Regularized Transformation-Optics Cloaking for the Helmholtz Equation: From Partial Cloak to Full Cloak

1.9 MB·English

by  Jingzhi Li

Regularized Transformation-Optics Cloaking for the
Helmholtz Equation: From Partial Cloak to Full Cloak
Jingzhi Li∗, Hongyu Liu†, Luca Rondi‡, Gunther Uhlmann§
Abstract
3
1 We develop a very general theory on the regularized approximate invisibility
0
cloaking for the wave scattering governed by the Helmholtz equation in any space
2
dimensions N 2 via the approach of transformation optics. There are four major

n ≥
ingredients in our proposed theory: 1). The non-singular cloaking medium is ob-
a
tained by the push-forwarding construction through a transformation which blows
J
up a subset K in the virtual space, where ε 1 is an asymptotic regularization
9 ε (cid:28)
2 parameter. Kε willdegeneratetoK0 asε +0,andinourtheoryK0 couldbeany
convex compact set in RN, or any set who→se boundary consists of Lipschitz hyper-

] surfaces, or a finite combination of those sets. 2). A general lossy layer with the
P
material parameters satisfying certain compatibility integral conditions is employed
A
rightbetweenthecloakedandcloakingregions. 3).Thecontentsbeingcloakedcould
.
h also be extremely general, possibly including, at the same time, generic mediums
t and, sound-soft, sound-hard and impedance-type obstacles, as well as some sources
a
m or sinks. 4). In order to achieve a cloaking device of compact size, particularly for
thecasewhenK isnot“uniformlysmall”,anassembly-by-components, the(ABC)
[ ε
geometry is developed for both the virtual and physical spaces and the blow-up

1
construction is based on concatenating different components.
v
Within the proposed framework, we show that the scattered wave field u cor-
3 ε
1 responding to a cloaking problem will converge to u0 as ε +0, with u0 being
→
0 the scattered wave field corresponding to a sound-hard K0. The convergence result
7 is used to theoretically justify the approximate full and partial invisibility cloaks,
.
1 depending on the geometry of K0. On the other hand, the convergence results are
0 conducted in a much more general setting than what is needed for the invisibility
3 cloaking, so they are of significant mathematical interest for their own sake. As for
1
applications, we construct three types of full and partial cloaks. Some numerical
:
v experiments are also conducted to illustrate our theoretical results.
i
X
∗Faculty of Science, South University of Science and Technology of China, Shenzhen 518055,
r P. R. China. Email: [email protected]
a
†Department of Mathematics and Statistics, University of North Carolina, Charlotte, NC 28223,
USA. Email: [email protected]
‡Dipartimento di Matematica e Geoscienze, Universita` degli Studi di Trieste, Trieste, Italy. Email:
[email protected]
§DepartmentofMathematics,UniversityofWashington,Seattle,WA98195,USAandFondationdes
Sciences Math´ematiques de Paris. Email: [email protected]
1
Keywords: wavescattering,Helmholtzequation,invisibilitycloaking,transforma-
tion optics, partial and full cloaks, asymptotic estimates
2010MathematicsSubjectClassification: 35Q60,35J05,31B10,35R30,78A40
1 Introduction
This paper is concerned with the invisibility cloaking for the wave scattering governed
by the Helmholtz equation via the approach of transformation optics [16, 17, 22, 36],
which is a rapidly growing scientific field with many potential applications. We refer to
[8, 14, 15, 33, 39, 40] and the references therein for discussions of the recent progress on
both the theory and experiments.
Let Ω and D be two bounded Lipschitz domains in RN, N 2, such that D (cid:98) Ω.
