In this paper we analyze a passive, broadband approximate cloaking scheme for the Helmholtz equation in R d for d = 2 or d = 3. Specifically, we are interested in making a bounded region approximately invisible to a far field observer and to probing by incident fields at arbitrary frequencies, independently of the material inside this region. Using ideas from transformation optics we achieve this by surrounding the region with a layer of an appropriate anisotropic material. By including a layer of extremely high conductivity adjacent to the region, we may without loss of generality assume that the region we want to cloak is "soft", that is, supports a homogeneous Dirichlet boundary condition. The approach of cloaking by mapping, also known as transformation optics, has been popularized by Pendry, Schuring and Smith [29] and Leonhardt [23] for Maxwell's equations. The basic idea is to make a singular change of variables which blows up a point (invisible to any probing incident wave) to a cloaked region. The same idea had previously been used by Greenleaf, Lassas and Uhlmann to create anisotropic objects that were invisible to EIT [15] (see also [14]). The singular nature of the perfect cloaks presents various difficulties: in practice this means they are hard to fabricate, and from the analysis point of view in some cases the rigorous definition of the corresponding electromagnetic fields is not obvious [12,33,34]. To avoid the use of singular materials in the cloak, regularized schemes have been suggested [20,21,30,31]. The trade-off is that such schemes only lead to approximate cloaking. We refer the reader to [2,8,13,16] for work on enhancement of approximate cloaks.
To design a passive approximate cloaking device, we blow up a small ball B ϵ of radius ϵ > 0 (the regularization parameter) to the ball B 1 of radius one, which represents the cloaked region (see Fig. 1 in Section 2). To be more precise we actually map B 2 \ B ϵ onto B 2 \ B 1 , keeping fixed the outer boundary ∂B 2 .
B 2 \ B 1 represents the cloak. As result of this change of variables one obtains an anisotropic layer in B 2 \ B 1 . We include a Drude-Lorentz-type term (see e.g. [19]) in the refractive index of the cloaking layer. This results in a frequency dependent and complex valued index of refraction which is consistent with causality. Since the cloaked region B 1 is "soft" we impose a zero Dirichlet boundary condition on the boundary ∂B 1 . As mentioned earlier, this Dirichlet condition may be viewed as a limit of a highly conducting layer, and it thus may be interpreted as "hiding" the contents of B 1 . A main focus of this paper is to establish estimates on the scattered field outside the cloak in terms of the small parameter ϵ > 0 and the probing frequencies. We remark that the choice of B 1 and B 2 \ B 1 for the cloaked region and the cloak, respectively, is made for convenience and one can use more general domains in the change of variables. We also note that, in the context of approximate cloaking for the Helmholtz equation (the frequency domain wave equation), the Drude-Lorentz model was previously used by Nguyen and Vogelius in [28]. The Drude-Lorentz model takes into account the effect of the oscillations of free electrons on the electric permittivity by means of a simple harmonic oscillator model. When viewed in (complex) frequency domain, the refractive index associated with the Drude-Lorentz model may be extended analytically to the whole upper half plane. It is well-known that an immediate consequence of this is causality for the associated non-local time-domain wave equation, see [19,32]. This property is most essential for the well-posedness (and the physical relevance) of this equation. Another well known consequence of this analyticity property are the so-called Kramers-Kronig relations between the real and the imaginary part of the refractive index (they are essentially related by Hilbert transforms). However, this fact is not explicitly used in our analysis.
We investigate two questions related to the scattering by the aforementioned cloak B 2 \ B 1 . The first one is whether, for a fixed ϵ > 0, there are wave numbers (proportional to frequencies) and incident fields for which the corresponding scattered field is zero, i.e., the cloak (and B 1 ) is perfectly invisible to this particular probing experiment. This question is related to the existence of real eigenvalues of the interior transmission eigenvalue problem defined on B 2 \ B 1 [3], for which that part of the eigenfunction, which corresponds to the incident field, is extendable as a solution to the Helmholtz equation in all of R d [6,7]. In particular, such non-scattering wave numbers, for which perfect cloaking is achieved for a particular incident field, form a subset of the real transmission eigenvalues. We prove that, real transmission eigenvalues do not exist for the inhomogeneity presented by the cloak, i.e., for the anisotropic inhomogeneity B 2 \ B 1 with the complex-valued frequency dependent Drude-Lorentz term and a homogeneous Dirichlet condition on the inner boundary ∂B 1 . In addition, we show that all the (complex) transmission eigenvalues, that lie outside a precisely characterized compact set of the lower half plane, form a