The relationship with the singularities of the limit problem is also described

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

BOUNDARY LAYERS FOR TRANSMISSION PROBLEMS WITH SINGULARITIES

ABDERRAHMAN MAGHNOUJI, SERGE NICAISE

Abstract. We study two-dimensional transmission problems for the Laplace operator for two diffusion coefficients. We describe the boundary layers of this problem and show that the layers appear only in the part where the coefficient is large. The relationship with the singularities of the limit problem is also described.

1. Introduction

We study two-dimensional transmission problems (also called interface problems) for the Laplace operator on polygonal domains consisting of different materials con- nected via an interface line. Dirichlet boundary conditions on the exterior boundary and standard transmission conditions are imposed. Such problems appear in diffu- sion problems where the conductivity of the materials are different on some parts of the domain. It is well known that the solutions of such problems have corner singularities due the jump of the coefficients [6, 7, 9, 10, 12, 13, 14]. On the other hand, for a homogeneous medium having a large diffusion coefficient, the solution exhibits boundary layers added to corner singularities. Their relationship and de- scription are well understood nowadays [1, 4, 5, 8, 11]. But to our knowledge, the description of such a phenomenon is not known for transmission problems where only one of the diffusion coefficients is large. Therefore in this paper we study a relatively simple example of a transmission problem that has corner singularities and boundary layers.

For a standard problem

−ε∆uε+uε=f in Ω, (1.1)

when Ω is a polygonal domain of the plane,f is smooth and ε >0 is a fixed (but small) parameter. An asymptotic expansion of uε is well known [1, 4, 8, 11] and may be written as

u_ε=w_ε+w^BL+w^CL+r_ε,

wherewεis the outer expansion,w^BLdescribes the boundary layer,w^CLdescribes the corner layer, andrεis a remainder that is estimated as a function ofεin some appropriate norms. Usually the termswε, w^BLandw^CLare explicit, which means

2000Mathematics Subject Classification. 35J25, 35B30.

Key words and phrases. Transmission problems; boundary layers; singularities.

c

2006 Texas State University - San Marcos.

Submitted September 26, 2005. Published January 31, 2006.

1

that, for numerical purposes for instance, the behaviour of uε is fully understood by the behaviour of the termswε, w^BL andw^CL.

The goal of the present paper is to reproduce a similar but simpler expansion for a transmission problem where on a part Ω₊ of the domain we consider the problem

−ε∆uε+uε=f in Ω+, and on the other part Ω−, the problem

−∆uε+uε=f in Ω₋,

with, of course, transmission conditions on the interface. By a simpler expansion, we mean thatwε,w^BL,w^CLwill be reduced to one term. As we shall see the situation is more complicated than in the standard case of problem (1.1). The main reason is that the solution of the limit problem has singularities in the domain Ω−. Let us further notice that surprisingly the solution of our problem has only layers in the domain Ω+.

In this paper, the spacesH^s(Ω), withs≥0, are the standard Sobolev spaces in Ω with normk · ks,Ωand semi-norm| · |s,Ω. The spaceH₀¹(Ω) is defined, as usual, by H₀¹(Ω) :={v∈H¹(Ω)/v= 0 on Γ}. L^p(Ω),p >1, are the usual Lebesgue spaces with normk · k_0,p,Ω (as usual we drop the indexpforp= 2). Finally, the notation a .b means the existence of a positive constant C, which is independent of the quantitiesaandbunder consideration and of the parameterε, such that a≤Cb.

This paper is organized as follows: In section 2 we start with a one-dimensional problem in order to describe and understand the typical phenomena. Section 3 is devoted to the introduction of the two-dimensional problem and to the (weak) convergence of the solution to the solution of the limit problem. We go on with the description of the boundary and corner layers in section 4, paying a particular attention to the interface layers due to the singularities. Finally in section 5 we give the expansion of the solution of our problem.

2. The one-dimensional case

Letε∈]0,1] be a fixed parameter. Consider the following transmission problem in ]−1,1[:

−ε²u⁰⁰_ε+u_ε= 1 in ]−1,0[,

−w⁰⁰_ε+w_ε= 0 in ]0,1[, uε(−1) =wε(1) = 0,

uε(0)−wε(0) = 0, ε²u⁰_ε(0)−w⁰_ε(0) = 0.

(2.1)

We remark that in this problem the small parameter ε appears only on ]−1,0[.

Consequently the formal limit problem is the non standard transmission problem u₀= 1 in ]−1,0[,

−w⁰⁰₀+w0= 0 in ]0,1[, u0(−1) =w0(1) = 0,

u₀(0)−w₀(0) = 0, w⁰₀(0) = 0.

