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The effective permittivity and permeability generated by a

cluster of moderately contrasting nanoparticles

X. Cao, M. Sini

RICAM-Report 2021-39

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BY A CLUSTER OF MODERATELY CONTRASTING NANOPARTICLES

XINLIN CAO AND MOURAD SINI

Abstract. In a 3Dbounded andC1,α-smooth domain Ω,α(0,1), we distribute a clus- ter of nanoparticles enjoying moderately contrasting relative permittivity and permeability which can be anisotropic. We show that the effective permittivity and permeability gener- ated by such cluster is explicitly characterized by the corresponding electric and magnetic polarization tensors of the fixed shape. The error of the approximation of the scattered fields corresponding to the cluster and the effective medium is inversely proportional to the dilution parametercr := aδ wherea is the maximum diameter of the nanoparticles andδ the minimum distance between them. The constant of the proportionality is given in terms of a priori bounds on the cluster of nanoparticles (i.e. upper and lower bounds on their permittivity and permeability parameters, upper bound on the dilution parametercr, the used incident frequency and the domain Ω).

A key point of the analysis is to show that the Foldy-Lax field appearing in the meso-scale approximation, derived in [14], is a discrete form of a (continuous) system of Lippmann- Schwinger equations with a related effective permittivity and permeability contrasts. To derive this, we prove that the Lippmann-Schwinger operator, for the Maxwell system, is invertible in the H¨older spaces. As a by-product, this shows a H¨older regularity property of the electromagnetic fields up to the boundary of the inhomogeneity.

Keywords: Maxwell system, anisotropic permittivity and permeability, effective medium theory, integral equations.

AMS subject classification: 35C15; 35C20; 35Q60

1. Introduction

1.1. Background and motivation. In this work, we are interested in estimating the elec- tromagnetic fields generated by a cluster of small particles enjoying moderate contrasts of both their electric permittivity and magnetic permeability as compared with the ones of the background. Such a topic, which enters in the general framework of understanding the inter- action between waves and matter, traces back at least as far as to the pioneering works of Rayleigh’s and Kirchhoff’s. At their time, it was known already that the diffracted wave by small scaled inhomogeneities is dominated by the first multi-poles (poles or dipoles) which are given by (polarized) point sources located inside the particles. In case of spherically shaped particles, Mie [30] derived the full expansion of the electromagnetic fields.These for- mal expansions were later mathematically justified, see for instance [21], in the framework of low frequencies expansions. A further step was achieved by Ammari and Kang, see [2,3], where the full expansion at any order and any shape is derived and justified.

These mentioned key works focused on a single or well-separated particles where their interactions is neglected . When the particles are close enough, i.e. behave as a cluster, the mutual interactions between them and the waves must be taken into account. Motivated by the propagation of acoustic waves in the presence of small bubbles, Foldy proposed, in his seminal work [22], a formal way how to handle the multiple interactions between the bubbles.

Under the assumption that the bubbles behave as point-like potentials, he states a close form of the scattered wave by simply eliminating the singularity on the locations of these potentials

1

arXiv:2111.02846v1 [math.AP] 4 Nov 2021

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(see [28] for more details). Later on, these formal representations of the scattered waves were justified by Berezin and Faddeev, see [10], in the framework of the Krein’s selfadjoint extension of symmetric operators. Another approach to give sense to Foldy’s formal method is the regularization method, see [1]. Let us mention here that the link between Foldy’s result, the one by Berezin and Faddeev and the one in [1] was made in a recent work [24].

There was no mention of Foldy’s work in Berezin and Faddeev’s work [10] nor in [1]. These two approaches give us, via the Krein’s resolvent representations or regularization, exact formulas to represent the scattered waves generated by singular potentials.

Inspired by these representations, it is natural to expect the dominant part of the fields, generated by a cluster of small scaled particles, to be reminiscent to the exact formulas described above with a difference that the scattering coefficients (which are also called the polarization tensors) entering into the close form dominating field are modeled by geometric or contrasts properties of the inhomogeneities. This is called the Foldy-Lax approximation or the point-interaction approximation.