countable set with no finite accumulation points outside this compact set. Supported by some computational evidence, we conjecture that a sequence of complex transmission eigenvalues accumulate at a point (as well as at its symmetric counterpart) on the boundary of this compact set. These points have imaginary part equal to -1/2, but real parts that depend on the resonant frequency of the Drude-Lorentz term. A complete analysis of the transmission eigenvalue problem for inhomogeneities with such a Drude-Lorentz term is still open. This eigenvalue problem, in addition to being non-selfadjoint, is nonlinear since the Drude-Lorentz term involves the eigenvalue parameter in a non-linear fashion, and thus the known approaches do not apply [3]. If the Drude-Lorentz term is not present, the existence of an infinite set of real transmission eigenvalues accumulating at +∞ for (anisotropic) inhomogeneities containing a Dirichlet obstacle is proven in [4,5]. Secondly, although perfect cloaking is impossible at any frequency (even for a single incident wave) we prove that one can achieve approximate cloaking over any given finite band of wave numbers for sufficiently small ϵ > 0. In particular, we prove that provided the Drude-Lorentz resonant frequency k ϵ is sufficiently large, more precisely k 2 ϵ > c * ϵ -3 for d = 3, and k 2 ϵ > c * | ln ϵ|/ϵ for d = 2, then for any fixed R the L 2 -norm of the scattered field in B R \ B 2 is of order ϵ in R 3 and of order 1/| ln ϵ| in R 2 , with a constant depending on the given band of wave numbers, c * and R. These estimates hold for a large class of incident waves, including plane waves and their superpositions (Herglotz waves). We note that point source waves with sources outside the cloak, as well as their superpositions would also be admissible. Furthermore, we prove that the far field pattern is uniformly O(ϵ) in R 3 and O(1/| ln ϵ|) in R 2 , with constants depending on the given band of wave numbers. These latter results are obtained by estimating the norm of the Lippmann-Schwinger volume integral over B 2 \ B 1 and using scattering estimates adapted from [27]. We should mention that cloaking via change of variables for the Helmholtz equation at any frequency is investigated in [24][25][26][27][28], but in these papers the region is cloaked to an active source compactly supported in the exterior of the cloak. The scattering problem with incident field cannot be written in this framework. In fact, in that case the scattered field may be viewed as satisfying an inhomogeneous Helmholtz equation with a source given by the incident field, but this source is supported inside the cloak. Finally let us mention that perfect cloaking for a penetrable obstacle and the quasi-static Helmholtz equation (i.e., the limit when the probing wavelength is much larger than the cloaking device) with incident plane waves is investigated in [9]. There, the authors show the impossibility of perfect broadband cloaking (see also [18]) and using the theory of Herglotz-Nevanlinna functions they derive lower bounds on the polarizability tensor,foot_0 in terms of the frequency band, and the geometry and dielectric contrast of the obstacle. These bounds show the limitations of broadband quasi-static cloaking and apply to a very wide class of passive cloaks. The fact that perfect quasi-static cloaking is possible only at a discrete set of frequencies is entirely consistent with the fact that we in the present context show that there are no real transmission eigenvalues. In contrast, our analysis demonstrates the possibility of broadband approximate cloaking in the context of the full Helmholtz equation and for a specific transformation-opticsbased cloaking scheme. We mention that the lower bounds of [9] are not directly comparable to the upper bounds derived here, as in the quasi-static regime the frequency dependence is in the principal part of the operator, whereas in our context it is in the zeroth order term. However, our analysis shows that general lower bounds in the spirit of [9] cannot be expected to hold for the full Helmholtz equation with a dissipative refractive index (such as the one given by the Drude-Lorentz model).
the open ball of radius r > 0 centered at the origin and let S r = ∂B r . For a small parameter ϵ > 0 consider the following continuous and piecewise smooth mapping:
For simplicity of notation we will suppress the dependence of F on the parameter ϵ. Note that F maps
, and that F (x) = x on S 2 . Now, we design a cloaking device, occupying B 2 \ B 1 , to approximately cloak the (soft) region B 1 . We incorporate a Drude-Lorentz type term to account for a more physically relevant nonlinear dependence of the index of refraction on wavenumber.
The constitutive material properties are thus given by
where I denotes the d × d identity matrix and σ ϵ is the Drude-Lorentz term given by
cf. [19], page 331. Here k ϵ > 1 2 represents the so-called resonant frequency of the Drude-Lorentz model. F * denotes the push-forward by the map F , defined by
for a matrix-valued function A, and for a scalar function q, respectively. The definition of the push-forward is motivated by the following change of variables property, which can be proven by straightforward calculations (cf. [15,20]).
