J Ér

Ôme Le Rousseau

Emmanuel Wend-Benedo

Zongo

STABILIZATION OF THE DAMPED PLATE EQUATION UNDER GENERAL BOUNDARY CONDITIONS

Keywords: Carleman estimates, stabilization, Lopatinskiȋ-Šapiro condition, resolvent estimate AMS 2020 subject classification: 35B45, 35J30, 35J40, 74K20, 93D15

We consider a damped plate equation on a smooth open bounded subset of R d , or a smooth compact manifold with boundary, along with general boundary operators fulfilling the Lopatinskiȋ-Šapiro condition. The damping term acts on a internal region without imposing a geometrical condition. We derive a resolvent estimate for the generator of associated semigroup that yields a logarithmic decay of the energy of the solution to the plate equation. The resolvent estimate is a consequence of a Carleman inequality obtained for the bi-Laplace operator involving a spectral parameter under the considered boundary conditions. The derivation goes first through microlocal estimates, then local estimates, and finally a global estimate.

     ∂ 2 t y + ∆ 2 y + α(x)∂ t y = 0 (t, x) ∈ R + × Ω, B 1 y |R + ×∂Ω = B 2 y |R + ×∂Ω = 0, y |t=0 = y 0 , ∂ t y |t=0 = y 1 , (1.1)
where α ≥ 0 and where B 1 and B 2 denote two boundary differential operators. The damping property is provided by +α(x)∂ t thus referred as the damping term. As introduced below ∆ 2 is the bi-Laplace operator, that is, the square of the Laplace operator. Here, it is associated with a smooth metric g to be introduced below; it is thus rather the bi-Laplace-Beltrami operator. This equation appears in models for the description of mechanical vibrations of thin domains. The two boundary operators are of order k j , j = 1, 2 respectively, yet at most of order 3 in the direction normal to the boundary. They are chosen such that the two following properties are fulfilled:

(1) the Lopatinskiȋ-Šapiro boundary condition holds (this condition is fully described in what follows); (2) along with the homogeneous boundary conditions given above the bi-Laplace operator is self-adjoint and nonnegative. This guarantees the conservation of the energy of the solution in the case of a damping free equation, that is, if α = 0. We are concerned with the decay of the energy of the solution in the case α is not identically zero. We shall prove that the damping term yields a stabilization property: the energy decays to zero as time t → ∞ and we shall prove that the decay rate is at least logarithmic.

1.1. Stabilization and control of Schrödinger and plate equations. In the case of the "hinged" boundary conditions y |R + ×∂Ω = ∆y |R + ×∂Ω = 0, the plate equation can be written as the product of two schrödinger equation since ∂ 2 t y+∆ 2 y = (-i∂ t +∆)(i∂ t +∆). As observed by Lebeau in [START_REF] Lebeau | Contrôle de l'équation de Schrödinger[END_REF] this allows one to transfer a control result obtained for the Schrödinger equation to the plates equations. In particular, Lebeau proved that controlability can be obtained for both equations if the control region fulfills the celebrated Rauch-Taylor condition, often coined GCC for geometrical control condition [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF][START_REF] Rauch | Exponential decay of solutions to hyperbolic equations in bounded domains[END_REF]. The GCC expresses that all rays of geometrical optics reach the control region. Yet, the GCC condition is not a necessary condition as expressed by the result of Jaffard [START_REF] Jaffard | Contrôle interne exact des vibrations d'une plaque rectangulaire[END_REF] on the controllability of the plate equation on a rectangle domain with an arbitrarily small control domain along with the "hinged" boundary conditions. Yet, the proof of [START_REF] Jaffard | Contrôle interne exact des vibrations d'une plaque rectangulaire[END_REF] relies on the generalization of Ingham type inequalities in [START_REF] Kahane | Pseudo-périodicité et séries de Fourier lacunaires[END_REF] and the very particular geometry that is considered.

For stabilization, GCC concerns the damping region, here given by the support of the function α. If GCC holds exponential stabilization holds. Note that in the same geometry as that of [START_REF] Jaffard | Contrôle interne exact des vibrations d'une plaque rectangulaire[END_REF] an exponential stabilization result is proven in [START_REF] Ramdani | Internal stabilization of the plate equation in a square: the continuous and the semi-discretized problems[END_REF], using similar techniques. However, if no geometrical condition is imposed on the damping region, if a general geometry is considered, and if different boundary conditions are considered, one cannot expect an exponential decay rate. A logarithmic decay rate is quite natural if one has in mind the equivalent result obtained for the wave equation in the works of G. Lebeau [START_REF] Lebeau | Équation des ondes amorties[END_REF] and G. Lebeau and L. Robbiano [START_REF] Lebeau | Stabilisation de l'équation des ondes par le bord[END_REF]. We also refer to Chapter 6 in [START_REF] Rousseau | Elliptic Carleman Estimates and Applications to Stabilization and Controllability I: Dirichlet Boundary Condition on Euclidean Space[END_REF] and Chapters 10 and 11 in [START_REF] Rousseau | Elliptic Carleman Estimates and Applications to Stabilization and Controllability II: General Boundary Conditons on Riemannian Manifold[END_REF] where the result of [START_REF] Lebeau | Équation des ondes amorties[END_REF][START_REF] Lebeau | Stabilisation de l'équation des ondes par le bord[END_REF] for the wave equation are reviewed and generalized in particular on the framework of general boundary conditions as those we consider here. See also the work of P. Cornilleau and L. Robbiano for a quite exotic boundary condition, namely the Zaremba condition.

Among the existing results available in the literature for plate type equations, many of them concern the "hinged" boundary conditions described above. In [START_REF] Ramdani | Internal stabilization of the plate equation in a square: the continuous and the semi-discretized problems[END_REF] where exponential stabilization is proven, the localized damping term involves the time derivative ∂ t y as in (1.1). Interior nonlinear feedbacks can be used for exponential stabilization [START_REF] Tebou | Well-posedness and stability of a hinged plate equation with a localized nonlinear structural damping[END_REF]. There, feedbacks are localized in a neighborhood of part of the boundary that fulfills multiplier-type conditions. A general analysis of nonlinear damping that includes the plate equation is provided in [START_REF] Alabau-Boussouira | Sharp energy estimates for nonlinearly locally damped PDEs via observability for the associated undamped system[END_REF] under multiplier-type conditions. For "hinged" boundary conditions also, with a boundary damping term, we cite [START_REF] Ammari | A sharp geometric condition for the boundary exponential stabilizability of a square plate by moment feedbacks only[END_REF] where, on a square domain, a necessary and sufficient condition is provided for exponential stabilization.

Note that under "hinged" boundary conditions the bi-Laplace operator is precisely the square of the Dirichlet-Laplace operator. This makes its mathematical analysis much easier, in particular where using spectral properties, and this explains why this type of boundary conditions appears very frequently in the mathematical literature.

A more challenging type of boundary condition is the so-called "clamped" boundary conditions, that is, u |∂Ω = 0 and ∂ ν u |∂Ω = 0, for which few results are available. We cite [START_REF] Alabau-Boussouira | Piecewise multiplier method and nonlinear integral inequalities for Petrowsky equation with nonlinear dissipation[END_REF], where a general analysis of nonlinearly damped systems that includes the plate equation under multiplier-type conditions is provided. In [START_REF] Alabau-Boussouira | Nonlinear damped partial differential equations and their uniform discretizations[END_REF], the analysis of discretized general nonlinearly damped system is also carried out, with the plate equation as an application. In [START_REF] Tebou | Well-posedness and stabilization of an Euler-Bernoulli equation with a localized nonlinear dissipation involving the p-Laplacian[END_REF], a nonlinear damping involving the p-Laplacian is used also under multiplier-type conditions. In [START_REF] Denk | A structurally damped plate equation with Dirichlet-Neumann boundary conditions[END_REF], an exponential decay is obtained in the case of "clamped" boundary conditions, yet with a damping term of the Kelvin-Voigt type, that is of the form ∂ t ∆y, that acts over the whole domain. In the case of the "clamped" boundary conditions, the logarithmic-type stabilization result we obtain here was proven in [START_REF] Rousseau | Spectral inequality and resolvent estimate for the bi-laplace operator[END_REF]. The present article thus stands as a generalization of the stabilization result of [START_REF] Rousseau | Spectral inequality and resolvent estimate for the bi-laplace operator[END_REF] if considering a whole class of boundary condition instead of specializing to a certain type. The present work contains in particular also the case of "hinged" boundary conditions. 1.2. Method. Following the works of [START_REF] Rousseau | Spectral inequality and resolvent estimate for the bi-laplace operator[END_REF][START_REF] Lebeau | Équation des ondes amorties[END_REF][START_REF] Lebeau | Stabilisation de l'équation des ondes par le bord[END_REF] we obtain a logarithmic decay rate for the energy of the solution to (1.1) by means of a resolvent estimate for the generator of the semigroup associated with the damped plate equation (1.1). In [START_REF] Rousseau | Spectral inequality and resolvent estimate for the bi-laplace operator[END_REF] this was achieved by means of a Carleman estimate for the elliptic operator D 4 s + ∆ 2 with an additional variable s in the case of the "clamped" boundary conditions. Extension of this strategy has however not been possible in the case of general boundary conditions. The proof of the resolvent estimate we seek is yet also possible by means of a Carleman estimate for the operator P σ = ∆ 2σ 4 where σ is a spectral parameter for the generator of the semigroup and we found that this method of proof extends to general boundary conditions.

Our first goal is thus the derivation of the Carleman inequality for the operator P σ near the boundary under the boundary conditions given by B 1 and B 2 .

Then, from the Carleman estimate one deduces an observation inequality for the operator P σ in the case of the prescribed boundary conditions. The resolvent estimate then follows from this observation inequality. 1.3. On Carleman estimates. A Carleman estimate is a weighted a priori inequality for the solutions of a partial partial differential equation, where the weight is of exponential type. For instance, for a partial differential operator P away from the boundary, it takes the form e τ ϕ u L 2 (Ω) ≤ C e τ ϕ P u L 2 (Ω) , for u ∈ C ∞ c (Ω) and τ ≥ τ 0 for ϕ well chosen and some τ 0 chosen sufficiently large. The exponential weight function involves a parameter τ that can be taken as large as desired, making Carleman inequalities very powerful estimates. Additional terms on the left-hand side of the inequality can be obtained, including higher-order derivatives of the function u, depending of course of the order of the operator P itself. For a second-order elliptic operator such as the Laplace operator one has τ 3/2 e τ ϕ u L 2 (Ω) + τ 1/2 e τ ϕ Du L 2 (Ω) + τ -1/2 |β|=2 e τ ϕ D β u L 2 (Ω)

≤ C e τ ϕ ∆u L 2 (Ω) , under the so-called sub-ellipticity condition; see Chapter 3 in [START_REF] Rousseau | Elliptic Carleman Estimates and Applications to Stabilization and Controllability I: Dirichlet Boundary Condition on Euclidean Space[END_REF]. Note that the power of the large parameter τ adds to 3/2 with the order of the derivative in each term on the left-hand side. In fact, in the calculus used to derive such estimates one power of τ is equivalent to a derivative of order one. Thus with this 3/2 compared with the order two of the operator one says that one looses a half-derivative in the estimate.

This type of estimate was used for the first time by T. Carleman [START_REF] Carleman | Sur une problème d'unicité pour les systèmes d'équations aux dérivées partielles à deux variables indépendantes[END_REF] to achieve uniqueness properties for the Cauchy problem of an elliptic operator. Later, A.-P. Calderón and L. Hörmander further developed Carleman's method [START_REF] Calderón | Uniqueness in the Cauchy problem for partial differential equations[END_REF][START_REF] Hörmander | On the uniqueness of the Cauchy problem[END_REF]. To this day, the method based on Carleman estimates remains essential to prove unique continuation properties; see for instance [START_REF] Zuily | Uniqueness and Non Uniqueness in the Cauchy Problem[END_REF] for an overview. On such questions, more recent advances have been concerned with differential operators with singular potentials, starting with the contribution of D. Jerison and C. Kenig [START_REF] Jerison | Unique continuation and absence of positive eigenvalues for Schrödinger operators[END_REF]. The reader is also referred to [START_REF] Santos Ferreira | Sharp L p Carleman estimates and unique continuation[END_REF][START_REF] Koch | Carleman estimates and unique continuation for second-order elliptic equations with nonsmooth coefficients[END_REF][START_REF] Koch | Sharp counterexamples in unique continuation for second order elliptic equations[END_REF][START_REF] Koch | Dispersive estimates for principally normal pseudodifferential operators[END_REF][START_REF] Sogge | Oscillatory integrals and unique continuation for second order elliptic differential equations[END_REF]. In more recent years, the field of applications of Carleman estimates has gone beyond the original domain; they are also used in the study of:

• Inverse problems, where Carleman estimates are used to obtain stability estimates for the unknown sought quantity (for instance coefficient, source term) with respect to norms on measurements performed on the solution of the PDE, see for instance [START_REF] Bukhgeim | Global uniqueness of class of multidimensional inverse problems[END_REF][START_REF] Imanuvilov | An inverse problem for the dynamical Lamé system with two sets of boundary data[END_REF][START_REF] Isakov | Inverse problems for partial differential equations[END_REF][START_REF] Kubo | Uniqueness in inverse hyperbolic problems-Carleman estimate for boundary value problems[END_REF]; Carleman estimates are also fundamental in the construction of complex geometrical optic solutions that lead to the resolution of inverse problems such as the Calderón problem with partial data [START_REF] Santos Ferreira | Limiting carleman weights and anisotropic inverse problems[END_REF][START_REF] Kenig | The Calderón problem with partial data[END_REF]. • Control theory for PDEs; Carleman estimates yield the null controllability of linear parabolic equations [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF] and the null controllability of classes of semi-linear parabolic equations [START_REF] Barbu | Exact controllability of the superlinear heat equation[END_REF][START_REF] Fernández-Cara | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF][START_REF] Fursikov | Controllability of evolution equations[END_REF]. They can also be used to prove unique continuation properties, that in turn are crucial for the treatment of low frequencies for exact controllability results for hyperbolic equations as in [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF].

For a function supported near a point at the boundary, in normal geodesic coordinates where Ω is locally given by {x d > 0} (see Section 1.4 below) the estimate can take the form

|β|≤2 τ 3/2-|β| e τ ϕ D β u L 2 (Ω) + |β|≤1 τ 3/2-|β| |e τ ϕ D β u |x d =0 + | L 2 (Ω)
≤ C e τ ϕ ∆u L 2 (Ω) . This is the type of estimate we seek here for the operator P σ , with some uniformity with respect to σ.

1.4. Geometrical setting. On Ω we consider a Riemannian metric g x = (g ij (x)), with associated cometric (g ij (x)) = (g x ) -1 . It stands as a bilinear form that act on vector fields,

g x (u x , v x ) = g ij (x)u i x v j x , u x = u i x ∂ x i , v x = v i x ∂ x i .
For x ∈ ∂Ω we denote by ν x the unit outward pointing normal vector at x, unitary in the sense of the metric g, that is

g x (ν x , ν x ) = 1 and g x (ν x , u x ) = 0 ∀u x ∈ T x ∂Ω.
We denote by ∂ ν the associated derivative at the boundary, that is, ∂ ν f (x) = ν x (f ). We also denote by n x the unit outward pointing conormal vector at x, that is,

n x = ν ♭ x , that is, (n x ) i = g ij ν j
x . Near a boundary point we shall often use normal geodesic coordinates where Ω is locally given by {x d > 0} and the metric g takes the form

g = dx d ⊗ dx d + 1≤i,j≤d-1 g ij dx i ⊗ dx j .
Then, the vector field ν x is locally given by (0, . . . , 0, -1). The same for the one form n x .

Normal geodesic coordinates allow us to locally formulate boundary problems in a half-space geometry. We write

R d + := {x ∈ R d , x d > 0} where x = (x ′ , x d ) with x ′ ∈ R d-1 , x d ∈ R.
We shall denote its closure by

R d + , that is, R d + = {x ∈ R d ; x d ≥ 0}. The Laplace-Beltrami operator is given by (∆ g f )(x) = (det g x ) -1/2 1≤i,j≤d ∂ x i (det g x ) 1/2 g ij (x)∂ x j f (x).
(1.2) in local coordinates. Its principal part is given by 1≤i,j≤d g ij (x)∂ x i ∂ x j and its principal symbol by 1≤i,j≤d g ij (x)ξ i ξ j . The bi-Laplace operator is P = ∆ 2 g . In the main text of the article we shall write ∆, ∆ 2 in place of ∆ g , ∆ 2 g .

1.5. Main results.

1.5.1. Stabilization result. Let (P 0 , D(P 0 )) be the unbounded operator on L 2 (Ω) given by the domain

D(P 0 ) = u ∈ H 4 (Ω); B 1 u |∂Ω = B 2 u |∂Ω = 0 , (1.3)
given by P 0 u = ∆ 2 u for u ∈ D(P 0 ). As written above the two boundary differential operators are such that (P 0 , D(P 0 )) is self-adjoint and nonnegative.

Let y(t) be a strong solution of the plate equation (1.1). A precise definition of strong solutions is given in Section 9.3. One has y 0 ∈ D(P 0 ) and y 1 ∈ D(P 1/2 0 ). Its energy is defined as

E(y)(t) = 1 2 ∂ t y(t) 2 L 2 (Ω) + (P 0 y(t), y(t)) L 2 (Ω) .
Theorem 1.1 (logarithmic stabilization for the damped plate equation).

There exists C > 0 such that for any such strong solution to the damped plate equation (1.1) one has

E(y)(t) ≤ C log(2 + t) 4 P 0 y 0 + αy 1 2 L 2 (Ω) + P 1/2 0 y 1 2 L 2 (Ω) .
A more precise and more general statement is given in Theorem 10.3 in Section 10.2. As explained in Section 1.2 the above stabilization result will be proven thanks to a Carleman estimate that we present now. 1.5.2. Carleman estimate. We state the main Carleman estimate for the operator P σ = ∆ 2σ 4 in normal geodesic coordinates as presented in Section 1.4. A point x 0 ∈ ∂Ω is considered and a weight function ϕ is assumed to be defined locally and such that

(1) ∂ d ϕ ≥ C > 0 locally. (2) (∆ ± σ 2 , ϕ)
) (P σ , B 1 , B 2 , ϕ) satisfies the Lopatinskiȋ-Šapiro condition of Defi- nition 4.1 at ̺ ′ = (x 0 , ξ ′ , τ, σ) for all (ξ ′ , τ, σ) ∈ R d-1 × [0, +∞) × [0, +∞) such that τ ≥ κ 0 σ, for some κ 0 > 0. 3
P σ = ∆ 2 -σ 4 ). Let κ ′ 0 > κ 0 > 0. Let x 0 ∈ ∂Ω.
Let ϕ be such that the properties above hold locally. Then, there exists W 0 a neighborhood of x 0 , C > 0, τ 0 > 0 such that

τ -1/2 e τ ϕ u 4,τ + | tr(e τ ϕ u)| 3,1/2,τ ≤ C e τ ϕ P σ u + + 2 j=1 |e τ ϕ B j v |x d =0 + | 7/2-k j ,τ , (1.4) for τ ≥ τ 0 , κ 0 σ ≤ τ ≤ κ ′ 0 σ, and u ∈ C ∞ c (W 0 + ). The volume norm is given by e τ ϕ u 4,τ = |β|≤4 τ 4-|β| e τ ϕ D β u L 2 (Ω) .
The trace norm is given by

| tr(e τ ϕ u)| 3,1/2,τ = 0≤n≤3 |∂ n ν (e τ ϕ u) |x d =0 + | 7/2-n,τ ,
where the norm

|.| 7/2-n,τ is the L 2 -norm in R d-1 after applying the Fourier multiplier (τ 2 + |ξ ′ | 2 ) 7/4-n/2
. These norms are well described in Section 2.3.

Observe that the Carleman estimate of Theorem 1.2 exhibits a loss of a half-derivative. A more precise statement is given in Theorem 7.4 in Section 7.2.

Remark 1.3. The condition κ 0 σ ≤ τ ≤ κ ′ 0 σ in Theorem 1.2 calls for a comment as it is not classical in the case of Carleman estimates, even in the presence of a spectral parameter. Usually one has only κ 0 σ ≤ τ and in applications similar to that we have in mind one chooses τ = κ 0 σ. The extended condition we make here thus does not appear as a potential limitation.

We postpone the technical explanation to Remark 7.2 below Proposition 7.1 where this condition appears first. 1.6. Some open questions.

1.6.1. Boundary damping. Here, we have considered a damping that acts in the interior of the domain Ω. The study of boundary damping, as in [START_REF] Lebeau | Stabilisation de l'équation des ondes par le bord[END_REF] for the wave equation, is also of relevance. Yet we foresee that it requires to specify more the used boundary operators. This was not our goal as we wished to treat general boundary operators here.

1.6.2. Spectral inequality. If the boundary operators B 1 and B 2 are well chosen, the bi-Laplace operator ∆ 2 can be selfadjoint on L 2 (Ω); see Section 9.1. Associated with the operator is then a Hilbert basis (φ j ) j∈N of L 2 (Ω). In the case of "clamped" boundary condition the following spectral inequality was proven in [START_REF] Rousseau | Spectral inequality and resolvent estimate for the bi-laplace operator[END_REF].

Theorem 1.4 (Spectral inequality for the "clamped" bi-Laplace operator). Let O be an open subset of Ω. There exists C > 0 such that

u L 2 (Ω) ≤ Ce Cµ 1/4 u L 2 (O) , µ > 0, u ∈ Span{φ j ; µ j ≤ µ}.
The proof of this theorem is based on a Carleman inequality for the fourth-order elliptic operator D 4 s + ∆ 2 , that is, after the addition of a variable s. Extending this strategy to the type of boundary conditions treated here was not successful because it is not guaranteed that having the Lopatinskiȋ-Šapiro condition for ∆ 2 , B 1 , and B 2 implies that the Lopatinskiȋ-Šapiro condition holds for D 4 s + ∆ 2 , B 1 , and B 2 . Yet, the Lopatinskiȋ-Šapiro condition is at the heart of the proof of our Carleman estimate. Proving a spectral estimate as in the above statement for the general boundary conditions considered here is an open question. 1.6.3. Quantification of the unique continuation property. For a secondorder operator like the Laplace operator ∆ and a boundary operator B of order k (yet of order at most one in the normal direction) such that the Lopatinskiȋ-Šapiro condition holds one can derive the following inequality that locally quantifies the unique continuation property up to the boundary; see [START_REF] Rousseau | Elliptic Carleman Estimates and Applications to Stabilization and Controllability I: Dirichlet Boundary Condition on Euclidean Space[END_REF]Lemma 9.2]. Proposition 1.5. Let x 0 ∈ ∂Ω and V be a neighborhood of x 0 where the Lopatinskiȋ-Šapiro condition holds. There exist W an open neighborhood of x 0 , ε ∈ (0, 1), δ ∈ (0, 1), and C > 0, such that

u H 1 (W ) ≤ C u 1-δ H 1 (V ) ∆u L 2 (V ) + |Bu| H 1-k (V ∩∂Ω) + u H 1 (Vε) δ , (1.5) for u ∈ H 2 (V ), with V ε = {x ∈ V ; dist(x, ∂Ω) > ε}.
This inequality can be obtained from a Carleman estimate as in Theorem 1.2 for the Laplace operator, yet with the large parameter τ allowed to be chosen as large as desired. This is exploited in an optimization procedure on the parameter τ in [τ 0 , +∞[ for some τ 0 . Note however that in the statement of Theorem 1.2 one has σ τ σ. Thus, the optimization procedure cannot be carried out in [τ 0 , +∞[. While having a result quantifying the unique continuation under Lopatinskiȋ-Šapiro-type conditions for the bi-Laplace operator similar to that of Proposition 1.5 can be expected, the Carleman estimate we obtain in the present article cannot be used, at least directly, for a proof as carried out in [START_REF] Rousseau | Elliptic Carleman Estimates and Applications to Stabilization and Controllability I: Dirichlet Boundary Condition on Euclidean Space[END_REF] or in former sources such as [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF]. 1.7. Outline. This article is organized as follows. In Section 2 we recall some basic aspects of pseudo-differential operators with a large parameters τ > 0 and some positivity inequality of Gårding type in particular for quadratic forms in a half-space or at the boundary. Associated with the large parameters are Sobolev like norms, also in a half-space or at the boundary.

In Section 3, the Lopatinskiȋ-Šapiro boundary condition is properly defined for an elliptic operator, we give examples focusing on the Laplace and bi-Laplace operator and we give a formulation in local normal geodesic coordinates that we shall mostly use throughout the article. For the bi-Laplace operator we provide a series of examples of boundary operators for which the Lopatinskiȋ-Šapiro boundary condition holds and moreover the resulting operator is symmetric. We also show that the algebraic conditions that characterize the Lopatinskiȋ-Šapiro condition is robust under perturbation. This last aspect is key in the understanding of how the Lopatinskiȋ-Šapiro condition get preserved under conjugation and the introduction of a spectral parameter. This is done in Section 4, where an analysis of the configuration of the roots of the conjugated bi-Laplace operator is performed. In Section 4.5 the Lopatinskiȋ-Šapiro condition for the conjugated operator is exploited to obtain a symbol positivity for a quadratic form to prepare for the derivation of a Carleman estimate.

