We consider solutions to the time-harmonic Maxwell problem in $\R^3$. For such solution we provide a rigorous derivation of the asymptotic expansions in the practically interesting situation, where a finite number of inhomogeneities of small diameter are imbedded in the entire space. Then, we describe the behavior of the electromagnetic energy caused by the presence of these inhomogeneities

Abdessatar Khelifi

Christian Daveau

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 346 (2008) 287–292http://france.elsevier.com/direct/CRASS1/Partial Differential EquationsOn the perturbation of the electromagnetic energydue to the presence of small inhomogeneitiesChristian Daveau a, Abdessatar Khelifi ba Université de Cergy-Pontoise, département de mathématique, CNRS UMR 8088, 2, avenue Adolphe-Chauvin, 95302 Cergy-Pontoise, Franceb Département de mathématiques, université des sciences de Carthage, Bizerte, 7021, TunisiaReceived 12 September 2007; accepted after revision 29 January 2008Presented by Olivier PironneauAbstractWe consider solutions to the time-harmonic Maxwell problem in R3. For such solution we propose a rigorous derivation ofthe asymptotic expansions in the interesting practical situation when a finite number of inhomogeneities of small diameter areembedded in the entire space. Then, we describe the behavior of the electromagnetic energy caused by the presence of theseinhomogeneities. To cite this article: C. Daveau, A. Khelifi, C. R. Acad. Sci. Paris, Ser. I 346 (2008).© 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.RésuméSur la perturbation de l’énergie électromagnétique due à la présence d’inhomogénéités de petits diamètres. Nous consi-dérons des solutions des équations de Maxwell dans R3 en présence d’un nombre fini d’inhomogénéités de petits diamètres. Pourde telles solutions, nous obtenons des formules asymptotiques rigoureuses. Puis, nous décrivons le comportement de l’énergieélectromagnétique. Pour citer cet article : C. Daveau, A. Khelifi, C. R. Acad. Sci. Paris, Ser. I 346 (2008).© 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Version française abrégéeL’objectif du travail décrit dans cette Note est de comprendre comment les solutions perturbées du probleme (3)se comportent quand un nombre fini d’inhomogénéités {zj + αBj } de petits diamètres sont introduites dans l’espaceentier R3. Ceci nous amène, également, à étudier l’évolution de l’énergie électromagnétique correspondante seloncette déformation du milieu de propagation.On suppose que dans R3 on a m inhomogénéités {zj + αBj }mj=1, où α est un petit paramètre, Bj ⊂ R3 est unouvert borné de diamètre borné indépendamment de j et les points {zj }mj=1 vérifient l’hypothèse (1). On suppose quela perméabilité magnétique μα et la permittivité électrique εα vérifient (2). Nous commençons notre analyse dans lasection 2 en obtenant rigoureusement des développements asymptotiques des champs électrique et magnétique, avecdes estimations d’erreurs uniformes dans R3. Ces formules asymptotiques sont construites par la méthode des déve-E-mail addresses: christian.daveau@math.u-cergy.fr (C. Daveau), abdessatar.khelifi@fsb.rnu.tn (A. Khelifi).1631-073X/$ – see front matter © 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2008.01.019288 C. Daveau, A. Khelifi / C. R. Acad. Sci. Paris, Ser. I 346 (2008) 287–292loppements asymptotiques raccordés. Concernant cette méthode, le lecteur peut consulter [7,10]. En domaine borné,on peut voir aussi les travaux [1,3,5]. Le terme d’ordre principal dans des développements asymptotiques analogues(mais pour le cas d’un domaine borné) a été obtenue par Vogelius et Volkov [14] et Ammari et al. [4]. Nos formulesasymptotiques utilisent des tenseurs de polarisation associés à des inhomogénéités électromagnétiques qui semblentêtre des généralisations naturelles des tenseurs qui ont été présentés par Schiffer et Szegö [13] et complètement étudiéspar beaucoup d’autres auteurs [2,6,8,11].1. Problem formulationWe suppose that there is a finite number of electromagnetic inclusions in R3, each of the form zj + αBj , whereBj ⊂ R3 is a bounded and C∞-domain containing the origin. The total collection of inhomogeneities is:Bα =m⋃j=1{zj + αBj }.The points zj ∈ R3, j = 1, . . . ,m, which determine the location of the inhomogeneities, are assumed to satisfy thefollowing inequality:|zj − zl |  c0 > 0, ∀j = l. (1)Let μ0 and ε0 denote the permeability and the permittivity of the free space, ω > 0 is a given frequency and k =ω√ε0μ0 > 0 is the wave number. We denote by x = (x1, x2, x3) the Cartesian coordinates in R3. We shall assumethat μ0 > 0 and ε0 > 0 are positive constants. Let μj > 0 and εj > 0 denote the permeability and the real permittivityof the j -th inhomogeneity, zj +αBj ; these are also assumed to be positive constants. Introduce the piecewise-constantmagnetic permeabilityμα(x) :={μ0, x ∈ R3 \ B̄α,μj , x ∈ zj + αBj , j = 1, . . . ,m. (2)The piecewise-constant electric permittivity εα(x) is defined analogously. If we allow the degenerate case α = 0, thenthe function μ0(x) (resp. ε0(x)) takes the constant value of μ0 (resp. ε0). We now suppose that the collection ofinclusions Bα is contained in an open subset Ω of R3 and that the source current density Js is supported in R3 \ Ω̄ .Let (Eα,Hα) ∈ R3 × R3 denote the time-harmonic electromagnetic fields, with pulsation ω, in the presence of theelectromagnetic inclusions Bα . These time-harmonic fields are solutions of the Maxwell’s equations, see [9,12]:⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩∇ × Eα = iωμαHα, in R3,∇ × Hα = −iωεαEα + Js , in R3,ν × Eα and ν × Hα are continuous across ∂(zj + αBj ),lim|x|→∞ |x|[∇ × Eα − ik x|x| × Eα]= 0 and lim|x|→∞|x|[∇ × Hα − ik x|x| × Hα]= 0.(3)Here ν stands for the outgoing unit normal to ∂(zj + αBj ). We eliminate the magnetic field from the above equationsby dividing the first equation in (3) by μα and taking the curl. We obtain a problem for Eα :⎧⎪⎪⎨⎪⎪⎩∇ × μ−1α ∇ × Eα − ω2εαEα = iωJs , in R3,ν × Eα is continuous across ∂(zj + αBj ),lim|x|→∞ |x|[∇ × Eα − ik x|x| × Eα]= 0.