbf{x}}+\int_{\Gamma}\widetilde{\mathbf{g}}\cdot\overline{\bm{\phi}}\,\mathrm{d}\sigma$
(2.10)
$\displaystyle=\int_{\Sigma_{N}}\mathcal{T}_{\bm{\nu}}(\mathbf{u_{\widetilde{\mathbf{f}}}})\cdot\overline{\bm{\phi}}\,\mathrm{d}\sigma-\int_{\Omega^{c}_{1}}(\mathbb{C}:\nabla\mathbf{\mathbf{u}_{\widetilde{\mathbf{f}}}}):\nabla\overline{\bm{\phi}}\,\mathrm{d}\mathrm{\mathbf{x}},\qquad\qquad\forall\bm{\phi}\in
H_{\Sigma_{D},\Sigma_{N}}^{1}(\Omega)^{n}.$

With the help of the first Betti identity, it’s clear to show that
$\mathbf{u}_{1}:=\mathbf{w}\big{|}_{\Omega_{1}}$ and
$\mathbf{u}_{2}:=\mathbf{w}\big{|}_{\Omega^{c}_{1}}+\mathbf{u}_{\,\widetilde{\mathbf{f}}}$
satisfies Problem (2.9). Conversely, multiplying these equations in (2.9) by a
test function and using transmission conditions, denote
$\mathbf{w}:=\mathbf{u}_{1}$ in $\Omega_{1}$ and
$\mathbf{w}:=\mathbf{u}_{2}-\mathbf{u}_{\widetilde{\mathbf{f}}}$ in
$\Omega^{c}_{1}$, we can directly show that $\mathbf{w}$ belongs to
$H_{\Sigma_{D},\Sigma_{N}}^{1}(\Omega)^{n}$ and fulfils (2.10), where
$(\mathbf{u}_{1},\mathbf{u}_{2})$ is a solution to Problem (2.9).

Let $a(\cdot,\cdot)$ and $\mathcal{F}(\cdot)$ be defined as follows:

$\displaystyle a(\mathbf{w},\bm{\phi})$
$\displaystyle:=\int_{\Omega_{1}}(\mathbb{C}:\nabla\mathbf{w}):\nabla\overline{\bm{\phi}}\,\mathrm{d}\mathrm{\mathbf{x}}+\int_{\Omega^{c}_{1}}(\mathbb{C}:\nabla\mathbf{w}):\nabla\overline{\bm{\phi}}\,\mathrm{d}\mathrm{\mathbf{x}},$
$\displaystyle\mathcal{F}(\bm{\phi})$
$\displaystyle:=-\int_{\Gamma}\widetilde{\mathbf{g}}\cdot\overline{\bm{\phi}}\,\mathrm{d}\sigma+\int_{\Sigma_{N}}\mathcal{T}_{\bm{\nu}}(\mathbf{u_{\,\widetilde{\mathbf{f}}}})\cdot\overline{\bm{\phi}}\,\mathrm{d}\sigma-\int_{\Omega^{c}_{1}}(\mathbb{C}:\nabla\mathbf{\mathbf{u}_{\,\widetilde{\mathbf{f}}}}):\nabla\overline{\bm{\phi}}\,\mathrm{d}\mathrm{\mathbf{x}}.$

Hence, we can rewrite (2.10) as the problem of finding $\mathbf{w}\in
H^{1}_{\Sigma_{D},\Sigma_{N}}(\Omega)^{n}$ such that

$a(\mathbf{w},\bm{\phi})=\mathcal{F}(\bm{\phi})\quad\quad\mbox{for
all}\quad\bm{\phi}\in H_{\Sigma_{D},\Sigma_{N}}^{1}(\Omega)^{n}.$

It is easy to see that $\mathcal{F}$ is a linear continuous operator from
$H^{1}_{\Sigma_{D},\Sigma_{N}}(\Omega)^{n}$ to $\mathbb{C}$. Combining Korn’s
inequality (cf.[14]) with the uniform strong convexity of $\mathbb{C}$, it
directly leads to the fact that the bilinear form $a(\cdot,\cdot)$ is strictly
coercive and bounded. As a consequence of the Lax-Milgram Lemma, there exists
a unique solution $\mathbf{w}\in H_{\Sigma_{D},\Sigma_{N}}^{1}(\Omega)^{n}$ of
(2.10) such that

$\displaystyle\|\mathbf{w}\|_{H_{\Sigma_{D},\Sigma_{N}}^{1}(\Omega)^{n}}$
$\displaystyle\leq\Big{\|}\mathbf{w}\big{|}_{\Omega_{1}}\Big{\|}_{H^{1}(\Omega_{1})^{n}}+\Big{\|}\mathbf{w}\big{|}_{\Omega^{c}_{1}}\Big{\|}_{H^{1}(\Omega^{c}_{1})^{n}}\leq\|\mathcal{F}\|\leq
C\big{(}\|\widetilde{\mathbf{f}}\|_{H^{\frac{1}{2}}(\Gamma)^{n}}+\|\widetilde{\mathbf{g}}\|_{H^{-\frac{1}{2}}(\Gamma)^{n}}\big{)}$
$\displaystyle\leq
C\big{(}\|\mathbf{f}\|_{H_{00}^{\frac{1}{2}}(\mathcal{S})^{n}}+\|\mathbf{g}\|_{H^{-\frac{1}{2}}_{0}(\mathcal{S})^{n}}\big{)}.$

The above results implies immediately that there exists a unique solution
$\mathbf{u}\in
H_{\Sigma_{D},\Sigma_{N}}^{1}(\Omega\backslash\overline{\mathcal{S}})^{n}$ to
Problem (2.6) and $\mathbf{u}$ can be estimated by $\mathbf{f}$ and
$\mathbf{g}$ with respect to $H_{00}^{\frac{1}{2}}(\mathcal{S})^{n}$ norm and
$H^{-\frac{1}{2}}_{0}(\mathcal{S})^{n}$ norm, respectively.

The proof is complete. ∎

##  3\. The inverse problem and main results

In this section, we are devoted to studying the uniqueness results of the
inverse dislocation problem, which is composed of identifying the dislocations
$\mathcal{S}$ and slips $\mathbf{f}$ and $\mathbf{g}$ over $\mathcal{S}$ by
observation data on an open set $\Sigma_{0}\subset\Sigma_{N}$. For our study,
we formulate the inverse problem as

$\Lambda_{\mathcal{S};\mathbf{f},\mathbf{g}}=\mathbf{u}\big{|}_{\Sigma_{0}},$

where $\mathbf{u}$ is the solution to Problem (2.6). That is,
$\Lambda_{\mathcal{S};\mathbf{f},\mathbf{g}}$ contains the elastic deformation
data caused by the dislocation $(\mathcal{S};\mathbf{f},\mathbf{g})$ and
observed on $\Sigma_{0}\subset\Sigma_{N}$. The inverse problem we are devoted
to can be formulated by

$\Lambda_{\mathcal{S};\mathbf{f},\mathbf{g}}(\mathbf{u}\big{|}_{\Sigma_{0},\Sigma_{N}})\rightarrow\mathcal{S},\mathbf{f},\mathbf{g}.$
(3.1)

The uniqueness results can be proved under some assumptions about the geometry
of the dislocation $\mathcal{S}$ and a priori information about Lamé
parameters $\lambda(\mathbf{x})$ and $\mu(\mathbf{x})$ of the elastic solid
$\Omega$, where $\lambda(\mathbf{x})$ and $\mu(\mathbf{x})$ are real constants
and satisfy the strong convexity condition (2.2).

In order to describe the geometry of $\mathcal{S}$, we next introduce some
notations for the geometric setup; see Fig. 3 for a schematic illustration.
Given $\mathbf{x}_{c}\in\mathbb{R}^{2}$ and constants
$\theta_{m},\,\theta_{M}\in(-\pi,\pi)$ such that
$\theta_{M}-\theta_{m}\in(0,\pi)$, we consider the following open sector

$\displaystyle\mathcal{K}_{\mathbf{x}_{c}}\,=\left\\{\mathbf{x}\in\mathbb{R}^{2}\big{|}\,{\bf
0}<\mathbf{x}=\mathbf{x}_{c}+(r\cos\theta,r\sin\theta)^{\top},\,\,\theta_{\min}<\theta<\theta_{\max},r>0\right\\}$
(3.2)

with boundaries

$\displaystyle\Gamma^{+}_{\mathbf{x}_{c}}$
$\displaystyle=\left\\{\mathbf{x}\in\mathbb{R}^{2}\big{|}\,\mathbf{x}=\mathbf{x}_{c}+(r\cos\theta_{M},r\sin\theta_{M})^{\top},\,r>0\right\\},$
$\displaystyle\Gamma^{-}_{\mathbf{x}_{c}}$
$\displaystyle=\left\\{\mathbf{x}\in\mathbb{R}^{2}\big{|}\,\mathbf{x}=\mathbf{x}_{c}+(r\cos\theta_{m},r\sin\theta_{m})^{\top},\,r>0\right\\}.$

The point $\mathbf{x}_{c}$ is said to be a planar corner point with opening
angle $\theta_{M}-\theta_{m}$ and boundaries $\Gamma^{\pm}_{\mathbf{x}_{c}}$.
Let

$\displaystyle\mathcal{C}_{\mathbf{x}_{c},h}$
$\displaystyle:=\mathcal{K}_{\mathbf{x}_{c}}\cap B_{h}(\mathbf{x}_{c}),\quad$
$\displaystyle\Gamma^{\pm}_{\mathbf{x}_{c},h}$
$\displaystyle:=\Gamma^{\pm}_{\mathbf{x}_{c}}\cap B_{h}(\mathbf{x}_{c}),$
$\displaystyle\Lambda^{\mathbf{x}_{c}}_{h}$
$\displaystyle:=\mathcal{K}_{\mathbf{x}_{0}}\cap\partial
B_{h}(\mathbf{x}_{c}),\quad$ $\displaystyle\Sigma_{\mathbf{x}_{c}}$
$\displaystyle:=\mathcal{C}_{\mathbf{x}_{c},h}\backslash\mathcal{C}_{\mathbf{x}_{c},h/2},$
(3.3)

where $B_{h}(\mathbf{x}_{c})$ denotes an open disk centered at
$\mathbf{x}_{c}$ of radius $h\in\mathbb{R}_{+}$. For the sake of brevity, we
use
$B_{h},\,\mathcal{K},\,\Gamma^{\pm},\,\mathcal{C}_{h},\,\Gamma^{\pm}_{h},\,\Lambda_{h}$
and $\Sigma$ to represent the corresponding notations at the origin.

Figure 2. Schematic illustration of a 2D corner/3D edge corner.

###  3.1. Main uniqueness results

Before giving the uniqueness results in Theorem 3.1–Theorem 3.3, we introduce
some admissible conditions about the dislocation for our subsequent study.

######  Definition 3.1.

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^{n}(n=2,3)$. We say
that $(\mathcal{S};\mathbf{f},\mathbf{g})$ belongs to the admissible class
$\mathcal{T}$ if the following conditions are fulfilled:

  * (1)

In $\mathbb{R}^{2}$, $\mathcal{S}\subset\mathbb{R}^{2}$ is an oriented
Lipschitz curve. There exists at least one planar corner point
$\mathbf{x}_{c}$ on $\mathcal{S}$ such that
$\Gamma^{\pm}_{\mathbf{x}_{c},h}\subset\mathcal{S}$, where
$\Gamma^{\pm}_{\mathbf{x}_{c},h}=\partial\mathcal{K}_{\mathbf{x}_{0}}\cap
B_{h}(\mathbf{x}_{c})$ and
$\mathcal{C}_{\mathbf{x}_{c},h}=B_{h}(\mathbf{x}_{c})\cap\mathcal{K}_{\mathbf{x}_{c}}=B_{h}(\mathbf{x}_{c})\cap\Omega_{1}$.
Here, $\mathcal{K}_{\mathbf{x}_{c}}$ and $B_{h}(\mathbf{x}_{c})$ are defined
in (3.2) and $\Omega_{1}={\mathrm{enclose}}(\mathcal{S})$ is given in (2.8).

  * (2)

In $\mathbb{R}^{3}$, $\mathcal{S}\subset\mathbb{R}^{3}$ is an oriented
Lipschitz surface. Suppose that $\mathcal{S}$ possesses at least one 3D edge
corner
$\mathbf{x}_{c}=(\mathbf{x}^{\prime}_{c},x_{3})^{\top}\in\mathbb{R}^{3}$,
where $\mathbf{x}^{\prime}_{c}\in\mathbb{R}^{2}$ is a planner corner point. In
other words, for sufficient small positive numbers $h$ and $M$, we have that
$\Gamma^{\pm}_{\mathbf{x}^{\prime}_{c},h}\times(-M,M)\subset\mathcal{S}$ and
$B^{\prime}_{h}(\mathbf{x}^{\prime}_{c})\times(-M,M)\cap\Omega_{1}=\mathcal{C}^{\prime}_{\mathbf{x}_{c},h}\times(-M,M)$,
where $\Gamma^{\pm}_{\mathbf{x}^{\prime}_{c},h}$ are two edges of a sectorial
corner at $\mathbf{x}^{\prime}_{c}$ and
$B^{\prime}_{h}(\mathbf{x}^{\prime}_{c})$ is an open disk centered at
$\mathbf{x}^{\prime}_{c}$ with radius $h$, which are defined in (3.3). The
opening angle of the sectorial corner at $\mathbf{x}^{\prime}_{c}$ is referred
to be the opening angle of the corresponding 3D edge corner.

  * (3)

In $\mathbb{R}^{2}$, let
$\mathbf{f}_{j}:=\mathbf{f}\big{|}_{\Gamma^{j}_{\mathbf{x}_{c},h}}$ and
$\mathbf{g}_{j}:=\mathbf{g}\big{|}_{\Gamma^{j}_{\mathbf{x}_{c},h}}$ satisfy
$\mathbf{f}_{j}\in C^{1,\alpha_{j}}(\Gamma^{j}_{\mathbf{x}_{c},h})^{2}$ and
$\mathbf{g}_{j}\in C^{\beta_{j}}(\Gamma^{j}_{\mathbf{x}_{c},h})^{2}$ with
$\alpha_{j},\beta_{j}$ being in $(0,1)$ and $j=+,-$, where $\mathbf{x}_{c}$
and $\Gamma^{\pm}_{\mathbf{x}_{c},h}$ are just the ones in $(1)$.

  * (4)

In $\mathbb{R}^{3}$, let $\mathbf{f}_{j}:=\mathbf{f}\big{|}_{\Gamma^{\prime
j}_{\mathbf{x}^{\prime}_{c},h}\times(-M,M)}$ and
$\mathbf{g}_{j}=\mathbf{g}\big{|}_{\Gamma^{\prime
j}_{\mathbf{x}^{\prime}_{c},h}\times(-M,M)}$ fulfill that $\mathbf{f}_{j}\in
C^{1,\alpha_{j}}(\Gamma^{\prime
j}_{\mathbf{x}^{\prime}_{c},h}\times(-M,M))^{3}$ and $\mathbf{g}_{j}\in
C^{\beta_{j}}(\Gamma^{\prime j}_{\mathbf{x}^{\prime}_{c},h}\times(-M,M))^{3}$
with $\alpha_{j},\beta_{j}$ being in $(0,1)$ and $j=+,-$, where
$\mathbf{x}^{\prime}_{c}$ and $\Gamma^{\prime\pm}_{\mathbf{x}^{\prime}_{c},h}$
are just the ones in $(2)$. Furthermore, $\mathbf{f}_{j}$ and $\mathbf{g}_{j}$
are are independent of $x_{3}$.

