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\markboth{\hfil Denseness of domains \hfil EJDE--2002/23}
{EJDE--2002/23\hfil Sasun Yakubov \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 23, pp. 1--13. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
  Denseness of domains of differential operators in Sobolev spaces.  
 %
\thanks{ {\em Mathematics Subject Classifications:} 26B35, 26D10.
\hfil\break\indent
{\em Key words:} Local rectification, local coordinates, normal system,
holomorphic  semigroup, \hfil\break\indent
infinitesimal operator, dense sets, Sobolev spaces
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Submitted December 25, 2001. Published February 27, 2002. \hfil\break\indent
Supported by the Israel Ministry of  Absorption.} }
\date{}
%
\author{Sasun Yakubov}
\maketitle

\begin{abstract} 
 Denseness of the domain of differential operators plays an essential
 role in many areas of differential equations and functional
 analysis. This, in turn, deals with dense sets in Soblev spaces.
 Denseness for functions of a single variable was formulated and
 proved, in a very general form, in the book by Yakubov and Yakubov 
 \cite[Theorem 3.4.2/1]{y1}. In the same book, denseness for
 functions of several variables was formulated.  However, the proof of
 such result is complicated and needs a series of constructions
 which are presented in this paper. We also prove some independent 
 and new results.
\end{abstract}

\newtheorem{theorem}{Theorem}[section]
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\newtheorem{corollary}[theorem]{Corollary}

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\section{Introduction} 

We denote by $\mathbb{R}^r$ the $r$-dimensional real Euclidean space. 
For a bounded (open) domian $G$ in $\mathbb{R}^r$ its boundary is
denoted by $\partial G$: $\partial G=\overline {G}/G$.

A bounded domain $G \subset \mathbb{R}^r$ is said to be a $C^\ell$,
where $\ell=1,2,\dots$, if there exists a finite number of open balls
$G_i$, $i=1,\dots,N$,
such that $\partial G \subset \cup_{i=1}^N G_i$,
$G_i \cap \partial G\ne \emptyset$, $i=1,\dots,N$, and if there exist
$\ell$-fold differentiable real vector-functions
$f^{(i)}(x)=(f_1^{(i)}(x),\dots,f_r^{(i)}(x))$
defined in $G_i$ such that $y=f^{(i)}(x)$ is a one-to-one mapping from
$G_i$ onto a bounded domain $\mathbb{R}^r$, where $G_i \cap \partial G$
is a part of the hyper-plane $\{y : y\in  \mathbb{R}^r; y_r=0\}$ and
$G_i \cap G$ is a simply
connected domain in the half-space $\mathbb{R}_+^r=\{(y',y_r): y'\in  \mathbb{R}^{r-1}; y_r>0\}$.
On the Jacobian of $f$ we assumed that 
$$
\frac {\partial(f_1^{(i)},\dots,f_r^{(i)})}{\partial(x_1,\dots,x_r)}\ne 0,
\quad x\in \overline G_i.
$$
In this case we write  $\partial G\in C^\ell$ and say that $\partial G$
admits a {\it local rectification}  by means of  smooth non-degenerate transformations of
coordinates.   The coordinates $f^{(i)}$ will be
called {\it local coordinates} in $G_i$.

Let 
\begin{equation}
L_{\nu }u=\sum_{|\alpha|=m_\nu }b_{\nu \alpha}(x')D^\alpha u(x')+
\sum_{p=0}^{m_\nu -1}K_{\nu p}\frac {\partial^p u(x')}{\partial n^p}, \quad
x'\in\partial G, \ \nu =1,\dots,m, \label{e1}
\end{equation}
where $D^\alpha:=D_1^{\alpha_1}\cdots
D_r^{\alpha_r}$, $D_j:=-i\frac{\partial}{\partial x_j}$, $j=1,\dots,r$,
$\alpha:=(\alpha_1,\dots,\alpha_r)$ is a multi-index,  $|\alpha|:=\sum_{j=1}^r\alpha_j$,
$x:=(x_1,\dots,x_r)$, $x':=(x'_1,\dots,x'_r)$, $n$ is a normal vector to the
boundary $\partial G$ at the point $x'\in\partial G$.
Then $L_{\nu }u$ is called {\it normal} if $m_j\ne m_k$ for $j\ne k$ and 
for any vector $\sigma$, normal to the boundary $\partial G$ at the 
point $x'\in\partial G$, 
$$
L_{\nu 0}(x',\sigma)=\sum_{|\alpha|=m_\nu }b_{\nu \alpha}(x')\sigma^\alpha\ne 0,\quad
\nu =1,\dots,m,
$$
and the operator $K_{\nu p}$ from $W_q^{m_\nu-p}(\partial G)$ into $L_q(\partial G)$ is compact,
where $ \sigma^\alpha=\sigma_1^{\alpha_1}\cdots\sigma_r^{\alpha_r}$,
 $\partial G\in C^\ell, \ q \in (1,\infty)$.

