A general perturbation formula for electromagnetic fields in presence of low volume scatterers

DOI:10.1051/m2an/2011015 www.esaim-m2an.org

A GENERAL PERTURBATION FORMULA FOR ELECTROMAGNETIC FIELDS IN PRESENCE OF LOW VOLUME SCATTERERS

Roland Griesmaier

Abstract. In several practically interesting applications of electromagnetic scattering theory like,e.g., scattering from small point-like objects such as buried artifacts or small inclusions in non-destructive testing, scattering from thin curve-like objects such as wires or tubes, or scattering from thin sheet-like objects such as cracks, the volume of the scatterers is small relative to the volume of the surrounding medium and with respect to the wave length of the applied electromagnetic fields. This smallness typically causes problems when solving direct scattering problems due to the need to discretize the objects and also when solving inverse scattering problems because small objects have very little effect on electromagnetic fields. In this paper we consider an asymptotic representation formula for scattered electromagnetic waves caused by low volume objects contained in some otherwise homogeneous three- dimensional bounded domain, assuming only that the scatterers are measurable and well-separated from the boundary of the domain. The formula yields a very general asymptotic model for electro- magnetic scattering due to low volume objects that can either be used to simulate the corresponding electromagnetic fields or as the foundation of efficient reconstruction methods for inverse scattering problems with low volume scatterers. Our analysis extends results originally obtained in [Y. Capde- boscq and M.S. Vogelius,A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction. Math. Model. Numer. Anal.37(2003) 159–173] for steady state voltage potentials to time-harmonic Maxwell’s equations.

Mathematics Subject Classification. 35C20, 35Q60, 35J20.

Received August 31, 2010.

Published online July 22, 2011.

1. Introduction

In this work we study time-harmonic electromagnetic waves in a smoothly bounded domain filled with a ho- mogeneous medium, which we call the background domain and accordingly the background medium. Supposing that this domain contains a penetrable object, the scatterer, that is a subdomain on which the electromagnetic properties of the medium differ from that of the background medium, we want to describe the influence of this object on electromagnetic fields under the additional assumption that its volume is small. Our main motivation to do so stems from inverse scattering problems, where one aims to recover the position and the shape of the scatterer from measurements of electromagnetic waves that are scattered by this object. If the volume of the object is small, it is well known that it has very little effect on electromagnetic fields,i.e., on the measurement

Keywords and phrases. Perturbation formulas, electromagnetic scattering, low volume scatterers, asymptotic expansions.

1 Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716, USA.[email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2011

data given in the inverse scattering problem. Therefore it is advisable to include all available a priori informa- tion on the structure of perturbations of electromagnetic fields due to low volume scatterers, when designing methods to reconstruct such objects from scattering data.

The aim of this article is to provide sucha prioriinformation for a very general class of low volume scatterers.

For this purpose we consider the asymptotic behavior of scattered electromagnetic fields, i.e., the difference between the fields with and without scatterer, as the volume of the scatterer tends to zero. Our only regularity assumptions on the scatterer are that it is measurable and well separated from the boundary of the background domain. Observing that away from the scatterer the scattered field decreases at the same rate as the volume of the scatterer, we derive an analytic expression for the leading order term in the corresponding asymptotic expansion. Due to the generality of this formula higher order terms or even their precise order in terms of the volume of the scatterer are not obtained. For time-harmonic Maxwell’s equations such an expansion has so far only been studied for the special case of diametrically small scatterers by Ammari, Vogelius, and Volkov [9]

(see also [28] for a corresponding result for perfectly conducting scatterers in an unbounded layered background medium and Ammari and Kang [4] or Ammari and Khelifi [6] for higher order expansions in a two-dimensional setting).

Our analysis extends the work of Capdeboscq and Vogelius [18], where a similar representation formula has been established for perturbations in electrostatic potentials due to low volume conductivity inhomogeneities (cf. [19–21] for further studies in this direction). This formula generalizes and unifies earlier results obtained for the special case of diametrically small conductivity inhomogeneities (see,e.g., Friedman and Vogelius [26], Cedio-Fengya, Moskow, and Vogelius [22], or Ammari and Kang [3]) as well as for conductivity inhomogeneities of small thickness (see,e.g., Berettaet al.[15,16]). Similarly, the general formula for time-harmonic Maxwell’s equations discussed in this article allows to recover the formula for diametrically small scatterers from [9] but also yields new asymptotic formulas for practically important cases like thin tubes or thin penetrable cracks mentioned before. Furthermore, our results can be extended to unbounded domains straightforwardly as done for diametrically small scatterers by Ammari and Volkov in [11].

