Unique solvability for high-frequency heterogeneous time-harmonic Maxwell equations via the Fredholm alternative theory

Mathematical Methods in the Applied Sciences, Vol. 21, 463—477 (1998)MOS subject classification: 35Q 60

Unique Solvability for High-frequency HeterogeneousTime-harmonic Maxwell Equations via the FredholmAlternative Theory

Ana Alonso* and Alberto Valli

Dipartimento di Matematica, Universita di Trento, 38050 Povo (Trento), Italy

Communicated by R. Leis

We consider the time-harmonic Maxwell equations in the high-frequency case for a heterogeneous medium,i.e., a medium which is composed by a conductor and a perfect insulator, or, in other words, a loaded cavity.As a consequence of a suitable compactness result, we prove that Fredholm alternative holds for sucha problem. Since the kernels of the considered operator and of its adjoint are proven to be trivial, a uniquesolution exists for each given datum. ( 1998 B. G. Teubner Stuttgart—John Wiley & Sons Ltd.

Math. Meth. Appl. Sci., Vol. 21, 463—477 (1998)

1. Introduction

The time-harmonic Maxwell equations are derived from the complete Maxwellequations assuming that both the electric field E and the magnetic field H are of theform

E(t, x)"Re[E(x) exp(iat)], H (t, x)"Re[H(x) exp(iat)],

where aO0 is a given angular frequency and E and H are complex valued vectorfunctions.

Making this assumption we are left with

iaeE"rotH!pE in ), (1.1)

iakH"!rotE in ), (1.2)

where )LR3 is a bounded domain, e is the dielectric coefficient, k is the magneticpermeability coefficient and p is the electric conductivity. In the general case ofanisotropic inhomogeneous media the coefficients e and k are 3]3 symmetric realmatrices, uniformly positive definite in ) and with entries in ¸= ()). On the otherhand, the conductivity p is a symmetric real matrix, uniformly positive definite and

*Correspondence to: A. Alonso, Dipartimento di Matematica, Universita di Trento, 38050 Povo (Trento),Italy

CCC 0170—4214/98/060463—15$17.50 Received 14 March 1997

( 1998 B. G. Teubner Stuttgart—John Wiley & Sons Ltd.

with coefficients in ¸= when we refer to a conductor, and equal to zero for aninsulator.

As k is non-singular it is possible to eliminate the magnetic field H by taking therotational of (1.2), thus obtaining from (1.1)

rot (k~1 rotE)!a2(e!ia~1p)E"0. (1.3)

We deal with the high-frequency model, i.e., we consider the complete equation (1.3),keeping the term a2eE, that on the contrary is neglected in the low-frequency model.

On the boundary, we impose the Dirichlet boundary condition for the electric field:

(n]E) D)"W on ), (1.4)

where W is a given tangential vector field defined on ), and n denotes the unitoutward normal vector on ).

Most often, it is assumed that a vector function Eª is known, satisfying(n]Eª ) D)"W on ). Then the resulting boundary value problem reads

rot (k~1 rot u)!a2 (e!ia~1p)u"F in ),

(n]u) D)"0 on ), (1.5)

where u"E!Eª and F"!rot (k~1 rotEª )#a2(e!ia~1p)Eª .We are interested in the case where the conductivity p is given by p (x)"

p8 (x)s)C)0(x), where )

0is an open subset of ) representing an insulator and s)C)0

is thecharacteristic function of )C)

0. In particular, the case )

0") corresponds to a per-

fect insulator and the case )0"0 corresponds to a conductor.

In this paper we study the solvability of (1.5) when )0

is a non-empty proper opensubset of ). This corresponds to the problem of a loaded cavity, namely, a cavity) with inserted a conductor )C)

0surrounded by the vacuum )

0. The cases )

0")

(the problem of the isolated cavity) and )0"0 (the problem of a conductor) have

been studied by Leis (see [7]).Let us make precise some notations. As usual we denote by Hk()), k*0, the

Sobolev space of real or complex valued functions belonging to ¸2()) together withall their distributional derivatives of order less than or equal to k. In particular,¸2())"H0 ()). The norm in Hk()) will be denoted by E ) E

k,) . It is well known thatthe trace space of H1()) over ) is given by the Sobolev space H1@2()). The spaceH~1@2()) is the dual space of H1@2()). The duality pairing will be denoted byS) , )T) .

