Singularities of Maxwell's system in non-hilbertian Sobolev spaces

Singularities of Maxwell’s system in non-Hilbertian Sobolev spaces

WIDEDCHIKOUCHE ANDSERGENICAISE

Abstract. We study the regularity of the solution of the regularized electric Maxwell problem in a polygonal domain with data inL^p()². Using a duality method, we prove a decomposition of the solution into a regular part in the non- Hilbertian Sobolev spaceW^2,p()²and an explicit singular one.

Mathematics Subject Classification (2000):35A20 (primary); 35Q60, 78A25 (secondary).

1.

Introduction

Let be a bounded Lipschitz domain in

R

^{d, d = 2 or 3.} The time harmonic Maxwell equations with perfect conductor boundary conditions can be written as

curlE−i kH=0 and curlH+i kE=J in ,

E×n=0 and H·n=0 on ∂, (1.1)

where E is the electric part andH the magnetic part of the electromagnetic field andk ≥ 0 is the frequency of the electromagnetic wave. The right hand sideJis the current density which in the absence of free electric charges is divergence free, namely

divJ=0 in .

Ifkis different from zero, eliminatingHby the relationH= _{i k}¹ curlE, we deduce from (1.1) that the electric fieldEsatisfies the second order system

curl curlE−k²E=i kJ in ,

E×n=0 on ∂. (1.2)

When is a non-convex polygonal or polyhedral domain, it is well know [5–8, 11, 18] that the natural energy space for problem (1.2) is not H¹()^d but only H₀(curl;)∩ H(div;), the space of square integrable vector fields with square Received November 13, 2007; accepted April 15, 2008.

integrable rotational and divergence satisfying the tangential boundary conditions E×n=0 on∂. Moreover it was shown in [11] that if the right-hand side belongs toL²()^d, then this solution (in H₀(curl;)∩H(div;)) is decomposed into the sum of a regular part inH²()^dand a singular part, only determined by the singular functions of the Laplace operator with Dirichlet boundary conditions.

Regularity results with data in L^p()², p = 2, seem to be useful for three- dimensional numerical purposes [4, 20] and when one studies non linear equations related to Maxwell’s system (like the MHD system, see [16]) by some kind of linearization or fixed point method. Hence it would be interesting to derive similar results than the ones cited above for data inL^p()², with p=2. In this paper, we start this analysis by considering the two-dimensional case. Namely using a duality method introduced by P. Grisvard in [15] for the Laplace operator, we obtain a decomposition (for the variational solution) into a regular part inW^2,p()²and an explicit singular one. In a forthcoming paper, we will treat the three-dimensional case by combining the two-dimensional results obtained here with the theory of the sums of operators in Banach spaces, see for instance [2, 9, 12, 13].

The paper is organized as follows. Section 2 is devoted to the setting of the problem, first we regularize the electric Maxwell problem by the addition of a graddiv term to the operator curlcurl in (1.2) which leads to a boundary value problem for the vectorial Laplace operator. Next we prove the existence and unique- ness of a variational solution in the space H₀(curl)∩ H(div)for data in L^p()² with p ≥ ⁴₃. We conclude this section by a regularity result for the divergence of the variational solution.

In Section 3, we prove a priori estimates which allow to deduce that the vec- torial Laplace operator is a semi-Fredholm operator in appropriate Sobolev spaces.

This will be done by a local method and a change of variables which reduces our problem to a boundary value problem in an infinite strip studied in Subsection 3.1.

There the solution is written explicitly by partial Fourier transform and thus esti- mated by applying Mikhlin’s multiplier theorem.

In Section 4, in order to prove that the vectorial Laplace operator is Fredholm, we show that its kernel is reduced to {0} and that the orthogonal of its range is finite dimensional. Finally we compute exactly the index of our problem by giving explicit functions which are not in the range of our operator. This will also lead to the desired decomposition into a regular part and a singular one.

2.

