Approximate transmission conditions for time-harmonic Maxwell equations in a domain with thin layer

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Maxwell equations in a domain with thin layer

Victor Péron, Clair Poignard

To cite this version:

Victor Péron, Clair Poignard. Approximate transmission conditions for time-harmonic Maxwell equa- tions in a domain with thin layer. [Research Report] RR-6775, INRIA. 2008, pp.37. �inria-00347971v3�

a p p o r t

d e r e c h e r c h e

0249-6399ISRNINRIA/RR--6775--FR+ENG

Thème NUM

Approximate transmission conditions for

time-harmonic Maxwell equations in a domain with thin layer

Victor Péron — Clair Poignard

N° 6775

Décembre 2008

Centre de recherche INRIA Bordeaux – Sud Ouest

time-harmonic Maxwell equations in a domain with thin layer

Victor P´eron ^∗, Clair Poignard^†

Th`eme NUM — Syst`emes num´eriques Equipes-Projets MC2´

Rapport de recherche n°6775 — D´ecembre 2008 — 34 pages

Abstract: We study the behavior of the electromagnetic field in a biological cell modelized by a medium surrounded by a thin layer and embedded in an ambient medium. We derive approximate transmission conditions in order to replace the membrane by these conditions on the boundary of the interior domain. Our approach is essentially geometric and based on a suitable change of variables in the thin layer. Few notions of differential calculus are given in order to obtain our asymptotic conditions in a simple way. This paper extends to time-harmonic Maxwell equations the previous works presented in [27, 29, 28, 4]. Asymptotic transmission conditions at any order are given in Appendix 1.

Key-words: Asymptotic analysis, Maxwell’s equations, differential calculus

∗Universit´e de Rennes1, I.R.M.A.R. & CNRS UMR 6625, Campus de Beaulieu, 35042 Rennes Cedex, France

†INRIA Bordeaux-Sud-Ouest, Institut de Math´ematiques de Bordeaux, CNRS UMR 5251

& Universit´e de Bordeaux1, 351 cours de la Lib´eration, 33405 Talence Cedex, France

´

equations de Maxwell en r´egime harmonique dans un milieu `a couche mince

R´esum´e : Nous ´etudions le comportement asymptotique du champ ´electroma- gn´etique dans une cellule biologique plong´ee dans un milieu ambiant. La cellule est compos´ee d’un cytoplasme entour´e d’une fine membrane. Nous obtenons des conditions de transmission sur le bord du cytoplasme ´equivalentes `a la couche mince. Notre approche est essentiellement g´eom´etrique et bas´ee sur un change- ment de variables ad´equat dans la couche mince. Quelques notions de calcul diff´erentiel sont rappel´ees afin d’obtenir directement notre d´eveloppement as- ymptotique. Par ailleurs des estimations d’erreur sont d´emontr´ees. En appen- dice, nous pr´esentons le d´eveloppement asymptotique `a tout ordre.

Mots-cl´es : Analyse Asymptotique, Equations de Maxwell, Calcul diff´erentiel

Contents

1 Introduction and motivations 4

2 Maxwell equations using differential forms 5 2.1 The considered problem . . . . 5 2.2 Regularized variational formulation. . . . 7 2.3 Main result . . . . 10

3 Geometry 13

3.1 Parameterization of Γ. . . . 13 3.2 The transmission conditions in local coordinates . . . . 14 3.3 Rescaling in the thin layer . . . . 15

4 Ansatz and formal expansion 15

4.1 Order 0 . . . . 16 4.2 Order 1 . . . . 17 4.3 Order 2 . . . . 18

5 Justification of the expansion 19

5.1 Regularity results . . . . 20 5.2 Convergence . . . . 21

1 Introduction and motivations

The electromagnetic modelization of biological cells has become extremely im- portant since several years, in particular in the biomedical research area. In the simple models [14, 16], the biological cell is a domain with thin layer composed of a conducting cytoplasm surrounded by a thin insulating membrane. When exposed to an electric field, a potential difference is induced across the cell mem- brane. This transmembrane potential (TMP) may be of sufficient magnitude to be biologically significant. In particular, if it overcomes a threshold value, complex phenomenons such as electropermeabilization or electroporation may occur [34, 33, 22, 21]. The electrostatic pressure becomes too high that the thin membrane is locally destructured: some exterior molecules might be internal- ized inside the cell. These process hold great promises particularly in oncology and gene therapy, to deliver drug molecules in cancer treatment. This is the reason why an accurate knowledge of the distribution of the electromagnetic field in the biological cell is necessary.

