Cloaking using complementary media for electromagnetic waves

0.23 MB·

by  Hoai-Minh Nguyen

Cloaking using complementary media for electromagnetic waves
Hoai-Minh Nguyen ∗
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1 January 11, 2017
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a
Abstract
J

9 Negativeindexmaterialsareartificialstructureswhoserefractiveindexhasnegativevalue
over some frequency range. The study of these materials has attracted a lot of attention in
]
P the scientific community not only because of their many potential interesting applications
A but also because of challenges in understanding their intriguing properties due to the sign-
changing coefficients in equations describing their properties. In this paper, we establish
.
h cloaking using complementary media for electromagnetic waves. This confirms and extends
t
a the suggestions in two dimensions of Lai et al. in [15] for the full Maxwell equations. The
m analysisisbasedonthereflectingandremovinglocalizedsingularitytechniques,three-sphere
[ inequalities, and the fact that the Maxwell equations can be reduced to a weakly coupled

second order elliptic equations.
1
v
Key words: negative index materials, cloaking, complementary media, localized resonance,
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3 electromagnetic waves.
3 AMS classifications: 35B34, 35B35, 35B40, 35J05, 78A25.
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0
. 1 Introduction
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0
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Negative index materials (NIMs) are artificial structures whose refractive index has negative
1
: value over some frequency range. These materials were investigated theoretically by Veselago
v
i in [35]. The existence of such materials was confirmed by Shelby, Smith, and Schultz in [34].
X
The study of NIMs has attracted a lot of attention in the scientific community not only because
r
a of their many potential interesting applications but also because of challenges in understanding
intriguing properties of these materials.
One of the interesting applications of NIMs is cloaking using complementary media, which
was inspired by the concept of complementary media, see [33, 15, 17, 23]. Cloaking using com-
plementary media was proposed and studied numerically by Lai et al. in [15] in two dimensions.
The idea of this cloaking technique is to cancel the light effect of an object usingits complemen-
tary media. Cloaking using complementary media was mathematically established by Nguyen
in [21] in the quasistatic regime. To this end, we introduced the removing localized singularity
technique and used a standard three-sphere inequality. The method used in [21] also works
for the Helmholtz equation. Nevertheless, it requires small size of the cloaked region for large
frequency due to the use of the (standard) three-sphere inequality. In [27], Nguyen and (H.
∗EPFL SBMATHAA CAMA, Station 8, CH-1015 Lausanne, hoai-minh.nguyen@epfl.ch
1
L.) Nguyen gave a mathematical proof of cloaking using complementary media in the finite
frequency regime for acoustic waves without imposing any condition on the size of the cloaked
region. To successfully apply the approach in [21], we established a new three-sphere inequality
for the Helmholtz equations which holds for arbitrary radii.
Another cloaking object technique using NIMs is cloaking an object via anomalous localized
resonance technique. This was suggested and studied by Nguyen in [24]. Concerning this
technique, an object is cloaked by the complementary property (or more precisely by the doubly
complementary property) of the medium. This cloaking technique is inspired by the work of
Milton and Nicorovici in [16]. In their work, they discovered cloaking a source via anomalous
localized resonance for constant radial plasmonic structures in the two-dimensional quasistatic
regime (see [8, 4, 13, 18, 19, 25] and the references therein for recent results in this direction).
Another interesting application of NIMs is superlensing,i.e., the possibility to beatthe Rayleigh
diffraction limit: no constraint between the size of the object and the wavelength is imposed,
see [20, 23] and references therein.
Two difficulties in the study of cloaking using complementary are as follows. Firstly, the
problemisunstable. Thiscanbeexplainedbythefactthattheequationsdescribingthephenom-
ena have sign-changing coefficients; hencethe ellipticity and the compactness are lost in general.
Indeed, this is the case for complementary media, see [22]. Secondly, localized resonance might
appear, i.e., the field explodes in some regions and remains bounded in some others. It is wor-
thy noting that the character of resonance associated with NIMs is quite complex; localized
resonance and complete resonance can occur, see [26].
Inthispaper,westudycloakingusingcomplementarymediaforelectromagnetic waves(The-
orem 1). This is a natural continuation of [21, 27] where the acoustic setting was investigated.
