On the half-space matching method for real wavenumberA.-S. Bonnet-Ben Dhia et
al.

# On the half-space matching method for real wavenumber††thanks: Submitted to
the editors DATE. This work was funded by …

Anne-Sophie Bonnet-Ben Dhia POEMS (CNRS-INRIA-ENSTA Paris), Institut
Polytechnique de Paris, Palaiseau, France (, ) anne-sophie.bonnet-
<EMAIL_ADDRESS><EMAIL_ADDRESS>Simon N. Chandler-Wilde
Department of Mathematics and Statistics, University of Reading, Whiteknights
PO Box 220, Reading RG6 6AX, UK<EMAIL_ADDRESS>Sonia
Fliss22footnotemark: 2

###### Abstract

The Half-Space Matching (HSM) method has recently been developed as a new
method for the solution of 2D scattering problems with complex backgrounds,
providing an alternative to Perfectly Matched Layers (PML) or other artificial
boundary conditions. Based on half-plane representations for the solution, the
scattering problem is rewritten as a system coupling (1) a standard finite
element discretisation localised around the scatterer and (2) integral
equations whose unknowns are traces of the solution on the boundaries of a
finite number of overlapping half-planes contained in the domain. While
satisfactory numerical results have been obtained for real wavenumbers, well-
posedness and equivalence of this HSM formulation to the original scattering
problem have been established only for complex wavenumbers. In the present
paper we show, in the case of a homogeneous background, that the HSM
formulation is equivalent to the original scattering problem also for real
wavenumbers, and so is well-posed, provided the traces satisfy radiation
conditions at infinity analogous to the standard Sommerfeld radiation
condition. As a key component of our argument we show that, if the trace on
the boundary of a half-plane satisfies our new radiation condition, then the
corresponding solution to the half-plane Dirichlet problem satisfies the
Sommerfeld radiation condition in a slightly smaller half-plane. We expect
that this last result will be of independent interest, in particular in
studies of rough surface scattering.

###### keywords:

Helmholtz equation, scattering, Sommerfeld radiation condition, integral
equation, domain decomposition, uniqueness, rough surface scattering

35J05, 35J25, 35P25, 45B05, 45F15, 65N30, 65N38, 78A45

##  1 Introduction and the scattering problem

###  1.1 The HSM method

Recently a new method, called the Half-Space Matching (HSM) method, has been
developed as an (exact) artificial boundary condition for two-dimensional
time-harmonic scattering problems. This method is based on explicit or semi-
explicit expressions for the outgoing solutions of radiation problems in half-
planes, these expressions established by using Fourier, generalized Fourier,
or Floquet transforms when the background is, respectively, homogeneous [6, 5]
(and possibly anisotropic [31, 3, 30]), stratified [26], or periodic [17, 18].
The domain exterior to a bounded region enclosing the scatterers is covered by
a finite number $\mathcal{N}$ of half-planes (at least three). The unknowns of
the formulation are the traces $\varphi^{1},\ldots,\varphi^{\mathcal{N}}$ of
the solution on the boundaries of these half-planes and the restriction of the
solution to the bounded region. The system of equations which couples these
unknowns is derived by writing compatibility conditions between the different
representations of the solution. This coupled system includes second-kind
integral equations on the infinite boundaries of the half-planes.

This new formulation is attractive and versatile as a method to truncate
computational domains in problems of scattering by localized inhomogeneities
in complex backgrounds, including backgrounds that may be different at
infinity in different directions. It has been employed successfully in
numerical implementations for various applications, like optical waveguides
(including cases with different stratifications in different parts of the
background domain) [26], or ultrasonic non-destructive testing (with an
anisotropic elastic background) [31, 30].

Up to now the theoretical and numerical analysis of the method has remained an
open question in the challenging, and practically relevant, non-dissipative
case when waves radiate out to infinity. But a rather complete analysis has
been carried out in the simpler dissipative case, when the solution (and its
traces) decay exponentially at infinity. In that case the analysis can be done
using an $L^{2}$ framework for the traces: the associated formulation has been
shown to be of Fredholm type and well-posed in a number of cases where the
background is homogeneous (but not necessarily isotropic) [6, 5], with the
sesquilinear form of the weak formulation coercive plus compact, enabling the
numerical analysis of the method [5]. This analysis fails in the non-
dissipative case, not least because of the slow decay at infinity of the
solution which results in non-$L^{2}$ traces.

###  1.2 Our main results

In this paper we address well-posedness of the HSM formulation in the non-
dissipative case for the scalar Helmholtz equation when the background is
homogeneous and isotropic, so that

(1) $-\Delta u-k^{2}u=0$

outside some ball. (Here, assuming ${\mathrm{e}}^{-{\mathrm{i}}\omega t}$ time
dependence with $\omega>0$, $k=\omega/c>0$ and $c>0$ are the constant
wavenumber and wave speed, respectively, outside the ball.) For such
configurations, it is well known that to achieve a well-posed scattering
problem the scattered field $u$ must satisfy the Sommerfeld radiation
condition

(2) $\frac{\partial u(\boldsymbol{x})}{\partial
r}-{\mathrm{i}}ku(\boldsymbol{x})=o\left(r^{-1/2}\right)\quad\text{as}\;r:=|\boldsymbol{x}|\rightarrow+\infty,$

uniformly with respect to $\widehat{\boldsymbol{x}}:=\boldsymbol{x}/r$.
Further, it is well known that, if $u$ satisfies (1) outside some ball and
(2), then

(3)
$u(\boldsymbol{x})=\frac{{\mathrm{e}}^{{\mathrm{i}}kr}}{r^{1/2}}\left(F(\widehat{\boldsymbol{x}})+O(r^{-1})\right),\quad\mbox{as}\quad
r\to\infty,$

uniformly in $\widehat{\boldsymbol{x}}:=\boldsymbol{x}/r$, where $F\in
C^{\infty}(S^{1})$, with $S^{1}$ the unit circle, is the so-called far-field
pattern (e.g., [11, Lemma 2.5]).

The main results of this paper (Theorems 5 and 1) are to establish well-
posedness for the HSM formulations, stated in detail as (30) and (57) below,
in the case that $k>0$ and the background is homogeneous and isotropic, so
that the scattered field $u$ satisfies (2) and (1) outside some ball.
Precisely, we show that the HSM formulations are well-posed and equivalent to
the original scattering problems provided we require that each half-plane
trace $\varphi^{j}$ is locally in $L^{2}$ and has the asymptotic behaviour
predicted by (3), meaning that

(4)
$\varphi^{j}(\boldsymbol{x})=c_{\pm}^{j}\,\frac{{\mathrm{e}}^{{\mathrm{i}}kr}}{r^{1/2}}\left(1+O(r^{-1})\right),$

for some constants $c_{\pm}^{j}$, as $\boldsymbol{x}$ tends to infinity in the
$\pm$ directions along the infinite boundary of the half-plane. Note that
condition (4) can be seen as a radiation condition for the half-plane trace
$\varphi^{j}$, playing the role in the HSM formulation for $k>0$ that the
Sommerfeld radiation condition (2) plays in the formulation of the original
scattering problem.

Since (i) the HSM system of equations is derived from the unique solution to
the scattering problem and (ii) the half-plane traces of the solution to the
scattering problem do satisfy (4) (because (3) holds), we will see that
existence of a solution holds by construction. The challenge remaining in
order to show well-posedness is to establish uniqueness. There are two
difficult aspects to this challenge. The first is to show that, if
$\varphi^{j}$ satisfies (4) and $u$ is the solution reconstructed from
$\varphi^{j}$ in the corresponding half-plane, so that $u$ satisfies (1) in
the half-plane and $u=\varphi^{j}$ on the half-plane boundary, then
$u(\boldsymbol{x})=O(r^{-1/2})$ as $r\to\infty$ and $u$ satisfies an
appropriate version of the Sommerfeld radiation condition (2). These
properties are established in §3.2. The second challenge, given that we assume
ab initio only that the traces $\varphi^{j}$ are locally $L^{2}$, is to show
that each trace $\varphi^{j}$ is locally in the trace space $H^{1/2}$, so that
the reconstructed solution $u$ is locally in $H^{1}$ as required. But this
difficulty arises already in the corresponding formulation in the dissipative
case, and the proof in that case is sketched in [6, §3.3]. As preparation for
the more difficult non-dissipative case, we expand on that argument in §2.2
below.

###  1.3 The significance of our results

The main significance of Theorems 5 and 1 is that these are the first well-
posedness results in the important non-dissipative case for the HSM method, a
method which, as discussed above, has already proved effective for computing
scattering by localised inhomogeneities in a range of complex backgrounds. Our
theorems, challenging as they are to prove, are for the simplest case when the
background is homogeneous and isotropic. However, we expect that these results
and formulations will be an important stepping stone to more complex cases,
and we discuss this further in concluding remarks to the paper.

Our main uniqueness result, Theorem 5, is also important because it is a
crucial ingredient in the proof of well-posedness in [4, §5] of the so-called
complex-scaled HSM method, proposed recently in [4]. This new formulation is a
version of the HSM method in the non-dissipative case that, in the spirit of
complex-scaling in PML (e.g., [14]), achieves the $L^{2}$ framework of the
dissipative case by analytically continuing the half-plane traces into the
complex plane, so that the original half-plane boundary is replaced by a path
in the complex plane on which the (analytically-continued) trace is
exponentially decreasing.

Additionally, we expect that the results that we establish (Lemma 1, §3.2) on
properties of the half-plane solution operator, that takes the trace on the
boundary of a half-plane and recovers the half-plane solution, will be of
independent interest. Indeed, there is large interest in so-called rough
surface scattering problems, where the Helmholtz equation (or more complicated
vector equations) are solved in a non-locally perturbed half-plane $D$ with
boundary or transmission conditions on the rough surface $\partial D$ (e.g.,
[13, 12, 1, 2, 23, 10, 27, 16, 32]). In this context it is usual, to ensure
uniqueness, to impose the so-called upwards propagating radiation condition
(e.g., [12, 2, 23, 10, 16, 32]), which is precisely a requirement that, in
some half-plane above the rough surface, the solution can be represented as
the action of this half-plane solution operator on some $L^{\infty}$ data on
the boundary of the half-plane. This rough surface scattering application, in
particular the use of the upwards propagating radiation condition, has driven
significant study of this half-plane solution operator (e.g. [8, 9], [13, §2],
[2], [16, §2.3]). The results in §3.2 contribute to this study, shedding light
on the relationship between the upwards propagating radiation condition and
the Sommerfeld radiation condition, complementing previous work, especially
[13, Theorem 2.9] and [2].

###  1.4 The scattering problem and the structure of the paper

Let us spell out in more detail the 2D scattering problem that we consider in
this paper. The propagation domain $\Omega$ is $\mathbb{R}^{2}$, or
$\mathbb{R}^{2}$ minus a set of bounded Lipschitz obstacles so that $\Omega$
is a Lipschitz domain. We seek $u$ that satisfies an inhomogeneous Helmholtz
equation,

(5) $-\Delta u-k^{2}\,\rho u=f\;\text{in}\;\Omega.\\\ $

Here $\rho$ is a function in $L^{\infty}(\Omega)$ such that $\rho-1$ is
compactly supported, so that, if $\rho$ is real-valued and bounded below by a
positive constant, (5) models propagation in a domain with a local
perturbation in wave speed111It is straightforward to incorporate more
elaborate local behaviour (e.g. local anisotropies, or local inhomogeneities
in density as well as wave speed), as long as the resulting problem remains
well-posed.. The source term $f\in L^{2}(\Omega)$ is also assumed to be
compactly supported, so that (1) holds outside some ball. We suppose that the
wavenumber is such that $k>0$ in the non-dissipative case, and such that
$\Re(k)>0$ and $\Im(k)>0$ in the dissipative case. In the case when
$\Omega\subsetneq\mathbb{R}^{2}$ we require also a standard (e.g. Dirichlet or
Neumann) boundary condition on $\Gamma:=\partial\Omega$. As is standard we
seek a solution $u\in H^{1}(\Omega)$ in the dissipative case and $u\in
H^{1}_{\mathrm{loc}}(\Omega):=\\{v_{|\Omega};v\in
H^{1}_{\mathrm{loc}}(\mathbb{R}^{2})\\}$ in the case $k>0$. In the non-
dissipative case we require also that $u$ satisfies the Sommerfeld radiation
condition (2).

To explain the HSM method, our uniqueness argument, and the role of the
radiation condition (4), we consider two specific instances of the problem
(5). To get the main ideas across and to prove the key uniqueness and well-
posedness result, Theorem 5, we focus first on the case when
$\Omega=\mathbb{R}^{2}\setminus\overline{{\mathcal{O}}}$, for some convex
polygon ${\mathcal{O}}$, with $\rho\equiv 1$ and $f\equiv 0$, so that (1)
holds in $\Omega$, imposing a Dirichlet boundary condition $u=g$ on $\Gamma$.
In this case, where $\Omega$ is the exterior of a convex polygon and (1) holds
in $\Omega$, the HSM formulation is a system of second kind integral
equations, in which the unknowns are the traces of $u$ on the finite number of
half-planes that abut the sides of the polygon ${\mathcal{O}}$. In §2 we
recall the HSM formulation for this problem in the dissipative case, in
particular how uniqueness is proved. In §3 we prove our main Theorem 5, i.e.,
we prove well-posedness in the non-dissipative case under the additional
radiation condition (4).