(2.2)

This limit problem has a solution w0 ≡ 0, but has no solution u0 since u0 = 1 does not satisfy the boundary conditionu0(−1) = 0 and the transmission condition u0(0)−w0(0) = 0. Therefore, we may expect thatuεwill develop boundary layers at 0 (transmission layer) and at−1 (standard boundary layer). We now justify this formal argument.

The exact solution of this problem (2.1) is u_ε(x) =αcoshx

ε +βsinhx ε + 1, wε(x) =γcoshx+δsinhx,

(2.3) where α, β, γ and δ are constants (depending on ε) determined in order to check the boundary and transmission conditions. This yields a 4×4 linear system that gives after resolution:

α+ 1 =γ, β =δ/ε,

γ=−δtanh 1, δ=−εψ(ε), (2.4)

where the functionψ is

ψ(ε) = cosh¹_ε−1 εtanh 1 cosh¹_ε+ sinh¹_ε.

Since one easily sees that ψ(ε) approaches 1 as ε approaches 0, we deduce that δ = −εψ(ε)∼ −ε as ε → 0. Due to the identities (2.3) and (2.4), we can show that, asεapproaches 0,uε→1 andwε→0, as well as

u⁰_ε(−1) =−α ε sinh1

ε +β ε cosh1

ε ∼1 ε, u⁰_ε(0) = β

ε ∼ −1 ε , w⁰_ε(1) =γsinh 1 +δcosh 1∼ −ε

cosh 1, w⁰_ε(0) =δ∼ −ε.

From these equivalences, we may say thatwεhas no layer, whileuεhas a standard boundary layer at −1 and a transmission layer at 0. We also refer to Figure 1 for an illustration of this fact.

1 0,5 0

-0,5 -1

0,75

0,5

0,25

0

x y

1 0,5 0

-0,5 -1

1

0,75

0,5

0,25

0

x y

Figure 1. Exact solutions forε= 0.1 (left) and ε= 0.05 (right).

Let us give a more precise result, that will also allow us to underline the fact that the transmission layer at 0 may be seen as a (Dirichlet) boundary layer.

Theorem 2.1. For any ε∈]0,1], the unique solution(uε, wε)of (2.1) satisfies u_ε(x) = 1−χ^b(x) exp −dist(x,−1)

ε

−χⁱ(x) exp −dist(x,0) ε

+r_ε(x), ∀x∈]0,1[, (2.5) whereχ^b andχⁱ are the two following cut-off functions:

χ^b= 1 on ]−1,−1 +η[, χⁱ= 1 on ]−η, η[, suppχ^b∩suppχⁱ=∅.

Moreover,

εkr⁰_εk_0,]−1,0[+krεk_0,]−1,0[+kwεk1,]0,1[ .(εe^−ηε +ε). (2.6) Proof. Let us define the functionsv^b:x7→ −exp −^dist(x,−1)_ε

, a solution of

−ε²v^b00+v^b= 0 in ]−1,0[, v^b(−1) + 1 = 0,

v^b(+∞) = 0, andvⁱ:x7→ −exp −^dist(x,0)_ε

, a solution of

−ε²vⁱ⁰⁰+vⁱ= 0, in ]−1,0[, vⁱ(0) + 1 = 0,

vⁱ(−∞) = 0.

Using these two problems and by substitution of (2.5) in (2.1), we see that (rε, wε) is solution of

−ε²r_ε⁰⁰+rε=gε in ]−1,0[, wε−w_ε⁰⁰= 0 in ]0,1[,

rε(−1) = 0, w_ε(1) = 0, rε(0) =wε(0), ε²r_ε⁰(0)−w⁰_ε(0) =−ε,

(2.7)

where

gε:=ε²

[χ^b; d²

dx²]e^−x+1ε + [χⁱ; d² dx²]e^xε

, the bracket [χ^b;_dx^d22] being defined as usual,

[χ^b; d²

dx²]h= d²

dx²(χ^bh)−χ^b d²

dx²h=h d²

dx²χ^b+ 2 d dxχ^b d

dxh.

The variational formulation of this problem is Z 0

−1

ε²r_ε⁰w⁰dx+ Z 1

0

w⁰_εw⁰dx+ Z 0

−1

r_εw dx+ Z 1

0

w_εw dx

= Z 0

−1

gεw dx−εw(0),∀w∈H₀¹(]−1,1[).

