The point-interaction approximation for the fields generated by contrasted bubbles at arbitrary fixed frequencies

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The point-interaction approximation for the fields generated by contrasted bubbles at arbitrary fixed frequencies Habib Ammari a , Durga Prasad Challa b,1 , Anupam Pal Choudhury c,2 , Mourad Sini c,∗,3 a Department of Mathematics, ETH Zürich, Rämistrasse 101, CH-8092 Zürich, Switzerland b Faculty of Mathematics, Indian Institute of Technology Tirupati, Tirupati, India c RICAM, Austrian Academy of Sciences, Altenbergerstrasse 69, A-4040, Linz, Austria

Received 8 November 2018; revised 4 February 2019

Abstract We deal with the linearized model of the acoustic wave propagation generated by small bubbles in the harmonic regime. We estimate the waves generated by a cluster of M small bubbles, distributed in a bounded domain , with relative densities having contrasts of the order a β , β > 0, where a models their relative maximum diameter, a  1. We provide useful and natural conditions on the number M, the minimum distance and the contrasts parameter β of the small bubbles under which the point interaction approximation (called also the Foldy-Lax approximation) is valid. With the regimes allowed by our conditions, we can deal with a general class of such materials. Applications of these expansions in material sciences and imaging are immediate. For instance, they are enough to derive and justify the effective media of the cluster of the bubbles for a class of gases with densities having contrasts of the order a β , β ∈ ( 32 , 2) and in this case we can handle any fixed frequency. In the particular and important case β = 2, we can handle any fixed frequency far or close (but distinct) from the corresponding

* Corresponding author.

E-mail addresses: [email protected] (H. Ammari), [email protected] (D.P. Challa), [email protected] (A.P. Choudhury), [email protected] (M. Sini). 1 This author was partially supported by the Austrian Science Fund (FWF): P28971-N32 and DST SERB MATRICS (Mathematical Research Impact Centric Support) MTR/2017/000539. 2 This author is supported by the Austrian Science Fund (FWF): P28971-N32. 3 This author is partially supported by the Austrian Science Fund (FWF): P28971-N32. https://doi.org/10.1016/j.jde.2019.03.010 0022-0396/© 2019 Elsevier Inc. All rights reserved.

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Minnaert resonance. The cluster of the bubbles can be distributed to generate volumetric metamaterials but also low dimensional ones as metascreens and metawires. © 2019 Elsevier Inc. All rights reserved. MSC: 35R30; 35C20 Keywords: Bubbly media; Foldy-Lax approximation; Effective medium theory

1. Introduction Diffusion by highly contrasted small particles is of fundamental importance in several branches of applied sciences as material sciences and imaging. We are interested in the models where these small particles have sizes at the micro scales as in the models related to gas bubbles. To describe properly the mathematical model we are dealing with in this work, let us 3 denote by {Ds }M s=1 a finite collection of small particles in R of the form Ds := δBs + zs , where Bs are open, bounded (with Lipschitz boundaries), simply connected sets in R3 containing the origin, and zs specify the locations of the particle. The parameter δ > 0 characterizes the smallness assumption on the particles. We shall further assume that the Lipschitz constants of the open sets Bs are uniformly bounded. Let us consider piecewise constant densities of the form 

ρ0 , x ∈ R3 ∪M l=1 Dl ,

ρδ (x) =

ρs , x ∈ Ds , s = 1, ..., M,

(1.1)

and piecewise constant bulk modulus in the analogous form  kδ (x) =

k0 , x ∈ R3 ∪M l=1 Dl , ks , x ∈ Ds , s = 1, ..., M,

(1.2)

where ρ0 , ρs , k0 , ks are positive constants. Thus ρ0 and k0 denote the density and bulk modulus of the background medium and ρs and ks denote the density and bulk modulus of the bubbles respectively. We are interested in the following problem describing the acoustic scattering by the collection of small bubbles Ds , s = 1, ..., M: ⎧ ⎪ ∇.( ρ10 ∇u) + ω2 k10 u = 0 in R3 ∪M ⎪ l=1 Dl , ⎪ ⎪ ⎪ ⎨∇.( 1 ∇u) + ω2 1 u = 0 in Ds , s = 1, ..., M, ρs

ks

⎪ ⎪ u|− − u|+ = 0, on ∂Ds , s = 1, ..., M, ⎪ ⎪ ⎪ 1 ∂u   ⎩  − 1 ∂us  = 0 on ∂Ds , s = 1, ..., M, s ρs ∂ν

−

ρ0 ∂ν

(1.3)

+

where ω > 0 is a given frequency and ν s denotes the external unit normal to ∂Ds . Here the total field u := uI + us , where uI denotes the incident field (we restrict to plane incident waves) and us denotes the scattered waves. The above set of equations has to be supplemented by the Sommerfeld radiation condition on us which we shall henceforth refer to as (S.R.C.).

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Keeping in mind the positivity of the bulk modulus, the above problem can be equivalently formulated as ⎧ ⎪ u + κ02 u = 0 in R3 ∪M ⎪ l=1 Dl , ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ u + κs u = 0 in Ds , s = 1, ..., M, ⎨ (1.4) u|− − u|+ = 0, on ∂Ds , s = 1, ..., M, ⎪   ⎪ ⎪ 1 ∂u  1 ∂u  ⎪ ⎪ ρs ∂ν s − − ρ0 ∂ν s + = 0 on ∂Ds , s = 1, ..., M, ⎪ ⎪ ⎪ s ⎩ ∂u 1 s ∂|x| − iκ0 u = o( |x| ), |x| → ∞ (S.R.C.), where κ02 = ω2 ρk00 and κs2 = ω2 ρkss . As in (1.3), the total field u := uI + us , with uI denoting the acoustic incident field and us denoting the acoustic scattered field. The scattered field us can be expanded as us (x, θ ) =

eiκ0 |x| ∞ ˆ θ ) + O(|x|−2 ), |x| → ∞, u (x, |x|

x where xˆ := |x| and u∞ (x, ˆ θ ) denotes the far-field pattern corresponding to the unit vectors x, ˆ θ, i.e. the incident and propagation directions respectively.

Definition 1.1. To describe the collection of small bubbles, we use the following parameters:  1. a := max diam(Dm ) = δ max diam(Bm ) , 2. d :=

1≤m≤M

min

m =j 1≤m,j ≤M

1≤m≤M

dmj , where dmj := dist (Dm , Dj ),

3. ωmax as the upper bound of the used wave numbers, i.e. ω ∈ (0, ωmax ], 4. there exist constants ζm ∈ (0, 1] such that Bζ3m a (zm ) ⊂ Dm ⊂ B 3a (zm ), 2

2

where Br3 denotes a ball of radius r and ζm are assumed to be uniformly bounded below by a positive constant. The distribution of the small bubbles is modeled as follows: 5. the number M := M(a) := O(a −s ) ≤ Mmax a −s with a given positive constant Mmax , 6. the minimum distance d := d(a) ≈ a t , i.e. dmin a t ≤ d(a) ≤ dmax a t , with given positive constants dmin and dmax , 7. the coefficients km , ρm satisfy the conditions: ρm ρm = Cρ a β , β > 0, ( i.e.  1), (1.5) ρ0 ρ0 keeping the relative speed of propagation uniformly bounded, i.e. 2 κm ρ m k0 ρ m k0 := = ∼ 1, as a  1. 2 km ρ 0 ρ 0 km κ0

Here the real numbers s, t and β are assumed to be non negative.

(1.6)

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We call the upper bounds of the Lipschitz character of Bm ’s, Mmax , dmin , dmax and ωmax the set of the a priori bounds. The scattering problem described above models the acoustic wave diffracted in the presence of small bubbles. In this case, the parameter β fixes the kind of medium we are considering, see [13,



(s−s  ) 1  14,11,24]. To state our results, let us first denote Aˆ l := |∂D  · νs  dσl (s ) dσl (s) l | ∂Dl ∂Dl |s−s | and define 2 ωM :=

8π kl (ρl − ρ0 )Aˆ l

.



2 1 Note that Aˆ l is negative, as Aˆ l = − |∂D dy dσl (s) by the divergence theorem, and l | ∂Dl Dl |s−y| 2 since ρl satisfy (1.5) and a  1, it follows that ωM is positive. In the case β = 2, to simplify the 2 is the same for all l = 1, . . . , M. For exposition of the results, we assume that the constant ωM example, this can hold if all the bubbles are identical in shape, and have the same density and bulk modulus. Let κ (x, y) :=

eiκ|x−y| , for x, y ∈ R3 , 4π|x − y|

denote the fundamental solution of the Helmholtz equation in three dimensions with a fixed wave ˆ , where xˆ = x . number κ. To k , we correspond its farfield pattern ∞ ˆ y) := e−iκ x·y κ (x, |x| The main results of this work are stated in the following theorem. Theorem 1.2. Under the conditions 0 ≤ t < 12 ,4 0 ≤ s ≤ 32 , β = 1 + γ , with 0 ≤ γ ≤ 1 and s + γ ≤ 2 we have the following expansions. 1. Assume that γ < 1 or γ = 1 with ω being away from ωM , i.e. |1 − constant l0 independent of a, a  1. Then u∞ (x, ˆ θ) =

M 

2 ωM | ≥ l0 ω2

with a positive

∞ ˆ zm )Qm + O(a 2−s + a 3−γ −s−2t ) κ0 (x,

(1.7)

m=1

under the additional condition on t : t ≥ 3s . 2. Assume that γ = 1 and the frequency ω is near ωM , i.e. 1 − Then u∞ (x, ˆ θ) =

M 

2 ωM ω2

= lM a h1 , h1 ∈ (0, 1), lM = 0.

∞ ˆ zm )Qm + O(a 2−s−2h1 + a 3−2t−2s−2h1 ) κ0 (x,

(1.8)

m=1

4 The condition that t < 1 , which is general enough for the applications described later, is imposed to keep more gen2 eralities regarding the other parameters describing the composite (i.e. s, β and h1 ), and bypass some technicalities in the calculations. For example, we required this condition to derive the apriori estimates of the densities in Proposition 3.14.

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under the additional conditions on t and h1 given by • t ≥ 3s and s + h1 ≤ 1, if lM < 0. • t ≥ 3s , t + h1 ≤ 1, s + h1 < min{ 32 − t, 2 − h1 }, if lM > 0. The vector (Qm )M m=1 is the solution of the following algebraic system Cm −1 Qm +



κ0 (zl , zm )Ql = −uI (zm ), m = 1, ..., M,

(1.9)

l =m

with Cm :=

2 |D | κm m ρm ρm −ρ0

−

1 2 ˆ 8π κm Am

and Aˆ m :=

1 |∂Dm |





∂Dm ∂Dm

(s − s  ) · νs  dσm (s  ) dσm (s). |s − s  |

(1.10)

The algebraic system (1.9) is invertible under one of the following conditions: 1. The coefficients Cm are negative and max |Cm | = O(a s ), as a  1. This condition holds if (a) γ < 1 or γ = 1 with ω being away from ωM and we have the relations 0 ≤ γ ≤ 1, γ + s ≤ 2 and 3s ≤ t ≤ 1. (b) γ = 1 and the frequency ω approaches ωM from below (lM < 0), i.e. ω < ωM , and we have the relations 3s ≤ t ≤ 1 and 1 − h1 − s ≥ 0. 2. The coefficients Cm are positive and one of the following conditions is fulfilled (a) max |Cm | = O(a t ), as a  1, and τ := min1≤j,m≤M, j =m cos(κ0 |zm − zj |) > 0. The first condition holds if γ = 1, the frequency ω approaches ωM from above (lM > 0), i.e. ω > ωM , and we have the relations 0 ≤ t ≤ 1 − h1 and s ≤ 1. (b) max |Cm | = O(a s ), as a  1. This condition holds if γ = 1 and the frequency ω approaches ωM from above (lM > 0), i.e. ω > ωM , and we have the relations 3s ≤ t ≤ 1 and 1 − h1 − s ≥ 0. The constants appearing in the error terms of (1.7) and (1.8) depend only on the a priori bounds mentioned above. In the case γ = 1 (i.e. β = 2), we assume that the constant Cρ , defined in (1.5) is larger than a certain constant depending only on those a priori bounds. Remark 1.3. Let the vector (Qm )M m=1 be the solution of the algebraic system Cm −1 Qm +



κ0 (zl , zm )Ql = −uI (zm ), m = 1, ..., M,

l =m

where Cm −1 is now defined as C−1 m =

1 2 + [κm

ρm 2 ρ0 (κm

− κ02 )]

Imd − iImd

2 + [κm

Jm ρm 2 ρ0 (κm

 − κ02 )]

+

2 1 s d (Jm ) − I I m m 2 4 κm κm

with Imd := |Dm |−1



   1 ρm 2 ρm 2 + − (κm − κ02 ) Aˆ m , −κm ρm − ρ0 8π ρ0

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Ims := i

κ3

m

4π

−

 1 2 (κ0 − κm )κm |Dm |−1 Aˆ m Capm 2 32π

and Jm :=

 1 ρ0 1 ˆ 2 (κ0 − κm )κm Am |Dm |−1 Capm Capm , 1+ ρm − ρ0 4π 8π

where Capm stands for the capacitance of Dm , see (3.4), which scales as Capm = O(a). As it is shown in Remark 3.16, if γ = 1 and the frequency ω is near ωM , i.e. 1 − h1 ∈ (0, 1],5 lM = 0, the improved estimate u∞ (x, ˆ θ) =

M 

2 ωM ω2

= lM a h1 ,

∞ ˆ zm )Qm + O(a 2−s−h1 + a 3−2t−2s−2h1 ) κ0 (x,

m=1

holds instead of (1.8). In addition, note that the case by Theorem

31.2.  h1 = 1 is not covered  1 2 ˆ s = i κm − m Observe now that, in the case γ = 1, Imd = |Dm |−1 ρmρ−ρ − κ + O(a), I A m 8π m m 4π 0  1 2 −1 −1 (κ − κm )κm |Dm | Aˆ m Capm ∼ 1 and Jm = O(a). Hence we can estimate Cm as 32π 2 0 C−1 m =

 ρ

κ3  1  1 2 ˆ  1 m m −1 2 −1 ˆ | − (κ − κ )κ |D | Cap κ − |D + i A A m m 0 m m m m m 2 κm ρm − ρ0 8π m 4π 32π 2 + O(a)

or C−1 m =



κ 2  Aˆ m  ωM 1 m −1 ˆ − 1 + i (κ − κ )|D | Cap − + O(a). A 0 m m m m 8π |Dm | ω2 4π 32π 2

Now, if in addition the frequency ω is near ωM , i.e. 1 − scaling, we derive C−1 m =−

2 ωM ω2

= lM a h1 , h1 > 0, lM = 0, then after

κ  lM Aˆ m h1 −1 1 m −1 ˆ +i (κ − κ )|B | Cap(B ) a − A 0 m m m m + O(a) 8π |Bm | 4π 32π 2

(1.11)



)   where Aˆ m := |∂B1m | ∂Bm ∂Bm (s−s |s−s  | · νs dσm (s ) dσm (s) ans Cap(Bm ) is the capacitance of Bm . We see that this approximation makes sense for any h1 in (0, 1] and while for h1 ∈ (0, 1), the first term is the most dominating, for h1 = 1 the first two terms should be taken into account. Related to the model (1.4), the asymptotic expansion is derived formally in [13,14] and justified mathematically in [11] in the case β = 2 when the frequency ω is close to the particular frequency ωM . The particular frequency ωM is shown to be an approximation, as a  1, of a resonance known as the Minnaert resonance, see [5]. The approximations provided in Theorem 1.2 5 The condition that h ∈ (0, 1] appears naturally to ensure the invertibility of the algebraic system, see Remark 3.13. 1

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are valid for a large class of bubbles. In particular, they can be used to derive the equivalent effective media at least in the following class: 1. If γ ∈ ( 12 , 1), which means that β ∈ ( 32 , 2), and for any frequency ω. 2. If γ = 1, i.e. β = 2, for any frequency ω away or very close (but distinct) from ωM . The error in the approximation in (1.7) is going to zero since t < (1.8) goes to zero provided s, h1 and t satisfy the condition s < min{2 − 2h1 ,

3 − 2t − 2h1 }. 2

1 2

and s ≤ 32 . The error term in

(1.12)

These conditions are fulfilled, in particular, if 1 s + h1 ≤ 1, h1 < 1 and t < . 2

(1.13)

These last conditions are the regimes in which one can derive the effective media. Actually, in the case lM > 0, one can also allow s + h1 > 1 (but s + h1 < 32 ) and hence generate potential walls which are completely reflecting any incident wave sent from outside of its support. This phenomenon is already observed and justified in the framework of acoustic waves in the presence of very large number of holes, see [17]. We also observe that, in our approximations, we can handle the case s = 0 and h1 = 1, see Remark 1.3. In this case, the error term goes also to zero as soon as t < 12 . Hence, as a → 0, and  M ∞ choosing t = 0 for instance, u∞ (x, ˆ θ) = M m=1  (x, zm )Qm and (Qm )m=1 is the solution of

 lM Aˆ m κm 1 −1 A ˆ m Cap(Bm ) . the algebraic system (1.9) with Cm −1 := − 8π|B + i − (κ − κ )|B | 0 m m 2 | 4π 32π m This limiting field describes the Foldy-Lax field generated by the interaction of the point-like scatterers, given by the centers of our small bubbles, with scattering strengths modeled by Cm’s, compare to [19,21] and see also [22]. This limiting field is also modeled by the Dirac-like potentials supported on the centers of the small bubbles, see [1]. What we have shown then is that if we inject into the background small bubbles characterized by ρρm0 ≈ a 2 and the frequency ω near to the Minnaert resonance, as follows

ω2 2 ωM

= 1 + l a, the generated fields behave as the

ones created by Dirac-like potentials (or point-like scatterers). This shows a way how to generate fields generated by singular potentials by injecting regular small bubbles. These singular potentials are supported on points. In [4], we show how to generate fields due to potentials supported by 2D surfaces (metasurfaces or metascreens) or 3D domains (metamaterials) by injecting in the background bubbles distributed on 2D surfaces or in 3D domains. The case of 1D curves (i.e. metawires) will be treated separately in a forthcoming work as the scattering by 1D curves is more delicate, see [12] and the references therein. Thanks to the kind of approximations we provide in Theorem 1.2, these different settings of the metamaterials can be treated in a unified way, namely as Dirac type potentials supported on curves, surfaces or domains. The analysis of the wave propagation in the presence of highly heterogeneous media is an object of extensive study, see [23]. An important situation is when the heterogeneity of the medium is modeled by the high relative contrasts as the relative densities or the relative bulk modulus described above. In the recent years there was an increase of interest in describing the effective macroscopic models generated by periodically distributed microscopic structures characterized

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by high contrasted media, see [20] and the references therein, based on homogenization techniques. Compared to these works, we need to study the expansions of the fields generated by contrasted small bubbles that are not necessarily periodically distributed. Besides, we focus on the precise description of the dominant parts of the fields as the cluster of the bubbles becomes dense. For this, we propose to derive the point-interaction approximation of those fields using integral equation methods coupled with asymptotic expansions tools. This point-interaction approach has its roots back to the Foldy approximation method known in several branches of physics and engineering, see [1,22]. The advantage of this approach is that we can characterize clearly the dominant fields generated by the interaction of the small bubbles between each other and also with the background medium. This approach was already tested and justified in our previous works [16,3] when the contrast is modeled by high surface impedances and in [11] where the case γ = 1 (i.e. β = 2) and the frequency ω is close to the resonance ωM . The purpose of our work here is to extend those last results to more general values of β and the whole range of frequencies ω. Related to this bubbles model, other results were derived very recently. In particular, in the case β = 2 and ω close to the resonance ωM , we find in [6] a mathematical framework for modeling metasurfaces with bubbles, in [7] a justification of the superfocusing of acoustic waves in the presence of gas bubbles and in [8] a justification of the bandgap opening due to periodically distributed bubbles. As compared to the results in [11,7], using our derived estimates, we can handle not only frequencies near the Minnaert resonance but arbitrary other fixed frequencies6 and any gas modeled by densities having contrasts of the order a β with β ∈ ( 32 , 2]. In the particular case when β = 2 and ω is close to the resonance ωM , we retrieve the results in [11]. Namely taking s + h1 = 1 in (1.8), we see that we can get the effective media with an additive coefficient changing sign depending if ω is lower or higher than ωM . The equivalent medium is absorbing if ω < ωM and reflecting if ω > ωM . In addition, and as discussed above, if ω is close to ωM and ω > ωM , i.e. with lM > 0 in (1.8), we can also take 1 < s + h1 < 32 . In these cases, the equivalent media behave as extremely reflecting media allowing no incident wave to penetrate inside it, see [17] for a different but related setting using holes. The quantification and justification of these results are reported in [4]. The rest of the paper is divided into the following sections. In section 2, we describe briefly the main steps of the proof. In section 3, we give the core proof of the result while in section 4, we provide the detailed proofs of the main tools used in section 2. 2. A brief description of the proof We recall that the single layer potential, double layer potential and the adjoint of the double layer potential are defined as κ SD φ(x) := m

κ (x, y)φ(y)dσm (y),

∂Dm



κ KD φ(x) := m

∂νy κ (x, y)φ(y)dσm (y),

∂Dm

6 This, of course, means that we are in the Rayleigh regime, i.e. ωa << 1 as a << 1.

(2.1)

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9

∂νx κ (x, y)φ(y)dσm (y).

∂Dm

The problem (1.4) is well-posed (see [9,10,18] for instance) and the total field can be represented as  u(x) =

uI +

M

κ0 l=1 SDl φl (x),

x ∈ R3 ∪M l=1 Dl ,

(2.2)

κs SD ψ (x), x ∈ Ds , s = 1, ..., M, s s

where the densities φl , ψl satisfy the system of integral equations  M     κi κ0 SDl φl  SD i ψ i −  l=1

= uI |∂Di , ∂Di

   κ0 M

1    ∂(S φ )  l ρ0 1 Dl κi ∗ κ0 ∗  I d + (KDi ) ψi (x) − − I d + (KDi ) φi (x) −  i ρi 2 2 ∂ν  l=1  l =i

=

∂uI |∂Di . ∂ν i

∂Di

As mentioned in the introduction, a characteristic of the bubbles’ model is that the relative speed of propagation is uniformly bounded, precisely, we assumed that the main steps of the proof, we assume in this section that

2 κm κ02

2 κm κ02

∼ 1, as a  1. To describe

= 1, as a  1. Under this condi-

tion, the system of integral equations simplifies as follows. For l = 1 . . . , M, κ

SD0l (ψl − φl ) −

M 

κ

SD0m φm = uI , on ∂Dl ,

(2.3)

m=1 m =l κ M  ∂(SD0m φm ) ∂uI ρ0 1 1 κ0 ∗ κ0 ∗ [ I d + (KDl ) ]ψl − [− I d + (KDl ) ]φl − = l , on ∂Dl . (2.4) ρl 2 2 ∂ν l ∂ν m=1 m =l

As both the left and right hand sides of (2.3), extended in Di , i = 1, . . . , M, satisfy the same Helmholtz equation, they are equal there. Taking the normal derivative from inside Dl , we have κ0 M   ρ0 1 ρ0 ∂(SDm φm )  ρ0 ∂uI  κ0 ∗ ) − φ ) − = I d + (KD (ψ   . l l l l ∂Dl ρl 2 ρl ∂ν ρl ∂ν l ∂Dl m=1 m =l

Next we use this in the second identity (2.4) to derive −

κ M  ∂(SD0m φm )  1 ρ0 + ρl  ∂uI κ = , on ∂Dl , φl − (KD0l )∗ φl −  ∂Dl 2 ρ 0 − ρl ∂ν l ∂ν l m=1 m =l

(2.5)

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which we rewrite as −

κ0 M  ∂(SD φ ) 1 ρ 0 + ρl  ∂uI κ m m  0 ∗ 0 ∗ ) φ − = l + [(KD0l )∗ − (KD ) ]φl , on ∂Dl . φl − (KD  l l l l ∂Dl 2 ρ 0 − ρl ∂ν ∂ν m=1 m =l

(2.6)

The aim now is to estimate ∂Dl φl dσl from (2.6) above, leading us to the final algebraic system. 0 (1) = − 1 , we obtain By integrating (2.6) on ∂Dl , and since KD 2 l   ∂uI 1 ρ 0 + ρl κ0 ∗ 0 ∗ − −1 φl dσl = dσ + ) − (K ) (K φl dσl l D D l l 2 ρ 0 − ρl ∂ν l ∂Dl

∂Dl

+

∂Dl

κ M  ∂(SD0m φm ) m=1 ∂D l m =l

∂ν l

(2.7) dσl .