In §4 we consider the more involved case where $f$ is non zero and/or
$\rho\not\equiv 1$, in which case the second kind integral equation
formulation is coupled to a local variational formulation in a bounded region
$\Omega_{b}$ containing $\partial\Omega$ and the supports of $f$ and $\rho-1$.
In this case we assume, to be specific, that the boundary condition on
$\partial\Omega$ is a homogeneous Neumann condition. (The changes needed, to
the formulation and well-posedness argument, to address other boundary
conditions, and/or inhomogeneities in the boundary condition, indeed to
include other types of compactly supported inhomogeneities, are
straightforward.) The main result in this case, the uniqueness and well-
posedness result Theorem 1, is proved by use of Theorem 5 and by adapting the
proof of [4, Proposition 6.1]. Section 5 provides a brief conclusion and gives
an indication of more complex configurations to which the same methods and
arguments are expected to apply.

##  2 The HSM method for complex wavenumber

###  2.1 The Half-Space Matching formulation

In this section, as preparation for studying the HSM method for real
wavenumber, we first recall what is known about the method in the dissipative
case. For this purpose, as discussed in §1.4, we consider the Dirichlet
problem for complex wavenumber ($\Im(k)>0$, $\Re(k)>0$) in the exterior
$\Omega$ of a convex polygon ${\mathcal{O}}$. Thus, for given data $g\in
H^{1/2}(\Gamma)$, where $\Gamma=\partial\Omega=\partial{\mathcal{O}}$, we
consider the Dirichlet problem

(6) $\begin{cases}-\Delta u-k^{2}\,u=0\;\text{in}\;\Omega,\\\\[4.0pt]
u=g\;\text{on}\;\Gamma.\end{cases}$

It is well known that Problem (6) has a unique solution $u\in H^{1}(\Omega)$.

If ${\mathcal{O}}$ is a polygon with ${\mathcal{N}}$ edges denoted
$\Gamma^{1},\ldots,\Gamma^{{\mathcal{N}}}$, the domain $\Omega$ is the union
of ${\mathcal{N}}$ overlapping half-planes $\Omega^{j}$,
$j=1,...,{\mathcal{N}}$, that abut the ${\mathcal{N}}$ edges of the polygon
${\mathcal{O}}$. The angle between $\Gamma^{j}$ and $\Gamma^{j+1}$ is denoted
as $\Theta^{j,j+1}$ or equivalently $\Theta^{j+1,j}$, where, here and in the
sequel, $j\in\mathbb{Z}/{\mathcal{N}}\mathbb{Z}$ where
$\mathbb{Z}/{\mathcal{N}}\mathbb{Z}$ is the ring of integers modulo
${\mathcal{N}}$. This convenient notation means that $j=0$ is equivalent to
$j={\mathcal{N}}$ and $j=-1$ to $j={\mathcal{N}}-1$. Note finally that,
because of the convexity, one has $0<\Theta^{j,j+1}<\pi$ for all
$j\in\mathbb{Z}/{\mathcal{N}}\mathbb{Z}$.

$\mathcal{O}$$\Sigma^{1}$$\Sigma^{2}$$\Sigma^{3}$$x^{1}_{1}$$x^{1}_{2}$$x^{2}_{1}$$x^{2}_{2}$$x^{3}_{1}$$x^{3}_{2}$$\Theta^{1,2}$$\Theta^{2,3}$$\Theta^{3,1}$

$\mathcal{O}$$\Sigma^{1}$$\Sigma^{2}$$\Sigma^{3}$$\Sigma^{4}$$x^{1}_{1}$$x^{1}_{2}$$x^{2}_{1}$$x^{2}_{2}$$x^{3}_{1}$$x^{3}_{2}$$x^{4}_{1}$$x^{4}_{2}$$\Theta^{1,2}$$\Theta^{2,3}$$\Theta^{3,4}$$\Theta^{4,1}$

Figure 1: Examples of polygons $\mathcal{O}$ for ${\mathcal{N}}=3$ and $4$ and
associated notations.

It is convenient to make use of local coordinate systems
$\boldsymbol{x}^{j}=(x^{j}_{1},x^{j}_{2})$ in each half-space $\Omega^{j}$.
The origin of all of them is the centroid $O$ of the polygon $\mathcal{O}$. We
define the Cartesian coordinate system $(O,x^{1}_{1},x^{1}_{2})$ such that the
axis $Ox^{1}_{1}$ is orthogonal to $\Gamma^{1}$ and directed into the exterior
of the polygon, while the axis $Ox^{1}_{2}$ is $\pi/2$ counter clockwise to
$Ox^{1}_{1}$. The other local coordinate systems $(O,x^{j}_{1},x^{j}_{2})$ are
defined recursively as follows:

(7) $\forall
j\in\mathbb{Z}/{\mathcal{N}}\mathbb{Z},\quad\begin{array}[]{|l}x^{j+1}_{1}:=\cos(\Theta^{j,j+1})\,x^{j}_{1}+\sin(\Theta^{j,j+1})\,x^{j}_{2},\\\\[5.0pt]
x^{j+1}_{2}:=-\sin(\Theta^{j,j+1})\,x^{j}_{1}+\cos(\Theta^{j,j+1})\,x^{j}_{2}.\end{array}$

Defining $a^{j}$, for $j=1,...,{\mathcal{N}}$, to be the distance of the
centroid of the polygon to the edge $\Gamma^{j}$, each half-plane $\Omega^{j}$
is defined in the local coordinate system $(O,x^{j}_{1},x^{j}_{2})$ as

$\Omega^{j}:=\\{x^{j}_{1}>a^{j}\\}\times\\{x^{j}_{2}\in\mathbb{R}\\},$

and its boundary, denoted by $\Sigma^{j}$, is given by

$\Sigma^{j}:=\\{x^{j}_{1}=a^{j}\\}\times\\{x^{j}_{2}\in\mathbb{R}\\}.$

As explained in the introduction, the formulation uses the representation of
the solution in each half-plane $\Omega^{j}$ in terms of its trace on
$\Sigma^{j}$. More precisely, let us denote

(8)
$\varphi^{j}:=u\big{|}_{\Sigma^{j}}\quad\text{for}\;j=1,...,{\mathcal{N}},$

so that

(9)
$u\big{|}_{\Omega^{j}}=U^{j}(\varphi^{j})\quad\text{for}\;j=1,...,{\mathcal{N}},$

where, for $j=1,...,{\mathcal{N}}$ and $\psi\in H^{1/2}(\Sigma^{j})$,
$U^{j}(\psi)\in H^{1}(\Omega^{j})$ is the unique solution of

(10) $\begin{array}[]{|lcr}-\Delta
U^{j}-k^{2}\,U^{j}=0&\text{in}&\Omega^{j},\\\\[4.0pt]
\;U^{j}=\psi&\text{on}&\Sigma^{j},\end{array}$

this solution being well-defined since $\Im(k)\neq 0$. We can express
$U^{j}(\psi)$ explicitly in terms of its trace $\psi$ using a Green’s function
representation:

(11) $U^{j}(\psi)({\boldsymbol{x}}^{j})=\int_{\Sigma^{j}}\frac{\partial
G^{j}({\boldsymbol{x}}^{j},{\boldsymbol{y}}^{j})}{\partial
n({\boldsymbol{y}}^{j})}\,\psi({\boldsymbol{y}}^{j})\,\mathrm{d}s({\boldsymbol{y}}^{j}),\quad{\boldsymbol{x}}^{j}\in\Omega^{j},$

where $G^{j}({\boldsymbol{x}}^{j},{\boldsymbol{y}}^{j})$ is the Dirichlet
Green’s function for $\Omega^{j}$ and $n({\boldsymbol{y}}^{j})$ is the unit
normal to $\Sigma^{j}$ that points into $\Omega^{j}$. Explicitly,
$G^{j}({\boldsymbol{x}}^{j},{\boldsymbol{y}}^{j})=\Phi({\boldsymbol{x}}^{j},{\boldsymbol{y}}^{j})-\Phi(\widetilde{\boldsymbol{x}}^{j},{\boldsymbol{y}}^{j})$,
with $\widetilde{\boldsymbol{x}}^{j}$ the image of ${\boldsymbol{x}}^{j}$ in
$\Sigma^{j}$, where $\Phi({\boldsymbol{x}},{\boldsymbol{y}})$ is the standard
fundamental solution of the Helmholtz equation defined by

(12)
$\Phi({\boldsymbol{x}},{\boldsymbol{y}}):=\frac{{\mathrm{i}}}{4}H_{0}^{(1)}(k|\boldsymbol{x}-\boldsymbol{y}|),\quad{\boldsymbol{x},\,\boldsymbol{y}\in\mathbb{R}^{2},\;\;\boldsymbol{x}\neq\boldsymbol{y}},$

where $H^{(1)}_{n}$ is the Hankel function of the first kind of order $n$.
This leads finally to

(13)
$U^{j}(\psi)({\boldsymbol{x}}^{j})=2\int_{\Sigma^{j}}\frac{\partial\Phi({\boldsymbol{x}}^{j},{\boldsymbol{y}}^{j})}{\partial
y_{1}^{j}}\,\psi({\boldsymbol{y}}^{j})\,\mathrm{d}s({\boldsymbol{y}}^{j}),\quad{\boldsymbol{x}}^{j}\in\Omega^{j},$

which can be rewritten as

(14)
$U^{j}(\psi)({\boldsymbol{x}}^{j})=\int_{\mathbb{R}}\mathcal{H}(k;x_{1}^{j}-a^{j},x_{2}^{j}-y_{2}^{j})\,\psi(a^{j},y_{2}^{j})\,\mathrm{d}y_{2}^{j},\quad{\boldsymbol{x}}^{j}=(x_{1}^{j},x_{2}^{j})\in\Omega^{j},$

where we have set

(15)
$\mathcal{H}(k;x_{1},x_{2}):=\frac{{\mathrm{i}}kx_{1}}{2}\frac{H^{(1)}_{1}(k|\boldsymbol{x}|)}{|\boldsymbol{x}|}.$

To derive a system of equations whose unknowns are the traces $\varphi^{j}$ of
the solution, it suffices to observe that the half-plane representations must
coincide where they coexist.

$\mathcal{O}$$\Sigma^{1}$$\Sigma^{2}$$\Theta^{1,2}$$u\big{|}_{\Omega^{1}}=U^{1}(\varphi^{1})$$u\big{|}_{\Omega^{2}}=U^{2}(\varphi^{2})$${\Omega^{1}}\cap{\Omega^{2}}$

$\Omega^{2}\cap\Sigma^{1}$$\Omega^{1}\cap\Sigma^{2}$$\varphi^{2}=U^{1}(\varphi^{1})$$\varphi^{1}=U^{2}(\varphi^{2})$${\Omega^{1}}\cap{\Omega^{2}}$

Figure 2: Compatibility condition in $\Omega^{1}\cap\Omega^{2}$.

For instance, in the overlapping zone $\Omega^{1}\cap\Omega^{2}$ (see Figure
2) we have

(16)
$u=U^{1}(\varphi^{1})=U^{2}(\varphi^{2})\quad\text{in}\;\Omega^{1}\cap\Omega^{2},$

and in particular

(17) $\varphi^{2}=U^{1}(\varphi^{1})\quad\text{on}\;\Omega^{1}\cap\Sigma^{2},$

which is a relation linking $\varphi^{1}$ and $\varphi^{2}$. By introducing
the operator $D_{1,2}:L^{2}(\Sigma^{1})\rightarrow
L^{2}(\Omega^{1}\cap\Sigma^{2})$ defined by

$D_{1,2}\,\psi:=U^{1}(\psi)\big{|}_{\Omega^{1}\cap\Sigma^{2}},\quad\psi\in
L^{2}(\Sigma^{1}),$

the relation (17) can be rewritten as

(18) $\varphi^{2}=D_{1,2}\varphi^{1}\quad\text{on}\;\Omega^{1}\cap\Sigma^{2}.$

From (16), we deduce similarly

$\varphi^{1}=U^{2}(\varphi^{2})\quad\text{on}\;\Omega^{2}\cap\Sigma^{1},$

another relation linking $\varphi^{1}$ and $\varphi^{2}$. Repeating this with
$1$ and $2$ replaced, respectively, by $j$ and $j+1$, we get $2\mathcal{N}$
equations linking the $\mathcal{N}$ traces. Let us introduce, for all
$j\in\mathbb{Z}/\mathcal{N}\mathbb{Z}$, the operators $D_{j,j\pm
1}:H^{1/2}(\Sigma^{j})\rightarrow H^{1/2}(\Omega^{j}\cap\Sigma^{j\pm 1})$
defined by

(19) $D_{j,j\pm 1}\,\psi:=U^{j}(\psi)\big{|}_{\Omega^{j}\cap\Sigma^{j\pm
1}},\quad\psi\in H^{1/2}(\Sigma^{j}).$

Then the compatibility relations between all the traces can be written as:

(20) $\forall
j\in\mathbb{Z}/\mathcal{N}\mathbb{Z},\quad\begin{array}[]{|lcl}\varphi^{j}=D_{j-1,j}\,\varphi^{j-1},\quad\text{on}\;\Omega^{j-1}\cap\Sigma^{j},\\\
\varphi^{j}=D_{j+1,j}\,\varphi^{j+1},\quad\text{on}\;\Omega^{j+1}\cap\Sigma^{j}.\end{array}$

This system of equations has to be completed with the Dirichlet boundary
condition, rewritten as

(21) $\varphi^{j}=g|_{\Gamma^{j}}\quad\text{on}\;\Gamma^{j},\quad
j=1,...,{\mathcal{N}}.$

Note that for all $j$ the operators $D_{j,j\pm 1}$ can be given explicitly by
using the representation (13) for $U^{j}$. They are integral operators and
more precisely double-layer potential operators (in the sense, e.g., of [15]
or [7]).

