# Stability estimate for an inverse problem for the time harmonic magnetic
Schrödinger operator from the near and far field pattern

Mourad Bellassoued  ,  Houssem Haddar  and  Amal Labidi  M. Bellassoued.
Université de Tunis El Manar, Ecole Nationale d’ingénieurs de Tunis, ENIT-
LAMSIN, B.P. 37, 1002 Tunis, Tunisia<EMAIL_ADDRESS>H.
Haddar, INRIA, ENSTA Paris Tech, Institut Polytechnique de Paris, Route de
Saclay, 91128 Palaiseau, France<EMAIL_ADDRESS>A. Labidi. Université
de Tunis El Manar, Ecole Nationale d’ingénieurs de Tunis, ENIT-LAMSIN, B.P.
37, 1002 Tunis, Tunisia<EMAIL_ADDRESS>
###### Abstract.

We derive conditional stability estimates for inverse scattering problems
related to time harmonic magnetic Schrödinger equation. We prove logarithmic
type estimates for retrieving the magnetic (up to a gradient) and electric
potentials from near field or far field maps. Our approach combines techniques
from similar results obtained in the literature for inhomogeneous inverse
scattering problems based on the use of geometrical optics solutions.

###### Key words and phrases:

Inverse medium scattering, inverse problems, stability estimate, magnetic
potential

###### 2010 Mathematics Subject Classification:

35R30, 81U40, 35J05

##  1\. Introduction and main results

###  1.1. Introduction

This paper is concerned with the inverse scattering problem of recovering the
magnetic and electric potentials in the magnetic Schrödinger model from near
field or far field measurements at a fixed frequency. The forward model is as
follows (see for instance [20, 10]). Let $D\subset\mathbb{R}^{3}$ be a bounded
open set with smooth boundary such that $\mathbb{R}^{3}\setminus D$ is
connected and let $A=(a_{1},a_{2},a_{3})\in W^{1,\infty}(\mathbb{R}^{3})^{3}$
be a real valued vector modeling the magnetic potential and $q\in
L^{\infty}(\mathbb{R}^{3})$ be a complex valued function with non negative
imaginary part modeling the electric potential such that
$\textrm{Supp}(A)\subset D$ and $\textrm{Supp}(q)\subset D$. The magnetic
Schrödinger operator we are considering is

$\mathcal{H}_{A,q}:=-(\nabla+iA)^{2}+q=-\Delta-Q_{A,q},$ (1.1)

where $Q_{A,q}$ is the first order operator given by

$Q_{A,q}v:=i\;\textrm{div}(Av)+iA\cdot\nabla v-(\left|A\right|^{2}+q)v,\quad
v\in H^{1}_{\mathrm{loc}}(\mathbb{R}^{3}).$ (1.2)

The direct scattering problem in the near field setting can be phrased as
follows. Let $B$ be a smooth bounded and simply connected domain (typically a
ball) containing $D$ with outward normal denoted by $\nu$ and let
$y\in\partial B$ be the location of a point source. The total field
$u(\cdot,y)$ generated by the point source satisfies

$\mathcal{H}_{A,q}u(\cdot,y)-k^{2}u(\cdot,y)=\delta_{y}\quad\mathrm{in}\;\mathbb{R}^{3},$
(1.3)

with $\delta_{y}$ denoting the Dirac distribution at $y$ and $k>0$ is the wave
number. The total field is decomposed into

$u_{A,q}(\cdot,y)=\Phi(\cdot,y)+u_{A,q}^{s}(\cdot,y)\quad\mathrm{in}\;\mathbb{R}^{3},$
(1.4)

where the scattered field $u_{A,q}^{s}(\cdot,y)\in
H^{2}_{\mathrm{loc}}(\mathbb{R}^{3})$ and satisfies the Sommerfeld radiation
condition at infinity

$\lim_{r\rightarrow\infty}r\Big{(}\partial_{r}u^{s}-iku^{s}\Big{)}=0,\quad
r=|x|$ (1.5)

uniformly with respect to $\hat{x}=\frac{x}{|x|}$. The incident field is given
by

$\Phi(x,y):=\frac{1}{4\pi}\frac{e^{ik\left|x-y\right|}}{\left|x-y\right|},\quad
x\neq y$ (1.6)

and is the fundamental solution of the Helmholtz equation, i.e. satisfying
(1.3) for $A=q=0$ together with the Sommerfeld radiation condition.

The first inverse problem that we shall investigate is to recover $A$ and $q$
from the knowledge $u_{A,q}^{s}(x,y)$ for all $(x,y)\in\partial
B\times\partial B$. Defining the near field operator
$\mathcal{N}_{A,q}:\;L^{2}(\partial B)\rightarrow L^{2}(\partial B),$ as

$\displaystyle\mathcal{N}_{A,q}h(x):=\int_{\partial
B}u^{s}_{A,q}(x,y)h(y)\,\mathrm{ds}(y),\quad x\in\partial B,$ (1.7)

where $u^{s}_{A,q}(\cdot,y)$ is given by (1.4) and satisfying (1.5), the
inverse problem in the near field setting can be equivalently stated as
identifying $A$ and $q$ from the knowledge of $\mathcal{N}_{A,q}$.

The direct scattering problem in the far field setting formally corresponds
with letting $|y|\to\infty$ in the direction $-d$ with $d\in\mathbb{S}^{2}$
(the unit sphere of $\mathbb{R}^{3}$) and can be phrased as follows: Given an
incident plane wave $u^{i}(x,d)=e^{ikx\cdot d}$, $x\in\mathbb{R}^{3}$, seek a
total field $u_{A,q}(\cdot,d)$ that satisfies

$\mathcal{H}_{A,q}u(\cdot,d)-k^{2}u(\cdot,d)=0\quad\mathrm{in}\;\mathbb{R}^{3}$
(1.8)

and can be decomposed into

$u_{A,q}(\cdot,d)=u^{i}(\cdot,d)+u_{A,q}^{s}(\cdot,d)\quad\mathrm{in}\;\mathbb{R}^{3},$
(1.9)

where the scattered field $u_{A,q}^{s}(\cdot,d)\in
H^{2}_{\mathrm{loc}}(\mathbb{R}^{3})$ and satisfies the Sommerfeld radiation
condition. The latter implies in particular that the scattered field has the
following asymptotic behavior as $|x|\to\infty$,

$u_{A,q}^{s}(x,d)=\frac{e^{ik|x|}}{|x|}\Big{(}u_{A,q}^{\infty}(\hat{x},d)+O\Big{(}\frac{1}{|x|}\Big{)}\Big{)},\quad$
(1.10)

where $u_{A,q}^{\infty}(\cdot,d)$ is the so-called far field pattern. The
second inverse problem that we shall consider is the identification of $A$ and
$q$ from the knowledge of $u_{A,q}^{\infty}(\hat{x},d)$ for all
$(\hat{x},d)\in\mathbb{S}^{2}\times\mathbb{S}^{2}$.

Our main goal for both settings is to investigate the conditional stability of
recovering the electric potential and the magnetic field from the given so-
called full aperture measurements. Our strategy relies the use of geometrical
optics solutions to the magnetic Schrödinger equation to derive estimates on
the Fourier coefficients of these potentials under additional regularity
assumptions. The stability for the electric potential requires the use of an
adapted version of the Helmholtz decomposition. Transferring the results from
the near field operator to the far field measurements necessitate a careful
study of the near field to far field mapping. The reciprocity between the
operators ${\mathcal{H}_{A,q}}-k^{2}$ and ${\mathcal{H}_{-A,q}}-k^{2}$
(supplemented with the radiation condition) and mixed reciprocity between far
field and near field play a central role in the proofs.

In the absence of the magnetic potential $A$, the study of the identifiability
of $q$ from full aperture measurements is one of the first foundational
problems in inverse scattering theory and we refer to [14, 16, 17, 23] for
pioneering uniqueness results under various (regularity) assumptions. In the
presence of a magnetic potential $A$, we remind that there is an obstruction
to uniqueness for both near field and far field settings (as has been noted in
[22] for instance). In fact, the scattered field outside a ball $B$ containing
$D$ is invariant under the gauge transformation of the magnetic potential.
Namely, given $\varphi\in W^{2,\infty}(\mathbb{R}^{3})$ with support compactly
embedded in $B$ and letting $\tilde{u}=u(x)e^{-i\varphi(x)}$ one easily
observes that

$\mathcal{H}_{A+\nabla\varphi,q}\tilde{u}:=-(\nabla+i(A+\nabla\varphi))^{2}\tilde{u}+q(x)\tilde{u}=e^{-i\varphi(x)}\mathcal{H}_{A,q}u.$
(1.11)

Since $\varphi=0$ outside $B$, $\tilde{u}$ then satisfies the same equation as
$u$, namely (1.4) (respectively (1.9)) in the near field setting (respectively
in the far field setting) with $\mathcal{H}_{A,q}$ replaced by
$\mathcal{H}_{A+\nabla\varphi,q}$. Let us denote by
$u^{s}_{A+\nabla\varphi,q}$ the scattered field associated with the potentials
$A+\nabla\varphi$ and $q$. From uniqueness of solutions to the above stated
scattering problems one easily deduces that for all $y\in\partial B$ and
$d\in\mathbb{S}^{2}$

$u^{s}_{A+\nabla\varphi,q}(\cdot,y)=(e^{-i\varphi(x)}-1)\Phi(\cdot,y)+e^{-i\varphi(x)}u_{A,q}^{s}(\cdot,y)\quad\mathrm{in}\;\mathbb{R}^{3},$
$u^{s}_{A+\nabla\varphi,q}(\cdot,d)=(e^{-i\varphi(x)}-1)u_{i}(\cdot,d)+e^{-i\varphi(x)}u_{A,q}^{s}(\cdot,d)\quad\mathrm{in}\;\mathbb{R}^{3}.$

This clearly shows that
$u^{s}_{A+\nabla\varphi,q}(\cdot,y)=u^{s}_{A,q}(\cdot,y)$ and
$u^{s}_{A+\nabla\varphi,q}(\cdot,d)=u^{s}_{A,q}(\cdot,d)$ outside $D$ and
therefore, the magnetic potential $A$ cannot be uniquely determined from far
field or near field measurements outside $B$. It indicates that the best we
can expect from the knowledge of the near field operator $\mathcal{N}_{A,q}$
or the far field $u_{A,q}^{\infty}$ is to identify $(A,q)$ modulo a gauge
transformation of $A$. When $\textrm{Supp}(A)\subset D$ is known, the problem
may be equivalently reformulated as whether the magnetic field defined by the
$2$-form associated with the vector $A$,

$\textrm{curl}\,A:=\frac{1}{2}\sum_{i,j=1}^{3}\left(\partial_{x_{j}}a_{i}-\partial_{x_{i}}a_{j}\right)dx_{j}\wedge
dx_{i},$ (1.12)

and the electric potential $q$ can be retrieved from far field or near field
measurements. The uniqueness for similarly stated inverse problems has been
established in [11] for $L^{\infty}$ regularity of the coefficients. It has
been studied in earlier works under more regularity assumptions in [15] and
for small perturbations in [22, 20]. We also quote the recent uniqueness
result in [13] for measurements associated with finite number of incident
waves but with full frequency range.

Concerning stability results with full aperture measurements, in [8] Hähner
and Hohage established logarithmic stability estimates for the case $A=0$.
These results improve previous ones due to Stefanov [21] by giving an explicit
exponent in the logarithmic estimate and using the $L^{2}$-norm for far field
patterns. In [9] Isaev and Novikov proved stability estimates with explicit
dependence on the wave number. We hereafter shall follow a similar approach as
in [8, 21] to study the case when one would like to simultaneously recover
$\textrm{curl}\,A$ and $q$ from full aperture measurements in the light of
geometrical optics solutions developed in [1, 19, 22, 24] for various context
in relation with the inverse problem we are interested in. We establish the
stability result first for the magnetic field in the case of near field data.
We then employ a carefully designed Helmholtz decomposition to infer the
stability result for the electric potential. The derivation of the results for
far field data are obtained after establishing some key properties relating
this data to the near field data.

For bounded domains, the inverse problem with full aperture measurements
corresponds with measuring the global Dirichlet to Neumann map. For this
problem Tzou proved in [24] $\mathrm{log}$-type stability estimate for
$H^{-1}$ norms of the coefficients, assuming that the magnetic potentials are
in $W^{2,\infty}$ and the electric potential are in $L^{\infty}$. We here
consider stability with respect to the $L^{\infty}$ norm with explicit link
between the additional needed regularity for the coefficients and the
logarithm exponent. Let us finally indicate that uniqueness and log-log
stability results with partial data have been also studied by many authors in
the literature (see for instance [24, 2, 18, 4, 1]) but are not addressed in
the present work.

###  1.2. Main stability results

We here state the main results of this paper concerning conditional log-
stability reconstruction of the magnetic field $\textrm{curl}\,A$ given by
(1.12) and the electric potential $q$ from knowledge of the full aperture far
field measurements, i.e., $u_{A,q}^{\infty}(\hat{x},d)$ for any
$(\hat{x},d)\in\mathbb{S}^{2}\times\mathbb{S}^{2}$ or from knowledge of the
near field operator $\mathcal{N}_{A,q}$.

Let us first indicate the required conditions for admissible compactly
supported magnetic potentials $A$ and electric potentials $q$. Let $M>0$ and
$\sigma>0$ be given. We define the class of admissible magnetic potentials
$\mathcal{A}_{\sigma}(M)$ by

$\mathcal{A}_{\sigma}(M):=\\{A\in
W^{2,\infty}(\mathbb{R}^{3},\mathbb{R}^{3}),\,\mathrm{Supp}(A)\subset
D,\,\cr\|A\|_{W^{2,\infty}}\leq
M,\,\textrm{and}\,\|\widehat{\textrm{curl}A}\|_{L^{1}_{\sigma}(\mathbb{R}^{3})}\leq
M\\},$ (1.13)

where $\widehat{v}$ denotes the Fourier transform of $v$ and
$L^{1}_{\tau}(\mathbb{R}^{3})$ is the weighted $L^{1}(\mathbb{R}^{3})$ space
with norm

$\|v\|_{L^{1}_{\tau}(\mathbb{R}^{3})}:=\int_{\mathbb{R}^{3}}(1+|\xi|^{2})^{\tau/2}|v(\xi)|d\xi.$

Given $M>0$ and $\gamma>0$, we define the class of admissible electric
potentials $\mathcal{Q}_{\gamma}(M)$ by

$\mathcal{Q}_{\gamma}(M):=\\{q\in
L^{\infty}(\mathbb{R}^{3},\mathbb{C}),\,\Im(q)\geq 0,\,\mathrm{Supp}(q)\subset
D,\,\cr\|q\|_{L^{\infty}(D)}\leq
M\,\textrm{and}\,\|\widehat{q}\|_{L^{1}_{\gamma}(\mathbb{R}^{3})}\leq M\\}.$
(1.14)

The first main result of this paper is the following log-stability for the
magnetic field $\mathrm{curl}\,A$ and the electric potential $q$ from the near
field measurements.

######  Theorem 1.1.

Let $M>0$, $\sigma>0$ and $\gamma>0$. Then there exists a constant $C>0$ such
that for any
$(A_{j},q_{j})\in\mathcal{A}_{\sigma}(M)\times\mathcal{Q}_{\gamma}(M)$,
$j=1,2$, we have

$\|\mathrm{curl}(A_{1}-A_{2})\|_{L^{\infty}}\leq
C\big{(}\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|^{1/2}+\big{(}\log^{-}(\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|)\big{)}^{-\frac{\sigma}{(\sigma+3)}}\big{)},$
$\|q_{2}-q_{1}\|_{L^{\infty}}\leq
C\big{(}\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|^{1/2}+\big{(}\log^{-}(\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|)\big{)}^{-\frac{\gamma\sigma}{(\sigma+3)(2\gamma+3)}}\big{)},$

where $\log^{-}(t):=\max(-\log(t),0)$ for $t>0$. Here $C$ depends only on $B$,
$M$, $\sigma$ and $\gamma$.