Let $E_0$ and $E_1$ be two Banach spaces continuously embedded into a
Banach space $E:\ E_0\subset E$, $E_1\subset E$. Such spaces are
called an {\it interpolation couple} and is denoted by $\{E_0,E_1\}$. 
Consider the Banach
space
\begin{gather*}
E_0+E_1:=\big\{ u=u_0+u_1 : u_j\in E_j,\; j=0,1\big\} \\
\|u\|_{E_0+E_1}:=\inf_{ u=u_0+u_1, \; u_j\in E_j }
(\|u_0\|_{E_0}+\|u_1\|_{E_1}).
\end{gather*}
Due to Triebel \cite[1.3.1]{t1},  the functional
$$
K(t,u):=\inf_{ u=u_0+u_1,\;u_j\in E_j }
\Big(\|u_0\|_{E_0}+t\|u_1\|_{E_1}\Big), \quad u\in E_0+E_1,
$$
is continuous on $(0,\infty)$ in $t$, and the following estimate holds$:$
$$\min\{1,t\}\|u\|_{E_0+E_1}\le K(t,u)\le\max\{1,t\}\|u\|_{E_0+E_1}.$$
An {\it interpolation space} for $\{E_0,E_1\}$ by the $K$-method is
defined as follows:
\begin{gather*}
(E_0,E_1)_{\theta,p}
:=\big\{ u\in E_0+E_1 : \|u\|_{(E_0,E_1)_{\theta,p}} <\infty,\;
0<\theta<1,\; 1\le p<\infty,\big\} \\
\|u\|_{(E_0,E_1)_{\theta,p}}:=\Big(\int_0^\infty t^{-1-\theta p}K^p(t,u) \,dt
\Big)^{1/p} \\
(E_0,E_1)_{\theta,\infty}
:=\big\{u\in E_0+E_1 : \|u\|_{(E_0,E_1)_{\theta,\infty}}<\infty ,\;
0<\theta<1\big\}\\
\|u\|_{(E_0,E_1)_{\theta,\infty}}:=\sup_{t\in(0,\infty)}t^{-\theta}K(t,u).
\end{gather*}
$W_q^\ell((0,\infty);E)$, $1\le q<\infty$, with $\ell$ integer, denotes
a Banach space of functions $u(x)$ with values from $E$ which have
generalized derivatives up to $\ell$-th order, inclusive, on $(0,1)$
and the norm $\|u\|_{W_q^\ell((0,\infty);E)}:=\sum_{k=0}^\ell
\big(\int_0^1\|u^{(k)}(x)\|_E^q \ dx\big)^{1/q}$ is finite.

Let the embedding $E_0 \subset E_1$ be continuous. Consider the Banach
space
 $W_q^\ell((0,\infty);E_0,E_1):=L_q((0,\infty);E_0)\cap W_q^\ell((0,\infty);E_1)$
with the norm
$$\|u\|_{W_q^\ell((0,\infty);E_0,E_1)}:=\|u\|_{L_q((0,\infty);E_0)}+
\|u^{(\ell)}\|_{L_q((0,\infty);E_1)}\,.$$

