Explicit bounds for the high-frequency time-harmonic Maxwell equations in heterogeneous media

We consider the time-harmonic Maxwell equations posed in $\mathbb{R}^3$. We prove a priori bounds on the solution for $L^\infty$ coefficients $\epsilon$ and $\mu$ satisfying certain monotonicity properties, with these bounds valid for arbitrarily-large frequency, and explicit in the frequency and properties of $\epsilon$ and $\mu$. The class of coefficients covered includes (i) certain $\epsilon$ and $\mu$ for which well-posedness of the time-harmonic Maxwell equations had not previously been proved, and (ii) scattering by a penetrable $C^0$ star-shaped obstacle where $\epsilon$ and $\mu$ are smaller inside the obstacle than outside. In this latter setting, the bounds are uniform across all such obstacles, and the first sharp frequency-explicit bounds for this problem at high-frequency.

where the frequency ω > 0, the sources J, K ∈ L 2 comp (R 3 ), and the coefficients ǫ and µ are 3 × 3 real, symmetric, positive-definite matrices (with the set of these matrices denoted by SPD) such that ǫ = ǫ 0 I and µ = µ 0 I outside a compact set, with ǫ 0 , µ 0 > 0. The fields E(x), H(x) additionally satisfy the Silver-Müller radiation condition The PDEs in (1.1) are understood in a distributional sense, and thus are well defined for ǫ, µ ∈ L ∞ . We are particularly interested in the case when ǫ and µ are discontinuous. Recall that when ǫ and µ have a single jump on a common interface, (1.1) corresponds to transmission by a penetrable obstacle (see the examples in §1.2 below) with the conditions that the tangential jumps across the interface of both E and H are zero (coming from the condition that E, H ∈ H loc (curl; R 3 )).

Statement of the main results
The main results of this paper give bounds on the solution of (1.1)-(1.2) that are explicit in both ω and properties of the coefficients ǫ and µ, and valid for arbitrarily-large ω.
Notation. Given M 1 , M 2 ∈ SPD we write M 1 M 2 to denote inequality in the sense of quadratic forms, namely Let 0 < ǫ min ≤ ǫ max < ∞ and 0 < µ min ≤ µ max < ∞ be such that ǫ min ǫ(x) ǫ max , µ min µ(x) µ max for almost every x ∈ R 3 , (1. 4) and let R > 0 be such that where B R denotes the ball of radius R centred at the origin. Then the solution E, H ∈ H loc (curl; R 3 ) of (1.1)-(1.2) exists, is unique, and satisfies where each instance of 8 on the right-hand side reduces to 4 if either K = 0 or J = 0.
Rotating the 2-d domain in Figure 1a around the vertical axis through the origin gives a 3-d domain Ω − satisfying the conditions in Example 1.3. This example therefore includes domains with inner and outer cusps.
A key feature of Theorem 1.2 for ǫ, µ as in Example 1.3 is that the bound (1.7) is then uniform across all such penetrable obstacles; §1.4.3 below discusses one important application of this feature in the theory of uncertainty quantification for the time-harmonic Maxwell equations. Example 1.4. (Transmission by a penetrable obstacle with self-intersecting surface.) Ω 1 , Ω 2 , and Ω 3 are the 2-d domains in Figure 1b rotated around the vertical axis through the origin, and ǫ| Ωj = ǫ j I for j ∈ {1, 2, 3}, where 0 < ǫ 1 ≤ ǫ 2 ≤ ǫ 3 = ǫ 0 , and similarly for µ. Example 1.5. (Transmission by a penetrable obstacle with infinitely many components accumulating towards a bounded limit surface.) ǫ| B j/(j+1) \B (j−1)/j = ǫ j for j ∈ N and   The ω-dependence of the bound (1.7) is the same as in the sharp bound on the solution to (1.1)-(1.2) when µ = µ 0 I and ǫ = ǫ 0 I. A simple way to see this sharpness is to let E = χE I and H = χH I where χ ∈ C ∞ comp (B R ) and E I and H I are the plane-wave solutions with |A| = |d| = 1 and A · d = 0. Then (E, H) is solution of the Maxwell problem (1.1)-(1.2) with J = ∇χ × H I and K = ∇χ × E I , and the L 2 norms of the solutions and sources are all independent of ω: These plane-wave solutions also show that the dependence on R in the bound (1.7) is sharp and that this bound cannot be improved, in general, by a factor larger than 32π 2 ≈ 316; indeed, fix R > 0 and define χ ∈ C 0,1 (R 3 ) by χ(x) := R sin(π|x|/R)(π|x|) −1 if |x| < R and χ(x) := 0 otherwise. χ is the first Laplace-Dirichlet eigenfunction of the ball (the spherical Bessel function j 0 ( π R |x|)): ∆χ + π 2 R 2 χ = 0 in B R and χ = 0 on ∂B R , so ∇χ . Choosing E, H as the cut-off plane waves above, we obtain a solution of the constant-coefficient problem It is well-known that the behaviour of solutions of the time-harmonic Maxwell equations in the limit ω → ∞ is dictated by the behaviour of the geometric optic rays, and the conditions on ǫ and µ in (1.6) ensure that all the rays starting in a neighbourhood of supp(ǫ−ǫ 0 I)∪supp(µ−µ 0 I) escape from that neighbourhood in a uniform time -see, e.g, [28, §7] -i.e., the problem is nontrapping. When ǫ and µ correspond to transmission through a penetrable obstacle (as in Example 1.3), the conditions (1.10) imply that the wave speed inside the obstacle is larger than the wave speed outside the obstacle, ruling out total internal reflection, and thus ruling out trapped rays.