One can easily check that (20)-(21) is equivalent to the original problem (6)
and so is well-posed. We recall here the proof of this result because its
general idea will be used in the results proved later in the paper.

######  Theorem 2.1.

In the case where $\Re(k)>0$ and $\Im(k)>0$, there exists, for each $g\in
H^{1/2}(\Gamma)$, a unique solution
$(\varphi^{1},...,\varphi^{\mathcal{N}})\in
H^{1/2}(\Sigma^{1})\times\ldots\times H^{1/2}(\Sigma^{{\mathcal{N}}})$ of
(20)-(21).

Existence was proven by construction of (20)-(21) from the solution $u$ of
(6), with $\varphi^{j}:=u|_{\Sigma^{j}}$ for $j=1,...,{\mathcal{N}}$.
Uniqueness follows from the following proposition since (6) is well-posed.

######  Proposition 2.2.

Suppose $\Re(k)>0$ and $\Im(k)>0$ and let
$(\varphi^{1},...,\varphi^{\mathcal{N}})$ $\in$ $H^{1/2}(\Sigma^{1})$
$\times\ldots\times H^{1/2}(\Sigma^{{\mathcal{N}}})$ be a solution to
(20)-(21) (with $g\in H^{1/2}(\Gamma)$). Then, for all
$j\in\mathbb{Z}/\mathcal{N}\mathbb{Z}$,

(22)
$U^{j}(\varphi^{j})=U^{j+1}(\varphi^{j+1})\quad\text{in}\;\Omega^{j}\cap\Omega^{j+1},$

and the function defined by $u=U^{j}(\varphi^{j})$ in $\Omega^{j}$,
$j=1,...,{\mathcal{N}}$, is the unique solution $u\in H^{1}(\Omega)$ of (6).

######  Proof 2.3.

To prove (22), let us set $v=U^{j}(\varphi^{j})-U^{j+1}(\varphi^{j+1})$. By
definition of the half-plane representations $U^{j}$ and $U^{j+1}$, it is
clear that $v\in H^{1}(\Omega^{j}\cap\Omega^{j+1})$ and that $v$ satisfies
$\Delta v+k^{2}v=0$ in $\Omega^{j}\cap\Omega^{j+1}$. Moreover, the
compatibility conditions (20) imply that $v$ vanishes on the boundary of
$\Omega^{j}\cap\Omega^{j+1}$. Using that $\Im(k)\neq 0$, one deduces that
$v=0$ by uniqueness of the solution of the Dirichlet problem in
$\Omega^{j}\cap\Omega^{j+1}$. The rest of the proof is straightforward.

It has been shown in previous papers [6, 5] that, for both mathematical
analysis and computation, it is more convenient to consider the Half-Space
Matching formulation in an $L^{2}$-framework, which means that $\varphi^{j}$
is sought in $L^{2}(\Sigma^{j})$ instead of $H^{1/2}(\Sigma^{j})$ so that the
formulation is

(23)
$\begin{array}[]{c}\text{Find}\;(\varphi^{1},...,\varphi^{\mathcal{N}})\in
L^{2}(\Sigma^{1})\times\ldots\times L^{2}(\Sigma^{{\mathcal{N}}})\;\text{such
that for }j\in\mathbb{Z}/\mathcal{N}\mathbb{Z}\\\\[5.0pt]
\begin{array}[]{|ll}\varphi^{j}=D_{j-1,j}\,\varphi^{j-1},&\text{on}\;\Sigma^{j}\cap\Omega^{j-1},\\\
\varphi^{j}=g|_{\Gamma^{j}},&\text{on}\;\Gamma^{j},\\\
\varphi^{j}=D_{j+1,j}\,\varphi^{j+1},&\text{on}\;\Sigma^{j}\cap\Omega^{j+1}.\end{array}\end{array}$

Let us emphasize that this makes sense since $D_{j,j\pm 1}\psi$ is well-
defined by (19) and (14) for all $\psi\in L^{2}(\Sigma^{j})$ and the operators
$D_{j,j\pm 1}$ are in fact continuous from $L^{2}(\Sigma^{j})$ to
$L^{2}(\Omega^{j}\cap\Sigma^{j\pm 1})$ [6]. It has been shown in [6] that
problem (23) is of Fredholm type. More precisely, by rewriting it in an
operator form, it is proven that the associated operator is Fredholm of index
0. This means that existence of a solution is equivalent to its uniqueness. We
refer the reader to [6] for more detail. Here, we focus on the question of
uniqueness. We need to prove that, for a solution of (23), $g=0$ implies
$\varphi^{j}=0$ for all $j=1,...,{\mathcal{N}}$. The sketch of the proof of
this uniqueness result, which is much less straightforward than in the
$H^{1/2}$ framework, has been given in [6]. We give a detailed presentation of
this proof in the following paragraph, in a form that will be directly used
for our main result, which is uniqueness for the case of a real wavenumber
$k$.

###  2.2 The uniqueness result in an $L^{2}$-framework

The difficulty in proving a uniqueness result for problem (23) comes from the
fact that the half-plane solutions $U^{j}(\varphi^{j})$ (with
$U^{j}(\varphi^{j})$ defined now by (11), (13), or (14)) are in general not in
$H^{1}(\Omega^{j})$, assuming only that $\varphi^{j}\in L^{2}(\Sigma^{j})$.
But we will take advantage of the following lemma which summarizes key
properties of the half-plane solutions with $L^{2}$ and/or $L^{\infty}$
Dirichlet boundary data.

######  Lemma 1.

Suppose that $\Im(k)\geq 0,\Re(k)>0$. Let $\phi\in
L^{2}(\mathbb{R})+L^{\infty}(\mathbb{R})$ and define $u(\boldsymbol{x})$, for
$\boldsymbol{x}=(x_{1},x_{2})\in\mathbb{R}^{2}_{+}:=(0,\infty)\times\mathbb{R}$,
by

(24)
$u(\boldsymbol{x}):=\int_{\mathbb{R}}\mathcal{H}(k;x_{1},x_{2}-t)\phi(t)\,\mathrm{d}t,$

where $\mathcal{H}$ is defined by (15). Then the integral is absolutely
convergent for $\boldsymbol{x}\in\mathbb{R}^{2}_{+}$,
$u\in{C}^{\infty}(\mathbb{R}^{2}_{+})$, and $\Delta u+k^{2}\,u=0$ in
$\mathbb{R}^{2}_{+}$. Moreover if $\phi$ is continuous on an open interval
$I\subset\mathbb{R}$, then $u\in C(\mathbb{R}^{2}_{+}\cup\Gamma)$ where
$\Gamma:=\\{(0,t),\;t\in I\\}$, and $u(0,t)=\phi(t),\;t\in I$. Finally, if
$\phi\in L^{2}(\mathbb{R})$ and $\Im(k)>0$, then $u\in
L^{2}(\mathbb{R}^{2}_{+})$ and there exists a constant $C_{\infty}>0$
independent of $\phi$ such that

(25) $\|u\|_{L^{2}(\mathbb{R}^{2}_{+})}\leq
C_{\infty}\|\phi\|_{L^{2}(\mathbb{R})},$

whereas if $\Im(k)=0$ then, for any $L>0$, $u\in L^{2}(\omega_{L})$ where
$\omega_{L}:=\\{\boldsymbol{x}\in\mathbb{R}^{2}_{+},\;x_{1}<L\\}$, and there
exists a constant $C_{L}>0$ independent of $\phi$ such that

(26) $\|u\|_{L^{2}(\omega_{L})}\leq C_{L}\|\phi\|_{L^{2}(\mathbb{R})}.$

######  Proof 2.4.

From asymptotics of the Hankel function $H^{(1)}_{1}$ for large and small
argument, it follows that, for every $k$ with $\Im(k)\geq 0$, $\Re(k)>0$,
there exists a $C>0$ such that

(27) $|\mathcal{H}(k;x_{1},x_{2})|\leq
C\frac{x_{1}}{|\boldsymbol{x}|^{2}}(1+|\boldsymbol{x}|^{1/2})e^{-\Im(k)|\boldsymbol{x}|}.$

It follows that, for $\boldsymbol{x}\in\mathbb{R}^{2}_{+}$, the integral in
(24) is absolutely convergent for any $\phi\in
L^{2}(\mathbb{R})+L^{\infty}(\mathbb{R})$. To see that
$u\in{C}^{\infty}(\mathbb{R}^{2}_{+})$ and satisfies the Helmholtz equation,
it suffices to argue as in [9, Theorem 3.2]. If $\phi$ is continuous on an
open interval $I\subset\mathbb{R}$, to see that $u\in
C(\mathbb{R}^{2}_{+}\cup\Gamma)$ and that $u(0,t)=\phi(t)$ for $t\in I$, it is
enough to show that $u\in C(\mathbb{R}^{2}_{+}\cup\tilde{\Gamma})$ and that
$u(0,t)=\phi(t)$ for $t\in\tilde{I}$ for every compact
$\tilde{\Gamma}=\\{(0,t),\;t\in\tilde{I}\\}\subset\Gamma$. So suppose
$\tilde{I}\subset I$ is compact and write $\phi=\phi_{1}+\phi_{2}$ where the
support of $\phi_{2}$ does not intersect $\tilde{I}$ and $\phi_{1}$ is bounded
and continuous. Correspondingly, we can split $u$ as $u=u_{1}+u_{2}$, where
$u_{j}$ is defined by (24) with $\phi$ replaced by $\phi_{j}$. It follows from
the definition that $u_{2}\in{C}(\mathbb{R}^{2}_{+}\cup\tilde{\Gamma})$ with
$u_{2}(0,t)=0$ for $t\in\tilde{I}$, while
$u_{1}\in{C}(\overline{\mathbb{R}^{2}_{+}})$ and $u_{1}(0,t)=\phi_{1}(t)$ for
$t\in\tilde{I}$ by [8, Theorem 3.1], so that $u(0,t)=\phi(t)$ for
$t\in\tilde{I}$. Concerning the $L^{2}$-estimates (25) and (26) let us recall
that for $\psi_{1}\in L^{1}(\mathbb{R})$ and $\psi_{2}\in L^{2}(\mathbb{R})$
the convolution product $\psi_{1}*\psi_{2}$ belongs to $L^{2}(\mathbb{R})$,
with the estimate (Young’s convolution inequality)

$\|\psi_{1}*\psi_{2}\|_{L^{2}(\mathbb{R})}\leq\|\psi_{1}\|_{L^{1}(\mathbb{R})}\|\psi_{2}\|_{L^{2}(\mathbb{R})}.$

For a given $x_{1}>0$, this implies that

$\int_{\mathbb{R}}|u(x_{1},x_{2})|^{2}dx_{2}\leq\left[\int_{\mathbb{R}}|\mathcal{H}(k;x_{1},x_{2})|dx_{2}\right]^{2}\|\phi\|_{L^{2}(\mathbb{R})}^{2}.$

From (27), we deduce that, for some $C^{\prime}>0$,

$\int_{\mathbb{R}}|\mathcal{H}(k;x_{1},x_{2})|dx_{2}\leq
C^{\prime}(1+x_{1})^{1/2}e^{-\Im(k)x_{1}},\quad x_{1}>0.$

One easily obtains the estimates (25) for $\Im(k)>0$ and (26) for $\Im(k)=0$.

Now to prove the uniqueness result, we need some results concerning $L^{2}$
solutions of the homogeneous Dirichlet problem in a domain with corners. We
gather in the next lemma the results that we need which are proved in [19,
Chapter 2] (see Theorem 2.3.3 and the proof of Theorem 2.3.7).

######  Lemma 2.5.

Let $Q$ be a bounded polygonal domain with $N$ vertices denoted by
$S_{1},\ldots,S_{N}$. Let us suppose that $Q$ has $M<N$ reentrant vertices
that can be supposed to be $S_{1},\ldots,S_{M}$ without loss of generality.
Let $P:=\\{S_{1},\ldots,S_{N}\\}$ and suppose that $w\in{C}(\bar{Q}\setminus
P)\cap{C}^{2}(Q)$ satisfies $\Delta w=0$ in $Q$ and $w=0$ on $\partial
Q\setminus P$. Then, if $w\in L^{2}(Q)$, there exists $\tilde{w}\in H^{1}(Q)$
and $M$ complex constants $c_{1},\ldots,c_{M}$ such that

$w(\boldsymbol{x})=\tilde{w}(\boldsymbol{x})+\sum_{m=1}^{M}c_{m}\,r_{m}^{-\pi/\alpha_{m}}\sin(\pi\theta_{m}/\alpha_{m}),\quad\boldsymbol{x}\in
Q,$

where, for each $m$, $\alpha_{m}$ is the interior angle of $Q$ at $S_{m}$ and
$(r_{m},\theta_{m})$ are the polar coordinates of $\boldsymbol{x}$ whose
origin is $S_{m}$ such that $\theta_{m}$ is the angle from one of the sides of
$\partial Q$ containing $S_{m}$. In particular if $M=0$ ($Q$ is convex) then
$w=0$.

The previous lemma will be used to prove that particular $L^{2}$ solutions of
the Helmholtz equations in unbounded domains with Dirichlet boundary
conditions are in fact $H^{1}$ in every bounded subset of the domain. To prove
that they are $H^{1}$ in the whole domain, we will resort to the following
result, whose proof follows the arguments of [29, Lemma 2.2].