The space H(rot; )) (respectively, H (div; ))) indicates the set of the real or complexvector functions w3(¸2()))3 such that rotw3(¸2()))3 (respectively, divw3¸2())). Set

H0(rot; )) :"Mw3H (rot; )) D (n]w) D)"0N,

H0 (rot; )) :"Mw3(¸2()))3 Drotw"0N,

H0(div; )) :"Mw3H(div; )) D(n )w) D)"0N,

H0 (div; )) :"Mw3(¸2()))3 Ddivw"0N.

464 A. Alonso and A. Valli

Math. Meth. Appl. Sci., Vol. 21, 463—477 (1998) ( 1998 B. G. Teubner Stuttgart—John Wiley & Sons Ltd.

Given f"(fbc (x))1)b,c)3

with real or complex coefficients in ¸=()), we call

H(f, div; )) :"Mw3(¸2()))3 Ddiv(fw)3(¸2()))3N,

H0(f, div; )) :"Mw3(¸2()))3 Ddiv(fw)"0N.

The norm in H(f, div; )) is given by

EvEH(f, div;))"(EvE20,)#Ediv(fv)E2

0,))1@2;

as a consequence, notice that the norm in H0(f, div; )) is indeed given by the(¸2()))3-norm.

The set of (real) Neumann harmonic fields will be denoted by

H(m) :"Ml3(¸2()))3 Drotl"0, divl"0, (l ) n) D)"0N.

As it is well known, this vector space has finite dimension M, the total number ofhandles of ) (see, e.g., [5, 9, 3]). In particular, this space is trivial if and only if ) issimply connected. We indicate by l

r, r"1,2, M, an orthonormal basis of H(m)

with respect to the (¸2()))3-scalar product.Finally, we denote by

H(e) :"Me3(¸2()))3 Drot e"0, div e"0, (e]n) D)"0N,

the set of (real) Dirichlet harmonic fields. The dimension of this vector space is P, thenumber of internal connected components !

iof ) (see, e.g., [9, 3]). In particular, this

space is trivial if and only if ) is connected. For constructing a basis it is sufficient toconsider the functions z

i3H1()), i"1,2, P, solutions to

*zi"0 in ),

zi D!i

"1 on !i,

zi D)C!

i"0 on )C!

i.

It is easily seen that the gradients of zifurnish a basis of H (e).

2. Variational formulation and statement of the main theorems

Let us make precise the variational formulation of (1.5). We introduce in H(rot; ))the bilinear form

ap(w, v) :"(k~1 rotw, rot v)!a2((e!ia~1p)w, v),

where ( ) , )) is the scalar product in (¸2 ()))3 for complex valued vector functions.Assuming that F3(¸2()))3, the variational formulation of (1.5) reads: find

u3H0(rot; )) : ap(u, v)"(F, v) ∀v3H

0(rot; )). (2.1)p

Heterogeneous time-harmonic Maxwell equations 465

( 1998 B. G. Teubner Stuttgart—John Wiley & Sons Ltd. Math. Meth. Appl. Sci., Vol. 21, 463—477 (1998)

We underline that in the sequel the real matrices k and e are assumed to besymmetric and uniformly positive definite in ), with coefficients in ¸= ()). Concerningthe conductivity, if p is a symmetric real matrix, uniformly positive definite in ) andwith coefficients in ¸=()). then it is easily seen that the bilinear form ap( ) , ) ) iscontinuous and coercive in H(rot; )) and, as a consequence of the Lax—Milgramlemma for complex Hilbert spaces, we can state the following theorem, which is alsovalid for unbounded domains (see, e.g., [7]).

Theorem 2.1. ¸et ) be an arbitrary domain in R3. If p is a symmetric real matrix withcoefficients in ¸=()) and uniformly positive definite in ), then there exists a uniquesolution of (2.1)p .

