T Chaumont-Frelet

E A Spence
email: a.spence@bath.ac.uk

Scattering by finely-layered obstacles: frequency-explicit bounds and homogenization

Keywords: Helmholtz equation, high frequency, transmission problem, homogenization AMS subject classifications: 35B27, 35J05, 35P25, 78M40

We consider the scalar Helmholtz equation with variable, discontinuous coefficients, modelling transmission of acoustic waves through an anisotropic penetrable obstacle. We first prove a well-posedness result and a frequency-explicit bound on the solution operator, with both valid for sufficiently-large frequency and for a class of coefficients that satisfy certain monotonicity conditions in one spatial direction, and are only assumed to be bounded (i.e., L ∞ ) in the other spatial directions. This class of coefficients therefore includes coefficients modelling transmission by penetrable obstacles with a (potentially large) number of layers (in 2-d) or fibres (in 3-d). Importantly, the frequency-explicit bound holds uniformly for all coefficients in this class; this uniformity allows us to consider highly-oscillatory coefficients and study the limiting behaviour when the period of oscillations goes to zero. In particular, we bound the H 1 error committed by the first-order bulk correction to the homogenized transmission problem, with this bound explicit in both the period of oscillations of the coefficients and the frequency of the Helmholtz equation; to our knowledge, this is the first homogenization result for the Helmholtz equation that is explicit in these two quantities and valid without the assumption that the frequency is small.

We now give the weak form of the variable coefficient Helmholtz equation

∇ • (A∇u) + k 2 nu = -f. (1.2)
Definition 1.1 (Helmholtz transmission problem) Given R > R 0 > 0,

• n ∈ L ∞ (R d , R) with supp(1 -n) ⊂ B R0 and such that 0 < n min ≤ n(x) ≤ n max < ∞ for almost every x ∈ R d , (

• A ∈ L ∞ (R d , SPD) with supp(I -A) ⊂ B R0 and such that 0 < A min A(x) A max < ∞ for almost every x ∈ R d , (1.4)

• F ∈ (H 1 (B R )) ,
and k > 0, we say that u ∈ H 1 (B R ) satisfies the Helmholtz transmission problem if b(u, w) = F (w) for all w ∈ H 1 (B R ), (1.5)

where b(v, w) := B R (A∇v) • ∇w -k 2 nvw -DtN k v, w Γ R , (1.6)
where •, • Γ R denotes the duality pairing on Γ R that is linear in the first argument and antilinear in the second.

An important special case of the Helmholtz transmission problem of Definition 1.1 is transmission through a (not necessarily connected) Lipschitz penetrable obstacle, given in the following lemma. We first introduce some notation. Let D in ⊂ R d , d = 2 or 3, be a bounded open set such that D out := R d \D in is connected. For any ϕ ∈ L 2 loc (R d ), we write ϕ in := ϕ| Din and ϕ out := ϕ| Dout . Let Γ := ∂D in = ∂D out . When Γ is Lipschitz, we let n denote the unit normal vector field on Γ pointing from D in into D out , ∂ n denote the corresponding Neumann trace on Γ, and ∂ n,A denote the corresponding conormal-derivative trace (for A ∈ L ∞ (R d , SPD)); we do not use any notation for the Dirichlet trace on Γ.

∆v out + k 2 v out = -f out in D out , ∇ • (A∇v in ) + k 2 nv in = -f in in D in , (1.7a
)

and v out = v in and ∂ n v out = ∂ n,A v in + g N on Γ, (1.7b)
References for proof. This follows by combining, e.g., [START_REF] Mclean | Strongly elliptic systems and boundary integral equations[END_REF]Lemma 4.19] (to obtain the PDEs in (1.7a) and the jump relations in (1.7b)) and, e.g., [START_REF] Graham | The Helmholtz equation in heterogeneous media: a priori bounds, well-posedness, and resonances[END_REF]Lemma 3.3] (to obtain that v satisfies radiation condition (1.1)).

Remark 1.3 (Choice of parameters for transmission by a single penetrable obstacle)

When n is constant, A is a constant scalar, and D in is connected, the classical transmission problem (1.7) is governed by two free parameters. By rescaling v in , v out , f in , f out and k, (1.7) covers all other problems involving transmission by a penetrable obstacle described by constant real (scalar) coefficients; see [59, Text after Definition 2.3]. (1.9)

We make three remarks: (i) The monotonicity conditions (1.8) and (1.9) (i.e. that A decreases with distance from x d = 0 and n increases) along with the assumptions that supp(I-A) and supp(1-n) are compact imply that A min ≥ 1 and n max ≤ 1. (ii) Condition 1.4 only imposes constraints on the behaviour of A and n in the x d direction, while the behaviour in the x directions is unconstrained.

(iii) The origin of the condition that (A) d (x) = 0 for = 1, . . . , d -1, is discussed in Remark 2.3. We are most interested in the following example of A and n satisfying Condition 1.4.

Example 1.5 (Example of A and n defined piecewise satisfying Condition 1.4)

• D in is as in §1.1 and such that Γ ∩ {x : x d ≥ 0} is the graph of a C 0 function f + : x → R, and Γ ∩ {x : x d ≤ 0} is the graph of a C 0 function f -: x → R.

• A is as in Definition 1.1 and furthermore

A(x) = 1 Din (x) A(x) + 1 Dout (x)I,
where 1 D denotes the indicator function of a set D and A ∈ L ∞ (D in , SPD) satisfies (1.8) (with A replaced by A), ( A) d (x) = 0 for = 1, . . . , d -1 and for all x ∈ D in , and A min ≥ 1.

• n is as in Definition 1.1 and furthermore

n(x) = 1 Din (x) n(x) + 1 Dout (x),
where n ∈ L ∞ (D in , R) satisfies (1.8) (with n replaced by n), and n max ≤ 1.

In Example 1.5, if Γ is Lipschitz, then the first bullet point can be replaced by

x d e d • n(x) ≥ 0 for all x ∈ Γ such that n(x) is defined, (1.10) and the boundary value problem is then transmission through a Lipschitz penetrable obstacle as in Lemma 1.2.

The following theorem bounds the norm of the solution operator of the Helmholtz transmission problem; i.e., the operator norm of the (linear) mapping (H 1 (B R ))

F → u ∈ H 1 (B R ), given by C sol (A, n, k, R, R 0 ) := sup

F ∈(H 1 k (B R )) F (H 1 k (B R )) =1 u H 1 k (B R ) , (1.11)
where, for a bounded open set D, F (v) .

(1.12)

The construction in [START_REF] Chandler-Wilde | Wave-number-explicit bounds in time-harmonic scattering[END_REF]Lemma 3.10] (see also [START_REF] Spence | Wavenumber-explicit bounds in time-harmonic acoustic scattering[END_REF]Lemma 4.12]) giving lower bounds on C sol with A = I and n = 1 shows that C sol must grow at least linearly with k as k → ∞.

Theorem 1.6 (Well-posedness under Condition 1.4) There exists C wave , C 1 > 0 such that the following holds. Given R > R 0 > 0 and A and n satisfying Condition 1.4, if kR 0 ≥ C wave then the Helmholtz transmission problem of Definition 1.1 exists, is unique, and

C sol (A, n, k, R, R 0 ) ≤ 1 n min
1 + 2(kR) 2 (1 + kR 0 )C 1 =: C layer (kR, kR 0 , n min ).

(1.13)

The key point is that, other than n min , all the terms on the right-hand side of (1.13) are independent of A and n. Explicit expressions for C wave and C 1 are available; see Remark 3.5 below. The notation C wave is chosen because this constant determines the minimum number of wavelengths in B R0 for the results of the theorem to hold.

Theorem 1.6 is obtained as a corollary of the following result.

Theorem 1.7 (A priori bound under Condition 1.4 for L 2 data) Under the assumptions of Theorem 1.6, if F (v) = B R f v and kR 0 ≥ C wave , then

u H 1 k (B R ) ≤ C 1 (1 + kR 0 )(kR)R f L 2 (B R ) .
(1.14)

Remark 1.8 (Plane-wave scattering) A common Helmholtz problem in applications is scattering of an incident field u inc (such as a plane wave or a point source), creating a scattered field u sca , satisfying the Sommerfeld radiation condition, with the total field u inc + u sca satisfying the Helmholtz equation with zero right-hand side (see, e.g., [START_REF] Moiola | Acoustic transmission problems: wavenumber-explicit bounds and resonance-free regions[END_REF]Definition 2.4]). This problem fits into the framework of Definition 1.1 with u = u sca + χu inc , where χ is a smooth cutoff function such that χ = 1 on D in and χ = 0 in a neighborhood of ∂B R , and f := (∆χ)u inc + 2∇u inc • ∇χ.

1.2.2 Discussion of the novelty of the well-posedness result in Theorem 1.6

The standard way to prove well-posedness of the variable-coefficient Helmholtz equation is to prove a unique continuation principle (UCP) and use Fredholm theory. In 2-d, a UCP for the Helmholtz equation holds with A ∈ L ∞ and n ∈ L p with p > 1 [START_REF] Alessandrini | Strong unique continuation for general elliptic equations in 2D[END_REF], therefore the well-posedness result of Theorem 1.6 for d = 2 is not new.

In constrast, the well-posedness result of Theorem 1.6 for d = 3 is new. Indeed, when d = 3 the UCP holds when A is piecewise Lipschitz [START_REF] Ball | On uniqueness for time harmonic anisotropic Maxwell's equations with piecewise regular coefficients[END_REF], [START_REF] Liu | Mosco convergence for H(curl) spaces, higher integrability for Maxwell's equations, and stability in direct and inverse EM scattering problems[END_REF]Proposition 2.11] (by using the UCP in the Lipschitz case [START_REF] Garofalo | Unique continuation for elliptic operators: A geometric-variational approach[END_REF][START_REF] Kazdan | Unique continuation in geometry[END_REF]) and n ∈ L 3/2 [START_REF] Jerison | Unique continuation and absence of positive eigenvalues for Schrödinger operators[END_REF][START_REF] Wolff | A property of measures in R N and an application to unique continuation[END_REF]. However, an example of an A ∈ C 0,α for all α < 1 for which the solution of the transmission problem is not unique at a particular k > 0 (and thus the UCP fails) is given in [START_REF] Filonov | Second-order elliptic equation of divergence form having a compactly supported solution[END_REF]Theorem 1]. The only other Helmholtz well-posedness result we are aware of that is valid for A with no smoothness assumptions other than being L ∞ is in [START_REF] Graham | The Helmholtz equation in heterogeneous media: a priori bounds, well-posedness, and resonances[END_REF]. There, well-posedness is proved for A ∈ L ∞ satisfying a radial-monotonicity condition (as opposed to the monotonicity in a single coordinate direction in (1.8)). The method of proof in [START_REF] Graham | The Helmholtz equation in heterogeneous media: a priori bounds, well-posedness, and resonances[END_REF] is similar to the present paper (see the discussion in §1.2.5 below).

Discussion of the novelty of the bound in Theorem 1.6

The bound (1.14) is novel in two ways.

(i) A standard way of obtaining frequency-explicit bounds on solutions of the high-frequency Helmholtz equation is to consider the billiard flow defined by the (semiclassical) principal symbol of the Helmholtz equation and use associated results on propagation of singularities under this flow [START_REF] Melrose | Singularities of boundary value problems[END_REF], [START_REF] Melrose | Singularities of boundary value problems[END_REF], [START_REF] Hörmander | The analysis of linear partial differential operators III: pseudo-differential operators[END_REF]Chapter 24]. However, this flow not well-defined for the class of A and n in Condition 1.4.

(ii) Even when the flow is defined, while there has been a large amount of work on proving bounds that are explicit in k, there are relatively few bounds in the literature that are explicit both in k and in the coefficients A and n (such as (1.13)/(1.14)).

Regarding (i): the flow is defined as the solution (x(t), ξ(t)) of the Hamiltonian system

ẋi (t) = ∂ ξi p x(t), ξ(t) , ξi (t) = -∂ xi p x(t), ξ(t) , (1.15)
where the Hamiltonian equals the semiclassical principal symbol of the Helmholtz equation, namely

p(x, ξ) := d i=1 d j=1 A ij (x)ξ i ξ j -n(x);
see, e.g., [START_REF] Zworski | Semiclassical analysis[END_REF]Page 281]. By the Picard-Lindelöf theorem, the flow exists and is unique if A, n are both C 1,1 , since the coefficients of the ODE system (1.15) are then Lipschitz. By the well-known examples of ODE non-uniqueness with non-Lipschitz coefficients, if A and n are rougher than C 1,1 , then the flow is not guaranteed to be well-defined. The flow can be defined piecewise, with results about the behaviour of singularities hitting the interface given in [55, Chapter 11], [START_REF] Miller | Refraction of high-frequency waves density by sharp interfaces and semiclassical measures at the boundary[END_REF], and k-explicit bounds on the solution of the Helmholtz transmission problem when A and/or n are discontinuous on a C ∞ strictly-convex interface given in [START_REF] Cardoso | Distribution of resonances and local energy decay in the transmission problem II[END_REF], [START_REF] Popov | Resonances near the real axis for transparent obstacles[END_REF], [START_REF] Popov | Distribution of the resonances and local energy decay in the transmission problem[END_REF]. However, it is not possible to defined the flow for the range of coefficients covered by Condition 1.4, which need only be L ∞ in the x directions. Regarding (ii): the first such bounds were proved for the exterior Helmholtz equation in [START_REF] Bloom | Estimates for solutions of reduced hyperbolic equations of the second order with a large parameter[END_REF] (for A variable and n = 1) and [START_REF] Bloom | A priori bounds for solutions of the Dirichlet problem for [∆ + λ 2 n(x)]u = f (x, λ) on an exterior domain[END_REF], (for n variable and A = I). Recent such bounds were proved in [START_REF] Moiola | Acoustic transmission problems: wavenumber-explicit bounds and resonance-free regions[END_REF][START_REF] Graham | The Helmholtz equation in heterogeneous media: a priori bounds, well-posedness, and resonances[END_REF][START_REF] Galkowski | Optimal constants in nontrapping resolvent estimates[END_REF][START_REF] Sauter | The Heterogeneous Helmholtz Problem with Spherical Symmetry: Green's Operator and Stability Estimates[END_REF] (and for problems posed on bounded domains with impedance boundary conditions [START_REF] Frelet | Approximation par éléments finis de problèmes d'Helmholtz pour la propagation d'ondes sismiques[END_REF]Chapter 2], [START_REF] Barucq | Stability analysis of heterogeneous Helmholtz problems and finite element solution based on propagation media approximation[END_REF][START_REF] Brown | Multiscale Petrov-Galerkin method for high-frequency heterogeneous Helmholtz equations[END_REF][START_REF] Ohlberger | A new heterogeneous multiscale method for the helmholtz equation with high contrast[END_REF][START_REF] Sauter | Stability estimate for the Helmholtz equation with rapidly jumping coefficients[END_REF][START_REF] Graham | The Helmholtz equation in heterogeneous media: a priori bounds, well-posedness, and resonances[END_REF][START_REF] Graham | Stability and finite element error analysis for the Helmholtz equation with variable coefficients[END_REF]), the renewed interest due to growing interest in the numerical analysis of Helmholtz equation with variable coefficients [START_REF] Chaumont-Frelet | On high order methods for the heterogeneous Helmholtz equation[END_REF][START_REF] Barucq | Stability analysis of heterogeneous Helmholtz problems and finite element solution based on propagation media approximation[END_REF][START_REF] Brown | Multiscale Petrov-Galerkin method for high-frequency heterogeneous Helmholtz equations[END_REF][START_REF] Ohlberger | A new heterogeneous multiscale method for the helmholtz equation with high contrast[END_REF][START_REF] Fang | A hybrid approach to solve the high-frequency Helmholtz equation with source singularity in smooth heterogeneous media[END_REF][START_REF] Ganesh | A coercive heterogeneous media Helmholtz model: formulation, wavenumberexplicit analysis, and preconditioned high-order FEM[END_REF][START_REF] Chaumont-Frelet | Wavenumber explicit convergence analysis for finite element discretizations of general wave propagation problem[END_REF][START_REF] Chaumont-Frelet | A multiscale hybrid-mixed method for the Helmholtz equation in heterogeneous domains[END_REF][START_REF] Pembery | The Helmholtz Equation in Heterogeneous and Random Media: Analysis and Numerics[END_REF][START_REF] Graham | Stability and finite element error analysis for the Helmholtz equation with variable coefficients[END_REF][START_REF] Galkowski | Optimal constants in nontrapping resolvent estimates[END_REF][START_REF] Peterseim | Computational high frequency scattering from high-contrast heterogeneous media[END_REF][START_REF] Lafontaine | Wavenumber-explicit convergence of the hp-FEM for the full-space heterogeneous Helmholtz equation with smooth coefficients[END_REF][START_REF] Gong | Domain decomposition preconditioners for high-order discretisations of the heterogeneous Helmholtz equation[END_REF][START_REF] Lafontaine | Decompositions of high-frequency Helmholtz solutions via functional calculus, and application to the finite element method[END_REF].

