# Stability estimates in a partial data inverse boundary value problem for
biharmonic operators with first order perturbation

Boya Liu  B. Liu, Department of Mathematics
North Carolina State University
NC 27695, USA<EMAIL_ADDRESS>
###### Abstract.

We establish logarithmic stability estimates for an inverse boundary value
problem for the biharmonic operator with first order perturbation from partial
boundary measurements. Our geometric setting is that of a bounded domain in
the Euclidean space of dimension three or higher. In particular, the
inaccessible portion of the boundary is flat, and we have knowledge of the
Dirichlet-to-Neumann map on the complement.

##  1\. Introduction and Statement of Results

Let $\Omega\subseteq\mathbb{R}^{n},n\geq 3$, be a bounded simply connected
domain with smooth boundary $\partial\Omega$. Consider a biharmonic operator
with first order perturbation

$\mathcal{L}_{A,q}(x,D):=\Delta^{2}+A(x)\cdot D+q(x),$ (1.1)

where $D=i^{-1}\nabla$, $A$ is a complex-valued vector field, called the
magnetic potential, and $q$ is a complex-valued function, called the electric
potential. The operator $\mathcal{L}_{A,q}$ arises when we consider the
equilibrium configuration of an elastic plate hinged along the boundary. In
physics and geometry, higher order operators occur in the study of the
Kirchhoff plate equation in the theory of elasticity and the study of the
Paneitz-Branson operator in conformal geometry. We refer readers to see [15,
Chapter 1] for more applications.

The operator $\mathcal{L}_{A,q}$ equipped with the domain

$\mathcal{D}(\mathcal{L}_{A,q})=\\{u\in
H^{4}(\Omega):u|_{\partial\Omega}=(\Delta u)|_{\partial\Omega}=0\\}$

is an unbounded closed operator on $L^{2}(\Omega)$ with purely discrete
spectrum, see [17, Chapter 11]. We shall make the following assumption.

Assumption 1: 0 is not an eigenvalue of the perturbed biharmonic operator
$\mathcal{L}_{A,q}:\mathcal{D}(\mathcal{L}_{A,q})\to L^{2}(\Omega)$.

Consider the boundary value problem with Navier boundary conditions

$\begin{cases}\mathcal{L}_{A,q}u=0\quad\text{in}\quad\Omega,\\\
u=f\quad\text{on}\quad\partial\Omega,\\\ \Delta
u=g\quad\text{on}\quad\partial\Omega,\end{cases}$ (1.2)

where $A\in W^{1,\infty}(\Omega,\mathbb{C}^{n})$ and $q\in
L^{\infty}(\Omega,\mathbb{C})$. Under Assumption 1, as explained in [21], for
any pair of functions $(f,g)\in H^{\frac{7}{2}}(\partial\Omega)\times
H^{\frac{3}{2}}(\partial\Omega)$, the boundary value problem (1.2) has a
unique solution $u\in H^{4}(\Omega)$. Here and in what follows
$H^{s}(\Omega)=\\{U|_{\Omega}:U\in H^{s}(\mathbb{R}^{n})\\}$,
$s\in\mathbb{R}$, is the standard $L^{2}$-based Sobolev space on $\Omega$, see
[1, Chapter 5]. Also, throughout this paper we shall denote
$H^{\alpha,\beta}(\partial\Omega)$ the product of two Sobolev spaces
$H^{\alpha}(\partial\Omega)\times H^{\beta}(\partial\Omega)$, equipped with
the norm

$\|(f,g)\|_{H^{\alpha,\beta}(\partial\Omega)}=\|f\|_{H^{\alpha}(\partial\Omega)}+\|g\|_{H^{\beta}(\partial\Omega)}.$

Let us next describe the geometric framework considered in this paper. We
assume that
$\Omega\subseteq\\{x=(x_{1},\dots,x_{n})\in\mathbb{R}^{n}:x_{n}<0\\}$, $n\geq
3$, and split its boundary $\partial\Omega$ in two non-empty subsets
$\Gamma_{0}:=\partial\Omega\cap\\{x\in\mathbb{R}^{n}:x_{n}=0\\}$, called the
inaccessible part, and its complement
$\Gamma=\partial\Omega\setminus\Gamma_{0}$. That is, the inaccessible part of
the boundary is flat.

We define the partial Dirichlet-to-Neumann map associated with the boundary
value problem (1.2) on $\Gamma$ as

$\displaystyle\Lambda_{A,q}^{\Gamma}:H^{\frac{7}{2}}(\Gamma)\times
H^{\frac{3}{2}}(\Gamma)\to H^{\frac{5}{2}}(\Gamma)\times
H^{\frac{1}{2}}(\Gamma),\quad\Lambda_{A,q}^{\Gamma}(f,g)=(\partial_{\nu}u|_{\Gamma},\partial_{\nu}(\Delta
u)|_{\Gamma}),$ (1.3)

where $\hbox{supp }f,\>\hbox{supp }g\subset\Gamma$, and $\nu$ is the outer
unit normal of $\partial\Omega$. Then the norm of $\Lambda_{A,q}^{\Gamma}$ is
given by

$\|\Lambda_{A,q}^{\Gamma}\|:=\sup\\{\|\Lambda_{A,q}^{\Gamma}(f,g)\|_{H^{\frac{5}{2},\frac{1}{2}}(\Gamma)}:\|(f,g)\|_{H^{\frac{7}{2},\frac{3}{2}}(\Gamma)}=1\\}.$

The purpose of this paper is to establish log-type stability estimates for the
magnetic potential $A$ and electric potential $q$ from the partial Dirichlet-
to-Neumann map $\Lambda_{A,q}^{\Gamma}$. The corresponding uniqueness result
was established in [35, Theorem 1.4].

The study of inverse boundary value problems concerning the first order
perturbation of polyharmonic operators $(-\Delta)^{m}$, $m\geq 2$, was
initiated in [23], where unique identification results from the Dirichlet-to-
Neumann map were obtained. We highlight that unique recovery of the first
order perturbation appearing in higher order elliptic operators differs from
analogous problems concerning the Laplacian. In the latter case, one can only
hope to recover $dA$, the exterior derivative of the first order perturbation
$A$, due to the gauge invariance of boundary measurements, see for instance
[22, 25, 28, 32] and the references therein among the extensive literature in
this direction. Here, if we view $A$ as a 1-form, then $dA$ is a 2-form given
by the formula

$dA=\sum_{1\leq j<k\leq
n}(\partial_{x_{j}}A_{k}-\partial_{x_{k}}A_{j})dx_{j}\wedge dx_{k}.$ (1.4)

For the polyharmonic operator, which is of order $2m$, a gauge invariance
occurs when we attempt to recover perturbations of order $2m-1$ from boundary
measurements, see a very recent result [30].

Before stating the main results of this paper, let us introduce the following
admissible sets for the coefficients $A$ and $q$. Given a constant $M>0$, we
define the class of admissible magnetic potentials $\mathcal{A}(\Omega,M)$ by

$\mathcal{A}(\Omega,M)=\\{A\in
W^{1,\infty}(\Omega)\cap\mathcal{E}^{\prime}(\overline{\Omega}):\|A\|_{H^{s}(\Omega)}\leq
M,s>\frac{n}{2}+1\\}.$

Notice that we assumed a priori bounds for the $H^{s}$-norm of $A$ above. We
shall explain the reason later in this section. We also define the class of
admissible electric potentials $\mathcal{Q}(\Omega,M)$ in a similar fashion by

$\mathcal{Q}(\Omega,M)=\\{q\in
L^{\infty}(\Omega):\|q\|_{L^{\infty}(\Omega)}\leq M\\}.$

The main results of this paper are as follows. First, we state a log-type
stability estimate for the magnetic potential $A$.

######  Theorem 1.1.

Let $\Omega\subseteq\\{x\in\mathbb{R}^{n}:x_{n}<0\\}$, $n\geq 3$, be a bounded
simply connected domain with smooth boundary $\partial\Omega$. Assume that
$\Gamma_{0}=\partial\Omega\cap\\{x\in\mathbb{R}^{n}:x_{n}=0\\}$ is nonempty,
and let $\Gamma=\partial\Omega\setminus\Gamma_{0}$. Let $M\geq 0$, and let
$A^{(j)}\in\mathcal{A}(\Omega,M)$ and $q_{j}\in\mathcal{Q}(\Omega,M)$,
$j=1,2$. Suppose that Assumption 1 holds for both operators
$\mathcal{L}_{A^{(1)},q_{1}}$ and $\mathcal{L}_{A^{(2)},q_{2}}$. Then there
exists a constant $C=C(\Omega,n,M,s)>0$ such that

$\|A^{(2)}-A^{(1)}\|_{L^{\infty}(\Omega)}\leq
C(\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{\frac{\eta^{2}}{2(1+s)^{2}}}+\big{|}\log\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|\big{|}^{-\theta_{1}}),$
(1.5)

where $\eta=\frac{1}{2}(s-\frac{n}{2})$ and
$0<\theta_{1}=\frac{\alpha^{2}\eta^{4}}{(n+2)^{2}(1+s)^{4}}<1$.

We shall also establish a log-type estimate for the electrical potential $q$,
which is given by the following theorem.

######  Theorem 1.2.

Under the same hypotheses as in Theorem 1.1, there exists a constant
$C=C(\Omega,n,M,s)>0$ such that

$\|q_{1}-q_{2}\|_{H^{-1}(\Omega)}\leq
C\big{(}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{\frac{\eta^{2}}{2(1+s)^{2}}}+\log\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|\big{|}^{-\theta_{2}}\big{)},$
(1.6)

where $\eta=\frac{1}{2}(s-\frac{n}{2})$ and
$0<\theta_{2}:=\frac{2\alpha^{2}\eta^{4}}{(n+2)^{3}(1+s)^{4}}<1$.

If the electric potentials $q_{1}$ and $q_{2}$ pose additional regularity and
a priori bounds, we have the following corollary of Theorem 1.2.

######  Corollary 1.3.

Let $\Omega\subseteq\\{x\in\mathbb{R}^{n}:x_{n}<0\\}$, $n\geq 3$, be a bounded
domain with smooth connected boundary. Assume that
$\Gamma_{0}:=\partial\Omega\cap\\{x\in\mathbb{R}^{n}:x_{n}=0\\}$ is nonempty,
and let $\Gamma=\partial\Omega\setminus\Gamma_{0}$. Let $M\geq 0$ and
$s>\frac{n}{2}$. Let $A^{(j)}\in\mathcal{A}(\Omega,M)$, and let $q_{j}\in
L^{\infty}(\Omega)$ be such that $\|q_{j}\|_{H^{s}(\Omega)}\leq M$, $j=1,2$.
Suppose that Assumption 1 holds for both operators
$\mathcal{L}_{A^{(1)},q_{1}}$ and $\mathcal{L}_{A^{(2)},q_{2}}$. Then there
exists a constant $C=C(\Omega,n,M,s)>0$ such that

$\|q_{1}-q_{2}\|_{L^{\infty}(\Omega)}\leq
C\bigg{(}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{\frac{\eta^{2}}{2(1+s)^{2}}}+C\log\big{|}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|\big{|}^{-\theta_{2}}\bigg{)}^{\frac{\eta}{1+s}}.$
(1.7)

Here the costants $\theta_{2}$ and $\eta$ are the same as in Theorem 1.2.

######  Remark 1.4.

To the best of our knowledge, estimates (1.5) and (1.6) are the first
stability estimates for partial data inverse boundary value problems for the
biharmonic operator with first order perturbations.

###  1.1. Previous literature

The study of inverse problems concerning biharmonic or polyharmonic operators
have attracted significant attentions in recent years. For the case that only
the zeroth order perturbation is considered, unique identifiability results
were obtained in [18, 19]. If we extend our consideration to include first
order perturbations, it was proved in [23] that the set of Cauchy data
determines sufficiently smooth first order perturbation of the polyharmonic
operator uniquely. There are extensive subsequent efforts to recover the first
order perturbation of the polyharmonic operator with lower regularity, see for
instance [4, 5, 10, 24] and the references therein. We remark that relaxing
the regularity of coefficients is crucial in inverse problems, since it
enables the imaging of rough medium. In addition to the aforementioned results
in the Euclidean setting, we refer readers to [6, 34] to see uniqueness
results in the setting of Riemannian manifolds.

In all the aforementioned results, measurements are made on the whole
boundary. However, in many applications, such as geophysics, it could be
either impossible or extremely cost consuming to perform measurements on the
entire boundary of a medium. Hence, it is of great interest and significance
to study partial data inverse problems, in which case measurements are
performed only on parts of the boundary. In the context of inverse boundary
value problems for biharmonic operators, it was established in [21] that the
Dirichlet-to-Neumann map, with the Neumann data measured on slightly more than
half of the boundary, determines the first and the zeroth order perturbations
of the biharmonic operator uniquely. Several partial data uniqueness results
were obtained in [35] to recover the first and the zeroth order perturbation
of the biharmonic operator from the set of Cauchy data, in bounded domains and
infinite slabs.

