Chenmin Sun

SEMI-CLASSICAL PROPAGATION OF SINGULARITIES FOR THE STOKES SYSTEM

Keywords: Stokes system, Propagation of singularities, Semiclassical analysis. AMS Classification: 35P20, 35S15, 35Q35

Semi-classical propagation of singularities for the Stokes system

Chenmin

Introduction

Let Ω ⊂ R d be a smooth bounded domain. Consider the eigenvalue problem of the Stokes operator

     -∆u k + ∇P k = λ 2 k u k , in Ω div u k = 0, in Ω u k | ∂Ω = 0 (1.1)
where u k ∈ (H 2 (Ω)) d ∩ V , u k L 2 = 1, are R d -valued normalized eigenfunctions and V = {u ∈ (H 1 0 (Ω)) d : div u = 0}. We collect several facts which are well-known in functional analysis:

• u k forms a orthonormal basis of

H = {u ∈ (L 2 (Ω)) d : div u = 0, u • ν| ∂Ω = 0}
The canonical projector Π : (L 2 (Ω)) d → H is called Leray projector.

• The pressure P k ∈ L 2 (Ω)/R satisfies Ω P k = 0. • ∇u k 2 L 2 = λ 2 k , u k H 2 ≤ Cλ 2 k , ∇P k L 2 ≤ Cλ 2 k , P k L 2 ≤ Cλ 2
k . We rephrase the system (1.1) by semi-classical reduction. Taking h k = λ -1 k and q k = λ -1 k P k , dropping the sub-index, we obtain the following h-dependent system

     -h 2 ∆u -u + h∇q = 0, in Ω h div u = 0, in Ω u| ∂Ω = 0
In this article, we will study the following generalization by adding a quasi-mode:

     -h 2 ∆u -u + h∇q = f, in Ω h div u = 0, in Ω u| ∂Ω = 0 (1.2)
with the following conditions:

u L 2 = 1, h∇u L 2 = O(1), h 2 ∇ 2 u L 2 = O(1), h∇q L 2 = O(1), f ∈ H, f L 2 = o(h).
When h is small, the corresponding solution u = u(h) can be interpreted as highfrequency quasi-mode as its mass, i.e., the L 2 norm, is essentially concentrated on the frequency scale h -1 . Before stating the main result, it is worth mentioning the eigenvalue problem of the Laplace operator in semi-classical version:

-h 2 ∆u -u = 0 in Ω u| ∂Ω = 0.

(1.3)

One method to capture the high-frequency behavior of the solutions of (1.3) is to use semi-classical defect measure associated to a bounded sequence (u k ) of L 2 (Ω) and to a sequence of positive scales h k converging to zero. This measure is aimed to describe quantitatively the oscillations of (u k ) at the frequency scale h -1 k . More precisely, for any bounded sequence (w k ) of L 2 (R d ), there exists a subsequence of (w k ) and a non-negative Radon measure µ on T * R d such that for any a(x, ξ) ∈ S(R 2d ), lim k→∞ (a(x,

h k D x )w k |w k ) L 2 (R d ) = µ, a .
When Ω is a bounded domain, the precise definition of defect measure corresponding to the boundary value problem will be given later.

Let us mention that a counterpart of semi-classical defect measure, micro-local defect measure, was introduced by P. Gérard [START_REF] Gérard | Microlocal defect measures[END_REF] and L. Tartar [START_REF] Tartar | H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations[END_REF] independently. These objects are widely used in the study of control and stabilization, scattering theory and quantum ergodicity, see for example [START_REF] Burq | Mesures de défaut de compacité, application au systeme de lamé[END_REF], [START_REF] Burq | Semi-classical estimates for the resolvent in nontrapping geometries[END_REF], [START_REF] Gérard | Ergodic properties of eigenfunctions for the dirichlet problem[END_REF].

In the context of semi-classical defect measure, the classical theorem of Melrose-Sjöstrand about propagation of singularities ( [START_REF] Melrose | Singularities of boundary value problems[END_REF], [START_REF] Melrose | Singularities of boundary value problems. ii[END_REF]) for hyperbolic equation can be rephrased as follows:

Theorem 1.1 ( [START_REF] Gérard | Ergodic properties of eigenfunctions for the dirichlet problem[END_REF]). Assume that Ω is a smooth, bounded domain with no infinite order of contact on the boundary. Suppose µ is the semi-classical defect measure associated to the pair (u k , h k ) where (u k ) is a sequence of solutions to (1.3) (with h = h k ) which are bounded in L 2 (Ω). Then µ is invariant under the Melrose-Sjöstrand flow.

We will give the precise definition of the Melrose-Sjöstrand flow and the associated concept of the order of contact in the second section. Intuitively, these flows are the generalization of geometric optics. No infinite order of contact means that the trajectory of the flow can not tangent to the boundary with an infinite order.

The main result of this paper is as follows.

Theorem 1.2. Assume that Ω is a smooth, bounded domain with no infinite order of contact on the boundary. Suppose (u k ) is a sequence of solutions to the quasimode problem (1.2) with semi-classical parameters h = h k . Assume that f k ∈ H, f k L 2 (Ω) = o(h k ) and u k converges weakly to 0 in L 2 (Ω). Assume that µ is a semi-classical measure associated to some subsequence of (u k , h k ), then supp(µ) is invariant under the Melrose-Sjöstrand flow.

We make some comments about the result. Firstly, the measure µ is Hermitian matrix-valued, and we have no information so far on the precise propagation for µ except for supp(µ). Secondly, since the eigenfunctions of Stokes operator converge weakly to 0 in L 2 (Ω), our results includes this special case.

The Propagation theorem for a given quasi-mode has many applications, in particular, it leads to the stabilization of the associated damped evolution system. In the context of the damped wave equation, it was shown that (see [START_REF] Rauch | Exponential decay of solutions to hyperbolic equations in bounded domains[END_REF], [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF], [START_REF] Lebeau | Equation des ondes amorties[END_REF]) under the geometric control condition, the energy decays exponentially. An application of Theorem 1.2 is the stabilization of a hyperbolic Stokes system, a model in the theory of linear elasticity introduced in [START_REF] Lions | On some hyperbolic equations with a pressure term, partial differential equations and related subjects[END_REF], under the geometric control condition. More precisely, consider the damped hyperbolic-Stokes system:

       ∂ 2 t u -∆u + ∇p + a(x)∂ t u = 0 in R × Ω, div u = 0 in R × Ω, u = 0 on R × ∂Ω, (u(0, x), ∂ t u(0, x)) = (u 0 , v 0 ) ∈ V × H, (1.4)
The energy

E[u](t) = 1 2 Ω (|∂ t u| 2 + |∇u| 2 )dx
is dissipative. In [START_REF] Chaves-Silva | On the stabilization of a hyperbolic stokes system under geometric control condition[END_REF], we use propagation Theorem 1.2 to show that the energy decays exponentially in time.

Let us describe briefly our strategy for the proof of Theorem 1.2. The pressure term q is harmonic and in heuristic, it can only have the influence to the solution near the boundary. Hence we will prove that the measure µ i is propagated in the same way as Laplace quasi-mode (semi-classical analogue of wave equation) along the rays inside the domain. When a ray reaches the boundary, we need a more careful analysis between the wave-like propagation phenomenon and the concentration phenomenon of the pressure. It is difficult to get a simple propagation formula near the boundary, comparing to the treatment of quasi-mode problem of Laplace operator as in [START_REF] Burq | Mesures de défaut de compacité, application au systeme de lamé[END_REF], [START_REF] Gérard | Ergodic properties of eigenfunctions for the dirichlet problem[END_REF]. We partition the phase space into elliptic region E, hyperbolic region H and glancing surface G. It turns out that no singularity accumulates near the elliptic region. For the hyperbolic region, we prove the propagation by the standard energy estimate, with an additional treatment when the incidence of the ray is right. Near the glancing surface, we will follow the arguments of Ivrii and Melrose-Sjöstrand. The main difference is that we will encounter two new cross terms essentially of the form (q|u) L 2 after certain micro-localization. To overcome this difficulty, we further micro-localize the solution according to the distance to the glancing surface G and treat them separately. For the part nearing G, we use the fact that the pressure decays fast away from the boundary while the solution can not concentrate too much near the boundary, provided that it is micro-localized close enough to the glancing surface. For the part away from G, it can be wellcontrolled by induction argument, using geometric properties of the generalized bicharacteristic flow.

Preliminary

2.1. Notations. We will sometimes drop the sub-index k for a sequence of functions (u k ) and semi-classical parameters h k . In this circumstance, the notion u X = O(1), o(1) as h → 0 should be understood as u k X = O(1), o(1) as k → ∞ (thus h k → 0) up to certain subsequence.

As in the introduction, we follow the notation in the context of the analysis of Stokes system (see [START_REF] Temam | Navier-Stokes equations: theory and numerical analysis[END_REF]) V = {u ∈ H 1 0 (Ω) N : div u = 0} and H = {u ∈ L 2 (Ω) N : div u = 0, u • ν| ∂Ω = 0}. In this paper we always use ν to denote the outward normal vector on ∂Ω.

For a manifold M , we let T M be its tangent bundle and T * M be the cotangent bundle with canonical projection π : T M ( or T * M ) → M. We will identify system (1.2) as a system on differential form

     h 2 ∆ H u -u + hdq = f in Ω hd * u = 0 in Ω u| ∂Ω = 0 (2.1)
where the unknown u ∈ Λ 1 (Ω) is 1-form, and d : Λ p (Ω) → Λ p+1 (Ω), d * : Λ p+1 (Ω) → Λ p (Ω) are exterior differential and divergence operator on forms, with respectively. Recall also that the Hodge Laplace operator is defined by

∆ H = dd * + d * d = (d + d * ) 2 .
In the tubular neighborhood of boundary, we can identify Ω locally as one side of the tubular neighborhood denoted by Y + = [0, 0 ) × X, X = {x ∈ R d-1 : |x | < 1}. We denote by ∂Y + = Y + | y=0 and Y 0 + = Y + | y>0 . For x ∈ Ω, we note x = (y, x ), where y ∈ [0, 0 ), x ∈ X, and x ∈ ∂Ω if and only if x = (0, x ). In this coordinate system, the Euclidean metric dx 2 can be written as matrices g = 1 0 0 g(y, x ) , g -1 = 1 0 0 g -1 (y, x ) ,

with |ξ | 2 g -1 (y,x )) = ξ , g -1 (y, x )ξ R d-1 = g jk ξ j ξ k be the induced metric on T * ∂Ω, parametrized by y. Note that |ξ | 2 g -1 (0,x ) = ξ , g -1 (0, x )ξ R d-1 = g jk ξ j ξ k is the natural norm on T * ∂Ω, dual of the norm on T ∂Ω, induced by the canonical metric on Ω. Write (x, ξ) = (y, x , η, ξ ) and denote by |ξ| the Euclidean norm on T * R d . For u, v ∈ Λ 1 (Y + ) with support in the local chart of turbulence neighborhood, we define the L 2 norms and inner product on [0, 0 ) × X via where for u = u 0 dy + u j dx j , v = v 0 dy + v j dx j , u|v = u 0 v 0 + u j v k g jk .

In certain situations we also use global notation for L 2 inner product:

(u|v) Ω := Ω u • vdx, (f |g) ∂Ω := ∂Ω f • gdσ(x).
We will identify the unknown vector fields u, v, etc. with their dual 1-forms. Formulation of differential form will simplify some calculations. In the tubular neighborhood, we write a vector field

L = L ⊥ ∂ ∂y + L , L = d-1 j=1 L ,j ∂ ∂x j
and we write L = (L ⊥ , L ). The normal component obeys the following convention: (a, 0) = -aν.

Following [START_REF] Chaves-Silva | Spectral inequality and optimal cost of controllability for the stokes system[END_REF], we will write down system (1.2) in the tubular neighborhood. For u = (u ⊥ , u ), equation (1.2) can be written as:

         (-h 2 ∆ -1)u + h∇ x q = f , (-h 2 ∆ g -1)u ⊥ + h∂ y q = f ⊥ , h div u + h √ det g ∂ y ( detgu ⊥ ) = 0 (2.2)
where

h 2 ∆ = h 2 ∂ 2 y -Λ 2 (y, x , hD x ) + hM (y, x , hD x ) + hM 1 (y, x )h∂ y , h 2 ∆ g = h 2 ∂ 2 y -Λ 2 (y, x , hD x ) + hM ⊥ (y, x , hD x ) + hN 1 (y, x )h∂ y , h div u = h √ det g N -1 j=1 ∂ x j ( det gu ,j ).
h 2 Λ 2 (y, x , hD x ) has the symbol λ 2 = |ξ | 2 α(y,•) , and M ,⊥ are both first-order matrix-valued semi-classical differential operators.

Geometric Preliminaries.

Denote by b T Ω the vector bundle whose sections are the vector fields X(p) on Ω with X(p) ∈ T p ∂Ω if p ∈ ∂Ω. Moreover, denote by b T * Ω the Melrose's compressed cotangent bundle which is the dual bundle of b T Ω. Let j : T * Ω → b T * Ω be the canonical map. In our geodesic coordinate system near ∂Ω, b T Ω is generated by the vector fields

∂ ∂x 1 , • • •, ∂ ∂x d-1
, y ∂ ∂y and thus j is defined by j(y, x ; η, ξ ) = (y, x ; v = yη, ξ ).

The principal symbol of operator

P h = -(h 2 ∆ + 1) is p(y, x , η, ξ ) = η 2 + |ξ | 2
g -1 (y,x ) -1. By Car(P ) we denote the characteristic variety of p:

Car(P ) := {(x, ξ) ∈ T * R d | Ω : p(x, ξ) = 0}, Z := j(Car(P )).

By writing in another way

p = η 2 -r(y, x , ξ ), r(y, x , ξ ) = 1 -|ξ | 2
g -1 (y,x ) , we have the decomposition

T * ∂Ω = E ∪ H ∪ G,
according to the value of r 0 := r| y=0 where

E = {r 0 < 0}, H = {r 0 > 0}, G = {r 0 = 0}.
The sets E, H, G are called elliptic, hyperbolic and glancing, with respectively. For a symplectic manifold S with local coordinate (z, ζ), a Hamiltonian vector field associated with a real function f is given by

H f = ∂f ∂ζ ∂ ∂z - ∂f ∂z ∂ ∂ζ .
Now for (x, ξ) ∈ Ω far away from the boundary, the Hamiltonian vector field associated to the characteristic function p is given by

H p = 2ξ ∂ ∂x .
We call the trajectory of the flow

φ s : (x, ξ) → (x + sξ, ξ)
bicharacteristic or simply ray, provided that the point x + sξ is still in the interior.

To classify different situations as a ray reaching the boundary, we need more accurate decomposition of the glancing set G. Let r 1 = ∂ y r| y=0 and define

G k+3 = {(x , ξ ) : r 0 (x , ξ ) = 0, H j r0 (r 1 ) = 0, ∀j ≤ k; H k+1 r0 (r 1 ) = 0}, k ≥ 0 G 2,± := {(x , ξ ) : r 0 (x , ξ ) = 0, ±r 1 (x , ξ ) > 0}, G 2 := G 2,+ ∪ G 2,-. Denote by G ∞ = G \ ∪ j≥2 G j .
We say that there is no infinite order of contact on the boundary if G ∞ = ∅.

By setting

H G p := H p + H 2 p y H 2 y p H y ,
H G p is a well-defined vector field tangent to G which is called the gliding vector field. Given a ray γ(s) with π(γ(0)) ∈ Ω and π(γ(s 0 )) ∈ ∂Ω be the first point that reaches the boundary. If γ(s 0 ) ∈ H, then η ± (γ(s 0 )) = ± r 0 (γ(s 0 )) are the two different roots of η 2 = r 0 at this point. Notice that the ray starting with direction η -will leave Ω, while the ray with direction η + will enter the interior of Ω. This motivates the following definition of broken bicharacteristic:

Definition 2.1 ([8]

). A broken bicharacteristic arc of p is a map:

s ∈ I \ B → γ(s) ∈ T * Ω \ {0},
where I is an interval on R and B is a discrete subset, such that

• If J is an interval contained in I \B, then s ∈ J → γ(s) is a bicharacteristic of p over Ω. • If s ∈ B,
then the limits γ(s + ) and γ(s -) exist and belongs to T *

x Ω \ {0} for some x ∈ ∂Ω, and the projections in T *

x ∂Ω \ {0} are the same hyperbolic point.

When a ray γ(s) reaches a point ρ 0 ∈ G, there are several situations. If ρ 0 ∈ G 2,+ , then the ray passes transversally over ρ 0 and enters T * Ω immediately. If ρ 0 ∈ G 2,- or ρ 0 ∈ G k for some k ≥ 3, then we can continue it inside T * ∂Ω as long as it can not leave the boundary along the trajectory of the Hamiltonian flow of H -r0 . We now give the precise definition.

Definition 2.2 ([8]

). A generalized bicharacteristic ray of p is a map:

s ∈ I \ B → γ(s) ∈ (T * Ω \ T * ∂Ω) ∪ G
where I is an interval on R and B is a discrete set of I such that p • γ = 0 and the following:

(1) γ(s) is differentiable and

dγ ds = H p (γ(s)) if γ(s) ∈ T * Ω \ T * ∂Ω or γ(s) ∈ G 2,+ . (2) Every s ∈ B is isolated, γ(s) ∈ T * Ω \ T * ∂Ω if s = t and |s -t| is small
enough, the limits γ(s ± ) exist and are different points in the same fibre of Moreover, there is also the continuous dependence with the initial data, namely the map (s, ρ) → (y(s, ρ), η 2 (s, ρ), x (s, ρ), ξ (s, ρ)) is continuous. We denote the flow map by γ(s, ρ).

T * ∂Ω. (3) γ(s) is differentiable and dγ ds = H G p (γ(s)) if γ(s) ∈ G \ G
Remark 2.4. Under the map j : T * Ω → b T * Ω, one could regard γ(s) as a continuous flow on the compressed cotangent bundle b T * Ω, and it is called the Melrose-Sjöstrand flow. We will also call each trajectory generalized bicharacteristic or simply ray in the sequel.

From the classical result of Melrose-Sjöstrand, a generalized bicharacteristic that does not meet G ∞ is uniquely defined (see Corollary 24.3.10 in [START_REF] Hörmander | The analysis of linear partial differential operators[END_REF]). Then, having G ∞ = ∅, meaning G j = ∅ for some j ≥ 2, implies the uniqueness of all generalized bicharacteristics and thus the existence of the Melrose-Sjöstrand flow. We refer to Example 24.3.11 in [START_REF] Hörmander | The analysis of linear partial differential operators[END_REF] where nonuniqueness occurs precisely in the case of a point in G ∞ . 2.3. definition of defect measure. We follow closely as in [START_REF] Burq | Semi-classical estimates for the resolvent in nontrapping geometries[END_REF] and one can find in [START_REF] Gérard | Ergodic properties of eigenfunctions for the dirichlet problem[END_REF] for a little different but comprehensive introduction.

We denote by S m the usual symbol class. Define the partial symbol class S m ∂ and the class of boundary h-pseudo-differential operators A m h as follows

S m ∂ := {a(y, x , ξ ) : sup α,β,y∈[0, 0] |∂ α x ∂ β ξ a(y, x , ξ )| ≤ C m,α,β (1 + |ξ |) m-β }. A m h =: Op comp h (S m ) + Op h (S m ξ ) := A m h,i + A m h,,∂ . Consider functions of the form a = a i + a ∂ with a i ∈ C ∞ c (Ω × R d
) which can be viewed as a symbol in S 0 , and

a ∂ ∈ C ∞ c (Y + × R d-1
) can be viewed as a symbol in S 0 ξ . We quantize a as follows: Take

ϕ i ∈ C ∞ c (Ω), equal to 1 near the x-projection of supp(a i ) and ϕ ∂ ∈ C ∞ c (R d ), equal to 1 near the x-projection of supp(a ∂ ). Define Op ϕi,ϕ ∂ h (a)f (y, x ) = 1 (2πh) d R 2d e i(x-z)ξ h a i (x, ξ)ϕ i (z)f (z)dzdξ + 1 (2πh) d-1 R 2(d-1) e i(x -z )ξ h a ∂ (y, x , ξ )ϕ ∂ (y, z )f (y, z )dz dξ .
According to the symbolic calculus, the operator Op ϕi,ϕ ∂ h (a) does not depend on the choice of functions ϕ i , ϕ ∂ , modulo operators of norms O L 2 loc →L 2 comp (h ∞ ), and we will use the notion Op h (a) in the sequel. Notice that the acting of tangential operator Op h (a ∂ ) can be viewed as pseudo-differential operator on the manifold ∂Ω, parametrized by the parameter y ∈ [0, 0 ). The bounded family of operators A m h,∂ is defined uniquely up to a family of operators with norms uniformly dominated by Ch, as h → 0. Moreover, for any family (A h ), such that

A h -Op h (a ∂ ) L 2 →L 2 = O(h),
the principal symbol σ(A) is determined uniquely as a function on T * ∂Ω, smoothly depending on y, i.e.

