Strongly Elliptic Boundary Integral Equationsfor Electromagnetic Transmission Problems

Martin Costabel Ernst P. Stephan

Abstract

For transmission problems for strongly elliptic differential operators, we study aboundary integral equation method which yields a strongly elliptic system of pseu-dodifferential operators and which therefore can be used for numerical computationswith Galerkin’s procedure. The method is shown to work for the vector Helmholtzequation in IR3 with electromagnetic transmission conditions. We propose a slightlymodified system of boundary values in order for the corresponding bilinear form tobe coercive over H1. We analyze the boundary integral equations with the calculusof pseudodifferential operators. Here the concept of the principal symbol is used toderive existence and regularity results for the solution.

1 Introduction

The usefulness of strongly elliptic pseudodifferential operators for numerical methods isnow well established ([9, 13, 22, 29, 30, 11]). For boundary value problems, there are manypapers dealing with the analytical as well as the computational aspects of various examplesfor this fact. The so-called “direct method” leads in a simple way from a strongly ellipticboundary value problem to a strongly elliptic system of pseudodifferential equations on theboundary. This method is now well-understood and has, by its simplicity, a wide rangeof applications also to mixed boundary value problems and to problems with irregularboundaries [3]. This is one advantage of these first kind integral equations compared tothe traditionally preferred Fredholm integral equations of the second kind. For the latterequations, there is no generally applicable method of derivation, and they loose theirFredholm properties as soon as the coefficients or the boundaries are not smooth. Anotheradvantage of the direct method is that their solutions have direct physical meaning andthat the appropriate norm is the energy norm. Therefore the corresponding boundaryelement methods share several nice properties of the usual finite element methods [31].

For acoustic transmission problems, the authors showed in [4] that for the case of theHelmholtz equation the direct method leads also to strongly elliptic boundary integralequations. This was done using the symbols of the operators defined by local Fouriertransformation in the case of smooth boundaries in any dimension respectively by localMellin transformation in the case of a two-dimensional domain with corners (see also [23]).

In the present paper, we show for a general class of strongly elliptic transmissionproblems how to derive strongly elliptic boundary integral equations by the direct method.

1

Then we apply this to electromagnetic transmission problems. To do this we have to havea bilinear form, connected with the boundary data through Green’s first theorem, thatis coercive over all of H1, whereas the usual energy form is not [8]. To achieve this,we use a modification of the boundary data that gives rise to a transmission problemequivalent to the original one. Then we compute the kernels of the integral operatorsand their principal symbols which show clearly how the proposed modification transformsthe system of operators into a strongly elliptic one. This kind of transformation of theboundary integral operators was first used by MacCamy and Stephan in [18] for the perfectconductor problem, i. e. the exterior boundary value problem for Maxwell’s equations withgiven electric data.

Under the assumption of uniqueness we derive existence of the solution of our integralequation on the transmission manifold. Since the general electromagnetic transmissionproblem is equivalent to our boundary integral equation we have an analytic solutionprocedure via the integral equation and in connection with Galerkin’s method we haveeven a numerical solution procedure by boundary elements. Our system of operators forthe transmission problem contains the systems that can be used for the electric and themagnetic boundary value problems and for the screen scattering problems with electric andwith magnetic boundary data. For all these cases we therefore have the strong ellipticityof the corresponding boundary integral equations. For the screen problems, these andthe corresponding symbols are the starting point of an analysis of the singularities at thescreen edge, see [24, 25].

2 The direct method for transmission problems

The “direct method” for strongly elliptic boundary value problems was studied in [5]. Herewe present a related analysis of transmission problems. We see that the case of the scalarHelmholtz equation studied in [4] is representative for a general class of transmissionproblems. A similar analysis is possible for very general combinations of boundary andtransmission conditions [28].

Let Ω1 be a bounded domain in IRn with boundary Γ ∈ C∞ and Ω2 := IRn \ Ω1. Weconsider the following transmission problem

P1u1 = 0 in Ω1 (2.1)

P2u2 = 0 in Ω2, (2.2)

u2 satisfies some “radiation condition ”, see (2.17),

R2γ2u2 −R1γ1u1 = u0 on Γ. (2.3)

Here, for j = 1, 2, Pj are elliptic differential operators of order 2m with C∞ coefficients,for simplicity both defined throughout IRn; γjuj are the Cauchy data of uj on Γ from Ωj:

γjuj = (uj, ∂nuj, . . . , ∂2m−1n uj)

∣∣∣Γ, (2.4)

where ∂n means the normal derivative with respect to the normal pointing from Ω1 to Ω2;

2

Rjγj are systems of 2m differential operators with C∞ coefficients:

(Rjγjuj)i =2m−1∑k=0

Rikj ∂

knuj

∣∣∣Γ, (2.5)

where Rikj are tangential differential operators of order

ord Rikj = i− k. (2.6)

We assume that Rj are Dirichlet systems of order 2m (see [17]), that means besides(2.6) that the lower triangular matrix

Rj = (Rikj )i,k=0,...,2m−1

is invertible, the inverse being also a tangential differential operator.In order to transform the transmission problem to a problem on the boundary, we

have to assume that we know fundamental solutions Gj for the differential operators Pj,i. e. two-sided inverses on the space E ′ of distributions with compact support on IRn. Thishypothesis immediately makes available a “second Green formula” and a “representationformula” as follows:Let

Pj =2m∑l=0

Pjl∂ln (2.7)

be the representation of Pj near Γ with differential operators Pjl of order 2m − l which

are tangential on Γ. Let uj ∈ C∞0 (Ωj), fj := Pjuj∣∣∣Ωj

, and u0j be the extension of uj by

zero outside Ωj. If we apply Pj in the distributional sense to u0j , the result differs from

f 0j by a distribution supported by Γ [6, (23.48.13.4)],[1]:

Pju0j = f 0

j + (−1)j2m−1∑k=0

2m−1−k∑l=0

Pj,k+l+1∂lnuj ⊗ ∂knδΓ (2.8)

Here appear multipole layers v ⊗ ∂kn defined by

<v ⊗ ∂knδΓ, φ>:=∫Γ

v∂′nkφ do for φ ∈ C∞0 (IRn), (2.9)

where do is the (n − 1)-dimensional surface measure on Γ, and ∂′n is the transpose ofthe differential operator ∂n. Thus (2.8) is a distributional formulation of Green’s sec-ond formula. Applying the relation GjPju

0j = u0

j to (2.8), we obtain the representationformula:

u0j = Gjf

0j + (−1)j

2m−1∑k=0

2m−1−k∑l=0

Kjk(Pj,k+l+1∂lnuj). (2.10)

Here the multipole potential operators Kjk are defined by

Kjkφ = Gj(φ⊗ ∂knδΓ) =∫Γ

∂′n(y)kGj(·, y)φ(y)do(y) for φ ∈ C∞(Γ) (2.11)

3

where Gj(x, y) is the kernel of Gj.If we introduce the matrices

Pj := (Pj,k+l+1)2m−1k,l=0 with Pjk := 0 for k > 2m;

Kj :=(∂inKjk

∣∣∣∂Ωj

)2m−1

i,k=0

(2.12)

then we find from (2.10) by taking Cauchy data

γjuj = γjGjf0j + (−1)jKjPjγjuj. (2.13)

The operatorCj : = (−1)jKjPj (2.14)

is the so-called “Calderon projector”. Its properties are known [21]:

Lemma 2.1 Cj is a pseudodifferential operator

Cj =(Cikj

)2m−1

i,k=0with orders ord Cik

j = i− k.

Thus it is a continuous operator from

Hs :=2m−1∏k=0

Hm−k− 12

+s(Γ) (2.15)

into itself for any s ∈ IR. If P1 = P2 then

C1 + C2 = 1. (2.16)

Here Hs(Γ) is the usual Sobolev space on Γ.From (2.13) follows immediately that Cj are projection operators: C2

j = Cj. For Ω1

we can take closures in Sobolev spaces in (2.10), whereas for the exterior domain Ω2 wehave (2.10) at first only for functions u2 with compact support. But we can use (2.10) for

ΩR2 := Ω2 ∩ x ∈ IRn | |x| ≤ R (R large enough).

