# Uniform stabilization of an acoustic system

Kaïs AMMARI  LR Analysis and Control of PDEs, LR 22ES03, Department of
Mathematics, Faculty of Sciences of Monastir, University of Monastir, 5019
Monastir, Tunisia and LMV/UVSQ/Paris-Saclay, France<EMAIL_ADDRESS>,
Fathi Hassine  LR Analysis and Control of PDEs, LR 22ES03, Department of
Mathematics, Faculty of Sciences of Monastir, University of Monastir, 5019
Monastir, Tunisia<EMAIL_ADDRESS>and  Luc ROBBIANO  Université
Paris-Saclay, UVSQ, CNRS, Laboratoire de Mathématiques de Versailles, 78000,
Versailles, France<EMAIL_ADDRESS>
###### Abstract.

We study the problem of stabilization for the acoustic system with a spatially
distributed damping. With imposing hypothesis on the structural properties of
the damping term, we identify exponential decay of solutions with growing
time.

###### Key words and phrases:

exponential stability, resolvent estimate, dissipative hyperbolic system,
acoustic equation

###### 2010 Mathematics Subject Classification:

35L04, 37L15

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries and main results
  3. 3 Tools
    1. 3.1 Riemannian geometry tools
    2. 3.2 Pseudo-differential tools
      1. 3.2.1 Composition
      2. 3.2.2 Commutator
      3. 3.2.3 Adjoint
      4. 3.2.4 Operators bounded on $L^{2}$
      5. 3.2.5 Gårding inequality
    3. 3.3 Uniform stabilization
    4. 3.4 Invariant form of acoustic system
  4. 4 Introduction of a semiclassical measure
    1. 4.1 Semiclassical measure
      1. 4.1.1 Hyperbolic region
      2. 4.1.2 Glancing region
      3. 4.1.3 Elliptic region
      4. 4.1.4 End of proof
    2. 4.2 The measure is not identically null
    3. 4.3 The semiclassical measure null on support of damping
  5. 5 Propagation of the support
    1. 5.0.1 Computations in interior
    2. 5.0.2 Limit on quantities on $\Omega$.
    3. 5.0.3 Computations at boundary, tangential symbol
    4. 5.0.4  Computations at boundary, tangential symbol with normal derivative
  6. A Geometric Control Condition and propagation
    1. A.1 Geometry
    2. A.2 Geometric Control Condition (GCC)
    3. A.3 Properties of measure
    4. A.4 Proof of Proposition 5.1
      1. A.4.1 Propagation in interior domain $\Omega$
      2. A.4.2 Propagation at boundary: hyperbolic points
      3. A.4.3 Propagation at gliding points
      4. A.4.4 Propagation at diffractive points
      5. A.4.5 Propagation at boundary: integral curves with high contact order
  7. B Estimate of boundary trace, case strictly diffractive
    1. B.1 Review on OIF and Airy functions
    2. B.2 Estimates at the boundary, glancing points

##  1\. Introduction

Sound waves propagate due to the compressibility of a medium. In fact, it is a
balance between the compressibility and the inertia of fluid that governs the
propagation of sound waves through it. The assumptions made for the acoustic
wave equation are gravitational forces is neglected, dissipative effects are
neglected, the medium (fluid) is homogeneous, isotropic and perfectly elastic.
In addition to these assumptions we also assume that particle velocity is
small, and there are only very small perturbations (fluctuations) to the
equilibrium pressure and density.

We consider the following acoustic damping system of equations:

(1.1) $\left\\{\begin{array}[]{l}u_{t}+\nabla r+b\,u=0,\,\hbox{ in
}\Omega\times{\mathbb{R}}^{+},\\\ r_{t}+\operatorname{div}u=0,\,\hbox{ in
}\Omega\times{\mathbb{R}}^{+},\\\ u\cdot n=0,\,\hbox{ on
}\Gamma\times{\mathbb{R}}^{+},\\\
u(0,x)=u^{0}(x),\,r(0,x)=r^{0}(x),\,x\in\Omega,\end{array}\right.$

where $\Omega$ is a bounded domain in ${\mathbb{R}}^{d},\,d\geq 2$, with a
smooth boundary $\Gamma$, $\operatorname{div}=\nabla\cdot$ is the divergence
operator and $b\in L^{\infty}(\Omega)$, with $b\geq 0$ on $\Omega$. As usual
$n$ denotes the unit outward normal vector along $\Gamma$.

The system of equations (1.1) is a linearization of the _acoustic equation_
governing the propagation of acoustic waves in a compressible medium, see
Lighthill [33, 34, 35], where $b\,u$ represents a damping term of Brinkman
type. This kind of damping arises also in the process of homogenization (see
Allaire [1]), and is frequently used as a suitable _penalization_ in fluid
mechanics models, see Angot, Bruneau, and Fabrie [5]. Our main goal is to
prove the exponential decay of solutions of (1.1) with growing time.

Let $L^{2}(\Omega)$ denote the standard Hilbert space of square integrable
functions in $\Omega$ and its closed subspace $L^{2}_{m}(\Omega)=\\{f\in
L^{2}(\Omega):\int_{\Omega}f(x)\,dx=0\\}$. To avoid abuse of notation, we
shall write $\|\cdot\|$ for the $L^{2}(\Omega)$-norm or the
$L^{2}(\Omega)^{d}$-norm.

Denoting $H=(L^{2}(\Omega))^{d}\times L^{2}_{m}(\Omega)$, we introduce the
operator

${\mathcal{A}}=\left(\begin{array}[]{ll}0&\nabla\\\
\operatorname{div}&0\end{array}\right):{\mathcal{D}}({\mathcal{A}})=\left\\{(u,r)\in
H,\,(\nabla r,\operatorname{div}u)\in H,\,u\cdot n_{|\Gamma}=0\right\\}\subset
H\rightarrow H,$

and

${\mathcal{B}}=\left(\begin{array}[]{ll}\sqrt{b}\\\
0\end{array}\right)\in{\mathcal{L}}((L^{2}(\Omega))^{d},H),\,{\mathcal{B}}^{*}=\left(\begin{array}[]{cc}\sqrt{b}&0\end{array}\right)\in{\mathcal{L}}(H,(L^{2}(\Omega))^{d}).$

We recall that for $u\in(L^{2}(\Omega))^{d}$ with $\operatorname{div}u\in
L^{2}(\Omega)$, $u\cdot n_{|\Gamma}$ make sens in $H^{-1/2}(\Gamma)$ (see
Girault-Raviart [21, Chap 1, Theorem 2.5]).

Accordingly, the problem (1.1) can be recasted in an abstract form:

(1.2)
$\left\\{\begin{array}[]{l}Z_{t}(t)+{\mathcal{A}}Z(t)+{\mathcal{B}}{\mathcal{B}}^{*}Z(t)=0,\,t>0,\\\
Z(0)=Z^{0},\end{array}\right.$

where $Z=(u,r)$, or, equivalently,

(1.3) $\left\\{\begin{array}[]{l}Z_{t}(t)={\mathcal{A}}_{d}Z(t),\,t>0,\\\
Z(0)=Z^{0},\end{array}\right.$

with ${\mathcal{A}}_{d}=-{\mathcal{A}}-{\mathcal{B}}{\mathcal{B}}^{*}$ with
${\mathcal{D}}({\mathcal{A}}_{d})={\mathcal{D}}({\mathcal{A}}).$

It can be shown (see [2]) that for any initial data
$(u^{0},r^{0})\in{\mathcal{D}}({\mathcal{A}})$ the problem (1.1) admits a
unique solution

$(u,r)\in C([0,\infty);{\mathcal{D}}({\mathcal{A}}))\cap C^{1}([0,\infty);H).$

Moreover, the solution $(u,r)$ satisfies, the energy identity

(1.4)
$E(0)-E(t)=\int_{0}^{t}\left\|\sqrt{b}\,u(s)\right\|_{(L^{2}(\Omega))^{d}}^{2}{\rm
d}s,\ \mbox{for all}\ t\geq 0$

with

(1.5) $E(t)=\frac{1}{2}\,\left\|(u(t),r(t))\right\|^{2}_{H},\,\forall\,t\geq
0,$

where we have denoted

$\left\langle(u,r),(v,p)\right\rangle_{H}=\int_{\Omega}\left(u(x).v(x)+r(x)p(x)\right)\,\mathrm{d}x,\,\left\|(u,r)\right\|_{H}=\sqrt{\int_{\Omega}\left(\left|u(x)\right|^{2}+r^{2}(x)\right)\,\mathrm{d}x}.$

Using (1.4) and a standard density argument, we can extend the solution
operator for data $(u^{0},r^{0})\in H$. Consequently, we associate with the
problem (1.1) (or to the abstract Cauchy problems (1.2) or (1.3)) a semigroup
that is globally bounded in $H$.

As the energy $E$ is nonincreasing along trajectories, we want to determine
the set of initial data $(u^{0},r^{0})$ for which

(1.6) $E(t)\to 0\ \mbox{as}\ t\to\infty.$

Such a question is of course intimately related to the structural properties
of the function $b$, notably to the geometry of the set $\omega$ on which the
damping is effective. In fact, when the damping term is globally distributed
Ammari, Feireisl and Nicaise [2] showed an exponential decay rate of the
energy by the means of an observability inequality associated with the
conservative problem of (1.1). Besides, it is also shown that if the damping
coefficient is not uniformly positive definite (i.e
$\displaystyle\inf_{x\in\Omega}b(x)=0$) then the system (1.1) may be no
exponentially stable. In this paper we consider a damping which is locally
distributed over the domain $\Omega$ with a geometrical control conditions. In
this article we prove uniform decay as given in [2] but under more restrictive
geometrical conditions on the damping coefficient. More precisely, we prove
exponential decay rates of the energy.

The paper is organized as follows. Section 2 summarizes some well known facts
concerning the acoustic system (1.1). In Section 3, we introduce some tools
and notation as Riemannian geometry, pseudo-differential operators, and
Gearhart-Huang-Prüss theorem allowing to prove exponential decay from
estimates on the resolvent. In Section 4 we introduce a semi-classical measure
and we prove some properties, first the measure is supported on characteristic
set and is not identically null. In Section 5 we prove local properties in a
neighborhood of boundary. From these results we prove in Section A global
propagation of measure support. In Section B we prove an estimate in a
neighborhood of a diffractive point. Section 5, A and B imply that the measure
is null if $\\{x,b(x)>0\\}$ satisfies a geometric condition (GCC)(see
Definition 2.1 and Section A.1, Section A.2 for more precise definition).

##  2\. Preliminaries and main results

We start with a simple observation that the problem (1.1) can be viewed as a
bounded (in $H$) perturbation of the conservative system

(2.7) $\left\\{\begin{array}[]{l}u_{t}+\nabla r=0,\,\hbox{ in
}\Omega\times{\mathbb{R}}^{+},\\\ r_{t}+\operatorname{div}u=0,\,\hbox{ in
}\Omega\times{\mathbb{R}}^{+},\end{array}\right.$

which can be recast as the standard _wave equation_

$r_{tt}-\Delta r=0.$

Consequently, the basic existence theory for (1.1) derives from that of (2.7).
Hence ${\mathcal{A}}_{d}$ generates a $C_{0}$-semigroup $(S(t))_{t\geq 0}$ in
$H$ that is even of contraction because ${\mathcal{A}}_{d}$ is dissipative
(see (1.4)).

The first main difficulty is that the operator ${\mathcal{A}}_{d}$ possesses a
non-trivial (and large) kernel that is left invariant by the evolution. Indeed
if $(u,r)$ belongs to $\ker{\mathcal{A}}_{d}$, then it is solution of the
“stationary” problem

(2.8) $\nabla r+bu=0,\ \operatorname{div}u=0,\,\hbox{ in }\Omega.$

Thus multiplying the first identity of (2.8) by $\overline{u}$ and integrating
over $\Omega$ yields

$\int_{\Omega}(\nabla r\cdot\overline{u}+b|u|^{2})\,\mathrm{d}x=0.$

By an integration by parts, using the fact that $u$ is solenoidal and the
boundary condition $u\cdot n=0$ on $\Gamma$, we get

$\int_{\Omega}\nabla
r\cdot\overline{u}\,\mathrm{d}x=\int_{\Omega}r\operatorname{div}\overline{u}\,\mathrm{d}x=0,$

and therefore we obtain

$\int_{\Omega}b|u|^{2}\,\mathrm{d}x=0.$

In other words, we have

$u=0\ \mbox{on}\ \operatorname{supp}b,$

and coming back to (2.8), we find

$\nabla r=0.$

Accordingly, we have shown that

$\ker{\mathcal{A}}_{d}=\\{(u,0)\in{\mathcal{D}}({\mathcal{A}})\ |\
\operatorname{div}u=0,\ u|_{\operatorname{supp}b}=0,\ u\cdot
n|_{\Gamma}=0\\}.$

For shortness set $E=\ker{\mathcal{A}}_{d}$ and introduce also its orthogonal
complement $H_{0}$ in $H$.

It is easy to check that

$\left<{\mathcal{A}}_{d}(w,s),(u,r)\right>_{H}=0\ \mbox{for any}\
(w,s)\in{\mathcal{D}}({\mathcal{A}}),\ (u,r)\in E;$

in particular, the semigroup associated with (1.1) leaves both $E$ and $H_{0}$
invariant. Consequently, the decay property (1.6) may only hold for initial
data emenating from the set $H_{0}$.

The following observation can be shown by a simple density argument:

######  Lemma 2.1.

The solution $(u,r)$ of (1.1) with initial datum in
${\mathcal{D}}(\mathcal{A}_{d})$ satisfies

(2.9) $E^{\prime}(t)=-\int_{\Omega}b\left|u\right|^{2}dx\leq 0.$

Therefore the energy is non-increasing and (1.4) holds for all initial datum
in $H$.

As already shown in the above, the strong stability result (1.6) may hold only
if we take the initial data

$(u^{0},r^{0})\in H_{0}=\ker[{\mathcal{A}}_{d}]^{\perp}.$

There are several ways how to show (1.6), here we make use of the following
result due to Arendt and Batty [6]:

######  Theorem 2.1.

Let $(T(t))_{t\geq 0}$ be a bounded $C_{0}$-semigroup on a reflexive Banach
space $X$. Denote by $A$ the generator of $(T(t))$ and by $\sigma(A)$ the
spectrum of $A$. If $\sigma(A)\cap i\mathbb{R}$ is countable and no eigenvalue
of $A$ lies on the imaginary axis, then
$\displaystyle\lim_{t\rightarrow+\infty}T(t)x=0$ for all $x\in X$.

In view of this theorem we need to identify the spectrum of
${\mathcal{A}}_{d}$ lying on the imaginary axis, and we have according to [2]:

  * •

Suppose that $|\omega|>0$. If $\lambda$ is a non-zero real number, then
$i\lambda$ is not an eigenvalue of ${\mathcal{A}}_{d}$.

  * •

Suppose that $|\omega|>0$. If $\lambda$ is a non-zero real number, then
$i\lambda$ belongs to the resolvent set $\rho({\mathcal{A}}_{d})$ of
${\mathcal{A}}_{d}$.

Now, Theorem 2.1 leads to

######  Corollary 2.1 ([2]).

Let $(u,r)$ be the unique semigroup solution of the problem (1.1) emanating
from the initial data $(u^{0},r^{0})\in H$. Let $P_{E}$ be the orthogonal
projection onto the space $E=\ker[{\mathcal{A}}_{d}]$ in $H$, and let

$(w,s)=P_{E}(u^{0},r^{0}).$

Then

$\|(u,r)(t,\cdot)-(w,s)\|_{H}\to 0\ \mbox{as}\ t\to\infty$

Next we give uniform decay result under the Geometric Control Condition (GCC).
A precise definition is given in Definition A.2 and this requires a
description of geometry. This is done in Section A. In the following a
definition for flat Laplace operator is given.

######  Definition 2.1.

Let $\Omega$ be a smooth open domain in $\mathbb{R}^{d}$. Let $\omega$ be an
open subset of $\Omega$. We assume the geodesic associated with $\Delta$ are
not tangent to infinity order to the boundary. We say that $\omega$ satisfies
(GCC) if every generalized bicharacteristics starting from $\Omega$ meet
$\omega$.

######  Remark 2.1.

We give in Annex A.1 a precise description of the generalized
bicharacteristics. Here we give an idea of this notion in case of flat Laplace
operator. In $\Omega$ a generalized bicharateristic is an integral curve of
$H_{p}$, it is a straight line for $p(\xi)=|\xi|^{2}$. When such a curve hit
the boundary, if the curve and the boundary are transverse, the generalized
bicharacteristic is extended following the law of reflection in geometrical
optics. If the curve is tangent to the boundary, essentially either the curve
stays in $\Omega$ except at intersection between curve and boundary and the
generalized bicharacteristic is the curve or the curve go out $\Omega$ and the
generalized bicharacteristic glide on boundary.

We now state the main results of this article.

######  Theorem 2.2.

Let $(u,r)$ be the unique semigroup solution of the problem (1.1) emanating
from the initial data $(u^{0},r^{0})\in H$. Let $P_{E}$ be the orthogonal
projection onto the space $E=\ker[{\mathcal{A}}_{d}]$ in $H$, and let
$(w,s)=P_{E}(u^{0},r^{0})$. We assume that the interior of the support of $b$,
$\omega=\left\\{x\in\Omega;b(x)>0\right\\}$, satisfies the geometric control
condition (GCC). Then,

(2.10) $\|(u,r)(t,\cdot)-(w,s)\|_{H}\leq
C\,e^{-ct}\,\|(u^{0},r^{0})\|_{H},\,\forall\,t\geq 0,$

for some $C,c>0$ independent of $(u^{0},r^{0})$.

Before proving the previous theorem, we need some tools in Riemannian
geometry, pseudo-differential calculus. Our goal is to apply characterization
of exponential decay by an estimate of the resolvent on imaginary axis. To
prove such an estimate we use semiclassical measure. In next section we gives
all the tools used in main part of the proof.

##  3\. Tools

In this section we give some tools on Riemannian geometry, pseudo-differential
calculus, stabilization and invariant form of acoustic system. If theses tools
are classical, this introduction allows us to fix the notation used in this
article.

###  3.1. Riemannian geometry tools

In this section we introduce notation used in this article. Let $M$ be a
manifold and let $g$ be a metric on $M$, $g$ is a inner product acting on the
vector fields. In local coordinates, $g=(g_{jk})_{1\leq j,k\leq d}$, where $g$
is symmetric, i.e. $g_{jk}=g_{kj}$ and $g$ depends on $x\in M$. If
$v=(v^{j})_{1\leq j\leq d}$ and $w=(w^{j})_{1\leq j\leq d}$ are vector fields,
for each $x\in M$ the inner product is given by $g_{jk}(x)v^{j}(x)w^{k}(x)$,
where we use here and in what follows, the Einstein notation i.e.
$g_{jk}v^{j}w^{k}=\displaystyle\sum_{1\leq j,k\leq d}g_{jk}v^{j}w^{k}$. We
denote by $(.,.)_{g}$ the $L^{2}$ inner product on $L^{2}(M)$. The local form
depends on the objects used, in what follows the inner product is considered
on functions or on vector fields. In local coordinates, if $f$ and $q$ are
functions on $M$,

(3.11) $(f,q)_{g}=\int_{M}f(x)\overline{q(x)}|g|^{1/2}dx,$

where $|g|=|\det g|$. In fact $|g|^{1/2}dx$ is the local form of the
Riemannian density associated with the metric $g$. It is a geometric object,
invariant under change of variables.

If $v$ and $w$ are vector fields, in local coordinates, we have

(3.12) $(v,w)_{g}=\sum_{1\leq j,k\leq
d}\int_{M}g_{jk}v^{j}(x)\overline{w^{k}(x)}|g|^{1/2}dx.$

In what follows, we denote some time $(v^{j},v^{k})_{g}$ instead of
$(v,w)_{g}$, in particular when for instance $v^{j}$ is given by a formula for
each $j$. The notation $(v^{j},v^{k})_{g}$ must be interpreted as the right
hand side of (3.12).

We denote by $(g^{jk})_{1\leq j,k\leq d}$ the inverse of $g$ and we have
$g^{jk}g_{k\ell}=\delta_{\ell}^{j}$, where $\delta_{\ell}^{j}$ is the
Kronecker symbol.

For a function $f$, we can define its covariant derivative given, in local
coordinates, by the following formula

(3.13) $(\nabla_{g}f)^{j}=g^{jk}\partial_{k}f,$

and $\nabla_{g}f$ is a vector field.

For a vector field $v$, we can define its divergence given, in local
coordinates, by the following formula

(3.14) $\operatorname{div}_{g}v=|g|^{-1/2}\partial_{k}(|g|^{1/2}v^{k}).$

The operators $\nabla_{g}$ and $\operatorname{div}_{g}$ are dual. If $f$, a
function, $v$, a vector field, one of both supported in interior of $M$, we
have the formula

(3.15) $(f,\operatorname{div}_{g}v)_{g}=-(\nabla_{g}f,v)_{g}.$

To study the problem in a neighborhood of boundary, we can use geodesic
coordinates. In these coordinates, the boundary is locally defined by
$x_{d}=0$, the domain by $x_{d}>0$, and the metric takes the following special
form, $g_{jd}=0$ for $1\leq j\leq d-1$, and $g_{dd}=1$. We denote by
$\tilde{g}_{jk}=g_{jk}$ for $1\leq j,k\leq d-1$, and $(\tilde{g}^{jk})_{1\leq
j,k\leq d-1}$ the inverse of $(\tilde{g}_{jk})_{1\leq j,k\leq d-1}$. It is the
restriction of the metric $g$ on the hypersurfaces $x_{d}=constant$. If we
write, for instance $\tilde{g}^{jk}\xi_{k}$ means the sum is on $k$ between
$1$ and $d-1$.

If $f$, a function, and $v$, a vector field, are supported in a neighborhood
of the boundary, the integration by parts formula takes the form

(3.16)
$(f,\operatorname{div}_{g}v)_{g}=-(\nabla_{g}f,v)_{g}-(f_{|x_{d}=0},v^{d}_{|x_{d}=0})_{\partial},$

where
$(q,r)_{\partial}=\int_{\mathbb{R}^{d-1}}q(x^{\prime})\overline{r(x^{\prime})}|g|_{|x_{d}=0}^{1/2}dx^{\prime}$.
Observe that in geodesic coordinates, the exterior unit normal at the boundary
is the vector $\nu=(0,\dots,0,-1)$, and $-v^{d}$ is the inner product between
$\nu$ and $v$. In fact we can give an invariant form to the integration
formula given above but we do not use this invariant form in this article. We
also use in local coordinates the following integration by parts formula.

(3.17)
$(w_{1},hD_{d}w_{2})=(hD_{d}w_{1},w_{2})-ih((w_{1})_{|x_{d}=0},(w_{2})_{|x_{d}=0})_{\partial}.$

In fact we can give an invariant form to the integration formula given above
but we do not use this invariant form in this article.

###  3.2. Pseudo-differential tools

We recall in this section the notation and main definitions on pseudo-
differential operators and semi-classical calculus. Let
$a(x,\xi^{\prime})\in\mathscr{C}(\mathbb{R}^{d}\times\mathbb{R}^{d-1})$, we
say $a\in S^{m}$ if for all $\alpha\in\mathbb{N}^{d}$ and
$\beta\in\mathbb{N}^{d-1}$ there exist $C_{\alpha,\beta}$ such that

(3.18)
$|\partial_{x}^{\alpha}\partial_{\xi^{\prime}}^{\beta}a(x,\xi^{\prime})|\leq
C_{\alpha,\beta}\langle\xi^{\prime}\rangle^{m-|\beta|},\text{ for all
}x\in\mathbb{R}^{d},\xi^{\prime}\in\mathbb{R}^{d-1},$

where $\alpha=(\alpha_{1},\dots,\alpha_{d})$,
$|\alpha|=\alpha_{1}+\cdots+\alpha_{d}$,
$\partial_{x}^{\alpha}=\partial_{x_{1}}^{\alpha_{1}}\cdots\partial_{x_{d}}^{\alpha_{d}}$,
and $\langle\xi^{\prime}\rangle=(1+|\xi^{\prime}|^{2})^{1/2}$.