σ(A) ∈ C ∞ ([0, 0 ) × T * ∂Ω).
When we deal with vector-valued functions, we could require the symbol a to be matrix-valued. Now for any sequence of vector-valued function w k , uniformly bounded in L 2 (Ω), there exists a subsequence (still use w k for simplicity), and a nonnegative definite Hermitian matrix-valued Radon measure µ i on T * Ω such that lim k→0

(Op h k (a i )w k |w k ) L 2 = µ i , a i := T * Ω tr (a i dµ i ).
For a proof, see for example [START_REF] Burq | Semi-classical estimates for the resolvent in nontrapping geometries[END_REF] or the textbook [START_REF] Zworski | Semiclassical analysis[END_REF], and the micro-local version was appeared in [START_REF] Gérard | Microlocal defect measures[END_REF].

From now on the symbols and operators will be scalar-valued unless otherwise specified. Suppose u k be a sequence of solutions to

-h 2 k ∆u k -u k + h k ∇q k = f k , (u k , f k ) ∈ (H 2 (Ω) ∩ V ) × H, h k divu k = 0, in Ω (2.3)
under the assumptions below:

u k L 2 (Ω) = O(1), f k ∈ H and f k L 2 (Ω) = o(h k ), h∇q k L 2 (Ω) = O(1), Ω q k dx = 0. (2.4)
Let µ i be the interior measure associated with (u k ) k . Then the following result shows that µ i is supported on Car(P ).

Proposition 2.5. Let a i ∈ C ∞ c (Ω × R d ) be equal to 0 near Car(P ), then we have lim k→∞ (Op h k (a i )u k |u k ) L 2 = 0.
Proof. Note that the symbol b(x, ξ) = ai(x,ξ) |ξ| 2 -1 ∈ S 0 is well-defined from the assumption on a i . From symbolic calculus, we have

Op h k (a i ) = B h k (-h 2 k ∆ -1) + O L 2 →L 2 (h k ). Therefore (B h k (-h 2 k ∆ -1)u k |u k ) L 2 = (B h k f k |u k ) L 2 -(B h k h k ∇q k |u k ) L 2 = o(1) + ([h k ∇, B h k ]q k |u k ) L 2 -(h k ∇B h k q k |u k ) L 2 = o(1), as k → ∞,
where in the last line we have used the symbolic calculus, integration by part, and Lemma 3.3. Now we denote by Z = j(Car(P )). Proposition 2.5 indicates that the interior defect measure µ i is supported on Z. To define the defect measure near the boundary, we have to check that if

a ∂ ∈ C ∞ c (U × R d-1
) vanishing near Z (i.e. a ∂ is supported in the elliptic region for all y small) then

lim k→∞ (Op h k (a ∂ )u k |u k ) L 2 = 0.
Indeed, this can be ensured by the analysis of the boundary value problem in the elliptic region, which will be given later. Now for any family of operators A h ∈ A 0 h , let a = σ(A h ) be the principal symbol of A h and we define κ(a) ∈ C 0 (Z) via κ(a)(ρ) := a(j -1 (ρ)). Note that Z is a locally compact metric space and the set

{κ(a) : a = σ(A h ), A h ∈ A 0 h }
is a locally dense subset of C 0 (Z). We then have the following proposition, which guarantees the existence of a Radon measure on Z:

Proposition 2.6. There exists a subsequence of u k , h k and a nonnegative definite Hermitian matrix-valued Radon measure µ on Z, such that

lim k→∞ (A h k u k |u k ) L 2 = µ, κ(a) , a = σ(A h ), ∀A h ∈ A 0 h .
The proof of this result can be found in [START_REF] Burq | Semi-classical estimates for the resolvent in nontrapping geometries[END_REF], see also [START_REF] Burq | Mesures de défaut de compacité, application au systeme de lamé[END_REF] and [START_REF] Gérard | Microlocal defect measures[END_REF] for its micro-local counterpart. Notice that if we write a = a i + a ∂ , then

(A k u k |u k ) → T * Ω tr (a i (ρ)dµ i (ρ)) + Z tr (a ∂ (ρ)dµ(ρ)).
The following result shows that information about frequencies higher than the scale h -1 k is not lost, and the measure µ contains the relevant information of the sequence (u k ).

Proposition 2.7. The sequence of solution (u k ) is h k -oscillating in the following sense:

lim R→∞ lim sup k→∞ |ξ|≥Rh -1 k | ψu k (ξ)| 2 dξ = 0, ∀ψ ∈ C ∞ c (Ω), lim R→∞ lim sup k→∞ 0 0 dy |ξ |≥Rh -1 k | ψu k (y, ξ )| 2 dξ = 0, ∀ψ ∈ C ∞ c (Ω),
where in the second formula, the Fourier transform is only taken for the x direction. Consequently, the total mass

u k 2 L 2 (Ω) converges to µ i , T * Ω + µ, 1 Z .
The proof will be given in appendix.

3.

A priori information about the system 3.1. Information about the trace. We consider the semi-classical Stokes system

-h 2 ∆u -u + h∇q = f, (u, f ) ∈ (H 2 (Ω) ∩ V ) × H h divu = 0, in Ω (3.1) Assume that u L 2 (Ω) = O(1), f L 2 (Ω) = o(h).
Taking inner product with u and doing integration by part, we have h∇u L 2 (Ω) = O(1). Since q ∈ L 2 (Ω)/R, we may assume that Ω qdx = 0. From the regularity theory of the steady Stokes system, (see [START_REF] Temam | Navier-Stokes equations: theory and numerical analysis[END_REF], page 33) and Poincaré's inequality, we have

h 2 ∇ 2 u L 2 (Ω) = O(1), q L 2 (Ω) = O(h -1 ), h∇q L 2 (Ω) = O(1).
The following is a direct consequence of trace theorem for q 0 = q| ∂Ω .

Lemma 3.1. q 0 H 1/2 (∂Ω) = O(h -1 ).
We also have the hidden regularity property for the normal derivative.

Lemma 3.2. h∂ ν u| ∂Ω = (h∂ ν u , 0) and h∂ ν u| ∂Ω L 2 (∂Ω) = O(1).

The proof of this lemma will be given in appendix. We will recover some information for low frequencies from the following lemma:

Lemma 3.3. Suppose u 0 in L 2 (Ω).
Then after extracting to subsequences, we have h∇q 0 weakly in L 2 (Ω) and hq → 0 strongly in H 1/2 (Ω).

Proof. We may assume that h∇q r weakly in L 2 (Ω). Then the Rellich theorem implies that hq → P strongly in L 2 (Ω), and thus ∇P = r with the property Ω P = 0. Moreover ∆P = 0 in Ω, in the distributional sense. Since the sequence (h 2 ∇ 2 u) is bounded in L 2 , then up to a subsequence, h 2 ∇ 2 u W weakly in L 2 . From the Rellich theorem, the sequence (h 2 u) converges strongly in L 2 and the strong limit must be 0, due to the fact that u 0, weakly in L 2 . Thus W = 0 and this implies that ∇P = 0. Finally, we must have P = 0 since it has zero mean value. The last assertion follows from the Rellich theorem.

3.2.

Semi-classical parametrix of the pressure term. In system (3.1), the family of pressures q satisfy the boundary value problem of the Laplace equation -h 2 ∆q = 0, in Ω, q| ∂Ω = q 0 with unknown boundary data q 0 . We denote by PI(q 0 ) the Poisson integral of the corresponding harmonic function with trace q 0 . Let N be the Dirichlet-Neumann operator satisfying

N q 0 = ∂ ν PI(q 0 )| ∂Ω .
Next we study the behaviour of the sequence of pressures q in the regime of frequency scale h -1 . We always fix the notation

λ(y, x , ξ ) = |ξ | g -1 (y,x ) ∼ |ξ |.
Let Y = (-0 , 0 ) y × X x and Y + = [0, 0 ) y × X x . We first have the L 2 bound of q, micro-locally away from ξ = 0. Lemma 3.4. Let (f h ) 0<h<1 be a h-dependent family of distributions such that

f h L 2 (R n ) = O(h -N ) for some N ∈ N. Assume that for any χ ∈ C ∞ c (R 2n ), vanishing near ξ = 0, we have χ(x, hD x )f h H 1 2 (R n ) = O(h -1 ). Then χ(x, hD x )f h L 2 (R n ) = O(h -1 2 ). Proof. Assume that {|ξ| ≤ 2δ 0 } ∩ supp(χ) = ∅. Take Φ ∈ C ∞ c (R n ) such that Φ(ξ) = 1, |ξ| ≤ δ 0 , Φ(ξ) = 0, |ξ| > 2δ 0 .
We write

χ(x, hD x )f = Φ(hD x )χ(x, hD x )f + (1 -Φ(hD x ))χ(x, hD x )f.
From the support property we have Φ(hD

x )χ(x, hD x )f = O H ∞ (h ∞ ). Thus (1 - Φ(hD x ))χ(x, hD x )f = O H 1 2 (h -1 ). Let b(ξ) = |ξ| 1/2 (1 -Φ(ξ)), and we have b(hD x )χ(x, hD x )f = O L 2 (h -1 2
). Since b(ξ) = 0 on supp(χ), we have

χ(x, hD x )f L 2 (R n ) ≤ C b(hD x )χ(x, hD x )f L 2 (R n ) +Ch χ(x, hD x )f L 2 (R n ) ≤ Ch -1 2 . Lemma 3.5. Given δ 0 > 0 and ϕ, ϕ ∈ C ∞ c (Y + ). For any χ δ0 ∈ C ∞ c (Y + × R d-1 ) such that χ δ0 vanishes if λ(y, x , ξ ) ≤ 2δ 0 , we have ϕOp h (χ δ0 )(ϕq) L 2 (R d + ) + h 1/2 ( ϕOp h (χ δ0 )(ϕq)) | y=0 L 2 (R d-1 ) ≤ C δ0,ϕ, ϕ . Proof. Write D j = 1 i ∂ ∂x j , we have hD j (ϕq) L 2 (R d + ) = O(1). Since ξ j |ξ | 2 χ δ0 (y, x , ξ ) ∈ S 0 ∂ , then for χ j = ξ j |ξ | 2 χ δ0 , we have ϕχ δ0 (y, x , hD x )(ϕq) = d-1 j=1 ϕχ j (y, x , hD x )hD j (ϕq) + O L 2 (R d + ) (1) = O L 2 (R d + ) (1),
where the implicit bound in big O depends on δ 0 , ϕ, ϕ. For the boundary term, we observe that ϕOp h (χ δ0 )(ϕq

)| y=0 = O H 1/2 (R d-1 ) (h -1 ) from trace theorem. Thus from Lemma 3.4, ϕOp h (χ δ0 )(ϕq)| y=0 = O L 2 (R d-1 ) (h -1/2 ).
We express semi-classical Laplace operator h 2 ∆ g in the geodesic coordinates of tubular neighborhood Y by

P 0 = h 2 ∂ 2 y + g ij ∂ i ∂ j + hM j (y, x )h∂ j + hH(y, x
)h∂ y where ∂ j = ∂ x j . We make the ansatz

q(y, x ) := 1 (2πh) d-1
a(y, h, x , ξ )e ix ξ h θ(ξ )dξ , then we calculate

P 0 ( q)(y, x , ξ ) = 1 (2πh) d-1 h 2 ∂ 2 y a + g jk (h 2 ∂ j ∂ k a -g jk ξ j ξ k a) e ix ξ h θ(ξ )dξ + 1 (2πh) d-1 ihg jk ξ k ∂ j a e ix ξ h θ(ξ )dξ + 1 (2πh) d-1
(h 2 M j ∂ j a + ihM j ξ j a) + h 2 H∂ y a e ix ξ h θ(ξ )dξ .

We next look for the formal semi-classical expansion a(y, h, x , ξ ) j≥0 h j a j (y, h, x , ξ ) with a j ∈ S -j ∂ and h k ∂ k y a j ∈ S -j+k

∂

. We obtain

P 0 q 1 (2πh) d-1 ((h 2 ∂ 2 y a 0 -g ij ξ i ξ j a 0 ) +h(ig jk ξ k ∂ j a 0 + iM j ξ j a 0 + h 2 H∂ y a 0 ) +h(h 2 ∂ 2 y a 1 -g jk ξ j ξ k a 1 ) +h 2 (g jk ∂ j ∂ k a 0 + M j ∂ j a 0 ) +h 2 (ig jk ξ k ∂ j a 1 + iM j ξ j a 1 + h 2 H∂ y a 1 ) +h 2 (h 2 ∂ 2 y a 2 -g jk ξ j ξ k a 2 ) + • ••)e ix ξ h θ(ξ )dξ . Pick ϕ 1,0 = ϕ 1 | ∂Ω , ϕ 1 ∈ C ∞ c (Y ).
For q 0 = ϕ 1,0 q 0 , we put

θ(ξ ) = F h ( q 0 (ξ )) = (2πh) -(d-1) R d-1
q 0 (x )e -ix ξ /h dx , a 0 (0, •) ≡ 1, a j (0, •) ≡ 0, ∀j ≥ 1, and we define the functions a j inductively as follows: firstly we define a 0 a 0 (y, x , ξ ) = e -yλ(y,x ,ξ ) h , λ(y, x , ξ ) =: g ij ξ i ξ j ∼ |ξ |, and the quantity

(h 2 ∂ 2 y -λ 2 )a 0 = h h 2 λ 2 y 2 λ 2 h 2 (∂ y λ) 2 + 2yλ h ∂ y λ -2∂ y λ e -yλ h
can be viewed as of order h. Next we set a j , j ≥ 1 implicitly by solving a sequence of linear ODEs:

h 2 ∂ 2 y a 1 -λ 2 a 1 = -h -1 (h 2 ∂ 2 y -λ 2 )a 0 -(ig jk ξ k ∂ j a 0 + iM j ξ j a 0 + h 2 H∂ y a 0 ). h 2 ∂ 2 y a n -g ij ξ i ξ j a n = -(g ij ∂ i ∂ j a n-2 + M j ∂ j a n-2 ) -(ig jk ξ k ∂ j a n-1 + iM j ξ j a n-1 + h 2 H∂ y a n-1 ), n ≥ 2.
Unfortunately, the functions a j constructed above are not symbols, since they have singularities when ξ = 0. This indicates that we can only obtain information of q(h) from such parametrix away from ξ = 0. We modify the construction above by setting

A 0 (y, x , ξ ) = e -yλ h ψ δ0 (λ)ϕ 2 (y, x ), (y, x , ξ ) ∈ R d + × R d-1 , with ψ δ0 = ψ(δ -1 0 •), ψ ∈ C ∞ (R +
) satisfying ψ(s) ≡ 1 when s ≥ 1 and ψ(s) = 0 when 0 < s ≤ 1 2 . We next modify other A j in the same manner. Indeed, the ODEs which define A j are linear ODEs in y variable. Thus for j ≥ 1, A j (y, x ξ ) ≡ 0 when λ(y, x , ξ ) ≤ δ0 2 . We define the particular class of symbols in S j ∂ . Definition 3.6.

E j ∂ := a ∈ S j ∂ : |(h∂ y ) l ∂ α x ,ξ a(y, x , ξ )| ≤ C l,α e - C l,α y h .
Lemma 3.7. The symbols constructed above can be chosen to satisfy A j ∈ E -j ∂ for all j ∈ N.

The proof will be given in appendix. In summary, we have A j≥0 h j A j , and a tangential symbol B δ0 (y, x , ξ ) compactly supported in λ(y, x , ξ ) ≤ δ0 2 , such that ϕP 0 A(y, x , hD x )(ϕ 1,0 q 0 ) =ϕB δ0 (y, x , hD x )(ϕ 1,0 q

0 ) + O H ∞ (h ∞ ), ϕ 0 A(0, x , hD x )(ϕ 1,0 q 0 ) =ϕ 0 Op h (ψ δ (λ))(ϕ 1,0 q 0 ) + O H ∞ (h ∞ ).
The following proposition states that the parametrix constructed above is an approximation of the pressure q in the relevant semi-classical scale. Proposition 3.8. There exists A ∈ S 0 ∂ with principal symbol

A 0 (y, x , ξ ) = exp - yλ(y, x , ξ ) h ψ δ0 (λ(y, x , ξ ))ϕ 1 (y, x ), which satisfies asymptotic expansion A ∼ j≥0 h j A j , A j ∈ E -j ∂ . Moreover, for any ϕ, ϕ 1 ∈ C ∞ c (Y + ), ϕ 1 | supp(ϕ) ≡ 1, we have ϕOp h (χ δ0 A j )(ϕ 1,0 q 0 ) = O L 2 (R d + ) (1
) for all j, and

ϕOp h (χ δ0 )(ϕ 1 q) = ϕOp h (χ δ0 A)(ϕ 1,0 q 0 ) + O L 2 (R d + ) (h 3/4 ), ϕOp h (χ δ0 )h∂ y (ϕ 1 q) = ϕOp h (χ δ0 λA)(ϕ 1,0 q 0 ) + O L 2 (R d + ) (h 3/4 ), ϕOp h (χ δ0 )h∂ y (ϕ 1 q) = ϕOp h (χ δ0 λA)(ϕ 1,0 q 0 ) + O H 2 3 (R d + ) (h 1/4 ),
where

ϕ 0 = ϕ| ∂Ω , ϕ 1,0 = ϕ 1 | ∂Ω , χ δ0,0 = χ δ0 | y=0 .
We postpone the proof of this proposition in appendix. A direct consequence is that the singularities of the family of pressures (q h ) must concentrate in a very thin strip near the boundary.

Lemma 3.9. With the same χ δ0 ∈ C ∞ c (Y + × R d-1 ) and ϕ 1 , ϕ ∈ C ∞ c (Y + ), for any 0 < y 0 < 0 , we have 0 y0 ϕOp h (χ δ0 )(ϕ 1 q) 2 L 2 (R d-1 ) dy ≤ C δ0 (e -cy 0 h + h),
where the constant C δ0 only depends on δ 0 and is independent of y 0 and h.

Proof. The second term appearing on the right hand side comes from all the possible remainder terms. It suffices to estimate the term

0 y0 ϕOp h (χ δ0 A 0 )(ϕ 1 q 0 ) 2 L 2 (R d-1 ) dy. Since ϕ 1,0 q 0 = O L 2 (R d-1 ) (h -1/2
), micro-locally, we have for each fixed y > 0 that

ϕOp h (χ δ0 A 0 )(ϕ 1,0 q 0 ) L 2 (R d-1 ) ≤Ch -1/2 |β|≤Cd h |β| 2 sup y>0,(x ,ξ ) |∂ β x ,ξ (χ δ0 A 0 )| + O(h ∞ ) ≤Ch -1/2 e -cy h 1 + 1≤m,n≤Cd h m/2 y h n + O(h ∞ ).
Squaring and Integrating the right hand side in y variable yields the desired conclusion.

Main Steps of the Proof

The proof of Theorem 1.2 can be divided into several steps according to different geometric situations. We want to show that for any given point

ρ 0 ∈ b T * Ω, if ρ 0 / ∈ supp µ, then γ(s, ρ 0 ) / ∈ supp(µ) for any s > 0. The first step is to show that if ρ 0 ∈ T * Ω, ρ 0 / ∈ supp(µ), then γ(s, ρ 0 ) / ∈ supp(µ) for all s > 0 provided that π(γ(•, ρ 0 )| [0,s] ) ∩ ∂Ω = ∅.
This can be summarized by the following proposition, in which we have stronger conclusion that the measure is also invariant under the flow.

Proposition 4.1. For any real-valued scalar function a ∈ C ∞ c (Ω × R d ) vanishing near ξ = 0, we have d ds µ, a • γ(s, •) = 0.
Proof. Let A = Op h (a) and P = -h 2 ∆ -1. Applying the equation and Lemma 3.3, we have i h

([P, A]u|u) Ω = i h (Au|P u) Ω - i h (AP u|u) Ω = i h (Au|f -h∇q) Ω - i h (A(f -h∇q)|u) Ω = - i h (Au|h∇q) Ω + i h (Ah∇q|u) Ω + o(1) = - i h ([A, hdiv ]u|q) Ω + i h ([A, h∇]q|u) Ω + o(1) = i(Op h (∇a) • u|q) Ω -i(Op h (∇a)q|u) Ω + o(1) = i(u|Op h (∇a)q) Ω -i(Op h (∇a)q|u) Ω + o(1). (4.1)
where we have used integration by part freely, thanks to the fact that A has compact support in x ∈ Ω. Now we claim that for any χ = a or a, vanishing near ξ = 0, we have (u|Op h (∇χ)q) Ω = o(1).

Indeed, q = O L 2 (Ω) (1) micro-locally away from ξ = 0 since h∇q = O L 2 (1). More- over, h 2 ∆(Op h (∇χ)q) = O L 2 (Ω) (h)
and this implies that Op h (∇χ)q = o L 2 (1) since the symbol of h 2 ∆Op h (∇χ) vanishes near ξ = 0 as well as x near the boundary. In view of the definition of µ, this completes the proof.