The case p"0 has been considered by Leis [7]. He proves that the Fredholmalternative theorem holds for problem (2.1)

0assuming that the domain ) is regular

enough to ensure the following compactness property:

Definition 2.2. A bounded domain )LR3 is said to have the compactness property ifboth immersions

H0(rot; ))WH(div; ))) (¸2()))3,

H(rot; ))WH0(div; ))) (¸2()))3

are compact.

In particular, this compactness property is true for Lipschtiz domains. A moregeneral result has been given by Witsch [12].

The existence theorem of Leis reads:

Theorem 2.3. ¸et )LR3 be a bounded domain having the compactness property. Ifp"0 and F3H0(div; )) then the Fredholm alternative holds for the boundary valueproblem (2.1)

0and the necessary and sufficient solution condition is (F, /)"0 for all

solutions /3H0(rot; )) of the homogeneous problem

(k~1 rot/, rot v)!a2(e/, v)"0 ∀v3H0(rot; )).

We want first to extend this result to the case p (x)"p8 (x)s)C)0(x), where )

0O0 is

a proper open subset of ) and s)C)0is the characteristic function of )C)

0. As we

already noticed, this corresponds to a medium which is heterogeneous, namely, it iscomposed by an insulator (the set )

0) together with a conductor (the set )C)

0); in

other words, we are considering the problem of the loaded cavity.The first result of this paper reads:

Theorem 2.4. ¸et )LR3 be a bounded domain with ¸ipschitz boundary. Assume thatp(x)"p8 (x)s)C)0

(x), where pJ is a symmetric real matrix which is uniformly positivedefinite in )C)

0and has coefficients in ¸= ()C)

0), and )

0O0 is a proper open subset of

). If F3H0(div; )) then the Fredholm alternative holds for the boundary value problem(2.1)p and the necessary and sufficient solution condition is (F, /)"0 for all solutions



/3H0(rot; )) of the adjoint homogeneous problem:

(k~1 rot/, rot v)!a2( (e#ia~1p)/, v)"0 ∀v3H0(rot; )). (2.2)

In Theorems 2.3 and 2.4 the assumption F3H0(div; )) is not restrictive. In fact, letus consider the general case F3(¸2()))3. Then divF3H~1()) and, due to the fact thate is uniformly positive definite in ), in both cases p"0 or p"p8 s)C)0

the boundaryvalue problem

div((e!ia~1p)+º)"divF in ),

º D)"0 on ),

has a unique solution º3H10()). Setting G :"F!(e!ia~1p)+º, which clearly

satisfies divG"0, we are left with the solvability of the following problem:

rot(k~1 rot u)!a2(e!ia~1p)u"G in ),

(n]u) D)"0 on ).

Then u"u!a~2+º is a solution of (2.1)p .By looking at the proof of Theorem 2.4 presented in the following sections, it can be

noticed that a slightly more general result indeed holds. In fact, we only need that p isa symmetric real matrix with coefficients belonging to ¸=()). The more restrictiveassumptions in Theorem 2.4 just describe a physically reasonable configuration;however, they play a crucial role in the following existence and uniqueness result,which is the main result presented in this paper.

Theorem 2.5. ¸et )LR3 be a bounded domain with ¸ipschitz boundary. Assume thatp(x)"p8 (x)s)C)0

(x), where pJ is a symmetric real matrix which is uniformly positivedefinite in )C)

0and has coefficients in ¸= ()C)

0), and )

0O0 is a proper open subset of

) with ¸ipschitz boundary. Assume moreover that the entries of the matrices e andk satisfy ebc , kbc3C2()) for each b, c"1, 2, 3. ¹hen both problems (2.1)p with F"0 and(2.2) have only the trivial solution, and consequently the boundary value problem (2.1)p isuniquely solvable. Moreover, accordingly to the Closed-Graph ¹heorem, there existsa constant CK '0 such that

EuE0,)#Erot uE

0,))CK EFE0,) .

