Coercive second-kind boundary integral equations for the Laplace Dirichlet problem on Lipschitz domains

We present new second-kind integral-equation formulations of the interior and exterior Dirichlet problems for Laplace's equation. The operators in these formulations are both continuous and coercive on general Lipschitz domains in $\mathbb{R}^d$, $d\geq 2$, in the space $L^2(\Gamma)$, where $\Gamma$ denotes the boundary of the domain. These properties of continuity and coercivity immediately imply that (i) the Galerkin method converges when applied to these formulations; and (ii) the Galerkin matrices are well-conditioned as the discretisation is refined, without the need for operator preconditioning (and we prove a corresponding result about the convergence of GMRES). The main significance of these results is that it was recently proved (see Chandler-Wilde and Spence, Numer. Math., 150(2):299-271, 2022) that there exist 2- and 3-d Lipschitz domains and 3-d starshaped Lipschitz polyhedra for which the operators in the standard second-kind integral-equation formulations for Laplace's equation (involving the double-layer potential and its adjoint) $\textit{cannot}$ be written as the sum of a coercive operator and a compact operator in the space $L^2(\Gamma)$. Therefore there exist 2- and 3-d Lipschitz domains and 3-d starshaped Lipschitz polyhedra for which Galerkin methods in $L^2(\Gamma)$ do $\textit{not}$ converge when applied to the standard second-kind formulations, but $\textit{do}$ converge for the new formulations.


Boundary integral equations for Laplace's equation
If an explicit expression for the fundamental solution of a linear PDE is known, then boundary value problems (BVPs) for that PDE can be converted to integral equations on the boundary of the domain. The main advantage of this procedure is that the dimension of the problem is reduced; indeed, the problem is converted from one on a d-dimensional domain to one on a (d − 1)dimensional domain. Futhermore, if the original domain is the exterior of a bounded obstacle, then the problem is reduced from one on a d-dimensional infinite domain, to one on a (d−1)-dimensional finite domain.
This reduction to the boundary has both theoretical and practical benefits: on the theoretical side, C. Neumann famously used boundary integral equations (BIEs) to prove existence of the solution of the Dirichlet problem for Laplace's equation in convex domains in [87] (see, e.g., the account in [72,Chapter 1]), and BIEs have a long history of use in the harmonic analysis literature to prove wellposedness of BVPs on rough domains (see, e.g., [25], [106], [16], [55, §2.1], [79], [75,Chapter 15], [102,Chapter 4], [77]). On the more practical side, numerical methods based on Galerkin, collocation, or numerical quadrature discretisation of BIEs, coupled with fast matrix-vector multiply and compression algorithms, and iterative solvers such as GMRES, provide spectacularly effective computational tools for solving a range of linear boundary value problems, for example in potential theory, elasticity, and acoustic and electromagnetic wave scattering (see, e.g., [92,4,61,11,27,24,8,110,48,94,18]   When Γ is Lipschitz, the integrals in D and D are defined as Cauchy principal values, in general only for almost all x ∈ Γ with respect to the surface measure ds. The definition of H on spaces larger than H 1 (Γ) is complicated (it must be understood either as a finite-part integral, or as the non-tangential limit of a potential; see [72,Chapter 7], [20, Page 113] respectively), but these details are not essential to the present paper. The standard mapping properties of S, D, D , and H on Sobolev spaces on Γ are recalled in Appendix B (see (B.3)).
The BIE operators involved in the standard first-and second-kind BIEs for the Dirichlet and Neumann problems for Laplace's equation are shown in Table 1.1; although we do not explicitly consider the Neumann problem in this paper, we use the information in this table in what follows. For the details of the right-hand sides and unknowns for the integral equations corresponding to the operators in Table 1.1, see, e.g., [94, §3.4], [72,Chapter 7], [100,Chapter 7], [20, §2.5]. Recall that the adjective "direct" in the table refers to equations where the unknown is either the Dirichlet or Neumann trace of the solution to the corresponding BVP, and the adjective "indirect" refers to equations where the unknown does not have immediate physical relevance.
Following [94,Pages 9 and 10], we call BIEs first kind where the unknown function only appears under the integral, and second kind where the unknown function appears outside the integrand as well as inside; by this definition, the BIEs in the first and third row of Table 1.1 are first kind, and in the second and fourth row second kind. An alternative definition of second kind BIEs is that, in addition to the unknown function appearing outside the integrand as well as inside, the BIO is Fredholm of index zero (i.e., the Fredholm alternative applies to the BIE); see, e.g., [4, §1.1.4]. Every BIE that we describe in the paper as second-kind is second-kind in both senses above. (1. 4) We say that the Galerkin method converges for the sequence (H N ) ∞ N =1 if, for every f ∈ H, the Galerkin equations (1.4) have a unique solution for all sufficiently large N and φ N → A −1 f as N → ∞. We say that (H N ) ∞ N =1 is asymptotically dense in H if, for every φ ∈ H,

The Galerkin method
(1.5) A necessary condition for the convergence of the Galerkin method is that (H N ) ∞ N =1 is asymptotically dense in H. Indeed, a standard necessary and sufficient condition (e.g., [46, Chapter II, Theorem 2.1]) for convergence of the Galerkin method is that (H N ) ∞ N =1 is asymptotically dense and that, for some N 0 ∈ N and C dis > 0, where φ = A −1 f and φ N is the unique solution of the Galerkin equations (1.4). We note that (1.7) is known as a quasioptimal error estimate. We now recap the main abstract theorem on convergence of the Galerkin method; this theorem uses the definition that an operator A : H → H is coercive 1 if there exists C coer > 0 such that (Aψ, ψ) H ≥ C coer ψ 2 H for all ψ ∈ H. (1.8) References for the proof. Part (a) was first proved in [71,Theorem 1]; see also [46, Chapter II, Theorem 4.1]. Part (b) was first proved in [71,Theorem 2], with this result building on results in [104]; see also [46, Chapter II, Lemma 5.1 and Theorem 5.1]. Part (c) is Céa's Lemma, first proved in [17].

1.3
The rationale for using second-kind BIEs posed in L 2 (Γ) The BIOs in Table 1.1 are coercive in the trace spaces H ±1/2 (Γ) (or certain subspaces of these) for Lipschitz Γ, thus insuring convergence of the associated Galerkin methods by Part (c) of Theorem 1.1; this coercivity theory was established for first-kind equations by Nédélec and Planchard [86], Le Roux [62], [63], and Hsiao and Wendland [52], and for second-kind equations by Steinbach and Wendland [101]. These arguments involve transferring boundedness/coercivity properties of the PDE solution operator to the associated boundary integral operators via the trace map and layer potentials; the generality of these arguments is why coercivity holds with Γ only assumed to be Lipschitz, and Costabel [29] highlighted how these ideas can be traced back to the work of Gauss and Poincaré. Despite convergence of the associated Galerkin methods, using the first-kind formulations in the trace spaces has the disadvantage that the condition numbers of the Galerkin matrices grow as the discretisation is refined; e.g., for the h-version of the Galerkin method (where convergence is obtained by decreasing the mesh-width h and keeping the polynomial degree fixed), the condition numbers grow like h −1 ; see, e.g. [94, §4.5]. Furthermore, using the second-kind formulations in the trace spaces has the disadvantage that the inner products on H ±1/2 (Γ) are non-local and nontrivial to evaluate; even if the basis functions φ N and ψ N in (1.4) have supports only on a subset of Γ, (Aφ N , ψ N ) H is an integral over all of Γ, and the calculation of the Galerkin matrix in this case is impractical.
For the second-kind BIEs, an attractive alternative to working in the trace spaces is to work in L 2 (Γ). When Γ is C 1 , D and D are compact in L 2 (Γ) by the results of Fabes, Jodeit, and Rivière [41, Theorems 1.2 and 1.9] and thus each of the second-kind BIOs 1 2 I ±D and 1 2 I ±D is the sum of a coercive operator and a compact operator, and convergence of the associated Galerkin methods in L 2 (Γ) is ensured by Part (b) of Theorem 1.1. Since the L 2 (Γ) norm is local, (Aφ N , ψ N ) H is an integral over the support of ψ N , and the Galerkin matrix is much more easily computable. Furthermore, when D and D are compact, the condition numbers of the Galerkin matrices of 1.4 Convergence of the Galerkin method in L 2 (Γ) for the standard second-kind integral equations on polyhedral and Lipschitz domains.
The disadvantage of using second-kind BIEs in L 2 (Γ) is that convergence of the Galerkin method is harder to establish when Γ is only Lipschitz, or Lipschitz polyhedral. Indeed, in these cases D and D are not compact; e.g., when Γ has a corner or edge their spectra are not discrete; see, e.g. [4, §8.1.3]. When Γ is only Lipschitz, D and D are bounded on L 2 (Γ) by the results on boundedness of the Cauchy integral on Lipschitz Γ of Coifman, McIntosh, and Meyer [25] (following earlier work by Calderón [15] on boundedness for Γ with small Lipschitz character). Verchota [106] showed that the operators 1 2 I ± D and 1 2 I ± D are Fredholm of index zero on L 2 (Γ); when Γ is connected, 1 2 I − D and 1 2 I − D are invertible on L 2 (Γ) and 1 2 I + D and 1 2 I + D invertible on L 2 0 (Γ), the set of φ ∈ L 2 (Γ) with mean value zero; see [106, Theorems 3.1 and 3.3(i)]. 2 A long-standing open question has been Can 1 2 I ± D and 1 2 I ± D be written as the sum of a coercive operator and a compact operator in the space L 2 (Γ) when Γ is only assumed to be Lipschitz? By Part (b) of Theorem 1.1, this question is equivalent to the question: does the Galerkin method applied to 1 2 I ± D and 1 2 I ± D in L 2 (Γ) converge for every asymptotically-dense sequence of subspaces when Γ is only assumed to be Lipschitz?
Until recently, this question was answered only in the following two cases, both in the affirmative: (i) Γ is a 2d curvilinear polygon with each side C 1,α for some 0 < α < 1 and with each corner angle in the range (0, 2π). (ii) Γ is Lipschitz, with sufficiently small Lipschitz character. Regarding (i): this result was announced by Shelepov in [95], with details of the proof given in 2 The invertibility of 1 2 I − D on L 2 (Γ) implies that the bilinear form of the associated least-squares formulation is coercive. This formulation, however, suffers from the same disadvantages as the variational formulation of 1 2 I − D in H −1/2 (Γ), including that computing the entries of the Galerkin matrix requires computing integrals over all of Γ, even when the basis functions have support on (small) subsets of Γ.   [22,Corollary 3.5]; for more discussion on both (i) and (ii), see [22, §1].
The recent paper [22] finally settled the question above negatively by giving examples of 2-d Lipschitz domains and 3-d starshaped Lipschitz polyhedra for which 1 2 I ± D and 1 2 I ± D cannot be written as the sum of a coercive operator and a compact operator in the space L 2 (Γ). The 3-d starshaped Lipschitz polyhedra are defined in [22,Definition 5.7], and called the open-book polyhedra; see Figure 1.1 for an example, where we use the notation that Ω θ,n is the open-book polyhedron with n pages and opening angle θ. Given > 0 there exists θ 0 ∈ (0, π] such that the essential numerical range of D in L 2 (Γ) contains the interval [− √ n/2 + , √ n/2 − ] [22,Theorem 1.3]. By the definition of the essential numerical range (see, e.g., [22,Equation 2.3]), this result implies that if θ is sufficiently small and n ≥ 2, then 1 2 I ± D and 1 2 I ± D cannot be written as the sum of a coercive operator and a compact operator in the space L 2 (Γ) when Γ = ∂Ω θ,n .
Nevertheless, Part (b) of Theorem 1.1 only shows that the Galerkin method applied to these domains does not converge for every asymptotically dense sequence (H N ) ∞ N =1 ⊂ L 2 (Γ), leaving opening the possibility that all Galerkin methods used in practice (based on boundary element method discretisation [100,94]) are in fact convergent. However, the following result from [22] clarifies that this is not the case. (1.10) We can apply this result when (H * N ) ∞ N =1 is a sequence of boundary element subspaces that is asymptotically dense in L 2 (Γ), in which case (H N ) ∞ N =1 , satisfying (1.10), is also a sequence of boundary element subspaces (since H N ⊂ H * M N ) and is also asymptotically dense in L 2 (Γ) (since H * N ⊂ H N ). In summary, the results of [22] show that there exist Lipschitz and polyhedral boundaries Γ for which there are Galerkin methods for solving BIEs involving 1 2 I ± D and 1 2 I ± D that do not converge, with these methods based on asymptotically-dense sequences (H N ) ∞ N =1 ⊂ L 2 (Γ) of boundary element subspaces.

