Monique Dauge
email: monique.dauge@univ-rennes1.fr

AND Ébastien Tordeux
email: sebastien.tordeux@insa-toulouse.fr

G R Égory Vial

SELF-SIMILAR PERTURBATION NEAR A CORNER: MATCHING VERSUS MULTISCALE EXPANSIONS FOR A MODEL PROBLEM

In this paper we consider the Laplace-Dirichlet equation in a polygonal domain perturbed at the small scale ε near a vertex. We assume that this perturbation is self-similar, that is, derives from the same pattern for all relevant values of ε. We construct and validate asymptotic expansions of the solution in powers of ε via two different techniques, namely the method of multiscale expansions and the method of matched asymptotic expansions. Then we show how the terms of each expansion can be split into a finite number of sub-terms in order to reconstruct the other expansion. Compared with the fairly general approach of Maz'ya, Nazarov and Plamenevskiȋ relying on multiscale expansions, the novelty of our paper is the rigorous validation of the method of matched asymptotic expansions, and its comparison with the multiscale method. The consideration of a model problem allows to simplify the exposition of these rather complicated two techniques.

The perturbations under consideration concern the space domain, they have the same small scale ε in every direction, and they are self-similar, which means that there exists a reference point x 0 and a pattern Ω such that the ε-perturbation is given by the set of point x such that (x -x 0 )ε -1 belongs to Ω. Although a local perturbation of the metric of a Riemannian manifold could be of interest as well, we only investigate in this paper the case when the perturbation involves the boundary of the domain. We are more particularly interested in the influence of corners, both in the unperturbed domain ω and in the perturbation pattern Ω.

An example of such a perturbation is given by rounded corners: Here the unperturbed domain ω is a domain with conical points, the perturbation pattern Ω is a smooth domain, and the limiting point x 0 of the perturbation is a conical point, see Fig. 1. The fillets in material engineering precisely enter this framework.

r * π/α ω 0 • π/α Ω R * O • εR * r * π/α U ε 0 • FIGURE 1.
Rounded corner: Domains ω, Ω and U ε .

The interesting and, at first glance puzzling, feature of such domains is the following: If one considers the solutions u ε of a common elliptic problem posed on such a family of domains U ε with rounded corners, each solution u ε is smooth, but the sequence u ε converges as ε → 0 to a limit solution in the corner domain ω which should contain singularities -we refer to the fundamental papers [START_REF] Kondrat | Boundary value problems for elliptic equations in domains with conical or angular points[END_REF][START_REF] Maz'ya | Estimates in L p and in Hölder classes and the Miranda-Agmon maximum principle for solutions of elliptic boundary value problems in domains with singular points on the boundary[END_REF] and to the books [START_REF] Grisvard | Boundary value problems in non-smooth domains[END_REF][START_REF] Dauge | Elliptic Boundary Value Problems in Corner Domains -Smoothness and Asymptotics of Solutions[END_REF][START_REF] Nazarov | Elliptic problems in domains with piecewise smooth boundaries[END_REF][START_REF] Kozlov | Elliptic boundary value problems in domains with point singularities[END_REF] for the vast topic of singularities.

Conversely, one can consider smooth limiting domains ω and associated patterns Ω with corners, see Fig. 2. In this case, the limiting solution is smooth and each solution u ε has corner singularities.

ω 0 • Ω 1 O • ε U ε 0 • FIGURE 2.
Corner perturbation originating from a smooth boundary point (α = 1).

More generally, self-similar perturbations may include numerous different situations: Let us mention for example small cracks originating from a boundary point of the limiting domain, see Fig. 5 p.30, and also small junctions between several connected components of ω, see Fig. 4 p. [START_REF] Vial | Analyse multi-échelle et conditions aux limites approchées pour un problème de couche mince dans un domaine à coin[END_REF].

For such singular perturbation problems, the method of matched asymptotic expansions is widely used. This method, spread by [START_REF] Vandyke | Perturbation methods in fluid mechanics[END_REF], consists in constructing two distinct complete expansions of the solution in different regions with different scalings, and to match them in an intermediate region. It has been used in [START_REF] Leguillon | Computation of singular solutions in elliptic problems and elasticity[END_REF] for the situation of Figure 1 (see also [START_REF] Il | Matching of asymptotic expansions of solutions of boundary value problems[END_REF] for a general framework). Although intuitive, this method is difficult to justify rigorously, see [START_REF] Tordeux | Méthodes asymptotiques pour la propagation des ondes dans les milieux comportant des fentes[END_REF][START_REF] Joly | Matching of asymptotic expansions for wave propagation in media with thin slots i: The asymptotic expansion[END_REF] for such a more recent justification in the case of thin slots.

An alternative is given by the multi-scale expansion technique, consisting of a superposition of terms via cut-off functions, which involve different scales. An optimal rigorous error analysis can be performed for such a method. This analysis was performed V.G. Maz'ya and coauthors in [START_REF] Maz | Asymptotic behavior of the solutions of elliptic boundary value problems in the case of variation of the domain near conic points[END_REF][START_REF] Maz ′ Ya | On the asymptotic behavior of solutions of elliptic boundary value problems with irregular perturbations of the domain[END_REF] and written in a very general framework in the monograph [START_REF] Maz'ya | Asymptotic theory of elliptic boundary value problems in singularly perturbed domains[END_REF].

In this paper we mainly investigate, as a model case, the solutions u ε of the Dirichlet problem for the Laplace operator set on a family of plane self-similar domains U ε . For each fixed ε, the regularity properties of u ε can be very different from those of their limit u 0 (more or less regular, depending on different configurations, see Fig. 1 and2, respectively). An asymptotic expansion of u ε as ε tends to 0 is the right way of understanding the mechanism of this transformation.

Our aim in this work is twofold (i) Provide the complete constructions and validations of the two different expansions provided by the two methods of multi-scale expansion and matched asymptotic expansions for the same simple example, so that everything is made explicit and as clear as possible, (ii) Compare the two expansions with each other, i.e. split each term of each expansion into sub-terms, and re-assemble them to reconstruct the terms of the other expansion. Our paper is organized as follows. In Section 2, we define the families of self-similar domains and the problems under consideration, and next we provide an outline of our results, giving the structure of the first terms of both expansions. In Section 3 we state some preliminary results on limit problems in spaces with asymptotics, which we call "supervariational problems". Section 4 is devoted to the method of multi-scale expansion, like in [START_REF] Costabel | A singularly perturbed mixed boundary value problem[END_REF][START_REF] Vial | Analyse multi-échelle et conditions aux limites approchées pour un problème de couche mince dans un domaine à coin[END_REF][START_REF] Caloz | Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer[END_REF], where optimal remainder estimates are proved. In Section 5, we present the method of matched asymptotic expansions, with the construction of the terms and matching conditions, and, by the technique of [START_REF] Tordeux | Méthodes asymptotiques pour la propagation des ondes dans les milieux comportant des fentes[END_REF][START_REF] Joly | Matching of asymptotic expansions for wave propagation in media with thin slots i: The asymptotic expansion[END_REF], the validation of the expansion by remainder estimates. Sections 4 and 5 may be read independently. We compare the expansions obtained by these two techniques in Section 6, providing formulas for the translation of the terms of each expansion into the terms of the other one. In Section 7 we mention how expansions can be generalized to other situations (more general domains, data, operators, etc...). We conclude in Section 8 with the definition of a "compound expansion" with the application to the study of the first singularity coefficient as ε → 0.

NOTATION, OUTLINE OF RESULTS

2.1. Self-similar perturbations. The families (U ε ) ε>0 under consideration are defined with the help of two domains, ω the limit (or unperturbed) domain, and Ω the pattern (or profile) of the perturbation. We denote by x and X the Cartesian coordinates in ω and Ω, respectively, and by 0 and O the corresponding origin of coordinates.

To simplify the exposition, we assume without restriction of the analysis that there is one perturbation and that the corresponding reference point x 0 coincides with the origin 0. Indeed ω and Ω do not "live" in the same world. The x coordinates are the slow variables and X = x ε are the fast variables. The junction set. The connection between ω and Ω is realized by a plane sector K with vertex at the origin. Let π α be the opening of K, including the situations of a half-plane (α = 1) or of a crack (α = 1 2 ). Thus K is a dilation invariant set and makes sense in both systems of coordinates x and X.

We denote by B ρ and B ρ the ball centered at the origin with radius ρ in the x and X coordinates, respectively. Let (r, θ) and (R, θ) be polar coordinates corresponding to variables x and X, respectively, and such that

K = x ∈ R 2 ; r > 0, θ ∈ (0, π α ) = X ∈ R 2 ; R > 0, θ ∈ (0, π α )
. The limit domain. Let ω be a bounded domain of R 2 , containing the origin 0 in its boundary ∂ω and we assume that there exists r * > 0 such that

ω ∩ B r * = K ∩ B r * .
The perturbing pattern. Let Ω be an unbounded domain of R 2 such that there exists R * > 0 for which

Ω ∩ ∁ R 2 B R * = K ∩ ∁ R 2 B R * .
The perturbed domains. Let ε 0 such that ε 0 R * = r * . For any ε < ε 0 , U ε denotes the bounded domain

(2.1) U ε = {x ∈ ω ; |x| > εR * } ∪ {x ∈ εΩ ; |x| < r * } .
The domain U ε coincides with the limit domain ω except in an ε-neighborhood of the origin, where its shape is given by the ε-dilation of the domain Ω, see Figures

U ε ∩ {εR * |x| r * } = K ∩ {εR * |x| r * }.
Note that Ω is the limit as ε → 0 of the domain U ε /ε, whereas ω is the limit of U ε .

For the most part of this work, we do not assume any particular regularity for ω and Ω, except the coincidence with the sector K in the matching regions.

2.2.

The Dirichlet problem and its singularities. As the simplest, and nevertheless typical, example of elliptic boundary value problem on a family (U ε ) of self-similarly perturbed domains, we consider the Laplace-Dirichlet problem. We are interested in asymptotic expansions with respect to ε of the solution u ε of the problem

(2.3) Find u ε ∈ H 1 0 (U ε ) such that -∆u ε = f | Uε in U ε .
Here f is a fixed function belonging to L 2 (R 2 ). We assume for simplicity 1 that

(2.4) f ≡ 0 in B r * .
Thus the support of f is contained in U ε \ B r * , which coincides with ω \ B r * , hence independent of ε. Without risk of misunderstanding, we denote simply by f the right hand side of (2.3). When ε tends to 0, we expect the solution u ε of (2.3) to converge to the solution u 0 of the limit problem (2.5) Find

u 0 ∈ H 1 0 (ω) such that -∆u 0 = f in ω. 1
This assumption may be removed, see section 7.1.

In the following, we will derive the full asymptotic expansion of u ε into powers of ε. The nature of the terms in this expansion depends on the asymptotics as r → 0 and R → ∞ of solutions to the Dirichlet problem on the limit domain ω and the pattern domain Ω, respectively. Both asymptotics involve the singular functions of the Laplace-Dirichlet problem in the sector K, which solve the homogeneous problem (2.6) s = 0 on ∂K and -∆s = 0 in K.