In Section 5 we derive an estimation of the boundary traces. This is precisely where the Lopatinskiȋ-Šapiro condition is used. The result is first obtained microlocally and we then apply a patching procedure.

To obtain the Carleman estimate for the bi-Laplace operator with spectral parameter ∆ 2σ 4 in Section 6 we first derive microlocal estimates for the operators ∆ ± σ 2 . Imposing σ to be non-zero, in the sense that σ τ , the previous estimates exhibits losses in different microlocal regions. Thus concatenating the two estimates one derives an estimate for ∆ 2σ 4 where losses do not accumulates. A local Carleman estimate with only a loss of a half-derivative is obtained. This is done in Section 7. With the traces estimation obtained in Section 5 one obtains the local Carleman estimate of Theorem 1.2.

For the application to stabilization we have in mind, in Section 8 we use a global weight function and derive a global version of the Carleman estimate for ∆ 2σ 4 on the whole Ω. This leads to an observability inequality.

In Section 9 we recall aspects of strong and weak solutions to the damped plate equation, in particular through a semigroup formulation. With the observability inequality obtained in Section 8 we derive in Section 10 a resolvent estimate for the generator of the plate semigroup that in turn implies the stabilization result of Theorem 1.1.

1.8. Some notation. The canonical inner product in C m is denoted by

(z, z ′ ) C m = m-1 k=0 z k z′ k , for z = (z 0 , • • • , z m-1 ) ∈ C m , z ′ = (z ′ 0 , • • • , z ′ m-1 ) ∈ C m .
The associated norm will be denoted

|z| 2 C m = m-1 k=0 |z k | 2 .
We shall use the notations a b for a ≤ Cb and a b for a ≥ Cb, with a constant C > 0 that may change from one line to another. We also write a ≍ b to denote a b a.

For an open set U of R d we set

U + = U ∩ R d + and C ∞ c (U + ) = {u = v |R d + ; v ∈ C ∞ c (R d ) and supp(v) ⊂ U}. (1.6) We set S (R d + ) = {u |R d + ; u ∈ S (R d )} with S (R d ) the usual Schwartz space in R d : u ∈ S (R d ) ⇔ u ∈ C ∞ (R d ) and ∀α, β ∈ N d sup x∈R d |x α D β x u(x)| < ∞.
We recall that the Poisson bracket of two smooth functions is given by

{f, g} = d j=1 ∂ ξ j f ∂ x j g -∂ x j f ∂ ξ j g .

Pseudo-differential calculus

In a half-space geometry motivated by the normal geodesic coordinates introduced in Section 1.4 we shall use ξ = (ξ ′ , ξ d ) ∈ R d-1 × R and we shall consider the operators

D d = -i∂ d and D ′ = -i∂ ′ , with ∂ ′ = (∂ x 1 , . . . , ∂ x d-1 ).
2.1. Pseudo-differential operators with a large a parameter. In this subsection we recall some notions on semi-classical pseudo-differential operators with large parameter τ ≥ 1. We denote by S m τ the space of smooth functions a(x, ξ, τ ) defined on R d × R d , with τ ≥ 1 as a large parameter, that satisfies the following : for all multi-indices α, β ∈ N d and m ∈ R, there exists C α,β > 0 such that

|∂ α x ∂ β ξ a(x, ξ, τ )| ≤ C α,β λ m-|β| τ , where λ 2 τ = τ 2 + |ξ| 2 , for all (x, ξ, τ ) ∈ R d × R d × [1, ∞). For a ∈ S m
τ , one defines the associated pseudo-differential operator of order m, denoted by A = Op(a),

Au(x) := 1 (2π) d R d e ix•ξ a(x, ξ, τ )û(ξ)dξ, u ∈ S (R d ).
One says that a is the symbol of A. We denote Ψ m τ the set of pseudodifferential operators of order m. We shall denote by D m τ the space of semi-classical differential operators, i.e, the case when the symbol a(x, ξ, τ ) is a polynomial of order m in (ξ, τ ).

2.2. Tangential pseudo-differential operators. Here, we consider pseudo-differential operators that only act in the tangential direction x ′ with x d behaving as a parameter. We shall denote by S m T,τ , the set of smooth functions b(x, ξ ′ , τ ) defined for τ ≥ 1 as a large parameter and satisfying the following: for all multi-indices α ∈ N d , β ∈ N d-1 and m ∈ R, there exists C α,β > 0 such that

|∂ α x ∂ β ξ ′ b(x, ξ ′ , τ )| ≤ C α,β λ m-|β| T,τ
, where

λ 2 T,τ = τ 2 + |ξ ′ | 2 , for all (x, ξ ′ , τ ) ∈ R d ×R d-1 ×[1, ∞). For b ∈ S m
T,τ , we define the associated tangential pseudo-differential operator B = Op T (b) of order m by

Bu(x) := 1 (2π) d-1 R d-1 e ix ′ •ξ ′ b(x, ξ ′ , τ )û(ξ ′ , x d )dξ ′ , u ∈ S (R d + ).
We define Ψ m T,τ as the set of tangential pseudo-differential operators of order m, and D m T,τ the set of tangential differential operators of order m. We also set

Λ m T,τ = Op T (λ m T,τ ).
Let m ∈ N and m ′ ∈ R. If we consider a of the form

a(x, ξ, τ ) = m j=0 a j (x, ξ ′ , τ )ξ j d , a j ∈ S m+m ′ -j T,τ , we define Op(a) = m j=0
Op T (a j )D j x d . We write a ∈ S m,m ′ τ and Op(a) ∈ Ψ m,m ′ τ .

Function norms.

For functions norms we use the notation . for functions defined in the interior of the domain and |.| for functions defined on the boundary. In that spirit, we shall use the notation

u + = u L 2 (R d + ) , (u, ũ) + = (u, ũ) L 2 (R d + ) , for functions defined in R d + and |w| ∂ = w L 2 (R d-1 ) , (w, w) ∂ = (w, w) L 2 (R d-1 ) ,
for functions defined on {x d = 0}, such as traces.

We introduce the following norms, for m ∈ N and m ′ ∈ R,

u m,m ′ ,τ ≍ m j=0 Λ m+m ′ -j T,τ D j x d u + , u m,τ = u m,0,τ ≍ m j=0 Λ m-j T,τ D j x d u + , for u ∈ S (R d + ). One has u m,τ ≍ |α|≤m τ m-|α| D α u + ,
and in the case m ′ ∈ N one has

u m,m ′ ,τ ≍ α d ≤m |α|≤m+m ′ τ m+m ′ -|α| D α u + , with α = (α ′ , α d ) ∈ N d .
The following argument will be used on numerous occasions: for m ∈ N, m ′ , ℓ ∈ R, with ℓ > 0,

u m,m ′ ,τ ≪ u m,m ′ +ℓ,τ , if τ is chosen sufficiently large.
At the boundary {x d = 0} we define the following norms, for m ∈ N and m ′ ∈ R, Here differential quadradic forms are defined either in a half-space or at the boundary. The results we present here without proof can be found in [START_REF] Bellassoued | Carleman estimates for elliptic operators with complex coefficients. Part I: Boundary value problems[END_REF] and [START_REF] Rousseau | Elliptic Carleman Estimates and Applications to Stabilization and Controllability II: General Boundary Conditons on Riemannian Manifold[END_REF].

| tr(u)| 2 m,m ′ ,τ = m j=0 |Λ m+m ′ -j T,τ D j x d u |x d =0 + | 2 ∂ , u ∈ S (R d + ).
2.4.1. Differential quadratic forms in a half-space. Definition 2.1 (interior differential quadratic form). Let u ∈ S (R d + ). We say that

Q(u) = N s=1 (A s u, B s u) + , A s = Op(a s ), B s = Op(b s ), ( 2.1)
is an interior differential quadratic form of type (m, r) with smooth coefficients, if for each s = 1, . . . N, we have

a s (̺) ∈ S m,r ′ τ and b s (̺) ∈ S m,r ′′ τ , with r ′ + r ′′ = 2r, ̺ = (x, ξ, τ ).
The principal symbol of the quadratic form Q is defined as the class of

q(̺) = N s=1 a s (̺)b s (̺) (2.2) in S 2m,2r τ /S 2m,2r-1 τ .
A result we shall use is the following microlocal Gårding inequality.

Proposition 2.2 (microlocal Gårding inequality). Let K be a compact set of R d + and let U be an conic open set of R d + × R d-1 × R + contained in K × R d-1 × R + . Let also χ ∈ S 0
T,τ be homogeneous of degree 0 and such that supp(χ) ⊂ U . Let Q be an interior differential quadratic form of type (m, r) with homogeneous principal symbol q ∈ S 2m,2r τ satisfying, for some C 0 > 0 and τ 0 > 0, Re q(̺) ≥ C 0 λ 2m τ λ 2r T,τ , for τ ≥ τ 0 , ̺ = (̺ ′ , ξ d ), ̺ ′ = (x, ξ ′ , τ ) ∈ U , and ξ d ∈ R.

For 0 < C 1 < C 0 and N ∈ N there exist τ * , C > 0, and

C N > 0 such that Re Q(Op T (χ)u) ≥ C 1 Op T (χ)u 2 m,r,τ -C| tr(Op T (χ)u)| 2 m-1,r+1/2,τ -C N u 2 m,-N,τ , for u ∈ S (R d + )
and τ ≥ τ * . We refer to [8, Proposition 3.5] and [START_REF] Rousseau | Elliptic Carleman Estimates and Applications to Stabilization and Controllability II: General Boundary Conditons on Riemannian Manifold[END_REF]Theorem 6.17] for a proof. We also state a local version of the result that follows from Proposition 2.2. Proposition 2.3 (Gårding inequality). Let U 0 be a bounded open subset of R d + and let Q be an interior differential quadratic form of type (m, r) with homogeneous principal symbol q ∈ S 2m,2r τ satisfying, for some C 0 > 0 and τ 0 > 0, Re q(̺)

≥ C 0 λ 2m τ λ 2r T,τ , for τ ≥ τ 0 , ̺ = (̺ ′ , ξ d ), ̺ ′ = (x, ξ ′ , τ ) ∈ U 0 × R d-1 × R + , and ξ d ∈ R. For 0 < C 1 < C 0 there exist τ * , C > 0 such that Re Q(u) ≥ C 1 u 2 m,r,τ -C| tr(u)| 2 m-1,r+1/2,
Q(u) = N s=1 (A s u |x d =0 + , B s u |x d =0 + ) ∂ , A s = a s (x, D, τ ), B s = b s (x, D, τ ), is a boundary differential quadratic form of type (m -1, r) with C ∞ coefficients, if for each s = 1, . . . N, we have a s (̺) ∈ S m-1,r ′ τ (R d + × R d ), b s (̺) ∈ S m-1,r ′′ τ (R d + ×R d ) with r ′ +r ′′ = 2r, ̺ = (̺ ′ , ξ d ) with ̺ ′ = (x, ξ ′ , τ
). The symbol of the boundary differential quadratic form Q is defined by

q(̺ ′ , ξ d , ξd ) = N s=1 a s (̺ ′ , ξ d )b s (̺ ′ , ξd ). For z = (z 0 , . . . , z ℓ-1 ) ∈ C ℓ and a(̺) ∈ S ℓ-1,t τ , of the form a(̺ ′ , ξ d ) = ℓ-1 j=0 a j (̺ ′ )ξ j d with a j (̺ ′ ) ∈ S ℓ-1+t-j T,τ we set Σ a (̺ ′ , z) = ℓ-1 j=0 a j (̺ ′ )z j . (2.3)
From the boundary differential quadratic form Q we introduce the following bilinear symbol Σ

Q : C m × C m → C Σ Q (̺ ′ , z, z ′ ) = N s=1 Σ a s (̺ ′ , z)Σ b s (̺ ′ , z ′ ), z, z ′ ∈ C m . (2.4) We let W be an open conic set in R d-1 × R d-1 × R + .
Definition 2.5. Let Q be a boundary differential quadratic form of type (m -1, r) with homogeneous principal symbol and associated with the bilinear symbol Σ Q (̺ ′ , z, z ′ ). We say that Q is positive definite in W if there exist C > 0 and R > 0 such that

Re Σ Q (̺ ′′ , x d = 0 + , z, z) ≥ C m-1 j=0 λ 2(m-1-j+r) T,τ |z j | 2 , for ̺ ′′ = (x ′ , ξ ′ , τ ) ∈ W , with |(ξ ′ , τ )| ≥ R, and z = (z 0 , . . . , z m-1 ) ∈ C m .
We have the following Gårding inequality.

Proposition 2.6. Let Q be a boundary differential quadratic form of type

(m -1, r), positive definite in W , an open conic set in R d-1 × R d-1 × R + , with bilinear symbol Σ Q (̺ ′ , z, z ′ ). Let χ ∈ S 0 T,τ be homogeneous of degree 0, with supp(χ |x d =0 + ) ⊂ W and let N ∈ N. Then there exist τ * ≥ 1, C > 0, C N > 0 such that Re Q(Op T (χ)u) ≥ C| tr(Op T (χ)u)| 2 m-1,r,τ -C N | tr(u)| 2 m-1,-N,τ , for u ∈ S (R d + ) and τ ≥ τ * .
2.5. Symbols and operators with an additional large parameter.

In this article, we shall use operators with a symbol that depends on an additional large parameter σ, say a(x, ξ, τ, σ). They will satisfy estimate of the form

|∂ α x ∂ β ξ a(x, ξ, τ, σ)| ≤ C α,β (τ 2 + |ξ| 2 + σ 2 ) (m-|β|)/2 . We observe that if τ σ one has λ 2 τ ≤ τ 2 + |ξ| 2 + σ 2 λ 2 τ .
Thus, as far as pseudo-differential calculus is concerned it is as if a ∈ S m τ and this property will be exploited in what follows.

Similarly if a = a(x, ξ ′ , τ, σ) fulfills a tangential-type estimate of the form

|∂ α x ∂ β ξ ′ a(x, ξ ′ , τ, σ)| ≤ C α,β (τ 2 + |ξ ′ | 2 + σ 2 ) (m-|β|)/2
, if one has τ σ one will be able to apply techniques adapted to symbols in S m T,τ and associated operators, like for instance the results on differential quadratic forms listed in Section 2.4.

Lopatinskiȋ-Šapiro boundary conditions for elliptic operator

Let P be an elliptic differential operator of order 2k on Ω, (k ≥ 1), with principal symbol p(x, ω) for (x, ω) ∈ T * Ω. One defines the following polynomial in z, p(x, ω ′ , z) = p(x, ω ′zn x ), for x ∈ ∂Ω, ω ′ ∈ T * x ∂Ω, z ∈ R and where n x denotes the outward unit pointing conormal vector at x (see Section 1.4). Here x and ω ′ are considered to act as parameters. We denote by ρ j (x, ω ′ ), 1 ≤ j ≤ 2k the complex roots of z → p(x, ω ′ , z). One sets

p+ (x, ω ′ , z) = Im ρ j (x,ω ′ )≥0 (z -ρ j (x, ω ′ )). Given boundary operators B 1 , • • • , B k in a neighborhood of ∂Ω, with principal symbols b j (x, ω), j = 1, • • • , k, one also sets bj (x, ω ′ , z) = b j (x, ω ′ -zn x ). Definition 3.1 (Lopatinskiȋ-Šapiro boundary condition). Let (x, ω ′ ) ∈ T * ∂Ω with ω ′ = 0. One says that the Lopatinskiȋ-Šapiro condition holds for (P, B 1 , • • • , B k ) at (x, ω ′ ) if for any polynomial f (z) with complex co- efficients, there exists c 1 , • • • , c k ∈ C and a polynomial g(z) with complex coefficients such that, for all z ∈ C, f (z) = 1≤j≤k c j bj (x, ω ′ , z) + g(z)p + (x, ω ′ , z).
We say that the Lopatinskiȋ-Šapiro condition holds for (P,

B 1 , • • • , B k ) at x ∈ ∂Ω if it holds at (x, ω ′ ) for all ω ′ ∈ T *
x ∂Ω with ω ′ = 0. The Lopatinskiȋ-Šapiro boundary condition is of great importance in the analysis of elliptic equations. In fact, the Lopatinskiȋ-Šapiro condition is equivalent to having the operator

L : H m+2k (Ω) → H m (Ω) ⊕ H m+2k-ℓ 1 -1/2 (∂Ω) ⊕ • • • ⊕ H m+2k-ℓ k -1/2 (∂Ω) u → (P u, B 1 u |∂Ω , . . . , B k u |∂Ω ),
to be Fredholm, if m ≥ 0 and if each operator B j is of order ℓ j . This is done in great generality in [START_REF] Hörmander | The Analysis of Linear Partial Differential Operators[END_REF]Chapter 20]. If P is the Laplace operator this is presented in great details in [START_REF] Rousseau | Elliptic Carleman Estimates and Applications to Stabilization and Controllability II: General Boundary Conditons on Riemannian Manifold[END_REF]Chapter 3].

Roughly speaking the Lopatinskiȋ-Šapiro condition states that modes exponentialy small in the interior of the domain Ω (as frequency increases) yet not small at the boudary cannot be estimated without the use of a proper boundary operator. No boundary operator is needed for mode that do not have this asympotic behavior. This is explained in [ The general boundary operators B 1 and B 2 are then given by

B ℓ (x, D) = 0≤j≤min(3,k ℓ ) B k ℓ -j ℓ (x, D ′ )(i∂ ν ) j , ℓ = 1, 2,
with B k ℓ -j ℓ (x, D ′ ) differential operators acting in the tangential variables. We denote by b 1 (x, ω) and b 2 (x, ω) the principal symbols of B 1 and

B 2 respectively. For (x, ω ′ ) ∈ T * ∂Ω, we set bℓ (x, ω ′ , z) = 0≤j≤min(3,k ℓ ) b k ℓ -j ℓ (x, ω ′ )z j , ℓ = 1, 2.
We recall that the principal symbol of P is given by p

(x, ω) = |ω| 4 g . One thus has p(x, ω ′ , z) = p(x, ω ′ -zn x ) = z 2 + |ω ′ | 2 g 2 . Therefore p(x, ω ′ , z) = (z -i|ω ′ | g ) 2 (z + i|ω ′ | g ) 2 .
According to the above definition we set p+ (x,

ω ′ , z) = (z -i|ω ′ | g ) 2 .
Thus, the Lopatinskiȋ-Šapiro condition holds at (x, ω ′ ) with ω ′ = 0 if and only if for any function

f (z) the complex number i|ω ′ | g is a root of the polynomial z → f (z) - c 1 b1 (x, ω ′ , z) -c 2 b2 (x, ω ′ , z
) and its derivative for some c 1 , c 2 ∈ C. This leads to the following determinant condition.

∂ z b1 ∂ z b2 (x, ω ′ , z = i|ω ′ | g ) = 0. (3.1)
Remark 3.3. With the determinant condition and homogeneity, we note that if the Lopatinskiȋ-Šapiro condition holds for (P, B 1 , B 2 ) at (x, ω ′ ) it also holds in a conic neighborhood of (x, ω ′ ) by continuity. If it holds at x ∈ Ω, it also holds in a neighborhood of x.

3.3.

Formulation in normal geodesic coordinates. Near a boundary point x ∈ ∂Ω, we shall use normal geodesic coordinates. These coordinates are recalled in Section 1.4. Then the principal symbols of ∆ and ∆ 2 are given by ξ 2 d + r(x, ξ ′ ) and (ξ 2 d + r(x, ξ ′ )) 2 respectively, where r(x, ξ ′ ) is the principal symbol of a tangential differential elliptic operator R(x, D ′ ) of order 2, with

r(x, ξ ′ ) = 1≤i,j≤d-1 g ij (x)ξ ′ i ξ ′ j and r(x, ξ ′ ) ≥ C|ξ ′ | 2 .
Here g ij is the inverse of the metric g ij . Below, we shall often write |ξ ′ | 2 x = r(x, ξ ′ ) and we shall also write

|ξ| 2 x = ξ 2 d + r(x, ξ ′ ), for ξ = (ξ ′ , ξ d ).
If b 1 (x, ξ) and b 2 (x, ξ) are the principal symbols of the boundary operators B 1 and B 2 in the normal geodesic coordinates then the Lopatinskiȋ-Šapiro condition for (P,

B 1 , B 2 ) with P = ∆ 2 at (x, ξ ′ ) reads det b 1 b 2 ∂ ξ d b 1 ∂ ξ d b 2 (x, ξ ′ , ξ d = i|ξ ′ | x ) = 0, if |ξ ′ | x = 0 according to Lemma 3.2.
If the Lopatinskiȋ-Šapiro condition holds at some x 0 , because of homogeneity, there exists

C 0 > 0 such that det b 1 b 2 ∂ ξ d b 1 ∂ ξ d b 2 (x 0 , ξ ′ , i|ξ ′ | x ) ≥ C 0 |ξ ′ | k 1 +k 2 -1 x , ξ ′ ∈ R d-1 . (3.2)
3.4. Stability of the Lopatinskiȋ-Šapiro condition. To prepare for the study of how the Lopatinskiȋ-Šapiro condition behaves under conjugation with Carleman exponential weight and the addition of a spectral parameter, we introduce some perturbations in the formulation of the Lopatinskiȋ-Šapiro condition for (P, B 1 , B 2 ) as written in (3.2).

Lemma 3.4. Let V 0 be a compact set of ∂Ω such that the Lopatinskiȋ-Šapiro condition holds for (P, B 1 , B 2 ) at every point x of V 0 . There exist

C 1 > 0 and ε > 0 such that det b 1 b 2 ∂ ξ d b 1 ∂ ξ d b 2 (x, ξ ′ + ζ ′ , ξ d = i|ξ ′ | x + δ) ≥ C 1 |ξ ′ | k 1 +k 2 -1 x , (3.3
)

for x ∈ V 0 , ξ ′ ∈ R d-1 , ζ ′ ∈ C d-1 , and δ ∈ C, if |ζ ′ | + |δ| ≤ ε|ξ ′ | x . Moreover one has det b 1 (x, ξ ′ + ζ ′ , ξ d = i|ξ ′ | x + δ) b 2 (x, ξ ′ + ζ ′ , ξ d = i|ξ ′ | x + δ) b 1 (x, ξ ′ + ζ ′ , ξ d = i|ξ ′ | x + δ) b 2 (x, ξ ′ + ζ ′ , ξ d = i|ξ ′ | x + δ) ≥ C 1 |δ -δ| |ξ ′ | k 1 +k 2 -1 x , (3.4
)

for x ∈ V 0 , ξ ′ ∈ R d-1 , ζ ′ ∈ C d-1 , and δ, δ ∈ C, if |ζ ′ | + |δ| + | δ| ≤ ε|ξ ′ | x .
Proof. From (3.2), since V 0 is compact having the Lopatinskiȋ-Šapiro condition holding at every point x of V 0 means there exists

C 0 > 0 such that det b 1 b 2 ∂ ξ d b 1 ∂ ξ d b 2 (x, ξ ′ , i|ξ ′ | x ) ≥ C 0 |ξ ′ | k 1 +k 2 -1 x , x ∈ V 0 , ξ ′ ∈ R d-1 .
(3.5) The first part is a consequence of the mean value theorem, homogeneity and (3.5) with say

C 1 = C 0 /2.
For the second part it is sufficient to assume that δ = δ since the result is obvious otherwise. For j = 1, 2 one writes the Taylor formula

b j (x, ξ ′ + ζ ′ , i|ξ ′ | x + δ) = b j (x, ξ ′ + ζ ′ , i|ξ ′ | x + δ) + ( δ -δ)∂ ξ d b j (x, ξ ′ + ζ ′ , i|ξ ′ | x + δ) + ( δ -δ) 2 1 0 (1 -s)∂ 2 ξ d b j (x, ξ ′ + ζ ′ , i|ξ ′ | x + δ s ) ds, with δ s = (1 -s)δ + s δ, yielding 1 δ -δ det b 1 (x, ξ ′ + ζ ′ , i|ξ ′ | x + δ) b 2 (x, ξ ′ + ζ ′ , i|ξ ′ | x + δ) b 1 (x, ξ ′ + ζ ′ , i|ξ ′ | x + δ) b 2 (x, ξ ′ + ζ ′ , i|ξ ′ | x + δ) = det b 1 b 2 ∂ ξ d b 1 ∂ ξ d b 2 (x, ξ ′ + ζ ′ , i|ξ ′ | x + δ) + ( δ -δ) 1 0 (1 -s)f (x, ξ ′ , ζ ′ , δ, s) ds, with f (x, ξ ′ , ζ ′ , δ, s) = det b 1 (x, ξ ′ + ζ ′ , i|ξ ′ | x + δ) b 2 (x, ξ ′ + ζ ′ , i|ξ ′ | x + δ) ∂ 2 ξ d b 1 (x, ξ ′ + ζ ′ , i|ξ ′ | x + δ s ) ∂ 2 ξ d b 2 (x, ξ ′ + ζ ′ , i|ξ ′ | x + δ s )
.

With homogeneity, if |ζ

′ | + |δ| + | δ| |ξ ′ | x one finds det b 1 (x, ξ ′ + ζ ′ , i|ξ ′ | x + δ) b 2 (x, ξ ′ + ζ ′ , i|ξ ′ | x + δ) ∂ 2 ξ d b 1 (x, ξ ′ + ζ ′ , i|ξ ′ | x + δ s ) ∂ 2 ξ d b 2 (x, ξ ′ + ζ ′ , i|ξ ′ | x + δ s ) |ξ ′ | k 1 +k 2 -2 x ,
Thus with |δ -δ| ≤ ε|ξ ′ | x , for ε > 0 chosen sufficiently small, using the first part of the lemma one obtains the second result.

