Asymptotic formulas for perturbations in the ... - CiteSeerX

Asymptotic Analysis 30 (2002) 331–350 IOS Press

331

Asymptotic formulas for perturbations in the eigenfrequencies of the full Maxwell equations due to the presence of imperfections of small diameter Habib Ammari a,∗ and Darko Volkov b a Centre

de Mathématiques Appliquées, CNRS UMR 7641 & École Polytechnique, 91128 Palaiseau cedex, France E-mail: [email protected] b Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA E-mail: [email protected] Abstract. We consider electromagnetic media that consist of a homogeneous (constant coefficient) electromagnetic material with a finite number of small dielectric imperfections. For such media, we provide a rigorous derivation of asymptotic formulas for perturbations in the eigenfrequencies of the full Maxwell equations caused by the presence of the dielectric imperfections. Keywords: Maxwell’s equations, eigenvalues, asymptotics, small dielectric imperfections

1. The main result Let Ω be a bounded, open subset of R3 , with a smooth boundary. For simplicity we take ∂Ω to be C ∞ , but this regularity condition could be considerably weakened. We suppose that Ω contains a finite number (C ∞ ) domain of imperfections, each of the form zj + ρBj , where Bj ⊂ R3 is a bounded, smooth m containing the origin. The total collection of imperfections thus takes the form Iρ = j=1 (zj + ρBj ). The points zj ∈ Ω, j = 1, . . . , m, that determine the location of the imperfections are assumed to satisfy l, 0 < d0 |zj − zl | ∀j = 0 < d0 dist(zj , ∂Ω) ∀j.

(1)

As a consequence of this assumption it follows immediately that m 6|Ω|/πd30 We also assume that ρ > 0, the common order of magnitude of the diameters of the imperfections, is sufficiently small that these are disjoint and that their distance to R3 \ Ω is larger than d0 /2. Let µ0 > 0 and ε0 > 0 denote the permeability and the permittivity of the background medium; we shall assume *

Corresponding author.

0921-7134/02/$8.00  2002 – IOS Press. All rights reserved

332

H. Ammari and D. Volkov / Asymptotic formulas for perturbations

that these are positive constants. Let µj > 0 and εj > 0 denote the permeability and the permittivity of the j-th inhomogeneity, zj + ρBj ; these are also assumed to be positive constants. Using this notation we introduce the piecewise constant magnetic permeability µρ (x) =

0 µ ,

µj ,

x ∈ Ω \ I ρ, x ∈ zj + ρBj , j = 1, . . . , m.

(2)

If we allow the degenerate case ρ = 0, then the function µ0 (x) equals the constant µ0 . The piecewise constant electric permittivity, ερ (x) is defined analogously. The eigenvalue problem for the full Maxwell equations in the presence of imperfections consists of finding ωρ such that there exists a nontrivial electric field Eρ that is solution to

∇×

1 ∇ × Eρ = ωρ2 ερ Eρ , µρ

∇ · (ερ Eρ ) = 0 in Ω,

(3)

with Eρ × ν = 0 on ∂Ω.

(4)

Eqs (3), (4) may alternatively be formulated as follows ∇ × (∇ × Eρ ) = ωρ2 µ0 ε0 Eρ ,

∇ · (Eρ ) = 0

in Ω \ I ρ ,

∇ × (∇ × Eρ ) = ωρ2 µj εj Eρ ,

∇ · (Eρ ) = 0 in zj + ρBj ,

Eρ × ν is continuous across ∂(zj + ρBj ), 1 1 (∇ × Eρ )+ × ν − j (∇ × Eρ )− × ν = 0 on ∂(zj + ρBj ), 0 µ µ ε0 Eρ+ · ν − εj Eρ− · ν = 0

on ∂(zj + ρBj ),

Eρ × ν = 0 on ∂Ω. Here ν denotes the outward unit normal to ∂(zj + ρBj ) (and to ∂Ω); superscripts “+” and “−” indicate the limiting values as we approach ∂(zj + ρBj ) from outside zj + ρBj , and from inside zj + ρBj , respectively. It is well known that all eigenfrequencies of the Maxwell equations are real, of finite multiplicity, have no finite accumulation points, and there are corresponding eigenfunctions which make up an orthonormal basis of L2 (Ω), see [6,10]. These results are shown to be still valid for a large class of domains Ω and electromagnetic parameters (ε, µ), see [4]. We find it relevant to mention that it is not clear how to view the Maxwell operator as a self-adjoint operator in L2 (Ω) × L2 (Ω). Difficulties arise since the Maxwell system is not elliptic and is not semi-bounded. To overcome these difficulties, we embed the Maxwell operator in an elliptic boundary value problem using a Hodge decomposition, see [4,19]. Let ω0 be an eigenfrequency of multiplicity n for the Maxwell equations in the absence of any imperfections. Then there exist n nonzero solutions (E0i )ni=1 to

∇ × ∇ × E0i = ω02 µ0 ε0 E0i ,

∇ · E0i = 0 in Ω,

(5)


333

with E0i × ν = 0

on ∂Ω,

(6)

such that Ω

ε0 E0i · E0l = δil ,

∀i, l = 1, . . . , n,

(7)

where δil is the Kronecker symbol. Before our main results we need to introduce more notations. Let γ k , 0 k m, be a set of positive constants. In effect, {γ k } will either be the set {εk } or the set {µk }. For any fixed 1 j0 m, let γ denote the coefficient given by γ(x) =

0 γ ,

x ∈ R3 \ B j0 , x ∈ Bj0 .

γ j0 ,

(8)

φl , 1 l 3, denotes the solution to ∇y · γ(y)∇y φl = 0 in R3 , φl − yl → 0 as |y| → ∞. This problem may alternatively be written   ∆φl = 0 in Bj0 , and in R3 \ Bj0 ,      φl is continuous across ∂Bj0 ,

γ0

 (∂ φ )+ − (∂ν φl )− = 0  j0 ν l  γ   

on ∂Bj0 ,

φl (y) − yl → 0 as |y| → +∞.

It is therefore obvious that the function φl only depends on the coefficients γ 0 and γ j0 through the ratio c = γ 0 /γ j0 . The existence and uniqueness of this φl can be established using single layer potentials with suitably chosen densities. It is essential here, that the constant c, by assumption, cannot be 0 or a negative real number. We now define the polarization tensor, M j0 (c) of the inhomogeneity Bj0 (with aspect ratio c) j0 (c) = c−1 Mkl

Bj0

∂yk φl dy.