For the sector opening π α , a generating set for all solutions of (2.6) on the sector K is given in polar coordinates (ρ, θ) by (see e.g. [START_REF] Kondrat | Boundary value problems for elliptic equations in domains with conical or angular points[END_REF][START_REF] Grisvard | Boundary value problems in non-smooth domains[END_REF])

(2.7) s pα (ρ, θ) = ρ pα sin(pαθ), ∀p ∈ Z * .
2.3. Outline of results. As a result of our two methods of analysis, this expansion is described by two different formulas, the first terms of which we present now.

• The powers of ε appearing in both formulas are the exponents pα of the singularities (2.7). • The remainders in the following formulas are of the form O H 1 (ε α ), which means that their norms in H 1 (U ε ) are uniformly bounded by Cε α as ε → 0.

Multi-Scale Expansion:

The method of Multi-Scale Expansion consists in looking for an expansion of u ε in powers of ε with "coefficients" v α (x) and V α ( x ε ) in slow and rapid variables respectively, and so that these terms are combined with each other by cut-off functions χ( x ε ) and ψ(x) in rapid and slow variables respectively. We choose a smooth function X → χ(X) which equals 1 except in a neighborhood of O and another smooth function x → ψ(x) with compact support and equal to 1 in a neighborhood of 0. The first step of the multi-scale expansion yields that (2.8)

u ε = χ( x ε ) v 0 (x) + O H 1 (ε α
), with v 0 = u 0 , which makes precise in what sense u 0 is the limit of u ε . Thanks to the cut-off χ( x ε ), the term χ( x ε ) u 0 (x) is well-defined on U ε and is zero on the boundary U ε in any configuration, cf. Figs 1 and 2 for instance. The next step of this method yields the two-term asymptotics (2.9)

u ε = χ( x ε ) v 0 (x) + ψ(x) ε α V α ( x ε ) + O H 1 (ε 2α
), which proves in particular that the remainder in (2.8) is optimal. The general terms in the multi-scale expansion are χ( x ε ) v pα (x) and ψ(x) V pα ( x ε ), for integers p = 2, . . ., see Theorem 4.1 for an optimal estimate of remainders. The slow terms v λ (x), λ = 2α, 3α, . . ., are also solution of variational problems in the limiting domain ω, while the profiles V λ (X), λ = α, 2α, . . . solve variational problems in the pattern Ω.

The cut-off functions are used with a scale opposite to the associated terms of the asymptotic expansion. As a consequence, the transition region where both terms v 0 (x) and V λ ( x ε ) contribute together to the asymptotics is the full domain (2.2) where U ε coincides with the sector K. A wide range of problems can be treated like this, cf. [START_REF] Maz'ya | Asymptotic theory of elliptic boundary value problems in singularly perturbed domains[END_REF]Ch.4]. The slow-rapid product Ansatz can also be compared with the homogenization and asymptotic expansions in periodic structures, see [START_REF] Oleinik | Mathematical Problems in Elasticity and Homogenization[END_REF].

Matched Asymptotic Expansions: The method of Matched Asymptotic Expansions consists in constructing two different expansions (the inner and outer expansions) of u ε in rapid variables (near the perturbation) and slow variables (outside the perturbation). A priori, none of these expansions is unique or valid everywhere. They have to be matched inside an intermediate zone contained in the region (2.2). The method consists in matching the asymptotics as X = x ε → ∞ of the inner expansion with the asymptotics as x → 0 of the outer expansion.

Following the approach of [START_REF] Gadyl | Surface potentials and the method of matching asymptotic expansions in the Helmholtz resonator problem[END_REF] or of [START_REF] Tordeux | Méthodes asymptotiques pour la propagation des ondes dans les milieux comportant des fentes[END_REF] it is possible to construct an asymptotics of u ε valid everywhere with the help of a smooth cut-off function ϕ at an intermediate scale ε δ , with a fixed δ ∈ (0, 1). Let ϕ be such that ϕ(ρ) = 0 for ρ 1 and ϕ(ρ) = 1 for ρ 2. By the method of Matched Asymptotic Expansions we find the following first terms, see Theorem 5.2, (2.10)

u ε = ϕ r/ε δ u 0 (x) + 1 -ϕ r/ε δ ε α U α ( x ε ) + O H 1 (ε 2αβ
), with β = min{δ, (1 -δ)}. The remainders are optimized if we choose δ = 1 2 . Here, again, the first term u 0 coincides with the limit u 0 . The general asymptotics involve outer terms u pα , defined in ω, and inner terms U pα defined in the pattern Ω. All of them are solution of what we call "super-variational problems", i.e. problems set in spaces larger than the variational spaces, see equations (3.9) and (3.17), and where standard formulations would have non-unique solutions.

Comparison: The terms v 2α , V α , u 2α and U α exchange with each other via two singular terms colinear to the singular functions s α and s -α , cf. (2.7). There holds, see Theorem 6.1

(2.11) U α (X) = V α (X) + χ(X) b 0 1 s α (X), X ∈ Ω, u 2α (x) = v 2α (x) + ψ(x) B 1 1 s -α (x), x ∈ ω.
Here b 0 1 and B 1 1 are the first coefficients of singularities for v 0 and V α , respectively. More generally, all terms of the matched inner and outer expansions can be reconstructed from the terms of the multi-scale expansion, and vice versa. The pros and contras of each method are

• The multiscale technique gives by construction a global approximation of the solution, with optimal estimates of the remainder, whereas twice as much terms are needed in the case of matched asymptotic expansions if one wants the same order for the remainder. • The matched asymptotic expansions method builds outer and inner terms which are canonical, i.e. they do depend only on the domains ω and Ω, and not on cut-off functions, as it is the case for the multiscale technique.

SUPER-VARIATIONAL PROBLEMS

In this section, we define the precise functional framework in which we will build the asymptotic expansions. The objects we define here are needed to derive rigorously both expansions.

All the terms in (2.8)-(2.10) appear as solutions of Dirichlet problems on ω or Ω. We first recall their variational framework before considering their solutions in larger spaces.

3.1. Variational problems. The variational space V (ω) for the Dirichlet problem on the bounded domain ω is H 1 0 (ω) and for f in its dual space, the variational formulation is

(3.1)    Find u ∈ V (ω) such that ω ∇u(x) ∇v(x) dx = ω f (x) v(x) dx ∀v ∈ V (ω).
Problem (3.1) has a unique solution. As a classical consequence of an angular Poincaré inequality, we find that the variational space is embedded into a weighted Sobolev space

(3.2) V (ω) = H 1 0 (ω) ⊂ W 1 0 (ω) := {u ∈ H 1 (ω) ; r -1 u ∈ L 2 (ω)}.
The variational space V (Ω) for the Dirichlet problem on the unbounded domain Ω is the weighted space

(3.3) V (Ω) = {U ∈ L 2 loc (Ω) ; R -1 U ∈ L 2 (Ω), ∇U ∈ L 2 (Ω), U| ∂Ω = 0}, where R = √ R 2 + 1.
Then, for f in the dual of V (Ω), the variational problem below has a unique solution

(3.4)    Find U ∈ V (Ω) such that Ω ∇U(X) ∇V (X) dX = Ω f (X) V (X) dX ∀V ∈ V (Ω).
One can refer for example to [START_REF] Caloz | Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer[END_REF] for more details.

Super-variational problems in ω.

Behavior at the origin. First, we introduce some functional spaces to specify the behavior near the origin. Definition 3.1. (i) Let V loc,0 (ω) be the space of distributions

V loc,0 (ω) = u ∈ D ′ (ω) ; ϕu ∈ H 1 0 (ω), ∀ϕ ∈ D(R 2 \ {0}) . (ii) For m ∈ N and s ∈ R let W m s (ω) be the weighted Sobolev space W m s (ω) = u ∈ D ′ (ω) ; r |β|-s-1 ∂ β x u ∈ L 2 (ω), ∀β, |β| m .
Then, we particularize the meaning of O(r s ) as follows:

Notation 3.2. For s ∈ R, the function u : ω → R is said to be a O r→0 (r s ) and we write u = O r→0 (r s ) if there exists a neighborhood V of 0 in R 2 such that ∀m, n ∈ N, ∃C > 0, |r m ∂ m r ∂ n θ u| Cr s in ω ∩ V .
Combining the change of variables x → (t = log r, θ) with Sobolev embeddings, we prove:

(3.5) u ∈ W m s (ω ∩ V ), ∀m ∈ N =⇒ u = O r→0 (r s
). Note that the converse implication is not true: the function x → r s is a O r→0 (r s ), but does not even belong to W 0 s (ω ∩ V ). For functions harmonic in a neighborhood of the corner 0, there holds:

Lemma 3.3. Let u ∈ V loc,0 (ω) such that ∆u = 0 in ω ∩ V for a neighborhood V of 0.
Then for any real number s, we have the implication

(3.6) u ∈ W 1 s (ω) =⇒ u = O r→0 (r s ). Proof. Let u ∈ W 1 s (ω)
satisfying the assumptions of the lemma. Let ρ ′ ∈ (0, r * ] such that the finite sector

K ρ ′ := ω ∩ B ρ ′ is contained in V . Let ρ ∈ (0, ρ ′ ), and m ∈ N be fixed. Let us prove that u belongs to W m+2 s (K ρ ), where K ρ = ω ∩ B ρ .
For this, we consider two sectorial annuli, A and A ′ , defined as

A = {x ∈ ω ρ 0 < |x| < ρ} and A ′ = {x ∈ ω ρ ′ 0 < |x| < ρ ′ }, with ρ ′ 0 < ρ 0 < ρ/2, whence A ⊂ A ′ . A standard local elliptic estimate reads, for u satisfying u ∈ W 1 s (K ρ ′ ), ∆u ∈ W m s+2 (K ρ ′ ), and u = 0 on ∂ω ∩ B ρ ′ -see [1], (3.7) u H m+2 (A ) C ∆u H m (A ′ ) + u H 1 (A ′ ) .
Applying this estimate to the functions u k (x) = u(2 -k x) and summing up over k the obtained inequalities (multiplied by 2 -sk ), we get the following estimate from dyadic partition equivalence

(3.8) u W m+2 s (Kρ) C ∆u W m s+2 (K ρ ′ ) + u W 1 s (K ρ ′ )
. The conclusion then follows from (3.5).

We can now state about the solvability of super-variational problems on ω, that is, in spaces containing some of the dual singular functions s -pα for p 1: If we know the dual singular part of a function u ∈ V loc,0 (ω) and its Laplacian ∆u, then this function is uniquely defined. Proposition 3.4. For any data f ∈ H -1 (ω), f ≡ 0 in a neighborhood of 0, and any finite sequence (a p ) 1 p P of real numbers, there exists a unique solution u to the "supervariational problem"

(3.9)      Find u ∈ V loc,0 (ω) such that -∆u = f in ω and u - P p=1 a p s -pα = O r→0 (1).
Remark 3.5. If the sequence of coefficients (a p ) p is empty, the problem (3.9) is nothing but the variational problem (3.1).