Examples of boundary operators yielding symmetry. We

give some examples of pairs of boundary operators B 1 , B 2 that (1) fulfill the Lopatinskiȋ-Šapiro condition, (2) yield symmetry for the bi-Lalace operator P = ∆ 2 , that is,

(P u, v) L 2 (Ω) = (u, P v) L 2 (Ω) for u, v ∈ H 4 (Ω) such that B j u |∂Ω = B j v |∂Ω = 0, j = 1, 2. We first recall that following Green formula (∆u, v) L 2 (Ω) = (u, ∆v) L 2 (Ω) + (∂ n u |∂Ω , v |∂Ω ) L 2 (∂Ω) -(u |∂Ω , ∂ n v |∂Ω ) L 2 (∂Ω) , (3.6)
which applied twice gives

(P u, v) L 2 (Ω) = (u, P v) L 2 (Ω) + T (u, v) with T (u, v) = (∂ n ∆u |∂Ω , v |∂Ω ) L 2 (∂Ω) -(∆u |∂Ω , ∂ n v |∂Ω ) L 2 (∂Ω) + (∂ n u |∂Ω , ∆v |∂Ω ) L 2 (∂Ω) -(u |∂Ω , ∂ n ∆v |∂Ω ) L 2 (∂Ω) . (3.7)
Using normal geodesic coordinates in a neighborhood of the whole boundary ∂Ω allows one to write ∆ = ∂ 2 n + ∆ ′ where ∆ ′ is the resulting Laplace operator on the boundary, that is, associated with the trace of the metric on ∂Ω. Since ∆ ′ is selfadjoint on ∂Ω this allows one to write formula (3.7) in the alternative forms

T (u, v) = (∂ 3 n u |∂Ω , v |∂Ω ) L 2 (∂Ω) -((∂ 2 n + 2∆ ′ )u |∂Ω , ∂ n v |∂Ω ) L 2 (∂Ω) + (∂ n u |∂Ω , (∂ 2 n + 2∆ ′ )v |∂Ω ) L 2 (∂Ω) -(u |∂Ω , ∂ 3 n v |∂Ω ) L 2 (∂Ω) , (3.8) or T (u, v) = ((∂ 3 n + 2∆ ′ ∂ n )u |∂Ω , v |∂Ω ) L 2 (∂Ω) -(∂ 2 n u |∂Ω , ∂ n v |∂Ω ) L 2 (∂Ω) + (∂ n u |∂Ω , ∂ 2 n v |∂Ω ) L 2 (∂Ω) -(u |∂Ω , (∂ 3 n + 2∆ ′ ∂ n )v |∂Ω ) L 2 (∂Ω) . (3.9)
We start our list of examples with the most basics ones.

∂ z b1 ∂ z b2 (x, ω ′ , z = i|ω ′ | g ) = det 1 |ω ′ | 2 g 0 -2i|ω ′ | g = -2i|ω ′ | g = 0,
if ω ′ = 0 and thus the Lopatinskiȋ-Šapiro condition holds by Lemma 3.2.

With the hinged boundary conditions observe that the bi-Laplace operator is precisely the square of the Dirichlet-Laplace operator. This makes its analysis quite simple and this explains why this type of boundary condition is often chosen in the mathematical literature. Observe that symmetry is then obvious without invoking the above formulae.

∂ z b1 ∂ z b2 (x, ω ′ , z = i|ω ′ | g ) = det 1 |ω ′ | g 0 -i = -i = 0.
Thus the Lopatinskiȋ-Šapiro condition holds by Lemma 3.2. Note that with the clamped boundary conditions the bi-Laplace operator cannot be seen as the square of the Laplace operator with some well chosen boundary condition as opposed to the case of the hinged boundary conditions displayed above. We have b1 (x, ω ′ , z) = -z 2 -2|ω ′ | 2 g and b2 (x, ω ′ , z) = iz 3 and det b1 b2

∂ z b1 ∂ z b2 (x, ω ′ , z = i|ω ′ | g ) = det -|ω ′ | 2 g |ω ′ | 3 g -2i|ω ′ | g -3i|ω ′ | 2 g = 5i|ω ′ | 4 g = 0, if ω ′ = 0
′ (x, ω ′ ) such that a ′ (x, ω ′ ) = 2|ω ′ | 3 g for ω ′ = 0, that is, a ′ (x, ω ′ ) = 2 for |ω ′ | g = 1. With (3.8) one finds T (u, v) = (-A ′ u |∂Ω , v |∂Ω ) L 2 (∂Ω) + (u |∂Ω , A ′ v |∂Ω ) L 2 (∂Ω) = 0,
in the case of homogeneous conditions, hence symmetry for P .

We have b1 (x, ω ′ , z) = -iz and b2 (x,

ω ′ , z) = iz 3 +a ′ (x, ω ′ ) with a ′ the principal symbol of A ′ . det b1 b2 ∂ z b1 ∂ z b2 (x, ω ′ , z = i|ω ′ | g ) = det |ω ′ | g |ω ′ | 3 g + a ′ (x, ω ′ ) -i -3i|ω ′ | 2 g = i a ′ (x, ω ′ ) -2|ω ′ | 3 g = 0, if ω ′ = 0 since a ′ (x, ω ′ ) = 2|ω ′ | 3
g by assumption implying that the Lopatinskiȋ-Šapiro condition holds by Lemma 3.2. (4) Take

B 1 u |∂Ω = u |∂Ω and B 2 u |∂Ω = (∂ 2 n + A ′ ∂ n )u |∂Ω with A ′
a symmetric differential operator of order less than or equal to one on ∂Ω, with homogeneous principal symbol a

′ (x, ω ′ ) such that a ′ (x, ω ′ ) = -2|ω ′ | g for ω ′ = 0, that is, a ′ (x, ω ′ ) = -2 for |ω ′ | g = 1.
This is a refinement of the case of hinged boundary conditions given in Example 3.5 above.

With (3.8) one finds

T (u, v) = (A ′ ∂ n u |∂Ω , ∂ n v |∂Ω ) L 2 (∂Ω) + (∂ n u |∂Ω , -A ′ ∂ n v |∂Ω ) L 2 (∂Ω) = 0,
in the case of homogeneous conditions, hence symmetry for P . We have b1 (x, ω ′ , z) = 1 and b2 (x,

ω ′ , z) = -z 2 -iza ′ (x, ω ′ ) with a ′ the principal symbol of A ′ . det b1 b2 ∂ z b1 ∂ z b2 (x, ω ′ , z = i|ω ′ | g ) = det 1 |ω ′ | 2 g + |ω ′ | g a ′ (x, ω ′ ) 0 -2i|ω ′ | g -ia ′ (x, ω ′ ) = -i a ′ (x, ω ′ ) + 2|ω ′ | g = 0, if ω ′ = 0 since a ′ (x, ω ′ ) = -2|ω ′ | g by assumption implying that the Lopatinskiȋ-Šapiro condition holds by Lemma 3.2. (5) Take B 1 u |∂Ω = (∂ 2 n + A ′ ∂ n )u |∂Ω and B 2 u |∂Ω = (∂ 3 n + 2∂ n ∆ ′ )u |∂Ω , with A ′ a
symmetric differential operator of order less than or equal to one on ∂Ω, with homogeneous principal symbol a

′ (x, ω ′ ) such that 2a ′ (x, ω ′ ) = -3|ω ′ | g for ω ′ = 0, that is, a ′ (x, ω ′ ) = -3/2 for |ω ′ | g = 1. With (3.9) one finds T (u, v) = (A ′ ∂ n u |∂Ω , ∂ n v |∂Ω ) L 2 (∂Ω) + (∂ n u |∂Ω , -A ′ ∂ n v |∂Ω ) L 2 (∂Ω) = 0,
in the case of homogeneous conditions, hence symmetry for P .

We have b1 (x, ω ′ , z) = -z 2iza ′ (x, ω ′ ) and b2 (x, ω ′ , z) = iz 3 + 2iz|ω ′ | 2 g and det b1 b2

∂ z b1 ∂ z b2 (x, ω ′ , z = i|ω ′ | g ) = det |ω ′ | 2 g + |ω ′ | g a ′ (x, ω ′ ) -|ω ′ | 3 g -2i|ω ′ | g -ia ′ (x, ω ′ ) -i|ω ′ | 2 g = -i|ω ′ | 3 g 2a ′ (x, ω ′ ) + 3|ω ′ | g = 0, if ω ′ = 0 since 2a ′ (x, ω ′ ) + 3|ω ′ | g =
0 by assumption implying that the Lopatinskiȋ-Šapiro condition holds by Lemma 3.2.

Lopatinskiȋ-Šapiro condition for the conjugated bi-Laplacian

Set P σ = ∆ 2σ 4 with σ ∈ [0, +∞) and denote by P σ,ϕ = e τ ϕ P σ e -τ ϕ the conjugate operator of P σ with τ ≥ 0 a large parameter and ϕ ∈ C ∞ (R d , R). We shall refer to ϕ as the weight function. The principal symbol of P σ in normal geodesic coordinates is given by

p σ (x, ξ) = (ξ 2 d + r(x, ξ ′ )) 2 -σ 4 .
Observe that e τ ϕ D j e -τ ϕ = D j + iτ ∂ j ϕ ∈ D 1 τ . So, after conjugation, the principal symbol becomes

p σ,ϕ (x, ξ, τ ) = p σ (x, ξ + iτ d x ϕ) = (ξ d + iτ ∂ d ϕ) 2 + r(x, ξ ′ + iτ d x ′ ϕ) 2 -σ 4 = (ξ d + iτ ∂ d ϕ) 2 + r(x, ξ ′ + iτ d x ′ ϕ) -σ 2 × (ξ d + iτ ∂ d ϕ) 2 + r(x, ξ ′ + iτ d x ′ ϕ) + σ 2
We write p σ,ϕ (x, ξ, τ ) = q 1 σ,ϕ (x, ξ, τ )q 2 σ,ϕ (x, ξ, τ ) with

q j σ,ϕ (x, ξ, τ ) = (ξ d + iτ ∂ d ϕ) 2 + r(x, ξ ′ + iτ d x ′ ϕ) + (-1) j σ 2 , j = 1, 2.
We consider two boundary operators B 1 and B 2 of order k 1 and k 2 with b j (x, ξ) for principal symbol, j = 1, 2. The associated conjugated operators

B j,ϕ = e τ ϕ B j e -τ ϕ , have respective principal symbols b j,ϕ (x, ξ, τ ) = b j (x, ξ + iτ dϕ), j = 1, 2.
We assume that the Lopatinskiȋ-Šapiro condition holds for (P 0 , B 1 , B 2 ) as in Definition 3.1 for any point (x, ω ′ ) ∈ T *

x ∂Ω. We wish to know if the Lopatinskiȋ-Šapiro condition can hold for (P σ , B 1 , B 2 , ϕ), as given by the following definition (in the chosen local normal geodesic coordinates for simplicity).

Definition 4.1. Let (x, ξ ′ , τ, σ) ∈ ∂Ω × R d-1 × [0, +∞) × [0, +∞) with (ξ ′ , τ, σ) = 0.
One says that the Lopatinskiȋ-Šapiro condition holds for (P σ , B 1 , B 2 , ϕ) at (x, ξ ′ , τ, σ) if for any polynomial f (z) with complex coefficients there exist c 1 , c 2 ∈ C and a polynomial ℓ(z) with complex coefficients such that, for all z ∈ C

f (z) = c 1 b 1,ϕ (x, ξ ′ , ξ d = z, τ ) + c 2 b 2,ϕ (x, ξ ′ , ξ d = z, τ ) + ℓ(z)p + σ,ϕ (x, ξ ′ , ξ d = z, τ ), with p + σ,ϕ (x, ξ ′ , ξ d = z, τ ) = Im ρ j (ξ ′ ,τ,σ)≥0 (z -ρ j (ξ ′ , τ, σ)),
where

ρ j (x, ξ ′ , τ, σ), j = 1, • • • , 4, are the complex roots of the polynomial z → p σ,ϕ (x, ξ ′ , ξ d = z, τ ).
In what follows, we shall assume that ∂ d ϕ > 0. Locally, one has

∂ d ϕ ≥ C 1 > 0, for some C 1 > 0.
4.1. Lopatinskiȋ-Šapiro condition and root positions. With the assumption that ∂ d ϕ > 0, for any point (x, ξ ′ , τ, σ) at most two roots lie in the upper complex closed half-plane (this is explained below). We then enumerate the following cases.

• Case 1 : No root lying in the upper complex closed half-plane, then p + σ,ϕ (x, ξ ′ , ξ d , τ ) = 1 and the Lopatinskiȋ-Šapiro condition of Definition 4.1 holds trivially at (x, ξ ′ , τ, σ).

• Case 2 : One root lying in the upper complex closed half-plane.

Let us denote by ρ + that root, then

p + σ,ϕ (x, ξ ′ , ξ d , τ ) = ξ d -ρ + . With Definition 4.1, for any choice of f , the polynomial ξ d → f (ξ d ) -c 1 b 1,ϕ (x, ξ ′ , ξ d , τ ) -c 2 b 2,ϕ (x, ξ ′ , ξ d , τ ) admits ρ + as a root for c 1 , c 2 ∈ C well chosen. Hence, the Lopatinskiȋ-Šapiro condition holds at (x, ξ ′ , τ, σ) if and only if b 1,ϕ (x, ξ ′ , ξ d = ρ + , τ ) = 0 or b 2,ϕ (x, ξ ′ , ξ d = ρ + , τ ) = 0.
Note that it then suffices to have

det b 1,ϕ b 2,ϕ ∂ ξ d b 1,ϕ ∂ ξ d b 2,ϕ (x, ξ ′ , ξ d = ρ + , τ ) = 0.
• Case 3 : Two different roots lying in the upper complex closed half-plane. Let denote by ρ + 1 and ρ + 2 these roots. One has

p + σ,ϕ (x, ξ ′ , ξ d , τ ) = (ξ d -ρ + 1 )(ξ d -ρ + 2
). The Lopatinskiȋ-Šapiro condition holds at (x, ξ ′ , τ, σ) if and only if the complex numbers ρ + 1 and ρ + 2 are the roots of the polynomial

ξ d → f (ξ d )-c 1 b 1,ϕ (x, ξ ′ , ξ d , τ )- c 2 b 2,ϕ (x, ξ ′ , ξ d , τ ), for c 1 , c 2 well chosen. This reads f (ρ + 1 ) = c 1 b 1,ϕ (x, ξ ′ , ξ d = ρ + 1 , τ ) + c 2 b 2,ϕ (x, ξ ′ , ξ d = ρ + 1 , τ ), f (ρ + 2 ) = c 1 b 1,ϕ (x, ξ ′ , ξ d = ρ + 2 , τ ) + c 2 b 2,ϕ (x, ξ ′ , ξ d = ρ + 2 , τ ).
Being able to solve this system in c 1 and c 2 for any f is equivalent to having

det b 1,ϕ (x, ξ ′ , ξ d = ρ + 1 , τ ) b 2,ϕ (x, ξ ′ , ξ d = ρ + 1 , τ ) b 1,ϕ (x, ξ ′ , ξ d = ρ + 2 , τ ) b 2,ϕ (x, ξ ′ , ξ d = ρ + 2 , τ ) = 0. (4.1)
• Case 4 : A double root lying in the upper complex closed halfplane. Denote by ρ + this root; one has

p + σ,ϕ (x, ξ ′ , ξ d , τ ) = (ξ d - ρ + ) 2 .
The Lopatinskiȋ-Šapiro condition holds at at (x, ξ ′ , τ, σ) if and only if for any choice of f , the complex number ρ + is a double root of the polynomial

ξ d → f (ξ d ) -c 1 b 1,ϕ (x, ξ ′ , ξ, τ ) - c 2 b 2,ϕ (x, ξ ′ , ξ, τ ) for some c 1 , c 2 ∈ C. Thus having the Lopatinskiȋ- Šapiro condition is equivalent of having the following determinant condition, det b 1,ϕ (x, ξ ′ , ξ d = ρ + , τ ) b 2,ϕ (x, ξ ′ , ξ d = ρ + , τ ) ∂ ξ d b 1,ϕ (x, ξ ′ , ξ d = ρ + , τ ) ∂ ξ d b 2,ϕ (x, ξ ′ , ξ d = ρ + , τ ) = 0. (4.2)
Observe that case 4 can only occur if σ = 0 (then one has (ξ ′ , τ ) = (0, 0)). If σ > 0 then only cases 1, 2, and 3 are possible. This is precisely stated in Lemma 4.7. This will be an important point in what follows.

We now state the following important proposition.

Proposition 4.2. Let x 0 ∈ ∂Ω. Assume that the Lopatinskiȋ-Šapiro condition holds for (P 0 , B 1 , B 2 ) at x 0 and thus in a compact neighborhood V 0 of x 0 (by Remark 3.3). Assume also that

∂ d ϕ ≥ C 1 > 0 in V 0 . There exist µ 0 > 0 and µ 1 > 0 such that if (x, ξ ′ , τ, σ) ∈ V 0 × R d-1 × [0, +∞) × [0, +∞) with (ξ ′ , τ, σ) = (0, 0, 0), |d x ′ ϕ(x)| ≤ µ 0 ∂ d ϕ(x) and σ ≤ µ 1 τ ∂ d ϕ(x),
then the Lopatinskiȋ-Šapiro condition holds for

(P σ , B 1 , B 2 , ϕ) at (x, ξ ′ , τ, σ).
The proof of Proposition 4.2 is given below. We first need to analyze the configuration of the roots of the symbol p σ,ϕ starting with each factor q j σ,ϕ , j = 1, 2. 4.2. Root configuration for each factor. We consider either factors ξ d → q j σ,ϕ (x, ξ ′ , ξ d , τ ). We recall that q j σ,ϕ (x, ξ, τ

) = (ξ d + iτ ∂ d ϕ) 2 + r(x, ξ ′ + iτ d x ′ ϕ) + (-1) j σ 2 , j = 1, 2. First, we consider the case r(x, ξ ′ + iτ d x ′ ϕ) + (-1) j σ 2 ∈ R -, that is, equal to -β 2 with β ∈ R . Then, the roots of ξ d → q j σ,ϕ (x, ξ ′ , ξ d , τ ) are given by -iτ ∂ d ϕ + β and -iτ ∂ d ϕ -β.
Both lie in the lower complex open half-plane.

Second, we consider the case r(x, ξ

′ + iτ d x ′ ϕ) + (-1) j σ 2 ∈ C \ R -.
There exists a unique α j ∈ C such that Re α j > 0 and

α 2 j (x, ξ ′ , τ, σ) = r(x, ξ ′ + iτ d x ′ ϕ(x)) + (-1) j σ 2 = r(x, ξ ′ ) -τ 2 r(x, d x ′ ϕ(x)) + (-1) j σ 2 + i2τ r(x, ξ ′ , d x ′ ϕ(x)) 2 , (4.3)
where r(x, ., .) denotes the symmetric bilinear form associated with the quadratic form r(x, .). Then, the two roots of ξ d → q j σ,ϕ (x, ξ ′ , ξ d , τ ) are given by

π j,1 (x, ξ ′ , τ, σ) = -iτ ∂ d ϕ(x) -iα j (x, ξ ′ , τ, σ) (4.4)
and

π j,2 (x, ξ ′ , τ, σ) = -iτ ∂ d ϕ(x) + iα j (x, ξ ′ , τ, σ). (4.5) One has Im π j,1 < 0 since ∂ d ϕ ≥ C 1 > 0. With Im π j,2 = -τ ∂ d ϕ + Re α j
one sees that the sign of Im π j,2 may change. The following lemma gives an algebraic characterization of the sign of Im π j,2 .

Lemma 4.3. Assume that ∂ d ϕ > 0. Having Im π j,2 (x, ξ ′ , τ, σ) < 0 is equivalent to having (∂ d ϕ) 2 r(x, ξ ′ ) + r(x, ξ ′ , d x ′ ϕ) 2 < τ 2 (∂ d ϕ) 2 |d x ϕ| 2 x + (-1) j+1 σ 2 (∂ d ϕ) 2 . Proof. From (4.4)-(4.5) one has Im π j,2 < 0 if and only if Re α j < τ ∂ d ϕ = |τ ∂ d ϕ|, that is, if and only if 4(τ ∂ d ϕ) 2 Re α 2 j -4(τ ∂ d ϕ) 4 + (Im α 2 j ) 2 < 0, by Lemma 4.4 below. With (4.3) this gives the result. Lemma 4.4. Let z ∈ C such that m = z 2 . For x 0 ∈ R such that x 0 = 0, we have | Re z| |x 0 | ⇐⇒ 4x 2 0 Re m -4x 4 0 + (Im m) 2 0.
The notation is to be understood as <, =, or > in each term of the equivalence respectively.

Proof. Let z = x+iy ∈ C. On the one hand we have z 2 = x 2 -y 2 +2ixy = m and Re m = x 2y 2 , Im m = 2xy. On the other hand we have

4x 2 0 Re m -4x 4 0 + (Im m) 2 = 4x 2 0 (x 2 -y 2 ) -4x 4 0 + 4x 2 y 2 = 4(x 2 0 + y 2 )(x 2 -x 2 0
), thus with the same sign as (

x 2 -x 2 0 ). Since | Re z| |x 0 | ⇔ x 2 -x 2 0 0 the conclusion follows.
With the following two lemmata we now connect the sign of Im π j,2 with the low frequency regime |ξ ′ | τ . Lemma 4.5. Assume there exists

K 0 > 0 such that |d x ′ ϕ| ≤ K 0 |∂ d ϕ|.
Then, there exists

C K 0 > 0 such that Im π j,2 (x, ξ ′ , τ, σ) < 0 if C K 0 |ξ ′ | + σ ≤ τ ∂ d ϕ, j = 0, 1.
Proof. With Lemma 4.3 having Im π j,2 < 0 reads

(∂ d ϕ) 2 r(x, ξ ′ )+r(x, ξ ′ , d x ′ ϕ) 2 < τ 2 (∂ d ϕ) 2 |d x ϕ| 2 x +(-1) j+1 σ 2 (∂ d ϕ) 2 . (4.6) On the one hand, since |d x ′ ϕ| ≤ K 0 |∂ d ϕ| one has (∂ d ϕ) 2 r(x, ξ ′ ) + r(x, ξ ′ , d x ′ ϕ) 2 ≤ K(∂ d ϕ) 2 |ξ ′ | 2 ,
for some K > 0 that depends on K 0 , using that |ξ ′ | x |ξ ′ |. On the other hand one has

τ 2 (∂ d ϕ) 2 |d x ϕ| 2 x + (-1) j+1 σ 2 (∂ d ϕ) 2 ≥ τ 2 (∂ d ϕ) 4 -σ 2 (∂ d ϕ) 2 . Thus (4.6) holds if one has τ 2 (∂ d ϕ) 4 -σ 2 (∂ d ϕ) 2 ≥ K(∂ d ϕ) 2 |ξ ′ | 2 , that is, τ 2 (∂ d ϕ) 2 ≥ K|ξ ′ | 2 + σ 2 .
Lemma 4.6. Let W be a bounded open set of R d and x 0 ∈ W . Assume that ∂ d ϕ > 0 in W and let κ 0 > 0. Then, there exists C > 0 such that

|ξ ′ | ≤ Cτ if Im π j,2 (x, ξ ′ , τ, σ) < 0 and κ 0 σ ≤ τ, x ∈ W. Proof. With Lemma 4.3 having Im π j,2 < 0 reads (∂ d ϕ) 2 r(x, ξ ′ ) + r(x, ξ ′ , d x ′ ϕ) 2 < τ 2 (∂ d ϕ) 2 |d x ϕ| 2 x + (-1) j+1 σ 2 (∂ d ϕ) 2 . In particular, this implies r(x, ξ ′ ) < τ 2 |d x ϕ| 2 x + (-1) j+1 σ 2 ≤ (sup W |d x ϕ| 2 x + 1/κ 2 0 )τ 2 .
The result follows since |ξ ′ | ≍ r(x, ξ ′ ).

As mentioned in Section 4.1, we have the following result.

Lemma 4.7. Assume that σ > 0. Then, π 1,2 (x, ξ ′ , τ, σ) = π 2,2 (x, ξ ′ , τ, σ). Moreover, the roots π 1,2 (x, ξ ′ , τ, σ) and π 2,2 (x, ξ ′ , τ, σ) cannot be both real.

Proof. With the forms of the roots given in (4.4)-(4.5

) if π 1,2 = π 2,2 then α 1 = α 2 , thus α 2 1 = α 2 2 implying σ 2 = 0. Assume now that π 1,2 ∈ R and π 2,2 ∈ R, that is, Im π 1,2 = Im π 2,2 = 0. This reads Re α j = τ ∂ d ϕ, giving | Re α j | = |∂ d ϕ|, for j = 1 and 2. With Lemma 4.4 one has 4(τ ∂ d ϕ) 2 Re α 2 j -4(τ ∂ d ϕ) 4 + (Im α 2 j ) 2 = 0, j = 1, 2.
Observing that Im α 2 1 = Im α 2 2 one thus obtains Re α 2 1 = Re α 2 2 , and the conclusion follows as for the first part.

|d x ′ ϕ| ≤ µ 0 ∂ d ϕ and σ ≤ µ 1 τ ∂ d ϕ.
First, we choose 0 < µ 0 ≤ 1 and 0 < µ 1 ≤ 1/2. Below both may be chosen much smaller. According to Lemma 4.5, with K 0 = 1 therein, for some (4.7)

C 2 = 2C K 0 > 0 if one has C 2 |ξ ′ | ≤ τ ∂ d ϕ
Our proof of Proposition 4.2 relies on the following lemma.