Lemma 2.1. Let F be as defined in (2.1)
The functions u and v satisfy the boundary relations
where ν denotes the unit outward normal vector on S 2 and the equality of the conormal derivatives is understood in the sense of distributions in
Let u i be an incident field at a given wave number k > 0 (we suppress the dependence of u i on k for the ease of notation), i.e.,
Given the incident wave u i and the "cloaked" soft obstacle B 1 , consider now the associated Helmholtz scattering problem. If A c and q c denote the constitutive material properties defined in (2.2), then the total field
of the form
where u t c is the transmitted field and u s c is the scattered field, which satisfies the Sommerfeld radiation condition
uniformly in x = x/|x| (cf. [10] for more details about the scattering problem). As u c and its conormal derivative are continuous across S 2 , the problem (2.6) can equivalently be written 2 Since similar formulas hold for F * (F -1 ) * B and F * (F -1 ) * p it follows that (F -1 ) * = (F * ) -1 , and for that reason we sometimes use the notation
(2.9)
As the scattered field u s c satisfies the constant coefficient Helmholtz equation, it is in fact real analytic and admits the following asymptotic behavior as r → ∞:
where the function u ∞ , defined on S 1 , is the so-called far field pattern of the scattered field u s c . It is wellknown that the vanishing of u ∞ on S 1 , implies the vanishing of the scattered field u s c in R d \ B 2 (cf. Rellich's Lemma in [10]). A non-trivial incident field u i and the wave number k > 0 for which the corresponding far field pattern vanishes are referred to as non-scattering incident field and a non-scattering wave number, respectively. If we regard u i as a function defined in B 2 , then from (2.9) it is clear that at a non-scattering wave number k > 0, there exist non-trivial functions w c = u t c and v = u i defined in B 2 \ B 1 and B 2 , respectively, such that
(2.11)
A wave number k for which (2.11) admits a non-trivial solution is called an interior transmission eigenvalue with the corresponding eigenfunction (w c , v). Thus, non-scattering wave numbers are necessarily real interior transmission eigenvalues [3]. Conversely, a real interior transmission eigenvalue k > 0 is a non-scattering wave number if the eigenvector v can be extended from B 2 to a solution of the Helmholtz equation in all of R d [7,6].
For clarity and the reader's convenience we now state the main results of our paper. The first theorem addresses the question whether our cloak provides a perfect cloaking of the region B 1 for even a single incident wave. (i) There are no interior transmission eigenvalues in R ∪ iR. (ii) k ∈ C is an interior transmission eigenvalue if and only if so is -k.
2 and let K be the shaded compact region in Fig. 2. The region K is symmetric about the imaginary axis, the slanted line segment of the boundary in the right half-plane has the equation Imk = -ℜek, the curved arc joining κ to k ϵ is given by ℜek = (Imk) 2 + Imk + k 2 ϵ . Let G denote the open set G = C \ K. Then those interior transmission eigenvalues which lie inside G form a discrete set (i.e., an at most countable set with no limit points in G).
Part (i) of Theorem 3.1 will be proven in Section 4. As a consequence we conclude that perfect cloaking/nonscattering is impossible at any wave number k > 0, since real transmission eigenvalues do not exist. One of the main ingredients in deriving this result is the property Imσ ϵ (k) > 0 for k > 0, i.e., passivity of the material due to dissipation (the electromagnetic energy gets absorbed by the material as the electromagnetic wave passes through). Therefore, the same conclusion will continue to hold for a more general σ ϵ , provided it exhibits passivity. It might be of interest to study the feasibility of perfect cloaking in a non-dissipative Drude-Lorentz model. Part (ii) is an immediate consequence of the symmetry relation
As a result q c (x, k) has the same symmetry property and k is a transmission eigenvalue of (2.11) with eigenfunction (w c , v), if and only if so is -k with eigenfunction (w c , v). The proof of part (iii) will be given in the Appendix since the discreteness of complex eigenvalues is not central to the cloaking discussion. The value κ is one of the poles of σ ϵ (k) (the other one is -κ). Numerical evidence, presented in Section 4.2, indicates that it is a limit point for the set of transmission eigenvalues of (2.11). Being bold, we venture Conjecture 3.2. (Finite accumulation point of transmission eigenvalues) Let κ be defined as in part (iii) of Theorem 3.1. Then κ is a limit point of transmission eigenvalues of (2.11).
We note that Theorem 3.1 asserts nothing about potential interior transmission eigenvalues in the set K \R.
Their nature is a completely open problem. Although perfect cloaking is impossible, we demonstrate that, under a suitable growth assumption on k ϵ , one can achieve approximate cloaking over any given finite band of wave numbers. We first state the main estimate on the scattered field including its explicit dependence on k (and ϵ). The broadband cloaking estimates follow as a corollary from this. We define
where
denotes the push-forward by the map F -1 , and we set
Let u s c be the scattered field from (2.9). There exists
and
where the implicit constants in (3.4) and (3.5) depend only on R, k 0 and C.
Remark 3.4. In the above theorem, H
(1) 0
denotes the Hankel function of the first kind of order 0. We also adopt the following notation: for two positive quantities A and B, we write A B, if there exists a constant d > 0 (independent of A and B) such that A ≤ dB.
Imposing a suitable lower bound on the resonant frequency k ϵ with respect to ϵ, the quantity M ϵ,k (for bounded k) becomes of order ϵ for d = 3, and of order 1/| ln ϵ| for d = 2 (cf. (5.21)) and Theorem 3.3 implies the following result:
. Furthermore, assume that the incident field u i satisfies
Let u s c be the scattered field from (2.9). There exists a constant
where the implicit constant depends only on k -, k + , R, c * and C R . Similarly, there exists a constant
where u ∞ is the far field pattern defined in (2.10), and the implicit constant depends only on k -, k + , c * and
Remark 3.6.