Then (2.10) for Ωj holds if and only if the terms coming from Ω2 \ΩR2 and from ∂ΩR

2 \Γ =x ∈ IRn | |x| = R tend to zero as R→∞. For f2 = 0 this means

limR→∞

∑k+l+1≤2m

∫|y|=R

∂′n(y)kG2(·, y)(P2,k+l+1∂

lnu2)(y) do(y)→ 0 for all x ∈ Ω2). (2.17)

In analogy to the classical situation of the Helmholtz and Maxwell equations we call thisa “radiation condition”.

We can now define the solution spaces for (2.1), (2.2):

Definition 2.2

Ls1 : =u1 ∈ Hm+s(Ω1) | P1u1 = 0 in Ω1

Ls2 : =

u2 ∈ Hm+s(Ω2) | P2u2 = 0 in Ω2 and u2 satisfies (2.17)

4

Note that the ellipticity of P2 implies u2 ∈ C∞(Ω2) so that the integral in (2.17) makessense. If we use the well-known mapping properties of the potential operators Kjk ([1],[6]), we obtain

Lemma 2.3 Let s ∈ IR. For v ∈ Hs the following are eqivalent:

(i) Cjv = v.

(ii) Cjg = v for some g ∈ Hs.

(iii) v = γjuj for some uj ∈ Lsj .In this case, uj is given by the representation formula

uj = KjPjv in Ωj (2.18)

where Kj is the vector (Kj0, . . . , Kj,2m−1).

Thus we can write problem (2.1)–(2.3) in the equivalent form

(1− C1)γ1u1 = 0 (2.19)

(1− C2)γ2u2 = 0 (2.20)

R2γ2u2 −R1γ1u1 = u0. (2.21)

This is a system of 6m equations on the boundary for the 4m unknowns γ1u1, γ2u2.We want to emphasize here that up to now everything remains true if we consider u1

and u2 as vector-valued functions with N components and P1 and P1 as (N ×N)-systemsof differential operators. The matrices Rj, Pj, Kj, Cj, etc., then have to be block matricesof (N ×N)-blocks. System (2.19), (2.20), (2.21) then is a (6mN × 4mN)-system.

From this system we can extract a quadratic subsystem by eliminating R2γ2u2 from(2.21) and multiplying (2.19), (2.20) by R1 and R2, respectively. We obtain

(1− C1)R1γ1u1 = 0 (2.22)

(1− C2)R1γ1u1 = −(1− C2)u0 (2.23)

withCj : = RjCjR

−1j (j = 1, 2). (2.24)

Now we subtract (2.22) from (2.23) and obtain the quadratic system

Hv = −(1− C2)u0 with H : = C1 − C2 (2.25)

for the unknown v = R1γ1u1.We have the following equivalence theorem

Theorem 2.4 Let u0 ∈ Hs be given.(i) If uj ∈ Lsj (j = 1, 2) solve the transmission problem (2.1), (2.2), (2.3) then v =R1γ1u1 ∈ Hs solves the equation (2.25).(ii) If v ∈ Hs solves (2.25) then with

v1 := C1v ; v2 := C2(v + u0) (2.26)

5

anduj := KjPjR−1

j vj in Ωj (see (2.18)), (2.27)

uj ∈ Lsj solve the problem (2.1), (2.2), (2.3).

Proof. (i) follows from the derivation of (2.25) above. For (ii) we use Lemma 2.3which gives from (2.26) that (for j = 1, 2) (1 − Cj)vj = 0, hence vj = Rjγjuj for someuj ∈ Lsj . It remains to show (2.3):From (2.26) and (2.25) follows

v2 − v1 = (C2 − C1)v + C2u0 = −Hv + C2u0 = (1− C2 + C2)u0 = u0.

Now we formulate the assumptions which will imply the strong ellipticity of the opera-tor H. They consist essentially of the strong ellipticity of the boundary value problems onΩ1 and Ω2 and of two other boundary value problems that are obtained from interchangingthe domains Ω1 and Ω2.

We require the existence of a “first Green formula” for Pj and Rj: Let

Rj :=

B0j

...

Bm−1j

Qm−1j

...

Q0j

, i. e.

Bikj = Rik

j for k = 0, . . . ,m− 1,

Qikj = Ri,2m−k−1

j for k = 0, . . . ,m− 1;(2.28)

and Bijv :=

m−1∑k=0

Bikj v

k for v ∈ C∞(Γ;C/ m), Qij correspondingly.

Assumption 2.5 For j = 1, 2, there exists a sesquilinear formΨj : (u, v) 7→ Ψj(u, v) on C∞0 (Ωj)× C∞0 (Ωj) such that

Re∫Ωj

uj · Pjuj dx = Re Ψj(uj, uj) + (−1)j Re∫Γ

m−1∑i=0

Bijγjuj ·Qi

jγjuj do (2.29)

for uj ∈ C∞0 (Ωj).

Assumption 2.6 a.)(Continuity) To every bounded subset K ⊂ IRn there exists C > 0such that (for j = 1, 2)

|Ψj(u, v)| ≤ C ‖u‖Hm(Ωj)‖v‖Hm(Ωj)

for all u, v ∈ C∞0 (Ωj ∩K).b.)(Garding’s inequality) To every bounded subset K ⊂ IRn there exist λ > 0, c ∈ IR,ε > 0 such that

Re Ψj(u, u) ≥ λ ‖u‖2Hm(Ωj) − c‖u‖2

Hm−ε(Ωj) (2.30)

for all u ∈ C∞0 (Ωj ∩K).

6

It is classical (see e. g. [17]) that these assumptions are satisfied for many stronglyelliptic boundary value problems. For example, let Pj be of second order, i. e.

Pj = −n∑

i,k=1

∂iaikj ∂k +

n∑k=1

bkj∂k + cj (2.31)

with smooth coefficients aikj , bkj , and cj; (∂k = ∂∂xk

).

Green’s first formula (2.29) holds with

Ψj(u, v) =∫Ωj

n∑i,k=1

∂iu · aikj ∂kv +n∑k=1

u · bkj∂kv + u · cjv

dx (2.32)

andBjγjuj = u

∣∣∣Γ

Qjγjuj = ∂νju∣∣∣Γ

(2.33)

with the conormal derivative

∂νju =n∑

i,k=1

niaikj ∂ku. (2.34)

Here u may be an N -vector and aikj , bkj , and cj (N×N)-matrices. Of course, the represen-tation (2.31) is not unique, and, contrary to the scalar case, in the vector-valued case thevalidity of Assumption 2.6 b.) depends on the choice of this divergence representation.This is what happens for the case of electromagnetic problems, see section 3.

The boundary integral in (2.29) corresponds to the natural duality on the “energyspace” H0 with respect to the L2(Γ;C/ 2m) scalar product:

For v, w ∈ C∞(Γ;C/ m) with v =

v0

...

v2m−1

, w =

w0

...

w2m−1

let

(v, w)H0 : =∫Γ

2m−1∑k=0

vk · w2m−1−k do. (2.35)

Then this extends by continuity to v, w ∈ H0 (cf. (2.15)), and we have

(Rjv,Rjw)H0 =∫Γ

(m−1∑k=0

Bkj v ·Qk

jw +m−1∑k=0

Qkjv ·Bk

jw

)do,

in particular for v = w

(Rjv,Rjv)H0 = 2 Re∫Γ

m−1∑k=0

Bkj v ·Qk

jv do. (2.36)

We need to consider two more boundary value problems defined by interchanging theinterior and exterior domains. Thus we write

P †1 := P2 on Ω1, P †2 := P1 on Ω2,

R†1 := R2, R†2 := R1.(2.37)

7

Assumption 2.7 The assumptions 2.5, 2.6 are satisfied if Pj is replaced by P †j and Rj

by R†j for j = 1, 2.

