A Carleman estimate in the neighborhood of a multi-interface and applications to control theory

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A Carleman estimate in the neighborhood of a multi-interface and applications to control theory

Rémi Buffe

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Rémi Buffe. A Carleman estimate in the neighborhood of a multi-interface and applications to control theory. 2018. �hal-01703306�

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A Carleman estimate in the neighborhood of a multi-interface and applications to control theory

Rémi Buffe February 7, 2018

Abstract

We prove a Carleman estimate in a neighborhood of a multi-interface, that is, near a point wheren manifolds intersect, under compatibility assumptions between the Carleman weighte^{τ ϕ}, the operators at the multi-interface, and the elliptic operators in the interior and the usual sub-ellipticity condition. The compatibility condition at the multi-interface is a version of the known covering condition for systems in the literature. This Carleman estimate is a generalization of the results obtained in [6,7]. Applications in control theory for second-order transmission problems are also considered, and provide a generalization of the known results for one-dimensional networks of strings [4,13,19,30]

to higher dimensions.

Keywords : Carleman estimate, elliptic operators, multiple interfaces, control, stabilization, Lopatinskii condition.

AMS 2010 Subject Classification : 35B45, 35G45, 35S15, 35Q93

1 Introduction

We consider n smooth d−dimensional manifolds Ω_k, d ≥ 2, i ∈ {1, . . . , n} with boundary ∂Ω_k. We assume that they share parts of their boundary. More precisely, we denote by ∂Ωⁱ_k, i = 0, . . . , n_k, the different connected components of ∂Ω_k. We assume that one of these connected components is shared by all manifolds. For instance, we assume that we have

∂Ω⁰₁=· · ·=∂Ω⁰_n.

We setΩ = Ω1∪ · · · ∪Ωn, and we callI the common connected component of their boundaries. Note that Ω is not a manifold because of its topology near I. We shall refer to I as an interface between the manifolds. It is itself a d−1 dimensional manifold. An example of such configuration is given in Figure1. On eachΩk, we consider an elliptic differential operatorPk of order2mk. In local coordinates (x1, . . . , xd)forΩk near a pointx⁰∈ I, whereΩk is given by{xd≥0}, the operatorPk reads

Pk(x, D) = X

|α|≤2mk

a^k_α(x)D^α,

where the coefficients a^k_α(x)are C^∞ complex valued functions,D_j =−i∂xj and D= (D₁, . . . , D_d), and α∈N^d is a multi-index. With their principal symbols given by

p_k(x, ξ) = X

|α|=2m_k

a^k_α(x)ξ^α, ξ∈R^d, x∈Ω_k, (1.1)

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the ellipticity assumption reads|p_k(x, ξ)| ≥C|ξ|^2m^k, withC >0. We setm=m₁+· · ·+m_n. For each manifoldΩ_k, we also consider a set of boundary operatorsT_k^`, l= 1, . . . , m, of order less than or equal toβ_k^`<2mk, that take the following form in local coordinates

T_k^`(x, D) = X

|α|≤β_k^`

t^`_α,k(x)D^α, `∈ {1, . . . , m}, k∈ {1, . . . , n},

where the coefficients t^`_α,k(x) are complex-valued functions defined in a neighborhood of I in Ω_k. We denote by

t^`_k(x, ξ) = X

|α|=β_k^`

t^`_α,kξ^α,

the principal symbol of the operatorT_k^`, for k∈ {1, . . . , n} and`∈ {1, . . . , m}. Note that we may have t^`_k = 0if the actual order ofT_`^j is less thanβ_k^`. We assume that the orders satisfy the relations

γ_`:= 2m_k−β_k^` = 2m_j−β_j^`, ∀k, j∈ {1, . . . , n}, `∈ {1, . . . , m}. (1.2) We shall consider the following elliptic system, coupled through the interface,







P_ku_k =f_k inΩ_k, k∈ {1, . . . , n},

n

X

k=1

T_k^`(x, D)u_k|_I =g_` onI, `∈ {1, . . . , m}. (1.3) The main purpose of this article is the proof of a Carleman estimate for solutions of such system in a neighborhood of a pointx⁰ of the interfaceI.