Lemma 4.8. There exists C 3 > 0 such that, for j = 1 or 2, for 0 < µ 0 ≤ 1, 0 < µ 1 ≤ 1/2, and for all (x, ξ ′ , τ, σ)

∈ V 0 × R d-1 × [0, +∞) × [0, +∞), one has |d x ′ ϕ(x)| ≤ µ 0 ∂ d ϕ(x), σ ≤ µ 1 τ ∂ d ϕ(x) and Im π j,2 (x, ξ ′ , τ, σ) ≥ 0 =⇒ α j -|ξ ′ | x + τ |d x ′ ϕ(x)| ≤ |ξ ′ | x C 3 (µ 0 + µ 2 1 ). Proof. With (4.7) one has τ |d x ′ ϕ| ≤ µ 0 τ ∂ d ϕ µ 0 |ξ ′ | x . (4.8)
With the first-order Taylor formula one has

α 2 j = r(x, ξ ′ + iτ d x ′ ϕ) + (-1) j σ 2 = r(x, ξ ′ ) + 1 0 d ξ ′ r(x, ξ ′ + iτ s d x ′ ϕ)(iτ d x ′ ϕ)ds + (-1) j σ 2 .
With (4.8) and homogeneity one has

d ξ ′ r(x, ξ ′ + iτ s d x ′ ϕ)(iτ d x ′ ϕ) µ 0 |ξ ′ | 2 x . One also has σ ≤ µ 1 τ ∂ d ϕ µ 1 |ξ ′ | x . Since r(x, ξ ′ ) = |ξ ′ | 2 x , this yields α 2 j = |ξ ′ | 2 x 1 + O(µ 0 + µ 2 1 ) and hence α j = |ξ ′ | x 1 + O(µ 0 + µ 2 1
) . This and (4.8) give the result.

Before proceeding, we make the following computation. For j = 1, 2 and ℓ = 1, 2 we write

b ℓ,ϕ (x, ξ ′ , ξ d = π j,2 , τ ) = b ℓ (x, ξ ′ + iτ d x ′ ϕ, π j,2 + iτ ∂ d ϕ) = b ℓ (x, ξ ′ + iτ d x ′ ϕ, iα j ) = b ℓ (x, ξ ′ + iτ d x ′ ϕ, i|ξ ′ | x + i(α j -|ξ ′ | x )). (4.9)
We use Lemma 3.4 and the value of ε > 0 given therein. We choose 0 < µ 0 ≤ 1 and 0 < µ 1 ≤ 1/2 such that

C 3 (µ 0 + µ 2 1 ) ≤ ε, (4.10)
with C 3 > 0 as given by Lemma 4.8. We now consider the root configurations that remain to consider according to the discussion in Section 4.1.

Case 2.

In this case, one root of p σ,ϕ lies in the upper complex closed half-plane.

We denote this root by ρ + . According to the discussion in Section 4.1 it suffices to prove that In this case Im π 1,2 > 0 and Im π 2,2 > 0. According to the discussion in Section 4.1 it suffices to prove that

det b 1,ϕ b 2,ϕ ∂ ξ d b 1,ϕ ∂ ξ d b 2,ϕ (x, ξ ′ , ξ d = ρ + , τ ) = 0. ( 4
det b 1,ϕ (x, ξ ′ , ξ d = π 1,2 , τ ) b 2,ϕ (x, ξ ′ , ξ d = π 1,2 , τ ) b 1,ϕ (x, ξ ′ , ξ d = π 2,2 , τ ) b 2,ϕ (x, ξ ′ , ξ d = π 2,2 , τ ) = 0. (4.12)
We use the second part of Lemma 3.4 with

ζ ′ = iτ d x ′ ϕ, δ = i(α 1 -|ξ| x )
, and δ = i(α 2 -|ξ| x ). With (4.9) and (4.10) with Lemma 4.8 and the second part of Lemma 3.4 one obtains (4.12). Case 4.

In this case (that only occurs if σ = 0) the Lopatinskiȋ-Šapiro condition holds also if one has (4.11). The proof is thus the same as for Case 2. This concludes the proof of Proposition 4.2.

4.4.

Local stability of the algebraic conditions. In Section 3 we saw that the Lopatinskiȋ-Šapiro condition for (P σ , B 1 , B 2 ) in Definition 3.1 exhibits some stability property. This was used in the statement of Proposition 4.2 that states how the Lopatinskiȋ-Šapiro condition for (P σ , B 1 , B 2 ) can imply the Lopatinskiȋ-Šapiro condition of Definition 4.1 for (P σ , B 1 , B 2 , ϕ), that is, the version of this condition for the conjugated operators.

A natural question would then be: does the Lopatinskiȋ-Šapiro condition for the conjugated operators enjoy the same stability property? The answer is yes. Yet, this is not needed in what follows. In fact, below one exploits the algebraic conditions listed in Section 4.1 once the Lopatinskiȋ-Šapiro condition is known to hold at a point ̺ 0′ = (x 0 , ξ 0′ , τ 0 , σ 0 ) in tangential phase space. One thus rather needs to know that these algebraic conditions are stable. Here also the answer is positive and is the subject of the present section.

As in Definition 4.1 for ̺ ′ = (x, ξ ′ , τ, σ) one denotes by ρ j (̺ ′ ) the roots of p σ,ϕ (x, ξ ′ , ξ d , τ ) viewed as a polynomial in ξ d .

Let ). One sets

̺ 0′ = (x 0 , ξ 0′ , τ 0 , σ 0 ) ∈ ∂Ω × R d-1 × [0, +∞) × [0, +∞
J + = j ∈ {1, 2, 3, 4}; Im ρ j (̺ 0′ ) ≥ 0}, J -= j ∈ {1, 2, 3, 4}; Im ρ j (̺ 0′ ) < 0},
and, for ̺ ′ = (x, ξ ′ , τ, σ),

κ + ̺ 0′ (̺ ′ ) = j∈J + ξ d -ρ j (̺ ′ ) , κ - ̺ 0′ (̺ ′ ) = j∈J - ξ d -ρ j (̺ ′ ) .
Naturally, one has

κ + ̺ 0′ (̺ 0′ , ξ d ) = p + σ,ϕ (x 0 , ξ 0′ , ξ d , τ 0 ) and κ - ̺ 0′ (̺ 0′ , ξ d ) = p -
σ,ϕ (x 0 , ξ 0′ , ξ d , τ 0 ). Moreover, there exists a conic open neighborhood U 0 of ̺ 0′ where both κ + ̺ 0′ (̺ ′ ) and κ - ̺ 0′ (̺ ′ ) are smooth with respect to ̺ ′ . One has

p σ,ϕ = p + σ,ϕ p - σ,ϕ = κ + ̺ 0′ κ - ̺ 0′ . Note however that for ̺ ′ = (x, ξ ′ , τ, σ) ∈ U 0 it may very well happen that p + σ,ϕ (x, ξ ′ , ξ d , τ ) = κ + ̺ 0′ (̺ ′ , ξ d ) and p - σ,ϕ (x, ξ ′ , ξ d , τ ) = κ - ̺ 0′ (̺ ′ , ξ d ).
The following proposition can be found in [8, proposition 1.8]. Proposition 4.9. Let the Lopatinskiȋ-Šapiro condition hold at

̺ 0′ = (x 0 , ξ 0′ , τ 0 , σ 0 ) ∈ ∂Ω×R d-1 ×[0, +∞)×[0, +∞) for (P σ , B 1 , B 2 , ϕ). Then, (1)
The polynomial ξ d → p + σ,ϕ (x 0 , ξ 0′ , ξ d , τ 0 ) is of degree less than or equal to two.

(2) There exists a conic open neighborhood U of ̺ 0′ such that the family

{b 1 ϕ (̺ ′ , ξ d ), b 2 ϕ (̺ ′ , ξ d )} is complete modulo κ + ̺ 0′ (̺ ′ , ξ d
) at every point ̺ ′ = (x, ξ ′ , τ, σ) ∈ U , namely for any polynomial f (ξ d ) with complex coefficients there exist c 1 , c 2 ∈ C and a polynomial ℓ(ξ d ) with complex coefficients such that, for all

ξ d ∈ C f (ξ d ) = c 1 b 1,ϕ (x, ξ ′ , ξ d , τ ) + c 2 b 2,ϕ (x, ξ ′ , ξ d , τ ) + ℓ(ξ d )κ + ̺ 0′ (̺ ′ , ξ d ). (4.13)
We emphasize again that the second property in Proposition 4.9 looks very much like the statement of Lopatinskiȋ-Šapiro condition for (P σ , B 1 , B 2 , ϕ) at ̺ ′ in Definition 4.1. Yet, it differs by having p + σ,ϕ (x, ξ ′ , ξ d , τ ) that only depends on the root configuration at ̺ ′ replaced by κ + ̺ 0′ (̺ ′ , ξ d ) whose structure is based on the root configuration at ̺ 0′ . Let m + be the common degree of p + σ,ϕ (̺ 0′ , ξ d ) and

κ + ̺ 0′ (̺ ′ , ξ d ) and m - be the common degree of p - σ,ϕ (̺ 0′ , ξ d ) and κ - ̺ 0′ (̺ ′ , ξ d ) for ̺ ′ ∈ U . One has m + + m -= 4 and thus m -≥ 2 for ̺ ′ ∈ U since m + ≤ 2.
Invoking the Euclidean division of polynomials, one sees that it is sufficient to consider polynomials f of degree less than or equal to m + -1 ≤ 1 in (4.13). Since the degree of b j,ϕ (̺ ′ , ξ d ) can be as high as 3 > m + -1 it however makes sense to consider f of degree less than or equal to m = 3. Then, the second property in Proposition 4.9 is equivalent to having

{b 1,ϕ (x, ξ ′ , ξ d , τ ), b 2,ϕ (x, ξ ′ , ξ d , τ )} ∪ 0≤ℓ≤3-m + {κ + ̺ 0′ (̺ ′ , ξ d )ξ ℓ d }
be a complete in the set of polynomials of degree less than or equal to m = 3. Note that this family is made of m ′ = 6m + = 2 + m - polynomials.

We now express an inequality that follows from Proposition 4.9 that will be key in the boundary estimation given in Proposition 5.1 below. 4.5. Symbol positivity at the boundary. The symbols b j,ϕ , j = 1, 2, are polynomial in ξ d of degree k j ≤ 3 and we may thus write them in the form

b j,ϕ (̺ ′ , ξ d ) = k j ℓ=0 b ℓ j,ϕ (̺ ′ )ξ ℓ d , with b ℓ j,ϕ homogeneous of degree k j -ℓ. The polynomial ξ d → κ + ̺ 0′ (̺ ′ , ξ d ) is of degree m + ≤ 2 for ̺ ′ ∈ U with U
given by Proposition 4.9. Similarly, we write

κ + ̺ 0′ (̺ ′ , ξ d ) = m + ℓ=0 κ +,ℓ ̺ 0′ (̺ ′ )ξ ℓ d ,
with κ +,ℓ ̺ 0′ homogeneous of degree m +ℓ. We introduce

e j,̺ 0′ (̺ ′ , ξ d ) = b j,ϕ (̺ ′ , ξ d ) if j = 1, 2, κ + ̺ 0′ (̺ ′ , ξ d )ξ j-3 d if j = 3, . . . , m ′ .
As explained above, all these polynomials are of degree less than or equal to three. If we now write

e j,̺ 0′ (̺ ′ , ξ d ) = 3 ℓ=0 e ℓ j,̺ 0′ (̺ ′ )ξ ℓ d ,
for j = 1, 2 one has e ℓ j,̺ 0′ (̺ ′ ) = b ℓ j,ϕ (̺ ′ ), with ℓ = 0, . . . , k j and e ℓ j,̺ 0′ (̺ ′ ) = 0 for ℓ > k j , and for j = 3, . . . , m ′ ,

e ℓ j,̺ 0′ (̺ ′ ) =      0 if ℓ < j -3, κ +,ℓ+3-j ̺ 0′ (̺ ′ ) if ℓ = j -3, . . . , m + + j -3 ≤ m + + m ′ -3, 0 if ℓ > m + + j -3.
In particular e ℓ j,̺ 0′ (̺ ′ ) is homogeneous of degree m + + jℓ -3. Note that m + + m ′ -3 = 3. We thus have the following result. Lemma 4.10. Set the m ′ ×(m+ 1) matrix M(̺ ′ ) = (M j,ℓ (̺ ′ )) 1≤j≤m ′ 0≤ℓ≤m with M j,ℓ (̺ ′ ) = e ℓ j,̺ 0′ (̺ ′ ). Then, the second point in Proposition 4.9 states that

M(̺ ′ ) is of rank m + 1 = 4 for ̺ ′ ∈ U . Recall that m ′ = m -+ 2 ≥ 4. We now set Σ e j,̺ 0′ (̺ ′ , z) = 3 ℓ=0 e ℓ j,̺ 0′ (̺ ′ )z ℓ = 3 ℓ=0 M j,ℓ (̺ ′ )z ℓ , z = (z 0 , . . . , z 3 ). ( 4
̺ 0′ = (x 0 , ξ 0′ , τ 0 , σ 0 ) ∈ ∂Ω × R d-1 × [0, +∞) × [0, +∞) for (P σ , B 1 , B 2 , ϕ
) and let U be as given by Proposition 4.9. Then, if ̺ ′ ∈ U there exists C > 0 such that

m ′ j=1 Σ e j,̺ 0′ (̺ ′ , z) 2 ≥ C|z| 2 C 4 , z = (z 0 , . . . , z 3 ) ∈ C 4 .
Proof. In C 4 define the bilinear form

Σ B (z, z ′ ) = m ′ j=1 Σ e j,̺ 0′ (̺ ′ , z)Σ e j,̺ 0′ (̺ ′ , z ′ ).
With (4.14) one has

Σ B (z, z ′ ) = M(̺ ′ )z, M(̺ ′ )z ′ C m ′ = t M(̺ ′ )M(̺ ′ )z, z ′ C 4 . As rank t M(̺ ′ )M(̺ ′ ) = rank M(̺ ′ ) = 4
by Lemma 4.10 one obtains the result.

Boundary norm estimate under Lopatinskiȋ-Šapiro condition

Near x 0 ∈ ∂Ω we consider two boundary operators B 1 and B 2 . As in Section 4 the associated conjugated operators are denoted by B j,ϕ , j = 1, 2 with respective principal symbols b j,ϕ (x, ξ, τ ).

The main result of this section is the following proposition for the fourth-order conjugated operator P σ,ϕ . It is key in the final result of the present article. It states that all traces are controlled by norms of B 1,ϕ v |x d =0 + and B 2,ϕ v |x d =0 + if the Lopatinskiȋ-Šapiro condition holds for (P, B 1 , B 2 , ϕ). Proposition 5.1. Let κ 0 > 0. Let x 0 ∈ ∂Ω, with Ω locally given by {x d > 0}. Assume that (P σ , B 1 , B 2 , ϕ) satisfies the Lopatinskiȋ-Šapiro condition of Definition 4.1 at

̺ ′ = (x 0 , ξ ′ , τ, σ) for all (ξ ′ , τ, σ) ∈ R d-1 × [0, +∞) × [0, +∞) such that τ ≥ κ 0 σ.
Then, there exist W 0 a neighborhood of x 0 , C > 0, τ 0 > 0 such that

| tr(v)| 3,1/2,τ ≤ C P σ,ϕ v + + 2 j=1 |B j,ϕ v |x d =0 + | 7/2-k j ,τ + v 4,-1,τ , for σ ≥ 0, τ ≥ max(τ 0 , κ 0 σ) and v ∈ C ∞ c (W 0 + ).
The notation of the function space

C ∞ c (W 0 + ) is introduced in (1.6).
For the proof of Proposition 5.1 we start with a microlocal version of the result.

A microlocal estimate.

Proposition 5.2. Let κ 1 > κ 0 > 0. Let x 0 ∈ ∂Ω, with Ω locally given by {x d > 0} and let W be a bounded open neighborhood of x 0 in R d . Let (ξ 0′ , τ 0 , σ 0 ) ∈ R d-1 × [0, +∞) × [0, +∞) nonvanishing with τ 0 ≥ κ 1 σ 0 and such that (P σ , B 1 , B 2 , ϕ) satisfies the Lopatinskiȋ-Šapiro condition of Definition 4.1 at ̺ 0′ = (x 0 , ξ 0′ , τ 0 , σ 0 ).

Then, there exists U a conic neighborhood of

̺ 0′ in W × R d-1 × [0, +∞) × [0, +∞) where τ ≥ κ 0 σ such that if χ ∈ S 0
T,τ , homogeneous of degree 0 in (ξ ′ , τ, σ) with supp(χ) ⊂ U , there exist C > 0 and τ 0 > 0 such that

| tr(Op T (χ)v)| 3,1/2,τ ≤ C 2 j=1 |B j,ϕ v |x d =0 + | 7/2-k j ,τ + P σ,ϕ v + + v 4,-1,τ + | tr(v)| 3,-1/2,τ , for σ ≥ 0, τ ≥ max(τ 0 , κ 0 σ) and v ∈ C ∞ c (W + ).
Proof. We choose a conic open neighborhood U 0 of ̺ 0′ according to Proposition 4.9 and such that

U 0 ⊂ W × R d-1 × [0, +∞) × [0, +∞) . Assume moreover that τ ≥ κ 0 σ in U 0 .
In Section 4.5 we introduced the symbols e j,̺

0′ (̺ ′ , ξ d ), j = 1, . . . , m ′ = m -+ 2 = 6 -m + . For a conic set V denote S V = {̺ ′ = (x, ξ ′ , τ, σ) ∈ V ; |(ξ ′ , τ, σ)| = 1}.
Consequence of the Lopatinskiȋ-Šapiro condition holding at ̺ 0′ , by Lemma 4.11 for all ̺ ′ ∈ S U 0 there exists C > 0 such that

m ′ j=1 Σ e j,̺ 0′ (̺ ′ , z) 2 ≥ C|z| 2 C 4 , z = (z 0 , . . . , z 3 ) ∈ C 4 . Let U 1 be a second conic open neighborhood of ̺ 0′ such that U 1 ⊂ U 0 . Since S U 1 is compact (recall that W is bounded), there exists C 0 > 0 such that m ′ j=1 Σ e j,̺ 0′ (̺ ′ , z) 2 ≥ C 0 |z| 2 C 4 , z = (z 0 , . . . , z 3 ) ∈ C 4 , ̺ ′ ∈ S U 1 .
Introducing the map N t ̺ ′ = (x, tξ ′ , tτ, tσ), for

̺ ′ = (x, ξ ′ , τ, σ) with t = |(ξ ′ , τ, σ)| -1 one has m ′ j=1 Σ e j,̺ 0′ (N t ̺ ′ , z) 2 ≥ C 0 |z| 2 C 4 , z = (z 0 , . . . , z 3 ) ∈ C 4 , ̺ ′ ∈ U 1 , (5.1)
since

N t ̺ ′ ∈ S U 1 . Now, for j = 1, 2 one has Σ e j,̺ 0′ (̺ ′ , z) = k j ℓ=0 e ℓ j,̺ 0′ (̺ ′ )z ℓ ,
with e ℓ j,̺ 0′ (̺ ′ ) homogeneous of degree k jℓ, and for 3 ≤ j ≤ m ′ one has

Σ e j,̺ 0′ (̺ ′ , z) = 3 ℓ=0 e ℓ j,̺ 0′ (̺ ′ )z ℓ ,
with e ℓ j,̺ 0′ (̺ ′ ) homogeneous of degree m + + jℓ -3. We define z ′ ∈ C 4 by z ′ ℓ = t ℓ-7/2 z ℓ , ℓ = 0, . . . , 3. One has Σ e j,̺ 0′ (N t ̺ ′ , z ′ ) = t k j -7/2 Σ e j,̺ 0′ (̺ ′ , z), j = 1, 2, and Σ e j,̺ 0′ (N t ̺ ′ , z ′ ) = t m + +j-13/2 Σ e j,̺ 0′ (̺ ′ , z), j = 3, . . . , m ′ .

Thus, from (5.1) we deduce

2 j=1 λ 2(7/2-k j ) T,τ Σ e j,̺ 0′ (̺ ′ , z) 2 + m ′ j=3 λ 2(13/2-m + -j) T,τ Σ e j,̺ 0′ (̺ ′ , z) 2 ≥ C 0 3 ℓ=0 λ 2(7/2-ℓ) T,τ |z ℓ | 2 , (5.2)
for z = (z 0 , . . . , z 3 ) ∈ C 4 , and

̺ ′ ∈ U 1 , since t ≍ λ -1 T,τ as τ σ in U 1 . We now choose U a conic open neighborhood of ̺ 0′ , such that U ⊂ U 1 . Let χ ∈ S 0
T,τ be as in the statement and let χ ∈ S 0 T,τ be homogeneous of degree 0, with supp( χ) ⊂ U 1 and χ ≡ 1 in a neighborhood of U , and thus in a neighborhood of supp(χ).

For j = 3, . . . , m ′ one has e j,̺ 0′ (̺ ′ , ξ

d ) = κ + ̺ 0′ (̺ ′ , ξ d )ξ j-3 d ∈ S m + +j-3,0 τ . Set E j = Op( χe j,̺ 0′
). The introduction of χ is made such that χe j,̺ 0′ is defined on the whole tangential phase-space. Observe that

B(w) = 2 j=1 |B j,ϕ w |x d =0 + | 2 7/2-k j ,τ + m ′ j=3 |E j w |x d =0 + | 2 13/2-m + -j,τ = 2 j=1 |Λ 7/2-k j T,τ B j,ϕ w |x d =0 + | 2 ∂ + m ′ j=3 |Λ 13/2-m + -j T,τ E j w |x d =0 + | 2 ∂
is a boundary quadratic form of type (3, 1/2) as in Definition 2.4. From Proposition 2.6 and (5.2) we have

| tr(u)| 2 3,1/2,τ 2 j=1 |B j,ϕ u |x d =0 + | 2 7/2-k j ,τ + m ′ j=3 |E j u |x d =0 + | 2 13/2-m + -j,τ + | tr(v)| 2 3,-N,τ . (5.3)
for u = Op T (χ)v and τ ≥ κ 0 σ chosen sufficiently large.

In U 1 one can write

p σ,ϕ = p + σ,ϕ p - σ,ϕ = κ + ̺ 0′ κ - ̺ 0′ , with κ + ̺ 0′ of degree m + and κ - ̺ 0′ of degree m -. In fact we set κ+ ̺ 0′ (̺ ′ ) = j∈J + ξ d -χρ j (̺ ′ ) , κ- ̺ 0′ (̺ ′ ) = j∈J - ξ d -χρ j (̺ ′ ) ,
with the notation of Section 4.4, thus making the two symbols defined on the whole tangential phase-space. In U , one has also

p σ,ϕ = κ+ ̺ 0′
κ-̺ 0′ . The factor κ-̺ 0′ is associated with roots with negative imaginary part. With Lemma A.1 given in Appendix A.1 one has the following microlocal elliptic estimate

Op T (χ)w m -,τ + | tr(Op T (χ)w)| m --1,1/2,τ Op T (κ - ̺ 0′ )Op T (χ)w + + w m -,-N,τ , for w ∈ S (R d + )
and τ ≥ κ 0 τ chosen sufficiently large. We apply this inequality to w = Op T (κ

+ ̺ 0′ )v. Since Op T (κ - ̺ 0′ )Op T (χ)Op T (κ + ̺ 0′ ) = Op T (χ)P σ,ϕ mod Ψ 4,-1 τ , one obtains | tr(Op T (χ)Op T (κ + ̺ 0′ )v)| m --1,1/2,τ P σ,ϕ v + + v 4,-1,τ . With [Op T (χ), Op T (κ + ̺ 0′ )] ∈ Ψ m + ,-1 τ one then has | tr(Op T (κ + ̺ 0′ )u)| m --1,1/2,τ P σ,ϕ v + + v 4,-1,τ + | tr(v)| 3,-1/2,τ ,
with u = Op T (χ)v as above, using that m + + m -= 4. Note that

| tr(Op T (κ + ̺ 0′ )u)| m --1,1/2,τ ≍ m --1 j=0 |D j d Op T (κ + ̺ 0′ )u |x d =0 + | m --j-1/2,τ m ′ j=3 |E j u |x d =0 + | 5/2+m --j,τ -| tr(v) |x d =0 + | 3,-1/2,τ ,
using that ξ j d κ+ ̺ 0′ = χe j+3,̺ 0′ in a conic neighborhood of supp(χ) and using that m -= m ′ -2. We thus obtain

m ′ j=3 |E j u |x d =0 + | 13/2-m + -j,τ P σ,ϕ v + + v 4,-1,τ + | tr(v)| 3,-1/2,τ , since 13/2 -m + = 5/2 + m -. With (5.3) then one finds | tr(u)| 3,1/2,τ 2 j=1 |B j,ϕ u |x d =0 + | 2 7/2-k j ,τ + P σ,ϕ v + + v 4,-1,τ + | tr(v)| 3,-1/2,τ .
In addition, observing that

|B j,ϕ u |x d =0 + | 7/2-k j ,τ |B j,ϕ v |x d =0 + | 7/2-k j ,τ + | tr(v)| 3,-1/2,τ ,
the result of Proposition 5.2 follows.