For the second step, we need prove that if

ρ 0 ∈ E, then µ = 0 in a neighborhood of ρ 0 . Proposition 4.2. µ1 E = 0. Furthermore, for ν, the semi-classical defect measure of the sequence (h k ∂ ν u k | ∂Ω , h k ), we have ν1 E = 0.
The third step consists of proving that after reflection near a hyperbolic point, the measure µ is still zero. Proposition 4.3. Suppose that ρ 0 / ∈ supp(µ) and there exists s 0 > 0 such that γ(s 0 , ρ 0 ) ∈ H and π(γ(s, ρ 0 )) ∈ Ω for all 0 ≤ s < s 0 . Then there exists δ > 0 such that

π(γ(•, ρ 0 )| [s0,s0+δ] ) ∩ supp(µ) = ∅.
Next step is to prove the propagation near a diffractive point.

Proposition 4.4. Suppose that ρ 0 / ∈ supp(µ) and there exists s 0 > 0 such that γ(s 0 , ρ 0 ) ∈ G 2,+ and π(γ(s, ρ 0 )) ∈ Ω for all 0 ≤ s < s 0 . Then γ(s 0 , ρ 0 ) / ∈ supp(µ).

To deal with higher order contact, we will use induction. First let us introduce Definition 4.5 (k-propagation property). For k ≥ 2, we say that k-propagation property holds, if along generalized ray γ(s, ρ 0 ), the following statement is true:

For some σ 0 > 0, if γ(•, ρ 0 )| [0,σ0) ∩ supp(µ) = ∅ (or γ(•, ρ 0 )| (-σ0,0] ∩ supp(µ) = ∅) and γ(σ 0 , ρ 0 ) ∈ 2≤j≤k G j (or γ(-σ 0 , ρ 0 ) ∈ 2≤j≤k G j ), then γ(σ 0 , ρ 0 ) / ∈ supp(µ) (or γ(-σ 0 , ρ 0 ) / ∈ supp(µ) ).
The last step for the proof of Theorem 1.2 can be reduced to Proposition 4.6. k-propagation property holds for all k ≥ 2.

Near E

This section is devoted to the proof of Proposition 4.2. We set Q(y, x , ξ

) := √ λ 2 -1 and define Q h = ϕOp h (Qχ 0 )ϕ 1 with χ 0 ∈ C ∞ c (R d-1 ξ
) with support near E in which 1 + δ < λ < C. With a bit abuse of notation, we refer q 0 , q to be ϕOp h (χ 0 )ϕ 1 q 0 , ϕOp h (χ 0 )ϕ 1 q and u to be ϕOp h (χ 0 )ϕ 1 u. In this manner, we can combine the parametrix in last section to write the system (1.2) as

(-h 2 ∂ 2 y + Q 2 h )u ,j + g jk h∂ x k (Op h (A 0 )q 0 ) = R ,j = O L 2 (R d + ) (h), (-h 2 ∂ 2 y + Q 2 h )u ⊥ + h∂ y (Op h (A 0 )q 0 ) = R ⊥ = O L 2 (R d + ) (h).
(

Note that the symbol A 0 (y, x , ξ ) is defined in last section which equals to e

-yλ h s- ince λ > 1. Take ψ ∈ C ∞ (R + ), with ψ| [0, 0] ≡ 1, ψ [2 0,∞) ≡ 0. Denoting the extend- ed distributions of u by w = (w , w ⊥ ) = (u , u ⊥ )ψ(y)1 y≥0 , we have from standard elliptic parametrix construction (see Appendix A) that modulo O H ∞ (R d + ) (h ∞ ), w ,j = E(y, x , hD y , hD x )(-ψ(y)g jk h∂ x k (Op h (A 0 )q 0 ) + hv j ⊗ δ y=0 + ψ(y)R ,j ), w ⊥ = E(y, x , hD y , hD x )(-hψ(y)∂ y (Op h (A 0 )q 0 ) + ψ(y)R ⊥ ). (5.2) where v = h∂ y u | y=0 = O L 2 x
(1). Recall that the principal symbol of E is given by

E 0 := χ 0 (ξ )ϕ(y, x ) η 2 + λ(y, x , ξ ) 2 -1 ,
Now we need a lemma which deals with the trace of error terms:

Lemma 5.1. Assume that R = ϕOp h (χ 0 )ϕ 1 R+O H ∞ (h ∞ ), then if ψ(y)R L 2 (R d + ) = O(h), we have E(y, x , hD y , hD x )(ψ(y)R)| y=0 L 2 (R d-1 x ) = O(h 1/3 ).
Proof. From the parametrix construction above, we know that

|∂ α y,x ,η,ξ E(y, x ; η, ξ )| ≤ C α η 2 + Q(y, x , ξ ) 2 .
Therefore, the symbols ηE(y, x ; η, ξ ) and λ(y, x , ξ )E(y, x ; η, ξ ) are uniformly bounded in S 0 . Thus E(y, x ; hD y , hD

x )(ψR) = O L 2 (R d + ) (h) = O H 1 (R d + ) (1)
, and from interpolation, we have E(y, x ; hD y , hD

x )(ψR) = O H 2/3 (R d + ) (h 1/3
). The conclusion then follows from the trace theorem that

H s (R d + ) → H s-1/2 (R d-1 ) is bounded for s > 1/2.
Proof of Proposition 4.2. Denote by F h (q 0 ) = θ the semi-classical Fourier transform of q 0 , we calculate

E(y, x , hD y , hD x )(ψ(y)h∂ y Op h (A 0 )q 0 ) = 1 (2πh) d e i(y-z)η h dzdη e ix ξ h θ(ξ )ϕ(y, x )χ 0 (ξ ) η 2 + Q 2 (y, x , ξ ) ψ(z)h∂ z (e -zλ(z,x ,ξ ) h )dξ + R 1 = - h (2πh) d ξ θ(ξ )e i(yη+x ξ ) h B 1 (η, x , ξ )ϕ(y, x )χ 0 (ξ ) η 2 + Q 2 (y, x , ξ ) dηdξ - h 2 (2πh) d e ix ξ h θ(ξ )dξ e iyη h B 0 (η, x , ξ )ϕ(y, x )χ 0 (ξ ) η 2 + Q(y, x , ξ ) 2 dη + R 1 , (5.3
)

with reminder term R 1 = O L 2 (R d ) (h), where λ 0 = λ| y=0 , B 1 (η, x , ξ ) = ∞ 0 ψ(z)e - iη+λ(z,x ,ξ ) h z λ(z, x , ξ ) ξ 1 h dz, and
B 0 (η, x , ξ ) = ∞ 0 ψ(hz)z(∂ z λ)(hz, x , ξ )e -(iη+λ(hz,x ,ξ ))z dz.
We notice that

K 0 (y, x , ξ ) := e iyη h B 0 (η, x , ξ )ϕ(y, x )χ 0 (ξ ) η 2 + Q(y, x , ξ ) 2 dη
is a bounded symbol in S 0 ξ . Thus the second term on the right hand side of (5

.3) is equal to R 2 = O C 0 (Ry;L 2 (R d-1 x
)) (h) and we may concentrate on the first term. Write

B 1 (η, x , ξ ) = ∞ 0 ψ(hz)e -(iη+λ(hz,x ,ξ ))z λ(hz, x , ξ ) ξ dz.
Taylor expansion gives

e -λ(hz,x ,ξ )z λ(hz, x , ξ )ψ(hz) =e -λ0(x ,ξ )z λ 0 (x , ξ ) + 1 0 d dt e -λ(htz,x ,ξ )z λ(htz, x , ξ )ψ(htz) dt =e -λ0(x ,ξ )z λ 0 (x , ξ ) + h 1 0 P t (z, x , ξ )e -λ(htz,x ,ξ )z dt with P t (z, x , ξ ) = -z 2 (λ∂ y λ)(htz, x , ξ )ψ(htz)+z(∂ y λ)(htz, x , ξ )ψ(htz)+zλ(htz, x , ξ )ψ (htz). Thus we have B 1 (η, x , ξ ) = λ 0 (x , ξ ) (iη + λ 0 (x , ξ )) ξ + h B 1 (η, x , ξ ),
where

B 1 (η, x , ξ ) = 1 0 ∞ 0 e -(iη+λ(htz,x ,ξ ))z 1 ξ P t (z, x , ξ )dzdt. Note that near a point in E, |∂ α x ∂ β ξ P t (z, x , ξ )| ≤ C αβ z 2 , independent of t, h, hence the symbol K 1 (y, x , ξ ) = e iyη h ϕ(y, x )χ 0 (ξ ) η 2 + Q(y, x , ξ ) 2 B 1 (η, x , ξ )dη ∈ S 0 ξ .
Therefore, the symbol in the principal term of E(y, x , hD y , hD x )(ψ(y)h∂ y Op h (A 0 )q 0 ) equals to

K 1 (y, x , ξ ) =λ 0 (x , ξ ) e iyη h ϕ(y, x )χ 0 (ξ ) (η 2 + Q(y, x , ξ ) 2 )(iη + λ 0 (x , ξ )) dη + h K 1 (y, x , ξ ) =2πλ 0 ϕ(y, x )χ 0 (ξ ) e -yQ h 2(λ 0 -Q)Q - e -yλ 0 h λ 2 0 -Q 2 + h K 1 (y, x , ξ ) =2πλ 0 ϕ(y, x )χ 0 (ξ ) e -yQ h -e -yλ 0 h 2(λ 0 -Q)Q + e -yλ 0 h 2Q(λ 0 + Q) + h K 1 (y, x , ξ ).
Note that

E 1 (y, x , ξ ) = 2πλ 0 ϕ(y, x )χ 0 (ξ ) e -yQ h -e -yλ 0 h 2(λ 0 -Q)Q + e -yλ 0 h 2Q(λ 0 + Q) > 0 near E, we have E(y, x , hD y , hD x )(ψ(y)h∂ y Op h (A 0 )q 0 ) = E 1 (y, x , hD x )q 0 + R 1 + R 2 , with R 2 = O C(Ry;L 2 (R d-1 x )) (h). We claim that R 1 = O H 1 (R d ) (1). Once it is justified, by interpolation, we have R 1 H 2/3 (R d ) = O(h 1/3
). To verify the claim, we note that the symbol of the reminder term R 1 is of the form hS -1 (in both η and ξ variables), hence the symbolic calculus yields

∂ y R 1 = O L 2 (R d ) (1), and ∂ x R 1 = O L 2 (R d ) (1).
We next calculate the parallel component

1 (2πh) d ψ(z)e i(y-z)η h dzdη e ix ξ h - zλ(z,x ,ξ ) h θ(ξ )g jk (z, x )ξ k ϕ(y, x )χ 0 (ξ ) η 2 + Q(y, x , ξ ) 2 dξ = 1 (2πh) d e i(x ξ +yη) h ξ k θ(ξ )ϕ(y, x )χ 0 (ξ ) η 2 + Q(y, x , ξ ) 2 dηdξ ∞ 0 ψ(z)e -iη+λ(z,x ,ξ ) h z g jk (z, x )dz = h (2πh) d e i(x ξ +yη) h ξ k θ(ξ )ϕ(y, x )χ 0 (ξ ) (η 2 + Q(y, x , ξ ) 2 ) B 2,jk (η, x , ξ )dηdξ =: E 2 (y, x , hD x )q 0 .
where

B 2,jk (η, x , ξ ) = ∞ 0 ψ(z)e -iη+λ(z,x ,ξ ) h z g jk (z, x ) 1 h dz. Define K 2,jk (y, x , ξ ) = e iyη h B 2,jk (η, x , ξ )ϕ(y, x )χ 0 (ξ ) η 2 + Q(y, x , ξ ) 2 dη,
and from similar argument we have

K 2,k (y, x , ξ ) = g jk (0, x ) e iyη h ϕ(y, x )χ 0 (ξ ) (η 2 + Q(y, x , ξ ) 2 )(iη + λ 0 (x , ξ )) + h K 2 (y, x , ξ )
and the principal symbol of E 2 (y, x , hD x ) is elliptic if λ 0 (ξ ) > 1 and y small enough. Finally,

E(y, x , hD y , hD x )(hv ⊗ δ y=0 ) = h (2πh) d F h (v)(ξ )e i(yη+x ξ ) h ϕ(y, x )χ 0 (ξ ) η 2 + Q(y, x , ξ ) 2 dξ dη + O L 2 (R d ) (h) = h (2πh) d F h (v)(ξ )e ix ξ h πe -yQ(y,x ,ξ ) h ϕ(y, x )χ 0 (ξ ) Q(y, x , ξ ) dξ + O L 2 (R d ) (h) = : E 3 (y, x , hD x )v + O L 2 (R d ) (h), (5.4)
and again, E 3 (y, x , hD x ) is elliptic near λ 0 (ξ ) > 1. Moreover, we deduce from the same argument as for R 1 that the reminder terms are indeed of

O H 2/3 (R d ) (h 1/3
). Now the boundary condition (w ⊥ , w )| y=0 = 0 and the trace theorem yield

E 1 (0, x , hD x )q 0 = O L 2 (R d-1 x ) (h 1/3 ), E 2 (0, x , hD x )q 0 + E 3 (0, x , hD x )v = O L 2 (R d-1 x ) (h 1/3
). Therefore, from the ellipticity of E 1 , E 2 , E 3 , the measure of pressure in the elliptic region vanishes, so does the measure of v, namely ν| E = 0. The proof of Proposition 4.2 is complete.

6. Near H We take ϕ 1 , ϕ ∈ C ∞ c (Y + ) such that ϕ 1 | supp(ϕ) ≡ 1. For any tangential symbol b ∈ C ∞ c (Y + × R d-1
), we define the pseudo-differential operator B h = ϕOp h (b)ϕ 1 , with compact support in Y + . We will change the notation of tangential variables (x , ξ ) to (x, ξ). We always work in local coordinate (y, x) and sometimes abuse the notation u = ϕ 1 u, q = ϕ 1 q as compactly supported functions in Y + . Note that q 0 , the trace of q is not bounded in L 2 in priori. Fortunately, it turns out that q 0 = O L 2 (1), micro-locally near a point in H.

6.1. L 2 bound of boundary datum. Take b(y, x, ξ), b 1 (y, x, ξ) ∈ C ∞ c ([0, 0 )×H), such that b 1 | [0, 0/2)×supp(b) ≡ 1. Let Q(y, x, ξ) = 1 -λ(y, x, ξ) 2 b 1 (y, x, ξ
). We will first factorize the operator (-h 2 ∆ -1) near a hyperbolic point. Lemma 6.1. For 0 ≤ y < 0 , we have

B h (-h 2 ∆ -1) = -(hD y -Q + h )(hD y -Q - h ) + R = -(hD y -Q - h )(hD y -Q + h ) + R , where R , R ∈ C ∞ ([0, 0 ], h ∞ Ψ -∞ (∂Ω)), and Q ± h , Q ± h have principal symbols ±Q(y, x, ξ). Proof.
The proof is quite standard, and we follow the construction in [START_REF] Burq | Mesures de défaut de compacité, application au systeme de lamé[END_REF] by translating word by word to the semi-classical setting. In local coordinate, we have

B h (-h 2 ∆ -1) = h 2 D 2 y + R(y, x, hD x ) + hM 1 (y, x )hD y + hM 0 (y, x)hD x with σ(R) = Q 2 . Set q + 1 = √ Q(y, x , ξ ), Q + 1 = Op h (q + 1 ) and Q - 1 = -Q + 1 -hM 1 . Direct calculation gives (hD y -Q + 1 )(hD y -Q - 1 ) = h 2 D 2 y -(Q + 1 ) 2 -hQ + 1 M 1 -(Q + 1 + Q - 1 )hD y - h i ∂ y (Q - 1 ) = h 2 D 2 y -(Q + 1 ) 2 + hM 1 hD y -h(Q + 1 M 1 -i∂ y (Q - 1 )). Thus B h (-h 2 ∆ -1) -(hD y -Q + 1 )(hD y -Q - 1 )
= hT 1 , with some operator T 1 , bounded in L 2 . Now for j ≥ 1, suppose that we have

B h (-h 2 ∆ -1) -(hD y -Q + j )(hD y -Q - j ) = h j T j , by setting Q ± j+1 := Q ± j + h j S ± j+1 with S + j+1 + S - j+1 = 0 and σ(S + j+1 ) = σ(T j ) 2σ(Q + j )
, we obtain that

B h (-h 2 ∆ -1) -(hD y -Q + j+1 )(hD y -Q - j+1 ) =h j T j + h j (S + j+1 Q - j + Q + j S - j+1 ) -h j (S + j+1 + S - j+1 )hD y - h j+1 i ∂ y (S - j+1 ) + h 2j S + j+1 S - j+1 =:h j+1 T j+1 ,
for some operator T j+1 bounded in L 2 . Thus the proof can be completed by induction.

Define w = ϕ 1 u -h∇(ϕ 1 q), w ± = B h (hD y -Q ± h )w and their boundary values

w ± 0 := w ± | y=0 . Note that ϕP h w = ϕf . Proposition 6.2. B h h∂ y w ⊥ L 2 (R d + ) = O(1), and consequently, w ± ⊥ L 2 (R d + ) = O(1).
Proof. From hdiv u = 0, we have ϕhdiv w = 0, hence

ϕ(h∂ y w ⊥ + hdiv w ) = O L 2 (R d + ) (h), where in local coordinates, div w = 1 det(g) d-1 j=1 ∂ xj ( det(g)w ,j ).
Therefore,

B h h∂ y w ⊥ L 2 (R d + ) ≤ O(h) + B h hdiv w L 2 (R d + ) = O(1).
Now we recall the following hyperbolic energy estimate.

Lemma 6.3. Suppose A h = Op h (a) is ellptic (with real-valued symbol a smoothly depending on t) of order 0 on a compact manifold M and w are solutions of the h-dependent equations

(hD t ± A h )w = g, (t, x) ∈ R × M.
Assume that for any compact time interval I and small h,

w L 2 (I×M ) ≤ C(I), g L 2 (I×M ) ≤ C(I)h,
then we have for all small h,

sup t∈I w(t) L 2 (M ) ≤ C(I ), ∀I ⊂ I compact.
Proof. By symmetry, it suffices to treat the case hD t -A h . Take χ(t) ∈ C ∞ c (I ), and we may assume that 0 ∈ I with χ(0) = 1. Multiplying by χ(t) to the equation, we have

(hD t -A h )(χw) = χg + [χ, hD t -A h ]w =: r = O L 2 (R×M ) (h).
We now calculate

h d dt (χw|χw)(t) L 2 (M ) = (ihD t χw|χw) L 2 (M ) + (χw|ihD t χw) L 2 (M ) = i(A h (χw) + r|χw) L 2 (M ) -i(χw|A h (χw) + r) L 2 (M ) = i((A h -A * h )χw|χw) L 2 (M ) + i(r|χw) L 2 (M ) -i(χw|r) L 2 (M )
Integrating the formula above from 0 to sup I , we finally have w(0

) 2 L 2 (M ) = O(1). Lemma 6.4. w ± 0 L 2 (R d-1 ) = O(1). Proof. From Proposition 6.2, we have (hD y -Q ∓ h )w ± ⊥ = O L 2 (R d + ) (h). Applying the previous lemma to w ± ⊥ , we have w ± 0,⊥ L 2 (R d-1 x ) = O(1)
. Combining the boundary condition, we have

B h (Q + h -Q - h )(h∂ y q)| y=0 = -B h (Q + h -Q - h )hN q 0 = w + 0,⊥ -w - 0,⊥ = O L 2 (R d-1 ) (1)
. Remark that in priori, N is a classical first order pseudo-differential operator, and we only have

B h hN q 0 L 2 (R d-1 ) ≤ B h H -1 →L 2 h N q 0 H -1 (R d-1 ) = O(h -1 ).
From the exact pricipal symbol of Q ± h , we have B h hN q 0 L 2 (R d-1 ) = O(1), and the constant in big O depends on the micro-local cut-off b(y, x , ξ ). As a consequence,

w ± 0,⊥ L 2 (R d-1 ) = O(1).
It remains to study w ± . Notice that their boundary values are [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF]. All terms are obviously bounded in L 2 (R d-1 ) except the trace of B h h∇ hD y q. To bound it, we use the support property of b and Proposition 3.8, hence B h h∇ hD

w ± 0, = B h (v -(hD y h∇ q)| y=0 ) -B h Q ± h h∇ q 0 , where v = (h∂ y u)| y=0 = O L 2 (R d-1 )
y q| y=0 = -B h h∇ hN q 0 = O L 2 (R d-1 x
) [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF]. Again by hyperbolic estimates, we have the following result:

Proposition 6.5. w ± L 2 (R d + ) = O(1). In particular, B h hD y w L 2 (R d + ) + B h hN q 0 L 2 (R d-1 x ) + B h h 2 ∆ 0 q 0 L 2 (R d-1 x ) = O(1)
, where ∆ 0 = ∆ ∂Ω is the Laplace-Beltrami operator on ∂Ω.