Setting of the problem

Let be a bounded polygonal domain of

R

² with a Lipschitz boundary =

∪^N_j=1j with j a linear segment, for all j = 1,2, . . . ,N. Let us denote by S_j the vertex ofdefined byS_j =j ∩j+1and byωj the opening ofatS_j.

Since we are interested in the regularity of the Maxwell system inand since this regularity has a local character, without loss of generality we can assume that is simply connected.

As usual, we denote by L^p() and by W^k,p(), k ∈

N

^{,p ∈ (1,∞) the} Lebesgue and Sobolev spaces. The usual norm ofW^k,p()is denoted by · k,p,. We consider the following electric Maxwell problem: givenJ ∈ L^p()², we look for a solutionE∈W^2,p()²of

curlcurlE−k²E=J in ,

E×n=0 on , (2.1)

wherendenotes the unit outer normal to. The notations curl andcurldistinguish between the scalar and vector curl operators:

curlu= ∂u₂

∂x −∂u₁

∂y whenu=(u₁,u₂)and curlϕ= ∂ϕ

∂y,−∂ϕ

∂x

, where we denote by(x,y)the Cartesian coordinates in

R

². We assume that

divJ=0 in .

With this hypothesis, it is clear that a solution of (2.1) satisfies divE =0 ifk =0, and thus, this is also a solution of







curlcurlE−graddivE−k²E=J in ,

E×n=0 on ,

divE=0 on .

(2.2)

The divergence free condition divE=0 on,is necessary to prove that conversely any solutionEto (2.2) satisfies (2.1).

Instead of problem (2.2), we will consider along this paper the simpler Maxwell problem (2.3)-(2.5) below since its corner singularities has the same principal parts as problem (2.2) (see [11]),

curlcurlE−graddivE = J in, (2.3)

E×n = 0 on, (2.4)

divE = 0 on. (2.5)

Since−=curlcurlE−graddivE, equation (2.3) is equivalent to the vectorial Laplace equation

−E=J in . 2.1. Variational formulation

Let us introduce the spacesH(curl;)andH(div;) H(curl;)=

u∈L²()²;curlu∈L²() , H(div;)=

u∈L²()²;divu∈L²() .

The variational formulation of problem (2.3)-(2.5) consists in looking for u ∈ X_N()solution of

∀v∈X_N(),

(curlucurlv+divudivv)=

J·v, (2.6) where X_N()is the closed subspace of H(curl;)∩H(div;)defined as

X_N()=

u∈H(curl;)∩H(div;); u×n=0 on .

As X_N()is compactly embedded into L²()² [21] (see also [3, Th. 2.8]), the form

a(u,v)=

(curlucurlv+divudivv),

is continuous and strongly coercive on X_N(). Moreover as X_N() is embed- ded into H^1/2()²[6, 10, 11], by the Sobolev embedding theorem, we deduce that X_N()is embedded into L^q()², for all 1≤q ≤4 and therefore the linear form v →

J·vis continuous on X_N()for all p ≥ ⁴₃.Applying the Lax-Milgram lemma we get the

Proposition 2.1. The variational problem(2.6)admits a unique solutionu∈X_N() for anyJ∈L^p()²such that p ≥ ⁴₃.

2.2. The regularity of the divergence

As for data inJ ∈ L²()²[11], we show that the divergence of the solutionuof problem (2.6) with p≥ ⁴₃ has a divergence with a better regularity.

Lemma 2.2. Letu ∈ X_N()be the solution of(2.6)withJ ∈ L^p()² such that p≥ ⁴₃. Then

divu∈W₀^1,r()with r =min{p,2}.

Proof. i) For p ≥ 2, the result was proved in Theorem 1.2 of [11], where it was shown that divu=q ∈H₀¹()is the unique solution of

∇q· ∇ϕ= −

J· ∇ϕ,∀ϕ∈ H₀¹(). (2.7) ii) For ⁴₃ ≤ p<2, we fix a sequence(J_n)n∈Nof elements inL²()²such that

J_n →JinL^p()²asn→ ∞. (2.8)

We consideru_n ∈ X_N()the solution of (2.6) with right hand sideJ_n. Then from (2.8), we know that

u_n →uin X_N()asn→ ∞. (2.9)

Applying the first item to u_n, we deduce that divu_n = q_n ∈ H₀¹()is the unique solution of

∇q_n· ∇ϕ= −

J_n· ∇ϕ,∀ϕ∈ H₀¹().