An exactly similar stability result can be deduced for far field measurements
if one uses the following very restrictive norm on the measurements. Let

$\Gamma:=\left\\{(\ell,m),\,\ell\in\mathbb{N}\cup\\{0\\},\,m\in[\\![-\ell,\ell]\\!]\right\\},$
(1.15)

and denote by $Y_{\ell}^{m}$, $(\ell,m)\in\Gamma$ the complete system of
special harmonics on $\mathbb{S}^{2}$. For a far field pattern, $u^{\infty}$
we denote by $\mu_{(\ell_{1},m_{1};\ell_{2},m_{2})}$,
$(\ell_{i},m_{i})\in\Gamma$, $i=1,2$ its Fourier coefficients given by

$\mu_{(\ell_{1},m_{1};\ell_{2},m_{2})}:=\int_{\mathbb{S}^{2}}\int_{\mathbb{S}^{2}}u^{\infty}(\hat{x},d)\overline{Y^{m_{1}}_{\ell_{1}}}(\hat{x})\overline{Y^{m_{2}}_{\ell_{2}}}(d)\;\mathrm{ds}(\hat{x})\;\mathrm{ds}(d).$
(1.16)

Let $a>0$ such that $D\subset\\{x\in\mathbb{R}^{3};|x|<a\\}$. Following [8],
we then introduce the following norm

$\|u^{\infty}\|_{\mathcal{F}}^{2}:=\sum_{(\ell_{1},m_{1})\in\Gamma}\sum_{(\ell_{2},m_{2})\in\Gamma}\Big{(}\frac{2\ell_{1}+1}{eka}\Big{)}^{2\ell_{1}}\Big{(}\frac{2\ell_{2}+1}{eka}\Big{)}^{2\ell_{2}}\left|\mu_{(\ell_{1},m_{1};\ell_{2},m_{2})}\right|^{2}.$
(1.17)

In Lemma 4.1 below we prove that this norm is finite for all far fields
$u_{A,q}^{\infty}$ with $A\in\mathcal{A}_{\sigma}(M)$ and
$q\in\mathcal{Q}_{\gamma}(M)$. Using Lemma 4.3 and Theorem 1.2, we immediately
get.

######  Theorem 1.2.

Let $M>0$, $\sigma>0$ and $\gamma>0$. Then there exists a constant $C>0$ such
that for any
$(A_{j},q_{j})\in\mathcal{A}_{\sigma}(M)\times\mathcal{Q}_{\gamma}(M)$,
$j=1,2$, we have

$\|\mathrm{curl}(A_{1}-A_{2})\|_{L^{\infty}}\leq
C\big{(}\|u^{\infty}_{A_{1},q_{1}}-u^{\infty}_{A_{2},q_{2}}\|_{\mathcal{F}}^{1/2}+\big{(}\log^{-}(\|u^{\infty}_{A_{1},q_{1}}-u^{\infty}_{A_{2},q_{2}}\|_{\mathcal{F}})\big{)}^{-\frac{\sigma}{(\sigma+3)}}\big{)},$

and

$\|q_{2}-q_{1}\|_{L^{\infty}}\leq
C\big{(}\|u^{\infty}_{A_{1},q_{1}}-u^{\infty}_{A_{2},q_{2}}\|_{\mathcal{F}}^{1/2}+\big{(}\log^{-}(\|u^{\infty}_{A_{1},q_{1}}-u^{\infty}_{A_{2},q_{2}}\|_{\mathcal{F}}))^{-\frac{\gamma\sigma}{(\sigma+3)(2\gamma+3)}}\big{)}.$

Here $C$ depends only on $D$, $a$, $M$, $\sigma$ and $\gamma$.

One can also obtain a slightly modified stability result using the $L^{2}$
norm of the measurements following the method in [8]. It is summarized in the
following theorem.

######  Theorem 1.3.

Let $M>0$, $\sigma>0$, $\gamma>0$ and $\varepsilon>0$. Then there exist two
constants $C>0$ and $\delta>0$ such that for all
$(A_{j},q_{j})\in\mathcal{A}_{\sigma}(M)\times\mathcal{Q}_{\gamma}(M)$,
$j=1,2$ verifying
$\|u^{\infty}_{A_{1},q_{1}}-u^{\infty}_{A_{2},q_{2}}\|_{L^{2}(\mathbb{S}^{2}\times\mathbb{S}^{2})}<\delta$
we have

$\|\mathrm{curl}(A_{1}-A_{2})\|_{L^{\infty}(D)}\leq
C\big{(}\log^{-}(\|u^{\infty}_{A_{1},q_{1}}-u^{\infty}_{A_{2},q_{2}}\|_{L^{2}(\mathbb{S}^{2}\times\mathbb{S}^{2})}\big{)}^{-\frac{\sigma}{\sigma+3}+\epsilon},$
$\|q_{2}-q_{1}\|_{L^{\infty}(D)}\leq
C\big{(}\log^{-}(\|u^{\infty}_{A_{1},q_{1}}-u^{\infty}_{A_{2},q_{2}}\|_{L^{2}(\mathbb{S}^{2}\times\mathbb{S}^{2})}))^{-\frac{\gamma\sigma}{(\sigma+3)(2\gamma+3)}+\epsilon}.$

Here $C$ depends only on $D$, $a$, $M$, $\sigma$, $\epsilon$, $\delta$ and
$\gamma$.

From Theorems 1.1 and 1.2 (or 1.3) we immediately derive the uniqueness
corollary.

######  Corollary 1.4.

Let $A_{1}$ and $A_{2}\in\mathcal{A}_{\sigma}(M)$ two vector fields, $q_{1}$
and $q_{2}\in\mathcal{Q}_{\gamma}(M)$ and $B\supset D$. Then, we have

$u^{\infty}_{A_{1},q_{1}}(\hat{x},d)=u^{\infty}_{A_{2},q_{2}}(\hat{x},d),\quad\forall(\hat{x},d)\in\mathbb{S}^{2}\times\mathbb{S}^{2},$

or

$u^{s}_{A_{1},q_{1}}(x,y)=u^{s}_{A_{2},q_{2}}(x,y),\quad\forall(x,y)\in\partial
B\times\partial B,$

implies $q_{1}=q_{2}$ and $\mathrm{curl}\,A_{1}=\mathrm{curl}\,A_{2}$ in $D$.

The remainder of this paper is organized as follows. In Section 2, we give a
brief outline of some basic properties of solutions to the Helmholtz equation,
the magnetic Lippmann-Schwinger equation and some properties of the near field
operator. In Section 3, we review the construction of the complex geometric
optics solutions due to Tzou [24] and we estimate the magnetic fields and the
electric potentials respectively from the near field operator. In section 4 we
establish a relation between the far field and the near field and then we
prove Theorems 1.2 and 1.3. An appendix is dedicated to some technical results
in relation with geometric optics solutions and the connection between far
field and near field data.

##  2\. Helmholtz equation and magnetic Lippmann-Schwinger equation

We here outline some results on the well-posedness of the direct scattering
problem using the formulation of the problem as a Lippmann-Schwinger equation
(see for instance [20, 7]) and prove a uniform bound with respect to the
potentials.

Throughout this section, we assume that $A\in
W^{1,\infty}(\mathbb{R}^{3},\mathbb{R}^{3})$ and $q\in
L^{\infty}(\mathbb{R}^{3},\mathbb{C})$ with $\mathrm{Supp}(A)$,
$\mathrm{Supp}(q)\subset D$ and $\Im(q)\geq 0$.

The function $v$ will be referring in this section to the incident wave (i.e.
$\Phi(\cdot,y)$ or $u^{i}(\cdot,d)$), the associated total field is denoted by
$u=u_{A,q}$ and the scattered field $u^{s}=u^{s}_{A,q}=u-v\in
H^{2}_{\mathrm{loc}}(\mathbb{R}^{3})$. Both scattering problems can then be
stated as solving for $u^{s}\in H^{2}_{\mathrm{loc}}(\mathbb{R}^{3})$
satisfying

$-\Delta u^{s}-k^{2}u^{s}=Q_{A,q}(u^{s}+v)\quad\mathrm{in}\;\mathbb{R}^{3},$
(2.18)

and the Sommerfeld radiation condition (1.5). For the study of this problem,
we only require that $v\in H^{1}(D)$. Convolution properties imply in
particular that $u^{s}$ can be represented as

$u^{s}(x)=\int_{D}\Phi(x,y)Q_{A,q}u(y)\,dy,\quad x\in\mathbb{R}^{3},\,\mbox{
with }u=u^{s}+v\mbox{ in }D.$ (2.19)

Let us introduce the integral operator $T_{A,q}:H^{1}(D)\to H^{1}(D)$ defined
by

$T_{A,q}w(x):=\int_{D}\Phi(x,y)Q_{A,q}w(y)dy,\quad x\in D.$ (2.20)

We remark that since $Q_{A,q}:H^{1}(D)\to L^{2}(D)$ is continuous (by
regularity assumptions on $A$ and $q$), and since the volume potential

$w\mapsto\int_{D}\Phi(\cdot,y)w(y)dy,$

continuously maps $L^{2}(D)$ into $H^{2}_{\mathrm{loc}}(\mathbb{R}^{3})$ (see
[3]), we deduce that $T_{A,q}$ is compact. Equation (2.19) implies in
particular that the total field $u\in H^{1}(D)$ and is a solution of the
Lippmann-Schwinger equation

$u-T_{A,q}u=v\quad\mbox{ in }H^{1}(D).$ (2.21)

Conversely, if $u\in H^{1}(D)$ satisfies (2.21), then one easily verifies
using the properties of volume potentials [3] that $u_{A,q}^{s}$ given by
(2.19) is in $H^{2}_{\mathrm{loc}}(\mathbb{R}^{3})$ and is a solution of the
scattering problem (2.18)-(1.5). The well posedness of the latter is then a
consequence of the following proposition.

######  Proposition 2.1.

The operator $I-T_{A,q}:H^{1}(D)\to H^{1}(D)$, with $I$ denoting the identity
operator on $H^{1}(D)$ is continuously convertible.

###### Proof.

The operator $I-T_{A,q}$ is of Fredholm type with index 0. It is therefore
sufficient to prove the injectivity of this operator. If $u-T_{A,q}u=0$, then
from the above equivalence, $u^{s}$ given by (2.19) with $v=0$, satisfies

$-\Delta u^{s}-k^{2}u^{s}=Q_{A,q}u^{s}\quad\mathrm{in}\;\mathbb{R}^{3}.$

Multiplying by $\overline{u^{s}}$ and integrating over a ball $B$ containing
$D$ implies after applying the Green’s theorem in both sides

$\int_{B}(|\nabla u^{s}|^{2}-k^{2}|u^{s}|^{2})dx-\int_{\partial
B}\partial_{r}u^{s}\,\overline{u^{s}}\mathrm{ds}(x)\cr=\int_{B}\big{(}iA\cdot(\nabla
u^{s}\overline{u^{s}}-\nabla\overline{u^{s}}u^{s})-(|A|^{2}+q)|u^{s}|^{2}\big{)}dx.$

Taking the imaginary part of the previous equality implies that
$\Im(\int_{\partial
B_{R}}\partial_{r}u^{s}\,\overline{u^{s}}\mathrm{ds}(x))\geq 0$ (since $A$ is
real valued and $\Im(q)\geq 0$). The Rellich lemma then implies that $u^{s}=0$
in $\mathbb{R}^{3}\setminus B$. We now observe that

$|\Delta u^{s}(x)|\leq(k^{2}+\|q\|_{\infty}+\|A\|^{2}_{\infty}+\|\nabla\cdot
A\|_{\infty})|u^{s}(x)|+2\|A\|_{\infty}|\nabla u^{s}(x)|,\,\mbox{ for a.e.
}x\in\mathbb{R}^{3}.$

Unique continuation theorem yields $u^{s}=0$ in $\mathbb{R}^{3}$ and therefore
$u=0$ in $D$. This proves the injectivity of $I-T_{A,q}$ and finishes the
proof of the proposition. ∎

It is also possible to prove the following uniform bound.

######  Proposition 2.2.

Let $A\in W^{1,\infty}(\mathbb{R}^{3},\mathbb{R}^{3})$ and $q\in
L^{\infty}(\mathbb{R}^{3},\mathbb{C})$ as above and such that
$\|A\|_{W^{1,\infty}}\leq M$ and $\|q\|_{L^{\infty}}\leq M$, for some constant
$M>0$. Then there exists a constant $C$ that only depends on $D$, $M$ and $k$
such that $\|(I-T_{A,q})^{-1}\|\leq C$. Here $\|\,\cdot\,\|$ denotes the norm
in $\mathcal{L}(H^{1}(D))$.

###### Proof.

We prove the result using a contradiction argument. Let us assume that, for
each $n\in\mathbb{N}$, there exists $A_{n}\in
W^{1,\infty}(\mathbb{R}^{3})^{3}$ and $q_{n}\in L^{\infty}(\mathbb{R}^{3})$ as
in the proposition such that

$\|(I-T_{A_{n},q_{n}})^{-1}\|\geq n.$

This implies in particular the existence of a non trivial function $v_{n}\in
H^{1}(D)$ such that the function $u_{n}\in H^{1}(D)$ satisfying
$u_{n}-T_{A_{n},q_{n}}u_{n}=v_{n}\mbox{ in }H^{1}(D)$ verifies

$\|u_{n}\|_{H^{1}(D)}\geq n\|v_{n}\|_{H^{1}(D)}.$

This gives for the normalized sequence
$\tilde{u}_{n}=\frac{u_{n}}{\|u_{n}\|_{H^{1}(D)}}$

$\tilde{u}_{n}-T_{A_{n},q_{n}}\tilde{u}_{n}=\frac{v_{n}}{\|u_{n}\|_{H^{1}(D)}}=:\tilde{v}_{n}\quad\textrm{in
}H^{1}(D),$ (2.22)

where $\|\tilde{v}_{n}\|_{H^{1}(D)}\leq\frac{1}{n}$. The associated scattered
field $\tilde{u}^{s}_{n}\in H^{2}_{\mathrm{loc}}(\mathbb{R}^{3})$ is defined
by

$\tilde{u}^{s}_{n}(x)=\int_{D}\Phi(x,y)Q_{A_{n},q_{n}}\tilde{u}_{n}(y)\,dy,\quad
x\in\mathbb{R}^{3},\,\mbox{ with
}\tilde{u}_{n}=\tilde{u}^{s}_{n}+\tilde{v}_{n}\mbox{ in }D.$ (2.23)

Since the sequence $(\tilde{u}_{n})$ is bounded in $H^{1}(D)$, the assumptions
on $A_{n}$ and $q_{n}$ imply that the sequence
$(Q_{A_{n},q_{n}}(\tilde{u}_{n}))$ is also bounded in $L^{2}(D)$. It yields in
particular, using (2.23), that the sequence $(\tilde{u}^{s}_{n})$ is bounded
in $H^{2}(D)$. Using the Rellich-Kondrachov compactness theorem, we infer
that, an extracted subsequence, that we keep denoting $(\tilde{u}^{s}_{n})$ is
a Cauchy sequence in $H^{1}(D)$. From
$\tilde{u}_{n}=\tilde{u}^{s}_{n}+\tilde{v}_{n}$ we deduce that
$(\tilde{u}_{n})$ is also a Cauchy sequence in $H^{1}(D)$ and therefore
converges to some $u$ in $H^{1}(D)$. Given the boundedness of the sequences
$(A_{n})$ and $(q_{n})$, by changing the original sequence (and without
corrupting the contradiction argument), one can assume that $(A_{n})$ and
$(q_{n})$ weak-* converge to $A$ and $q$ respectively in
$W^{1,\infty}(\mathbb{R}^{3})^{3}$ and $L^{\infty}(\mathbb{R}^{3})$. One then
easily verifies that $Q_{A_{n},q_{n}}(\tilde{u}_{n})$ weakly converges in
$L^{2}(D)$ to $Q_{A,q}({u})$. Consequently, from (2.20), we get that
$T_{A_{n},q_{n}}\tilde{u}_{n}$ strongly convergences to $T_{A,q}\tilde{u}$ in
$H^{1}(D)$. Passing to the limit in (2.22) implies that $u\in H^{1}(D)$
verifies $u-T_{A,q}u=0$. The limits $A$ and $q$ obviously verify the
hypothesis of Proposition 2.1 and therefore $u=0$. This contradicts
$\|u\|_{H^{1}(D)}=\|\tilde{u}_{n}\|_{H^{1}(D)}=1$. ∎

Let us observe for later use that, thanks to (2.19), the far field associated
with the scattered wave verifying (2.18) can be expressed as

$u_{A,q}^{\infty}(\hat{x}):=\frac{1}{4\pi}\int_{\mathbb{R}^{3}}e^{-ik\hat{x}\cdot
y}Q_{A,q}u(y)dy,\quad\hat{x}\in\mathbb{S}^{2},$ (2.24)

where $u$ is the solution of (2.21).