Using the expansions of the fundamental solution, we derive the following estimates:



0 ∗ ) φl (s)dσ(s) = (KD0l )∗ − (KD l κ

1 2 κ 8π 0

∂Dl

⎡



⎢ φl (s) ⎣

∂Dl

−



∂Dl

i 3 κ |Dl | 4π 0



⎤ (s − t) ⎥ · νt dt ⎦ dσl (s) |s − t|

(2.8)

  φl (s)dσl (s) + O a 5 φl  ,

∂Dl



κ

∂(SD0m φm ) ∂ν l

∂Dl

⎡

⎢ = −κ02 |Dl | ⎣κ0 (zl , zm )

φm (s) dσm (s)

∂Dm

+∇t κ0 (zl , zm ) ·

⎤

⎥ (s − zm )φm (s) dσm (s)⎦

∂Dm

− κ02 ∇x κ0 (zl , zm ) ·

(2.9)



(x − zl )dx

Dl

∂Dl



 φm (s) dσm (s) + O

∂Dm

  ∂uI = − κ02 |Dl | uI (zl ) + O a 4 . l ∂ν

 a6 φm  , m = l, 3 dml (2.10)

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11

We set Al to be the function defined on ∂Dl by Al (s) := ∂Dl (s−t) |s−t| · νt dσl (t). It is clear that

1 2 Al scales as a . We denote its average as Aˆ l := |∂Dl | ∂Dl Al (s) dσl (s) and write

⎡ ⎢ φl (s) ⎣

∂Dl



∂Dl

= Aˆ l

⎤ (s − t) ⎥ · νt dσl (t)⎦ dσl (s) |s − t|



φl dσl (s) +

∂Dl

(2.11) φl (s)(Al (s) − Aˆ l ) dσl (s).

∂Dl

In addition, we have the estimate Al − Aˆ l L2 (∂Dl ) = O(a 3 ) and we can use an approximation

of ∂Dl φl (s)(Al (s) − Aˆ l ) dσl (s) of the form

φl (s)(Al (s) − Aˆ l ) dσl (s) = O(a 3 φl ).

(2.12)

∂Dl

However, the following more precise approximation is derived from the original system (2.5), or (2.6): 

M   ∇s κ0 (zl , zm )Qm Al (s) − Aˆ l φl dσl (s)= − Rl ·

(2.13)

m=1 m =l

∂Dl

⎞

⎛

⎜ 4 3 +O⎜ ⎝a + a

where Rl :=

∂Dl

M  m=1 m =l

⎟ a3 φm  + a 5 φl ⎟ ⎠, 3 dml



 −1 0 ˆ l ) (s)νl (s) dσl (s) and λl := (A (·) − A λl I d + KD l l

1 ρ0 +ρl 2 ρ0 −ρl . The value

Rl

scales as a 4 .

Also, the term ∂Dm (s − zm )φm (s) dσm (s) can be estimated either directly, but roughly, as (s − zm )φm (s) dσm (s) = O(a 2 φm )

(2.14)

∂Dm

or more precisely, by using the original system (2.5), as (s − zm )φm (s) dσm (s) = a 3−γ + O([a 3 + lot]φm ),

(2.15)

∂Dm

where the abbreviation lot denotes the lower order terms. To give an idea on what to keep and what to neglect in these approximations, let us consider the regime where γ = 1 and s < 1.

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12



2 l As both Al and − 12 ρρ00 +ρ −ρl − 1 scale as a , the first term of (2.8) cannot be neglected. Putting the second term of (2.8), the second and the third terms of (2.9) as error terms and plugging (2.8)-(2.9)-(2.10)-(2.11)-(2.13), we derive the algebraic system, for the vector (Ql )M 1 , with Ql :=

φ dσ (s), l l ∂Dl "

# M  Aˆ  1 ρ0 + ρl l 2 −1 − κ0 (zl , zm )Qm κ0 Ql + κ02 |Dl | − 2 ρ 0 − ρl 8π ⎛   ⎜ = −κ02 |Dl | uI (zl ) + O a 4 + O ⎜ ⎝

m=1 m =l

M  m=1 m =l

a5 3 dml

⎞

(2.16)

⎟ φm ⎟ ⎠.

By a careful counting of the bubbles, distributed in a bounded domain , we show that  M 1 −3 ln(d)). m=1 3 = O(d m =l dml

We show that the system (2.5) is invertible and that φl  = O(a −1 ) for γ = 1. In addition, as the domain  where we inject thes bubbles is bounded hence its volume will be of the order d −3 , hence a s ∼ d −3 and then d ∼ a 3 . With these estimates, we see that the remaining terms of  a 4 ln(d) a5 order O(a 4 ) and O( M ∼ a 3 a 1−s ln(a) are dominated by the first term m=1 3 φm ) ∼ d3 m =l dml

κ02 |Dl | uI (zl )

which is of the order a 3 (as s < 1). s i Now, from (2.1),

we see that for x, |x| >> 1, the scattered field u (x) := u(x) − u (x) given M s by u (x) = i=1 ∂Di k (x, y)φi (z)dσi (y) can be estimated as: us (x) =

M   k (x, zi )Qi + ∇k (x, zi ) (y − zi )φi (z)dσi (y) + O(a 3 φi ) . i=1

(2.17)

∂Di

Using (2.15)7 and the fact that φi  = O(a −1 ), we obtain u (x) = s

M 

k (x, zi )Qi + a 2−s .

(2.18)

i=1

Finally, combining (2.18) and the algebraic system (2.16), via its invertibility property, we derive the final estimates of the scattered fields. However, if we take into account the terms appearing in (2.8)-(2.9)-(2.10)) and the first order terms in the approximation of (2.13) and (2.15) respectively, then we can handle, not only the case γ = 1 and s < 1, but a wider range of the parameters as described in our main theorem. The basic system (2.5) is derived under the conditions κm = κ0 , m = 1, ..., M. In this case it involves only the densities φm ’s. If these conditions are not satisfied, then the corresponding system involves also the densities ψm ’s. Hence, we need to handle this coupling. However, apart from additional highly technical issues, the scheme of the proof is as described above. 7 If we use (2.14) instead of (2.15), then we will have an error of the order a 1−s instead of a 2−s in (2.18).

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13

3. Proof of Theorem 1.2 3.1. Representation of the solutions As discussed in section 2, the problem (1.4) is well-posed and the total field can be represented as  u(x) =

uI +

M

κ0 l=1 SDl φl (x),

x ∈ R3 ∪M l=1 Dl ,

(3.1)

κs SD ψ (x), x ∈ Ds , s = 1, ..., M, s s

where φl , ψl are appropriate densities. Observe that we represent the field inside each Ds , s = 1, . . . , M, using a single density. This simplifies the presentation of the computations. Using (3.1), the jump conditions across the boundary of the obstacles in (1.4) can be reformulated as follows. From the third identity in (1.4), we derive that on ∂Ds ,       0 = u (x) − u (x) = uI  +

−

∂Ds

(x) +

M 

    κ κs SD0l φl  (x) − SD ψ  (x). s s +

l=1

−

Since the single-layer potentials are continuous, this implies that   κs SD ψ  s s

∂Ds

(x) −

M 

  κ SD0l φl 

l=1

∂Ds

  (x) = uI 

∂Ds

(x).

(3.2)

Again M 0  1 ∂u  1 ∂(SDl φl )  1 ∂uI   (x) =  (x) +  (x), + ρ0 ∂ν s + ρ0 ∂ν s ∂Ds ρ0 ∂ν s κ

l=1

κs 1 ∂u  1 ∂(SDs ψs )   (x) =  (x), − ρs ∂ν s − ρs ∂ν s

and therefore from the fourth identity in (1.4), we derive that   1 ∂u  1 ∂u  0= (x) − (x) ρ0 ∂ν s + ρs ∂ν s −

   κ0 M  1 ∂(SD φl )   1 l   (x) + = ρ0 ∂ν s ∂Ds ρ0 ∂ν s  l=1  l =s ∂uI

(x) ∂Ds

1 1 1 1 κ0 ∗ κs ∗ φs (x) + (KD ) φs (x) − ψs (x) − (KD ) ψs (x) − s s 2ρ0 ρ0 2ρs ρs and hence

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14

 1 ∂uI  (x) ρ0 ∂ν s ∂Ds

   κ0 M



  ∂(S φ )  l 1 1 1 1 1 Dl κ0 ∗ κs ∗  ) (x) + ) (x) − I d + (K − I d + (KD φ ψ =− s s Ds  s s ρ0 2 ρs 2 ρ ∂ν  l=1 0  l =s

(x). ∂Ds

(3.3)

The following proposition guarantees the existence of densities φl , ψl satisfying (3.2) and (3.3). Proposition 3.1. For each M M $ $ (Fi , Gi ) ∈ Y := [H 1 (∂Di ) × L2 (∂Di )], i=1

i=1

there exists a unique solution M $

(φi , ψi ) ∈ X :=

i=1

M $

[L2 (∂Di ) × L2 (∂Di )]

i=1

to the system of integral equations  M     κ0 κi SDl φl  SD i ψ i −  l=1

= Fi , ∂Di

1   ρ0 1 κ0 ∗ κi ∗ ) (x) − − ) I d + (KD I d + (K ψ φi (x) − i Di i ρi 2 2

Proof. We outline a proof of this proposition in section 4.1.

 

 κ M  ∂(SD0l φl )   ∂ν i  l=1  l =i ∂Di

= Gi .

2

Note that in order to prove  the representation (3.1), we apply the above proposition with ∂uI  I Fi := u |∂Di and Gi := ∂ν i  respectively, where i = 1, . . . , M. ∂Di

3.2. Integral identities Let us recall that the capacitance Capl is defined as Capl :=



0 SD l

−1

∂Dl

It is known that Capm ≈ δ and hence Capm ≈ a.

(1)(t) dσl (t).

(3.4)

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15

We note that the system of integral equations (3.2)-(3.3) can be rewritten as follows. For l = 1 . . . , M, κ0 SD (ψl l

− φl ) −

M 

κ0 κ0 κl SD φ = uI + [SD − SD ]ψl , on ∂Dl , m m l l

(3.5)

m=1 m =l κ M  ∂(SD0m φm ) ρ0 1 1 κ κ [ I d + (KD0l )∗ ]ψl − [− I d + (KD0l )∗ ]φl − ρl 2 2 ∂ν l m=1 m =l

=

∂uI ∂ν l

+

(3.6)

ρ0 κ κl ∗ [(KD0l )∗ − (KD ) ]ψl , on ∂Dl . l ρl

Our first step is to convert the system of double integral equations (3.5)-(3.6) for (φl , ψl )M l=1 into a system of single integral equation for (φl )M . For this, we need to estimate first the source κ0 κl κ0 ∗ κl ∗ l=1 terms [SD − S ]ψ and [(K ) − (K ) ]ψ . l l Dl Dl Dl l In the following two lemmas, we collect some approximations that we shall use.  −1 0 Lemma 3.2. The functions SD l







n ∂Dl | · −t| φl (t) dσl (t)

the following approximate behavior. % % % %  % % −1 % % 0 | · −t|n φl (t)dσl (t) % % SDl % % % % ∂Dl

 −1   0 n exhibit and SD ) (· − z l l

  = O a n+1 φl L2 (∂Dl ) ,

(3.7)

= O(a n ).

(3.8)

L2 (∂Dl )

%  −1  % % % 0 (· − zl )n % % SDl

L2 (∂Dl )

Proof. We refer to section 4.2 for a proof of this lemma.

2

Lemma 3.3. The following approximations hold true.

 κ κm SD0m − SD ψm (x) m

=

i (κ0 − κm ) 4π

ψm (s) dσm (s) +

∞ n n n)  i (κ − κm 0

n=2

∂Dm

4πn! ∂Dm

&

 κ κm ∗ ) (KD0m )∗ − (KD ψm (s)dσm (s) m ∂Dm

=

1 2 2 ) (κ − κm 8π 0



∂Dm

⎡ ⎢ ψm (s) ⎣



∂Dm

|x − s|n−1 ψm (s) dσm (s), '(

* + =:Err1m =O a 2 ψm 

(3.9)

)

⎤ (s − t) ⎥ · νt dt ⎦ dσm (s) |s − t|

(3.10)

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16

i 3 − )|Dm | (κ 3 − κm 4π 0

ψm (s)dσm (s) +

,

*

+ [=O a 5 ψm  ]

∂Dm



Err2m & '( )

κ0 ∂(SD φ ) m m ∂ν l

∂Dl

⎡

⎢ = −κ02 |Dl | ⎣κ0 (zl , zm )





⎤

⎥ (s − zm )φm (s) dσm (s)⎦

+∇t κ0 (zl , zm ) · ∂Dm

− κ02 ∇x κ0 (zl , zm ) ·



(x − zl )dx

Dl

∂Dl

,

(3.11)

φm (s) dσm (s)

∂Dm



φm (s) dσm (s) −

Err4m & '( ) ,

∂Dm

=O

a6 3 dml

, m = l,

φm 

∂uI = − κ02 |Dl | uI (zl ) + Err5l , & '( ) ∂ν l * +

(3.12)

=O a 4

-

 −1 i(κ − κ ) 0 l 0 ψl − φl = SD Ql l 4π ⎛ ⎞   −1  , iκ l 0 ⎝ Capl κ0 (zl , zm ) + (s − zl ) · ∇s κ0 (zl , zm ) Qm ⎠ 1− + SD l 4π m = l ⎛ ⎞  −1  −1  0 0 ⎝ ⎠ + SD ∇  (z , z ) · V uI (zl ) + SD t κ l m m 0 l l m = l

+ Err6l

⎞   a3 ⎠ φ  , in L2 , := O ⎝a + a 2 φl  + m d3 ⎛

m = l

where Vm :=



∂Dm (s

− zm )φm dσm (s).

∂Dl

+

,

ψl − 

φl = Capl

i(κ0 − κl ) Ql 4π

-

∂Dl

 m = l

"

, iκl Capl κ0 (zl , zm ) Capl 1 − 4π

#    −1 0 (· − zl )(s)dσl (s) Qm S Dl + ∇s κ0 (zl , zm ) · ∂Dl

(3.13)

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⎛



+ Capl ⎝

17

⎞ ∇t κ0 (zl , zm ) · Vm ⎠ + Capl uI (zl )

m = l

+ Err7l

⎞   a4 ⎠ . φ  := O ⎝a 2 + a 3 φl  + m d3 ⎛

(3.14)

m = l

Proof. We defer the proof of the lemma to section 4.3.

2

We shall also sometimes use (3.10) with κm = 0. In this case, we shall refer to Err2m as Err3m . 3.3. A way of counting the number of bubbles In the following sections, we will repeatedly need to estimate sums of the form M −k i=1,i =j f (zi , zj ) with functions f involving inverse power of distances, i.e. |zi − zj | , k ∈ R+ . Here zj is the center of the small bubble Dj . Following [16], we describe here a way how to handle these sums by a proper counting. To do it, for any m = 1, . . . , M, fixed, we distinguish between the points zj , j = m, by keeping them into different layers based on their distance from Dm . Let  be a bounded domain, say of unit volume. The way how the bubbles Dm, m = 1, ..., M, are distributed in  can be described as follows. Recall that M = O(a −s ). We shall divide  into [a −s ] cubes m , m = 1, ..., [a −s ],8 such that each m contains some of the Dj ’s (and maybe non). It is natural then to assume that the number of bubbles in each m , for m = 1, ..., [a −s ], is uniformly bounded in terms of m. We take the cubes m ’s to be equal, up to translations, and distributed periodically in , i.e. the set of m ’s is a uniform grid of . Observe that even if the m ’s are periodically distributed, the bubbles are still not periodically distributed as in each m we can have a different number of bubbles. Let us emphasize that any distribution of the bubbles can be structured with a periodic set of m ’ as above. The case when the number of bubbles in each m is exactly one, i.e. the distribution of the bubbles is periodic, is the situation dealt with in the homogenization theory. For each m, 1 ≤ m ≤ [a −s ], m is the cube whose sides have sizes which can be estimated as a ( 2 + d α ) with 0 ≤ α ≤ 1. Observe that α = 1 means that the corresponding m contains at most one bubble Dm . Recall that d is of the order a t , then we have the relation 3αt = s, see below. To estimate the aforementioned sums, we need to count the number of the bubbles. The idea is as follows. As the number of bubbles in each cube m is uniformly bounded in terms of m, we reduce the question to counting the m ’s. To this end, we use the periodic structure of the cubes m ’s. Precisely, for a given m, we take the cube m as the ‘starting cube’ and then the other cubes are attached to it as to form different layers of cubes (i.e. as a Rubik cube with m in the center). Hence, starting from m , the total cubes upto the nth layer consists of (2n + 1)3 cubes for n = 0, . . . , [d −α ]. It is clear then that the number of bubbles located in the nth, n = 0 layer will be at most [(2n + 1)3 − (2n − 1)3 ] and their distance from Dj is more than nd α . 8 For a given real and positive number x, we denote by [x], the unique integer n such that n ≤ x ≤ n + 1.

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18

With this way of counting, we deduce that for j fixed M 

−k

|zi − zj |

≤

i=1,i =j

−α



−k

|zm − zj |

+

zj , zm ∈j ,zm =zj

d  

|zi − zj |−k

(3.15)

l=1 zi ∈l l =j

where 

|zm − zj |−k = O(d −k )

zj , zm ∈j ,zm =zj

and −α

−α

d  

−k

|zi − zj |

≤

d 

l=1 zi ∈l l =j

−α

α −k

((2l + 1) − (2l − 1) )(ld ) 3

3

d  = O( l 2 (ld α )−k )

l=1

l=1 −α

= O(d

−α k

d 

l 2−k ).

l=1

Hence −α

M 

−k

|zi − zj |

= O(d

−k

) + O(d

i=1,i =j

−α k

d 

l 2−k ).

(3.16)

l=1

 −k = M − 1. By the previous formulas Observe that for k = 0, it is obvious that M i=1,i =j |zi − zj | M we have i=1,i =j |zi − zj |−k = O(1) + O(d −3α ). Recalling the way we counted, d 3α is the volume of the j ’. Hence d −3α is of the order of the number of the bubbles, i.e. d −3α = O(M). As we set M = O(a −s ), we will be using the formula 3αt = s, and then t ≥ 3s , as α ≤ 1. For k > 0, we obtain the following formulas: 1. If k < 3, then M 

−α

−k

|zi − zj |

= O(d

i=1,i =j

−k

) + O(d

−α k

d 

l 2−k ) = O(d −k ) + O(d −3α ).

(3.17)

l=1

2. If k = 3, then M  i=1,i =j

|zi − zj |−k = O(d −k ) + O(d −3α | ln(d)|).

(3.18)

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19

3. If k > 3, then M 

−α

−k

|zi − zj |

= O(d

−k

) + O(d

d 

−α k

i=1,i =j

l 2−k ) = O(d −k ) + O(d −αk ).

(3.19)

l=1

M

|zi − zj |−k = O(M d −k ) = a −s−k t .  −k = O(a −t ) + Let us take k = 1 as an example. We have shown above that M i=1,i =j |zi − zj | O(d −3α ) = O(a −t ) + O(a −3αt ) = O(a −t ) + O(a −s ) instead of O(a −s−t ). Hence, in this case, we gain an order of a −t as soon as s ≥ t. This kind of reasoning will be used in the next sections. If we do not count properly, then we would have

i=1,i =j

3.4. The a priori approximation of the system of integral equations κ

κ

κ

Our first step is to approximate the source term [SD0l − SDll ]ψl in (3.5), that is, SD0l (ψl − φl ) − M κ0 κ0 κl κ0 I m=1 SDm φm = u + [SDl − SDl ]ψl by single layers of the form SDl gl , where gl is the unique m =l

solution of the problem κ

SD0l gl (s) =

i (κ0 − κl ) 4π

ψl (t)dσl (t), on ∂Dl .

(3.20)

∂Dl

Hence κ0 κl κ0 [SD − SD ]ψl = SD g + Err1l . l l l l

Let us define κ0 H 1 := SD (ψl l

− φl ) −

M 

κ0 κ0 SD φ − uI − SD g, m m l l

m=1 m =l

and g˜ l such that κ0 SD g˜ = Err1l . l l

(3.21)

Using (3.20) and (3.21) in (3.9), it follows that H 1 satisfies  κ ( + κ02 )(H 1 − SD0l g˜ l ) = 0 in Dl , κ

H 1 = SD0l g˜ l on ∂Dl . κ

By the maximum principle, we can immediately conclude that H 1 = SD0l g˜ l in Dl . Taking the normal derivative from inside Dl , we have κ0 κ0 κ M  ∂[SD (ψl − φl )]  g ) ∂(SD ∂(SD0m φm )  ∂uI  l l l  − = + + Erl ,     ∂Dl ∂Dl ∂Dl ∂ν l ∂ν l ∂ν l ∂Dl ∂ν l m=1 m =l

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20

∂[SD0 g˜ l ]  κ

where Erl :=



l

∂ν l

∂Dl

. This gives us

κ0 M   ρ0 1 ρ0 ∂(SDm φm )  κ I d + (KD0l )∗ (ψl − φl ) −  ∂Dl ρl 2 ρl ∂ν l m=1 m =l

= ρ0 ρl Erl .

where Er2,l :=

−

 ρ0 ∂uI  ρ0 1 κ0 ∗ + ) I d + (K gl + Er2,l ,  Dl ρl ∂ν l ∂Dl ρl 2 Next we use this in the second identity (3.6) to derive

M  

ρ   ∂(S κ0 φm )   1 ρ0 ρ0 0  Dm κ + 1 φl − − 1 (KD0l )∗ φl − −1  ∂Dl 2 ρl ρl ρl ∂ν l m=1 m =l

=

ρ

0

ρl

 ∂uI   ρ0 1 ρ0  κ0 ∗ κ0 ∗ κl ∗ + ) ) − (KD ) ]ψl + Er2,l , I d + (K gl − [(KD  D l l l l ∂ν ∂Dl ρl 2 ρl

−1

which, in turn, implies that

−

κ M  ∂(SD0m φm )  1 ρ0 + ρl  κ φl − (KD0l )∗ φl −  ∂Dl 2 ρ 0 − ρl ∂ν l m=1 m =l

  ∂uI  ρl −1 1 κ0 ∗ κ0 ∗ κl ∗ = l + 1− [(KDl ) − (KDl ) ]ψl + [ I d + (KDl ) ]gl + Er3,l , on ∂Dl , ∂ν ρ0 2

where Er3,l :=



ρ0 ρl

−1

−1

Er2,l . Hence, we have

1 ρ0 + ρl  0 ∗ ) φl φl − (KD l 2 ρ 0 − ρl   ∂uI  ρl −1 1 κ0 ∗ κ0 ∗ κl ∗ = l + 1− [(KDl ) − (KDl ) ]ψl + [ I d + (KDl ) ]gl ∂ν ρ0 2

−

+

M  m=1 m =l

κ ∂(SD0m φm )   ∂D ∂ν l

l

(3.22)

κ0 ∗ 0 ∗ + [(KD ) − (KD ) ]φl + Er3,l , on ∂Dl . l l

Our aim henceforth would be to derive a suitable approximate system for ∂Dl φl dσl from (3.22) above, leading us to the final algebraic system. To do so, we start by integrating (3.22) on ∂Dl , 0 (1) = − 1 , it follows that and since KD 2 l

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−

21



1 ρ0 + ρ l −1 2 ρ 0 − ρl

∂uI dσl + ∂ν l

= ∂Dl

φl dσl ∂Dl



κ (KD0l )∗

0 ∗ − (KD ) l

κ M   ∂(SD0m φm ) dσl φl dσl + ∂ν l m=1 ∂D l m =l

∂Dl



ρl −1 + 1− ρ0

(3.23)

  1 κ0 ∗ κ0 ∗ κl ∗ [(KDl ) − (KDl ) ]ψl + [ I d + (KDl ) ]gl dσl + Er4,l , 2

∂Dl

where Er4,l :=

∂Dl

Er3,l . For s ∈ ∂Dl , let us define Al (s) := ∂Dl

(s − t) · νt dσl (t). |s − t|

From the definition, it easily follows that Al L2 (∂Dl ) = O(a 3 ). Next we observe that, , , - - ρl −1 ρl −1 1 κ0 ∗ Er4,l = 1 − Erl = 1 − [ I d + (KD ) ]g˜ l l ρ0 ρ0 2 ∂Dl

∂Dl

, - ρl −1 κ0 ∗ 0 ∗ = 1− [−(KD ) + (KD ) ]g˜ l l l ρ0 ∂Dl

, ρl −1 1 2 = 1− g˜ l (s)Al (s)dσl (s) + O(a 4 g˜ l ), κ0 ρ0 8π ∂Dl

where the last step follows from (3.10) applied to g˜ l .

In order to check the behavior of the Er4,l , let us consider its dominating term ∂Dl g˜ l (s)Al (s)dσl (s). From the definition of g˜ l , it can be observed that the dominating term  −1 

 κ0 of g˜ l is SD | · −t|ψ (t)dσ (t) and hence it’s sufficient to consider l l ∂Dl l 

 0 −1 

∂Dl SDl ∂Dl | · −t|ψl (t)dσl (t) (s) Al (s)dσl (s). Using (3.7), we can deduce that   ⎛ ⎞      −1 ⎜ ⎟   0 | · −t|ψ (t)dσ (t) (s) A (s) dσ (s) SD   ⎝ ⎠ l l l l l    ∂Dl ∂Dl % ⎞% ⎛ % % % % −1   ⎟% ⎜ % 0 Al L2 (∂Dl ) =O a 5 ψl  . ≤ % SD l ⎝ | · −t|ψl (t)dσl (t)⎠% % % % 2 % ∂Dl

(3.24)

L (∂Dl )

* + κ0 −1 In addition, g˜ l  ≤ (SD ) Err1l H 1 (∂Dl ) = O(aψl ). Therefore Er4,l = O a 5 ψl  . l In the following lemma, we note down some estimates for gl that we shall use.