An operator is associated with a symbol by the following formula

(3.19)
$\operatorname{op_{sc}}(a)u=\frac{1}{(2\pi)^{d-1}}\int_{\mathbb{R}^{d-1}}e^{ix^{\prime}\xi^{\prime}}a(x,h\xi^{\prime})\hat{u}(\xi^{\prime},x_{d})d\xi^{\prime},$

where $u$ is in Schwartz space, and

$\hat{u}(\xi^{\prime},x_{d})=\int_{\mathbb{R}^{d-1}}e^{-ix^{\prime}\xi^{\prime}}u(x^{\prime},x_{d})dx^{\prime}.$

The symbolic calculus is given by the following results. Let $a\in S^{m}$ and
$b\in S^{q}$.

####  3.2.1. Composition

There exists $c\in S^{m+q}$ such that
$\operatorname{op_{sc}}(a)\operatorname{op_{sc}}(b)=\operatorname{op_{sc}}(c)$
and there exists $c_{m+q-1}\in S^{m+q-1}$ such that $c=ab+hc_{m+q-1}.$

####  3.2.2. Commutator

There exists $d\in S^{m+q-1}$ such that
$[\operatorname{op_{sc}}(a),\operatorname{op_{sc}}(b)]=h\operatorname{op_{sc}}(d)$
and there exists $d_{m+q-2}\in S^{m+q-2}$ such that
$d=-i\\{a,b\\}+hd_{m+q-2}$.

We also have an exact formula if $a$ is a differential operator, and we shall
use the following formula
$[-ih\partial_{d},\operatorname{op_{sc}}(b)]=-ih\operatorname{op_{sc}}(\\{\xi_{d},b\\})$.
Observe that $\\{\xi_{d},b\\}=\partial_{d}b$ which is a tangential symbol.

####  3.2.3. Adjoint

There exists $\tilde{c}\in S^{m}$ such that
$\operatorname{op_{sc}}(a)^{*}=\operatorname{op_{sc}}(\tilde{c})$ and there
exists $\tilde{c}_{m-1}\in S^{m-1}$ such that $\tilde{c}=\bar{a}+hc_{m-1}$.

####  3.2.4. Operators bounded on $L^{2}$

If $a\in S^{0}$ then there exists $C>0$ such that
$\|\operatorname{op_{sc}}(a)u\|_{L^{2}}\leq C\|u\|_{L^{2}}$ for all $u$ is in
Schwartz space. Observe from this formula we can extend by density
$\operatorname{op_{sc}}(a)$ to an bounded operator on $L^{2}$. We keep the
same notation if $\operatorname{op_{sc}}(a)$ act on a function depending only
on $x^{\prime}$. In this case the estimate take the following form
$|\operatorname{op_{sc}}(a)u|_{L^{2}}\leq C|u|_{L^{2}}$. When no confusion is
possible, we shall write $\|.\|$ instead of
$\|.\|_{L^{2}}=\|.\|_{L^{2}(\mathbb{R}^{d})}$ and $|.|$ instead of
$|.|_{L^{2}}=|.|_{L^{2}(\mathbb{R}^{d-1})}$.

The constant $C$ in the estimation of $L^{2}$-norm depends on a finite numbers
of semi-norms of the symbol $a$. We have a precise result for $h$ sufficiently
small.

Let $a\in S^{0}$, there exists $C>0$ depending on a finite numbers of semi-
norms of the symbol $a$ such that

(3.20) $\|\operatorname{op_{sc}}(a)u\|\leq 2\|a\|_{\infty}\|u\|+C\|u\|h,\text{
for all }h\in(0,1),$

where
$\|a\|_{\infty}=\sup_{x\in\mathbb{R}^{d},\xi^{\prime}\in\mathbb{R}^{d-1}}|a(x,\xi^{\prime})|$.
Observe that $2$ in this estimate is arbitrary and may be replaced by
$1+\delta$ for all $\delta>0$ but in this case $C$ also depends on $\delta$.

For $s\in\mathbb{R}$, let
$\Lambda^{s}=\operatorname{op_{sc}}(\langle\xi^{\prime}\rangle^{s})$. We
deduce from $L^{2}$ bound an estimation for operators with symbol $a\in
S^{m}$, we have

$\|\operatorname{op_{sc}}(a)u\|\lesssim\|\Lambda^{m}u\|.$

We can define $H^{s}_{sc}$-norm by $\|u\|_{H^{s}_{sc}}=\|\Lambda^{s}u\|$.

####  3.2.5. Gårding inequality

We state the Gårding inequality adapted to our purpose. We assume there exist
$K\subset\mathbb{R}^{d}$ a compact, $W\subset\mathbb{R}^{d-1}$ satisfying for
a fix $R_{0}>0$, if $|\xi|\geq R_{0}$ then for all $\lambda\geq 1$,
$\lambda\xi\in W$. Let $m\in\mathbb{R}$ and $a\in S^{m}$, we assume there
exists $C_{0}>0$ such that $a$ satisfies $a(x,\xi^{\prime})\geq
C_{0}|\xi^{\prime}|^{m}$ for all $x\in K$, $\xi^{\prime}\in
W\cap\\{\xi^{\prime},\ |\xi^{\prime}|\geq R_{0}\\}$. Let $\chi\in S^{0}$
supported in $K\times W$, there exist $C_{1}>0$ and $C_{2}>0$ such that

(3.21)
$\operatorname{Re}(\operatorname{op_{sc}}(a)\operatorname{op_{sc}}(\chi)u,\operatorname{op_{sc}}(\chi)u)\geq
C\|\Lambda^{m/2}\operatorname{op_{sc}}(\chi)u\|^{2}-C_{2}h\|\Lambda^{(m-1)/2}u\|^{2},$

for all $u$ in Schwartz space.

###  3.3. Uniform stabilization

The proof of the first part of Theorem 2.2, is based on Gearhart-Huang-Prüss
result we recall below.

######  Theorem 3.1.

Let $A$ be a generator of a semigroup $e^{tA}$ on $H$. We assume the following
conditions

  1. (1)

$\exists M>0,\ \|e^{tA}\|_{{\mathcal{L}}(H)}\leq M$ for all $t\geq 0$,

  2. (2)

$A-i\mu I$ is invertible on $H$ for all $\mu\in\mathbb{R}$,

  3. (3)

$\exists M>0,\ \|(A-i\mu I)^{-1}\|_{{\mathcal{L}}(H)}\leq M$.

Then there exist $\lambda>0$ and $C>0$, such that $\
\|e^{tA}\|_{{\mathcal{L}}(H)}\leq Ce^{-\lambda t}$ for all $t\geq 0$.

This result is classical and we refer to [18], [39], [25, Theorem 3], [16] for
a proof. In our case, the first condition is satisfied as energy is non
increasing. The first item is a consequence of the energy decaying. The second
is not true for $\mu=0$ as ${\mathcal{A}}_{d}$ as a non trivial kernel. To
avoid this difficulty we only have to consider ${\mathcal{A}}_{d}$ as an
operator on $\tilde{H}=H/\ker{\mathcal{A}}_{d}$. On this space $A-i\mu I$ is
invertible for all $\mu\in\mathbb{R}$, see [3] for a proof. In [3] we prove
the following estimate $\|({\mathcal{A}}_{d}-i\mu
I)^{-1}\|_{{\mathcal{L}}(\tilde{H})}\leq Me^{C|\mu|}$ for some constants $M>0$
and $C>0$. In particular, for all $C>0$, there exists $M>0$ such that
$\|(A-i\mu I)^{-1}\|_{{\mathcal{L}}(H)}\leq M$ for every $\mu\in[-C,C]$. This
means, only large $|\mu|$ have to be considered to prove the third item of
Theorem 3.1.

###  3.4. Invariant form of acoustic system

The generator of semigroup of acoustic system has been defined above (see
(1.2) and (1.3)). It is useful to introduce the following system

(3.22) $\left\\{\begin{array}[]{l}-\nabla_{g}r-i\mu u-bu=f,\,\hbox{ in
}\Omega,\\\ -\operatorname{div}_{g}u-i\mu r=q,\,\hbox{ in
}\Omega,\end{array}\right.$

where $(f,q)\in H=(L^{2}(\Omega))^{d}\times L^{2}_{m}(\Omega)$, and
$(u,r)\in{\mathcal{D}}({\mathcal{A}})$. From geometric point of view, $u$ is a
vector field or equivalently a 1-contravariant tensor and $g$ is the euclidian
metric. The form (3.22) allows us to change variables and to follow the
different geometric quantities, $g$ and $u$. With the previous notation it is
equivalent to estimate $({\mathcal{A}}_{d}-i\mu)^{-1}$ and estimate $(u,r)$ by
$(f,q)$. Observe the space $H$ is decomposed in direct sum
$\ker{\mathcal{A}}_{d}\oplus(H/\ker{\mathcal{A}}_{d})$, and it is easy to see
that an estimate of $({\mathcal{A}}_{d}-i\mu)^{-1}$ on $H$ for $\mu\neq 0$
implies an estimate on $({\mathcal{A}}_{d}-i\mu)^{-1}$ restricted to
$(H/\ker{\mathcal{A}}_{d})$. In what follows, we do not consider the space
$(H/\ker{\mathcal{A}}_{d})$ as it must be introduced only to avoid the problem
coming from the fact that ${\mathcal{A}}_{d}$ has a non trivial null space. In
geodesic coordinates the system keep the same form, with the metric $g$
satisfying properties view in Section 3.1.

To use semi-classical tools, we set $h=1/|\mu|$ and multiplying System (3.22)
by $ih$ we obtain

(3.23) $\left\\{\begin{array}[]{l}-ih\nabla_{g}r+\varpi u-ihbu=ihf,\,\hbox{ in
}\Omega,\\\ -ih\operatorname{div}_{g}u+\varpi r=ihq,\,\hbox{ in
}\Omega,\end{array}\right.$

where $\varpi=\mu/|\mu|=\pm 1$. In the following we only consider case
$\varpi=+1$, in fact changing $u$ in $-u$ and $q$ in $-q$, in System (3.23)
with $\varpi=+1$, we have the system with $\varpi=-1$ and $b$ is changed in
$-b$. Observe that the sign of $b$ is important to have the decay of energy
but plays no role in the proof of the estimate for resolvent. Nevertheless $b$
should be non negative or non positive, here it is the case.

##  4\. Introduction of a semiclassical measure

The goal is to prove the following estimate, there exists $C>0$ such that

(4.24) $\|u\|_{(L^{2}(\Omega))^{d}}+\|r\|_{L^{2}(\Omega)}\leq
C(\|f\|_{(L^{2}(\Omega))^{d}}+\|q\|_{L^{2}(\Omega)}),$

for all $(f,q)\in(L^{2}(\Omega))^{d}\times L^{2}(\Omega)$ and
$(u,r)\in(L^{2}(\Omega))^{d}\times L^{2}(\Omega)$ solution of System (3.22) or
equicalently System (3.23).

We prove Estimate (4.24) by contradiction. If Estimate (4.24) is false, there
exist a sequence $(h_{j})_{j\in\mathbb{N}}$ and $(f_{h_{j}},q_{h_{j}})$ and
$(u_{h_{j}},r_{h_{j}})$ satisfying Estimate (4.24) and Equation (3.23) with
$h_{j}$ instead of $h$ such that

(4.25) $\displaystyle h_{j}\to 0\text{ as }j\to+\infty,$
$\displaystyle\|u_{h_{j}}\|_{(L^{2}(\Omega))^{d}}+\|r_{h_{j}}\|_{L^{2}(\Omega)}=1,$
$\displaystyle\|f_{h_{j}}\|_{(L^{2}(\Omega))^{d}}+\|q_{h_{j}}\|_{L^{2}(\Omega)}\to
0\text{ as }j\to+\infty.$

As view above Estimate (4.24) is true for $|\mu|$ bounded, thus only $|\mu|$
goes to $+\infty$ can give such sequences. We can always normalized
$(u_{h_{j}},r_{h_{j}})$ by homogeneity. In what follows, we do not write the
index $j$ and we write $h$ instead of $h_{j}$. The reader must keep in mind
that $h$ is in fact a sequence. Finally with these reductions and notation, we
have sequences $h$, $(u_{h},r_{h})$ and $(f_{h},q_{h})$, such that

(4.26)
$\left\\{\begin{array}[]{l}-ih\nabla_{g}r_{h}+u_{h}-ihbu_{h}=ihf_{h},\,\hbox{
in }\Omega,\\\ -ih\operatorname{div}_{g}u_{h}+r_{h}=ihq_{h},\,\hbox{ in
}\Omega,\\\ u_{h}.n=0\text{ on }\partial\Omega\end{array}\right.$

and

(4.27)
$\displaystyle\|u_{h}\|_{(L^{2}(\Omega))^{d}}+\|r_{h}\|_{L^{2}(\Omega)}=1,$
$\displaystyle\|f_{h}\|_{(L^{2}(\Omega))^{d}}+\|q_{h}\|_{L^{2}(\Omega)}\to
0\text{ as }h\to 0.$

###  4.1. Semiclassical measure

To define a semiclassical measure associated with
$\gamma_{h}=({u}_{h},{r}_{h})$ we have to prove that $\gamma_{h}$ converges
weakly to 0. To prove that we decompose $\gamma_{h}$ in a Littlewood-Paley
type series.

In the following we use functional calculus, we refer to [40] in particular
Theorem VIII.6 for precise statements.

Let $\psi$ be a $C^{\infty}(\mathbb{R})$ cutoff function such the support of
$\psi$ is contained in $|s|\leq 2$ and $\psi(s)=1$ for $|s|\leq 1$. We have

$1=\lim_{N\to\infty}\psi(2^{-N}s)=\psi(s)+\sum_{k=1}^{\infty}(\psi(2^{-k}s)-\psi(2^{-k+1}s))=\psi(s)+\sum_{k=1}^{\infty}\varphi(2^{-k}s),$

where $\varphi(s)=\psi(s)-\psi(2s)$.

Here we use the notation introduced in Introduction for ${\mathcal{A}}$ and
${\mathcal{B}}$. As $i{\mathcal{A}}$ is selfadjoint, we have

(4.28) $\displaystyle Id$
$\displaystyle=\psi(iB^{-1}h{\mathcal{A}})+\sum_{k=1}^{\infty}\varphi(2^{-k}iB^{-1}h{\mathcal{A}})$
$\displaystyle=\psi(iBh{\mathcal{A}})+\sum_{k=1}^{\infty}\varphi(2^{-k}iB^{-1}h{\mathcal{A}})+\psi(iB^{-1}h{\mathcal{A}})-\psi(iBh{\mathcal{A}}),$

where $B>0$ will be chosen sufficiently large.

######  Lemma 4.1.

Let $(u_{h},r_{h})$ satisfying (4.26), let $B>0$ be sufficiently large, let
$\gamma_{h}=(u_{h},r_{h})^{T}$, we have

$\displaystyle\|\psi(iBh{\mathcal{A}})\gamma_{h}\|\leq
Ch(\|f_{h}\|+\|q_{h}\|+\|\gamma_{h}\|)$
$\displaystyle\sum_{k=1}^{\infty}\|\varphi(2^{-k}iB^{-1}h{\mathcal{A}})\gamma_{h}\|\leq
Ch(\|f_{h}\|+\|q_{h}\|+\|\gamma_{h}\|).$

###### Proof.

We can write (4.26),

$-ih{\mathcal{A}}\gamma_{h}+\gamma_{h}-ih{\mathcal{B}}{\mathcal{B}}^{*}\gamma_{h}=hF_{h}$

with previous notation ${\mathcal{B}}^{*}=(\sqrt{b},0)$ and
$F_{h}=i\begin{pmatrix}f_{h}\\\ q_{h}\end{pmatrix}$. Applying
$\psi(iBh{\mathcal{A}})$ on both side we have

$-iB^{-1}\tilde{\psi}(iBh{\mathcal{A}})\gamma_{h}+\psi(iBh{\mathcal{A}})\gamma_{h}-ih\psi(iBh{\mathcal{A}}){\mathcal{B}}{\mathcal{B}}^{*}\gamma_{h}=h\psi(iBh{\mathcal{A}})F_{h},$

where $\tilde{\psi}(s)=s\psi(s)$. We obtain

$\|\psi(iBh{\mathcal{A}})\gamma_{h}\|\lesssim
h\|\gamma_{h}\|+h\|F_{h}\|+B^{-1}\|\tilde{\psi}(iBh{\mathcal{A}})\gamma_{h}\|.$

As $\tilde{\psi}(s)=s\chi(s)\psi(s)$ where $\chi$ is a smooth cutoff function,
such that $\chi(s)=1$ for $|s|\leq 2$. We have

$\|\tilde{\psi}(iBh{\mathcal{A}})\gamma_{h}\|\lesssim\|\psi(iBh{\mathcal{A}})\gamma_{h}\|.$

This proves the first estimate choosing $B$ sufficiently large for absorbing
this term with the left hand side.

We now apply $\varphi(2^{-k}iB^{-1}h{\mathcal{A}})$ on both side of equation
on $\gamma_{h}$, we have

$-i2^{k}B\tilde{\varphi}(2^{-k}iB^{-1}h{\mathcal{A}})\gamma_{h}+\varphi(2^{-k}iB^{-1}h{\mathcal{A}})\gamma_{h}-ih\varphi(2^{-k}iB^{-1}h{\mathcal{A}}){\mathcal{B}}{\mathcal{B}}^{*}\gamma_{h}=h\varphi(2^{-k}iB^{-1}h{\mathcal{A}})F_{h},$

where $\tilde{\varphi}(s)=s\varphi(s)$ and we obtain the estimate

(4.29)
$2^{k}B\|\tilde{\varphi}(2^{-k}iB^{-1}h{\mathcal{A}})\gamma_{h}\|\lesssim\|\varphi(2^{-k}iB^{-1}h{\mathcal{A}})\gamma_{h}\|+h\|F_{h}\|+h\|\gamma_{h}\|.$

On the support of $\varphi$, we have $1/2\leq|s|\leq 2$, thus $\varphi(s)\leq
2|s|\varphi(s)$. From functional calculus $\|\tilde{\varphi}(\alpha
ih{\mathcal{A}})\|=\||\alpha ih{\mathcal{A}}|\varphi(\alpha
ih{\mathcal{A}})\|$ for all $\alpha>0$. We deduce from (4.29) and functional
calculus.

$2^{k}B\|\varphi(2^{-k}iB^{-1}h{\mathcal{A}})\gamma_{h}\|\lesssim\|\varphi(2^{-k}iB^{-1}h{\mathcal{A}})\gamma_{h}\|+h\|F_{h}\|+h\|\gamma_{h}\|.$

If $B$ is sufficiently large, we can absorb the term
$\|\varphi(2^{-k}iB^{-1}h{\mathcal{A}})\gamma_{h}\|$ by the left hand side and
summing over $k$ we obtain the second estimate of the lemma. ∎

######  Lemma 4.2.

The sequence $(u_{h},r_{h})$ converges weakly to $0$.

###### Proof.

Let $\chi$ be defined by $\chi(s)=\psi(B^{-1}s)-\psi(Bs)$. From (4.28) and
Lemma 4.1, we have

(4.30) $\gamma_{h}=\chi(ih{\mathcal{A}})\gamma_{h}+O(h).$

For $v\in L^{2}(\Omega)^{d+1}$, we have

$(\chi(ih{\mathcal{A}})\gamma_{h},v)=(\gamma_{h},\chi(ih{\mathcal{A}})v).$

The sequence $(\gamma_{h})_{h}$ is bounded and $(\chi(ih{\mathcal{A}})v)_{h}$
converges to 0. Indeed from functional calculus

$\|\chi(ih{\mathcal{A}})v\|^{2}=\int\chi^{2}(h\lambda)(dE_{\lambda}v,v),$

where $(E_{\lambda})_{\lambda}$ is the family of projector associated to
$i{\mathcal{A}}$. In particular $(dE_{\lambda}v,v)$ is a bounded measure. As
$\chi^{2}(h\lambda)$ converges every where to 0 and $\chi^{2}\leq 1$,
dominated convergence theorem implies
$\int\chi^{2}(h\lambda)(dE_{\lambda}v,v)$ converges to 0. This proves the
result ∎

As $u_{h}$ and $r_{h}$ are bounded functions in $L^{2}(\Omega)$ we extend
these functions by 0 in exterior of $\Omega$, and we denote respectively by
$\underline{u}_{h}$ and $\underline{r}_{h}$ these extensions. Clearly
$((\underline{u}_{h},\underline{r}_{h})$ converges weakly to 0 in
$(L^{2}(\mathbb{R}^{d}))^{d}\times L^{2}(\mathbb{R}^{d})$.

Let $a_{j}^{k}(x,\xi)$, $b^{\ell}(x,\xi)$, $c(x,\xi)$ symboles of order 0. We
shall consider $b=(b^{\ell})$ as a vector field (or a contravariant vector
field) and $a=(a_{j}^{k})$ as a 1-covariant, 1-contravariant tensor. We assume
$a$ symmetric in the following sense, $g^{\ell j}a_{j}^{k}=g^{kj}a_{j}^{\ell}$
and this relation is equivalent to $g_{\ell j}a^{j}_{k}=g_{kj}a^{j}_{\ell}$.
Both formulas are useful.

As $((\underline{u}_{h},\underline{r}_{h})$ converges weakly to 0, up to a
subsequence of $h$, it is possible to define the semiclassical measure by the
following relation,

$\displaystyle((\operatorname{op_{sc}}(a_{j}^{k})\underline{u}_{h}^{j}),(\underline{u}_{h}^{\ell}))_{g}+((\operatorname{op_{sc}}(b^{\ell})\underline{r}_{h}),(\underline{u}_{h}^{j}))_{g}+(g_{jk}\operatorname{op_{sc}}(b^{j})\underline{u}_{h}^{k},\underline{r}_{h})_{g}+(\operatorname{op_{sc}}(c)\underline{r}_{h},\underline{r}_{h})_{g}$
(4.31) converges to
$\displaystyle\langle\mu_{j}^{k},a_{k}^{j}\rangle+2\langle\nu_{j},b^{j}\rangle+\langle
m,c\rangle\text{ as }h\text{ goes to 0,}$

where the measure $(\mu_{j}^{k})_{1\leq j,k\leq d}$ is symmetric in the sense
$g^{\ell j}\mu_{j}^{k}=g^{kj}\mu_{j}^{\ell}$ which is related with the
symmetry on $(a_{j}^{k})_{1\leq j,k\leq d}$. From (4.26), the functions
$\underline{u}_{h}$ and $\underline{r}_{h}$ satisfy the system

(4.32)
$\begin{cases}&\underline{u}_{h}-ih\nabla_{g}\underline{r}_{h}=hf_{h}-ih(r_{h})_{|\partial\Omega}\nabla_{g}(\chi_{\Omega}),\\\
&-ih\operatorname{div}_{g}\underline{u}_{h}+\underline{r}_{h}=hq_{h},\end{cases}$

where $\nabla_{g}(\chi_{\Omega})$ is a measure on $\partial\Omega$. We shall
give in what follows a form of $\nabla_{g}(\chi_{\Omega})$ in adapted local
coordinates. As $u_{h}.n=0$ at boundary, second equation does not have measure
terms at the right hand side.

