COMPACT IMBEDDINGS IN ELECTROMAGNETISM WITH INTERFACES BETWEEN CLASSICAL MATERIALS AND META-MATERIALS

INTERFACES BETWEEN CLASSICAL MATERIALS AND META-MATERIALS

LUCAS CHESNEL^∗ AND PATRICK CIARLET JR.^†

Abstract. In a meta-material, the electric permittivity and/or the magnetic permeability can be negative in given frequency ranges. We investigate the solution of the time-harmonic Maxwell equations in a composite material, made up of classical materials, and meta-materials with negative electric permittivity, in a two-dimensional bounded domainΩ. We study the imbedding of the space of electric fields intoL²(Ω)². In particular, we extend the famous result of Weber, proving that it is compact. This result is obtained by studying the regularity of the fields. We first isolate their most singular part, using a decompositionà laBirman-Solomyak. With the help of the Mellin transform, we prove that this singular part belongs to H^s(Ω)², for somes > 0. Finally, we show that the compact imbedding result holds as soon as no ratio of permittivities between two adjacent materials is equal to−1.

Key words. Maxwell’s equations, interface problem, meta-material, compact imbedding, regu- larity of fields, Mellin transform.

AMS subject classifications.

1. Introduction. We consider the solution of the time-harmonic Maxwell equa- tions in a composite material. A composite material is modelled by non constant electric permittivity ε and magnetic permeability µ. The variations of εand µ can be smooth, or piecewise smooth. Recently, some new composites appeared, includ- ing classical materials and meta-materials. A meta-material exhibits special prop- erties. In given frequency ranges, it can behave like a material with negative elec- tric permittivity or/and negative magnetic permeability. Examples of meta-materials [25, 26, 27, 12] include superconductors, left-handed materials, etc. Due to the sign change between a classical material and a meta-material, the usual mathematical ap- proaches fail to resolve the corresponding electromagnetic models. In other words, these composites raise challenging questions, both from the mathematical and numer- ical points of view.

In this paper, we focus on an essential tool to study time-harmonic Maxwell equa- tions in a bounded (connected) domain Ω of R^d (d is the space dimension), that is the compact imbedding of the space of electric fields inL²(Ω)^d. This result is indeed a key ingredient to solve the two instances of time-harmonic equations, namely the source problem (sustained vibrations) and the eigenvalue problem (free vibrations).

If the domain of interest is surrounded by a perfect conductor, the following functional space for electric fields,X_N(Ω, ε), appears. It is made up of vector fieldsv that belong toL²(Ω)^d, and such thatcurlv∈L²(Ω)^d, div(εv)∈L²(Ω)andv×n= 0 on∂Ω, where nis the unit outward normal vector to ∂Ω.

Our main objective is to find an extension of the Weber compact imbedding theorem in the case of a composite material including classical and negative meta- materials. In the landmark paper [28], Weber proved that XN(Ω, ε) is compactly

∗Laboratoire POEMS, UMR 7231 CNRS/ENSTA/INRIA, ENSTA ParisTech, 32, boulevard Vic- tor, 75739 Paris Cedex 15, France;email: [email protected]

†Laboratoire POEMS, UMR 7231 CNRS/ENSTA/INRIA, ENSTA ParisTech, 32, boulevard Vic- tor, 75739 Paris Cedex 15, France;email: [email protected]

1

imbedded intoL²(Ω)^d whenεis a real function, bounded above and below by strictly positive constants, ford= 2,3.

In [9], the second author and co-workers prove this result for d = 3, in the following setting. The domainΩis partitioned into two subdomainsΩ₁andΩ₂, with a smooth interface (between the subdomains), and withεexhibiting a sign-changeat the crossing of the interface. This new result is proved under the assumption that the contrastκ_ε (which is just κ_ε =ε_|Ω1/ε_|Ω2 whenε is constant in each subdomains) is such that κε ∈/

κⁱ_ε; κ^s_ε

, with −1 ∈ κⁱ_ε;κ^s_ε

. The proof is direct, in the sense that the classical result stating that the imbeddings ofH^s(Ω)into L²(Ω) are compact for s >0, is not used. Our method of proof, on the other hand, is to study thea priori regularity of elements of XN(Ω, ε). In this respect, it follows what has been done in proposition 3.7 of [1], or section 3 of [14], or in part 3.8 of [23], but it relies on different mathematical tools. More precisely, we shall prove that the space of electric field X_N(Ω, ε) is (continuously) imbedded into spaces like H^s(Ω)^d, with some ad hoc s >0 with the help of the Mellin transform, so that one can obtain the desired compactness result by using the classical result mentioned above. Along the way, we shall use several results of [16].

This compactness result is one of the two key ingredient result to prove that the source problem is well-posed, within the Fredholm, or isomorphism+compact, frame- work. The T-coercivity method (see [10]), or the three-field variational formulation (see [9]), then help obtaining the second key ingredient, coerciveness, so well-posedness follows. These two techniques provide some guidelines as to how the problem can be discretized.

In this paper, we shall study the two-dimensional case (d= 2). To simplify the proofs, we assume that the domain Ω has a connected boundary ∂Ω, and that εis piecewise constant. If that were not the case, the regularity results of elements of X_N(Ω, ε) should remain valid, under reasonable assumptions, such as a piecewise smooth ε, with |ε| bounded above and below by strictly positive constants, or such as a multiply-connected boundary, which adds a finite dimensional vector subspace of curl-free and divergence-free elements ofX_N(Ω, ε), whose elements can be studied separately.

We begin by the notations in section 2. We shall consider the case of a polygonal domain Ω, partitioned into polygonal subdomains. In the next section, we prove a continuous splitting result for elements of X_N(Ω, ε), which allows us to isolate the most singular part, expressed as a gradient of a scalar field. Then, in section 4, we study the regularity of this singular part, which leads to the compact imbedding result (§5). We generalize this result in section 6 by assuming that the domain and subdomains are curvilinear polygons. We conclude by providing a counter-example for a symmetric cavity composed of two materials whose permittivities are opposite.