Several papers in the bioelectromagnetic research area deal with numerical electromagnetic modelizations of biological cells [23, 32, 30]. Actually the main difficulties of the numerical computations lie in the thinness of the membrane (the relative thickness of the membrane is of order 1/1000 compared with the size of the cell) and in the high contrast between the electromagnetic parameters of the cytoplasm and the membrane. We present here an asymptotic method to replace the thin membrane by appropriate transmission conditions on the boundary of the cytoplasm.

In previous papers [27, 29, 28, 4], an asymptotic analysis is proposed to compute the electric potential in domains with thin layer, using the electroqua- sistatic approximation¹. However it is not clear that the magnetic effects of the field may be neglected. This is the reason why we present in this paper an asymptotic analysis for time-harmonic Maxwell equations in a domain with thin layer.

Our analysis is closed to those performed in [27, 29, 28]. Roughly speaking, it is based on a suitable change of variables in the membrane in order to write the explicit dependence of the studied differential operator in terms of the small parameter (the thinness of the membrane). Since we consider Maxwell equations in time-harmonic regime, in accordance with Flanders [15], Warnicket al.[35, 36] and Lassaset al. [17, 18], we choose the differential calculus formalism to perform our change of variables in a simple way. Basic notions of differential forms with explicit formulae are recalled in Appendix 2.

Throughout this paper, we consider a material composed of an interior do- main surrounded by a thin membrane. This material, representing a biological cell, is embedded in an ambient medium submitted to an electric current den- sity. We study the asymptotic behavior of the electromagnetic field in the three domains (the ambient medium, the thin layer and the cytoplasm) for the thick- ness of the membrane tending to zero. We derive appropriate transmission conditions on the boundary of the cytoplasm in order to remove the thin layer from the problem. Actually, the influence of the membrane is approached by these transmission conditions. To justify our asymptotic expansion, we provide

1The electroquasistatic approximation consists in considering that the electric field comes from a potential:E=−∇V. In this approximation the curl part of the electric field vanishes and the magnetic field is neglected.

piecewise estimates of the error between the exact solution and the approximate solution.

The paper is structured as follows. In Section 2, we present the studied problem in the differential calculus formalism and the main result of the paper.

Section 3 is devoted to the geometry: we perform our change of variables and we write the problem in the so-called local coordinates. In Section 4 we derive formally our asymptotic expansion, which is rigorously proved in Section 5. In Appendix 1, we give recurrence formulae to obtain asymptotics at any order.

Appendix 2 is devoted to the basic notions of differential forms used in the paper.

2 Maxwell equations using differential forms

In the following we use the conventions of differential calculus formalism. Defi- nitions and basic properties of the differential calculus are given in Appendix 2.

We recall here the usual notations.

Notation 2.1. Let k be an integer. For a compact, connected and oriented Riemanian manifold (M,g) of Rⁿ we denote by Ω^k(M) the space of k–forms defined on M.

• The exterior product between two differential forms ω and η is denoted by ω∧η.

• The inner product on Ω^k(M)is denoted by

·,·

Ω^k.

• The Hodge star operator is denoted by⋆.

• The interior product of a differential formω with a smooth vector fieldY is writtenint(Y)ω.

Let us denote the exterior differential and codifferential operators respectively by d,δ. The Laplace-Beltrami operator∆is defined by∆ =−dδ−δd.

L²Ω^k(M) is the space of the square integrable k−f orms of M while for s ∈ R, H^sΩ^k(M) is the usual Sobolev space of k-forms. Let HΩ^k(d, M) and HΩ^k(δ, M)denote

HΩ^k(d, M) =

ω∈L²Ω^k(M) : dω∈L²Ω^k+1(M) , (1) HΩ^k(δ, M) =

ω∈L²Ω^k(M) : δω ∈L²Ω^k−1(M) , (2) that are Banach spaces when associated with their respective norms

kωkHΩ^k(d,M)=kωkL²Ω^k(M)+kdωkL²Ω^k+1(M), kωkHΩ^k(δ,M)=kωkL²Ω^k(M)+kδωkL²Ω^k−1(M).

We also denote by HΩ^k(d, δ, M) the space HΩ^k(d, M)∩HΩ^k(δ, M) equipped with the norm

kωkHΩ^k(d,δ,M)=kωkL²Ω^k(M)+kdωkL²Ω^k+1(M)+kδωkL²Ω^k−1(M). H^s(M)andL²(M)denotes the respective spaces H^sΩ⁰(M)andL²Ω⁰(M).