Motivation for the present paper and summary of the main results
Given the negative results of [22] about convergence of the Galerkin method for the standard second-kind formulations, a natural question is therefore Do there exist second-kind BIE formulations in L 2 (Γ) of Laplace's equation where, with Γ only assumed to be Lipschitz, the operators are continuous, invertible, and can be written as the sum of a coercive operator and a compact operator?
In this paper we answer this question in the affirmative for the Laplace interior and exterior Dirichlet problems. We present new BIE formulations that are continuous and in fact coercive (i.e., not just the sum of a coercive and a compact operator) in the space L 2 (Γ), with Γ only assumed to be Lipschitz; thus convergence of the Galerkin method in L 2 (Γ) for every asymptotically-dense sequence (H N ) ∞ N =1 , plus the explicit error estimate (1.9), is ensured by Part (c) of Theorem 1.1. Furthermore, the strong property of coercivity allows us to prove that, if the Galerkin matrices are preconditioned by a specified diagonal matrix, then the number of GMRES iterations required to solve the associated linear systems to a prescribed accuracy does not increase as the discretisation is refined and N increases.
Outline of the paper. §1.6 defines more precisely the Laplace BVPs we consider. §1.7 recaps results about a non-standard layer potential introduced in [16] and its non-tangential limits. §2 states the main results. §3 discusses the ideas behind the main results, and the links to other work in the literature. §4 proves the main results, except the parts of the proofs that are related to the wellposedness and regularity of the Laplace oblique Robin problem, with these given in §5. §6 presents results for the Helmholtz exterior Dirichlet problem (with these results corollaries of the Laplace results in §2).

Notation and statement of the BVPs
Let Ω − ⊂ R d , d ≥ 2, be a bounded (not necessarily connected) Lipschitz open set, and let Ω + := R d \ Ω − and Γ := ∂Ω − . Let n be the outward-pointing unit normal vector to Ω − (so n points out of Ω − and into Ω + ).
We make three remarks.
(i) Recall that, by elliptic regularity (see, e.g., [72,Theorem 4.16]), the solution of the IDP and EDP are C ∞ in Ω − and Ω + respectively. Therefore, the pointwise conditions at infinity imposed in the EDP make sense.
Definition 1.6 (Laplace EDP formulated via non-tangential limits) With Ω − and Ω + as above, assume further that Ω + is connected. Given g D ∈ L 2 (Γ), we say that u ∈ C 2 (Ω + ) with u * ∈ L 2 (Γ) satisfies the EDP if ∆u = 0 in Ω + , γ + u = g D on Γ, and u(x) = O(1) when d = 2 and Existence and uniqueness of the solutions of these formulations of the IDP and EDP go back to the work of Dahlberg [31], and are given explicitly in, e.g., [106, Corollary 3.2 and Lemma 3.7], [16, §3]. The following equivalence result is proved in Appendix D. Similarly, if g D ∈ H 1/2 (Γ), then the solution of the EDP in the sense of Definition 1.4 is the solution of the EDP in the sense of Definition 1.6, and vice versa.

Recap of results about layer potentials and their non-tangential limits
Recall that the surface gradient operator on Γ is the unique operator such that, when v ∈ C 1 (Ω − ), ∇v = n(n · ∇v) + ∇ Γ (γ − v) on Γ (and similarly for v ∈ C 1 (Ω + )); for an explicit expression for ∇ Γ in terms of a parametrisation of Γ, see, e.g., [20,Equation A.14].
The following results all rely on the harmonic-analysis results in [25] and [106] (see also the accounts in [75,Chapter 15], [102,Chapter 4], [55, Chapter 2, Section 2]). Define where the integral is understood in the principal-value sense. By [106,Theorem 1.6], ∇ Γ S : L 2 (Γ) → L 2 (Γ), with this mapping continuous, and (∇ Γ S)φ = ∇ Γ (Sφ). The following potential was introduced in [16, §2]; given Z ∈ (L ∞ (Γ)) d that is real-valued (which we assume throughout), let and let where the integral in (1.15) is understood in the principal-value sense. The results of [25] and [106] imply that K Z : Observe that (i) when Z = n, K Z = D, K Z = D, and (1.16) is the usual jump relation for the double-layer potential, and (ii) we can rewrite K Z as In a similar way to how the L 2 adjoint of D is D (see, e.g., [75,Chapter 15, text around Equation 4 .10]), the L 2 adjoint of K Z is The significance of the operator K Z is that it appears in the inner product of Z with the nontangential limit of ∇S, where S is the single-layer potential defined for φ ∈ L 2 (Γ) by Indeed, by [106, Theorems 1.6 and 1.11] (see also [75,Theorem 5], [20,Equation 2.30]), for almost every x ∈ Γ, so that We focus on the case d ≥ 3, since the question of whether or not there exist BIE formulations of the Laplace IDP and EDP that are coercive, or coercive up to a compact perturbation, on Lipschitz domains is more pressing when d = 3 than d = 2 (because of the existing convergence theory for ± 1 2 I + D and ± 1 2 I + D on curvilinear polygons [95,19,96] but negative results for these operators for certain 3-d starshaped polyhedra [22] recapped in §1.4). Results for d = 2 are given in §2.3.

The interior Dirichlet problem
Given Z ∈ (L ∞ (Γ)) d and α ∈ R, define the integral operators A I,Z,α , A I,Z,α , and B I,Z,α by A I,Z,α := 1 2 (Z · n)I − K Z + αS, A I,Z,α := 1 2 (Z · n)I − K Z + αS, (2.1) with the subscript I standing for "interior", and the superscript indicating that A I,Z,α is the L 2 adjoint of A I,Z,α .
is the solution of the Laplace IDP of Definition 1.5.
(iv) Coercivity up to compact perturbation for all α ∈ R. If Z ∈ (C(Γ)) d and there exists c > 0 such that Z(x) · n(x) ≥ c for almost every x ∈ Γ, (2.6) then, for all α ∈ R, both A I,Z,α and A I,Z,α are the sum of a coercive operator and a compact operator on L 2 (Γ).
For any bounded Lipschitz open set Ω − there exists Z ∈ (C 0,1 (Γ)) d such that (2.6) holds; for completeness, and because a formula for Z is needed for implementation, we include this result and a concrete, constructive proof in §A.1 below. The combination of this result and Parts (iii) and (vi) of Theorem 2.1 imply that, for any bounded Lipschitz open set, there exists a BIE formulation of the Laplace IDP that is continuous and coercive in L 2 (Γ).
is the solution of the Laplace EDP of Definition 1.6.
, and these mappings are continuous.
(iv) Coercivity up to compact perturbation for all α ∈ R. If Z ∈ (C(Γ)) d and there exists c > 0 such that (2.6) holds, then, for all α ∈ R, both A E,Z,α and A E,Z,α are the sum of a coercive operator and a compact operator on L 2 (Γ).
(vi) Coercivity for sufficiently large α. If Z ∈ (C 0,1 (Γ)) d with Lipschitz constant L Z and (2.7) holds, then both A E,Z,α and A E,Z,α are coercive on L 2 (Γ) with coercivity constant c/2 (with c defined by (2.6)), in that (2.8) holds with A I,Z,α replaced by either A E,Z,α or A E,Z,α .
Similar to the case of the IDP, the result in §A.1 and Parts (iii) and (vi) of Theorem 2.2 imply that, for any bounded Lipschitz open set Ω − such that Ω + is connected, there exists a BIE formulation of the Laplace EDP that is continuous and coercive in L 2 (Γ).

