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MULTISCALED ASYMPTOTIC EXPANSIONS FOR THE ELECTRIC POTENTIAL: SURFACE CHARGE

DENSITIES AND ELECTRIC FIELDS AT ROUNDED CORNERS

PATRICK CIARLET, JR.and SAMIR KADDOURI Laboratoire POEMS, UMR 2706 CNRS/ENSTA/INRIA,

Ecole Nationale Sup´´ erieure de Techniques Avanc´ees, 32, boulevard Victor, 75739 Paris Cedex 15, France

patrick.ciarlet@ensta.fr

Received 28 July 2006 Communicated by N. Bellomo

We are interested in computing the charge density and the electric ﬁeld at the rounded tip of an electrode of small curvature. As a model, we focus on solving the electrostatic problem for the electric potential. For this problem, Peek’s empirical formulas describe the relation between the electric ﬁeld at the surface of the electrode and its curvature radius. However, it can be used only for electrodes with either a purely cylindrical, or a purely spherical, geometrical shape. Our aim is to justify rigorously these formulas, and to extend it to more general, two-dimensional, or three-dimensional axisymmetric, geometries. With the help of multiscaled asymptotic expansions, we establish an explicit formula for the electric potential in geometries that coincide with a cone at inﬁnity. We also prove a formula for the surface charge density, which is very simple to compute with the Finite Element Method. In particular, the meshsize can be chosen indepen- dently of the curvature radius. We illustrate our mathematical results by numerical experiments.

0. Introduction

Electrostatic phenomena are of great interest in many domestic and industrial applications. For instance, processing the smokes in thermal power stations: the principle is to charge the particles in suspension with an ionizing device, which then aggregate on the grounded inner walls so that they can ﬁrst be gathered, and later be disposed of. Let us also mention electrostatic spray painting. Below, we describe in detail this second application, which motivates our mathematical studies.

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When an object (a “target”) is painted with the help of a gun (electrode here- after), the paint is made of microscopic droplets, a fraction of which actually reaches the target, while the rest dissolves into the atmosphere. In order to improve the method (greater fraction reaching the object, less pollution), one can apply a high voltage between the electrode and the target. Thanks to the applied voltage, the paint droplets are charged electrically. The diﬀerence between the electric poten- tials (the voltage) produces a stationary electric ﬁeldE=−∇φ. Then, the charged droplets (or charges hereafter) follow the lines of force, from the electrode to the target. An electric corona discharge is produced at the onset, and the phenomena are then considered to be stationary. The electric ﬁeld solves Coulomb’s law

div(εE) =. (0.1)

In the above, ε is the electric permittivity of the medium, and is the charge density. Assuming that the medium is homogeneous, Eq. (0.1) can be rewritten

∆φ=

ε. (0.2)

In order to “close” the model with the potentialφas the main unknown, one has to state precisely what the (computational) domain is, and also to add suitable boundary conditions.

On the surface of the electrode and of the target, it is natural to prescribe a constant value ofφ1and respectivelyφ0, with for instanceφ0= 0), i.e. a Dirichlet boundary condition.

Then, in a ﬁrst approach, one can close the domain by putting artiﬁcial bound- aries, on which a zero-ﬂux condition is imposed (a Neumann boundary condition):

the resulting computational domain is bounded. This amounts to saying that there is no loss of charges through these artiﬁcial boundaries, which is reasonable. Some numerical experiments have been carried out on this model by Timouyaset al.,23,17 where the quantity of interest was the maximum (absolute value of the) surface charge density on the electrode.

On the other hand, at the charges level, one can safely assume that the domain of interest is unbounded. In order to deﬁne a well-posed problem, one has to prescribe the behavior of the potential at the inﬁnity: this deﬁnes a second approach, set in an unbounded domain.