The proof of this result is based on the fact that each solution of (2.1)p for F"0 orof (2.2) is vanishing in )C)

0, therefore, its restriction to )

0is a solution of the problem

w3H0(rot; )

0) : (k~1D)0

rotw, rot v0))0

!a2 (eD)0w, v

0))0

"0 ∀v03H

0(rot; )

0)

(see Proposition 5.1). The unique continuation principle proven in [7] (see also[8, p. 168]) is the main tool in the proof of Theorem 2.5.

We end this section by noticing that, after having completed this paper, we havebeen informed that the proofs of Theorems 2.4 and 2.5 are briefly presented in



[4, pp. 17—19], by a somehow different (though similar) approach. However, hisassumptions are more restrictive than ours, as there it is required that the boundary) is connected, and that the coefficients e and k are scalar positive constants.

3. A compactness result.

The main point in the proof of Theorem 2.3 is to show that the immersion

H(e, div; ))WH0(rot; )))(¸2 ()))3

is compact. Set now g :"e!ia~1p. Notice that the matrix e is real, symmetric anduniformly positive definite in ). On the contrary g is complex and not hermitian. Westart by proving the following lemma.

Lemma 3.1. If )LR3 is a bounded domain with ¸ipschitz boundary, then theimmersion

H(g, div; ))WH0(rot; )))(¸2()))3

is compact.

Proof. The proof extends the one of Weber [11], which is valid for a simply connecteddomain ) with connected boundary ) and for a real-valued symmetric and uniformlypositive definite matrix.

Let MvnNn3NLH(g, div; ))WH

0(rot; )) be such that

EvnE0,)#Erot v

nE0,)#Ediv(gv

n)E

0,))1 ∀n3N. (3.1)

Let us introduce the complex function »n, which is the unique solution of the

Dirichlet boundary value problem

»n3H1

0()) : b(»

n, /) :"(g+»

n, +/)"(gv

n, +/) ∀/3H1

0()). (3.2)

Notice that the bilinear form b ( ) , )) satisfies

Db(/, /) D*Reb (/, /)"(e+/ , +/)*e0E+/E2

0,

as e and p are real and symmetric, and e is uniformly positive definite in ). Hence weapply the Lax—Milgram lemma for complex Hilbert spaces and we can assure theexistence and uniqueness of »

n, and moreover, there exists a constant C

0'0 such that

E+»nE0,))C

0EgE

¸= ())EvnE0,) ∀n3N.

Therefore, from (3.1) and the Poincare inequality we deduce that there exists a con-stant K

0'0 such that

E»nE1,))K

0∀n3N. (3.3)



By applying the Rellich theorem we can find a subsequence of M»nNnwhich is strongly

convergent in ¸2()) and, since Ediv(gvn)E

0,))1, a subsequence of MvnNn

such thatMdiv(gv

nk)N

kis weakly convergent in ¸2()). Summing up, we have a subsequence Mn

kNk

such that

(div(g(vnk!v

nl)), »

nk!»

nl) )P0 as k, lPR.

As M»nNnLH1

0())

(div(g(vnk!v

nl)), »

nk!»

nl)"(div(g+(»

nk!»

nl) ), »

nk!»

nl)

"!(g+(»nk!»

nl), + (»

nk!»

nl)).

In particular,

Re[(g+(»nk!»

nl), + (»

nk!»

nl))]P0 as k, lPR.

Since Re[(g+(»nk!»

nl), + (»

nk!»

nl))]"(e+(»

nk!»

nl), + (»

nk!»

nl)) and e is uniform-

ly positive definite we obtain that

e0E+»

nk!+»

nlE0,)P0 as k, lPR,

hence,

M+»nkNk

is strongly convergent in (¸2()))3. (3.4)

Let us split now vn"+»

n#v2

n. From (3.1) and (3.3) we have Ev2

nE0,))1#K

0.

Moreover, div(gv2n)"0, rot v2

n"rot v

nand (n]v2

n) D)"(n]v

n) D)!(n]+»

n) D)"0,

hence Mv2nNn3NLH0 (g, div; ))WH

0(rot; )) and there exists a constant K

1'0 such

that

Ev2nE0,)#Erot v2

nE0,))K

1∀n3N. (3.5)

For each n3N we introduce the complex numbers

an, i

:"S(gv2n) n) D!

i, 1T!i

, i"1,2 , P.