Proof. It remains to prove

B h h 2 ∆ 0 q 0 L 2 (R d-1 x ) = O(1). Indeed, B h h∂ y w ⊥ =B h h∂ y u ⊥ -h 2 B h ∂ 2 y q =h 2 B h ∂ y u ⊥ + h 2 B h 1 det(g) 1≤j,k≤d-1 ∂ j (g jk ∂ k q 0 ) + ∂ y det(g) det(g) ∂ y q . Thus B h h∂ y w ⊥ | y=0 = B h h 2 ∆ 0 q 0 + O L 2 (R d-1 ) (1), thanks to h∂ y u ⊥ = 0 and B h hN q 0 = O L 2 (R d-1 ) (h -1 ). From w ± 0,⊥ = B h hD y w ⊥ | y=0 + B h Q ± h hN q 0 = O L 2 (R d-1 ) (1), we deduce that B h hD y w ⊥ | y=0 = O L 2 (R d-1 ) (1)
, and these yield

B h h 2 ∆ 0 q 0 L 2 (R d-1 ) = O(1). Corollary 6.6. B h h∇q L 2 (R d + ) = o(1)
. Proof. We will go back to the global notation in this calculation. It suffices to show that [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF] since there are only change of bounded weight in the integral with respect to the measure det(g)dydx and dydx in local coordinate, and the former allows us to apply integration by part and the structure of the equation in a simple way. We calculate

B h h∇q = ϕOp h (b)ϕ 1 h∇q = o L 2 (Ω)
(B h h∇q|B h h∇q) L 2 (Ω) =([B h , h∇]q|B h h∇q) L 2 (Ω) + (h∇B h q|B h h∇q) L 2 (Ω) =o(1) -(A h B h q|hdivB h h∇q) L 2 (Ω) +(B h hq 0 |B h (h∂ ν q)| ∂Ω ) L 2 (∂Ω) =o(1) + (B h hq 0 |B h hN q 0 ) L 2 (∂Ω) ,
where we have used the fact that hq = o L 2 (Ω) (1) and ∆q = 0 in the calculation. Now from Lemma 3.3, we know that hq 0 weakly in H 1 (Ω) and hq 0 → 0 strongly in L 2 (∂Ω). The last term is o(1) since B h hN q 0 = O L 2 (∂Ω) (1). 6.2. propagation estimate. In this subsection, we prove Proposition 4.3. We factorize

-h 2 ∆ -1 as (hD y -Q ± h )(hD y -Q ∓ h ) + R ± near z 0 ∈ H and choose Q ± h with principal symbols ±Q(y, x, ξ) = √ 1 -λ 2 b 1 (y, x, ξ)
, as in the previous subsection. Take ψ ∈ C ∞ c ([0, 0 )) with ψ ≡ 1 in a neighborhood of y = 0. By an abuse of notation, we introduce The proof will be divided into several lemmas. First we calculate

w ± = B ± h (hD y -Q ∓ h )w, where B ± h have principal symbols ψ(y)b ± (y, x, ξ). Here, b ± are solutions of ∂b ± ∂y ∓ H Q(y,x,ξ) b ± = 0, b ± | y=0 = b 0 , (6.1
(hD y -Q ± h )w ± = [hD y -Q ± h , B ± h ](hD y -Q ∓ h )w + B ± h (hD y -Q ± h )(hD y -Q ∓ h )w, and
[hD y -Q ± h , B ± h ] = h i Op h (∂ y b ± ∓ H Q b ± )ψ(y) + h i ψ (y)B ± h + R .
The first operator vanishes thanks to the definition of b ± , and the remainder term

R = O L 2 (R d + ) (h 2 ). Therefore we have R (hD y -Q ∓ h )w L 2 (R d + ) = O(h 2
), and consequently

(hD y -Q ± h )w ± = h i ψ (y)w ± + g ± , with g ± = o L 2 (R d + ) (h
). Lemma 6.8. Let µ ± be the semi-classical defect measure of w ± and b is defined as above. Suppose that b ± µ ± 1 0<y≤ 0 = 0, then we must have b ± µ ± ≡ 0 and µ ± 0 = 0, where µ ± 0 is the defect measure of w ± 0 = w ± | y=0 .

Proof. For y 0 = 0 /2, we first claim that w ± (y 0 ) L 2 x = o(1). Indeed, from the assumption and compactness, the measure µ ± vanishes in a small neighborhood of semi-bicharacteristic curve γ ± . Thus

w ± L 2 ([y0, 0]×R d-1 ) = o(1)
, provided that we choose supp(b 0 ) small enough in the definition of w ± . Finally, repeating the argument in the proof of Lemma 6.3, we have

-h w ± (y 0 ) 2 L 2 (R d-1 ) = i 0 y0 ((Q ± h -(Q ± h ) * )χw ± |χw ± ) L 2 (R d-1 ) (y)dy + o(h).
The claim then follows.

Integrating the identity

h d dy (w ± |w ± ) L 2 (R d-1 ) = (i(Q ± h -(Q ± h ) * )w ± |w ± ) L 2 (R d-1 ) + 2h(ψ (y)w ± |w ± ) L 2 (R d-1 ) + 2Im(w ± |g ± ) L 2 (R d-1 )
from y = z < y 0 to y = y 0 , we have

w ± (z) 2 L 2 (R d-1 ) ≤ C y0 z w ± (y) 2 L 2 (R d-1 ) dy + o(1)
.

Using y0 0 w ± (y) 2 L 2 (R d-1 ) dy = o(1), we obtain that w ± 0 L 2 (R d-1 ) = o(1)
. This completes the proof of Lemma 6.8. Remark 6.9. Away from the boundary, the defect measure of u equals to the defect measure of w, and it propagates along the bicharacteristic curves γ ± . Since we can decompose w into w + and w -near a hyperbolic point, we call w + (w -)the incoming wave and the out-coming wave. Thus the above proposition asserts that if we have no singularity of w + (w -) along incoming wave(out-coming wave) near the boundary but strictly away from the boundary, then there is no singularity of the boundary data of incoming wave(out-coming wave).

Changing the role of y = y 0 and y = 0 in the proof of Lemma 6.8, we conclude that if µ ± 0 = 0, then b ± µ ± = 0. To finish the proof of Proposition 6.7, we need understand how the singularity transfers form boundary data of in-coming wave to the boundary data of out-coming wave. Lemma 6.10.

µ ± 0 = 0 implies that µ ∓ 0 1 ξ =0 = 0. Consequently, µ ∓ 1 ξ =0 = 0.
Proof. By symmetry, we only need to deduce µ - 0 1 ξ =0 = 0 from µ + 0 = 0. For δ > 0, we define

b 0,δ (x, ξ) = b 0 (x, ξ) 1 -ψ λ(0, x, ξ) δ , with some ψ ∈ C ∞ c (R), ψ| [-2,2] ≡ 1.
We define b ± δ (y, x, ξ) by solving ODE (6.1) with initial data b 0,δ . Let B ± δ,h be the associated semi-classical PdO of b ± δ . From compactness and continuous dependence of the initial data, we have that δ < λ(y, x, ξ) < c 0 < 1 on supp(b δ (y)) for 0 ≤ y ≤ 0 , since on

Y + × R d-1 , λ(y,x,ξ) |ξ| ∼ 1.
Note that the solutions of the transport equation (6.1) are given by

b ± (y, x, ξ) = b 0 • γ ± (y) -1 (x, ξ), b ± δ (y, x, ξ) = b 0,δ • γ ± (y) -1 (x, ξ), we have that b ± δ b ± is a smooth function with compact support in Y + × R d-1 . Denote by w ± δ = B ± δ,h (hD y -Q ∓ h
)w, and µ ± δ its semi-classical defect measure, we have

µ ± δ = µ ± b ± δ b ± 2 . In particular, µ ± δ,0 = µ ± 0 1 -ψ λ0 δ 2
and supp(µ ± δ ) ⊂ supp(µ ± ). On the boundary, B + δ,h and B - δ,h coincide and will be denoted by B 0 δ,h . Taking the trace of w ± δ , we have

w + δ,0, = -iB 0 δ,h v + ih 2 B 0 δ,h ∂ y (∇q) | y=0 + B 0 δ,h Q + h h∇ q 0 , w + δ,0,⊥ = iB 0 δ,h h 2 ∂ 2 y q| y=0 + B 0 δ,h Q + h h∂ y q| y=0 , where v = h∂ y u| y=0 = O L 2
x (1). Similarly, we have

w - δ,0, = -iB 0 δ,h v + ih 2 B 0 δ,h ∂ y (∇q) | y=0 + B 0 δ,h Q - h h∇ q 0 , w - δ,0,⊥ = iB 0 δ,h h 2 ∂ 2 y q| y=0 + B 0 δ,h Q - h h∂ y q| y=0 . Notice that σ(Q + h ) = -σ(Q - h ), we write α = -B 0 δ,h h 2 ∆ 0 q 0 , β = B 0 δ,h Q + h hN q 0 , hence w ± δ,0,⊥ = iα ∓ β + O L 2 x (h). From the assumption w + δ,0,⊥ L 2 = o(1), we have that iα -β 2 L 2 = o(1)
, and this implies that α 2 + β 2 -2Im(α|β) = o(1). We claim that Im(α|β) = o(1).

Indeed, from Proposition 6.5 and the ellipticity of the Dirichlet-Neumann operator N , we have that

q 0 = O L 2 (R d-1 x ) (1
), micro-locally away from ξ = 0. Now from the trace theorem and Proposition 3.8, we have

β = A δ,h q 0 + O L 2 (R d-1 x ) (h 1/3
) for some PdO with real-valued principal symbol a δ , compactly supported and vanishing when λ(y, x, ξ) ≤ δ/4. Similarly,

α = A δ,h q 0 + o L 2 (R d-1 x ) (1)
for some PdO with real-valued principal symbol a δ . Thus Im(α|β) L 2 = o(1), since all the principal symbols involved in the inner product are real-valued. Now from α L 2 = o(1), β L 2 = o(1), one deduce that the terms on the righthand side of w ± δ,0, involving pressure are also o L 2

x (1), and [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF]. Therefore µ - δ,0 = 0 and consequently µ - δ = 0 from Lemma 6.8. This implies that µ - 0 1 λ>δ = µ -1 λ>δ = 0. Since δ > 0 is arbitrary, we have that b -µ -1 ξ =0 = 0. Moreover, Corollary 6.6 implies that µ1 ξ =0 = 0. This completes the proof of Lemma 6.10. Now we finish the proof of Proposition 6.7 by showing the following lemma. Lemma 6.11. µ + = 0 implies that µ -= 0.

v = o L 2 x (1) follows since w - δ,0, = o L 2 (R d-1 )
Proof. We only need deal with ξ = 0. Take ψ to be a cut-off function which equals to 1 near the origin. Pick any > 0, we define the operator

B ,± h = Op h ( ψ(λ(y, x, ξ)/ ))B ± h . Applying divergence equation for w ± B ,± h hdiv w ± + B ,± h h∂ y w ± ⊥ = O L 2 (R d + ) (h), we have B ,± h h∂ y w ± ⊥ L 2 (R d + ) ≤ B ,± h hdiv w ± L 2 (R d + ) + R (h) with R (h) → 0,
as h → 0 for each fixed > 0. By estimating the operator norm from its symbol, we have

B ,± h h∂ y w ± ⊥ L 2 (R d + ) ≤ C + R (h), and lim sup h→0 + B ,± h h∂ y w ± ⊥ L 2 (R d + ) ≤ C . Using the equation hD y w ± ⊥ = Q ± h w ± ⊥ + O L 2 (R d + ) (h), we have lim sup h→0 + B ,± h Q ± h w ± ⊥ L 2 (R d + ) ≤ C .
Finally let → 0, we have µ ± ⊥ 1 ξ=0 = 0. Therefore µ - ⊥ = 0. As a consequence of the proof of Lemma 6.8, µ ± 0,⊥ 1 ξ=0 = 0. Now let µ α , µ β be the defect measures of α = -B 0 h h 2 ∆ 0 q 0 , β = B 0 h Q h hN q 0 , and let µ iα±β be the defect measure of iα ± β. Denote also by µ αβ the limit corresponding to the quadratic form (A h α|β). Similarly for µ βα . Note that µ αβ = µ βα . From

µ iα+β , 1 ξ=0 = µ α , 1 ξ=0 + µ β , 1 ξ=0 -2 Im µ αβ , 1 ξ=0 = 0 µ iα+β , 1 ξ=0 = µ α , 1 ξ=0 + µ β , 1 ξ=0 + 2 Im µ αβ , 1 ξ=0 = 0, we have that µ α 1 ξ=0 = µ β 1 ξ=0 = 0.
Next we consider parallel components. The key claim is that the measure corresponding to B 0 h Q ± h h∇ q 0 vanishes on the set {ξ = 0}. Indeed, from Lemma 3.3 and the trace theorem, hq 0 → 0 strongly in L 2 (∂Ω). From the ellipticity of N , there exists a classical pseudo-differential operator E of order -1 such that EN = I + R, where R is a classical smoothing operator. Our goal is to show that lim

→0 lim sup h→0 B 0 h B ,0 h Q ± h h∇ q 0 L 2 (R d-1 ) = 0.
From symbolic calculus and the strong convergence of hq 0 in L 2 (R d-1

x ), it suffices to prove lim

→0 lim sup h→0 h∇ B 0 h B ,0 h Q ± h q 0 L 2 (R d-1 ) = 0. (6.2)
We write

h∇ B 0 h B ,0 h Q ± h q 0 =h∇ B 0 h B ,0 h Q ± h EN q 0 -h∇ B 0 h B ,0 h Q ± h Rq 0 =∇ EB 0 h B ,0 h Q ± h hN q 0 + h∇ [B 0 h B ,0 h Q ± h , E]N q 0 -h∇ B 0 h B ,0 h Q ± h Rq 0 . (6.3)
Here we are taking the commutator between a semi-classical PdO and a classical PdO, hence the semi-classical symbolic calculus is not applicable. Yet, it is not difficult to check that for any

a ∈ C ∞ c (T * ∂Ω), E ∈ S -1 x,ξ , [a(x, hD x ), E(x, D x )] = hOp(S -1 ) + Op(S -2 ),
where the implicit constants only depend on the semi-norms of the symbols a(x, ξ) and E(x, ξ). Notice that h∇ B 0 h , B ,0 h , Q ± h are uniformly bounded operators in L 2

x with respect to h, thus

∇ B 0 h B ,0 h Q ± h R, ∇ B 0 h Op(S -2 )N , h∇ B 0 h Op(S -1
)N are uniformly bounded operators in L 2

x with respect to h. Thus from the strong convergence of hq 0 , the last two terms on the right hand side of (6.3) are killed when we let h → 0 first. Thus (6.2) follows from the vanishing of the measure of ±β = B 0 h Q ± h hN q 0 on the set {ξ = 0}. Combining the assumption that µ + 0, 1 ξ=0 = 0, we deduce that µ - 0, 1 ξ=0 = 0. The proof of Proposition 6.7 is now complete.

Near G 2,+

In this section, we follow the strategy of V. Ivrii (see [START_REF] Ivrii | Microlocal analysis and precise spectral asymptotics[END_REF] or [START_REF] Hörmander | The analysis of linear partial differential operators[END_REF]) to prove Proposition 4.4. Denote by G = det(g) and P h = h 2 ∆ H -1, we have Lemma 7.1. In local coordinate Y + , we have

P h = -h 2 g √ G ∂ y √ Gg -1 ∂ y + R h = h 2 D 2 y + Op h (r) + O L 2 →L 2 (h),
where R h is a matrix-valued second order differential operator in x with scalar principal symbol r(y, x, ξ) = 1 -λ(y, x, ξ) 2 , which is self-ajoint with respect to the

(•|•) L 2 (Y+) .
The proof will be given in the appendix.

To simplify the notations, in a fix local coordinate in Y + , we will identify u = ϕ 1 u, q = ϕ 1 q and all the operators B by ϕBϕ 1 .

Proposition 7.2. For any tangential operator B with scalar principal symbol b(y, x, ξ) vanishing near ξ = 0, we have

lim sup h→0 BhD y u L 2 (Y+) ≤ sup ρ∈supp (b) |r(ρ)| 1/2 |b(ρ)|. Proof. We calculate (BhD y u|BhD y u) Y+ =([B, hD y ]u|BhD y u) Y+ + (hD y Bu|BhD y u) Y+ =O(h) + (Bu|Bh 2 D 2 y u) L 2 (Y+) =O(h) -(Bu|BRu) L 2 (Y+) + (Bu|BP h u) Y+ =O(h) -(Bu|BRu) Y+ -(Bu|Bhdq) Y+ .
Integrating by part and using symbolic calculus, we have

(Bu|Bhdq) Y+ =(Bu|hdBq) Y+ + (Bu|[B, hd]q) Y+ = -([hd * , B]u|Bq) Y+ + (Bu|[B, hd]q) Y+ =O(h),
thanks to the fact that B has scalar-valued principal symbol.

The proof of Proposition 4.4 is based on the following integration by part result. Proposition 7.3. Given real scalar-valued tangential symbols a 0 , a 1 , there exist tangential operators A 0 , A 1 (constructed in the local coordinate) with real, scalarvalued principal symbol a 0 , a 1 and A = A 1 hD y + A 0 , such that for any 1-form w with compact support in Y + , we have

2 h
Im(P h w|Aw) Y+ = (A 1 hD y w|hD y w) ∂Y+ + Re

2 j=0 (C j (hD y ) j w|w) Y+ + O(h),
where the tangential operators C j have scalar-valued principal symbol c j (y, x, ξ) and 2 j=0 c j (y, x, ξ)η j = {p, a}.

Proof. We first calculate We want to construct operators A 0 , A 1 such that

I = 1 ih - h 2 √ G gh∂ y ( √ Gg -1 h∂ y w) Aw Y+ - 1 ih Aw - h 2 √ G gh∂ y ( √ Gg -
A * 1 = A 1 + O(h 2 ) and A * = A + O(h 2 ). Assume that a 1 a (0) 1 + h i a (1) 1
with real-valued a (j)

1 (not necessarily scalar-valued). From

Y+ A 1 u|v R d-1 √ Gdydx = Y+ g -1 A 1 u, v R d-1 √ Gdydx, the symbol of A * 1 is equal to the symbol of g √ G Op h ( a 1 * ) √ Gg -1 , which can be expressed by b 1 (y, x, ξ) k≥0 h i k b (k) 1 (y, x, ξ), with b (k) 1 (y, x, ξ) = 1 j=0 |α|+|β|+j=k (-1) j ∂ β ξ g √ G ∂ α ξ ∂ α x a (j) 1 • ∂ β x ( √ Gg -1 ), k ≥ 1.
We have that b

(0) 1 = a (0) 1 , b (1)
1 = -a

1

+ |α|+|β|=1 ∂ β ξ g √ G ∂ α ξ ∂ α x a (0) 1 • ∂ β x ( √ Gg -1 )
We set

a (0) 1 = a 1 , a (1) 1
= 1 2 |α|+|β|=1 ∂ β ξ g √ G ∂ α ξ ∂ α x a (0) 1 • ∂ β x ( √ Gg -1 ), thus A * 1 = A 1 + O(h 2 ). Note that a (1) 1
is matrix-valued while a (0) 1 is real and scalar-valued.

The construction of A 0 is similar. We observe that (hD y ) * = hD y +h g √ G D y ( √ Gg -1 ) and set

a 0 = a (0) 0 + h i a (1)
0 . A * 0 has symbol which can be expanded as

b 0 k≥0 h i k b (k) 0 with b (0) 0 = a (0) 0 and b (k) 0 (y, x, ξ) = 1 j=0 |α|+|β|+j=k (-1) j ∂ β ξ g √ G ∂ α ξ ∂ α x a (j) 0 • ∂ β x ( √ Gg -1 ), k ≥ 1. (7.1) Note that (hD y ) * A * 1 -A * 1 hD y =[(hD y ) * , A * 1 ] + A * 1 (hD y ) * -A * 1 hD y = h i (∂ y A * 1 ) + h i g √ G ∂ y ( √ Gg -1 ), A * 1 + h i A * 1 g √ G ∂ y ( √ Gg -1 ),
and its symbol can be expanded as

k≥0 h i k κ k (y, x, ξ)
with κ 0 = 0 and

κ 1 =∂ y b 1 + b 1 g √ G ∂ y ( √ Gg -1 ), κ k = |α|=k-1 1 i |α|+1 {∂ α ξ , ∂ α x } g √ G ∂ y ( √ Gg -1 ), b 1 + h i ∂ y b 1 + b 1 g √ G ∂ y ( √ Gg -1 ) + h i |α|≥1 h |α| i |α| {∂ α ξ , ∂ α x } g √ G ∂ y ( √ Gg -1 ), b 1 , k ≥ 2, where {∂ α ξ , ∂ α x }(f 1 , f 2 ) = ∂ α ξ f 1 ∂ α x f 2 -∂ α ξ f 2 ∂ α x f 1 . We set b (0) 0 = a 0 and a (1) 0 such that a (1) 0 = b (1)
0 + κ 1 (it has a solution thanks to (7.1)). Finally, we construct A j by ϕ 1 Op h ( a j )ϕ 1 in local coordinate and it can be easily verified that

A * 1 = A 1 + O L 2 →L 2 (h 2 ), A * = A + O L 2 →L 2 (h 2 ). Therefore I = (hD y w|A 1 hD y w) ∂Y+ + 1 ih (hD y w|[hD y , A]w) Y+ - 1 ih ([hD y , A]w|hD y w) Y+ +O(h). We next calculate 1 ih (R h w|Aw) Y+ - 1 ih (Aw|R h w) Y+ = 1 ih ((A * R h -R * h A)w|w) Y+ = 1 ih ([A, R h ]w|w) Y+ + O(h), since R h is self-ajoint and A * -A = O L 2 →L 2 (h 2 )
. Moreover, the principal symbol of 1 ih [A, R h ] is {r, a}. This completes the proof of Proposition 7.3. Now assume that we are working near a diffractive point ρ ∈ G 2,+ in Y + where

∂ y r ≥ c 0 > 0
The following lemma is a semi-classical version of Lemma 24.4.5 in [START_REF] Hörmander | The analysis of linear partial differential operators[END_REF]. The proof is slightly more complicated, due to the different equation that we are considering. Lemma 7.4. Let B j = ϕB j ϕ 1 , with real, scalar-valued tangential principal symbols b j , j = 0, 1, 2, compactly supported and 2 j=0 b j (y, x, ξ)η j = -ψ(y, x, η, ξ) 2 when η 2 = r(y, x, ξ),

with some smooth function ψ ∈ C ∞ c (R d × (R d \ {0})). Assume that dr = 0, ∂ y r > 0, on {y = r = 0} ∩ 2 j=1
supp(b j ).