Proof. Let the smooth cut-off function ψ satisfy ψ(x) = 1 for |x| < r * /2 and ψ(x) = 0 for |x| > r * . We set v = ψ p a p s -pα , which obviously satisfies

(3.10) v ∈ V loc,0 (ω), and ∆v = 0 in ω ∩ B r * /2 .
Hence, the problem to find w such that -∆w = f + ∆v in ω admits a unique variational solution w ∈ V (ω) = H 1 0 (ω). Moreover, (3.2) gives that w belongs to W 1 0 (ω), and by localization near point 0, w is a O r→0 (1) thanks to (3.6); the function u = w + v meets then the requirements.

On the other hand, every solution of the Laplace-Dirichlet equation can be expanded near the corner point 0 in terms of the singular functions, compare with the results in e.g. [START_REF] Kondrat | Boundary value problems for elliptic equations in domains with conical or angular points[END_REF][START_REF] Maz'ya | Estimates in L p and in Hölder classes and the Miranda-Agmon maximum principle for solutions of elliptic boundary value problems in domains with singular points on the boundary[END_REF][START_REF] Maz'ya | On the coefficients in the asymptotic of solutions of the elliptic boundary problem in domains with conical points[END_REF][START_REF] Grisvard | Boundary value problems in non-smooth domains[END_REF]. Proposition 3.6. Let s 0 be a real number. We define P as the integer part of s/α. For any u ∈ V loc,0 (ω) for which there is a neighborhood V of 0 such that

(3.11) ∆u = 0 in ω ∩ V and u = O r→0 (r -s ),
there exist a unique finite sequence (a p ) 1 p P and a unique sequence

(b p ) p∈N * (generically infinite) such that for all N ∈ N * (3.12) u(x) = P p=1 a p s -pα (r, θ) + N p=1 b p s pα (r, θ) + O r→0 (r (N +1)α ).
Notation 3.7. In the situation of Proposition 3.6, we write

(3.13) u(x) ≃ r→0 P p=1 a p s -pα (r, θ) + ∞ p=1 b p s pα (r, θ).
Proof. One can prove this lemma using the Mellin transform, see [START_REF] Kondrat | Boundary value problems for elliptic equations in domains with conical or angular points[END_REF]. In the particular case we are interested in, an argument based on separation of variables via angular Fourier series also leads to the result.

In accordance with the literature on corner asymptotics [START_REF] Maz'ya | On the coefficients in the asymptotic of solutions of the elliptic boundary problem in domains with conical points[END_REF][START_REF] Dauge | Coefficients des singularités pour des problèmes aux limites elliptiques sur un domaine à points coniques. I. Résultats généraux pour le problème de Dirichlet[END_REF][START_REF] Costabel | A quasi-dual function method for extracting edge stress intensity functions[END_REF] we can call the sum a p s -pα the dual singular part of u, whereas b p s pα represents the asymptotics of the variational part of u and can be called primal singular part of u.

In the particular case of an opening angle equal to π, i.e. α = 1, the asymptotics of the variational part contains polynomials only -it is a Taylor expansion, but the dual singular part is actually singular. More generally, if the opening angle has the form π n with a positive integer n, i.e. α = n, the asymptotics of the variational part is polynomial and can be regarded as regular.

3.3.

Super-variational problems in Ω. Behavior at infinity. We give for the pattern domain Ω similar definitions and results as in the previous section, r → 0 being replaced with R → +∞.

Definition 3.8. (i) Let V loc,∞ (Ω) be the space of distributions V loc,∞ (Ω) = U ∈ D ′ (Ω) ; ϕU ∈ H 1 0 (Ω), ∀ϕ ∈ D(R 2 ) . (ii) For m ∈ N and s ∈ R let W m s (Ω) be the weighted Sobolev space W m s (Ω) = U ∈ D ′ (Ω) ; R |β|-s-1 ∂ β X U ∈ L 2 (Ω), ∀β, |β| m , where R = √ R 2 + 1.
In the following, we shall say that W is a neighborhood of infinity if there exists a ball B R of radius R such that

(3.14) ∁ R 2 B R ⊂ W.
We introduce, similarly to Notation 3.2 Notation 3.9. For s ∈ R, the function U : Ω → R is said to be a O R→∞ (R s ) and we write

U = O R→∞ (R s ) if there exists a neighborhood W of infinity such that ∀m, n ∈ N, ∃C > 0, |R m ∂ m R ∂ n θ U(R, θ)| CR s in Ω ∩ W.
We have the implication

(3.15) u ∈ W m s (Ω ∩ W ), ∀m ∈ N =⇒ u = O R→∞ (R s ).
Thanks to a similar shift result as for Lemma 3.3, we get Lemma 3.10. Let U ∈ V loc,∞ (Ω) such that ∆U = 0 in Ω ∩ W for a neighborhood W of infinity. Then for any real number s, we have the implication

(3.16) U ∈ W 1 s (Ω) =⇒ U = O R→∞ (R s ).
The following two propositions are the counterparts of Propositions 3.4 and 3.6. The dual singular functions at infinity in Ω are now the s pα for positive integers p.

Proposition 3.11. For any F ∈ H -1 (Ω) with compact support in Ω and any finite sequence (A p ) 1 p P of real numbers, there exists a unique solution U to the "super-variational problem"

(3.17)

     Find U ∈ V loc,∞ (Ω) such that -∆U = F in Ω and U - P p=1 A p s pα = O R→∞ (1).
Proof. It is very similar to Proposition 3.4, the suitable variational space being here V (Ω) = W 1 0 (Ω).

Remark 3.12. If the sequence of coefficients (A p ) is empty, the problem (3.17) is nothing but the variational problem (3.4).

Proposition 3.13. Let s 0 be a real number. We define P as the integer part of s/α. For any U ∈ V loc,∞ (Ω) for which there is a neighborhood W of infinity such that

(3.18) ∆U = 0 in Ω ∩ W and U = O R→∞ (R s ),
there exist a unique finite sequence (A p ) 1 p P and a unique sequence

(B p ) p∈N * (generi- cally infinite) such that for all N ∈ N * (3.19) U(X) = P p=1 A p s pα (R, θ) + N p=1 B p s -pα (R, θ) + O R→∞ (R -(N +1)α ).
Notation 3.14. In the situation of Proposition 3.13, we write

(3.20) U(X) ≃ R→∞ P p=1 A p s pα (R, θ) + ∞ p=1 B p s -pα (R, θ).

MULTI-SCALE EXPANSION

The multi-scale expansion in the domain U ε is composed of two different types of terms: the slow terms involving the original variable x, and the profiles appearing in the rapid scaled variable x ε . They are superposed via cut-off functions according to the Ansatz (4.1)

u ε (x) = χ( x ε ) n ℓ=0 ε ℓα v ℓα (x) + ψ(x) n ℓ=0 ε ℓα V ℓα ( x ε ) + O(ε nα ),
where the functions χ and ψ are smooth and satisfy

(4.2)      χ(X) = 1 for |X| > 2R * and χ(X) = 0 for |X| < 3R * 2 , ψ(x) = 1 for |x| < r * 2 and ψ(x) = 0 for |x| > r * .
The first sum in (4.1) has its support away from an ε-neighborhood of the limit point 0 and, conversely, the second brings a contribution in a neighborhood of 0 (independent of ε). The transition region is the common support of the two sums which, thanks to (2.2), satisfies for any ε ε 0 /2,

U ε ∩ supp χ • ε ∩ supp ψ ⊂ {x ∈ U ε , εR * |x| r * } = {x ∈ K, εR * |x| r * }.
The construction principles of the terms is as follows: v ℓα and V ℓα are solutions of variational problems in slow variables x ∈ ω and fast variables X ∈ Ω. The cut-off by χ( x ε ) = χ(X) and ψ(x) introduces an error in fast and slow variables. These errors can be corrected with the help of the expansions as r → 0 of the terms v ℓα and as R → ∞ of the terms V ℓα . Both expansions in homogeneous terms do make sense in fast and slow variables simultaneously, which allows us to bridge the terms in the two sums in (4.1).

4.1.

The construction of the first terms. Step 0. Let v 0 = u 0 be the solution of the limit variational problem (2.5). Since v 0 is defined on the domain ω, and not on U ε , we choose to consider the truncated function v 0 = χ( x ε )v 0 instead. We note that v 0 satisfies the Dirichlet boundary condition v 0 = 0 on ∂U ε and belongs to H 1 0 (U ε ). We consider the first remainder r 0 ε defined as

u ε (x) = χ( x ε )v 0 (x) + r 0 ε (x). Thus the support of ∆r 0 ε is contained in the support of ∇χ( x ε ). Using the commutator [∆, ϕ] defined by [∆, ϕ]f := ∆(ϕf ) -ϕ∆f , we find (4.3) -∆r 0 ε (x) = ∆, χ( • ε ) v 0 (x) = 2∇ x v 0 (x) • ∇ x χ( x ε ) + v 0 (x)∆ x χ( x ε
) . Since f ≡ 0 in a neighborhood of 0, according to Proposition 3.6 (and using Notation 3.7) there exists a sequence b 0 p p 1 such that v 0 expands as r → 0 as

(4.4) v 0 (x) ≃ r→0 ∞ p=1 b 0 p s pα (x).
We insert the expansion (4.4) into (4.3). For each of its terms we use the fundamental relation which allows to convert the commutator in fast variables

(4.5) ∆, χ( x ε ) s pα (x) = ε -2 ε pα ∆ X , χ s pα ( x ε
). Thus the remainder (4.3) can be written as

(4.6) ∆r 0 ε (x) ≃ r→0 -ε -2 ∞ p=1 ε pα b 0 p ∆ X , χ s pα ( x ε ).
To complete step 0, we set V 0 = 0 and we are going to consider the further terms for p = 1, . . . , as right hand sides of a problem on Ω in the fast variable X = x ε .

Step 1. The first term in the remainder asymptotics (4.6) is

(4.7) ε -2 ε α b 0 1 ∆ X , χ s α (X).
This function is smooth with compact support. Let V α be the solution of the variational problem in Ω, cf. (3.4),

(4.8) Find V α ∈ V (Ω) such that -∆ X V α = b 0 1 ∆ X , χ s α in Ω. Then it is clear that ∆ x ε α V α ( x ε
) coincides with the function (4.7). Therefore a better start for the asymptotic expansion of u ε reads

χ( x ε )v 0 (x) + ψ(x)ε α V α ( x ε
), which satisfies the Dirichlet boundary conditions on ∂U ε , and the associated remainder r α ε is defined as

u ε (x) = χ( x ε )v 0 (x) + ψ(x)ε α V α ( x ε ) + r α ε (x)
. Since ψ ≡ 1 on the support of the right hand side (4.7), we find (4.9)

∆r α ε (x) = -∆, χ( x ε ) v 0 (x) -b 0 1 s α (x) -∆, ψ ε α V α ( x ε ). Again, the commutator ∆, χ( x ε ) v 0 (x) -b 0 1 s α (x)
will be converted in rapid variables, and since (4.10)

v 0 (x) -b 0 1 s α (x) ≃ r→0 ∞ p=2 b 0 p s pα ( x ε ),
we have gained one power of ε α . Next, we express the other part of the remainder (4.9) in slow variables. Thanks to Lemma 3.10, we have V α (X) = O R→∞ [START_REF] Agmon | Lectures on elliptic boundary value problems[END_REF]. Thus Proposition 3.13 yields that V α expands at infinity as

(4.11) V α (X) ≃ R→∞ ∞ p=1 B 1 p s -pα (X).
Since ∆s -pα = 0, we find (4.12)

∆, ψ ε α V α ( x ε ) ≃ ε→0 ∞ p=1 ε (1+p)α B 1 p ∆, ψ s -pα (x).
The terms in (4.12) start with ε 2α . They can be compensated by the solution of problems in ω. We set v α = 0.