2.1.3
The new formulations of the IDP and EDP for d ≥ 3 on domains that are star-shaped with respect to a ball When Ω − is star-shaped with respect to a ball, the coercivity results in Theorems 2.1 and 2.2 take a particularly simple form.
(ii) D is star-shaped with respect to the ball B κ (x 0 ) if it is star-shaped with respect to every point in B κ (x 0 ).
From now on, if a domain D is star-shaped with respect to x 0 , we assume (without loss of generality) that x 0 = 0.  Since the proof is so short, we include it here. We highlight that Corollary 2.6 is the first time convergence of the Galerkin method for a BIE posed in L 2 (Γ) used to solve a boundary-value problem for Laplace's equation has been proved with the only assumption on Γ that it is Lipschitz; the same is true if Γ is assumed to be Lipschitz polyhedral in 3-d.
Remark 2.7 (Bounding the best approximation and Galerkin errors) For 3-d Lipschitz polyhedra the smoothness of the solution, in particular its singularities at corners and edges, is well understood (see, e.g., [108]) for the direct formulations (2.3) and (2.11), where the solution of the integral equation is φ = ∂ ± n u. Moreover, it is well understood how to design effective h-and hpboundary element approximation spaces H N based on graded, anisotropic meshes so as to obtain optimal best approximation error estimates (see, e.g., [108,34,35,68,69]), indeed exponential convergence of min ψ∈H N φ − ψ L 2 (Γ) as a function of M N := dim(H N ) if the Dirichlet data g D is the restriction to Γ of an analytic function (see [69,Theorem 3.1]). Further, by Part (a) of Corollary 2.6 and the quasioptimality (1.7), the same rates of convergence follow for the Galerkin error φ − φ N L 2 (Γ) as long as α > 0.

Solution of the Galerkin linear systems of the new formulations
Let H N = span{ψ N 1 , . . . , ψ N M N }, with M N = dim(H N ) and {ψ N 1 , . . . , ψ N M N } a basis for H N . The Galerkin method (1.4) applied to (2.4) is then equivalent to the linear system with A † Z,α := A I,Z,α , and with the Galerkin solution φ N given by The Galerkin method applied to (2.3), (2.12), or (2.11), respectively, is also equivalent to (2.15) with A † Z,α := A I,Z,α , A E,Z,α , or A E,Z,α in (2.16) and with correspondingly different definitions of the right-hand side components b i .
In each case, whether A † Z,α = A I,Z,α , A I,Z,α , A E,Z,α , or A E,Z,α , the matrix A defined in (2.16) is non-symmetric, and a popular method for solving such non-symmetric linear systems is the generalised minimum residual method (GMRES) [93], which we now briefly recall. Consider the abstract linear system Cx = d in C M N , where C ∈ C M N ×M N is invertible. Let x 0 be an initial guess for x, and define the corresponding initial residual r 0 := Cx 0 − d and the corresponding standard Krylov spaces by K m (C, r 0 ) := span C j r 0 : j = 0, . . . , m − 1 .
For m ≥ 1, define the mth GMRES iterate x m to be the unique element of K m such that its residual r m := Cx m − d satisfies the minimal residual property The main result of this subsection (Theorem 2.11 below) is a result about the convergence of GMRES applied to (2.15) preconditioned by diagonal matrices. This result is proved under the following assumption in which (and subsequently) for every v N ∈ H N we denote by v ∈ C M N the Remark 2.9 (Relation of C 1 and C 2 in (2.17) to the mass matrix) Let M N be the mass matrix defined by i.e., D is an orthonormal basis of H N , then Assumption 2.8 is satisfied with D N equal the identity matrix.
Lemma 4.15 below shows that Assumption 2.8 is satisfied (and specifies the matrices D N ) when (H N ) ∞ N =1 are piecewise-polynomial subspaces allowing discontinuities across elements, under mild constraints on the sequence of meshes; in particular Lemma 4.15 covers nodal basis functions on highly anisotropic meshes, such as the meshes highlighted in Remark 2.7. We highlight that the assumption that discontinuities are allowed is made so that we can assume in the proof that each basis function is supported on only one element, but we expect the result to hold more generally. In particular, if d = 3 and the sequence of meshes is regular, shape-regular, and quasi-uniform (in the sense of [ Let y m be the mth iterate when the linear system is solved using GMRES with zero initial guess. (Since D N is diagonal, the cost of calculating the Theorem 2.11 (Convergence of GMRES applied to the linear system (2.19)) Assume that Z is Lipschitz, there exists c > 0 such that (2.6) holds, α satisfies (2.7), and Assumption 2.8 holds. With C 1 and C 2 as in (2.17), and where The key point about Theorem 2.11 is that both the bound on the number of iterations (2.23) and the terms on the right-hand side of (2.24) other than the best-approximation error are independent of the dimension M N . Therefore, the number of iterations required to solve systems involving D −1/2 N AD −1/2 N to a prescribed accuracy does not increase as the discretisation is refined and M N increases. The same property holds when the conjugate-gradient method is applied to sequences of M N ×M N symmetric, positive-definite matrices whose condition number is bounded independently of M N . Remark 2.12 (Bounds on the condition number) Recall that, in general, a bound on the condition number for a nonnormal matrix cannot be used to rigorously prove results about the convergence of GMRES applied to that matrix; see, e.g., [67,Page 165], [36,Page 3]. We have no reason to expect that A is normal, so to prove Theorem 2.11 we crucially use the coercivity of A I,Z,α .
Nevertheless, since there is a long history of studying the condition numbers of second-kind integral equations posed on L 2 (Γ), we record that in the course of proving Theorem 2.11 we prove that, where C :  shows that constructing the Galerkin matrix requires evaluating integrals involving the operators above, and evaluating integrals of the form the integral (2.27) can therefore be evaluated in terms of integrals only involving D and (by the discussion above regarding (2.26)) S.

New boundary integral equations for the Laplace interior and exterior Dirichlet problems on general 2-d Lipschitz domains
The biggest difference in going from d ≥ 3 to d = 2 is that the single-layer potential is no longer o(1) at infinity, and is only O(1) for a restricted class of densities; see (4.23), (4.24) below. In this section, we first outline what parts of the d ≥ 3 results in §2.1 immediately carry over to d = 2.
We then present modifications of the integral equations in Theorem 2.1 and Theorem 2.2 that are coercive for general 2-d Lipschitz domains when α is sufficiently large. Inspecting the proof of Theorem 2.1 in §4, we see that Parts (i), (ii), (iii), and (iv) hold when d = 2 (i.e., everything apart from invertibility (v) and coercivity for sufficiently large α (vi)).
Similarly, inspecting the proof of Theorem 2.2 in §4, we see that Parts (i), (iii), (iv), and (v) hold when d = 2 (i.e., everything apart from the indirect formulation (ii) and coercivity for sufficiently large α (vi)), although, firstly, αu ∞ must be added to the right-hand side of the BIE (2.11), where u ∞ is the limit of u at infinity 5 and, secondly, Part (v) holds when d = 2 provided the constant a in the fundamental solution (1.1) is not equal to the capacity of Γ, Cap Γ (defined in, e.g., [72, Page 263]), which holds, in particular, if a > diam(Γ). Let i.e., P Γ φ is the mean value of φ. Observe that P 2 Γ = P Γ and P Γ = P Γ . Let Q Γ := I − P Γ . We give two theorems: the first for general 2-d Lipschitz domains, the second for 2-d star-shaped Lipschitz domains.
is the solution of the Laplace IDP of Definition 1.5.
(iii) EDP direct formulation. Let u be the solution of the Laplace EDP of Definition 1.4 with d = 2 and g D ∈ H 1 (Γ). Let (2.33) is the solution of the Laplace EDP of Definition 1.6.

Theorem 2.15 (New integral equations for 2-d star-shaped domains)
(i) IDP direct formulation. Let u be the solution of the Laplace IDP of Definition 1.3 with d = 2 and g D ∈ H 1 (Γ). Then ∂ − n u satisfies is the solution of the Laplace IDP of Definition 1.5.
(iiii) EDP direct formulation. Let u be the solution of the Laplace EDP of Definition 1.3 with d = 2 and g D ∈ H 1 (Γ). Then ∂ + n u satisfies is the solution of the Laplace EDP of Definition 1.6.

How the BIEs arise
The indirect BIE (2.4) for the IDP arises from imposing the boundary condition on the ansatz u = (K Z − αS)φ via taking the nontangential limit. Similarly, the indirect BIE (2.12) for the EDP arises from the ansatz u = (K Z + αS)φ. For the indirect BIEs for d = 2 in Theorem 2.14, the idea is the same, except now a) the density in the ansatz is not a general L 2 (Γ) function (so that Sφ has the correct behaviour at infinity), and b) extra terms are added to the ansatz to ensure that the resulting BIE is still coercive on L 2 (Γ). For the direct BIE (2.3) for the IDP, recall that u = S∂ − n u − Dγ − u by Green's integral representation. The direct BIE (2.3) then arises from considering Similarly, the direct BIE (2.11) for the EDP arises from considering 3) can be obtained by adding (i) (Z · n) multiplied by the standard direct second-kind BIE and (iii) α multiplied by (3.2). Similar considerations hold for the direct BIE (2.11), and the 2-d direct BIEs of Theorems 2.14 and 2.15, where, additionally, one uses that P Γ (∂ ± n u) = 0 (see Lemma 4.10).