######  Proposition 4.1.

The measures $\mu_{j}^{k}$, $\nu_{k}$ and $m$ are supported on the
characteristic set

$\\{(x,\xi)\in\mathbb{R}^{d}\times\mathbb{R}^{d},\
(g^{jp}(x)\xi_{p}\xi_{j}-1)=0\text{ and }x\in\overline{\Omega}\\},$

and satisfy the following equations

$\displaystyle\nu_{k}=-\xi_{k}m,$
$\displaystyle\mu_{k}^{j}=g^{jp}\xi_{p}\xi_{k}m\text{ for all }1\leq j,k\leq
d.$

Obviously, the measures are null in a neighborhood of $(x,\xi)$ if
$x\in\mathbb{R}^{d}\setminus\overline{\Omega}$ as
$(\underline{u}_{h},\underline{r}_{h})$ is null in a neighborhood of $x$. We
describe first the semiclassical measure in $\Omega$. The description in a
neighborhood of the boundary will be done in the following. For symbols
$a_{j}^{k}$ compactly supported in $\Omega$, we consider

(4.33)
$I_{1}(h)=\big{(}(\operatorname{op_{sc}}(a_{j}^{k})(\underline{u}_{h}^{j}-ih(\nabla_{g}\underline{r}_{h})^{j})),(\underline{u}_{h}^{\ell})\big{)}_{g}+\big{(}(\operatorname{op_{sc}}(a_{j}^{k})\underline{u}_{h}^{j}),(\underline{u}_{h}^{\ell}-ih(\nabla_{g}\underline{r}_{h})^{\ell})\big{)}_{g}.$

From System (4.32), we have $I_{1}(h)$ goes to 0 as $h$. Observe that
$-ih(\nabla_{g})^{j}\underline{r}_{h}=\operatorname{op_{sc}}(g^{jk})\xi_{k})\underline{r}_{h}+hV\underline{r}_{h}$
where $V$ is bounded and
$-ih\operatorname{div}_{g}w=\operatorname{op_{sc}}(\xi_{k})w^{k}+hWw$ where
$W$ is bounded. In what follows we denote by $O(h)$ all terms bounded by $h$
in $L^{2}$-norm. From integration by parts formula and symbolic calculus, we
have

$\displaystyle\big{(}(\operatorname{op_{sc}}(a_{j}^{k})(-ih(\nabla_{g}\underline{r}_{h})^{j})),(\underline{u}_{h}^{\ell})\big{)}_{g}=\big{(}(\operatorname{op_{sc}}(a_{j}^{k}g^{jp}\xi_{p})\underline{r}_{h}),(\underline{u}_{h}^{\ell})\big{)}_{g}+O(h)$
$\displaystyle\big{(}(\operatorname{op_{sc}}(a_{j}^{k})\underline{u}_{h}^{j}),(-ih(\nabla_{g}\underline{r}_{h})^{\ell})\big{)}_{g}=\big{(}-ih\operatorname{div}_{g}(\operatorname{op_{sc}}(a_{j}^{k})\underline{u}_{h}^{j}),\underline{r}_{h}\big{)}_{g}=(\operatorname{op_{sc}}(a_{j}^{k}\xi_{k})\underline{u}_{h}^{j},\underline{r}_{h})_{g}+O(h).$

Then

$I_{1}(h)=2\big{(}(\operatorname{op_{sc}}(a_{j}^{k})\underline{u}_{h}^{j}),(\underline{u}_{h}^{\ell})\big{)}_{g}+\big{(}(\operatorname{op_{sc}}(a_{j}^{k}g^{jp}\xi_{p})\underline{r}_{h}),(\underline{u}_{h}^{\ell})\big{)}_{g}+\big{(}\operatorname{op_{sc}}(a_{j}^{k}\xi_{k})\underline{u}_{h}^{j},\underline{r}_{h}\big{)}_{g}+O(h).$

We can apply (4.1) as, by the symmetry on $a$, we have
$g_{qk}a_{j}^{k}g^{jp}\xi_{p}=g_{jk}a_{q}^{k}g^{jp}\xi_{p}=\delta_{k}^{p}a_{q}^{k}\xi_{p}=a_{q}^{p}\xi_{p}$
thus, the condition on the $b^{j}$ is satisfied. We obtain

$2\langle\mu_{k}^{j},a_{j}^{k}\rangle+2\langle\nu_{k},a_{j}^{k}g^{jp}\xi_{p}\rangle=0,$

for all symbols $a_{j}^{k}$ compactly supported in $\Omega$, then

(4.34) $\mu_{k}^{j}+g^{jp}\xi_{p}\nu_{k}=0\text{ for all }1\leq j,k\leq d.$

For symbols $b^{k}$ compactly supported in $\Omega$, we consider

(4.35) $\displaystyle I_{2}(h)$
$\displaystyle=\big{(}(\operatorname{op_{sc}}(b^{k})(-ih\operatorname{div}_{g}\underline{u}_{h}+\underline{r}_{h})),(\underline{u}_{h}^{\ell})\big{)}_{g}+\big{(}\operatorname{op_{sc}}(g_{jk}b^{j})(\underline{u}_{h}^{k}-ih(\nabla_{g}\underline{r}_{h})^{k}),\underline{r}_{h}\big{)}_{g}$
$\displaystyle\quad+\big{(}(\operatorname{op_{sc}}(b^{q})\underline{r}_{h}),(\underline{u}_{h}^{k}-ih(\nabla_{g}\underline{r}_{h})^{k})\big{)}_{g}+\big{(}\operatorname{op_{sc}}(g_{jk}b^{j})\underline{u}_{h}^{k},-ih\operatorname{div}_{g}\underline{u}_{h}+\underline{r}_{h}\big{)}_{g}.$

From System (4.32), we have $I_{2}(h)$ goes to 0 as $h$. From integration by
parts formula and symbolic calculus, we have

$\displaystyle I_{2}(h)$
$\displaystyle=\big{(}(\operatorname{op_{sc}}(b^{k}\xi_{j})\underline{u}_{h}^{j}),(\underline{u}_{h}^{\ell})\big{)}_{g}+\big{(}(\operatorname{op_{sc}}(b^{k})\underline{r}_{h}),(\underline{u}_{h}^{\ell})\big{)}_{g}+\big{(}\operatorname{op_{sc}}(g_{jk}b^{j})\underline{u}_{h}^{k},\underline{r}_{h}\big{)}_{g}$
$\displaystyle\quad+\big{(}\operatorname{op_{sc}}(g_{jk}g^{kq}b^{j}\xi_{q})\underline{r}_{h},\underline{r}_{h}\big{)}_{g}+\big{(}(\operatorname{op_{sc}}(b^{q})\underline{r}_{h}),(\underline{u}_{h}^{k})\big{)}_{g}+\big{(}\operatorname{op_{sc}}(b^{q}\xi_{q})\underline{r}_{h},\underline{r}_{h})_{g}$
$\displaystyle\quad+\big{(}(\operatorname{op_{sc}}(g_{jk}b^{j}g^{pq}\xi_{p})\underline{u}_{h}^{k}),(\underline{u}_{h}^{\ell})\big{)}_{g}+\big{(}\operatorname{op_{sc}}(g_{jk}b^{j})\underline{u}_{h}^{k},\underline{r}_{h}\big{)}_{g}+O(h)$
$\displaystyle=\big{(}(\operatorname{op_{sc}}(b^{k}\xi_{j}+g_{jq}b^{q}g^{pk}\xi_{p})\underline{u}_{h}^{j}),(\underline{u}_{h}^{\ell})\big{)}_{g}+2\big{(}(\operatorname{op_{sc}}(b^{k})\underline{r}_{h}),(\underline{u}_{h}^{\ell})\big{)}_{g}+2\big{(}\operatorname{op_{sc}}(g_{jk}b^{j})\underline{u}_{h}^{k},\underline{r}_{h}\big{)}_{g}$
$\displaystyle\quad+2\big{(}\operatorname{op_{sc}}(b^{j}\xi_{j})\underline{r}_{h},\underline{r}_{h}\big{)}_{g}+O(h).$

To apply (4.1) we have to respect some symmetry. We have

$g_{k\ell}(b^{k}\xi_{j}+g_{jq}b^{q}g^{pk}\xi_{p})=g_{k\ell}b^{k}\xi_{j}+g_{jq}b^{q}\xi_{\ell},$

which is symmetric with respect $j$ and $\ell$. We then obtain, as

$\langle\mu_{k}^{j},b^{k}\xi_{j}+g_{jq}b^{q}g^{pk}\xi_{p}\rangle+4\langle\nu_{k},b^{k}\rangle+2\langle
m,b^{k}\xi_{k}\rangle=\langle\xi_{j}\mu_{k}^{j},b^{k}\rangle+\langle
g_{jq}g^{pk}\xi_{p}\mu_{k}^{j},b^{q}\rangle+4\langle\nu_{k},b^{k}\rangle+2\langle\xi_{k}m,b^{k}\rangle,$

the following equation

(4.36)
$\xi_{j}\mu_{k}^{j}+g_{jk}g^{pq}\xi_{p}\mu_{q}^{j}+4\nu_{k}+2\xi_{k}m=0\text{
for all }1\leq k\leq d.$

For symbols $c$ compactly supported in $\Omega$, we consider

(4.37) $\displaystyle
I_{3}(h)=\big{(}\operatorname{op_{sc}}(c)(-ih\operatorname{div}_{g}\underline{u}_{h}+\underline{r}_{h}),\underline{r}_{h}\big{)}_{g}+\big{(}\operatorname{op_{sc}}(c)\underline{r}_{h},-ih\operatorname{div}_{g}\underline{u}_{h}+\underline{r}_{h}\big{)}_{g}.$

From System (4.32), we have $I_{3}(h)$ goes to 0 as $h$ and integrating by
parts we obtain

(4.38) $\displaystyle
I_{3}(h)=2\big{(}\operatorname{op_{sc}}(c)\underline{r}_{h},\underline{r}_{h}\big{)}_{g}+\big{(}\operatorname{op_{sc}}(c\xi_{j})\underline{u}_{h}^{j},\underline{r}_{h}\big{)}_{g}+\big{(}\operatorname{op_{sc}}(cg^{kj}\xi_{j})\underline{r}_{h},\underline{u}_{h}^{j}\big{)}_{g}+O(h).$

Observe that $g_{pk}g^{kj}c\xi_{j}=c\xi_{p}$ we may apply (4.1) and we obtain

(4.39) $\displaystyle m+g^{kj}\xi_{j}\nu_{k}=0.$

From (4.34), (4.36) and (4.39) we obtain the following system on measure
$\mu_{k}^{j}$, $\nu_{k}$ and $m$.

(4.40) $\begin{cases}\mu_{k}^{j}+g^{jp}\xi_{p}\nu_{k}=0\text{ for all }1\leq
j,k\leq d,\\\
\xi_{j}\mu_{k}^{j}+g_{jk}g^{pq}\xi_{p}\mu_{q}^{j}+4\nu_{k}+2\xi_{k}m=0\text{
for all }1\leq k\leq d,\\\ m+g^{kj}\xi_{j}\nu_{k}=0.\end{cases}$

From symmetry on $(\mu_{j}^{k})_{1\leq j,k\leq d}$ we may write the second
equation

$\xi_{j}\mu_{k}^{j}+g_{jk}g^{jq}\xi_{p}\mu_{q}^{p}+4\nu_{k}+2\xi_{k}m=0\text{
for all }1\leq k\leq d,$

thus we have

$2\xi_{j}\mu_{k}^{j}+4\nu_{k}+2\xi_{k}m=0\text{ for all }1\leq k\leq d.$

Replace $\mu_{k}^{j}$ in this equation from the first equation of (4.40) we
have

(4.41) $-g^{jp}\xi_{p}\xi_{j}\nu_{k}+2\nu_{k}+\xi_{k}m=0\text{ for all }1\leq
k\leq d.$

Multiplying this equation by $g^{kq}\xi_{q}$ and from the third equation of
(4.40) which is $g^{jp}\xi_{p}\nu_{k}=-m$, this yields

$g^{jp}\xi_{p}\xi_{j}m-2m+g^{kq}\xi_{q}\xi_{k}m=0.$

Then

(4.42) $(g^{jp}\xi_{p}\xi_{j}-1)m=0,$

which implies $m$ supported on $(g^{jp}\xi_{p}\xi_{j}-1)=0$. If
$(g^{jp}\xi_{p}\xi_{j}-1)\neq 0$, from (4.40) and as $m=0$ we have the
following system

(4.43) $\begin{cases}\mu_{k}^{j}+g^{jp}\xi_{p}\nu_{k}=0\text{ for all }1\leq
j,k\leq d,\\\ \xi_{j}\mu_{k}^{j}+2\nu_{k}=0\text{ for all }1\leq k\leq d,\\\
g^{kj}\xi_{j}\nu_{k}=0.\end{cases}$

Multiplying the first equation by $g^{k\ell}\xi_{\ell}$ we obtain

$g^{k\ell}\xi_{\ell}\mu_{k}^{j}+g^{k\ell}\xi_{\ell}g^{jp}\xi_{p}\nu_{k}=0\text{
for all }1\leq j\leq d.$

From the third equation of (4.43) and the symmetry
$g^{k\ell}\mu_{k}^{j}=g^{kj}\mu_{k}^{\ell}$, we have

$g^{kj}\xi_{\ell}\mu_{k}^{\ell}=0\text{ for all }1\leq j\leq d,$

and as $(g^{kj})_{1\leq j,k\leq d}$ is invertible we obtain
$\xi_{\ell}\mu_{j}^{\ell}=0\text{ for all }1\leq j\leq d$ and the second
equation of (4.43) implies $\nu_{k}=0$ for $1\leq k\leq d$ and the first
$\mu_{k}^{j}=0$ for $1\leq k,j\leq d$. Thus all the measures $\mu_{k}^{j}$,
$\nu_{k}$ and $m$ are supported on $(g^{jp}\xi_{p}\xi_{j}-1)=0$.

Now from (4.41) as on the support of measures $g^{jp}\xi_{p}\xi_{j}=1$, we
have

$\nu_{k}=-\xi_{k}m,$

and from the first equation of (4.40) we have

$\mu_{k}^{j}=g^{jp}\xi_{p}\xi_{k}m\text{ for all }1\leq j,k\leq d.$

This achieves the proof of Proposition 4.1 in $\Omega$.

Now we have to prove same result at the boundary. The proof is a consequence
of the two following lemmas.

######  Lemma 4.3.

Let $a$ be a symbol compactly supported, then there exists $C>0$, such that,
we have

$\|\operatorname{op_{sc}}(a)(w\otimes\delta_{x_{d}=0})\|\leq
Ch^{-1/2}|w|_{L^{2}(\mathbb{R}^{d-1})},$

for all $w$ in the Schwartz space of $\mathbb{R}^{d-1}_{x^{\prime}}$.

######  Lemma 4.4.

Let $\underline{u}_{h}$ and $\underline{r}_{h}$ solution of System (4.32) then
$\displaystyle\lim_{h\to 0}h|\Lambda^{1/2}({r}_{h})_{|x_{d}=0}|^{2}=0$.

With these lemmas we can prove Proposition 4.1. Let us write System (4.32) in
geodesic coordinates. We have

(4.44)
$\begin{cases}&\underline{u}_{h}-ih\nabla_{g}\underline{r}_{h}=hf_{h}-ih(r_{h})_{|x_{d}=0}\otimes\delta_{x_{d}=0},\\\
&-ih\operatorname{div}_{g}\underline{u}_{h}+\underline{r}_{h}=hq_{h}.\end{cases}$

Following the proof given above, in $I_{1}(h)$ (see (4.33)), we have two extra
terms to control,

$(\operatorname{op_{sc}}(a)(h(r_{h})_{|x_{d}=0}\otimes\delta_{x_{d}=0}),\underline{u}_{h}^{j})\text{
and}$
$(\operatorname{op_{sc}}(a)(\underline{u}_{h}^{j}),h(r_{h})_{|x_{d}=0}\otimes\delta_{x_{d}=0})=(\underline{u}_{h}^{j},\operatorname{op_{sc}}(a)^{*}(h(r_{h})_{|x_{d}=0}\otimes\delta_{x_{d}=0})).$

For $I_{2}(h)$ (see (4.35)) we have to control same type of terms where
$\underline{u}_{h}^{j}$ is replaced by $\underline{r}_{h}$. The term
$I_{3}(h)$ (see (4.37)) does not have an extra boundary term. As all these
terms write analogous, we only treat one term. We apply Lemma 4.3, we have

$|(\operatorname{op_{sc}}(a)(h(r_{h})_{|x_{d}=0}\otimes\delta_{x_{d}=0}),\underline{u}_{h}^{j})|\lesssim
h^{1/2}\|\underline{u}_{h}^{j}\||(r_{h})_{|x_{d}=0}|\to 0\text{ as }h,$

from Lemma 4.4.

###### Proof of Lemma 4.3.

We can assume that
$a(x,\xi)=a_{1}(x^{\prime},\xi^{\prime})\chi(x_{d})\ell(\xi_{d})$, where
$a_{1}$, $\chi$ and $\ell$ are compactly supported, as all symbols can be
approximated by a sum of such symbols. From definition of operators associated
with a symbol, we have

$\operatorname{op_{sc}}(a)(w\otimes\delta_{x_{d}=0})=\chi(\operatorname{op_{sc}}(a_{1})w)\otimes\operatorname{op_{sc}}(\ell)\delta_{x_{d}=0}.$

A direct computation in Fourier variables gives
$\|\operatorname{op_{sc}}(\ell)\delta_{x_{d}=0}\|_{L^{2}(\mathbb{R})}\lesssim
h^{-1/2}$, and as $a_{1}\in S^{-N}$ for all $N\in\mathbb{N}$,
$|\operatorname{op_{sc}}(a_{1})w|\lesssim|w|_{L^{2}(\mathbb{R}^{d-1})}$, which
implies the lemma. ∎

###### Proof of Lemma 4.4.

We work in a neighborhood of a boundary point in geodesic coordinates, we can
write $x_{0}=(x_{0}^{\prime},0)$. We use the notation $\tilde{g}$ defined in
Section 3.1. The system written in interior is given by

(4.45)
$\begin{cases}&u_{h}^{k}+\operatorname{op_{sc}}(\tilde{g}^{jk}\xi_{j})r_{h}=h\tilde{f}_{h}^{k},\text{
for }k=1,...,d-1,\\\ &u_{h}^{d}+hD_{d}r_{h}=h\tilde{f}_{h}^{d}\\\
&hD_{d}u_{h}^{d}+\displaystyle\sum_{j=1}^{d-1}\operatorname{op_{sc}}(\xi_{j})u_{h}^{j}+r_{h}=h\tilde{q}_{h},\end{cases}$

where $\tilde{f}_{h}$ and $\tilde{q}_{h}$ are bounded in $L^{2}$. Indeed
System (4.45) comes from System (4.26) and pseudo-differential calculus. From
(3.13) and (3.14), we have
$-ih\nabla_{\tilde{g}}r_{h}=\operatorname{op_{sc}}(\tilde{g}^{jk})r_{h}+O(h)$,
$hbu_{h}=O(h)$, and
$-hi\operatorname{div}_{g}u_{h}=hD_{d}u_{h}^{d}+\displaystyle\sum_{j=1}^{d-1}\operatorname{op_{sc}}(\xi_{j})u_{h}^{j}+O(h)$
where we have denoted by $O(h)$ terms bounded by $Ch$ in $L^{2}$-norm.

Using the first equations, we can replace $u_{h}^{j}$ in the last equation of
(4.45). We then obtain the 2 by 2 system

(4.46) $\begin{cases}&u_{h}^{d}+hD_{d}r_{h}=h\tilde{f}_{h}^{d}\\\
&hD_{d}u_{h}^{d}-\displaystyle\sum_{j=1}^{d-1}\operatorname{op_{sc}}(\xi_{j})\operatorname{op_{sc}}(\tilde{g}^{jk}\xi_{k})r_{h}+r_{h}=hq_{h}+h\displaystyle\sum_{j=1}^{d-1}\operatorname{op_{sc}}(\xi_{j})\tilde{f}_{h}^{j}.\end{cases}$

From symbolic calculus,
$\sum_{j=1}^{d-1}\operatorname{op_{sc}}(\xi_{j})\operatorname{op_{sc}}(\tilde{g}^{jk}\xi_{k})=\operatorname{op_{sc}}(R(x,\xi^{\prime}))+h\operatorname{op_{sc}}(r_{1})$
where $r_{1}$ is of order 1, and
$R(x,\xi^{\prime})=\tilde{g}^{jk}\xi_{k}\xi_{j}$. System (4.46) can be recast
as

(4.47)
$\begin{cases}&u_{h}^{d}+\operatorname{op_{sc}}(\xi_{d})r_{h}=h\tilde{f}_{h}^{d}\\\
&\operatorname{op_{sc}}(\xi_{d})u_{h}^{d}-\operatorname{op_{sc}}(R(x,\xi^{\prime})-1)r_{h}=hG,\end{cases}$

where
$G=q_{h}+\displaystyle\sum_{j=1}^{d-1}\operatorname{op_{sc}}(\xi_{j})\tilde{f}_{h}^{j}+\operatorname{op_{sc}}(r_{1})r_{h}$.

We define three types of points, hyperbolic points, the set where
$R(x,\xi^{\prime})-1<0$, glancing points, the set where
$R(x,\xi^{\prime})-1=0$ and elliptic points where $R(x,\xi^{\prime})-1>0$.
Observe that in both cases hyperbolic and glancing, $\xi^{\prime}$ is bounded.
In what follows we made some computations useful for hyperbolic or glancing
cases.

Let $\chi(x,\xi^{\prime})$ be compactly supported on a neighborhood of
$(x_{0},\xi^{\prime}_{0})$. In particular $\chi\in S^{-N}$ for all
$N\in\mathbb{N}$. We denote by $w_{h}=\operatorname{op_{sc}}(\chi)r_{h}$ and
$v_{h}=\operatorname{op_{sc}}(\chi)u_{h}^{d}$, in particular
$(v_{h})_{|x_{d}=0}=0$ from boundary conditions. From symbolic calculus we
have
$hD_{d}\operatorname{op_{sc}}(\chi)z=\operatorname{op_{sc}}(\chi)hD_{d}z+A$
where $\|A\|\lesssim h\|z\|$. We also have

$\operatorname{op_{sc}}(\chi)\operatorname{op_{sc}}(R(x,\xi^{\prime})-1)=\operatorname{op_{sc}}(R(x,\xi^{\prime})-1)\operatorname{op_{sc}}(\chi)+[\operatorname{op_{sc}}(\chi),\operatorname{op_{sc}}(R(x,\xi^{\prime})-1)],$

and the symbol of
$[\operatorname{op_{sc}}(\chi),\operatorname{op_{sc}}(R(x,\xi^{\prime})-1)]$
is in $hS^{-N}$ for all $N\in\mathbb{N}$. We then have

$\|[\operatorname{op_{sc}}(\chi),\operatorname{op_{sc}}(R(x,\xi^{\prime})-1)]z\|\lesssim
h\|z\|.$

Let $\tilde{\chi}$ compactly supported and $\tilde{\chi}=1$ on the support of
$\chi$. The symbol of operator defined by
$\operatorname{op_{sc}}(R(x,\xi^{\prime})-1)\operatorname{op_{sc}}(\chi)$ is
$(R(x,\xi^{\prime})-1)\chi+ha_{0}$ with $a_{0}\in hS^{-N}$ for all
$N\in\mathbb{N}$. And the symbol of
$\operatorname{op_{sc}}(\tilde{\chi}^{2}(R(x,\xi^{\prime})-1))\operatorname{op_{sc}}(\chi)$
is $(R(x,\xi^{\prime})-1)\chi+ha_{1}$ with $a_{1}\in hS^{-N}$ for all
$N\in\mathbb{N}$. Then we have

$\|(\operatorname{op_{sc}}(R(x,\xi^{\prime})-1)\operatorname{op_{sc}}(\chi)-\operatorname{op_{sc}}(\tilde{\chi}^{2}(R(x,\xi^{\prime})-1))\operatorname{op_{sc}}(\chi))z\|\lesssim
h\|z\|.$

Applying $\operatorname{op_{sc}}(\chi)$ to System (4.47), using the previous
analysis, we obtain the following system:

(4.48) $\begin{cases}&hD_{d}w_{h}+v_{h}=h\tilde{f}_{h}\\\
&hD_{d}v_{h}-\operatorname{op_{sc}}(\tilde{\chi}^{2}(R(x,\xi^{\prime})-1))w_{h}=h\tilde{G}_{h}\\\
&(v_{h})_{|x_{d}=0}=0,\end{cases}$

where
$\|\tilde{G}_{h}\|\lesssim\|q_{h}\|+\displaystyle\sum_{j=1}^{d-1}\|\tilde{f}_{h}^{j}\|+\|r_{h}\|+\|u_{h}^{d}\|$.