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22

Lemma 3.4. Similar to (3.10), gl satisfies the estimate



0 ∗ ) gl = (KD0l )∗ − (KD l κ

1 2 κ 8π 0

∂Dl

⎡



⎢ gl (s) ⎣

∂Dl

−

⎤



(s − t) ⎥ · νt dt ⎦ dσl (s) |s − t|

∂Dl

(3.25)



i 3 κ |Dl | 4π 0

gl (s)dσl (s) + Err8l . & '( ) * + =O a 5 gl 

∂Dl

Also ⎡ gl (s) =

i ⎢ (κ0 − κl ) ⎣Capl uI (zl ) + 4π

⎤

φl +

 m = l

∂Dl

⎥ 0 −1 κ0 (zl , zm ) Qm Capl ⎦ (SD ) (1) (s) l (3.26)

⎞

⎛



 a3 ⎠ , in L2 . φ  + Err9l := O ⎝a 2 + a 2 φl  + m d2 m = l ml 2

Proof. We defer the proof of this result to section 4.4. Now using Lemma 3.3, (3.23) can be rewritten as ρl ρ l − ρ0



M 

 2 φl dσl (s) + φm (s) dσm (s) κ0 |Dl | κ0 (zl , zm ) m=1 m =l

∂Dl



∂Dm

 (s − zm )φm (s) dσm (s)

+ ∇t κ0 (zl , zm ) · ∂Dm

+ κ02 ∇x κ0 (zl , zm ) ·



(x − zl )dx

Dl

= − κ02 |Dl | uI (zl ) + Err5l +

−

i 3 κ |Dl | 4π 0

(3.27)

 ∂Dm

1 2 κ 8π 0



⎡

⎢ φl (s) ⎣

∂Dl



φm (s) dσm (s) + Err4m

∂Dl

⎤ (s − t) ⎥ · νt dσl (t)⎦ dσl (s) |s − t|

φl (s)dσl (s) + Err3l

∂Dl

⎡ ⎤  (s − t) ρl −1 1 2 ⎢ ⎥ + 1− ψl (s) ⎣ (κl − κ02 ) · νt dσl (t)⎦ dσl (s) ρ0 8π |s − t| 

∂Dl

∂Dl



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23

  ρl −1 i ρl −1 3 3 − 1− ψl (s)dσl (s) − 1 − Err2l (κl − κ0 )|Dl | ρ0 4π ρ0 ⎡

∂Dl

⎤   (s − t) ρl −1 1 2 ⎢ ⎥ + 1− gl (s) ⎣ κ0 · νt dσl (t)⎦ dσl (s) ρ0 8π |s − t| ∂Dl

∂Dl

  ρl −1 ρl −1 i 3 gl (s)dσl (s) + 1 − Err8l + Er4,l . κ0 |Dl | − 1− ρ0 4π ρ0 ∂Dl

Dividing by |Dl | and making use of (3.13), (3.14) and (3.26) in (3.27), we obtain ρl |Dl |−1 ρ l − ρ0

(3.28)

φl (s) dσl (s)

∂Dl

+ κ02

M 



φm (s) dσm (s) + ∇t κ0 (zl , zm ) ·

κ0 (zl , zm )

m=1 m =l

∂Dm

+ κ02 |Dl |−1

M 

∂Dm

∇x κ0 (zl , zm ) ·

m=1 m =l

= − κ02 uI (zl ) +

 (s − zm )φm (s) dσm (s)



(x − zl )dx

Dl

1 2 κ |Dl |−1 8π 0

 φm (s) dσm (s) ∂Dm

φl (s)Al (s)dσl (s) −

i 3 κ 4π 0

∂Dl

ρl −1 1 2 + 1− φl (s)Al (s)dσl (s) (κl − κ02 )|Dl |−1 ρ0 8π

φl (s)dσl (s)

∂Dl



∂Dl

ρl −1 1 2 (κ − κ02 )|Dl |−1 + 1− ρ0 8π l 

+

 , m = l

+





0 SD l

−1



i(κ0 − κl ) Ql 4π

∂Dl

 iκl Capl κ0 (zl , zm ) + (· − zl ) · ∇s κ0 (zl , zm ) Qm 1− 4π -

 ∇t κ0 (zl , zm ) · Vm (s)Al (s)dσl (s)

m = l

    −1 ρl −1 1 2 0 I u (z ) + Err6 (κl − κ02 )|Dl |−1 SD + 1− l l Al (s)dσl (s) l ρ0 8π ∂Dl

ρl −1 i − 1− φl (s)dσl (s) (κl3 − κ03 ) ρ0 4π 

∂Dl

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24

" ,   ρl −1 i i(κ0 − κl ) iκl 3 3 − 1− (κ − κ0 ) Capl Ql + Capl κ0 (zl , zm ) Capl 1 − ρ0 4π l 4π 4π m = l

+ ∇s κ0 (zl , zm ) ·



0 SD l

−1

  (· − zl )(s)dσl (s) Qm + Capl ∇t κ0 (zl , zm ) · Vm

#

m = l

∂Dl

  ρl −1 i (κl3 − κ03 ) Capl uI (zl ) + Err7l − 1− ρ0 4π

  −1 1 2 i ρl κ0 |Dl |−1 (κ0 − κl ) Capl uI (zl ) + 1− ρ0 8π 4π + Ql +

 m = l

∂Dl

 0 −1 κ0 (zl , zm ) Qm Capl (SD ) (1) (s)Al (s)dσl (s) l

 ρl −1 1 2 Err9l · Al (s) dσl (s) + Er5,l , κ0 |Dl |−1 + 1− ρ0 8π ∂Dl

where Er5,l = |Dl |−1 Er4,l " + |Dl |−1 Err5l − ⎛

M  m=1 m =l

= O(a 2 ψl ) + O ⎝a + ⎛ = O(a 2 ψl ) + O ⎝a + ⎛ = O ⎝a +

 a3 d3 m = l ml

(3.29) #



 ρl −1 Err4m + Err3l + 1 − (−Err2l + Err8l ) ρ0

 a3 d3 m = l ml  a3 d3 m = l ml

⎞ φm  + a 2 ψl  + a 2 φl  + a 2 gl ⎠ ⎞ φm  + a 2 ψl  + a 2 φl  + a 3 ψl ⎠ ⎞

φm  + a 2 φl  + a 2 ψl ⎠ .

Collecting the error terms together, we can rewrite (3.28) further as |Dl |−1

,

∂Dl

1 ρl + ρl − ρ0 8π



 ρ0 (κl2 − κ02 ) − κ02 Al (s) φl (s)dσl (s) ρ l − ρ0

+

i ρ0 i 3 κ0 Ql + (κ 3 − κl3 )Ql 4π ρl − ρ0 4π 0

+

1 ρ0 (κ 3 − κ03 )(κ0 − κl )Capl Ql ρl − ρ0 16π 2 l

(3.30)

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i ρ0 + (κ 2 − κ02 )(κ0 − κl )|Dl |−1 Ql ρl − ρ0 32π 2 l

25

    −1 0 SD 1 (s)Al (s)dσl (s) l

∂Dl

⎞   −1 ρ0 i ⎟ ⎜ 0 + (κ0 − κl )κ02 |Dl |−1 ⎝ SD (1) (s)Al (s)dσl (s)⎠ Ql l 2 ρl − ρ0 32π ⎛

∂Dl

,  i ρ0 iκl + (κ03 − κl3 ) Capl κ0 (zl , zm ) Capl 1 − ρl − ρ0 4π 4π m = l

+ ∇s κ0 (zl , zm ) ·



0 SD l

∂Dl

1 2 ρ0 + (κ − κ02 )|Dl |−1 ρl − ρ0 8π l

 (· − zl )(s)dσl (s) Qm

−1



0 SD l

−1

·

∂Dl



 ,  iκl Qm 1 − Capl κ0 (zl , zm ) + (· − zl ) · ∇s κ0 (zl , zm ) (s)Al (s)dσl (s) 4π

m = l

⎡ ⎤  i ρ0 κ 2 (κ0 − κl )|Dl |−1 ⎣ κ0 (zl , zm ) Qm Capl ⎦ · + ρl − ρ0 32π 2 0 m = l



0 −1 (SD ) (1) (s)Al (s)dσl (s) + l

 κ02 κ0 (zl , zm ) Qm

m=1 m =l

∂Dl

+ κ02 |Dl |−1

M 

M 

∇x κ0 (zl , zm ) ·



m=1 m =l

 (x − zl )dx Qm

Dl

   −1    1 2 ρ0 2 −1 0 + ∇t κ0 (zl , zm ) · Vm (κl − κ0 )|Dl | SD 1 (s)Al (s)dσl (s) l ρl − ρ0 8π m = l

+

+

i ρ0 (κ 3 − κl3 )Capl ρl − ρ0 4π 0



∂Dl

∇t κ0 (zl , zm ) · Vm

m = l

M   κ02 ∇t κ0 (zl , zm ) · Vm m=1 m =l

= − κ02 uI (zl ) +

1 2 ρ0 (κ − κl2 )|Dl |−1 uI (zl ) ρl − ρ0 8π 0

  −1 * + 0 1 (s)Al (s)dσl (s) SD l

∂Dl

+

i ρ0 (κ 3 − κ03 )Capl uI (zl ) ρl − ρ0 4π l

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26

i ρ0 + κ 2 (κl − κ0 )|Dl |−1 Capl uI (zl ) ρl − ρ0 32π 2 0



0 −1 (SD ) (1) (s)Al (s)dσl (s) + Er6,l , l

∂Dl

where  ρl −1 1 2 2 −1 Er6,l = Er5,l + 1 − Err6l · Al (s)dσl (s) (κ − κ0 )|Dl | ρ0 8π l

(3.31)

∂Dl

  ρl −1 i ρl −1 1 2 Err9l · Al (s)dσl (s) − 1 − κ0 |Dl |−1 (κ 3 − κ03 )Err7l + 1− ρ0 8π ρ0 4π l ∂Dl

⎛ = O ⎝a + ⎛

 a3 d3 m = l ml

⎞

⎛

φm  + a φl  + a ψl ⎠ + O ⎝a + a φl  + 2

2

2

 a3 d3 m = l ml

⎞ φm ⎠

⎞ ⎛ ⎞  a3  a4 + O ⎝a 2 + a 2 φl  + φm ⎠ + O ⎝a 2 + a 3 φl  + φm ⎠ 2 d d3 m = l ml m = l ml ⎛ ⎞  a3 = O ⎝a + φm  + a 2 φl  + a 2 ψl ⎠ . 3 d m = l ml In the following lemma, we state some identities concerning the integrals with Al in the integrand. Lemma 3.5. The following identities hold true.   −1 0 [ SD (1)](s)Al (s) dσl (s) = −8π |Dl |, l

(3.32)

∂Dl

  −1 0 [ SD l (· − zl )](s)Al (s) dσl (s) = −8π (x − zl ) dx. ∂Dl

Dl

Proof. Observe that,     −1 −1 0 0 [ SD (1)](s)A (s) dσ (s) = −2 [ SD (1)](s) l l l l ∂Dl

Dl ∂Dl

1 dσl (s) dy |s − y|

  −1 0 = −8π [ SD (1)](s)0 (s, y) dσl (s) dy l Dl ∂Dl



= −8π

0 SD l Dl

,

0 SD l

−1

(1) (y) dy.

(3.33)

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H. Ammari et al. / J. Differential Equations ••• (••••) •••–••• 0 Now, by denoting f := SD l

27

,  −1 0 (1) , we can observe that f = 0 in Dl and f = 1 on SD l

¯ we can conclude ∂Dl . Since the boundary integral equation has a unique solution f = 1 in D, that   −1 0 [ SD (1)](s)Al (s) dσl (s) = −8π |Dl |. l ∂Dl

Similarly, we can prove (3.33).

2

Let us denote the average of Al as Aˆ l :=

1 |∂Dl |

Al (s) dσl (s). ∂Dl

It is easy to see from the definitions that both Al and Aˆ l scale as O(a 3 ) in L2 norm and therefore Al − Aˆ l L2 (∂Dl ) = O(a 3 ).

(3.34)

Remark 3.6. While dealing with the difference of Al and Aˆ l , one is tempted to believe that the difference behaves better than Al and such a result might be possible to derive using Poincaré type inequalities. But such a result is not true in this case. Indeed from the definition of Al , we note that 1 Al (s) = −2 dy. (3.35) |s − y| Dl

Now, using the Poincaré type inequality mentioned in [2, Proposition 3.2], we can conclude that there exists a positive constant CA independent of a such that 2     1   2 2 A − Aˆ l L2 (∂D ) ≤ CA a |∇Al (x)| dx ≤ 4CA a  dy  dx l  |x − y|2    Dl Dl Dl 2     1   ≤ 4CA a 6  dξ  dη = C˜ A a 6 ,  |ξ − η|2   Bl Bl

where 2     1   dξ C˜ A := 4CA   dη.  |ξ − η|2    Bl Bl

2

(3.36)

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28

Now by making use of (3.32) and splitting Al as Aˆ l + (Al − Aˆ l ), we can rewrite (3.30) as " |Dl |−1

,

1 ρl + ρl − ρ0 8π



 i ρ0 ρ0 i 3 (κl2 − κ02 ) − κ02 Aˆ l + κ + (κ 3 − κl3 ) ρ l − ρ0 4π 0 ρl − ρ0 4π 0

ρ0 1 i ρ0 (κ 3 − κ03 )(κ0 − κl )Capl − (κ 2 − κ02 )(κ0 − κl ) ρl − ρ0 16π 2 l ρl − ρ0 4π l # i ρ0 (κ0 − κl )κ02 Ql − ρl − ρ0 4π     1 ρ0 2 2 2 −1 + (κl − κ0 ) − κ0 |Dl | Al (s) − Aˆ l (s) φl (s)dσl (s) 8π ρl − ρ0

+

"

(3.37)

∂Dl

, ρ0 i iκl iκl ρ0 (κl2 − κ02 ) 1 − (κ03 − κl3 )Capl 1 − Capl − Capl ρl − ρ0 4π 4π ρ l − ρ0 4π # M  i 2 ρ0 κ0 (zl , zm )Qm κ0 (κ0 − κl )Capl + κ02 − ρl − ρ0 4π ,

+

m=1 m =l

"

i ρ0 + (κ 3 − κl3 ) ρl − ρ0 4π 0

  −1 0 (· − zl )(s)dσl (s) SD l

∂Dl

−

ρ0 (κl2 − κ02 )|Dl |−1 ρ l − ρ0

(x − zl )dx



Dl

+ κ02 |Dl |−1

Dl

" +

κ02

(x − zl )dx



#

·

M 

∇x κ0 (zl , zm )Qm

m=1 m =l

i ρ0 ρ0 − (κl2 − κ02 ) + (κ 3 − κl3 )Capl ρ l − ρ0 ρl − ρ0 4π 0

"

#

M 

∇t κ0 (zl , zm ) · Vm

m=1 m =l

i ρ0 ρ0 (κ02 − κl2 ) + (κ 3 − κ03 )Capl ρ l − ρ0 ρl − ρ0 4π l # i 2 ρ0 − κ (κl − κ0 )Capl uI (zl ) + Er6,l . ρl − ρ0 4π 0

= − κ02 −

To proceed further, we shall use the following result describing the structure of Vm . Let zl :=

1 s dσ l (s). |∂Dl | ∂Dl Proposition 3.7. The term Vm can be written as Vm := Vmdom + Vmrem ,

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29

where Vmdom :=

M  n=1 n =m

 1 −1 2 (zm − zm ) λm − κ0 |Dm |κ0 (zm , zn )Qn 2

1 2 1 κ (zm − zm )(λm − )−1 Aˆ m Qm 8π 0 2 1 2 1 ρm −1 − (zm − zm )(λm − )−1 Aˆ m Qm κ0 1 − 8π ρ0 2

−

−

(3.38)

M  1 2 1 ρm −1 (zm − zm )(λm − )−1 Aˆ m Capm κ0 (zm , zn )Qn κ0 1 − 8π ρ0 2 n=1 n =m

−

1 2 1 ρm −1 (zm − zm )(λm − )−1 κ0 1 − 8π ρ0 2 M   0 −1 (Am (s) − Aˆ m ) κ0 (zm , zn )Qn (SD ) (s)dσm (s), m n=1 n =m

∂Dm

and Vmrem satisfies the estimate |Vmrem | = O(a 3−γ ) + O



a 3 + a 4−γ +

 a 5−γ a 5−γ a4 a4 a 6−γ a 6−γ  + + + + + φ . d2 d 3α d 2 d 3α d3 d 3α+1 (3.39)

Proof. We refer to section 4.6 for a proof of this result.

2

dom Remark 3.8. We shall also sometimes, for the sake of the analysis, split the term Vmdom into Vm,1 dom , where and Vm,2

1 2 1 κ0 (zm − zm )(λm − )−1 Aˆ m Qm 8π 2 1 2 1 ρm −1 − (zm − zm )(λm − )−1 Aˆ m Qm , κ 1− 8π 0 ρ0 2

dom Vm,1 := −

(3.40)

and dom Vm,2 :=

M  n=1 n =m

−

 1 −1 2 (zm − zm ) λm − κ0 |Dm |κ0 (zm , zn )Qn 2

M  1 ρm −1 1 2 (zm − zm )(λm − )−1 Aˆ m Capm κ0 (zm , zn )Qn κ0 1 − 8π ρ0 2 n=1 n =m

(3.41)

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30

−

1 2 1 ρm −1 (zm − zm )(λm − )−1 κ0 1 − 8π ρ0 2 M   0 −1 (Am (s) − Aˆ m ) κ0 (zm , zn )Qn (SD ) (s)dσm (s). m n=1 n =m

∂Dm

dom , the terms in V dom contain a summation and this makes a Note that unlike the terms in Vm,1 m,2 difference in the manner we deal with these terms and therefore we chose to distinguish between them. 2

Next we need to deal with the quantities Vm and link them to Qm to derive a closed system involving only Qm . For this, we use the splitting of Vm into its dominant and remainder parts, Vmdom and Vmrem respectively, to obtain " |Dl |−1

,

1 ρl + ρl − ρ0 8π



 i ρ0 ρ0 i 3 (κl2 − κ02 ) − κ02 Aˆ l + κ0 + (κ 3 − κl3 ) ρ l − ρ0 4π ρl − ρ0 4π 0

ρ0 1 i ρ0 (κl3 − κ03 )(κ0 − κl )Capl − (3.42) (κ 2 − κ02 )(κ0 − κl ) 2 ρl − ρ0 16π ρl − ρ0 4π l # i ρ0 2 (κ0 − κl )κ0 Ql − ρl − ρ0 4π " , , i ρ0 iκl iκl ρ0 3 3 2 2 + (κ − κ0 ) 1 − (κ − κl )Capl 1 − Capl − Capl ρl − ρ0 4π 0 4π ρ l − ρ0 l 4π # M  i 2 ρ0 2 κ0 (zl , zm )Qm κ0 (κ0 − κl )Capl + κ0 − ρl − ρ0 4π +

"

m=1 m =l

i ρ0 (κ 3 − κl3 ) ρl − ρ0 4π 0

+

  −1 0 (· − zl )(s)dσl (s) SD l

∂Dl

ρ0 (κl2 − κ02 )|Dl |−1 − ρ l − ρ0

(x − zl )dx



Dl

+ κ02 |Dl |−1

Dl

" +

κ02

" =

− κ02

(x − zl )dx



#

·

M 

∇x κ0 (zl , zm )Qm

m=1 m =l

i ρ0 ρ0 − (κl2 − κ02 ) + (κ 3 − κl3 )Capl ρ l − ρ0 ρl − ρ0 4π 0 # ρ0 2 2 − (κ − κl ) uI (zl ) ρ l − ρ0 0

#

M  m=1 m =l

∇t κ0 (zl , zm ) · Vmdom

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−

1 8π



31

   ρ0 (κl2 − κ02 ) − κ02 |Dl |−1 Al (s) − Aˆ l (s) φl (s)dσl (s) ρ l − ρ0 ∂Dl

⎛

⎞

M ⎜ ⎟ 1 + O ((κ0 − κl )a) + O ⎜ |Vmrem |⎟ ⎝ ⎠ + Er6,l . 2 d m=1 ml m =l



 We next have to deal with the term ∂Dl Al (s) − Aˆ l (s) φl (s)dσl (s) in the above identity. The following lemma provides an approximation of this term in terms of Ql . Proposition 3.9. We have the following estimate 

M   ∇s κ0 (zl , zm )Qm Al (s) − Aˆ l φl dσl (s) = −Rl ·

(3.43)

m=1 m =l

∂Dl

−

    i ρ0 Capl κ0 (zl , zm )Qm 8π |Dl | + Aˆ l Capl (κ0 − κl ) Ql + ρl − ρ0 4π m = l ⎛ ⎞

M  ⎜ 4 ⎟ a3 3 a +O⎜ + a φm  + a 5 φl  + a 5 ψl ⎟ ⎝ ⎠, 3 d m=1 ml m =l

where Rl :=



∂Dl

0 λl I d + KD l

−1

 (Al (·) − Aˆ l ) (s)νl (s) dσl (s) and λl :=

Proof. We defer the proof of this result to section 4.5.

1 ρ0 +ρl 2 ρ0 −ρl .

2

Making use of Proposition 3.9 we can simplify (3.42) as below. " |Dl |−1

,

1 ρl + ρl − ρ0 8π



 i ρ0 ρ0 i 3 (κl2 − κ02 ) − κ02 Aˆ l + κ + (κ 3 − κl3 ) ρ l − ρ0 4π 0 ρl − ρ0 4π 0

1 i ρ0 ρ0 (κ 3 − κ03 )(κ0 − κl )Capl − (3.44) (κ 2 − κ02 )(κ0 − κl ) ρl − ρ0 16π 2 l ρl − ρ0 4π l # ρ0 i − (κ0 − κl )κ02 Ql ρl − ρ0 4π " , , i ρ0 iκl iκl ρ0 3 3 2 2 + (κ − κ0 ) 1 − (κ − κl )Capl 1 − Capl − Capl ρl − ρ0 4π 0 4π ρ l − ρ0 l 4π # M  ρ0 i 2 2 − κ0 (zl , zm )Qm κ0 (κ0 − κl )Capl + κ0 ρl − ρ0 4π +

m=1 m =l

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32

"

i ρ0 + (κ 3 − κl3 ) ρl − ρ0 4π 0

  −1 0 (· − zl )(s)dσl (s) SD l

∂Dl

ρ0 (κl2 − κ02 )|Dl |−1 − ρ l − ρ0

(x − zl )dx



Dl

+ κ02 |Dl |−1



(x − zl )dx



#

+

∇x κ0 (zl , zm )Qm

i ρ0 ρ0 − (κl2 − κ02 ) + (κ 3 − κl3 )Capl ρ l − ρ0 ρl − ρ0 4π 0

κ02

"

1 − 8π



#

M 

∇t κ0 (zl , zm ) · Vmdom

m=1 m =l

⎛

#

= − κ02 −

−

M  m=1 m =l

Dl

"

·

⎜ ρ0 (κ02 − κl2 ) uI (zl ) + O ((κ0 − κl )a) + O ⎜ ⎝ ρ l − ρ0

M  m=1 m =l

⎞ ⎟ 1 |Vmrem |⎟ ⎠ + Er6,l 2 dml

"  M  ρ0 2 2 2 −1 (κl − κ0 ) − κ0 |Dl | ∇s κ0 (zl , zm )Qm − Rl · ρ l − ρ0 m=1 m =l

    i ρ0 Capl κ0 (zl , zm )Qm 8π |Dl | + Aˆ l Capl (κ0 − κl ) Ql + ρl − ρ0 4π m = l

⎛

⎞

# M  ⎟ ⎜ 4 a3 3 5 5 ⎟ ⎜ φm  + a φl  + a ψl ⎠ . + O ⎝a + a d3 m=1 ml m =l

We can rewrite it further as, by making use of (3.8) and the expression for Vmdom , " −1

|Dl |

,

1 ρl + ρl − ρ0 8π



 i ρ0 ρ0 i 3 2 2 2 ˆ (κl − κ0 ) − κ0 Al + κ0 + (κ 3 − κl3 ) ρ l − ρ0 4π ρl − ρ0 4π 0

 i 2 ρ0 (κl − κ02 )(κ0 − κl ) + (κ0 − κl )κ02 (3.45) ρl − ρ0 4π #    i ρ0  ρ0 2 2 2 −1 − (κ0 − κl ) (κ − κ0 ) − κ0 |Dl | 8π |Dl | + Aˆ l Capl Ql 32π 2 ρ l − ρ0 l ρ l − ρ0 " , i ρ0 iκl ρ0 2 2 2 + κ0 − (κl − κ0 ) 1 − Capl + (κ 3 − κl3 )Capl ρ l − ρ0 4π ρl − ρ0 4π 0 −

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−

  i 2 ρ0 i ρ0 2 2 2 −1 (κ − κ ) (κ − κ ) − κ κ0 (κ0 − κl )Capl − 0 l 0 0 |Dl | · ρl − ρ0 4π ρ l − ρ0 l 32π 2 # M   ρ0  κ0 (zl , zm )Qm 8π|Dl | + Aˆ l Capl Capl ρ l − ρ0 m=1 m =l

" +

−

ρ0 (κl2 − κ02 )|Dl |−1 ρ l − ρ0



 

(x − zl )dx + κ02 |Dl |−1 (x − zl )dx

Dl

− " +

33

κ02

1 8π

 M  ρ0 (κl2 − κ02 ) − κ02 |Dl |−1 Rl · ∇x κ0 (zl , zm )Qm ρ l − ρ0 m=1 m =l

# i ρ0 ρ0 2 2 3 3 − (κ − κ0 ) + (κ − κl )Capl κ02 · ρ l − ρ0 l ρl − ρ0 4π 0

M  m=1 m =l



Dl

#

−

1 (zm − zm )(λm − )−1 ∇t κ0 (zl , zm )· 2

M  1  ρm −1 ˆ 1 ˆ κ0 (zm , zn )Qn Am Qm − 1− Am Qm + |Dm | 8π 8π ρ0 n=1 n =m

M  1  ρm −1 ˆ Am Capm − κ0 (zm , zn )Qn 1− 8π ρ0 n=1 n =m

M   1  ρm −1   0 −1 − k0 (zm , zn )Qn (Am (s) − Aˆ m )(SD ) (1)(s) dσ (s) 1− m m 8π ρ0 n=1 n =m

"

∂Dm

⎛

#

= − κ02 −

⎞

M ⎜ ⎟ 1 ρ0 (κ02 − κl2 ) uI (zl ) + O ((κ0 − κl )a) + O ⎜ |Vmrem |⎟ ⎝ ⎠ + Er6,l 2 ρ l − ρ0 d m=1 ml

⎛

m =l

⎞

M  ⎜ ⎟ a3 a + +O⎜ φm  + a 2 φl  + a 2 ψl ⎟ ⎝ ⎠ 3 d m=1 ml m =l

⎛

⎜ = − κl2 uI (zl ) + O ((κ0 − κl )a) + O ⎜ ⎝

⎞ M  m=1 m =l

⎟ 1 |Vmrem |⎟ ⎠ 2 dml

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34

⎛ ⎜ + Er6,l + O ⎜ ⎝a +

⎞ M 

a3

d3 m=1 ml m =l

⎟ φm  + a 2 φl  + a 2 ψl ⎟ ⎠ + O(ρl (κ0 − κl )),

where to get the last identity, we use the fact that ρ0 (κ 2 − κl2 ) = ρ l − ρ0 0

, 1−

ρl ρ0

-−1

(κl2 − κ02 ) = κl2 − κ02 + O (ρl (κ0 − κl )) .