The other BVPs solved by the new BIEs
The BIEs introduced in §2 to solve the Dirichlet problem can be used to solve other Laplace BVPs. Although the focus of this paper is on solving the Dirichlet problem, we highlight this fact here since these other BVPs affect the properties of the new BIEs.
For example, the BIO A E,Z,α used to solve the EDP in Theorem 2.2 can also be used to solve the Laplace interior oblique Robin problem, i.e., the problem of finding u in Ω − satisfying ∆u = 0 and

The use of similar BIEs by Calderón [16] and Medková [74]
Calderón [16]  In both [16] and [74], the BIOs are proved to be Fredholm of index zero on L 2 (Γ); see [16,Page 39] (where the result is proved to hold on a slightly wider range of L p (Γ) spaces) and [ Our coercivity results are proved using the identity arising from multiplying ∆u by Z · ∇u + αu (see Lemma 4.6 below); our use of a multiplier that is a linear combination of u and a derivative of u is inspired by the use of such multipliers by Morawetz [82,84,83], and the particular identity we use also appears as [60, Equation 2.28]. As recalled in §1.3, the idea of proving coercivity of Laplace BIOs in the trace spaces goes back to Nédélec and Planchard [86], Le Roux [62], Hsiao and Wendland [52], and Steinbach and Wendland [101], with this method based on using Green's identity (i.e. multiplying ∆u by u). The idea of proving coercivity of second-kind BIOs in L 2 (Γ) using Rellich-type identities was introduced in [97] for a particular Helmholtz BIE on star-shaped domains and then further developed in [98] for the standard second-kind Helmholtz BIE on smooth convex domains. The main differences between [97,98] and the present paper are that (i) [97,98] only consider direct BIEs for the exterior Helmholtz Dirichlet problem whereas the present paper considers direct and indirect BIEs for the interior and exterior Laplace Dirichlet problems and (ii) [97,98] only prove coercivity under geometric restrictions on Γ (which is somewhat expected for the high-frequency Helmholtz equation; see [7], [23, §6.3.2]), namely star-shapedness with respect to a ball for [97] and strict convexity and a piecewise analytic C 3 boundary for [98], whereas the present paper proves coercivity of Laplace BIOs for general Lipschitz domains.

Combined-potential ansatz for solutions of Laplace's equation
A key difference between the indirect BIEs in the present paper and those in [16] is that ours arise from the ansatz u = (K Z − αS)φ for the solution of the Laplace IDP, whereas [16] poses the ansatz u = K Z φ. We saw in the discussion above that the presence of the parameter α -i.e., the fact that we use a combined-potential ansatz -is crucial for proving coercivity of our BIOs.
The combined-potential ansatz is also crucial to proving uniqueness for cases where coercivity does not hold. Indeed, using a linear combination of double-and single-layer potentials to find solutions of the Helmholtz equation is standard, and goes back to [10,64,88], with the motivation to ensure uniqueness at all wavenumbers (see §6). Using such a combination for Laplace's equation is less common, but this was done by D. Mitrea in [76, Theorem 5.1] and subsequently by Medková in [73]. The rationale for this combined ansatz is similar, namely that the standard indirect second-kind equations (based on a double-layer-potential ansatz) have non-trivial null spaces for multiply connected domains (with these characterised in [59,76]) but the BIOs resulting from a combined double-and single-layer potential ansatz are invertible no matter the topology of Ω − ; see [ .2) and (3.1). If g D ∈ H 1 (Γ), then ∂ − n u ∈ L 2 (Γ) (by Theorem C.1), and then the mapping properties (B.3a) of S and D imply that both sides of (3.2) are in H 1 (Γ). Taking the surface gradient, ∇ Γ , of (3.2) yields the (vector) integral equation in (L 2 (Γ)) d Taking (Z · n) times the scalar equation ( For Part (ii) of Theorem 2.1, first recall that K Z φ and Sφ are both in C 2 (Ω − ) and satisfy Laplace's equation (for K Z this was recalled in §1.7). When φ ∈ L 2 (Γ), Sφ ∈ H 3/2 (Ω − ) by (B.2) and then (Sφ) * ∈ L 2 (Γ) by Part (iii) of Theorem C.2. As recalled in §1.7, (K Z φ) * ∈ L 2 (Γ) by [106], and thus u defined by (2.5) satisfies u * ∈ L 2 (Γ). To show that φ satisfies the BIE (2.4), we take the non-tangential limit of (2.5), using (1.16) and that, by Lemma C.3, γ − (Sφ) = γ − (Sφ), where γ − (Sφ) is given by the first jump relation in which is proved in a similar way to the bound on the double-layer potential in [94,Equation 3.23]. Part (iii) of both Theorems 2.1 and 2.2 follows from the mapping properties (B.3) with s = 1/2, and from the fact, recalled in §1.7, that K Z : L 2 (Γ) → L 2 (Γ) and ∇ Γ S : L 2 (Γ) → L 2 (Γ), with these mappings continuous.
4.2 Proofs of Part (iv) of Theorems 2.1 and 2.2 (coercivity up to a compact perturbation).
Thus there exists a compact operator C : Part (iv) of both Theorems 2.1 and 2.2 follow by combining Lemma 4.1 with the assumption (2.6) and the fact that S is compact on L 2 (Γ) (via the mapping property in (B.3a) with s = 1/2).
Proof of Lemma 4.1.
Since y); the definitions of K Z (1.15) and K Z (1.18) then imply that, for all φ ∈ L 2 (Γ), If Z ∈ (C 0,β (Γ)) d for β > 0, then the kernel of the integral on the right-hand side of (4.4) is weakly singular, and thus the operator is compact on L 2 (Γ) by, e.g., the combination of [ [25] and [106] (as discussed in §1.7). Let K Z = Z · T; then 4.3 Proof of Theorem 2.5 (coercivity for Ω − that are star-shaped with respect to a ball) Theorem 2.5 is an immediate consequence of combining (i) the following special case of Lemma 4.1, (ii) the definitions of A I,Z,α and A I,Z,α in (2.1) and A E,Z,α and A E,Z,α in (2.9), and (iii) the inequality (Sφ, φ) L 2 (Γ) ≥ 0 for all φ ∈ L 2 (Γ). The inequality in (iii) is well-known, following from Green's identity, and is a special case of Lemma 4.4 below with Z = 0.

Lemma 4.2 (Key lemma for coercivity for star-shaped
where P Γ is defined by (2.28).
Proof of Lemma 4.2.
Remark 4.3 (Link with the work of Fabes, Sand, and Seo [40]) The analogue of (4.5) when Γ is the graph of a function (i.e., the boundary of a hypograph) appears in the first sentence after the first displayed equation on [40,Page 133]. Indeed, the analogue of the operator K Z for the hypograph with Z = e d (i.e., the unit vector pointing in the x d direction) arises in [40] when they apply the Rellich identity (4.9) below with u = Sφ, as part of their proof that λI − D is invertible on L 2 (Γ) for λ ∈ R with |λ| ≥ 1/2.

