\documentclass[reqno]{amsart} \usepackage{graphicx,amssymb,mathrsfs} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2006(2006), No. 48, pp. 1--16.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2006 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2006/48\hfil $L^p$-resolvent estimates] {$L^p$-resolvent estimates and time decay for generalized thermoelastic plate equations} \author[R. Denk, R. Racke\hfil EJDE-2006/48\hfilneg] {Robert Denk, Reinhard Racke} \address[Robert Denk]{Department of Mathematics and Statistics, University of Konstanz, 78457 Konstanz, Germany} \email{robert.denk@uni-konstanz.de} \address[Reinhard Racke]{ Department of Mathematics and Statistics, University of Konstanz, 78457 Konstanz, Germany} \email{reinhard.racke@uni-konstanz.de} \date{} \thanks{Submitted September 13, 2005. Published April 11, 2006.} \subjclass[2000]{35M20, 35B40, 35Q72, 47D06, 74F05} \keywords{Analytic semigroup in $L^p$; polynomial decay rates; Cauchy problem} \begin{abstract} We consider the Cauchy problem for a coupled system generalizing the thermoelastic plate equations. First we prove resolvent estimates for the stationary operator and conclude the analyticity of the associated semigroup in $L^p$-spaces, $1
0$, and $\alpha, \beta \in [0,
1]$ are parameters of the ``$\alpha$-$\beta$-system" (\ref{eq1.1})
(\ref{eq1.2}). The constants $a, b, d, g$ are positive and assumed
to be equal to one in the sequel w.l.o.g. For $\eta = 2$ and
$\alpha = \beta = 1/2$ we have the thermoelastic plate equations
(\cite{La89}),
\begin{gather}
u_{tt}+ a \Delta^2 u + b \Delta \theta 0, \label{eq1.4}\\
\theta_{t} - g \Delta \theta - b \Delta
u_{t} = 0 \label{eq1.4a}
\end{gather}
which has been widely discussed in
particular for bounded reference configurations $\Omega \ni x$,
see the work of Kim \cite{Ki92}, Mu\~noz Rivera \& Racke
\cite{MR95}, Liu \& Zheng \cite{LZ97}, Avalos \& Lasiecka
\cite{AL97}, Lasiecka \& Triggiani \cite{LTa, LTb, LTc, LTd} for
the question of exponential stability of the associated semigroup
(for various boundary conditions), and Russell \cite{Ru93}, Liu \&
Renardy \cite{LR95}, Liu \& Liu \cite{LL97}, Liu \& Yong
\cite{LY98} for proving its analyticity, see also the book of Liu
\& Zheng \cite{LZ99} for a survey. In our paper \cite{MR96} we
introduced the more general $\alpha$-$\beta$-system (\ref{eq1.1}),
(\ref{eq1.2}), in a general Hilbert space $\mathcal{H}$, $S$
self-adjoint, and also proved for $\beta =1/2$ polynomial decay
rates of $L^\infty$-norms $\|(- \Delta)^{\eta/2} u (t, \cdot),
u_t(t, \cdot), \theta(t, \cdot)\| _{L^\infty (\Omega)}$
of the solutions for $\Omega = \mathbb{R}^n$ or $\Omega$ being an exterior
domain, $\mathcal{H} = L^2(\Omega)$ essentially.
It was demonstrated that the
$\alpha$-$\beta$-system may also describe viscoelastic equations of
memory type with even non convolution type kernels for $(\beta =
1/2, \alpha = 0)$, and that it captures features of second-order
thermoelasticity for $(\beta = 1/2, \alpha = 1/2)$.
In \cite{MR96} the region $D$ of parameters where the system
has a smoothing property,
\begin{equation} D = \{(\beta, \alpha) | 1 - 2 \beta < \alpha < 2
\beta, \alpha> 2\beta - 1\}; \label{eq1.5}
\end{equation}
see Figure 1.
\noindent
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%\vspace*{1cm}
\caption{Area of smoothing}
\end {figure}
The $\alpha$-$\beta$-system was independently introduced by Ammar
Khodja \& Benabdallah \cite{AB1}. In particular they proved the
analyticity of the
associated semigroup for $\alpha = 1$ and, if and only if,
$3/4 \leq \beta \leq 1$.