There are different ways to derive these dominant fields. We can cite the variation method, via the maximum principle, as proposed by Maz’ya and Movchan [29] or integral equations as proposed firstly by Ramm [35] for some particular models and scaling regimes, and developed since then by many authors [16–18], [32–34] and [36–38]. We also cite the method of matched asymptotic expansions, see [11,27] and the references therein. Regarding the Maxwell system, apart from [37,38] which are derived more formally with questionable issues, few results are known. The first complete results are derived in [13] for the conductive particles and in [14]

for particles enjoying moderate contrasts of the permittivity and permeability as compared to the ones of the background. The important issue in those two works is that the meso- scaled regime could be handled. In this regime the maximum radius of the particles aand the minimum distance δ between them could be of the same order. The dilution parameter cr := δa indicates how dense the particles can be distributed. The main result in [14] is to have derived the Foldy-Lax approximation in the meso-scale regime with an error of approximation inversely proportional to cr. In addition, the close form of the dominating field is provided taking into account both the electric permittivity and magnetic permeability contrasts, allowing, eventually, their anisotropy and non-symmetry.

The work under consideration is related to the effective medium theory for Maxwell. The goal is to derive the effective permittivity and permeability that provides the same scat- tered electromagnetic field as the one derived in [14]. In the case of periodically distributed small inhomogeneities, the homogenization applies, see [9,25] and the references therein, and provides the equivalent media with averaged materials. In our analysis, we do not rely on ho- mogenization techniques. Rather, our starting point is the Foldy-Lax approximation derived in [14]. The key point of the analysis is to show that the Foldy-Lax field appearing in the meso-scale approximation is a discrete form of a (continuous) system of Lippmann-Schwinger equations with a related effective permittivity and permeability contrasts. To derive this, we prove an invertibility result for Lippmann-Schwinger operator, for Maxwell, in the H¨older spaces. As a by-product, the corresponding electromagnetic fields has a H¨older regularity up to the boundary of the inhomogeneity.

In this work, we discuss electromagnetic material with moderate and positive contrasts.

The study of the wave propagation in highly contrasting or negative materials is highly attractive and we witness a rapid growth of the number of published works in the recent years. The reason is that for special large scales, we can have resonances (as Minnaert for the acoustics and Mie (or dielectric) or plasmonic resonances in electromagnetism). So far, most of the results concern the acoustic model in the presence of bubbles. The observation, first made in [7], is that when the contrast of both the mass density and bulk modulus have a

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similar but large contrasts, as compared to the ones of the background (the air, for instance), then we have a resonance, named Minnaert resonance. This is a resonance in the sense that if the used incident frequency is close to it then the scattering coefficient (the polarization tensor) becomes large. Such an enhancement can be used to generate volumetric and low dimensional metamaterials (materials with single or double negativity), see [4–8]. As far as the effective medium theory is concerned, few results are known for the electromagnetism with highly contrasting or negative permittivity of permeability. We can cite [12,26,39] who assume periodicity and derive the equivalent coefficients, via homogenization, for dielectric nanoparticles. However, we do believe that in the near future this gap will be filled.

1.2. The electromagnetic scattered fields generated by a cluster of nanoparticles.

We denote by D := ∪Mm=1Dm a collection of M connected, bounded and Lipschitz-smooth particles of R3. We deal with the electromagnetic wave propagation from this collection of particles when excited by time-harmonic electromagnetic plane wave at a given frequency ω.The nature of the wave propagation is described by the following Maxwell system:









∆× E −iωµH= 0, in R3\∂D,

∆× H+iωE =σE, in R3\∂D, E=Ein+Es,

ν× E|+=ν× E|, ν× H|+ =ν× H| on ∂D,

(1.1)

whereE andHdenote the total electric and magnetic fields,andµare the electric permit- tivity and the magnetic permeability, respectively and σ is the corresponding conductivity.