Step 2. Next we define v 2α as the solution of the problem in slow variables in ω

(4.13) Find v 2α ∈ H 1 0 (ω) such that -∆ x v 2α = B 1 1 ∆, ψ s -α
, and V 2α as the solution of the problem in fast variables in Ω (compare with (4.8))

(4.14) Find V 2α ∈ V (Ω) such that -∆ X V 2α = b 0 2 ∆, χ s 2α . 4.2.
The general construction. The construction is done by induction. Let us assume the asymptotic expansion built up to order n -1, i.e. (4.15)

u ε (x) = χ( x ε ) n-1 ℓ=0 ε ℓα v ℓα (x) + ψ(x) n-1 ℓ=1 ε ℓα V ℓα ( x ε ) + r (n-1)α ε (x),
with v ℓα ∈ H 1 0 (ω) and V ℓα ∈ V (Ω) whose Laplacians vanish in a neighborhood of zero and ∞, respectively. For ℓ = 0, . . . , n -1, we expand the term v ℓα into singular functions at the corner point (see Proposition 3.6)

(4.16) v ℓα (x) ≃ r→0 +∞ p=1 b ℓ p s pα (x),
and, we also expand the profiles V ℓα into dual singular functions at infinity (see Proposition 3.13) (4.17)

V ℓα (X) ≃ R→+∞ +∞ p=1 B ℓ p s -pα (X).
The definitions for the next terms v nα and V nα generalize (4.13) and (4.14). The function

v nα ∈ H 1 0 (ω) solves (4.18) ∆v nα (x) = -∆ ψ(x) n-1 ℓ=1 B ℓ n-ℓ s -(n-ℓ)α (x) ,
and,

V nα ∈ V (Ω) satisfies (4.19) ∆V nα (X) = -∆ χ(X) n-1 ℓ=0 b ℓ n-ℓ s (n-ℓ)α (X) .
Let us calculate the residual: The Laplacian of the remainder is given by (4.20)

∆r (n-1)α ε (x) = ∆u ε - n-1 ℓ=0 ε ℓα ∆ χ( x ε )v ℓα (x) + ∆ ψ(x)V ℓα ( x ε ) .
Next, we expand this relation using (4.16), (4.17), and relations (4.18), (4. [START_REF] Maz'ya | Asymptotic theory of elliptic boundary value problems in singularly perturbed domains[END_REF] with n replaced by 1, 2, . . . , n -1. We obtain (4.21)

∆r (n-1)α ε = - n-1 ℓ=0 ε ℓα ∆, χ( • ε ) v ℓα n-1-ℓ + ∆, ψ V ℓα n-1-ℓ ( • ε ) , with v ℓα k (x) := v ℓα (x) - k p=1 b ℓ p s pα (x) ≃ r→0 +∞ p=k+1 b ℓ p s pα (x) (4.22a) V ℓα k (X) := V ℓα (X) - k p=1 B ℓ p s -pα (X) ≃ R→∞ +∞ p=k+1 B ℓ p s -pα (X). (4.22b)
The leading term of the remainder ∆r (n-1)α ε corresponds to the lowest terms in the sums in the right hand sides of identities (4.22), and is therefore

∆ n-1 ℓ=0 ε ℓα b ℓ n-ℓ s (n-ℓ)α (x)χ( x ε ) + ∆ n-1 ℓ=1 ε ℓα B ℓ n-ℓ s -(n-ℓ)α ( x ε )ψ(x)
which leads after scaling to, compare with (4.18) and (4.19):

ε nα ∆ n-1 ℓ=0 b ℓ n-ℓ s (n-ℓ)α ( x ε )χ( x ε ) + ∆ n-1 ℓ=1 B ℓ n-ℓ s -(n-ℓ)α (x)ψ(x) .
4.3. Optimal error estimate.

Theorem 4.1. The solution u ε of problem (2.3) admits the following multiscale expansion into powers of ε (recall that π/α is the opening angle of ω at 0):

(4.23) u ε (x) = χ( x ε ) n ℓ=0 ε ℓα v ℓα (x) + ψ(x) n ℓ=0 ε ℓα V ℓα ( x ε ) + r nα ε (x),
where the terms v ℓα and V ℓα do not depend on ε, and are defined in ω and Ω by Equations (4.18) and (4.19), respectively. Moreover, the remainder r nα ε satisfies the following estimate

(4.24) r nα ε H 1 (Uε)
Cε (n+1)α .

Proof. A basic technique to estimate the remainder consists in investigating the Laplace-Dirichlet problem it solves. By construction, r nα ε satisfies the homogeneous Dirichlet condition and belongs to H 1 0 (U ε ). By uniform coercivity, there exists C 0 > 0 such that (4.25)

r nα ε H 1 (Uε) C 0 ∆r nα ε H -1 (Uε)
, ∀ε ε 0 . Since ∆r nα ε has the expression (4.21) (with n -1 replaced by n) we have to estimate each of its terms in H -1 (U ε )-norm.

• For all v, the commutator of ∆ and χ(

• ε ) is given by (4.26) [∆, χ( • ε )]v (x) = 2ε -1 ∇v(x) • (∇χ)( x ε ) + ε -2 v(x)(∆χ)( x ε ). Hence, the support of [∆, χ( • ε )]v is included in the annulus 3R * ε/2 r 2R * ε. For v ℓα k , which is a O r→0 (r (k+1)α ), one obtains the L ∞ -bound (4.27) ∆, χ( • ε ) v ℓα k L ∞ (Uε) C ε (k+1)α-2 . εR * r * 2 U ε 0 • • x ε •x r ε ψ ≡ 1 ψ ≡ 0 εR * 2εR * U ε 0 • • x ε •x χ( • ε ) ≡ 1 χ( • ε ) ≡ 0 FIGURE 3. Point x ε , distance r ε
, and supports of cut-off functions ψ and χ( • ε ).

Let us choose X 0 such that X 0 ∈ ∂Ω and |X 0 | = R * (such a point does exist since Ω coincides with K in the region R > R * ). Then the point x ε = X 0 /ε belongs to ∂U ε , see Figure 3. Moreover, if we set

r ε (x) = |x -x ε | we find that r ε is equivalent to r in the support of [∆, χ( • ε )], uniformly in ε. Since x ε ∈ ∂U ε , there holds w r ε L 2 (Uε) C 1 w H 1 (Uε) , ∀w ∈ H 1 0 (U ε ),
with a constant C 1 independent of ε < ε 0 /2 and w. Let w ∈ H 1 0 (U ε ). We deduce via Hölder inequality

∆, χ( • ε ) v ℓα k , w = r ε ∆, χ( • ε ) v ℓα k , w r ε C ∆, χ( • ε ) v ℓα k L ∞ (Uε) r ε L 2 (Uε) w H 1 (Uε) C ε (k+1)α-2 ε 2 w H 1 (Uε) . Hence (4.28) ∆, χ( • ε ) v ℓα k H -1 (Uε) C ε (k+1)α .
• Using that the function V ℓα k is a O R→∞ (R (k+1)α ), we easily deduce the estimate C ε (k+1)α , and using the a priori estimate (4.25), we obtain the bound (4.24), which ends the proof.

[∆, ψ]V ℓα k ( • ε ) L 2 (Uε) C ε (k+1)α . Hence (4.29) [∆, ψ]V ℓα k ( • ε ) H -1 (Uε) C ε (k+1)α .

MATCHING OF ASYMPTOTIC EXPANSIONS

5.1. Formal derivation of the asymptotic expansions. We will represent the solution u ε as a formal series in each zone of interest, that is the corner expansion (or inner expansion) near the origin 0 and the outer expansion away from 0. We write these two formal series in the form (5.1)

u ε (x) ≃ +∞ ℓ=-∞ ε ℓα U ℓα ( x ε ) and u ε (x) ≃ +∞ ℓ=-∞ ε ℓα u ℓα (x).
This Ansatz is suggested by the homogeneity of the singular functions, see (2.7). We will give a sense to the infinite sums in terms of asymptotic expansions later on.

Since the H 1 -norm of u ε is uniformly bounded with respect to ε, we know that all the u ℓα and U ℓα for ℓ < 0 are just zero. Moreover, it is clear that the terms of the asymptotic expansions must satisfy (5.2)

    

-∆u 0 = f in ω and u 0 = 0 on ∂ω, ∀ℓ > 0, ∆u ℓα = 0 in ω and u ℓα = 0 on ∂ω \ {0}, ∀ℓ 0, ∆U ℓα = 0 in Ω and U ℓα = 0 on ∂Ω. Now we need to ensure the matching of the two formal series in the transition zone (5.3) ε ≪ r ≪ 1.

To do so, we expand the terms u ℓα and U ℓα . Thanks to Propositions 3.6 and 3.13 -note that r ≪ 1 and r ε ≫ 1 -these expansions read 2

(5.4)

           u ℓα (x) = +∞ p=1 a ℓ p s -pα (r, θ) + b ℓ p s pα (r, θ) , U ℓα (X) = +∞ p=1 A ℓ p s pα (R, θ) + B ℓ p s -pα (R, θ) .
2 Note that we do not use the boldface notation for the coefficients b 0 p , because we do not yet know whether they coincide with the coefficients b 0 p already defined in Section 4. The coincidence will be shown in Section 6.

We use the homogeneity of the functions s pα and transform the rapid variable r ε into the slow one r. Ensuring the equality of the two formal series (5.1), we get (5.5)

                                 +∞ ℓ=-∞ ε ℓα +∞ p=1 a ℓ p s -pα (r, θ) + b ℓ p s pα (r, θ) = +∞ ℓ=-∞ ε ℓα +∞ p=1 A ℓ p s pα ( r ε , θ) + B ℓ p s -pα ( r ε , θ) = +∞ ℓ=-∞ ε ℓα +∞ p=1 ε -pα A ℓ p s pα (r, θ) + ε pα B ℓ p s -pα (r, θ) = +∞ ℓ=-∞ ε ℓα +∞ p=1 A ℓ+p p s pα (r, θ) + B ℓ-p p s -pα (r, θ) .
Identifying the terms of the two series leads to Moreover, U 0 is chosen to be 0. Let b 0 be the sequence of numbers defined by Proposition 3.6 and B 0 be zero:

(5.9) b 0 = b 0 p p∈N * and B 0 = B 0 p p∈N * = 0.