2. Notations. LetΩbe a bounded, open, connected polygonal set ofR², with connected boundary ∂Ω. The unit outward normal vector to ∂Ω is denoted by n, whereasτ is a unit tangent vector to∂Ω.

We assume thatΩis partitioned intoN open, polygonal subsetsΩ_j:

Ω =

N

[

j=1

Ω_j withΩ_i∩Ω_j =∅ifi6=j.

We callP the partition. The interface between subdomainsΣ :=∪_i6=j(∂Ωi∩∂Ωj)is made up of straight edges. Without loss of generality, we consider convex polygonsΩj, as they can be further subdivided into convex polygons if necessary. Then, to avoid unnecessary technical difficulties, we assume that the N polygons of the partition P can be labelled so thatΩjandΩj+1share at least one common edge, forj= 1.. N−1.

For j = 1.. N, (∂Ω^k_j)k=1.. n_j denote the set of edges of Ωj, and(n^k_j)k=1.. n_j are the corresponding unit outward normal vectors.

Let A denote the set of edges on the boundary ∂Ω (a side of ∂Ω can be divided into two or more edges, due to the partition), or on the interfaceΣ: define Aext :=

{A∈ A |A⊂∂Ω} the set of exterior edges, and Aint := A\Aext the set of inte- rior edges. Similarly, let S denote the set of vertices (new vertices can appear on

∂Ω, due to the partition), andSext :={S ∈ S |S∈∂Ω} the set of exterior vertices, S_int:=S\S_extthe set of interior vertices.

In what follows, the characteristic quantitiesεandµdefining the material, func- tions, etc. arecomplex valued. Let us consider more precisely ε ∈L^∞(Ω), constant over each subdomain Ωj, with εj := ε|Ω_j 6= 0. As a particular case, ε can be real valued and exhibit a sign-shift between two neighboring subdomains. For a < b, we define the closed strip of the complex planeB(a;b) :={λ∈C|a≤ <(λ)≤b}.

Using differentiation in the sense of distributions, one classically defines the oper- ators curl and div acting overL²(Ω) :=L²(Ω)×L²(Ω), and the operatorcurl acting overL²(Ω). Forv= (vx, vy)∈L²(Ω),

curlv= ∂vy

∂x −∂vx

∂y ∈ D⁰(Ω), divv=∂vx

∂x +∂vy

∂y ∈ D⁰(Ω).

Forv∈L²(Ω),

curlv= ∂v

∂y,−∂v

∂x

!

∈ D⁰(Ω)× D⁰(Ω).

Let us introduce the functional spaces H(curl; Ω) :=

u∈L²(Ω)|curlu∈L²(Ω) , H(div;ξ; Ω) :=

u∈L²(Ω)|div(ξu)∈L²(Ω) , whereξbelongs toL^∞(Ω). Next, let

XN(Ω, ε) :={u∈H(curl; Ω)∩H(div;ε; Ω)|u·τ = 0on∂Ω}. Endowed with the normkuk_X

N(Ω):=

kuk²_L2(Ω)+kcurluk²_L2(Ω)+kdiv(εu)k²_L2(Ω)

¹₂ , X_N(Ω, ε) is a Hilbert space. Following Grisvard [18, 19], for any edge ∂Ω^k_j, we let He¹²(∂Ω^k_j) denote the set of elements of H¹²(∂Ω^k_j) whose continuation by 0 to ∂Ω_j belongs toH¹²(∂Ωj).

For u∈L²(Ω) (respectively u ∈ L²(Ω)), we use the notation uj :=u|Ω_j (resp.

uj :=u|Ω_j). To study the regularity of scalar fields, we consider P H^s(Ω,P) :=

u∈L²(Ω)|u_j∈H^s(Ω_j), j= 1.. N fors >0.

On P H^s(Ω,P), we introduce the broken norm kuk_{P H}s(Ω) :=

PN

j=1kujk²_Hs(Ωj)

¹₂ . We recall that, fors <1/2,kuk_{P H}^s_(Ω)' kuk_Hs(Ω). The vector valued counterpart is

PH^s(Ω,P) :=P H^s(Ω,P)×P H^s(Ω,P).

So, we can introduce the subspace of piecewise smooth elements ofX_N(Ω, ε), which writes

HN(Ω, ε) :=XN(Ω, ε)∩PH¹(Ω,P).

Finally, we define families of weighted functional (Sobolev) spaces. For a 2-index α= (α₁, α₂)∈N², we let

|α|:=α1+α2 and ∂_x^α:= ∂^|α|

∂^α1x ∂^α2y. GivenO any open subset ofR²,m∈Nandγ∈R, we introduce

V_γ^m(O) :=n

v∈L²_loc(O)|r^|α|−m+γ∂_x^αv∈L²(O),∀α∈N²,|α| ≤mo whereris the distance to some given point ofO, with norm

kvk_Vm γ (O):=



 X

|α|≤m

r^|α|−m+γ∂_x^αv

2 L²(O)





1 2

.

To conclude,V^◦m_γ(O)is the closure ofD(O)in V_γ^m(O).

3. A continuous splitting of fields of X_N(Ω, ε). Let us prove a continuous splitting of fields that belong to XN(Ω, ε). This kind of result can be traced back to the works of Birman-Solomyak [5, 4, 6, 7]. We give here a proof that follows the lines of those of theorems 3.4 and 3.5 of [14]. By assumption, the partitionP ofΩis labelled so thatΩj and Ωj+1 share at least one common edge, forj = 1.. N−1: we call(Aj)_{j=1.. N−1} these edges.

Proposition 3.1. Let u ∈ XN(Ω, ε). There exist u1 ∈ HN(Ω, ε) and u2 ∈ XN(Ω, ε)with(curlu2,1)L²(Ω_j)= 0 forj = 1.. N, such that

u=u₁+u₂. In addition, one has the stability estimate

ku1k_PH1(Ω)+ku2k_X

N(Ω)≤Ckuk_X

N(Ω) (3.1)

where the constant C is independent ofu.

