\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 166, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/166\hfil Analytic semigroups]
{Analytic semigroups generated by an operator matrix in 
$L^2(\Omega)\times L^2(\Omega)$}

\author[S. Badraoui \hfil EJDE-2012/166\hfilneg]
{Salah Badraoui}

\address{Salah Badraoui \newline
Laboratoire LAIG, Universit\'e du 08 Mai 1945-Guelma \\
BP 401, Guelma 24000, Algeria}
\email{sabadraoui@hotmail.com}

\thanks{Submitted June 15, 2012. Published September 28, 2012.}
\subjclass[2000]{35B40, 35B45, 35K55, 35K65}
\keywords{Analytic semigroup; infinitesimal generator;
 operator matrix; \hfill\break\indent dissipative operator; dual space;
 adjoint operator; strongly elliptic operator}

\begin{abstract}
 This article concerns the generation of analytic semigroups by an
 operator matrix in the space  $L^2(\Omega)\times L^2(\Omega)$.
 We assume that one of the diagonal elements is strongly elliptic and
 the other is weakly elliptic, while the sum of the non-diagonal
 elements is weakly elliptic.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

 The theory of semigroups of linear operators has applications in
many branches of analysis as evolution equations: parabolic and hyperbolic
equations and systems with various boundary conditions, harmonic analysis and
ergodic theory. In the theory of evolution equations, it is usually shown that
a given differential operator  $A$  is the infinitesimal generator of a
strongly continuous semigroup in a certain concrete Banach space of
functions $X$. This provides us with the existence and uniqueness of a
solution of the initial value problem
\begin{gather*}
\frac{\partial u(x,t)}{\partial t}+Au(x,t)=0\\
u(x,0) =u_0(x)
\end{gather*}
in the sense of the Banach space  $X$.

 This article  concerns the generation of analytic semigroups by an
operator matrix in the space  $L^2(\Omega)\times L^2(\Omega)$, where
$\Omega$  is a bounded open set in  $\mathbb{R}^N$, with smooth boundary
$\partial\Omega$. Passo and Mottoni \cite{p1} proved that the operator matrix
\begin{equation}
\mathcal{M}=\begin{pmatrix}
a_{11}\Delta & a_{12}\Delta\\
a_{21}\Delta & a_{22}\Delta
\end{pmatrix}  \label{e1.1}
\end{equation}
 with domain  $D(\mathcal{M})=(  H^2(\Omega)\cap H_0^1(\Omega))  ^2$
generates an analytic semigroup on  $L^2(\Omega)\times L^2(\Omega)$  provided  $a_{11}, a_{22}\geq0$,
$a_{11}+a_{22}>0$, $a_{11}a_{22}>a_{12}a_{21}$.

Also, de Oliveira  \cite{o1} proved that the operator matrix
\begin{equation}
\mathcal{M}=\begin{pmatrix}
a_{11}\Delta & \dots & a_{1n}\Delta\\
\vdots & \vdots & \vdots\\
a_{n1}\Delta & \dots & a_{nn}\Delta
\end{pmatrix}  \label{e1.2}
\end{equation}
with domain  $D(\mathcal{M})=( H^2(\Omega)\cap H_0^1(\Omega)) ^n$
generates an analytic semigroup on  $(L^2(\Omega)) ^n$ provided that all eigenvalues of the matrix
\[
\begin{pmatrix}
a_{11} & \dots & a_{1n}\\
\vdots & \vdots & \vdots\\
a_{n1} & \dots & a_{nn}
\end{pmatrix}
\]
have positive real part.