(2.8)

Since this left-hand side is trivially coercive on H₀¹(]−1,1[), by the Lax-Milgram lemma, this problem has a unique solution rε ∈H¹(]−1,0[) and wε ∈H¹(]0,1[) such thatrε(−1) =wε(1) = 0,andrε(0) =wε(0) (this means that the functionkε

defined byrεon ]−1,0[ and bywεon ]0,1[ belongs toH₀¹(]−1,1[)). Moreover by taking as test function in (2.8)w=kεwe obtain

ε²kr_ε⁰k²_0,]0,1[+krεk²_0,]−1,0[+kw⁰_εk²_0,]0,1[+kwεk²_0,]0,1[

≤ kgεk_0,]−1,0[krεk_0,]−1,0[+ε|wε(0)|. (2.9) It then remains to estimate theL²-norm of g_ε. The properties ofχ^b and χⁱ imply that suppgε ⊂[−1 +η,−η]. Since on this intervale^−(x+1)ε ≤e^−ηε and e^xε ≤e^−ηε, we obtain

kgεk_0,]−1,0[.εe^−ηε .

On the other hand, the identities (2.3) and (2.4) imply that

|w_ε(0)|.ε ∀ε∈]0,1[.

These two estimates in (2.9) yield

ε²kr_ε⁰k²_0,]−1,0[+krεk²_0,]−1,0[+kw_ε⁰k²_0,]0,1[+kwεk²_0,]0,1[.εe^−ηε krεk_0,]−1,0[+ε². The desired estimate (2.6) follows from Young’s inequality.

Note that the estimate (2.6) is optimal: Direct calculations yieldkwεk0,]0,1[∼ε.

The above theorem gives an explicit expansion ofuε, which also shows thatuε

has two layers (at 0 and −1). It further says that the natural energy norm of the remainderr_εis of orderε. Finally it says thatw_ε has no layer and that its natural energy norm is of orderε.

The goal of the next sections is to show similar results for a polygonal domain on the plane.

3. The two-dimensional problem

Ω₋ Ω+

A B

Figure 2. The domains Ω+ and Ω₋

Let Ω+ and Ω−be two polygonal domains ofR²with respective boundary ∂Ω+

and ∂Ω− having in common a segment Σ = [A, B], see Figure 2. Denote by A1, A2, . . . , AN the vertices of ∂Ω+ enumerated clockwise and so that A1 = A and A₂ =B. Denote further byω_j the interior angle of Ω₊ at the vertex A_j, for anyj∈ {1,2, . . . , N}and letϕ_j the interior angle of Ω₋ at the vertexA_j,j= 1,2.

For further purposes we denote by Ω = Ω₊∪Ω₋∪Σ. Moreover for a functionu defined in Ω, we denote byu₊ (resp. u₋) the restriction ofuto Ω₊ (resp. Ω₋).

Forε∈]0,1[, f_± ∈ C^∞( ¯Ω_±) andh∈ C^∞( ¯Σ), we consider the transmission prob- lem in Ω: Findu^ε solution of

−ε²∆u^ε₊+u^ε₊=f+ in Ω+,

−∆u^ε₋+u^ε₋=f₋ in Ω₋, u^ε₊= 0 on∂Ω₊\Σ, u^ε₋= 0 on∂Ω−\Σ, u^ε₊−u^ε₋= 0 on Σ, ε²∂u^ε₊

∂ν −∂u^ε₋

∂ν =h on Σ,

(3.1)

where ν denotes the outward normal vector along Σ oriented outside Ω₊. The variational formulation of this problem consists in finding a unique solution u^ε ∈ H₀¹(Ω) of

Z

Ω₊

(ε²∇u^ε₊· ∇v₊+u^ε₊v₊) + Z

Ω−

(∇u^ε₋· ∇v₋+u^ε₋v₋)

= Z

Ω+

f v+ Z

Σ

hv,∀v∈H₀¹(Ω).

(3.2)

Since this left-hand side is a coercive and continuous bilinear form onH₀¹(Ω), this problem has a unique solution thanks to the Lax-Milgram lemma.

We now look at the limit of the problem and ofu^ε asεgoes to zero. As before the formal limit problem is

u⁰₊=f₊ in Ω₊,

−∆u⁰₋+u⁰₋=f₋ in Ω₋, u⁰₊= 0 on∂Ω₊\Σ, u⁰₋= 0 on∂Ω₋\Σ, u⁰₊−u⁰₋= 0 on Σ,

−∂u⁰₋

∂ν =h on Σ.

(3.3)

As in dimension 1, in this limit problem u⁰₋ may be seen as the (unique) solution of a mixed Dirichlet-Neumann problem in Ω₋, and since f+ does not satisfy the Dirichlet boundary conditionf+= 0 on ∂Ω+\Σ, and f+=u⁰₋ on Σ, the solution u^ε₊ should develop boundary layers along ∂Ω+. This will be proved in details in the remainder of this paper. Let us first state a weak convergence.