At this stage, for an arbitrary elementv∈L^q()², we use its Helmholtz decompo- sition

v= ∇r+curlψ, wherer ∈ H₀¹()is solution of

∇r· ∇ϕ= −

v· ∇ϕ,∀ϕ∈H₀¹(),

andψ ∈H¹().This decomposition follows from [14, Theorem I.3.1] sinceq>2.

Moreover from the elliptic regularity of the Laplace operatorrbelongs toW^1,q() with the estimate

r1,q,≤Cv0,q,, (2.10) for someC >0. This is a consequence of [17, Theorems 8.1 and 8.2] since there is no singular exponents ^k_ω^π

j between 0 and 1−²_q. Hence we may write

∇q_n·v=

∇q_n· ∇r = −

J_n· ∇r, since by Green’s formula

∇q_n·curlψ = −

q_n·divcurlψ+q_n;curlψ·n = 0 (sinceq_n =0 on the boundary). By H¨older’s inequality we obtain that

∇q_n·v

≤ J_n0,p,∇r0,q,, and by (2.10), we get

∇q_n·v

≤CJ_n0,p,v0,q,. Taking the supremum onv∈L^q()², we have shown that

∇q_n0,p,≤CJ_n0,p,.

This estimate and (2.8) and (2.9) allow to conclude that divu∈W₀^1,p().

Remark 2.3. According to the previous theorem and [17, Theorems 8.1 and 8.2], for p> 2, if 1− ²_p = ^k_ω^π_j for all j =1, . . . ,N,k ∈

N

^{, and ifu}∈ X_N()is the solution of (2.6) withJ∈L^p()², then divusatisfies

divu=vR +vS,

wherevR ∈W₀^1,p()andvS ∈W₀^1,2()is the singular part of divuand is of the form

vS = N

j=1

k∈N: 0<kπ

ωj<1−2 p

c_kr

kπωj j sin

kπθj

ωj

,

wherec_k ∈

R

^{and(rj, θj)}are polar coordinates centered atS_j such thatθj =0 on j+1andθj =ωj onj.

3.

The semi-Fredholm property

In this section, we prove that our problem (2.3) -(2.5) has the semi-Fredholm prop- erty by proving an a priori estimate. For that purpose, we proceed locally (Subsec- tion 3.2) in order to reduce our problem to a boundary value problem in an infinite strip, studied in Subsection 3.1. Such a strip has a smooth boundary, and we can use partial Fourier transform and Mikhlin’s multiplier theorem to state the a priori estimate.

3.1. A priori estimate in an infinite strip

For a positive real numberω, letBbe the infinite strip

B=

R

^{×]0, ω[= {(x,y)∈}

R

^{2; x ∈}

R

^{, 0< y< ω}.}

For a real numberrwe consider the operatorLdefined by Lu=∂x²u+∂y²u+2r∂xu+r²u.

Givenf ∈L^p(B)², we look atu=(u₁,u₂) ∈W^2,p(B)²a solution of







Lu_i = f_i in B,i =1,2, u₁cosα+u₂sinα=0 on F_α, α=0, ω, sinα ∂yu₁−cosα ∂yu₂=0 on F_α, α=0, ω,

(3.1)

where forα=0 orω

F_α = {(x, α);x∈

R

^}.

We are looking for a sufficient condition betweenr andω that guarantees the a priori estimate

u2,p,B ≤c_pf0,p,B, (3.2) for some positive constantc_p.

For that purpose, we compute explicitly a solutionuof (3.1) by applying partial Fourier transform inxand then using Miklin’s multiplier theorem.