(3.46)

Let us write " Jl :=

κ02

, i ρ0 iκl ρ0 2 2 − (κl − κ0 ) 1 − Capl + (κ 3 − κl3 )Capl ρ l − ρ0 4π ρl − ρ0 4π 0

(3.47)

ρ0 i 2 κ (κ0 − κl )Capl ρl − ρ0 4π 0 #    ρ0  i ρ0 2 2 2 −1 − (κ0 − κl ) (κ − κ0 ) − κ0 |Dl | 8π |Dl | + Aˆ l Capl Capl 32π 2 ρ l − ρ0 l ρ l − ρ0

−

= κl2 + O (ρl (κ0 − κl )) + iO ((κ0 − κl )a) , (using (3.46)), where using (3.46), it follows that the real part Jlr of Jl satisfies Jlr = κ02 −

  ρ0 ρl (κl2 − κ02 ) = κl2 + (κl2 − κ02 ) + O ρl2 (κ0 − κl ) . ρ l − ρ0 ρ0

Let "

  i 1 ρ0 ρ0 ρl i 3 2 2 2 ˆ Il := |Dl | + (κ − κ0 ) − κ0 Al + κ + (κ 3 − κl3 ) ρl − ρ0 8π ρl − ρ0 l 4π 0 ρl − ρ0 4π 0  ρ0 i 2 − (κl − κ02 )(κ0 − κl ) + (κ0 − κl )κ02 (3.48) ρl − ρ0 4π #    i ρ0  ρ0 2 2 2 −1 − (κ0 − κl ) (κ − κ0 ) − κ0 |Dl | 8π |Dl | + Aˆ l Capl 32π 2 ρ l − ρ0 l ρ l − ρ0 " ∞ ,  #   ρl n 1 ρl i 3 −1 2 2 2 ˆ + (κl − κ0 ) − κ0 Al + − κ = |Dl | ρl − ρ0 8π ρ0 4π 0 n=0 " - # ∞ ,  ρl n i − 1+ (κ 3 − κl3 ) ρ0 4π 0 n=1 " - # ∞ ,   ρl n i 2 (κl − κ02 )(κ0 − κl ) + (κ0 − κl )κ02 + 1+ ρ0 4π −1

,

n=1

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35

"" # - # ∞ ,  i ρl n 2 2 2 − (κ0 − κl ) 1 + (κl − κ0 ) + κ0 |Dl |−1 · ρ0 32π 2 n=1 " # , ∞ n    ρl 1+ 8π|Dl | + Aˆ l Capl ρ0 n=1   = Il + O a −1 (κ0 − κl )ρl2 + iO ((κ0 − κl )ρl ) , where   1 ρl i i 3 ρl + −κl2 − (κl2 − κ02 ) Aˆ l + κ − (κ 3 − κl3 ) ρl − ρ0 8π ρ0 4π 0 4π 0  i i 2 (κ0 − κl )κl2 |Dl |−1 (8π |Dl | + Aˆ l Capl ) (κl − κ02 )(κ0 − κl ) + (κ0 − κl )κ02 − + 4π 32π 2     ρ 1 ρl l = |Dl |−1 + −κl2 − (κl2 − κ02 ) Aˆ l ρl − ρ0 8π ρ0 3

κ  1 +i l − (κ0 − κl )κl2 |Dl |−1 Aˆ l Capl , (3.49) 2 4π 32π  



ρ0 (κl2 − κ02 )|Dl |−1 (x − zl )dx + κ02 |Dl |−1 (x − zl )dx Fl := − ρ l − ρ0

Il := |Dl |−1

,

−

1 8π



Dl



Dl

ρ0 (κ 2 − κ02 ) − κ02 |Dl |−1 Rl ρ l − ρ0 l

= Fl + O((κ0 − κl )ρl a)

(3.50)

(using (3.46)),

where Fl := κl2 |Dl |−1



 κ2 (x − zl )dx + l |Dl |−1 Rl . 8π

Dl

We can observe that Jl is not scaling. Also, let us denote the dominating term of Il by Il := |Dl |−1 d



1 2ˆ ρl − κ Al , ρl − ρ0 8π l

and the remaining terms (of O(1)) by Il := − s

Then

κ3  1 ρl 2 1 2 −1 ˆ (κl − κ02 )Aˆ l |Dl |−1 + i l − (κ − κ )κ |D | Cap A 0 l l l l l . 8π ρ0 4π 32π 2

(3.51)

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36

∞ −1  1 r 1 = Jl + iJli = r 2 Jlr − iJli (−1)n Jl (Jl )

"

=

#

∞ Jli  1 (−1)n 1 − i Jlr Jlr



n=0

= = =

1 + O((κ0 − κl )a) Jlr

Jli Jlr

2n



n=0

Jli Jlr

2n

(3.52)

[since Jli = O((κ0 − κl )a)]

1 1 + O((κ0 − κl )a) = 2 + O((κ0 − κl )ρl ) + O((κ0 − κl )a) + O((κ0 − κl )ρl ) κl

κl2

1 + O((κ0 − κl )a), κl2

and hence Fl Fl = 2 + O(a 2 ). Jl κl

(3.53)

Also note that using (3.46), we have " ρ 1 3 1 2 1 0 Jli = κl Capl (κl2 − κ02 ) + (κ − κl3 )Capl − κ (κ0 − κl )Capl ρl − ρ0 4π 4π 0 4π 0 #     1 ρ0 − (κ0 − κl ) (κ 2 − κ02 ) − κ02 |Dl |−1 8π |Dl | + Aˆ l Capl Capl 32π 2 ρ l − ρ0 l  ρ0 1 1 2 ˆ −1 2 = (κ − κ )κ |D | Cap (κ0 − κl )κl2 Capl + + O(aρl (κ0 − κl )) A 0 l l l l l ρl − ρ0 4π 32π 2 

= Jli + O((κ0 − κl )aρl ), where

(3.54)

 1 ρ0 1 ˆ (κ0 − κl )κl2 1 + Al |Dl |−1 Capl Capl . ρl − ρ0 4π 8π



Jli :=

(3.55)

Therefore, since Jli = O((κ0 − κl )a)), we can write ∞

−1  I Il = Il Jlr + iJli = rl 2 Jlr − iJli (−1)n Jl (Jl ) d

s

"

#

∞ Jli  (−1)n Jlr



n=0



Jli Jlr

2n

2n

=

I l + Il Jlr

=

 i 2   Jli 1 1 s d d d (Jl ) − iI − I I I + l l l l [κl2 + ρρ0l (κl2 − κ02 )] [κl2 + ρρ0l (κl2 − κ02 )] κl2 κl4

1−i

+ O((κ0 − κl )a).

n=0

Jli Jlr

(3.56)

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37

Using (3.46)-(3.50), we can rewrite (3.45) as M M  Il F  Ql + κ0 (zl , zm )Qm + l · ∇x κ0 (zl , zm )Qm Jl Jl m=1 m = l

m=1 m = l

" # i 1 2 ρ0 ρ0 2 2 3 3 + (κ − κ0 ) + (κ − κl )Capl κ02 · κ − J l 0 ρ l − ρ0 l ρl − ρ0 4π 0 M  m=1 m =l

−

1 (zm − zm )(λm − )−1 ∇t κ0 (zl , zm )· 2

M  1  ρm −1 ˆ 1 ˆ Am Qm + |Dm | κ0 (zm , zn )Qn Am Qm − 1− 8π 8π ρ0 n=1 n =m

−

M  1  ρm −1 ˆ κ0 (zm , zn )Qn 1− Am Capm 8π ρ0 n=1 n =m

−

M   1  ρm −1   0 −1 κ0 (zm , zn )Qn (Am (s) − Aˆ m )(SD ) (1)(s) dσ (s) 1− m m 8π ρ0 n=1 n =m

∂Dm

M  κ2 1 = − l uI (zl ) + O Er6 + O(a) + O(a 2 φl ) + Jl Jl

⎛

m=1 m = l

⎞



a3 3 dml

 φm 

M ⎜ ⎟ 1 2 + O(a 2 ψl ) + O ⎜ |Vmrem |⎟ ⎝ ⎠ + O((κ0 − κl )ρl ) + O((κ0 − κl )ρl φl ) 2 d m=1 ml m =l

+ O((κ0 − κl )aρl φl ) + O((κ0 − κl )a 2 ρl (

M   1 φ )) . m 2 d ml m=1

(3.57)

m =l

Now let us assume that ρl  a 1+γ , with γ ≥ 0. Then from (3.39), we can deduce that M  a 3−γ  1 a 3−γ  rem |V | = O + m d2 d 3α d2 m=1 ml m =l

+O

 a 3

a3 a 4−γ a 4−γ a4 a4 a4 a 5−γ a 5−γ a 5−γ + 2 + 3α + 4 + 2+3α + 6α + 4 + 2+3α + 6α 3α d d d d d d d d d   6−γ 6−γ 6−γ a a a + 5 + 3+3α + 6α+1 φ . (3.58) d d d d2

+

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38

Let us define C−1 l :=

1

ρ [κl2 + ρ l (κl2 −κ02 )] 0

Il − iIl d

d

 Jli ρ [κl2 + ρ l (κl2 −κ02 )] 0



+

1 κl2

Il − Il s

d



(Jli )2 κl4

 . Then, using

(3.46), (3.52) and (3.58), we can rewrite (3.57) as C−1 l Ql +

M 

κ0 (zl , zm )Qm +

m=1 m = l

+ κ02

M  m=1 m =l

Fl κl2

·

M 

∇x κ0 (zl , zm )Qm

m=1 m = l

M

 1 1 ˆ (zm − zm )(λm − )−1 ∇t κ0 (zl , zm ) · − κ0 (zm , zn )Qn Am Qm + |Dm | 2 4π n=1 n =m

M   1  ρm −1   0 −1 − κ0 (zm , zn )Qn Am (s)(SD ) (1)(s) dσ (s) 1− m m 8π ρ0 n=1 n =m

(3.59)

∂Dm

 a 3−γ a 3−γ  a3 a3 a 4−γ a 4−γ a4 a4 = −uI (zl ) + O a + 2 + 3α + a 2 + 3 + 3α+1 + 2 + 3α + 4 + 2+3α d d d d d d d d  a4 a 5−γ a 5−γ a 5−γ a 6−γ a 6−γ a 6−γ  + 6α + 4 + 2+3α + 6α + 5 + 3+3α + 6α+1 φ . d d d d d d d Remark 3.10. We note that the dominant part of C−1 l is given by

d

Il κl2

=

|Dl |−1 κl2

ρl ρl −ρ0

−

When γ < 1 or the frequency is away from the resonance, the sign of the real part C−1 is given by the sign of the term l r (C−1 l )

< 0, ∀ l = 1, . . . , M.

|Dl |−1 κl2

·

ρl ρl −ρ0

which is negative. Therefore in this case,

When the frequency ω is near the resonance, we can write 1 − we can write Il

d

κl2

=

Aˆ l |Dl |−1 ρkll 8π

"



1 2 ˆ 8π κl Al . r (C−1 l ) of

* ωM +2 ω

= lM a h1 . In this case,

# 2  Aˆ l |Dl |−1 ρkll ωM − 1 = − lM a h1 . 2 ω 8π

−1 r r Therefore if lM > 0, (C−1 l ) > 0, ∀ l = 1, . . . , M and if lM < 0, (Cl ) < 0, ∀ l = 1, . . . , M. 2

Lemma 3.11. For m = 1, · · · , M, let us define  a 3−γ a 3−γ  a3 a3 a 4−γ a 4−γ a4 Ym := −uI (zm ) + O a + 2 + 3α + a 2 + 3 + 3α+1 + 2 + 3α + 4 d d d d d d d   4 4 5−γ 5−γ 5−γ 6−γ 6−γ 6−γ a a a a a a a a + 2+3α + 6α + 4 + 2+3α + 6α + 5 + 3+3α + 6α+1 φ . d d d d d d d d Then the algebraic system (3.59) is invertible provided

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39

r • in the case when (C−1 l ) > 0, ∀ l = 1, . . . , M,

min Crm

1≤m≤M

( max |Cm |)2

≥

1≤m≤M

+ Ca 2−γ

.

    1 2 F  . 3τ  C MMmax 1 + 1 + max  m  2 4 5α 5π d 1≤m≤M κm d d 

MMmax

1 1 + d 4 d 5α

1

2

 + Ca 3−γ MMmax

1 1 + d 2 d 3α

1  2

1 1 2 1 + , d 4 d 5α (3.60)

r • in the case when (C−1 l ) < 0, ∀ l = 1, . . . , M,

    1  1 .  Fm  . 1 2 1 2 1 1   ≥ C MM + + max MM + C max max 2  1≤m≤M  κm ( max |Cm |)2 d 2 d 3α d 4 d 5α min |Crm |

1≤m≤M

1≤m≤M

+ Ca

2−γ

.

 MMmax

1 1 + d 4 d 5α

1

2

 + Ca

3−γ

MMmax

1 1 + d 2 d 3α

1  2

1 1 2 1 + , d 4 d 5α (3.61)

and the solution vector Qm , m = 1, ..., M, satisfies either the estimate M 

|Qm |2

m=1

≤ 4 ( max |Cm |)2 1≤m≤M

M 

|Ym |2

m=1

    1  3τ  Fm  . 1 2 1   C MMmax 4 + 5α − + max 2  max |Cm | 5π d 1≤m≤M  κm d d

min |Crm |  1≤m≤M 1≤m≤M

1 1 2 1 + Ca MMmax 4 + 5α d d  1  1 −2 1 2 1 1 2 1 + Ca 3−γ MMmax 2 + 3α + |C | , max m 4 1≤m≤M d d d d 5α 2−γ

.



where τ := minl = m cos(κ|zm − zl |) ≥ 0, or the estimate M 

|Qm |2

m=1

≤ 4 ( max |Cm |)2 1≤m≤M

M  m=1

|Ym |2

(3.62)

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40

    1  1  . 2 F  . 1 2 1  C MMmax 1 + 1 + max  m − C MMmax 2 + 3α  2 1≤m≤M κm max |Cm | d d d 4 d 5α

min |Crm |  1≤m≤M 1≤m≤M

1 1 2 1 + Ca MMmax 4 + 5α d d  1  1 −2 1 2 1 1 2 1 + Ca 3−γ MMmax 2 + 3α + |C | , max m 1≤m≤M d d d 4 d 5α 2−γ



.

(3.63)

where C is a constant (depending on the Lipschitz characters of the obstacles) uniformly bounded with respect to a. Proof. We refer to section 4.7 for a proof of this lemma.

2

Remark 3.12. From (3.62), we conclude that M 

|Qm |

m=1

≤ 2M ( max |Cm |) max |Ym | 1≤m≤M

1≤ m≤ M

    1  3τ  Fm  . 1 2 1   − + max C MMmax 4 + 5α 2  max |Cm | 5π d 1≤m≤M  κm d d

min  1≤m≤M

|Crm |

1≤m≤M

 1 . 1 2 1 MMmax 4 + 5α d d  1  1 −1 1 2 1 1 2 1 3−γ + Ca MMmax 2 + 3α + |C | , max m 1≤m≤M d d d 4 d 5α + Ca 2−γ

(3.64)

while (3.63) yields M 

|Qm |

m=1

≤ 2M ( max |Cm |) max |Ym | 1≤m≤M

1≤ m≤ M

    1  1  .  Fm  . 1 2 1 2 1 1   + max  2  C MMmax 4 + 5α − C MMmax 2 + 3α 1≤m≤M κm max |Cm | d d d d

min  1≤m≤M

|Crm |

1≤m≤M

1 1 2 1 + Ca MMmax 4 + 5α d d  1  1 −1 1 2 1 1 2 1 3−γ + Ca MMmax 2 + 3α + |C | . 2 max m 1≤m≤M d d d 4 d 5α 2−γ

.



(3.65)

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41

Remark 3.13. We note that if     1

3τ 2 F  .  C MMmax 1 + 1 ( max |Cm |) + max  m  2 4 5α 1≤m≤M 5πd 1≤m≤M κm d d 1    1  1 . 1 2 1 2 1 1 2 1 1 2−γ 3−γ + Ca MMmax 4 + 5α + Ca MMmax 2 + 3α + = O(1), d d d d 4 d 5α d then the condition (3.60) holds. Since in this case the frequency is near the resonance ωM (and lM > 0), this is equivalent to the condition a 1−h1 · O(a −t + a 1−s−t + a 2−γ −s−t + a 3−γ −2s−t ) = O(1).

(3.66)

This leads to the additional conditions 2 − h1 − s − t ≥ 0 and 1 − t − h1 ≥ 0 provided γ + s ≤ 2. As in this case γ = 1 and hence s ≤ 1, the conditions reduce to 1 − t − h1 ≥ 0 and s ≤ 1. Note that (3.66) also gives rise to the condition 2(a) in Theorem 1.2. Similarly if     1  1

.  Fm  . 1 2 1 2 1 1   ( max |Cm |) C MMmax 2 + 3α + max  2  C MMmax 4 + 5α 1≤m≤M 1≤m≤M κm d d d d  1  1  1 . 1 2 1 2 1 1 2 1 1 + Ca 2−γ MMmax 4 + 5α + Ca 3−γ MMmax 2 + 3α + = O(1), d d d d 4 d 5α d then the condition (3.61) holds true. When the frequency is away from the resonance ωM , and as we take 3αt = s with α ∈ (0, 1], this is equivalent to the condition a 2−γ · O(a −s + a 1−s−t + a 2−γ −s−t + a 3−γ −2s−t ) = O(1), which holds if γ + s ≤ 2, 3s ≤ t ≤ 1 and 0 ≤ γ ≤ 1. This also gives rise to the condition 1(a) in Theorem 1.2. If the frequency is near the resonance ωM (and lM < 0), and as we take 3αt = s with α ∈ (0, 1], then the condition is equivalent to a 1−h1 · O(a −s + a 1−s−t + a 2−γ −s−t + a 3−γ −2s−t ) = O(1),

(3.67)

which holds provided 1 − h1 − s ≥ 0, γ + s ≤ 2, 0 ≤ t ≤ 1, 0 ≤ γ ≤ 1 and 3s ≤ t . Again as here also γ = 1 and hence s ≤ 1, these conditions reduce to 1 − h1 − s ≥ 0 and 3s ≤ t ≤ 1. Also note that (3.67) gives rise to the condition 1(b) in Theorem 1.2. The condition 2(b) follows from the fact that a condition similar to (3.67) can also be derived when the coefficients Cm are positive. 2

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42

From (3.64) or (3.65) and the definition of Ym , we can derive the a priori estimate M 

|Qm |

m=1



a 3−γ a 3−γ  a3 a3 a 4−γ a 4−γ = O M max |Cm | 1 + a + 2 + 3α + a 2 + 3 + 3α+1 + 2 + 3α d d d d d d a4 a4 a4 a 5−γ a 5−γ a 5−γ a 6−γ + + + + + + d 4 d 2+3α d 6α d4 d 2+3α d 6α d5   6−γ 6−γ a a + 3+3α + 6α+1 φ d d

  3−γ −2t + a 3−γ −s + a 2 + a 3−3t + a 3−s−t = O M max |Cm | 1 + a + a +

+ a 4−γ −2t + a 4−γ −s + a 4−4t + a 4−s−2t + a 4−2s + a 5−γ −4t

  + a 5−γ −s−2t + a 5−γ −2s + a 6−γ −5t + a 6−γ −s−3t + a 6−γ −2s−t φ

    (3.68) = O M max |Cm | 1 + a + a 2 + a 3−3t + a 3−s−t + a 4−2s φ , assuming that 0 ≤ t < 12 , 0 ≤ s ≤ 32 , 0 ≤ γ ≤ 1 and s + γ ≤ 2. Using (3.68), we also note that Fl  · ∇x κ0 (zl , zm )Qm κl2 m = l

(3.69)

M a   =O 2 |Qm | d m=1 

   1−2t =O a M max |Cm | 1 + a + a 2 + a 3−3t + a 3−s−t + a 4−2s φ ,

−

M 1 2 1 (zm − zm )(λm − )−1 ∇t κ0 (zl , zm )Aˆ m Qm κ0 4π 2

(3.70)

m=1 m =l

=

M  a 2−γ 

d2

 |Qm |

m=1



   = O a 2−γ −2t M max |Cm | 1 + a + a 2 + a 3−3t + a 3−s−t + a 4−2s φ , and κ02

M  m=1 m =l

M

 1 (zm − zm )(λm − )−1 ∇t κ0 (zl , zm ) · |Dm | κ0 (zm , zn )Qn 2 n=1 n =m

(3.71)

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−

43

M   1  ρm −1   0 −1 κ0 (zm , zn )Qn Am (s)(SD ) (1)(s) dσ (s) 1− m m 8π ρ0

⎛

⎛

n=1 n =m

∂Dm

⎞

⎞

  , - M M  1  1 1 1 3−γ ⎝ ⎠ = O ⎝a |Qn |⎠ = O a + |Qn | d d 3 d 3α+1 d2 m =l ml n=1 n=1 

   = O a 3−γ −s−t M max |Cm | 1 + a + a 2 + a 3−3t + a 3−s−t + a 4−2s φ . 3−γ

Therefore we can rewrite (3.59) as C−1 l Ql

+

M 

(3.72)

κ0 (zl , zm )Qm

m=1 m = l

    = −uI (zl ) + O a + a 3−γ −s + a 2 + a 3−3t + a 3−s−t + a 4−2s φ  + O (a 1−2t + a 2−γ −2t + a 3−γ −s−t )M max |Cm |·  

 1 + a + a 2 + a 3−3t + a 3−s−t + a 4−2s φ . 3.5. The final approximations Proposition 3.14. Let the parameters t, s, γ satisfy the conditions 0 ≤ t < 12 , 0 ≤ s ≤ 32 , 0 ≤ γ ≤ 1, s + γ ≤ 2, 3s ≤ t and let M max |Cm | = O(a −h ), h < 12 . Then for every l, we have φl  = O(a −γ ) + O(a −γ −h ). Proof. We refer to section 4.8 for a proof of this result.

2

Remark 3.15. In case γ < 1 or when the frequency is away from the resonance, we have max |Cm | = O(a 2−γ ) whence it follows using s +γ ≤ 2 that M max |Cm | = O(a 2−γ −s ) = O(1). Therefore in this case, we have h ≤ 0 and then φl  = O(a −γ ). In contrast with this case, near the resonance maxl |Cl | = O(a 1−h1 ) where h1 ≥ 0. Therefore M maxl |Cl | = O(a 1−s−h1 ) = O(a −h ). Now if lM < 0, we need the condition 1 − h1 − s ≥ 0, see Remark 3.13, which implies that h ≤ 0 and then φl  = O(a −γ ). But if lM > 0, we have only the condition 1 − h1 ≥ t and s ≤ 1 and it is possible to have −1 + s + h1 > 0 and hence h > 0. 2 Combining (3.72) and Proposition 3.14, we deduce that under the conditions 0 ≤ t < 12 , 0 ≤ s ≤ 32 , 0 ≤ γ ≤ 1, s + γ ≤ 2 and 1 − 2t − h > 0 with h < 12 ,9 the vectors (Ql )M l=1 satisfy 9 Here, we do the computations when h ≥ 0 in which case φ  = O(a −γ −h ). For the case h < 0, we have φ  = l l O(a −γ ).