Theorem 3.1. There exists a subsequence ofuε, still denoted byuε, such that the pair (u^ε₊, u^ε₋) converges in L²(Ω+)×H¹(Ω₋) to (u⁰₊, u⁰₋) as ε goes to 0, where u⁰₊=f₊ andu⁰₋ is the unique variational solution of the mixed Dirchlet-Neumann

problem

−∆u⁰₋+u⁰₋=f₋ inΩ₋, u⁰₋= 0 on∂Ω₋\Σ,

∂u⁰₋

∂ν =−h on Σ.

(3.4)

Before proving this theorem, let us introduce some notation and give a density result. Let us introduce the following bilinear and linear forms:

a(u, v) = Z

Ω₊

∇u₊· ∇v₊, b(u, v) =

Z

Ω−

∇u₋· ∇v₋+ Z

Ω

uv, F(v) =

Z

Ω

f v+ Z

Σ

hv.

(3.5)

Let us define the space

W ={w∈L²(Ω) :w−∈H¹(Ω−) andw−= 0 on∂Ω−\Σ}, which is a Hilbert space, equipped with the normkwk²_W =b(w, w).

Lemma 3.2. H₀¹(Ω) is dense in W.

Proof. Letw∈W. Sincew₋∈H˜^1/2(Σ), by [3, Theorem 1.5.2.3] (trace Theorem), there exists ˜w+∈H¹(Ω+) such that

˜

w+=w₋ on Σ,

˜

w₊= 0 on∂Ω₊\Σ.

Since w₊−w˜₊ belongs to L²(Ω₊) and since H₀¹(Ω₊) is dense in L²(Ω₊), there exists a sequence of functionswⁿ₊∈H₀¹(Ω₊),n∈Nsuch that

kw₊ⁿ −(w+−w˜+)k0,Ω₊ →0 asn→ ∞. (3.6) For all positive integern, we introduce the function ˜wⁿ defined in Ω as follows

˜

wⁿ₊=w₊ⁿ+ ˜w+,

˜

wⁿ₋=w₋.

From the boundary condition satisfied by ˜w+, ˜wⁿ belongs to H₀¹(Ω). Moreover from the definition of ˜wⁿ and owing to (3.6), we have

kw˜ⁿ−wkW =kw₊ⁿ−(w+−w˜+)k0,Ω₊→0.

Proof of Theorem 3.1. From (3.2) and the definition ofu⁰, we see thatu^ε∈H₀¹(Ω) andu⁰∈W are the respective solution of

ε²a(u^ε, v) +b(u^ε, v) =F(v),∀v∈H₀¹(Ω), (3.7) b(u⁰, w) =F(w),∀w∈W. (3.8) Step 1. u^εis weakly convergent to u⁰ inW. We first remark that

ku^εk²_W =b(u^ε, u^ε)≤b(u^ε, u^ε) +ε²a(u^ε, u^ε).

Now takingv=u^ε in (3.7) andw=u^εin (3.8) we obtain

ε²a(u^ε, u^ε) +b(u^ε, u^ε) =b(u⁰, u^ε). (3.9) Using Cauchy-Schwarz’s inequality, we directly have

|b(u⁰, u^ε)| ≤ ku⁰kWku^εkW. These three properties imply that

ku^εkW ≤ ku⁰kW. (3.10)

Therefore, there existsw∈W and a subsequence ofu^ε, still denoted byu^ε, weakly convergent towin W.

Now for any fixedv∈H₀¹(Ω), using successively (3.9) and (3.10) we may write

|a(u^ε, v)| ≤ k∇u^ε₊k0,Ω+k∇vk0,Ω+

≤ε⁻¹(ε²k∇u^ε₊k²_0,Ω++b(u^ε, u^ε))¹²k∇vk0,Ω+

=ε⁻¹b(u⁰, u^ε)¹²k∇vk0,Ω₊

≤ε⁻¹ku⁰k_W¹² ku^εk_W¹² k∇vk0,Ω₊

≤ε⁻¹ku⁰kWk∇vk0,Ω₊. This last estimate implies that

ε→0limε²a(u^ε, v) = 0, ∀v∈H₀¹(Ω).

Therefore, passing to the limit in (3.7), we obtain

ε→0limb(u^ε, v) =F(v) =b(u⁰, v), ∀v∈H₀¹(Ω).

SinceH₀¹(Ω) is dense inW,we conclude that

b(u⁰, v) =b(w, v), ∀v∈W.

Sinceb(·,·) is the inner product ofW, we deduce that u⁰=w.

Step 2. u^εis strongly convergent tou⁰in W. ku^ε−u⁰k²_W =b(u^ε−u⁰, u^ε−u⁰)

=b(u^ε, u^ε)−b(u⁰, u^ε)−b(u^ε−u⁰, u⁰).

Taking into account (3.9), we obtain

ku^ε−u⁰k²_W ≤ −b(u^ε−u⁰, u⁰).