The coefficients, µ, σ can be either real or complex tensors or scalar valued functions. Here, Ein, being an entire solution of the Maxwell equation, is an incident field and Es stands for the associated scattered field. The surrounding background of D is homogeneous with the constant parameters 0, µ0 and null conductivity σ. Furthermore, the scattered wave fields Es and Hs satisfy the Sylver-M¨uller radiation conditions

q

µ0−10 Hs(x)× x

|x|− Es(x) =O( 1

|x|2).

Letk:=ω√

0µ0,E :=

q

0µ−10 E and H= q

0µ−10 H with µr:= µµ

0 and r := +iσ/ω

0 , we then derive from (1.1) that













curlE−ikµrH = 0, curlH+ikrE = 0 inD, curlE−ikH = 0, curlH+ikE = 0 in R3\D, E=Ein+Es,

ν×E|+=ν×E|, ν×H|+=ν×H|, on∂D, Hs×|x|x −Es=O(|x|12), as|x| → ∞,

(1.2)

whereEinandHinare the relative incident wave fields fulfilling the whole system inR3 that (curlEin−ikHin= 0,

curlHin+ikEin= 0.

Let the incident waves (Ein, Hin) be typical plane waves of the form Ein =Ein(x, θ) = P eikθ·x for x ∈ R3 and Hin = Hin(x, θ) = P ×θeikθ·x/ik, where P is the constant vector modeling the polarization direction and θ fulfilling |θ| = 1 and P ·θ = 0 is the incident direction.

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LetDm =aBm+zm, m= 1,· · ·, M, be the connected small components of the mediumD, which are characterized by the parameter a >0 and the locations zm. Each Bm containing the origin is a bounded Lipschitz domain and fulfills that Bm ⊂ B01. In fact, we have the following definition.

Definition 1.1. Definea:= max

1≤m≤Mdiam(Dm), andδ:= min

1≤m,j≤M m6=j

δmj := min

1≤m,j≤M m6=j

dist(Dm, Dj).

Our analysis in this work is based on [14]. Therefore, we first recall the needed assumptions stated there regarding the model (1.2). More details can be found in [14].

I. Assumptions on the exterior domain. The electric permittivity 0 and the magnetic permeabilityµ0 of the background are real, scalar and constants such that

k=ω√

0µ0 >0.

II.Assumptions on the permittivity and permeability of each particle. Assume thatrand µr satisfy the following conditions:

(1) r and µr are W1,∞-regular and both positive definite real symmetric matrices.

(2) Define the contrasts with respect to r and µr as CK :=K−I,

whereK =r, µr and I ∈R3×R3 is the identity matrix. CK is a real valued 3×3- tensor with the corresponding derivatives satisfy the essentially uniformly bounded condition as

kCKk

W1,∞(∪Mm=1Dm) ≤c, (1.3) as well as the essentially uniformly coercive conditions:

< Cr(x)ξ·ξ¯

≥c|ξ|2,

Cµr(x)ξ·ξ¯≥cµ− |ξ|2, (1.4) forx∈D, wherec andcµ− are two positive constants.

III.Assumptions on the distribution of the cluster of particles. Let Ω be a bounded domain of unit volume, containing the particlesDm,m= 1,2,· · ·, M, with∂Ω which isC1,α-regular for α ∈ (0,1). We divide Ω into [δ−3] subdomains Ωm, m = 1,2,· · · ,[δ−3], distributed periodically, such that Dm ⊂Ωm. Then we see that M =O(δ−3). Let each Ωm be a cube of volumeδ3. Furthermore, the minimum distance defined in Definition 1.1fulfills that

δ=cra, (1.5)

wherecr is a constant satisfying the lower bound cr = δ

a ≥max

1, 2ckc2 max(c, cµ−)

,

withc being a positive constant independent on the parameters of the model. See Fig1and Fig 2 for the schematic illustrations of the global distribution as well as the local relation between any two particles.