The stationary conservation equation on charges is

divJ= 0, (0.3)

whereJdenotes the current density. Under some suitable assumptions,5 it can be linked to the electric ﬁeld as

J=KE, (0.4)

whereK is the constantcharge mobility. This leads ﬁnally to

∇φ· ∇=2

ε. (0.5)

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This conservation equation for the charge density has also be supplemented with a boundary condition around the tip of the electrode, where (all) charges are emitted:

it can be modelled as an injection law.2,5In this setting the boundary value ofis a function of the norm of the electric ﬁeld, denoted byE. This is a more delicate topic. Accepting Kaptzov’s hypothesis, which states that the electric ﬁeld remains constant at the surface of the electrode after the discharge occurs, allows to simplify the model. One can simply choose near the tip of the electrode a boundary condition =f(E−Ec), with Ec the threshold ﬁeld, and f such that f(x) = 0 if x≤ 0, and f(x)>0 if x >0. See Refs. 2 and 5 for examples. In the air, the value of the thresholdEc is described by Peek’s heuristic formulas2:

In a cylindrical geometry, Ec = 3.1×104d

1 + 0.308

drc

kV·cm−1. (0.6)

In a spherical geometry,

Ec= 3.1×104d

1 + 0.308

2 drc

kV·cm−1. (0.7)

In the above,rcdenotes the curvature radius of the electrode (expressed in centime- ters), and d the ratio (P/P0)

(T /T0),withT0andP0respectively the reference temperature and pressure, and T and P the actual temperature and pressure. It is crucial to note that Peek’s heuristic formulas are valid only around speciﬁc electrodes, i.e.

thin, either cylindrical or spherical, electrodes.

Then, the full model — coupled in φ, — consists of Eqs. (0.2) and (0.5), together with the boundary conditions presented before.

Solving numerically the problem inis standard, for example with the help of the method of characteristics.2,5

In this paper, we focus instead on the theoretical and numerical solutions of the problem on the electrostatic potential, that is

∆u=f (0.8)

with a homogeneous Dirichlet boundary condition on the boundary and a prescribed behavior at the inﬁnity. Among others, we propose to justify mathematically Peek’s formulas and to provide a rigorous and easily computable formula for the charge density, at the tip of the electrode. We extend those results to other, less speciﬁc geometries (still, under the assumption that at inﬁnity the geometry coincides with that of a cone).

The outline is as follows. First, we describe in some details the functional set- ting. To that aim, we introduce a series of weighted Sobolev spaces around the unbounded, sharp (at the vertex), cone, and around the unbounded, rounded (at the tip), cones. Second, we solve the problem around the unbounded sharp cone:

we establish existence and uniqueness, together with somea priori regularity esti- mates. The third and fourth parts are devoted to a detailed solution of the problem

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around rounded cones, either in Cartesian, or in axisymmetric coordinates. Using multiscale expansions, we establish rigorously the asymptotic — when the cur- vature radius goes to zero — behavior of the solution. In particular, we address the problem of mathematically justifying Peek’s formulas. Moreover, we can char- acterize the behavior of the trace of the normal derivative at the tip, which is exactly the maximum surface charge density on the electrode. Among others, the multiscale expansion formula can be computed accurately with a Finite Element approximation, on meshes with a meshsize which is independent of the curvature radius. Finally, we conclude by a series of numerical experiments, and we compare our numerical method to a “standard” discretized integral representation.

The results of this paper have been announced in Ref. 10.

1. Notations and Functional Spaces 1.1. Preliminary notations

Rn is the Euclidean space of dimensionn,withn= 2,3.

Forε >0 andx= (x1, . . . , xn)Rn,λ= (λ1, . . . , λn)Nn, let us introduce ρ(x) :=

1 +|x|2, ρε(x) :=ερ

x ε

=

ε2+|x|2, where|x|=

x21+· · ·+x2n 1/2 ,

λu = |λ|u

∂xλ11· · ·∂xλnn, where|λ|=λ1+· · ·+λn. Forε= 0, we introduceρ0=ρ(note thatρ0(x)= limε→0ρε(x)).

Given an open subsetAofRn andσ >0,σAis equal to:σA:={σm; m∈A}. Bσ(x) is the open ball centered atxwith radiusσ. It is calledBσ whenx=O.

GivenA,B two topological spaces withA⊂B topologically and algebraically, AB is the closure ofAinB.

If the open subset under study is axisymmetric (n = 3), it is denoted by ˘A.

Then,A stands for its trace in a meridian half-plane, the location of any point of Abeing given by the coordinates (r, z), withr=

x21+x22,z=x3.

The set of compactly supported, C-class, functions deﬁned in A is denoted by C0 (A). Then, C0(A) is the set of restrictions (to A) of elements of C0(Rn).

Finally,D(A) is the set of distributions deﬁned onA, andL2loc(A) is the space of measurable and locally square-integrable functions onA.