(9)

It is quite easy to see that the tensor Mkl (c) is symmetric; since c is a positive real number, it is furthermore positive definite, see [5,7]. Our main results are summarized in the following theorem. Theorem 1.1. Suppose ω0 is an eigenfrequency of multiplicity n for the Maxwell equations in absence of inhomogeneities, and let (E0i )ni=1 denote the corresponding eigenfunctions defined in (5)–(7). For ρ small enough there exist n eigenfrequencies (ωρi )ni=1 (counted according to multiplicity) for the Maxwell

334


equations (3), (4) in presence of small imperfections that converge to ω0 as ρ approaches zero. Then the following asymptotic formula pertaining to the convergence in average of the inverse of the squared eigenvalues of the perturbed problem, that is the convergence of, n1 ni=1 1/(ωρi )2 , holds:

m n n 0

1 1 ε0 1 3ε 1 − = ρ −1 n i=1 (ωρi )2 ω02 ω02 n i=1 j=1 εj

+ ρ3

m n 1 1 µ0 −1 4 ω0 µ0 n i=1 j=1 µj

Mj

0 µ

µj

M

j

0 ε

E0i (zj )

εj

· E0i (zj )

∇ × E0i (zj ) · ∇ × E0i (zj ) + O ρ4 .

(10)

The term O(ρ4 ) is bounded by Cρ4 , where the constant C depends on the eigenfrequency ω0 , the domains j j m (Bj )m j=1 , Ω, the constants d0 , (µ , ε )j=0 , but is otherwise independent of the location of the set of points m (zj )j=1 . The asymptotic formula (10) generalizes those in [2], where only two-dimensional Maxwell equations with TE (and TM) symmetries were considered and those derived by Ozawa [12–17] for the Laplace operator. In view of the asymptotic results derived in [1] a similar approach may be applied to the Lamé system in the presence of small elastic inhomogeneities of different Lamé coefficients. This will be discussed in a forthcoming paper. For reasons of brevity we restrict a significant part of the proof of our asymptotic formula to the case of a single inhomogeneity (m = 1). The proof in the case of multiple inhomogeneities may be derived by a fairly straightforward iteration of the arguments we present, however we leave the details to the reader. 2. Preliminary results Throughout this paper, we shall use quite standard L2 -based Sobolev spaces to measure regularity. The notation H s is used to denote those functions who along with all their derivatives of order less than or equal to s are in L2 . H01 denotes the closure of C0∞ in the norm of H 1 . Sobolev spaces with negative indices are in general defined by duality, using L2 -inner product. We shall only need one such space, namely H −1 , which is defined as the dual of H01 . The standard notation H(curl, Ω) is used to denote those −1/2 functions that, along with their curls, are in L2 (Ω). T Hdiv (∂Ω) denotes the space of tangential vector −1/2 fields on ∂Ω that lie in H −1/2 (∂Ω) and whose surface divergences also lie in H −1/2 (∂Ω). T Hcurl (∂Ω) −1/2 −1/2 denotes its dual: T Hcurl (∂Ω) = (T Hdiv (∂Ω)) , see [18]. Let us also introduce the two Hilbert spaces

Xρ (Ω) = u ∈ L2 (Ω), ∇ · (ερ u) = 0 in Ω , and

Zρ (Ω) = u ∈ L2 (Ω), ∇ × u ∈ L2 (Ω), ∇ · (ερ u) = 0 in Ω, u × ν = 0 on ∂Ω , which are respectively equipped with the norms

(u, v)Xρ (Ω) =

Ω

ερ u · v,


335

and

(u, v)Zρ (Ω) =

Ω

ερ u · v +

Ω

∇ × u · ∇ × v.

It will turn out to be convenient to first establish two auxiliary results. By adapting a compactness result, stated and proved in Proposition 3 in Appendix 1 from [3], the following compactness result for Zρ (Ω) holds. Lemma 2.1. Suppose (ρl ) is a sequence that converges to 0, or is constant, and suppose (ul ) is a sequence with ul ∈ Zρl (Ω), such that ul L2 (Ω) + ∇ × ul L2 (Ω) C. Then (ul ) has a strongly convergent subsequence in L2 (Ω). Proof. The boundary condition for the vector fields involved in [3] is that their normal component vanishes on the boundary. It can be easily realized that a similar compactness result hold for vector fields whose tangential component has to be zero. We leave the details to the reader. The second lemma asserts that the inner product equivalent to the Zρ (Ω)-norm.

1 Ω µρ ∇

× u · ∇ × v generates a norm which is

Lemma 2.2. The following inequalities hold for all u in Zρ (Ω): C1 u2Zρ (Ω)

Ω

1 |∇ × u|2 C2 u2Zρ (Ω) , µρ

(11)

where C1 and C2 are constants that are independent of ρ. Proof. It suffices to prove that there exists a constant C that is independent of ρ such that, for any u ∈ Zρ (Ω), uH(curl,Ω) C∇ × uL2 (Ω) .

(12)

We argue by contradiction. Suppose (12) fails to be true. Then there is a bounded sequence (ρl ) of positive numbers and a sequence (uρl ) in Zρl (Ω) such that ul L2 (Ω) l∇ × ul L2 (Ω) .

(13)

By extraction of a subsequence we can assume that ρl converges to a nonnegative number ρ∗ . It is also possible to assume that ||ul ||L2 (Ω) = 1. From Lemma 2.1 it follows that there is a subsequence of (ul ), also denoted by (ul ) for ease of notation, that is strongly convergent in L2 (Ω) and weakly convergent in H(curl, Ω) to a vector field u∗ with u∗ L2 (Ω) = 1. We infer from (13) that ∇ × u∗ = 0, thus u∗ is equal to a gradient of a function ϕ that exhibits the following properties ∇ · (ερ∗ ∇ϕ) = 0 in Ω

and

∇ϕ × ν = 0 on ∂Ω.

It follows that ∇ϕ = 0 which contradicts u∗ L2 (Ω) = 1.

336


We will need the following existence and uniqueness result. Lemma 2.3. For any f ∈ Xρ (Ω), there exists a unique uρ ∈ Zρ (Ω) solution of

∇×

1 ∇ × uρ = ερ f , µρ

∇ · (ερ uρ ) = 0

in Ω,

(14)

with uρ × ν = 0,

on ∂Ω.