Then we have the conclusion, by the weak convergence inW ofu^ε tou⁰. From this Theorem we may seeu⁰ as the first term of the outer expansion ofu^ε. Let us now pass to the description of the boundary layers.

4. Boundary layers

In the sequel let Lε denote the operator Lε = I−ε²∆. In this section, we define in Ω₊, the boundary layerv^b_j along Γ_j = [A_j−1, A_j],j= 2,3, . . . , N and the interface layervⁱ along Σ, such that if V_j denote a small neighbourhood of Γ_j, we have

Lε(u^ε₊−f+−v_j^b) =ε²O(ε) inVj∩Ω+,

f₊+v_j^b= 0 on Γ_j (4.1)

and

Lε(u^ε₊−f+−vⁱ) =ε²O(ε) inV1∩Ω+,

f₊+vⁱ=u⁰₋ on Σ, (4.2)

when O(ε) denote as usual a function ofε bounded in a neighbourhood of ε= 0.

Note that the situation is not the same along Σ due to the lack of regularity ofu⁰₋ (see below).

4.1. Some notation and definitions. We denote by (x, y) the Cartesian coordi- nates of the plane with origin at A₁ and such that Γ₁ ⊂ {(x,0), x >0}. Similarly we denote by (x_j, y_j) the Cartesian coordinates of the plane with origin atA_j and such that Γ_j ⊂ {(x_j,0), x_j>0}.

We now fix two cut-off functionsχ¹_j, χ²_j ∈ C₀^∞(R) satisfying suppχ¹_j ⊂[−a_j, a_j], and

χ¹_j(x) = 1 on ]0, lj[,

wherelj is the length of Γj, and suppχ²_j ⊂[−b, b], as well as χ²_j(y) = 1 on ]−b

2,b 2[, for a sufficiently small fixedb >0.

Now we can introduce the cut-off function along Γ_j by

χ^b_j(x, y) =χ¹_j(x)χ²_j(y). (4.3) We finally takeχⁱ=χ^b₁.

Now we assume that f₊ is the restriction to Ω₊ of a smooth function ˜f₊ ∈ C^∞(R²) and that Ω+ is convex, i.e., 0 < ωj < π, for all j = 1, . . . , N. This last assumption is simply made to simplify the construction of corner layers. Using the method of [11], we probably can treat the non convex case.

4.2. Construction of v^b_j. They are standard, see for instance [4, 5]. For j = 2, . . . , N,v^b_j is the unique solution of the problem

v_j^b−ε²v^b_j⁰⁰= 0 in yj >0, v_j^b=−f˜+(xj, .) atyj = 0,

v^b_j = 0 aty_j = +∞.

It is explicitly given by

v^b_j(x_j, y_j) =−f˜₊(x_j,0)e^−yj/ε. (4.4)

Since ωj < π, the function χ^b_jv^b_j is well defined in Ω+ and satisfies the conditions (4.1). Moreover it has the regularityC^∞( ¯Ω₊) and for any (x_j, y_j)∈Ω₊,

Lε(χ^b_jv_j^b)(xj, yj) = (I−ε²∆_(xj,yj))(χ^b_jv^b_j)(xj, yj)

=ε²χ^b_j(xj, yj)e^−yjε(∂x_j)²f˜+(xj,0) +ε²[χ^b_j; ∆_(xj,yj)]v_j^b, where we recall that [χ; ∆_(xj,yj)]v:=χ∆_(xj,yj)v−∆_(xj,yj)(χv). Since

|χ^b_je^−yjε(∂_xj)²f₊(x_j,0)|.1,

|[χ^b_j ; ∆_(xj,yj)]v^b_j|.1 εe^−2εb, we deduce that

kLε(χ^b_jv_j^b)k0,Ω₊.ε²+εe^−2εb. (4.5) 4.3. Construction ofvⁱ. In general the solutionu⁰₋of problem (3.4) has only the regularityu⁰₋∈H¹(Ω₋). Consequently if we proceed as in the previous subsection, namely if we take

vⁱ(x1, y1) = (˜u⁰₋−f˜+)(x1,0)e^−y1/ε,

the regularity of vⁱ is not sufficient to obtain an estimate similar to (4.5). To overcome this difficulty, we shall use the decomposition ofu⁰₋ into a regular part and singular one.

For j = 1,2, we recall that the singular exponents associated with the mixed Dirichlet-Neumann problem nearA_j are given by (see [3, 2])

Λ_j={λ_k = π 2ϕ_j +kπ

ϕ_j, k∈Z}.