1.2.4 Discussion about the k-dependence in the bound (1.14) The key points are the following.

• There exist (fixed) A and n such that C sol grows super-algebraically in k as k → ∞ due to trapped rays [START_REF] Ralston | Trapped rays in spherically symmetric media and poles of the scattering matrix[END_REF][START_REF] Popov | Resonances near the real axis for transparent obstacles[END_REF][START_REF] Capdeboscq | On the scattered field generated by a ball inhomogeneity of constant index[END_REF][START_REF] Capdeboscq | On the scattered field generated by a ball inhomogeneity of constant index in dimension three. In Multi-scale and high-contrast PDE: from modelling, to mathematical analysis, to inversion[END_REF]. The polynomial bound on C sol in (1.13) shows that this behaviour is ruled out by Condition 1.4, which is consistent with the physical understanding of what causes rays to be trapped for the transmission problem.

• The recent results of [START_REF] Sauter | Stability estimate for the Helmholtz equation with rapidly jumping coefficients[END_REF] show that, in 1-d, C sol can grow exponentiallly through a sequence n j , k j , with k j → ∞ as j → ∞; these examples involve piecewise constant n (i.e. a 1-d layered medium), and the key point is that to get the exponential growth the width of the pieces need to be tied to k in a delicate way. This behaviour is ruled out for the layered obstacles included in Condition 1.4 via the polynomial bound on C sol in (1.13), which is uniform in A and n satisfying Condition 1.4. Exponential growth of C sol through a sequence n j , k j , with k j → ∞ as j → ∞, in d > 1 is still possible through layered obstacles if the layers are radial -this is shown in [START_REF] Sauter | The Heterogeneous Helmholtz Problem with Spherical Symmetry: Green's Operator and Stability Estimates[END_REF], with radially symmetric n chosen based on the 1-d results of [START_REF] Sauter | Stability estimate for the Helmholtz equation with rapidly jumping coefficients[END_REF].

We now give more detail on these two points.

The case A and n fixed and k → ∞. For the Helmholtz transmission problem of Definition 1.1, super-algebraic growth of C sol through a sequence of k j s has been proved in the following two cases.

(i) A and n are as in Lemma 1.2 with D in a C ∞ convex domain with strictly positive curvature, and the jumps of A and n have a certain combination of signs (e.g., moving outwards, A jumps down with n fixed, or n jumps up with A fixed) so that rays can be totally internally reflected when hitting Γ from inside D in [START_REF] Popov | Resonances near the real axis for transparent obstacles[END_REF] (see also [START_REF] Capdeboscq | On the scattered field generated by a ball inhomogeneity of constant index[END_REF], [START_REF] Capdeboscq | On the scattered field generated by a ball inhomogeneity of constant index in dimension three. In Multi-scale and high-contrast PDE: from modelling, to mathematical analysis, to inversion[END_REF], [START_REF] Alberti | Lectures on elliptic methods for hybrid inverse problems[END_REF]Chapter 5] for similar results in the specific case when the obstacle is a ball). The solutions corresponding to the trapped rays are known as "whispering-gallery modes"; see, e.g., [START_REF] Babich | Short-Wavelength Diffraction Theory[END_REF] and the references therein.

(ii) A = I and n is C ∞ and spherically symmetry with 2n(r 1 ) + (∂n/∂r)(r 1 ) < 0 for some r 1 [START_REF] Ralston | Trapped rays in spherically symmetric media and poles of the scattering matrix[END_REF] (see also [START_REF] Graham | The Helmholtz equation in heterogeneous media: a priori bounds, well-posedness, and resonances[END_REF]Theorem 7.7(ii)]) 1 ; this condition on n causes great circles on r = r 1 to be stable trapped rays; see [START_REF] Ralston | Trapped rays in spherically symmetric media and poles of the scattering matrix[END_REF]Page 572].

Neither of the situations in (i) or (ii) are allowed under Condition 1.4. Indeed, Condition 1.4 implies that A min ≥ 1 and n max ≤ 1 (as noted just below the condition), and these prevent A and n having the "bad" jumps in (i). Furthermore, the monotonicity condition on n (1.9) prevents n from satisfying the condition in (ii).

The case when A and n can depend on k and k → ∞. For (1.2) posed on a 1-d interval with either Dirichlet or impedance boundary conditions at either end, and at least one of the ends having impedance boundary conditions, bounds on the Helmholtz solution where A and n have finite number of jumps are given in [START_REF] Chaumont-Frelet | On high order methods for the heterogeneous Helmholtz equation[END_REF], [START_REF] Graham | Stability and finite element error analysis for the Helmholtz equation with variable coefficients[END_REF]. These results have C sol ≤ exp C(A, n) (see [START_REF] Graham | Stability and finite element error analysis for the Helmholtz equation with variable coefficients[END_REF]Theorem 5.4 and 5.10]), but C sol → ∞ if the number of jumps goes to infinity The fact that C sol is independent of k is consistent with the fact that, for fixed A and n geometric-optic rays are not trapped, since although A and n can jump across interfaces, the 1-d nature of the problem means that no rays can get trapped moving tangent to the interface (like in Point (i) above).

The paper [START_REF] Sauter | Stability estimate for the Helmholtz equation with rapidly jumping coefficients[END_REF] considers the 1-d case with A = 1 and n variable and proves

• there exist sequences n j , k j (with k j → ∞ as j → ∞) such that C sol grows exponentially as j → ∞ [78, Remark 14],

• if the number of jumps k then C sol ≤ exp C(n)k , but C(n) doesn't blow up with number of jumps [START_REF] Sauter | Stability estimate for the Helmholtz equation with rapidly jumping coefficients[END_REF]Prop. 18], and

• for a class of n that oscillate between two values, with an arbitrarily-large number of jumps, C sol ≤ exp C(n)k and C(n) doesn't blow up with number of jumps [START_REF] Sauter | Stability estimate for the Helmholtz equation with rapidly jumping coefficients[END_REF]Theorem 22].

The paper [START_REF] Sauter | The Heterogeneous Helmholtz Problem with Spherical Symmetry: Green's Operator and Stability Estimates[END_REF] generates the results of [START_REF] Sauter | Stability estimate for the Helmholtz equation with rapidly jumping coefficients[END_REF] to radially-symmetric n in 3-d (see [START_REF] Sauter | The Heterogeneous Helmholtz Problem with Spherical Symmetry: Green's Operator and Stability Estimates[END_REF]Theorem 3.7] for the upper bounds and [START_REF] Sauter | The Heterogeneous Helmholtz Problem with Spherical Symmetry: Green's Operator and Stability Estimates[END_REF]Lemma 4.3] for the examples of exponential blow-up). (iv) Proving the bound (1.14) for these smooth coefficients, importantly with the constants C wave and C 1 independent of , using a novel Morawetz-type identity ((2.2) below) involving a particular vector field and associated arguments introduced recently in [START_REF] Chandler-Wilde | High-frequency bounds for the helmholtz equation under parabolic trapping and applications in numerical analysis[END_REF] (see §3.1). The key point is that the vector field (defined by (3.4)) equals e d x d in a neighbourhood of supp(I -A) and supp (1 -n), where e d denotes the unit vector in the x d direction; this vector field is constant in the x directions, and so "does not see" the behaviour of A and n in these directions, hence why no constraint is imposed on this behaviour.

(v) Using approximation arguments from [START_REF] Graham | The Helmholtz equation in heterogeneous media: a priori bounds, well-posedness, and resonances[END_REF] to prove the bound (1.14) for the original A and n (see Lemma 2.10); these approximation arguments were inspired by similar arguments in [START_REF] Thomas | Analysis of Rough Surface Scattering Problems[END_REF] in the setting of rough-surface scattering (with this thesis recently made available as [START_REF] Baden-Riess | PHD Thesis: Existence, Uniqueness & Explicit Bounds for Scattering By Rough Surfaces[END_REF]).

Main result II: homogenization

An attractive feature of Theorem 1.6 is that it does not constrain the variations of A or n along the x hyperplane. In particular, highly-oscillatory coefficients are allowed, and the results in this section concern the limiting behaviour when the period of oscillations goes to zero. Theorem 1.6 would also allow one to do homogenization under assumptions other than periodicity (see, e.g., [START_REF] Cioranescu | An introduction to homogenization[END_REF]Chapter 13]), but for simplicity we only consider a periodic setting here.

Statement of the result

With ε > 0 a (small) oscillation period, we define oscillatory coefficients n ε and A ε created by repeating functions n and A on a grid of size ε inside a bounded Lipschitz domain D in . Definition 1.9 (Oscillatory coefficients in D in ) Let Y := (0, 1) d (i.e., the unit cube in

R d ). If n ∈ L ∞ (Y, R) and A ∈ L ∞ (Y, SPD), let n ε (x) := n x ε and A ε (x) := A x ε
for all x ∈ R d , where {x/ε} j = (x j /ε) mod 1. Then, given a bounded Lipschitz domain D in satisfying (1.10), let

n ε := 1 Din n ε + 1 Dout and A ε := 1 Din A ε + 1 Dout I. (1.16)
We now impose constraints on n and A so that n ε and A ε satisfy Condition 1.4 (and so Theorem 1.6 applies) -these are the first three points in Condition 1.10 below. We also impose additional constraints on n and A so that the correctors in the homogenization argument (see §4 below) have sufficient regularity -these constraints are in the final point in Condition 1.10.

Condition 1.10 (Admissible periodic patterns) n ∈ L ∞ (Y ) and A ∈ L ∞ (Y, SPD) are ad- missible periodic patterns if • n and A are independent of y d • A d = 0 for = 1, . . . , d -1, • A min ≥ 1 and n max ≤ 1,
• either (a) n and A only depend on y 1 (or, when d = 3, only on y 2 ) and A is piecewise C 0,1 .

(b) the periodic extension of A is piecewise C 1,1 and that of n is piecewise H 1 on a partition of R d that consists of subdomains with C 2,1 boundaries.

The class of coefficients covered by Condition 1.10 corresponds to obstacles made of thin layers in 2D, and of thin layers or fibers in 3D.

Remark 1.11 (Physical relevance of finely-layered and finely-fibred materials) Many key applications motivating homogenized theory are based on "composite" or "meta" materials. Such materials are based on blending two (or more) materials in a periodic manner at a fine scale, and thus are modelled by piecewise-constant periodic patterns, as covered by Condition 1.10. Particular instances of layered and fibred media arise when considering the propagation of elastic waves through rocks [START_REF] Carcione | Long-wave anisotropy in stratified media: A numerical test[END_REF] and electromagnetic waves in thin-film coatings [START_REF] Keçebaş | Enhancing the spectral reflectance of refractory metals by mutlilayer optical thin-film coatins[END_REF] or optical fibres [START_REF] Upendar | Orders of magnitude loss redution in photonic bandgap fibers by engineering the core surround[END_REF]. Corollary 1.12 Let C wave be as in Theorem 1.6. If n and A satisfy Condition 1.10 and kR 0 ≥ C wave , then, for all ε > 0, the solution of the variational problem (1.5) with coefficients n ε and A ε exists and is unique, and, with C layer defined by (1.13),

C sol (A ε , n ε , k, R, R 0 ) ≤ C layer (kR, kR 0 , n min ).
(1.17)

Crucially, the bound (1.17) is uniform in ε, and this uniformity allows us to extract a convergent subsequence u ε . In fact, by classical homogenization theory [START_REF] Bensoussan | Asymptotic analysis for periodic structures[END_REF][START_REF] Cioranescu | An introduction to homogenization[END_REF], in the limit ε → 0, the whole sequence u ε to u 0 weakly in H 1 (B R ) and strongly in L 2 (B R ), where u 0 is the solution to the transmission problem of Definition 1.1 with constant "homogenized" coefficients A H and n H . These homogenized coefficients are obtained from A and n by averaging formulas given by (4.7) and (4.2) below; note that these are the same formulas for the coercive case (see, e.g., [START_REF] Bensoussan | Asymptotic analysis for periodic structures[END_REF]Chapter 1], [START_REF] Cioranescu | An introduction to homogenization[END_REF]Chapter 6]).

Recall that a domain D ⊂ R d is star-shaped with respect to the point

x 0 ∈ R d if, whenever x ∈ D, the line segment [x 0 , x] ∈ D.
Lemma 1.13 (Well-posedness of the homogenized problem) Let C wave be as in Theorem 1.7. Given n and A satisfying Condition 1.10, let n H and A H be defined by (4.2) and (4.7), respectively, and let u 0 ∈ H 1 (B R ) be the solution of the variational problem (1.5) with coefficients n H and A H (we then say that u 0 is the solution of the homogenized problem).

Then, if kR 0 ≥ C wave , u 0 exists and is unique, and

C sol (A H , n H , k, R, R 0 ) ≤ C layer (kR, kR 0 , n min ). (1.18)
Furthemore, if D in is additionally star-shaped with respect to a point, then, for all k > 0, u 0 exists and is unique, and

C sol (A H , n H , k, R, R 0 ) ≤ 1 n min   1 + kR 4 √ n min 1 + 1 n min 1 + d -1 2kR 2   . (1.19)
Theorem 1.14 (k-and ε-explicit homogenization error estimate in H 1 k (B R )) Let C wave and C layer be as in Theorem 1.6. Let n and A satisfy Condition 1.10 and assume that Γ is C 1,1 . Given f ∈ L2 (B R ), let u ε be the solution of the variational problem (1.5) with coefficients A ε and n ε and F (v) := B R f v. Let u 0 be the solution of the variational problem (1.5) with this F and coefficients A H and n H , and let u ε 1 be defined in terms of u 0 by (4.9) and (4.1) below. If kR 0 ≥ C wave then u ε , u 0 , and u ε 1 exist and are unique, and there exist C 2 , C 3 > 0 (depending only on A, n, and D in ) such that

k u ε -u 0 -εu ε 1 H 1 k (Din) + k u ε -u 0 H 1 k (Dout) ≤ C 2 C layer (kR, kR 0 , n min ) (kε) 1/2 + kε u 0 H 2 (B R ) + k u 0 H 1 k (B R ) (1.20) ≤ C 3 C layer (kR, kR 0 , n min ) (kε) 1/2 + kε C sol (A H , n H , k, R, R 0 ) f L 2 (B R ) . (1.21) Since u ε H 1 k (B R ) ≤ u ε -u 0 H 1 k (B R ) + u 0 H 1 k (B R )
, the bound on u ε -u 0 from Theorem 1.14 and the definition of C sol,H then imply the following bound on C sol (A ε , n ε , k, R, R 0 ). Corollary 1.15 (Improved bound on C sol,ε for kε sufficiently small (depending on k)) Let C wave and C layer be as in Theorem 1.6. Let n and A satisfy Condition 1.10 and assume that Γ is C 1,1 . Then there exists C 5 (depending only on A, n, and D in ) such that if kR 0 ≥ C wave then

C sol (A ε , n ε , k, R, R 0 ) ≤ C 5 n min 1 + (kε) 1/2 + kε C layer (kR, kR 0 , n min ) C sol (A H , n H , k, R, R 0 ). (1.22)
Corollary 1.15 shows that if kε is sufficiently small (depending on k), then the problem with coefficients A ε and n ε inherits the behaviour of the solution operator for the homogenized problem. The first thing to highlight about Theorem 1.14 is that it is a k-explicit analogue of the H 1 error bound in [START_REF] Cakoni | On the homogenization of a scalar scattering problem for highly oscillating anisotropic media[END_REF]Theorem 2]. More precisely, [START_REF] Cakoni | On the homogenization of a scalar scattering problem for highly oscillating anisotropic media[END_REF]Theorem 2] proves the bound

u ε -u 0 -εu ε 1 H 1 k (Din) + u ε -u 0 H 1 k (Dout) ≤ C(k)ε 1/2 u 0 H 2 (B R )
, for some unspecified C(k), with this bound valid for sufficiently small kR and then arbitrarily small ε/R. In contrast, in Theorem 1.14, kR can be arbitrarily large, and ε/R can be tied to kR. We recall that (i) a main novelty of [START_REF] Cakoni | On the homogenization of a scalar scattering problem for highly oscillating anisotropic media[END_REF]Theorem 2] is that it uses boundary correctors for homogenization of a transmission problem (as opposed to using boundary correctors for problems on bounded domains with Dirichlet or Neumann boundary conditions; see, e.g., [START_REF] Bensoussan | Asymptotic analysis for periodic structures[END_REF][START_REF] Moskow | First-order corrections to the homogenised eigenvalues of a periodic composite medium. A convergence proof[END_REF][START_REF] Moskow | First order corrections to the homogenized eigenvalues of a periodic composite medium. The case of Neumann boundary conditions[END_REF][START_REF] Griso | Error estimate and unfolding for periodic homogenization[END_REF][START_REF] Zhikov | On operator estimates for some problems in homogenization theory[END_REF][START_REF] Griso | Interior error estimate for periodic homogenization[END_REF][START_REF] Onofrei | Error estimates for periodic homogenization with non-smooth coefficients[END_REF][START_REF] Jikov | Homogenization of differential operators and integral functionals[END_REF][START_REF] Kenig | Homogenization of elliptic systems with Neumann boundary conditions[END_REF][START_REF] Suslina | Homogenization of the Neumann problem for elliptic systems with periodic coefficients[END_REF][START_REF] Kenig | Periodic homogenization of Green and Neumann functions[END_REF][START_REF] Semin | On the homogenization of the Helmholtz problem with thin perforated walls of finite length[END_REF]), and (ii) [START_REF] Cakoni | On the homogenization of a scalar scattering problem for highly oscillating anisotropic media[END_REF]Theorem 2]

also proves the L 2 error estimate u ε -u 0 L 2 (B R ) ≤ C(k)ε u 0 H 2 (B R ) ,
but for brevity we have not pursued here obtaining a k-explicit analogue of this result.

