On Quasi-static Cloaking Due to Anomalous Localized Resonance in \mathbb{R}^{3}

# On Quasi-static Cloaking Due to Anomalous Localized Resonance in R3

Hongjie Li School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, P. R. China.    Jingzhi Li Department of Mathematics, South University of Science and Technology of China, Shenzhen 518055, P. R. China. Email: li.jz@sustc.edu.cn    Hongyu Liu Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR, and HKBU Institute of Research and Continuing Education, Virtual University Park, Shenzhen, P. R. China. Email: hongyu.liuip@gmail.com
###### Abstract

This work concerns the cloaking due to anomalous localized resonance (CALR) in the quasi-static regime. We extend the related two-dimensional studies in [2, 11] to the three-dimensional setting. CALR is shown not to take place for the plasmonic configuration considered in [2, 11] in the three-dimensional case. We give two different constructions which ensure the occurrence of CALR. There may be no core or an arbitrary shape core for the cloaking device. If there is a core, then the dielectric distribution inside it could be arbitrary.

Key words. anomalous localized resonance, plasmonic material, invisibility cloaking

AMS subject classifications. 35R30, 35B30

## 1 Introduction

This work concerns the invisibility cloaking due to anomalous localized resonance (CALR) in the quasi-static regime, which has gained growing interest in the literature; see [1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13] and the references therein. Let and be bounded domains in , , such that . , and signify, respectively, the core, shell and matrix of a cloaking device, which hosts a dielectric object as follows

 ϵ(x)=⎧⎪⎨⎪⎩ϵc(x),x∈Σ,ϵs(x),x∈Ω∖¯¯¯¯Σ,ϵm(x),x∈Rd∖¯¯¯¯Ω. \hb@xt@.01(1.1)

In most of the existing studies, one takes and . is negatively valued, which denotes the plasmonic material parameter. Let denote the loss parameter and consider a material distribution given as

 ϵη(x)=ϵ(x)+iηχD(x),x∈Rd, \hb@xt@.01(1.2)

where is given in (LABEL:eq:struc), and denotes the characteristic function of the domain , with or . For time-harmonic wave propagation in the quasi-static regime, the wave pressure satisfies the following equation

 {∇x⋅(ϵη(x)(∇xuη(x)))=f(x),x∈Rd,uη(x)=o(1),d=2;  O(|x|−1),d=3,|x|→+∞, \hb@xt@.01(1.3)

where denotes a source term that is compactly supported in and satisfies

 ∫Rdf(x) dx=0. \hb@xt@.01(1.4)

Define

 Eη=Eη(uη,ϵη,f):=∫Dη2|∇xuη(x)|2 dx, \hb@xt@.01(1.5)

where is the solution to (LABEL:eq:m1). denotes the rate at which the energy of the wave field is dissipated into heat. Then anomalous localized resonance (ALR) is said to occur if there holds

 Eη→+∞as  η→+0. \hb@xt@.01(1.6)

In what follows, we sometimes simply refer to ALR as resonance. If in addition to (LABEL:eq:cond1), one further has that

 |uη(x)/√Eη|→0as  η→+0when  |x|>a, \hb@xt@.01(1.7)

where is such that the central ball contains , then it is said that CALR occurs. Here and also in what follows, with denotes a central ball of radius in , . By (LABEL:eq:cond2), it is readily seen that if CALR occurs, then both the source and the cloaking device are invisible to the wave observation made from the outside of . If (LABEL:eq:cond1) is replaced by

 limsupη→+0Eη=+∞, \hb@xt@.01(1.8)

then it is said that weak CALR occurs. We refer to [2], [4] and [11] for more discussions on the anomalous localized resonance and its connection to invisibility cloaking.