Proof. For anyw∈XN(Ω, ε), we definemj(w) := (curlw,1)_L2(Ωj), forj= 1.. N.

To prove our claim, let us build explicitly a family (f_j)_{j=1.. N−1} of elements of HN(Ω, ε), such that

u₂=u−

N−1

X

i=1

c_if_i withc_i =

i

X

k=1

m_k(u), i= 1.. N−1 (3.2)

automatically fulfills the conditionsmj(u2) = 0,j = 1.. N.

For j = 1.. N −1, let nj be the unit normal vector to Aj, going from Ωj to Ωj+1, andτj such that (τj,nj)is a direct, orthonormal basis. Let Mj be a given interior point of Aj, and let rj be the distance toMj. Next, define f_j :=Cjζj(rj)τj, with a positive-valued function ζj ∈ D(R⁺), equal to 1 in a neighborhood of 0, with support such that suppf_j ∩ S = ∅ and (suppf_j ∩ A) ⊂ Aj. The constant Cj

is chosen so that (curlf_j,1)_L2(Ωj) = R

A_jf_j ·τj = 1. Note that, by construction, f_j|Ω_j·τ−f_j|Ω_j+1·τ= 0andεf_j|Ω_j·nj−εf_j|Ω_j+1·nj = 0. Sincef_j is smooth on Ωj andΩj+1 (and supported inΩj∪Ωj+1), we infer thatf_j∈HN(Ω, ε).

Now, let u2 =u−PN−1

i=1 cif_i ∈ XN(Ω, ε)be defined as in (3.2), and let us check thatmj(u2) = 0,j= 1.. N. First, for1≤j≤N,1≤i≤N−1, we remark that

m_j(f_i) =







1 ifj=i

−1 ifj=i+ 1 0 else. So, we have

m₁(u₂) =m₁(u)−

N−1

X

i=1

c_im₁(f_i) =m₁(u)−c₁= 0, and, forj= 2.. N−1,

mj(u2) =mj(u)−

N−1

X

i=1

cimj(f_i) =mj(u)−cj+c_j−1= 0.

On the other hand,

mN(u2) = mN(u) +c_N−1 =

N

X

i=1

mi(u) = 0because u·τ= 0 on∂Ω.

Finally, let u₁ := PN−1

i=1 c_if_i ∈ H_N(Ω, ε). For i = 1.. N −1, there holds |c_i| ≤ Pi

k=1|mk(u)| ≤Pi k=1

p|Ωk| kcurluk_L2(Ω_k) ≤Pi k=1

p|Ωk| kuk_X

N(Ω). As a conse- quence,ku1k_PH1(Ω)≤C₁kuk_X

N(Ω). This leads to (3.1), as ku2k_X

N(Ω)≤ ku1k_X

N(Ω)+kuk_X

N(Ω)≤C₂ku1k_PH1(Ω)+kuk_X

N(Ω). Theorem 3.2. Let u∈X_N(Ω, ε). There exist u₀ ∈H_N(Ω, ε) andϕ∈H₀¹(Ω) with div(ε∇ϕ)∈L²(Ω), such that

u=u₀+∇ϕ.

Furthermore, one has the stability estimate

ku₀k_PH1(Ω)+k∇ϕk_L2(Ω)+kdiv(ε∇ϕ)k_L2(Ω)≤C kuk_X

N(Ω)

where the constant C is independent ofu.

Proof. Letu∈X_N(Ω, ε). Thanks to proposition 3.1, there existu₁∈H_N(Ω, ε) andu₂∈X_N(Ω, ε)such thatu=u₁+u₂, with(curlu₂,1)_L2(Ω_j)= 0, forj= 1.. N.

In addition, we have the estimate

ku₁k_PH1(Ω)+ku₂k_X

N(Ω)≤C₁ kuk_X

N(Ω).

Let us begin by a study of u2. Thanks to the assumption (curlu2,1)_L2(Ωj) = 0, for every j = 1.. N, there exist one, and only one up to a constant, φj in H¹(Ωj) that solves

−∆φj =curlcurlφj = curlu2 inΩj

∂φj

∂n_j = 0 on∂Ωj. (3.3)

All polygons Ωj are convex, so φj ∈ H²(Ωj) (cf. theorem 2.4.3 of [19]). Next, we define v ∈ L²(Ω) by v|Ω_j = curlφj, for j = 1.. N: it belongs to PH¹(Ω,P), and kvk_PH1(Ω)≤C2 kcurlu2k_L2(Ω). In addition, forj = 1.. N,k= 1.. nj, there holds

v·τ^k_j = ∂φj

∂n^k_j = 0 (3.4)

where τ^k_j is such that (τ^k_j,n^k_j) is a direct, orthonormal basis. Consequently, curlv belongs toL²(Ω), andw:=u2−v ∈H(curl; Ω)is curl-free. According to theorem 2.9 of [17, chapter 1] and thanks to the connectedness of ∂Ω, there exists one, and only oneϕ₁∈H₀¹(Ω) such that

w=∇ϕ1. So far,uhas been split as

u=u1+v+∇ϕ1

where u₁ ∈ H_N(Ω, ε), v ∈ PH¹(Ω) and ϕ₁ ∈ H₀¹(Ω). However, except when ε is constant over Ω, div(εv) does not belong to L²(Ω). Using either theorem 1.5.2.8 of [18] or theorem 1 of [3], one can lift the normal trace of v_j on∂Ω_j, resulting in r∈P H²(Ω,P)∩H₀¹(Ω)such that rj ∈H²(Ωj)∩H₀¹(Ωj)that fulfills

∂r_j

∂n^k_j|_∂Ωk

j = v_j·n^k_j

|_∂Ωk

j, k= 1.. n_j, j= 1.. N and krk_{P H}2(Ω)≤C3 kvk_PH1(Ω).

