Prof Bernhard Ruf

Zongo André

Zongo Inoussa

Harouna Ouedraogo

Sogo Sanon

Zongo Pierre

Alima

Keywords: quasi-linear operators, bifurcation, bifurcation from infinity, multiple solutions, Carleman estimates; stabilization, Lopatinskiȋ-Šapiro, resolvent estimate. Résumé operateurs quasi-linéaires, bifurcation, bifurcation à l'infini, solutions multiples, stabilisation, inégalités de Carleman, inégalité de la reslovente, Lopatinskiȋ-Šapiro. Riassunto operatori quasi lineari, biforcazione, biforcazione da infinito, soluzioni multipli, stabilizzazione, stima di Carleman, stima del resolvente, Lopatinskiȋ-Šapiro

Looking back over these last three years, I feel very impressed when I realize how many interesting people have crossed my life. People that, in many ways, have motivated me and this work. This work would not have been possible without the financial support of the borsa di studio dottorato di ricerca di Università degli studi di Milano (Dipartimento di Matematica Federigo Enriques). It is a pleasure for me to express deep

General introduction

This thesis is done in cotutelle and consists of two parts: Part A and Part B. The first part is related to the analysis of a nonlinear eigenvalue value problem for quasi-linear operators and its applications to bifurcation. Those results were obtained during my stay at the Università degli Studi di Milano in Italy. The second part is related to control theory, more especially the stabilization of the damped plate equation. The latter results were obtained during my stay at the Université Sorbonne Paris Nord in France. The goal of Part A is to study the asymptotic behavior of the spectrum and the existence of multiple solutions of the following eigenvalue problem

   -∆ p u -∆u = λu in Ω, u = 0 on ∂Ω, (1.0.1)
where p ∈ (1, ∞)\{2} is a real number. The goal of Part B is to answer the following question: How fast does the energy of the following hyperbolic equation decrease?

       ∂ 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 ,
where α ≥ 0 and where B 1 and B 2 denote two boundary differential operators.

Part A: Nonlinear eigenvalue problems

Nonlinear eigenvalue problems arise in many areas of computational science and engineering, including acoustics, control theory, fluid mechanics and structural engineering. We give the formulation of such a problems.

Description

Nonlinear eigenvalue problems are in general described by equation of the form

G(λ, u) = 0 λ ∈ K, u ∈ X, (1.1.1)
where K = R or C, and X is a Banach space that can be the n-space R n of C n .

In equation (1.1.1), G is a continuous map of K × X into X. It is assumed that G(λ, 0) = 0 for all λ, that is, u = 0 solves trivially equation (1.1.1) for all scalars λ. Then, one looks for those λ ′ s, i.e, the eigenvalues of G, such that equation (1.1.1) has a solution u ̸ = 0 (an eigenvector or eigenfunction of G corresponding to λ).

A typical example of nonlinear eigenvalue problem is the so-called p-Laplace equation    -div(|∇u| p-2 ∇u) = λ|u| p-2 u in Ω u = 0 on ∂Ω, (1. 1.2) where p > 1 and Ω is a bounded open set in R n . Let X be the Sobolev space W 1,p 0 (Ω). Let X ′ = W -1,p ′ (Ω) be the dual space of X. A weak solution of the p-Laplace equation (1.1.2) is a function u ∈ X such that

Ω |∇u| p-2 ∇u • ∇v dx = λ Ω |u| p-2 uv dx,
for all v ∈ W 1,p 0 (Ω). The proof of the existence of countably many eigenvalues and eigenfunctions of equation (1.1.2) relies on the Lusternik-Schnirelmann theory of critical points for an even functional on a symmetric manifold. Presentations of this theory in both finite and infinite dimensional spaces can be found among others in [START_REF] Ambrosetti | Nonlinear analysis and semilinear elliptic problems[END_REF][START_REF] Kung-Ching | Methods in Nonlinear Analysis[END_REF].

Bifurcation from the eigenvalues of the p-Laplacian

For instance to study the bifurcation phenomena associated with the p-Laplacian under Dirichlet boundary conditions, one considers the following problem

   -∆ p u = λ|u| p-2 u + h(x, u, λ) in Ω u = 0 on ∂Ω, (1.2.1)
where h : Ω×R×R → R satisfies a Carathéodory condition in the two variables (x, t) and h(x, t, λ) = o(|t| p-1 ) near t = 0, uniformly a.e with respect to x and uniformly with respect to λ on bounded sets. We recall that -∆ p u := -div(|∇u| p-2 ∇u) for all p > 1. By a solution of (1.2.1) we understand a couple (λ, u) ∈ R × W 1,p 0 (Ω) satisfying the following integral equality in the weak sense,

Ω |∇u| p-2 ∇u • ∇v dx = λ Ω |u| p-2 uv dx + Ω h(x, u, λ)vdx,
for all v ∈ W 1,p 0 (Ω). We say that (λ * , 0) is a bifurcation point of (1.2.1) if in any neighborhood of (λ * , 0) in R × W 1,p 0 (Ω) there is a nontrivial solution of (1.2.1). One speaks of bifurcation from trivial solution. In [START_REF] Ma | Global bifurcation from the eigenvalues of the p-Laplacian[END_REF] Proposition 2.1, it is shown with a compactness argument that a necessary condition for (λ * , 0) to be a bifurcation point of (1.2.1) is that λ * be an eigenvalue of    -∆ p u = λ|u| p-2 u in Ω u = 0 on ∂Ω.

Let λ 1 (p) denote the first eigenvalue of (1.1.2). We note that λ 1 (p) can be characterized variationally as

λ 1 (p) = inf Ω |∇u| p dx : u ∈ W 1,p 0 (Ω), Ω |u| p dx = 1 .
In [START_REF] Ma | Global bifurcation from the eigenvalues of the p-Laplacian[END_REF] Theorem 1.1, it is shown that (λ 1 (p), 0) is a bifurcation point of (1.2.1). This result is well known in the case p = 2 (see, [START_REF] Rabinowitz | Some aspects of nonlinear eigenvalue problems[END_REF]). The key ingredient in the proof is the index formula which is proved via a suitable homotopic deformation from a general p > 1 to p = 2.

Introducing the following change of variable w = u/∥u∥ 2 1,p for u ∈ W With this transformation, we have that the pair (λ * , ∞) is a bifurcation point for the problem (1.2.1) if and only if the pair (λ * , 0) is a bifurcation point for the problem (1.2.2). One says that the point (λ * , ∞) is a bifurcation point from infinity for problem (1.2.1).

We write equation (1.1.1) in the form

G(λ, u) = u -λLu -K(λ, u), (1.2.3)
where L is a compact linear operator and K : R × X → X is compact with K = o(∥u∥ X ) at u = 0 uniformly on bounded λ intervals. In this context, Krasnoselski [START_REF] Krasnoselski | Topological Methods in the Theory of Nonlinear Integral Equations[END_REF] has shown that if µ is a real characteristic value of L of odd multiplicity, then (µ, 0) is a bifurcation point for G. With a suitable change of variable, one shows that if L is compact and linear, µ is a real characteristic value of L of odd multiplicity, K = o(∥u∥ X ) at u = ∞ uniformly on bounded λ intervals and is appropriately compact, then µ is a bifurcation point for u = λLu + K(λ, u). We will address a problem in the form (1.2.3), which corresponds to the first results concerning Part A given in Section 3.3.1.

Main results of Part A

In this section we give the main results concerning the bifurcation from the trivial solutions and from infinity that are fully presented in Section 3.7. In addition we state the result about the existence of multiple solutions, also presented in Section 3.8.

Bifurcation results

We set S λ (u) = u -λ(-∆ p -∆)u, for u ∈ L 2 (Ω) ⊂ W -1,p ′ (Ω) and λ > 0. We recall that λ D k stands for the k-th Dirichlet eigenvalue of the Laplacian. with odd multiplicity is a bifurcation point in R * + × W 1,p 0 (Ω) of S λ (u) = 0, in the sense that in any neighborhood of (λ D k , 0) in R * + × W 1,p 0 (Ω) there exists a nontrivial solution of S λ (u) = 0.

Introducing a suitable change of variable in equation (1.0.1), leads us to set Sλ (u) = u -λ(-∥u∥ γ 1,2 ∆ p -∆)u with γ = 4 -p for 1 < p < 2, λ > 0 and u ∈ B r (0) ⊂ L 2 (Ω) ⊂ W -1,2 (Ω). With the change of variable, Theorem 1.3.2 is equivalent to the following theorem.

Theorem 1.3.3. The pair (λ D 1 , 0) is a bifurcation point in R + × L 2 (Ω) of Sλ (u) = 0, for 1 < p < 2.

A more general result of Theorem 1.3.3 is the following. Theorem 1.3.4. The pair (λ D k , 0) (k > 1) is a bifurcation point of Sλ (u) = 0 for 1 < p < 2 if λ D k is of odd multiplicity.

Multiple solutions

We have obtained the following result.

Theorem 1.3.5. Let 1 < p < 2 or 2 < p < ∞, and suppose that λ ∈ (λ D k , λ D k+1 ) for any k ≥ 1. Then the equation (1.0.1) has at least k pairs of nontrivial solutions.

Part B: Control theory

Control theory is an important subject in science engineering. Control theory deals with the behaviour of dynamic systems and how to control such systems. More precisely it deals with a dynamical system on which one can act by using suitable controls. We refer to the book of Jean-Michel Coron [START_REF] Coron | Control and nonlinearity[END_REF] for more details on the notions of control theory.

Some different notions of control

We consider a system of differential or partial differential equation of the form

   ∂ t u = K(u, f ) u(0) = u 0 , (1.4.1)
where ∀t ≥ 0, u(t) is the sought solution belonging to a certain state-space H (a Banach or Hilbert space) K an operator that describes the system, and f a source term acting on the system. Natural questions are : knowing f and the initial data u 0 , is it possible to recover the solution u? Is this continuous with respect to the data of the problem ? In control theory, problematics are different. Here, we consider the question of controllability: given the initial data u 0 as well as a couple of state-time target (u T , T ), can we find a control f belonging to some control space X such that the solution u to (1.4.1) satisfies u(T ) = u T ? This means, controlling the solution so that it reaches a desired state at the desired time. We recall some notions of controllability. Definition 1.4.1. (exact controllability) Let T > 0. We say that the control system (1.4.1) is exactly controllable in time T, if for every u 0 ∈ H and for any target state u T ∈ H, there exists a control f ∈ L 2 ((0, T ); X ) such that the solution u of the Cauchy problem (1.4.1) satisfies u(T ) = u T . Definition 1.4.2. (approximate controllability) Let T > 0. We say that the control system (1.4.1) is approximately controllable in time T, if for every u 0 ∈ H and for any target state u T ∈ H, and for every ε > 0, there exists a control f ∈ L 2 ((0, T ); X ) such that the solution u of the Cauchy problem (1.4.1) satisfies ∥u(T ) -u T ∥ H < ε.

Definition 1.4.3. (null controllability) Let T > 0. We say that the control system (1.4.1) is null controllable in time T if for every u 0 ∈ H, there exists a control f ∈ L 2 ((0, T ); X ) such that the solution u of the Cauchy problem (1.4.1) associated to f satisfies u(T ) = 0.

In the case of a system of linear ordinary differential equations in finite dimension of the form   

∂ t u = Au + Bf u(0) = u 0 , (1.4.2)
where A and B are matrices of respective sizes n × n and n × m, u(t) ∈ R n , and f (t) ∈ R m , these three notions of controllability are equivalent. This is not the case in the context of partial differential equations. A necessary and sufficient exact controllability criterion for the finite-dimensional system (1.4.2) exists: the so-called Kalman criterion. In this theorem, we remark that the criterion does not depend on the chosen time T > 0, so we can deduce the following: if the system is controllable in some time T > 0, then it is controllable for any time T ′ > 0. However, this criterion does not specify how the control depends on the time. In the context of partial differential equations, some control systems are not exactly controllable in any time T . Some are exactly controllable if T is chosen sufficiently large. Then one speaks of minimal time of control. This is in articular the case of hyperbolic equations since the influence of the control is limited by the finite speed of propagation. These notions of control have some weakness: they depend strongly on the initial data, and do not depend on the state of the system when 0 < t < T. It is often important to consider problems with feedback, where the control acts on the system by responding continuously to the system (which does not undergo any interruption in time). It is particularly the case when studying stablization problems. Such systems can be written as

   ∂ t u = K(u, f (u)) u(0) = u 0 .
(1. 4.3) Defining an energy depending on the solution u of (1.4.3), we are then interested in the decay properties of this energy. When the control function f depends on the solution u and when the system becomes dissipative (for instance if absorbing boundary conditions or damped terms are involved), the energy is a positive time-decreasing function. Therefore, we study the long time asymptotic behavior of the energy. In particular, the choice of various regularity for the initial data and/or geometrical hypotheses gives different estimates for the decay rate of the energy.

In general the proof of the controllability of a linear system relies on the proof of an observability criterium.

From control to observability

Here, we present how the null controllability can be reduced to obtaining a functional inequality, known as observability, by a duality argument. These inequalities play a central role in the study of control and stabilization of partial differential equations. Indeed, constructing a control is in general a difficult problem. On the other hand, there are many existing tools that can be used to show functional inequalities, such as energy inequalities, Carleman inequalities, Ingham inequalities, or Gårding's inequalities.

We place ourselves in the following case

   ∂ t u = Au + Bf, u(0) = u 0 ∈ H, (1.5.1)
where A is an operator acting on a Hilbert space H with domain D(A) ⊂ H. We suppose that A is the generator of a strongly continuous semigroup, denoted S(t). We assume that the control operator B acts on the control space X (also a Hilbert space) and in addition we assume that B ∈ L(X , H) for simplicity. Note that this condition can be relaxed, then we must replace the continuity of the operator B by the so-called admissibility condition (see [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF] for more details). This is particularly useful if one wishes to control a partial differential equation from the boundary of the domain. One acts on the system by means of the operator B, and in general, it restricts the possibilities of action. The definition of the null controllability of system (1.5.1) is given by Definition 1.4. [START_REF] Alinhac | Pseudo-differential Operators and the Nash-Moser Theorem[END_REF]. With the assumptions made on S(t) and B, we can write the solution of (1.5.1) with the Duhamel formula u(t) = S(t)u 0 + t 0 S(t -s)Bf (s)ds.

(1.5.2)

Let B * ∈ L(H, X ) be the adjoint of B (the dual spaces H ′ and X ′ are identified to H and X respectively). We introduce here the following dual system on

(0, T ),    ∂ t v = -A * v, on (0, T ) v(T ) = v T ∈ H, (1.5.3)
where, A * is the adjoint operator of A in H. The generator of the adjoint semigroup S * (t) = (e tA ) * is A * , and the solution of (1.5.3) can be written v(t) = e (T -t)A * v T , where e (T -t)A * = (e (T -t)A ) * . Note that (1.5.3) is homogeneous. The question of the observability is the following : Is it possible, by observing only the quantity B * v(t), to know the energy of the system (1.5.3) at time t = 0, i.e, ∥v(0)∥ 2 H ? The observability notion of system (1.5.3) is given by the following definition.

Definition 1.5.1. One says that the system (1.5.3) is observable at time T > 0 if there exists a constant C Obs,T > 0 such that for every v T ∈ H, the solution of (1.5.3) satisfies

∥S * (T )v T ∥ H ≤ C Obs,T ∥B * v(t)∥ L 2 ((0,T );X ) = C Obs,T T 0 ∥B * v(t)∥ 2 X dt 1/2
.

(1.5.4)

We note that S * (t)v T = v(0), where v is the solution of (1.5.3). The constant C Obs,T is called the constant of observability. This notion of observability has its own interest because it appears in many concrete situations when one would like to know the state of a system on which one can only make partial measurements. This is the case for instance in meteorology, in imaging or, more generally, in the field of inverse problems. Another interest of the observability for system (1.5.3) lies in its link with the null controllability of the initial system (1.5.1). We then have the following result, proved by S. Dolecki and D. L. Russell [START_REF] Dolecki | A general theory of observation and control[END_REF], and J.-L. Lions [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF].

Theorem 1.5.2. The null controllability at time T > 0 of the system (1.5.3) is equivalent to the inequality (1.5.4) with a constant C Obs,T > 0. Moreover if (1.5.4) holds with constant C Obs,T , then one can find a control f ∈ L 2 ((0, T ), X ) satisfying ∥f ∥ L 2 ((0,T ),X ) ≤ C Obs,T ∥u 0 ∥ H .

We also note that the exact controllability of system (1.5.1) is equivalent to the observability of system (1.5.3). More precisely, in this case the operator S * (T ) is replaced by the identity Id in the left part of inequality (1.5.4).

Example of stabilization: case of the wave equation

We give an example concerning the stabilization of the wave equation. Let Ω be a smooth open bounded subset of R d and ω an open subset of Ω. We consider the following wave equation

       ∂ 2 t u -∆u + α(x)∂ t u = 0 on [0, ∞) × Ω u | t=0 = u 0 , ∂ t u | t=0 = u 1 , u | ∂Ω = 0, (1.6.1)
where α(x) is a nonnegative bounded function that satisfies α(x) ≥ C > 0, for x ∈ ω. Multiplying (1.6.1) by ∂ t u and integrating over Ω, we obtain

Ω ∂ 2 t u ∂ t u dx + Ω ∇ x u∇ x ∂ t u dx + Ω α(x)∂ t u ∂ t u dx = 0, which implies that 1 2 d dt ∥∂ t u∥ 2 L 2 (Ω) + ∥∇ x u∥ 2 L 2 (Ω) = -∥α(x) 1/2 ∂ t u∥ 2 L 2 (Ω) .
Introducing the H 1 -energy

E(u, t) := 1 2 ∥∂ t u∥ 2 L 2 (Ω) + ∥∇ x u∥ 2 L 2 (Ω) ,
one finds d dt E(u, t) = -∥α(x) 1/2 ∂ t u∥ 2 L 2 (Ω) ≤ 0.

One calls α(x)∂ t u the damping term; it is the responsible for the decay of the energy. One refers to (1.6.1) as the damped wave equation. One proves that the energy decays to zero. A natural question is thus the study of the rate of convergence of this energy, i.e., to obtain an estimate of the type

E(u, t) ≤ h(t)G(u 0 , u 1 ),
where h is a decreasing function that tends to zero at infinity and G a function.

The weak stabilization consists in showing that for any (u 0 , u 1 ) in a suitable space, lim t→+∞ E(u, t) = 0 and the strong stabilization consists to show, under suitable conditions the existence of C > 0 and K > 0 such that for any (u 0 , u 1 ) in a suitable space, we have a uniform and exponential decay rate E(u, t) ≤ Ke -Ct E(u, 0).

The exponential decay of the energy can be achieved if the so-called geometrical control condition (GCC) is fulfilled. This result was first proven in dimension one or on manifolds without boundary by [80, J. Rauch and M. Taylor]. The geometrical control condition expresses that all bicharacteristics (or rays of geometrical optics ) reach the damping region in a finite time. The generalization of this exponential decay result to domains with boundaries in the case of homogeneous Dirichlet and Neumann conditions was proven [57,58, C. Bardos , G. Lebeau and J. Rauch]. Bicharacteristics are then replaced by so-called generalized bicharacteristics that obey the laws of reflection at a boundary. There are introduced by R. Melrose and J. Sjöstrand [START_REF] Melrose | Singularities of boundary value problems[END_REF][START_REF] Melrose | Singularities of boundary value problems[END_REF] to describe the propagation of singularities. The proof in [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF] is based on this description of propagation of singularities.

Note that exponential decay of the energy is equivalent to having an observability estimate for the wave equation (without damping), the observation being located in the region where the damping acts; we refer to the work of A. Haraux [START_REF] Haraux | Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps[END_REF]. For the wave equation, stabilization (and equivalently observability) can also be expressed by means of the so-called Hautus test for the resolvent; we refer to the works of D. Russel and G. Weiss [START_REF] Russell | A general necessary condition for exact observability[END_REF] and L. Miler [START_REF] Miller | Refraction of high-frequency waves density by sharp interfaces and semiclassical measures at the boundary[END_REF]. Under weaker geometrical conditions one can obtain a polynomial decay rate of the damped wave equation. We shall be interested in such weaker decay rates here.

Carleman estimates

In 1939, T. Carleman introduced some energy estimates with exponential weights to prove a uniqueness result for some elliptic partial differential equations (PDE) with smooth coefficients in dimension two [START_REF] Carleman | Sur un problème d'unicité pour les systèmes d'équations aux dérivées partielles à deux variables indépendantes[END_REF]. This type of estimate, now referred to as Carleman estimates, were generalized by L. Hörmander and others for a large class of differential operators in arbitrary dimensions (see, [START_REF] Hörmander | Linear Partial Differential Operators[END_REF][START_REF] Hörmander | The Analysis of Linear Partial Differential Operators IV[END_REF][START_REF] Zuily | Uniqueness and Non Uniqueness in the Cauchy Problem[END_REF]).

Carleman estimates are weighted a priori inequalities for the solutions of a partial partial differential equation of the form

∥e τ φ u∥ L 2 (Ω) ≤ ∥e τ φ P u∥ L 2 (Ω) , (1.7.1)
where P is a differential operator, u a function, φ a function called the weight function and τ > 0 a large parameter.

The interest of such inequalities is the presence of the weight function φ which allows to "propagate" information from areas where φ is large to the whole domain, by means of the large parameter τ > 0, known as large Carleman parameter. 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[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.

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 (e.g. coefficient, source term) with respect to norms on measurements performed on the solution of the PDEs see e.g. [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] Kubo | Uniqueness in inverse hyperbolic problems-Carleman estimate for boundary value problems[END_REF][START_REF] Isakov | Inverse problems for partial differential equations[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; through unique continuation properties, Carleman estimates are used for the exact controllability of hyperbolic equations [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF]. They also 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].

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 4.5.2 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 = ∆ 2 -σ 4 , with some uniformity with respect to σ. The main results concerning Part B are the following.

Main results of Part B

We state the main Carleman estimate for the operator P σ in normal geodesic coordinates as presented in Section 4.5.2.

Carleman estimate

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) satisfies the sub-ellipticity condition of Definition 4.9.1 locally. This is a necessary and sufficient condition for a Carleman estimate to hold for a second-order operator -∆±σ 2 to hold regardless of boundary conditions [START_REF] Rousseau | Elliptic Carleman estimates and applications to stabilization and controllability[END_REF]Chapters 3 and 4].

(

) (P σ , B 1 , B 2 , φ) satisfies the Lopatinskiȋ-Šapiro condition of Definition 4.7.1 at ϱ ′ = (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
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 1.8.1 (Carleman estimate). 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 u |x d =0 + | 7/2-k j ,τ , (1.8.1) 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 .
Observe that the Carleman estimate of Theorem 1.8.1 exhibits a loss of a half-derivative. A more precise statement is given in Theorem 4.10.3 in Section 4.10.2.

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 (1.8.2)
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 (4.5.1). A precise definition of strong solutions is given in Section 4.12.3. 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 1.8.2 (logarithmic stabilization for the damped plate equation).

There exists C > 0 such that for any such strong solution to the damped plate equation (4.5.1) 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 4.13.3 in Section 4.13.2.

Some elements of functional analysis

General facts

We deal with functions u defined almost everywhere (a.e.) on an open bounded domain

Ω ⊂ R d . A point x ∈ Ω is written x = (x 1 , • • • , x d ).
The boundary of Ω will be denoted by ∂Ω. We suppose that ∂Ω is sufficiently smooth, let say at least of class C 2 .

2.1.1 Functions space and definitions.

{u n } n∈N is a Cauchy in X ⇒ ∃u ∈ X such that u n → u as n → ∞.
For functions u (measurable) defined a.e. in Ω, the following spaces will be considered.

(i) The Lebesgue spaces L p (Ω), 1 ≤ p ≤ ∞, where

L p (Ω) = u : Ω → R ; Ω |u| p dx < ∞ , for 1 ≤ p < ∞. We denote by ∥u∥ p = Ω |u(x)| p dx 1 p the norm on L p (Ω).
For p = ∞, by L ∞ (Ω) we denote the space of measurable functions u which are essentially bounded over Ω equipped with the norm

∥u∥ ∞ = inf{M ∈ R + : |u(x)| ≤ M a.e. in Ω}. (2.1.1)
The norms that make L p (Ω) Banach space are respectively ∥.∥ p and ∥.∥ ∞ .

The space L p (Ω) is reflexive for 1 < p < ∞ with its dual denoted by

L p ′ (Ω) with 1 p + 1 p ′ = 1.
(ii) By C ∞ c (Ω) we denote the set of all functions u defined and infinitely differentiable on Ω such that their support supp(u) is compact and satisfies supp(u) ⊂ Ω. Recall that supp(u) is the closure of the set {x ∈ Ω; u(x) ̸ = 0}.

(iii) For α a multi-index, |α| ≥ 1, the function v α is called a weak (or distributional) derivative of u (of order α) if the identity

Ω v α (x)φ(x) dx = (-1) |α| Ω u(x)D α φ(x) dx holds true for every φ ∈ C ∞ c (Ω). Then v α is denoted by D α u.
(iv) Sobolev spaces. For k ∈ N and 1 ≤ p < ∞ we denote by W k,p (Ω) the set of all functions u ∈ L p (Ω) for which the weak derivatives D α u with |α| ≤ k exists in the weak sense and belongs to L p (Ω) as well.

The Sobolev space W k,p (Ω) is a Banach space (uniformly convex and hence reflexive if 1 < p < ∞) if equipped with the norm

∥u∥ k,p =   |α|≤k ∥D α u∥ p p   1 p . (2.1.2)
Further, the space W k,p 0 (Ω) is defined as the closure of C ∞ c (Ω) with respect to the norm ∥.∥ k,p . For Ω bounded the expression

∥u∥ k,p =   |α|=k ∥D α u∥ p p   1 p
is a norm on W k,p 0 (Ω) equivalent to the one defines in (2.1.2). This is based on the Poincaré inequality, see below Proposition 2.1.4.

Some definitions

Definition 2.1.2. Let X be a Banach space. A sequence {u n } n∈N ⊂ X converges weakly to u ∈ X, in which case we write u

n ⇀ u in X if f (u n ) → f (u) for all f ∈ X ′ (the dual space of X). In addition if u n ⇀ u in X, then u n is bounded in X and ∥u∥ X ≤ lim inf n→∞ ∥u n ∥ X .
Definition 2.1.3. One says that the sequence u n converges strongly to u in X, in which case we note u n → u in X, if u n , u ∈ X and if

lim n→∞ ∥u n -u∥ X = 0. Definition 2.1.4. Let x be an element of the open subset U ⊂ X. The mapping F : U → Y is Fréchet-differentiable at x ∈ U if there exists a linear operator A ∈ L c (X, Y ), such that F (x + h) -F (x) -Ah = o(∥h∥) with F ′ (x) = A.
We use the notation r(h) = o(∥h∥ X ) for the mapping r : X → Y if and only if

lim h→0 ∥r(h)∥ Y ∥h∥ X = 0,
which means that for every ε > 0 there exists δ > 0 such that if ∥h∥ X < δ then ∥r(h)∥ Y < ε∥h∥ X .

Definition 2.1.5. Let X, Y be a Banach spaces and U ⊂ X a sub-open set of X. Let F : U → Y be a mapping and x ∈ X. We say that F is Gâteauxdifferentiable at x if there exists A ∈ L c (X, Y ), such that

lim t→0 F (x + th) -F (x) t = Ah, for all h ∈ U .
The mapping A is uniquely determined. It is call Gâteaux-derivative of F at x and is denoted by

F ′ G (x). If Y = R, F is said to be a functional.
Definition 2.1.6. Let X be a Banach space, U ⊆ X an open set and assume that I :

U → R is differentiable. A critical point of I is a point u ∈ U such that I ′ (u) = 0. As I ′ (u)
is an element of the dual space X ′ , this means of course ⟨I ′ (u), v⟩ = 0 for all v ∈ X. Then they satisfy the integration by parts formula

Ω u(x) ∂v ∂x i (x) dx = - Ω v(x) ∂u ∂x i (x) dx + ∂Ω u(x)v(x)ν i (x) dS. ( 2
, q ∈ [1, ∞] such that 1 p + 1 q = 1. Let f ∈ L p (Ω), g ∈ L q (Ω), then ∥f g∥ 1 ≤ ∥f ∥ p ∥g∥ q .
Proposition 2.1.10. (Poincaré inequality) Let Ω be a bounded domain.

There exists a positive constant C p such that, for every u ∈ W

i. W 1,p (Ω) ⊂⊂ L q (Ω), ∀ q ∈ [1, p ⋆ ), where 1 p ⋆ = 1 p -1 d , if p < d. ii. W 1,p (Ω) ⊂⊂ L q (Ω), ∀ q ∈ [p, ∞), if p = d. iii. W 1,p (Ω) ⊂⊂ C(Ω) if p > d.
We have the following theorem.

Theorem 2.1.12. Suppose that f : For almost all x ∈ Ω we set f

U → R has a continuous Gâteaux-derivative on U. Then f is Fréchet-differentiable and f ∈ C 1 (U, R).
(x) = lim inf n→∞ f n (x) < ∞. Then f ∈ L 1 (Ω) and Ω f dx ≤ lim inf n→∞ Ω f n dx.

Linear operators in Banach spaces

Here, X and Y will denote Banach spaces with their norms denoted by

∥ • ∥ X , ∥ • ∥ Y or simply ∥ • ∥ if no confusion is possible.
An operator A from X to Y is a linear map on its domain. One denotes by D(A) the domain of A. An operator from X to Y is thus characterized by its domain and how it acts on this domain. Operators defined this way are usually referred to as unbounded operators. One writes (A, D(A)) to denote the operator along with its domain. The set of linear operators from X to Y is denoted by L (X, Y ). The set of linear and continuous operator from X to Y will be denoted by L c (X, Y ) If D(A) is dense in X the operator A is said to be densely defined. If D(A) = X one says that the operator A is on X to Y.

The range of the operator A is denoted by Ran(A), that is

Ran(A) = {Ax : x ∈ D(A)} ⊂ Y
and its kernel, Ker(A), is the set of all x ∈ D(A) such that Ax = 0. The graph of A, G (A) is given by

G (A) = {(x, Ax) : x ∈ D(A)} ⊂ X × Y.
We naturally endow X × Y with the norm ∥(x, y)∥ X×Y = ∥x∥ 2 X + ∥y∥ 2 Y which makes X × Y a Banach space. One says that A is a closed operator if its graph G (A) is a closed subset of X × Y for this norm. The so-called graph norm on D(A) is given by

∥x∥ 2 D(A) = ∥(x, Ax)∥ 2 X×Y = ∥x∥ 2 X + ∥Ax∥ 2 Y .
The operator A is closed if and only if the space D(A) is complete for the graph norm

∥ • ∥ D(A) . If a linear operator A from X to Y is injective, one can define the operator A -1 from Y to X such that D(A -1 ) = Ran(A), Ran(A -1 ) = D(A), A -1 A = Id D(A) , AA -1 = Id Ran(A) .
One says that A is invertible and A -1 is called the inverse operator.

Definition 2.1.14. An linear operator A from X to Y is said to be continuous if it is continuous at every x ∈ D(A) or equivalently if it is continuous at x = 0. This is equivalent to having K > 0 such that ∥Ax∥ Y ≤ K∥x∥ X for all x ∈ D(A).

One says that A is a bounded operator.