Let $G$ be an open set of $\mathbb{R}^{r}$, in particular, $G=\mathbb{R}^{r}$ and $G=\mathbb{R}_+^{r}$.
Then, $W_q^m(G)$ is a Banach space of functions $u(x)$ that have
generalized derivatives on $G$ up to the $m$-th order inclusive, for
which the following norm is finite$:$
$$
\|u\|_{W_q^m(G)}:=\Big(\sum_{|\alpha|\le m}\|D^\alpha
u\|_{L_q(G)}^q\Big)^{1/q}.
$$

Let $s_0$ and $s_1$ be non-negative integers, $0<\theta<1$, $1<p<\infty$,
$1\le q\le\infty$ and $s=(1-\theta)s_0+\theta s_1$. From
Triebel \cite[Theorem 4.3.2/1, formula 2.4.2/16]{t1} it follows that if
$s=(1-\theta)s_0+\theta s_1=(1-\theta')s'_0+\theta' s'_1$, then,
$$
(W_p^{s_0}(G), W_p^{s_1}(G))_{\theta,q}=
(W_p^{s'_0}(G), W_p^{s'_1}(G))_{\theta',q}.
$$
Consider the space
$$
B_{p,q}^s(G):=(W_p^{s_0}(G),W_p^{s_1}(G))_{\theta,q},
$$
where $ s_0, s_1$  are non-negative integers,  $0<\theta<1$, $1<p<\infty$,
$1\le q\le\infty$  and  $s=(1-\theta)s_0+\theta s_1$. For $s$ positive and
not an integer, set
$$
W_p^s(G):= B_{p,p}^s(G):=(W_p^{s_0}(G),W_p^{s_1}(G))_{\theta,p}\,.
$$
The closure of a set $M$ of $E$ by the norm of $E$ is
denoted by $\overline M|_E$ and sometimes by $\overline M$. The set $M$ is
called {\it dense} in $E$ if $\overline M|_E=E$.

The following two equalities are application of
Theorem \ref{thm5} in this paper. For $(r-1)/q<2$, we have:
\begin{gather*}
\overline{W_q^3\big(G;u|_{\partial G}=0,\frac {\partial^2u}{\partial n^2}\Big|_{\partial G}+u(x'_0)=0\big)}
\Big|_{W_q^2(G)}=W_q^2(G;u|_{\partial G}=0),\\
\overline{W_q^4\big(G;u|_{\partial G}=0,\frac {\partial^2u}{\partial n^2}\Big|_{\partial G}+u(x'_0)=0\big)}
 \Big|_{W_q^3(G)} \\
=W_q^3\big(G;u|_{\partial G}=0,\frac {\partial^2u}{\partial n^2}\Big|_{\partial G}+u(x'_0)=0\big)\,,
\end{gather*}
where $x'_0 \in \partial G$ is a fixed point of the boundary $\partial G$.
These equalities are simple but are new. Such equalities without the term
$u(x'_0)$ follow from interpolation theorems of Grisvard-Seeley type
(see,  Grisvard \cite{g1}, Seeley \cite{s1}).
The following equality is also known:
$$
\overline{C_0^\infty(G)} \Big|_{W_q^k(G)}=
W_q^k\Big(G;u|_{\partial G}=\frac {\partial u}{\partial n}\Big|_{\partial G}=\cdots=\frac {\partial^{k-1}u}{\partial n^{k-1}}
\Big|_{\partial G}=0\Big).
$$
Moreover, one can add first derivatives of the function $u$ at some fixed
points of the boundary $\partial G$, and integral terms with  $u(x')$
in addition to $u(x'_0)$ in boundary conditions.