≥
Let σ = σ(x) = (σij(x)) RN×N, x RN, be a symmetric-matrix valued measurable
∈ ∈
function such that, for some λ, 0 < λ 1, we have
≤
N
(cid:88)
λ ξ 2 σij(x)ξ ξ λ−1 ξ 2 for any ξ RN and for a.e. x RN. (1.1)
i j
(cid:107) (cid:107) ≤ ≤ (cid:107) (cid:107) ∈ ∈
i,j=1
Let q = q +iq = q(x), x RN, be a complex-valued bounded measurable function
1 2
∈
with real and imaginary parts q and q respectively, such that, for some λ, 0 < λ 1,
1 2
≤
we have
q (x) λ, q (x) 0 for a.e. x RN. (1.2)
1 2
≥ ≥ ∈
Furthermore, we assume that q(x) = q := 1 and σij(x) = σij := δ for x RN Ω,
0 0 ij ∈ \
where δ denotes the Kronecker delta function. In the following, (1.1) and (1.2) will be
ij
referred to as the regular conditions on σ and q, and λ is called the regular constant.
Next, we introduce the time-harmonic wave scattering governed by the Helmholtz
equation whose weak solution is u = u(x,d,k), x RN, where d SN−1, k R ,
+
∈ ∈ ∈
(cid:40)
div(σ u)+k2qu = 0 in RN,
∇ (1.3)
u(x,d,k) eikx·d satisfies the radiation condition.
−
The last statement in (1.3) means that if one lets us(x,d,k) = u(x,d,k) eikx·d, then
−
lim rN2−1 (cid:18)∂us(x) ikus(x)(cid:19) = 0, r = x . (1.4)
r→∞ ∂r − (cid:107) (cid:107)
In the physical situation, (1.3) can be used to describe the time-harmonic acoustic
scattering due to an inhomogeneous acoustical medium (Ω;σ,q) located in an otherwise
uniformly homogeneous space (RN Ω;σ ,q ). σ and q, respectively, denote the den-
0 0
\
sity tensor and acoustic modulus of the acoustical medium, and u(x) denotes the wave
pressure with U(x,t) := u(x)e−ikt representing the wave field satisfying the scalar wave
equation
N (cid:18) (cid:19)
(cid:88) ∂ ∂
q(x)U (x,t) σij(x) U(x,t) = 0 in RN R.
tt
− ∂x ∂x ×
i j
i,j=1
2
The function ui(x) := eikx·d is an incident plane wave with k denoting the wave number
and d SN−1 denoting the impinging direction. u(x) is called the total wave field and
∈
us(x) is called scattered wave field, which is the perturbation of the incident plane wave
caused by the presence of the inhomogeneity (Ω;σ,q) in the whole space. Indeed, it is
easilyseenthatifthereisnopresenceoftheinhomogeneity, us willbevanishing. Forthe
particularcasewithN = 2,(1.3)canalsobeusedtodescribethetransverse-electric(TE)
polarized electromagnetic (EM) wave propagation with the presence of an infinitely long
cylindrical EM inhomogeneity (Ω;σ,q) (see, e.g., [7]). In this case, ε := q, γ := k q
(cid:60) (cid:61)
and µ := σ−1 denote, respectively, the electric permittivity, conductivity and magnetic
permeability, where and denote taking the respective real and imaginary parts. We
(cid:60) (cid:61)
refer to [9, 32] for related physical background. In the rest of the paper, in order to ease
the exposition, we stick to the terminologies with the acoustic scattering.
We recall that by a weak solution to (1.3) we mean that u H1 (RN) and that it
∈ loc
satisfies
(cid:90)
σ u ϕ k2quϕ = 0 for any ϕ C∞(RN).
0
RN ∇ · − ∈
The limit in (1.4) has to hold uniformly for every direction xˆ = x/ x SN−1 and
(cid:107) (cid:107) ∈
is also known as the Sommerfeld radiation condition which characterizes the radiating
nature of the scattered wave field us (cf. [9, 32]). There exists a unique weak solution
u(x,d,k) = u−χ + u+χ H1 (RN) to (1.3), and we refer to Appendix in [27]
Ω RN\Ω ∈ loc
for a convenient proof. We remark that, if the coefficients are regular enough, (1.3)
corresponds to the following classical transmission problem

N (cid:18) (cid:19)
 (cid:88) ∂ ∂
 σij u−(x,d,k) +k2qu−(x,d,k) = 0 x Ω,

 i,j=1 ∂xi ∂xj ∈

 ∆u+(x,d,k)+k2u+(x,d,k) = 0 x RN Ω,
 ∈ \
 u−(x) = u+(x), (cid:88)N (ν σij∂u−)(x) = (ν u+)(x) x ∂Ω, (1.5)
i
 i,j=1 ∂xj ·∇ ∈

 u+(x,d,k) = eikx·d+us(x,d,k) x RN Ω,
 rl→im∞rN2−1 (cid:18)∂u∂sr(x) −ikus(x)(cid:19) = 0 r =∈ (cid:107)x(cid:107)\,
where ν = (ν )N is the outward unit normal vector to ∂Ω.