Under these assumptions, we can infer the strong ellipticity of our boundary integraloperator H:

Theorem 2.8 Let the Assumptions 2.5, 2.6, 2.7 be satisfied. Then there exists a compactoperator C : H0 → H0 and a constant β > 0 such that

Re(v, (H + C)v)H0 ≥ β ‖v‖2H0 for all v ∈ H0. (2.38)

Here ‖v‖2H0 =

2m−1∑k=0

‖vk‖2Hm−k−1/2(Γ).

Proof. We first consider the special case

P1 = P2, R1 = R2. (2.39)

In this case, one finds from (2.16), (2.24)

C1 + C2 = 1. (2.40)

It suffices to show (2.38) for v ∈ C∞(Γ;C/ m) ⊂ H0. Define uj (j = 1, 2) by therepresentation formula (2.27):

uj = χKjPjR−1j v in Ωj, (2.41)

where χ ∈ C∞0 (IRn) satisfies χ ≡ 1 on a neighborhood of Ω1. Then from (2.24) and (2.14)we see that for the Cauchy data holds

Rjγjuj = (−1)jCjv. (2.42)

Thus, by definition of H (2.25)

Hv = −(R1γ1u1 +R2γ2u2). (2.43)

By (2.40) we havev = R2γ2u2 −R1γ1u1. (2.44)

Thus the trace lemma gives

‖v‖2H0 ≤ C(‖u1‖2

Hm(Ω1) + ‖u2‖2Hm(Ω2)). (2.45)

By Garding’s inequality (2.30), the right hand side can be further estimated by

C

λRe(Ψ1(u1, u1) + Ψ2(u2, u2)) +

cC

λ(‖u1‖2

Hm−ε(Ω1) + ‖u2‖2Hm−ε(Ω2)) (2.46)

Note that supp u2 ⊂ K where K = supp χ is bounded.

8

From (2.41) follows that the ‖·‖Hm−ε-terms in (2.46) can be estimated by ‖T1v‖H0 witha compact operator T1 : H0 → H0. Such an estimate is also possible for ‖P2u2‖Ht(Ω2) forany t ∈ IR, because P2u2 = 0 where χ ≡ 1 or χ ≡ 0 holds, thus P2u2 ∈ C∞0 (IRn), and itssupport has a positive distance from Γ, so that the kernel of the operator PjχKjPjR−1

j

defining it is smooth. Thus

|∫

Ω2

u2 · P2u2dx| ≤ ‖T2v‖2H0 for some compact T2 : H0 → H0.

Now we apply Green’s first theorem (2.29) to (2.46) and use (2.36). We obtain withP1u1 = 0

Re Ψj(uj, uj) = Re∫Ωj

uj · Pjujdx−(−1)j

2(Rjγjuj, Rjγjuj)H0 ,

hence

‖v‖2H0 ≤ ‖T1v‖2

H0 + ‖T2v‖2H0 +

C

2λ(R1γ1u1, R1γ1u1)H0 − (R2γ2u2, R2γ2u2)H0 . (2.47)

On the other hand we find from (2.43), (2.44)

(v,Av)H0 = (−R1γ1u1 +R2γ2u2,−R1γ1u1 −R2γ2u2)H0

= (R1γ1u1, R1γ1u1)H0 − (R2γ2u2, R2γ2u2)H0

+(R1γ1u1, R2γ2u2)H0 − (R2γ2u2, R1γ1u1)H0

= (R1γ1u1, R1γ1u1)H0 − (R2γ2u2, R2γ2u2)H0 + 2i Im(R1γ1u1, R2γ2u2)H0 .

By taking real parts, we conclude from (2.47)

‖v‖2H0 ≤ ‖T1v‖2

H0 + ‖T2v‖2H0 +

C

2λRe(v,Av)H0 .

After subsuming all compact parts into a single one, we arrive at (2.38) .

Now we give up hypothesis (2.39). By what we have shown so far, we know thatGarding’s inequality (2.38) holds in particular for the case where P2 and R2 are replacedby P †2 and R†2, respectively, because by (2.37) the hypothesis (2.39) is then satisfied. Wedenote the corresponding Calderon projector by C†2, and the corresponding boundaryintegral operator by

H†1 := C1 − C†2.

Similarly, if we replace P1 and R1 by P †1 and R†1, then (2.39) is satisfied and thereforeGarding’s inequality holds. We denote the corresponding Calderon projector by C†1 andthe boundary integral operator by

H†2 := C†1 − C2.

Now from (2.40) followsC1 + C†2 = 1 = C†1 + C2,

9

henceH†1 = 2C1 − 1 ; H†2 = 1− 2C2

and finally

H = C1 − C2 =1

2(H†1 +H†2). (2.48)

Therefore, the Garding inequality (2.38) for H follows by adding the two Garding inequal-ities for H†1 and H†2.

3 A coercive bilinear form for electromagnetic prob-

lems

The time-harmonic scattering of electromagnetic fields by a penetrable body Ω1 in thecase of isotropic homogeneous materials is described by Maxwell’s equations (see [20]):

curl ~E = iωµ ~H; curl ~H = −iωε ~E in Ω1 ∪ Ω2 (3.1)

with the transmission conditions

[~n× ~E]Γ = 0; [~n× ~H]Γ = 0 on Γ (3.2)

and the radiation condition

ωµx

|x|× ~Hsc + k ~Esc = o(|x|−1); ~Esc = O(|x|−1) as |x| → ∞. (3.3)

Here [~v]Γ := ~v∣∣∣Ω2

− ~v∣∣∣Ω1

denotes the jump of ~v across Γ.

We assume that the coefficient functions ε, µ, and k are constant on Ω1 and on Ω2,and ω is the constant frequency:

ε = εj, µ = µj, k = kj on Ωj (j = 1, 2); k2 = ω2εµ.

We further assume [20]

Re εj > 0, Im εj ≥ 0; arg kj ∈ [0, π); argω ∈ [0, π); µj > 0 (3.4)

The total fields ~E and ~H are decomposed into

~E = ~Ein + ~Esc; ~H = ~Hin + ~Hsc,

where the incoming fields ~Ein and ~Hin are supposed to satisfy (3.1), and the scattered

fields ~Esc and ~Hsc appear in the radiation condition (3.3). In [16], [20] the unique solv-ability of (3.1)–(3.3) is shown.

If we define

~u = ~u1 = ~E in Ω1 ; ~u = ~u2 = ~Esc in Ω2 ; ~u0 = ~Ein, (3.5)

10

then the components of ~u satisfy the Helmholtz equation:(∆ + k2

1

)~u1 = 0 in Ω1;

(∆ + k2

2

)~u2 = 0 in Ω2. (3.6)

We consider the transmission conditions (compare [16])(~v> := −~n× (~n× ~v) denote the tangential components of ~v)

~u1> − ~u2> = ~u0>;

λ1 div ~u1 − λ2 div ~u2 = λ2 div ~u0;

ε1~n · ~u1 − ε2~n · ~u2 = ε2~n · ~u0;1µ1~n× curl ~u1 − 1

µ2~n× curl ~u2 = 1

µ2~n× curl ~u0

(3.7)

and the radiation condition

x

|x|× curl ~u2 −

x

|x|div ~u2 + ik2~u2 = o(|x|−1) as |x| → ∞. (3.8)

The coefficients λj 6= 0 in (3.7) are specified later on.Let us now study the equivalence of the transmission problems (3.1)–(3.3) and (3.6)–

(3.8). Because we only need standard applications of Green’s formulas, we do not specifythe precise smoothness requirements. Even weak solutions in H1

loc(Ωj) are allowed.We first show that (3.1)–(3.3) imply (3.6)–(3.8), for any choice of λ1, λ2:By the definition (3.5), it is clear that Maxwell’s equations (3.1) imply the Helmholtz

equations (3.6), and div ~E = 0 shows that the radiation condition (3.8) follows from(3.3). Also the first, second, and fourth of the transmission conditions (3.7) are satisfied.In order to show the third condition in (3.7), we choose a test function ϕ ∈ C∞0 (IR3) andobtain

0 =∫Γ

gradϕ · [~n× ~H]Γ do = −∫

Ω1∪Ω2

gradϕ · curl ~H dx

= iω∫

Ω1∪Ω2

gradϕ · ε ~E dx = −iω∫Γϕ[ε~n · ~E]Γ do.