2.1 The considered problem

Let Γ be a compact oriented surface of R³ without boundary. Consider the smooth connected bounded domain O^c with boundary Γ;O^c is surrounded by

a thin layer O^mε with constant thickness ε. This material with thin layer is embedded in an ambient smooth connected domainO^εe with compact oriented boundary. We denote byOtheε-independent domain defined by

O=O^εe∪ O^mε ∪ O^c.

Moreover, we denote by Γε the boundary of O^c∪ O^mε (see Figure 1). Letµ^c, µ^m and µ^e be three positive constants and letq^e, q^c and q^m be three complex numbers. Define the two piecewise functionsµandq onO by

∀x∈ O, µ(x) =







µ^e, inO^εe, µ^m, inO^εm, µc, inO^c,

∀x∈ O, q(x) =







q^e,inO^eε, q^m,inO^εm, qc,inO^c.

The function µ is the dimensionaless permeability of O while the function q denotes its dimensionaless complex permittivity².

O^c ε

Γε

Γ O^εm

Oe^ε

O

qc

µc

qmµm

qe

µe

Figure 1: Geometry of the problem

Let d0 >0 such that for each point q of Γ, the normal lines of Γ passing through q, with center at q and length 2d0 are disjoints. In the following, we suppose that ε ∈ (0, d0). We denote by O^ed0 the set of points x ∈ O^εe at the distance greater than d0 of Γ. We suppose that the ambient medium is submitted to a current density J compactly supported in O^de0. All along the paper the following hypothesis holds.

Hypothesis 2.2. (i)There existsc1, c2>0 such that for all x∈ O,

c16−ℑ(q(x))6c2, 0<ℜ(q(x))6c2. (3)

2Using the notations of the electrical engineeering community,q=ω²` ǫ−i^σ_ω´

, whereωis the frequency,ǫthe permittivity andσthe conductivity of the domain [3].

(ii)The source current density J satisfies

supp(J)⋐O^ed0, J∈L²Ω¹(O), δJ= 0, in O.

Maxwell equations describe the behavior of the electromagnetic field inO. Denote by Eand Hthe 1–forms representing respectively the electric and the magnetic fields in O in time-harmonic regime. Denote by N∂O the outward normal vector field to ∂O. In the following and accordingly Remark 2.2 of Appendix 2, the normal vector field and the corresponding normal 1–form are identified. Maxwell equations in time-harmonic regime write [17, 18, 35]

dE= i⋆(µH), dH=−i⋆(qE+J),in O, (4a)

N∂O∧E|^∂O = 0,on∂O. (4b)

Using the idempotence of⋆inR³, we may infer the vector wave equation onE

⋆d 1

µ⋆dE

−qE=J,inO, N∂O∧E|^∂O= 0,on∂O. Sinceµis a scalar function³ofO, we infer

δ 1

µdE

−qE=J,in O, N∂O∧E|^∂O = 0,on∂O. (5) Remark 2.3. DenoteEin Euclidean coordinates byExdx+Eydy+Ezdz and similarly forHandJ. Problem (4) and problem (5) write now

curlE= iµH, curlH=−i (qE+J),in O, N∂Ω×E|^∂Ω=O,on∂O, and

curl 1

µcurlE

−qE=J,inO, N∂Ω×E|∂Ω= 0,on∂O.

△

2.2 Regularized variational formulation.

Our functional spaceX(q) is defined as X(q) =

u∈HΩ¹(d,O), δ(qu)∈L²(O), N∂O∧u|∂O = 0 , associated with its graph norm

kukX(q)=kukHΩ¹(d,O)+kδ(qu)kL²(O).

Define the sesquilinear form aq in X(q) adapted to a regularized variational formulation of the problem (5) by

aq(u, v) = Z

O

1 µ

du,dv

Ω²+

δ(qu), δ(qv)

Ω⁰−q u, v

Ω¹

dvolO. Using inequalities (3), the following lemma holds.

3Ifµis a tensor the previous equation (5) becomesδ`

⋆µ⁻¹⋆dE´

−qE=J.

Lemma 2.4. There exists a constant c0 > 0 and α ∈ C such that for all ε∈(0, d0),

ℜ αaq(u, u)

>c0kuk²_X(q). (6) For allε∈(0, d0), we consider the variational problem: find E∈X(q) such that

∀u∈X(q), aq(E, u) = Z

O

J, u

Ω¹dvolO. (7)

Using Hypothesis 2.2 the following theorem holds.