(15)

Furthermore, uρ Zρ (Ω) Cf Xρ (Ω) ,

(16)

where the constant C is independent of ρ. Proof. The natural weak formulation of the boundary value problem (14), (15) is that uρ be in Zρ (Ω) and satisfy aρ (uρ , v) = lρ (v),

∀v ∈ Zρ (Ω),

(17)

where aρ (u, v) denotes the sesquilinear form

aρ (u, v) = Ω

1 ∇ × u · ∇ × v, µρ

and the continuous linear functional lρ is defined by lρ (v) = Ω ερ f · v. From (11) it follows immediately by the Lax–Milgram lemma that the variational problem (17) has a unique solution uρ ∈ Zρ (Ω) satisfying estimate (16). Let the continuous linear operator Rρ : X0 (Ω) → Xρ (Ω) be defined by Rρ (f ) = (1/ερ )f . Define the operator Tρ : Xρ (Ω) → Xρ (Ω) by Tρ (f ) = uρ where uρ is the unique solution of (14), (15) given by Lemma 2.3. We begin with establishing some properties of the operator Tρ . Lemma 2.4. The following properties of the operator Tρ hold. (a) Tρ : Xρ (Ω) → Xρ (Ω) is a self-adjoint compact positive operator; (b) Tρ is (uniformly) bounded; (c) The set (1/ωρ2 ), where (ωρ2 ) is the set of eigenfrequencies of the Maxwell equations (3), (4), is the set of eigenvalues of Tρ ; (d) For any uniformly bounded sequence fρ ∈ Xρ (Ω), there exists a subsequence fρ and f0 ∈ X0 (Ω) such that Tρ (fρ ) − Rρ (f0 )Xρ (Ω) → 0 as ρ → 0; (e) Tρ (fρ ) − Rρ T0 (f0 )Xρ (Ω) → 0 if fρ − Rρ f0 Xρ (Ω) → 0 for any fρ ∈ Xρ (Ω) and f0 ∈ X0 (Ω).


337

Proof. (a) it is easy to see that Tρ : Xρ (Ω) → Xρ (Ω) is a self-adjoint positive operator. Its compactness follows from the compact embedding of Zρ (Ω) into L2 (Ω), see, for example, [21]; (b) follows from estimate (16); (c) if Eρ ∈ Zρ (Ω) is a nonzero solution to the Maxwell equations (3), (4) then Tρ (Eρ ) = (1/ωρ2 )Eρ and so, 1/ωρ2 is an eigenvalue of Tρ . The converse is also immediate. Point (d) can be easily obtained by using once again estimate (16) and applying Lemma 2.1. The proof of point (e) is slightly more delicate. Since *ρ fρ is divergence free, the following equality is true for all v in H(curl, Ω) such that v × ν = 0 on ∂Ω

Ω

1 ∇ × Tρ (fρ ) · ∇ × v = µρ

Ω

*ρ fρ · v.

(18)

Thus, using Lemma 2.2, Tρ (fρ ) is bounded in H(curl, Ω). A subsequence Tρ (fρ ) converges weakly to an element g in H(curl, Ω), this convergence is strong in L2 (Ω) due to Lemma 2.1. At the limit we obtain

Ω

1 ∇×g·∇×v = µ0

Ω

f0 · v

(19)

which proves that g = R0 T0 (f0 ), which does not depend on the choice of the subsequence ρ . In addition, it is clear that Rρ T0 (f0 ) − R0 T0 (f0 ) tends to 0 in L2 (Ω), which concludes the proof of (e). Now, direct application of classical theorems [8, Theorems 11.4 and 11.5, pp. 343, 344] on spectral properties of a sequence of operators satisfying properties (a), (b), (d), and (e) that have been stated in Lemma 2.4 allow us in view of point (c) to describe in the following lemma the relative asymptotic properties and the deviation of the eigenfrequencies for the Maxwell equations (3), (4) in the presence of small imperfections to the eigenfrequencies for the homogeneous Maxwell equations (5), (6). Lemma 2.5. Let ω0 be an eigenfrequency of multiplicity n for (5), (6), with the basis of eigenfunctions (E0i )ni=1 : E0i ∈ X0 (Ω),

∇× ∇

× E0i

E0i × ν = 0

Ω

=

ω02 µ0 ε0 E0i ,

(20)

∇ · E0i = 0

in Ω,

(21)

on ∂Ω,

(22)

ε0 E0i · E0l = δil .

(23)

For ρ small enough there exist n eigenfrequencies (ωρi )ni=1 (counted according to multiplicity) of the Maxwell equations (3), (4) in the presence of small imperfections and n nonzero solutions (Eρi )ni=1 to: Eρi ∈ Xρ (Ω), ∇×

2 1 ∇ × Eρi = ωρi ερ Eρi , µρ

(24)

∇ · ερ Eρi = 0

in Ω,

(25)

Eρi × ν = 0 on ∂Ω,

(26)

ερ Eρi · Eρl = δil ,

(27)

Ω

such that, for any i = 1, . . . , m, ωρi → ω0 as ρ → 0.

338


3. Proof of the asymptotic formulas 3.1. The intermediate fields uiρ For i = 1, . . . , n, define uiρ to be the unique solution in Zρ (Ω) of

∇×

1 ∇ × uiρ = E0i , µρ

∇ · ερ uiρ = 0

in Ω,

(28)

with uiρ × ν = 0 on ∂Ω.