Remark 1.2. In the above condition III, without loss of generality, we assume that the particles are distributed periodically in Ω. However, the periodicity is used only for simplicity of the exposition. It is not required to derive our results. We refer to [15] for the case of non- periodic distribution of the particles.

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j zj

Dj

zm

Dm m

Figure 1. A schematic illustration for the global distribution of the particles in Ω.

m j

Dm Dj

a

δ δ

Figure 2. A schematic illustration for the local relation between any two particles.

Remark 1.3. Since the shape of Ω is arbitrary, the intersecting part between the set of the cubes and ∂Ω is not necessarily to be empty, unless Ω is a cube. It is known that for any m= 1,· · ·, M, the volume of Ωmsatisfies|Ωm|=δ3, which implies that the maximum radius of Ωm is of orderδ, and thus the intersecting surfaces with ∂Ω possess the area of order δ2. Since the total area of ∂Ω is of order one, we can derive that the number of such particles will not exceed of orderδ−2, and therefore the volume of this set is of orderδ, for sufficiently small a.

1.3. Main results. Our starting point is the following Foldy-Lax approximation (or point- interaction approximation) of the electromagnetic fields generated by a cluster of small scaled inhomogeneties in the mesoscale regime derived in [14].

Theorem 1.4. [14, Theorem 2.1]. Let Ω be a bounded domain with Lipschitz boundary.

Under the assumptions (I, II,III) above, we have the following expansion:

E(ˆx) =

M

X

m=1

k2

4πe−ikˆx·zmxˆ×(Rmr×x) +ˆ ik

4πe−ikˆx·zmxˆ× Qµmr

+O(kc( 1 c

+ 1 cµ−

)c−7r ), (1.6)

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with (Rmr,Qµmr) fulfilling the following invertible linear algebraic system [PDrm]−1Rmr =

M

X

j=1 j6=m

h

Πk(zm, zj)Rjr +ik∇Φk(zm, zj)× Qµjri

+Ein(zm),

[PDµrm]−1Qµmr =

M

X

j=1 j6=m

h

Πk(zm, zj)Qµjr −ik∇Φk(zm, zj)× Rjri

+Hin(zm) for m= 1,· · ·, M. cr, cµr are the positive constants introduced in (1.4)

Here, [PDr

m] and [PDµr

m] are respectively the electric polarization tensor and the magnetic polarization tensor associated with (1.2), which satisfy the scaling property

[PDrm] =a3[PBrm], [PDµrm] =a3[PBµrm], (1.7) as well as the boundedness property

λra3|V|2 ≤[PDr

m]V ·V ≤λ+ra3|V|2, λµra3|V|2 ≤[PDµr

m]V ·V ≤λ+µra3|V|2, (1.8) with positive constantsλr, λµr, for any vector V. Then we denote

λ(

rr):= min{λr, λµr}, andλ+(

rr):= max{λ+r, λ+µr}. (1.9) IV. Assumption on the shape of Dm’s. We assume that the Bm’s have the same shapes and we denote

[PBr

m] := [P0r], [PBµr

m] := [P0µr]. (1.10)

Actually, we just need the shapes to have the same polarization tensors.

Now, we are in a position to state the main contribution of this work.

Theorem 1.5. Let Ω be a bounded domain with C1,α regularity for α∈(0,1). Suppose that the wave number k fulfills

g(α, k)creg(α,Ω)c−3r c(rr) <1, where

g(α, k) :=k3+α+k3+k2+k+ 1, and c(rr) is defined by

c(rr):= max{k[P0r]kL(Ω),k[P0µr]kL(Ω)},

and creg(α,Ω) is a positive constant depending on α and Ω, see (2.31). Then under the assumptions (I, II,III, IV) above, we have the following expansion:

E(ˆx)−Eeff(ˆx) =O kλ+(

rr)(c0r +cµ0r)(kcPr

0c0r,−1+cµPr

0cµ0r,−1)c−6r

, (1.11)

where λ+(

rr), cPr

0, cµPr

0, c0r,−1, cµ0r,−1, c0r and cµ0r are positive constants depending on the a- prior bounds for r, µr and Ω and they are given by (1.9), (2.21) and (3.6), respectively.