Proof of
with compact support, and α ∈ R satisfies the lower bound We first show how the coercivity results of Theorems 2.1 and 2.2 are a consequence of Lemma 4.4 combined with the following lemma.
Lemma 4.5 Given Z ∈ (C 0,1 (Γ)) d with non-zero Lipschitz constant, there exists a compactly supported Z ext ∈ (C 0,1 (R d )) d with the same Lipschitz constant as Z and such that Z ext | Γ = Z.
The proof of Lemma 4.5 is given in Appendix A. Note that, by the Kirszbraun theorem [57], [105], Z ∈ (C 0,1 (Γ)) d can be extended to a function Z ext ∈ (C 0,1 (R d )) d with the same (non-zero) Lipschitz constant, so to prove Lemma 4.5 we only need to show that there exists an extension with compact support.
Proof of Part (vi) of Theorems 2.1 and 2.2 assuming Lemmas 4.4 and 4.5. Given Z, by Lemma 4.5 there exists a compactly-supported Lipschitz extension of Z to R d with the same Lipschitz constant; call this Z. This Z then satisfies the assumptions of Lemma 4.4, and the inequality (2.7) then ensures that (4.7) holds (where we have used the inequality A 2 2 ≤ i j |(A) ij | 2 to show that sup x D Z(x) 2 ≤ dL Z ). Thus (4.8) holds (with K Z replaced by K Z ) and the coercivity results follow from the definitions of A I,Z,α and A I,Z,α (2.1) and A E,Z,α and A E,Z,α (2.9) and the inequality (2.6) on Z · n.
The proof of Lemma 4.4 is based on the following identity. The relationship of this identity to other similar identities in the literature is discussed in §3.4, and we note, in particular, that this identity appears as [60,Equation 2.28]; for completeness we include the short proof.
where Zv := Z · ∇v + αv . (4.10) Proof. Splitting Zv into its component parts, we see that the identity (4.9) is the sum of the identities To prove (4.12), expand the divergence on the right-hand side. The identity (4.11) is obtained by combining the identities which can both be proved by expanding the divergences on the right-hand sides.
For the proof of Lemma 4.4, we need the identity (4.9) integrated over a Lipschitz domain when v is the single-layer potential. As a step towards this, we prove the following lemma.   H 1 (D), and the usual product rule for differentiation holds for such functions. Thus F = 2 Zv∇v − |∇v| 2 Z is in (H 1 (D)) d and then (4.9) implies that ∇ · F is given by the integrand on the left-hand side of (4.16). Furthermore, Proof of Lemma 4.4. As discussed in §3, our strategy is to mimic the classic method of "trans-ferring" coercivity properties of the PDE formulation to the BIOs in the trace spaces, but with Green's identity (4.17) replaced by the integrated version of the Rellich-type identity (4.9). That is, we apply the integrated version of (4.9), namely (4.16), with v replaced by u = Sφ (with φ ∈ L 2 (Γ)), and D first equal to Ω − , and then equal to Ω + ∩ B R , where R > sup x∈Ω − |x|. At this stage we let Z be a general real-valued W 1,∞ (R 3 ) vector field with compact support, and let α be an arbitrary real constant. That (4.16) holds with v replaced by u = Sφ can be justified by using the results of [54] and [20, Appendix A] recapped in §C. Indeed, when φ ∈ L 2 (Γ), u = Sφ ∈ H 3/2 (D) when D = Ω − or Ω + ∩ B R by the first mapping property in (B.2); then u ∈ V (D) by Corollary C.5 and (4.16) holds by Lemma 4.7. 6 We have therefore established that (4.16) holds when D = Ω − or Ω + ∩ B R and u = Sφ for φ ∈ L 2 (Γ). That is, with the identity (4.9) written as ∇ · Q = P , where (remembering that ∆u = 0 and α is a constant) If R is chosen large enough so that supp Z ⊂ B R , then where we have used the fact that u is C ∞ in a neighbourhood of Γ R (either by elliptic regularity or directly by the definition of the single-layer potential (1.19)) to justify writing ∂u/∂r in place of some appropriate trace. Adding (4.18) and (4.19) yields as |x| → ∞, uniformly in all directions x/|x|. If d = 2 then Recalling the definition of P Γ (2.28) and the assumption that P Γ φ = 0 when d = 2, we see that, by (4.22) (4.25) The expressions for Q ± · n (4.21) and the single-layer potential jump relations (4.2) then imply that . (4.26) A key identity to help one see this is which can be established using a 2 − b 2 = (a − b)(a + b) and the jump relations (4.2) for ∂ ± n u. Combining (4.25), (4.26), and (2.28), we therefore have that Using the Cauchy-Schwarz inequality and the definition of the matrix 2-norm for the term involving 2 ∂ i Z j ∂ i u∂ j u = 2 ∇u · (D Z ∇u), and then standard results about integrals for both this term and the term involving ∇ · Z, we find that the right-hand side of (4.27) is Therefore, choosing α to satisfy the lower bound (4.7) establishes the lemma with the + sign in (4.8). Multiplying (4.27) by −1 and letting α → −α we see again that if α satisfies (4.7) then this modified right-hand side is ≥ 0, which establishes the lemma with the − sign in (4.8). 2 This is because, despite the additional terms in the analogue of (4.22) coming from Z no longer having compact support, it turns out that Γ R Q R ds = O(R 2−d ) as R → ∞ as before. Letting α = (d − 2)/2 in (4.28) and recalling the definition (1.18) of K Z , we obtain the second equality in (4.5).
Remark 4.9 (Link with Verchota's proof of invertibility of 1 2 I − D on L 2 (Γ)) Verchota's proof that 1 2 I − D is invertible on L 2 (Γ) when Γ is Lipschitz in [106, Theorem 3.1] relies on the inequalities which hold for all φ ∈ L 2 (Γ) for d ≥ 3 and for all φ ∈ L 2 (Γ) with P Γ φ = 0 for d = 2, and where the omitted constants depend only on the Lipschitz character of Ω − . (Note that [106, Theorem 2.1] proves the slightly weaker result that The inequalities in (4.29) can be obtained by applying the following Dirichlet-to-Neumann and Neumann-to-Dirichlet map bounds with u = Sφ and using the jump relations (4.2).
The link with our proofs of coercivity of our new BIEs comes from the fact that the bounds (4.30) and (4.31) can be proved using the identity (4.9) with α = 0 and Z the vector field of Lemma A.1; see, e.g., [

gives
where u ∞ is the limit of the solution of the EDP at infinity and we use Green's integral representation u(x) = −S∂ + n u(x) + Dγ + u(x) + u ∞ for x ∈ Ω + and d = 2. The BIEs (2.30) and (2.34) then follow by applying Q Γ = I − P Γ to the equations in (4.32) and then using that P Γ ∂ ± n u = 0 by Lemma 4.10, so that ∂ ± n u = Q Γ ∂ ± n u. For Part (ii), taking the non-tangential limit of u defined by (2.32) and using the jump relations (1.16) and (4.2) (similar to the proof of Part (ii) of Theorem 2.1) and the fact that Q Γ = I − P Γ , we obtain that γ − u = g D if the BIE (2.31) holds. Exactly as in the analogous proof for d ≥ 3 in §4.1, K Z ψ and Sψ with ψ ∈ L 2 (Γ) are in C 2 (Ω − ), have non-tangential maximal functions in L 2 (Γ), and satisfy Laplace's equation; therefore u defined by (2.32) inherits these properties.
The proof of Part (iv) is very similar to the proof of Part (ii), except that now need to show that u defined by (2.36) satisfies u(x) = O(1) as |x| → ∞; these asymptotics follow from the first bound in (4.23) (since P Γ Q Γ φ = 0) and the bound (4.3).
Since P Γ ∂ ± n u = 0 by Lemma 4.10, the BIEs (2.37) and (2.39) follow. The proofs of Parts (ii) and (iv) follow in the same way as the proofs of Parts (ii) and (iv) of Theorem 2.14, namely by taking non-tangential limits of u, using the jump relations (1.16) and (4.2), and using the asymptotics (4.3) for the exterior problem.
Part (v) follows immediately from using the second equation in (4.6).

Proof of the results in §2.2.2 (the conditioning results)
4.6.1 Proof of Theorem 2.11.
Theorem 2.11 is a special case of the following general theorem about GMRES applied to Galerkin linear systems of a continuous and coercive operator on a Hilbert space. We first establish some notation. As in §1.   .17), let β ∈ [0, π/2) be defined such that The first step in proving Theorem 4.11 is to establish the following relationship between the error φ − φ m N H , the GMRES relative residual r m 2 / r 0 2 , and the Galerkin error φ − φ N H . Lemma 4.12 Suppose that A : H → H is coercive (i.e., there exists C coer > 0 such that (1.8) holds) and Assumption 2.8 holds with · L 2 (Γ) in (2.17) replaced by · H . If C 1 and C 2 are as in (2.17) and φ m N is defined by (2.20), then The right-hand side of (4.36) contains the relative residual r m 2 / r 0 2 . The following bound, from [5], gives sufficient conditions on m for this relative residual to be controllably small; recall that this bound is an improvement of the so-called "Elman estimate" from [33,32].

Theorem 4.13 (Elman-type estimate for GMRES from [5]) Let C be an
is the field of values, also called the numerical range, of C. Let β ∈ [0, π/2) be such that (observe that cos β is indeed ≤ 1 by the definition of W (C)) and, given β, let Let r m be the mth GMRES residual, as defined in §2.2.2. Then Proof of Lemma 4.12. We first use continuity and coercivity of A to obtain bounds on the norm of D −1/2 N AD −1/2 N and its inverse. By the definition (2.16), Using this, along with the norm equivalence (2.17), we find that, for all v, w ∈ C M N , Letting The inequalities (4.39) and (4.40) then imply that with the second inequality and the Lax-Milgram theorem then implying that and then combining this with (4.43) we obtain Combining this last inequality with the triangle inequality, we obtain that ; note that here it is important that H is a Hilbert space over C, so that continuity and coercivity of A control W (A) (which involves A applied to vectors in C M N ).

Conditions under which Assumption 2.8 holds.
Our result about the convergence of GMRES applied to the Galerkin matrices of the new formulations, namely Theorem 2.11, is proved under Assumption 2.8, which is an assumption about the sequence of finite-dimensional subspaces (H N ) ∞ N =1 and their associated bases. Recall from §2.2.2 that Assumption 2.8 holds, indeed with D N the identity matrix, for any sequence (H N ) ∞ N =1 (and in any dimension d ≥ 2) provided that the bases we choose are orthonormal. But many standard implementations of boundary element approximation methods use non-orthogonal bases, particularly bases of so-called nodal basis functions (e.g., [2,47], [100, Page 216], [94, Pages 205 and 280]. We show as Lemma 4.15 below that Assumption 2.8 holds (moreover specifying the diagonal matrices D N ) under mild constraints on the sequence of meshes when the approximation space allows discontinuities across elements. In particular, Lemma 4.15 holds when nodal basis functions are used, including for sequences of highly anisotropic meshes.
To specify the conditions under which Assumption 2.8 holds, we recall the notion of a surface mesh on Γ, and aspects of the standard implementation of boundary element methods, including the notation of a reference element (for the moment, until we indicate otherwise, our results hold for any dimension d ≥ 2). Following, e.g., [94, Defn. 4.1.2], we call G a mesh of Γ if G is a set of finitely many disjoint, relatively open, topologically regular 7 subsets of Γ that cover Γ in the sense that Γ = ∪ τ ∈G τ , and are such that the relative boundary of each τ ∈ G has zero surface measure. We call the elements of G the (boundary) elements of the mesh and, for τ ∈ G, set h τ := diam(τ ) and s τ := |τ |, where |τ | denotes the (d − 1)-dimensional surface measure of τ , and set h := max τ ∈G h τ .
Consider now the case that we keep R, p, and the bases {ψ τ 1 , . . . , ψ τ P τ }, τ ∈ R, fixed but use a sequence of meshes G N , N ∈ N, with associated approximation spaces H N := S p G N that are such that h N := max τ ∈G N h τ → 0 as N → ∞, i.e. we consider the h-version of the boundary-element method. Lemma 4.15 below applies in this regime under the following assumption on the constants g ± τ defined by (4.45) (this assumption is the first half, Equation 3.5a, of [47, Assumption 3.1]). Assumption 4.14 There exists a constant c 1 ≥ 1 such that, for every N ∈ N and τ ∈ G N , equivalently, there exists a constant c 2 ≥ 1 such that, for every N ∈ N and τ ∈ G N , We make two remarks about Assumption 4.14.
For the following lemma, recall that the matrix A is given by (2.16) with A † Z,α equal to one of A I,Z,α , A I,Z,α , A E,Z,α , or A E,Z,α .