Step ℓ. We denote by a ℓ = a ℓ p p∈N * and A ℓ = A ℓ p p∈N * the two finite sequences of real numbers such that (5.10)

a ℓ p = B ℓ-p p if 1 p ℓ -1 and a ℓ p = 0 if p ℓ, A ℓ p = b ℓ-p p if 1 p ℓ and A ℓ p = 0 if p ℓ + 1.
The functions u ℓα and U ℓα are defined via Propositions 3.4 and 3.11 as the unique solutions of the problems (5.11)

    
Find u ℓα ∈ V loc,0 (ω) such that

∆u ℓα = 0 in ω and u ℓα - ℓ-1 p=1 a ℓ p s -pα = O r→0 (1),

and

(5.12)

     Find U ℓα ∈ V loc,∞ (Ω) such that ∆U ℓα = 0 in Ω and U ℓα - ℓ p=1 A ℓ p s pα = O R→∞ (1).
Finally, we define the sequences b ℓ and B ℓ associated with u ℓα and U ℓα in Propositions 3.6 and 3.13

Global error estimates.

The main idea to prove error estimates is to define a global approximation

u ε nα ∈ H 1 0 (U ε ) of u ε by the formula (5.14) u ε nα (x) = ϕ r η(ε) n ℓ=0 ε ℓα u ℓα (x) + 1 -ϕ r η(ε) n ℓ=1 ε ℓα U ℓα ( x ε ),
where ϕ is a smooth cut-off function with ϕ(ρ) = 0 for ρ < 1 and ϕ(ρ) = 1 for ρ > 2 and η is a smooth function of ε such that Theorem 5.2. There exists a constant C such that

(5.16) u ε -u ε nα H 1 (Uε) C η(ε) (n+1)α + ε η(ε) (n+1)α .
Remark 5.3. One can optimize the estimate (5.16) by choosing the best η: For η(ε) = ε 1/2 , there exists a constant C such that (5.17)

u ε -u ε nα H 1 (Uε) C ε (n+1)α/2 .
Proof. First, we denote by e ε nα the approximation error at step n e ε nα (x) = u ε nα (x) -u ε (x) and by E ε nα the corresponding matching error

E ε αn (x) = n ℓ=0 ε ℓα u ℓα (x) -U ℓα ( x ε ) .
Of course, the matching error makes sense and is small only in the intermediate region; we shall express the H 1 -norm of e ε nα over U ε in terms of E ε nα in this region. By harmonicity of u ε , u ℓα , and U ℓα , we obtain

∆ e ε nα (x) = 2 η(ε) [∇ϕ]( r η(ε) ) ∇E ε nα (x) + 1 [η(ε)] 2 [∆ϕ]( r η(ε) ) E ε nα (x).
Since e ε nα belongs to H 1 0 (U ε ), the Green formula leads to

             Uε ∇ e ε nα 2 dx = 2 η(ε) Uε [∇ϕ]( r η(ε) ) ∇E ε nα e ε nα + 1 [η(ε)] 2 Uε [∆ϕ]( r η(ε) ) E ε nα e ε nα dx C [η(ε)] 2 E ε nα ∞,η(ε) + η(ε) ∇E ε nα ∞,η(ε) e ε nα 1,η(ε) , with the notation, for p ∈ [1, ∞] (5.18) u p,η(ε) = u L p ( {x∈ω ; η(ε) r 2η(ε)} ) .
Using a Poincaré inequality on U ε (uniform with respect to ε), we get

e ε nα 2 H 1 (Uε) C (η(ε)) 2 E ε nα ∞,η(ε) + η(ε) ∇E ε nα ∞,η(ε) × e ε nα 1,η(ε) .
The conclusion follows from the following two lemmas (proved below).

Lemma 5.4. There exists a constant C such that for all u ∈ H 1 0 (U ε ), the norm u 1,η(ε) , defined in (5.18), can be estimated as follows

(5.19) u 1,η(ε) C [η(ε)] 2 u H 1 (Uε) .
Lemma 5.5. There exists a constant C such that -for the definition of the norms, see (5.18),

(5.20)

E ε nα ∞,η(ε) C η(ε) (n+1)α + ε η(ε) (n+1)α , (5.21) ∇E ε nα ∞,η(ε) C 1 η(ε) η(ε) (n+1)α + ε η(ε) (n+1)α . Proof. [lemma 5.4] For all u ∈ H 1 0 (U ε ) and for all r ∈ [η(ε), 2η(ε)] π α 0 |u(r, θ)| dθ π α 0 θ 0 ∂u ∂θ (r, θ ′ ) dθ ′ dθ π α π α 0 ∂u ∂θ (r, θ) dθ.
Hence, we have

2η(ε) η(ε) π α 0 |u(r, θ)| rdr dθ π α 2η(ε) η(ε) π α 0 ∂u ∂θ (r, θ) rdr dθ π α 2η(ε) η(ε) π α 0 2η(ε) r ∂u ∂θ (r, θ) rdr dθ C η(ε) ∇u 1,η(ε) .
We conclude using the Cauchy-Schwarz inequality that

u 1,η(ε) C η(ε) ∇u 1,η(ε) C [η(ε)] 2 ∇u 2,η(ε) .
Proof. [lemma 5.5] We will give the proof of (5.20). Inequality (5.21) can be obtained using the same technique. The first step is to expand the u ℓα and U ℓα using (3.12) and (3.19). By definition of u ℓα and U ℓα -see (5.11) and (5.12), and taking (5.10) into account one finds

u ℓα (x) = ℓ p=1 B ℓ-p p s -pα (r, θ) + n-ℓ p=1 b ℓ p s pα (r, θ) + O r→0 (r (n+1-ℓ)α ), U ℓα (X) = ℓ p=1 b ℓ-p p s pα (R, θ) + n-ℓ p=1 B ℓ p s -pα (R, θ) + O R→∞ (R (ℓ-n-1)α ).
Since η(ε) tends to 0 and η(ε)/ε tends to +∞ when ε tends to 0, one has for η(ε) r 2η(ε)

(5.22)            u ℓα (x) - ℓ p=1 B ℓ-p p s -pα (r, θ) - n-ℓ p=1 b ℓ p s pα (r, θ) C [η(ε)] (n+1-ℓ)α , U ℓα ( x ε ) - ℓ p=1 b ℓ-p p s pα ( r ε , θ) - n-ℓ p=1 B ℓ p s -pα ( r ε , θ) C[ ε η(ε) ] (n+1-ℓ)α .
Let S be given by (5.23)

S = n ℓ=0 ε ℓα ℓ p=1 B ℓ-p p s -pα (r, θ) + n-ℓ p=1 b ℓ p s pα (r, θ) - n ℓ=0 ε ℓα ℓ p=1 b ℓ-p p s pα ( r ε , θ) + n-ℓ p=1 B ℓ p s -pα ( r ε , θ) .
From (5.22) and triangle inequalities, we obtain

(5.24) E ε nα (r, θ) -S ∞,η(ε) C n ℓ=0 ε ℓα [η(ε)] (n+1-ℓ)α + n ℓ=0 ε ℓα [ ε η(ε) ] (n+1-ℓ)α C n ℓ=0 [ ε η(ε) ] ℓα [η(ε)] (n+1)α + n ℓ=0 η(ε) ℓα [ ε η(ε) ] (n+1)α C [η(ε)] (n+1)α + [ ε η(ε) ] (n+1)α .
Now it remains to show that S = 0. By definition -see (2.7) -the singular functions s ±pα satisfy the homogeneity property

s -pα ( r ε , θ) = ε pα s -pα (r, θ) and s pα (r, θ) = ε pα s pα ( r ε , θ).
Therefore, S is given by

S = n ℓ=0 ℓ p=1 ε (ℓ-p)α B ℓ-p p s -pα ( r ε , θ) + n ℓ=0 n p=1 ε ℓα b ℓ p s pα (r, θ) - n ℓ=0 ℓ p=1 ε (ℓ-p)α b ℓ-p p s pα (r, θ) - n ℓ=0 n p=1 ε ℓα B ℓ p s -pα ( r ε , θ).
The change of variables ℓ -p → ℓ in the first and third terms leads to S = 0.

Local error estimates.

In this paragraph B r will denote the ball of radius r and of center O. Starting from the global error estimates obtained in (5.17), it is easy to get estimates far from the corner and near the corner Theorem 5.6. For any r 0 > 0, there exists C > 0 such that

(5.25) u ε (r, θ) - n ℓ=0 ε ℓα u ℓα (r, θ) H 1 (ω\Br 0 ) = O(ε (n+1)α ).
For any R 0 > 0, there exists C > 0 such that

(5.26) u ε (εR, θ) - n ℓ=0 ε ℓα U ℓα (R, θ) H 1 (Ω∩B R 0 ) = O(ε (n+1)α ).
Proof. To prove (5.25), we remark that, for ε small enough, the only contribution comes from the terms u ℓα

(5.27)

u ε nα = n ℓ=1 ε ℓα u ℓα in U ε \ B r 0 = ω \ B r 0 .
Consequently,

(5.28)

u ε -u ε nα H 1 (ω\Br 0 ) u ε -u ε (2n+2)α H 1 (ω\Br 0 ) + u ε (2n+2)α -u ε nα H 1 (ω\Br 0 ) u ε -u ε (2n+2)α H 1 (Uε) + u ε (2n+2)α -u ε nα H 1 (ω\Br 0 )
. On the other hand, it follows from (5.27)

(5.29) u ε (2n+2)α -u ε nα = 2n+2 ℓ=n+1 ε ℓα u ℓα in ω \ B r 0 ,
and, since the u ℓα 's do not depend on ε

(5.30) u ε (2n+2)α -u ε nα H 1 (ω\Br 0 ) C ε (n+1)α .
Due to (5.17), one finally has

(5.31) u ε -u ε (2n+2)α H 1 (ω\Br 0 ) C ε (n+1)α .
The estimate (5.25) follows from (5.27), (5.28), (5.30) and (5.31). The same technique leads to (5.26) as well. A scaling is needed (R = r/ε) to recover a domain independent of ε.

Remark 5.7. Due to estimates (5.25) and (5.26), the outer and corner expansions are unique. Moreover, as the remainders are of the same orders as the first neglected term in the outer and corner expansions, these estimates are optimal. The outer and corner expansions can be seen as Taylor expansions of the exact solution expressed in the (r, θ) or (r/ε, θ) coordinates.