Moving from the acoustic setting to electromagnetic one makes the analysis more delicate due
to the lack of the information of the whole gradient in the Maxwell equations. Besides extend-
ing the removing localized singualirty technique in [21, 20] with a simplification in [25], and
using the reflecting technique in [23] with roots in [17], the analysis also involves recent stability
result for Maxwell equations in [23], the weakly coupled second order elliptic equations prop-
erty of Maxwell equations, and new three-sphere inequalities. Using the concept of reflecting
complementary in [23], one can establish cloaking using complementary media for the Maxwell
equations for a general class of schemes (Proposition 1 in Section 4).
Let us now describe in details a scheme to cloak an arbitrary object using complementary
media for the Maxwell equations. A more general class of schemes is considered in Section 4.
Let B denote the ball centered at the origin and of radius r in R3 unless specified otherwise
r
and let h·,·,i denote the Euclidean scalar product in R3. Assume that the cloaked region is the
annulus B \B in R3 for some r > 0 in which the medium is characterized by a pair of two
2r2 r2 2
matrix-valued functions (ε ,µ ) of the permittivity ε and the permeability µ of the region.
O O O O
The assumption on the cloaked region by all means imposes no restriction since any bounded
set is a subset of such a region provided that the radius and the origin are appropriately chosen.
We assume that ε and µ are uniformly elliptic, i.e.,
O O
1 1
|ξ|2 ≤ ε (x)ξ,ξ ≤ Λ|ξ|2 and |ξ|2 ≤ µ (x)ξ,ξ ≤ Λ|ξ|2 ∀ξ ∈ R3, a.e. x ∈B \B .
Λ O Λ O r2 r1
(1.1)
(cid:10) (cid:11) (cid:10) (cid:11)
In this paper, we use a scheme in [21] with roots in the work of Lai et al. [15]. Following [21],
2
the cloak contains two parts. The first one, in B \B , makes use of complementary media
r2 r1
to cancel the effect of the cloaked region and the second one, in B , is to fill the space which
r1
“disappears” from the cancellation by the homogeneous medium. Concerning the first part,
instead of B \B , we consider B \B for some r > 0 as the cloaked region in which the
2r2 r2 r3 r2 3
medium is given by
ε ,µ in B \B ,
O O 2r2 r2
ε ,µ =
O O
( (cid:0) I,I (cid:1) in Br3 \B2r2.
(cid:0) (cid:1)
(This extension first seems teo bee technical(cid:0)but(cid:1)is later proved to be necessary in the acoustic
setting, see [24].) The (reflecting) complementary medium in B \B is then given by
r2 r1
F−1ε ,F−1µ , (1.2)
∗ O ∗ O
where F : B \B¯ → B \B¯ is the(cid:0)Kelvin transform(cid:1) with respect to ∂B , i.e.,
r2 r1 r3 r2 e e r2
r2
F(x) = 2 x. (1.3)
|x|2
Here
∇T(x)a(x)∇TT(x)
T a(y) = , (1.4)
∗
J(x)
where x = T−1(y) and J(x) = det∇T(x) for a diffeomorphism T. It follows that
r = r2/r . (1.5)
1 2 3
Note that inthedefinition of T given in (1.4), J(x) := det∇T(x)not |det∇T(x)|as often used
∗
in the acoustic setting. This convention is very suitable for the electromagnetic setting when a
change of variables is used (see (2.44) of Lemma 8). With this convention, one can easily verify
that F−1ε and F−1µ are negative symmetric matrices since det∇F(x) < 0. This clarifies the
∗ ∗
point that one uses NIMs to construct the complementary medium for the cloaked object.
Concerning the second part, the medium in B is given by
r1
(r2/r2)I,(r2/r2)I . (1.6)
3 2 3 2
As seen later, mathematically, it is re(cid:0)quired to have (3.3(cid:1)) in the proof of Theorem 1.
Taking into account the loss, the medium in the whole space R3 is thus characterized by
(ε ,µ ) defined as follows (see Figure 1 for the case δ = 0)
δ δ
ε ,µ in B \B ,
O O r3 r2
 F∗−1εO +(cid:0)iδI,F∗−(cid:1)1µO +iδI in Br2 \Br1,
(ε ,µ ) =  e e (1.7)
δ δ 
 (cid:0) e(r32/r22)I,(r32/re22)I (cid:1) in Br1,
 (cid:0) I,I (cid:1) in R3\Br3.



Physically, εδ and µδ are thepermittivity a(cid:0)nd p(cid:1)ermeability of the medium, k denotes the fre-
quency, and the imaginary parts of ε and µ in B \B describe the dissipative property of
δ δ r2 r1
this (negative index) region.