Theorem 2.5(Equivalent problems). Let Hypothesis 2.2 hold.

(i)There is at most one solutionE∈X(q)to problem (7).

(ii)The solution Esatisfies (5)in a weak sense

δdE−µqE=J,in O^eε∪ O^εm∪ O^c, N∂O∧E|^∂O= 0, with the divergence condition

δ(qE) = 0,in O (8)

and the following equalities⁴ hold forS ∈ {Γ,Γε} [NS ∧E]S = 0,

1

µint(NS)dE

S

= 0, [qint(NS)E]S = 0. (9) (iii) If (E,H) ∈ L²Ω¹(O)2

is solution to problem (4) then E ∈ X(q) sat- isfies (5). Conversely, if E ∈ X(q) satisfies (5) then the couple of 1–forms (E,−(i/µ)⋆dE)belongs to L²Ω¹(O)2

and satisfies problem (4).

Proof. The proof is based on an idea of Costabelet al..

(i) Accordingly estimate (6), a straightforward application of the well-known Lax-Milgram theorem leads to existence and uniqueness of the solutionEto the regularized variational problem (7).

(ii) The proof is precisely worked out in full details in [5, 6] in a very slightly different configuration. We just give here the sketch of the proof. The first transmission conditions (9) easily come from Green formula of Proposition 2.8.

The divergence conditions straightforwardly come from δ(qE) = 0, using (5) andδ²≡0. The other implication is obvious. We then prove that this solution satisfiesδ(qE) = 0.

Denote byH∆(O) the space of functionsφ∈H₀¹(O) such thatδ(qdφ) belongs toL²(O). Integrations by parts imply

∀φ∈H∆(O), aq(E,dφ) = Z

O

δ(qE), δ(qdφ) +φ

Ω⁰dvolO.

Sinceℑ(q)≤ −c1 <0, the functionδ(qdφ) +φruns through the wholeL²(O) space forφrunning throughH∆ (O). Moreover, sinceδ(J) vanishes we have

Z

O

J,dφ

Ω¹dvolO = 0,

4For an oriented surfaceS without boundary and for a differential formudefined in a neighborhood ofS we denote by [u]S the jump acrossS. NS denotes the outward normal toS

from which we infer thatδ(qE) identically vanishes inL²(O) according to (7).

Therefore the solutionEof problem (7) solves problem (5).

(iii) If (E,H) solves problem (4) we straightforwadly infer (5), since⋆ is idem- potent and sinceµis a scalar function. Conversely, defining Hby

H=−i µ⋆dE, we infer that (E,H) solves problem (4).

Denote byO^ethe domainO^e=O \ O^c.Define ˜µand ˜qby

∀x∈ O, µ(x) =˜

(µ^c, inO^c,

µ^e, inO^e, ∀x∈ O, q(x) =˜

(q^c,in O^c, q^e,in O^e. LetE⁰∈X(˜q) be the “background” solution defined by

∀u∈X(˜q), aq˜(E⁰, u) = Z

O

J, u

Ω¹dvolO, which means in a weak sense

δ 1

˜ µdE⁰

−q˜E⁰=J,inO, N∂O∧E⁰|∂O = 0. (10) We have the following regularity result.

Proposition 2.6. Let Hypothesis 2.2 hold. Moreover let s ≥0 and J belong to H^sΩ¹(O^ed0). Then the 1-form E⁰ exists and is unique in X(˜q). Moreover, denoting byE^c,0 andEe^e,0 its respective restrictions to O^c andO^e, we have

e

E^e,0∈H^2+sΩ¹(O^e), E^c,0∈H^2+sΩ¹(O^c).

Proof. The 1–formE⁰satisfies (10). The proof of the existence and the unique- ness ofE⁰inX(˜q) is very similar to those performed in Theorem 2.5 by replacing X(q) byX(˜q) andaq byaq˜. SinceδJvanishes, we inferδ(˜qE⁰) = 0 and therefore E⁰ satisfies

−∆E⁰−µ˜˜qE⁰=J, in O^e∪ O^c, N∂O∧E⁰|^∂O = 0, with transmission conditions

NΓ∧dE⁰

Γ = 0,

˜

qint(NΓ)E⁰

Γ= 0, 1

˜

µint(NΓ)dE⁰

Γ

= 0, δ(˜qE⁰)

Γ= 0.

