# Stabilization of the damped plate equation under general boundary conditions

Jérôme Le Rousseau  Université Sorbonne Paris Nord, Laboratoire Analyse,
Géométrie et Applications, LAGA, CNRS, UMR 7539, F-93430, Villetaneuse,
France<EMAIL_ADDRESS>and  Emmanuel Wend-Benedo Zongo  Dip. di
Matematica, Via Saldini 50, 20133 Milano, Italy and Université Sorbonne Paris
Nord, Laboratoire Analyse, Géométrie et Applications, LAGA, CNRS, UMR 7539,
F-93430, Villetaneuse, France<EMAIL_ADDRESS>
(Date: today)

###### Abstract.

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

Keywords: Carleman estimates; stabilization; Lopatinskiĭ-Šapiro condition;
resolvent estimate.

###### Contents

  1. 1 Introduction
    1. 1.1 On the stabilization of waves and plates
    2. 1.2 Method
    3. 1.3 On Carleman estimates
    4. 1.4 Geometrical setting
    5. 1.5 Main results
      1. 1.5.1 Carleman estimate
      2. 1.5.2 Stabilization result
    6. 1.6 Some open questions
      1. 1.6.1 Boundary damping
      2. 1.6.2 Spectral inequality
      3. 1.6.3 Quantification of the unique continuation property
    7. 1.7 Outline
    8. 1.8 Some notation
  2. 2 Pseudo-differential calculus: notation, definitions, and some properties
    1. 2.1 Pseudo-differential operators with a large a parameter on $\mathbb{R}^{d}$
    2. 2.2 Tangential pseudo-differential operators with a large a parameter
    3. 2.3 Function norms
    4. 2.4 Differential quadratic forms
      1. 2.4.1 Differential quadratic forms in a half-space
      2. 2.4.2 Boundary differential quadratic forms
    5. 2.5 Symbols and operators with an additional large parameter
  3. 3 Lopatinskiĭ-Šapiro boundary conditions for an elliptic operator
    1. 3.1 Some examples
    2. 3.2 Case of the bi-Laplace operator
    3. 3.3 Formulation in normal geodesic coordinates
    4. 3.4 Stability of the Lopatinskiĭ-Šapiro condition under perturbation
    5. 3.5 Examples of boundary operators yielding symmetry
  4. 4 Lopatinskiĭ-Šapiro condition for the conjugated bi-Laplacian with spectral parameter
    1. 4.1 Discussion on the Lopatinskiĭ-Šapiro condition according to the position of the roots
    2. 4.2 Root configuration for each factor
    3. 4.3 Proof of Proposition 4.2
    4. 4.4 Local stability of the algebraic conditions associated with the Lopatinskiĭ-Šapiro condition
    5. 4.5 Symbol positivity at the boundary
  5. 5 Estimate for the boundary norm under Lopatinskiĭ-Šapiro condition
    1. 5.1 A microlocal estimate
    2. 5.2 Proof of Proposition 5.1
  6. 6 Microlocal estimate for each second-order factors composing $P_{\sigma,\varphi}$
    1. 6.1 Sub-ellipticity
    2. 6.2 Lopatinskiĭ-Šapiro condition for the second-order factors
    3. 6.3 Microlocal estimates for a second-order factor
      1. 6.3.1 Case (i): one root lying in the upper complex half-plane.
      2. 6.3.2 Case (ii): one real root.
      3. 6.3.3 Case (iii): both roots lying in the lower complex half-plane.
  7. 7 Local Carleman estimate for the fourth-order operator
    1. 7.1 A first estimate
    2. 7.2 Final estimate
  8. 8 Global Carleman estimate and observability
    1. 8.1 A global Carleman estimate
    2. 8.2 Observability inequality
  9. 9 Solutions to the damped plate equations
    1. 9.1 The unbounded operator associated with the bi-Laplace operator
    2. 9.2 The plate semigroup generator
    3. 9.3 Strong and weak solutions to the damped plate equation
  10. 10 Resolvent estimates and applications to stabilization
    1. 10.1 Resolvent estimate
    2. 10.2 Stabilization result
  11. A Some technical results and proofs
    1. A.1 A perfect elliptic estimate
    2. A.2 Basic resolvent estimation
    3. A.3 Basic estimation for the resolvent set

##  1\. Introduction

Let $\Omega$ be a bounded connected open subset in $\mathbb{R}^{d}$, or a
smooth bounded connected $d$-dimensional manifold, with smooth boundary
$\partial\Omega$, where we consider a damped plate equation

$\begin{cases}\partial^{2}_{t}y+\Delta^{2}y+\alpha(x)\partial_{t}y=0&(t,x)\in\mathbb{R}_{+}\times\Omega,\\\
B_{1}y_{|\mathbb{R}_{+}\times\partial\Omega}=B_{2}y_{|\mathbb{R}_{+}\times\partial\Omega}=0,\\\
y_{|t=0}=y^{0},\ \ \partial_{t}y_{|t=0}=y^{1},\end{cases}$ (1.1)

where $\alpha\geq 0$ and where $B_{1}$ and $B_{2}$ denote two boundary
differential operators. The damping property is provided by
$+\alpha(x)\partial_{t}$ thus referred as the damping term. As introduced
below $\Delta^{2}$ is the bi-Laplace operator, that is, the square of the
Laplace operator. Here, it is associated with a smooth metric $g$ to be
introduced below; it is thus rather the bi-Laplace-Beltrami operator. This
equation appears in models for the description of mechanical vibrations of
thin domains. The two boundary operators are of $k_{j}$, $j=1,2$ respectively,
yet at most of order $3$ in the direction normal to the boundary. They are
chosen such that the two following properties are fulfilled:

  1. (1)

the Lopatinskiĭ-Šapiro boundary condition holds (this condition is fully
described in what follows);

  2. (2)

along with the homogeneous boundary conditions given above the bi-Laplace
operator is self-adjoint and nonnegative. This guarantees the conservation of
the energy of the solution in the case of a damping free equation, that is, if
$\alpha=0$.

We are concerned with the decay of the energy of the solution in the case
$\alpha$ is not identically zero. We shall prove that the damping term yields
a stabilization property: the energy decays to zero as time $t\to\infty$ and
we shall prove that the decay rate is at least logarithmic.

###  1.1. On the stabilization of waves and plates

If no geometrical condition is imposed on the damping region, here as given by
the support of the function $\alpha$ one cannot expect a exponential decay
rate as observed in the case this region fulfills the celebrated Rauch-Taylor
condition, often coined GCC for geometrical control condition [40, 6]. The GCC
expresses that all rays of geometrical optics reach the damping region
$\\{\alpha>0\\}$ in a finite time. Here, with no such condition, a logarithmic
decay rate is quite natural if one has in mind the equivalent result obtained
for the wave equation in the works of G. Lebeau [36] and G. Lebeau and L.
Robbiano [38]. We also refer to Chapter 6 in [33] and Chapters 10 and 11 in
[34] where the result of [36, 38] are reviewed and generalized in particular
on the framework of general boundary conditions as those we consider here. See
also the work of P. Cornilleau and L. Robbiano for a quite exotic boundary
condition, namely the Zaremba condition.

Among the existing results available in the literature for plate type
equations, many of them concern the “hinged” boundary conditions, that is,
$u_{|\partial\Omega}=0$ and $\Delta u_{|\partial\Omega}=0$. We first mention
these result. An important result obtained in [25] on the controllability of
the plate equation on a rectangle domain with an arbitrarily small control
domain. The method relies on the generalization of Ingham type inequalities in
[27]. An exponential stabilization result, in the same geometry, can be found
in [39], using similar techniques. In [39] the localized damping term involves
the time derivative $\partial_{t}y$ as in (1.1). Interior nonlinear feedbacks
can be used for exponential stabilization [43]. There, feedbacks are localized
in a neighborhood of part of the boundary that fulfills multiplier-type
conditions. A general analysis of nonlinear damping that includes the plate
equation is provided in [2] under multiplier-type conditions. For “hinged”
boundary conditions also, with a boundary damping term, we cite [4] where, on
a square domain, a necessary and sufficient condition is provided for
exponential stabilization.

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

A more challenging type of boundary condition is the so-called “clamped”
boundary conditions, that is, $u_{|\partial\Omega}=0$ and
$\partial_{\nu}u_{|\partial\Omega}=0$, for which few results are available. We
cite [1], where a general analysis of nonlinearly damped systems that includes
the plate equation under multiplier-type conditions is provided. In [3], the
analysis of discretized general nonlinearly damped system is also carried out,
with the plate equation as an application. In [44], a nonlinear damping
involving the p-$Laplacian$ is used also under multiplier-type conditions. In
[15], an exponential decay is obtained in the case of “clamped” boundary
conditions, yet with a damping term of the Kelvin-Voigt type, that is of the
form $\partial_{t}\Delta y$, that acts over the whole domain. In the case of
the “clamped” boundary conditions, the logarithmic-type stabilization result
we obtain here was proven in [35]. The present article thus stands as a
generalization of the stabilization result of [35] if considering a whole
class of boundary condition instead of specializing to a certain type. The
present work contains in particular also the case of “hinged” boundary
conditions.

###  1.2. Method

Following the works of [36, 38, 35] we obtain a logarithmic decay rate for the
energy of the solution to (1.1) by means of a resolvent estimate for the
generator of the semigroup associated with the damped plate equation (1.1).
This estimate follows from a Carleman inequality derived for the operator
$P_{\sigma}=\Delta^{2}-\sigma^{4}$ where $\sigma$ is a spectral parameter for
the generator of the semigroup.

Our first goal is thus the derivation of the Carleman inequality for the
operator $P_{\sigma}$ near the boundary under the boundary conditions given by
$B_{1}$ and $B_{2}$.

Then, from the Carleman estimate one deduces an observation inequality for the
operator $P_{\sigma}$ in the case of the prescribed boundary conditions. The
resolvent estimate then follows from this observation inequality.

###  1.3. On Carleman estimates

A Carleman estimate is a weighted a priori inequality for the solutions of a
partial partial differential equation, where the weight is of exponential
type. For instance, for a partial differential operator $P$ away from the
boundary, it takes the form

$\displaystyle{\|e^{\tau\varphi}u\|}_{L^{2}(\Omega)}\leq
C{\|e^{\tau\varphi}Pu\|}_{L^{2}(\Omega)},$

for $u\in\mathscr{C}^{\infty}_{c}(\Omega)$ and $\tau\geq\tau_{0}$ for
$\varphi$ well chosen and some $\tau_{0}$ chosen sufficiently large. The
exponential weight function involves a parameter $\tau$ that can be taken as
large as desired, making Carleman inequalities very powerful estimates.
Additional terms on the left-hand side of the inequality can be obtained,
including higher-order derivatives of the function $u$, depending of course of
the order of the operator $P$ itself. For a second-order elliptic operator
such as the Laplace operator one has

$\displaystyle\tau^{3/2}{\|e^{\tau\varphi}u\|}_{L^{2}(\Omega)}+\tau^{1/2}{\|e^{\tau\varphi}Du\|}_{L^{2}(\Omega)}+\tau^{-1/2}\sum_{|\beta|=2}{\|e^{\tau\varphi}D^{\beta}u\|}_{L^{2}(\Omega)}\leq
C{\|e^{\tau\varphi}\Delta u\|}_{L^{2}(\Omega)},$

under the so-called sub-ellipticity condition; see Chapter 3 in [33]. Note
that the power of the large parameter $\tau$ adds to $3/2$ with the order of
the derivative in each term on the left-hand side. In fact, in the calculus
used to derive such estimates one power of $\tau$ is equivalent to a
derivative of order one. Thus with this $3/2$ compared with the order two of
the operator one says that one looses a half-derivative in the estimate.

This type of estimate was used for the first time by T. Carleman [13] to
achieve uniqueness properties for the Cauchy problem of an elliptic operator.
Later, A.-P. Calderón and L. Hörmander further developed Carleman’s method
[12, 20]. To this day, the method based on Carleman estimates remains
essential to prove unique continuation properties; see for instance [45] for
an overview. On such questions, more recent advances have been concerned with
differential operators with singular potentials, starting with the
contribution of D. Jerison and C. Kenig [26]. There, Carleman estimates rely
on $L^{p}$-norms rather than $L^{2}$-norms as in the estimates above. The
proof of such $L^{p}$ Carleman estimates is very delicate. The reader is also
referred to [42, 29, 30, 16, 31]. In more recent years, the field of
applications of Carleman estimates has gone beyond the original domain; they
are also used in the study of:

  * •

Inverse problems, where Carleman estimates are used to obtain stability
estimates for the unknown sought quantity (for instance coefficient, source
term) with respect to norms on measurements performed on the solution of the
PDE, see for instance [11, 24, 32, 23]; Carleman estimates are also
fundamental in the construction of complex geometrical optic solutions that
lead to the resolution of inverse problems such as the Calderón problem with
partial data [28, 17].

  * •

Control theory for PDEs; Carleman estimates yield the null controllability of
linear parabolic equations [37] and the null controllability of classes of
semi-linear parabolic equations [19, 5, 18]. They can also be used to prove
unique continuation properties, that in turn are crucial for the treatment of
low frequencies for exact controllability results for hyperbolic equations as
in [6].

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

$\displaystyle\sum_{|\beta|\leq
2}\tau^{3/2-|\beta|}{\|e^{\tau\varphi}D^{\beta}u\|}_{L^{2}(\Omega)}+\sum_{|\beta|\leq
1}\tau^{3/2-|\beta|}{|e^{\tau\varphi}D^{\beta}u_{|x_{d}=0^{+}}|}_{L^{2}(\Omega)}\leq
C{\|e^{\tau\varphi}\Delta u\|}_{L^{2}(\Omega)}.$

This is the type of estimate we seek here for the operator $P_{\sigma}$, with
some uniformity with respect to $\sigma$.

###  1.4. Geometrical setting

On $\Omega$ we consider a Riemannian metric $g_{x}=(g_{ij}(x))$, with
associated cometric $(g^{ij}(x))=(g_{x})^{-1}$. It stands as a bilinear form
that act on vector fields,

$\displaystyle g_{x}(u_{x},v_{x})=g_{ij}(x)u_{x}^{i}v_{x}^{j},\quad
u_{x}=u_{x}^{i}\partial_{x_{i}},\ v_{x}=v_{x}^{i}\partial_{x_{i}}.$

For $x\in\partial\Omega$ we denote by $\nu_{x}$ the unit outward pointing
normal vector at $x$, unitary in the sense of the metric $g$, that is

$\displaystyle g_{x}(\nu_{x},\nu_{x})=1\ \ \text{and}\ \
g_{x}(\nu_{x},u_{x})=1\quad\forall u_{x}\in T_{x}\partial\Omega.$

We denote by $\partial_{\nu}$ the associated derivative at the boundary, that
is, $\partial_{\nu}f(x)=\nu_{x}(f)$. We also denote by $n_{x}$ the unit
outward pointing conormal vector at $x$, that is, $n_{x}=\nu_{x}^{\flat}$,
that is, $(n_{x})_{i}=g_{ij}\nu_{x}^{j}$.

Near a boundary point we shall often use normal geodesic coordinates where
$\Omega$ is locally given by $\\{x_{d}>0\\}$ and the metric $g$ takes the form

$\displaystyle g=dx^{d}\otimes dx^{d}+\sum\limits_{1\leq i,j\leq
d-1}g_{ij}dx^{i}\otimes dx^{j}.$

Then, the vector field $\nu_{x}$ is locally given by $(0,\dots,0,-1)$. The
same for the one form $n_{x}$.

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

$\mathbb{R}^{d}_{+}:=\\{x\in\mathbb{R}^{d},\ x_{d}>0\\}\qquad\text{where}\
x=(x^{\prime},x_{d})\ \text{with}\
x^{\prime}\in\mathbb{R}^{d-1},x_{d}\in\mathbb{R}.$

We shall naturally denote its closure by $\overline{\mathbb{R}^{d}_{+}}$, that
is, $\overline{\mathbb{R}^{d}_{+}}=\\{x\in\mathbb{R}^{d};x_{d}\geq 0\\}$.

The Laplace-Beltrami operator is given by

$\displaystyle(\Delta_{g}f)(x)=(\det g_{x})^{-1/2}\sum_{1\leq i,j\leq
d}\partial_{x_{i}}\big{(}(\det
g_{x})^{1/2}g^{ij}(x)\partial_{x_{j}}f\big{)}(x).$ (1.2)

in local coordinates. Its principal part is given by $\sum_{1\leq i,j\leq
d}g^{ij}(x)\partial_{x_{i}}\partial_{x_{j}}$ and its principal symbol by
$\sum_{1\leq i,j\leq d}g^{ij}(x)\xi_{i}\xi_{j}$.

The bi-Laplace operator is $P=\Delta^{2}_{g}$. In the main text of the article
we shall write $\Delta$, $\Delta^{2}$ in place of $\Delta_{g}$,
$\Delta_{g}^{2}$.

###  1.5. Main results

####  1.5.1. Carleman estimate

We state the main Carleman estimate for the operator $P_{\sigma}$ in normal
geodesic coordinates as presented in Section 1.4. A point
$x^{0}\in\partial\Omega$ is considered and a weight function $\varphi$ is
assumed to be defined locally and such that

  1. (1)

$\partial_{d}\varphi\geq C>0$ locally.

  2. (2)

$(\Delta\pm\sigma^{2},\varphi)$ satisfies the sub-ellipticity condition of
Definition 6.1 locally. This is a necessary and sufficient condition for a
Carleman estimate to hold for a second-order operator $\Delta\pm\sigma^{2}$,
regardless of boundary conditions [33, Chapters 3 and 4].

  3. (3)

$(P_{\sigma},B_{1},B_{2},\varphi)$ satisfies the Lopatinskiĭ-Šapiro condition
of Definition 4.1 at $\varrho^{\prime}=(x^{0},\xi^{\prime},\tau,\sigma)$ for
all
$(\xi^{\prime},\tau,\sigma)\in\mathbb{R}^{d-1}\times[0,+\infty)\times[0,+\infty)$
such that $\tau\geq\kappa_{0}\sigma$, for some $\kappa_{0}>0$. This means that
the Lopatinskiĭ-Šapiro condition holds after the conjugation of the operator
$P_{\sigma}$ and the boundary operators $B_{1}$ and $B_{2}$ by the weight
function $\exp(\tau\varphi)$.

######  Theorem 1.1 (Carleman estimate).

Let $\kappa_{0}^{\prime}>\kappa_{0}>0$. Let $x^{0}\in\partial\Omega$. Let
$\varphi$ be such that the properties above hold locally. Then, there exists
$W^{0}$ a neighborhood of $x^{0}$, $C>0$, $\tau_{0}>0$ such that

$\displaystyle\tau^{-1/2}{\|e^{\tau\varphi}u\|}_{4,\tau}+{|\operatorname{tr}(e^{\tau\varphi}u)|}_{3,1/2,\tau}\leq
C\big{(}{\|e^{\tau\varphi}P_{\sigma}u\|}_{+}+\sum\limits_{j=1}^{2}{|e^{\tau\varphi}B_{j}v_{|x_{d}=0^{+}}|}_{7/2-k_{j},\tau}\big{)},$
(1.3)

for $\tau\geq\tau_{0}$,
$\kappa_{0}\sigma\leq\tau\leq\kappa_{0}^{\prime}\sigma$, and
$u\in\overline{}\mathscr{C}^{\infty}_{c}(W^{0}_{+})$.

The volume norm is given by

$\displaystyle{\|e^{\tau\varphi}u\|}_{4,\tau}=\sum_{|\beta|\leq
4}\tau^{4-|\beta|}{\|e^{\tau\varphi}D^{\beta}u\|}_{L^{2}(\Omega)}.$

The trace norm is given by

$\displaystyle{|\operatorname{tr}(e^{\tau\varphi}u)|}_{3,1/2,\tau}=\sum_{0\leq
n\leq 3}{|\partial_{\nu}^{n}(e^{\tau\varphi}u)_{|x_{d}=0^{+}}|}_{7/2-n,\tau},$

where the norm ${|.|}_{7/2-n,\tau}$ is the $L^{2}$-norm in $\mathbb{R}^{d-1}$
after applying the Fourier multiplier
$(\tau^{2}+|\xi^{\prime}|^{2})^{7/4-n/2}$. These norms are well described in
Section 2.3.

Observe that the Carleman estimate of Theorem 1.1 exhibits a loss of a half-
derivative. A more precise statement is given in Theorem 7.3 in Section 7.2.

####  1.5.2. Stabilization result

Let $(\mathsf{P}_{0},D(\mathsf{P}_{0}))$ be the unbounded operator on
$L^{2}(\Omega)$ given by the domain

$\displaystyle D(\mathsf{P}_{0})=\big{\\{}u\in H^{4}(\Omega);\
B_{1}u_{|\partial\Omega}=B_{2}u_{|\partial\Omega}=0\big{\\}},$ (1.4)

given by $\mathsf{P}_{0}u=\Delta^{2}u$ for $u\in D(\mathsf{P}_{0})$. As
written above the two boundary differential operators are such that
$(\mathsf{P}_{0},D(\mathsf{P}_{0}))$ is self-adjoint and nonnegative.

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

$\displaystyle\mathcal{E}(y)(t)=\frac{1}{2}\big{(}{\|\partial_{t}y(t)\|}_{L^{2}(\Omega)}^{2}+(\mathsf{P}_{0}y(t),y(t))_{L^{2}(\Omega)}\big{)}.$

######  Theorem 1.2 (logarithmic stabilization for the damped plate equation).

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

$\displaystyle\mathcal{E}(y)(t)\leq\frac{C}{\big{(}\log(2+t)\big{)}^{4}}\big{(}{\|\mathsf{P}_{0}y^{0}+\alpha
y^{1}\|}_{L^{2}(\Omega)}^{2}+{\|\mathsf{P}_{0}^{1/2}y^{1}\|}_{L^{2}(\Omega)}^{2}\big{)}.$

A more precise and more general statement is given in Theorem 10.3 in Section
10.2.

###  1.6. Some open questions

####  1.6.1. Boundary damping

Here, we have considered a damping that acts in the interior of the domain
$\Omega$. The study of boundary damping, as in [38] for the wave equation, is
also of relevance. Yet we foresee that it requires to specify more the used
boundary operators. This was not our goal here as we wished to treat general
boundary operators here.

