 2.13 (Bounds for non-Lipschitz Ω − ) In formulating the EDP we assumed that Γ D := ∂Ω − was Lipschitz. We see below that our proofs of the bounds in Theorems 2.5 and 2.7 also hold in the non-Lipschitz case; i.e. with the only assumptions on Ω − that it is a C 0 bounded open set that is star-shaped with respect to the origin (the BVP is then understand as the variational problem (3.5) below). These arguments were inspired by the arguments in [22], where the bound (1.9) was proved for such Ω − when A ≡ I and n ≡ 1.
Remark 2.14 (H 2 -regularity) The standard finite-element-error analysis of the Helmholtz equation is most comfortable when the solution of the adjoint BVP is in H 2 (H 2 loc for exterior problems), with a bound on the H 2 -norm of the adjoint BVP playing a key role (for the reasons behind this see, e.g., [83, Theorem 6.32 and Remarks 26 and 31] and the references therein). When ∂Ω − ∈ C 1,1, A ∈ C 0,1 (Ω +, R d×d ), and g D ∈ H 3/2 (Γ D ), the solution of the EDP (Definition 2.1) is in H 2 loc (Ω + ), and the bound in Theorem 2.5 can then be combined with [  and applying Theorem 2.5/2.7, we see that if k η L ∞ (ΩR) is sufficiently small, then bounds similar to those in Theorem 2.5/2.7 hold, since one can absorb the contribution from the k 2 ηu term in (2.12) into the µ 2 k 2 u 2 L 2 (ΩR) appearing on the left-hand side of (1.9).
In Theorems 2.5 and 2.7 the right-hand side of the PDE, f, is in L 2 (Ω + ) with compact support. A standard argument appearing in, e.g., [22,Text between   Let Ω −, A, and n satisfy the requirements in the definition of the EDP (Definition 2.1), along with the requirements in Condition 2.4/2.6. Given F ∈ (H 1 0,D (Ω R )) ′ let u be the solution of the variational formulation of the EDP (3.5) below. Then u exists, is unique, and satisfies the bound for all k > 0, where C 1 is given by (1.10). Thus, the inf-sup constant of the sesquilinear form a(·, ·) defined by (3.6) satisfies the lower bound

17)
for all k > 0, where (ii) If Ω −, A, n, f, and g D satisfy the requirements in the definition of the EDP (Definition 2.1) along with the requirements in Condition 2.18, then the solution of the EDP exists and is unique. Let L D := max x∈ΓD |x|, and let aL D be the radius of the ball with respect to which Ω − is starshaped. Given R > 0 with both supp(1 − n) and supp f compactly contained in Ω R, for all k ≥ 3/8R −1, where (2.19) and Observe that, given k 0 > 0, each of C j, j = 2,..., 5 can be bounded above, independently of k, for k ≥ k 0.
Analogues of Remarks 2.13, 2.14, and 2.15, and Corollary 2.16 hold for the results of Theorem 2.19. The only exception is that the assumption that Ω − is Lipschitz cannot be removed (as described in Remark 2.13) from Part (ii) of Theorem 2.19, since the bound (2.18) involves the normal derivative on the boundary, and one needs the domain to be Lipschitz for the normal derivative to be well-defined.
Remark 2.20 (Star-shaped vs. star-shaped with respect to a ball) In Conditions 2.4 and 2.17 we assumed that g D ≡ 0 and Ω − is star-shaped, and we obtained bounds under these conditions in Theorems 2.5 and 2.19(i) (respectively) using Morawetz identities. Morawetz identities can be used to obtain bounds when g D ≡ 0 if Ω − is star-shaped with respect to a ball, but when A ≡ I the calculations involving the surface gradient are more involved than the case A ≡ I (compare (4.9) and (4.12) below); we therefore exclude the case g D ≡ 0 from Theorems 2.5 and 2.19(i) for brevity.
On the other hand, Condition 2.18 (where A ≡ I) allows g D ≡ 0 and Ω − that are star-shaped with respect to a ball. Recall that g D ≡ 0 is needed to cover scattering of a plane-wave; indeed, in this case, u in Definition 2.1 is the scattered field and g D equals minus the trace of the incident plane wave (which is not zero); see, e.g., [21,Page 107]. Note that if Ω − is only star-shaped and additionally g D ≡ 0, then the bound (2.18) holds with all the norms over Γ D removed.

Remark 2.21 (Bounds on the full-space problem)
Taking the obstacle Ω − to be the empty set, changing Ω R to B R, and ignoring all the norms on Γ D, the results of Theorem 2.19 apply to the full-space problem.

Remark 2.22 (Dimensional arguments)
From dimensional arguments, we expect the factors C 1, C 2, and C 3 to have the dimension of (length) 2, and C 4 and C 5 to have dimension of length (see, e.g., [60, Remarks 3.6 and 3.8]). This is the case provided that we (a) interpret the R + 1 in C 2 as having the dimension of length, and (b) interpret the (A max ) 2 /k 2 in C 2 as having an extra (length) −2 factor (and thus being dimensionless). These apparent discrepancies arise because we choose a distance to be equal to one in the proof of the bound (2.17): we apply the Morawetz identity in a ball of radius R + 1, as opposed to R + b for some arbitrary b with dimensions of length. Thus in (a) above we have R + 1 instead of R + b and in (b) we have

Discussion of previous work
This discussion focuses on the exterior Dirichlet and full-space problems; nevertheless, there is a substantial literature on well-posedness and k-explicit bounds for the Helmholtz equation when either the obstacle Ω − is unbounded or at least one of supp(I − A), supp(1 − n), and supp f is not compact (e.g. scattering by infinite rough surfaces and/or infinite rough layers); see [61, Remark 3.9] and the references therein for an overview of results about these problems.
Existence and uniqueness results. As discussed in §1, the unique continuation principle (UCP) gives uniqueness of the solution of (1.1) when A is Lipschitz and n ∈ L ∞.
The Baire-category argument of [4] uses the fact that a UCP is known for the time-harmonic Maxwell system with Lipschitz coefficients [68] to prove uniqueness of the solution of the timeharmonic Maxwell problem posed in R 3 with piecewise-Lipschitz coefficients, provided that the subdomains on which the coefficients are defined satisfy [4, Assumption 1]. This assumption allows a large class of subdomains (including any bounded finite collection), but does not allow the subdomain boundaries to concentrate from below on a surface in Ω +, and thus the subdomains in Figure 1b are ruled out (see [4, Figure 1]).
The argument in [4] equally applies to the Helmholtz equation with piecewise-Lipschitz A and piecewise-L ∞ n, proving uniqueness (and hence, by Fredholm theory, well-posedness) of the BVPs corresponding to Condition 2.8 (see Figure 1a) and Condition 2.10 (see Figure 1c), but not of the BVP corresponding to Condition 2.9 (see Figure 1b).
Bounds on the EDP/full-space problem when A and n are continuous. The paper [9] considers A ∈ C 1, n ≡ 1, A → I uniformly as r → ∞, [11] considers A ≡ I, n ∈ C 1, n → 1 as r → ∞, and both prove weighted estimates rather than cut-off resolvent estimates. The paper [9] uses Morawetz identities with the vector field x, and hence requires Ω − to be star-shaped with respect to ball. The paper [11] considers a more general class of domains than star-shaped [11, Definition 2.1] by modifying vector field x in a neighbourhood of the obstacle (see also [10]). The results of [9] are obtained under inequalities about norms of A in Ω + that essentially guarantee that (1.7) holds; the results