Cloaking using complementary media for the Helmholtz equation and a three spheres inequality for second order elliptic equations

Cloaking using complementary media was suggested by Lai et al. This was proved in the quasistatic regime by H. M. Nguyen. One of the difficulties in the study of this problem is the appearance of the localized resonance, i.e., the fields blow up in some regions and remain bounded in some others as the loss goes to 0. To this end, the author introduced the technique of removing localized singularity and used a standard three spheres inequality. This method also works for the Helmholtz equation. However, it requires small size of the cloaked region for large frequency due to the use of the (standard) three spheres inequality. In this paper, we give a proof of cloaking using complementary media in the finite frequency regime without imposing any condition on the cloaked region; hence the cloak works for all frequency. To successfully apply the approach of H.M. Nguyen, we establish a new three spheres inequality. A modification of the cloaking setting to obtain illusion optics is also discussed.


Introduction
Negative index materials (NIMs) were investigated theoretically by Veselago in [36]. The existence of such materials was confirmed by Shelby, Smith, and Schultz in [35]. The study of NIMs has attracted a lot attention in the scientific community thanks to their interesting properties and applications. One of the appealing one is cloaking using complementary media.
Cloaking using NIMs or more precisely cloaking using complementary media was suggested by Lai et al in [11]. Their work was inspired from the notion of complementary media suggested by Pendry and Ramakrishna in [32]. Cloaking using complementary media was established in [21] in the quasistatic regime using slightly different schemes from [11]. Two difficulties in the study of cloaking using complementary media are as follows. Firstly, this problem is unstable since the equations describing the phenomenon have sign changing coefficients, hence the ellipticity is lost. Secondly, the localized resonance, i.e., the field blows up in some regions and remains bounded in some others, might appear. To handle these difficulties, in [21] the author introduced the technique of removing localized singularity and used a standard three spheres inequality. The approach in [21] also involved the reflecting technique introduced in [18]. The method in [21] also works for the Helmholtz equation; however since the largest radius in the (standard) three spheres inequality is small as frequency is large (see Section 2 for further discussion), the size of the cloaked region is required to be small for large frequency.
In this paper, we present a proof of cloaking using complementary media in the finite frequency regime. Our goal is not to impose any condition on the size of the cloaked region (Theorem 1); hence the cloak works for all frequency. To successfully apply the approach in [21], we establish a new three spheres inequality for the second order elliptic equations which holds for arbitrary radius (Theorem 2 in Section 2). This inequality is inspired from the unique continuation principle and its proof is in the spirit of Protter in [34]. A modification of the cloaking setting to obtain illusion optics is discussed in in Section 4 (Theorem 3). This involves the idea of superlensing in [19]. Cloaking using complementary media for electromagnetic waves is investigated in [23].
Let us describe the problem more precisely. Assume that the cloaked region is the annulus B γr 2 \ B r 2 for some r 2 > 0 and 1 < γ < 2 in which the medium is characterized by a matrix a and a function σ. The assumption on the cloaked region by all means imposes no restriction since any bounded set is a subset of such a region provided that the radius and the origin are appropriately chosen. The idea suggested by Lai et al. in [11] in two dimensions is to construct a complementary media in B r 2 \ B r 1 for some 0 < r 1 < r 2 .
In this paper, instead of taking the schemes of Lai et al., we use a scheme from [21] which is inspired but different from the ones from [11]. Following [21], the cloak contains two parts. The first one, in B r 2 \ B r 1 , makes use of complementary media to cancel the effect of the cloaked region and the second one, in B r 1 , is to fill the space which "disappears" from the cancellation by the homogeneous media. Concerning the first part, instead of B γr 2 \ B r 2 , we consider B r 3 \ B r 2 with r 3 = 2r 2 (the constant 2 considered here is just a matter of simple representation) as the cloaked region in which the medium is given bŷ The complementary media in B r 2 \ B r 1 is given by where F : B r 2 \B r 1 → B r 3 \B r 2 is the Kelvin transform with respect to ∂B r 2 , i.e., Here where x = T −1 (y) and J(x) = | det DT (x)| for a diffeomorphism T . It follows that Concerning the second part, the medium in B r 1 is given by The reason for this choice will be explained later.
With the loss, the medium is characterized by s δ A, s 0 Σ, where and Physically, the imaginary part of s δ A is the loss of the medium (more precisely the loss of the medium in B r 2 \ B r 1 ). Here and in what follows, we assume that, for some Λ ≥ 1. In what follows, we assume in addition that We can verify that medium s 0 A is of reflecting complementary property, a concept introduced in [18, Definition 1], by considering diffeomorphism G : R d \B r 3 → B r 3 \ {0} which is the Kelvin transform with respect to ∂B r 3 , i.e., It is important to note that x. This is the reason for choosing A in (1.3). Let Ω be a smooth open subset of R d (d = 2, 3) such that B r 3 ⊂⊂ Ω. Given f ∈ L 2 (Ω), let u δ , u ∈ H 1 0 (Ω) be respectively the unique solution to div(s δ A∇u δ ) + s 0 k 2 Σu δ = f in Ω, (1.10) and ∆u + k 2 u = f in Ω. (1.11) As in [18], we assume that equation (1.11) with f = 0, has only zero solution in H 1 0 (Ω). (1.12) Our result on cloaking using complementary media is: and let u and u δ in H 1 0 (Ω) be the unique solution to (1.10) and (1.11) resp. There exists γ 0 > 1, depending only on Λ and the Lipschitz constant ofâ such that if 1 < γ < γ 0 then For an observer outside B r 3 , the medium in B r 3 looks like the homogeneous one by (1.13) (and also (1.11)): one has cloaking. Remark 1. The case k = 0 was established in [21].
The proof of Theorem 1 is given in Section 3. It is based on the removing localized singularity technique introduced in [21] and uses a new three sphere inequality (Theorem 2) discussed in the next section. The discussion on illusion optics is given in Section 4.