Since polyharmonic operators are of order $2m$, it is natural to consider
inverse problems of determining second or higher order perturbations from
boundary measurements, and some progress have been made in this direction. For
instance, the authors of [7, 8] proved unique determination results for
perturbations of up to second order appearing in a polyharmonic operator.
Beyond second order perturbations, it was proved in [9] that the Dirichlet-to-
Neumann map, with the Neumann data measured on roughly half of the boundary,
determines anisotropic perturbations of up to order $m$ appearing in
polyharmonic operators. Also, recovery of multiple isotropic perturbations,
given as either a function or a vector field, of a polyharmonic operator from
partial boundary measurements was established in [16].

Turning attention to the stability question, the majority of known literature
for polyharmonic operators considers only the zeroth order perturbation. A
log-type estimate was obtained in [13] when measurements are made on the
entire boundary. In the realm of partial data results, a log-type estimate was
proved in [12] when the inaccessible part of the boundary is contained in a
hyperplane. Additionally, a log-log-type estimate was established in [13] in
the scenario that the Neumann data is measured on slightly more than half of
the boundary. We remark that the stability estimates probed in [12, 13]
indicate that the inverse problems considered are severely ill-posed. This
phenomenon means that small errors in measurements may result in exponentially
large errors in the reconstruction of the unknown potential, which makes it
very challenging to design reconstruction algorithms with high resolution in
practice. However, as observed numerically in [14], if a frequency $k$ is
introduced to the operator, then stability may increase when $k$ becomes
large. For the perturbed biharmonic operator $\mathcal{L}_{A,q}$, with $A=0$,
the author derived a Hölder-type stability estimate for the potential $q$ at
high frequencies from partial boundary measurements in [26], where the
inaccessible part of the boundary is flat.

Once stability estimates are established for the zeroth order perturbation of
the polyharmonic operator, the natural next step is to derive stability
estimates for higher order perturbations. To the best of our knowledge,
stability estimates for higher order perturbations of the biharmonic operator
were obtained only when full data is considered. We refer readers to see a
very recent result [27] for log-type stability estimates for the first and
zeroth perturbations of the biharmonic operator $\mathcal{L}_{A,q}$, as well
as [3] for stability estimates concerning perturbations of up the second order
of polyharmonic operators.

###  1.2. Outline of the proof of Theorem 1.1 and Theorem 1.2

The general strategy of the proof follows from the methods introduced in [2]
using complex geometric optics (CGO) solutions. To elaborate, we shall
construct a solution $v\in H^{4}(\Omega)$ to the equation
$\mathcal{L}_{A^{(1)},q_{1}}^{\ast}v=0$ in $\Omega$, as well as a solution
$u_{2}\in H^{4}(\Omega)$ satisfying the equation
$\mathcal{L}_{A^{(2)},q_{2}}u_{2}=0$ in $\Omega$. Due to the restrictions that
$\hbox{supp }u|_{\partial\Omega},\hbox{supp }(\Delta
u)|_{\partial\Omega}\subseteq\Gamma$ from the definition of the partial
Dirichlet-to-Neumann map (1.3), we need to construct solutions $v$ and $u_{2}$
such that $v|_{\Gamma_{0}}=(\Delta v)|_{\Gamma_{0}}=0$ and
$u_{2}|_{\Gamma_{0}}=(\Delta u_{2})|_{\Gamma_{0}}=0$. We shall accomplish this
by employing the reflection argument originated in [20]. This approach has
been successfully implemented to solve many partial data inverse problems when
the inaccessible part of the boundary is flat. We refer readers to see for
instance [11, 12, 22, 26, 29, 35].

Another important part of the proof is the integral inequality (3.14), whose
derivation requires knowledge of the partial Dirichlet-to-Neumann map. This
integral inequality, in conjunction with CGO solutions (3.12) and (3.13),
results in two Fourier transforms, one for the difference of magnetic fields
$dA^{(2)}-dA^{(1)}$, and the other for the difference of electric potentials
$q_{2}-q_{1}$, as well as some error terms. We then apply an estimate for the
remainder terms of CGO solutions (3.10) and a quantitative version of the
Riemann-Lebesgue lemma (Lemma 2.2) to control those error terms. Subsequently,
we use the definition of the $H^{-1}$-norm in terms of the Fourier transform,
as well as judicious choice of various parameters to deduce an estimate for
$\|dA\|_{H^{-1}(\Omega)}$.

The main difficulty to establish stability estimates for the magnetic
potential $A$ is as follows. As outlined above, we can only obtain an estimate
for the magnetic field $dA$ from CGO solutions and the integral inequality.
However, as uniqueness results have been achieved for the magnetic potential
$A$, we must establish stability estimates for $A$ in a suitable norm rather
than for $dA$. To do so, we shall utilize a specific Hodge decomposition of
vector fields [31, Theorem 3.3.2]: There exists uniquely determined vector
field $A^{\mathrm{sol}}$ and function $\varphi$ such that
$A=A^{\mathrm{sol}}+\nabla\varphi$, where the solenoidal part
$A^{\mathrm{sol}}$ and the potential part $\nabla\varphi$ satisfy
$\operatorname{div}A^{\mathrm{sol}}=0$ and $\varphi|_{\partial\Omega}=0$. Here
$\operatorname{div}A^{\mathrm{sol}}$ is the divergence of$A^{\mathrm{sol}}$.
Then an application of the Morrey’s inequality yields

$\|A^{\mathrm{sol}}\|_{L^{\infty}(\Omega)}\leq\|dA\|_{L^{\infty}(\Omega)}.$

Notice that we need an estimate for $\|dA\|_{L^{\infty}(\Omega)}$ in order to
apply the Morrey’s inequality, but we only established one for
$\|dA\|_{H^{-1}(\Omega)}$ from earlier parts of the proof. To medicate this,
as we shall detail in (3.40), we apply the Sobolev embedding theorem and the
interpolation inequality [17, Theorem 7.22] to obtain an estimate for
$\|dA\|_{L^{\infty}(\Omega)}$. We point out that this is the reason why we
assumed an a priori bound for the norm $\|A\|_{H^{s}(\Omega)}$,
$s>\frac{n}{2}+1$, in the definition of $\mathcal{A}(\Omega,M)$. This is also
the reason why we have a log-type estimate for $\|A\|_{L^{\infty}(\Omega)}$,
not $\|A\|_{H^{-1}(\Omega)}$, in Theorem 1.1.

To prove an estimate for $\|A\|_{L^{\infty}(\Omega)}$, we still need to
establish an estimate for $\|\nabla\varphi\|_{L^{\infty}(\Omega)}$. This is
accomplished in Lemma 3.4 by utilizing the CGO solutions (3.12) and (3.13)
again. From here, estimate (1.5) follows immediately by combining the
estimates for $\|A^{\mathrm{sol}}\|_{L^{\infty}(\Omega)}$ and
$\|\nabla\varphi\|_{L^{\infty}(\Omega)}$.

This paper is organized as follows. In Section 2We state a result pertaining
the existence of CGO solutions, as well as the quantitative version of the
Riemann-Lebesgue lemma, which will be applied in the proof. Section 3 is
devoted to proving Theorem 1.1. Finally, we verify Theorem 1.2 and Corollary
1.3 in Section 4.

##  2\. Preliminary Results

In this section we collect some results that are necessary to derive stability
estimates (1.5) and (1.6). Let us start by stating a result concerning the
existence of CGO solutions to the equation $\mathcal{L}_{A,q}u=0$ in $\Omega$
of the form

$u(x,\zeta,h)=e^{\frac{x\cdot\zeta}{h}}(a(x,\zeta)+r(x,\zeta,h)).$

Here $\zeta\in\mathbb{C}^{n}$ is a complex vector satisfying
$\zeta\cdot\zeta=0$, $h>0$ is a small semiclassical parameter, $a$ is a smooth
amplitude, and $r$ is a correction term that vanishes in a suitable sense in
the limit $h\to 0$. We refer readers to [21, 23, 35] for detailed discussions
and proofs. In the following proposition, we equip the domain $\Omega$ with a
semiclassical Sobolev norm

$\|u\|^{2}_{H^{1}_{\mathrm{scl}}(\Omega)}=\|u\|^{2}_{L^{2}(\Omega)}+\|hDu\|^{2}_{L^{2}(\Omega)}.$

######  Proposition 2.1.

Let $A\in W^{1,\infty}(\Omega,\mathbb{C}^{n})$ and $q\in
L^{\infty}(\Omega,\mathbb{C})$. Let $\zeta\in\mathbb{C}^{n}$ be a complex
vector such that $\zeta\cdot\zeta=0,\>\zeta=\zeta^{(0)}+\zeta^{(1)}$, where
$\zeta^{(0)}$ is independent of $h>0$,
$|\mathrm{Re}\zeta^{(0)}|=|\mathrm{Im}\zeta^{(0)}|=1$, and
$\zeta^{(1)}=\mathcal{O}(h)$ as $h\to 0$. Then for all $h>0$ small enough,
there exists a solution $u\in H^{4}(\Omega)$ to the equation
$\mathcal{L}_{A,q}u=0$ in $\Omega$ of the form

$\displaystyle
u(x,\zeta,h)=e^{\frac{ix\cdot\zeta}{h}}(a(x,\zeta^{(0)})+r(x,\zeta,h)),$

where $a\in C^{\infty}(\overline{\Omega})$ satisfies the transport equation

$(\zeta^{(0)}\cdot\nabla)^{2}a=0\quad\text{in}\quad\Omega,$ (2.1)

and $\|r\|_{H^{1}_{\mathrm{scl}}(\Omega)}=\mathcal{O}(h)$ as $h\to 0$.

We shall utilize the following quantitative version of the Riemann-Lebesgue
lemma to control the error terms involving Fourier transforms. This result was
originally proved in [12, Lemma 2.4].

######  Lemma 2.2.

Let $f\in L^{1}(\mathbb{R}^{n})$ and suppose that there exist constants
$\delta>0$, $C_{0}>0$ and $\alpha\in(0,1)$ such that for $|y|<\delta$, we have

$\|f(\cdot-y)-f(y)\|_{L^{1}(\mathbb{R}^{n})}\leq C_{0}|y|^{\alpha}.$ (2.2)

Then there exist constant $C=C(C_{0},\|f\|_{L^{1}},n,\delta,\alpha)>0$ and
$\varepsilon_{0}>0$ such that the inequality

$|\mathcal{F}(f)(\xi)|\leq
C(e^{-\frac{\varepsilon^{2}|\xi|^{2}}{4\pi}}+\varepsilon^{\alpha})$ (2.3)

holds for any $0<\varepsilon<\varepsilon_{0}$.

##  3\. Estimate for the Magnetic Potential

The goal of this section is to prove Theorem 1.1. We shall begin this section
by constructing CGO solutions vanishing on the inaccessible part $\Gamma_{0}$
of the boundary, followed by establishing an estimate for the magnetic field
$dA$. We shall then utilize a special Hodge decomposition to verify estimate
(1.5).

###  3.1. Construction of CGO solutions

In this subsection we utilize the reflection arguments originally developed in
[20] to construct CGO solutions $u_{2}$ and $v$ to the equations
$\mathcal{L}_{A^{(2)},q_{2}}u_{2}=0$ and
$\mathcal{L}_{A^{(1)},q_{1}}^{\ast}v=0$ in $\Omega$, respectively. In
particular, the solutions $u_{2}$ and $v$ satisfy

$u_{2}|_{\Gamma_{0}}=(\Delta u_{2})|_{\Gamma_{0}}=0,\quad
v|_{\Gamma_{0}}=(\Delta v)|_{\Gamma_{0}}=0.$ (3.1)

Let us first introduce some notations. In what follows we shall denote
$x^{\ast}=(x^{\prime},-x_{n})$ for all
$x=(x^{\prime},x_{n})\in\mathbb{R}^{n-1}\times\mathbb{R}$, and denote
$f^{\ast}(x)=f(x^{\ast})$ for any function $f$.

We shall proceed to construct CGO solutions as follows. For a given non-zero
vector $\xi\in\mathbb{R}^{n}$, we choose unit vectors
$\mu^{(1)},\mu^{(2)}\in\mathbb{R}^{n}$ in an appropriate way, which we shall
specify later. The vectors $\xi$, $\mu^{(1)}$, and $\mu^{(2)}$ will be used to
construct CGO solutions. Let us write $\xi=(\xi^{\prime},\xi_{n})$, where
$\xi^{\prime}=(\xi_{1},\dots,\xi_{n-1})\in\mathbb{R}^{n-1}$, and assume first
that $\xi^{\prime}\neq 0$. We then define
$e(1)=(\frac{\xi^{\prime}}{|\xi^{\prime}|},0)$, $e(n)=(0,\dots,0,1)$, and
complete $e(1),\dots,e(n)$ to an orthonormal basis in $\mathbb{R}^{n}$, which
we shall denote by $\\{e(1),e(2),\dots,e(n)\\}$. In this basis, the vector
$\xi$ has the coordinate representation
$\widetilde{\xi}=(|\xi^{\prime}|,0,\dots,0,\xi_{n})$. If $\xi^{\prime}=0$, we
let $\\{e(1),\dots,e(n)\\}$ be the standard basis in $\mathbb{R}^{n}$.