i i=1
Furthermore, u(x) admits the following asymptotic development as x +
(cid:107) (cid:107) → ∞
(cid:32) (cid:33)
eik(cid:107)x(cid:107) (cid:18) x (cid:19) 1
u(x,d,k) = eikx·d+ u ,d,k + . (1.6)
x N2−1 ∞ (cid:107)x(cid:107) O x N2+1
(cid:107) (cid:107) (cid:107) (cid:107)
In (1.6), u (xˆ,d,k) with xˆ := x/ x SN−1 is known as the far-field pattern or the
∞
(cid:107) (cid:107) ∈
scattering amplitude, which depends on the impinging direction d and wave number k of
the incident wave ui(x) := eikx·d, observation direction xˆ, and obviously, also the under-
lyingscatteringobject(Ω;σ,q). Inthefollowing, weshallalsowriteu (xˆ,d;(Ω;σ,q))to
∞
3
indicate such dependences, noting that we consider k to be fixed and we drop the depen-
dence on k. An important inverse scattering problem arising from practical applications
is to recover the medium (Ω;σ,q) by knowing of u (xˆ,d). This inverse problem is of
∞
fundamental importance to many areas of science and technology, such as radar and
sonar, geophysical exploration, non-destructive testing, and medical imaging to name
just a few; see [9, 18] and the references therein. In this work, we shall be mainly con-
cerned with the invisibility cloaking for the inverse scattering problem, which could be
generally introduced as follows.
Definition 1.1. Let Ω and D be bounded Lipschitz domains such that D (cid:98) Ω. Ω D
\
and D represent, respectively, the cloaking region and the cloaked region. Let Γ and Γ(cid:48)
be two subsets of SN−1. (Ω D;σ ,q ) is said to be an (ideal/perfect) invisibility cloaking
c c
\
device for the region D if
u (xˆ,d;(Ω;σ ,q )) = 0 for xˆ Γ, d Γ(cid:48), (1.7)
∞ e e
∈ ∈
where the extended object
(cid:40)
σ ,q in D,
a a
(Ω;σ ,q ) =
e e
σ ,q in Ω D,
c c
\
with (D;σ ,q ) denoting a target medium. If Γ = Γ(cid:48) = SN−1, then it is called a full
a a
cloak, otherwise it is called a partial cloak with limited apertures Γ of observation angles,
and Γ(cid:48) of impinging angles.
By Definition 1.1, we have that the cloaking layer (Ω D;σ ,q ) makes the target
c c
\
medium (D;σ ,q ) invisible to the exterior scattering measurements when the detecting
a a
waves come from the aperture Γ(cid:48) and the observations are made in the aperture Γ.