Obtaining ω-explicit bounds on solutions of the Helmholtz equation under the nontrapping hypothesis is a classic topic. These bounds then imply results about the location of resonances and the local-energy decay of solutions of the corresponding wave equation; see the overview in [23, §4.6]. For certain nontrapping geometries and coefficients, such ω-explicit bounds can be obtained by multiplying the PDE by a carefully-chosen test function and integrating by parts [8,9,57,58,65]; this is the method we use in the Maxwell case -see the discussion in §1.4.5 below. For smooth geometries and coefficients, the propagation-of-singularities results of [47,48] and the parametrix argument of [71] prove the sharp bound on the solution under the general nontrapping hypothesis (see the recent presentation in [23,Theorem 4.43]). For Helmholtz transmission problems, i.e., (1.12) with discontinuous A and n, the propagation-of-singularities results are much more complicated, and ω-explicit bounds on the Helmholtz transmission problem proved using propagation of singularities only exist for smooth obstacles with strictly positive curvature; see [14]. For the Maxwell equations, the propagation-of-singularities results analogous to [47,48] were proved in [73] for the constant-coefficient Maxwell equations in the exterior of a perfectly-conducting obstacle. To the best of our knowledge, there do not yet exist corresponding results for the Maxwell transmission problem.
As stated above, the bounds in the present paper are proved by multiplying the PDE by a carefully-chosen test function and integrating by parts (see the discussion in §1.4.5 below). Perhaps surprisingly, the present paper appears to be the first time this technique has been applied to the time-harmonic Maxwell transmission problem, and Theorem 1.2 therefore contains the first ωexplicit bounds on the solution of this problem.
We highlight that if the monotonicity conditions (1.6) or (1.10) on ǫ and µ are violated then the Maxwell solution operator can grow exponentially through a sequence of ωs; this is proved for the Helmholtz solution operator in [68] (for smooth coefficients) and [13,67], [1,Chapter 5] (for discontinuous coefficients such that the wave speed outside is higher than the wave speed inside).

The novelty of the well-posedness result in Theorem 1.2
The class of coefficients for which existence and uniqueness of the Maxwell solution is proved in Theorem 1.2 contains configurations for which existence and uniqueness of the Maxwell solution had not yet been established. Indeed, the general arguments of [66] prove existence of a solution to (1.1)-(1.2) for J, K ∈ L 2 comp (R 3 ) once uniqueness is established; see [66,Theorem 2.10]. The Bairecategory argument of [5] uses the fact that a UCP is known for the time-harmonic Maxwell system with Lipschitz coefficients [61] to prove uniqueness of the solution of the time-harmonic Maxwell problem posed in R 3 with piecewise-Lipschitz coefficients, provided that the subdomains on which the coefficients are defined satisfy [5, Assumption 1]/[41, (i)-(iii) in statement of Proposition 2.11]. This assumption allows a large class of subdomains (including any bounded finite collection), but does not allow the subdomain boundaries to concentrate "from below" on a bounded surface in R 3 , and thus the subdomains corresponding to Example 1.5 are ruled out; see [5, Figure 1] (in the notation of [41,Proposition 2.13], ∂B 1 (0) ⊂ C and thus R 3 \ C is not connected). Theorem 1.2 is therefore the first time existence and uniqueness of the Maxwell solution with ǫ, µ as in Example 1.5 has been proved.  [35] as the basis of a frequency-explicit analysis of UQ algorithms for the high-frequency Helmholtz transmission problem (following the analysis at fixed frequency in [34]). Theorem 1.2 can therefore form the basis of the Maxwell analogue of [35].