######  Lemma 2.6.

Let $\Pi\subset\mathbb{R}^{2}$ be an unbounded open set such that
$\Pi_{R}:=\\{\boldsymbol{x}\in\Pi;|\boldsymbol{x}|<R+1\\}$ is a Lipschitz
domain for all sufficiently large $R>0$, and $v\in L^{2}(\Pi)$ a function such
that $v_{|B}\in H^{1}(B)$ for all bounded open subsets $B$ of $\Pi$. If $v$
satisfies the Helmholtz equation $\Delta v+k^{2}v=0$ in $\Pi$ and its trace is
zero on $\partial\Pi$, then $v\in H^{1}(\Pi)$.

######  Proof 2.7.

Let us define the truncation function
$\chi_{R}(\boldsymbol{x}):=F_{R}(|\boldsymbol{x}|)$ where $F_{R}\in
C^{\infty}(\mathbb{R}^{+})$ is chosen such that $0\leq F_{R}\leq 1$, $F_{R}=1$
on $[0,R]$, $F_{R}=0$ on $[R+1,+\infty)$ and there exists a constant $c>0$
independent of $R$ such that $|F_{R}^{\prime}|\leq c\sqrt{F_{R}}$ (this can be
achieved by defining $F_{R}$ so that $F_{R}>0$ on $(R,R+1)$ and vanishes
quadratically at $R+1$). One has for all $R>0$

$\int_{\Pi}(\Delta v+k^{2}v)\overline{v}\chi_{R}\,d\boldsymbol{x}=0.$

Since $v\in H^{1}(\Pi_{R};\Delta):=\left\\{u\in H^{1}(\Pi_{R});\Delta u\in
L^{2}(\Pi_{R})\right\\}$ we can apply Green’s first identity in $\Pi_{R}$.
Using that $v$ vanishes on $\partial\Pi$, this yields

$\int_{\Pi}|\nabla v|^{2}\chi_{R}\,d\boldsymbol{x}\leq
k^{2}\|v\|^{2}_{L^{2}(\Pi)}+\left|\int_{\Pi}\overline{v}\nabla
v\cdot\nabla\chi_{R}\,d\boldsymbol{x}\right|.$

Since $|\nabla\chi_{R}|\leq c\sqrt{\chi_{R}}$, applying the Cauchy-Schwarz
inequality gives

$\int_{\Pi}|\nabla v|^{2}\chi_{R}\,d\boldsymbol{x}\leq
k^{2}\|v\|^{2}_{L^{2}(\Pi)}+c\|v\|_{L^{2}(\Pi)}\sqrt{\int_{\Pi}|\nabla
v|^{2}\chi_{R}\,d\boldsymbol{x}},$

from which we deduce the existence of a constant $C>0$ independent of $R$ such
that

$\int_{\Pi}|\nabla v|^{2}\chi_{R}\,d\boldsymbol{x}\leq
C\|v\|^{2}_{L^{2}(\Pi)}.$

This being true for all $R$, we can conclude that $v\in H^{1}(\Pi)$.

We are now able to prove the following well-posedness result.

######  Theorem 2.8.

If $\Re(k)>0$ and $\Im(k)>0$ then there exists, for each $g\in
H^{1/2}(\Gamma)$, a unique solution
$(\varphi^{1},...,\varphi^{\mathcal{N}})\in
L^{2}(\Sigma^{1})\times\ldots\times L^{2}(\Sigma^{{\mathcal{N}}})$ of (23).

This is a direct consequence of the following proposition and Theorem 2.1.

######  Proposition 2.9.

Suppose $\Re(k)>0$ and $\Im(k)>0$ and let
$(\varphi^{1},...,\varphi^{\mathcal{N}})$ $\in$ $L^{2}(\Sigma^{1})$
$\times\ldots\times L^{2}(\Sigma^{{\mathcal{N}}})$ be a solution of (23) (with
$g\in H^{1/2}(\Gamma)$). Then $\varphi^{j}\in H^{1/2}(\Sigma^{j})$,
$j=1,...,{\mathcal{N}}$.

######  Proof 2.10.

The proof consists of two steps.

  1. 1.

Prove the relations (22) as in Proposition 2.2 but noting that here the
functions $(\varphi^{1},...,\varphi^{\mathcal{N}})$ that solve (23) are a
priori supposed to be only in $L^{2}$.

  2. 2.

By noting that the traces on $\Sigma^{1},\ldots,\Sigma^{\mathcal{N}}$ of the
unique solution of (6) form a solution of (23) (and are obviously in
$H^{1/2}$), it suffices to prove that, if $g=0$, $\varphi^{j}=0$ for
$j=1,...,{\mathcal{N}}$. To prove this uniqueness result, we show that, in the
case that $(\varphi^{1},...,\varphi^{\mathcal{N}})$ satisfies (23) with $g=0$,
the function defined (thanks to step 1) by $u=U^{j}(\varphi^{j})$ in
$\Omega^{j}$ ($U^{j}(\varphi^{j})$ defined by (13)) is in $H^{1}(\Omega)$,
with zero trace. Therefore it is equal to 0 everywhere, as the unique solution
of (6) for $g=0$. We deduce finally that each $\varphi^{j}$, which we show is
the trace of $u$ on $\Sigma^{j}$, is zero.

Step 1. As in the proof of Proposition 2.2, we introduce
$v=U^{j}(\varphi^{j})-U^{j+1}(\varphi^{j+1})$, where the $\varphi^{j}\in
L^{2}(\Sigma^{j})$ satisfy (23), in particular satisfy (20). A priori, we just
know, thanks to Lemma 1, that $v\in L^{2}(\Omega^{j}\cap\Omega^{j+1})\cap
C^{\infty}(\Omega^{j}\cap\Omega^{j+1})$ and that $v$ satisfies $\Delta
v+k^{2}v=0$ in $\Omega^{j}\cap\Omega^{j+1}$. Note that it follows from (20)
and Lemma 1 that the $\varphi^{j}$’s are in fact continuous on
$\Sigma^{j}\setminus\overline{\Gamma^{j}}$, so that, from Lemma 1,
$U^{j}(\varphi^{j})\in C(\overline{\Omega^{j}}\setminus\overline{\Gamma^{j}})$
and $U^{j}(\varphi^{j})=\varphi^{j}$ on
${\Sigma^{j}\setminus\overline{\Gamma^{j}}}$. As a consequence, $v$ is
continuous in $\overline{\Omega^{j}\cap\Omega^{j+1}}$, except maybe at
$S_{j}$, the intersection of $\Sigma^{j}$ and $\Sigma^{j+1}$, and, thanks to
(20), $v$ vanishes on
$\partial(\Omega^{j}\cap\Omega^{j+1})\setminus\\{S_{j}\\}$. It follows by
standard reflection and elliptic regularity arguments that $v\in
C^{\infty}\left(\overline{\Omega^{j}\cap\Omega^{j+1}}\setminus\\{S_{j}\\}\right)$.

To conclude that $v=0$, we only need to prove that $v\in
H^{1}(\Omega^{j}\cap\Omega^{j+1})$. Firstly, we show that $v$ is $H^{1}$ in
all bounded subdomains of $\Omega^{j}\cap\Omega^{j+1}$. Let us introduce a
bounded convex polygon $\mathcal{O}^{\prime}$ such that
$\overline{\mathcal{O}}\subset\mathcal{O}^{\prime}$ (see figure 3). Then
$Q:=\mathcal{O}^{\prime}\cap\Omega^{j}\cap\Omega^{j+1}$ is a convex polygon.
Let us consider the following Dirichlet problem: find $\tilde{v}\in H^{1}(Q)$
such that

$\begin{array}[]{|ll}-\Delta\tilde{v}=k^{2}v&\mbox{in }Q,\\\
\tilde{v}=v&\mbox{on }\partial Q.\end{array}$

Since $v\in L^{2}(Q)$ and the trace of $v$ on $\partial Q$ is in
$H^{1/2}(\partial Q)\cap C(\partial Q)$ (indeed is supported in $\partial
Q\cap\partial\mathcal{O}^{\prime}$, where $v$ is ${C}^{\infty}$), this has a
unique solution. Further, by standard elliptic regularity arguments,
$\tilde{v}\in C^{2}(Q)$ and (see [24], Corollary 7.11.7) $\tilde{v}\in
C(\bar{Q})$. One can finally apply Lemma 2.5 to the function $w=v-\tilde{v}\in
C(\bar{Q}\setminus\\{S_{j}\\})\cap C^{2}(Q)\cap L^{2}(Q)$. Since $Q$ is
convex, we conclude that $w=0$, and consequently that $v|_{Q}=\tilde{v}\in
H^{1}(Q)$. As $v\in
C^{\infty}\left(\overline{\Omega^{j}\cap\Omega^{j+1}}\setminus\\{S_{j}\\}\right)$,
we deduce that $v$ is $H^{1}$ in all open bounded subdomains of
$\Omega^{j}\cap\Omega^{j+1}$. Since also $v\in
L^{2}(\Omega^{j}\cap\Omega^{j+1})$, satisfies the Helmholtz equation in
$\Omega^{j}\cap\Omega^{j+1}$ and vanishes on
$\partial(\Omega^{j}\cap\Omega^{j+1})$, Lemma 2.6 implies that $v\in
H^{1}(\Omega^{j}\cap\Omega^{j+1})$. Thus $v=0$ so that (22) holds.

Step 2. The second step consists in proving that, if $g=0$, the function
defined by $u=U^{j}(\varphi^{j})$ in $\Omega^{j}$ is equal to 0 everywhere,
and deduce from this that each $\varphi^{j}=0$. For $j=1,...,{\mathcal{N}}$,
as above let $S_{j}:=\Sigma^{j}\cap\Sigma^{j+1}$ so that
$P:=\\{S_{1},S_{2},...,S_{\mathcal{N}}\\}$ is the set of corners of
$\partial\Omega$ (i.e. the set of vertices of the polygon $\mathcal{O}$).
Since, for each $j$, $\varphi^{j}$ is such that $\varphi^{j}=g=0$ on
$\Gamma^{j}$, and $\varphi^{j}\in C(\Sigma^{j}\setminus P)$ (see Step 1), it
follows from Lemma 1 that $u\in C^{\infty}(\Omega^{j})\cap
L^{2}(\Omega^{j})\cap C(\overline{\Omega^{j}}\setminus P)$, $u=\varphi^{j}$ on
$\Sigma^{j}\setminus P$, in particular $u={\varphi^{j}=}0$ on $\Gamma^{j}$,
and $\Delta u+k^{2}u=0$ in $\Omega^{j}$. Consequently $u\in
C^{\infty}(\Omega)\cap L^{2}(\Omega)\cap C(\overline{\Omega}\setminus P)$,
$\Delta u+k^{2}u=0$ in $\Omega$, and $u=0$ on $\partial\Omega\setminus P$.

Again, to conclude that $u=0$, we only need to prove that $u\in
H^{1}(\Omega)$. We proceed as in the first step of the proof. Introducing the
bounded polygon $\Omega^{\prime}:=\mathcal{O}^{\prime}\setminus\mathcal{O}$,
we denote by $\tilde{u}$ the unique solution in $H^{1}(\Omega^{\prime})\cap
C(\overline{\Omega^{\prime}})\cap C^{2}(\Omega^{\prime})$ of the Dirichlet
problem

$\begin{array}[]{|ll}-\Delta\tilde{u}=k^{2}u&\mbox{in }\Omega^{\prime},\\\
\tilde{u}=u&\mbox{on }\partial\Omega^{\prime}.\end{array}$

Then we apply Lemma 2.5 to the function $w=u-\tilde{u}\in
C(\overline{\Omega^{\prime}}\setminus P)\cap C^{2}(\Omega^{\prime})\cap
L^{2}(\Omega^{\prime})$. But, in contrast to the argument in the first step,
the polygon $\Omega^{\prime}$ is not convex and has $\mathcal{N}$ reentrant
corners at the vertices $S_{1}$, $S_{2}$, …, $S_{\mathcal{N}}$. Consequently,
we deduce from Lemma 2.5 that there exists a function $\tilde{w}\in
H^{1}(\Omega^{\prime})$ and $\mathcal{N}$ complex coefficients
$c_{1},...,c_{\mathcal{N}}$ such that

$w=\tilde{w}+\sum_{j=1}^{\mathcal{N}}c_{j}\,r_{j}^{-\pi/\alpha_{j}}\sin(\pi\theta_{j}/\alpha_{j})\quad\mbox{
in }\Omega^{\prime},$

where, as explained in the lemma, $\alpha_{j}$ is the interior angle of
$\Omega^{\prime}$ at $S_{j}$ and $(r_{j},\theta_{j})$ are polar coordinates
centered at $S_{j}$. The definition of $\theta_{j}$ is such that
$\sin(\pi\theta_{j}/\alpha_{j})$ vanishes on
$\Sigma^{j}\cap\partial\mathcal{O}$ and $\Sigma^{j+1}\cap\partial\mathcal{O}$
but note that it does not vanish on $\Sigma^{j}\cap\Omega^{j+1}$. Since
$\varphi^{j}\in L^{2}(\Sigma^{j})$, $\tilde{u}\in
C(\overline{\Omega^{\prime}})$, and $w=u-\tilde{u}=\varphi^{j}-\tilde{u}$ on
$\Sigma^{j}\cap\Omega^{\prime}$, we must have $w\in
L^{2}(\Sigma^{j}\cap\Omega^{\prime})$, which is possible only if $c_{j}=0$.
Indeed, since $\alpha_{j}<2\pi$, we have $\pi/\alpha_{j}>1/2$ so that
$r_{j}^{-\pi/\alpha_{j}}$ is not square integrable near $r_{j}=0$. Summing up,
we have $w=\tilde{w}$ and $u=\tilde{u}+\tilde{w}\in H^{1}(\Omega^{\prime})$.
Since $u\in{C}^{\infty}(\Omega)$, one concludes that $u$ is $H^{1}$ in all
open bounded subdomains of $\Omega$, and finally, thanks to Lemma 2.6, that
$u\in H^{1}(\Omega)$. We have shown that $u\in H^{1}(\Omega)$ and $u$
satisfies (6) with $g=0$. Thus $u=0$ and, since $\varphi^{j}=u$ on
$\Sigma^{j}\setminus P$, we have that $\varphi^{j}=0$ for all $j$, which ends
the proof.