One efficient way of constructing the invisibility cloak that has received significant
attentions in recent years is the so-called transformation optics [16, 17, 22, 36]. By
taking advantage of the push-forward properties of the material parameters σ and q,
the transformation optics approach via a blow-up transformation in constructing an
(ideal) invisibility cloak can be simply described as follows. Let (Ω;σ ,q ) be selected
0 0
for constructing the cloaking device, and let P Ω be a point. (Ω P;σ ,q ) lives in the
0 0
∈ \
so-called virtual space. Suppose that there exists a transformation F which blows up the
point P to an open subset D within Ω. The homogeneous virtual space is then pushed-
forward to form the cloaking layer (Ω D;σ ,q ). The cloaking layer together with a
c c
\
filling-intargetmedium(D;σ ,q )formsthecloakingdevice, whichlivesintheso-called
a a
physical space. Due to the transformation invariance of the Helmholtz equation, it can
beheuristicallyarguedthatthescatteringamplitudeinthephysicalspaceisthesameas
the scattering amplitude in the virtual space. Since the scatterer in the virtual space is
a singular point P, whose scattering effect is negligible, this implies that the scattering
amplitude in the physical space is also vanishing. Here, we would like to emphasize
that from a practical viewpoint, the target medium should be arbitrary or as general
as possible, and this viewpoint shall be adopted throughout our current study. The
4
blow-up-a-point construction yields singular cloaking materials, namely, the material
parameters violate the regular conditions. The singular media present a great challenge
forboththeoreticalanalysisandpracticalfabrications(cf. [11,29]). Inordertoavoidthe
singularstructure, severalregularizedconstructionshavebeendeveloped. In[10,12,38],
a truncation of singularities has been introduced. In [20, 21, 26], the ‘blow-up-a-point’
transformationin[17,22,36]hasbeenregularizedtobecomethe‘blow-up-a-small-region’
transformation. In the current study, we shall adopt the latter one for the construction
of our cloaking device. Nevertheless, as pointed out in [19], the truncation-of-singularity
construction and the blow-up-a-small-region construction are equivalent to each other.
Hence, all the obtained results in this work equally hold for the truncation-of-singularity
construction. Instead of ideal/perfect invisibility, one would consider approximate/near
invisibility for a regularized construction; that is, one intends to make the corresponding
scattering amplitude due to a regularized cloaking device as small as possible depending
on an asymptotically small regularization parameter ε R . This is the main subject
+
∈
of study for the present paper.
Due to its practical importance, the approximate cloaking has recently been ex-
tensively studied. In [5, 21], approximate cloaking schemes were developed for EIT
(electric impedance tomography) which might be regarded as optics at zero frequency.
In [6, 4, 20, 23, 28, 26, 34, 35], various near-cloaking schemes were presented for scalar
waves governed by the Helmholtz equation. In all the aforementioned work, the con-
structions of the cloaking layer (Ω D;σε,qε) are based on blowing up a uniformly small
c c
\
neighborhood P of a singular point P; namely, P degenerates to the single point P
ε ε
as ε +0. In order to stabilize and enhance the accuracy of approximation of the
→
near-cloaks, various mechanisms have been developed in those literatures. Particularly,
we would like to note that, in [20], it is shown that the regularized approximate cloak is
unstable due to the existence of cloak-busting inclusions, and the authors propose to in-
corporate a special lossy layer to stabilize the approximation. A different lossy layer was
proposed and investigated in [23, 28]. The cloaking of impenetrable obstacles, which
could be taken as lossy mediums with extreme material parameters, were considered
in [5, 6, 4] and [26]. Also, we would like to point out that, in all those studies, the
approximate full invisibility cloaks were obtained.
Inthepresentwork, wedevelopaverygeneraltheoryontheregularizedapproximate
invisibility cloaking for the wave scattering governed by the Helmholtz equation in any
space dimensions N 2 via the approach of transformation optics. First, the non-
≥
singular cloaking medium is obtained by the push-forwarding construction through a
transformation which blows up a subset K in the virtual space, with K degenerating
ε ε
to K as ε +0. In our theory, K could be very general. It could be any convex
0 0
→
compact subset in RN, or any set whose boundary consists of Lipschitz hypersurfaces,
or a finite combination of those sets. For example, in R3, it could be a single point,
or a line segment, or a bounded planar subset. This includes all the existing studies
in the literature by blowing up ‘point-like’ regions as a very special case. Second, in
order to stabilize the approximation process, a lossy layer with the material parameters
satisfying certain mild compatibility integral conditions is employed right between the
5
cloaked and cloaking regions. The lossy layer is also very general and could be variable
and even be anisotropic. Third, the proposed cloaking scheme is shown to be capable
of nearly cloaking an very general content, possibly including, at the same time, generic
passive mediums, and sound-soft, sound-hard, and impedance-type obstacles, and some
active sources or sinks as well. Finally, in order to achieve a cloaking device of compact
size, particularly for the case when K is not ‘point-like’, assembled-by-components
ε
(ABC) geometry is developed for both the virtual and physical spaces and the blow-
up construction is based on concatenating different components. Within the proposed
framework,weshowthatthescatteredwavefieldu correspondingtoacloakingproblem
ε
will converge to u as ε +0, with u being the scattered wave field corresponding to a
0 0
→
sound-hard K . The convergence result is used to theoretically justify the approximate
0
full and partial invisibility cloaks, depending on the geometry of K . On the other
0
hand, the convergence results are conducted in a much more general setting than what
is needed for the invisibility cloaking, so they are of significant mathematical interest for
their own sake. As for applications, we construct three types of full and partial cloaks.