 In this paper we consider the linear operator
\begin{equation}
A(x,D)=\begin{pmatrix}
A_{11}(x,D) & A_{12}(x,D)\\
A_{21}(x,D) & A_{22}(x,D)
\end{pmatrix} \label{e1.3}
\end{equation}
where every element $A_{hl}$  is a symmetric second order differential
operator given by
\begin{equation}
A_{hl}(x,D)u=-{\sum_{j,k=1}^N}\frac{\partial}{\partial x_{j}
}\Big(  a_{jk}(x)\frac{\partial u}{\partial x_{k}}\Big)   \label{e1.4}
\end{equation}
and  one of the diagonal operators $A_{11}$  or  $A_{22}$  is
strongly elliptic and the other diagonal operator is weakly elliptic and the
sum of the non-diagonal operators  $A_{12}(x,D)+A_{21}(x,D)$  is also weakly
elliptic. Under these assumptions we show that this operator matrix generates
an analytic semigroup on  $L^2(\Omega)\times L^2(\Omega)$.

\section{Preliminaries}

 Let us consider the differential operator
\begin{equation}
A(x,D)=\begin{pmatrix}
A_{11}(x,D) & A_{12}(x,D)\\
A_{21}(x,D) & A_{22}(x,D)
\end{pmatrix}  \label{e2.1}
\end{equation}
where
\begin{equation}
A_{hl}(x,D)u=-{\sum_{j,k=1}^N}\frac{\partial}{\partial x_{j}}\Big(
a_{jk}^{hl}(x)\frac{\partial u}{\partial x_{k}}\Big)\quad x\in
\overline{\Omega},\; h,l=1,2   \label{e2.2}
\end{equation}
under the following assumptions:
\begin{itemize}
\item[(H1)] The operators $A_{hl}$ $(h, l=1, 2)$  are
symmetric; i.e.,
\begin{equation}
a_{kj}^{hl}(x)=a_{jk}^{hl}(x),\quad x\in\overline{\Omega},
\text{ for all } j,\; k=1,\dots N  \label{e2.3}
\end{equation}


\item[(H2)] The operators $A_{hl}$ $(h=1,2)$  are regular;
i.e.,
\begin{equation}
a_{jk}^{hl}(x)\in C^1(  \overline{\Omega};\mathbb{R})  ,\quad
h,l=1,2\text{  and }j,k=1,\dots N  \label{e2.4}
\end{equation}


\item[(H3)] One of the diagonal operator  $A_{11}$  or  $A_{22}$
 is strongly elliptic; i.e., there is a  constant  $\mu>0$  such that for
all  $\xi=(  \xi_{j})  _{j=1}^N\in\mathbb{R}^N$  and all
 $x\in\Omega$,
\begin{equation}
{\sum_{j,k=1}^N}a_{jk}^{mm}(x)\xi_{j}\xi_{k}\geq\mu{\sum
_{j=1}^N}\xi_{j}^2=\mu| \xi| ^2,\text{
}m=1\text{  or  }m=2 \label{e2.5}
\end{equation}


\item[(H4)]  The other diagonal operator  $A_{ll}$  ($l=2$  if
 $m=1$  and $l=1$  if  $m=2$) is  weakly elliptic; i.e., for all
$\xi=(  \xi_{j})  _{j=1}^N\in\mathbb{R}^N$  and all
 $x\in\Omega$
\begin{equation}
{\sum_{j,k=1}^N}a_{jk}^{ll}(x)\xi_{j}\xi_{k}\geq0,  \label{e2.6}
\end{equation}


\item[(H5)]  The sum non-diagonal operators  $A_{12}+A_{21}$  is
weakly elliptic; i.e., for all  $\xi=(  \xi_{j})  _{j=1}^N
\in\mathbb{R}^N$  and all  $x\in\Omega$
\begin{equation}
{\sum_{j,k=1}^N}(  a_{jk}^{12}+a_{jk}^{21})  (x)\xi_{j}
\xi_{k}\geq0.  \label{e2.7}
\end{equation}