In [12], Carleman introduced weighted inequalities to prove uniqueness of the Cauchy problem associated with an elliptic operator. The method has been developped later by Hörmander [17]. Carleman estimates for general elliptic operators has been derived away from boundaries in [18]. Recently, for boundaries and interfaces problems, Carleman estimates has been obtained in [6, 7, 29], under Lopatinskii-type assumptions.. The purpose of this article is to generalize these results to systems such as (1.3).

Many Carleman estimates near boundaries and interfaces has been already obtained. For instance, local Carleman estimates near boundaries for Dirichlet [25], Neumann [26], Ventcel [10] conditions has been already obtained by using pseudo-differential methods. Near smooth interfaces, away from boundaries, one can cite for instance [5,15,24,20,23,14]. For interfaces that meet the boundary, very few are known.

We mention out the works pof [8,9] where the authors obtained a Carleman estimate for medias with a stratified structure. The Carleman estimate we prove, in a neighborhood of a point of the mult-interface I, has the following form

τ⁻¹

n

X

k=1

||e^{τ ϕ}^kuk||+

n

X

k=1

|e^{τ ϕ}^|Itruk|²≤CXⁿ

k=1

||e^{τ ϕ}^kPkuk||+

m

X

`=1

n

X

k=1

T_k^`u_k|_I

2 .

where ϕ_k is a weight function onΩ_k, τ is the usual Carleman large parameter, and tru_k denotes the successive normal derivatives ofu_kat the multi-interfaceI, that istru_k= (u_k|_I, ∂_νu_k|_I, . . . , ∂^2m_ν ^k⁻¹u_k|_I).

As usual with Carleman estimates, in the interior, some compatibilty conditions between the operators Pk and the weight ϕk are introduced, the so-called sub-ellipticity condition (see Section 2.1). At the multi-interface, we also impose some compatibility conditions between the weightsϕk, the boundary op- eratorsT_k^`and the interior operatorsPk. These conditions are a natural generalization of the well-known Lopatinskii condition (see Section2.2) .

Carleman estimates are a powerful tool to prove control and stabilization properties. In the last section of this article, we provide an application of the Carleman estimate we obtain to a natural transmission

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Ω₁ Ω₂

Ω₃ Ω₄

∂Ω¹₁

I =∂Ω⁰_j, j= 1, . . . ,4

Figure 1: A sphere meets a surface and creates 4 smooth manifolds with boundaries. The shared interface is a connected component of∂Ωi, for alli∈ {1, . . . , n}.

problem. Under assumption on the control region, we prove the null-controllability result for the associated parabolic problem, and a stabilization result for the associated hyperbolic one. This generalizes what has been done for one-dimensional networks of strings in [4,13,19,30], and the references therein, to higher dimensions.

This article is organized as follows. In Section2we introduce the geometrical setting and the relevant hypothesis on the operators and the weight functions. In Section 3 we prove that we can find proper weight functions for which these conditions are satisfied in the special case of second-order operators with transmission condition through the interface. The general Carleman estimate is proven in Section4.

Finally, we state in Section5 some of the immediate applications of the Carleman estimate of Theorem 2.6to null-controllability of heat equation and to logarithmic stabilization of the damped wave equation, provided a natural geometric condition is satisfied (see Section5.1, and also Section (3.2)).

2 Statement of the problem and geometrical configurations

Letx⁰∈ I be a point of the interface. In each manifoldΩk, we consider an open bounded neighborhood U˜k of x⁰ in Ωk where local coordinates as above can be used. More precisely, for k = 1, . . . , n, we use local charts:

φk : Ωk⊃U˜k −→V_k⁺=Vk∩ {xd ≥0}, φk diffeomorphism, for allk∈ {1, . . . , n}, (2.1) withV_k an open set ofR^d, satisfyingφ_k(x⁰) = 0andφ_k(I ∩U˜_k) ={x_d = 0} ∩V_k for allk∈ {1, . . . , n}.