Second, the bounds on the Helmholtz equation in [28] that are explicit in properties of the coefficients were used in [64] to prove the first well-posedness results about the Helmholtz equation with random coefficients (see [64, §1.1]). Inputting the bounds from Theorem 1.2 into the general framework of [64, §2], one can obtain the analogous results for the time-harmonic Maxwell equations.
i.e., the total fields are bounded (uniformly in ω) in terms of the incident fields. The basic ingredient of the proofs of Theorems 1.1 and 1.2 is the identity for a suitable scalar function β. The bound in Theorem 1.1 arises from integrating this identity over B R , ensuring that the non-divergence terms on the right-hand side control the appropriate weighted H(curl; B R ) norm of E (observe from the right-hand side of (1.15) how the conditions ǫ + (x · ∇)ǫ > 0 and (µ + (x · ∇)µ) > 0 then arise), and show that the term on ∂B R has the appropriate sign using the fact that E satisfies the Silver-Müller radiation condition (1.2). The bound in Theorem 1.2 is then obtained from Theorem 1.1 using approximation arguments similar to those in [28], which in turn were inspired by analogous arguments in the setting of rough-surface scattering in [70] (with this thesis recently made available as [4]).
To connect (1.14) with other identities in the literature, it is convenient to consider the case when iωµH = ∇ × E (i.e., K = 0 in (1.1)) and (1.14) then becomes Observe that the left-hand side of (1.15) involves the second-order form of the time-harmonic Maxwell equations multiplied by a linear combination of (µ −1 ∇ × E) × x and E. For the Helmholtz equation, Morawetz pioneered the use of multipliers that are a linear combination of a derivative of u and u itself [57,58], with the key insight being that this linear combination could deal with the contribution "at infinity" -in our case on ∂B R -using the radiation condition (for more on this, see the more-recent presentation and discussion in [ [56], with similar multipliers used in control theory by [38,39,62], and in general relativity by [3,10,43]. For the second-order form of the time-harmonic Maxwell equations, multipliers involving (∇ × E) × Z for a vector field Z have been well-used; see, e.g., [29,40,49,52,60] However, the time-harmonic Maxwell equations posed in R 3 with the Silver-Müller radiation condition seem not to have been studied using the multiplier technique before, and, correspondingly, we have not been able to find in the literature the identity (1.15)/(1.14), involving the linear combination of multipliers needed to deal with the radiation condition.

Bounds in unweighted norms
The L 2 norms on the left-and right-hand sides of the bound (1.7) are weighted with the coefficients ǫ, µ. Alternatively, one can repeat the arguments leading to (1.7) and work in unweighted norms.
The analogue of (1.7) is then where now with both assumed > 0. Remark 3.5 below discusses in more detail how to obtain (1.16).

The analogous results for the interior impedance problem
In §4, we prove results analogous to Theorems 1.1 and 1.2 for the Maxwell interior impedance problem in Lipschitz domains that are star-shaped with respect to a ball. This problem is: given a bounded Lipschitz open set Ω that is star-shaped with respect to a ball (i.e., star-shaped with respect to each point in a ball of non-zero radius) and has outward-pointing unit vectorn, ǫ, µ ∈ W 1,∞ (Ω; SPD), ϑ ∈ L ∞ (∂Ω) uniformly positive, J, K ∈ L 2 (Ω), and g ∈ L 2 T (∂Ω), find and where E T denotes the tangential trace of E, defined for smooth vector fields by v T := (n × v) ×n. There are two reasons we prove these results. First, the interior impedance problem is a ubiquitous model problem in the numerical analysis of the time-harmonic Maxwell equations; see, e.g., [55,Chapter 7], [17,25,26,33,42,46,63,72]. Second, domain-decomposition methods for timeharmonic wave problems (including the Helmholtz and Maxwell equations) often use impedance boundary conditions on the subdomains, following the work of [7,20] in the Helmholtz context; see, e.g., [11,12]. Such impedance boundary conditions are then the starting point for so-called optimised Schwarz methods; see, e.g., [2,21,22]. Just as results about the Helmholtz interior impedance problem can be used to analyse these methods in the Helmholtz context (see, e.g., the heterogeneous analysis in [27]), we expect the bounds in §4 to play the analogous role in the analysis of Maxwell domain-decomposition methods.
The proofs of Theorems 4.1 and 4.2 below are very similar to those of Theorems 1.1 and 1.2. In particular, recall from §1.4.5 that, for the problem in R 3 , multiplying the PDE by a linear combination of (µ −1 ∇ × E) × x and E allows one to deal with the contribution on ∂B R using the Silver-Müller radiation condition; for the interior impedance problem, this linear combination allows one to deal with the contribution on ∂Ω using the impedance boundary condition.