3.1.1. Explicit solution by partial Fourier transform

After applying partial Fourier transform inx, problem (3.1) becomes











v_i−ρ²vi =g_i in ]0, ω[,i =1,2, v1(ξ,0)=v₂(ξ,0)=0,

v1(ξ, ω)cosω+v2(ξ, ω)sinω=0, sinω v₁(ξ, ω)−cosω v₂(ξ, ω)=0,

(3.3)

for allξ ∈

R

, where we have setvi =u_i,g_i = f_i, andρ=sgnξ(−ξ+ir), where sgnξ = 1 ifξ ≥0 and−1 else. Any solution of the differential equation in (3.3) takes the form

vj(ξ,y)= 1 2ρ

_ω

0

e^ρ|y−z|g_j(ξ,z)d z+a_je^ρy+a_−je^ρ(ω−y),

for somea_j,a_−j ∈

C

^{,j =1,}2. The boundary conditions in problem (3.3) provides a linear system of 4 equations with 4 unknowns which admits a unique solution if and only if its determinantdis different from zero. We have

d =4e^2ρωsin(1−iρ)ωsin(1+iρ)ω.

Henced =0 for allξ ∈

R

if and only if r = kπ

ω +1,∀k∈

Z

^andr = kπ

ω −1,∀k ∈

Z

. (3.4) Under this assumption, the solution of (3.3) is given by

v1(ξ,y)= 1 2ρ

_ω

0

e^ρ|y−z|+e^ρ(y+z)

g₁(ξ,z)d z + 1

2ρd _ω

0

e^{ρ(2ω+y−z)}−e^{ρ(2ω−y−z)} (cos 2ω−e^2ρω)g₁(ξ,z)+sin 2ωg₂(ξ,z) d z

− 1 2ρd

_ω

0

e^ρ(2ω+y+z)−e^{ρ(2ω−y+z)} (cos 2ω+e^2ρω)g₁(ξ,z)−sin 2ωg₂(ξ,z) d z, (3.5)

v2(ξ,y)= 1 2ρ

_ω

0

e^ρ|y−z|+e^ρ(y+z)

g₂(ξ,z)d z

− 1 2ρd

_ω

0

e^{ρ(2ω+y−z)}−e^{ρ(2ω−y−z)} sin 2ωg₁(ξ,z)−(cos 2ω−e^2ρω)g₂(ξ,z) d z + 1

2ρd _ω

0

e^ρ(2ω+y+z)−e^{ρ(2ω−y+z)} sin 2ωg₁(ξ,z)−(cos 2ω−e^2ρω)g₂(ξ,z) d z. (3.6) 3.1.2. L^pbounds for the solution

We shall now use the last two identities to show the existence of a constantC such that

u0,p,B ≤Cf0,p,B. (3.7) For this purpose, we need the following lemma (see [15, Lemma 4.2.1.3]).

Lemma 3.1. Letξ, y, z→K(ξ,y,z)be a smooth(inξ)function such that

ymax∈]0,ω[

_ω

0

maxξ∈R{|K(ξ,y,z)| + |ξ||∂_ξK(ξ,y,z)|}d z <+∞ (3.8) and

z∈]max0,ω[

_ω

0

maxξ∈R{|K(ξ,y,z)| + |ξ||∂_ξK(ξ,y,z)|}d y <+∞; (3.9) then the mapping f →u defined by

ˆ

u(ξ,y)= _ω

0

K(ξ,y,z)fˆ(ξ,z)d z is continuous in L^p(B)for p such that1< p<∞.

Now we have to check that each term of the right hand side of (3.5) and of (3.6) satisfies the conditions (3.8) and (3.9) from the previous lemma. Indeed, we can see that the kernels involved in the expression ofvi have the form ^e_ρ^ρβ or ^e_ρ^ρβ_d, whereβis a function depending only on the variablesyandzsuch that

β(y,z)= |y−z|orβ(y,z)=ay+bz+c≥0 ∀y,z∈ [0, ω], wherea,b∈ {−1,1}andc∈ {0,2ω,4ω}.