\section{Functions of several real variables}

\begin{lemma} \label{lm1}
Let $\varphi_j \in W_q^{\ell-j-\frac 1q}(\mathbb{R}^{r-1}),$
where  $\ell, j$ are integer numbers,
 $0\le j\le \ell-1$, $q \in (1,\infty)$.
Then, there exist functions $u_j(y_r,\lambda)=u_j(y',y_r,\lambda)$, $\lambda>0$ belonging to
the Banach space
$W_q^\ell(\mathbb{R}_+^{r})=W_q^\ell((0,\infty);W_q^\ell(\mathbb{R}^{r-1}),
L_q(\mathbb{R}^{r-1}))$
and satisfying
\begin{equation}
\frac {\partial^j  u_j(y',0,\lambda)} {\partial y_r^j} =\varphi_j(y'),
\quad y'\in \mathbb{R}^{r-1}, \label{e2}
\end{equation}
such that the following estimate  holds
\begin{equation}
\sum_{k=0}^{\tilde\ell} \lambda^{\tilde\ell-k} \|u_j\|_{W_q^k(\mathbb{R}^r_+)}
\le C\Bigl(\|\varphi_j\|_{W_q^{\tilde\ell-j-\frac 1q}(\mathbb{R}^{r-1})}
+\lambda^{\tilde\ell-j-\frac 1q} \|\varphi_j\|_{L_q(\mathbb{R}^{r-1})} \Bigr),
\label{e3}
\end{equation}
where \ $0\le j\le \tilde \ell-1, \ \tilde\ell\le \ell$.
\end{lemma}

\paragraph{Proof}
In the  Banach space $E=L_q(\mathbb{R}^{r-1})$ consider the operator
$A=(-\Lambda+I)^2$. By virtue of Lemma \ref{lmA1}, for $k=1,2,\dots$,
$$
 D(A^{k})= D(\Lambda^{2k})=W_q^{2k}(\mathbb{R}^{r-1}),\quad
 D(A^{\frac k2})= D(\Lambda^{k})=W_q^{k}(\mathbb{R}^{r-1}).
$$
Consider the functions
\begin{equation}
u_j(y_r,\lambda)=\text{e}^{-y_r(A+\lambda^2 I)^{1/2}}g_j, \label{e4}
\end{equation}
where $g_j \in E$. Since
$$
u_{jy_r}^{(j)}(y_r,\lambda)=(-1)^j\text{e}^{-y_r(A+\lambda^2 I)^{1/2}}
(A+\lambda^2 I)^{j/2}g_j,
$$
the functions in (\ref{e4}) satisfy (\ref{e2}) if
$(-1)^j (A+\lambda^2 I)^{\frac j2}g_j=\varphi_j$.
Consequently,
\begin{equation}
u_j(y_r,\lambda)=(-1)^j \text{e}^{-y_r(A+\lambda^2 I)^{1/2}}
(A+\lambda^2 I)^{-j/2}\varphi_j. \label{e5}
\end{equation}
Since
\begin{gather*}
A^{\frac k2}u_j(y_r,\lambda)=(-1)^j A^{k/2}(A+\lambda^2 I)^{-\frac j2}
\text{e}^{-y_r(A+\lambda^2 I)^{1/2}}\varphi_j,\\
u_{j }^{(k)}(y_r,\lambda)=(-1)^{k+j} (A+\lambda^2 I)^{\frac {k-j}2}
\text{e}^{-y_r(A+\lambda^2 I)^{1/2}}\varphi_j,
\end{gather*}
for $k\le \ell$,  we have
\begin{align*}
\lambda^{(\ell-k)q} \|u_j\|^q_{W_q^k(\mathbb{R}^r_+)}
=&\lambda^{(\ell-k)q} \|u_j\|^q_{W_q^k((0,\infty);W_q^k(\mathbb{R}^{r-1}),L_q(\mathbb{R}^{r-1}))}\\
=&\lambda^{(\ell-k)q} \|u_j\|^q_{W_q^k((0,\infty);E(A^{\frac k2}),E)}\\
\le& C \lambda^{(\ell-k)q}\Big( \|A^{\frac k2}u_j\|^q_{L_q((0,\infty);E)}
+ \|u_j^{(k)}\|^q_{L_q((0,\infty);E)}\Big)\\
\le& C \lambda^{(\ell-k)q}
\int_{0}^{\infty} \Bigl( \|A^{\frac k2}(A + \lambda^2 I)^{-\frac j2}
\hbox{\rm e}^{-y_r(A + \lambda^2 I)^{\frac 12}}\varphi_j \|_E^q\\
&+\|(A + \lambda^2 I)^{\frac {k-j}2}
\hbox{\rm e}^{-y_r(A + \lambda^2 I)^{\frac 12}}\varphi_j \|_E^q \Bigr)dy_r\\
\le &C\lambda^{(\ell-k)q}\|(A + \lambda^2 I)^{-\frac {\ell-k}2}\|^q_{B(E)}
(\|A^{\frac k2}(A + \lambda^2 I)^{-\frac k2}\|^q_{B(E)}+1)\\
&\times \int_{0}^{\infty}  \|(A + \lambda^2 I)^{\frac {\ell-j}2}
\hbox{\rm e}^{-y_r(A + \lambda^2 I)^{\frac 12}}\varphi_j \|_E^q dy_r.
\end{align*}
By virtue of Lemma \ref{lmA2}  and Theorem \ref{thmA3}, we have
\begin{equation}
\lambda^{(\ell-k)q} \|u_j\|^q_{W_q^k(\mathbb{R}^r_+)}
\le C\Bigl(\|\varphi_j\|_{(E,E(A^\ell))_{\frac {\ell-j}{2\ell}-\frac 1{2\ell q},q}}^q
+\lambda^{(\ell-j)q- 1} \|\varphi_j\|^q_E \Bigr). \label{e6}
\end{equation}
Since
\begin{equation}
(E,E(A^\ell))_{\frac {\ell-j}{2\ell}-\frac 1{2\ell q},q}=
(L_q(\mathbb{R}^{r-1}),W_q^{2\ell}(\mathbb{R}^{r-1}))_{\frac {\ell-j}{2\ell}-\frac 1{2\ell q},q}=
W_q^{\ell-j-\frac 1q}(\mathbb{R}^{r-1}), \label{e7}
\end{equation}
 from (\ref{e6}) and (\ref{e7}) it follows that a function defined by (\ref{e5}) belongs
to the space
$W_q^\ell(\mathbb{R}_+^{r})=W_q^\ell((0,\infty);W_q^\ell(\mathbb{R}^{r-1}),
L_q(\mathbb{R}^{r-1}))$
and  estimate (\ref{e3}) holds. \hfill$\Box$