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44

C−1 l Ql +

M 

κ0 (zl , zm )Qm

m=1 m = l

  = −uI (zl ) + O a + a 2−γ −h + a 3−γ −3t−h + a 3−γ −s−t−h + a 4−γ −2s−h 

+ O (a 1−2t + a 2−γ −2t + a 3−γ −s−t )a −h 1 + a + a 2−γ −h + a 3−γ −h−3t (3.73)  + a 3−γ −h−s−t + a 4−γ −h−2s   = −uI (zl ) + O a + a 2−γ −h + a 3−γ −h−3t + a 3−γ −h−s−t + a 4−γ −h−2s + a 1−2t−h   = −uI (zl ) + O a 4−γ −h−2s + a 1−2t−h , where C−1 l =

 i 2 

d  Jli 1 1 s  d d (Jl ) − iI − I I I + l l l l [κl2 + ρρ0l (κl2 − κ02 )] [κl2 + ρρ0l (κl2 − κ02 )] κl2 κl4

(3.74)

with Il := Il + Il  := |Dl |−1 d

s

   1 ρl 2 ρl 2 2 + −κl − (κl − κ0 ) Aˆ l ρl − ρ0 8π ρ0

κ3  1 2 −1 ˆ +i l − (κ − κ )κ |D | Cap A 0 l l l l l 4π 32π 2

and 

Jli =

 1 ρ0 1 ˆ (κ0 − κl )κl2 1 + Al |Dl |−1 Capl Capl . ρl − ρ0 4π 8π

We recall that we have the representation of the scattered field of the form us (x, θ ) =  κ0 M m=1 SDm φm . From this we can deduce that the far-field can be approximated as u∞ (x, ˆ θ) =

M 

∞ ˆ zm )Qm + κ0 (x,

m=1

M 

∇∞ ˆ zm ) · Vm + O(Ma 3−γ ). κ0 (x,

m=1

Now under the assumptions on t, s, γ , h, using (3.39), we see that M 

∇∞ ˆ zm ) · Vmrem κ0 (x,

m=1

= O(M|Vmrem |)

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45

 = O a 3−γ −s + a 4−2γ −s−h + a 5−2γ −s−h−2t + a 5−2γ −h−2s + a 4−γ −s−h−2t + a 4−γ −h−2s + a 6−2γ −s−h−3t + a 6−2γ −h−2s−t   = O a 3−γ −s + a 4−2γ −s−h + a 1−h + a 2−s−h .



Similarly M 

dom ∇∞ ˆ zm ) · Vm,1 κ0 (x,

m=1

=−

, , M  κ02  ρm −1 1 −1 ˆ ∇∞ ( x, ˆ z ) 1 + 1 − − z ) λ − (z Am Qm m m m m κ0 8π ρ0 2 

=O a

m=1

2−γ



M 

∇∞ ˆ zm )Qm κ0 (x,

,

m=1 M 

dom ∇∞ ˆ zm ) · Vm,2 κ0 (x,

M 

= κ02

m=1

m=1 m =l



1  ρm −1 |Dm | − 1− 8π ρ0

1 (zm − zm )(λm − )−1 ∇∞ ˆ zm )· κ0 (x, 2

⎜ 0 −1 ⎜ Am (s)(SD ) (1)(s) dσ (s) m ⎝ m

∂Dm

= κ02

M  m=1 m =l

M  n=1 n =m

⎟ κ0 (zm , zn )Qn ⎟ ⎠

1 (zm − zm )(λm − )−1 · ∇∞ ˆ zm )· κ0 (x, 2

|Dm | − 

= −κ02

⎞

⎛

1  ρm −1 1− 8π ρ0



 0 −1 Am (s)(SD ) (1)(s) dσ (s) m m

∂Dm

  I 4−γ −h−2s − C−1 + a 1−2t−h m Qm − u (zm ) + O a

M  m=1 m =l

1 (zm − zm )(λm − )−1 · ∇∞ ˆ zm ) κ0 (x, 2



1  ρm −1 0 −1 Am (s)(SD ) (1)(s) dσm (s) C−1 1− |Dm | − m Qm m 8π ρ0 



∂Dm

+ O(a 3−γ −s ) + O a 3−γ −s a 4−γ −h−2s + a 1−2t−h

 .

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46

   ∞ ˆ z )Q Note that the first term in the right-hand side above is of order O a M m m . m=1 ∇κ0 (x, Thus under the conditions 0 ≤ t < 12 , 0 ≤ s ≤ 32 , 0 ≤ γ ≤ 1, s + γ ≤ 2 and 1 − 2t − h > 0 with h < 12 , using (3.68) and Proposition 3.14, we have the following estimate for the far-field: u∞ (x, ˆ θ) =

M 

(3.75)

∞ ˆ zm )Qm κ0 (x,

m=1

m=1

, λm −

, M  κ02  ρm −1 ∞ − ∇κ0 (x, ˆ zm ) 1 + 1 − (zm − zm )· 8π ρ0

1 2

-−1

Aˆ m Qm − κ02

M  m=1 m =l

1  ρm −1 1− |Dm | − 8π ρ0



1 (zm − zm )(λm − )−1 · ∇∞ ˆ zm )C−1 κ0 (x, m Qm 2

 0 −1 Am (s)(SD ) (1)(s) dσ (s) m m

∂Dm

   + O(a 3−γ −s ) + O a 3−γ −s a 4−γ −h−2s + a 1−2t−h   + O a 3−γ −s + a 4−2γ −s−h + a 1−h + a 2−s−h =

M  m=1

 ∞ ˆ zm )Qm κ0 (x, 

+O a

M 

 ∇∞ ˆ zm )Qm κ0 (x,

m=1



+ O a 3−γ −s a 4−γ −h−2s + a 1−2t−h =

M 



+ O(a 3−γ −s )

  + O a 3−γ −s + a 4−2γ −s−h + a 1−h + a 2−s−h

∞ ˆ zm )Qm + O(a 1−h + a 5−2s−γ −2h ) + O(a 3−γ −s ) κ0 (x,

m=1

     + O a 3−γ −s a 4−γ −h−2s + a 1−2t−h + O a 4−2γ −s−h + a 2−s−h . Next let us observe that for l = 1, . . . , M, we can write C−1 l = O(a −1+γ ). ˜ l )M satisfy the algebraic system Let the vectors (Q l=1 Il

d

˜ + Q 2 l

κl

M 

d

Il κl2

κ0 (zl , zm )Q˜ m = −uI (zl ).

+ Reml , where |Reml | =

(3.76)

m=1 m = l

˜ M Note that the dominant terms in the systems satisfied by (Ql )M l=1 and (Ql )l=1 are same and therefore the algebraic system (3.76) is invertible under the same conditions.

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47

˜ l − Ql )M satisfies the algebraic system From (3.73) and (3.76), we observe that the vector (Q l=1 Il

d

(Q˜ l − Ql ) + 2

κl

M 

κ0 (zl , zm )(Q˜ m − Qm ) = Reml Ql + O(a 4−γ −2s−h + a 1−2t−h ).

m=1 m = l

Therefore  2 " M # M  κ2      l 2 2 4−γ −2s−h 1−2t−h 2 |Q˜ l − Ql | ≤ C max  d  |Reml | |Ql | + |O(a +a )| l I  l l=1 l=1 l=1 " # M M   2 2 2 4−γ −2s−h 1−2t−h 2 ≤ C(max |Cl |) max |Reml | |Ql | + |O(a +a )| 

M 

2

l

l

l=1

l=1

2 ≤ CM(max |Cl |)4 (max |Reml |)2 O(1 + a 3−s−t−γ −h + a 4−γ −2s−h )

l

l

2

+ CM(max |Cl |)2 O(a 4−γ −2s−h + a 1−2t−h ) , l

where C is some generic constant. Hence M 

2 ˜ l − Ql | |Q

l=1

≤M

M 

2

|Q˜ l − Ql |2 ≤ CM 2 (max |Cl |)4 (max |Reml |)2 O(1 + a 3−s−t−γ −h + a 4−γ −2s−h ) l

l=1

l

2

+ CM 2 (max |Cl |)2 O(a 4−γ −2s−h + a 1−2t−h ) , l

which gives M 

˜ l − Ql | |Q

(3.77)

l=1

 ≤ CM(max |Cl |)2 (max |Reml |) O(1 + a 3−s−t−γ −h + a 4−γ −2s−h ) l

l

 + CM(max |Cl |) O(a 4−γ −2s−h + a 1−2t−h ) l



≤ CM(max |Cl |)2 a −1+γ O(1 + a 3−s−t−γ −h + a 4−γ −2s−h ) + O(a 4−γ −2s−2h + a 1−2t−2h ). l

Using (3.77) in (3.75), we obtain

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48 M 

u∞ (x, ˆ θ) =

˜m + ∞ ˆ zm )Q κ0 (x,

m=1

M 

∞ ˆ zm )(Qm − Q˜ m ) + O(a 1−h + a 5−2s−γ −2h ) κ0 (x,

m=1





+ O(a 3−γ −s ) + O a 3−γ −s a 4−γ −h−2s + a 1−2t−h M 

=





(3.78) 

+ O a 4−2γ −s−h + a 2−s−h

˜ m + O(a 1−h + a 5−2s−γ −2h ) ∞ ˆ zm )Q κ0 (x,

m=1

     + O(a 3−γ −s ) + O a 3−γ −s a 4−γ −h−2s + a 1−2t−h + O a 4−2γ −s−h + a 2−s−h 

+ M(max |Cl |)2 a −1+γ O(1 + a 3−s−t−γ −h + a 4−γ −2s−h ) l

+ O(a

4−γ −2s−2h

+ a 1−2t−2h ).

Finally from (3.78), we can conclude the following: • We recall that when γ < 1 or we are away from the resonance, we have M maxl |Cl | = O(a 2−γ −s ) and hence h = γ + s − 2 ≤ 0 (see Remark 3.15). Also maxl |Cl | = O(a 2−γ ). Therefore (3.78) reduces to u∞ (x, ˆ θ) =

M 

∞ ˆ zm )Q˜ m + O(a + a 5−2s−γ ) κ0 (x,

(3.79)

m=1

     + O a 4−2γ −s + a 2−s + O(a 3−γ −s ) + O a 3−γ −s a 4−γ −2s + a 1−2t

+ a 3−γ −s O(1 + a 3−s−t−γ + a 4−γ −2s ) + O(a 4−γ −2s + a 5−2γ −2s−2t ) =

M 

∞ ˆ zm )Q˜ m + O(a 2−s + a 4−γ −2s + a 5−2γ −2s−2t ) κ0 (x,

m=1

=

M 

∞ ˆ zm )Q˜ m + O(a 2−s + a 3−γ −s−2t ), κ0 (x,

m=1

where in the last line we use the fact s + γ ≤ 2. • We recall that near the resonance maxl |Cl | = O(a 1−h1 ) and therefore O(a −h ) = M maxl |Cl | = O(a −s+1−h1 ) leading to the condition h = −1 + s + h1 . Also since γ = 1, we obtain that M(maxl |Cl |)2 a −1+γ = O(a s−2h ). Then from (3.78), we conclude that u∞ (x, ˆ θ) =

M  m=1

(3.80)

∞ ˆ zm )Q˜ m + O(a 1−h + a 5−2s−γ −2h ) κ0 (x,

     + O(a 3−γ −s ) + O a 3−γ −s a 4−γ −h−2s + a 1−2t−h + O a 4−2γ −s−h + a 2−s−h

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49

+ a s−2h O(1 + a 3−s−t−γ −h + a 4−γ −2s−h ) + O(a 4−γ −2s−2h + a 1−2t−2h ) =

M 

∞ ˆ zm )Q˜ m + O(a 1−h + a 2−s−2h + a 1−2t−2h + a s−2h ) κ0 (x,

m=1

=

M 

∞ ˆ zm )Q˜ m + O(a 2−s−h1 + a 4−3s−2h1 + a 3−2t−2s−2h1 + a 2−s−2h1 ) κ0 (x,

m=1

=

M 

∞ ˆ zm )Q˜ m + O(a 3−2t−2s−2h1 + a 2−s−2h1 ) κ0 (x,

m=1

where in the last line we use the fact that s ≤ 1 (as we have s + γ ≤ 2 and γ = 1). 1. When lM < 0, then we have seen that we need that h ≤ 0, see Remark 3.15 and Remark 3.13). Hence in this case we need s + h1 ≤ 1. 2. When lM > 0, we need only h < 12 which means that s + h1 < 32 . Note that the error term in (3.80) goes to zero provided s, h1 and t satisfy the condition s < min{2 − 2h1 ,

3 − 2t − 2h1 }. 2

(3.81)

These conditions are fulfilled if 1 s + h1 ≤ 1, h1 < 1 and t < . 2

(3.82)

These last conditions are the regimes in which one can derive the effective media. Actually, in the case lM > 0, one can allow s + h1 > 1 (but s + h1 < min{ 32 − t, 2 − h1 }) and hence generate very large effective potentials. Remark 3.16. Note that in the algebraic system (3.76), unlike that in the case of (3.73), we have d

only used the (leading) term the algebraic system

Il κl2

˜ M instead of C−1 l . Now suppose that the vectors (Ql )l=1 satisfy

˜ C−1 l Ql +

M 

κ0 (zl , zm )Q˜ m = −uI (zl ).

m=1 m = l

Then proceeding as in the derivation of (3.80), when the frequency is near the Minnaert resonance, we can obtain the improved estimate u∞ (x, ˆ θ) =

M 

∞ ˆ zm )Q˜ m + O(a 3−2t−2s−2h1 + a 2−s−h1 ). κ0 (x,

m=1

The error term in the above estimate goes to zero under the same conditions as in the case of (3.80). This can be observed if we take Reml = 0 and continue as in the derivation of (3.80) d

from the algebraic system (3.76), now obviously replacing

Il κl2

by C−1 l .

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50

4. Proofs of auxiliary results 4.1. Proof of Proposition 3.1: Let us define operators T , T0 : X → Y by   φ1 , ψ1 , ..., φM , ψM  →



κ1 SD ψ 1 1

−

M 

κ SD0l φl

l=1

    

,

∂D1

1   ρ0 1 κ0 ∗ κ1 ∗ ) − − ) I d + (KD I d + (K ψ φ1 − 1 D1 1 ρ1 2 2 

κM SD ψ − M M

M  l=1

   κ SD0l φl  

 

 κ M  ∂(SD0l φl )   ∂ν 1  l=1  l =1 ∂D

, ..., 1

,

∂DM

M

1    ρ0 1 κ0 ∗ κM ∗ ) − − ) − I d + (KD I d + (K ψ φ M M D M M ρM 2 2

l=1 l =M

 



 κ ∂(SD0l φl )   ∂ν M   

∂DM

and 

 φ1 , ψ1 , ..., φM , ψM

→

   0 0 SD ψ − S φ  1 1 D1 1

1   ρ0 1 0 ∗ 0 ∗ ) − − ) I d + (KD I d + (K ψ φ1 , ..., 1 D1 1 ∂D1 ρ1 2 2

1      ρ0 1  0 0 0 ∗ 0 ∗ ψ − S φ , ) − − ) I d + (K I d + (K SD ψ φM .  M DM M DM DM M M ∂DM ρM 2 2 ,

We note that the operator T − T0 : X → Y is a compact operator. To see this, first of all we observe that the typical terms in (T − T0 )(φ1 , ψ1 , ..., φM , ψM ) are of the form −

M 

κ0 κi κ0 0 0 SD φ + (SD − SD )ψi − (SD − SD )φi i i l l i i

l=1 l =i

and −

κ0 M  ∂(SD φl ) l

l=1 l =i

∂ν i

+

  ρ 0 κi ∗ κ0 ∗ 0 ∗ 0 ∗ ) − (K ) − (K ) (KDi ) − (KD ψ φi . i Di Di i ρi

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51

κ

κi 0 ), (S 0 − S 0 ) are compact and The first term gives rise to a compact operator since (SD − SD Di Di i i κ also for l = i, SD0l φl is compact. A similar argument works for the second term as well, and therefore it follows that T − T0 is a compact operator. Also note that the operator T0 is invertible. This follows directly as in the proof of Theorem 11.4, Page 189, of [9], observing that T0 is diagonal with a 2 × 2 operator, at the diagonal, of the form

(φl , ψl ) →



  0 0 SD ψ − S φ  l l D l l

∂Dl

1    ρ0 1 0 ∗ 0 ∗ ) − − ) I d + (KD I d + (K ψ φl . l D l l ρl 2 2

,

Therefore to prove that T is invertible, it is sufficient to prove that T is injective, as an application of the Fredholm alternative. Let us therefore suppose that T (φ1 , ψ1 , ..., φM , ψM ) = 0. Then M u(x) =

κ0 l=1 SDl φl (x),

x ∈ R3 ∪M l=1 Dl

κs SD ψ (x), x ∈ Ds s s

is the unique solution to the problem under consideration. Now ∂u  ∂u  ρ0 ρ0 ¯ = (|∇u|2 − κi2 |u|2 ) dx  u¯ dσ =  udσ ∂ν + ρi ∂ν − ρi ∂Di

∂Di

Di

which is real and therefore

∂u   u¯ dσ = 0, ∂ν +

Im ∂Di

which in turn implies that u = 0 in R3  ∪M l=1 Dl . Again, u solves the equation ( + κi2 )u = 0 in Di , u=

∂u = 0 on ∂Di ∂ν i

and hence by unique continuation property, u = 0 in Di . Arguing similarly for each obstacle Di , we obtain that u = 0 in R3 . In particular, this implies that M 

κ0 SD φ (x) = 0 on ∂Di . l l

l=1

  κ0 2 Since + κ02 ( M l=1 SDl φl ) = 0 in Di and κ0 is not a Dirichlet eigenvalue for − on Di (which holds since a  1), it follows that

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52

M 

κ

SD0l φl = 0 in Di

l=1

and therefore since this is true for each i, we have M 

κ

SD0l φl = 0 in R3 .

(4.1)

l=1 ∂SD0 φl  κ

Now for a fixed i, we know that whenever l = i, we have

l ∂ν i

obtain using (4.1), φi =

 = +

∂SD0 φl  κ

l

∂ν i

 and therefore we −

κ κ ∂SD0i φi  ∂SD0i φi  −   = 0 on ∂Di . ∂ν i + ∂ν i −

Next we prove that ψi = 0 on ∂Di , for all i. Since we already know that φi = 0 ∀ i, it follows κi from the form of the solution u and the zero jump condition that SD ψ = 0 on ∂Di . Using this i i   κi κi 2 3 ψ = 0 in R3 Di . fact and the fact that + κi SDi ψi = 0 in R Di , we obtain that SD i i κi Recalling the fact that we have already proved that u = 0 in Di and hence SD ψ = 0 on ∂Di , it i i κi 3 therefore follows that SDi ψi = 0 in R . Now we can proceed just in the case of φl by using the κ Neumann jump of SDii ψi across Di to obtain that ψi = 0.

4.2. Proof of Lemma 3.2  −1 0 In order to prove (3.7), we first set SD l δ n+2

|ˆs − tˆ|n φˆ l (tˆ) d σˆ l (tˆ) =

∂Bl









n ∂Dl | · −t| φl (t)dσl (t)

=: χl . As



 0 0 |s − t|n φl (t) dσl (t) = SD χ χ ˆ (s) = δ S (ˆs ), l l Bl l

(4.2)

∂Dl

then χˆ l = δ n+1 (SB0 l )−1



 | · −tˆ|n φˆ l (tˆ) d σˆ l (tˆ) ,

∂Bl

where [SB0 l χˆ l ](ˆs ) := Therefore



1 ˆ ∂Bl 4π|ˆs −tˆ| χˆ l (t )

d σˆ l (tˆ) and d σˆ l denotes the surface measure on ∂Bl .

% % 1 % % n+1 0 −1 χl L2 (∂Dl ) = χˆ l L2 (∂Bl ) = δ (SBl ) L(H 1 (∂Bl ),L2 (∂Bl )) % | · −tˆ|n φˆ l (tˆ) d σˆ l (tˆ)% 1 H (∂Bl ) δ ∂Bl

= O(δ whence it follows that

n+1

φˆ l L2 (∂Bl ) ) = O(δ φl L2 (∂Dl ) ), n

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% % % %  % % −1 % % 0 | · −t|n φl (t)dσl (t) % % SDl % % % % ∂Dl

53

  = χl L2 (∂Dl ) = O δ n+1 φl L2 (∂Dl )

L2 (∂Dl )

  = O a n+1 φl L2 (∂Dl ) .

  0 )−1 (· − z )n = η . Then Similarly, to prove (3.8), we write (SD l l l



 0 0 δ n sˆ n = (s − zl )n = SD η η ˆ (s) = δ S s ), Bl l (ˆ l l which implies ηˆ l = δ n−1 (SB0 l )−1 (·n ), and hence 1 ηl L2 (∂Dl ) = ηˆ l L2 (∂Bl ) = δ n−1 (SB0 l )−1 (·n )L2 (∂Bl ) δ   0 −1 n ⇒(SD ) ) (· − z L2 (∂Dl ) = ηl L2 (∂Dl ) = δ n (SB0 l )−1 (·n )L2 (∂Bl ) = O(a n ). l l 4.3. Proof of Lemma 3.3 In order to prove (3.9), we proceed as follows.

 κ0 κm [κ0 − κm ](x, s)ψm (s) dσm (s) SDm − SDm ψm (x) = ∂Dm

i = (κ0 − κm ) 4π



∂Dm

ψm (s) dσm (s) +

∞ n n n)  i (κ − κm 0

n=2

&

4πn!

|x − s|n−1 ψm (s) dσm (s)

∂Dm

'( ) =:Err1m ⎛ ⎞ ∞ n n i ⎜ (κ0 + κm ) n−1 ⎟ = ψm (s) dσm (s) + O ⎝ |ψm (s)| dσm (s)⎠ (κ0 − κm ) a 4π 4πn! n=2 ∂Dm ∂Dm   " # 2 2 2 κ0 i a κm = ψm (s) dσm (s) + O (κ0 − κm ) + ψm  4π 4π 1 − κ0 a 1 − κm a ∂Dm   i = ψm (s) dσm (s) + O a 2 ψm  , (4.3) (κ0 − κm ) 4π ∂Dm

whence (3.9) follows. Note that here we use the fact that |κ0 a|, |κm a| < 12 , which holds since a  1.

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54

In order to prove (3.10), we first observe that

κ ∗ (KD ) ψ(s)dσl (s) =

∂Dl



"

κ ψ(s)(KD )(1)(s)dσl (s) =

∂Dl

=

ψ(s)

∂Dl

"

# ∂ κ (s, t)dσl (t) dσl (s) ∂νt

∂Dl

#

∇t κ (s, t) · νt dσl (t) dσl (s).

ψ(s)

∂Dl

∂Dl

Then we can write

κ0 ∗ ) (KD l

∂Dl

"

=

∇t ∂Dl

∂Dl

ψl (s)dσl (s) =

∞ n n  i (κ − κ n )



⎡ ⎢ ψl (s) ⎣

∂Dl 0

l

4π n!

1 = (κ02 − κl2 ) 8π



(n − 1)

⎢ ψl (s) ⎣

∂Dl

+

l

n=4

4π n!

0

· νt dσl (t) dσl (s) − t) |s − t|

n−2 (s

(n − 1)|s − t|

#



· νt dσl (t) dσl (s)

" 



(s − t) |s − t|n−2 |s − t|

ψl (s) ∂Dl



# · νt dσl (t) dσl (s)

⎤

(s − t) i ⎥ · νt dσl (t)⎦ dσl (s) − (κ 3 − κl3 )|Dl | |s − t| 4π 0

(n − 1) ∂Dl

&

|s − t|

∂Dl

∂Dl

∞ n n  i (κ − κ n )

#



n−1

(s − t) ⎥ · νt dσl (t)⎦ dσl (s) |s − t|

⎡

∂Dl

⎤



∂Dl

∞ n n  i (κ − κ n ) n=3

0

4π n!

n=2

∂Dl

1 2 (κ − κl2 ) 8π 0 +

− κln )

4π n!

#  ∇t κ0 − κl (s, t) · νt dσl (t) dσl (s) 

ψl (s)

∂Dl ∞ n n  i (κ0

l

ψl (s)

"



n=1

" 

=





ψl (s)

∂Dl

=

κl ∗ − (KD ) l

" 

|s − t|

ψl (s) ∂Dl

− t) |s − t|

n−2 (s



#



· νt dσl (t) dσl (s),

'(

=:Err2l =O(a 5 ψl )

where the last step follows from the divergence theorem. Thus (3.10) follows. Next we establish (3.11). For this, we first observe that κ0 ∂SDm φm  ∂ν l ∂Dl

 ∂κ0 (s, t) (t) dσ (t) dσl (s) φ (s) dσl (s) = m m ∂ν l (s) ∂Dl

∂Dm

∂Dm

∂Dl

ψl (s)dσl (s)

∂Dl

 ∂κ0 (s, t) = (s) φm (t)dσm (t) dσ l ∂ν l (s)

)

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=



∂Dm

55

 κ0 (x, t) dx φm (t) dσm (t)

Dl

= −κ02



∂Dm

 κ0 (x, t) dx φm (t) dσm (t).