5.2. Proof of Proposition 5.1. As mentioned above the proof relies on a patching procedure of microlocal estimates given by Proposition 5.2. Let 0 < κ ′ 0 < κ 0 . We set

Γ d-1 +,κ 0 = {(ξ ′ , τ, σ) ∈ R d-1 × [0, +∞) × [0, +∞); τ ≥ κ 0 σ, }, and
S d-1 +,κ 0 = {(ξ ′ , τ, σ) ∈ Γ d-1 +,κ 0 ; |(ξ ′ , τ, σ)| = 1}. Consider (ξ 0′ , τ 0 , σ 0 ) ∈ S d-1
+,κ 0 . Since the Lopatinskiȋ-Šapiro condition holds at ̺ 0′ = (x 0 , ξ 0′ , τ 0 , σ 0 ), we can invoke Proposition 5.2:

(1) There exists a conic open neighborhood

U ̺ 0′ of ̺ 0′ in W × R d-1 × [0, +∞) × [0, +∞) where τ ≥ κ ′ 0 σ;
(2) For any χ ̺ 0′ ∈ S 0 T,τ homogeneous of degree 0 supported in U ̺ 0′ the estimate of Proposition 5.2 applies to Op T (χ ̺ 0′ )v for τ ≥ max(τ ̺ 0′ , κ 0 σ).

Without any loss of generality we may choose U ̺ 0′ of the form

U ̺ 0′ = O ̺ 0′ × Γ ̺ 0′ , with O ̺ 0′ ⊂ W an open neighborhood of x 0 and Γ ̺ 0′ a conic open neighborhood of (ξ 0′ , τ 0 , σ 0 ) in R d-1 × [0, +∞) × [0, +∞) where τ ≥ κ ′ 0 σ. Since {x 0 } × S d-1
+,κ 0 is compact we can extract a finite covering of it by open sets of the form of U ̺ 0′ . We denote by Ũi , i ∈ I with |I| < ∞, such a finite covering. This is also a finite covering of

{x 0 } × Γ d-1 +,κ 0 . Each Ũi has the form Ũi = O i × Γ i , with O i an open neighborhood of x 0 and Γ i a conic open set in R d-1 × [0, +∞) × [0, +∞) where τ ≥ κ ′ 0 σ. We set O = ∩ i∈I O i and U i = O × Γ i , i ∈ I. Let W 0 be an open neighborhood of x 0 such that W 0 ⋐ O. The open sets U i give also an open covering of W 0 × S d-1
+,κ 0 and W 0 × Γ d-1 +,κ 0 . With this second covering we associate a partition of unity χ i , i ∈ I, of W 0 × S d-1

+,κ 0 , where each χ i is chosen smooth and homogeneous of degree one for |(ξ ′ , τ, σ)| ≥ 1, that is:

i∈I χ i (̺ ′ ) = 1 for ̺ ′ = (x, ξ ′ , τ, σ) in a neighborhood of W 0 × Γ d-1
+,κ 0 , and

|(ξ ′ , τ, σ)| ≥ 1. Let u ∈ C ∞ c (W 0 + )).
Since each χ i is in S 0 T,τ and supported in U i , Proposition 5.2 applies:

| tr(Op T (χ i )v)| 3,1/2,τ ≤ C i 2 j=1 |B j,ϕ v |x d =0 + | 7/2-k j ,τ + P σ,ϕ v + + v 4,-1,τ + | tr(v)| 3,-1/2,τ , (5.4)
for some C i > 0, for σ ≥ 0, τ ≥ max(τ i , κ 0 σ) for some τ i > 0.

We set χ = 1-

i∈I χ i . One has χ ∈ S -∞ T,τ microlocally in a neighborhood of W 0 × Γ d-1
+,κ 0 . Thus, considering the definition of Γ d-1 +,κ 0 , if one imposes τ ≥ κ 0 σ, as we do, then χ ∈ S -∞ T,τ locally in a neighborhood of W 0 .

For any N ∈ N using that supp(v) ⊂ W 0 one has

| tr(v)| 3,1/2,τ ≤ i∈I | tr(Op T (χ i )v)| 3,1/2,τ + | tr(Op T ( χ)v)| 3,1/2,τ i∈I | tr(Op T (χ i )v)| 3,1/2,τ + | tr(v)| 3,-N,τ i∈I | tr(Op T (χ i )v)| 3,1/2,τ + v 4,-N,τ .
Summing estimates (5.4) together for i ∈ I we thus obtain

| tr(v)| 3,1/2,τ 2 j=1 |B j,ϕ v |x d =0 + | 7/2-k j ,τ + P σ,ϕ v + + v 4,-1,τ + | tr(v)| 3,-1/2,τ ,
for τ ≥ max(max i τ i , κ 0 σ). Therefore, by choosing τ ≥ κ 0 σ sufficiently large one obtains the result of Proposition 5.1.

Microlocal estimates for second-order factors

We recall that

P σ = ∆ 2 -σ 4 = (-∆ -σ 2 )(-∆ + σ 2 ) with σ ≥ 0. Set Q j σ = -∆ + (-1) j σ 2 ; then P σ = Q 1 σ Q 2 σ . We also set Q = -∆, that is, Q = Q 1 0 = Q 2 0 .
The principal symbols of Q j σ and Q are given by q j σ (x, ξ) = ξ 2 d + r(x, ξ ′ ) + (-1) j σ 2 and q(x, ξ) = ξ 2 d + r(x, ξ ′ ), (6.1) respectively. The conjugated operator P σ,ϕ = e τ ϕ P σ e -τ ϕ reads

P σ,ϕ = Q 1 σ,ϕ Q 2 σ,ϕ , with Q j σ,ϕ = e τ ϕ Q j σ e -τ ϕ . We set Q j s = Q j σ,ϕ + (Q j σ,ϕ ) * 2 and Q a = Q j σ,ϕ -(Q j σ,ϕ ) * 2i ,
both formally selfadjoint and such that Q j σ,ϕ = Q j s + iQ a . Note that Q a is independent of σ. Their respective principal symbols are

q j s (x, ξ, τ, σ) = ξ 2 d -(τ ∂ d ϕ) 2 + r(x, ξ ′ ) + (-1) j σ 2 -τ 2 r(x, d x ′ ϕ), q a (x, ξ, τ ) = 2τ ξ d ∂ d ϕ + 2τ r(x, ξ ′ , d x ′ ϕ).
Note that Q j s and Q a take the forms

Q j s = D 2 d + T j s , Q a = τ (∂ d ϕD d + D d ∂ d ϕ) + T a , (6.2)
where T j s , T a ∈ D 2 T,τ are such that (T j s ) * = T j s and T * a = T a . Naturally, the principal symbol of Q j σ,ϕ is q j σ,ϕ (x, ξ, τ ) = q j s (x, ξ, τ, σ) + iq a (x, ξ, τ ).

The principal symbol of Q j σ,ϕ = e τ ϕ Q j σ e -τ ϕ is

q j σ,ϕ (x, ξ, τ, σ) = (ξ d + iτ ∂ d ϕ) 2 + r(x, ξ ′ ) + (-1) j σ 2 -τ 2 r(x, d x ′ ϕ) + 2iτ r(x, ξ ′ , d x ′ ϕ).
As in Section 4.2 we let α j ∈ C be such that

α j (x, ξ ′ , τ, σ) 2 = r(x, ξ ′ + iτ d x ′ ϕ) + (-1) j σ 2 = r(x, ξ ′ ) -τ 2 r(x, d x ′ ϕ) + 2iτ r(x, ξ ′ , d x ′ ϕ) + (-1) j σ 2 ,
and Re α j ≥ 0. Note that uniqueness in the choice of α j holds except if r(x, ξ ′ + iτ d x ′ ϕ) + (-1) j σ 2 ∈ R -; this lack of uniqueness in that case is however not an issue in what follows. One has

q j σ,ϕ (x, ξ ′ , ξ d , τ ) = (ξ d + iτ ∂ d ϕ) 2 + α j (x, ξ ′ , τ, σ) 2 = ξ d + iτ ∂ d ϕ + iα j (x, ξ ′ , τ, σ) × ξ d + iτ ∂ d ϕ -iα j (x, ξ ′ , τ, σ) .
We recall from (4.4)-(4.5) that we write

q j σ,ϕ (x, ξ ′ , ξ d , τ ) = (ξ d -π j,1 )(ξ d - π j,2 ) with π j,1 = -iτ ∂ d ϕ -iα j (x, ξ ′ , τ, σ) and π j,2 = -iτ ∂ d ϕ + iα j (x, ξ ′ , τ, σ).
The roots π j,k , k = 1, 2 are functions of x, ξ ′ , τ and σ.

We denote by B a boundary operator of order k that takes the form

B(x, D) = B k (x, D ′ ) + B k-1 (x, D ′ )D d , with B k (x, D ′ ) and B k-1 (x, D ′ ) tangential differential operators of order k and k -1 respectively. The boundary operator B(x, D) has b(x, ξ) = b k (x, ξ ′ ) + b k-1 (x, ξ ′ )ξ d for principal symbol. The conjugate boundary operator B ϕ = e τ ϕ Be -τ ϕ is then given by B ϕ (x, D, τ ) = B k ϕ (x, D ′ , τ ) + B k-1 ϕ (x, D ′ , τ )(D d + iτ ∂ d ϕ) = Bk ϕ (x, D ′ , τ ) + B k-1 ϕ (x, D ′ , τ )D d , with Bk ϕ (x, D ′ , τ ) = B k ϕ (x, D ′ , τ ) + iτ B k-1 ϕ (x, D ′ , τ )∂ d ϕ. The principal symbol of B ϕ (x, D, τ ) is b ϕ (x, ξ, τ ) = bk ϕ (x, ξ ′ , τ ) + b k-1 ϕ (x, ξ ′ , τ )ξ d , where b k-1 ϕ (x, ξ ′ , τ ) is homogeneous of degree k-1 in λ T,τ and bk ϕ (x, ξ ′ , τ ) = b k ϕ (x, ξ ′ , τ ) + iτ b k-1 ϕ (x, ξ ′ , τ )∂ d ϕ is homogeneous of degree k in λ T,τ .
6.1. Sub-ellipticity. Set

q s (x, ξ, τ ) = ξ 2 d + r(x, ξ ′ ) -(τ ∂ d ϕ) 2 -r(x, τ d x ′ ϕ) = |ξ| 2 x -|τ dϕ| 2
x , where |ξ| 2

x = ξ 2 d + r(x, ξ ′ ). One has q j s = q s + (-1) j σ 2 . Observe that {q j s , q a } = {q s , q a }. Definition 6.1 (Sub-ellipticity). Let W be a bounded open subset of R d and ϕ ∈ C ∞ (W ) such that |d x ϕ| > 0. Let j = 1 or 2. We say that the couple (Q j σ , ϕ) satisfies the sub-ellipticity condition in W if there exist C > 0 and τ 0 > 0 such that for σ > 0

∀ (x, ξ) ∈ W × R d , τ ≥ τ 0 σ, q j
σ,ϕ (x, ξ, τ ) = 0 ⇒ {q j s , q a }(x, ξ, τ ) = {q s , q a }(x, ξ, τ ) ≥ C > 0. Remark 6.2. Note that with homogeneity the sub-ellipticity property also reads

∀ (x, ξ) ∈ W × R d , τ ≥ τ 0 σ, q j σ,ϕ (x, ξ, τ ) = 0 ⇒ {q j s , q a }(x, ξ, τ ) ≥ Cλ 3 τ . Proposition 6.3. Let W be a bounded open subset of R d and ψ ∈ C ∞ (R d ) such that ψ ≥ 0 and |d x ψ| ≥ C > 0 on W . Let τ 0 > 0.
Then, there exists γ 0 ≥ 1 such that (Q j σ , ϕ) satisfies the sub-ellipticity condition on W for τ ≥ τ 0 σ for ϕ = e γψ , with γ ≥ γ 0 , for both j = 1 and 2.

Proof. We note that |d x ϕ(x)| = 0.

The proof is slightly different whether one considers the symbol q 1 σ,ϕ

or the symbol q 2 σ,ϕ . Case 1: proof for q 1 σ,ϕ . Assume that q 1 σ,ϕ = 0. Thus |ξ| 2 x -|τ dϕ| 2 x -σ 2 = 0 implying |ξ| ∼ σ + γτ ϕ. On the one hand by Lemma 3.55 in [START_REF] Rousseau | Elliptic Carleman Estimates and Applications to Stabilization and Controllability I: Dirichlet Boundary Condition on Euclidean Space[END_REF], one has

{q s , q a }(x, ξ, τ ) = τ (γ 2 ϕ)(γϕ) 2 (H q ψ(x, β)) 2 + 4τ 2 q(x, dψ(x)) 2 + (γϕ) 3 1 2i {q ψ , q ψ }(x, β, τ ), (6.3)
with β = ξ/(γϕ), and where H q denotes the Hamiltonian vector field associated with the symbol q as defined in (6.1). Here, q ψ denotes the principal symbol of e τ ψ Qe -τ ψ , that is,

q ψ (x, ξ, τ ) = q(x, ξ + iτ d x ψ(x)) = (ξ d + iτ ∂ d ψ(x)) 2 + r(x, ξ ′ + iτ d x ′ ψ(x)).
On the other hand, one has (H q ψ(x, β)) 2 + 4τ 2 q(x, dψ(x)) 2 τ 2 and since 1 2i {q ψ , q ψ }(x, β, τ ) is homogeneous of degree 3 in (β, τ ), we obtain {q s , q a } ≥ Cγ(γτ ϕ) 3

-C ′ (γτ ϕ + |β|γϕ) 3 = Cγ τ 3 -C ′′ (τ + |ξ|) 3 , with τ = γτ ϕ. Yet, one has |ξ| ∼ σ + τ implying {q s , q a } ≥ Cγ τ 3 -C ′′′ (τ + |ξ|) 3 ≥ Cγ τ 3 -C (4) (τ + σ) 3 .
Since ψ ≥ 0 and γ ≥ 1 one has ϕ ≥ 1 implying τ 0 σ ≤ τ τ and thus {q s , q a }(x, ξ, τ ) ≥ τ 3 (Cγ -C (5) ).

It follows that for γ chosen sufficiently large one finds {q s , q a }(x, ξ, τ ) ≥ C > 0.

Case 2: proof for q 2 σ,ϕ . Assume that q 2 σ,ϕ = 0. Then |ξ| 2

x + σ 2 = |τ dϕ| 2

x implying |ξ| + σ ∼ τ |dϕ| ∼ τ . The same computation as in Case 1 gives

{q s , q a }(x, ξ, τ ) ≥ Cγ τ 3 -C ′ (τ + |ξ|) 3 .
Here |ξ| + τ τ yielding

{q s , q a }(x, ξ, τ ) ≥ (Cγ -C ′′ )τ 3 .
The remaining part of the proof is the same. Lemma 6.4. Let j = 1 or 2. Let (Q j σ , ϕ) have the sub-ellipticity property of Definition 6.1 in W . For µ > 0 one sets t(̺) = µ((q j s ) 2 + q 2 a )(̺) + τ {q j s , q a }(̺) with ̺ = (x, ξ, τ, σ)

∈ W × R d × [0, ∞) × [0, ∞). Let τ 0 > 0.
Then, for µ chosen sufficiently large and τ ≥ τ 0 σ one has t(̺) ≥ Cλ 4 τ for some C > 0.

The proof of Lemma 6.4 uses the following lemma. Lemma 6.5. Consider two continuous functions, f and g, defined in a compact set L , and assume that f ≥ 0 and moreover f (y) = 0 ⇒ g(y) > 0 for all y ∈ L .

Setting h µ = µf + g, we have h µ ≥ C > 0 for µ > 0 chosen sufficiently large.

Proof of Lemma 6.4. Consider the compact set

L = {(x, ξ, τ, σ); x ∈ W , |ξ| 2 + τ 2 + σ 2 = 1, τ ≥ τ 0 σ}.
Applying the result of Lemma 6.5 to t(̺) on L with f = (q j s ) 2 + q 2 a and g = τ {q j s , q a } we find for t(̺) ≥ C on L for some C > 0 for µ chosen sufficiently large. Since t(̺) is homogeneous of degree 4 in the variables (ξ, τ, σ) it follows that t(̺) ≥ C(σ 2 + τ 2 + |ξ| 2 ) 2 λ 4 τ .

6.2. Lopatinskiȋ-Šapiro condition for the second-order factors.

Above, in Section 4, the Lopatinskiȋ-Šapiro condition is addressed for the fourth-order operator P σ,ϕ . Here, we consider the two second-order factors Q j σ,ϕ . With the roots π j,1 and π j,2 defined in (4.4)-(4.5) one sets

q j,+ σ,ϕ (x, ξ ′ , ξ d , τ ) = Im π j,k ≥0 k=1,2 ξ d -π j,k (x, ξ ′ , τ, σ) .
Definition 6.6. Let j = 1, 2. Let x ∈ ∂Ω, with Ω locally given by

{x d > 0}. Let (ξ ′ , τ, σ) ∈ R d-1 × [0, +∞) × [0, +∞) with (ξ ′ , τ, σ) = 0.
One says that the Lopatinskiȋ-Šapiro condition holds for (Q j σ , B, ϕ) at ̺ ′ = (x, ξ ′ , τ, σ) if for any polynomial f (ξ d ) with complex coefficients there exist c ∈ C and a polynomial ℓ(ξ d ) with complex coefficients such that, for all

ξ d ∈ C f (ξ d ) = cb ϕ (x, ξ ′ , ξ d , τ ) + ℓ(ξ d )q j,+ σ,ϕ (x, ξ ′ , ξ d , τ ). (6.4) Remark 6.7.
With the Euclidean division of polynomials, we see that it suffices to consider the polynomial f (ξ d ) to be of degree less than that of q j,+ σ,ϕ (x, ξ ′ , ξ d , τ ) in (6.4). Thus, in any case, the degree of f (ξ d ) can be chosen less than or equal to one. Lemma 6.8. Let j = 1 or 2. Let x ∈ ∂Ω and (ξ ′ , τ, σ) ∈ R d-1 × [0, +∞) × [0, +∞) with (ξ ′ , τ, σ) = 0. The Lopatinskiȋ-Šapiro condition holds for (Q j σ , B, ϕ) at (x, ξ ′ , τ, σ) if and only if [START_REF] Alabau-Boussouira | Piecewise multiplier method and nonlinear integral inequalities for Petrowsky equation with nonlinear dissipation[END_REF]

either q j,+ σ,ϕ (x, ξ ′ , ξ d , τ ) = 1; (2) or q j,+ σ,ϕ (x, ξ ′ , ξ d , τ ) = ξ d -π and b ϕ (x, ξ ′ , π, τ ) = 0. Proof. If q j,+ σ,ϕ (x, ξ ′ , ξ d , τ ) = ξ d -π j,1 (x, ξ ′ , τ, σ) ξ d -π j,2 (x, ξ ′ , τ, σ)
, that is, both roots π j,1 (x, ξ ′ , τ, σ) and π j,2 (x, ξ ′ , τ, σ) are in the upper complex half-plane, then condition (6.4) cannot hold, since by Remark 6.7 it means that the vector space of polynomials of degree less than or equal to one would be generated by the single polynomial b ϕ (x, ξ ′ , ξ d , τ ). Suppose that q j,+ σ,ϕ (x, ξ ′ , ξ d , τ ) = ξ dπ that is one the root π j,1 (x, ξ ′ , τ, σ) and π j,2 (x, ξ ′ , τ, σ) has a nonnegative imaginary part and the other root has a negative imaginary part. Then, the Lopatinskiȋ-Šapiro condition holds at (x, ξ ′ , σ, τ ) if for any f (ξ d ), the polynomial ξ d → f (ξ d )cb ϕ (x, ξ ′ , ξ d , τ ) admits π as a root for some c ∈ C. A necessary and sufficient condition is then b ϕ (x, ξ ′ , ξ d = π, τ ) = 0. Finally if q j,+ σ,ϕ (x, ξ ′ , ξ d , τ ) = 1, that is, both roots π j,1 (x, ξ ′ , τ, σ) and π j,2 (x, ξ ′ , τ, σ) lie in the lower complex half-plane, then condition (6.4) trivially holds.

Microlocal estimates.

Here, for j = 1 or 2, we establish estimates for the operator Q j σ in a microlocal neighborhood of point at the boundary where (Q j σ , B, ϕ) satisfies the Lopatinskiȋ-Šapiro condition (after conjugaison) of Definition 6.6.

The quality of the estimation depends on the position of the roots. We shall assume that ∂ d ϕ > 0. Thus, from the form of the roots π j,1 and π j,2 given in (4.4)-(4.5), the root π j,1 always lies in the lower complex half-plane. The sign of Im π j,2 may however vary. Three cases can thus occur:

(1) The root π j,2 at the considered point lies in the upper complex half-plane. (2) The root π j,2 at the considered point is real.

(3) The root π j,2 at the considered point lies in the lower complex half-plane. Proposition 6.9. Let j = 1 or 2 and κ 1 > κ 0 > 0. Let x 0 ∈ ∂Ω, with Ω locally given by {x d > 0} and let W be a bounded open neighborhood of

x 0 in R d . Let ϕ be such that ∂ d ϕ ≥ C > 0 in W and such that (Q j σ , ϕ) satisfies the sub-ellipticity condition in W . Let ̺ 0′ = (x 0 , ξ 0′ , τ 0 , σ 0 ) with (ξ 0′ , τ 0 , σ 0 ) ∈ R d-1 ×[0, +∞) ×[0, +∞) nonvanishing with τ 0 ≥ κ 1 σ 0 and such that (Q j σ , B, ϕ
) satisfies the Lopatinskiȋ-Šapiro condition of Definition 6.6 at ̺ 0′ .

(1) Assume that Im π j,2 (̺ 0′ ) > 0. Then, there exists U a conic neighborhood of

̺ 0′ in W × R d-1 × [0, +∞) × [0, +∞)
where τ ≥ κ 0 σ such that if χ ∈ S 0 T,τ , homogeneous of degree 0 in (ξ ′ , τ, σ) with supp(χ) ⊂ U , there exist C > 0 and τ 0 > 0 such that

Op T (χ)v 2,τ + | tr(Op T (χ)v)| 1,1/2,τ ≤ C Q j σ,ϕ v + + |B ϕ v |x d =0 + | 3/2-k,τ + v 2,-1,τ , (6.5
)

for σ ≥ 0, τ ≥ max(τ 0 , κ 0 σ) and v ∈ C ∞ c (W + ). (2) Assume that Im π j,2 (̺ 0′ ) = 0. Then, there exists U a conic neigh- borhood of ̺ 0′ in W × R d-1 × [0, +∞) × [0, +∞) where τ ≥ κ 0 σ such that if χ ∈ S 0 T,τ , homogeneous of degree 0 in (ξ ′ , τ, σ) with supp(χ) ⊂ U , there exist C > 0 and τ 0 > 0 such that τ -1/2 Op T (χ)v 2,τ + | tr(Op T (χ)v)| 1,1/2,τ ≤ C Q j σ,ϕ v + + |B ϕ v |x d =0 + | 3/2-k,τ + v 2,-1,τ , (6.6
)

for σ ≥ 0, τ ≥ max(τ 0 , κ 0 σ) and v ∈ C ∞ c (W + ). (3) Assume that Im π j,2 (̺ 0′ ) < 0. Then, there exists U a conic neigh- borhood of ̺ 0′ in W × R d-1 × [0, +∞) × [0, +∞) where τ ≥ κ 0 σ such that if χ ∈ S 0 T,τ , homogeneous of degree 0 in (ξ ′ , τ, σ) with supp(χ) ⊂ U , there exist C > 0 and τ 0 > 0 such that Op T (χ)v 2,τ + | tr(Op T (χ)v)| 1,1/2,τ ≤ C Q j σ,ϕ v + + v 2,-1,τ , (6.7
)

for σ ≥ 0, τ ≥ max(τ 0 , κ 0 σ) and v ∈ C ∞ c (W + ). The notation of the function space C ∞ c (W + ) is introduced in (1.6).
6.3.1. Case (i): one root lying in the upper complex half-plane. One has Im π j,2 (̺ 0′ ) > 0 and Im π j,1 (̺ 0′ ) < 0.