\begin{theorem} \label{thm2}
 Let $\varphi_j \in W_q^{\ell-j-\frac 1q}(\mathbb{R}^{r-1})$, $0\le j \le m-1$,
 $m\le \ell$.
Then, there exist functions $u(y_r,\lambda)=u(y',y_r,\lambda)$, $\lambda>0,$ belonging to
the Banach space
$W_q^\ell(\mathbb{R}_+^{r})=W_q^\ell((0,\infty);W_q^\ell(\mathbb{R}^{r-1}),
L_q(\mathbb{R}^{r-1}))$
and satisfying
\begin{equation}
\frac {\partial^j  u(y',0,\lambda)} {\partial y_r^j}=\varphi_j(y'), \quad y'\in \mathbb{R}^{r-1},
\; j=0,\dots,m-1,\label{e8}
\end{equation}
such that the following estimate  holds
\begin{equation}
\sum_{k=0}^{\tilde\ell} \lambda^{\tilde\ell-k} \|u\|_{W_q^k(\mathbb{R}^r_+)}
\le C\sum_{j=0}^{m-1} \Bigl(\|\varphi_j\|_{W_q^{\tilde\ell-j-\frac 1q}(\mathbb{R}^{r-1})}
+\lambda^{\tilde\ell-j-\frac 1q} \|\varphi_j\|_{L_q(\mathbb{R}^{r-1})} \Bigr), \label{e9}
\end{equation}
where $m\le \tilde \ell\le  \ell$.
\end{theorem}