For any pair of vectors $\eta^{(1)},\eta^{(2)}\in\mathbb{R}^{n}$, we denote by
$\widetilde{\eta}^{(1)},\widetilde{\eta}^{(2)}$ their coordinate
representations in the basis $\\{e(1),\dots,e(n)\\}$. Then we have

$\eta^{(1)}\cdot\eta^{(2)}=\widetilde{\eta}^{(1)}\cdot\widetilde{\eta}^{(2)},\quad\eta^{(1)}_{n}=\widetilde{\eta}^{(1)}_{n},\quad\eta^{(2)}_{n}=\widetilde{\eta}^{(2)}_{n}.$

Let $\mu^{(1)},\mu^{(2)}$ be vectors in $\mathbb{R}^{n}$ such that

$\widetilde{\mu}^{(1)}=\bigg{(}-\frac{\xi_{n}}{|\xi|},0,\dots,0,\frac{|\xi^{\prime}|}{|\xi|}\bigg{)},\quad\widetilde{\mu}^{(2)}=(0,1,0,\dots,0).$
(3.2)

Then direct computations show that

$\mu^{(1)}\cdot\mu^{(2)}=\mu^{(1)}\cdot\xi=\mu^{(2)}\cdot\xi=0,\quad|\mu^{(1)}|=|\mu^{(2)}|=1.$
(3.3)

We now set

$\displaystyle\zeta_{1}=\frac{h\xi}{2}+\sqrt{1-h^{2}\frac{|\xi|^{2}}{4}}\mu^{(1)}+i\mu^{(2)},\quad\zeta_{2}=-\frac{h\xi}{2}+\sqrt{1-h^{2}\frac{|\xi|^{2}}{4}}\mu^{(1)}-i\mu^{(2)},$
(3.4)

where $0<h\ll 1$ is a small semiclassical parameter such that
$1-h^{2}\frac{|\xi|^{2}}{4}\geq 0$. It follows from direct computations that
$\zeta_{j}\cdot\zeta_{j}=0$, $j=1,2$, and
$\frac{\zeta_{2}-\overline{\zeta_{1}}}{h}=-\xi$. Moreover, we see that
$\zeta_{1}=\mu^{(1)}+i\mu^{(2)}+\mathcal{O}(h)$ and
$\zeta_{2}=\mu^{(1)}-i\mu^{(2)}+\mathcal{O}(h)$ as $h\to 0$. From the
definition of $\zeta^{(0)}$ given in Proposition 2.1, we have

$\zeta_{1}^{(0)}=\mu^{(1)}+i\mu^{(2)}\quad\text{ and
}\quad\zeta_{2}^{(0)}=\mu^{(1)}-i\mu^{(2)}.$

In order to fulfill the conditions in (3.1), similar to [20, 22, 29, 35], we
reflect $\Omega$ with respect to the hyperplane $\\{x_{n}=0\\}$ and denote
this reflection by

$\Omega^{\ast}:=\\{x^{\ast}:x=(x^{\prime},x_{n})\in\Omega\\},\quad
x^{\prime}=(x_{1},\dots,x_{n-1})\in\mathbb{R}^{n-1}.$

We also reflect the magnetic potential $A=(A_{1},\dots,A_{n})$ and the
electric potential $q$ with respect to $\\{x_{n}=0\\}$. Specifically, we shall
make even extensions for $A_{j}$, $j=1,\dots,n-1$, and make an odd extension
for $A_{n}$. That is, we set

$\widetilde{A}_{j}(x)=\begin{cases}A_{j}(x^{\ast}),\>x_{n}<0,\\\
A_{j}(x),\>x_{n}>0,\end{cases}j=1,\dots,n-1,\quad\widetilde{A}_{n}(x)=\begin{cases}A_{n}(x),\>x_{n}<0,\\\
-A_{n}(x^{\ast}),\>x_{n}>0.\end{cases}$ (3.5)

In the same way, we extend electric potential $q$ evenly to write

$\widetilde{q}(x)=\begin{cases}q(x),\quad x_{n}<0,\\\ q(x^{\ast}),\quad
x_{n}>0.\end{cases}$ (3.6)

As explained in [35, Section 5], we have $\widetilde{A}^{(j)}\in
W^{1,\infty}(\Omega\cup\Omega^{\ast};\mathbb{C}^{n})$ and $q_{j}\in
L^{\infty}(\Omega\cup\Omega^{\ast};\mathbb{C})$, $j=1,2$. Furthermore, we get
$\widetilde{A}^{(j)}\in\mathcal{A}(\Omega\cup\Omega^{\ast},2M)$ and
$q_{j}\in\mathcal{Q}(\Omega\cup\Omega^{\ast},2M)$.

Let $B=B(0,\widetilde{R})$ be a ball in $\mathbb{R}^{n}$ centered at the
origin of radius $\widetilde{R}\geq 1$ such that
$\Omega\cup\Omega^{\ast}\subset\subset B$. By Proposition 2.1, there exists a
solution $v\in H^{4}(B)$ to the equation
$\mathcal{L}_{\widetilde{A}^{(1)},\widetilde{q}_{1}}^{\ast}\widetilde{v}=0$ in
$B$ of the form

$\widetilde{v}(x,\zeta_{1},h)=e^{\frac{ix\cdot\zeta_{1}}{h}}(a_{1}(x,\mu^{(1)}+i\mu^{(2)})+r_{1}(x,\zeta_{1},h)).$
(3.7)

There also exists a solution $u_{2}\in H^{4}(B)$ to the equation
$\mathcal{L}_{\widetilde{A}^{(2)},\widetilde{q}_{2}}\widetilde{u_{2}}=0$ in
$B$ given by

$\widetilde{u_{2}}(x,\zeta_{2},h)=e^{\frac{ix\cdot\zeta_{2}}{h}}(a_{2}(x,\mu^{(1)}-i\mu^{(2)})+r_{2}(x,\zeta_{2},h)).$
(3.8)

Here the amplitudes $a_{j}\in C^{\infty}(\overline{B})$, $j=1,2$, solve the
transport equations

$((\mu^{(1)}+i\mu^{(2)})\cdot\nabla)^{2}a_{1}(x,\mu^{(1)}+i\mu^{(2)})=0\>\text{
and
}((\mu^{(1)}-i\mu^{(2)})\cdot\nabla)^{2}a_{2}(x,\mu^{(1)}-i\mu^{(2)})=0\>\text{
in }B,$ (3.9)

respectively, and the correction terms $r_{j}\in H^{4}(B)$, $j=1,2$, satisfy
the estimate

$\|r_{j}\|_{H^{1}_{\mathrm{scl}}(B)}=\mathcal{O}(h),\quad h\to 0.$ (3.10)

By the interior elliptic regularity, we have
$\widetilde{v},\widetilde{u_{2}}\in H^{4}(\Omega\cup\Omega^{\ast})$ and

$\|r_{j}\|_{H^{1}_{\mathrm{scl}}(\Omega\cup\Omega^{\ast})}=\mathcal{O}(h),\quad
h\to 0.$ (3.11)

We now proceed to construct CGO solutions in $\Omega$ satisfying the
conditions in (3.1). To that end, we set

$v(x)=\widetilde{v}(x)-\widetilde{v^{\ast}}(x)\quad\text{and}\quad
u_{2}(x)=\widetilde{u_{2}}(x)-\widetilde{u_{2}^{\ast}}(x),$

where $\widetilde{v}$ and $\widetilde{u_{2}}$ are CGO solutions given by (3.7)
and (3.8), respectively. Then we have $v,u_{2}\in H^{4}(\Omega)$. Furthermore,
direct computations yield that $v$ and $u_{2}$ satisfy the conditions in
(3.1). Hence, the explicit forms of desired CGO solutions $v$ and $u_{2}$ in
$\Omega$ are given by

$v(x,\zeta_{1},h)=e^{\frac{ix\cdot\zeta_{1}}{h}}(a_{1}(x,\zeta^{(0)}_{1})+r_{1}(x,\zeta_{1},h))-e^{\frac{ix\cdot\zeta_{1}^{\ast}}{h}}(a_{1}^{\ast}(x,\zeta^{(0)}_{1})+r_{1}^{\ast}(x,\zeta_{1},h))$
(3.12)

and

$u_{2}(x,\zeta_{2},h)=e^{\frac{ix\cdot\zeta_{2}}{h}}(a_{2}(x,\zeta^{(0)}_{2})+r_{2}(x,\zeta_{2},h))-e^{\frac{ix\cdot\zeta_{2}^{\ast}}{h}}(a_{2}^{\ast}(x,\zeta^{(0)}_{2})+r_{2}^{\ast}(x,\zeta_{2},h)).$
(3.13)

###  3.2. Derivation of an estimate for the magnetic field

The goal of this subsection is to derive an estimate for the magnetic field
$d(A^{(2)}-A^{(1)})$. Our starting point is the following integral inequality.

######  Lemma 3.1.

Let $A^{(j)}\in W^{1,\infty}(\Omega,\mathbb{C}^{n})$, $q_{j}\in
L^{\infty}(\Omega,\mathbb{C})$, and let $\Lambda_{A^{(j)},q_{j}}^{\Gamma}$ be
the partial Dirichlet-to-Neumann map associated with the operator
$\mathcal{L}_{A^{(j)},q_{j}}$, $j=1,2$. Then there exist a constant $C>0$ such
that the following estimate

$\bigg{|}\int_{\Omega}[(A^{(2)}-A^{(1)})\cdot
Du_{2}+(q_{2}-q_{1})u_{2}]\overline{v}dx\bigg{|}\leq
Ce^{\frac{9R}{h}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|$
(3.14)

holds for any functions $u_{2},v\in H^{4}(\Omega)$ that satisfy the equations
$\mathcal{L}_{A^{(2)},q_{2}}u_{2}=0$ and
$\mathcal{L}_{A^{(1)},q_{1}}^{\ast}v=0$ in $\Omega$.

###### Proof.

We start by recalling the Green’s formula, see [17]:

$\displaystyle\int_{\Omega}(\mathcal{L}_{A,q}u)\overline{v}dx-\int_{\Omega}u\overline{\mathcal{L}_{A,q}^{\ast}v}dx=$
$\displaystyle-i\int_{\partial\Omega}\nu\cdot
uA\overline{v}dS+\int_{\partial\Omega}\partial_{\nu}(\Delta
u)\overline{v}dS-\int_{\partial\Omega}(\Delta u)\overline{\partial_{\nu}v}dS$
(3.15) $\displaystyle+\int_{\partial\Omega}\partial_{\nu}u(\overline{\Delta
v})dS-\int_{\partial\Omega}u\overline{\partial_{\nu}(\Delta v)}dS$

holds for all $u,v\in H^{4}(\Omega)$. Here
$\mathcal{L}_{A,q}^{\ast}=\mathcal{L}_{\overline{A},q-i\operatorname{div}\overline{A}}$
is the $L^{2}$-adjoint of $\mathcal{L}_{A,q}$ and $\nu$ is the outward unit
normal to the boundary $\partial\Omega$.