COMPARISON OF THE TWO EXPANSIONS

In Section 5, starting from the outer and corner (matched) expansions, we were able to build a global asymptotic expansion for the solution u ε of problem (2.3), see expression (5.14). Using the multiscale technique, we proved in Section 4 another asymptotic expansion, which is also valid in the whole domain U ε . The global error estimates given in Theorems 5.2 and 4.1 allow to compare these expansions. Theorem 6.1. The expansions (5.14) and (4.23) compare in the following way:

• The terms u nα and v nα coincide away from the corner point i.e. for r r * ; • The profiles U nα and V nα coincide in the corner region i.e. for R R * /2. More precisely, we have the identities

(6.1)            v nα (x) = u nα (x) -ψ(x) n-1 p=1 a n p s -pα (x), V nα (X) = U nα (X) -χ(X) n p=1 A n p s pα (X).
where the coefficients a n p and A n p , are those defined in Section 5.2. Proof. The first two statements follow directly from the optimal estimates, (5.25), (5.26), (4.23), and (4.24), via localization. To get formulas (6.1), we start from problem (4.19) which defines V nα . We set

U nα (X) = V nα (X) + χ(X) n-1 ℓ=0 b ℓ n-ℓ s (n-ℓ)α (X) (6.2) = V nα (X) + χ(X) n p=1 b n-p p s pα (X). (6.3) From the definition of V nα (see (4.19)), U nα satisfies ∆ U nα = 0 in Ω. Hence, one has (6.4)        U nα -U nα ∈ C ∞ (Ω), ∆[ U nα -U nα ] = 0 in Ω, U nα (X) -U nα (X) = 0 for R R * /2.
Since U nα -U nα is harmonic, it is analytic in Ω. Hence, by unique continuation Theorem,

U nα = U nα . Moreover, as V nα is a O R→∞ (1), one has A n p = b n-p p (6.5) U nα (X) = V nα (X) + χ(X) n p=1 A n p s pα (X).
The same argumentation can be done for u nα . Remark 6.2. As can be seen in ( 6.3), another formula linking the two expansions is (6.6)

           u nα (x) = v nα (x) + ψ(x) n-1 p=1 B n-p p s -pα (x), U nα (X) = V nα (X) + χ(X)
n ℓ=0 ε ℓα V ℓα ( x ε ) = n ℓ=0 ε ℓα U ℓα ( x ε ) -χ( x ε ) ℓ p=1 A ℓ p s pα ( x ε ) = n ℓ=0 ε ℓα U ℓα ( x ε ) -χ( x ε ) n ℓ=0 ℓ p=1 ε (ℓ-p)α A ℓ p s pα (x) = n ℓ=0 ε ℓα U ℓα ( x ε ) -χ( x ε ) n j=0 ε jα n-j p=0 A p+ℓ p =b ℓ p s pα (x).
The second term involves the slow variable and will contribute to the terms (u ℓα ) in the intermediate region.

EXTENSIONS AND GENERALIZATIONS

The above results can be more or less easily generalized to other situations of interest. For the convenience of the reader, we briefly present a few of them:

(1) Smooth right hand sides f without condition of support, (2) Neumann boundary conditions,

Small holes and multiple junctions, (4) Helmholtz operator.

7.1. Smooth data without condition of support. Until now we have assumed that the right hand side f of problem (2.3) is zero in a neighborhood of the limit point 0 of the ε-perturbation. This assumption can be relaxed by considering functions f which are restrictions to U ε of a C ∞ function f defined on a neighborhood of ∪ ε ε 0 U ε . In this case we can write

(7.1) f (x) ≃ r→0 +∞ q=0 f q (r, θ) with f q (x) = ε q f q ( x ε ).
The asymptotic expansion (7.1) of the right hand-side introduces new terms with integer exponents in the asymptotics of u 0 = v 0 as r → 0: Instead of the infinite expansion (4.4) we have now

(7.2) v 0 (r, θ) ≃ r→0 ∞ p=1 b 0 p s pα (r, θ) + ∞ q=1 T q (r, θ),
where T q (r, θ) is a sum of terms of the form r q ϕ 0 (θ) and r q log rϕ 1 (θ) (with ϕ 1 = 0 if α is not rational). In turn, these new terms induce new factors with integer powers of ε in rapid and slow expansions. If α is not a rational number, the expansion of u ε takes the form, compare with (4.23):

(7.3) u ε = p, q∈N pα+q<s ε pα+q χ( x ε ) v pα+q (x) + ψ(x) V pα+q ( x ε ) + O H 1 (ε s ).
The asymptotics obtained by matched asymptotics expansion contains the same powers of ε as (7.3). In the case where α is rational, logarithms may appear via the scale ε pα+q log ε.

For more details, we refer to [START_REF] Maz'ya | Asymptotic theory of elliptic boundary value problems in singularly perturbed domains[END_REF][START_REF] Caloz | Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer[END_REF].

7.2. Neumann boundary conditions. Instead of (2.3) let us consider the problem

(7.4) Find u ε ∈ H 1 (U ε ) such that ∀v ∈ H 1 (U ε ), Uε ∇u ε • ∇v dx = Uε f v dx.
Its solvability needs the compatibility condition (7.5)

Uε f dx = 0, ∀ε < ε 0 .

Let us assume that f ≡ 0 in a neighborhood of 0 and that ω f dx = 0. This implies condition (7.5) for ε small enough. To ensure uniqueness, we require (7.6)

Uε u ε dx = 0, ∀ε < ε 0 .
The construction of the multiscale expansion for u ε relies on the solution of variational Neumann problems in ω and Ω. In the unbounded domain Ω, the variational space V (Ω) is defined as

(7.7) {U ∈ D ′ (Ω) ; ∇U ∈ L 2 (Ω), (1 + R) -1 (log(2 + R)) -1 U ∈ L 2 (Ω)}.
The bilinear form (U, V ) -→ Ω ∇U • ∇V dx is continuous and coercive on the quotient space V (Ω)/R, see [2] or [START_REF] Caloz | Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer[END_REF]Prop. 3.22]. Therefore, like in ω, the solution of the Neumann problem in Ω with right hand side F requires the compatibility condition Ω F dX = 0. Thus new features have to be taken into account:

(1) Compatibility conditions. The right hand sides which occur during the construction have the form ∆, ψ s -pα in ω and ∆ X , χ s pα in Ω, with the Neumann singularities s pα = ρ pα cos pαθ. Since s pα is harmonic, these right hand sides are equal to ∆(ψ s -pα ) and ∆ X (χ s pα ), respectively. The functions ψ s -pα and χ s pα satisfying the Neumann boundary condition on ∂ω and ∂Ω, respectively, we can show that the compatibility conditions are fulfilled for all integer p 1.

(2) The role of constants. (i) The asymptotic expansion of v 0 at O starts with b 0 0 s 0 , which is a constant. The associated problem in fast variables is, cf. (4.8)

(7.8) -∆V 0 = b 0 0 ∆ X χ in Ω and ∂ n V 0 = 0 on ∂Ω.
We choose the solution V 0 = b 0 0 (1 -χ). Thus, in particular, ψ(x)V 0 ( x ε ) = V 0 ( x ε ): The cut-off by ψ does not introduce any error. Let us notice that the function χ( x ε ) + ψ(x)(1 -χ)( x ε ) represents the extension by 1 from ω to U ε . (ii) For problems in Ω giving V pα , p 1, we choose the variational solution which tends to 0 as R → ∞.

(3) The condition for uniqueness. By construction the slow variable terms v pα have a zero integral on ω. Using their asymptotics as r → 0 we find that

Uε χ( x ε ) v pα (x) dx = β p ε 2 , β p ∈ R.
For fast variable terms we find

Uε ψ(x) V pα ( x ε ) dx = β ′ p ε 2 , β ′ p ∈ R.
We compensate the possible non-zero integral of the multiscale expansion by a series of constant functions -with values γ p,n ∈ R, p ∈ N, n ∈ N * -associated with the gauge functions ε pα+2n . Taking the equality Uε dx = meas ω + γε 2 into account, the γ p,n are defined by forcing the formal equality (7.9)

+∞ p=0 ε pα+2 β p + β ′ p + meas ω + γε 2 +∞ p=0 +∞ n=1 ε 2n+pα γ p,n = 0.
In the end we obtain a multiscale expansion of the form (7.10)

u ε = p∈N pα<s ε pα χ( x ε ) v pα (x) + ψ(x) V pα ( x ε ) + n∈N * pα+2n<s γ p,n ε 2n + O H 1 (ε s ).
7.3. Small holes. The set K = R 2 may also be convenient as a junction set: It allows to consider the case of small holes (or small cracks) of size ε inside U ε . This is indeed the first case considered in the book [19, sec. 2.4.1]. Let us consider the Dirichlet case. Then we are in a situation which shares common features with the Dirichlet case investigated in the most part of this paper, and the Neumann case considered above.

Indeed, the limit problem in ω is uniquely solvable. But the limit problem in Ω is not coercive on the subspace of W 1 0 (Ω) with zero trace on ∂Ω. The correct variational space is the subspace of the space (7.7) with zero trace on ∂Ω. Nevertheless, arguments are slightly different from the Neumann case (like in [START_REF] Caloz | Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer[END_REF], the asymptotic behavior log R as R → ∞ has to be considered). The outcome of the analysis is the appearance of terms (log ε) -1 . Finally, cut-off functions χ( x ε ) and ψ(x) can be simply omitted since U ε is a subset of ω and of εΩ.

7.4. Domains with multiply connected junction sectors. This is the situation where the family of domains (U ε ) is defined like in Section 2.1, where we relax the assumption on that the set K is a plane sector. Our results extend to the case where K is a finite disjoint union of plane sectors K 1 , . . . , K m with common vertex. Accordingly, we relax the assumption on ω which is still open and bounded, but can be multiply connected. The unbounded open set Ω can also be multiply connected. The open sets U ε have still to be connected. If m = 2, this requires that either ω or Ω should be connected. Of course, the interesting case occurs when Ω is connected, see Figure 4.

ω π/α 1 π/α 2 0 • π/α 1 π/α 2 Ω • O U ε π/α 1 π/α 2 • 0 FIGURE 4. Example of domains ω, Ω and U ε in the multiply connected case (α 1 = 3, α 2 = 1).
The generalization of our expansions to this situation is straightforward. We denote by π/α 1 , . . . , π/α m the openings of the sectors K 1 , . . . , K m . The multiscale expansion of u ε solution of the Dirichlet problem (2.3) with a right hand side f ≡ 0 in a neighborhood of O is as follows. For all real number s > 0 there holds (7.11)

u ε = p 1 ,...,pm∈N p 1 α 1 +•••+pmαm<s ε p 1 α 1 +•••+pmαm χ( x ε ) v p 1 α 1 +•••+pmαm (x) + ψ(x) V p 1 α 1 +•••+pmαm ( x ε ) + O H 1 (ε s ).
Here V 0 = 0, and v α j = 0 for j = 1, . . . , m. The matched asymptotics exansion is similar.

7.5. Non homogeneous operators : Helmholtz equation. The investigation of the Helmholtz operator in a singularly perturbed domain is of major importance for applications, see [START_REF] Joly | Matching of asymptotic expansions for wave propagation in media with thin slots i: The asymptotic expansion[END_REF][START_REF] Tordeux | Méthodes asymptotiques pour la propagation des ondes dans les milieux comportant des fentes[END_REF] for an example of wave propagation in a domain with thin slots. We want to give here the key arguments to derive and justify the matched asymtptotic of the solution of the following model problem, posed in the domain U ε defined in (2.1):

(7.12) Find u ε ∈ H 1 0 (U ε ) such that -∆u ε -k 2 u ε = f | Uε in U ε ,
where for the sake of simplicity we suppose that (i) α is not a rational, (ii) -k 2 is not an eigenvalue of the Dirichlet Laplacian in the limit domain ω.

The first assumption avoids the occurrence of a logarithmic gauge function in the asymptotic expansions. The second leads to a well-posed limit problem.