\end{itemize}
 We give now some definitions which will be used in the sequel.
 We define the operator $A$  with domain
\begin{equation}
D(  A)  =(  H^2(\Omega)\cap H_0^1(\Omega))  ^2,
\label{e2.8}
\end{equation}
as
\begin{equation}
Au\equiv \begin{pmatrix}
A_{11} & A_{12}\\
A_{21} & A_{22}
\end{pmatrix}
 u=A(x,D)u\equiv \begin{pmatrix}
A_{11}(x,D)u_{1}+A_{12}(x,D)u_{2}\\
A_{21}(x,D)u_{1}+A_{22}(x,D)u_{2}
\end{pmatrix}
  \label{e2.9}
\end{equation}
where  $u=\operatorname{col}(u_{1},u_{2})$.
The following results are well known; see, for instance
\cite[page 213]{p2}.

\begin{theorem} \label{thm2.1}
The operator $A_{hl}$  with domain
\begin{equation}
D(  A_{hl})  =H^2(\Omega)\cap H_0^1(\Omega) \label{e2.10}
\end{equation}
and defined by
\begin{equation}
A_{hl}u=A_{hl}(x,D)u \label{e2.11}
\end{equation}
is closed.
\end{theorem}


\begin{theorem} \label{thm2.2}
Let $1\leq p<\infty$,  $L_{n}^p(\Omega)={\prod_{j=1}^n}L^p(\Omega)$,
and  $(L_{n}^p(\Omega))'$  the dual space of
$L_{n}^p(\Omega)$. Then, to every
$\varphi\in(  L_{n}^p(\Omega)) '$  there corresponds unique
 $v=(v_{1} ,\dots,v_{n})\in L_{n}^q(\Omega)$  such that for every
$u=(u_{1},\dots,u_{n})\in L_{n}^p(\Omega)$:
\begin{equation}
\varphi(u)={\sum_{j=1}^n}\langle u_{j},vj\rangle
\label{e2.12}
\end{equation}
Moreover,
$\| \varphi;(  L_{n}^p(  \Omega)  ) '\| =\| v;L_{n}^q(  \Omega)  \|$,
where $q$  is the conjugate exponent of  $p$  and
$\langle u_{k},v_{k}\rangle =\int_{\Omega} u_{k}(x)v_{k}(x)dx$.
Therefore,  $(  L_{n}^p(  \Omega)  ) '\sim L_{n}^q(\Omega)$.
\end{theorem}

For a proof of the above theorem, see \cite[page 47]{a1}.

\begin{definition} \label{def2.1} \rm
Let  $X$ be a Banach space and let
 $X^{\ast}$ be its dual. For every  $x\in X$, the duality set is defined by
\begin{equation}
J(x)=\{  x^{\ast}\in X^{\ast}: \langle x^{\ast},x\rangle =\| x\| ^2=\| x^{\ast
}\| ^2 \}  \label{e2.13}
\end{equation}
\end{definition}

\section{Main results}

\begin{theorem} \label{thm3.1}
 Assume that \eqref{e2.1}-\eqref{e2.11} hold. Then, the
operator $A$  generates a strongly continuous semigroup of contractions
on the space $X=L^2(\Omega)\times L^2(\Omega)$  endowed with the norm
$\| u\| =(  \| u_{1}\| _{2}
^2+\| u_{2}\| _{2}^2)  ^{1/2}$, where
$u=(u_{1},u_{2})$  and  $\| u_{1}\| _{2}^2=\int_{\Omega}| u_{1}(x)| ^2dx$.
\end{theorem}

To prove this theorem we will need some lemmas.