####  4.1.1. Hyperbolic region

We treat now the hyperbolic points

${\mathcal{H}}=\\{(x,\xi^{\prime})\in\mathbb{R}^{d}\times\mathbb{R}^{d-1},R(x,\xi^{\prime})-1<0\\}.$

Let $\chi$ be compactly supported in ${\mathcal{H}}$ and $\tilde{\chi}$ as
defined above. Let
$\lambda(x,\xi^{\prime})=\tilde{\chi}(x,\xi^{\prime})\sqrt{1-R(x,\xi^{\prime})}$.
From symbolic calculus, we have
$\operatorname{op_{sc}}(\lambda^{2})\operatorname{op}(\chi)=-\operatorname{op_{sc}}(\tilde{\chi}^{2}(R-1))+h\operatorname{op_{sc}}(a_{0})$
where $a_{0}$ is of order 0. System (4.48) writes

(4.49) $\begin{cases}&hD_{d}w_{h}+v_{h}=h\tilde{f}_{h}\\\
&hD_{d}v_{h}+\operatorname{op_{sc}}(\lambda^{2})w_{h}=h\tilde{G}_{h}\\\
&(v_{h})_{|x_{d}=0}=0,\end{cases}$

where $(\tilde{f}_{h},\tilde{G}_{h})$ has been modified but are still bounded
in $L^{2}$ uniformly with respect $h$. In what follows we keep alway
$(\tilde{f}_{h},\tilde{G}_{h})$ for $L^{2}$ bounded terms.

Let $z_{k}=v_{h}+(-1)^{k}\operatorname{op_{sc}}(\lambda)w_{h}$ for $k=1,2$. We
have from (4.49)

$\displaystyle hD_{d}z_{k}$
$\displaystyle=-\operatorname{op_{sc}}(\lambda^{2})w_{h}-(-1)^{k}\operatorname{op_{sc}}(\lambda)v_{h}+h\tilde{G}_{h}+(-1)^{k}\operatorname{op_{sc}}(\lambda)h\tilde{f}_{h}+h\operatorname{op_{sc}}(r_{0})w_{h}$
$\displaystyle=-(-1)^{k}\operatorname{op_{sc}}(\lambda)z_{k}+h\tilde{G}_{h}+(-1)^{k}\operatorname{op_{sc}}(\lambda)h\tilde{f}_{h}+h\operatorname{op_{sc}}(r_{0}^{\prime})w_{h},$

where $r_{0}$ and $r_{0}^{\prime}$ are of order 0 and come from
$[hD_{d},\operatorname{op_{sc}}(\lambda)]$ and formula
$\operatorname{op_{sc}}(\lambda)\operatorname{op_{sc}}(\lambda)=\operatorname{op_{sc}}(\lambda^{2})+h\operatorname{op_{sc}}(r_{0}^{\prime\prime})$
where $r_{0}^{\prime\prime}$ is of order 0. Then we obtain
$hD_{d}z_{k}+(-1)^{k}\operatorname{op_{sc}}(\lambda)z_{k}=hH_{k}$ where
$H_{k}$ bounded in $L^{2}$ uniformly with respect $h$. To obtain an estimation
for $z_{k}$ we apply the following lemma.

######  Lemma 4.5.

Let $a\in S^{1}$. We assume $a$ supported on a neighborhood of $x_{d}=0$.
There exists $C>0$, such that for all $w$ satisfying
$hD_{d}w+\operatorname{op_{sc}}(a)w=hf$, we have $|w_{|x_{d}=0}|\leq
C(\|f\|+\|w\|)$.

###### Proof.

In one hand, we have

(4.50) $|\operatorname{Im}(hf,w)|\leq h\|f\|\|w\|.$

In other hand, we have

$\displaystyle
2i\operatorname{Im}(hD_{d}+\operatorname{op_{sc}}(a)w,w)=(hD_{d}w,w)-(w,hD_{d}w)+(\operatorname{op_{sc}}(a)w,w)-(w,\operatorname{op_{sc}}(a)w).$

We now apply integration by parts formula (3.17) and as $a$ is real valued,
$\operatorname{op_{sc}}(a)^{*}=\operatorname{op_{sc}}(a)+h\operatorname{op_{sc}}(r_{0})$
where the symbol $r_{0}$ is of order 0, we obtain

$\displaystyle
2i\operatorname{Im}(hD_{d}+\operatorname{op_{sc}}(a)w,w)=ih|w_{|x_{d}=0}|^{2}+h(\operatorname{op_{sc}}(r_{0})w,w),$

and from (4.50) above, we deduce Lemma 4.5. ∎

From Lemma 4.5, applied to $z_{k}$ we obtain,

(4.51) $|(z_{k})_{|x_{d}=0}|\lesssim\|z_{k}\|+\|H_{k}\|$

From boundary condition, $(v_{h})_{|x_{d}=0}=0$, and as
$\|z_{k}\|\lesssim\|v_{h}\|+\|w_{h}\|$ then we obtain

(4.52)
$|(\operatorname{op_{sc}}(\lambda)w_{h})_{|x_{d}=0}|\lesssim\|v_{h}\|+\|w_{h}\|+\|H_{k}\|.$

Let $\chi_{2}$ be compactly supported in ${\mathcal{H}}$ and such that
$\chi_{2}\tilde{\chi}=\tilde{\chi}$. We have by symbolic caculus

$\operatorname{op_{sc}}(\chi_{2}(1-R(x,\xi^{\prime}))^{-1/2})\operatorname{op_{sc}}(\lambda)=\operatorname{op_{sc}}(\tilde{\chi})+h\operatorname{op_{sc}}(a_{0}),$

where $a_{0}\in S^{0}$. We can deduce

$|(\operatorname{op_{sc}}(\tilde{\chi})w_{h})_{|x_{d}=0}|\lesssim\|v_{h}\|+\|w_{h}\|+\|H_{k}\|+h|(w_{h})_{|x_{d}=0}|.$

As $\tilde{\chi}=1$ on the support of $\chi$, we have
$\operatorname{op_{sc}}(\tilde{\chi})w_{h}=w_{h}+h\operatorname{op_{sc}}(a_{0})r_{h},$
where $a_{0}\in S^{0}$. We thus obtain

(4.53)
$|(w_{h})_{|x_{d}=0}|\lesssim\|v_{h}\|+\|w_{h}\|+\|H_{k}\|+h|(r_{h})_{|x_{d}=0}|.$

The estimate above is better in hyperbolic region than the statement of Lemma
4.4. Observe that in hyperbolic region we do not need boundary condition as by
linear combination between $z_{1}$ and $z_{2}$ we can estimate both
$(w_{h})_{|x_{d}=0}$ and $(v_{h})_{|x_{d}=0}$.

####  4.1.2. Glancing region

We treat now the glancing region. Let ${\varepsilon}>0$ we define

${\mathcal{G}}_{\varepsilon}=\\{(x,\xi^{\prime})\in\mathbb{R}^{d}\times\mathbb{R}^{d-1},|R(x,\xi^{\prime})-1|<{\varepsilon}\\}.$

Let $\chi$ compactly supported in ${\mathcal{G}}_{\varepsilon}$. We keep same
notation as in hyperbolic region, i.e.
$w_{h}=\operatorname{op_{sc}}(\chi)r_{h}$ and
$v_{h}=\operatorname{op_{sc}}(\chi)u_{h}^{d}$ satisfying System (4.48). We
have for $0\leq x_{d}\leq{\varepsilon}^{-1/2}h$,

$\displaystyle|v_{h}(x^{\prime},x_{d})|^{2}\leq\left(\int_{0}^{{\varepsilon}^{-1/2}h}|\partial_{x_{d}}v_{h}(x^{\prime},\sigma)|d\sigma\right)^{2}.$

Integrating with respect $x_{d}$, we obtain

$\displaystyle\int_{0}^{{\varepsilon}^{-1/2}h}|v_{h}(x^{\prime},x_{d})|^{2}dx_{d}\leq{\varepsilon}^{-1/2}h\left(\int_{0}^{{\varepsilon}^{-1/2}h}|\partial_{x_{d}}v_{h}(x^{\prime},\sigma)|d\sigma\right)^{2}\leq{\varepsilon}^{-1}h^{2}\int_{0}^{{\varepsilon}^{-1/2}h}|\partial_{x_{d}}v_{h}(x^{\prime},\sigma)|^{2}d\sigma.$

From System (4.48) and with an integration with respect $x^{\prime}$ we obtain

(4.54)
$\displaystyle\int_{\mathbb{R}^{d-1}}\int_{0}^{{\varepsilon}^{-1/2}h}|v_{h}(x^{\prime},x_{d})|^{2}dx_{d}dx^{\prime}\leq{\varepsilon}^{-1}\|\operatorname{op_{sc}}(R-1)w_{h}\|^{2}+h^{2}\|h\tilde{G}_{h}\|^{2}.$

From
$w_{h}(x^{\prime},0)=w_{h}(x^{\prime},x_{d})-\int_{0}^{x_{d}}\partial_{x_{d}}w_{h}(x^{\prime},\sigma)d\sigma$,
we have for $x_{d}\in(0,{\varepsilon}^{-1/2}h)$

$\displaystyle|w_{h}(x^{\prime},0)|^{2}\lesssim|w_{h}(x^{\prime},x_{d})|^{2}+\left(\int_{0}^{{\varepsilon}^{-1/2}h}|\partial_{x_{d}}w_{h}(x^{\prime},\sigma)|d\sigma\right)^{2}.$

With an integration with respect $x_{d}$ and Cauchy-Schwarz inequality, we
obtain

$\displaystyle{\varepsilon}^{-1/2}h|w_{h}(x^{\prime},0)|^{2}\lesssim\int_{0}^{{\varepsilon}^{-1/2}h}|w_{h}(x^{\prime},x_{d})|^{2}dx_{d}+{\varepsilon}^{-1}\int_{0}^{{\varepsilon}^{-1/2}h}|h\partial_{x_{d}}w_{h}(x^{\prime},x_{d})|^{2}dx_{d}.$

With an integration with respect $x^{\prime}$ and from System (4.48), this
yields

$\displaystyle{\varepsilon}^{-1/2}h|(w_{h})_{|x_{d}=0}|^{2}\lesssim\|w_{h}\|^{2}+{\varepsilon}^{-1}\int_{\mathbb{R}^{d-1}}\int_{0}^{{\varepsilon}^{-1/2}h}|v_{h}(x^{\prime},x_{d})|^{2}dx_{d}dx^{\prime}+{\varepsilon}^{-1}h^{2}\|\tilde{f}_{h}^{d}\|^{2}.$

Applying (4.54), we have

(4.55)
$\displaystyle{\varepsilon}^{-1/2}h|(w_{h})_{|x_{d}=0}|^{2}\lesssim\|w_{h}\|^{2}+{\varepsilon}^{-2}\|\operatorname{op_{sc}}(R-1)w_{h}\|^{2}+{\varepsilon}^{-1}h^{2}\|\tilde{G}_{h}^{d}\|^{2}+{\varepsilon}^{-1}h^{2}\|\tilde{f}_{h}^{d}\|^{2}.$

By symbolic calculus,
$\operatorname{op_{sc}}(R-1)w_{h}=\operatorname{op_{sc}}((R-1)\chi)r_{h}+h\operatorname{op_{sc}}(a_{0})r_{h}$
where $a_{0}\in S^{0}$. As by assumption $|(R-1)\chi|\leq{\varepsilon}$, for
all ${\varepsilon}>0$, from (3.20) there exists $C_{\varepsilon}>0$ such that

$\|\operatorname{op_{sc}}(R-1)w_{h}\|^{2}\leq
4{\varepsilon}^{2}\|r_{h}\|^{2}+C_{\varepsilon}h^{2}\|r_{h}\|^{2}.$

Then from (4.55) we obtain

(4.56) $\limsup_{h\to 0}h|(w_{h})_{|x_{d}=0}|^{2}\lesssim{\varepsilon}^{1/2}.$

####  4.1.3. Elliptic region

The last zone to consider is the elliptic region. Let
${\mathcal{E}}=\\{(x,\xi^{\prime})\in\mathbb{R}^{d}\times\mathbb{R}^{d-1},R(x,\xi^{\prime})-1>0\\}$.
As ${\mathcal{E}}$ is not compact in $\mathbb{R}^{d}\times\mathbb{R}^{d-1}$ we
have to treat the large $|\xi^{\prime}|$. Let $\chi$ be a smooth function from
$\mathbb{R}^{d}\times\mathbb{R}^{d-1}$ to $\mathbb{R}$, satisfying,
$\chi(x,\xi^{\prime})=1$ for $(x,\xi^{\prime})$ such that
$R(x,\xi^{\prime})\geq 1+{\varepsilon}$ and $|x|\leq\delta$,
$\chi(x,\xi^{\prime})=0$ for $|x|\geq 2\delta$. We assume a 0-homogeneity
condition that is $\chi(x,\lambda\xi^{\prime})=\chi(x,\xi^{\prime})$ for
$(x,\xi^{\prime})$ such that $R(x,\xi^{\prime})\geq 1+{\varepsilon}$ and
$\lambda\geq 1$. The arguments below are close to the one used in hyperbolic
region but we take care to large $|\xi^{\prime}|$.

We denote by $w_{h}=\operatorname{op_{sc}}(\chi)r_{h}$ and
$v_{h}=\operatorname{op_{sc}}(\chi)u_{h}^{d}$. From symbolic calculus we have
$hD_{d}\operatorname{op_{sc}}(\chi)z=\operatorname{op_{sc}}(\chi)hD_{d}z+A$
where $\|A\|\lesssim h\|z\|$. Indeed, the symbol of
$[hD_{d},\operatorname{op_{sc}}(\chi)]$ is $hD_{d}\chi\in hS^{0}$.

We also have

$\operatorname{op_{sc}}(\chi)\operatorname{op_{sc}}(R(x,\xi^{\prime})-1)=\operatorname{op_{sc}}(R(x,\xi^{\prime})-1)\operatorname{op_{sc}}(\chi)+[\operatorname{op_{sc}}(\chi),\operatorname{op_{sc}}(R(x,\xi^{\prime})-1)],$

and the symbol of
$[\operatorname{op_{sc}}(\chi),\operatorname{op_{sc}}(R(x,\xi^{\prime})-1)]$
is in $hS^{1}$.

We then have
$\|[\operatorname{op_{sc}}(\chi),\operatorname{op_{sc}}(R(x,\xi^{\prime})-1)]z\|\lesssim
h\|\Lambda^{1}z\|$. Let $\tilde{\chi}$ be a smooth function, supported on
${\mathcal{E}}$ and $\tilde{\chi}=1$ on the support of $\chi$. We assume
$\tilde{\chi}(x,\xi^{\prime})=1$ for $(x,\xi^{\prime})$ such that
$R(x,\xi^{\prime})\geq 1+{\varepsilon}/2$ and $|x|\leq 2\delta$,
$\tilde{\chi}(x,\xi^{\prime})=0$ for $|x|\geq 3\delta$ and a 0-homogeneity
condition that is $\tilde{\chi}(x,\lambda\xi^{\prime})=\chi(x,\xi^{\prime})$
for $(x,\xi^{\prime})$ such that $R(x,\xi^{\prime})\geq 1+{\varepsilon}/2$ and
$\lambda\geq 1$. We then have $\chi\tilde{\chi}=\chi$.

The symbol of
$\operatorname{op_{sc}}(R(x,\xi^{\prime})-1)\operatorname{op_{sc}}(\chi)$ is
$(R(x,\xi^{\prime})-1)\chi+ha_{1}$ where $a_{1}$ is in $S^{1}$. The symbol of
$\operatorname{op_{sc}}(\tilde{\chi}(R(x,\xi^{\prime})-1))\operatorname{op_{sc}}(\chi)$
is $(R(x,\xi^{\prime})-1)\chi+h\tilde{a}_{1}$ where $\tilde{a}_{1}$ is in
$S^{1}$. We can then replace
$\operatorname{op_{sc}}(R(x,\xi^{\prime})-1)\operatorname{op_{sc}}(\chi)$ by
$\operatorname{op_{sc}}(\tilde{\chi}^{2}(R(x,\xi^{\prime})-1))\operatorname{op_{sc}}(\chi)$
modulo a term $ha_{1}$ where $a_{1}$ is in $S^{1}$.

Applying $\operatorname{op_{sc}}(\chi)$ to System (4.47), using the previous
analysis, we obtain the following:

(4.57) $\begin{cases}&hD_{d}w_{h}+v_{h}=h\tilde{f}_{h}\\\
&hD_{d}v_{h}-\operatorname{op_{sc}}(\tilde{\chi}^{2}(R(x,\xi^{\prime})-1))w_{h}=h\tilde{G}_{h}\\\
&(v_{h})_{|x_{d}=0}=0,\end{cases}$

where
$\|\Lambda^{-1}\tilde{G}_{h}\|\lesssim\|\Lambda^{-1}G\|+\|r_{h}\|+\|u^{d}_{h}\|$
with notation used in (4.47).

Let
$\lambda(x,\xi^{\prime})=\tilde{\chi}(x,\xi^{\prime})\sqrt{R(x,\xi^{\prime})-1}$.
The symbol of
$\operatorname{op_{sc}}((1-\tilde{\chi}^{2})(R-1))\operatorname{op_{sc}}(\chi)$
is in $h^{N}S^{-N}_{1,0}$ for all $N>0$ as $(1-\tilde{\chi}^{2})\chi=0$ the
asymptotic expansion of the symbol is null. We may write system (4.57)

(4.58) $\begin{cases}&hD_{d}w_{h}+v_{h}=h\tilde{f}_{h}\\\
&hD_{d}v_{h}-\operatorname{op_{sc}}(\lambda^{2})w_{h}=h\tilde{G}_{h}\\\
&(v_{h})_{|x_{d}=0}=0,\end{cases}$

where $\tilde{G}_{h}$ satisfy
$\|\Lambda^{-1}\tilde{G}_{h}\|\lesssim\|\Lambda^{-1}G\|+\|r_{h}\|+\|u^{d}_{h}\|$
estimations as above.

We set $z_{k}=v_{h}+(-1)^{k}i\operatorname{op_{sc}}(\lambda)w_{h}$ for
$k=1,2$. We have

$\displaystyle hD_{d}z_{k}$
$\displaystyle=hD_{d}v_{h}+(-1)^{k}ihD_{d}\operatorname{op_{sc}}(\lambda)w_{h}$
$\displaystyle=\operatorname{op_{sc}}(\lambda^{2})w_{h}+h\tilde{G}_{h}+(-1)^{k}i\operatorname{op_{sc}}(\lambda)hD_{d}w_{h}+h\operatorname{op_{sc}}(a_{1})w_{h}$
$\displaystyle=(-1)^{k+1}i(\operatorname{op_{sc}}(\lambda)v_{h}+(-1)^{k}i\operatorname{op_{sc}}(\lambda^{2})w_{h})+h\tilde{G}_{h}+(-1)^{k}i\operatorname{op_{sc}}(\lambda)h\tilde{f}_{h}+h\operatorname{op_{sc}}(a_{1})w_{h}$
$\displaystyle=(-1)^{k+1}i\operatorname{op_{sc}}(\lambda)(v_{h}+(-1)^{k}i\operatorname{op_{sc}}(\lambda)w_{h})+h\tilde{G}_{h}+(-1)^{k}i\operatorname{op_{sc}}(\lambda)h\tilde{f}_{h}+h\operatorname{op_{sc}}(a_{1})w_{h},$

as the symbol of $[D_{d},\operatorname{op_{sc}}(\lambda)]$ is in $S^{1}$ and
$\operatorname{op_{sc}}(\lambda)^{2}=\operatorname{op_{sc}}(\lambda^{2})$
modulo an operator with symbol in $hS^{1}$. In previous formula $a_{1}$ is a
symbol in $S^{1}$, changing line from line eventually. Then $z_{k}$ satisfies
the following equation

(4.59)
$hD_{d}z_{k}+(-1)^{k}i\operatorname{op_{sc}}(\lambda)z_{k}=h\tilde{G}_{h}+(-1)^{k}i\operatorname{op_{sc}}(\lambda)h\tilde{f}_{h}+h\operatorname{op_{sc}}(a_{1})w_{h}=hH_{h},$

where we have
$\|\Lambda^{-1}H_{h}\|\lesssim\|\tilde{f}_{h}\|+\|\Lambda^{-1}G\|+\|r_{h}\|+\|u^{d}_{h}\|$.