The main obstacle for obtaining homogenization results about the Helmholtz equation for large kR is that the first step of the homogenization procedure relies on a uniform-in-ε bound on the H 1 norm of u ε , which allows one to extract a weakly-converging subsequence. Such estimate is immediate for the coercive PDE -∇ • (A ε u) = f , since the coercivity constant only depends on A min , which is independent of ε. In Appendix C, we recall how the b(•, •) defined by (1.6) is coercive if kR is sufficiently small, with both the coercivity constant (and hence C sol ) and the constant for "sufficiently small" independent of ε and depending on A min and n max ; see Lemma C.1 below. Therefore, homogenization results for the Helmholtz equation can be obtained for kR sufficiently small, and this is implicitly what [START_REF] Cakoni | On the homogenization of a scalar scattering problem for highly oscillating anisotropic media[END_REF] do. 2 In contrast, if kR is sufficiently large, then b(•, •) is not coercive (see, e.g., [82, §6.1.6]); furthermore, at least when A ε and n ε are smooth, [START_REF] Galkowski | Optimal constants in nontrapping resolvent estimates[END_REF] shows that C sol depends on global properties of A ε and n ε (more precisely, C sol is proportional to the length of the longest ray in B R ). Therefore, obtaining a uniform-in-ε bound on C sol for arbitrary A ε and n ε and arbitrary kR is very challenging. The present paper bypasses this fundamental problem by considering the restricted class of A ε and n ε of Condition 1.10, for which Theorem 1.6 gives the required bound that is uniform in ε and valid for large kR. Another place where this problem is bypassed is [START_REF] Bouchitté | Homogenization near resonances and artificial magnetism from dielectrics[END_REF]: this paper considers homogenization of particular 2-d Helmholtz transmission problem (where the contrast in the coefficients depends on ε). The result of [START_REF] Bouchitté | Homogenization near resonances and artificial magnetism from dielectrics[END_REF] is that, via a contradiction argument, the sequence u ε is bounded in L 2 and two-scale converges to the homogenized solution (albeit without an error estimate); this result is valid for fixed k, excluding a countable set, but crucially not assumed to be small.

Finally, we note that another difference between the analysis in the present paper and that of [START_REF] Cakoni | On the homogenization of a scalar scattering problem for highly oscillating anisotropic media[END_REF] is that, for simplicity, [START_REF] Cakoni | On the homogenization of a scalar scattering problem for highly oscillating anisotropic media[END_REF] assume that A, n, and D in are C ∞ , but the present paper makes much weaker regularity assumptions on A, n, and D in . In particular, we allow for piecewise smooth A and n; recall from Remark 1.11 that these are particularly important in applications. The bound (1.20) shows that the "relative error"

k u ε -u 0 -εu ε 1 H 1 k (Din) + k u ε -u 0 H 1 k (Dout) u 0 H 2 (B R ) + k u 0 H 1 k (B R ) (1.23)
is controllably small if C layer (kR, kR 0 , n min )(kε) 1/2 is small, independently of k and ε, i.e., if kε ≤ c(kR) -6 (with c independent of k and ε) by the definition of C layer (kR, kR 0 , n min ) (1.13). While this is certainly pessimistic, to our knowledge it is the first homogenization result for the Helmholtz equation that is explicit in both ε and k. Note that C layer (kR, kR 0 , n min ) kR (see the discussion after (1.12)), and therefore even if we obtained a sharper upper bound on C layer (kR, kR 0 , n min ), the bound (1.20) would still only show that (1.23) is small when kε ≤ c(kR) -1 . (Note that the factor of C layer (kR, kR 0 , n min ) in the bound on (1.23) from (1.20) arises from the standard treatment of u ε -u 0 minus the correctors as a solution of the PDE -in our case the Helmholtz equation; see (4.27) and (4.28) below.)

The bound (1.22) shows that if C layer (kR, kR 0 , n min )(kε) 1/2 is bounded independently of k and ε, then

C sol (A ε , n ε , k, R, R 0 ) C sol (A H , n H , k, R, R 0 ). (1.24)
From (1.17) and (1.18), both

C sol (A ε , n ε , k, R, R 0 ) and C sol (A H , n H , k, R, R 0 ) are ≤ C layer (kR, kR 0 , n min ).
Therefore, the bound (1.24) is only interesting when

C sol (A H , n H , k, R, R 0 )
C layer (kR, kR 0 , n min ). A concrete case when this is true is when D in is star-shaped with respect to a point, since then

C sol (A H , n H , k, R, R 0 )
kR by (1.19) (which is the smallest possible growth of C sol with kR), whereas C layer (kR, kR 0 , n min ) (kR) 3 by its definition (1.13).

Preliminary results

Morawetz-type identities and associated results

When writing these identities, it is convenient to use the notation that a, b :=

d j=1 a j b j for a, b ∈ C d .
Here and in the rest of the paper, we use the convention that all indices are lowered, and repeated indices i, j, and , are summed over, but not repeated indices d.

Lemma 2.1 (Morawetz-type identity) Let D ⊂ R d . Let v ∈ C 2 (D), A ∈ C 1 (D, Sym), n ∈ C 1 (D, R), Z ∈ C 1 (D, R d ), α ∈ C 2 (D, R), and β ∈ C 1 (D, R). Let L A,n v := ∇ • (A∇v) + k 2 n v and Zv := Z • ∇v -ikβv + αv. (2.1)
Then, in D,

2 Zv L A,n v = ∇ • 2 Zv A∇v + Z k 2 n|v| 2 -A∇v, ∇v -∇α|v| 2 -(2α -∇ • Z) A∇v, ∇v + (Z • ∇)A ∇v, ∇v -(∇ • Z -2α)n + Z • ∇n k 2 |v| 2 -2 A∇v, ∂Z ∇v -2 ikv A∇v, ∇β -2 v (A -I)∇v, ∇α + ∆α|v| 2 , (2.2)
where ∂Z is the derivative matrix of Z, i.e. (∂Z

) ij = ∂ i Z j .
Proof. Splitting Zv up into its component parts, we see that the identity (2.2) is the sum of the following four identities:

2 Z • ∇v L A,n v = ∇ • 2 Z • ∇v A∇v + Z k 2 n |v| 2 -A∇v, ∇v + ∇ • Z A∇v, ∇v -2 A∇v, ∂Z ∇v + (Z • ∇)A ∇v, ∇v -(∇ • Z n + Z • ∇n)k 2 |v| 2 , (2.3) 2 ikβv L A,n v = ∇ • 2 ikβv A∇v -2 ikv A∇v, ∇β , (2.4
)

2 αv L A,n v = ∇ • 2 {αv A∇v} + 2αk 2 n|v| 2 -2α A∇v, ∇v -2 v A∇v, ∇α . (2.5) and 0 = -∇ • ∇α|v| 2 + 2 v∇α • ∇v + ∆α|v| 2 . (2.6)
To prove (2.4), (2.5), and (2.6), expand the divergences on the right-hand sides (remembering that α and β are real and that A is symmetric, so Aξ, ξ is real for any ξ ∈ C d ).

The basic ingredient of (2.3) is the identity

(Z • ∇v)∇ • (A∇v) = ∇ • (Z • ∇v)A∇v -A∇v, ∂Z ∇v -(Z • ∇)∇v • A∇v. (2.7)
To prove this, expand the divergence on the right-hand side and use the fact that the second derivatives of v commute. We would like each term on the right-hand side of (2.7) to either be single-signed or be the divergence of something. To deal with the final term we use the identity

2 (Z • ∇)∇v • A∇v = ∇ • Z A∇v, ∇v -(∇ • Z) A∇v, ∇v -(Z • ∇)A ∇v, ∇v , (2.8)
which can be proved by expanding the divergence on the right-hand side and using the fact that A is symmetric. Therefore, taking twice the real part of (2.7) and using (2.8) yields

2 (Z • ∇v)∇ • (A∇v) =∇ • 2 (Z • ∇v) A∇v -Z A∇v, ∇v + (∇ • Z) A∇v, ∇v + (Z • ∇)A ∇v, ∇v -2 A∇v, ∂Z ∇v . (2.9) Now add k 2 times 2 (Z • ∇v)nv = ∇ • Zn|v| 2 -(∇ • Z)n|v| 2 -Z • ∇n|v| 2
(which is the analogue of (2.8) with the vector ∇v replaced by the scalar v and the matrix A replaced by the scalar n) to (2.9) to obtain (2.3).

Remark 2.2 (Bibliographic remarks on Morawetz-type identities for L A,n ) Multiplying ∆v by a derivative of v goes back to Rellich [START_REF] Rellich | Darstellung der Eigenwerte von ∆u + λu = 0 durch ein Randintegral[END_REF][START_REF] Rellich | Über das asymptotische Verhalten der Lösungen von ∆u + λu = 0 in unendlichen Gebieten[END_REF], and multiplying ∇ • (A∇v) by a derivative of v goes back to Hörmander [START_REF] Hörmander | Uniqueness theorems and estimates for normally hyperbolic partial differential equations of the second order[END_REF] and Payne and Weinberger [START_REF] Payne | New bounds for solutions of second order elliptic partial differential equations[END_REF] (e.g., the identity (2.2) with Z = x and n, α, and β all equal zero appears as [69, Equation 2.4]).

In the context of the Helmholtz equation, the identity (2.2) with A = I, n = 1, x replaced by a general vector field, and α and β replaced by general scalar fields was the heart of Morawetz's paper [START_REF] Morawetz | Decay for solutions of the exterior problem for the wave equation[END_REF] (following the earlier work [START_REF] Morawetz | The decay of solutions of the exterior initial-boundary value problem for the wave equation[END_REF][START_REF] Morawetz | An inequality for the reduced wave operator and the justification of geometrical optics[END_REF]); hence why we call (2.2) a "Morawetz-type" identity. The identity (2.2) with Z = x, A variable, and n = 1 was used by Bloom in [START_REF] Bloom | Estimates for solutions of reduced hyperbolic equations of the second order with a large parameter[END_REF], and the identity (2.2) with Z = x, A = I, and variable n was used in Bloom and Kazarinoff in [START_REF] Bloom | A priori bounds for solutions of the Dirichlet problem for [∆ + λ 2 n(x)]u = f (x, λ) on an exterior domain[END_REF]. To our knowledge, the present paper is the first time that the identity (2.2), with A, n, Z, β, and α all variable, has appeared in the literature.

2 A∇v, ∂Z ∇v = 2 ∂ d v d =1 (A d )∂ v , (2.10)
hence we impose that (A) d = 0 for = 1, . . . , d -1.

For notational convenience, given A, n, Z, β, and α, we let

Q(v) := 2 Zv A∇v + Z k 2 n|v| 2 -A∇v, ∇v -∇α|v| 2 , (2.11)
and

P (v) :=2 Zv L A,n v + (2α -∇ • Z) A∇v, ∇v -(Z • ∇)A ∇v, ∇v + (∇ • Z -2α)n + Z • ∇n k 2 |v| 2 + 2 A∇v, ∂Z ∇v + 2 iku A∇v, ∇β + 2 v (A -I)∇v, ∇α -∆α|v| 2 , (2.12) so that the identity (2.2) becomes ∇ • Q(v) = P (v).
. Let v ∈ H 2 (B R ), A ∈ C 1 (B R , SPD), n ∈ C 1 (B R , R), Z ∈ C 1 (B R , R d ), β ∈ C 1 (B R , R), and α ∈ C 2 (B R , R). Suppose further that, in a neighbourhood of Γ R , A = I, n = 1, Z = x, and α is constant. Then B R P (v) = Γ R R |∂ r v| 2 -|∇ S u| 2 + k 2 |u| 2 -2k Γ R βv ∂ r v + 2α Γ R v ∂ r v (2.

13)

where P (v) is defined by (2.12).

Proof. We first assume that v ∈ C ∞ (B R ). Given a Lipschitz domain D with outward-pointing unit normal vector ν, recall that the divergence theorem

D ∇ • F = ∂D F • ν is valid when F ∈ C 1 (D, C d ) [
B R ∇ • Q = B R P .
We now need to show that Q • x equals the integrand on the right-hand side of (2.13).

The assumptions that A = I, n = 1, Z = x, and α is constant in a neighbourhood of Γ R imply that, in this neighbourhood,

Q(v) := 2 (x • ∇v + ikβv + αv)∇v + x k 2 |v| 2 -|∇v| 2 . (2.14) Since v ∈ C ∞ (B R ), ∇v = x∂ r v + ∇ S v on Γ R .
Using this, along with the fact that x • ∇ S v = 0, in (2.14), we find that Q • x equals the integrand on the right-hand side of (2.13).

We have therefore proved the result under the additional condition Page 77] and (ii) the fact that (2.13) is continuous in v with respect to the topology of

that v ∈ C ∞ (B R ). The result for v ∈ H 2 (B R ) then follows from (i) the density of C ∞ (B R ) in H 2 (B R ) [53,
H 2 (B R ). (Regarding (ii), recall that, since Γ R ∈ C 1,1 , both ∂ r and ∇ S map H 2 (B R ) → L 2 (Γ R
) by properties of the trace map [START_REF] Mclean | Strongly elliptic systems and boundary integral equations[END_REF]Theorem 3.37].)

The following lemma deals with the contribution from Γ R in (2.13) when v equals a solution of the Helmholtz equation satisfying the Sommerfeld radiation condition.

Let α ∈ R with 2α ≥ d -1. Then, for R > R 0 , Γ R R |∂ r u| 2 -|∇ S u| 2 + k 2 |u| 2 -2kR Γ R ū ∂ r u + 2α Γ R ū ∂ r u ≤ 0, (2.15)
where ∇ S is the surface gradient on r = R.

References for proof of Lemma 2. The idea behind the proof of Lemma 2.5 is that, when 2α = d -1, the result follows integrating over R d \ B R the following identity, which first appeared as [63, Equation 1.2],

2 {M α vLv} = ∇ • 2 M α v∇v + k 2 |v| 2 -|∇v| 2 x + 2α -(d -1) k 2 |v| 2 -|∇v| 2 -|∇v| 2 - ∂v ∂r - ∂v ∂r -ikv 2 . (2.16)
where

M α v := r ∂v ∂r -ikv + α r v , (2.17)
α ∈ R, and ∂v/∂r = x • ∇v/r. In particular, with (2.16) written as ∇ • Q(v) = P (v), the term on the left-hand side of (2.15) is Γ R Q(u); this is why the constant in (2.15) is denoted by α, to emphasise its connection with the function α in (2.1)/(2.17). Lemma 2.5 therefore implies that if the multiplier Z (2.1) in the integrated identity (2.13) is equal to M α (with 2α ≥ d -1) outside a large ball, then the right-hand side of (2.13) is ≤ 0.

Background results for the Helmholtz transmission problem

(i) -DtN k φ, φ Γ R ≥ 0 for all φ ∈ H 1/2 (Γ R ), (2.18)
where •, • Γ R denotes the duality pairing on Γ R .

(ii) There exists C DtN > 0, independent of k, such that

DtN k u, v Γ R ≤ C DtN u H 1 k (D R ) v H 1 k (D R ) for all u, v ∈ H 1 (D R ). ( 2
1 (B R ).
Proof of Lemma 2.7. This follows from the Cauchy-Schwarz inequality and the inequalites (1.3), (1.4), (2.19).

Lemma 2.8 (Well-posedness from an a priori bound) Suppose that, under the assumption of existence, the solution of the transmission problem (1.5) with

F (v) = B R f v and f ∈ L 2 (B R ) satisfies u H 1 k (B R ) ≤ C(A, n, k, R) f L 2 (B R ) for all k ≥ k 0 , (2.20)
for some C(A, n, k, R) > 0 and k 0 > 0. Then the Helmholtz transmission problem of Definition 1.1 is well-posed: i.e., given F ∈ (H 1 (B R )) , let u satisfy the variational problem (1.5); then, if k ≥ k 0 , u exists, is unique, and the map F → u is continuous (i.e., C sol < ∞)

Proof. The inequalities (2.18), (1.3), and (1.4) imply that, for any

v ∈ H 1 (B R ), b(v, v) ≥ A min ∇v 2 L 2 (B R ) -k 2 n max v 2 L 2 (B R ) ≥ A min v 2 H 1 k (B R ) -k 2 (n max + A min ) v 2 L 2 (B
> 0. Given F ∈ (H 1 (B R ))
, let u satisfy the variational problem (1.5). Then u exists, is unique, and satisfies the bound

u H 1 k (B R ) ≤ 1 min(A min , n min ) 1 + 2n max k C(A, n, k, R) F (H 1 k (B R ))
for all k ≥ k 0 .

References for the proof. This is proved in [

A L 2 (B R ,C d×d ) := |A(•)| 2 L 2 (B R ) for A ∈ C d×d .
Lemma 2.10 (Solution of variational problems by approximation) Given coefficients A and n satisfying the conditions in Definition 1.1, assume that there exist sequences of coefficients (A ) ∞ =0 and (n ) ∞ =0 satisfying, for every , the conditions in Definition 1.1 (possibly with different limits n min , n max , A min , and A max in the inequalities (1.3) and (1.4)), and such that (i)

A -A L 2 (B R ,C d×d ) ≤ 1 and n -n L 2 (B R ) ≤ 1 for all ,
and (ii) there exists an 0 > 0 and C(k, R) > 0 such that, for every ≥ 0 , the solution of the variational problem (1.5) with F (v) = B R f v and coefficients A and n exists, is unique, and satisfies the bound

u H 1 k (B R ) ≤ C(k, R) f L 2 (B R ) .
(

F (H 1 k (B R )) (with a different C(k, R))
, thereby covering more general variational problems. In the rest of the paper, however, we only use Lemma 2.10 as stated, i.e. applied to variational problems with right-hand sides of the form B R f v.