Let $u_{j}$, $j=1,2$, be solutions to the boundary value problem (1.2) with
coefficients $A^{(j)},q_{j}$, respectively. By a direct computation, we see
that $u:=u_{1}-u_{2}$ solves the following equation

$\begin{cases}\mathcal{L}_{A^{(1)},q_{1}}u=(A^{(2)}-A^{(1)})\cdot
Du_{2}+(q_{2}-q_{1})u_{2}\quad\text{ in }\quad\Omega,\\\ u=0\quad\text{ on
}\quad\partial\Omega,\\\ \Delta u=0\quad\text{ on
}\quad\partial\Omega.\end{cases}$ (3.16)

Multiplying both sides of the first equation in (3.16) by $\overline{v}$ and
applying the Green’s formula (3.15), we obtain the integral identity

$\int_{\Omega}[(A^{(2)}-A^{(1)})\cdot
Du_{2}+(q_{2}-q_{1})u_{2}]\overline{v}dx=\int_{\Gamma}\partial_{\nu}(\Delta(u_{1}-u_{2}))\overline{v}dS+\int_{\Gamma}\partial_{\nu}(u_{1}-u_{2})(\overline{\Delta
v})dS.$ (3.17)

Let us now estimate the right-hand side of (3.17). To that end, we apply the
Cauchy-Schwartz inequality and the trace theorem to deduce that

$\displaystyle\bigg{|}$
$\displaystyle\int_{\Gamma}\partial_{\nu}(\Delta(u_{1}-u_{2}))\overline{v}dS+\int_{\Gamma}\partial_{\nu}(u_{1}-u_{2})(\overline{\Delta
v})dS\bigg{|}$ (3.18)
$\displaystyle\leq\|\partial_{\nu}(\Delta(u_{1}-u_{2}))\|_{L^{2}(\Gamma)}\|v\|_{L^{2}(\Gamma)}+\|\partial_{\nu}(u_{1}-u_{2})\|_{L^{2}(\Gamma)}\|\Delta
v\|_{L^{2}(\Gamma)}$ $\displaystyle\leq
C(\|\partial_{\nu}(\Delta(u_{1}-u_{2}))\|_{L^{2}(\Gamma)}\|v\|_{H^{1}(\Omega)}+\|\partial_{\nu}(u_{1}-u_{2})\|_{L^{2}(\Gamma)}\|\Delta
v\|_{H^{1}(\Omega)})$ $\displaystyle\leq
C(\|\partial_{\nu}(\Delta(u_{1}-u_{2}))\|_{H^{\frac{1}{2}}(\Gamma)}+\|\partial_{\nu}(u_{1}-u_{2})\|_{H^{\frac{5}{2}}(\Gamma)})(\|v\|_{H^{1}(\Omega)}+\|\Delta
v\|_{H^{1}(\Omega)})$ $\displaystyle\leq
C\|(\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma})(f,g)\|_{H^{\frac{5}{2},\frac{1}{2}}(\Gamma)}(\|v\|_{H^{1}(\Omega)}+\|\Delta
v\|_{H^{1}(\Omega)})$ $\displaystyle\leq
C\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|\|(f,g)\|_{H^{\frac{7}{2},\frac{3}{2}}(\Gamma)}(\|v\|_{H^{1}(\Omega)}+\|\Delta
v\|_{H^{1}(\Omega)})$ $\displaystyle\leq
C\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|(\|u_{2}\|_{H^{4}(\Omega)}+\|\Delta
u_{2}\|_{H^{2}(\Omega)})(\|v\|_{H^{1}(\Omega)}+\|\Delta v\|_{H^{1}(\Omega)}).$

To proceed, we need to estimate the $H^{s}$-norms of $u_{2},v$, and their
derivatives appearing in the inequality above. Since $\Omega$ is a bounded
domain, there exists a constant $R>0$ such that $\Omega\subseteq B(0,R)$.
Thus, we have the inequality $\big{|}e^{\frac{ix\cdot\zeta_{j}}{h}}\big{|}\leq
e^{\frac{2R}{h}}$, where we have utilized the fact that $|\zeta_{j}|\leq 2$.
By the same computations as in [13, Section 4], we obtain the estimates

$\|v\|_{H^{1}(\Omega)}\leq\frac{C}{h}e^{\frac{2R}{h}},\quad\|\Delta
v\|_{H^{1}(\Omega)}\leq\frac{C}{h}e^{\frac{2R}{h}},\quad\|u\|_{H^{4}(\Omega)}\leq\frac{C}{h^{4}}e^{\frac{2R}{h}},\quad\|\Delta
u_{2}\|_{H^{2}(\Omega)}\leq\frac{C}{h}e^{\frac{2R}{h}}.$ (3.19)

Therefore, (3.18) and (3.19) yield

$\displaystyle\bigg{|}$
$\displaystyle\int_{\Gamma}\partial_{\nu}(\Delta(u_{1}-u_{2}))\overline{v}dS+\int_{\Gamma}\partial_{\nu}(u_{1}-u_{2})(\overline{\Delta
v})dS\bigg{|}$ $\displaystyle\leq
C\|\Lambda_{A^{(1)},q_{1}}-\Lambda_{A^{(2)},q_{2}}\|\big{(}\frac{C}{h}e^{\frac{2R}{h}}+\frac{C}{h^{4}}e^{\frac{2R}{h}}\big{)}\big{(}\frac{C}{h}e^{\frac{2R}{h}}+\frac{C}{h}e^{\frac{2R}{h}}\big{)}$
$\displaystyle\leq\frac{C}{h^{5}}e^{\frac{4R}{h}}\|\Lambda_{A^{(1)},q_{1}}-\Lambda_{A^{(2)},q_{2}}\|.$

From here, we obtain the claimed estimate (3.14) from the inequality above by
using the fact that $\frac{1}{h}\leq e^{\frac{R}{h}}$. This completes the
proof of Lemma 3.1. ∎

We are now ready to state and prove an estimate for magnetic field.

######  Proposition 3.2.

Let $\Omega\subseteq\\{x\in\mathbb{R}^{n}:x_{n}<0\\}$, $n\geq 3$, be a bounded
domain with smooth connected boundary $\partial\Omega$. Suppose that
$\Gamma_{0}=\partial\Omega\cap\\{x\in\mathbb{R}^{n}:x_{n}=0\\}$ is non-empty,
and let $\Gamma=\partial\Omega\setminus\Gamma_{0}$. Let
$A^{(j)}\in\mathcal{A}(\Omega,M)$ and $q_{j}\in\mathcal{Q}(\Omega,M)$,
$j=1,2$. Then there exists constants $C>0$ and $h_{0}>0$ such that the
estimate

$\|d(A^{(2)}-A^{(1)})\|_{H^{-1}(\Omega)}\leq
Ce^{\frac{10R}{h}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|+Ch^{\frac{\alpha}{n+2}}$
(3.20)

holds for $0<h<h_{0}$ and some $\alpha\in(0,1)$. Here $R>0$ is a constant such
that $\Omega\subseteq B(0,R)$.

###### Proof.

We shall derive (3.20) by estimating the Fourier transform of
$d(A^{(2)}-A^{(1)})$. To that end, we extend $A^{(j)}$ and $q_{j}$, $j=1,2$,
by zero to $\mathbb{R}^{n}\setminus\Omega$ and denote the extensions by the
same letters. It follows from [1, Chapter 1, Theorem 3.4.4] that $A^{(j)}\in
W^{1,\infty}(\mathbb{R}^{n};\mathbb{C}^{n})$. For convenience, we shall denote
$A:=A^{(2)}-A^{(1)}$ and $q:=q_{2}-q_{1}$ for the remainder of this section.

We now substitute the CGO solutions $v$ and $u_{2}$, given by (3.12) and
(3.13), respectively, into the left-hand side of the integral inequality
(3.14), multiply the resulted expression by $h$, and let $h\to 0$. This
requires us to compute the quantities $\overline{v}Du_{2}$ and
$\overline{v}u_{2}$, which contain expressions of the form

$e^{\frac{ix\cdot(\zeta_{2}-\overline{\zeta_{1}})}{h}}=e^{-ix\cdot\xi},\quad
e^{\frac{ix\cdot(\zeta_{2}^{\ast}-\overline{\zeta_{1}^{\ast}})}{h}}=e^{-ix^{\ast}\cdot\xi},\quad
e^{\frac{ix\cdot(\zeta_{2}^{\ast}-\overline{\zeta_{1}})}{h}}=e^{-ix\cdot\xi_{+}},\quad
e^{\frac{ix\cdot(\zeta_{2}-\overline{\zeta_{1}^{\ast}})}{h}}=e^{-ix\cdot\xi_{-}},$
(3.21)

where

$\xi_{\pm}=\bigg{(}\xi^{\prime},\pm\frac{2}{h}\sqrt{1-h^{2}\frac{|\xi|^{2}}{4}}\frac{|\xi^{\prime}|}{|\xi|}\bigg{)}.$
(3.22)

Hence, we write the first term on the left-hand side of (3.14) as

$\displaystyle h\int_{\Omega}A\cdot Du_{2}\overline{v}dx$ (3.23)
$\displaystyle=$ $\displaystyle\int_{\Omega}A\cdot
i(\zeta_{2}e^{-ix\cdot\xi}\overline{a_{1}}a_{2}+\zeta_{2}^{\ast}e^{-ix^{\ast}\cdot\xi}\overline{a_{1}^{\ast}}a_{2}^{\ast})dx+\int_{\Omega}A\cdot
i(-\zeta_{2}e^{-ix\cdot\xi_{-}}\overline{a_{1}^{\ast}}a_{2}-\zeta_{2}^{\ast}e^{-ix\cdot\xi_{+}}\overline{a_{1}}a_{2}^{\ast})dx$
$\displaystyle+\int_{\Omega}A\cdot w_{1}dx$ $\displaystyle:=$ $\displaystyle
I_{1}+I_{2}+I_{3},$

where

$\displaystyle w_{1}$
$\displaystyle=i\zeta_{2}\big{[}e^{-ix\cdot\xi}(\overline{a_{1}}r_{2}+a_{2}\overline{r_{1}}+\overline{r_{1}}r_{2})-e^{-ix\cdot\xi_{-}}(\overline{a_{1}^{\ast}}r_{2}+a_{2}\overline{r_{1}^{\ast}}+\overline{r_{1}^{\ast}}r_{2})\big{]}$
$\displaystyle+i\zeta_{2}^{\ast}\big{[}-e^{-ix\cdot\xi_{+}}(\overline{a_{1}}r_{2}^{\ast}+a_{2}^{\ast}\overline{r_{1}}+\overline{r_{1}}r_{2}^{\ast})+e^{-ix^{\ast}\cdot\xi}(\overline{a_{1}^{\ast}}r_{2}^{\ast}+\overline{r_{1}^{\ast}}a_{2}^{\ast}+\overline{r_{1}^{\ast}}r_{2}^{\ast})\big{]}$
$\displaystyle+h\big{[}e^{-ix\cdot\xi}(\overline{a_{1}}Da_{2}+\overline{a_{1}}Dr_{2}+\overline{r_{1}}Da_{2}+\overline{r_{1}}Dr_{2})-e^{-ix\cdot\xi_{-}}(\overline{a_{1}^{\ast}}Da_{2}+\overline{a_{1}^{\ast}}Dr_{2}+\overline{r_{1}^{\ast}}Da_{2}+\overline{r_{1}^{\ast}}Dr_{2})$
$\displaystyle-e^{-ix\cdot\xi_{+}}(\overline{a_{1}}Da_{2}^{\ast}+\overline{a_{1}}Dr_{2}^{\ast}+\overline{r_{1}}Da_{2}^{\ast}+\overline{r_{1}}Dr_{2}^{\ast})+e^{-ix^{\ast}\cdot\xi}(\overline{a_{1}^{\ast}}Da_{2}^{\ast}+\overline{a_{1}^{\ast}}Dr_{2}^{\ast}+\overline{r_{1}^{\ast}}Da_{2}^{\ast}+\overline{r_{1}^{\ast}}Dr_{2}^{\ast})\big{]}.$

We now investigate the limit of each integral in (3.23) as $h\to 0$. To
analyze $I_{1}$, we recall the form of $\zeta_{2}$ in (3.4) and the definition
of $\widetilde{A}$ from (3.5), in conjunction with making a change of
variables, to deduce that

$I_{1}=i\zeta_{2}\cdot\int_{\Omega\cup\Omega^{\ast}}\widetilde{A}e^{-ix\cdot\xi}\overline{a_{1}}a_{2}dx\to
i(\mu^{(1)}-i\mu^{(2)})\cdot\int_{\Omega\cup\Omega^{\ast}}\widetilde{A}e^{-ix\cdot\xi}\overline{a_{1}}a_{2}dx,\quad
h\to 0.$ (3.24)

By replacing the vector $\mu^{(2)}$ in (3.24) by $-\mu^{(2)}$, we have

$I_{1}\to
i(\mu^{(1)}+i\mu^{(2)})\cdot\int_{\Omega\cup\Omega^{\ast}}\widetilde{A}e^{-ix\cdot\xi}\overline{a_{1}}a_{2}dx,\quad
h\to 0.$ (3.25)

Hence, in view of (3.24) and (3.25), we see that

$I_{1}\to
i\mu\cdot\int_{\Omega\cup\Omega^{\ast}}\widetilde{A}e^{-ix\cdot\xi}\overline{a_{1}}a_{2}dx,\quad
h\to 0,$ (3.26)

for any $\mu\in\mathrm{Span}\\{\mu^{(1)},\mu^{(2)}\\}$ and all
$\xi\in\mathbb{R}^{n}$ satisfying (3.3). By choosing $a_{1}=a_{2}=1$, which
clearly satisfy the respective transport equation in (3.9), we get

$I_{1}\to
i\mu\cdot\int_{\Omega\cup\Omega^{\ast}}\widetilde{A}e^{-ix\cdot\xi}dx=i\mathcal{F}(d\widetilde{A})(\xi),\quad
h\to 0,$ (3.27)

where $d\widetilde{A}$ is a 2-form defined by (1.4).

Turning attention to $I_{2}$, we argue similarly as above to obtain

$\displaystyle I_{2}$
$\displaystyle\to-i\mu\cdot\int_{\Omega\cup\Omega^{\ast}}\widetilde{A}(e^{-ix\cdot\xi_{-}}+e^{-ix\cdot\xi_{+}})dx$
(3.28)
$\displaystyle=-i\mathcal{F}(d\widetilde{A})(\xi_{-})-i\mathcal{F}(d\widetilde{A})(\xi_{+}),\quad
h\to 0.$

for any $\mu\in\mathrm{Span}\\{\mu^{(1)},\mu^{(2)}\\}$ and all
$\xi\in\mathbb{R}^{n}$ that satisfies (3.3). By Lemma 2.2 and the definition
of $\xi_{\pm}$ given by (3.22), there exist constant $C>0$,
$\varepsilon_{0}>0$, and $0<\alpha<1$ such that the estimate

$|I_{2}|\leq
C\big{(}e^{-\frac{\varepsilon^{2}}{\pi}\cdot\frac{|\xi^{\prime}|^{2}}{h^{2}|\xi|^{2}}}+\varepsilon^{\alpha}\big{)}$
(3.29)

is valid for any $0<\varepsilon<\varepsilon_{0}$.