We set

$\displaystyle
I=\operatorname{Re}(\Lambda^{-1}(hD_{d}z_{k}+(-1)^{k}i\operatorname{op_{sc}}(\lambda)z_{k}),\Lambda^{-1}(-1)^{k}i\operatorname{op_{sc}}(\lambda)z_{k}),$

and in one hand we have

(4.60) $\displaystyle|I|\lesssim h\|\Lambda^{-1}H_{h}\|\|z_{k}\|,$

and in other hand

(4.61) $\displaystyle 2I$
$\displaystyle=2\|\Lambda^{-1}\operatorname{op_{sc}}(\lambda)z_{k}\|^{2}$
$\displaystyle\quad+\operatorname{Re}(hD_{d}\Lambda^{-1}z_{k},\Lambda^{-1}(-1)^{k}i\operatorname{op_{sc}}(\lambda)z_{k})+(\Lambda^{-1}(-1)^{k}i\operatorname{op_{sc}}(\lambda)z_{k},hD_{d}\Lambda^{-1}z_{k})$
$\displaystyle=I_{1}+\operatorname{Re}I_{2}+\operatorname{Re}I_{3}.$

To compute $I_{2}+I_{3}$ we integrate by parts applying (3.17) and by symbolic
calculus we have
$\operatorname{op_{sc}}(\lambda)^{*}=\operatorname{op_{sc}}(\lambda)+h\operatorname{op_{sc}}(a_{0})$,
the following commutator results,
$[\Lambda^{-1},\operatorname{op_{sc}}(\lambda)]=h\operatorname{op_{sc}}(a_{-1})$,
$[hD_{d},\Lambda^{-1}]=0$ and
$[hD_{d},\operatorname{op_{sc}}(\lambda)]=h\operatorname{op_{sc}}(a_{1})$where
$a_{j}\in S^{j}_{1,0}$. Then we have

$\displaystyle(\Lambda^{-1}(-1)^{k}i\operatorname{op_{sc}}(\lambda)z_{k},hD_{d}\Lambda^{-1}z_{k})$
$\displaystyle=(hD_{d}\Lambda^{-1}(-1)^{k}i\operatorname{op_{sc}}(\lambda)z_{k},\Lambda^{-1}z_{k})-ih(\Lambda^{-1}(-1)^{k}i\operatorname{op_{sc}}(\lambda)(z_{k})_{|x_{d}=0},\Lambda^{-1}(z_{k})_{|x_{d}=0})_{\partial},$

and

$\displaystyle(hD_{d}\Lambda^{-1}(-1)^{k}i\operatorname{op_{sc}}(\lambda)z_{k},\Lambda^{-1}z_{k})$
$\displaystyle=(\Lambda^{-1}(-1)^{k}i\operatorname{op_{sc}}(\lambda)hD_{d}z_{k},\Lambda^{-1}z_{k})+h(\operatorname{op_{sc}}(a_{0})z_{k},\Lambda^{-1}z_{k})$
$\displaystyle=(\Lambda^{-1}(-1)^{k}ihD_{d}z_{k},\operatorname{op_{sc}}(\lambda)^{*}\Lambda^{-1}z_{k})$
$\displaystyle\quad+((-1)^{k}i[\Lambda^{-1},\operatorname{op_{sc}}(\lambda)]hD_{d}z_{k},\Lambda^{-1}z_{k})$
$\displaystyle\quad+h(\operatorname{op_{sc}}(a_{0})z_{k},\Lambda^{-1}z_{k})$
$\displaystyle=(\Lambda^{-1}(-1)^{k}ihD_{d}z_{k},\operatorname{op_{sc}}(\lambda)\Lambda^{-1}z_{k})$
$\displaystyle\quad+h(\Lambda^{-1}hD_{d}z_{k},\operatorname{op_{sc}}(a_{-1})z_{k})+h(\operatorname{op_{sc}}(a_{0})z_{k},\Lambda^{-1}z_{k})$
$\displaystyle=-I_{2}+h(\Lambda^{-1}hD_{d}z_{k},\operatorname{op_{sc}}(a_{-1})z_{k})+h(\operatorname{op_{sc}}(a_{0})z_{k},\Lambda^{-1}z_{k}).$

From the two previous computations we obtain

$\displaystyle I_{2}+I_{3}$
$\displaystyle=-ih(\Lambda^{-1}(-1)^{k}i\operatorname{op_{sc}}(\lambda)(z_{k})_{|x_{d}=0},\Lambda^{-1}(z_{k})_{|x_{d}=0})_{\partial}$
$\displaystyle\quad+h(\Lambda^{-1}hD_{d}z_{k},\operatorname{op_{sc}}(a_{-1})z_{k})+h(\operatorname{op_{sc}}(a_{0})z_{k},\Lambda^{-1}z_{k}).$

Then from previous computations and (4.61) we obtain

(4.62) $\displaystyle
2I=2\|\Lambda^{-1}\operatorname{op_{sc}}(\lambda)z_{k}\|^{2}+h\operatorname{Re}(\Lambda^{-1}(-1)^{k}\operatorname{op_{sc}}(\lambda)(z_{k})_{|x_{d}=0},\Lambda^{-1}(z_{k})_{|x_{d}=0})_{\partial}+R$
$\displaystyle\text{where }|R|\lesssim
h\|\Lambda^{-1}hD_{d}z_{k}\|\|\Lambda^{-1}z_{k}\|+h\|\Lambda^{-1}z_{k}\|\|z_{k}\|.$

From Equation (4.59) we have

$\|\Lambda^{-1}hD_{d}z_{k}\|\lesssim\|\Lambda^{-1}H_{h}\|+\|z_{k}\|.$

We obtain from the previous estimates, (4.61), (4.62), and using that
$(z_{1})_{|x_{d}=0}=-(z_{2})_{|x_{d}=0}$,

(4.63)
$\displaystyle\sum_{k=1,2}\|\Lambda^{-1}\operatorname{op_{sc}}(\lambda)z_{k}\|^{2}+h\operatorname{Re}(\Lambda^{-1}\operatorname{op_{sc}}(\lambda)(z_{2})_{|x_{d}=0},\Lambda^{-1}(z_{2})_{|x_{d}=0})_{\partial}$
$\displaystyle\qquad\lesssim
h\sum_{k=1,2}(\|\Lambda^{-1}z_{k}\|\|z_{k}\|+\|\Lambda^{-1}z_{k}\|\|\Lambda^{-1}H_{h}\|).$

As $v_{h}=(z_{1}+z_{2})/2$ and
$\operatorname{op_{sc}}(\lambda)w_{h}=(z_{2}-z_{1})/2i$, we obtain

(4.64)
$\displaystyle\|\Lambda^{-1}\operatorname{op_{sc}}(\lambda)v_{h}\|^{2}+\|\Lambda^{-1}\operatorname{op_{sc}}(\lambda)^{2}w_{h}\|^{2}+h\operatorname{Re}(\Lambda^{-1}\operatorname{op_{sc}}(\lambda)(z_{2})_{|x_{d}=0},\Lambda^{-1}(z_{2})_{|x_{d}=0})_{\partial}$
$\displaystyle\qquad\lesssim
h\sum_{k=1,2}(\|\Lambda^{-1}z_{k}\|\|z_{k}\|+\|\Lambda^{-1}z_{k}\|\|\Lambda^{-1}H_{h}\|).$
$\displaystyle\qquad\lesssim h(\|z_{k}\|^{2}+\|\Lambda^{-1}H_{h}\|^{2}).$

By Gårding inequality (3.21) there exist $C_{0}>0$ and $C_{1}>0$ such that

$\|\Lambda^{-1}\operatorname{op_{sc}}(\lambda)v_{h}\|^{2}\geq
C_{0}\|v_{h}\|^{2}-C_{1}h\|u_{h}^{d}\|^{2},$

as $v_{h}=\operatorname{op_{sc}}(\chi)u_{h}^{d}$, the symbol of
$\operatorname{op_{sc}}(\lambda)^{*}\Lambda^{-2}\operatorname{op_{sc}}(\lambda)$
is $(1+|\xi^{\prime}|)^{-2}(R(x,\xi^{\prime})-1)+ha_{0}$ and
$(1+|\xi^{\prime}|)^{-2}(R(x,\xi^{\prime})-1)\geq C>0$ on the support of
$\chi$.

By analogous arguments, we have

$\|\Lambda^{-1}\operatorname{op_{sc}}(\lambda)^{2}w_{h}\|^{2}\geq
C_{0}\|\Lambda^{1}w_{h}\|^{2}-C_{1}h\|\Lambda^{1/2}r_{h}\|^{2}.$

As $r_{h}=w_{h}+\operatorname{op_{sc}}(1-\chi)r_{h}$ and
$\Lambda^{s}\operatorname{op_{sc}}(1-\chi)$ is bounded on $L^{2}$ for all
$s\in\mathbb{R}$ as $\chi$ is compactly supported, we have

$\|\Lambda^{-1}\operatorname{op_{sc}}(\lambda)^{2}w_{h}\|^{2}\geq
C^{\prime}_{0}\|\Lambda^{1}w_{h}\|^{2}-C_{1}h\|r_{h}\|^{2},$

and

$\|z_{k}\|\lesssim\|u_{h}^{d}\|+\|\Lambda
r_{h}\|\lesssim\|u_{h}^{d}\|+\|\Lambda^{1}w_{h}\|+\|r_{h}\|.$

For trace term in (4.64), as
$(z_{2})_{|x_{d}=0}=\operatorname{op_{sc}}(\lambda)\operatorname{op_{sc}}(\chi)r_{h}$,
we can apply Gårding inequality to obtain

$\operatorname{Re}(\Lambda^{-1}\operatorname{op_{sc}}(\lambda)(z_{2})_{|x_{d}=0},\Lambda^{-1}(z_{2})_{|x_{d}=0})_{\partial}\geq
C_{0}|\Lambda^{1/2}(w_{h})_{|x_{d}=0}|^{2}-C_{1}h|(r_{h})_{|x_{d}=0}|^{2}.$

Applying the previous estimates and (4.64), we obtain

(4.65)
$\displaystyle\|\Lambda^{1}w_{h}\|^{2}+\|v_{h}\|^{2}+h|\Lambda^{1/2}(w_{h})_{|x_{d}=0}|^{2}\lesssim
h(\|u_{h}^{d}\|^{2}+\|r_{h}\|^{2}+\|\Lambda^{-1}H_{h}\|^{2}+h|(r_{h})_{|x_{d}=0}|^{2}).$

####  4.1.4. End of proof

Taking three cutoffs on each region, hyperbolic, glancing and elliptic,
respectively $\chi_{H}$, $\chi_{G}$ and $\chi_{E}$, we obtain

$\displaystyle h|\Lambda^{1/2}(r_{h})_{|x_{d}=0}|^{2}$
$\displaystyle\qquad\lesssim
h|\Lambda^{1/2}(\operatorname{op_{sc}}(\chi_{H})r_{h})_{|x_{d}=0}|^{2}+h|\Lambda^{1/2}(\operatorname{op_{sc}}(\chi_{G})r_{h})_{|x_{d}=0}|^{2}+h|\Lambda^{1/2}(\operatorname{op_{sc}}(\chi_{E})r_{h})_{|x_{d}=0}|^{2}.$

From (4.53), we have, as $\chi_{H}$ is compactly supported,

$|\Lambda^{1/2}(\operatorname{op_{sc}}(\chi_{H})r_{h})_{|x_{d}=0}|^{2}\lesssim
C+h|(r_{h})_{|x_{d}=0}|^{2}\lesssim C+h|\Lambda^{1/2}(r_{h})_{|x_{d}=0}|^{2}.$

From (4.65), we have

$|\Lambda^{1/2}(\operatorname{op_{sc}}(\chi_{E})r_{h})_{|x_{d}=0}|^{2}\lesssim
C+h|\Lambda^{1/2}(r_{h})_{|x_{d}=0}|^{2},$

as $\|\Lambda^{-1}H_{h}\|$ is bounded from (4.59).

We obtain

$h|\Lambda^{1/2}(r_{h})_{|x_{d}=0}|^{2}\lesssim
Ch+h|\Lambda^{1/2}(\operatorname{op_{sc}}(\chi_{G})r_{h})_{|x_{d}=0}|^{2}+h^{2}|\Lambda^{1/2}(r_{h})_{|x_{d}=0}|^{2},$

then for all $h$ sufficiently small we have

$h|\Lambda^{1/2}(r_{h})_{|x_{d}=0}|^{2}\lesssim
Ch+h|\Lambda^{1/2}(\operatorname{op_{sc}}(\chi_{G})r_{h})_{|x_{d}=0}|^{2}.$

From (4.56) we have

$\limsup_{h\to
0}h|\Lambda^{1/2}(r_{h})_{|x_{d}=0}|^{2}\lesssim{\varepsilon}^{1/2},$

for all ${\varepsilon}>0$, this proves Lemma 4.4. ∎

In the following we prove that the measure defined is not null and it is null
on the set $\\{x,\ b(x)>0\\}$.

###  4.2. The measure is not identically null

######  Proposition 4.2.

The measure $m$ is not identically null.

###### Proof.

From (4.26), multiplying the first equation by $u_{h}$, integrating by parts
(see (3.16)) and taking account the boundary condition
$(u^{d}_{h})_{|x_{d}=0}=0$, we have

$O(h)=(-ih\nabla_{g}r_{h}+u_{h},u_{h})_{g}=\|u_{h}\|^{2}_{g}+(r_{h},-ih\operatorname{div}_{g}u_{h})_{g}.$

From second equation of (4.26) we replace $-ih\operatorname{div}_{g}u_{h}$ by
$r_{h}$ and we obtain

$O(h)=\|u_{h}\|^{2}_{g}-\|r_{h}\|^{2}_{g}.$

As from (4.25), we have $\|u_{h}\|+\|r_{h}\|=1$, we have $\|r_{h}\|=1/2+O(h)$
and $\|u_{h}\|=1/2+O(h)$. From the first equation of (4.26), we obtain
$\|h\nabla_{g}r_{h}\|\lesssim 1$.

We prove now that $m$ is not null. Let $\phi$ be a smooth function such that
$\phi(s)=1$ for $|s|\leq 1$ and supported in $|s|\leq 2$. For $R>0$, we set
$\phi_{R}(\xi)=\phi(|\xi|/R)$. For $\alpha\in(0,1/2)$ we have

(4.66) $\|\operatorname{op_{sc}}(1-\phi_{R})\underline{r}_{h}\|\lesssim
R^{-\alpha}\|\operatorname{op_{sc}}(\langle\xi\rangle^{\alpha})\underline{r}_{h}\|\lesssim
R^{-\alpha}(\|r_{h}\|_{L^{2}(\Omega)}+\|h\nabla_{g}r_{h}\|_{L^{2}(\Omega)})\lesssim
R^{-\alpha}.$

(See for instance [13, Lemma 4.3]).

Then for $h$ sufficiently small and $R$ sufficiently large,

$(\operatorname{op_{sc}}(\phi_{R})\underline{r}_{h},\underline{r}_{h})\geq
1/4,$

and from the definition of $m$ (see (4.1)), $m$ is not null. this gives the
result. ∎

###  4.3. The semiclassical measure null on support of damping

In this section we prove the following proposition.

######  Proposition 4.3.

Let $m$ be the semiclassical measure defined in Proposition 4.1, we have
$b(x)m=0$.

###### Proof.

From System (4.26) we have

(4.67)
$\displaystyle(ihf_{h},u_{h})_{g}+(ihq_{h},r_{h})_{g}=(-ih\nabla_{g}r_{h}+u_{h}-ihbu_{h},u_{h})_{g}+(-ih\operatorname{div}_{g}u_{h}+r_{h},r_{h})_{g}.$

Integrating by parts, using (3.16) and taking account the boundary condition
$(u^{d}_{h})_{|x_{d}=0}=0$, we have

(4.68)
$\displaystyle(-ih\nabla_{g}r_{h},u_{h})_{g}+(-ih\operatorname{div}_{g}u_{h},r_{h})_{g}=2\operatorname{Re}(-ih\nabla_{g}r_{h},u_{h})_{g}.$

From the two previous formula, we obtain

(4.69)
$\displaystyle\operatorname{Im}\big{(}(ihf_{h},u_{h})_{g}+(ihq_{h},r_{h})_{g}\big{)}=ih(bu_{h},u_{h})_{g}.$

As the left hand side is a $o(h)$, we obtain $(bu_{h},u_{h})_{g}\to 0$ as
$h\to 0$. Then from Proposition 4.1 and Proposition 5.3 below, we have
$\langle m\sum_{k=1}^{d}g^{kk}\xi_{k}^{2},b\rangle=0$.

As $b$ and $m$ are non negative,
$\displaystyle\sum_{k=1}^{d}g^{kk}\xi_{k}^{2}\gtrsim\displaystyle\sum_{j,k=1}^{d}g^{kj}\xi_{k}\xi_{j}=1$
on the support of $m$, we obtain the result. ∎

##  5\. Propagation of the support

The goal is to prove the following proposition.

######  Proposition 5.1.

We assume that $\omega$ satisfy GCC. Let $m$ the semiclassical measure
constructed from $(u_{h},r_{h})_{h}$ and satisfying (4.26), we have $m=0$.

First in this section we give some formulas yielding to the propagation
results. This is stated in the following proposition.

######  Proposition 5.2.

We denote by $p(x,\xi)=g^{jk}\xi_{j}\xi_{k}$ and
$R(x,\xi^{\prime})=\tilde{g}^{jk}\xi_{k}\xi_{j}$.

We have $\\{p,m\\}=0$ in $\Omega\times\mathbb{R}^{d}$.

In a neighborhood of $\partial\Omega$ i.e. $x_{d}=0$ in geodesic coordinates,
let $a(x,\xi^{\prime})$ be a tangential symbol compactly supported in a
neighborhood of the boundary, we have,

(5.70) $\displaystyle\langle\\{p,m\\},a\rangle=0,$ (5.71)
$\displaystyle\langle\\{p,m\\},a\xi_{d}\rangle=-\lim_{h\to
0}(\operatorname{op_{sc}}(a(R-1))(r_{h})_{|x_{d}=0},(r_{h})_{|x_{d}=0})_{\partial}.$

Observe that we do not know if $(r_{h})_{|x_{d}=0}$ is bounded in $L^{2}$,
Lemma 4.4 is too weak. But the quantity
$(\operatorname{op_{sc}}(a(R-1))(r_{h})_{|x_{d}=0},(r_{h})_{|x_{d}=0})_{\tilde{g}}$
has a limit. A priori the measure $m$ is not defined on functions in
$(x,\xi^{\prime})$ but as $m$ is supported on $g^{jp}\xi_{j}\xi_{p}-1=0$,
which is a bounded set on $\xi$, it is easy to extend $m$ to functions in
$(x,\xi^{\prime})$. At boundary we assume the metric written in geodesic
coordinates (see Section 3.1).

####  5.0.1. Computations in interior

We introduce the following quantity, where $a$ is a symbol compactly supported
in $\Omega\times\mathbb{R}^{d}$. In what follows we shall choose $b^{j}_{k}$
depending on $a$ but the choice of $b^{j}_{k}$ is more natural after the
computations below. Note that $b^{j}_{k}$ are also compactly supported in
$\Omega\times\mathbb{R}^{d}$. In interior $\underline{u}_{h}$ and
$\underline{r}_{h}$ are $u_{h}$ and $r_{h}$. Rigorously we need to introduce a
cutoff to do that but this introduces error terms of order $O(h^{\infty})$.

(5.72) $\displaystyle A$
$\displaystyle=(\operatorname{op_{sc}}(a)(u_{h}-ih\nabla_{g}r_{h}),u_{h})_{g}+(\operatorname{op_{sc}}(a)(r_{h}-ih\operatorname{div}_{g}u_{h}),r_{h})_{g}$
$\displaystyle\quad-(\operatorname{op_{sc}}(a)u_{h},u_{h}-ih\nabla_{g}r_{h})_{g}-(\operatorname{op_{sc}}(a)r_{h},r_{h}-ih\operatorname{div}_{g}u_{h})_{g}$
$\displaystyle\quad+(ih\operatorname{op_{sc}}(b^{j}_{k})(u_{h}^{k}-ih(\nabla_{g}r_{h})^{k}),u_{h}^{\ell})_{g}.$

Here and in what follows, we use Einstein notation, see Section 3.1.

Obviously, the assumptions on System (5.76) imply that $h^{-1}A\to 0$ as $h\to
0$.

We can simplify $A$ and we have,

(5.73) $\displaystyle A$
$\displaystyle=-(ih\operatorname{op_{sc}}(a)\nabla_{g}r_{h},u_{h})_{g}-(ih\operatorname{op_{sc}}(a)\operatorname{div}_{g}u_{h},r_{h})_{g}$
$\displaystyle\quad+(\operatorname{op_{sc}}(a)u_{h},ih\nabla_{g}r_{h})_{g}+(\operatorname{op_{sc}}(a)r_{h},ih\operatorname{div}_{g}u_{h})_{g}$
$\displaystyle\quad+(ih\operatorname{op_{sc}}(b^{j}_{k})u_{h}^{k},u_{h}^{\ell})_{g}+(h\operatorname{op_{sc}}(b^{j}_{k})(h\nabla_{g}r_{h})^{k},u_{h}^{\ell})_{g}.$

We integrate by parts (see (3.15)) and we write by $R$ terms estimated by
$h(\|u_{h}\|^{2}+\|r_{h}\|^{2})$ playing no role in the analysis. Observe that
no boundary term appears as $a$ is supported far away the boundary and we have

(5.74) $\displaystyle h^{-1}A$
$\displaystyle=(i([\nabla_{g},\operatorname{op_{sc}}(a)]r_{h},u_{h})_{g}+(i[\operatorname{div}_{g},\operatorname{op_{sc}}(a)]u_{h},r_{h})_{g}$
$\displaystyle\quad+(i\operatorname{op_{sc}}(b^{j}_{k})u_{h}^{k},u_{h}^{\ell})_{g}+(h\operatorname{op_{sc}}(b^{j}_{k})(\nabla_{g}r_{h})^{k}),u_{h}^{\ell})_{g}$
$\displaystyle=(i\operatorname{op_{sc}}\\{g^{jk}\xi_{k},a\\}r_{h},(u_{h})^{\ell})_{g}+(i\operatorname{op_{sc}}\\{\xi_{k},a\\}u_{h}^{k},r_{h})_{g}$
$\displaystyle\quad+(i\operatorname{op_{sc}}(b^{j}_{k})u_{h}^{k},u_{h}^{\ell})_{g}+(i\operatorname{op_{sc}}(b^{j}_{k}g^{k\mu}\xi_{\mu})r_{h},u_{h}^{\ell})_{g}+R.$

To use Formula (4.1) defining the semi-classical measure, we need a symmetry
property. Let $b^{j}=\\{g^{jk}\xi_{k},a\\}+b^{j}_{k}g^{k\ell}\xi_{\ell}$. The
symmetry required is $g_{pj}b^{j}=\\{\xi_{p},a\\}$. Then we have

$g_{pj}b^{j}=g_{pj}\\{g^{jk}\xi_{k},a\\}+g_{pj}b^{j}_{k}g^{k\ell}\xi_{\ell},$

and

$\\{\xi_{p},a\\}=\\{g_{pj}g^{jk}\xi_{k},a\\}=g_{pj}\\{g^{jk}\xi_{k},a\\}+g^{jk}\xi_{k}\\{g_{pj},a\\}.$

Then

$g_{pj}b^{j}=\\{\xi_{p},a\\}-g^{jk}\xi_{k}\\{g_{pj},a\\}+g_{pj}b^{j}_{k}g^{k\ell}\xi_{\ell}.$

Let $b_{k}^{j}=g^{jq}\\{g_{qk},a\\}$, we have

$g_{pj}b^{j}_{k}g^{k\ell}\xi_{\ell}=g_{pj}g^{jq}\\{g_{qk},a\\}g^{k\ell}\xi_{\ell}=\\{g_{pk},a\\}g^{k\ell}\xi_{\ell}=g^{jk}\xi_{k}\\{g_{pj},a\\},$

changing the summation indexes. This choice of $b_{k}^{j}$ allows to apply
(4.1).

With the previous definition of $b^{j}_{k}$, Formula (5.74) writes

$\displaystyle h^{-1}A$
$\displaystyle=(i\operatorname{op_{sc}}\\{g^{jk}\xi_{k},a\\}r_{h},(u_{h})^{\ell})_{g}+(i\operatorname{op_{sc}}\\{\xi_{k},a\\}u_{h}^{k},r_{h})_{g}$
$\displaystyle\quad+(i\operatorname{op_{sc}}(g^{jq}\\{g_{qk},a\\})u_{h}^{k},u_{h}^{\ell})_{g}+(i\operatorname{op_{sc}}(g^{jq}\\{g_{qk},a\\}g^{k\mu}\xi_{\mu})r_{h},u_{h}^{\ell})_{g}+R.$

Then $ih^{-1}A$ converges to
$-\langle\mu_{j}^{k},g^{jq}\\{g_{qk},a\\}\rangle-2\langle\nu_{j},\\{g^{jk}\xi_{k},a\\}+g^{jq}\\{g_{qk},a\\}g^{k\mu}\xi_{\mu}\rangle=0$.
From Proposition 4.1, $\mu_{j}^{k}=g^{kp}\xi_{p}\xi_{j}m$ and
$\nu_{j}=-m\xi_{j}$, we thus obtain

(5.75) $-\langle g^{kp}\xi_{p}\xi_{j}m,g^{jq}\\{g_{qk},a\\}\rangle+2\langle
m\xi_{j},\\{g^{jk}\xi_{k},a\\}+g^{jq}\\{g_{qk},a\\}g^{k\mu}\xi_{\mu}\rangle=\langle
m,F\rangle=0,$

where

$F=-g^{kp}\xi_{p}\xi_{j}g^{jq}\\{g_{qk},a\\}+2\xi_{j}\\{g^{jk}\xi_{k},a\\}+2\xi_{j}g^{jq}\\{g_{qk},a\\}g^{k\mu}\xi_{\mu}.$

Observing that
$0=\\{g^{jq}g_{qk},a\\}=g^{jq}\\{g_{qk},a\\}+g_{qk}\\{g^{jq},a\\}$, we have

$\displaystyle F$
$\displaystyle=g^{kp}g_{qk}\xi_{p}\xi_{j}\\{g^{jq},a\\}+2\xi_{j}\\{g^{jk}\xi_{k},a\\}-2g_{qk}\xi_{j}\\{g^{jq},a\\}g^{k\mu}\xi_{\mu}$
$\displaystyle=\xi_{q}\xi_{j}\\{g^{jq},a\\}+2\xi_{j}\\{g^{jk}\xi_{k},a\\}-2\xi_{j}\xi_{q}\\{g^{jq},a\\}$
$\displaystyle=\xi_{q}\xi_{j}\\{g^{jq},a\\}+2g^{jq}\xi_{j}\\{\xi_{q},a\\}.$

As

$\\{g^{jq}\xi_{j}\xi_{q},a\\}=g^{jq}\xi_{j}\\{\xi_{q},a\\}+g^{jq}\xi_{q}\\{\xi_{j},a\\}+\xi_{j}\xi_{q}\\{g^{jq},a\\}=2g^{jq}\xi_{j}\\{\xi_{q},a\\}+\xi_{j}\xi_{q}\\{g^{jq},a\\},$

by symmetry of $g^{jq}$. Then $F=\\{g^{jq}\xi_{j}\xi_{q},a\\}$. Then from
(5.75) we have $\langle m,\\{g^{jq}\xi_{j}\xi_{q},a\\}\rangle=0$ or
$\langle\\{g^{jq}\xi_{j}\xi_{q},m\\},a\rangle=0$. This proves Proposition 5.2
in $\Omega\times\mathbb{R}^{d}$.