This situation is rather more technical since this operator is not self-similar :

(7.13) ∆ x + k 2 = 1 ε 2 ∆ X + ε 2 k 2 .
A second difficulty comes from the loss of coercivity. The proofs of existence and convergence need then to be modified, see for example [START_REF] Maz'ya | Asymptotic theory of elliptic boundary value problems in singularly perturbed domains[END_REF]Ch.4] and [START_REF] Joly | Matching of asymptotic expansions for wave propagation in media with thin slots i: The asymptotic expansion[END_REF][START_REF] Tordeux | Méthodes asymptotiques pour la propagation des ondes dans les milieux comportant des fentes[END_REF]. Some preliminaries on super-variational problems. According to a common usage, we denote by J pα and Y pα the Bessel function of first and second kind of order pα, respectively, see for instance [START_REF] Lebedev | Special functions and their applications[END_REF]. In the neighborhood of 0, this solution can be expanded as follows u(x) ≃ Remark 7.3. If condition (i) is not satisfied, the expansions of J pα and Y pα do not only include terms of the form r µ but also terms of the form r µ (ln r) ν .

                       For m = 0, . . . , M, find U m ∈ V loc,∞ (Ω) such that ∆U m + k 2 U m-2 = 0 in Ω, (with convention U m = 0 if m < 0) U m (X) = P p=1 m ℓ=0 A m-ℓ p J pα,ℓ (
Definition of the matched asymtptotic expansions. The gauge functions appearing in the asymptotic expansions of u ε are of the form ε m+nα , ie. we look for the two asymptotic expansions with the form

u ε (x) ≃ (m,n)∈N 2 ε m+nα U m,n ( x ε ) and u ε (x) ≃ (m,n)∈N 2 ε m+nα u m,n (x).
The coefficients of these expansions can be defined hierarchically as the unique solutions of the coupled problem The global approximation is defined by

           Find u m,n ∈ V loc,0 (ω) such that ∆u m,n + k 2 u m,n = 0 in ω, (or -f if m = n = 0) u m,n - n p=1 a m,n p Y pα (kr) sin (pαθ) = O r→0 (1),                      Find U m,n ∈ V loc,∞ (Ω) such that ∆U m,n + k 2 U m-2,n = 0 in Ω, (with U m,n = 0 if m < 0) U m,n (X) - n p=1 m ℓ=0 A m-ℓ,n p J pα,ℓ (
(7.15)        u ε N (x) = ϕ r η(ε) (m,n)∈I N ε m+nα u m,n (x) + 1 -ϕ r η(ε) (m,n)∈I N ε m+nα U m,n ( x ε ),
where ϕ is a smooth cut-off function with ϕ(ρ) = 0 for ρ < 1 and ϕ(ρ) = 1 for ρ > 2 and η is smooth and satisfies There exists a constant C such that (7.17)

u ε -u ε N H 1 (Uε) C η(ε) N + ε η(ε) N .

CONCLUSION: A PRACTICAL TWO-TERM EXPANSION

In order to investigate the influence of singular perturbations of the domain on a local functional ϕ ε acting over the solution u ε , it is valuable to use a compound version of the asymptotic expansion, in between multiscale and matched asymptotic expansions. 8.1. Compound expansion. Indeed, using (2.9) and the relation (2.11) between the profiles V α and U α , we get

u ε = χ( x ε ) u 0 (x) + ψ(x) ε α U α ( x ε ) -χ( x ε ) A s α ( x ε ) + O H 1 (ε 2α
), which can be written, thanks to the homogeneity of the singular function s α u ε = χ( x ε ) u 0 (x) -A ψ(x)s α (x) + ψ(x) ε α U α ( x ε ) + O H 1 (ε 2α ). Let us introduce the first "canonical" profile U α Ω as the solution of the super-variational Dirichlet problem on Ω (8.1)

Find U α Ω ∈ V loc,∞ (Ω) such that ∆U α Ω = 0 in Ω and U α Ωs α = O R→∞ (1). We have U α = A U α Ω and, hence (8.2) u ε = χ( x ε ) [u 0 (x) -A ψ(x)s α (x)] + ψ(x)ε α A U α Ω ( x ε ) + O H 1 (ε 2α ). In (8.2), only canonical objects are involved: the limit term u 0 , its first singularity coefficient A, and the first profile U α Ω . The contribution near the corner is fully contained in the profile AU α Ω , whereas the "far-field" information is carried out by u 0 -Aψs α , corresponding to the limit term without its first singularity. In a sense, the strongest singularity of u 0 is "chopped off" for ε > 0 via the cut-off χ( x ε ), and is replaced with the profile AU α Ω , which connects the local geometry around O with the plane sector of opening α at infinity. 8.2. Application: asymptotics of coefficients of singularities. An interesting application of expansion (8.2) is the determination of Stress Intensity Factors at the tip of a short crack emanating from a sharp of a rounded V-notch, see [START_REF] Leguillon | Crack onset at a v-notch. Influence of the notch tip radius[END_REF]. More generally, the question is the determination of the asymptotic behavior of the coefficients of singularities of u ε associated with the corners (or cracks) of the domain U ε inside its perturbed region. The functional ϕ ε (u ε ) is then defined as the value of this coefficient of singularity, corresponding to a corner whose position depends on ε.

Indeed, to each corner point (or crack tip) d of the perturbation pattern Ω corresponds a corner point (or crack tip) d ε of the perturbed domain U ε . In Figure 5, two such points are involved, both associated with angle 2π.

ω • 0 Ω 1 O • d • U ε 0 • d ε • ε FIGURE 5.
Crack tips: Domains ω, Ω and U ε .

The solution u ε of the Laplace-Dirichlet problem (2.3) is singular at point d ε , with the following first order approximation (8.3) u ε (x) = γ ε r µ ε sin(µθ ε ) + O ε (r µ ε ), as r ε → 0, where (r ε , θ ε ) denote the polar coordinates around d ε . The exponent µ is the singular exponent corresponding to d ε (µ = π/ϑ for a corner of opening ϑ, µ = 1/2 for a crack). The functional ϕ ε is defined as ϕ ε (u ε ) = γ ε .

Our results allow to give an asymptotic expansion of the singular coefficient γ ε as ε → 0: we still denote by α the singular exponent associated with the limit problem in ω. Using (8.2), we get

γ ε = ε α-µ AΓ + O(ε α-µ ).
It is worth noticing that a coefficient associated with a stronger singularity than the limit singularity (µ < α) will go to 0 whereas it will blow up to infinity for weaker singularities. It has to be linked to the appearance of singularities discussed above. In the case of Figure 5 we have α = 1/2 and µ = 1/2. The coefficient associated with the v-notch cracks is a O(1).

The above analysis also applies in the framework of elasticity, and is the foundation of the investigation in [START_REF] Leguillon | Crack onset at a v-notch. Influence of the notch tip radius[END_REF]. We stress that a rigorous derivation of (8.8) with an optimal estimate for the remainder requires more effort in studying the singular-regular expansion of u ε .

Expansion (8.2) could also be used to investigate the asymptotic behavior of other local functionals ϕ ε (u ε ) relating, for example, to the maximal values of the stress tensor in elasticity.

Remark 5 . 1 . 5 . 2 .

 5152 and aℓ p = 0 if p > ℓ, A ℓ p = b ℓ-p p if p ℓ and A ℓ p = 0 if p > ℓ,knowing that b ℓ-p p = B ℓ-p p = 0 for p > ℓ, since the terms u nα and U nα are 0 for n < 0. Here, we have chosen to derive the matching relations without any knowledge of the matched asymptotic technique. However, one can derive the relations (5.7) using the Van Dyke principle, see[START_REF] Vandyke | Perturbation methods in fluid mechanics[END_REF]. Definition of the asymptotic terms. For ℓ ∈ N, the functions u ℓα and U ℓα are defined inductively. The following algorithm defines step by step u ℓα : ω → R, U ℓα : Ω → R, b ℓ = b ℓ p p∈N * , and B ℓ = B ℓ p p∈N * for ℓ ∈ N.Step 0. u 0 ∈ V loc,0 (ω) is defined via Proposition 3.4 (in the particular case of Remark 3.5) as the unique function satisfying(5.8) ∆u 0 = -f in ω and u 0 = O r→0 (1).

( 5 .

 5 13) b ℓ = b ℓ p p∈N * and B ℓ = B ℓ p p∈N * .

  p = b ℓ p , ∀ℓ ∈ N, ∀p ∈ N * . Remark 6.3. The mechanism to switch from expansion (4.23) to expansion (5.14) consists in using the homogeneity of the singular functions s pα to pass them from fast variables into slow variables:

Proposition 7 . 1 .

 71 Under condition (ii), for any finite sequence (a p ) 1 p P of real numbers, there exists a unique solution u to the "super-variational problem" Find u ∈ V loc,0 (ω) such that ∆u + k 2 u = 0 in ω and u(x) = P p=1 a p Y pα (kr) sin (pαθ) + O r→0 (1).

Proposition 7 . 2 .

 72 pα (kr) sin (pαθ) + +∞ p=1 b p J pα (kr) sin (pαθ).Let J pα,ℓ and Y pα,ℓ be the coefficients of the generalized Taylor series of the Bessel functions J pα and Y pα (the coefficients for odd ℓ are all zero):J pα (z) = z pα ℓ∈N J pα,ℓ z ℓ and Y pα (z) = 1 z pα ℓ∈NY pα,ℓ z ℓ Under condition (i), for any finite sequence (A m p ) 1 p P, 0 m M of real numbers, there exists a unique solution (U m ) 0 m M of the "super-variational system"

YJY

  kR) pα+ℓ sin (pαθ) pα,ℓ (kR) -pα+ℓ sin (pαθ) + O R→∞[START_REF] Agmon | Lectures on elliptic boundary value problems[END_REF], where (B m p ) 0 p +∞, 0 m M are the coefficents such that in the neighborhood of infinity there holds pα,ℓ (kR) pα+ℓ sin (pαθ) pα,ℓ (kR) -pα+ℓ sin (pαθ).

Y 4 .

 4 kR) pα+ℓ sin (pαθ) pα,ℓ (kR) -pα+ℓ sin (pαθ) = O R→∞ (1), together with the matching conditions a m,n p = B m,n-p p and A m,n p = b m,n-p p if 1 p n, where the coefficients b m,n p are defined through the sub-variational expansion of u m,n : u m,n (x) ≃ Let I N be the set of indices corresponding to gauge functions of order lower than N (7.14) I N = (m, n) ∈ N 2 : m + nα N .

(8. 4 )

 4 u ε (x) = ε α A U α Ω ( x ε ) + higher order profiles, if |x| εr * . But the first canonical profile U α Ω has a singularity at point d, associated with exponent µ (8.5) U α Ω (X) = ΓR µ d sin(µΘ d ) + O(R µ d ), as R d → 0, where (R d , Θ d ) are the polar coordinates around point d. We have the relation (8.6) r ε = εR d and θ ε = Θ d .Back to the variable x, relations (8.4) to (8.6) lead to (8.7)u ε (x) = ε α-µ AΓr µ ε sin(µθ ε ) + O(ε α-µ r µ ε ), if |x| εr * . Putting (8.3) and (8.7) together, we obtain the expression of the singular coefficient γ ε(8.8)


S Kaushik

B Cerino

Y Yan

E Trouvé

L Ravanel

F Magnin

SAR COHERENCE MATRIX AS A TOOL TO UNDERSTAND BEHAVIOURS OF ICE APRONS

Keywords: InSAR coherence, TerraSAR-X, PAZ, Ice apron, Mont-Blanc massif

This paper focuses on understanding the temporal behaviour of Ice Aprons (IAs) in the Mont-Blanc massif using high resolution SAR coherence matrix. InSAR coherence is estimated between all possible pairs of TerraSAR-X images acquired in 2009 and 2011 as well as between PAZ images acquired in 2020, both at 11-day interval. The results show that coherence values in summer are higher in 2020 than in 2009 and 2011. Coherence matrices are also computed for different regions of a glacial system. The results are compared with those of IAs to understand the differences in their temporal and physical behaviours. In summer, all IAs show an increase in coherence values, while other glacier regions show very low or no coherence. This information could be useful for automatic classification methods, where IAs could be classified separately as a different class from the other types of glaciers.