In fact, the results about the interior impedance problem are actually harder to prove that the results about the problem in R 3 . The difficulty comes in integrating the Morawetz-type identity over the domain. For the problem in R 3 , we integrate the identity over B R , and E has sufficient regularity for this integration when ǫ, µ ∈ W 1,∞ because of interior regularity of solutions of the Maxwell equations. In contrast, for the interior impedance problem, we integrate the identity over Ω, and the regularity of the solution of the interior impedance problem when Ω is only Lipschitz is more delicate. This exactly parallels the Helmholtz case, where, at least for constant-coefficient problems, justifying integrating Morawetz identities over Lipschitz domains follows from the density result [18,Theorem 1], which uses the harmonic-analysis results of [36,37]. In Theorem A.1 we generalise [18, Theorem 1] to ǫ, µ = I using the harmonic-analysis results of [50,51]. This result is used to justify integrating the Morawetz-type identity over Lipschitz domains in Part (i) of Lemma 2.4.

Morawetz-type identities 2.1 The identities in pointwise form
In this section, Sym denotes the set of 3 × 3 real, symmetric matrices.
We write (1.14) as i.e., Q β is defined by and R β and P β are defined analogously. Observe that, when E and H satisfy (1.1) the term R β depends linearly on the data J.
The following lemma describes a special case of (1.14) with ǫ and µ constant and scalar, and β = r √ ǫµ; in this case P β is the sum of (i) terms that vanish when ∇ · [ǫE] = 0 and ∇ · [µH] = 0, and (ii) terms that are non-negative. 3 . Let ǫ 0 , µ 0 be real-valued constants and recall that r := | x|. Then To prove the bound (1.7) on the solution of the transmission problem, the plan is to use the identity (1.14) in B R and then the identity (2.2) in R 3 \ B R to deal with the contribution from infinity.
Proof of Lemma 2.1. The identity (1.14) is the sum of the identity for the "Rellich multipliers", i.e. the test fields ǫE × x and µH × x, and the identity for the parts of the multipliers containing β Since µH · H and ǫE · E are real, the left-hand side of (2.4) equals which equals the right-hand side of (2.4), and thus we only need to prove (2.3). The left-hand side of (2.3) equals We now claim that for all v ∈ C 2 (D) 3 and all α ∈ C 1 (D, SPD), Summing (2.5) with v = E and α = ǫ to the same expression with v = H and α = µ, we arrive at (2.3). Therefore, we only need to show that (2.5) holds. Proceeding in a similar way to that in the proof of [52, Lemma 5.3.1], we find that standard vector calculus identities give (2.6) The identity can be proved by expanding in components the divergence on its right-hand side. Using (2.7) in (2.6) we find (2.5), and the proof is complete.
To separate tangential and normal traces on boundaries we use the following identity. 11) where v N := (v ·n)n and v T : Proof. By the symmetry of α and the decomposition and the result follows.

The identities in integrated form
Our next result is an integrated version of the identity (1.14). To state this result it is convenient to define the space where D is a bounded Lipschitz open set with outward-pointing unit normal vectorn and A ∈ W 1,∞ (D, SPD). We make three remarks about this space.  . We first assume that ǫ, µ, and β are as in the statement of the theorem, but E, H ∈ C ∞ (D) 3 Recall that the product of an H 1 (D) function and a W 1,∞ (D) function is in H 1 (D), and the usual product rule for differentiation holds for such functions. This result implies that Q β (2.1) is in H 1 (D) 3 and then (1.14) implies that ∇ · Q β is given by the integrand on the left-hand side of (2.13). The divergence theorem then implies that (2.13) holds, where the fact that Q β ·n equals the integrand on the right-hand side of (2.13) follows from the identity (2.11) and the identity Integrating (2.2) over B R1 \ B R and using the divergence theorem, we obtain We therefore only need to prove that |Q r √ ǫ0µ0 · x| = o R→∞ (R −2 ) uniformly in x so that lim R→∞ | ∂BR Q r √ ǫ0µ0 · x| = 0. On ∂B R ,n = x, and therefore E N = (E· x) x and E T = ( x×E)× x. By the definition of Q r √ ǫ0µ0 (2.1) and the identity (2.11), Taking the tangential and normal components of the Silver-Müller radiation conditions (1.2), uniformly in x and the proof is complete.
The Helmholtz analogue of Lemma 2.5 first appeared implicitly in [57,58], and first appeared explicitly in [ j . Then, by the Cauchy-Schwarz inequality and the inequality 2ab ≤ ε −1 a 2 + εb 2 for all a, b, ε > 0, The result (3.1) then follows from the definition of the weighted norms (1.3).