Then one can chose compactly supported, tangential operators Ψ j , j = 0, 1 with real, scalar-valued principal symbols ψ j , j = 0, 1, satisfying ψ 0 (y, x, ξ) = ψ(y, x, 0, ξ), ψ 1 (y, x, ξ) = ∂ η (y, x, 0, ξ) when η = r(y, x, ξ) = 0, so that for any solution u of P h u = f -h∇q, hdivu = 0 with u| y=0 = 0, we have

Re 2 j=0 (B j (hD y ) j v|v) Y+ + Ψ 0 v + Ψ 1 hD y v 2 L 2 (Y+) + (ΘP h v|v) Y+ =o (1) (7.2)
as h → 0, where v = ϕOp h (χ)ϕ 1 u and χ ∈ C ∞ c (Y + × R d-1
) has support near ρ ∈ G 2,+ . Θ is a tangential operator, depending on ψ j , b j whose principal symbols are scalar-valued.

The proof is based on the following elementary lemma, for which the proof can be found as Lemma 24.4.3 in [8], Lemma 7.5. Let X be an open subset of R n + = {x ∈ R n : x 1 ≥ 0}, and let r ∈ C ∞ (X). Assume that r is real-valued, that dr = 0 when r = 0 and that ∂r ∂x1 > 0 when r = x 1 = ∂r ∂xj = 0 for j = 1. Let

F (t, x) = 2 j=0 f j (x)t j
be a quadratic polynomial in t with coefficients in C ∞ (X) such that

F (t, x) = -ψ(t, x) 2 when t 2 = r(x),
where ψ ∈ C ∞ (R × X). Then one can find ψ 0 , ψ 1 , θ ∈ C ∞ (X) such that ψ 0 (x) = ψ(0, x), ψ 1 (x) = ∂ψ ∂t (0, x) when r(x) = 0, and

F (t, x) + (ψ 0 (x) + tψ 1 (x)) 2 ≤ θ(x)(t 2 -r(x)), ∀t ∈ R, x ∈ X.
Proof of Lemma 7.4. Choose C ∞ functions ψ 0 (y, x, ξ) and ψ 1 (y, x, ξ) as in Lemma 7.5, such that ψ j (y, x, ξ) = ∂ j η ψ| y=0 , j = 0, 1 when η = r(y, x, ξ) = 0 and

2 j=0 b j η j + (ψ 0 + ηψ 1 ) 2 ≤ θ(y, x, ξ)(η 2 -r).
Since ψ 0 , ψ 1 and each b j are compactly supported in variables (y, x, ξ), we may assume that θ is smooth and with compact support. Define Θ = ϕOp h (θ)ϕ 1 , Ψ j = ϕOp h (ψ j )ϕ 1 , j = 0, 1 and consider the quantity Re

2 j=0 (B j (hD y ) j v|v) Y+ + ((Ψ 0 + Ψ 1 hD y ) 2 v|v) Y+ -(ΘhD y v|hD y v) Y+ + (ΘR h v|v) Y+ .
The expression above can be written under the form below

2 j=0 C j (hD y ) j v|v Y+ ,
where the tangential operators C j have real, scalar-valued principle symbol. Moreover, 2 j=0 c j (y, x, ξ)η j ≤ 0.

However, since the symbol is not bounded in η and we can not apply sharp Gårding inequality directly. To resolve this issue, we extend each c j to c j ∈ C m c (R × R 2d-2 ) who agrees with c j on y ≥ 0 up to order m, any given order, of derivatives. This is possible since any order of y derivative of all the symbols has continuous limit as y → 0. We still use the notation c j in what follows. Let v = v1 y≥0 and we use the boundary condition v| y=0 = 0 and calculate

2 j=0 C j (hD y ) j v v Y+ = 2 j=0 C j (hD y ) j v v Y+ = ψ hD y A 2 j=0 C j (hD y ) j v v Y+ + 1 -ψ hD y A 2 j=0 C j (hD y ) j v v Y+ =:I + II,
for any big number A > 0. Now we apply sharp Gårding inequality to the first term to get

I ≤ C A h,
with some constant C A depending on A. For the second term, the principle symbol is supported in the elliptic region and we define Ξ(y, x, η, ξ)

:= 1 -ψ η A 2 j=0 c j (y, x, ξ)η j η 2 -r(y, x, ξ) ∈ S 0 (R 2d ),
hence we can bound

|II| ≤O(h) + C Ξ(y, x, hD y , hD x )χ(y, x, hD x )P h u 1 -ψ 2hD y A v Y+ =O(h) + C Ξ(y, x, hD y , hD x )χ(y, x, hD x )(hw ⊗ δ y=0 ) 1 -ψ 2hD y A v Y+ +C Ξ(y, x, hD y , hD x )χ(y, x, hD x )(1 y≥0 h∇q) 1 -ψ 2hD y A v Y+ ,
with w = hD y u| y=0 . Note that to obtain the expression above, one can not use symbolic calculus to deal with commutator between semi-classical tangential symbol and the classical symbol. However, since P h is a differential operator, we can compute its commutator with χ(y, x, hD x ) explicitly. Now from Proposition 8.17, the limsup of the third term on the right hand side when h → 0 can be bounded by (A) with lim A→∞ (A) = 0. Here we can use the flat metric to estimate the L 2 norm. The second term on the right hand side can be bounded by

Ch (1 -h 2 ∆ y,x ) -s 2 (w ⊗ δ y=0 ) L 2 (R d ) (1 -h 2 ∆ y,x ) s 2 v L 2 (R d ) ≤ Ch 1-s ,
for any s ∈ 1 2 , 1 . Here we have used the fact that δ y=0 ∈ H -s (R y ) for any s > 1 2 and h s v is bounded in H s (R d ) since v| y=0 = 0 and h∇ y,x v is bounded in L 2 (R d ). Therefore, for any A > 0, we have showed that lim sup

h→0 |II| ≤ (A),
and this completes the proof of Lemma 7.4.

Adapting to the notations in this section, Proposition 4.4 can be rephrased as follows Proposition 7.6. Suppose that ρ ∈ G 2,+ , and

ρ 0 ∈ T * Ω approaching to ρ such that ∂ y r(ρ 0 ) ≥ 1 2 ∂ y r(ρ) ≥ c 0 . Let γ -= [ρ 0 ,
ρ] be a segment of the generalized ray issued from ρ 0 to ρ (the trajectory under the canonical projection is tangent to the boundary at ρ). Then if ρ 0 / ∈ supp(µ), we have ρ / ∈ supp(µ).

Proof. Take a small neighborhood Γ 0 of ρ 0 such that Γ 0 ∩ supp(µ) = ∅. Take a small neighborhood W 0 ⊂ Ω × R d-1 such that ∂r ∂y (y, x, ξ) ≥ c 0 /4 > 0. Shrinking W 0 if necessary, we assume that each point (y, x, ξ) ∈ W 0 with r(y, x, ξ) ≥ 0 can be connected by a (possibly broken) ray issued from Γ 0 with at most one reflection or tangency at ∂Ω. It suffices to prove the following statement:

For any χ ∈ C ∞ c (Ω × R d-1
) with supp(χ) ⊂ W 0 , small enough, we have

ϕOp h (χ)ϕ 1 u = o L 2 (1), h → 0.
As in [START_REF] Hörmander | The analysis of linear partial differential operators[END_REF], we construct test functions which satisfy the following properties:

Lemma 7.7. There exists a(y, x, η, ξ) = a 0 (y, x, ξ)

+ a 1 (y, x, ξ)η, a j ∈ C ∞ c (W 0 )
with the following properties:

(1) a 1 (0, x, ξ) = -t(x, ξ) 2 , for some t ∈ C ∞ c (T * ∂Y + ), (2) For some large M ≥ 0, when p = η 2 -r(y, x, ξ) = 0, we have

{p, a} + aM = -ψ(y, x, η, ξ) 2 + ω(y, x, ξ)(η -r 1/2 (y, x, ξ)), a = s 2 , where s ∈ C ∞ (Y + × (R d \ {0})),ψ ∈ C ∞ c (Y + × R d \ {0}) and ω ∈ C ∞ c (W 0 ). Moreover, r| supp(ω) > 0.
The construction is exactly the same as in [START_REF] Hörmander | The analysis of linear partial differential operators[END_REF] and will be given in the appendix D for the sake of completeness. Now we take χ ∈ C ∞ c (W 0 ) with χ ≡ 1, in a neighborhood of supp(a 0 )∪supp(a 1 ). Let v = ϕOp h (χ)ϕ 1 u, and we calculate

(P h v|Av) Y+ =(ϕOp h (χ)ϕ 1 P h u|Av) Y+ + ([P h , ϕOp h (χ)ϕ 1 ]u|Av) Y+ =(ϕOp h (χ)ϕ 1 f |Av) Y+ -(ϕOp h (χ)ϕ 1 hdq|Av) Y+ +([P h , ϕOp h (χ)ϕ 1 ]u|Av) Y+ .
Here we have used the differential form to calculate the inner product. Notice that {p, χ} = 0 on supp(a j ) and

f = o L 2 (h), hD y u ⊥ | y=0 = 0, thus 2 h Im(P h v|Av) Ω = o(1) - 2 h Im([ϕOp h (χ)ϕ 1 , hd]q|AϕOp h (χ)ϕ 1 u) Y+ + 2 h Im(Op h (χ)q|hd * (AϕOp h (χ)ϕ 1 u)) Y+ . (7.3) From Proposition 7.3, 2 j=0 (C j (hD y ) j v|v) Y+ = -(A 1 hD y v|hD y v) ∂Y+ - 2 h Im([ϕOp h (χ)ϕ 1 , hd]q|Av) Y+ + 2 h Im(ϕOp h (χ)ϕ 1 q|hd * (AϕOp h (χ)ϕ 1 u)) Y+ + o(1). (7.4)
Since the principal symbol of A is scalar-valued, by using d * u = 0, we have

2 h Im(ϕOp h (χ)ϕ 1 q|hd * (AϕOp h (χ)ϕ 1 u)) Y+ = (ϕOp h (χ)ϕq|Υu) Y+ + O(h) and - 2 h Im([ϕOp h (χ)ϕ 1 , hd]q|Av) Y+ = (Υ 2 q|Av) Y+ + O(h),
where Υ = Υ 0 + Υ 1 hD y , and Υ j are matrix-valued tangential pseudo-differential operators with principal symbols supported in supp(χ). Applying Lemma 7.4 to the function

2 j=0 c j η j + aM -ω(η -r 1/2 ) = -ψ 2 ,
we have Re

2 j=0 C j (hD y ) j u|u Y+ -Re(Φ(hD y -Q + )v|ϕ(y, x, hD x )v) Y+ + Re(M v|Av) Y+ + (ΘP h v|v) Y+ + Ψ 0 v + Ψ 1 hD y v 2 L 2 (Y+) ≤o(1) + Ch v 2 L 2 (Y+) , (7.5)
where the compact supported tangential operator Φ has scalar-valued principal symbol φ ∈ C ∞ c (W 0 ) and r| supp(φ) > 0, φ = 1 in a neighborhood of supp ω. Q + is the operator constructed in the hyperbolic region with principal symbol r 1/2 . This is possible since in the proof of Lemma 7.7, we indeed have r ≥ δ 2 |ξ| 2 on the support of ω. Note that the principal symbol of A is positive on η 2 -r = 0, we can apply Lemma 7.4 again to the term (M v|Av) Ω and bound it from below by

o(1) -|(Θ 1 P h v|v) Y+ |.

Thus we have

-(A 1 hD y v|hD y v) ∂Y+ + Ψ 0 v + Ψ 1 hD y v 2 L 2 (Y+) ≤ o(1) + Ch v 2 L 2 (Y+) + C|(ΘP v |v) Y+ | + C|(Θ 1 P h v|v) Y+ | + Re(ϕOp h (φ)ϕ 1 (hD y -Q + )v|ϕOp h (ω)ϕ 1 v) Y+ + |(Υ 2 q|Av) Y+ | + |(ϕOp h (χ)ϕ 1 q|Υu) Y+ |. (7.6)
The terms on the left hind side are essentially positive from the sharp Gårding (semi-classical, see [START_REF] Zworski | Semiclassical analysis[END_REF]) inequality, hence we only need to control the terms on the right hind side Indeed, micro-locally on supp(φ), r δ 2 > 0, hence in the region where λ 2 (y, x, ξ) < 1, we could construct Q + , Q -micro-locally such that

P h = (hD y -Q -)(hD y -Q + ) + O(h ∞ )
as we have done in the hyperbolic case. From symbolic calculus and Corollary 6.6, we have

(hD y -Q -)(hD y -Q + )u = O L 2 y,x (h) + h∇q = o L 2 y,x (1)
, micro-locally on supp(φ). Therefore the measure µ concentrates on {η = -√ r}∪{η = √ r}. For any point ρ 1 ∈ supp(φ)∩supp(µ), with η(ρ 1 ) = -r(ρ 1 ) < 0, the backward generalized ray issued from ρ 1 must enter Γ 0 without meeting any point in G 2,+ , since along the backward flow, η is decreasing. Consequently, away from the boundary, u = o L 2 (Y+) (1) and hence (hD y -Q + )u = o L 2 (Y+) (1), micro-localized near η = -√ r, due to the fact that hD y -Q -is micro-locally elliptic near η = √ r. Near the boundary and some point ρ 1 ∈ H ∩ supp(φ), any point can be connected backwardly to Γ 0 by at most transversal reflection. Thus (7.7) holds true.

It remains to control the last two terms involving pressure. We just treat one of them, and the other can be treated in the same way. Pick ϕ 0 ∈ C ∞ c ((-2, 2)) which is equal to 1 on (-1, 1). Define χ (y, x, ξ) = χ(y, x, ξ)ϕ 0 r(y, x, ξ) -1 .

We fix any > 0, small enough, and write

(Υ 2 q|Av) Y+ =(Υ 2 q|AϕOp h (χ )ϕ 1 u) Y+ +(Υ 2 q|AϕOp h (χ -χ )ϕ 1 u) Y+ =:I h, + II h, . (7.8)
We first deal with I h, . Notice that from Proposition 7.2, we have lim sup

h→0 hD y ϕOp h (χ )ϕ 1 u L 2 (Y+) ≤ C 1/2 .
Applying Cauchy Schwartz, we have

y0 0 ϕOp h (χ )ϕ 1 u 2 L 2 (∂Y+, √ Gdx) dy ≤ Ch -2 y0 0 y 0 hD y ϕOp h (χ )ϕ 1 u(s, x)ds 2 dxdy ≤ Cy 2 0 h 2 hD y ϕOp h (χ ϕ 1 u) 2 L 2
x,y . By choosing θ ∈ (0, 1/2) and y 0 = h -θ , we estimate

|I h, | ≤ y0 0 + 0 y0 |(Υ 2 q|AϕOp h (χ )ϕ 1 u) L 2 (∂Y+, √ Gdx) |dy ≤ C 1 2θ ( hD y Op h (χ )u 2 L 2 x,y + O(h)) + Ce -c θ ,
where we have used Lemma 3.9. Note that Lemma 3.9 is applicable even when the micro-local cut-off χ δ0 is matrix-valued. In summary we have lim sup

h→0 |I h, | ≤ C( 1-2θ + e -c θ ).
We now turn to the estimates of II h, . This can be done from geometric argument. Let S := {(y, x, ξ) : r(y, x, ξ) ≥ , y ≤ 4 /c 0 } ∩ W 0 . We claim that for any ray γ with γ(0) ∈ Γ 0 and Γ(s 0 ) ∈ S , γ| [0,s0] ∩ G 2,+ = ∅.

Indeed, by contradiction, assume that for some γ and s 1 ∈ [0, s 0 ], we have

ρ 1 = γ(s 1 ) ∈ G 2,+ . After time s 1 , along γ we have ẏ = 2η, η = ∂ y r ≥ c 0 /4, with y(s 1 ) = η(s 1 ) = 0, η(s 0 ) ≥ √ . This implies that s 0 -s 1 ≥ 4 √ /c 0 and y(s 0 ) ≥ c 0 T 2 /4 ≥ 4 δ0
c0 . The claim then follows. Now we write II = (ϕ 0 (c

0 y/ ) Υ 2 q|AϕOp h (χ -χ )ϕ 1 u) Y+ + ((1 -ϕ 0 (c 0 y/ )) Υ 2 q|AϕOp h (χ -χ )ϕ 1 u) Y+ .
From the discussion above, the first term on the right hand side above tends to 0 as h → 0 for any fixed > 0, while the second term is controlled from above by

0 δ 0 4C |C(y, x, hD x )q| 2 dxdy
for some zero order semi-classical tangential operator with principal symbol c(y, x, ξ) such that supp c ∩ {ξ = 0} = ∅. Applying Lemma 3.9, we have lim sup h→0 |II | = 0 is true for any > 0. Notice that the left hand side of (7.5) is independent of , we have lim sup

h→0 ((-A 1 hD y v|hD y v) ∂Y+ + Ψ 0 v + Ψ 1 hD y v 2 L 2 (Y+) ) = 0.
From the construction of a 0 , a 1 and the corresponding expression of ψ 0 , ψ 1 , we can choose another different a 0 , a 1 , such that the function ψ 0 + ψ 1 η is independent of ψ 0 + ψ 1 η on supp(χ)(see appendix D). It follows then

v L 2 (Y+) + hD y v L 2 (Y+) = o(1), h → 0,
and this completes the proof of Proposition 7.6.

8. Near G 2,-and G k for k ≥ 3

This section is devoted to the proof of Proposition 4.6. Before proving it, we need some preparation. In what follows, we take tangential operators

A = ϕOp h (a)ϕ 1 , A * = A + O L 2 (∂Y+) (h 2 ). Proposition 8.1. 1 h (([P, A]u|u) Y+ = 1 h (Au|P u) Y+ - 1 h (AP u|u) Y+ + O(h).
Proof. The proof goes in exactly the same way and much simpler than the diffractive case, and we omit it here.

Recall that r 0 = r| y=0 and r 1 = ∂ y r| y=0 . Direct calculation gives

H p a = 2η ∂a ∂y + ∂r ∂y ∂a ∂η + H -r a.
Pick ρ 0 ∈ G 2,-⊂ T * ∂Ω {0} and a small neighborhood U ⊂ T * ∂Ω {0} of ρ 0 . Let L ⊂ U be a co-dimension 1 hypersurface containing ρ 0 in T * ∂Ω and transversal to the vector field H -r0 . For small positive numbers δ, τ > 0, define

L ± (δ, τ ; ρ 0 ) := {exp(tH -r0 )(ρ) ∈ U : ρ ∈ L, dist (ρ, ρ 0 ) ≤ δ 2 , 0 ≤ ±t ≤ τ }.
When there is no risk of confusion, we write it simply as L ± (δ, τ ). Define also

F ± (δ, τ ) := {(y, x, ξ) : 0 ≤ y ≤ δ 2 , (x, ξ) ∈ L ± (δ, τ )}, F (δ, τ ) = F + (δ, τ ) ∪ F -(δ, τ ).
Let C 1 > 0 sufficiently large and δ 0 > 0, τ 0 > 0 sufficiently small so that δ < δ 0 , τ < τ 0

|r(y, x, ξ)| ≤ 1 2 C 2 1 δ 2 (8.1)
in F (δ, τ ) for all 0 < δ ≤ δ 0 , 0 < τ ≤ τ 0 . With the same constant C 1 , we further define the sets

V ± (δ, τ ) :={(y, x, η, ξ) : 0 ≤ y ≤ δ 2 /2, (x, ξ) ∈ L ± (δ, τ )} ∪{(y, x, η, ξ) : δ 2 /2 ≤ y ≤ δ 2 , (x, ξ) ∈ L ± (δ, τ ), |η| ≤ C 1 δ}, W ± (δ, τ ) :={(y, x, η, ξ) : 0 ≤ y ≤ δ 2 /2, (x, ξ) ∈ L ± (δ, τ )} ∪{(y, x, η, ξ) : δ 2 /2 ≤ y ≤ δ 2 , (x, ξ) ∈ L ± (δ, τ ), |η| ≤ 2C 1 δ}, V (δ, τ ) := V + (δ, τ ) ∪ V -(δ, τ ), W (δ, τ ) = W + (δ, τ ) ∪ W -(δ, τ
). We need test functions constructed in [START_REF] Melrose | Singularities of boundary value problems[END_REF]: Lemma 8.2 ( [START_REF] Melrose | Singularities of boundary value problems[END_REF]). Let I = [0, 0 ) y . There exist σ > 0, δ 0 > 0, τ 0 > 0, small enough with the hierarchy δ 0 σ 1, and families of smooth functions

a δ ∈ C ∞ c (I × U ), g δ , h δ ∈ C ∞ (Y + × R η × R d-1 ξ \ {0}
), where 0 < δ ≤ δ 0 , satisfying the following properties:

(1)

a δ ≥ 0, supp(a δ ) ⊂ F + (δ, σδ) ∪ F -(δ, δ 2 ) .
(2) a δ (0, exp(tH -r0 (ρ 0 ))) = 0, ∀0 ≤ t < δσ.