1.2. Classes of domains

Let us ﬁrst introduce the classes of domains of interest, forε >0 given. We consider either Cartesian geometries, set inR2, or axisymmetric geometries, set inR3.

Cartesian case: for 1/2< α <1, let Ω denote theunbounded, sharp(at its vertex), coneofR2with vertexO and angleπ/α (see Fig. 1), and let Γ be its boundary.

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z

O r φ

π/β

cε/2 γ

γ ε

ε a

b

ε

O Ω

−ε

Oε

ε

π/α x

x 2

1

Fig. 1. Domains Ω and Ωε: Cartesian (n= 2) and axisymmetric (n= 3) geometries.

Letϕ:R−→Rbe the mapping

x−→







cx, x <−1,

−c

2(x2+ 1), |x| ≤1,

−cx, x >1.

(1.1)

We set the constant cto the valuec=(tan[π/(2α)])−1. In this way, the map- ping ϕ globally belongs to C1. In addition, it is C2 everywhere, except at the coupling pointsx=±1. Then, letω be the open subset ofR2

ω:=

(x1, x2)R2|x2> ϕ(x1) . Next, we introduce theunbounded, rounded(at the tip), cone

ε:=εω=

(x1, x2)R2x2 ε > ϕ

x1 ε

. Note that Ω1=ω.

The boundary and vertex of Ωεare respectively called ΓεandOε, withOε= (0,(cε)/2).

Axisymmetric case: for 1 < β < 2, let Ω denote the 2D, unbounded, sharp, half-cone with vertex O and angle π/β, located in the half plane R2+ :=

(r, z)R2, r >0

(see Fig. 1). Then, let ˘Ω be the 3Dunbounded sharp cone, generated by the rotation of Ω around the axis (Oz).

In the axisymmetric case, the mapping ϕ is still deﬁned by (1.1), with the constant cset to c=(tan[π/β])−1.

The open subsetω ofR2+ is equal to ω:=

(r, z)R2+|z > ϕ(r) .

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By rotation around the (Oz) axis, one generates the open subset ofR3 called ˘ω.

As above, one deﬁnes the 2D open set Ωε and then the 3D unbounded rounded coneΩ˘ε by rotation. Let Γa (resp. Γεa) be the part of the boundary of Ω (resp.

ε) located on the axisr= 0 (see Fig. 1).

Forε= 0, we introduce Ω0= Ω in 2D, and ˘Ω0= ˘Ω in 3D.

Remark 1.1. For practical reasons, we choose a parabolic shape. However, one can check that any shape deﬁned with a smooth functionϕsuch thatϕ(0)<0 and ϕ(0) = 0 can be used.

1.3. Sobolev spaces in unbounded domains

We deﬁne in this subsection the many functional spaces required to carry out the subsequent analysis. Note that we have to deal both with the behavior of solutions near the vertex of the domain, and with the behavior of the same solution at the inﬁnity. As in the case of the Dirichlet problem set in an exterior domain, the weighted Sobolev spaces are an appropriate setting. But, in addition, we have to take into account the unbounded character of its boundary. One has in particular to be careful in the deﬁnition of the trace spaces. Finally, we have to deal with either sharp or rounded (at the tip) cones.

Let us consider the ﬁrst family of Sobolev spaces below.

Definition 1.1. Givenm∈N,β∈R, introduce

around rounded cones, givenε >0, the functional space:

Wβm(Ωε) :={u∈L2loc(Ωε)|wλελu∈L2(Ωε) for|λ| ≤m},

around the sharp cone (ε= 0), the functional space:

Wβm(Ω) :={u∈L2loc(Ω) |wλλu∈L2(Ω) for |λ| ≤m}. In the above, the weightswλε andwλ are respectively deﬁned by

wλε(x) :=ρε(x)β+|λ|−m, wλ(x) :=ρ(x)β+|λ|−m.

The weights are introduced for three reasons. First, they allow to control the behavior at the inﬁnity (as previoulsy mentioned). Second, they are chosen in such a way that compact embeddings and coercivity inequalities (Poincar´e-like) hold.

Third, using them in the deﬁnition of theWβmspaces does not modify the behavior near the originO: in this way, local properties are similar to those of elements of Sobolev spaces set in bounded domains. More precisely, one has:

Hm(Ωε)⊂Wβm(Ωε)⊂Hlocm(Ωε).