The anomalous localized resonance phenomenon was first observed in [13] and connected to invisibility cloaking in [12]. Recently, a mathematical theory was developed in [2] by Ammari et al to rigorously explain the CALR observed in [13] and [12]. In their study, and is given by (LABEL:eq:d1) with

 ϵm≡1,ϵc≡1,ϵs≡−1andD=Ω∖¯¯¯¯Σ. \hb@xt@.01(1.9)

Moreover, they let and , with . For the above plasmonic configuration, the solution to (LABEL:eq:m1) in [2] was shown to have a spectral representation associated with a Neumann-Poincaré-type operator. Using the spectral representation, it is shown that there exists a critical radius

 r∗=√r3e/ri, \hb@xt@.01(1.10)

such that when a generic source term lies within , then CALR occurs; and when lies outside , then CALR does not occur. Here and also in what follows, when we say that CALR does not occur, it actually means that weak CALR does not occur. Later on, the CALR was considered from a variational perspective in [11] by taking and in (LABEL:eq:d1) with

 ϵm≡1,ϵc≡1,ϵs≡−1andD=R2,Ω=Bre. \hb@xt@.01(1.11)

By using the primal and dual variational principles, it is shown in [11] that for a large class of sources, if , then resonance always occurs; whereas if and , then there exists the critical radius in (LABEL:eq:critical) for the occurrence and nonoccurrence of resonance.

The aim of this work is to extend the related results in [2, 11] to the three dimensional setting. Indeed, the three-dimensional CALR was considered in the literature and the situation becomes much more complicated. In [3], it is shown that if one takes a similar structure as that in (LABEL:eq:ammari) but with , then CALR does not occur. The same conclusion was draw in [10] without the quasi-static approximation. In [1], the anomalous localized resonance is shown to take place by using a folded geometry where the plasmonic material is spatially variable. In this paper, we show that CALR does not occur for the configuration (LABEL:eq:kohn) in . Then, we show that by properly choosing the plasmonic parameters, CALR can still happen, at least approximately. We follow both the spectral and variational arguments developed, respectively, in [2] and [11].

The rest of the paper is organized as follows. In Sections 2, by using the variational argument, we show the nonoccurrence and occurrence of ALR by taking the loss parameter to be given over the whole space . In Section 3, by following the relevant study in [2], we consider the occurrence and nonoccurrence of CALR by taking the loss parameter to be given only in the plasmonic layer.

## 2 Variational perspective on ALR in three dimensions

Henceforth, we let and be a constant. Throughout the present section, we let the source term be a real-valued distributional functional such that it is supported at a distance from the origin, and has a zero mean :

 f=FH2⌊∂Bq,F:∂Bq→R,  F∈L2(∂Bq)  and  ∫∂BqFdH2=0, \hb@xt@.01(2.1)

where denotes the two-dimensional Hausdorff measure restricted to the set . Moreover, in this section, we let

 (Ω;ϵη) be given by (LABEL:eq:d1) with Ω=Bre and D=R3, \hb@xt@.01(2.2)

and without loss of generality, it is assumed that and . Indeed, the subsequent results derived in this section can be easily extended to the general case by a direct scaling argument. For the solution to (LABEL:eq:m1) with given in (LABEL:eq:d2), we set

 uη=vη+i1ηwηwithvη,wη:R3→R. \hb@xt@.01(2.3)

It is straightforward to verify that

 ∇⋅(ϵ∇vη)−△wη=f  in\ \ R3, \hb@xt@.01(2.4) ∇⋅(ϵ∇wη)+η2△vη=0in\ \ R3. \hb@xt@.01(2.5)

Accordingly, the energy can be represented with and as

 Eη(uη)=η2∫R3|∇uη|2=η2∫R3|∇vη|2+12η∫R3|∇wη|2. \hb@xt@.01(2.6)

The following variational principles were proved in [11] when , and can be extended to the three-dimensional case for the present study by straightforward modifications. Define

 Iη(v,w):=η2∫R3|∇v|2+12η∫R3|∇w|2,v,w∈H1loc(R3). \hb@xt@.01(2.7)

Consider the optimization problem

 min(v,w)∈H1loc(R3)×H1loc(R3)Iη(v,w)subject to ∇⋅(ϵ∇v)−Δw=f, \hb@xt@.01(2.8)

where , are assumed to be real-valued. (LABEL:eq:pv) is referred to as the primal variational problem, and the minimizing pair is attainable at such that is a solution to (LABEL:eq:m1). Similarly, we define

 Jη(v,ψ):=∫R3f⋅ψ−η2∫R3|∇v|2−η2∫R3|∇ψ|2,v,w∈H1loc(R3), \hb@xt@.01(2.9)

and consider the following optimization problem

 max(v,ψ)∈H1loc(R3)×H1loc(R3)Jη(v,ψ)subject to ∇⋅(ϵ∇ψ)+ηΔv=0, \hb@xt@.01(2.10)

where are assumed to be real-valued. (LABEL:eq:dv) is referred to as the dual variational problem, and the maximizing pair is attainable at such that is a solution to (LABEL:eq:m1).