$\mathcal{O}$$Q$$\Sigma^{1}$$\Sigma^{2}$${\Omega^{1}}\cap{\Omega^{2}}$ Figure
3: Illustration for the proof of Theorem 2.8 in the case $j=1$ and
${\mathcal{N}}=3$. The convex polygon $\mathcal{O}^{\prime}$ is indicated in
gray and the boundaries of $\mathcal{O}$ and $Q$ with dotted contours.

##  3 The HSM method for real wavenumber

###  3.1 The Half-Space Matching formulation

In this section we consider the Dirichlet problem of the previous section, but
now with a real wavenumber $k>0$. We seek $u\in H^{1}_{\mathrm{loc}}(\Omega)$
such that

(28) $\begin{cases}-\Delta u-k^{2}\,u=0\;\text{in}\;\Omega,\\\\[4.0pt]
u=g\;\text{on}\;\Gamma,\end{cases}$

for a given $g\in H^{1/2}(\Gamma)$, and such that $u$ satisfies the Sommerfeld
radiation condition (2), which it is convenient to rewrite as

(29) $\lim_{R\to+\infty}R^{1/2}\sup\left\\{\left|\frac{\partial
u(\boldsymbol{x})}{\partial
r}-{\mathrm{i}}ku(\boldsymbol{x})\right|;|\boldsymbol{x}|=R\right\\}=0.$

It is well known that this problem has a unique solution.

Let us now give a Half-Space Matching formulation of problem (28-29). If we
set as before $\varphi^{j}:=u\big{|}_{\Sigma^{j}}$ for
$j=1,...,{\mathcal{N}}$, one can easily check by a Green’s identity argument
that the half-plane representations (9), with $U^{j}(\varphi^{j})$ given by
(11) (equivalently (13) or (14)), still hold. A main difference with the case
of a complex wavenumber is that now the solution of (28-29) decays only
slowly, like $r^{-1/2}$ as $r\to\infty$, so that we cannot expect that
$\varphi^{j}\in L^{2}(\Sigma^{j})$. However $\varphi^{j}\in
L^{2}(\Sigma^{j})+L^{\infty}(\Sigma^{j})$ so that, for $x^{j}_{1}>a^{j}$, the
integral

$\int_{\mathbb{R}}\mathcal{H}(k;x_{1}^{j}-a^{j},x_{2}^{j}-y_{2}^{j})\,\varphi^{j}(a^{j},y_{2}^{j})\,\mathrm{d}y_{2}^{j}$

is still convergent by Lemma 1. Then one can check, exactly as in the case of
a complex wavenumber, that the $\varphi^{j}$’s satisfy the following Half-
Space Matching equations:

(30)
$\begin{array}[]{c}\begin{array}[]{|ll}\varphi^{j}=D_{j-1,j}\,\varphi^{j-1},&\text{on}\;\Sigma^{j}\cap\Omega^{j-1},\\\
\varphi^{j}=g|_{\Gamma^{j}},&\text{on}\;\Gamma^{j},\\\
\varphi^{j}=D_{j+1,j}\,\varphi^{j+1},&\text{on}\;\Sigma^{j}\cap\Omega^{j+1},\end{array}\quad
j\in\mathbb{Z}/\mathcal{N}\mathbb{Z}\end{array}$

where

(31) $D_{j,j\pm
1}\,\varphi^{j}:=U^{j}(\varphi^{j})\big{|}_{\Omega^{j}\cap\Sigma^{j\pm 1}},$

with $U^{j}(\varphi^{j})$ given by (11). The idea is again, for numerical
purposes, to replace problem (28-29) by the system of equations (30). But in
contrast to the case with $\Im(k)>0$, it is not clear in which space the
functions $\varphi^{j}$’s must be sought to make (30) uniquely solvable. In
what follows, we will provide a framework which ensures existence and
uniqueness of the solution of (30). More precisely, in this framework, the
unique solution of (30) is given by $\varphi^{j}:=u\big{|}_{\Sigma^{j}}$,
where $u$ is the unique solution of (28-29).

Let us first recall, as just remarked above, that $U^{j}(\varphi)$ is well-
defined when $\varphi\in L^{2}(\Sigma^{j})+L^{\infty}(\Sigma^{j})$ by Lemma 1,
and, indeed, we will require that the solution of (30) satisfies
$\varphi^{j}\in L^{2}(\Sigma^{j})+L^{\infty}(\Sigma^{j})$ for
$j=1,\ldots,{\mathcal{N}}$. But, as discussed in §1.2, our uniqueness result
requires an additional property: analogously to requiring that $u\in
H^{1}_{\mathrm{loc}}(\Omega)$ satisfies the Sommerfeld radiation condition
when $k$ is real, we will require, for $j=1,...,{\mathcal{N}}$, that
$\varphi^{j}$ has the following asymptotics playing the role of a radiation
condition:

(32)
$\varphi^{j}(a^{j},t)=\left\\{\begin{array}[]{cc}c_{+}^{j}\,{\mathrm{e}}^{{\mathrm{i}}k|t|}|t|^{-1/2}\left(1+O(|t|^{-1})\right),&\mbox{as}\quad
t\to+\infty,\\\
c_{-}^{j}\,{\mathrm{e}}^{{\mathrm{i}}k|t|}|t|^{-1/2}\left(1+O(|t|^{-1})\right),&\mbox{as}\quad
t\to-\infty,\end{array}\right.$

for some constants $c_{\pm}^{j}\in\mathbb{C}$. This obviously holds when
$\varphi^{j}:=u\big{|}_{\Sigma^{j}}$, where $u$ is the unique solution of
(28-29), since, as discussed in §1.2, $u$ satisfies (3).

The main result of the present paper is Theorem 5 stating that the radiation
condition (32) is strong enough to ensure uniqueness for (30) for $k$ real. We
will prove it by following the same steps as in the case of a complex
wavenumber, proving first the compatibility of half-plane representations in
the overlaps between half-planes, and then uniqueness for the reconstructed
solution in the exterior domain $\Omega$. To establish these results, we will
use classical uniqueness results for the Helmholtz equation on unbounded
domains when the solution satisfies the Sommerfeld radiation condition at
infinity. A preliminary question is: can we prove, using (32), that the half-
plane representations (14) satisfy a Sommerfeld condition at infinity? We will
establish, in the next subsection, a slightly weaker result which will be
sufficient for the proof of Theorem 5.

###  3.2 Properties of the half-plane solution

We consider in this subsection the half-space
$\mathbb{R}^{2}_{+}:=(0,\infty)\times\mathbb{R}$. Let us prove a preliminary
lemma for the solution of the Dirichlet problem in $\mathbb{R}^{2}_{+}$ given
by (24), in the case that the Dirichlet data decays sufficiently rapidly at
infinity.

######  Lemma 1.

Suppose that $\phi\in L^{\infty}(\mathbb{R})$ satisfies, for some $p>1$,

(33) $|\phi(t)|\leq(1+|t|)^{-p},\quad t\in\mathbb{R},$

and define $u(\boldsymbol{x})$, for $\boldsymbol{x}\in\mathbb{R}^{2}_{+}$, by

(34)
$u(\boldsymbol{x}):=\int_{\mathbb{R}}\mathcal{H}(k;x_{1},x_{2}-t)\phi(t)\,\mathrm{d}t.$

Then there exists a constant $C(p)>0$ such that

(35) $|u(\boldsymbol{x})|\leq
C(p)(1+|\boldsymbol{x}|)^{-1/2},\quad\boldsymbol{x}\in\mathbb{R}^{2}_{+}.$

######  Proof 3.1.

We will use the bound (27), where $C>0$, there and throughout the remainder of
the proof, denotes a constant independent of $\boldsymbol{x}$ (and also
independent of $p$), not necessarily the same at each occurrence. It will be
convenient for the proof to denote by $u[\phi]$ the function $u$ defined by
(34). Then defining $\chi_{x_{2}}\in L^{\infty}(\mathbb{R})$, for
$x_{2}\in\mathbb{R}$, to be the characteristic function of the interval

$I_{x_{2}}:=\left\\{t\in\mathbb{R};|x_{2}-t|\leq(1+|x_{2}|)/2\right\\},$

we have $u[\phi]=u[\chi_{x_{2}}\phi]+u[(1-\chi_{x_{2}})\phi]$. We will
establish the bound (35) for each term of the sum (cf. the argument between
(4.16) and (4.17) in [1]).

For the first term we have

$|u[\chi_{x_{2}}\phi](\boldsymbol{x})|\leq\int_{\mathbb{R}}|\mathcal{H}(k;x_{1},x_{2}-t)|\chi_{x_{2}}(t)|\phi(t)|\mathrm{d}t,$

which gives

(36) $|u[\chi_{x_{2}}\phi](\boldsymbol{x})|\leq
2\|\chi_{x_{2}}\phi\|_{L^{\infty}(\mathbb{R})}\int_{0}^{\frac{1+|x_{2}|}{2}}|\mathcal{H}(k;x_{1},s)|\mathrm{d}s.$

Now note that if $|t|<|x_{2}|/3$ and $|x_{2}|\geq 3$, then
$|t-x_{2}|>2|x_{2}|/3\geq(1+|x_{2}|)/2$, so that $\chi_{x_{2}}(t)=0$. Thus,
for $|x_{2}|\geq 3$, (33) implies that

$\|\chi_{x_{2}}\phi\|_{L^{\infty}(\mathbb{R})}\leq\sup_{|t|\geq|x_{2}|/3}|\phi(t)|\leq(1+|x_{2}|/3)^{-p},$

while also
$\|\chi_{x_{2}}\phi\|_{L^{\infty}(\mathbb{R})}\leq\|\phi\|_{L^{\infty}(\mathbb{R})}\leq
1$, for every $x_{2}\in\mathbb{R}$, so that

(37) $\|\chi_{x_{2}}\phi\|_{L^{\infty}(\mathbb{R})}\leq
C(1+|x_{2}|)^{-p},\quad x_{2}\in\mathbb{R}.$

Moreover, it follows from (27) that

$\displaystyle\int_{0}^{\frac{1+|x_{2}|}{2}}|\mathcal{H}(k;x_{1},s)|\mathrm{d}s$
$\displaystyle\leq$ $\displaystyle
Cx_{1}\int_{0}^{\frac{1+|x_{2}|}{2}}\left[(x_{1}^{2}+s^{2})^{-1}+(x_{1}^{2}+s^{2})^{-3/4}\right]\mathrm{d}s$
$\displaystyle=$ $\displaystyle
C\int_{0}^{\frac{1+|x_{2}|}{2x_{1}}}\left[(1+t^{2})^{-1}+x_{1}^{1/2}(1+t^{2})^{-3/4}\right]\mathrm{d}t$
$\displaystyle\leq$ $\displaystyle\displaystyle
C(1+x_{1})^{1/2}\int_{0}^{\frac{1+|x_{2}|}{2x_{1}}}(1+t^{2})^{-3/4}\mathrm{d}t$
$\displaystyle\leq$ $\displaystyle\displaystyle
C(1+x_{1})^{1/2}\min\left(1,\frac{1+|x_{2}|}{2x_{1}}\right).$

Combining this with (36) and (37), we get the following estimate:

$|u[\chi_{x_{2}}\phi](\boldsymbol{x})|\leq
C(1+|x_{2}|)^{-p}(1+x_{1})^{1/2}\min\left(1,\frac{1+|x_{2}|}{2x_{1}}\right),$

which gives the expected bound (recalling that $p>1$)

$|u[\chi_{x_{2}}\phi](\boldsymbol{x})|\leq
C(1+|\boldsymbol{x}|)^{-1/2},\quad\boldsymbol{x}\in\mathbb{R}^{2}_{+}.$

To see that the same bound holds for $u[(1-\chi_{x_{2}})\phi]$, we note that

$|u[(1-\chi_{x_{2}})\phi](\boldsymbol{x})|\leq
C\sup_{t\in\mathbb{R}}|\mathcal{H}(k;x_{1},x_{2}-t)(1-\chi_{x_{2}})(t)|\int_{\mathbb{R}}|\phi(t)|\,\mathrm{d}t.$

Observing that, for $\boldsymbol{x}\in\mathbb{R}^{2}_{+}$ and
$t\in\mathbb{R}\setminus I_{x_{2}}$,

$(x_{1}^{2}+(x_{2}-t)^{2})^{1/2}\geq(x_{1}^{2}+(1+|x_{2}|)^{2}/4)^{1/2}\geq
C(1+|\boldsymbol{x}|),$

and using (27) and (33), it follows that, for
$\boldsymbol{x}\in\mathbb{R}^{2}_{+}$,

(38) $|u[(1-\chi_{x_{2}})\phi](\boldsymbol{x})|\leq
C\frac{x_{1}}{(1+|\boldsymbol{x}|)^{3/2}}\int_{\mathbb{R}}|\phi(t)|\,\mathrm{d}t\leq\frac{C}{p-1}(1+|\boldsymbol{x}|)^{-1/2}.$

It follows from standard interior elliptic regularity results that if
$u\in{C}^{2}(\mathbb{R}^{2}_{+})$ satisfies

$\Delta u+k^{2}u=0\quad\mbox{in }\mathbb{R}^{2}_{+},$

then for every $\varepsilon>0$ and $\boldsymbol{x}\in\mathbb{R}^{2}_{+}$ with
$x_{1}>\varepsilon$, so that
$\\{\boldsymbol{y}\in\mathbb{R}^{2}:|\boldsymbol{x}-\boldsymbol{y}|\leq\varepsilon\\}\subset\mathbb{R}^{2}_{+}$:

(39) $|\nabla u(\boldsymbol{x})|\leq
C\>\frac{1+k^{2}\varepsilon^{2}}{\varepsilon}\max_{|\boldsymbol{x}-\boldsymbol{y}|\leq\varepsilon}|u(\boldsymbol{y})|$

where $C$ is an absolute constant (see, e.g. [13, Lemma 2.7]). Combining this
estimate with the previous lemma, we have immediately the following corollary:

######  Corollary 2.

Suppose that $\phi\in L^{\infty}(\mathbb{R})$ satisfies (33) for some $p>1$.
Then there exists a constant $C(p)>0$ such that for all $\varepsilon>0$, the
function $u$ defined by (34) satisfies

(40) $|\nabla u(\boldsymbol{x})|\leq
C(p)\>\frac{1+k^{2}\varepsilon^{2}}{\varepsilon}(1+|\boldsymbol{x}|)^{-1/2},\quad
x_{1}>\varepsilon,\,x_{2}\in\mathbb{R}.$

Now, using the previous results, we will establish a radiation condition for
$u$ defined by (34) when its trace $\varphi$ satisfies itself a radiation
condition.

