X Claeys

R Hiptmair

Integral Equations for Acoustic Scattering by Partially Impenetrable Composite Objects

Keywords: Mathematics Subject Classification (2010). Primary 45A05; Secondary 65R20 integral equation, scattering, wave propagation, Helmholtz, junction points

We study direct first-kind boundary integral equations arising from transmission problems for the Helmholtz equation with piecewise constant coefficients and Dirichlet boundary conditions imposed on a closed surface. We identify necessary and sufficient conditions for the occurrence of so-called spurious resonances, that is, the failure of the boundary integral equations to possess unique solutions.

Following rA. Buffa and R. Hiptmair, Regularized combined field integral equations, Numer. Math., 100 (2005), pp. 1-19s we propose a modified version of the boundary integral equations that is immune to spurious resonances. Via a gap construction it will serve as the basis for a universally well-posed stabilized global multi-trace formulation that generalizes the method of rX. Claeys and R. Hiptmair, Multi-trace boundary integral formulation for acoustic scattering by composite structures, Communications on Pure and Applied Mathematics, 66 (2013), pp. 1163-1201s to situations with Dirichlet boundary conditions.

Introduction

We are concerned with boundary integral equations (BIE) describing the propagation of acoustic waves in so-called composite media composed of parts with linear and spatially homogenous material properties. Such media are rather common in mathematical models in engineering and well-posed BIE are important as foundation for boundary element methods (BEM), a well established and widely used technique for computational acoustics.

The bulk of mathematical investigations on BIE has addressed the case of only two different homogeneous media, with one occupying a bounded volume in space, see, for instance, [START_REF] Costabel | A direct boundary integral equation method for transmission problems[END_REF], [START_REF] Mclean | Strongly Elliptic Systems and Boundary Integral Equations[END_REF]Ch. 9], [START_REF] Sauter | Boundary Element Methods[END_REF]Sect. 3.9], and the monographs [START_REF] Nédélec | Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems[END_REF][START_REF] Colton | Inverse Acoustic and Electromagnetic Scattering Theory[END_REF]. Apparently, the first profound mathematical derivation and analysis of particular direct BIEs for acoustics with composite media was given in [START_REF] Petersdorff | Boundary integral equations for mixed Dirichlet, Neumann and transmission problems[END_REF]. Of course, boundary element methods for composite scattering had been devised before in computational engineering, notably the so-called Poggio-Miller-Chew-Harrington-Wu-Tsai (PMCHWT) integral equations [START_REF] Poggio | Integral equation solution of three-dimensional scattering problems[END_REF][START_REF] Chang | A surface formulation or characteristic modes of material bodies[END_REF][START_REF] Wu | Scattering from arbitrarily-shaped lossy dielectric bodies of revolution[END_REF][START_REF] Harrington | Boundary integral formulations for homogeneous material bodies[END_REF] for electromagnetic scattering.

The BIEs proposed in [START_REF] Petersdorff | Boundary integral equations for mixed Dirichlet, Neumann and transmission problems[END_REF] may be dubbed a single trace formulation (STF), because they involve a single pair of Cauchy data on each interface as unknowns. They can legitimately be regarded as the standard direct BIEs for transmission problems, because they immediately arise from fundamental Calderón identities and the transmission conditions are imposed strongly through the trial trace spaces. If all participating media are penetrable, the BIEs of STF are well-posed in natural trace spaces, see [START_REF] Claeys | Multi-trace boundary integral equations[END_REF]Sect. 3.2], [START_REF] Claeys | Multi-trace boundary integral formulation for acoustic scattering by composite structures[END_REF]Prop. A.1]. However, if impenetrable media are admitted, the standard STF may be affected by the notorious spurious resonance phenomenon, that is, for particular combinations of wave numbers the BIE may fail to possess unique solutions. This has not been properly addressed in [START_REF] Petersdorff | Boundary integral equations for mixed Dirichlet, Neumann and transmission problems[END_REF] and in Section 4 we provide a detailed analysis of when the STF becomes vulnerable to spurious resonances. In short, spurious resonances can occur, if an impenetrable part is completely surrounded by another homogeneous medium, see Theorem 4.8.

To restore unconditional well-posedness of the STF, we adapt the classical idea of combined field integral equations (CFIE), both in its indirect and direct version, cf. [START_REF] Brakhage | Ueber das Dirichletsche Außenraumproblem für die Helmholtzsche Schwingungsgleichung[END_REF][START_REF] Leis | Zur Dirichletschen Randwertaufgabe des Aussenraumes der Schwingungsgleichung[END_REF][START_REF] Panich | On the question of the solvability of the exterior boundary-value problems for the wave equation and Maxwell's equations[END_REF] for the former, and [START_REF] Burton | The application of integral methods for the numerical solution of boundary value problems[END_REF] for the latter. Sloppily speaking, CFIEs exploit the capacity of (approximate) absorbing boundary conditions to ensure unique solvability of time-harmonic wave propagation problems even on bounded domains, whereas a discrete set of resonant frequencies will always haunt pure Dirichlet or Neumann boundary conditions. The simplest choice of approximate absorbing boundary conditions is plain impedance or Robin boundary conditions with non-zero purely imaginary impedance, see [START_REF] Sauter | Boundary Element Methods[END_REF]Sect. 3.4.9]. Yet, in this work, we rely on regularized or modified versions of CFIEs from [START_REF]Regularized combined field integral equations[END_REF][START_REF] Steinbach | Stable boundary element domain decomposition methods for the Helmholtz equation[END_REF], which are compatible with variational formulations in natural trace spaces. The corresponding extensions of the single trace boundary integral equations are studied in Section 5.

Another drawback of the classical STF-BIEs, when used as the foundation for low-order Galerkin boundary element discretization, is their failure to be amenable to the powerful and popular Calderón preconditioning techniques [START_REF] Hiptmair | Operator preconditioning[END_REF][START_REF] Steinbach | The construction of some efficient preconditioners in the boundary element method[END_REF][START_REF] Christiansen | Des préconditionneurs pour la résolution numérique des équations intégrales de frontiére de l'acoustique[END_REF]. For lucid explanations refer to [START_REF] Claeys | Multi-trace boundary integral equations[END_REF]Sect. 4]. Lately, this shortcoming of the STF has prompted the development of so-called multitrace formulations (MTF) for scattering at composite objects. They feature four unknown traces at (some) material interfaces and come in two flavors: global MTFs as introduced in [START_REF] Claeys | A single trace integral formulation of the second kind for acoustic scattering[END_REF][START_REF] Claeys | Multi-trace boundary integral formulation for acoustic scattering by composite structures[END_REF][START_REF] Claeys | Multi-trace boundary integral equations[END_REF][START_REF] Claeys | Electromagnetic scattering at composite objects : a novel multi-trace boundary integral formulation[END_REF] and [START_REF] Claeys | Multi-trace boundary integral equations[END_REF]Sect. 5], and local MTFs presented in [START_REF] Hiptmair | Multiple traces boundary integral formulation for Helmholtz transmission problems[END_REF][START_REF] Hiptmair | Domain decomposition for boundary integral equations via local multi-trace formulations[END_REF]. They all have in common that they enforce the transmission conditions only weakly, in contrast to the STF. Thus, trial and test spaces can neatly be split into contributions of different sub-domains, and, in the spirit of domain decomposition, this paves the way for local preconditioning.

Thus far, all mathematical analyses of MTFs eschew non-penetrable media, except for [START_REF] Claeys | Novel multi-trace boundary integral equations for transmission boundary value problems[END_REF], which is confined to pure diffusion problems. Only in computational engineering some recent variants of local MTF for computational electromagnetics [START_REF] Peng | Computations of electromagnetic wave scattering from penetrable composite targets using a surface integral equation method with multiple traces[END_REF][START_REF] Peng | Integral equation based domain decomposition method for solving electromagnetic wave scattering from nonpenetrable objects[END_REF] include CFIE ideas in order to treat impenetrable, that is, perfectly electrically conducting, bodies. In this article, in Section 5, we propose a CFIE-type extension of the global MTF introduced in [START_REF] Claeys | Multi-trace boundary integral formulation for acoustic scattering by composite structures[END_REF]. It naturally emerges from single trace CFIEs appealing to the "gap idea" described in [START_REF] Claeys | Multi-trace boundary integral formulation for acoustic scattering by composite structures[END_REF]Sect. 5] and [START_REF] Claeys | Multi-trace boundary integral equations[END_REF]Sect. 5.2]. The new global multi-trace CFIEs inherit unconditional stability and turn out to be a compact perturbation of the previously known global MTF. Thus, the customary Calderón preconditioning technique [START_REF] Claeys | Multi-trace boundary integral equations[END_REF]Sect. 4] can be applied to them.

Discretization, for instance, by Galerkin boundary element methods, will not be addressed in this article. However, coercivity of variational formulations in spaces of Cauchy traces together with uniqueness of solutions, immediately allows to conclude quasi-optimality of conforming Galerkin BEM, see [START_REF] Demkowicz | Asymptotic convergence in finite and boundary element methods: Part 1, Theoretical results[END_REF], [START_REF] Wendland | Boundary element methods for elliptic problems[END_REF], and [START_REF] Sauter | Boundary Element Methods[END_REF]Sect 4.2.3]. Hence, our theory paves the way for predicting the convergence of all varieties of Galerkin BEM for both single-and multi-trace CFIE provided that the smoothness of Cauchy traces of the exact field solution is known. 2 Setting of the problem

List of notations

In the present article, we consider a partition R d n j0 Ω j Ω Σ where Ω Σ and the Ω j for j $ 0 are open, bounded, and mutually disjoint, and each Ω Σ , Ω j is a Lipschitz domain [START_REF] Mclean | Strongly Elliptic Systems and Boundary Integral Equations[END_REF]Def. 3.28].

In addition, we assume that Ω Σ , R d zΩ Σ , and each Ω j are connected. An important consequence of these assumptions is that Ω Σ does not contain any hole which rules out the presence of an internal resonant cavity. We set Γ : n j0 fΩ j (the "skeleton") and Σ : fΩ Σ .

(2.1)

As in Figure 1 there may exist points where three or more sub-domains abut, which is precisely the situation that we wish to tackle. We consider the following transmission problem for the Helmholtz equation: Find U

H 1 loc pR d zΩ Σ q 1 such that 4 ¡∆U ¡ κ 2 j U 0 in Ω j U ¡ U inc is κ 0 -outgoing in Ω 0 (2.2a) 4 U | fΩj ¡ U | fΩ k 0 f nj U | fΩj ¡ f n k U | fΩ k 0 on fΩ j fΩ k (2.2b) t U | Σ 0 . (2.2c)
In equation (2.2a), the outgoing condition refers to Sommerfeld's radiation condition, i.e. if ω R d is any bounded open subset, we shall say that

V H 1 loc p∆, R d zωq is κ-outgoing if lim ρÑV » fBρ |f r V ¡ iκV | 2 dσ ρ 0
where B ρ is the ball of center 0 and radius ρ, dσ ρ is the surface measure on fB ρ , and f r refers to the radial derivative. Sommerfeld's radiation is presented in detail for example in [27, §.2.6.5] or [25, §.4.4]. For the sake of simplicity and clarity, we asume that all wave numbers are positive κ j ¡ 0 , j 0, . . . , n .

(2. As it involves transmission conditions, and since we will be interested in the derivation of boundary integral equations adapted to this problem, we need to introduce suitable trace operators. According to [32, Thm. 2.6.8 and Thm. 2.7.7], for every subdomain Ω j , j 0 . . . n, there exist continuous trace operators γ j d : H 1 loc pΩ j q Ñ H 1{2 pfΩ j q and γ j n : H 1 loc p∆, Ω j q Ñ H ¡1{2 pfΩ j q (so-called Dirichlet and Neumann traces) by density defined through γ j d pϕq : ϕ| fΩj and γ j n pϕq : n j ¤ ∇ϕ| fΩj dϕ C V pΩ j q . (2.4)

We use similar notations for traces on Σ with n Σ fixing the orientation of the Neumann trace, see Figure 1. Both traces can be merged into the interior Cauchy trace operators γ j pvq :

γ j d pvq γ j n pvq & dv H 1 loc p∆, Ω j q .
(2.5)

Traces from the exterior of Ω j spawn the exterior Cauchy trace operators γ j c : H 1 loc p∆, R d zΩ j q Ñ H 1{2 pfΩ j q¢H ¡1{2 pfΩ j q, whose Neumann trace is still based on the normal n j .

Remark 2.1. Forgoing generality in favor of clarity and brevity, we focus on the rather simple problem (2.2) as typical specimen of transmission problem describing acoustic scattering. Straightforward extensions of the approach in this article can cope with the following situations: several impenetrable subdomains (not just one), Neumann (instead of Dirichlet) boundary conditions imposed on Σ, wave-numbers κ j with non-vanishing imaginary part, piecewise constant coefficients in the second-order part of the differential operator as in [START_REF] Claeys | Multi-trace boundary integral equations[END_REF], more general source terms (for example, general inhomogeneous transmission and boundary conditions).