The asymptotic formulas for diametrically small conductivity inhomogeneities and scatterers known so far form the foundation of several efficient reconstruction methods for inverse conductivity problems (see, e.g., Ammari, Moskow, and Vogelius [7], Ammari and Seo [8] or Br¨uhl, Hanke, and Vogelius [17]) and inverse scattering problems for Maxwell’s equations (see, e.g., Ammari et al. [2], Iakovleva et al. [33], Volkov [42], or [28,29,31,32]). In addition the general formula for electrostatic potentials from [18] has recently been used to investigate inverse conductivity problems for wires and tubes (see Berettaet al. [13] or [30]). Similarly, the general formula for electromagnetic fields considered here gives a new approach to develop efficient reconstruction methods for practically interesting inverse scattering problems, such as, e.g., inverse scattering for penetrable cracks (cf.,e.g., Beretta and Francini [14] and Park and Lesselier [39] for an earlier study in this direction in a two-dimensional setting) or inverse scattering for thin tubular scatterers.

The arguments we use to establish the asymptotic representation formula rest on those applied in [18], suitably modified for the non-coercive Maxwell’s equations and vector-valued functions. Among others the proof involves a representation formula for the scattered electromagnetic field in terms of Green’s functions, energy estimates for electromagnetic fields obtained by duality arguments, so-called corrector potentials and polarization tensors, and an integration by parts technique that goes back to Murat and Tartar [38]. Applying two types of Helmholtz decompositions to the electromagnetic fields the gradient parts can be treated similarly to [18], while the main contribution of this work is the analysis of the divergence free parts that requires different techniques and is slightly more involved. In case of homogeneous background media, the polarization tensors turn out to be equivalent to the polarization tensor appearing in the electrostatic case, and thus earlier results on the properties of these tensors (see, e.g., [5,18,19,21]) immediately carry over to our setting. For historical remarks on polarization tensors and alternative approaches to establish asymptotic expansions of similar type as considered here we refer to the monographs by Ammari and Kang [5] and Il’in [34].

For several particular geometries of the scatterer explicit characterizations of the polarization tensor are known. Then the asymptotic representation formula yields a fast way to approximate the scattered field due

to such low volume objects without meshing the scatterer itself. This is, e.g., the case for small ellipsoidal scatterers (see [5]), thin cylindrical scatterers (see [13]), or thin sheet-like scatterers (see [15,16]).

This article is organized as follows. In the next section we specify our mathematical setting and state the main result of this work, the asymptotic perturbation formula for the electromagnetic field. In Section 3, we collect some estimates for the scattered magnetic field, estimates for corresponding asymptotic corrector potentials, and the definitions of the electric and magnetic polarization tensors that will be used in the proof of the asymptotic formula. This proof is carried out in Section4. In Section5we discuss three particular examples and comment on how the corresponding formulas can be used to solve inverse scattering problems. Finally, in Section 6we outline possible generalizations of our findings.

2. The mathematical setting

Suppose Ω⊂R³ is a bounded domain with smooth boundary∂Ω and unit outward normalν. We consider a homogeneousbackground medium with constant electric conductivity σ0 ≥0, constant electric permittivity ε0>0, and constant magnetic permeabilityμ0>0. A time-harmonicmagnetic background field in this medium at a frequencyω/2π >0 corresponding to boundary data

g∈H^−1/2_div (∂Ω) :={f ∈H^−1/2(∂Ω,C³)|div_∂Ωf ∈H^−1/2(∂Ω,C)} is governed by the boundary value problem

curl 1

ε˜0

curlH₀

−ω²μ0H₀= 0 in Ω, (2.1a)

1 ε˜0

(curlH₀)×ν=g on∂Ω. (2.1b)

Here and in the following we write ˜ε0 := ε0+ iσ0/ω. Accordingly the electric background field is given by E₀= (i/ωε˜0)curlH₀.

