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Coupled Domain-Boundary Variational Formulations for Hodge–Helmholtz Operators

We couple the mixed variational problem for the generalized Hodge–Helmholtz or Hodge–Laplace equation posed on a bounded 3D Lipschitz domain with the first-kind boundary integral equations arising from the latter when constant coefficients are assumed in the unbounded complement. Recently developed Calderón projectors for the relevant boundary integral operators are used to perform a symmetric coupling. We prove stability of the coupled problem away from resonant frequencies by establishing a generalized Gårding inequality (T-coercivity). The resulting system of equations describes the scattering of monochromatic electromagnetic waves at a bounded inhomogeneous isotropic body possibly having a “rough” surface. The low-frequency robustness of the potential formulation of Maxwell’s equations makes this model a promising starting point for Galerkin discretization.

We assume for simplicity that Ω s has trivial cohomology, in other words that its first and second Betti numbers are zero [2,Sec. 4.4]. Qualitatively, this means that it doesn't feature handles nor interior voids: it is homeomorphic to a ball.
Remark 1. The hypothesis that the second Betti number is zero is only used to prove injectivity of the coupling problem for Hodge-Laplace operators. It can be dropped without any changes to the following development for couplings involving the Hodge-Helmholtz operator (non-static electromagnetic transmission problems). The hypothesis that the first Betti number is zero is used in Sect. 5 to guarantee the existence of a certain "scalar potential lifting" that greatly simplifies the Fredholm arguments.

Our Contributions
In the following, we couple the mixed formulation of the weak variational problem associated to (3a) with the first-kind boundary integral equation (BIE) arising from (3b) using these recently developed Calderón projectors for the Hodge-Helmholtz and Hodge-Laplace operators. The proof of the well-posedness of the coupled problem relies on T-coercivity (c.f. [14]) and is given in Sect. 5.2. It draws on and integrates several fundamental results of the theory of first-kind boundary integral operators on Lipschitz domains and of the mathematical analysis of Maxwell's equations: M. Costabel's symmetric coupling approach linking volume variational equations with BIEs [17], T-coercivity for electromagnetic variational problems via Hodge-type decompositions [15,23], mixed variational formulations of boundary value problems for Hodge-Laplace operators [3].

Preliminaries
Let Ω ∈ {Ω s , Ω }. As usual, L 2 (Ω) and L 2 (Ω) denote the Hilbert spaces of square integrable scalar and vector-valued functions defined over Ω. We denote their inner products using round brackets, e.g. (·, ·) Ω . Similarly, H 1 (Ω) and H 1 (Ω) refer to the corresponding Sobolev spaces. We write C ∞ 0 (Ω) for the space of smooth compactly supported functions in Ω, but denote by D(Ω) 3 the analogous space of vector fields to simplify notation. The Banach spaces equipped with the natural graph norms will be important. The variational space for the primal variational formulation of the classical and generalized Hodge-Helmholtz/Laplace operator is given by A subscript is used to identify spaces of locally integrable functions or vector fields, e.g. U ∈ L 2 loc (Ω) if and only if φU is square-integrable for all φ ∈ C ∞ 0 (R 3 ). Dual spaces, e.g. H 1 0 (Ω s ) = H −1 (Ω s ), and dual operators, e.g. (γ − ) , are written with primes. We use an asterisk to indicate spaces of functions with zero mean, e.g. Based on the continuous and surjective extensions the traces previously introduced can also be extended by continuity to the relevant Sobolev spaces. We denote the duality pairing between H 1/2 (Γ) and H −1/2 (Γ) by ·, · Γ , but use ·, · τ for the duality pairing between the trace spaces H −1/2 (curl Γ , Γ) and H −1/2 (div Γ , Γ) [10, Lem. 5.6]. The duality pairings enter Green's formulas (+ for Ω = Ω s ) which hold for all P ∈ H 1 (Ω), W ∈ H(div, Ω), U, V ∈ H(curl, Ω) and E ∈ H(curl 2 , Ω). As explained in [15,Sec. 3], a theory of differential equations for the Hodge-Helmholtz/Laplace problem in three dimensions entails partitioning our collection of traces into two dual pairs. Accordingly, we now introduce the continuous and surjective mappings where They admit continuous right-inverses, i.e. lifting maps from the trace spaces into X(Δ, Ω) [15,Lem. 3.2].
In literature the pair of traces involved in T N is labelled as magnetic, while the pair in T D is referred to as electric-simply because one recovers the magnetic field by taking the curl of the potential U. However, our choice 7 Page 6 of 29 E. Schulz, R. Hiptmair IEOT of subscripts is motivated by the analogy between this pair of product traces and the classical Dirichlet and Neumann boundary conditions for secondorder elliptic BVPs. The trace spaces H D and H N are put in duality using the sum of the inherited component-wise duality parings. That is, for p = (p, q) ∈ H N and η = (η, ζ) ∈ H D , we define p, η := p, η τ + q, ζ Γ .
We indicate with curly brackets the average of a trace and with square brackets its jump • over the interface Γ, • = R, D, t, τ , or n. Corresponding notation is used for the product traces.
Warning. Notice the sign in the jump [γ] = γ − − γ + , which is often taken to be the opposite in literature!