Dl

We can write this as κ0 ∂SDm φm  ∂ν l ∂Dl

(s) dσl (s)



= −κ02

κ0 (zl , t)|Dl |φm (t)dσm (t)

∂Dm



∇x κ0 (zl , t) ·

− κ02



∂Dm

− κ02



∂Dm

&

 (x − zl )dx φm (t)dσm (t)

Dl

 (κ0 (x, t) − κ0 (zl , t) − (x − zl ) · ∇x κ0 (zl , t))dx φm (t)dσm (t)

Dl

'( 6 κ =:G10 =O( a3 ml dml

) φm )

which further implies that κ0 ∂SDm φm  ∂ν l ∂Dl

(s) dσl (s)



= −κ02

κ0 (zl , zm )|Dl |φm (t)dσm (t)

∂Dm



|Dl |∇y κ0 (zl , zm ) · (t − zm )φm (t)dσm (t)

− κ02 ∂Dm



− κ02 &

 κ |Dl | κ0 (zl , t) − κ0 (zl , zm ) − ∇t κ0 (zl , zm ) · (t − zm ) φm (t)dσm (t) −G10ml

∂Dm

'( κ

=:G20 =O( ml

− κ02 ∂Dm

∇x κ0 (zl , zm ) ·

Dl

) a6 3 dml

φm )

 (x − zl )dx φm (t)dσm (t)

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56

− κ02 &

[∇x κ0 (zl , t) − ∇x κ0 (zl , zm )] ·

∂Dm



 (x − zl )dx φm (t)dσm (t)

Dl

)

'( 6 κ =:G30 =O( a3 ml dml

φm )

and therefore κ0 ∂SDm φm  ∂ν l

(4.4)

(s) dσl (s)

∂Dl



= −κ02 κ0 (zl , zm )|Dl |

φm (t)dσm (t)

∂Dm



− κ02 |Dl |∇t κ0 (zl , zm ) · − κ02 ∇x κ0 (zl , zm ) ·



(t − zm )φm (t)dσm (t)

∂Dm

(x − zl )dx

Dl

 ∂Dm

φm (t)dσm (t) − [Gκ10ml + Gκ20ml + Gκ30ml ], & '( ) =:Err4m =O(

a6 3 dml

φm )

whence (3.11) follows. The proof of (3.12) follows easily from the observation

∂uI 2 I 2 = − κ |D | u (z ) −κ (uI (y) − uI (zl ) dy . l l 0 0 ∂ν l Dl '( ) &

∂Dl

(4.5)

=:Err5l =O(a 4 )

We shall next derive the approximations (3.13) and (3.14). First of all, we note that

dκ

eiκ0 |s−y| − 1 φl (y) dσl (y) = 4π |s − y|

SDl0 φl (s) =

∂Dl n=1

∂Dl



iκ0 = 4π

 ∞ 1 (iκ0 )n φl (y) dσl (y) + |s − y|n−1 φl (y) dσl (y) 4π n! ∂Dl n=2

∂Dl

 iκ =

0

4π Ql iκ0 4π Ql

+ O(a 2 φl ), in L∞ and H 1 , + O(a 3 φl ), in L2 .

From (3.5), we have that on ∂Dl , ψl (s) =



 κ −1 SDll

 ∞ 1 (iκ0 )n |s − y|n−1 φl (y) dσl (y) (4.6) 4π n!

" u + I

M  m=1

# κ SD0m φm

(s)

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57

⎡ ⎤ M −1 ⎢  κ ⎥ 0 ⎢uI + S κ0 φl + = SD SD0m φm ⎥ Dl ⎣ ⎦ (s) l 

m=1 m =l

⎡  −1 ⎢ 0 ⎢ + SD ⎣ l

⎛ ⎞⎤ , M  −1 n ⎜  κ ⎟⎥ dκ 0 0 ⎜uI + S κ0 φl + ⎟⎥ (s) (−1)n SDll SD S φ m D D ⎝ ⎠⎦ l m l

∞  n=1

m=1 m =l

⎡

⎤

M  −1 ⎢  ⎥ κ 0 ⎢uI (zl ) + (s − zl ) · ∇uI (zl ) + O(a 2 ) + S 0 φl + S dκ0 φl + = SD SD0m φm ⎥ Dl Dl ⎣ ⎦ (s) l m=1 m =l

⎡  −1 ⎢ 0 ⎢ + SD ⎣ l

⎛ ⎞⎤ , M  −1 n ⎜  κ ⎟⎥ dκ 0 ⎜uI + S 0 φl + S dκ0 φl + ⎥ (−1)n SDll SD SD0m φm ⎟ Dl Dl ⎝ ⎠⎦ (s), l

∞  n=1

where the approximation above is in

eiκl |s−t| −1 ∂Dl 4π|s−t| φ(t) dσl (t). Using (3.8) and (4.6), we can write this as

m=1 m =l

a

point-wise

sense

dκ

SDll φ(s) :=

and

 −1    −1  d  dκl κ0 0 I 0 2 ψl (s) = φl (s) + SD (z ) + S − S u S l Dl Dl Dl φl (s) + O(a) + O(a φl ) l ⎡ ⎤ ⎡ ⎛ ⎞⎤ , ∞ M M  −1 n ⎜  ⎢ 0 −1  ⎥  0 −1 ⎢ ⎟⎥ κ0 κ ⎥+ S ⎢ (−1)n S dκl S 0 ⎜ ⎥ S S φ SD0m φm ⎟ +⎢ m D D D Dm Dl ⎣ ⎦ ⎣ ⎝ ⎠⎦ l l l m=1 m =l

n=1

m=1 m =l

dκ

dκ

κ

κ

with error estimates in L2 sense. This further implies, using (SDl0 − SDll )φl = (SD0l − SDll )φl and (3.9), that  −1   i(κ − κ )  −1 0 l 0 I 0 ψl (s) = φl (s) + SD (z ) + (1)(s) Ql SD u l l l 4π ⎞ ⎛ ∞ n (κ n − κ n )  −1  i 1 ⎟ ⎜ 0 l 0 |s − y|n−1 φl (y) dσl (y)⎠ (s) + SD ⎝ l 4π n! ∂Dl n=2

⎡ ⎤ ⎡ ⎛ ⎞⎤ , ∞ M M n       −1 ⎢ −1 ⎢ 0 −1  κ0 ⎥ ⎜  κ0 ⎟⎥ 0 ⎢ (−1)n S dκl S 0 ⎜ ⎥ +⎢ SDm φm ⎥ SDm φm ⎟ Dl Dl ⎣ SDl ⎦ + SDl ⎣ ⎝ ⎠⎦ m=1 m =l

n=1

+ O(a) + O(a 2 φl ), in L2 , and hence, keeping only the first terms in the two infinite series,

m=1 m =l

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 −1   i(κ − κ )  −1 0 l 0 I 0 ψl (s) = φl (s) + SD (z ) + (1)(s) Ql SD u l l l 4π ⎞ ⎛ i 2 (κ02 − κl2 )  0 −1 ⎜ ⎟ + SDl ⎝ | · −y|φl (y) dσl (y)⎠ (s) 8π ∂Dl

⎡ ⎤ ⎡ ⎛ ⎞⎤ M M ⎢ 0 −1  κ0 ⎥  0 −1 ⎢ dκl  0 −1 ⎜  κ0 ⎟⎥ ⎢S ⎜ ⎥ +⎢ SDm φm ⎥ SDm φm ⎟ ⎣ SD l ⎦ − SDl ⎣ D l SDl ⎝ ⎠⎦ m=1 m =l

⎛ ⎜ +O⎜ ⎝

M 

a3

d2 m=1 lm m =l

m=1 m =l

⎞

⎟ 2 φm ⎟ ⎠ + O(a) + O(a φl ).

Using (3.7), we can write this as ⎡ ⎤ M  −1   i(κ − κ )  −1   ⎢ 0 −1  κ0 ⎥ 0 l 0 0 ψl (s) = φl (s) + SD (1)(s) + ⎢ SDl SDm φm ⎥ Ql SD uI (zl ) + ⎣ ⎦ l l 4π ⎡

⎛

⎞⎤

⎛

⎞

m=1 m =l

M M  −1 ⎢ d  −1 ⎜  ⎜ ⎟ ⎟⎥ a3 κ0 0 ⎢S κl S 0 ⎜ ⎟⎥ + O ⎜ S φ φm ⎟ − SD m Dl Dm ⎝ ⎠ ⎣ Dl ⎝ ⎠⎦ l 2 d m=1 m=1 lm m =l

m =l

+ O(a) + O(a φl ) in L . 2

2

(4.7)

For a better understanding of the dominating terms in (4.7), we next estimate the last two terms. Using Taylor series expansion, for s ∈ ∂Dl and t ∈ ∂Dm , we can write 1 κ0 (s, t) = κ0 (zl , t) + (s − zl ) · ∇s κ0 (zl , t) + (s − zl )2 · ∇s ∇s κ0 (zl , t) 2  |α| + (s − zl )α (1 − β)|α|−1 Dsα κ0 (zl + β(s − zl ), t)dβ α! 1

|α|=3

0

= κ0 (zl , zm ) + (t − zm ) · ∇t κ0 (zl , zm ) + (s − zl ) · ∇s κ0 (zl , zm ) 1 + (s − zl )2 · ∇s ∇s κ0 (zl , t) 2  |α| + (t − zm )α (1 − r)|α|−1 Dtα κ0 (zl , zm + r(t − zm )) dr α! 1

|α|=2

0

(4.8)

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 |α| + (s − zl ) · (t − zm )α (1 − r)|α|−1 Dtα ∇s κ0 (zl , zm + r(t − zm )) dr α! 1

|α|=1

+

0

 |α| (s − zl )α α!

|α|=3

1

(1 − β)|α|−1 Dsα κ0 (zl + β(s − zl ), t) dβ,

0

which gives us κ0 (SD φ )(s) = κ0 (zl , zm ) Qm + ∇t κ0 (zl , zm ) · Vm + (s − zl ) · ∇s κ0 (zl , zm ) Qm m m  1 + (s − zl )2 · ∇s ∇s κ0 (zl , t) 2

(4.9)

∂Dm

 |α| + (t − zm )α (1 − r)|α|−1 Dtα κ0 (zl , zm + r(t − zm )) dr α! 1

|α|=2

0

 |α| α + (s − zl ) · (1 − r)|α|−1 Dtα ∇s κ0 (zl , zm + r(t − zm )) dr (t − zm ) α! 1

|α|=1

0

 1  |α| α |α|−1 α + (1 − β) Ds κ0 (zl + β(s − zl ), t) dβ φm (t)dσm (t). (s − zl ) α! |α|=3

0

Using (3.8) and (4.9), we have  −1 κ 0 (SD0m φm )(s) SD l

(4.10)

 −1  −1 0 0 (1)(s) + ∇  (z , z ) · V (1)(s) SD = κ0 (zl , zm ) Qm SD t κ l m m 0 l l  (· − zl ) (s) · ∇s κ0 (zl , zm ) Qm  "  −1 1 0 + SDl (· − zl )2 · ∇s ∇s κ0 (zl , t) 2 +



0 SD l

−1

∂Dm

 |α| α + (1 − r)|α|−1 Dtα κ0 (zl , zm + r(t − zm )) dr (t − zm ) α! 1

|α|=2

0

 |α| + (s − zl ) · (t − zm )α (1 − r)|α|−1 Dtα ∇s κ0 (zl , zm + r(t − zm )) dr α! 1

|α|=1

0

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60

#  1  |α| α |α|−1 α + (1 − β) Ds κ0 (zl + β(· − zl ), t) dβ φm (t)dσm (t) (s). (· − zl ) α! |α|=3

0

Therefore 

0 SD l

−1

κ

(SD0m φm )(s)

(4.11)

 −1  −1 0 0 (1)(s) + ∇  (z , z ) · V (1)(s) SD = κ0 (zl , zm ) Qm SD t κ l m m 0 l l   

 −1 a3 0 (· − zl ) (s) · ∇s κ0 (zl , zm ) Qm + O φm  in L2 , + SDl 3 dml , where the last term in (4.10), which is of order O a dml

= O(1). To estimate the other term in (4.7), we note that

a4 4 φm  dml

-

, , is absorbed in O

a3 3 φm  dml

 −1 d  −1 κ κ 0 0 SDll SD (SD0m φm )(s) SD l l   −1 iκ  −1 l κ 0 0 (SD0m φm )(t) dσl (t) SD = SD l l 4π ∂Dl

  ∞  −1 1 (iκl )n κ0 n−1 0 (SDm φm )(t) dσl (t) |s − t| SDl + 4π n! ∂Dl n=2

=



0 SD l

+



−1

iκl 4π

)2

(iκl 8π

  −1 κ 0 (SD0m φm )(t) dσl (t) SD l ∂Dl

 −1 κ0 0 |s − t| SD (SD φ )(t) dσl (t) l m m

∂Dl

  ∞  −1 1 (iκl )n κ0 n−1 0 (SDm φm )(t) dσl (t) |s − t| SDl + 4π n! ∂Dl n=3

=



0 SD l

−1



iκl 4π

  −1 κ 0 (SD0m φm )(t) dσl (t) SD l ∂Dl

(iκl )2 + 8π

|s − t|



0 SD l

−1

 κ (SD0m φm )(t) dσl (t)

∂Dl

Using (4.11), we can further write this as

+ O(

a3 φm ), in L2 . 2 dml

as

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61

 −1 dκ κ 0 SDll SD (SD0m φm )(s) l  iκl  0 −1 = S κ0 (zl , zm ) Qm Capl + ∇t κ0 (zl , zm ) · Vm Capl 4π Dl       −1 a4 0 + ∇s κ0 (zl , zm ) Qm · (· − zl ) (t)dσl (t) + O φm  SDl 3 dml L∞

0 SD l

−1

∂Dl

(iκl )2  0 −1 SD l + 8π

∂Dl



=



  −1  a3  κ0 0 |s − t| SDl (SDm φm )(t) dσl (t) + O 2 φm  dml

iκl κ0 (zl , zm ) Qm Capl + ∇t κ0 (zl , zm ) · Vm Capl 4π        −1 −1 a4 0 0 + ∇s κ0 (zl , zm ) Qm · (· − zl ) (t)dσl (t) + O φ  (1) SDl SD m l 3 dml L∞ ∂Dl

(iκl )2  0 −1 + SD l 8π



∂Dl

  −1  a3  κ 0 0 |s − t| SD (S φ )(t) dσ (t) + O φ  , in L2 , m l m Dm l 2 dml

where by (·)L∞ we mean the point-wise error estimate. Therefore, using (3.8) and the fact Vm = O(a 2 φm ), we get 

0 SD l

−1

 −1  −1 iκl dκ κ0 0 0 SDll SD (S φ )(s) = (z , z ) Q Cap (1)(s) (4.12)  SD m κ l m m l 0 Dm l l 4π    −1 (iκl )2  0 −1 a3 κ0 0 + |s − t| SD (S φ )(t) dσ (t) + O( φm ) SD l m l D l 2 m 8π dml ∂Dl

=

 −1 a3 iκl 0 (1)(s) + O( φm ), in L2 , κ0 (zl , zm ) Qm Capl SD l 2 4π dml

where we have also used the estimate % % %  %  % % −1  −1 % % 0 κ0 0 | · −t| SD (S φ )(t))dσ (t) (s) % % SDl m l Dm l % % % % ∂Dl

, =O

a4 φm  , dml

L2 (∂Dl )

the proof of which follows by a similar argument as in the proof of Lemma 3.2. Now, by making use of (4.11)-(4.12) in (4.7), we obtain  −1   i(κ − κ )  −1 0 l 0 I 0 ψl (s) =φl (s) + SD (z ) + (1)(s) Ql SD u l l l 4π

(4.13)

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62

+

M 

"

 −1  −1 0 0 (1)(s) + ∇  (z , z ) · V (1)(s) SD κ0 (zl , zm ) Qm SD t κ l m m 0 l l

m=1 m =l

#

 −1   −1 iκl 0 0 + SD (· − z ) (s) · ∇  (z , z ) Q − (z , z ) Q Cap (1)  SD l s κ l m m κ l m m l 0 0 l l 4π ⎛ ⎞ M ⎜ ⎟ a3 2 + O(a) + O(a 2 φl ) + O ⎜ φm ⎟ ⎝ ⎠ , in L , 3 d m=1 lm m =l

whence (3.13) follows. The approximation (3.14) can then be obtained by integrating (4.13) on ∂Dl as follows. (ψl − φl ) = Capl uI (zl ) + ∂Dl

+

M 

i(κ0 − κl ) Ql Capl 4π

(4.14)

" κ0 (zl , zm ) Qm Capl + ∇t κ0 (zl , zm ) · Vm Capl

m=1 m =l

iκl − κ (zl , zm ) Qm Capl2 + ∇s κ0 (zl , zm ) Qm · 4π 0 ⎛

⎞

#   −1 0 (· − zl )(s)dσl (s) S Dl

∂Dl

M  ⎜ 2 ⎟ a4 3 a +O⎜ + a φ  + φm ⎟ l ⎝ ⎠, 3 d m=1 lm

&

'(

m =l

)

=:Err7l

where Err7l is in the point-wise sense. 4.4. Proof of Lemma 3.4 The proof of (3.25) follows as in the case of (3.10). To prove (3.26), we note that from the definition of gl , we have ⎛ gl (s) =

i κ0 −1 ⎜ ) ⎝ (κ0 − κl )(SD l 4π



⎞ ⎟ ψl ⎠ (s)

∂Dl

⎛ ⎞ # ∞    n ⎜ i dκ ⎟ 0 −1 0 −1 0 −1 = (κ0 − κl ) (SD ) + (−1)n (SD ) ) SDl0 (SD ⎝ ψl ⎠ (s) l l l 4π "

n=1

∂Dl

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⎛ =

i 0 −1 ⎜ ) ⎝ (κ0 − κl )(SD l 4π

63

⎞



⎟ ψl ⎠ (s) + O(a 2 ψl ), in L2 ,

∂Dl

where we use (3.7) applied to ψl with n = 0 and the fact that for any f ∈ L2 (∂Dl ), dκ SDl0 f H 1 (∂Dl ) = O(a 2 f L2 (∂Dl ) ). Using (4.13) and (4.14), we can write this as ⎡ gl (s) =

i ⎢ (κ0 − κl ) ⎣ 4π



⎤

⎞ 3  a ⎥ 0 −1 ψl ⎦ (SD ) (1) (s) + O ⎝a 2 + a 2 φl  + φm ⎠ , in L2 l dml ⎛

m = l

∂Dl

⎡ i ⎢ = (κ0 − κl ) ⎣Capl uI (zl ) + 4π ⎛ + O ⎝a 2 + a 2 φl  +

⎤





φl +

∂Dl

m = l

(4.15)

⎥ 0 −1 κ0 (zl , zm ) Qm Capl ⎦ (SD ) (1) (s) l

⎞

 a3 φm ⎠ , in L2 , 2 d m = l ml

whence (3.26) follows. 4.5. Proof of Proposition 3.9 First of all, we note that using the definition of gl and g˜ l and (3.9), (3.22) can be rewritten as 0 ∗ −1 φl = − [λl I d + (KD ) ] l

∂uI ∂ν l

+

κ0 M  ∂(SD φ ) κ m m  0 ∗ + [(KD0l )∗ − (KD ) ]φl (4.16)  l ∂Dl ∂ν l

m=1 m =l

 ρl −1 κl ∗ κ0 ∗ + 1− [(KD ) − (KD ) ]ψl l l ρ0   −1    ρl −1 1 κ κ κ κl + 1− [ I d + (KD0l )∗ ] SD0l SD0l − SD ψl , on ∂Dl . l ρ0 2 Then we have 

 Al (s) − Aˆ l φl (s) dσl (s)

∂Dl

"  −1   ∂uI 0 =− (s) λl I d + KDl Al (·) − Aˆ l (s) ∂ν l ∂Dl

(4.17)

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64

    −1   ρl −1 1 κl ∗ κ0 ∗ κ0 ∗ κ0 κ0 κl + 1− SDl − SDl ψl (s) [(KDl ) − (KDl ) ]ψl (s) + [ I d + (KDl ) ] SDl ρ0 2 # κ M  ∂(SD0m φm )  κ0 ∗ 0 ∗ +  (s) + [(KDl ) − (KDl ) ]φl (s) dσl (s). ∂Dl ∂ν l m=1 m =l

To estimate the terms in the right hand side of equation (4.17), we make use of (3.34) and the  −1 0 maps L20 to L20 and is uniformly facts that Al − Aˆ l ∈ L20 , and the operator λl I d + KD l bounded with respect to a and proceed as follows. By Cauchy-Schwarz inequality and as Al − Aˆ l L2 (∂Dl ) = O(a 3 ), we have 

0 λl I d + KD l

−1

 ∂uI (Al (·) − Aˆ l ) (s) l (s)dσl (s) = O(a 4 ), ∂ν

(4.18)

∂Dl κl ∗ κ0 ∗ and as (KD ) − (KD ) L2 (∂Dl ) = O(a 2 ), we get l l



0 λl I d + KD l

−1

 κ κ (Al (·) − Aˆ l ) (s) [(KDll )∗ − (KD0l )∗ ]ψl (s)dσl (s)

(4.19)

∂Dl

= O(a 3 · a 2 ψl ) = O(a 5 ψl ). Similarly, we have   −1 κ 0 0 ∗ (Al (·) − Aˆ l ) (s) [(KD0l )∗ − (KD ) ]φl (s)dσl (s) = O(a 5 φl ). (4.20) λl I d + KD l l ∂Dl κ

Next for s ∈ ∂Dl , we use the expansion

∂(SD0m φm ) (s) = ∇x κ0 (zl , zm ) · νl (s)Qm + O ∂ν l

,

a3 3 φm  dml

-

with the error estimate in the L2 sense, to deduce that  κ M  −1  ∂(SD0m φm ) 0 ˆ (A (·) − A ) (s) (s)dσl (s) λl I d + KD l l l ∂ν l m=1 m =l

∂Dl

= Rl ·

⎛

⎞

M 

M ⎜ 3 ⎟ a3 a ∇s κ0 (zl , zm )Qm + O ⎜ φm ⎟ ⎝ ⎠, 3 d m=1 m=1 ml m =l

m =l

where the term Rl behaves as O(a 4 ) and is defined as   −1 0 Rl := (Al (·) − Aˆ l ) (s)νl (s) dσl (s). λl I d + KD l ∂Dl

(4.21)

(4.22)

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65

The remaining term can be dealt with in the following manner. First we write   −1  −1    κ0 κl 0 ˆ l (s) [ 1 I d + (K κ0 )∗ ] S κ0 (·) − A − S λl I d + KD A S l Dl Dl Dl Dl ψl (s) dσl (s) l 2 ∂Dl

=

1 2

− λl

 

0 λl I d + KD l

−1    −1   κ κ κl Al (·) − Aˆ l (s) SD0l SD0l − SD ψl (s) dσl (s) l

∂Dl

   −1   κ κ κl + Al (s) − Aˆ l SD0l SD0l − SD ψl (s) dσl (s) l ∂Dl

+



0 λl I d + KD l

−1   κ 0 ∗ ) ] Al (·) − Aˆ l (s) [(KD0l )∗ − (KD l

∂Dl

 −1   κ κ κl SD0l SD0l − SD ψl (s) dσl (s) l =

1 2

− λl

 

0 λl I d + KD l

−1   Al (·) − Aˆ l (s)

∂Dl

,  ∞ −1    −1 dκ n  κ0 κl 0 0 0 (−1)n SD S − SD SD SD ψl (s)dσl (s) Dl l l l l n=0

" ,  - #  ∞   −1 dκ n n 0 + (−1) SDl0 Al (·) − Aˆ l (s) I d + SDl n=1

∂Dl

 −1   κ0 κl 0 − S SD S Dl Dl ψl (s)dσl (s) l +



0 λl I d + KD l

−1   κ 0 ∗ ) ] Al (·) − Aˆ l (s) [(KD0l )∗ − (KD l

∂Dl

 −1   κ κ κ SD0l SD0l − SDll ψl (s)dσl (s) =

ρl O(a 4 ψl ) + O(a 5 ψl ) + O(a 6 ψl ) ρ l − ρ0    −1   κ0 κl 0 − S Al (s) − Aˆ l SD S + Dl Dl ψl (s)dσl (s) l ∂Dl

=

     −1 i 0 ψl (t) dσl (t) (1)(s)dσl (s) + O(a 5 ψl ) (κ0 − κl ) Al (s) − Aˆ l SD l 4π ∂Dl

∂Dl

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66

+



Al (s) − Aˆ l



⎤ ⎡ ∞ n n n) −1  i (κ − κ ⎥ ⎢ l 0 0 |t − s|n−1 ψl (t) dσl (t)⎦ dσl (s), SD ⎣ l 4πn! n=2

∂Dl

∂Dl

where we use Lemma 3.3 and the fact that ρl ∼ a 1+γ , γ ≥ 0. Using (3.7) and (3.32), we can then deduce   −1  −1    1 κ κ κ κ 0 λl I d + KD Al (·) − Aˆ l (s) [ I d + (KD0l )∗ ] SD0l SD0l − SDll ψl (s) dσl (s) l 2 ∂Dl

     −1 i 0 ψl (t)dσl (t) (1)(s) dσl (s) (κ0 − κl ) Al (s) − Aˆ l SD = l 4π ∂Dl

∂Dl

+ O(a ψl ) + O(a ·a ψl )   i ψl (t) dσl (t) + O(a 5 ψl ). = − (κ0 − κl ) 8π|Dl | + Aˆ l Capl 4π 5

3

2

(4.23)

∂Dl

Now by substituting (4.18)-(4.23) in (4.17) and using (3.14), we have   Al (s) − Aˆ l φl (s)dσl (s)

(4.24)

∂Dl

= −Rl ·

M 

∇s κ0 (zl , zm )Qm

m=1 m =l

  ρl −1 i ˆ + 1− ψl (s) dσl (s) (κ0 − κl ) 8π|Dl | + Al Capl ρ0 4π 

∂Dl

⎞

⎛ ⎜ 4 3 +O⎜ ⎝a + a

= −Rl ·

M 

M 

a3

d3 m=1 ml m =l

⎟ φm  + a 5 φl  + a 5 ψl ⎟ ⎠

∇s κ0 (zl , zm )Qm

m=1 m =l

     ρl −1 i + 1− Capl κ0 (zl , zm )Qm 8π |Dl | + Aˆ l Capl (κ0 − κl ) Ql + ρ0 4π m = l ⎞ ⎛ M  ⎟ ⎜ 4 a3 3 a + a φm  + a 5 φl  + a 5 ψl ⎟ +O⎜ ⎠, ⎝ 3 d m=1 ml m =l

whence (3.43) follows.