Next, let Ω₀⊂⊂Ω be well separated from∂Ω,i.e., dist(Ω₀, ∂Ω)≥d0for some constantd0>0, and denote by (Dρn)_n∈Na family of measurable subsets of Ω satisfying lim_n→∞|D_ρn|= 0, where|D_ρn|denotes the Lebesgue measure ofDρn. EachDρnis considered as a scatterer contained in Ω causing a discontinuous permeability and (complex-valued) permittivity

μρn(x) :=

μ1(x), x∈Dρn,

μ0, x∈Ω\Dρn, ε˜ρn(x) :=

ε˜1(x), x∈Dρn, ε˜0, x∈Ω\Dρn,

where μ1 ∈ C^∞(Ω,R) satisfies 0 < cμ1 ≤ μ1 ≤ Cμ1 < ∞ and ˜ε1 := ε1+ i(σ1/ω) with ε1 ∈ C^∞(Ω,R) and σ1∈C^∞(Ω,R) such that 0< cε1 ≤ε1≤Cε1 <∞and 0≤σ1≤Cσ1 <∞for some constants cμ1, Cμ1, cε1, Cε1, andCσ1. The magnetic field in presence of these scatterers corresponding to the same boundary data as above satisfies

curl 1

ε˜ρn

curlH_ρn

−ω²μρnH_ρn= 0 in Ω, (2.2a)

1 ε˜ρn

(curlH_ρn)×ν=g on∂Ω. (2.2b)

Introducing the sesquilinear formsa0 andaρn,n∈N, onH(curl,Ω)×H(curl,Ω), where H(curl,Ω) :={u∈L²(Ω,C³)|curlu∈L²(Ω,C³)},

by

a0(u,v) :=

Ω

1 ε˜0

curlu·curlvdx−ω²

Ω

μ0u·vdx, aρn(u,v) :=

Ω

1 ε˜ρn

curlu·curlvdx−ω²

Ωμρnu·vdx,

the weak formulations of the boundary value problems (2.1) and (2.2) ask to findH₀,H_ρn∈H(curl,Ω) such that

a0(H₀,v) =

∂Ωg·vds, for allv ∈H(curl,Ω), (2.3)

aρn(H_ρn,v) =

∂Ω

g·vds, for allv ∈H(curl,Ω), (2.4)

respectively. Throughout we assume that (2.3) has a unique solution. In this case H_0H(curl,Ω)≤Cg_H−1/2

div (∂Ω)

and regularity results for Maxwell’s equations (cf., e.g., Weber [43]) guarantee that H₀ is smooth in Ω₀. It has been shown in [9] for the special case of diametrically small scatterers that uniqueness of solutions to (2.3) implies existence and uniqueness of solutions to (2.4) provided that|D_ρn|is small enough,i.e.,nis large enough.

The proof of this result in [9] carries over to our setting straightforwardly.

Proposition 2.1. Assume that (2.3) has a unique solution. Then, there exists an upper bound R > 0 such that for anyg∈H^−1/2_div (∂Ω)and anyn∈Nwith |D_ρn| ≤R the variational problem (2.4)has a unique solution H_ρn ∈H(curl,Ω)satisfying

H_{ρnH(curl,Ω)}≤Cg_H^−1/2

div (∂Ω), where the constant C is independent ofn.

We are interested in the asymptotic behavior ofν×(H_ρn−H₀)

∂Ωas |Dρn| →0. Following [18] we start by observing that for alln∈Nthe positive regular Borel measure

μn(E) :=

E|Dρn|⁻¹χρndx, E⊂Ω Borel measurable, (2.5) satisfies|μ_n| ≤1, where|μ_n|denotes the total variation ofμnandχρnis the characteristic function ofDρn. This means that the sequence (μn)_n∈Nis bounded in the spaceM(Ω,C) of complex regular Borel measures on Ω. By Riesz’s representation theoremM(Ω,C) is isomorphic to the dual space ofC0(Ω,C) of continuous functions on Ω that vanish on∂Ω (cf.,e.g., Rudin [40], Thm. 6.19). Thus the Banach-Alaoglu Theorem (cf.,e.g., Rudin [41], Thm. 3.15) guarantees the existence of a subsequence, also denoted by (Dρn)_n∈N, and a complex regular Borel measureμsuch that for everyφ∈C0(Ω,C),

n→∞lim

Ωφdμn =

Ωφdμ. (2.6)