INTRODUCTION

Glaciers in the Mont-Blanc massif (MBM) constitute a living subject in numerous studies using remote sensing [START_REF] Jauvin | Integration of corner reflectors for the monitoring of mountain glacier areas with sentinel-1 time series[END_REF] [START_REF] Prébet | A data-adaptive eof-based method for displacement signal retrieval from insar displacement measurement time series for decorrelating targets[END_REF], because they are considered as important indicators of water resources and climate change. An increase in temperature directly results in permafrost warming [START_REF] Magnin | Modelling rock wall permafrost degradation in the mont blanc massif from the lia to the end of the 21st century[END_REF] and usually increases glacier retreat. However, the local impact of global warming on the behaviour of ice aprons (IAs) is still not clear. The IAs are small ice features (less than 0.1 km 2 ) located on very steep slope (> 40 • ) and mostly in high altitude [START_REF] Guillet | Variations in surface area of six ice aprons in the mont-blanc massif since the little ice age[END_REF]. IAs correspond to likely oldest ice regions in the MBM and can thus be considered as a glacial heritage [START_REF] Guillet | Investigation of a cold-based ice apron on a high-mountain permafrost rock wall using ice texture analysis and micro-14c dating: a case study of the triangle du tacul ice apron (mont blanc massif, france)[END_REF]. The number of IAs in the MBM is significant, but studies for understanding these IAs are complicated because of their small size, difficult access and complex links to local climate variables [START_REF] Helfricht | Local extremes in the lidar-derived snow cover of alpine glaciers[END_REF]. In fact, all major global glacier inventories even fail to consider the IAs as a specific sub-class of glaciers and often show them as part of larger glacial systems [START_REF] Benn | Glaciers and Glaciation[END_REF]. As a result, very few IAs over the world have been studied. Initial definitions of IAs Corresponding author made the hypothesis that IAs are temporally stable ice bodies with no significant evolution along the year. But recent studies have shown a high sensitivity of IAs to climate change. [START_REF] Guillet | Variations in surface area of six ice aprons in the mont-blanc massif since the little ice age[END_REF] showed that IAs have been continuously losing their surface areas and thickness since the end of the Little Ice Age (LIA). But all previous studies have relied on historical photographs and recent high resolution optical images. Studies of IAs with advanced tools such as high resolution SAR images have never been attempted before and thus seem of particular interest.

In this paper, we aim to understand the temporal behavior of IAs in the MBM by means of high resolution SAR images. The interest of high resolution SAR images for studying IAs is twofold: 1) the high resolution (on the order of ≈ 2.5 m) allows efficiently capturing small areas covered by IAs; 2) SAR coherence matrix provides a useful way to survey the temporal evolution of targets under observation, since the coherence values constitute a relevant indicator of surface changes. We compute coherence matrices over IAs from TerraSAR-X images acquired in 2009 and in 2011 and PAZ images acquired in 2020. Since TerraSAR-X and PAZ have very similar acquisitions (in terms of wave frequency, revisit time, etc.), a joint analysis of both data sets enables us to consistently follow the temporal behaviors of IAs. Moreover, coherence matrices for other land classes in the glacial system such as accumulation zones, ablation zone, hanging glaciers, forest and urban areas are also computed in order to highlight the difference in behaviour of IAs compared to other glacier regions.

The main contribution of this paper lies on the first deployment of high resolution SAR images to understand temporal behaviors of IAs in the MBM, the considered targets being important for climate research but poorly studied until now. This study constitutes a preliminary step towards the physical interpretation of IAs. The paper is organised as follows: the study area and data processing are described in Section 2 and Section 3 respectively. Results and discussion are given in Section 4. Final conclusions are drawn in Section 5.

STUDY AREA

The MBM is spread over three countries, France, Italy and Switzerland. It is dominated by the Mont-Blanc summit (4809 m a.s.l) and composed of lots of glacier basins, but in this study we focus on the IAs spread across the entire massif. In the chosen study area, there are two main glaciers: Mer de Glace and Argentière glaciers (Fig. 1). The IAs are spread in the massif as shown in Fig. 1. The revisit time of TerraSAR-X and PAZ satellites is 11 days.

The shapefiles of all IAs in the MBM are obtained by accurately digitizing the boundaries on very high resolution orthophotos (0.2 m resolution). Due to geometrical distortions (e.g. layover, shadow and foreshortening) related to the specific SAR acquisition geometry, all IAs cannot be studied. Only IAs avoiding these distortions have been selected by using a visibility map built from the Digital Elevation Model (DEM) obtained from Pleiades stereo image pairs and the temporal mean of the 28 SAR images acquired by PAZ in 2020 [START_REF] Kaushik | Visibility Analysis of Glaciers on Steep Slopes in the European Alps using TerraSAR-X/PAZ Data[END_REF].

For each year and for each IA, coherence matrices are estimated between all image pairs of TerraSAR-X and of PAZ images. The coherence indicates the temporal correlation/decorrelation of IAs that is strongly related to stability/changes (as a result of melting or accumulation) of IAs' surface over time. A comparison is then performed between different years in order to highlight the IAs' surface changes over time, even though there are gaps between the three considered years due to data availability. Further, the aim is to compare the behaviours of IAs with those of other regions commonly associated with the mountain environment. For this, we focus on the PAZ data set acquired in 2020 (because of the regular acquisitions) and compute the coherence matrices for different land cover classes such as ablation zone, accumulation zone, urban area, forest and hanging glaciers.

RESULTS AND DISCUSSIONS

Coherence matrix comparison between different years

An example of the coherence matrix obtained for an IA located on the North face of the Aiguille d'Argentière with PAZ data set in 2020 is shown in Fig. 2. A seasonal effect is observed, i.e. the coherence values are much higher in the summer months (from the beginning of June to the beginning of September 2020) than in other periods of the year. This is related to the melting of the IA cover, leading to the exposure of bedrock surfaces below the IA. In other periods of the year, coherence is lost due to regular snow covers. These high coherence values in summer make it possible to perform further analysis (e.g. elevation change, movement detection) with Interferometry SAR. 3, that is to say, the coherence matrices are built between image pairs from June to September. From this comparison, it is safe to say that coherence values have globally increased in 2020 compared to in 2009 and 2011 (except for the IA "Tour"). High coherence is observed for the IA "Tour" for the entire period of observation. The IAs, "Argentière", "A Neuve" and "Jorasses", present high coherence in 2020 but not in 2009 and 2011. This increased summer coherence is interesting, a plausible explanation for this could be attributed to the exposure of rock surfaces beneath the IAs as a result of the melting in summers. This is consistant with the fact that IAs have been continuously losing their surface areas and thickness since the end of the little ice age. This could further be a strong indicator of the rapid melting of IAs as a result of the changing climate.

Comparison of coherence matrices for IAs and other land classes

Coherence matrices for different land classes, including IAs, glacier ablation and accumulation zones, urban areas, forest and hanging glaciers, obtained with PAZ data set in 2020 are shown in Fig. 4. The ablation and accumulation zones present similar behaviors and always present very low coherence along the year, because of continuous surface change in these areas. As expected, long term and seasonal correlation is observed for urban area and forest, due to permanently and seasonally stable features. Global low coherence is observed for hanging glaciers, but seasonal correlation can still be observed, coherence values being slightly higher in summer. Regarding IAs, those presented in this study, always show a higher coherence in summer and thus a relatively stable seasonal variation. Therefore, we can consider IAs, in general, as a separate class whose temporal behaviors are different from those of other zones in the glacial system, even though it happens that some IAs present low coherence along the year, similar to accumulation and ablation zones. Further investigations are necessary to relate the temporal behaviors of IAs to their physical characteristics such as location, orientation, morphology, etc. in order to better classify IAs and other glacial zones.

CONCLUSIONS

This paper deploys the high resolution SAR coherence matrix to understand the temporal behaviors of IAs in the MBM. According to the obtained results, we get the following conclusions: first, all IAs do not have the same temporal behaviors. Some of them stay weakly coherent over time, while others present high seasonal coherence. Besides the external climate conditions, their locations and orientations play an important part in their temporal evolution. Second, IAs do not always present temporal stability; they can present seasonal variations. In particular, in summer, high coherence is observed for a large number of studied IAs. This is related to the exposure of rock surfaces as IAs are melting. Considering that the thickness of IAs is of the order of a few meters, it is plausible that some IAs completely disappear in summer months and regain the mass again in winter. If this is the case, these ice features cannot be classified as IAs any more and further reconsideration of their definition needs to be contemplated. Third, different temporal behaviors of IAs from other land classes such as accumulation zones, ablation zones, hanging glaciers are observed. We can consider IAs as a separate class from other zones in a glacial system.

Future works include joint analyses of SAR coherence matrices and meteorological data (e.g. temperature, precipitation) in order to understand the physical behaviors of IAs and provide further evidences to prove their sensitivity to climate changes.

Fig. 1 .

 1 Fig. 1. Locations of the studied IAs in the MBM (in red). The background image is an ortho-photograph with 50 m resolution (source: IGN Geoportail).

  3. DATA PROCESSING Two high resolution X-band SAR image data sets are used for this study: 1. 51 TerraSAR-X images acquired in 2009 and in 2011, mostly during the 2 summer seasons, 2. 28 PAZ images acquired in 2020.

Fig. 2 .

 2 Fig. 2. Coherence matrix for the IA located on the North face of Aiguille d'Argentiere with PAZ data set in 2020. A value of 1 shows highest coherence, while values close to 0 show no coherence.

Fig. 3

 3 Fig. 3 presents the coherence matrices for 7 IAs (named by their location) in the years of 2009, 2011 and 2020. Since we were limited with continuous data availability for TerraSAR-X in 2009 and 2011, a comparison of only the

Fig. 3 .

 3 Fig. 3. Coherence matrix comparison for the years of 2009, 2011 and 2020 in case of 7 IAs. Each column corresponds to an IA named by its location.

Fig. 4 .

 4 Fig. 4. Coherence matrix for different land classes with PAZ data set. a) Ice aprons, b) Ablation zones, c) Accumulation zones, d) Urban areas, e) Forest, f) Hanging glaciers. The different matrices represent different locations in the study area.

Acknowledgement

The authors would also like to thank the German Aerospace Agency (DLR) for the TerraSAR-X images (Project MTH0232) and the Spanish Instituto Nacional de Tecnica Aerospacial (INTA) for the PAZ images (Project AO-001-051).