(i) Note that k ϵ → ∞ as ϵ → 0. Therefore, the higher degree of invisibility is achieved at the cost of using "extreme" materials throughout the cloak. These may be hard to manufacture.
(ii) Most likely the implicit constants in the estimates (3.7) and (3.8) go to infinity as the length of the frequency band, k +k -, goes to infinity. (iii) The results of the above two theorems do not use the radial geometry in any essential way and carry over to the non-radial setting as well. (iv) The assumption (3.6) (or (3.3)) is satisfied by incident plane waves as well as by their superpositions, the so-called Herglotz waves u i := u g given by
It is also satisfied by radiating point sources (outside of B 2 ) and their appropriate superpositions.
In this section we study the interior transmission eigenvalue problem. We first eliminate the anisotropy A c in the formulation (2.11) by using a change of variables to arrive at a new interior transmission eigenvalue problem, which has the same eigenvalues as (2.11). Then we reformulate the resulting problem in terms of a fourth order PDE, following [4] (see also [3]). Using this new formulation we prove part (i) of Theorem 3.1. Furthermore, in Section 4.2 we present numerical evidence supporting Conjecture 3.2 in two dimension.
In the interior transmission eigenvalue problem (2.11) let us change the variables in w c , while leaving v unchanged. Namely, let
where F is defined by (2.1). Using the properties of the map F (namely that F (x) = x on S 2 , F maps
along with Lemma 2.1, we obtain that w, v solve the following transmission problem:
where
Let us introduce the notation
Remark 4.1.
(i) We note that the trace (on S ϵ ) of a function w ∈ L 2 ∆ (O) makes sense as an element of H -1 2 (S ϵ ) by duality, using the identity
where ϕ ∈ H 2 (O) is such that ϕ = 0 in a neighborhood of S 2 , and ϕ = 0 and ∂ϕ/∂ν = τ on S ϵ . (ii) Similarly we note that for a function u ∈ H 1 ∆ (O) the normal derivative ∂ ν u (on S 2 ) makes sense as an element of H -1 2 (S 2 ) by duality, using the formula
where ϕ ∈ H 1 (O) is such that ϕ = 0 on S ϵ , and ϕ = ψ on S 2 .
We can reformulate (4.1) as a fourth order problem. Indeed, given a weak solution w, v of (4.1), let us set
It is clear that
Dividing both sides of the above equation by 1q (note that 1
Note that as v ∈ L 2 (B 2 ) solves the Helmholtz equation, by local elliptic regularity v ∈ H 1 (B ϵ ). But as u is continuous across S ϵ , we conclude that u ∈ H 1 (B 2 ) ∩ H 1 ∆ (O). Incorporating the boundary conditions on S 2 we introduce the Hilbert space of functions
where
), and is defined as described in the earlier remark. Thus, given a non-trivial weak solution w, v of (4.1), the function u ∈ X, given by (4.2), is a non-trivial weak solution of (4.4). Conversely, if u ∈ X is a non-trivial weak solution of (4.4), then
)
and yield a non-trivial weak solution of (4.1). Integration by parts easily yields a variational formulation of (4.4), namely: find u ∈ X such that
Before excluding the existence of real and purely imaginary transmission eigenvalues we need the following formulas for the map F : Lemma 4.2. Let F be given by (2.1), and set x = x/|x|, then
where I is the d × d identity matrix and for any two vectors a, b ∈ R d , a ⊗ b denotes the matrix whose (i, j)-th element is a i b j . In particular,
Proof. The formulas for DF (x) inside B ϵ and outside of B 2 are trivial. In the region B 2 \ B ϵ it is a direct consequence of the identity
Finally, using the identity det
, which concludes the proof.
Lemma 4.3. There are no non-trivial solutions to (2.11) for k ∈ R ∪ iR, i.e., there are no transmission eigenvalues for (2.11) in R ∪ iR.
Proof. First suppose k = iτ with τ ∈ R is a transmission eigenvalue. The above discussion shows that the problem (4.4) has a non-trivial solution u ∈ X for this value of k. Using the variational formulation (4.7) with ϕ = u we get
If τ ≥ 0 the above quantity is obviously positive. For τ < 0, it is still positive due to the assumption 2k ϵ > 1. Thus q(x, iτ ) -1 > 0 for all τ ∈ R and x ∈ O. For τ ̸ = 0 we now conclude from (4.9) that u = 0 in B 2 , contradicting the non-triviality of u for τ ̸ = 0. For τ = 0 we conclude from (4.9) that ∆u = 0 in O.