Another ingredient of the asymptotic perturbation formula established in Theorem2.2below, is thedyadic Green’s functionfor time-harmonic Maxwell’s equations corresponding to the homogeneous background medium,

G(x,y) := Φ_k(x−y)I3+ 1

k²∇xdiv_x(Φ_k(x−y)I3), x,y∈R³, x=y,

where Φ_k(x−y) := e^ik|x−y|/4π|x−y| is the fundamental solution of the Helmholtz equation with wave number k :=

ω²ε˜0μ0 (if ˜ε0 ∈/ R, thenk is taken to have positive imaginary part) andI3 denotes the 3×3- identity matrix. Here and throughout this work we let scalar operators operate on vectors component-wise and vector operators on matrices column by column. Note that for any y∈R³ the dyadic Green’s function is the distributional solution of

curl_xcurl_xG(x,y)−k²G(x,y) =δy(x), x∈R³.

In the following theoremL²(Ω,K^3×3;μ),K=RorK=C, denotes the space of real or complex matrix-valued functions on Ω that are square integrable with respect to the regular Borel measureμ, respectively.

Theorem 2.2. Suppose (Dρn)_n∈N is a sequence of measurable subsets of Ω₀⊂⊂Ω as introduced above and assume that|Dρn| ≤R for alln∈N, whereR is the upper bound from Proposition 2.1. Given g∈H^−1/2_div (∂Ω) let H₀ and H_ρn, n ∈ N, denote the corresponding solutions of (2.3) and (2.4), respectively. Then, there exists a subsequence, also denoted by(Dρn)_n∈N, a positive regular Borel measure μand matrix-valued functions M^ε˜∈L²(Ω,C^3×3;μ) andM^μ ∈L²(Ω,R^3×3;μ), called electricand magnetic polarization tensors, respectively, such that for y∈∂Ω,

ν(y)×(H_ρn−H₀)(y)−2

∂Ω

ν(y)×curl_xG(·,y) ν×(H_ρn−H₀) ds

=|D_ρn|2˜ε0

−

Ω

ε˜1

ε˜0

1 ε˜0

− 1 ε˜1

ν(y)×curl_xG(·,y) M^ε˜curlH₀dμ

+ω²

Ω

(μ0−μ1)

ν(y)×G(·,y) M^μH₀dμ

+o(|D_ρn|). (2.7) The subsequence(Dρn)_n∈Nand the functionsM^ε˜andM^μare independent ofg. The last term on the right hand side of (2.7)satisfies

n→∞lim o(|Dρn|)_L^∞_(∂Ω,C³)/|Dρn|= 0 for any g∈H^−1/2_div (∂Ω), uniformly on bounded subsets ofH^−1/2_div (∂Ω).

Although in this work we consider the magnetic field only, we note that an asymptotic perturbation formula similar to (2.7) can be established for the electric field as well.

Remark 2.3(Polarization Tensors). Before we give a precise definition of the electric and magnetic polarization tensor in Section3.3below, we recall in the following two important properties of these matrix valued functions.

It has been shown in [18], Section 4, that the magnetic polarization tensor M^μ (and similarly the electric polarization tensor M^ε˜, provided that ˜ε0,ε˜1 ∈R, i.e., σ0 =σ1 = 0) is symmetric and positive definite in the sense that forμ-a.e.x∈Ω,

M^μ(x) =M^μ(x), (2.8a)

min

1,μ1(x) μ0(x)

|ξ|²≤ξM^μ(x)ξ≤max

1,μ1(x) μ0(x)

|ξ|² (2.8b)

for allξ∈R³. For dissipative media, following the proof for real-valued coefficients in [18], Section 4, we obtain that forμ-a.e.x∈Ω the electric polarization tensorM^˜ε(x) satisfies

M^˜ε(x) =M^˜ε(x),

i.e., it is symmetric but not Hermitian, and

Re(˜ε1−ε˜0)−

(˜ε1−ε˜0)² ε˜1

|ξ|²≤ξRe

(˜ε1−ε˜0)M^ε˜(x) ξ≤Re(˜ε1−ε˜0)|ξ|²,

Im(˜ε1−ε˜0)−

(˜ε1−ε˜0)² ε˜1

|ξ|²≤ξIm

(˜ε1−ε˜0)M^ε˜(x) ξ≤Im(˜ε1−ε˜0)|ξ|² for allξ∈R³,i.e., it is uniformly bounded.