(3) a δ > 0 on supp(a δ ) if 0 < δ < δ ≤ δ 0 and a δ is independent of y for 0 ≤ y < δ 2 /2. (4) g δ + h δ = -H p a δ .

(5) in W (δ, τ ), g δ ≥ 0 and g δ > 0 when a δ = 0.

(6) For any m > 1 and any multiple index α ∈ N d , |g

-1 m δ ∂ α g δ | = O δ (1)
, locally uniformly on W (δ, τ 0 ), where the implicit constant inside O δ (1) depends on α, m and δ.

(7) supp(h δ ) ⊂ I × L -(δ, δ 2 ) × R η , and supp(g δ ) ∪ supp(h δ ) ⊂ supp(a δ ), g δ , h δ are independent of η for 0 ≤ y ≤ δ 2 /2.
For the convenience of the reader, we will recall the proof in the appendix D.

According to the lemma, we have ∂(g

1/2 δ ) = 2g -1/2 δ ∂g δ = O(1), this implies that g 1/2 δ ∈ C ∞ (W (δ, τ )). Set b δ := g 1/2 δ ∈ C ∞ (W (δ, τ ))
. Note b δ may not be smooth with compact support. We need split it into two parts as follows: Let

φ 1 ∈ C ∞ (R) such that φ 1 ≡ 1 if 0 ≤ y ≤ δ 2 4 and φ 1 ≡ 0 if y > 3δ 2 8 . Let φ 2 ∈ C ∞ (Ω × R d \ {0}
) with compact support in x, ξ, η variables, such that φ 2 ≥ 0 and φ 2 ≡ 0 whenever y ≤ δ 2 4 or |η| > 2C 1 δ. Indeed, we can choose κ ∈ C ∞ c (R), non-negative, smooth and with compact support, such that κ(z) ≡ 0 when |z| > 2C 1 δ and κ(z) ≡ 1 when |z| ≤ 3 2 C 1 δ. Now let φ 2 (y, x, η, ξ

) 2 = (1 -φ 1 (y) 2 )κ(δ -1 η)χ δ (y, x, ξ) with χ δ | supp(a δ ) ≡ 1, supp(χ δ ) ⊂ F + (δ, σδ) ∪ F -(δ, σδ). We observe that W ± (δ, τ ) ∩ supp (1 -φ 2 1 -φ 2
2 ) ⊂ (y, x, η, ξ) :

δ 2 4 ≤ y ≤ δ 2 , |η| > 3 2 C 1 δ, .
We finally put b δ,j :=

φ j b δ , j = 1, 2. Note that b δ,1 ∈ C ∞ c (F (δ, τ )) is a tan- gential symbol (since for y ≥ δ 2 2 , supp(g δ ) ⊂ supp(a δ ) is compact) while b δ,2 ∈ C ∞ c (W (δ, τ )
) is a usual interior symbol with compact support in T * Ω. 8.1. Gliding case. The propagation of support of µ near a gliding point in G 2,- can be stated as follows:

Proposition 8.3. Suppose ρ 0 ∈ G 2,-and L + (δ 0 , τ 0 ) ∪ L -(δ 0 , τ 0 ) ⊂ G 2,-
for some δ 0 , τ 0 > 0. Then for any σ > 0 with σδ 0 < τ 0 , such that if

{(y, x, η, ξ) : 0 ≤ y ≤ δ 2 , (x, ξ) ∈ L -(δ, δ 2 ; ρ 0 )} ∩ supp(µ) = ∅ for some 0 < δ ≤ δ 0 , then exp(tH -r0 )(ρ 0 ) / ∈ supp(µ) for any t ∈ [0, σδ).
We need several lemmas.

Lemma 8.4. Suppose a ∈ C ∞ c (R 2d ), b ∈ C ∞ ([0, 1], C ∞ c (R 2(d-1)
)) with the following support property:

a(y, x, η, ξ) ≡ 0 if y ≤ c 0 < 1 or |η| > C 0 |ξ|.
Then the usual symbolic calculus for a(y, x, hD y , hD x )b(y, x, hD x ) still valid. In particular, a(y, x, hD y , hD x )b(y, x, hD x ) = c(y, x, hD y , hD x ) + O L 2 →L 2 (h), with c(y, x, η, ξ) = a(y, x, η, ξ)b(y, x, ξ)

We postpone the proof in the appendix F. Lemma 8.5. Given any ρ 1 ∈ G, there exist δ 1 > 0, τ 1 > 0, σ 1 > 0 with δ 1 σ 1 and σ 1 δ 1 < τ 1 such that if ρ ∈ T * ∂Ω and dist(ρ, ρ 1 ) ≤ δ 2 for some 0 < δ ≤ δ 1 , then dist(γ(s, ρ), γ(s, ρ 1 )) ≤ Cδ 2 for |s| ≤ σ 1 δ. In particular, γ(s, ρ) ∈ W (δ, τ 1 ) for all |s| ≤ σ 1 δ.

Proof. Write γ(s, ρ) and exp(sH -r0 )(ρ) in coordinate as γ 1 (s) = (y(s), η(s), x(s), ξ(s)) and γ 2 (s) = (ỹ(s), η(s), x(s), ξ(s)).

From ẏ = 2η, η = O(1), we have y(s) ≤ Cs 2 and the same estimate holds for ỹ(s). Let

d(s) = |x(s) -x(s)| 2 + |ξ(s) -ξ(s)| 2 ,
and then ḋ(s) ≤ C d(s). This implies d(s) ≤ C 1 δ 2 for all |s| ≤ σ 1 δ. By the same argument, we have dist(exp(sH -r0 )(ρ), exp(sH -r0 )(ρ 1 )) ≤ Cδ 2 . The conclusion then follows from the triangle inequality.

We will see the crucial role of ρ 0 ∈ G 2,-in the following lemma: Lemma 8.6. Assume that δ 1 , τ 1 are parameters given in the previous lemma. Suppose that -C 0 ≤ ∂ y r(ρ) ≤ -c 0 < 0 for all ρ ∈ W (δ 1 , τ 1 ). Define S = W (δ 1 , τ 1 ) ∩ {r ≥ , y ≤ } for sufficiently small > 0. Then along any ray γ(s, ρ 1 ) in W (δ 1 , τ 1 ) with ρ 1 ∈ S , if y(γ(-t, ρ 1 )) = 0 for some 0 ≤ t ≤ τ 1 , we have r(y(γ(-t, ρ 1 )) ≥ c 1 , where c 1 depends only on W (δ 1 , τ 1 ).

Proof. Assume ρ 1 = (y 1 , x 1 , η 1 , ξ 1 ) ∈ S and γ(s, ρ 1 ) = (y(s), x(s); η(s), ξ(s)). Let

s 3 = inf{0 ≤ s ≤ τ 1 : y(-s) = 0}. For s ∈ [-s 3 , 0], ẏ = 2η, -C 0 ≤ η = ∂ y r ≤ -c 0 . There are two possibilities. If η 1 ≥ √ , then η(-s + 3 ) ≥ η 1 > 0 since η < 0. Otherwise, η 1 ≤ - √
, and we denote by

s 2 = inf{s ∈ [0, s 1 ] : η(-s) = 0}. From η 1 = 0 -s2 ηds ≥ -C 0 s 2 , we have s 2 ≥ |η1| C0 . Moreover, y 1 -y(-s 2 ) = 2η 1 s 2 - 0 -s2 ds 0 s ÿds ≤ 2η 1 s 2 + C 0 s 2 2 ≤ - |η 1 | 2 C 0 . Now from y(-s 2 ) = y(-s 2 ) -y(-s 3 ) = - -s2 -s3 ds -s2 s ÿds ≤ C 0 |s 3 -s 2 | 2 , we have |s 3 -s 2 | 2 ≥ y(-s 2 ) C 0 ≥ (y(-s 2 ) -y 1 ) C 0 ≥ |η 1 | 2 C 2 0
and finally

η(-s + 3 ) = - -s2 -s3 ηds ≥ c 0 |s 3 -s 2 | ≥ c 0 √ C 0 .
The proof of Lemma 8.6 is then complete by applying the argument above between any two adjacent zeros of s → y(γ(-s, ρ 1 )).

Proof of Proposition 8.3. For any δ > 0, we define the operator

N δ = 1 ih [P, A δ ] with principal symbol n δ = -H p a δ = g δ + h δ . Define operators B δ ,j := Op h (b δ ,j ), j = 1, 2, N δ,3 = Op h ((1 -φ 2 1 -φ 2 2 )n δ ). Write h δ ,j = φ 2 j h δ , H δ ,j = Op h (h δ ,j ), j = 1, 2.
The proposition will follow if we can show that for any δ < δ,

lim h→0 2 j=1 B δ ,j u 2 L 2 (Y+) = 0 (8.2)
We remark that h δ ,1 , b δ ,1 are both tangential symbols while h δ ,2 , b δ ,2 are interior symbols vanishing near the boundary. Observe also that N δ ,3 is interior pseudodifferential operator with symbol vanishing near the boundary as well as on p -1 (0), thanks to the fact that in

W (δ , τ ), |r(y, x, ξ)| ≤ 1 2 C 2 1 δ 2 . Thus N δ ,3 u = o L 2 (Y+) (1
) as h → 0 for δ > 0 small enough. Moreover, from the assumption on the support of µ near the original point ρ 0 we have

H δ ,j u = o L 2 y,x (1). Now set M δ ,j = φ 2 j N δ ,j -B * δ ,j B δ ,j -H δ ,j , j = 1, 2. From symbolic calculus, we deduce that M δ ,1 = O L 2 →L 2 (h)
and it has the principal symbol depending only on y, x, ξ. Note that by definition of M δ ,2 , we will encounter the composition of tangential operator with interior operator Op h (φ 22 ). Since φ 2 has support far away form y = 0 and η = 0, the symbolic calculus still valid thanks to Lemma 8.4. Therefore M δ ,2 = O L 2 →L 2 (h) is an interior operator. Finally, we obtain that

N δ = N δ ,3 + 2 j=1 (B * δ ,j B δ ,j + H δ ,j ) + O L 2 (Y+)→L 2 (Y+) (h).
Combining all the analysis above and applying Proposition 8.1, we have

2 j=1 B δ ,j u 2 L 2 (Y+) ≤ o(1) + 2 h Im([A δ , hd]q|u) Y+ + 1 h Im(q|hd * (A δ u)) Y+ =o(1) + |(q|Υ 1 u) Y+ | + |(Υ 2 q|u) Y+ | (8.3)
where Υ 1 , Υ 2 are compactly supported matrix-valued tangential operators with principal symbols vanishing outside supp(a δ ).

To finish the proof, we need show that the right hand side of (8.

3) is o(1) as h → 0. Pick χ ∈ C ∞ c (R) such that χ(s) ≡ 1 if 0 ≤ s ≤ 1 2 and χ(s) ≡ 0 is s ≥ 1. Let χ (y, x, ξ) = χ( -1 r(y, x, ξ)). Denote by I h, = (Υ 1 ϕOp h (χ )ϕ 1 u|q) Y+ , II h, = (Υ 1 (1 -ϕOp h (χ )ϕ 1 )u|q) Y+ .
The treatment of I h, is exactly the same as in the diffractive case, so we have lim →0 lim sup h→0 I h, = 0.

For II h, , we may assume that the interval of the integration over y variable is [0, ], since for y ≥ we can use the rapid decreasing of q as in the treatment of I h, . According to Lemma 8.5 and Lemma 8.6, the measure of Υ 1 (1 -ϕOp h (χ )ϕ 1 )u vanishes, since all the backward generalized rays starting from each point in S will enter the small neighborhood of ρ 0 ∈ G 2,-by at most reflection at boundary. From the propagation theorem in the hyperbolic case(Proposition 4.3), the proof of Proposition 8.3 is complete.

Remark 8.7. We remark that as a consequence of Proposition 8.3, the measure of q(or h∇q) also vanishes along exp(tH -r0 ) for t ∈ [0, σδ). 8.2. high order contact. In this subsection we will use a new coordinate system in a neighborhood

W k of ρ k ∈ G k in [0, 0 ] × T * ∂Ω: (y, η, x, ξ) → (y, η, z, ζ), z = (z 1 , z ), ζ = (ζ 1 , ζ ) with p = η 2 -r, r = ζ 1 + yr 1 (z, ζ) + O(y 2 ), ζ 1 = r 0 , where r 0 = r| y=0 , r 1 = ∂ y r| y=0 . This is possible since d x,ξ r 0 = 0, if ξ = 0. Along the generalized bicharacteristic curve γ(s), (z, ζ) satisfies ż = -∂ ζ r(y(s), z(s), ζ(s)), ζ = ∂ z r(y(s), z(s), ζ(s)).
This implies that in W k , -ż1 ∼ 1 > 0, as y → 0, and thus s → z 1 (s) is strictly decreasing. Moreover, ζ1 ∼ y∂ z1 r 1 , as y → 0.

Suppose now k ≥ 3, we have locally that

G k := {(z, ζ) : ζ 1 = 0, ∂ l z1 r 1 (z, ζ) = 0, ∀l ≤ k -3, ∂ k-2 z1 r 1 (z, ζ) = 0}. Define Σ k := {(z, ζ) : ∂ k-3 z1 r 1 (z, ζ) = 0, ∂ k-2 z1 r 1 (z, ζ) = 0}.
From implicit function theorem, Σ k is locally a hypersurface and we can write it as

Σ k = {(z, ζ) : z 1 = Θ k (z , ζ)}.
G k can be viewed locally as a closed subset of Σ k . Since the map s → z 1 (s) is bijective, we may assume that along each ray, z 1 (0) = Θ k (z (0), ζ(0)), and

z 1 (s) < Θ k (z (s), ζ(s)), s > 0; z 1 (s) > Θ k (z (s), ζ(s)), s < 0.
We see that all the generalized rays are transversal to the codimension 2 manifold(locally) Σ k . Moreover, a ray passes Σ k if and only if y(0) = 0 and ζ 1 (0) = 0. Now we define the set near ρ k :

Σ ± k := {(y, η, z, ζ) ∈ Car(P ) ∩ W k : z 1 ∓ Θ k (z , ζ) > 0}.
Note that the gliding rays exp(sH -r0 ) intersect transversally to Σ k and H -r0 = -∂ z1 inside T * ∂Ω. Thus we can re-parametrize the gliding flows by z 1 . Moreover, Σ ± k ∩ G j = ∅, ∀j ≥ k, provided that we choose W k small enough. In other word, z 1 gives a foliation of

T * ∂Ω near Σ k for small |z 1 -Θ k (z , ζ)|.
The following proposition is a long time extension of Proposition 8.3, adapted to the notations introduced above.

Proposition 8.8. Suppose ρ 0 ∈ G 2,-near ρ k ∈ Σ k with coordinate (z, ζ), z 1 > Θ k (z , ζ).
Then there exists δ 0 > 0, sufficiently small such that if

{(y, x, η, ξ) : 0 ≤ y ≤ δ 2 , (x, ξ) ∈ L -(δ, δ 2 ; ρ 0 )} ∩ supp(µ) = ∅ for 0 < δ ≤ δ 0 , then exp(sH -r0 )(ρ 0 ) / ∈ supp(µ) for any s < z 1 -Θ k (z , ζ).
In other words, each generalized ray, issued from gliding set outside supp(µ) does not carry any singularity until it reaches some point in G k for k ≥ 3.

Proof. The proof is purely topological. For each ρ 0 = (z, ζ) / ∈ supp(µ) and z 1 > 0, let s 1 := sup{s : s ≤ z 1 -Θ k (z , ζ), exp(s H -r0 ) / ∈ supp (µ), ∀s ∈ [0, s)}. The existence of s 1 is guaranteed by Proposition 8.3. It remains to show that

s 1 = z 1 -Θ k (z , ζ). By contradiction, suppose s 1 < z 1 -Θ(z , ζ), then the point ρ 1 = (z 1 -s 1 , z , ζ) is in G 2,
-. We can apply Proposition 8.3 again to obtain that for some small σ 1 > 0, exp(θσH -r0 )(ρ 1 ) / ∈supp (µ) for any θ ∈ [0, 1]. This is a contradiction of the choice of s 1 .

As a consequence, we have Corollary 8.9. Suppose ρ 0 ∈ G 2,-and ρ 0 / ∈ supp(µ). Let γ(s) be the generalized ray passing ρ 0 with γ(0) = ρ 0 . Then γ(s) / ∈ supp(µ) for any s ∈ [-s 0 , s 0 ], provided that γ| [-s0,s0] ⊂ G 2,-.

Combining the analysis near a diffractive point and a gliding point, we have already established the k-propagation property for k = 2. We will argue by induction to prove k-propagation property for all k ≥ 3. To this end, we need an intermediate step. Let us first introduce a notation Γ(ρ 0 ; δ) := {(y, x; z, ζ) : 0 ≤ y ≤ δ 2 , (z, ζ) ∈ L -(δ, δ 2 ; ρ 0 )} and a definition Definition 8.10 (k-pre-propagation property). For k ≥ 2, we say that k-prepropagation property holds, if the following statement is true:

For any ρ k ∈ G k , there exists a neighborhood V k of ρ k in T * ∂Ω, and 0 < δ k σ k 1, depending on V k , such that for any

ρ 0 ∈ G 2,-∪ 3≤j≤k G j ∩ V k , if Γ(ρ 0 ; δ) ∩ supp(µ) = ∅ for some 0 < δ < δ k , then exp(sH -r0 )(ρ 0 ) / ∈ supp(µ) for all s ∈ [0, σ k δ).
The key step is the following inductive proposition. Proposition 8.11. Suppose k ≥ 3 and (k -1)-propagation property holds true, then k-pre-propagation property also holds true.

We do some preparation before proving this proposition. Select a neighborhood W k of ρ k ∈ G k in T * ∂Ω (and contained in W k ) with compact closure such that |∂ k-2 z1 r 1 (ρ)| ≥ c 0 > 0 for all ρ ∈ W k . By abusing the notation, we will refer G k to be G k ∩ W k in the sequel. According to the asymptotic behaviour of the flow exp(sH -r0 ) as s → 0, we have for any given (z

1 = Θ k (z 0 , ζ 0 ), z 0 , ζ 0 ) ∈ G k , r 1 • exp(sH -r0 )(z 0 , ζ 0 ) = b k (z 0 , ζ 0 )s k-2 + O(s k-1 ),
where b k = 0 can be viewed as a function of points in G k . From compactness, we can choose σ > 0, θ > 0 depending only on W k such that for all ρ ∈ G k ,

|b k (ρ)| ≥ θ > 0, |r 1 • exp(sH -r0 )(ρ)| ≥ 1 2 |b k s k-2 |, ∀s ∈ [-σ, 0) ∪ (0, σ].
Now we define a smaller neighborhood V k of ρ k such that for any ρ 0 ∈ V k , and

δ k > 0, σ k > 0, exp(sH -r0 )(L ± (δ k , δ 2 k ; ρ 0 )) ⊂ W k for all |s| ≤ σ k δ k . Moreover, |r 1 • exp(sH -r0 )(ρ 0 )| ≤ δ k . We also put W k = [0, δ 2 k ] × W k , V k = [0, δ 2 k ] × V k . Choosing a cut-off ãδ ∈ C ∞
c with ãδ ≡ 1 near ρ k , we define S δ, := supp (ã δ ) ∩ {y ≤ , r ≥ } for any 0 < δ . Note that near S δ, (thus near ρ k ∈ G k , k ≥ 3) we have |r 1 | ≤ δ k , and this implies that ζ 1 , near S δ, . We divide the proof of Proposition 8.11 into several lemmas. Lemma 8.12. Given any generalized ray γ(s) = (y(s), η(s), z(s), ζ(s)) with γ(s 0 ) ∈ Γ(ρ 0 ; δ) ∩ G 2,-and γ(s 1 ) ∈ S δ, . Assume that γ| [s0,s1] ⊂ Car(P ) ∩ W k , then γ(s) / ∈ G k for all s ∈ [s 0 , s 1 ].