(29)

Notice that Tρ Rρ E0i = uiρ . Since T0 R0 E0i = (1/(ω02 *0 ))E0i it seems that uiρ − (1/(ω02 *0 ))E0i is small in norm. To obtain our main result, it is in fact crucial not only to prove that uiρ − (1/(ω02 *0 ))E0i is indeed small in norm but also to give a first order expansion of that difference. As stated earlier, we restrict our proof to the case of a single inhomogeneity (m = 1). We suppose that this inhomogeneity is centered at the origin, so it is of the form ρB. Assumption (1) simply becomes 0 ∈ Ω with 0 < d0 dist(0, ∂Ω). The general case may be verified by a fairly direct iteration of the argument we present here, adding one inhomogeneity at a time. Let q ∗ be the unique (scalar) solution to  ∗  ∆q = 0 in R3 \ B and in B,      ε0 ∂ν q ∗ + − ε1 ∂ν q ∗ − = − ε0 − ε1 E0i (0) · ν

on ∂B,

q ∗ is continuous across ∂B,     ∗   lim q (y) = 0. |y|→+∞

The reader is referred to [3] for the existence and uniqueness of q ∗ . The following asymptotic estimate can be derived for ∇q ∗ , see [3]:

1 . ∇q (y) = O |y|3 ∗

= Ω/ρ in the rescaling y = x/ρ. We will need the approximation q ∗ of q ∗ in Ω defined as Denote Ω ρ follows:  ∗ \ B and in B, ∆q = 0 in Ω    ρ   q ∗ is continuous across ∂B, ρ + − 0  ∇qρ∗ · ν − ε1 ∇qρ∗ · ν = − ε0 − ε1 E0i (ρy) · ν ε     ∗ q (y) = q ∗ (y) on ∂Ω. ρ

across ∂B,


339

We form an equation for the difference (qρ∗ − q ∗ )( xρ ) in order estimate it. We have:  ∗ ∆ qρ − q ∗ = 0 in ρB and in Ω \ ρB,      q ∗ − q ∗ is continuous across ∂(ρB),    ρ0 +  ∗ ∗ 1 ∗

∇y q ρ − ∇y q ∗ · ν

∇y q ρ − ∇y q · ν − ε   across ∂(ρB),     x    qρ∗ − q ∗ = 0 on ∂Ω. ρ ε

−

= − ε0 − ε1 E0i (ρy) − E0i (0) · ν

(30)

Since (qρ∗ − q ∗ )( xρ ) lies in H01 (Ω) and, recalling that E0i is divergence free, we get the following identity −

2 ε ∇ qρ∗ − q ∗ − 0

Ω\ρB

= ρB

ρB

2 ε1 ∇ qρ∗ − q ∗

(31)

0 ε − ε1 E0i (x) − E0i (0) · ∇ qρ∗ − q ∗ .

Since E0i is a smooth vector field, the supremum of (E0i (x) − E0i (0)) for x in ρB is of order ρ thus the right-hand side of (31) is of order ρ5/2 . Consequently, the following lemma holds. Lemma 3.1. ∗ ∇y q − q ∗ x ρ ρ

L2 (Ω)

Cρ5/2 .

(32)

Now, let S denote the space

S = u:

u 2 3 2 3 ∈ L R , ∇ × u ∈ L R , ∇ · u = 0 in R3 \ B, ∇ · u = 0 in B 2 1/2 (1 + r )

and ε0 (u · ν)+ = ε1 (u · ν)− across ∂B .

We introduce the vector field e∗ by the requirements that e∗ ∈ S and R3 \B

1 ∇ × e∗ · ∇ × v + µ0

B

1 ∇ × e∗ · ∇ × v = µ1

1 1 − µ0 µ1

∂B

∇ × E0i (0) × ν · v,

for any v ∈ S. For existence and uniqueness of such a vector field we refer the reader to [3] where it is also proven that e∗ satisfies the following properties   ∆e∗ = 0 in R3 \ B and in B,      ∇ · e∗ = 0 in R3 \ B and in B,      1  1 ∗ + ∗ −

1 1 ∇×e ×ν − 1 ∇×e × ν = − 0 − 1 ∇ × E0i (0) × ν 0 µ µ µ µ + −    across ∂B, ε0 e∗ · ν = ε1 e∗ · ν     ∗ × ν is continuous across ∂B,  e     ∗ e (y) = O |y|−1 uniformly as |y| → +∞.

across ∂B,

340


It is also explained in [3] how the following two asymptotic estimates can be derived: e∗ (y) = O

1 , |y|2

∇ × e∗ (y) = O

1 |y|3

uniformly as |y| → +∞.

(33)

by: We will need to use the field e∗ρ (y) defined for y in Ω  ∗   ∆eρ = 0 in Ω \ B and in B,    \ B and in B,  ∇ · e∗ρ = 0 in Ω      1 1 1 1  ∗ + ∗ − i   0 ∇ × eρ × ν − 1 ∇ × eρ × ν = − − ∇ × E (0) ×ν 0 µ µ µ0 µ1 + −  across ∂B, ε0 e∗ρ · ν = ε1 e∗ρ · ν     ∗ × ν is continuous across ∂B,  e  ρ     1    e∗ (y) × ν = − ∇q ∗ (y) × ν on ∂Ω. ρ

ρ

across ∂B,

ρ

Back to the x variable, (e∗ρ − e∗ )( xρ ) satisfies in Ω:  ∗ ∗ 0 in ρB and in Ω \ ρB,   ∆ eρ − e =   ∗ ∗   ∇ · eρ − e = 0 in ρB and in Ω \ ρB,     ∗ ∗ 1  1 ∗ + ∗ − 0

∇ × eρ − e

×ν−

1

∇ × eρ − e

× ν = 0 across ∂(ρB),

(34)

µ µ  −  0 ∗ ∗ + 1 ∗  e − e · ν = ε e − e∗ · ν across ∂(ρB), ε  ρ ρ       e∗ − e∗ × ν = −e∗ × ν − 1 ∇q ∗ x × ν on ∂Ω.  ρ

ρ

ρ

ρ

In order to estimate e∗ρ − e∗ we need the following lemma. −1/2

Lemma 3.2. There is a constant C depending only on Ω such that for all w in T Hcurl (∂Ω) there is a vector field w in H(curl, Ω) that satisfies ν × (ν × w) = w on ∂Ω together with w

H(curl,Ω)

CwT H −1/2 (∂Ω) curl

and ∇ · (ερ w) = 0. Proof. This is a well known result except for the “∇ · (ερ w) = 0” requirement. Let us then pick an extension w 2 of w that does not necessarily satisfy this last requirement. Then find q in H01 (Ω) such that

∇ · (ερ ∇q) = ∇ · ερ w 2 . Necessarily

∇qL2 (Ω) w2 L2 (Ω) . It suffices then to set w = w2 − ∇q.


341

Analogously to Lemma 3.1 we have: Lemma 3.3. (e∗ρ − e∗ ) is well defined in H(curl, Ω) and satisfies ∗ e − e∗ x ρ ρ

Cρ3/2 .