3
Complementary layer Filling space part
(F−1ε˜ ,F−1µ˜ ) (r2/r2)I,(r2/r2)I
∗ O ∗ O 3 2 3 2
(cid:0) (cid:1)
r
3
r
1 r
2
(I,I)
Cloaked object
(ε ,µ )
O O
Figure 1: Cloaking scheme for an object (ε ,µ ) in B \ B . Two parts are used: the
O O 2r2 r2
complementary one in B \B (the red region) which is the complementary medium of the
r2 r1
medium (ε˜ ,µ˜ ) in B \B and the filling space part in B (the blue region)
O O r3 r2 r1
Given(acurrent)j ∈ L2(R3) 3withcompactsupport,let(E ,H ),(E,H) ∈ [H (curl,R3)]2
δ δ loc
be respectively the unique outgoing solutions to the Maxwell systems
(cid:2) (cid:3)
∇×E = ikµ H in R3
δ δ δ
(1.8)
( ∇×Hδ = −ikεδEδ +j in R3,
and
∇×E = ikH in R3
(1.9)
( ∇×H = −ikE +j in R3.
For an open subset Ω of R3, the following standard notations are used:
H(curl,Ω):= u ∈[L2(Ω)]3; ∇×u∈ [L2(Ω)]3 ,
n o
kukH(curl,Ω) := kukL2(Ω)+k∇×ukL2(Ω),
and
H (curl,Ω) := u∈ [L2 (Ω)]3; ∇×u∈ [L2 (Ω)]3 .
loc loc loc
Recall that a solution (E,H) ∈ [H (cnurl,R3\B )]2 (for some R > 0) ofothe system
loc R
∇×E = ikH in R3\B ,
R
( ∇×H = −ikE in R3\BR,
is said to satisfy the outgoing condition (or the Silver-Mu¨ller radiation condition) if
E ×x+rH = O(1/r), (1.10)
4
as r = |x|→ +∞.
We extend (ε ,µ ) by (I,I) in B and still denote this extension by (ε ,µ ). We also
O O r2 O O
assume that
e e (εO,µO) is C2 in Br3. e e (1.11)
Condition (1.11) is required for the use of the unique continuation principle and three-sphere
e e
inequalities for Maxwell equations.
Cloaking effect of scheme (3.10) (see Figure 1) is mathematically confirmed in the following
main result of this paper.
Theorem 1. Let R > 0, j ∈ L2(R3) 3 with suppj ⊂⊂ B \B and let (E ,H ), (E,H) ∈
0 R0 r3 δ δ
H (curl,R3) 2 be the unique outgoing solution to (1.8) and (1.9) respectively. Given 0 < γ <
loc (cid:2) (cid:3)
1/2, there exists a positive constant ℓ = ℓ(γ) > 0, depending only on the elliptic constant of ε
O
(cid:2) (cid:3)
and µO in B2r2 \Br2 and k(εO,µO)kW2,∞(B4r2) such that if r3 > ℓr2 then
e
e k(Eδ,eHδe)−(E,H)kH(curl,BR\Br3) ≤ CRδγkjkL2, (1.12)
for some positive constant C independent of j and δ.
R
For an observer outside B , the medium in B looks like the homogeneous one by (1.9):
r3 r3
one has cloaking.
ThestartingpointoftheproofofTheorem1istousereflections(see(3.1)and(3.2))toobtain
Cauchy problems. We then explore the construction of the cloaking device (its complementary
property), use various three-sphere inequalities (Lemmas 6 and 7), and the removing localized
singularity technique to deal with the localized resonance. Using reflections is also the starting
point in the study of stability of Helmholtz equations with sign changing coefficients in [22]
(see also [6, 12, 31] for different approaches) and also plays an important role in the study of
superlensing applications of hyperbolic metamaterials in [7]. A numercial algorithm used for
NIMs in the spirit [22] is considered in [1]. Various techniques developed to study NIMs were
explored in the context of interior transmission eigenvalues in [28]. The study of NIMs in time
domain is recently investigated in [10].
The paper is organized as follows. The proof of Theorem 1 is given in Section 3 after
presenting several useful results in Section 2. In Section 4, we present a class of cloaking
schemes via the concept of reflecting complementary media.
2 Preliminaries
In this section, we present several results which are used in the proof of Theorem 1. We first
recall a known result on the trace of H(curl,D) (see [3, 9]).