These points would entail only minor adjustments in our analysis. We refer the reader to [START_REF] Claeys | Multi-trace boundary integral formulation for acoustic scattering by composite structures[END_REF][START_REF] Claeys | Multi-trace boundary integral equations[END_REF] for more details on how to deal with more complex situations. In [START_REF] Claeys | Multi-trace boundary integral equations[END_REF] electromagnetic scattering problems are treated alongside their acoustic counterparts in a unified setting. Following this policy and the CFIE ideas of [START_REF] Buffa | A coercive combined field integral equation for electromagnetic scattering[END_REF], the considerations of this article could also be generalized to electromagnetic wave propagation.

Trace spaces

We want to recast Problem (2.2) into variational boundary integral equations, so that these are immune to spurious resonances. We aim for BIE set in natural trace spaces. The most fundamental trace space we can introduce consist is the multi-trace space [11, Sect. 2.1], the Cartesian product of local traces:

HpΓq : HpfΩ 0 q ¢ ¤ ¤ ¤ ¢ HpfΩ n q where HpfΩ j q : H 1 2 pfΩ j q ¢ H ¡ 1 2 pfΩ j q .

(3.1)

We endow each HpfΩ j q with the norm given by }pv, qq} HpfΩj q : p}v} 2 H 1{2 pfΩjq }q} 2 H ¡1{2 pfΩjq q 1{2 , and equip HpΓq with the norm naturally associated with the cartesian product }u} HpΓq :

¡ }u 0 } 2 HpfΩ0q ¤ ¤ ¤ }u n } 2 HpfΩnq © 1{2
for u pu 0 , . . . , u n q HpΓq 2 . We write ¤, ¤ j for the duality pairing between H 1 2 pfΩ j q and H ¡ 1 2 pfΩ j q. We also need a bilinear duality pairing for HpfΩ j q

and HpΓq; we opt for the skew-symmetric version ru, vs :

n j0 ru j , v j s j where ¢ u j p j , ¢ v j q j & j : u j , q j j ¡ v j , p j j . (3.2)
This particular choice of a duality pairing is well adapted to the forthcoming analysis. Note that under the duality pairing r , s, the space HpΓq is its own topological dual, and it is easy to show, using the duality between H 1{2 pfΩ j q and H ¡1{2 pfΩ j q, that the pairing r , s induces an isometric isomorphism be- (3.3)

2 Functions in Dirichlet trace spaces like H 1 2 pfΩ j q will be denoted by u, v, w, whereas we use p, q, r for Neumann traces. Small fraktur font symbols u, v, w are reseved for Cauchy traces, with an integer subscript indicating restriction to a subdomain boundary. Capital letters will be used to designate scalar functions on domains, whereas small bold letters will be used for vector fields.

We also write HpΣq : H 1{2 pΣq ¢ H ¡1{2 pΣq and equip this space of Cauchy traces with the norm }pv, qq} 2 HpΣq : }v} 2 H 1{2 pΣq }q} 2 H ¡1{2 pΣq . Analogous to (3.2), on this space we shall consider the following skew-symmetric duality pairing

¢ u p ¢ v q & Σ : u, q Σ ¡ v, p Σ .
(3.4)

Single-trace spaces

Next, as in [11, Sect. 2.2], [START_REF] Claeys | Multi-trace boundary integral equations[END_REF]Sect. 3.1], we introduce subspaces of traces that respect the transmission conditions (2.2b) across interfaces. We first focus on traces of Dirichlet/Neumann type introducing

X 1 2 pΓq : 2 pu j q n j0 Π n j0 H 1 2 pfΩ j q § § hV H 1 pR d q s.t. V | fΩj u j dj @ , X ¡ 1 2 pΓq : 2 pp j q n j0 Π n j0 H ¡ 1 2 pfΩ j q § § hq Hpdiv, R d q s.t. n j ¤ q| fΩj p j dj @ . (3.5)
The Cartesian product (up to some permutation of indices) X 1{2 pΓq¢X ¡1{2 pΓq yields the single-trace space XpΓq HpΓq defined by XpΓq :

3 u ¢ u j p j n j0 § § § pu j q n j0 X 1 2 pΓq, pp j q n j0 X ¡ 1 2 pΓq A . (3.6)
Observe that a function U H 1 p∆, Ω 0 q ¢ ¤ ¤ ¤ ¢ H 1 p∆, Ω n q satisfies the transmission conditions (2.2b), if and only if pγ j pUqq n j0 XpΓq. In particular, if U H 1 p∆, R d zΩ Σ q then pγ j pUqq n j0 XpΓq. Indeed, from an intuitive point of view, the space XpΓq can be viewed as the space of traces of functions that satisfy the transmission conditions (2.2b). Thus, in the sequel, we will use this space to enforce transmission conditions.

Since every x Σ also belongs to some fΩ j , j 0, . . . , n, functions in X ¨1{2 pΓq can be expected to induce traces in H ¨1{2 pΣq. This is made precise in the following proposition. Proposition 3.1. For every element pu j q n j0 X 1{2 pΓq, there exists a unique u Σ H 1{2 pΣq such that V | Σ u Σ for any V H 1 pR d q that satisfies V | fΩj u j , j 0 . . . n. Moreover the linear operator T d : XpΓq Ñ H 1{2 pΣq defined by T d p pu j , p j q n j0 q : u Σ is continuous and surjective. Similarly, for every element pp j q n j0 X ¡1{2 pΓq, there exists a unique p Σ H ¡1{2 pΣq such that n Σ ¤ p| Σ p Σ for any p Hpdiv, R d q that satisfies n j ¤ p| fΩj p j , j 0 . . . n . Moreover the linear mapping T n : XpΓq Ñ H ¡1{2 pΣq defined by T n p pu j , p j q n j0 q : p Σ is continuous and surjective.

Proof: We prove only the first part of the proposition, as the proof of the second part follows along the same lines. Assume that u Σ H ¡1{2 pΣq satisfies V | Σ u Σ for one particular V H 1 pR d q such that V | fΩj u j , dj 0 . . . n. If V I H 1 pR d q also satisfies V | fΩj u j , j 0 . . . n, then V and V I coincide on Σ since Σ n j0 fΩ j . Hence u Σ V I | Σ . This proves the uniqueness of

u Σ .
Let us construct the map T d explicitely. First, for every subdomain Ω j we consider a continuous lifting operator E j : H 1{2 pfΩ j q Ñ H 1 pΩ j q satisfying γ j

d ¤ E j pv j q v j . Then define E : X 1{2 pΓq Ñ L 2 pR d zΩ Σ q by combining the E j according to Ep pu j q n j0 q| Ωj : E j pu j q, j 0 . . . n. Actually EpX 1{2 pΓqq H 1 pR d zΩ Σ q. Indeed, note that γ k d ¤ Ep pu j q n j0 q u k for all k 0 . . . n and for any choice of the u j 's. Choose u : pu j q n j0 arbitrarily in X 1{2 pΓq. There exists V H 1 pR d q such that γ j d pV q u j γ j d pEpuqq, which implies γ j d pV ¡ Epuqq 0. From this we conclude Epuq ¡ V H 1 pR d zΩ Σ q and finally Epuq V H 1 pR d zΩ Σ q H 1 pR d zΩ Σ q. Now consider any continuous extension operator r

E : H 1 pR d zΩ Σ q Ñ H 1 pR d q such that r
EpV q| R d zΩΣ V . Whenever u pu j , p j q n j0 belongs to XpΓq, we have in particular pu j q n j0 X 1{2 pΓq, so we can define

T d pUq : γ Σ d ¥ r E ¥ E ¨ pu j q n j0
¨for any u pu j , p j q n j0 XpΓq .

With this definition, T d is clearly continuous. In addition, it fulfills the other requirements: setting V r E ¥ E pu j q n j0 ¨we have V H 1 pR d q and V | fΩj u j , j 0 . . . n, by construction. In particular, this implies that u Σ V | Σ T d pUq. l

The following elementary result generalizes [11, Eq. (2.

2)] and [12, Theorem 3.1] and it will be crucial for many manipulations.

Proposition 3.2. Define the continuous operator T : XpΓq Ñ HpΣq by the formula Tpuq pT d puq, T n puqq. Then we have ru, vs ¡ rTpuq, Tpvqs Σ du, v XpΓq .

Proof: According to the explicit expression of r , s and r , s Σ given by (3.2) and (3.4), it suffices to show that, whenever u pu j , p j q n j0 XpΓq and v pv j , q j q n j0 XpΓq, we have n j0 u j , q j j ¡ T d puq, T n pvq Σ and n j0 v j , p j j ¡ T n puq, T d pvq Σ .

We will prove only the first identity above, as the second can be shown in exactly the same manner, exchanging the roles of u and v. First of all note that pu j q n j0 X 1{2 pΓq since u XpΓq, and pq j q n j0 X ¡1{2 pΓq since v XpΓq. In addition, according to Proposition 3.1, there exist G H 1 pR d q and h Hpdiv, R d q such that G| fΩj u j , G| Σ T d puq and n j ¤ h| fΩj q j , n Σ ¤ h| Σ T n pvq .

As a consequence, applying Green's formula in each Ω j , Ω Σ and then in R d , we obtain

T d puq, T n pvq Σ n j0 u j , q j j » ΩΣ divphq G h ¤ ∇G dx n j0 » Ωj divphq G h ¤ ∇G dx » R d divphq G h ¤ ∇G dx 0 . l 3.

Review of potential operators

In this paragraph we recapitulate well-known results concerning the integral representation of solutions of the homogeneous Helmholtz equation in Lipschitz domains. Detailed proofs can be found, for example, in [START_REF] Sauter | Boundary Element Methods[END_REF]Chap.3].

Let the function G κ pxq designate the κ-outgoing fundamental solution for the Helmholtz operator ¡∆ ¡ κ 2 . For each subdomain Ω j , for any u pu, pq HpfΩ j q and any x R d zfΩ j , define the single/double layer potential operators by3 (3.7)

The operator G j κ defined above maps continuously HpfΩ j q into H 1 loc p∆, Ω j q¢ [START_REF] Sauter | Boundary Element Methods[END_REF]Thm 3.1.16]. In particular G j κ can be applied to a pair of traces, i.e. Cauchy traces, of the form u γ j pV q. This potential operator can be used to write a representation formula for solutions of the homogeneous Helmholtz equation, see [START_REF] Sauter | Boundary Element Methods[END_REF]Thm 3.1.6].

H 1 loc p∆, R d zΩ j q, see
Proposition 3.3. Let U H 1 loc pΩ j q satisfy ∆U κ 2 j U 0 in Ω j . In addition, assume that U is κ j -outgoing, if j 0. Then we have the representation formula

G j κj pγ j pUqqpxq 5 U pxq for x Ω j , 0 for x R d zΩ j . Similarly, if V H 1 loc pR d zΩ j q satisfies ∆V κ 2 j V 0 in R d zΩ j
, as well as a radiation condition in the case j $ 0, then we have G j κj pγ j pV qqpxq ¡V pxq for x R d zΩ j , and G j κj pγ j pV qqpxq 0 for x Ω j .

The potential operator G j κ also satisfies a remarkable identity, known as jump relations, describing the relationship of traces of G j κj puq from both sides of fΩ j . Using the jump operator for Cauchy traces rγ j s : γ j ¡ γ j c , they can concisely be expressed as rγ j s ¤ G j κj pu j q u j du j HpfΩ j q , j 0, . . . , n .

(3.8)

We refer the reader to [START_REF] Sauter | Boundary Element Methods[END_REF]Thm.3.3.1] (the jump formulas are often given in the form of four equations in literature). Proposition 3.3 shows that, if U is solution to a homogeneous Helmholtz equation in Ω j (and is κ j -outgoing, if j 0) then γ j ¥ G j κj ¨pγ j pUqq γ j pUq. C κ pfΩ j q : tγ j pUq | U H 1 loc pΩ j q and ∆U κ 2 U 0 in Ω j , U κ j -outgoing, if j 0 u . Then γ j ¥G j κ : HpfΩ j q Ñ HpfΩ j q is a continuous projector, called the interior Calderón projector of Ω j , whose range coincides with C κ pfΩ j q, i.e. for any u j HpfΩ j q γ j ¤ G j κ pu j q u j ðñ u j C κ pfΩ j q .

For a detailed proof of this proposition, see [START_REF] Sauter | Boundary Element Methods[END_REF]Prop. 3.6.2]. This characterization of Cauchy traces of (outgoing) Helmholtz solutions is instrumental for deriving direct boundary integral equations for the subdomains Ω j . The next lemma gives another caracterization of the space of Cauchy data, which was established in [START_REF] Claeys | Multi-trace boundary integral formulation for acoustic scattering by composite structures[END_REF]Lemma 6.2].

Lemma 3.5. Consider any j 0, . . . n, and any κ ¡ 0. Then for any u j HpfΩ j q we have u j C κ pfΩ j q ðñ ru j , v j s j 0 dv j C κ pfΩ j q .