Lemma 4.15 (Conditions under which Assumption 2.8 holds)
Suppose that, while keeping R, p, and the bases {ψ τ 1 , . . . , ψ τ P τ }, τ ∈ R, fixed, we use a sequence of meshes G N , N ∈ N, with associated approximation spaces H N := S p G N and bases (4.48) that are such that h N := max τ ∈G N h τ → 0 as N → ∞ and Assumption 4.14 holds. Then the following is true.
Part (ii) of Lemma 4.15 is proved for d = 3 using the coercivity results of Part (vi) of both Theorems 2.1 and 2.2. An analogous result holds for d = 2 using the coercivity results of Part (v) of Theorem 2.14, but we omit this for brevity. (4.48),
A regularity result of Nečas [85] (stated as Theorem C.1 below) implies that either of the requirements ∂ − n u ∈ L 2 (Γ) and γ − u ∈ H 1 (Γ) in Definition 5.1 can be removed; similarly in Definition 5.2.
The IORP and EORP can also be formulated in terms of non-tangential maximal functions and non-tangential limits (similar to the case of the Dirichlet problem discussed in §1.6). We now give this alternative formulation for the IORP and prove that it is equivalent to Definition 5.1; this equivalence is necessary to use results from the harmonic-analysis literature on the standard Laplace oblique derivative problem (see Theorem 5.13 below). The alternative formulation for the EORP and proof of equivalence to Definition 5.2 are completely analogous and are omitted.
Proof of Theorem 5.6. This is very similar to the proof of Theorem 5.5, except that now we can also consider d = 2, since (by definition) there are no conditions at infinity imposed on the solution of the IORP. and Proof. We first prove (5.8). Suppose A E,Z,α φ = g with φ, g ∈ L 2 (Γ) and let u := Sφ. Then γ + u = γ − u = P −,α,Z ItD g by the first jump relation in (4.2) and Theorem 5.6. By the second jump relation in (4.2), the definition of P + DtN , and the boundary condition (5.1), which implies (5.8). The proof of (5.9) is then very similar, using Theorem 5.5 instead of Theorem 5.6.
Corollary 5.9 (Existence for the IORP and invertibility of A E,Z,α ) If the assumptions of Theorem 5.8 hold and a = Cap Γ when d = 2, then A E,Z,α is invertible and the IORP has exactly one solution.
Corollary 5.11 (Existence for the EORP and invertibility of A I,Z,α ) If the assumptions of Theorem 5.10 hold and d ≥ 3, then the EORP has exactly one solution and A I,Z,α is invertible.
The following result is standard in the theory of potential theory on Lipschitz domains; see, e.g., [107,Page 203].
We can now prove Theorems 5.8 and 5.10 and Corollaries 5.9 and 5.11.
Proof of Theorem 5.8. Suppose that u satisfies the IORP with g = 0. To show that u = 0 it is enough to show that u ≤ 0 in Ω − , since this implies, by the same argument applied to −u, that also u ≥ 0, and hence u = 0. By Theorem 5.14, u ∈ C 1 (Ω − ) (indeed ∇u is Hölder continuous). By the maximum principle, since u ∈ C 2 (Ω − ) ∩ C(Ω − ) is harmonic in Ω − , the maximum value of u in Ω − is attained at some point x 0 ∈ Γ. Since u ∈ C 1 (Ω − ) it follows from (5.1) with g = 0 that Since Z is continuous and Z·n ≥ c ≥ 0 almost everywhere on Γ, x 0 −hZ(x 0 ) ∈ Ω − for all sufficiently small h > 0 by Lemma 5.12, so that Z(x 0 ) · ∇u(x 0 ) ≥ 0 since x 0 is the global maximum. Since α(x 0 ) > 0, it follows that u(x 0 ) ≤ 0, so that u ≤ 0 in Ω − .
Proof of Theorem 5.10. Suppose that u satisfies the EORP with g = 0. As in the proof of Theorem 5.8, it is enough to show that u ≤ 0 in Ω + . We recall that, when d = 2, the condition that u is bounded on Ω + implies that, for some u ∞ ∈ R, uniformly in x/|x|, and that Equation 6.11]. By Corollary 5.15, u ∈ C 1 (Ω + ). By the maximum principle, since u ∈ C 2 (Ω + ) ∩ C(Ω + ) is harmonic in Ω + , the maximum value of u in Ω + is attained on Γ or, when d = 2, u(x) ≤ u ∞ for x ∈ Ω + . If the maximum is attained on Γ, the result that u ≤ 0 follows by arguing as in the proof of Theorem 5.8. Therefore, it is sufficient to prove that the maximum is attained on Γ when d = 2. If u(x) ≤ u ∞ for x ∈ Ω + , then (5.11) implies that u(x) = u ∞ for |x| ≥ R if Γ ⊂ B R , so that the maximum is attained in Ω + . The maximum principle (see, e.g., [58, Theorem 6.8]) then implies that u is constant in Ω + , so that the maximum is also attained on Γ.
The following proofs of Corollaries 5.9 and 5.11 use the fact that, when α ∈ L ∞ (Γ), Z is continuous, and (2.6) (i.e., the first lower bound in (5.10)) holds, then A I,Z,α and A E,Z,α are Fredholm of index zero by Parts (iii) and (iv) of Theorem 2.1 and 2.2 respectively. Although these two theorems are for d = 3, Parts (iii) and (iv) also hold when d = 2 (as noted at the beginning of §2.3).
Proof of Corollary 5.9. If we can prove invertibility of A E,Z,α , then existence of a solution to the IORP follows from Theorem 5.5. Since A E,Z,α is Fredholm of index zero on L 2 (Γ), by the Fredholm alternative (see, e.g., [72,Theorem 2.27]), to prove invertibility it is sufficient to prove injectivity. Assume that A E,Z,α φ = 0 for φ ∈ L 2 (Γ). By Theorem 5.6, u := Sφ satisfies the IORP, and by Theorem 5.8 u = 0 in Ω − . Therefore γ − u = 0 and the first jump relation in (4.2) implies that Sφ = 0. Lemma B.1 then implies that φ = 0 and the proof is complete.
Proof of Corollary 5.11. This is very similar to that of Corollary 5.9 except that now we only work in d ≥ 3, since Theorem 5.5 requires d ≥ 3.
Remark 5.16 (The results of [65]) Although not directly used to prove the results in this section, the results of [65] concern the Laplace IORP in Lipschitz domains with Hölder continuous Z and g, and we comment here on their relevance to the results above. The results of [65] give an alternative route for obtaining uniqueness of the IORP (i.e., proving Theorem 5.8). Indeed, in the proof of Theorem 5.8, once we have established that u ∈ C 1 (Ω − ) (by using Theorem 5.14), then uniqueness follows from [65,Theorem 3.2]. The reason we argue as we do in the proof of Theorem 5.8 is that this argument easily carries over to the proof of uniqueness for the EORP (Theorem 5.10), whereas [65, Theorem 3.2] concerns only the IORP.
Remark 5.17 (Additional uniqueness results for the EORP with Z = x) The coercivity result of Theorem 2.5 allows us to extend the range of α for which the EORP is unique when Z = x and d ≥ 3. Indeed, Theorem 2.5 implies that A I,Z,α is injective when Z = x, α(x) ≥ −(d − 2)/2 for almost every x ∈ Γ, and d ≥ 3. Then, using Theorem 5.5 and arguing as at the end of the proof of Corollary 5.9, we see that the solution of the EORP is unique under these conditions. This result proves uniqueness for certain non-positive values of α, which are not covered by Theorem 5.10.
Remark 5.20 (Link with the work of Medkova [74]) In [74,Theorem 5.23.5], the solution of the IORP is sought as (5.13) without the final term on the right-hand side, resulting in the BIE (A E,Z,α Q Γ + βP Γ )φ = g; this BIO is then proved to be invertible on L 2 (Γ) if β = α and α is sufficiently large [74,Theorem 5.23.5]. The advantage of including the final term on the right-hand side of (5.13) is that, by Theorem 2.14, the resulting BIO T E,Z,α,β is not just invertible when α is sufficiently large, but also coercive by Part (v) of Theorem 2.14.
6 New formulations of the Helmholtz exterior Dirichlet problem 6