Explicit calculations lead to

d≥e^−2ξωcosh²ξω|sinω(1+r)sinω(1−r)| ≥c, where

c= |sinω(1+r)sinω(1−r)|

4 .

Let us set

K₁(ξ,y,z)= e^ρβ

ρ , K₂(ξ,y,z)= e^ρβ ρd. It is easy to check that

|K₁(ξ,y,z)| ≤ 1

r, |ξ||∂_ξK₁(ξ,y,z)| ≤β+1

r, |K₂(ξ,y,z)| ≤ 1 r c. In addition, we have

|ξ||∂_ξK₂(ξ,y,z)| ≤ 1 c

β+1

r

+ d

d

. It remains to estimate ^d_d. We compute it explicitly and obtain

d d

=

2ω−iω sin(2rω+2iξω)

sin[(1+r)ω+iξω]sin[(1−r)ω−iξω]

≤ 2ω

1+1 c

.

It follows that (3.8) and (3.9) hold for each kernel involved in (3.5) and (3.6) and consequently the estimate (3.7) holds.

3.1.3. W^2,pbounds of the solution

Now we are able to state the a priori estimate (3.2).

Theorem 3.2. Under the hypothesis(3.4), there exists a constant c_p >0such that (3.2)holds for allu∈W^2,p(B)²solution of(3.1).

Proof. Letη_α(α=0, ω)be two cut-off functions such that (a) ηα ∈C^∞(

R

)with a compact support,

(b) η0 =1 (respectivelyη_ω =1) in a neighbourhood ofF₀(respectively F_ω) and vanishes nearF_ω(respectively F₀).

(c) η0+η_ω =1.

Let us setw_α =(w_α1, w_α2) =η_αu. Then the functionw_αis solution of the next boundary value problem on

R

²_α,+ ^(with

R

²0,+ =

R

²₊ ^{= (0,∞)×}

R

^and

R

²_ω,+ ⁼ (−∞, ω)×

R

^):

Lw_αi = g_αi in

R

²_α,+^{, (3.10)} w_α1(x, α)cosα+w_α2(x, α)sinα = 0 ∀x∈

R

^{, (3.11)} sinα∂yw_α1(x, α)−cosα∂yw_α2(x, α) = 0 ∀x∈

R

, (3.12) where

g_αi =η_αLu_i +2∂yη_α∂yu_i+∂_y²η_αu_i, i =1,2.

This problem can be looked as a system of partial differential equations as follows (see [19])

2 i=1

L_{i j}(x, ∂)w_αj =g_αi in

R

²_α,+^{,∀i =1,2, (3.13)} whereL_{i j} =δi jL.

It is clear that the system (3.13) is properly elliptic in

R

² in the sense of Douglis-Nirenberg. Moreover we easily check that the boundary conditions (3.11), (3.12) cover the system (3.10) on

R

. Therefore applying Theorem 10.5 of [1] (see also [19, Corollary 1.48]), we conclude the existence of a constantC_α > 0 such that

2 i=1

w_αi2,p,B ≤C_α 2

i=1

g_αi0,p,B+²

i=1

w_αi0,p,B

.

Adding this estimate forα = 0 with the one forα = ωand using an interpolation inequality, there exists a constantC >0 such that

2 i=1

u_i2,p,B ≤C 2 i=1

f_i0,p,B+ u_i0,p,B ,

which leads with (3.7) to the estimate (3.2).

3.2. Bounds in a polygon

Let us denote by P^2,p() the weighted Sobolev space of all distributions u ∈

D

()satisfying

r^|α|−2D^αu ∈L^p()

for all |α| ≤ 2, where r(x,y) denotes the distance from the point(x,y) to the vertices of. P^2,p()is a Banach space for the norm

u_P2,p()=

|α≤2

r^|α|−2D^αu₀^p_,p, ¹_p

.