Lemma 1. Let D be a smooth open bounded subset of R3 and set Γ = ∂D. The tangential trace
operator
γ :H(curl,D) → H−1/2(div ,Γ)
0 Γ
u 7→ u×ν
5
is continuous. Moreover, for all φ∈ H−1/2(div ,Γ), there exists u∈ H(curl,D) such that
Γ
γ0(u) = φ and kukH(curl,D) ≤ CkφkH−1/2(divΓ,Γ),
for some positive constant C independent of φ.
Here
H−1/2(div ,Γ) := φ ∈ [H−1/2(Γ)]3; φ·ν = 0 and div φ ∈ H−1/2(Γ)
Γ Γ
n o
kφkH−1/2(divΓ,Γ) := kφkH−1/2(Γ) +kdivΓφkH−1/2(Γ).
Thenext resultimplies thewell-posedness and a prioriestimates of (E ,H )definedin (1.8).
δ δ
Lemma 2. Let k > 0, 0 < δ < 1, R > 0, D ⊂ B be a smooth bounded open subset of R3. Let
0 R0
ε,µ be two real measurable matrix-valued functions defined in R3 such that ε,µ are uniformly
elliptic and piecewise C1 in R3, and
ε = µ = I in R3\B . (2.1)
R0
Set, for δ > 0,
(−ε+iδI,−µ+iδI) if x ∈ D,
(ε ,µ ) = (2.2)
δ δ
( (ε,µ) otherwise.
Let j ∈ L2(R3) with suppj ⊂ B . There exists a unique outgoing solution (E ,H ) ∈
R0 δ δ
[H (curl,R3)]2 to the Maxwell system
loc
∇×E = ikµ H in R3,
δ δ δ
(2.3)
( ∇×Hδ = −ikεδEδ +j in R3.
Moreover,
1
k(Eδ,Hδ)k2H(curl,BR) ≤ CR δkjkL2k(Eδ,Hδ)kL2(suppj)+kjk2L2 . (2.4)
(cid:18) (cid:19)
Here C denotes a positive constant depending on R, R , ε, µ but independent of j and δ.
R 0
Consequently, we have
C
R
k(Eδ,Hδ)kH(curl,BR) ≤ δ kjkL2. (2.5)
Proof. The existence of (E ,H ) can be derived from the uniqueness of (E ,H ) as usual. The
δ δ δ δ
uniqueness of (E ,H ) can be deduced from the estimates of (E ,H ). Estimate (2.5) is a direct
δ δ δ δ
consequence of (2.4). We hence only give the proof of (2.4). We have, by (2.3),
∇×(µ−1∇×E )−k2ε E = ikj in R3.
δ δ δ δ
Set
1
Mδ = δkjkL2k(Eδ,Hδ)kL2(suppj)+kjk2L2.
6
Multiplying the equation by E¯, integrating in B , and using the fact that suppj ⊂ B , we
δ R R0
have, for R > R ,
0
hµ−1∇×E ,∇×E i− (µ−1∇×E )×ν,E −k2 hε E ,E i = hikj,E i.
δ δ δ δ δ δ δ δ δ δ
ZBR Z∂BR ZBR ZBR
(cid:10) (cid:11)
Since µ = I and so ∇×E = ikH in R3\B , we derive that, for R > R ,
δ δ δ R0 0
hµ−1∇×E ,∇×E i+ hikH ,E ×νi−k2 hε E ,E i = hikj,E i.
δ δ δ δ δ δ δ δ δ
ZBR Z∂BR ZBR ZBR
LettingR → +∞,usingtheoutgoingcondition(E (x)×ν(x) = −H (x)+O(1/R2)forx ∈ ∂B ),
δ δ R
and considering the imaginary part, we obtain
kE k2 ≤ CM . (2.6)
δ H(curl,D) δ
This implies, by Lemma 1, with the notation Γ = ∂D,
kE ×νk2 ≤ CM . (2.7)
δ H−1/2(divΓ,Γ) δ
Using the equations of (E ,H ) in D, we derive from (2.6) that
δ δ
kH k2 ≤ CM ; (2.8)
δ H(curl,D) δ
which yields, by Lemma 1 again,
kH ×νk2 ≤CM . (2.9)
δ H−1/2(divΓ,Γ) δ
Let Dc be the unbounded connected component of R3\D¯ and let Dc be the complement of Dc
1 2 1
in R3\D¯, i.e., Dc = (R3\D¯)\Dc 1. We have
2 1
∇×E = ikµH in Dc,
δ δ 1
( ∇×Hδ = −ikεEδ +j in D1c.