The positive number

K = sup x∈D(A) x̸ =0 ∥Ax∥ Y ∥x∥ X
is called the bound of A, and denoted by ∥A∥ L (X,Y ) or simply ∥A∥. Note that linear operator from X to Y that fails to be continuous are such that

sup x∈D(A) x̸ =0 ∥Ax∥ Y ∥x∥ X = +∞.
This justifies the name unbounded for general linear operators from X to Y .

Theorem 2.1.15. (closed-graph theorem) Let A be such that D(A) is a closed linear subspace in X. Then, A is bounded if and only if A is a closed operator.

For a proof we refer to [START_REF] Kato | Perturbation Theory for Linear Operators[END_REF].

Spectrum of linear operator in a Banach space

We consider here a linear operator from X to itself. One says that λ ∈ C is in the resolvent set ρ(A) of an linear operator A from X to X if the operator λId -A is injective and the inverse operator (λId -A) -1 has a dense domain

D((λId -A) -1 ) = Ran(λId -A) in X and is bounded. If λ ∈ ρ(A) then we set R λ (A) = (λId -A) -1 .
The spectrum is then simply the complement set of ρ(A) in C. We denote by sp(A) the spectrum of A.

The spectrum of a linear operator is othen separated in three disjoints sets (a) The point spectrum that gathers all λ ∈ C such that the operator λId -A is not injective. Such a complex number λ is called an eigenvalue of A and the (finite or infinite) dimension of the kernel ker(λId -A) is the the geometric multiplicity associated with this eigenvalue. An element of ker(λId -A) is called an eigenvector or an eigenfunction in the case the Banach X is a function space.

(b) The continuous spectrum that gathers all λ ∈ C such that the operator λId -A is injective, has a dense domain but its inverse R λ (A) is not bounded.

(c) The residual spectrum that gathers all λ ∈ C such that the operator λId -A is injective but does not have a dense image.

Monotone mappings

Here X ′ denote the dual space of X.

Definition 2.1.16. Let X be a real Banach space. A mapping B :

X → X ′ is called hemi-continuous at x 0 in X , if for all y in X, ∀s n ↓ 0 with x 0 + s n y ∈ X, imply that B(x 0 + s n y) ⇀ Bx 0 . It is called demi-continuous at x 0 ∈ X, if for all x n ∈ E, x n → x 0 in X implies that Bx n ⇀ Bx 0 ,
where ⇀ is the weak convergence.

We observe that "continuous" ⇒ "demi-continuous" ⇒ "hemi-continuous" .

Definition 2.1.17. Let X be a real Banach space. A mapping B : X → X ′ is called monotone if ⟨Bx -By, x -y⟩ ≥ 0, for all x, y ∈ X.

An important property for monotone mappings reads as follows.

Lemma 2.1.18. Let E be a convex subset of a real Banach space X. If B : E ⊂ X → X ′ is hemi-continuous and monontone, then for any sequence x n ∈ E with x n ⇀ x ∈ E and lim⟨Bx n , x n -x⟩ ≤ 0, we have

lim⟨Bx n , x n -y⟩ ≥ ⟨Bx, x -y⟩ ∀y ∈ E.
Proof. Since A is monotone, we have ⟨Bx n -Bx, x n -x⟩ ≥ 0 and this implies,

0 = lim⟨Bx, x n -x⟩ ≤ lim⟨Bx n , x n -x⟩ ≤ lim⟨Bx n , x n -x⟩ ≤ 0 (by assumption).
We then obtain that lim⟨Bx n , x n -x⟩ = 0.

(2.1.5)

Again by monotonicity and (2.1.5), for all z ∈ E we have

lim⟨Bx n , x -z⟩ = lim⟨Bx n , x n -z⟩ ≥ lim⟨Bz, x n -z⟩ = ⟨Bz, x -z⟩. (2.1.6)
For all z ∈ E and s k ∈ (0, 1) we set z = z k := (1 -s k )x + s k y and substituting this in (2.1.6), it follows that

lim⟨Bx n , x -y⟩ ≥ ⟨Bz k , x -y⟩. (2.1.7)
Now thanks to the hemi-continuity, the right hand side of (2.1.7) converges to ⟨Bx, x -y⟩ as s k → 0. Combining (2.1.5)-(2.1.7) together with the last fact, it follows that lim⟨Bx n , x n -y⟩ ≥ ⟨Bx, x -y⟩.

The following notion on pseudo-monotonicity is abstracted from the combination of the monotonicity and the hemi-continuity. Definition 2.1.19. Let X be a reflexive Banach space and let E ⊂ X be a nonempty closed convex subset. An operator

B : E → X ′ is called pseudo- monotone, if (a) For all finite-dimensional linear subspace L ⊂ X, B| L∩E : L ∩ E → X ′ is demi-continuous. (b) For all sequence {x n } ⊂ E with x n ⇀ x ∈ E, the condition lim⟨Bx n , x n - x⟩ ≤ 0 implies lim⟨Bx n , x n -y⟩ ≥ ⟨Bx, x -y⟩, ∀y ∈ E.
One sees that a hemi-continuous monotone operator is pseudo-monotone. Moreover, a completely continuous mapping B : X → X ′ ( that is, for any

x n ⇀ x in X, we have Bx n → Bx in X ′ ) is pseudo-monotone.
Theorem 2.1.20. (F. Browder) Suppose that X is a reflexive Banach space, and that B : X → X ′ is pseudo-monotone and coercive, i.e., lim ∥x∥→+∞ ⟨Bx,x⟩ ∥x∥ = +∞. Then A is surjective.

Proof. We need to show that ∀z ∈ X ′ , ∃x 0 ∈ X such that Bx 0 = z. We define T : x → Bx -z. Then, T is hemi-continous and monotone, and so T is pseudomonotone. In addition T satisfies ⟨T x, x⟩ > 0 as ∥x∥ > R 0 , for some R 0 > 0, provided by the coerciveness of B. We apply the Hartman-Stampacchia theorem [START_REF] Kung-Ching | Methods in Nonlinear Analysis[END_REF]Theorem 2.5.7] to conclude the existence of x 0 ∈ X satisfying ⟨T x 0 , x 0 -y⟩ = ⟨Bx 0 -z, x 0 -y⟩ ≤ 0 for all y ∈ X. Since y is arbitrary in the linear space X, it follows that Bx 0 = z. Corollary 2.1.21. Suppose that H is a real Hilbert space, and that B is a continuous strongly operator, i.e., ∃C > 0 such that (Bx -By, x -y) ≥ C∥x -y∥ 2 ∀x, y ∈ H.

(2.1.8)

Then B is a homeomorphism.

Proof. We clearly have that B is pseudo-monotone and coercive. As the consequence of Theorem 2.1.20, B is surjective. The injectivity of B as well as the continuity of B -1 follows from the inequality (2.1.8).

Fredholm operators

We shall denote by B(X, Y ) the set of bounded operators A on X to Y, that is, such that D(A) = X. Once we speak of a bounded operator A : X → Y without any mention of its domain, this means that D(A) = X, that is, A is on X to Y. Let A be a linear closed operator from X to Y. The nullity of A, denoted nul(A), is defined as the dimension of ker(A) and the deficiency of A, denoted def(A), is defined as the dimension of Y /Ran(A). Both nul(A) and def(A) take value in N ∪ {∞}. iii. both nul(A) and def(A) are finite.

One then sets the index of A as ind(A) = nul(A) -def(A).

Characterization of bounded Fredholm operators

We denote by F rB(X, Y ) the space of Fredholm operators that are bounded on X into Y. The following result states that those operators are the operators in B(X, Y ) that have an inverse up to remainder operators that are compact.

SA = Id X + K p , AS = Id Y + K q , (2.1.9)
where K p ∈ B(X, X) and K q ∈ B(Y, Y ) are compact operators. In particular, S is Fredholm and ind(A) = -ind(S).

For a proof we refer to [START_REF] Rousseau | Elliptic Carleman estimates and applications to stabilization and controllability[END_REF]Theorem 11.7].

Semigroup theory

Semigroup theory is at the centre of the understanding of many evolution equations that can be put in the following form

d dt x(t) + Ax(t) = f (t), t > 0, x(0) = u 0 , (2.2.1)
with x(t) and x 0 in a proper function space, usually a Banach space, denoted by X, if not a Hilbert, with A an unbounded operator on X, with dense domain, and f a function of the time variable t taking values in [0, ∞). We will only review the case of a homogeneous equation, that is f ≡ 0. For general references on semigroups we refer to [START_REF] Davies | One-parameter semigroups. ol. 15 of London Mathematical Society Mono-graphs[END_REF][START_REF] Hille | Functional analysis and semi-groups[END_REF][START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF].

Strongly continuous semigroups

∥S(t)∥ L (X) ≤ M e ωt , for 0 ≤ t < ∞. (2.2.7)
Proof. We first show that ∥S(t)∥ L (X) is bounded. More precisely, we show that there is an µ > 0 such that

sup t∈[0,µ] ∥S(t)∥ L (X) < ∞.
Suppose that ∥S(t)∥ L (X) is not bounded, that is, sup

t∈[0,µ] ∥S(t)∥ L (X) = +∞, for all µ = 1 k with k ∈ N. Therefore for all k ∈ N, there exists t k ∈ [0, 1 k ] such that sup k≥1 ∥S(t k )∥ L (X) = +∞. From the Uniform boundedness theorem ∃x ∈ X : sup k≥1 ∥S(t k )x∥ L (X) = +∞, i.e, ∥S(t k )x∥ L (X) is unbounded. On the other hand, for all x ∈ X, R + ∋ t → S(t)x ∈ X is continuous at 0; that is, ∀ε > 0, ∃N > 0 : |t| < N ⇒ ∥S(t)u -u∥ L (X) < ε.
In particular, letting ε = 1, it follows that ∥S(t)x -x∥ L (X) < 1 and hence we have the estimates

∥S(t)x∥ L (X) -∥x∥ L (X) ≤ ∥S(t)x∥ L (X) -∥x∥ L (X) ≤ ∥S(t)x -x∥ L (X) < 1.
This implies that ∥S(t)x∥ L (X) ≤ 1 + ∥x∥ L (X) . But we have 0 ≤ t k ≤ 1 k and then t k → 0 as k → ∞, and taking N = ε,

∃k 0 ∈ N : |t k | < N ; ∀k ≥ k 0 ⇒ ∥S(t k )x∥ L (X) ≤ 1 + ∥x∥ L (X) .
(2.2.8)

Therefore sup

k≥k 0 ∥S(t k )x∥ L (X) ≤ 1 + ∥x∥ L (X) . For k ∈ {1, . . . , k 0 -1} set K = max ∥S(t k )x∥ L (X)
∥S(t)∥ L (X) < ∞. Let M = sup t∈[0,µ] ∥S(t)∥ L (X) , we have M ≥ 1 since ∥S(0)∥ L (X) = 1.
Set ω = log M µ ≥ 0. Given t ≥ 0 with t > δ, we have t = kµ + δ, where 0 ≤ δ < µ and therefore by the semigroup property

∥S(t)∥ L (X) = ∥S(µ) k S(δ)∥ L (X) ≤ ∥S(µ) k ∥ L (X) ∥S(δ)∥ L (X) ≤ M M k = M M t-δ µ .
This implies that

∥S(t)∥ L (X) ≤ M M t µ = M e ωµ× t µ = M e ωt . Theorem 2.2.3. If (S(t)) t≥0 is a C 0 -semigroup then for all u ∈ X, (t, u) → S(t)u is continous from [0, ∞) × X into X.
A C 0 -semigroup is said to be bounded if there exists M ≥ 1 such that ∥S(t)∥ L (X) ≤ M, for t ≥ 0. In the case M = 1, one says that the the C 0semigroup is of contraction. We define the unbounded linear operator A from X to X, with domain

D(A) = {x ∈ X; lim t→0 + t -1 (x -S(t)x) exists}, (2.2.10)
and given by Ax = lim

t→0 + t -1 (x -S(t)x), x ∈ D(A). (2.2.11)
The domain D(A) is equiped with the graph norm

∥x∥ D(A) = ∥x∥ X + ∥Ax∥ X .
Since A is closed one finds that (D(A), ∥ • ∥ D(A) ) is complete. This operator (A, D(A)) is called the generator of the C 0 -semigroup .

We have the following theorem whose proof can be found in [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF]. For a proof we refer to [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF].

Observe that if S(t) is a C 0 -semigroup and z ∈ C then e zt S(t) statisfies (2.2.6).

We have the following proposition.

Proposition 2.2.6. Let S(t) be a C 0 -semigroup and z ∈ C. Then e zt S(t) is also a C 0 -semigroup and its generator is A -zId X .

Note that because of the uniqueness of the generator of a C 0 -semigroup [76, Theorem 1.2.6], conversely, if A generates a C 0 -semigroup, then A -zId X is the generator of a C 0 -semigroup, namely e zt S(t).

The Hille-Yosida theorem

The natural question is to wonder if an unbounded operator on X is the generator of C 0 -semigroup. The Hille-Yosida theorem is central in the semigroup theory, providing a clear answer to this question. We refer to [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF]Theorem 1.3.1] for a proof. Theorem 2.2.7. Let (A, D(A)) be a linear unbounded operator on a Banach space X. It generates a C 0 -semigroup of contraction if and only if (a) A is closed and D(A) is dense in X.

(b) The resolvent set ρ(A) of A contains (-∞, 0) and we have the following estimate

∥R λ (A)∥ L (X) ≤ 1/|λ|, λ < 0, R λ (A) = (λId X -A) -1 .
This result is limited to contraction C 0 -semigroups. The following corollary provides a charcaterization of all generators of C 0 -semigroups, we refer to [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF]Theorem 1.5.3] for a proof. Corollary 2.2.8. Let (A, D(A)) be a linear unbounded operator on a Banach space X. It generates a C 0 -semigroup S(t) such that ∥S(t)∥ L (X) ≤ M e ωt , for some M ≥ 1 and ω ∈ R, if and only if i. A is closed and D(A) is dense in X.

ii. The resolvent set ρ(A) of A contains (-∞, -ω) and we have the following estimate

∥R λ (A) n ∥ L (X) ≤ 1/|ω + λ| n , λ < -ω, n ∈ N * , R λ (A) = (λId X -A) -1 .
The Hille-Yosida theorem has the following simple consequence.

Corollary 2.2.9. Let (A, D(A)) be the generator of a bounded C 0 -semigroup S(t), that is, ∥S(t)∥ L (X) ≤ M, for t ≥ 0, for some M > 0. Then, its spectrum satisfies sp(A) ⊂ {z ∈ C : Re z ≥ 0}.

Proof. Let c ∈ R, the C 0 -semigroup e ict S(t) is generated by A -icId X . Since e ict S(t) satisfies ∥e ict S(t)∥ L (X) ≤ M, for t ≥ 0, the conclusion follows from Corollary 2.2.8 in the case ω = 0.

The Lumer-Phillips theorem

The Lumer-Phillips theorem provides another characterization of generators of contraction semigroups.

Let X ′ be the dual space of X equipped with the strong topology. For x ∈ X we set

F(x) = {φ ∈ X ′ : φ(x) = ⟨φ, x⟩ X ′ ,X = ∥φ∥ 2 X ′ = ∥x∥ 2 X },
which is not empty by the Hahn-Banach theorem.

Definition 2.2.10. A linear unbounded operator (A, D(A)) on X is said to be monotone (or accretive) if for all x ∈ D(A), x ̸ = 0, there exists φ ∈ F(x) such that Re⟨φ, Ax⟩ X ′ ,X ≥ 0.

Definition 2.2.11. A linear unbounded operator (A, D(A)) on X is said to be maximal monotone if it is monotone and if moreover there exists λ 0 > 0 such that the range of λ 0 Id X + A, Ran(λ 0 Id X + A) = X.

The Lumer-Phillips theorem reads as follows.

Theorem 2.2.12. Let (A, D(A)) be a linear unbounded operator. It generates a C 0 -semigroup of contraction if and only if 1) A has a dense domain.

2) A is maximal monotone

A proof based on the Hille-Yosida theorem directly follows from the two lemmata below.

Remark 2.2.13. Observe that there is no need to assume that the operator A is closed in the converse part of the Lumer-Phillips theorem as in the Hille-Yosida theorem. In fact, as proven below, a maximal monotone operator is closed. In the case of a reflexive Banach space, the dense domain assumption may be dropped in the converse part of the Lumer-Phillips theorem: a maximal monotone operator has a dense domain, see [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF]Theorem 1.4.6] and also [START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Differential Equations[END_REF]Proposition 7.1] for Hilbert space case. This lemma gives a characterization of monotone operators. We refer to [START_REF] Rousseau | Elliptic Carleman estimates and applications to stabilization and controllability[END_REF]Lemma 12.13] for a proof. Lemma 2.2.14. An unbounded operator (A, D(A)) on X is monotone if and only if

∥(λId X + A)x∥ X ≥ λ∥x∥ X , x ∈ D(A) and λ > 0.
(2.2.16)

The value of λ 0 > 0 in Definition 2.2.11 is not of great significance. In fact, we have the following result.

Lemma 2.2.15. Let A be a maximal monotone operator on X. Then, A is closed and for all λ > 0 the operator λId X + A is bijective from D(A) onto X. Moreover, if λ > 0, its inverse (λId X + A) -1 is a bounded operator and we have the following estimation ∥(λId

X + A) -1 ∥ L (X) ≤ 1/λ.
Proof. Let λ > 0, for x ∈ X, we have (λId + A)x = 0 and ∥(λId X + A)x∥ X = 0 implies that λ∥x∥ X = 0 thanks to Lemma 2.2.14. It follows that x = 0 and λId X + A is injective. As A is maximal monotone, there exists λ 0 > 0 such that Ran(λ 0 Id X + A) = X, so λ 0 Id X + A is surjective. Its inverse (λ 0 Id X + A) -1 is thus well defined on X. By Lemma 2.2.14, we have ∥(λ 0 Id X + A) -1 ∥ L (X) ≤ 1 λ 0 . By the closed-graph theorem (see, Theorem 2.1.15), the graph of (λ 0 Id X + A) -1 is closed in X × X and so is the graph of A. We now prove that if λId X + A is surjective then so is λ ′ Id X + A for any λ ′ such that λ > λ ′ /2 > 0. By induction, starting with λ = λ 0 we then reach to the conclusion that λId X + A is surjective for any λ > 0 and then the boundedness of its inverse follows from Lemma 2.2.14. Let λ, λ ′ > 0 be such that λId X + A is surjective and 2λ > λ ′ > 0. Let y ∈ X. We want to find x ∈ X such that (λ ′ Id X +A)x = y. This reads as (λId X +A)x = y + (λ -λ ′ )x and thus we have x = (λId X + A) -1 (y + (λ -λ ′ )x), meaning that we seek fixed point for the bounded map M : x → (λId X + A) -1 (y + (λ -λ ′ )x). By the computation above, we have ∥(λId

X + A) -1 ∥ L (X) ≤ 1 λ , we then find ∥M (x) -M (x ′ )∥ X ≤ |1 -λ ′ λ |∥x -x ′ ∥ X . Since 0 < |1 -λ ′ λ | < 1,
the Banach contraction fixed point theorem applies.

Part A: Nonlinear eigenvalue problems

Calculus of variations

The calculus of variations uses variations, which are small changes in functions and functionals, to find extrema(maxima, minima or other critical values) of functionals: mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives.

Functionals and critical points

Let X be a Banach space. A functional on X is a continuous real valued map I : X → R. In general, one could consider functionals defined on open subsets of X. But we will deal with functionals defined on all of X. In the applications, critical points turn out to be weak solutions of differential equations. Definition 3.1.1. We say that u ∈ X is a local minimum, respectively maximum of the functional I ∈ C(E, R) if there exists a neighbourhood V of u such that

I(u) ≤ I(v), respectively I(u) ≥ I(v), ∀ v ∈ V \{u}. (3.1.1)
If the inequalities in (3.1.1) are strict we say that u is a strict local minimum respectively maximum. If (3.1.1) holds for every u ∈ X, not only on V \{u}, u is a global minimum respectively maximum.

Next, we state some results dealing with the existence of maxima or minima. We restrict ourself to classical result dealing with functionals which are coercive and wealky lower-semi-continuous (shortly : w.l.s.c.).

Let us recall that

I ∈ C(X, R) is coercive if lim ∥u∥ X →+∞ I(u) = +∞.
The functional I is w.l.s.c. if for every sequence u n ∈ X such that u n ⇀ u one has that

I(u) ≤ lim inf n→+∞ I(u n ).
Lemma 3.1.2. Let X be a reflexive Banach space and let I : X → R be coercive and w.l.s.c. Then I is bounded from below on X, namely there exists a ∈ R such that I(u) ≥ a for all u ∈ X.

Proof. Suppose by contradiction that there exists u n ∈ X such that I(u n ) → -∞. Since I is coercive there exists M > 0 such that ∥u∥ X ≤ M. Hence (u n ) n is bounded in X. But since X is a reflexive Banach space (u n ) n has a subsequence u n (without relabelling) which converges weakly to some u ∈ X. Since I is w.l.s.c. we infer that 0 < I(u) ≤ lim inf n→+∞ I(u n ) = -∞, contradiction, proving the lemma.

Remark 3.1.3. The same arguments show that a w.l.s.c. functional is bounded from below on any ball B ρ = {u ∈ X : ∥u∥ X ≤ ρ}. Theorem 3.1.4. Let X be a reflexive Banach space and let I : X → R be coercive and w.l.s.c. Then I has a global minimum, namely there exists u ∈ X such that

I(u) = min{I(v) : v ∈ X}. If I is differentiable at u, then I ′ (u) = 0.
Proof. From Lemma 3.1.2, it follows that m = inf{I(u) : u ∈ X} is finite. Let u n be a minimizing sequence, namely such that I(u n ) → m. Again, the coercivity of I implies that ∥u n ∥ X ≤ M ′ , and u n ⇀ u for some u ∈ X. Since I is w.l.s.c. it follows that I(u) ≤ lim inf n→+∞ I(u n ) = m. Thus I achieves its infimum at u : I(u) = m.

Bifurcation: definition and necessary conditions

Let X, Y be Banach spaces. We deal with an equation like

S(λ, u) = 0 (3.1.2)
where S : R × X → Y is such that

S(λ, 0) = 0 ∀ λ ∈ R.
The solution u = 0 will be called trivial solution of (3.1.2). The set

Σ = {(λ, u) ∈ R × X : u ̸ = 0, S(λ, u) = 0}
will be called the set of nontrivial solutions of (3.1.2). The following phenomenon has been observed: a branch of solutions u(λ) depending on λ, either disappeared or split into several branches, as λ approaches some critical values. This kind of phenomenon is called bifurcation. Many problems arising in applications can be modelled in this way.

For example, the following algebraic equation

u 3 -λu = 0, λ ∈ R
has a solution u = 0 for all λ ∈ R. For λ ≤ 0, this is the unique solution, but for λ > 0 we have two more branches of solutions u = ± √ λ. Bifurcation phenomena occur frequently in nature. Early in 1744, Euler observed the bending of a rod pressed along the direction of its axis. Let θ be the angle between the real axis and the tangent of the central line of the rod, and let λ be the pressure. The length of the rod is normalized to be π. We obtain the following differential equation with the two free end point conditions

   θ + λ sin θ = 0 θ(0) = θ(π) = 0. (3.1.3)
Obviously, θ = 0 is always a solution of the ordinary differential equation (3.1.3).

Actually the solution is unique, if λ is not large. As λ increasingly passes through a certain value λ 0 , it is shown by experiment that there exists a bending solution θ ̸ = 0.

The same phenomenon occurs in the bending of plates, shells etc. Also, bifurcation occurs in the study of thermodynamics, rotation of fluids, solitary waves, superconductivity and lasers, etc. Mathematically, we describe the bifurcation by the following:

Definition 3.1.5

. (bifurcation from trivial solution)

A bifurcation point for (3.1.2) is a number λ * ∈ R such that (λ * , 0) belongs to the closure of Σ. In other words, λ * is a bifurcation point if there exist sequences

λ n ∈ R, u n ∈ X\{0} such that (i) S(λ n , u n ) = 0, (ii) (λ n , u n ) → (λ * , 0).
The main purpose of the theory of bifurcation is to establish conditions for finding bifurcation points and in general, to study the structure of Σ. If S ∈ C 1 (R × X, Y ) a necessary condition for λ * to be a bifurcation point can be immediately deduced from the implicit function theorem.

Definition 3.1.6. (bifurcation from infinity)

We say that λ * ∞ is a bifurcation point from infinity for (3.1.2) if there exist λ n → λ * ∞ and u n ∈ X, such that ∥u n ∥ X → ∞ and (λ n , u n ) ∈ Σ. We state a remarkable bifurcation result due to M. A. Krasnoselski [START_REF] Krasnoselski | Topological Methods in the Theory of Nonlinear Integral Equations[END_REF].

Theorem 3.1.7. (Krasnoselski bifurcation theorem) Let X be a Banach space and let T ∈ C 1 (X, X) be a compact operator such that T (0) = 0 and T ′ (0) = 0. Moreover, let A ∈ L (X) also be compact. Then every characteristic value λ * of A with odd (algebraic) multiplicity is a bifurcation point for u = λAu + T (u).

For the proof of Theorem 3.1.7, we refer to [START_REF] Ambrosetti | Nonlinear analysis and semilinear elliptic problems[END_REF].

Palais-Smale condition

Definition 3.1.8. Let X be a Banach space and F : X → R be a differentiable functional. A sequence {u n } n ⊆ X such that

{F (u n )} n is bounded (in R) and F ′ (u n ) → 0 (in X ′ ) as n → ∞,
is called a Palais-Smale sequence for F. Remark 3.1.9. In a Hilbert space E we can identify the differential with the gradient through the scalar product. Therefore, the second property of a Palais-Smale sequence reads ∇F (u n ) → 0 in E.

We recall that the convergence takes place in the strong topology of E. Definition 3.1.10. Let X be a Banach space and let F : X → R be a differentiable functional. We say that F satisfies the Palais-Smale condition (shortly: F satisfies (PS)) if every Palais-Smale sequence for F has a converging subsequence (in X).

The following lemma shows that the search for critical points can be split into two independent parts : the existence of Palais-Smale sequences, which will follow from topological reasons, and the convergence of these sequences, which is a compactness problem. Lemma 3.1.11. Let X be a Banach space and let F : X → R be C 1 functional. If there exists a Palais-Smale sequence for F and F satisfies (PS), then F has a critical point.

In the following section we discuss the notion of degree.

Topological degree

The reader who meets the notion of topological degree (shortly, degree) for the first time, could maybe start by asking the following question : what is the topological degree ? As a rough answer, the degree is a tool, precisely a number which gives information about the solution of particular equations. The degree was introduced by L. Brouwer in finite dimensional spaces and extended by J. Leray and J. Schauder to infinite dimensional spaces. The Leray-Schauder degree is an important topological tool in the study of nonlinear partial differential equations while the Brouwer degree is a powerful tool in algebraic topology. The nontriviality of the degree ensures the existence of a fixed point of the compact mapping in the domain. It enjoys the properties of homotopy invariance and additivity, which make the topological tool more convenient in application, and provides more information on fixed points. There is a very broad literature dealing with degree, among those we cite the following books [START_REF] Amann | Lectures on some Fixed Point Theorems[END_REF][START_REF] Ambrosetti | Nonlinear analysis and semilinear elliptic problems[END_REF][START_REF] Brown | A Topological Introduction to Nonlinear Analysis[END_REF][START_REF] Deimling | Nonlinear Functional Analysis[END_REF][START_REF] Kung-Ching | Methods in Nonlinear Analysis[END_REF].

Brouwer degree and its properties

Let us assume that [START_REF] Alabau-Boussouira | Piecewise multiplier method and nonlinear integral inequalities for Petrowsky equation with nonlinear dissipation[END_REF] Ω is an open bounded set in R d , with boundary ∂Ω,

(2) h is a continuous function map from Ω to R d , the components of h will be denoted by h i ,

(3) p is a point in R d such that p / ∈ h(∂Ω).
To each triple (h, Ω, p) satisfying ( 1)-( 3), one can associate an integer deg(h, Ω, p), called the degree of h (with respect to Ω and p), with the following properties.

(P1) Normalization: if Id R d denotes the identity map in R d , then deg(Id R d , Ω, p) =      1 if p ∈ Ω, 0 if p / ∈ Ω.
(P2) Solution property: if deg(h, Ω, p) ̸ = 0 then there exists y ∈ Ω such that h(y) = p.

(P3) deg(h, Ω, p) = deg(h -p, Ω, 0). (P4) Decomposition: if Ω 1 ∩ Ω 2 = ∅ then deg(h, Ω 1 ∪ Ω 2 , p) = deg(h, Ω 1 , p) + deg(h, Ω 2 , p).
An outline of the procedure usually followed to define the degree, omitting the consistency of the definition and the verification of (P1)-(P4) is given by the following. Consider a C 1 map h and a regular value p. Let us recall that p is said to be regular value for h, if the Jacobian J h (x) ̸ = 0 for every x ∈ h -1 (p). The Jacobian is the determinant of the matrix h ′ (x) with entries

e ij = ∂h i ∂x j .
If p is a regular value then the set h -1 (p) is finite and one can define the degree by setting

deg(h, Ω, p) = x∈h -1 (p) sign[J h (x)], (3.2.1)
where, for a ∈ R\{0}, we set

sign[a] =      +1 if a > 0, -1 if a < 0.
We see that the degree defined in (3.2.1) verifies the properties of (P1)-(P4) of the degree defined above. An important property of the degree defined above is the invariance by homotopy. An homotopy is a map

H = H(λ, x) such that H ∈ C([0, 1] × Ω, R d ).
An homotopy is admissible (with respect to

Ω an p), if H(λ, x) ̸ = p for all (λ, x) ∈ [0, 1] × ∂Ω. (P5) Homotopy invariance : if H is an admissible homotopy, then deg(H(λ, •), Ω, p) is constant with respect to λ ∈ [0, 1]. In particular, if f (x) = H(0, x) and g(x) = H(1, x) then deg(f, Ω, p) = deg(g, Ω, p).
As an immediate consequence of the homotopy invariance, we can deduce the following Proof. Consider the homotopy defined by

H(λ, x) = λg(x) + (1 -λ)f (x).
One has g(x) = f (x) for all x ∈ ∂Ω and thus H(λ, x) = g(x) ̸ = p. Hence H is admissible and the homotopy invariance yields :

deg(f, Ω, p) = deg(H(0, •), Ω, p) = deg(H(1, •), Ω, p) = deg(g, Ω, p).
We list below some further properties of the degree.

(P6) Continuity : if h k → h uniformly in Ω, then deg(h k , Ω, p) → deg(h, Ω, p). Moreover, deg(h, Ω, p) is continuous with respect to p. (P7) Excision property: let Ω 0 ⊂ Ω be an open set such that f (x) ̸ = p, for all x ∈ Ω\Ω 0 . Then deg(f, Ω, p) = deg(f, Ω 0 , p).

The Leray-Schauder degree

The Leray-Schauder degree is defined for mappings of the form Id -C, where C is a compact mapping from the closure of an open bounded subset of a Banach space X.