Applying the integrated Morawetz identity (2.13) with D = B R is justified by Part (ii) of Lemma 2.4, and then substituting in ∇ · [ǫE] = (iω) −1 ∇ · J, ∇ · [µH] = (−iω) −1 ∇ · K and (1.1), and recalling that ǫ = ǫ 0 I and µ = µ 0 I on ∂B R , we obtain that, for all β ∈ C 1 (B R ), 3) The choice β = R √ ǫ 0 µ 0 implies both that ∇β = 0 and that the terms integrated over ∂B R on the right-hand side of (3.3) equal −Q R √ ǫ0µ0 (2.16), thanks to (2.11); therefore, using the inequality (2.15), (3.3) becomes   Proof. Let J, K ∈ L 2 (B R ). Let p, q ∈ H 1 0 (B R ) be the unique solutions of the variational problems iω(ǫ∇p, ∇v) L 2 (BR) = (J, ∇v) L 2 (BR) and iω(µ∇q, ∇v) L 2 (BR) = (K, ∇v) L 2 (BR) (3.5) for all v ∈ H 1 0 (B R ); i.e., iω∇p is the orthogonal projection of ǫ −1 J in the (ǫ·, ·) L 2 (BR) inner product, so that and similarly (3.7) Extend p and q to functions on R 3 via extension by zero. Then ∇p, ∇q ∈ H(curl; B R ) and ∇p, ∇q ∈ H loc (curl; R 3 \ B R ). By the definition of the weak derivative and integration by parts, a piecewise H(curl) function is in H(curl) if and only if its tangential trace is continuous across the relevant interface. Since (∇p) T is the surface gradient of the trace of p (which is zero), ∇p ∈ H loc (curl, R 3 ), and similarly for ∇q. Therefore E − ∇p, H − ∇q ∈ H loc (curl; R 3 ) and satisfy the Silver-Müller radiation condition Combining this with (and the analogous inequality for H), we obtain for all v ∈ C 3 , and the analogous inequality with ǫ and µ swapped, to find The result (1.7) then follows by using (3.6) and (3.7). If K = 0, then q = 0, and we don't need to use the inequality (a + b) 2 ≤ 2(a 2 + b 2 ) for a, b > 0 in (3.8). The 8 in the term involving J in both (3.9) and (1.7) then reduces to 4. Similarly if J = 0, then the 8 in the term involving K in both (3.9) and (1.7) reduces to 4.
δ δ ⊲⊳ δ R Figure 2: The shaded region represents the set ⊲⊳ δ ⊂ B R defined in (3.13). The parameter 0 < δ < min{1/2, R/2} is both the radius of the inner ball and the opening of the two cones.
Finally, f δ : R 3 → R defined by f δ (ρ cos θ sin ϕ, ρ sin θ sin ϕ, ρ cos ϕ) := f ⋆ δ (ρ, θ, ϕ) is smooth, takes values in the interval [f min , f L ∞ ], satisfies f δ = f min on ⊲⊳ δ , and approximates f both almost everywhere and in L 2 (B R ): Moreover, if f is radially monotonic in the sense that ess inf ≥ 0, f ⋆ δ is monotonic in ρ by the definition of convolution, and f δ is radially monotonic in pointwise sense: The arguments extends to matrix-valued fields ǫ satisfying the assumptions in the assertion. The field ǫ ⋆ : R 3 → SPD is defined similarly to f ⋆ with diagonal value ǫ min I in place of f min . The smooth field ǫ ⋆ δ is obtained by componentwise mollification of ǫ ⋆ , and the pullback ǫ δ is defined as f δ . Then ǫ δ → ǫ with the same argument for the scalar case and each component of ǫ δ is in C ∞ (R 3 ). The monotonicity follows: for all x ∈ R 3 , v ∈ C 3 , h > 0, where the term in the integral is non-negative because ǫ((1 + h)x) ǫ(x) a.e., ǫ ǫ min a.e. (since Π ǫ ∈ L ∞ (R 3 , SPD)) and because of the shape of the region where ǫ ⋆ = ǫ min I. In particular ǫ δ is positive definite and ǫ min ǫ δ ǫ max . Since ǫ δ is smooth and radially monotonic, (x · ∇)ǫ δ 0 and (3.12) follows.
The proof of Lemma 3.4 also corrects the proof of [28, Theorem 2.7], where it was assumed that standard mollification in Cartesian coordinates preserves radial monotonicity.