(3.9)

From this we find [ε~n · ~E]Γ = 0, which is the third condition in (3.7).

Conversely, assume that (3.6)–(3.8) are satisfied. Further assume that ~u0 satisfies

(∆ + k22)~u0 = 0 and div ~u0 = 0, ~E is defined by (3.5), and ~H is defined by ~H = 1

iωµcurl ~E.

Then (3.1)–(3.3) will be satisfied if and only if div ~u1 ≡ 0 and div ~u2 ≡ 0. Thereforewe define

ρ := ρ1 := k−21 div ~u1 in Ω1; ρ := ρ2 := k−2

2 div ~u2 (= k−22 div(~u0 + ~u2)) in Ω2.

Then ρ satisfies(∆ + k2) ρ = 0 in Ω1 ∪ Ω2 (3.10)

andλ1k

21ρ1 = λ2k

22ρ2 on Γ. (3.11)

11

Furthermore, from the radiation condition (3.8) follows [14], [24] that u2 can be repre-sented in Ω2 by the Stratton-Chu representation formula (see Lemma 4.3 below). Thisimplies in particular that ρ satisfies a Sommerfeld type radiation condition

x

|x|· grad ρ− ik2ρ = o(|x|−1); ρ = O(|x|−1) as |x| → ∞. (3.12)

In addition, a second transmission condition for ρ holds:Define ~E0 := ~E + grad ρ in Ω1 ∪ Ω2. Then

div ~E0 = div ~E + ∆ρ = div ~u− k2ρ = 0 in Ω1 ∪ Ω2.

Hence

curl ~H =1

iωµcurl curl ~E0 = −iωε ~E0.

Thus the pair ( ~E0, ~H) satisfies Maxwell’s equations (3.1) and the transmission condition

[~n× ~H]Γ = 0. We conclude as above (3.9) that [ε~n · ~E0]Γ = 0. Subtracting [ε~n · ~E]Γ = 0,we find

ε1∂nρ1 = ε2∂nρ2 on Γ. (3.13)

Thus we have reduced the question of equivalence of the two transmission problemsto the question of unique solvability of the scalar transmission problem (3.10)–(3.13).

Sufficient conditions for this uniqueness are well-known. For example, from Proposi-tion 4.7 in [4] it follows that either one of the following two conditions imply ρ ≡ 0:

k2 > 0 and Imλ1λ2ε1ε2k21 ≥ 0 and Imλ1λ2ε1ε2 ≤ 0; (3.14)

Im k2 > 0 or k2 = 0, and if there exist four numbers α, β, γ, δ ≥ 0

with − αλ1ε1 − βλ2ε2 + γλ1ε1k21 + δλ2ε2k

22 = 0,

then at least one of the numbers α, β, γ, δ has to be zero.

(3.15)

Remarkable special cases of these conditions are:

(i) If λ1 = ε1 and λ2 = ε2, then ρ ≡ 0 follows.

(ii) If λ1 =1

µ1ε1

and λ2 =1

µ2ε2

, then ρ ≡ 0 follows.

(iii) If all coefficients ε, λ, and k2 are real , then ρ ≡ 0 follows.

(iv) Periodic eddy current problem:

Here ε1 =iσ

ω, ω > 0, and σ > 0 is the electric conductivity in Ω1. Also ε2 > 0,

hencek2

1 = iωµ1σ; k22 > 0.

Therefore, (3.14) reduces to the conditions

Imλ2

λ1

≤ 0 and Reλ2

λ1

≥ 0. (3.16)

12

Note that Imk2

1

k22

> 0, so that in this case the natural choice

λ = k−2 (3.17)

does not nessecarily imply ρ ≡ 0. The choices (i) or (ii) above, however, will alsowork in this case.

(v) The choice (3.17) always leads to a solution of the transmission problem (3.1)–

(3.3). Namely, in this case define ~E0 as above by ~E0 = ~E + grad ρ, then the pair

( ~E0, ~H) satisfies Maxwell’s equations (3.1) as well as the transmission conditions(3.2). However, as seen in (iv) above, then the solution of the problem (3.6)–(3.8)might be non-unique.

Besides the “physical” transmission problem, we also consider for mathematical sim-plicity the corresponding problem where all the coefficients λ, ε, and µ are equal to unity,i. e. the transmission conditions

~u1> − ~u2> = ~u0>;

div ~u1 − div ~u2 = div ~u0;

~n · ~u1 − ~n · ~u2 = ~n · ~u0;

~n× curl ~u1 − ~n× curl ~u2 = ~n× curl ~u0.

(3.18)

Now we want to apply the theory developed in section 2 to the present case. We havethe differential operators

Pj = −(∆ + k2j ) = curl curl− grad div−k2

j . (3.19)

This representation leads to the conormal derivatives (cf. (2.31), (2.34))

∂ν~u := ∂ν1~u = ∂ν2~u = −~n× curl ~u+ ~n · div ~u (3.20)

and to the well-known [14] Green formula (cf. (2.29), (2.32))

−∫Ωj

~uj · (∆ + k2j )~wjdx =

∫Ωj

(curl ~uj · curl ~wj + div ~uj div ~wj − k2j~uj · ~wj)dx

+(−1)j∫Γ

~uj · (−~n× curl ~wj + ~n div ~wj) do (3.21)

for ~uj, ~wj ∈ C∞0 (Ωj;C/3).

So we have the sesquilinear forms

Ψj(~u, ~w) =∫Ωj

(curl ~u · curl ~w + div ~u div ~w − k2j~u · ~w)dx (3.22)

and the boundary operators

R~u := R1~u = R2~u =

~u

∂ν~u

∣∣∣Γ

;B~u := B1~u = B2~u = ~u

∣∣∣Γ

;

Q~u := Q1~u = Q2~u = ∂ν~u∣∣∣Γ.

(3.23)

13

By separating tangential and normal components and defining (on Γ) the Cauchy data

v := ~n · ~u; ~v := ~u> := ~u− ~n(~n · ~u) = −~n× (~n× ~u);

ψ := div ~u; ~ψ := −~n× curl ~u,(3.24)

Green’s first formula (3.21) reads∫Ωj

~u1 · Pj~u2 dx = Ψj(~u1, ~u2) + (−1)j

∫Γ

(v1ψ2 + ~v1 · ~ψ2) do. (3.25)

The corresponding second Green’s formula (obtained by antisymmetrizing (3.25)) andrepresentation formula (“Stratton-Chu formula”, see section 4)(by inserting a fundamentalsolution in the latter) are also well known. Hence one can derive boundary integralequations by the method described in section 2. Assumptions 2.5 and 2.6 a.) are satisfied.Assumption 2.6 b.) does not hold, however:For harmonic vector fields ~u we have

Ψj(~u, ~u) = −∫Ωj

k2j |~u|2 dx

which cannot be estimated from below by ‖~u‖2H1(Ωj ;CI 3). This well-known fact ([8], [24])

implies here that the boundary integral operator H is not strongly elliptic. There is,however, a strongly elliptic boundary integral operator for the transmission problem (3.6),(3.18), (3.8) which gives as solution of the corresponding equation (2.25) not directly theset of Cauchy data as defined by (3.18) or (3.24), but a modified set. This modified setof Cauchy data is entirely equivalent to the original one in the sense that each set canbe computed from the other one by application of tangential differential operators. Wepropose the following modification:

ψ′ := ψ − div~v = div ~u− div> ~u~ψ′ := ~ψ + grad> v = −~n× curl ~u− ~n× (~n× (grad(~n · ~u))).

(3.26)

Thus the “electric” boundary operators are modified by ~v

ψ′

=

~v

ψ − div> ~v

⇐⇒ ~v

ψ

=

~v

ψ′ + div> ~v

(3.27)

and the “magnetic” boundary operators by v~ψ′

=

v~ψ + grad> v

⇐⇒ v~ψ

=

v~ψ′ − grad> v

. (3.28)

This modification of the Cauchy data was suggested by corresponding modificationsof boundary integral operators in the case of boundary value problems in [18] leading tostrongly elliptic operators.