3. Preliminary convergence estimates

In this section we derive energy estimates for the difference H_ρn−H₀, n∈N, of the magnetic fields with and without scatterers and for the difference of corresponding asymptotic corrector potentials introduced below.

Moreover, we define the electric and magnetic polarization tensors used in this work and discuss their relation to the polarization tensor appearing in the asymptotic representation formula for electrostatic potentials from [18,19,21]. Throughout we assume that (Dρn)_n∈Nis a sequence of measurable subsets of Ω₀⊂⊂Ω as in Section2 such that|Dρn| ≤R, whereRis the upper bound from Proposition2.1, and that (2.6) is satisfied.

3.1. Two decompositions of H(curl , Ω)

The energy estimates will be formulated in terms of two types of Helmholtz decompositions of H(curl,Ω) related to the variational formulations (2.3) and (2.4), respectively. Following [9], we define the spacesY :=

∇H¹(Ω,C),

Y^⊥₀ :={u∈H(curl,Ω)|div(μ0u) = 0 in Ω, ν·u= 0 on∂Ω}, and for alln∈N,

Y^⊥_ρn:={u∈H(curl,Ω)|div(μρnu) = 0 in Ω, ν·u= 0 on∂Ω}.

It is well known that the subspaceY is closed inH(curl,Ω). Furthermore, Y^⊥₀ and Y^⊥_ρn are the orthogonal complements of Y in H(curl,Ω) with respect to the inner products

u,v0:=

Ω

μ0u·vdx and u,vρn :=

Ω

μρnu·vdx, u,v∈H(curl,Ω), respectively. This yields decompositions

H(curl,Ω) =Y ⊕Y^⊥₀ =Y ⊕Y^⊥_ρn, n∈N, and corresponding orthogonal projections

P0:H(curl,Ω)→Y^⊥₀ and Pρn :H(curl,Ω)→Y^⊥_ρ

n, (3.1)

given byP0u:=u− ∇p0 andPρnu=u− ∇p_ρn, wherep0, pρn ∈H¹(Ω,C) :={φ∈H¹(Ω,C)|

∂Ωφds= 0}

satisfy

Ω

μ0∇p0· ∇φdx=

Ω

μ0u· ∇φdx for allφ∈H¹(Ω,C), (3.2a)

Ω

μρn∇pρn· ∇φdx=

Ω

μρnu· ∇φdx for allφ∈H¹(Ω,C), (3.2b)

respectively. Forv∈Y^⊥₀ the Friedrichs inequality

u_L²_(Ω,C³₎≤Ccurlu_L²_(Ω,C³₎ (3.3) follows directly from [37], Corollary 3.51 in the book of Monk and using the compactness result [9], Proposition 3 it can be shown that (3.3) holds foru∈Y^⊥_ρ

n as well.

Accordingly we can decompose

H₀=:h₀+∇q0 and H_ρn =:h_ρn+∇q_ρn, n∈N, (3.4) such that h₀ ∈Y^⊥₀, h_ρn ∈Y^⊥_ρ

n, and q0, qρn ∈H¹(Ω,C). Combining the weak formulations (2.3), (2.4), and (3.2) it follows immediately thatq0 andqρn are weak solutions of

div(μ0∇q0) = 0 in Ω, (3.5a)

μ0∂q0

∂ν = 1

ω²div_∂Ωg on∂Ω, (3.5b)

and

div(μρn∇q_ρn) = 0 in Ω, (3.6a)

μρn

∂qρn

∂ν = 1

ω²div_∂Ωg on∂Ω, (3.6b)

respectively. Thus, regularity results for elliptic equations show that the solution q0 is smooth in Ω₀ (cf.

McLean [36], Thm. 4.18).