Traces and weighted Sobolev spaces

We repeatedly use the following standard trace result (see, e.g., [START_REF] Mclean | Strongly elliptic systems and boundary integral equations[END_REF]Theorem 3.37]).

Lemma 2.11 If D is a bounded C -1,1 domain and 1/2 < s ≤ , then the trace operator is bounded

H s (D) → H s-1/2 (∂D).
We use also the multiplicative trace inequality for Lipschitz D, i.e. that there exists C tr > 0 such that

v 2 L 2 (∂D) ≤ C tr 1 D v 2 L 2 (D) + v L 2 (D) |v| H 1 (D)
for all v ∈ H 1 (D), (2.23) where D is the diameter of D (see [42, Theorem 1.5.1.10, last formula on Page 41]) and thus, with

• H 1 k (D) defined by (1.12), k 1/2 v L 2 (∂D) v H 1 k (D)
for all v ∈ H 1 (D).

(2.24)

We define • H s k (∂D) for s = 1 by

φ 2 H 1 k (∂D) := ∇ ∂D φ 2 L 2 (∂D) + k 2 φ 2 L 2 (∂D) ,
and for 0 < s < 1 by interpolation, choosing the specific norm given by the complex interpolation method (equivalently, by real methods of interpolation appropriately defined and normalised; see, e.g., [START_REF] Chandler-Wilde | Interpolation of Hilbert and Sobolev spaces: quantitative estimates and counterexamples[END_REF]Remark 3.6]). We then define the norms on H s (∂D) and H s k (∂D) for -1 ≤ s < 0 by duality,

φ H s (∂D) := sup 0 =ψ∈H -s (∂D) | φ, ψ ∂D | ψ H -s (∂D) and φ H s k (∂D) := sup 0 =ψ∈H -s (∂D) | φ, ψ ∂D | ψ H -s k (∂D)
, for φ ∈ H s (∂D), where φ, ψ ∂D denotes the standard duality pairing that reduces to (φ, ψ) ∂D , the inner product on L 2 (∂D), when ψ ∈ L 2 (∂D). In the terminology of [20, Remark 3.8], with the norms we have selected, {H s k (∂D) : -1 ≤ s ≤ 1} are exact interpolation scales. Finally, we need the weighted analogue of Lemma 2.11 with s = 1; see, e.g., [START_REF] Nédélec | Acoustic and electromagnetic equations: integral representations for harmonic problems[END_REF]Theorem 5.6.4].

Lemma 2.12 (Trace and extension in k-weighted spaces) If D is Lipschitz, then given k 0 > 0 there exists C, depending on k 0 but independent of k, such that

v H 1/2 k (∂D) ≤ C v H 1 k (D)
for all v ∈ H 1 (D) and k ≥ k 0 .

Furthermore, there exists an extension operator E : H 1/2 (∂D) → H 1 (D) such that, given k 0 > 0 there exists C , depending on k 0 but independent of k, such that

E(φ) H 1 k (D) ≤ C φ H 1/2 k (∂D)
for all φ ∈ H 1/2 (Γ).

(2.25)

3 Proof of Theorem 1.6 (the well-posedness result)

We follow the steps outlined in §1.2.5. In §3.1 we prove the analogue of Theorem 1.7 for smooth coefficients satisfying Condition 3.1. In §3.2 we show how coefficients satisfying Condition 1.4 can be approximated by coefficients satisfying Condition 3.1. Theorem 1.7 then follows from using Lemma 2.10. Having proved Theorem 1.7, Theorem 1.6 then follows from Lemma 2.9.

3.1 The analogue of Theorem 1.7 for a class of smooth coefficients

• A ∈ C ∞ (R d , SPD), n ∈ C ∞ (R d , SPD),
• there exist A min , A max , n min , and n max such that A and n satisfy (1.4) and (1.3), respectively,

• there exists

0 < R 0 < R such that supp(I -A) ⊂ B R0 and supp(1 -n) ⊂ B R0 , • x d ∂A ∂x d (x) 0 and x d ∂n ∂x d (x) ≥ 0 for all x ∈ R d , (3.1)
• (A d )(x) = 0 for = 1, . . . , d -1 and for all x ∈ R d .

Observe that the monotonicity conditions in (3.1) are the continuous versions of (1.8) and (1.9) from Condition 1.4. In addition, the conditions (3.1) along with the assumptions supp(I-A) ⊂ B R0 and supp(1 -n) ⊂ B R0 imply that A min ≥ 1 and n max ≤ 1. • β = R; and

• 2α = ∇ • Z -q χ for some q ∈ [0, 1] (to be fixed later in the proof).

These choices allow us to use the inequality of Lemma 2.5 in (2.13) and obtain that B R P (u) ≤ 0. The expression for P (u) (2.12) can then be simplified, and bounded below using the monotonicity assumptions (3.1) on A and n, with the result that

B R 2|∂ d u| 2 1 -χ + |∇u| 2 (2 -q)χ + qk 2 |u| 2 χ + 2r|∂ r u| 2 χ dx -2 B R x d ∂ d ū∂ r uχ ≤ -2kR B R f ū + B R f 2x d ∂ d ū 1 -χ + 2r∂ r ūχ + 2αū + B R ∆α|u| 2 . (3.2)
Observe that when χ = 0 (i.e. in B R0 ) we only have control of |∂ d u| 2 in the first integral on the left-hand side, but in supp(χ) we have control of

|∇u| 2 + k 2 |u| 2 .
The rest of the argument consists of 1. getting rid of the sign-indefinite "cross" term in the second integral on the left-hand side of (3.2);

2. using the following Poincaré-Friedrichs-type inequality [START_REF] Chandler-Wilde | High-frequency bounds for the helmholtz equation under parabolic trapping and applications in numerical analysis[END_REF]Lemma 2.7] to put |u| 2 back in B R0 (i.e. where χ = 0),

B 2R |v| 2 ≤ 8 B √ 13R \B 2R |v| 2 + 4R 2 B √ 13R |∂ d v| 2 for all R > 0 and v ∈ H 1 (R d );
3. using the following consequence of Green's identity and the inequality (2.18) to put |∇u| 2 back in B R0 ,

min A min , 1 B R |∇u| 2 ≤ max n max , 1 B R |u| 2 + B R f u; (3.3)
4. bounding the term on the right-hand side of (3.2) involving the Laplacian of α;

5. using the inequality 2ab ≤ δa 2 + δ -1 b 2 (for a, b, δ > 0) on the other terms on the right-hand side of (3.2).

Regarding 1: this involves imposing a constraint on the growth of χ (see Point (ii) in Definition 3.3 and the discussion below this definition). Regarding 2: here is the place where we lose powers of k compared to the nontrapping estimate, since, in the Poincaré-Friedrichs-type inequality, |∂ d u| 2 is bounded below by |u| 2 without a corresponding factor of k 2 .

The paper [START_REF] Chandler-Wilde | High-frequency bounds for the helmholtz equation under parabolic trapping and applications in numerical analysis[END_REF] considers the constant-coefficient Helmholtz equation posed outside a class of obstacles, the prototypical example of which is two aligned cubes (so that there exist weaklytrapped rays between the two components of the obstacle). The arguments in [START_REF] Chandler-Wilde | High-frequency bounds for the helmholtz equation under parabolic trapping and applications in numerical analysis[END_REF] use the same vector field Z we use here, and, in fact, [START_REF] Chandler-Wilde | High-frequency bounds for the helmholtz equation under parabolic trapping and applications in numerical analysis[END_REF] obtains exactly the inequality (3.2) in the course of its arguments. The details of the Steps 1-5 above are therefore exactly as in [START_REF] Chandler-Wilde | High-frequency bounds for the helmholtz equation under parabolic trapping and applications in numerical analysis[END_REF], and so we do not repeat the details below, instead making precise reference to the relevant results in [START_REF] Chandler-Wilde | High-frequency bounds for the helmholtz equation under parabolic trapping and applications in numerical analysis[END_REF].

3.1.3 The proof of Lemma 3.2 Definition 3.3 (The vector field Z) Given 0 < R 0 < R 1 < ∞ such that supp(I -A) ⊂ B R0 and supp(1 -n) ⊂ B R0 and χ ∈ C 3 [0, ∞) with (i) χ(r) = 0 for 0 ≤ r ≤ R 0 , χ(r) = 1, for r ≥ R 1 , 0 < χ(r) < 1, for R 0 < r < R 1 ; and (ii) 0 ≤ rχ (r) < 4, for r > 0; let Z(x) := e d x d 1 -χ(r) + x χ(r), x ∈ R d . (3.4)
The requirement (ii) on χ is needed to control the term 2 B R x d ∂ d ū∂ r uχ (r) dx on the lefthand side of (3.2), ensuring that this left-hand side is bounded below by a multiple of B R (|∂ d u| 2 + χ|∇u| 2 )dx. This requirement imposes a constraint on R 1 /R 0 ; indeed, [START_REF] Chandler-Wilde | High-frequency bounds for the helmholtz equation under parabolic trapping and applications in numerical analysis[END_REF]Remark 1.5] shows that if χ satisfies Points (i) and (ii) in Definition 3.3, then R 1 /R 0 > e 1/4 ≈ 1.284, and, conversely, if R 1 /R 0 > e 1/4 , then there exists a χ ∈ C 3 [0, ∞) satisfying Points (i) and (ii) in Definition 3.3.

2α := ∇ • Z -q χ(r), (3.5)
for some q ∈ [0, 1], with χ as in Definition 3.

B R P (u) ≤ 0, (3.6)
where P (u) is defined by (2.12). We first claim that, in B R ,

P (u) ≥ -2 (Zu f ) -qχ|∇u| 2 + qχk 2 |u| 2 + 2 A∇u, ∂Z ∇u -∆α|u| 2 . (3.7)
Indeed, this follows from using (i) the conditions on ∂A/∂x d and ∂n/∂x d (3.1), noting that when A = I and n = 1, Z = x d e d , (ii) the definitions of α (3.5) and β = R, (iii) the fact that when χ = 0, A = I and n = 1, (iv) the fact that α is constant when A = I.

To deal with the term A∇u, ∂Z ∇u , we first observe that (with the summation convention for the indices i and j but not d)

∇v, ∂Z ∇v = ∂ i Z j ∂ i v∂ j v = |∂ d v| 2 1 -χ(r) + |∇v| 2 χ(r) + r|∂ r v| 2 -x d ∂ d v∂ r v χ (r).
P (u) ≥ -2 Zu f out + 2|∂ d u| 2 1 -χ(r) + |∇u| 2 (2 -q)χ(r) + qk 2 |u| 2 χ(r) + 2r|∂ r u| 2 χ (r) -2 x d ∂ d u∂ r u χ (r) -∆α|u| 2 .
and (3.2) then follows from (3.6) by expanding the Zu f term using that

Z • ∇u = x d ∂ d u 1 -χ(r) + r∂ r uχ(r).
Proof 3.20] is a condition of the form kR 0 ≥ C wave with C wave dependent only on χ, we need to show that (R 0 ) 2 m α (R) is a dimensionless quantity depending only on χ, where m α (R) := sup x∈B R ∆α(x) and 2α = ∇ • Z -q χ (as in §3.1.2), with Z defined by (3.4). By definition, α is constant for R ≥ R 1 (see [START_REF] Chandler-Wilde | High-frequency bounds for the helmholtz equation under parabolic trapping and applications in numerical analysis[END_REF]Equation 3.15]), and thus m α (R) = m α (R 1 ). The only constraint on R 1 is that R 1 /R 0 is sufficiently large (see the discussion below Definition 3.3); therefore, we can choose R 1 to be proportional to R 0 , and thus (R 0 ) 2 m α (R) is a dimensionless quantity depending only on χ as claimed.

3.2

Approximation of A and n satisfying Condition 1.4 by A δ and n δ satisfying Condition 3.1 Lemma 3.6 (Approximation of A and n satisfying Condition 1.4) Given A and n satisfying Condition 1.4, there exist (A ) ∞ =0 , (n ) ∞ =0 such that, for sufficiently large, (i) A and n satisfy Condition 3.1, and (ii)

A -A L 2 (B R ,C d×d ) ≤ 1 and n -n L 2 (B R ) ≤ 1 .
Proof. The idea of the proof is to mollify A and n to approximate them by families of C ∞ functions A δ and n δ that satisfy Condition 3.1. However, before mollification, we first must approximate A and n by functions that are constant in the x d direction in a neighbourhood of x d = 0. To see why this is necessary, suppose that, in some compact set, n is only a function of x d , and equals C for 0 ≤ x d < c and C -x d for -c < x d < 0 (observe that this function satisfies the monotonicity assumption (1.9) in -c < x d < c). Then, with n δ the standard mollification of n, given δ > 0, there exists an x * d > 0 such that (∂n δ /∂x d )| x d =x * d ≤ 0, violating the requirement on n δ in (3.1). Let 0 < R 0 < R be such that supp(I -A) and supp(1 -n) are both ⊂ B R0 . Given δ 1 > 0, let

A δ1 (x) := A max I, for x ∈ (x , x d ) : |x | ≤ R 0 , |x d | ≤ δ 1 , A(x), otherwise, (3.11)
and let

n δ1 (x) := n min I, for x ∈ (x , x d ) : |x | ≤ R 0 , |x d | ≤ δ 1 , n(x), otherwise,
Observe that (i), A δ1 and n δ1 satisfy Condition 1.4, (ii) supp(I -A δ1 ) and supp(1 -n δ1 ) are both ⊂ B R1(δ1) , where R 1 (δ 1 ) := R 2 0 + δ 2 1 , and (iii) there exists C d > 0 (depending only on the dimension d) such that

A -A δ1 L 2 (B R ,C d×d ) ≤ C d (R 0 ) d-1 δ 1 A max and n -n δ1 L 2 (B R ) ≤ C d (R 0 ) d-1 δ 1 n max . (3.12) Let ψ ∈ C ∞ 0 (R d ) be defined by ψ(x) := C exp (|x| 2 -1) -1 if |x| < 1, 0 if |x| ≥ 1,
where C is chosen so that

R d ψ(x)dx = 1. Define ψ δ (x) := δ -d ψ(x/δ), so that ψ δ (x) = 0 if |x| > δ and R d ψ δ (x)dx = 1. Let ( A δ1 ) δ (x) := ( A δ1 * ψ δ )(x) = |y|<δ A δ1 (x -y)ψ δ (y) dy,
where the convolution is understood element-wise, and similarly

( n δ1 ) δ (x) := ( n δ1 * ψ δ )(x) = |y|<δ n δ1 (x -y)ψ δ (y) dy.
Standard properties of mollifiers (see, e.g., [29, §C.4 Theorem 6]) imply that

• ( A δ1 ) δ , ( n δ1 ) δ ∈ C ∞ (R d ), • given R > 0, both ( A δ1 ) -( A δ1 ) δ L 2 (B R ;C d×d ) and ( n δ1 ) -( n δ1 ) δ L 2 (B R )
→ 0 as δ → 0, and

• n min ≤ ( n δ1 ) δ ≤ n max and A min ( A δ1 ) δ A max for all x ∈ R d .
Since ( A δ1 ) δ is formed from ( A δ1 ) by elementwise convolution, the condition (( A δ1 ) δ ) d (x) = 0 for = 1, . . . , d -1 and for all x ∈ R d in Condition 3.1 follows from the corresponding condition on A δ1 , which follows from the corresponding condition on A in Condition 1.4. Furthermore, since supp(I-A δ1 ) and supp(1-n δ1 ) ⊂ B R1(δ1) , the definitions of A δ and n δ imply that that supp(I-A δ ) and supp(1

-n) ⊂ B R1(δ1)+δ which is ⊂ B R for δ < R -R 1 (δ 1 ) = R -R 2 0 + δ 2 1
. We now show that, when δ is sufficiently small, ( A δ1 ) δ and ( n δ1 ) δ satisfy the monotonicity conditions (3.1) (with A replaced by ( A δ1 ) δ and n replaced by ( n δ1 ) δ ). Observe that

( A δ1 ) δ (x + he d ) -( A δ1 ) δ (x) = |y|<δ A δ1 (x + he d -y) -A δ1 (x -y) ψ δ (y)dy. ( 3.13)
We claim that, when δ ≤ δ 1 , (i) when x d > 0, the integrand on the right-hand side of (3.13) is 0 for all h ≥ 0, and (ii) when x d < 0, the integrand on the right-hand side of (3.13) is 0 for all h ≥ 0. To see (i), first recall that, from its definition (3.11)

, A δ1 (z) is constant in the z d direction when |z d | ≤ δ 1 . If x d ≥ δ 1 then x d -y d > 0,
and the integrand on the right-hand side of (3.13) is 0 by (1.8)

. If 0 < x d < δ 1 , x d -y d is no longer > 0 for all |y| < δ, but x d -y d > -δ 1 ,
and the fact that A δ1 (z) is constant in the z d direction when |z d | ≤ δ 1 implies that the integrand on the right-hand side of (3.13) is either = 0 (when

x d -y d + h ≤ δ 1 ) or 0 by (1.8) (when x d -y d + h > δ 1
). The proof of (ii) is similar. We have therefore shown that, for δ sufficiently small,

x d ( A δ1 ) δ (x + he d ) -( A δ1 ) δ (x) 0 for all h ≥ 0; since ( A δ1 ) δ ∈ C ∞ (R d ) this implies that x d (∂( A δ1 ) δ /∂x d )(x) 0.
In an essentially-identical way (with inequality in the sense of quadratic forms replaced by standard inequality), we find that

x d (∂( n δ1 ) δ /∂x d )(x) ≥ 0.
We have therefore shown that ( A δ1 ) δ and ( n δ1 ) δ satisfy Condition 3.