(3.9)

Applying traces to potentials yields boundary integral operators. In our compact notation, the crucial local boundary integral operators are

A j κj : tγ j u ¥ G j κj : 1 2 pγ j γ j c q ¥ G j κj ¢ ¡ K j V j W j K I j , j 0, . . . , n . (3.10)
We adopted the notations of [START_REF] Sauter | Boundary Element Methods[END_REF]Sect. 3.1] for the atomic boundary integral operators, the double layer operators K j , the single layer operators V j , the adjoint double layer operators K I j , and the hypersingular boundary integral operators W j .

The operators A j κj satisfy an intriguing symmetry property, which seems to be well known in literature, see for example [4, Thm 3.9] (that concerns the Maxwell case, though). Since, apparently, the proof for acoustic waves is not published, we give it for the sake of completeness. Lemma 3.6. For any j 0, . . . , n, and any wave number κ j we have,

A j κj pu j q, v j % j A j κj pv j q, u j % j du j , v j HpfΩ j q .
Proof: This result is just a consequence of the jump formulas (3.8), as well as of Lemma 3.5 applied repeatedly in Ω j and R d zΩ j :

A j κj u j , v j % j (3.10) tγ j u G j κj u j , v j % j (3.8) tγ j u G j κj u j , rγ j s G j κj v j % j (3.9) ¡ γ j G j κj u j , γ j c G j κj v j % j γ j c G j κj u j , γ j G j κj v j % j (3.9) ¡ rγ j s G j κj u j , tγ j u G j κj v j % j (3.8) tγ j u G j κj v j , u j % j (3.10) A j κj v j , u j % j .
l

Another symmetry of potentials and their traces applies to the coupling between different subdomains:

Lemma 3.7. Take two arbitrary subdomains Ω j , Ω k with j $ k, any wave number κ 0 . We have

γ j G k κ0 pv k q, v j % j γ k G j κ0 pv j q, v k $ k dv j HpfΩ j q, dv k HpfΩ k q .
Proof: First of all, applying Lemma in 3.5 in Ω j yields

γ j G k κ0 pv k q, v j % j γ j G k κ0 pv k q, rγ j s G j κ0 pv j q % j ¡ γ j G k κ0 pV k q, γ j c G j κ0 pV j q % j .
Consider two Cauchy traces w j pw j q q n q0 , w k pw k q q n q0 , defined by the following formulas (with α j, k)

w α q : γ q G α κ0 pv α q for q $ α, w α α : γ α c G α κ0 pv α q .
With these notations

γ j G k κ0 pu k q, γ j c G j κ0 pv j q % j w k j , w j j % j .
Observe that w j , w k XpΓq. As a consequence, we can apply Proposition 3.2 and obtain

w k j , w j j % j ¡ Tpw k q, Tpw j q $ Σ ¡ q0...n q$j w k q , w q j $ q .
In addition, note that w j q , w k q C κ0 pfΩ q q for q $ j, k, and similarly Tpw j q,Tpw k q C κ0 pfΩ Σ q. Now we apply Lemma 3.5 on fΩ q for q $ j, k and on fΩ Σ , which shows that all the terms vanish on the right hand side of (3.2), except the one associated to q k. This yields

w k j , w j j % j w k k , w j k % k
. Finally we conclude the proof by applying Lemma 3.5 once more in Ω k to obtain

w k k , w j k % k γ k c G k κ0 pv k q, γ k G j κ0 pv j q % k ¡ rγ k s G k κ0 pv k q, γ k G j κ0 pv j q % k γ k G j κ0 pv j q, v k $ k .
l

Since we will also use potential operators SL Σ κ , DL Σ κ and G Σ κ that are defined by (3.7) with Ω j replaced by Ω Σ , we would like to mention that all the above results also hold for the subdomain Ω Σ .

Classical single-trace formulation of the first kind

Now we present a first direct boundary integral formulation for Problem (2.2). This first formulation was already introduced and analysed in [START_REF] Petersdorff | Boundary integral equations for mixed Dirichlet, Neumann and transmission problems[END_REF]. Since it is pivotal for our later developments, we recall its derivation and main properties.

Boundary and transmission conditions

The classical single-trace formulation takes into account the homogenous Dirichlet boundary conditions (2.2c) on Σ by incorporating them into the variational space. Set u : pγ j pUqq n j0 where U is the unique solution to Problem (2.2). To arrive at an integral equation formulation, one first enforces the transmission conditions across the interfaces, and the Dirichlet boundary conditions on Σ by demanding that u X 0 pΓq where X 0 pΓq :

t u XpΓq | T d puq 0 u . (4.1)
Note that in the case n 0 where R d Ω 0 Ω Σ and Γ Σ, this space is simply given by X 0 pΓq t0u ¢ H ¡1{2 pΣq. Thanks to the continuity of T d : XpΓq Ñ H 1{2 pΣq, the space X 0 pΓq is a closed subspace of XpΓq. In addition, the function U H 1 loc p∆, R d zΩ Σ q satisfies the boundary and transmission conditions in (2.2), if and only if pγ j pUqq n j0 X 0 pΓq. In order to impose these conditions in a variational manner, one may rely on the following elementary characterization of X 0 pΓq. Lemma 4.1. For any u HpΓq, we have, u X 0 pΓq ðñ ru, vs 0 dv X 0 pΓq. Proof: Let u X 0 pΓq. Take any element v X 0 pΓq. Denote by u, v H 1{2 pΣq and p, q H ¡1{2 pΣq the traces such that Tpuq pu, pq and Tpvq pv, qq. According to the definition of X 0 pΓq we must have u v 0. Applying Proposition 3.2, we obtain ru, vs ¡ rTpuq, Tpvqs Σ 0, q Σ ¡ 0, p Σ 0 . Now assume that u HpΓq satisfies ru, vs 0, for all v X 0 pΓq. It is a direct consequence of Proposition 7.1 in [START_REF] Claeys | Multi-trace boundary integral formulation for acoustic scattering by composite structures[END_REF] that actually u XpΓq (note that notations are different in [START_REF] Claeys | Multi-trace boundary integral formulation for acoustic scattering by composite structures[END_REF]). Let u H 1{2 pΣq and p H ¡1{2 pΣq satisfy Tpuq pu, pq. Take any trace q H ¡1{2 pΣq and consider q Hpdiv, R d q such that n Σ ¤q| Σ q. Finally denote q j : n j ¤q| fΩj and set v p0, q j q n j0 . Clearly v XpΓq since q Hpdiv, R d q, and T d pvq 0, T n pvq q by construction, so v X 0 pΓq. Finally we obtain 0 ru, vs ¡ rTpuq, Tpvqs Σ ¡ u, q Σ . Since this holds for every q H ¡1{2 pΣq, we finally conclude that u T d puq 0, which implies u X 0 pΓq. l

Integral formulation

Define u inc : pγ 0 pU inc q, 0, . . . , 0q. According to the characterization of Cauchy data given by Proposition 3.4, the trace u : pγ 0 U, . . . , γ n U q of a solution U of the boundary transmission problem (2.2) satisfies p¡Id{2 Aqpu ¡ u inc q 0 , where the operator A : HpΓq Ñ HpΓq is defined subdomain-wise by Apuq : p A j κj pu j q q n j0 p tγ j u ¤ G j κj pu j q q n j0 " " " " " " " " " !

A 0 κ0 0 ¤ ¤ ¤ 0 0 A 1 κ1 . . . . . . . . . . . . . . . 0 0 ¤ ¤ ¤ 0 A n κn ( 0 0 0 0 0 0 0 0 0 ) ¤ " " " " " " " " " ! u 0 . . . . . . u n ( 0 0 0 0 0 0 0 0 0 ) , (4.2)
for u pu 0 , . . . , u n q HpΓq. Summing up, Problem (2.2) spawns the boundary integral equations Let the isometric isomorphism Θ : HpΓq Ñ HpΓq be defined by 4 Θpvq : p¡v j , q j q n j0 for v pv j , q j q n j0 HpΓq. There exists a compact operator K : HpΓq Ñ HpΓq, and a constant β ¡ 0 such that | rpA Kqv, Θpvqs | ¥ β}v} 2 HpΓq dv HpΓq .

u X 0 pΓq such that p¡Id{2 Aqpu ¡ u inc q 0 . ( 4
A direct consequence of this proposition is that the operator A : X 0 pΓq Ñ X 0 pΓq is of Fredholm type with index 0. As a consequence, dimpkerpAqq is finite and will depend on the wave numbers κ 0 , κ 1 , . . . , κ n . Fredholm alternative arguments [START_REF] Sauter | Boundary Element Methods[END_REF]Sect. 2.1.4]

Spurious resonances

As mentioned in the introduction, an important drawback of Formulation (4.4), is the possibility that kerpAq $ t0u, which is commonly referred to as "spurious resonance phenomenon" in literature. Of course, this is highly undesirable, because, in case kerpAq $ t0u, then (4.4) is not well posed, whereas Problem (2.2) always has a unique solution. In this section, we examine in what situations spurious resonances can occur. First of all, we need to establish an auxiliary result.

Lemma 4.5. Let u pu 0 , . . . , u n q X 0 pΓq satisfy rApuq, vs 0 for all v X 0 pΓq, and set W j pxq G j κj pu j qpxq, x R d zΩ j Then, for each j 0 . . . n, we have W j 0 on any connected component of R d zΩ j that does not coincide with Ω Σ .

Proof: The proof takes the cue from [37, Sect. 2] and combines elements of the proofs of [START_REF] Claeys | Multi-trace boundary integral equations[END_REF]Lemma 3.4], [START_REF] Claeys | Multi-trace boundary integral formulation for acoustic scattering by composite structures[END_REF]Prop. A.1]. We split it into three steps.

Let u satisfy the assumptions of the lemma and define W j as above. By the definition of A and Lemma 4.1 (ñ) we conclude rApuq, vs n j0 γ j c pW j q, v j $ j 0 dv X 0 pΓq . Appealing to Lemma 4.1 (ð), this implies that w : pγ j c pW j qq n j0 X 0 pΓq .

(4.7)
Next we establish that W j 0 in any unbounded connected component of R d zΩ j . To see this, note that for any j 0 . . . n, we have ∆W j κ 2 j W j 0 in R d zΩ j and W j is κ j -outgoing (radiating). Take ρ ¡ 0 large enough to ensure that R d zΩ 0 B ρ , where B ρ R d denotes the ball centered at 0 with radius ρ. Applying Green's formula in B ρ zΩ j for j 0 . . . n yields » fBρ

W j f r W j dσ » BρzΩj |∇W j | 2 ¡ κ 2 j |W j | 2 dx » fΩj γ j d,c pW j qγ j n,c pW j q dσ 0 » BρzΩ0 |∇W 0 | 2 ¡ κ 2 0 |W 0 | 2 dx » fΩ0 γ 0
d,c pW 0 qγ 0 n,c pW 0 q dσ In the equations above, f r refers to the radial derivative. Take the imaginary part of the identity above, and sum over j 0 . . . n, taking into account that w : pγ j c pW j qq 0¤j¤n XpΓq. This yields

n j1 Im 2 » fBρ W j f r W j dσ @ Im 2 n j0 » fΩj γ j d,c pW j qγ j n,c pW j q dσ @ 1 2
Imt rw, ws u 0.

In the last equality above we used Lemma 4.1. By construction, the functions W j are κ j -outgoing radiating, so that 0 lim rÑV

³ fBρ |f r W j ¡ iκ j W j | 2 dσ.
As a consequence we obtain

n j1 1 κ j » fBρ |f r W j | 2 κ 2 j |W j | 2 dσ n j1 1 κ j » fBρ |f r W j ¡ iκ j W j | 2 dσ ¡ 2 n j1 Im 3 » fBρ W j f r W j dσ A n j1 1 κ j » fBρ |f r W j ¡ iκ j W j | 2 dσ ÝÑ ρÑ V 0 .
This shows in particular that lim ρÑV ³ fBρ |W j | 2 dσ 0 for all j 1 . . . n. As a consequence, we can apply Rellich Lemma, see Lemma 2.11 in [START_REF] Colton | Integral equation methods in scattering theory[END_REF], which implies that W j 0 in the unbounded component of R d zΩ j , j 1 . . . n.

Consider an arbitrary j t0, . . . nu, and let O j be a bounded connected component of R d zΩ j with O j $ Ω Σ . Since (i) Ω Σ , Ω 0 , . . . , Ω n form a partition of R d , (ii) all these domains are connected, and (iii) R d zΩ Σ is connected, we find that Σ fO j , there is a t1, . . . , nu, j, such that Ω l O j and |fΩ fΩ j | ¡ 0. A typical situation is depicted in Figure 2. Hence, there exists x j fO j fΩ and an open ball D Bpx j , ρ j q, ρ j ¡ 0, such that D fO j D fΩ $ r .

Ω 2 Ω 3 Ω 1 Ω Σ D Figure 2.
Geometrical situation for part of the proof of Lemma 4.5. Here j 1, O 1 Ω Σ Ω 2 Ω 3 and 3. Since both O j and Ω j are connected and bounded, the set R d zO j is unbounded and connected. Thus, it is entirely contained in the unbounded connected component of R d zΩ that we denote by U . From part of the proof we know that W 0 in U .