We claim that there exists a constant C1'0 such that +P

i/1Da

n, iD)C

1for all n3N; in

fact,

Dan, i

D"DS(gv2n) n) D!

i, z

i D!iT!i

D"D (div(gv2n), z

i)#(gv2

n, +z

i) D

)EgE¸=())Ev2

nE0,)E+z

iE0,) ,

where M+ziNPi/1

is the basis of H (e). Therefore, using (3.5) we find

P+i/1

Dan, i

D)EgE¸=())K1

P+i/1

E+ziE0,)": C

1.

Now, we introduce a function wn3H(e) such that

S(wn) n) D!

i, 1T!i

"an, i

, i"1,2 , P.



It is sufficient to write wn"+P

j/1bn, j

+ziwhere the vector b

n3CP is the solution of the

linear system:

P+j/1

bn, jTA

zj

nBD!i

, 1U!i

"an, i

, i"1,2,P. (3.6)

The real matrix G"(gij)1)i,j)P

, where gi,j

:"S(zj/n) D!

i, 1T!i

, is symmetric andpositive definite: in fact,

TAz

jnBD!

i

, 1U!i

"TAz

jnBD)

, zi D)U)

"P)+zj)+z

i.

Therefore, the constants bn, j

are uniquely determined by (3.6) and there existsa constant C

2'0 such that

P+j/1

Dbn, j

D)C2

P+j/1

Dan, j

D;

setting K2

:"C1C

2max

1)j)PE+z

jE0,) we have at once

EwnE0,))K

2∀n3N. (3.7)

The difference gv2n!w

nsatisfies div(gv2

n!w

n)"0 and S((gv2

n!w

n) ) n) D!

i, 1T!

i"0

for i"1,2 , P, then there exists a unique function An3(¸2()))3 such that

rotAn"gv2

n!w

nin ),

divAn"0 in ),

(An) n) D)"0 on ),

(An, l

r)"0 r"1,2,M, (3.8)

where MlrNMr/1

is an orthonormal basis of H (m) (see [10] and Appendix, Lemma A.2).As ) satisfies the compactness property, there exists a constant C

3'0 such that

E/E0,))C

3(Erot/E

0,)#Ediv /E0,))

for all /3¼ :"H(rot; ))WH0(div; ))WH(m)o (see [10, 1]). Hence,

EAnE0,)#ErotA

nE0,))(C

3#1)Egv2

n!w

nE0,)

)(C3#1)(EgE

¸= ())Ev2nE0,)#Ew

nE0,) )

)(C3#1)(EgE

¸= ())K1#K

2)": K

3. (3.9)

From (3.5), (3.7), (3.9) and the compactness assumption we can select a subsequenceMn

kNksuch that

Mv2nkNkis weakly convergent in H (rot; )),

MwnkNkis strongly convergent in (¸2()))3,

MAnkNkis strongly convergent in (¸2()))3,



therefore,

(v2nk!v2

nl, w

nk!w

nl)P0 as k, lPR

and

(rot(v2nk!v2

nl), A

nk!A

nl)P0 as k, lPR.

From Mv2nNnLH

0(rot; )) it follows:

(rot(v2nk!v2

nl), A

nk!A

nl)"(v2

nk!v2

nl, rot(A

nk!A

nl))

"(v2nk!v2

nl, g(v2

nk!v2

nl)!(w

nk!w

nl)),

thus,

(v2nk!v2

nl, g (v2

nk!v2

nl))P0 as k, lPR.

In particular,

Re[(v2nk!v2

nl, g (v2

nk!v2

nl))]"(v2

nk!v2

nl, e (v2

nk!v2

nl))P0 as k, lPR,

having used the symmetry of e and p. Since e is uniformly positive definite in ) we canconclude

Ev2nk!v2

nlE0,)P0 as k, lPR.

Therefore, we have finally obtained

Mv2nkNkis strongly convergent in (¸2()))3, (3.10)

and from this result and (3.4) Lemma 3.1 is proven.