Theorem 3.3. Assume that 3− 2

p = kπ

ωj,∀k∈

Z

^{and1− 2}_p ^{= k}_ω^π

j,∀k∈

Z

^{, j =1, . . . ,N. (3.14)} Then there exists a constant C_p>0such that

E_P2,p()²≤C_p

J0,p,+ E1,p, , (3.15) for every solutionE∈P^2,p()²of(2.3)-(2.5).

Proof. Let us fix a partition of unity{ηj}, j =0, ...,N onsuch that N

j=0

ηj =1 and satisfying the following assumptions:

(a) the functionηjbelongs to

D

(

R

²⁾and fulfills the boundary conditions∂nηj =0 on.

(b) the support ofη0does not contain any of the vertices of.

(c) the support ofηj (j = 0)contains S_j and does not contain any other vertex;

in addition it does not intersectkfork= jandk= j+1.

By [1, Theorem 10.5] (see also [19, Theorem 1.49]), there exists a positive constant C₀such that

η0E_i2,p, ≤C₀{(η0E_i)0,p,+ η0E_i0,p,}. (3.16) It remains to estimateηjEfor j =1, ...,N. For that purpose we fix j and use local polar coordinates with origin at S_j, and such thatθ =0 onj+1whileθ =ωj on j. We denote byGthe infinite sector defined by

G= {(rcosθ,rsinθ);r >0,0< θ < ωj}.

With this notation, the functionF=ηjE(the extension by zero ofηjEoutside the support ofηj) satisfies (2.4) and (2.5), as well as

F_i0,p,G ≤K{J_i0,p,+ E_i1,p,}, i =1,2. (3.17) In polar coordinates, the boundary condition (2.4) takes the form

E₁ =0 forθ =0, E₁cosωj +E₂sinωj =0 forθ =ωj

while (2.5) becomes

∂_θE₂=0, forθ =0, sinωj∂_θE₁−cosωj∂_θE₂=0, forθ =ωj. Let us set fori =1,2

W_i(t, θ)=e^−2tqF_i(e^t, θ), (3.18) K_i(t, θ)=e^2tp(F_i)(e^t, θ). (3.19) ThenW = (W₁,W₂)is solution of the following boundary value problem in the stripB=

R

^{×]0, ωj[}











∂_t²W_i+D²_θW_i+ ⁴_q∂tW_i +_q⁴2W_i = K_i in B,i =1,2,

W₁=∂_θW₂=0 on θ =0,

W₁cosωj+W₂sinωj =0 on θ =ωj, sinωj∂θW₁−cosωj∂θW₂=0 on θ =ωj.

(3.20)

In other words,W is solution of problem (3.1) (studied in the previous section) with r = _q² andω=ωj. Therefore, by Theorem 3.2, there exits a constantC_p>0 such that

2 i=1

W_i2,p,B ≤C_p 2 i=1

K_i0,p,B.

Performing the inverse change of variables in (3.18) and (3.19) leads to 2

i=1

F_iP^2,p()≤C_p 2

i=1

F_i0,p,,

This last inequality together with (3.16) and (3.17) imply (3.15).

Now, we shall derive the consequences of the estimate (3.15). Let us set F_p=

E∈ P^2,p()²satisfying (2.4) and (2.5) ,

which is a Banach space for the norm ofP^2,p()². We also introduce the operator B_p: F_p −→L^p()²:E−→E.

Proposition 3.4. Assume that(3.14) holds, then the operator B_p has a finite-di- mensional kernel and a closed range.

Proof. Consequence of Theorem 3.3 and of Peetre’s Lemma.

4.

The Fredholm alternative

4.1. The kernel

Proposition 4.1. kerB_p= {0}.

Proof. E∈kerB_pmeans thatE∈ F_pandE=0. Applying Green’s formula, we

get

(curlE)²d x +

(divE)²d x =0, which implies that

curlE=0,divE=0 in.

Therefore there existsφ∈H₀¹()such thatE=gradφ. Since divE=0 in, we conclude thatφis harmonic. Henceφ=0 and thenE=gradφ=0.