Since the Lopatinskiȋ-Šapiro condition holds for (Q j σ , B, ϕ) at ̺ 0′ , by Lemma 6.8 one has

b ϕ (x 0 , ξ 0′ , ξ d = π j,2 (̺ 0′ ), τ 0 ) = b x 0 , ξ 0′ + iτ 0 d x ′ ϕ(x 0 ), iα j (̺ 0′ ) = 0.
As the roots π j,1 and π j,2 are locally smooth with respect to ̺ ′ = (x, ξ ′ , τ, σ) and homogeneous of degree one in (ξ ′ , τ, σ), there exists U a conic neighborhood of

̺ 0′ in W × R d-1 × [0, +∞) × [0, +∞) and C 1 > 0, C 2 > 0 such that S U = {̺ ′ ∈ U ; |ξ ′ | 2 + τ 2 + σ 2 = 1} is compact and τ ≥ κ 0 σ, Im π j,2 (̺ ′ ) ≥ C 2 λ T,τ , and Im π j,1 (̺ ′ ) ≤ -C 1 λ T,τ , and b ϕ (x, ξ ′ , ξ d = π j,2 (̺ ′ ), τ ) = 0, (6.8)
if ̺ ′ = (x, ξ ′ , τ, σ) ∈ U . We let χ ∈ S 0 T,τ and χ ∈ S 0 T,τ be homogeneous of degree zero in the variable (ξ ′ , τ, σ) and be such that supp( χ) ⊂ U and χ ≡ 1 on a neighborhood of supp(χ). From the smoothness and the homogeneity of the roots, one has χπ j,k ∈ S 1 T,τ , k = 1, 2. We set

L 2 = D d -Op T ( χπ j,2 ) and L 1 = D d -Op T ( χπ j,1 ).
The proof of Estimate (6.5) is based on three lemmata that we now list. Their proofs are given at the end of this section.

The following lemma provides an estimate for L 2 and boundary traces. Lemma 6.10. There exist C > 0 and τ 0 > 0 such that for any N ∈ N, there exists

C N > 0 such that tr Op T (χ)w 1,1/2,τ ≤ C B ϕ Op T (χ)w |x d =0 + 3/2-k,τ + L 2 Op T (χ)w |x d =0 + 1/2,τ + C N | tr(w)| 1,-N,τ ,
for τ ≥ max(τ 0 , κ 0 σ) and w ∈ S (R d + ). The proof of Lemma 6.10 relies on the Lopatinskiȋ-Šapiro condition. The following lemma gives an estimate for L 1 . Lemma 6.11. Let χ ∈ S 0 T,τ , homogeneous of degree 0, be such that supp(χ) ⊂ U and s ∈ R. There exist C > 0, τ 0 > 0 and N ∈ N such that

Op T (χ)w 1,s,τ + |Op T (χ)w |x d =0 + | s+1/2,τ ≤ C L 1 Op T (χ)w 0,s,τ + w 0,-N,τ , for w ∈ S (R d + )
and τ ≥ max(τ 0 , κ 0 σ). The proof of Lemma 6.11 is based on a multiplier method and relies on the fact that the root π j,1 that appears in the principal symbol of L 1 lies in the lower complex half-plane.

The following lemma gives an estimate for L 2 .

Lemma 6.12. Let χ ∈ S 0 T,τ , homogeneous of degree 0, be such that supp(χ) ⊂ U and s ∈ R. There exist C > 0, τ 0 > 0 and N ∈ N such that

Op T (χ)w 1,s,τ ≤ C L 2 Op T (χ)w 0,s,τ + |Op T (χ)w |x d =0 + | s+1/2,τ + w 0,-N,τ , for w ∈ S (R d + )
and τ ≥ max(τ 0 , κ 0 σ). Note that this estimate is weaker than that of Lemma 6.11 Observing that

L 1 Op T (χ)L 2 = Op T (χ)L 1 L 2 mod Ψ 1,0 τ = Op T (χ)Q j σ,ϕ mod Ψ 1,0
τ , and applying Lemma 6.11 to w = L 2 v with s = 0, one obtains

Op T (χ)L 2 v 1,τ + |Op T (χ)L 2 v |x d =0 + | 1/2,τ L 1 Op T (χ)L 2 v + + v 1,-N,τ Q j σ,ϕ v + + v 1,τ
, for τ ≥ κ 0 σ chosen sufficiently large. We set u = Op T (χ)v, and using the trace inequality

|w |x d =0 + | s,τ w s+1/2,τ , w ∈ S (R d + )
and s > 0, we have

L 2 u 1,τ + |L 2 u |x d =0 + | 1/2,τ Op T (χ)L 2 v 1,τ + |Op T (χ)L 2 v |x d =0 + | 1/2,τ + v 1,τ + |v |x d =0 + | 1/2,τ Op T (χ)L 2 v 1,τ + |Op T (χ)L 2 v |x d =0 + | 1/2,τ + v 1,τ .
Therefore, we obtain

L 2 u 1,τ + |L 2 u |x d =0 + | 1/2,τ Q j σ,ϕ v + + v 1,τ . With Lemma 6.10, one has the estimate tr(u) 1,1/2,τ + L 2 u 1,τ |B ϕ u |x d =0 + | 3/2-k,τ + Q j σ,ϕ v + + v 2,-1,τ
, for τ ≥ κ 0 σ chosen sufficiently large using the following trace inequality

| tr(w)| m,s,τ w m+1,s-1/2,τ , w ∈ S (R d + )
and m ∈ N, s ∈ R. With Lemma 6.12 for s = 1 one obtains

u 1,1,τ + tr(u) 1,1/2,τ + L 2 u 1,τ |B ϕ u |x d =0 + | 3/2-k,τ + Q j σ,ϕ v + + v 2,-1,τ
, for τ ≥ κ 0 σ chosen sufficiently large. Finally, we write

D d u 1,τ ≤ L 2 u 1,τ + Op T ( χπ j,2 )u 1,τ L 2 u 1,τ + u 1,1,τ , yielding u 2,τ + tr(u) 1,1/2,τ |B ϕ u |x d =0 + | 3/2-k,τ + Q j σ,ϕ v + + v 2,-1,τ . As u = Op T (χ)v, with a commutator argument we obtain |B ϕ u |x d =0 + | 3/2-k,τ |B ϕ v |x d =0 + | 3/2-k,τ + | tr(v)| 1,-1/2,τ |B ϕ v |x d =0 + | 3/2-k,τ + v 2,-1,τ .
yielding (6.5) and thus concluding the proof of Proposition 6.9 in Case (i).

We now provide the proofs the three key lemmata used above.

Proof of Lemma 6.10. Set

T (w) = |B ϕ w |x d =0 + | 2 3/2-k,τ + |L 2 w |x d =0 + | 2 1/2,τ = |Λ 3/2-k T,τ B ϕ w |x d =0 + | 2 ∂ + |Λ 1/2 T,τ L 2 w |x d =0 + | 2 ∂
. This is a boundary differential quadratic form of type (1, 1/2) in the sense of Definition 2.4. The associated bilinear symbol is given by

Σ T (̺ ′ , z, z ′ ) = λ 3-2k T,τ bk ϕ (x, ξ ′ , τ )z 0 + b k-1 ϕ (x, ξ ′ , τ )z 1 bk ϕ (x, ξ ′ , τ )z ′ 0 + b k-1 ϕ (x, ξ ′ , τ )z ′ 1 + λ T,τ z 1 -χπ j,2 (̺ ′ )z 0 z′ 1 -χπ j,2 (̺ ′ )z ′ 0 , with z = (z 0 , z 1 ) ∈ C 2 and z ′ = (z ′ 0 , z ′ 1 ) ∈ C 2 , yielding Σ T (̺ ′ , z, z) = λ 3-2k T,τ bk ϕ (x, ξ ′ , τ )z 0 + b k-1 ϕ (x, ξ ′ , τ )z 1 2 + λ T,τ z 1 -χπ j,2 (̺ ′ )z 0 2 .
which yields for ε chosen sufficiently small,

|u |x d =0 + | s+1/2,τ + u 0,s+1,τ L 1 u 0,s,τ + w 0,-N,τ . (6.9)
Finally, we write D d u 0,s,τ ≤ L 1 u 0,s,τ + Op T ( χπ j,1 )u 0,s,τ L 1 u 0,s,τ + u 0,s+1,τ . (6.10) Putting together (6.9) and (6.10), the result of Lemma 6.11 follows.

Proof of Lemma 6.12. We let u = Op T (χ)w. Performing an integration by parts, one has

2 Re L 2 u, -iΛ 2s+1 T,τ u + = 2 Re D d -Op T ( χπ j,2 ) u, -iΛ 2s+1 T,τ u + = Re i Op T ( χπ j,2 ) * Λ 1 T,τ -Λ 2s+1 T,τ Op T ( χπ j,2 ) u, u + -Re(Λ 2s+1 T,τ u |x d =0 + , u |x d =0 + ) ∂ . Note that Re(Λ 2s+1 T,τ u |x d =0 + , u |x d =0 + ) ∂ = |u |x d =0 + | 2 s+1/2,τ . Next, the operator i Op T ( χπ j,2 ) * Λ 2s+1 T,τ -Λ 2s+1 T,τ Op T ( χπ j,2 ) has the fol- lowing real principal symbol ϑ(̺ ′ ) = 2 Im π j,2 (̺ ′ )λ 2s+1 T,τ . and since Im π j,2 (̺ ′ ) ≥ C 2 λ T,τ > 0 in U one obtains ϑ(̺ ′ ) λ 2s+2
T,τ in U . Since U is neighborhood of supp(χ), the microlocal Gårding inequality of Theorem 2.49 in [START_REF] Rousseau | Elliptic Carleman Estimates and Applications to Stabilization and Controllability I: Dirichlet Boundary Condition on Euclidean Space[END_REF] (the proof adapts to the case with parameter σ as explained in Section 2.5 since σ τ ) yields

2 Re L 2 u, iΛ 2s+1 T,τ u + ≥ -|u |x d =0 + | 2 s+1/2,τ + C Λ s+1 T,τ u 2 + -C N w 2 0,-N,τ
, for τ ≥ κ 0 σ chosen sufficiently large. The end of the proof is then similar to that of Lemma 6.11. 6.3.2. Case (ii): one real root. One has Im π j,2 (̺ 0′ ) = 0 and Im π j,1 (̺ 0′ ) < 0.

Since the Lopatinskiȋ-Šapiro condition holds for (Q j σ , B, ϕ) at ̺ 0′ , by Lemma 6.8 one has

b ϕ (x 0 , ξ 0′ , ξ d = π j,2 (̺ 0′ ), τ 0 ) = b x 0 , ξ 0′ + iτ 0 d x ′ ϕ(x 0 ), iα j (̺ 0′ ) = 0.
As the roots π j,1 and π j,2 are locally smooth with respect to ̺ ′ = (x, ξ ′ , τ, σ) and homogeneous of degree one in (ξ ′ , τ, σ), there exists U a conic neighborhood of

̺ 0′ in W × R d-1 × [0, +∞) × [0, +∞) and C 1 > 0, C 2 > 0 such that S U = {̺ ′ ∈ U ; |ξ ′ | 2 + τ 2 + σ 2 = 1} is compact and τ ≥ κ 0 σ, π j,1 (̺ ′ ) = π j,2 (̺ ′ ), Im π j,2 (̺ ′ ) ≥ -C 2 λ T,τ , and Im π j,1 (̺ ′ ) ≤ -C 1 λ T,τ , and b ϕ (x, ξ ′ , ξ d = π j,2 (̺ ′ ), τ ) = 0, (6.11) if ̺ ′ = (x, ξ ′ , τ, σ) ∈ U .
We let χ ∈ S 0 T,τ and χ ∈ S 0 T,τ be homogeneous of degree zero in the variable (ξ ′ , τ, σ) and be such that supp( χ) ⊂ U and χ ≡ 1 on supp(χ). From the smoothness and the homogeneity of the roots, one has χπ j,k ∈ S 1 T,τ , k = 1, 2. We set

L 2 = D d -Op T ( χπ j,2 ) and L 1 = D d -Op T ( χπ j,1
). Lemma 6.10 and Lemma 6.11 also apply in Case (ii) and we shall use them. In addition to these two lemmata we shall need the following lemma. Lemma 6.13. There exist C > 0, τ 0 > 0 such that

τ -1/2 w 2,τ ≤ C Q j σ,ϕ w + + | tr(w)| 1,1/2,τ , for τ ≥ max(τ 0 , κ 0 σ) and w ∈ C ∞ c (W + ).
Proving Lemma 6.13 is fairly classical, based on writing Q j σ,ϕ = Q j s + iQ a and on an expansion of Q j σ,ϕ w 2 + and some integration by parts. We provide the details in the proof below as the occurence of the parameter σ is not that classical. Lemma 6.13 expresses the loss of a half-derivative if one root, here π j,2 , is real.

Observing that

L 1 Op T (χ)L 2 = Op T (χ)L 1 L 2 mod Ψ 1,0 τ = Op T (χ)Q j σ,ϕ mod Ψ 1,0
τ , and applying Lemma 6.11 to w = L 2 v, one obtains

|Op T (χ)L 2 v |x d =0 + | 1/2,τ L 1 Op T (χ)L 2 v + + v 1,-N,τ Q j σ,ϕ v + + v 1,τ
, for τ ≥ κ 0 σ chosen sufficiently large. We set u = Op T (χ)v, and using the trace inequality

|w |x d =0 + | s,τ w s+1/2,τ , w ∈ S (R d + )
and s > 0, we have

|L 2 u |x d =0 + | 1/2,τ |Op T (χ)L 2 v |x d =0 + | 1/2,τ + |v |x d =0 + | 1/2,τ |Op T (χ)L 2 v |x d =0 + | 1/2,τ + v 1,τ .
Therefore, we obtain

|L 2 u |x d =0 + | 1/2,τ Q j σ,ϕ v + + v 1,τ .
On the one hand, together with Lemma 6.10, one has the estimate

tr(u) 1,1/2,τ |B ϕ u |x d =0 + | 3/2-k,τ + Q j σ,ϕ v + + v 2,-1,τ , (6.12)
for τ ≥ κ 0 σ chosen sufficiently large using the following trace inequality

| tr(w)| m,s,τ w m+1,s-1/2,τ , w ∈ S (R d + )
and m ∈ N, s ∈ R. On the other hand, with Lemma 6.13 one has

τ -1/2 u 2,τ Q j σ,ϕ u + + | tr(u)| 1,1/2,τ , again for τ ≥ κ 0 σ chosen sufficiently large and since [Q j σ,ϕ , Op T (χ)] ∈ Ψ 1,0 τ one finds τ -1/2 u 2,τ Q j σ,ϕ v + + v 1,τ + | tr(u)| 1,1/2,τ , (6.13)
Now, with ε > 0 chosen sufficiently small one computes (6.12)+ε×(6.13) and obtains

τ -1/2 u 2,τ + | tr(u)| 1,1/2,τ Q j σ,ϕ v + + |B ϕ u |x d =0 + | 3/2-k,τ + v 2,-1,τ . As u = Op T (χ)v, with a commutator argument we obtain |B ϕ u |x d =0 + | 3/2-k,τ |B ϕ v |x d =0 + | 3/2-k,τ + | tr(v)| 1,-1/2,τ |B ϕ v |x d =0 + | 3/2-k,τ + v 2,-1,τ .
yielding (6.6) and thus concluding the proof of Proposition 6.9 in Case (ii).

We now provide a proof of Lemma 6.13.

Proof of Lemma 6.13. We recall that Q j σ,ϕ = Q j s + iQ a , yielding

Q j σ,ϕ w 2 + = Q j s w 2 + + Q a w 2 + + 2 Re(Q j s w, iQ a w) + . (6.14)
With the integration by parts formula

(f, D d g) + = (D d f, g) + -i(f |x d =0 + , g |x d =0 + ) ∂ ,
and the forms of Q j s and Q a given in (6.2) one has (f,

Q j s g) + = (Q j s f, g) + -i(f |x d =0 + , D d g |x d =0 + ) ∂ -i(D d f |x d =0 + , g |x d =0 + ) ∂ , and (f, Q a g) + = (Q a f, g) + -2τ i(∂ d ϕf |x d =0 + , g |x d =0 + ) ∂ , yiedling (Q a w, Q j s w) + = (Q j s Q a w, w) + -i(Q a w |x d =0 + , D d w |x d =0 + ) ∂ -i(D d Q a w |x d =0 + , w |x d =0 + ) ∂ , (Q j s w, Q a w) + = (Q a Q j s w, w) + -2iτ (∂ d ϕQ j s w |x d =0 + , w |x d =0 + ) ∂ .
This gives 2 Re(Q j s w, iQ a w)

+ = i([Q j s , Q a ]w, w) + + τ A(w), (6.15)
with

A(w) = τ -1 (Q a w, D d w) ∂ + τ -1 (D d Q a -2τ ∂ d ϕQ j s )w, w ∂ . (6.16)
We have the following lemma adapted from Lemma 3.25 in [START_REF] Rousseau | Elliptic Carleman Estimates and Applications to Stabilization and Controllability I: Dirichlet Boundary Condition on Euclidean Space[END_REF].

Lemma 6.14. The operators

Q a ∈ τ D 1 and D d Q a -2τ ∂ d ϕQ j s ∈ D 3
τ can be cast in the following forms

Q a = 2τ ∂ d ϕD d + 2r(x, D ′ , τ d x ′ ϕ) mod τ D 0 ,
and

D d Q a -2τ ∂ d ϕQ j s = -2τ ∂ d ϕ R(x, D ′ ) + (-1) j σ 2 -(τ ∂ d ϕ) 2 -r(x, τ d x ′ ϕ) + 2r(x, D ′ , τ d x ′ ϕ)D d mod τ Ψ 1,0
τ . With this lemma we find From (6.14) and (6.15) one writes

A(w) = 2(∂ d ϕD d w |x d =0 + , D d w |x d =0 + ) ∂ + 2(r(x, D ′ , d x ′ ϕ)w |x d =0 + , D d w |x d =0 + ) ∂ + 2 r(x, D ′ , d x ′ ϕ)D d w |x d =0 + , w |x d =0 + ∂ -2 ∂ d ϕ R(x, D ′ ) + (-1) j σ 2 w |x d =0 + , w |x d =0 + ∂ + 2 ∂ d ϕ (τ ∂ d ϕ) 2 + r(x, τ d x ′ ϕ) w |x d =0 + , w |x d =0 + ∂ + (Op(c 0 )w |x d =0 + , D d w |x d =0 + ) ∂ + Op(c 0 )D d + Op(c 1 ) w |x d =0 + , w |x d =0 + ∂ , (6.17
Q j σ,ϕ w 2 + + τ | tr(w)| 2 1,0,τ Q j s w 2 + + Q a w 2 + + Re(i[Q j s , Q a ]w, w) + . (6.19)
We now use the following lemma whose proof is given below. Lemma 6.15. There exists C, C ′ > 0, µ > 0 and τ 0 > 0 such that

µ Q j s w 2 + + Q a w 2 + + τ Re(i[Q j s , Q a ]w, w) + ≥ C w 2 2,τ -C ′ | tr(w)| 2 1,1/2,τ , for τ ≥ max(τ 0 , κ 0 σ) and w ∈ C ∞ c (W + )
Let µ > 0 be as in Lemma 6.15 and let τ > 0 be such that µτ -1 ≤ 1. From (6.19) one then writes

Q j σ,ϕ w 2 + + τ | tr(w)| 2 1,0,τ τ -1 µ Q j s w 2 + + Q a w 2 + + iτ ([Q j s , Q a ]
w, w) + , which with Lemma 6.15 yields the result of Lemma 6.13 using that

τ | tr(w)| 1,0,τ | tr(w)| 1,1/2,τ . Proof of Lemma 6.15. One has [Q j s , Q a ] ∈ τ D 2 τ . Writing τ Re i[Q j s , Q a ]w, w + = Re iτ -1 [Q j s , Q a ]
w, τ 2 w + , it can be seen as a interior differential quadratic form of type (2, 0) as in Definition 2.1. Therefore

T (w) = µ Q j s w 2 + + Q a w 2 + + τ Re i[Q j s , Q a ]
w, w + is also an interior differential quadratic form of this type with principal symbol given by t(̺) = µ|q j σ,ϕ (̺)| 2 + τ {q j s , q a }(̺), ̺ = (x, ξ, τ, σ).

Let τ 0 > 0. By Lemma 6.4, the sub-ellipticity property of (

Q j σ , ϕ) implies t(̺) λ 4 τ , ̺ ∈ W × R d × [0, +∞) × [0, +∞), τ ≥ τ 0 σ,
for µ > 0 chosen sufficiently large. The Gårding inequality of proposition 2.3 yields

T (w) ≥ C w 2 2,τ -C ′ | tr(w)| 2
1,1/2,τ , for some C, C ′ > 0 and for τ ≥ κ 0 σ chosen sufficiently large. 6.3.3. Case (iii): both roots lying in the lower complex half-plane. The result in the present case is a simple consequence of the general result given in Lemma A.1 whose proof can be found in [START_REF] Bellassoued | Carleman estimates for elliptic operators with complex coefficients. Part I: Boundary value problems[END_REF]. In the second order case however, the proof does not require the same level of technicality.

One has Im π j,1 (̺ 0′ ) < 0 and Im π j,2 (̺ 0′ ) < 0. As the roots π j,1 and π j,2 depend continuously on the variable

̺ ′ = (x, ξ ′ , τ, σ), there exists U a conic open neighborhood of ̺ 0′ in W × R d-1 × [0, +∞) × [0, +∞) and C 0 > 0 such that τ ≥ κ 0 σ, Im π j,1 (̺ ′ ) ≤ -C 0 λ T,τ , and Im π j,2 (̺ ′ ) ≤ -C 0 λ T,τ , if ̺ ′ = (x, ξ ′ , τ, σ) ∈ U .
Let χ ∈ S 0 T,τ be as in the statement of Proposition 6.9 and set u = Op T (χ)v.

We recall that Q j σ,ϕ = Q j s + iQ a , yielding

Q j σ,ϕ u 2 + = Q j s u 2 + + Q a u 2 + + 2 Re(Q j s u, iQ a u) + . We set L(u) = Q j s u 2 + + Q a u 2 +
. This is an interior differential quadratic form in the sense of Definition 2.1. Its principal symbol is given by ℓ(̺) = (q j s )(̺) 2 + q a (̺) 2 , ̺ = (x, ξ, τ, σ).

For ε ∈ (0, 1) we write

Q j σ,ϕ u 2 + ≥ εL(u) + 2 Re(Q j s u, iQ a u) + . (6.20)
For concision we write ̺ = (̺ ′ , ξ d ) with ̺ ′ = (x, ξ ′ , τ, σ). The set

L = {̺ = (̺ ′ , ξ d ); ̺ ′ ∈ U , ξ d ∈ R, and |ξ| 2 + τ 2 + σ 2 = 1}
is compact recalling that W is bounded. On L one has |q j σ,ϕ (̺)| ≥ C > 0. By homogeneity one has

|q j σ,ϕ (̺)| λ 2 τ , ̺ ′ ∈ U , ξ d ∈ R, if τ ≥ τ 0 σ, (6.21)
for some τ 0 > 0. Therefore

ℓ(̺) λ 4 τ , ̺ ′ ∈ U , ξ d ∈ R, if τ ≥ τ 0 σ. (6.22)
By the Gårding inequality of Proposition 2.2 one obtains

Re L(u) ≥ C u 2 2,τ -C ′ | tr(u)| 2 1,1/2 -C N v 2 2,-N,τ , (6.23)
for τ ≥ κ 0 σ chosen sufficiently large.

From the proof of Lemma 6.13 one has 2 Re(Q j s u, iQ a u)

+ = i([Q j s , Q a ]u, u) + + τ A(u) (6.24)
with the boundary quadratic form A given in (6.16)-(6.17). On the one hand, one has

[Q j s , Q a ] ∈ τ D 2 τ and therefore | Re([Q j s , Q a ]u, u) + | τ u 2 2,-1,τ τ -1 u 2 2,τ . (6.25)
On the other hand, we have the following lemma that provides a microlocal positivity property for the boundary quadratic form A. A proof is given below. Lemma 6.16. There exist C, C N and τ 0 > 0 such that

τ Re A(u) ≥ C| tr(u)| 2 1,1/2,τ -C N | tr(v)| 2 1,-N,τ , for u = Op T (χ)v, for τ ≥ max(τ 0 , κ 0 σ).
With (6.24)-(6.25), and Lemma 6.16 one obtains 2 Re(Q j s u, iQ a u)

+ ≥ C| tr(u)| 2 1,1/2,τ -C ′ τ -1 u 2 2,τ -C N | tr(v)| 2 1,-N,τ ≥ C| tr(u)| 2 1,1/2,τ -C ′ τ -1 u 2 2,τ -C ′ N v 2 2,-N,τ , (6.26)
with a trace inequality, for τ ≥ κ 0 σ chosen sufficiently large.

With (6.20), (6.23), and (6.26) one obtains

Q j σ,ϕ u 2 + ≥ εC u 2 2,τ -C ′ ε| tr(u)| 2 1,1/2 -C N ε v 2 2,-N,τ + C| tr(u)| 2 1,1/2,τ -C ′ τ -1 u 2 2,τ -C ′ N v 2 2,-N,τ
. With ε chosen sufficiently small and τ ≥ κ 0 σ sufficiently large one obtains for any N ∈ N

u 2,τ + | tr(u)| 1,1/2,τ Q j σ,ϕ u + + v 2,-N,τ . With a commutator argument, as u = Op T (χ)v one finds Q j σ,ϕ u + Q j σ,ϕ v + + v 2,-1,τ
, yielding estimate (6.7) and thus concluding the proof of Proposition 6.9 in Case (iii).