Also Liu \& Liu \cite{LL97} and Liu \& Yong \cite{LY98} studied general
$\alpha$-$\beta$-systems in the Hilbert space case (``bounded
domains''), in particular in \cite{LY98} they obtained analyticity in the
region
\begin{equation}
\widetilde {\mathfrak{A}}:= \{(\beta, \alpha) | \alpha > \beta, \alpha \leq 2
\beta - 1/2\}.
\label{eq1.6}
\end{equation}
Shibata \cite{Sh04} obtained the analyticity in $L^p$-spaces,
$ 1 < p < \infty$,
for the classical thermoelastic plate; i.e.,
for $(\beta, \alpha) = (1/2, 1/2)$.
All but the last one of the above mentioned papers work in Hilbert spaces,
none
can replace $L^2 (\Omega)$ by $L^p (\Omega)$,
$1 < p < \infty$ if $(\beta,\alpha) \neq (1/2, 1/2))$, and none
gives (polynomial) decay rates --- if
$\beta$ is different from $1/2$. So our goals and new contributions are
\begin{itemize}
\item To discuss the $\alpha$-$\beta$-system in $L^p (\mathbb{R}^n)$-spaces,$1
< p <
\infty$, and to describe the region ${\mathfrak{A}}$ of parameters $(\beta,
\alpha)$
of analyticity of the semigroup, and
\item To obtain sharp polynomial decay rates for $\| S^{1/2} u(t,
\cdot), u_t(t,\cdot),\theta (t, \cdot) \|_{L^q(\Omega)}$ for $2 \leq q \leq \infty$, and
$(\beta, \alpha)$ in the analyticity region ${\mathfrak{A}}$, but also for $
1/4 \leq \beta \leq 3/4$ while $\alpha = 1/2$ (exemplarily).
\end{itemize}
We shall obtain the following region of analyticity
\begin{equation}
{\mathfrak{A}} = \{(\beta, \alpha)| \; \alpha \geq \beta, \; \alpha
\leq 2 \beta - 1/2\}, \label{eq1.7}
\end{equation}
see Figure 2
(cp.(\ref{eq1.6})) in proving resolvent estimates in $L^p$-spaces
using the theory of parameter-elliptic mixed-order systems by Denk,
Mennicken \& Volevich \cite{dmv98}.
\noindent
\begin{figure}[htb]
\centering
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\end{picture}
\vspace*{1cm}
\caption{Area of analyticity}
\end {figure}
The polynomial decay estimates will be obtained in applying the
Fourier transform and analysing the arising characteristic
polynomial
\begin{equation}
P (\xi, \lambda) := \lambda^3 + \rho^\alpha
\lambda^2 + (\rho^{2 \beta} + \rho) \lambda + \rho^{1 + \alpha}
\label{eq1.8}
\end{equation}
carefully, where $\rho := | \xi |^{2\eta}$. In
particular we describe the asymptotic expansion as $\lambda \to 0$
(and $\lambda \to \infty$).
The results here will also be basic for
further investigations of boundary value
problems in exterior domains.
We remark that the transformation to a first-order system via
$(S^{1/2}u,u_t,\theta)$ immediately transfers for the classical case
(\ref{eq1.4}), (\ref{eq1.4a}) to the hinged boundary conditions
$$
u=\Delta u =\theta =0 \quad \mbox{on the boundary}.
$$
For other boundary conditions like the Dirichlet type ones
$$
u=\partial_\nu u =\theta =0,
$$
where $\partial_\nu$ stands for the normal derivative at the boundary,
other transformations like $(u,u_t,\theta)$ will be more appropriate,
cp. \cite{Sh04}. We stress that we here obtain information on the Cauchy
problem independent of any, and useful for any boundary conditions.
Still boundary values problems require a sophisticated
analysis as a future task.
The paper is organized as follows: In Section 2 we review the relevant parts of
the theory of parameter-elliptic mixed-order systems. The application to
the
$\alpha$-$\beta$-system is given in Section 3. In Section 4 we prove
the decay estimates for solutions as time tends to infinity.