To prove Proposition 5.2 at boundary, we have to use tangential operator, we
consider System (4.26), instead System (4.32) and to conclude we have to prove
an analogous result as Formula (4.1) where one integrate on $\Omega$ instead
on $\mathbb{R}^{d}$.

Let us recall System (4.26). The system satisfies by $u_{h}$ and $r_{h}$ is
the following.

(5.76) $\begin{cases}&u_{h}-ih\nabla_{g}r_{h}=hf_{h},\text{ in }\Omega\\\
&-ih\operatorname{div}_{g}u_{h}+r_{h}=hq_{h},\text{ in }\Omega\\\
&u_{h}.n=0\text{ on }\partial\Omega\end{cases}$

where $\|f_{h}\|+\|q_{h}\|\to 0$ as $h\to 0$, and $\|u_{h}\|+\|r_{h}\|$
bounded uniformly with respect $h$.

####  5.0.2. Limit on quantities on $\Omega$.

In the following result, we prove that the measure defined for
$(\underline{u_{h}},\underline{r_{h}})$ also appear in the limit of some
quantities computing on $x_{d}>0$.

######  Proposition 5.3.

We assume
$a_{j}^{k},b^{j},c\in\mathscr{C}^{\infty}(\mathbb{R}^{d}_{x}\times\mathbb{R}^{d-1}_{\xi^{\prime}})$
compactly supported, satisfying $g^{\ell j}a_{j}^{k}=g^{kj}a_{j}^{\ell}$ and
supported (in $x$) on a neighborhood of a point of the boundary. We assume the
metric $g$ written in geodesic coordinates. Then

  1. (1)

$(\operatorname{op_{sc}}(a_{j}^{k})u_{h}^{j},u_{h}^{k})_{g}$ converges to
$\langle\mu_{j}^{k},a_{k}^{j}\rangle=\langle
g^{kp}\xi_{p}\xi_{j}m,a_{k}^{j}\rangle$ as $h$ to 0.

  2. (2)

$((\operatorname{op_{sc}}(b^{j})r_{h}),(u_{h}^{k}))_{g}+(g_{jk}\operatorname{op_{sc}}(b^{j})u_{h}^{k},r_{h})_{g}$
converges to $2\langle\nu_{j},b^{j}\rangle=-2\langle\xi_{j}m,b^{j}\rangle$ as
$h$ to 0.

  3. (3)

$(\operatorname{op_{sc}}(c)r_{h},r_{h})_{g}$ converges to $\langle m,c\rangle$
as $h$ to 0.

  4. (4)

$(\operatorname{op_{sc}}(c)(-ih\partial_{d})u_{h}^{d},u_{h}^{d})_{g}$
converges to
$\langle\mu_{d}^{d},c\xi_{d}\rangle=\langle\xi_{d}^{2}m,c\xi_{d}\rangle$ as
$h$ to 0.

  5. (5)

$(\operatorname{op_{sc}}(c)(-ih\partial_{d})r_{h},u_{h}^{d})_{g}+(\operatorname{op_{sc}}(c)(-ih\partial_{d})u_{h}^{d},r_{h})_{g}$
converges to $2\langle\nu_{d},c\rangle=-2\langle\xi_{d}m,c\rangle$ as $h$ to
0.

  6. (6)

$(\operatorname{op_{sc}}(c)(-ih\partial_{d})r_{h},r_{h})_{g}$ converges to
$\langle m,c\xi_{d}\rangle$ as $h$ to 0.

In previous formulas, the integration in the inner product is taken on
$\Omega$ that is, on local coordinates, on $x_{d}>0$.

###### Proof.

From Proposition 4.1, $m$ is supported on $(g^{jp}(x)\xi_{p}\xi_{j}-1)=0$. The
$m$ is compactly supported and the expressions given at the limit make sens.

As $r_{h}$ and $h\nabla_{g}r_{h}$ are in $L^{2}(x_{d}>0)$, the extension
$\underline{r}_{h}$ is in $H^{s}_{sc}$ for $s<1/2$ and

$\|\underline{r}_{h}\|_{H^{s}_{sc}}\lesssim\|r_{h}\|_{L^{2}(x_{d}>0)}+\|h\nabla_{g}r_{h}\|_{L^{2}(x_{d}>0)}.$

To prove the three first item, we observe that we can replace $u_{h}$ and
$r_{h}$ by the extension. Then we introduce
$\operatorname{op_{sc}}(\phi(\xi_{d}/R))$ and Proposition 4.1 gives the result
as for instance
$\langle\mu_{j}^{k},a_{k}^{j}\phi(\xi_{d}/R)\rangle=\langle\mu_{j}^{k},a_{k}^{j}\rangle$
for $R$ sufficiently large. We have to estimate the remainder terms with
$\operatorname{op_{sc}}(1-\phi(\xi_{d}/R))$. To do that we have to distinguish
different cases. To prove the first item, if $j\leq d-1$ or $k\leq d-1$ we
have in the first case to estimate
$(\operatorname{op_{sc}}(a_{j}^{k})\operatorname{op_{sc}}(1-\phi(\xi_{d}/R))\underline{u}_{h}^{j},\underline{u}_{h}^{k})_{g}$.
As $u^{j}=ih(\nabla_{g}r_{h})^{j}+hf_{h}^{j}$ we have

$(\operatorname{op_{sc}}(a_{j}^{k})\operatorname{op_{sc}}(1-\phi(\xi_{d}/R))\underline{u}_{h}^{j},\underline{u}_{h}^{k})_{g}=(\operatorname{op_{sc}}(a_{j}^{k})\operatorname{op_{sc}}(1-\phi(\xi_{d}/R))(ih(\nabla_{g}\underline{r}_{h})^{j}+h\underline{f}_{h}^{j}),\underline{u}_{h}^{k})_{g}.$

From the form of the metric $g$, $(\nabla_{g}r_{h})^{j}$ does not contain term
$\partial_{d}r_{h}$ and as $a_{j}^{k}$ is compactly supported and by symbolic
calculus we can write
$\operatorname{op_{sc}}(a_{j}^{k})(-ih\partial_{\ell})=\operatorname{op_{sc}}(a_{j}^{k}\xi_{\ell})$
for $\ell\leq d-1$ modulo $h$ times a bounded operator on $L^{2}$. Obviously
$\operatorname{op_{sc}}(a_{j}^{k}\xi_{\ell})$ is bounded on $L^{2}$ and the
term $\operatorname{op_{sc}}(1-\phi(\xi_{d}/R)\underline{r}_{h}$ can be
estimate from estimate (4.66)

As this term goes to 0 as $R$ to $\infty$ and
$(\operatorname{op_{sc}}(a_{j}^{k})u_{h}^{j},u_{h}^{k})_{g}$ does not depend
on $R$, we obtain the result.

If $j=d$ and $k=d$, we have $u_{h}^{d}=ih\partial_{d}r_{h}+hf_{h}^{d}$. Then,
from (5.76), using $(u_{h}^{d})_{|x_{d}=0}$, symbolic calculus and integrating
by parts, we have

$\displaystyle 2(\operatorname{op_{sc}}(a)u_{h}^{d},u_{h}^{d})$
$\displaystyle=(\operatorname{op_{sc}}(a)(-ih\partial_{d})r_{h},u_{h}^{d})+(\operatorname{op_{sc}}(a)u_{h}^{d},(-ih\partial_{d})r_{h})+O(h)$
$\displaystyle=(\operatorname{op_{sc}}(a)r_{h},(-ih\partial_{d})u_{h}^{d})+(\operatorname{op_{sc}}(a)(-ih\partial_{d})u_{h}^{d},r_{h})+O(h)$
$\displaystyle=(\operatorname{op_{sc}}(a)r_{h},(ih\operatorname{div}_{\tilde{g}}){\tilde{u}}_{h}-r_{h})+(\operatorname{op_{sc}}(a)\big{(}(ih\operatorname{div}_{\tilde{g}}){\tilde{u}}_{h}-r_{h}\big{)},r_{h})+O(h)$
$\displaystyle=(r_{h},\operatorname{op_{sc}}(a)^{*}\big{(}(ih\operatorname{div}_{\tilde{g}}){\tilde{u}}_{h}-r_{h}\big{)})+(\operatorname{op_{sc}}(a)\big{(}(ih\operatorname{div}_{\tilde{g}}){\tilde{u}}_{h}-r_{h}\big{)},r_{h})+O(h).$

As $(\operatorname{op_{sc}}(a)(ih\operatorname{div}_{\tilde{g}})$ and
$\operatorname{op_{sc}}(a)^{*}(ih\operatorname{div}_{\tilde{g}})$ are
tangential bounded operators, we can conclude as the previous terms, extending
the $L^{2}$ functions to $\mathbb{R}^{d}$ by 0 in $x_{d}<0$ and introducing a
cutoff $\operatorname{op_{sc}}(\phi(\xi_{d}/R))$. Terms
$(\operatorname{op_{sc}}(a)r_{h},r_{h})$ and
$((\operatorname{op_{sc}}(b^{j})r_{h}),(u_{h}^{k}))_{g}+(g_{jk}\operatorname{op_{sc}}(b^{j})u_{h}^{k},r_{h})_{g}$
can be treated following same method, as we can introduce the cutoff
$\operatorname{op_{sc}}(1-\phi(\xi_{d}/R))$ on terms $r_{h}$.

To treat the second and third items, we can follow the proof of first item,
where we have already computed terms as
$(\operatorname{op_{sc}}(a)r_{h},u_{h}^{j})$ for $j\leq d-1$ or
$(\operatorname{op_{sc}}(a)r_{h},u_{h}^{d})$.

For the forth item we can replace $-ih\partial_{d}u_{h}^{d}$ by
$ih\operatorname{div}_{\tilde{g}}{\tilde{u}}_{h}-r_{h}$ and we obtain
analogous terms treated in first item.

For firth item, we can replace $ih\partial_{d}r_{h}$ by $u_{h}^{d}$ and
$-ih\partial_{d}u_{h}^{d}$ by
$ih\operatorname{div}_{\tilde{g}}{\tilde{u}}_{h}-r_{h}$. We can treat terms
obtained as previous terms.

For sixth item we replace $ih\partial_{d}r_{h}$ by $u_{h}^{d}$ and we find
terms already treated. ∎

####  5.0.3. Computations at boundary, tangential symbol

Let $a(x,\xi^{\prime})$ be a tangential symbol compactly supported in a
neighborhood of a point of boundary. We introduce the same $A$ as in Formula
(5.72), where $a$ is now a tangential symbol. We obtain by the same
simplification Formula (5.73). We integrate by parts and two terms have to be
consider $(\operatorname{op_{sc}}(a)r_{h},ih\operatorname{div}_{g}u_{h})_{g}$
and $(\operatorname{op_{sc}}(a)u_{h},ih\nabla_{g}r_{h})_{g}$, as other terms
do not give boundary terms. Applying (3.16) the first gives

$(\operatorname{op_{sc}}(a)r_{h},ih\operatorname{div}_{g}u_{h})_{g}=(ih\nabla_{g}\operatorname{op_{sc}}(a)r_{h},u_{h})_{g}-(\operatorname{op_{sc}}(a)(r_{h})_{|x_{d}=0},ih(u_{h}^{d})_{|x_{d}=0})_{\partial},$

and the second gives

$(\operatorname{op_{sc}}(a)u_{h},ih\nabla_{g}r_{h})_{g}=(ih\operatorname{div}_{g}\operatorname{op_{sc}}(a)u_{h},r_{h})_{g}-(\operatorname{op_{sc}}(a)(u_{h}^{d})_{|x_{d}=0},ih(r_{h})_{|x_{d}=0})_{\partial},$

and as $(u^{d}_{h})_{|x_{d}=0}=0$, in both cases, the boundary term
disappears. From that, the rest of the proof does not change and we find the
same conclusion with $a$ a tangential symbol,
$\langle\\{g^{jq}\xi_{j}\xi_{q},m\\},a\rangle=0$.

####  5.0.4.  Computations at boundary, tangential symbol with normal
derivative

We need computation with symbol as $a(x,\xi^{\prime})\xi_{d}$, but we cannot
compute directly considering previous formula. For instance in the first term
of (5.72) we should have a term $\partial_{d}f_{h}$ if we replace $a$ by
$a\xi_{d}$ and if we put $\partial_{d}$ at the right hand side of inner
product we have term $\partial_{d}u_{h}$ and the tangential part of this term
is not in $L^{2}$. We have to introduce another quantity.

From System (4.26), we obtain the following $2\times 2$ system, replacing
$u_{h}^{j}$ in function of $r_{h}$ and $f_{h}^{j}$ as we do to obtain System
(4.46), we have

(5.77)
$\begin{cases}&-ih\partial_{d}r_{h}+u_{h}^{d}-ihbu_{h}^{d}=h\tilde{f}_{h}^{d}\\\
&-ih|g|^{-1/2}\partial_{d}(|g|^{1/2}u_{h}^{d})-ih\operatorname{div}_{\tilde{g}}(ih\nabla_{\tilde{g}}r_{h})+r_{h}=h\tilde{q}_{h},\end{cases}$

where $\tilde{f}_{h}^{d}=if_{h}^{d}$,
$\tilde{q}_{h}=iq_{h}+ih\operatorname{div}_{\tilde{g}}{\tilde{f}_{h}}+ih\operatorname{div}_{\tilde{g}}(b\tilde{u})$,
$\tilde{f}_{h}=(f^{j}_{h})_{j=1,\dots,d-1}$, and
$\tilde{u}_{h}=(u^{j}_{h})_{j=1,\dots,d-1}$.

For a symbol $a(x,\xi^{\prime})$ compactly supported, we introduce the
following quantity

(5.78) $\displaystyle A$
$\displaystyle=(\operatorname{op_{sc}}(a)(u_{h}^{d}-ih\partial_{d}r_{h}),ih\operatorname{div}_{\tilde{g}}(ih\nabla_{\tilde{g}}r_{h})-r_{h})_{g}$
$\displaystyle\quad-(\operatorname{op_{sc}}(a)(ih\operatorname{div}_{\tilde{g}}(ih\nabla_{\tilde{g}}r_{h})-r_{h}),u_{h}^{d}-ih\partial_{d}r_{h})_{g}$
$\displaystyle\quad-(\operatorname{op_{sc}}(a)(-ih|g|^{-1/2}\partial_{d}(|g|^{1/2}u_{h}^{d}))-ih\operatorname{div}_{\tilde{g}}(ih\nabla_{\tilde{g}}r_{h})+r_{h},u_{h}^{d})_{g}$
$\displaystyle\quad+(\operatorname{op_{sc}}(a)u_{h}^{d},-ih|g|^{-1/2}\partial_{d}(|g|^{1/2}u_{h}^{d})-ih\operatorname{div}_{\tilde{g}}(ih\nabla_{\tilde{g}}r_{h})+r_{h})_{g}.$

We can simplify some terms with $u_{h}^{d}$, and we obtain

(5.79) $\displaystyle A$
$\displaystyle=(\operatorname{op_{sc}}(a)(-ih\partial_{d}r_{h}),ih\operatorname{div}_{\tilde{g}}(ih\nabla_{\tilde{g}}r_{h})-r_{h})_{g}$
$\displaystyle\quad-(\operatorname{op_{sc}}(a)(ih\operatorname{div}_{\tilde{g}}(ih\nabla_{\tilde{g}}r_{h})-r_{h}),-ih\partial_{d}r_{h})_{g}$
$\displaystyle\quad-(\operatorname{op_{sc}}(a)(-ih|g|^{-1/2}\partial_{d}(|g|^{1/2}u_{h}^{d})),u_{h}^{d})_{g}$
$\displaystyle\quad+(\operatorname{op_{sc}}(a)u_{h}^{d},-ih|g|^{-1/2}\partial_{d}(|g|^{1/2}u_{h}^{d}))_{g}.$

We have to compute $A$ from two different manners. First we integrate by parts
applying (3.16), we have

$\displaystyle(\operatorname{op_{sc}}(a)(-ih\partial_{d}r_{h}),ih\operatorname{div}_{\tilde{g}}(ih\nabla_{\tilde{g}}r_{h})-r_{h})_{g}=((ih\operatorname{div}_{\tilde{g}}(ih\nabla_{\tilde{g}})-1)\operatorname{op_{sc}}(a)(-ih\partial_{d})r_{h},r_{h})_{g},$
$\displaystyle-(\operatorname{op_{sc}}(a)(ih\operatorname{div}_{\tilde{g}}(ih\nabla_{\tilde{g}}r_{h})-r_{h}),-ih\partial_{d}r_{h})_{g}=(ih\partial_{d}(\operatorname{op_{sc}}(a)(ih\operatorname{div}_{\tilde{g}}(ih\nabla_{\tilde{g}}r_{h})-r_{h})),r_{h})_{g}$
$\displaystyle\quad+ih(\operatorname{op_{sc}}(a)(ih\operatorname{div}_{\tilde{g}}(ih\nabla_{\tilde{g}}r_{h})-r_{h})_{|x_{d}=0},(r_{h})_{|x_{d}=0})_{\partial},$
$\displaystyle(\operatorname{op_{sc}}(a)u_{h}^{d},-ih|g|^{-1/2}\partial_{d}(|g|^{1/2}u_{h}^{d}))_{g}=(-ih|g|^{-1/2}\partial_{d}(|g|^{1/2}\operatorname{op_{sc}}(a)u_{h}^{d}),u_{h}^{d})_{g},$

as $(u_{h}^{d})_{|x_{d}=0}=0$ in the last computation, the boundary term is
null.

From (5.79) and previous computations, we have

(5.80) $\displaystyle A$
$\displaystyle=((ih\operatorname{div}_{\tilde{g}}(ih\nabla_{\tilde{g}})-1)\operatorname{op_{sc}}(a)(-ih\partial_{d})r_{h},r_{h})_{g}$
$\displaystyle\quad-(-ih\partial_{d}(\operatorname{op_{sc}}(a)(ih\operatorname{div}_{\tilde{g}}(ih\nabla_{\tilde{g}}r_{h})-r_{h})),r_{h})_{g}$
$\displaystyle\quad+ih(\operatorname{op_{sc}}(a)(ih\operatorname{div}_{\tilde{g}}(ih\nabla_{\tilde{g}}r_{h})-r_{h})_{|x_{d}=0},(r_{h})_{|x_{d}=0})_{\partial}$
$\displaystyle\quad-(\operatorname{op_{sc}}(a)(-ih|g|^{-1/2}\partial_{d}(|g|^{1/2}u_{h}^{d})),u_{h}^{d})_{g}$
$\displaystyle\quad+(-ih|g|^{-1/2}\partial_{d}(|g|^{1/2}\operatorname{op_{sc}}(a)u_{h}^{d}),u_{h}^{d})_{g}$
$\displaystyle=(K_{1}r_{h},r_{h})_{g}+(K_{2}u_{h}^{d},u_{h}^{d})_{g}+ih(\operatorname{op_{sc}}(a)(ih\operatorname{div}_{\tilde{g}}(ih\nabla_{\tilde{g}}r_{h})-r_{h})_{|x_{d}=0},(r_{h})_{|x_{d}=0})_{\partial},$

where

$\displaystyle
K_{1}=(ih\operatorname{div}_{\tilde{g}}(ih\nabla_{\tilde{g}})-1)\operatorname{op_{sc}}(a)(-ih\partial_{d})-ih\partial_{d}(\operatorname{op_{sc}}(a)(ih\operatorname{div}_{\tilde{g}}(ih\nabla_{\tilde{g}})-1))$
$\displaystyle
K_{2}=-ih|g|^{-1/2}\partial_{d}|g|^{1/2}\operatorname{op_{sc}}(a)-\operatorname{op_{sc}}(a)(-ih|g|^{-1/2})\partial_{d}|g|^{1/2}=[-ih|g|^{-1/2}\partial_{d}|g|^{1/2},\operatorname{op_{sc}}(a)].$

The principal symbol of $K_{2}$ is $-ih\\{\xi_{d},a\\}$ which is a tangential
symbol. We write

$\displaystyle K_{1}$
$\displaystyle=[ih\operatorname{div}_{\tilde{g}}(ih\nabla_{\tilde{g}})-1,\operatorname{op_{sc}}(a)](-ih\partial_{d})$
$\displaystyle\quad+\operatorname{op_{sc}}(a)[ih\operatorname{div}_{\tilde{g}}(ih\nabla_{\tilde{g}})-1,-ih\partial_{d}]+[\operatorname{op_{sc}}(a),-ih\partial_{d}](ih\operatorname{div}_{\tilde{g}}(ih\nabla_{\tilde{g}})-1).$

The principal symbol of $K_{1}$ is then

$\displaystyle-
ih\\{R(x,\xi^{\prime})-1,a(x,\xi^{\prime})\\}\xi_{d}-iha(x,\xi^{\prime})\\{R(x,\xi^{\prime})-1,\xi_{d}\\}-ih\\{a(x,\xi^{\prime}),\xi_{d}\\}(R(x,\xi^{\prime})-1)$
$\displaystyle=-ih\\{R(x,\xi^{\prime}),a(x,\xi^{\prime})\\}\xi_{d}-iha(x,\xi^{\prime})\\{R(x,\xi^{\prime}),\xi_{d}\\}-ih\\{a(x,\xi^{\prime}),\xi_{d}\\}(R(x,\xi^{\prime})-1).$

Then from Proposition 5.3, we have

$\displaystyle\lim_{h\to
0}h^{-1}\big{(}(K_{1}r_{h},r_{h})_{g}+(K_{2}u_{h}^{d},u_{h}^{d})_{g}\big{)}$
$\displaystyle=\langle
m,-i\\{R(x,\xi^{\prime}),a(x,\xi^{\prime})\\}\xi_{d}-ia(x,\xi^{\prime})\\{R(x,\xi^{\prime}),\xi_{d}\\}-i\\{a(x,\xi^{\prime}),\xi_{d}\\}(R(x,\xi^{\prime})-1)-i\xi_{d}^{2}\\{\xi_{d},a(x,\xi^{\prime})\\}\rangle$
$\displaystyle=\langle
m,-i\\{R(x,\xi^{\prime}),\xi_{d}a(x,\xi^{\prime})\\}-2i\xi_{d}^{2}\\{\xi_{d},a(x,\xi^{\prime})\\}\rangle,$

as $m$ is null on $\xi_{d}^{2}+R(x,\xi^{\prime})-1$. Observing that

$\\{\xi_{d}^{2}+R(x,\xi^{\prime}),\xi_{d}a(x,\xi^{\prime})\\}=2\xi_{d}^{2}\\{\xi_{d},a(x,\xi^{\prime})\\}+\\{R(x,\xi^{\prime}),\xi_{d}a(x,\xi^{\prime})\\},$

we obtain

(5.81) $\displaystyle\lim_{h\to
0}h^{-1}\big{(}(K_{1}r_{h},r_{h})_{g}+(K_{2}u_{h}^{d},u_{h}^{d})_{g}\big{)}=\langle
m,-i\\{\xi_{d}^{2}+R(x,\xi^{\prime}),\xi_{d}a(x,\xi^{\prime})\\}\rangle.$

Second we compute the limit of $h^{-1}A$, using $2\times 2$ system. From
(5.77) and (5.78) we have

$\displaystyle A$
$\displaystyle=(\operatorname{op_{sc}}(a)(h\tilde{f}_{h}^{d}+ihbu_{h}^{d}),ih\operatorname{div}_{\tilde{g}}(ih\nabla_{\tilde{g}}r_{h})-r_{h})_{g}$
$\displaystyle\quad-(\operatorname{op_{sc}}(a)(ih\operatorname{div}_{\tilde{g}}(ih\nabla_{\tilde{g}}r_{h})-r_{h}),ihbu_{h}^{d}+h\tilde{f}_{h}^{d})_{g}-(h\operatorname{op_{sc}}(a)\tilde{q}_{h},u_{h}^{d})_{g}+(\operatorname{op_{sc}}(a)u_{h}^{d},h\tilde{q}_{h})_{g}.$

Observe that $\partial_{j}\operatorname{op_{sc}}(a)$ and
$\operatorname{op_{sc}}(a)\partial_{j}$ for $j=1,\dots d-1$ are tangential
operators bounded on $L^{2}$ as $a(x,\xi^{\prime})$ is compactly supported.
Some terms in $\tilde{q}_{h}$ can be write
$\operatorname{op_{sc}}(a)\operatorname{div}_{\tilde{g}}\tilde{f}_{h}$ or
$\operatorname{op_{sc}}(a)^{*}\operatorname{div}_{\tilde{g}}\tilde{f}_{h}$,
these terms are bounded on $L^{2}$ and disappear passing to the limit. Then we
have

$\displaystyle A$
$\displaystyle=(ih\operatorname{op_{sc}}(a)(bu_{h}^{d}),ih\operatorname{div}_{\tilde{g}}(ih\nabla_{\tilde{g}}r_{h})-r_{h})_{g}-(\operatorname{op_{sc}}(a)(ih\operatorname{div}_{\tilde{g}}(ih\nabla_{\tilde{g}}r_{h})-r_{h}),ihbu_{h}^{d})_{g}$
$\displaystyle\quad-(h\operatorname{op_{sc}}(a)ih\operatorname{div}_{\tilde{g}}(b\tilde{u}_{h}),u_{h}^{d})_{g}+(\operatorname{op_{sc}}(a)u_{h}^{d},ih^{2}\operatorname{div}_{\tilde{g}}(b\tilde{u}_{h}))_{g}+o(h).$

Integrating by parts, the two last terms, we have

$\displaystyle-(h\operatorname{op_{sc}}(a)ih\operatorname{div}_{\tilde{g}}(b\tilde{u}_{h}),u_{h}^{d})_{g}+(\operatorname{op_{sc}}(a)u_{h}^{d},ih^{2}\operatorname{div}_{\tilde{g}}(b\tilde{u}_{h}))_{g}$
$\displaystyle\quad=-(h\operatorname{op_{sc}}(a)ih\operatorname{div}_{\tilde{g}}(b\tilde{u}_{h}),u_{h}^{d})_{g}+(ih^{2}b\nabla_{\tilde{g}}\operatorname{op_{sc}}(a)u_{h}^{d},\tilde{u}_{h})_{g}$
$\displaystyle=h(\operatorname{op_{sc}}(M)u_{h},u_{h})_{g},$

where $M$ is a matrix of symbole supported on the support of $b$. This term
gives 0 passing to the limit in $h$ as $bm=0$ from Proposition 4.3.