Obviously, fU fΩ as well as fO j fΩ j . Moreover we know that D fO j D fU has positive measure. Since w pγ k c pW k qq n k0 X 0 pΓq according to (4.7) from Part of the proof, we deduce that on DfO j fU fΩ j fΩ holds γ j d,c pW j q γ d,c pW q 0 γ j n,c pW j q ¡γ n,c pW q 0 on D fO j fU . This means that γ j c pW j q 0 on fO j D. As ∆W j κ 2 j W j 0 in O j , by analytic continuation this implies W j 0 in O j according to Lemma 2.2 in [START_REF] Petersdorff | Boundary integral equations for mixed Dirichlet, Neumann and transmission problems[END_REF].

l Our final goal is to find sufficient and necessary conditions, under which the assumptions of Lemma 4.5 imply u 0. The next result teaches that we need to examine the functions W j outside Ω j . Lemma 4.6. Let u X 0 pΓq satisfy rA u, vs 0 for all v X 0 pΓq. Set W j pxq : G j κj pu j qpxq, x fΩ j , and assume that γ j c pW j q 0 for all j 0 . . . n. Then u 0.

Proof: We have γ j pW j q rγ j pW j qs rγ j s¤G j κj pu j q u j so pγ j pW j qq n j0 u X 0 pΓq. Moreover, by construction ∆W j κ 2 j W j 0 in Ω j for all j 0, . . . , n. We conclude that V H 1 loc pR d zΩ Σ q defined by V | Ωj W j | Ωj satisfies all the equations of Problem (2.2) without incident field, U inc 0. Since this transmission problem is well-posed V must vanish. Hence γ j pW j q 0 for all j 0 . . . n, and finally u j rγ j pW j qs 0 for all j 0 . . . n, i.e. u 0. l

The previous lemma together with Lemma 4.5 sends the message that kerpAq $ t0u can occur only if Ω Σ agrees with a connected component of the complement of some subdomain. Now we describe a simple setting in which this is the case.

Example ([33, Sect. 3.1]). Consider the case where n 0, so that the scatterer reduces to a single impenetrable part R d Ω 0 Ω Σ , and Γ fΩ 0 Σ, see Figure 3. In this geometrical setting we have X 0 pΓq t0u ¢ H ¡1{2 pΣq.

Choose κ 0 R such that there exists V H 1 p∆, Ω Σ qzt0u that satisfies ∆V κ 2 0 V 0 in Ω Σ , and V 0 on fΩ Σ . The existence of such non-trivial functions V is a classical consequence of spectral theory. Formulation (4.4) then reduces to the well-known single-layer integral formulation of the first

kind: seek p H ¡ 1 2 pΓq such that q, ptγ 0 d u ¥ SL 0 κ0 qppq 0 ¡ q, γ 0 d pU inc q 0 dq H ¡ 1 2 pΓq . (4.8)
Note that rγ 0 d s ¤ SL 0 κ0 0 according to (3.8), so we have tγ 0

d u ¤ SL 0 κ0 γ 0 d,c ¤ SL κ0 .
Coming back to the function V considered above, we have γ 0 d,c pV q 0 and γ 0 n,c pV q $ 0. In addition, a direct application of Proposition 3.3 yields V pxq ¡ SL 0 κ0 p γ 0 n,c pV q qpxq for x Ω Σ , so tγ 0 d u ¤ SL 0 κ0 p γ 0 n,c pV q q γ 0 d,c ¤ SL 0 κ0 p γ 0 n,c pV q q 0, which means that p : γ 0 n,c pV q 0 solves (4.8), although U inc 0.

Ω Σ Ω 0 Figure 3.
Homogeneous impenetrable scatterer giving rise to an exterior Dirichlet problem for the Helmholtz equation. We have assumed that R d zΩ Σ is connected. Therefore it is evident, that, if Ω Σ coincides with a bounded component of R d zΩ j , the boundary Σ of Ω Σ must be contained in fΩ j .

Ω 1 Ω Σ Ω 0
Corollary 4.7. Assume that Σ fΩ j for all j 0 . . . n. Then, for any choice of wave numbers satisfying (2.3), we have kerpAq t0u

The insights we have gained so far are not exactly intuitive as demonstrated by the following example.

Example. Consider Problem (2.2) where n 1, so that R d Ω 0 Ω 1 Ω Σ . Assume that κ 0 κ 1 so that the interface fΩ 0 fΩ 1 is "artificial". In fact, we face the very same scattering problem as in Example 4.3 above. Suppose that mespΣ fΩ 0 q ¡ 0 and mespΣ fΩ 1 q ¡ 0 like in Figure 4. Then, no matter what the value of κ 0 (even if κ 0 Sp∆, Ω Σ q), there is no spurious resonance!

The following lemma generalizes the observation made in Example 4.3.

In the interest of a concise statement we introduce the (discrete) set of interior Dirichlet eigenvalues of ¡∆ on Ω Σ : Sp∆, Ω Σ q : Σ fΩ j for a j t0, . . . , nu and

4 κ ¡ 0 | hV H 1 p∆, Ω Σ qzt0u : ¡∆V κ 2 V in Ω Σ , V 0 on fΩ Σ B . ( 4
κ j Sp∆, Ω Σ q D G F G E
.

Proof: Without loss of generality assume that Σ fΩ 0 (the proof below can easily be adapted to the case Σ fΩ j for j $ 0). There exists a connected component O 0 of R d zΩ 0 such that Σ fO 0 . We necessarily have Σ fO 0 , otherwise Σ would admit a boundary as a Lipschitz manifold of dimension d¡1, and this is not possible since Σ fΩ Σ . The set R d zO 0 is connected, it is contained in R d zΩ Σ , and it is maximal as a connected subset of R d zΩ Σ . As a consequence R d zO 0 R d zΩ Σ since R d zΩ Σ is assumed to be connected. In conclusion, Ω Σ is exactly one bounded connected component of R d zΩ 0 . In particular, Ω Σ is separated from the other subdomains Ω j , j 1, . . . , n:

Ω Σ n j1 Ω j r. (4.10)
Assume first that κ 0 Sp∆, Ω Σ q. As in Example 4.3, consider a function V H 1 pΩ Σ qzt0u such that ∆V κ 2 0 V 0 in Ω Σ , and V 0 on Σ. Consider u 0 pu 0 , p 0 q HpfΩ 0 q with u 0 0, p 0 0 on fΩ 0 zΣ, and p 0 γ Σ n pV q 0 on Σ. Applying Proposition 3.3 to V , we see that

G 0 κ0 pu 0 qpxq SL Σ κ0 pp 0 qpxq 0 for x Ω 0 R d zΩ Σ , so that γ 0 d SL Σ
κ0 pp 0 q 0. Now set u pu 0 , 0, . . . , 0q X 0 pΓqzt0u. For any v pv 0 , . . . , v n q X 0 pΓq we have rApuq, vs γ 0 SL 0 κ0 pp 0 q, v 0 $ 0 γ 0 d SL 0 κ0 pp 0 q, q 0 0 0 , where v 0 p0, q 0 q on Σ. Hence, u KerpAqzt0u. Now assume that κ 0 Sp∆, Ω Σ q. We have to confirm that necessarily u 0. Thanks to Lemma 4.5 W j 0 in R d zΩ j for j 1 . . . n, and W 0 0 in R d zpΩ 0 Ω Σ q, which implies γ j c pW j q 0 for j 1 . . . n, and γ 0 c pW 0 q| fΩ0zΣ 0 . Now let us show that γ 0 c pW 0 q 0 on Σ as well, i.e. γ Σ pW 0 q 0. We already know that, with w from (4.7), T d pwq γ Σ d pW 0 q 0 since w X 0 pΓq.

According to Proposition 3.3, we have

W 0 pxq ¡ G 0 κ0 pγ 0 c pW 0 qqpxq G Σ κ0 pTpwqqpxq SL Σ κ0 p T n pwq qpxq for all x Ω Σ R d zΩ 0 . So we conclude that 0 γ Σ d pW 0 q γ Σ d ¤SL Σ κ0 p T n pwq q.
It is well known, see for example [32, Thm. 3.9.1], that Kerpγ Σ d SL κ0 q t0u, if κ 0 Sp∆, Ω Σ q, hence we finally conclude that T n pwq γ 0 n,c W 0 0, which means γ 0 c pW 0 q 0. To finish the proof we apply Lemma 4.6. l

5 Single-trace combined field integral equation

We have discovered that the STF (4.4) is free of spurious resonnance except for the situation Σ fΩ j . As a remedy we are going to devise an augmented STF taking the cue from the CFIE approach already mentioned in the Introduction. We will not restrict ourselves to geometries that allow spurious resonances because, if Σ is largely contained in fΩ j with the exception of a small section, discretizations of the STF may already suffer from poor conditioning. Thus, even when spurious resonances cannot occur, the CFIE augmentation may enhance numerical stability! The classical CFIEs resort to simple complex combinations of Dirichlet and Neumann traces, ignoring the fact that they belong to different function spaces. This compounds the difficulties of a rigorous analysis of the resulting boundary integral equations. We refer to the discussion in [START_REF]Regularized combined field integral equations[END_REF]Sect. 3.1 ]. This problem can be overcome by using regularized CFIE that rely on compact operators which map between Dirichlet and Neumann traces. This was first employed for theoretical investigations [START_REF] Panich | On the question of the solvability of the exterior boundary-value problems for the wave equation and Maxwell's equations[END_REF] and, more recently, used for the design of stable Galerkin boundary element methods [START_REF] Buffa | On the acoustic single layer potential: stabilization and fourier analysis[END_REF][START_REF]Regularized combined field integral equations[END_REF][START_REF] Hiptmair | Stabilized FEM-BEM coupling for Helmholtz transmission problems[END_REF][START_REF]Stabilized FEM-BEM coupling for Maxwell transmission problems[END_REF][START_REF] Steinbach | Modified combined field integral equations for electromagnetic scattering[END_REF][START_REF] Buffa | A coercive combined field integral equation for electromagnetic scattering[END_REF]. Our approach is inspired by [START_REF]Regularized combined field integral equations[END_REF].

Transformed traces

The principle of regularized CFIE boils down to enforcing generalized impedance (Robin type) boundary conditions for potentials on Σ. As in [3, Sect. 3.2], these impedance boundary conditions rely on a regularizing linear operator M : H ¡1{2 pΣq Ñ H 1{2 pΣq that satisfies

(i) M is compact, (5.1a) (ii)
Imt ϕ, M ϕ Σ u ¡ 0 dϕ H ¡1{2 pΣqzt0u .

(

5.1b)

There exist many operators satisfying (i)-(ii). Indeed if M is any second order strongly coercive real symmetric surface differential operator on Σ, then M ¡ı M matches these conditions. The particular choice M ¡ı p∆ Σ Idq ¡1

will be further commented in §5. [START_REF] Buffa | Boundary element methods for Maxwell transmission problems in Lipschitz domains[END_REF]. Based on M we define the space of traces complying with generalized impedance boundary conditions

X M pΓq : 2 u XpΓq | T d puq M T n puq @ . (5.2)
Appealing to the duality of the trace spaces H ¡1{2 pΣq and H 1{2 pΣq we can define the adjoint regularizing operator M ¦ : H ¡1{2 pΣq Ñ H 1{2 pΣq by the formula q, M ¦ p Σ : p, M q Σ dp, q H ¡1{2 pΣq .

( As T n is surjective, the second assertion of the lemma follows.

l

The regularizing operator will enter the definition of a trace transformation R : XpΓq Ñ XpΓq that realizes an isomorphism of the form "identity + compact". Its definition involves a continuous extension operator

E Σ : H 1 2 pΣq Ñ H 1 pR d q that furnishes a right inverse of the trace γ Σ d . Then we define R Id C , C : γ j d ¥ E Σ ¥ M ¥ T n , 0 ¨¨n j0 , (5.5)
where C : XpΓq Ñ XpΓq inherits compactness from M. Lemma 5.2. R is an isomorphism and we have Rp X 0 pΓq q X M pΓq.

Proof: Observe that C 2 0, so that R ¡1 Id ¡ C, which proves the first statement. Now let γ d : H this shows that Rp X 0 pΓq q X M pΓq. We show in the same manner that pId ¡ Cqp X M pΓq q X 0 pΓq, which finishes the proof. l

Remark 5.3. If Σ fΩ j for some j t0, . . . , nu, we can pick an extension

E Σ that is local in the sense that supppE uq Ω Σ Σ Ω j , u H 1{2 pΣq .
(5.6)

Direct single trace CFIE

The STF (4.4) is a direct BIE in the sense that its unknowns are Cauchy traces of the solution of the transmission problem (2.2). This property is preserved by the CFIE augmentation proposed in this section.