\begin{lemma} \label{lem3.1}
For every  $\lambda>0$  and  $u\in D(A)$ we have
\begin{equation}
\lambda\| u\| \leq\| (  \lambda I+A)u\| \label{e3.1}
\end{equation}
\end{lemma}

\begin{proof} We denote the pairing between
$L_{2}^2(\Omega)$  and  itself by  $\langle ,\rangle $.
If $u=\operatorname{col}(u_{1},u_{2})\in D(A)\backslash\{  0\}  $
 then the function $u^{\ast}=\operatorname{col}(u_{1}^{\ast},u_{2}^{\ast})$
is in the duality map $J(u)$  (see Definition \ref{def2.1} and Theorem \ref{thm2.1}),
where $u_{h}^{\ast}=\overline{u_{h}}$  for
$h=1,2$.
We have
\begin{equation}
\langle Au,u^{\ast}\rangle =\langle A_{11}u_{l},u_{1}^{\ast
}\rangle +\langle A_{22}u_{2},u_{2}^{\ast}\rangle
+\langle A_{12}u_{2},u_{1}^{\ast}\rangle +\langle A_{21}
u_{1},u_{2}^{\ast}\rangle \label{e3.2}
\end{equation}
Integration by parts yields
\begin{align*}
\langle A_{hh}u_{h},u_{h}^{\ast}\rangle
&  =-{\int_{\Omega}}{\sum_{j,k=1}^N}\frac{\partial}{\partial x_{j}
}(  a_{jk}^{hh}(x)\frac{\partial u_{h}}{\partial x_{k}})
\overline{u_{h}}dx\\
& ={\int_{\Omega}}{\sum_{j,k=1}^N}a_{jk}^{hh}
(x)\frac{\partial u_{h}}{\partial x_{k}}\frac{\partial\overline{u_{h}}
}{\partial x_{j}}dx\,.
\end{align*}
Denoting
\[
\frac{\partial u_{h}}{\partial x_{j}}=\alpha_{hj}+i\beta_{hj},\quad
h=1,2,\; j=1,\dots,N
\]
where  $\alpha_{hj}, \beta_{hj}\in\mathbb{R}$, we find that
\begin{equation}
\langle A_{hh}u_{h},u_{h}^{\ast}\rangle =
\int_{\Omega} \sum_{j,k=1}^N
a_{jk}^{hh}(x)(\alpha_{hk}\alpha_{hj}+\beta_{hk}\beta_{hj})dx,\quad
h=1,2\,. \label{e3.3}
\end{equation}
Also, integrating by parts we have
\begin{equation}
\begin{aligned}
\langle A_{12}u_{2},u_{1}^{\ast}\rangle
&  =  \int_{\Omega}\sum_{j,k=1}^N
a_{jk}^{12}(x)(  \alpha_{1j}\alpha_{2k}+\beta_{1j}\beta_{2k})
dx \\
& \quad +i\Big( \int_{\Omega}\sum_{j,k=1}^N
a_{jk}^{12}(x)(  \alpha_{1j}\beta_{2k}-\alpha_{2k}\beta_{1j})
dx\Big)
\end{aligned} \label{e3.4}
\end{equation}
and
\begin{equation}
\begin{aligned}
\langle A_{21}u_{1},u_{2}^{\ast}\rangle
 &  =\int_{\Omega} \sum_{j,k=1}^N
a_{jk}^{21}(x)(  \alpha_{1k}\alpha_{2j}+\beta_{1k}\beta_{2j})
dx \\
& \quad +i\Big( \int_{\Omega} \sum_{j,k=1}^N
a_{jk}^{21}(x)(  \alpha_{2j}\beta_{1k}-\alpha_{1k}\beta_{2j})
dx\Big)
\end{aligned} \label{e3.5}
\end{equation}
Then substituting \eqref{e3.3}--\eqref{e3.5} into \eqref{e3.2} yields
\begin{equation}
\begin{aligned}
\langle Au,u^{\ast}\rangle
&  = \sum_{h=1}^2 \int _{\Omega}  \sum_{j,k=1}^N
a_{jk}^{hh}(x)(  \alpha_{hk}\alpha_{hj}+\beta_{hk}\beta_{hj})
dx \\
&\quad  +\int_{\Omega} \sum_{j,k=1}^N
(a_{jk}^{12}+a_{jk}^{21})(x)(  \alpha_{1j}\alpha_{2k}+\beta_{1j}
\beta_{2k})  dx \\
&\quad  +i\Big\{\int_{\Omega} \sum_{j,k=1}^N
(  a_{jk}^{12}-a_{jk}^{21})  (x)(  \alpha_{1j}\beta
_{2k}-\alpha_{2j}\beta_{1k})  dx\Big\}
\end{aligned} \label{e3.6}
\end{equation}
 Set
\begin{equation}
| \alpha_{h}| ^2= \sum_{j=1}^N \int_{\Omega} \alpha_{hj}^2dx, \quad
\ | \beta_{h}| ^2=\sum_{j=1}^N\int_{\Omega}
\beta_{hj}^2dx,\quad h=1,\text{ }2 \label{e3.7}
\end{equation}
 Then from \eqref{e3.6}--\eqref{e3.7} and using  (H3)-(H5), we have that the real
part of $\langle Au,u^{\ast}\rangle $ satisfies
\begin{equation}
\operatorname{Re}\langle Au,u^{\ast}\rangle \geq2\mu(
 \sum_{h=1}^2 | \alpha_{h}| ^2+
 \sum_{h=1}^2
| \beta_{h}| ^2)  \geq0 \label{e3.8}
\end{equation}
 From \eqref{e3.8}, the linear operator  $-A$  is dissipative. It follows
that for every  $\lambda>0$  and  $u\in D(A)$  we have
$\lambda \| u\| \leq\| (  \lambda I+A)u\| $ (see \cite[page 14]{p2}.
\end{proof}