From Proposition 5.3, item (2), choosing $b^{d}=iab(R-1)$ and $b^{j}=0$ for
$j=1,\dots d-1$, we obtain

(5.82) $\lim_{h\to 0}h^{-1}A=\langle
m,-2i\xi_{d}a(x,\xi^{\prime})b(x)(R(x,\xi^{\prime})-1)\rangle=0,$

from Proposition 4.3 and from (5.80), (5.81) and (5.82), we obtain that

$(\operatorname{op_{sc}}(a)(ih\operatorname{div}_{\tilde{g}}(ih\nabla_{\tilde{g}}r_{h})-r_{h})_{|x_{d}=0},(r_{h})_{|x_{d}=0})_{\partial},$

has a limit when $h$ goes to 0, and we have

$\langle\\{\xi_{d}+R(x,\xi^{\prime}),m\\},\xi_{d}a(x,\xi^{\prime})\rangle=\lim_{h\to
0}(-\operatorname{op_{sc}}(a)(ih\operatorname{div}_{\tilde{g}}(ih\nabla_{\tilde{g}}r_{h})-r_{h})_{|x_{d}=0},(r_{h})_{|x_{d}=0})_{\partial}.$

As the principal symbol of
$\operatorname{op_{sc}}(a)(ih\operatorname{div}_{\tilde{g}}(ih\nabla_{\tilde{g}})-1)$
is $a(x,\xi^{\prime})(R(x,\xi^{\prime})-1)(x,\xi^{\prime})$, we obtain the
last result of Proposition 5.2.

Proposition 5.1 is a consequence of Proposition 5.2. To prove that we follows
works [19, 20, 28, 29, 7, 9, 38]. To be complete, we give the proof in Annex
A. This proof follows the one given in [13].

##  Appendix A Geometric Control Condition and propagation

###  A.1. Geometry

Here we give the geometrical notion we use in this article. This framework
comes from Melrose and Sjöstrand [36, 37] and the reader may also find in
Hörmander [24, Chapter 24] more informations and proofs. The characterisation
of symplectic sub-manifold is probably classical and more details can be found
in Grigis [22].

Assumption on the symbol. We recall the definition of the symbol of $P$ given
in Proposition 5.2,

(1.83) $\displaystyle p(x,\xi)=\sum_{1\leq j,k\leq
d}g^{jk}(x)\xi_{j}\xi_{k}-1,$

where $g^{jk}$ are $\mathscr{C}^{\infty}(\overline{\Omega})$. Locally in a
neighborhood of the boundary we can define $\Omega$ by $\varphi>0$ with
$d\varphi\neq 0$. We can also choose coordinates (i.e. _normal geodesic
coordinates_) such that $\varphi(x)=x_{d}$ and
$p(x,\xi)=\xi_{d}^{2}+R(x,\xi^{\prime})-1$ where $x=(x^{\prime},x_{d})$ and
$\xi=(\xi^{\prime},\xi_{d})$.

Symplectic sub-manifold $\Sigma$. We can define a symplectic manifold
$\Sigma$, contained into $T^{*}\mathbb{R}^{d}\cap\\{\varphi=0\\}$. We set
$\Sigma=\\{(x,\xi),\ \varphi(x)=0\text{ and }\\{\varphi,p\\}(x,\xi)=0\\}$. The
set $\Sigma$ is a symplectic manifold as $\\{\varphi,\\{\varphi,p\\}\\}\neq
0$. In coordinates $(x^{\prime},x_{d})$, we have
$\Sigma=\\{x_{d}=0,\xi_{d}=0\\}$, this manifold is isomorphic to
$T^{*}\partial\Omega$ and described by coordinates
$(x^{\prime},\xi^{\prime})$.

The Hamiltonian vector field $H_{p}$ is not a vector field on $\Sigma$, but
for all $X$ a vector field on $T^{*}\mathbb{R}^{d}$, we can find unique
fonctions $\alpha$ and $\beta$ such $X+\alpha H_{\varphi}+\beta
H_{\\{\varphi,p\\}}$ is a vector field on $\Sigma$. For $H_{p}$ we denote the
associated vector $H^{\prime}_{p}$ and an elementary computation leads to

$\displaystyle
H^{\prime}_{p}=H_{p}+\frac{\\{p,\\{p,\varphi\\}}{\\{\varphi,\\{\varphi,p\\}}H_{\varphi}.$

In coordinates $(x^{\prime},x_{d})$, $H^{\prime}_{p}$ only depends on $R$ and
we have

$\displaystyle
H^{\prime}_{p}=H^{\prime}_{R}=\sum_{j=1}^{d-1}(\partial_{\xi_{j}}R(x^{\prime},0,\xi^{\prime})\partial_{x_{j}}-\partial_{x_{j}}R(x^{\prime},0,\xi^{\prime})\partial_{\xi_{j}}).$

In particular the integral curves associated with $H^{\prime}_{p}$ starting
from a point into $\Sigma$ stay into $\Sigma$. In coordinates
$(x^{\prime},x_{d})$, we denote the integral curve starting from
$(x^{\prime},\xi^{\prime})$, either $\gamma_{g}(s;x^{\prime},\xi^{\prime})$,
either $\gamma_{g}(x^{\prime},\xi^{\prime})$, if $s$ is implicit or
$\gamma_{g}(s)$, if $(x^{\prime},\xi^{\prime})$ is implicit.

Description and topology of $T^{*}_{b}\Omega$. Let
$T^{*}_{b}\Omega=T^{*}\partial\Omega\cup T^{*}\Omega$, this set is equipped
with the following topology.

First if $\rho\in T^{*}\Omega$, a set $V$ is a neighborhood of $\rho$ if $V$
contains an open set $W$ of $T^{*}\Omega$ such that $\rho\in W$.

Second if $\rho=(x_{0}^{\prime},\xi_{0}^{\prime})\in T^{*}\partial\Omega$, a
set $V$ is a neighborhood of $\rho$ if $V$ contains a set

$\displaystyle\\{(x^{\prime},\xi^{\prime})\in T^{*}\partial\Omega,\
|x^{\prime}_{0}-x^{\prime}|+|\xi^{\prime}_{0}-\xi^{\prime}|\leq{\varepsilon}\\}$
$\displaystyle\cup\\{(x,\xi)\in T^{*}\Omega,\
|x^{\prime}_{0}-x^{\prime}|+|\xi^{\prime}_{0}-\xi^{\prime}|\leq{\varepsilon}\text{
and }(x_{d},\xi_{d})\in U\cap\\{x_{d}>0\\}\\},$

where ${\varepsilon}>0$ and $U$ is a neighborhood of
$\\{(x_{d},\xi_{d})\in\mathbb{R}^{2},\ x_{d}=0\\}$ in $\mathbb{R}^{2}$.

In local coordinates where $\Omega$ is defined by $x_{d}>0$, we define
$j:T^{*}\overline{\Omega}\to T^{*}_{b}\Omega$ by $j(x,\xi)=(x,\xi)$ if
$x\in\Omega$, and $j(x,\xi)=(x^{\prime},\xi^{\prime})$ if $x_{d}=0$. The map
$j$ is continuous for the topology given above. We can define more
intrinsically $j$ with the previous notation where $\Omega$ is given by
$\varphi(x)>0$. For $(x,\xi)\in T^{*}\overline{\Omega}$, $j(x,\xi)=(x,\xi)$ if
$x\in\Omega$ and $j(x,\xi)=(x,\xi
dx-(\\{p,\varphi\\}/H^{2}_{\varphi}p)d\varphi)$, if $\varphi(x)=0$. We verify,
in this last case that $j(x,\xi)\in\\{\varphi=\\{p,\varphi\\}=0\\}$, as
$\\{p,\varphi\\}(x,d\varphi)=\\{\varphi,\\{\varphi,p\\}\\}$.

As usually we define the map $\pi:T^{*}_{b}\Omega\to\overline{\Omega}$, in
local coordinates, as $\pi(x,\xi)=x$, if $(x,\xi)\in T^{*}\Omega$ and
$\pi(x^{\prime},\xi^{\prime})=(x^{\prime},0)$, if
$(x^{\prime},\xi^{\prime})\in T^{*}\partial\Omega$.

Bicharacteristic and generalized flow.

For $(x,\xi)\in T^{*}\mathbb{R}^{d}$, we denote by $\gamma(s;x,\xi)$ the
integral curve of $H_{p}$ starting from $(x,\xi)$. We use the same short
notation $\gamma(s)$ and $\gamma(x,\xi)$ as above.

Now we define the generalized bicharacteristic denoted by $\Gamma(s,\rho)$ for
$\rho\in T^{*}_{b}\Omega$. To describe this curve in a neighborhood of the
boundary we use the coordinates $(x^{\prime},x_{d},\xi^{\prime},\xi_{d})$ and
we identify $\Sigma^{\prime}$ and $T^{*}\partial\Omega$ and locally
$\Omega=\\{x\in\mathbb{R}^{d},\ x_{d}>0\\}$. Moreover, we assume
$\rho\in\text{char}(P)=\\{(x,\xi)\in T^{*}\Omega,\
p(x,\xi)=0\\}\cup\\{(x^{\prime},\xi^{\prime})\in T^{*}\partial\Omega,\
R(x^{\prime},0,\xi^{\prime})-1\leq 0\\}$.

Now we define the curve $\Gamma(s,\rho)$ locally for each $(s_{0},\rho)$ and
we use the group property of the flow, namely
$\Gamma(s+t,\rho)=\Gamma(s,\Gamma(t,\rho))$ to extend this function for every
$s\in\mathbb{R}$.

For $\Gamma(s_{0},\rho)\in T^{*}\Omega$,
$\Gamma(s,\rho)=\gamma(s-s_{0};\Gamma(s_{0},\rho))$ if
$\gamma(s-s_{0};\Gamma(s_{0},\rho))\in T^{*}\Omega$ for $s-s_{0}$ in an
interval containing 0. In particular, this defined $\Gamma(s,\rho)$ at least
for $s$ in a neighborhood of $s_{0}$ as $\gamma(s-s_{0};\Gamma(s_{0},\rho))$
stay in $T^{*}\Omega$ for small $|s-s_{0}|$. Observe that
$p(\gamma(s-s_{0};\Gamma(s_{0},\rho)))=0$.

For $\rho=(x_{0}^{\prime},\xi^{\prime}_{0})\in T^{*}\partial\Omega$, we have
to distinguish different cases, first if
$R(x^{\prime}_{0},0,\xi^{\prime}_{0})<1$ and second if
$R(x^{\prime}_{0},0,\xi^{\prime}_{0})=1$, where $\Gamma(s,\rho)$ depends on
the properties of $\gamma(s;x^{\prime}_{0},0,\xi^{\prime}_{0},0)$. In what
follows we only define the flow in a neighborhood of $s=0$. We can extend the
flow by the group property.

We begin to treat hyperbolic points and we recall
${\mathcal{H}}=\\{(x^{\prime}_{0},\xi^{\prime}_{0})\in T^{*}\partial\Omega,\
R(x^{\prime}_{0},0,\xi^{\prime}_{0})<1\\}.$ If
$R(x^{\prime}_{0},0,\xi^{\prime}_{0})<1$, let
$\xi^{\pm}=\pm\sqrt{1-R(x^{\prime}_{0},0,\xi^{\prime}_{0})}$. Let
$\gamma(s;x_{0},\xi_{0})=(x(s;x_{0},\xi_{0}),\xi(s;x_{0},\xi_{0}))$, as
$\dot{x}_{d}=2\xi_{d}$, we have
$x_{d}(s;x^{\prime}_{0},0,\xi^{\prime}_{0},\xi^{+})>0$ for $s>0$ sufficiently
small, and $x_{d}(s;x^{\prime}_{0},0,\xi^{\prime}_{0},\xi^{-})>0$ for $s<0$
sufficiently small. Then we set $\Gamma(0,\rho)=\rho$,
$\Gamma(s,\rho)=\gamma(s;x_{0}^{\prime},0,\xi_{0}^{\prime},\xi^{+})$ for $s>0$
sufficiently small and
$\Gamma(s,\rho)=\gamma(s;x_{0}^{\prime},0,\xi_{0}^{\prime},\xi^{-})$ for $s<0$
sufficiently small. Observe that for $s\neq 0$ sufficiently small,
$\Gamma(s,\rho)\in T^{*}\Omega$ and $p(\Gamma(s,\rho))=0$.

######  Definition A.1 (Finite contact with the boundary).

Let $(x^{\prime}_{0},\xi^{\prime}_{0})$ be such that
$R(x^{\prime}_{0},0,\xi^{\prime}_{0})=1$. We say that the bicharacteristic
$\gamma(s;x^{\prime}_{0},0,\xi^{\prime}_{0},0)=\gamma(s)$ _does not have an
infinite contact with the boundary_ if there exists $k\in\mathbb{N},\ k\geq
2,\ \alpha\neq 0$ such that
$x_{d}(s;x^{\prime}_{0},0,\xi^{\prime}_{0},0)=x_{d}(s)=\alpha
s^{k}+{\mathcal{O}}(s^{k+1})$ in a neighborhood of $s=0$. We denote by
${\mathcal{G}}^{k}$ the set of such points.

For $k=2$ we distinguish two cases.

  * •

The diffractive points, and we denote

${\mathcal{G}}_{d}=\\{(x^{\prime},\xi^{\prime})\in T^{*}\partial\Omega,\
R(x^{\prime},0,\xi^{\prime})=1,\
\partial_{x_{d}}R(x^{\prime},0,\xi^{\prime})<0\\}.$

In this case $\alpha$ defined above is positive.

  * •

The gliding points, and we denote

${\mathcal{G}}_{g}=\\{(x^{\prime},\xi^{\prime})\in T^{*}\partial\Omega,\
R(x^{\prime},0,\xi^{\prime})=1,\
\partial_{x_{d}}R(x^{\prime},0,\xi^{\prime})>0\\}.$

In this case $\alpha$ defined above is negative.

######  Remark A.1.

By Taylor’s theorem and as $x_{d}(0)=0$ and $\dot{x}_{d}(0)=2\xi_{d}(0)=0$, we
always have $x_{d}(s)={\mathcal{O}}(s^{2})$.

We have four cases to treat.

  1. -

$k$ even, $\alpha>0$. In this case $x_{d}(s)>0$ for $s\neq 0$ sufficiently
small. We define $\Gamma(0,\rho)=\rho$ and
$\Gamma(s,\rho)=\gamma(s;x_{0}^{\prime},0,\xi^{\prime}_{0},0)\in T^{*}\Omega$
for $s\neq 0$ sufficiently small.

  2. -

$k$ even, $\alpha<0$. In this case $x_{d}(s)<0$ for $s\neq 0$ sufficiently
small. We define $\Gamma(s)=\gamma_{g}(s,\rho)\in T^{*}\partial\Omega$ for $s$
sufficiently small.

  3. -

$k$ odd, $\alpha>0$. In this case $x_{d}(s)>0$ for $s>0$ sufficiently small
and $x_{d}(s)<0$ for $s<0$ sufficiently small. We define
$\Gamma(s,\rho)=\gamma_{g}(s,\rho)\in T^{*}\partial\Omega$ for $s\leq 0$
sufficiently small, and
$\Gamma(s,\rho)=\gamma(s;x_{0}^{\prime},0,\xi^{\prime}_{0},0)\in T^{*}\Omega$
for $s>0$, sufficiently small.

  4. -

$k$ odd, $\alpha<0$. In this case $x_{d}(s)<0$ for $s>0$ sufficiently small
and $x_{d}(s)>0$ for $s<0$ sufficiently small. We define
$\Gamma(s,\rho)=\gamma(s;x_{0}^{\prime},0,\xi^{\prime}_{0},0)\in T^{*}\Omega$
for $s<0$ sufficiently small, and $\Gamma(s,\rho)=\gamma_{g}(s,\rho)\in
T^{*}\partial\Omega$ for $s\geq 0$, sufficiently small.

This local description of $\Gamma(s)$ allows to extend $\Gamma(s)$ for every
$s\in\mathbb{R}$. The function $\Gamma(s,\rho)$ defined on
$\mathbb{R}\times\text{char}(P)$ is continuous for the topology of
$\mathbb{R}\times T^{*}_{b}\Omega$, where the topology of $T^{*}_{b}\Omega$ is
defined above.

###  A.2. Geometric Control Condition (GCC)

We may give here a precise definition of (GCC).

######  Definition A.2.

Let $\Omega$ be a smooth open domain in $\mathbb{R}^{d}$. Let $\omega$ be an
open subset of $\Omega$. We assume the bicaracteristics do not have contact of
infinite order with the boundary (see Definition A.1). We say that $\omega$
satisfies (GCC) if for every $\rho\in T^{*}_{b}\Omega$ there exists $s>0$ such
that $\pi(\Gamma(s,\rho))\in\omega$.

######  Remark A.2.

By symmetry the assumption $s>0$ is not relevant as if $\rho=(x,\xi)$ and
$\tilde{\rho}=(x,-\xi)$ we have $\Gamma(s,\rho)=\Gamma(-s,\tilde{\rho})$.

By compactness argument if (GCC) is satisfied, there exist $s_{0}$ such that
for every $\rho\in T^{*}_{b}\Omega$, there exist $s\in[0,s_{0}]$ such that
$\pi(\Gamma(s,\rho))\in\omega$. The value of $s_{0}$ play a role for control
problems as $s_{0}$ is related with the minimal time of control, but is not
relevant for stabilization problems.

###  A.3. Properties of measure

In this section, we give several properties on measure $m$, in particular in
neighborhood of boundary. This description is related to the property of point
in $T^{*}(\partial\Omega)$.

In next lemma, we decompose the measure into an interior measure and a measure
supported at boundary. Moreover this last measure is supported on $\xi_{d}=0$.

######  Lemma A.1.

There exists a non negative Radon measure $m^{\partial}$ on $x_{d}=\xi_{d}=0$
such that
$m=1_{x_{d}>0}m+m^{\partial}\otimes\delta_{x_{d}=0}\otimes\delta_{\xi_{d}=0}$.
Furthermore $m^{\partial}$ is supported on $R(x^{\prime},0,\xi^{\prime})-1=0$.

###### Proof.

We apply Formula (5.71). We observe that
$H_{p}=2\xi_{d}\partial_{x_{d}}-(\partial_{x_{d}}R)\partial_{\xi_{d}}+H^{\prime}_{R}$
where
$H^{\prime}_{R}=\displaystyle\sum_{j=1}^{d-1}\big{(}(\partial_{\xi_{j}}R)\partial_{x_{j}}-(\partial_{x_{j}}R)\partial_{\xi_{j}}\big{)}$.
We have
$H_{p}(a\xi_{d})=2\xi_{d}^{2}(\partial_{x_{d}}a)-(\partial_{x_{d}}R)a+\xi_{d}H^{\prime}_{R}a$.
Let
$a=a^{\varepsilon}={\varepsilon}\chi(x_{d}/{\varepsilon})\ell(x,\xi^{\prime})$,
where $\chi\in\mathscr{C}_{0}^{\infty}(\mathbb{R})$, such that $\chi(0)=0$ and
$\chi^{\prime}(0)=1$,
$\ell\in\mathscr{C}_{0}^{\infty}(\mathbb{R}^{d}\times\mathbb{R}^{d-1})$. We
have

$\displaystyle H_{p}(\xi_{d}a^{\varepsilon})$
$\displaystyle=2\xi_{d}^{2}\chi^{\prime}(x_{d}/{\varepsilon})\ell(x,\xi^{\prime})+2{\varepsilon}\chi(x_{d}/{\varepsilon})\xi_{d}^{2}\partial_{x_{d}}\ell-(\partial_{x_{d}}R){\varepsilon}\chi(x_{d}/{\varepsilon})\ell(x,\xi^{\prime})+\xi_{d}{\varepsilon}\chi(x_{d}/{\varepsilon})H^{\prime}_{R}\ell.$

Clearly $H_{p}(\xi_{d}a^{\varepsilon})$ is uniformly bounded on the support of
$m$ and $H_{p}(\xi_{d}a^{\varepsilon})\to 2\xi_{d}^{2}\chi^{\prime}(0)\ell
1_{x_{d}=0}$ everywhere as ${\varepsilon}\to 0$. Then by Lebesgue’s dominated
convergence theorem we have $\langle
m,H_{p}(\xi_{d}a^{\varepsilon})\rangle\to\langle m,2\xi_{d}^{2}\ell
1_{x_{d}=0}\rangle$ as ${\varepsilon}\to 0$. As $\chi(0)=0$, the right hand
side of (5.71) is 0 for every $h$. Then $\langle m,2\xi_{d}^{2}\ell
1_{x_{d}=0}\rangle=0$. This means that $1_{x_{d}=0}m$ is supported on
$\xi_{d}=0$. We denote $m^{\partial}$ the measure $1_{\xi_{d}=0}1_{x_{d}=0}m$.
As $m=1_{x_{d}>0}m+1_{x_{d}=0}m$, we have
$m=1_{x_{d}>0}m+m^{\partial}\otimes\delta_{x_{d}=0}\otimes\delta_{\xi_{d}=0}$.
We have by Proposition 4.1, $(\xi_{d}^{2}+R(x,\xi^{\prime})-1)1_{x_{d}=0}m=0$,
then $(R(x^{\prime},0,\xi^{\prime})-1)m^{\partial}=0$. This gives the
conclusion of Lemma. ∎

The Hamiltonian of the interior measure is a priori a distribution of order
one supported on $x_{d}=0$. The following lemma says that this quantity is a
measure if the Hamiltonian vector field is transverse to the boundary.