  * (5)

Let $\mathcal{V}_{\mathcal{S}}$ signify the set of 2D corners/3D edge corners
of $\mathcal{S}$. In $\mathbb{R}^{3}$, denote
$\mathbf{g}=(\mathbf{g}^{(1,2)},g^{3})$. Either the following assumptions
$\mathcal{A}_{1}$ or $\mathcal{A}_{2}$ is satisfied,

$\displaystyle{\it\mbox{ Assumption}\,\,\mathcal{A}_{\rm 1}}:\,\,$
$\displaystyle\forall\mathbf{x}_{c}\in\mathcal{V}_{\mathcal{S}},\quad$
$\displaystyle\mathbf{f}_{-}(\mathbf{x}_{c})\neq\mathbf{f}_{+}(\mathbf{x}_{c}),$
$\displaystyle{\it\mbox{ Assumption}\,\,\mathcal{A}_{\rm 2}}:\,\,$
$\displaystyle\forall\mathbf{x}_{c}\in\mathcal{V}_{\mathcal{S}},\quad$
$\displaystyle\left\\{{\small\begin{array}[]{ll}\mathbf{g}_{+}(\mathbf{x}_{c})\neq
W_{\mathbf{x}_{c}}\mathbf{g}_{-}(\mathbf{x}_{c}),\qquad\qquad\mbox{ if
$n=2$},\\\\[14.22636pt] \mathbf{g}^{(1,2)}_{+}(\mathbf{x}_{c})\neq
W_{\mathbf{x}^{\prime}_{c}}\mathbf{g}^{(1,2)}_{-}(\mathbf{x}_{c})\,\qquad\,\mbox{
if $n=3$},\\\ \mbox{or}\quad g^{3}_{+}(\mathbf{x}_{c})\neq 0\\\ \mbox{or}\quad
g^{3}_{-}(\mathbf{x}_{c})\neq 0,\end{array}}\right.$

where

$W_{\mathbf{x}_{c}}=\begin{bmatrix}-\cos\theta_{\mathbf{x}_{c}}&-\sin\theta_{\mathbf{x}_{c}}\\\
-\sin\theta_{\mathbf{x}_{c}}&\cos\theta_{\mathbf{x}_{c}}\end{bmatrix}\,\mbox{and}\,\,W_{\mathbf{x}^{\prime}_{c}}=\begin{bmatrix}-\cos\theta_{\mathbf{x}^{\prime}_{c}}&-\sin\theta_{\mathbf{x}^{\prime}_{c}}\\\
-\sin\theta_{\mathbf{x}^{\prime}_{c}}&\cos\theta_{\mathbf{x}^{\prime}_{c}}\end{bmatrix}.$

Here, $\theta_{\mathbf{x}_{c}}$ and $\theta_{\mathbf{x}^{\prime}_{c}}$ denote
the opening angle at the 2D corner/3D edge corner point $\mathbf{x}_{c}$ of
$\mathcal{S}$.

######  Remark 3.1.

The admissible conditions that $\mathcal{S}$ possesses at least one planar
corner/3D edge corner in Definitions 3.1 can be easily fulfilled in generic
physical scenarios. For example, $\mathcal{S}$ is a piecewise linear fault
curve/surface. In what follows, $(\mathcal{S};\mathbf{f},\mathbf{g})$ is said
to be an admissible dislocation with slip vectors $\mathbf{f}$ and
$\mathbf{g}$ if it fulfils the conditions in Definitions 3.1.

The uniqueness results in [1, 3] for determining $\mathcal{S}$ only focused on
the case that $\mathcal{S}$ is open and there is only the displacement
discontinuity on the fault curve/surface. Those methodology developed in [1,
3] cannot deal with the case that $\mathcal{S}$ is closed. In our study, we
can handle these two situations and also allow the occurrence of
discontinuities in the both displacement and traction fields on the
dislocation. In Theorem 3.1, we obtain a local uniqueness result for the
inverse dislocation problem (3.1) by using a single displacement measurement
on $\Sigma_{0}$ whose proof is postponed in Section 5.

######  Theorem 3.1.

Let $(\mathcal{S}_{1};\mathbf{f}^{1},\mathbf{g}^{1})$ and
$(\mathcal{S}_{2};\mathbf{f}^{2},\mathbf{g}^{2})$ belong to $\mathcal{T}$.
Assume that
$\mathrm{supp}(\mathbf{g}^{i})=\mathrm{supp}(\mathbf{f}^{i})=\overline{\mathcal{S}}_{i}$
and $\mathbf{u}_{i}$ is the unique solution to Problem (2.6) in
$H^{1}_{\Sigma_{D},\Sigma_{N}}(\Omega\backslash\overline{\mathcal{S}_{i}})$
with respect with to $\mathbf{g}=\mathbf{g}^{i}$, $\mathbf{f}=\mathbf{f}^{i}$
and $\mathcal{S}=\mathcal{S}_{i}$, respectively, for $i=1,2$. If
$\mathbf{u}_{1}\big{|}_{\Sigma_{0}}=\mathbf{u}_{2}\big{|}_{\Sigma_{0}}$, then
$\mathcal{S}_{1}\Delta\mathcal{S}_{2}:=(\mathcal{S}_{1}\backslash\mathcal{S}_{2})\cup(\mathcal{S}_{2}\backslash\mathcal{S}_{1})$
cannot contain a planar corner/3D edge corner $\mathbf{x}_{c}$.

In Theorem 3.2 and 3.3 we derive the global uniqueness results for the inverse
dislocation problem (3.1) on the determination of the dislocation surface
$\mathcal{S}$ and $\mathbf{f}$, $\mathbf{g}$ from a given single displacement
measurement on $\Sigma_{0}$. The proofs of Theorems 3.2 and 3.3 are postponed
in Section 5.

We first consider the case that $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ are
closed.

######  Theorem 3.2.

Let $(\mathcal{S}_{1};\mathbf{f}^{1},\mathbf{g}^{1})$ and
$(\mathcal{S}_{2};\mathbf{f}^{2},\mathbf{g}^{2})$ belong to $\mathcal{T}$,
where $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ are closed. Assume that
$\Omega_{1}={\mathrm{enclose}}(\mathcal{S}_{1})$ and
$\Omega_{2}={\mathrm{enclose}}(\mathcal{S}_{2})$ are two convex polygons in
$\mathbb{R}^{2}$ or two convex polyhedrons in $\mathbb{R}^{3}$, where
$\mathcal{S}_{i}=\bigcup_{k=1}^{m_{i}}\Pi_{i,k}$ $(i=1,2)$. Here $\Pi_{i,k}$
is the $k$-th edge or surface of the polygon or polyhedron $\Omega_{i}$. Let
$\mathbf{u}_{i}$ be the unique solution to Problem (2.6) in
$H^{1}_{\Sigma_{D},\Sigma_{N}}(\Omega\backslash\overline{\mathcal{S}})$ with
respect with to $\mathbf{g}^{i}$ and $\mathbf{f}^{i}$, respectively, for
$i=1,2$. If
$\mathbf{u}_{1}\big{|}_{\Sigma_{0}}=\mathbf{u}_{2}\big{|}_{\Sigma_{0}}$, then
$\mathcal{S}_{1}=\mathcal{S}_{2}$, namely $m_{1}=m_{2}:=m,\
\Pi_{1,k}=\Pi_{2,k}:=\Pi_{k}\ (k=1,\ldots,m)$. Furthermore, assume that
$\mathbf{f}^{i}$ and $\mathbf{g}^{i}$ are piecewise-constant functions on
$\Pi_{k}$. Then we have

$(\mathbf{f}^{1}-\mathbf{f}^{2})\big{|}_{\Pi_{k+1}}=(\mathbf{f}^{1}-\mathbf{f}^{2})\big{|}_{\Pi_{k}},\,\,(\mathbf{g}^{1}-\mathbf{g}^{2})\big{|}_{\Pi_{k+1}}=W_{\mathbf{x}_{k}}(\mathbf{g}^{1}-\mathbf{g}^{2})\big{|}_{\Pi_{k}},\,k=1,\cdots,m-1$
(3.4)

and

$(\mathbf{f}^{1}-\mathbf{f}^{2})\big{|}_{\Pi_{1}}=(\mathbf{f}^{1}-\mathbf{f}^{2})\big{|}_{\Pi_{m}},\quad(\mathbf{g}^{1}-\mathbf{g}^{2})\big{|}_{\Pi_{1}}=W_{\mathbf{x}_{m}}(\mathbf{g}^{1}-\mathbf{g}^{2})\big{|}_{\Pi_{m}},$
(3.5)

where
$W_{\mathbf{x}_{k}}=\begin{bmatrix}-\cos\theta_{\mathbf{x}_{k}}&-\sin\theta_{\mathbf{x}_{k}}\\\
-\sin\theta_{\mathbf{x}_{k}}&\cos\theta_{\mathbf{x}_{k}}\end{bmatrix}$ is
similarly defined as $W_{\mathbf{x}_{c}}$ in Definition 3.1 and
$\theta_{\mathbf{x}_{k}}$ corresponds to the opening angle for the 2D
corners/3D edge corners $\mathbf{x}_{k}$, $k=1,2,\cdots,m$.

In Theorem 3.3, we investigate the unique determination of a polygonal curve
or a polyhedral surface $\mathcal{S}$, where $\mathcal{S}$ is open. Before
that we introduce the corresponding definition.

######  Definition 3.2.

Suppose that $\mathcal{S}\subset\mathbb{R}^{n}$ $(n=2,3)$ is open. Under rigid
motion, let $\mathcal{S}\subset\mathbb{R}^{2}$ be the graph of a function
$f(x_{1})$, where $x_{1}\in[a,b]$. If $[a,b]=\cup_{i=1}^{\ell}[a_{i},a_{i+1}]$
with $l\geq 3$, $a_{i}<a_{i+1}$, $a_{1}=a$ and $a_{\ell}=b$, which is
piecewise linear polynomial on each piece $[a_{i},a_{i+1}]$, then
$\mathcal{S}\subset\mathbb{R}^{2}$ is referred as a polygonal curve. Under
rigid motion, let $\mathcal{S}\subset\mathbb{R}^{3}$ be the graph of a
function $f(x_{1},x_{3})$, where
$(x_{1},x_{3})\in[a_{1},a_{2}]\times[b_{1},b_{2}]$. If $f(x_{1},c)$ with
$c\in[b_{1},b_{2}]$ being fixed satisfies $f(x_{1},c)=g(x_{1})$, where the
graph of $g(x_{1})$ is a polygonal curve like the ones in $\mathbb{R}^{2}$,
then $\mathcal{S}\subset\mathbb{R}^{3}$ is referred as a polyhedral surface.
See Fig. 3 for a schematic illustration.

Figure 3. Schematic illustration of a polygonal curve or a polyhedral surface.

######  Theorem 3.3.

Assume that $(\mathcal{S}_{1};\mathbf{f}^{1},\mathbf{g}^{1})$ and
$(\mathcal{S}_{2};\mathbf{f}^{2},\mathbf{g}^{2})$ belong to $\mathcal{T}$,
where $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ are open. Let $\mathbf{u}_{i}$
be the unique solution to Problem (2.6) in
$H^{1}_{\Sigma_{D},\Sigma_{N}}(\Omega\backslash\overline{\mathcal{S}_{i}})$
with respect to $\mathbf{g}^{i}$, $\mathbf{f}^{i}$ and $\mathcal{S}_{i}$,
where $\mathbf{f}^{i}\in H^{\frac{1}{2}}_{00}(\mathcal{S}_{i})$ and
$\mathbf{g}^{i}\in H^{-\frac{1}{2}}_{0}(\mathcal{S}_{i})$ with
$\mathrm{supp}(\mathbf{g}^{i})=\mathrm{supp}(\mathbf{f}^{i})=\overline{\mathcal{S}}_{i}$,
$i=1,2$. Suppose that the curves/surfaces $\mathcal{S}_{1}$ and
$\mathcal{S}_{2}$ are polygonal curves in $\mathbb{R}^{2}$ or polyhedral
surfaces in $\mathbb{R}^{3}$. If

$\displaystyle\mathbf{u}_{1}\big{|}_{\Sigma_{0}}=\mathbf{u}_{2}\big{|}_{\Sigma_{0}},$
(3.6)

then

$\mathcal{S}_{1}=\mathcal{S}_{2},\quad\mathbf{f}^{1}=\mathbf{f}^{2}\,\,\,\mbox{and}\,\,\,\mathbf{g}^{1}=\mathbf{g}^{2}.$

##  4\. Local results of slips $\mathbf{f}$ and $\mathbf{g}$ at the corners on
$\mathcal{S}$

In this section, we shall proceed to derive several auxiliary propositions
which describe the local results of slip vectors $\mathbf{f}$ and $\mathbf{g}$
at the corner points of $\mathcal{S}$. These auxiliary results play a key role
in establishing our main results in Theorems 3.1, 3.2 and 3.3. To derive these
propositions, we shall introduce two kinds of the so-called CGO (complex
geometrical optics) solutions satisfying different Lamé /acoustic equations.

###  4.1. Local results for 2D case

This subsection is devoted to analyzing the local behaviours of slip vectors
$\mathbf{f}$ and $\mathbf{g}$ around a planar corner. Next, we introduce the
first kind of CGO solution $\mathbf{u}_{0}$ introduced in [5], which is given
by

$\mathbf{u}_{0}(\mathbf{x})=\begin{pmatrix}\exp(-s\sqrt{z})\\\ {\rm
i}\exp(-s\sqrt{z})\end{pmatrix}:=\begin{pmatrix}u^{0}_{1}(\mathbf{x})\\\
u^{0}_{2}(\mathbf{x})\end{pmatrix}\quad\mbox{in}\quad\Omega,\quad\mathbf{x}=(x_{1},x_{2})^{\top},$
(4.1)

where $z=x_{1}+{\rm i}\,x_{2}$, $s\in\mathbb{R}_{+}$ and
$\Omega\cap(\mathbb{R}_{-}\cup\\{0\\})=\emptyset$. Here the complex square
root of $z$ is defined as

$\sqrt{z}=\sqrt{|z|}\left(\cos\frac{\theta}{2}+{\rm
i}\sin\frac{\theta}{2}\right),$

where $-\pi<\theta<\pi$ is the argument of $z$. Furthermore, it yields that
$\mathcal{L}\,\mathbf{u}_{0}={\bf 0}$ in $\Omega$.

Some significant properties and regularity results of the CGO solution given
in (4.1) need to be reviewed, which is beneficial for the subsequent analysis.

######  Lemma 4.1.

[5, Proposition 3.1] Let $\mathbf{u}_{0}$ be given as above. Then we have the
following properties

$\int_{\mathcal{K}}u^{0}_{1}(\mathbf{x}){\rm d}\mathbf{x}=6{\rm
i}(e^{-2\theta_{M}{\rm i}}-e^{-2\theta_{m}\mathrm{i}})s^{-4}$ (4.2)

and

$\int_{\mathcal{K}}|u^{0}_{j}(\mathbf{x})||\mathbf{x}|^{\alpha}{\rm
d}\mathbf{x}\leq\frac{2(\theta_{M}-\theta_{m})\Gamma(2\alpha+4)}{\delta_{\mathcal{K}}^{2\alpha+4}}s^{-2\alpha-4},\quad
j=1,2,$ (4.3)

where $\mathcal{K}$ is defined in Section 3, $\alpha,h>0$,
$\delta_{\mathcal{K}}=\min\limits_{\theta_{m}<\theta<\theta_{M}}\cos\frac{\theta}{2}$
is a positive constant.