In the case of a bounded domain Ωε, for a given ε > 0, one ﬁnds Wβm(Ωε) = Hm(Ωε).

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Remark 1.2. The relevance of the use ofε-dependent weights will appear when we derive error bounds for the asymptotic expansions.

Proposition 1.1. The functional spaceWβm(Ωε) is a Hilbert space, endowed with the scalar product

(u, v)Wm

β(Ωε):=

λ∈N2,0≤|λ|≤m

ε

wλε(x)2λu∂λvdx.

By truncation,15one can recover properties of the spacesHm(Ωε).

Proposition 1.2. C0(Ωε)is dense in Wβm(Ωε).

One can then consider the closure ofC0(Ωε) inWβm(Ωε).

Definition 1.2. IntroduceWβm(Ωε) :=C0(Ωε)W

βm(Ωε)

. Classically, it follows that, ifv∈Wβ1(Ωε), then ∆v(

Wβ1(Ωε)).

The spaceWβm(Ωε) coincides locally with the “usual” spaceHm(Ωε): traces can therefore be deﬁned locally. Since the space of choice for the Laplace problem is W01(Ωε), let us focus on the properties of traces of its elements. From Ref. 15, it is known that the behavior at inﬁnity is comparable to those of traces in a half-plane.

So, let us introduce Definition 1.3.

W012ε) :=

u∈L2locε) u

√ρε ∈L2ε),

Γε×Γε

|u(σ)−u(σ)|2

|σ−σ|2 dσdσ <∞

. The dual space of W01/2ε) is denoted by W0−1/2ε). The next proposition provides a characterization of traces of elements of W01(Ωε).

Proposition 1.3. Givenε≥0,the trace mapping γ:u∈ C0(Ωε)→u

ε ∈ C0ε)

can be extended by continuity to a linear,continuous and surjective mapping,from W01(Ωε)toW01/2ε). Its kernel isγ−1({0}) =

W01(Ωε).

To build suitable Green formulas, we need some other functional spaces.

Definition 1.4. ConsiderW(div,ε) :=

u∈L2(Ωε)2, ρεdivu∈L2(Ωε) . W(div,ε) is a Hilbert space, endowed with the scalar product (u,v)div,Ωε:=

(u,v)L2(Ωε)2+ (ρεdivu, ρεdivv)L2(Ωε).

Proposition 1.4. The spaceC0(Ωε)2 is dense in W(div,ε).

Proof. Cf. Ref. 16.

The normal trace of elements ofW(div,ε) on Γε can then be introduced.

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Proposition 1.5. Givenε≥0,the mapping γn:u∈ C0(Ωε)2u.n|Γε ∈ C0ε) can be extended by continuity to a linear and continuous mapping,from W(div,Ω) toW0−1/2ε). In addition,the following integration by parts formula holds:

u∈W(div,ε), v∈W01(Ωε),

(u,∇v)L2(Ωε)2+ (divu, v)L2(Ωε)=

W0−1/2ε)u·n|Γε, v|ΓεW1/2

0 ε). (1.2) Proof. Cf. Ref. 16.

1.4. Axisymmetric setting

Given ε 0, m N, β R, we can deﬁne similarly the spaces Wβm( ˘Ωε) in an axisymmetric domain ˘Ωε. The trace space of W01( ˘Ωε) is now characterized by the Definition 1.5. Introduce

W01/2(˘Γε) :=

u∈L2loc(˘Γε) u

√ρε ∈L2(˘Γε),

Γ˘ε×Γ˘ε

|u(σ)−u(σ)|2

|σ−σ|3 dσdσ <∞

. Let us consider the dimension reduction, from n= 3 to n= 2. Forµ R, let L2µ(Ωε) be the space of mesurable and square-integrable functions over Ωε, with respect to the measurerµdrdz. In particular, there holds the

Lemma 1.1. Let u∈ Wβ2( ˘Ωε) be invariant by rotation around the axis r= 0. If D2udenotes its Hessian,then there holds

˘ε

ρε(x)|D2u(x)|2d ˘Ωε

= 2π

ε

ρε(r, z) 2u

∂r2 2+1

r

∂u

∂r

2+ 2u

∂r∂z

2+2u

∂z2 2

rdrdz.

This helps to deﬁne the ad hocfunctional spaces in the meridian half-plane.