Let (rj, θj) be the polar coordinates centred at Aj and such thatθj= 0 on Σ, and θj=−ωj on the other edge of Ω₋ havingAjas extremity. Forλk ∈Λj, we denote

Sj,λ_k(rj, θj) =r^λ_j^ksinλk(ϕj+θj), −ϕj < θj < ωj. (4.6) Recall that this function satisfies

∆Sj,λ_k = 0, Sj,λ_k(rj,−ϕj) = 0,

∂

∂θSj,λ_k(rj,0) = 0.

(4.7)

According to [3, Corollary 4.4.3.8], the solutionu⁰₋ ∈H¹(Ω₋) of (3.4) admits the decomposition

u⁰₋=u⁰_−,r+ X

j=1,2

η_j X

λk∈Λj,0<λk<2

C_j,λkS_j,λk, (4.8) where u⁰_−,r ∈ H³(Ω₋∩ V1), Cj,λ_k are real constants and ηj is a (radial) cut-off function equal to 1 in neighbourhood of Aj and equal to zero outside another neighbourhood ofAj, j= 1,2. Using this expansion, we can define

vⁱ=vⁱ_r+vⁱ_s (4.9)

where

vⁱ_r(x, y) = (˜u⁰_−,r−f˜₊)(x,0)e^−y/ε, vⁱ_s(x, y) = X

j=1,2

ηj

X

λ_k∈Λ_j,0<λ_k<2

Cj,λ_kSj,λ_ke^−y/ε, (4.10)

where ˜u⁰_−,r(·) is an extension to the real line of u⁰_−,r(·,0). Since u⁰_−,r(·,0) belongs toH⁵²(Σ), this extension may be chosen inH⁵²(R) and by the Sobolev embedding Theorem,vⁱ_r∈ C²( ¯Ω+). Therefore, as in the previous subsection, we have

kLε(χⁱv_rⁱ)k0,Ω₊ .ε²+εe^−2εb. (4.11) On the other hand, Leibniz’s rule yields

∆(χⁱη_jS_j,λke^−y/ε)

=χⁱη_jS_j,λk∆e^−y/ε+ 2∇(χⁱη_jS_j,λk)· ∇e^−y/ε+ ∆(χⁱη_jS_j,λk)e^−y/ε

=e^−y/ε{1

ε²χⁱηjSj,λ_k−2 ε

∂(χⁱηjSj,λ_k)

∂y +χⁱηj∆Sj,λ_k−[χⁱηj; ∆]Sj,λ_k}, and therefore (reminding ∆Sj,λ_k = 0)

L_ε(χⁱη_jS_j,λke^−y/ε) =ε²e^−y/ε2 ε

∂

∂y(χⁱη_jS_j,λk) + [χⁱη_j; ∆]S_j,λk . From this identity, we deduce that

kLε(χⁱv_sⁱ)k0,Ω₊.ε^1+λ+εe^−b/(2ε), (4.12) where λ= mink=1,2min{λk :λk ∈Λk}. Asvⁱ=v_rⁱ+vⁱ_s,the estimates (4.11) and (4.12) lead to

kL_ε(χⁱvⁱ)k_0,Ω+.ε^1+λ+εe^−2εb. (4.13) At this stage if we set

U+:=f++

N

X

j=2

χ^b_jv^b_j+χⁱvⁱ in Ω+, (4.14) then we may write (sinceLεu^ε₊=f+)

Lε(u^ε₊−U+) =ε²∆f+− Lε(

N⁺

X

j=2

χ^b_jv_j^b)− Lε(χⁱvⁱ).

And by (4.5) and (4.13), we arrive at

kLε(u^ε₊−U+)k0,Ω₊ .ε^1+λ+εe^−2εb. (4.15) At this stage we can say thatU₊approachesu⁺_ε in the interior of Ω₊, satisfies the Dirichlet boundary condition in the interior of Γ_j, j = 2, . . . , N and the correct interface condition in the interior of Σ. But the correct boundary/interface condi- tions are not satisfied near the cornersAj. Therefore, corner correctors have to be introduced.

4.4. Corner correctors. For all j = 1, . . . , N consider polar coordinates (rj, θj) centered atA_j and such that Γ_j⊂ {(rj,0), r_j >0} and therefore

Γj−1⊂ {(rjcosωj, rjsinωj), rj >0}

(here and below the index are considered moduloN, i.e. 0=N). Denote Sj={(rj, θj), rj >0,0< θj < ωj},

Γ˜_j−1={(rj, ω_j), r_j>0}, Γ˜_j ={(r_jcosω_j, r_jsinω_j), r_j>0},

and letRj>0 be fixed sufficiently small so that

suppχ^b_j−1∩suppχ^b_j∩Sj⊂B(Aj,Rj

2 ), B(Aj, Rj)∩B(Ak, Rk) =∅ ifk6=j.