Sometimes we will also use the slightly different decompositions H₀=:h^(ρ₀ⁿ⁾+∇q₀^(ρn) and H_ρn =:h⁽⁰⁾_ρ

n +∇q_ρ⁽⁰⁾_n, n∈N, (3.7) withh^(ρ₀ⁿ⁾∈Y^⊥_ρn,h⁽⁰⁾_ρn ∈Y^⊥₀, andq^(ρ₀ⁿ⁾, qρ⁽⁰⁾n ∈H¹(Ω,C). Note thatq₀^(ρn) andqρ⁽⁰⁾n are weak solutions of

div

μρn∇q^(ρ₀ⁿ⁾ = div(μρnH₀) in Ω, (3.8a)

μρn

∂q^(ρ₀ⁿ⁾

∂ν = 1

ω²div_∂Ωg on∂Ω, (3.8b)

and

div

μ0∇q_ρ⁽⁰⁾_n = div(μ0H_ρn) in Ω, (3.9a)

μ0∂qρ⁽⁰⁾n

∂ν = 1

ω²div_∂Ωg on∂Ω, (3.9b)

respectively.

3.2. Estimates for the magnetic field

To estimate the difference H_ρn−H₀, n ∈ N, we consider its gradient part and its divergence free part according to the decompositions (3.4) and (3.7) separately. As already mentioned in the introduction, the analysis for the gradient part in this and the following sections follows closely the corresponding analysis for the electrostatic case from [18], while the divergence free parts require different arguments and techniques.

Throughout we use generic constants C andCη, η ∈[1/5,1/2], the values of which might change from line to line.

Lemma 3.1. Let q0 andqρn be as in (3.4). Then, there exists a constant C such that for anyn∈N, q_ρn−q0_H¹_(Ω,C)≤C|D_ρn|^1/2g_H−1/2

div (∂Ω), (3.10)

and for any η∈[1/5,1/2]there exists a constantCη such that

qρn−q0_L²_(Ω,C)≤Cη|Dρn|^1−ηg_H^−1/2

div (∂Ω). (3.11)

Proof. Sinceq0andqρn satisfy (3.5) and (3.6), respectively, this lemma is a special case of [18], Lemma 1.

Lemma 3.2. Let h₀,h^(ρ₀ⁿ⁾,h_ρn, andh⁽⁰⁾_ρn be as in (3.4)and (3.7), respectively. Then, there exists a constant C such that for anyn∈N,

h_ρn−h^(ρ₀ⁿ⁾_H(curl,Ω)≤C|Dρn|^1/2g_H^−1/2

div (∂Ω), (3.12a)

h₀−h^(ρ₀ⁿ⁾_H(curl,Ω)≤C|Dρn|^1/2g_H^−1/2

div (∂Ω), (3.12b)

h_ρn−h⁽⁰⁾_{ρnH(curl,Ω)}≤C|Dρn|^1/2g_H−1/2

div (∂Ω). (3.12c)

Furthermore, for any η∈[1/5,1/2]there exists a constantCη such that h⁽⁰⁾_ρ

n −h_0L²_(Ω,C3)≤Cη|D_ρn|^1−ηg_H−1/2

div (∂Ω). (3.13)

Proof. Step 1(proof of (3.12a)). From the weak formulations (2.3) and (2.4) we find thatH₀=h^(ρ₀ⁿ⁾+∇q^(ρ₀ⁿ⁾ andH_ρn =h_ρn+∇qρn satisfy

Ω

1 ε˜ρn

curl(H_ρn−H₀)·curludx−ω²

Ωμρn(H_ρn−H₀)·udx

=

Ω

1 ε˜0 − 1

ε˜ρn

curlH₀·curludx−ω²

Ω

(μ0−μρn)H₀·udx for allu∈H(curl,Ω). The regularity results for weak solutions of Maxwell’s equations mentioned before show that H_0W^1,∞_(Ω0,C³₎≤Cg_H−1/2

div (∂Ω), and we may estimate

Ω

1 ε˜0

− 1 ε˜ρn

curlH₀·curludx−ω²

Ω

(μ0−μρn)H₀·udx

≤C|Dρn|^1/2

curlH_0L^∞(Ω0,C³)curlu_L²_(Ω,C³₎+H_0L^∞(Ω0,C³)u_L²_(Ω,C³₎

≤C|Dρn|^1/2g_H−1/2

div (∂Ω)u_H(curl,Ω). So, using orthogonality we obtain thath_ρn−h^(ρ₀ⁿ⁾satisfies

aρn(h_ρn−h^(ρ₀ⁿ⁾,v) =lρn(v) for allv∈Y^⊥_ρn, (3.14) wherelρn is a bounded conjugate linear form onY^⊥_ρn such that

sup

vH(curl,Ω)=1

|l_ρn(v)| ≤C|D_ρn|^1/2g_H−1/2 div (∂Ω).