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67

4.6. Proof of Proposition 3.7 We note that using (4.16), we can write

(s − zl )p φl dσl (s) = −

∂Dl

0 ∗ −1 (s − zl )p [λl I d + (KD ) ] l

∂Dl

'(

&

p

∂uI dσl (s) ∂ν l )

(4.25)

:=El

−

M 

∂(SD0m φm )  dσl (s)  ∂Dl ∂ν l κ

(s − zl )p [λl I d

m=1 ∂D l m =l

&

0 ∗ −1 + (KD ) ] l

'(

)

p

:=Bφl

−

0 ∗ −1 0 ∗ (s − zl )p [λl I d + (KD ) ] [(KD0l )∗ − (KD ) ]φl dσl (s) l l κ

∂Dl

&

'(

)

p

=:Kφl



ρl −1 − 1− ρ0



κl ∗ 0 ∗ −1 (s − zl )p [λl I d + (KD ) ] [(KD ) − (KD0l )∗ ]ψl dσl (s) l l κ

∂Dl

'(

&

)

p

=:Kψl

 ρl −1 − 1− ρ0  −1   κ0 ∗ κ0 κ0 κl 0 ∗ −1 1 (s − zl )p [λl I d + (KD ) ] [ ) ] S − S I d + (K S ψl dσl (s) . D D D D l l l l l 2 ∂Dl & '( ) p

=:KGgl

1 Let us recall that zl = |∂D s dσl (s). By definition, we have (s − zl ) ∈ (L20 (∂Dl ))3 . In the l | ∂Dl sequel, we will repeatedly use this property. p To estimate El , using zl we write

p El

:=

0 ∗ −1 (s − zl )p [λl I d + (KD ) ] l

∂uI dσl (s) ∂ν l

∂Dl

= ∂Dl

&

0 ∗ −1 (s − zl )p [λl I d + (KD ) ] l

'( I

∂uI dσl (s) ∂ν l )

(4.26)

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68

+

0 ∗ −1 (zl − zl )p [λl I d + (KD ) ] l

∂Dl

'(

&

∂uI dσl (s) . ∂ν l )

II

To estimate the term I I , we proceed as follows. We note that

0 ∗ −1 (zl − zl )p [λl I d + (KD ) ] l

II = ∂Dl



= (zl − zl )p ∂Dl

∂uI dσl (s) ∂ν l

(4.27)

I 0 ∗ −1 ∂u [λl I d + (KD ) ] dσl (s), l ∂ν l) & '( f

and we can write 0 ∗ [λl I d + (KD ) ]f = l

∂uI . ∂ν l

Now integrating over ∂Dl , we find that [λl I d

0 ∗ + (KD ) ]f (s) l

dσl (s) =

∂Dl

∂uI (s) dσl (s) = −κ02 ∂ν l

∂Dl

uI (x) dx Dl

 1 0 2 f (s) dσl (s) = f (s) [λl I d + KD ](1) dσ (s) = −κ uI (x) dx ⇒ λl − l 0 l 2

∂Dl



f (s) dσl (s) = − λl −

⇒

∂Dl

1 −1 2

∂Dl

Dl

κ02

uI (x) dx.

(4.28)

Dl

Using this in (4.27), it follows that  1 −1 2 I I = −(zl − zl )p λl − κ0 uI (x) dx. 2

(4.29)

Dl

In order to estimate I , we note that, I= ∂Dl

0 ∗ −1 (s − zl )p [λl I d + (KD ) ] l

∂uI dσl (s) = ∂ν l



0 −1 [λl I d + KD ] (s − zl )p l

∂uI dσl (s). ∂ν l

∂Dl

−1 0 We recall that λI d + KD : L20 (∂Dl ) → L20 (∂Dl ) is uniformly bounded with respect to λ for l |λ| ∈ [ 12 , +∞). Using the fact that (s − zl ) is mean-free, we can now conclude that

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% ∂uI % % % 0 −1 |I | ≤ [λl I d + KD ] (s − z )  2 % l p L (∂Dl ) % l ∂ν l L2 (∂Dl ) % ∂uI % % % ≤ C(s − zl )p L2 (∂Dl ) % l % 2 = O(a 3 ). ∂ν L (∂Dl )

69

(4.30)

Combining (4.29) and (4.30), we can therefore write p −El

= (zl − zl )p



1 −1 2 κ0 λl − 2

uI (x) dx + O(a 3 )

(4.31)

Dl

 1 −1 2 I κ0 u (zl )|Dl | + O(a 3 ) + O(a 5 ρl−1 ) = O(a 3−γ ), = (zl − zl )p λl − 2 as λl − 12 ∼ ρl and we assume that ρl  a 1+γ , with γ ≥ 0. p Next let us estimate the term KGgl .

p

0 ∗ −1 (s − zl )p [λl I d + (KD ) ] l

KGgl = ∂Dl



0 −1 [λl I d + KD ] (s − zl )p · l

= ∂Dl

=

1 2

− λl

+ ∂Dl



+





+

κ

  −1   κ κ κl SD0l SD0l − SD ψl (s) dσl (s) l

 −1   κ0 κ0 κl 0 −1 [λl I d + KD ] (s − z ) · S − S S l p Dl Dl Dl ψl (s) dσl (s) l



 −1 0 ∗ −1 0 (s − zl )p [λl I d + (KD ) ] SD l l

∂Dl

dκ SDl0



0 SD l

−1 -n   κ κ SD0l − SDll ψl (s) dσl (s)

n=0



I d + (KD0l )∗

  −1   κ κ κl SD0l SD0l − SD ψl (s) dσl (s) l

 −1   κ0 ∗ κ0 κ0 κl 0 −1 0 ∗ [λl I d + KD ] (s − z ) · [(K ) − (K ) ] S − S S l p Dl Dl Dl Dl Dl ψl (s) dσl (s) l

∞ , 

∂Dl

2

κ

 −1   κ κ κl (s − zl )p SD0l SD0l − SD ψl (s) dσl (s) l

1 ρl 2 ρ l − ρ0

+

1

2

I d + (KD0l )∗

∂Dl

∂Dl

=

1

0 SD l

−1

(s − zl )p ·

∞ ,  −1 -n    dκ κ0 κl 0 − S S SDl0 SD Dl Dl ψl (s) dσl (s) l n=0

 −1   κ0 ∗ κ0 κ0 κl 0 ∗ −1 0 ∗ (s − zl )p [λl I d + (KD ) ] [(K ) − (K ) ] S − S S ψl (s)dσl (s). D D D D D l l l l l l

∂Dl

Then using (3.8), we can deduce that

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70

KGgl = O(ρl a 2 ρl−1 a −1 a 2 ψl ) + O(a a 2 ψl ) + O(a 2 ρl−1 a 2 a −1 a 2 ψl ) p

(4.32)

= O(a 3 ψl ) + O(a 5 ρl−1 ψl ). p

Next, we shall estimate the term Kφl .

p

0 ∗ −1 0 ∗ (s − zl )p [λl I d + (KD ) ] [(KD0l )∗ − (KD ) ]φl (s) dσl (s) l l κ

Kφl = ∂Dl



0 ∗ −1 0 ∗ (s − zl )p [λl I d + (KD ) ] [(KD0l )∗ − (KD ) ]φl (s) dσl (s) l l κ

= ∂Dl

&

'(

)

p,1 l

Kφ



0 ∗ −1 0 ∗ (zl − zl )p [λl I d + (KD ) ] [(KD0l )∗ − (KD ) ]φl (s) dσl (s) . l l κ

+ ∂Dl

'(

&

)

p,2 l

Kφ

Now, the first term in the right hand side above can be estimated in the following manner using the fact that (s − zl )p ∈ L20 (∂Dl ). p,1 Kφl

=

0 ∗ −1 0 ∗ (s − zl )p [λl I d + (KD ) ] [(KD0l )∗ − (KD ) ]φl (s) dσl (s) l l κ

∂Dl

=

0 −1 0 ∗ [λl I d + KD ] (s − zl )p · [(KD0l )∗ − (KD ) ]φl (s) dσl (s) l l κ

∂Dl

= O(a 2 a 2 φl ) = O(a 4 φl ). p,2

To estimate the second term Kφl , we observe that

p,2

Kφl = (zl − zl )p

κ0 ∗ 0 −1 0 ∗ [λl I d + KD ] (1) · [(KD ) − (KD ) ]φl (s) dσl (s) l l l

∂Dl



= (zl − zl )p

κ0 0 0 −1 [KD − KD ][λl I d + KD ] (1) · φl (s) dσl (s). l l l

∂Dl 0 ]−1 (1). Then Let us denote h := [λl I d + KD l p,2



Kφl = (zl − zl )p ∂Dl

 φl (s) (∇t κ0 (s, t) − ∇t 0 (s, t)) · ν l (t)h(t) dσl (t) dσl (s) ∂Dl

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∞

  κ nin (s − t)  l φl (s) − 0 (n − 1)|s − t|n−2 · ν (t)h(t)dσl (t) dσl (s) 4πn! |s − t|

= (zl − zl )p ∂Dl

=

1 2 κ (zl − zl )p 8π 0 iκ03 (zl − zl )p 12π



n=2

∂Dl



∂Dl

+

(s − t)  φl (s) · ν l (t)h(t) dσl (t) dσl (s) |s − t| ∂Dl

 φl (s) [(s − t) · ν l (t)]h(t)dσl (t) dσl (s)

∂Dl

+ (zl − zl )p ∂Dl

71

∂Dl

∞

  κ nin (s − t)  l φl (s) − 0 (n − 1)|s − t|n−2 · ν (t)h(t)dσl (t) dσl (s). 4πn! |s − t| n=4

∂Dl

Note that the third term in the right hand side of the above identity is of order O(a 6 ρl−1 φl ). To understand the first two terms better, we now proceed as follows. Let us write

[(s − t) · ν l (t)]h(t)dσl (t) =

∂Dl

∂Dl

0 ∗ −1 [λl I d + (KD ) ] [(s − t) · ν l (t)] dσl (t). l '( ) & f1

0 )∗ ]f = (s − t) · ν l (t) and hence Now [λl I d + (KD 1 l



0 ∗ [λl I d + (KD ) ]f1 (t) dσl (t) = l

∂Dl



=⇒ λl −



∂Dl

1

f1 (t) dσl (t) =

2 ∂Dl

0 f1 (t)[λl I d + KD ](1) dσl (t) l

∂Dl



= =⇒

(s − t) · ν l (t)dσl (t)

(s − t) · ν l (t)dσl (t) = −3|Dl |

∂Dl

 1 −1 [(s − t) · ν l (t)]h(t)dσl (t) = −3 λl − |Dl |. 2

∂Dl

Therefore iκ03 (zl − zl )p 12π



∂Dl

=−

 φl (s) [(s − t) · ν l (t)]h(t)dσl (t) dσl (s) ∂Dl

iκ03  1 −1 |Dl |(zl − zl )p φl (s)dσl (s). λl − 4π 2 ∂Dl

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72

Next, we consider the term we write



1 2 8π κ0 (zl − zl )p ∂Dl

(s − t) l · ν (t)h(t) dσl (t) = |s − t|

∂Dl



∂Dl

φl (s) ∂Dl

(s−t) |s−t|

 · ν l (t)h(t) dσl (t) dσl (s). Again

  0 ∗ −1 (s − t) [λl I d + (KD ) ] · ν l (t) dσl (t), l |s − t| '( ) & f2

then arguing as in the previous case, we obtain ∂Dl

  s −t 1 −1 1 −1 f2 (t) dσl (t) = λl − Al (s) · ν l (t) dσl (t) = λl − 2 |s − t| 2 ∂Dl

and hence using (3.43), it follows that 1 2 κ (zl − zl )p 8π 0



∂Dl

=

(s − t)  φl (s) · ν l (t)h(t) dσl (t) dσl (s) |s − t| ∂Dl

1 1 2 κ (zl − zl )p (λl − )−1 8π 0 2

Al (s)φl (s) dσl (s)

∂Dl

=

1 1 2 1 1 2 κ0 (zl − zl )p (λl − )−1 Aˆ l Ql + κ0 (zl − zl )p (λl − )−1 8π 2 8π 2



(Al − Aˆ l )(s)φl (s) dσl (s)

∂Dl

=

1 1 2 κ (zl − zl )p (λl − )−1 Aˆ l Ql 8π 0 2 M

 1 2 1 + ∇s κ0 (zl , zm )Qm κ0 (zl − zl )p (λl − )−1 − Rl · 8π 2 m=1 m =l

−

    i ρ0 Capl κ0 (zl , zm )Qm 8π |Dl | + Aˆ l Capl (κ0 − κl ) Ql + ρl − ρ0 4π m = l

⎛ ⎜ 4 3 +O⎜ ⎝a + a

M  m=1 m =l

a3 3 dml

⎞

⎟ φm  + a 5 φl  + a 5 ψl ⎟ ⎠ .

Combining the estimates above, we finally have p

−Kφl =

iκ03  1 −1 1 2 1 |Dl |(zl − zl )p Ql − λl − κ0 (zl − zl )p (λl − )−1 Aˆ l Ql 4π 2 8π 2 M

 1 2 1 + ∇s κ0 (zl , zm )Qm κ0 (zl − zl )p (λl − )−1 Rl · 8π 2 m=1 m =l

(4.33)

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+

73

    i ρ0 Capl κ0 (zl , zm )Qm 8π |Dl | + Aˆ l Capl (κ0 − κl ) Ql + ρl − ρ0 4π m = l

⎛

⎞

M  ⎜ 4 ⎟ a3 3 5 5 ⎟ + O(a 4 φl ) + O(a 6 ρ −1 φl ) a −O⎜ + a φ  + a φ  + a ψ  m l l l ⎝ ⎠ 3 d m=1 ml m =l

=−

1 2 1 κ (zl − zl )p (λl − )−1 Aˆ l Ql 8π 0 2 ⎛

⎜ 4−γ +O⎜ + a 3−γ ⎝a

M  m=1 m =l

+ O(a 4−γ φl ) + O(

a3 3 dml

⎞

⎟ φm  + a 5−γ φl  + a 5−γ ψl ⎟ ⎠

M M   a 5−γ a 5−γ φ ) + O( φm ). m 2 dml dml m=1 m=1 m =l

m =l

p

Proceeding as in the case of Kφl , we can write p −Kψl

iκ03  1 −1 = |Dl |(zl − zl )p ψl (s) dσl (s) λl − 4π 2 ∂Dl

−

−

1 2 1 κ (zl − zl )p (λl − )−1 Aˆ l 8π 0 2 1 1 2 κ0 (zl − zl )p (λl − )−1 8π 2



ψl (s) dσl (s)

∂Dl

(Al − Aˆ l )(s)ψl (s) dσl (s)

∂Dl

+ O(a 4 ψl ) + O(a 6 ρl−1 ψl ), and therefore using (3.13), (3.14) and (3.43), it follows that −Kψl = O(a 6 ρl−1 ψl ) + O(a 4 ψl ) p



iκ 3 1 −1 1 2 ˆ  i(κ0 − κl ) 0 + λl − (zl − zl )p |Dl | − κ0 Al Ql + Capl uI (zl ) + Ql Capl 2 4π 8π 4π M   iκl + κ (zl , zm )Qm Cl2 κ0 (zl , zm )Qm Capl + ∇t κ0 (zl , zm ) · Vm Capl − 4π 0 m=1 m =l

+ ∇s κ0 (zl , zm )Qm ∂Dl

M    a4 0 −1 2 3 (SD ) (· − z )(s) dσ (s) + O(a + a φ  + φ ) l l l m l d3 m=1 ml m =l

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74

+

M

 1 2 1 ∇s κ0 (zl , zm )Qm κ0 (zl − zl )p (λl − )−1 Rl · 8π 2 m=1 m =l

+

    i ρ0 Capl κ0 (zl , zm )Qm 8π |Dl | + Aˆ l Capl (κ0 − κl ) Ql + ρl − ρ0 4π m = l

⎛ ⎜ 4 3 −O⎜ ⎝a + a

−

1 2 1 κ (zl − zl )p (λl − )−1 8π 0 2 

0 + SD l



0 + SD l

−1

⎛ ⎝

−1

⎝

 ,



m = l

⎛

a3

d3 m=1 ml m =l

⎞ ⎟ φm  + a 5 φl  + a 5 ψl ⎟ ⎠

 −1 , i(κ − κ ) 0 l 0 (Al (s) − Aˆ l ) SD Ql l 4π

∂Dl

m = l

⎛



M 

⎞  iκl Capl κ0 (zl , zm ) + (s − zl ) · ∇s κ0 (zl , zm ) Qm ⎠ 1− 4π ⎞

 −1 0 ∇t κ0 (zl , zm ) · Vm ⎠ + SD uI (zl ) l

⎞   a3 ⎠ , + O ⎝a + a 2 φl  + φ  m d3 m = l

where the last error estimate mentioned above is in the sense of L2 , while the others are pointwise. Therefore we can write , ρl −1 p 1  1 ρl −1 − 1− Kψl = − κ02 1 − (zl − zl )(λl − )−1 Aˆ l Ql ρ0 8π ρ0 2 −

M  1 2 1 ρl −1 (zl − zl )(λl − )−1 Aˆ l Capl κ0 (zl , zm )Qm κ0 1 − 8π ρ0 2 m=1 m =l

−

1 2 1 ρl −1 (zl − zl )(λl − )−1 κ0 1 − 8π ρ0 2 ∂Dl

M   0 −1 (Al (s) − Aˆ l ) κ0 (zl , zm )Qm (SD ) (s)dσl (s) l m=1 m =l

+ O(a 5−γ ψl ) + O(a 4 ψl ) + O(a 4−γ φl ) + O(

M  a 5−γ φm ) dml

m=1 m =l

(4.34)

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+ O(a 3−γ ) + O(

75

M M   a 5−γ a 6−γ φ ) + O( φm ). m 2 3 dml dml m=1 m=1 m =l

m =l

p

Finally, we deal with the term Bφl in the following manner. As in the previous cases, using zl , we first write p

Bφl =

M 

κ

0 ∗ −1 (s − zl )p [λl I d + (KD ) ] l

∂(SD0m φm ) ∂ν l

m=1 ∂D l m =l

=

M 

0 ∗ −1 (s − zl )p [λl I d + (KD ) ] l

m=1 ∂D l m =l

&

κ0 ∂(SD φ ) m m

∂ν l

(4.35)

dσl (s)

dσl (s)

'(

)

p,1 Bφ l

+

M 

κ

0 ∗ −1 (zl − zl )p [λl I d + (KD ) ] l

m=1 ∂D l m =l

&

∂(SD0m φm ) ∂ν l

dσl (s) .

'(

)

p,2 l

Bφ

The first term can be dealt with as p,1 Bφl

=

M 

(s − zl )p [λl I d

0 ∗ −1 + (KD ) ] l

κ0 ∂(SD φ ) m m

∂ν l

m=1 ∂D l m =l

=

M 

dσl (s)

M 

κ

[λl I d

0 −1 + KD ] (s l

− zl )p ·

∂(SD0m φm ) ∂ν l

m=1 ∂D l m =l

dσl (s) =

 O

m=1 m =l

 a4 φm  , 2 dml

using the fact that (s − zl )p ∈ L20 . To deal with the second term, we note that



κ

(zl − zl )p [λl I d

0 ∗ −1 + (KD ) ] l

∂(SD0m φm ) ∂ν l

dσl (s) = (zl − zl )p

∂Dl

∂Dl

where f3 satisfies κ

[λl I d

0 ∗ + (KD ) ]f3 l

=

∂(SD0m φm ) ∂ν l

.

From this we can deduce, as in the earlier cases and using (3.11), that

f3 (s) dσl (s),

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76



κ  ∂(SD0m φm ) 1 −1 f3 (s) dσl (s) = λl − dσl (s) 2 ∂ν l

∂Dl

∂Dl

   1 −1 − κ02 |Dl | κ0 (zl , zm )Qm + ∇t κ0 (zl , zm ) · Vm = λl − 2

  (x − zl )dx Qm − Err4m . − κ02 ∇x κ0 (zl , zm ) · Dl

Therefore p

−Bφl = −

M  m=1 m =l

   1 −1 (zl − zl )p λl − − κ02 |Dl | κ0 (zl , zm )Qm + ∇t κ0 (zl , zm ) · Vm 2

− κ02 ∇x κ0 (zl , zm ) ·



  (x − zl )dx Qm − Err4m + O

Dl

=

M  m=1 m =l



(4.36) 

a4 φm  2 dml

 1 −1 2 (zl − zl ) λl − κ0 |Dl |κ0 (zl , zm )Qm 2 M 

+

m=1 m =l

a 6−γ a 5−γ O( 2 φm ) + O( 3 φm ) + O dml dml



  a4 φ  . m 2 dml

Using (4.31), (4.32), (4.33), (4.34), (4.36) in (4.25), we can write Vl =

(s − zl )φl dσl (s) = Vldom + Vlrem ,

(4.37)

∂Dl

where Vldom :=

M  m=1 m =l

 1 −1 2 1 2 1 (zl − zl ) λl − κ0 |Dl |κ0 (zl , zm )Qm − κ0 (zl − zl )(λl − )−1 Aˆ l Ql 2 8π 2

−

1 2 1 ρl −1 (zl − zl )(λl − )−1 Aˆ l Ql κ0 1 − 8π ρ0 2

−

M  1 2 1 ρl −1 (zl − zl )(λl − )−1 Aˆ l Capl κ0 (zl , zm )Qm κ0 1 − 8π ρ0 2

−

1 2 1 ρl −1 (zl − zl )(λl − )−1 κ0 1 − 8π ρ0 2

m=1 m =l

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77

M   0 −1 (Al (s) − Aˆ l ) κ0 (zl , zm )Qm (SD ) (s)dσl (s) l m=1 m =l

∂Dl dom dom + Vl,2 , = Vl,1

with dom Vl,1 := −

dom := Vl,2

1 2 1 1 2 1 ρl −1 (zl − zl )(λl − )−1 Aˆ l Ql , κ0 (zl − zl )(λl − )−1 Aˆ l Ql − κ0 1 − 8π 2 8π ρ0 2

M  m=1 m =l

−

 1 −1 2 (zl − zl ) λl − κ0 |Dl |κ0 (zl , zm )Qm 2

M  1 2 1 ρl −1 (zl − zl )(λl − )−1 Aˆ l Capl κ0 (zl , zm )Qm κ0 1 − 8π ρ0 2 m=1 m =l

−

1 2 1 ρl −1 (zl − zl )(λl − )−1 κ0 1 − 8π ρ0 2 M   0 −1 (Al (s) − Aˆ l ) κ0 (zl , zm )Qm (SD ) (s)dσl (s), l m=1 m =l

∂Dl

and by Vlrem , we denote the rest of the terms. The remainder Vlrem satisfies the estimate |Vlrem | = O(a 3−γ ) + O(a 4−γ φl ) + O(a 3 φl ) M M     a 5−γ a 5−γ +O φ  + O φ  m m 2 dml dml m=1 m=1 m =l

+O

m =l

M  a 6−γ m=1 m =l

3 dml

M    a4 φm  + O φ  m d2 m=1 ml m =l

  a 5−γ a 5−γ a4 a4 a 6−γ a 6−γ  = O(a 3−γ ) + O a 3 + a 4−γ + 2 + 3α + 2 + 3α + 3 + 3α+1 φ , d d d d d d where we use the fact that ρl  a 1+γ , with γ ≥ 0 and 0 ≤ α ≤ 1. Also note that in the above estimate, we have majorised φi  by φ. 4.7. Proof of the invertibility of the algebraic system We rewrite (3.59) in the following compact form; (CI + B + B + R1 + R2 )Q = Y,

(4.38)

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78

where Q, Y ∈ C M×1 and CI , B, B , R1 , R2 ∈ C M×M are defined as 

if if

l = m , l=m

Fl ∇  (z , z ), κl2 x κ0 l m

if

l = m

0,

if

l=m

0,

if

l = m

C−1 l ,

if

B(l, m) := 

B (l, m) :=

⎧ ⎨ ⎩

κ0 (zl , zm ), 0,



CI (l, m) := * Q := Q1

Q2

...