######  Lemma 3.

Suppose that $\phi\in{C}(\mathbb{R})$ satisfies

(41)
$\phi(t)=\left\\{\begin{array}[]{cc}c_{+}\,{\mathrm{e}}^{{\mathrm{i}}k|t|}|t|^{-1/2}+O(|t|^{-p}),&\mbox{as}\quad
t\to+\infty,\\\
c_{-}\,{\mathrm{e}}^{{\mathrm{i}}k|t|}|t|^{-1/2}+O(|t|^{-p}),&\mbox{as}\quad
t\to-\infty,\end{array}\right.$

for some constants $c_{+},c_{-}\in\mathbb{C}$ and $p>1$. Define $u$ by (34) in
$\mathbb{R}^{2}_{+}$. Then the following properties hold:

  * (i)

there exists a constant $C>0$ such that

(42) $|u(\boldsymbol{x})|\leq
C(1+|\boldsymbol{x}|)^{-1/2},\quad\boldsymbol{x}\in\mathbb{R}^{2}_{+};$

  * (ii)

for all $\varepsilon>0$,

(43) $\lim_{R\to+\infty}R^{1/2}\sup\left\\{\left|\frac{\partial
u(\boldsymbol{x})}{\partial
r}-{\mathrm{i}}ku(\boldsymbol{x})\right|;x_{1}>\varepsilon\mbox{ and
}|\boldsymbol{x}|=R\right\\}=0.$

######  Proof 3.2.

We use the same notation $u[\phi]$ for (34) as in the proof of Lemma 1. We
prove this proposition by writing $\phi$ as $\phi=\phi^{(1)}+\phi^{(2)}$, and
correspondingly writing the integral (34) as
$u[\phi]=u[\phi^{(1)}]+u[\phi^{(2)}]$, choosing this splitting as follows:
$\phi^{(1)}$ is the trace on the boundary of a solution of the Helmholtz
equation in $\mathbb{R}^{2}_{+}$ that clearly satisfies (42) and (43), and
$\phi^{(2)}(t)$ decays sufficiently rapidly as $|t|\to\infty$ so that, thanks
to Lemma 1 and Corollary 2, $u[\phi^{(2)}]$ also satisfies (42) and (43).

In more detail, from the asymptotics of the Hankel function $H_{0}^{(1)}$ it
follows that, for some non-zero $c^{*}\in\mathbb{C}$,

(44)
$\Phi(\boldsymbol{x},(0,t))=c^{*}\,{\mathrm{e}}^{{\mathrm{i}}k|t-x_{2}|}\,|t|^{-1/2}+O(|t|^{-3/2})\quad\mbox{as}\quad|t|\to\infty,$

for every $\boldsymbol{x}\in\mathbb{R}^{2}$. Pick
$\boldsymbol{z}=(z_{1},z_{2}),\boldsymbol{z}^{\prime}=(z^{\prime}_{1},z^{\prime}_{2})\in\mathbb{R}^{2}$
with $k(z_{2}-z^{\prime}_{2})/\pi\not\in\mathbb{Z}$, and with
$z_{1},z^{\prime}_{1}<0$. Define $\phi^{(1)}\in C(\mathbb{R})$ by

(45)
$\phi^{(1)}(t):=(c/c^{*})\Phi(\boldsymbol{z},(0,t))+(c^{\prime}/c^{*})\Phi(\boldsymbol{z}^{\prime},(0,t))\;\text{
for}\;t\in\mathbb{R}.$

Then $\phi^{(1)}(t)$ has the same leading asymptotic behaviour as $\phi(t)$ as
$t\to\pm\infty$ provided

(46)
$c{\mathrm{e}}^{-{\mathrm{i}}kz_{2}}+c^{\prime}{\mathrm{e}}^{-{\mathrm{i}}kz^{\prime}_{2}}=c_{+}\quad\mbox{and}\quad
c{\mathrm{e}}^{{\mathrm{i}}kz_{2}}+c^{\prime}{\mathrm{e}}^{{\mathrm{i}}kz^{\prime}_{2}}=c_{-}.$

Let us choose $c$ and $c^{\prime}$ so that these two linear constraints hold;
this is possible since $k(z_{2}-z^{\prime}_{2})/\pi\not\in\mathbb{Z}$ implies
that the determinant
$\exp({\mathrm{i}}k(z^{\prime}_{2}-z_{2}))-\exp({\mathrm{i}}k(z_{2}-z^{\prime}_{2})=2{\mathrm{i}}\sin(k(z_{2}-z^{\prime}_{2}))\neq
0$ . Then it is clear from an application of Green’s second identity that, for
$\boldsymbol{x}\in\mathbb{R}^{2}_{+}$,
$u[\phi^{(1)}](\boldsymbol{x})=(c/c^{*})\Phi(\boldsymbol{z},\boldsymbol{x})+(c^{\prime}/c^{*})\Phi(\boldsymbol{z}^{\prime},\boldsymbol{x})$
(see, e.g., the proof that (ii)$\Rightarrow$(v)$\Rightarrow$(i) in [13,
Theorem 2.9]). Hence $u[\phi^{(1)}]$ satisfies items (i) and (ii).

It remains to show that the same is true for $u[\phi^{(2)}]$ where
$\phi^{(2)}=\phi-\phi^{(1)}$. It follows from (41) and (44)–(46) that, for
some $c_{0}>0$, where $p^{\prime}:=\min(p,3/2)$,

(47) $|\phi^{(2)}(t)|\leq c_{0}(1+|t|)^{-p^{\prime}},\quad t\in\mathbb{R}.$

Item (i) is therefore a direct consequence of Lemma 1. To see that
$u[\phi^{(2)}]$ also satisfies the radiation condition (ii), choose
$q\in(1,p^{\prime})$ and, for each $A>0$, let $\chi_{A}$ be the characteristic
function of the interval $[-A,A]$. Write
$u[\phi^{(2)}]=u[\chi_{A}\phi^{(2)}]+u[(1-\chi_{A})\phi^{(2)}]$. Now, by (47),

$|(1-\chi_{A}(t))\phi^{(2)}(t)|\leq c_{0}(1+|t|)^{-q}(1+A)^{q-p^{\prime}}$

so that, by (35) and (40) (applied with $p$ replaced by $q$), there exists a
constant $C(q)>0$ depending only on $q$ such that

$|u[(1-\chi_{A})\phi^{(2)}](\boldsymbol{x})|\leq
C(q)c_{0}(1+A)^{q-p^{\prime}}(1+|\boldsymbol{x}|)^{-1/2},\quad\boldsymbol{x}\in\mathbb{R}^{2}_{+},$

and from Corollary 2, for all $\varepsilon>0$, there exists a constant
$C(q,\varepsilon)>0$ depending only on $q$ and $\varepsilon$ such that

$|\nabla u[(1-\chi_{A})\phi^{(2)}](\boldsymbol{x})|\leq
C(q,\varepsilon)c_{0}(1+A)^{q-p^{\prime}}(1+|\boldsymbol{x}|)^{-1/2},\quad
x_{1}>\varepsilon,\,x_{2}\in\mathbb{R}.$

Let us set for any function $v$ defined on $\mathbb{R}^{2}_{+}$:

$M_{R,\varepsilon}(v):=R^{1/2}\sup\left\\{\left|\frac{\partial
v(\boldsymbol{x})}{\partial
r}-{\mathrm{i}}kv(\boldsymbol{x})\right|;x_{1}>\varepsilon\mbox{ and
}|\boldsymbol{x}|=R\right\\}.$

It follows from the estimates above that, for all $R>0$,

$M_{R,\varepsilon}(u[(1-\chi_{A})\phi^{(2)}])\leq\max(C(q),C(q,\varepsilon))c_{0}(1+A)^{q-p^{\prime}}.$

Thus, for all $\varepsilon>0$, given $\eta>0$ we can choose $A>0$ such that
$M_{R,\varepsilon}(u[(1-\chi_{A})\phi^{(2)}])\leq\eta/2$ for all $R>0$. But
also, for each $A>0$, it is standard that
$u[\chi_{A}\phi^{(2)}](\boldsymbol{x})$, a double-layer potential supported on
a bounded interval, satisfies the Sommerfeld radiation condition, in
particular that $M_{R,\varepsilon}(u[\chi_{A}\phi^{(2)}])\leq\eta/2$ for $R$
large enough. Thus, $M_{R,\varepsilon}(u[\phi^{(2)}])\leq\eta$ for all
sufficiently large $R$, which concludes the proof.

######  Corollary 4.

Suppose that $\phi=\phi_{1}+\phi_{2}$ where $\phi_{1}\in L^{2}(\mathbb{R})$ is
compactly supported and $\phi_{2}\in{C}(\mathbb{R})$. If $\phi$ satisfies (41)
and $u$ is defined by (34), then

  * (i)

there exist two constants $C>0$ and $R>0$ such that

(48) $|u(\boldsymbol{x})|\leq
C(1+|\boldsymbol{x}|)^{-1/2},\quad\boldsymbol{x}\in\mathbb{R}^{2}_{+},\;|\boldsymbol{x}|>R;$

  * (ii)

for all $\varepsilon>0$, (43) holds.

######  Proof 3.3.

Again, we write $u[\phi]=u[\phi_{1}]+u[\phi_{2}]$. The properties for
$u[\phi_{2}]$ have been established in the previous lemma, while $u[\phi_{1}]$
is a double-layer potential supported on a bounded interval, for which such
properties are standard, e.g. [11, Theorem 2.14, Lemma 2.5].

###  3.3 Proof of the uniqueness result

We are now able to prove the first main result of the paper.

######  Theorem 5.

Suppose $k>0$. There exists, for each $g\in H^{1/2}(\Gamma)$, a unique
solution $(\varphi^{1},...,\varphi^{\mathcal{N}})\in L^{2}_{\rm
loc}(\Sigma^{1})\times\ldots\times L^{2}_{\rm loc}(\Sigma^{\mathcal{N}})$
satisfying (30) such that, for all $j$, $\varphi^{j}$ satisfies the radiation
condition (32).

Existence was proven by the construction above of a solution of (30) from the
unique solution $u$ of (28,2), namely $\varphi^{j}:=u|_{\Sigma^{j}}$,
$j=1,...,{\mathcal{N}}$. Note in particular that the Sommerfeld radiation
condition (2) for $u$ implies the far-field asymptotics (3), which implies in
turn that each trace $\varphi^{j}$ satisfies the radiation condition (32).
Uniqueness follows from the following proposition since (28, 2) is well-posed.

######  Proposition 3.4.

Suppose $k>0$ and let $(\varphi^{1},...,\varphi^{\mathcal{N}})\in L^{2}_{\rm
loc}(\Sigma^{1})\times\ldots\times L^{2}_{\rm loc}(\Sigma^{\mathcal{N}})$
satisfying (30) (with $g\in H^{1/2}(\Gamma)$) be such that, for all $j$,
$\varphi^{j}$ satisfies the radiation condition (32). Then the compatibility
equations (22) hold for all $j\in\mathbb{Z}/\mathcal{N}\mathbb{Z}$, the
function defined by $u=U^{j}(\varphi^{j})$ in $\Omega^{j}$,
$j=1,...,{\mathcal{N}}$, is the unique solution $u\in H^{1}_{\rm loc}(\Omega)$
of (28, 2), and $\varphi^{j}=u|_{\Sigma^{j}}$ for each $j$.