H(curl,Ω)

Proof. We first prove that ν × e∗ x ρ

−1/2

T Hdiv

and (∂Ω)

1 ν × ∇q ∗ −1/2 ρ TH (∂Ω) ρ div

are of order ρ2 and ρ3/2 , respectively. Let S be a sphere strictly containing Ω. There is a constant C such that any smooth tangential vector field v on ∂Ω can be extended as a smooth vector field in S \ Ω such that ν × v = 0 on ∂S and vL2 (S\Ω) + ∇ × vL2 (S\Ω) Cv × νT H −1/2 (∂Ω) . div

Then an integration by parts yields

∗ ∗ x ∗ x e × ν · ν × (ν × v) ∇x × e · v + e · ∇x × v ρ ρ ∂Ω S\Ω S\Ω 2 1/2 2 1/2 ∗ x ∇x × e∗ x dx e dx C + v × νT H −1/2 (∂Ω) . ρ ρ div S\Ω

S\Ω

But R3 \Ω

2 1/2 1/2 ∇x × e∗ x dx =ρ ρ

R3 \Ω

1/2 ∇y e∗ (y)2 dy Cρ2 ,

and R3 \Ω

2 1/2 ∗ x 3/2 e dx =ρ ρ

R3 \Ω

1/2 ∗ 2 e (y) dy Cρ2 ,

which shows that ν × e∗ x ρ

−1/2 T Hdiv (∂Ω)

Cρ2 .

Now we want to estimate the term ρ1 ν × ∇y qρ∗ T H −1/2 (∂Ω) . We start out by estimating div

1 ν × ∇y q ∗ (y) −1/2 . T Hdiv (∂Ω) ρ

342


An integration by parts yields

∇y q (y) × ν · (ν × ν × v) ∗

S\Ω

∂Ω

∇y q

C

S\Ω

∗

x ∇x × v ρ

2 1/2 ∇y q ∗ x dx v × νT H −1/2 (∂Ω) . ρ div

But R3 \Ω

2 1/2 3/2 ∇y q ∗ x dx =ρ ρ

R3 \Ω

1/2 ∇y q ∗ (y)2 dy Cρ3 ,

which shows that 1 ν × (∇y q) x ρ ρ

−1/2

T Hdiv

Cρ2 .

(35)

(∂Ω)

To estimate ρ1 ν × ∇y (qρ − q)(y)T H −1/2 (∂Ω) , we extend this time the tangential field v in Ω instead div

of S \ Ω. Using an integration by parts followed by (32), we can write

∂Ω

∗ ∗ ∇y qρ − q (y) × ν · (ν × ν × v)

∗ ∗ x ∇y q ρ − q ∇x × v ρ Ω

1/2 ∗ x 2 ∇y q − q ∗ dx C v × νT H −1/2 (∂Ω) Cρ5/2 v × νT H −1/2 (∂Ω) . ρ ρ div div

(36)

Ω

Combining (35) with (36) we conclude 1 ν × ∇y q ∗ x ρ ρ ρ

−1/2 T Hdiv (∂Ω)

Cρ3/2 .

Next pick lρ in H(curl, Ω) such that ∇ · (ερ lρ ) = 0, and satisfying

lρ (x) × ν = − e∗

x ρ

1 x + ∇qρ∗ ρ ρ

×ν

on ∂Ω,

and lρ L2 (Ω) + ∇ × lρ L2 (Ω) Cρ3/2 .

(37)

We can then set the following definition for (e∗ρ − e∗ ): find (e∗ρ − e∗ − lρ ) in Zρ (Ω) such that Ω

1 ∇ × e∗ρ − e∗ − lρ · ∇ × w = − µρ

Ω

∇ × lρ · ∇ × w,


343

for all w in Zρ (Ω). It can easily be verified that ∇ × ( µ1ρ ∇ × (e∗ρ − e∗ )) = 0 inside Ω and that ∗ 1 x eρ − e∗ × ν = − e∗ + ∇qρ∗ ×ν ρ ρ

on ∂Ω.

From Lemma 2.3 and (37) we infer ∗ ∗ ∗ 3/2 e − e∗ 2 2 , ρ L (Ω) + ∇x × eρ − e L (Ω) Cρ

(38)

which is what Lemma 3.3 is claiming.

We now proceed to construct an asymptotic expansion of uiρ . Set

Rρ (x) = uiρ (x) −

1 x E0i (x) + ρe∗ρ ρ ω02 ε0

+ ∇y qρ∗

x

ρ

.

Notice that Rρ lies in the space Zρ (Ω). For any v in Zρ (Ω), we can perform the following integration by parts: Ω

1 (∇ × Rρ ) · (∇ × v) = − µρ

∂ρB

+ ∂ρB

ρB

1 = 2 0 ω0 ε

+

1 1 x ∇ × uiρ (x) − 2 0 E0i (x) + ρe∗ρ 1 µ ρ ω0 ε

×ν

+

×ν

1 1 x ∇ × uiρ (x) − 2 0 E0i (x) + ρe∗ρ 0 µ ρ ω0 ε

·v −

·v

µ0 1 1 − E0i · v 2 µ1 ω0 ε0

ρB

1 1 − 1 ∇ × E0i (x) − ∇ × E0i (0) · ∇ × v. 0 µ µ

(39)

Since E0i is a smooth function in Ω, (∇ × E0i (x) − ∇ × E0i (0)) is of order ρ for x in ρB and therefore the last term in (39) is of order ρ5/2 . Due to Lemma 2.3 we derive, Rρ H(curl,Ω) Cρ5/2 . Recalling (38) and (32), we obtain i i ∗ x ∗ x u (x) − 1 E0 (x) + ρe + ∇y q Cρ5/2 , ρ 2 0 ρ ρ L2 (Ω) ω0 ε

(40)

i ∗ x ∇x × ui (x) − 1 E (x) + ρe Cρ5/2 . ρ 0 2 2 0 ρ ω0 ε L (Ω)

(41)

and

Recalling (33), (40) can be rewritten as follows.