As a straightforward corollary of Proposition 2.2, the continuity properties
of volume potentials and (2.24), we have the following uniform estimates for
$u^{s}$ solution of (2.18)-(1.5) and associated far field.

######  Corollary 2.3.

Let $A\in W^{1,\infty}(\mathbb{R}^{3},\mathbb{R}^{3})$ and $q\in
L^{\infty}(\mathbb{R}^{3})$ as above such that $\|A\|_{W^{1,\infty}}\leq M$
and $\|q\|_{L^{\infty}}\leq M$ for some constant $M>0$. Then there exists a
constant $C$ that depends only on $M$, $D$ and $k$ such that

$\|u_{A,q}^{s}\|_{H^{2}(D)}\leq C\|v\|_{H^{1}(D)}\quad\mbox{ and
}\quad\|u_{A,q}^{\infty}\|_{L^{2}(\mathbb{S}^{2})}\leq C\|v\|_{H^{1}(D)}$

for all $v\in H^{1}(D)$, where $u_{A,q}^{s}\in
H^{2}_{\mathrm{loc}}(\mathbb{R}^{3})$ and is solution of the scattering
problem (2.18)-(1.5). Moreover, for any compact $K$ there exists a constant
$C$ that depends only on $M$, $D$, $K$ and $k$ such that for all $v\in
H^{1}(D)$

$\|u_{A,q}^{s}\|_{H^{2}(K)}\leq C\|v\|_{H^{1}(D)}.$

Armed with with above, let us define for later use the linear and continuous
solution operator $\mathcal{M}_{A,q}$ by

$\begin{array}[]{ccccc}\mathcal{M}_{A,q}&:&H^{1}(D)&\rightarrow&H^{2}_{\mathrm{loc}}(\mathbb{R}^{3}),\\\
&&v&\mapsto&\mathcal{M}_{A,q}v:=u_{A,q}^{s}\end{array}$ (2.25)

where $u_{A,q}^{s}$ is the solution of (2.18)-(1.5).

We use this data to define the near field operator
$\mathcal{N}_{A,q}:\;L^{2}(\partial B)\rightarrow L^{2}(\partial B),$ as

$\displaystyle\mathcal{N}_{A,q}h(x):=\int_{\partial
B}u^{s}_{A,q}(x,y)h(y)\,\mathrm{ds}(y),\quad x\in\partial B,$ (2.26)

where $u^{s}_{A,q}(\cdot,y):=\mathcal{M}_{A,q}\Phi(\cdot,y)$, $y\in\partial
B$. We first remark that

$\|\mathcal{N}_{A,q}\|\leq\|u^{s}_{A,q}\|_{L^{2}(\partial B\times\partial
B)},$

and therefore it is sufficient to study the stability of
$\mathcal{N}_{A,q}\mapsto(A,q)$ in order to infer stability results in terms
of near field measurements.

We second observe that, after introducing the single-layer operator
$\mathcal{S}:\;L^{2}(\partial B)\rightarrow H^{1}(D)$ defined by

$\displaystyle\mathcal{S}h(x):=\int_{\partial
B}\Phi(x,y)h(y)\,\mathrm{ds}(y),\quad x\in D,$ (2.27)

one has by linearity and continuity properties of the mapping
$\mathcal{M}_{A,q}$ the following identity

$\mathcal{N}_{A,q}h=(\mathcal{M}_{A,q}\mathcal{S}h)|_{\partial B}.$ (2.28)

This equality states that $\mathcal{N}_{A,q}h$ is nothing but the near field
measurements on $\partial B$ generated by an incident field $v:=\mathcal{S}h$.

From properties and jump relations for single-layer potential (see [3])
$v=\mathcal{S}h$ with density $h\in L^{2}(\partial B)$ is defined in
$\mathbb{R}^{3}$, satisfies the Helmholtz equation in
$\mathbb{R}^{3}\setminus\partial{B}$, the Sommerfeld radiation condition (1.5)
and the following continuity and jump properties across $\partial B$.

$\begin{array}[]{lll}v^{-}(x)=v^{+}(x)=v(x)&\textrm{on}\,\,\partial
B,\cr\partial_{\nu}v^{-}(x)-\partial_{\nu}v^{+}(x)=h(x)&\textrm{on}\,\,\partial
B,\end{array}$ (2.29)

where $v^{+}$ and $v^{-}$ respectively denote the restriction of $v$ to
$\mathbb{R}^{3}\setminus\overline{B}$ and $B$. In order to exploit the
information encoded into the identity (2.28), one can easily check the
following lemma by using (2.29).

######  Lemma 2.4.

Let $A\in W^{1,\infty}(\mathbb{R}^{3},\mathbb{R}^{3})$, $q\in
L^{\infty}(\mathbb{R}^{3})$ as above, and $h\in L^{2}(\partial B)$. Set
$v=\mathcal{S}h$ and $u^{s}=\mathcal{M}_{A,q}v$. Then the total field
$u=v+u^{s}$ is solution to the transmission problem

$\begin{array}[]{llll}\mathcal{H}_{A,q}u(x)=k^{2}u(x)&\textrm{in
}\,\mathbb{R}^{3}\backslash\partial B,\cr u^{+}(x)=u^{-}(x)&\textrm{on
}\,\partial B,\\\
\partial_{\nu}u^{-}(x)-\partial_{\nu}u^{+}(x)=h(x),&\textrm{on }\,\partial
B\cr\end{array}$ (2.30)

together with the Sommerfeld radiation condition (1.5).

We finally point out the following identity that will be useful in the sequel
and that can be easily derived from (2.19) and Fubini’s theorem

$\int_{\partial
B}\left(\mathcal{N}_{A,q}f\right)g\mathrm{ds}(x)=\int_{D}Sg\,Q_{A,q}(\mathcal{M}_{A,q}\mathcal{S}f+\mathcal{S}f)dx\quad\mbox{
for all }f,g\in L^{2}(\partial B).$ (2.31)

We establish now the following result that proves that the transpose operator
associated with $\mathcal{N}_{A,q}$ is equal to $\mathcal{N}_{-A,q}$. In order
to ease the writing, we indicate two useful formulas that we shall use a few
times. The first one is a consequence of the Green’s theorem and states that

$\int_{B}\left(\mathcal{H}_{A,q}u_{1}\,u_{2}-u_{1}\mathcal{H}_{-A,q}u_{2}\right)dx=\int_{\partial
B}\left(u_{1}\partial_{r}u_{2}-u_{2}\partial_{r}u_{1}\right)\mathrm{ds}(x),$
(2.32)

for all $u_{1},\,u_{2}\in H^{2}(B)$. The second one is a classical consequence
of the Green’s theorem and the Rellich lemma and states that [3]

$\int_{\partial
B}\left(u_{1}\partial_{r}u_{2}-u_{2}\partial_{r}u_{1}\right)\mathrm{ds}(x)=0,$
(2.33)

for all $u_{1}$, $u_{2}\in H^{2}_{\mathrm{loc}}(\mathbb{R}^{3}\setminus{B})$
satisfying the Helmholtz equation $\Delta u+k^{2}u=0$ in
$\mathbb{R}^{3}\setminus\overline{B}$ and the Sommerfeld radiation condition
(1.5).

######  Lemma 2.5.

Let $A\in W^{1,\infty}(\mathbb{R}^{3},\mathbb{R}^{3})$, $q\in
L^{\infty}(\mathbb{R}^{3})$ as above. Let
$y,z\in\mathbb{R}^{3}\setminus\overline{D}$ and set

$u^{s}_{A,q}(\cdot,y):=\mathcal{M}_{A,q}\Phi(\cdot,y),\quad\textrm{and}\,\,u^{s}_{-A,q}(\cdot,z):=\mathcal{M}_{-A,q}\Phi(\cdot,z).$

Then we have the following reciprocity relation,

$u^{s}_{A,q}(z,y)=u^{s}_{-A,q}(y,z).$

This reciprocity implies in particular that
$(\mathcal{N}_{A,q})^{t}=\mathcal{N}_{-A,q}$, i.e.,

$\int_{\partial
B}f\left(\mathcal{N}_{-A,q}g\right)\mathrm{ds}(x)=\int_{\partial
B}\left(\mathcal{N}_{A,q}f\right)g\,\mathrm{ds}(x)\quad\mbox{ for all }f,g\in
L^{2}(\partial B).$ (2.34)

###### Proof.

From equation (2.19)

$u^{s}_{A,q}(z,y)=\int_{D}\Phi(z,t)Q_{A,q}\,(u^{s}_{A,q}(t,y)+\Phi(t,y))dt.$

On the other hand, applying (2.32) and (2.33) to $u_{1}=u^{s}_{A,q}(\cdot,y)$
and $u_{2}=u^{s}_{-A,q}(\cdot,z)$ implies

$\int_{B}\left(\mathcal{H}_{A,q}u^{s}_{A,q}(t,y)\,u^{s}_{-A,q}(t,z)-u^{s}_{A,q}(t,y)\mathcal{H}_{-A,q}u^{s}_{-A,q}(t,z)\right)dt=0.$

Using (2.18) yields

$\int_{D}\left(Q_{A,q}\Phi(t,y)\,u^{s}_{-A,q}(t,z)-u^{s}_{A,q}(t,y)Q_{-A,q}\Phi(t,z)\right)dt=0.$

Using the Green’s theorem we obtain (since
$y,z\in\mathbb{R}^{3}\setminus\overline{D}$ and $A$ has compact support in
$D$)

$\int_{B}\left(\Phi(t,y)\,Q_{-A,q}u^{s}_{-A,q}(t,z)-Q_{A,q}u^{s}_{A,q}(t,y)\Phi(t,z)\right)dt=0.$

When then conclude, since $\Phi(z,t)=\Phi(t,z)$

$u^{s}_{A,q}(z,y)=\int_{D}\big{(}\Phi(t,y)\,Q_{-A,q}u^{s}_{-A,q}(t,z)+\Phi(t,z)Q_{A,q}\Phi(t,y)\big{)}dt.$

Applying the Green’s theorem to the second term in the integral finally shows
that

$u^{s}_{A,q}(z,y)=\int_{D}\Phi(t,y)\,Q_{-A,q}\big{(}u^{s}_{-A,q}(t,z)+\Phi(t,z)\big{)}dt=u^{s}_{-A,q}(y,z).$

Identity (2.34) is a direct consequence of the reciprocity relation and the
Fubini theorem. ∎

##  3\. Stability analysis for near field data

The aim of this section is to prove the stability estimates given in Theorem
1.1. The first step will be to use the properties of the near fields to prove
an orthogonality identity, which relates the difference of potentials to the
difference of near field operators. Then we will use a special family of
solutions called complex geometric optics solutions (CGO-solutions) to
estimate the Fourier transform of the difference of the magnetic fields and
the difference of the electric potentials.

Consider two pairs of potentials $(A_{j},q_{j})\in
W^{1,\infty}(\mathbb{R}^{3},\mathbb{R}^{3})\times
L^{\infty}(\mathbb{R}^{3},\mathbb{C})$, $j=1,2$, satisfying the same
assumptions as $(A,q)$ in the beginning of the previous section. We set

$A(x):=(A_{2}-A_{1})(x),\quad q(x):=(q_{2}-q_{1})(x),\quad
x\in\mathbb{R}^{3},$ (3.1)

and introduce the first order operator $\mathcal{P}_{A_{1},A_{2},q}$ defined
by

$\mathcal{P}_{(A_{1},A_{2},q)}v:=i\textrm{div}(Av)+iA\cdot\nabla
v+(\left|A_{2}\right|^{2}-\left|A_{1}\right|^{2}+q)v,\quad v\in
H^{1}(\mathbb{R}^{3}),$ (3.2)

here we remark that the coefficients of the first order operator
$\mathcal{P}_{(A_{1},A_{2},q)}$ are supported in $D$.

###  3.1. An orthogonality identity and a key integral inequality

First, we present an orthogonality identity, which relates the difference of
potentials to the difference of near field operators.

######  Lemma 3.1.

Let $f_{1},\,f_{2}\in L^{2}(\partial B)$ and set

$u^{s}_{1}:=\mathcal{M}_{-A_{1},q_{1}}\mathcal{S}f_{1},\quad\quad
u^{s}_{2}:=\mathcal{M}_{A_{2},q_{2}}\mathcal{S}f_{2}\quad\mbox{ in
}\mathbb{R}^{3},$ (3.3)

and for $j=1,2$,

$v_{j}:=\mathcal{S}f_{j}\quad\mbox{ in }D\quad\mbox{ and }\quad
u_{j}:=v_{j}+u^{s}_{j}\quad\mbox{ in }D.$ (3.4)

Then the following identity holds true.

$\int_{\partial
B}\left(\mathcal{N}_{A_{1},q_{1}}f_{2}-\mathcal{N}_{A_{2},q_{2}}f_{2}\right)\,f_{1}\,\mathrm{ds}(x)=\int_{D}\mathcal{P}_{(A_{1},A_{2},q)}u_{1}\,u_{2}\,dx.$
(3.5)

###### Proof.

Using (2.34) we first observe that

$\int_{\partial
B}\left(\mathcal{N}_{A_{1},q_{1}}f_{2}-\mathcal{N}_{A_{2},q_{2}}f_{2}\right)\,f_{1}\,\mathrm{ds}(x)=\int_{\partial
B}\left(\mathcal{N}_{-A_{1},q_{1}}f_{1}\right)\,f_{2}-\left(\mathcal{N}_{A_{2},q_{2}}f_{2}\right)\,f_{1}\,\mathrm{ds}(x).$

We deduce from (2.31) that

$\int_{\partial
B}\left(\mathcal{N}_{A_{1},q_{1}}f_{2}-\mathcal{N}_{A_{2},q_{2}}f_{2}\right)\,f_{1}\,\mathrm{ds}(x)=\int_{D}(v_{2}\,Q_{-A_{1},q_{1}}u_{1}-v_{1}\,Q_{A_{2},q_{2}}u_{2})\,dx.$
(3.6)

Applying the Green’s theorem and (2.33) to $u^{s}_{1}$ and $u^{s}_{2}$ implies

$\int_{B}\left(u^{s}_{2}\,(-\Delta u^{s}_{1}-k^{2}u^{s}_{1})-u^{s}_{1}(-\Delta
u^{s}_{2}-k^{2}u^{s}_{2})\right)dx=0,$

which yields according to (2.18),

$\int_{D}\left(u^{s}_{2}\,(Q_{-A_{1},q_{1}}u_{1})-(Q_{A_{2},q_{2}}u_{2})u^{s}_{1}\right)dx=0.$

Adding the left hand side of this equality to the right hand side of (3.6)
shows that

$\int_{\partial
B}\left(\mathcal{N}_{A_{1},q_{1}}f_{2}-\mathcal{N}_{A_{2},q_{2}}f_{2}\right)\,f_{1}\,\mathrm{ds}(x)=\int_{D}(u_{2}\,Q_{-A_{1},q_{1}}u_{1}-u_{1}\,Q_{A_{2},q_{2}}u_{2})\,dx.$
(3.7)

The result of the lemma follows from (3.7) after integrating by parts in the
right hand side and observing that

$Q_{-A_{1},q_{1}}u_{1}-Q_{-A_{2},q_{1}}u_{1}=\mathcal{P}_{(A_{1},A_{2},q)}u_{1}.$

This completes the proof. ∎

We now prove the fundamental integral inequality, which relates the difference
of two magnetic potentials and electric potential in $D$ to the difference
between their corresponding near pattern fields. This integral inequality will
be the starting point in the proof of the stability estimate for the
corresponding inverse problem.

######  Lemma 3.2.

There is a constant $C>0$ that only depends on $B$ and $k$ such that

$\left|\int_{B}[iA\cdot(u_{1}\nabla u_{2}-u_{2}\nabla
u_{1})-(\left|A_{2}\right|^{2}-\left|A_{1}\right|^{2}+q)u_{1}u_{2}]dx\right|\cr\leq
C\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|\|u_{1}\|_{H^{2}(B)}\|u_{2}\|_{H^{2}(B)}$
(3.8)

for all $u_{1}\in H^{2}(B)$ satisfying
$\mathcal{H}_{-A_{1},q_{1}}u_{1}=k^{2}u_{1}$ in $B$ and all $u_{2}\in
H^{2}(B)$ satisfying $\mathcal{H}_{A_{2},q_{2}}u_{2}=k^{2}u_{2}$ in $B$.

