boundary integral equations for the transmission ... · boundary integral equations for the...

Boundary Integral Equations for theTransmission Eigenvalue Problem for

Maxwell’s Equations

Fioralba CakoniDepartment of Mathematical Sciences, University of Delaware

In collaboration with

Houssem HaddarINRIA Saclay Ile de France / CMAP Ecole Polytechnique

Research supported by grants from AFOSR and NSF

Scattering by an Inhomogeneous Media

i sEE

!

D

∇×∇× Es − k2Es = 0 in R3 \ D∇×∇× E − k2N(x)E = 0 in Dν × E = ν × (Es + E i ) on Γ

ν ×∇× E = ν ×∇× (Es + E i ) on Γ

lim|x|→∞

(∇× Es × x − ik |x |Es) = 0

N =ε(x)

ε0is relative electric permittivity, k the wave number, and E i

incident electric field.

Question: Is there an incident wave E i that does not scatter?

The answer to this question leads to the transmission eigenvalueproblem.

Transmission Eigenvalues

If there exists a nontrivial solution to the transmission eigenvalueproblem

∇×∇× E − k2N(x)E = 0 in D∇×∇× E0 − k2E0 = 0 in D

ν × E = ν × E0 on Γ

ν × (∇× E) = ν × (∇× E0) on Γ

such that E0 can be extended outside D as a solution to∇×∇× E0 − k2E0 = 0, then the scattered field due to this extendedfield as incident wave is identically zero.

Values of k for which this problem has a non trivial solution arereferred to as transmission eigenvalues.

Transmission EigenvaluesIn general such an extension of E0 does not exits!

Since superposition of plane waves, so-called electric Herglotz wavefunctions

Eg(x) := ik∫Ω

eikx·dg(d)ds(d)), g ∈ L2t (Ω), Ω := d : |d | = 1

or superposition of point sources

Sϕ(x) := ∇×∇×∫Λ

ϕ(y)Φ(x , y)dsy , ϕ ∈ L2t (Λ),

where Λ is a surface in R3 \ D and Φ(x , y) = 14π

eik|x−y|

|x−y| , are dense inW ∈ L2(Ω) : ∇×∇×W − k2W = 0

,

at a transmission eigenvalue there is an incident field that producesan arbitrarily small scattered field

Motivation

The transmission eigenvalue problem is non-selfadjoint andnon-linear.

Why study the transmission eigenvalue problem?

Fredholm property of the interior transmission problem. It arisesin important questions such as uniqueness of inverse problemsfor inhomogenous media.

Discreteness of transmission eigenvalues. Methods for solvingthe inverse problem for inhomogeneous media such as linearsampling method and factorization method fail at a transmissioneigenvalue.

Existence of transmission eigenvalues

Real transmission eigenvalues can be determined from thescattering data.Transmission eigenvalues carry information about materialproperties.

Historical Overview

The transmission eigenvalue problem in scattering theory wasintroduced by Kirsch (1986) and Colton-Monk (1988)

Research was focused on the discreteness of transmissioneigenvalues for variety of scattering problems:Colton-Kirsch-Päivärinta (1989) and Rynne-Sleeman (1991).

The first proof of existence of at least one transmissioneigenvalues for large contrast Päivärinta-Sylvester (2009).

The existence of an infinite set of transmission eigenvalues isproven by Cakoni-Gintides-Haddar (2010).

The determination of real transmission eigenvalues fromscattering data by Cakoni-Colton-Haddar (2010) improved forsimple scattering problems by Kirsch-Lechleiter (2013).

Since the appearance of these papers there has been anexplosion of interest in the transmission eigenvalue problem.

Special issue of Inverse Problems on Transmission EigenvaluesOctober 2013.


In a "natural" variational form this problem reads∫D

(∇× E) · (∇× E′) dx −

∫D

(∇× E0) · (∇× E′0) dx

−k2∫

DNE · E ′ dx + k2

∫D

E0 · E′0 dx = 0

for all E ′,E ′0 ∈ X (D), where

X (D) := (w , v) ∈ H(curl ,D)× H(curl ,D) | ν × w = ν × v on Γ.

Chesnel, Inverse Problems, 2012 – uses >- coercivity to provediscreteness of TE for media ∇× A∇× E − k2NE = 0, providedA− I and N − I are bounded away from zero in a neighborhood of Γ.Existence of TE in this case is proven under stronger assumptions onA and N (Cakoni - Kirsch, (2010)).