The following critical estimate can be obtained by using Laplace transform and
the exponential function with negative power.

######  Lemma 4.2.

For any $\alpha>0$, if $\omega(\theta)>0$, then we have

$\lim\limits_{s\rightarrow+\infty}\int^{h}_{0}r^{\alpha}e^{-s\sqrt{r}\omega(\theta)}\mathrm{d}\mathrm{r}={\mathcal{O}}(s^{-2\alpha-2}).$

We next recall some critical lemmas about the regularity of the CGO solution
$\mathbf{u}_{0}$ defined in (4.1).

######  Lemma 4.3.

[10, Lemma 2.3] Let $\mathcal{C}_{\mathbf{x}_{c},h}$ be defined in (3.3) and
$\mathbf{u}_{0}$ be given in (4.1). Then $\mathbf{u}_{0}\in
H^{1}(\mathcal{C}_{\mathbf{x}_{c},h})^{2}$ and
$\mathcal{L}\,\mathbf{u}_{0}=\mathbf{0}$ in $\mathcal{C}_{\mathbf{x}_{c},h}$.
Furthermore, it holds that

$\big{\|}\mathbf{u}_{0}\big{\|}_{L^{2}(\mathcal{C}_{\mathbf{x}_{c},h})^{2}}\leq\sqrt{\theta_{M}-\theta_{m}}e^{-s\sqrt{\Theta}}h$

and

$\Big{\|}|\mathbf{x}|^{\alpha}\mathbf{u}_{0}\Big{\|}_{L^{2}(\mathcal{C}_{\mathbf{x}_{c},h})^{2}}\leq
s^{-2(\alpha+1)}\frac{2\sqrt{(\theta_{M}-\theta_{m})\Gamma(4\alpha+4)}}{(2\delta_{\mathcal{K}})^{2\alpha+2}},$

where $\Theta\in[0,h]$ and $\delta_{\mathcal{K}}$ is defined in (4.3).

######  Lemma 4.4.

[9, Lemma 2.8] Let $\Gamma^{\pm}_{h}$ and $u^{0}_{1}(\mathbf{x})$ be
respectively defined in (3.3) and (4.1) with $\mathbf{x}_{c}$ coinciding with
the origin. We have

$\displaystyle\begin{split}\int_{\Gamma^{+}_{h}}u^{0}_{1}(\mathbf{x}){\rm
d}\sigma&=2s^{-2}\left(\mu(\theta_{M})^{-2}-\mu(\theta_{M})^{-2}e^{-s\sqrt{h}\mu(\theta_{M})}\right.\\\
&\left.\hskip
99.58464pt-\mu(\theta_{M})^{-1}s\sqrt{h}e^{-s\sqrt{h}\mu(\theta_{M})}\right),\\\
\int_{\Gamma^{-}_{h}}u^{0}_{1}(\mathbf{x}){\rm
d}\sigma&=2s^{-2}\left(\mu(\theta_{m})^{-2}-\mu(\theta_{m})^{-2}e^{-s\sqrt{h}\mu(\theta_{m})}\right.\\\
&\left.\hskip
99.58464pt-\mu(\theta_{m})^{-1}s\sqrt{h}e^{-s\sqrt{h}\mu(\theta_{m})}\right),\end{split}$

where
$\mu(\theta):=\cos(\theta/2)+\mathrm{i}\sin(\theta/2)=e^{\mathrm{i}\theta/2}$.

To prove our main Theorems 3.1, 3.2 and 3.3, the next critical auxiliary
proposition is needed. Since $\mathcal{L}$ is invariant under rigid motion, in
what follows, the underlying corner point $\mathbf{x}_{c}$ coincides with the
origin.

######  Proposition 4.1.

Under the same setup about $\mathcal{C}_{h}$ and $\Gamma^{\pm}_{h}$ given in
(3.3) with $\mathbf{x}_{c}$ coinciding with the origin. Let $\mathbf{v}\in
H^{1}(\mathcal{C}_{h})^{2}$ and $\mathbf{w}\in H^{1}(\mathcal{C}_{h})^{2}$
satisfy

$\displaystyle\begin{cases}\mathcal{L}\,\mathbf{v}=\mathbf{0},\qquad\mathcal{L}\,\mathbf{w}=\mathbf{0}&\mbox{in}\quad\mathcal{C}_{h},\\\
\mathbf{v}-\mathbf{w}=\mathbf{f}_{+},\,\mathcal{T}_{\bm{\nu}}\,\mathbf{v}-\mathcal{T}_{\bm{\nu}}\,\mathbf{w}=\mathbf{g}_{+}&\mbox{on}\quad\Gamma^{+}_{h},\\\
\mathbf{v}-\mathbf{w}=\mathbf{f}_{-},\,\mathcal{T}_{\bm{\nu}}\,\mathbf{v}-\mathcal{T}_{\bm{\nu}}\,\mathbf{w}=\mathbf{g}_{-}&\mbox{on}\quad\Gamma^{-}_{h}\\\
\end{cases}$

with $\mathbf{f}_{j}\in H^{\frac{1}{2}}(\Gamma_{h}^{j})^{2}\cap
C^{1,\alpha_{j}}(\Gamma_{h}^{j})^{2}$ and $\mathbf{g}_{j}\in
H^{-\frac{1}{2}}(\Gamma_{h}^{j})^{2}\cap C^{\beta_{j}}(\Gamma_{h}^{j})^{2}$,
where $j=+,-$ and $\alpha_{+},\alpha_{-},\beta_{+},\beta_{-}\in(0,1)$. Then we
have the following continuities at the vertex point, that is to say,

$\displaystyle\mathbf{g}_{+}({\bf 0})=W\,\mathbf{g}_{-}({\bf
0})\quad\mbox{and}\quad\mathbf{f}_{+}({\bf 0})=\mathbf{f}_{-}({\bf 0}),$ (4.4)

where
$W=\begin{bmatrix}-\cos(\theta_{M}-\theta_{m}),&-\sin(\theta_{M}-\theta_{m})\\\
-\sin(\theta_{M}-\theta_{m}),&+\cos(\theta_{M}-\theta_{m})\end{bmatrix}.$

###### Proof.

Thanks to the symmetric role of $(\Re\mathbf{v},\Re\mathbf{w})$ and
$(\Im\mathbf{v},\Im\mathbf{w})$. We just have to prove that the corresponding
results hold for $(\Re\mathbf{v},\Re\mathbf{w})$. By a similar proof process,
it can be proved that those results are still valid for
$(\Im\mathbf{v},\Im\mathbf{w})$, hence for $(\mathbf{v},\mathbf{w})$. Due to
Betti’s second formula, we have the following integral identity

$\displaystyle\int_{\Gamma^{+}_{h}}\Re\mathbf{g}_{+}\cdot\mathbf{u}_{0}-\mathcal{T}_{\bm{\nu}}\mathbf{u}_{0}\cdot\Re\mathbf{f}_{+}\,\mathrm{d}\sigma+\int_{\Gamma^{-}_{h}}\Re\mathbf{g}_{-}\cdot\mathbf{u}_{0}-\mathcal{T}_{\bm{\nu}}\mathbf{u}_{0}\cdot\Re\mathbf{f}_{-}\,\mathrm{d}\sigma$
$\displaystyle=\int_{\Lambda_{h}}\mathcal{T}_{\bm{\nu}}\big{(}\Re\mathbf{v}-\Re\mathbf{w}\big{)}\cdot\mathbf{u}_{0}-\mathcal{T}_{\bm{\nu}}\mathbf{u}_{0}\cdot\big{(}\Re\mathbf{v}-\Re\mathbf{w}\big{)}\,\mathrm{d}\sigma.$
(4.5)

Since $\mathbf{f}_{j}\in C^{1,\alpha_{j}}(\Gamma_{h}^{j})^{2}$ and
$\mathbf{g}_{j}\in C^{\beta_{j}}(\Gamma_{h}^{j})^{2}$ for $j=+,-$, we have the
expansions as follows

$\displaystyle\mathbf{f}_{j}(\mathbf{x})$ $\displaystyle=\mathbf{f}_{j}({\bf
0})+\delta\mathbf{f}_{j}(\mathbf{x}),\,\,\quad\big{|}\delta\mathbf{f}_{j}(\mathbf{x})\big{|}\leq
A_{j}|\mathbf{x}|^{1+\alpha_{j}},$ (4.6)
$\displaystyle\mathbf{g}_{j}(\mathbf{x})$ $\displaystyle=\mathbf{g}_{j}({\bf
0})+\delta\mathbf{g}_{j}(\mathbf{x}),\quad\big{|}\delta\mathbf{g}_{j}(\mathbf{x})\big{|}\leq
B_{j}|\mathbf{x}|^{\beta_{j}},$ (4.7)

where $A_{j}$ and $B_{j}$ are positive. From the expression of
$\mathbf{u}_{0}$, it is easy to imply that

$\displaystyle\frac{\partial u^{0}_{1}}{\partial
r}=-\frac{s}{2\sqrt{r}}e^{-sr^{1/2}\mu(\theta)+\mathrm{i}\frac{\theta}{2}}\quad\mbox{and}\quad\frac{\partial
u^{0}_{1}}{\partial\theta}=-\frac{\mathrm{i}s\sqrt{r}}{2}e^{-sr^{1/2}\mu(\theta)+\mathrm{i}\frac{\theta}{2}},$

where $\mu(\cdot)$ is given by Lemma 4.9. Thus, we directly obtain

$\displaystyle\frac{\partial u^{0}_{1}}{\partial
x_{1}}=-\frac{s}{2\sqrt{r}}e^{-sr^{1/2}\mu(\theta)-\mathrm{i}\frac{\theta}{2}}\quad\mbox{and}\quad\frac{\partial
u^{0}_{1}}{\partial
x_{2}}=-\frac{\mathrm{i}s\sqrt{r}}{2}e^{-sr^{1/2}\mu(\theta)-\mathrm{i}\frac{\theta}{2}}.$

Notice that $u^{0}_{2}(\mathbf{x})=\mathrm{i}u^{0}_{1}(\mathbf{x})$, we get

$\nabla\mathbf{u}_{0}=-\frac{s}{2\sqrt{r}}e^{-s\sqrt{r}\mu(\theta)-\frac{\theta}{2}\mathrm{i}}\begin{bmatrix}1&\mathrm{i}\\\
\mathrm{i}&-1\end{bmatrix},$

Therefore, one can prove that

$\displaystyle\int_{\Gamma^{+}_{h}}\mathcal{T}_{\bm{\nu}_{M}}\mathbf{u}_{0}\,\mathrm{d}\sigma$
$\displaystyle=\int_{\Gamma_{h}^{+}}-\frac{s}{2r^{\frac{1}{2}}}e^{-s\sqrt{r}\mu(\theta_{M})-\mathrm{i}\frac{\theta_{M}}{2}}\begin{bmatrix}1&\mathrm{i}\\\
\mathrm{i}&-1\end{bmatrix}\cdot\begin{bmatrix}-\sin\theta_{M}\\\
\cos\theta_{M}\end{bmatrix}\mathrm{d}\sigma$ $\displaystyle=-\frac{\mu
s}{2}e^{\mathrm{i}\theta_{M}/2}\begin{bmatrix}\mathrm{i}\\\
-1\end{bmatrix}\int_{0}^{h}r^{-\frac{1}{2}}e^{-s\sqrt{r}\mu(\theta_{M})}\mathrm{d}r$
$\displaystyle=-\mu se^{\mathrm{i}\theta_{M}/2}\begin{bmatrix}\mathrm{i}\\\
-1\end{bmatrix}\int_{0}^{\sqrt{h}}e^{-s\,t\,\mu(\theta_{M})}\mathrm{d}t$
$\displaystyle=\mu\big{(}e^{-s\,\sqrt{h}\,\mu(\theta_{M})}-1\big{)}\begin{bmatrix}\mathrm{i}\\\
-1\end{bmatrix}.$ (4.8)

By the similar arguments, we can derive

$\displaystyle\int_{\Gamma^{-}_{h}}\mathcal{T}_{\bm{\nu}_{m}}\mathbf{u}_{0}\,\mathrm{d}\sigma$
$\displaystyle=\mu\big{(}e^{-s\sqrt{h}\mu(\theta_{m})}-1\big{)}\begin{bmatrix}-\mathrm{i}\\\
1\end{bmatrix}.$ (4.9)

From Lemma 4.9, we obtain

$\displaystyle\int_{\Gamma^{+}_{h}}\mathbf{u}_{0}\,{\rm
d}\sigma=2s^{-2}\left(\mu^{-2}(\theta_{M})-\mu^{-2}(\theta_{M})e^{-s\sqrt{h}\mu(\theta_{M})}-\mu^{-1}(\theta_{M})s\sqrt{h}e^{-s\sqrt{h}\mu(\theta_{M})}\right)\begin{bmatrix}1\\\
\mathrm{i}\end{bmatrix},$ (4.10)
$\displaystyle\int_{\Gamma^{-}_{h}}\mathbf{u}_{0}\,{\rm
d}\sigma=2s^{-2}\left(\mu^{-2}(\theta_{m})-\mu^{-2}(\theta_{m})e^{-s\sqrt{h}\mu(\theta_{m})}-\mu^{-1}(\theta_{m})s\sqrt{h}e^{-s\sqrt{h}\mu(\theta_{m})}\right)\begin{bmatrix}1\\\
\mathrm{i}\end{bmatrix},$ (4.11)

Substituting (4.1)–(4.11) into (4.5), the following integral identity holds,

$\displaystyle\Re\mathbf{f}_{+}({\bf
0})\cdot\mu\Big{(}e^{-s\sqrt{h}\mu(\theta_{M})}-1\Big{)}\begin{bmatrix}\mathrm{i}\\\
-1\end{bmatrix}+\Re\mathbf{f}_{-}({\bf
0})\cdot\mu\Big{(}e^{-s\sqrt{h}\mu(\theta_{m})}-1\Big{)}\begin{bmatrix}-\mathrm{i}\\\
+1\end{bmatrix}$ $\displaystyle-\Re\mathbf{g}_{+}({\bf
0})\cdot\begin{bmatrix}1\\\
\mathrm{i}\end{bmatrix}2s^{-2}\mu^{-2}(\theta_{M})-\Re\mathbf{g}_{-}({\bf
0})\cdot\begin{bmatrix}1\\\
\mathrm{i}\end{bmatrix}2s^{-2}\mu^{-2}(\theta_{m})=\sum\limits_{j=1}^{7}R_{j},$
(4.12)

where

$\displaystyle R_{1}$ $\displaystyle=-2s^{-2}\Re\mathbf{g}_{+}({\bf
0})\cdot\begin{bmatrix}1\\\
\mathrm{i}\end{bmatrix}\Big{(}\mu^{-1}(\theta_{M})\,s\sqrt{h}\,e^{-s\sqrt{h}\,\mu(\theta_{M})}+\mu^{-2}(\theta_{M})\,e^{-s\sqrt{h}\,\mu(\theta_{M})}\Big{)},$
$\displaystyle R_{2}$ $\displaystyle=-2s^{-2}\Re\mathbf{g}_{-}({\bf
0})\cdot\begin{bmatrix}1\\\
\mathrm{i}\end{bmatrix}\big{(}\mu^{-1}(\theta_{m})\,s\sqrt{h}\,e^{-s\sqrt{h}\,\mu(\theta_{m})}+\mu^{-2}(\theta_{m})\,e^{-s\sqrt{h}\,\mu(\theta_{m})}\big{)},$
$\displaystyle R_{3}$
$\displaystyle=\int_{\Lambda_{h}}\mathcal{T}_{\bm{\nu}}\big{(}\Re\mathbf{v}-\Re\mathbf{w}\big{)}\cdot\mathbf{u}_{0}-\mathcal{T}_{\bm{\nu}}\mathbf{u}_{0}\cdot\big{(}\Re\mathbf{v}-\Re\mathbf{w}\big{)}\,\mathrm{d}\sigma,$
$\displaystyle R_{4}$
$\displaystyle=-\int_{\Gamma^{+}_{h}}\Re\delta\mathbf{f}_{+}\cdot\mathcal{T}_{\bm{\nu}_{M}}\mathbf{u}_{0}\,\mathrm{d}\sigma,\quad\,\,R_{5}=-\int_{\Gamma^{-}_{h}}\Re\delta\mathbf{f}_{-}\cdot\mathcal{T}_{\bm{\nu}_{m}}\mathbf{u}_{0}\,\mathrm{d}\sigma,$
$\displaystyle R_{6}$
$\displaystyle=\int_{\Gamma^{+}_{h}}\Re\delta\mathbf{g}_{+}\cdot\mathbf{u}_{0}\,\mathrm{d}\sigma,\qquad\qquad
R_{7}=\int_{\Gamma^{-}_{h}}\Re\delta\mathbf{g}_{-}\cdot\mathbf{u}_{0}\,\mathrm{d}\sigma.$