###### Proof.

Let $u_{j}$, $j=1,2$, be given as in the lemma. Let $u^{+}_{j}\in
H^{1}_{\textrm{loc}}(\mathbb{R}^{3}\setminus\overline{B})$ be the outgoing
solution to the following exterior Dirichlet problem (see for instance [3])

$\left\\{\begin{array}[]{lll}\Delta u^{+}_{j}+k^{2}u^{+}_{j}=0&\mbox{ in
}\mathbb{R}^{3}\setminus\overline{B},\\\ u^{+}_{j}=u_{j}&\mbox{ on }\partial
B,\end{array}\right.$

and $u^{+}_{j}$ satisfies the Sommerfeld radiation condition (1.5). Elliptic
regularity infers that $u^{+}_{j}\in
H^{2}_{\textrm{loc}}(\mathbb{R}^{3}\setminus\overline{B})$ and in particular,
by trace theorems [12, Theorem 2.1 in Chapter 4],

$\|\partial_{\nu}u^{+}_{j}\|_{L^{2}(\partial
B)}\leq\tilde{C}\|u_{j}\|_{H^{2}(B)},$ (3.9)

for some constant $\tilde{C}$ that only depends on $B$ and $k$. Let us extend
the functions $u_{j}$ as follows

$u_{j}(x):=\left\\{\begin{array}[]{lll}u^{-}_{j}(x)=u_{j}(x)&\textrm{if}\;x\in
B,\\\
u^{+}_{j}(x)&\textrm{if}\;x\in\mathbb{R}^{3}\setminus\overline{B},\end{array}\right.$

and set

$f_{j}:=\partial_{\nu}u_{j}^{-}-\partial_{\nu}u_{j}^{+}\quad\textrm{on}\;\partial
B.$ (3.10)

By trace theorems and (3.9) we have that $f_{j}\in L^{2}(\partial B)$ and

$\|f_{j}\|_{L^{2}(\partial B)}^{2}\leq C\|u_{j}\|_{H^{2}(B)}^{2},$ (3.11)

for some constant $C$ that only depends on $B$ and $k$. One can easily check
that $u_{1}$ and $u_{2}$ satisfy the following transmission problems
respectively

$\left\\{\begin{array}[]{lll}\mathcal{H}_{-A_{1},q_{1}}u_{1}=k^{2}u_{1}&\textrm{
in }\mathbb{R}^{3}\backslash\partial B,\\\ u^{-}_{1}=u^{+}_{1}&\textrm{ on
}\partial B,\\\ \partial_{\nu}u^{-}_{1}-\partial_{\nu}u^{+}_{1}=f_{1}&\textrm{
on }\partial
B,\end{array}\right.\qquad;\qquad\left\\{\begin{array}[]{lll}\mathcal{H}_{A_{2},q_{2}}u_{2}=k^{2}u_{2}&\textrm{
in }\mathbb{R}^{3}\backslash\partial B,\\\ u^{-}_{2}=u^{+}_{2}&\textrm{ on
}\partial B,\\\ \partial_{\nu}u^{-}_{2}-\partial_{\nu}u^{+}_{2}=f_{2}&\textrm{
on }\partial B.\end{array}\right.$

Moreover, $u_{j}^{+}$, $j=1,2$, satisfy the Sommerfeld radiation condition
(1.5). Consider now the functions

$v_{j}(x):=\mathcal{S}f_{j}=\int_{\partial
B}\Phi(x,y)f_{j}(y)\,\mathrm{ds}(y)\quad x\notin\partial B,\quad j=1,2.$

Therefore, $u^{s}_{j}:=u_{j}-v_{j}$, $j=1,2$, are the same as in Lemma 2.4 and
then satisfies Lemma 2.5, i.e they verify (3.3)-(3.4). Consequently, identity
(3.5) holds, namely

$\int_{D}\mathcal{P}_{(A_{1},A_{2},q)}u_{1}\,u_{2}\,dx=\int_{\partial
B}\left(\mathcal{N}_{A_{1},q_{1}}f_{2}-\mathcal{N}_{A_{2},q_{2}}f_{2}\right)\,f_{1}\,\mathrm{ds}(x).$

In view of (3.2), we get

$\int_{B}[iA\cdot(u_{1}\nabla u_{2}-u_{2}\nabla
u_{1})-(\left|A_{2}\right|^{2}-\left|A_{1}\right|^{2}+q)u_{1}u_{2}]dx\cr=\int_{\partial
B}\left(\mathcal{N}_{A_{1},q_{1}}f_{2}-\mathcal{N}_{A_{2},q_{2}}f_{2}\right)\,f_{1}\,\mathrm{ds}(x).$
(3.12)

Consequently,

$\left|\int_{B}[iA\cdot(u_{1}\nabla u_{2}-u_{2}\nabla
u_{1})-(\left|A_{2}\right|^{2}-\left|A_{1}\right|^{2}+q)u_{1}u_{2}]dx\right|\cr\leq\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|\|f_{1}\|_{L^{2}(\partial
B)}\|f_{2}\|_{L^{2}(\partial B)}.$ (3.13)

Identity (3.8) immediately follows from (3.13) and (3.11). ∎

###  3.2. Complex geometric optics solutions-CGO

The main strategy of the proof of stability estimate on determining the
magnetic field and the electric potential from the near field data is the use
of complex geometrical optics solutions in (3.8) to estimate the Fourier
coefficients of the difference of two magnetic fields
$\mathrm{curl}(A_{2}-A_{1})$ and the difference of two potential
$q_{2}-q_{1}$. We therefore first outline some known results about these
special solutions extracted from the literature [1, 19, 22, 24]. Let
$\omega=\omega_{1}+i\omega_{2}$ be a vector with
$\omega_{1},\omega_{2}\in\mathbb{S}^{2}$ and $\omega_{1}\cdot\omega_{2}=0$. We
define the operator $N_{\omega}:=\omega\cdot\nabla$. Since this operator can
be interpreted as the $\overline{\partial}$ operator in the complex plane
defined by $\omega_{1}$ and $\omega_{2}$ one can construct an inverse operator
that can be formally defined by

$N^{-1}_{\omega}(g)(x)=\frac{1}{(2\pi)^{3}}\int_{\mathbb{R}^{3}}e^{-ix\cdot\xi}\Big{(}\frac{\hat{g}(\xi)}{\omega\cdot\xi}\Big{)}d\xi,$
(3.14)

for a compactly supported distribution $g$ (for instance). Some key mapping
properties of $N^{-1}_{\omega}$ are summarized in the appendix. We remark that
if $\rho\in\mathbb{C}^{3}$ satisfies $\rho\cdot\rho=0$, then $\rho=s\omega$
with $s=\frac{|\rho|}{\sqrt{2}}$ and $\omega$ is as above. With this notation
for $\rho$ we have the following Lemma where the proof can be found in [1]
which require more regularity than $W^{1,\infty}$ for the magnetic potentials.

######  Lemma 3.3.

Let $A\in W^{2,\infty}(D)$ and $q\in L^{\infty}(D)$ such that
$\|A\|_{W^{2,\infty}}\leq M$, $\|q\|_{L^{\infty}}\leq M$ for some positive
constant $M$, and $\mathrm{Supp}(A)$, $\mathrm{Supp}(q)\subset D$. There
exists $s_{0}>0$ such that for any $s\geq s_{0}$, $\rho=s\omega$ satisfying
$\rho\cdot\rho=0$, there exist complex geometrical solutions $u(\cdot,\rho)\in
H^{2}(B)$ of the form

$u(x,\rho)=e^{ix\cdot\rho}(e^{i\varphi(x,\omega)}+r(x,\rho)),$

to the equation $\mathcal{H}_{A,q}u=k^{2}u$ in $B$, where
$\varphi(x,\omega)=N^{-1}_{\omega}(-\omega\cdot A)$ and

$\|r(\cdot,\rho)\|_{H^{m}(B)}\leq Cs^{m-1},\quad 0\leq m\leq 2\quad\mbox{ and
}\quad\|u(\cdot,\rho)\|_{H^{2}(B)}\leq Cs^{2}e^{\Lambda s},$

where $C$, $\Lambda$ and $s_{0}$ depend only on $B$, $k$ and $M$.

In the remainder of this section we consider two pairs of potentials
$(A_{j},q_{j})\in W^{2,\infty}\times L^{\infty}$, $j=1,2$, with
$\mathrm{Supp}(A)$, $\mathrm{Supp}(q)\subset D$, $\Im(q_{j})\geq 0$ and
satisfying

$\|A_{j}\|_{W^{2,\infty}}\leq M,\quad\|q_{j}\|_{L^{\infty}}\leq M,\quad
j=1,2,$ (3.15)

for some $M>0$ fixed and set as previously

$A(x):=(A_{2}-A_{1})(x),\quad q(x):=(q_{2}-q_{1})(x),\quad
x\in\mathbb{R}^{3}.$ (3.16)

Let $\xi\in\mathbb{R}^{3}$, $\omega_{1}$, $\omega_{2}\in\mathbb{S}^{2}$ be
three mutually orthogonal vectors in $\mathbb{R}^{3}$. For each
$s>\left|\xi\right|/2$, let

$\rho_{1}=s\Big{(}i\omega_{2}+\big{(}\frac{\xi}{2s}-\sqrt{1-\frac{|\xi|^{2}}{4s^{2}}}\omega_{1}\big{)}\Big{)}:=s\omega^{*}_{1}(s),$
(3.17)
$\rho_{2}=s\Big{(}-i\omega_{2}+\big{(}\frac{\xi}{2s}+\sqrt{1-\frac{|\xi|^{2}}{4s^{2}}}\omega_{1}\big{)}\Big{)}:=s\omega^{*}_{2}(s).$
(3.18)

For $s\geq s_{0}>0$ for some $s_{0}$ sufficiently large (that only depends on
$B$ and $k$), Lemma 3.3 guarantees the existence of the geometrical optics
solutions: $u_{1}\in H^{2}(B)$ verifying
$\mathcal{H}_{-A_{1},q_{1}}u_{1}=k^{2}u_{1}$ in $B$ and $u_{2}\in H^{2}(B)$
verifying $\mathcal{H}_{A_{2},q_{2}}u_{2}=k^{2}u_{2}$ in $B$ and such that

$u_{j}(x)=e^{ix\cdot\rho_{j}}(e^{i\varphi_{j}(x,\omega_{j}^{*})}+r_{j}(x,\rho_{j})),$
(3.19)

where $r_{j}(\cdot,\rho_{j})$, $j=1,2$, satisfies

$\|r_{j}(\cdot,\rho_{j})\|_{H^{m}(D)}\leq Cs^{m-1},\quad 0\leq m\leq 2,$
(3.20)

and where
$\varphi_{1}(x,\omega_{1}^{*})=N_{\omega_{1}^{*}}^{-1}(\omega_{1}^{*}\cdot
A_{1})$ and
$\varphi_{2}(x,\omega_{2}^{*})=N_{\omega_{2}^{*}}^{-1}(-\omega_{2}^{*}\cdot
A_{2})$ are a solutions of

$\omega_{1}^{*}\cdot\nabla\varphi_{1}=\omega_{1}^{*}\cdot
A_{1},\quad\omega_{2}^{*}\cdot\nabla\varphi_{2}=-\omega_{2}^{*}\cdot A_{2}.$
(3.21)

Furthermore, according to (3.19), (3.17) and (3.18), there exist $C$ and
$\Lambda>0$ such that

$\|u_{1}u_{2}\|_{L^{1}(B)}\leq C,\quad\text{and}\quad\|u_{j}\|_{H^{2}(B)}\leq
Cs^{2}e^{\Lambda s},\text{for }\,j=1,2.$ (3.22)

###  3.3. Stability estimate for the magnetic potential

We derive in this section a stability estimate for the magnetic fields. First,
we will use Lemma 3.1 and the complex geometric solutions constructed as above
to estimate the Fourier transform of the difference of the magnetic fields
$\textrm{curl}\,A$. Second, we exploit the condition
$\|\widehat{\mathrm{curl}A}\|_{L^{\sigma}}$, $\sigma>0$, is a priori bounded
to prove the stability estimate.

######  Lemma 3.4.

Let $u_{j}$, $j=1,2$, be the functions given by (3.19) and set
$\omega=\omega_{1}+i\omega_{2}$. Then for any $\left|\xi\right|\leq s$, we
have the following identity

$i\int_{D}A(x)\cdot\left(u_{2}\nabla u_{1}-u_{1}\nabla
u_{2}\right)dx=2s\int_{D}\overline{\omega}\cdot
A(x)e^{ix\cdot\xi}dx+\mathcal{R}(\xi,s),$ (3.23)

with $|\mathcal{R}(\xi,s)|\leq C\left<\xi\right>$, where $C$ is independent of
$s$, $\xi$ and $M$, with the short notation
$\left<\xi\right>:=\sqrt{|\xi|^{2}+1}$.

This Lemma is a straightforward extension and adaptation of a similar lemma in
[1] and, for shake of completeness, we provide the proof in the Appendix.

In what follows, for $A_{1}$ and $A_{2}\in W^{2,\infty}(\mathbb{R}^{3})$ as
above, we introduce the notation

$a_{j}(x)=(A_{2}-A_{1})(x)\cdot e_{j}=A(x)\cdot e_{j},\quad j=1,2,3,\quad
x\in\mathbb{R}^{3},$ (3.24)

where $(e_{1},e_{2},e_{3})$ is the canonical basis of $\mathbb{R}^{3}$ and set
for $j,\ell=1,2,3$,

$b_{j\ell}(x):=\frac{\partial a_{\ell}}{\partial x_{j}}(x)-\frac{\partial
a_{j}}{\partial x_{\ell}}(x),\quad x\in\mathbb{R}^{3},$ (3.25)

the components of $\textrm{curl}\,A$ and

$\hat{b}_{j\ell}(\xi)=\int_{\mathbb{R}^{3}}e^{ix\cdot\xi}b_{j\ell}(x)dx,$

the associated Fourier coefficients. We then have following estimate for the
Fourier transform of the difference of the magnetic fields.

######  Lemma 3.5.

For any $s\geq s_{0}$ and $\xi\in\mathbb{R}^{3}$ such that $|\xi|\leq s$ the
following estimate holds true,

$|\hat{b}_{j\ell}(\xi)|\leq C\left<\xi\right>\left(e^{\Lambda
s}\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|+s^{-1}\left<\xi\right>\right),$
(3.26)

for $j,\ell=1,2,3$, where $C$ and $\Lambda$ are positive constants independent
of $s$, $\xi$ and $M$.

###### Proof.

Let $\xi\in\mathbb{R}^{3}$ such that $|\xi|\leq s$. Let
$\omega=\omega_{1}+i\omega_{2}$, where $\omega_{j}$, $j=1,2$ are as above and
consider $u_{j}$, $j=1,2$ the solutions given by (3.19). By using (3.8) and
(3.23), we get for $\left|\xi\right|\leq s$

$2s\big{|}\int_{B}\overline{\omega}\cdot Ae^{ix\cdot\xi}dx\big{|}\cr\leq
C\Big{(}\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|\|u_{1}\|_{H^{2}(B)}\|u_{2}\|_{H^{2}(B)}+\|u_{1}u_{2}\|_{L^{1}(B)}+\left<\xi\right>\Big{)}.$
(3.27)

Then, we obtain by (3.22)

$\big{|}\int_{B}\overline{\omega}\cdot A(x)e^{ix\cdot\xi}dx\big{|}\leq
C\left(e^{\Lambda
s}\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|+s^{-1}\left<\xi\right>\right).$
(3.28)

The reasoning above remains valid if we change $\omega_{1}$ by $-\omega_{1}$
and therefore we also have

$\big{|}\int_{B}-\omega\cdot A(x)e^{ix\cdot\xi}dx\big{|}\leq C\left(e^{\Lambda
s}\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|+s^{-1}\left<\xi\right>\right).$
(3.29)

Outside the plane ${\xi_{j}e_{k}-\xi_{\ell}e_{j}}=0$, we choose
$\omega_{2}=\frac{\xi_{j}e_{\ell}-\xi_{\ell}e_{j}}{\left|\xi_{j}e_{\ell}-\xi_{\ell}e_{j}\right|}$
which is indeed an orthogonal unitary direction to $\xi$. Multiplying both
sides of (3.28) and (3.29) by $\left|\xi_{j}e_{\ell}-\xi_{\ell}e_{j}\right|$,
adding them together and using the triangular inequality to eliminate
$\omega_{1}$ imply

$\big{|}\int_{\mathbb{R}^{3}}e^{ix\cdot\xi}(\xi_{j}a_{\ell}(x)-\xi_{\ell}a_{j}(x))dx\big{|}\leq
C\left<\xi\right>\left(e^{\Lambda
s}\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|+s^{-1}\left<\xi\right>\right),$
(3.30)

for ${\xi_{j}e_{\ell}-\xi_{\ell}e_{j}}\neq 0$. The inequality extends to all
$|\xi|\leq s$ by regularity of both sides in terms of $\xi$ and proves (3.26).
This end the proof. ∎

We are now in position to prove the main stability result for the magnetic
potential from the near field data using (3.26) and regularity assumptions to
estimate Fourier coefficients for large $|\xi|$. More precisely, we assume now
that for $j=1,2$, $A_{j}\in W^{2,\infty}(D)$ with $\|A\|_{W^{2,\infty}(D)}\leq
M$, and

$\int_{\mathbb{R}^{3}}\left<\xi\right>^{\sigma}|\widehat{\mathrm{curl}A_{j}}(\xi)|\,d\xi<M$
(3.31)

for some $\sigma>0$, where $\widehat{\mathrm{curl}A_{j}}$ denotes the Fourier
transform of $\mathrm{curl}A_{j}$.