####  1.6.2. Spectral inequality

If the boundary operators $B_{1}$ and $B_{2}$ are well chosen, the bi-Laplace
operator $\Delta^{2}$ can be selfadjoint on $L^{2}(\Omega)$; see Section 9.1.
Associated with the operator is then a Hilbert basis
$(\phi_{j})_{j\in\mathbb{N}}$ of $L^{2}(\Omega)$. In the case of “clamped”
boundary condition the following spectral inequality was proven in [35].

######  Theorem 1.3 (Spectral inequality for the “clamped” bi-Laplace
operator).

Let $\O$ be an open subset of $\Omega$. There exists $C>0$ such that

$\displaystyle{\|u\|}_{L^{2}(\Omega)}\leq
Ce^{C\mu^{1/4}}{\|u\|}_{L^{2}(\O)},\qquad\mu>0,\quad
u\in\operatorname{Span}\\{\phi_{j};\ \mu_{j}\leq\mu\\}.$

The proof of this theorem is based on a Carleman inequality for the fourth-
order elliptic operator $D_{s}^{4}+\Delta^{2}$, that is, after the addition of
a variable $s$. Extending this strategy to the type of boundary conditions
treated here was not successful because it is not guaranteed that having the
Lopatinskiĭ-Šapiro condition for $\Delta^{2}$, $B_{1}$, and $B_{2}$ implies
that the Lopatinskiĭ-Šapiro condition holds for $D_{s}^{4}+\Delta^{2}$,
$B_{1}$, and $B_{2}$. Yet, the Lopatinskiĭ-Šapiro condition is at the heart of
the proof of our Carleman estimate. Proving a spectral estimate as in the
above statement for the general boundary conditions considered here is an open
question.

####  1.6.3. Quantification of the unique continuation property

For a second-order operator like the Laplace operator $\Delta$ and a boundary
operator $B$ of order $k$ (yet of order at most one in the normal direction)
such that the Lopatinskiĭ-Šapiro holds one can derive the following inequality
that locally quantifies the unique continuation property up to the boundary;
see [33, Lemma 9.2].

######  Proposition 1.4.

Let $x^{0}\in\partial\Omega$ and $V$ be a neighborhood of $x^{0}$ where the
Lopatinskiĭ-Šapiro condition holds. There exist $W$ an open neighborhood of
$x^{0}$, $\varepsilon\in(0,1)$, $\delta\in(0,1)$, and $C>0$, such that

$\displaystyle{\|u\|}_{H^{1}(W)}\leq
C{\|u\|}_{H^{1}(V)}^{1-\delta}\Big{(}{\|\Delta
u\|}_{L^{2}(V)}+{|Bu|}_{H^{1-k}(V\cap\partial\Omega)}+{\|u\|}_{H^{1}(V_{\varepsilon})}\Big{)}^{\delta},$
(1.5)

for $u\in H^{2}(V)$, with $V_{\varepsilon}=\\{x\in V;\
\operatorname{dist}(x,\partial\Omega)>\varepsilon\\}$.

This inequality can be obtained from a Carleman estimate as in Theorem 1.1 for
the Laplace operator, yet with the large parameter $\tau$ allowed to be chosen
as large as desired. This is exploited in an optimization procedure on the
parameter $\tau$ in $[\tau_{0},+\infty[$ for some $\tau_{0}$. Note however
that in the statement of Theorem 1.1 one has
$\sigma\lesssim\tau\lesssim\sigma$. Thus, the optimization procedure cannot be
carried out in $[\tau_{0},+\infty[$. While having a result quantifying the
unique continuation under Lopatinskiĭ-Šapiro-type conditions for the bi-
Laplace operator similar to that of Proposition 1.4 can be expected, the
Carleman estimate we obtain in the present article cannot be used, at least
directly, for a proof as carried out in [33] or in former sources such as
[37].

###  1.7. Outline

This article is organized as follows. In Section 2 we recall some basic
aspects of pseudo-differential operators with a large parameters $\tau>0$ and
some positivity inequality of Gårding type in particular for quadratic forms
in a half-space or at the boundary. Associated with the large parameters are
Sobolev like norms, also in a half-space or at the boundary.

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

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

To obtain the Carleman estimate for the bi-Laplace operator with spectral
parameter $\Delta^{2}-\sigma^{4}$ in Section 6 we first derive microlocal
estimates for the operators $\Delta\pm\sigma^{2}$. Imposing $\sigma$ to be
non-zero, in the sense that $\sigma\gtrsim\tau$, the previous estimates
exhibits losses in different microlocal regions. Thus concatenating the two
estimates one derives an estimate for $\Delta^{2}-\sigma^{4}$ where losses do
not accumulates. A local Carleman estimate with only a loss of a half-
derivative is obtained. This is done in Section 7. With the traces estimation
obtained in Section 5 one obtains the local Carleman estimate of Theorem 1.1.

For the application to stabilization we have in mind, in Section 8 we use a
global weight function and derive a global version of the Carleman estimate
for $\Delta^{2}-\sigma^{4}$ on the whole $\Omega$. This leads to an
observability inequality.

In Section 9 we recall aspects of strong and weak solutions to the damped
plate equation, in particular through a semigroup formulation. With the
observability inequality obtained in Section 8 we derive in Section 10 a
resolvent estimate for the generator of the plate semigroup that in turn
implies the stabilization result of Theorem 1.2.

###  1.8. Some notation

The canonical inner product in $\mathbb{C}^{m}$ is denoted by
$(z,z^{\prime})_{\mathbb{C}^{m}}=\sum\limits_{k=0}^{m-1}z_{k}\bar{z^{\prime}}_{k},$
for $z=(z_{0},\cdots,z_{m-1})\in\mathbb{C}^{m},$
$z^{\prime}=(z^{\prime}_{0},\cdots,z^{\prime}_{m-1})\in\mathbb{C}^{m}.$ The
associated norm will be denoted
$|z|^{2}_{\mathbb{C}^{m}}=\sum\limits_{k=0}^{m-1}|z_{k}|^{2}.$

We shall use the notations $a\lesssim b$ for $a\leq Cb$ and $a\gtrsim b$ for
$a\geq Cb$, with a constant $C>0$ that may change from one line to another. We
also write $a\asymp b$ to denote $a\lesssim b\lesssim a.$

For an open set $U$ of $\mathbb{R}^{d}$ we set $U_{+}=U\cap\mathbb{R}^{d}_{+}$
and

$\overline{}\mathscr{C}^{\infty}_{c}(U_{+})=\\{u=v_{|\mathbb{R}^{d}_{+}};v\in\mathscr{C}^{\infty}_{c}(\mathbb{R}^{d})\text{
and }\operatorname{supp}(v)\subset U\\}.$ (1.6)

We set
$\overline{\mathscr{S}}(\mathbb{R}^{d}_{+})=\\{u_{|\mathbb{R}^{d}_{+}};\
u\in\mathscr{S}(\mathbb{R}^{d})\\}$ with $\mathscr{S}(\mathbb{R}^{d})$ the
usual Schwartz space in $\mathbb{R}^{d}$:

$\displaystyle u\in\mathscr{S}(\mathbb{R}^{d})\ \ \Leftrightarrow\ \
u\in\mathscr{C}^{\infty}(\mathbb{R}^{d})\ \text{and}\
\forall\alpha,\beta\in\mathbb{N}^{d}\sup\limits_{x\in\mathbb{R}^{d}}|x^{\alpha}D_{x}^{\beta}u(x)|<\infty.$

We recall that the Poisson bracket of two smooth functions is given by

$\\{f,g\\}=\sum\limits_{j=1}^{d}\left(\partial_{\xi_{j}}f\partial_{x_{j}}g-\partial_{x_{j}}f\partial_{\xi_{j}}g\right).$

##  2\. Pseudo-differential calculus: notation, definitions, and some
properties

In a half-space geometry motivated by the normal geodesic coordinates
introduced in Section 1.4 we shall use
$\xi=(\xi^{\prime},\xi_{d})\in\mathbb{R}^{d-1}\times\mathbb{R}$ and we shall
consider the operators $D_{d}=-i\partial_{d}$ and
$D^{\prime}=-i\partial^{\prime}$, with
$\partial^{\prime}=(\partial_{x_{1}},\dots,\partial_{x_{d-1}})$.

###  2.1. Pseudo-differential operators with a large a parameter on
$\mathbb{R}^{d}$

In this subsection we recall some notions on semi-classical pseudo-
differential operators with large parameter $\tau\geq 1.$ We denote by
$S_{\tau}^{m}$ the space of smooth functions $a(x,\xi,\tau)$ defined on
$\mathbb{R}^{d}\times\mathbb{R}^{d},$ with $\tau\geq 1$ as a large parameter,
that satisfies the following : for all multi-indices
$\alpha,\beta\in\mathbb{N}^{d}$ and $m\in\mathbb{R}$, there exists
$C_{\alpha,\beta}>0$ such that

$\displaystyle|\partial_{x}^{\alpha}\partial^{\beta}_{\xi}a(x,\xi,\tau)|\leq
C_{\alpha,\beta}\lambda_{\tau}^{m-|\beta|},\ \ \text{where}\ \
\lambda_{\tau}^{2}=\tau^{2}+|\xi|^{2},$

for all $(x,\xi,\tau)\in\mathbb{R}^{d}\times\mathbb{R}^{d}\times[1,\infty).$
For $a\in S_{\tau}^{m},$ one defines the associated pseudo-differential
operator of order $m$, denoted by $A=\mathrm{Op}(a)$,

$\displaystyle
Au(x):=\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}e^{ix\cdot\xi}a(x,\xi,\tau)\hat{u}(\xi)d\xi,\qquad
u\in\mathscr{S}(\mathbb{R}^{d}).$

One says that $a$ is the symbol of $A$. We denote $\Psi_{\tau}^{m}$ the set of
pseudo-differential operators of order $m.$ We shall denote by
$\mathscr{D}_{\tau}^{m}$ the space of semi-classical differential operators,
i.e, the case when the symbol $a(x,\xi,\tau)$ is a polynomial function of
order $m$ in $(\xi,\tau).$

###  2.2. Tangential pseudo-differential operators with a large a parameter

Here, we consider pseudo-differential operators that only act in the
tangential direction $x^{\prime}$ with $x_{d}$ behaving as a parameter. We
shall denote by $S_{\mbox{\tiny${\mathsf{T}}$},\tau}^{m}$, the set of smooth
functions $b(x,\xi^{\prime},\tau)$ defined for $\tau\geq 1$ as a large
parameter and satisfying the following: for all multi-indices
$\alpha\in\mathbb{N}^{d}$, $\beta\in\mathbb{N}^{d-1}$ and $m\in\mathbb{R}$,
there exists $C_{\alpha,\beta}>0$ such that

$\displaystyle|\partial_{x}^{\alpha}\partial^{\beta}_{\xi^{\prime}}b(x,\xi^{\prime},\tau)|\leq
C_{\alpha,\beta}\lambda_{\mbox{\tiny${\mathsf{T}}$},\tau}^{m-|\beta|},\ \
\text{where}\ \
\lambda_{\mbox{\tiny${\mathsf{T}}$},\tau}^{2}=\tau^{2}+|\xi^{\prime}|^{2},$

for all
$(x,\xi^{\prime},\tau)\in\mathbb{R}^{d}\times\mathbb{R}^{d-1}\times[1,\infty).$
For $b\in S_{\mbox{\tiny${\mathsf{T}}$},\tau}^{m}$, we define the associated
tangential pseudo-differential operator
$B=\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(b)$ of order $m$ by

$\displaystyle
Bu(x):=\frac{1}{(2\pi)^{d-1}}\int_{\mathbb{R}^{d-1}}e^{ix^{\prime}\cdot\xi^{\prime}}b(x,\xi^{\prime},\tau)\hat{u}(\xi^{\prime},x_{d})d{\xi^{\prime}},\qquad
u\in\overline{\mathscr{S}}(\mathbb{R}^{d}_{+}).$

We define $\Psi_{\mbox{\tiny${\mathsf{T}}$},\tau}^{m}$ as the set of
tangential pseudo-differential operators of order $m,$ and
$\mathscr{D}_{\mbox{\tiny${\mathsf{T}}$},\tau}^{m}$ the set of tangential
differential operators of order $m.$ We also set

$\displaystyle\Lambda_{\mbox{\tiny${\mathsf{T}}$},\tau}^{m}=\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\lambda_{\mbox{\tiny${\mathsf{T}}$},\tau}^{m}).$

Let $m\in\mathbb{N}$ and $m^{\prime}\in\mathbb{R}.$ If we consider $a$ of the
form

$\displaystyle
a(x,\xi,\tau)=\sum\limits_{j=0}^{m}a_{j}(x,\xi^{\prime},\tau)\xi_{d}^{j},\quad
a_{j}\in S_{\mbox{\tiny${\mathsf{T}}$},\tau}^{m+m^{\prime}-j},$

we define
$\mathrm{Op}(a)=\sum\limits_{j=0}^{m}\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(a_{j})D^{j}_{x_{d}}.$
We write $a\in S_{\tau}^{m,m^{\prime}}$ and
$\mathrm{Op}(a)\in\Psi_{\tau}^{m,m^{\prime}}.$

###  2.3. Function norms

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

$\displaystyle{\|u\|}_{+}={\|u\|}_{L^{2}(\mathbb{R}^{d}_{+})},\quad(u,\tilde{u})_{+}=(u,\tilde{u})_{L^{2}(\mathbb{R}^{d}_{+})},$

for functions defined in $\mathbb{R}^{d}_{+}$ and

$\displaystyle{|w|}_{\partial}={\|w\|}_{L^{2}(\mathbb{R}^{d-1})},\quad(w,\tilde{w})_{\partial}=(w,\tilde{w})_{L^{2}(\mathbb{R}^{d-1})},$

for functions defined on $\\{x_{d}=0\\}$, such as traces.

We introduce the following norms, for $m\in\mathbb{N}$ and
$m^{\prime}\in\mathbb{R},$

$\displaystyle{\|u\|}_{m,m^{\prime},\tau}\asymp\sum\limits_{j=0}^{m}{\|\Lambda_{\mbox{\tiny${\mathsf{T}}$},\tau}^{m+m^{\prime}-j}D_{x_{d}}^{j}u\|}_{+},$
$\displaystyle{\|u\|}_{m,\tau}={\|u\|}_{m,0,\tau}\asymp\sum\limits_{j=0}^{m}{\|\Lambda_{\mbox{\tiny${\mathsf{T}}$},\tau}^{m-j}D_{x_{d}}^{j}u\|}_{+},$

for $u\in\overline{\mathscr{S}}(\mathbb{R}^{d}_{+})$. One has

$\displaystyle{\|u\|}_{m,\tau}\asymp\sum\limits_{|\alpha|\leq
m}\tau^{m-|\alpha|}{\|D^{\alpha}u\|}_{+},$

and in the case $m^{\prime}\in\mathbb{N}$ one has

$\displaystyle{\|u\|}_{m,m^{\prime},\tau}\asymp\sum\limits_{\begin{subarray}{c}\alpha_{d}\leq
m\\\ |\alpha|\leq
m+m^{\prime}\end{subarray}}\tau^{m+m^{\prime}-|\alpha|}{\|D^{\alpha}u\|}_{+},$

with $\alpha=(\alpha^{\prime},\alpha_{d})\in\mathbb{N}^{d}$.

The following argument will be used on numerous occasions: for
$m\in\mathbb{N},$ $m^{\prime},\ell\in\mathbb{R},$ with $\ell>0,$

$\displaystyle{\|u\|}_{m,m^{\prime},\tau}\ll{\|u\|}_{m,m^{\prime}+\ell,\tau},$

if $\tau$ is chosen sufficiently large.

At the boundary $\\{x_{d}=0\\}$ we define the following norms, for
$m\in\mathbb{N}$ and $m^{\prime}\in\mathbb{R}$,

$\displaystyle{|\operatorname{tr}(u)|}_{m,m^{\prime},\tau}^{2}=\sum\limits_{j=0}^{m}{|\Lambda_{\mbox{\tiny${\mathsf{T}}$},\tau}^{m+m^{\prime}-j}D^{j}_{x_{d}}u_{|x_{d}=0^{+}}|}_{\partial}^{2},\qquad
u\in\overline{\mathscr{S}}(\mathbb{R}^{d}_{+}).$

###  2.4. Differential quadratic forms

####  2.4.1. Differential quadratic forms in a half-space

######  Definition 2.1 (interior differential quadratic form).

Let $u\in\overline{\mathscr{S}}(\mathbb{R}^{d}_{+})$. We say that

$Q(u)=\sum_{s=1}^{N}(A^{s}u,B^{s}u)_{+},\qquad A^{s}=\mathrm{Op}(a^{s}),\
B^{s}=\mathrm{Op}(b^{s}),$ (2.1)

is an interior differential quadratic form of type $(m,r)$ with smooth
coefficients, if for each $s=1,\dots N$, we have $a^{s}(\varrho)\in
S_{\tau}^{m,r^{\prime}}$ and $b^{s}(\varrho)\in
S_{\tau}^{m,r^{\prime\prime}}$, with $r^{\prime}+r^{\prime\prime}=2r$,
$\varrho=(x,\xi,\tau)$.

The principal symbol of the quadratic form $Q$ is defined as the class of

$q(\varrho)=\sum_{s=1}^{N}a^{s}(\varrho)\overline{b^{s}}(\varrho)$ (2.2)

in $S_{\tau}^{2m,2r}/S_{\tau}^{2m,2r-1}$.

A result we shall use is the following microlocal Gårding inequality.

######  Proposition 2.2 (microlocal Gårding inequality).

Let $K$ be a compact set of $\overline{\mathbb{R}^{d}_{+}}$ and let
$\mathscr{U}$ be an conic open set of
$\overline{\mathbb{R}^{d}_{+}}\times\mathbb{R}^{d-1}\times\mathbb{R}_{+}$
contained in $K\times\mathbb{R}^{d-1}\times\mathbb{R}_{+}$. Let also $\chi\in
S_{\mbox{\tiny${\mathsf{T}}$},\tau}^{0}$ be homogeneous of degree $0$, be such
that $\operatorname{supp}(\chi)\subset\mathscr{U}$. Let $Q$ be an interior
differential quadratic form of type $(m,r)$ with homogeneous principal symbol
$q\in S_{\tau}^{2m,2r}$ satisfying, for some $C_{0}>0$ and $\tau_{0}>0$,

$\operatorname{Re}q(\varrho)\geq
C_{0}\lambda_{\tau}^{2m}\lambda_{\mbox{\tiny${\mathsf{T}}$},\tau}^{2r},\quad\text{for}\
\tau\geq\tau_{0},\ \ \varrho=(\varrho^{\prime},\xi_{d}),\
\varrho^{\prime}=(x,\xi^{\prime},\tau)\in\mathscr{U},\ \xi_{d}\in\mathbb{R}.$

For $0<C_{1}<C_{0}$ and $N\in\mathbb{N}$ there exist $\tau_{\ast}$, $C>0$, and
$C_{N}>0$ such that

$\displaystyle\operatorname{Re}Q(\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\chi)u)$
$\displaystyle\geq
C_{1}{\|\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\chi)u\|}_{m,r,\tau}^{2}-C{|\operatorname{tr}(\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\chi)u)|}_{m-1,r+1/2,\tau}^{2}-C_{N}{\|u\|}_{m,-N,\tau}^{2},$

for $u\in\overline{\mathscr{S}}(\mathbb{R}^{d}_{+})$ and
$\tau\geq\tau_{\ast}$.

We refer to [8, Proposition 3.5] and [34, Theorem 6.17] for a proof. A local
version of the result is the following one that follows from Proposition 2.2.

######  Proposition 2.3 (Gårding inequality).

Let $U_{0}$ be a bounded open subset of $\overline{\mathbb{R}^{d}_{+}}$ and
let $Q$ be an interior differential quadratic form of type $(m,r)$ with
homogeneous principal symbol $q\in S_{\tau}^{2m,2r}$ satisfying, for some
$C_{0}>0$ and $\tau_{0}>0$,

$\operatorname{Re}q(\varrho)\geq
C_{0}\lambda_{\tau}^{2m}\lambda_{\mbox{\tiny${\mathsf{T}}$},\tau}^{2r},\quad\text{for}\
\tau\geq\tau_{0},\ \ \varrho=(\varrho^{\prime},\xi_{d}),\
\varrho^{\prime}=(x,\xi^{\prime},\tau)\in
U_{0}\times\mathbb{R}^{d-1}\times\mathbb{R}_{+},\ \xi_{d}\in\mathbb{R}.$

For $0<C_{1}<C_{0}$ there exist $\tau_{\ast}$, $C>0$such that

$\displaystyle\operatorname{Re}Q(u)$ $\displaystyle\geq
C_{1}{\|u\|}_{m,r,\tau}^{2}-C{|\operatorname{tr}(u)|}_{m-1,r+1/2,\tau}^{2},$

for $u\in\overline{\mathscr{S}}(\mathbb{R}^{d}_{+})$ and
$\tau\geq\tau_{\ast}$.

####  2.4.2. Boundary differential quadratic forms

######  Definition 2.4.

Let $u\in\overline{\mathscr{S}}(\mathbb{R}^{d}_{+})$. We say that

${Q}(u)=\sum_{s=1}^{N}(A^{s}u_{|x_{d}=0^{+}},B^{s}u_{|x_{d}=0^{+}})_{\partial},\quad
A^{s}=a^{s}(x,D,\tau),\ B^{s}=b^{s}(x,D,\tau),$

is a boundary differential quadratic form of type $(m-1,r)$ with
$\mathscr{C}^{\infty}$ coefficients, if for each $s=1,\dots N$, we have
$a^{s}(\varrho)\in
S_{\tau}^{m-1,r^{\prime}}(\overline{\mathbb{R}^{d}_{+}}\times\mathbb{R}^{d})$,
$b^{s}(\varrho)\in
S_{\tau}^{m-1,r^{\prime\prime}}(\overline{\mathbb{R}^{d}_{+}}\times\mathbb{R}^{d})$
with $r^{\prime}+r^{\prime\prime}=2r$, $\varrho=(\varrho^{\prime},\xi_{d})$
with $\varrho^{\prime}=(x,\xi^{\prime},\tau)$. The symbol of the boundary
differential quadratic form ${Q}$ is defined by

${q}(\varrho^{\prime},\xi_{d},\tilde{\xi}_{d})=\sum_{s=1}^{N}a^{s}(\varrho^{\prime},\xi_{d})\overline{b^{s}}(\varrho^{\prime},\tilde{\xi}_{d}).$

For $\mathbf{z}=(z_{0},\dots,z_{\ell-1})\in\mathbb{C}^{\ell}$ and
$a(\varrho)\in S_{\tau}^{\ell-1,t}$, of the form
$a(\varrho^{\prime},\xi_{d})=\sum_{j=0}^{\ell-1}a_{j}(\varrho^{\prime})\xi_{d}^{j}$
with $a_{j}(\varrho^{\prime})\in
S_{\mbox{\tiny${\mathsf{T}}$},\tau}^{\ell-1+t-j}$ we set

$\Sigma_{a}(\varrho^{\prime},\mathbf{z})=\sum_{j=0}^{\ell-1}a_{j}(\varrho^{\prime})z_{j}.$
(2.3)

From the boundary differential quadratic form ${Q}$ we introduce the following
bilinear symbol $\Sigma_{{Q}}:\mathbb{C}^{m}\times\mathbb{C}^{m}\to\mathbb{C}$

$\Sigma_{{Q}}(\varrho^{\prime},\mathbf{z},\mathbf{z}^{\prime})=\sum_{s=1}^{N}\Sigma_{a^{s}}(\varrho^{\prime},\mathbf{z})\overline{\Sigma_{b^{s}}}(\varrho^{\prime},\overline{\mathbf{z}}^{\prime}),\quad\mathbf{z},\mathbf{z}^{\prime}\in\mathbb{C}^{m}.$
(2.4)

We let $\mathscr{W}$ be an open conic set in
$\mathbb{R}^{d-1}\times\mathbb{R}^{d-1}\times\mathbb{R}_{+}$.