Definition 1.6. Consider, forε≥0, mN, β R,

Wβ,am(Ωε) :={u∈L21,loc(Ωε)β+|λ|−mε r,zλ u∈L21(Ωε),|λ| ≤m}.

With the help of Lemma 1.1, and according to Refs. 6 and 9, one ﬁnds that the dimension reduction works in the following manner.

Proposition 1.6. Forε≥0, βRandm∈ {0,1},the spaceWβ,am(Ωε)is exactly the space of traces in the meridian half-plane of invariant by rotation elements of Wβm( ˘Ωε). For m= 2,the relevant trace space is

Wβ,a,+2 (Ωε) :={u∈Wβ,a2 (Ωε)βεru∈L2−1(Ωε)}.

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1.5. Sobolev spaces on the unbounded sharp cone

For a ﬁxed x∈Ω,when ε→0,the value of the weight wλε(x) goes to|x|β+|λ|−m. Based on this simple observation, and from Refs. 19 and 22 it follows that the Sobolev spaces thus weighted are the “natural” ones to describe the solutions to elliptic problems in a sharp cone. The corresponding deﬁnitions are

Definition 1.7. Givenm∈N, β R,introduce Vβm(Ω) :=

u∈L2loc(Ω)

µ∈N2,|µ|≤m

|x|2(β+|µ|−m)|∂µu(x)|2dx <

. Note that there holds

u∈Vβm(Ω)⇐⇒u∈L2loc(Ω), ∀µ∈N2, |µ| ≤m, |x||µ|−mµ(|x|βu)∈L2(Ω).

1.6. The problems to be solved

In the next sections, for ε≥0,we shall solve the Laplace equation with Dirichlet boundary condition in two-dimensional or three-dimensional domains.

In 2D:

∆uε=f in Ωε,

uε=g on Γε. In 3D:

∆˘uε=f in ˘Ωε,

˘

uε=g on ˘Γε. (1.3) We make the following simplifying assumptions: ﬁrst, that g = 0, and second that f vanishes in a (ﬁxed) neighborhood of the origin O. See Ref. 19 (Chap. 7, Sec. 7.1), or Remark 3.3, for a more general assumption on f. In 3D, we further assume thatf is invariant by rotation.

We call (1.3), withg= 0,theDirichlet problemhereafter.

2. Dirichlet Problem in the Sharp Cone 2.1. Setting of the problem

For the Cartesian case, existence and uniqueness results19,22in the spacesVβm(Ω) come from the study of the Dirichlet problem in the strip R×]0,πα[. As a matter of fact, one easily checks that a one-to-one mapping from Ω to this strip is deﬁned by the change of variable t = log(r). In particular, one ﬁnds that the Laplace operator maps Vβm+2(Ω) onto Vβm(Ω), except for speciﬁc values of the index β.

More precisely, one has

Theorem 2.1. Letβ R, m≥2, f∈Vβm−2(Ω).

If β−m+ 1∈/ Zα,then the Dirichlet problem (1.3) set in the sharp cone has one, and only one solutionu0∈Vβm(Ω), which satisﬁes the a priori estimate

u0Vβm ≤cfVm−2

β ,

with a constantc independent of the right-hand sidef.

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Remark 2.1. For our purpose, it is important to note that, in the case whenm= 2 and if β is such that|β−1| < α, then the solution to (1.3) belongs toVβ2(Ω) for anyf ∈Vβ0(Ω) (see Refs. 19 and 22 for details).

For the axisymmetric case, let us introduce G,the intersection between ˘Ω and the unit sphereS2. In other words, ˘Ω = {ρσ, ρ >0, σ∈G}. Let us call (ρ, θ, φ) the spherical coordinates. The Laplace–Beltrami operator ∆GonGis deﬁned as follows:

G := 1 sinφ

∂φ

sinφ

∂φ

+ 1

sin2φ

2

∂θ2. OnG,we consider the surface gradient

Gu:=



 1 sinφ

∂u

∂θ

∂u

∂φ



.

Then, we deﬁne thead hocHilbert space (and accompanying scalar product) H1(G) :=

u∈L2(G)|∇Gu∈L2(G)2 , (u, v)H1(G):=

G

1

4uv+Gu· ∇Gv

dσ (with dσ= sinφdφdθ).