#### End of the proof of the stability for the magnetic field

We derive now the stability estimate for the magnetic fields in
$L^{\infty}$-norm. Let $s_{0}>1$ be as in Lemma 3.5 and $s$ and $R$ be two
parameters satisfying $s\geq R\geq s_{0}$. From (3.26) we get

$\displaystyle\int_{\mathbb{R}^{3}}|\hat{b}_{j\ell}(\xi)|\,d\xi$
$\displaystyle=\int_{\left<\xi\right>\leq
R}|\hat{b}_{j\ell}(\xi)|\,d\xi+\int_{\left<\xi\right>\geq
R}|\hat{b}_{j\ell}(\xi)|\,d\xi$ $\displaystyle\leq CR^{2}\left(e^{\Lambda
s}\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|+s^{-1}R\right)+2MR^{-\sigma}.$

Choosing $R=s^{1/(\sigma+3)}$ we deduce that for $s_{0}$ sufficiently large
(depending only on $B$, $k$, $M$ and $\sigma$),

$\|b_{j\ell}\|_{L^{\infty}(\mathbb{R}^{3})}\leq
C^{\prime}\big{(}e^{\Lambda^{\prime}s}\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|+s^{-\sigma/{\sigma+3}}\big{)},$
(3.32)

for some positive constants $C^{\prime}$ and $\Lambda^{\prime}$ and all $s\geq
s_{0}$. Now if
$\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|\leq\varepsilon_{0}$,
for some $\varepsilon_{0}>0$, such that $-\log(\epsilon_{0})\geq
2\Lambda^{\prime}s_{0}$, then taking
$s=\frac{-1}{2\Lambda^{\prime}}\log(\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|)$
in (3.32) implies

$\|b_{j\ell}\|_{L^{\infty}(\mathbb{R}^{3})}\leq
C^{\prime}\big{(}\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|^{1/2}+\big{(}\frac{-1}{2\Lambda^{\prime}}\log(\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|)\big{)}^{-\sigma/{\sigma+3}}\big{)}.$
(3.33)

We also observe that this type of inequality holds true if
$\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|\geq\varepsilon_{0}$
since in that case we can simply write

$\|b_{j\ell}\|_{L^{\infty}(\mathbb{R}^{3})}\leq
M\leq(M/\sqrt{\epsilon_{0}})\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|^{1/2}.$
(3.34)

The proof of the first estimate of Theorem 1.1 is then completed.

Using the above result, we are able to prove the second main result related to
the stability for the electric potential.

###  3.4. Stability estimate for the electric potential

In this section, we are going to use the complex geometric optics solutions
and the stability estimate we already obtained for the magnetic field in order
to retrieve a stability result for the electric potential. There are, however,
some difficulties with this. In fact, in order to isolate the integral of the
difference $q=q_{2}-q_{1}$ we would need to control the norm of the difference
$A=A_{2}-A_{1}$. Unfortunately, we can only estimate the difference of the
magnetic fields $\mathrm{curl}(A)$. To overcome this difficulty we will use
the Helmholtz decomposition and write $A=H-\nabla\vartheta$, with the fact
that $\mathrm{div}\,H=0$ and we are able to estimate the norm of
$\nabla\vartheta$.

######  Lemma 3.6.

Let $p>3$. There exist $\vartheta\in W^{3,p}(B)\cap H^{1}_{0}(B)$ and a
positive constant $C$ such that

$\|\vartheta\|_{W^{3,p}(B)}\leq C\|A\|_{W^{2,p}(D)},$ (3.35)

and

$\|A+\nabla\vartheta\|_{W^{1,p}(B)}\leq C\|\mathrm{curl}(A)\|_{L^{p}(D)}.$
(3.36)

Moreover, if $B^{\prime}$ is a ball containing $\overline{D}$ and such that
$\overline{B^{\prime}}\subset B$, then

$\|\vartheta\|_{W^{2,p}(B\backslash B^{\prime})}\leq
C\|\mathrm{curl}(A)\|_{L^{p}(D)}.$ (3.37)

###### Proof.

Let $\vartheta$ solves the following elliptic boundary value problem in the
ball $B$

$\left\\{\begin{array}[]{lll}-\Delta\vartheta&=\mathrm{div}(A)&\textrm{in}\,B,\cr\vartheta&=0&\textrm{on}\,\partial
B.\end{array}\right.$ (3.38)

Since the source term $\mathrm{div}(A)$ belongs to $W^{1,p}(B)$ by the
elliptic regularity (see [6, Theorem 2.5.1.1 in Chapter 2]), we have
$\vartheta\in W^{3,p}(B)\cap H^{1}_{0}(B)$. Moreover, there exist $C>0$ such
that

$\|\vartheta\|_{W^{3,p}(B)}\leq C\|\mathrm{div}(A)\|_{W^{1,p}(D)}\leq
C\|A\|_{W^{2,p}(D)},$ (3.39)

this ends the prove of (3.35). To prove (3.36), we consider the vector field
$H\in W^{2,p}(B)$ defined by

$H=A+\nabla\vartheta.$ (3.40)

By (3.38) and (3.40), $H$ satisfies

$\mathrm{div}(H)=0,\quad\mathrm{curl}(H)=\mathrm{curl}(A)\quad\textrm{in}\,B,\quad\textrm{and}\quad
H\wedge\nu=0\quad\textrm{on}\,\partial B.$ (3.41)

By the $L^{p}$-$\mathrm{div}$-$\mathrm{curl}$ estimate, we get

$\|H\|_{W^{1,p}(B)}\leq C\|\mathrm{curl}(H)\|_{L^{p}(B)},$ (3.42)

and we conclude (3.36). Finally, to prove (3.37) we consider a cut-off
function $\chi_{0}\in C^{\infty}(\mathbb{R}^{n})$ such that $\chi_{0}=1$ in
$B\backslash B^{\prime}$ and $\chi_{0}=0$ in $\overline{D}$. Set
$\vartheta_{0}=\chi_{0}\vartheta$, we have by (3.38)

$\left\\{\begin{array}[]{lll}-\Delta\vartheta_{0}=[\Delta,\chi_{0}]\vartheta&\textrm{in}\,B,\cr\vartheta_{0}=0&\textrm{on}\,\partial
B,\end{array}\right.$ (3.43)

where we have used $A=0$ outside $D$. Thus, we obtain

$\|\vartheta_{0}\|_{W^{2,p}(B)}\leq C\|\nabla\vartheta\|_{L^{p}(B\backslash
D)},$ (3.44)

since the first order operator $[\Delta,\chi_{0}]$ is supported in
$B^{\prime}\backslash D$. Then we deduce that

$\|\vartheta_{0}\|_{W^{2,p}(B)}\leq C\|A+\nabla\vartheta\|_{L^{p}(B\backslash
D)}\leq C\|\mathrm{curl}(A)\|_{L^{p}(D)}.$ (3.45)

This ends the proof. ∎

Applying now Morrey’s inequality, we obtain for some positive constant $C$

$\|A+\nabla\vartheta\|_{L^{\infty}(B)}\leq
C\|\mathrm{curl}(A)\|_{L^{\infty}(D)},$ (3.46)

and

$\|\vartheta\|_{W^{1,\infty}(B\backslash B^{\prime})}\leq
C\|\mathrm{curl}(A)\|_{L^{\infty}(D)}.$ (3.47)

We prove now the following inequality which is a slight modification of the
previous estimate (3.8) given in Lemma 3.2. This inequality is based on the
invariance of the near field operator under gauge transformation for the
magnetic potential that was explained in the introduction.

######  Lemma 3.7.

Let $B$ denote an open ball containing $\overline{D}$ and let $\varphi\in
W^{2,\infty}(B)$ with $\mathrm{Supp}(\varphi)\subset B$. Then there exist a
constant $C$ that only depends on $B$ and such that

$\left|\int_{B}e^{-i\varphi}\big{[}i(A+\nabla\varphi)\cdot(u_{1}\nabla
u_{2}-u_{2}\nabla
u_{1})-((A+\nabla\varphi)\cdot(A_{1}+A_{2})+q)u_{1}u_{2}\big{]}dx\right|\\\
\leq
C\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|\|u_{1}\|_{H^{2}(B)}\|u_{2}\|_{H^{2}(B)},$
(3.48)

for all $u_{1}\in H^{2}(B)$ satisfying
$\mathcal{H}_{-A_{1},q_{1}}u_{1}=k^{2}u_{1}$ in $B$ and all $u_{2}\in
H^{2}(B)$ satisfying $\mathcal{H}_{A_{2},q_{2}}u_{2}=k^{2}u_{2}$ in the ball
$B$.

###### Proof.

Define $\tilde{A}_{1}=A_{1}-\frac{1}{2}\nabla\varphi$ and
$\tilde{A}_{2}=A_{2}+\frac{1}{2}\nabla\varphi$. Consider $u_{1}$, $u_{2}\in
H^{2}(B)$ as in the lemma. Let denote $\tilde{u}_{j}=e^{-i\varphi/2}u_{j}$,
then by (1.11) we get

$\displaystyle\mathcal{H}_{-\tilde{A}_{1},q_{1}}\tilde{u}_{1}$
$\displaystyle=e^{-i\varphi/2}\mathcal{H}_{-A_{1},q_{1}}u_{1}=k^{2}\tilde{u}_{1},$
(3.49) $\displaystyle\mathcal{H}_{\tilde{A}_{2},q_{2}}\tilde{u}_{2}$
$\displaystyle=e^{-i\varphi/2}\mathcal{H}_{A_{2},q_{2}}u_{1}=k^{2}\tilde{u}_{2},\quad\textrm{in}\,B.$
(3.50)

Moreover, due the gauge invariance of the scattered field and since
$\varphi_{|\partial B}=0$, we have from (2.26) that

$\mathcal{N}_{\tilde{A}_{j},q_{j}}=\mathcal{N}_{A_{j},q_{j}},\quad j=1,2.$
(3.51)

With these solution at hand, we use the gauge invariance (3.51) and (3.8) with
$\tilde{A}_{j}$ and $\tilde{u}_{j}$ instead of $A_{j}$ and $u_{j}$,
respectively, to obtain

$\left|\int_{B}[i\tilde{A}\cdot(\tilde{u}_{1}\nabla\tilde{u}_{2}-\tilde{u}_{2}\nabla\tilde{u}_{1})-(|\tilde{A}_{2}|^{2}-|\tilde{A}_{1}|^{2}+q)\tilde{u}_{1}\tilde{u}_{2}]dx\right|\cr\leq
C\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|\|\tilde{u}_{1}\|_{H^{2}(B)}\|\tilde{u}_{2}\|_{H^{2}(B)}.$
(3.52)

From the identity (3.52), we get (3.48). This ends the proof. ∎

Lemma 3.7, allows us then to obtain an estimate to $A$ added with a gradient
term. By adding $\nabla\vartheta$, we would thus get an estimate with
controlled terms. Unfortunately, we cannot directly add $\nabla\vartheta$,
because of the requirement that $\mathrm{Supp}(\varphi)\subset B$ in Lemma
3.7. We can solve this difficulty by using a cutoff argument.

Now, we will fix $\varphi=\chi\vartheta$, for $\chi\in C_{0}^{\infty}(B)$ such
that $\chi=1$ in $\overline{B}^{\prime}$ and $\vartheta\in W^{2,\infty}(B)$
given by Lemma 3.6 which satisfying (3.46) and (3.47).

######  Lemma 3.8.

Let $u_{j}$, $j=1,2$ be the functions given by (3.19) for some $s_{0}>0$. Then
there exist a positive constants $C$ and $\Lambda$ such that

$\left|\int_{B}e^{-i\varphi}q(x)u_{1}u_{2}dx\right|\leq Ce^{2\Lambda
s}\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|+s\|\mathrm{curl}(A)\|_{L^{\infty}(D)}$
(3.53)

for all $s>s_{0}$.

###### Proof.

Let $u_{j}$, $j=1,2$ be the functions given by (3.19) for some $s_{0}>0$.
Adding and subtracting the same terms we get

$\displaystyle\int_{B}e^{-i\varphi}q(x)u_{1}u_{2}dx=$ $\displaystyle\int
e^{-i\varphi}(A+\nabla\vartheta)\cdot(A_{1}+A_{2})u_{1}u_{2}dx$ (3.54)
$\displaystyle+\int_{B}ie^{-i\varphi}(A+\nabla\vartheta)\cdot(u_{1}\nabla
u_{2}-u_{2}\nabla u_{1})dx$ (3.55)
$\displaystyle+\int_{B}e^{-i\varphi}(A+\nabla\vartheta)\cdot\nabla\varphi
u_{1}u_{2}dx+\mathcal{R}$ (3.56) $\displaystyle:=$
$\displaystyle\mathcal{J}_{1}+\mathcal{J}_{2}+\mathcal{J}_{3}+\mathcal{R},$
(3.57)

where $\mathcal{R}$ denotes the integral

$\displaystyle\mathcal{R}=-\int_{B}$ $\displaystyle
e^{-i\varphi}\big{[}i(A+\nabla\vartheta)\cdot(u_{1}\nabla u_{2}-u_{2}\nabla
u_{1})$
$\displaystyle\qquad+\Big{(}(A+\nabla\vartheta)\cdot(A_{1}+A_{2})-q+(A+\nabla\vartheta)\cdot\nabla\varphi\Big{)}u_{1}u_{2}\big{]}\,dx.$

First, we have

$\displaystyle\left|\mathcal{J}_{2}\right|\leq
C\|A+\nabla\vartheta\|_{L^{\infty}(B)}(\|u_{1}\nabla
u_{2}\|_{L^{1}(B)}+\|u_{2}\nabla u_{1}\|_{L^{1}(B)}).$ (3.58)

Then, we deduce from (3.46) and (3.22) that

$\left|\mathcal{J}_{2}\right|\leq Cs\|\mathrm{curl}(A)\|_{L^{\infty}(D)}.$
(3.59)

Similarly, we have

$\left|\mathcal{J}_{1}\right|\leq
C\|A+\nabla\vartheta\|_{L^{\infty}(B)}\|u_{1}u_{2}\|_{L^{1}(B)}$ (3.60)

and then, we have

$\left|\mathcal{J}_{1}\right|\leq C\|\mathrm{curl}(A)\|_{L^{\infty}(D)}.$
(3.61)

Finally, by the same arguments, we have

$\left|\mathcal{J}_{3}\right|\leq C\|\mathrm{curl}(A)\|_{L^{\infty}(D)}.$
(3.62)

Now recall that $\varphi=\chi\vartheta$ and set
$\tilde{\varphi}=(1-\chi)\vartheta$. Since
$\nabla\vartheta=\nabla\varphi+\nabla\tilde{\varphi}$ we obtain, by Lemma 3.7,
that

$\left|\mathcal{R}\right|\leq
C\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|\|u_{1}\|_{H^{2}(B)}\|u_{2}\|_{H^{2}(B)}\cr+\left|\int_{B}ie^{-i\varphi}\nabla\tilde{\varphi}\cdot(u_{1}\nabla
u_{2}-u_{2}\nabla
u_{1})+e^{-i\varphi}(\nabla\varphi+A_{1}+A_{2})\cdot\nabla\tilde{\varphi}\,u_{1}u_{2}dx\right|.$
(3.63)

By the same arguments used previously, we get

$\left|\mathcal{R}\right|\leq Ce^{2\Lambda
s}\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|+s\|\nabla\tilde{\varphi}\|_{L^{\infty}(B)}.$
(3.64)

Since by (3.47) we have

$\|\nabla\tilde{\varphi}\|_{L^{\infty}(B)}\leq\|\nabla\vartheta\|_{L^{\infty}(B\backslash
B^{\prime})}\leq C\|\mathrm{curl}(A)\|_{L^{\infty}(D)},$ (3.65)

we conclude (3.53) from (3.58), (3.61), (3.62) and (3.65). ∎

We now state the following integral identity for the electric potential which
is proved in the Appendix.