To each vertexAj we associate a radial cut-off function χ^c_j such that χ^c_j(r) =

(1 ifr < ^R₂^j, 0 ifr > Rj.

In the sectorSj,according to the definition of the functionU+we may write U+(x, y) =f+(x, y) +χ^b_j−1(x_j−1, y_j−1)v_j−1^b (x_j−1, y_j−1) +χ^b_j(xj, yj)v^b_j(xj, yj),

(4.16) where for shortness we writev^b₁ =vⁱ, χ^b₁ =χⁱ. By construction of the boundary layersv_j^b, we then have

U+

_∂S

j =

(χ^b_jv^b_j on ˜Γj−1, χ^b_j−1v^b_j−1 on ˜Γ_j. Now we introduce the changes of coordinates

Ψ_j: (r_j, θ_j)7−→(x_j, y_j) = (r_jcosθ_j, r_jsinθ_j), Φj: (xj, yj)7−→(xj−1, yj−1).

Using the fact that ˜Γ_j−1(resp. ˜Γj) is parametrized by (xj, yj) = (rjcosωj,rjsinωj) (resp. (x_j, y_j) = (r_j,0)) and using the definition of v_j^b andvⁱ, we see that

U+|∂S_j=

(g¹_j(rj) exp −^rjsin_ε^ωj

) on ˜Γj−1, g²_j(r_j) exp −^rjsin_ε^ωj

on ˜Γ_j,

where, except in the case j =k= 1 andj =k= 2, the functionsg_j^k are smooth, while in the exceptional case, due to (4.9) and (4.10), we have

g¹₁(r₁) =g_1,r¹ (r₁) +g¹_1,s(r₁), (4.17) g²₂(r₂) =g_2,r² (r₂) +g²_2,s(r₂), (4.18) g¹_1,r(r1) =χⁱ◦Ψ1(r1, ω1)vⁱ_r(r1cosω1,0), (4.19) g_1,s¹ (r₁) =χⁱ◦Ψ₁(r₁, ω₁)η₁(r₁) X

λ_k∈Λ1,0<λ_k<2

C_1,λkS_1,λk(r₁, ω₁), g²_2,r(r2) =χⁱ◦Φ⁻¹₂ ◦Ψ2(r2, ω2)vⁱ_r(−r2cosω2+l1,0),

g_2,s² (r2) =χⁱ◦Φ⁻¹₂ ◦Ψ2(r2, ω2)η2(r2) X

λk∈Λ2,0<λk<2

C2,λ_kS2,λ_k(r2, ω2).

The boundary condition imposed at v_j^b on Γ_j implies v_j^b(A_j) = v^b_j−1(A_j) =

−f+(Aj),j= 3, . . . , N. On the other handu⁰₋ ∈H¹(Ω₋) and satisfies the Dirichlet condition on∂Ω−\Σ. By the continuity ofu⁰₋ (due to the expansion (4.8)) we get u⁰₋(A1) =u⁰₋(A2) = 0, and consequentlyv^b₁(Aj) =−f+(Aj), j= 1,2. All together the next compatibility conditions are satisfied

g¹_j(0) =g_j²(0) ∀j= 1, . . . , N. (4.20)

Now we look for explicit functions u^c_j defined in the cone Sj and satisfying the boundary conditions

u^c_j=−g¹_j on ˜Γ_j−1, u^c_j =−g_j² on ˜Γj.

Since the term g¹_1,r and g²_2,r are sufficiently smooth (namely H^5/2), they can be treated as the functionsg_j^k, for j >2. As a consequence we splitu^c_j =u^c_j,r+u^c_j,s, whereu^c_j,s= 0 forj6= 1,2 and

u^c_1,s(r1, θ1) =

(0 ifθ1= 0,

−g¹_1,s(r₁) ifθ₁=ω₁, (4.21) u^1c₂ (r2, θ2) =

(0 ifθ2=ω2,

−g²_2,s(r2) ifθ2= 0, (4.22) and

u^c_j,r =−ˆg_j¹ on ˜Γ_j−1, (4.23) u^c_j,r=−ˆg²_j on ˜Γj. (4.24) where ˆg^k_j = g_j^k except if j = k = 1 and j = k = 2; in that last cases, we take ˆ

g₁¹=g_1,r¹ , ˆg²₂=g_2,r² .

For our purpose, we introduce the functions σj,λ_k(rj, θj) =







r_j^λksin(λ_k(ϕ_j+ω_j)

ωj θ_j if sin(λ_kω_j) = 0,

S_j,λk(r_j,ω_j)

sinλ_kω_j sin(λkθj) if sin(λkωj)6= 0,

so that it fulfilsσ_j,λk(r_j,0) = 0 andσ_j,λk(r_j, ω_j) =S_j,λk(r_j, ω_j). Note that the first choice is also valid in the (generic) case sin(λkωj)6= 0, but in this case the second choice gives rise to a harmonic function.