14

We shall show now that with these modified boundary operators the correspondingbilinear forms Ψ′j which are now defined by Green’s first formula, satisfy Garding’s in-equality, i. e. Assumption 2.6 b.). Then all hypotheses of Theorem 2.8 are satisfied andthis theorem proves the strong ellipticity of the corresponding boundary integral operator,which corresponds to the transmission problem (3.6), (3.18), (3.8).

We shall come back to the “physical” transmission conditions (3.7) in section 5.

Theorem 3.1 Define the sesquilinear form Ψ′j (j = 1, 2) by∫Ωj

~u1 · Pj~u2 dx = Ψ′j(~u1, ~u2) + (−1)j

∫Γ

(v1ψ′2 + ~v1 · ~ψ′2) do, (3.29)

where Pj are defined by (3.19) and v, ψ′, ~v, ~ψ′ by (3.26). Then Ψ′j is coercive over H1(Ωj;C/3),

i. e. there exist λ > 0, C ∈ IR such that

Re Ψ′j(~u, ~u) ≥ λ‖~u‖2H1(Ωj ;CI 3) − C‖~u‖2

L2(Ωj ;CI 3) (3.30)

for all ~u ∈ C∞0 (Ωj;C/3).

Proof. There holds for ~u ∈ C∞0 (Ωj;C/3) on Γ:

div ~u− div> ~u = div(~u− ~u>) = div((~n · ~u)~n)

= ∂n(~n · ~u) + (~n · ~u) div ~n;

grad(~n · ~u)− ~n× curl ~u = ∂n~u+ (~u · grad)~n+ ~u× curl~n.

This gives

vψ′ + ~v · ~ψ′ = (div ~u− div> ~u)(~n · ~u) + (grad(~n · ~u)− ~n× curl ~u) · ~u>= (∂n(~n · ~u))(~n · ~u) + div ~n(~n · ~u)(~n · ~u) + (∂n~u)~u>

+~u> · (~u · grad)~n+ ~u> · (~u× curl~n)

= ∂n~u · (~n(~n · ~u)) + ~u · ∂n~n(~n · ~u) + (∂n~u)~u>

+ div ~n|~n · ~u|2 + ~u> · (~u · grad)~n+ ~u> · (~u× curl~n)

= (∂n~u) · ~u+ b(~u) (3.31)

withb(~u) = (~n · ~u)~u · ∂n~n+ |~n · ~u|2 div ~n+ ~u> · ((~u · grad)~n+ ~u× curl~n).

In particular (this requires Γ ∈ C2)

|b(~u)(x)| ≤ C|~u(x)|2 for x ∈ Γ,

hence|∫Γ

b(~u)do| ≤ C‖~u∣∣∣Γ‖2L2(Γ) ≤ Cε‖~u‖2

H12+ε(Ωj)

(ε > 0). (3.32)

15

From the definition (3.29) and (3.31) now follows with Green’s first formula for the scalarpotential equation:

Ψ′j(~u, ~u) = −∫Ωj

~u ·∆~udx− (−1)j∫Γ

~u · ∂n~udo

−∫Ωj

k2j |~u|2dx− (−1)j

∫Γ

b(~u)do

= ‖~u‖2H1(Ωj ;CI 3) − ‖~u‖2

L2(Ωj ;CI 3)

−k2j‖~u‖2

L2(Ωj ;CI 3) − (−1)j∫Γ

b(~u)do.

Together with (3.32) for some ε ∈ (0, 12) this gives (3.30).

4 The integral operators and their symbols

In this section we derive a boundary integral equation procedure to solve the transmissionproblem (3.6), (3.18), (3.8).

First we give some additional notation. For s ∈ IR we denote by IHs(Ωj) = Hs(Ωj;C/3)

resp. IHs(Γ) = Hs(Γ;C/ 3) the Sobolev spaces formed by vector fields ~u with componentswhich belong to Hs(Ωj) resp. Hs(Γ). As indicated by (3.24) we can decompose IHs(Γ)into two subspaces generated by the tangential fields to Γ and the normal fields to Γ,

IHs(Γ) = THs(Γ)⊕NHs(Γ)

with

THs(Γ) = ~u ∈ IHs(Γ) | (~n · ~u) = 0 , NHs(Γ) = ~u ∈ IHs(Γ) | ~u = ~nv, v ∈ Hs(Γ) .

The most general case where (3.6), (3.18), (3.8) can be converted into a variationalproblem (see Theorem 3.1) is when

(~u0>, div ~u0, ~n× curl ~u0, ~n · ~u0) ∈ TH12 (Γ)×H−

12 (Γ)× TH−

12 (Γ)×H

12 (Γ) =: H0.

Then we look for ~u ∈ Lj (j = 1, 2) where

L1 := ~u ∈ IH1(Ω1) | (∆ + k21)~u = 0 in Ω1

L2 := ~u ∈ IH1(Ω2) | (∆ + k22)~u = 0 in Ω2, ~u satisfies (3.8) .

(4.1)

According to (3.24), (3.25), the Cauchy data for (3.6), (3.18), (3.8) are defined asfollows:

Definition 4.1 Let ~u ∈ Lj, j = 1, 2. Then the Cauchy data (~v, ψ, ~ψ, v) ∈ H0 of ~u aredefined via (3.25) by the traces

~v := −~n× (~n× ~u)∣∣∣Γ, ψ := div ~u

∣∣∣Γ, ~ψ := −~n× curl ~u

∣∣∣Γ, v := ~n · ~u

∣∣∣Γ.

16

Before we give our solution procedure let us briefly recall the idea of layer potentialsby introducing the fundamental solution

Ψj(x, y) =eikj |x−y|

4π|x− y|(4.2)

of (∆ + k2j ) ~u = 0 in Ωj; j = 1, 2.

Definition 4.2 Let ~u ∈ C∞(Γ;C/ 3). Then for any complex number kj,(0 ≤ arg kj < π) and for x ∈ Ωj we define

VΩj~u(x) : = −2

∫Γ

Ψj(x, y) ~u(y) do(y),

KΩj~u(x) : = 2 curlx

∫Γ

Ψj(x, y) ~u(y) do(y)(4.3)

The same definition of the single and double layer potential is valid for arbitrary distri-butions ~u on Γ since for x 6∈ Γ the above kernel Ψj is a C∞-function on Γ.

With these potentials there holds the Stratton-Chu representation formula for thesolution of the homogeneous Helmholtz equation in Ωj (for classical solutions see [14], forweak solutions see [24]).

Lemma 4.3 For ~u ∈ Lj with Cauchy data (~v, ψ, ~ψ, v) ∈ H0 and for x ∈ Ωj,j = 1, 2, there holds

~u(x) =(−1)j

2

(− curlVΩj

(~n× ~v) + VΩj(~nψ) + VΩj

~ψ + gradVΩjv)

(x). (4.4)

In order to formulate the boundary values (jump relations) for the potential (4.4) wedefine the following boundary integral operators:

Definition 4.4 Let ~u be a C∞ vector field on Γ. Then for x ∈ Γ and xj ∈ Ωj

Vj~u(x) : = −2∫Γ

Ψj(x, y) ~u(y) do(y),

~Kj~u(x) : = 2 curl>

∫Γ

Ψj(x, y) (~n× ~u)(y) do(y), (4.5)

Dj~u(x) : = limxj→x

(~n× curl curlVΩj(~n× ~u))(xj),

and, correspondingly, for u ∈ C∞(Γ)

Vju(x) : = −2∫Γ

u(y) Ψj(x, y) do(y),

Kju(x) : = −2∫Γ

u(y) ∂n(y)Ψj(x, y) do(y), (4.6)

K ′ju(x) : = −2∫Γ

u(y) ∂n(x)Ψj(x, y) do(x).