From the expression of $\mu(\cdot)$ given by Lemma 4.9, it is direct to obtain
that $\mu^{-2}(\theta_{M})$, $\mu^{-1}(\theta_{M})$, $\mu^{-2}(\theta_{m})$
and $\mu^{-1}(\theta_{m})$ are bounded. For sufficient large $s$, we have

$\displaystyle\big{|}R_{1}\big{|}={\mathcal{O}}(s^{-1}e^{-c_{1}s})\quad\mbox{and}\quad\big{|}R_{2}\big{|}$
$\displaystyle={\mathcal{O}}(s^{-1}e^{-c_{2}s}),$ (4.13)

where $c_{1}$ and $c_{2}$ are positive constants not depending on $s$.
Considering the estimates of $\delta\mathbf{f}_{+}$ in (4.6), the expression
of $\mathcal{T}_{\bm{\nu}_{M}}\mathbf{u}_{0}$ in (4.1) and Lemma 4.9, we get

$\displaystyle\big{|}R_{4}\big{|}\leq\,c_{4}\,\int_{0}^{h}r^{\alpha_{+}+1}e^{-s\sqrt{r}\cos\frac{\theta_{M}}{2}}\,\mathrm{d}r={\mathcal{O}}(s^{-2\alpha_{+}-2}).$
(4.14)

Similarly, we get the following estimates

$\displaystyle\big{|}R_{5}\big{|}={\mathcal{O}}(s^{-2\alpha_{-}-2}),\quad\big{|}R_{6}\big{|}={\mathcal{O}}(\tau^{-2\beta_{+}-2}),\quad\big{|}R_{7}\big{|}={\mathcal{O}}(\tau^{-2\beta_{-}-2}).$
(4.15)

For the term $R_{3}$, by virtue of Cauchy-Schwarz inequality, trace theorem
and Lemma 4.3, we obtain

$\displaystyle\big{|}R_{3}\big{|}$
$\displaystyle\leq\|\mathbf{u}_{0}\|_{L^{2}(\Lambda_{h})^{2}}\|\mathcal{T}_{\bm{\nu}}(\Re\mathbf{v}-\Re\mathbf{w})\|_{L^{2}(\Lambda_{h})^{2}}+\|\Re\mathbf{v}-\Re\mathbf{w}\|_{L^{2}(\Lambda_{h})^{2}}\|\mathcal{T}_{\bm{\nu}}\mathbf{u}_{0}\|_{L^{2}(\Lambda_{h})^{2}}$
$\displaystyle\leq\Big{(}\|\mathbf{u}_{0}\|_{L^{2}(\Lambda_{h})^{2}}+\|\mathcal{T}_{\bm{\nu}}\mathbf{u}_{0}\|_{L^{2}(\Lambda_{h})^{2}}\Big{)}\big{\|}\mathbf{v}-\mathbf{w}\big{\|}_{H^{1}(\mathcal{C}_{h})^{2}}$
$\displaystyle\leq
c_{3}\big{\|}\mathbf{u}_{0}\big{\|}_{H^{1}(\mathcal{C}_{h})^{2}}\leq
c_{3}e^{-s\sqrt{h}\delta_{\mathcal{K}}},$ (4.16)

where $c_{3}$ is positive and $\delta_{\mathcal{K}}$ is defined in (4.3).
Together (4.13)–(4.16) with the identity (4.1), as $s\rightarrow+\infty$, one
clearly implies that

$\begin{bmatrix}+\mathrm{i}\\\
-1\end{bmatrix}\cdot\Big{(}\Re\mathbf{f}_{+}({\bf 0})-\Re\mathbf{f}_{-}({\bf
0})\Big{)}=0.$ (4.17)

Since $\Re\mathbf{f}_{+}({\bf 0})$ and $\Re\mathbf{f}_{-}({\bf 0})$ belong to
$\mathbb{R}$, we derive that

$\Re\mathbf{f}_{+}({\bf 0})=\Re\mathbf{f}_{-}({\bf 0}).$ (4.18)

Substituting (4.18) into (4.1), and then multiplying the new equation by
$s^{2}$, one gets that

$\displaystyle\mu\,s^{2}\Re\mathbf{f}_{+}({\bf
0})\cdot\begin{bmatrix}\mathrm{i}\\\
-1\end{bmatrix}\bigg{(}e^{-s\sqrt{h}\mu(\theta_{M})}-e^{-s\sqrt{h}\mu(\theta_{m})}\bigg{)}-2\begin{bmatrix}1\\\
\mathrm{i}\end{bmatrix}\cdot\bigg{(}\frac{\Re\mathbf{g}_{+}({\bf
0})}{\mu^{2}(\theta_{M})}+\frac{\Re\mathbf{g}_{-}({\bf
0})}{\mu^{2}(\theta_{m})}\bigg{)}=\sum\limits_{j=1}^{7}s^{2}\,R_{j},$

Let $s$ tend to $+\infty$, we have

$\displaystyle\begin{bmatrix}1\\\
\mathrm{i}\end{bmatrix}\cdot\bigg{(}\Re\mathbf{g}_{+}({\bf
0})\mu^{-2}(\theta_{M})+\Re\mathbf{g}_{-}({\bf
0})\mu^{-2}(\theta_{m})\bigg{)}=0.$

Noting that $\mu^{-2}(\theta_{M})\neq\mu^{-2}(\theta_{m})$ and
$\frac{\mu^{2}(\theta_{M})}{\mu^{2}(\theta_{m})}=e^{\mathrm{i}(\theta_{M}-\theta_{m})}$,
then let $\Re\mathbf{g}_{+}({\bf 0}):=\begin{bmatrix}a_{11}\\\
a_{21}\end{bmatrix}$ and $\Re\mathbf{g}_{-}({\bf 0}):=\begin{bmatrix}a_{12}\\\
a_{22}\end{bmatrix}$, where $a_{ij}\in\mathbb{R}$ for $i,j=1,2$, the above
equation can be rewritten as follows

$\begin{cases}a_{11}+a_{12}\cos(\theta_{M}-\theta_{m})-a_{22}\sin(\theta_{M}-\theta_{m})=0,\\\
a_{21}+a_{12}\sin(\theta_{M}-\theta_{m})+a_{22}\cos(\theta_{M}-\theta_{m})=0,\end{cases}$

which is

$\mathbf{g}_{+}({\bf 0})=W\mathbf{g}_{-}({\bf 0}).$

Here,

$W=\begin{bmatrix}-\cos(\theta_{M}-\theta_{m}),&-\sin(\theta_{M}-\theta_{m})\\\
-\sin(\theta_{M}-\theta_{m}),&\cos(\theta_{M}-\theta_{m})\end{bmatrix}\quad\mbox{and}\quad\det(W)\neq
0.$

The proof is complete. ∎

###  4.2. Local results for 3D case

As discussed in Remark 4.2 in [10], the regularity result on the underlying
elastic displacement around a general polyhedral corner in $\mathbb{R}^{3}$ is
challenging. So in this subsection, we shall restrict ourselves to the 3D edge
corner. We introduce a dimension reduction operator $\mathcal{P}$ to study the
relevant continuities about slip vectors $\mathbf{f}$ and $\mathbf{g}$ at a 3D
edge corner. In what follows, we suppose that the dislocation $\mathcal{S}$ is
a Lipschitz surface possessing at least one 3D edge corner
$\mathbf{x}_{c}=(\mathbf{x}^{\prime}_{c},x^{3}_{c})^{\top}\in\mathcal{S}\subset\mathbb{R}^{3}$.

The next definition shall state a dimension reduction operator, which is
beneficial to derive a crucial auxiliary proposition similar to Proposition
4.1 at a 3D edge corner.

######  Definition 4.1.

Let $\mathcal{C}^{\prime}_{\mathbf{x}^{\prime}_{c},h}$ be defined as
$\mathcal{C}_{\mathbf{x}_{c},h}$ in (3.3) with the vertex
$\mathbf{x}^{\prime}_{c}$. For a given function $\phi$ in the domain
$\mathcal{C}^{\prime}_{\mathbf{x}^{\prime}_{c},h}\times(-M,M)$ with $M>0$. For
any fixed $x^{3}_{c}\in(-M,M)$, we assume that $\phi\in
C^{\infty}_{0}(x^{3}_{c}-L,x^{3}_{c}+L)$ is a nonnegative function
$\phi\not\equiv\emptyset$. Write
$\mathbf{x}=(\mathbf{x}^{\prime},x_{3})^{\top}\in\mathbb{R}^{3}$. The
dimension reduction operator $\mathcal{P}$ is defined as follows

$\mathcal{P}(\mathbf{h})(\mathbf{x}^{\prime})=\int_{x^{3}_{c}-L}^{x_{c}^{3}+L}\phi(x_{3})\mathbf{h}(\mathbf{x}^{\prime},x_{3})\mathrm{d}x_{3},$

where $\mathcal{C}^{\prime}_{\mathbf{x}^{\prime}_{c},h}$ is defined in (3.3)
with the vertex $\mathbf{x}_{c}^{\prime}$.

Before deriving the main results of this subsection, we review some important
properties of such operator.

######  Lemma 4.5.

[10, Lemma 3.1] Let $\mathbf{h}\in
H^{m}(\mathcal{C}^{\prime}_{\mathbf{x}^{\prime}_{c},h}\times(-M,M))^{3}$,
$m=1,2$. Then

$\mathcal{P}(\mathbf{h})(\mathbf{x}^{\prime})\in
H^{m}(\mathcal{C}^{\prime}_{\mathbf{x}^{\prime}_{c},h}))^{3}.$

Similarly, if $\mathbf{h}\in
C^{\delta}\big{(}\overline{\mathcal{C}^{\prime}_{\mathbf{x}^{\prime}_{c},h}}\times[-M,M]\big{)}^{3}$
with $\delta\in(0,1)$, then

$\mathcal{P}(\mathbf{h})(\mathbf{x}^{\prime})\in
C^{\delta}\big{(}\overline{\mathcal{C}^{\prime}_{\mathbf{x}^{\prime}_{c},h}}\big{)}^{3}.$

Noting that the three-dimensional isotropic elastic operator $\mathcal{L}$
defined in (2.1)–(2.3) can be rewritten as

$\displaystyle\mathcal{L}$
$\displaystyle=\begin{bmatrix}\lambda\Delta+(\lambda+\mu)\partial_{1}^{2}&(\lambda+\mu)\partial_{1}\partial_{2}&(\lambda+\mu)\partial_{1}\partial_{3}\\\
(\lambda+\mu)\partial_{1}\partial_{2}&\lambda\Delta+(\lambda+\mu)\partial_{2}^{2}&(\lambda+\mu)\partial_{2}\partial_{3}\\\
(\lambda+\mu)\partial_{1}\partial_{3}&(\lambda+\mu)\partial_{2}\partial_{3}&\lambda\Delta+(\lambda+\mu)\partial_{3}^{2}\end{bmatrix}$
$\displaystyle=\widetilde{\mathcal{L}}+\begin{bmatrix}\lambda\partial^{2}_{3}&0&(\lambda+\mu)\partial_{1}\partial_{3}\\\
0&\lambda\partial_{3}^{2}&(\lambda+\mu)\partial_{2}\partial_{3}\\\
(\lambda+\mu)\partial_{1}\partial_{3}&(\lambda+\mu)\partial_{2}\partial_{3}&\lambda\partial_{3}^{2}+(\lambda+\mu)\partial_{3}^{2}\end{bmatrix},$

where

$\widetilde{\mathcal{L}}=\begin{bmatrix}\lambda\Delta^{\prime}+(\lambda+\mu)\partial_{1}^{2}&(\lambda+\mu)\partial_{1}\partial_{2}&0\\\
(\lambda+\mu)\partial_{1}\partial_{2}&\lambda\Delta^{\prime}+(\lambda+\mu)\partial_{2}^{2}&0\\\
0&0&\lambda\Delta^{\prime}\end{bmatrix}=\begin{bmatrix}\mathcal{L}_{\mathcal{P}}&0\\\
0&\lambda\Delta^{\prime}\end{bmatrix}$ (4.19)

with $\Delta^{\prime}=\partial_{1}^{2}+\partial_{2}^{2}$ being the Laplace
operator with respect to the $\mathbf{x}^{\prime}$-variables. Here, the
operator $\mathcal{L}_{\mathcal{P}}$ is the two-dimensional isotropic elastic
operator with respect to the $\mathbf{x}^{\prime}$-variables.

For any fixed $x_{c}^{3}\in(-M,M)$ and sufficient small $L>0$ such that
$(x_{c}^{3}-L,x_{c}^{3}+L)\subset(-M,M)$. At this moment, we have
$\mathcal{L}=\widetilde{\mathcal{L}}$. Since $\mathcal{L}$ is invariant under
rigid motion, in what follows we assume that
$\mathbf{x}^{\prime}_{c}=\mathbf{0}$ in $\mathbb{R}^{2}.$ Hence let
$\mathcal{C}^{\prime}_{\mathbf{x}^{\prime}_{c},h}$ and
$\Gamma^{\pm}_{\mathbf{x}^{\prime}_{c},h}$ be defined in (3.3) with
$\mathbf{x}^{\prime}_{c}$ coinciding with the origin in 2D case. By some
tedious calculations, we have the following lemma.