We shall make use the variational principles introduced above to prove the resonance and non-resonance results. In doing so, the spherical harmonic functions for , and will be needed and they form an orthonormal basis to ; see [9]. In the rest of the current section, for ease of notations, we write instead of to signify the spherical harmonic functions of order . Set

 fqk(x):=YkH2⌊∂Bq,x∈∂Bq.

Hence, the source in (LABEL:eq:source) can be written as

 f(x)=∞∑k=1αkfqk(x),αk=∫S2f(q^x)⋅¯¯¯¯¯¯¯Yk(^x) ds(^x),x∈∂Bq, \hb@xt@.01(2.11)

where and also in what follows, for and , denotes the spherical coordinates. Moreover, in the subsequent arguments, we let and denote two generic positive constants that may change from one inequality to another, but should be clear from the context.

The following proposition will be needed and can be proved by direct verifications.

###### Proposition 2.1

Consider the PDE for : ,

 ∇x⋅(A(x)∇xψ(x))=0,ψ(x)=O(|x|−1)  as  |x|→∞. \hb@xt@.01(2.12)

where

 A(x)={−1−1k,  r

with any . Then there exists a non-trivial solution which achieves its maximum value at a point with , given by

Moreover, one has that

 ∫R3∣∣∇^ψk∣∣2=−∫|x|=re¯¯¯¯¯¯^ψk⋅[∂^ψk∂r]=C(k+2)re2k+1, \hb@xt@.01(2.14)

where denotes the jump of the normal flux of the function across .

### 2.1 Non-resonance result

In this subsection, we consider the non-resonance for the standard plasmonic configuration in (LABEL:eq:d2) with , and , . This is exactly the one considered in [11] for the two-dimensional case. We have

###### Theorem 2.1

Let be given in (LABEL:eq:d2) with , and , . Let be given in (LABEL:eq:source2). Then ALR does not occur.

Proof. We make use of the primal variational principle (LABEL:eq:pv) to prove the theorem. To that end, we first construct the test functions and that satisfy the PDE constraint in (LABEL:eq:pv) for . For , we set

 ^vk(x)=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩  rkYk(^x),|x|≤1,(12k+1rk+2k2k+1r−k−1)Yk(^x),1<|x|≤re,(4k+4k2+r2k+1er2k+1e(2k+1)2rk+2k(−1+r2k+1e)(2k+1)2r−k−1)Yk(^x),re<|x|≤q,q2k+1(1+4k(k+1)r−1−2ke)+2k(−1+r2k+1e)(2k+1)2r−k−1Yk(^x),q<|x|. \hb@xt@.01(2.15)

It is straightforward to verify that is continuous over , and satisfies

 ∇x⋅(ϵ(x)∇x^vk(x))=0for  x∈R3∖∂Bq.

However, across , has a jump in its normal flux as follows,

 [ν⋅∇^vk]|∂Bq=−(4k(k+1)+r2k+1e)qk−1r2k+1e(2k+1)Yk, \hb@xt@.01(2.16)

where denotes the exterior unit normal vector to . Therefore, if we let

 λk=−αk⋅(2k+1)⋅q⋅re4k(k+1)+r2k+1e⋅q−k⋅r2ke, \hb@xt@.01(2.17)

 ∇⋅(ϵ∇(λk^vk))=αkfqkinR3. \hb@xt@.01(2.18)

Hence, by setting

 vη=∞∑k=1λk^vk, \hb@xt@.01(2.19)

with and , respectively, given in (LABEL:eq:decomp2) and (LABEL:eq:decomp1), then by (LABEL:eq:decomp4), one sees that there holds

 ∇⋅(ϵ∇vη)=f. \hb@xt@.01(2.20)

Finally, we let , then by virtue of (LABEL:eq:decomp5), clearly satisfies the PDE constraint in (LABEL:eq:pv). Hence, by the primal variational principle, together with straightforward calculations (though a bit tedious), one has that

 Eη(uη)≤Iη(vη,wη)=η2∫R3|∇vη|2=η2∞∑k=1|λk|2∫R3|∇^vk|2≤Cη∞∑k=1|αk|2k2q2k⋅q2kk2≤Cη∥F∥2L2(∂Bq).