As in Section 4.2 let u pu j q n j0 X 0 pΓq denote the Cauchy traces of the solution U of Problem (2.2) i.e. u j γ j puq, j 0, . . . , n. We have seen that it satisfies the integral equation (4.3). The derivation of a direct combined field integral equation starts from this identity and, as before, casts it into a weak form similar to (4.4). However, this time we employ test functions

r v X M pΓq instead of taking v X 0 pΓq! We end up with: seek u X 0 pΓq such that p¡Id{2 Aqu, r v $ n j0 γ j c G j κj pu j q,r v j % j ¡ u inc , r v $ dr v X M pΓq .
(5.7)

Thanks to Lemmas 5.2 and 4.1, an equivalent reformulation of (5.7) is rp¡Id{2 Aqu, pId Cqvs rA u, vs cpu, vq ¡

u inc , pId Cqv $ dv X 0 pΓq , (5.8)
where we define the compact bilinear form c : XpΓq ¢ XpΓq Ñ C according to cpw, vq : rp¡Id{2 Aqw, C vs , w, v XpΓq .

(5.9)

Compactness of c is an immediate consequence of the compactness of C :

XpΓq Ñ XpΓq. We may also introduce the unique element r u inc HpΓq such that rr u inc , vs ¡ u inc , pId Cqv $ . This makes it possible to write the direct single trace CFIE in variational form: 5 seek u X 0 pΓq such that rA u, vs cpu, vq ¡ rr u inc , vs dv X 0 pΓq .

(5.10) Below we write a M for the bilinear form from (5.10). Obviously, (5.10) amounts to a compact perturbation of (4.4) so that it preserves many key properties. In particular, it satisfies a generalized Gårding inequality analogous to Proposition 4.2.

Corollary 5.4. Recall the isomorphism Θ : HpΓq Ñ HpΓq from Proposition 4.2, defined by Θpvq p¡v j , q j q n j0 for v pv j , q j q n j0 HpΓq. The bilinear form a M on the left side of (5.10) satisfies

| a M pv, Θpvqq kpv, Θpvqq| ¥ β }v} 2
HpΓq dv X 0 pΓq , with a compact sesqui-linear form k : XpΓq ¢ XpΓq Ñ C. Denote A M : X 0 pΓq Ñ X 0 pΓq I the operator induced by a M . The previous proposition shows that A M is of Fredholm type with index 0. Thanks to Fredholm alternative arguments injectivity of A M is sufficient for stability of the variational problem (5.10) (in the sense of an inf-sup condition like (4.5)).

Lemma 5.5. For any choice of the wave numbers κ 0 , . . . κ n satisfying (2.3), KerpA M q is trivial.

Proof: By and large, the proof runs parallel to that of Lemma 4.5 and Theorem 4.8. Thus, some parts will only be sketched and for details the reader may refer to Section 4.3.

Pick u pu 0 , u 1 , . . . , u n q X 0 pΓq such that it solves (5.7)/(5.8) with u inc 0. As in Section 4.3 we set W j pxq G j κj pu j qpxq and w : γ j c W j ¨n j0 HpΓq, cf. (4.7). Since (5.7) with u inc 0 implies w, r v $ 0 for all r v X M pΓq, Lemma 5.1 confirms w X M ¦pΓq.

We exploit (5.1b) and exactly as in Step of the proof of Lemma 4.5 we show that W j 0 in any unbounded connected component of R d zΩ j .

The arguments employed in

Step of the proof of Lemma 4.5 completely carry over to the present situation and confirm that W j 0 in any connected component of R d zΩ j that does not coincide with Ω Σ . This is the counterpart of the statement of Lemma 4.5 for (5.10).

If Σ fΩ j for every j 0, . . . , n, we find w 0 as explained when justifying Corollary 4.7. Then apply Lemma 4.6 and the proof is finished.

Assume Σ fΩ j for some j 0, . . . , n. By above arguments all W k , k j, vanish on R d zΩ k . However, W j may not vanish on Ω Σ , recall Step of the proof of Theorem 4.8. However, from w X M ¦pΓq we conclude

γ Σ d pW j q M ¦ γ Σ n pW j q .
Thus, In Ω Σ the function W j satisfies ∆W j κ 2 j W j 0 in Ω Σ and γ Σ d pW j q M ¦ γ Σ n pW j q. By Green's formula, we obtain as in [3] 0 Im

3 » ΩΣ |∇W j | 2 ¡ κ 2 j |W j | 2 dx A Im 3 » Σ γ Σ n pW j q ¤ M ¦ ¤γ Σ n pW j q dσ A .
According to property (5.1b) of M ¦ , this implies γ Σ n pW j q 0, hence γ Σ d pW j q M ¦ γ Σ n pW j q 0. Finally this yields γ j c pW j q 0 and W j 0 in Ω Σ , so that we know w 0. Appealing to Lemma 4.6 finishes the proof. l

As in Section 4.2, via Fredholm theory, from this lemma we conclude that (5.10) always possesses a unique solution.

Remark 5.6. In the case n 0 of a single impenetrable scatterer the spaces and operators reduce to

X 0 pΓq t0u ¢ H 1 2 pΣq , A (4.2) A 0 κ0 , C (5.5) ¢ M ¥ T n 0 . (5.11)
As a consequence, with (3.10) the variational equation (5.8) becomes: seek

u p0, p 0 q X 0 pΓq ¢ ¡ Id 2 ¢ ¡ K 0 V 0 W 0 K I 0 ¢ 0 p 0 , ¢ Id ¡ ¢ 0 M 0 0 ¢ 0 q 0 & ¡ ¢ u inc p inc , ¢ Id ¡ ¢ 0 M 0 0 ¢ 0 q 0 &
for all q 0 H ¡ 1 2 pΣq. Owing to (3.2) and with u inc pu inc , p inc q this is equivalent to finding p 0 H ¡ 1 2 pΣq such that V 0 p 0 , q 0 p¡Id{2 K I 0 qp 0 , M q 0 u inc , q 0 p inc , M q 0 õ V 0 M ¦ p¡Id{2 K I 0 q ¨p0 , q 0 u inc M ¦ p inc , q 0 , for all q 0 H ¡ 1 2 pΣq. This agrees with the regularized CFIE from [3, Sect. 4].

Indirect CFIE

Both the STF (4.4) and the regularized CFIE (5.10) are direct BIE, since their unique solutions provide true Cauchy traces of the solution U of (2.2). If the solution of a BIE does not agree with traces of the solution of the related boundary value problem, it is classified as indirect. In [3, Sect. 3] a regularized indirect CFIE was devised for the simple situation n 0. In this section we adapt this approach to the STF. We obtain a variational equation that is dual to the direct CFIE introduced in the previous section.

The indirect CFIE stems from a representation of the solution to Problem (2.2) in the following form

U pxq G 0 κ0 pũ 0 qpxq U inc pxq for x Ω 0 , U pxq G j κj pũ j qpxq for x Ω j , j 1 . . . n,
where ũ pũ j q n j0 X M pΓq .

(5.12)

Admittedly, existence of such a representation of U is not obvious at first glance. Assume for a moment that such a representation can be found. Then the boundary and transmission conditions of Problem (2.2) can be expressed as pγ j pUqq n j0 X 0 pΓq. Using Lemma 4.1 and representation (5.12) yields

γ 0 pU inc q, v 0 $ 0 n j0
γ j G j κj pũ j q, v j % j 0 dv pv j q n j0 X 0 pΓq . (5.13) Definition (3.10) together with the jump relations (3.8) give the equivalent statement

p 1 2 Id Aqũ, v & ¡ γ 0 pU inc q, v 0 $ 0 dv pv j q n j0 X 0 pΓq . (5.14)
Thanks to Lemma 5.2 there exists u pu 0 , . . . , u n q X 0 pΓq such that ũ R u pId Cqu. Plugging this into (5.14), and taking account of the definition of u inc and Lemma 4.1, we obtain rApuq, vs

pA 1 2 Idq C u, v & ¡ u inc , v
$ dv X 0 pΓq .

(5.15)

Clearly, this equation is a perturbed version of Formulation (4.4). Introduce the bilinear form c I pw, vq :

pA 1 2 Idq C w, v & , (5.16)
the variational problem of the indirect single trace CFIE can be stated as:

5
seek u X 0 pΓq such that rA u, vs c I pu, vq ¡ ru inc , vs dv X 0 pΓq .

(5.17)

Lemma 5.7. We have c I pw, vq cpv, wq for all v, w X 0 pΓq.

Proof: This is an immediate consequence of the definitions (5.9), (5.16), of Lemma 3.6, and of the skew-symmetry of the pairing r¤, ¤s. l

Corollary 5.8. For any choice of the wave numbers κ 0 , . . . κ n satisfying (2.3), the indirect single trace CFIE (5.17) has a unique solution.

Proof: Lemma 5.7 tells us that the bilinear forms of (5.17 The previous proposition makes clear that Formulation (5.17) is always well posed. Now, assume that u is defined as the solution to Formulation (5.17). Undo the substitution made above by setting ũ R ¡1 u pId ¡ Cqu.

Then, by construction, the function U defined by (5.12) solves Problem (2.2) and coincides with its unique solution. Ultimately, this proves that a representation according to (5.12) can always be found for a solution of Problem (2.2). In addition, by means of (5.12) the field can be recovered. Remark 5.9. In the case n 0 already discussed in Remark 5.6 the variational problem (5.17) boils down to the indirect CFIE derived in [START_REF]Regularized combined field integral equations[END_REF]Sect. 3].

Mixed variational formulations

A convenient concrete choice for an operator M satisfying (5.1a) and

(5.1b) was proposed in [START_REF]Regularized combined field integral equations[END_REF]Sect. 4], namely M p∆ Σ Idq ¡1 : H ¡1 pΣq Ñ H 1 pΣq, where ∆ Σ stands for the Laplace-Beltrami operator on the closed surface Σ. The variational definition of this operator reads:

M ϕ H 1 pΣq : d Σ pM ϕ, v Σ q ¡ı ϕ, v Σ Σ dv Σ H 1 pΣq, ϕ H ¡1 pΣq , (5.18)
with sesqui-linear form (grad Σ is the surface gradient on Σ)

d Σ pz, vq : » Σ grad Σ z ¤ grad Σ v z v dS , z, v H 1 pΣq . (5.19)
Compactness of M : H ¡1{2 pΣq Ñ H 1{2 pΣq is immediate from the continuity M : H ¡1 pΣq Ñ H 1 pΣq and the compact embeddings H ¡1{2 pΣq H ¡1 pΣq and H 1 pΣq H 1{2 pΣq. This operator is also symmetric in the sense that M ϕ, ψ Σ M ψ, ϕ Σ , ϕ, ψ H ¡1 pΣq .

(5.20)

The bilinear forms of the variational formulations (5.10) and (5.15) of single-trace CFIEs involve evaluations of M. With Galerkin boundary element discretization in mind, it is desirable to avoid these and rely on the variational definition of M instead. As in [3, Sect. 4.2 & Sect. 3.2], this can be achieved by introducing auxiliary variables. In light of Lemma 5.7 we will restrict the discussion to the direct formulation (5.8). Using (4.2), the bilinear form c from (5.9) can be rewritten as (w, v X 0 pΓq) of C. The step uses that T d ¥ E Σ Id and the adjoint operator pγ j d E Σ q I : H ¡1{2 pfΩ j q Ñ H ¡1{2 pΣq. In we apply (5.20). These manipulations suggest that we introduce the new unknown

cpw, vq n j0 ¡Id{2 A j κj ¨wj , C v ¨j% j ¡ 1 2 rw, C vs n j0 A j κj w j , C v ¨j% j 1 2 rT w, TpC vqs Σ ¡ n j0 A j κj w j ¨n, γ j d E Σ MpT n vq j ¡ 1 2 T n pwq, MpT n vq Σ ¡ n j0 pγ j d E Σ q I A j κj w j ¨n, MpT n vq Σ e M ¡ ¡ 1 2 T n pwq ¡ n j0 pγ j d E Σ q I A j
z Σ : M ¡ ¡ 1 2 T n pwq ¡ n j0 pγ j d E Σ q I A j κj w j ¨n© H 1 pΣq , (5.21)
which satisfies

d Σ pz Σ , v Σ q ¡ 1 2 T n w ¡ n j0 pγ j d E Σ q I A j κj w j ¨n, v Σ Σ ¡ 1 2 T n w, v Σ Σ ¡ n j0 A j κj w j ¨n, γ j d E Σ pv Σ q j dv Σ H 1 pΣq . (5.22)
By means of z Σ we can express cpw, vq z Σ , T n v Σ , which converts the variational problem (5.10) of the direct single-trace CFIE into mixed form:

seek u X 0 pΓq, z Σ H 1 pΓq such that, dv X 0 pΓq, dv Σ H 1 pΣq, rA u, vs z Σ , T n v Σ ¡ rr u inc , vs , d 1 2 T n puq n °j0 pA j κj u j q n , γ j d E Σ pv Σ q h Σ d Σ pz Σ , v Σ q 0 . (5.23)
This variational problem inherits coercivity from (5.10), because the compact embedding H 1 pΣq H 1{2 pΣq renders the off-diagonal operators of (5.23) compact. Uniqueness also carries over from (5.10). Moreover, (5.23) is amenable to Galerkin discretization by means of standard boundary elements, for instance, piecewise linear continuous functions on a triangular surface mesh of Σ for the approximation of z Σ .