######  Lemma 4.6.

Under the setup about $\mathcal{C}^{\prime}_{\mathbf{x}^{\prime}_{c},h}$ and a
3D edge corner as above, suppose that $\mathbf{f}_{j}\in
C^{1,\alpha_{j}}\big{(}\Gamma^{\prime
j}_{\mathbf{x}^{\prime}_{c},h}\times(-M,M)\big{)}^{3}\bigcap
H^{\frac{1}{2}}\big{(}\Gamma^{\prime
j}_{\mathbf{x}^{\prime}_{c},h}\times(-M,M)\big{)}^{3}$ and $\mathbf{g}_{j}\in
C^{\beta_{j}}\big{(}\Gamma^{\prime
j}_{\mathbf{x}^{\prime}_{c},h}\times(-M,M)\big{)}^{3}\bigcap
H^{-\frac{1}{2}}\big{(}\Gamma^{\prime
j}_{\mathbf{x}^{\prime}_{c},h}\times(-M,M)\big{)}^{3}$ do not depend on
$x_{3}$ for $j=+,-$, where $\alpha_{+}$, $\alpha_{-}$, $\beta_{+}$,
$\beta_{-}\in(0,1)$. Denote
$\mathbf{v}=(\mathbf{v}^{(1,2)},\,\,v_{3})^{\top}$,
$\mathbf{w}=(\mathbf{w}^{(1,2)},\,\,w_{3})^{\top}$,
$\mathbf{f}_{\pm}=(\mathbf{f}^{(1,2)}_{\pm},\,\,f^{3}_{\pm})^{\top}$ and
$\mathbf{g}_{\pm}=(\mathbf{g}^{(1,2)}_{\pm},\,\,g^{3}_{\pm})^{\top}$. Then the
transmission eigenvalue problem for $(\mathbf{v},\mathbf{w})\in
H^{1}\big{(}\mathcal{C}^{\prime}_{\mathbf{x}^{\prime}_{c},h}\times(-M,M)\big{)}^{3}\times
H^{1}\big{(}\mathcal{C}^{\prime}_{\mathbf{x}^{\prime}_{c},h}\times(-M,M)\big{)}^{3}:$

$\begin{cases}\mathcal{L}\,\mathbf{v}=\mathbf{0},\hskip
15.0pt\mathcal{L}\,\mathbf{w}=\mathbf{0}&\mbox{in}\quad\mathcal{C}^{\prime}_{\mathbf{x}^{\prime}_{c},h}\times(-M,M),\\\\[5.69054pt]
\mathbf{v}-\mathbf{w}=\mathbf{f}_{+},\,\mathcal{T}_{\bm{\nu}}\,\mathbf{v}-\mathcal{T}_{\bm{\nu}}\,\mathbf{w}=\mathbf{g}_{+}&\mbox{on}\quad\Gamma^{\prime+}_{\mathbf{x}^{\prime}_{c},h}\times(-M,M),\\\\[5.69054pt]
\mathbf{v}-\mathbf{w}=\mathbf{f}_{-},\,\mathcal{T}_{\bm{\nu}}\,\mathbf{v}-\mathcal{T}_{\bm{\nu}}\,\mathbf{w}=\mathbf{g}_{-}&\mbox{on}\quad\Gamma^{\prime-}_{\mathbf{x}^{\prime}_{c},h}\times(-M,M)\end{cases}$

can be reduced to be

$\begin{cases}\widetilde{\mathcal{L}}\,\mathcal{P}(\mathbf{v})(\mathbf{x}^{\prime})=\mathcal{G}_{1}(\mathbf{x}^{\prime}),\hskip
8.0pt\widetilde{\mathcal{L}}\,\mathcal{P}(\mathbf{w})(\mathbf{x}^{\prime})=\mathcal{G}_{2}(\mathbf{x}^{\prime}),&\mathbf{x}^{\prime}\in\mathcal{C}^{\prime}_{\mathbf{x}^{\prime}_{c},h},\\\\[5.69054pt]
\mathcal{P}(\mathbf{v})(\mathbf{x}^{\prime})=\mathcal{P}(\mathbf{w})(\mathbf{x}^{\prime})+\mathcal{P}(\mathbf{f}_{+})(\mathbf{x}^{\prime}),&\mathbf{x}^{\prime}\in\Gamma^{\prime+}_{\mathbf{x}^{\prime}_{c},h},\\\\[5.69054pt]
\mathcal{P}(\mathbf{v})(\mathbf{x}^{\prime})=\mathcal{P}(\mathbf{w})(\mathbf{x}^{\prime})+\mathcal{P}(\mathbf{f}_{-})(\mathbf{x}^{\prime}),&\mathbf{x}^{\prime}\in\Gamma^{\prime-}_{\mathbf{x}^{\prime}_{c},h},\\\\[5.69054pt]
\mathcal{R}^{+}_{1}=\mathcal{R}^{+}_{2}+\mathcal{P}(\mathbf{g}_{+})(\mathbf{x}^{\prime}),&\mathbf{x}^{\prime}\in\Gamma^{\prime+}_{\mathbf{x}^{\prime}_{c},h},\\\\[5.69054pt]
\mathcal{R}^{-}_{1}=\mathcal{R}^{-}_{2}+\mathcal{P}(\mathbf{g}_{-})(\mathbf{x}^{\prime}),&\mathbf{x}^{\prime}\in\Gamma^{\prime-}_{\mathbf{x}^{\prime}_{c},h},\end{cases}$
(4.20)

where

$\displaystyle\mathcal{G}_{1}=-\int^{x_{c}^{3}+L}_{x_{c}^{3}-L}\phi^{\prime\prime}(x_{3})\begin{bmatrix}\lambda\mathbf{v}^{(1,2)}(\mathbf{x})\\\
(2\lambda+\mu)v_{3}(\mathbf{x})\end{bmatrix}\mathrm{d}x_{3}+(\lambda+\mu)\int_{x_{c}^{3}-L}^{x_{c}^{3}+L}\phi^{\prime}(x_{3})\begin{bmatrix}\nabla
v_{3}(\mathbf{x})\\\
\partial_{1}v_{1}(\mathbf{x})+\partial_{2}v_{2}(\mathbf{x})\end{bmatrix}\mathrm{d}x_{3},$
$\displaystyle\mathcal{G}_{2}=-\int^{x_{c}^{3}+L}_{x_{c}^{3}-L}\phi^{\prime\prime}(x_{3})\begin{bmatrix}\lambda\mathbf{w}^{(1,2)}(\mathbf{x})\\\
(2\lambda+\mu)w_{3}(\mathbf{x})\end{bmatrix}\mathrm{d}x_{3}+(\lambda+\mu)\int_{x_{c}^{3}-L}^{x_{c}^{3}+L}\phi^{\prime}(x_{3})\begin{bmatrix}\nabla
w_{3}(\mathbf{x})\\\
\partial_{1}w_{1}(\mathbf{x})+\partial_{2}w_{2}(\mathbf{x})\end{bmatrix}\mathrm{d}x_{3},$
$\displaystyle\mathcal{R}^{\mathrm{+}}_{1}=\begin{bmatrix}\mathcal{T}_{\bm{\nu}_{M}}\mathcal{P}(\mathbf{v}^{(1,2)})+\lambda\mathcal{P}(\partial_{3}v_{3})\bm{\nu}_{M}\\\\[2.84526pt]
\mu\partial_{\bm{\nu}_{M}}\mathcal{P}(v_{3})+\mu\begin{bmatrix}\mathcal{P}(\partial_{3}v_{1})\\\
\mathcal{P}(\partial_{3}v_{2})\end{bmatrix}\cdot\bm{\nu}_{M}\end{bmatrix},\,\mathcal{R}^{\mathrm{+}}_{2}=\begin{bmatrix}\mathcal{T}_{\bm{\nu}_{M}}\mathcal{P}(\mathbf{w}^{(1,2)})+\lambda\mathcal{P}(\partial_{3}w_{3})\bm{\nu}_{M}\\\\[2.84526pt]
\mu\partial_{\bm{\nu}_{M}}\mathcal{P}(w_{3})+\mu\begin{bmatrix}\mathcal{P}(\partial_{3}w_{1})\\\
\mathcal{P}(\partial_{3}w_{2})\end{bmatrix}\cdot\bm{\nu}_{M}\end{bmatrix}\,\mbox{on}\,\Gamma^{\prime\mathrm{+}}_{\mathbf{x}^{\prime}_{c},h},$
$\displaystyle\mathcal{R}^{\mathrm{-}}_{1}=\begin{bmatrix}\mathcal{T}_{\bm{\nu}_{m}}\mathcal{P}(\mathbf{v}^{(1,2)})+\lambda\mathcal{P}(\partial_{3}v_{3})\bm{\nu}_{m}\\\\[2.84526pt]
\mu\partial_{\bm{\nu}_{m}}\mathcal{P}(v_{3})+\mu\begin{bmatrix}\mathcal{P}(\partial_{3}v_{1})\\\
\mathcal{P}(\partial_{3}v_{2})\end{bmatrix}\cdot\bm{\nu}_{m}\end{bmatrix},\,\mathcal{R}^{\mathrm{-}}_{2}=\begin{bmatrix}\mathcal{T}_{\bm{\nu}_{m}}\mathcal{P}(\mathbf{w}^{(1,2)})+\lambda\mathcal{P}(\partial_{3}w_{3})\bm{\nu}_{m}\\\\[2.84526pt]
\mu\partial_{\bm{\nu}_{m}}\mathcal{P}(w_{3})+\mu\begin{bmatrix}\mathcal{P}(\partial_{3}w_{1})\\\
\mathcal{P}(\partial_{3}w_{2})\end{bmatrix}\cdot\bm{\nu}_{m}\end{bmatrix}\,\mbox{on}\,\Gamma^{\prime\mathrm{-}}_{\mathbf{x}^{\prime}_{c},h}.$

Here, $\bm{\nu}_{M}$ and $\bm{\nu}_{m}$ denote the exterior unit normal vector
to $\Gamma^{\prime+}_{\mathbf{x}^{\prime}_{c},h}$ and
$\Gamma^{\prime+}_{\mathbf{x}^{\prime}_{c},h}$, respectively,
$\mathcal{T}_{\bm{\nu}}$ is the two-dimensional boundary traction operator.

By applying the decomposition of $\widetilde{\mathcal{L}}$ given in (4.19), it
is direct to obtain the following results. Here, we omit the proof.

######  Lemma 4.7.

Under the same setup in Lemma 4.6, the transmission system (4.20) is
equivalent to the following two PDE systems

$\begin{cases}\mathcal{L}_{\mathcal{P}}\,\mathcal{P}(\mathbf{v}^{(1,2)})(\mathbf{x}^{\prime})=\mathcal{G}^{(1,2)}_{1}(\mathbf{x}^{\prime})\quad&\mbox{in}\quad\mathcal{C}^{\prime}_{\mathbf{x}^{\prime}_{c},h},\\\\[5.69054pt]
\mathcal{L}_{\mathcal{P}}\,\mathcal{P}(\mathbf{w}^{(1,2)})(\mathbf{x}^{\prime})=\mathcal{G}^{(1,2)}_{2}(\mathbf{x}^{\prime})\quad&\mbox{in}\quad\mathcal{C}^{\prime}_{\mathbf{x}^{\prime}_{c},h},\\\\[5.69054pt]
\mathcal{P}(\mathbf{v}^{(1,2)})(\mathbf{x}^{\prime})=\mathcal{P}(\mathbf{w}^{(1,2)})(\mathbf{x}^{\prime})+\mathcal{P}(\mathbf{f}_{+}^{(1,2)})(\mathbf{x}^{\prime})\quad&\mbox{on}\quad\Gamma^{\prime+}_{\mathbf{x}^{\prime}_{c},h},\\\\[5.69054pt]
\mathcal{P}(\mathbf{v}^{(1,2)})(\mathbf{x}^{\prime})=\mathcal{P}(\mathbf{w}^{(1,2)})(\mathbf{x}^{\prime})+\mathcal{P}(\mathbf{f}_{-}^{(1,2)})(\mathbf{x}^{\prime})\quad&\mbox{on}\quad\Gamma^{\prime-}_{\mathbf{x}^{\prime}_{c},h},\\\\[5.69054pt]
\mathcal{R}^{(1,2)}_{1}=\mathcal{R}^{(1,2)}_{2}+\mathcal{P}(\mathbf{g}_{+}^{(1,2)})(\mathbf{x}^{\prime})\quad&\mbox{on}\quad\Gamma^{\prime+}_{\mathbf{x}^{\prime}_{c},h},\\\\[5.69054pt]
\mathcal{R}^{(1,2)}_{1}=\mathcal{R}^{(1,2)}_{2}+\mathcal{P}(\mathbf{g}_{-}^{(1,2)})(\mathbf{x}^{\prime})\quad&\mbox{on}\quad\Gamma^{\prime-}_{\mathbf{x}^{\prime}_{c},h}\end{cases}$
(4.21)

and

$\begin{cases}\lambda\Delta^{\prime}\,\mathcal{P}(v_{3})(\mathbf{x}^{\prime})=\mathcal{G}^{(3)}_{1}(\mathbf{x}^{\prime})&\mbox{in}\quad\mathcal{C}^{\prime}_{\mathbf{x}^{\prime}_{c},h},\\\\[5.69054pt]
\lambda\Delta^{\prime}\,\mathcal{P}(w_{3})(\mathbf{x}^{\prime})=\mathcal{G}^{(3)}_{2}(\mathbf{x}^{\prime})&\mbox{in}\quad\mathcal{C}^{\prime}_{\mathbf{x}^{\prime}_{c},h},\\\\[5.69054pt]
\mathcal{P}(v_{3})(\mathbf{x}^{\prime})=\mathcal{P}(w_{3})(\mathbf{x}^{\prime})+\mathcal{P}(f^{3}_{+})(\mathbf{x}^{\prime})&\mbox{on}\quad\Gamma^{\prime+}_{\mathbf{x}^{\prime}_{c},h},\\\\[5.69054pt]
\mathcal{P}(v_{3})(\mathbf{x}^{\prime})=\mathcal{P}(w_{3})(\mathbf{x}^{\prime})+\mathcal{P}(f^{3}_{-})(\mathbf{x}^{\prime})&\mbox{on}\quad\Gamma^{\prime-}_{\mathbf{x}^{\prime}_{c},h},\\\\[5.69054pt]
\partial_{\bm{\nu}_{M}}\mathcal{P}(v_{3})(\mathbf{x}^{\prime})=\partial_{\bm{\nu}_{M}}\mathcal{P}(w_{3})(\mathbf{x}^{\prime})+\frac{1}{\mu}\mathcal{P}(g^{3}_{+})(\mathbf{x}^{\prime})\qquad\quad&\mbox{on}\quad\Gamma^{\prime+}_{\mathbf{x}^{\prime}_{c},h},\\\\[5.69054pt]
\partial_{\bm{\nu}_{m}}\mathcal{P}(v_{3})(\mathbf{x}^{\prime})=\partial_{\bm{\nu}_{m}}\mathcal{P}(w_{3})(\mathbf{x}^{\prime})+\frac{1}{\mu}\mathcal{P}(g^{3}_{-})(\mathbf{x}^{\prime})\qquad\quad&\mbox{on}\quad\Gamma^{\prime-}_{\mathbf{x}^{\prime}_{c},h},\end{cases}$
(4.22)