4.2. The range

In this subsection, we shall study the range ofB_p. Since the operator is closed, we shall instead investigate its annihilatorN_qwhich is a subspace ofL^q()²(¹_p+_q¹=1).

Let us recall that

N_q = {F∈L^q()²satisfying(4.1)below}.

E·Fd xd y=0 ∀E∈F_p. (4.1) Lemma 4.2. LetF∈N_q, thenFis solution of

F=0 in , (4.2)

F×n=0 on , (4.3)

divF=0 on . (4.4)

Proof. (4.2) follows by applying (4.1) withE∈

D

^{()2, we get}

E·Fd xd y=0,∀E∈

D

^()2, and consequentlyF_i =0,i =1,2,in the distributional sense.

Now for a fixed 1≤ j ≤ N, givenϕj ∈ ˜W^2−1p,p(j)andψj ∈ ˜W^1−1p,p(j), Lemma 5.2 below ensures the existence ofE ∈ P^2,p()²such that (see Section 5 for the definitions and properties used in this proof)

E·n=ϕj, E×n=0, curlE=ψj, divE=0 onj, E=0, curlE=0, divE=0 onk,∀k= j.

Consequently,Ebelongs toF_pand then

E·Fd xd y=0, forF∈N_q. Applying Corollary 5.4 to this functionEand toF∈N_q, we obtain

curlE,F×n_j + E·n,divF_j =0, i.e.,

ψj,F×nj + ϕj,divFj =0.

If we letψj vary inW˜^1−1p,p(j)andϕj vary inW˜^2−1p,p(j), we obtain (4.3) and (4.4).

Let us set

M_q = {E∈L^q()²satisfying(4.2)−(4.4)}.

Lemma 4.2 shows that N_q is a subspace of M_q, the next lemma ensures that these two spaces coincide. In other words,N_qis completely characterized as the subspace ofL^q()²of the solutions of the adjoint problem (4.2)-(4.4).

Lemma 4.3. N_q =M_q.

Proof. LetF ∈ M_q andE ∈ F_p. ThereforeF ∈ D(,L^p())² and thus we can apply Corollary 5.4 to get

E·Fd xd y =0.

We shall now study the behavior of the elements of M_q. First of all, we show that they are regular far from the vertices.

Lemma 4.4. LetF∈M_q, thenF∈C^∞(\V)², where V is any neighborhood of the vertices of.

Proof. The smoothness ofFinside follows from the fact thatF_i is a harmonic function infor eachi =1,2. It remains to prove thatFis also smooth near any of thej. For that purpose, we proceed as in [15, proof of Lemma 4.4.2.1]: We fix j and perform a change of coordinates such thatj is on the axis{y=0}andis abovej. Then we introduce a cut-off functionϕ ∈C^∞(), whose support does not intersect any of the sidesk withk = j. In addition,ϕdoes not depend on y for small values of y. Let us setW=ϕF. The functionWbelongs toL^q(

R

²₊^)and

is solution of 









−W_i +W_i = f_i in

R

²₊, W₁=0 on {y=0},

∂W₂

∂y =0 on {y=0}, where

f_i = {ϕF_i−2∇ϕ∇F_i −(ϕ)F_i˜}.

We remark that the boundary conditions inW₁,W₂are decoupled. Therefore, each of the functionsW₁andW₂can be looked as a solution of a boundary value problem which is similar to the one in Lemma 4.4.2.1 in [15]; so we deduce that W_i ∈ C^∞(\V), fori =1,2.

In order to specify the behavior ofv ∈ M_q near the vertexS_j, we define the unbounded operatorj inH_j =L²(]0, ωj[)²as follows

jϕ= −ϕ, whereϕ∈ D(j), the domain ofj being given by

D(j)= {ϕ=(ϕ1, ϕ2) ∈H²(]0, ωj[)²satisfying (4.5) below}







ϕ1(0)=ϕ₂(0)=0,

cosωjϕ1(ωj)+sinωjϕ2(ωj)=0, sinωjϕ₁(ωj)−cosωjϕ₂(ωj)=0.