Finally, we estimate $I_{3}$. Note that estimate (3.11) yields

$\|Dr_{j}\|_{L^{2}(\Omega\cup\Omega^{\ast})}=\mathcal{O}(1),\quad h\to 0.$
(3.30)

Since the amplitudes $a_{j}\in C^{\infty}(\overline{\Omega})$, $j=1,2$, we
utilize estimates (3.10) and (3.30), in conjunction with the Cauchy-Schwartz
inequality, to deduce that

$|I_{3}|\leq Ch,\quad h\to 0.$ (3.31)

We next proceed to estimate the second term on the left-hand side of the
integral inequality (3.14). To that end, after substituting the CGO solutions
$v$ and $u_{2}$ given by (3.12) and (3.13), respectively, into (3.14), we have

$\displaystyle h\int_{\Omega}qu_{2}\overline{v}dx$ $\displaystyle=$
$\displaystyle
h\int_{\Omega}q[e^{-ix\cdot\xi}(\overline{a_{1}}a_{2}+\overline{a_{1}}r_{2}+a_{2}\overline{r_{1}}+\overline{r_{1}}r_{2})+e^{-ix^{\ast}\cdot\xi}(\overline{a_{1}^{\ast}}a_{2}^{\ast}+\overline{a_{1}^{\ast}}r_{2}^{\ast}+a_{2}^{\ast}\overline{r_{1}^{\ast}}+\overline{r_{1}^{\ast}}r_{2}^{\ast})$
$\displaystyle\quad-e^{-ix\cdot\xi_{-}}(\overline{a_{1}^{\ast}}a_{2}+\overline{a_{1}^{\ast}}r_{2}+a_{2}\overline{r_{1}^{\ast}}+\overline{r_{1}^{\ast}}r_{2})-e^{-ix\cdot\xi_{+}}(\overline{a_{1}}a_{2}^{\ast}+\overline{a_{1}}r_{2}^{\ast}+a_{2}^{\ast}\overline{r_{1}}+\overline{r_{1}}r_{2}^{\ast})]dx.$

As $a_{j}\in C^{\infty}(\overline{\Omega})$, $j=1,2$, we apply estimate (3.10)
and the Cauchy-Schwartz inequality to conclude that the following estimate is
valid

$\bigg{|}h\int_{\Omega}(q_{2}-q_{1})u_{2}\overline{v}dx\bigg{|}\leq Ch,\quad
h\to 0.$ (3.32)

Therefore, by combining (3.27), (3.29), (3.31), and (3.32), we deduce from the
integral inequality (3.14) that

$|\mathcal{F}(d\widetilde{A})(\xi)|\leq
C\big{(}e^{\frac{9R}{h}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|+h+e^{-\frac{\varepsilon^{2}}{4\pi}\cdot\frac{4|\xi^{\prime}|^{2}}{h^{2}|\xi|^{2}}}+\varepsilon^{\alpha}\big{)},\quad
h\to 0.$ (3.33)

We are now ready to derive a stability estimate $\|dA\|_{H^{-1}(\Omega)}$.
With a parameter $\rho>0$ to be specified later, we consider the set

$E(\rho)=\\{\xi\in\mathbb{R}^{n}:|\xi^{\prime}|\leq\rho,\>|\xi_{n}|\leq\rho\\}.$
(3.34)

Then an application of the Parseval’s formula gives us

$\|d\widetilde{A}\|_{H^{-1}(\mathbb{R}^{n})}^{2}\leq\int_{E(\rho)}\frac{|\mathcal{F}(d\widetilde{A})(\xi)|^{2}}{1+|\xi|^{2}}d\xi+\int_{\mathbb{R}^{n}\setminus
E(\rho)}\frac{|\mathcal{F}(d\widetilde{A})(\xi)|^{2}}{1+|\xi|^{2}}d\xi.$
(3.35)

Let us estimate the terms on the right-hand side of (3.35). For the second
term, by the Plancherel theorem, we have

$\displaystyle\int_{\mathbb{R}^{n}\setminus
E(\rho)}\frac{|\mathcal{F}(d\widetilde{A})(\xi)|^{2}}{1+|\xi|^{2}}d\xi$
$\displaystyle\leq C\int_{\mathbb{R}^{n}\setminus
E(\rho)}\frac{|\mathcal{F}(d\widetilde{A})(\xi)|^{2}}{1+|\rho|^{2}}d\xi$
(3.36) $\displaystyle\leq
C\frac{1}{\rho^{2}}\|d\widetilde{A}\|_{L^{2}(\mathbb{R}^{n})}$
$\displaystyle\leq C\frac{1}{\rho^{2}}.$

We now turn to estimate the first term. Observe that the inequality
$|\xi|^{2}\leq 2\rho^{2}$ holds for all $\xi\in E(\rho)$. Hence, we get

$e^{-\frac{\varepsilon^{2}}{\pi}\cdot\frac{|\xi^{\prime}|^{2}}{h^{2}|\xi|^{2}}}\leq
e^{-\frac{\varepsilon^{2}}{\pi}\cdot\frac{|\xi^{\prime}|^{2}}{2h^{2}\rho^{2}}}.$

Since the set $\\{\xi\in\mathbb{R}^{n}:|\xi^{\prime}|=0\\}$ is of zero
Lebesgue measure, we estimate the first term on the right-hand side of (3.35)
as

$\displaystyle\int_{E(\rho)}\frac{|\mathcal{F}(d\widetilde{A})(\xi)|^{2}}{1+|\xi|^{2}}d\xi$
(3.37) $\displaystyle\leq
C[e^{\frac{18R}{h}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{2}+h^{2}+\varepsilon^{2\alpha}]\int_{E(\rho)}\frac{1}{1+|\xi|^{2}}d\xi+C\int_{E(\rho)}\frac{e^{-\frac{\varepsilon^{2}}{\pi}\cdot\frac{|\xi^{\prime}|^{2}}{h^{2}\rho^{2}}}}{1+|\xi|^{2}}d\xi$
$\displaystyle\leq
C\rho^{n}[e^{\frac{18R}{h}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{2}+h^{2}+\varepsilon^{2\alpha}]+C\int_{-\rho}^{\rho}\int_{B^{\prime}(0,\rho)}\frac{e^{-\frac{\varepsilon^{2}}{\pi}\cdot\frac{|\xi^{\prime}|^{2}}{h^{2}\rho^{2}}}}{1+|\xi|^{2}}d\xi^{\prime}d\xi_{n}.$

where $B^{\prime}(0,\rho)$ is a ball in $\mathbb{R}^{n-1}$ centered at the
origin with radius $\rho$.

We next follow the arguments in [12, Section 3] closely to estimate the
integral

$\int_{-\rho}^{\rho}\int_{B^{\prime}(0,\rho)}\frac{e^{-\frac{\varepsilon^{2}}{\pi}\cdot\frac{|\xi^{\prime}|^{2}}{h^{2}\rho^{2}}}}{1+|\xi^{2}}d\xi^{\prime}d\xi_{n}.$

To that end, by choosing $\varepsilon>0$ such that $\varepsilon=\sqrt{h}$ and
using the polar coordinates, we obtain

$\displaystyle\int_{-\rho}^{\rho}\int_{B^{\prime}(0,\rho)}\frac{e^{-\frac{\varepsilon^{2}}{\pi}\cdot\frac{|\xi^{\prime}|^{2}}{h^{2}\rho^{2}}}}{1+|\xi|^{2}}d\xi^{\prime}d\xi_{n}$
$\displaystyle\leq
2\rho\int_{B^{\prime}(0,\rho)}e^{-\frac{\varepsilon^{2}}{\pi}\cdot\frac{|\xi^{\prime}|^{2}}{h^{2}\rho^{2}}}d\xi^{\prime}$
(3.38) $\displaystyle\leq C\rho\int_{0}^{\rho}r^{n-2}e^{-\frac{1}{\pi
h\rho^{2}}r^{2}}dr$
$\displaystyle=C\rho^{n}h^{\frac{n-1}{2}}\int_{0}^{h^{-\frac{1}{2}}}u^{n-2}e^{-\frac{1}{\pi}u^{2}}du$
$\displaystyle\leq
C\rho^{n}h^{\frac{n-1}{2}}\int_{0}^{\infty}u^{n-2}e^{-\frac{1}{\pi}u^{2}}du\leq
C\rho^{n}h^{\frac{n-1}{2}},$

where we have made a change of variables $r=h^{\frac{1}{2}}\rho u$.

Therefore, by combining estimates (3.36)–(3.38), we deduce from (3.35) that

$\|d\widetilde{A}\|_{H^{-1}(\mathbb{R}^{n})}^{2}\leq
C\rho^{n}e^{\frac{18R}{h}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{2}+C\rho^{n}h^{2}+C\rho^{n}h^{\alpha}+C\rho^{n}h^{\frac{n-1}{2}}+C\frac{1}{\rho^{2}}.$

Furthermore, since $0<h\ll 1$, $\frac{n-1}{2}\geq 1$, and $0<\alpha<1$, we
have

$\|d\widetilde{A}\|_{H^{-1}(\mathbb{R}^{n})}^{2}\leq
C\rho^{n}e^{\frac{18R}{h}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{2}+C\rho^{n}h^{\alpha}+C\frac{1}{\rho^{2}}.$

Let us now choose $\rho$ such that $\rho^{n}h^{\alpha}=\frac{1}{\rho^{2}}$,
i.e., $\rho=h^{-\frac{\alpha}{n+2}}$. Then we obtain

$\|d\widetilde{A}\|_{H^{-1}(\mathbb{R}^{n})}^{2}\leq
Ch^{-\frac{n\alpha}{n+2}}e^{\frac{18R}{h}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{2}+Ch^{\frac{2\alpha}{n+2}}.$

Since $\frac{n\alpha}{n+2}<2$, we get

$\|d\widetilde{A}\|_{H^{-1}(\mathbb{R}^{n})}^{2}\leq
Ce^{\frac{20R}{h}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{2}+Ch^{\frac{2\alpha}{n+2}}.$
(3.39)

From here, the claimed estimate (3.20) follows immediately from (3.39) by an
application of the inequality

$\|dA\|_{H^{-1}(\Omega)}\leq\|d\widetilde{A}\|_{H^{-1}(\mathbb{R}^{n})}.$

This completes the proof of Proposition 3.2. ∎

For later purposes we need an estimate for $\|dA\|_{L^{\infty}(\Omega)}$.
Recall that we assumed an a priori bound for $\|A\|_{H^{s}(\Omega)}$,
$s>\frac{n}{2}+1$, in the definition of the admissible set
$\mathcal{A}(\Omega,M)$. Thus, there exists a constant $\eta>0$ such that
$s=\frac{n}{2}+2\eta$. Hence, we apply the Sobolev embedding theorem and the
interpolation theorem to derive the following estimate:

$\displaystyle\|dA\|_{L^{\infty}(\Omega)}$ $\displaystyle\leq
C\|dA\|_{H^{\frac{n}{2}+\eta}(\Omega)}$ (3.40) $\displaystyle\leq
C\|dA\|_{H^{-1}(\Omega)}^{\frac{\eta}{1+s}}\|dA\|_{H^{s}(\Omega)}^{\frac{1+s-\eta}{1+s}}$
$\displaystyle\leq C\|dA\|_{H^{-1}(\Omega)}^{\frac{\eta}{1+s}}$
$\displaystyle\leq
Ce^{\frac{10R}{h}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{\frac{\eta}{1+s}}+Ch^{\frac{\alpha\eta}{(n+2)(1+s)}}.$

###  3.3. Estimating the magnetic potential

In this subsection we verify estimate (1.5), thus complete the proof of
Theorem 1.1. The key step is to apply a special Hodge decomposition of a
vector field [31, Theorem 3.3.2].

######  Lemma 3.3.

Let $\Omega\subseteq\mathbb{R}^{n}$, $n\geq 3$, be a bounded domain with
smooth boundary $\partial\Omega$, and let $A\in
W^{1,\infty}(\Omega,\mathbb{C}^{n})$ be a complex-valued vector field. Then
there exist uniquely determined vector field $A^{\mathrm{sol}}\in
W^{1,\infty}(\Omega,\mathbb{C}^{n})$ and function $\varphi\in
W^{2,\infty}(\Omega,\mathbb{C})$ such that

$A=A^{\mathrm{sol}}+\nabla\varphi,\quad\operatorname{div}A^{\mathrm{sol}}=0,\quad\varphi|_{\partial\Omega}=0.$
(3.41)

Furthermore, the following inequalities

$\|\varphi\|_{W^{2,\infty}(\Omega)}\leq
C\|A\|_{W^{1,\infty}(\Omega)},\quad\|A^{\mathrm{sol}}\|_{W^{1,\infty}(\Omega)}\leq
C\|A\|_{W^{1,\infty}(\Omega)},$ (3.42)

are valid, where the constant $C$ is independent of $A$.