######  Proof 3.5.

For each $j$, $\varphi^{j}$ can be decomposed as
$\varphi^{j}=\varphi^{j}_{1}+\varphi^{j}_{2}$ with
$\varphi^{j}_{1}=\varphi^{j}\chi_{A}^{j}$ and
$\varphi^{j}_{2}=\varphi^{j}(1-\chi_{A}^{j})$, $\chi_{A}^{j}$ being the
characteristic function of
$\\{(x_{1}^{j}=a^{j},x_{2}^{j}),x_{2}^{j}\in(-A,A)\\}$. Since $\varphi^{j}\in
L^{2}_{\rm loc}(\Sigma^{j})$ satisfies (32), we have that $\varphi^{j}_{1}\in
L^{2}(\Sigma^{j})$ with compact support and, for $A$ large enough,
$\varphi^{j}_{2}\in L^{\infty}(\Sigma^{j})$. From Lemma 1, and by the
definition (31) of the operator $D_{j,j\pm 1}$, since
$(\varphi^{1},...,\varphi^{\mathcal{N}})$ satisfies (30), we deduce, provided
we choose $A$ large enough, that $\varphi^{j}_{2}\in C(\Sigma^{j})$ for all
$j$.

We follow now the same steps as in the proof of Proposition 2.9.

The first step consists in proving equations (22). Again, we introduce
$v=U^{j}(\varphi^{j})-U^{j+1}(\varphi^{j+1})$ and one can show by using Lemma
1 as in the proof of Proposition 2.9 that $v\in
L^{2}_{\mathrm{loc}}(\Omega^{j}\cap\Omega^{j+1})\cap
C^{\infty}(\Omega^{j}\cap\Omega^{j+1})\cap
C(\overline{\Omega^{j}\cap\Omega^{j+1}}\setminus\\{S_{j}\\})$, satisfies
$\Delta v+k^{2}v=0$ in $\Omega^{j}\cap\Omega^{j+1}$, and vanishes on
$\partial(\Omega^{j}\cap\Omega^{j+1})\setminus\\{S_{j}\\}$. Moreover,
proceeding exactly as in the proof of Proposition 2.9, one can show that $v\in
H^{1}_{\mathrm{loc}}(\Omega^{j}\cap\Omega^{j+1})$. We would like, by applying
Corollary 4 to $U^{j}(\varphi^{j})$ and $U^{j+1}(\varphi^{j+1})$, to deduce
that $v$ also satisfies the Sommerfeld radiation condition in
$\Omega^{j}\cap\Omega^{j+1}$. The issue is that Corollary 4 does not ensure
that $v$ satisfies the Sommerfeld radiation condition up to the boundary. But
this difficulty can be overcome; since $v=0$ on
$\partial(\Omega^{j}\cap\Omega^{j+1})$, reflection arguments and standard
elliptic regularity results combined with the estimate (48) imply that, for
$R$ large enough, there exists a constant $C>0$ such that:

(49) $|v(\boldsymbol{x})|+|\nabla v(\boldsymbol{x})|\leq
C(1+|\boldsymbol{x}|)^{-1/2},\quad\boldsymbol{x}\in\Omega^{j}\cap\Omega^{j+1},|\boldsymbol{x}|>R.$

Then, denoting
$\Sigma^{j}_{R}:=\\{\boldsymbol{x}\in\Omega^{j}\cap\Omega^{j+1}:|\boldsymbol{x}|=R\\}$
and $\displaystyle
d^{j}(\boldsymbol{x}):=\min_{\boldsymbol{y}\in\partial(\Omega^{j}\cap\Omega^{j+1})}|\boldsymbol{x}-\boldsymbol{y}|$
, one has by (49) for all $R$ large enough,

$\int_{\boldsymbol{x}\in\Sigma^{j}_{R},d^{j}(\boldsymbol{x})<1}\left|\frac{\partial
v(\boldsymbol{x})}{\partial
r}-{\mathrm{i}}kv(\boldsymbol{x})\right|^{2}ds(\boldsymbol{x})\leq
C(1+R)^{-1},$

where the constant $C$ is independent of $R$. Combined with (43) with
$\varepsilon=1$ applied to $U^{j}(\varphi^{j})$ and $U^{j+1}(\varphi^{j+1})$,
this proves that

(50) $\lim_{R\to+\infty}\int_{\Sigma^{j}_{R}}\left|\frac{\partial
v(\boldsymbol{x})}{\partial
r}-{\mathrm{i}}kv(\boldsymbol{x})\right|^{2}ds(\boldsymbol{x})=0,$

i.e. $v$ satisfies the Sommerfeld radiation condition in a standard integrated
form. Since $v\in H^{1}_{\rm loc}(\Omega^{j}\cap\Omega^{j+1})$ satisfies
$\Delta v+k^{2}v=0$ in $\Omega^{j}\cap\Omega^{j+1}$, the Sommerfeld radiation
condition (50), and vanishes on
$\partial(\Omega^{j}\cap\Omega^{j+1})\setminus\\{S_{j}\\}$, one can conclude
that $v=0$ by uniqueness of such problems in conical domains [20].

The second step of the proof is simpler. Arguing as in the proof of
Proposition 2.9, it is enough to prove, in the case $g=0$, that the function
defined by $u=U^{j}(\varphi^{j})$ in $\Omega^{j}$, $j=1,...,{\mathcal{N}}$, is
equal to 0 everywhere, and to deduce from this that each $\varphi^{j}=0$.
Proceeding exactly as in that proof, one can show that $u\in
H^{1}_{\mathrm{loc}}(\Omega)\cap C(\overline{\Omega}\setminus P)$, where $P$
is the set of corners of $\partial\Omega$, that $u$ is a solution of (28) with
$g=0$, and that $\varphi^{j}=u|_{\Sigma^{j}}$ on $\Sigma^{j}\setminus P$.
Moreover, thanks to the overlaps between the halfspaces $\Omega^{j}$, one can
deduce from Corollary 4 applied to each $U^{j}(\varphi^{j})$ that $u$
satisfies the Sommerfeld condition (2). This implies that $u=0$ so that
$\varphi^{j}=0$ for all $j$,

Theorem 5 is a (new) well-posedness result for (30) in the case of real $k$.
And, indeed, numerical solution of (30) works well [6, 5] for real $k$.
However, there are still significant gaps in our theoretical understanding of
this formulation. In particular, while (23) for complex wavenumber $k$ can be
written formally in operator form satisfying a Fredholm property, in the case
when $k$ is real we know of no function space setting for which this
formulation makes sense, where the $D_{j,j+1}$ are well-defined bounded linear
operators.

##  4 General configurations

Let us now recall how to extend the HSM formulation to the general problem
presented in the introduction and extend the previous uniqueness result to
this new formulation. More precisely, for a real wavenumber $k>0$ we consider,
as discussed in §1.4, the isotropic Helmholtz equation (5) in a Lipschitz
domain $\Omega$, where $\rho-1\in L^{\infty}(\Omega)$ and $f\in L^{2}(\Omega)$
are compactly supported and $\Omega$ is $\mathbb{R}^{2}$ or $\mathbb{R}^{2}$
minus a set of Lipschitz obstacles which are supposed to lie in a bounded
domain. To ensure uniqueness we impose some constraint on $\rho$, e.g. that
$\rho$ is real-valued or that $\Im(\rho)\geq 0$. As announced in §1.4 the
problem we consider is to seek $u\in H^{1}_{\mathrm{loc}}(\Omega)$ that
satisfies (5), the Sommerfeld radiation condition (2), and a boundary
condition on $\Gamma=\partial\Omega$. To be specific we will impose the
homogeneous Neumann condition

(51) $\frac{\partial u}{\partial n}=0\quad\mbox{on}\quad\Gamma,$

but the modifications to use other boundary conditions and/or include
inhomogeneous terms are straightforward.

It is well known that this problem is uniquely solvable; in particular, with
one of the above constraints on $\rho$, uniqueness follows by a Green’s
identity, the Rellich lemma, and unique continuation arguments (e.g., [15,
Theorem 8.7]).

Let $\mathcal{O}$ be the interior of a convex polygon containing the supports
of all these perturbations. The half-space matching method has been mainly
presented for the case where $\mathcal{O}$ is a square (see [6]). But, whereas
in the problem in §3 the polygon ${\mathcal{O}}$ is given and the number of
trace unknowns is imposed by the number of sides of ${\mathcal{O}}$, here we
have freedom to choose the polygon $\mathcal{O}$. In particular, choosing
$\mathcal{O}$ to be a triangle has the advantage of minimising the number of
trace unknowns. In the sequel, we suppose that $\mathcal{O}$ is a polygon with
$\mathcal{N}$ edges and we adopt the same notations as introduced in §2 for
the edges $\Gamma^{1},\ldots,\Gamma^{\mathcal{N}}$, the angles between the
edges, the $\mathcal{N}$ overlapping half-planes
$\Omega^{1},\ldots\Omega^{\mathcal{N}}$, their boundaries
$\Sigma^{1},\ldots,\Sigma^{\mathcal{N}}$, and their associated local
coordinate systems.

Let us now introduce a bounded Lipschitz domain $\Omega_{b}\subset\Omega$
containing $\overline{\mathcal{O}}\cap\Omega$ such that
$\partial\Omega_{b}\setminus\Gamma$ is connected (for examples see Figure 4),
and a partition $\Gamma_{b}^{1},\ldots,\Gamma_{b}^{\mathcal{N}}$ of
$\partial\Omega_{b}\setminus\Gamma$ such that, for all
$j=1,...,{\mathcal{N}}$, $\Gamma_{b}^{j}$ is connected and
$\Gamma_{b}^{j}\subset\Omega^{j}$ with
$\overline{\Gamma_{b}^{j}}\cap\Sigma^{j}=\emptyset$. There is, of course, no
unique choice for such a partition, but the uniqueness result given in Theorem
1 below implies that the solution of the HSM formulation that we will write
down is independent of the choice of the partition.

$\Gamma$$\Omega_{b}$$\Gamma^{j}_{b}$$\Omega^{j}$$\Sigma^{j}$

$\Gamma$$\Omega_{b}$$\Gamma^{j}_{b}$$\Omega^{j}$$\Sigma^{j}$

Figure 4: Two examples of possible geometries for our general configuration:
$\mathcal{O}$ is the triangle bounded by the lines $\Sigma^{1}$, $\Sigma^{2}$,
and $\Sigma^{3}$, and $\Omega_{b}$, shaded in darker gray, is a larger
triangle in the left figure and a disc in the right figure.

The unknowns of the HSM formulation are the traces
$\varphi^{1},...,\varphi^{\mathcal{N}}$ of the solution $u$ on the infinite
lines $\Sigma^{1},...,\Sigma^{\mathcal{N}}$, and the restriction
$u_{b}:=u|_{\Omega^{b}}$ to the bounded domain $\Omega_{b}$. Let us derive the
equations linking $\varphi^{1},...,\varphi^{\mathcal{N}}$ and $u_{b}$. On the
one hand, one can show as in the case of scattering by a polygon (see §3.1)
that the $\varphi^{j}$’s satisfy the equations

(52)
$\begin{array}[]{c}\begin{array}[]{|ll}\varphi^{j}=D_{j-1,j}\,\varphi^{j-1},&\text{on}\;\Sigma^{j}\cap\Omega^{j-1},\\\
\varphi^{j}=D_{j+1,j}\,\varphi^{j+1},&\text{on}\;\Sigma^{j}\cap\Omega^{j+1},\end{array}\quad
j\in\mathbb{Z}/\mathcal{N}\mathbb{Z},\end{array}$

where the operators $D_{j,j\pm 1}$ are defined in (31). Compared to the
problem of scattering by a polygon, the conditions for the traces on
$\Gamma^{j}$ (linked to the given data $g$ in the case of scattering by a
polygon) have to be replaced by an equality between $\varphi^{j}$ and $u_{b}$
on $\Gamma^{j}$:

$\varphi^{j}=u_{b}\;\text{on}\;\Gamma^{j},\;j=1,...,{\mathcal{N}}.$

As in the case of scattering by a polygon, since the wavenumber is real, the
traces $\varphi^{j}$ are not in $L^{2}$ but they decay like
$|y_{2}^{j}|^{-1/2}$ at infinity and we will require, as in §3.1, that, for
$j=1,...,{\mathcal{N}}$, $\varphi^{j}$ satisfies the asymptotic condition (32)
at $\pm\infty$.