Then, we introduce the closure of smooth surface ﬁeldsH01(G) :=C0(G)H

1(G)

. It is standard knowledge that the bilinear form onH01(G)×H01(G), deﬁned by

A(u, v) :=

G

Gu· ∇Gvdσ is symmetric and coercive. As a consequence, one has

Theorem 2.2. There exist a countable sequence of positive real numbers)≥1 (sorted by increasing values), which goes to +∞, and a countable set)≥1 of elements ofH01(G)such that

)≥1 is a Hilbert basis of L2(G);

/√

Λ)≥1 is a Hilbert basis of H01(G);

• −GΨ= ΛΨ,for all≥1.

In addition, the ﬁrst eigenvalueΛ1 is simple,andΛ1>0.

As in the Cartesian case, the change of variablest= log(r) transforms problem (1.3), set in ˘Ω,in another one, set inR×G.

Theorem 2.3. Let β∈Rsuch that|β−1|<

Λ1+1

4, f∈Vβ0( ˘Ω). The Dirichlet problem(1.3)set in the 3D sharp cone has one, and only one solutionu0∈Vβ2( ˘Ω), which satisﬁes the a priori estimate

u0V2

β ≤cfV0

β, with a constantc independent of the right-hand sidef.

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Proof. See Refs. 13, 19 and 16.

2.2. Cartesian case

In this subsection, we propose a variational setting for the Dirichlet problem, set in the unbounded, sharp cone Ω. Recall that the spaceW01(Ω) combines the local regularity of the spaceHloc1 ,with a prescribed behavior at inﬁnity, which allows to recover Poincar´e inequalities in unbounded domains. More precisely, there holds the Proposition 2.1. There exists a constantC >0, such that one has

∀u∈W01(Ω),

|u(x)|2

1 +|x|2dx≤C

|∇u|2dx. (2.1)

The semi-norm|u|W01= (

|∇u|2dx)1/2is a norm on W01(Ω),which is equivalent to the full norm.

As a consequence, the Lax–Milgram Theorem can be applied to the variational formulation of the Dirichet problem, set inW01(Ω).

Proposition 2.2. Given f ∈L2loc(Ω) such that ρf ∈L2(Ω),there exists one, and only one solution u0 to the Dirichlet problem (1.3) which belongs to W01 (Ω), and satisﬁes(with a constantC independent of f)

u0W01 ≤CfW10.

Remark 2.2. The Dirichlet problem is also well-posed withf (W01(Ω)). Thanks to the property thatf vanishes in a neighborhood ofO, f also belongs to the space V10(Ω), and sou0∈V12(Ω).

Whenf vanishes in a neighborhood ofO,the asymptotic behavior ofu0,that is when r→0, is governed by its local regularity. We follow here Grisvard13,14: con- sider the polar coordinates (r, θ) with respect to the vertexO,such that Γ+ (resp.

Γ) is a subset of= 0} (resp.= απ}). Let (ϕ)≥1, ϕ(θ) =

2α/πsin(αθ) be the orthonormal basis of L2(]0,π

α[), which is associated to the 1D operator

θ := d22 with homogeneous Dirichlet boundary conditions at θ = 0 and θ = π/α. For N, we introduce the dual functions sD(r, θ) = r−αϕ(θ), as well as the primal onesφ(r, θ) =rαϕ(θ). One has

Proposition 2.3. There exists coeﬃcients)≥1such that one can write,for any N N,the expansion (in increasing powers ofr):

u0(r, θ) = N

=1

λφ(r, θ) +O(r(N+1)α), r→0. (2.2) In addition, each coeﬃcient λ depends linearly on the right-hand sidef:

λ= 1 2α

f sDdx.

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Remark 2.3. For each∈ {1, . . . , m}such thatα∈N,a log term appears in the expansion13: more precisely,φ(r, θ) =drα(logrsin(αθ) +θcos(αθ)),with d a normalizing factor. Anyway, all computations can still be carried out in this case.

We note that sinceαbelongs to ]1/2,1[,this cannot happen for2.

2.3. Axisymmetric case

Since we consider here the axisymmetric case, it is assumed that the right-hand sidef is invariant by rotation, in addition to vanishing in a neighborhood of the originO. The solution to the Dirichlet problem (1.3) set in the meridian half-plane is invariant by rotation and satisﬁes







+u0=f in Ω, u0= 0 on Γb,

∂u0

∂n = 0 on Γa.