1 for δ ≤ min{δ 1 , R - R 2 0 + δ 2 1 }
, where the first term in the minimum ensures that ( A δ1 ) δ and ( n δ1 ) δ satisfy the monotonicity condition (3.1), and the second term in the minimum ensures that supp(I -( A δ1 ) δ ) and supp(1

-( n δ1 ) δ ) are both ⊂ B R .
We now define A to be ( A δ1 ) δ for specific δ 1 and δ, and similarly for n . Indeed, from the properties of mollifiers above we have that, given δ 1 > 0, ε > 0, there exists

δ * = δ * (A, d, δ 1 , ε) > 0 such that ( A δ1 ) δ -A δ1 L 2 (B R ,C d×d ) ≤ ε 2 for all δ ≤ δ * (A, d, δ 1 , ε). (3.14) Set δ 1 := (2C d (R 0 ) d-1 A max ) -1 ; then, if δ ≤ δ * (A, d, δ 1 , -1
), the inequalities (3.12) and (3.14) imply that

A -( A δ1 ) δ L 2 (B R ;C d×d ) ≤ A -A δ1 L 2 (B R ;C d×d ) + A δ1 -( A δ1 ) δ L 2 (B R ;C d×d ) , ≤ C d (R 0 ) d-1 δ 1 A max + 1 2 ≤ 1 2 + 1 2 = 1 .
Recall from above that ( A δ1 ) δ and ( n δ1 ) δ satisfy Condition 3. 4 Proofs of Lemma 1.13, Theorem 1.14, and Corollary 1.15

1 when δ ≤ min{δ 1 , R -R 2 0 + δ 2 1 }. We therefore set δ := min δ 1 , R -R 2 0 + δ 2 1 , δ * (A, d, δ 1 , -1 ) and A := ( A δ1 ) δ ; then A -A L 2 (B R ;R d×d ) ≤ -
If v : D in × Y → C, let v ε : D in → C denote the function v ε (x) := v x, x ε ,
where {y} = y mod 1. (As a special case of this, if v :

Y → C, then v ε (x) := v({x/ε}).) Given such a v, ∂v ε ∂x j = ∂v ∂x j + 1 ε ∂v ∂y j ε . (4.1)
From here on, when we use the notation , the omitted constant only depends on n, A, and D in .

The homogenized problem

The homogenized coefficients

The homogenized coefficient n H is the mean value of n over the periodic cell Y , i.e., (recalling that |Y | = 1)

n H := Y n. (4.2)
The definition of A H requires the auxiliary functions χ j , j = 1, 2, 3, and matrices C and B.

Let H 1 (Y ) be the subspace of H 1 per (Y ) consisting of functions with zero mean. The function

χ j ∈ H 1 (Y ) is defined as the solution of -∇ y • A∇ y χ j = - ∂( A) j ∂y (4.3)
where the index on the right-hand side is summed over, and the derivative on the right-hand side is understood in a distributional sense. Lemma A.2 shows that χ j , j = 1, 2, 3, exists, is unique, and is in H 1+s (Y ) ∩ C 1 (P) (i.e., C 1 on each element of the partition P) for some s > 0, with

χ j H 1+s (Y ) 1 and χ j W 1,∞ (P) 1, (4.4)
where both the omitted constant and s depend only on A. The matrix C is defined by

( C) j := ∂ χ ∂y j , 1 ≤ j, ≤ d. (4.5)
Because of (4.4), ( C) j ∈ H s (Y ) ∩ C 0 (P) for some s > 0 with however, χ d = 0 since the right-hand side of (4.4) is

C H s (Y ) 1 and C L ∞ (Y ) 1. With B := A(I -C), ( 4
A H = Y B = Y A - Y A C,
- d =1 ∂( A) d ∂y = - ∂( A) dd ∂y d = 0, because A is independent of y d . If D in is star-shaped,
F (v) = B R f v, then u 0 H 1 k (B R ) ≤ 2 √ n min R 2 + 1 n min R + d -1 2k 2 f L 2 (B R )
for all k > 0;

the bound (1.19) follows from combining this with Lemma 2.9.

Lemma 4.1 (Shift property for the homogenized problem)

If Γ is C 1,1 , then for all g ∈ L 2 (B R ), if v ∈ H 1 (B R ) satisfies -∇ • A H ∇v = g in B R then v ∈ H 2 (D in ) with |v| H 2 (Din) g L 2 (B R ) .
Proof.

Since
F (v) = B R f v satisfies u 0 H 2 (B R ) + k u 0 H 1 k (B R ) C sol,H f L 2 (B R ) . (4.8)
Proof. The bound on u H 1 k (B R ) in (4.8) follows from the definition of C sol,H . The bound on u H 2 (B R ) in (4.8) follows from Lemma 4.1 with v = u, g = n H k 2 u + f , and α = 0, and the fact that C sol,H 1 (as noted below (1.12)).

The first-order, second-order, and boundary correctors

Periodic homogenization relies on the formal asymptotic expansion

u ε (x) = u 0 (x) + εu ε 1 (x) + ε 2 u ε 2 (x) + . . . ,
where u 0 is the homogenized solution and u 1 , u 2 are "two scale" functions usually called "correctors". Since, in our case, the homogenization process only takes place in D in , we need an additional corrector, called the "boundary corrector", to deal with the interface Γ.

The first-order corrector

The first-order corrector u 1 : D in × Y → C is defined by

u 1 (x, y) := -χ j (y) ∂u 0 ∂x j (x). ( 4
and k u ε 1 L 2 (Din) + (∇ x u 1 ) ε L 2 (Din) k u 0 H 1 k (B R ) + |u 0 | H 2 (Din) . (4.11) Furthermore, u ε 1 H 1 k (B R ) 1 + (kε) -1 |u 0 | H 2 (Din) + k u 0 H 1 k (B R ) , (4.12)
and, if kR 0 1,

u ε 1 H 1/2 k (Γ) 1 + (kε) -1/2 k u 0 H 1 (Din) + u 0 H 2 (Din) . (4.13)
Proof. By the definitions of C (4.5) and of u 1 (4.9),

∂u 1 ∂y = - ∂ χ j ∂y ∂u 0 ∂x j = -C∇u 0 ,
which is (4.10). By the definition of u 1 (4.9) and the bound on χ j (4.4),

u ε 1 L 2 (Din) ≤ χ j L ∞ (Y ) ∂u 0 ∂x j L 2 (Din) |u 0 | H 1 (Din)
and

(∇ x u 1 ) ε L 2 (Din) ≤ χ j L ∞ (Y ) ∂u 0 ∂x j H 1 (Din) |u 0 | H 2 (Din) ,
and then the bound (4.11) follows. By (4.1) and (4.10),

∇(u ε 1 ) = (∇ x u 1 ) ε + 1 ε (∇ y u 1 ) ε = (∇ x u 1 ) ε - 1 ε C ε ∇u 0 . Therefore ∇(u ε 1 ) L 2 (Din) |u 0 | H 2 (Din) + ε -1 |u 0 | H 1 (Din) (kε) -1 |u 0 | H 2 (Din) + k u 0 H 1 k (B R ) .
Then, the bound (4.12) follows from combining this with the bound on u ε 1 L 2 (Din) in (4.11). Let the operator T : L 2 (Γ) → L 2 (Γ) be defined by T v := -χ ε j v j for all v ∈ L 2 (Γ); then

T v L 2 (Γ) v L 2 (Γ) for all v ∈ L 2 (Γ).
In addition, T :

H 1 (Γ) → H 1 (Γ) with T v H 1 (Γ) ε -1 v L 2 (Γ) + v H 1 (Γ) for all v ∈ H 1 (Γ).
By interpolation (see, e.g., [53, Appendix B]), T then maps

H 1/2 (Γ) → H 1/2 (Γ) with T v H 1/2 (Γ) ε -1/2 v L 2 (Γ) + |v| H 1/2 (Γ) .
We now observe that

u ε 1 | Γ = T ((∇u 0 )| Γ ), so that u ε 1 H 1/2 (Γ) ε -1/2 ∇u 0 L 2 (Γ) + |∇u 0 | H 1/2 (Γ) . (4.14)
By Lemma 2.11,

∇u 0 H 1/2 (Γ) u 0 H 2 (Din) , (4.15)
and, by (2.23),

∇u 0 2 L 2 (Γ) 1 R 0 u 0 2 H 1 (Din) + u 0 H 1 (Din) u 0 H 2 (Din) (4.16)
1 k 1 + 1 kR 0 k 2 u 0 2 H 1 (Din) + u 0 2 H 2 (Din) . (4.17)
Using (4.15) and (4.16) in (4.14), and recalling the assumption that kR 0 1, the result (4.13) follows.

The second-order corrector

The second-order corrector arises since n ε varies (in addition to A ε varying); that corrector is formally defined by

u 2 (x, y) := k 2 µ(y)u 0 (x),
where µ is the unique element of H 1 (Y ) such that

-∇ y • A∇ y µ = n -n H . (4.18) By Lemma A.2, µ ∈ C 1 (P) with µ W 1,∞ (P) 1.
We can actually bypass referring explicitly to this second-order corrector in the rest of proof of Theorem 1.14, and directly establish the following result (where, both here and in the rest of the paper, (•, •) Din denotes the L 2 (D in ) inner product, and similarly for inner products over other domains).

Lemma 4.4 (Bound on the contribution from the second-order corrector)

k 2 ( n ε -n H )u 0 , v Din ε k u 0 H 1 k (Din) v H 1 k (Din) for all v ∈ H 1 (D in ). (4.19)
Proof. Since both A and µ are independent of x, by (4.18),

-∇ • ( A∇ y µ) ε = - 1 ε ∇ y • ( A∇ y µ) ε = 1 ε ( n ε -n H ) ∈ L ∞ (D in ).
Therefore,

-∇ • (u 0 ( A∇ y µ) ε ) = -u 0 ∇ • ( A∇ y µ) ε -∇u 0 • ( A∇ y µ) ε = 1 ε ( n ε -n H )u 0 -∇u 0 • ( A∇ y µ) ε .
By Condition 1.10 and Lemma A.2, A ∈ C 0 (P) and µ ∈ C 1 (P). Therefore, the function ( A∇ y µ) ε is piecewise continuous on a partition of D in consisting of a finite number of open sets, so that its trace on ∂D in is well-defined. In addition,

( A∇ y µ) ε L ∞ (Din) ≤ A L ∞ (Y ) ∇ y µ L ∞ (Y ) 1 and similarly ( A∇ y µ) ε L ∞ (∂Din) 1. By Green's first identity in D in , Din u 0 ( A∇ y µ) ε • ∇v = Γ u 0 ( A∇ y µ) ε • νv - Din ∇ • (u 0 A∇ y µ) ε v = Γ u 0 ( A∇ y µ) ε • νv + 1 ε Din ( n ε -n H )u 0 v - Din ∇u 0 • ( A∇ y µ) ε v,
and thus

1 ε Din ( n ε -n H )u 0 v = Γ u 0 ( A∇ y µ) ε • νv - Din ∇u 0 • ( A∇ y µ) ε v - Din u 0 ( A∇ y µ) ε • ∇v
To estimate each term in the right-hand side, we use the Cauchy-Schwarz inequality on each L 2 inner-product, so that

|(( n ε -n H )u 0 , v) Din | ≤ ε ( A∇ y µ) ε L ∞ (Din) u 0 L 2 (Din) |v| H 1 (Din) + ε ( A∇ y µ) ε L ∞ (Din) |u 0 | H 1 (Din) v L 2 (Din) + ε ( A∇ y µ) ε L ∞ (Γ) u 0 L 2 (Γ) v L 2 (Γ) ε u 0 L 2 (Din) |v| H 1 (Din) + |u 0 | H 1 (Din) v L 2 (Din) + u 0 L 2 (Γ) v L 2 (Γ) .
Multiplying by k 2 and then using (2.24), we have

k 2 |(( n ε -n H )u 0 , v) Din | kε k u 0 L 2 (Din) |v| H 1 (Din) + k|u 0 | H 1 (Din) v L 2 (Din) + k u 0 L 2 (Γ) v L 2 (Γ) kε u 0 H 1 k (Din) v H 1 k (Din)
, which is the result (4.19).

Boundary corrector

The following lemma is proved using Theorem B.6 from Appendix B. A similar result (without the explicit k-dependence) appears in [68, Lemma 2.2] and [13, Page 2536].

Lemma 4.5 (Existence of vector potential) There exists a vector potential

q : D in × Y → C d such that ∇ y × q = ( B -A H )∇u 0 , (4.20)
(∇ x × q) ε L 2 (Din) |u 0 | H 2 (Din) (4.21)
and, if kR 0 1,

∇ × q ε • ν H -1/2 (Γ) (1 + (kε) -1/2 ) k u 0 H 1 (Din) + u 0 H 2 (Din) . (4.22)
Remark 4.8 ( Difference between our first-order and boundary correctors and those in [START_REF] Cakoni | On the homogenization of a scalar scattering problem for highly oscillating anisotropic media[END_REF] .) In [START_REF] Cakoni | On the homogenization of a scalar scattering problem for highly oscillating anisotropic media[END_REF], the first-order corrector is defined by u ε 1 in D in and zero in D out , and is therefore not in H 1 (B R ) because of the jump on Γ. Analogously, the boundary corrector θ ε defined by [START_REF] Cakoni | On the homogenization of a scalar scattering problem for highly oscillating anisotropic media[END_REF]Equation 18] has a jump on Γ and is not in H 1 (B R ). In contrast, both our first-order corrector u 1,ε (4.25) and our boundary corrector θ ε (4.23) are defined so that they are in H 1 (B R ).

Proof of Theorem 1.14 using Lemma 4.7. By (in this order) the triangle inequality, the definition of u 1,ε (4.25), the triangle inequality again and the bound (2.25), the bounds (4.26) and (4.24), and the fact that C layer (kR, kR 0 , n min ) 1,

k u ε -u 0 -εu ε 1 H 1 k (Din) + k u ε -u 0 H 1 k (Dout) ≤ k u ε -u 0 -εu 1,ε H 1 k (B R ) + kε E(u ε 1 ) H 1 k (B R ) , k u ε -u 0 -ε(u 1,ε -θ ε ) H 1 k (B R ) + kε θ ε H 1 k (B R ) + kε u ε 1 H 1/2 k (Γ) , C layer (kR, kR 0 , n min )kε u 0 H 2 (B R ) + k u 0 H 1 k (B R ) + C layer (kR, kR 0 , n min )kε u ε 1 H 1/2 k (Γ) + ∇ × q ε • ν H -1/2 (Γ) .
The bound (1.20) then follows from the bounds (4.22) and (4.13); the bound (1.21) then follows from the bound (4.8).

Proof of Lemma 4.7

Since

φ H 1 k (B R ) = sup ψ∈(H 1 k (B R )) ψ =0 φ, ψ ψ (H 1 k (B R ))
(see, e.g., [53, Page 76 and Theorem 3.30]), if we can show that

(u ε -(u 0 + ε(u 1,ε -θ ε ), ψ) B R M ψ (H 1 k (B R )) (4.27)
for all ψ ∈ L 2 (B R ), with

M := C layer (kR, kR 0 , n min )ε u 0 H 2 (B R ) + k u 0 H 1 k (B R ) , then the bound (4.26) follows since L 2 (B R ) is dense in (H 1 (B R )) . With ψ an arbitrary element of L 2 (B R ), let W ε ∈ H 1 (B R ) be the solution of the adjoint problem b ε (v, W ε ) = (v, ψ) B R for all v ∈ H 1 (B R ). (4.28)
We now claim that b

ε (W ε , v) = (ψ, v) B R for all v ∈ H 1 (B R ); indeed, this follows from that fact that b ε (•, •) is the sesquilinear form b(•, •) (1.

6) with coefficients

A ε and n ε , and the fact that DtN k φ, ψ Γ = DtN k ψ, φ Γ (from Green's second identity and the Sommerfeld radiation condition (1.1); see, e.g., [START_REF] Spence | Overview of Variational Formulations for Linear Elliptic PDEs[END_REF]Lemma 6.13]). Then, by (1.17),

W ε H 1 k (B R ) ≤ C layer (kR, kR 0 , n min ) ψ (H 1 k (B R )) . (4.29)
The following lemmas involve different components of the inner product on the left-hand side of (4.27); we highlight that the first lemma is the point where the existence of the vector-potential (4.20) is used.