######  Definition 2.5.

Let ${Q}$ be a boundary differential quadratic form of type $(m-1,r)$ with
homogeneous principal symbol and associated with the bilinear symbol
$\Sigma_{{Q}}(\varrho^{\prime},\mathbf{z},\mathbf{z}^{\prime})$. We say that
${Q}$ is positive definite in $\mathscr{W}$ if there exist $C>0$ and $R>0$
such that

$\operatorname{Re}\Sigma_{{Q}}(\varrho^{\prime\prime},x_{d}=0^{+},\mathbf{z},\mathbf{z})\geq
C\sum_{j=0}^{m-1}\lambda_{\mbox{\tiny${\mathsf{T}}$},\tau}^{2(m-1-j+r)}|z_{j}|^{2},$

for $\varrho^{\prime\prime}=(x^{\prime},\xi^{\prime},\tau)\in\mathscr{W}$,
with $|(\xi^{\prime},\tau)|\geq R$, and
$\mathbf{z}=(z_{0},\dots,z_{m-1})\in\mathbb{C}^{m}$.

We have the following Gårding inequality.

######  Proposition 2.6.

Let ${Q}$ be a boundary differential quadratic form of type $(m-1,r)$,
positive definite in $\mathscr{W}$, an open conic set in
$\mathbb{R}^{d-1}\times\mathbb{R}^{d-1}\times\mathbb{R}_{+}$, with bilinear
symbol $\Sigma_{{Q}}(\varrho^{\prime},\mathbf{z},\mathbf{z}^{\prime})$. Let
$\chi\in S_{\mbox{\tiny${\mathsf{T}}$},\tau}^{0}$ be homogeneous of degree 0,
with $\operatorname{supp}(\chi_{|x_{d}=0^{+}})\subset\mathscr{W}$ and let
$N\in\mathbb{N}$. Then there exist $\tau_{\ast}\geq 1$, $C>0$, $C_{N}>0$ such
that

$\operatorname{Re}{Q}(\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\chi)u)\geq
C{|\operatorname{tr}(\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\chi)u)|}_{m-1,r,\tau}^{2}-C_{N}{|\operatorname{tr}(u)|}_{m-1,-N,\tau}^{2},$

for $u\in\overline{\mathscr{S}}(\mathbb{R}^{d}_{+})$ and
$\tau\geq\tau_{\ast}$.

###  2.5. Symbols and operators with an additional large parameter

In this article, we shall use operators with a symbol that depends on an
additional large parameter $\sigma$, say $a(x,\xi,\tau,\sigma)$. They will
satisfy estimate of the form

$\displaystyle|\partial^{\alpha}_{x}\partial^{\beta}_{\xi}a(x,\xi,\tau,\sigma)|\leq
C_{\alpha,\beta}(\tau^{2}+|\xi|^{2}+\sigma^{2})^{(m-|\beta|)/2}.$

We observe that if $\tau\gtrsim\sigma$ one has

$\displaystyle\lambda_{\tau}^{2}\leq\tau^{2}+|\xi|^{2}+\sigma^{2}\lesssim\lambda_{\tau}^{2}.$

Thus, as far as pseudo-differential calculus is concerned it is as if $a\in
S_{\tau}^{m}$ and this property will be exploited in what follows.

Similarly if $a=a(x,\xi^{\prime},\tau,\sigma)$ fulfills a tangential-type
estimate of the form

$\displaystyle|\partial^{\alpha}_{x}\partial^{\beta}_{\xi^{\prime}}a(x,\xi^{\prime},\tau,\sigma)|\leq
C_{\alpha,\beta}(\tau^{2}+|\xi^{\prime}|^{2}+\sigma^{2})^{(m-|\beta|)/2},$

if one has $\tau\gtrsim\sigma$ one will be able to apply techniques adapted to
symbols in $S_{\mbox{\tiny${\mathsf{T}}$},\tau}^{m}$ and associated operators,
like for instance the results on differential quadratic forms listed in
Section 2.4.

##  3\. Lopatinskiĭ-Šapiro boundary conditions for an elliptic operator

Let $P$ be an elliptic differential operator of order $2k$ on $\Omega$,
$(k\geq 1)$, with principal symbol $p(x,\omega)$ for $(x,\omega)\in
T^{*}\Omega$. One defines the following polynomial in $z,$

$\tilde{p}(x,\omega^{\prime},z)=p(x,\omega^{\prime}-zn_{x}),$

for $x\in\partial\Omega,$ $\omega^{\prime}\in T^{*}_{x}\partial\Omega,$
$z\in\mathbb{R}$ and where $n_{x}$ denotes the outward unit pointing conormal
vector at $x$ (see Section 1.4). Here $x$ and $\omega^{\prime}$ are considered
to act as parameters. We denote by $\rho_{j}(x,\omega^{\prime}),$ $1\leq j\leq
2k$ the complex roots of $\tilde{p}.$ One sets

$\tilde{p}^{+}(x,\omega^{\prime},z)=\prod\limits_{\operatorname{Im}\rho_{j}(x,\omega^{\prime})\geq
0}(z-\rho_{j}(x,\omega^{\prime})).$

Given boundary operators $B_{1},\cdots,B_{k}$ in a neighborhood of
$\partial\Omega,$ with principal symbols $b_{j}(x,\omega)$, $j=1,\cdots,k,$
one also sets
$\tilde{b}_{j}(x,\omega^{\prime},z)=b_{j}(x,\omega^{\prime}-zn_{x}).$

######  Definition 3.1 (Lopatinskiĭ-Šapiro boundary condition).

Let $(x,\omega^{\prime})\in T^{*}\partial\Omega$ with $\omega^{\prime}\neq 0.$
One says that the Lopatinskiĭ-Šapiro condition holds for
$(P,B_{1},\cdots,B_{k})$ at $(x,\omega^{\prime})$ if for any polynomial
function $f(z)$ with complex coefficients, there exists
$c_{1},\cdots,c_{k}\in\mathbb{C}$ and a polynomial function $g(z)$ with
complex coefficients such that, for all $z\in\mathbb{C},$

$f(z)=\sum\limits_{1\leq j\leq
k}c_{j}\tilde{b_{j}}(x,\omega^{\prime},z)+g(z)\tilde{p}^{+}(x,\omega^{\prime},z).$

We say that the Lopatinskiĭ-Šapiro condition holds for
$(P,B_{1},\cdots,B_{k})$ at $x\in\partial\Omega$ if it holds at
$(x,\omega^{\prime})$ for all $\omega^{\prime}\in T^{*}_{x}\partial\Omega$
with $\omega^{\prime}\neq 0.$

###  3.1. Some examples

For instance the Lopatinskiĭ-Šapiro condition holds in the following cases

  * •

$P=-\Delta$ on $\Omega,$ with the Dirichlet boundary condition,
$Bu_{|\partial\Omega}=u_{|\partial\Omega}.$

  * •

$P=\Delta^{2}$ on $\Omega,$ along with the so-called clamped boundary
conditions, i.e, $B_{1}u_{|\partial\Omega}=u_{|\partial\Omega}$ and
$B_{2}u_{|\partial\Omega}=\partial_{\nu}u_{|\partial\Omega},$ where $\nu$ is
the normal outward pointing unit vector to $\partial\Omega$; see Section 1.4.

  * •

$P=\Delta^{2}$ on $\Omega,$ along with the so-called hinged boundary
conditions, i.e, $B_{1}u_{|\partial\Omega}=u_{|\partial\Omega}$ and
$B_{2}u_{|\partial\Omega}=\Delta u_{|\partial\Omega}.$

###  3.2. Case of the bi-Laplace operator

With $P=\Delta^{2}$ on $\Omega$, along with the general boundary operators
$B_{1}$ and $B_{2}$ of orders $k_{1}$ and $k_{2}$ respectively, we give a
matrix criterion of the Lopatinskiĭ-Šapiro condition. The general boundary
operators $B_{1}$ and $B_{2}$ are then given by

$B_{\ell}(x,D)=\sum\limits_{0\leq
j\leq\min(3,k_{\ell})}B_{\ell}^{k_{\ell}-j}(x,D^{\prime})(i\partial_{\nu})^{j},\qquad\ell=1,2,$

with $B_{\ell}^{k_{\ell}-j}(x,D^{\prime})$ differential operators acting in
the tangential variables. We denote by $b_{1}(x,\omega)$ and $b_{2}(x,\omega)$
the principal symbols of $B_{1}$ and $B_{2}$ respectively. For
$(x,\omega^{\prime})\in T^{*}\partial\Omega,$ we set

$\tilde{b}_{\ell}(x,\omega^{\prime},z)=\sum\limits_{0\leq
j\leq\min(3,k_{\ell})}b_{\ell}^{k_{\ell}-j}(x,\omega^{\prime})z^{j},\qquad\ell=1,2.$

We recall that the principal symbol of $P$ is given by
$p(x,\omega)=|\omega|^{4}_{g}.$ One thus has

$\tilde{p}(x,\omega^{\prime},z)=p(x,\omega^{\prime}-zn_{x})=\big{(}z^{2}+|\omega^{\prime}|_{g}^{2}\big{)}^{2}.$

Therefore
$\tilde{p}(x,\omega^{\prime},z)=(z-i|\omega^{\prime}|_{g})^{2}(z+i|\omega^{\prime}|_{g})^{2}$.
According to the above definition we set
$\tilde{p}^{+}(x,\omega^{\prime},z)=(z-i|\omega^{\prime}|_{g})^{2}.$ Thus, the
Lopatinskiĭ-Šapiro condition holds at $(x,\omega^{\prime})$ with
$\omega^{\prime}\neq 0$ if and only if for any function $f(z)$ the complex
number $i|\omega^{\prime}|_{g}$ is a root of the polynomial function $z\mapsto
f(z)-c_{1}\tilde{b}_{1}(x,\omega^{\prime},z)-c_{2}\tilde{b}_{2}(x,\omega^{\prime},z)$
and its derivative for some $c_{1},c_{2}\in\mathbb{C}$. This leads to the
following determinant condition.

######  Lemma 3.2.

Let $P=\Delta^{2}$ on $\Omega$, $B_{1}$ and $B_{2}$ be two boundary operators.
If $x\in\partial\Omega$, $\omega^{\prime}\in T_{x}^{*}\partial\Omega$, with
$\omega^{\prime}\neq 0$, the Lopatinskiĭ-Šapiro condition holds at
$(x,\omega^{\prime})$ if and only if

$\det\begin{pmatrix}\tilde{b}_{1}&\tilde{b}_{2}\\\\[2.0pt]
\partial_{z}\tilde{b}_{1}&\partial_{z}\tilde{b}_{2}\end{pmatrix}(x,\omega^{\prime},z=i|\omega^{\prime}|_{g})\neq
0.$ (3.1)

######  Remark 3.3.

With the determinant condition and homogeneity, we note that if the
Lopatinskiĭ-Šapiro condition holds for $(P,B_{1},B_{2})$ at
$(x,\omega^{\prime})$ it also holds in a conic neighborhood of
$(x,\omega^{\prime})$ by continuity. If it holds at $x\in\Omega,$ it also
holds in a neighborhood of $x$.

###  3.3. Formulation in normal geodesic coordinates

Near a boundary point $x\in\partial\Omega$, we shall use normal geodesic
coordinates. These coordinates are recalled at the beginning of Section 2.
Then the principal symbols of $\Delta$ and $\Delta^{2}$ are given by
$\xi_{d}^{2}+r(x,\xi^{\prime})$ and $(\xi_{d}^{2}+r(x,\xi^{\prime}))^{2}$
respectively, where $r(x,\xi^{\prime})$ is the principal symbol of a
tangential differential elliptic operator $R(x,D^{\prime})$ of order 2, with

$r(x,\xi^{\prime})=\sum\limits_{1\leq i,j\leq
d-1}g^{ij}(x)\xi^{\prime}_{i}\xi^{\prime}_{j}\ \ \text{and}\ \
r(x,\xi^{\prime})\geq C|\xi^{\prime}|^{2}.$

Here $g^{ij}$ is the inverse of the metric $g_{ij}.$ Below, we shall often
write $|\xi^{\prime}|^{2}_{x}=r(x,\xi^{\prime})$ and we shall also write
$|\xi|^{2}_{x}=\xi_{d}^{2}+r(x,\xi^{\prime}),$ for
$\xi=(\xi^{\prime},\xi_{d}).$

If $b_{1}(x,\xi)$ and $b_{2}(x,\xi)$ are the principal symbols of the boundary
operators $B_{1}$ and $B_{2}$ in the normal geodesic coordinates then the
Lopatinskiĭ-Šapiro condition for $(P,B_{1},B_{2})$ with $P=\Delta^{2}$ at
$(x,\xi^{\prime})$ reads

$\det\begin{pmatrix}b_{1}&b_{2}\\\\[2.0pt]
\partial_{\xi_{d}}b_{1}&\partial_{\xi_{d}}b_{2}\end{pmatrix}(x,\xi^{\prime},\xi_{d}=i|\xi^{\prime}|_{x})\neq
0,$

if $|\xi^{\prime}|_{x}\neq 0$ according to Lemma 3.2. If the Lopatinskiĭ-
Šapiro condition holds at some $x^{0}$, because of homogeneity, there exists
$C_{0}>0$ such that

$\left|\det\begin{pmatrix}b_{1}&b_{2}\\\\[2.0pt]
\partial_{\xi_{d}}b_{1}&\partial_{\xi_{d}}b_{2}\end{pmatrix}\right|(x^{0},\xi^{\prime},i|\xi^{\prime}|_{x})\geq
C_{0}|\xi^{\prime}|_{x}^{k_{1}+k_{2}-1},\qquad\xi^{\prime}\in\mathbb{R}^{d-1}.$
(3.2)

###  3.4. Stability of the Lopatinskiĭ-Šapiro condition under perturbation

To prepare for the study of how the Lopatinskiĭ-Šapiro condition behaves under
conjugation with Carleman exponential weight and the addition of a spectral
parameter, we introduce some perturbations in the formulation of the
Lopatinskiĭ-Šapiro condition for $(P,B_{1},B_{2})$ as written in (3.2).

######  Lemma 3.4.

Let $V^{0}$ be a compact set of $\partial\Omega$ be such that the Lopatinskiĭ-
Šapiro condition holds for $(P,B_{1},B_{2})$ at every point $x$ of $V^{0}$.
There exist $C_{1}>0$ and $\varepsilon>0$ such that

$\left|\det\begin{pmatrix}b_{1}&b_{2}\\\\[2.0pt]
\partial_{\xi_{d}}b_{1}&\partial_{\xi_{d}}b_{2}\end{pmatrix}\right|(x,\xi^{\prime}+\zeta^{\prime},\xi_{d}=i|\xi^{\prime}|_{x}+\delta)\geq
C_{1}|\xi^{\prime}|_{x}^{k_{1}+k_{2}-1},$ (3.3)

for $x\in V^{0}$, $\xi^{\prime}\in\mathbb{R}^{d-1}$,
$\zeta^{\prime}\in\mathbb{C}^{d-1}$, and $\delta\in\mathbb{C}$, if
$|\zeta^{\prime}|+|\delta|\leq\varepsilon|\xi^{\prime}|_{x}$. Moreover one has

$\left|\det\begin{pmatrix}b_{1}(x,\xi^{\prime}+\zeta^{\prime},\xi_{d}=i|\xi^{\prime}|_{x}+\delta)&b_{2}(x,\xi^{\prime}+\zeta^{\prime},\xi_{d}=i|\xi^{\prime}|_{x}+\delta)\\\\[4.0pt]
b_{1}(x,\xi^{\prime}+\zeta^{\prime},\xi_{d}=i|\xi^{\prime}|_{x}+\tilde{\delta})&b_{2}(x,\xi^{\prime}+\zeta^{\prime},\xi_{d}=i|\xi^{\prime}|_{x}+\tilde{\delta})\end{pmatrix}\right|\geq
C_{1}|\delta-\tilde{\delta}|\,|\xi^{\prime}|_{x}^{k_{1}+k_{2}-1},$ (3.4)

for $x\in V^{0}$, $\xi^{\prime}\in\mathbb{R}^{d-1}$,
$\zeta^{\prime}\in\mathbb{C}^{d-1}$, and $\delta,\tilde{\delta}\in\mathbb{C}$,
if
$|\zeta^{\prime}|+|\delta|+|\tilde{\delta}|\leq\varepsilon|\xi^{\prime}|_{x}$.

###### Proof.

From (3.2), since $V^{0}$ is compact having the Lopatinskiĭ-Šapiro condition
holding at every point $x$ of $V^{0}$ means there exists $C_{0}>0$ such that

$\left|\det\begin{pmatrix}b_{1}&b_{2}\\\\[2.0pt]
\partial_{\xi_{d}}b_{1}&\partial_{\xi_{d}}b_{2}\end{pmatrix}\right|(x,\xi^{\prime},i|\xi^{\prime}|_{x})\geq
C_{0}|\xi^{\prime}|_{x}^{k_{1}+k_{2}-1},\qquad x\in
V^{0},\xi^{\prime}\in\mathbb{R}^{d-1}.$ (3.5)

The first part is a consequence of the mean value theorem, homogeneity and
(3.5) with say $C_{1}=C_{0}/2$.

For the second part it is sufficient to assume that $\delta\neq\tilde{\delta}$
since the result is obvious otherwise. For $j=1,2$ one writes the Taylor
formula

$\displaystyle
b_{j}(x,\xi^{\prime}+\zeta^{\prime},i|\xi^{\prime}|_{x}+\tilde{\delta})$
$\displaystyle=b_{j}(x,\xi^{\prime}+\zeta^{\prime},i|\xi^{\prime}|_{x}+\delta)+(\tilde{\delta}-\delta)\partial_{\xi_{d}}b_{j}(x,\xi^{\prime}+\zeta^{\prime},i|\xi^{\prime}|_{x}+\delta)$
$\displaystyle\quad+(\tilde{\delta}-\delta)^{2}\int_{0}^{1}(1-s)\partial_{\xi_{d}}^{2}b_{j}(x,\xi^{\prime}+\zeta^{\prime},i|\xi^{\prime}|_{x}+\delta_{s})\,ds,$

with $\delta_{s}=(1-s)\delta+s\tilde{\delta}$, yielding

$\displaystyle\frac{1}{\tilde{\delta}-\delta}$
$\displaystyle\det\begin{pmatrix}b_{1}(x,\xi^{\prime}+\zeta^{\prime},i|\xi^{\prime}|_{x}+\delta)&b_{2}(x,\xi^{\prime}+\zeta^{\prime},i|\xi^{\prime}|_{x}+\delta)\\\\[4.0pt]
b_{1}(x,\xi^{\prime}+\zeta^{\prime},i|\xi^{\prime}|_{x}+\tilde{\delta})&b_{2}(x,\xi^{\prime}+\zeta^{\prime},i|\xi^{\prime}|_{x}+\tilde{\delta})\end{pmatrix}$
$\displaystyle\qquad\quad=\det\begin{pmatrix}b_{1}&b_{2}\\\\[2.0pt]
\partial_{\xi_{d}}b_{1}&\partial_{\xi_{d}}b_{2}\end{pmatrix}(x,\xi^{\prime}+\zeta^{\prime},i|\xi^{\prime}|_{x}+\delta)$
$\displaystyle\qquad\quad\quad+(\tilde{\delta}-\delta)\int_{0}^{1}(1-s)\det\begin{pmatrix}b_{1}(x,\xi^{\prime}+\zeta^{\prime},i|\xi^{\prime}|_{x}+\delta)&b_{2}(x,\xi^{\prime}+\zeta^{\prime},i|\xi^{\prime}|_{x}+\delta)\\\\[2.0pt]
\partial_{\xi_{d}}^{2}b_{1}(x,\xi^{\prime}+\zeta^{\prime},i|\xi^{\prime}|_{x}+\delta_{s})&\partial_{\xi_{d}}^{2}b_{2}(x,\xi^{\prime}+\zeta^{\prime},i|\xi^{\prime}|_{x}+\delta_{s})\end{pmatrix}ds.$

With homogeneity, if
$|\zeta^{\prime}|+|\delta|+|\tilde{\delta}|\lesssim|\xi^{\prime}|_{x}$ one
finds

$\displaystyle\left|\det\begin{pmatrix}b_{1}(x,\xi^{\prime}+\zeta^{\prime},i|\xi^{\prime}|_{x}+\delta)&b_{2}(x,\xi^{\prime}+\zeta^{\prime},i|\xi^{\prime}|_{x}+\delta)\\\\[2.0pt]
\partial_{\xi_{d}}^{2}b_{1}(x,\xi^{\prime}+\zeta^{\prime},i|\xi^{\prime}|_{x}+\delta_{s})&\partial_{\xi_{d}}^{2}b_{2}(x,\xi^{\prime}+\zeta^{\prime},i|\xi^{\prime}|_{x}+\delta_{s})\end{pmatrix}\right|\lesssim|\xi^{\prime}|_{x}^{k_{1}+k_{2}-2},$

Thus with $|\delta-\tilde{\delta}|\leq\varepsilon|\xi^{\prime}|_{x}$, for
$\varepsilon>0$ chosen sufficiently small, using the first part of the lemma
one obtains the second result. ∎

###  3.5. Examples of boundary operators yielding symmetry

We give some examples of pairs of boundary operators $B_{1},B_{2}$ that
fulfill (1) the Lopatinskiĭ-Šapiro condition and (2) yield symmetry for the
bi-Lalace operator $P=\Delta^{2}$, that is,

$\displaystyle(Pu,v)_{L^{2}(\Omega)}=(u,Pv)_{L^{2}(\Omega)}$

for $u,v\in H^{4}(\Omega)$ such that
$B_{j}u_{|\partial\Omega}=B_{j}v_{|\partial\Omega}=0$, $j=1,2$.