(4)We obtain the magnetic field Hα from Eα through the formula, Hα = 1iωμα ∇ × Eα . The electromagnetic energy isdefined by:Eα := 12∫R3(εα∣∣Eα(x)∣∣2 + (μα)−1∣∣Hα(x)∣∣2)dx. (5)The finiteness of the electromagnetic energy Eα requires that both electric and magnetic field belong to a space offields with square integrable curls:H(curl;R3) := {a ∈ L2(R3)3; curl a ∈ L2(R3)3}.C. Daveau, A. Khelifi / C. R. Acad. Sci. Paris, Ser. I 346 (2008) 287–292 289It can be shown that there exists a unique solution Eα ∈ H(curl;R3) to the problem (4), and this solution satisfiesthe following Lippman–Schwinger integral representation formula:Lemma 1.1. Let Eα be the solution of (4). Then, the following integral representation formula holdsEα(x) = E0(x) +m∑j=1∫zj +αBj(−iω(μj − μ0)∇′ × G(x, x′) · Hα(x′) + ω2μ0(εj − ε0)G(x, x′) · Eα(x′))dx′.(6)The 3 × 3 matrix valued function G means the Green’s function solution to⎧⎨⎩∇ × ∇ × G(x, x′) − k2G(x, x′) = I3δ(x − x′), in R3,lim|x|→∞ |x|[∇ × G(x, x′) − ik x|x| × G(x, x′)]= 0,where I3 is the 3 × 3 identity matrix. In the above notation the curl operator, ∇×, acts on matrices column by columnand ∇′× denotes the curl operator with respect to the second variable x′.The proof of Lemma 1.1 consists of multiplying the first equation in (4) by G(x, x′) · v (v ∈ R3) and integrating byparts over the domain Ω which contains the set of inclusions {zj + αBj }mj=1.2. Asymptotic behaviorAs mentioned in the introduction, our analysis will involve the polarization tensors M ∈ R3×3. We recall that thesetensors are defined byM(qj/q0;Bj ) :=∫Bj∇vq(x)dx, (7)where {qj } (resp. q0) denote either the set {εj } or {μj } for 1  j  m (resp. denote either ε0 or μ0) and where thefunctions vq , which depend on the contrast qj/q0, are solutions of the following problem,{∇ · q(x)∇vq(x) = 0, in R3,vq(x) − x → 0 as x → ∞, (8)whereq(x) ={q0, x ∈ R3\Bj ,qj , x ∈ Bj .The following proposition holds:Proposition 2.1. Suppose that (1) and (7) are satisfied and let (Eα,Hα) be the solution of (3), then we have(i) The electric field Eα satisfies the following uniform expansionEα(x) = E0(x) + α3m∑j=1[−iω(μj − μ0)∇′ × G(x, zj ) · M(μj/μ0;Bj )H0(zj )+ ω2μ0(εj − ε0)G(x, zj ) · M(εj /ε0;Bj )E0(zj )] + O(α4). (9)(ii) The magnetic field Hα satisfies the following uniform expansionHα(x) = H0(x) + α3m∑j=1[iω(εj − ε0)∇′ × G(x, zj ) · M(εj /ε0;Bj )E0(zj )− ω2ε0(μj − μ0)G(x, zj ) · M(μj /μ0;Bj )H0(zj )] + O(α4). (10)290 C. Daveau, A. Khelifi / C. R. Acad. Sci. Paris, Ser. I 346 (2008) 287–292Proof. For simplicity, we may focus on the case of a single inhomogeneity, i.e. m = 1. The derivation of the asymp-totic expansions for any fixed number m of well separated inhomogeneities follows by iteration of the arguments thatwe will present for the case m = 1. In order to simplify the notations we assume that the single inhomogeneity has theform z +αB and denote the electric permittivity and magnetic permeability inside z +αB by ε∗ and μ∗, respectively.For the problem (3) stated in section 1 the asymptotic expansion of the solution (Eα,Hα) which is uniformly valid inspace is constructed by the method of matched asymptotic expansions for α small, as done in [3,7,10]. To reveal thenature of the perturbations in the electric