Lemma 4.9 (The terms in the inner product (4.27) involving u ε -u 0 )

(u ε -u 0 , ψ) B R = -k 2 ((n H -n ε )u 0 , W ε ) Din -( A C) ε ∇u 0 , ∇W ε Din + ε (∇ x × q) ε , ∇W ε Din -ε (∇ × q ε ) • ν, W ε Γ (4.30)
By (4.1) and (4.10),

∇(u ε 1 ) = (∇ x u 1 ) ε + 1 ε (∇ y u 1 ) ε = (∇ x u 1 ) ε - 1 ε C ε ∇u 0 ; therefore, A ε ∇(u ε 1 ) = ( A∇ x u 1 ) ε - 1 ε ( A C) ε ∇u 0 ,
and (4.34) (and hence (4.33)) follows.

Proof of Lemma 4.7. By the discussion at the start of the subsection, it is sufficient to prove (4.27). By the definitions of W ε (4.28) and θ ε (4.23)

ε(θ ε , ψ) = ε b ε (θ ε , W ε ) = ε b out (E(u ε 1 ), W ε ) + ε((∇ × q ε ) • ν, W ε ) Γ . (4.35)
We now rewrite

(u ε -(u 0 + ε(u 1,ε -θ ε )), ψ) B R
using (4.30), (4.33), and (4.35), to obtain that

(u ε -(u 0 + ε(u 1,ε -θ ε )), ψ) B R = -k 2 (n H -n ε )u 0 , W ε Din -( A C) ε ∇u 0 , ∇W ε Din + ε (∇ x × q) ε , ∇W ε Din -ε (∇ × q ε ) • ν, W ε Γ -ε b out (E(u ε 1 ), W ε ) + ( A C) ε ∇u 0 , ∇W ε Din + k 2 ε n ε u ε 1 , W ε Din -ε ( A∇ x u 1 ) ε , ∇W ε Din + ε b out (E(u ε 1 ), W ε ) + ε (∇ × q ε ) • ν, W ε Γ = -k 2 ((n H -n ε )u 0 , W ε ) Din + ε((∇ x × q) ε , ∇W ε ) Din -k 2 ε( n ε u ε 1 , W ε ) Din + ε(( A∇ x u 1 ) ε , ∇W ε ) Din .
In particular, we see that the terms ε((∇ × q ε ) • ν, W ε ) Γ and ε b out (E(u ε 1 ), W ε ), present in the right-hand sides of (4.30) and (4.33), respectively, have been removed thanks to the definition of the boundary corrector θ ε (4.23) and its consequence (4.35).

We now bound each of the four terms on the right-hand side. By (4.19),

k 2 (n H -n ε )u 0 , W ε Din kε u 0 H 1 k (B R ) W ε H 1 k (Din) ,
and by (4.21),

ε (∇ x × q) ε , ∇W ε Din ε u 0 H 2 (B R ) W ε H 1 k (Din) . Using the fact that n ε L ∞ (Y )
1 (by Definition 1.9), the Cauchy-Schwarz inequality, and the bound (4.11), we have

k 2 ε ( n ε u ε 1 , W ε ) Din ε k u ε 1 L 2 (Din) k W ε L 2 (Din) ε u 0 H 2 (B R ) +k u 0 H 1 k (B R ) W ε H 1 k (Din) .
Similarly, using that A L ∞ (Y ) 1 (again by Definition 1.9), and the bound (4.11), we have

ε|(( A∇ x u 1 ) ε , ∇W ε ) Din | ≤ ε A ε L ∞ (Din) (∇ x u 1 ) ε L 2 (Din) |W ε | H 1 (Din) ε u 0 H 2 (B R ) + k u 0 H 1 k (B R ) W ε H 1 k (Din) .
Combining these bounds, we have

|(u ε -(u 0 + ε(u 1,ε -θ ε )), ψ) B R | ε u 0 H 2 (B R ) + k u 0 H 1 k (B R ) W ε 1,k,Din
and then using (4.29) we obtain (4.27).

Proof of Corollary 1.15

By Lemma 2.9, it is sufficient to prove that, if

F (v) = B R f v, then u ε H 1 k (B R ) C sol,H k 1 + C layer (kR, kR 0 , n min ) (kε) 1/2 + kε f L 2 (B R ) . (4.36)
By the triangle inequality, Theorem 1.14, and the definition of C sol,H (1.11),

u ε H 1 k (B R ) ≤ u ε -u 0 H 1 k (B R ) + u 0 H 1 k (B R ) , ≤ C sol,H k C 4 C layer (kR, kR 0 , n min ) (kε) 1/2 + kε f L 2 (B R ) + ε u ε 1 H 1 k (Din) + C sol,H k f L 2 (B R ) , (4.37)
The bound on u ε 1 (4.12) and the bound (4.8) imply that

ε u ε 1 H 1 k (Din) ε 1 + (kε) -1 |u 0 | H 2 (Din) + k u 0 H 1 k (B R ) C sol,H k 1 + kε f L 2 (B R ) ; (4.38)
the bound (4.36) then follows by combining (4.37) and (4.38).

Lemma A.2 If Condition 1.10 holds, then χ j , j = 1, 2, 3, defined by (4.3) exists, is unique, and satisfies χ j ∈ H 1+s (Y ) ∩ C 1 (P), for some s > 0, and µ defined by (4.18) exists, is unique, and satisfies µ ∈ C 1 (P). Furthermore, if A and n do not depend on y , then neither do χ j and µ.

Proof. We first claim that both χ j and µ satisfy equations of the form (A.1), (A.2) with f, 1 = 0. For µ this is immediate from (4.18) and the fact that, by the definition of n H (4.2), (n H -n, 1) Y = 0. For χ j , observe that, by its definition (4.3), for all v ∈ H 1 (Y ),

- ∂( A) j ∂y , v = ( A) j , ∂v ∂y Y , so that - ∂( A) j ∂y , 1 = 0.
Therefore, by Lemma A.1, both χ j defined by (4.3) and µ defined by (4.18) exist and are unique.

The lemma also implies that if A and n do not depend on y , then neither do χ j and µ.

We now give the details of the proof of the regularity for χ j ; the proof of the regularity of µ is very similar. Our goal is to use 1. that, for functions depending on r variables, then, if t > 0, there exists C > 0 such that

v C 0,t (Y ) v H r/2+t (Y ) (A.4)
i.e., Sobolev embedding (see, e.g., [START_REF] Mclean | Strongly elliptic systems and boundary integral equations[END_REF]Theorem 3.26]), and 2. an elliptic regularity shift.

Regarding 2.: such a shift is well-known, at least in the non-periodic case. To go from the periodic case to the non-periodic case, we extend the functions u, f 0 and f j in (A.1) and (A. Therefore, if A is piecewise C 0,1 , then d( A) 11 /dy 1 is piecewise L 2 , and χ j is piecewise H 2 by elliptic regularity (see, e.g., [START_REF] Mclean | Strongly elliptic systems and boundary integral equations[END_REF]Theorem 4.20]). By (A.4), χ j is C 0 and piecewise C 1 . By the definition of the Slobodeckii seminorm, if 0 ≤ s < 1/2, then a function that is locally H s is in H s (Y ). Therefore, ∇ χ j ∈ H s (Y ) for s < 1/2; thus χ j ∈ H 1+s (Y ) for s < 1/2. Finally, for (b), by integrating by parts piecewise the right-hand side of the variational problem defining χ j , we find that, for all v ∈ H 1 (Y ),

( A∇ χ j , ∇v) Y = ω∈P ( A) j , ∂v ∂y ω = ω∈P ( A) j ν ω, , v ∂ω - ∂( A) j ∂y , v ω = Γ∈∂P [[( A) j ]]ν Γ, , v Γ - ω∈P ∂( A) j ∂y , v ω
where ∂P denotes the set of interfaces of the partition, and [[•]] denotes the jump of a function across an interface Γ. By elliptic regularity (see, e.g. [START_REF] Mclean | Strongly elliptic systems and boundary integral equations[END_REF]Theorem 4.20]), since each Γ is C 2,1 , the periodic extension of A is piecewise C 1,1 and hence piecewise H 2 , we have

χ j H 3 (P) ∂( A) j ∂y H 1 (P) + Γ∈∂P [[( A) j ]] • ν ,Γ H 3/2 (Γ) A H 2 (P) ,
by continuity of the trace map H 2 (D) → H 3/2 (D) for a C 1,1 domain D (see, e.g., [START_REF] Mclean | Strongly elliptic systems and boundary integral equations[END_REF]Theorem 3.37]). By (A.4) with r = 2, ∇ χ j ∈ C 0,t (P) for all 0 < t < 1, and thus certainly χ j ∈ C 1 (P). Finally, exactly as in (a), since ∇ χ j is locally H s for s ≤ 2, it is in H s (Y ) for s < 1/2; thus χ j ∈ H 1+s (Y ) for s < 1/2.

B Periodic vector potentials

The goal of this appendix is to prove Theorem B.6 below, from which Lemma 4.5 follows.

B.1 Periodic Sobolev spaces

In this section C ∞ (R d ) is the space of smooth complex vector-valued functions; i.e., C ∞ (R

d ) := C ∞ (R d ; C d ). Y = (0, 1) d is the d-dimensional unit cube. The space C ∞ per (Y ) consists of those functions in C ∞ (R d ) that are Y -periodic, i.e. φ ∈ C ∞ (R d
) and φ(y + e j ) = φ(y) for all y ∈ R d , where, for 1 ≤ j ≤ d, e j denotes the j th vector in the canonical basis of R d . For φ ∈ C ∞ per (Y ),

φ 2 div,Y := φ 2 L 2 (Y ) + ∇ y • φ 2 L 2 (Y ) , φ 2 curl,Y := φ 2 L 2 (Y ) + ∇ y × φ 2 L 2 (Y ) , φ 2 H 1 (Y ) := φ 2 L 2 (Y ) + ∇ y φ 2 L 2 (Y ) ,

B.2 Fourier series

(∇ y • φ) α = 2πi α • φ α so that ∇ y • φ = 2πi α∈Z d (α • φ α )e α . (B.1)
Similarly if φ ∈ H per (curl, Y ) then integration by parts (see, e.g., [START_REF] Monk | Finite element methods for Maxwell's equations[END_REF]Theorem 3.29]) gives

(∇ y × φ) α = 2πi α × φ α so that ∇ y × φ = 2πi
(Y ) = d j=1 φ j 2 H 1+s (Y ) . (B.6) Lemma B.1 Let 0 ≤ s ≤ 1. If φ ∈ H per (div, Y ) with ∇ y • φ = 0 and ∇ y × φ ∈ H s (Y ), then φ ∈ H 1+s (Y ) with |φ| H 1+s (Y ) ≤ |∇ y × φ| H s (Y ) . (B.7) An immediate consequence of Lemma B.1 is that if φ ∈ H per (div, Y ) ∩ H per (curl, Y ) then φ ∈ H 1 per (Y ). (B.8)
Proof of Lemma B.1. The identity

|α| 2 v = (α • v)α + (α × v) × α,
holds for all α ∈ Z d and v ∈ C d . Using this with v = φ α , along with (B.1) and (B.2), we have

2πi|α| 2 φ α = 2πi(α • φ α )α + 2πi(α × φ α ) × α = (∇ y • φ) α α + (∇ y × φ) α × α = (∇ y × φ) α × α, since ∇ y • φ = 0 by assumption. Therefore 2π|α||φ α | = |(∇ y × φ) α × α| |α| ≤ |(∇ y × φ) α |.
Recalling the definition (B.3), we have

|φ| 2 H 1+s (Y ) = (2π) 2(1+s) α∈Z d |α| 2(1+s) |φ α | 2 = (2π) 2s α∈Z d |α| 2s (2π|α||φ α |) 2 ≤ (2π) 2s α∈Z d |α| 2s |(∇ y × φ) α | 2 = |∇ y × φ| 2 H s (Y ) .

B.4 Univariate vector potentials

Lemma B.2 For all φ ∈ H (div, Y ) with ∇ y • φ = 0, there exists a unique q ∈ H 1 (Y ) such that ∇ y × q = φ and ∇ y • q = 0. (B.9)

In addition,

q H 1 (Y ) ≤ 1 + 1 π φ L 2 (Y ) . (B.10)
The analogue of Lemma B.2 in bounded domains appears in, e.g., [START_REF] Girault | Finite element methods for Navier-Stokes equations: theory and algorithms[END_REF]Theorem 3.4]. Proof of Lemma B.2. Under the assumption that the solution q ∈ H 1 (Y ) exists and is unique, the inequality (B.7) with s = 0 shows that

|q| H 1 (Y ) ≤ ∇ y × q L 2 (Y ) = φ L 2 (Y ) .
Since the mean value of q vanishes and Y is convex, the Poincaré inequality

q L 2 (Y ) ≤ 1 π |q| H 1 (Y ) ,
holds, and (B.10) immediately follows. We now show that if the solution to (B.10) exists, it is necessarily unique. It is sufficient to prove that if φ = 0, then q = 0. If φ = 0 then ∇ y × q = 0, and |q| H 1 (Y ) = 0 by (B.7); therefore q is constant. But since q ∈ H 1 (Y ), its mean value vanishes, showing that q = 0. It remains to prove existence; given φ ∈ H (div, Y ), define the vector q α by q 0 := 0,

q α := - 1 2πi α × φ α |α| 2 , α ∈ Z d \ {0}. (B.11) Since α∈Z d |q α | 2 ≤ (2π) -1 α |φ α | 2 < ∞, the results recapped in §B.2 imply that q := α∈Z d
q α e α ∈ L 2 (Y ).

Since (q, 1) Y = q 0 = 0, q ∈ L 2 (Y ). It remains to show that q satisfies (B.9), since then the fact that q ∈ H 1 (Y ) will then follow from φ ∈ L 2 (Y ) and the relation (B.8). First, since α • (α × v) = 0 for all v ∈ C d , α • q α = 0 for all α ∈ Z d ; therefore (B.1) implies that ∇ y • q = 0. Next, the fact that ∇ y • φ = 0 implies by (B.1) that α • φ α = 0 for all α ∈ Z d \ {0}. Therefore, (i) If q ∈ H 1 (Y ) is the unique solution to (B.9), then q does not depend on the y d variable.

|α| 2 φ α = (α • φ α )α + (α × φ α ) × α = (α × φ α ) × α = 2πi|α| 2 (α × q α ), and therefore 2πi(α × q α ) = φ α (B.12) for all α ∈ Z d \ {0}. For α = 0, 2πi(α × q α ) = 0 = φ α , (
(ii) If, in addition, φ ∈ H s (Y ) for some s > 0, then q ∈ L ∞ (Y ) and there exists C emb,s > 0 such that

q L ∞ (Y ) ≤ C emb,s 1 + 1 π φ H s (Y ) . (B.15)
Proof. (i) We consider the case d = 3; the case d = 2 is very similar (and easier). For y ∈ R 3 , we write y = (y , y 3 ) for y ∈ Y = (0, 1) 2 . Similarly, for ψ ∈ R 3 we write ψ = (ψ , ψ 3 ). We introduce the notation that, for ψ (y ) a two-dimensional vector independent of y 3 , div y ψ := ∂ y1 ψ 1 +∂ y2 ψ 2 and curl y ψ := ∂ y1 ψ 2 -∂ y2 ψ 1 . Furthermore, if ψ 3 (y ) is a scalar field that does not depend on y 3 , then curl y ψ 3 := (∂ y2 ψ 3 , -∂ y1 ψ 3 ) (i.e. curl y is the rotation of the gradient of ψ). These definitions imply that, if ψ = (ψ , ψ 3 ) is a vector field that does not depend on y 3 , then

curl y ψ = (curl y ψ 3 , curl y ψ ). (B.16)
Now, since φ is independent of y 3 and ∇ y • φ = 0, div y φ = 0. Observe that, with the notation φ = (φ , φ 3 ), both φ and φ 3 have mean value zero. Using this with (B.10), we find that

(p 3 ) α = |α| -2 (α 2 , -α 1 ) • (φ ) α , α = 0, 0, α = 0, and (p ) α = |α| -2 (α 2 , -α 1 )(φ d ) α , α = 0, 0, α = 0, (
q 2 H 1+s (Y ) = q 2 H 1 (Y ) + |q| 2 H 1+s (Y ) ≤ 1 + 1 π 2 φ 2 L 2 (Y ) + |φ| 2 H s (Y ) ≤ 1 + 1 π 2 φ 2 H s (Y ) , (B.18)
Our goal is now to use Sobolev embedding to bound q L ∞ (Y ) by q H 1+s (Y ) . Recall that v L ∞ v H d/2+s for 0 < s < 1 (see, e.g., [START_REF] Mclean | Strongly elliptic systems and boundary integral equations[END_REF]Theorem 3.26]), indicating that for d = 3 we require the H 3/2+s norm of q. However, since q and its components are only functions of y ∈ Y ⊂ R d-1 , and d -1 ≤ 2, we only require the H 1+s norm of q. Given 0 < s < 1, there exists a constant C emb,s such that

ψ L ∞ (Y ) ≤ C emb,s ψ H 1+s (Y ) for all ψ ∈ H 1+s (Y ). (B.19)
Using (in this order) the definition of the L ∞ (Y ) norm (B.14), the bound (B. [START_REF] Chandler-Wilde | Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering[END_REF]), the fact that (by Part (i)) q ∈ H 1+s (Y ) and does not depend on the y d variable, and the property (B.6), we have

q 2 L ∞ (Y ) ≤ d j=1 q j 2 L ∞ (Y ) ≤ C 2 emb,s d j=1 q j 2 H 1+s (Y ) = C 2 emb,s q 2 H 1+s (Y ) . (B.