We first recall that following Green formula

$\displaystyle(\Delta u,v)_{L^{2}(\Omega)}=(u,\Delta
v)_{L^{2}(\Omega)}+(\partial_{n}u_{|\partial\Omega},v_{|\partial\Omega})_{L^{2}(\partial\Omega)}-(u_{|\partial\Omega},\partial_{n}v_{|\partial\Omega})_{L^{2}(\partial\Omega)},$
(3.6)

which applied twice gives
$(Pu,v)_{L^{2}(\Omega)}=(u,Pv)_{L^{2}(\Omega)}+T(u,v)$ with

$\displaystyle T(u,v)$ $\displaystyle=(\partial_{n}\Delta
u_{|\partial\Omega},v_{|\partial\Omega})_{L^{2}(\partial\Omega)}-(\Delta
u_{|\partial\Omega},\partial_{n}v_{|\partial\Omega})_{L^{2}(\partial\Omega)}$
$\displaystyle\quad+(\partial_{n}u_{|\partial\Omega},\Delta
v_{|\partial\Omega})_{L^{2}(\partial\Omega)}-(u_{|\partial\Omega},\partial_{n}\Delta
v_{|\partial\Omega})_{L^{2}(\partial\Omega)}.$ (3.7)

Using normal geodesic coordinates in a neighborhood of the whole boundary
$\partial\Omega$ allows one to write $\Delta=\partial_{n}^{2}+\Delta^{\prime}$
where $\Delta^{\prime}$ is the resulting Laplace operator on the boundary,
that is, associated with the trace of the metric on $\partial\Omega$. Since
$\Delta^{\prime}$ is selfadjoint on $\partial\Omega$ this allows one to write
formula (3.5) in the alternative forms

$\displaystyle T(u,v)$
$\displaystyle=(\partial_{n}^{3}u_{|\partial\Omega},v_{|\partial\Omega})_{L^{2}(\partial\Omega)}-((\partial_{n}^{2}+2\Delta^{\prime})u_{|\partial\Omega},\partial_{n}v_{|\partial\Omega})_{L^{2}(\partial\Omega)}$
$\displaystyle\quad+(\partial_{n}u_{|\partial\Omega},(\partial_{n}^{2}+2\Delta^{\prime})v_{|\partial\Omega})_{L^{2}(\partial\Omega)}-(u_{|\partial\Omega},\partial_{n}^{3}v_{|\partial\Omega})_{L^{2}(\partial\Omega)},$
(3.8)

or

$\displaystyle T(u,v)$
$\displaystyle=((\partial_{n}^{3}+2\Delta^{\prime}\partial_{n})u_{|\partial\Omega},v_{|\partial\Omega})_{L^{2}(\partial\Omega)}-(\partial_{n}^{2}u_{|\partial\Omega},\partial_{n}v_{|\partial\Omega})_{L^{2}(\partial\Omega)}$
$\displaystyle\quad+(\partial_{n}u_{|\partial\Omega},\partial_{n}^{2}v_{|\partial\Omega})_{L^{2}(\partial\Omega)}-(u_{|\partial\Omega},(\partial_{n}^{3}+2\Delta^{\prime}\partial_{n})v_{|\partial\Omega})_{L^{2}(\partial\Omega)}.$
(3.9)

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

######  Example 3.5 (Hinged boundary conditions).

This type of conditions refers to
$B_{1}u_{|\partial\Omega}=u_{|\partial\Omega}$ and
$B_{2}u_{|\partial\Omega}=\Delta u_{|\partial\Omega}$. With (3.5) one finds
$T(u,v)=0$ in the case of homogeneous conditions, hence symmetry.

Note that the hinged boundary conditions are equivalent to having
$B_{1}u_{|\partial\Omega}=u_{|\partial\Omega}$ and
$B_{2}u_{|\partial\Omega}=\partial_{n}^{2}u_{|\partial\Omega}$. With the
notation of Section 3 this gives $\tilde{b}_{1}(x,\omega^{\prime},z)=1$ and
$\tilde{b}_{2}(x,\omega^{\prime},z)=(-iz)^{2}=-z^{2}$. It follows that

$\det\begin{pmatrix}\tilde{b}_{1}&\tilde{b}_{2}\\\\[2.0pt]
\partial_{z}\tilde{b}_{1}&\partial_{z}\tilde{b}_{2}\end{pmatrix}(x,\omega^{\prime},z=i|\omega^{\prime}|_{g})=\det\begin{pmatrix}1&|\omega^{\prime}|^{2}_{g}\\\\[2.0pt]
0&-2i|\omega^{\prime}|_{g}\end{pmatrix}=-2i|\omega^{\prime}|_{g}\neq 0,$

if $\omega^{\prime}\neq 0$ and thus the Lopatinskiĭ-Šapiro condition holds by
Lemma 3.2.

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

######  Example 3.6 (Clamped boundary conditions).

This type of conditions refers to
$B_{1}u_{|\partial\Omega}=u_{|\partial\Omega}$ and
$B_{2}u_{|\partial\Omega}=\partial_{n}u_{|\partial\Omega}$. With (3.5) one
finds $T(u,v)=0$ in the case of homogeneous conditions, hence symmetry. With
the notation of Section 3 this gives $\tilde{b}_{1}(x,\omega^{\prime},z)=1$
and $\tilde{b}_{2}(x,\omega^{\prime},z)=-iz$. It follows that

$\det\begin{pmatrix}\tilde{b}_{1}&\tilde{b}_{2}\\\\[2.0pt]
\partial_{z}\tilde{b}_{1}&\partial_{z}\tilde{b}_{2}\end{pmatrix}(x,\omega^{\prime},z=i|\omega^{\prime}|_{g})=\det\begin{pmatrix}1&|\omega^{\prime}|_{g}\\\\[2.0pt]
0&-i\end{pmatrix}=-i\neq 0.$

Thus the Lopatinskiĭ-Šapiro condition holds by Lemma 3.2.

Note that with the clamped boundary conditions the bi-Laplace operator cannot
be seen as the square of the Laplace operator with some well chosen boundary
condition as opposed to the case of the hinged boundary conditions displayed
above.

######  Examples 3.7 (More examples).

  1. (1)

Take $B_{1}u_{|\partial\Omega}=\partial_{n}u_{|\partial\Omega}$ and
$B_{2}u_{|\partial\Omega}=\partial_{n}\Delta u_{|\partial\Omega}$. With these
boundary conditions the bi-Laplace operator is precisely the square of the
Neumann-Laplace operator. The symmetry property is immediate and so is the
Lopatinskiĭ-Šapiro condition.

  2. (2)

Take
$B_{1}u_{|\partial\Omega}=(\partial_{n}^{2}+2\Delta^{\prime})u_{|\partial\Omega}$
and $B_{2}u_{|\partial\Omega}=\partial_{n}^{3}u_{|\partial\Omega}$. With (3.5)
one finds $T(u,v)=0$ in the case of homogeneous conditions, hence symmetry.

We have $\tilde{b}_{1}(x,\omega^{\prime},z)=-z^{2}-2|\omega^{\prime}|_{g}^{2}$
and $\tilde{b}_{2}(x,\omega^{\prime},z)=iz^{3}$ and

$\det\begin{pmatrix}\tilde{b}_{1}&\tilde{b}_{2}\\\\[2.0pt]
\partial_{z}\tilde{b}_{1}&\partial_{z}\tilde{b}_{2}\end{pmatrix}(x,\omega^{\prime},z=i|\omega^{\prime}|_{g})=\det\begin{pmatrix}-|\omega^{\prime}|_{g}^{2}&|\omega^{\prime}|^{3}_{g}\\\\[2.0pt]
-2i|\omega^{\prime}|_{g}&-3i|\omega^{\prime}|^{2}_{g}\end{pmatrix}=5i|\omega^{\prime}|^{4}_{g}\neq
0,$

if $\omega^{\prime}\neq 0$ and thus the Lopatinskiĭ-Šapiro condition holds by
Lemma 3.2.

  3. (3)

Take $B_{1}u_{|\partial\Omega}=\partial_{n}u_{|\partial\Omega}$ and
$B_{2}u_{|\partial\Omega}=(\partial_{n}^{3}+A^{\prime})u_{|\partial\Omega}$,
with $A^{\prime}$ a symmetric differential operator of order less than or
equal to three on $\partial\Omega$, with homogeneous principal symbol
$a^{\prime}(x,\omega^{\prime})$ such that $a^{\prime}(x,\omega^{\prime})\neq
2|\omega^{\prime}|_{g}^{3}$ for $\omega^{\prime}\neq 0$, that is,
$a^{\prime}(x,\omega^{\prime})\neq 2$ for $|\omega^{\prime}|_{g}=1$.

With (3.5) one finds

$\displaystyle
T(u,v)=(-A^{\prime}u_{|\partial\Omega},v_{|\partial\Omega})_{L^{2}(\partial\Omega)}+(u_{|\partial\Omega},A^{\prime}v_{|\partial\Omega})_{L^{2}(\partial\Omega)}=0,$

in the case of homogeneous conditions, hence symmetry for $P$.

We have $\tilde{b}_{1}(x,\omega^{\prime},z)=-iz$ and
$\tilde{b}_{2}(x,\omega^{\prime},z)=iz^{3}+a^{\prime}(x,\omega^{\prime})$ with
$a^{\prime}$ the principal symbol of $A^{\prime}$.

$\det\begin{pmatrix}\tilde{b}_{1}&\tilde{b}_{2}\\\\[2.0pt]
\partial_{z}\tilde{b}_{1}&\partial_{z}\tilde{b}_{2}\end{pmatrix}(x,\omega^{\prime},z=i|\omega^{\prime}|_{g})=\det\begin{pmatrix}|\omega^{\prime}|_{g}&|\omega^{\prime}|^{3}_{g}+a^{\prime}(x,\omega^{\prime})\\\\[2.0pt]
-i&-3i|\omega^{\prime}|^{2}_{g}\end{pmatrix}=i\big{(}a^{\prime}(x,\omega^{\prime})-2|\omega^{\prime}|^{3}_{g}\big{)}\neq
0,$

if $\omega^{\prime}\neq 0$ since $a^{\prime}(x,\omega^{\prime})\neq
2|\omega^{\prime}|^{3}_{g}$ by assumption implying that the Lopatinskiĭ-Šapiro
condition holds by Lemma 3.2.

  4. (4)

Take $B_{1}u_{|\partial\Omega}=u_{|\partial\Omega}$ and
$B_{2}u_{|\partial\Omega}=(\partial_{n}^{2}+A^{\prime}\partial_{n})u_{|\partial\Omega}$
with $A^{\prime}$ a symmetric differential operator of order less than or
equal to one on $\partial\Omega$, with homogeneous principal symbol
$a^{\prime}(x,\omega^{\prime})$ such that
$a^{\prime}(x,\omega^{\prime})\neq-2|\omega^{\prime}|_{g}$ for
$\omega^{\prime}\neq 0$, that is, $a^{\prime}(x,\omega^{\prime})\neq-2$ for
$|\omega^{\prime}|_{g}=1$. This is a refinement of the case of hinged boundary
conditions given in Example 3.5 above.

With (3.5) one finds

$\displaystyle
T(u,v)=(A^{\prime}\partial_{n}u_{|\partial\Omega},\partial_{n}v_{|\partial\Omega})_{L^{2}(\partial\Omega)}+(\partial_{n}u_{|\partial\Omega},-A^{\prime}\partial_{n}v_{|\partial\Omega})_{L^{2}(\partial\Omega)}=0,$

in the case of homogeneous conditions, hence symmetry for $P$.

We have $\tilde{b}_{1}(x,\omega^{\prime},z)=1$ and
$\tilde{b}_{2}(x,\omega^{\prime},z)=-z^{2}-iza^{\prime}(x,\omega^{\prime})$
with $a^{\prime}$ the principal symbol of $A^{\prime}$.

$\det\begin{pmatrix}\tilde{b}_{1}&\tilde{b}_{2}\\\\[2.0pt]
\partial_{z}\tilde{b}_{1}&\partial_{z}\tilde{b}_{2}\end{pmatrix}(x,\omega^{\prime},z=i|\omega^{\prime}|_{g})=\det\begin{pmatrix}1&|\omega^{\prime}|^{2}_{g}+|\omega^{\prime}|_{g}a^{\prime}(x,\omega^{\prime})\\\\[2.0pt]
0&-2i|\omega^{\prime}|_{g}-ia^{\prime}(x,\omega^{\prime})\end{pmatrix}=-i\big{(}a^{\prime}(x,\omega^{\prime})+2|\omega^{\prime}|_{g}\big{)}\neq
0,$

if $\omega^{\prime}\neq 0$ since
$a^{\prime}(x,\omega^{\prime})\neq-2|\omega^{\prime}|_{g}$ by assumption
implying that the Lopatinskiĭ-Šapiro condition holds by Lemma 3.2.

  5. (5)

Take
$B_{1}u_{|\partial\Omega}=(\partial_{n}^{2}+A^{\prime}\partial_{n})u_{|\partial\Omega}$
and
$B_{2}u_{|\partial\Omega}=(\partial_{n}^{3}+2\partial_{n}\Delta^{\prime})u_{|\partial\Omega}$,
with $A^{\prime}$ a symmetric differential operator of order less than or
equal to one on $\partial\Omega$, with homogeneous principal symbol
$a^{\prime}(x,\omega^{\prime})$ such that
$2a^{\prime}(x,\omega^{\prime})\neq-3|\omega^{\prime}|_{g}$ for
$\omega^{\prime}\neq 0$, that is, $a^{\prime}(x,\omega^{\prime})\neq-3/2$ for
$|\omega^{\prime}|_{g}=1$. With (3.5) one finds

$\displaystyle
T(u,v)=(A^{\prime}\partial_{n}u_{|\partial\Omega},\partial_{n}v_{|\partial\Omega})_{L^{2}(\partial\Omega)}+(\partial_{n}u_{|\partial\Omega},-A^{\prime}\partial_{n}v_{|\partial\Omega})_{L^{2}(\partial\Omega)}=0,$

in the case of homogeneous conditions, hence symmetry for $P$.

We have
$\tilde{b}_{1}(x,\omega^{\prime},z)=-z^{2}-iza^{\prime}(x,\omega^{\prime})$
and $\tilde{b}_{2}(x,\omega^{\prime},z)=iz^{3}+2iz|\omega^{\prime}|_{g}^{2}$
and

$\displaystyle\det\begin{pmatrix}\tilde{b}_{1}&\tilde{b}_{2}\\\\[2.0pt]
\partial_{z}\tilde{b}_{1}&\partial_{z}\tilde{b}_{2}\end{pmatrix}(x,\omega^{\prime},z=i|\omega^{\prime}|_{g})$
$\displaystyle=\det\begin{pmatrix}|\omega^{\prime}|^{2}_{g}+|\omega^{\prime}|_{g}a^{\prime}(x,\omega^{\prime})&-|\omega^{\prime}|^{3}_{g}\\\\[2.0pt]
-2i|\omega^{\prime}|_{g}-ia^{\prime}(x,\omega^{\prime})&-i|\omega^{\prime}|^{2}_{g}\end{pmatrix}$
$\displaystyle=-i|\omega^{\prime}|^{3}_{g}\big{(}2a^{\prime}(x,\omega^{\prime})+3|\omega^{\prime}|_{g}\big{)}\neq
0,$

if $\omega^{\prime}\neq 0$ since
$2a^{\prime}(x,\omega^{\prime})+3|\omega^{\prime}|_{g}\neq 0$ by assumption
implying that the Lopatinskiĭ-Šapiro condition holds by Lemma 3.2.

##  4\. Lopatinskiĭ-Šapiro condition for the conjugated bi-Laplacian with
spectral parameter

Set $P_{\sigma}=\Delta^{2}-\sigma^{4}$ with $\sigma\in[0,+\infty)$ and denote
by $P_{\sigma,\varphi}=e^{\tau\varphi}P_{\sigma}e^{-\tau\varphi}$ the
conjugate operator of $P_{\sigma}$ with $\tau\geq 0$ a large parameter and
$\varphi\in\mathscr{C}^{\infty}(\mathbb{R}^{d},\mathbb{R})$. We shall refer to
$\varphi$ as the weight function. The principal symbol of $P_{\sigma}$ in
normal geodesic coordinates is given by

$p_{\sigma}(x,\xi)=(\xi_{d}^{2}+r(x,\xi^{\prime}))^{2}-\sigma^{4}.$

Observe that
$e^{\tau\varphi}D_{j}e^{-\tau\varphi}=D_{j}+i\tau\partial_{j}\varphi\in\mathscr{D}_{\tau}^{1}.$
So, after conjugation, the principal symbol becomes

$\displaystyle p_{\sigma,\varphi}(x,\xi,\tau)$
$\displaystyle=p_{\sigma}(x,\xi+i\tau d_{x}\varphi)$
$\displaystyle=\left((\xi_{d}+i\tau\partial_{d}\varphi)^{2}+r(x,\xi^{\prime}+i\tau
d_{x^{\prime}}\varphi)\right)^{2}-\sigma^{4}$
$\displaystyle=\left((\xi_{d}+i\tau\partial_{d}\varphi)^{2}+r(x,\xi^{\prime}+i\tau
d_{x^{\prime}}\varphi)-\sigma^{2}\right)\left((\xi_{d}+i\tau\partial_{d}\varphi)^{2}+r(x,\xi^{\prime}+i\tau
d_{x^{\prime}}\varphi)+\sigma^{2}\right)$

We write
$p_{\sigma,\varphi}(x,\xi,\tau)=q_{\sigma,\varphi}^{1}(x,\xi,\tau)q_{\sigma,\varphi}^{2}(x,\xi,\tau)$
with

$\displaystyle
q_{\sigma,\varphi}^{j}(x,\xi,\tau)=(\xi_{d}+i\tau\partial_{d}\varphi)^{2}+r(x,\xi^{\prime}+i\tau
d_{x^{\prime}}\varphi)+(-1)^{j}\sigma^{2},\ \ j=1,2.$

We consider two boundary operators $B_{1}$ and $B_{2}$ of order $k_{1}$ and
$k_{2}$ with $b_{j}(x,\xi)$ for principal symbol, $j=1,2$. The associated
conjugated operators

$\displaystyle B_{j,\varphi}=e^{\tau\varphi}B_{j}e^{-\tau\varphi},$

have respective principal symbols

$\displaystyle b_{j,\varphi}(x,\xi,\tau)=b_{j}(x,\xi+i\tau d\varphi),\quad
j=1,2.$

We assume that the Lopatinskiĭ-Šapiro condition holds for
$(P_{0},B_{1},B_{2})$ as in Definition 3.1 for any point
$(x,\omega^{\prime})\in T^{*}_{x}\partial\Omega.$ We wish to know if the
Lopatinskiĭ-Šapiro condition can hold for $(P_{\sigma},B_{1},B_{2},\varphi),$
as given by the following definition (in local coordinates for simplicity).

######  Definition 4.1.

Let
$(x,\xi^{\prime},\tau,\sigma)\in\partial\Omega\times\mathbb{R}^{d-1}\times[0,+\infty)\times[0,+\infty)$
with $(\xi^{\prime},\tau,\sigma)\neq 0$. One says that the Lopatinskiĭ-Šapiro
condition holds for $(P_{\sigma},B_{1},B_{2},\varphi)$ at
$(x,\xi^{\prime},\tau,\sigma)$ if for any polynomial function $f(\xi_{d})$
with complex coefficients there exist $c_{1},c_{2}\in\mathbb{C}$ and a
polynomial function $\ell(\xi_{d})$ with complex coefficients such that, for
all $\xi_{d}\in\mathbb{C}$

$f(\xi_{d})=c_{1}b_{1,\varphi}(x,\xi^{\prime},\xi_{d},\tau)+c_{2}b_{2,\varphi}(x,\xi^{\prime},\xi_{d},\tau)+\ell(\xi_{d})p_{\sigma,\varphi}^{+}(x,\xi^{\prime},\xi_{d},\tau),$

with

$p_{\sigma,\varphi}^{+}(x,\xi^{\prime},\xi_{d},\tau)=\prod\limits_{\operatorname{Im}\rho_{j}(\xi^{\prime},\tau,\sigma)\geq
0}(\xi_{d}-\rho_{j}(\xi^{\prime},\tau,\sigma)),$

where $\rho_{j}(x,\xi^{\prime},\tau,\sigma),$ $j=1,\cdots,4,$ denote the
complex roots of $p_{\sigma,\varphi}(x,\xi^{\prime},\xi_{d},\tau)$ viewed as a
polynomial in $\xi_{d}$.

In what follows, we shall assume that $\partial_{d}\varphi>0.$ Locally, one
has $\partial_{d}\varphi\geq C_{1}>0$, for some $C_{1}>0$.

###  4.1. Discussion on the Lopatinskiĭ-Šapiro condition according to the
position of the roots

With the assumption that $\partial_{d}\varphi>0,$ for any point
$(x,\xi^{\prime},\tau,\sigma)$ at most two roots lie in the upper complex
closed half-plane (this is explained below). We then enumerate the following
cases.

  1. •

Case 1 : No root lying in the upper complex closed half-plane, then
$p_{\sigma,\varphi}^{+}(x,\xi^{\prime},\xi_{d},\tau)=1$ and the Lopatinskiĭ-
Šapiro condition of Definition 4.1 holds trivially at
$(x,\xi^{\prime},\tau,\sigma)$.