In particular, the function $\varphi$ in Lemma 3.3 satisfies

$\begin{cases}\Delta\varphi=\operatorname{div}A\quad\text{in}\quad\Omega,\\\
\varphi=0\quad\text{on}\quad\partial\Omega.\end{cases}$

Hence, by the Morrey’s inequality and [33, Lemma 6.2], we get from estimate
(3.40) that

$\|A^{\mathrm{sol}}\|_{L^{\infty}(\Omega)}\leq
C\|dA\|_{L^{\infty}(\Omega)}\leq
Ce^{\frac{10R}{h}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{\frac{\eta}{1+s}}+Ch^{\frac{\alpha\eta}{(n+2)(1+s)}},$
(3.43)

where $s=\frac{n}{2}+2\eta$.

Thanks to (3.43), we may estimate $\|A\|_{L^{\infty}(\Omega)}$ by

$\|A\|_{L^{\infty}(\Omega)}\leq\|A^{\mathrm{sol}}\|_{L^{\infty}(\Omega)}+\|\nabla\varphi\|_{L^{\infty}(\Omega)}\leq\|dA\|_{L^{\infty}(\Omega)}+\|\nabla\varphi\|_{L^{\infty}(\Omega)}.$

Therefore, in order to verify estimate (1.5), we need to estimate
$\|\nabla\varphi\|_{L^{\infty}(\Omega)}$. This is accomplished in the
following lemma.

######  Lemma 3.4.

Let $\varphi\in W^{2,\infty}(\Omega)$ be the function from Lemma 3.3, and let
$\eta>0$ be a constant such that $s=\frac{n}{2}+2\eta$. Then there exists a
constant $h_{0}>0$ such that the estimate

$\|\nabla\varphi\|_{L^{\infty}(\Omega)}\leq
Ce^{\frac{11R}{h}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{\frac{\eta^{2}}{(1+s)^{2}}}+Ch^{\frac{2\alpha\eta^{2}}{(n+2)^{2}(1+s)^{2}}}$
(3.44)

holds for all $h<h_{0}$ and some $\alpha\in(0,1)$. Here $R>0$ is a constant
such that $\Omega\subseteq B(0,R)$.

###### Proof.

We shall follow similar ideas as in the proof of [27, Lemma 3.6]. We extend
$\varphi$ by zero to $\mathbb{R}^{n}\setminus\Omega$ and denote the extension
by the same letter. Our starting point is the analysis of the integrals
appearing in (3.23) as $h\to 0$. By arguing similarly as in the proof of
Proposition 3.2, we get

$\bigg{|}(\mu^{(1)}-i\mu^{(2)})\cdot\int_{\Omega\cup\Omega^{\ast}}\widetilde{A}e^{-ix\cdot\xi}\overline{a_{1}}a_{2}dx\bigg{|}\leq
C\big{(}e^{\frac{9R}{h}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|+e^{-\frac{\varepsilon^{2}}{4\pi}\cdot\frac{4|\xi^{\prime}|^{2}}{h^{2}|\xi|^{2}}}+\varepsilon^{\alpha}+h\big{)}.$
(3.45)

Let us now choose $a_{1}=1$ and $a_{2}$ such that

$((\mu^{(1)}-i\mu^{(2)})\cdot\nabla)a_{2}(x,\mu^{(1)}-i\mu^{(2)})=1\quad\text{
in }\quad\Omega.$ (3.46)

It is straightforward to verify that this choice of $a_{1}$ and $a_{2}$
satisfies the respective transport equation in (3.9).

In view of Lemma 3.3 applied to the domain $\Omega\cup\Omega^{\ast}$, we may
write $\widetilde{A}=\widetilde{A}^{\mathrm{sol}}+\nabla\widetilde{\varphi}$,
where the function $\widetilde{\varphi}\in
W^{2,\infty}(\Omega\cup\Omega^{\ast})$ satisfies
$\widetilde{\varphi}|_{\partial(\Omega\cup\Omega^{\ast})}=0$. This allows us
to rewrite the left-hand side of (3.45) as

$\displaystyle(\mu^{(1)}-i\mu^{(2)})\cdot\int_{\Omega\cup\Omega^{\ast}}\widetilde{A}e^{-ix\cdot\xi}a_{2}dx$
$\displaystyle=(\mu^{(1)}-i\mu^{(2)})\cdot\int_{\Omega\cup\Omega^{\ast}}\widetilde{A}^{\mathrm{sol}}e^{-ix\cdot\xi}a_{2}dx+(\mu^{(1)}-i\mu^{(2)})\cdot\int_{\Omega\cup\Omega^{\ast}}\nabla\widetilde{\varphi}e^{-ix\cdot\xi}a_{2}dx.$

Since $a_{2}\in C^{\infty}(\overline{\Omega})$, we get

$\bigg{|}(\mu^{(1)}-i\mu^{(2)})\cdot\int_{\Omega\cup\Omega^{\ast}}\widetilde{A}^{\mathrm{sol}}e^{-ix\cdot\xi}a_{2}dx\bigg{|}\leq
C\|\widetilde{A}^{\mathrm{sol}}\|_{L^{\infty}(\Omega\cup\Omega^{\ast})}.$

On the other hand, as $a_{2}$ satisfies (3.46), the function
$\widetilde{\varphi}$ satisfies
$\widetilde{\varphi}|_{\partial(\Omega\cup\Omega^{\ast})}=0$, and
$\mu^{(1)}\cdot\xi=\mu^{(2)}\cdot\xi=0$, we integrate by parts to obtain

$(\mu^{(1)}-i\mu^{(2)})\cdot\int_{\Omega\cup\Omega^{\ast}}\nabla\widetilde{\varphi}e^{-ix\cdot\xi}a_{2}dx=-\int_{\Omega\cup\Omega^{\ast}}\widetilde{\varphi}e^{-ix\cdot\xi}dx=-\mathcal{F}(\widetilde{\varphi})(\xi).$

Therefore, it follows from estimate (3.45) that

$|\mathcal{F}(\widetilde{\varphi})(\xi)|\leq
C\big{(}e^{\frac{9R}{h}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|+h+e^{-\frac{\varepsilon^{2}}{4\pi}\cdot\frac{4|\xi^{\prime}|^{2}}{h^{2}|\xi|^{2}}}+\varepsilon^{\alpha}+\|\widetilde{A}^{\mathrm{sol}}\|_{L^{\infty}(\Omega\cup\Omega^{\ast})}\big{)}.$

To estimate
$\|\widetilde{A}^{\mathrm{sol}}\|_{L^{\infty}(\Omega\cup\Omega^{\ast})}$, by
the Morrey’s inequality, we have

$\|\widetilde{A}^{\mathrm{sol}}\|_{L^{\infty}(\Omega\cup\Omega^{\ast})}\leq\|d\widetilde{A}\|_{L^{\infty}(\mathbb{R}^{n})}.$

On the other hand, by arguing similarly as in estimate (3.40), we get from
estimate (3.39) that

$\|d\widetilde{A}\|_{L^{\infty}(\mathbb{R}^{n})}\leq
Ce^{\frac{10R}{h}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{\frac{\eta}{1+s}}+Ch^{\frac{\alpha\eta}{(n+2)(1+s)}}.$
(3.47)

Due to $0<h\ll 1$ and $\frac{\alpha\eta}{(n+2)(1+s)}<1$, we obtain

$|\mathcal{F}(\widetilde{\varphi})(\xi)|\leq
C\big{(}e^{\frac{10R}{h}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{\frac{\eta}{1+s}}+h^{\frac{\alpha\eta}{(n+2)(1+s)}}+e^{-\frac{\varepsilon^{2}}{4\pi}\cdot\frac{4|\xi^{\prime}|^{2}}{h^{2}|\xi|^{2}}}+\varepsilon^{\alpha}\big{)}.$
(3.48)

We next proceed similarly as in the proof of Proposition 3.2 to estimate
$\|\varphi\|_{H^{-1}(\Omega)}$. Let $\rho$ be a parameter that we shall choose
later. Then we apply the Parseval’s formula to write

$\|\widetilde{\varphi}\|_{H^{-1}(\mathbb{R}^{n})}^{2}\leq\int_{E(\rho)}\frac{|\mathcal{F}(\widetilde{\varphi})(\xi)|^{2}}{1+|\xi|^{2}}d\xi+\int_{\mathbb{R}^{n}\setminus
E(\rho)}\frac{|\mathcal{F}(\widetilde{\varphi})(\xi)|^{2}}{1+|\xi|^{2}}d\xi,$

where the set $E(\rho)$ is given by (3.34). By following the same computations
as in (3.36), we have

$\int_{\mathbb{R}^{n}\setminus
E(\rho)}\frac{|\mathcal{F}(\widetilde{\varphi})(\xi)|^{2}}{1+|\xi|^{2}}d\xi\leq
C\frac{1}{\rho^{2}}.$ (3.49)

By choosing $\varepsilon=\sqrt{h}$, utilizing estimate (3.38), in conjunction
with the inequalities $0<h\ll 1$, $\frac{n-1}{2}\geq 1$, and
$\frac{\eta}{(n+2)(1+s)}<1$, we deduce from (3.48) that

$\int_{E(\rho)}\frac{|\mathcal{F}(\widetilde{\varphi})(\xi)|^{2}}{1+|\xi|^{2}}d\xi\leq
C\rho^{n}e^{\frac{20R}{h}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{\frac{2\eta}{1+s}}+C\rho^{n}h^{\frac{2\alpha\eta}{(n+2)(1+s)}}$
(3.50)

for some $\alpha\in(0,1)$. Combining (3.49) and (3.50), we conclude that

$\|\widetilde{\varphi}\|_{H^{-1}(\mathbb{R}^{n})}^{2}\leq
C\rho^{n}e^{\frac{20R}{h}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{\frac{2\eta}{1+s}}+C\rho^{n}h^{\frac{2\alpha\eta}{(n+2)(1+s)}}+C\frac{1}{\rho^{2}}.$

Choosing $\rho>0$ such that
$\rho^{n}h^{\frac{2\alpha\eta}{(n+2)(1+s)}}=\frac{1}{\rho^{2}}$, that is,
$\rho=h^{-\frac{2\alpha\eta}{(n+2)^{2}(1+s)}}$, we get

$\displaystyle\|\widetilde{\varphi}\|_{H^{-1}(\mathbb{R}^{n})}^{2}\leq$
$\displaystyle
Ch^{-\frac{2n\alpha\eta}{(n+2)^{2}(1+s)}}e^{\frac{20R}{h}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{\frac{2\eta}{1+s}}+Ch^{\frac{4\alpha\eta}{(n+2)^{2}(1+s)}}$
$\displaystyle\leq$ $\displaystyle
Ce^{\frac{22R}{h}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{\frac{2\eta}{1+s}}+Ch^{\frac{4\alpha\eta}{(n+2)^{2}(1+s)}}.$

Here we have utilized the inequality $\frac{2n\alpha\eta}{(n+2)^{2}(1+s)}<2$.
Thus, by similar arguments as the end of the proof of Proposition 3.2, we have

$\|\widetilde{\varphi}\|_{H^{-1}(\mathbb{R}^{n})}\leq
Ce^{\frac{11R}{h}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{\frac{\eta}{1+s}}+Ch^{\frac{2\alpha\eta}{(n+2)^{2}(1+s)}}.$
(3.51)

Since $A=\widetilde{A}$ in $\Omega$, we see that
$\varphi|_{\Gamma}=\widetilde{\varphi}|_{\Gamma}=0$. Furthermore, as
$\varphi|_{\Gamma_{0}}=0$, we obtain the inequality

$\|\varphi\|_{H^{-1}(\Omega)}\leq\|\widetilde{\varphi}\|_{H^{-1}(\mathbb{R}^{n})},$

which implies that

$\|\varphi\|_{H^{-1}(\Omega)}\leq
Ce^{\frac{11R}{h}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{\frac{\eta}{1+s}}+Ch^{\frac{2\alpha\eta}{(n+2)^{2}(1+s)}}.$

From here, we have the claimed estimate (3.44) by similar computations as in
estimate (3.40). This completes the proof of Lemma 3.4. ∎

We are now ready to verify estimate (1.5). In view of estimates (3.43) and
(3.44), we have

$\displaystyle\|A\|_{L^{\infty}(\Omega)}\leq$
$\displaystyle\|A^{\mathrm{sol}}\|_{L^{\infty}(\Omega)}+\|\nabla\varphi\|_{L^{\infty}(\Omega)}$
(3.52) $\displaystyle\leq$ $\displaystyle
e^{\frac{10R}{h}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{\frac{\eta}{1+s}}+Ch^{\frac{\alpha\eta}{(n+2)(1+s)}}+Ce^{\frac{11R}{h}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{\frac{\eta^{2}}{(1+s)^{2}}}$
$\displaystyle+Ch^{\frac{2\alpha\eta^{2}}{(n+2)^{2}(1+s)^{2}}}$
$\displaystyle\leq$ $\displaystyle
Ce^{\frac{11R}{h}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{\frac{\eta^{2}}{(1+s)^{2}}}+Ch^{\frac{2\alpha\eta^{2}}{(n+2)^{2}(1+s)^{2}}}$
$\displaystyle\leq$ $\displaystyle
Ce^{\frac{11R}{h^{1/\alpha}}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{\frac{\eta^{2}}{(1+s)^{2}}}+Ch^{\frac{2\alpha\eta^{2}}{(n+2)^{2}(1+s)^{2}}},$

where we have used the inequality $\frac{1}{h}<\frac{1}{h^{\frac{1}{\alpha}}}$
in the last step.