B.5 Bivariate vector potentials

We now consider functions φ :

D in × Y → C such that φ(x, y) = M(y)v(x), (B.21a)
where M is a Y -periodic matrix-valued function with zero mean value that does not depend on y d and is such that

M j ∈ H s (Y ) ∩ L ∞ (Y ) (B.21b)
for some fixed s > 0, and v ∈ H 2 (D in ). We further assume that

∇ y • φ(x, •) = 0
in the sense of distribution for a.e. x ∈ D in .

Lemma B.4 Assume that φ satisfies (B.21) and additionally that v ∈ C ∞ (D in ). Then, for all x ∈ D in , there exists a unique q(x,

•) ∈ H 1 (Y ) such that ∇ y × q(x, •) = φ(x, •) and ∇ y • q(x, •) = 0 (B.22) for all y ∈ Y . Furthermore, q ∈ C ∞ (D in , H 1 (Y )) with (∂ β x q)(x, •) H 1 (Y ) ≤ 1 + 1 π M(•)∂ β x v(x, •) L 2 (Y ) (B.23) and q(x, •) ∈ C ∞ (D in , L ∞ (Y )) with (∂ β x q)(x, •) L ∞ (Y ) ≤ C emb,s 1 + 1 π M H s (Y ) |(∂ β v)(x)| (B.24)
for all β ∈ Z d + and x ∈ D in . In particular,

(∂ β x q) ε L 2 (Din) ≤ C emb,s 1 + 1 π M H s (Y ) ∂ β v L 2 (Din) , (B.25)
and

(∂ β x q) ε L 2 (Γ) ≤ C emb,s 1 + 1 π M H s (Y ) ∂ β v L 2 (Γ) , (B.26) for all β ∈ Z d + .
Proof. The existence and uniqueness of the solution q(x, •) ∈ H 1 (Y ) for all x ∈ D in is a direct consequence of Theorem B.3. Now, set δ > 0 and define D δ in ⊂ D in as the subset of D in of distance ≥ δ from Γ. With x ∈ D δ in and x ∈ D in , ∇ y × (q(x, •) -q(x , •)) = φ(x, •) -φ(x , •) ∇ y • q(x, •) -q(x , •) = 0, so that, by (B.10), q(x, •) -q(x ,

∇ × q ε • ν H -1 (Γ) ≤ q ε × ν L 2 (Γ) ≤ C emb,s 1 + 1 π M H s (Y ) v L 2 (Γ) , (B.29)
and

∇ × q ε • ν L 2 (Γ) ≤ 1 ε M L ∞ (Y ) v L 2 (Γ) + C emb,s 1 + 1 π M H s (Y ) |v| H 1 (Γ) . (B.30)
In addition, if kR 0 1,

∇ × q ε • ν H -1/2 (Γ) (1 + (kε) -1/2 ) k v L 2 (Din) + |v| H 1 (Din) (B. 31
)
(where the omitted constants depend on M H s (Y ) and M L ∞ (Y ) ).

Proof. Once we have shown that q exists, is unique, and satisfies the bounds (B.25) and (B.26) for |β| ≤ 1 the bound (B.28) is a direct consequence of (B.25), and the bound on q ε × ν L 2 (Γ) in (B.29) is a direct consequence of (B.26). We now prove the bound

∇ × q ε • ν H -1 (Γ) ≤ q ε × ν L 2 (Γ) (B.32)
to complete the proof of (B.29). Consider a test function φ ∈ C ∞ (D in ). Observe that ∇ × q ε • ν, φ Γ = ∇ × q ε , ∇φ Din = -q ε × ν, ∇φ Γ , with the first equality holding by the divergence theorem and the fact that ∇ • (∇ × q ε ) = 0, and the second equality holding by the Green-type identity in, e.g., [START_REF] Monk | Finite element methods for Maxwell's equations[END_REF]Theorem 3.29] and the fact that ∇ × (∇φ) = 0. The bound (B.32) then follows since γ(C ∞ (D in )) is dense in H 1 (Γ) (see, e.g., [START_REF] Chandler-Wilde | Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering[END_REF]Page 276]).

To prove (B.30), we observe that, by (4.1),

∇ × q ε = (∇ x × q) ε + 1 ε (∇ y × q) ε = (∇ x × q) ε + 1 ε M ε v.
Since ( M) j ∈ L ∞ (Y ) for 1 ≤ j, ≤ d, (B.30) follows from this last equality by taking norms and using (B.26).

To prove (B.31), we let T : L 2 (Γ) → H -1 (Γ) be the operator T : v → ∇ × q ε • ν. By interpolation (see, e.g., [53, Appendix B]) using the bounds (B.29) and (B.30) (similar to in the proof of Lemma 4.3), we have

∇ × q ε • ν H -1/2 (Γ) ε -1/2 v L 2 (Γ) + |v| H 1/2 (Γ) for all v ∈ H 1 (D in ).
By the multiplicative trace inequality (2.23),

ε -1 v 2 L 2 (Γ) 1 Rε v 2 L 2 (Din) + ε -1 v L 2 (Din) |v| H 1 (Din) 1 Rε v 2 L 2 (Din) + k ε v 2 L 2 (Din) + 1 kε |v| 2
H 1 (Din)

1 kε 1 + 1 kR k 2 v 2 L 2 (Din) + 1 kε |v| 2 H 1 (Din) ,
and then (B.31) follows from the assumption that kR 0 1. It therefore remains to show that q exists, is unique, and satisfies the bounds (B.25) and (B.26) for |β| ≤ 1. Given v ∈ H 2 (D in ) there exist v n ∈ C ∞ (D in ) such that v n → v in H 2 (D in ) as n → ∞. Let q n ∈ C ∞ (D in , H 1 (Y )) be the solution of ∇ y × q n (x, y) = M(y)v n (x), ∇ y • q n (x, y) = 0, which exists by Lemma B.4. By linearity, ∇ y × q n -q m (x, y) = M(y) v n -v m (x), ∇ y • q n -q m (x, y) = 0.

The definition of the Bochner norm and the bound (B.23) imply that

q n -q m H 2 (Din,H 1 (Y )) ≤ 1 + 1 π M L ∞ (Y ) v n -v m H 2 (Din)
for all n, m. Therefore q n is Cauchy sequence in H 2 (D in , H 1 (Y )) and converges to some q ∈ H 2 (D in , H 1 (Y )). By continuity of weak derivative operators, ∇ y × q = φ and ∇ y • q = 0 for a. e. x ∈ D in . Furthermore, by (B.24), Theorem B.6 Let Γ be C 1,1 . Assume that u 0 ∈ H 2 (D in ), M ∈ H (Y ) ∩ L ∞ (Y ), for some s > 0, and that M does not depend on y d . Assume further that ∇ y • ( Mw) = 0 for all w ∈ C d independent of y. Then there exists a unique vector potential q ∈ H 1 (D in , H 1 (Y )) such that ∇ y × q(x, •) = M(y)∇u 0 (x) and ∇ y • q(x, •) = 0 for a.e. x ∈ D in . Futhermore, C Bounding C sol (A ε , n ε , k, R, R 0 ) for kR sufficiently small

Din (∂ β x q n )(x, •) 2 L ∞ (Y ) dx ≤ (C emb,s ) 2 1 + 1 π 2 M 2 H s (Y ) Din |(∂ β v n )(x)| 2 dx and Din ∂ β x (q n -q m )(x, •) 2 L ∞ (Y ) dx ≤ (C emb,s ) 2 1 + 1 π 2 M 2
(∇ x × q) ε L 2 (
As discussed in §1.3.2, in [13, Bottom of Page 2539 and top of Page 2540] C sol (A ε , n ε , k, R, R 0 ) is assumed to be independent of ε and to only depend on n min , n max , A min , and A max . We now show that this is true if kR is sufficiently small. By [START_REF] Melenk | Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions[END_REF]Lemma 3.3], given k 0 , R 0 > 0, there exist

C DtN = C DtN (k 0 R 0 ) such that -DtN k φ, φ Γ R ≥ C DtN R -1 φ 2 L 2 (Γ R )
for all φ ∈ H 1/2 (Γ R ), k ≥ k 0 , and R ≥ R

(v, v) ≥ A min ∇v 2 L 2 (B R ) -k 2 n max v 2 L 2 (B R ) + R -1 C DtN v 2 L 2 (Γ R ) , ≥ min{A min , C DtN } ∇v 2 L 2 (B R ) + 1 R v 2 L 2 (Γ R ) -k 2 n max v 2 L 2 (B R ) , ≥ min{A min , C DtN } 2 ∇v 2 L 2 (B R ) + 1 R v 2 L 2 (Γ R ) + n max -1 + min{A min , C DtN } 2n max (kR) 2 C PF k 2 v 2 L 2 (B R ) ,
and the result (C.4) follows.

1

 1 Definition of the Helmholtz transmission problemFor R > 0, let B R := {x ∈ R d | |x| < R} and let Γ R := ∂B R = {x ∈ R d | |x| = R}. Let DtN k : H 1/2 (Γ R ) → H -1/2 (Γ R )be the Dirichlet-to-Neumann map for the equation ∆u + k 2 u = 0 posed in the exterior of B R with the Sommerfeld radiation condition ∂u ∂r (x) -iku(x) = o 1 r (d-1)/2 (1.1) as r := |x| → ∞, uniformly in x := x/r. The definition of DtN k in terms of Hankel functions and polar coordinates (when d = 2)/spherical polar coordinates (when d = 3) is given in, e.g., [21, Equations 3.5 and 3.6] [66, §2.6.3], [54, Equations 3.7 and 3.10]. Let Sym denote the set of d × d real, symmetric matrices and let SPD denote the set of d × d real, symmetric, positive-definite matrices. Given M 1 , M 2 ∈ SPD we write M 1 M 2 to indicate inequality in the sense of quadratic forms, namely M 1 v • v ≤ M 2 v • v for all v ∈ C d . For a nonnegative scalar m and M ∈ SPD we write m M if mI M, and M m if M mI, where I is the identity matrix.

Lemma 1 . 2 (

 12 Definition 1.1 includes transmission by a Lipschitz penetrable obstacle) Let D in , D out , and Γ be as above, and assume further that Γ is Lipschitz. Let f ∈ L 2 (R d ) with suppf ⊂ B R , and let g N ∈ L 2 (Γ). If A and n in Definition 1.1 are such that supp(I -A) = supp(1 -n) = D in and F (w) = B R f w -Γ g N w, then the solution u of the Helmholtz transmission problem of Definition 1.1 is the restriction to B R of v ∈ H 1 loc (R d ) satisfying the Sommerfeld radiation condition (1.1) (with u replaced by v) and

Figure 1 . 1 : 5 1. 2

 1152 Figure 1.1: An example of a domain D in satisfying the first bullet point in Example 1.5

Figure 1 . 2 :

 12 Figure 1.2: Sketches of admissible periodic patterns, with the black lines representing discontinuities in the coefficients. Left-panel: if the coefficients in this 2-d domain are constant in the vertical direction, then they satisfy (a) in Condition 1.10. Right-panel: horizontal cross section of a 3-d domain satisfying (b) in Condition 1.10.

1. 3 . 2

 32 Discussion of the novelty and context of Theorem 1.14.

1. 3 . 3

 33 Discussion of the k-and ε-dependence of the bounds in Theorem 1.14 and Corollary 1.15

Remark 2 . 3 (

 23 The origin of the condition (A) d = 0 for = 1, . . . , d -1 in Condition 1.4) Our argument below requires that the term 2 A∇v, ∂Z ∇v be bounded below by a multiple of |∂ d v| 2 when Z = e d x d . Under this choice of Z,

Lemma 2 . 4 (

 24 The Morawetz-type identity (2.2) integrated over a ball) Recall the notation that B R := {x : |x| < R} and Γ R := ∂B R . On Γ R , let ∂ r denote the normal derivative and let ∇ S denote the surface gradient

Lemma 2 . 5 (

 25 Inequality on Γ R for outgoing Helmholtz solutions) Let u be a solution of the homogeneous Helmholtz equation in R d \ B R0 , for some R 0 > 0, satisfying the Sommerfeld radiation condition (1.1).

5 .

 5 See [21, Lemma 2.1] or [83, Lemma 2.4] for the proof when d = 2, 3 and [59, Lemma 4.4] for general d ≥ 2.

Lemma 2 . 6 (

 26 Two key properties of DtN k )

. 22 )

 22 Then the solution of the variational problem (1.5) with F (v) = B R f v and with coefficients A and n exists, is unique, and satisfies the bound (2.22).References for the proof. The proof is contained in[38, Pages 2901 and 2902] (with A δ , n δ replaced by A , n ). Lemma 2.10 holds with the f L 2 (B R ) in the bound (2.22) replaced by

3. 1 . 1

 11 Statement of the result Condition 3.1 (A particular class of smooth coefficients)

Lemma 3 . 2 (

 32 Bound for coefficients satisfying Condition 3.1) There exist C wave , C 1 > 0 such that the following holds. Given R > R 0 > 0, A and n satisfying Condition 3.1, and f ∈ L 2 (B R ), if kR 0 ≥ C wave , then the variational problem (1.5) with F (v) = B R f v and coefficients A and n has a unique solution that satisfies the bound (1.14).3.1.2 Overview of the ideas behind the proof of Lemma 3.2The basic idea is to use the integrated Morawetz identity (2.13) in B R with v = u,• Z given by the vector field in Definition 3.3 below; the key point is that Z transitions from being equal to x d e d in B R0 (and hence in a neighbourhood of supp(I -A) and supp(1 -n) to being equal to x in R d \ B R1 for some 0 < R 0 < R 1 < R, with this transition controlled by a radial function χ transitioning from 0 (in B R0 ) to 1 (in R d \ B R1 );

Lemma 3 . 4

 34 Let u be the solution of the transmission problem of Definition 1.1, with F (v) = B R f v and coefficients A and n satisfying Condition 3.1. Let α be defined by

(3. 8 )

 8 Next, when A = I, χ = 0 so Z = x d e d ; therefore ((∂Z)∇u) i = δ id ∂ d u, so that (2.10) above holds. The assumption that (A d ) = 0 for = 1, . . . , d -1 implies that, when A = I, A∇u, ∂Z∇u = (A) dd |∂ d u| 2 . The fact that A min ≥ 1 (noted after Condition 3.1) implies that (A) dd ≥ 1, and so when A = I, A∇u, ∂Z ∇u ≥ |∂ d u| 2 . (3.9) Since χ = 0 when A = I, we can combine (3.8) and (3.9) to obtain that A∇v, ∂Z ∇v ≥ |∂ d v| 2 1 -χ(r) + |∇v| 2 χ(r) + r|∂ r v| 2 -x d ∂ d v∂ r v χ (r). (3.10) Combining (3.7) and (3.10) we have

  1 and A satisfies Condition 3.1. The definition of n follows in an essentially-identical way.

  it is sufficient to show that ( C) d = 0 for = 1, . . . , d -1. By the definition of C (4.5), ( C) d := ∂ χ ∂y d = 0 since χ is independent of y d , and ( C) d := ∂ χ d ∂y ;

. 9 ) 4 . 3 (

 943 Lemma Key properties of the first-order corrector)∇ y u 1 = -C∇u 0 for a.e. y ∈ Y ,(4.10)

  2) by periodicity and multiply by a cutoff function, and obtain a problem posed in a bounded domain in Ω ⊂ R d with Y ⊂⊂ Ω. We can then just use the standard interior regularity shift result; see, e.g.,[START_REF] Mclean | Strongly elliptic systems and boundary integral equations[END_REF] Theorem 4.20].We need to check that both the coefficient A of the PDE (4.3) and the right-hand side -∂( A) j /∂y have sufficient regularity under either (a) or (b) in Condition 1.10.If (a) holds, then both A and χ j only depend on y 1 , so that (4.3) becomes -

For α ∈ Z d , let |α| 2 2 j

 22 and e α (y) := e 2πiα•y for all y ∈ R d . For φ ∈ L 2 per (Y ), we define φ α ∈ Z d by (φ α ) j := φ j , e α L 2 (Y ) for all α ∈ Z d . Then, by the L 2 theory of Fourier series (see, e.g., [77, §4.26, Page 91] for the results in one dimension, and, e.g., [37, Prop. 3.1.15 and 3.1.16, Page 170] for the results in arbitrary dimensions) α∈Z d |φ α | 2 < ∞ and φ = α∈Z d φ α e α , where the sum converges in L 2 (Y ). Conversely, if { ψα } α∈Z d is a set of coefficients satisfying α | ψα | 2 < ∞ then there exists a unique ψ ∈ L 2 (Y ) such that ψ α = ψα and ψ := α∈Z d ψ α e α , where the sum converges in L 2 (Y ). If φ ∈ H per (div, Y ) then, by the divergence theorem (see, e.g., [60, Part 2 of Theorem 3.24])

B. 13 )

 13 since φ ∈ H (div, Y ). Combining (B.12), (B.13), and (B.2), we find that ∇ y × q = φ, which concludes the proof.The next lemma involves the L ∞ (Y ) and L ∞ (Y ) norms defined byq L ∞(Y ) := ess sup y∈Y |q(y)| 2 and q L ∞ (Y ) := ess sup y∈Y |q(y)|. (B.14) Theorem B.3 Let φ ∈ H (div, Y ) with ∇ y • φ = 0, and assume that φ does not depend on the y d variable.