Proof of Lemma 6. [START_REF] Santos Ferreira | Sharp L p Carleman estimates and unique continuation[END_REF]. With (6.17) one sees that it suffices to consider the following boundary quadratic form

Ã(w) = 2(∂ d ϕD d w |x d =0 + , D d w |x d =0 + ) ∂ + 2(r(x, D ′ , d x ′ ϕ)w |x d =0 + , D d w |x d =0 + ) ∂ + 2 r(x, D ′ , d x ′ ϕ)D d w |x d =0 + , w |x d =0 + ∂ -2 ∂ d ϕ R(x, D ′ ) + (-1) j σ 2 w |x d =0 + , w |x d =0 + ∂ , + 2 ∂ d ϕ (τ ∂ d ϕ) 2 + r(x, τ d x ′ ϕ) w |x d =0 + , w |x d =0 + ∂ ,
in place of A. It is of type [START_REF] Alabau-Boussouira | Piecewise multiplier method and nonlinear integral inequalities for Petrowsky equation with nonlinear dissipation[END_REF]0) in the sense of Definition 2.4. Its principal symbol is given by a

0 (̺ ′ , ξ d , ξ ′ d ) = (1, ξ d )A(̺ ′ ) t (1, ξ d ) with A(̺ ′ ) = 2 A 11 (̺ ′ ) r(x, ξ ′ , d x ′ ϕ) |x d =0 + r(x, ξ ′ , d x ′ ϕ) |x d =0 + ∂ d ϕ |x d =0 + ,
where

A 11 (̺ ′ ) = -(∂ d ϕ) r(x, ξ ′ ) + (-1) j σ 2 -(τ ∂ d ϕ) 2 -r(x, τ d x ′ ϕ) |x d =0 +
with ̺ ′ = (x, ξ ′ , τ, σ). The associated bilinear symbol introduced in (2.4) is given by

Σ A (̺ ′ , z, z ′ ) = zA(̺ ′ ) t z′ , z = (z 0 , z 1 ) ∈ C 2 , z ′ = (z ′ 0 , z ′ 1 ) ∈ C 2 . One computes det A(̺ ′ ) = -4 (∂ d ϕ) 2 r(x, ξ ′ ) + (-1) j σ 2 -(τ ∂ d ϕ) 2 -r(x, τ d x ′ ϕ) + r(x, ξ ′ , d x ′ ϕ) 2 |x d =0 + .
With Lemma 4.3 one sees that Im π j,2 < 0 is equivalent to having det A(̺ ′ ) > 0. We thus have

det A(̺ ′ ) ≥ C > 0, for ̺ ′ = (x, ξ ′ , τ, σ) ∈ S U , with S U = {̺ ′ ∈ U ; ξ d ∈ R, |ξ| 2 + τ 2 + σ 2 = 1} since S U is compact. Since ∂ d ϕ |x d =0 + ≥ C ′ > 0 then one finds that Re Σ A (̺ ′ , z, z) ≥ C(|z 0 | 2 + |z 1 | 2 ), ̺ ′ = (x, ξ ′ , τ, σ) ∈ U , |(ξ ′ , τ, σ)| = 1.

By homogeneity one obtains

Re Σ A (̺ ′ , z, z) ≥ C(λ 2 T,τ |z 0 | 2 + |z 1 | 2 ), ̺ ′ = (x, ξ ′ , τ, σ) ∈ U , |(ξ ′ , τ, σ)| ≥ 1.
With Proposition 2.6, having in mind what is exposed in Section 2.5 since we have τ ≥ κ 0 σ here, one obtains

Re Ã(u) ≥ C| tr(u)| 2 1,0,τ -C N | tr(v)| 2 1,-N,τ , for u = Op T (χ)v, for τ ≥ κ 0 σ chosen sufficiently large.
Here, we have Im π j,2 < 0 and thus |ξ ′ | τ by Lemma 4.6. Thus one has

τ | tr(u)| 2 1,0,τ | tr(u)| 2 1,1/2,τ -| tr(v)| 2 1,-N,τ
, by the microlocal Gårding inequality, for instance invoking Proposition 2.6 for a boundary quadratic form of type (1, 1/2). This concludes the proof.

Local Carleman estimate for the fourth-order operator

In Proposition 5.1, we proved that a norm of all traces at the boundary could be estimates from the values taken by the boundary operators. That result relied on the Lopatinskiȋ-Šapiro condition. Here we aim to moveover estimate a volume norm. The strategy is to estimate this norm from all traces at the boundary and not from the boundary operators. The proof is then much simpler. Yet combined with Proposition 5.1 an estimation of both volume and trace norms will be obtained from the the values taken by the boundary operators.

Let ̺ 0′ = (x 0 , ξ 0′ , τ 0 , σ 0 ) with (ξ 0′ , τ 0 , σ 0 ) ∈ R d-1 × [0, +∞) × [0, +∞) nonvanishing with κ 1 σ 0 ≤ τ 0 ≤ κ ′ 1 σ 0 .
Then, there exists U a conic neighborhood of

̺ 0′ in W × R d-1 × [0, +∞) × [0, +∞) where κ 0 σ ≤ τ ≤ κ ′ 0 σ such that if χ ∈ S 0
T,τ , homogeneous of degree 0 in (ξ ′ , τ, σ) with supp(χ) ⊂ U , there exist C > 0 and τ 0 > 0 such that

τ -1/2 Op T (χ)v 4,τ ≤ C P σ,ϕ v + + | tr(v)| 3,1/2,τ + v 4,-1,τ , (7.1) for τ ≥ τ 0 , κ 0 σ ≤ τ ≤ κ ′ 0 σ, and v ∈ C ∞ c (W +
). The proof of Proposition 7.1 is based on the microlocal results of Proposition 6.9.

Remark 7.2. An important aspect is that here we have σ τ ; it is key to have only a loss of a half-derivative. Losses are due to lack of ellipticity, that is, having root(s) of p σ,ϕ lying on the real axis. If σ = 0 the operator is a square and two roots can lie and the real axis yielding a loss of a full derivative.

Note that the issue of having two potential roots on the real axis and thus a loss of a full derivative is independant of the choice of boundary operators.

Having σ > 0 implies that only one root of p σ,ϕ can lie on the real axis. In the proof we shall write P σ as a factor of two operators and we apply the Carleman estimates of Proposition 7.1 to each one. Each estimates exhibits a loss of a half-derivative. If naively concatenated the two estimates indeed yield a Carleman estimate with a loss of a full derivative. In fact the loss of each estimate occurs in different microlocal regions and a microlocal concatenation allows one to have a loss of only a half derivative.

Having σ > 0 was not sufficient for us to separate microlocal regions if τ increases because of homogeneity issues. This explains the introduction of the condition is σ τ . We do not exclude that σ > 0 could suffice to reach a similar estimate; we could not prove it.

Proof. We shall concatenate the estimates of Proposition 6.9 for Q 1 σ,ϕ and Q 2 σ,ϕ with the boundary operator B simply given by the Dirichlet trace operator, Bu

|x d =0 + = u |x d =0 + .
One has b(x, ξ) = 1 and b ϕ (x, ξ ′ , ξ d , τ ) = 1. Since ∂ d ϕ > 0 then Im π j,1 < 0. Thus, either q j,+ σ,ϕ (x, ξ ′ , ξ d , τ ) = 1 or q j,+ σ,ϕ (x, ξ ′ , ξ d , τ ) = ξ dπ j,2 . With Lemma 6.8 one sees that the Lopatinskiȋ-Šapiro holds for (Q 1 σ,ϕ , B, ϕ) and (Q 2 σ,ϕ , B, ϕ) at ̺ 0′ . Proposition 6.9 thus applies. Let U j be the conic neighborhood of ̺ 0′ obtained invoking this proposition for Q j σ,ϕ , for j = 1 or 2. In U j one has τ ≥ κ 0 σ. We set

U = U 1 ∩ U 2 ∩ {τ ≤ κ ′ 0 σ},
and we consider χ ∈ S 0 T,τ , homogeneous of degree 0 in (ξ ′ , τ, σ) with supp(χ) ⊂ U .

Since in U one has σ > 0 then π 1,2 and π 2,2 cannot be both real by Lemma 4.7. Proposition 6.9 thus implies that we necessarily have the following two estimates

τ -ℓ 1 Op T (χ)w 2,τ Q 1 σ,ϕ w + + |w |x d =0 + | 3/2,τ + w 2,-1,τ , (7.2)
and

τ -ℓ 2 Op T (χ)w 2,τ Q 2 σ,ϕ w + + |w| 3/2,τ + w 2,-1,τ , (7.3)
with either (ℓ 1 , ℓ 2 ) = (1/2, 0) or (ℓ 1 , ℓ 2 ) = (0, 1/2), for w ∈ C ∞ c (W + ) and τ ≥ κ 0 σ chosen sufficiently large.

Let us assume that (ℓ 1 , ℓ 2 ) = (1/2, 0). The other case can be treated similarly. Writing P

σ,ϕ = Q 2 σ,ϕ Q 1 σ,ϕ , with (7.3) one has Op T (χ)Q 1 σ,ϕ v 2,τ P σ,ϕ v + + |Q 1 σ,ϕ v |x d =0 + | 3/2,τ + v 4,-1,τ P σ,ϕ v + + | tr(v)| 2,3/2,τ + v 4,-1,τ . Since [Op T (χ), Q 1 σ,ϕ ] ∈ Ψ 1,0 τ one finds Q 1 σ,ϕ Op T (χ)v 2,τ P σ,ϕ v + + | tr(v)| 3,1/2,τ + v 4,-1,τ . (7.4)
For k = 0, 1 or 2, one writes

Q 1 σ,ϕ Op T (χ)D k d Λ 2-k T,τ v + + | tr(Op T (χ)D k d Λ 2-k T,τ v)| 1,1/2,τ + Op T (χ)D k d Λ 2-k T,τ v 2,-1,τ Q 1 σ,ϕ Op T (χ)v 2,τ + | tr(v)| 3,1/2,τ + v 4,-1,τ , since [Q 1 σ,ϕ Op T (χ), D k d Λ 2-k T,τ ] ∈ Ψ 4,-1 τ . Let χ ∈ S 0
T,τ be homogeneous of degree zero in the variable (ξ ′ , τ, σ) and be such that supp( χ) ⊂ U and χ ≡ 1 on a neighborhood of supp(χ). With (7.2), from (7.4) one thus obtains

τ -1/2 Op T ( χ)Op T (χ)D k d Λ 2-k T,τ v 2,τ P σ,ϕ v + + | tr(v)| 3,1/2,τ + v 4,-1,τ . Since Op T ( χ)Op T (χ)D k d Λ 2-k T,τ = Λ 2-k T,τ D k d Op T (χ) mod Ψ 2,-1 τ one deduces τ -1/2 D k d Op T (χ)v 2,2-k,τ P σ,ϕ v + + | tr(v)| 3,1/2,τ + v 4,-1,τ .
Using that k = 0, 1 or 2, the result follows.

Consequence of this microlocal result is the following local result by means of a patching procedure as for the proof of Proposition 5.1 in Section 5.2. Proposition 7.3. Let κ ′ 0 > κ 0 > 0. Let x 0 ∈ ∂Ω, with Ω locally given by {x d > 0} and let W be a bounded open neighborhood of x 0 in R d . Let ϕ be such that ∂ d ϕ ≥ C > 0 in W and such that (Q j σ , ϕ) satisfies the sub-ellipticity condition in W for both j = 1 and 2.

Then, there exists W 0 a neighborhood of x 0 , C > 0, τ 0 > 0 such that

τ -1/2 v 4,τ ≤ C P σ,ϕ v + + | tr(v)| 3,1/2,τ , (7.5
) Assume that (P σ , B 1 , B 2 , ϕ) satisfies the Lopatinskiȋ-Šapiro condition of Definition 4.1 at ̺ ′ = (x 0 , ξ ′ , τ, σ) for all (ξ ′ , τ, σ) ∈ R d-1 × [0, +∞) × [0, +∞) such that τ ≥ κ 0 σ.

for τ ≥ τ 0 , κ 0 σ ≤ τ ≤ κ ′ 0 σ, and v ∈ C ∞ c (W + ). 7.2.
Then, there exists W 0 a neighborhood of x 0 , C > 0, τ 0 > 0 such that

τ -1/2 e τ ϕ u 4,τ + | tr(e τ ϕ u)| 3,1/2,τ ≤ C e τ ϕ P σ u + + 2 j=1 |e τ ϕ B j v |x d =0 + | 7/2-k j ,τ , (7.6
)

for τ ≥ τ 0 , κ 0 σ ≤ τ ≤ κ ′ 0 σ, and u ∈ C ∞ c (W 0 + ).
The notation of the function space

C ∞ c (W 0 + ) is introduced in (1.6).
For the application of this theorem, one has to design a weight function that yields the two important properties: sub-ellipticity and the Lopatinskiȋ-Šapiro condition. Sub-ellipticity is obtained by means of Proposition 6.3; the Lopatinskiȋ-Šapiro condition by means of Proposition 4.2.

Proof of Theorem 7.4. The assumption of the theorem allows one to invoke both Propositions 5.1 and 7.3 yielding the existence of a neighborhood W 0 of x 0 where, by Proposition 5.1, one has

| tr(v)| 3,1/2,τ P σ,ϕ v + + 2 j=1 |B j,ϕ v |x d =0 + | 7/2-k j ,τ + v 4,-1,τ , (7.7)
Then, with Proposition 4.2, for κ 0 > 0 chosen sufficiently large one finds that the Lopatinskiȋ-Šapiro condition holds for (P σ , B 1 , B 2 , ϕ) at any (x, ξ ′ , τ, σ) for any x ∈ ∂Ω, ξ ′ ∈ T * x ∂Ω ≃ R d-1 , τ > 0, and σ > 0 such that τ ≥ κ 0 σ, for κ 0 chosen sufficiently large, using that ∂Ω is compact.

Thus for any x ∈ ∂Ω the local estimate of Theorem 7.4 applies. A similar result applies in the neighborhood of any point of Ω \ O 0 .

With the weight function ϕ constructed above, following the patching procedure described in the proof of Theorem 3.34 in [START_REF] Rousseau | Elliptic Carleman Estimates and Applications to Stabilization and Controllability I: Dirichlet Boundary Condition on Euclidean Space[END_REF], one obtains the following global estimate Remark 8.1. Observe that inequality (8.1) also holds for third-order perturbations of P σ . Below, we shall use it for a second-order perturbation P σiσ 2 α = ∆ 2σ 4iσ 2 α. 8.2. Observability inequality. By density one finds that inequality 8.1 holds for u ∈ H 4 (Ω).

τ -1/2 e τ ϕ u 4,τ + | tr(e τ ϕ u)| 3,1/2,τ e τ ϕ P σ u L 2 (Ω) + 2 j=1 |e τ ϕ B j u |∂Ω | 7/2-k j ,τ + τ -1/2 e τ ϕ χ 0 u 4,τ , (8.1) for τ ≥ τ 0 , κ 0 σ ≤ τ ≤ κ ′ 0 σ,
Let

C 0 > sup Ω ϕ -1. Since 1 ≤ ϕ ≤ sup Ω ϕ one obtains u H 4 (Ω) e C 0 τ P σ u L 2 (Ω) + 2 j=1 |B j u |∂Ω | H 7/2-k j (∂Ω) + u H 4 (O 1 ) . (8.2) for τ ≥ τ 0 , κ 0 σ ≤ τ ≤ κ ′ 0 σ. With the ellipticity of P 0 one has u H 4 (O 1 ) P 0 u L 2 (O) + u L 2 (O) , since O 1 ⋐ O.
This can be proven by the introduction of a parametrics for P 0 . One thus obtain

u H 4 (O 1 ) P σ u L 2 (Ω) + (1 + σ 4 ) u L 2 (O) ,
and thus with (8.2) one obtains the following observability result. This last property is very natural to define a nonnegative energy for the plate equation given in (9.1). We first review some properties of the unbounded operator associated with the bi-Laplace operator and the two homogeneous boundary conditions based on the assumptions made here. Second, the well-posedness of the plate equation is reviewed by means of the a semigroup formulation. This semigroup formalism is also central in the stabilization result in Sections 10.1-10.2. 9.1. The unbounded bi-Laplace operator. Associated with P and the boundary operators B 1 and B 2 is the operator (P 0 , D(P 0 )) on L 2 (Ω), with domain

D(P 0 ) = u ∈ L 2 (Ω); P u ∈ L 2 (Ω), B 1 u |∂Ω = B 2 u |∂Ω = 0 ,
and given by P 0 u = P u ∈ L 2 (Ω) for u ∈ D(P 0 ). The definition of D(P 0 ) makes sense since having P u ∈ L 2 (Ω) for u ∈ L 2 (Ω) implies that the traces ∂ k ν u |∂Ω are well defined for k = 0, 1, 2, 3. Since the Lopatinskiȋ-Šapiro condition holds on ∂Ω one has D(P 0 ) ⊂ H 4 (Ω) (see for instance Theorem 20.1.7 in [START_REF] Hörmander | The Analysis of Linear Partial Differential Operators[END_REF]) and thus one can also write D(P 0 ) as in (1.3). From the assumed nonnegativity in (9.4) above one finds that P 0 + Id is injective. Since the operator

L ′ : H 4 (Ω) → L 2 (Ω) ⊕ H 7/2-k 1 (∂Ω) ⊕ H 7/2-k 2 (∂Ω) u → (P u + u, B 1 u |∂Ω , B 2 u |∂Ω )
is Fredholm and has the same zero index as L defined in (9.3), one finds that L ′ is surjective. Thus Ran(P 0 + Id) = L 2 (Ω). One thus concludes that P 0 is maximal monotone. From the assumed symmetry property (9.2) and one finds that P 0 is selfadjoint, using that a symmetric maximal monotone operator is selfadjoint (see for instance Proposition 7.6 in [START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Differential Equations[END_REF]).

The resolvent of P 0 + Id being compact on L 2 (Ω), P 0 has a sequence of eigenvalues with finite multiplicities. With the assumed nonnegativity (9.4) they take the form of a sequence

0 ≤ µ 0 ≤ µ 1 ≤ • • • ≤ µ k ≤ • • •
that grows to +∞. Associated with this sequence is (φ j ) j∈N a Hilbert basis of L 2 (Ω). Any u ∈ L 2 (Ω) reads u = j∈N u j φ j , with u j = (u, φ j ) L 2 (Ω) . We define the Sobolev-like scale

H k B (Ω) = {u ∈ L 2 (Ω); (µ k/4 j u j ) j ∈ ℓ 2 (C)} for k ≥ 0. (9.5) One has D(P 0 ) = H 4 B (Ω) and L 2 (Ω) = H 0 B (Ω). Each H k B (Ω), k ≥ 0, is equipped with the inner product and norm (u, v) H k B (Ω) = j∈N (1 + µ j ) k/2 u j v j . u 2 H k B (Ω) = j∈N (1 + µ j ) k/2 |u j | 2 ,
is also an inner-product on H 2 B (Ω), that yields an equivalent norm. (2) If 0 is an eigenvalue, that is, dim ker(P 0 ) = n ≥ 1 then (φ 0 , . . . , φ n-1 ) is a orthonormal basis of ker(P 0 ) for the L 2 -inner product. From a classical unique continuation property, since α(x) > 0 for x in an open subset of Ω one sees that

(u, v) → (αu, v) L 2 (Ω) (9.10)
is also an inner product on the finite dimensional space ker(P 0 ) ⊂ L 2 (Ω). We introduce a second basis (ϕ 0 , . . . , ϕ n-1 ) of ker(P 0 ) orthonormal with respect to this second inner product. In what follows, we treat the more difficult case where dim ker(P 0 ) = n ≥ 1. The case ker(P 0 ) = {0} is left to the reader. 9.2. The plate semigroup generator. Set H = H 2 B (Ω) ⊕ L 2 (Ω) with natural inner product and norm (u 0 , u 1 ), (v 0 , v 1 ) H = (u 0 , v 0 ) H 2 B (Ω) + (u 1 , v 1 ) L 2 (Ω) , (9.11)

(u 0 , u 1 ) 2 H = u 0 2 H 2 B (Ω) + u 1 2 L 2 (Ω) . (9.12)
Define the unbounded operator

A = 0 -1 P 0 α(x) , (9.13)
on H with domain given by D(A) = D(P 0 ) ⊕ H 2 B (Ω). This domain is dense in H and A is a closed operator. One has N = ker(A) = t (u 0 , 0); u 0 ∈ ker(P 0 ) .

The important result of this section is the following proposition. The understanding of this generator property relies on the introduction of a reduced function space associated with ker(P 0 ), following for instance the analysis of [START_REF] Lebeau | Stabilisation de l'équation des ondes par le bord[END_REF]. It will be also important in the derivation of a precise resolvent estimate in Section 10.1. If ker(P 0 ) = {0}, that is, µ 0 > 0, this procedure is not necessary. For v ∈ ker(P 0 ), v = 0, we introduce the linear form ker(F ϕ j ), (9.15) with the basis (ϕ 0 , . . . , ϕ n-1 ) of ker(P 0 ) introduced above. If (v, 0) ∈ ker(A), with 0 = v ∈ ker(P 0 ), note that F v (v, 0) = 1. We set Θ j = t (ϕ j , 0), j = 0, . . . , n -1 and

Π N V = n-1 j=0 F ϕ j (V )Θ j , for V ∈ H,
and Π Ḣ = Id H -Π N . We obtain that Π N and Π Ḣ are continuous projectors associated with the direct sum H = Ḣ ⊕ N and Ḣ = ker(Π N ). (9.16)

Note that Ḣ and N are not orthogonal in H. Yet, it is important to note the following result.

Lemma 9.2. We have Ran(A) ⊂ Ḣ.

Proof. Let U = t (u 0 , u 1 ) = AV with V = t (v 0 , v 1 ) ∈ D(A). One has u 0 = -v 1 ∈ H 2 B (Ω) and u 1 = P 0 v 0 + αv 1 ∈ L 2 (Ω). If 0 = ϕ ∈ ker(P 0 ) one writes (αϕ, ϕ) L 2 (Ω) F ϕ (U) = (-αv 1 , ϕ) L 2 (Ω) + (P 0 v 0 + αv 1 , ϕ) L 2 (Ω) = (P 0 v 0 , ϕ) L 2 (Ω) = (v 0 , P 0 ϕ) L 2 (Ω) = 0. using that v 0 , ϕ ∈ D(P 0 ), that (P 0 , D(P 0 )) is selfadjoint, and that ϕ ∈ ker(P 0 ). The conclusion follows from the definition of Ḣ in (9.15).

The space Ḣ inherits the natural inner product and norm of H given in (9.11). Yet one finds that the inner product ((u 0 , u 1 ), (v 0 , v 1 )) Ḣ = P 0 u 0 , v 0 As for the decomposition of H given in (9.16) note that D( Ȧ) and N are not orthogonal.

H -2 B ,H 2 B + (u 1 , v 1 ) L 2 (Ω) , (9.17
Lemma 9.3. Let z ∈ C be such that Re z < 0. We have

(z Id Ḣ -Ȧ)U Ḣ ≥ | Re z| U Ḣ, U ∈ D( Ȧ).
The proof of this lemma is quite classical. It is given in Appendix A.2. With the previous lemma, with the Hille-Yosida theorem one proves the following result. Similarly to Lemma 9.3 one has the following result with a similar proof. For a strong solution, the natural energy is given by E(y)(t) = 1 2 ∂ t y(t) 2 L 2 (Ω) + (P 0 y(t), y(t)) L 2 (Ω) .

Observe that if y 0 ∈ ker(P 0 ) then y(t) = y 0 is solution to (9.1) with y 1 = 0. This is consistent with the form of the semigroup S(t) given in (9.20). Such a solution is independent of the evolution variable t, and thus, despite damping, there is no decay. However, note that such a solution is 'invisible' for the energy defined in (9.24). In fact, for a strong solution to (9.1) as given by Proposition 9.6 one has

E(y)(t) = 1 2 Ẏ (t) 2 Ḣ, (9.25)
with Ẏ (t) as defined below (9.21) and . Ḣ defined in (9.18) = Re(∂ t y(t), (∂ 2 t + P 0 )y(t)) L 2 (Ω) = -Re(∂ t y(t), α∂ t y(t)) L 2 (Ω) ≤ 0 since α ≥ 0. Thus, the energy of a strong solution is nonincreasing. To understand the decay of the energy one has to focus on the properties of the semigroup Ṡ(t) and its generator ( Ȧ, D( Ȧ)) on Ḣ. This is done in Section 10.1.

For a weak solution y(t) ∈ C 0 [0, +∞); H

Resolvent estimates and applications to stabilization

Here we use the observability inequality of Theorem 8.2 to obtain a resolvent estimate for the plate semigroup generator that allows one to deduce a stabilization result for the damped plate equation. This a sequence of argument comes from the seminal works of Lebeau [START_REF] Lebeau | Équation des ondes amorties[END_REF] and Lebeau-Robbiano [START_REF] Lebeau | Stabilisation de l'équation des ondes par le bord[END_REF]. 10.1. Resolvent estimate. We prove a resolvent estimate for the unbounded operator ( Ȧ, D( Ȧ)) that acts on Ḣ. First, we establish that {Re z ≤ 0} lies in the resolvent set of Ȧ. Proof. By Proposition 10.1 iσ Id -Ȧ is indeed invertible. Observe that it then suffices to prove the resolvent estimate (10.1) for |σ| ≥ σ 0 for some σ 0 > 0. Let U = t (u 0 , u 1 ) ∈ D( Ȧ) and F = t (f 0 , f 1 ) ∈ Ḣ be such that (iσ Id -Ȧ)U = F . This reads

f 0 = iσu 0 + u 1 , f 1 = -P 0 u 0 + (iσ -α)u 1 .
which gives (P 0σ 2iσα)u 0 = f with f = (iσα)f 0f 1 . Computing the L 2 -inner product with u 0 one finds ((P 0σ 2 )u 0 , u 0 ) L 2 (Ω)iσ(αu 0 , u 0 ) L 2 (Ω) = (f, u 0 ) L 2 (Ω) . ε -1 e 2C|σ| 1/2 f L 2 (Ω) + ε u 0 L 2 (Ω) . Thus, with ε chosen sufficiently small one obtains

u 0 H 4 (Ω)
e C|σ| 1/2 f L 2 (Ω) .