######  Lemma 3.9.

Let $u_{j}$, $j=1,2$ be the functions given by (3.19) for some $s_{0}>0$. Then
for all $\xi\in\mathbb{R}^{3}$ and $s\geq\textrm{max}(s_{0},|\xi|/2)$, we have
the following identity

$\int_{D}e^{-i\varphi}q(x)u_{1}u_{2}\,dx=\int_{D}q(x)e^{ix\cdot\xi}\,dx+\mathcal{R}^{\prime}(\xi,s),$
(3.66)

where $\mathcal{R}^{\prime}(\xi,s)$, satisfy

$\left|\mathcal{R}^{\prime}(\xi,s)\right|\leq
C\big{(}\|\mathrm{curl}(A)\|_{L^{\infty}(D)}+s^{-1}\left<\xi\right>\big{)}.$
(3.67)

The constants $C$ and $s_{0}$ depend only on $B$, $M$ and $k$.

This identity allows to obtain the following estimates for the Fourier
coefficients

$\hat{q}(\xi):=\int_{\mathbb{R}^{3}}e^{ix\cdot\xi}q(x)\,dx.$

From Lemma 3.8 and Lemma 3.9 we deduce the following estimate.

######  Lemma 3.10.

There exists $s_{0}>0$ such that for all $s\geq s_{0}$ and
$\xi\in\mathbb{R}^{3}$ with $|\xi|\leq s$ the following estimate holds true,

$|\widehat{q}(\xi)|\leq C\big{(}e^{\Lambda
s}\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|+s\|\mathrm{curl}(A)\|_{L^{\infty}(D)}+s^{-1}\left<\xi\right>\big{)}.$
(3.68)

The constants $s_{0}$, $C$ and $\Lambda$ depend only on $B$, $M$ and $k$.

With the help of the previous lemma, we are now in position to prove the
stability result for the electric potential under the assumption

$\int_{\mathbb{R}^{3}}\left<\xi\right>^{\gamma}|\hat{q}_{j}(\xi)|\,d\xi<M,$
(3.69)

for some $\gamma>0$.

#### End of the proof of the stability estimate for the electric potential

Let $s_{0}>1$ be as in Lemma 3.10 and $s$ and $R$ be two parameters satisfying
$s\geq R\geq s_{0}$. From (3.68) and (3.69) we get

$\displaystyle\int_{\mathbb{R}^{3}}|\hat{q}(\xi)|\,d\xi$
$\displaystyle=\int_{\left<\xi\right>\leq
R}|\hat{q}(\xi)|,d\xi+\int_{\left<\xi\right>\geq R}|\hat{q}(\xi)|\,d\xi$
$\displaystyle\leq CR\left(e^{\Lambda
s}\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|+s\|\textrm{curl}(A)\|_{L^{\infty}}+Rs^{-1}\right)+2MR^{-\gamma}.$

Choosing $R=s^{1/(\gamma+2)}$, we deduce that for $s_{0}$ sufficiently large
(depending only on $B$, $k$ $M$ and $\gamma$),

$\|q\|_{L^{\infty}(\mathbb{R}^{3})}\leq
C^{\prime}\big{(}e^{\Lambda^{\prime}s}\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|+s^{(\gamma+3)/(\gamma+2)}\|\textrm{curl}(A)\|_{L^{\infty}}+s^{-\gamma/{(\gamma+2)}}\big{)}$
(3.70)

for some positive constants $C^{\prime}$ and $\Lambda^{\prime}$ and all $s\geq
s_{0}$. Observe now that (3.32) implies in particular (after eventually
changing the constants $C^{\prime}$, $\Lambda^{\prime}$ and $s_{0}$)

$\|\textrm{curl}A\|_{L^{\infty}(D)}\leq
C^{\prime}\big{(}e^{\Lambda^{\prime}s^{\kappa}}\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|+s^{-\kappa\sigma/(\sigma+3)}\big{)},$
(3.71)

for all $\kappa\geq 1$. Choosing $\kappa$ such that

$-\frac{\kappa\sigma}{\sigma+3}+\frac{\gamma+3}{\gamma+2}=-\frac{\gamma}{\gamma+2}\Leftrightarrow\kappa=\frac{(2\gamma+3)(\sigma+3)}{\sigma(\gamma+2)},$

we obtain by substituting (3.71) in (3.70)

$\|q\|_{L^{\infty}(\mathbb{R}^{3})}\leq
C^{\prime}\big{(}e^{\Lambda^{\prime}s^{\kappa}}\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|+s^{-\gamma/{(\gamma+2)}}\big{)},$
(3.72)

with possibly different constants $C^{\prime}$ and $\Lambda^{\prime}$.

Now if
$\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|\leq\varepsilon_{0}$,
for some $\varepsilon_{0}>0$, such that $-\log(\epsilon_{0})\geq
2\Lambda^{\prime}s_{0}^{\kappa}$, then taking
$s^{\kappa}=\frac{-1}{2\Lambda^{\prime}}\log(\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|)$
in (3.32) implies

$\displaystyle\|q\|_{L^{\infty}(\mathbb{R}^{3})}\leq
C^{\prime}\big{(}\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|^{1/2}+\big{(}\frac{-1}{2\Lambda^{\prime}}\log(\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|)\big{)}^{-\gamma/{\kappa(\gamma+2)}}\big{)}.$
(3.73)

We also observe that this type of inequality holds true if
$\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|\geq\varepsilon_{0}$
since in that case we can simply write

$\|q\|_{L^{\infty}(\mathbb{R}^{3})}\leq
M\leq(M/\sqrt{\epsilon_{0}})\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|^{1/2}.$
(3.74)

The proof of the second part of Theorem 1.2 is then completed.

##  4\. Stability analysis for far field data

In order to prove stability estimates with far field measurements we shall
exploit the relation between the far field $u_{A,q}^{\infty}(\hat{x},d)$,
$(\hat{x},d)\in\mathbb{S}^{2}\times\mathbb{S}^{2}$ and the operator
$\mathcal{N}_{A,q}$ given by (1.7) with $B=\\{x\in\mathbb{R}^{3},|x|<a\\}$ for
some sufficiently large $a>0$ so that $D\subset B$. We here assume that the
hypothesis of Section 2 hold. We recall that the far field pattern can be
expressed as

$u_{A,q}^{\infty}(\hat{x},d)=\frac{1}{4\pi}\int_{D}e^{-ik\hat{x}\cdot
y}Q_{A,q}u_{A,q}(y,d)dy.$ (4.1)

We denote by $\mu_{(\ell_{1},m_{1};\ell_{2},m_{2})}$,
$(\ell_{i},m_{i})\in\Gamma$, $i=1,2$ the Fourier coefficients of
$u_{A,q}^{\infty}$ given by

$\mu_{(\ell_{1},m_{1};\ell_{2},m_{2})}:=\int_{\mathbb{S}^{2}}\int_{\mathbb{S}^{2}}u_{A,q}^{\infty}(\hat{x},d)\overline{Y^{m_{1}}_{\ell_{1}}}(\hat{x})\overline{Y^{m_{2}}_{\ell_{2}}}(d)\mathrm{ds}(\hat{x})\;\mathrm{ds}(d),$
(4.2)

For proving the first lemma we need the following well known results about the
asymptotic of spherical Bessel functions $j_{\ell}$ and spherical Hankel
functions of the first kind $h_{\ell}^{(1)}$ [21, Appendix A]:

$\left|j_{\ell}(kr)\right|\leq\alpha\Big{(}\frac{ekr}{2\ell+1}\Big{)}^{\ell}\frac{1}{2\ell+1},\quad
0\leq r\leq a,\quad\ell\in\mathbb{N}\cup\\{0\\},$ (4.3)

and

$\left|h_{\ell}^{(1)}(kr)\right|\leq\alpha\Big{(}\frac{2\ell+1}{ekr}\Big{)}^{\ell},\quad
0<r\leq a,\quad\ell\in\mathbb{N}\cup\\{0\\},$ (4.4)

where $\alpha$ is a constant that only depend on $a$ and $k$. We also recall
the following equality that comes from the addition formula [3],

$\int_{\mathbb{S}^{2}}Y^{m_{2}}_{\ell_{2}}(\hat{z})\Phi(x,r\hat{z})\mathrm{ds}(\hat{z})=ikj_{\ell_{2}}(kr)h^{(1)}_{\ell_{2}}(k|x|)Y^{m_{2}}_{\ell_{2}}(\hat{x}),\quad|x|>r,$
(4.5)

together with the Funk-Hecke formula

$\int_{\mathbb{S}^{2}}e^{-ikx\cdot\hat{z}}Y^{m_{2}}_{\ell_{2}}(\hat{z})\mathrm{ds}(\hat{z})=\frac{4\pi}{i^{\ell_{2}}}j_{\ell_{2}}(k|x|)Y^{m_{2}}_{\ell_{2}}(\hat{x}),\hskip
14.22636ptx\in\mathbb{R}^{3}.$ (4.6)

######  Lemma 4.1.

Let $A\in W^{1,\infty}(\mathbb{R}^{3},\mathbb{R}^{3})$ and $q\in
L^{\infty}(\mathbb{R}^{3},\mathbb{C})$ be as in Section 2 and such that
$\|A\|_{W^{1,\infty}}\leq M$ and $\|q\|_{L^{\infty}}\leq M$ for some constant
$M>0$. Let $\mu_{(\ell_{1},m_{1};\ell_{2},m_{2})}$ denote the Fourier
coefficients of the far field patterns $u_{A,q}^{\infty}$ as defined in (4.2).
Then there exists a constant $C>0$ that only depends on $D$, $a$, $k$ and $M$
such that

$\left|\mu_{(\ell_{1},m_{1};\ell_{2},m_{2})}\right|^{2}\leq
C\Big{(}\frac{eka}{2\ell_{1}+1}\Big{)}^{2\ell_{1}+3}\Big{(}\frac{eka}{2\ell_{2}+1}\Big{)}^{2\ell_{2}+3}$
(4.7)

and

$\sum_{(\ell_{1},m_{1})\in\Gamma}\sum_{(\ell_{2},m_{2})\in\Gamma}\Big{(}\frac{2\ell_{1}+1}{eka}\Big{)}^{2\ell_{1}+1}\Big{(}\frac{2\ell_{2}+1}{eka}\Big{)}^{2\ell_{2}+1}\left|\mu_{(\ell_{1},m_{1};\ell_{2},m_{2})}\right|^{2}\leq
C.$

###### Proof.

We only need to prove (4.7). According to (4.2) and (4.1), we obtain

$\displaystyle\mu_{(\ell_{1},m_{1};\ell_{2},m_{2})}$
$\displaystyle=\frac{1}{4\pi}\int_{B}\big{(}\int_{\mathbb{S}^{2}}Q_{A,q}u(y,d)\overline{Y^{m_{2}}_{\ell_{2}}}(d)\mathrm{ds}(d)\big{)}\big{(}\int_{\mathbb{S}^{2}}e^{-ik\hat{x}\cdot
y}\overline{Y^{m_{1}}_{\ell_{1}}}(\hat{x})\mathrm{ds}(\hat{x})\big{)}dy$ (4.8)
$\displaystyle:=\frac{1}{4\pi}\int_{B}w_{\ell_{2},m_{2}}(y)v_{\ell_{1},m_{1}}(y)dy.$
(4.9)

With the help of the Funk-Hecke formula (4.6) we compute

$v_{\ell_{1},m_{1}}(y):=\int_{\mathbb{S}^{2}}e^{-ik\hat{x}\cdot
y}\overline{Y^{m_{1}}_{\ell_{1}}}(\hat{x})\mathrm{ds}(\hat{x})=\frac{4\pi}{i^{\ell_{1}}}j_{\ell_{1}}(k\left|y\right|)\overline{Y_{\ell_{1}}^{m_{1}}}(\hat{y}),\quad
y\in\mathbb{R}^{3}.$ (4.10)

Then by (4.3), we get

$\|v_{\ell_{1},m_{1}}\|_{L^{2}(B)}^{2}\leq
C\int_{0}^{a}\left|j_{\ell_{1}}(kr)\right|^{2}r^{2}dr\leq
C\Big{(}\frac{eka}{2\ell_{1}+1}\Big{)}^{2\ell_{1}+3}.$ (4.11)

Using again the Funk-Hecke formula, we obtain

$\int_{\mathbb{S}^{2}}u(y,d)\overline{Y^{m_{2}}_{\ell_{2}}}(d)\mathrm{ds}(d)=T_{A,q}\Big{(}\int_{\mathbb{S}^{2}}u(y,d)\overline{Y^{m_{2}}_{\ell_{2}}}(d)\mathrm{ds}(d)\Big{)}+(-1)^{\ell_{2}}v_{\ell_{2},m_{2}},$
(4.12)

and we deduce that

$w_{\ell_{2},m_{2}}(x)=(-1)^{\ell_{2}}Q_{A,q}\left((I-T_{A,q})^{-1}v_{\ell_{2},m_{2}}\right)(x).$
(4.13)

Using the fact that $Q_{A,q}$ is a first order operator supported in $D$, then
we obtain from Proposition 2.2,

$\|w_{\ell_{2},m_{2}}\|_{L^{2}(B)}\leq
C\|(I-T_{A,q})^{-1}v_{\ell_{2},m_{2}}\|_{H^{1}(D)}\leq
C\|v_{\ell_{2},m_{2}}\|_{H^{1}(D)}.$ (4.14)

We note that there is a constant $C>0$ such that the inequality

$\|u\|_{H^{1}(D)}\leq C\|u\|_{L^{2}(B)}$ (4.15)

holds true for all $u\in H^{1}_{loc}(\mathbb{R}^{3})$ satisfying the Helmholtz
equation $\Delta u+k^{2}u=0$ in $\mathbb{R}^{3}$. We can then estimate

$\|w_{\ell_{2},m_{2}}\|_{L^{2}(B)}\leq
C\|v_{\ell_{2},m_{2}}\|^{2}_{H^{1}(D)}\leq\|v_{\ell_{2},m_{2}}\|^{2}_{L^{2}(B)}\leq
C\Big{(}\frac{eka}{2\ell_{2}+1}\Big{)}^{2\ell_{2}+3},$ (4.16)

we conclude, from (4.8), (4.11) and (4.16), that

$\displaystyle\left|\mu_{(\ell_{1},m_{1};\ell_{2},m_{2})}\right|^{2}$
$\displaystyle\leq
C\|v_{\ell_{1},m_{1}}\|_{L^{2}(B)}^{2}\|w_{\ell_{2},m_{2}}\|^{2}_{L^{2}(B)}$
(4.17) $\displaystyle\leq
C\Big{(}\frac{eka}{2\ell_{2}+1}\Big{)}^{2\ell_{2}+3}\Big{(}\frac{eka}{2\ell_{1}+1}\Big{)}^{2\ell_{1}+3}.$
(4.18)

This competes the proof. ∎

The following lemma makes the link between $u^{s}_{A,q}(\cdot,y)$ and
$u^{\infty}_{A,q}(\cdot,d)$. The proof follows similar ideas as in Stefanov
[21] for $A=0$ but uses different arguments since we do not rely on the
properties of the Green function for $A\neq 0$ when $x\sim y$. Our proof would
apply to more general contexts since we mainly rely on the reciprocity
relation in Lemma 2.5. The proof is given in Appendix B.

######  Lemma 4.2.

The scattered field associated with point sources can be expanded as

$u^{s}_{A,q}(x,y)=-\frac{k^{2}}{4\pi}\underset{\underset{(\ell_{2},m_{2})\in\Gamma}{(\ell_{1},m_{1})\in\Gamma}}{\sum}i^{\ell_{1}-\ell_{2}}\mu_{(\ell_{1},m_{1};\ell_{2},m_{2})}h^{(1)}_{\ell_{1}}(k|x|)h^{(1)}_{\ell_{2}}(k|y|)Y^{m_{1}}_{\ell_{1}}\left(\hat{x}\right)Y^{m_{2}}_{\ell_{2}}\left(\hat{y}\right),$
(4.19)

uniformly for $|x|,|y|\geq a$, with $\hat{x}=x/|x|$ and $\hat{y}=y/|y|$.