Note that this is possible as soon as vj·n^k_j

|_∂Ωk

j ∈He¹²(∂Ω^k_j): as a matter of fact, there holdsv_j·n^k_j = ∂φj

∂τ^k_j ∈He¹²(∂Ω^k_j), because ∂φj

∂n^k_j⁰ = 0for allk⁰ = 1.. n_j. Define nextu3:=v− ∇r. This field is such that

i) u3∈L²(Ω) ii) u3|Ω_j ∈H¹(Ωj)

iii) curlu3=curlv=curlu2∈L²(Ω) iv) u₃·τ|∂Ω=v·τ|∂Ω− ∇r·τ|∂Ω =

cf. (3.4)

−∇r·τ|∂Ω =

r∈H₀¹(Ω)

0.

Now, as ε_ju₃|Ωj·n^k_j

|_∂Ωk

j = 0,k= 1.. n_j, j= 1.. N, one obtains div(εu₃)∈L²(Ω).

Combining those arguments allows us to conclude thatu3∈HN(Ω, ε) =XN(Ω, ε)∩ PH¹(Ω,P).

Letϕ:=ϕ₁+r∈H₀¹(Ω) andu₀:=u₁+u₃. By construction u=u₀+∇ϕ

with div(ε∇ϕ) =div(εu)−div(εu0)∈L²(Ω). Also

ku0k_PH1(Ω) ≤ ku1k_PH1(Ω)+kvk_PH1(Ω)+k∇rk_PH1(Ω)

≤ ku1k_PH1(Ω)+ (1 +C3)kvk_PH1(Ω)

≤ ku₁k_PH1(Ω)+C₂(1 +C₃)kcurlu₂k_L2(Ω)

≤ C1(1 +C2(1 +C3))kuk_X

N(Ω)

k∇ϕk_L2(Ω) ≤ ku−u₀k_L2(Ω)

≤ kuk_X

N(Ω)+ku0k_L2(Ω)

≤ (1 +C4)kuk_X

N(Ω)

and kdiv(ε∇ϕ)k_L2(Ω) ≤ kdiv(εu)k_L2(Ω)+kdiv(εu₀)k_L2(Ω)

≤ kdiv(εu)k_L2(Ω)+C5 ku0k_PH1(Ω)

≤ (1 +C4C5)kuk_X

N(Ω)

whereC₄:=C₁(1 +C₂(1 +C₃))et C₅:= max

j |εj|.

4. A study of the regularity. Recall that Ω is a bounded, open, connected polygonal set ofR², with connected boundary∂Ω. In this section, we show that, under some assumptions that will be described later on, the scalar potential ϕ ∈ H₀¹(Ω) that appears in the splitting of theorem 3.2 is actually “more regular thanH¹”. More precisely, we prove that there existsσ₀>1, which depends only onΩ, on the partition and onε, such that one hasϕ∈ ∩s < σ₀H^s(Ω).

Let us consider the unbounded operatorF inL²(Ω):

D(F) :=

u∈H₀¹(Ω)|div(ε∇u)∈L²(Ω)

F u := div(ε∇u) . (4.1)

Now, we study the regularity of an elementu∈D(F). From time to time, we shall use the notationf :=F u=div(ε∇u)∈L²(Ω).

Let us begin by some well-known results. Classically (chapter 2, volume 1 of [22];

theorem 2.1.3 of [19], or [2]), the following interior regularity holds.

Theorem 4.1. Let Obe an open subset ofΩsuch thatO ∩ S=O ∩ A=∅. Then ubelongs toH²(O), with estimate

kuk_H2(O)≤C

kdiv(ε∇u)k_L2(Ω)+k∇uk_L2(Ω)

where the constant C is independent ofu.

Furthermore, theorem 2.1.4 of [19] provides the following regularity result around exterior edges.

Theorem 4.2. Let O be an open subset ofΩ such thatO ∩ S =O ∩ A_int =∅.

Thenubelongs toH²(O), with estimate kuk_H2(O)≤C

kdiv(ε∇u)k_L2(Ω)+k∇uk_L2(Ω)

where the constant C is independent ofu.

4.1. Regularity at interior edges: preliminaries. Let M be an interior point ofA∈ A_int. Assume thatΩ₁ andΩ₂ are the two subdomains of P such that Ω₁∩Ω₂=A. Considerdsmall enough so thatB(M, d)∩S=∅and(B(M, d)∩A)⊂A

where B(M, d) denotes the open ball of centreM and radiusd. Evidently, the fact that the value ofεjumps atApreventsa prioriuto belong toH²(B(M, d)). However, using a technique based on partial Fourier transform alongside one direction, we shall prove thatuj ∈H²(B(M, d)∩Ωj), j = 1,2. The proof of this result can be found in [16] (theorem 2.1). We will give a slightly different version using the T-coercivity approach (see [10, 8] for more details concerning this technique).

Let us begin by some geometric notations. Let(r, θ)denote the polar coordinates with respect to M. The angular coordinateθ is chosen arbitrarily. Letχ ∈ D(R⁺) be such that0≤χ≤1, equal to 1 on[0;d], with support contained in[0;dM]. Here dM > dis small enough so thatB(M, dM)∩S =∅and(B(M, dM)∩A)⊂A. Define the radial cutoff functionχ_M : (r, θ)7→χ(r)and the infinite stripsI:=R×I,Ij:=R×Ij, j= 1,2, with respectivelyI:= ]−dM;d_M[,I₁:= ]−dM; 0[andI₂:= ]0;d_M[. Without loss of generality, we suppose thatA⊂R× {0}, (B(M, d_M)∩Ω_j)⊂ I_j,j= 1,2.