  2. •

Case 2 : One root lying in the upper complex closed half-plane. Let us denote
by $\rho^{+}$ that root, then
$p_{\sigma,\varphi}^{+}(x,\xi^{\prime},\xi_{d},\tau)=\xi_{d}-\rho^{+}.$ With
Definition 4.1, for any choice of $f$, the polynomial function $\xi_{d}\mapsto
f(\xi_{d})-c_{1}b_{1,\varphi}(x,\xi^{\prime},\xi_{d},\tau)-c_{2}b_{2,\varphi}(x,\xi^{\prime},\xi_{d},\tau)$
admits $\rho^{+}$ as a root for $c_{1},c_{2}\in\mathbb{C}$ well chosen. Hence,
the Lopatinskiĭ-Šapiro condition holds at $(x,\xi^{\prime},\tau,\sigma)$ if
and only if

$\displaystyle b_{1,\varphi}(x,\xi^{\prime},\xi_{d}=\rho^{+},\tau)\neq 0\ \
\text{or}\ \ b_{2,\varphi}(x,\xi^{\prime},\xi_{d}=\rho^{+},\tau)\neq 0.$

Note that it then suffices to have

$\det\begin{pmatrix}b_{1,\varphi}&b_{2,\varphi}\\\\[2.0pt]
\partial_{\xi_{d}}b_{1,\varphi}&\partial_{\xi_{d}}b_{2,\varphi}\end{pmatrix}(x,\xi^{\prime},\xi_{d}=\rho^{+},\tau)\neq
0.$

  3. •

Case 3 : Two different roots lying in the upper complex closed half-plane. Let
denote by $\rho^{+}_{1}$ and $\rho^{+}_{2}$ these roots. One has
$p_{\sigma,\varphi}^{+}(x,\xi^{\prime},\xi_{d},\tau)=(\xi_{d}-\rho^{+}_{1})\xi_{d}-\rho^{+}_{2}).$
The Lopatinskiĭ-Šapiro condition holds at $(x,\xi^{\prime},\tau,\sigma)$ if
and only if the complex numbers $\rho^{+}_{1}$ and $\rho^{+}_{2}$ are the
roots of the polynomial function $\xi_{d}\mapsto
f(\xi_{d})-c_{1}b_{1,\varphi}(x,\xi^{\prime},\xi_{d},\tau)-c_{2}b_{2,\varphi}(x,\xi^{\prime},\xi_{d},\tau),$
for $c_{1},c_{2}$ well chosen. This reads

$\begin{cases}f(\rho^{+}_{1})=c_{1}b_{1,\varphi}(x,\xi^{\prime},\xi_{d}=\rho^{+}_{1},\tau)+c_{2}b_{2,\varphi}(x,\xi^{\prime},\xi_{d}=\rho^{+}_{1},\tau),\\\
f(\rho^{+}_{2})=c_{1}b_{1,\varphi}(x,\xi^{\prime},\xi_{d}=\rho^{+}_{2},\tau)+c_{2}b_{2,\varphi}(x,\xi^{\prime},\xi_{d}=\rho^{+}_{2},\tau).\end{cases}$

Being able to solve this system in $c_{1}$ and $c_{2}$ for any $f$ is
equivalent to having

$\det\begin{pmatrix}b_{1,\varphi}(x,\xi^{\prime},\xi_{d}=\rho^{+}_{1},\tau)&b_{2,\varphi}(x,\xi^{\prime},\xi_{d}=\rho^{+}_{1},\tau)\\\\[4.0pt]
b_{1,\varphi}(x,\xi^{\prime},\xi_{d}=\rho^{+}_{2},\tau)&b_{2,\varphi}(x,\xi^{\prime},\xi_{d}=\rho^{+}_{2},\tau)\end{pmatrix}\neq
0.$ (4.1)

  4. •

Case 4 : A double root lying in the upper complex closed half-plane. Denote by
$\rho^{+}$ this root; one has
$p_{\sigma,\varphi}^{+}(x,\xi^{\prime},\xi_{d},\tau)=(\xi_{d}-\rho^{+})^{2}.$
The Lopatinskiĭ-Šapiro condition holds at at $(x,\xi^{\prime},\tau,\sigma)$ if
and only if for any choice of $f$, the complex number $\rho^{+}$ is a double
root of the polynomial function $\xi_{d}\mapsto
f(\xi_{d})-c_{1}b_{1,\varphi}(x,\xi^{\prime},\xi,\tau)-c_{2}b_{2,\varphi}(x,\xi^{\prime},\xi,\tau)$
for some $c_{1},c_{2}\in\mathbb{C}.$ Thus having the Lopatinskiĭ-Šapiro
condition is equivalent of having the following determinant condition,

$\det\begin{pmatrix}b_{1,\varphi}(x,\xi^{\prime},\xi_{d}=\rho^{+},\tau)&b_{2,\varphi}(x,\xi^{\prime},\xi_{d}=\rho^{+},\tau)\\\\[4.0pt]
\partial_{\xi_{d}}b_{1,\varphi}(x,\xi^{\prime},\xi_{d}=\rho^{+},\tau)&\partial_{\xi_{d}}b_{2,\varphi}(x,\xi^{\prime},\xi_{d}=\rho^{+},\tau)\end{pmatrix}\neq
0.$ (4.2)

Observe that case 4 can only occur if $\sigma=0$ (then one has
$(\xi^{\prime},\tau)\neq(0,0)$). If $\sigma>0$ then only cases 1, 2, and 3 are
possible. This is precisely stated in Lemma 4.7. This will be an important
point in what follows.

We now state the following important proposition.

######  Proposition 4.2.

Let $x^{0}\in\partial\Omega$. Assume that the Lopatinskiĭ-Šapiro condition
holds for $(P_{0},B_{1},B_{2})$ at $x^{0}$ and thus in a compact neighborhood
$V^{0}$ of $x^{0}$ (by Remark 3.3). Assume also that $\partial_{d}\varphi\geq
C_{1}>0$ in $V^{0}$. There exist $\mu_{0}>0$ and $\mu_{1}>0$ such that if
$(x,\xi^{\prime},\tau,\sigma)\in
V^{0}\times\mathbb{R}^{d-1}\times[0,+\infty)\times[0,+\infty)$ with
$(\xi^{\prime},\tau,\sigma)\neq(0,0,0)$,

$\displaystyle|d_{x^{\prime}}\varphi(x)|\leq\mu_{0}\partial_{d}\varphi(x)\ \
\text{and}\ \ \sigma\leq\mu_{1}\tau\partial_{d}\varphi(x),$

then the Lopatinskiĭ-Šapiro condition holds for
$(P_{\sigma},B_{1},B_{2},\varphi)$ at $(x,\xi^{\prime},\tau,\sigma).$

The proof of Proposition 4.2 is given below. We first need to analyze the
configuration of the roots of the symbol $p_{\sigma,\varphi}$ starting with
each factor $q_{\sigma,\varphi}^{j}$, $j=1,2$.

###  4.2. Root configuration for each factor

We consider either factors $\xi_{d}\mapsto
q_{\sigma,\varphi}^{j}(x,\xi^{\prime},\xi_{d},\tau)$. We recall that

$\displaystyle
q_{\sigma,\varphi}^{j}(x,\xi,\tau)=(\xi_{d}+i\tau\partial_{d}\varphi)^{2}+r(x,\xi^{\prime}+i\tau
d_{x^{\prime}}\varphi)+(-1)^{j}\sigma^{2},\ \ j=1,2.$

First, we consider the case $r(x,\xi^{\prime}+i\tau
d_{x^{\prime}}\varphi)+(-1)^{j}\sigma^{2}\in\mathbb{R}^{-}$, that is, equal to
$-\beta^{2}$ with $\beta\in\mathbb{R}$ . Then, the roots of $\xi_{d}\mapsto
q_{\sigma,\varphi}^{j}(x,\xi^{\prime},\xi_{d},\tau)$ are given by

$\displaystyle-i\tau\partial_{d}\varphi+\beta\quad\text{and}\quad-i\tau\partial_{d}\varphi-\beta.$

Both lie in the lower complex open half-plane.

Second, we consider the case $r(x,\xi^{\prime}+i\tau
d_{x^{\prime}}\varphi)+(-1)^{j}\sigma^{2}\in\mathbb{C}\setminus\mathbb{R}^{-}$.
There exists a unique $\alpha_{j}\in\mathbb{C}$ such that
$\operatorname{Re}\alpha_{j}>0$ and

$\displaystyle\alpha_{j}^{2}$ $\displaystyle=r(x,\xi^{\prime}+i\tau
d_{x^{\prime}}\varphi)+(-1)^{j}\sigma^{2}$
$\displaystyle=r(x,\xi^{\prime})-\tau^{2}r(x,d_{x^{\prime}}\varphi)+(-1)^{j}\sigma^{2}+i2\tau\tilde{r}(x,\xi^{\prime},d_{x^{\prime}}\varphi)^{2},$
(4.3)

where $\tilde{r}(x,.,.)$ denotes the symmetric bilinear form associated with
the quadratic form $r(x,.)$. Then, the two roots of $\xi_{d}\mapsto
q_{\sigma,\varphi}^{j}(x,\xi^{\prime},\xi_{d},\tau)$ are given by

$\displaystyle\pi_{j,1}=-i\tau\partial_{d}\varphi-i\alpha_{j}\quad\text{and}\quad\pi_{j,2}=-i\tau\partial_{d}\varphi+i\alpha_{j}.$
(4.4)

One has $\operatorname{Im}\pi_{j,1}<0$ since $\partial_{d}\varphi\geq
C_{1}>0$. With
$\operatorname{Im}\pi_{j,2}=-\tau\partial_{d}\varphi+\operatorname{Re}\alpha_{j}$
one sees that the sign of $\operatorname{Im}\pi_{j,2}$ may change. The
following lemma gives an algebraic characterization of the sign of
$\operatorname{Im}\pi_{j,2}$.

######  Lemma 4.3.

Assume that $\partial_{d}\varphi>0$. Having $\operatorname{Im}\pi_{j,2}<0$ is
equivalent to having

$(\partial_{d}\varphi)^{2}r(x,\xi^{\prime})+\tilde{r}(x,\xi^{\prime},d_{x^{\prime}}\varphi)^{2}<\tau^{2}(\partial_{d}\varphi)^{2}|d_{x}\varphi|_{x}^{2}+(-1)^{j+1}\sigma^{2}(\partial_{d}\varphi)^{2}.$

###### Proof.

From 4.4 one has $\operatorname{Im}\pi_{j,2}<0$ if and only if
$\operatorname{Re}\alpha_{j}<\tau\partial_{d}\varphi=|\tau\partial_{d}\varphi|$,
that is, if and only if

$\displaystyle
4(\tau\partial_{d}\varphi)^{2}\operatorname{Re}\alpha_{j}^{2}-4(\tau\partial_{d}\varphi)^{4}+(\operatorname{Im}\alpha_{j}^{2})^{2}<0,$

by Lemma 4.4 below. With (4.2) this gives the result. ∎

######  Lemma 4.4.

Let $z\in\mathbb{C}$ such that $m=z^{2}.$ For $x_{0}\in\mathbb{R}$ such that
$x_{0}\neq 0$, we have

$\displaystyle|\operatorname{Re}z|\lesseqqgtr|x_{0}|\qquad\Longleftrightarrow\qquad
4x_{0}^{2}\operatorname{Re}m-4x_{0}^{4}+(\operatorname{Im}m)^{2}\lesseqqgtr
0.$

###### Proof.

Let $z=x+iy\in\mathbb{C}.$ On the one hand we have $z^{2}=x^{2}-y^{2}+2ixy=m$
and $\operatorname{Re}m=x^{2}-y^{2},$ $\operatorname{Im}m=2xy.$ On the other
hand we have

$\displaystyle
4x_{0}^{2}\operatorname{Re}m-4x_{0}^{4}+(\operatorname{Im}m)^{2}$
$\displaystyle=4x_{0}^{2}(x^{2}-y^{2})-4x_{0}^{4}+4x^{2}y^{2}$
$\displaystyle=4(x_{0}^{2}+y^{2})(x^{2}-x_{0}^{2}),$

thus with the same sign as $(x^{2}-x_{0}^{2})$. Since
$|\operatorname{Re}z|\lesseqqgtr|x_{0}|\ \Leftrightarrow\
x^{2}-x_{0}^{2}\lesseqqgtr 0$ the conclusion follows. ∎

With the following two lemmata we now connect the sign of
$\operatorname{Im}\pi_{j,2}$ with the low frequency regime
$|\xi^{\prime}|\lesssim\tau$.

######  Lemma 4.5.

Assume there exists $K_{0}>0$ such that $|d_{x^{\prime}}\varphi|\leq
K_{0}|\partial_{d}\varphi|$. Then, there exists $C_{K_{0}}>0$ such that
$\operatorname{Im}\pi_{j,2}<0$ if
$C_{K_{0}}|\xi^{\prime}|+\sigma\leq\tau\partial_{d}\varphi$, $j=0,1$.

###### Proof.

With Lemma 4.3 having $\operatorname{Im}\pi_{j,2}<0$ reads

$(\partial_{d}\varphi)^{2}r(x,\xi^{\prime})+\tilde{r}(x,\xi^{\prime},d_{x^{\prime}}\varphi)^{2}<\tau^{2}(\partial_{d}\varphi)^{2}|d_{x}\varphi|_{x}^{2}+(-1)^{j+1}\sigma^{2}(\partial_{d}\varphi)^{2}.$
(4.5)

On the one hand, since $|d_{x^{\prime}}\varphi|\leq
K_{0}|\partial_{d}\varphi|$ one has

$(\partial_{d}\varphi)^{2}r(x,\xi^{\prime})+\tilde{r}(x,\xi^{\prime},d_{x^{\prime}}\varphi)^{2}\leq
K(\partial_{d}\varphi)^{2}|\xi^{\prime}|^{2},$

for some $K>0$ that depends on $K_{0}$, using that
$|\xi^{\prime}|_{x}\eqsim|\xi^{\prime}|$. On the other hand one has

$\displaystyle\tau^{2}(\partial_{d}\varphi)^{2}|d_{x}\varphi|_{x}^{2}+(-1)^{j+1}\sigma^{2}(\partial_{d}\varphi)^{2}\geq\tau^{2}(\partial_{d}\varphi)^{4}-\sigma^{2}(\partial_{d}\varphi)^{2}.$

Thus (4.5) holds if one has

$\displaystyle\tau^{2}(\partial_{d}\varphi)^{4}-\sigma^{2}(\partial_{d}\varphi)^{2}\geq
K(\partial_{d}\varphi)^{2}|\xi^{\prime}|^{2},$

that is, $\tau^{2}(\partial_{d}\varphi)^{2}\geq
K|\xi^{\prime}|^{2}+\sigma^{2}$. ∎

######  Lemma 4.6.

Let $W$ be a bounded open set of $\mathbb{R}^{d}$ and $x^{0}\in W$. Assume
that $\partial_{d}\varphi>0$ in $\overline{W}$ and let $\kappa_{0}>0$. Then,
there exists $C>0$ such that

$\displaystyle|\xi^{\prime}|\leq C\tau\quad\text{if}\
\operatorname{Im}\pi_{j,2}(x,\xi^{\prime},\tau,\sigma)<0\ \text{and}\
\kappa_{0}\sigma\leq\tau,\quad x\in W.$

###### Proof.

With Lemma 4.3 having $\operatorname{Im}\pi_{j,2}<0$ reads

$(\partial_{d}\varphi)^{2}r(x,\xi^{\prime})+\tilde{r}(x,\xi^{\prime},d_{x^{\prime}}\varphi)^{2}<\tau^{2}(\partial_{d}\varphi)^{2}|d_{x}\varphi|_{x}^{2}+(-1)^{j+1}\sigma^{2}(\partial_{d}\varphi)^{2}.$

In particular, this implies

$r(x,\xi^{\prime})<\tau^{2}|d_{x}\varphi|_{x}^{2}+(-1)^{j+1}\sigma^{2}\leq(\sup_{W}|d_{x}\varphi|_{x}^{2}+1/\kappa_{0}^{2})\tau^{2}.$

The result follows since $|\xi^{\prime}|\asymp r(x,\xi^{\prime})$. ∎

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

######  Lemma 4.7.

Assume that $\sigma>0$. Then, $\pi_{1,2}\neq\pi_{2,2}$. Moreover, the roots
$\pi_{1,2}$ and $\pi_{2,2}$ cannot be both real.

###### Proof.

With the forms of the roots given in (4.4) if $\pi_{1,2}=\pi_{2,2}$ then
$\alpha_{1}=\alpha_{2}$, thus $\alpha_{1}^{2}=\alpha_{2}^{2}$ implying
$\sigma^{2}=0$.

Assume now that $\pi_{1,2}\in\mathbb{R}$ and $\pi_{2,2}\in\mathbb{R}$, that
is, $\operatorname{Im}\pi_{1,2}=\operatorname{Im}\pi_{2,2}=0$. This reads
$\operatorname{Re}\alpha_{j}=\tau\partial_{d}\varphi$, giving
$|\operatorname{Re}\alpha_{j}|=|\partial_{d}\varphi|$, for $j=1$ and $2$. With
Lemma 4.4 one has

$\displaystyle
4(\tau\partial_{d}\varphi)^{2}\operatorname{Re}\alpha_{j}^{2}-4(\tau\partial_{d}\varphi)^{4}+(\operatorname{Im}\alpha_{j}^{2})^{2}=0,\qquad
j=1,2.$

Observing that
$\operatorname{Im}\alpha_{1}^{2}=\operatorname{Im}\alpha_{2}^{2}$ one thus
obtains $\operatorname{Re}\alpha_{1}^{2}=\operatorname{Re}\alpha_{2}^{2}$, and
the conclusion follows as for the first part. ∎

###  4.3. Proof of Proposition 4.2

Here, according to the statement of Proposition 4.2 we consider

$\displaystyle|d_{x^{\prime}}\varphi|\leq\mu_{0}\partial_{d}\varphi\ \
\text{and}\ \ \sigma\leq\mu_{1}\tau\partial_{d}\varphi.$

First, we choose $0<\mu_{0}\leq 1$ and $0<\mu_{1}\leq 1/2$. Below both may be
chosen much smaller. According to Lemma 4.5, with $K_{0}=1$ therein, for some
$C_{2}=2C_{K_{0}}>0$ if one has
$C_{2}|\xi^{\prime}|\leq\tau\partial_{d}\varphi$ then all four roots of
$\xi_{d}\mapsto p_{\sigma,\varphi}(x,\xi^{\prime},\xi_{d},\tau)$ lie in the
lower complex open half-plane. If so, we face Case 1 as in the discussion of
Section 4.1 and the Lopatinskiĭ-Šapiro condition holds. To carry on with the
proof of Proposition 4.2 we now only have to consider having

$\displaystyle\tau\partial_{d}\varphi\leq C_{2}|\xi^{\prime}|.$ (4.6)

Our proof of Proposition 4.2 relies on the following lemma.

######  Lemma 4.8.

There exists $C_{3}>0$ such that, for $j=1$ or $2$, for $0<\mu_{0}\leq 1$,
$0<\mu_{1}\leq 1/2$, and for all
$(x,\xi^{\prime},\tau,\sigma)\in\overline{V^{0}}\times\mathbb{R}^{d-1}\times[0,+\infty)\times[0,+\infty),$
one has

$|d_{x^{\prime}}\varphi|\leq\mu_{0}\partial_{d}\varphi,\ \
\sigma\leq\mu_{1}\tau\partial_{d}\varphi\ \ \text{and}\ \
\operatorname{Im}\pi_{j,2}\geq 0\ \ \Longrightarrow\ \
\big{|}\alpha_{j}-|\xi^{\prime}|_{x}\big{|}+\tau|d_{x^{\prime}}\varphi|\leq|\xi^{\prime}|_{x}C_{3}(\mu_{0}+\mu_{1}^{2}).$

###### Proof.

With (4.6) one has

$\displaystyle\tau|d_{x^{\prime}}\varphi|\leq\mu_{0}\tau\partial_{d}\varphi\lesssim\mu_{0}|\xi^{\prime}|_{x}.$
(4.7)

With the first-order Taylor formula one has

$\displaystyle\alpha_{j}^{2}$ $\displaystyle=r(x,\xi^{\prime}+i\tau
d_{x^{\prime}}\varphi)+(-1)^{j}\sigma^{2}$
$\displaystyle=r(x,\xi^{\prime})+\int_{0}^{1}d_{\xi^{\prime}}r(x,\xi^{\prime}+i\tau
s\,d_{x^{\prime}}\varphi)(i\tau d_{x^{\prime}}\varphi)ds+(-1)^{j}\sigma^{2}.$

With (4.7) and homogeneity one has

$\displaystyle\big{|}d_{\xi^{\prime}}r(x,\xi^{\prime}+i\tau
s\,d_{x^{\prime}}\varphi)(i\tau
d_{x^{\prime}}\varphi)\big{|}\lesssim\mu_{0}|\xi^{\prime}|_{x}^{2}.$

One also has
$\sigma\leq\mu_{1}\tau\partial_{d}\varphi\lesssim\mu_{1}|\xi^{\prime}|_{x}$.
Since $r(x,\xi^{\prime})=|\xi^{\prime}|_{x}^{2}$, this yields
$\alpha_{j}^{2}=|\xi^{\prime}|_{x}^{2}\big{(}1+O(\mu_{0}+\mu_{1}^{2})\big{)}$
and hence
$\alpha_{j}=|\xi^{\prime}|_{x}\big{(}1+O(\mu_{0}+\mu_{1}^{2})\big{)}$. This
and (4.7) gives the result. ∎

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

$\displaystyle b_{\ell,\varphi}(x,\xi^{\prime},\xi_{d}=\pi_{j,2},\tau)$
$\displaystyle=b_{\ell}(x,\xi^{\prime}+i\tau
d_{x^{\prime}}\varphi,\pi_{j,2}+i\tau\partial_{d}\varphi)=b_{\ell}(x,\xi^{\prime}+i\tau
d_{x^{\prime}}\varphi,i\alpha_{j})$
$\displaystyle=b_{\ell}(x,\xi^{\prime}+i\tau
d_{x^{\prime}}\varphi,i|\xi^{\prime}|_{x}+i(\alpha_{j}-|\xi^{\prime}|_{x})).$
(4.8)

We use Lemma 3.4 and the value of $\varepsilon>0$ given therein. We choose
$0<\mu_{0}\leq 1$ and $0<\mu_{1}\leq 1/2$ such that

$\displaystyle C_{3}(\mu_{0}+\mu_{1}^{2})\leq\varepsilon,$ (4.9)

with $C_{3}>0$ as given by Lemma 4.8.