\section{Remarks on mixed order systems}
The theory of mixed order systems usually deals with matrices of
partial differential operators. As the generalized thermoelastic
plate equation leads to a matrix with pseudo-differential operators
with constant symbols, we will formulate the definitions and results
for such matrices. It is also possible to consider general
pseudo-differential operators (see, for instance, the book of Grubb
\cite{grubb} in this context). However, for the present case such
general framework is not necessary, and we will deal only with
Fourier multipliers.
In the following, the letter $\mathcal F$ stands for the Fourier
transform in $\mathbb{R}^n$, acting in the Schwartz space of tempered
distributions $\mathcal F\colon \mathscr S'(\mathbb{R}^n)\to \mathscr
S'(\mathbb{R}^n)$. For a symbol $a(\xi)$ (belonging to some symbol class),
the pseudo-differential operator $a(D)$ is defined by $a(D) :=
\mathcal F^{-1} a(\xi) \mathcal F$. If $a(\xi)$ is homogeneous with
respect to $\xi$ of non-negative degree $\mu$, then the
pseudo-differential operator $a(D)$ has order $\mu$. In an obvious
way, for $r>0$ the order of symbols like $|\xi|^r$ and $1+|\xi|^r$
are equal to $r$.
In the following we will consider operator matrices
$A(D) = (A_{ij}(D))_{i,j=1,\dots,n}$ where every entry $A_{ij}(D)$
is a Fourier multiplier of the form
\[
A_{ij}(D) = |D|^{\alpha_{ij}} = \mathcal F^{-1} |\xi|^{\alpha_{ij}}
\mathcal F.
\]
In this case, $\alpha_{ij} = {\rm ord} A_{ij}(D)$. For a permutation
$\pi:\{ 1,\dots,n\}\to\{1,\dots,n\}$ we define
\[
R(\pi) := \alpha_{1 \pi(1)} + \dots + \alpha_{n \pi(n)}.
\]
We set $R := \max_\pi R(\pi)$. Then there exist real numbers
$s_1,\dots,s_n$ and $r_1,\dots,r_n$ such that
\[
\alpha_{ij} \le s_i + t_j,\quad
\sum_{i=1}^n (s_i+t_i) = R.
\]
For differential operators, this was shown by Volevich in
\cite{v63}. The case of non-integer orders follows in exactly the
same way.
\begin{definition}\label{2.1}\rm
The matrix $A(D)$ and the
corresponding symbol $A(\xi)$ are called elliptic in the sense of
Douglis-Nirenberg (or elliptic mixed order system) if
\begin{itemize}
\item[(i)] $A(\xi)$ is non-degenerate, i.e. $R = \deg\det A(\xi)$.
\item[(ii)] $\det A(\xi)\not=0$ for all $\xi\in \mathbb{R}^n\setminus\{0\}$.
\end{itemize}
\end{definition}
For an elliptic matrix $A(\xi)$ the principal part is defined as
\[
A_{ij}^0(\xi) := \begin{cases}
A_{ij}(\xi) & \text{ if } \mathop{\rm ord}
A_{ij} = s_i+t_j,\\
0 & \text{otherwise.}\end{cases}
\]
Note that the numbers $s_i$ and $t_j$ are defined up to translations
of the form
\begin{equation}
(s_1,\dots,s_n,t_1,\dots,t_n)\mapsto (s_1-\kappa, \dots, s_n-\kappa,
t_1+\kappa, \dots, t_n+\kappa). \label{eq2.1}
\end{equation}
On the diagonal we have the orders
\begin{equation}
r_i := s_i + t_i \quad (i=1,\dots,n). \label{eq2.2}
\end{equation}
Now let $A(D)$ be an elliptic mixed order system of the form
indicated above. To solve the Cauchy problem $\frac{d}{dt} U = A(D)
U$, one can consider the parameter-dependent symbol $\lambda -
A(\xi)$ where the complex parameter $\lambda$ belongs to some sector
in the complex plane. In the homogeneous case, this is the standard
approach to parabolic equations, see, e.g., \cite{AV64}. Here the
homogeneity of the determinant $\det(\lambda - A(\xi))$ as a
function of $\lambda$ and $\xi$ is essential for resolvent
estimates.