It follows that, see e.g., [23, Lemma 5],
k(Eδ,Hδ)k2H(curl,BR∩D1c) ≤ CR kjkL2 +k(Eδ,Hδ)kH−1/2(div∂D1c,∂D1c) , (2.10)
(cid:0) (cid:1)
We deduce from (2.7) that
k(E ,H )k2 ≤ C M , (2.11)
δ δ H(curl,BR∩D1c) R δ
and, by Lemma 3 below, we derive from (2.7) and (2.9) that
k(E ,H )k2 ≤ CM . (2.12)
δ δ H(curl,D2c) δ
1We will apply Lemma 2 with D=Br2\Br1; in this case D1c =R3\B¯r2 and D2c =Br1.
7
A combination of (2.6), (2.8), (2.11), and (2.12) yields
k(E ,H )k ≤ C M ; (2.13)
δ δ H(curl,BR) R δ
which is (2.4). (cid:3)
In the proof of Lemma 2 we use the following result (see [23, Lemma 3]) whose proof follows
directly from the unique continuation principle for the Maxwell equations (see [5, 29]) via a
contradiction argument.
Lemma 3. Let k > 0, D be a smooth bounded open subset of R3, f,g ∈ [L2(D)]3, and h ,h ∈
1 2
H−1/2(div ,∂D), and let ε and µ be two piecewise C1, symmetric uniformly elliptic matrix-
∂D
valued functions defined in D. Assume that (E,H) ∈[H(curl,D)]2 is a solution to
∇×E = ikµH+f in D,
 ∇×H = −ikεE +g in D,


 H×ν =h ; E ×ν = h on ∂D.
1 2

Then 
k(E,H)kH(curl,D) ≤ C k(f,g)kL2(D) +k(h1,h2)kH−1/2(divΓ,∂D) , (2.14)
(cid:16) (cid:17)
for some positive constant C depending on D, ε, µ, and k but independent of f, g, h , and h .
1 2
Wenextstatearesultontheexistence,uniqueness,andstabilityresultsonMaxwellequations
with elliptic coefficients (see [23, Lemma 4]).
Lemma 4. Let k > 0, D be a smooth bounded open subset of R3, f,g ∈ [L2(R3)]3, h ,h ∈
1 2
H−1/2(div ,∂D). Assume that D¯,suppf,suppg ⊂ B for some R > 0. Let ε,µ be two
∂D R0 0
symmetric uniformly elliptic matrix-valued functions defined in R3 such that (ε,µ) = (I,I) in
R3\B and (ε,µ) is piecewise C1. There exists a unique solution (E,H) ∈ H(curl,B \
R0 R>0 R
2
∂D) of the system
(cid:2)T
(cid:3) ∇×E = ikµH+f in R3\∂D,
 ∇×H = −ikεE +g in R3\∂D, (2.15)


 [H×ν]= h ; [E ×ν]= h on ∂D.
1 2

Moreover, 
k(E,H)kH(curl,BR\∂D) ≤ CR k(f,g)kL2 +k(h1,h2)kH−1/2(divΓ,∂D) , (2.16)
(cid:16) (cid:17)
for some positive constant C depending on R, R , D, ε, µ, and k, but independent of f,g,h ,
R 0 1
and h .
2
Hereafter [·] denotes the jump across the boundary ∂D, i.e., [u] = u| −u| for an appro-
ext int
priate function u.
Remark 1. In the proof of Lemma 4, one uses a new compactness criterion for the Maxwell
equations in [23] with roots in [14].
8
We next present a known result which reveals a connection between Maxwell equations with
weakly coupled elliptic systems.
Lemma 5. Let D be an open subset of R3, ε,µ be two matrix-valued functions defined in D,
and let (E,H) ∈ H1(D) 2 be a solution of the system
(cid:2) (cid:3)
∇×E = ikµH in D,
(2.17)
( ∇×H = −ikεE in D.
Then, for 1 ≤ a ≤ 3,
div(µ∇H )+div(∂ µH−ikµǫaεE) = 0 in D, (2.18)
a a
div(ε∇E )+div(∂ εE +ikεǫaµH)= 0 in D. (2.19)
a a
Here ǫa (1 ≤ a,b,c ≤ 1) denotes the usual Levi Civita permutation, i.e.,
bc
sign (abc) if abc is a permuation,
ǫa = (2.20)
bc
( 0 otherwise.