Proof. Take Γ + (ρ 0 ; δ) := Γ(ρ 0 ; δ) ∩ Σ + k and identify points in Σ ± k as their projection to (y, x, ξ). Let F k (may be empty) be the union of generalized rays issued from Γ + (ρ 0 ; δ) which meet G k . Note that along both real trajectories γ(s) and exp(sH -r0 ), s → z 1 is strictly decreasing, it suffices to show that F k ∩ S δ, ⊂ Σ + k since generalized rays intersect with Σ k transversally,.

We argue by contradiction. Assume that some ray in F k satisfies γ(s 0 ) ∈ Γ + (ρ 0 ; δ), γ(0) ∈ G k , and γ(s 1 ) ∈ S δ, for s 0 < 0 < s 1 . Write exp(sH -r0 )(γ(0)) = (z(s), ζ(s)), and

r 1 • exp(sH -r0 )(z (0), ζ(0)) = r 1 (z(s), ζ(s)) = b k s k-2 + O(s k-1 ), s → 0,
More precisely, we have

|b k (z (0), ζ(0))| ≥ θ > 0, |r 1 (z(s), ζ(s))| ≥ 1 2 |b k s k-2 |, ∀s ∈ [-σ, 0) ∪ (0, σ].
After shrinking support of a δ if necessary, we may assume that s 1 < σ. According to the parity of k and the sign of b k , there are several situations. If b k < 0, then for any k we have γ(s) ∈ G 2,-for all s ∈ (0, σ). This is impossible since r •γ(s 1 ) ≥ . Otherwise b k > 0, in this case we have r 1 (z(s), ζ(s)) ≥ b k s k-2 /2, for all s ∈ (0, σ), and (8.4) thanks to ∂ ζ r| y=0 = ∂ z r| y=0 = 0. Taking the difference with real trajectory γ(s) = (y(s), η(s); z(s), ζ(s)), we have

(∂ z1 r 1 )(z(s), ζ(s)) =(∂ z1 r 1 ) • exp(sH -r0 (z (0), ζ(0)) = -∂ s (r 1 • exp(sH -r0 (z (0), ζ(0))) = -(k -2)b k s k-3 + O(s k-2 ) ≤ 0, ∀s ∈ [0, σ),
∂ z1 r 1 (y(s), z(s), ζ(s)) -∂ z1 r 1 (z(s), ζ(s)) = (∂ z1 r 1 (0, z(s), ζ(s)) -∂ z1 r 1 (z(s), ζ(s))) + (∂ z1 r 1 (z(s), ζ(s)) -∂ z1 r 1 (z(s), ζ(s))) +(∂ z1 r 1 (y(s), z(s), ζ(s)) -∂ z1 r 1 (0, z(s), ζ(s))).
Using the fact that (z(0), ζ(0)) = (z(0), ζ(0)) and y(s) = O(s 2 ), we have

∂ z1 r 1 (y(s), z(s), ζ(s)) -∂ z1 r 1 (z(s), ζ(s)) = O(s).
This together with (8.4) yield ζ1 ≤ y∂ z1 r 1 (y(s), z(s), ζ(s)) + C 0 y 2 ≤ C 0 (y 2 + ys), ẏ = 2η,

η 2 = ζ 1 + yr 1 (z, ζ) + O(y 2 ), (ζ 1 (0), y(0)) = (0, 0), (8.5)
where the constant C 0 and the implicit constant inside the big O only depends on supp(a δ ).

Applying the formula H k p y(0) = 2(H -r0 ) k-2 r 1 = 2b k (k -2)! > 0 and Taylor expansion, we have

y(s) = 2b k k(k -1) s k + O(s k+1 ) ≥ b k k(k -1) s k , s ∈ (0, σ), ẏ(s) = 2b k k -1 s k-1 + O(s k ) > b k k -1 s k-1 > 0, s ∈ (0, σ). (8.6)
Injecting in (8.5), we have ζ1 (s) ≤ C 0 ( 2 + s) for all s > 0 small as long as y(s) ≤ and γ(s) / ∈ S δ, . For these s,

ζ 1 (s) ≤ C 0 ( 2 s + s 2 /2).
Setting s 2 = inf{0 ≤ s ≤ s 1 : γ(s) ∈ S δ, }, we know that along the flow, 2 √ = 2η(s 2 ) = ẏ(s 2 ), and this implies that 1) . In summary, we have

s 2 ∼ 1 2(k-1) since y(s) > if s > 1 2(k-
≤ r • φ s2 ≤ 2C 0 1+ 1 2(k-1) + δ k + C 1 2 .
However, this contradicts to r = The goal is to show that the last two terms on the right hand side tend to 0 as h → 0. We denote by γ(s) the gliding ray exp(sH -r0 ) such that γ(s 0 ) = ρ 0 for some s 0 < 0. Suppose γ(0) = ρ ∈ G k for some k ≥ 3 and γ(s) ∈ G 2,-for s ∈ (s 0 , 0). In view of Corollary 8.9, we may assume that ρ 0 is close enough to ρ, and

ζ 1 + yr 1 + O(y 2 ), provided that δ k < 1, δ k < 1.
|s 0 | is small. Pick χ ∈ C ∞ c (R) such that χ(s) ≡ 1 if 0 ≤ s ≤ 1 2 and χ(s) ≡ 0 if s ≥ 1. For any > 0, let χ (y, x, ξ) = χ( -1 r(y, x, ξ)). Let I h, = 2 h (hd * (ϕOp h (χ )ϕ 1 u)|q) Y+ , II h, = 2 h (hd * (1 -ϕOp h (χ )ϕ 1 )u|q) Y+ .
The treatment of I h, is exactly the same as in the diffractive case, we have lim

→0 lim sup h→0 I h, = 0.
For II h, , again, we only concern about the integration over [0, ] in y variable. From Lemma 8.12, any ray entering S δ, can at most pass G j for j < k. Applying (k -1)propagation property, we deduce that for any cut-off ϕ with supp(ϕ ) ⊂ S δ, , supp(ϕ )∩ supp(µ) = ∅. Therefore lim h→0 II h, = 0 for any > 0. This completes the proof of Lemma 8.13.

Lemma 8.14. The conclusion of Proposition 8.11 holds if ρ 0 ∈ G j for some 3 ≤ j ≤ k.

Proof. Taking a micro-local cut-off ψ(y, x, ξ) with support near ρ 0 , we have

ϕOp h (ψ)ϕ 1 u L 2 (Y+) = o(1)
from the assumption that ρ 0 / ∈ supp(µ). Note that along the flow of H -r0 and on supp(1 -ψ) ∩ V k we have |r 1 (0, x, ξ)| ≥ c(ψ, δ) > 0. Hence from Corollary 8.9, if exp(tH -r0 )(ρ 0 ) ∈ G 2,-for all t ∈ (0, σδ), and then exp(tH -r0 )(ρ 0 ) / ∈ supp(µ). Otherwise exp(tH -r0 )(ρ 0 ) ∈ G 2,+ for all t ∈ (0, σδ), we claim that we still have exp(tH -r0 )(ρ 0 ) / ∈supp(µ) from geometric consideration. Indeed, by considering the backward generalized ray, we conclude that for any s 0 ∈ (0, σ k δ), there exists ρ ∈ W k , so that γ(s 0 , ρ) = exp(s 0 H -r0 )(ρ 0 ) where γ(s, ρ) is the generalized ray issued from ρ. From this fact we must have γ(s, ρ) / ∈ G k for s ∈ (0, s 0 ), since any ray intersecting with G k will enter T * Ω or G 2,-immediately, provided that the neighborhood W k is chosen to be small enough. By (k -1)propagation property, if suffices to show that ρ / ∈ supp(µ). Therefore, by definition of Γ(ρ 0 ; δ), we only need to show that

ρ ∈ {(y, z, ζ) : 0 ≤ y ≤ δ 2 , |(z, ζ) -ρ 0 | ≤ δ 2 }.
We will prove this by comparing two flows exp(sH -r0 )(ρ 0 ) = (z(s), ζ(s)) and γ(s, ρ) = (y(s), η(s), z(s), ζ(s)). Taking the difference of the two, we have Proof. Up to re-parameter the flow, we may assume that ρ 0 ∈ G k and γ(s) is the generalized ray such that γ(0) = ρ 0 . We also denote γ(s) by γ(s, ρ 0 ) in view of flow map. Suppose γ(s 0 ) / ∈supp(µ) for some s 0 < 0 and γ| [s0,0) ∩supp(µ) = ∅. Our goal is to show that ρ 0 / ∈supp(µ). Let σ k-1 > 0 be the required length in the definition of (k -1)-propagation property.

d ds (z 1 (s) -z1 (s)) = -∂ ζ1 r(y(s), z(s), ζ(s)) + ∂ ζ1 r(0, z(s), ζ(s)) = O(y(s)), d ds (z (s) -z (s)) = O(y(s)), d ds (ζ(s) -ζ(s)) = O((y (
Let

δ k > 0, σ k > 0 and V k , neighborhood of ρ 0 ∈ G k in T * ∂Ω and V k , neigh- borhood of ρ 0 in [0, 0 ] × T * ∂Ω,
as in the definition of k-pre-propagation property which satisfy the conditions in the paragraph in front of Lemma 8.12. Note in particular that we have V k ∩ G j = ∅ for all j > k. Without loss of generality, we may assume that |s 0 | < min{σ k-1 , σ k } and γ(s 0 ) ∈ V k , since otherwise we can choose s 0 < 0, |s 0 | small enough and replace γ(s 0 ) by γ(s 0 ).

Let Γ 0 ⊂ V k be a neighborhood of γ(s 0 ) so that Γ 0 ∩ supp(µ) = ∅. For δ 1 > 0 small with δ 1 σ k , we set ρ 1 = exp -σ k δ1 2 H -r0 (ρ 0 ) and define

U δ1 := {ρ = (y, η, z, ζ) ∈ Car(P ) : 0 ≤ y ≤ δ 2 1 , |(z, ζ) -ρ 1 | ≤ δ 2 1 }.
From continuous dependence of the generalized bicharacteristic flow, we have U δ1 ⊂ γ(s 0 , Γ 0 ), provided that δ 1 small enough . Now we claim that for possibly smaller δ 1 > 0, we have

γ(s 1 , U δ1 ) ∩ j≥k G j = ∅, ∀s 1 ∈ (s 0 , 0).
Indeed, it suffices to prove that γ(s 1 , U δ1 ) ∩ G k = ∅ since there are no point of G j in V k for j > k. Firstly, from the transversality of the flow exp(sH -r0 ) and Σ k , we deduce that at ρ 1 , z 1 > Θ k (z , ζ). By choosing δ 1 smaller, there exists 1 > 0, such that for all ρ ∈ U δ1 ,

z 1 > Θ k (z , ζ) + 1 holds. In particular, U δ1 ⊂ Σ + k . We calculate d ds Θ k (z (s), ζ(s)) = ∂Θ k ∂z dz ds + ∂Θ k ∂ζ dζ ds = - ∂Θ k ∂z ∂r ∂ζ + ∂Θ k ∂ζ ∂r ∂z . Note that in V k , we can write r = ζ 1 + yr 1 (z, ζ) + O(y 2 ), hence d ds Θ k (z (s), ζ(s)) = O(y(s)).
Next we argue by contradiction, assume that for some s 1 ∈ (s 0 , 0) and ρ ∈ U Combining with ż1 ∼ -1, we have

Θ k (z (s 1 ), ζ(s 1 )) ≤Θ k (z (0), ζ(0)) + C 0 s1 |s -s 1 |ds <z 1 (0) + Cs 2 1 =z 1 (s 1 ) + 0 s1 dz 1 ds ds + Cs 2 1 ≤z 1 (s 1 ) -C 1 |s 1 | + Cs 2 1 ≤z 1 (s 1 ),
provided that |s 0 | is chosen to be small enough. This implies that γ(s 1 , ρ) ∈ Σ + k , which is a contradiction.

From (k -1)-propagation property, we know that U δ1 ∩ supp(µ) = ∅. Therefore, applying k-pre-propagation property with respect to ρ 1 and U δ1 , we deduce that ρ 0 / ∈ supp(µ), and this completes the proof of Proposition (8.15).

A.Proof of Lemma 3.2

Proof of Lemma 3.2. The first assertion follows from hdivu = 0 and Dirichlet boundary condition, while we apply a multiplier method to prove the second. From the geometric assumption on Ω, we can find a vector field L ∈ C 1 (Ω) such that L| ∂Ω = ν(see [START_REF] Miranda | Introduction to exact control theory[END_REF], page 36). In global coordinate system, we write L = L j (x)∂ xj .

By using the equation, we have

Ω Lu • f dx = Ω Lu • (-h 2 ∆u -u + h∇q)dx, - Ω Lu • udx = - Ω L j (x)∂ xj u i u i dx = - Ω ∂ xj L j (x)u i u i dx + Ω div L(x)|u| 2 dx = Ω L j (x)u i (x)∂ xj u i dx + Ω div L(x)|u| 2 dx = Ω Lu • udx + Ω div L(x)|u| 2 dx,
and thus

Ω Lu • udx = - 1 2 Ω div L(x)|u| 2 dx = O(1). Next we calculate h Ω Lu • ∇qdx = -h Ω u i ∂ xj (L j ∂ xi q) dx = -h Ω u • L(∇q)dx -h Ω (div L(x))u • ∇qdx = -h Ω u • [L, ∇]qdx -h Ω div L(x)u • ∇qdx = O(1), -h 2 Ω Lu i ∆u i dx = -h 2 ∂Ω ∂ ν u i 2 dσ + h 2 Ω ∇L(∇u i , ∇u i )dx + h 2 Ω L j (x)∂ 2 xj x k u i ∂ x k u i = -h 2 ∂Ω ∂ ν u i 2 dσ + h 2 Ω ∇L(x)(∇u i , ∇u i )dx + h 2 Ω ∂ xj L j ∂ x k u i ∂ x k u i dx -h 2 Ω div L(x)∇u i • ∇u i (x)dx, h 2 Ω ∂ xj L j ∂ x k u i ∂ x k u i dx = h 2 ∂Ω L • ν ∂ ν u i 2 dσ -h 2 Ω L j (x)∂ x k u i ∂ 2 xj x k u i dx, -h 2 Ω Lu i ∆u i dx = - h 2 2 ∂Ω ∂ ν u i 2 dσ+ Ω ∇L(x)(h∇u i , h∇u i )dx- h 2 2 Ω div L(x)|∇u i | 2 dx. Observing that Ω Lu • f dx = o(1), we have ∂Ω |h∂ ν u| 2 dσ = O(1).

B.Standard elliptic theory

The differential operator is given by

P h = Op h (η 2 + λ(y, x , ξ ) 2 -1 + hm(y, x , η, ξ )),
where the principal symbol p = η 2 + λ 2 -1 is scalar while m is matrix valued. When micro-locally near the region p > 0, we want to construct the parametrix of the inverse of P . Denote by U the tubular neighborhood (two sided) of ∂Ω. Take

ϕ ∈ C ∞ c (U ), χ 0 ∈ C ∞ (R d-1
) and the support of ϕ is contained in a coordinate patch near the boundary. We put

E 0 := Op h χ 0 (ξ )ϕ(y, x ) η 2 + λ(y, x , ξ ) 2 -1 ,
and we define matrix valued PdO E l , l ≥ 1 inductively via

E 1 × p = - |α|=1 1 i ∂ α ξ ,η E 0 × ∂ α x ,y p -E 0 × m, E n × p = - |α|+k=n,k =n 1 i |α| ∂ α ξ ,η E k × ∂ α x ,y p - |α|+k=n-1 1 i |α| ∂ α ξ ,η E k × ∂ α x ,y m. (8.8)
For any N ∈ N, we set

E N = N j=0 h j E j ,
and then

E N • P = χ 0 (hD x )ϕ(y, x )Id + R N , R N L 2 →L 2 = O(h N +1 ).
Proposition 8.16. The sequence of solutions (u k ) is h k -oscillating in the following sense:

lim R→∞ lim sup k→∞ |ξ|≥Rh -1 k | ϕu k (ξ)| 2 dξ = 0 ∀ψ ∈ C ∞ c (Ω), lim R→∞ lim sup k→∞ 0 0 dy |ξ |≥Rh -1 k | ϕu k (y, ξ )| 2 dξ = 0, ∀ψ ∈ C ∞ c (Ω),
where in the second formula, the support of ϕ is contained in some local coordinate patch near the boundary, and the Fourier transform is only taken for the x direction.

Proof. We drop the subindex k in the proof. For the first formula, one can use the equation of u to obtain

(-h 2 ∆ -1)(ϕu) = g = O L 2 (1), and
|ξ|≥Rh -1 | ϕu(ξ)| 2 dξ ≤ h|ξ|≥R | g(ξ)| 2 |h 2 |ξ| 2 -1| 2 dξ ≤ C (R 2 -1) 2 .
For the second formula, it will be sufficient to show that lim [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF]. We apply the parametrix construction above with χ 0 (ξ ) = 1 -χ ξ R . Let e N (y, x , η, ξ ) be the symbol of the operator E N , which is meromorphic in η with poles η ± 0 = ±i λ 2 (y, x , ξ ) 2 -1. Moreover,

R→∞ lim sup k→∞ 1 -χ hD x R (ϕu) L 2 = 0 for some χ ∈ C ∞ c (-1, 1). We write w = u1 y≥0 , g = g1 y≥0 , v = h∂ y u| y=0 = O L 2 (y=0)
|∂ α e N (y, x , η, ξ )| ≤ C N,α R . (8.9)
We take ϕ to be a slight enlargement of ϕ such that ϕϕ = ϕ. Then

E N • ϕP w = 1 -χ hD x R (ϕw) + R N w.
From the jump formula, P w = g + hv(x ) ⊗ δ y=0 . We have

1 -χ hD x R (ϕw) = E N ϕ g + ϕhv ⊗ δ y=0 + O L 2 y,x (h N +1 ).
From symbolic calculus,

E N ( ϕ g) L 2 y,x ≤ |α|≤Cd sup (y,x ,η,ξ )∈R 2d
|∂ α e N (y, x , η, ξ )| + Ch, and it vanishes after the taking the limit h → 0 and then R → ∞, thanks to (8.9). We next write

E N ( ϕhv ⊗ δ y=0 ) = π (2πh) d-1 R 2(d-1) e i(x -z )•ξ h ω N (y, x , ξ ) ϕ(0, z )v(z )dz dξ , with ω N (y, x , ξ ) = ∞ -∞
e N (y, x , η, ξ )e iyη h dη.

To calculate ω N for y > 0, we deform the contour of integral in η in the half plane Im η > 0. From the Residue formula, we have ω N (y, x , ξ ) = 2πiRes(e N (y, x , η, ξ ); iη + 0 )e iyη + 0 h . The principal symbol of ω N is given by

π exp - yQ(y, x , ξ ) h ϕ(y, x ) 1 -χ ξ R 2Q(y, x , ξ ) , Q(y, x , ξ ) = λ(y, x , ξ ) 2 -1. Therefore lim sup h→0 E N ( ϕ2hv ⊗ δ y=0 ) L 2 y,x ≤C N,d ∞ 0 |α|≤Cd sup (x ,ξ ) |∂ α x ,ξ ω N (y, x , ξ )| v L 2 x dy ≤ C R ,
where we have used the point-wise estimate

|∂ α x ,ξ ω N (y, x , ξ )| ≤ C α e -y √ R 2 -1 2Rh √ R 2 -1 1 + y h |α| . Given χ(y, x , ξ ) ∈ C ∞ c ([0, 0 )×R 2d-2
), the following proposition can be deduced in the same manner. Proposition 8.17.

Let w k = χ(y, x , hD x )(ϕu k ), w k = 1 y≥0 u k . Then for χ 1 ∈ C ∞ c (R), 0 ≤ χ 1 ≤ 1, χ 1 = 1 near 0, we have lim R→∞ lim sup k→∞ 1 -χ 1 h k D y R w k L 2 (R d ) = 0.
Proof. We have P w

= g + hv(x ) ⊗ δ y=0 with g L 2 y,x = O(1), v L 2 x = O(1)
. Note that the functions g, v here may not coincide with the functions in the proof of Proposition 2.7. We define

E N = N j=0 h j E j + R N , R N L 2 →L 2 = O(h N +1 ), with E 0 = Op h ϕ(y, x ) 1 -χ 1 η R η 2 + λ(y, x , ξ ) 2 -1
Id and E l , l ≥ 1 as in (8.8). This implies that For this, we first notice that W is smooth and in H s (R + ) for all s ≥ 0. To prove the exponential decay, we pick θ (z) = e 2δ 0 z 1+ z with δ 0 > 0 to be chosen later. One observe easily that 0 < θ (z) ≤ 2δ 0 θ (z) for z ≥ 0. We multiply by θ W to the both sides of the equation and integrate it, then

∞ 0 (θ W ) W + θ V W 2 dz = ∞ 0 θ W Gdz.
The desired functions a 0 , a 1 are chosen to be the remainders when f is divided by p = η 2 -r(y, x, ξ), thanks to the Malgrange preparation theorem: f (y, x, η, ξ) = (η 2 -r(y, x, ξ))g(y, x, η, ξ) + a 1 (y, x, ξ)η + a 0 (y, x, ξ).