(4.5)

This is a nonnegative self-adjoint operator with a discrete spectrum. Let us denote by ϕj an eigenfunction and by λ²_j the corresponding eigenvalue. We thus have ϕ_j ∈ D(j)with

−ϕ_j,i =λ²_jϕj,i, i =1,2.

By solving the last differential equation and taking into account the boundary con- ditions (4.5), we get

ϕj,m(θ)=(sin(λj,mθ),−cos(λj,mθ)), λj,m−1= mπ

ωj , m ∈

Z

, or

˜

ϕj,m(θ)=(sin(λ˜j,mθ),cos(λ˜j,mθ)), λ˜j,m+1= mπ

ωj , m ∈

Z

. Remark 4.5. For covering all the linearly independent eigenfunctions ofj, it suffices to consider only the functionsϕj,mcorresponding toλj,m. Indeed we may notice thatλ˜j,m = −λj,−mand thatϕ˜_j,m = −ϕj,−m. This means that

Sp(j)= {λ²:λ−1∈E_j}, where

E_j = mπ

ωj , m ∈

Z

.

The following theorem gives near each corner S_j, an expansion for the elements of M_q in series of the eigenfunctions ofj.

Theorem 4.6. Ifv ∈ M_q, then for all j ∈ {1, ...,N}, and for every ρ > 0fixed sufficiently small,

v(r_je^iθj)=

λj,m>⁻²_q

αj,mr^λ_j^j,mϕ_j,m(θj)

+

λj,m<_q²

˜

αj,mr^−λ_j ^j,mϕ_j,m(θj)in D_j,ρ,

(4.6)

where(r_j, θj) denotes the polar coordinates with origin at S_j, D_j,ρ = {r_je^iθj : 0<r_j < ρ,0< θj < ωj},αj,mandα˜j,mare real numbers given by

αj,mr^λ_j^j,m+ ˜αj,mr^−λ_j ^j,m= _ωj

0

{v1(r_je^iθj)sin(λj,mθj)−v2(r_je^iθj)cos(λj,mθj)}dθj. In addition, there exists a constant L >0depending only onvandρsuch that

|αj,m| ≤Lρ^{−λj,m−2q}(λj,mq+2)^1q, (4.7)

whenλj,m≥ _q² and

| ˜αj,m| ≤Lρ^λj,m−q2(−λj,mq+2)^q1, (4.8) whenλj,m≤ ⁻_q².

Proof. Let us fix j ∈ {1,2, ...,N}and for shortness drop the index j. Thanks to Lemma 4.4,vis smooth forr >0 and consequently the function

θ−→v(r e^iθ)

is differentiable inr with values inH_j = L²(]0, ωj[)², for eachr ∈]0, ρ],ρ > 0 chosen sufficiently small.

Writing (4.2)-(4.4) in polar coordinates, we notice that for each r ∈]0, ρ], v(r e^iθ)∈D(j)and thatvis solution of

∂²v

∂r² +1 r

∂v

∂r − 1

r²jv=0, 0<r < ρ. (4.9) Since the sequence(ϕj,m)is a basis ofH_j,vcan be expanded in the series

v(r e^iθ)=

m∈Z

λj,m−1=mπ

ωj

vm(r)ϕ_j,m(θ),

where

vm(r)= _ωj

0

v1(r e^iθ)sinλj,mθ−v2(r e^iθ)cosλj,mθ

dθ. (4.10)

However, equation (4.9) implies that v_m(r)+ 1

rv_m (r)−λ²_j,m

r² vm(r)=0, 0<r < ρ.

Consequently, we have

vm(r)=αj,mr^λj,m+ ˜αj,mr^−λj,m (4.11) becauseλj,m =0.

Asvbelongs toL^q(D_ρ)², it follows from identity (4.10) that

|vm(r)|^q ≤C _ωj

0

|v1(r e^iθ)|^qdθ+ _ωj

0

|v2(r e^iθ)|^qdθ

,