Setting V =∩ⁿ_k=1V_k onR^d andV⁺ ={x_d≥0} ∩V, we considerU_k =ϕ⁻¹_k (V⁺), and U =∪ⁿ_k=1U_k in Ω so that we have

φk : Ωk⊃Uk −→V⁺, φk diffeomorphism, for allk∈ {1, . . . , n}, andφ_k|_I =φ_`|_I, as sketched in Figure2.

Throughout this article, two assumptions shall play a central role. The first one is the sub-ellipticity assumption, that is classical when working with Carleman estimates, and the second one is the covering condition (also known as the Shapiro-Lopatinskii condition, or complementing condition).

2.1 The sub-ellipticity condition

For two functionsf(x, ξ)andg(x, ξ)in C^∞(R^d×R^d), we recall that their Poisson bracket is given by {f, g}(x, ξ) =∂ξf(x, ξ)∂xg(x, ξ)−∂xf(x, ξ)∂ξg(x, ξ).

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•x⁰ U

x⁰

xd

V

φk

0

Figure 2: Local configuration and local charts in the neighborhood of the interface.

From symbolic calculus, {f, g} is the principal symbol of the commutator i[Op(f),Op(g)]. For k ∈ {1, . . . , n}, we letϕk be a realC^∞ function onΩk, and we consider the conjugated operators

P_k,ϕ_k =e^{τ ϕ}^kP_ke^{−τ ϕ}^k, (2.2)

with principal symbolpk,ϕ(x, ξ, τ) =pk(x, ξ+iτ dϕk(x))in a semi-classical sense. In general,Pk,ϕ_kis not elliptic. The sub-ellipticity condition states that [P_k,ϕ^∗

k, Pk,ϕ_k] is elliptic positive wherePk,ϕ_k fails to be elliptic. More precisely, we write the following definition.

Definition 2.1. Let k∈ {1, . . . , n}. We say that (Pk, ϕk) satisfies the sub-ellipticity condition onUk if dϕk6= 0 onUk and if

pk,ϕ(x, ξ, τ) = 0 =⇒n

Repk,ϕ,Impk,ϕ

o

(x, ξ, τ) = 1 2i

n

p_k,ϕ, pk,ϕ

o

(x, ξ, τ)>0, (2.3) for allx∈Uk, τ >0, and allξ∈R^d,ξ6= 0.

For the case of elliptic operators of order two, it is well-known that the following lemma gives a construction of such a weight function.

Lemma 2.2. Let U be an open subset ofR^d,M be an elliptic operator of order two, andψ be a smooth function onU such that∇ψ6= 0 onU. We setϕ_λ=e^λψ, forλ >0. Then, there existsλ₀>0such that (M, ϕ_λ) satisfies the sub-ellipticity condition of Definition2.1, for allλ≥λ₀.

The proof of this lemma can be found for instance in [16]. Note that there exist operators such that the sub-ellipticity condition cannot be achieved, independently of the choice ofϕ. This is for instance the case for the Bi-Laplace operator. Indeed, in this case, we havepk,ϕ=q²_k,ϕ, whereqk is the principal symbol of the Laplace operator. The Poisson bracket is then given by

1 2i

n

p_k,ϕ, pk,ϕ

o

= 1

2i ∂ξp_k,ϕ∂xpk,ϕ−∂xp_k,ϕ∂ξpk,ϕ

= 2|qk,ϕ|²

i ∂ξq_k,ϕ∂xqk,ϕ−∂xq_k,ϕ∂ξqk,ϕ

,

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and we see that this cannot be positive on the characteristic set ofp_k,ϕ.