Here, Eeff is the far-field pattern corresponding to the following electromagnetic scattering problem for the effective medium as a→0:













curlU˚r −ik˚µrV˚µr = 0, curlV˚µr +ik˚rU˚r = 0 in D, curlU˚r −ikV˚µr = 0, curlV˚µr+ikU˚r = 0 in R3\D, U˚r =U˚sr+U˚inr,

ν×U˚r|+=ν×U˚r|, ν×V˚µr|+=ν×V˚µr|, on ∂D, Vs˚µr×|x|x −U˚sr =O(|x|12), as |x| → ∞,

(1.12)

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The electric permittivity and magnetic permeability of the effective medium are given by

˚r =I+c−3r [P0r], ˚µr =I+c−3r [P0µr].

Remark 1.6. The assumption in (II. (1)) can be changed into “r and µr are real positive definite matrices with arbitrary<r ”. In particular,r andµr do not need to be symmetric.

In this case, for m=,1,2,· · · , M, we define Amf := 1

Dm Z

Dm

f dv,

for any f ∈L2(Dm). Then if eitherr orµr is not symmetric, with (Am

r)Mm=1 and (Amµ

r)Mm=1 standing for the respective average over eachDm, the Foldy-Lax approximation for the far- field keeps the same form as (1.6). But the invertible linear algebraic system with repsect to (Rmr,Qµmr) now becomes

[TA

m rT

Dm ]−1Rmr =

M

X

j=1 j6=m

h

Πk(zm, zj)Rjr +ik∇Φk(zm, zj)× Qµjri

+Ein(zm),

[TA

m µrT

Dm ]−1Qµmr =

M

X

j=1 j6=m

h

Πk(zm, zj)Qµjr −ik∇Φk(zm, zj)× Rjri

+Hin(zm),

form= 1,2,· · · , M, where [TA

m rT

Dm ]−1 and [TA

m µrT

Dm ]−1 are given by [TA

m rT

Dm ]−1:=AmC

rT[PA

m rT

Dm ]−1(AmC

r)−1, and

[TA

m µrT

Dm ]−1:=AmC

µrT[PA

m µrT

Dm ]−1(AmC

µr)−1.

We refer [14, Theorem 2.1] for more detailed discussions. Based on (1.6), we show that the corresponding effective medium under this circumstances still fulfills (1.12), with the new electric permittivity ˚r and ˚µr being represented as

˚

r =I+c−3r [T0ArT], µ˚r =I+c−3r [T0AµrT], under the similar assumption as IV.

1.4. A brief description of the arguments. Here, we briefly describe the whole steps and ideas of the proof of Theorem1.5.

Recall the Foldy-Lax approximation given in [14, Theorem 2.1]. By the variable substitu- tion Um:= [PDr

m]−1Rmr, Vm := [PDµr

m]−1Qµmr, we can rewrite the far-field expansion as E(ˆx) =

M

X

m=1

k2

4πe−ikˆx·zmxˆ×(a3[P0r]Um×x) +ˆ ik

4πe−ikˆx·zmxˆ×a3[P0µr]Vm

+O(kc( 1 c

+ 1 cµ−

)c−7r ), (1.13)

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where (Um, Vm)Mm=1 are solutions to the invertible linear algebraic system Umưa3

M

X

j=1 j6=m

k(zm, zj)[P0r]Uj +ik∇Φk(zm, zj)×([P0µr]Vj)] =Ein(zm),

Vmưa3

M

X

j=1 j6=m

k(zm, zj)[P0µr]Vjưik∇Φk(zm, zj)×([P0r]Uj)] =Hin(zm), (1.14)

and c,cư and cµư are defined by (1.3) and (1.4), respectively.