where

$\displaystyle\mathcal{G}^{(1,2)}_{1}=-\lambda\int^{x_{c}^{3}+L}_{x_{c}^{3}-L}\phi^{\prime\prime}(x_{3})\mathbf{v}^{(1,2)}(\mathbf{x})\mathrm{d}x_{3}+(\lambda+\mu)\int_{x_{c}^{3}-L}^{x_{c}^{3}+L}\phi^{\prime}(x_{3})\nabla
v_{3}(\mathbf{x})\mathrm{d}x_{3},$
$\displaystyle\mathcal{G}^{(1,2)}_{2}=-\lambda\int^{x_{c}^{3}+L}_{x_{c}^{3}-L}\phi^{\prime\prime}(x_{3})\mathbf{w}^{(1,2)}(\mathbf{x})\mathrm{d}x_{3}+(\lambda+\mu)\int_{x_{c}^{3}-L}^{x_{c}^{3}+L}\phi^{\prime}(x_{3})\nabla
w_{3}(\mathbf{x})\mathrm{d}x_{3},$
$\displaystyle\mathcal{G}^{(3)}_{1}=-(2\lambda+\mu)\int^{x_{c}^{3}+L}_{x_{c}^{3}-L}\phi^{\prime\prime}(x_{3})v_{3}(\mathbf{x})\mathrm{d}x_{3}+(\lambda+\mu)\int_{x_{c}^{3}-L}^{x_{c}^{3}+L}\phi^{\prime}(x_{3})(\partial_{1}v_{1}+\partial_{2}v_{2})\mathrm{d}x_{3},$
$\displaystyle\mathcal{G}^{(3)}_{2}=-(2\lambda+\mu)\int^{x_{c}^{3}+L}_{x_{c}^{3}-L}\phi^{\prime\prime}(x_{3})w_{3}(\mathbf{x})\mathrm{d}x_{3}+(\lambda+\mu)\int_{x_{c}^{3}-L}^{x_{c}^{3}+L}\phi^{\prime}(x_{3})(\partial_{1}w_{1}+\partial_{2}w_{2})\mathrm{d}x_{3},$
$\displaystyle\mathcal{R}^{+(1,2)}_{1}=\mathcal{T}_{\bm{\nu}_{M}}\mathcal{P}(\mathbf{v}^{(1,2)})+\lambda\mathcal{P}(\partial_{3}v_{3})\,\bm{\nu}_{M},\,\,\mathcal{R}^{+(1,2)}_{2}=\mathcal{T}_{\bm{\nu}_{M}}\mathcal{P}(\mathbf{w}^{(1,2)})+\lambda\mathcal{P}(\partial_{3}w_{3})\hskip
10.0pt\mbox{on}\,\,\Gamma^{\prime+}_{\mathbf{x}^{\prime}_{c},h},$
$\displaystyle\mathcal{R}^{-(1,2)}_{1}=\mathcal{T}_{\bm{\nu}_{m}}\mathcal{P}(\mathbf{v}^{(1,2)})+\lambda\mathcal{P}(\partial_{3}v_{3})\,\bm{\nu}_{m},\,\,\,\mathcal{R}^{-(1,2)}_{2}=\mathcal{T}_{\bm{\nu}_{m}}\mathcal{P}(\mathbf{w}^{(1,2)})+\lambda\mathcal{P}(\partial_{3}w_{3})\hskip
12.0pt\mbox{on}\,\,\Gamma^{\prime-}_{\mathbf{x}^{\prime}_{c},h}.$

To obtain the continuity of $\mathcal{P}(f_{+}^{3})$ and
$\mathcal{P}(f_{-}^{3})$, $\mathcal{P}(g_{+}^{3})$ and
$\mathcal{P}(g_{-}^{3})$ at a 3D edge corner in the conventional or rotational
sense. We also use the CGO solution $u_{0}$ introduced in [4]. To be more
precise, such CGO solution possesses a similar form and properties like Lemma
4.1–Lemma 4.9 (cf. [9]). For the reader’s sake, we include these properties
here.

######  Lemma 4.8.

[4, Lemma 2.2] Let
$\mathbf{x}^{\prime}=(x_{1},x_{2})^{\top}=r(\cos\theta,\sin\theta)^{\top}\in\mathbb{R}^{2}$
and $s\in\mathbb{R}_{+}$,

$u_{0}(s\mathbf{x}^{\prime}):=\exp(-\sqrt{sr}\mu(\theta)),$ (4.23)

where $\mu(\cdot)$ is defined in Lemma 4.9. Then $s\longmapsto
u_{0}(s\mathbf{x}^{\prime})$ decays exponentially in $\mathbb{R}_{+}$ and

$\Delta^{\prime}u_{0}=0\quad\mbox{in}\quad\mathbb{R}^{2}\backslash\mathbb{R}^{2}_{0,-}$
(4.24)

where
$\mathbb{R}^{2}_{0,-}:=\\{\mathbf{x}^{\prime}\in\mathbb{R}^{2}|\mathbf{x}^{\prime}=(x_{1},x_{2});x_{1}\leq
0,x_{2}=0\\}$. Moreover,

$\int_{\mathcal{K_{\mathcal{P}}}}u_{0}(s\mathbf{x}^{\prime}){\rm
d}\mathbf{x}^{\prime}=6{\rm i}(e^{-2\theta_{M}{\rm
i}}-e^{-2\theta_{m}\mathrm{i}})s^{-2}$ (4.25)

and for $\alpha,\,s>0$ and $h>0$

$\displaystyle\int_{\mathcal{K}_{\mathcal{P}}}|u_{0}(s\mathbf{x}^{\prime})||\mathbf{x}^{\prime}|^{\alpha}{\rm
d}\mathbf{x}^{\prime}$
$\displaystyle\leq\frac{2(\theta_{M}-\theta_{m})\Gamma(2\alpha+4)}{\delta_{\mathcal{K}_{\mathcal{P}}}^{2\alpha+4}}s^{-\alpha-2},$
$\displaystyle\int_{\mathcal{K}_{\mathcal{P}}\backslash
B_{h}}|u_{0}(s\mathbf{x}^{\prime})|{\rm d}\mathbf{x}^{\prime}$
$\displaystyle\leq\frac{6(\theta_{M}-\theta_{m})}{\delta_{\mathcal{K}_{\mathcal{P}}}^{4}}s^{-2}e^{-\frac{\sqrt{sh}}{2}\delta_{\mathcal{K}_{\mathcal{P}}}},$
(4.26)

where $\mathcal{K}_{\mathcal{P}}$ is defined like $\mathcal{K}$ in Section 3
and
$\delta_{\mathcal{K}_{\mathcal{P}}}=\min\limits_{\theta_{m}<\theta<\theta_{M}}\cos\frac{\theta}{2}$
is a positive constant.

######  Lemma 4.9.

[9, Lemma 2.4] For any $\alpha>0$, if $\omega(\theta)>0$, then we have

$\lim\limits_{s\rightarrow+\infty}\int^{h}_{0}r^{\alpha}e^{-\sqrt{sr}\omega(\theta)}\mathrm{d}\mathrm{r}={\mathcal{O}}(s^{-\alpha-1}).$

######  Lemma 4.10.

[9, Lemma 2.3] Let $\mathcal{C}^{\prime}_{\mathbf{x}^{\prime}_{c},h}$ be
defined as in (3.3) and $u_{0}$ be given in (4.23). Then $u_{0}\in
H^{1}(\mathcal{C}^{\prime}_{\mathbf{x}^{\prime}_{c},h})^{2}$ and
$\Delta^{\prime}u_{0}=0$ in
$\mathcal{C}^{\prime}_{\mathbf{x}^{\prime}_{c},h}$. Furthermore, it holds that

$\big{\|}u_{0}\big{\|}_{L^{2}(\mathcal{C}^{\prime}_{\mathbf{x}^{\prime}_{c},h})^{2}}\leq\frac{\sqrt{\theta_{M}-\theta_{m}}e^{-2\sqrt{s\Theta}\delta_{\mathcal{K}_{\mathcal{P}}}}h^{2}}{2}$

and

$\Big{\|}|\mathbf{x}^{\prime}|^{\alpha}u_{0}\Big{\|}^{2}_{L^{2}(\mathcal{C}^{\prime}_{\mathbf{x}^{\prime}_{c},h})^{2}}\leq
s^{-2(\alpha+1)}\frac{2(\theta_{M}-\theta_{m})\Gamma(4\alpha+4)}{(4\delta_{\mathcal{K}_{\mathcal{P}}})^{2\alpha+2}},$

where $\Theta\in[0,h]$ and $\delta_{\mathcal{K}_{\mathcal{P}}}$ is given in
Lemma 4.8.

We proceed to derive a key proposition to establish one main geometric result,
which is a three-dimensional result similar to Proposition 4.1.

######  Proposition 4.2.

Consider the same setup in Lemma 4.6 with $\mathbf{x}_{c}$ coinciding with the
origin. Assume that $\mathbf{f}_{j}\in C^{1,\alpha_{j}}\big{(}\Gamma^{\prime
j}_{h}\times(-M,M)\big{)}^{3}\bigcap H^{\frac{1}{2}}\big{(}\Gamma^{\prime
j}_{h}\times(-M,M)\big{)}^{3}$ and $\mathbf{g}_{j}\in
C^{\beta_{j}}\big{(}\Gamma^{\prime j}_{h}\times(-M,M)\big{)}^{3}\bigcap
H^{-\frac{1}{2}}\big{(}\Gamma^{\prime j}_{h}\times(-M,M)\big{)}^{3}$ are
independent of $x_{3}$, where $\alpha_{j},\beta_{j}$ are in $(0,1)$ and
$j=+,-$. Let $\mathbf{v}\in
H^{1}\left(\mathcal{C}^{\prime}_{h}\times(-M,M)\right)^{3}$ and $\mathbf{w}\in
H^{1}\left(\mathcal{C}^{\prime}_{h}\times(-M,M)\right)^{3}$ satisfy the PDE
system (4.20). Let $\mathbf{g}_{j}({\bf 0})=(\mathbf{g}^{(1,2)}_{j}({\bf
0})^{\top},g^{3}_{j}({\bf 0}))^{\top}$ for $j=+,-$. Then we have

$\mathbf{f}_{+}({\bf 0})=\mathbf{f}_{-}({\bf
0}),\quad\mathbf{g}^{(1,2)}_{+}({\bf 0})=W\mathbf{g}^{(1,2)}_{-}({\bf
0})\quad{and}\quad g_{+}^{3}({\bf 0})=g_{-}^{3}({\bf 0})=0,$ (4.27)

where
$W=\begin{bmatrix}-\cos(\theta_{M}-\theta_{m}),&-\sin(\theta_{M}-\theta_{m})\\\
-\sin(\theta_{M}-\theta_{m}),&+\cos(\theta_{M}-\theta_{m})\end{bmatrix}$,
$\theta_{M}$ and $\theta_{m}$ are the arguments corresponding to the
boundaries $\Gamma^{\prime+}_{h}$ and $\Gamma^{\prime-}_{h}$ respectively.

###### Proof.

Similar to the proof of Proposition 4.1, we only consider the corresponding
proofs for $(\Re\mathcal{P}(\mathbf{v}),\Re\mathcal{P}(\mathbf{w}))$. We
follow similar arguments in the proof of Proposition 4.1 with some necessary
modifications. We divide the proof into two parts.

Part I. We first shall prove that

$\Re\mathcal{P}(\mathbf{f}_{+}^{(1,2)})({\bf
0})=\Re\mathcal{P}(\mathbf{f}_{-}^{(1,2)})({\bf
0})\quad\mbox{and}\quad\Re\mathcal{P}(\mathbf{g}_{+}^{(1,2)})({\bf
0})=W\Re\mathcal{P}(\mathbf{g}^{(1,2)}_{-})({\bf 0}).$ (4.28)

In this part, we consider the PDE system (4.21). Noting that
$\mathbf{f}_{j}\in C^{1,\alpha_{j}}\big{(}\Gamma^{\prime
j}_{h}\times(-M,M)\big{)}^{3}$ and $\mathbf{g}_{j}\in
C^{\beta_{j}}\big{(}\Gamma^{\prime j}_{h}\times(-M,M)\big{)}^{3}$ for $j=+,-$.
Since $\mathbf{f}_{j}$ and $\mathbf{g}_{j}$ do not depend on $x_{3}$, we have
the following expansions

$\displaystyle\mathcal{P}(\mathbf{f}_{j})(\mathbf{x}^{\prime})$
$\displaystyle=\mathcal{P}(\mathbf{f}_{j})({\bf
0})+\delta\mathcal{P}(\mathbf{f}_{j})(\mathbf{x}^{\prime}),\quad$
$\displaystyle\big{|}\delta\mathcal{P}(\mathbf{f}_{j})(\mathbf{x}^{\prime})\big{|}$
$\displaystyle\leq A_{j}|\mathbf{x}^{\prime}|^{1+\alpha_{j}},$
$\displaystyle\mathcal{P}(\mathbf{g}_{j})(\mathbf{x}^{\prime})$
$\displaystyle=\mathcal{P}(\mathbf{g})({\bf
0})+\delta\mathcal{P}(\mathbf{g})(\mathbf{x}^{\prime}),\quad$
$\displaystyle\big{|}\delta\mathcal{P}(\mathbf{g})(\mathbf{x}^{\prime})\big{|}$
$\displaystyle\leq B_{j}|\mathbf{x}^{\prime}|^{\beta_{j}}.$ (4.29)

By a series of similar derivations in Proposition 4.1, we deduce the following
integral identity

$\displaystyle\mu\,\Re\mathcal{P}(\mathbf{f}_{+}^{(1,2)})({\bf
0})\cdot\begin{bmatrix}\mathrm{i}\\\
-1\end{bmatrix}\Big{(}e^{-s\sqrt{h}\,\mu(\theta_{M})}-1\Big{)}+\mu\,\Re\mathcal{P}(\mathbf{f}_{-}^{(1,2)})({\bf
0})\cdot\begin{bmatrix}-\mathrm{i}\\\
+1\end{bmatrix}\Big{(}e^{-s\sqrt{h}\,\mu(\theta_{m})}-1\Big{)}$
$\displaystyle-2s^{-2}\Re\mathcal{P}(\mathbf{g}_{+}^{(1,2)})({\bf
0})\cdot\begin{bmatrix}1\\\
\mathrm{i}\end{bmatrix}\mu^{-2}(\theta_{M})-2s^{-2}\Re\mathcal{P}(\mathbf{g}_{-}^{(1,2)})({\bf
0})\cdot\begin{bmatrix}1\\\
\mathrm{i}\end{bmatrix}\mu^{-2}(\theta_{m})=\sum\limits_{j=1}^{8}Q_{j},$
(4.30)

where $\mu(\cdot)$ is given by Lemma 4.9. These $\Re\mathbf{f}_{j}({\bf 0})$,
$\Re\mathbf{g}_{j}({\bf 0})$, $\Re\delta\mathbf{f}_{j}$,
$\Re\delta\mathbf{g}_{j}$, $\Re\mathbf{v}$ and $\Re\mathbf{w}$ in
$R_{1}$–$R_{7}$ given by (4.1) are replacing by
$\Re\mathcal{P}(\mathbf{f}_{j}^{(1,2)})({\bf 0})$,
$\Re\mathcal{P}(\mathbf{g}_{j}^{(1,2)})({\bf 0})$,
$\delta\Re\mathcal{P}(\mathbf{f}_{j}^{(1,2)})$,
$\delta\Re\mathcal{P}(\mathbf{g}_{j}^{(1,2)})$,
$\Re\mathcal{P}(\mathbf{v}^{(1,2)})$ and $\Re\mathcal{P}(\mathbf{w}^{(1,2)})$
for $j=+,-$. In addition,

$\displaystyle Q_{8}$
$\displaystyle=-\int_{\mathcal{C}^{\prime}_{h}}\big{(}\Re\mathcal{G}^{(1,2)}_{1}(\mathbf{x}^{\prime})-\Re\mathcal{G}^{(1,2)}_{2}(\mathbf{x}^{\prime})\big{)}\cdot\mathbf{u}_{0}\,\mathrm{d}{\mathbf{x}^{\prime}}.$