We now prove Theorem 1.2. In this proof, we use the weighted norm on L 2 (B R ) × L 2 (B R ) corresponding to the left-hand side of (1.7), i.e.,

|||(E, H)|||
Proof of Theorem 1.2. By [66, Theorem 2.10] it is sufficient to show that the bound (1.7) holds under the assumption that the solution of (1.1) exists. Without loss of generality, we assume that J and K are compactly supported in B R . Indeed, once the bound (1.7) is proved for such J and K, since compactly-supported functions are dense in L 2 (B R ) and the bound is independent of the supports of J and K, the bound (1.7) holds for all J, K ∈ L 2 (B R ).
With ǫ δ and µ δ as in Lemma 3.4, observe that (3.15) and thus, combining (3.14) and (3.15), we obtain  .7), where we have abbreviated the constants on the right-hand side of (1.7) to C 1 and C 2 to keep the notation concise. The crucial point is that the C 1 and C 2 corresponding to ǫ δ and µ δ can be taken to be the C 1 and C 2 corresponding to ǫ and µ by (3.10) and (3.12); thus, in particular, C 1 and C 2 are independent of δ. We now claim that the approximation properties (3.11) and (3.19) imply that given ǫ, µ, E, H (and associated J, K), ω > 0, and ζ > 0 one can choose η = η(ζ) > 0 and δ = δ(ζ, η) > 0 such that the difference between the right-hand side of (3.19) and C 1 J 2

(3.20)
Once this claim is established, without loss of generality, we can further assume that η ≤ ζ, and reduce δ (if necessary) so that the last term on the right-hand side of (3.16) is ≤ ζ/4. Then combining (3.16) and (3.20), and using that η ≤ ζ, we obtain that
We now complete the proof by establishing the claim above. First observe that and, similarly, We now need to make the terms on the right-hand sides of (3.23) and (3.24) that are not J 2 L 2 (BR;ǫ −1 δ ) and K 2 L 2 (BR;µ −1 δ ) , respectively, small. We first deal with the terms that involve E − E η and H − H η (and thus will be made small by choosing η small). By (3.10), Therefore, by (3.14) and the bounds (1.4) on ǫ and µ, given ζ > 0, we can choose η > 0 such that We now deal with the remaining terms on the right-hand sides of (3.23) and (3.24), which will be made small by making δ small. By (3.11) (and arguing similarly to above using (3.10) to deal with the norms weighted by ǫ −1 δ and µ −1 δ ), given ζ and η, we can choose δ > 0 such that (3.28) Combining (3.23)-(3.28), we obtain that , and the claim (and hence also the result) is proved.

Proof of Corollary 1.6
Let χ(x) := max{0, min{1, R−|x| R−Rscat }} (i.e., χ is piecewise-linear in the radial direction); then χ L ∞ (BR) = 1 and ∇χ L ∞ (BR) = 1 R−Rscat . Furthermore, since the tangential components of ∇χ are continuous on the boundary of the support of ∇χ (which is a spherical shell), ∇χ ∈ H(curl; R 3 ). Let E := χE I + E S = E T − (1 − χ)E I , and similarly for H. Since E = E S and H = H S for |x| ≥ R, E, H satisfy the Silver-Müller condition (1.2). Furthermore E, H ∈ H loc (curl; R 3 ) and satisfy (1.1) with J := ∇χ × H I and K := ∇χ × E I . Applying (1.7) to E and H, we obtain where we have used the fact that ǫ and µ are scalar-valued on B R \B Rscat to "pull out" ∇χ L ∞ (BR) from the weighted norms of H I and E I on the right-hand side. The bound (1.13) then follows using the inequality E T 2 L 2 (BR;ǫ) ≤ 2 E 2 L 2 (BR;ǫ) + 2 (1 − χ)E I 2 L 2 (BR;ǫ) , its analogue with E replaced by H, and the bounds on χ above.
Remark 3.6. (Small contrast limit.) The bound (1.13) is sharp in its ω dependence, but does not show that the scattered fields E S and H S vanish in the small-contrast limit, i.e., for One can easily bound the scattered fields E S and H S observing that they solve the Maxwell problem (1.1)-(1.2) with J = iω(ǫ 0 I − ǫ)E I and K = iω(µ − µ 0 I)H I , however the resulting bound is suboptimal in its ω dependence. We note that [54, Cor. 3.1] obtained a bound for the Helmholtz scattering problem in this latter way; its wavenumber dependence can easily be improved by adapting the proof of Corollary 1.6 to the Helmholtz setting.  where |ǫ| and |µ| denote the point values of the matrix norms (induced by the Euclidean vector norm). Then, given J, K ∈ L 2 (Ω) and g ∈ L 2 (∂Ω), the solution E, H ∈ H imp (curl; Ω) of (1.18)-(1.19) exists, is unique, and satisfies where the instances of 8 on the right-hand side reduce to 4 if K = 0.