Now, we localize the study of the regularity with the help of χM. Let f˜be the extension of f = F u = div(ε∇u) by 0 to I. Consider u := χMu; u belongs to H₀¹(B(M, dM)), so its extensionw:= ˜uby0 toI belongs toH¹(I). Consider next

p:=ε(˜u∆χM + 2∇˜u· ∇χM) + ˜f χM, (4.2) which belongs to L²(I), with compact support. According to its definition, w is a solution to the transmission problem in the infinite stripI

(Pstrip)

Findw∈H¹(I) such that

εj∆wj =pj inIj, j= 1,2 w_j= 0 on∂Ij∩∂I, j= 1,2 w₁−w₂= 0 on∂I1 ∩ ∂I2

ε₁∂_yw₁−ε₂∂_yw₂= 0 on∂I₁ ∩ ∂I₂.

4.2. Regularity at interior edges: Fourier transform. Applying the Fourier transform with respect to xto the equations of (Pstrip) for λ ∈ Ri, one finds that y7→w(λ, y) :=ˆ R+∞

−∞ e^−λxw(x, y)dxis a solution to

( ˆPstrip)

ε_j ∂²_y+λ² ˆ

w_j(λ, y) = ˆp_j(λ, y) inIj, j= 1,2 ˆ

w1(λ,−dM) = ˆw2(λ, dM) = 0 ˆ

w1(λ,0) = ˆw2(λ,0)

ε1∂ywˆ1(λ,0) =ε2∂ywˆ2(λ,0).

Let us introduce the symbol related to( ˆPstrip)

Lstrip(λ) : D(Lstrip) −→ L²(I) v 7−→ (ε λ²+dyε dy)v whereD(Lstrip) :=

v∈H₀¹(I)|v_j∈H²(I_j), j= 1,2, ε₁v⁰₁(0) =ε₂v⁰₂(0) .

Lemma 4.3. Assume that ε₁+ε₂ 6= 0. Then Lstrip(λ) is an isomorphism from D(L_strip)toL²(I)for allλ∈Ri.

Proof. Denote respectively(·,·),(·,·)₁,(·,·)₂the scalar products of L²(I),L²(I₁) and L²(I₂). Define τ :=iλ∈R. If ψ is a measurable function onI, we will denote ψj:=ψ|I_j,j= 1,2. Let us introduce the sesquilinear form associated withLstrip(λ).

Forv,ψ∈H₀¹(I), it is defined by a(v, ψ) :=

2

X

j=1

ε_j(v⁰_j, ψ⁰_j)_j+τ²ε_j(v_j, ψ_j)_j .

Let us distinguish two cases according to the value of the ratioε2/ε1.

•Ifε2/ε1∈C\R⁻, then the formais coercive onH₀¹(I).

Thus, for every g ∈ H⁻¹(I), there exists a unique v ∈ H₀¹(I) such that (ε λ² + dyε dy)v=g inI. If moreovergbelongs toL²(I)thend²_yv1∈L²(I1)sov1∈H²(I1).

Similarly, v2 ∈ H²(I2). Since Lsym(λ) is continuous from D(Lstrip) to L²(I), one can use the open mapping theorem to conclude thatLstrip is an isomorphism for all λ∈Ri.

•Ifε2/ε1∈R⁻∗, the formais no more coercive onH₀¹(I). Note that, asε1+ε26= 0,we know thatε2/ε16=−1.

To address this difficulty, we use the T-coercivity method, cf. [10, 8]. Introduce the operators R₁, R₂ such that (R₁v₁)(y) = v₁(−y), (R₂v₂)(y) = v₂(−y) and the isomorphisms (T₁◦T₁=T₂◦T₂=Id) ofH₀¹(I)respectively defined by

T₁v:=

v1 onI1

−v2+ 2R1v1 onI2

and T₂v:=

v1−2R2v2 onI1

−v2 onI2

. For allv∈H₀¹(I), one can write, using Young’s inequality, for allη >0,

|ε⁻¹₁ a(v, T₁v)| = |(v₁⁰, v₁⁰)₁+τ²(v₁, v₁)₁+|ε₂/ε₁|((v⁰₂, v₂⁰)₂+τ²(v₂, v₂)₂) +2(ε2/ε1)((v⁰₂,(R1v1)⁰)2+τ²(v2,(R1v1))2)|

≥ (1−η⁻¹|ε2/ε1|)((v⁰₁, v₁⁰)1+τ²(v⁰₁, v₁⁰)1) +|ε2/ε1|(1−η)((v₂⁰, v⁰₂)2+τ²(v₂⁰, v⁰₂)2).

Thus, if|ε2/ε1|<1, taking η such that|ε2/ε1|< η <1, one infers the existence of a constantC independent ofτ such that

|a(v, T1v)| ≥C((v⁰, v⁰) +τ²(v, v)),∀v∈H₀¹(I). (4.3) Since T1 is an isomorphism of H₀¹(I), this proves that for everyg ∈ H⁻¹(I), there exists a uniquev∈H₀¹(I)such that(ε λ²+d_yε d_y)v=g.

One proceeds similarly in the case|ε2/ε₁|>1, working withT₂. Finally, we conclude as in the caseε₂/ε₁∈C\R⁻.

Remark 4.4. Notice that theT-coercivity method can not be used to deal with the caseε₂+ε₁ = 0, for whichε₂/ε₁=−1. Now, let us study the norm ofL_sym(λ)⁻¹, which is an operator fromL²(I)to D(Lsym).

Lemma 4.5. Assume thatε1+ε26= 0. Then there exists a constantCindependent of λ∈Ri such that

P2

j=1kvjk_H2(Ij)+|λ|²kvk_L2(I)≤CkLstrip(λ)vk_L2(I), for allv∈D(Lstrip).

Proof. We prove this result in the caseε2/ε1∈R⁻∗\{−1}, the caseε2/ε1∈C\R⁻ being easier to tackle. Consider v ∈ D(Lsym) and denote g = Lsym(λ)v. Suppose

|ε2/ε1| <1. According to (4.3), one can writeC|λ|²(v, v)≤a(v, T1v) = −(g, T1v), withC >0 independent of λ. Noticing that T₁ is a continuous operator from L²(I) toL²(I), one obtains the estimation

|λ|²kvk_L2(I)≤Ckgk_L2(I) (4.4) whereC does not depend onλ. On the other hand, sinceε_j(λ²+d²_y)v_j=g_j,j= 1,2, one has

d²_yv_j

L²(Ij)≤Ckgk_L2(I), j= 1,2. (4.5)

From (4.3), one can also writeC(v⁰, v⁰)≤a(v, T1v), hence

kdyvk_L2(I)≤Ckgk_L2(I). (4.6) Using (4.4), (4.5) and (4.6), we can finally assert the existence ofC >0 independent ofλ∈Risuch that

2

X

j=1

kvjk_H2(Ij)+|λ|²kvk_L2(I)≤C kgk_L2(I), (4.7)

for allv∈D(Lsym).