For mixed order systems of the form $\lambda-A(D)$, however, the
definition of parabolicity and parameter-ellipticity is not obvious.
In \cite{dmv98} a definition of this notion and several equivalent
descriptions can be found. We will recall and slightly generalize
some definitions and main results of \cite{dmv98}.
We start with the definition of the Newton polygon associated to
$A(\xi)$. In the case considered in the present paper, the
determinant $P(\xi,\lambda) := \det(\lambda - A(\xi))$ is a
polynomial in $\lambda\in\mathbb{C}$ and $|\xi|$ for $\xi\in\mathbb{R}^n$ which can
be written in the form
\begin{equation}
P(\xi,\lambda) = \sum_{\gamma,k} a_{\gamma k} |\xi|^\gamma
\lambda^k. \label{eq2.3}
\end{equation}
Here the exponents $\gamma$ of $|\xi|$ are, in general, non-integer.
The Newton polygon $N(P)$ of $P(\xi,\lambda)$ is defined
as the convex hull of all points $(\gamma,k)$ for which the
coefficient $a_{\gamma k}$ in (\ref{eq2.3}) does not vanish, and the
projections of these points onto the coordinate axes. For instance,
consider the symbol $\lambda^3 + |\xi|^3 \lambda^2 +
|\xi|^{9/2}\lambda$. The associated Newton polygon is the convex
hull of the points $(0,3), (3,2), (\frac92,1), (\frac92,0)$ and
$(0,0)$. A Newton polygon is called regular if it has no edges
parallel to one of the axes but not belonging to this axis.
In the following, let $\mathscr L$ be a closed sector in the
complex plane with vertex at the origin. The constant $C$ stands
for an unspecified constant which may vary from line to line but
which is independent of the free variables. The following definition
is a slight modification of the definition in \cite{dmv98}.
\begin{definition}\label{2.2} \rm
a) Let $N(P)$ be the Newton polygon of the symbol \eqref{eq2.3}.
Then this symbol is called parameter-elliptic in $\mathscr L$ if
there exists a $\lambda_0>0$ such that the inequality
\begin{equation}
|P(\xi,\lambda) | \ge C W_P(\xi,\lambda) \quad (\lambda\in\mathscr
L,\; |\lambda|\ge\lambda_0,\;\xi\in\mathbb{R}^n) \label{eq2.4}
\end{equation}
holds where $W_P$ denotes the weight function associated to $P$:
\begin{equation}
W_P(\xi,\lambda) := \sum_{\gamma, k} |\xi|^\gamma |\lambda|^k. \label{eq2.5}
\end{equation}
The last sum runs over all indices $(\gamma,k)$ which are vertices
of the Newton polygon $N(P)$.
b) The mixed order system $\lambda-A(D)$ is called
parameter-elliptic in $\mathscr L$ if the Newton polygon of
$P(\xi,\lambda) := \det(\lambda-A(D))$ is regular and if $P$ is
parameter-elliptic in $\mathscr L$.
\end{definition}
There are several equivalent descriptions of parameter-ellipticity
for mixed order systems (see \cite{dmv98}). The Newton polygon
approach is a geometric description of the various homogeneities
contained in the determinant $P(\xi,\lambda) =
\det(\lambda-A(\xi))$. For $r>0$ and a polynomial of the form
(\ref{eq2.3}) define the $r$-order of $P$ by
\[
d_r(P) := \max\{ \gamma+rk: a_{\gamma k}\not=0\}.
\]
The $r$-principal part $P_r(\xi,\lambda)$ is given by
\[
P_r(\xi,\lambda) := \sum_{\gamma+rk= d_r(P)} a_{\gamma k} |\xi|^\gamma
\lambda^k.
\]
The following result is a straightforward
generalization of Theorem 2.2 in \cite{dmv98} where polynomial
entries were considered.
\begin{theorem}\label{2.3}
Let $A(D)$ be a mixed order system and
$P(\xi,\lambda) = \det(\lambda-A(\xi))$.
Then the following statements are equivalent.
\begin{itemize}
\item[(a)] The operator matrix $\lambda - A(D)$ is parameter-elliptic
in $\mathscr L$.