\paragraph{Proof} By virtue of Lions and Magenes \cite[Theorem 1.3.2]{l1},
if
\begin{equation}
U_j(y_r,\lambda)=\sum_{p=1}^m c_{pj}u_j(py_r,\lambda), \label{e10}
\end{equation}
where
\begin{equation}
u_j(py_r,\lambda)=(-1)^j \text{e}^{-py_r(A+\lambda^2 I)^{1/2}}
(A+\lambda^2 I)^{-\frac j2}\varphi_j, \label{e11}
\end{equation}
the operator $A$ is defined in the proof of Lemma \ref{lm1}, and the complex numbers
$c_{pj}$ satisfy the systems
\begin{equation}
\sum_{p=1}^m p^k c_{pj}=\begin{cases}
0, & k\ne j, \\
1, & k=j,
\end{cases} \quad k=0,\dots,m-1, \label{e12}
\end{equation}
then
\begin{equation}
\frac {\partial^k  U_j(0,\lambda)} {\partial y_r^k}=\begin{cases}
0, & k\ne j, \\
\varphi_j, & k=j.
\end{cases} \label{e13}
\end{equation}
Consequently, the function
\begin{equation}
u(y_r,\lambda)=\sum_{j=0}^{m-1}U_j(y_r,\lambda)
=\sum_{j=0}^{m-1}\sum_{p=1}^m c_{pj}u_j(py_r,\lambda) \label{e14}
\end{equation}
belongs to the space $W_q^\ell(\mathbb{R}_+^{r})=W_q^\ell((0,\infty);W_q^\ell(\mathbb{R}^{r-1}),
L_q(\mathbb{R}^{r-1}))$ and
satisfies (\ref{e8}). From (\ref{e3}) and  (\ref{e10})--(\ref{e14}) for the function $u(y_r,\lambda)$, it
follows estimate (\ref{e9}). \hfill$\Box$


\begin{corollary} \label{coro3}
 For $\lambda>0,$ there exists a continuous operator which is a continuation
 of $\mathbb{R}(\lambda):\ (\varphi_0,\dots,\varphi_{ m-1})
\to \mathbb{R}(\lambda)(\varphi_0,\dots,\varphi_{ m-1})$ from \\
${\dot+}^{m-1}_{j=0}$
$W_q^{\ell-j-\frac 1q}(\mathbb{R}^{r-1})$
into
 $W_q^\ell(\mathbb{R}^r_+)$, such that
$$
\frac {\partial^j  \mathbb{R}(\lambda)(\varphi_0,\dots,\varphi_{m-1})(y',0)}
{\partial y_r^j}=\varphi_j(y'),
 \quad y'\in \mathbb{R}^{r-1}, \ j=0,\dots,m-1
$$
and
\begin{multline*}
\sum_{k=0}^{\tilde\ell} \lambda^{\tilde\ell-k}
\|\mathbb{R}(\lambda)(\varphi_0,\dots,\varphi_{m-1})\|_{W_q^k(\mathbb{R}^r_+)}\\
\le C\sum_{j=0}^{m-1} \Bigl(\|\varphi_j\|_{W_q^{\tilde\ell-j-\frac 1q}(\mathbb{R}^{r-1})}
+\lambda^{\tilde\ell-j-\frac 1q} \|\varphi_j\|_{L_q(\mathbb{R}^{r-1})} \Bigr).
\end{multline*}
\end{corollary}

\paragraph{Proof} Define a continuation operator as
$$
\mathbb{R}(\lambda)(\varphi_0,\dots,\varphi_{m-1}):=
\sum_{j=0}^{m-1}\sum_{p=1}^m c_{pj}u_j(py_r,\lambda)
$$
and apply Theorem \ref{thm2}. \hfill$\Box$


\begin{theorem} \label{thm4} Let the following conditions be satisfied:
\begin{enumerate}
\item $b_{\nu \alpha}\in C^{\ell-m_\nu }(\overline G)$,  operators $K_{\nu p}$
 from $W_q^{m_\nu-p}(\partial G)$ into $L_q(\partial G)$ and
from $W_q^{\ell-p}(\partial G)$ into $W_q^{\ell-m_\nu}(\partial G)$
 are compact, where $\ell\ge\max\{ m_\nu \} +1$, $q\in(1,\infty),$  \ $\partial G \in
C^\ell$.
\item System (\ref{e1}) is normal.
\item $f_\nu \in W_q^{\ell-m_\nu-\frac 1q}(\partial G)$,  $\nu=1,\dots,m$.
\end{enumerate}
Then, there exist functions $u(x,\lambda)$, $\lambda>0,$ belonging to the Sobolev space
$W_q^\ell(G)$ and satisfying
\begin{equation}
L_\nu  u(x',\lambda)=f_\nu(x'), \quad x'\in \partial G, \; \nu=1,\dots,m,
\label{e15}
\end{equation}
where $L_\nu$ are defined in (\ref{e1}), such that the following estimate holds
\begin{equation}
\sum_{k=0}^{\tilde\ell} \lambda^{\tilde\ell-k} \|u\|_{W_q^k(G)}
\le C\sum_{\nu=1}^{m} \Bigl(\|f_\nu\|_{W_q^{\tilde\ell-m_\nu-\frac 1q}(\partial G)}
+\lambda^{\tilde\ell-m_\nu-\frac 1q} \|f_\nu\|_{L_q(\partial G)} \Bigr), \label{e16}
\end{equation}
where  $\max\{ m_\nu \} +1\le \tilde\ell\le \ell$.
\end{theorem}