QM

+

* and Y := Y1

Y2

,

(4.40)

, l=m + . . . YM ,

(4.41)

and R1 = PP1 , R2 = PP2 with the matrices P, P1 , P2 defined as  ∇t κ0 (zl , zm ), if l = m P(l, m) := , 0, if l = m ⎧

⎪ κ02 (zm − zm )(λm − 12 )−1 |Dm | ⎪ ⎪ ⎨ 

0 )−1 (s)dσ (s)  (z , z ), P1 (m, n) := − 1 (1 − ρm )−1 A (s)(S m m κ0 m n Dm ∂Dm ⎪ 8π ρ0 ⎪ ⎪ ⎩ 0,  P2 (m, n) :=

0, 1 2 κ0 (zm − 4π

− zm )(λm −

1 −1 ˆ 2 ) Am ,

(4.39)

if

m = n

if

m=n

(4.42)

(4.43)

if

m = n ,

if

m=n (4.44)

.

(4.45)

Our strategy here is to follow the methodology in [15]. In this direction, we multiply the system with Qr and Qi , add the resulting identities and then use the fact that the matrices CI , B are self-adjoint to derive the inequality CI r Qr , Qr  + Br Qr , Qr  + B Qr , Qr  + CI r Qi , Qi  + Bi Qi , Qi  + B Qi , Qi  (4.46) r

r

+ B Qr , Qi  − B Qi , Qr  + R1 r Qr , Qr  + R1 r Qi , Qi  + R1 i Qr , Qi  − R1 i Qi , Qr  i

i

+ R2 r Qr , Qr  + R2 r Qi , Qi  + R2 i Qr , Qi  − R2 i Qi , Qr   = Yr , Qr  + Yi , Qi  ≤ 2

M 

m=1

 12  |Ym |2

M 

 12 |Qm |2

.

m=1

r Let us first consider the case when (C−1 l ) > 0, ∀ l = 1, . . . , M. In this case, proceeding as in [15], we can obtain

Br Qr , Qr  + Br Qi , Qi  ≥ −

M 3τ  |Qm |2 , 5π d m=1

where τ := min1≤j,m≤M, j =m cos(κ0 |zm − zj |) and is assumed to be non-negative.

(4.47)

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79

We can also observe that CI Q , Q  + CI Q , Q r

r

r

r

i

i

M 

r  ≥ min(C−1 m ) m

|Qm | ≥

min (Cm )r  M

1≤m≤M

2

m=1

( max |Cm |)2 1≤m≤M

|Qm |2 .

(4.48)

m=1

r r We would like to note that (C−1 m ) and (Cm ) have the same sign. Next using Cauchy-Schwarz inequality, we can write

B Qr , Qr  + B Qi , Qi  ≥ −B 2 r

r

M 

|Qm |2 ,

m=1 i

i



B Q , Q  ≥ −B 2 Q Q  ≥ −B 2 r

i

r

i

M 

|Qm |2 ,

m=1

− B Qi , Qr  ≥ −B 2 Qi Qr  ≥ −B 2 i

i

M 

|Qm |2 ,

m=1

and therefore B Qr , Qr  + B Qi , Qi  + B Qr , Qi  − B Qi , Qr  ⎡ ⎤1 2    M M ⎥   Fm  κ0 + 1 ⎢  1 ⎢ ⎥ ≥ −3 max  2  |Qm |2 4 ⎦ 1≤m≤M κm 4π ⎣ d ml m,l=1 m=1 r

r

i

i

(4.49)

m =l

    F  κ0 + 1 .  ≥ −3 max  m C MMmax 2  4π 1≤m≤M κm ⎡ ⎤1 −α 2 M [d ]  1 1 3 3 ⎣ + ⎦ [(2n + 1) − (2n − 1) ] |Qm |2 d4 n4 d 4α n=1

m=1

    1 M  Fm  . 1 2 1   ≥ − max  2  C MMmax 4 + 5α |Qm |2 1≤m≤M κm d d

,

m=1

where C denotes a generic constant that is bounded in terms of a. Similarly we derive R1 r Qr , Qr  + R1 r Qi , Qi  + R1 i Qr , Qi  − R1 i Qi , Qr   1  1 M 1 2 1 1 2 1 + |Qm |2 ≥ −Ca 3−γ MMmax 2 + 3α d d d 4 d 5α m=1

and

(4.50)

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80

R2 r Qr , Qr  + R2 r Qi , Qi  + R2 i Qr , Qi  − R2 i Qi , Qr   1 M . 1 2 1 2−γ ≥ −Ca MMmax 4 + 5α |Qm |2 . d d

(4.51)

m=1

Making use of (4.47)-(4.51) in (4.46), the required estimate (3.62) follows. r Let us next consider the case when (C−1 l ) < 0, ∀ l = 1, . . . , M. In this case, we multiply the identity (4.46) with −1 and note that using Cauchy-Schwarz inequality, we can still write  −Yr , Qr  − Yi , Qi  ≤ 2

M 

 12  |Ym |2

m=1

 12

M 

|Qm |2

.

m=1

As in the previous case, we derive the estimates −B Qr , Qr  − B Qi , Qi  − B Qr , Qi  + B Qi , Qr      1  M 2 F  .  C MMmax 1 + 1 ≥ − max  m |Qm |2 , 2  1≤m≤M κm d 4 d 5α r

r

i

i

m=1

−R1 Q , Q  − R1 Q , Q  − R1 Q , Q  + R1 Q , Q   1  1 M 1 2 1 1 2 1 ≥ −Ca 3−γ MMmax 2 + 3α + |Qm |2 d d d 4 d 5α r

r

r

r

i

i

i

r

i

i

i

r

m=1

and −R2 r Qr , Qr  − R2 r Qi , Qi  − R2 i Qr , Qi  + R2 i Qi , Qr  ≥ −Ca 2−γ

.

 MMmax

1 1 + d 4 d 5α

1  M 2

|Qm |2 .

m=1

To deal with the terms −Br Qr , Qr  and −Br Qi , Qi , in contrast to the earlier case, we use Cauchy-Schwarz inequality to derive 

.

−B Q , Q  − B Q , Q  ≥ −C MMmax r

r

r

r

i

i

1 1 + 3α 2 d d

1  M 2

|Qm |2 .

m=1

Also −CI Q , Q  − CI Q , Q  ≥ min r

r

r

r

i

i

1≤m≤M

r (−(C−1 m ) )

M 

, |Qm | = min 2

1≤m≤M

m=1

≥

|Crm | |Cm |2

- M m=1

min |Crm |  M 1≤m≤M ( max |Cm |)2 1≤m≤M

|Qm |2

m=1

|Qm |2 .

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81

Combining the above estimates, we can now conclude (3.63). We would like to remark that the r argument in this case can be applied also to the case when (C−1 l ) > 0, ∀ l = 1, . . . , M, but the estimate would be worse than that already achieved. We note that in view of Remark 3.10, these two are the only possible cases and hence the proof is complete. 4.8. Proof of Proposition 3.14 First of all, we note that using (3.9), (3.20) and (3.21), we can rewrite (3.22) on ∂Dl as κ M

  ∂(SD0m φm )  ∂uI κ0 ∗ + λl I d + (KDl ) φl +  ∂Dl ∂ν l ∂ν l

(4.52)

m=1 m =l

,   −1   ρl −1 1 κ κ κ κ κ κ =− 1− SD0l − SDll ψl [(KDll )∗ − (KD0l )∗ ]ψl + [ I d + (KD0l )∗ ] SD0l ρ0 2 ,   −1   ρl −1 κl ∗ κ0 ∗ κ0 ∗ κ0 κ0 κl 0 ∗ =− 1− S D l − SD l ψl [(KDl ) − (KDl ) ] + [(KDl ) − (KDl ) ] SDl ρ0 ,   −1   ρl −1 1 κ0 κ0 κl 0 ∗ − 1− ) ] S − S S [ I d + (KD Dl Dl D l ψl l ρ0 2 = O(a 2 ψl ) + O(a 2 · a −1 · a 2 ψl ) + O(a −1 ·a 2 ψl ) = O(a 2 ψl ) + O(aψl ) in L2 '( ) & & '( ) ⎛

∈L20

⎤⎞  a  a2 = O ⎝a 2 ⎣1 + φl  + φm  + φm ⎦⎠ 2 dml d ml m = l m = l ⎛ ⎡ ⎤⎞  a  a2 + O ⎝a ⎣1 + φl  + φm  + φm ⎦⎠ in L2 2 dml d m = l m = l ml & '( ) ⎡

∈L20

∈L20

= E1,l + E2,l . Now let us consider the system [λl I d + (KD0l )∗ ]φl + κ

κ M  ∂(SD0m φm )  ∂uI  = − + E1,l + E2,l .   ∂Dl ∂ν l ∂ν l ∂Dl

(4.53)

m=1 m =l

We can further rewrite it as (L + K)φ = −∂ν uI + E1 + E2 , M where L := (Llm )M l,m=1 and K := (K lm )l,m=1 , with

(4.54)

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82

 Llm =

[λl I d + (KD0l )∗ ],

l=m

0,

else

κ

, ∂ν uI :=

 , K lm =

∂uI ∂uI ... M 1 ∂ν ∂ν

∂ κ0 S , ∂ν l Dm

l = m

0,

else

,

(4.55)

-T ,

(4.56)

φ := (φ1 . . . φM )T ,

(4.57)

and Ei := (Ei,1 ... Ei,M )T , i = 1, 2. Let us also set + * cκ0 φ := (cκ0 φ)1 . . . (cκ0 φ)M where (cκ0 φ)l (s) :=

M  ∂ κ (s, zm )Qm , ∂ν l 0

m=1 m =l

  ∇ 1 cκ0 φ := (∇ 1 cκ0 φ)1 . . . (∇ 1 cκ0 φ)M where

(∇

1

cκ0 φ)l (s) :=

M 

∇s ∇t κ0 (s, zm ) · ν l (s) · Vm ,

m=1 m =l

  ∇ 2 cκ0 φ := (∇ 2 cκ0 φ)1 . . . (∇ 2 cκ0 φ)M where (∇ 2 cκ0 φ)l (s) :=

M  m=1 m =l

(s − zl ) ·

∂ ∇t ∇s k0 (s, zm ) · Vm , ∂ν l

and ´ := Kφ − c φ − [cκ − c ]φ − ∇ 1 c φ − [∇ 1 cκ − ∇ 1 c ]φz − [∇ 2 cκ − ∇ 2 c ]φ, Kφ 0 0 0 0 0 0 0 0  with (∇ 1 c0 φz )l := (∇ 1 cκ0 φ)l (zl ) = m =l ∇s ∇t κ0 (zl , zm ) · ν l (s) · Vm . Using the above notations and the fact that the matrix L is invertible, we can write φ as φ = − L−1 ∂ν uI − L−1 c0 φ − L−1 [cκ0 − c0 ]φ − L−1 ∇ 1 c0 φ − L−1 [∇ 1 cκ0 − ∇ 1 c0 ]φz (4.58) ´ + L−1 E1 + L−1 E2 . − L−1 [∇ 2 cκ0 − ∇ 2 c0 ]φ − L−1 Kφ

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Now using the fact that c0 φ, ∇ 1 c0 φ, [∇ 1 cκ0 −∇ 1 c0 ]φz and E2 are mean-free and L−1 doesn’t   scale while acting on mean-free vectors but in general L−1 L(L2 ,L2 ) = O ρl−1 , we obtain −1

(L

, ∂ν u )l  = O I

a , |ρl |

⎛ ⎞    1 1 |Qm |⎠ , a (L−1 c0 φ + L−1 [cκ0 − c0 ]φ)l  = O ⎝ 2 + d |ρl | ⎛ (L−1 ∇ 1 c0 φ)l  = O ⎝

 1 d3 m = l ml

⎞

m =l

a|Vm |⎠ ,

⎛

⎞  1 (L−1 [∇ 1 cκ0 − ∇ 1 c0 ]φz )l  = O ⎝ a|Vm |⎠ , dml ⎛

m = l

⎞  1 1 (L−1 [∇ 2 cκ0 − ∇ 2 c0 ]φ)l  = O ⎝ a 2 |Vm |⎠ , 2 |ρl | dml ⎛

m = l

⎞  a4 1 ´ l = O ⎝ (L−1 Kφ) φ⎠ , max |ρl | l d4 m = l ml ⎛

⎤⎞ ⎡ 2  a  a2 a ⎣1 + φl  + (L−1 E1 )l  = O ⎝ φm  + φm ⎦⎠ , 2 |ρl | dml d m = l m = l ml and ⎛ ⎡

⎤⎞  a  a2 (L−1 E2 )l  = O ⎝a ⎣1 + φl  + φm  + φm ⎦⎠ . 2 dml d ml m = l m = l Therefore we can write ⎞ ⎛    a 1 1 |Qm |⎠ +O⎝ 2 + a |ρl | d |ρl |

, φl  = O

m =l

⎛



+O⎝



⎞ 1 1 a|Vmrem | + a 2 |Vmrem | ⎠ 3 2 |ρl | dml dml



⎞ 1 1 dom dom ⎠ a|Vm,1 |+ a 2 |Vm,1 | 3 2 |ρl | dml dml

m = l

⎛ +O⎝



m = l

1

1

(4.59)

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84

⎛ +O⎝



m = l

⎛



⎛ ⎞ ⎞  a4 1 1 1 dom dom ⎠ +O⎝ a|Vm,2 |+ a 2 |Vm,2 | φ⎠ max 3 2 4 |ρl | dml |ρl | l d dml ml m = l 1

⎤⎞  a  a2 ⎣1 + φl  + +O⎝ φm  + φm ⎦⎠ |ρl | d d2 m = l ml m = l ml ⎡

a2

⎛ ⎡

⎤⎞  a  a2 + O ⎝a ⎣1 + φl  + φm  + φm ⎦⎠ , d d2 m = l ml m = l ml

where we have ignored the contribution of the term L−1 [∇ 1 cκ0 − ∇ 1 c0 ]φz since the term L−1 ∇ 1 c0 φ is clearly more singular. To deal with the terms involving V rem , we note that  1 d3 m = l ml

a|Vmrem |

(4.60)

  1 a 5−γ a 5−γ a4 a4 a 6−γ a 6−γ  = a O a 3−γ + a 3 + a 4−γ + 2 + 3α + 2 + 3α + 3 + 3α+1 φ d d d d d d d3 m = l ml 

 a 6−γ a 6−γ a5 a5 a 7−γ a 7−γ  = O a 4−γ + a 4 + a 5−γ + 2 + 3α + 2 + 3α + 3 + 3α+1 φ d d d d d d , 1 1 O + d 3 d 3α+1

a4  a 4−γ a 4−γ a4 a 5−γ a 5−γ a 6−γ a 6−γ a 6−γ + + + + + + + + =O d3 d 3α+1 d 3 d 3α+1 d3 d 3α+1 d 3α+3 d 6α+1 d5  a5 a5 a5 a 7−γ a 7−γ a 7−γ  + 5 + 3α+3 + 6α+1 + 6 + 3α+4 + 6α+2 φ d d d d d d 

= O a 4−γ −3t + a 4−γ −s−t + a 4−3t + a 4−s−t + a 5−γ −3t + a 5−γ −s−t + a 6−γ −5t + a 6−γ −s−3t   + a 6−γ −2s−t + a 5−5t + a 5−s−3t + a 5−2s−t + a 7−γ −6t + a 7−γ −s−4t + a 7−γ −2s−2t φ .  1 1 a 2 |Vmrem | 2 |ρl | dml

(4.61)

m = l

= =

 1 a 5−γ a 5−γ a4 a4 a 6−γ a 6−γ  a 2  3−γ 3 + a + a 4−γ + 2 + 3α + 2 + 3α + 3 + 3α+1 φ O a |ρl | d d d d d d d2 m = l ml  1  5−γ 5 a 7−γ a 7−γ a6 a6 a 8−γ a 8−γ  + a + a 6−γ + 2 + 3α + 2 + 3α + 3 + 3α+1 φ O a |ρl | d d d d d d , 1 1 + 3α O 2 d d

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85



 a 6−2γ a 6−2γ a 5−γ a 5−γ a 7−2γ a 7−2γ  = O a 4−2γ + a 4−γ + a 5−2γ + + 3α + 2 + 3α + + 3α+1 φ 2 3 d d d d d d , 1 1 + O d 2 d 3α  a 4−2γ a 4−2γ a 4−γ 4−γ a 5−2γ a 5−2γ a 6−2γ a 6−2γ a 6−2γ =O + 3α + + 3α + + 3α + + 3α+2 + 6α 2 2 2 4 d d d d d d d d d   5−γ 5−γ 5−γ 7−2γ 7−2γ 7−2γ a a a a a a + 4 + 3α+2 + 6α + + + φ d d d 3α+3 d 6α+1 d d5 

= O a 4−2γ −2t + a 4−2γ −s + a 4−γ −2t + a 4−γ −s + a 5−2γ −2t + a 5−2γ −s + a 6−2γ −4t + a 6−2γ −s−2t + a 6−2γ −2s + a 5−γ −4t + a 5−γ −s−2t + a 5−γ −2s + a 7−2γ −5t + a 7−2γ −s−3t   + a 7−2γ −2s−t φ . Now if we assume that 0 ≤ t < 12 , 0 ≤ s ≤ 32 , 0 ≤ γ ≤ 1, the estimate  m = l



1

a|Vmrem | + 3 dml

s 3

≤ t and s + γ ≤ 2, then we can derive

 3 1 1 2 rem a |Vm | = O(a + a 2 φ). 2 |ρl | dml

(4.62)

dom , we can deduce that For the terms involving Vm,1

 1 d3 m = l ml

dom a|Vm,1 | = a 4−γ φO(

1 1 + 3α+1 ) = O([a 4−γ −3t + a 4−γ −s−t ]φ), 3 d d

 1 1 1 1 dom a 2 |Vm,1 | = a 4−2γ φO( 2 + 3α ) = O([a 4−2γ −2t + a 4−2γ −s ]φ). 2 |ρl | dml d d

(4.63)

(4.64)

m = l

Assuming that 0 ≤ t < 12 , 0 ≤ s ≤ 32 , 0 ≤ γ ≤ 1, 



m = l

1

dom a|Vm,1 |+ 3 dml

s 3

≤ t and s + γ ≤ 2, we can derive the estimate

 1 1 2 dom a |Vm,1 | = O(aφ). 2 |ρl | dml

(4.65)

dom , we can write Similarly for the terms involving Vm,2

⎞   M M  1 1  1  1 dom 3−γ 4−γ a|Vm,2 | = O ⎝a a |Qn |⎠ = O a + |Qn | 3 3 d d 4 d 3α+2 dml dml

 1 m = l

⎛

m =l

n=1

n=1

(4.66) = O([a 4−γ −4t + a 4−γ −s−2t ]

M  n=1

|Qn |),

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86

⎛ ⎞ M  1 1   1 1 dom a 2 |Vm,2 | = O ⎝a −1−γ a 3−γ a 2 |Qn |⎠ 2 2 |ρl | dml d dml

m = l

m =l

 =O a

4−2γ

n=1

 M  1 1 ( 3 + 3α+1 ) |Qn | d d

(4.67)

n=1

= O([a 4−2γ −3t + a 4−2γ −s−t ]

M 

|Qn |).

n=1

Again assuming that 0 ≤ t < 12 , 0 ≤ s ≤ 32 , 0 ≤ γ ≤ 1, estimate 



m = l

1

dom a|Vm,2 |+ 3 dml

s 3

≤ t and s + γ ≤ 2, we can derive the

 M  1 1 1 2 dom 2+ a |V | = O(a |Qn |). m,2 2 |ρl | dml n=1

Using (3.68), (4.62), (4.65), (4.68) in (4.59), we can deduce that for l = 1, . . . , M,   φl  = O(a −γ ) + O [a 1−2t + a −γ ]M max |C m |    + O [a 1−2t + a −γ ]M max |C m | a 2 + a 3−3t + a 3−s−t + a 4−2s φ 3

+ O(a) + O(a 2 )φ + O(a)φ  1

   + O a 2 + M max |C m | 1 + a + a 2 + a 3−3t + a 3−s−t + a 4−2s φ ⎛ ⎞  1 ⎠ φ + O ⎝a 3−γ d4 m =l ml ⎛ ⎞ ⎞⎤⎞  1  1 ⎣1 + φ + aφ ⎝ ⎠ + a 2 φ ⎝ ⎠⎦⎠ . 2 dml dml ⎡

⎛

+ O ⎝a 1−γ

⎛

m =l

m =l

Now if M max |C m | = O(a −h ), then we obtain      φl  = O(a −γ ) + O a −γ · a −h + O a −γ −h a 2 + a 3−3t + a 3−s−t + a 4−2s φ 3

+ O(a) + O(a 2 )φ + O(a)φ 

   1 + O a −h+ 2 + 1 + a + a 2 + a 3−3t + a 3−s−t + a 4−2s φ ⎛ ⎞  1 ⎠ φ + O ⎝a 3−γ 4 d ml m =l

(4.68)

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⎛

⎛

⎛ ⎞ ⎞⎤⎞  1  1 ⎣1 + φ + aφ ⎝ ⎠ + a 2 φ ⎝ ⎠⎦⎠ 2 dml dml ⎡

+ O ⎝a 1−γ

m =l

= O(a

−γ

87

) + O(a

−γ −h

) + O((a

2−γ −h

m =l

+a

3−3t−γ −h

+a

3−s−t−γ −h

+ a 4−2s−γ −h )φ)

+ O(a 1−2t )φ + O(a 1−γ )φ + O(a 2−γ −s )φ. Therefore provided h < 12 , φ = O(a −γ ) + O(a −γ −h ) + O(a 0+ )φ + O(a 1−2t )φ + O(a 1−γ )φ + O(a 2−γ −s )φ, whence it follows that φ = O(a −γ ) + O(a −γ −h ). Note that if γ = 1 or γ + s = 2, to deduce the last step we need to assume that the constant Cρ in (1.5) is large enough. References [1] S. Albeverio, F. Gesztesy, R. Høegh-Krohn, H. Holden, Solvable Models in Quantum Mechanics, second edition, AMS Chelsea Publishing, Providence, RI, 2005, with an appendix by Pavel Exner. [2] G. Alessandrini, A. Morassi, E. Rosset, Detecting cavities by electrostatic boundary measurements, Inverse Probl. 18 (5) (2002) 1333–1353. [3] A. Alsaedi, B. Ahmad, D.P. Challa, M. Kirane, M. Sini, A cluster of many small holes with negative imaginary surface impedances may generate a negative refraction index, Math. Methods Appl. Sci. 39 (13) (2016) 3607–3622. [4] H. Ammari, D.P. Challa, A.P. Choudhury, M. Sini, The equivalent media generated by bubbles of high contrasts: volumetric metamaterials and metasurfaces, arXiv:1811.02912. [5] H. Ammari, B. Fitzpatrick, D. Gontier, H. Lee, H. Zhang, Minnaert resonances for acoustic waves in bubbly media, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 35 (7) (2018) 1975–1998. [6] H. Ammari, B. Fitzpatrick, D. Gontier, H. Lee, H. Zhang, Sub-wavelength focusing of acoustic waves in bubbly media, Proc. R. Soc. A 473 (2017) 20170469. [7] H. Ammari, B. Fitzpatrick, D. Gontier, H. Lee, H. Zhang, A mathematical and numerical framework for bubble meta-screens, SIAM J. Appl. Math. 77 (2017) 1827–1850. [8] H. Ammari, B. Fitzpatrick, H. Lee, S. Yu, H. Zhang, Subwavelength phononic bandgap opening in bubbly media, J. Differ. Equ. 263 (2017) 5610–5629. [9] H. Ammari, H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements, Lect. Notes Math., vol. 1846, Springer-Verlag, Berlin, 2004, x+238 pp. [10] H. Ammari, H. Kang, Polarization and Moment Tensors, with Applications to Inverse Problems and Effective Medium Theory, Appl. Math. Sci., vol. 162, Springer, New York, 2007. [11] H. Ammari, H. Zhang, Effective medium theory for acoustic waves in bubbly fluids near Minnaert resonant frequency, SIAM J. Math. Anal. 49 (2017) 3252–3276. [12] J. Behrndt, R.L. Frank, C. Kuehn, V. Lotoreichik, J. Rohleder, Spectral theory for Schroedinger operators with δ-interactions supported on curves, Ann. Henri Poincaré 18 (2017) 1305–1347. [13] R. Caflisch, M. Miksis, G. Papanicolaou, L. Ting, Effective equations for wave propagation in a bubbly liquid, J. Fluid Mech. 153 (1985) 259–273. [14] R. Caflisch, M. Miksis, G. Papanicolaou, L. Ting, Wave propagation in bubbly liquids at finite volume fraction, J. Fluid Mech. 160 (1986) 1–14. [15] D.P. Challa, M. Sini, On the justification of the Foldy-Lax approximation for the acoustic scattering by small rigid bodies of arbitrary shapes, Multiscale Model. Simul. 12 (1) (2014) 55–108. [16] D.P. Challa, M. Sini, Multiscale analysis of the acoustic scattering by many scatterers of impedance type, Z. Angew. Math. Phys. 67 (3) (2016), Art. 58.

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