Working withT2, the case|ε2/ε1|>1 can be handled similarly.

SinceLstrip(λ) ˆw(λ,·) =p(λ,·), thanks to lemma 4.5, one obtains

2

X

j=1

kwˆ_j(λ,·)kH²(I_j)+|λ|²kw(λ,ˆ ·)kL²(I)≤Ckˆp(λ,·)kL²(I), ∀λ∈Ri.

Above,C >0is independent ofλ.

With the help of the Parseval identity (see the lemma 5.2.4 of [21]), one deduces that w_j∈H²(I_j),j= 1,2. Besides, there holds the estimate

kw1k_H2(I1)+kw2k_H2(I2)≤Ckpk_L2(I).

Using the expression (4.2) ofpand noticing thatχM = 1overB(M, d), one concludes

2

X

j=1

kujk_H2(B(M,d)∩Ω_j)≤C

kdiv(ε∇u)k_L2(Ω)+k∇uk_L2(Ω)

.

Let us summarize this result with the

Theorem 4.6. LetObe an open subset ofΩsuch thatO ∩ S=∅ and(O ∩ A)⊂ A= Ωi∩Ωj. Under the assumptionεi+εj6= 0,ui belongs toH²(O ∩Ωi),uj belongs toH²(O ∩ Ωj), with estimate

kuik_H2(O ∩Ω_i)+kujk_H2(O ∩Ωj)≤C

kdiv(ε∇u)k_L2(Ω)+k∇uk_L2(Ω)

where the constant C is independent ofu.

4.3. Regularity at boundary vertices: preliminaries. To carry out the study in the neighborhood of boundary vertices, we shall follow the method of proof given in [16], which relies itself on the founding paper of Kondrat’ev [20].

As in the study of the regularity at interior edges, let us begin by some geometric notations. ForS∈ Sext, let(r, θ)denote the polar coordinates with respect toS. The angular coordinateθis chosen in such a way that, ford >0small enough, there holds

B(S, d)∩Ω ={(rcosθ, rsinθ)|(r, θ)∈[0;d]×[0;θmax]}. One has alwaysθ_max≤2π.

Next, letd_S >0be small enough, so thatB(S, d_S)∩ S={S}. Letχ∈ D(R⁺)be such that 0≤χ≤1, equal to1 on

0;^d₂^S

, with support contained in [0;d_S]. Then, the radial cutoff function χ_S : (r, θ)7→χ(r)is such that suppχ_S ∩ S ={S}, where suppχ_S is the support ofχ_S.

Define Ω := Ω˜ ∩B(S, dS). We number again the JS subdomains that have S as one of their boundary vertices, from 1 to JS. Further, if we denote by Ω˜j :=

Ωj∩B(S, dS), j = 1.. JS, the new numbering is such that Ω˜j and Ω˜j+1 share one edge, forj= 1..(J_S−1), whose angleθgrows withj. Then, forj = 1.. J_S,δσ_jdenotes the interior opening ofΩ˜j (with δσj ≤πbecause Ω˜j is convex), and we setσ0:= 0, and σj :=σ_j−1+δσj, j = 1.. JS. By definition, there holds P

j=1.. JSδσj = θmax. Finally, we define the intervalsGj:= ]σ_j−1;σj[,j= 1.. JS, andG:= ]0;σJ_S[, and the unbounded angular sectors

Γ :=

(x, y)∈R²|r >0, θ∈G , Γ_j :=

(x, y)∈R²|r >0, θ∈G_j , j= 1.. J_S.

Forv∈L²( ˜Ω)(resp. v∈L²(Γ)), we writev_j:=v|Ω˜j (resp. v_j :=v|_Γj),j= 1.. J_S. We introduceΛε, S, the set of singular exponents related to the vertex S, which we define as the set of complex numbersλ∈Csuch that there is a non-zeroJS-tuple (φλ, j)^J_j=1^S ∈QJS

j=1H²(Gj)which fulfills the conditions below:

∂²_θ+λ²

φλ, j= 0 in Gj, j= 1.. JS

φλ,1(0) =φλ, J_S(σJ_S) = 0

φλ, j(σj) =φλ, j+1(σj) j = 1..(JS−1) εj∂θφλ, j(σj) =εj+1∂θφλ, j+1(σj) j = 1..(JS−1).

Next, we localize the study of the regularity, usingχS. Letf˜be the extension of f by 0 to Γ. Next consideru:=χSu: ubelongs to H₀¹( ˜Ω), so its extension w:= ˜u by0toΓ belongs toH¹(Γ). Consider next

p:=ε(˜u∆χS+ 2∇˜u· ∇χS) + ˜f χS, (4.8) which belongs to L²(Γ), with compact support. According to its definition, w is a solution to the transmission problem

(Psector)

Findw∈H¹(Γ) such that

ε_j∆w_j =p_j inΓ_j, j= 1.. J_S

w₁= 0 on∂Γ₁∩∂Γ

w_JS = 0 on∂Γ_JS ∩∂Γ

w_j−w_j+1= 0 on∂Γ_j ∩∂Γ_j+1, j= 1..(J_S−1) ε_j∂_θw_j−ε_j+1∂_θw_j+1 = 0 on∂Γ_j ∩∂Γ_j+1, j= 1..(J_S−1).

4.4. Regularity at boundary vertices: Mellin transform. For the defini- tion of the Mellin transform, let us recall a classical lemma (see §2, chapter 2 of [24], or Annex AA of [15]).