.1 Statement of the new formulations
As a corollary of Theorem 2.2, we obtain results about BIE formulations of the Helmholtz exterior Dirichlet problem. We state the results in this subsection but defer the more substantial proofs to §6.3 (and see Remark 6.12).
Existence and uniqueness of the solution of the Helmholtz EDP for Lipschitz Ω − is shown in, e.g., [20,Theorem 2.10]. Theorem 6.4 below considers g D ∈ H 1 (Γ); this holds, for example, for the sound-soft scattering problem when u corresponds to the scattered field and g D is the Dirichlet trace of the (C ∞ ) incident field -see, e.g., [20, Proof of Theorem 2.12].
The solution of the EDP in the sense of Definition 6.1 exists and is unique by [103,Theorem 5.6,Part (ii)]. The following equivalence result is proved in Appendix D.  19) and (1.14), respectively, with Φ replaced by Φ k .
Given Z ∈ (L ∞ (Γ)) d and η ∈ L ∞ (Γ), in analogy with (2.9), define the integral operators A k,η,Z A k,η,Z , and B k,η,Z by Regarding notation, since we are only dealing with the exterior Helmholtz Dirichlet problem, we omit the subscript E present in the analogous operators for Laplace's equation, but add a subscript k to highlight the k dependence. We use the notation −iη rather than α for consistency with standard notation for Helmholtz BIEs -see (6.7) below -and allow η to be a function, rather than just a constant, to make a link to existing results -see Theorem 6.9 below.
Theorem 6.4 (New integral equations for the Helmholtz EDP) (i) Direct formulation. Let u be the solution of the Helmholtz EDP of Definition 6.1 with additionally g D ∈ H 1 (Γ). Then ∂ + n u satisfies is the solution of the Helmholtz EDP of Definition 6.2.
(iv) Coercivity up to compact perturbation for all η. If Z ∈ (C(Γ)) d and there exists c > 0 such that (2.6) holds, then, for all η ∈ L ∞ (Γ), both A k,η,Z and A k,η,Z are the sum of a coercive operator and a compact operator on L 2 (Γ).
If Z = n then A k,η,Z = A k,η , A k,η,Z = A k,η , and B k,η,Z = B k,η where A k,η := are the operators appearing in the standard "combined field" or "combined potential" BIEs for the Helmholtz EDP; see, e.g., [20,Equations 2.68 and 2.69]. Recall from §1.4 that there exist Lipschitz polyhedra such that each of A k,η and A k,η is not the sum of a coercive operator and a compact operator on L 2 (Γ). By Part (b) of Theorem 1.1, to prove that the Galerkin method converges when applied to either of the BIEs (6.5) or (6.6), it is sufficient to show that A k,η,Z and A k,η,Z are injective. Analytic Fredholm theory and the fact that A k,η,Z is a compact perturbation of A E,Z,α imply that A k,η,Z is invertible for all except at most a discrete set of wavenumbers. Lemma 6.5 If Z is Lipschitz, then there exists a discrete set D ⊂ C (i.e. each element of D is isolated) such that A k,η,Z is invertible on L 2 (Γ) for all k ∈ C\D (d ≥ 3 and odd) or k ∈ (C\R − )\D Exactly as in the Laplace case, injectivity of A k,η,Z and A k,η,Z is equivalent to uniqueness of the solution of an interior oblique Robin problem.
Corollary 6.8 For η ∈ L ∞ (Γ) and Z ∈ (L ∞ (Γ)) d , the operator A k,η,Z : L 2 (Γ) → L 2 (Γ) is injective if and only if the interior oblique impedance problem with g = 0 has only the trivial solution, and is surjective if and only if the interior oblique impedance problem has a solution for every g ∈ L 2 (Γ).
Whereas BVPs involving oblique derivatives have been well-studied for Laplace's equation (see the references in Theorem 5.13 and also [66], [74, § §5.23, 5.24, 6.19]), there do not appear to be any results in the literature on the unique solvability of the interior oblique impedance problem for the Helmholtz equation for all wavenumbers and general Lipschitz Ω − . The following theorem collects three situations in which the solution of the interior oblique impedance problem is known to be unique, and hence, by the results above, the Galerkin method applied to either of the BIEs (6.5) or (6.6) is provably convergent for every asymptotically-dense sequence of subspaces. Theorem 6.9 The solution of the Helmholtz interior oblique impedance problem is unique in the following situations.
(i) Ω − is Lipschitz and star-shaped with respect to a ball centred (without loss of generality) at the origin, k > 0, Z(x) = x, and (ii) Ω − is Lipschitz and is a finite union of domains as in (i), k > 0, and, on each connected part of Γ, where x j is the point from which that connected component of Ω − is star-shaped with respect to a ball, Z(x) = x − x j and for all unit tangent vectors t(x) at the point x.
The following theorem is the Helmholtz analogue of Theorem 5.7.
Theorem 6.10 Assume that the solution of the Helmholtz interior oblique impedance problem of Definition 6.6 exists and is unique, and let P −,η,Z

ItD
: L 2 (Γ) → H 1 (Γ) denote the map that takes g to γ − u, where u is as in Definition 6.6. Let P + DtN : H 1 (Γ) → L 2 (Γ) denote the exterior Dirichletto-Neumman map for the Helmholtz exterior Dirichlet problem. Then, as an operator on L 2 (Γ), Remark 6.11 (The star-combined operator) When Ω − is star-shaped with respect to a ball, one can choose Z and η so that A k,η,Z is coercive (not just coercive up to a compact perturbation). Indeed if Z(x) = x and η(x) is given by (6.10), then [97,Theorem 1.1] showed that A k,η,Z is coercive for all k > 0 with coercivity constant given by κ in (2.14) 9 . This operator was named the star-combined operator and given the notation A k . Note that the coercivity of the star-combined operator implies the uniqueness result about the interior oblique impedance problem in Part (i) of Theorem (6.9).

Discussion of the Helmholtz results and related literature
The ideas behind the results in §6.1. The proofs of Parts (i), (ii), and (iii) of Theorem 6.4 are very similar to their Laplace analogues in Theorem 2.2. The proof of Part (iv) (coercivity up to a compact perturbation on L 2 (Γ)) follows from Part (iv) of Theorem 2.2 (the analogous result for A E,Z,α ) since the difference A k,η,Z − A E,Z,α is compact (for any α ∈ R). Indeed, The bounds on Φ k (x, y) − Φ(x, y) in, e.g., [20,Equation 2.25] imply that D k − D and ∇ Γ (S k − S) map L 2 (Γ) continuously to H 1 (Γ), and are thus compact. Furthermore, the mapping property (B.3b) holds with S replaced by S k (again because of properties of Φ k (x, y) − Φ(x, y)) and thus S k is compact on L 2 (Γ). The link between A k,η,Z and the Helmholtz interior oblique impedance problem expressed in Theorem 6.7 and Corollary 6.8 is proved in an analogous way to the link between A E,Z,α and the Laplace IORP in Theorem 5.6 and Corollary 5.9.
Discussion of our results in the context of related literature. The summary is that there does not yet exist a BIE posed in L 2 (Γ) for solving the Helmholtz EDP that, for all Lipschitz Ω − and all k > 0, is bounded, invertible, and the sum of a coercive operator and a compact operator. The standard BIOs A k,η and A k,η (6.7) fail to be the sum of a coercive operator and a compact operator on the Lipschitz domains and 3-d starshaped Lipschitz polyhedra given in [22]. The new BIOs A k,η,Z and A k,η,Z have not been proved to be injective for all k > 0. The formulations of [12,13,37,38] are invertible for all k > 0 and the sum of a coercive operator and a compact operator, but only from H −1/2 (Γ) → H 1/2 (Γ). We now give more detail on all these points.
The motivation for considering second-kind Laplace BIEs over first-kind Laplace BIEs is primarily based on the good conditioning of second-kind BIEs. For Helmholtz BIEs, however, there is an additional complication compared to Laplace BIEs: the BIEs in the exterior Helmholtz analogues of (3.2) and (3.1), namely are not uniquely solvable for all values of k. Indeed, S k is not injective when k 2 is an Dirichlet eigenvalue of −∆ in Ω − ; this is because, from the first equation in (6.14), the Neumann trace of a Dirichlet eigenfunction of −∆ in Ω − satisfies S k ∂ − n u = 0. Similarly, 1 2 I + D k is not injective when k 2 is a Neumann eigenvalue of −∆ in Ω − ; this is because the operator 1 2 I + D k appears in the indirect formulation of the interior Neumann problem (see Table 1.1) and thus cannot be injective when k 2 is a Neumann eigenvalue.
The standard remedy (going back to [10,64,88] in the context of indirect BIEs and [14] in the context of direct BIEs) is to take a linear combination of the BIEs in (6.14) and observe that, for η ∈ C \ {0}, ∂ + n u satisfies A k,η ∂ + n u = B k,η g D (6.15) with A k,η and B k,η defined in (6.7). The mapping properties (B.3b) imply that, for |s| ≤ 1/2, Since D k − D and S k are both compact on L 2 (Γ) when Γ is Lipschitz, A k,η is the sum of 1 2 I + D and a compact operator. Therefore the question of whether or not A k,η is the sum of a coercive operator and a compact operator is equivalent to the analogous question for 1 2 I + D . The results recalled in §1. 4 imply that the answer to this question is yes if Γ is C 1 , Ω − is a 2-d curvilinear polygon with C 1,α sides, or Γ is a Lipschitz domain with sufficiently-small Lipschitz character; furthermore the answer to this question is no if Γ is one of the 2-and 3-d Lipschitz domains or 3-d starshaped Lipschitz polyhedra described in [22] (see [22, §1.2.1] for more detail).
The lack of a convergence theory for the Galerkin method applied to either A k,η or A k,η on L 2 (Γ) motivated [12,13,37,38] to introduce modifications of A k,η and A k,η . For direct BIEs, the idea in these papers is to choose an operator M : , is injective, and is the sum of a coercive operator and a compact operator (with this last property following from the facts that S : H −1/2 (Γ) → H 1/2 (Γ) is coercive and A k,η,M + iηS k is compact); different choices of M were then proposed and analysed in [12,13,37,38], and the corresponding interior boundary value problems proved to have a unique solution (so that the operator A k,η,M is injective). Convergence of the Galerkin method then follows from Part (b) of Theorem 1.1. However, the BIE involving A k,η,M is now a first-kind equation (in contrast to the second-kind equation (6.15)), and so one expects the conditioning of the Galerkin linear systems to worsen as the meshes are refined (as discussed in §1.3).