Boundary Potentials
By exploiting the radiating fundamental solution for the scalar Helmholtz operator −Δ − ν 2 Id, it is shown in [15,Sec. 4.2] that a distributional solution U ∈ L 2 (R 3 ) such that U| Ωs ∈ X(Δ, Ω s ) and U| Ω ∈ X loc (Δ, Ω ) of the homogeneous (scaled) Hodge-Helmholtz/Laplace equation (3b) with constant coefficients η > 0, κ ≥ 0, stated in the whole of R 3 with radiation conditions at infinity as considered in Sect. 1, affords a representation formula Lettingκ = κ/ √ n, the Hodge-Helmholtz single layer potential is explicitly given by where the Helmholtz scalar single-layer, vector single-layer and the regular potentials are written individually for respectively. The expression (8) is derived with (9a)-(9c) understood as duality pairings. However, if the essential supremum of p, q and div Γ (p) is bounded, then they can safely be computed as improper integrals [15,Rmk. 4.2]. These classical potentials satisfy and the identity [28, Lem.
Ultimately, we will resort to a Fredholm alternative argument to prove well-posedness of the coupled system. It is therefore evident that the compactness properties of the boundary integral operators introduced in the next lemma will be extensively used both explicitly and implicitly-notably through exploiting the results found in [15,Sec. 6].
The Hodge-Helmholtz double layer potential is given for boundary data We recognize in (13) in which appears the matrix-valued fundamental solution [15] and detailed in [21, App. A]. This surface potential satisfies 7 The mapping properties of the potentials curlΨ κ (· × n) and Υ κ are detailed in [15,Sec. 5].