In addition to the sub-ellipticity condition of the couples(P_k, ϕ_k), when considering systems as (1.3) with general operators and boundary conditions, we have to impose some compatibility properties between the operators in the interior and at the boundary, even for well-posedness properties. For Carleman estimates, this is also the case, for the conjugated operator.

2.2 The covering condition

Forx∈ I, we shall denote byTx(I)andT_x^∗(I)respectively the tangent and cotangent spaces toI above x, andT^∗(I) ={(x, ν), x∈ I, ν∈T_x^∗(I)}. We moreover define the conormal spaces, forx∈ I,

N_x,k^∗ (I) ={ν∈T_x^∗(Ωk), ν(Y) = 0, ∀Y ∈Tx(I)}, and

N_k^∗(I) ={(x, ν), x∈ I, ν∈N_x,k^∗ (I)}.

For allk∈ {1, . . . , n}, we define a boundary quadrupleρk= (x, Y, νk, τ), where

x∈ I ∩U , Y ∈T_x^∗(I), νk∈N_x,k^∗ (I), andτ >0. (2.4) Moreover, we imposeνk to be an inward pointing conormal vector, that is,ν= (0, . . . ,0, t)witht >0in the local chart (Uk, φk)given in (2.1). We shall denote by L^∗,+_k (I)the set of suchρk. We define for a fixedx∈ I,

L^∗_x(I) :={(x, Y, ν1, . . . , νn, τ), ∀k∈ {1, . . . , n}, ρk= (x, Y, νk, τ)∈ L^∗,+_k (I)}, (2.5) and we also consider

L^∗(I) :={(x, Y, ν₁, . . . , ν_n, τ), (Y, ν₁, . . . , ν_n, τ)∈ L^∗_x(I), x∈ I}. (2.6) We introduce a weight function ϕdefined onU, which is smooth in each Ω_k and continuous across the interfaceI. We shall denoteϕ_k =ϕ_|_Ω

k. Forρ_k ∈ L^∗,+_k (I)we introduce, for allλ∈C,

˜

pk,ϕ(ρk, λ) =pk(x, Y +λνk+iτ dϕk(x)).

Considering the following factorization, as a polynomial inλ∈C,

˜

p_k,ϕ(ρ_k, λ) =c_kp˜⁻_k,ϕ(ρ_k, λ)˜p⁰_k,ϕ(ρ_k, λ)˜p⁺_k,ϕ(ρ_k, λ), (2.7) withp˜⁰_k,ϕ(ρk, λ) = Π_Im_σj

k(ρ_k)=0(λ−σ_k^j(ρk))^µ^j^k andp˜^±_k,ϕ(ρk, λ) = Π_±_Im_σj

k(ρ_k)>0(λ−σ_k^j(ρk))^µ^j^k, where(σ^j_k)j

are the complex roots ofp˜k,ϕas a polynomial in λ, with multiplicityµ^j_k. It should be kept in mind that this factorization depends heavily on the choice ofρk. We set

˜

κk,ϕ(ρk, λ) := ˜p⁺_k,ϕ(ρk, λ)˜p⁰(ρk, λ).

We also define at the boundary, for allλ,

t^`_k,ϕ(ρ_k, λ) =t^`_k(x, Y +λν_k+iτ dϕ_k(x)).

Definition 2.3. We say that (T_k^`, ϕ)k∈{1,...,n},`∈{1,...,m} covers (P_k, ϕ)k∈{1,...,n} at ρ⁰ ∈ L^∗(I) if for all polynomials f_k ∈ C[λ] there exists c₁, . . . , c_m ∈ C and polynomials q_k ∈ C[λ] such that, for all k∈ {1, . . . , n},

f_k(λ) =

m

X

l=1

c_`t^`_k,ϕ(ρ⁰_k, λ) +q_k(λ)κ_k,ϕ(ρ⁰_k, λ).