Leray and Schauder extend as follows the Brouwer degree to compact perturbations of the identity in a Banach space X. If Ω ⊂ X is an open bounded set, h : Ω → X is compact, and p / ∈ (Id -h)(∂Ω), the Leray-Schauder degree deg LS (Id -h, Ω, p) of Id -h in Ω over p is constructed from the Brouwer degree by approximating the compact map h over Ω by mappings h ε with range in a finite-dimensional subspace X ε (containing p) of X, and showing that the Brouwer degrees deg B ((Id-h ε )|X ε , Ω∩X ε , p) stabilize for sufficiently small positive ε to a common value defining deg LS (Id -h, Ω, p). This topological degree "algebraically counts" the number of fixed point of h(•) -p in Ω, and for h of class C 1 , and Id -h ′ (a) invertible for each fixed point h(•) -p in Ω, Leray and Schauder show that

deg LS (Id -h, Ω, p) = a∈(Id-h) -1 (p) (-1) σ j (a) ,
where σ j (a) is the sum of the algebraic multiplicities of the eigenvalues h ′ (a).

The Leray-Schauder degree conserves the basic properties of Brouwer degree.

if p / ∈ (Id -h)(∂Ω 1 ) ∪ (Id -h)(∂Ω 2 ), then deg LS (Id -h, Ω, p) = deg LS (Id -h, Ω 1 , p) + deg LS (Id -h, Ω 2 , p). (b) (Existence) If deg LS (Id -h, Ω, p) ̸ = 0, then p ∈ (Id -h)(Ω).
(c) (Homotopy invariance) Let Ω ⊂ X be a bounded open set, and let

F : R × Ω → X be compact. If x -F (λ, x) ̸ = p for all (λ, x) ∈ R × ∂Ω, then deg LS (Id -F (λ, •), Ω, p) is independent of λ.

The Krasnoselski genus

The genus was introduced by M. A. Krasnoselski [START_REF] Krasnoselski | Topological Methods in the Theory of Nonlinear Integral Equations[END_REF]. Let E be a infinite dimensional Hilbert space. We say that a subset O ⊂ E is symmetric if it is symmetric with respect to the origin of E, namely

u ∈ O ⇒ -u ∈ O.
Let Γ be the class of all the symmetric subsets A ⊆ E\{0} which are closed in E\{0}.

Definition 3.3.1. Let A ∈ Γ.
The genus of A is defined as the least integer number n ∈ N such that there exists ψ : A → R n continuous, odd and such that ψ(x) ̸ = 0 for all x ∈ A.

The genus of A is usually denoted by γ(A). If such a number does not exists, we set γ(A) = ∞ and, if A = ∅, we conventionally set γ(A) = 0. Remark 3.3.2. An equivalent way to define the genus γ(A) is to take the minimal integer d such that there exists an odd map

ψ ∈ C(A, R d \{0}). Actually, such a ψ can be extended to a map ψ ∈ C(E, R d ). If ψ * is the odd part of ψ, namely ψ * (u) = 1 2 ψ(u) -ψ(-u) ,
ψ * verifies the properties required in the above definition. Then γ(A) = +∞ .

Proof. Let n ∈ N be any positive integer, and let ψ : A → R n be continuous and odd map. The infinite dimensional sphere contains the k-sphere

S k ⊂ R k+1 ; thus by Borsuk-Ulam theorem it follows that, for k > n 0 ∈ ψ(S k ) ⇒ 0 ∈ ψ(A).
Since A contains every finite dimensional sphere, for every n ∈ N we can take k = n + 1 and obtain that 0 is in the image. This proves that the genus is +∞.

Remark 3.3.5. In a similar way, one shows that γ(∂Ω) = n, where Ω ⊂ R n is an open bounded symmetric subset such that 0 ∈ Ω. In particular,

γ(S n-1 ) = n.
The following proposition gives some properties of the genus.

Proposition 3.3.6. Let A and B be elements of the class Γ.

(i) The set A is empty if and only if the genus γ(A) = 0.

(ii) If ψ : A → B is a continuous odd map, then γ(A) ≤ γ(B). In particular

A ⊆ B ⇒ γ(A) ≤ γ(B).
(iii) The genus is subadditive, namely, γ(A ∪ B) ≤ γ(A) + γ(B).

(iv) If A is compact then γ(A) < +∞ and there exists a symmetric neighbourhood U A of A such that γ(U A ) = γ(A).

Existence of multiple critical points of functionals

The genus can be used to prove existence results of critical points of functional provided that the functional is even and M ∈ Γ, where M is a the constraint set. Here, we consider the functional J ∈ C 1 (E, R). For any positive integer, we define

Γ m = {A ⊂ M : A ∈ Γ, A is compact and γ(A) ≥ m} and σ m = inf A∈Γm sup u∈A J(u).
We explicitly remark that σ m < +∞ and σ m ≤ σ m+1 . Moreover, if J is bounded from below on M , then σ 1 > -∞ and hence any σ m is finite. If we deal with problems without constraints, namely if we are looking for stationary points of

J ∈ C 1 (E, R) on E, we understand Γ = {A ∈ E\{0} : A is symmetric}, that Γ m = {A ⊂ Γ, A is compact and γ(A) ≥ m},
and that σ m is defined as above.

We state the following general result which holds both in the case of critical points of J constrained on M and in the case without constraints. We refer to [6, Proposition 10.8] for a proof.

Proposition 3.3.7. Each finite σ m is a critical level for J ∈ C 1 (E, R) (or a critical level for J on M ) provided (P S) σm holds. Moreover, if σ = σ m = σ m+1 = • • • = σ m+n ∈ R for some integer n ≥ 1, then γ(Z σ ) ≥ n + 1
, where Z σ denotes the set of critical points of J at the critical level σ.

The following theorem, known as Clark's theorem asserts the existence of a sequence of negative critical values tending to 0 for even coercive functionals. Theorem 3.3.8. (Clark, [START_REF] Clark | variant of the Lusternik-Schnirelman theory[END_REF]) Let X be a Banach space and G ∈ C 1 (X, R) satisfying the Palais-Smale condition with G(0

) = 0. Let Γ k = { A ∈ Σ : γ(A) ≥ k } with Σ = { A ⊂ X ; A = -A and A closed }. If c k = inf A∈Γ k sup u∈A G(u) ∈ (-∞, 0), then c k is a critical value.

Nonlinear eigenvalue problems for quasi-linear operators and applications to bifurcation

In this part we present the results obtained in [START_REF] Zongo | Nonlinear eigenvalue problems and bifurcation for quasi-linear elliptic operators[END_REF].

Setting of the problem

We consider Ω ⊂ R N (N ≥ 2) an open bounded domain with smooth boundary ∂Ω. A classical result in the theory of eigenvalue problems guarantees that the problem

-∆u = λu in Ω, u = 0 on ∂Ω (3.4.1)
possesses a nondecreasing sequence of eigenvalues and a sequence of corresponding eigenfunctions which define a Hilbert basis in L 2 (Ω) [see, [START_REF] Henrot | Extremum problems for eigenvalues of elliptic operators[END_REF]]. Moreover, it is known that the first eigenvalue of problem (3.4.1) is characterized in the variational point of view by,

λ D 1 := inf u∈W 1,2 0 (Ω)\{0} Ω |∇u| 2 dx Ω u 2 dx .
Suppose that p > 1 is a given real number and consider the nonlinear eigenvalue problem with Neumann boundary condition

-∆ p u = λu in Ω, ∂u ∂ν = 0 on ∂Ω (3.4.2)
where ∆ p u := div(|∇u| p-2 ∇u) stands for the p-Laplace operator and λ ∈ R. This problem was considered in [START_REF] Fȃrcȃseanu | On the set of eigenvalues of some PDEs with homogeneous Neumann boundary condition[END_REF], and using a direct method in calculus of variations

(if p > 2) or a mountain-pass argument (if p ∈ ( 2N N +2 , 2
)) it was shown that the set of eigenvalues of problem (3.4.2) is exactly the interval [0, ∞). Indeed, it is sufficient to find one positive eigenvalue, say -∆ p u = λu. Then a continuous family of eigenvalues can be found by the reparametrization u = αv,

satisfying -∆ p v = µ(α)v, with µ(α) = λ α p-2 .
We consider the so-called (p, 2)-Laplace operator [see, [START_REF] Liang | Some properties and applications related to the (2,p)-Laplacian operator. al. Boundary Value Problems[END_REF]] with Dirichlet boundary conditions. More precisely, we analyze the following nonlinear eigenvalue problem,

-∆ p u -∆u = λu in Ω, u = 0 on ∂Ω (3.4.3)
where p ∈ (1, ∞)\{2} is a real number. We recall that if 1 < p < q, then L q (Ω) ⊂ L p (Ω) and as a consequence, one has W 1,q 0 (Ω) ⊂ W 1,p 0 (Ω). We will say that λ ∈ R is an eigenvalue of problem (3.4.3)

if there exists u ∈ W 1,p 0 (Ω)\{0} (if p > 2 ), u ∈ W 1,2 0 (Ω)\{0} (if 1 < p < 2) such that Ω |∇u| p-2 ∇u • ∇v dx + Ω ∇u • ∇v dx = λ Ω u v dx, (3.4.4) for all v ∈ W 1,p 0 (Ω) (if p > 2), v ∈ W 1,2 0 (Ω) (if 1 < p < 2).
In this case, such a pair (u, λ) is called an eigenpair, and λ ∈ R is called an eigenvalue and u ∈ W 1,p 0 (Ω)\{0} is an eigenfunction associated to λ. We say that λ is a "first eigenvalue", if the corresponding eigenfunction u is positive or negative. The operator -∆ p -∆ appears in quantum field theory [see, [START_REF] Benci | Solitons in several space dimensions: Derrick's problem and infinitely many solutions[END_REF]], where it arises in the mathematical description of propagation phenomena of solitary waves. We recall that a solitary wave is a wave which propagates without any temporal evolution in shape. The operator -∆ p -∆ is a special case of the so called (p, q)-Laplace operator given by -∆ p -∆ q which has been widely studied; for some results related to our studies, see e.g., [START_REF] Bobkov | On positive solutions for (p, q)-Laplace equations with two parameters[END_REF][START_REF] Bobkov | Remarks on minimizers for (p, q)-Laplace equations with two parameters[END_REF][START_REF] Cingolani | Nontrivial solutions for p-Laplace equations with right-hand side having p-linear growth at infinity[END_REF][START_REF] Marano | Some recent results on the Dirichlet problem for (p, q)-Laplace equations[END_REF][START_REF] Tanaka | On bifurcation from infinity[END_REF] . We investigate the nonlinear eigenvalue problem (3.4.3) when p > 2, and 1 < p < 2 respectively. In particular, we show in section 3.5 that the set of the first eigenvalues is given by the interval (λ D 1 , ∞), where λ D 1 is the first Dirichlet eigenvalue of the Laplacian. We show that the first eigenvalue of (3.4.3) can be obtained variationally, using a Nehari set for 1 < p < 2, and a minimization for p > 2. Also in the same section, we recall some results of [START_REF] Fȃrcȃseanu | On the set of eigenvalues of some PDEs with homogeneous Neumann boundary condition[END_REF], [START_REF] Mihailescu | An eigenvalue problem possessing a continuous family of eigenvalues plus an isolated eigenvalue[END_REF] and [START_REF] Mihailescu | Continuous spectrum for a class of nonhomogeneous differential operators[END_REF]. In section 3.6, we prove that the eigenfunctions associated to λ belong to L ∞ (Ω), the first eigenvalue λ D 1 of problem (3.4.3) is simple and the corresponding eigenfunctions are positive or negative. In addition, in section 3.6.3 we show a homeomorphism property related to -∆ p -∆. In section 3.7, we prove that λ D 1 is a bifurcation point for a branch of first eigenvalues from zero if p > 2, and λ D 1 is a bifurcation point from infinity if p < 2. Also the higher Dirichlet eigenvalues λ D k are bifurcation points (from 0 if p > 2, respectively from infinity if 1 < p < 2 ), if the multiplicity of λ D k is odd. Finally in section 3.8, we prove by variational methods that if λ ∈ (λ D k , λ D k+1 ) then there exist at least k nonlinear eigenvalues using Krasnoselski's genus.

The spectrum of the nonlinear problem

We start with the discussion of the properties of the spectrum of the nonlinear eigenvalues problem (3.4.3). Remark 3.5.1. Any λ ≤ 0 is not an eigenvalue of problem (3.4.3). Indeed, suppose by contradiction that λ = 0 is an eigenvalue of equation (3.4.3), then relation (3.4.4)

with v = u 0 gives Ω |∇u 0 | p dx + Ω |∇u 0 | 2 dx = 0.
Consequently |∇u 0 | = 0, therefore u 0 is constant on Ω and u 0 = 0 on Ω. And this contradicts the fact that u 0 is a nontrivial eigenfunction. Hence λ = 0 is not an eigenvalue of problem (3.4.3). Now it remains to show that any λ < 0 is not an eigenvalue of (3.4.3). Suppose by contradiction that λ < 0 is an eigenvalue of (3.4.3), with u λ ∈ W 1,p 0 (Ω)\{0} the corresponding eigenfunction. The relation (3.4.4)

with v = u λ implies 0 ≤ Ω |∇u λ | p dx + Ω |∇u λ | 2 dx = λ Ω u 2 λ dx < 0.
Which yields a contradiction and thus λ < 0 cannot be an eigenvalue of problem (3.4.3).

Lemma 3.5.2. Any λ ∈ (0, λ D 1 ] is not an eigenvalue of (3.4.3).

For the proof see also [START_REF] Fȃrcȃseanu | On the set of eigenvalues of some PDEs with homogeneous Neumann boundary condition[END_REF].

Proof. Let λ ∈ (0, λ D 1 ), i.e., λ D 1 > λ. Let's assume by contradiction that there exists a λ ∈ (0, λ D 1 ) which is an eigenvalue of (3.4.3) with u λ ∈ W 1,2 0 (Ω)\{0} the corresponding eigenfunction. Letting v = u λ in relation (3.4.4), we have on the one hand,

Ω |∇u λ | p dx + Ω |∇u λ | 2 dx = λ Ω u 2 λ dx
and on the other hand,

λ D 1 Ω u 2 λ dx ≤ Ω |∇u λ | 2 dx. (3.5.1)
By subtracting both side of (3.5.1) by λ Ω u 2 λ dx, we obtain

(λ D 1 -λ) Ω u 2 λ dx ≤ Ω |∇u λ | 2 dx -λ Ω u 2 λ dx, (λ D 1 -λ) Ω u 2 λ dx ≤ Ω |∇u λ | 2 dx -λ Ω u 2 λ dx + Ω |∇u λ | p dx = 0. Therefore (λ D 1 -λ) Ω u 2 λ dx ≤ 0, which is a contradiction. Hence, we conclude that λ ∈ (0, λ D 1 )
is not an eigenvalue of problem (3.4.3). In order to complete the proof of the Lemma 3.5.2 we shall show that λ = λ D 1 is not an eigenvalue of (3.4.3). By contradiction we assume that λ = λ D 1 is an eigenvalue of (3.4.3). So there Proof. Let λ ∈ (λ D 1 , ∞), and define the energy functional

exists u λ D 1 ∈ W 1,2 0 (Ω)\{0} such that relation (3.4.4) holds true. Letting v = u λ D 1 in relation (3.4.4), it follows that Ω |∇u λ D 1 | p dx + Ω |∇u λ D 1 | 2 dx = λ D 1 Ω u 2 λ D 1 dx. But λ D 1 Ω u 2 λ D 1 dx ≤ Ω |∇u λ D 1 | 2 dx, therefore Ω |∇u λ D 1 | p dx + Ω |∇u λ D 1 | 2 dx ≤ Ω |∇u λ D 1 | 2 dx ⇒ Ω |∇u λ D 1 | p dx ≤ 0. Using relation (2.1.4), we have u λ D 1 = 0, which is a contradiction since u λ D 1 ∈ W 1,2 0 (Ω)\{0}. So λ = λ D 1 is not an eigenvalue of (3.4.3).
J λ : W 1,2 0 (Ω) → R by J λ (u) = Ω |∇u| 2 dx + 2 p Ω |∇u| p dx -λ Ω u 2 dx.
One shows that J λ ∈ C 1 (W 1,2 0 (Ω), R) (see, [START_REF] Liang | Some properties and applications related to the (2,p)-Laplacian operator. al. Boundary Value Problems[END_REF]) with its derivatives given by

⟨J ′ λ (u), v⟩ = 2 Ω ∇u•∇v dx+2 Ω |∇u| p-2 ∇u•∇v dx-2λ Ω u v dx , ∀ v ∈ W 1,2 0 (Ω).
Thus we note that λ is an eigenvalue of problem (3.4.3) if and only if J λ possesses a nontrivial critical point. Considering J λ (ρe 1 ), where e 1 is the L 2 -normalized first eigenfunction of the Laplacian, we see that

J λ (ρe 1 ) ≤ λ D 1 ρ 2 + Cρ p -λρ 2 → -∞, as ρ → +∞.
Hence, we cannot establish the coercivity of J λ on W 1,2 0 (Ω) for p ∈ (1, 2), and consequently we cannot use a direct method in calculus of variations in order to determine a critical point of J λ . To overcome this difficulty, the idea will be to analyze the functional J λ on the so called Nehari manifold defined by

N λ := u ∈ W 1,2 0 (Ω)\{0} : Ω |∇u| 2 dx + Ω |∇u| p dx = λ Ω u 2 dx .
Note that all non-trivial solutions of (3.4.3) lie on N λ . On N λ the functional J λ takes the form

J λ (u) = Ω |∇u| 2 dx + 2 p Ω |∇u| p dx -λ Ω u 2 dx = ( 2 p -1) Ω |∇u| p dx > 0.
We have seen in Lemma 3.5.2, that any λ ∈ (0, λ D 1 ] is not an eigenvalue of problem (3.4.3); see also [START_REF] Fȃrcȃseanu | On the set of eigenvalues of some PDEs with homogeneous Neumann boundary condition[END_REF]. It remains to prove the :

Claim : Every λ ∈ (λ D 1 , ∞) is a first eigenvalue of problem (3.4.
3). Indeed, we will split the proof of the claim into four steps.

Step 1. Here we will show that N λ ̸ = ∅ and every minimizing sequence for J λ on

N λ is bounded in W 1,2 0 (Ω). Since λ > λ D 1 there exists v λ ∈ W 1,2 0 (Ω) such that Ω |∇v λ | 2 dx < λ Ω v 2 λ dx.
Then there exists t > 0 such that tv λ ∈ N λ . In fact

Ω |∇(tv λ )| 2 dx + Ω |∇(tv λ )| p dx = λ Ω (tv λ ) 2 dx ⇒ t 2 Ω |∇v λ | 2 dx + t p Ω |∇v λ | p dx = t 2 λ Ω v 2 λ dx ⇒ t = Ω |∇v λ | p dx λ Ω v 2 λ dx -Ω |∇v λ | 2 dx 1 2-p > 0.
With such t we have tv λ ∈ N λ and N λ ̸ = ∅.

Note that for

u ∈ B r (v λ ), r > 0 small, the inequality λ Ω |u| 2 dx > Ω |∇u| 2 dx remains valid, and then t(u)u ∈ N λ for u ∈ B r (v λ ). Since t(u) ∈ C 1 we conclude that N λ is a C 1 -manifold. Let {u k } ⊂ N λ be a minimizing sequence of J λ | N λ , i.e. J λ (u k ) → m = inf w∈N λ J λ (w). Then λ Ω u 2 k dx - Ω |∇u k | 2 dx = Ω |∇u k | p dx → 2 p -1 -1 m as k → ∞. (3.5.2) Assume by contradiction that {u k } is not bounded in W 1,2 0 (Ω), i.e. Ω |∇u k | 2 dx → ∞ as k → ∞. It follows that Ω u 2 k dx → ∞ as k → ∞, thanks to relation (3.5.2). We set v k = u k ∥u k ∥ 2 . Since Ω |∇u k | 2 dx < λ Ω u 2 k dx, we deduce that Ω |∇v k | 2 dx < λ, for each k and ∥v k ∥ 1,2 < √ λ. Hence {v k } ⊂ W 1,2 0 (Ω) is bounded in W 1,2 0 (Ω). Therefore there exists v 0 ∈ W 1,2 0 (Ω) such that v k ⇀ v 0 in W 1,2 0 (Ω) ⊂ W 1,p 0 (Ω) and v k → v 0 in L 2 (Ω). Dividing relation (3.5.2) by ∥u k ∥ p 2 , we get Ω |∇v k | p dx = λ Ω u 2 k dx - Ω |∇u k | 2 dx ∥u k ∥ p 2 → 0 as k → ∞, since λ Ω u 2 k dx - Ω |∇u k | 2 dx → 2 p -1 -1 m < ∞ and ∥u k ∥ p 2 → ∞ as k → ∞. On the other hand, since v k ⇀ v 0 in W 1,p 0 (Ω), we have Ω |∇v 0 | p dx ≤ lim k→∞ inf Ω |∇v k | p dx = 0 and consequently v 0 = 0. It follows that v k → 0 in L 2 (Ω), which is a contradiction since ∥v k ∥ 2 = 1. Hence, {u k } is bounded in W 1,2 0 (Ω).
Step 2. m = inf w∈N λ J λ (w) > 0. Indeed, assume by contradiction that m = 0. Then, for {u k } as in step 1, we have

0 < λ Ω u 2 k dx - Ω |∇u k | 2 dx = Ω |∇u k | p dx → 0, as k → ∞. (3.5.3) By Step 1, we deduce that {u k } is bounded in W 1,2 0 (Ω). Therefore there exists u 0 ∈ W 1,2 0 (Ω) such that u k ⇀ u 0 in W 1,2 0 (Ω) and W 1,p 0 (Ω) and u k → u 0 in L 2 (Ω). Thus Ω |∇u 0 | p dx ≤ lim k→∞ inf Ω |∇u k | p dx = 0. And consequently u 0 = 0, u k ⇀ 0 in W 1,2 0 (Ω) and W 1,p 0 (Ω) and u k → 0 in L 2 (Ω). Writing again v k = u k ∥u k ∥ 2 we have 0 < λ Ω u 2 k dx - Ω |∇u k | 2 dx ∥u k ∥ 2 2 = ∥u k ∥ p-2 2 Ω |∇v k | p dx, therefore Ω |∇v k | p dx = ∥u k ∥ 2-p 2     λ∥u k ∥ 2 2 ∥u k ∥ 2 2 -Ω |∇u k | 2 dx ∥u k ∥ 2 2     = ∥u k ∥ 2-p 2 λ - Ω |∇v k | 2 dx → 0 as k → ∞, since ∥u k ∥ 2 → 0 and p ∈ (1, 2), and
{v k } is bounded in W 1,2 0 (Ω). Next since v k ⇀ v 0 in W 1,2 0 (Ω) ⊂ W 1,p 0 (Ω), we deduce that Ω |∇v 0 | p dx ≤ lim k→∞ inf Ω |∇v k | p dx = 0 and we have v 0 = 0. And it follows that v k → 0 in L 2 (Ω) which is a contradiction since ∥v k ∥ 2 = 1 for each k. Hence, m is positive.
Step 3. There exists

u 0 ∈ N λ such that J λ (u 0 ) = m.
Let {u k } ⊂ N λ be a minimizing sequence, i.e., J λ (u k ) → m as k → ∞.

Thanks to

Step 1, we have that {u k } is bounded in W 1,2 0 (Ω). It follows that there exists u 0 ∈ W 1,2 0 (Ω) such that u k ⇀ u 0 in W 1,2 0 (Ω) and W 1,p 0 (Ω) and strongly in L 2 (Ω). The results in the two steps above guarantee that

J λ (u 0 ) ≤ lim k→∞ inf J λ (u k ) = m. Since for each k we have u k ∈ N λ , we have Ω |∇u k | 2 dx + Ω |∇u k | p dx = λ Ω u 2
k dx for all k.

(3.5.4)

Assuming u 0 ≡ 0 on Ω implies that Ω u 2 k dx → 0 as k → ∞,
Ω |∇u k | 2 dx → 0 as k → ∞. Combining
this with the fact that u k converges weakly to 0 in W 1,2 0 (Ω), we deduce that u k converges strongly to 0 in W 1,2 0 (Ω) and consequently in W 1,p 0 (Ω). Hence we infer that

λ Ω u 2 k dx - Ω |∇u k | 2 dx = Ω |∇u k | p dx → 0, as k → ∞.
Next, using similar argument as the one used in the proof of Step 2, we will reach to a contradiction, which shows that u 0 ̸ ≡ 0. Letting k → ∞ in relation (3.5.4), we deduce that

Ω |∇u 0 | 2 dx + Ω |∇u 0 | p dx ≤ λ Ω u 2 0 dx.
If there is equality in the above relation then u 0 ∈ N λ and m ≤ J λ (u 0 ). Assume by contradiction that

Ω |∇u 0 | 2 dx + Ω |∇u 0 | p dx < λ Ω u 2 0 dx. (3.5.5) Let t > 0 be such that tu 0 ∈ N λ , i.e., t =     λ Ω u 2 0 dx - Ω |∇u 0 | 2 dx Ω |∇u 0 | p dx     1 p-2 .
We note that t ∈ (0, 1) since 1 < t p-2 (thanks to (3.5.5)). Finally, since tu 0 ∈ N λ with t ∈ (0, 1) we have

0 < m ≤ J λ (tu 0 ) = 2 p -1 Ω |∇(tu 0 )| p dx = t p 2 p -1 Ω |∇u 0 | p dx = t p J λ (u 0 ) ≤ t p lim k→∞ inf J λ (u k ) = t p m < m for t ∈ (0, 1),
and this is a contradiction which assures that relation (3.5.5) cannot hold and consequently we have u 0 ∈ N λ . Hence m ≤ J λ (u 0 ) and m = J λ (u 0 ).

Step 4. We conclude the proof of the claim. Let u ∈ N λ be such that J λ (u) = m (thanks to Step 3). Since u ∈ N λ , we have

Ω |∇u| 2 dx + Ω |∇u| p dx = λ Ω u 2 dx,
and

Ω |∇u| 2 dx < λ Ω u 2 dx. Let v ∈ ∂B 1 (0) ⊂ W 1,2 0
(Ω) and ε > 0 be very small such that u + δv ̸ = 0 in Ω for all δ ∈ (-ε, ε) and

Ω |∇(u + δv)| 2 dx < λ Ω (u + δv) 2 dx; this is equivalent to λ Ω u 2 dx - Ω |∇u| 2 dx > δ 2 Ω ∇u • ∇v dx -2λ Ω uv dx + δ 2 Ω |∇v| 2 dx -λ Ω v 2 dx ,
which holds true for δ small enough since the left hand side is positive while the function

h(δ) := |δ| 2 Ω ∇u • ∇v dx -2λ Ω uv dx +δ 2 Ω |∇v| 2 dx -λ Ω v 2 dx
dominates the term from the right hand side and h(δ) is a continuous function (polynomial in δ) which vanishes in δ = 0. For each δ ∈ (-ε, ε), let t(δ) > 0 be given by

t(δ) =     λ Ω (u + δv) 2 dx - Ω |∇(u + δv)| 2 dx Ω |∇(u + δv)| p dx     1 p-2 , so that t(δ) • (u + δv) ∈ N λ . We have that t(δ) is of class C 1 (-ε, ε) since t(δ)
is the composition of some functions of class C 1 . On the other hand, since u ∈ N λ we have t(0) = 1.

Define ι : (-ε, ε) → R by ι(δ) = J λ (t(δ)(u + δv)) which is of class C 1 (-ε, ε
) and has a minimum at δ = 0. We have

ι ′ (δ) = [t ′ (δ)(u + δv) + vt(δ)]J ′ λ (t(δ)(u + δv)) ⇒ 0 = ι ′ (0) = J ′ λ (t(0)(u))[t ′ (0)u + vt(0)] = ⟨J ′ λ (u), v⟩ since t(0) = 1 and t ′ (0) = 0. This shows that every λ ∈ (λ D 1 , ∞) is an eigenvalue of problem (3.4.3).
In the next theorem we consider the case p > 2. For similar results for the Neumann case, [see, [START_REF] Mihailescu | An eigenvalue problem possessing a continuous family of eigenvalues plus an isolated eigenvalue[END_REF]]. The proof of Theorem 3.5.4 will follow as a direct consequence of the lemmas proved below:

Lemma 3.5.5. Let λ 1 (p) := inf u∈W 1,p 0 \{0}        1 p Ω |∇u| p dx + 1 2 Ω |∇u| 2 dx 1 2 Ω u 2 dx        . Then λ 1 (p) = λ D 1 , for all p > 2.
Proof. We clearly have λ 1 (p) ≥ λ D 1 since a positive term is added. On the other hand, consider u n = 1 n e 1 (where e 1 is the first eigenfunction of -∆), we get

λ 1 (p) ≤ 1 2n 2 Ω |∇e 1 | 2 dx + 1 pn p Ω |∇e 1 | p dx 1 2n 2 Ω |e 1 | 2 dx → λ D 1 as n → ∞.
Lemma 3.5.6. For each λ > 0, we have

lim ∥u∥ 1,p →∞ 1 2 Ω |∇u| 2 dx + 1 p Ω |∇u| p dx - λ 2 Ω u 2 dx = ∞.
Proof. Clearly

1 p Ω |∇u| p dx + 1 2 Ω |∇u| 2 dx ≥ 1 p Ω |∇u| p dx.
On the one hand, using Poincaré's inequality with p = 2, we have

Ω u 2 dx ≤ C 2 (Ω) Ω |∇u| 2 dx, ∀u ∈ W 1,p 0 (Ω) ⊂ W 1,2 0
(Ω) and then applying the Hölder inequality to the right hand side term of the previous estimate, we obtain

Ω |∇u| 2 dx ≤ |Ω| p-2 p ∥u∥ 2 1,p , so Ω u 2 dx ≤ D∥u∥ 2 1,p , where D = C 2 (Ω)|Ω| p-2 p . Therefore for λ > 0, 1 2 Ω |∇u| 2 dx + 1 p Ω |∇u| p dx - λ 2 Ω u 2 dx ≥ C∥u∥ p 1,p - λ 2 D∥u∥ 2 1,p , (3.5.6)
and the the right-hand side of (3.5.6) tends to ∞, as ∥u∥ 1,p → ∞, since p > 2.

Lemma 3.5.7.