In (4.2), the norm on ∂Ω on the left-hand side can be replaced by the more natural, smaller quantity ǫ 1/2 E T 2 L 2 (∂Ω) = ∂Ω ǫE T · E T involving the square-root matrix of ǫ (which is SPD).

Proof of Theorem 4.1
Lemma 4.3. (Bound for C 1 coefficients and right-hand sides in H(div; Ω).) Suppose that Ω is a bounded Lipschitz open set that is star-shaped with respect to a ball centred at the origin with radius ρR Ω , where R Ω := sup{|x|, x ∈ Ω}. Suppose that ǫ, µ ∈ C 1 (Ω; SPD) satisfy (1.4) and ϑ ∈ L ∞ (∂Ω) is uniformly positive. Let M ϑ be defined by (4.1). Then, given J, K ∈ H(div; Ω), the solution E, H ∈ H imp (curl; Ω) of (1.18)-(1.19) exists, is unique, and satisfies the bound  19) with J, K ∈ H(div; Ω) and g ∈ L 2 T (∂Ω) exists, is unique, and satisfies the bound (4.3). Then the solution to (1.18)-(1.19) with J, K ∈ L 2 (Ω) and g ∈ L 2 T (∂Ω) exists, is unique, and satisfies (4.2). Proof. This is almost identical to the proof of Lemma 3.3. The main new ingredient is the fact that (∇p) T = (∇q) T = 0, which holds since (∇p) T and (∇q) T are the surface gradients of the traces of p and q, respectively, and these traces are zero, since p, q ∈ H 1 0 (Ω). This fact shows that E − ∇p and H − ∇q satisfy the impedance boundary condition (1.19) (since H ×n = H T ×n), and also means that the correct norms on ∂Ω appear in the analogue of (3.8).
Proof. The impedance boundary condition (4.5) implies that

so (4.4) becomes
Then condition (4.6) implies that (compare to (3.3)). If β = R Ω M ϑ (which satisfies the condition (4.6)) then Lemma 4.7 implies that the integrals over ∂Ω are bounded above by Proceeding as in the proof of Lemma 3.2 and using Lemma 3.1 with this value of Ξ, one obtains the assertion (4.3).

Proof of Theorem 4.2
By [55, §4.5] it is sufficient to show that the bound (1.7) holds under the assumption that the solution of (1.18) exists. Let By the density of C ∞ (Ω) 3 in H imp (curl; Ω) (see, e.g., [55,Theorem 3.54]), given η > 0, ǫ, µ, E, and H, there exists E η , H η ∈ C ∞ (Ω) 3 such that Exactly as in the proof of Theorem 1.2, E η and H η satisfy the PDEs (3.17) and (3.18), and furthermore By Lemma 3.4, ǫ δ and µ δ satisfy the conditions of Theorem 4.1, and thus, by the bound (4.2), where we have abbreviated the constants on the right-hand side of (1.7) to C 1 , C 2 , and C 3 to keep the notation concise. As in the proof of Theorem 1.2, the crucial point is that the C 1 , C 2 , C 3 corresponding to ǫ δ and µ δ can be taken to be the C 1 , C 2 , C 3 corresponding to ǫ and µ by (3.10) and (3.12). We now claim that the approximation properties (3.11) and (4.8) imply that given ǫ, µ, E, H (and associated J, K, g), ω > 0, and ζ > 0 one can choose η = η(ζ) > 0 and δ = δ(ζ, η) > 0 such that the difference between the right-hand side of (3.19) and C 1 J 2 L 2 (Ω;ǫ −1 ) + C 2 K 2 L 2 (Ω;µ −1 ) + C 3 ϑ −1/2 g 2 L 2 (∂Ω) is ≤ ζ/4. The proof of this claim is almost exactly the same as the proof of the analogous claim in the proof of Theorem 1.2, except now the first step of choosing η depending on ζ also involves the terms on ∂Ω arising from (H − H η ) T and (E − E η ) T as well as the terms in Ω in (3.25) and (3.26). Once this claim is established, without loss of generality, we can further assume that η ≤ ζ (also exactly as in the proof of Theorem 1.2).
The proof then proceeds exactly as in the proof of Theorem 1.2, in particular using (3.15) with B R replaced by Ω. The end result is that and since ζ > 0 was arbitrary, the result follows.