17

Using the well-known jump relations for smooth layers ([10], [2]) and approximatingthe Cauchy data in H0 by smooth functions, we find for the traces of the potential (4.4)a system of second kind Fredholm integral equations on the boundary Γ:

2(−1)j(−~n× (~n× ~u))∣∣∣Γ

= ((−1)j − ~Kj)~v − ~n× (~n× Vj(~nψ)) + Vj ~ψ + grad> Vjv

2(−1)j div ~u∣∣∣Γ

= ((−1)j −Kj)ψ + Vj div> ~ψ − k2jVjv

2(−1)j(−~n× curl ~u)∣∣∣Γ

= Dj~v − ~n× curlVj(~nψ) + (−1)j ~ψ − ~n× ~Kj(~n× ~ψ)

2(−1)j~n · ~u∣∣∣Γ

= −~n · curlVj(~n× ~v) + ~n · Vj(~nψ) + ~n · Vj ~ψ + ((−1)j +K ′j)v

(4.7)The right hand side of (4.7) defines (up to a factor 2(−1)j) the Calderon projection op-erator Cj (cf. Lemma 2.1 and equation (2.24)) for the problem (3.6), (3.18), (3.8). Thusit is a matrix of pseudodifferential operators whose principal symbol we shall computenow. As is known from the calculus of pseudodifferential operators, the principal symbolgives easy criteria for continuity of the operators in Sobolev spaces, for their ellipticity(Fredholm properties) and also strong ellipticity. By introducing a basis of orthonormalcoordinates in the cotangent bundle T ∗(Γ) (see [7, p. 255]), we obtain for the pseudodiffer-ential operators on the closed, smooth, bounded manifold Γ the same principal symbols asin the half-space case. Thus we may simply assume that Γ is a plane. Then the principalsymbols of the pseudodifferential operators in (4.7) are easily obtained by use of Fouriertransformation.

Let Ω1 coincide with IR−3 := x ∈ IR3 | x = (x1, x2, x3), x3 < 0 and Ω2 with IR+3 :=

x ∈ IR3 |, x3 > 0 and ~n = (0, 0, 1) yielding ~n × ~a = (−a2, a1) for any vector ~a =(a1, a2, a3) ∈ IR3.

In the following lemma we list the principal symbols of those pseudodifferential oper-ators on Γ which arise in our solution procedure. The orders of the operators are thosegiven by Lemma 2.1, namely −1, 0, or + 1, respectively. Thus, for example, the operator~Kj has a vanishing principal symbol, because in our Agmon-Douglis-Nirenberg-elliptic

system (4.7) , ~Kj is considered as an operator of order 0, whereas actually it is a pseu-dodifferential operator of order −1.

Lemma 4.5 For any ξ ∈ IR2, ξ 6= (0, 0) with |ξ| =√ξ2

1 + ξ22 there holds for the principal

symbols σm of order m:

σ−1(Vj)(ξ) = − 1|ξ| ; σ0(Kj)(ξ) = 0 = σ0(K ′j)(ξ); σ0(~n× ·)(ξ) =

0 −1

1 0

;

σ1(grad>)(ξ) = i

ξ1

ξ2

; σ1(~n · curl)(ξ) = i(−ξ2, ξ1); σ1(div>)(ξ) = i(ξ1, ξ2);

σ1(Dj)(ξ) = − 1

|ξ|

ξ22 −ξ1ξ2

−ξ1ξ2 ξ21

; σ0(−~n× curlVj~n·)(ξ) =i

|ξ|

ξ1

ξ2

;

σ0( ~Kj)(ξ) =

0 0

0 0

;

18

σ0(−~n× curlVj~n div>)(ξ) = − 1

|ξ|

ξ21 ξ1ξ2

ξ1ξ2 ξ22

= σ0(− grad> ~n · curlVj(~n× ·))(ξ)= −σ0(− grad> ~n · Vj~n div>)(ξ).

Proof. The general calculus of pseudodifferential operators ([7], [26]) shows that thepseudodifferential operators on a smooth manifold Γ have the same principal symbols asthe corresponding operators of the half-space case.

Furthermore, exchanging points of integration and restriction in the boundary integraloperators causes perturbations of lower order only. In particular, we know from [19] thatWj~u := ~nVj(~n · ~u)− Vj(~n · ~n)~u defines a pseudodifferential operator of order -2. Similarly,σ−1(~n× Vj(~n·))(ξ) = σ−1(Vj(~n× (~n·)))(ξ) = 0.

From [13], [19] follows σ−1(Vj)(ξ) = −1/|ξ| by taking the Fourier transform of thefundamental solution of the Laplacian 1/(4π|x − y|) which is the leading term in theTaylor series expansion of (4.2).

From the identity ∆~u = grad div ~u − curl curl ~u we obtain for any smooth tangentialfield ~v

Dj~v = k2j (~n× Vj(~n× ~v)) + ~n× gradVj div>(~n× ~v).

Thusσ1(Dj)(ξ) = σ1(~n× grad>)(ξ) · σ−1(Vj)(ξ) · σ1(div>(~n× ·))(ξ)

yields the result for σ1(Dj)(ξ). The other symbols are computed similarly.

Applying the general results of section 2, we see that the Calderon projector from thesystem (4.7) has the form

Cj =1

2(1 + (−1)jAj) (4.8)

where the operator Aj is given by

Aj

~v

ψ~ψ

v

=

− ~Kj~v + (Vj(~nψ))> + Vj ~ψ + grad> Vjv

−Kjψ + Vj div> ~ψ − k2jVjv

Dj~v − ~n× curlVj(~nψ)− ~n× ~Kj(~n× ~ψ)

−~n · curlVj(~n× ~v) + ~n · Vj(~nψ) + ~n · Vj ~ψ +K ′jv

(4.9)

The operator H from (2.25) is given by

H = C1 − C2 = −1

2(A1 + A2). (4.10)

19

Hence with Lemma 4.5 its principal symbol is given by

σ(H)(ξ) = −σ(Aj)(ξ) =1

|ξ|

0 0 0 1 0 iξ1

0 0 0 0 1 iξ2

0 0 0 iξ1 iξ2 0

ξ22 −ξ1ξ2 −iξ1 0 0 0

−ξ1ξ2 ξ21 −iξ2 0 0 0

−iξ1 −iξ2 1 0 0 0

(4.11)

Thus, H : Hs → Hs is continuous for any real s where

Hs : = THs+ 12 (Γ)×Hs− 1

2 (Γ)× THs− 12 (Γ)×Hs+ 1

2 (Γ).

We remark that the off-diagonal blocks of σ(H)(ξ) in (4.11), namely the matrices

E =1

|ξ|

1 0 iξ1

0 1 iξ2

iξ1 iξ2 0

, M =1

|ξ|

ξ2

2 −ξ1ξ2 −iξ1

−ξ1ξ2 ξ21 −iξ2

−iξ1 −iξ2 1

, (4.12)

are also the principal symbols of the integral operators which arise via the direct methodfor the “electric” and “magnetic” boundary value problems, respectively [24], [25].

On H0 the natural duality is given by

(

~v

ψ~ψ

v

,

~w

χ

~χ

w

)H0 : =∫Γ

~v · ~χ+ ψw + ~ψ · ~w + vχ do (4.13)

for smooth elements in H0 (cf. (2.35)); this corresponds also to Green’s formula (3.25).Therefore, with respect to this sesquilinear form, the operator H in (4.10) is strongly

elliptic, i. e. satisfies a Garding inequality , if and only if the matrices E and M in (4.12)define positive definite quadratic forms on C/ 3, i. e. their selfadjoint parts are positivedefinite matrices. That this is not the case can easily be seen as follows: From equations(4.11)–(4.13) we see that in the half-space case, i. e. on the symbol level there holds for

Ψ := (~v, ψ, ~ψ, v)>

(Ψ, HΨ)H0 =∫(~v, ψ)M

~v

ψ

+ (~ψ, v)E

~ψ

v

dξ1 dξ2 . (4.14)

Furthermore, the selfadjoint parts of both M and E are singular matrices:

1

2(E + E>) =

1

|ξ|

1 0 0

0 1 0

0 0 0

, 1

2(M +M>) =

1

|ξ|

ξ2

2 −ξ1ξ2 0

−ξ1ξ2 ξ21 0

0 0 1

.