Multi-trace Combined Field Integral Equations

As pointed out in the Introduction, a shortcoming of the classical singletrace formulation (4.4) and also of its stabilized versions(5.10) and (5.17) is the tight coupling between subdomains implicit in the use of the single trace variational space X 0 pΓq, which contains the transmission conditions "in strong form". This limits flexibility in using Galerkin trial spaces locally on the subdomains. More severely, it turned out to be a big obstacle to the use of operator preconditioning techniques. We skip a detailed explanation here and recommend that the reader study [START_REF] Claeys | Multi-trace boundary integral equations[END_REF]Sect. 4]. We only quote the conclusion drawn in [START_REF] Claeys | Multi-trace boundary integral equations[END_REF] and [START_REF] Claeys | Novel multi-trace boundary integral equations for transmission boundary value problems[END_REF] that switching to variational formulations posed on decoupled local trace spaces will pave the way for effective operator preconditioning.

This has been the main motivation behind the development of so-called multi-trace formulations (MTFs). Here the expression "multi-trace" refers to a family of BIE where the unknowns are doubled on each interface that separates two (bounded) subdomains. In [START_REF] Claeys | Multi-trace boundary integral formulation for acoustic scattering by composite structures[END_REF] and [START_REF] Claeys | Multi-trace boundary integral equations[END_REF]Sect. 5] a global MTF was devised based on the classical STF (4.4). In this section we are going to derive and study its CFIE counterpart related to the formulations that we have established in Sections 4 and 5.

The gap idea

The global MTF was discovered through a heuristic geometric limit process, which is elucidated and justified in [START_REF] Claeys | Multi-trace boundary integral formulation for acoustic scattering by composite structures[END_REF]Sect. 5], [START_REF] Claeys | Multi-trace boundary integral equations[END_REF]Sect. 5.2], and [START_REF] Claeys | Novel multi-trace boundary integral equations for transmission boundary value problems[END_REF]Sect. 4.2]. Tersely speaking, we imagine an (infinitely) narrow gap between bounded subdomains Ω j , j 1, . . . , N , including Ω Σ . This gap is filled with the same ambient medium as Ω 0 , see Figure 5 for an illustration. For this arrangement where all bounded subdomains are isolated from each other we consider variational single trace formulations. Sloppily speaking, the corresponding global MTFs then boil down to STFs applied to gap configurations with vanishing gap.

Ω 1 Ω 2 Ω Σ Ω 0 Ω 1 Ω 2 Ω Σ Ω 0 Figure 5.
Illustration of the gap idea (gap highlighted)

Recalling Theorem 4.8, the alert reader will have noticed that the gap configuration as in Figure 5 (right) is exactly the situation, in which spurious resonances may afflict the classical STF (4.4), because Σ fΩ 0 . More precisely, uniqueness of solutions will be lost, if κ 0 Sp∆, Ω Σ q, where the latter set comprises the interior Dirichlet eigenvalues for ¡∆ on Ω Σ , see (4.9). Thus, (E1). we expect that the standard global MTF will suffer from spurious resonances whenever κ 0 Sp∆, Ω Σ q.

On the other hand, (E2). we also expect that the MTFs we obtain from pursuing the gap construction for CFIE extensions of the STF, will be stable for all combinations of wave numbers. This hope relies on Lemmas 5.5 and Corollary 5.8. In the sequel we give rigorous justifications of our conjectures. We are not going to employ vanishing gap arguments, which entail difficult geometric limit processes, but directly scrutinize the variational formulations as in [START_REF] Claeys | Multi-trace boundary integral formulation for acoustic scattering by composite structures[END_REF].

In the gap setting (Figure 5, right) we face a partitioning fΩ 0 n j1 fΩ j Σ so that, in this special case, the variational space X 0 pΓq from (4.1) for the STF variational formulations is clearly isomorphic to a product of Cauchy trace spaces on the subdomain boundary and Neumann traces on Σ:

p HpΓq : HpfΩ 1 q ¢ ¤ ¤ ¤ ¢ HpfΩ n q ¢ H ¡ 1 2 pΣq . (6.1)
This space will supply the functional framework for the global MTF, including for general configurations (such as in Figure 5, left). The main difference between p HpΓq and the space HpΓq introduced in (3.1) is that the former does not contain any contribution from fΩ 0 . Instead, it comprises contributions from Σ, via a trace chosen in H ¡1{2 pΣq. We equip the new space p

HpΓq with a norm defined by

}p u} 2 p HpΓq : }p u 1 } 2 HpfΩ1q ¤ ¤ ¤ }p u n } 2 HpfΩnq }p Σ } 2 H ¡1{2 pΣq
for all p u pp u 1 , . . . , p u n , p Σ q. Clearly, the dual space of p HpΓq with respect to local L 2 -type duality pairings is q HpΓq : HpfΩ 1 q ¢ ¤ ¤ ¤ ¢ HpfΩ n q ¢ H 1{2 pΣq. In concrete terms, for p u pp u 1 , . . . , p u n , u Σ q q HpΓq, and p v pp v 1 , . . . , p v n , q Σ q p HpΓq the underlying duality pairing between p HpΓq and q HpΓq is defined by the bilinear form p u, p v :

n j1 p u j , p v j $ j u Σ , q Σ Σ .
Routine verifications show that this bilinear form is non-degenerate and satisfies inf-sup conditions. We will use it to derive variational formulations.

Multi-trace formulations (MTFs)

Guided by the gap idea, and the STF (4.4) in gap settings, we can embark on the lengthy manipulations elaborated in [START_REF] Claeys | Multi-trace boundary integral formulation for acoustic scattering by composite structures[END_REF]Sect. 8] and [START_REF] Claeys | Multi-trace boundary integral equations[END_REF]Eq. (5.8)]. Since no new complications arise in the presence of essential boundary conditions, we omit the details. In the end we arrive at a multitrace formulation for the transmission boundary value problem with Dirichlet boundary conditions on Σ: where p f p p f 1 , . . . , p f n , f Σ q q HpΓq is defined by p f j γ j pU inc q and f Σ γ Σ d pU inc q, and p

A : p HpΓq Ñ q
HpΓq is a continuous linear operator defined by p A :

" " " " " " " " " " " " ! A 1 κ1 A 1 κ0 γ 1 G 2 κ0 ¤ ¤ ¤ γ 1 G n κ0 γ 1 SL Σ κ0 γ 2 G 1 κ0 A 2 κ2 A 2 κ0 ¤ ¤ ¤ γ 2 G n κ0 γ 2 SL Σ κ0 . . . . . . . . . . . . . . . γ n G 1 κ0 γ n G 2 κ0 ¤ ¤ ¤ A n κn A n κ0 γ n SL Σ κ0 γ Σ d G 1 κ0 γ Σ d G 2 κ0 ¤ ¤ ¤ γ Σ d G n κ0 V Σ κ0 ( 0 0 0 0 0 0 0 0 0 0 0 0 ) (6.3)
Definitions of the potentials SL Σ κ and G j κ0 can be found in (3.7), and

V Σ κ0 : γ Σ d SL Σ
κ0 is a single layer boundary integral operator on Σ. Hence, with p u pp u 1 , . . . , p u n , p Σ q, p v pp v 1 , . . . , p v n , q Σ q, the bilinear form of (6.2) boils down to

p App uq, p v n j1 pA j κj A j κ0 qpp u j q,p v j % j n j1 i1 i$j γ j G i pp u i q,p v j $ j n j1 γ j SL Σ κ0 pp Σ q,p v j % j n j1 γ Σ d G j κ0 pp u j q, q Σ Σ n j1 γ j SL Σ κ0 pp Σ q,p v j % j γ Σ d SL Σ κ0 pp Σ q, q Σ Σ (6.4) Remark 6.1.
The key observation is that all building blocks of p A and the terms in (6.4) remain well defined, even if we dispense with a gap between the subdomains Ω j , j ¥ 1 and Ω Σ . Thus, p A and the multi-trace variational problem (6.2) remain meaningful in the generic setting with junction points depicted in Figure 5, left, and introduced in Section 2. The gap idea instills confidence that (6.2) will inherit all properties of the single-trace problem (4.4) on isolated subdomains. In the next section, we are going to provide a rigorous foundation for this intuition.

Analysis of standard MTF

We consider the standard global MTF variational problem (6.2)/(6.4) in the general "non-gap" setting with possible junction points (Figure 5, left). Obviously, the bilinear form pp u, p vq Ñ p App uq, p v is continuous on p HpΓq. Also let us point out a symmetry property of this bilinear form that will be useful later. Due to the definition of p A from (6.3), the next result is a direct consequence of Lemma 3.6 Proof: By construction, the function U defined by (6.6) satisfies ∆U κ 2 j U 0 in Ω j for j 0 . . . n, and the radiation conditions at V (with respect to κ 0 ).

The only property we have to verify is the transmission conditions (2.2b), that is, pγ j pUqq n j0 X 0 pΓq. Owing to Lemma 4.1 this is equivalent to showing that for all v pv j q n j0 X 0 pΓq we have °n j0 γ j pUq, v j $ j 0 which, see (6.6), is equivalent to

γ 0 U inc ¡ γ 0 SL Σ κ0 pp Σ q ¡ n j1 γ 0 G j κ0 pp u j q,p v 0 ' 0 n j1 γ j G j κ0 pp u j q, v j $ j 0 . (6.7)
We fix some v X 0 pΓq, and denote v : pv 1 , . . . , v n , T n pvqq p HpΓq. For the remainder of the proof it is important to remember that T d pvq 0. From the jump relations (3.8) and (3.10) we can conclude A j κj

A j κ0 γ j G j κj γ j c G j κ0 .
We use this identity and infer from (6.2) and (6.4) with p v v

0 p Ap u, v ¡ p f, v n j1 ¡ γ j G j κj pp u j q, v j % j γ j c G j κ0 pp u j q, v j $ j n i1 i j γ j G i κ0 pp u i q, v j $ j γ Σ D G j κ0 pp u j q, T n pvq Σ © (6.8a) n j1 γ j SL Σ κ0 pp Σ q, v j % j γ Σ d SL Σ κ0 pp Σ q, T n pvq Σ (6.8b) ¡ n j1 γ j U inc , v j $ j ¡ γ Σ d U inc , T n pvq Σ . (6.8c)
For j 1, . . . , n, evidently G j κ0 pp u j q H 1 loc p∆, R d zΩ j q. As a consequence, when we take the trace on fΩ j from outside, we have z : γ 0 G j κ0 pp u j q, . . . , γ j c G j κ0 pp u j q, . . . , γ n G j κ0 pp u j q ¨ XpΓq. Thus, we can invoke Proposition 3.2, and find rz, vs ¡ rTpzq, Tpvqs Σ , which means

γ j c G j κ0 pp u j q, v j $ j n °i1 i j γ i G j κ0 p u j , v i $ i γ Σ d G j κ0 pp u j q, T n pvq Σ ¡ γ 0 G j κ0 p u j , v 0 $ 0 . (6.9)
In the same vein, we can set y : γ 0 SL Σ κ0 pp Σ q, . . . , γ n SL Σ κ0 pp Σ q ¨ XpΓq, which, again by Proposition 3.2, satisfies ry, vs ¡ rTpyq, Tpvqs Σ , equivalent

to n j1 γ j SL Σ κ0 pp Σ q, v j % j γ Σ d SL Σ κ0 pp Σ q, T n pvq Σ ¡ γ 0 SL Σ κ0 pp Σ q, v 0 % 0 . (6.10)
Similarly, since ∆U inc κ 2 0 U inc 0 everywhere, Proposition 3.2 yields °n j0

γ j U inc , v j $ j ¡ γ Σ U inc , T v $ Σ ¡ γ Σ d U inc , T n v Σ .
Obviously, we aim to use this last identity to tackle (6.8c), (6.9) (summed over j 1, . . . , n) to simplify (6.8a), and (6.10) to replace (6.8b). Thus we arrive at

0 n j1 γ j G j κ0 pp u j q, v j $ j ¡ n j1 γ 0 G j κ0 pp u j q, v 0 $ 0 ¡ γ 0 SL Σ κ0 pp Σ q, v 0 % 0 γ 0 U inc , v 0 $ Σ , (6.11)
which agrees the equation (6.7)! Since v was chosen arbitrarily in X 0 pΓq, this finishes the proof.

l

The gap construction hints that the operators p A defined in (6.3) will enjoy coercivity analogous to the assertions of Theorem 4.2. This is confirmed by the next result, which generalizes [START_REF] Claeys | Multi-trace boundary integral formulation for acoustic scattering by composite structures[END_REF]Thm. 10.4]. Proposition 6.3. Define the operators θ j : HpfΩ j q Ñ HpfΩ j q by θ j pv, q Σ q p¡v, q Σ q, and let Φ : p HpΓq Ñ p HpΓq denote the operator Φpp vq pθ 1 pp v 1 q, . . . , θ n pp v n q, qq for p v pp v 1 , . . . , p v n , qq. There exists a compact operator K :

p HpΓq Ñ q
HpΓq, and a constant β ¡ 0 such that § § § p p

A Kqp v, Φpp vq § § § ¥ β}p v} 2
p HpΓq for all p v p HpΓq .