Lemma 4.7. Let γ∈Rands∈N.

If v∈V_γ^s(Γ), then one can define its Mellin transform ˆ

v(λ,·) :=Mv(λ,·) = Z +∞

0

r^−λv(r,·)dr r

for λ ∈ C such that <(λ) = s−γ −1. Moreover, one has η 7→ v(ξˆ +iη,·) ∈ L²(R, H^s(G)), whereξ:=s−γ−1.

In the problem of interest,wbelongs to H¹(Γ), with a compact support. So, it also belongs toV_γ¹(Γ), for allγ >0. As a consequence, one can define its Mellin transform

ˆ

w(λ,·)on the complex lines{λ∈C| <(λ) =−γ}, for allγ >0. It follows thatw(λ,ˆ ·) is well-defined over the complex half-plane{λ∈C| <(λ)<0}.

Next, we shall useh:=r²p, wherepis defined in (4.8). Sincep∈L²(Γ)and moreover pis compactly supported,hbelongs toV_−2+γ⁰ (Γ), for allγ≥0. In this case, its Mellin transformθ7→ˆh(λ, θ)is well-defined over the complex half-plane{λ∈C| <(λ)<1}.

Multiplying byr²the volume PDEs in(Psector), and then carrying out the Mellin transform forλ∈Csuch that<(λ)<0, one finds thatθ7→w(λ, θ)ˆ is a solution to

( ˆPsector)

εj ∂_θ²+λ² ˆ

wj(λ, θ) = ˆhj(λ, θ) inGj, j= 1.. JS

ˆ

w1(λ,0) = ˆwJ_S(λ, σJ_S) = 0 ˆ

wj(λ, σj) = ˆwj+1(λ, σj) j= 1..(JS−1) εj∂θwˆj(λ, σj) =εj+1∂θwˆj+1(λ, σj) j= 1..(JS−1).

Next, consider the Mellin symbol related to( ˆP_sector)

L(λ) : D(L) −→

JS

Y

j=1

L²(Gj) v 7−→ εj d²_θ+λ²

vj

J_S j=1

whereD(L) :=

v∈H₀¹(G)|vj ∈H²(Gj), εjv_j⁰(σj) =εj+1v_j+1⁰ (σj), j= 1..(JS−1) . Lemma 4.8. Let λ ∈ B(−1; 1). Then, L(λ) is a bijective map from D(L) to QJS

j=1L²(Gj)if, and only if, there holdsλ /∈Λε, S.

Proof. •Ifλ∈Λε, S, thenL(λ)is not one-to-one, so it is not bijective.

• If λ /∈ Λε, S, then L(λ) is one-to-one. Let us prove that L(λ) is also onto in this case. Let q= (qj)^J_j=1^S ∈QJS

j=1L²(Gj), and let us build a preimagev ∈D(L)ofq by L(λ).

Consider first the problems, set inGj

(P_j^a) Findv_j^a∈H₀¹(G_j) such that:

εj d²_θ+λ²

v^a_j =qj in L²(Gj),

j = 1.. J_S. For a given indexj, the problem(P_j^a)is well-posed within the Fredholm framework. But, the operatorTj :=−d²_θofL²(Gj), with domainH₀¹(Gj)∩H²(Gj), is self-adjoint with compact resolvent. Its spectrum is equal to

k²π²/(σj−σ_j−1)², k∈N^∗ . The first (smallest) eigenvalue ofTj,π²/(σj−σ_j−1)², is therefore larger than or equal to1 becauseσj−σ_j−1≤π. Then, forλ∈ B(−1; 1),λ² does not belong to the spec- trum of Tj. For the solution of problem (P_j^a) (for j = 1.. JS), uniqueness follows.

Within the Fredholm framework, this shows that, for allλ∈ B(−1; 1), problem(P_j^a) has one, and only one, soution v_j^a,j = 1.. JS. Moreover, results on the regularity of solutions to elliptic PDEs indicate thatv_j^a∈H²(G_j)⊂ C¹(G_j), j= 1.. J_S.

Next, let us consider the problem

(P^b)

Find(v_j^b)^J_j=1^S ∈

JS

Y

j=1

H¹(G_j) such that:

εj d²_θ+λ²

v_j^b= 0 inGj, j= 1.. JS

v₁^b(0) =v_J^b

S(σJ_S) = 0

v_j^b(σj) =v_j+1^b (σj) j= 1..(JS−1) εjv^b_j⁰(σj)−εj+1v^b_j+1⁰(σj) =−αj j= 1..(JS−1)

where the right-hand sideαj is equal toεjv^a_j⁰(σj)−εj+1v_j+1^{a 0}(σj), j= 1..(JS−1).

If(v_j^b)^J_j=1^S solves(P^b), then the volume PDEs imply thatv_j^b(θ) =Aje^{i λθ}+Bje^{−i λθ} (resp. v_j^b(θ) =A_jθ+B_j) forλ6= 0(resp. λ= 0). Writing the transmission conditions (there are2 (JS−1)of them), together with the two Dirichlet boundary conditions at 0and atσJ_S, one builds an algebraic set of2JS linear equations, with2JS unknowns.

SinceL(λ)is one-to-one, it follows that(P^b)has one, and only one, solution.

Finally, define v by v|G_j = v_j^a+v_j^b, j = 1.. JS. It belongs to D(L), and moreover L(λ)v=q. This proves that the operatorL(λ)is onto.

Next, we state a result on the norm ofL(λ)⁻¹.

Lemma 4.9. Assume that εj +εj+1 6= 0, j = 1..(JS −1). Consider two real numbersα,β withα < β. Then there existη0>0 andC >0 independent of λsuch that,

J_S

X

j=1

kd²_θvjk_L2(Gj)+|λ| kdθvk_L2(G)+|λ|²kvk_L2(G)≤CkL(λ)vk_L2(G), for allv∈D(L)and all λ∈Csuch that|=(λ)|> η0 andα <<(λ)< β.