Integral Operators
In this section, we extend the analysis performed in [11,23] for the classical electric wave equation to the boundary integral operators arising from Hodge-Helmholtz and Hodge-Laplace problems.
The well-known Caldéron identities are obtained from (7) upon taking the classical compounded traces on both sides and utilizing the jump relations given in [15,Thm. 5.1]. The operator forms of the interior and exterior Caldéron projectors defined on H D × H N , which we denote P − κ and P + κ respectively, enter the Caldéron identites: Note that P − κ + P + κ = Id and that the range of P + κ coincides with the kernel of P − κ and vice-versa [11,Sec. 5]. As a consequence of the jump relations (15a)-(15b), the representation formula (7) and the existence of trace liftings, the pair of "magnetic" and "electric" traces ( η p) ∈ H D × H N is valid interior or exterior Cauchy data, if and only if it lies in the kernel of P + κ or P − κ respectively (c.f. [32,Lem. 6.18], [11,Thm. 8] and [15,Prop. 5.2]). Inspecting Eqs. (16a)-(16b) reveals that the Caldéron projectors share a common structure. They can be written as and where the Caldéron operator A κ : An analog of the operator matrix A k was found convenient in the study of the boundary integral equations of electromagnetic scattering problems [11,Sec. 6]. It is known from [15] that the off-diagonal blocks A DN κ and A ND κ of A k independently satisfy generalized Gårding inequalities making them of Fredholm type with index 0. Injectivity holds when κ 2 lies outside a discrete set of "forbidden resonant frequencies" accumulating at infinity [15,Sec. 3]. More explanations will be given in Section 3. In the static case κ = 0, the dimensions of ker ({T N } · SL 0 ) and ker ({T D } · DL 0 ) agree with the zeroth and first Betti number of Γ, respectively [15,Sec. 7].
We will extend this result to the integral operators defined for the scaled Hodge-Helmholtz/Laplace equation to better characterize the structure of (17). The symmetry we are about to reveal in the diagonal blocks A NN κ and A DD κ of the Caldéron projectors will be crucial in the derivation of the main T-coercivity estimate of this work. It will be exploited for complete cancellation, up to compact terms, of the operators lying on the off-diagonal of the block operator matrix associated to the coupled variational system introduced in Sect. 3. The following lemmas are required.

Lemma 2.1. There is a compact linear operator
Proof. This proof utilizes a strategy found in [23,Lem. 5.4] and [9,Thm. 3.9]. Let ρ > 0 be such that B ρ is an open ball containing Ω s . We will indicate with a hat (e.g. γ) the traces taken over the boundary ∂B ρ of that ball and use Green's formula to compare the following terms.
On the one hand, using the scalar Helmholtz equation (10a) and recalling thatκ = κ/ √ η, we have and similarly, On the other hand, using (10a) together with the scaled Hodge-Helmholtz equation (14), we also have Equations (19) and (20) together yield Similarly, the terms involving the exterior traces satisfy From the first row of the jump properties [15, Sec. 5] we obtain, by gathering the above results, integrating by parts again and using the fact that curl • ∇ ≡ 0, Fortunately, when restricted to domains away from Γ, the potentials are C ∞ -smoothing. Hence, their evaluation on ∂B ρ , the highlighted terms in (22), induce compact operators. This shows that for some compact operator The jump identities (21b) for the potentials yield formulas of the form {γ • }K = γ ± • K ± (1/2)Id, where • = n, D and K = ∇ψκ, Υ κ accordingly. Substituting each one-sided trace involved in the two leftmost duality pairings of (23) for the integral operators using these equations completes the proof.

Lemma 2.2. For all
Proof. In the following calculations, the boundary integrals are to be understood as duality pairings. Since p ∈ L 2 t (Γ) is a tangent vector field lying in the image of γ t , the tangential trace operator can safely be dropped in expanding these integrals using the definitions of Sect. 2.2. On the one hand, this leads to

Coupled Problem
In this section, we derive a variational formulation for the system (3a)-(4b) which couples a mixed variational formulation defined in the interior domain to a boundary integral equation of the first kind that arises in the exterior domain.
As proposed in [3], we introduce a new variable P = −div ( (x)U) into Eq. (3a) to dispense with trial spaces contained in H(curl, Ω s ) ∩ H(div, Ω s ). Applying Green's formula (6c) in Ω s , we obtain for all V ∈ H (curl, Ω s ), Q ∈ H 1 (Ω s ). The volume integrals in these equations enter the interior bi-linear form related to the one supplied for the Hodge-Laplace operator in [4,Sec. 3.2]. We aim to couple (26) with the BIEs replacing the PDEs in Ω . We use the transmission conditions (4a)-(4b) to couple (25) to the variational equation which involves a functional bounded over the test space. The exterior Calderón projector can be used to express the so-called Dirichlet-to-Neumann operator in two different ways.
1. Introducing the jump conditions into the first exterior Calderón identity given on the first line of (16b) along with a new unknown p = T + N (U ext ) yields a variational system for all (V Q) ∈ H (curl, Ω s ) × H 1 (Ω s ) and a ∈ H N , resembling the original Johnson-Nedélec coupling [6]. The new functional appearing on the right hand side of (27) is defined as 2. Following the exposition of Costabel in [17], we also retain the second exterior Calderón identity -in which we again introduce the jump conditions to eliminate the dependence on the exterior solution-and insert the resulting Eq. (27) to obtain the symmetrically coupled problem. Again, the right hand side of our system of equations has to be modified to include a new bounded linear functional We arrive at the following variational problem.