\begin{lemma} \label{lem3.2}
The operator $A$  is closed.
\end{lemma}

\begin{proof} The adjoint operator of $A$  is
\begin{equation}
A^{\ast}=\begin{pmatrix}
A_{11}^{\ast} & A_{21}^{\ast}\\
A_{12}^{\ast} & A_{22}^{\ast}
\end{pmatrix}  \label{e3.9}
\end{equation}
where $A_{hl}^{\ast}$  is the adjoint operator of $A_{hl}$, for
$h,l=1, 2$.
As the domain $D(A^{\ast})=D(A)$  is dense in
$L_{2}^2 (\Omega)$, then the operator  $(  A^{\ast})  ^{\ast}$  is closed
(see \cite[page 28]{b1}). Also, as  $L^2(\Omega)$  is reflexive, then
$L^2(\Omega)\times L^2(\Omega)$  is reflexive
(see \cite[page 8]{a1});
whence $(  A^{\ast})  ^{\ast}=A$  \cite[page 46]{b1}. We
finally conclude that  $A$  is closed.
\end{proof}

\begin{lemma} \label{lem3.3}
for every  $\lambda>0$, the operator $\lambda I+A$ is bijective.
\end{lemma}

\begin{proof} From   \eqref{e3.1} it follows that $\lambda I+A$
is injective.
As in lemma \ref{lem3.1}, we can prove that for every every
$\lambda>0$  and  $u\in D(A)$,
\begin{equation}
\lambda\| u\| \leq\| (  \lambda I+A)
^{\ast}u\| \label{e3.10}
\end{equation}
then the operator  $(  (  \lambda I+A)  ^{\ast})  ^{\ast}=\lambda I+A$
is surjective (see \cite[page 30]{b1}).
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm3.1}]
The domain  $D(A)$  of  $A$  contains
 $C_0^{\infty}(\Omega)\times C_0^{\infty}(\Omega)$  and
it is therefore dense in  $X\equiv L^2(\Omega)\times L^2(\Omega)$. Also,
$A$  is closed and as a consequence of Lemmas \ref{lem3.1} and \ref{lem3.3}
 we have
\begin{equation}
\| (  \lambda I+A)  ^{-1}\| \leq\frac{1}{\lambda
},\quad \text{for all  }\lambda>0 \label{e3.11}
\end{equation}
The Hille-Yosida theorem \cite[page 8]{p2} now implies that
$-A$  is the infinitesimal generator of a  strongly continuous semigroup
of contractions on  $L^2(\Omega)\times L^2(\Omega)$.
\end{proof}