Some numerical experiments are also conducted to illustrate our theoretical results.
It is interesting to note that in addition to the blow-up-a-single-point construction,
thecloakingconstructionsbasedonblowingupanarccurveoraplanarrectanglearealso
proposed and investigated in [13, 24], and they respectively yield the so-called electro-
magnetic wormholes and carpet-cloaking. As discussed earlier, the regularized blow-up-
a-single-point construction, namely the blow-up-a-small-region construction, has been
extensively studied in the literature. Using the general framework developed in the
present work, one can easily construct the regularized electromagnetic wormholes and
carpet-cloaking by employing non-singular materials. In this paper, we focus entirely on
the transformation optics approach in achieving the cloaks. We would like to mention
in passing other promising cloaking techniques which we did not consider in the present
study including the one based on anomalous localized resonance [3, 31], and another one
based on special (object-dependent) coatings [2].
The rest of the paper is organized as follows. In the next section, we present the
general blow-up construction of the proposed regularized cloaks and give some relevant
discussions. Section3isdevotedtotheconvergenceanalysisinthevirtualspace. Section
4 is on the application of the results obtained in Section 3 to the construction of full
and partial cloaks in the physical space. In Section 5, we develop the ABC-geometry for
both the virtual and physical spaces, and construct three types of full and partial cloaks
that are new to the literature. Finally, in Section 6, we give some numerical simulations.
2 General construction of the regularized cloaks
In this section, we shall give the general construction of a regularized cloaking device
via the transformation optics approach based on a blow-up mapping between the virtual
and the physical spaces. The main purpose of this section is to pave the way for our
convergence analysis study in the virtual space that shall be conducted in the next
section. We first give a definition of an admissible acoustic configuration.
6
Foranyx RN,N 2,wedenotex = (x(cid:48),x ) RN−1 Randx = (x(cid:48)(cid:48),x ,x )
N N−1 N
∈ ≥ ∈ × ∈
RN−2 R R. For any r > 0 and any x RN, B (x) denotes the Euclidean ball con-
r
× × ∈
tained in RN with radius r and center x, whereas B(cid:48)(x(cid:48)) denotes the Euclideean ball
r
contained in RN−1 with radius r and center x(cid:48). Moreover, B = B (0) and B(cid:48) = B(cid:48)(0).
r r r r
Finally, for any E RN, we denote B (E) = (cid:83) B (x).
⊂ r x∈E r
Definition 2.1. We say that K RN is a scatterer if K is compact and G = RN K is
⊂ \
connected.
A scatterer K B , for some R > 0, is regular if the immersion W1,2(B K)
R R+1
⊂ \ →
L2(B K) is compact.
R+1
\
We say that a scatterer K is Lipschitz-regular if, for some positive constants r, L
1
and L , for any x ∂K there exists a bi-Lipschitz function Φ : B (x) RN such that
2 x r
∈ →
the following properties hold. First, for any z , z B (x) we have
1 2 r
∈
L z z Φ (z ) Φ (z ) L z z .
1 1 2 x 1 x 2 2 1 2
(cid:107) − (cid:107) ≤ (cid:107) − (cid:107) ≤ (cid:107) − (cid:107)
Second, Φ (x) = 0 and Φ (∂K B (x)) π = y RN : y = 0 .
x x r N
∩ ⊂ { ∈ }
A scatterer K is said to be Lipschitz if, for some positive constants r and L, the
following assumptions hold.