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In the above, the 2D operator ∆+ is deﬁned as

+:= 2

∂r2 +1 r

∂r+ 2

∂z2, or

+:= 2

∂ρ2 +2 ρ

∂ρ+cot(φ) ρ2

∂φ+ 1 ρ2

2

∂φ2 in spherical coordinates (ρ, θ, φ).

Propositions 2.1 and 2.2 are still valid, so thatu0 exists, and is unique.

Let us consider for a moment that f is not invariant by rotation. The domain Ω can include a conical˘ geometrical singularity.a In addition, when the aperture angleπ/βis greater than a limit value, the behavior of the solution to the Dirichlet problem becomes singular in the sense that ˘u0 does not belong to Hloc2 ( ˘Ω) (see for instance Refs. 6, 4 and 9). This value is related to the zero of the Legendre functionP1/20 , where t Pν0(t) is the Legendre polynomial of order 0 and index ν > 0, which is bounded att = 0. More precisely, let us deﬁne βc ]1,2[, such thatP1/20 (cos (π/βc)) = 0,and assume that the aperture angle π/β is larger than π/βc: in this case, the solution can be singular. Moreover, if one performs a Fourier expansion in the azimuthal coordinateθ,it can be proven9that the singular behav- ior is contained in the modek= 0. So, albeit this mode is the only one when the solution is invariant by rotation, it is nonetheless exactly the one which behaves singularly.

As we shall see, the regularity of the solution is related to the value of the index ν >0,and to the zero(s) of the Legendre polynomialsPν0. As a matter of fact, the

aBy deﬁnition,3,4for a bounded axisymmetric domain, a geometrical singularity is located on the boundary of the domain, and is either a re-entrant circular edge, or a conical re-entrant corner (with a circular basis), both of which yield a non-convex domain.

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solutionu0 in a neighborhood ofO can be expressed as an inﬁnite sum, using the eigenfunctions of the 1D operator derived from (2.3): consider

+φ := d2

2 cot(φ) d dφ.

Its domain (taking into account the boundary conditions on φ= 0 andφ=π/β) is described below. Following Ref. 3, we introduce the functional space

H:=

u∈L2loc 0,π β

!

,(sinφ)1/2u∈L2 0,π β

!

. (2.4)

The domain of ∆+φ is D(∆+φ) :=

u∈ H

+φu∈ H, u(0) =u π

β

= 0

. (2.5)

Theorem 2.4. There exists a Hilbert basis(u)≥1ofH,made of eigenfunctions of

+φ, which are associated to the eigenvalues(sorted by increasing values) (λ)≥1, withλ R+ for all≥1andλ+∞when→+∞. All eigenfunctions(u)≥1 belong to

H1:=

u∈ H, u∈ H, u π

β

= 0

λ)≥1 is a Hilbert basis of H1.

Now, let (ν)≥1be the sequence of non-negative real numbers, sorted by increas- ing values, such thatPν0[cos(π/β)] = 0. The eigenpairs of the operator ∆+φ are

λ=ν+ 1), u(φ) = Pν0

(cosφ) Pν0

(cos·)H, 1.

Whenπ/β > π/βc,according to the tables of solutions to f ind ν >0such that Pν0[cos(π/β)] = 0,

which can be found in Ref. 1, there holdsν1<1/2 and ν>1/2 for all >1. The fact thatν1is strictly smaller than 1/2 implies that the ﬁrst term in the expansion of the solutionu0to (2.3) generates a term in ˘u0,which does not belong toHloc2 ( ˘Ω) (except if it vanishes. . . ).

ForN,deﬁne the primal and dual functions, resp.sD(ρ, φ) =ρ−νu(φ) and φc(ρ, φ) =ρνu(φ). Than, the expansion ofu0whenρ→0 is as follows.

Proposition 2.4. There exist coeﬃcients)≥1such that one can write,for any N N,the expansion (in increasing powers ofρ):

u0(ρ, φ) = N

=1

λφc(ρ, φ) +O(ρνN+1), ρ→0. (2.6) Each coeﬃcient λ depends linearly on the right-hand sidef:

λ= 1 1 + 2ν

f(ρ, φ)sD(ρ, φ)ρ2sinφdφ.

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3. Asymptotic Expansions Around Rounded Cones

In the previous section, we examined the behavior of the solutionu0to the Dirichlet problem in the sharp cone (for n = 2,3). We shall now derive the asymptotic expansions (in powers ofε) of the solutionuε to the Dirichlet problem, around the rounded cones (forn= 2,3). We assume that the right-hand sidef isindependent ofεand, as before, that it vanishes in a neighborhood ofO.