Three spheres inequalities
Let v be an holomorphic function defined in B R 3 , Hadamard in [8] proved the following famous three spheres inequality: A three spheres inequality for general elliptic equations was proved by Landis [13] using Carleman type estimates. Landis proved [13, where M is elliptic, symmetric, and of class C 2 , b, c ∈ C 1 , and c ≤ 0, then there is a constant for some α ∈ (0, 1) depending only on R 2 /R 1 , R 2 /R 3 , the ellipticity constant of M , and the regularity constants of M , b, and c. The assumption c ≤ 0 is crucial and this is discussed in the next paragraph. Another proof was obtained by Agmon [1] in which he used the logarithmic convexity. Garofalo and Lin in [6] established similar results where the L ∞ -norm is replaced by the L 2 -norm, and M is of class using the frequency function. A typical example of (2.2) when c > 0 is the Helmholtz equation: where J n is the Bessel function of order n. By taking R 1 , R 2 , and R 3 such that J n (kR 1 ) = 0 = J n (kR 2 ), one reaches the fact that neither (2.4) nor (2.3) is valid. The same conclusion holds in the higher dimensional case by similar arguments. In the case c > 0, (2.4) holds under the smallness of R 3 (see e.g., [2, Theorem 4.1]); this condition is equivalent to the smallness of c for a fixed R 3 by a scaling argument.
In this paper, we establish a new type of three spheres inequalities without imposing the smallness condition on R 3 . This inequality will play an important role in the proof of Theorem 1. Define Here and in what follows, ν denotes the outward normal vector on a sphere.
Our result on three spheres inequalities is: Then, for any λ 0 > 1 with R 2 ∈ (λ 0 R 1 , R 3 /λ 0 ), there exist a constant C and q ≥ 1, depending on the elliptic and the Lipschitz constant of M , C also depends on c 1 , c 2 , R * , R * , d, and λ 0 such In Theorem 2, one does not impose any smallness condition on R 1 , R 2 , R 3 and the exponent α is independent of c 1 and c 2 . The proof of Theorem 2 is inspired from the approach of Protter in [34]. Nevertheless, different test functions are used. The ones in [34] are too concentrated at 0 and not suitable for our purpose. The connection between three spheres inequalities and the unique continuation principle, and the application of three spheres inequalities for the stability of Cauchy problems can be found in [2].
The proof of Theorem 2 is presented in the next two subsections.

Preliminaries
This section contains several lemmas used in the proof of Theorem 2. These lemmas are in the spirit of [34]. Nevertheless, the test functions used here are different from there. Let 0 < R 1 < R 3 < +∞. In this section, we assume that M is a Lipschitz symmetric matrix-valued function defined in B R 3 \ B R 1 and satisfies The first lemma is: for some positive constant C depending only on d.
Proof. An integration by parts gives Using the symmetry of M , we have 2 In what follows, the repeated summation is used.
The second lemma is for some positive constant C depending only on d.

An integration by parts gives
Here We next estimate P and Q. A computation yields This implies Similarly, A combination of (2.16) and (2.17) yields Here we used the fact that p ≥ p Λ,L and |β|R −p 3 ≥ 2. On the other hand, using Cauchy's inequality, we have It follows from (2.18) that the conclusion follows.
Using Lemmas 1 and 2, we can prove the following result.
There exists a positive constant p Λ,L ≥ 1 such that if p ≥ p Λ,L and |β|R −p 3 ≥ 2 then for some positive constant C depending only on d.
Proof. By considering the real part and the imaginary part of v separately, one might assume that v is real. Set z = e βr −p v equivalently v = e −βr −p z.