The same calculations as performed in Proposition 2.1 of Costabel et al. [6]

imply that the set of the above transmission and boundary conditions covers⁵ the Laplacian in O^c and in O^e, in the sense of Definition 1.5 at page 125 of Lions and Magenes [20]. Therefore we infer the piecewise elliptic regularity of E⁰, which ends the proof of the lemma.

5Accordingly the appendix of the paper of Li and Vogelius [19] the regularity ofE⁰ may also be obtained by using a reflection to reduce the problem to an elliptic system with com- plementing boundary conditions in the sense of Agmonet al.[1, 2].

The following estimates hold

Proposition 2.7. Under Hypothesis 2.2, there exists C > 0 such that for all ε∈(0, d0)

kEkX(q)6C, (11)

kE−E⁰kHΩ¹(d,O)6C√

ε. (12)

Proof. Using (6), estimates (11) are obvious since ℑ(q) 6 −c1 < 0. Prove now (12). We first mention that E⁰ belongs to H²Ω¹(̟) for ̟ ∈ {O^e,O^c}, according to Proposition 2.6; hence E⁰ ∈ L^∞Ω¹(̟) and dE⁰ ∈ L^∞Ω²(̟).

Denoting byU=E−E⁰ we infer Z

O

1

µhdU,dUiΩ²−qhU,UiΩ¹dvolO =q^m Z

O_m^εhE⁰,UiΩ¹dvolO_m^ε

− 1 µ^m

Z

O^ε_m

hdE⁰,dUiΩ²dvolO_m^ε.

Therefore using (11) and using the assumption (3) onq, we infer kdUkL²Ω²(O)+kUkL²Ω¹(O)6C√

ε.

2.3 Main result

Consider the inclusion J : Γ −→ O, and J^∗ its pull-back J^∗ : Ω^k(O) −→

Ω^k(Γ), k ∈ {0,1,2,3}. Denote by dΓ and δΓ the exterior differential and the codifferential operators defined on Ω^k(Γ). DefineSandTby

S= (qm−qe)J^∗(E⁰)|Γ⁺+ 1

µ^m − 1 µ^e

δΓdΓ J^∗(E⁰)

|Γ⁺, (13) T=qc

1 q^m− 1

q^e

d

int(NΓ)E⁰|Γ⁻

+µ^m−µ^e

µ^c int(NΓ) dE⁰

Γ⁻. (14) The explicit expressions ofSandT in local coordinates are given in Section 5.

LetE¹ be the 1–forms defined by

δdE¹−µ˜˜qE¹= 0,inO^e∪ O^c, N∂O∧E¹|∂O= 0, with the following transmission conditions on Γ

1

µ^eint(NΓ)dE¹|Γ⁺− 1

µ^cint(NΓ)dE¹|Γ⁻=S, (15) NΓ∧E¹|Γ⁺−NΓ∧E¹|Γ⁻ =NΓ∧T. (16) The aim of this paper is to prove the following theorem.

Theorem 2.8. Under Hypothesis 2.2, if moreover the current densityJbelongs toH³Ω¹(O^ed0), there existsε0>0and a constantC, independent onεsuch that

∀ε∈(0, ε0), kE−(E⁰+εE¹)kHΩ¹(d,δ,Oc)6Cε²,

and for any domain ̟ compactly embedded in O^e, there exists ε̟ > 0 and a constantC̟>0 independent onεsuch that

∀ε∈(0, ε̟), kE−(E⁰+εE¹)kHΩ¹(d,δ,̟)6C̟ε².

Remark 2.9. It is possible to give a precise behavior of Ein a neighborhood of Γ by defining a 1–form in the thin membrane (see Theorem 5.3). △ In this paper we choose to deal with differential forms, in accordance with Flanders [15]. This point of view has the convenience of considering both electric and magnetic fields as 1–forms, i.e. they are physically similar in accordance with electrical engineering considerations [3]. It is also possible to derive our asymptotics by tensor calculus considerations, as used in linear elasticity of thin shell [7, 12, 13]. This approach is worked out in full details in the thesis [25] of the first author.