Every λ ∈ (λ D 1 , ∞) is a first eigenvalue of problem (3.4.3). Proof. For each λ > λ D 1 define F λ : W 1,p 0 (Ω) → R by F λ (u) = 1 2 Ω |∇u| 2 dx + 1 p Ω |∇u| p dx - λ 2 Ω u 2 dx , ∀u ∈ W 1,p 0 (Ω).
Standard arguments shows that F λ ∈ C 1 (W 1,p 0 (Ω), R) [see, [START_REF] Liang | Some properties and applications related to the (2,p)-Laplacian operator. al. Boundary Value Problems[END_REF]] with its derivative given by

⟨F ′ λ (u), φ⟩ = Ω (|∇u| p-2 + 1)∇u • ∇φ dx -λ Ω uφ dx,
for all u, φ ∈ W 1,p 0 (Ω). Estimate (3.5.6) shows that F λ is coercive in W 1,p 0 (Ω). On the other hand, F λ is also weakly lower semi-continuous on W 1,p 0 (Ω) since F λ is a continuous convex functional, (see [8, Proposition 1.5.10 and Theorem 1.5.3]) . Then we can apply a calculus of variations result, in order to obtain the existence of a global minimum point of F λ , denoted by θ λ , i.e., F λ (θ λ ) = min

W 1,p 0 (Ω) F λ . Note that for any λ > λ D 1 there exists u λ ∈ W 1,p 0 (Ω) such that F λ (u λ ) < 0 . Indeed, taking u λ = re 1 , we have F λ (re 1 ) = r 2 2 (λ D 1 -λ) + r p p Ω |∇e 1 | p dx < 0 for r > 0 small. But then F λ (θ λ ) ≤ F λ (u λ ) < 0, which means that θ λ ∈ W 1,p 0 (Ω)\{0}. On the other hand, we have ⟨F ′ λ (θ λ ), φ⟩ = 0, ∀φ ∈ W 1,p 0 (Ω) (θ λ is a critical point of F λ ) with θ λ ∈ W 1,p 0 (Ω)\{0} ⊂ W 1,2 0 (Ω)\{0}. Consequently each λ > λ D 1 is an eigenvalue of problem (3.4.3). Proposition 3.5.8. The first eigenfunctions u λ 1 associated to λ ∈ (λ D 1 , ∞) are positive or negative in Ω. Proof. Let u λ 1 ∈ W 1,p 0 (Ω)\{0} be an eigenfunction associated to λ ∈ (λ D 1 , ∞), then Ω |∇u λ 1 | p dx + Ω |∇u λ 1 | 2 dx = λ Ω |u λ 1 | 2 dx, which means u λ 1 achieves the infi- mum in the definition of λ D 1 .
On the other hand we have

∥∇|u λ 1 |∥ 1,p = ∥∇u λ 1 ∥ 1,p and ∥∇|u λ 1 |∥ 1,2 = ∥∇u λ 1 ∥ 1,2 , since |∇|u λ 1 || = |∇u λ 1 | almost everywhere. It follows that |u λ
1 | achieves also the infimum in the definition of λ D 1 , and therefore by the Harnack inequality [see, [START_REF] Evans | Partial Differential Equations ,Second Edition[END_REF]], we have |u λ 1 (x)| > 0 ∀x ∈ Ω and consequently u λ 1 is either positive or negative in Ω.

A similar result of Theorem 3.6.1 was proved in [START_REF] Lê | Eigenvalue problems for the p-Laplacian[END_REF] in the case of the p-Laplacian.

3.6 Properties of eigenfunctions and the operator -∆ p -∆

Boundedness of the eigenfunctions

We shall prove boundedness of eigenfunctions and use this fact to obtain C 1,α smoothness of all eigenfunctions of the quasi-linear problem (3.4.3). The latter result is due to [START_REF] Lê | Eigenvalue problems for the p-Laplacian[END_REF]Theorem 4.4], which originates from [START_REF] Benedetto | C 1+α local regularity of weak solutions of degenerate elliptic equations[END_REF] and [START_REF] Tolksdorf | Regularity for a more general class of quasilinear elliptic equations[END_REF].

Theorem 3.6.1. Let (u, λ) ∈ W 1,p 0 (Ω) × R ⋆ + be an eigensolution of the weak formulation (3.4.4). Then u ∈ L ∞ (Ω).

Proof. By Morrey's embedding theorem it suffices to consider the case p ≤ N. Let us assume first that u > 0.

For M ≥ 0 define w M (x) = min{u(x), M }. Letting g(x) = x if x ≤ M M if x > M (3.6.1)
we have g ∈ C(R) piecewise smooth function with g(0

) = 0. Since u ∈ W 1,p 0 (Ω) and g ′ ∈ L ∞ (Ω), then g•u ∈ W 1,p 0 (Ω) and w M ∈ W 1,p 0 (Ω)∩L ∞ (Ω) (see, Theorem B.3 in [56]). For k > 0, define φ = w kp+1 M then ∇φ = (kp + 1)∇w M w kp M and φ ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω). Using φ as a test function in (3.4.4), one obtains (kp+1) Ω |∇u| p-2 ∇u • ∇w M w kp M dx + Ω ∇u • ∇w M w kp M dx = λ Ω u w kp+1 M dx.
On the other hand using the fact that w kp+1 M ≤ u kp+1 , it follows that

(kp+1) Ω |∇u| p-2 ∇u • ∇w M w kp M dx + Ω ∇u • ∇w M w kp M dx ≤ λ Ω |u| (k+1)p dx.
We have ∇(

w k+1 M ) = (k + 1)∇w M w k M ⇒ |∇w k+1 M | p = (k + 1) p w kp M |∇w M | p .
Since the integrals on the left are zero on {x : u(x) > M } we can take u = w M in the previous inequality, and it follows that

(kp + 1) Ω |∇w M | p w kp M dx + Ω |∇w M | 2 w kp M dx ≤ λ Ω |u| (k+1)p dx. Replacing |∇w M | p w kp M by 1 (k+1) p |∇w k+1 M | p , we have kp + 1 (k + 1) p Ω |∇w k+1 M | p dx + (kp + 1) Ω |∇w M | 2 w kp M dx ≤ λ Ω |u| (k+1)p dx, which implies that kp + 1 (k + 1) p Ω |∇w k+1 M | p dx ≤ λ Ω |u| (k+1)p dx and then Ω |∇w k+1 M | p dx ≤ λ (k + 1) p kp + 1 Ω |u| (k+1)p dx. (3.6.2)
By Sobolev's embedding theorem, there is a constant c 1 > 0 such that

∥w k+1 M ∥ p ⋆ ≤ c 1 ∥w k+1 M ∥ 1,p . (3.6.3)
where p ⋆ is the Sobolev critical exponent. Consequently, we have

∥w M ∥ (k+1)p ⋆ ≤ ∥w k+1 M ∥ 1 k+1 p ⋆ , (3.6.4)
and therefore

∥w M ∥ (k+1)p ⋆ ≤ c 1 ∥w k+1 M ∥ 1,p 1 k+1 = c 1 k+1 1 ∥w k+1 M ∥ 1 k+1
1,p .

(3.6.5)

But by (3.6.2),

∥w k+1 M ∥ 1,p ≤ λ (k + 1) p kp + 1 1 p ∥u∥ k+1 (k+1)p (3.6.6)
and we note that we can find a constant c 2 > 0 such that

λ (k+1) p kp+1 1 p √ k+1 ≤ c 2 , independently of k and consequently ∥w M ∥ (k+1)p ⋆ ≤ c 1 k+1 1 c 1 √ k+1 2 ∥u∥ (k+1)p . (3.6.7)
Letting M → ∞, Fatou's lemma implies

∥u∥ (k+1)p ⋆ ≤ c 1 k+1 1 c 1 √ k+1 2 ∥u∥ (k+1)p . (3.6.8) Choosing k 1 , such that (k 1 + 1)p = p ⋆ , then ∥u∥ (k 1 +1)p ⋆ ≤ c 1 k 1 +1 1 c 1 √ k 1 +1 2 ∥u∥ p ⋆ . Next we choose k 2 such that (k 2 + 1)p = (k 1 + 1)p ⋆ , then taking k 2 = k in inequality (3.6.8), it follows that ∥u∥ (k 2 +1)p ⋆ ≤ c 1 k 2 +1 1 c 1 √ k 2 +1 2 ∥u∥ (k 1 +1)p ⋆ .
(3.6.9)

By induction we obtain

∥u∥ (kn+1)p ⋆ ≤ c 1 kn+1 1 c 1 √ kn+1 2 ∥u∥ (k n-1 +1)p ⋆ , (3.6.10)
where the sequence {k n } is chosen such that (k n + 1)p = (k n-1 + 1)p ⋆ , k 0 = 0. One gets k n + 1 = ( p ⋆ p ) n . As p p ⋆ < 1, there is C > 0 (which depends on c 1 and c 2 ) such that for any n = 1, 2, . . .

∥u∥ rn ≤ C∥u∥ p ⋆ (3.6.11)
with r n = (k n + 1)p ⋆ → ∞ as n → ∞. We note that (3.6.11) follows by iterating the previous inequality (3.6.10). We will indirectly show that u ∈ L ∞ (Ω). Suppose u ̸ ∈ L ∞ (Ω), then there exists ε > 0 and a set A of positive measure in Ω such that |u(x)| > C∥u∥ p ⋆ + ε = K, for all x ∈ A. We then have,

lim n→∞ inf ∥u∥ rn ≥ lim n→∞ inf A K rn 1/rn = lim n→∞ inf K|A| 1/rn = K > C∥u∥ p ⋆ ,
(3.6.12) which contradicts (3.6.11). If u changes sign, we consider u = u + -u -where u + = max{u, 0} and u -= max{-u, 0}.

(3.6.13)

We have u + , u -∈ W 1,p 0 (Ω). For each M > 0 define w M = min{u + (x), M } and take again φ = w kp+1 M as a test function in (3.4.4). Proceeding the same way as above we conclude that u

+ ∈ L ∞ (Ω). Similarly we have u -∈ L ∞ (Ω). Therefore u = u + -u -is in L ∞ (Ω).

Simplicity of the eigenvalues

We prove an auxiliary result which will imply uniqueness of the first eigenfunction. Let

I(u, v) = ⟨-∆ p u, u p -v p u p-1 ⟩ + ⟨-∆u, u 2 -v 2 u ⟩ + ⟨-∆ p v, v p -u p v p-1 ⟩ + ⟨-∆v, v 2 -u 2 v ⟩,
for all (u, v) ∈ D I , where

D I = {(u 1 , u 2 ) ∈ W 1,p 0 (Ω)×W 1,p 0 (Ω) : u i > 0 in Ω and u i ∈ L ∞ (Ω) for i = 1, 2} if p > 2,
and

D I = {(u 1 , u 2 ) ∈ W 1,2 0 (Ω)×W 1,2 0 (Ω) : u i > 0 in Ω and u i ∈ L ∞ (Ω) for i = 1, 2} if 1 < p < 2.
Proposition 3.6.2. For all (u, v) ∈ D I , we have I(u, v) ≥ 0. Furthermore I(u, v) = 0 if and only if there exists α ∈ R ⋆ + such that u = αv.

Proof. We first show that I(u, v) ≥ 0. We recall that (if

2 < p < ∞) ⟨-∆ p u, w⟩ = Ω |∇u| p-2 ∇u • ∇w dx for all w ∈ W 1,p 0 (Ω) ⟨-∆u, w⟩ = Ω ∇u • ∇w dx for all w ∈ W 1,p 0 (Ω). and (if 1 < p < 2) ⟨-∆ p u, w⟩ = Ω |∇u| p-2 ∇u • ∇w dx for all w ∈ W 1,2 0 (Ω) ⟨-∆u, w⟩ = Ω ∇u • ∇w dx for all w ∈ W 1,2 0 (Ω). Let us consider β = u p -v p u p-1 , η = v p -u p v p-1 , ξ = u 2 -v 2 u and ζ = v 2 -u 2 v
as test functions in (3.4.4) for any p > 1. Straightforward computations give,

∇ u p -v p u p-1 = 1 + (p -1) v u p ∇u -p v u p-1 ∇v ∇ v p -u p v p-1 = 1 + (p -1) u v p ∇v -p u v p-1 ∇u ∇ u 2 -v 2 u = 1 + v u 2 ∇u -2 v u ∇v ∇ v 2 -u 2 v = 1 + u v 2 ∇v -2 u v ∇u. Therefore ⟨-∆ p u, u p -v p u p-1 ⟩ = Ω -p v u p-1 |∇u| p-2 ∇u • ∇v + 1 + (p -1) v u p |∇u| p dx = Ω p v u p-1 |∇u| p-2 (|∇u||∇v| -∇u • ∇v) + 1 + (p -1) v u p |∇u| p dx - Ω p v u p-1 |∇u| p-1 |∇v| dx and ⟨-∆u, u 2 -v 2 u ⟩ = Ω 2 v u (|∇u||∇v| -∇u • ∇v) + 1 + v u 2 |∇u| 2 -2 v u |∇u||∇v| dx.
By symmetry we have

⟨-∆ p v, v p -u p v p-1 ⟩ = Ω -p u v p-1 |∇v| p-2 ∇v • ∇u + 1 + (p -1) u v p |∇v| p dx = Ω p u v p-1 |∇v| p-2 (|∇v||∇u| -∇v • ∇u) + 1 + (p -1) u v p |∇v| p dx - Ω p u v p-1 |∇v| p-1 |∇u| dx and ⟨-∆v, v 2 -u 2 v ⟩ = Ω 2 u v (|∇v||∇u| -∇v • ∇u) + 1 + u v 2 |∇v| 2 -2 u v |∇v||∇u| dx.
Thus

I(u, v) = Ω p v u p-1 |∇u| p-2 (|∇u||∇v| -∇u • ∇v) + 1 + (p -1) v u p |∇u| p dx -p v u p-1 |∇u| p-1 |∇v| dx + Ω p u v p-1 |∇v| p-2 (|∇v||∇u| -∇v • ∇u) + 1 + (p -1) u v p |∇v| p dx -p u v p-1 |∇v| p-1 |∇u| dx + Ω 2 v u (|∇u||∇v| -∇u • ∇v) + 1 + v u 2 |∇u| 2 -2 v u |∇u||∇v| dx + Ω 2 u v (|∇v||∇u| -∇v • ∇u) + 1 + u v 2 |∇v| 2 -2 u v |∇v||∇u| dx. So I(u, v) = Ω F ( v u , ∇v, ∇u) dx + Ω G( v u , |∇v|, |∇u|) dx,
where

F (t, S, R) = p t p-1 |R| p-2 (|R||S| -R • S) + t 1-p |S| p-2 (|R||S| -R • S) + 2 {t (|R||S| -R • S)} + 2 t -1 (|R||S| -R • S)
and

G(t, s, r) = (1 + (p -1)t p ) r p + 1 + (p -1)t -p s p + (1 + t 2 )r 2 + (1 + t -2 )s 2 -pt p-1 r p-1 s -pt 1-p s p-1 r -2trs -2t -1 rs, for all t = v u > 0, R = ∇u, S = ∇v ∈ R N and r = |∇u|, s = |∇v| ∈ R + . We clearly have that F is non-negative. Now let us show that G is non-negative.
Indeed, we observe that

G(t, s, 0) = 1 + (p -1)t -p s p + (1 + t -2 )s 2 ≥ 0 and G(t, s, 0) = 0 ⇒ s = 0. If r ̸ = 0, by setting z = s tr we obtain G(t, s, r) = t p r p (z p -pz + (p -1)) + r p ((p -1)z p -pz p-1 + 1) + t 2 r 2 (z 2 -2z + 1) + r 2 (z 2 -2z + 1),
and G can be written as

G(t, s, r) = r p (t p f (z) + g(z)) + r 2 (t 2 h(z) + k(z)), with f (z) = z p -pz+(p-1), g(z) = (p-1)z p -pz p-1 +1, h(z) = k(z) = z 2 -2z+1 ∀p > 1.
We can see that f, g, h and k are non-negative. Hence G is non-negative and thus I(u, v) ≥ 0 for all (u, v) ∈ D I . In addition since f, g, h and k vanish if and only if z = 1, then G(t, s, r) = 0 if and only if s = tr. Consequently, if

I(u, v) =

Invertibility of the operator -∆ p -∆

To simplify some notations, here we set X = W 1,p 0 (Ω) and its dual

X ⋆ = W -1,p ′ (Ω), where 1 p + 1 p ′ = 1.
For the proof of the following lemma, we refer to [START_REF] Lindqvist | Notes on the p-Laplace equation[END_REF]. Lemma 3.6.4. Let p > 2. Then there exist two positive constants c 1 , c 2 such that, for all x 1 , x 2 ∈ R n , we have :

(i) (x 2 -x 1 ) • (|x 2 | p-2 x 2 -|x 1 | p-2 x 1 ) ≥ c 1 |x 2 -x 1 | p (ii) ||x 2 | p-2 x 2 -|x 1 | p-2 x 1 | ≤ c 2 (|x 2 | + |x 1 |) p-2 |x 2 -x 1 | Proposition 3.6.5. For p > 2, the operator -∆ p -∆ is a global homeomor- phism.
The proof is based on the previous Lemma 3.6.4.

Proof. Define the nonlinear operator

A : X → X ⋆ by ⟨Au, v⟩ = Ω ∇u • ∇v dx + Ω |∇u| p-2 ∇u • ∇v dx for all u, v ∈ X.
To show that -∆ p -∆ is a homeomorphism, it is enough to show that A is a continuous strongly monotone operator, [see Theorem 2.1.21].

For p > 2, for all u, v ∈ X, by (i), we get

⟨Au -Av, u -v⟩ = Ω |∇(u -v)| 2 dx + Ω |∇u| p-2 ∇u -|∇v| p-2 ∇v • ∇(u -v) dx ≥ Ω |∇(u -v)| 2 dx + c 1 Ω |∇(u -v)| p dx ≥ c 1 ∥u -v∥ p 1,p
Thus A is a strongly monotone operator. We claim that A is a continuous operator from X to X ⋆ . Indeed, assume that u n → u in X. We have to show that ∥Au n -Au∥ X ⋆ → 0 as n → ∞. Indeed, using (ii) and Hölder's inequality and the Sobolev embedding theorem, one has

|⟨Au n -Au, w⟩| ≤ Ω |∇u n | p-2 ∇u n -|∇u| p-2 ∇u |∇w| dx + Ω |∇(u n -u)||∇w| dx ≤ c 2 Ω (|∇u n | + |∇u|) p-2 |∇(u n -u)||∇w| dx + Ω |∇(u n -u)||∇w| dx ≤ c 2 Ω (|∇u n | + |∇u|) p dx p-2/p Ω |∇(u n -u)| p dx 1/p Ω |∇w| p dx 1/p + c 3 ∥u n -u∥ 1,2 ∥w∥ 1,2 ≤ c 4 (∥u n ∥ 1,p + ∥u∥ 1,p ) p-2 ∥u n -u∥ 1,p ∥w∥ 1,p + c 5 ∥u n -u∥ 1,p ∥w∥ 1,p .
Thus ∥Au n -Au∥ X ⋆ → 0, as n → +∞, and hence A is a homeomorphism.

Bifurcation of eigenvalues

In the next subsection we show that for equation (3.4.3) there is a branch of first eigenvalues bifurcating from

(λ D 1 , 0) ∈ R + × W 1,p 0 (Ω).
u = λ(-∆ p -∆) -1 u for u ∈ W -1,p ′ (Ω). (3.7.1) We set S λ (u) = u -λ(-∆ p -∆) -1 u, (3.7.2) u ∈ L 2 (Ω) ⊂ W -1,p ′ (Ω) and λ > 0. By Σ = {(λ, u) ∈ R + × W 1,p 0 (Ω)/ u ̸ = 0 , S λ (u) = 0},
we denote the set of nontrivial solutions of (4.1). A bifurcation point for (4.1) is a number λ ⋆ ∈ R + such that (λ ⋆ , 0) belongs to the closure of Σ. This is equivalent to say that, in any neighbourhood of (λ ⋆ , 0) in R + × W 1,p 0 (Ω), there exists a nontrivial solution of S λ (u) = 0. Our goal is to apply the Krasnoselski bifurcation Theorem 3.1.7. We state our bifurcation result.

Theorem 3.7.1. Let p > 2. Then every eigenvalue λ D k with odd multiplicity is a bifurcation point in R + × W 1,p 0 (Ω) of S λ (u) = 0, in the sense that in any neighbourhood of (λ D k , 0) in R + × W 1,p 0 (Ω) there exists a nontrivial solution of S λ (u) = 0.
Proof. We write the equation S λ (u) = 0 as

u = λAu + T λ (u), where Au = (-∆) -1 u and T λ (u) = [(-∆ p -∆) -1 -(-∆) -1 ](λu)
, where we consider

(-∆ p -∆) -1 : L 2 (Ω) ⊂ W -1,p ′ (Ω) → W 1,p 0 (Ω) ⊂⊂ L 2 (Ω)
and

(-∆) -1 : L 2 (Ω) ⊂ W -1,2 (Ω) → W 1,2 0 (Ω) ⊂⊂ L 2 (Ω).
For p > 2, the mapping

(-∆ p -∆) -1 -(-∆) -1 : L 2 (Ω) ⊂ W -1,p ′ (Ω) → W 1,p 0 (Ω) ⊂⊂ L 2 (Ω)
is compact thanks to Rellich-Kondrachov theorem. We clearly have A ∈ L(L 2 (Ω)) and T λ (0) = 0. Now we have to show that

(1) T λ ∈ C 1 .
(

) T ′ λ (0) = 0. 2
In order to show (1) and (2), it suffices to show that

(a) -∆ p -∆ : W 1,p 0 (Ω) → W -1,p ′ (Ω) is continuously differentiable in a neigh- borhood u ∈ W 1,p 0 (Ω). (b) (-∆ p -∆) -1 is a continuous inverse operator.
According to Proposition 3.6.5, -∆ p -∆ is a homeomorphism, hence (-∆ p -∆) -1 is continuous and this shows (b). We also recall that in section 3.6.2, we have shown that λ D 1 is simple.

Let us show (a). We claim that -∆ p : W 1,p 0 (Ω) → W -1,p ′ (Ω) is Gâteaux differentiable. Indeed, for φ ∈ W 1,p 0 (Ω) we have

⟨-∆ p (u + δv), φ⟩ -⟨-∆ p u, φ⟩ = ⟨|∇(u + δv)| p-2 ∇(u + δv), ∇φ⟩ -⟨|∇u| p-2 ∇u, ∇φ⟩ = ⟨ |∇(u + δv)| 2 p-2 2 ∇(u + δv), ∇φ⟩ -⟨|∇u| p-2 ∇u, ∇φ⟩ = ⟨ |∇u| 2 + 2δ⟨∇u, ∇v⟩ + δ 2 |∇v| 2 p-2 2 ∇(u + δv), ∇φ⟩ -⟨|∇u| p-2 ∇u, ∇φ⟩ = ⟨[|∇u| p-2 + (p -2)|∇u| 2( p-2 2 -1) δ⟨∇u, ∇v⟩ + O(δ 2 )]∇(u + δv), ∇φ⟩ -⟨|∇u| p-2 ∇u, ∇φ⟩ = ⟨[|∇u| p-2 + (p -2)|∇u| p-4 δ⟨∇u, ∇v⟩ + O(δ 2 )]∇(u + δv), ∇φ⟩ -⟨|∇u| p-2 ∇u, ∇φ⟩ = (p -2)δ|∇u| p-4 ⟨∇u, ∇v⟩⟨∇u, ∇φ⟩ + δ⟨|∇u| p-2 ∇v, ∇φ⟩ + O(δ 2 ) = δ[(p -2)|∇u| p-4 ⟨∇u, ∇v⟩⟨∇u, ∇φ⟩ + ⟨|∇u| p-2 ∇v, ∇φ⟩ + O(δ)]. Define ⟨B(u)v, φ⟩ = (p -2)|∇u| p-4 ⟨∇u, ∇v⟩⟨∇u, ∇φ⟩ + ⟨|∇u| p-2 ∇v, ∇φ⟩ and let (u n ) n≥0 ⊂ W 1,p 0 (Ω). Assume that u n → u, as n → ∞ in W 1,p 0 (Ω).
We have

⟨B(u n )v -B(u)v, φ⟩ = (p -2) |∇u n | p-4 ⟨∇u n , ∇v⟩⟨∇u n , ∇φ⟩ -|∇u| p-4 ⟨∇u, ∇v⟩⟨∇u, ∇φ⟩ + ⟨|∇u n | p-2 ∇v, ∇φ⟩ -⟨|∇u| p-2 ∇v, ∇φ⟩. Therefore, |⟨B(u n )v -B(u)v, φ⟩| ≤ (p -2) |∇u n | p-4 ⟨∇u n , ∇v⟩⟨∇u n , ∇φ⟩ -|∇u| p-4 ⟨∇u, ∇v⟩⟨∇u, ∇φ⟩ + |∇u n | p-2 -|∇u| p-2 |⟨∇v, ∇φ⟩|.
By assumption, we can assume that, up to subsequences,

( * ) ∇u n → ∇u in (L p (Ω)) N as n → ∞ and
( * * ) ∇u n (x) → ∇u(x) almost everywhere as n → ∞.

Then |∇u n | p-4 ⟨∇u n , ∇v⟩⟨∇u n , ∇φ⟩ → |∇u| p-4 ⟨∇u, ∇v⟩⟨∇u, ∇φ⟩ as n → ∞ and consequently ⟨B(u n )v, φ⟩ → ⟨B(u)v, φ⟩as n → ∞. Thus, we find that -∆ p -∆ ∈ C 1 and thanks to the Inverse function theorem (-∆ p -∆) -1 is differentiable in a neighborhood of u ∈ W 1,p 0 (Ω). Therefore according to the Krasnoselski bifurcation Theorem, we obtain that λ D k is a bifurcation point at zero.

Bifurcation from infinity : the case 1 < p < 2

We recall the nonlinear eigenvalue problem we are studying,

-∆ p u -∆u = λu in Ω, u = 0 on ∂Ω. (3.7.3)
Under a solution of (3.7.3) (for 1 < p < 2), we understand a pair (λ, u) ∈

R + ⋆ × W 1,2 0 (Ω) satisfying the integral equality, Ω |∇u| p-2 ∇u • ∇φ dx + Ω ∇u • ∇φ dx = λ Ω uφ dx for every φ ∈ W 1,2 0 (Ω). ( 3
For u ∈ W 1,2 0 (Ω), u ̸ = 0, we set v = u/∥u∥ 2-1 2 p 1,2 . We have ∥v∥ 1,2 = 1 ∥u∥ 1-1 2 p 1,2 and |∇v| p-2 ∇v = 1 ∥u∥ (2-1 2 p)(p-1) 1,2 |∇u| p-2 ∇u.
Introducing this change of variable in (3.7.4), we find that,

∥u∥ (2-1 2 p)(p-2) 1,2 Ω |∇v| p-2 ∇v•∇φ dx+ Ω ∇v•∇φ dx = λ Ω vφ dx for every φ ∈ W 1,2 0 (Ω). (3.7.5)
But, on the other hand, we have

∥v∥ p-4 1,2 = 1 ∥u∥ (1-1 2 p)(p-4) 1,2 = 1 ∥u∥ (2-1 2 p)(p-2) 1,2 .
Consequently it follows that equation (3.7.5) is equivalent to

∥v∥ 4-p 1,2 Ω |∇v| p-2 ∇v•∇φ dx+ Ω ∇v•∇φ dx = λ Ω vφ dx for every φ ∈ W 1,2 0 (Ω). (3.7.6)
This leads to the following nonlinear eigenvalue problem (for 1 < p < 2)

-∥v∥ 4-p 1,2 ∆ p v -∆v = λv in Ω, v = 0 on ∂Ω. (3.7.7)
The proof of Theorem 3.7.2 follows immediately from the following remark, and the proof that (λ D 1 , 0) is a bifurcation of (3.7.7). Let us consider a small ball B r (0) := { w ∈ W 1,2 0 (Ω)/ ∥w∥ 1,2 < r }, and consider the operator

T := -∥ • ∥ 4-p 1,2 ∆ p -∆ : W 1,2 0 (Ω) → W -1,2 (Ω).
Proposition 3.7.4. Let 1 < p < 2. There exists r > 0 such that the mapping

T : B r (0) ⊂ W 1,2 0 (Ω) → W -1,2
(Ω) is invertible, with a continuous inverse.

Proof. In order to prove that the operator T is invertible with a continuous inverse, we again rely on Theorem 2.1.21. We show that there exists δ > 0 such that

⟨T (u) -T (v), u -v⟩ ≥ δ∥u -v∥ 2 1,2 , for u, v ∈ B r (0) ⊂ W 1,2 0 (Ω)
with r > 0 sufficiently small. Indeed, using that -∆ p is strongly monotone on W 1,p 0 (Ω) on the one hand and the Hölder inequality on the other hand, we have

⟨T (u) -T (v), u -v⟩ = ∥∇u -∇v∥ 2 2 + ∥u∥ 4-p 1,2 (-∆ p u) -∥v∥ 4-p 1,2 (-∆ p v), u -v = ∥u -v∥ 2 1,2 + ∥u∥ 4-p 1,2 ((-∆ p u) -(-∆ p v), u -v) + ∥u∥ 4-p 1,2 -∥v∥ 4-p 1,2 (-∆ p v, u -v) ≥ ∥u -v∥ 2 1,2 -∥u∥ 4-p 1,2 -∥v∥ 4-p 1,2 ∥∇v∥ p-1 p ∥∇(u -v)∥ p ≥ ∥u -v∥ 2 1,2 -∥u∥ 4-p 1,2 -∥v∥ 4-p 1,2 C∥v∥ p-1 1,2 ∥u -v∥ 1,2 . (3.7.8)
Now, we obtain by the Mean Value Theorem that there exists θ ∈ [0, 1] such that

∥u∥ 4-p 1,2 -∥v∥ 4-p 1,2 = d dt ∥u + t(v -u)∥ 2 1,2 2-1 2 p | t=θ (v -u) = (2 - 1 2 p) ∥u + θ(v -u)∥ 2 1,2 1-1 2 p 2 (u + θ(v -u), v -u) 1,2 ≤ (4 -p)∥u + θ(v -u)∥ 2-p 1,2 ∥u + θ(v -u)∥ 1,2 ∥u -v∥ 1,2 = (4 -p)∥u + θ(v -u)∥ 3-p 1,2 ∥u -v∥ 1,2 ≤ (4 -p) ((1 -θ)∥u∥ 1,2 + θ∥v∥ 1,2 ) 3-p ∥u -v∥ 1,2 ≤ (4 -p)r 3-p ∥u -v∥ 1,2 .
Hence, continuing with the estimate of equation (3.7.8), we get

⟨T (u) -T (v), u -v⟩ ≥ ∥u -v∥ 2 1,2 (1 -(4 -p)r 3-p Cr p-1 ) = ∥u -v∥ 2 1,2 (1 -C ′ r 2 ),
and thus the claim, for r > 0 small enough. Hence, the operator T is strongly monotone on B r (0) and it is continuous, and hence the claim follows.

Clearly the mappings

T τ = -∆ -τ ∥ • ∥ γ 1,2 ∆ p : B r (0) ⊂ W 1,2 0 (Ω) → W -1,2 (Ω), 0 ≤ τ ≤ 1
are also local homeomorphisms for 1 < p < 2 with γ = 4 -p > 0. Consider now the homotopy maps

H(τ, y) := (-τ ∥ • ∥ γ 1,2 ∆ p -∆) -1 (y), y ∈ T τ (B r (0)) ⊂ W -1,2 (Ω).
Then we can find a ρ > 0 such that the ball

B ρ (0) ⊂ 0≤τ ≤1 T τ (B r (0)) and H(τ, •) : B ρ (0) ∩ L 2 (Ω) → W 1,2 0 (Ω) ⊂⊂ L 2 (Ω)
are compact mappings. Set now

Sλ (u) = u -λ(-∥u∥ γ 1,2 ∆ p -∆) -1 u.
Notice that Sλ is a compact perturbation of the identity in L 2 (Ω). We have 0 / ∈ H([0, 1]×∂B r (0)). So it makes sense to consider the Leray-Schauder topological degree of H(τ, •) on B r (0). And by the property of the invariance by homotopy, one has deg(H(0, •), B r (0), 0) = deg(H(1, •), B r (0), 0).

(3.7.9)

Theorem 3.7.5. The pair

(λ D 1 , 0) is a bifurcation point in R + × L 2 (Ω) of Sλ (u) = 0, for 1 < p < 2.
Proof. Suppose by contradiction that (λ D 1 , 0) is not a bifurcation for Sλ . Then, there exist δ 0 > 0 such that for all r ∈ (0, δ 0 ) and ε ∈ (0, δ 0 ),

Sλ (u) ̸ = 0 ∀ |λ D 1 -λ| ≤ ε, ∀ u ∈ L 2 (Ω), ∥u∥ 2 = r. (3.7.10)
Taking into account that (3.7.10) holds, it follows that it make sense to consider the Leray-Schauder topological degree deg( Sλ , B r (0), 0) of Sλ on B r (0).