We shall say that (T_k^`, ϕ)k,` covers (Pk, ϕ)k atx if this holds for all ρ⁰ ∈ L^∗_x(I), and we shall say that (T_k^`, ϕ)k,` covers(Pk, ϕ)k if this holds for allρ⁰∈ L^∗(I)

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Observe that in the above definition, we can restrict the degree of the polynomialsf_k to be less than or equal to2m_k−1, using the Euclidean division. If we writem⁺_k =d^◦κ_k,ϕandm⁻_k = 2m_k−m⁺_k, then d^◦qk is at most equal to2mk−m⁺_k −1 =m⁻_k −1. We can reformulate the above definition as follows.

Introducing, fork∈ {1, . . . , n}

e^`_k(ρ⁰_k, λ) =

t^`_k,ϕ(ρ⁰_k, λ) if`∈ {1, . . . , m}

λ^`−(m+1)κk,ϕ(ρ⁰_k, λ) if`∈ {m+ 1, . . . , m+m⁻_k}, (2.8) and the linear map

Φ :C^m×C^m

−

1 × · · · ×C^m

−

n →C2m₁−1[λ]× · · · ×C2m_n−1[λ]











 c₁

... cm





,





 q1,1

... q_1,m⁻

k





, . . . ,





 qn,1

... q_n,m−

n











7→





m

X

`=1

c_`e^`₁+

m+m⁻₁

X

`=m+1

q_1,`−me^`₁, . . . ,

m

X

`=1

c_`e^`_n+

m+m⁻_n

X

`=m+1

q_n,`−me^`_n



,

Definition2.3is precisely equivalent to the surjectivity of the above mapΦ.

Remark 2.4. Like the sub-ellipticity condition, the covering condition is unvariant under a change of coordinates. In particular, this definition does not depend on the sizes of the conormal vectors νk.

2.3 On the well-posedness of systems satisfying the covering condition

The well-posedness of elliptic systems satisfying the covering condition is a known fact for a non- conjugated system (that is, in the case τ ϕ = 0). We refer for instance to [1, 2]. In their setting, they consider a more general system. It can be related to our problem as explained as follows. Near a point ofI, we can choose a coordinates system for eachΩ_k and thus the problem can be reformulated in R^d+, as in Section2. This choice of coordinates has no impact on the covering condition as it is invariant under a change of coordinates. In [1, 2], the authors consider systems of mixed-order elliptic operators in the half-space, that are said to be properly elliptic, i.e the rootsσi ofdet(A(x,Θ1+zΘ2)), viewed as a polynomial inz∈Csatsifies

#{Im(σi)>0}= #{Imσ_i<0}, forΘ₁, Θ₂∈R^d, linearly independant.

In our case, the considered matrix A is diagonal, thus the determinant is the product of the symbols pk(x,Θ1+zΘ2). If the system is properly elliptic, we obtain thatm=m⁺₁ +· · ·+m⁺_n =m⁻₁ +· · ·+m⁻_n. Hence,

dim(C^m×C^m

−

1 × · · · ×C^m

−

n) = dim(C2m1−1[λ]× · · · ×C2mn−1[λ]),

and thenΦis onto if and only ifΦis one-to-one. The covering condition stated in [1,2] precisely implies that the map Φ is one-to-one, and the covering condition in the present work implies that the map Φ is onto. Thus, in the setting of properly elliptic diagonal system of operators, those two conditions are consistent. However, in the present article, we ask for the conjugated operator to satisfy the covering condition, and moreover, as we shall see, the conjugated system may not be elliptic anymore. We give the following definition.

Definition 2.5. Let (T_k^`, Pk)be a system of operators such as (1.3)satisfying

(T_k^`,0)_k,` covers(P_k,0)_k. (2.9) We say that the weight function ϕpreserves the covering condition for(T_k^`, Pk)if we also have

(T_k^`, ϕ)k,` covers(Pk, ϕ)k.

An interesting question then arises: given a system (T_k^`, Pk)that satisfies (2.9), can we always find a weight function that preserve the covering condition for (T_k^`, Pk)? We give a positive anwer for the special case of transmission conditions for Laplace-Beltrami operators in Section3.