20

Note that the matrices E and M themselves are non-singular which corresponds tothe ellipticity of the corresponding pseudodifferential operator. Likewise, the operator Hin (4.10) is an elliptic pseudodifferential operator and hence a Fredholm operator in thespace Hs, s ∈ IR.

Thus, using standard arguments from the calculus of elliptic pseudodifferential oper-ators, one can derive a-priori estimates for H and thus obtain regularity results for thesolution Ψ ∈ Hs of the system

HΨ := −1

2(A1 + A2) Ψ =

1

2(1− A2) Ψ0 (4.15)

for given Ψ0 ∈ Hs with Aj in (4.9), j = 1, 2.We note that equation (4.15) follows from (2.22)–(2.25) together with the Calderon

projector (4.8). Therefore, application of Theorem 2.4 shows that the solution of (4.15) isthe set of the Cauchy data of the refracted field in (3.6), (3.18), (3.8), (see also [24], [4]).

According to the general results in [22], the least squares method for (4.15) — withregular finite elements on the interface manifold Γ — converges with quasi-optimal order.However, its convergence rate is considerably smaller compared with that of the Galerkinmethod. Therefore, we are more interested in a suitable Galerkin procedure. Unfortu-nately, H is not strongly elliptic with respect to the energy form (·, ·)H0 in (4.13) — as wehave shown above. But strong ellipticity is necessary and sufficient for the convergenceof general Galerkin procedures [12], [27], [29], [30].

In order to obtain a strongly elliptic boundary integral operator for the transmissionproblem (3.6), (3.18), (3.8), we modify the Cauchy data as in (3.26) and insert them intothe system (4.7). For the new Cauchy data

Ψ′ := (~v, ψ′, ~ψ′, v)> ∈ H0, ψ′ := ψ − div> ~v, ~ψ′ := ~ψ + grad> v (4.16)

with ~v, ψ, ~ψ, v as in Definition 4.1, the system (4.7) takes the form

2(−1)j~v = ((−1)j − ~Kj)~v + (Vj(~n div> ~v))> + (Vj(~nψ′))>

+Vj ~ψ′ − Vj grad> v + grad> Vjv

2(−1)jψ′ = Lj~v + ((−1)j −Kj)ψ′ − div>(Vj~nψ

′))>

+Vj div> ~ψ′ − div> Vj ~ψ

′ +Mjv

2(−1)j ~ψ′ = Nj~v + grad> ~n · Vj(~nψ′)− ~n× curlVj(~nψ′)

+(−1)j ~ψ′ − ~n× ~Kj(~n× ~ψ′) + grad> ~n · Vj ~ψ′ +Rjv

2(−1)jv = −~n · curlVj(~n× ~v) + ~n · Vj(~n div> ~v)

+~n · Vj(~nψ′) + ~n · Vj ~ψ′ − ~n · Vj grad> v + ((−1)j +K ′j)v

(4.17)

with

Lj~v := div>Kj~v −Kj div> ~v − div(Vj(~n div> ~v))>

Mjv := div> Vj grad> v − Vj div> grad> v − k2jVjv − div> grad> Vjv

Nj~v := Dj~v − ~n× curlVj(~n div> ~v)

+ grad>(−~n · curlVj(~n× ~v)) + grad> ~n · Vj(~n div> ~v)

Rjv := ~n×Kj(~n× grad> v)− grad> ~n · Vj grad> v + grad>K′jv.

(4.18)

21

Therefore the operator H ′ from (2.25) has the form

H ′ = −1

2(A′1 + A′2) (4.19)

where

A′j

~v

ψ′

~ψ′

v

=

− ~Kj~v + (Vj(~n div> ~v))> + (Vj(~nψ′))> + Vj ~ψ

′ − Vj grad> v + grad> Vjv

Lj~v −Kjψ′ − div>(Vj~nψ

′))> + Vj div> ~ψ′ − div> Vj ~ψ

′ +Mjv

Nj~v + grad> ~n · Vj(~nψ′)− ~n× curlVj(~nψ′)−

−~n× ~Kj(~n× ~ψ′) + grad> ~n · Vj ~ψ′ +Rjv

−~n · curlVj(~n× ~v) + ~n · Vj(~n div> ~v) + ~n · Vj(~nψ′)++~n · Vj ~ψ′ − ~n · Vj grad> v +K ′jv

(4.20)

Hence with Lemma 4.5 and equation (4.11) the principal symbol of H ′ is given by

σ(H ′)(ξ) =1

|ξ|

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 |ξ|2

|ξ|2 0 0 0 0 0

0 |ξ|2 0 0 0 0

0 0 1 0 0 0

(4.21)

The principal symbol shows that H ′ is continuous from Hs into itself for any real s.Furthermore its off-diagonal blocks

E ′ =

|ξ|−1 0 0

0 |ξ|−1 0

0 0 |ξ|

, M ′ =

|ξ| 0 0

0 |ξ| 0

0 0 |ξ|−1

, (4.22)

obviously define positive quadratic forms on C/ 3. We note that E ′ and M ′ are used in [24],[25] to derive the “edge behaviour” of the “electric” and the “magnetic” fields, respectively,since the components of the fields are decoupled in first order, i. e. they are only coupledvia compact perturbations in (4.20).

From standard results on pseudodifferential operators [26], [30] we deduce that theform (4.21) of the principal symbol σ(H ′) implies the coerciveness of H ′ in the sense of aGarding inequality in H0.

Lemma 4.6 There exists a real γ > 0 such that (with the duality (4.13))

Re(Ψ, H ′Ψ)H0 ≥ γ ‖Ψ‖2H0 − |k(Ψ,Ψ)| (4.23)

for all Ψ ∈ H0 with a compact bilinear form k(·, ·) on H0 ×H0.

22

Proof. The arguments following (4.13) show that in the half-space case (4.14) holdswith E ′ and M ′ instead of E and M yielding (4.23) due to (4.22). The compact bilinearform k(·, ·) arises in (4.23) since H ′ acts on functions on a bounded manifold Γ whichcauses a compact perturbation to the half-space situation.

Now we concentrate on the connection between our strongly elliptic pseudodifferentialoperator H ′ and the original interface problem (3.6), (3.18), (3.8).

Following (2.22)–(2.25) we obtain for (3.6), (3.18), (3.8) the system of integral equa-tions

H ′Ψ′ = −(1− C ′2)Ψ′0 with H ′ := C ′1 − C ′2, (4.24)

where Ψ′0 := (~v0, ψ′0,~ψ′0, v0)> are the modified Cauchy data (4.16) of the incident field ~u0

and C ′j = 12(1 + (−1)jA′j) is the Calderon projector (corresponding to the wave number

kj) of A′j in (4.20).Application of Theorem 2.4 to (4.24) yields the following equivalence between the

transmission problem (3.6), (3.18), (3.8) and the boundary integral equations (4.24). Theproof is identical to the proof of Theorem 2.4 and is therefore omitted.

Theorem 4.7 Let ~u0 ∈ Hs be given.

(i) If ~uj ∈ Lj (j = 1, 2) as defined in (4.1) solve the transmission problem (3.6), (3.18),(3.8), then

((~u1)>, div ~u1 − div>(~u1)>,−~n× curl ~u1 + grad>(~n · ~u1), ~n · ~u1)> ∈ Hs

solves the equation (4.24).

(ii) If Ψ′ := (~v, ψ′, ~ψ′, v)> ∈ Hs solves (4.24) with Ψ′0 := (~v0, ψ′0,~ψ′0, v0)>, i. e.

−1

2(A′1 + A′2)Ψ′ =

1

2(A′2 − 1)Ψ′0,

then withΨ1 := C ′1Ψ′, Ψ2 := C ′2(Ψ′ + Ψ′0) (4.25)

and~uj := KjPjR−1

j Ψj in Ωj (see (2.18)), (4.26)

~uj ∈ Lj solve the transmission problem (3.6), (3.18), (3.8).