For any x ∂K, there exists a function ϕ : RN−1 R, such that ϕ(0) = 0 and
∈ →
which is Lipschitz with Lipschitz constant bounded by L, such that, up to a rigid change
of coordinates, we have x = 0 and
B (x) ∂K y B (x) : y = ϕ(y(cid:48)) .
r r N
∩ ⊂ { ∈ }
We say that x ∂K belongs to the interior of ∂K if there exists δ, 0 < δ r,
∈ ≤
such that B (x) ∂K = y B (x) : y = ϕ(y(cid:48)) . Otherwise we say that x belongs
δ δ N
∩ { ∈ }
to the boundary of ∂K. We remark that the boundary of ∂K might be empty and
that, if x ∂K belongs to the interior of ∂K, then K may lie at most on one side of
∈
∂K, that is B (x) K = B (x) ∂K, or B (x) K = y B (x) : y ϕ(y(cid:48)) , or
δ δ δ δ N
∩ ∩ ∩ { ∈ ≥ }
B (x) K = y B (x) : y ϕ(y(cid:48)) .
δ δ N
∩ { ∈ ≤ }
For any x belonging to the boundary of ∂K, we assume that there exists another
function ϕ : RN−2 R, such that ϕ (0) = 0 and which is Lipschitz with Lipschitz
1 1
→
constant bounded by L, such that, up to the previous rigid change of coordinates, we
have x = 0 and
B (x) ∂K = y B (x) : y = ϕ(y(cid:48)), y ϕ (y(cid:48)(cid:48)) .
r r N N−1 1
∩ { ∈ ≤ }
Finally, for any x ∂K, let e (x),...,e (x) be the unit vectors representing the
1 N
∈
orthonormal base of the coordinate system for which the previous representations hold.
Then we assume that e (x) is a Lipschitz function of x ∂K, with Lipschitz constant
N
∈
bounded by L, and e (x) is a Lipschitz function of x, as x varies on the boundary of
N−1
∂K, with Lipschitz constant bounded by L.
Properties of Lipschitz scatterers are thoroughly investigated in [30, Section 4]. Let
usjustnoticethataLipschitzscattererK isLipschitz-regular. Furthermore,aLipschitz-
regular scatterer K B is regular and the immersion W1,2(B K) L2(∂K) is
R R+1
⊂ \ →
7
compact. Notice that for a connected component K˜ of K with empty interior, with a
slight abuse of notation, with ∂K˜ we denote two copies of K˜, (K˜+,K˜−), and by L2(∂K˜)
we denote the couple (u+,u−) L2(K˜+) L2(K˜−). In such a way we can define the
∈ ×
trace of a function u W1,2(B K) on both sides of K˜.
R+1
∈ \
Definition 2.2. We fix k > 0. An acoustic configuration = (K ,K ,K ,s,σ,q,h,H)
1 2 3
C
is admissible if the following assumptions hold.
There exist three scatterers K , K , K which are pairwise disjoint (possibly some or
1 2 3
allofthemmaybetheemptyset)andsuchthatK isregularandK isLipschitz-regular.
2 3
We set K = K K K .
1 2 3
∪ ∪
Let s = s +is = s(x), x ∂K , be a complex-valued bounded N−1-measurable
1 2 3
∈ H
function, with real and imaginary part s and s respectively, such that
1 2
s (x) 0 and s (x) 0 for N−1-a.e. x ∂K .
1 2 3
≤ ≥ H ∈
Again, on a connected component K˜ of K with empty interior, s L∞(∂K˜) means
3
∈
(s+,s−) L∞(K˜+) L∞(K˜−).
∈ ×
Let σ = σ(x), x RN K, be an N N symmetric matrix whose entries are real-
∈ \ ×
valued measurable functions such that, for some λ, 0 < λ 1, we have
≤
λ ξ 2 σ(x)ξ ξ λ−1 ξ 2 for any ξ RN and for a.e. x RN K.
(cid:107) (cid:107) ≤ · ≤ (cid:107) (cid:107) ∈ ∈ \
Let q = q + iq = q(x), x RN K, be a complex-valued bounded measurable
1 2
∈ \
function, with real and imaginary part q and q respectively, such that, for some λ,
1 2
0 < λ 1, we have
≤
q (x) λ and q (x) 0 for a.e. x RN K.