Let us introduce a cut-oﬀ function ξ ∈ C0(R+), such that 0 ξ 1, and moreoverξ≡1 in [0, a] andξ≡0 in [2a,+[ (a >0 is given). Then, forγ >0,let us consider the scaled functionχγ deﬁned byx→ξ(|x|/γ); forγ= 1,the function will be denoted simply byχ.

3.1. Rounded corners in two-dimensional domains

As for the sharp cone, the inequality (2.1) and the integration by parts formula (1.2) remain valid in Ωε. There follows

Proposition 3.1. Given f ∈W10(Ωε),there exists one, and only one solution uε, which belongs toW12(Ωε), to the problem (1.3). Moreover,it satisﬁes

uεW12≤CεfW10.

Existence and uniqueness of the solution are still valid, provided thatf belongs toWβ0(Ωε),under the condition|β−1|< α.

A priori, Cε depends on the domain Ωε, so the index ε. As we shall see, it is not the casea posteriori.

Theorem 3.1. (Stability)The constantCε of Proposition3.1can be chosen inde- pendently ofε.

Proof. Let us consider the operator

A−1ε :W10(Ωε)→W12(Ωε)

which to f W10(Ωε) associates the solution u W12(Ωε) of the Dirichlet prob- lem (1.3). The idea of the proof is to show that its operator norm is bounded, independently ofε.

Let us write f(x) =f1(x) +ε−2f2

x ε

, withf1:=

1−χε f, f2 ·

ε

:=ε2χεf. (3.1) Consider next the two problems:

∆u1=f1 in Ω, u1= 0 on Γ, (3.2)

∆u2=f2 in ω, u2= 0 on∂ω. (3.3) One can easily ﬁnd the bounds:

f1V10(Ω)+f2W10(ω)≤cfW10(Ωε),

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withcindependent ofε. In particular,u1andu2exist, and are unique, respectively in V12(Ω) andW12(ω) respectively. Moreover, one has

u1V12(Ω)+u2W12(ω)≤cfW10(Ωε).

Deﬁne (with a continuation ofu1by zero to Ωε\Ω in the ﬁrst term, still calledu1) Uε= (1−χε)u1+χu2

· ε

.

LetRεdenote the operator which tof ∈W10(Ωε) associatesUε∈W12(Ωε). We prove ﬁrst that the operator norm of Rε is bounded independently ofε. As a matter of fact

UεW12(Ωε)(1−χε)u1W12(Ωε)+"""χu2 ·

ε

"""

W12(Ωε)

.

Let us bound the ﬁrst term of the right-hand side. Since 1−χε vanishes on{x∈ε| |x| ≤aε},one ﬁnds

(1−χε)u12W2

1(Ωε)=

|λ|≤2

|x|≥aε

ε2+|x|2 |λ|−1λ(1−χε)u1)(x)2 dx.

Terms which do not include a derivative of (1−χε) are bounded by u1V12(Ω). For the other terms (|λ| ≥ 1), the integral is actually carried out over the region Cε={x∈ε|aε <|x|<2aε}. In this region, one has|x| ε,so (ε2+|x|2)|λ|−1

|x|2(|λ|−1). Therefore

(1−χε)u1W2

1(Ωε)≤C1u1V12(Ω), withC1 independent ofε.

For the second term, we perform the change of variablesy=x/εto recover

""

"χu2 ·

ε

"""

W12(Ωε)≤C2u2W12(ω).

As expected, the above allows to prove thatRεW10→W12 ≤C,withCindependent ofε.

Next, let Tε = (A−1ε )−1Rε−I. In this way, the norm A−1ε = Rε(I+Tε)−1 will be bounded independently of ε,provided that the norm ofTε is small. So, let f ∈W10(Ωε) and compute:

Tεf =∆Uε−f,

=−χεf+ ∆χεu1+χε∆u1+ 2∇χε· ∇u1

∆χu2 ·

ε −χ∆

# u2

· ε

\$2# u2

· ε

\$· ∇χ,

= ∆χεu1+ 2∇χε· ∇u1∆χu2 ·

ε

2# u2

· ε

\$· ∇χ.

Note that the last equality comes from the fact that the respective supports ofχε and of

1−χε are disjoint for small enoughε.

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