\item[(b)] There exist constants $C>0$, $\lambda_0>0$ such that
\begin{equation}
\big| P(\xi,\lambda) \big| \ge C \prod_{i=1}^n (|\xi|^{r_i} +
|\lambda|) \quad (\lambda\in {\mathscr L}, |\lambda|\ge \lambda_0,
\xi\in\mathbb{R}^n). \label{eq2.6}
\end{equation}
Here the numbers $r_i$ are defined in (\ref{eq2.2}).
\item[(c)] For every $r>0$,
\begin{equation}
P_r(\xi,\lambda)\not=0 \quad (\lambda\in {\mathscr
L}\setminus\{0\},\; \xi\in\mathbb{R}^n\setminus\{0\}). \label{eq2.7}
\end{equation}
\end{itemize}
\end{theorem}
The condition of parameter-ellipticity is equivalent to a uniform
estimate of the entries of the inverse matrix, see \cite{dmv98},
Proposition 3.10. Applying Plancherel's theorem, we immediately
obtain $L_2$-estimates for the solution. When we deal with
$L_p$-spaces, we want to apply Michlin's theorem. For this we need
another estimate which is contained in the following theorem.
\begin{theorem}\label{2.4}
Let $P(\xi,\lambda)$ be parameter-elliptic in the sector $\mathscr
L$ and assume, for simplicity, that $N(P)$ is regular. Let
$(\sigma,\kappa)\in\mathbb{R}^2$ be a point belonging to the Newton polygon
$N(P)$. Then there exists a $\lambda_0>0$ and for every
$\alpha\in\mathbb{N}_0^n$ a constant $C_\alpha= C_\alpha$ such that
\begin{equation}
\Big|\partial_\xi^\alpha \Big( \frac{|\xi|^\gamma
|\lambda|^\kappa}{P(\xi,\lambda)}\Big)\Big| \le
C_\alpha|\xi|^{-\alpha}\quad (\xi\in \mathbb{R}^n\setminus\{0\},\;
\lambda\in \mathscr L, \; |\lambda|\ge \lambda_0). \label{eq2.8}
\end{equation}
If $P(\xi,\lambda)\not=0$ for all $\xi\in\mathbb{R}^n$ and
$\lambda\in\mathscr L$ with $|\lambda|\ge\epsilon$ for some
$\epsilon>0$, then inequality \eqref{eq2.8} holds for all $\xi\in
\mathbb{R}^n\setminus\{0\}$ and all $\lambda\in\mathscr L$ with
$|\lambda|\ge \epsilon$.
\end{theorem}
\begin{proof}
For a point $(\sigma,\kappa)\in N(P)$ we have, by convexity and
Jensen's inequality,
\[
|\xi|^\sigma |\lambda|^\kappa \le W_P(\xi,\lambda).
\]
Let $\alpha=0$. By definition of parameter-ellipticity, there exists
a $\lambda_0>0$ such that
\[
\big| \;|\xi|^\gamma \lambda^\kappa \big|\le C
|P(\xi,\lambda)|\quad ( \lambda\in\mathscr L,\;|\lambda|\ge
\lambda_0,\; \xi\in\mathbb{R}^n).
\]
Thus, the case $\alpha=0$ follows directly
from the definition of parameter-ellipticity.
Now let $|\alpha|=1$ and assume, without loss of generality, that
$\partial_\xi^\alpha = \partial_{\xi_1}$. We have
\begin{equation}
\big|\partial_{\xi_1}\big( |\xi|^\gamma \lambda^\kappa\big)\big|
= \Big|\lambda^\kappa \Big( \sum_{i=1}^n \xi_i^2\Big)^{\frac\gamma
2-1}\cdot\frac\gamma 2\cdot 2\xi_1\Big|
= \big|\gamma \cdot \lambda^\kappa \xi_1|\xi|^{\gamma-2}\big|
\le C |\lambda|^k |\xi|^{\gamma-1}.