20 )

 20 The result (B.15) then follows from combining (B.20) and (B.[START_REF] Keçebaş | Enhancing the spectral reflectance of refractory metals by mutlilayer optical thin-film coatins[END_REF]

2 L 2 M 2 H

 222 •) H 1 (Y ) ≤ 1 + 1 π φ(x, •) -φ(x , •) L 2 (Y ) )∇ x v(x, •) L 2 (Y ) ,and, by (B.[START_REF] Capdeboscq | On the scattered field generated by a ball inhomogeneity of constant index in dimension three. In Multi-scale and high-contrast PDE: from modelling, to mathematical analysis, to inversion[END_REF],q(x, •) -q(x , •) L ∞ (Y ) ≤ C emb,s 1 + 1 π φ(x, •) -φ(x , •) H s (Y ) , ≤ C emb,s 1 + 1 π |x -x | sup x∈Din M(•)∇ x v(x, •) H s (Y ) .Since the above inequalities hold for any x ∈ D in , we deduce that supx∈D δ in ∇ x q(x, •) H 1 (Y ) ≤ 1 + 1 π sup x∈Din M(•)∇ x v(x, •) L 2 (Y ) ,and supx∈D δ in ∇ x q(x, •) L ∞ (Y ) ≤ C emb,s 1 + 1 π sup x∈Din M(•)∇ x v(x, •) H s (Y ) .But now, since the upper bounds are independent of δ, we obtain (B.23) and (B.24) for |β| = 1 by letting δ → 0. We have there proved that q∈ C 1 (D in , H 1 (Y )) ∩ C 1 (D in , L ∞ (Y )); we can therefore differentiate (B.22) with respect to x j , to obtain that∂ xj q ∈ C 0 (D in , H 1 (Y )) ∩ C 0 (D in , L ∞ (Y )).By linearity, we can then repeating the above argument with q := ∂ xj q and φ = ∂ xj φ. We then obtain (B.23) and (B.24) by induction on |β|.By integrating over D in and using (B.24), we have(∂ β x q) ε 2 L 2 (Din) = Din x q) (x, •) ∞ (Y ) dx, (B.27) ≤ (C emb,s ) 2 1 + 1 π s (Y ) Din |(∂ β v)(x)|2 dx, and (B.25) follows. The bound (B.26) follows similarly, but integrating over Γ. Theorem B.5 Assume that φ : D in × Y → C satisfies the three conditions in (B.21) with v ∈ H 2 (D in ). Then there exists a unique vector potential q ∈ H 2 (D in , H 1 (Y )) such that ∇ y × q(x, •) = φ(x, •) and ∇ y • q(x, •) = 0 for a.e. x ∈ D in . Futhermore, (∇ x × q) ε L 2 (Din) ≤ C emb,s 1 + 1 π M H s (Y ) |v| H 1 (Din) , (B.28)

q 2 H 2 (β x q 2 L 2 (

 2222 Din,H 1 (Y )) := |β|≤2 ∂ Din,H 1 (Y ))

Hπ 2 M 2 H

 22 s (Y ) Din |∂ β (v n -v m )(x)| 2 dx for |β| ≤ 2 and n, m ∈ N. Therefore q n is a Cauchy sequence in H 2 (D in , L ∞ (Y )) and Din (∂ β x q)(x, •) 2 L ∞ (Y ) dx ≤ (C emb,s ) 2 1 + 1 s (Y ) Din |(∂ β v)(x)| 2 dx (B.33) for |β| ≤ 2. The bounds (B.25) and (B.26) for |β| ≤ 2 therefore follow from (B.33) using similar reasoning to in (B.27).

R -2 v 2 L 2 ( 2 L 2 (Γ R ) + ∇v 2 L 2 ( 2 L 2 (B R ) + n max 2 k 2 v 2 L 2 (

 2222222222 0 . (C.1)By, e.g.,[START_REF] Toselli | Domain Decomposition Methods: Algorithms and Theory[END_REF] Corollary A.15], there existsC PF = C PF (Ω -) ('PF' standing for 'Poincaré-Friedrichs') such that B R ) ≤ C PF R -1 v B R ) for all v ∈ H 1 (B R ). (C.2) Lemma C.1 (Coercivity of b(•, •) for kR sufficiently small) If kR ≤ 3n max C PF min{A min , C DtN } , (C.3) then b(v, v) ≥ min{A min , C DtN } 2 ∇v B R ) for all v ∈ H 1 (B R ). (C.4)The boundC sol (A ε , n ε , k, R, R 0 ) ≤ min min{A min , C DtN } , n max-1 under the assumption (C.3) then immediately follows from the Lax-Milgram lemma and the definition of C sol (1.11) Proof of Lemma C.1. Using the definition of b(•, •) (1.6) and the inequalities (C.1) and (C.2), we have b

  Lemma 2.1], and (ii) is proved in [54, Lemma 3.3]. Lemma 2.7 (Continuity of the sesquilinear form) The sesquilinear form a(•, •) of the transmission problem defined by (1.6) is continuous on H

.19) References for the proof. (i) is proved in [66, Theorem 2.6.4, p.97], [54, Lemma 3.3], or [21,

  Gårding inequality. A bound on the solution in terms of the data such as (2.20), under the assumption of existence, shows that the solution of the boundary value problem (if it exists) is unique. Since b(•, •) is continuous and satisfies a Gårding inequality, the result then follows from Fredholm theory; see, e.g., [53, Theorems 2.27 and 2.34], [29, §6.2.3].Lemma 2.9 (Well-posedness and bound for f ∈ L 2 implies well-posedness and bound for F ∈ (H 1 ) .) Assume that A and n are such that, given f ∈ L 2 (B R ), the solution of the transmission problem of Definition 1.1 with F (v) = B R f v exists, is unique, and satisfies the bound (2.20) for some C(A, n, k, R) > 0 and k 0

R ) ; (2.21) i.e. b(•, •) satisfies a

  of Lemma 3.2. The inequality (3.2) in Lemma 3.4 is identical to the inequality that is the result of [22, Lemma 3.1]. The results [22, Lemmas 3.2, 3.3, 3.4, 3.5] (discussed in §3.1.2) then go through ad verbatim (with the value of q fixed in [22, Lemma 3.5]); in seeing this, observe that, since A min ≥ 1 and n max ≤ 1, the factors in front of the integrals of |∇u| 2 and |u| 2 in (3.3) are both one, as they are in the analogous inequality [22, Equation 2.11]. Lemma 3.2 is then a consequence of [22, Lemma 3.5] (see [22, Proof of Theorem 1.10 from Lemma 3.5]).

Remark 3.5 (Explicit expressions for C wave , and k 0 ) C 1 is given in terms of χ in [22, Equation 3.22] (see the discussion in [22, Proof of Theorem 1.10 from Lemma 3.5]).

C wave is given in terms of the function χ in Definition 3.3 in

[START_REF] Chandler-Wilde | High-frequency bounds for the helmholtz equation under parabolic trapping and applications in numerical analysis[END_REF] Equation 3.20]

. Indeed, to see that

[START_REF] Chandler-Wilde | High-frequency bounds for the helmholtz equation under parabolic trapping and applications in numerical analysis[END_REF] Equation

  .6) the homogenized matrix coefficient A H is the mean value of B over the periodic cell Y , i.e., H ∈ SPD by, e.g., [7, Remarks 2.6 and 2.7],[27, §6.3]. We now claim that B ∈ H s (Y )∩C 0 (P)∩L ∞ (Y ) for some 0 < s < 1/2; indeed, this follows from the facts that C ∈ H s (Y ), A is (at least) piecewise C 0,1 (by Condition 1.10), and a piecewise H s function for s < 1/2 is in H s (Y ) (by the definition of the Slobodeckij seminorm).4.1.2 Basic properties of the homogenized problemWe now prove Lemma 1.13, i.e., well-posedness of the homogenized problem.Proof of Lemma 1.13. We first show that if n and A are admissible periodic patterns (in the sense of Definition 1.10), then n H and A H satisfy Condition 1.4; once this is established, the bound (1.18) follows from Theorem 1.6.Since both n H and A H are constant in D in , they are clearly independent of x d ; the monotonicity conditions (1.8) and (1.9) therefore hold if n H ≤ 1 and A H I. Recall from the discussion below Condition 1.4 that n max ≤ 1 and A min ≥ 1. Since n H is obtained from n through simple averaging (4.2), n H ≤ 1. While the averaging process is more complicated for A H , [27, Theorem 13.7] implies that A H I.Therefore, to show that n H and A H satisfy Condition 1.4, it only remains to show that A H d, = 0 for = 1, . . . , d -1. By (4.7),

	(A H ) j =	( B) j ,	1 ≤ j, ≤ d;	(4.7)
	Y
	note that A

  since A H I and n H ≤ 1, A H and n H satisfy [38, Condition 2.6] with µ 1 = A min = 1 and µ 2 = n min (see the example in [38, Condition 2.10]) and so, by [38, Theorem 2.7], if

  A H is constant in both D in and D out , Lemma 4.1 holds if Γ is C 1,1 by, e.g.,[START_REF] Mclean | Strongly elliptic systems and boundary integral equations[END_REF] Theorem 4.18]. (If Γ has edges or corners, similar shift properties hold but with the angles of edges/corners constrained by the values of A H , see, e.g.,[START_REF] Costabel | Singularities of Maxwell interface problems[END_REF].)We abbreviate C sol (A H , n H , k, R, R 0 ) by C sol,H in the rest of the paper. Lemma 4.2 (Bounds on u 0 ) Assume that n and A satisfy Condition 1.10, and Γ is C 1,1 . Then, for all f ∈ L 2 (B R ), the solution u 0 of (1.5) with coefficients n H and A H and

  and H per (div, Y ), H per (curl, Y ) and H 1 per (Y ) are the closures of C ∞ per (Y ) in L 2 (Y ) with respect to the • div,Y , • curl,Y and • H 1 (Y ) norms, respectively. The spaces C ∞ (Y ), H (div, Y ), H (curl, Y ) and H 1 (Y ) are the subspaces of C ∞ per (Y ), H per (div, Y ), H per (curl, Y ) and H 1 per (Y ), respectively, consisting of functions with zero mean. Since the map L 2 (Y ) → C d defined by φ → 1 |Y | Y φ is bounded, C ∞ (Y ) is dense in H (div, Y ), H (curl, Y ), and H 1 (Y ).

  Sobolev norms defined by Fourier coefficientsFor s ∈ R + , we define the Sobolev semi-norms on vector-and scalar-valued functions by

	B.3 |φ| 2 H s (Y ) := (2π) s	|α| 2s |φ α | 2 and |φ| 2 H s (Y ) := (2π) s	|α| 2s |φ α | 2 ;	(B.3)
			α∈Z d			α∈Z d
	where | • | 2 denotes the Euclidean norm on C d (and the modulus on C); observe that then
						d
					|φ| 2 H s (Y ) =	|φ j | 2 H s (Y )	(B.4)
						j=1
	Let
		m				t
	φ	2 H m (Y ) :=	|φ| 2 H s (Y ) , m ∈ Z + , and φ	2 H t (Y ) :=	|φ| 2 H s (Y ) + |φ| 2 H t (Y ) , t ∈ R + \ Z + ,
		s=0				s=0
						(B.5)
	and similarly for scalar-valued functions. We use below the particular consequence of (B.4) and
	(B.5) that
				φ	2 H 1+s
						(α × φ α )e α .	(B.2)
						α∈Z d

  Using the lower-dimensional analogues of the Fourier-series argument used to prove Lemma B.2, one can show that there exists a unique p d ∈ H (curl, Y ) and a unique p ∈ H (curl, Y ) such that curl y p 3 = φ , curl y p = φ 3 , div y p = 0. (B.17) Indeed, the Fourier coefficients of p 3 and p are given by

  Din) |u| H 2 (Din) , q ε × ν L 2 (Γ) |u 0 | H 1 (Γ) , (B.34)and, if kR 0 1,∇ × q ε • ν H -1/2 (Γ) (1 + (kε) -1/2 ) k u 0 H 1 (Din) + u 0 H 2 (Din) (B.35) (where the omitted constants depend on M H s (Y ) and M L ∞ (Y ) ). Proof. By applying Theorem B.5 with v = ∇u 0 , we see that q exists and the bounds in (B.34), (B.35) hold when u 0 ∈ H 3 (D in ). By the density of H 3 (D in ) in H 2 (D in ), and the fact that righthand sides of the bounds (B.34) are controlled by u 0 H 2 (Din) , q and the bounds in (B.34), (B.35) hold for u 0 ∈ H 2 (D in ).

Strictly speaking,[START_REF] Ralston | Trapped rays in spherically symmetric media and poles of the scattering matrix[END_REF] proves the existence of a sequence of resonances exponentially close to the real axis, but then the "resonances to quasimodes" result of[START_REF] Stefanov | Resonances near the real axis imply existence of quasimodes[END_REF] implies super-algebraic growth through a sequence of real k j s.

The paper[START_REF] Cakoni | On the homogenization of a scalar scattering problem for highly oscillating anisotropic media[END_REF] cites[START_REF] Cakoni | Qualitative Approach to Inverse Scattering Theory[END_REF] Theorem 5.26] as a reference for C sol being independent of ε and depending on Aε and nε only through Amax, A min , nmax, and n min (see[13, Bottom of Page 2539 and top of Page 2540]). Actually,[START_REF] Cakoni | Qualitative Approach to Inverse Scattering Theory[END_REF] Theorem 5.26] records how Fredholm theory proves an a priori bound on the solution of the transmission problem; however this theory (relying on abstract functional analysis arguments) is unable to give the constant in this bound. The arguments in[START_REF] Cakoni | On the homogenization of a scalar scattering problem for highly oscillating anisotropic media[END_REF] hold, however, with the results of Appendix C replacing reference to[START_REF] Cakoni | Qualitative Approach to Inverse Scattering Theory[END_REF] Theorem 5.26], and under the explicit assumption that kR is sufficiently small.

Acknowledgements. The authors thank Kirill Cherednichenko (University of Bath), Daniel Peterseim (Universität Augsburg), and Barbara Verfürth (KIT) for useful discussions about the homogenisation literature. EAS was supported by EPSRC grant EP/R005591/1.

Proof of Lemma 4.5 using Theorem B.6.

To apply Theorem B.6, we need to show that (i) ( B -A H ) ∈ H s (Y ) ∩ L ∞ (Y ) for some s > 0, (ii) ( B -A H ) does not depend on y d , and (iii) ∇ y • (( B -A H )w) = 0 for all w ∈ C d independent of y.

Regarding (i): this is shown under (4.7). Regarding (ii): A H is constant, and therefore independent of y d . A is independent of y d by Condition 1.10, therefore so are the χ j by Lemma A.2, therefore so is C by (4.5), and therefore so is B by (4.6). Regarding (iii): by the definition of B (4.6), for w ∈ C d independent of y, We now define the boundary corrector θ ε as the unique element of

C layer (kR, kR 0 , n min ) 1 + (kε

By (1.17),

, so that it remains to estimate the dual norm of ψ. By the definition of θ ε (4.23), continuity of b out , and Lemma 2.11, for all

, and the inequality (4.24) follows. The second inequality then follows from using the bounds (4.13) and (4.22).

Proof of Theorem 1.14

The proof relies on the following error esimate involving the boundary corrector θ ε , whose proof is postponed until Section 4.4.

Lemma 4.7 (Error estimate with a boundary corrector)

where the extension operator E is as in Lemma 2.12. Then,

Proof. By the definition of W ε (4.28),

Since u ε and u 0 are the solutions to the oscillatory and homogenized problems, respectively,

Lemma 4.4 bounds the second term on the right-hand side; we therefore focus on the first term.

Since

and thus

By the definition of q (4.20), ((A H -B)∇u 0 ) ε = -(∇ y × q) ε , and thus

combining this with (4.31) yields

We now focus on the term involving (∇ y × q) ε . By (4.1),

ε and thus, by the divergence theorem,

where we have crucially used that ∇ • ∇ × q ε = 0; the result (4.30) then follows by combining this last equality with (4.32).

Lemma 4.10 (The term in the inner product (4.27) involving εu 1,ε )

Proof. By the definitions of W ε (4.28) and u 1,ε (4.25),

and so to prove the result (4.33), we only need to prove that

A Regularity of periodic cell problems

The goal of this section is to show that the functions χ j and µ appearing in the homogenization section have the required regularity, namely that χ j ∈ H 1+s (Y ) ∩ C 1 (P), for some s > 0, and that µ ∈ C 1 (P). Recall that H 1 (Y ) is the subspace of H 1 per (Y ) consisting of functions with zero mean.

Furthermore, if A(y) does not depend on y for some ∈ {1, . . . , d} and

where f 0 , f j ∈ L 2 (Y ) are functions that are independent of y , then u does not depend on y .

Proof. Given f ∈ H 1 (Y ) , by the Poincaré-Wirtinger inequality and the Lax-Milgram lemma applied in the space H 1 (Y ), there exists a unique u ∈ H 1 (Y ) such that

, with φ 0 ∈ C the mean value of φ. Then φ = φ -φ 0 ∈ H 1 (Y ), so that, by (A.3) and the fact that f, φ 0 = 0, A∇u, ∇φ Y = A∇u, ∇ φ Y = f, φ = f, φ ; the result (A.1) follows. Since functions in H 1 (Y ) can be periodically extended, the translation operation τ h, : v(y) → v(y + he ) is well-defined. Then, with the summation convention,

so that u = τ h, u for all h ∈ R; therefore u does not depend on y .