Let $\widetilde{h}_{0}=\min\\{h_{0},\varepsilon_{0}^{2}\\}$ and
$\delta=(e^{-\frac{11R}{\widetilde{h}_{0}^{1/\alpha}}})^{\frac{2(1+s)^{2}}{\eta^{2}}}$.
Let us assume first that
$\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|<\delta$.
We then choose

$h=\bigg{\\{}\frac{1}{11R}\big{|}\log\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|\big{|}\bigg{\\}}^{-\frac{\alpha\eta^{2}}{2(1+s)^{2}}}.$
(3.53)

With this choice of $h$, we have

$\frac{1}{h^{\frac{1}{\alpha}}}=\frac{1}{11R}\big{|}\log\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|\big{|}^{\frac{\eta^{2}}{2(1+s)^{2}}}.$

Hence, by substituting (3.53) into (3.52), we get the estimate

$\displaystyle\|A\|_{L^{\infty}(\Omega)}$ $\displaystyle\leq
C\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{\frac{\eta^{2}}{2(1+s)^{2}}}+C\big{|}\log\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|\big{|}^{-\frac{\alpha^{2}\eta^{4}}{(n+2)^{2}(1+s)^{4}}}.$
(3.54)

When
$\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|\geq\delta$,
we have

$\|A\|_{L^{\infty}(\Omega)}\leq
2M\leq\frac{2CM}{\delta^{\frac{\eta^{2}}{2(1+s)^{2}}}}\delta^{\frac{\eta^{2}}{2(1+s)^{2}}}\leq\frac{2CM}{\delta^{\frac{\eta^{2}}{2(1+s)^{2}}}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{\frac{\eta^{2}}{2(1+s)^{2}}}.$

Thus, we have verified estimate (1.5).

To complete the proof of Theorem 1.1, we need to show that the choice of $h$
in (3.53) implies that $h<h_{0}$, and $1-\frac{h^{2}}{4}|\xi|^{2}>0$ for
$\xi=(\xi^{\prime},\xi_{n})\in\mathbb{R}^{n}$ such that $|\xi^{\prime}|<\rho$
and $\xi_{n}<\rho$. To verify $h<h_{0}$, we have the following inequalities:

$\displaystyle\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|$
$\displaystyle<\delta=(e^{-\frac{11R}{\widetilde{h}_{0}^{1/\alpha}}})^{\frac{2(1+s)^{2}}{\eta^{2}}}\ll
1$
$\displaystyle\implies\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{\frac{\eta^{2}}{2(1+s)^{2}}}$
$\displaystyle<e^{-\frac{11R}{\widetilde{h}_{0}^{1/\alpha}}}$
$\displaystyle\implies\log\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{\frac{\eta^{2}}{2(1+s)^{2}}}$
$\displaystyle<-\frac{11R}{\widetilde{h}_{0}^{1/\alpha}}$
$\displaystyle\implies\big{|}\log\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|\big{|}^{\frac{\eta^{2}}{2(1+s)^{2}}}$
$\displaystyle>\frac{11R}{\widetilde{h}_{0}^{1/\alpha}}$
$\displaystyle\implies\frac{1}{11R}\big{|}\log\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|\big{|}^{\frac{\eta^{2}}{2(1+s)^{2}}}$
$\displaystyle>\frac{1}{\widetilde{h}_{0}^{1/\alpha}}$
$\displaystyle\implies\frac{1}{h^{\frac{1}{\alpha}}}$
$\displaystyle>\frac{1}{\widetilde{h}_{0}^{1/\alpha}}$ $\displaystyle\implies
h$ $\displaystyle<\widetilde{h}_{0}<h_{0}.$

We now turn to check the inequality $1-\frac{h^{2}}{4}|\xi|^{2}>0$. To that
end, we have

$h^{2}\frac{|\xi|^{2}}{4}<h^{2}\frac{\rho^{2}}{2}<\frac{1}{2}h^{2}h^{-\frac{2\alpha\eta}{(n+2)^{2}(1+s)}}=\frac{1}{2}h^{2-\frac{2\alpha\eta}{(n+2)^{2}(1+s)}}<1,$

where we have utilized the inequalities $|\xi|^{2}<2\rho^{2}$ and
$\frac{2\alpha\eta}{(n+2)^{2}(1+s)}<2$. This completes the proof of Theorem
1.1.

##  4\. Estimates for the Electric Potential

This section is devoted to proving Theorem 1.2 and Corollary 1.3. For
convenience, in what follows we shall again denote $A=A^{(2)}-A^{(1)}$ and
$q=q_{2}-q_{1}$. We begin this section by recalling the integral identity
(3.17):

$\int_{\Omega}(A\cdot
Du_{2}+qu_{2})\overline{v}dx=\int_{\Gamma}\partial_{\nu}(\Delta(u_{1}-u_{2}))\overline{v}dS+\int_{\Gamma}\partial_{\nu}(u_{1}-u_{2})(\overline{\Delta
v})dS.$

Here $v$ is a solution to the equation $\mathcal{L}^{\ast}_{A^{(1)},q_{1}}v=0$
in $\Omega$, and $u_{j}$, $j=1,2$, satisfies the equation
$\mathcal{L}_{A^{(j)},q_{j}}u_{j}=0$ in $\Omega$.

Let us now estimate the right-hand side of the integral identity above. To
that end, we apply Lemma 3.1 to obtain

$\bigg{|}\int_{\Gamma}\partial_{\nu}(\Delta(u_{1}-u_{2}))\overline{v}dS+\int_{\Gamma}\partial_{\nu}(u_{1}-u_{2})(\overline{\Delta
v})dS\bigg{|}\leq
Ce^{\frac{9R}{h}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|.$

To estimate the left-hand side of the integral identity, by the Cauchy-
Schwartz inequality, estimate (3.19), and the inequality $\frac{1}{h}\leq
e^{\frac{R}{h}}$, we have

$\displaystyle\bigg{|}\int_{\Omega}A\cdot Du_{2}\overline{v}dx\bigg{|}$
$\displaystyle\leq\|A\|_{L^{\infty}(\Omega)}\|Du_{2}\|_{L^{2}(\Omega)}\|v\|_{L^{2}(\Omega)}$
$\displaystyle\leq
C\|A\|_{L^{\infty}(\Omega)}\cdot\frac{1}{h^{4}}e^{\frac{2R}{h}}\cdot\frac{1}{h}e^{\frac{2R}{h}}$
$\displaystyle\leq Ce^{\frac{9R}{h}}\|A\|_{L^{\infty}(\Omega)}.$

Hence, by combining the previous two estimates, we obtain

$\bigg{|}\int_{\Omega}qu_{2}\overline{v}dx\bigg{|}\leq
Ce^{\frac{9R}{h}}\big{(}\|A\|_{L^{\infty}(\Omega)}+\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|\big{)}.$
(4.1)

We next substitute the CGO solutions (3.12) and (3.13) into the left-hand side
of (4.1) and pass to the limit $h\to 0$. To that end, by a direct computation,
we get

$\displaystyle\int_{\Omega}qu_{2}\overline{v}dx$ (4.2)
$\displaystyle=\int_{\Omega}q(e^{-ix\cdot\xi}\overline{a_{1}}a_{2}+e^{-ix^{\ast}\cdot\xi}\overline{a_{1}^{\ast}}a_{2}^{\ast})dx+\int_{\Omega}q(-e^{-ix\cdot\xi_{-}}\overline{a_{1}^{\ast}}a_{2}-e^{-ix\cdot\xi_{+}}\overline{a_{1}}a_{2}^{\ast})dx+\int_{\Omega}qw_{2}dx$
$\displaystyle=I_{1}+I_{2}+I_{3},$

where $\xi_{\pm}$ is given by (3.22), and

$\displaystyle w_{2}=$ $\displaystyle
e^{-ix\cdot\xi}(\overline{a_{1}}r_{2}+a_{2}\overline{r_{1}}+\overline{r_{1}}r_{2})+e^{-ix^{\ast}\cdot\xi}(\overline{a_{1}^{\ast}}r_{2}^{\ast}+a_{2}^{\ast}\overline{r_{1}^{\ast}}+\overline{r_{1}^{\ast}}r_{2}^{\ast})$
$\displaystyle-e^{-ix\cdot\xi_{-}}(\overline{a_{1}^{\ast}}r_{2}+a_{2}\overline{r_{1}^{\ast}}+\overline{r_{1}^{\ast}}r_{2})-e^{-ix\cdot\xi_{+}}(\overline{a_{1}}r_{2}^{\ast}+a_{2}^{\ast}\overline{r_{1}}+\overline{r_{1}}r_{2}^{\ast}).$

Let us now analyze each integral in (4.2). We extend $q$ by zero on
$\mathbb{R}^{n}\setminus\Omega$ and denote the extension by the same letter.
For $I_{3}$, since $q\in\mathcal{Q}(\Omega,M)$, and the amplitudes $a_{j}\in
C^{\infty}(\overline{\Omega})$, $j=1,2$, we utilize estimate (3.11) to deduce

$|I_{3}|\leq Ch,\quad h\to 0.$ (4.3)

Turning attention to $I_{1}$, by choosing $a_{1}=a_{2}=1$ and making a change
of variables, we have

$I_{1}=\int_{\Omega\cup\Omega^{\ast}}e^{-ix\cdot\xi}\widetilde{q}dx=\mathcal{F}(\widetilde{q})(\xi),$
(4.4)

where the function $\widetilde{q}$ is the even extension of $q$ given by
(3.6).

We finally estimate $I_{2}$. To that end, we set $a_{1}=a_{2}=1$ again to
obtain

$I_{2}=\int_{\Omega\cup\Omega^{\ast}}q(-e^{-ix\cdot\xi_{+}}-e^{-ix\cdot\xi_{-}})dx=-\mathcal{F}(\widetilde{q})(\xi_{+})-\mathcal{F}(\widetilde{q})(\xi_{-}).$

Hence, it follows from Lemma 2.2 and the definition of $\xi_{\pm}$ given by
(3.22) that there exist constants $C>0$, $\varepsilon_{0}>0$, and $0<\alpha<1$
such that the estimate

$|I_{2}|\leq
C\big{(}e^{-\frac{\varepsilon^{2}}{\pi}\cdot\frac{|\xi^{\prime}|^{2}}{h^{2}|\xi|^{2}}}+\varepsilon^{\alpha}\big{)}$
(4.5)

is valid for any $0<\varepsilon<\varepsilon_{0}$.

Hence, it follows from (4.1)–(4.5) that

$|\mathcal{F}(\widetilde{q})|(\xi)\leq
C\big{(}e^{\frac{9R}{h}}\|A\|_{L^{\infty}(\Omega)}+e^{\frac{9R}{h}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|+e^{-\frac{\varepsilon^{2}}{4\pi}\cdot\frac{4|\xi^{\prime}|^{2}}{h^{2}|\xi|^{2}}}+\varepsilon^{\alpha}+h\big{)}.$
(4.6)

Furthermore, by substituting estimate (3.52) into (4.6) and applying the
equality $s=\frac{n}{2}+2\eta$, we get

$\displaystyle|\mathcal{F}(\widetilde{q})|(\xi)\leq$ $\displaystyle
C\big{(}e^{\frac{20R}{h}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{\frac{\eta^{2}}{(1+s)^{2}}}+h^{\frac{2\alpha\eta^{2}}{(n+2)^{2}(1+s)^{2}}}+e^{-\frac{\varepsilon^{2}}{4\pi}\cdot\frac{4|\xi^{\prime}|^{2}}{h^{2}|\xi|^{2}}}+\varepsilon^{\alpha}+h\big{)}.$
(4.7)

We are now ready to establish a stability estimate for
$\|\widetilde{q}\|_{H^{-1}(\mathbb{R}^{n})}$. Let $\rho>0$ be a parameter to
be chosen later, and consider again the set $E(\rho)$ given by (3.34). By the
Parseval’s formula, we write

$\|\widetilde{q}\|_{H^{-1}(\mathbb{R}^{n})}^{2}\leq\int_{E(\rho)}\frac{|\mathcal{F}(\widetilde{q})(\xi)|^{2}}{1+|\xi|^{2}}d\xi+\int_{\mathbb{R}^{n}\setminus
E(\rho)}\frac{|\mathcal{F}(\widetilde{q})(\xi)|^{2}}{1+|\xi|^{2}}d\xi.$ (4.8)

Arguing similarly as in (3.36), we get

$\int_{\mathbb{R}^{n}\setminus
E(\rho)}\frac{|\mathcal{F}(\widetilde{q})(\xi)|^{2}}{1+|\xi|^{2}}d\xi\leq
C\frac{1}{\rho^{2}}.$ (4.9)

We next follow similar arguments as in Section 3 to estimate first term on the
right-hand side of (4.8), and provide the proof for the sake of completeness.
To that end, we have

$\int_{E(\rho)}\frac{|\mathcal{F}(\widetilde{q})(\xi)|^{2}}{1+|\xi|^{2}}d\xi\leq
C\rho^{n}\big{(}e^{\frac{40R}{h}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{\frac{2\eta^{2}}{(1+s)^{2}}}+h^{\frac{4\alpha\eta^{2}}{(n+2)^{2}(1+s)^{2}}}+h^{\frac{n-1}{2}}+h^{\alpha}+h^{2}\big{)},$
(4.10)

where we have applied estimate (3.38) and chosen $\varepsilon>0$ such that
$\varepsilon=\sqrt{h}$.