344


Lemma 3.4. There exists a constant C that is independent of ρ and the set of points (zj )m j=1 such that i i ∗ x u (x) − 1 E (x) + ∇ q Cρ5/2 . y 0 ρ 2 0 ρ L2 (Ω) ω0 ε

(42)

Approximating the two following integrals in the lemma below will prove to be useful in presenting our final result in a form utilizing polarization tensors rather than integrals involving uiρ . Lemma 3.5. Let uiρ be the unique solution of (28), (29). The following asymptotic formulas hold:

zj +ρBj

0 ρ3 1 j ε M E0i (zj ) · E0i (zj ) + O ρ4 , 2 j j ε ω0 ε

uiρ · E0i =

(43)

and

zj +ρBj

∇ × uiρ · ∇ × E0i =

0 ρ3 1 j µ M ∇ × E0i (zj ) · ∇ × E0i (zj ) + O ρ4 . 2 0 j µ ω0 ε

(44)

Here, the terms O(ρ4 ) are bounded by Cρ4 , where the constant C is independent of ρ and the set of points (zj )m j=1 . Proof. Upon insertion of (42) we get ρB

uiρ · E0i = ρ3

B

uiρ (ρy) · E0i (ρy) dy = ρ3

1 = ρ 2 0 |B|E0i (0) · E0i (0) + ω0 ε

1 i E0 (ρy) + ∇y q ∗ (y) dy · E0i (0) + O ρ4

2 0 B ω0 ε

∗

∇y q (y) dy ·

3

B

E0i (0)

+ O ρ4 .

· E0i (0)

ε0 ε0 = 1 M 1 E0i (0) · E0i (0). ε ε

From the definition of the function q ∗ it follows immediately that ψ(y) = q ∗ (y) + E0i (0) · y is the unique solution to  ∆ψ = 0 in B, and in R3 \ B,       ψ is continuous across ∂B, 0

ε   (∂ ψ)+ − (∂ν ψ)− = 0  1 ν  ε   i

on ∂B,

ψ(y) − E0 (0) · y → 0 as |y| → +∞,

and therefore |B|E0i (0)

·

E0i (0)

∗

+ B

∇y q (y) dy

· E0i (0)

= B

∇y ψ(y) dy


345

Here M (ε0 /ε1 ) is the polarization tensor defined by (9). Similarly, we obtain ρB

∇×

= ρ3

uiρ

·∇

× E0i

3

=ρ

B

∇ × uiρ (ρy) · ∇ × E0i (ρy) dy

1 ∇ × E0i (ρy) + ∇ × e∗ (y) dy · ∇ × E0i (0) + O ρ4

2 0 B ω0 ε

= ρ3 = ρ3

1 |B|∇ × E0i (0) · ∇ × E0i (0) + 2 ω0 ε0 1

ω02 ε0

M

0 µ

B

∇ × e∗ (y) dy · ∇ × E0i (0) + O ρ4

∇ × E0i (0) · ∇ × E0i (0) + O ρ4 .

µ1

The reader is referred to Section 7 in [3] for more details on these calculations.

3.2. Proof of main result Lemma 3.6. Let Pρi , 1 i n, denote the orthogonal projection in Xρ (Ω) on the eigenspace of Tρ corresponding to the eigenvalue 1/(ωρi )2 , let P0 denote the orthogonal projection in X0 (Ω) on the eigenspace of T0 corresponding to the eigenvalue 1/ω02 , and denote Pρ = ni=1 Pρi . We then have the following estimate relating Pρ to P0 Pρ Rρ (f ) − P0 R0 (f )

L2 (Ω)

Cρ3/2 f X0 (Ω)

(45)

for all f in X0 (Ω). Proof. Let Γ be the circle in the complex plane centered at ω02 of radius δ and δ is small enough so that the Maxwell equations (5)–(7) have no other eigenvalue inside Γ . For z ∈ Γ , let wρ be the unique solution in Zρ (Ω) to the following equations:  1   ∇ × ∇ × w  ρ − zερ wρ = f 

µρ w ∇ · (ε ρ ρ ) = 0 in Ω,    

in Ω,

wρ × ν = 0 on ∂Ω,

where f ∈ X0 (Ω). Note that

(I − zTρ )wρ = Tρ

f . *ρ

Analogously, we define for any f ∈ X0 (Ω) and z ∈ Γ , w0 as the unique solution in Z0 (Ω) to the following equations:  1 0     ∇ × µ0 ∇ × w0 − zε w0 = f ∇ · ε0 w0 = 0 in Ω,    

w0 × ν = 0 on ∂Ω.

in Ω,

346


The following estimate is derived in the appendix wρ − w0 H(curl,Ω) Cρ3/2 f L2 (Ω) ,

(46)

uniformly in z ∈ Γ . The following calculation can be justified using classical results on symmetric compact operators as developed in [9] for example. Provided that 0 < δ < ω02 the image of Γ under the conformal mapping of the complex plane z → 1/z is the circle Γ centered at ω02 /(ω04 − δ2 ) and of radius δ/(ω04 − δ2 ). Note that 1/ω02 is on the inside of Γ , and that this mapping from Γ to Γ reverses the orientation of parameterization. Setting ζ = 1/z, for ρ small enough, 1 2iπ

wρ dz = Γ

=

1 2iπ

Γ

n

(I − zTρ )−1 Tρ

2

ωρi Pρi Tρ

i=1

f *ρ

=

1 f dz = − *ρ 2iπ

n

2

ωρi Tρ Pρi

i=1

f *ρ

Γ

1 f (ζI − Tρ )−1 Tρ dζ ζ *ρ

= Pρ

f . *ρ

(47)

We have used the fact that Tρ and Pρi commute. Similarly 1 2iπ

Γ

w0 dz = ω02 P0 T0

f *0

= P0

Combining (46)–(48) we arrive at (45).

f . *0

(48)

The usual way to get to the average of the 1/(ωρi )2 is to introduce linear operators on a finite dimensional space and use the fact that the trace is independent of the choice of a basis. Such a method is put into practice in [11] for example. We can claim in view of (45) that for ρ small enough Pρ Rρ is one to one from R(P0 ) into R(Pρ ), where R denotes the range. But these two finite dimensional spaces have the same dimension, thus Pρ Rρ is an isomorphism from R(P0 ) onto R(Pρ ). Set Tρ = (Pρ Rρ )−1 Tρ Pρ Rρ

and

T = T0|R(P0 )

The trace of a finite dimensional linear operator is independent of the choice of a basis. Consequently n n

1 1 1 1 0 − Tρ = 1 trace − = T T0 − Tρ E0i , E0i X0 (Ω) 2 i 2 n i=1 (ωρ ) n n i=1 ω0

=

n 1 1 i i E ,E n i=1 ω02 0 0

X0 (Ω)

− Tρ E0i , E0i X0 (Ω) .