We observe that

I -(λ D 1 -ε)H(τ, •) | ∂Br(0) ̸ = 0 for τ ∈ [0, 1]. ( 3
v ∈ ∂B r (0) ⊂ L 2 (Ω) such that v -(λ D 1 -ε)H(τ, v) = 0, for some τ ∈ [0, 1]. One concludes that then v ∈ W 1,2
0 (Ω), and then that

-∆v -τ ∥v∥ γ 1,2 ∆ p v = (λ D 1 -ε)v.
However, we get the contradiction,

(λ D 1 -ε)∥v∥ 2 2 = ∥∇v∥ 2 2 + τ ∥v∥ γ 1,2 ∥∇v∥ p p ≥ ∥∇v∥ 2 2 ≥ λ D 1 ∥v∥ 2 2 .
By the contradiction assumption, we have

deg(I -(λ D 1 + ε)H(1, •), B r (0), 0) = deg(I -(λ D 1 -ε)H(1, •), B r (0), 0). (3.7.12)
By homotopy using (3.7.9), we have

deg(I -(λ D 1 -ε)H(1, •), B r (0), 0) = deg(I -(λ D 1 -ε)H(0, •), B r (0), 0) = deg(I -(λ D 1 -ε)(-∆) -1 , B r (0), 0) = 1 (3.7.13)
Now, using (3.7.13) and (3.7.12), we find that

deg(I -(λ D 1 + ε)H(1, •), B r (0), 0) = deg(I -(λ D 1 -ε)H(0, •), B r (0), 0) = 1 (3.7.14)
Furthermore, since λ D 1 is a simple eigenvalue of -∆, it is well-known [see [START_REF] Ambrosetti | Nonlinear analysis and semilinear elliptic problems[END_REF]] that

deg(I -(λ D 1 + ε)(-∆) -1 , B r (0), 0) = deg(I -(λ D 1 + ε)H(0, •), B r (0), 0) = -1 (3.7.15)
In order to get contradiction (to relation (3.7.14)), it is enough to show that,

deg(I -(λ D 1 + ε)H(1, •), B r (0), 0) = deg(I -(λ D 1 + ε)H(0, •), B r (0), 0), (3.7.16)
r > 0 sufficiently small. We have to show that

I -(λ D 1 + ε)H(τ, •) | ∂Br(0) ̸ = 0 for τ ∈ [0, 1].
Suppose by contradiction that there is

r n → 0, τ n ∈ [0, 1] and u n ∈ ∂B rn (0) such that u n -(λ D 1 + ε)H(τ n , u n ) = 0 or equivalently -τ n ∥u n ∥ γ 1,2 ∆ p u n -∆u n = (λ D 1 + ε)u n . (3.7.17)
Dividing the equation (3.7.17) by ∥u n ∥ 1,2 , we obtain

-τ n ∥u n ∥ γ+p-1 1,2 ∆ p u n ∥u n ∥ 1,2 -∆ u n ∥u n ∥ 1,2 = (λ D 1 + ε) u n ∥u n ∥ 1,2 ,
and by setting v n = un ∥un∥ 1,2 , it follows that

-τ n ∥u n ∥ γ+p-1 1,2 ∆ p v n -∆v n = (λ D 1 + ε)v n . (3.7.18) But since ∥v n ∥ 1,2 = 1, we have v n ⇀ v in W 1,2 0 (Ω) and v n → v in L 2 (Ω)
. Furthermore, the first term in the left hand side of equation (3.7.18) tends to zero in W -1,p ′ (Ω) as r n → 0 and hence in W -1,2 (Ω). Equation (3.7.17) then implies that

v n → v strongly in W 1,2 0 (Ω) since -∆ : W 1,2 0 (Ω) → W -1,2 (Ω) is a homeomorphism and thus v with ∥v∥ 1,2 = 1 solves -∆v = (λ D 1 + ε)v, which is impossible because λ D 1 +ε is not the first eigenvalue of -∆ on W 1,2 0 (Ω) for ε > 0.
Therefore, by homotopy it follows that

deg(I -(λ D 1 + ε)H(1, •), B r (0), 0) = deg(I -(λ D 1 + ε)H(0, •), B r (0), 0).
Now, thanks to (3.7.15), we find that

deg(I -(λ D 1 + ε)H(1, •), B r (0), 0) = -1,
which contradicts equation (3.7.14).

Theorem 3.7.6. The pair

(λ D k , 0) (k > 1) is a bifurcation point of Sλ (u) = 0, for 1 < p < 2 if λ D k is of odd multiplicity.
Proof. Suppose by contradiction that (λ D k , 0) is not a bifurcation for Sλ . Then, there exist δ 0 > 0 such that for all r ∈ (0, δ 0 ) and ε ∈ (0, δ 0 ),

Sλ (u) ̸ = 0 ∀ |λ D k -λ| ≤ ε, ∀ u ∈ L 2 (Ω), ∥u∥ 2 = r. (3.7.19)
Taking into account that (3.7.19) holds, it follows that it make sense to consider the Leray-Schauder topological degree deg( Sλ , B r (0), 0) of Sλ on B r (0). We show that

I -(λ D k -ε)H(τ, •) | ∂Br(0) ̸ = 0 for τ ∈ [0, 1]. ( 3
v ∈ ∂B r (0) ⊂ L 2 (Ω) such that v -(λ D k -ε)H(τ, v) = 0, for some τ ∈ [0, 1]. This is equivalent of having -∆v -(λ D k -ε)v = τ ∥v∥ γ 1,2 ∆ p v. ( 3
⟨[-∆-(λ D k -ε)](P + v+P -v), P + v-P -v⟩ = τ ∥P + v+P -v∥ γ 1,2 ⟨∆ p [P + v+P -v], P + v-P -v⟩ ⇕ -∥∇P -v∥ 2 2 -(λ D k -ε)∥P -v∥ 2 2 + ∥∇P + v∥ 2 2 -(λ D k -ε)∥P + v∥ 2 2 = τ ∥P + v + P -v∥ γ 1,2 × ⟨∆ p [P + v + P -v], P + v -P -v⟩. But ⟨∆ p [P + v+P -v], P + v-P -v⟩ = - Ω |∇(P + v+P -v)| p-2 ∇(P + v+P -v)•∇(P + v-P -v) dx,
and using the Hölder inequality, the embedding W 1,2 0 (Ω) ⊂ W 1,p 0 (Ω) and the fact that P + v and P -v don't vanish simultaneously, there is some positive constant C ′ > 0 such that

∥P + v-P -v∥ 1,2 ≤ C ′ (∥P + v∥ 2 1,2 +∥P -v∥ 2 1,2 ) = C ′ ∥P + v-P -v∥ 2 1,2 , since (P + v, P -v) 1,2 = 0, we have ⟨∆ p [P + v + P -v], P + v -P -v⟩ ≤ ∥P + v + P -v∥ p-1 1,p ∥P + v -P -v∥ 1,p ≤ C ′ ∥P + v + P -v∥ p-1 1,2 ∥P + v -P -v∥ 2 1,2 ≤ C ′ ∥P + v + P -v∥ p+1 1,2 , since ∥P + v -P -v∥ 2 1,2 = ∥P + v + P -v∥ 2 1,2 .
On the other hand, thanks to the Poincaré inequality as well as the variational characterization of eigenvalues we find

-∥∇P -v∥ 2 2 -(λ D k -ε)∥P -v∥ 2 2 ≥ 0 and ∥∇P + v∥ 2 2 -(λ D k -ε)∥P + v∥ 2 2 ≥ 0,
we can bound from below these two inequalities together by ∥∇P + v∥ 2 2 +∥∇P -v∥ 2 2 . Finally, we have

∥v∥ 2 1,2 = ∥∇P + v∥ 2 2 + ∥∇P -v∥ 2 2 ≤ τ C ′ ∥P + v + P -v∥ γ+p+1 1,2 , with γ = 4 -p, ⇕ ∥v∥ 2 1,2 ≤ C ′′ ∥v∥ γ+p+1 1,2 ⇔ 1 ≤ C ′′ r 3 → 0,
for r taken small enough. This shows that (3.7.20) holds. By the contradiction assumption, we have

deg(I -(λ D k + ε)H(1, •), B r (0), 0) = deg(I -(λ D k -ε)H(1, •), B r (0), 0). (3.7.22)
By homotopy using (3.7.20), we have

deg(I -(λ D k -ε)H(1, •), B r (0), 0) = deg(I -(λ D k -ε)H(0, •), B r (0), 0) (3.7.23) = deg(I -(λ D k -ε)(-∆) -1 , B r (0), 0) = (-1) β ,
where β is the sum of algebraic multiplicities of the eigenvalues λ D k -ε < λ. Similarly, if β ′ denotes the sum of the algebraic multiplicities of the characteristic values of (-∆) -1 such that λ > λ D k + ε, then

deg(I -(λ D k + ε)H(1, •), B r (0), 0) = (-1) β ′ (3.7.24) But since [λ D k -ε, λ D k + ε]
contains only the eigenvalue λ D k , it follows that β ′ = β + α, where α denotes the algebraic multiplicity of λ D k . Consequently, we have

deg(I -(λ D k + ε)H(1, •), B r (0), 0) = (-1) β+α = (-1) α deg(I -(λ D k + ε)H(1, •), B r (0), 0) = -deg(I -(λ D k + ε)H(1, •), B r (0), 0),
since λ D k is with odd multiplicity. This contradicts (3.7.22).

Multiple solutions

In this section we prove multiciplity results by distinguishing again the two cases 1 < p < 2 and p > 2. Let Γ be the class of all the symmetric subsets A ⊆ X\{0} which are closed in X\{0}. Proof. Case 1:

1 < p < 2.
In this case we will avail of [START_REF] Ambrosetti | Nonlinear analysis and semilinear elliptic problems[END_REF]Proposition 10.8]. We consider the energy functional I λ : W 1,2 0 (Ω)\{0} → R associated to the problem (3.4.3) defined by

I λ (u) = 2 p Ω |∇u| p dx + Ω |∇u| 2 dx -λ Ω u 2 dx.
The functional I λ is not bounded from below on W 1,2 0 (Ω), so we consider again the natural constraint set, the Nehari manifold on which we minimize the functional I λ . The Nehari manifold is given by

N λ := {u ∈ W 1,2 0 (Ω)\{0} : ⟨I ′ λ (u), u⟩ = 0}.
On N λ , we have

I λ (u) = ( 2 p -1)
Ω |∇u| p dx > 0. We clearly have that, I λ is even and bounded from below on N λ . Now, let us show that every (PS) sequence for I λ has a converging subsequence on N λ . Let (u n ) n be a (P S) sequence, i.e, |I λ (u n )| ≤ C, for all n, for some C > 0 and

I ′ λ (u n ) → 0 in W -1,2
(Ω) as n → +∞. We first show that the sequence (u n ) n is bounded on N λ . Suppose by contradiction that this is not true, so

Ω |∇u n | 2 dx → +∞ as n → +∞. Since I λ (u n ) = ( 2 p -1) Ω |∇u n | p dx we have Ω |∇u n | p dx ≤ c. On N λ , we have 0 < Ω |∇u n | p dx = λ Ω u 2 n dx - Ω |∇u n | 2 dx, (3.8.1)
and hence

Ω u 2 n dx → +∞. Let v n = un ∥un∥ 2 then Ω |∇v n | 2 dx ≤ λ and hence v n is bounded in W 1,2 0 (Ω). Therefore there exists v 0 ∈ W 1,2 0 (Ω) such that v n ⇀ v 0 in W 1,2 0 (Ω) and v n → v 0 in L 2 (Ω). Dividing (3.8.1) by ∥u n ∥ p 2 , we have λ Ω u 2 n dx - Ω |∇u n | 2 dx ∥u n ∥ p 2 = Ω |∇v n | p dx → 0, since λ Ω u 2 n dx - Ω |∇u n | 2 dx = ( 2 p -1) -1 I λ (u n ), |I λ (u n )| ≤ C and ∥u n ∥ p 2 → +∞. Now, since v n ⇀ v 0 in W 1,2 0 (Ω) ⊂ W 1,p 0 (Ω), we infer that Ω |∇v 0 | p dx ≤ lim inf n→+∞ Ω |∇v n | p dx = 0,
and consequently v 0 = 0. So v n → 0 in L 2 (Ω) and this is a contradiction since

∥v n ∥ 2 = 1. So (u n ) n is bounded on N λ .
Next, we show that u n converges strongly to u in W 1,2 0 (Ω). To do this, we will use the following vectors inequality for

1 < p < 2 (|x 2 | p-2 x 2 -|x 1 | p-2 x 1 ) • (x 2 -x 1 ) ≥ C ′ (|x 2 | + |x 1 |) p-2 |x 2 -x 1 | 2 ,
for all x 1 , x 2 ∈ R N and for some C ′ > 0, [see [START_REF] Lindqvist | Notes on the p-Laplace equation[END_REF]].

We have

Ω u 2 n dx → Ω u 2 dx and since I ′ λ (u n ) → 0 in W -1,2 (Ω), u n ⇀ u in W 1,2
0 (Ω), we also have I ′ λ (u n )(u n -u) → 0 and I ′ λ (u)(u n -u) → 0 as n → +∞. On the other hand, one has

⟨I ′ λ (u n ) -I ′ λ (u), u n -u⟩ = 2 Ω |∇u n | p-2 ∇u n -|∇u| p-2 ∇u • ∇(u n -u) dx + 2 Ω |∇(u n -u)| 2 dx -2λ Ω |u n -u| 2 dx ≥ C ′ Ω (|∇u n | + |∇u|) p-2 |∇(u n -u)| 2 dx + 2 Ω |∇(u n -u)| 2 dx -2λ Ω |u n -u| 2 dx ≥ 2 Ω |∇(u n -u)| 2 dx -2λ Ω |u n -u| 2 dx ≥ ∥u n -u∥ 2 1,2 -λ Ω |u n -u| 2 dx.
Therefore ∥u n -u∥ 1,2 → 0 as n → +∞ and u n converges strongly to u in W 1,2 0 (Ω).

Let Σ ′ = {A ⊂ N λ : A closed and -A = A} and Γ j = {A ∈ Σ ′ : γ(A) ≥ j}, where γ(A) denotes the Krasnoselski's genus. We show that Γ j ̸ = ∅. Set E j = span{e i , i = 1, . . . , j}, where e i are the eigenfunctions associated to the problem (3.4.1). Let λ ∈ (λ D j , λ D j+1 ), and consider v ∈ S j := {v ∈ E j : Ω |v| 2 dx = 1}. Then set In this case, we will rely on Theorem 3.3.8.

ρ(v) = Ω |∇v| p dx λ Ω v 2 dx -Ω |∇v| 2 dx 1 2-p . Then λ Ω v 2 dx-Ω |∇v| 2 dx ≥ λ Ω v 2 dx- j i=1 Ω λ i |e i | 2 dx ≥ (λ-λ j ) Ω |v| 2 dx > 0. Hence, ρ(v)v ∈ N λ ,
Let us consider the C 1 energy functional I λ : W 1,p 0 (Ω) → R defined as

I λ (u) = 2 p Ω |∇u| p dx + Ω |∇u| 2 dx -λ Ω |u| 2 dx.
We want to show that

-∞ < σ j = inf {A∈Σ ′ ,γ(A)≥j} sup u∈A I λ (u) (3.8.2)
is a critical point for I λ , where

Σ ′ = {A ⊆ S j }, where S j = {v ∈ E j : Ω |v| 2 dx = 1}.
We clearly have that I λ (u) is an even functional for all u ∈ W 1,p 0 (Ω), and also

I λ (u) is bounded from below on W 1,p 0 (Ω) since I λ (u) ≥ C∥u∥ p 1,p -C ′ ∥u∥ 2 1,p .
We show that I λ (u) satisfies the (PS) condition. Let {u n } be a Palais-Smale sequence, i.e., |I λ (u n )| ≤ M for all n, M > 0 and

I ′ λ (u n ) → 0 in W -1,p ′ (Ω) as n → ∞. We first show that {u n } is bounded in W 1,p 0 (Ω). We have M ≥ |C∥u n ∥ p 1,p -C ′ ∥u n ∥ 2 1,p | ≥ C∥u n ∥ p-2 1,p -C ′ ∥u n ∥ 2 1,p ,
and so {u n } is bounded in W 1,p 0 (Ω). Therefore, u ∈ W 1,p 0 (Ω) exists such that, up to subsequences that we will denote by (u n ) n we have u n ⇀ u in W 1,p 0 (Ω) and u n → u in L 2 (Ω). We will use the following inequality for v 1 , v 2 ∈ R N : there exists R > 0 such that

|v 1 -v 2 | p ≤ R |v 1 | p-2 v 1 -|v 2 | p-2 v 2 (v 1 -v 2 ),
for p > 2 [see [START_REF] Lindqvist | Notes on the p-Laplace equation[END_REF]]. Then we obtain

⟨I ′ λ (u n ) -I ′ λ (u), u n -u⟩ = 2 Ω |∇u n | p-2 ∇u n -|∇u| p-2 ∇u • ∇(u n -u) dx + 2 Ω |∇u n -∇u| 2 dx -2λ Ω |u n -u| 2 dx ≥ 2 R Ω |∇u n -∇u| p dx + 2 Ω |∇u n -∇u| 2 dx -2λ Ω |u n -u| 2 dx ≥ 2 R ∥u n -u∥ p 1,p -2λ Ω |u n -u| 2 dx.
Therefore ∥u n -u∥ 1,p → 0 as n → +∞, and so u n converges to u in W 1,p 0 (Ω).

Next, we show that there exists sets A j of genus j = 1, . . . , k such that sup

u∈A j I λ (u) < 0.
Consider E j = span{e i , i = 1, . . . , j} and S j = {v ∈ E j : Ω |v| 2 dx = 1}.

For any s ∈ (0, 1), we define the set A j (s) := s(S j ∩ E j ) and so γ(A j (s)) = j for j = 1, . . . , k. We have, for any s ∈ (0, 1) sup

u∈A j I λ (u) = sup v∈S j ∩E j I λ (sv) ≤ sup v∈S j ∩E j s p p Ω |∇v| p dx + s 2 2 Ω |∇v| 2 dx - λs 2 2 Ω |v| 2 dx ≤ sup v∈S j ∩E j s p p Ω |∇v| p dx + s 2 2 (λ j -λ) < 0
for s > 0 sufficiently small, since Ω |∇v| p dx ≤ c j , where c j denotes some positive constant. Finally, we conclude that σ λ,j (j = 1, . . . , k) are critical values thanks to Clark's Theorem.

Part B: Control theory

For more details about what we recall in this chapter we refer to [START_REF] Rousseau | Elliptic Carleman estimates and applications to stabilization and controllability[END_REF].

Pseudo-differential operators, oscillatory integrals, parametrices

Pseudo-differential operators with large parameter

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 functions norms we also use the notation ∥ • ∥ for functions defined in the interior of the domain and | • | for functions defined on the boundaries.

To motivate the form of the pseudo-differential operators, which we will present below, we first formulate differential operators by the means of the Fourier transformation. Suppose that q(x, ξ, τ ) is a polynomial in (ξ, τ ) of order less than or equal to m, with x, ξ ∈ R d , and τ ≥ 1. We write it in the form q(x, ξ, τ ) = |α|+k≤m a α (x)ξ α τ k , and we set q(x, D, τ )u = |α|+k≤m a α (x)τ k D α u. For u in the Schawrtz space S (R d ), we denote by û the Fourier transform, that is

û(ξ) = R d e -ix•ξ u(x)dx.
Observing that

D α u = 1 (2π) d R d e ix•ξ ξ α û(ξ) dξ for u ∈ S (R d ), we write q(x, D, τ )u = 1 (2π) d |α|+k≤m τ k R d e ix•ξ a α (x)ξ α û(ξ)dξ, that is q(x, D, τ )u = 1 (2π) d R d e ix•ξ q(x, ξ, τ )û(ξ)dξ.
We also note that |q(x, ξ, τ

)| ≲ (τ + |ξ|) m (4.1.1)
and for all α, β ∈ N d ,

|∂ α x ∂ β ξ q(x, ξ, τ )| ≤ C αβ (τ + |ξ|) m-|β| , (4.1.2)
for |β| ≤ m and ∂ α x ∂ β ξ q(x, ξ, τ ) = 0 for |β| > m. We wish to generalize such differential operators Q(x, D, τ ) that involves a large parameter such as τ, to the case of more general functions q(x, ξ, τ ).

Semi-classical calculus acting on R d

Here, 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 for u ∈ S (R d + ). We say that a is the symbol of A. We denote Ψ m τ the set of pseudo-differential 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 function of order m in (ξ, τ ).

|∂ 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, ∞).
We have the following properties for symbols.

Proposition 4.1.1. i. If a ∈ S m τ , then ∂ α x ∂ β ξ a ∈ S m-|β| τ . ii. If a ∈ S m τ , and b ∈ S m ′ τ then ab ∈ S m+m ′ τ . iii. If m ≤ m ′ , then S m τ ⊂ S m ′ τ . Proof. Let α, β, α ′ , β ′ ∈ N d .
i. We have

∂ α ′ x ∂ β ′ ξ ∂ α x ∂ β ξ a(x, ξ, τ ) = ∂ α ′ x ∂ β ′ ξ ∂ α x ∂ β ξ a(x, ξ, τ ) = ∂ α ′ +α x ∂ β ′ +β ξ a(x, ξ, τ ) ≤ C α ′′ ,β ′′ λ m-(|β|+|β ′ |) τ ≤ C α ′ +α,β ′ +β λ (m-|β|)-|β ′ | τ . Hence ∂ α x ∂ β ξ a ∈ S m-|β| .
ii. For all multi-indices δ, γ there exists C δ,γ , C ′ δ,γ > 0 such that

∂ δ x ∂ γ ξ a(x, ξ, τ ) ≤ C δ,γ λ m-|γ| τ , ∂ δ x ∂ γ ξ b(x, ξ, τ ) ≤ C ′ δ,γ λ m ′ -|γ| τ .
In addition, using Leibniz formula, we have

∂ α x ∂ β ξ (a(x, ξ, τ )b(x, ξ, τ )) ≤ γ≤β β γ ∂ α x ∂ β-γ ξ a(x, ξ, τ )∂ γ ξ b(x, ξ, τ ) ≤ γ≤β β γ δ≤α α δ ∂ α-δ x ∂ β-γ ξ a(x, ξ, τ ) ∂ δ x ∂ γ ξ b(x, ξ, τ ) ≤ γ≤β δ≤α β γ α δ C α-δ,β-γ λ m-|β-γ| τ C ′ δ,γ λ m ′ -|γ| τ ≤ γ≤β δ≤α β γ α δ C ′ δ,γ C α-δ,β-γ λ m+m ′ -|β| τ , since |β| ≤ |β -γ| + |γ|, ≤ Cλ m+m ′ -|β| τ , with C = γ≤β δ≤α β γ α δ C ′ δ,γ C α-δ,β-γ . Thus ab ∈ S m+m ′ τ . Let m ≤ m ′ . We have λ m τ ≤ λ m ′ τ and λ m-|β| τ ≤ λ m ′ -|β| τ . So, if a ∈ S m τ , it follows that for all α, β ∈ N d , there exists C α,β > 0 such that ∂ α x ∂ β ξ a(x, ξ, τ ) ≤ C α,β λ m-|β| ≤ C α,β λ m ′ -|β| τ . Thus a ∈ S m τ .
Remark 4.1.2. Formally we can define

Au(x) = 1 (2π) d R d R d e i(x-y)•ξ a(x, ξ, τ )u(y, x d )dy dξ.
Such a double integral may not have a meaning in the classical sense, e.g. Lebesgue integration. Yet it has a very precise definition and meaning in the sense of so-called oscillatory integrals.

Remark 4.1.3. The pseudo-differential operators defined above apply to functions defined in the whole R d , through the use of the Fourier transformation.

Below, we shall introduce tangential pseudo-differential operators that can act on functions defined on a half-space. Proof. For all 1 ≤ ℓ ≤ d, we have

D x ℓ (Op(a)u)(x) = 1 (2π) d R d e ix•ξ ( ξ ℓ a(x, ξ, τ ) + D x ℓ a(x, ξ, τ ) ∈S m+1 τ )û(ξ)dξ,
and by induction we find that for α multi-index,

D α x (Op(a)u)(x) = 1 (2π) d R d e ix•ξ b(x, ξ, τ ) ∈S m+|α| τ û(ξ) dξ.
We also have

x ℓ (Op(a)u)(x) = (2π) -d R d D ξ ℓ e ix•ξ a(x, ξ, τ )û(ξ) dξ = -(2π) -d R d e ix•ξ (D ξ ℓ a(x, ξ, τ )û(ξ) + a(x, ξ, τ )D ξ ℓ û(ξ)) dξ,
and by induction,

x α (Op(a)u)(x) = (-1) |α| (2π) -d R d e ix•ξ β≤α α β D β ξ ℓ a(x, ξ, τ ) ∈S m-|β| τ D α-β ξ ℓ û(ξ) dξ.
Combining the two formulae, for all multi-indices α et β, we obtain the existence of semi-norms p j ∈ S m (R d × R d ) and q j , q ′ j in S (R d ) such that

|x α D α x (Op(a)u)(x)| ≤ C α,β (τ ) j p j (a)q j (û) ≤ C α,β (τ ) j p j (a)q ′ j (û),
by Proposition 4.1.1 and this shows the continuity result.

The action of pseudo-differential operators can extended to temperate distributions.

Oscillatory integrals

Oscillatory integral are useful for the definition of the Schwartz kernel of pseudodifferential operators (and many other operators) and also for the understanding of the pseudo-differential calculus. For a review of Schwartz kernel we refer to [START_REF] Rousseau | Elliptic Carleman estimates and applications to stabilization and controllability[END_REF]. We shall give a precise meaning to integrals of the form

1 (2π) d R d ×R d e i(x-y)•ξ a(x, ξ, τ )u(y) dydξ, u ∈ S (R d ), x ∈ R d , for a ∈ S +∞ τ (R d × R d
) and view the Schwartz kernel of Op(a),

K τ (x, y) = 1 (2π) d R d
e i(x-y)•ξ a(x, ξ, τ )u(y) dξ, as a distribution. Actually these two integrals are perfectly well defined if

|a(x, ξ, τ )| ≲ ⟨ξ⟩ m , with m < -d where ⟨•⟩ = (1+|•| 2 ) 1/2 . This holds in particular if a ∈ S m τ (R d ×R d
) and m < -d. Yet, for m ≥ -d, the meaning of the two integrals may not be clear according to the classical integration theories. This type of integral is called oscillatory because of the phase term e i(x-y)•ξ . We shall in fact introduce more general phase functions, to be denoted by Φ here, and give a precise meaning to the following type of integral (3) for all α, β, there exists C α,β > 0 such that

|ξ| |β| |∂ α x ∂ β ξ Φ(x, ξ)| ≤ C α,β |ξ|, x ∈ R p , ξ ∈ R d .
(4) there exists C > 0 such that

|d x Φ| 2 + |ξ| 2 |d ξ Φ| 2 ≥ C|ξ| 2 , x ∈ R p , ξ ∈ R d .
Then, the functional

I Φ (a, u, τ ) = R p ×R d e iΦ(x,ξ) a(x, ξ, τ )u(x)dξdx,
that is well defined for u ∈ S (R p ) and a ∈ S -d-ε τ (R p × R d ), ε > 0, can be extended in a unique manner by continuity to all a ∈ S m τ (R p × R d ), for all m ∈ R. Moreover as a distribution in S ′ (R p ), the map u → I Φ (a, u, τ ), is of order less or equal k for all k > m + d.

Note that

Φ : R d × R d × R d → R (x, y, ξ) → (x -y) • ξ
satisfies the assumption made on the phase function in Theorem 4.2.1. Thus for

a ∈ S m τ (R d × R d ) the map w → (2π) -d R d ×R d ×R d e i(x-y)•ξ a(x, ξ, τ )w(x, y) dξdydx is a distribution in S (R d × R d )
in the x, y variables of order less or equal k for all k > m + d. This allows one to write the Schwartz kernel of the operator Op(a) as

K τ (x, y) = 1 (2π) d R d e i(x-y)•ξ a(x, ξ, τ )u(y) dξ,
and with the kernel theorem (see [START_REF] Rousseau | Elliptic Carleman estimates and applications to stabilization and controllability[END_REF]Theorem 8.45]) we have for u, v ∈ S (R d ),

⟨Op(a)u(x), v(y)⟩ S ′ (R d ),S (R d ) = ⟨K τ , u ⊗ v⟩ S ′ (R 2d ),S (R 2d ) = (2π) -d R d ×R d ×R d e i(x-y)•ξ a(x, ξ, τ ) v(x)u(y) dξdxdy.
We recall that the kernel theorem states that Op(a)u(

x) = ⟨K τ (x, •), u(•)⟩ S ′ (R d ),S (R d ) ∈ S ′ (R d ) for u ∈ S (R d ).
Actually, in the present case we have Op(a)u(x) ∈ S (R d ) by Proposition 4.1.4. Observe that Theorem 4.2.1 gives a precise meaning to the formula

Op(a)u(x) = ⟨K τ (x, •), u(•)⟩ S ′ (R d ),S (R d ) = (2π) -d R d ×R d e i(x-y)•ξ a(x, ξ, τ )u(y) dξ dy,
for any given value x by considering the phase Φ x (y, ξ) = (x -y) • ξ with the variable x as a parameter.

Remark 4.2.2. Let a ∈ S m (R p × R d ) and χ ∈ C ∞ c (R) be such that χ(0) = 1. By [62, Proposition 2.58] one has that χ(ελ τ ) converges to a in S m ′ τ (R p × R d )
for m < m ′ and we conclude by Theorem 4.2.1 that we have

I Φ (a, u, τ ) = R p ×R d e iΦ(x,ξ) a(x, ξ, τ )u(x)dξdx = lim ε→0 R p ×R d e iΦ(x,ξ) χ(ελ τ )a(x, ξ, τ )u(x)dξdx for u ∈ S (R d ), i.e, R d e iΦ(x,ξ) a(x, ξ, τ )dξ = lim ε→0 R d e iΦ(x,ξ) χ(ελ τ )a(x, ξ, τ )dξ,
in the sense of distributions.

We say that the oscillatory integral is regularized in this limiting process.

Remark 4.2.3. Regularization allows one to generalize to oscillatory integrals the usual calculus rules for absolutely convergent integrals: integration by parts, homogeneous change of variables, the Fubini theorem, limits and differentiations under the sum sign.

Tangential semi-classical calculus

Here, we consider pseudo-differential operators that only act in the tangential direction x ′ with x d 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, 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 , (4.2.1) for all (x, ξ ′ , τ ) ∈ R d × R d-1 × [1, ∞). For b ∈ S m T,τ , we define a tangential pseudo-differential operator B = Op T (b) of order m by b(x, D ′ , τ )u(x) = Bu(x) := 1 (2π) d-1 R d-1 e ix ′ •ξ ′ b(x, ξ ′ , τ )û(ξ ′ , x d )dξ ′ (4.2.2) = 1 (2π) d-1 R d-1 R d-1 e i(x ′ -y ′ )•ξ ′ b(x, ξ ′ , τ )u(y ′ , x d )dy ′ dξ ′ for 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 ).
Additional classes of symbols.