1 2
≥ ≥ ∈ \
Furthermore, we assume that
σ(x) I and q(x) 1 for any x outside a compact subset.
≡ ≡
The source term (h,H) L2(RN K,C CN) and it has compact support.
∈ \ ×
Finally, we require that the following scattering problem has only a trivial weak
solution
 div(σ u)+k2qu = 0 in RN K,
 ∇ \
 u = 0 on ∂K1,
σ u ν = 0 on ∂K , (2.1)
2
∇ ·
 σ u ν su = 0 on ∂K3,
 ∇ · −
u satisfies the radiation condition,
where ν denotes the exterior normal to G = RN K.
\
Physically speaking, we have that K is a sound-soft scatterer, K is sound-hard
1 2
and we have an impedance boundary condition on K . The admissible configuration
3
represents the scattering object for our study, which consists of the impenetrable
C
obstacle (K,s), the passive medium (σ,q), and the active source/sink (h,H). We note
the following result that can be proved in a standard way (cf. [27]).
8
Proposition 2.1. Let ui be an entire solution to the Helmholtz equation ∆u+k2u = 0
in RN. Let us consider an admissible configuration in RN. Then there exists a unique
C
weak solution to the following scattering problem
 div(σ u)+k2qu = h+div(H) in RN K,
 ∇ − \
 u = 0 on ∂K1,
σ u ν = 0 on ∂K , (2.2)
2
∇ ·
 σ u ν su = 0 on ∂K3,
 u∇ u·i sa−tisfies the radiation condition.
−
If we take ui(x) = eikx·d, x RN, to be the plane wave, then the solution u to
∈
(2.2) has exactly the same asymptotic development as that in (1.6), and we denote by
u (xˆ,d; ), xˆ SN−1, the far-field pattern of the scattered field us.
∞
C ∈
Next, we present a lemma with some key ingredients of the transformation optics,
the proofs of which are available in [14, 20, 26]. The main remark is that the definition
of an admissible configuration is stable under bi-Lipschitz transformations.
Lemma 2.1. Let Ω and Ω(cid:101) be two Lipschitz domains in RN and x(cid:101)= F(x) : Ω Ω(cid:101) be a
→
bi-Lipschitz and orientation-preserving mapping.
Let = (Ω;K ,K ,K ,s,σ,q,h,H) be an admissible configuration in Ω. For sim-
1 2 3
C
plicity we assume that K and the supports of h and H are contained in Ω. We let the
push-forwarded configuration be defined as
(cid:101)= (Ω(cid:101);K(cid:101)1,K(cid:101)2,K(cid:101)3,s(cid:101),σ(cid:101),q(cid:101),(cid:101)h,H(cid:101)) = F∗(Ω;K1,K2,K3,s,σ,q,h,H) :=
C
(F(Ω);F(K ),F(K ),F(K ),F s,F σ,F q,F h,F H). (2.3)
1 2 3 ∗ ∗ ∗ ∗ ∗
Notice that K(cid:101)1 is a scatterer, whereas K(cid:101)2 is a regular scatterer and K(cid:101)3 is a Lipschitz-
regular scatterer. Moreover,
(cid:12)
s(x) (cid:12)
s(cid:101)(x(cid:101)) = F∗s(x(cid:101)) := (cid:12) , (2.4)
det(D F(x)) (cid:12)
| τ | x=F−1(x(cid:101))
where D F denotes the tangential component of the Jacobian matrix of F. We also have
τ
DF(x) σ(x) DF(x)T(cid:12)(cid:12)
σ(cid:101)(x(cid:101)) = F∗σ(x(cid:101)) := · · (cid:12) ,
det(DF(x)) (cid:12)
| | x=F−1(x(cid:101)) (2.5)
(cid:12)
q(x) (cid:12)
q(cid:101)(x(cid:101)) = F∗q(x(cid:101)) := (cid:12) ,
det(DF(x)) (cid:12)
| | x=F−1(x(cid:101))
where DF denotes the Jacobian matrix of F. Finally,
(cid:12)
h(x) (cid:12)
(cid:101)h(x(cid:101)) = F∗h(x(cid:101)) := (cid:12) ,
det(DF(x)) (cid:12)
| | x=F−1(x(cid:101)) (2.6)
(cid:12)
DF(x) H(x)(cid:12)
H(cid:101)(x(cid:101)) = F∗H(x(cid:101)) := · (cid:12) .