(49)

We now want to introduce uiρ in Tρ E0i , E0i X0 (Ω) . We first split the latter in two terms, the second of these two terms will be proved to be of lower order. For ease of notation we only treat the case of a single


347

inhomogeneity centered at 0.

Tρ E i , E i 0

0 X0 (Ω)

= Ω

= Ω

*0 (Pρ Rρ )−1 Tρ Pρ Rρ E0i · E0i

*0 Rρ−1 Tρ Rρ E0i · E0i +

*0 (Pρ Rρ )−1 Tρ Pρ Rρ − Rρ−1 Tρ Rρ E0i · E0i .

Ω

(50)

Recalling that Tρ Rρ E0i = uiρ we have Ω

Rρ−1 Tρ Rρ E0i · E0i =

1 = 2 ω 0 µ0 1 = 2 ω0 =

1 ω02

Ω

Ω

Ω

Ω

*ρ uiρ · E0i = *0

Ω

i

uiρ · E0i + *1 − *0

uiρ · ∇ × ∇ × E0 + *1 − *0

ρB

1 1 1 1 ∇ × uiρ · ∇ × E0i + 2 − 1 0 µρ µ ω0 µ

E0i · E0i +

1 1 1 − 1 2 0 µ ω0 µ

ρB

ρB

uiρ · E0i

uiρ · E0i

ρB

∇×

uiρ

·∇×

E0i

∇ × uiρ · ∇ × E0i + *1 − *0

+ *1 − *0

ρB

ρB

uiρ · E0i

uiρ · E0i .

Finally we want to estimate the order of

Aρ = Ω

*0 (Pρ Rρ )−1 Tρ Pρ Rρ − Rρ−1 Tρ Rρ E0i · E0i .

Since Tρ and Pρ commute as linear operators of Xρ (Ω), this term is equal to Ω

*0 (Pρ Rρ )−1 Pρ − Rρ−1 uiρ · E0i .

Since uiρ = Rρ *ρ uiρ , Aρ is again equal to Ω

*0 (Pρ Rρ )−1 Pρ Rρ − I *ρ uiρ · E0i .

But since P0 is an orthogonal projection and E0i lies in the range of P0 , the following equality holds Ω

*0 (P0 − I)*ρ uiρ · E0i = 0.

Therefore

Aρ =

Ω

*0 (Pρ Rρ )−1 Pρ Rρ − P0 *ρ uiρ · E0i .

We use again the fact that P0 is an orthogonal projection to write Ω

*0 (I − P0 )(Pρ Rρ )−1 Pρ Rρ *ρ uiρ · (Pρ Rρ )−1 Pρ Rρ *ρ uiρ = 0.

(51)

348


Since P0 = P0 (Pρ Rρ )−1 Pρ Rρ we rewrite Aρ as Ω

*0 (Pρ Rρ )−1 Pρ Rρ − P0 *ρ uiρ · E0i − ω02 (Pρ Rρ )−1 Pρ Rρ *ρ uiρ .

Finally, remarking that ((Pρ Rρ )−1 Pρ Rρ − P0 )E0i = 0, Aρ is equal to Ω

*0 (Pρ Rρ )−1 Pρ Rρ − P0

*ρ uiρ −

E0i ω02

· E0i − ω02 (Pρ Rρ )−1 Pρ Rρ *ρ uiρ .

We notice that for all f in X0 (Ω)

(Rρ − R0 )P0 (f )

L2 (Ω)

= ρB

1 1 − *1 *0

2

1/2 P0 (f )2 .

(52)

But all the elements in the space R(P0 ) are smooth and any of their norm is bounded by the L2 norm. We infer that (Rρ − R0 )P0 (f )

L2 (Ω)

Cρ3/2 P0 (f )L2 (Ω) Cρ3/2 f L2 (Ω) .

Now as a consequence of (45) we may write for all f in X0 (Ω) Pρ Rρ (f ) − Rρ P0 (f )

L2 (Ω)

Cρ3/2 f X0 (Ω) .

(53)

Since the operators Pρ are uniformly bounded in Xρ (Ω) and Pρ2 = Pρ , it follows that Pρ Rρ (f ) − Pρ Rρ P0 (f )

L2 (Ω)

Cρ3/2 f X0 (Ω) .

(54)

Finally by definition of (Pρ Rρ )−1 we can write the following estimate (Pρ Rρ )−1 Pρ Rρ (f ) − P0 (f )

L2 (Ω)

Cρ3/2 f X0 (Ω) .

(55)

We are now ready to estimate the order of the term Aρ . First remember that we have proved i *ρ ui − E0 ρ 2 ω 0

Cρ3/2 .

(56)

X0 (Ω)

Combining (55) and (56), we find i (Pρ Rρ )−1 Pρ Rρ − P0 *ρ ui − E0 ρ 2 ω 0

L2 (Ω)

Cρ3 .

(57)


349

Next i E − ω 2 (Pρ Rρ )−1 Pρ Rρ *ρ ui 2 0 0 ρ L (Ω) P0 E i − ω 2 *ρ ui 2 + (Pρ Rρ )−1 Pρ Rρ − P0 ω 2 *ρ ui 0

ρ

0

L (Ω)

C E0i − ω02 *ρ uiρ L2 (Ω) + Cρ3/2 *ρ uiρ L2 (Ω) Cρ3/2 .

0

ρ L2 (Ω)

(58)

Combining (57) and (58) we find the estimate |Aρ | Cρ9/2 . We conclude using (49)–(51) that n 1 1 1 − 2 n i=1 (ωρi )2 ω0

=

n 1 1 *0 1 − 0 2 1 n i=1 ω0 µ µ

ρB

∇ × uiρ · ∇ × E0i + *0 − *1 *0

ρB

uiρ · E0i + O ρ9/2 .