We now consider the root configurations that remain to consider according to
the discussion in Section 4.1.

Case 2.
In this case, one root of $p_{\sigma,\varphi}$ lies in the upper complex
closed half-plane. We denote this root by $\rho^{+}$. According to the
discussion in Section 4.1 it suffices to prove that

$\det\begin{pmatrix}b_{1,\varphi}&b_{2,\varphi}\\\\[2.0pt]
\partial_{\xi_{d}}b_{1,\varphi}&\partial_{\xi_{d}}b_{2,\varphi}\end{pmatrix}(x,\xi^{\prime},\xi_{d}=\rho^{+},\tau)\neq
0.$ (4.10)

In fact, one has $\rho^{+}=\pi_{j,2}$ with $j=1$ or $2$. We use the first part
of Lemma 3.4 with $\zeta^{\prime}=i\tau d_{x^{\prime}}\varphi$ and
$\delta=i(\alpha_{j}-|\xi|_{x})$. With (4.3) and (4.9) with Lemma 4.8 and the
first part of Lemma 3.4 one obtains (4.10).

Case 3.
In this case $\operatorname{Im}\pi_{1,2}>0$ and
$\operatorname{Im}\pi_{2,2}>0$. According to the discussion in Section 4.1 it
suffices to prove that

$\det\begin{pmatrix}b_{1,\varphi}(x,\xi^{\prime},\xi_{d}=\pi_{1,2},\tau)&b_{2,\varphi}(x,\xi^{\prime},\xi_{d}=\pi_{1,2},\tau)\\\\[4.0pt]
b_{1,\varphi}(x,\xi^{\prime},\xi_{d}=\pi_{2,2},\tau)&b_{2,\varphi}(x,\xi^{\prime},\xi_{d}=\pi_{2,2},\tau)\end{pmatrix}\neq
0.$ (4.11)

We use the second part of Lemma 3.4 with $\zeta^{\prime}=i\tau
d_{x^{\prime}}\varphi$, $\delta=i(\alpha_{1}-|\xi|_{x})$, and
$\tilde{\delta}=i(\alpha_{2}-|\xi|_{x})$. With (4.3) and (4.9) with Lemma 4.8
and the second part of Lemma 3.4 one obtains (4.11).

Case 4.
In this case (that only occurs if $\sigma=0$) the Lopatinskiĭ-Šapiro condition
holds also if one has (4.10). The proof is thus the same as for Case 2. This
concludes the proof of Proposition 4.2. ∎

###  4.4. Local stability of the algebraic conditions associated with the
Lopatinskiĭ-Šapiro condition

In Section 3 we saw that the Lopatinskiĭ-Šapiro condition for
$(P_{\sigma},B_{1},B_{2})$ in Definition 3.1 exhibits some stability property.
This was used in the statement of Proposition 4.2 that states how the
Lopatinskiĭ-Šapiro condition for $(P_{\sigma},B_{1},B_{2})$ can imply the
Lopatinskiĭ-Šapiro condition of Definition 4.1 for
$(P_{\sigma},B_{1},B_{2},\varphi)$, that is, the version of this condition for
the conjugated operators.

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

As in Definition 4.1 for $\varrho^{\prime}=(x,\xi^{\prime},\tau,\sigma)$ one
denotes by $\rho_{j}(\varrho^{\prime})$ the roots of
$p_{\sigma,\varphi}(x,\xi^{\prime},\xi_{d},\tau)$ viewed as a polynomial in
$\xi_{d}$.

Let
$\varrho^{0\prime}=(x^{0},\xi^{0\prime},\tau^{0},\sigma^{0})\in\partial\Omega\times\mathbb{R}^{d-1}\times[0,+\infty)\times[0,+\infty)$.
One sets

$\displaystyle J^{+}=\big{\\{}j\in\\{1,2,3,4\\};\
\operatorname{Im}\rho_{j}(\varrho^{0\prime})\geq 0\\},\qquad
J^{-}=\big{\\{}j\in\\{1,2,3,4\\};\
\operatorname{Im}\rho_{j}(\varrho^{0\prime})<0\\}$

and, for $\varrho^{\prime}=(x,\xi^{\prime},\tau,\sigma)$,

$\displaystyle\kappa^{+}_{\varrho^{0\prime}}(\varrho^{\prime})=\prod_{j\in
J^{+}}\big{(}\xi_{d}-\rho_{j}(\varrho^{\prime})\big{)},\qquad\kappa^{-}_{\varrho^{0\prime}}(\varrho^{\prime})=\prod_{j\in
J^{-}}\big{(}\xi_{d}-\rho_{j}(\varrho^{\prime})\big{)}.$

Naturally, one has
$\kappa^{+}_{\varrho^{0\prime}}(\varrho^{0\prime},\xi_{d})=p_{\sigma,\varphi}^{+}(x^{0},\xi^{0\prime},\xi_{d},\tau^{0})$
and
$\kappa^{-}_{\varrho^{0\prime}}(\varrho^{0\prime},\xi_{d})=p_{\sigma,\varphi}^{-}(x^{0},\xi^{0\prime},\xi_{d},\tau^{0})$.
Moreover, there exists a conic neighborhood $\mathscr{U}_{0}$ of
$\varrho^{0\prime}$ where both
$\kappa^{+}_{\varrho^{0\prime}}(\varrho^{\prime})$ and
$\kappa^{-}_{\varrho^{0\prime}}(\varrho^{\prime})$ are smooth with respect to
$\varrho^{\prime}$. One has

$\displaystyle
p_{\sigma,\varphi}=p_{\sigma,\varphi}^{+}p_{\sigma,\varphi}^{-}=\kappa^{+}_{\varrho^{0\prime}}\kappa^{-}_{\varrho^{0\prime}}.$

Note however that for
$\varrho^{\prime}=(x,\xi^{\prime},\tau,\sigma)\in\mathscr{U}_{0}$ it may very
well happen that
$p_{\sigma,\varphi}^{+}(x,\xi^{\prime},\xi_{d},\tau)\neq\kappa^{+}_{\varrho^{0\prime}}(\varrho^{\prime},\xi_{d})$
and
$p_{\sigma,\varphi}^{-}(x,\xi^{\prime},\xi_{d},\tau)\neq\kappa^{-}_{\varrho^{0\prime}}(\varrho^{\prime},\xi_{d})$.

The following proposition can be found in [8, proposition 1.8].

######  Proposition 4.9.

Let the Lopatinskiĭ-Šapiro condition hold at
$\varrho^{0\prime}=(x^{0},\xi^{0\prime},\tau^{0},\sigma^{0})\in\partial\Omega\times\mathbb{R}^{d-1}\times[0,+\infty)\times[0,+\infty)$
for $(P_{\sigma},B_{1},B_{2},\varphi)$. Then,

  1. (1)

The polynomial $\xi_{d}\mapsto
p_{\sigma,\varphi}^{+}(x^{0},\xi^{0\prime},\xi_{d},\tau^{0})$ is of degree
less than or equal to two.

  2. (2)

There exists a conic neighborhood $\mathscr{U}$ of $\varrho^{0\prime}$ such
that
$\\{b^{1}_{\varphi}(\varrho^{\prime},\xi_{d}),b^{2}_{\varphi}(\varrho^{\prime},\xi_{d})\\}$
is complete modulo $\kappa^{+}_{\varrho^{0\prime}}(\varrho^{\prime},\xi_{d})$
at every point $\varrho^{\prime}=(x,\xi^{\prime},\tau,\sigma)\in\mathscr{U}$,
namely for any polynomial function $f(\xi_{d})$ with complex coefficients
there exist $c_{1},c_{2}\in\mathbb{C}$ and a polynomial function
$\ell(\xi_{d})$ with complex coefficients such that, for all
$\xi_{d}\in\mathbb{C}$

$f(\xi_{d})=c_{1}b_{1,\varphi}(x,\xi^{\prime},\xi_{d},\tau)+c_{2}b_{2,\varphi}(x,\xi^{\prime},\xi_{d},\tau)+\ell(\xi_{d})\kappa^{+}_{\varrho^{0\prime}}(\varrho^{\prime},\xi_{d}).$
(4.12)

We emphasize again that the second property in Proposition 4.9 looks very much
like the statement of Lopatinskiĭ-Šapiro condition for
$(P_{\sigma},B_{1},B_{2},\varphi)$ at $\varrho^{\prime}$ in Definition 4.1.
Yet, it differs by having
$p_{\sigma,\varphi}^{+}(x,\xi^{\prime},\xi_{d},\tau)$ that only depends on the
root configuration at $\varrho^{\prime}$ replaced by
$\kappa^{+}_{\varrho^{0\prime}}(\varrho^{\prime},\xi_{d})$ whose structure is
based on the root configuration at $\varrho^{0\prime}$.

Let $m^{+}$ be the common degree of
$p_{\sigma,\varphi}^{+}(\varrho^{0\prime},\xi_{d})$ and
$\kappa^{+}_{\varrho^{0\prime}}(\varrho^{\prime},\xi_{d})$ and $m^{-}$ be the
common degree of $p_{\sigma,\varphi}^{-}(\varrho^{0\prime},\xi_{d})$ and
$\kappa^{-}_{\varrho^{0\prime}}(\varrho^{\prime},\xi_{d})$ for
$\varrho^{\prime}\in\mathscr{U}$. One has $m^{+}+m^{-}=4$ and thus $m^{-}\geq
2$ for $\varrho^{\prime}\in\mathscr{U}$ since $m^{+}\leq 2$.

Invoking the Euclidean division of polynomials, one sees that it is sufficient
to consider polynomials $f$ of degree less than or equal to $m^{+}-1\leq 1$ in
(4.12). Since the degree of $b_{j,\varphi}(\varrho^{\prime},\xi_{d})$ can be
as high as $3>m^{+}-1$ it however makes sense to consider $f$ of degree less
than or equal to $m=3$. Then, the second property in Proposition 4.9 is
equivalent to having

$\displaystyle\\{b_{1,\varphi}(x,\xi^{\prime},\xi_{d},\tau),b_{2,\varphi}(x,\xi^{\prime},\xi_{d},\tau)\\}\cup\bigcup_{0\leq\ell\leq
3-m^{+}}\\{\kappa^{+}_{\varrho^{0\prime}}(\varrho^{\prime},\xi_{d})\xi_{d}^{\ell}\\}$

be a complete in the set of polynomials of degree less than or equal to $m=3$.
Note that this family is made of $m^{\prime}=6-m^{+}=2+m^{-}$ polynomials.

We now express an inequality that follows from Proposition 4.9 that will be
key in the boundary estimation given in Proposition 5.1 below.

###  4.5. Symbol positivity at the boundary

The symbols $b_{j,\varphi}$, $j=1,2$, are polynomial in $\xi_{d}$ of degree
$k_{j}\leq 3$ and we may thus write them in the form

$\displaystyle
b_{j,\varphi}(\varrho^{\prime},\xi_{d})=\sum\limits_{\ell=0}^{k_{j}}b_{j,\varphi}^{\ell}(\varrho^{\prime})\xi_{d}^{\ell},$

with $b_{j,\varphi}^{\ell}$ homogeneous of degree $k_{j}-\ell$.

The polynomial
$\xi_{d}\mapsto\kappa^{+}_{\varrho^{0\prime}}(\varrho^{\prime},\xi_{d})$ is of
degree $m^{+}\leq 2$ for $\varrho^{\prime}\in\mathscr{U}$ with $\mathscr{U}$
given by Proposition 4.9. Similarly, we write

$\displaystyle\kappa^{+}_{\varrho^{0\prime}}(\varrho^{\prime},\xi_{d})=\sum\limits_{\ell=0}^{m^{+}}\kappa^{+,\ell}_{\varrho^{0\prime}}(\varrho^{\prime})\xi_{d}^{\ell},$

with $\kappa^{+,\ell}_{\varrho^{0\prime}}$ homogeneous of degree $m^{+}-\ell$.
We introduce

$\displaystyle
e_{j,\varrho^{0\prime}}(\varrho^{\prime},\xi_{d})=\begin{cases}b_{j,\varphi}(\varrho^{\prime},\xi_{d})&\text{if}\
j=1,2,\\\
\kappa^{+}_{\varrho^{0\prime}}(\varrho^{\prime},\xi_{d})\xi_{d}^{j-3}&\text{if}\
j=3,\dots,m^{\prime}.\end{cases}$

As explained above, all these polynomials are of degree less than or equal to
three. If we now write

$\displaystyle
e_{j,\varrho^{0\prime}}(\varrho^{\prime},\xi_{d})=\sum_{\ell=0}^{3}e_{j,\varrho^{0\prime}}^{\ell}(\varrho^{\prime})\xi_{d}^{\ell},$

for $j=1,2$ one has
$e_{j,\varrho^{0\prime}}^{\ell}(\varrho^{\prime})=b_{j,\varphi}^{\ell}(\varrho^{\prime})$,
with $\ell=0,\dots,k_{j}$ and
$e_{j,\varrho^{0\prime}}^{\ell}(\varrho^{\prime})=0$ for $\ell>k_{j}$, and

$\displaystyle\text{for}\ j=3,\dots,m^{\prime},\quad
e_{j,\varrho^{0\prime}}^{\ell}(\varrho^{\prime})=\begin{cases}0&\text{if}\
\ell<j-3,\\\
\kappa^{+,\ell+3-j}_{\varrho^{0\prime}}(\varrho^{\prime})&\text{if}\
\ell=j-3,\dots,m^{+}+j-3\leq m^{+}+m^{\prime}-3=3,\\\ 0&\text{if}\
\ell>m^{+}+j-3.\end{cases}$

In particular $e_{j,\varrho^{0\prime}}^{\ell}(\varrho^{\prime})$ is
homogeneous of degree $m^{+}+j-\ell-3$. We thus have the following result.

######  Lemma 4.10.

Set the $m^{\prime}\times(m+1)$ matrix
$M(\varrho^{\prime})=(M_{j,\ell}(\varrho^{\prime}))_{{1\leq j\leq
m^{\prime}}\atop{0\leq\ell\leq m}}$ with
$M_{j,\ell}(\varrho^{\prime})=e_{j,\varrho^{0\prime}}^{\ell}(\varrho^{\prime})$.
Then, the second point in Proposition 4.9 states that $M(\varrho^{\prime})$ is
of rank $m+1=4$ for $\varrho^{\prime}\in\mathscr{U}$.

Recall that $m^{\prime}=m^{-}+2\geq 4$.

We now set

$\displaystyle\Sigma_{e_{j,\varrho^{0\prime}}}(\varrho^{\prime},\mathbf{z})=\sum_{\ell=0}^{3}e_{j,\varrho^{0\prime}}^{\ell}(\varrho^{\prime})z_{\ell}=\sum_{\ell=0}^{3}M_{j,\ell}(\varrho^{\prime})z_{\ell},\qquad\mathbf{z}=(z_{0},\dots,z_{3}).$
(4.13)

in agreement with the notation introduced in (2.3) in Section 2.4.2. One has
the following positivity result.

######  Lemma 4.11.

Let the Lopatinskiĭ-Šapiro condition hold at
$\varrho^{0\prime}=(x^{0},\xi^{0\prime},\tau^{0},\sigma^{0})\in\partial\Omega\times\mathbb{R}^{d-1}\times[0,+\infty)\times[0,+\infty)$
for $(P_{\sigma},B_{1},B_{2},\varphi)$ and let $\mathscr{U}$ be as given by
Proposition 4.9. Then, if $\varrho^{\prime}\in\mathscr{U}$ there exists $C>0$
such that

$\displaystyle\sum_{j=1}^{m^{\prime}}\big{|}\Sigma_{e_{j,\varrho^{0\prime}}}(\varrho^{\prime},\mathbf{z})\big{|}^{2}\geq
C|\mathbf{z}|_{\mathbb{C}^{4}}^{2},\qquad\mathbf{z}=(z_{0},\dots,z_{3})\in\mathbb{C}^{4}.$

###### Proof.

In $\mathbb{C}^{4}$ define the bilinear form
$\Sigma_{\mathscr{B}}(\mathbf{z},\mathbf{z}^{\prime})=\sum_{j=1}^{m^{\prime}}\Sigma_{e_{j,\varrho^{0\prime}}}(\varrho^{\prime},\mathbf{z})\overline{\Sigma_{e_{j,\varrho^{0\prime}}}(\varrho^{\prime},\mathbf{z}^{\prime})}$.
With (4.13) one has

$\displaystyle\Sigma_{\mathscr{B}}(\mathbf{z},\mathbf{z}^{\prime})=\big{(}M(\varrho^{\prime})\mathbf{z},M(\varrho^{\prime})\mathbf{z}^{\prime}\big{)}_{\mathbb{C}^{m^{\prime}}}=\big{(}\phantom{}^{t}\overline{M(\varrho^{\prime})}M(\varrho^{\prime})\mathbf{z},\mathbf{z}^{\prime}\big{)}_{\mathbb{C}^{4}}.$

As
$\operatorname{rank}\phantom{}^{t}\overline{M(\varrho^{\prime})}M(\varrho^{\prime})=\operatorname{rank}M(\varrho^{\prime})=4$
by Lemma 4.10 one obtains the result. ∎

##  5\. Estimate for the boundary norm under Lopatinskiĭ-Šapiro condition

Near $x^{0}\in\partial\Omega$ we consider two boundary operators $B_{1}$ and
$B_{2}$. As in Section 4 the associated conjugated operators are denoted by
$B_{j,\varphi}$, $j=1,2$ with respective principal symbols
$b_{j,\varphi}(x,\xi,\tau)$.

The main result of this section is the following proposition for the fourth-
order conjugated operator $P_{\sigma,\varphi}$. It is key in the final result
of the present article. It states that all traces are controlled by norms of
$B_{1,\varphi}v_{|x_{d}=0^{+}}$ and $B_{2,\varphi}v_{|x_{d}=0^{+}}$ if the
Lopatinskiĭ-Šapiro condition holds for $(P,B_{1},B_{2},\varphi)$.

######  Proposition 5.1.

Let $\kappa_{0}>0$. Let $x^{0}\in\partial\Omega$, with $\Omega$ locally given
by $\\{x_{d}>0\\}$. Assume that $(P_{\sigma},B_{1},P_{2},\varphi)$ satisfies
the Lopatinskiĭ-Šapiro condition of Definition 4.1 at
$\varrho^{\prime}=(x^{0},\xi^{\prime},\tau,\sigma)$ for all
$(\xi^{\prime},\tau,\sigma)\in\mathbb{R}^{d-1}\times[0,+\infty)\times[0,+\infty)$
such that $\tau\geq\kappa_{0}\sigma$.

Then, there exist $W^{0}$ a neighborhood of $x^{0}$, $C>0$, $\tau_{0}>0$ such
that

$\displaystyle{|\operatorname{tr}(v)|}_{3,1/2,\tau}\leq
C\big{(}{\|P_{\sigma,\varphi}v\|}_{+}+\sum\limits_{j=1}^{2}{|B_{j,\varphi}v_{|x_{d}=0^{+}}|}_{7/2-k_{j},\tau}+{\|v\|}_{4,-1,\tau}\big{)},$

for $\sigma\geq 0$, $\tau\geq\max(\tau_{0},\kappa_{0}\sigma)$ and
$v\in\overline{}\mathscr{C}^{\infty}_{c}(W^{0}_{+})$.

The notation of the function space
$\overline{}\mathscr{C}^{\infty}_{c}(W^{0}_{+})$ is introduced in (1.6).

For the proof of Proposition 5.1 we start with a microlocal version of the
result.

###  5.1. A microlocal estimate

######  Proposition 5.2.

Let $\kappa_{1}>\kappa_{0}>0$. Let $x^{0}\in\partial\Omega$, with $\Omega$
locally given by $\\{x_{d}>0\\}$ and let $W$ be a bounded open neighborhood of
$x^{0}$ in $\mathbb{R}^{d}$. Let
$(\xi^{0\prime},\tau^{0},\sigma^{0})\in\mathbb{R}^{d-1}\times[0,+\infty)\times[0,+\infty)$
nonvanishing with $\tau^{0}\geq\kappa_{1}\sigma^{0}$ and such that
$(P_{\sigma},B_{1},P_{2},\varphi)$ satisfies the Lopatinskiĭ-Šapiro condition
of Definition 4.1 at
$\varrho^{0\prime}=(x^{0},\xi^{0\prime},\tau^{0},\sigma^{0})$.

Then, there exists $\mathscr{U}$ a conic neighborhood of $\varrho^{0\prime}$
in $W\times\mathbb{R}^{d-1}\times[0,+\infty)\times[0,+\infty)$ where
$\tau\geq\kappa_{0}\sigma$ such that if $\chi\in
S_{\mbox{\tiny${\mathsf{T}}$},\tau}^{0}$, homogeneous of degree $0$ in
$(\xi^{\prime},\tau,\sigma)$ with
$\operatorname{supp}(\chi)\subset\mathscr{U}$, there exist $C>0$ and
$\tau_{0}>0$ such that

$\displaystyle{|\operatorname{tr}(\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\chi)v)|}_{3,1/2,\tau}\leq
C\Big{(}\sum\limits_{j=1}^{2}{|B_{j,\varphi}v_{|x_{d}=0^{+}}|}_{7/2-k_{j},\tau}+{\|P_{\sigma,\varphi}v\|}_{+}+{\|v\|}_{4,-1,\tau}+{|\operatorname{tr}(v)|}_{3,-1/2,\tau}\Big{)},$

for $\sigma\geq 0$, $\tau\geq\max(\tau_{0},\kappa_{0}\sigma)$ and
$v\in\overline{}\mathscr{C}^{\infty}_{c}(W_{+})$.

###### Proof.

We choose a conic neighborhood $\mathscr{U}_{0}$ of $\varrho^{0\prime}$
according to Proposition 4.9 and such that $\mathscr{U}_{0}\subset
W\times\mathbb{R}^{d-1}\times[0,+\infty)\times[0,+\infty)$ . Assume moreover
that $\tau\geq\kappa_{0}\sigma$ in $\mathscr{U}_{0}$.

In Section 4.5 we introduced the symbols
$e_{j,\varrho^{0\prime}}(\varrho^{\prime},\xi_{d})$,
$j=1,\dots,m^{\prime}=m^{-}+2=6-m^{+}$. Set
$\mathbb{S}_{\overline{\mathscr{U}_{0}}}=\\{\varrho^{\prime}=(x,\xi^{\prime},\tau,\sigma)\in\overline{\mathscr{U}_{0}};\
|(\xi^{\prime},\tau,\sigma)|=1\\}$.