Remark. In (4.26), ~uj is given by the Stratton-Chu formula (4.4) applied to

(~vj, ψj, ~ψj, vj) which are connected with Ψj via (4.16), (3.27), (3.28). By Garding’s in-equality (4.23), H ′ is a Fredholm operator of index zero from H0 into itself and thus alsofrom Hs into itself for any s. Therefore we obtain existence of a solution of (4.24) assoon as we know its uniqueness, and Theorem 4.7 then implies the existence of a solutionof the transmission problem. For the question of unique solvability of the transmissionproblem (3.6), (3.18), (3.8) and therefore of our boundary integral equation (4.24) we referto [2, 4, 15, 20, 24]. In the case of the “physical” transmission conditions (3.7), we discussthis question in more detail in section 5. From the discussion in [4] and [24] follows theresult:

23

Proposition 4.8 Assume that the homogeneous transmission problem (3.6), (3.18), (3.8)and an associated homogeneous adjoint problem with interchanged wave numbers haveonly the trivial solution in Lj (defined in (4.1)). Then for given Ψ0 ∈ Hs there existsexactly one solution Ψ′ ∈ Hs of the integral equation (4.24) yielding exactly one solutionof (3.6),(3.18),(3.8) via (4.26).

5 Integral equations for the electromagnetic trans-

mission problem

In this section we study the integral equations corresponding to the “physical” transmis-sion conditions (3.7). Instead of repeating all the arguments of the preceding section, weonly point out the necessary modifications.

We define, according to (3.7), the “physical” Cauchy data on Γ:

vj := εjvj = εj~n · ~uj; ~vj := ~vj = ~uj>;

ψj := λjψj = λj div ~uj;~ψj := 1

µj~ψj = − 1

µj~n× curl ~uj.

(5.1)

We want to use the results of the previous section. Therefore we write

Ψ := (~v, ψ, ~ψ, v)>; Ψj := (~vj, ψj,~ψj, vj)

>.

Thus we have with the obvious block notation

Ψj = BjΨ ; Bj = diag (1, λj,1µj, εj). (5.2)

Now we can insert these Cauchy data into the representation formula as before and obtaininstead of (4.8) the Calderon projectors

Cj =1

2(1 + (−1)jAj); Aj = BjAjB

−1j . (5.3)

The boundary integral operators Aj are given explicitly in (4.9). The procedure describedin the previous section then leads to a boundary integral equation corresponding to (2.25).The matrix of integral operators is given by

H = C1 − C2 = −1

2(A1 + A2) = −1

2(B1A1B

−11 +B2A2B

−12 ). (5.4)

From Lemma 4.5 and (4.11) we find the principal symbol (in block form)

σ(H)(ξ) =

0 E

M 0

(5.5)

with

E =1

|ξ|

µe 0 iξ1εe

0 µe iξ2εe

iξ1λe iξ2λe 0

, M =1

|ξ|

ξ2

2µm −ξ1ξ2µm −iξ1λm

−ξ1ξ2µm ξ21µm −iξ2λm

−iξ1εm −iξ2εm νm

, (5.6)

24

Here we used the abbreviations

µe = 12(µ1 + µ2); εe = 1

2( 1ε1

+ 1ε2

); λe = 12(λ1µ1 + λ2µ2); νe = 1

2(λ1ε1

+ λ2ε2

);

µm = 12( 1µ1

+ 1µ2

); εm = 12(ε1 + ε2); λm = 1

2( 1λ1µ1

+ 1λ2µ2

); νm = 12( ε1λ1

+ ε2λ2

).(5.7)

Again, as in the previous section, the matrices E and M are, in general, not positive,hence the operator H will not be strongly elliptic. Therefore we modify the Cauchy data(5.1) in analogy to (3.26), (4.16):

Ψj′ := (~vj, ψj

′, ~ψj′, vj

′)>

with

ψj′ := ηψj − div> ~vj = ηλjψj

′ + (ηλj − 1) div> ~vj = ηλjψj − div> ~vj;~ψj′ := ~ψj + ϑ grad> vj = 1

µj~ψj′ + (ϑεj − 1

µj) grad> vj = 1

µj~ψj + ϑεj grad> vj;

vj′ := ϑvj = ϑεjvj.

(5.8)Here we introduced two new complex parameters η and ϑ which will be fixed later on(see (5.11) and (5.13)). It turns out that they can always be chosen in such a way thatthe resulting boundary integral operator is strongly elliptic.

It is quite obvious how to insert these constants into the system of integral equations(4.17). Therefore we need not repeat this explicit representation here. In short notation,the modified system of integral operators is

H ′ = C ′1 − C ′2 = −1

2(A′1 + A′2) = −1

2(B′1A

′1B′1−1 +B′2A

′2B′2−1) (5.9)

with A′j as defined in (4.20) and, according to (5.8),

B′j =

1 0 0 0

(ηλj − 1) div> ηλj 0 0

0 0 µ−1j (ϑεj − µ−1

j ) grad>

0 0 0 ϑεj

with the obvious block notation.

For the computation of the principal symbols, we can take advantage of (4.21), (4.22).The result is

σ(H ′)(ξ) =

0 E ′

M ′ 0

with E ′ =

1

2(E ′1 + E ′2); M ′ =

1

2(M ′

1 + M ′2) and

Ej′=

1

|ξ|

µj 0 iaξ1

0 µj iaξ2

ibξ1 ibξ2 c|ξ|2

; Mj′=(E ′j)−1

,

25

wherea = (ϑεj)

−1 − µj; b = µj(ηλj − 1); c = a+ b+ µj.

For simplicity, we now make the choices,

λj = (µjεj)−1 (compare case (ii) of section 3), (5.10)

andϑ = (η)−1. (5.11)

This gives b = a, c = c, hence one obtains with dj = 2 Re(ηε−1j )− µj, ej = |µjεjη−1 − 1|2

1

2(E ′j + (E ′j)

>) =1

|ξ|

µj 0 0

0 µj 0

0 0 dj|ξ|2

;

1

2(M ′

j + (M ′j)>) =

1

µj|ξ|

|ξ|2 − ejξ2

1 −ejξ1ξ2 0

−ejξ1ξ2 |ξ|2 − ejξ22 0

0 0 |µjεj/η|2

.Because we assumed µj > 0, the positivity of both matrices depends only on the

positivity of dj:

Lemma 5.1 The operator H ′ is strongly elliptic, i. e. there exists a γ > 0 such that

Re(Ψ, H ′Ψ)H0 ≥ γ‖Ψ‖2H0 − |k(Ψ,Ψ)| (5.12)

for all Ψ ∈ H0 with a compact bilinear form k(·, ·) on H0 ×H0,if and only if

2 Re η( 1ε1

+ 1ε2

) > µ1 + µ2

and

2 Re η(ε1 + ε2) > µ1|ε1|2 + µ2|ε2|2.(5.13)

Remark. The condition (5.13) can always be satisfied by a suitable choice of η, underthe assumption (3.4) and also for the eddy current problem (see case (iv) in section 3)even by a large enough real η.

Let us write the system of integral equations as

H ′Ψ′ = −(1− C ′2) Ψ′0. (5.14)

We summarize the results of this section in the following theorem.

Theorem 5.2 Let the assumptions (3.4), (5.10), (5.11), and (5.13) for the coefficients be

satisfied. Then for given Ψ0 := (~v0, ψ0, ~ψ0, v0)> ∈ H0, the transmission problem (3.6)–(3.8)has a unique solution ~u with ~uj ∈ Lj, j = 1, 2. This solution corresponds via (5.1), (5.8)to the unique solution Ψ′ ∈ H0 of the system (5.14) of boundary integral equations. Theboundary integral equations are a strongly elliptic system of pseudodifferential equationson Γ.

26

Acknowledgement. We like to thank the referee for his helpful remarks on thechoice of the appropriate transmission conditions.

The second author was supported by the NSF grant DMS-8603954.

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Dr. Martin CostabelFachbereich MathematikTechnische Hochschule DarmstadtSchlogartenstr. 76100 DarmstadtGermany

Prof. Dr. Ernst P. StephanSchool of MathematicsGeorgia Institute of TechnologyAtlanta, Ga. 330332USA

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