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2.4 Statement of the main result

The main result of this article is the following Carleman estimate in the neighborhood of the interface.

We define the following space of smooth functions defined in a neighborhood of the point of interest at the interface.

C^∞₀ (Uk) ={w∈C^∞(Ωk), supp(w)⊂Uk is compact inΩk}. (2.10) Theorem 2.6. Let x0 ∈ I and let ϕ be a weight function on U such that ϕ ∈ C⁰(U) and such that ϕ_|_Uk ∈C^∞(Uk). In addition, assume that (Pk, ϕk)satisfies the sub-ellipticity condition of Definition2.1 in a neighborhood of x0 inΩk, for allk∈ {1, . . . , n}, and assume that (T_k^`, ϕ)k∈{1,...,n},`∈{1,...,m} covers (P_k, ϕ)k∈{1,...,n} atx₀∈ I (see Definition 2.3). Then, there exists a neighborhoodU˜ of x₀ in Ω,C >0, andτ₀>0 such that

τ⁻¹

n

X

k=1

||e^{τ ϕ}^kuk||²_2m_k_,τ +

n

X

k=1

|e^{τ ϕ}^|Itruk|²_2m

k−1,1/2,τ

≤C

n

X

k=1

||e^{τ ϕ}^kPkuk||²_L2(Ωk)+

m

X

`=1

n

X

k=1

T_k^`u_k|_I

2 γ_`−1/2,τ

!

, (2.11) for allτ ≥τ₀, for allu∈C₀^∞( ˜U)such thatu_k:=u_|

Uk∩U˜ ∈C^∞₀ (U_k∩U˜).

This Carleman estimate is local in a neighborhood of a point of the interface. Local Carleman estimates can be generally patched together to obtain a global one. For such issue, we refer to Section5.1. The various norms are presented in Section4.1.3. Let us simply mention that

1. ||.||_s,τ is equivalent toτ^s||.||_L2+||.||_Hs in someΩ_k;

2. |.|s,τis equivalent toτ^s|.|_L2(I)+|.|_Hs(I). Norms on the interface are denoted by|.|for easier reading;

3. onI, forr∈Nands∈R, we mean

|trw|_r,s,τ =

r

X

`=0

|∂^`_νw_|_I|_r+s−`,τ,

using the notation of 2).

2.5 Conditions in a neighborhood of the interface

Letx0∈ I. We use the local chart introduce in (2.1) and the neighborhood V⁺={xd≥0} ∩V inR^d. We shall often use the notationx= (x⁰, xd)andξ= (ξ⁰, ξd). We also introduce

C^∞₀ (V⁺) ={u_|

V+, u∈C₀^∞(V)}, (2.12)

the set of restrictions toV⁺ of smooth and compactly supported functions in V. Note that Definition (2.12) is consistent with (2.10).

In this coordinate system, the operatorsPk reads

P_k= X

|α|≤2m_k

a^k_α(x)D^α, k= 1, . . . , n,

forxin V⁺, and near the boundaryV ∩ {x_d= 0}, the boundary operators read T_k^`= X

|α|≤β_k^`

t^`_α_kD^α, k= 1, . . . , n, `= 1, . . . , m,

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withβ_k^`∈ {1, . . . ,2m_k−1}andm=m₁+· · ·+m_n. We callp_k(x, ξ)andt^`_k(x, ξ)their principal symbols.

We also introduce the symbols

pk,ϕ(x, ξ, τ) =pk(x, ξ+iτdϕk), t^`_k,ϕ(x, ξ, τ) =t^`_k(x, ξ+iτdϕk),

which are respectively the principal symbols of the conjugated operatorsPk,ϕ:=e^{τ ϕ}^kPke^{−τ ϕ}^kandT_k,ϕ^` :=

e^{τ ϕ}^kT_k^`e^{−τ ϕ}^k. In this coordinate system, we make the following identification L^∗,+_k (I)∼={ρ= (x, ξ, τ)∈({xd= 0} ∩V)×R^d+×R⁺}.