This implies
Applying Lemmas 1 and 2, we have A combination of (2.20) and (2.21) yields, since p ≥ p Λ,L , The conclusion follows.
We also have There exists a positive constant p Λ,L ≥ 1 such that if p ≥ p Λ,L and |β|R −p 3 ≥ 2 then for some positive constant C depending only on d, Λ, and L.

Combining the inequalities of Lemmas 3 and 4, we obtain
Lemma 5. Let d ≥ 2, β ∈ R, and v ∈ H 2 (B R 3 \ B R 1 ). There exists a positive constant p Λ,L ≥ 1 such that if p ≥ p Λ,L and |β| ≥ 2R −p 3 then for some positive constant C depending only on d, Λ, and L.
The conclusion now follows from Lemmas 3 and 4. The details are left to the reader.

Proof of Theorem 2
Let 1 < λ < λ 0 , (which will be defined later) and set Here and in what follows, [·] denotes the jump across a sphere and ν denotes the unit outward normal vector on a sphere. It follows that and Here and in what follows in this proof, C denotes a positive constant depending only on the elliptic and the Lipschitz constant of M , c 1 , c 2 , λ 0 , R * , R * , and d. Set Let ϕ 1 , ϕ 3 ∈ C 2 c (R d ) be such that Applying Lemma 5, we obtain, for |β| > 2(γR 3 ) p , The proof is now quite standard and divided into two cases.

Cloaking using complementary media. Proof of Theorem 1
This section containing three subsections is devoted to the proof of Theorem 1. In the first subsection, we present two useful lemmas. The proof of Theorem 1 is given in the second subsection.

Preliminaries
In this section, we present two lemmas which will be used in the proof of Theorems 1 and 3. The first lemma is on a change of variables and follows from [18,Lemma 1].
be a complex function, and K : B R 2 \B R 1 → B R 3 \B R 2 be the Kelvin transform with respect to ∂B R 2 , i.e., The second lemma is a stability estimate for solutions of (1.10).

1)
for some positive constant C independent of δ and f . Lemma 7 is a variant of [18, Lemma 1]. The case k = 0 and its variant in the case k > 0 were considered in [21] and [19] respectively. The proof is similar to the one of [18,Lemma 1]. For the convenience of the reader, we present the proof.

Proof of Theorem 1
We use the approach in [21] with some modifications from [19] so that the same proof also give the result on illusion optics (Theorem 3 in Section 4). However, instead of applying the standard three sphere inequality as in [21], we use Theorem 2.
Hence U = u. Since the limit is unique, we have the convergence for the family (U δ ) as δ → 0.
Case 2: d = 3. Definê j n (t) = 1 · 3 · · · (2n + 1)j n (t) andŷ n = − y n (t) 1 · 3 · · · (2n − 1) , where j n and y n are the spherical Bessel and Neumann functions of order n. Then, as n large enough, (see, e.g., [5, (2.37 Thus one can represent U 2,δ of the form for a n m , b n m ∈ C andx = x/|x|. The proof now follows similarly as in the case d = 2. The details are left to the reader.

Remark 2.
In the proof, we use essentially the fact (A, Σ) = (I, 1) in B r 3 \B γr 2 to use separation of variables in this region. In fact, this condition is not necessary by using the technique of separation of variables for a general structure in [20].
Remark 3. The construction of the cloak given by (1.4) is not restricted to the Kelvin transforms F (and G). In fact, one can extend this construction to a general class of reflections considered in [18].
Remark 4. The condition (F * A, F * Σ) = (A, Σ) in B r 3 \ B r 2 is necessary to ensure that cloaking can be achieved and the localized resonance might take place see [24] (see also [4] for related results).
Remark 5. Cloaking can also be achieved via schemes generated by changes of variables [7,14,31]. Resonance might also appear in this context but for specific frequencies see [9,16]. It is shown in [16] that in the resonance case cloaking might not be achieved and the field inside the cloaked region can depend on the field outside. Cloaking can also be achieved in the time regime via change of variables [26,27].

Illusion optics using complementary media
We next discuss briefly how to obtain illusion optics in the spirit of Lai et al. in [12]. The scheme used here is a combination of the ones used for cloaking and superlensing in [21,19] and is slightly different from [12]. More precisely, set m = r 2 3 /r 2 2 .
Proof. The proof is similar to the one of Theorem 1. Note that in the proof of Theorem 1, we do not use the information of the medium inside B r 2 /m . The details are left to the reader.