Consequence of the Lopatinskiĭ-Šapiro condition holding at $\varrho^{0\prime}$
for all $\varrho^{\prime}\in\mathbb{S}_{\overline{\mathscr{U}_{0}}}$, by Lemma
4.11 there exists $C>0$ such that

$\displaystyle\sum_{j=1}^{m^{\prime}}\big{|}\Sigma_{e_{j,\varrho^{0\prime}}}(\varrho^{\prime},\mathbf{z})\big{|}^{2}\geq
C|\mathbf{z}|_{\mathbb{C}^{4}}^{2},\qquad\mathbf{z}=(z_{0},\dots,z_{3})\in\mathbb{C}^{4}.$

Since $\mathbb{S}_{\overline{\mathscr{U}_{0}}}$ is compact (recall that $W$ is
bounded), there exists $C_{0}>0$ such that

$\displaystyle\sum_{j=1}^{m^{\prime}}\big{|}\Sigma_{e_{j,\varrho^{0\prime}}}(\varrho^{\prime},\mathbf{z})\big{|}^{2}\geq
C_{0}|\mathbf{z}|_{\mathbb{C}^{4}}^{2},\qquad\mathbf{z}=(z_{0},\dots,z_{3})\in\mathbb{C}^{4},\
\varrho^{\prime}\in\mathbb{S}_{\overline{\mathscr{U}_{0}}}.$

Introducing the map $N_{t}\varrho^{\prime}=(x,t\xi^{\prime},t\tau,t\sigma)$,
for $\varrho^{\prime}=(x,\xi^{\prime},\tau,\sigma)$ with
$t=|(\xi^{\prime},\tau,\sigma)|^{-1}$ one has

$\displaystyle\sum_{j=1}^{m^{\prime}}\big{|}\Sigma_{e_{j,\varrho^{0\prime}}}(N_{t}\varrho^{\prime},\mathbf{z})\big{|}^{2}\geq
C_{0}|\mathbf{z}|_{\mathbb{C}^{4}}^{2},\qquad\mathbf{z}=(z_{0},\dots,z_{3})\in\mathbb{C}^{4},\
\varrho^{\prime}\in\mathscr{U}_{0},$ (5.1)

since $N_{t}\varrho^{\prime}\in\mathbb{S}_{\overline{\mathscr{U}_{0}}}$. Now,
for $j=1,2$ one has

$\displaystyle\Sigma_{e_{j,\varrho^{0\prime}}}(\varrho^{\prime},\mathbf{z})=\sum_{\ell=0}^{k_{j}}e_{j,\varrho^{0\prime}}^{\ell}(\varrho^{\prime})z_{\ell},$

with $e_{j,\varrho^{0\prime}}^{\ell}(\varrho^{\prime})$ homogeneous of degree
$k_{j}-\ell$, and for $3\leq j\leq m^{\prime}$ one has

$\displaystyle\Sigma_{e_{j,\varrho^{0\prime}}}(\varrho^{\prime},\mathbf{z})=\sum_{\ell=0}^{3}e_{j,\varrho^{0\prime}}^{\ell}(\varrho^{\prime})z_{\ell},$

with $e_{j,\varrho^{0\prime}}^{\ell}(\varrho^{\prime})$ homogeneous of degree
$m^{+}+j-\ell-3$. We define $\mathbf{z}^{\prime}\in\mathbb{C}^{4}$ by
$z^{\prime}_{\ell}=t^{\ell-7/2}z_{\ell}$, $\ell=0,\dots,3$. One has

$\displaystyle\Sigma_{e_{j,\varrho^{0\prime}}}(N_{t}\varrho^{\prime},\mathbf{z}^{\prime})=t^{k_{j}-7/2}\Sigma_{e_{j,\varrho^{0\prime}}}(\varrho^{\prime},\mathbf{z}),\qquad
j=1,2,$

and

$\displaystyle\Sigma_{e_{j,\varrho^{0\prime}}}(N_{t}\varrho^{\prime},\mathbf{z}^{\prime})=t^{m^{+}+j-13/2}\Sigma_{e_{j,\varrho^{0\prime}}}(\varrho^{\prime},\mathbf{z}),\qquad
j=3,\dots,m^{\prime}.$

Thus from (5.1) we deduce

$\displaystyle\sum_{j=1}^{2}\lambda_{\mbox{\tiny${\mathsf{T}}$},\tau}^{2(7/2-k_{j})}\big{|}\Sigma_{e_{j,\varrho^{0\prime}}}(\varrho^{\prime},\mathbf{z})\big{|}^{2}+\sum_{j=3}^{m^{\prime}}\lambda_{\mbox{\tiny${\mathsf{T}}$},\tau}^{2(13/2-m^{+}-j)}\big{|}\Sigma_{e_{j,\varrho^{0\prime}}}(\varrho^{\prime},\mathbf{z})\big{|}^{2}\geq
C_{0}\sum_{\ell=0}^{3}\lambda_{\mbox{\tiny${\mathsf{T}}$},\tau}^{2(7/2-\ell)}|z_{\ell}|^{2},$
(5.2)

for $\mathbf{z}=(z_{0},\dots,z_{3})\in\mathbb{C}^{4}$, and
$\varrho^{\prime}\in\mathscr{U}_{0}$, since
$t\asymp\lambda_{\mbox{\tiny${\mathsf{T}}$},\tau}^{-1}$ as $\tau\gtrsim\sigma$
in $\mathscr{U}_{0}$.

We now choose $\mathscr{U}$ a conic open neighborhood of $\varrho^{0\prime}$,
such that $\overline{\mathscr{U}}\subset\mathscr{U}_{0}$. Let $\chi\in
S_{\mbox{\tiny${\mathsf{T}}$},\tau}^{0}$ be as in the statement and let
$\tilde{\chi}\in S_{\mbox{\tiny${\mathsf{T}}$},\tau}^{0}$ be homogeneous of
degree $0$, with $\operatorname{supp}(\tilde{\chi})\subset\mathscr{U}_{0}$ and
$\tilde{\chi}\equiv 1$ in a neighborhood of $\overline{\mathscr{U}}$, and thus
in a neighborhood of $\operatorname{supp}(\chi)$.

For $j=3,\dots,m^{\prime}$ one has
$e_{j,\varrho^{0\prime}}(\varrho^{\prime},\xi_{d})=\kappa^{+}_{\varrho^{0\prime}}(\varrho^{\prime},\xi_{d})\xi_{d}^{j-3}\in
S_{\tau}^{m^{+}+j-3,0}$. Set
$E_{j}=\mathrm{Op}(\tilde{\chi}e_{j,\varrho^{0\prime}})$. The introduction of
$\tilde{\chi}$ is made such that $\tilde{\chi}e_{j,\varrho^{0\prime}}$ is
defined on the whole tangential phase-space. Observe that

$\displaystyle\mathscr{B}(w)$
$\displaystyle=\sum\limits_{j=1}^{2}{|B_{j,\varphi}w_{|x_{d}=0^{+}}|}_{7/2-k_{j},\tau}^{2}+\sum\limits_{j=3}^{m^{\prime}}{|E_{j}w_{|x_{d}=0^{+}}|}_{13/2-m^{+}-j,\tau}^{2}$
$\displaystyle=\sum\limits_{j=1}^{2}{|\Lambda_{\mbox{\tiny${\mathsf{T}}$},\tau}^{7/2-k_{j}}B_{j,\varphi}w_{|x_{d}=0^{+}}|}_{\partial}^{2}+\sum\limits_{j=3}^{m^{\prime}}{|\Lambda_{\mbox{\tiny${\mathsf{T}}$},\tau}^{13/2-m^{+}-j}E_{j}w_{|x_{d}=0^{+}}|}_{\partial}^{2}$

is a boundary quadratic form of type $(3,1/2)$ as in Definition 2.4. From
Proposition 2.6 and (5.2) we have

$\displaystyle{|\operatorname{tr}(u)|}_{3,1/2,\tau}^{2}\lesssim\sum\limits_{j=1}^{2}{|B_{j,\varphi}u_{|x_{d}=0^{+}}|}_{7/2-k_{j},\tau}^{2}+\sum\limits_{j=3}^{m^{\prime}}{|E_{j}u_{|x_{d}=0^{+}}|}_{13/2-m^{+}-j,\tau}^{2}+{|\operatorname{tr}(v)|}_{3,-N,\tau}^{2}.$
(5.3)

for $u=\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\chi)v$ and
$\tau\geq\kappa_{0}\sigma$ chosen sufficiently large.

In $\mathscr{U}_{0}$ one can write

$\displaystyle
p_{\sigma,\varphi}=p_{\sigma,\varphi}^{+}p_{\sigma,\varphi}^{-}=\kappa^{+}_{\varrho^{0\prime}}\kappa^{-}_{\varrho^{0\prime}},$

with $\kappa^{+}_{\varrho^{0\prime}}$ of degree $m^{+}$ and
$\kappa^{-}_{\varrho^{0\prime}}$ of degree $m^{-}$. In fact we set

$\displaystyle\tilde{\kappa}^{+}_{\varrho^{0\prime}}(\varrho^{\prime})=\prod_{j\in
J^{+}}\big{(}\xi_{d}-\tilde{\chi}\rho_{j}(\varrho^{\prime})\big{)},\qquad\tilde{\kappa}^{-}_{\varrho^{0\prime}}(\varrho^{\prime})=\prod_{j\in
J^{-}}\big{(}\xi_{d}-\tilde{\chi}\rho_{j}(\varrho^{\prime})\big{)},$

with the notation of Section 4.4, thus making the two symbols defined on the
whole tangential phase-space. In $\mathscr{U}$, one has also

$\displaystyle
p_{\sigma,\varphi}=\tilde{\kappa}^{+}_{\varrho^{0\prime}}\tilde{\kappa}^{-}_{\varrho^{0\prime}}.$

The factor $\tilde{\kappa}^{-}_{\varrho^{0\prime}}$ is associated with roots
with negative imaginary part. With Lemma A.1 given in Appendix A.1 one has the
following microlocal elliptic estimate

$\displaystyle{\|\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\chi)w\|}_{m^{-},\tau}+{|\operatorname{tr}(\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\chi)w)|}_{m^{-}-1,1/2,\tau}\lesssim{\|\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\tilde{\kappa}^{-}_{\varrho^{0\prime}})\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\chi)w\|}_{+}+{\|w\|}_{m^{-},-N,\tau},$

for $w\in\overline{\mathscr{S}}(\mathbb{R}^{d}_{+})$ and
$\tau\geq\kappa_{0}\tau$ chosen sufficiently large. We apply this inequality
to
$w=\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\tilde{\kappa}^{+}_{\varrho^{0\prime}})v$.
Since

$\displaystyle\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\tilde{\kappa}^{-}_{\varrho^{0\prime}})\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\chi)\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\tilde{\kappa}^{+}_{\varrho^{0\prime}})=\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\chi)P_{\sigma,\varphi}\mod\Psi_{\tau}^{4,-1},$

one obtains

$\displaystyle{|\operatorname{tr}(\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\chi)\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\tilde{\kappa}^{+}_{\varrho^{0\prime}})v)|}_{m^{-}-1,1/2,\tau}\lesssim{\|P_{\sigma,\varphi}v\|}_{+}+{\|v\|}_{4,-1,\tau}.$

With
$[\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\chi),\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\tilde{\kappa}^{+}_{\varrho^{0\prime}})]\in\Psi_{\tau}^{m^{+},-1}$
one then has

$\displaystyle{|\operatorname{tr}(\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\tilde{\kappa}^{+}_{\varrho^{0\prime}})u)|}_{m^{-}-1,1/2,\tau}\lesssim{\|P_{\sigma,\varphi}v\|}_{+}+{\|v\|}_{4,-1,\tau}+{|\operatorname{tr}(v)|}_{3,-1/2,\tau},$

with $u=\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\chi)v$ as above, using that
$m^{+}+m^{-}=4$. Note that

$\displaystyle{|\operatorname{tr}(\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\tilde{\kappa}^{+}_{\varrho^{0\prime}})u)|}_{m^{-}-1,1/2,\tau}$
$\displaystyle\asymp\sum_{j=0}^{m^{-}-1}{|D_{d}^{j}\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\tilde{\kappa}^{+}_{\varrho^{0\prime}})u_{|x_{d}=0^{+}}|}_{m^{-}-j-1/2,\tau}$
$\displaystyle\gtrsim\sum_{j=3}^{m^{\prime}}{|E_{j}u_{|x_{d}=0^{+}}|}_{5/2+m^{-}-j,\tau}-{|\operatorname{tr}(v)_{|x_{d}=0^{+}}|}_{3,-1/2,\tau},$

using that
$\xi_{d}^{j}\tilde{\kappa}^{+}_{\varrho^{0\prime}}=\tilde{\chi}e_{j+3,\varrho^{0\prime}}$
in a conic neighborhood of $\operatorname{supp}(\chi)$ and using that
$m^{-}=m^{\prime}-2$. We thus obtain

$\displaystyle\sum_{j=3}^{m^{\prime}}{|E_{j}u_{|x_{d}=0^{+}}|}_{13/2-m^{+}-j,\tau}\lesssim{\|P_{\sigma,\varphi}v\|}_{+}+{\|v\|}_{4,-1,\tau}+{|\operatorname{tr}(v)|}_{3,-1/2,\tau},$

since $13/2-m^{+}=5/2+m^{-}$. With (5.3) then one finds

$\displaystyle{|\operatorname{tr}(u)|}_{3,1/2,\tau}\lesssim\sum\limits_{j=1}^{2}{|B_{j,\varphi}u_{|x_{d}=0^{+}}|}_{7/2-k_{j},\tau}^{2}+{\|P_{\sigma,\varphi}v\|}_{+}+{\|v\|}_{4,-1,\tau}+{|\operatorname{tr}(v)|}_{3,-1/2,\tau}.$

In addition, observing that

$\displaystyle{|B_{j,\varphi}u_{|x_{d}=0^{+}}|}_{7/2-k_{j},\tau}\lesssim{|B_{j,\varphi}v_{|x_{d}=0^{+}}|}_{7/2-k_{j},\tau}+{|\operatorname{tr}(v)|}_{3,-1/2,\tau},$

the result of Proposition 5.2 follows. ∎

###  5.2. Proof of Proposition 5.1

As mentioned above the proof relies on a patching procedure of microlocal
estimates given by Proposition 5.2.

Let $0<\kappa_{0}^{\prime}<\kappa_{0}$. We set

$\displaystyle\Gamma^{d-1}_{+,\kappa_{0}}=\\{(\xi^{\prime},\tau,\sigma)\in\mathbb{R}^{d-1}\times[0,+\infty)\times[0,+\infty);\
\tau\geq\kappa_{0}\sigma,\\},$

and

$\displaystyle\mathbb{S}^{d-1}_{+,\kappa_{0}}=\\{(\xi^{\prime},\tau,\sigma)\in\Gamma^{d-1}_{+,\kappa_{0}};\
|(\xi^{\prime},\tau,\sigma)|=1\\}.$

Consider
$(\xi^{0\prime},\tau^{0},\sigma^{0})\in\mathbb{S}^{d-1}_{+,\kappa_{0}}$. Since
the Lopatinskiĭ-Šapiro condition holds at
$\varrho^{0\prime}=(x^{0},\xi^{0\prime},\tau^{0},\sigma^{0})$, we can invoke
Proposition 5.2:

  1. (1)

There exists a conic open neighborhood $\mathscr{U}_{\varrho^{0\prime}}$ of
$\varrho^{0\prime}$ in
$W\times\mathbb{R}^{d-1}\times[0,+\infty)\times[0,+\infty)$ where
$\tau\geq\kappa_{0}^{\prime}\sigma$;

  2. (2)

For any $\chi_{\varrho^{0\prime}}\in S_{\mbox{\tiny${\mathsf{T}}$},\tau}^{0}$
homogeneous of degree $0$ supported in $\mathscr{U}_{\varrho^{0\prime}}$ the
estimate of Proposition 5.2 applies to
$\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\chi_{\varrho^{0\prime}})v$ for
$\tau\geq\max(\tau_{\varrho^{0\prime}},\kappa_{0}\sigma)$.

Without any loss of generality we may choose $\mathscr{U}_{\varrho^{0\prime}}$
of the form
$\mathscr{U}_{\varrho^{0\prime}}=\mathscr{O}_{\varrho^{0\prime}}\times\Gamma_{\varrho^{0\prime}}$,
with $\mathscr{O}_{\varrho^{0\prime}}\subset W$ an open neighborhood of
$x^{0}$ and $\Gamma_{\varrho^{0\prime}}$ a conic open neighborhood of
$(\xi^{0\prime},\tau^{0},\sigma^{0})$ in
$\mathbb{R}^{d-1}\times[0,+\infty)\times[0,+\infty)$ where
$\tau\geq\kappa_{0}^{\prime}\sigma$.

Since $\\{x^{0}\\}\times\mathbb{S}^{d-1}_{+,\kappa_{0}}$ is compact we can
extract a finite covering of it by open sets of the form of
$\mathscr{U}_{\varrho^{0\prime}}$. We denote by $\tilde{\mathscr{U}}_{i}$,
$i\in I$ with $|I|<\infty$, such a finite covering. This is also a finite
covering of $\\{x^{0}\\}\times\Gamma^{d-1}_{+,\kappa_{0}}$.

Each $\tilde{\mathscr{U}}_{i}$ has the form
$\tilde{\mathscr{U}}_{i}=\mathscr{O}_{i}\times\Gamma_{i}$, with
$\mathscr{O}_{i}$ an open neighborhood of $x^{0}$ and $\Gamma_{i}$ a conic
open set in $\mathbb{R}^{d-1}\times[0,+\infty)\times[0,+\infty)$ where
$\tau\geq\kappa_{0}^{\prime}\sigma$.

We set $\mathscr{O}=\cap_{i\in I}\mathscr{O}_{i}$ and
$\mathscr{U}_{i}=\mathscr{O}\times\Gamma_{i}$, $i\in I$.

Let $W^{0}$ be an open neighborhood of $x^{0}$ such that
$W^{0}\Subset\mathscr{O}$. The open sets $\mathscr{U}_{i}$ give also an open
covering of $\overline{W^{0}}\times\mathbb{S}^{d-1}_{+,\kappa_{0}}$ and
$\overline{W^{0}}\times\Gamma^{d-1}_{+,\kappa_{0}}$. With this second covering
we associate a partition of unity $\chi_{i}$, $i\in I$, of
$\overline{W^{0}}\times\mathbb{S}^{d-1}_{+,\kappa_{0}}$, where each $\chi_{i}$
is chosen smooth and homogeneous of degree one for
$|(\xi^{\prime},\tau,\sigma)|\geq 1$, that is:

$\displaystyle\sum\limits_{i\in
I}\chi_{i}(\varrho^{\prime})=1\qquad\text{for}\
\varrho^{\prime}=(x,\xi^{\prime},\tau,\sigma)\ \text{in a neighborhood of}\
\overline{W^{0}}\times\Gamma^{d-1}_{+,\kappa_{0}},\ \text{and}\
|(\xi^{\prime},\tau,\sigma)|\geq 1.$

Let $u\in\overline{}\mathscr{C}^{\infty}_{c}(W^{0}_{+}))$. Since each
$\chi_{i}$ is in $S_{\mbox{\tiny${\mathsf{T}}$},\tau}^{0}$ and supported in
$\mathscr{U}_{i}$, Proposition 5.2 applies:

$\displaystyle{|\operatorname{tr}(\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\chi_{i})v)|}_{3,1/2,\tau}\leq
C_{i}\Big{(}\sum\limits_{j=1}^{2}{|B_{j,\varphi}v_{|x_{d}=0^{+}}|}_{7/2-k_{j},\tau}+{\|P_{\sigma,\varphi}v\|}_{+}+{\|v\|}_{4,-1,\tau}+{|\operatorname{tr}(v)|}_{3,-1/2,\tau}\Big{)},$
(5.4)

for some $C_{i}>0$, for $\sigma\geq 0$,
$\tau\geq\max(\tau_{i},\kappa_{0}\sigma)$ for some $\tau_{i}>0$.

We set $\tilde{\chi}=1-\sum\limits_{i\in I}\chi_{i}$. One has $\tilde{\chi}\in
S_{\mbox{\tiny${\mathsf{T}}$},\tau}^{-\infty}$ microlocally in a neighborhood
of $\overline{W^{0}}\times\Gamma^{d-1}_{+,\kappa_{0}}$. Thus, considering the
definition of $\Gamma^{d-1}_{+,\kappa_{0}}$, if one imposes
$\tau\geq\kappa_{0}\sigma$, as we do, then $\tilde{\chi}\in
S_{\mbox{\tiny${\mathsf{T}}$},\tau}^{-\infty}$ locally in a neighborhood of
$\overline{W^{0}}$.

For any $N\in\mathbb{N}$ using that $\operatorname{supp}(v)\subset W^{0}$ one
has

$\displaystyle{|\operatorname{tr}(v)|}_{3,1/2,\tau}$
$\displaystyle\leq\sum\limits_{i\in
I}{|\operatorname{tr}(\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\chi_{i})v)|}_{3,1/2,\tau}+{|\operatorname{tr}(\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\tilde{\chi})v)|}_{3,1/2,\tau}$
$\displaystyle\lesssim\sum\limits_{i\in
I}{|\operatorname{tr}(\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\chi_{i})v)|}_{3,1/2,\tau}+{|\operatorname{tr}(v)|}_{3,-N,\tau}$
$\displaystyle\lesssim\sum\limits_{i\in
I}{|\operatorname{tr}(\mathrm{Op}_{\mbox{\tiny${\mathsf{T}}$}}(\chi_{i})v)|}_{3,1/2,\tau}+{\|v\|}_{4,-N,\tau}.$

Summing estimates (5.4) together for $i\in I$ we thus obtain

$\displaystyle{|\operatorname{tr}(v)|}_{3,1/2,\tau}\lesssim\sum\limits_{j=1}^{2}{|B_{j,\varphi}v_{|x_{d}=0^{+}}|}_{7/2-k_{j},\tau}+{\|P_{\sigma,\varphi}v\|}_{+}+{\|v\|}_{4,-1,\tau}+{|\operatorname{tr}(v)|}_{3,-1/2,\tau},$

for $\tau\geq\max(\max_{i}\tau_{i},\kappa_{0}\sigma)$. Therefore, by choosing
$\tau\geq\kappa_{0}\sigma$ sufficiently large one obtains the result of
Proposition 5.1. ∎

##  6\. Microlocal estimate for each second-order factors composing
$P_{\sigma,\varphi}$

We recall that
$P_{\sigma}=\Delta^{2}-\sigma^{4}=(-\Delta-\sigma^{2})(-\Delta+\sigma^{2})$
with $\sigma\geq 0$. Set $Q_{\sigma}^{j}=-\Delta+(-1)^{j}\sigma^{2}$; then
$P_{\sigma}=Q_{\sigma}^{1}Q_{\sigma}^{2}.$ We also set $Q=-\Delta$, that is,
$Q=Q^{1}_{0}=Q^{2}_{0}$.