That is, the resonance does not occur.

The proof is complete.

### 2.2 ALR with no core

Theorem LABEL:thm:nr1 indicates that the standard plasmonic structure does not induce the ALR. In order for ALR to take place, one has to devise different plasmonic structures. Next, we first consider a construction without a core, which ensures that ALR can always occur.

###### Theorem 2.2

Consider the configuration described in (LABEL:eq:d2) with . Let be given in (LABEL:eq:source2) and assume that for some . Set . Then ALR occurs.

Proof. Since , we first assume that . Then we choose

 vη=0andψη=ληR¯¯¯¯¯¯¯¯^ψk0(x) \hb@xt@.01(2.21)

where satisfies and will be further chosen below. Clearly, by Proposition LABEL:prop:1, the pair satisfies the constraint . By the dual variational principle, we have

where the two positive constants and depend only on and . Choosing with , we obtain for .

Next, if , then by choosing

 vη=0andψη=ληI¯¯¯¯¯¯¯¯^ψk0(x)

and using a similar argument as the previous case, one can show that resonance occurs.

The proof is complete.

### 2.3 Approximate ALR with an arbitrary shape core

In this subsection, we assume that is a simply connected domain with a -smooth boundary. Let us consider a configuration given by (LABEL:eq:d2). We assume that , , is a symmetric-positive-definite matrix-valued function satisfying

 τ0I3×3≤ϵc(x)≤τ−10I3×3,x∈Σ, \hb@xt@.01(2.22)

where and is the identity matrix. Fix and let . Let be the smallest integer satisfying

 r−k(η)e<η≤r−k(η)+1e, \hb@xt@.01(2.23)

and let , be given by

 ϵs=−1−k(η)−1,k(η)∈N. \hb@xt@.01(2.24)

Consider the source given in (LABEL:eq:source2). We shall prove that

###### Theorem 2.3

Consider the configuration described in (LABEL:eq:d2) with given in (LABEL:eq:assump1) and in (LABEL:eq:assump2). Let the source be given by (LABEL:eq:source2). Let and assume that the source satisfies

 k−1|αk|2(r3eq2)k→+∞as  k→+∞, \hb@xt@.01(2.25)

Then for any , there exists a sufficiently small such that one has . That is, approximate ALR occurs.

###### Remark 2.4

Noting that and , we see that the condition (LABEL:eq:ca8) indicates that as long as the Fourier coefficient of the source in (LABEL:eq:source2) does not decay very quickly as , then approximate ALR occurs.

Proof. [Proof of Theorem LABEL:thm:r2] We make use of the dual variational principle to construct a sequence satisfying

 ∇⋅(ϵ∇ψη)+ηΔvη=0 withJη(vη,ψη)→+∞.

First, we set

 ψη(x):=ληR¯¯¯¯¯¯¯¯¯¯¯^ψk(η)(x),x∈R3, \hb@xt@.01(2.26)

where is to be chosen below. Let be the solution to By the standard elliptic estimate, one has

 η∥∇vη∥2L2(R3)≤Cη−1∥∇⋅(ϵ∇ψη)∥2H−1(R3)≤˜Cη−1λ2ηk(η), \hb@xt@.01(2.27)

where and are two positive constants depending on and in (LABEL:eq:assump1). Next, by straightforward calculations, we have