(59)

Finally, Lemma 3.5 applied to (59) yields our main result. Appendix We begin by deriving two estimates from [3]. In this paragraph we will make references to [3] and use the same notations, that is in this paragraph only, the notations from [3] will overrule the current notations of this paper. Note that the function q lies in the space R. It follows from the definition of q and an integration by parts that R3 \B

0

µ ∇q

2

+ B

1

µ ∇q

2

= B

0 µ − µ1 ∇q · H0 (0)

consequently ∇q

L2 (R3 )

C H0 (0)

and using the first estimate in Proposition 1 in [3]

Hρ − H0 L2 (Ω) C H0 (0)ρ3/2 , where C does not depend on the boundary data g. The results from Section 3 of [3] can be carried out to our case although we now have a tangential zero boundary condition instead of the normal zero boundary condition in [3]. For ease of notation, we assume that there only is a single inhomogeneity centered at 0. Supposing that f is smooth, which makes w0 smooth, we can construct two auxiliary fields akin to those from Section 5 of [3]. We then arrive at the equivalent of the two estimates in Proposition 1 in [3] and just as shown above, we derive

wρ − w0 L2 (Ω) Cρ3/2 w0 (0).

(60)

350


The H 2 (Ω) norm of w0 depends only on f L2 (Ω) . Recall that H 2 (Ω) is continuously embedded in C 0 (Ω) since ∂Ω was chosen to be smooth. We derive from (60) wρ − w0 L2 (Ω) Cρ3/2 f L2 (Ω) .

(61)

Now estimate (61) can be inferred by density if f has only an X0 (Ω) regularity. Acknowledgements This work is partially supported by ACI Jeunes Chercheurs (0693) from the Ministry of Education and Scientific Research, France. References [1] C. Alves and H. Ammari, Boundary integral formulae for the reconstruction of imperfections of small diameter in an elastic medium, SIAM J. Appl. Math. 62 (2002), 94–106. [2] H. Ammari and S. Moskow, Asymptotic expansions for eigenvalues in the presence of small inhomogeneities, Math. Methods Appl. Sci., submitted. [3] H. Ammari, M. Vogelius and D. Volkov, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of imperfections of small diameter. II. The full Maxwell equations, J. Math. Pures Appl. 80 (2001), 769–814. [4] M.Sh. Birman and M.Z. Solomyak, L2 theory of the Maxwell operator in an arbitrary domain, Russian Math. Surveys 42 (1987), 75–96. [5] D.J. Cedio-Fengya, S. Moskow and M. Vogelius, Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction, Inverse Problems 14 (1998), 553–595. [6] R. Dautray and J.L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 3, Spectral Theory and Applications, Springer, Berlin, 1990. [7] A. Friedman and M. Vogelius, Identification of small imperfections of extreme conductivity by boundary measurements: a theorem on continuous dependence, Arch. Rational Mech. Anal. 105 (1989), 299–326. [8] V.V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization of Differential Operators and Integral Functions, Springer, 1994. [9] T. Kato, Perturbation Theory for Linear Operators, Springer, 1980. [10] R. Leis, Initial Boundary Value Problems in Mathematical Physics, Teubner and Wiley, Stuttgart, 1986. [11] J.E. Osborn, Spectral approximations for compact operators, Math. Comp. 29 (1975), 712–725. [12] S. Ozawa, Eigenvalues of the Laplacian under singular variation of domains – the Robin problem with obstacle of general shape, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996), 124–125. [13] S. Ozawa, Asymptotic property of an eigenfunction of the Laplacian under singular variation of domains – the Neumann condition, Osaka J. Math. 22 (1985), 639–655. [14] S. Ozawa, Spectra of domains with small spherical Neumann boundary, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1983), 259–277. [15] S. Ozawa, An asymptotic formula for the eigenvalues of the Laplacian in a three-dimensional domain with a small hole, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1983), 243–257. [16] S. Ozawa, Electrostatic capacity and eigenvalues of the Laplacian, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1983), 53–62. [17] S. Ozawa, Singular variation of domains and eigenvalues of the Laplacian, Duke Math. J. 48 (1981), 767–778. [18] L. Paquet, Probèmes mixtes pour le système de Maxwell, Ann. Fac. Sci. Toulouse 4 (1982), 103–141. [19] R. Picard, On the boundary value problems of electro- and magnetostatics, Proc. Roy. Soc. Edinburgh 92 (1982), 165–174. [20] M. Vogelius and D. Volkov, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities, Math. Model. Numer. Anal. 34 (2000), 723–748. [21] D. Volkov, An inverse problem for the time harmonic Maxwell equations, Ph.D. Thesis, Rutgers University, New Brunswick, NJ, 2001.

Asymptotic formulas for perturbations in the ... - CiteSeerX

Asymptotic formulas for perturbations in the ... - CiteSeerX

Suggest Documents

Asymptotic equivalence in Lee's moment formulas for the implied ...

Asymptotic Formulas for the Eigenvalues of the Timoshenko Beam

Perturbations of the Asymptotic Region of the Schwarzschild ... - Core

Asymptotic Completeness and S-Matrix for Singular Perturbations

Asymptotic formulas for elliptic integrals by John L. Gustafson

Asymptotic formulas for coefficients of Kac-Wakimoto characters

On asymptotic effects of boundary perturbations in exponentially ...

ASYMPTOTIC BEHAVIOUR FOR THE NONLINEAR ... - CiteSeerX

General Smoothing Formulas for Markov-Modulated ... - CiteSeerX

Control-Lyapunov Universal Formulas for Restricted ... - CiteSeerX

ON CHARACTERS AND DIMENSION FORMULAS FOR ... - CiteSeerX

Explicit formulas for local Mellin transforms - CiteSeerX

Convenient formulas for quantization efficiency - CiteSeerX

CORRELAnON BETWEEN DIFFERENT FORMULAS FOR ... - CiteSeerX

Availability Formulas and Performance Measures for ... - CiteSeerX

DIMENSION FORMULAS FOR THE FREE

Mixing formulas in time domain - CiteSeerX

How Perturbations to the Freshwater Environment ... - CiteSeerX

The complexity of symmetry-breaking formulas - CiteSeerX

ASYMPTOTIC THEORY FOR LOCAL TIME DENSITY ... - CiteSeerX

sufficient conditions for polynomial asymptotic behaviour ... - CiteSeerX

ASYMPTOTIC THEORY FOR ZERO ENERGY ... - CiteSeerX

ASYMPTOTIC THEORY FOR ZERO ENERGY ... - CiteSeerX

ASYMPTOTIC BEHAVIOR FOR A SEMILINEAR ... - CiteSeerX