The Cauchy boundary conditions on S 2 now imply that u = 0 in O, and the continuity of u across S ϵ in combination with the fact that ∆u = 0 in B ϵ yields that u = 0 in all of B 2 , contradicting the non-triviality of u also for τ = 0. Assume now that k ∈ R \ {0} is a transmission eigenvalue; again let ϕ = u in the variational formulation (4.7) and take the imaginary part of the resulting equation to conclude that
Therefore ∆u + k 2 u = 0 in O. Using the boundary conditions u = ∂ ν u = 0 on S 2 , we conclude that u = 0 in O. Since k ̸ = 0 also conclude from the boundary conditions of (4.4) that u -= ∂ - ν u = 0 on S ϵ . The fact that ∆u + k 2 u = 0 in B ϵ now implies that u = 0 in B ϵ , and thus u = 0 in all of B 2 . This contradicts the non-triviality of u.
In this section we assume that d = 2 and consider the transmission eigenvalue problem after change of variables, i.e., the problem (4.1). In polar coordinates (r, θ) we can expand the functions v and w as follows:
where α n , β n , γ n are complex constants, J n is the Bessel function of order n and A n , B n (which also depend on k and ϵ) are linearly independent solutions of
The boundary conditions of (4.1) can be rewritten as
To obtain a nontrivial solution (v, w) (i.e., to ensure that k is an interior transmission eigenvalue) we need that there exists some n ∈ Z such that
where
The functions A n , B n can be expressed in terms of the Whittaker functions as follows:
where
We show some numerical evidence that κ is a limit point of transmission eigenvalues. We conjecture that for each n = 1, 2, ... there exists k n ∈ C \ {κ} such that f (n, k n ) = 0 and k n → κ as n → ∞. In other words, κ is a limit point of the transmission eigenvalues {k n }.
For each of the values n = 1, n = 7, and n = 12 (see Figs.
In this section we prove Theorems 3.3 and 3.5. The first observation is that the anisotropy in (2.9) can be eliminated, if we change the variables in the transmitted field u t c , but leave the incident and scattered fields unchanged. Namely, let
where F is given by (2.1), then u t is defined in B 2 \ B ϵ . Invoking Lemma 2.1 and using the facts that F = Id on S 2 , F maps S ϵ onto S 1 and that F -1 * A c = I, we see that (2.9) can be equivalently rewritten as
u s satisfies the outgoing radiation condition
where
the problem (5.2) can be rewritten as
uu i satisfies the outgoing radiation condition.
(5.5)
Here we used that the boundary conditions on S 2 from (5.2) simply become u = ∂ ν u = 0 on S 2 , i.e., u and its normal derivative are continuous across S 2 .
Consider the fundamental solution of the Helmholtz equation in free space: for any x ̸ = y
(5.6)
We incorporate the homogeneous Dirichlet boundary condition of (5.5) into the fundamental solution, i.e., we let Φ 0 k be the Green's function for the Helmholtz equation in the region R d \ B ϵ with the Dirichlet boundary condition in S ϵ . For any fixed y
satisfies the outgoing radiation condition .
(5.7)
Clearly we can write
where the function Ψ k (•, y) is the unique solution to the following exterior Dirichlet boundary value problem for the Helmholtz equation
x ∈ S ϵ Ψ k (•, y) satisfies the outgoing radiation condition .
(5.8)
Note that the boundary data -Φ k (x, y) is smooth, hence the function Ψ k (x, y) is smooth for x ∈ R d \ B ϵ and for any fixed y as above. Next, let us introduce the volume integral operator
(5.9)
Then the solution u of (5.5) satisfies the integral equation
where u is is the scattered field from the ball B ϵ due to the incident field u i , i.e., it is the unique solution of
u is satisfies the outgoing radiation condition.
(5.11)
The equation (5.10) is known as the Lippmann-Schwinger equation for the scattering problem (5.5) written in terms of the Green's function Φ 0 k . It can be derived the same way as done for example in [10] (without Dirichlet boundary conditions and using the kernel Φ k ). In Lemma 5.1 (see also (5.23) and (5.24)) we prove that for any fixed interval of wave numbers [k -, k + ], 0 < k -< k + < ∞, and any fixed R > 2 there exists an ϵ 0 > 0 (depending on k + and R) such that
for any k ∈ [k -, k + ], and ϵ < ϵ 0 . Therefore the operator I -T is invertible on L 2 (B R \ B ϵ ) and the integral equation (5.10) has a unique solution u R ∈ L 2 (B R \ B ϵ ). Furthermore u R = u| B R \B ϵ where u is the solution of (5.5). This follows from the fact that u| B R \B ϵ is in L 2 (B R \ B ϵ ) and as already noted satisfies the integral equation (5.10). It now follows immediately from (5.10), and the fact that the domain of integration for the operator T is B 2 \ B ϵ , that the solution to (5.5) is given by
Note that due to the mapping properties of the volume potential
The above argument shows that solving (5.5) is equivalent to solving the Lippmann-Schwinger equation (5.10) on B R \ B ϵ (for any bounded set of wave numbers [k -, k + ] and ϵ sufficiently small).