Here, we shall often write ϱ = (x, ξ, τ ), ϱ ′ = (x, ξ ′ , τ ) and

X = R d or R d + . Definition 4.2.4. Let a(ϱ) ∈ C ∞ (X × R d ),
with τ as a large parameter in [1, +∞), and m ∈ N, r ∈ R. We say that a ∈ S m,r τ (X × R d ) if

a(ϱ) = m j=0 a j (ϱ ′ )ξ j d , a j ∈ S m-j+r T,τ (X × R d-1 ), for x ∈ X, ξ ∈ R d , τ ∈ [1, +∞), and ξ d ∈ R.
We also simply write a ∈ S m,r τ .

If U is a conic open set of X × R d × [1, +∞) we say that a ∈ S m,r τ microlocally for ϱ ′ ∈ U if each a j is in S m-j+r T,τ
microlocally in U , j = 0, . . . , m. Note that we have

S m,r τ ⊂ S m+m ′ ,r-m ′ τ , m, m ′ ∈ N, r ∈ R.
The principal symbol of a denoted by σ(a) is

σ(a)(ϱ) = m j=0 σ(a j )(ϱ ′ )ξ j d ,
which is a representative of the class of a in S m,r τ /S m,r-1 τ . Note that S m,r τ ̸ ⊂ S m+r τ . Indeed, consider a(x, ξ, τ ) = λ T,τ ξ d for λ T,τ ≥ 1. We have a ∈ S 1,1 τ ⊂ S 2,0 τ and yet a ̸ ∈ S 2 τ . In fact observe that differentiating with respect to ξ ′ yields

|∂ α ξ ′ a(x, ξ, τ )| ≤ C α λ 1-|α| T,τ |ξ d |.
An estimate of the form of (4.2.1) is however not achieved for |α| ≥ 2.

We recall that a ∼ j∈N a m-j ∈ S m τ,ph if a m-j ∈ S m-j τ is homogeneous of degree m -j with respect to (ξ, τ ) for j ∈ N. Additionally we give the definition of tangential polyhomogeneous symbols that are characterized by an asymptotic expansion where each term is positively homogeneous with respect to (ξ ′ , τ ).

Definition 4.2.5. We shall say that a ∈ S m T,τ,ph (X × R d-1 ) or simply S m T,τ,ph if there exists a j ∈ S m T,τ , homogeneous of degree m -j in (ξ ′ , τ

) for |(ξ ′ , τ )| ≥ r 0 , with r 0 ≥ 0, such that a ∼ j≥0 a j , in the sense that a - N j=0 a j ∈ S m-N -1 T,τ .
A representative of the principal symbol is then given by the first term in the expansion. We denote it by σ(a). We have

S m τ,ph ⊂ S m τ , S m T,τ,ph ⊂ S m T,τ .
Thus, for m ∈ N and r ∈ R, we say that a(ϱ) ∈ S m,r τ,ph (X × R d ) or simply S m,r τ,ph , if

a(ϱ) = m j=0 a j (ϱ ′ )ξ j d , with a j ∈ S m-j+r τ,ph
.

A representative of the principal symbol is given by m j=0 σ(a j )(ϱ ′ )ξ j d and is homogeneous of degree m in (ξ, τ ).

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 . 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 .

Sobolev norms with parameter

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 + ), where ∥ • ∥ + = ∥ • ∥ L 2 (R d + ) . We also denote (u, v) + = (u, v) L 2 (R d + )
and

(u | x d =0 + , v | x d =0 + ) ∂ = (u | x d =0 + , v | x d =0 + ) L 2 (R d-1
) . We have

∥u∥ m,τ ≍ |α|≤m τ m-|α| ∥D α u∥ + ,
and in the case m ′ ∈ N we have

∥u∥ m,m ′ ,τ ≍ α d ≤m |α|≤m+m ′ τ m+m ′ -|α| ∥D α u∥ + , with α = (α ′ , α d ) ∈ N d .
Remark 4.2.6. We have that for some C > 0,

∥u∥ m,s,τ ≤ Cτ -ℓ ∥u∥ m,s+ℓ,τ , u ∈ S (R d + ),
for m ∈ N, s ∈ R and ℓ ≥ 0. This implies that ∥u∥ m,s,τ ≪ ∥u∥ m,s+ℓ,τ for τ sufficiently large.

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. For a sufficiently smooth function u defined in R d + we set, for m ∈ N,

tr(u) = (u |x d =0 + , D d u |x d =0 + , . . . , D m d u |x d =0 + )
on {x d = 0} and we define the following norm for m ∈ N and m ′ ∈ R,

| 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 + ),
where Let m ∈ N and s ∈ R. For some C > 0, we have

| • | ∂ = | • | L 2 (R d-1
| tr(u)| m,s,τ ≤ C∥u∥ m+1,s-1/2,τ , u ∈ S (R d + ).
From pseudo-differential symbol calculus we obtain the following inequalities. A consequence of this result is the following corollary.

∥u∥ m,r,τ ≤ ∥u∥ m+m ′ ,r-m ′ ,τ , u ∈ S (R d + ). Proof. Set u = Λ m ′ T,τ v with v = Λ -m ′ T,τ u. Since we have λ m ′ T,τ ∈ S m ′ T,τ = S 0,m ′ τ ⊂ S m ′ ,0 τ , it follows that ∥u∥ m,r,τ = ∥Λ m ′ T,τ v∥ m,r,τ ≤ C∥v∥ m+m ′ ,r,τ = C∥Λ -m ′ T,τ u∥ m+m ′ ,r,τ = C∥u∥ m+m ′ ,r-m ′ ,τ ,
thanks to Proposition (4.2.9).

Differential quadratic forms

Quadratic forms in a half space Definition 4.2.11 (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 ), (4.2.3)
is an interior differential quadratic form of type (m, r) with smooth coefficients, if for each s = 1, . . . N , we have

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

q(ϱ) = N s=1 a s (ϱ)b s (ϱ) (4.2.4) in S 2m,2r τ /S 2m,2r-1 τ .
A result we shall use is the following microlocal Gårding inequality.

Proposition 4.2.12 (microlocal Gårding inequality). Let K be a compact set of

R d + and let U be a 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, be 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 r 0 > 0,

Re q(ϱ) ≥ C 0 λ 2m τ λ 2r T,τ , for τ ≥ r 0 , ϱ = (ϱ ′ , ξ d ), ϱ ′ = (x, ξ ′ , τ ) ∈ U , 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 [11, Proposition 3.5] and [63, Theorem 6.17] for a proof. A local version of the result is the following one that follows from Proposition 4.2.12. Proposition 4.2.13 (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 r 0 > 0,

Re q(ϱ) ≥ C 0 λ 2m τ λ 2r T,τ , for τ ≥ r 0 , ϱ = (ϱ ′ , ξ d ), ϱ ′ = (x, ξ ′ , τ ) ∈ U 0 ×R d-1 ×R + , ξ d ∈ R. For 0 < C 1 < C 0 there exist τ * , C > 0such that Re Q(u) ≥ C 1 ∥u∥ 2 m,r,τ -C| tr(u)| 2 m-1,r+1/2,τ , for u ∈ S (R d + ) and τ ≥ τ * .

Boundary differential quadratic forms

Definition 4.2.14. Let u ∈ S (R d + ). We say that

Q(u) = N s=1 (A s u, B s u) ∂ , A s = a s (x, D, τ ), B s = b s (x, D, τ ), (4.2.5)
is a boundary differential quadratic form of type (m-1, r) with C ∞ coefficients, if for each s = 1, . . . N , we have a s (ϱ)

∈ S m-1,r ′ τ (R d + × R d ), b s (ϱ) ∈ S m-1,r ′′ τ (R d + × R d ) with r ′ + r ′′ = 2r, ϱ = (ϱ ′ , ξ d ) with ϱ ′ = (x, ξ ′ , τ
). The symbol of the boundary differential quadratic form Q is defined by

B(ϱ ′ , ξ d , ξd ) = N s=1 a s (ϱ ′ , ξ d )b s (ϱ ′ , ξd ). For z = (z 0 , . . . , z m-1 ) ∈ C m and a(ϱ) ∈ S m-1,r τ , of the form a(ϱ ′ , ξ d ) = m-1 j=0 a j (ϱ ′ )ξ j d with a j (ϱ ′ ) ∈ S m-1+r-j T,τ
we set

Σ a (ϱ ′ , z) = ℓ-1 j=0 a j (ϱ ′ )z j . (4.2.6)
From the boundary differential quadratic form Q we introduce the following bilinear symbol

Σ Q : C m × C m → C Σ Q (ϱ ′ , z, z ′ ) = N s=1 Σ a s (ϱ ′ , z)Σ b s (ϱ ′ , z ′ ), z, z ′ ∈ C m . ( 4
(ϱ ′ , z, z ′ ). We say that Q is positive definite in W if there exist C > 0 and R > 0 such that Re Σ Q (ϱ ′′ , x d = 0 + , z, z) ≥ C m-1 j=0 λ 2(m-1-j+r) T,τ |z j | 2 ,
for ϱ ′′ = (x ′ , ξ ′ , τ ) ∈ W , and z = (z 0 , . . . , z m-1 ) ∈ C m .

Proposition 4.2.16. 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 (ϱ ′ , 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,r-N,τ , for u ∈ S (R d + ) and τ ≥ τ * .

Parametrices

Elliptic operators can be inverted up to some regularizing operator.

Proposition 4.2.17. Let m ∈ R and let p ∈ S m τ be elliptic, i.e, for some C > 0 and R > 0,

|p(x, ξ, τ )| ≥ Cλ m τ , x ∈ R d , ξ ∈ R d , τ ∈ [τ 0 , +∞), λ τ ≥ R.
For any N ∈ N there exist q N ∈ S -m τ and r N , r ′ N ∈ S -N τ such that

q N • p = 1 + r N , p • q N = 1 + r ′ N . Moreover q N is unique in S -m τ /S -m-N τ . There exist also p ∈ S -m τ and r ∞ , r ′ ∞ ∈ S -∞ τ such that q • p = 1 + r ∞ , p • q = 1 + r ′ ∞ , with q unique in S -m τ /S -∞ τ .
This proposition is a particular case of the microlocal version [62, Proposition 2.34] whose proof can be found in [START_REF] Rousseau | Elliptic Carleman estimates and applications to stabilization and controllability[END_REF].

Standard pseudo-differential operators

We refer to as standard pseudo-differential operators, the operators that do not depend on a large parameter.

|∂ α x ∂ β ξ a(x, ξ)| ≤ C αβ ⟨ξ⟩ m-|β| , x ∈ R d , ξ ∈ R d , (4.3.1)
where ⟨ξ⟩ = (1 + |ξ| 2 ) 1/2 . We then write a ∈ S m (R d × R d ).

We also define

S -∞ (R d × R d ) = ∩ r∈R S r (R d × R d ) and S +∞ (R d × R d ) = ∪ r∈R S r (R d ×R d ).
We shall often simply write S m (respectively S -∞ , S +∞ ) when no confusion is possible. For a ∈ S m we call principal symbol, σ(a) the equivalence class of a in S m /S m-1 . With this symbol classes we can define standard pseudo-differential operators.

Definition 4.3.2. (standard pseudo-differentiel operators) If a ∈ S m (R d × R d ), we set a(x, D)u(x) = Op(a)u(x) := 1 (2π) d R d e ix•ξ a(x, ξ)û(ξ)dξ = 1 (2π) d R d ×R d e i(x-y)•ξ a(x, ξ)u(y)dydξ for u ∈ S (R d ).
We denote by Ψ m the set of these pseudo-differential operators.

Remark 4.3.3. Note that a(x, ξ, 1) ∈ S m if a(x, ξ, τ ) ∈ S m τ . So we can recover standard pseudo-differential operators by setting τ to be a fix value (e.g τ = 1) in the case of the pseudo-differential operators with large parameter τ.

(x, ξ ′ ) ∈ S m T (R d × R d-1 ) if we have |∂ α x ∂ β ξ ′ a(x, ξ ′ )| ≤ C αβ ⟨ξ ′ ⟩ m-|β| , x ∈ R d , ξ ′ ∈ R d-1
,

where ⟨ξ ′ ⟩ = (1 + |ξ ′ | 2 ) 1/2 .
We denote by Ψ m T the set of associated operators, that is

a(x, D ′ )u(x) = Op T (a)u(x) := 1 (2π) d-1 R d-1 e ix ′ •ξ ′ a(x, ξ ′ )û(ξ ′ , x d )dξ ′ = 1 (2π) d-1 R d-1 ×R d-1 e i(x ′ -y ′ )•ξ ′ a(x, ξ ′ )u(y ′ , x d )dy ′ dξ ′ .

Fredholm properties for fourth order elliptic operators

It is well-known that elliptic problems are well-posed only if the boundary conditions are chosen appropriately. By well-posedness one usually means that the solution exists and is unique in some space, and it depends continuously on data and parameters, or more generally that the associated operator is at least Fredholm. The property which the boundary conditions should satisfy to have a well-posed problem in some Sobolev spaces for a elliptic boundary value problem is called the Lopatinskiȋ-Šapiro condition.

In this section we will point out the relation between the operators which are Fredholm type and that satisfy the Lopatinskiȋ-Šapiro condition.

On a smooth compact Riemmannian manifold (M, g), with boundary, we consider P = ∆ 2 g , where ∆ g denotes the Laplace-Beltrami operator. We denote by p(x, ω) its principal symbol for (x, ω) ∈ T * M. One defines the following polynomial in z, p(x, ω ′ , z) = p(x, ω ′ -zn x ),

for x ∈ ∂M, ω ′ ∈ T *
x ∂M, z ∈ R and where n x denotes the outward pointing conormal vector at x, unitary in the sense of the metric g. We denote by ρ j (x, ω ′ ), 1 ≤ j ≤ 4 the complex roots of p. One sets

p+ (x, ω ′ , z) = Im ρ j (x,ω ′ )≥0 (z -ρ j (x, ω ′ )).
Given boundary operators B 1 , B 2 in a neighborhood of ∂M, with principal symbols b j (x, ω), j = 1, 2 one also sets bj (x, ω ′ , z) = b j (x, ω ′ -zn x ). We note that the boundary operators B 1 and B 2 are of order k 1 and k 2 respectively. Definition 4.4.1. (Lopatinskiȋ-Šapiro condition) Let (x, ω ′ ) ∈ T * ∂M with ω ′ ̸ = 0. One says that the Lopatinskiȋ-Šapiro condition condition holds for (P, B 1 , B 2 ) at (x, ω ′ ) if for any polynomial function f (z) with complex coefficients, there exists c 1 , c 2 ∈ C and a polynomial function g(z) with complex coefficients such that, for all z ∈ C,

f (z) = 1≤j≤2 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 2 ) at x ∈ ∂M if it holds at (x, ω ′ ) for all ω ′ ∈ T *

x ∂M with ω ′ ̸ = 0. Observe that the Lopatinskiȋ-Šapiro condition is written here without any use of local coordinates. It is then a geometrical condition.

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.
For m ∈ N, we study the Fredholm property of the operator

L : H m+4 (M) → H m (M) ⊕ H (m+7/2) (∂M) u → (P u, B 1 u| ∂M , B 2 u| ∂M ), (4.4.1)
where In order to show Theorem 4.4.2 one needs to establish the following result. We note that by Theorem 2.1.23 this implies that the Lopatinskiȋ-Šapiro condition is sufficient for the Fredholm property of L to hold.

H (m+7/2) (∂M) = H m+7/2-k 1 (∂M) ⊕ H m+7/2-
M : H m (M) ⊕ H (m+7/2) (∂M) → H m+4 (M) such that M L = Id H m+4 (M) + K s and LM = Id H m (M)⊕H (m+7/2) (∂M) + K r ,
where both operators

K s : H m+4 (M) → H m+5 (M) K r : H m (M) ⊕ H (m+7/2) (∂M) → H m+1 (M) ⊕ H (m+9/2) (∂M) (4.4.2)
are bounded.

By the Rellich-Kondrachov theorem (see [START_REF] Rousseau | Elliptic Carleman estimates and applications to stabilization and controllability[END_REF]Theorem 30.7]) K s is compact from H m+4 (M) into itself and K r from H m (M) ⊕ H (m+7/2) (∂M) into itself. Proposition 4.4.3 is contained in Theorem 20.1.7 of [START_REF] Hörmander | The Analysis of Linear Partial Differential Operators[END_REF] and its proof is performed based on the analysis of the fourth order operator in a half-space. For instance the case of the Laplacian we refer to Proposition 15.2 in [START_REF] Rousseau | Elliptic Carleman estimates and applications to stabilization and controllability[END_REF]. Theorem 4.4.2 states in particular that L is not Fredholm if the Lopatinskiȋ-Šapiro condition does not hold.

Stabilization of the damped plate equation under general boundary conditions

Here, we present the results obtained in [START_REF] Rousseau | Stabilization of the damped plate equation under general boundary conditions[END_REF] concerning the Part B.

Setting of the problem and some notations

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

       ∂ 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 , (4.5.1)
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 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 preservation 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. Among the existing results available in the literature for plate type equations, many of them concern the "hinged" boundary conditions, that is, u |∂Ω = 0 and ∆u |∂Ω = 0. We first mention these result. An important result obtained in [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. The method relies on the generalization of Ingham type inequalities in [START_REF] Kahane | Pseudo-périodicité et séries de Fourier lacunaires[END_REF]. An exponential stabilization result, in the same geometry, can be found 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. In [START_REF] Ramdani | Internal stabilization of the plate equation in a square: the continuous and the semi-discretized problems[END_REF] the localized damping term involves the time derivative ∂ t y as in (4.5.1). 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 multipliertype 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 result thus stands as a generalization of the stabilisation 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. It contains in particular also the case of "hinged" boundary conditions.

Method

Following the works of [START_REF] Lebeau | Équation des ondes amorties[END_REF][START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF][START_REF] Rousseau | Spectral inequality and resolvent estimate for the bi-laplace operator[END_REF] we obtain a logarithmic decay rate for the energy of the solution to (4.5.1) which is obtained by means of a resolvent estimate for the generator of the semigroup associated with the damped plate equation (4.5.1). This estimate follows from a Carleman inequality derived for the operator P σ = ∆ 2 -σ 4 where σ is a spectral parameter for the generator of the semigroup. 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.

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 ) = 1 ∀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 naturally denote its closure by For an open set U of R d we set

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). ( 4
U + = U ∩ R d + and C ∞ c (U + ) = {u = v |R d + ; v ∈ C ∞ c (R d ) and supp(v) ⊂ U }. (4.5.3) 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)| < ∞.

Observations concerning the need for boundary conditions

On (0, +∞) we consider the first-order differential operator L = D s -λρ, with ρ ∈ C and λ > 0. The parameter λ is intended to become large. Let u ∈ S (R d + ). One aims to achieve the following estimate

∥u∥ L 2 (R + ) ≲ ∥Lu∥ L 2 (R + ) . (4.5.4)
First, we assume that Im ρ < 0 and we compute

Re(Lu, iu) L 2 (R + ) = Re(D s u, iu) L 2 (R + ) -λ Im ρ∥u∥ 2 L 2 (R + ) = Re(D s u, iu) L 2 (R + ) + λ| Im ρ|∥u∥ 2 L 2 (R + ) = 1 2 |u(0)| 2 + λ| Im ρ|∥u∥ 2 L 2 (R + ) , (4.5.5) since (D s u, iu) L 2 (R + ) = R + D s u (iu) ds = -i R + D s u u ds = - R + ∂ s u u ds = - 1 2 R + ∂ s |u| 2 ds.
Using the Cauchy-Schwartz inequality and the Young inequality we obtain for any ε > 0

|u(0)| 2 +λ∥u∥ 2 L 2 (R + ) ≲ ∥Lu∥ L 2 (R + ) ∥u∥ L 2 (R + ) ≲ (λε) -1 ∥Lu∥ 2 L 2 (R + ) +(λε)∥u∥ 2 L 2 (R + ) .
With ε > 0 chosen sufficiently small one concludes that

λ 1/2 |u(0)| + λ∥u∥ L 2 (R + ) ≲ ∥Lu∥ L 2 (R + ) . (4.5.6)
In this first case, we obtain (4.5.4) but, better yet, we estimate also the trace of u at s = 0 + . Second, assume that Im ρ > 0. Consider the cut-off χ ∈ C ∞ c (R) with χ(s) = 1 for |s| ≤ 1 and χ(s) = 0 for s ≥ 2. We set u(s) = χ(s)e iλρs and we observe that Lu(s) = -iχ ′ (s)e iλρs . On the one has

∥u∥ 2 L 2 (R + ) = 2 0 χ 2 (s)e -2λ Im ρs ds ≥ 1 0 e -2λ Im ρs ds = 1 2λ Im ρ 1 -e -2λ Im ρ .
On the other hand, one has

∥Lu∥ 2 L 2 (R + ) = 2 0 (χ ′ (s)) 2 e -2λ Im ρs ds ≲ 2 1 e -2λ Im ρs ds = 1 2λ Im ρ e -2λ Im ρ -e -4λ Im ρ = 1 2λ Im ρ 1 -e -2λ Im ρ e -2λ Im ρ .
This ruins any hope of having an estimate of the form (4.5.4). Yet, by computing Re(Lu, -iu) L 2 (R + ) and arguing as above for (4.5.4)-(4.5.6), one obtains the following estimate

λ∥u∥ L 2 (R + ) ≲ ∥Lu∥ L 2 (R + ) + λ 1/2 |u(0)|. (4.5.7)
Consider now the Laplace-Beltrami operator -∆ g in the normal geodesic coordinates, that is -∆ g = D 2 d + R(x, D x ′ ), where R(x, D x ′ ) is a second-order differential operator. To simplify, we assume that R(x, D x ′ ) is a constant coefficient operator. Then, up to a Fourier transformation in the x ′ variables, one obtains for the principal part the operator

P = D 2 d + R(ξ ′ ) = D 2 d + |ξ ′ | 2 R(ξ ′ ),
where R(ξ ′ ) = R(ξ ′ /|ξ ′ |), which we write

P = L + L -L ± = D d ± |ξ ′ | R(ξ ′ ) 1/2 .
With λ = |ξ ′ | > 0 and x d = s, and the two cases considered above, one finds that for the factor L -and estimate as in (4.5.6) can be obtained. For the operator L + , one can only obtain an estimation as in (4.5.7). Combined together one obtains an estimate either of the form

λ 2 ∥u∥ L 2 (R + ) + λ 1/2 |D d u(0)| ≲ ∥ P u∥ L 2 (R + ) + λ 3/2 |u(0)|,
or the form

λ 2 ∥u∥ L 2 (R + ) + λ 3/2 |u(0)| ≲ ∥ P u∥ L 2 (R + ) + λ 1/2 |D d u(0)|.
This simple example shows that if an elliptic operator can be written as a

Symbols and operators with an additional large parameter

We shall often 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 4.2.1.

Outline

In Section 4.6, the Lopatinskiȋ-Šapiro boundary condition are 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 coordinated that we shall mostly use throughout the manuscript. For the bi-Laplace operator we provide a series of examples of boundary operators for which the Lopatinskiȋ-Šapiro boundary conditions holds and moreover the resulting operator is symmetric. We also show that the algebraic conditions that characterize the Lopatinskiȋ-Šapiro condition are 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 4.7, where an analysis of the configuration of the roots of the conjugated bi-Laplace operator is performed. In Section 4.7.5 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 4.8 we derive a 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 ∆ 2 -σ 4 in Section 4.9 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 ∆ 2 -σ 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 4.10. With the traces estimation obtained in Section 4.8 one obtains the local Carleman estimate of Theorem 1.8.1.

For the application to stabilization we have in mind, in Section 4.11 we use a global weight function and derive a global version of the Carleman estimate for ∆ 2 -σ 4 on the whole Ω. This leads to an observability inequality.

In Section 4.12 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 4.11 we derive in Section 4.13 a resolvent estimate for the generator of the plate semigroup that in turn implies the stabilization result of Theorem 1.8.2.

Lopatinskiȋ-Šapiro boundary conditions for an 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 4.5.2). Here x and ω ′ are considered to act as parameters. We denote by ρ j (x, ω ′ ), 1 ≤ j ≤ 2k the complex roots of p. 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 4.6.1 (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 function f (z) with complex coefficients, there exists c 1 , • • • , c k ∈ C and a polynomial function 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.

Some examples

For instance 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 4.5.2.

• 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. 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 function 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.

Lemma 4.6.2. 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

∂ z b1 ∂ z b2 (x, ω ′ , z = i|ω ′ | g ) ̸ = 0. (4.6.1)
Remark 4.6.3. 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.

Formulation in normal geodesic coordinates

Near a boundary point x ∈ ∂Ω, we shall use normal geodesic coordinates. These coordinates are recalled at the beginning of Section 4.5.2. 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 4.6.2.
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 . (4.6.2)

Stability of the Lopatinskiȋ-Šapiro condition under perturbation

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 (4.6.2).

Lemma 4.6.4. Let V 0 be a compact set of ∂Ω be 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 , (4.6.3)
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 , (4.6
.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 (4.6.2), 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 .
(4.6.5) The first part is a consequence of the mean value theorem, homogeneity and (4.6.5) 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) 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 )
ds.

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 fulfill (1) the Lopatinskiȋ-Šapiro condition and (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.
∂ 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 4.6.2. 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. ii. Take

B 1 u |∂Ω = (∂ 2 n + 2∆ ′ )u |∂Ω and B 2 u |∂Ω = ∂ 3 n u |∂Ω .
With (4.6.8) one finds T (u, v) = 0 in the case of homogeneous conditions, hence symmetry.

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 and thus the Lopatinskiȋ-Šapiro condition holds by Lemma 4.6.2.

iii. 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

′ (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 (4.6.8) 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 4.6.2. iv. 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 4.6.5 above.

With (4.6.8) 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 4.6.2.

v. 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 (4.6.9) 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 2 -iza ′ (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 4.6.2.

Lopatinskiȋ-Šapiro condition for the conjugated bi-Laplacian with spectral parameter

Set P σ = ∆ 2 -σ 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 4.6.1 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 local coordinates for simplicity).

Definition 4.7.1. 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 function f (ξ d ) with complex coefficients there exist c 1 , c 2 ∈ C and a polynomial function 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 )p + σ,φ (x, ξ ′ , ξ d , τ ), with p + σ,φ (x, ξ ′ , ξ d , τ ) = Im ρ j (ξ ′ ,τ,σ)≥0 (ξ d -ρ j (ξ ′ , τ, σ)),
where

ρ j (x, ξ ′ , τ, σ), j = 1, • • • , 4, denote the complex roots of p σ,φ (x, ξ ′ , ξ d , τ ) viewed as a polynomial in ξ d .
In what follows, we shall assume that

∂ d φ > 0. Locally, one has ∂ d φ ≥ C 1 > 0, for some C 1 > 0.

Discussion on the Lopatinskiȋ-Šapiro condition according to the position of the roots

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 4.7.1 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 4.7.1, for any choice of f , the polynomial function

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

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 = r(x, ξ ′ + iτ d x ′ φ) + (-1) j σ 2 = r(x, ξ ′ ) -τ 2 r(x, d x ′ φ) + (-1) j σ 2 + i2τ r(x, ξ ′ , d x ′ φ) 2 , (4.7.3)
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 = -iτ ∂ d φ -iα j and π j,2 = -iτ ∂ d φ + iα j . (4.7.4)
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 4.7.3. Assume that ∂ d φ > 0. Having Im π j,2 < 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 4.7.4 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 4.7.4 below. With (4.7.3) this gives the result.

Lemma 4.7.4. 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.
Proof. Let z = x + iy ∈ C. On the one hand we have z 2 = x 2 -y 2 + 2ixy = m and Re m = x 2 -y 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 4.7.5. 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 < 0 if C K 0 |ξ ′ | + σ ≤ τ ∂ d φ, j = 0, 1.
Proof. With Lemma 4.7.3 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 . (4.7.5)
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 (4.7.5) 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 4.7.6. 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 4.7.3 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 4.7.1, we have the following result.

Lemma 4.7.7. Assume that σ > 0. Then, π 1,2 ̸ = π 2,2 . Moreover, the roots π 1,2 and π 2,2 cannot be both real.

Proof. With the forms of the roots given in (4.7.4

) 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
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. First, we choose 0 < µ 0 ≤ 1 and 0 < µ 1 ≤ 1/2. Below both may be chosen much smaller. According to Lemma 4.7.5, with K 0 = 1 therein, for some Lemma 4.7.8. 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)

C 2 = 2C K 0 > 0 if one has C 2 |ξ ′ | ≤ τ ∂ d φ
∈ V 0 × R d-1 × [0, +∞) × [0, +∞), one has |d x ′ φ| ≤ µ 0 ∂ d φ, σ ≤ µ 1 τ ∂ d φ and Im π j,2 ≥ 0 =⇒ α j -|ξ ′ | x +τ |d x ′ φ| ≤ |ξ ′ | x C 3 (µ 0 +µ 2 1 ).
Proof. With (4.7.6) one has

τ |d x ′ φ| ≤ µ 0 τ ∂ d φ ≲ µ 0 |ξ ′ | x . (4.7.7)
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 (4.7.7) 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 (4.7.7) gives 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 )). (4.7.8)
We use Lemma 4.6.4 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, (4.7.9)
with C 3 > 0 as given by Lemma 4.7.8. We now consider the root configurations that remain to consider according to the discussion in Section 4.7.1.

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 4.7.1 it suffices to prove that Naturally, one has κ + ϱ 0′ (ϱ 0′ , ξ d ) = p + σ,φ (x 0 , ξ 0′ , ξ d , τ 0 ) and κ - ϱ 0′ (ϱ 0′ , ξ d ) = p - σ,φ (x 0 , ξ 0′ , ξ d , τ 0 ). Moreover, there exists a conic neighborhood U 0 of ϱ 0′ where both κ + ϱ 0′ (ϱ ′ ) and κ - ϱ 0′ (ϱ ′ ) are smooth with respect to ϱ ′ . One has

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
p σ,φ = p + σ,φ p - σ,φ = κ + ϱ 0′ κ - ϱ 0′ .
Note however that for ϱ ′ = (x, ξ ′ , τ, σ) ∈ U 0 it may very well happen that

p + σ,φ (x, ξ ′ , ξ d , τ ) ̸ = κ + ϱ 0′ (ϱ ′ , ξ d ) and p - σ,φ (x, ξ ′ , ξ d , τ ) ̸ = κ - ϱ 0′ (ϱ ′ , ξ d ).
The following proposition can be found in [11, proposition 1.8].