Symmetrically Coupled Problem
Find U := (U, P) ∈ H (curl, Ω s ) × H 1 (Ω s ) and p ∈ H N such that Remark 2. Part of the justification for using mixed formulations for problems involving the Hodge-Helmholtz/Laplace operator is the need to avoid trial spaces contained in H(curl, Ω s )∩H(div, Ω s ), because the latter doesn't allow for viable discretizations using finite elements [4]. While from (27) the issue seems to reappear after using the Caldéron identities, the benefits of the introduced new unknown P ∈ H 1 (Ω s ) in the mixed formulation conveniently In the following proposition, we call forbidden resonant frequencies the interior "Dirichlet" (or electric) eigenvalues of the scaled Hodge-Laplace operator with constant coefficient η = μ 0 2 0 . That is, κ 2 is a forbidden frequency if there exists a non-trivial solution U = 0 in X(Δ, Ω) to in Ω s , We refer the reader to [15], where the spectrum of the scaled Hodge-Laplace operator is completely characterized. See for e.g. [13,16,18,29,30] for an overview of the issue of spurious resonances in electromagnetic and acoustic scattering models based on integral equations. Proposition 3.1. Suppose that κ 2 ∈ C avoids forbidden resonant frequencies.
By retaining an interior solution U ∈ H (curl, Ω s ) and producing U ext ∈ X loc (Δ, Ω ) using the representation formula (7) for the obtained Cauchy data , a solution to (30) solves the transmission system (3a)-(4b) in the sense of distribution.
Proof. The proof follows the approach in [23, Lem. 6.1]. Since D(Ω s ) 3 × C ∞ 0 (Ω s ) is a subset of the volume test space, any solution to the problem (30) solves (3a) in Ω s in the sense of distribution. It follows that (25) holds for all admissible V, which reduces (30) to the variational system . We recognize in the equivalent operator equation the interior Caldéron projector (16a) whose image is the space of valid Cauchy data for the homogeneous (scaled) Hodge-Laplace/Helmholtz interior equation with constant coefficient η. In particular, p − q = T − N Ũ for some vector-fieldŨ ∈ X (Δ, Ω s ) satisfying If κ 2 = 0, we rely on the hypothesis that κ 2 doesn't belong to the set of forbidden resonant frequencies to guarantee injectivity of the above boundary value problem [15,Sec. 3] [21,Sec. 3]. Otherwise, the second Betti number of Ω s being zero implies that zero is not a Dirichlet eigenvalue [2,Sec. 4.5.3]. We conclude thatŨ = 0 is the unique trivial solution to (32). Therefore, for the right hand side of (31) to exhibit valid Neumann data, it must be that p = q. Now, the null space of the interior Caldéron projector P − κ coincides with valid Cauchy data for the exterior boundary value problem (3b) complemented with the radiation conditions at infinity introduced in Sect. 1. In particular ( p, ξ) is valid Cauchy data for that exterior Hodge-Helmholtz or Hodge-Laplace problem and U ext = SL κ ( p) + DL κ ξ solves (3b) and (4b) by construction. The fact that p = T + N (U ext ) solves (4a) is confirmed by the earlier observation that p = q. Remark 3. We show in [30], where the kernel of the coupled problem is completely characterized, that when κ 2 happens to be a resonant frequency, the interior solution U remains unique. This is no longer true for p however, which is in general only unique up to Neumann traces of interior Dirichlet eigenfunctions of −Δ η associated to the eigenvalue κ 2 . Fortunately, this kernel vanishes under the exterior representation formula obtained from (7).
A very useful property of this pair of decompositions is stated is shown in [23,Lem. 8.1] and [23,Lem. 8.2]: The operators and are compact. Another benefit of this pair of regular decompositions will become explicit in the poof of Lemma 5.9 found in the next section.
It follows from [15,Lem. 6.4] that div Γ : is a continuous bijection. The bounded inverse theorem guarantees the existence of a continuous inverse (div Γ )