The main ingredients of the proofs of Theorems 3.3 and 3.5 are ϵ-explicit estimates for the scattered field u is and the operator T in appropriate Sobolev spaces. We state these estimates in the two lemmata below, however, for clarity of exposition their proofs are postponed to subsequent sections (see Section 5.3 and Section 5.4, respectively). Lemma 5.1. Let T be defined by (5.9), and let M ϵ,k and a(k) be defined by (3.1) and (3.2), respectively. Suppose R > 1, k 0 > 0, and 0 < ϵk < k 0 . Then for any u ∈ L 2 (B 2 \ B ϵ )
where the implicit constant depends only on R and k 0 . Lemma 5.2. Let u is be defined by (5.11), let R > 1, and k 0 > 0. Assume 0 < ϵk < k 0 and that u i satisfies (3.3), then
and
where the implicit constants depend only on R, k 0 and C (the constant from the inequality (3.3) for the incident field u i ).
With the help of the above lemmata we now prove the following scattering estimate:
Theorem 5.3. Let M ϵ,k and a(k) be defined by (3.1) and (3.2), respectively. Suppose R > 2, k 0 > 0, and 0 < ϵk < k 0 , and suppose u i satisfies (3.3). Let u be the solution to (5.5). There exists a constant c =
where the implicit constants depend only on R, k 0 and C (the constant from the inequality (3.3)).
Remark 5.4. As an immediate corollary we obtain Theorem 3.3, because u -
Proof. Consider the Lippmann-Schwinger equation (5.10) in the space L 2 (B R \ B ϵ ). Lemma 5.1 implies that there exists a constant
) and using (5.10) and the Neumann series expansion we obtain
Upon summation of the geometric series, the above equation implies the bound
where in the last step we used Lemma 5.2. This concludes the proof of the inequality (5.13).
To prove (5.14), we take d = 2. From the Lippmann-Schwinger equation uu i = u is + T u, and hence, using Lemma 5.2 we have,
From (5.13) with d = 2 we have
where in the last step we used the assumption that k 2 a(k)M ϵ,k < c. A combination of the last two estimates and insertion of r = C 1 k 2 a(k)M ϵ,k leads to (5.14).
Before proceeding to the proof of Theorem 3.5, we first estimate the far field pattern u ∞ , given by (2.10), in terms of the L 2 (B 5 \ B 2 )-norm of the scattered field u s , given by (5.1) and (2.9).
In the following, by using the term "an absolute implicit constant", we signify that, the inequality in question holds with a positive constant independent of the involved parameters.
Lemma 5.5. With an absolute implicit constant, for any |x| = 1 and k > 0,
(5.15)
Proof. The far field pattern has the following representation [10]:
where
and S 4 is the d -1-sphere of radius 4 centered at the origin (note that one could use any d -1 manifold circumscribing B 2 in its interior). Using Hölder's inequality and the duality
with the pivot space L 2 , we can bound
, where in the second step we used trace estimates. Next we bound the H -1 2 -norm of ∂ ν u s . Given any φ ∈ H 1 2 (S 4 ), consider its extension to B 4 \ B 3 via a bounded right inverse of the trace operator:
(5.16)
As this defines a bounded operator from H 1 2 (S 3 ∪S 4 ) to H 1 (B 4 \B 3 ), we have that with an absolute implicit constant
.
(5.17)
Now using the fact that u s satisfies the Helmholtz equation in B 4 \ B 3 , we obtain
where ⟨•, •⟩ denotes the dual pairing between H -1 2 (S 4 ) and H 1 2 (S 4 ). Using the Hölder's inequality and (5.17) we arrive at
Using that k + k 2 + k 3 k + k 3 , we obtain the bound
multiplication by ψ 2 u s and integration by parts leads to
Consequently,
Combining with (5.18) we obtain
which concludes the proof.
We are ready to establish the following broadband approximate cloaking estimates: Assume further that u i satisfies the estimate (3.6). Let u be the solution to (5.5) with q given by (5.3). There exists a constant
where the implicit constant depends only on k -, k + , R, c * and C R (the constants from (3.6)). Furthermore, there exists a constant
where the implicit constant depends only on k -, k + , c * and C 5 .
Remark 5.7.
(i) Since uu i = u s c outside of B 2 Theorem 3.5 follows as an immediate corollary of the above result. (ii) For d = 3 the following proof can be easily modified to show we can bound uu i up to the inner boundary S ϵ , i.e., ∥u -
Proof. We note that since u i is a solution to
, with a constant that only depends on k + . Due to (3.6) we thus conclude that u i , k ∈ Γ, satisfies the condition (3.3) as well (for ϵ < 1) with a constant that only depends on C 2 and k + . We proceed to estimate M ϵ,k . In view of (3.1), (5.3) and Lemma 4.2,
then for any k
where in the last step we used that k
k is positive and increasing on [0, 1 2 ], and it is positive and decreasing on [
where in the second inequality we have used that
We now conclude that there exist positive constants c 0 , C 0 depending only on c * and k + , such that
Let us further assume ϵ < 1/k + so that 0 < ϵk < 1 for k ∈ Γ. By Theorem 5.3 there exists a constant
and consequently (5.22) can be applied for all k ∈ Γ. Using the hypothesis (3.6) and (5.21) we conclude that for ϵ small enough
where the implicit constant depends only on k + , R, c * and C R .