The principal symbols of $Q_{\sigma}^{j}$ and $Q$ are given by

$\displaystyle
q_{\sigma}^{j}(x,\xi)=\xi_{d}^{2}+r(x,\xi^{\prime})+(-1)^{j}\sigma^{2}\ \
\text{and}\ \ q(x,\xi)=\xi_{d}^{2}+r(x,\xi^{\prime}),$ (6.1)

respectively. The conjugated operator
$P_{\sigma,\varphi}=e^{\tau\varphi}P_{\sigma}e^{-\tau}$ reads

$\displaystyle
P_{\sigma,\varphi}=Q_{\sigma,\varphi}^{1}Q_{\sigma,\varphi}^{2},\ \
\text{with}\ \
Q_{\sigma,\varphi}^{j}=e^{\tau\varphi}Q_{\sigma}^{j}e^{-\tau\varphi}.$

We set

$\displaystyle
Q_{s}^{j}=\frac{Q_{\sigma,\varphi}^{j}+(Q_{\sigma,\varphi}^{j})^{*}}{2}\ \
\text{and}\ \
Q_{a}=\frac{Q_{\sigma,\varphi}^{j}-(Q_{\sigma,\varphi}^{j})^{*}}{2i},$

both formally selfadjoint and such that
$Q_{\sigma,\varphi}^{j}=Q_{s}^{j}+iQ_{a}$. Note that $Q_{a}$ is independent of
$\sigma$. Their respective principal symbols are

$\displaystyle
q_{s}^{j}(x,\xi,\tau,\sigma)=\xi_{d}^{2}-(\tau\partial_{d}\varphi)^{2}+r(x,\xi^{\prime})+(-1)^{j}\sigma^{2}-\tau^{2}r(x,d_{x^{\prime}}\varphi),$
$\displaystyle
q_{a}(x,\xi,\tau)=2\tau\xi_{d}\partial_{d}\varphi+2\tau\tilde{r}(x,\xi^{\prime},d_{x^{\prime}}\varphi).$

Note that $Q_{s}^{j}$ and $Q_{a}$ take the forms

$\displaystyle Q_{s}^{j}=D_{d}^{2}+T_{s}^{j},\qquad
Q_{a}=\tau(\partial_{d}\varphi D_{d}+D_{d}\partial_{d}\varphi)+T_{a},$ (6.2)

where $T_{s}^{j},T_{a}\in\mathscr{D}_{\mbox{\tiny${\mathsf{T}}$},\tau}^{2}$
are such that $(T_{s}^{j})^{*}=T_{s}^{j}$ and $T_{a}^{*}=T_{a}$. Naturally,
the principal symbol of $Q_{\sigma,\varphi}^{j}$ is

$\displaystyle
q_{\sigma,\varphi}^{j}(x,\xi,\tau)=q_{s}^{j}(x,\xi,\tau,\sigma)+iq_{a}(x,\xi,\tau).$

The principal symbol of
$Q_{\sigma,\varphi}^{j}=e^{\tau\varphi}Q_{\sigma}^{j}e^{-\tau\varphi}$ is

$\displaystyle
q_{\sigma,\varphi}^{j}(x,\xi,\tau,\sigma)=(\xi_{d}+i\tau\partial_{d}\varphi)^{2}+r(x,\xi^{\prime})+(-1)^{j}\sigma^{2}-\tau^{2}r(x,d_{x^{\prime}}\varphi)+2i\tau\tilde{r}(x,\xi^{\prime},d_{x^{\prime}}\varphi).$

As in Section 4.2 we let $\alpha_{j}\in\mathbb{C}$ be such that

$\displaystyle\alpha_{j}(x,\xi^{\prime},\tau,\sigma)^{2}$
$\displaystyle=r(x,\xi^{\prime}+i\tau
d_{x^{\prime}}\varphi)+(-1)^{j}\sigma^{2}$
$\displaystyle=r(x,\xi^{\prime})-\tau^{2}r(x,d_{x^{\prime}}\varphi)+2i\tau\tilde{r}(x,\xi^{\prime},d_{x^{\prime}}\varphi)+(-1)^{j}\sigma^{2},$

and $\operatorname{Re}\alpha_{j}\geq 0$. Note that uniqueness in the choice of
$\alpha_{j}$ holds except if $r(x,\xi^{\prime}+i\tau
d_{x^{\prime}}\varphi)+(-1)^{j}\sigma^{2}\in\mathbb{R}^{-}$; this lack of
uniqueness in that case is however not an issue in what follows. One has

$\displaystyle q_{\sigma,\varphi}^{j}(x,\xi^{\prime},\xi_{d},\tau)$
$\displaystyle=(\xi_{d}+i\tau\partial_{d}\varphi)^{2}+\alpha_{j}(x,\xi^{\prime},\tau,\sigma)^{2}$
$\displaystyle=\big{(}\xi_{d}+i\tau\partial_{d}\varphi+i\alpha_{j}(x,\xi^{\prime},\tau,\sigma)\big{)}\big{(}\xi_{d}+i\tau\partial_{d}\varphi-i\alpha_{j}(x,\xi^{\prime},\tau,\sigma)\big{)}.$

We recall from (4.4) that we write
$q_{\sigma,\varphi}^{j}(x,\xi^{\prime},\xi_{d},\tau)=(\xi_{d}-\pi_{j,1})(\xi_{d}-\pi_{j,2})$
with

$\displaystyle\pi_{j,1}=-i\tau\partial_{d}\varphi-i\alpha_{j}(x,\xi^{\prime},\tau,\sigma)\
\ \text{and}\ \
\pi_{j,2}=-i\tau\partial_{d}\varphi+i\alpha_{j}(x,\xi^{\prime},\tau,\sigma).$

The roots $\pi_{j,k}$, $k=1,2$ are functions of $x,\xi^{\prime},\tau$ and
$\sigma.$

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

$B(x,D)=B^{k}(x,D^{\prime})+B^{k-1}(x,D^{\prime})D_{d},$

with $B^{k}(x,D^{\prime})$ and $B^{k-1}(x,D^{\prime})$ tangential differential
operators of order $k$ and $k-1$ respectively. The boundary operator $B(x,D)$
has $b(x,\xi)=b^{k}(x,\xi^{\prime})+b^{k-1}(x,\xi^{\prime})\xi_{d}$ for
principal symbol. The conjugate boundary operator
$B_{\varphi}=e^{\tau\varphi}Be^{-\tau\varphi}$ is then given by

$\displaystyle B_{\varphi}(x,D,\tau)$
$\displaystyle=B^{k}_{\varphi}(x,D^{\prime},\tau)+B^{k-1}_{\varphi}(x,D^{\prime},\tau)(D_{d}+i\tau\partial_{d}\varphi)$
$\displaystyle=\hat{B}^{k}_{\varphi}(x,D^{\prime},\tau)+B^{k-1}_{\varphi}(x,D^{\prime},\tau)D_{d},$

with
$\hat{B}^{k}_{\varphi}(x,D^{\prime},\tau)=B^{k}_{\varphi}(x,D^{\prime},\tau)+i\tau
B^{k-1}_{\varphi}(x,D^{\prime},\tau)\partial_{d}\varphi$. The principal symbol
of $B_{\varphi}(x,D,\tau)$ is

$\displaystyle
b_{\varphi}(x,\xi,\tau)=\hat{b}^{k}_{\varphi}(x,\xi^{\prime},\tau)+b^{k-1}_{\varphi}(x,\xi^{\prime},\tau)\xi_{d},$

where $b^{k-1}_{\varphi}(x,\xi^{\prime},\tau)$ is homogeneous of degree $k-1$
in $\lambda_{\mbox{\tiny${\mathsf{T}}$},\tau}$ and
$\hat{b}^{k}_{\varphi}(x,\xi^{\prime},\tau)=b^{k}_{\varphi}(x,\xi^{\prime},\tau)+i\tau
b^{k-1}_{\varphi}(x,\xi^{\prime},\tau)\partial_{d}\varphi$ is homogeneous of
degree $k$ in $\lambda_{\mbox{\tiny${\mathsf{T}}$},\tau}$.

###  6.1. Sub-ellipticity

Set

$\displaystyle
q_{s}(x,\xi,\tau)=\xi_{d}^{2}+r(x,\xi^{\prime})-(\tau\partial_{d}\varphi)^{2}-r(x,\tau
d_{x^{\prime}}\varphi)=|\xi|^{2}_{x}-|\tau d\varphi|_{x}^{2},$

where $|\xi|^{2}_{x}=\xi_{d}^{2}+r(x,\xi^{\prime})$. One has
$q_{s}^{j}=q_{s}+(-1)^{j}\sigma^{2}$. Observe that
$\\{q_{s}^{j},q_{a}\\}=\\{q_{s},q_{a}\\}$.

######  Definition 6.1 (Sub-ellipticity).

Let $W$ be a bounded open subset of $\mathbb{R}^{d}$ and
$\varphi\in\mathscr{C}^{\infty}(\overline{W})$ such that $|d_{x}\varphi|>0$.
Let $j=1$ or $2$. We say that the couple $(Q_{\sigma}^{j},\varphi)$ satisfies
the sub-ellipticity condition in $\overline{W}$ if there exist $C>0$ and
$\tau_{0}>0$ such that for $\sigma>0$

$\displaystyle\forall\leavevmode\nobreak\
(x,\xi)\in\overline{W}\times\mathbb{R}^{d},\leavevmode\nobreak\
\tau\geq\tau_{0}\sigma,\leavevmode\nobreak\
q_{\sigma,\varphi}^{j}(x,\xi,\tau)=0\Rightarrow\\{q_{s}^{j},q_{a}\\}(x,\xi,\tau)=\\{q_{s},q_{a}\\}(x,\xi,\tau)\geq
C>0.$

######  Remark 6.2.

Note that with homogeneity the sub-ellipticity property also reads

$\displaystyle\forall\leavevmode\nobreak\
(x,\xi)\in\overline{W}\times\mathbb{R}^{d},\leavevmode\nobreak\
\tau\geq\tau_{0}\sigma,\leavevmode\nobreak\
q_{\sigma,\varphi}^{j}(x,\xi,\tau)=0\Rightarrow\\{q_{s}^{j},q_{a}\\}(x,\xi,\tau)\geq
C\lambda_{\tau}^{3}.$

######  Proposition 6.3.

Let $W$ be a bounded open subset of $\mathbb{R}^{d}$ and
$\psi\in\mathscr{C}^{\infty}(\mathbb{R}^{d})$ such that $\psi\geq 0$ and
$|d_{x}\psi|\geq C>0$ on $\overline{W}.$ Let $\tau_{0}>0$. Then, there exists
$\gamma_{0}\geq 1$ such that $(Q_{\sigma}^{j},\varphi)$ satisfies the sub-
ellipticity condition on $\overline{W}$ for $\tau\geq\tau_{0}\sigma$ for
$\varphi=e^{\gamma\psi}$, with $\gamma\geq\gamma_{0}$, for both $j=1$ and $2$.

###### Proof.

We note that $|d_{x}\varphi(x)|\neq 0$.

The proof is slightly different whether one considers the symbol
$q_{\sigma,\varphi}^{1}$ or the symbol $q_{\sigma,\varphi}^{2}.$

Case 1: proof for $q_{\sigma,\varphi}^{1}$. Assume that
$q_{\sigma,\varphi}^{1}=0$. Thus $|\xi|_{x}^{2}-|\tau
d\varphi|_{x}^{2}-\sigma^{2}=0$ implying $|\xi|\sim\sigma+\gamma\tau\varphi$.
On the one hand by Lemma 3.55 in [33], one has

$\displaystyle\\{q_{s},q_{a}\\}(x,\xi,\tau)$
$\displaystyle=\tau(\gamma^{2}\varphi)(\gamma\varphi)^{2}\left((H_{q}\psi(x,\beta))^{2}+4\tau^{2}q(x,d\psi(x))^{2}\right)$
$\displaystyle\quad+(\gamma\varphi)^{3}\frac{1}{2i}\\{\overline{q_{\psi}},q_{\psi}\\}(x,\beta,\tau),$
(6.3)

with $\beta=\xi/(\gamma\varphi)$, and where $H_{q}$ denotes the Hamiltonian
vector field associated with the symbol $q$ as defined in (6.1). Here,
$q_{\psi}$ denotes the principal symbol of $e^{\tau\psi}Qe^{-\tau\psi}$, that
is,

$\displaystyle q_{\psi}(x,\xi,\tau)=q(x,\xi+i\tau
d_{x}\psi(x))=(\xi_{d}+i\tau\partial_{d}\psi(x))^{2}+r(x,\xi^{\prime}+i\tau
d_{x^{\prime}}\psi(x)).$

On the other hand, one has
$(H_{q}\psi(x,\beta))^{2}+4\tau^{2}q(x,d\psi(x))^{2}\gtrsim\tau^{2}$ and since
$\frac{1}{2i}\\{\overline{q_{\psi}},q_{\psi}\\}(x,\beta,\tau)$ is homogeneous
of degree $3$ in $(\beta,\tau),$ we obtain

$\displaystyle\\{q_{s},q_{a}\\}\geq
C\gamma(\gamma\tau\varphi)^{3}-C^{\prime}(\gamma\tau\varphi+|\beta|\gamma\varphi)^{3}=C\gamma\tilde{\tau}^{3}-C^{\prime\prime}(\tilde{\tau}+|\xi|)^{3},\
\ \text{with}\ \tilde{\tau}=\gamma\tau\varphi.$

Yet one has $|\xi|\sim\sigma+\tilde{\tau}$ implying

$\displaystyle\\{q_{s},q_{a}\\}\geq
C\gamma\tilde{\tau}^{3}-C^{\prime\prime\prime}(\tilde{\tau}+|\xi|)^{3}\geq
C\gamma\tilde{\tau}^{3}-C^{(4)}(\tilde{\tau}+\sigma)^{3}.$

Since $\psi\geq 0$ and $\gamma\geq 1$ one has $\varphi\geq 1$ implying
$\tau_{0}\sigma\leq\tau\lesssim\tilde{\tau}$ and thus

$\displaystyle\\{q_{s},q_{a}\\}(x,\xi,\tau)\geq\tilde{\tau}^{3}(C\gamma-C^{(5)}).$

It follows that for $\gamma$ chosen sufficiently large one finds
$\\{q_{s},q_{a}\\}(x,\xi,\tau)\geq C>0.$

Case 2: proof for $q_{\sigma,\varphi}^{2}$. Assume that
$q_{\sigma,\varphi}^{2}=0$. Then $|\xi|_{x}^{2}+\sigma^{2}=|\tau
d\varphi|_{x}^{2}$ implying $|\xi|+\sigma\sim\tau|d\varphi|\sim\tilde{\tau}.$
The same computation as in Case 1 gives

$\displaystyle\\{q_{s},q_{a}\\}(x,\xi,\tau)\geq
C\gamma\tilde{\tau}^{3}-C^{\prime}(\tilde{\tau}+|\xi|)^{3}.$

Here $|\xi|+\tilde{\tau}\lesssim\tilde{\tau}$ yielding

$\displaystyle\\{q_{s},q_{a}\\}(x,\xi,\tau)\geq(C\gamma-C^{\prime\prime})\tilde{\tau}^{3}.$

The remaining part of the proof is the same. ∎

######  Lemma 6.4.

Let $j=1$ or $2$. Let $(Q_{\sigma}^{j},\varphi)$ have the sub-ellipticity
property of Definition 6.1 in $\overline{W}$. For $\mu>0$ one sets
$t(\varrho)=\mu((q_{s}^{j})^{2}+q_{a}^{2})(\varrho)+\tau\\{q_{s}^{j},q_{a}\\}(\varrho)$
with
$\varrho=(x,\xi,\tau,\sigma)\in\overline{W}\times\mathbb{R}^{d}\times[0,\infty)\times[0,\infty)$.
Let $\tau_{0}>0$. Then, for $\mu$ chosen sufficiently large and
$\tau\geq\tau_{0}\sigma$ one has $t(\varrho)\geq C\lambda_{\tau}^{4}$ for some
$C>0$.

The proof of Lemma 6.4 uses the following lemma.

######  Lemma 6.5.

Consider two continuous functions, $f$ and $g$, defined in a compact set
$\mathscr{L},$ and assume that $f\geq 0$ and moreover

$f(y)=0\Rightarrow g(y)>0\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{for all}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ y\in\mathscr{L}.$

Setting $h_{\mu}=\mu f+g,$ we have $h_{\mu}\geq C>0$ for $\mu>0$ chosen
sufficiently large.

###### Proof of Lemma 6.4.

Consider the compact set

$\displaystyle\mathscr{L}=\\{(x,\xi,\tau,\sigma);\ x\in\overline{W},\
|\xi|^{2}+\tau^{2}+\sigma^{2}=1,\ \tau\geq\tau_{0}\sigma\\}.$

Applying the result of Lemma 6.5 to $t(\varrho)$ on $\mathscr{L}$ with
$f=(q_{s}^{j})^{2}+q_{a}^{2}$ and $g=\tau\\{q_{s}^{j},q_{a}\\}$ we find for
$t(\varrho)\geq C$ on $\mathscr{L}$ for some $C>0$ for $\mu$ chosen
sufficiently large. Since $t(\varrho)$ is homogeneous of degree $4$ in the
variables $(\xi,\tau,\sigma)$ it follows that $t(\varrho)\geq
C(\sigma^{2}+\tau^{2}+|\xi|^{2})^{4}\gtrsim\lambda_{\tau}^{4}.$ ∎

###  6.2. Lopatinskiĭ-Šapiro condition for the second-order factors

Above, in Section 4, the Lopatinskiĭ-Šapiro condition is addressed for the
fourth-order operator $P_{\sigma,\varphi}$. Here, we consider the two second-
order factors $Q_{\sigma,\varphi}^{j}$.

With the roots $\pi_{j,1}$ and $\pi_{j,2}$ defined in (4.4) one sets

$\displaystyle
q_{\sigma,\varphi}^{j,+}(x,\xi^{\prime},\xi_{d},\tau)=\prod\limits_{{\operatorname{Im}\pi_{j,k}\geq
0}\atop{k=1,2}}\big{(}\xi_{d}-\pi_{j,k}(x,\xi^{\prime},\tau,\sigma)\big{)}.$

######  Definition 6.6.

Let $j=1,2$. Let $x\in\partial\Omega$, with $\Omega$ locally given by
$\\{x_{d}>0\\}$. Let
$(\xi^{\prime},\tau,\sigma)\in\mathbb{R}^{d-1}\times[0,+\infty)\times[0,+\infty)$
with $(\xi^{\prime},\tau,\sigma)\neq 0$. One says that the Lopatinskiĭ-Šapiro
condition holds for $(Q_{\sigma}^{j},B,\varphi)$ at
$\varrho^{\prime}=(x,\xi^{\prime},\tau,\sigma)$ if for any polynomial function
$f(\xi_{d})$ with complex coefficients there exist $c\in\mathbb{C}$ and a
polynomial function $\ell(\xi_{d})$ with complex coefficients such that, for
all $\xi_{d}\in\mathbb{C}$

$f(\xi_{d})=cb_{\varphi}(x,\xi^{\prime},\xi_{d},\tau)+\ell(\xi_{d})q_{\sigma,\varphi}^{j,+}(x,\xi^{\prime},\xi_{d},\tau).$
(6.4)

######  Remark 6.7.

With the Euclidean division of polynomials, we see that it suffices to
consider the polynomial function $f(\xi_{d})$ to be of degree less than that
of $q_{\sigma,\varphi}^{j,+}(x,\xi^{\prime},\xi_{d},\tau)$ in (6.4). Thus, in
any case, the degree of $f(\xi_{d})$ can be chosen less than or equal to one.

######  Lemma 6.8.

Let $j=1$ or $2$. Let $x\in\partial\Omega$ and
$(\xi^{\prime},\tau,\sigma)\in\mathbb{R}^{d-1}\times[0,+\infty)\times[0,+\infty)$
with $(\xi^{\prime},\tau,\sigma)\neq 0$. The Lopatinskiĭ-Šapiro condition
holds for $(Q_{\sigma}^{j},B,\varphi)$ at $(x,\xi^{\prime},\tau,\sigma)$ if
and only if

  1. (1)

either $q_{\sigma,\varphi}^{j,+}(x,\xi^{\prime},\xi_{d},\tau)=1$;

  2. (2)

or $q_{\sigma,\varphi}^{j,+}(x,\xi^{\prime},\xi_{d},\tau)=\xi_{d}-\pi$ and
$b_{\varphi}(x,\xi^{\prime},\pi,\tau)\neq 0.$

###### Proof.

If
$q_{\sigma,\varphi}^{j,+}(x,\xi^{\prime},\xi_{d},\tau)=(\xi_{d}-\pi_{j,1})(\xi_{d}-\pi_{j,2})$,
that is, both roots $\pi_{j,1}$ and $\pi_{j,2}$ are in the upper complex half-
plane, then condition (6.4) cannot hold, since by Remark 6.7 it means that the
vector space of polynomials of degree less than or equal to one would be
generated by the single polynomial $b_{\varphi}(x,\xi^{\prime},\xi_{d},\tau)$.
Suppose that
$q_{\sigma,\varphi}^{j,+}(x,\xi^{\prime},\xi_{d},\tau)=\xi_{d}-\pi$ that is
one the root $\pi_{j,1}$ and $\pi_{j,2}$ has a nonnegative imaginary part and
the other root has a negative imaginary part. Then, the Lopatinskiĭ-Šapiro
condition holds at $(x,\xi^{\prime},\sigma,\tau)$ if for any $f(\xi_{d})$, the
polynomial function $\xi_{d}\mapsto
f(\xi_{d})-cb_{\varphi}(x,\xi^{\prime},\xi_{d},\tau)$ admits $\pi$ as a root
for some $c\in\mathbb{C}$. A necessary and sufficient condition is then