A Density of C ∞ (D) 3 in the space V (D, A) Throughout this appendix D ⊂ R d , d ≥ 2 is a bounded Lipschitz open set. We are mainly interested in the case d = 3, but some results hold for general d ≥ 2, and thus we specify the values of d explicitly in the statements. In this appendix only, we use γ to denote the standard Dirichlet trace operator γ : H 1 (D) → H 1 2 (∂D). The goal of this appendix is to prove the following density result.
Let V scal (D; A) (subscript "scal" for scalar) be defined by where A ∈ C 0,1 (D, SPD) = W 1,∞ (D, SPD) and ∂ n,A v is the conormal derivative defined such that ∂ n,A w =n · (A∇w) for w ∈ H 2 (D).
The following theorem of Nečas says that either of the conditions on ∂D in (A.1) can be removed.
(iv) for each i there exists a rigid motion (i.e. a composition of a rotation and translation) r i : Without loss of generality we take the rigid motions {r i } to be a family of pure rotations. Let {χ i } be a partition of unity subordinate to {W i }, i.e.
For x ∈ supp χ i ∩ ∂D and ε > 0, we define the segment where b i is the unit vector that satisfies b i = r −1 i (e d ); observe that ℓ i,ε (x) ⊂ D for all x ∈ W i ∩ ∂D and sufficiently small ε > 0. Observe that this definition implies that there exists ε * (independent of i) such that ℓ i,ε (x) ⊂ Θ(x) for all x ∈ supp χ i ∩ ∂D and 0 < ε ≤ ε * . for all x ∈ ∂D for whichn is defined. for some C > 0 (independent of u and g).
References for the proof. When A ∈ C 1 (D, SPD), this result, without the statement that u ∈ H 1 (D), is [ The next lemma relates the conormal derivative defined by (A.4) to the standard conormal derivative in H −1/2 (∂D) defined via Green's identity (denoted by ∂ n,A u).
Using the change of variable y = x − t b i for y ∈ ∂D i,t and x ∈ ∂D i , observing thatn(y) =n(x) in this case, and recalling that A ∈ C 0,1 (D, SPD) and ∇u ∈ C(D) (so we can omit the trace operator from A∇u), we have With φ t (x) := φ(x − t b i ), observe that φ t (x) − φ(x) → 0 if φ is continuous at x, and thus (by density of continuous functions in L 2 ) φ t − φ → 0 in L 2 . Therefore, as t → 0, the right-hand side of (A.7) tends to where for the last equality we have used the facts that supp v i ⊂ supp χ i ⊂ W i and D ∩ W i = D i ∩ W i .
For the left-hand side of (A.7), observe that for some C > 0 (dependent on v i ). Therefore, taking the limit as t → 0 in the left-hand side of (A.7) using the dominated convergence theorem (noting that L 2 (supp χ i ∩ ∂D i ) ⊂ L 1 (supp χ i ∩ ∂D i )) along with the inclusion (A.3) and the definition of ∂u/∂n A (A.4), we find that the left-hand side of (A.7) tends to ∂Di ∂u/∂n A γv i , and the proof is complete.
Combining Theorem A.7 and Lemma A.8 gives the following corollary. Then u ∈ C 1,s (D) for all 0 < s < 1, (A∇u) * ∈ L 2 (∂D), and ∂ n,A u = ∂u ∂n A . (A.8) We are now in a position to prove that C ∞ (D) is dense in V scal (D; A), from which Theorem A.1 follows. The idea of the proof is to decompose a general element w ∈ V scal (D; A) as the sum of a term v that is the restriction of the solution of a non-homogeneous PDE in R d (which enjoys interior elliptic regularity) and a term u that is the solution of a homogeneous PDE in D (which enjoys the regularity provided by Corollary A.9). Proof. By Part (i) of Theorem A.2, we can omit the H 1 (∂D) term from the definition of the norm on V scal (D; A) (A.2), and we do for the rest of this proof.
Let G i := D i ∩ supp χ i and define u t u t (x) := u(x − t b i ) for 0 < t ≤ ε * , (A.10) where ε * is as in (A.3). By interior regularity u t ∈ H 2 (G i ). Using again the fact that C ∞ (D) is dense in H 2 , and the fact that the H 2 norm controls the V scal (D; A) norm (by (A.9)), by the triangle inequality, to obtain the result it is sufficient to prove that u − u t V scal (D;A) → 0 as t → 0.
It only remains to prove that ∂ n,A u − ∂ n,A u t L 2 (∂Di) → 0 as t → 0. Since u t ∈ H 2 (G i ), for where we have dropped the trace operator since A ∈ C 1 (D, SPD) and ∇u ∈ C(D) by (A.8). Therefore,