Proof. The proof is quite simple as follows. Using the fact, for 1 ≤a ≤ 3,
∂ H = ∇H +ǫa(∇×H) and ∂ E = ∇E +ǫa(∇×E),
a a a a
we derive from (2.17) that, for 1 ≤a ≤ 3,
∂ H = ∇H −ikǫaεE and ∂ E = ∇E +ikǫaµH in D. (2.21)
a a a a
Since
div(µH)= 0 in D,
it follows that, for 1 ≤ a ≤ 3,
0 = ∂ div(µH)= div(µ∂ H)+div(∂ µH) in D.
a a a
This implies, by the first identity of (2.21),
div(µ∇H )+div(∂ µH−ikµǫaεE) = 0 in D;
a a
which is (2.18). Similarly, we obtain (2.19). (cid:3)
Hadamard proved the following three-circle inequality: Assume that ∆v = 0 in BR∗\BR∗ ⊂
R2 and 0 < R < R < R < R < R∗. Then
∗ 1 2 3
kvkL∞(∂BR2) ≤ kvkαL∞(∂BR1)kvk1L−∞α(∂BR3),
with α = ln(R /R )/ln(R /R ). Here is its variant which is useful in the proof of Theorem 1.
3 2 3 1
9
Lemma 6. Let d= 2, 3, k, R∗, R∗ > 0, and let v ∈ H1(BR∗\BR∗) be a solution to the equation
∆v+k2v = 0 in B \B ⊂ Rd. We have, for R ≤ R < R < R ≤ R∗,
R3 R1 ∗ 1 2 3
kvkH(∂BR2) ≤ CkvkαH(BR1)kvk1H−(αBR3), (2.22)
where α= ln(R /R )/ln(R /R ) and C is a positive constant depending only on k, R , and R∗.
3 2 3 1 ∗
Here
kvkH(∂Br) := kvkH1/2(∂Br)+k∂rvkH−1/2(∂Br).
Before giving the proof of Lemma 6, we recall some properties of the spherical Bessel and
Neumann functions and the Bessel and Neumann functions of large order. We first introduce,
for n ≥ 1,
y (t)
ˆj (t) = 1·3···(2n+1)j (t) and yˆ = − n , (2.23)
n n n
1·3···(2n−1)
and for n ≥ 0,
πi
Jˆ (r)= 2nn!J (r) and Yˆ (r)= Y (r), (2.24)
n n n 2n(n−1)! n
where j and y are the spherical Bessel and Neumann functions, and J and Y are the Bessel
n n n n
and Neumann functions of order n respectively. Then, see, e.g., [11, (2.37), (2.38), (3.57), and
(3.58)]), as n → +∞,
ˆj (r) = rn 1+O(1/n) yˆ (r) = r−n−1 1+O(1/n) , (2.25)
n n
Jˆ (t) = tn 1+(cid:2) O(1/n) , (cid:3) and Yˆ (t) = t(cid:2)−n 1+O(1/(cid:3)n) . (2.26)
n n
One also has, see, e.g., [11, ((cid:2)2.36) and (3(cid:3).56)], (cid:2) (cid:3)
1
j (r)y′(r)−j′(r)y (r)= (2.27)
n n n n r2
and
2
J (r)Y′(r)−J′(r)Y (r) = . (2.28)
n n n n πr
We are ready to give
Proof of Lemma 6. By rescaling, one can assume that k = 1. We consider the case d= 2 and
d = 3 separately.
Case 1: d=3. Since ∆v+v =0 in B \B , v can be represented in the form
R3 R1
∞ n
v = anˆj (|x|)+bnyˆ (|x|) Yn(xˆ) in B \B ,
m n m n m R3 R1
Xn=1mX=−n(cid:16) (cid:17)
for an ∈ C and xˆ = x/|x| where Yn is the spherical harmonic function of degree n and of order
m m
m. In what follows in this proof, C denotes a positive constant depending only on R and R∗
∗
and can change from one place to another and a ∼b means that a ≤ Cb and b ≤ Ca. Using the
fact (Yn) is an orthonormal basis of L2(∂B ) and
m 1
∆ Yn+n(n+1)Yn = 0 on ∂B ,
∂B1 m m 1
10