\label{eq2.9}
\end{equation}
In the same way we can estimate
\begin{equation}
\Big|\partial_{\xi_1} P(\xi,\lambda)\Big|\le W_P(\xi,\lambda) \cdot
|\xi|^{-1}. \label{eq2.10}
\end{equation}
We write
\[ \Big|\partial_{\xi_1} \Big( \frac{|\xi|^\gamma
\lambda^\kappa}{P(\xi,\lambda)}\Big)\Big| =
\Big|\frac{P(\xi,\lambda)\partial_{\xi_1}(|\xi|^\gamma
\lambda^\kappa)-|\xi|^\gamma
\lambda^\kappa\partial_{\xi_1}P(\xi,\lambda)}
{P(\xi,\lambda)^2}\Big|\] and obtain the first statement of the
theorem by (\ref{eq2.9}), (\ref{eq2.10}) and the definition of
parameter-ellipticity (\ref{eq2.4}).
The case of higher derivatives (general $\alpha$) follows by
iteration.
Now assume that $P(\xi,\lambda)\not=0$ for all $\xi\in\mathbb{R}^n$ and
$\lambda\in\mathscr L,\; |\lambda|\ge \epsilon$. By the regularity
of $N(P)$, we can write
\[
P(\xi,\lambda) = a_{\gamma_0,0} |\xi|^{\gamma_0} +
\sum_{\gamma,k,\; k>0} a_{\gamma k} |\xi|^\gamma \lambda^k.
\]
In the last sum, only exponents of $|\xi|$
appear with $\gamma<\gamma_0$. We obtain
\[ \lim_{|\xi|\to\infty} \frac{P(\xi,\lambda)}{|\xi|^{\gamma_0}} =
a_{\gamma_0,0}\not=0\quad (\lambda\in \mathscr L, \; \epsilon \le
|\lambda|\le \lambda_0).\]
In the same way,
\[ W_P(\xi,\lambda) = |\xi|^{\gamma_0} +
\sum_{\gamma,k,\; k>0} |\xi|^\gamma \lambda^k,
\]
and
\[
\lim_{|\xi|\to\infty} \frac{W_P(\xi,\lambda)}{|\xi|^{\gamma_0}} =
1\quad (\lambda\in \mathscr L, \; \epsilon \le |\lambda|\le
\lambda_0).
\]
Now we use $P(\xi,\lambda)\not=0$ and a compactness
argument to see that
\begin{align*}
|P(\xi,\lambda)| & \ge C |\xi|^{\gamma_0} && (\xi\in\mathbb{R}^n,
\lambda\in\mathscr L,\,\epsilon\le |\lambda|\le \lambda_0),\\
|P(\xi,\lambda)| & \ge C W_P(\xi,\lambda) && (\xi\in\mathbb{R}^n,
\lambda\in\mathscr L,\,\epsilon\le |\lambda|\le \lambda_0).
\end{align*}
>From these inequalities we obtain (\ref{eq2.8}) for all
$\lambda\in\mathscr L$ with $|\lambda|\ge \epsilon$ in the same way
as in the first part of the proof.
\end{proof}
\begin{remark}\label{2.5} \rm
As we can see from the proof of the
preceding theorem, we can also estimate
\[
\Big|\partial_\xi^\alpha \Big( \frac{\xi^\beta
\lambda^\kappa}{P(\xi,\lambda)}\Big)\Big| \le
C_\alpha|\xi|^{-\alpha}\quad (\xi\in \mathbb{R}^n\setminus\{0\},\;
\lambda\in \mathscr L,\,|\lambda|\ge\lambda_0) \] where now $\beta$
is a multi-index such that $(|\beta|,\kappa)$ belongs to the Newton
polygon.
\end{remark}
\section{Resolvent estimates for the generalized thermoelastic plate
equation}
To apply the results mentioned above to the generalized linear
thermoelastic plate equation (\ref{eq1.1}), (\ref{eq1.2}) we rewrite
this equation as a first-order system, setting $U := (S^{1/2}u, \;
u_t,\; \theta)^t$. We get
\[
U_t = A(D) U := \begin{pmatrix} 0 & S^{1/2} & 0 \\ -S^{1/2} & 0 &
S^\beta\\ 0 & -S^\beta & -S^\alpha\end{pmatrix} \; U.