Remark 2.10. [The tensor calculus formulation] Since we are confident that our result might be very useful for bioelectromagnetic computations, and since the electrical engineering community may feel uncomfortable with the differential calculus formalism, we translate our result with the help of “usual” differen- tial operators. Denote by ∇^Γ and∇^Γ· the respective gradient and divergence operators on Γ. Define RotΓ and rotΓ by

∀f ∈C^∞(Γ), RotΓf = (∇^Γf)×NΓ,

∀f ∈(C^∞(Γ))³, rotΓf =∇Γ·(f×NΓ). Then (E^k)k=0,1 (seen as vector field) satisfy the following equations

curl curlE^k−µ˜˜qE^k=δ^k₀J,in O^e∪ O^c, N∂O×E^k|∂O = 0, with the following transmission conditions on Γ

NΓ×E⁰|Γ⁺=NΓ×E⁰|Γ⁻, 1

µ^e NΓ×curlE⁰

|Γ⁺= 1

µ^c NΓ×curlE⁰

|Γ⁻, (17)

NΓ×E¹|Γ⁺×NΓ =NΓ×E¹|Γ⁻×NΓ+q^c 1

q^m − 1 q^e

∇^Γ E⁰|Γ⁻·NΓ +µ^m−µ^e

µ^c curlE⁰×NΓ

|Γ⁻,

(18) 1

µ^e curlE¹×NΓ

|Γ⁺ = 1

µ^c curlE¹×NΓ

|Γ⁻+ (q^m−q^e) NΓ×E⁰×NΓ

|Γ⁺

+ 1

µ^m − 1 µ^e

RotΓrotΓ NΓ×E⁰×NΓ

|Γ⁺.

△ Remark 2.11. [The impedance boundary condition of Engquist–N´ed´elec [11]]

LetJbe supported inO^c(and divergence free) and suppose thatO^εeis a perfectly conducting domain. Thereforeq^e= +∞andµ^e= 0. An homogeneous Dirichlet condition is imposed on Γε

NΓε×E|^Γε= 0.

We are now in the same configuration as the problem studied by Engquist and N´ed´elec [11] at page 18. According to (17)–(18), writing the condition satisfied

by E⁰+εE¹ and neglecting the terms in ε², we infer the following boundary condition for the first-order approximate fieldE_a

NΓ×E_a|Γ⁻×NΓ=−ε q^c

q^m∇^Γ(E_a|Γ⁻·NΓ) +µ^m

µ^c (curlE_a×NΓ)|Γ⁻

. Recall that according to Maxwell equations, curlE= iµcHand curlH=−iqcE. ThereforeqcE·NΓ = i curlH·NΓ. According to the definition of∇Γ· (see for example equation (2.22) page 5 of [11]), we infer⁶

∇^Γ·(H×NΓ) = curlH·NΓ =−iq^cE·NΓ, (19) and the impedance boundary condition follows

NΓ×E_a|Γ⁻×NΓ=−iε 1

q^m∇^Γ(∇^Γ·(H_a×NΓ)) +µ^m(H_a×NΓ)|Γ⁻

. Observe that this is the impedance boundary condition of given in [11] page 19, since they took the normal interior to their domain Ω∞, hencen=−NΓ. △ We point out few arguments to enlight the convenience of differential calculus formalism.

(i) Anisotropy. For sake of simplicity, we deal here with isotropic materials, however the anisotropic case may be interesting for applications. In this case, µandqare matrices and the vector wave equation becomes

δ ⋆µ⁻¹⋆ dE

−qE=J,inO N∂O∧E|^∂O= 0,on∂O, and the following transmission conditions hold onS ∈ {Γ,Γε}

int(NS) ⋆µ⁻¹⋆dE

S = 0, [NS ∧E]S = 0.

To obtain the approximate transmission conditions equivalent to the thin layer, we just have to write the tensor⋆µ⁻¹⋆in local coordinates, with the help of the explicit formulae given in Appendix 2. The calculations are more tedious but we are confident that the reader has all the tools to perform the analysis.

(ii) Non-constant thickness. We consider here a thin layer with constant thickness. As mentionned in Section 1 a high electric field may occur a dramat- ically local decreasing of the thickness of the membrane, possibly leading to the apparition of pores. Hence the thickness of the membrane is no more constant with respect to the tangential variable. As performed in [28], the change of variables would lead to additional terms in the transmission conditions. These terms would come from the fact that the coefficientsgi3of the matrix (gij) given by (20) do not vanish. The derivation of the asymptotics would be more tedious but, once again, we are confident that all the tools are given in the present paper to perform the calculation.

In the case of a rough thin layer, the present analysis may not be applied.

We have to introduce appropriate correctors as performed in [4].

6Using differential forms and since dN= 0 equality (2.05) implies

int(N^Γ)E⁰|Γ−=− 1 iqc

int(N^Γ)δ`

⋆H⁰´

=− 1 iqc

δ^Γ`

int(N)⋆H⁰|^Γ´

, which is exactly equality (19).