Proof. This nice result is proved in [16] (lemma 3.6). For the sake of clarity, we reformulate the (slightly modified) proof. Below,C >0 designate constant numbers independent of the functions and of λ. Define the two rectangles R1 := {(s, θ) ∈ R²|1/2< s <2 andθ∈G} andR2 :={(s, θ)∈R²|1/4< s < 4andθ ∈G}. One hasR1⊂R2. Considerζa smooth cutoff function which only depends onssuch that ζ= 1onR1 and suppζ⊂R2. Forv∈D(L), definew: (s, θ)7→ζ(s)e^λsv(θ). Using the proof of the theorem 4.6 (cf. §4.2), one can first write

JS

X

j=1

kwjkH²(R1∩Γj)≤C(

JS

X

j=1

kεj∆w_jkL²(R2∩Γj)+k∇wk_L²_(R2)). (4.9) On the one hand, in the cartesian coordinate system (s, θ), one has ∇w(s, θ) = ((d_sζ(s)+λζ(s))e^λsv(θ), ζ(s)e^λsd_θv(θ))and∆(ζ(s)e^λsv(θ))(s, θ) =ζ(s)(d²_θ+λ²)e^λsv(θ)+

2λd_sζ(s)e^λsv(θ) +d²_sζ(s)e^λsv(θ). Therefore, one obtains for|λ| 6= 0

JS

X

j=1

kεj∆w_jkL²(R2∩Γj)+k∇wk_L²_(R2)

≤ C(kL(λ)vk_L2(G)+kd_θvk_L2(G)+|λ| kvk_L2(G)).

(4.10)

Sinceds(e^λsv(θ)) =λe^λsv(θ); dθ(e^λsv(θ)) =e^λsdθv(θ); d²_s(e^λsv(θ)) =λ²e^λsv(θ);

d²_θ(e^λsv(θ)) =e^λsd²_θv(θ); d²_s,θ(e^λsv(θ)) =λe^λsdθv(θ), one has on the other hand

JS

X

j=1

kwjkH²(R1∩Γj)≥C(

JS

X

j=1

kd²_θv_jkL²(G_j)+|λ| kdθvkL²(G)+|λ|²kvkL²(G)). (4.11)

Plugging (4.10) and (4.11) in (4.9), one finds PJS

j=1kd²_θvjk_L2(Gj)+|λ| kdθvk_L2(G)+

|λ|²kvkL²(G)≤C(kL(λ)vkL²(G)+kdθvkL²(G)+|λ| kvkL²(G)).That is equivalent to

J_S

X

j=1

kd²_θvjk_L2(Gj)+ (|λ| −C)kdθvk_L2(G)+|λ|(|λ| −C)kvk_L2(G)≤CkL(λ)vk_L2(G).

Takingη0= 2C, one obtains the result of the lemma.

With the help of the analytic Fredholm theorem, one deduces the

Corollary 4.10. Assume thatεj+εj+16= 0,j= 1..(JS−1). Then there exist two real numbers αS, βS with 0 < αS < 1, 0 < βS < 1 such that (B(−αS;βS)∩ Λ_{ε, S})⊂Ri. In addition, the cardinality of B(−αS;β_S)∩Λ_{ε, S} is finite.

Using the Parseval identity (see the lemma 6.1.4 of [21]), one can now state an isomorphism result between weighted spaces (theorem 3.7 of [16]).

Theorem 4.11. Assume thatε_j+ε_j+1 6= 0,j= 1..(J_S−1). Letγ∈Rbe such that {λ∈C| <(λ) = 1−γ} ∩Λ_{ε, S} = ∅. Then, for all P ∈ V_γ⁰(Γ), there exists one, and only one, solution W ∈V^◦1γ−1(Γ) to the transmission problem

(P_sector^γ )

Find W ∈V_γ−1¹ (Γ)such that:

ε_j∆W_j=P_j inΓ_j, j= 1.. J_S

W₁= 0 on∂Γ₁∩∂Γ

W_JS = 0 on∂Γ_JS ∩∂Γ

Wj−Wj+1= 0 on∂Γj ∩∂Γj+1, j= 1..(JS−1) εj∂θWj−εj+1∂θWj+1= 0 on∂Γj ∩∂Γj+1, j= 1..(JS−1).

This solution can be expressed as W : (r, θ)7→W(r, θ) =M⁻¹

L(λ)⁻¹Hˆ

(r, θ) = 1 2iπ↑

Z

<(λ)=1−γ

r^λL(λ)⁻¹Hˆ(λ, θ)dλ withH :=r²P.

Moreover,W_j ∈V_γ²(Γ_j),j= 1.. J_S, with the continuity estimate

kWk_V1 γ−1(Γ)+

J_S

X

j=1

kWjk_V2

γ(Γ_j)≤C kPk_V0 γ(Γ).

We already noticed that the right-hand side pthat appears in(Psector) belongs to L²(Γ)with compact support, sop∈V_γ⁰(Γ)for allγ≥0.

Definition 4.12. Assume that εj+εj+16= 0, j = 1..(JS −1). Let γ≥0 such that {λ∈C| <(λ) = 1−γ} ∩Λε, S=∅. We denote by w^1−γ the solution to (P_sector^γ ) with right-hand side equal top.

According to previous results, we know thatw^1−γ can be expressed as w^1−γ : (r, θ)7→w^1−γ(r, θ) =M⁻¹

L(λ)⁻¹ˆh

(r, θ) = 1 2iπ↑

Z

<(λ)=1−γ

r^λL(λ)⁻¹ˆh(λ, θ)dλ .

Moreover,w^1−γ belongs toV^◦1_γ−1(Γ), andw^1−γ_j ∈V_γ²(Γ_j),j= 1.. J_S.

It turns out that, ifε_j+ε_j+16= 0,j= 1..(J_S−1), thenw^βS (whereβ_Sappears in corollary 4.10) is well-defined, and moreover it is more regular thanH¹. As a matter