\]
The symbol of this system is given by
\[
A(\xi) = \begin{pmatrix} 0 & \rho^{1/2} & 0\\ -\rho^{1/2} & 0 &
\rho^{\beta} \\ 0 & -\rho^{\beta} & - \rho^{\alpha}
\end{pmatrix}
\]
with $\rho := |\xi|^{2\eta}$. Thus we have ${\rm ord} A_{ij}(\xi)
\le s_i+t_j$ with
\begin{equation}
s := 2\eta\cdot \begin{pmatrix} \frac12\\ 2\beta-\alpha \\
\beta\end{pmatrix},\quad t := 2\eta\cdot\begin{pmatrix}
\frac12+\alpha-2\beta\\ 0\\
\alpha-\beta\end{pmatrix}. \label{eq3.1}
\end{equation}
Consequently, the weight vector is given by
\begin{equation}
\begin{pmatrix} r_1\\ r_2\\ r_3\end{pmatrix}
= 2\eta\begin{pmatrix} 1+\alpha-2\beta\\ 2\beta-\alpha\\
\alpha\end{pmatrix}. \label{eq3.2}
\end{equation}
With the order vectors $s$ and $t$ defined as above, the matrix
$A(\xi)$ coincides with its principal part. The determinant of this
system equals
\begin{equation}
P(\xi,\lambda) := \det(\lambda-A(\xi)) = \lambda^3
+ \lambda^2 \rho^\alpha + \lambda(\rho^{2\beta} + \rho) +
\rho^{1+\alpha}. \label{eq3.3}
\end{equation}
>From (\ref{eq3.3}) we can see that the Newton polygon $N(P)$ is the
convex hull of the points
\[
(0,3),\; (2\eta\alpha,2),\; (4\eta\beta,1),\; (2\eta+2\eta\alpha,0),\;
(0,0)
\]
(see Figure 3).
\begin{figure}[htb]
\begin{center}
\setlength{\unitlength}{0.7mm}
\begin{picture}(120,90)(-5,-2)
\put(0,0){\vector(1,0){115}}
\put(0,0){\vector(0,1){70}}
\put(-5,19){1}
\put(-5,39){2}
\put(-5,59){3}
\put(-5,-1){0}
\put(-1,-7){0}
\put(-2,20){\line(1,0){4}}
\put(-2,40){\line(1,0){4}}
\put(-2,60){\line(1,0){4}}
\put(-0.8,59.2){{$\scriptstyle \bullet$}}
\put(36.7,39.2){{$\scriptstyle \bullet$}}
\put(-0.8,-0.8){{$\scriptstyle \bullet$}}
\put(69.2,19.2){{$\scriptstyle \bullet$}}
\put(86.7,-0.8){{$\scriptstyle \bullet$}}
\put(70,-2){\line(0,1){4}}
\put(37.5,-2){\line(0,1){4}}
\put(50,-2){\line(0,1){4}}
\put(87.5,-2){\line(0,1){4}}
\put(36,-7){$2\eta\alpha$}
\put(68,-7){$4\eta\beta$}
\put(49,-7){$2\eta$}
\put(82,-7){$2\eta+2\eta\alpha$}
\put(0,60){\rotatebox{-28}{\line(1,0){43}}}
\put(37.5,40){\rotatebox{-31.5}{\line(1,0){38}}}
\put(70,20){\rotatebox{-49}{\line(1,0){27}}}
\put(-23,65){{\small power of $\lambda$}}
\put(110,-7){{\small power of $\xi$}} \put(25,20){$N(P)$}
\end{picture}
\end{center}
\caption{The Newton polygon of the mixed order system $\lambda-A(\xi)$}
\end{figure}
\begin{lemma}\label{3.1} Assume that $(\beta,\alpha)\in\mathfrak A$,
i.e. that
\begin{equation}
\alpha\ge\beta\quad\text{and}\quad 2\beta-\alpha\ge\frac12.
\label{eq3.4}
\end{equation}
Then the matrix $\lambda-A(D)$ is parameter-elliptic in $\mathbb{C}_+ :=
\{\lambda \in \mathbb{C}: \mathop{\rm Re}\lambda \ge 0\}$.
\end{lemma}
\begin{proof} We will check the conditions of Theorem \ref{2.3} (c).
Let us first assume $\alpha>\beta$ and $2\beta-\alpha>\frac12$. Then
we have $r_1