Let us now consider d = 2. The function
Similarly, as before we conclude that, for k ∈ Γ,
| ln ϵ| .
(1) 0 (t)| is decreasing and H
(1) 0 (t) ∼ 2 iπ | ln t| as t → 0 (cf. [22]). Hence we have the following basic estimates:
These readily imply the inequality
| ln ϵ| .
Since by assumption ϵk + < 1 we get
where the last inequality holds, provided ϵ < 1/k 2 + . Putting everything together we conclude that for ϵ sufficiently small max k∈Γ ∥u -
1
| ln ϵ| .
The corresponding estimates for the far field pattern readily follow from Lemma 5.5.
In this section we show that Lemma 5.2 is a direct consequence of the following result due to Nguyen and Vogelius [27] (see also [24]):
Then for any β ≥ 1
where the implicit constant depends only on k 0 and D but is independent of β and k. Furthermore, for
where the implicit constant depends only on k 0 , D, and R but is independent of β and k.
Remark 5.9. The estimate (5.25) for the L 2 -norm of u and (5.26), in the case R = 2, is proven in Lemma 3 of [27] under the assumption that k 0 is sufficiently small (see also the beginning of the proof of Lemma 4). The subsequent Remark 4 of [27] explains that these estimates hold without any smallness assumption on k 0 . The extension of (5.26) to any R > 1 is immediate. Finally, the extension from an L 2 estimate of u to an H 1 estimate, as in (5.25), is guaranteed by Lemma 4 of [27].
As a straightforward consequence of Lemma 5.8 (with D = B 1 2 ⊂ B 1 ) we obtain the following corollary.
Corollary 5.10.
) and u be the outward radiating solution of the problem (i) We will see in the proof of Corollary 5.10 below that one can also obtain the following bounds up to the inner boundary S ϵ :
) , (5.28) where again the implicit constants depend only on R and k 0 . These estimates for d = 3 are as good as the bound in (5.27), in terms of being of the same order ϵ. However, for d = 2 the smallness in ϵ is lost. (ii) For scattering estimates in other frequency regimes (e.g. the high frequency case) we refer to [27], and also to [17] concerning asymptotically precise estimates for a small circular inhomogeneity and d = 2.
Proof. Let u ϵ (y) = u(2ϵy), then u ϵ is the radiating solution of the problem
Let us start with the case d = 2. By scaling the norm and using the estimate (5.26) of Lemma 5.8 we obtain
) .
(5.
To prove bounds up to the inner boundary S ϵ we first note that
) .
(5.30)
The estimate (5.25) implies the bound
2
with an implicit constant that depends only on R and k 0 . In combination with (5.30) this now leads to the bounds (5.28).
To conclude the proof of Lemma 5.2 we apply the above corollary and the first estimate in (5.28) to the function u is . That way we obtain the desired estimates of Lemma 5.2, but with the additional factor
2
on the right-hand sides of the inequalities. It thus remains to prove that the above quantity is bounded by a constant depending only on k 0 . To this end, the standard trace estimate and a rescaling of the norms give
For the last inequality we used the assumption (3.3).
Let us split the operator T into two parts:
where
and
with Ψ k given by (5.8). Thus, to bound T u on L 2 (B R \ B ϵ ), it suffices to bound T 1 u and T 2 u. We start by deriving some estimates for the fundamental solution, Φ k , in Lemma 5.12 below. These are then used in Lemma 5.13 to obtain bounds for T 1 u. To bound T 2 u we need L 2 (B ϵ )-norm bounds for T 1 u and ∇T 1 u, with explicit dependence on the small parameter ϵ. Parts (ii) and (iv) of Lemma 5.13 serve that purpose, and this is where the estimates on the derivatives of the fundamental solution from part (ii) of Lemma 5.12 will be used. The bound for T 2 u is given in Lemma 5.14. Finally, Lemma 5.1 is a direct consequence of Lemmas 5.13 and 5.14.
Lemma 5.12. Let Φ k be given by (5.6).
(i) Let R, r > 0. With implicit constants depending only on R and r,
(ii) Let R, r > 0. With an absolute implicit constant (i.e. independent of all the involved parameter R, r and k) Consequently, we immediately obtain
This concludes the proof of part (i). Let us turn to gradient bounds. Direct calculation shows that
xy |x -y| 3 , and hence
Let us start by bounding I 2 . Using that |x -y| > 1 2k it is clear that I 2 1 with implicit constant depending only on r. This bound can be improved when k is large. Indeed, to get a better bound in that case, observe that
where the last inequality follows from (5.35). Combining, the two estimates, we have (with an implicit constant depending only on r)
Let us turn to bounding I 1 . Dropping B r from the integration and changing the variables z = yx inside the integral, we get
where in the last step we used that ln 2 |z| has an integrable singularity at z = 0. This bound can be improved when k is small. Dropping B 1 2k (x) from the integral I 1 and using the inequality ln (5.37)
The last inequality is easily established, based on the estimate
Combining the two estimates (5.36) and (5.37), we arrive at
Finally, a combination of the bounds for I 1 and I 2 yields that
For gradient bounds in 2d we use the asymptotic relations
iπt , as t → 0 , and H
(1) 0
along with the bound
we obtain, with the help of (5.35), that
with an implicit constant independent of r and x.