Hence, (3.12a) is a consequence of the well-posedness of (3.14), which follows directly from the proof of [9], Lemma 1 recalling that we assumed|Dρn| ≤R.

Step 2(proof of (3.12b)). Since div(μ0h₀) = 0 in Ω andμ0ν·h₀= 0 on∂Ω there exists a vector potentialz₀∈ H₀(curl,Ω) satisfying div(μ0z₀) = 0 in Ω such thatμ0h₀=curlz₀ (cf.[37], Thm. 3.41). Moreover (see [37], Thm. 3.38), z₀ = curl(A) for some A ∈H(curl,Ω), which means z₀ ∈curl(H(curl,Ω)). Analogously, we can findz^(ρ₀ⁿ⁾∈curl(H(curl,Ω)) such thatμρnh^(ρ₀ⁿ⁾=curlz^(ρ₀ⁿ⁾. Now, following an idea used in [37], p. 173, where it was attributed to Arnold et al. [12], we introduce sesquilinear forms αon H(curl,Ω)×H(curl,Ω) andβ oncurl(H(curl,Ω))×H(curl,Ω) by

α(u,v) :=

Ω

μρnu·vdx and β(w,v) :=−

Ω

w·curlvdx, respectively, and observe that (h₀−h^(ρ₀ⁿ⁾,z₀−z^(ρ₀ⁿ⁾) solves the mixed variational problem

α(h₀−h^(ρ₀ⁿ⁾,v) +β(z₀−z^(ρ₀ⁿ⁾,v) =

Ω

(μρn−μ0)h₀·vdx, (3.15a)

β(w,h₀−h^(ρ₀ⁿ⁾) = 0 (3.15b)

for allv∈H(curl,Ω) andw∈curl(H(curl,Ω)). Note thatαandβare bounded,αis coercive, andβsatisfies the Babuˇska-Brezzi condition

v∈H(curl,Ω)sup

|β(w,v)|

v_H(curl,Ω) = sup

v∈H(curl,Ω)

|

Ωw·curlvdx|

v_H(curl,Ω) ≥ w²_L2(Ω,C³)

u_H(curl,Ω) ≥Cw_L²_(Ω,C³₎

for allw∈curl(H(curl,Ω)), whereu∈Y^⊥₀ ⊂H(curl,Ω) has been chosen such thatw=curlu, and we used (3.3) to estimate u_L²_(Ω,C³₎≤Cw_L²_(Ω,C³₎. Therefore (cf.[37], Thm. 2.25), solutions to (3.15) are unique and (h₀−h^(ρ₀ⁿ⁾,z₀−z^(ρ₀ⁿ⁾) satisfies

h₀−h^(ρ₀ⁿ⁾_H(curl,Ω)+z₀−z^(ρ₀ⁿ⁾_L²_(Ω,C³₎≤ sup

u∈H(curl,Ω)

Ω(μρn−μ0)h₀·udx u_H(curl,Ω)

≤C|D_ρn|^1/2h_0L^∞_(Ω0,C³₎≤C|D_ρn|^1/2g_H−1/2 div (∂Ω). The last inequality follows from the definition of h₀ = H₀− ∇q0 and the interior regularity of q0 and H₀ mentioned before.

Step 3 (proof of (3.13)). Similar to Step 1 we find by subtracting the weak formulations (2.3) and (2.4) that for allu∈H(curl,Ω),

Ω

1 ε˜0

curl(H_ρn−H₀)·curludx−ω²

Ω

μ0(H_ρn−H₀)·udx

=

Ω

1 ε˜0 − 1

ε˜ρn

curlH_ρn·curludx−ω²

Ω

(μ0−μρn)H_ρn·udx. (3.16) Using a duality argument inspired by the proof of [18], Lemma 1 we denote by z ∈ Y^⊥₀ the solution of the adjoint problem

Ω

1 ε˜0

curlz·curludx−ω²

Ω

μ0z·udx=

Ω

μ0(h⁽⁰⁾_ρn −h₀)·udx (3.17) for allu∈Y^⊥₀. Uniqueness of solutions to (2.3) implies existence and uniqueness of a solution to (3.17) with

z_H(curl,Ω)≤Ch⁽⁰⁾_ρn −h_0L²_(Ω,C³₎.