\begin{theorem} \label{thm3.2}
The semigroup  generated in theorem \ref{thm3.1} is also analytic.
\end{theorem}

\begin{proof} Let  $X$  be a Banach space and let
$X^{\ast}$  be its dual. If  $A:X\to X$ is a linear operator
in $X$, the numerical range of $A$ is the set
\begin{equation}
\mathcal{N}(A)=\{  \langle x^{\ast},Ax\rangle : x\in
D(A),\; x^{\ast}\in X^{\ast},\; \langle x^{\ast},x\rangle
=\| x\| =\| x^{\ast}\| =1\}
\label{e3.12}
\end{equation}
If we put
\begin{equation}
| a_{jk}^{hl}(x)| \leq M, \quad \text{for all }h, l=1,2\text{ and  }
j,k=1,\dots,N \label{e3.13}
\end{equation}
 we get from  \eqref{e3.6} that the imaginary part of
$\langle Au,u^{\ast}\rangle $
\begin{equation}
| \operatorname{Im}\langle Au,u^{\ast}\rangle |
\leq M\Big(\sum_{h=1}^2
| \alpha_{h}| ^2+  \sum_{h=1}^2
| \beta_{h}| ^2\Big)  \label{e3.14}
\end{equation}
and hence from \eqref{e3.8} and \eqref{e3.14}, we find that
\begin{equation}
\frac{| \operatorname{Im}\langle Au,u^{\ast}\rangle
| }{| \operatorname{Re}\langle Au,u^{\ast
}\rangle | }\leq\frac{M}{2\mu} \label{e3.15}
\end{equation}
 We observe by \eqref{e3.8} and  \eqref{e3.15} that the numerical range
$\mathcal{N}(-A)$  of  $-A$  is contained in the set
$N_{\varphi}=\{  \lambda:| \arg\lambda| >\pi-\varphi\}$
where $\varphi=\arctan(NM/(2\mu))  $,
$0<\varphi <\pi/2$. Choosing
$\varphi<\theta<\pi/2$  and denoting
\begin{equation}
\mathcal{S}_{\theta}=\{  \lambda: | \arg\lambda|
<\pi-\theta\}  \label{e3.16}
\end{equation}
 It follows that there is a constant
$C_{\theta}=\sin (\theta-\varphi)>0$  for which the distance of
 $\lambda$  from  $\mathcal{N}(-A)$
\[
d(\lambda,\overline{\mathcal{N}(-A)})\geq C_{\theta}| \lambda
|, \quad \text{for  }\lambda\in\mathcal{S}_{\theta}
\]
Since  $\lambda>0$  is in the resolvent set  $\rho(-A)$  of
the operator  $-A$  by Theorem \ref{thm3.1}, it follows from
\cite[Theorem 1.3.9]{p2} that  $S_{\theta}\subset\rho(-A)$  and that
\begin{equation}
\| (  \lambda I+A)  ^{-1}\|
\leq\frac{1} {C_{\theta}| \lambda| },\quad \text{for all }\lambda
\in\mathcal{S}_{\theta} \label{e3.17}
\end{equation}
 Whence by \cite[Theorem 2.5.2]{p2}, the operator $-A$  is
the infinitesimal generator of an analytic semigroup on the space
$X=L^2(\Omega)\times L^2(\Omega)$.
\end{proof}