 Eη(uη)≥Jη(vη,ψη)=∫∂Bqf⋅ψη−η2∫R3|∇ψη|2−η2∫R3|∇vη|2≥Rλη∫∂Bqf⋅¯¯¯¯¯¯¯¯¯¯¯^ψk(η)−η2∫R3|∇^ψk(η)|2−η2∫R3|∇vη|2≥R∫∂Bqαk(η)Yk(η)⋅ληq−k(η)r2k(η)+1e¯¯¯¯¯¯¯¯¯¯¯¯Yk(η)−Cηλ2η(2k(η)+1)r2k(η)+1e−Cη−1λ2ηk(η)≥˜CRαk(η)ληq−k(η)−1r2k(η)+1e−Cηλ2η(2k(η)+1)r2k(η)+1e−Cη−1λ2ηk(η)≥ληrk(η)e[˜CRαk(η)(req)k(η)+1−Cλη(2k(η)+1)ηrk(η)+1e−Cληηrk(η)ek(η)]. \hb@xt@.01(2.28)

By (LABEL:eq:assump2), we see that and , and hence the last two terms in the last inequality in (LABEL:eq:ca3) are of comparable order. Therefore, one further has from (LABEL:eq:ca3) that

 Eη(uη)≥ληrkηe[˜CRαk(η)(req)k(η)+1−Cληk(η)]. \hb@xt@.01(2.29)

Using a completely similar argument by taking

 ψη(x):=ληI¯¯¯¯¯¯¯¯¯¯¯^ψk(η)(x),x∈R3, \hb@xt@.01(2.30)

one can show that

 Eη(uη)≥ληrkηe[˜CIαk(η)(req)k(η)+1−Cληk(η)]. \hb@xt@.01(2.31)

We choose to be

 λη=˜C2Ck(η)Rαk(η)(req)k(η)+1, \hb@xt@.01(2.32)

where can be replaced by . Then one has

 Eη(uη)≥ληrk(η)e[12˜CRαk(req)k(η)+1]=14Ck(η)(˜CRαk(η))2(req)2(r3eq2)k(η). \hb@xt@.01(2.33)

Clearly, the estimate (LABEL:eq:ca7) also holds with replaced by . The proof can be immediately concluded by noting (LABEL:eq:ca8) and (LABEL:eq:ca7).

### 2.4 Sensitivity and critical radius

By Theorem LABEL:thm:r2, we see that for any given , one can determine a sufficiently small and a sufficiently large according to (LABEL:eq:assump2) and (LABEL:eq:ca7), such that the configuration with given in (LABEL:eq:assump3) is “almost” resonant in the sense that . By (LABEL:eq:ca8) and (LABEL:eq:assump2), one has that as , and . Clearly, both and depend on , and hence of the plasmonic configuration depends on as well. It is natural to ask what would happen if one fixes the integer in (LABEL:eq:assump3). That is, in (LABEL:eq:assump3) is replaced by an integer , which can be as large as possible, but fixed. Next we show that resonance does not occur in such a case, and this indicates that the resonance is very sensitive to the plasmonic parameter.

###### Theorem 2.5

Let be any fixed positive integer and let . Let be given in (LABEL:eq:d2) with and . Suppose that the source is given in (LABEL:eq:source2). Then ALR does not occur.

Before giving the proof of Theorem LABEL:thm:sensitivity, we present another theorem whose proof would be more general than the one needed for Theorem LABEL:thm:sensitivity.

###### Theorem 2.6

Let be given in (LABEL:eq:d2) with and . Suppose that is given in (LABEL:eq:source2) and . Let be chosen according to (LABEL:eq:assump2). Then if with given in (LABEL:eq:critical), then ALR does not occur.

By Theorem LABEL:thm:r2, we see that ALR occurs if is chosen according to (LABEL:eq:assump3), namely, it is variable depending on the asymptotic parameter , and the source is located within the critical radius and satisfies the generic condition (LABEL:eq:ca8). However, by Theorem LABEL:thm:sensitivity, it is pointed out that the resonance phenomenon is very sensitive with respect to the plasmonic parameter , and if it is independent of the asymptotic parameter , then resonance does not occur. Theorem LABEL:thm:crc further shows that for the case with the variable plasmon parameter in Theorem LABEL:thm:r2, the resonance phenomenon is localized.

Proof. [Proof of Theorem LABEL:thm:crc] We make use of the primal variational principle to prove the theorem. To that end, we first construct test functions and that satisfy the PDE constraint in (LABEL:eq:pv).