Then we have the following Lemma showing the Lipschitz continuity of the
mapping $u_{A,q}^{\infty}\mapsto\mathcal{N}_{A,q}$ when $u_{A,q}^{\infty}$ is
endowed with the norm (1.17).

######  Lemma 4.3.

Let $A_{j}$ and $q_{j}$, $j=1,2$ be as in Section 2. Then

$\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|\leq\alpha^{2}\frac{k^{2}}{4\pi}\|u_{A_{1},q_{1}}^{\infty}-u_{A_{2},q_{2}}^{\infty}\|_{\mathcal{F}},$
(4.20)

###### Proof.

For $j=1,2$, denote by $\mu^{j}_{(\ell_{1},m_{1};\ell_{2},m_{2})}$ the Fourier
coefficients associated with $u_{A_{j},q_{j}}^{\infty}$ as above. We get from
(4.19)

$\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|^{2}\cr\leq\left(\frac{k^{2}}{4\pi}\right)^{2}\\!\\!\sum_{\begin{subarray}{c}(\ell_{1},m_{1})\in\Gamma\\\
(\ell_{2},m_{2})\in\Gamma\end{subarray}}|h_{\ell_{1}}^{(1)}(ka)|^{2}|h_{\ell_{2}}^{(1)}(ka)|^{2}|\mu^{1}_{(\ell_{1},m_{1};\ell_{2},m_{2})}-\mu^{2}_{(\ell_{1},m_{1};\ell_{2},m_{2})}|^{2}.$

The estimate then follows using (4.4) ∎

From (4.20) and Theorem 1.1 we easily derive Theorem 1.2. Following the same
arguments as in [8] one can obtain a stability result using only the $L^{2}$
norm of the far field. In fact, identity (4.19) and the uniform bound of
Corollary 2.3 allows us to reproduce exactly the same arguments as in [8,
Section 4] to state the following continuity result.

######  Lemma 4.4.

Let $M>0$ and $0<\theta<1$ be given. Let $A_{j}\in
W^{1,\infty}(\mathbb{R}^{3},\mathbb{R}^{3})$ and $q_{j}\in
L^{\infty}(\mathbb{R}^{3})$ be as in Section 2 such that
$\|A_{j}\|_{W^{1,\infty}}\leq M$ and $\|q_{j}\|_{L^{\infty}}\leq M$. Then
there exists a constant $\rho>0$ that only depends on $M$, $k$, $a$ and
$\theta$ and a constant $\omega$ that only depends on $a$ and $k$ such that

$\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|\leq\rho^{2}\exp\Big{(}-\big{(}-\ln\frac{\|u_{A_{1},q_{1}}^{\infty}-u_{A_{2},q_{2}}^{\infty}\|_{L^{2}(\mathbb{S}^{2}\times\mathbb{S}^{2})}}{\omega\rho}\big{)}^{\theta}\Big{)},$

where $\mathcal{N}_{A_{j},q_{j}}$, $j=1,2$ denote here the near field
operators associated with $B=\\{x\in\mathbb{R}^{3},|x|<2a\\}$.

Using the result of this lemma and Theorem 1.1 one can prove Theorem 1.3 as
follows.

#### Proof of Theorem 1.3

According to Lemma 4.4

$\displaystyle-\textrm{ln}\left(\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|\right)$
$\displaystyle\geq-\textrm{ln}(\rho^{2})+\Big{(}-\textrm{ln}\frac{\|u^{\infty}_{A_{1},q_{1}}-u^{\infty}_{A_{2},q_{2}}\|_{L^{2}}}{\omega\rho}\Big{)}^{\theta}$
$\displaystyle\geq\frac{1}{2}\Big{(}-\textrm{ln}\frac{\|u^{\infty}_{A_{1},q_{1}}-u^{\infty}_{A_{2},q_{2}}\|_{L^{2}}}{\omega\rho}\Big{)}^{\theta},$

for sufficiently small
$\|u^{\infty}_{A_{1},q_{1}}-u^{\infty}_{A_{2},q_{2}}\|_{L^{2}}$ such that
$\|u^{\infty}_{A_{1},q_{1}}-u^{\infty}_{A_{2},q_{2}}\|_{L^{2}}\leq\omega\rho
e^{-2(\textrm{ln}(\rho^{2}))^{\frac{1}{\theta}}}$. Then, if we further suppose
that $\|u^{\infty}_{A_{1},q_{1}}-u^{\infty}_{A_{2},q_{2}}\|_{L^{2}}\leq
e/(\omega\rho)$, then

$\displaystyle\left(-\textrm{ln}\left(\|\mathcal{N}_{A_{1},q_{1}}-\mathcal{N}_{A_{2},q_{2}}\|\right)\right)^{-\frac{\sigma}{\sigma+3}}\leq
2^{\frac{\sigma(1-\theta)}{\sigma+3}}\left(-\textrm{ln}\left({\|u^{\infty}_{A_{1},q_{1}}-u^{\infty}_{A_{2},q_{2}}\|_{L^{2}}}\right)\right)^{-\frac{\sigma\theta}{\sigma+3}}.$

Using the first inequality in Theorem 1.1 and choosing $\theta$ such that
$\theta\frac{\sigma}{\sigma+3}=\frac{\sigma}{\sigma+3}-\epsilon$, where
$0<\epsilon<\frac{\sigma}{\sigma+3}$, yield the first inequality of Theorem
1.3 related to $\|\mathrm{curl}A_{1}-\mathrm{curl}A_{2}\|_{L^{\infty}(D)}$.
The estimate for $\|q_{1}-q_{2}\|_{L^{\infty}(D)}$ is derived analogously.

##  Appendix A Proof of Lemmas 3.4 and 3.9

This Appendix is devoted to the proof of the Lemma 3.4. At first, we recall
the following three lemmas proved in [24] on the properties of the operator
$N_{\omega}^{-1}$, $\omega\in\mathbb{S}^{2}+i\mathbb{S}^{2}$, given by (3.14).
The first Lemma, due to Salo [19] in the reconstruction methods and similar to
the one appearing in Eskin and Ralston [5] and Sun [22], shows that a relation
between a non-linear and linear Fourier transform of $\omega\cdot A$ for a
vector field $A$.

######  Lemma A.1.

Let $\omega=\omega_{1}+i\omega_{2}$ with
$\omega_{1},\omega_{2}\in\mathbb{S}^{2}$ and $\xi\in\mathbb{R}^{3}$, such that
$\xi$, $\omega_{1}$ and $\omega_{2}$ be three mutually orthogonal vectors in
$\mathbb{R}^{3}$. Let $A\in W^{2,\infty}(D)^{3}$ with $\mathrm{Supp}(A)\subset
D$. Then we have the following equality

$\int_{\mathbb{R}^{3}}\omega\cdot
A(x)e^{ix\cdot\xi}e^{iN^{-1}_{\omega}(-\omega\cdot
A)(x)}dx=\int_{\mathbb{R}^{3}}\omega\cdot A(x)e^{ix\cdot\xi}dx,$ (A.1)

where $A$ is extended by $0$ outside $D$.

######  Lemma A.2.

Let $g\in W^{n,\infty}(\mathbb{R}^{3})$, $n\geq 0$, with
$\textrm{Supp}(g)\subseteq D$. Then $N^{-1}_{\omega}(g)\in
W^{n,\infty}(\mathbb{R}^{3})$ and satisfies

$\|N_{\omega}^{-1}(g)\|_{W^{n,\infty}(\mathbb{R}^{3})}\leq
C\|g\|_{W^{n,\infty}(D)},$

where $C$ depends only on $D$.

Finally, we have the following result which gives the dependence of
$N^{-1}_{\omega}(-\omega\cdot A)$ on the parameter $\omega$.

######  Lemma A.3.

Let $A\in W^{2,\infty}(D)$ with $\mathrm{Supp}(A)\subset D$ such that
$\|A\|_{W^{2,\infty}}\leq M$ and let
$\theta,\,\theta^{\prime}\in\mathbb{S}^{2}+i\mathbb{S}^{2}$ such that
$|\theta-\theta^{\prime}|<1$ such that
$\Re(\theta)\cdot\Im(\theta)=\Re(\theta^{\prime})\cdot\Im(\theta^{\prime})=0$.
Then, we have the following inequality

$\|N^{-1}_{\theta}(-\theta\cdot
A)-N^{-1}_{\theta^{\prime}}(-\theta^{\prime}\cdot A)\|_{L^{\infty}(D)}\leq
C\left|\theta-\theta^{\prime}\right|,$ (A.2)

where $C$ depends only in $D$ and $M$.

###### Proof of Lemma 3.4.

By using (3.19), we have for $j=1,2$

$\nabla
u_{j}=e^{ix\cdot\rho_{j}}(i(\rho_{j}+\nabla\varphi_{j})e^{i\varphi_{j}}+i\rho_{j}r_{j}+\nabla
r_{j}),$ (A.3)

where $\varphi_{1}$ and $\varphi_{2}$ are given by

$\varphi_{1}(x,\omega_{1}^{*})=N_{\omega_{1}^{*}}^{-1}(\omega_{1}^{*}\cdot
A_{1}),\quad\varphi_{2}(x,\omega_{1}^{*})=N_{\omega_{2}^{*}}^{-1}(-\omega_{2}^{*}\cdot
A_{2}),$ (A.4)

where $A_{j}$, $j=1,2$ are extended by $0$ outside $D$.
Therefore, direct calculation gives

$u_{2}\nabla u_{1}-u_{1}\nabla
u_{2}=i(\rho_{1}-\rho_{2})e^{i(\varphi_{1}+\varphi_{2})}e^{ix\cdot(\rho_{1}+\rho_{2})}+\Psi_{1}(x,\rho_{1},\rho_{2})+\Psi_{2}(x,\rho_{1},\rho_{2}),$
(A.5)

where $\Psi_{1}$ and $\Psi_{2}$ are given by

$\displaystyle\Psi_{1}(x,\rho_{1},\rho_{2})$
$\displaystyle=i(\rho_{1}-\rho_{2})\left[r_{1}e^{i\varphi_{2}}+r_{2}e^{i\varphi_{1}}+r_{1}r_{2}\right]e^{ix\cdot(\rho_{1}+\rho_{2})},$
$\displaystyle\Psi_{2}(x,\rho_{1},\rho_{2})$
$\displaystyle=\big{[}i(\nabla\varphi_{1}-\nabla\varphi_{2})e^{i(\varphi_{1}+\varphi_{2})}+(\nabla
r_{1}e^{i\varphi_{2}}-\nabla r_{2}e^{i\varphi_{1}})$
$\displaystyle\qquad\qquad+(ir_{2}\nabla\varphi_{1}e^{i\varphi_{1}}-ir_{1}\nabla\varphi_{2}e^{i\varphi_{2}})+(r\nabla
r_{1}-r_{1}\nabla r_{2})\big{]}e^{ix\cdot(\rho_{1}+\rho_{2})}.$

Using Lemma A.2 and the fact that $A_{j}$, $j=1,2$, is compactly supported in
$\mathbb{R}^{3}$, we obtain

$\|\varphi_{j}(\cdot,\omega_{j}^{*})\|_{L^{\infty}(D)}\leq
C\|A_{j}\|_{L^{\infty}(\mathbb{R}^{3})}\leq CM.$ (A.6)

This implies that

$\|\Psi_{1}(\cdot,\rho_{1},\rho_{2})\|_{L^{1}(D)}\leq C.$ (A.7)

From (A.5), we obtain

$i\int_{\mathbb{R}^{3}}A\cdot\left(u_{2}\nabla u_{1}-u_{1}\nabla
u_{2}\right)dx=\int_{\mathbb{R}^{3}}A\cdot(\rho_{2}-\rho_{1})e^{ix\cdot\xi}e^{i(\varphi_{1}+\varphi_{2})}dx+\mathcal{R}_{1}(\xi,s),$
(A.8)

where

$\mathcal{R}_{1}(\xi,s)=i\int_{\mathbb{R}^{3}}A\cdot(\Psi_{1}(x,\rho_{1},\rho_{2})+\Psi_{2}(x,\rho_{1},\rho_{2}))dx,\quad\xi=\rho_{1}+\rho_{2}.$
(A.9)

Let now compute the first integral in the right hand side of (A.8). By using
(3.17) and (3.20), we have for $\omega=\omega_{1}+i\omega_{2}$

$\int_{\mathbb{R}^{3}}A\cdot(\rho_{2}-\rho_{1})e^{ix\cdot\xi}e^{i(\varphi_{1}+\varphi_{2})}dx=2s\int_{\mathbb{R}^{3}}\overline{\omega}\cdot
Ae^{ix\cdot\xi}e^{i(\varphi_{1}+\varphi_{2})}dx\cr+2s\Big{(}\sqrt{1-\frac{|\xi|^{2}}{4s^{2}}}-1\Big{)}\int_{\mathbb{R}^{3}}\omega_{1}\cdot
Ae^{ix\cdot\xi}e^{i(\varphi_{1}+\varphi_{2})}dx,$ (A.10)

Let $\psi_{1}=N_{\overline{\omega}}^{-1}(\overline{\omega}\cdot A_{1})$, and
$\psi_{2}=N^{-1}_{\overline{\omega}}(-\overline{\omega}\cdot A_{2})$, then we
have

$\psi_{1}(x)+\psi_{2}(x)=N^{-1}_{\overline{\omega}}(-\overline{\omega}\cdot(A_{2}-A_{1}))=N^{-1}_{\overline{\omega}}(-\overline{\omega}\cdot
A).$

We insert $e^{i(\psi_{1}+\psi_{2})}$ in (A.10), then we have

$\int_{\mathbb{R}^{3}}A(x)\cdot(\rho_{2}-\rho_{1})e^{ix\cdot\xi}e^{i(\varphi_{1}+\varphi_{2})}dx=\mathcal{J}(\xi,s)+\mathcal{R}_{2}(\xi,s)+\mathcal{R}_{3}(\xi,s),$
(A.11)

where

$\displaystyle\mathcal{J}(\xi,s)$
$\displaystyle=2s\int_{\mathbb{R}^{3}}\overline{\omega}\cdot
A(x)e^{ix\cdot\xi}e^{i(\psi_{1}+\psi_{2})}dx,$
$\displaystyle\mathcal{R}_{2}(\xi,s)$
$\displaystyle=2s\int_{\mathbb{R}^{3}}\overline{\omega}\cdot
A(x)e^{ix\cdot\xi}\left(e^{i(\varphi_{1}+\varphi_{2})}-e^{i(\psi_{1}+\psi_{2})}\right)dx,$
$\displaystyle\mathcal{R}_{3}(\xi,s)$
$\displaystyle=2s\Big{(}\sqrt{1-\frac{|\xi|^{2}}{4s^{2}}}-1\Big{)}\int_{\mathbb{R}^{3}}\omega_{1}\cdot
A(x)e^{ix\cdot\xi}e^{i(\varphi_{1}+\varphi_{2})}dx.$

By using the Lemma A.1, we obtain

$\mathcal{J}(\xi,s)=2s\int_{\mathbb{R}^{3}}\overline{\omega}\cdot
A(x)e^{ix\cdot\xi}dx.$ (A.12)

On the other hand, we have

$\displaystyle\mathcal{R}_{2}(\xi,s)=-2s\int_{\mathbb{R}}^{3}e^{-ix\cdot\xi}\overline{\omega}\cdot
A\Big{(}e^{i\varphi_{1}}\big{(}e^{i\varphi_{2}}-e^{i\psi_{2}}\big{)}-e^{i\psi_{2}}\big{(}e^{i\psi_{1}}-e^{i\varphi_{1}}\big{)}\Big{)}dx.$
(A.13)

Using the dependence of $N^{-1}_{\omega}(-\omega\cdot A)$ on the parameter
$\omega$ given in Lemma A.3 and the fact that $\textrm{Supp}A\subset D$, we
get

$\displaystyle\left|e^{i\varphi_{2}}-e^{i\psi_{2}}\right|$ $\displaystyle\leq
C\left|N_{\omega_{2}^{*}}^{-1}(-\omega_{2}^{*}\cdot
A_{2})-N_{\overline{\omega}}^{-1}(-\overline{\omega}\cdot A_{2})\right|\leq
C\left|\omega_{2}^{*}-\overline{\omega}\right|,$ (A.14)
$\displaystyle\left|e^{i\psi_{1}}-e^{i\varphi_{1}}\right|$ $\displaystyle\leq
C\left|N_{\overline{\omega}}^{-1}(\overline{\omega}\cdot
A_{1})-N_{-\omega_{1}^{*}}^{-1}(-\omega_{1}^{*}\cdot A_{1})\right|\leq
C\left|\overline{\omega}+\omega_{1}^{*}\right|.$ (A.15)