det(DF(x)) (cid:12)
| | x=F−1(x(cid:101))
9
Then u H1(Ω K) solves the Helmholtz equation
∈ \
 (σ u)+k2qu = h+ H in Ω K,
 ∇· ∇ − ∇· \
 u = 0 on ∂K ,
1
σ u ν = 0 on ∂K ,
 2
 σ∇u·ν su = 0 on ∂K ,
3
∇ · −
if and only if the pull-back field u(cid:101)= (F−1)∗u := u F−1 H1(Ω(cid:101) K(cid:101)) solves
◦ ∈ \

 ∇(cid:101) ·(σ(cid:101)∇u(cid:101))+k2q(cid:101)u(cid:101)= −(cid:101)h+∇(cid:101) ·H(cid:101) in Ω(cid:101)\K(cid:101),
 u(cid:101)= 0 on ∂K(cid:101)1,
 σ(cid:101)∇u(cid:101)·ν = 0 on ∂K(cid:101)2,
 σ(cid:101) u(cid:101) ν s(cid:101)u(cid:101)= 0 on ∂K(cid:101)3.
∇ · −
We have made use of and (cid:101) to distinguish the differentiations respectively in x- and
∇ ∇
x-coordinates.
(cid:101)
As a consequence of Lemma 2.1, one can directly verify that if F : Ω Ω is a
→
bi-Lipschitz map with F = Identity, then the push-forward of an admissible configu-
∂Ω
|
ration is again an admissible configuration and
u∞(xˆ,d; ) = u∞(xˆ,d; (cid:101)), for any xˆ,d SN−1. (2.7)
C C ∈
The observation (2.7) is of critical importance for our following general construction
of the cloaking scheme. Finally, let us point out that the following quantities remain
unchanged under the push-forward
(cid:90) (cid:90) (cid:90) (cid:90)
(qi)−1 h 2 = (q(cid:101)i)−1 (cid:101)h 2, i = 1,2, and (σ)−1H H = (σ(cid:101))−1H(cid:101) H(cid:101).
| | | | · ·
Ω Ω(cid:101) Ω Ω(cid:101)
With the above preparations, we are ready to present the general construction of our
proposed cloaks. By a bit abuse of notation, we let from now on Ω, D and Σ be the
closures of three bounded Lipschitz domains in RN such that RN Ω is connected and
◦ ◦ \
Σ D(cid:98)Ω. Let K0 be a compact subset in RN, and Kε be an ε-neighborhood of K0.
⊂
Both K and K shall be made precise in the next section. We assume there exists a
0 ε
bi-Lipschitz and orientation-preserving mapping F such that
ε
F = Identity on RN Ω; F(1) on Ω K ; F(2) on K (2.8)
ε ε ε ε ε
\ \
with
F(2)(K ) = Σ, F(2)(K ) = D, F(1)(Ω K ) = Ω D. (2.9)
ε ε/2 ε ε ε ε
\ \
Let,inthephysicalspace, (cid:101)ε = (Σ1,Σ2,Σ3,s,σ(cid:101)ε,q(cid:101)ε,h,H)beanadmissibleconfiguration
C
asdescribedinwhatfollows. SetΣ(cid:101) = Σ1 Σ2 Σ3. Thefollowingpropertiesarerequired.
∪ ∪
First, Σ(cid:101) Σ and h and H are supported in Σ. Second, we have
⊂
(Ω D;σε,qε) = (F ) (Ω K ;I,1); σε(x) = I, qε(x) = 1 for a.e. x RN Ω, (2.10)
(cid:101) (cid:101) ε ∗ ε (cid:101) (cid:101)
\ \ ∈ \
10