######  Lemma A.2.

We have $H_{p}(m1_{x_{d}>0})=\delta_{x_{d}=0}\otimes m_{0}$, where $m_{0}$ is
a distribution of order 1 defined on $x_{d}=0$. Moreover, in a neighborhood
where $H_{p}x_{d}>0$, $m_{0}$ is a non negative Radon measure, and in a
neighborhood where $H_{p}x_{d}<0$, $m_{0}$ is a non positive Radon measure.

###### Proof.

The support of $H_{p}(m1_{x_{d}>0})$ as a distribution if $x_{d}\geq 0$ and
$H_{p}(m)=0$ on $x_{d}>0$. Then $H_{p}(m1_{x_{d}>0})$ is supported on
$x_{d}=0$. This implies that there exist $n\geq 0$ and $m_{j}$ distributions
on $x_{d}=0$ such that
$H_{p}(m1_{x_{d}>0})=\displaystyle\sum_{j=0}^{n}\delta_{x_{d}=0}^{(j)}\otimes
m_{k}$. Suppose that $n\geq 1$ and let
$\chi\in\mathscr{C}_{0}^{\infty}(\mathbb{R})$, be such that $\chi^{(k)}(0)=0$
for $k=0,\dots n-1$ and $\chi^{(n)}(0)=1$. Let
$b\in\mathscr{C}_{0}^{\infty}(\mathbb{R}^{d}\times\mathbb{R}^{d-1})$, we have

$\displaystyle-\langle
m1_{x_{d}>0},H_{p}\big{(}{\varepsilon}^{n}\chi(x_{d}/{\varepsilon})b(x,\xi^{\prime})\big{)}\rangle$
$\displaystyle=\langle
H_{p}m,{\varepsilon}^{n}\chi(x_{d}/{\varepsilon})b(x,\xi^{\prime})\rangle$
$\displaystyle=\langle\sum_{j=0}^{n}\delta_{x_{d}=0}^{(j)}\otimes
m_{k},{\varepsilon}^{n}\chi(x_{d}/{\varepsilon})b(x,\xi^{\prime})\rangle$
(1.84) $\displaystyle=(-1)^{n}\langle
m_{n},b(x^{\prime},0,\xi^{\prime})\rangle.$

We also have

$\displaystyle
H_{p}\big{(}{\varepsilon}^{n}\chi(x_{d}/{\varepsilon})b(x,\xi^{\prime})\big{)}=2\xi_{d}{\varepsilon}^{n-1}\chi^{\prime}(x_{d}/{\varepsilon})b(x,\xi^{\prime})+{\varepsilon}^{n}\chi(x_{d}/{\varepsilon})H_{p}b(x,\xi^{\prime}),$

which is bounded uniformly with respect ${\varepsilon}$ and supported in a
fixed compact set. Then if $n\geq 2$,
$H_{p}\big{(}{\varepsilon}^{n}\chi(x_{d}/{\varepsilon})b(x,\xi^{\prime})\big{)}\to
0$ everywhere as ${\varepsilon}\to 0$ and (A.3), Lebesgue’s dominated
convergence theorem imply that $m_{n}=0$ for $n\geq 2$. If $n=1$,
$H_{p}\big{(}{\varepsilon}\chi(x_{d}/{\varepsilon})b(x,\xi^{\prime})\big{)}\to
2\xi_{d}1_{x_{d}=0}b(x,\xi^{\prime})$ everywhere as ${\varepsilon}\to 0$.
Lebesgue’s dominated convergence theorem and (A.3) imply that

$\displaystyle\langle
m1_{x_{d}>0},2\xi_{d}1_{x_{d}=0}b(x,\xi^{\prime})\rangle=\langle
m_{1},b(x^{\prime},0,\xi^{\prime})\rangle,$

as $\langle m1_{x_{d}>0},2\xi_{d}1_{x_{d}=0}b(x,\xi^{\prime})\rangle=0$ we
find that $m_{1}=0$. Then we have $H_{p}(m1_{x_{d}>0})=\delta_{x_{d}=0}\otimes
m_{0}$, where $m_{0}$ is a distribution of order 1.

If $H_{p}x_{d}\not=0$, let
$\gamma(s;x,\xi)=(x(s;x^{\prime},\xi),\xi(s;x^{\prime},\xi))$ be the flow
associated with $H_{p}$. We verify that the map
$(s,x^{\prime},\xi)\to(x(s;x^{\prime},\xi),\xi(s;x^{\prime},\xi))$ locally is
one to one and transforms $\partial_{s}$ in $H_{p}$. Moreover, $s=0$ is
transformed in $x_{d}=0$ and if $H_{p}x_{d}>0$, $s>0$ is transformed in
$x_{d}>0$, if $H_{p}x_{d}<0$, $s<0$ is transformed in $x_{d}>0$. In
coordinates $(s,x^{\prime},\xi)$ the equation
$H_{p}(m1_{x_{d}>0})=\delta_{x_{d}=0}\otimes m_{0}$ is transformed in
$\partial_{s}(m1_{s>0})=\delta_{s=0}\otimes m_{0}$ if $H_{p}x_{d}>0$ and
$\partial_{s}(m1_{s<0})=\delta_{s=0}\otimes m_{0}$ if $H_{p}x_{d}<0$, where we
keep the notations $m,m_{0}$ in variables $(s,x^{\prime},\xi)$ for the images
of $m,m_{0}$. If $H_{p}x_{d}>0$, we have $m1_{s>0}=(1_{s>0}ds)\otimes m_{0}$
and if $H_{p}x_{d}<0$, we have $m1_{s<0}=-(1_{s<0}ds)\otimes m_{0}$. As $m$ is
non negative, we obtain that $m_{0}$ is a nonnegative measure in the first
case and a nonpositive measure in the second case. ∎

At the hyperbolic region the measure $m_{0}$ has a particular structure given
by next lemma.

######  Lemma A.3.

Let $(x_{0}^{\prime},\xi_{0}^{\prime})$ be a hyperbolic point (i.e.
$R(x^{\prime}_{0},0,\xi^{\prime}_{0})<1$). Locally in a neighborhood of
$(x_{0}^{\prime},\xi_{0}^{\prime})$, there exists $m^{+}$ a non negative
measures on
$\mathbb{R}^{d-1}_{x^{\prime}}\times\mathbb{R}^{d-1}_{\xi^{\prime}}$ such that

$m_{0}=m^{+}\otimes\delta_{\xi_{d}=\sqrt{1-R(x^{\prime},0,\xi^{\prime})}}-m^{+}\otimes\delta_{\xi_{d}=-\sqrt{1-R(x^{\prime},0,\xi^{\prime})}}.$

###### Proof.

As $m$ is supported on $\xi_{d}^{2}+R(x,\xi^{\prime})-1=0$, we have
$m=m1_{x_{d}>0}$ in a neighborhood of hyperbolic points and
$\delta_{x_{d}=0}\otimes m_{0}$ is supported on
$\xi_{d}^{2}+R(x,\xi^{\prime})-1=0$. Then $m_{0}$ is supported on
$\xi_{d}=\pm\sqrt{1-R(x^{\prime},0,\xi^{\prime})}$. Moreover
$H_{p}=2\xi_{d}\partial_{x_{d}}+X$ where $X$ is a vector field tangent to
$x_{d}=0$, in particular $H_{p}x_{d}=2\xi_{d}$ on $x_{d}=0$. This implies that
$H_{p}x_{d}>0$ if $\xi_{d}=\sqrt{1-R(x^{\prime},0,\xi^{\prime})}$ and
$H_{p}x_{d}<0$ if $\xi_{d}=-\sqrt{1-R(x^{\prime},0,\xi^{\prime})}$. From Lemma
A.2, we obtain
$m_{0}=m^{+}\otimes\delta_{\xi_{d}=\sqrt{1-R(x^{\prime},0,\xi^{\prime})}}-m^{-}\otimes\delta_{\xi_{d}=-\sqrt{1-R(x^{\prime},0,\xi^{\prime})}}$.
From (5.70) we have $\langle H_{p}m,a\rangle=0$ which implies $\langle
m^{+}-m^{-},a_{|x_{d}=0}\rangle=0$ as $a_{|x_{d}=0}$ describes every
$\mathscr{C}_{0}^{\infty}$ functions supported in a neighborhood of
$(x_{0}^{\prime},\xi_{0}^{\prime})$, this implies that $m^{+}=m^{-}$ in a
neighborhood of $(x_{0}^{\prime},\xi_{0}^{\prime})$. ∎

We recall that

$\mathcal{G}_{d}=\\{(x^{\prime},\xi^{\prime}),\
R(x^{\prime},0,\xi^{\prime})=1\text{ and
}\partial_{x_{d}}R(x^{\prime},0,\xi^{\prime})<0\\}$

is the set of diffractive points.

######  Lemma A.4.

We have $1_{\mathcal{G}_{d}}m^{\partial}=0$.

###### Proof.

We apply Proposition 5.2 and Formula (5.71). We have to choose an adapted
function $a$. Let $\chi\in\mathscr{C}^{\infty}$ be such that $\chi(\sigma)=0$
if $|s|\geq 2$, $\chi(\sigma)=1$ if $|\sigma|\leq 1$. We apply (5.71) with
$a(x,\xi^{\prime})=\chi((1-R(x,\xi^{\prime}))/{\varepsilon})\ell(x,\xi^{\prime})\chi(x_{d}/{\varepsilon})$,
where ${\varepsilon}>0$ will be chosen in what follows and $\ell$ is supported
in a neighborhood of a point of
$\partial\Omega\times\mathbb{R}\times\mathbb{R}^{d-1}$. We recall that
$H_{p}=2\xi_{d}\partial_{x_{d}}-(\partial_{x_{d}}R)\partial_{\xi_{d}}+H^{\prime}_{R}$
(see the proof of Lemma A.1). We have

(1.85) $\displaystyle H_{p}(\xi_{d}a)$
$\displaystyle=2\xi_{d}^{2}\Big{(}-(\partial_{x_{d}}R)\chi^{\prime}((1-R(x,\xi^{\prime}))/{\varepsilon})\ell(x,\xi^{\prime})\chi(x_{d}/{\varepsilon})/{\varepsilon}$
$\displaystyle\quad+\chi((1-R(x,\xi^{\prime}))/{\varepsilon})(\partial_{x_{d}}\ell(x,\xi^{\prime}))\chi(x_{d}/{\varepsilon})$
$\displaystyle\quad+\chi((1-R(x,\xi^{\prime}))/{\varepsilon})\ell(x,\xi^{\prime})\chi^{\prime}(x_{d}/{\varepsilon})/{\varepsilon}\Big{)}$
$\displaystyle\quad-(\partial_{x_{d}}R)\chi((1-R(x,\xi^{\prime}))/{\varepsilon})\ell(x,\xi^{\prime})\chi(x_{d}/{\varepsilon})$
$\displaystyle\quad+\chi((1-R(x,\xi^{\prime}))/{\varepsilon})\chi(x_{d}/{\varepsilon})\xi_{d}H^{\prime}_{R}\ell(x,\xi^{\prime}).$

We claim

(1.86) $\displaystyle H_{p}(a\xi_{d})\text{ is uniformly bounded on
}\xi_{d}^{2}+R(x,\xi^{\prime})=1,$ $\displaystyle
H_{p}(a\xi_{d})\to-(\partial_{x_{d}}R(x^{\prime},0,\xi^{\prime}))1_{R(x^{\prime},0,\xi^{\prime})=1}1_{x_{d}=0}\ell(x^{\prime},0,\xi^{\prime})\text{
as }{\varepsilon}\to 0$ $\displaystyle\text{ for all }(x,\xi),\text{ such that
}\xi_{d}^{2}+R(x,\xi^{\prime})=1.$

As $m$ is supported on $\xi_{d}^{2}+R(x,\xi^{\prime})-1=0$, then
$\xi_{d}^{2}/{\varepsilon}=(1-R(x,\xi^{\prime}))/{\varepsilon}$ on the support
of $m$, this implies that the three first terms in (1.85) are bounded. It is
easy to prove that they converge to $0$ as ${\varepsilon}$ to $0$. The fourth
term is bounded and converges to

$-(\partial_{x_{d}}R(x^{\prime},0,\xi^{\prime}))1_{R(x^{\prime},0,\xi^{\prime})=1}1_{x_{d}=0}\ell(x^{\prime},0,\xi^{\prime}).$

In the last term as $|R(x,\xi^{\prime})-1|/{\varepsilon}$ is bounded, thus
$|\xi_{d}|$ is bounded by $C\sqrt{\varepsilon}$ and then this term converges
to 0 as ${\varepsilon}\to 0$. This proves (1.86). From that we can conclude
that $\langle m,H_{p}(\ell\xi_{d})\rangle$ converges to

(1.87) $\langle
m,-(\partial_{x_{d}}R(x^{\prime},0,\xi^{\prime}))1_{R(x^{\prime},0,\xi^{\prime})=1}1_{x_{d}=0}\ell(x^{\prime},0,\xi^{\prime})\rangle\\\
=\langle
m^{\partial},-(\partial_{x_{d}}R(x^{\prime},0,\xi^{\prime}))1_{R(x^{\prime},0,\xi^{\prime})=1}\ell(x^{\prime},0,\xi^{\prime})\rangle,$

as ${\varepsilon}\to 0$.

Let

$\displaystyle A_{\varepsilon}$ $\displaystyle=-\lim_{h\to
0}(\operatorname{op_{sc}}(a(R-1))(r_{h})_{|x_{d}=0},(r_{h})_{|x_{d}=0})_{\partial}$
$\displaystyle=-\lim_{h\to
0}(\operatorname{op_{sc}}(\chi((1-R(x,\xi^{\prime}))/{\varepsilon})\ell(x,\xi^{\prime})(R(x,\xi^{\prime})-1))(r_{h})_{|x_{d}=0},(r_{h})_{|x_{d}=0})_{\partial}.$

From Proposition B.3, $A_{\varepsilon}$ goes to 0 as ${\varepsilon}\to 0$.
Then from (1.87) $1_{\mathcal{G}_{d}}m^{\partial}=0$ as $\ell$ can be chosen
arbitrary. ∎

This lemma describes how the support of the boundary measure propagates along
the boundary.

######  Lemma A.5.

Let $(x^{\prime}_{0},\xi^{\prime}_{0})\in T^{*}\partial\Omega$. Let
$(x,\xi)\in T^{*}\Omega$, we denote $\gamma{(x,\xi)}$ the integral curve of
$H_{p}$ starting from $(x,\xi)$. We assume that $m$ is locally supported in a
neighborhood of $(x^{\prime}_{0},0,\xi^{\prime}_{0},0)$, in the set
$\\{(x,\xi),\ \xi_{d}^{2}+R(x,\xi^{\prime})=1,\text{ such that
}\gamma{(x,\xi)}\text{ hits }x_{d}=0,\ \ \xi_{d}=0\\}$. In particular $m_{0}$
is supported on $\xi_{d}=0$ and
$m_{0}=\tilde{m}_{0}\otimes\delta_{\xi_{d}=0}+\tilde{m}_{1}\otimes\delta_{\xi_{d}=0}^{\prime}$,
where $\tilde{m}_{0}$ is a distribution and $\tilde{m}_{1}$ is a Radon
measure. Then $H^{\prime}_{R}m^{\partial}+\tilde{m}_{0}=0$, $\tilde{m}_{1}=0$
and
$H_{p}m=-\partial_{x_{d}}R(x^{\prime},0,\xi^{\prime})m^{\partial}\otimes\delta_{x_{d}=0}\otimes\delta_{\xi_{d}=0}^{\prime}$,
in a neighborhood of $(x^{\prime}_{0},0,\xi^{\prime}_{0},0)$.

###### Proof.

We have
$m=1_{x_{d}>0}m+m^{\partial}\otimes\delta_{x_{d}=0}\otimes\delta_{\xi_{d}=0}$.
Then we have

(1.88) $\displaystyle H_{p}m$
$\displaystyle=m_{0}\otimes\delta_{x_{d}=0}+2\xi_{d}m^{\partial}\otimes\delta_{x_{d}=0}^{\prime}\otimes\delta_{\xi_{d}=0}-(\partial_{x_{d}}R(x^{\prime},0,\xi^{\prime}))m^{\partial}\otimes\delta_{x_{d}=0}\otimes\delta_{\xi_{d}=0}^{\prime}$
$\displaystyle\quad+H^{\prime}_{R}m^{\partial}\otimes\delta_{x_{d}=0}\otimes\delta_{\xi_{d}=0}$
$\displaystyle=m_{0}\otimes\delta_{x_{d}=0}-(\partial_{x_{d}}R(x^{\prime},0,\xi^{\prime}))m^{\partial}\otimes\delta_{x_{d}=0}\otimes\delta_{\xi_{d}=0}^{\prime}+H^{\prime}_{R}m^{\partial}\otimes\delta_{x_{d}=0}\otimes\delta_{\xi_{d}=0},$

as $\xi_{d}m^{\partial}\delta_{x_{d}=0}^{\prime}\otimes\delta_{\xi_{d}=0}=0$.
As $m_{0}$ is of order 1 and supported on $\xi_{d}=0$, we have
$m_{0}=\tilde{m}_{0}\otimes\delta_{\xi_{d}=0}+\tilde{m}_{1}\otimes\delta_{\xi_{d}=0}^{\prime}$,
where $\tilde{m}_{j}$ are distributions.

To prove that $\tilde{m}_{1}$ is a Radon measure, we test $m_{0}$ on
$\varphi=\xi_{d}\psi(x^{\prime},\xi^{\prime})\chi(\xi_{d}/{\varepsilon})$,
where $\chi$ is defined in the proof of Lemma A.4. The supremum of $\varphi$
goes to 0 with ${\varepsilon}$. The first derivatives of $\varphi$ are
estimated by the supremum of
$\psi(x^{\prime},\xi^{\prime})\chi(\xi_{d}/{\varepsilon})$,
$\psi(x^{\prime},\xi^{\prime})\chi^{\prime}(\xi_{d}/{\varepsilon})\xi_{d}/{\varepsilon}$
and $\xi_{d}\partial\psi(x^{\prime},\xi^{\prime})\chi(\xi_{d}/{\varepsilon})$.
When ${\varepsilon}\to 0$ the supremum on $\varphi$ is estimated by supremum
of $\psi(x^{\prime},\xi^{\prime})$. We also have $\langle
m_{0},\varphi\rangle$ converging to
$\langle\tilde{m}_{1},\psi(x^{\prime},\xi^{\prime})\rangle$ as
${\varepsilon}\to 0$. We deduce that
$\langle\tilde{m}_{1},\psi(x^{\prime},\xi^{\prime})\rangle$ is estimated by
the supremum of $\psi(x^{\prime},\xi^{\prime})$, this implies that
$\tilde{m}_{1}$ is a Radon measure.

By Proposition 5.2, Formula (5.70) and if
$a\in\mathscr{C}_{0}^{\infty}(\mathbb{R}^{d}\times\mathbb{R}^{d-1})$ is
supported in $\partial\Omega$ and in a neighborhood of
$(x^{\prime}_{0},0,\xi^{\prime}_{0},0)$, we have $\langle
H_{p}m,a(x,\xi^{\prime})\rangle=0$.

Let $a\in\mathscr{C}_{0}^{\infty}(\mathbb{R}^{d}\times\mathbb{R}^{d-1})$, be
such that
$a_{|x_{d}=0}=\ell\in\mathscr{C}_{0}^{\infty}(\mathbb{R}^{d-1}\times\mathbb{R}^{d-1})$,
and $\chi\in\mathscr{C}_{0}^{\infty}(\mathbb{R})$ as above. From (1.88) we
obtain

$0=\langle
H_{p}m,a\chi(x_{d}/{\varepsilon})\rangle=\langle\tilde{m}_{0}+H^{\prime}_{R}m^{\partial},\ell\rangle.$

We deduce that $H^{\prime}_{R}m^{\partial}+\tilde{m}_{0}=0$, which gives the
first conclusion of Lemma. We deduce from (1.88)

(1.89) $\displaystyle
H_{p}m=(\tilde{m}_{1}-(\partial_{x_{d}}R(x^{\prime},0,\xi^{\prime}))m^{\partial})\otimes\delta_{x_{d}=0}\otimes\delta_{\xi_{d}=0}^{\prime}.$

We then can write

(1.90) $\displaystyle\langle
m,H_{p}a(x,\xi)\rangle=\langle\tilde{m}_{1}-(\partial_{x_{d}}R(x^{\prime},0,\xi^{\prime}))m^{\partial},\partial_{\xi_{d}}a(x^{\prime},0,\xi^{\prime},0)\rangle,$

for $a\in\mathscr{C}_{0}^{\infty}(\mathbb{R}^{d}\times\mathbb{R}^{d})$.

Now we choose an adapted $a$ to apply (1.90). Let
$\chi\in\mathscr{C}_{0}^{\infty}(\mathbb{R})$ as above. Let
$\ell\in\mathscr{C}_{0}^{\infty}(x^{\prime},\xi^{\prime})$ be supported in a
neighborhood of $(x^{\prime}_{0},\xi^{\prime}_{0})$. We set
$a(x,\xi)=\xi_{d}\ell(x^{\prime},\xi^{\prime})\chi(\xi_{d}/{\varepsilon})\chi(x_{d}/{\varepsilon})$,
where ${\varepsilon}>0$. We have

$\displaystyle H_{p}a(x,\xi)$
$\displaystyle=2\ell(x^{\prime},\xi^{\prime})\chi(\xi_{d}/{\varepsilon})\chi^{\prime}(x_{d}/{\varepsilon})\xi_{d}^{2}/{\varepsilon}$
$\displaystyle\quad-\ell(x^{\prime},\xi^{\prime})(\partial_{x_{d}}R(x,\xi^{\prime}))\chi(x_{d}/{\varepsilon})\big{(}\chi(\xi_{d}/{\varepsilon})+\chi^{\prime}(\xi_{d}/{\varepsilon})\xi_{d}/{\varepsilon}\big{)}$
$\displaystyle\quad+\xi_{d}H^{\prime}_{R}(\ell(x^{\prime},\xi^{\prime}))\chi(\xi_{d}/{\varepsilon})\chi(x_{d}/{\varepsilon}).$

As
$\chi^{\prime}(x_{d}/{\varepsilon})\chi(\xi_{d}/{\varepsilon})\xi_{d}^{2}/{\varepsilon}$,
$\chi(x_{d}/{\varepsilon})\chi(\xi_{d}/{\varepsilon})\xi_{d}/{\varepsilon}$
and $\xi_{d}\chi(\xi_{d}/{\varepsilon})\chi(x_{d}/{\varepsilon})$ are
uniformly bounded and goes to 0 as ${\varepsilon}$ goes to 0, and
$\chi(\xi_{d}/{\varepsilon})\chi(x_{d}/{\varepsilon})$ goes to
$1_{x_{d}=0,\xi_{d}=0}$ and is uniformly bounded. We have $\langle
m,H_{p}a\rangle$ goes to $\langle
m,-1_{x_{d}=0,\xi_{d}=0}(\partial_{x_{d}}R(x^{\prime},0,\xi^{\prime}))\ell(x^{\prime},\xi^{\prime})\rangle$
as ${\varepsilon}$ goes to 0 and we have

(1.91) $\displaystyle\langle
m,-1_{x_{d}=0,\xi_{d}=0}\partial_{x_{d}}R(x^{\prime},0,\xi^{\prime})\ell(x^{\prime},\xi^{\prime})\rangle=-\langle