Proof: Since a change of the wave numbers κ 0 , κ 1 , . . . , κ n only induces a compact perturbation of p A [32, Lemma 3.9.8], it suffices to prove the result for the case where κ 0 ¤ ¤ ¤ κ n ı where ı c ¡1. Take any p v pp v 1 , . . . , p v n , q Σ q p HpΓq. Denote W j pxq : G j κ0 pp v j qpxq for j 1 . . . n, and W n 1 pxq : G Σ κ0 pp v n 1 q where p v n 1 : p0, q Σ q HpΣq.

For the sake of concise notations, in the remainder of this proof, we will write r¤, ¤s

A n 1 κ0 pp v n 1 q, θ n 1 pp v n 1 q $ n 1 n j1 2 Re A j κ0 pp v j q, θ j pp v j q $ j n 1 j1 n 1 q1 q$j Re γ q G j κ0 pp v j q, θ q pp v q q $ q .
Proceeding exactly as in the proof of Proposition 10.3 in [START_REF] Claeys | Multi-trace boundary integral formulation for acoustic scattering by composite structures[END_REF], and in particular applying Proposition 10.1 and 10.2 of [START_REF] Claeys | Multi-trace boundary integral formulation for acoustic scattering by composite structures[END_REF], we have

Re p

App vq, Φpp vq

n 1 q0 n j1 » Ωq |∇W j | 2 |W j | 2 dx n 1 q0 » Ωq § § §∇ ¡ n 1 j1 W j © § § § 2 § § § n 1 j1 W j § § § 2 dx ¥ n 1 q0 n j1 }W j } 2 H 1 pΩqq .
(6.12)

In addition, we have HpfΩ l q Impγ l c G l κ l qC κ0 pfΩ l q according to [11, Lemma A.2]. Combining (6.15) and (6.16) we obtain that γ l c G l κ l pp u l q, v l % l 0 for all v l HpfΩ l q .

(6.17)

Finally, we conclude that γ l c G l κ l pp u l q 0 for all l 1 . . . n. As a consequence, we obtain from the jump relations p u j rγ j s G j κj pp u j q γ j G j κj pp u j q ¡ γ j c G j κj pp u j q 0.

Since p

App uq 0, from the bottom row of 6.3 we finally obtain that γ

Σ d SL Σ κ0 pp Σ q 0. Hence p Σ Kerpγ Σ d SL Σ κ0 q. Recall that the single layer operator γ Σ d SL Σ κ0
is a Fredholm operator with index 0, and it is an ismorphism (i.e. admits a trivial kernel) if and only if κ 0 Sp∆, Ω Σ q, see [32, Thm 3.9.1]. From this we conclude that, if κ 0 Sp∆, Ω Σ q, then p Σ 0, and Kerp p Aq t0u. In case κ 0 Sp∆, Ω Σ q, then SL Σ κ0 pp Σ qpxq 0 for all x R d zΩ Σ , so that γ l SL Σ κ0 pp Σ q 0 dl 1 . . . n, hence p0, . . . , 0, p Σ q Kerp p Aq.

l

Comparing Proposition 4.8, Proposition 4.7 and Proposition 6.4, we see that if Formulation (4.4) suffers spurious resonnances, then so does Formulation (6.2). On the other hand, we point out that for any geometric arrangement with Ω Σ r, there are certain κ 0 where Formulation (6.2) breaks down, while Formulation (4.4) remains well posed.

Direct multi-trace CFIE

Since we expect spurious resonances for (6.2), recall (E1), we also study multi-trace counterparts of CFIE formulations. The focus will be first on the direct single-trace CFIE proposed in Section 5.2 and its variational formulation on p HpΓq. By the structure of (5.10), we need only elaborate how to adapt the compact bilinear form c from (5.9).

Again we take inspiration from geometrical configurations involving a gap between the different scatterers (Figure 5, left). In gap configurations there exists a natural isomorphism p HpΓq ! X 0 pΓq, we look for p c : p HpΓq ¢ p HpΓq Ñ C such that p cpp u, p vq cpu, vq, where we have the correspondences p u Ø u and p v Ø v in the isomorphism mapping p HpΓq onto X 0 pΓq. Observe that c defined by (5.9) can be re-written as cpu, vq n j0 γ j c G j κj pu j q, Cpv j q % j , u, v X 0 pΓq . (6.18)

In the gap situation (i.e. the situation of disjoint subdomains), the extension operator E Σ can be chosen to map into functions, whose support is inside Ω 0 , which means that γ j d ¥ E Σ 0 for j 0, and that, essentially, C maps into H 1{2 pΣq. This brings about a substantial simplification of the operator C and leads to cpu, vq

γ 0 c G 0 κ0 pu 0 q, pC vq 0 $ 0 γ Σ n G 0 κ0 pu 0 q, M T n pvq Σ , u, v X 0 pΓq . (6.19)
For any pv, qq H 1{2 pfΩ j q ¢ H ¡1{2 pfΩ j q, denote θ j pv, qq : pv, ¡qq. Since pu j q n j0 X 0 pΓq and fΩ 0 Σ fΩ 1 ¤ ¤ ¤ fΩ n , the trace u 0 is equal to θ j pu j q on each fΩ j , j 1 . . . n, and equal to p0, ¡p Σ q on Σ. This yields

G 0 κ0 pu 0 q ¡ SL Σ κ0 pp Σ q ¡ °n j0 G j κ0 pu j q. Hence p cpp u, p vq ¡ M ¦ γ Σ n SL Σ κ0 pp Σ q ¨, q Σ Σ ¡ n j1 M ¦ γ Σ n G j κ0 pu j q ¨, q Σ Σ , (6.20)
for p u pu 1 , . . . , u n , p Σ q p HpΓq, p v pv 1 , . . . , v n , q Σ q p HpΓq. From (5.10), 6.3, and (6.20) we deduce the operator p A M : p

HpΓq Ñ q

HpΓq defined as

p A M : " " " " " " " " " " " " ! A 1 κ1 A 1 κ0 ¤ ¤ ¤ γ 1 G n κ0 γ 1 SL Σ κ0 γ 2 G 1 κ0 ¤ ¤ ¤ γ 2 G n κ0 γ 2 SL Σ κ0 . . . . . . . . . . . . γ n G 1 κ0 ¤ ¤ ¤ A n κn A n κ0 γ n SL Σ κ0 γ Σ d ¡ M ¦ γ Σ n ¨G1 κ0 ¤ ¤ ¤ γ Σ d ¡ M ¦ γ Σ n ¨Gn κ0 γ Σ d ¡ M ¦ γ Σ n ¨SL Σ κ0 ( 0 0 0 0 0 0 0 0 0 0 0 0 ) (6.

21) Similar considerations yield an expression in q

HpΓq for the right hand side of the direct single trace CFIE in the gap setting; we find

p f M : γ 1 U inc , . . . , γ n U inc , γ Σ d U inc ¡ M ¦ pγ Σ n U inc q ¨ q HpΓq . (6.22)
Then the direct multi-trace CFIE in variational form and in the gap setting reads:

6 9 8 9 7 find p u p HpΓq such that p A M pp uq, p v p f M , p v dp v p HpΓq , (6.23)
Although we have derived Formulation (6.23) in a gap setting where all scatterers were distant from each other, this formulation still makes sense in a general geometric configuration (such as in Figure 5, left). We justifies in the next paragraph the validity of (6.23) for a general setting. In addition, we give rigorous arguments for conjecture (E2) on Page 27, where we claimed that the direct global multi-trace CFIE (6.23) is immune to spurious resonances for any choice of wave numbers κ j .

Obviously, thanks to the compactness of M, see (5.1), the operator p A M from (6.21) is a compact perturbation of p A from 6.3, and the bilinear form of (6.2) is a compact perturbation of that of (6.23). The next result exhibits the precise relationship between the solution to (6.23) and the solution to (6.2). Proposition 6.5. A solution of the global multi-trace CFIE (6.23) is also a solution of the standard global MTF (6.2).

Proof: Take a solution p u pp u 1 , . . . , p u n , p Σ q p HpΓq of (6.23). Consider the function W pxq : U inc pxq ¡ SL Σ κ0 pp Σ qpxq ¡ °n j1 G j κ0 pp u j qpxq. Take test traces p v p HpΓq of the form p v p0, . . . , 0, q Σ q, where q Σ H ¡1{2 pΣq is arbitrary.

Formulation (6.23) yields

» Σ q γ Σ d pWq ¡ M ¦ γ Σ n pWq ¨dσ 0 dq H ¡

l

A corollary of the previous result is that, if U solves (6.23), then the unique solution to Problem (2.2) is given by (6.6). This justifies considering (6.23) for general geometric configurations. Now, since p c is compact, Proposition 6.3 implies that the bilinear form of (6.23) also satisfies a generalized Garding inequality. A consequence of the above proposition is that the operator p A M is of Fredholm type with index 0. One advantage of Formulation (6.23) over Formulation (6.2) is the absence of spurious resonnances, which is proved by the following result. Proposition 6.7. For any choice of wave numbers κ j ¡ 0, the global multitrace CFIE (6.23) possesses a unique solution.

Proof: Pick an element p u Kerp p A M q. This means that p u is a solution of (6.23) where p f M 0. As a consequence of Proposition 6.5, we have p u Kerp p Aq, so that, by Proposition 6.4, p u p0, . . . , 0, p Σ q for some p Σ H ¡1{2 pΣq. Coming back to (6.23), and choosing p v p HpΓq of the form p v p0, . . . , 0, q Σ q with some q Σ H ¡1{2 pΣq, we obtain

» Σ q Σ γ Σ d SL Σ κ0 pp Σ q ¡ M ¦ γ Σ n SL Σ κ0 pp Σ q ¨¨dσ 0 .
It was established in [START_REF]Regularized combined field integral equations[END_REF]Lemma 4.1] that the operator γ

Σ d SL Σ κ0 ¡ M ¦ γ Σ n SL Σ κ0
is injective for all κ 0 ¡ 0. So we conclude that p Σ 0 which finishes the proof.

l Corollary 6.8. For any choice of the wave numbers κ 0 , . . . κ n satisfying (2.3), Formulation (6.23) is well posed i.e. p A M : p

HpΓq Ñ q

HpΓq is an isomorphism.

Proof: Since p A M is a Fredholm operator with index 0, this holds true if and only if it is injective, which is the statement of Proposition 6.7. l

Indirect multi-trace CFIE

Of course, there is a multi-trace version also of the indirect CFIE presented in Section 5.3. Since developments are largely parallel to that for the direct CFIE, we do not give details. As is clear from (5.17), which serves as the starting point, the operator of the indirect multi-trace CFIE will be a perturbed version of p A. More precisely, the potential operator SL Σ κ0 is replaced with SL Σ κ0 DL Σ κ0 ¤ M. As in Section 6.2 the perturbation is encoded in a bilinear form p c ¦ : p

HpΓq ¢ p

HpΓq Ñ C, defined by p c ¦ pp u, p vq :

n j1 γ j DL Σ κ0 pM p Σ q, v j % j γ Σ d DL Σ κ0 pM p Σ q, q Σ Σ , (6.24)
for p u pu 1 , . . . , u n , p Σ q p HpΓq and p v pv 1 , . . . , v n , q Σ q p HpΓq. This bilinear form inherits compactness from M is. It can be used to state the indirect global multi-trace CFIE in variational form Compared to Formulation (6.2), this variational problem features an additional compact term. The next proposition gives a precise description of the relation between the solutions of (6.25) and the solutions to (2.2). Proposition 6.9. If p u pp u 1 , . . . , p u n , p Σ q p HpΓq is a solution of (6.25), then U L 2 loc pR d zΩ Σ q defined by (for j 1, . . . , n) U pxq G j κj pp u j qpxq , x Ω j U pxq U inc pxq ¡ SL Σ κ0 pp Σ qpxq ¡ DL Σ κ0 pM ¦ p Σ qpxq U pxq U inc pxq ¡ SL Σ κ0 pp Σ qpxq¡ n j1 G j κ0 pp u j qpxq, x Ω 0 , (6.26)

is the unique solution of the transmission boundary value problem (2.2).