Proofs of the results in §6.1
Proof of Theorem 6.4. The proof that the Neumann trace ∂ + n u of the solution of the Helmholtz EDP satisfies the integral equation (6.5) is very similar to the analogous arguments for the Laplace integral equations; see the proofs of Theorems 2.1 and 2.2 above. As recalled in §6.2, the difference A k,η,Z − A E,Z,α is compact; therefore, Part (iv) of Theorem 2.2 implies that, for any η ∈ C, A k,η,Z is the sum of a coercive operator and a compact operator.
Proof of Lemma 6.5. By the definitions of A E,Z,α (2.9) and A k,η,Z (6.3), By Part (v) of Theorem 2.2, A E,Z,α is invertible if α satisfies (2.7). Furthermore, for all k ∈ C (d ≥ 3 and odd) or k ∈ C \ R − (d even), K Z,k − K Z − iηS k − αS is compact on L 2 (Γ); indeed, as discussed in the proof of Theorem 6.4 above, S k , S 0 , D k − D , and ∇ Γ S k − ∇ Γ S are all compact operators on L 2 (Γ). The result then follows from analytic Fredholm theory; see, e.g., [26,Theorem 8.26].
Proof of Theorem 6.7. The proof is similar to the proofs of Theorems 5.6 and 5.5. If u = S k φ with φ ∈ L 2 (Γ) then, by the jump relations (4.2), the fact that S k : L 2 (Γ) → H 1 (Γ) by (B.3a) and the fact that S k satisfies the same mapping properties as S in (B.2), u satisfies the oblique impedance BVP if and only if the boundary condition (6.8) holds. However, the jump relations (4.2) imply that this equation is precisely (6.9). On the other hand, if u satisfies the oblique impedance problem then g 0 := ∂ − n u − iηγ − u ∈ L 2 (Γ). Recall the interior impedance BVP with η ∈ R \ {0} and g = g 0 has exactly one solution (see, e.g., [20,Theorem 2.3]). Clearly this solution is u. Further, recalling first that A k,k,n = A k,k , which is invertible by [20,Theorem 2.27], and second that the interior impedance problem is the special case of the interior oblique impedance problem when Z = n, we see that u = S k φ with φ = (A k,k ) −1 g 0 ∈ L 2 (Γ) by the first part of this theorem.
Proof of Corollary 6.8. Surjectivity follows from Lemma 6.7. Injectivity follows provided that u = S k φ = 0 in Ω − only if φ = 0. Assume that φ ∈ L 2 (Γ) and u = S k φ is zero in Ω − . Then, by the jump relations (4.2), u satisfies the Helmholtz exterior Dirichlet problem in Ω + with zero Dirichlet data, and hence is zero by uniqueness of the solution of this BVP (see, e.g., [20,Corollary 2.9]). The jump relations (4.2) then imply that φ = ∂ − n u − ∂ + n u = 0, as required.
Proof of Theorem 6.9.
(i) In this situation, A k,η,Z is coercive (not just coercive up to a compact perturbation) by [97, Theorem 1.1]; see Remark 6.11.
(ii) This is proved in [44,Theorem 5.6] by showing that the corresponding A k,η,Z is injective; for related results about this A k,η,Z see [45, Appendix A].
(iii) Let := k −1 be the semiclassical parameter. In the notation of [43, §4], the boundary condition (6.8) then becomes where N := Z · n and D = −Z · ∇ Γ /i + c 1 ; see [43,Equation 4.1]. We now check that the hypotheses of [43,Theorem 4.6] hold, and then the result then follows from [43,Theorem 4.6] (in fact, this theorem proves that, under these conditions, the interior oblique derivative problem is well-posed, and the bound on the solution in terms of the data has the same k dependence as the bound for the standard interior impedance problem). In the notation of [43], σ(N ) = Z · n, σ(D) = −Z · ξ + c 1 , m 1 = 0, and m 0 = 1. We now spell out the construction from [49] (in slightly modified form), calculate bounds on the Z that we construct, and then show how both the constant α (required to satisfy (2.7)/(4.7)) and Functions θ 1 , ..., θ M satisfying (A.4) exist by, e.g., [50,Theorem 2.17], and indeed can be chosen so that equality rather than inequality holds in (A.4) and so that each θ m ∈ C ∞ comp (R d ), in which case we say that (θ 1 , ..., θ M ) is a partition of unity for Γ subordinate to the cover (B a1 (x 1 ), ..., B a M (x m )). We construct explicitly functions satisfying (A.4) below; a choice of Z that satisfies (A.1) is then since, for almost all x ∈ Γ, Furthermore, for unit vectors a and b, writing Z m in terms of its components as Z m = (Z m,1 , ..., Z m,d ), Therefore, with D Z the matrix with (i, j)th element ∂ i Z j and · 2 denoting the matrix 2-norm, D Z 2 ≤ Θ . Combining this with (A.7), we see that the inequality for α (4.7) holds if We now choose a specific form for the partition-of-unity functions θ m , and hence obtain a bound on Θ L ∞ (R d ) . The right hand side of (A.2) is still a cover for Γ if we reduce the size of the balls slightly, i.e., (A.2) implies that, for some µ ∈ (0, 1), Assuming that (A.9) holds for some µ ∈ (0, 1), one way of constructing the functions θ m to satisfy (A.4) is to set with χ ∈ C 0,1 [0, ∞) chosen so that 0 ≤ χ(t) ≤ 1, for t ≥ 0, χ(t) = 1, for 0 ≤ t ≤ µ, while χ(t) = 0 for t ≥ 1. (Observe that, with these choices of θ m , Z is no longer smooth, but only Lipschitz.) where M * ∈ N is the smallest integer such that every x ∈ Γ is in at most M * balls B am (x m ). Furthermore, for x ∈ R d , then, by (A.7), if we make the simple choice that With A I,Z,α defined by (2.1), assume that Then, by Theorem 2.1 combined with (A.8) and (A.12), A I,Z,α is coercive with coercivity constant c given by (A.6). Furthermore, by (A.11), A I,Z,α is bounded with (A. 16) A.2 Proof of Lemma 4.5 Proof. By the Kirszbraun theorem [57,105], Z ∈ (C 0,1 (Γ)) d can be extended to a function Z ext ∈ (C 0,1 (R d )) d with the same (non-zero) Lipschitz constant. Let R > 0 be such that Ω − ⊂ B R . For a > 1 and R * > aR, let χ(r) :=      1, R < r < aR, (R * − r)/(R * − aR), aR < r < R * , 0, r > R * , and let Z ext (x) := Z ext (x), |x| ≤ R, Z ext (R x) χ(|x|), |x| > R, where x := x/|x|, so that supp Z ext = B R * .
Let L be the Lipschitz constant of Z ext . We now prove that if a and R * are both large enough then Z ext (x) − Z ext (y) ≤ L|x − y| for all x, y ∈ R d , (A. 17) i.e. the Lipschitz constant of Z ext does not exceed that of Z ext . First observe that there exists a 0 > 1 such that if a ≥ a 0 and at least one of |x| and |y| are ≥ aR then Case 2: |x| ≤ R, R ≤ |y| ≤ aR. The key point here is that |x − y| ≥ |x − R y| so that Case 3: |x| ≤ R, aR ≤ |y| ≤ R * . Now If a ≥ a 0 , then which is ensured by (A. 19) since a ≥ a 0 > 1.
Case 5: R ≤ |x| ≤ aR, R ≤ |y| ≤ aR. Similar to Case 2, the key point here is that |x − y| ≥ |R x − R y| so that Z ext (x) − Z ext (y) = Z ext (R x) − Z ext (R y) ≤ L|R x − R y| ≤ L|x − y|.
Case 9: aR ≤ |x| ≤ R * , |y| ≥ R * . Now Since (R * − |x|) ≤ |x − y|, it is sufficient to prove that the right-hand side of this last expression is ≤ L(R * − |x|), and this is ensured by the inequality (A.19).

C Recap of harmonic-analysis results
In this appendix we recap results on the behaviour of solutions to Laplace's or Poisson's equation near the boundary of the domain. For simplicity, these results are stated for a bounded Lipschitz domain D with boundary ∂D. Analogues of the results then hold with D = Ω − and D = Ω + , where in the latter case spaces such as H 1 (D) become H 1 loc (Ω + ) (since these results do not assume any particular behaviour at infinity).  [72,Theorem 4.24].) If u ∈ H 1 (D) and ∆u ∈ L 2 (D), then ∂ n u ∈ L 2 (Γ) iff γu ∈ H 1 (Γ). Given x ∈ Γ, let Θ(x) be the non-tangential approach set to x from D defined, for some sufficiently large C > 1, as in (1.11). Given u ∈ C 2 (D) with ∆u = 0, let the non-tangential maximal function of u, u * , be defined by (1.12), and let the non-tangential limit of u, γu, be defined by (1.13).
The reverse inclusion is proved similarly: given v ∈ H 3/2 (D) with ∆v ∈ L 2 (D), define v as before. Since N (∆v) ∈ H 2 (D), v ∈ H 3/2 (D). Since ∆ v = 0, v ∈ C 2 (D) by elliptic regularity, and then ∂ n v ∈ L 2 (∂D) and γ v ∈ H 1 (Γ) by Lemma C.4; thus v ∈ V (D). The result that v ∈ V (D) then follows from the definition of v and the fact that N (∆v) ∈ H 2 (D) ⊂ V (D) We also need the following results in L p (∂D) for p = 2 (as opposed to the L 2 -based results above).
D Proofs of Theorems 1.7 and 6.3 Proof of Theorem 1.7. We prove the result for the IDP when d = 3; the proof for the EDP is very similar. The proof for the IDP when d = 2 is also similar, with use of [74,Theorem 5.15.2] replaced by use of [74,Theorem 5.15.3].
If u is the solution of the IDP in the sense of Definition 1.3 then, since u ∈ H 1 (Ω − ), γ − u = γ − u by Lemma C.3 and thus γ − u = g D . Furthermore, since u ∈ H 1/2 (Ω − ), u * ∈ L 2 (Γ) by Part (i) of Theorem C.2. Finally, by elliptic regularity u ∈ C 2 (D). Therefore u is a solution of the IDP in the sense of Definition 1.5 To prove the converse, let v := (−D + S)φ for φ ∈ L 2 (Γ), with D the double-layer potential defined by (B.1) and S the single-layer potential defined by (1.19). Now γ + v = ( Proof of Theorem 6.3. The proof is similar to the proof of Theorem 1.7. The main difference is that now we define v := (D k − ikS k )φ, where the Helmholtz single-and double-layer potentials S k and D k are defined by (1.19) and (B.1) with Φ replaced by Φ k . Now γ + v = ( 1 2 I + D k − ikS k )φ, where we have used that (i) γ + Dφ = ( 1 2 I + D k )φ by [103, §4], [20,Page 111], and (ii) γ + Sφ = γ + Sφ = S k φ by Lemma C.3 and the Helmholtz analogue of the first jump relation in (4.2). The proof then follows the same steps as in the proof of Theorem 1.7, using uniqueness of the Helmholtz EDP in the sense of Definition 6.2 (see [103,Theorem 5.6, Part (ii)]) and the fact that A k,k := 1 2 I + D k − ikS k : H s+1/2 (Γ) → H s+1/2 (Γ) is bounded and invertible for |s| ≤ 1/2 by [20, Theorem 2.27].