Similar to Proposition 4.1, we have the following estimates

$\displaystyle\big{|}Q_{1}\big{|}={\mathcal{O}}(s^{-1}e^{-c^{\prime}_{1}\,s}),\quad$
$\displaystyle\big{|}Q_{2}\big{|}={\mathcal{O}}(s^{-1}e^{-c^{\prime}_{2}\,s}),\quad$
$\displaystyle\big{|}Q_{3}\big{|}={\mathcal{O}}(e^{-c_{3}^{\prime}s}),\quad$
$\displaystyle\big{|}Q_{4}\big{|}={\mathcal{O}}(s^{-2\alpha_{+}-2}),$
$\displaystyle\big{|}Q_{5}\big{|}={\mathcal{O}}(s^{-2\alpha_{-}-2}),\quad$
$\displaystyle\big{|}Q_{6}\big{|}={\mathcal{O}}(s^{-2\beta_{+}-2}),\quad$
$\displaystyle\big{|}Q_{7}\big{|}={\mathcal{O}}(s^{-2\beta_{-}-2}),$

where these constants $c^{\prime}_{1},\,c^{\prime}_{2}$ and $c^{\prime}_{3}$
do not depend on $s$, $\alpha_{+},\alpha_{-},\,\beta_{+},\,\beta_{-}\in(0,1)$.
From the expressions of $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ defined in
(4.20), denote
$\Re\mathcal{G}^{(1,2)}_{1}-\Re\mathcal{G}^{(1,2)}_{2}:=\mathbf{h}_{1}+\mathbf{h}_{2}$.
Here,
$\mathbf{h}_{1}=-\int_{-L}^{+L}\phi^{\prime\prime}(x_{3})\begin{bmatrix}\lambda(\Re
v_{1}-\Re w_{1})\\\ \lambda(\Re v_{2}-\Re w_{2})\end{bmatrix}\mathrm{d}x_{3}$
and
$\mathbf{h}_{2}=(\lambda+\mu)\int_{-L}^{+L}\phi^{\prime}(x_{3})\begin{bmatrix}\partial_{1}(\Re
v_{3}-\Re w_{3})\\\ \partial_{2}(\Re v_{3}-\Re
w_{3})\end{bmatrix}\mathrm{d}x_{3}$. By the regularities of $\mathbf{v}$ and
$\mathbf{w}$ in $\mathcal{C}^{\prime}_{h}\times(-M,M)$, we directly obtain
that $\Re\mathcal{G}^{(1,2)}_{1}\in H^{1}(\mathcal{C}^{\prime}_{h})^{2}$ and
$\Re\mathcal{G}^{(1,2)}_{2}\in L^{2}(\mathcal{C}^{\prime}_{h})^{2}$. By using
the Cauchy-Schwarz inequality and the first equation in Lemma (4.3), we prove

$\displaystyle\bigg{|}\int_{\mathcal{C}^{\prime}_{h}}(\Re\mathcal{G}^{(1,2)}_{1}-\Re\mathcal{G}^{(1,2)}_{2})\cdot\mathbf{u}_{0}\,\mathrm{d}x_{3}\,\bigg{|}=\bigg{|}\int_{\mathcal{C}^{\prime}_{h}}(\mathbf{h}_{1}+\mathbf{h}_{2})\cdot\mathbf{u}_{0}\,\mathrm{d}x_{3}\,\bigg{|}$
$\displaystyle\,\,\leq\|\mathbf{h}_{1}\|_{L^{2}(\mathcal{C}^{\prime}_{h})^{2}}\|\mathbf{u}_{0}\|_{L^{2}(\mathcal{C}^{\prime}_{h})^{2}}+\|\mathbf{h}_{2}\|_{L^{2}(\mathcal{C}^{\prime}_{h})^{2}}\|\mathbf{u}_{0}\|_{L^{2}(\mathcal{C}^{\prime}_{h})^{2}}$
$\displaystyle\,\,\leq C^{\prime}e^{-c^{\prime}_{8}s},$

where $C^{\prime}>0$ and $c^{\prime}_{8}>0$ do not depend on $s$. From (4.30),
we have

$\begin{bmatrix}-\mathrm{i}\\\
1\end{bmatrix}\cdot\Big{(}\Re\mathcal{P}(\mathbf{f}_{+}^{(1,2)})({\bf
0})-\Re\mathcal{P}(\mathbf{f}_{-}^{(1,2)})({\bf
0})\Big{)}=0\quad\mbox{as}\quad s\rightarrow+\infty.$

Noticing that $\Re\mathcal{P}(\mathbf{f}_{+}^{(1,2)})({\bf 0})$ and
$\Re\mathcal{P}(\mathbf{f}_{-}^{(1,2)})({\bf 0})$ are real, we can prove

$\Re\mathcal{P}(\mathbf{f}_{+}^{(1,2)})({\bf
0})=\Re\mathcal{P}(\mathbf{f}_{-}^{(1,2)})({\bf 0}).$

Substituting the above equation into (4.30), then multiplying the new identity
by $s^{2}$, we get

$\displaystyle\mu\,s^{2}\Re\mathcal{P}(\mathbf{f}_{+}({\bf
0}))\cdot\begin{bmatrix}\mathrm{i}\\\
-1\end{bmatrix}\Big{(}e^{-s\sqrt{h}\mu(\theta_{M})}-e^{-s\sqrt{h}\mu(\theta_{m})}\Big{)}-2\begin{bmatrix}1\\\
\mathrm{i}\end{bmatrix}\cdot\bigg{(}\frac{\Re\mathcal{P}(\mathbf{g}_{+}({\bf
0}))}{\mu^{2}(\theta_{M})}+\frac{\Re\mathcal{P}(\mathbf{g}_{-}({\bf
0}))}{\mu^{2}(\theta_{m})}\bigg{)}=\sum\limits_{j=1}^{8}s^{2}\,Q_{j}.$

We note that the first term on the left hand of the last equation is bounded
by $s^{2}e^{-c^{\prime}_{7}s}$ with
$c^{\prime}_{7}=\sqrt{h}\min\\{\cos\frac{\theta_{M}}{2},\cos\frac{\theta_{m}}{2}\\}>0$.
As $s$ tends to $+\infty$, we obtain

$\begin{bmatrix}1\\\
\mathrm{i}\end{bmatrix}\cdot\bigg{(}\frac{\Re\mathcal{P}(\mathbf{g}_{+}({\bf
0}))}{\mu^{2}(\theta_{M})}+\frac{\Re\mathcal{P}(\mathbf{g}_{-}({\bf
0}))}{\mu^{2}(\theta_{m})}\bigg{)}=0.$

By the same method of proving the first equation in (4.4), we can prove that
the second equation in (4.28) holds.

Part II. We next shall prove that

$\Re\mathcal{P}(f_{+}^{3})({\bf 0})=\Re\mathcal{P}(f_{-}^{3})({\bf
0})\quad\mbox{and}\quad\Re\mathcal{P}(g_{+}^{3})({\bf
0})=\Re\mathcal{P}(g_{-}^{3})({\bf 0})=0.$ (4.31)

In this part, we consider the PDE system (4.22). Let us deduce some similar
operations above by using the CGO solution $u_{0}$ given in (4.23), we set up
an integral identity as follows

$\displaystyle\frac{1}{\lambda}\int_{\mathcal{C}^{\prime}_{h}}\big{(}\Re\mathcal{G}^{(3)}_{1}(\mathbf{x}^{\prime})-\Re\mathcal{G}^{(3)}_{2}(\mathbf{x}^{\prime})\big{)}\,$
$\displaystyle
u_{0}\mathrm{d\mathbf{x}^{\prime}}=\int_{\Lambda^{\prime}_{h}}\partial_{\bm{\nu}}\mathcal{P}(v_{3}-w_{3})\,u_{0}-\partial_{\bm{\nu}}u_{0}\,\mathcal{P}(v_{3}-w_{3})\mathrm{d\sigma}$
$\displaystyle+\int_{\Gamma^{{}^{\prime}+}_{h}}{\bm{\nu}_{M}}\cdot\big{(}\Re\mathcal{P}(f^{(1,2)}_{+})(\mathbf{x}^{\prime})+\frac{1}{\mu}\Re\mathcal{P}(g^{(3)}_{+})\big{)}\,u_{0}-\Re\mathcal{P}(f^{(3)}_{+})\,\partial_{\bm{\nu}_{M}}u_{0}\mathrm{d\sigma}$
$\displaystyle+\int_{\Gamma^{{}^{\prime}-}_{h}}{\bm{\nu}_{m}}\cdot\big{(}\Re\mathcal{P}(f^{(1,2)}_{-})(\mathbf{x}^{\prime})+\frac{1}{\mu}\Re\mathcal{P}(g^{(3)}_{-})\big{)}\,u_{0}-\Re\mathcal{P}(f^{(3)}_{-})\,\partial_{\bm{\nu}_{m}}u_{0}\mathrm{d\sigma}.$

Due to the expansions of $\mathbf{f}_{\pm}$ and $\mathbf{g}_{\pm}$ in (4.29),
the above integral identity can be reduced into

$\displaystyle\lambda\,\mu\,\mathrm{i}\,\Big{(}\Re\mathcal{P}(f^{3}_{+})(0)-\Re\mathcal{P}(f^{3}_{-})(0)\Big{)}-2\lambda\,s^{-1}\bigg{(}\frac{\Re\mathcal{P}(g^{3}_{+})(0)}{\mu^{2}(\theta_{M})}+\frac{\Re\mathcal{P}(g^{3}_{-})(0)}{\mu^{2}(\theta_{m})}\bigg{)}=\sum^{9}_{j=1}M_{j},$

where

$\displaystyle M_{1}$
$\displaystyle=2\lambda\,s^{-1}\Re\mathcal{P}(g^{3}_{+})({\bf
0})\Big{(}\mu^{-1}(\theta_{M})\,\sqrt{sh}\,e^{-\sqrt{sh}\,\mu(\theta_{M})}+\mu^{-2}(\theta_{M})\,e^{-\sqrt{sh}\,\mu(\theta_{M})}\Big{)},$
$\displaystyle M_{2}$
$\displaystyle=2\lambda\,s^{-1}\Re\mathcal{P}(g^{3}_{-})({\bf
0})\Big{(}\mu^{-1}(\theta_{m})\,\sqrt{sh}\,e^{-\sqrt{sh}\,\mu(\theta_{m})}+\mu^{-2}(\theta_{m})\,e^{-\sqrt{sh}\,\mu(\theta_{m})}\Big{)},$
$\displaystyle M_{3}$
$\displaystyle=-\lambda\mu\,\int_{\Lambda^{\prime}_{h}}\partial_{\bm{\nu}}u_{0}\,\,\Re\mathcal{P}(v_{3}-w_{3})(\mathbf{x}^{\prime})-\partial_{\bm{\nu}}u_{0}\,\,\Re\mathcal{P}(v_{3}-w_{3})(\mathbf{x}^{\prime})\,\mathrm{d}\sigma,$
$\displaystyle M_{4}$
$\displaystyle=\lambda\,\mu\,\mathrm{i}\,\bigg{(}\frac{\Re\mathcal{P}(f^{3}_{-})(0)}{e^{\sqrt{sh}\,\mu(\theta_{m})}}-\frac{\Re\mathcal{P}(f^{(3)}_{+})(0)}{e^{\sqrt{sh}\,\mu(\theta_{M})}}\bigg{)},\,\,M_{5}=\mu\int_{\mathcal{C}^{\prime}_{h}}(t_{1}+t_{2})\,u_{0}\mathrm{d}{\mathbf{x}^{\prime}},$
$\displaystyle M_{6}$
$\displaystyle=\lambda\mu\,\int_{\Gamma^{\prime+}_{h}}\delta\Re\mathcal{P}(f^{3}_{+})(\mathbf{x}^{\prime})\,\,\partial_{\bm{\nu}_{M}}u_{0}\,\mathrm{d}\sigma,\qquad
M_{7}=\lambda\mu\,\int_{\Gamma^{\prime-}_{h}}\delta\Re\mathcal{P}(f^{3}_{-})(\mathbf{x}^{\prime})\,\,\partial_{\bm{\nu}_{m}}u_{0}\,\mathrm{d}\sigma,$
$\displaystyle M_{8}$
$\displaystyle=-\lambda\int_{\Gamma^{\prime+}_{h}}\delta\Re\mathcal{P}(g^{3}_{+})(\mathbf{x}^{\prime})\,\,u_{0}\,\mathrm{d}\sigma,\qquad\quad\,\,\,\,M_{9}=-\lambda\int_{\Gamma^{\prime-}_{h}}\delta\Re\mathcal{P}(g^{3}_{-})(\mathbf{x}^{\prime})\,\,u_{0}\,\mathrm{d}\sigma.$

Here,

$\displaystyle t_{1}$
$\displaystyle=-(2\lambda+\mu)\Re\int_{-L}^{L}\phi^{\prime\prime}(\mathbf{x}^{\prime})(v_{3}-w_{3})\mathrm{d}x_{3},$
$\displaystyle t_{2}$
$\displaystyle=(\lambda+\mu)\Re\int_{-L}^{L}\phi^{\prime}(\mathbf{x}^{\prime})\big{(}\partial_{1}(v_{1}-w_{1})+\partial_{2}(v_{2}-w_{2})\big{)}\mathrm{d}x_{3}.$

Using those estimates list in Lemma 4.8–Lemma 4.10, we have

$\displaystyle\big{|}M_{1}\big{|}={\mathcal{O}}(e^{-q_{1}\sqrt{s}}),$
$\displaystyle\big{|}M_{2}\big{|}={\mathcal{O}}(e^{-q_{2}\sqrt{s}}),$
$\displaystyle\big{|}M_{3}\big{|}={\mathcal{O}}(s^{-1}e^{-1/2hs}),\,\big{|}M_{4}\big{|}={\mathcal{O}}(s^{-q_{4}\sqrt{s}}),$
$\displaystyle\big{|}M_{6}\big{|}={\mathcal{O}}(s^{-\alpha_{+}-1}),$
$\displaystyle\big{|}M_{7}\big{|}={\mathcal{O}}(e^{-\alpha_{-}-1}),$
$\displaystyle\big{|}M_{8}\big{|}={\mathcal{O}}(e^{-\beta_{+}-1}),\qquad\big{|}M_{9}\big{|}={\mathcal{O}}(e^{-\beta_{-}-1}),$

where these above constants do not depend on $s$. Using the similar technique
of estimating of $Q_{8}$, we get

$\big{|}M_{5}\big{|}={\mathcal{O}}(e^{-q_{5}s}),\quad\mbox{where}\quad
q_{5}>0.$

Let $s\rightarrow+\infty$, the first equation in (4.31) holds clearly, that is
to say,

$\Re\mathcal{P}(f_{+}^{3})({\bf 0})=\Re\mathcal{P}(f_{-}^{3})({\bf 0}).$

Substituting the above equation into (4.30) and multiplying the new identity
by $s^{2}$, then letting $s\rightarrow+\infty$, one can obtain that

$\frac{\Re\mathcal{P}(g^{(3)}_{+})(0)}{\mu^{2}(\theta_{M})}+\frac{\Re\mathcal{P}(g^{(3)}_{-})(0)}{\mu^{2}(\theta_{m})}=0.$

It’s worth noting that
$\frac{\mu^{2}(\theta_{M})}{\mu^{2}(\theta_{m})}=e^{\mathrm{i}(\theta_{M}-\theta_{m})}$
and $\theta_{M}-\theta_{m}\in(0,\pi)$. Hence,

$\Re\mathcal{P}(g^{3}_{+})({\bf 0})=\Re\mathcal{P}(g^{3}_{-})({\bf 0})=0.$

Thanks to the symmetric role of $(\Re\mathbf{v},\Re\mathbf{w})$ and
$(\Im\mathbf{v},\Im\mathbf{w})$, the two equations (4.28) and (4.31) directly
lead to the results list in (4.27).