We do not give the proof of this result as it is identical to the proof of Proposition 6.2. The only difference is that SL Σ κ0 pp Σ q has to be replaced by SL Σ κ0 pp Σ q DL Σ κ0 pM p Σ q. Now let us underline the close relationship between (6.25) and (6.23), that are dual to each other in the sense of the following lemma. Proof: We already know that p App uq, p v p App vq, p u , according to (6.5), so we have to show that p cpp u, p vq p c ¦ pp v, p uq. Take two elements p u pu 1 , . . . , u n , p Σ q and p v pv 1 , . . . , v n , q Σ q in p HpΓq. We have

p cpp u, p vq ¡ M ¦ γ Σ n SL Σ κ0 pp Σ q, q Σ Σ ¡ n j1
M ¦ γ Σ n G j κ0 pu j q, q Σ Σ (6.27) We examine successively each term in the sum above. v Σ pM q, 0q HpΣq and u Σ p0, pq HpΣq. Applying symmetry property given by Lemma Similarly we have ¡xM ¦ γ Σ n G j κ0 pu j q, qy Σ rγ Σ G j κ0 pu j q, v Σ s Σ . We can apply Lemma 3.7 (taking Ω Σ as one of the subdomains) to obtain rγ Σ G j κ0 pu j q, v Σ s Σ rγ j G Σ κ0 pv Σ q, u j s j which can be written in the present case ¡xM ¦ γ Σ n G j κ0 pu j q, qy Σ rγ j DL Σ κ0 pM qq, u j s j HpΓq refer to the continuous operator associated to the bilinear form in the left-hand side of (6.25). The previous lemma, combined with the inf-sup conditions satisfied by p A, shows that p A I M is bijective if and only if p A M is bijective, which is systematically true according to Proposition 6.6. In addition, since p A and p

A I

M only differ by a compact contribution, Proposition 6.3 implies that the bilinear form associated to Formulation (6.25) satisfies a generalized Garding inequality. We sum up all these results in the next proposition. HpΓq is an isomorphism.

1 n 5 )

 15 Ω i material sub-domains R d , Ω 0 unbounded, see Fig. number of (bounded) sub-domains with penetrable medium Σ : fΩ Σ Boundary where homogeneous Dirichlet boundary conditions are imposed Γ union of interfaces (skeleton), see (2.1) γ j d , γ j n Dirichlet and Neumann trace operators on fΩ j , see (2.4) γ j Cauchy trace operator defined in (2.HpfΩ j q Cauchy trace space associated with fΩ j , see (3.1) HpΓq Multi-trace space as defined in (3.1) ¤, ¤ j Duality pairing between Dirichlet and Neumann traces on fΩ j r¤, ¤s self-duality pairing on HpΓq X ¨1 2 pΓq, XpΓq single trace Dirichlet/Neumann/Cauchy spaces, see (3.5), (3.6) T d , T n , T restriction of single trace functions onto Σ, see Propositions 3.1, 3.2 SL j κ single layer potential defined on fΩ j DL j κ double layer potential defined on fΩ j G j κ total potential defined on fΩ j C κ pfΩ j q space of Cauchy data on fΩ j A j κj boundary integral operator on fΩ j B i,j non-local "remote" coupling boundary integral operators X 0 pΓq single trace space with vanishing Dirichlet data on Σ, see (4.1)

tween

  HpΓq and its dual HpΓq I , equivalent to the inf-sup condition inf vHpΓq sup uHpΓq | ru, vs | }u} HpΓq }v} HpΓq 1.

  px ¡ yq dσpyq , DL j κ puqpxq : ¡ » fΩj upyq n j pyq ¤ ∇ y G κ px ¡ yq ¨dσpyq , G j κ puqpxq : DL j κ puqpxq SL j κ ppqpxq , x fΩ j .

Figure 4 .

 4 Figure 4. Situation without spurious resonances, cf. Corollary 4.7

  ) and (5.10) are adjoint to each other. As a consequence, Corollary 5.4 and Lemma 5.5 carry over to (5.17) verbatim. A Fredholm alternative argument clinches the case. .

  l

κj w j ¨n© , T n pvq i Σ

 i In step we appeal to Proposition 3.2 for the first term and use the notation p¤q n to extract the Neumann component of Cauchy traces. We also exploit that C v has vanishing Neumann component and the definition(5.5)

p

  App uq, p v p f, p v dp v p HpΓq , (6.2)

n 1 : 1 κ0 : G Σ κ0 , A n 1 κ0:

 111 r¤, ¤s Σ , G n tγ Σ u G Σ κ0 and Ω n 1 : Ω Σ . Then we have Re p App vq, Φpp vq Re

Corollary 6 . 6 .

 66 The assertion of Proposition 6.3 holds with p A replaced with p A M .

p

  App uq, p v p c ¦ pp u, p vq p f, p v dp v p HpΓq .(6.25)

Lemma 6 . 10 .

 610 The bilinear forms of the direct global multi-trace CFIE (6.23) and its indirect counterpart (6.25) are adjoint to each other: p App uq, p v p cpp u, p vq p App vq, p u p c ¦ pp v, p uq dp u, p v p HpΓq .

  3.6 inΩ Σ yields ¡xγ Σ n SL Σ κ0 ppq, M qy Σ xγ Σ d,c DL Σ κ0 pMqq, py Σ .(6.28)

(6. 29 )

 29 according to the explicit expression of v Σ . Plugging (6.28) and (6.29) into the explicit expression of p c given by (6.27), and comparing with the definition of p c ¦ , this concludes the proof.

Proposition 6 . 11 .

 611 The assertion of Proposition 6.3 holds with p A replaced by p A I M . In addition, for any choice of the wave numbers κ 0 , . . . κ n satisfying (2.3), Formulation (6.25) is well posed i.e. it admits a unique solution and p

  1 pωq | ∆v L 2 pωqu. If Hpωq is any one of these spaces, H loc pωq : tv | ϕ v Hpωq dϕFigure 1. Geometric setting for the Helmholtz transmission problem for composite media with impenetrable Ω Σ .

			n 1	Ω 0 = exterior domain For each j the vector nj
	n 2	Ω 2	Ω 1	refers to the normal vec-tor on Ωj directed to-
	n 0			ward the exterior of Ωj, and nΣ denotes the vec-
				tor normal to Σ directed
		Ω Σ	n Σ	toward the exterior of ΩΣ. The existence of
				such vector fields is guar-
				anteed by Rademacher's
				theorem [32,Thm. 2.7.1].
				3)
	Then Problem (2.2) admits a unique solution U , as proved in [37, Sect. 2].

1

We follow the usual notations; given some open subset ω R d , we define H 1 pωq : tv L 2 pωq | ∇v L 2 pωqu with }v} 2 L 2 pωq : }v} 2 L 2 pωq }∇v} 2 L 2 pωq , and H 1 p∆, ωq : tv H C V K pR d qu, where C V K pR d q refers to the space of C V function with compact support.

  This actually provides a caracterization of solutions of the homogeneous Helmholtz equation, cf. [11, Prop. 3.2], [27, Thm. 3.1.3], [32, Sect. 3.6]. Proposition 3.4. Define the space of Cauchy data

  bear out that injectivity of A already ensures stability of the variational problem (4.4). If kerpAq t0u then there is α ¡ 0 such that

	Corollary 4.3. inf uX0pΓq	sup vX0pΓq	| rApuq, vs | }u} HpΓq }v} HpΓq	¡ α and
	inf vX0pΓq	sup uX0pΓq	| rApuq, vs | }u} HpΓq }v} HpΓq	¡ 0 .

(4.5)

The link between the STF variational formulation

(4.4)

and the transmission boundary value problem (2.2) has been established in [37, §4.1]: Proposition 4.4. Provided that kerpAq t0u, the traces u pγ j pUqq N j0 solve (4.4), if and only if U L 2 loc pR d zΩ Σ q is solution to (2.2), where U pxq is defined by U pxq : U inc pxq G 0 κ0 pu 0 qpxq for x Ω 0 , U pxq : G j κj pu j qpxq for x Ω j , j 1, . . . , n . (4.6)

  .[START_REF]Regularized combined field integral equations[END_REF] It is immediate that M ¦ satisfies (5.1), if and only if M does. As a consequence, we can define the space X M ¦pΓq analogously to (5.2). It can be used to obtain a weak characterization of X M pΓq: Lemma 5.1. For any u HpΓq, we have u X M pΓq ðñ ru, vs 0 dv

X M ¦pΓq.

Proof: (ñ) From Proposition 3.2 and (3.4) we obtain the identity ru, vs T n puq,

T d pvq Σ ¡ T d puq, T n pvq Σ , u, v XpΓq . (5.4) For u X M pΓq we infer ru, vs T n puq, T d pvq Σ ¡ M T n puq, T n pvq Σ T n puq, T d pvq ¡ M ¦ T n pvq looooooooooomooooooooooon 0 Σ 0 dv X M ¦pΓq .

(ð) To begin with, as in the proof of

[START_REF] Claeys | Multi-trace boundary integral equations[END_REF] Thm. 3

.1], we conclude with (5.4) that u XpΓq. Then, for v X M ¦pΓq, (5.4) becomes ru, vs T n puq, M ¦ T n pvq Σ ¡ T d puq, T n pvq Σ pT d puq ¡ M T n puqq, T n pvq Σ .

  1 pR d q Ñ XpΓq refer to the global trace operator defined by γ d puq pγ j d puqq n j0 . Since T d ¥γ d ¥ E Σ Id and T n pC uq 0, we easily see that for u X 0 pΓq T d pR uq ¡ M T n pR uq T d puq

	lo omo on
	0

M T n puq ¡ M T n puq 0 .

  and Lemma 3.7: If p u pp u 1 , . . . , p u n , p Σ q p HpΓq solves (6.2) then U L 2 loc pR d zΩ Σ q defined by U pxq G j κj pp u j qpxq for x Ω j , j 1 . . . n U pxq U inc pxq ¡ SL Σ κ0 pp Σ qpxq ¡ °n j1 G j κ0 pp u j qpxq for x Ω 0

	p App uq, p v p App vq, p u	dp u, p v p HpΓq .	(6.5)
	Now, extending Proposition 4.4 to the standard global MTF, the following
	proposition exhibits the precise relationship between Formulation (6.2) and
	Problem (2.2). Corresponding results for the pure transmission problem can
	be found in [11, Sect. 9].
	Proposition 6.2. (6.6)
	is the unique solution of Problem (2.2).

  1 2 pΓq ,which implies γ Σ d pWq M ¦ γ Σ n pWq. Since we have ∆W κ 2 0 W 0 in Ω Σ , pψq 0. We conclude that γ Σ d pψq M ¦ γ Σn pψq 0. This corresponds to the equation of (6.2) associated with the last line of (6.3). Since the only difference between (6.23) and (6.2) is this equation, we are done with the proof.

	applying Green's formula provides 0 Imt » |∇W| 2 ¡ κ 2 0 |W| 2 dxu 2 Imt	»	γ Σ n pWq M γ Σ n pWqdσu ,
	ΩΣ	Σ
	hence γ Σ

n

We point out that in order to maintain symmetry of formulas our choice of signs differs from what is commonly adopted in the literature.

We use overbars to designate complex conjugation.

This work received financial support from Fondation ISAE, and from the French Ministry of Defense via DGA-MRIS..

Note that pa 1 ¤ ¤ ¤ a k q 2 ¤ k pa 2 1 . . . a 2 k q for any a 1 , . . . a k R. Applying this elementary identity to (6.12) allows to conclude that 2pn 1q Re p App vq, Φpp vq

Now, since ¡∆W j W j 0 in Ω q for any j, q, and since, by the jump relations (3.8), p v j rγ j pW j qs, the continuity of trace operations yields }p v j } HpfΩj q ¤ C °n 1 q0 }W j } H 1 pΩqq . Combining this with (6.12) and (6.13) concludes the proof.

l

A direct consequence of the previous proposition is that the operator p A is Fredholm with index 0. Hence it is an isomorphism if it is injective, which can fail only in case of spurious resonnance, since Problem (2.2) is well posed. Recalling the gap idea and the characterization of the kernel of A from Theorem 4.8, the following result about spurious resonances of the global MTF is not surprising, cf. Section 6.1. Proposition 6.4. Kerp p Aq t p0, . . . , 0, pq | p Kerpγ Σ d SL Σ κ0 q u. As a consequence, for any choice of wave numbers κ j , the operator p A is a bijection if and only if κ 0 Sp∆, Ω Σ q.

Proof: Since p A is Fredholm with index 0, it is a bijection, if and only if it is injective. Assume that p u pp u 1 , . . . , p u n , p Σ q p HpΓq satisfies p App uq 0. In this case Proposition 6.2 applies with U inc 0. Since Problem (2.2) is well posed this shows that, in Formula (6.6), U 0 as well, so we conclude that G j κj pp u j qpxq 0 for x Ω j , and finally γ j G j κj pp u j q 0 dj 1, . . . , n . (6.14) Now pick an arbitrary l 1 . . . n, and an arbitrary v l C κ l pfΩ l q from the space of Cauchy data defined in (3.4). We have p v : p0, . . . , 0, v l , 0 . . . , 0q p HpΓq. We can apply (6.2) in the form p App uq, p v 0, take into account the definition of p A, see (6.4), use (6.14), which yields 0

.15)

In the computations above, we used the identity

Next, as v l C κ l pfΩ l q, Lemma 3.5 show that the following terms vanish γ l G l κ0 pp u l q, v l % l 0 , γ l SL Σ κ0 ppq, v l % l 0 .

(6.16)