Due to the inequalities $0<h\ll 1$, $\frac{n-1}{2}\geq 1$, $0<\alpha<1$, and
$\frac{4\eta^{2}}{(n+2)^{2}(1+s)^{2}}<1$, we get

$\displaystyle\int_{E(\rho)}\frac{|\mathcal{F}(\widetilde{q})(\xi)|^{2}}{1+|\xi|^{2}}d\xi$
$\displaystyle\leq
C\rho^{n}\big{(}e^{\frac{40R}{h}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{\frac{2\eta^{2}}{(1+s)^{2}}}+h^{\frac{4\alpha\eta^{2}}{(n+2)^{2}(1+s)^{2}}}\big{)}.$
(4.11)

Thus, it follows from (4.9) and (4.11) that

$\displaystyle\|\widetilde{q}\|_{H^{-1}(\mathbb{R}^{n})}^{2}\leq$
$\displaystyle
C\rho^{n}\big{(}e^{\frac{40R}{h}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{\frac{2\eta^{2}}{(1+s)^{2}}}+h^{\frac{4\alpha\eta^{2}}{(n+2)^{2}(1+s)^{2}}}\big{)}+C\frac{1}{\rho^{2}}.$
(4.12)

By choosing $\rho=h^{-\frac{4\alpha\eta^{2}}{(n+2)^{3}(1+s)^{2}}}$, which
equates $\rho^{n}h^{\frac{4\alpha\eta^{2}}{(n+2)^{2}(1+s)^{2}}}$ and
$\frac{1}{\rho^{2}}$, we get

$\displaystyle\|\widetilde{q}\|_{H^{-1}(\mathbb{R}^{n})}^{2}\leq$
$\displaystyle
Ch^{\frac{-4n\alpha\eta^{2}}{(n+2)^{2}(1+s)^{2}}}e^{\frac{40R}{h}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{\frac{2\eta^{2}}{(1+s)^{2}}}+Ch^{\frac{8\alpha\eta^{2}}{(n+2)^{3}(1+s)^{2}}}.$

Since $\frac{4n\alpha\eta^{2}}{(n+2)^{2}(1+s)^{2}}<4$, we have

$\displaystyle\|\widetilde{q}\|_{H^{-1}(\mathbb{R}^{n})}$ $\displaystyle\leq
Ce^{\frac{22R}{h}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{\frac{\eta^{2}}{(1+s)^{2}}}+Ch^{\frac{4\alpha\eta^{2}}{(n+2)^{3}(1+s)^{2}}}$
(4.13) $\displaystyle\leq
Ce^{\frac{22R}{h^{1/\alpha}}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{\frac{\eta^{2}}{(1+s)^{2}}}+Ch^{\frac{4\alpha\eta^{2}}{(n+2)^{3}(1+s)^{2}}},$

where we have utilized the inequality $\frac{1}{h}<\frac{1}{h^{1/\alpha}}$ in
the last step.

Let $\widetilde{h}_{0}=\min\\{h_{0},\varepsilon_{0}^{2}\\}$ and
$\delta=(e^{-\frac{22R}{\widetilde{h}_{0}^{1/\alpha}}})^{\frac{2(1+s)^{2}}{\eta^{2}}}$.
We assume first that
$\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|<\delta$.
We then choose

$h=\bigg{\\{}\frac{1}{22R}\big{|}\log\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|\big{|}\bigg{\\}}^{-\frac{\alpha\eta^{2}}{2(1+s)^{2}}}.$

Arguing similarly as in Section 3, we conclude that this choice of $h$
satisfies $h<h_{0}$ and $1-\frac{h^{2}}{4}|\xi|^{2}>0$ for
$\xi=(\xi^{\prime},\xi_{n})\in\mathbb{R}^{n}$ such that $|\xi^{\prime}|<\rho$
and $\xi_{n}<\rho$.

We then substitute this choice of $h$ into (4.13) to obtain

$\|\widetilde{q}\|_{H^{-1}(\mathbb{R}^{n})}\leq
C\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{\frac{\eta^{2}}{(1+s)^{2}}}+C\log\big{|}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|\big{|}^{-\frac{2\alpha^{2}\eta^{4}}{(n+2)^{3}(1+s)^{4}}}.$

From here, the claimed estimate (1.6) follows from the estimate above and the
inequality

$\|q\|_{H^{-1}(\Omega)}\leq\|\widetilde{q}\|_{H^{-1}(\mathbb{R}^{n})}.$

When
$\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|\geq\delta$,
we utilize the continuous inclusions

$L^{\infty}(\Omega)\hookrightarrow L^{2}(\Omega)\hookrightarrow
H^{-1}(\Omega)$

to obtain

$\|q\|_{H^{-1}(\Omega)}\leq\|q\|_{L^{\infty}(\Omega)}\leq
2M\leq\frac{2CM}{\delta^{\frac{\eta^{2}}{2(1+s)^{2}}}}\delta^{\frac{\eta^{2}}{2(1+s)^{2}}}\leq\frac{2CM}{\delta^{\frac{\eta^{2}}{2(1+s)^{2}}}}\|\Lambda_{A^{(1)},q_{1}}^{\Gamma}-\Lambda_{A^{(2)},q_{2}}^{\Gamma}\|^{\frac{\eta^{2}}{2(1+s)^{2}}}.$

This completes the proof of Theorem 1.2.

Finally, we observe that estimate (1.7) follows from estimate (1.6) via
similar computations as in estimate (3.40). This completes the proof of
Corollary 1.3.

## References

  * [1] M. Agranovich.  Sobolev Spaces, Their Generalizations and Elliptic Problems in Smooth and Lipschitz Domains.  Springer Monographs in Mathematics. Springer Cham, 2015.
  * [2] G. Alessandrini.  Stable determination of conductivity by boundary measurements.  Applicable Analysis, 27:153–172, 1988.
  * [3] N. Aroua and M. Bellassoued.  Stable determination of a second order perturbation of the polyharmonic operator by boundary measurements.  Journal of Mathematical Analysis and Applications, 522(2):126965, 2023.
  * [4] Y. Assylbekov.  Inverse problems for the perturbed polyharmonic operator with coefficients in Sobolev spaces with non-positive order.  Inverse Problems, 32(10):105009, 2016.
  * [5] Y. Assylbekov and K. Iyer.  Determining rough first order perturbations of the polyharmonic operator.  Inverse Problems and Imaging, 13(5):1045–1066, 2019.
  * [6] Y. Assylbekov and Y. Yang.  Determining the first order perturbation of a polyharmonic operator on admissible manifolds.  Journal of Differential Equations, 262(1):590–614, 2017.
  * [7] S. Bhattacharyya and T. Ghosh.  Inverse boundary value problem of determining up to a second order tensor appear in the lower order perturbation of a polyharmonic operator.  Journal of Fourier Analysis and Applications, 25:661–683, 2019.
  * [8] S. Bhattacharyya and T. Ghosh.  An inverse problem on determining second order symmetric tensor for perturbed biharmonic operator.  Mathematische Annalen, 384:1–33, 2022.
  * [9] S. Bhattacharyya, V. Krishnan, and S. Sahoo.  Unique determination of anisotropic perturbations of a polyharmonic operator from partial boundary data.  arXiv preprint arXiv:2111.07610, 2021.
  * [10] R. Brown and L. Gauthier.  Inverse boundary value problems for polyharmonic operators with non-smooth coefficients.  Inverse Problems and Imaging, 16(4):943–966, 2022.
  * [11] P. Caro.  On an inverse problem in electromagnetism with local data: stability and uniqueness.  Inverse Problems and Imaging, 5(2):297–322, 2011.
  * [12] A. Choudhury and H. Heck.  Stability of the inverse boundary value problem for the biharmonic operator: Logarithmic estimates.  Journal of Inverse and Ill-posed Problems, 25(2):251–263, 2017.
  * [13] A. P. Choudhury and V. Krishnan.  Stability estimates for the inverse boundary value problem for the biharmonic operator with bounded potentials.  Journal of Mathematical Analysis and Applications, 431(1):300–316, 2015.
  * [14] D. Colton, H. Haddar, and M. Piana.  The linear sampling method in inverse electromagnetic scattering theory.  Inverse Problems, 19:S105–S137, 2003.
  * [15] F. Gazzola, H. Grunau, and G. Sweers.  Polyharminic Boundary Value Problems.  Springer-Verlag, Berlin, 2010.
  * [16] T. Ghosh and V. Krishnan.  Determination of lower order perturbations of the polyharmonic operator from partial boundary data.  Applicable Analysis, 95(11):2444–2463, 2016.
  * [17] G. Grubb.  Distributions and Operators, volume 252 of Graduate Texts in Mathematics.  Springer, 2009.
  * [18] M. Ikehata.  A special green’s function for the biharmonic operator and its application to an inverse boundary value problem.  Computers & Mathematics with Applications, 22(4-5):53–66, 1991.
  * [19] V. Isakov.  Completeness of products of solutions and some inverse problems for pde.  Journal of Differential Equations, 92(2):305–316, 1991.
  * [20] V. Isakov.  On uniqueness in the inverse conductivity problem with local data.  Inverse Problems and Imaging, 1(1):95–105, 2007.
  * [21] K. Krupchyk, M. Lassas, and G. Uhlmann.  Determining a first order perturbation of the biharmonic operator by partial boundary measurements.  Journal of Functional Analysis, 262(4):1781–1801, 2012.
  * [22] K. Krupchyk, M. Lassas, and G. Uhlmann.  Inverse problems with partial data for a magnetic Schrödinger operator in an infinite slab and on a bounded domain.  Communications in Mathematical Physics, 312:87–126, 2012.
  * [23] K. Krupchyk, M. Lassas, and G. Uhlmann.  Inverse boundary value problems for the perturbed polyharmonic operator.  Transactions of the American Mathematical Society, 366(1):95–112, 2014.
  * [24] K. Krupchyk and G. Uhlmann.  Inverse boundary problems for polyharmonic operators with unbounded potentials.  Journal of Spectral Theory, 6(1):145–183, 2016.
  * [25] B. Liu.  Global identifiability of low regulity fluid parameters in acoustic tomography of moving fluid.  SIAM Journal on Mathematical Analysis, 50(6):6348–6364, 2018.
  * [26] B. Liu.  Stability estimates in a partial data inverse boundary value problem for biharmonic operators at high frequencies.  Inverse Problems and Imaging, 14(5):783–796, 2020.
  * [27] Y. Ma and G. Liu.  Stability estimates for the inverse boundary value problem for the first order perturbation of the biharmonic operator.  Journal of Mathematical Analysis and Applications, 523:127025, 2023.
  * [28] G. Nakamura, Z. Sun, and G. Uhlmann.  Global identifiability for an inverse problem for the Schrödinger equation in a magnetic field.  Mathematische Annalen, 303(1):377–388, 1995.
  * [29] L. Potenciano-Machado.  Optimal stability estimates for a magnetic Schrödinger operator with local data.  Inverse Problems, 33:095001, 2017.
  * [30] S. Sahoo and M. Salo.  The linearized Calderón problem for polyharmonic operators.  Journal of Differential Equations, 360:407–451, 2023.
  * [31] V. Sharafutdinov.  Integral Geometry of Tensor Fields, volume 1 of Inverse and Ill-Posed Problems Series.  De Gruyter, Berlin, New York, 1994.
  * [32] Z. Sun.  An inverse boundary value problem for Schrödinger operators with vector potentials.  Transactions of the American Mathematical Society, 338(2):953–969, 1993.
  * [33] L. Tzou.  Stability estimates for coefficients of magnetic Schrödinger equation from full and partial boundary measurements.  Communications in Partial Differential Equations, 33(11):1911–1952, 2008.
  * [34] L. Yan.  Inverse boundary problems for biharmonic operators in transversally anisotropic geometries.  SIAM Journal on Mathematical Analysis, 53(6):6617–6653, 2021.
  * [35] Y. Yang.  Determining the first order perturbation of a bi-harmonic operator on bounded and unbounded domains from partial data.  Journal of Differential Equations, 257(10):3607–3639, 2014.