Taking into account (A.13), (A.14) and using that
$1-\sqrt{1-|\xi|^{2}/s^{2}}\leq|\xi|^{2}/4s^{2}$, for all $|\xi|\leq 2s$, we
conclude

$\big{|}\mathcal{R}_{2}(\xi,s)\big{|}\leq Cs\frac{|\xi|^{2}}{4s^{2}}\leq
C|\xi|.$ (A.16)

By the same way, we find $|\mathcal{R}_{3}(\xi,s)|\leq C|\xi|$, for some
positive constant which is independent of $\xi$ and $s$.
The proof is completed. ∎

###### Proof of Lemma 3.9.

By a direct calculation, we have

$u_{1}u_{2}=e^{ix\cdot\xi}e^{i(\varphi_{1}+\varphi_{2})}+e^{ix\cdot\xi}\big{(}e^{i\varphi_{2}}r_{1}+e^{i\varphi_{1}}r_{2}+r_{1}r_{2}\big{)}.$
(A.17)

We use the identity (A.17) and we insert $e^{ix\cdot\xi}q(x)$, we obtain

$\int_{D}e^{-i\varphi}q(x)u_{1}u_{2}dx=\int_{D}q(x)\;e^{ix\cdot\xi}\,dx+\mathcal{R}^{\prime}_{1}(\xi,s)+\mathcal{R}^{\prime}_{2}(\xi,s),$
(A.18)

where

$\displaystyle\mathcal{R}^{\prime}_{1}(\xi,s)$
$\displaystyle=\int_{D}q(x)\;e^{ix\cdot\xi}\,e^{i\varphi_{1}}\left(e^{i(\varphi_{2}-\varphi)}-e^{-i\varphi_{1}}\right)dx,$
$\displaystyle\mathcal{R}^{\prime}_{2}(\xi,s)$
$\displaystyle=\int_{D}e^{-i\varphi}q(x)e^{ix\cdot\xi}\big{(}e^{i\varphi_{2}}r_{1}+e^{i\varphi_{1}}r_{2}+r_{1}r_{2}\big{)}dx.$

Let $\psi_{3}=N^{-1}_{\omega_{2}^{*}}(-\omega_{2}^{*}\cdot A_{1}).$ We insert
$e^{i\psi_{3}}$ in $\mathcal{R}^{\prime}_{1}$ and obtain from Lemmas A.2 and
A.3

$\displaystyle\left|\mathcal{R}^{\prime}_{1}(\xi,s)\right|$ $\displaystyle\leq
C\|e^{i(\varphi_{2}-\varphi)}-e^{i\psi_{3}}\|_{L^{\infty}(D)}+\|e^{i\psi_{3}}-e^{-i\varphi_{1}}\|_{L^{\infty}(B)}$
(A.19) $\displaystyle\leq
C\big{(}\|N^{-1}_{\omega_{2}^{*}}(-\omega_{2}^{*}\cdot(A_{2}+\nabla\varphi))-N^{-1}_{\omega_{2}^{*}}(-\omega_{2}^{*}\cdot
A_{1})\|_{L^{\infty}(B)}$ (A.20)
$\displaystyle\qquad\qquad+\|N^{-1}_{\omega_{2}^{*}}(-\omega_{2}^{*}\cdot
A_{1})-N^{-1}_{-\omega_{1}^{*}}(\omega_{1}^{*}\cdot
A_{1})\|_{L^{\infty}(B)}\big{)}$ (A.21) $\displaystyle\leq
C\left(\|A+\nabla\varphi\|_{L^{\infty}(B)}+\left|\omega_{2}^{*}+\omega_{1}^{*}\right|\right)$
(A.22) $\displaystyle\leq
C\left(\|A+\nabla\vartheta\|_{L^{\infty}(B)}+\|\vartheta\|_{W^{1,\infty}(B\backslash
B^{\prime})}+\left|\omega_{2}^{*}+\omega_{1}^{*}\right|\right).$ (A.23)

Using (3.46) and (3.47), we obtain

$\left|\mathcal{R}^{\prime}_{1}(\xi,s)\right|\leq
C\left(\|\mathrm{curl}(A)\|_{L^{\infty}(B)}+s\left<\xi\right>^{-1}\right).$
(A.24)

Moreover from (3.20), we get $\left|\mathcal{R}^{\prime}_{2}(\xi,s)\right|\leq
Cs^{-1}$. Collecting this with (A.24) and (A.18) we obtain the desired result.
∎

##  Appendix B Proof of Lemma 4.2

This appendix is devoted to the proof of Lemma 4.2. We first establish the
following duality result. The coefficients $A$ and $q$ are as in Section 2.

######  Lemma B.1.

Let $v_{i}\in H^{1}(B)$, $i=1,2$ such that $\Delta v_{i}+k^{2}v_{i}=0$ in $B$.
Then, the following identity

$\int_{B}Q_{A,q}v_{1}(I-T_{-A,q})^{-1}v_{2}dx=\int_{B}Q_{-A,q}v_{2}(I-T_{A,q})^{-1}v_{1}dx,$
(B.25)

holds where $Q_{A,q}$ and $T_{A,q}$ are given by (1.2) and (2.20)
respectively.

###### Proof.

We associate to $v_{1}$ (respectively $v_{2}$) a total field $u_{A,q}$
(respectively $u_{-A,q}$) and a scattered field $u^{s}_{-A,q}$ (respectively
$u^{s}_{-A,q}$). Applying (2.33) to $u_{1}=u^{s}_{A,q}$ and
$u_{2}=u^{s}_{-A,q}$ implies

$\int_{B}\left(\mathcal{H}_{A,q}u^{s}_{A,q}\,u^{s}_{-A,q}-u^{s}_{A,q}\mathcal{H}_{-A,q}u^{s}_{-A,q}\right)dx=0.$

Making use of $u_{A,q}=u^{s}_{A,q}+v_{1}$ and $u_{-A,q}=u^{s}_{-A,q}+v_{2}$,
it follows by direct calculation

$\displaystyle\int_{B}\left(Q_{A,q}v_{1}u_{-A,q}-Q_{-A,q}v_{2}\right)dx=\int_{B}\left(Q_{A,q}v_{1}v_{2}-Q_{-A,q}v_{2}v_{1}\right)dx.$

Moreover, by integrating by parts, we get

$\int_{B}\left(Q_{A,q}v_{1}v_{2}-Q_{-A,q}v_{2}v_{1}\right)dx=0.$

Due to $u_{A,q}=(I-T_{A,q})^{-1}v_{1}$ and $u_{-A,q}=(I-T_{-A,q})^{-1}v_{2}$,
we obtain the desired result. ∎

######  Proposition B.2.

For $k>0$ fixed, we have

$\displaystyle u^{s}_{A,q}(x,y)$
$\displaystyle=\frac{1}{4\pi}\frac{e^{ik|x|}}{|x|}\frac{e^{ik|y|}}{|y|}u^{\infty}_{A,q}\left(\widehat{x},-\widehat{y}\right)+\frac{1}{|x||y|}\left(\frac{1}{|x|}+\frac{1}{|y|}\right)\Lambda(x,y),\hskip
14.22636ptx\neq y,$ (B.26)

where $\Lambda(x,y)$ is uniformly bounded as $|x|\longrightarrow\infty$ and
$|y|\rightarrow\infty$.

###### Proof.

Using the asymptotics of $\Phi(\cdot,z)$ and $\nabla_{z}\Phi(\cdot,z)$ for
$z\in D$, we get

$\displaystyle u_{A,q}^{s}(x,y)$
$\displaystyle=\int_{D}Q_{-A,q}\Phi(x,z)u_{A,q}(z,y)dz$ (B.27)
$\displaystyle=\frac{e^{ik|x|}}{4\pi|x|}w\left(y,-\widehat{x}\right)+\frac{O(1)}{|x|^{2}}\|u_{A,q}(.,y)\|_{L^{2}(D)},\hskip
14.22636pt|x|\rightarrow\infty,$ (B.28)

uniformly with respect to $y\in\mathbb{R}^{3}\setminus B$, where

$\displaystyle w(y,d):=\int_{D}Q_{-A,q}e^{ikz\cdot d}u_{A,q}(z,y)dz.$

Using now Lemma B.1 and integrating by parts, we get

$w(y,d)=-\int_{D}\Phi(z,y)Q_{-A,q}u_{-A,q}(z,d)dz,$ (B.29)

and therefore $w(\cdot,d)=u_{-A,q}^{s}(\cdot,d)$. Consequently, (B.28) and
(4.1) imply that for $|x|$, $|y|\rightarrow\infty$

$\displaystyle u_{A,q}^{s}(x,y)$
$\displaystyle=\frac{1}{4\pi}\frac{e^{ik|x|}}{|x|}\frac{e^{ik|y|}}{|y|}u^{\infty}_{-A,q}\left(\widehat{y},-\widehat{x}\right)+\frac{O(1)}{|x|^{2}}\|u_{A,q}(\cdot,y)\|_{L^{2}(D)}$
(B.30)
$\displaystyle\qquad+\frac{O(1)}{|x||y|^{2}}\|u_{-A,q}(\cdot,-\hat{x})\|_{L^{2}(D)}.$
(B.31)

Let us observe that according to Corollary 2.3 (and the asymptotic behaviour
of $\Phi(x,y)$ and $\nabla_{x}\Phi(x,y)$ with $x\in D$ and
$|y|\rightarrow\infty$) we get that

$\|u_{A,q}(\cdot,y)\|_{L^{2}(D)}=\frac{O(1)}{|y|}\quad\mbox{ as
}|y|\rightarrow\infty,$

and $\|u_{-A,q}(\cdot,-\hat{x})\|_{L^{2}(D)}$ is uniformly bounded with
respect to $\hat{x}$. We finally obtain the desired result by noticing the
reciprocity relation
$u^{\infty}_{A,q}(d,\theta)=u^{\infty}_{-A,q}(-\theta,-d)$ where
$\theta,d\in\mathbb{S}^{2}$ (which is also a consequence of Lemma B.1) or
using Lemma 2.5. ∎

Now, let us expand the scattering amplitude $u^{\infty}_{A,q}(d,\theta)$ in
spherical harmonics

$u^{\infty}_{A,q}(d,\theta)=\underset{(\ell_{1},m_{1})\in\Gamma}{\sum}\;\underset{(\ell_{2},m_{2})\in\Gamma}{\sum}\mu_{\ell_{1}m_{1}\ell_{2}m_{2}}Y^{m_{1}}_{\ell_{1}}(d)Y^{m_{2}}_{\ell_{1}}(\theta),$
(B.32)

where $\mu_{\ell_{1}m_{1}\ell_{2}m_{2}}$ is given by (4.2).

#### Proof of Lemma 4.2

Making use of the addition formula [3],

$\displaystyle\Phi(x,z)$
$\displaystyle=\underset{\ell,m}{\sum}\varepsilon_{\ell,m}(z)h^{(1)}_{\ell}(k|x|)Y^{m}_{\ell}\left(\widehat{x}\right),\hskip
14.22636pt|x|>|z|,$ (B.33) $\displaystyle\nabla_{y}\Phi(x,z)$
$\displaystyle=\underset{\ell,m}{\sum}\varepsilon^{\prime}_{\ell,m}(z)h^{(1)}_{\ell}(k|x|)Y^{m}_{\ell}\left(\widehat{x}\right),\hskip
14.22636pt|x|>|z|,$ (B.34)

where
$\varepsilon_{\ell,m}(z)=ikj_{\ell}(k|z|)Y^{m}_{\ell}\left(\widehat{z}\right)$
and
$\varepsilon^{\prime}_{\ell,m}=ik\nabla\left(j_{\ell}(k|z|)Y^{m}_{\ell}\left(\widehat{z}\right)\right)$,
it follows from (B.27) that for $y\in\mathbb{R}^{3}\setminus D$ and uniformly
for $|x|\geq a$

$\displaystyle u^{s}_{A,q}(x,y)$
$\displaystyle=\underset{(\ell_{1},m_{1})\in\Gamma}{\sum}\alpha_{\ell_{1}m_{1}}(y)h^{(1)}_{\ell_{1}}(k|x|)Y^{m_{1}}_{\ell_{1}}\left(\widehat{x}\right).$

Similarly, for $x\in\mathbb{R}^{3}\setminus D$ and uniformly for $|y|\geq a$

$\displaystyle u^{s}_{-A,q}(y,x)$
$\displaystyle=\underset{(\ell_{2},m_{2})\in\Gamma}{\sum}\beta_{\ell_{2}m_{2}}(x)h^{(1)}_{\ell_{2}}(k|y|)Y^{m_{2}}_{\ell_{2}}\left(\widehat{y}\right).$

We observe that

$\alpha_{\ell_{1}m_{1}}(y)h^{(1)}_{\ell_{1}}(ka)=\int_{{\mathbb{S}}^{2}}u^{s}_{A,q}(a\hat{x},y)\overline{Y^{m_{1}}_{\ell_{1}}}\left(\widehat{x}\right)ds(\hat{x}).$

Using the reciprocity relation of Lemma 2.5 we then get uniformly for $|y|\geq
a$

$\alpha_{\ell_{1}m_{1}}(y)=\underset{(\ell_{2},m_{2})\in\Gamma}{\sum}\gamma_{\ell_{1}m_{1}\ell_{2}m_{2}}h^{(1)}_{\ell_{2}}(k|y|)Y^{m_{2}}_{\ell_{2}}\left(\widehat{y}\right),$

where

$\gamma_{\ell_{1}m_{1}\ell_{2}m_{2}}=\frac{1}{h^{(1)}_{\ell_{1}}(ka)}\int_{{\mathbb{S}}^{2}}\beta_{\ell_{2}m_{2}}(a\hat{x})\overline{Y^{m_{1}}_{\ell_{1}}}\left(\widehat{x}\right)ds(\hat{x}).$

This yields in particular that for $|x|\geq a$ and $|y|\geq a$

$u^{s}_{A,q}(x,y)=\sum_{\begin{subarray}{c}(\ell_{1},m_{1})\in\Gamma\\\
(\ell_{2},m_{2})\in\Gamma\end{subarray}}\gamma_{\ell_{1}m_{1}\ell_{2}m_{2}}h^{(1)}_{\ell_{1}}(k|x|)Y^{m_{1}}_{\ell_{1}}\left(\widehat{x}\right)h^{(1)}_{\ell_{2}}(k|y|)Y^{m_{2}}_{\ell_{2}}\left(\widehat{y}\right).$
(B.35)

Observe that from Proposition B.2, we have in particular that $u^{s}_{A,q}\in
L^{2}(S_{R}\times S_{R})$ for $R$ sufficiently large, with
$S_{R}=\\{|x|=R\\}$. The orthonormality of products of spherical harmonics in
$L^{2}({\mathbb{S}}^{2}\times{\mathbb{S}}^{2})$ implies in particular that the
series (B.35) is convergent in $L^{2}({\mathbb{S}}^{2}\times{\mathbb{S}}^{2})$
and

$\int_{{\mathbb{S}}^{2}\times{\mathbb{S}}^{2}}u^{s}_{A,q}(R\hat{x},R\hat{y})\overline{Y^{m_{1}}_{\ell_{1}}}\left(\widehat{x}\right)\overline{Y^{m_{2}}_{\ell_{2}}}\left(\widehat{y}\right)ds(\hat{x})ds(\hat{y})=\gamma_{\ell_{1}m_{1}\ell_{2}m_{2}}h^{(1)}_{\ell_{1}}(kR)h^{(1)}_{\ell_{2}}(kR).$

We recall that ([3]),

$h^{(1)}_{\ell}(r)=(-i)^{\ell+1}\frac{e^{ir}}{r}+O\left(r^{-2}\right),$

Proposition B.2 and the identity
$Y^{m_{2}}_{\ell_{2}}(-\hat{y})=(-1)^{\ell_{2}}Y^{m_{2}}_{\ell_{2}}(\hat{y})$
then imply, by integrating on $S_{R}\times S_{R}$ and letting $R\to\infty$
that

$\gamma_{\ell_{1}m_{1}\ell_{2}m_{2}}=\frac{-k^{2}}{4\pi}i^{\ell_{1}+\ell_{2}}(-1)^{\ell_{2}}\mu_{\ell_{1}m_{1}\ell_{2}m_{2}}.$

This yields the desired result. We remark that it also implies from (4.7) and
(4.4) that the series (B.35) is absolutely convergent together with its first
and second derivatives with respect to $x$ or $y$.

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