Based on (1.14), we introduce the Lippmann-Schwinger equation with respect to two constant 3×3-tensors [CTr] and [CµTr] as

U˚r(x)ư Z

Πk(x, z)cư3r [CTr]U˚r+ik∇Φk(x, z)×(cư3r [CµTr]V˚µr)

dz=Ein(x, θ), V˚µr(x)ư

Z

Πk(x, z)cư3r [CµTr]V˚µrưik∇Φk(x, z)×(cư3r [CTr]U˚r)

dz =Hin(x, θ), (1.15) which models the unique solution to the electromagnetic scattering problem of equivalent medium (1.12). More explanations for [CTr] and [CµTr] will be presented later in Section3 by Definition3.1. The corresponding far-field possesses the form

Eeff(ˆx) = Z

k2

4πeưikˆx·zxˆ×(cư3r [CTr]U˚r(z)×x) +ˆ ik

4πeưikˆx·zxˆ×cư3r [CµTr]V˚µr(z)

dz.

(1.16) Our main result is to derive the following error estimate

E(ˆx)ưEeff(ˆx) =O kλ+(

rr)(c0r +cµ0r)(kcPr

0c0r,ư1+cµPr

0cµ0r,ư1)cư6r

, a→0. (1.17) We show that [CTr] = [P0r] +O(cư3r ) and [CµTr] = [P0µr] +O(cư3r ). Therefore, in terms of (1.17),Eeff(ˆx) in (1.16) reduces, keeping the same notation, to

Eeff(ˆx) = Z

k2

4πeưikˆx·zxˆ×(cư3r [P0r]U˚r ×x) +ˆ ik

4πeưikˆx·zxˆ×cư3r [P0µr]V˚µr

dz. (1.18) In order to derive (1.17), the first step is to investigate the regularity of the solution (U˚r, Vµ˚r) of (1.15). We prove that if the bounded domain Ω isC1,αregular, then (U˚r, Vµ˚r)∈ C0,α(Ω)×C0,α(Ω), forα ∈(0,1), provided that

g(α, k)creg(α,Ω)cư3r c(rr) <1,

where g(α, k) :=k3+α+k3+k2+k+ 1, and creg(α,Ω) is a positive constant depending on α and Ω. c(rr) is a positive constant related to [P0r] and [P0µr].

Next, with the help of counting lemma and the above regularity property, we derive the most important estimate of the difference between the solutions (Um, Vm) to the lin- ear algebraic system and the solutions (U˚r, Vµ˚r) of the Lippmann-Schwinger equation.

Precisely, we estimate the l2-norm of the vectors (I ưKr)|S1

m|

R

SmU˚(x)dxư Um, and (I ưKµr)|S1

m|

R

SmV˚(x)dxưVm,, form= 1,· · ·, M, as

M

X

m=1

|(IưKr) 1

|Sm| Z

Sm

U˚(x)dxưUm|2

!1/2

=O

λ+(

rr)c0r,ư1(c0r +cµ0r)cư

9

r 2aư32

,

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and

M

X

m=1

|(I−Kµr) 1

|Sm| Z

Sm

V˚(x)dx−Vm|2

!1/2

=O

λ+(

rr)cµ0r,−1(c0r+cµ0r)c

9

r 2a32

, whereKr andKµr are bounded linear operators associated with the constant tensors [CTr] and [CµTr] respectively; see Definition 3.1 in Section 3 for more detailed discussions. To derive these estimates, with tedious but needed computations, we systematically utilize the properties of the Newtonian operator, as Cald´eron-Zygmund inequality, and the invertibility property of the algebraic system in Theorem1.4. Based on these estimates, we finally derive the error estimate for E−Eeff.