4. Proof of Theorem 2.4

We introduce the auxiliary bilinear form

aJ (w, v) :"(k~1 rotw, rot v)#(gw, v)

"ap(w, v)#(1#a2)(gw, v). (4.1)

Since k~1 and e are uniformly positive definite in ), the bilinear from aJ ( ) , ) ) is coercivein H (rot; )). Therefore, for each f3(¸2()))3 there exists a unique Mf3 H

0(rot; )) such

that

aJ (Mf, v)"(f, v) ∀v3H0(rot; )), (4.2)

i.e.

rot(k~1 rotMf )#gMf"f in ),

(n]Mf ) D)"0 on ). (4.3)



In this way, we have defined the continuous linear operator

M : (¸2()))3PH0(rot; )).

Since from (4.3) it follows that div(gMf )"div f, we obtain

M(H0(div; )))LH0 (g, div; ))WH0(rot; )).

If we define the continuous linear operator Mg : (¸2()))3PH0(rot; )) by setting

Mgg :"M(gg),

it follows at once that

Mg :H0 (g, div; ))PH0(g, div; ))WH0(rot; )).

The boundary value problem (2.1)p can also be written as

u3H0(rot; )) : aJ (u, v)!(1#a2) (gu, v)"(F, v) ∀v3H

0(rot; )).

Assuming that F3H0(div; )), it is verified at once that the solution u indeed belongsto H0 (g, div; )). Therefore, under the assumptions of Theorem 2.4, the boundary valueproblem (2.1)p finally reads

u3H0 (g, div; ))WH0(rot; )) : u!(1#a2)Mgu"MF. (4.4)

From Lemma 3.1 we know that the inclusion

H0 (g, div; ))WH0(rot; )))H0 (g, div; ))

is compact, consequently, the operator

Mg :H0 (g, div; ))PH0(g, div; ))

is compact. Then the Fredholm alternative holds for (4.4) in the Hilbert spaceH0 (g, div; )) and

R(I!(1#a2)Mg)"N(I!(1#a2)M*g )o, (4.5)

where the symbol o denotes the orthogonal space with respect to the scalar product ofH0 (g, div; )), which, as we already noticed, is the (¸2 ()))3-scalar product. Further,M*g is the adjoint operator of Mg in H0 (g, div; )), defined as (M*g /, w) :"(/, Mgw) foreach /, w3H0(g, div; )).

Let us denote now by M* the adjoint operator of M in (¸2()))3 defined as(M*f, g)"(f, Mg) for each f, g3(¸2()))3. Denote, moreover, by g* the matrix

g*ij

:"gji; since the coefficients of the symmetric matrices e and p are real functions we

have g*"e#ia~1p. We find that M*f is coincident with the unique solution Sf of theboundary value problem

Sf3H0(rot; )) : (k~1 rotSf, rot v)#(g*Sf, v)"(f, v) ∀v3H

0(rot; )). (4.6)



In fact, given f3(¸2()))3 we have

(M*f, w)"(f, Mw)"(k~1 rotSf, rotMw)#(g*Sf, Mw)

"(k~1 rotMw, rotSf )#(gMw, Sf )

"(w, Sf )"(Sf, w) ∀w3(¸2()))3.

Using the adjoint operator M*, the necessary and sufficient solution condition (4.5)reads

(MF, w)"(F, M*w)"0 ∀w3N(I!(1#a2)M*g ),

or, in other words, F must be orthogonal to the space M*[N (I!(1#a2)M*g )].We claim that this condition is equivalent to the one stated in Theorem 2.4. Firstly,

we verify that M*g"g*M*, as for all w3H0 (g, div; )) we have

(M*g/, w)"(/, Mgw)"(/, M(gw))"(M*/, gw)"(g*M*/, w).

Moreover, we see that the set M*[N (I!(1#a2)g*M*)] is equal to the set ofsolutions of (2.2). In fact, w3N (I!(1#a2)g*M*) if and only if w"(1#a2)g*M*w.Then for all v3H

0(rot; ))

(1#a2) (g*M*w, v)"(w, v)"(k~1 rotM*w, rot v)#(g*M*w, v),

therefore,

(k~1 rotM*w, rot v)!a2(g*M*w, v)"0 ∀v3H0(rot; )).