and magnetic fields, we use the local variable ξ = x−zαand we set the fieldeα(ξ) = Eα(z + αξ) and hα(ξ) = Hα(z + αξ). We expect that Eα(x) and Hα(x) will appreciably differ from E0(x)and H0(x), respectively for x near z, but will be close to E0(x) and H0(x) for x far from z. Therefore, in the spiritof matched asymptotic expansions, we shall represent each of the fields Eα and Hα by two different expansions, aninner expansion for x near z, and an outer expansion for x far from z. The outer expansion is:Eα(x) = E0(x) + ατ1E1(x) + ατ2E2(x) + · · · , Hα(x) = H0(x) + ατ1H1(x) + ατ2H2(x) + · · · (11)for |x − z|  α, where 0 < τ1 < τ2 < · · · , and (Ej ,Hj )j1 are to be found.We write the inner expansions asEα(z + αξ) = eα(ξ) = e0(ξ) + αe1(ξ) + α2e2(ξ) + · · · , for |ξ | = O(1),Hα(z + αξ) = hα(ξ) = h0(ξ) + αh1(ξ) + α2h2(ξ) + · · · , for |ξ | = O(1), (12)where (ej , hj )j1 are to be found.We try to equate the inner and the outer expansions in some ‘transition region’, as done in [3], within which thestretched variable ξ is large and x − z is small. In this domain the matching conditions areE0(x) + ατ1E1(x) + ατ2E2(x) + · · · ∼ e0(ξ) + αe1(ξ) + α2e2(ξ) + · · · ,H0(x) + ατ1H1(x) + ατ2H2(x) + · · · ∼ h0(ξ) + αh1(ξ) + α2h2(ξ) + · · · . (13)Observing thatεα(x − zα)≡ ε0, and μα(x − zα)≡ μ0for |x − z|  α, we find that the stretched coefficients ε(ξ) and μ(ξ) can be defined byε(ξ) ={ε0, ξ ∈ R3 \ B̄,ε∗, ξ ∈ B, and μ(ξ) ={μ0, ξ ∈ R3 \ B̄,μ∗, ξ ∈ B.Inserting relation (12) into the Maxwell’s equations (3) and formally equating coefficients of α−1, we get from (13)that ⎧⎨⎩∇ × e0(ξ) = 0, in R3,∇ · (ε(ξ)e0(ξ)) = 0, in R3,e0(ξ) → E0(z), as |ξ | → +∞,(14)and ⎧⎨⎩∇ × h0(ξ) = 0, in R3,∇ · (μ(ξ)h0(ξ)) = 0, in R3,h0(ξ) → H0(z), as |ξ | → +∞.(15)Therefore,e0(ξ) = ∇vε(ξ)(E0(z)), h0(ξ) = ∇vμ(ξ)(H0(z)),where the function vq , for q(ξ) = ε(ξ) or μ(ξ), is defined in (8).To finish, we use the definition (7) and we insert the inner expansions into the Lippman–Schwinger integral repre-sentation formula (6) for the field Eα to find (9). The proof of (10) can be deduced similarly. C. Daveau, A. Khelifi / C. R. Acad. Sci. Paris, Ser. I 346 (2008) 287–292 291To evaluate the influence of the presence of the inhomogeneities on the electromagnetic energy defined by (5), weintroduce its density per unit of volume which is denoted by ℵα . According to Poynting’s theorem, the energy densityℵα is given by:ℵα = ∇ · α + Js · Eα, (16)where α is Poynting’s vector,α = Eα × Hαμ0. (17)The following theorem justifies the behavior of the electromagnetic energy in the entire space.Theorem 2.2. Suppose that (1) and (16) are satisfied and suppose that the inclusion Bj is a ball for eachj ∈ {1, . . . ,m}. Then, the following uniform expansion holdsℵα(x) − ℵ0(x) = α3m∑j=13|Bj |(εj − ε0εj + 2ε0[k2Js ·(G(x, zj ) · E0(zj ))+ 1μ0∇ · [iωε0E0 × (∇′ × G(x, zj ) · E0(zj )) + k2(G(x, zj ) · E0(zj )) × H0(zj )]]− μj − μ0μj + 2μ0[iωμ0Js ·(∇′ × G(x, zj ) · H0(zj )) + 1μ0∇ · [iωμ0(∇′ × G(x, zj ) · H0(zj ))+ k2E0 × G(x, zj ) · H0(zj )]]) + O(α5).Proof. Recall the following identity: ∇ · (A × B) = B · ∇ × A − A · ∇ × B. Relation (17) immediately gives∇ · (α) = 1μ0[Hα · ∇ × Eα − Eα · ∇ × Hα]. (18)Next, under the assumption that inclusion Bj is a ball for all j ∈ {1, . . . ,m}, it was proved in [6] that polarizationtensors (7) are simplifiedM(μj/μ0;Bj ) = 3μ0μj + 2μ0 |Bj |I3, and M(εj/ε0;Bj ) =3ε0εj + 2ε0 |Bj |I3. (19)Then, according to [5] the correction of order four in (9) is zero and the remainder is in fact O(α5). Using this resultand inserting (19) into (9), one obtain the following expansions:Eα(x) = E0(x) + α3m∑j=1[−iωμ0 3(μj − μ0)μj + 2μ0 |Bj |∇′ × G(x, zj ) · H0(zj )+ k2 3(εj − ε0)εj + 2ε0 |Bj |G(x, zj ) · E0(zj )]+ O(α5). (20)In a similar fashion, we obtain the expansion for the magnetic field:Hα(x) = H0(x) + α3m∑j=1[iωε03(εj − ε0)εj + 2ε0 |Bj |∇′ × G(x, zj ) · E0(zj )− k2 3(μj − μ0)μj + 2μ0 |Bj |G(x, zj ) · H0(zj )]+ O(α5). (21)Using relations (20) and (21), the following holds:292 C. Daveau, A. Khelifi / C. R. Acad. Sci. Paris, Ser. I 346 (2008) 287–292Hα · ∇ × Eα = H0 · ∇ × E0 + α3{m∑j=1c1H0 · ∇ ×(∇′ × G(x, zj )H0) + c2H0 · ∇ × (G(x, zj )E0)+m∑j=1c′1(∇′ × G(x, zj )E0) · (∇ × E0) + c′2(G(x, zj )H0) · (∇ × E0)}+ O(α5),where the constants c1, c2, c′1 and c′2 are given by⎧⎪⎪⎨⎪⎪⎩c1iωμ0= c′2k2= −3 μj − μ0μj + 2μ0 |Bj |,c2k2= c′1iωε0= 3 εj − ε0εj + 2ε0 |Bj |.In the same manner we find the expansion of the term Eα · ∇ × Hα ; therefore (18) becomes∇ · (α) = ∇ · (0) + α3m∑j=11μ0[c1∇ ·((∇′ × G(x, zj )H0) × H0) + c2∇ · (G(x, zj )E0 × H0)+ c′1∇ ·(E0 ×(∇′ × G(x, zj )E0)) + c′2∇ · (E0 × G(x, zj )H0)] + O(α5).The proof is achieved by expanding the term Js · Eα in (16) at order 5 with respect to α. By using Theorem 2.2, we can prove the main result:Theorem 2.3. Suppose that (1) is satisfied and suppose that the inclusion Bj is a ball for each j ∈ {1, . . . ,m}. Then,there exists a positive constant C such that the following estimate holds:|Eα − E0|  Cα3,as α → 0. The constant C is independent of α and the set of points {zj }mj=1, but this constant C depends on |Bj |.References[1] H. Ammari, E. Iakovleva, D. Lesselier, Two numerical methods for recovering small inclusions from the scattering amplitude at a fixedfrequency, SIAM J. Sci. Comput. 27 (2005) 130–158.[2] H. Ammari, H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements, Lecture Notes in Mathematics, vol. 1846,Springer-Verlag, Berlin, 2004.[3] H. Ammari, A. Khelifi, Electromagnetic scattering by small dielectric inhomogeneities, J. Math. Pures Appl. 82 (2003) 749–842.[4] H. Ammari, M.S. Vogelius, D. Volkov, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomo-geneities of small diameter II. The full Maxwell equations, J. Math. Pures Appl. 80 (2001) 769–814.[5] H. Ammari, D. 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On the perturbation of the electromagnetic energy due to the presence of small inhomogeneities

Daveau, Christian

Khelifi, Abdessatar

Numérisation de Documents Anciens Mathématiques

http://www.numdam.org/item/10.1016/j.crma.2008.01.019.pdf

On the perturbation of the electromagnetic energy due to the presence of small inhomogeneities

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