∎

##  5\. The proofs of Theorem 3.1–Theorem 3.3

###### Proof of Theorem 3.1.

We prove this theorem by contradiction. Assume that there exists a planar
corner/3D edge corner $\mathbf{x}_{c}$ on
$\mathcal{S}_{1}\Delta\mathcal{S}_{2}$. Without loss of generality, we assume
that $\mathbf{x}_{c}$ coincides with the origin $\bf 0$, $\bf
0\in\mathcal{S}_{1}$ and $\bf 0\notin\mathcal{S}_{2}$. Denote
$\mathbf{w}=\mathbf{u}_{1}-\mathbf{u}_{2}$.

$\bf Case~{}1$: $n=2$. We note that $\Gamma^{\pm}_{h}=\mathcal{S}_{1}\cap
B_{h}$, $\mathcal{C}_{h}=\Omega_{1}\cap B_{h}$ and
$\mathcal{C}_{h}\cap\Omega_{2}\neq\emptyset$ for sufficient small
$h\in\mathbb{R}_{+}$, where $\Omega_{1}$ and $\Omega_{2}$ are defined in a
similar way as (2.8). Since
$\mathbf{u}_{1}\big{|}_{\Sigma_{0}}=\mathbf{u}_{2}\big{|}_{\Sigma_{0}}$ and
$\Sigma_{0}\subset\Sigma_{N}$, we know that

$\mathbf{w}=\mathbf{u}_{1}-\mathbf{u}_{2}=\mathcal{T}_{\bm{\nu}}\mathbf{u}_{1}-\mathcal{T}_{\bm{\nu}}\mathbf{u}_{2}={\bf
0}\quad\mbox{on}\quad\Sigma_{0}.$

Let $\mathbf{w}^{-}$ and $\mathbf{w}^{+}$ represent
$\mathbf{w}\big{|}_{\Omega_{1}}$ and
$\mathbf{w}\big{|}_{\Omega\backslash\overline{\Omega}_{1}}$, respectively.
With the help of the unique continuation principle properly and the fact that
$\mathbf{u}_{2}$ is real analytic in $B_{h}$, it is clear to obtain

$\mathcal{L}\,\mathbf{w}^{-}={\bf
0}\,\,\mbox{in}\,\,\mathcal{C}_{h},\,\,\mathbf{w}^{-}\big{|}_{\Gamma_{h}^{j}}=\mathbf{u}_{1}\big{|}_{\Gamma_{h}^{j}}-\mathbf{u}_{2}\big{|}_{\Gamma_{h}^{j}}=-\mathbf{f}^{1}_{j},\,\,\mathcal{T}_{\bm{\nu}}\mathbf{u}_{1}\big{|}_{\Gamma_{h}^{j}}-\mathcal{T}_{\bm{\nu}}\mathbf{u}_{2}\big{|}_{\Gamma_{h}^{j}}=-\mathbf{g}^{1}_{j},\,\,j=+,-.$
(5.1)

From Proposition 4.1, we directly imply that $\mathbf{f}^{1}_{+}({\bf
0})=\mathbf{f}^{1}_{-}({\bf 0})$ and $\mathbf{g}^{1}_{+}({\bf
0})=W\mathbf{g}^{1}_{-}({\bf 0})$, where $W$ is given in Proposition 4.1. This
must contradict to the admissibility condition $(5)$ in Definition 3.1.

$\bf Case~{}2$: $n=3$. It is noted that ${\bf 0}=({\bf
0}^{\prime},0)^{\top}\in\Gamma^{\prime\pm}_{h}\times(-M,+M)\subset\mathcal{S}_{1}$
is a 3D edge corner point and
$\mathcal{C}^{\prime}_{h}\times(-M,+M)\subset\Omega_{1}$ and
$\mathcal{C}^{\prime}_{h}\times(-M,+M)\cap\Omega_{2}=\emptyset$, where
$\Gamma^{\prime\pm}_{h}$ and $\mathcal{C}^{\prime}_{h}$ are defined in (3.3)
and $\Omega_{1},\,\Omega_{2}$ are given by (2.8). Similar to the previous
case, we get

$\begin{cases}\mathcal{L}\,\mathbf{w}^{-}=\mathbf{0},&\mbox{in}\quad\mathcal{C}^{\prime}_{h}\times(-M,M),\\\
\mathbf{w}=-\mathbf{f}^{1}_{+}&\mbox{on}\quad\Gamma^{\prime+}_{h}\times(-M,M),\\\
\mathbf{w}=-\mathbf{f}^{1}_{-}&\mbox{on}\quad\Gamma^{\prime-}_{h}\times(-M,M),\\\
\mathcal{T}_{\bm{\nu}_{M}}\mathbf{w}=-\mathbf{g}^{1}_{+}&\mbox{on}\quad\Gamma^{\prime+}_{h}\times(-M,M),\\\
\mathcal{T}_{\bm{\nu}_{m}}\mathbf{w}=-\mathbf{g}^{1}_{-}&\mbox{on}\quad\Gamma^{\prime-}_{h}\times(-M,M).\end{cases}$

These results that $\mathbf{f}^{1}_{+}({\bf 0})=\mathbf{f}^{1}_{-}({\bf 0})$,
$\mathbf{g}^{1,(1,2)}_{+}({\bf 0})=W\mathbf{g}^{1,(1,2)}_{-}({\bf 0})$ and
$g^{1,3}_{+}({\bf 0})=g^{1,3}_{+}({\bf 0})=0$ are yielded by Proposition 4.2,
where $W$ is given in Definition 3.1. This contradicts to the admissibility
condition $(5)$ in Definition 3.1.

The proof is complete. ∎

###### Proof of Theorem 3.2.

Firstly, we shall prove $\mathcal{S}_{1}=\mathcal{S}_{2}$ by contradiction.
Assume that $\Omega_{1}\neq\Omega_{2}$. Since $\Omega_{1}$ and $\Omega_{2}$
are both convex polygons or polyherons, there must exist a corner
$\mathbf{x}_{c}$ belonging to $\Omega_{1}\Delta\Omega_{2}$, which contradicts
Theorem 3.1, thus we have $\Omega_{1}=\Omega_{2}$. It directly leads to
$\mathcal{S}_{1}=\mathcal{S}_{2}$. In what follows, we shall first prove (3.4)
and (3.5) for the 2D case.

Denote $\hat{\Omega}:=\Omega_{1}=\Omega_{2}$ and
$\mathcal{S}:=\mathcal{S}_{1}=\mathcal{S}_{2}$. Let
$\mathbf{w}=\mathbf{u}_{1}-\mathbf{u}_{2}$. Since
$\mathbf{u}_{1}=\mathbf{u}_{2}$ on $\Sigma_{0}\subset\Sigma_{N}$, as a direct
consequence, we have $\mathbf{w}=0$ and $\mathcal{T}_{\bm{\nu}}\mathbf{w}=0$
on $\Sigma_{0}$, where $\mathbf{w}=\mathbf{u}_{1}-\mathbf{u}_{2}$. By virtue
of the unique continuation principle properly again, one obtains

$\mathbf{w}^{+}\big{|}_{\Gamma_{h}^{\pm}}=\mathcal{T}_{\bm{\nu}}\mathbf{w}^{+}\big{|}_{\Gamma_{h}^{\pm}}={\bf
0},$

where $\mathbf{w}^{+}$ represents
$\mathbf{w}\big{|}_{\Omega\backslash\overline{\hat{\Omega}}}$. Since
$(\mathcal{S};\mathbf{f}^{1},\mathbf{g}^{1})$ and
$(\mathcal{S};\mathbf{f}^{2},\mathbf{g}^{2})$ are admissible, we get

$\mathcal{L}\,\mathbf{w}^{-}={\bf
0}\,\,\mbox{in}\,\,\mathcal{C}_{h},\quad\mathbf{w}^{-}\big{|}_{\Gamma_{h}^{j}}=\mathbf{f}^{2}_{j}-\mathbf{f}^{1}_{j}\,\,\,\mbox{and}\,\,\,\mathcal{T}_{\bm{\nu}}\mathbf{w}^{-}\big{|}_{\Gamma_{h}^{j}}=\mathbf{g}^{2}_{j}-\mathbf{g}^{1}_{j}\quad\mbox{for}\quad
j=+,-,$ (5.2)

where $\mathbf{w}^{-}$ signifies $\mathbf{w}\big{|}_{\hat{\Omega}}$. From
Proposition 4.1 and (5.2), we obtain the following local uniqueness

$\mathbf{f}^{2}_{+}({\bf 0})-\mathbf{f}^{1}_{+}({\bf
0})=\mathbf{f}^{2}_{-}({\bf 0})-\mathbf{f}^{1}_{-}({\bf
0})\quad\mbox{and}\quad\mathbf{g}^{1}_{+}({\bf 0})-\mathbf{g}^{2}_{+}({\bf
0})=W\Big{(}\mathbf{g}^{1}_{+}({\bf 0})-\mathbf{g}^{2}_{+}({\bf 0})\Big{)},$

Furthermore, since $\mathbf{f}_{i}$ and $\mathbf{g}_{i}$ ($i=1,2$) are
piecewise constant valued functions, we immediately obtain (3.4) and (3.5). By
a similar method used in 2D case, we can prove (3.4) and (3.5) in 3D case,. ∎

###### Proof of Theorem 3.3.

The argument of proving this theorem is similar to the one used in the proof
of Theorem 3.2, where we only need some necessary modifications. Let
$\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ are two different polygonal curves in
$\mathbb{R}^{2}$ or polyhedral surfaces in $\mathbb{R}^{3}$. From Definition
3.2, we can directly imply that $\mathcal{S}_{1}\Delta\mathcal{S}_{2}$
contains a planar or 3D edge corner can be derived by Definition 3.2. Under
the condition (3.6), adopting a similar argument in proving Theorem 3.2, we
can show that $\mathcal{S}_{1}=\mathcal{S}_{2}$.

Set $\mathbf{w}=\mathbf{u}_{1}-\mathbf{u}_{2}$. We have that
$\mathbf{w}=\mathcal{T}_{\bm{\nu}}\mathbf{w}=\bf 0$ on $\Sigma_{0}$ for
$\mathbf{u}_{1}=\mathbf{u}_{2}$ on $\Sigma_{0}\subset\Sigma_{N}$. By using the
unique continuation property again, we conclude that $\mathbf{w}=\bf 0$ in
$\Omega\backslash\overline{\mathcal{S}}$. Hence, it is direct to imply that

$\bf
0=[\mathbf{w}]_{\mathcal{S}_{1}}=[\mathbf{w}]_{\mathcal{S}_{2}}\,\Rightarrow\,[\mathbf{u}_{1}]_{\mathcal{S}_{1}}=[\mathbf{u}_{2}]_{\mathcal{S}_{2}}\,\mbox{and}\,[\mathcal{T}_{\bm{\nu}}\mathbf{u}_{1}]_{\mathcal{S}_{1}}=[\mathcal{T}_{\bm{\nu}}\mathbf{u}_{2}]_{\mathcal{S}_{2}}\,\Rightarrow\,\mathbf{f}_{1}=\mathbf{f}_{2}\,\mbox{and}\,\mathbf{g}_{1}=\mathbf{g}_{2}.$

The proof is complete. ∎

## Acknowledgment

The work of H. Diao is supported by National Natural Science Foundation of
China (No. 12371422) and the Fundamental Research Funds for the Central
Universities, JLU (No. 93Z172023Z01). 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). The work of Q. Meng is supported
by the Hong Kong RGC Postdoctoral Fellowship (No. 9061028).

## References

  * [1] A. Aspri, E. Beretta and A. Mazzucato, Dislocations in a layered elastic medium with application to fault detection, J. Eur. Math. Soc., to appear, 2022.
  * [2] A. Aspri, E. Beretta, M. De Hoop and A. Mazzucato, Fault detection in an anisotropic elastic medium, arXiv:2004.00321, 2020.
  * [3] A. Aspri, E. Beretta, A. Mazzucato and M. De Hoop, Analysis of a model of elastic dislocations in geophysics, Arch. Ration. Mech. Anal., 236:71-111, 2020.
  * [4] E. Blåsten, Nonradiating sources and transmission eigenfunctions vanish corners and edges, SIAM J. Math. Anal., 50:6255-6270, 2018.
  * [5] E. Blåsten and Y-H. Liu, Radiating and non-radiating sources in elasticity, Inverse problems, 35:015005, 2021.
  * [6] M. Bonafede and E. Rivalta, The tensile dislocation problem in a layered elastic medium, J. Int., 136: 341-356, 1999.
  * [7] M. Cessenat, Mathematical methods in electromagnetism:linear theory and applications, World Scientfic, River Edge, NJ. 1998.
  * [8] S. Cohen, Convenient formulas for determining dip-slip fault parameters from geophysical observables, Bull. Seismol. Soc. Am., 86:1642-1644, 1996.
  * [9] H. Diao, X. Cao and H. Liu, On the geometric structures of transmission eienfunctions with a conductive boundary condition and applications, Commum. PDE., 46:630–679, 2021.
  * [10] H. Diao, H. Liu and B. Sun, On a local geometrical property of the generalized elastic transmission eigenfunctions and application, Inverse problems, 37:105015, 2021.
  * [11] J. D. Eshelby, Dislocation theory for geophysical applications, Philos. Trans. R. Soc. A., 274:331-338, 1973.
  * [12] P. Hähner, On the uniqueness of the shape of a penetrable, anisotropic obstacle, J. Comput. Appl. Math., 116 (2000), 167–181.
  * [13] Z. Jiang et. al., GPS constrained coseismic source and slip distribution of the 2013 Mw6.6 Lushan, China, earthquake and its teconic implications, Geophys. Res. Lett., 41:407-413, 2014.
  * [14] W. Mclean, Strongly elliptic systems and boundary integral equations, Cambridge: Cambridge University press, 2010.
  * [15] J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. III, Springer-Verlay, New York-Heidelverg, 1973. Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 183.
  * [16] E. Rivalta, W. Mangiavillano and M. Bonafede, The edge dislocation problem in a lyered elastic medium, Geophys. J. Int., 149:508-523, 2002.
  * [17] P. Segall, Earthquake and volcano deformation, Princeton University Press, Princeton, 2010.
  * [18] M. Simons, Y. Fialko and L. Rivera, Coseismic deformation from the 1999 M w 7.1 Hector Mine, California, earthquake as inferred from InSAR and GPS observations, Bull. Seismol. Soc. Am., 92:1390-1402, 2002.
  * [19] L. Tartar, An introduction to Navier-Stokes equation and oceanography, Volume 1 of Lecture Notes of the unione Matematica Italian. Springer-Verlag, Berlin; UMI, Bologna, 2006.
  * [20] C. Xu, X. Xu and G. Yu, Landslides triggered by slipping-fault-generated earthquake on a plateau: an example of the 14 April 2010, Ms 7.1 , Yushu, China earthquake, Landslides, 10:421-431, 2013.