The rest of this paper is organized as follows. In Section 2, we first recall the Foldy-Lax approximation of the electromagnetic fields introduced in [14] for the discrete case with slightly different representations, which is more appropriate for our later comparison with the continuous model. In addition, we derive the C0,α-regularity of the solution (U˚r, Vµ˚r) appearing in the, limiting, far-field (1.18). Section 3 is devoted to prove our main result by investigating the equivalent medium as a → 0 and evaluate the difference between the discrete form of the far-field E(ˆx) and the corresponding asymptotic form Eeff(ˆx), which shows the error estimate ofEeff(ˆx)−E(ˆx). Finally, in Section4, we give the proof of one of the essential propositions used in Section3.

2. Lippmann-Schwinger equation and the C0,α regularity to its solution 2.1. Mesoscale Approximation for the Electromagnetic Fields: Discrete form. In this subsection, based on [14], we recall the discrete form of the Foldy-Lax Approximation as well as the corresponding linear algebraic system as the following proposition.

Proposition 2.1. [14, Theorem 2.1] Under the assumptions (I, II, III), the mesoscale electromagnetic scattering problem (1.2)has one and only one solution and the corresponding far-field fulfills the following expansion:

E(ˆx) =

M

X

m=1

k2

4πe−ikˆx·zmxˆ×(Rmr×x) +ˆ ik

4πe−ikˆx·zmxˆ× Qµmr

+O(kc( 1 c

+ 1 cµ−

)c−7r ).

(2.1) where (Rmr,Qµmr)Mm=1 are the solutions to the following invertible linear algebraic system

[PDr

m]−1Rmr =

M

X

j=1 j6=m

k(zm, zj)Rjr+ik∇Φk(zm, zj)× Qµjr] +Ein(zm),

[PDµr

m]−1Qµmr =

M

X

j=1 j6=m

k(zm, zj)Qµjr−ik∇Φk(zm, zj)× Rjr] +Hin(zm). (2.2) Furthermore, (Rmr,Qµmr)Mm=1 satisfy the estimates for k >1 as,

M

X

m=1

|Rmr|2

!12

≤ 9λ+(

rr)a3 8

 1 3

M

X

m=1

|Hin(zm)|2

!12 +

M

X

m=1

|Ein(zm)|2

!12

,

M

X

m=1

|Qµmr|2

!12

≤ 9λ+(

rr)a3 8

M

X

m=1

|Hin(zm)|2

!12 +1

3

M

X

m=1

|Ein(zm)|2

!12

, (2.3)

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provided cr=δ/a≥3kλ+(

rr).

The proof of deriving (2.2) and (2.3) can be founded directly in [14, Proposition 5.1 and Proposition 5.2]. The corresponding far-field representation here (2.1) is slightly different from the form presented in [14, Theorem 2.1].

Now, based on the scaling property (1.7) as well as the constant assumption for the polarization tensors (1.10), we rewrite the linear algebraic system (2.2) for (Rmr,Qµmr)Mm=1 as [PDrm]−1Rmr −a3

M

X

j=1 j6=m

h

Πk(zm, zj)[P0r][PDr

j]−1Rjr +ik∇Φk(zm, zj)×[P0µr][PDµr

j]−1Qµjri

=Ein(zm),

[PDµr

m]−1Qµmr −a3

M

X

j=1 j6=m

h

Πk(zm, zj)[P0µr][PDµr

j]−1Qµjr −ik∇Φk(zm, zj)×[P0r][PDr

j]−1Rjri

=Hin(zm).

(2.4) Denote

Um:= [PDrm]−1Rmr, Vm := [PDµrm]−1Qµmr. (2.5) Then (2.4) becomes

Um−a3

M

X

j=1 j6=m

k(zm, zj)[P0r]Uj +ik∇Φk(zm, zj)×([P0µr]Vj)] =Ein(zm),

Vm−a3

M

X

j=1 j6=m

k(zm, zj)[P0µr]Vj−ik∇Φk(zm, zj)×([P0r]Uj)] =Hin(zm). (2.6)

In the following discussions, when it comes to the discrete far-field approximation, we mainly use the representation (2.1) as well as (2.6).