Proposition 4.7.9. Let the Lopatinskiȋ-Šapiro condition hold at

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

ii. There exists a conic neighborhood U of ϱ 0′ such that

{b 1 φ (ϱ ′ , ξ d ), b 2 φ (ϱ ′ , ξ d )} is complete modulo κ + ϱ 0′ (ϱ ′ , ξ d )
at every point ϱ ′ = (x, ξ ′ , τ, σ) ∈ U , namely for any polynomial function f (ξ d ) with complex coefficients there exist c 1 , c 2 ∈ C and a polynomial function 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 + ϱ 0′ (ϱ ′ , ξ d ). (4.7.12)
We emphasize again that the second property in Proposition 4.7.9 looks very much like the statement of Lopatinskiȋ-Šapiro condition for (P σ , B 1 , B 2 , φ) at ϱ ′ in Definition 4.7.1. Yet, it differs by having p + σ,φ (x, ξ ′ , ξ d , τ ) that only depends on the root configuration at ϱ ′ replaced by κ + ϱ 0′ (ϱ ′ , ξ d ) whose structure is based on the root configuration at ϱ 0′ . Let m + be the common degree of p + σ,φ (ϱ 0′ , ξ d ) and κ + ϱ 0′ (ϱ ′ , ξ d ) and m -be the common degree of p - σ,φ (ϱ 0′ , ξ d ) and κ - ϱ 0′ (ϱ ′ , ξ d ) for ϱ ′ ∈ U . One has m + +m -= 4 and thus m -≥ 2 for ϱ ′ ∈ 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 (4.7.12). Since the degree of b j,φ (ϱ ′ , 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 4.7.9 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 + ϱ 0′ (ϱ ′ , 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 ′ = 6 -m + = 2 + m -polynomials.

We now express an inequality that follows from Proposition 4.7.9 that will be key in the boundary estimation given in Proposition 4.8.1 below.

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 (ϱ ′ , ξ d ) = k j ℓ=0 b ℓ j,φ (ϱ ′ )ξ ℓ d , with b ℓ j,φ homogeneous of degree k j -ℓ. The polynomial ξ d → κ + ϱ 0′ (ϱ ′ , ξ d
) is of degree m + ≤ 2 for ϱ ′ ∈ U with U given by Proposition 4.7.9. Similarly, we write

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

e j,ϱ 0′ (ϱ ′ , ξ d ) =    b j,φ (ϱ ′ , ξ d ) if j = 1, 2, κ + ϱ 0′ (ϱ ′ , 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,ϱ 0′ (ϱ ′ , ξ d ) = 3 ℓ=0 e ℓ j,ϱ 0′ (ϱ ′ )ξ ℓ d ,
for j = 1, 2 one has e ℓ j,ϱ 0′ (ϱ ′ ) = b ℓ j,φ (ϱ ′ ), with ℓ = 0, . . . , k j and e ℓ j,ϱ 0′ (ϱ ′ ) = 0 for ℓ > k j , and

for j = 3, . . . , m ′ , e ℓ j,ϱ 0′ (ϱ ′ ) =        0 if ℓ < j -3, κ +,ℓ+3-j ϱ 0′ (ϱ ′ ) if ℓ = j -3, . . . , m + + j -3 ≤ m + + m ′ -3 = 3, 0 if ℓ > m + + j -3.
In particular e ℓ j,ϱ 0′ (ϱ ′ ) is homogeneous of degree m + + j -ℓ -3. We thus have the following result. Recall that m ′ = m -+ 2 ≥ 4.

We now set

Σ e j,ϱ 0′ (ϱ ′ , z) = 3 ℓ=0 e ℓ j,ϱ 0′ (ϱ ′ )z ℓ = 3 ℓ=0 M j,ℓ (ϱ ′ )z ℓ , z = (
= (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 4.7.9. Then, if ϱ ′ ∈ U there exists C > 0 such that

m ′ j=1 Σ e j,ϱ 0′ (ϱ ′ , 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,ϱ 0′ (ϱ ′ , z)Σ e j,ϱ 0′ (ϱ ′ , z ′ ). With (4.7.13) one has

Σ B (z, z ′ ) = M (ϱ ′ )z, M (ϱ ′ )z ′ C m ′ = t M (ϱ ′ )M (ϱ ′ )z, z ′ C 4 .
As

rank t M (ϱ ′ )M (ϱ ′ ) = rank M (ϱ ′ ) =
′ = (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 (4.5.3). For the proof of Proposition 4.8.1 we start with a microlocal version of the result.

A microlocal estimate

Proposition 4.8.2. 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 , P 2 , φ) satisfies the Lopatinskiȋ-Šapiro condition of Definition 4.7.1 at ϱ 0′ = (x 0 , ξ 0′ , τ 0 , σ 0 ).

Then, there exists U a conic neighborhood of

ϱ 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 neighborhood U 0 of ϱ 0′ according to Proposition 4.7.9 and such that

U 0 ⊂ W × R d-1 × [0, +∞) × [0, +∞) . Assume moreover that τ ≥ κ 0 σ in U 0 .
In Section 4.7.5 we introduced the symbols e j,ϱ

0′ (ϱ ′ , ξ d ), j = 1, . . . , m ′ = m -+ 2 = 6 -m + . Set S U 0 = {ϱ ′ = (x, ξ ′ , τ, σ) ∈ U 0 ; |(ξ ′ , τ, σ)| = 1}.
Consequence of the Lopatinskiȋ-Šapiro condition holding at ϱ 0′ for all ϱ ′ ∈ S U 0 , by Lemma 4.7.11 there exists C > 0 such that

m ′ j=1 Σ e j,ϱ 0′ (ϱ ′ , z) 2 ≥ C|z| 2 C 4 , z = (z 0 , . . . , z 3 ) ∈ C 4 .
Since S U 0 is compact (recall that W is bounded), there exists C 0 > 0 such that

m ′ j=1 Σ e j,ϱ 0′ (ϱ ′ , z) 2 ≥ C 0 |z| 2 C 4 , z = (z 0 , . . . , z 3 ) ∈ C 4 , ϱ ′ ∈ S U 0 .
Introducing the map N t ϱ ′ = (x, tξ ′ , tτ, tσ), for ϱ ′ = (x, ξ ′ , τ, σ)

with t = |(ξ ′ , τ, σ)| -1 one has m ′ j=1 Σ e j,ϱ 0′ (N t ϱ ′ , z) 2 ≥ C 0 |z| 2 C 4 , z = (z 0 , . . . , z 3 ) ∈ C 4 , ϱ ′ ∈ U 0 , (4.8.1)
since N t ϱ ′ ∈ S U 0 . Now, for j = 1, 2 one has

Σ e j,ϱ 0′ (ϱ ′ , z) = k j ℓ=0 e ℓ j,ϱ 0′ (ϱ ′ )z ℓ ,
with e ℓ j,ϱ 0′ (ϱ ′ ) homogeneous of degree k j -ℓ, and for 3 ≤ j ≤ m ′ one has

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

Σ e j,ϱ 0′ (N t ϱ ′ , z ′ ) = t k j -7/2 Σ e j,ϱ 0′ (ϱ ′ , z), j = 1, 2,
and Σ e j,ϱ 0′ (N t ϱ ′ , z ′ ) = t m + +j-13/2 Σ e j,ϱ 0′ (ϱ ′ , z), j = 3, . . . , m ′ .

Thus from (4.8.1) we deduce

2 j=1 λ 2(7/2-k j ) T,τ Σ e j,ϱ 0′ (ϱ ′ , z) 2 + m ′ j=3 λ 2(13/2-m + -j) T,τ Σ e j,ϱ 0′ (ϱ ′ , z) 2 ≥ C 0 3 ℓ=0 λ 2(7/2-ℓ) T,τ |z ℓ | 2 , (4.8.2)
for z = (z 0 , . . . , z 3 ) ∈ C 4 , and ϱ ′ ∈ U 0 , since t ≍ λ -1 T,τ as τ ≳ σ in U 0 . We now choose U a conic open neighborhood of ϱ 0′ , such that U ⊂ U 0 . 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 0 and χ ≡ 1 in a neighborhood of U , and thus in a neighborhood of supp(χ).

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

′ , ξ d ) = κ + ϱ 0′ (ϱ ′ , ξ d )ξ j-3 d ∈ S m + +j-3,0 τ
. Set E j = Op( χe j,ϱ 0′ ). The introduction of χ is made such that χe j,ϱ 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
| 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 . (4.8.3)
for u = Op T (χ)v and τ ≥ κ 0 σ chosen sufficiently large.

In U 0 one can write

p σ,φ = p + σ,φ p - σ,φ = κ + ϱ 0′ κ - ϱ 0′ ,
with κ + ϱ 0′ of degree m + and κ - ϱ 0′ of degree m -. In fact we set

κ+ ϱ 0′ (ϱ ′ ) = j∈J + ξ d -χρ j (ϱ ′ ) , κ- ϱ 0′ (ϱ ′ ) = j∈J - ξ d -χρ j (ϱ ′ ) ,
with the notation of Section 4.7.4, thus making the two symbols defined on the whole tangential phase-space. In U , one has also

p σ,φ = κ+ ϱ 0′ κ- ϱ 0′ .
The factor κϱ 0′ is associated with roots with negative imaginary part. With Lemma 6.1.1 given in Appendix 6.1 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 - ϱ 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

+ ϱ 0′ )v. Since Op T (κ - ϱ 0′ )Op T (χ)Op T (κ + ϱ 0′ ) = Op T (χ)P σ,φ mod Ψ 4,-1 τ , one obtains | tr(Op T (χ)Op T (κ + ϱ 0′ )v)| m --1,1/2,τ ≲ ∥P σ,φ v∥ + + ∥v∥ 4,-1,τ . With [Op T (χ), Op T (κ + ϱ 0′ )] ∈ Ψ m + ,-1 τ one then has | tr(Op T (κ + ϱ 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 + ϱ 0′ )u)| m --1,1/2,τ ≍ m --1 j=0 |D j d Op T (κ + ϱ 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+ ϱ 0′ = χe j+3,ϱ 0′ in a conic neighborhood of supp(χ) and using that m -= m ′ -2. We thus obtain In addition, observing that

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 ,
|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 4.8.2 follows.

Proof of Proposition 4.8.1

As mentioned above the proof relies on a patching procedure of microlocal estimates given by Proposition 4.8.2. 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 ϱ 0′ = (x 0 , ξ 0′ , τ 0 , σ 0 ), we can invoke Proposition 4.8.2:

(1) There exists a conic open neighborhood U ϱ 0′ of ϱ 0′ in W ×R d-1 ×[0, +∞)× [0, +∞) where τ ≥ κ ′ 0 σ;

(2) For any χ ϱ 0′ ∈ S 0 T,τ homogeneous of degree 0 supported in U ϱ 0′ the estimate of Proposition 4.8.2 applies to Op T (χ ϱ 0′ )v for τ ≥ max(τ ϱ 0′ , κ 0 σ).

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

U ϱ 0′ = O ϱ 0′ ×Γ ϱ 0′ , with O ϱ 0′ ⊂ W an open neighborhood of x 0 and Γ ϱ 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 ϱ 0′ . We denote by Ũi , i ∈ I with |I| < ∞, such a finite covering. This is also a finite covering of +,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

{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.
+,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 (ϱ ′ ) = 1 for ϱ ′ = (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 4.8.2 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 , (4.8.4)
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 (4.8.4) 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 4.8.1.

4.9 Microlocal estimate for each second-order factors composing P σ,φ

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 ′ ), (4.9.1) respectively. The conjugated operator P σ,φ = e τ φ P σ e -τ 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 , (4.9.2)
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 4.7.2 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 (4.7.4) 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 ) + τ b k-1 φ (x, ξ ′ , τ )∂ d φ is homogeneous of degree k in λ T,τ .

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 4.9.1 (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 4.9.2. 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 4.9.3. 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[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 ), (4.9.3)
with β = ξ/(γφ), and where H q denotes the Hamiltonian vector field associated with the symbol q as defined in (4.9.1). 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 4.9.4. Let j = 1 or 2. Let (Q j σ , φ) have the sub-ellipticity property of Definition 4.9.1 in W . For µ > 0 one sets t(ϱ) = µ((q j s ) 2 + q 2 a )(ϱ) + τ {q j s , q a }(ϱ)

with ϱ = (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(ϱ) ≥ Cλ 4 τ for some C > 0.

The proof of Lemma 4.9.4 uses the following lemma.

Lemma 4.9.5. 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 4.9.4. 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 4.9.5 to t(ϱ) on L with f = (q j s ) 2 + q 2 a and g = τ {q j s , q a } we find for t(ϱ) ≥ C on L for some C > 0 for µ chosen sufficiently large. Since t(ϱ) is homogeneous of degree 4 in the variables (ξ, τ, σ) it follows that t(ϱ) ≥ C(σ 2 + τ 2 + |ξ| 2 ) 4 ≳ λ 4 τ .

4.9.2 Lopatinskiȋ-Šapiro condition for the second-order factors

Above, in Section 4.7, 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 (4.7.4) 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 4.9.6. 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 ϱ ′ = (x, ξ ′ , τ, σ) if for any polynomial function f (ξ d ) with complex coefficients there exist c ∈ C and a polynomial function 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 ). ( 4
.9.4) Remark 4.9.7. With the Euclidean division of polynomials, we see that it suffices to consider the polynomial function f (ξ d ) to be of degree less than that of q j,+ σ,φ (x, ξ ′ , ξ d , τ ) in (4.9.4). Thus, in any case, the degree of f (ξ d ) can be chosen less than or equal to one. Lemma 4.9.8.

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 i. either q j,+ σ,φ (x, ξ ′ , ξ d , τ ) = 1; ii. 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 )(ξ d -π j,2
), that is, both roots π j,1 and π j,2 are in the upper complex half-plane, then condition (4.9.4) cannot hold, since by Remark 4.9.7 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 d -π that is one the root π j,1 and π j,2 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 function 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 and π j,2 lie in the lower complex half-plane, then condition (4.9.4) trivially holds.

Microlocal estimates for a second-order factor

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 4.9.6.

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 (4.7.4), 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: i. The root π j,2 at the considered point lies in the upper complex half-plane.

ii. The root π j,2 at the considered point is real.

iii. The root π j,2 at the considered point lies in the lower complex half-plane. Proposition 4.9.9. 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 ϱ 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 4.9.6 at ϱ 0′ . i. Assume that Im π j,2 (ϱ 0′ ) > 0. Then, there exists U a conic neighborhood of

ϱ 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 , (4
.9.5)

for σ ≥ 0, τ ≥ max(τ 0 , κ 0 σ) and v ∈ C ∞ c (W + ).
ii. Assume that Im π j,2 (ϱ 0′ ) = 0. Then, there exists U a conic neighborhood of ϱ 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 , (4.9.6
)

for σ ≥ 0, τ ≥ max(τ 0 , κ 0 σ) and v ∈ C ∞ c (W + ).
iii. Assume that Im π j,2 (ϱ 0′ ) < 0. Then, there exists U a conic neighborhood of ϱ 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 , (4
.9.7)

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 (4.5.3).

4.9.3.1 Case (i): one root lying in the upper complex half-plane.

One has Im π j,2 (ϱ 0′ ) > 0 and Im π j,1 (ϱ 0′ ) < 0.

Since the Lopatinskiȋ-Šapiro condition holds for (Q j σ , B, φ) at ϱ 0′ , by Lemma 4.9.8 one has

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

ϱ 0′ in W × R d-1 × [0, +∞) × [0, +∞) and C 1 > 0, C 2 > 0 such that S U = {ϱ ′ ∈ U ; |ξ ′ | 2 + τ 2 + σ 2 = 1} is compact and τ ≥ κ 0 σ, Im π j,2 (ϱ ′ ) ≥ C 2 λ T,τ , and Im π j,1 (ϱ ′ ) ≤ -C 1 λ T,τ , and b φ (x, ξ ′ , ξ d = π j,2 (ϱ ′ ), τ ) ̸ = 0, (4.9.8) if ϱ ′ = (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 ).
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 4.9.10, 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 4.9.12 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 (4.9.5) and thus concluding the proof of Proposition 4.9.9 in Case (i).

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

for τ ≥ max(τ 0 , κ 0 σ) and w ∈ C ∞ c (W + ).

Proving Lemma 4.9.13 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 4.9.13 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 4.9.11 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 4.9.10, 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 , (4
.9.12)

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 4.9.13 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 , (4.9.13)
Now, with ε > 0 chosen sufficiently small if one computes (4.9.12) + ε × (4.9.13) one 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 (4.9.6) and thus concluding the proof of Proposition 4.9.9 in Case (ii).

We now provide a proof of Lemma 4.9.13.

Proof of Lemma 4.9.13. 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) + . (4.9.14)
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 (4.9.2) 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) (4.9.15) with A(w) = τ -1 (Q a w, D d w) ∂ + τ -1 (D d Q a -2τ ∂ d φQ j s )w, w ∂ . ( 4
.9.16)

We have the following lemma adapted from Lemma 3.25 in [START_REF] Rousseau | Elliptic Carleman estimates and applications to stabilization and controllability[END_REF].

Lemma 4.9.14. 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

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 -(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 + ∂ , (4.9.17)
with Op(c 0 ), Op(c 0 ) ∈ D 0 and Op(c 1 ) ∈ D We now use the following lemma whose proof is given below.

Lemma 4.9.15. 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 4.9.15 and let τ > 0 be such that µτ -1 ≤ 1. From (4.9.19) 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 4.9.15 yields the result of Lemma 4.9.13 using that τ | tr(w)| 1,0,τ ≲ | tr(w)| 1,1/2,τ .

Proof of Lemma 4.9.15. 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 4.2.11. 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(ϱ) = µ|q j σ,φ (ϱ)| 2 + τ {q j s , q a }(ϱ), ϱ = (x, ξ, τ, σ).
Let τ 0 > 0. By Lemma 4.9.4, the sub-ellipticity property of (Q j σ , φ) implies

t(ϱ) ≳ λ 4 τ , ϱ ∈ W × R d × [0, +∞) × [0, +∞), τ ≥ τ 0 σ,
for µ > 0 chosen sufficiently large. The Gårding inequality of proposition 4.2.13 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).

4.9.3.3 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 6.1.1 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 (ϱ 0′ ) < 0 and Im π j,2 (ϱ 0′ ) < 0. As the roots π j,1 and π j,2 depend continuously on the variable ϱ ′ = (x, ξ ′ , τ, σ), there exists U a conic open neighborhood of ϱ 0′ in W × R d-1 × [0, +∞) × [0, +∞) and C 0 > 0 such that τ ≥ κ 0 σ, Im π j,1 (ϱ ′ ) ≤ -C 0 λ T,τ , and Im π j,2 (ϱ ′ ) ≤ -C 0 λ T,τ , if ϱ ′ = (x, ξ ′ , τ, σ) ∈ U .

Let χ ∈ S 0 T,τ be as in the statement of Proposition 4.9.9 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 4.2.11. Its principal symbol is given by ℓ(ϱ) = (q j s )(ϱ) 2 + q a (ϱ) 2 , ϱ = (x, ξ, τ, σ).

For ε ∈ (0, 1) we write ∥Q j σ,φ u∥ for τ ≥ κ 0 σ chosen sufficiently large. From the proof of Lemma 4.9.13 one has 2 Re(Q j s u, iQ a u) + = i([Q j s , Q a ]u, u) + + τ A(u) (4.9.24)

with the boundary quadratic form A given in (4.9.16)-(4.9.17).

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 . (4.9.25)

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. 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 (4.9.7) and thus concluding the proof of Proposition 4.9.9 in Case (iii).

Proof of Lemma 4.9.16. With (4.9.17) one sees that it suffices to consider the following boundary quadratic form Since in U one has σ > 0 then π 1,2 and π 2,2 cannot be both real by Lemma 4.7.7. Proposition 4.9.9 thus implies that we necessarily have the following two estimates τ -ℓ 1 ∥Op T (χ)w∥ 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 4.8.1 in Section 4.8.2.

iii. dψ ̸ = 0 in Ω \ O 0 . Then, by Proposition 4.9.3, for γ chosen sufficiently large, one finds that φ = exp(γψ) is such that a i. φ = 1 and ∂ ν φ < -C 0 < 0 on ∂Ω;

ii. φ > 1 in Ω;

iii. (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.

Then, with Proposition 4.7.2, 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 4.10.3 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[END_REF], one obtains the following global estimate 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) + Observe that inequality (4.11.1) also holds for third-order perturbations of P σ . Below, we shall use it for a second-order perturbation P σ -iσ 2 α = ∆ 2 -σ 4 -iσ 2 α.

Observability inequality

By density one finds that inequality 4.11.1 holds for u ∈ H 4 (Ω).

Let C 0 > sup Ω φ - (ii) the operator P is symmetric under homogeneous boundary conditions, that is, (P u, v) L 2 (Ω) = (u, P v) L 2 (Ω) , (4.12.2) Proposition 4.12.6 (strong solutions of the damped plate equation). For (y 0 , y

= 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 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) . (4.12.24)

Observe that if y 0 ∈ ker(P 0 ) then y(t) = y 0 is solution to (4.12.1) with y 1 = 0. This is consistent with the form of the semigroup S(t) given in (4.12.20). 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 (4.12.24). In fact, for a strong solution to (4.12.1) as given by Proof. By Proposition 4.13.1 iσId -Ȧ is indeed invertible. Observe that it then suffices to prove the resolvent estimate (4.13.1) 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 (iσId-Ȧ)U = F . This reads

f 0 = iσu 0 + u 1 , f 1 = -P 0 u 0 + (iσ -α)u 1 .
which gives (P 0 -σ 2 -iσα)u 0 = f with f = (iσ -α)f 0 -f 1 . Computing the L 2 -inner product with u 0 one finds ((P 0 -σ 2 )u 0 , u 0 ) L 2 (Ω) -iσ(αu 0 , u 0 ) L 2 (Ω) = (f, u 0 ) L 2 (Ω) .

As α ≥ 0, computing the imaginary part one obtains Thus, with ε chosen sufficiently small one obtains ∥u 0 ∥ H 4 (Ω) ≲ e C|σ| 1/2 ∥f ∥ L 2 (Ω) .

σ∥α 1/2 u 0 ∥ 2 L 2 (Ω) = -Im(f, u 0 ) L 2 (Ω) .
Since u 1 = f 0 -iσu 0 and f = (iσ -α)f 0 -f 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 ∥U ∥ 2 Ḣ = (P 0 u 0 , u 0 ) L 2 (Ω) + ∥u 1 ∥ L 2 (Ω) ≲ e C|σ| 1/2 ∥F ∥ Ḣ, which concludes the proof of the resolvent estimate (4.13.1).

Stabilization result

As an application of the resolvent estimate of Theorem 4.13.2, we give a logarithmic stabilization result of the damped plate equation (4.5.1).

For the plate generator (A, D(A)) its iterated domains are inductively given by

D(A n+1 ) = {U ∈ D(A n ); AU ∈ D(A n )}.
With Proposition 4.12.6, 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 (4.12.1). 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 4.12.2. Moreover, by (4.12.25) the energy of y(t) is given by the square of the Ḣ-norm of Ẏ (t).

With the resolvent estimate of Theorem 4.13.2, with the result of Theorem 1.5 in [START_REF] Batty | Non-uniform stability for bounded semi-groups on Banach spaces[END_REF] one obtains the following bound for the energy of y(t):

E(y)(t) = ∥ Ẏ (t)∥ 2 Ḣ ≤ C log(2 + t) 4n ∥A n Y 0 ∥ Ḣ. ( 4

.13.2)

We have thus obtain the following theorem.

Theorem 4.13.3 (logarithmic stabilisation for the damped plate equation).

Assume that conditions (i) to (iv) of Section 4.12 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 (4.5.1) has the logarithmic energy decay given by (4.13.2).

Here, ω is a nonempty open subset of ω, B 1 and B 2 are boundary operators chosen that satisfies the Lopatinskiȋ-Šapiro condition. The function v is the control and lies in L 2 ((0, ∞) × Ω). It only acts on the solution u in ω. The question of null controllability for this controlled parabolic equation is the following:

For a given initial data u 0 ∈ L 2 (Ω), for a given time T > 0, can one find v ∈ L 2 ((0, T ) × Ω) such u(T ) = 0.

The answer to this question rely on the derivation of a spectral inequality. If the boundary operators B 1 and B 2 are well chosen, the bi-Laplace operator ∆ 2 g can be selfadjoint on L 2 (Ω); see Section 4.12.1. 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 5.2.1. (Spectral inequality for the "clamped" bi-Laplace operator).

Let ω be an open subset of Ω. There exists C > 0 such that ∥u∥ L 2 (Ω) ≤ Ce Cµ 1/4 ∥u∥ L 2 (ω) , µ > 0, u ∈ Span{φ j ; µ j ≤ µ}.

The proof of this theorem is based on a Carleman inequality for the fourthorder elliptic operator D 4 s + ∆ 2 g , that is, after the addition of a variable s. Extending this strategy to the type of boundary conditions treated in Part B was not successful so far because it is not guaranteed that having the Lopatinskiȋ-Šapiro condition for ∆ 2 g , B 1 , and B 2 implies that the Lopatinskiȋ-Šapiro condition holds for D 4 s + ∆ 2 g , B 1 , and B 2 . Yet, the Lopatinskiȋ-Šapiro 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. Next, as a follow up of the result in Part B, we aim to address the polyharmonic case, that is Q = P k , k ∈ N with k boundary operators B 1 , . . . , B k . This can lead to applications similar to the results of Theorem 1.8.2. For the polyharmonic case, the Lopatinskiȋ-Šapiro condition can be formulated as follows. Let Q = (-∆ g ) k be an elliptic differential operator of order 2k on Ω, (k ≥ 1), with principal symbol q(x, ω) for (x, ω) ∈ T * M. One defines the following polynomial in z, q(x, ω ′ , z) = q(x, ω ′ -zn x ), for x ∈ ∂M, ω ′ ∈ T * x ∂M, z ∈ R and n x denotes the outward pointing conormal vector at x, unitary in the sense of the metric g. Here x and ω ′ act as parameters. We denote by r j (x, ω ′ ), 1 ≤ j ≤ 2k the complex roots of q. One sets q+ (x, ω ′ , z) = Im r j (x,ω ′ )≥0 (z -r j (x, ω ′ )).

Given boundary operators B 1 , . . . , B k in a neighborhood of ∂M, with principal symbols b j (x, ω), j = 1, . . . , k, one also sets bj (x, ω ′ , z) = b j (x, ω ′ -zn x ). According to Definition 4. We recall that the principal symbol of Q is given by q(x, ω) = |ω| 2k g . We set q(x, ω ′ , z) = q(x, ω ′ -zn x ) = (|ω

′ -zn x | 2 ) k = (z 2 + |ω ′ | 2 ) k ,
where (n x , ω ′ ) g = 0. Therefore q(x, ω ′ , z) = (z -i|ω

  1,p 0 (Ω) with u ̸ = 0, into equation (1.2.1), this leads to the equation    -∆ p w = λ|w| p-2 w + ∥w∥

Theorem 1 . 3 . 1 .

 131 (bifurcation from zero) Let p > 2. Then every eigenvalue λ D k

Theorem 1 . 3 . 2 .

 132 (bifurcation from infinity) The pair (λ D 1 , 0) is a bifurcation point from infinity for the problem (1.0.1).

Theorem 1 . 4 . 4 .

 144 (Kalman rank condition in finite dimension) The time invariant linear control system (1.4.2) is controllable in time T if and only if rank B | AB | . . . | A n-1 B = n.

2. 1 . 1 .

 11 3 well-known results Theorem 2.1.7. (Gauss-Green) Let U ⊂ R d be a given open set and Ω any smooth subregion within U. Then we have Ω div(F ) dx = ∂Ω F • ν dS, where F denote the flux density (F ∈ C 1 (U , R d )) and ν the unit outer normal field. Corollary 2.1.8. (Integration by parts) Let Ω be a regular open set of class C 1 . Let u and v be two C 1 (Ω) functions with bounded support in the closed set Ω.

Lemma 2 . 1 .

 21 13. (Fatou's lemma) Let {f n } be a sequence of functions in L 1 (Ω) that satisfy i. for all n, f n ≥ 0 a.e ii. sup n Ω f n < ∞.

Theorem 3 . 2 . 1 .

 321 (Dependence on the boundary values) Let f, g ∈ C(Ω, R d ) be such that f (x) = g(x) for all x ∈ ∂Ω and let p ∈ f (∂Ω) = g(∂Ω). Then deg(f, Ω, p) = deg(g, Ω, p).

Theorem 3 . 2 . 2 .

 322 The Leray-Schauder degree has the following properties. (a) (Additivity) If Ω = Ω 1 ∪ Ω 2 , where Ω 1 and Ω 2 are open and disjoint, and

Remark 3 . 3 . 3 .Lemma 3 . 3 . 4 .

 333334 The definition of the genus does not change if we require ψ to be function with values in the sphere S n-1 instead of R n \{0} since we can compose with the projection proj(x) := x |x| • Let E = L 2 (R d) and let A = S E be the unit sphere in L 2 (R d ).

Theorem 3 . 5 . 3 .

 353 Assume p ∈ (1, 2). Then the set of first eigenvalues of problem (3.4.3) is given by (λ D 1 , ∞), where λ D 1 denotes the first eigenvalue of -∆ on Ω.

  and by relation (3.5.4) we obtain that

Theorem 3 . 5 . 4 .

 354 For p > 2, the set of first eigenvalues of problem (3.4.3) is given by (λ D 1 , ∞).

  0 then we have ∇u • ∇v = |∇u||∇v| and u|∇v| = v|∇u| almost everywhere in Ω. This is equivalent to (u∇v -v∇u) 2 = 0, which implies that u = αv with α ∈ R ⋆ + . Theorem 3.6.3. The first eigenvalues λ of equation (3.4.3) are simple, i.e. if u and v are two positive first eigenfunctions associated to λ, then u = v. Proof. By proposition 3.6.2, we have u = αv. Inserting this into the equation (3.4.3) implies that α = 1.

3. 7 . 1

 71 Bifurcation from zero : the case p > 2 By proposition 3.6.5, equation (3.4.3) is equivalent to

  .7.4) We now state the main theorem concerning the bifurcation from infinity. Theorem 3.7.2. The pair (λ D 1 , ∞) is a bifurcation point from infinity for the problem (3.7.3).

Remark 3 . 7 . 3 .

 373 With this transformation, we have that the pair (λ D 1 , ∞) is a bifurcation point for the problem (3.7.3) if and only if the pair (λ D 1 , 0) is a bifurcation point for the problem (3.7.7).

  .7.11) Proving (3.7.11) garantee the well posedness of deg(I -(λ D 1 ±ε)H(τ, •), B r (0), 0) for any τ ∈ [0, 1]. Indeed, by contradiction suppose that there exists

  .7.20) Proving (3.7.20) garantee the well posedness of deg(I -(λ D k ±ε)H(τ, •), B r (0), 0) for any τ ∈ [0, 1]. Indeed, consider the projections P -and P + onto the spaces span{e 1 , . . . , e k-1 } and span{e k , e k+1 , . . . }, respectively, where e 1 . . . , e k , e k+1 , . . . denote the eigenfunctions associated to the Dirichlet problem (3.4.1). Suppose by contradiction that relation (3.7.20) does not hold. Then there exists

.7. 21 )

 21 Replacing v by P + v +P -v, and multiplying equation (3.7.21) by P + v -P -v in the both sides, we obtain

Theorem 3 . 8 . 1 .

 381 Let 1 < p < 2 or 2 < p < ∞, and suppose that λ ∈ (λ D k , λ D k+1 ) for any k ∈ N * . Then equation (3.4.3) has at least k pairs of nontrivial solutions.

  and then ρ(S j ) ∈ Σ ′ , and γ(ρ(S j )) = γ(S j ) = j for 1 ≤ j ≤ k, for any k ∈ N * . It is then standard (see[START_REF] Ambrosetti | Nonlinear analysis and semilinear elliptic problems[END_REF] Proposition 10.8]) to conclude thatσ λ,j = inf γ(A)≥jsup u∈A I λ (u), 1 ≤ j ≤ k, for any k ∈ N * yields k pairs of nontrivial critical points for I λ , which gives rise to k nontrivial solutions of problem (3.4.3). Case 2: p > 2.