It is possible to write

∇×∇× E − k2NE = 0 in D∇×∇× E0 − k2E0 = 0 in D

ν × E = ν × E0 on Γ

ν × (∇× E) = ν × (∇× E0) on Γ

E ,E0 ∈ L2(D), for the difference W = E − E0 ∈ H0(curl2,D) as

(∇×∇×−k2)(N − I)−1(∇×∇×−k2N)W = 0

i.e. in the variational form∫D

(N − I)−1(∇×∇×W−k2NW )(∇×∇×W′−k2W

′) dx = 0, W ′ ∈ U0(D)

where H0(curl2,D) := U ∈ H0(curl ,D) : such that ∇× U ∈ H0(curl ,D)


DefinitionTransmission eigenvalues are values of k ∈ C for which thetransmission eigenvalue problem has non-zero solutions E ∈ L2(D),E0 ∈ L2(D) such that (E − E0) ∈ H0(curl ,D) and∇× (E − E0) ∈ H0(curl ,D).

Theorem (Cakoni-Gintides-Haddar, SIAM J. Math. Anal. (2010))

Assume that either N − I or I − N is positive definite uniformly in D.Then:

the set of all transmission eigenvalues is at most discrete.

there exists an infinite set of real transmission eigenvaluesaccumulating at +∞.

Note: The interior transmission problem with nonhomgeneousboundary data satisfies the Fredholm alternative.

Integral Equation Formulation

We use an alternative approach to study the transmission eigenvalueproblem based on integral equations. The goal is:

to relax the assumption on the sign of the contrast N − I.

to provide an alternative approach suitable for computation oftransmission eigenvalues.

The integral equation formulation of the transmission eigenvalueproblem was first introduced for the scalar case in

Cossonnière-Haddar, Surface integral formulation of the interiortransmission problem, J. Int. Eqn. Appl. (to appear)


Throughout here we assume that Γ is smooth enough!

For the moment we consider N = nI where n > 1 or 0 < n < 1 isconstant. Denote by k1 :=

√nk and

Φk (x , y) =1

4πeik|x−y|

|x − y |.

From Stratton-Chu formula we have

E(x) := curl∫

Γ

(E × ν)Φk1 (·, x) ds

+1

k12∇∫

Γ

divΓ · (curl E × ν)Φk1 (·, x) ds +

∫Γ

(curl E × ν)Φk1 (·, x) ds

with similar expression for E0 where we replace k1 by k .


Define the boundary integral operators:

Tk : H−1/2(div , Γ)→ H−1/2(curl , Γ)

Tk (ψ) := γΓ

(k∫

Γ

ψ(y)Φk (·, y) ds +1k

gradΓ

∫Γ

divΓψ(y) Φk (·, y) ds)

and

Kk : H−1/2(div , Γ)→ H−1/2(curl , Γ)

Kk (ψ) := γΓ

(curl

∫Γ

ψ(y)Φk (·, y) ds)

where γΓu := ν × (u × ν)


Taking the tangential traces we have

γΓE = Kk1 (E × ν) +1k1

Tk1 (curl E × ν) =12γΓE

γΓ(curl E) = Kk1 (curl E × ν) + k1Tk1 (E × ν) =12γΓ(curl E)

with similar expression for E0 where we replace k1 by k .

Recall that k1 :=√

nk

Next consider the difference E − E0 and the fact that the Cauchy datacoincide on Γ.


We formally obtain the following system of integral equations for

M := E × ν = E0 × ν and J := (∇× E)× ν = (∇× E0)× ν

L(k)

M

J

=

k1Tk1 − kTk Kk1 − Kk

Kk1 − Kk1k1

Tk1 − 1k Tk

M

J

=

0

0

Recall the Helmholtz orthogonal decomposition of tangential fields

U = curlΓp +∇Γq


Note that E ∈ L2(D), E0 ∈ L2(D) and hence ∇×∇× E ∈ L2(D),∇×∇× E0 ∈ L2(D). Hence M ∈ H−1/2

t (Γ) and

J ∈ H(Γ) :=

u ∈ H−3/2t (Γ) such that divΓ u ∈ H−1/2(Γ)

.

The dual H∗(Γ) (with respect to L2-inner product) is

H∗(Γ) :=

u ∈ H−1/2t (Γ) such that curlΓ u ∈ H1/2(Γ)

.