Let be such that . Let be of the following form

 vη=∑k≠k(η)vη,k+vη,k(η), \hb@xt@.01(2.34)

where , , satisfies

 ∇⋅(ϵ∇vη,k)=αkfqk, \hb@xt@.01(2.35)

and satisfies

 ∇⋅(ϵ∇vη,k(η))=αk(η)fqk(η)on\ \ ∂Bq. \hb@xt@.01(2.36)

Define to be

 ^vk(x):=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩  rkYk(^x), |x|≤1,(1+k+k(η)(1+2k)(1+k(η))rk+k+2kk(η)(1+2k)(1+k(η))r−k−1)Yk(^x), 1<|x|≤re,(r−2k−1e((−k(k+1)+k(η)2+k(η))r2k+1e+k(k+1)(2k(η)+1)2)(2k+1)2k(η)(k(η)+1)rk+k(2k(η)+1)(k+k(η)+1)(r2k+1e−1)(2k+1)2k(η)(k(η)+1)r−k−1)Yk(^x), re<|x|≤q,(r−2k−1e[−(k+k(η)+1)r2k+1e((k−k(η))q2k+1+2kk(η)+k)+k(k+1)(2k(η)+1)2q2k+1(2k+1)2k(η)(k(η)+1)+k(2k(η)+1)(k+k(η)+1)r4k+2e](2k+1)2k(η)(k(η)+1))r−k−1Yk(^x), |x|>q. \hb@xt@.01(2.37)

Using in (LABEL:eq:crc4), we then set

 vη,k=λk^vk,k≠k(η), \hb@xt@.01(2.38)

where

 \hb@xt@.01(2.39)

By straightforward calculations, though a bit tedious, one can verify that defined in (LABEL:eq:crc6) satisfies (LABEL:eq:crc2). Next, we define

 ˜Vk(η)(x)={rk(η)Yk(η)(^x),|x|≤q,r−k(η)−1q2k(η)+1Yk(η)(^x),|x|>q,

and set

 vη,k(η)=λk(η)˜Vk(η),λk(η)=−αk(η)q1−k(η)2k(η)+1. \hb@xt@.01(2.40)

It is directly verified that satisfies (LABEL:eq:crc3). Finally, we set

 −Δwη=−∇⋅(ϵ∇vη)+f=−∇⋅(ϵ∇vη,k(η))+αk(η)fqk(η)=−λk(η)[ν⋅ϵ∇˜Vk(η)]∣∣∂B1H2⌊∂B1−λk(η)[ν⋅ϵ∇˜Vk(η)]∣∣∂BreH2⌊∂Bre \hb@xt@.01(2.41)

Clearly, and satisfy the PDE constraint in (LABEL:eq:pv), and hence by the primal variational principle,

 Eη(uη)≤Iη(Rvη,Rwη)≤Iη(vη,wη), \hb@xt@.01(2.42)

where is defined in (LABEL:eq:v1).

We proceed to calculate the energy in (LABEL:eq:crc9) and show that it is bounded as , which readily implies that ALR does not occur. First, by (LABEL:eq:crc6), one can verify that

 \hb@xt@.01(2.43)

Hence we have the following estimate

 η∫R3∣∣∇vη,k∣∣2=η|λk|2∫R3|∇^vk|2≤Cη|αk|2(k(η))121q2kk2kq2k=Cη|αk|2(k(η))12k. \hb@xt@.01(2.44)

Since , we have , which together with (LABEL:eq:crc10) implies that

 η∑k≠k(η)∫R3∣∣∇vη,k∣∣2≤C∑k≠k(η)|αk|2≤C∥F∥2L2(∂Bq). \hb@xt@.01(2.45)

By (LABEL:eq:crc7), one can also calculate that

 η∫R3∣∣∇vη,k(η)∣∣2≤Cη∣∣λk(η)∣∣2∫R3∣∣∇˜Vk(η)∣∣2≤Cη∣∣αk(η)∣∣2 \hb@xt@.01(2.46)

Next we estimate the energy due to , and by (LABEL:eq:crc8) one has

 1η∫R3|∇wη|2≤C1η∥∇⋅(ϵ∇vη,k(η))−αk(η)fqk(η)∥2H−1≤C1η|λk(η)|2r