Recalling that inY^⊥₀ the norm · _H¹_(Ω,C³₎is equivalent to · _H(curl,Ω) (cf.Dautray and Lions [25], Thm. 3, p. 209), this shows thatz_H¹_(Ω,C³₎≤Ch⁽⁰⁾_ρn−h_0L²_(Ω,C³₎. Similarly, sincecurlzsatisfiescurl

(1/ε˜0)curlz = μ0

ω²z+h_ρn−h^(ρ₀ⁿ⁾ ,i.e.,curlz∈H(curl,Ω), div(curl(z)) = 0 in Ω, andν×curlz = 0 on∂Ω, we find that curlz∈H¹(Ω,C³) andcurlz_H¹_(Ω,C³₎≤Ch⁽⁰⁾_ρn −h_0L²_(Ω,C³₎ as well. Substituting thiszinto (3.16), using orthogonality, (3.17), and H¨older’s inequality we obtain that

Ω

|h⁽⁰⁾_ρn −h₀|²dx=

Ω

1 ε˜0

curl(h⁽⁰⁾_ρn −h₀)·curlzdx−ω²

Ω

μ0(h⁽⁰⁾_ρn −h₀)·zdx

=

Ω

1 ε˜0− 1

ερn

curlH_ρn·curlzdx−ω²

Ω

(μ0−μρn)H_ρn·zdx

≤C

curlH_ρnL^q_(Dρn,C³₎curlz_L^p_(Dρn,C³₎+H_ρnL^q_(Dρn,C³₎z_L^p_(Dρn,C³₎ , where 1≤p, q≤ ∞are such that 1/p+ 1/q= 1. Assuming that 2≤p≤6,i.e., 6/5≤q≤2, the boundedness of the embedding of H¹(Ω,C³) intoL^p(Ω,C³) (cf. Adams [1], Thm. 5.4) implies that

curlz_L^p_(Dρn,C³)≤Ccurlz_H¹_(Ω,C³₎ and z_L^p_(Dρn,C³)≤Cz_H¹_(Ω,C³₎. Thus,

h⁽⁰⁾_ρn −h_0L²_(Ω,C³₎≤C

curlH_ρnL^q_(Dρn,C³₎+H_ρnL^q_(Dρn,C³₎ . (3.18) Since 6/5≤q≤2, we can use the triangle inequality and embedL²(Dρn,C³) andL^∞(Dρn,C³) intoL^q(Dρn,C³) applying H¨older’s inequality (cf.Gilbarg and Trudinger [27], Eq. (7.8), p. 146) to see that

H_ρnL^q_(Dρn,C³)+curlH_ρnL^q_(Dρn,C³)≤ |Dρn|^1/q−1/2H_ρn−H_0H(curl,Ω)+|Dρn|^1/qH_0W^1,∞_(Ω0,C³₎. (3.19) Recalling that H_0W^1,∞_(Ω0,C³₎≤Cg_H−1/2

div (∂Ω), we use (3.4) to obtain

H_ρn−H_0H(curl,Ω)≤ h_ρn−h_0H(curl,Ω)+q_ρn−q0_H¹_(Ω,C), which together with (3.10), (3.12a), and (3.12b) gives

H_ρn−H_0H(curl,Ω)≤C|Dρn|^1/2g_H^−1/2

div (∂Ω). Therefore, combining (3.19) and (3.18) we find that

h⁽⁰⁾_ρn −h_0L²_(Ω,C³₎≤C|Dρn|^1/qg_H^−1/2

div (∂Ω)

and writingη:= 1−1/q yields (3.13).

Step 4 (proof of (3.12c)). As in Step 2 we can findz_ρn,z⁽⁰⁾_ρn ∈curl(H(curl,Ω)) such thatμρnh_ρn=curlz_ρn andμ0h⁽⁰⁾_ρn =curlz⁽⁰⁾_ρn. Then (h_ρn−h⁽⁰⁾_ρn,z_ρn−z⁽⁰⁾_ρn) satisfies

α(h_ρn−h⁽⁰⁾_ρn,v) +β(z_ρn−z⁽⁰⁾_ρn,v) =

Ω

(μ0−μρn)h⁽⁰⁾_ρn ·vdx, β(w,h_ρn−h⁽⁰⁾_ρn) = 0