On the support of f , we observe that φ 0 (y, x, ξ) = |(y, x, ξ) -(0, x 0 , ξ 0 )| 2 ≤ 3δ, η |ξ| + φ 0 ≤ δ, which implies η ≤ δ|ξ|. Moreover, on supp(f ) ∩ supp(∂χ 2 (δ -1 φ 0 )), we have φ 0 ≥ 2δ, φ 0 + φ 1 η ≤ δ, and these imply η ≤ -δ|ξ|, hence r(y, x, ξ) = η 2 ≥ δ 2 |ξ| 2 , when p = η 2 -r = 0. Direct calculation yields

H p f + f M |ξ| + ψ 2 = χ 0 1 - φ δ H p χ 2 φ 0 δ 2 -1 -χ 1 η δ|ξ| 2 N, with N = χ 2 φ 0 δ 2 χ 0 1 - φ δ H p φ δ -χ 0 1 - φ δ M |ξ| ∈ C ∞ , ψ = χ 1 η δ|ξ| N 1/2 .
Here N ≥ 0 on supp(ψ) if we choose δ > 0 small enough. Observe that when η = r 1/2 ≥ 0, we have χ 0 (1 -δ -1 φ)H p χ 2 δ -1 φ 0 2 = 0, 1 -χ 1 η δ|ξ| 2 N = 0.

We then define a function ϕ(y, x, ξ) = -

χ 0 1 -φ δ H p χ 2 φ0 δ 2 -1 -χ 1 η δ|ξ| 2 N 2r 1/2
| η=-r 1/2 1 r(y,x,ξ)>0

and then H p f + f M |ξ| + ψ 2 = ϕ(η -r 1/2 ), when p = η 2 -r = 0.

Therefore, on p = 0, we have

H p a + aM |ξ| + ψ 2 = ϕ(η -r 1/2 ).
It is left to check the smoothness of functions ϕ, ψ and ρ. Indeed, on the support of ψ,|φ 1 η| ≤ 2δ, φ 0 ≤ 3δ, and then 1 -φ δ ≤ 3. Notice that χ0(t) χ 0 (t) = t 2 , we have

N 1/2 = χ 2 φ 0 δ χ 0 1 - φ δ H p φ δ G M |ξ|δ H p φ , 1 - φ δ ∈ C ∞ ,
since the function G(a, t) = √ 1 -at 2 ∈ C ∞ for t ≤ 3, |a| 1. This implies that ψ ∈ C ∞ , provided that δ is chosen small enough.

For ϕ, the smoothness comes from the fact that on the support of provided that δ, σ 1. In the calculation above, we have used the fact that χ(u) = O(χ (u)) on the support of g δ . Thus (5) in Lemma 8.2 follows.

χ 0 1 - φ δ H p χ 2 φ 0 δ 2 | η 2 =r -1 -χ 1 η δ|ξ| 2 N
(6) follows from the construction of χ.

To check [START_REF] Gérard | Ergodic properties of eigenfunctions for the dirichlet problem[END_REF], we observe that supp(g δ )∪supp(h δ ) ⊂supp(a δ ). Moreover, from the construction, g δ , h δ are independent of η whenever 0 ≤ y < δ 2 2 . Finally, to check the support of h δ , we write h δ = -H p β t δ 2 χ(u). Since β is independent of y, η, we have

H p β t δ 2 = H -r β t δ 2
, which is supported on I × L -(δ, δ 2 ) × R η , thanks to supp β ⊂ [-1, -

P h = -h 2 g √ G ∂ ∂y √ Gg -1 ∂ y + R h = h 2 D 2 y + Op h (r) + O L 2 →L 2 (h).
Moreover, R h is a matrix-valued second order differential operator in x with scalar principal symbol r(y, x, ξ) = 1 -λ(y, x, ξ) 2 , which is self-ajoint with respect to the

(•|•) L 2 (Y+) .
Proof. Denote by y = x 0 , ∂ 0 = ∂ y , ∂ j = ∂ xj , j = 1, 2, • • • d -1. Let u ∈ Λ 1 (Y + ) and w ∈ Λ 2 (Y + ) written in the form u = u 0 dx 0 + u j dx j , w = w 0j dx 0 ∧ dx j + w jk dx j ∧ dx k .

We have from direct calculation that du =(∂ 0 u j -∂ j u 0 )dx 0 ∧ dx j + ∂ k u j dx j ∧ dx k ,

d * u = - 1 √ G ∂ 0 (u 0 √ G) - 1 √ G ∂ j (g jk u k √ G), d * w = 1 √ G ∂ k (w 0j g jk √ G)dx 0 - 1 √ G g jl ∂ 0 (w 0k g kl √ G) + ∂ m (
√ Gw pk (g pl g km -g pm g kl ) dx j

From direct calculation, h 2 ∆ H u = h 2 (dd * + d * d)u = v 0 dx 0 + v j dx j + R h u with Now it is reduced to prove the uniform L 2 boundness of the operator

v 0 = -h 2 ∂ 2 0 u 0 -h 2 ∂ 0 ( √ G) √ G ∂ 0 u 0 , v j = -h 2 ∂ 2 0 u l -
T h u = R d K h (

  det(g)dx dy, (u|v) L 2 (Y+) := (u|v) Y+ := 0 0 X u|v det(g)dx dy, u(y, •) 2 L 2 (∂Y+) := X u(y, •)|u(y, •) det(g)dx , (u|v) L 2 (∂Y+) (y) := X u(y, •)|v(y, •) det(g)dx ,

6 . 7 .

 67 ) where b 0 is another micro-localization near z 0 with b 1 | supp(b0) = 1, and H Q b = {Q, b}. Note that the compact support of ψ(y)b ± can be chosen arbitrarily close to the semi-bicharacteristic curves γ ± corresponding to the principal symbol p. Moreover, b ± are invariant along γ ± . Under these notations, Proposition 4.3 can be rephrased as follows Proposition Let µ be the defect measure of u. If b + µ1 0<y≤ 0 = 0 (b -µ1 0<y≤ 0 = 0), then we have b -µ1 0<y≤ 0 = 0 (b + µ1 0<y≤ 0 = 0). Moreover, we have in fact b + µ = b -µ = 0 in this case.

  . The term |(ΘP h v|v) Ω | + |(Θ 1 P h v|v) Ω | = o(1) follows from the equation and symbolic calculus since the principal symbols of Θ and Θ 1 are scalar-valued. Next we claim that Re(ϕOp h (φ)ϕ 1 (hD y -Q + )v|ϕOp h (ω)ϕ 1 v) Y+ = o(1), h → 0. (7.7)

Lemma 8 . 13 . 2 j=1B δ,j u 2 L 2 ( 2 h

 8132222 The conclusion of Proposition 8.11 holds if ρ 0 ∈ G 2,- Proof. Adapting the notations and argument in the proof of Proposition 8.3, we have Y+) ≤ o(1) + Im([A δ , hd]q|u) Y+ + 1 h Im(q|hd * (A δ u)) Y+ . (8.7)

Note that |η| 2 =≤ Cσ 3 k δ 3 ≤ δ 2 .

 232 |r| = O(1) and y(s 0 ) = 0, z(s 0 ) = z(s 0 ), ζ(s 0 ) = ζ(s 0 ), we have y(s) ≤ C(s -s 0 ) 2 for all s ∈ [0, s 0 ]. Hence y(0) ≤ Cσ 2 k δ 2 < δ 2 , provided that σ 2 k < 1/C. Moreover, |(z(0), ζ(0)) -ρ 0 | ≤ Cs 3 0 This completes the proof of Lemma 8.14 as well as Proposition 8.11. Proposition 8.15. Suppose that (k-1)-propagation property holds. Then k-prepropagation property implies k-propagation property.

  δ1 we have γ(s 1 , ρ) ∈ G k and γ(s, ρ) / ∈ G k for any 0 > s > s 1 . In this case we have |y(s)| ≤ C|s -s 1 | for all s ∈ [s 1 , 0]. Therefore we must have d ds Θ k (z (s), ζ(s)) ≤ C|s -s 1 |.

1 -χ 1 ( 1 + η 2 )( 1 + y 2 E- d 2 W dz 2 +

 1121222 hD y R (ϕw) = E N ( ϕg + 2h ϕv ⊗ δ y=0 ) -R N w.Consequently, we havelim R→∞ lim sup h→0 E N ( ϕg) L 2 y,x = 0. E N (h ϕv ⊗ δ y=0 )(y, x ) = π (2πh) d-1 R 2(d-1) e i(x -z )ξ h a(y, x , ξ ) ϕ(0, z )v(z )dz dξ , with a(y, x , ξ ) = e N (y, x , η, ξ )| ≤ C α 1 -χ 1 η R N (h ϕv ⊗ δ y=0 ) L 2 y,x = 0.C.Proof of technical results in section 3Proof of Lemma 3.7. The proof can be reduced to the point-wise estimate of the solution F (y) of the ODE:-h 2 d 2 F dy 2 + λ(y) 2 F (y) = G(y), F (0) = F (0) = 0, with 0 < c 1 ≤ λ(y) 2 ≤ c 2 , G ∈ C ∞ ([0, ∞)), and |G(y)| ≤ Ce -cy h for all y ≥ 0. By rescaling z = y h , it reduces to prove the exponential decay of the solution W of the ODE: V (z)W (z) = g(z), W (0) = W (0) = 0, with 0 < c 1 ≤ V (z) ≤ c 2 , g ∈ C ∞ ([0, ∞)) and |g(z)| ≤ Ce -cz for all z ≥ 0.

1 0∂: ih 1 0C

 11 ∂ 0 (g kl √ G)∂ 0 u k and the R h u consists only the tangential derivatives ∂ j . Hence in the matrix form,v = L h u := -h 2 g easily verified that L * h = L h , thus R * h = R h . F.Proof of Lemma 8.4Proof of Lemma 8.4. For our need, it suffices to prove the last assertion. We first let A h = a(y, x, hD y , hD x ) and B h = b(y, x, hD x ), thenA h B h u(y, x) = 1 (2πh) d e i(x-x )ξ+i(y-y )η h ϕ(y , y, x, η, ξ)u(y , x )dy dx dξdη, where ϕ(y , y, x, η, ξ) = 1 (2πh) d-1 e i(x-z)(ξ -ξ) h a(y, x, η, ξ )b(y , z, ξ)dξ dz.Talor expansion gives ϕ(y , y, x, η, ξ) = ϕ(y, y, x, η, ξ) + (y -y) y ϕ(ty + (1 -t)y, y, x, η, ξ)dt.Denote by c(y, x, η, ξ) = ϕ(y, y, x, η, ξ), it is obvious that c is an interior symbol, since it can be viewed as a tangential symbol for fixed η, and we have (1 + |ξ|) m (1 + |ξ| + |η|) m for all m ∈ R on the support of c, thanks to the support property of a. Now we note C h = c(y, x, hD y , hD x ), and writeA h B h u = C h u + R h u, where R h u(y, x) = d e i(x-x )ξ+i(y-y )η h(y -y )∂ y c t (y , y, x, η, ξ)u(y , x )dy dx dξdη d e i(x-x )ξ+i(y-y )η h ∂ η ∂ y c t (y , y, x, η, ξ)u(y , x )dy dx dξdη=t u(y, x)dt, with c t (y , y, x, η, ξ) = ϕ(ty + (1 -t)y, y, x, η, ξ). Notice that ∂ η ∂ y c t (y , y, x, η, ξ) = 1 (2πh) d-1 e i(x-z)(ξ -ξ) h ∂ η a(y, x, η, ξ )(∂ y b)(ty +(1-t)y, z, ξ)dξ dz.We need to be careful here since ∂ y b only exists for y > 0 and at the point y = 0, the right derivative (∂ m y ) + b(0) := lim y→0 + ∂ m b(y) exists for any order m. Since we are dealing with Dirichlet boundary condition, we always apply a tangential operator B(y, x, hD x ) to functions u(y, x) with u| y=0 = 0 in the trace sense. We could thus extend u(y , x ) by u(y , x )1 y ≥0 in y in the expression of the form 1 (2πh) d e i(x-x )ξ +i(y-y )η h ϕ(y , y, x, η, ξ)u(y , x )dy dx dξdη. Therefore, we have sup y,y ≥0,0<t<1,z,ξ |∂ α z ∂ β ξ b(ty + (1 -t)y, z, ξ)| ≤ C m,α,β , ∀m ∈ N, α, β ∈ N d-1 .

  2,+ . Remark 2.3. The definition above does not depend on the choice of local coordinate, and in the geodesic coordinate system, the map s → (y(s), η 2 (s), x (s), ξ (s))

	is always continuous and
		s → (x (s), ξ (s))
	is always differentiable and satisfies the ordinary differential equations
	dx dt	= -	∂r ∂ξ	,	dξ dt	=	∂r ∂x	,

the map s → y(s) is left and right differentiable with derivative 2η(s ± ) for any s ∈ B (hyperbolic point).

  | η 2 =r , we have r ≥ δ 2 |ξ| 2 . Moreover, ϕ has compact support.Finally, from the definition of a, we have Finally we defineh δ = -H p a δ -g δ . Note that supp (a δ ) = (y, s, t) : -δ 2 ≤ t,hence it is clear that (1)(2)(3)(4) in Lemma 8.2 are satisfied. Since r = r 0 + O(y), when H -r acts on functions independent of η we haveH p = ∂ t + O(y)∂ s + O(y)∂ t + O(δ)∂ y ,due to the bound |η| = O(δ). Therefore, we have

	a 1 (y, x, ξ) = 3t 4σδ + |s| 2 δ 4 + f -g δ = β t δ 2 1 δσ = β t δ 2 3 4δσ a with u = ∼ β t δ 2 1 δσ χ (u), f (y, x, η, ξ) -f (y, x, -η, ξ) 2η y δ 2 . t σδ + |s| 2 | η= √ δ 4 + f r(y,x,ξ) ∈ C ∞ c , y δ 2 ≤ 3 4 , + O(δ 2 ) |s| δ 4 + O(δ 2 ) 1 δσ + O(δ) 1 δ 2 χ (u) + O(1)χ(u) + O δ σ + O 1 δ + O(1)))χ (u)

0 (y, x, ξ) = f (y, x, η, ξ) + f (y, x, -η, ξ) 2 | η=± √ r(y,x,ξ) ∈ C ∞ c .

  y , x , y, x)u(y , x )dy dx , Notice that H t (y , y, x, η, ξ) is a tangential symbol, parametrized by (y , y, η). Moreover, it is compactly supported in (y, x, η, ξ) variables, uniformly in the first variable y . Thus, ∂ m η ∂ α ξ H t (y -hw, y, x, η, ξ) has compact support in (η, ξ) and|∂ m η ∂ α ξ H t (y -hw, y, x, η, ξ)| ≤ C m,αfor any m ∈ N and α ∈ N d-1 . Thus, doing integration by part in the expression of k h , we have sup (y,x) |k h (y, x, w, v)| ≤ C(1 + |w| + |v|) -(d+1) .

	with kernel K h (y , x , y, x) = where H t (y , y, x, η, ξ) = 1 y ,y≥0 From Schur's test, we need to show 1 (2πh) d R d 1 (2πh) d-1 sup (y,x)∈R d + R d To this end, we define k h (y, x, w, v) := 1 (2π) d R d hence, T h u(y, x) = 1 h d R d + k h y, x, e i(x-x )ξ+i(y-y )η h e e ivξ+iwη H t (y -hw, y, x, η, ξ)dηdξ, H t (y , y, x, η, ξ)dηdξ, y -y h , x -x h u(y , x )dy dx . Therefore, we obtain R d + |K h (y , x , y, x)| dy dx = 1 h d R d + k h y, x, y -y h , x -x h dy dx = R d + |K h (y, x, y , x )|dydx = 1 h d R d + k h y, x, y -y h , x -x h dydx ≤ R d sup (y,x) |k h (y, x, w, v)|dwdv |k and ≤ C 2 .
	R d

izζ h a 1 (y, x, η, ξ+ζ)b 1 (ty +(1-t)y, x-z, ζ)dzdζ. + |K h (y , x , y, x)|dy dx ≤ C 1 < ∞, sup (y ,x )∈R d + R d + |K h (y , x , y, x)|dydx ≤ C 2 < ∞, with C 1 , C 2 independent of h and t. h (y, x, w, v)|dwdv ≤ C 1 ,

Acknowledgement. This work is supported by the European Research Council, ERC-2012-ADG, project number 320845: Semi classical Analysis and Partial Differential Equations. It is a part of Phd thesis of the author. He would like to thank his advisor, Gilles Lebeau, for his encouragement and many fruitful discussions.

Notice that θ W W ≥ -2δ 0 θ |W ||W | ≥ -δ 0 θ (|W | 2 + |W | 2 ), by choosing

we have that

thanks to Cauchy-Schwartz and the fact that θ ≤ e 2δ0z , uniformly in . From the dominating convergence theorem, we have W e δ0z ∈ L 2 (R + ) and W e δ0z ∈ L 2 (R + ). Finally, by elliptic regularity, we have that W e δ0z ∈ L ∞ (R + ),W e δ0z ∈ L ∞ (R + ).

Proof of Proposition 3.8. We choose

We first claim that

Indeed, we can write

and χ δ0 has similar support property as χ δ0 . Thus from symbolic calculus, we have for each fixed y > 0,

thanks to Lemma 3.5. Integrating over y > 0 yields (8.10). By taking supp(χ δ0 ) small such that ϕ 2 χ δ0 = χ δ0 , we have that

. The trace of w satisfies

and w satisfies the equation (we use ϕ 2 = ϕ 1 ϕ 2 here)

. Multiply by w = ϕ 1 w to the both sides of (8.11) and integrate it, we have

w(y, x )P 0 wdx dy = O(h).

Doing integration by part for the left hand side, we have

This implies that

). Using this smallness and redoing the integration by part, we can improve each bound in the procedure above and obtain that

). To conclude, we observe that

where we have used symbolic calculus and Lemma 3.5 several times. Plugging into (8.11), we have that

We decompose w = w 1 + w 2 with P 0 w 1 = P 0 w, w 1 | y=0 = 0 and P 0 w 2 = 0, w 2 | y=0 = w| y=0 . From elliptic regularity of boundary value problem, we have 4 ). Observe that the error terms on the right hand side of (8.12) can be also bounded by

) by interpolation. This completes the proof.

D.Construction of test functions

We first give the detailed construction of a = a 0 + a 1 η used in the first step of the proof of Proposition7.6, which follows closely to [START_REF] Hörmander | The analysis of linear partial differential operators[END_REF].

Consider the smooth functiiton χ 0 (t) = e -1/t 1 t>0 . We work in the local coordinate (y, x, ξ), and assume that (0, x 0 , ξ 0 ) ∈ G 2,+ with

We calculate

provided that |η| ≤ c 0 for some c 0 > 0 and W 0 is chosen small enough such that ∂r ∂y ≥ 4c on it. The positivity then follows from the direct calculation:

We next take

we deduce that ∂f ∂η = -φ1 δ χ 2 φ0 δ

1/2 , one can show that t is a smooth function with compact support. The last observation is that a = f > 0 on p = 0, hence

We give some more calculations: Let ψ 0 = ψ| y=0 , ψ 1 = ∂ψ ∂η | y=0 , when η = r = 0. Thus at (x 0 , ξ 0 ),

Observe that near (x 0 , ξ 0 ), we have ψ1 ψ0 ∼ -

, provided that δ is small enough. Now if we make a different choice of δ > 0, the difference between two ratios ψ1 ψ0 and ψ1 ψ0 is non-zero. This implies that we can choose a further cut-off χ near (0, x 0 , ξ 0 ) such that ϕOp h (χ)ϕ 1 u L 2 (Y+) = o(1) and ϕOp h (χ)hD y ϕ

Next we recall the proof of Lemma 8.2, which is essentially given in [START_REF] Melrose | Singularities of boundary value problems[END_REF].

Proof of Lemma 8.2. : From the transversality, we can choose a new coordinate (s, t) in U such that ρ 0 = (0, 0) and H -r0 = ∂ t in this coordinate.

Step 1. Consider the function χ(u) = e 1 u-3/4 1 u<3/4 . It is easy to check that χ is smooth and non-increasing with the property:

Step 2. Next we choose β ∈ C ∞ (R) such that β ≥ 0 vanishing on (-∞, -1) and strictly increasing on (-1, - 1 2 ), equaling to 1 on (-1 2 , ∞). We modify β such that ∂ N β = O(β 1/m ), ∀N ∈ N, m > 1, locally uniformly.

Step 3. Choose f ∈ C ∞ (R) so that f vanishes on (-∞, 1/2) and is strictly increasing and convex on (1/2, ∞) with f (1) > 1.

Now we set a δ = β 3t 4δ