Lemma 5.13. Let T 1 be defined by (5.31), R > 1, ϵ < 1 and M ϵ,k be given by (3.1). Then for any u ∈
where all the implicit constants are independent of u, ϵ and k. The implicit constants in (i) and (iii) depend only on R; and those in (ii) and (iv) are absolute constants.
Proof. We set Ω R = B R \ B ϵ , in particular Ω 2 = B 2 \ B ϵ . Note that, by Hölder's inequality we have
which implies the estimate
.
(5.38)
Using part (i) of Lemma 5.12, we obtain that
where the implicit constant depends only on R. This concludes the proof of part (i).
The proof of (ii) proceeds analogously, with B ϵ in place of Ω R , and the conclusion follows from the estimate
The above direct estimation argument cannot be used to bound the L 2 -norm of ∇T 1 u, as ∇T 1 u is an integral operator whose kernel is not square integrable. However, we can obtain bounds using interpolation. To this end, differentiating inside the integral we have
Clearly,
Using part (ii) of Lemma 5.12, we get
, and noting that ∇ x Φ k (x, y) = -∇ y Φ k (y, x), we similarly get sup
, where the implicit constants depend only on R. Thus we obtain that T g
2 )M ϵ,k , where C is a constant depending only on R. The Marcinkiewicz interpolation theorem [11] now implies that T 1 maps L 2 (Ω 2 ) into L 2 (Ω R ) with the operator norm bound
2 )M ϵ,k , which concludes the proof of part (iii). The proof of part (iv) proceeds analogously, with B ϵ in place of Ω R . Part (ii) of Lemma 5.12 implies the estimate sup 2 ), with an absolute implicit constant. We then conclude that
where C is an absolute constant. Again using the Marcinkiewicz interpolation theorem we obtain
2 ) 1 2 .
Lemma 5.14. Let T 2 be defined by (5.32), k 0 > 0 and R > 1. Suppose 0 < ϵk < k 0 and let M ϵ,k be given by (3.1). Then for any u ∈ L 2 (B 2 \ B ϵ )
where the implicit constant depends only on R and k 0 .
Proof. Let v = T 2 u and f = -T 1 u then, using that T u vanishes on S ϵ , we conclude that v is the outward radiating solution to the problem
As before we introduce the notation f ϵ (x) = f (2ϵx). Using Corollary 5.10 (and the remark following) specifically the first estimate of (5.28) we now get, for d = 3,
)
where in the last step we used the parts (ii) and (iv) of Lemma 5.13. To conclude the proof, it remains to observe that ϵ(1 + √ k) √ ϵ, due to the bound ϵk < k 0 . Similarly, for d = 2 we have
Proof of Lemma A.1. We start by showing the discreteness in the sets R R,h , L R,h,h 0 , which are open, connected and disjoint. Let λ = λ 1 + iλ 2 ∈ C with λ 1 , λ 2 > 0 to be chosen later. Consider the bounded sesquilinear forms on X (cf. (4.5)) given by
In terms of A k and B k , the variational form of the interior transmission eigenvalue problem (4.7) reads:
A k (u, ϕ) + B k (u, ϕ) = 0 for all ϕ ∈ X. Since B k yields a compact operator, the discreteness of these eigenvalues, in the regions where both A k and B k depend analytically on k, will follow from the Analytic Fredholm Theory, as in [10, Section 8.5], once we prove that λ = λ(R, h, h 0 ) can be chosen such that A k becomes coercive [4]. For shorthand let us introduce the notation Let m, M be such that
We consider cases:
• It remains to see that the above supremum is finite. The first term inside the supremum is bounded and establishing the boundedness of the second term amounts to showing that
is bounded. Clearly, in the case (I) this is bounded with a constant depending only on R and k ϵ , and in the case (II) it is bounded with a constant depending on R, k ϵ and h 0 . Finally, the coercivity follows upon applying Poincare's inequality as X ⊂ H 1 0 (B 2 ).
• Let k = a + ib be such that |k| < R and Clearly this supremum is finite as ℜeγ k > h 2 . It remains to prove the discreteness in the set U R,h . This can be done analogously, only now λ in the sesquilinear forms must be chosen to be a real and negative number with very large absolute value. Coercivity then follows by deriving a lower bound on |ℜeA k (u, u)|.
The analogue of the far field pattern in the quasi-static regime.