LemmaFor a fixed k, the linear operatorL(k) : H−1/2

t (Γ)×H(Γ)→ H1/2t (Γ)×H∗(Γ) is bounded. The family of

operators L(k) depends analytically on k ∈ C \ R−.


The following statements are equivalent:

1 There exists E ,E0 ∈ L2(D), E − E0 ∈ H(curl2,D) a non trivialsolution of TEP.

2 There exists M ∈ H−1/2t (Γ) and J ∈ H(Γ) nonzero such that

L(k)

M

J

= 0, and either E∞0 (M, J) = 0 or E∞1 (M, J) = 0

where

E∞1 (M, J)(x) = x ×(

14π

curl∫

Γ

M e−ik1x·y dsy

+1

4πk12∇∫

Γ

divΓJ e−ik1x·y ds +

∫Γ

J e−ik1x·y dsy

)× x

with same expression for E∞0 (M, J) where we replace k1 by k .


Sk1 − Sk is smoothing of order 3 where

Skψ :=

∫Γ

ψ(y)Φk (·, y) dsy

Kk1 − Kk is a smoothing operator of order 2.

(see Cossonnière-Haddar, Hsiao-Wendland)

k1Tk1 − kTk = (k12Sk1 − k2Sk ) + gradΓ (Sk1 − Sk ) divΓ

1k1

Tk1 −1k

Tk = (Sk1 − Sk ) + gradΓ (

1k1

2 Sk1 −1k2 Sk

) divΓ


Define H0(Γ) the space of u ∈ H(Γ) such that divΓu = 0

Lemma

L(i |k |) is strictly coercive in H−1/2t (Γ)×H0(Γ).

U ∈ L2(R3), ∇× U ∈ L2(R3), ∇×∇× U ∈ L2(R3) satisfying(∇×∇×+|k1|2

) (∇×∇×+|k |2

)U = 0 in R3 \ Γ

[ν ×∇×∇× U] = M(k12 − k2) on Γ

[ν ×∇×∇×∇× U] = J(k12 − k2) on Γ

for M ∈ H−1/2t (Γ), J ∈ H0(Γ).

Here [·] is the jump across Γ


Lemma

Restricted to H−1/2t (Γ)×H0(Γ)

L(k) +k1

2 − k2

|k1|2 − |k |2L(i |k |)

is compact.

Using the Helmholtz orthogonal decomposition

J = P + Q, P = curlΓp, Q = ∇Γq

and writing L(k) as a 3× 3- matrix operator acting on M ∈ H−1/2t (Γ),

P ∈ H0(Γ), Q with divΓ ∈ H−1/2(Γ) we can now show:

Theorem

L(k) : H−1/2t (Γ)×H(Γ)→ H1/2

t (Γ)×H∗(Γ) is Fredholm operator ofindex zero. Its kernel fails to be trivial for at most a discrete set ofvalues k ∈ C \ R−.

Discreteness of Transmission Eigenvalues

If 0 < n 6= 1 is constant, then the set of transmission eigenvaluesis discrete.

This approach does not provide existence of transmissioneigenvalues.

In the context of using transmission eigenvalues to obtaininformation on material properties, one needs to solve thetransmission eigenvalue problem for homogeneous media.This approach provides an alternative computational frameworkto the finite element method for a forth order curl equation.(see Kleefeld, Inverse Problems (2013) for the scalar case)

Media with Contrast that Changes Sign

Ln1 (k) is the operator weanalyzed replacing n by n1

A−112 (k) is the operator

corresponding the thetransmission problem withinterface Γ2.

ZΓ2→Γ1 and ZΓ1→Γ2 are compactoperators.

L(k) = Ln1 (k) + ZΓ2→Γ1A−112 (k)ZΓ1→Γ2

In the general case if N is such that in a neighborhood of Γ is positiveconstant greater or less than one, than

L(k) = Ln1 (k) + compact perturbation.

Media with Contrast that Changes Sign

In general to show that there is k ∈ C which is not a transmissioneigenvalue one needs to work directly on the the transmissioneigenvalue problem in the PDE form. It is possible to show thereis a k real large enough not transmission eigenvalue(see Sylvester, SIAM J. Math Anal. (2012) for the scalar case).

Under the above assumption on N the set of transmissioneigenvalues is discrete.

The existence of transmission eigenvalues is open forelectromagnetic scattering problem for inhomogenouos mediawith contrast changing sign. Progress is recently made in thisdirection for the scalar problem by Lakshtanov-Vainberg, andRobbiano, both to appear in the special issue of InverseProblems.

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