Conversely, if / is a solution of (2.2), we have

(k~1 rot/, rot v)#(g*/, v)"(1#a2)(g*/, v) ∀v3H0(rot; )),

hence /"M*[(1#a2)g*/]. Setting w :"(1#a2)g*/, it is easily verified thatw3N(I!(1#a2)g*M*), as (1#a2)g*M*w"(1#a2)g*/"w.

5. Existence and uniqueness results

In this section we want to show that the space of solutions of both problems (2.1)pfor F"0 and (2.2) is trivial, hence, from Theorem 2.4, the boundary-value problem(2.1)p admits a unique solution for each vector field F3(¸2()))3.

Let us start by proving the following result

Proposition 5.1. ¸et the assumptions of ¹heorem 2.4 be satisfied. ¸et / be a solution to(2.2). ¹hen / D)C)0

"0, therefore, / D)0is a solution of the problem

w3H0(rot; )

0) : (k~1D)

0rotw, rot v

0))0

!a2 (eD)0w, v

0))0

"0 ∀v03H

0(rot; )

0). (5.1)

¹he same is true for any solution of the homogeneous problem (2.1)p .



Proof. Take in (2.2) the test function v"/. Then we have

(k~1 rot/, rot/)!a2 (e/, /)!ia(p/, /)"0,

hence, the imaginary part !a(p/, /) is vanishing. In other words,

(p8 / D)C)0, / D)C)0

))C)0"0,

and the thesis follows as p8 is uniformly positive definite in )C)0.

The proof for the homogeneous equation (2.1)p is analogous. K

Problem (5.1) has been studied by Leis [7], under the assumption that )0

satisfiesthe compactness property stated in Definition 2.2. Any solution w to (5.1) turns out tobe a solution of the problem

A0w"a2w in )

0, (5.2)

where A0

:"e~1D)0rot(k~1D)

0rot). Moreover, in H

0(rot; )

0) the eigenvalue problem

A0x"qx has a countably infinite set of non-negative eigenvalues q

jPR, hence for

a2Oqjthe set of solutions of (5.1) is trivial. We have thus proven the following.

Theorem 5.2. ¸et the assumptions of ¹heorem 2.4 be satisfied, and assume moreoverthat )

0has a ¸ipschitz boundary. If a2Oq

j, where q

jare the eigenvalues of the operator

A0

in H0(rot; )

0), the space of solutions of (2.2) is trivial, and problem (2.1)p admits

a unique solution.

This is the best that can be obtained for the case of the isolated cavity (see [7]), andin electrical engineering applications it is indeed well known that for that physicalconfiguration there exist resonant values of the frequency a.

On the contrary, in the case of the loaded cavity it is expected that such values donot exist. This is indeed true, as asserted in Theorem 2.5 and proven here below.

Proof of ¹heorem 2.5. Let / be a solution of problem (2.1)p with F"0 or of problem(2.2). As it is well known (see, for instance, [2], where a similar situation is considered),this implies that on the interface ! :")

0W ()C)

0) the following conditions are

satisfied (in a suitable week sense):

(n!]/ D)0) D!"(n!]/ D )C)0

) D!"0,

[n!](k~1D)0rot/ D)

0)] D!"[n!](k~1D)C)

0rot/ D )C)0

)] D!"0, (5.3)

n! being the unit normal vector on !.From Proposition 5.1 we know that /D)0

satisfies

e~1D)0rot(k~1D)

0rot/ D)

0)"a2/ D)

0in )

0.

Clearly, the same is true for /D)C)0"0 in )C)

0, and using the interface conditions

(5.3) we can infer that / is indeed a solution of the problem

A/"a2/ in ),

where A :"e~1 rot(k~1 rot). Therefore, from the unique continuation principle (see[8, Theorem 8.17] it follows that /"0. K



Appendix

In the proof of Lemma 3.1 we use two results that are well known for regulardomains. They can be easily generalized to Lipschitz domains.