  For a ∈ S m τ , the define pseudo-differential operator of order m, denoted by A = Op(a) is a(x, D, τ )u(x) = Au(x) := 1 (2π) d R d e ix•ξ a(x, ξ, τ )û(ξ)dξ (4.1.3)

Proposition 4 . 1 . 4 .

 414 Let a ∈ S m τ . We have Op(a) : S (R d ) → S (R d ) continuously.

e

  iΦ(x,ξ) a(x, ξ, τ )dξ, in the sense of distributions for a ∈ S m τ (R p × R d ) possibly with p ̸ = d. For a proof of the following theorem we refer to[START_REF] Alinhac | Pseudo-differential Operators and the Nash-Moser Theorem[END_REF].

Theorem 4 . 2 . 1 .( 1 )

 4211 Let p, d ∈ N and let Φ : R p × R d → C be C ∞ and such that Im Φ ≥ 0, (2) Φ is homogeneous of degree 1 in ξ, for |ξ| ≥ 1,

Proposition 4 . 2 . 9 .

 429 If a ∈ S m,r τ , with m ∈ N and r ∈ R, then for m ′ ∈ N and r ∈ R there exists C > 0 such that ∥Op(a)u∥ m ′ ,r ′ ,τ ≤ C∥u∥ m+m ′ ,r+r ′ ,τ , u ∈ S (R d + ).

Corollary 4 . 2 . 10 .

 4210 Let m, m ′ ∈ N and r ∈ R. There exists C > 0 such that

Definition 4 . 3 . 1 .

 431 (standard symbols) Let d ∈ N and let a(x, ξ) ∈ C ∞ (R d ×R d )with m ∈ R, be such that for all multi-indices α, β we have

Definition 4 . 3 . 4 .

 434 (Tangential symbols and operators) We say that a

Proposition 4 . 4 . 3 .

 443 Let m ∈ N. Assume that (P, B 1 , B 2 ) fulfills theLopatinskiȋ-Šapiro condition on ∂M. There exists a bounded linear operator

  product of several factors of the form D d -λρ with Im ρ < 0 yield an estimate without requiring any boundary term while factors of the form D d -λρ with Im ρ > 0 require a boundary term. The number of factors of the second kind yield the number of required boundary conditions. In that framework, if given some boundary operators, the Lopatinskiȋ-Šapiro condition states their compatibility with the different factors D d -λρ with Im ρ > 0.

Example 4 . 6 . 6 (

 466 Clamped boundary conditions). This type of conditions refers to B 1 u |∂Ω = u |∂Ω and B 2 u |∂Ω = ∂ n u |∂Ω . With (4.6.8) one finds T (u, v) = 0 in the case of homogeneous conditions, hence symmetry. With the notation of Section 4.6 this gives b1 (x, ω ′ , z) = 1 and b2 (x, ω ′ , z) = -iz. It follows that det b1 b2

Examples 4 . 6 . 7 (

 467 More examples). i. 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.

  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 4.7.4 one has

4. 7 . 3

 73 Proof of Proposition 4.7.2 Here, according to the statement of Proposition 4.7.2 we consider |d x ′ φ| ≤ µ 0 ∂ d φ and σ ≤ µ 1 τ ∂ d φ.

  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 4.7.1 and the Lopatinskiȋ-Šapiro condition holds. To carry on with the proof of Proposition 4.7.2 we now only have to consider havingτ ∂ d φ ≤ C 2 |ξ ′ |. (4.7.6)Our proof of Proposition 4.7.2 relies on the following lemma.

.7. 10 )

 10 In fact, one has ρ + = π j,2 with j = 1 or 2. We use the first part of Lemma 4.6.4 with ζ ′ = iτ d x ′ φ and δ = i(α j -|ξ| x ). With (4.7.8) and (4.7.9) with Lemma 4.7.8 and the first part of Lemma 4.6.4 one obtains (4.7.10).

Lemma 4 . 7 . 10 .

 4710 Set the m ′ × (m + 1) matrix M (ϱ ′ ) = (M j,ℓ (ϱ ′ ))1≤j≤m′ 0≤ℓ≤m with M j,ℓ (ϱ ′ ) = e ℓ j,ϱ 0′ (ϱ ′ ).Then, the second point in Proposition 4.7.9 states that M (ϱ ′ ) is of rank m + 1 = 4 for ϱ ′ ∈ U .

Proposition 4 . 8 . 1 .

 481 Let κ 0 > 0. Let x 0 ∈ ∂Ω, with Ω locally given by {x d > 0}. Assume that (P σ , B 1 , P 2 , φ) satisfies the Lopatinskiȋ-Šapiro condition of Definition 4.7.1 at ϱ

since 13 / 2 2 j=1

 1322 -m + = 5/2 + m -. With (4.8.3) then one finds | tr(u)| 3,1/2,τ ≲ |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 .

  The open sets U i give also an open covering of W 0 × S d-1

2 +

 2 ≥ εL(u) + 2 Re(Q j s u, iQ a u) + .(4.9.20)For concision we write ϱ = (ϱ ′ , ξ d ) with ϱ ′ = (x, ξ ′ , τ, σ). The setL = {ϱ = (ϱ ′ , ξ d ); ϱ ′ ∈ 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 (ϱ)| ≥ C > 0.By homogeneity one has|q j σ,φ (ϱ)| ≳ λ 2 τ , ϱ ′ ∈ U , ξ d ∈ R, if τ ≥ τ 0 σ, (4.9.21)for some τ 0 > 0. Thereforeℓ(ϱ) ≳ λ 4 τ , ϱ ′ ∈ U , ξ d ∈ R, if τ ≥ τ 0 σ. (4.9.22) By the Gårding inequality of Proposition 4.2.12 one obtains Re L(u) ≥ C∥u∥ 2 2,τ -C ′ | tr(u)| 2 1,1/2 -C N ∥v∥ 2 2,-N,τ , (4.9.23)

Lemma 4 . 9 . 16 .φ u∥ 2 +

 49162 There exist C, C N and τ 0 > 0 such thatτ 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 (4.9.24)-(4.9.25), and Lemma 4.9.16 one obtains2 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 ,(4.9.26)with a trace inequality, for τ ≥ κ 0 σ chosen sufficiently large. With (4.9.20), (4.9.23), and (4.9.26) one obtains∥Q j σ,≥ 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 .

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 -(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 (1, 0) in the sense of Definition 4.2.14. Its principal symbol is given by a 0 (ϱ ′ , ξ d , ξ ′ d ) = (1, ξ d )A(ϱ ′ ) t (1, ξ d ) with A(ϱ ′ ) = -(∂ d φ) r(x, ξ ′ ) + (-1) j σ 2 -(τ ∂ d φ) 2 -r(x, τ d x ′ φ) |x d =0 + r(x, ξ ′ , d x ′ φ) |x d =0 + r(x, ξ ′ , d x ′ φ) |x d =0 + ∂ d φ |x d =0 + , with ϱ ′ = (x, ξ ′ , τ, σ).The associated bilinear symbol introduced in (4.2.7) is given byΣ A (ϱ ′ , z, z ′ ) = zA(ϱ ′ ) t z′ , z = (z 0 , z 1 ) ∈ C 2 , z ′ = (z ′ 0 , z ′ 1 ) ∈ C 2 .One computesdet A(ϱ ′ ) = -(∂ 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 4.7.3 one sees that Im π j,2 < 0 is equivalent to having det A(ϱ ′ ) > 0. We thus havedet A(ϱ ′ ) ≥ C > 0, for ϱ ′ = (x, ξ ′ , τ, σ) ∈ S U , with S U = {ϱ ′ ∈ 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 (ϱ ′ , z, z) ≥ C(|z 0 | 2 + |z 1 | 2 ), ϱ ′ = (x, ξ ′ , τ, σ) ∈ U , |(ξ ′ , τ, σ)| = 1.By homogeneity one obtainsRe Σ A (ϱ ′ , z, z) ≥ C(λ 2 T,τ |z 0 | 2 + |z 1 | 2 ), ϱ ′ = (x, ξ ′ , τ, σ) ∈ U , |(ξ ′ , τ, σ)| ≥ 1.With Proposition 4.2.16, having in mind what is exposed in Section 4.5.4 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 4.7.6. 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 4.2.16 for a boundary quadratic form of type (1, 1/2). This concludes the proof.

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 .(4.10.4)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 0T,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 (4.10.2), from (4.10.4) one thus obtainsτ -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 Ψ2,-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 .

2 j=1

 2 |e τ φ B j u |∂Ω | 7/2-k j ,τ + τ -1/2 ∥e τ φ χ 0 u∥ 4,τ ,(4.11.1) for τ ≥ τ 0 , κ 0 σ ≤ τ ≤ κ ′ 0 σ, 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.

Remark 4 .

 4 11.1.

Remark 4 . 11 . 3 .

 4113 With Remark 4.11.1 the result of Theorem 4.11.2 hold forP σ = ∆ 2 -σ 4 replaced by P σ -iσ 2 α = ∆ 2 -σ 4 -iσ 2 α.

4. 12

 12 Solutions to the damped plate equationsHere, we review some aspects of the solutions of the damped plate equation :y + P y + α(x)∂ t y = 0 (t, x) ∈ R + × Ω, B 1 y |R + ×∂Ω = B 2 u |R + ×∂Ω = 0, y |t=0 = y 0 , ∂ t y |t=0 = y 1 ,(4.12.1)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 4.6.1 is fulfilled for (P, B 1 , B 2 ) on ∂Ω;

2 B

 2 (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 equivalentlyy ∈ C 0 [0, +∞); H 2 B (Ω) ∩ C 1 [0, +∞); L 2 (Ω) ∩ C 2 [0, +∞); H -2 B (Ω) .

Theorem 4 . 13 . 2 .

 4132 Let O be an open subset of Ω such that α ≥ δ > 0 on O. Then, for σ ∈ R the unbounded operator iσId -Ȧ is invertible on Ḣ and for there exist C > 0 such that∥(iσId -Ȧ) -1 ∥ L( Ḣ) ≤ Ce C|σ| 1/2 , σ ∈ R.(4.13.1)

2 L 2 ( 2 L 2 (Ω) ∥u 0 ∥ 1 / 2 L 2 ( 2 L 2 (Ω) ∥u 0 ∥ 1 / 2 L 2 (

 222212222122 Since α ≥ δ > 0 in O by assumption and since we consider |σ| ≥ σ 0 one hasδσ 0 ∥u 0 ∥ O) ≤ ∥f ∥ L 2 (Ω) ∥u 0 ∥ L 2 (Ω) .Applying Theorem 4.11.2 (with Remark 4.11.3) one has∥u 0 ∥ H 4 (Ω) ≲ e C|σ| 1/2 ∥f ∥ L 2 (Ω) + ∥u 0 ∥ L 2 (O) .Thus, we obtain∥u 0 ∥ H 4 (Ω) ≲ e C|σ| 1/2 ∥f ∥ L 2 (Ω) + ∥f ∥ 1/Ω) ,for |σ| ≥ σ 0 . With Young inequality we write, for ε > 0,e C|σ| 1/2 ∥f ∥ 1/Ω) ≲ ε -1 e 2C|σ| 1/2 ∥f ∥ L 2 (Ω) + ε∥u 0 ∥ L 2 (Ω) .

  6.1, along with the general boundary operators B 1 , B 2 . . . , B k of orders d 1 , d 2 , . . . , d ℓ respectively for ℓ = 1, . . . , k, we give a matrix criterion of the Lopatinskiȋ-Šapiro condition. The general boundary operators B 1 , B 2 . . . , B k is then given byB ℓ (x, D) = 0≤j≤min(2k-1,d ℓ ) (-i) j B d ℓ -j ℓ (x, D ′ )D j d , ℓ = 1 . . . , k.We denote by b 1 (x, ω), . . . , b k (x, ω) the principal symbols of B 1 , . . . , B k respectively. For (x, ω ′ ) ∈ T * ∂M, we setb ℓ (x, ω ′ , z) = 0≤j≤min(2k-1,d ℓ ) (-i) j b d ℓ -j ℓ (x, ω ′ )z j d , ℓ = 1 . . . , k.

  ′ |) k (z + |ω ′ |) k and we set q+ (x, ω ′ , z) = (z + |ω ′ |) k . Thus the Lopatinskiȋ-Šapiro condition holds at (x, ω ′ ) with ω ′ ̸ = 0 if and only if for any polynomial function f (z) the complex number i|ω ′ | g is a root of the polynomial function z → f (z) -c 1 b1 (x, ω ′ , z) -• • •c k bk (x, ω ′ , z) and its derivative up to order k -1 for some c 1 , . . . , c k ∈ C. This leads to the following determinant condition det ω ′ , z = i|ω ′ | g ) ̸ = 0.(5.2.2)

  2.1.1.1 Functions spaces Definition 2.1.1. A normed space X is a called a Banach space if it is complete, i.e., if every Cauchy sequence is convergent, that is,

  Theorem 2.1.11. (Rellich-Kondrachov embedding) Suppose that Ω ⊂ R d is bounded and of class C 1 . Then we have following compact injections.

	1,p 0 (Ω),
	∥u∥ p ≤ C p ∥∇u∥ p .	(2.1.4)

  Theorem 2.1.23. Let A ∈ B(X, Y ). It is Fredholm if and only if there exists S ∈ B(X, Y ) such that

  since we have a finite number of S(t k )x. Then, for k ∈ {1, . . . , k 0 -1} we have

	sup ∥S(t k )x∥ L (X) ≤ K.	(2.2.9)
	Combining (2.2.8) and (2.2.9) we get sup	∥S(t k )x∥ L (X) ≤ 1 + ∥x∥ L (X) + K < ∞
	≥1
	and this a contradiction. Hence there is an µ > 0 such that
	sup
	t∈[0,µ]

  Definition 4.2.15. 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

.2.7)

  ij (x)ξ i ξ j .The bi-Laplace operator is P = ∆ 2 g . In what follows we shall write ∆, ∆ 2 in place of ∆ g , ∆ 2g .

	in local coordinates. Its principal part is given by	g ij (x)∂ x i ∂ x j and its
	1≤i,j≤d
	principal symbol by
	1≤i,j≤d
		.5.2)

g

  z 0 , . . . , z 3 ). (4.7.13) Let the Lopatinskiȋ-Šapiro condition hold at ϱ 0′

	in agreement with the notation introduced in (4.2.6) in Section 4.2.1. One has
	the following positivity result.
	Lemma 4.7.11.

  4 by Lemma 4.7.10 one obtains the result. Near x 0 ∈ ∂Ω we consider two boundary operators B 1 and B 2 . As in Section 4.7 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 fourthorder 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).

	4.8 Estimate for the boundary norm under Lopatin-
	skiȋ-Šapiro condition

  2,τ ≲ ∥Q 1 σ,φ w∥ + + |w |x d =0 + | 3/2,τ + ∥w∥ 2,-1,τ , (4.10.2) and τ -ℓ 2 ∥Op T (χ)w∥ 2,τ ≲ ∥Q 2 σ,φ w∥ + + |w| 3/2,τ + ∥w∥ 2,-1,τ , (4.10.3)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. WritingP σ,φ = Q 2 σ,φ Q 1 σ,φ ,with (4.10.3) 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 .

  1. Since 1 ≤ φ ≤ sup Ω φ one obtains ∥u∥ H 4 (Ω) ≲ e C 0 τ ∥P σ u∥ L 2 (Ω) + |∂Ω | H 7/2-k j (∂Ω) + ∥u∥ H 4 (O 1 ) . (4.11.2) 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 (4.11.2) one obtains the following observability result. Theorem 4.11.2 (observability inequality). Let P σ = ∆ 2 -σ 4 and let B 1 and B 2 be two boundary operators of order k 1 and k 2 as given in Section 4.6.2. Assume that the Lopatinskiȋ-Šapiro condition of Definition 4.6.1 holds. Let O be an open set of Ω. There exists C > 0 such that ∥u∥ H 4 (Ω) ≤ Ce C|σ| 1/2 ∥P σ u∥ L 2 (Ω) +

	2	j=1 |B j u 2 |B
	j=1

j u |∂Ω | H 7/2-k j (∂Ω) + ∥u∥ L 2 (O) , for u ∈ H 4 (Ω).

  1 ) ∈ H 4 B (Ω) × H 2 B (Ω) there exists a uniquey ∈ 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 . (4.12.22) Moreover, there exists C > 0 such that ∥y(t)∥ H 4 B (Ω) + ∥∂ t y(t)∥ H 2 B (Ω) ≤ C ∥y 0 ∥ H 4 B (Ω) + ∥y 1 ∥ H 2 B (Ω) , t ≥ 0.

			(4.12.23)
	With Y (t) as above, for such a solution y(t) one has
	d dt	Y (t) + AY (t) = 0,	Y

Acknowledgements

We first recall that following Green formula (4.6.6) which applied twice gives (P u, v) L 2 (Ω) = (u, P v) L 2 (Ω) + T (u, v) with

(4.6.7)

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 (4.6.7) in the alternative forms

or

We start our list of examples with the most basics ones.

if ω ′ ̸ = 0 and thus the Lopatinskiȋ-Šapiro condition holds by Lemma 4.6.2.

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.

Being able to solve this system in c 1 and c 2 for any f is equivalent to having

• Case 4 : A double root lying in the upper complex closed half-plane. 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 (x, ξ ′ , τ, σ) if and only if for any choice of f , the complex number ρ + is a double root of the polynomial function 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,

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 4.7.7. This will be an important point in what follows.

We now state the following important proposition.

Proposition 4.7.2. 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 4.6.3). Assume also that

then the Lopatinskiȋ-Šapiro condition holds for

The proof of Proposition 4.7.2 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.

Case 3.

In this case Im π 1,2 > 0 and Im π 2,2 > 0. According to the discussion in Section 4.7.1 it suffices to prove that 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 4.7.1 once the Lopatinskiȋ-Šapiro condition is know to hold at a point ϱ 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 4.7.1 for ϱ ′ = (x, ξ ′ , τ, σ) one denotes by ρ j (ϱ ′ ) the roots of p σ,φ (x, ξ ′ , ξ d , τ ) viewed as a polynomial in ξ d .

Let

and, for ϱ ′ = (x, ξ ′ , τ, σ),

The proof of Estimate (4.9.6) 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 4.9.10. There exist C > 0 and τ 0 > 0 such that for any N ∈ N, there exists C N > 0 such that

The proof of Lemma 4.9.10 relies on the Lopatinskiȋ-Šapiro condition.

The following lemma gives an estimate for L 1 .

Lemma 4.9.11. 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

The proof of Lemma 4.9.11 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 4.9.12. 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

Note that this estimate is weaker than that of Lemma 4.9.11

Observing that

and applying Lemma 4.9.11 to w = L 2 v with s = 0, one obtains

Proof of Lemma 4.9.10. Set

This is a boundary differential quadratic form of type (1, 1/2) in the sense of Definition 4.2.14. The associated bilinear symbol is given by

One has

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 ϱ ′ ∈ U 1 one has

by (4.9.8). From the homogeneity of b k-1 φ (x, ξ ′ , τ ) and bk φ (x, ξ ′ , τ ) in ϱ ′ , it follows that there exists some C > 0 such that

The result of Lemma 4.9.10 thus follows from Proposition 4.2.16, having in mind what is exposed in Section 4.5.4 since we have τ ≥ κ 0 σ here.

Proof of Lemma 4.9.11. We let u = Op T (χ)w. Performing an integration by parts, one has

Since U is neigborhood of supp(χ), the microlocal Gårding inequality of Theorem 2.49 in [START_REF] Rousseau | Elliptic Carleman estimates and applications to stabilization and controllability[END_REF] (the proof adapts to the case with parameter σ as explained in Section 4.5.4 since σ ≲ τ ) yields

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

which yields for ε chosen sufficiently small,

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 . (4.9.10)

Putting together (4.9.9) and (4.9.10), the result of Lemma 4.9.11 follows.

Proof of Lemma 4.9.12. We let u = Op T (χ)w. Performing an integration by parts, one has

Note that Re(Λ 2s+1

T,τ

Op T ( χπ j,2 ) has the following real principal symbol

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[END_REF] (the proof adapts to the case with parameter σ as explained in Section 4.5.4 since σ ≲ τ ) yields

for τ ≥ κ 0 σ chosen sufficiently large. The end of the proof is then similar to that of Lemma 4.9.11.

4.9.3.2 Case (ii): one real root.

One has Im π j,2 (ϱ 0′ ) = 0 and Im π j,1 (ϱ 0′ ) < 0.

Since the Lopatinskiȋ-Šapiro condition holds for (Q j σ , B, φ) at ϱ 0′ , by Lemma 4.9.8 one has

As the roots π j,1 and π j,2 are locally smooth with respect to ϱ ′ = (x, ξ ′ , τ, σ) and homogeneous of degree one in (ξ ′ , τ, σ), there exists U a conic neighborhood of

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

Lemma 4.9.10 and Lemma 4.9.11 also apply in Case (ii) and we shall use them. In addition to these two lemmata we shall need the following lemma. Lemma 4.9.13. There exist C > 0, τ 0 > 0 such that

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 subellipticity condition in W for both j = 1 and 2. Let ϱ 0′ = (x 0 , ξ 0′ , τ 0 , σ 0 ) with

)

An important aspect is that here we have σ ≳ τ ; this explains that only one root of p σ,φ can lie on the real axis and thus only one half derivative is lost in this estimate. The proof of Proposition 4.10.1 is based on the microlocal results of Proposition 4.9.9.

Proof. We shall concatenate the estimates of Proposition 4.9.9 for

with the boundary operator B simply given by the Dirichlet trace operator,

. With Lemma 4.9.8 one sees that the Lopatinskiȋ-Šapiro holds for (Q 1 σ,φ , B, φ) and (Q 2 σ,φ , B, φ) at ϱ 0′ . Proposition 4.9.9 thus applies. Let U j be the conic neighborhood of ϱ 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

and we consider χ ∈ S 0 T,τ , homogeneous of degree 0 in (ξ ′ , τ, σ) with supp(χ) ⊂ U .

Proposition 4.10.2. 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

)

Final estimate

Combining the local results of Section 4.8 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 1.8.1. For a precise statement we write the following theorem.

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

)

The notation of the function space

3). 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 4.9.

for σ ≥ 0, τ ≥ max(τ 1 , κ 0 σ) for some τ 1 > 0. With the second proposition one has

)

. Combined together (4.10.7) and (4.10.8) yield

Since ∥v∥ 4,-1,τ ≪ τ -1/2 ∥v∥ 4,τ for τ large one obtains

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

Global Carleman estimate and observability

Using the local Carleman estimate of Theorem 4.10.3 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 4.13 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.

A global Carleman estimate

Assume that the Lopatinskiȋ-Šapiro condition of Definition 4.6.1 holds for

With Proposition 3.31 and Remark 3.32 in [START_REF] Rousseau | Elliptic Carleman estimates and applications to stabilization and controllability[END_REF] there exists ψ ∈ C ∞ (Ω) such that i. ψ = 0 and ∂ ν ψ < -C 0 < 0 on ∂Ω;

ii. ψ > 0 in Ω;

Examples of such conditions are given in Section 4.6.5.

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

(4.12.4)

This last property is very natural to define a nonnegative energy for the plate equation given in (4.12.1). 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 4.13.1-4.13.2.

The unbounded operator associated with the 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

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 (1.8.2). From the assumed nonnegativity in (4.12.4) above one finds that P 0 + id is injective. Since the operator

is Fredholm and has the same zero index as L defined in (4.12.3), 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 (4.12.2) 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 (4.12.4) they take the form of a sequence

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

One has

is equipped with the inner product and norm

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 H k B (Ω), k ∈ R. One has P 0 : H k B (Ω) → H k-4 B (Ω) continuously with

(4.12.6)

In particular, for u ∈ H 4 B (Ω) = D(P 0 ) and v ∈ H 2 B (Ω) one has

(4.12.8)

Note that

.9)

with the operator P viewed as un unbounded operator on L 2 (Ω). We make the following observations. i. If ker(P 0 ) = {0} then

is also an inner-product on H 2 B (Ω), that yields an equivalent norm.

ii. 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

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 plate semigroup generator

Set H = H 2 B (Ω) ⊕ L 2 (Ω) with natural inner product and norm

.12.11)

(4.12.12)

Define the unbounded operator

.12.13) 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

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 4.13.1. 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

We set

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

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 ). (4.12.16)

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

Lemma 4.12.2. We have Ran(A) ⊂ Ḣ.

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 (4.12.15).

The space Ḣ inherits the natural inner product and norm of H given in (4.12.11). Yet one finds that the inner product

and associated norm

yields an equivalent norm on Ḣ by a Poincaré-like argument. We introduce the unbounded operator Ȧ on Ḣ given by the domain

As for the decomposition of H given in (4.12.16) note that D( Ȧ) and N are not orthogonal.

Lemma 4.12.3. Let z ∈ C be such that Re z < 0. We have

The proof of this lemma is quite classical. It is given in Appendix 6.2.

With the previous lemma, with the Hille-Yosida theorem one proves the following result. Lemma 4.12.4. The operator ( Ȧ, D( Ȧ)) generates a semigroup of contraction Ṡ(t) = e -t Ȧ on Ḣ.

If we set

we find that S(t) is a semigroup on H generated by (A, D(A)), thus proving Proposition 4.12.1. If Y 0 ∈ D(A), the solution of the semigroup equation

We set

The adjoint of Ȧ has domain D( Ȧ * ) = D(A) and is given by

.

Similarly to Lemma 4.12.3 one has the following result with a similar proof.

Lemma 4.12.5. Let z ∈ C be such that Re z < 0. We have

Strong and weak solutions to the damped plate equation

For y(t) a solution to the damped plate equation (4.12.1) 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 4.12.1 allows one to go beyond this formal observation and one obtains the following wellposedness result for strong solutions of the damped plate equation.

Resolvent estimates and applications to stabilization

Here we use the observability inequality of Theorem 4.11.2 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].

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. 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 6.3.

Perspectives

Fučik spectrum

As an extension of the results of Part A, we are interested of investigating the Fučik spectrum of the (p, 2)-Lapalcian. The Fučik spectrum was introduced by S. Fučik [START_REF] Fučik | Boundary value problems with jumping nonlinearities[END_REF] and N. Dancer [START_REF] Dancer | On the Dirichlet problem for weakly non-linear elliptic partial differential equations[END_REF] in the 70's. For the Laplacian it is defined as the set Σ ⊂ R × R of the points (α, β) for which there exists a nontrivial solution of the problem

where Ω is a bounded domain in R N (N ≥ 2), Bu stands for the considered boundary conditions and u ± = max{±u, 0}. We aim to investigate the set of pairs (α,

has nontrivial solution, with p ∈ (1, ∞) \ {2}. We clearly see that if α = β we recover the case of equation (3.4.3). We expect solutions branches of (5.1.2) to bifurcate from the Fučik eigenvalues of (5.1.1). Next, our goal is to study the Fučik spectrum of the following problem

where m and n are positive bounded weights for p ∈ (1, ∞) \ {2}.

Controllability and stablization

On a bounded regular open set of R d if given a positive fourth-order elliptic operator P (or on a Riemannian manifold (M, g) with P = ∆ 2 g ), one can consider the following controlled parabolic equation

(5.2.1)

6. Appendix

A perfect elliptic estimate

Here we consider a(ϱ ′ , ξ d ) polynomial in the ξ d variable and such that its root have negative imaginary parts microlocally.

d , and where the coefficients a j are homogeneous in (ξ ′ , τ, σ). Moreover, assume that a 0 (ϱ ′ ) = 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(ϱ ′ , ξ d ) have a negative imaginary part for ϱ ′ ∈ U .

Let χ(ϱ ′ ) ∈ 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

where p and q are both homogeneous with p ∈ S k,0 τ and q ∈ S k-1,0 τ . We set P = Op(p) and Q = Op(q) and we introduce the following quadratic form of type (k, 0)

τ . The Hermite theorem (see, [START_REF] Bellassoued | Carleman estimates for elliptic operators with complex coefficients. Part I : boundary value problems[END_REF]Proposition 3.13]) implies that a(ϱ ′ , ξ d ) and b(ϱ ′ , ξ d ) have distinct real roots for all ϱ ′ ∈ U . Hence, on the compact set In addition, by the Generalized Green's formula (see, [START_REF] Bellassoued | Carleman estimates for elliptic operators with complex coefficients. Part I : boundary value problems[END_REF]Proposition 3.15]), we have 2 Re(P w, iQw) + = I p,q (w) + B p,q (w) + R(v),

where I p,q is an interior quadratic form of type (k, -1/2), B p,q is the boundary quadratic form of type (k -1, 1/2) and R(w) is the remainder term which is a quadratic form that satisfies [START_REF] Bellassoued | Carleman estimates for elliptic operators with complex coefficients. Part I : boundary value problems[END_REF]. Then we deduce that 2 Re(P w, iQw) + ≳ B p,q (w) -∥w∥ k,-1/2,τ.

Again by the Hermite theorem ( [11, Proposition 3.13]), the bilinear form associated to B p,q denoted Σ Bp,q is positive. Then by homogeneity we find that

Then the Gårding inequality of Lemma 3.9 in [START_REF] Bellassoued | Carleman estimates for elliptic operators with complex coefficients. Part I : boundary value problems[END_REF] gives, for any N ∈ N,

Basic resolvent estimation

Here we provide a proof of Lemma 4.12.3 Let U = t (u 0 , u 1 ) ∈ D( Ȧ). With (4.12.17) We write

B ,H 2 B -(P 0 u 0 , u 1 ) L 2 (Ω) -(αu 1 , u 1 ) L 2 (Ω)

= z∥U ∥ 2 Ḣ + 2i Im(u 1 , P 0 u 0 ) L 2 (Ω) -(αu 1 , u 1 ) L 2 (Ω) .

Computing the real part one obtains -Re((zId Ḣ -Ȧ)U, U ) Ḣ = -Re(z)∥U ∥ 2 Ḣ + (αu 1 , u 1 ) L 2 (Ω) . which yields the conclusion of Lemma 4.12.3.

Basic estimation for the resolvent set

Here we provide a proof of Proposition 4.13.1. Let z ∈ C. We consider the two cases.

Case 1: Re z < 0. By Lemma 4.12.3 zId Ḣ -Ȧ is injective. Moreover, as its adjoint zId Ḣ -Ȧ * is injective and satisfies ∥(zId Ḣ -Ȧ * )U ∥ Ḣ ≳ ∥U ∥ Ḣ for U ∈ D( Ȧ) by Lemma 4.12.5 the map zId Ḣ -Ȧ is surjective (see for instance [?, Theorem 2.20]). The estimation of Lemma 4.12.3 then gives the continuity of the operator (zId Ḣ -Ȧ) -1 on Ḣ.

Case 2: Re z = 0. We start by proving the injectivity of zId 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 + (z -α)u 1 = 0. (6.3.1)

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 (6.2.1) we obtain 0 = Re((zId 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 zId Ḣ -Ȧ is surjective, the result then follows from the closed graph theorem as Ȧ is a closed operator. We write zId Ḣ -Ȧ = T + Id Ḣ with T = (z -1)Id Ḣ -Ȧ. By the first part of 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 zId H -Ȧ is injective as proven above. Consequently, zId H -Ȧ is surjective. This concludes the proof of Proposition 4.13.1.