\section{Generalization}

The above results are also true for the operator
\[
A(x,D)=\begin{pmatrix}
A_{11}(x,D) & A_{12}(x,D) & \cdots & A_{1n}(x,D)\\
A_{21}(x,D) & A_{22}(x,D) & \cdots & A_{2n}(x,D)\\
\vdots & \vdots & \vdots & \vdots\\
A_{n1}(x,D) & A_{n2}(x,D) & \cdots & A_{nn}(x,D)
\end{pmatrix},
\]
 where
\[
A_{hl}(x,D)u=- \sum_{j,k=1}^N
\frac{\partial}{\partial x_{j}}
\Big(  a_{j,k}^{hl}(x)\frac{\partial u}{\partial x_{k}}\Big) ,
\quad x\in\overline{\Omega},\; h,l=1,\dots,N,
\]
under the following assumptions:
\begin{itemize}

\item[(A1)] The operators $A_{hh}$ $(h=1,\dots,n)$ are
symmetric; i.e.,
\[
a_{kj}^{hh}(x)=a_{jk}^{hh}(x),x\in\overline{\Omega},\quad \text{for all }
j,k=1,\dots,N.
\]


\item[(A2)] The operators $A_{hl}$ $(h=1,\dots,n)$ are
regular; i.e., for all $h,l=1,\dots,n$
\[
a_{jk}^{hl}(x)\in C^1(  \overline{\Omega};\mathbb{R}) ,\quad
j,k=1,\dots,N.
\]


\item[(A3)] There exists  $m\in\{  1,\dots,n\}  $
such that the diagonal operator $A_{mm}$  is strongly elliptic;
i.e., there is a  constant  $\mu>0$  such that for all 
 $\xi=(\xi_{j})  _{j=1}^N\in \mathbb{R}^N$  and all  $x\in\Omega$,
\[
 \sum_{j,k=1}^N
a_{jk}^{mm}(x)\xi_{j}\xi_{k}\geq\mu \sum_{j=1}^N \xi_{j}^2=\mu| \xi| ^2
\]


\item[(A4)] The other diagonal operators  $A_{ll}$  
($l\neq m$) are weakly elliptic; i.e., for all  
$\xi=(  \xi_{j})_{j=1}^N\in\mathbb{R}^N$  and all  $x\in\Omega$ 
\[
\sum_{j,k=1}^N a_{jk}^{ll}(x)\xi_{j}\xi_{k}\geq0,\quad
\text{for all }l\neq m.
\]


\item[(A5)] The operators sums  $A_{hl}+A_{lh}$  
$(h\neq l)$ are weakly elliptic; i.e., for all  
$\xi=(  \xi_{j})  _{j=1}^N\in\mathbb{R}^N$  and all  $x\in\Omega$
\[
 \sum_{j,k=1}^N(  a_{jk}^{hl}+a_{jk}^{lh})(x)\xi_{j}\xi_{k}\geq0.
\]
\end{itemize}

By examining the proof of the Theorem \ref{thm3.2}, we note that the above results
remain true if we assume only that one
of the operators  $A_{hh}$ $(h=1,\dots,n)$, $A_{hl}+A_{lh}$ $(h\neq l$,
 $h,l=1,\dots,n)$  is strongly elliptic and the rest of them are 
 all weakly elliptic.

\begin{thebibliography}{0}


\bibitem{a1} R. A. Adams;
\emph{Sobolev spaces}, Academic press, New York, 1975.

\bibitem{b1} H. Brezis;
\emph{Analyse fonctionnelle Th\'eorie et applications},
Dunod, 1999.

\bibitem{o1}L. A. F. de Oliveira;
\emph{On reaction diffusion equations}, Electron. J.
Differential Equations, Vol. 1998 (1998), No. 24, pp. 1-10.

\bibitem{p1} R. D. Passo, P. Mottoni;
\emph{ On a paper by A. I. Lee and J. M. Hille}, 
J. Math. Anal. Appl., 106, pp. 184-187, 1985.

\bibitem{p2} A. Pazy;
\emph{Semigroups of linear operators and applications to partial
differential equations}, Springer, New York, 1983.

\end{thebibliography}

\end{document}