On the other hand, we can derive a variational formulation for $u_{b}$ in
$\Omega_{b}$. Since $-\Delta u_{b}-k^{2}\rho u_{b}=f$ in $\Omega_{b}$, $f$ is
supported in $\mathcal{O}^{\prime}:=\mathcal{O}\cap\Omega$, and $u$ satisfies
(51), the following Green’s identity holds for all $v_{b}\in
H^{1}(\Omega_{b})$, where $n$ is the normal pointing out of $\Omega_{b}$:

(53) $\int_{\Omega_{b}}\left(\nabla u_{b}\cdot\overline{\nabla
v_{b}}-k^{2}\rho
u_{b}\overline{v_{b}}\right)-\int_{\partial\Omega_{b}\setminus\Gamma}\frac{\partial
u_{b}}{\partial
n}\overline{v_{b}}\,=\int_{\mathcal{O}^{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\prime}}f\overline{v_{b}}.$

The last idea is to replace the normal derivative on each part
$\Gamma_{b}^{j}$ of $\partial\Omega_{b}\setminus\Gamma$ by an integral
representation as a function of $\varphi^{j}$. Indeed, since, for all
$j=1,...,{\mathcal{N}}$, $\Gamma_{b}^{j}\subset\Omega^{j}$, we must have

(54) $\frac{\partial u_{b}}{\partial n}-{\mathrm{i}}ku_{b}=\frac{\partial
U^{j}(\varphi^{j})}{\partial n}-{\mathrm{i}}kU^{j}(\varphi^{j})\mbox{ on
}\Gamma^{j}_{b},\;j=1,...,{\mathcal{N}},$

where $U^{j}(\varphi^{j})$ is the restriction of the solution $u$ to the half-
plane $\Omega^{j}$ and is expressed in terms of $\varphi^{j}$ in (11). Let us
emphasize that our choice of imposing equality of Robin traces instead of
normal derivatives is so that later we have uniqueness for the formulation for
all $k>0$. Note also that, since
$\overline{\Gamma_{b}^{j}}\cap{\Sigma^{j}}=\emptyset$, the Robin trace of
$U^{j}(\varphi^{j})$ on $\Gamma^{j}_{b}$ is ${C}^{\infty}$ (see Lemma 1). To
express (54) succinctly we introduce the Dirichlet-to-Robin (DtR) operators
$\Lambda^{j}$, given by

(55) $\Lambda^{j}\varphi:=\left(\frac{\partial U^{j}(\varphi)}{\partial
n}-{\mathrm{i}}kU^{j}(\varphi)\right)\Big{|}_{\Gamma^{j}_{b}}.$

We note, by Lemma 1, that $\Lambda^{j}\varphi$ is well-defined (at least) for
all functions $\varphi\in L^{2}(\Sigma^{j})+L^{\infty}(\Sigma^{j})$. One can
use (11) to write $\Lambda^{j}\varphi$ explicitly, for
$j=1,...,{\mathcal{N}}$, as

$\Lambda^{j}\varphi(\boldsymbol{x}^{j})=\int_{\mathbb{R}}\left(\nabla\mathcal{H}(k;x_{1}^{j}-a^{j},x_{2}^{j}-y_{2}^{j})\cdot
n(x_{1}^{j},x_{2}^{j})\right.\\\
\left.-{\mathrm{i}}k\mathcal{H}(k;x_{1}^{j}-a^{j},x_{2}^{j}-y_{2}^{j})\right)\,\varphi(a^{j},y_{2}^{j})\,\mathrm{d}y_{2}^{j},\quad{\boldsymbol{x}}^{j}=(x_{1}^{j},x_{2}^{j})\in\Gamma^{j}_{b}.$

With this notation, the equality (54) can be written as

(56) $\frac{\partial u_{b}}{\partial
n}-{\mathrm{i}}ku_{b}=\Lambda^{j}\varphi^{j}\mbox{ on
}\Gamma^{j}_{b},\;j=1,...,{\mathcal{N}}.$

Our complete formulation reads as follows:

(57)
$\begin{array}[]{c}\begin{array}[]{|lcl}\varphi^{j}=D_{j-1,j}\,\varphi^{j-1},&\text{on}\;\Sigma^{j}\cap\Omega^{j-1},\\\
\varphi^{j}=u_{b}\;\text{on}\;\Gamma^{j},\\\
\varphi^{j}=D_{j+1,j}\,\varphi^{j+1},&\text{on}\;\Sigma^{j}\cap\Omega^{j+1},\\\
\end{array}\quad j\in\mathbb{Z}/\mathcal{N}\mathbb{Z},\\\\[20.0pt] \forall
v_{b}\in H^{1}(\Omega_{b}),\hfill\\\
\displaystyle\int_{\Omega_{b}}\left(\nabla u_{b}\cdot\overline{\nabla
v_{b}}-k^{2}\rho
u_{b}\overline{v_{b}}\right)-{\mathrm{i}}k\sum_{j=0}^{3}\int_{\Gamma^{j}_{b}}u_{b}\overline{v_{b}}-\sum_{j=0}^{3}\int_{\Gamma^{j}_{b}}\Lambda^{j}\varphi^{j}\overline{v_{b}}=\int_{\mathcal{O}^{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\prime}}f\overline{v_{b}},\end{array}$

where the operators $D_{j,j\pm 1}$ are defined in (31), and the operators
$\Lambda^{j}$ in (55).

The following theorem gives the well-posedness of this formulation.

######  Theorem 1.

Suppose $k>0$. There exists, for each $f\in L^{2}({\mathcal{O}}^{\prime})$, a
unique solution $(\varphi^{1},...,\varphi^{\mathcal{N}},u_{b})\in L^{2}_{\rm
loc}(\Sigma^{1})\times\ldots\times L^{2}_{\rm loc}(\Sigma^{\mathcal{N}})\times
H^{1}(\Omega_{b})$ of (57) such that, for all $j=1,...,{\mathcal{N}}$,
$\varphi^{j}$ satisfies the radiation condition (32).

Existence holds by the construction above of a solution to (57) from the
unique solution of (5, 2, 51). Uniqueness follows from the following
proposition since (5, 2, 51) is well-posed.

######  Proposition 4.1.

Suppose $k>0$ and let $(\varphi^{1},...,\varphi^{\mathcal{N}},u_{b})\in
L^{2}_{\rm loc}(\Sigma^{1})\times\ldots\times L^{2}_{\rm
loc}(\Sigma^{\mathcal{N}})\times H^{1}(\Omega_{b})$ be a solution of (57)
(with $f\in L^{2}({\mathcal{O}}^{\prime})$) such that, for all
$j=1,...,{\mathcal{N}}$, $\varphi^{j}$ satisfies the radiation condition (32).
Then, for all $j\in\mathbb{Z}/\mathcal{N}\mathbb{Z}$, the compatibility
equations (22) hold and $u_{b}=U^{j}(\varphi^{j})$ in
$\Omega_{b}\cap\Omega^{j}$. Moreover, the function defined by
$u=U^{j}(\varphi^{j})$ in $\Omega^{j}$, for $j=1,...,{\mathcal{N}}$, and by
$u=u_{b}$ in $\Omega_{b}$ is the unique solution $u\in H^{1}_{\rm
loc}(\Omega)$ of (5, 2, 51). Further, $\varphi^{j}=u|_{\Sigma^{j}}$ for each
$j$.

######  Proof 4.2.

Let $(\varphi^{1},...,\varphi^{\mathcal{N}},u_{b})\in L^{2}_{\rm
loc}(\Sigma^{1})\times\ldots\times L^{2}_{\rm loc}(\Sigma^{\mathcal{N}})\times
H^{1}(\Omega_{b})$ be a solution of (57) such that, for all
$j=1,...,{\mathcal{N}}$, $\varphi^{j}$ satisfies the radiation condition (32),
and let us denote by $u_{\infty}\in H^{1}_{\rm
loc}(\mathbb{R}^{2}\setminus\overline{\mathcal{O}})$ the unique solution of
(28, 2) with $g=u_{b}|_{\partial\mathcal{O}}\in H^{1/2}(\partial\mathcal{O})$.
Then as shown in §3, the set of traces of $u_{\infty}$ on
$\Sigma^{1},\ldots,\Sigma^{\mathcal{N}}$ solves (30), i.e. the first three
equations of (57). It follows from Theorem 5 and from (57) that
$\varphi^{j}=u_{\infty}|_{\Sigma^{j}}$ and $u_{\infty}=U^{j}(\varphi^{j})$ in
$\Omega^{j}$, for $j=1,...,{\mathcal{N}}$. In particular, the compatibility
equations (22) hold and $\varphi^{j}=u_{\infty}=u_{b}$ on $\Gamma^{j}$ for
$j=1,...,{\mathcal{N}}$. Moreover, we deduce from the last equation of (57)
that $-\Delta u_{b}-k^{2}\rho\,u_{b}=f$ in $\Omega_{b}$, that

$\frac{\partial u_{b}}{\partial n}=0\quad\mbox{on}\quad\Gamma,$

and that (56) holds. By definition (55) of the DtR operators, we have then

(58) $\Lambda^{j}\varphi^{j}=\left(\frac{\partial u_{\infty}}{\partial
n}-{\mathrm{i}}ku_{\infty}\right)\Big{|}_{\Gamma^{j}_{b}},\quad
j=1,...,{\mathcal{N}}.$

Consequently, $v:=u_{b}-u_{\infty}$ belongs to
$H^{1}(\Omega_{b}\backslash\overline{\mathcal{O}})$ and satisfies

(59) $\begin{array}[]{|lcr}\Delta
v+k^{2}v=0&\text{in}&\Omega_{b}\backslash\overline{\mathcal{O}},\\\\[4.0pt]
v=0&\text{on}&\partial\mathcal{O},\\\\[4.0pt] \displaystyle\frac{\partial
v}{\partial
n}-{\mathrm{i}}kv=0&\text{on}&\partial\Omega_{b}\backslash\Gamma,\end{array}$

the last of these equations following from (56) and (58). But, for every
$k>0$, this homogeneous problem has no solution except $v=0$. (To see this
apply Green’s identity (cf. (53)) in
$\Omega_{b}\backslash\overline{\mathcal{O}}$ to deduce that
$\int_{\partial\Omega_{b}\setminus\Gamma}|v|^{2}=0$, so that $v=\partial
v/\partial n=0$ on $\partial\Omega_{b}\setminus\Gamma$, and apply Holmgren’s
uniqueness theorem [11, p. 104].) Thus $u_{b}=u_{\infty}$ in
$\Omega_{b}\backslash\overline{\mathcal{O}}$ (in particular
$u_{b}=u_{\infty}=U^{j}(\varphi^{j})$ in $\Omega_{b}\cap\Omega^{j}$,
$j=1,...,{\mathcal{N}}$) so that the function

$w:=\left\\{\begin{array}[]{lcr}u_{b}&\text{in}&\Omega_{b},\\\\[4.0pt]
u_{\infty}&\text{in}&\mathbb{R}^{2}\backslash\overline{\mathcal{O}},\end{array}\right.$

is well-defined, and is the unique solution $u\in H^{1}_{\rm loc}(\Omega)$ of
(5, 2, 51).

##  5 Conclusion

The objective of this paper was to prove the well-posedness of the HSM
formulation applied to the Helmholtz equation (5) in the case of a real
wavenumber $k>0$. This objective has been achieved: we have proved well-
posedness provided we impose the radiation condition (32) at infinity on the
trace-unknowns $\varphi^{j}$ of the HSM equations, this radiation condition
analogous to the standard Sommerfeld radiation condition for the original
boundary value problem. Let us recall that the results of the present paper
also complete a proof of well-posedness of the complex-scaled HSM formulation
presented in [4, §5].

An open question is whether the radiation condition (32) on the trace-
unknowns, while natural, is necessary for uniqueness. The answer is not at all
clear to us. But one piece of evidence that suggests that this radiation
condition may not in fact be needed is that one achieves accurate results in
numerical experiments by simply truncating the trace unknowns, setting, for
each $j$, $\varphi^{j}=0$ outside some sufficiently large finite section of
$\Sigma^{j}$, not making any use of the radiation condition (32) (see [6, Fig.
7]).

Let us finish by mentioning that our main results should be extendable to more
complex configurations with obstacles extending to infinity. For instance, one
might consider the general configuration of §4 but add a screen that is a
semi-infinite line $\gamma^{1}$, choosing the half-planes $\Omega^{j}$ so that
$\gamma^{1}$ is orthogonal to $\Sigma^{1}$ and
$\gamma^{1}\cap\Omega^{j}=\emptyset$, for all $j\neq 1$ (see the left hand
part of Figure 5). In this case $\Omega^{1}\setminus\gamma^{1}$ has two
connected components that are quarter-planes. If $\gamma^{1}$ is sound soft or
sound hard, i.e. the solution satisfies a Dirichlet or Neumann homogeneous
boundary condition on $\gamma^{1}$, one can derive an expression for
$U^{1}(\varphi^{1})$, the solution in $\Omega^{1}$ given Dirichlet data
$\varphi^{1}$ on $\Sigma^{1}$, by solving separately in each quarter-plane,
combining formula (13) with reflection arguments.

More interesting is the case where two parallel semi-infinite lines
$\gamma^{j}_{\pm}$ are introduced, so that now, for some $j$, $\Omega^{j}$ has
three connected components, two of them quarter-planes and one of them a semi-
infinite strip which is a waveguide (this is the case for $\Omega^{1}$ and
$\Omega^{3}$ in the right hand part of Figure 5). One can derive an expression
for $U^{j}(\varphi^{j})$ in each quarter-plane as above, and an expression in
the waveguide by solving as a modal series expansion.

For each of these configurations, and for many other variations on these
geometries, we can write down HSM formulations, and we expect that one should
be able to prove well-posedness of the HSM formulation and equivalence with
other formulations by adapting the results proved in the present paper.

$\Sigma^{1}$$\gamma^{1}$

$\Sigma^{1}$$\Sigma^{3}$$\gamma^{1}_{+}$$\gamma^{1}_{-}$$\gamma^{3}_{-}$$\gamma^{3}_{+}$

Figure 5: Two geometries that could be considered.

Note that if only one semi-infinite line is introduced, as in the left hand
side of Figure 5, and no other obstacles or inhomogeneities are present, one
recovers the famous Sommerfeld half-plane problem [28, §38] that can be solved
using the Wiener-Hopf method (or other techniques). Analytical methods,
notably the Wiener-Hopf method, have been extended to a variety of more
complex configurations with waveguides and wedges, e.g., [25, 22, 21], though
not to all configurations that should be treatable by the extensions of the
HSM method to unbounded obstacles that we suggest above.

An interesting question is whether our uniqueness and well-posedness results
for the HSM formulation can be extended to open waveguides. (The HSM method
has been used to compute numerical solutions by Ott [26] in such cases where
the medium in each half-space $\Omega^{j}$ is stratified in the $x_{2}^{j}$
direction.) The difficulty is that the corresponding Green’s function is no
longer available in closed form, which makes the study of the properties of
the half-plane solution more intricate. A further difference with the cases
studied in this paper, and the extensions suggested above, is that, due to the
possible existence of guided waves in the open waveguides, the traces
$\varphi^{j}$ may have an oscillatory non-decaying behavior at infinity that
should be taken into account in a modified radiation condition, replacing
(32).

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