Well-Posedness of the Coupled Variational Problem
We use the direct decompositions introduced in Sect. 4 to prove that the bilinear form associated to the coupled system (3.1) of Sect. 3 satisfies a generalized Gårding inequality. The coupled variational problem (30) translates into the operator equation associated with the Hodge-Helmholtz/Laplace volume contribution to the system, the operator can be represented by the block operator matrix shown here in "variational arrangement". The symmetry revealed in Sect. 2.3 makes explicit much of the structure of the above operator. We have introduced colors to better highlight the contribution of each individual block in the following sections.
Our goal is to design an isomorphism X of the test space and resort to compact perturbations of G κ • X −1 to achieve an operator block structure with diagonal blocks that are elliptic over the splittings of Sect. 4 and offdiagonal blocks that fit a skew-symmetric pattern. Stability of the coupled system can then be obtained from the next theorem. An overline indicates component-wise complex conjugation.
The authors of [9] refer to (36) as "Generalized Gårding inequality", because generalizes the classical Gårding inequality for a bilinear form b associated with uniformly elliptic operator of even order 2 : Assuming that (36) holds with X = Id, a simple proof of the stability estimate u V ≤ C f V , obtained for the unique solution of the operator equation Au = f when A is injective is given in [32,Thm. 3.15]. A proof of the general case can be deduced from [22]. T-coercivity theory is a reformulation of the Banach-Necas-Babuska theory. The former relies on the construction of explicit inf-sup operators at the discrete and continuous levels, whereas the later develops on an abstract inf-sup condition [14]. In deriving the following results, it will be convenient to denote U := (U, P) ∈ H (curl, Ω s ) × H 1 (Ω) and p := (p, q) ∈ H N . We indicate with a hat equality up to a compact perturbation (e.g.=).

Space Isomorphisms
In this section, we take up the challenge of finding a suitable isomorphism X. We build it separately for the function spaces in Ω s and on the boundary Γ. Crucial hints are offered by the construction of the sign-flip isomorphism for the classical electric wave equation in [11].
We start with devising an isomorphism Ξ of the volume function spaces and show that the upper-left diagonal block of G κ satisfy a generalized Gårding inequality.
Under the assumption that the first Betti number of Ω s is zero, there exists a bijective "scalar potential lifting" S : N(curl, Ω s ) → H 1 * (Ω s ) satisfying ∇S (U) = U. The Poincaré-Friedrichs inequality guarantees that this map is continuous.
Notice that since it also follows from the Poincaré-Friedrichs inequality that ∇ : H 1 * (Ω s ) → N(curl, Ω s ) is injective, S • ∇ : H 1 (Ω s ) → H 1 * (Ω s ) is a bounded projection onto the space of Lebesgue measurable functions having zero mean. Its nullspace consists of the constant functions in Ω s .
Since the considerations of Sect. 4 readily yield that V ⊥ = U ⊥ , we conclude that V = U. In turn, it follows that ∇P = ∇Q and mean(P ) = mean(Q). Therefore, Ξ is injective.
We now turn to the design of an isomorphism for the Neumann trace space H N and prove that the lower-right block A ND κ of G κ satisfies a generalized Gårding inequality. p ⊥ − p 0 − λ (div Γ ) † Q * q −τ (div Γ (p) + λ mean (q)) , p = (p, q) , has a continuous inverse. In other words, Ξ Γ is an isomorphism of Banach spaces.