Since u 1 = f 0iσu 0 and f = (iσα)f 0f 1 we finally obtain that

u 0 H 4 (Ω) + u 1 L 2 (Ω)
e C|σ| 1/2 f 0 L 2 (Ω) + f 1

L 2 (Ω)
e C|σ| 1/2 F Ḣ.

Since u 0 ∈ H 4 (Ω) one has (P 0 u 0 , u 0 ) L 2 (Ω) ≤ u 0 H 4 (Ω) u 0 L 2 (Ω) ≤ u 0 2

H 4 (Ω)
and thus one finally obtains With Proposition 9.6, for Y 0 = t (y 0 , y 1 ) ∈ D(A n ) then the first component of Y (t) = S(t)Y 0 is precisely the solution to (9.1). One has Y (t) = Ẏ (t) + Π N Y 0 with Ẏ (t) = Ṡ(t)Π ḢY 0 with the semigroup Ṡ(t) defined in Section 9.2. Moreover, by (9.25) the energy of y(t) is given by the square of the Ḣ-norm of Ẏ (t). the proof, T is invertible with a bounded inverse. The operator T is unbounded on Ḣ. We denote by T the restriction of T to D( Ȧ) equipped with the graph-norm associated with Ȧ. The operator T is bounded. It is also invertible. It is thus a bounded Fredholm operator of index ind T = 0. Similarly, we denote by ι the injection of D( Ȧ) into Ḣ and à the restriction of Ȧ on D( Ȧ) viewed as a bounded operator. We have zι -à = T + ι. Since ι is a compact operator, we obtain that zι -à is also a bounded Fredholm operator of index 0. Hence, zι -à is surjective since z Id H -Ȧ is injective as proven above. Consequently, z Id H -Ȧ is surjective. This concludes the proof of Proposition 10.1.

3 . 1 . 3 . 2 .

 3132 35, Section 2.1.1]. Some examples. The Lopatinskiȋ-Šapiro condition holds in the following cases • P = -∆ on Ω, with the Dirichlet boundary condition, Bu |∂Ω = u |∂Ω . • P = ∆ 2 on Ω, along with the so-called clamped boundary conditions, i.e, B 1 u |∂Ω = u |∂Ω and B 2 u |∂Ω = ∂ ν u |∂Ω , where ν is the normal outward pointing unit vector to ∂Ω; see Section 1.4. • P = ∆ 2 on Ω, along with the so-called hinged boundary conditions, i.e, B 1 u |∂Ω = u |∂Ω and B 2 u |∂Ω = ∆u |∂Ω . Case of the bi-Laplace operator. With P = ∆ 2 on Ω, along with the general boundary operators B 1 and B 2 of orders k 1 and k 2 respectively, we give a matrix criterion of the Lopatinskiȋ-Šapiro condition.

Lemma 3 . 2 .

 32 Let P = ∆ 2 on Ω, B 1 and B 2 be two boundary operators. If x ∈ ∂Ω, ω ′ ∈ T * x ∂Ω, with ω ′ = 0, the Lopatinskiȋ-Šapiro condition holds at (x, ω ′ ) if and only if det b1 b2

Example 3 . 5 (

 35 Hinged boundary conditions). This type of conditions refers to B 1 u |∂Ω = u |∂Ω and B 2 u |∂Ω = ∆u |∂Ω . With (3.7) one finds T (u, v) = 0 in the case of homogeneous conditions, hence symmetry. Note that the hinged boundary conditions are equivalent to having B 1 u |∂Ω = u |∂Ω and B 2 u |∂Ω = ∂ 2 n u |∂Ω . With the notation of Section 3 this gives b1 (x, ω ′ , z) = 1 and b2 (x, ω ′ , z) = (-iz) 2 = -z 2 . It follows that det b1 b2

Example 3 . 6 (

 36 Clamped boundary conditions). This type of conditions refers to B 1 u |∂Ω = u |∂Ω and B 2 u |∂Ω = ∂ n u |∂Ω . With (3.8) one finds T (u, v) = 0 in the case of homogeneous conditions, hence symmetry. With the notation of Section 3 this gives b1 (x, ω ′ , z) = 1 and b2 (x, ω ′ , z) = -iz. It follows that det b1 b2

Examples 3 . 7 (

 37 More examples). (1) Take B 1 u |∂Ω = ∂ n u |∂Ω and B 2 u |∂Ω = ∂ n ∆u |∂Ω . With these boundary conditions the bi-Laplace operator is precisely the square of the Neumann-Laplace operator. The symmetry property is immediate and so is the Lopatinskiȋ-Šapiro condition. (2) Take B 1 u |∂Ω = (∂ 2 n + 2∆ ′ )u |∂Ω and B 2 u |∂Ω = ∂ 3 n u |∂Ω . With (3.8) one finds T (u, v) = 0 in the case of homogeneous conditions, hence symmetry.

  and thus the Lopatinskiȋ-Šapiro condition holds by Lemma 3.2. (3) Take B 1 u |∂Ω = ∂ n u |∂Ω and B 2 u |∂Ω = (∂ 3 n + A ′ )u |∂Ω , with A ′ a symmetric differential operator of order less than or equal to three on ∂Ω, with homogeneous principal symbol a

4. 3 .

 3 Proof of Proposition 4.2. Here, according to the statement of Proposition 4.2 we consider

  then all four roots of ξ d → p σ,ϕ (x, ξ ′ , ξ d , τ ) lie in the lower complex open half-plane. If so, we face Case 1 as in the discussion of Section 4.1 and the Lopatinskiȋ-Šapiro condition holds. To carry on with the proof of Proposition 4.2 we now only have to consider having τ ∂ d ϕ ≤ C 2 |ξ ′ |.

. 11 )

 11 In fact, one has ρ + = π j,2 with j = 1 or 2. We use the first part of Lemma 3.4 with ζ ′ = iτ d x ′ ϕ and δ = i(α j -|ξ| x ). With (4.9) and (4.10) with Lemma 4.8 and the first part of Lemma 3.4 one obtains (4.11). Case 3.

  ) with Op(c 0 ), Op(c 0 ) ∈ D 0 and Op(c 1 ) ∈ D 1 T,τ . Observe that one has |A(w)| | tr(w)| 2 1,0,τ . (6.18)

7. 1 .

 1 A first estimate. Proposition 7.1. Let κ ′ 0 > κ ′ 1 > κ 1 > κ 0 > 0. Let x 0 ∈ ∂Ω,with Ω locally given by {x d > 0} and let W be a bounded open neighborhood of x 0 in R d . Let ϕ be such that ∂ d ϕ ≥ C > 0 in W and such that (Q j σ , ϕ) satisfies the sub-ellipticity condition in W for both j = 1 and 2.

  Final estimate. Combining the local results of Section 5 for the estimation of the boundary norm under the Lopatinskiȋ-Šapiro condition and the previous local result without any prescribed boundary condition we obtain the Carleman estimate of Theorem 1.2. For a precise statement we write the following theorem.

Theorem 7 . 4 (

 74 local Carleman estimate for P σ ). Let κ ′ 0 > κ 0 > 0. Let x 0 ∈ ∂Ω, with Ω locally given by {x d > 0} and let W be a bounded open neighborhood of x 0 in R d . Let ϕ be such that ∂ d ϕ ≥ C > 0 in W and such that (Q j σ , ϕ) satisfies the sub-ellipticity condition in W for both j = 1 and 2.

  and u ∈ C ∞ (Ω), and where χ 0 ∈ C ∞ c (O) such that χ 0 ≡ 1 in a neighborhood of O 1 . Here, . s,τ and |.| s,τ , the Sobolev norms with the large parameter τ , are understood in Ω and ∂Ω respectively.

Theorem 8 . 2 (

 82 observability inequality). Let P σ = ∆ 2σ 4 and let B 1 and B 2 be two boundary operators of order k 1 and k 2 as given in Section 3.2. Assume that the Lopatinskiȋ-Šapiro condition of Definition 3.1

Proposition 9 . 1 .

 91 The operator (A, D(A)) generates a bounded semigroup S(t) = e -tA on H.

F

  v : H → C (9.14) (u 0 , u 1 ) → (αv, v) -1 L 2 (Ω) (αu 0 , v) L 2 (Ω) + (u 1 , v) L 2 (Ω) , We set Ḣ = v∈ker(P 0 ) v =0 ker(F v ) = 0≤j≤n-1

) and associated norm (u 0 , u 1 ) 2 ḢH - 2 B ,H 2 B+ u 1 2 L 2 (

 22222 = P 0 u 0 , u 0 Ω) ,(9.18) yields an equivalent norm on Ḣ by a Poincaré-like argument.We introduce the unbounded operator Ȧ on Ḣ given by the domain D( Ȧ) = D(A)∩ Ḣ and such that ȦV = AV for V ∈ D( Ȧ). We then haveA = Ȧ•Π Ḣ. Observe that D( Ȧ) = Π Ḣ D(A) since N = ker(A) ⊂ D(A).Thus, one has D(A) = D( Ȧ) ⊕ N . (9.19)

Lemma 9 . 4 .

 94 The operator ( Ȧ, D( Ȧ)) generates a semigroup of contraction Ṡ(t) = e -t Ȧ on Ḣ.If we setS(t) = Ṡ(t) • Π Ḣ + Π N ,(9.20) we find that S(t) is a semigroup on H generated by (A, D(A)), thus proving Proposition 9.1. If Y 0 ∈ D(A), the solution of the semigroup equation d dt Y (t) + AY (t) = 0 readsY (t) = S(t)Y 0 = Ṡ(t) • Π ḢY 0 + Π N Y 0 . (9.21) We set Ẏ (t) = Π ḢY (t) = Ṡ(t) • Π ḢY 0 .The adjoint of Ȧ has domain D( Ȧ * ) = D(A) and is given by

Lemma 9 . 5 . 9 . 3 . 4 B (Ω) + y 1 H 2 B

 9593412 Let z ∈ C be such that Re z < 0. We have (z Id Ḣ -Ȧ * )U Ḣ ≥ | Re z| U Ḣ, U ∈ D( Ȧ * ) = D( Ȧ). Strong and weak solutions to the damped plate equation. For y(t) a solution to the damped plate equation (9.1) one has Y (t) = t (y(t), ∂ t y(t)) formally solution to d dt Y (t) + AY (t) = 0 and conversely. The semigroup S(t) generated by A as given by Proposition 9.1 allows one to go beyond this formal observation and one obtains the following well-posedness result for strong solutions of the damped plate equation. Proposition 9.6 (strong solutions of the damped plate equation).For (y 0 , y 1 ) ∈ H 4 B (Ω) × H 2 B (Ω) there exists a unique y ∈ C 0 [0, +∞); H 4 B (Ω) ∩ C 1 [0, +∞); H 2 B (Ω) ∩ C 2 [0, +∞); L 2 (Ω) such that ∂ 2 t y + P y + α∂ t y = 0 in L ∞ ([0, +∞); L 2 (Ω)), y |t=0 = y 0 , ∂ t y |t=0 = y 1 . (9.22)Moreover, there exists C > 0 such thaty(t) H 4 B (Ω) + ∂ t y(t) H 2 B (Ω) ≤ C y 0 H (Ω) , t ≥ 0. (9.23)With Y (t) as above, for such a solution y(t) one hasd dt Y (t) + AY (t) = 0, Y (0) = Y 0 = t (y 0 , y 1 ),that is,Y (t) = S(t)Y 0 ∈ C 0 [0, +∞); D(A) ∩ C 1 [0, +∞); H 2 B (Ω) ⊕ L 2 (Ω). A weak solution to the damped plate equation is simply associated with an initial data (y 0 , y 1 ) ∈ H 2 B (Ω) × L 2 (Ω) and given by the first coordinate of Y (t) = S(t)Y 0 . Then one hasY (t) ∈ C 0 [0, +∞); H ∩ C 1 [0, +∞); L 2 (Ω) ⊕ H -2 B (Ω) .or equivalently y ∈ C 0 [0, +∞); H 2 B (Ω) ∩ C 1 [0, +∞); L 2 (Ω) ∩ C 2 [0, +∞); H -2 B (Ω) .

Proposition 10 . 1 .

 101 The spectrum of ( Ȧ, D( Ȧ)) is contained in {z ∈ C; Re(z) > 0}. The proof of this proposition is rather classical based on a unique continuation argument and a Fredholm index argument for a compact perturbation. It is given in Appendix A.3. Theorem 10.2. Let O be an open subset of Ω such that α ≥ δ > 0 on O. Then, for σ ∈ R the unbounded operator iσ Id -Ȧ is invertible on Ḣ and for there exist C > 0 such that (iσ Id -Ȧ) -1 L( Ḣ) ≤ Ce C|σ| 1/2 , σ ∈ R. (10.1)

0 H 4 2 L 2 2 L 2

 042222 As α ≥ 0, computing the imaginary part one obtainsσ α 1/2 u 0 2 L 2 (Ω) = -Im(f, u 0 ) L 2 (Ω). Since α ≥ δ > 0 in O by assumption and since we consider |σ| ≥ σ 0 one hasδσ 0 u 0 2 L 2 (O) ≤ f L 2 (Ω) u 0 L 2 (Ω). Applying Theorem 8.2 (with Remark 8.3) one hasu (Ω) e C|σ| 1/2 f L 2 (Ω) + u 0 L 2 (O), replacing |σ| by |σ| 2 therein. Thus, we obtainu 0 H 4 (Ω) e C|σ| 1/2 f L 2 (Ω) + f 1/(Ω) u 0 1/2L 2 (Ω) , for |σ| ≥ σ 0 . With Young inequality we write, for ε > 0, e C|σ| 1/2 f 1/(Ω) u 0 1/2 L 2 (Ω)

U 2 Ḣ 2 L 2 2 .

 2222 = (P 0 u 0 , u 0 ) L 2 (Ω) + u 1 (Ω) e C|σ| 1/2 F 2 Ḣ,which concludes the proof of the resolvent estimate (10.1). 10.Stabilization result. As an application of the resolvent estimate of Theorem 10.2, we give a logarithmic stabilization result of the damped plate equation (1.1).For the plate generator (A, D(A)) its iterated domains are inductively given byD(A n+1 ) = {U ∈ D(A n ); AU ∈ D(A n )}.

  Let Ω be a bounded connected open subset in R d , or a smooth bounded connected d-dimensional manifold, with smooth boundary ∂Ω, where we consider a damped plate equation

	1. Introduction
	1. Introduction
	1.1. Stabilization and control of Schrödinger and plate equations
	1.2. Method
	1.3. On Carleman estimates
	1.4. Geometrical setting
	1.5. Main results
	1.6. Some open questions
	1.7. Outline
	1.8. Some notation
	2. Pseudo-differential calculus
	2.1. Pseudo-differential operators with a large a parameter
	2.2. Tangential pseudo-differential operators

  satisfies the sub-ellipticity condition of Definition 6.1 locally. This is a necessary and sufficient condition for a Carleman estimate to hold for a second-order operator ∆ ± σ 2 , regardless of boundary conditions[START_REF] Rousseau | Elliptic Carleman Estimates and Applications to Stabilization and Controllability I: Dirichlet Boundary Condition on Euclidean Space[END_REF] Chapters 3 and 4]. (

  This means that the Lopatinskiȋ-Šapiro condition holds after the conjugation of the operator P σ and the boundary operators B 1 and B

2 by the weight function exp(τ ϕ).

Theorem 1.2 (Carleman estimate for

  2 B (Ω) ∩ C 1 [0, +∞); L 2 (Ω) the energy is defined by B that coincides with (9.24) for a strong solution. The stabilization result we are interested in only concerns strong solutions (see Section 10.2). Thus, we shall not mention weak solutions in what follows.

	E(y)(t) =	1 2	∂ t y(t) 2 L 2 (Ω) + P 0 y(t), y(t) H -2 B ,H 2

One has Σ T (̺ ′ , z, z) ≥ 0. For z = (0, 0) if Σ T (̺ ′ , z, z) = 0 then z 1 = χπ j,2 (̺ ′ )z 0 , bk ϕ (x, ξ ′ , τ )z 0 + b k-1 ϕ (x, ξ ′ , τ )z 1 = 0, implying that z 0 = 0 and b ϕ x, ξ ′ , ξ d = χπ j,2 (̺ ′ ), τ = bk ϕ (x, ξ ′ , τ ) + b k-1 ϕ (x, ξ ′ , τ ) χπ j,2 (̺ ′ ) = 0. Let U 1 ⊂ U be a conic open set such that supp(χ) ⊂ U 1 and χ = 1 in a conic neighborhood of U 1 . Then, for ̺ ′ ∈ U 1 one has b ϕ x, ξ ′ , ξ d = χπ j,2 (̺ ′ ), τ = b ϕ x, ξ ′ , ξ d = π j,2 (̺ ′ ), τ = 0, by (6.8). From the homogeneity of b k-1 ϕ (x, ξ ′ , τ ) and bk ϕ (x, ξ ′ , τ ) in ̺ ′ , it follows that there exists some C > 0 such that

The result of Lemma 6.10 thus follows from Proposition 2.6, having in mind what is exposed in Section 2.5 since we have τ ≥ κ 0 σ here.

Proof of Lemma 6.11. We let u = Op T (χ)w. Performing an integration by parts, one has

has the following real principal symbol

T,τ in U . Since U is neighborhood of supp(χ), the microlocal Gårding inequality of Theorem 2.49 in [START_REF] Rousseau | Elliptic Carleman Estimates and Applications to Stabilization and Controllability I: Dirichlet Boundary Condition on Euclidean Space[END_REF] (the proof adapts to the case with parameter σ as explained in Section 2.5 since σ τ ) yields

, for τ ≥ κ 0 σ chosen sufficiently large. With the Young inequality one obtains

With the Proposition 7.3 one also has

for τ ≥ τ ′ 1 and κ 0 σ ≤ τ ≤ κ ′ 0 σ for some τ ′ 1 > 0. Consider σ > 0 and τ ≥ max(τ 1 , τ ′ 1 ) such that κ 0 σ ≤ τ ≤ κ ′ 0 σ. Combined together (7.7) and (7.8) yield

If we set v = e τ ϕ u then the conclusion follows.

Global Carleman estimate and observability

Using the local Carleman estimate of Theorem 7.4 we prove a global version of this estimate. This allows us to obtain an observability inequality with observation in some open subset O of Ω. In turn in Section 10 we use this latter inequality to obtain a resolvent estimate for the plate semigroup generator that allows one to deduce a stabilization result for the damped plate equation.

Then, by Proposition 6.3, for γ chosen sufficiently large, one finds that ϕ = exp(γψ) is such that a

(1) ϕ = 1 and ∂ ν ϕ < -C 0 < 0 on ∂Ω;

(2) ϕ > 1 in Ω;

(3) (Q j σ , ϕ) satisfies the sub-ellipticity condition in Ω\O 0 , for j = 1, 2, for τ ≥ τ 0 σ for τ 0 chosen sufficiently large.

holds. Let O be an open set of Ω.

There exists C > 0 such that

Solutions to the damped plate equations

Here, we review some aspects of the solutions of the damped plate equation whose form we recall from the introduction:

where P = ∆ 2 and α ≥ 0, positive on some open subset of Ω. The boundary operators B 1 and B 2 of orders k j , j = 1, 2, less than or equal to 3 in the normal direction are chosen so that (i) the Lopatinskiȋ-Šapiro condition of Definition 3.1 is fulfilled for (P, B 1 , B 2 ) on ∂Ω; (ii) the operator P is symmetric under homogeneous boundary conditions, that is,

Examples of such conditions are given in Section 3.5.

With the assumed Lopatinskiȋ-Šapiro condition the operator

is Fredholm.

(iii) We shall further assume that the Fredholm index of the operator L is zero.

The previous symmetry property gives (P u, u) L 2 (Ω) ∈ R. We further assume the following nonnegativity property:

(iv) For u ∈ H 4 (Ω) such that B j u |∂Ω = 0 on ∂Ω, j = 1, 2 one has (P u, u) L 2 (Ω) ≥ 0.

(9.4) yielding a Hilbert space structure. The space

k and the injection is compact. Note that one uses (1 + µ j ) k/2 in place of µ k/2 j since ker(P 0 ) may not be trivial. Note that if k = 0 one recovers the standard L 2 -inner product and norm.

Using L 2 (Ω) as a pivot space, for k > 0 we also define the space H -k B (Ω) as the dual space of H k B (Ω). One finds that any u ∈ H -k B (Ω) takes the form of the following limit of L 2 -functions

for some (u j ) j ⊂ C such that (1 + µ j ) -k/4 u j j ∈ ℓ 2 (C), with the limit occurring in H k B (Ω) ′ with the natural dual strong topology. Moreover, one has

One can then extend (or restrict) the action of P 0 on any space

with convergence in

Note that

with the operator P viewed as un unbounded operator on L 2 (Ω).

We make the following observations.

(1) If ker

We have thus obtain the following theorem.

Theorem 10.3 (logarithmic stabilization for the damped plate equation). Assume that conditions (i) to (iv) of Section 9 hold. Let n ∈ N, n ≥ 1. Then, there exists C > 0 such that for any Y 0 = t (y 0 , y 1 ) ∈ D(A n ) the associated solution y(t) of the damped plate equation (1.1) has the logarithmic energy decay given by (10.2).

Note that for n = 1 using the form of A and (9.9) one recovers the statement of Theorem 1.1 in the introductory section.

Appendix A. Some technical results and proofs A.1. A perfect elliptic estimate. Here we consider a(̺ ′ , ξ d ) polynomial in the ξ d variable and such that its root have negative imaginary parts microlocally.

and where the coefficients a j are homogeneous in (ξ ′ , τ, σ). Moreover, assume that a 0 (̺ ′ ) = 1. Set A = Op(a).

Let U be a conic open subset of W × R d-1 × [0, +∞) × [0, +∞) where τ ≥ κ 0 σ and such that all the roots of a(̺ ′ , ξ d ) have a negative imaginary part for ̺ ′ ∈ U .

Let χ(̺ ′ ) ∈ S 0 T,τ be homogeneous of degree zero and such that supp(χ) ⊂ U and N ∈ N. Then there exist C > 0, C N > 0, and τ 0 > 0 such that

and τ ≥ max(τ 0 , κ 0 σ).

We refer to [START_REF] Bellassoued | Carleman estimates for elliptic operators with complex coefficients. Part I: Boundary value problems[END_REF] for a proof (see Lemma 4.1 therein and its proof that adapts to the presence of the parameter σ with σ τ in a straightforward manner).

A.2. Basic resolvent estimation. Here we provide a proof of Lemma 9.3 Let U = t (u 0 , u 1 ) ∈ D( Ȧ). With (9.17) We write

As α ≥ 0 and Re z < 0, this gives

which yields the conclusion of Lemma 9.3.

A.3. Basic estimation for the resolvent set. Here we provide a proof of Proposition 10.1. Let z ∈ C. We consider the two cases.

Case 1: Re z < 0. By Lemma 9.3 z Id Ḣ -Ȧ is injective. Moreover, as its adjoint z Id Ḣ -Ȧ * is injective and satisfies (z Id Ḣ -Ȧ * )U Ḣ U Ḣ for U ∈ D( Ȧ) by Lemma 9.5 the map z Id Ḣ -Ȧ is surjective (see for instance [START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Differential Equations[END_REF]Theorem 2.20]). The estimation of Lemma 9.3 then gives the continuity of the operator (z Id Ḣ -Ȧ) -1 on Ḣ.

Case 2: Re z = 0. We start by proving the injectivity of z Id Ḣ -Ȧ. Let thus U = t (u 0 , u 1 ) ∈ D( Ȧ) be such that zU -ȦU = 0. This gives zu 0 + u 1 = 0, -P 0 u 0 + (zα)u 1 = 0.

(A.2)

First, if z = 0 one has u 1 = 0, and then P 0 u 0 = 0. Thus, u 0 ∈ ker(P 0 ) given U ∈ N = ker(A). From the definition of Ḣ this gives U = 0. Second, if now z = 0, using (A.1) we obtain 0 = Re((z Id Ḣ -Ȧ)U, U) Ḣ = -(αu 1 , u 1 ) L 2 (Ω) .

As α ≥ 0, this implies that u 0 vanishes a.e. on supp(α). Observe that

The function u 0 is thus an eigenfunction for P 0 that vanishes on an open set. With the unique continuation property we obtain that u 0 vanishes in Ω and u 1 as well.

If we now prove that z Id Ḣ -Ȧ is surjective, the result then follows from the closed graph theorem as Ȧ is a closed operator. We write z Id Ḣ -Ȧ = T + Id Ḣ with T = (z -1) Id Ḣ -Ȧ. By the first part of