Lemma A.1. ¸et )LR3 be a bounded domain with ¸ipschitz boundary. ¹hen thereexists a constant C'0 such that

E/E0,))C(Erot/E

0,)#Ediv/E0,)).

for all /3» :"H0(rot; ))WH(div; ))WH(e)o.

Proof. Proceeding by contradiction, we can find a sequence /n3» with E/

nE0,)"1

and

(Erot/nE0,)#Ediv/

nE0,))(

1

n.

Due to the compactness property we can select a subsequence /nk

which is conver-gente to /K in (¸2()))3 and weakly in ». Clearly, E/K E

0,)"1 and /K 3H(e)o. On theother hand, we see that both rot/

nand div/

nare vanishing, therefore, rot/K "0 and

div/K "0. Moreover, for the continuity of the tangential trace from H(rot; )) into(H~1@2()))3, we obtain /K 3H(e), and consequently /K "0. This is a contradiction andthe proof is complete. K

Now we can extend to the case of a Lipschitz domain an existence theorem provedby Saranen [10] for regular domains.

Lemma A.2. ¸et )LR3 be a bounded domain with ¸ipschitz boundary. ¹hen, beinggiven g3H0(div; )) such that S(g ) n)D!

i, 1T!

i"0, i"1,2 , P, there exists a unique

w3¼ :"H(rot; ))WH0(div; ))WH(m)o such that

rotw"g in ),

divw"0 in ),

(w ) n) D)"0 on ),

(w, lr)"0 r"1,2 , M.

Proof. The proof of uniqueness is trivial. Concerning existence, from Lemma A.1 weknow that the bilinear form

c(w, v) :"(rotw, rot v)#(divw, div v)

is coercive in », thus there exists a unique q3» such that

c(q, v)"(g, v) ∀v3». (A.1)



We claim that rot q is the solution we are looking for. Let us start to prove thatrot rot q"g. Given f3¸2()) there exists a unique function u3H1

0()) such that

*u"f. Taking in (A.1) the test function v"+u3», from the assumptiong3H0 (div; )) it is easy to see that (div q, f )"(g, +u)"0 for all f3¸2()), therefore,div q"0.

Take now v3H0(rot; ))WH(div; )) and denote by v

eits (¸2()))3-orthogonal

projection on H (e). The conditions S(g ) n)D!i, 1T!

i"0 and div g"0 imply that

g3H(e)o, hence we find

c(q, v)"c (q, v!ve)#c (q, v

e)

"(g, v!ve)"(g, v) ∀v3H

0(rot; ))WH(div; )).

We can take now a test function v3(C=0

()))3, and we have at once that rot rot q"g.

Setting w :"rot q, it is now easy to show that

rotw"rot rot q"g in ),

divw"div rot q"0 in ),

(w ) n)D)"(rot q ) n)D)"0 on ),

(w, lr)"(rot q, l

r)"(q, rotl

r)#S(n]q)D) , l

r D)T)"0 r"1,2, M.

The third equation is based on the fact that rot q3H0(div; )) provided that

q3H0(rot; )) (see [6, p. 35]). K

Acknowledgements

It is a pleasure to thank Paolo Fernandes for some useful conversations on thesubject of this paper.

References

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2. Alonso, A. and Valli, A., ‘A domain decomposition Approach for Heterogeneous Time-HarmonicMaxwell Equations’, Comput. Meth. Appl. Mech. Eng., 143, 97—112 (1997).

3. Bendali, A., Dominguez, J.M. and Gallic, S., ‘A variational approach for the vector potential formula-tion of the Stokes and Navier—Stokes problems in three-dimensional domains’, J. Math. Anal. Appl.,107, 537—560 (1985).

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menes successifs de bifurcations’, Ann. Scuola Norm. Sup. Pisa, 5 (IV), 29—63 (1978).6. Girault, V. and Raviart, P.-A., Finite Element Methods for Navier—Stokes Equations. ¹heory and

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Unique solvability for high-frequency heterogeneous time-harmonic Maxwell equations via the Fredholm alternative theory