Lemma 4.1. Let

u^c_1,s(rj, θj) =−χⁱ◦Ψj(r1, ω1)η1(r1) X

λ_k∈Λ₁,0<λ_k<2

C1,λ_kσ1,λ_k, (4.25) u^c_2,s(rj, θj) =−χⁱ◦Φ⁻¹_j ◦Ψj(r2, ω2)η2(r2) X

λ_k∈Λ2,0<λ_k<2

C2,λ_kσ2,λ_k. (4.26) Then they respectively satisfy (4.21) and (4.22) and by setting αj = sinωj,

ke^{−αj rjε} u^c_j,sk0,Sj+εke^{−αj rjε} ∇u^c_j,sk0,Sj .ε^1+λ, j= 1,2. (4.27) Moreover

∆(χ^c_je^{−αj rjε} u^c_1,s(rj, θj))∈L^p(Sj), for allp∈[1,_2−λ² ), whereλ= mink=1,2min{λk :λk∈Λk}.

Proof. For simplicity, let us set ˆ

χⁱ(r1) =χⁱ◦Ψ1(r1, ω1)η1(r1) ifj= 1, ˆ

χⁱ(r₂) =χⁱ◦Φ⁻¹₂ ◦Ψ₂(r₂, ω₂)η₂(r₂) ifj= 2.

Since the functione^−rεαjD^γσ_j,λk behaves likee^−rεαjr_j^λk−|γ|at 0 and at∞, we have kχˆⁱe^−rεαjD^γσ_j,λkk0,Sj .kχˆⁱr^λk−|γ|e^−rjεαk0,Sj. (4.28) For|γ| ≤1< λ_k+ 1, by the scaling ρ_j =^r_ε^j, we obtain

kχˆⁱr^λk−γe^−rjεαk²_0,Sj . Z ∞

0

r^{2(λk−|γ|)}e^−2rjεαr dr

=ε^{2(λk−|γ|+1)} Z ∞

0

ρ^2(λk−γ)e^−2ραρ dρ .ε^{2(λk−|γ|+1)}.

(4.29)

The estimate (4.27) follows directly from (4.28) and (4.29). The regularity of

∆(χ^c_je^{−αj rjε} u^c_1,s(rj, θj))∈L^p(Sj) is proved in a similar manner.

Lemma 4.2. There existsu^c_j,r∈H¹(S_j)satisfying (4.23) and (4.24) and such that kχ^c_je^{−αj rjε} u^c_j,rk0,Sj +εkχ^c_je^{−αj rjε} ∇u^c_j,rk0,Sj .ε. (4.30) Moreover

∆(χ^c_je^{−αj rjε} u^c_1,r(rj, θj))∈L^p(Sj), for allp∈[1,2).

Proof. We simply take

u^c_j,r(r, θ) = (ˆg_j¹(r)−ˆg²_j(r))θ ωj

+ ˆg_j²(r),

which clearly satisfies (4.23) and (4.24). As ˆg_j¹∈H˜⁵²(˜Γ_j−1), ˆg²_j ∈H˜⁵²(˜Γ_j) and are equal to zero forr > R_j, we deduce thatχ^c_ju^c_j,r, χ^c_j^∂u

c j,r

∂r ∈L^∞(S_j) and χ^c_j1

r

∂u^c_j,r

∂θ =χ^c_j(ˆg_j¹(r)−gˆ¹_j(0) r

1

ω_j −ˆg_j²(r)−gˆ_j²(0) r

1

ω_j)∈L^∞(Sj).

Consequently it holds

kχ^c_je^{−αj rε} u^c_j,rk0,S_j +εkχ^c_je^−αrε ∇u^c_j,rk0,S_j .ke^−αrε k0,S_j.

By the change of variableρ= ^r_ε, one has ke^−αrε k_0,Sj .εand the estimate (4.30)

follows. The second assertion is proved similarly.

5. The full decomposition

We are now ready to formulate the main result of this paper.

Theorem 5.1. Assume thatf₊ is the restriction toΩ₊of a smooth function f˜₊∈ C^∞(R²)and thatΩ₊ is convex. Write for shortness

Uc=

N

X

k=1

χ^c_ke^{−sinωkrkε} u^c_k.

Then the unique solutionu^ε∈H₀¹(Ω) of (3.1) admits the splitting u^ε₊ =f₊+

N

X

j=2

χ^b_jv^b_j+χⁱvⁱ+U_c+r₊^ε in Ω₊, u^ε₋ =u⁰₋+r^ε₋ in Ω₋,

(5.1)