2.2. Lippmann-Schwinger equation and the C0,α regularity of the associated so- lutions. In this subsection, we first construct the Lippmann-Schwinger equation of the elec- tromagnetic scattering problem for the equivalent medium and investigate the regularity of the solutions, which is of significant use in the proof of our main theorem.

Suppose [CTr] and [CµTr] are two 3×3-tensors, which can be denoted by two positive definite real symmetric matrices. Inspired by (2.6), let us consider the Lippmann-Schwinger equation with respect toc−3r [CTr] andc−3r [CµTr] as

U˚r(x)− Z

Πk(x, z)c−3r [CTr]U˚r +ik∇Φk(x, z)×(c−3r [CµTr]V˚µr)

dz=Ein(x, θ), V˚µr(x)−

Z

Πk(x, z)c−3r [CµTr]V˚µr−ik∇Φk(x, z)×(c−3r [CTr]U˚r)

dz =Hin(x, θ). (2.7) forx ∈Ω, where cr >1 is the constant defined in (1.5). Indeed, we can see by Remark 3.3 from the explicit forms of [CTr] and [CµTr] that they are two constant tensors since [P0r] and [P0µr] are constant. It is direct to verify that (U˚r, V˚µr) satisfies the following electromagnetic

(12)

scattering problem of the equivalent medium













curlU˚r −ik˚µrV˚µr = 0, curlV˚µr +ik˚rU˚r = 0 inD, curlU˚r −ikV˚µr = 0, curlV˚µr +ikU˚r = 0 inR3\D, U˚r =U˚sr+U˚inr,

ν×U˚r|+ =ν×U˚r|, ν×V˚µr|+=ν×V˚µr|, on∂D, Vs˚µr ×|x|x −Us˚r =O(|x|12), as|x| → ∞,

(2.8)

where ˚r and ˚µr are formulated by

˚r =I+c−3r [CTr], ˚µr=I+c−3r [CµTr]. (2.9) It is obvious that ˚r and ˚µrare constant and symmetric under Assumption IV by the repre- sentations (3.3) and (3.4).

Recall the Green’s function for the Helmhotz operator in [14, Section 3]

Φk(x, z) = 1 4π

eik|x−z|

|x−z|, forx6=z, and the corresponding electromagnetic dyadic Green’s function

Πk(x, z) =k2Φk(x, z)I+∇xxΦk(x, z) =k2Φk(x, z)I− ∇xyΦk(x, z) forx6=z.

For any essentially bounded tensorA, denote the corresponding vector Newtonian operator as

Sk,A(W) :=

Z

Φk(x, z)AW(z)dz forW ∈L2(Ω). (2.10) Therefore, in order to investigate the regularity of (U˚r, V˚µr), it suffices to study the regularity of (E, H) solution to the following equivalent Lippmann-Schwinger equation

E H

− (k2+∇div)Sk,C˚r ikcurlSk,C˚µr

−ikcurlSk,C˚r (k2+∇div)Sk,C˚µr

! E H

= Ein

Hin

, (2.11) where Sk,C˚r,Sk,C˚µr are the Newtonian operators defined by (2.10) and (Ein, Hin) are the incident plane waves introduced in Subsection1.2.

We present the main regularity result as follows.

Theorem 2.2. Let Ω be a bounded domain withC1,α regularity for α∈(0,1). Consider the electromagnetic scattering problem (1.2). There exists a positive constantcreg(α,Ω), depend- ing only on α and Ω, such that if

g(α, k)creg(α,Ω)c−3r c(rr) <1, (2.12) where

g(α, k) :=k3+α+k3+k2+k+ 1 and c(rr) := max{k[P0r]kL(Ω),k[P0µr]kL(Ω)}, then we have

(E, H)∈C0,α(Ω)×C0,α(Ω).

Proof. We split the proof of Theorem2.2into the following three steps:

(i) Construction of the coupled surface-volume system of integral equations.

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