We furthermore allow one to identify all the L^∗,+_k (I) to one another. Moreover, as Definition 2.3 is coordinate invariant (see Remark2.4) we can identifyL^∗(I)withL^∗,+_k (I)for allk∈ {1, . . . , n}. We also set the cotangent bundle overI with parameterτ as

L^T(I)∼={ρ⁰ = (x, ξ⁰, τ)∈({xd= 0} ∩V)×R^d−1×R⁺}.

For ρ⁰ ∈ V⁺ ×R^d−1×R⁺, we consider the complex roots σ_k,1(ρ⁰), . . . , σ_k,`_k(ρ⁰) of multiplicity µ_k,j (satisfying µ_k,1+· · ·+µ_k,`_k = 2m_k) of the symbol p_k,ϕ(ρ⁰, ξ_d), viewed as a polynomial in ξ_d. Let ρ⁰₀∈V⁺×R^d−1×R⁺ be fixed in what follows. From Lemma A.2 in [Bellassoued- Le Rousseau], there exists a conic neighborhoodV ofρ⁰₀ inV⁺×R^d×R⁺ and three smooth and homogeneous polynomials p^±_k(ρ⁰, ξd),p⁰_k(ρ⁰, ξd)inξd, such that

1. each polynomial is of constant degree forρ⁰∈ V;

2. we have the following factorization

pk,ϕ(ρ⁰, ξd) =a(ρ⁰)p⁺_k,ϕ(ρ⁰, ξd)p⁻_k,ϕ(ρ⁰, ξd)p⁰_k,ϕ(ρ⁰, ξd), ρ⁰∈ V, (2.13) where a(ρ⁰)is the leading coefficient;

3. the roots ofp⁺_k,ϕ(ρ⁰, ξd)andp⁻_k,ϕ(ρ⁰, ξd)have positive and negative imaginary parts respectively for allρ⁰∈ V;

4. at ρ⁰ =ρ⁰₀ we have p^±_k,ϕ(ρ⁰₀, ξd) = Y

±Imσk,j(ρ⁰₀)>0

(ξd−σk,j(ρ⁰₀))^µ^k,j, p⁰_k,ϕ(ρ⁰₀, ξd) = Y

Imσk,j(ρ⁰₀)=0

(ξd−σk,j(ρ⁰₀))^µ^k,j.

Note that the decomposition thus depends on ρ⁰₀. Note also that for ρ⁰ ∈ V, the sign of the imaginary part of the roots ofp⁰_k,ϕ(ρ⁰, ξd)is not prescribed. However, atρ⁰=ρ⁰₀those imaginary parts vanish.

We set κk,ϕ(ρ⁰, ξd) = p⁺_k,ϕ(ρ⁰, ξd)p⁰_k,ϕ(ρ⁰, ξd). In fact, while the roots σk,j(ρ⁰) are only continuous, since they can cross each other, we shall work with the polynomials given in (2.13); they are smooth, which permits the use of semi-classical calculus. We can reformulate the covering condition at ρ⁰₀ as follows: for all family of polynomials(fk)k∈{1,...,n}∈C[ξd]ⁿ, there exists(c1, . . . , cm)∈C^mand a family of polynomials(q_k)k∈{1,...,n}∈C[ξ_d]ⁿ such that

fk(ξd) =

m

X

`=1

c`t^`_k,ϕ(ρ⁰₀, ξd) +qk(ξd)κk,ϕ(ρ⁰₀, ξd), ∀k∈ {1, . . . , n}.

In this setting, the covering condition can also be stated in a more convenient way, such as follows.

Settingm⁻_k =d^◦p⁻_k,ϕ, we write

κ_k,ϕ(ρ⁰, ξ_d) =

2m_k−m⁻_k

X

i=0

κⁱ_k,ϕ(ρ⁰)ξ_dⁱ,