\paragraph{Proof}
Consider the balls $G_i$, $i=1,\dots,N$, from $\mathbb{R}^r$,
 which cover the  $\partial G$, i.e.,
$$
\partial G \subset  \cup_{i=1}^N G_i; \quad G_i \cap \partial G\ne 0,
 \quad i=1,\dots,N.
$$
Let $\{\theta_i(x)\}$ be a partition of unity subordinate to a cover of
$\partial G$ by $\{G_i\}$ (see, e. g., Lions  and  Magenes \cite[2.5.1]{l1}).
The functions $\theta_i(x)$ have the following properties:
\begin{enumerate}
\item The support of the function $\theta_i(x)$ belongs to the set $G_i$,
i.e., $\theta_i(x)=0$ outside of $G_i$;
\item Functions $\theta_i(x)$ are infinitely differentiable on $\mathbb{R}^r$;
\item $0\le\theta_i(x)\le 1$; $\sum_{i=1}^N\theta_i(x)\equiv1$, $x\in \partial G $.
\end{enumerate}

In  $ G_i$ we introduce a system of curvilinear
coordinates   $y_1(x'), \dots,$  $y_r(x')$, where $x'\in \partial G$.
Assume that $ y_r(x')=n(x')$ is the normal vector, while
$y_1(x'), \dots, y_{r-1}(x')$ are tangential vectors on   $\partial G$.
The operators $L_\nu$ may be expressed in these curvilinear
coordinates as
\begin{align}
\tilde L_\nu \tilde u:=&c_{\nu }(y',0)\frac {\partial^{m_\nu}\tilde u(y',0)}
{\partial y_r^{m_\nu}} + \sum_{|\alpha|\le m_\nu,|\alpha_r|<m_\nu }
c_{\nu \alpha}(y',0) \frac {\partial^{\alpha}\tilde u(y',0)}
{\partial y_1^{\alpha_1} \cdots \partial y_{r-1}^{\alpha_{r-1}} 
\partial y_r^{\alpha_r}} \nonumber\\
&+\sum_{p=0}^{m_\nu -1}\tilde K_{\nu p}\frac {\partial^p\tilde u(y',0)}{\partial y_r^p},
\quad \nu =1,\dots,m, \ (y',0)\in f^{(i)}(G_i\cap \partial G),\label{e17}
\end{align}
where
\begin{gather*}
\tilde u(y):=u((f^{(i)})^{-1}(y)), \quad y\in f^{(i)}(G_i\cap G),\\
\tilde K_{\nu p}\frac {\partial^p\tilde u(y',0)}{\partial y_r^p}:=
\Big(K_{\nu p}\frac {\partial^p u(x')}{\partial n^p}\Big)((f^{(i)})^{-1}(y',0)), \quad
(y',0)=f^{(i)}(x'),
\end{gather*}
where $c_{\nu},\ c_{\nu \alpha} \in C^{\ell-m_\nu }(f^{(i)}(G_i\cap G))$,
and $f^{(i)}$ is defined in a similar way as in the beginning of the paper.
It holds that $c_{\nu }(y',0)\ne 0$
for $(y',0)\in f^{(i)}(G_i\cap \partial G)$.

 We look for  functions $u(x,\lambda) \in W_q^\ell(G)$, $\lambda>0,$ satisfying the
relations
\begin{equation}
\tilde L_\nu \tilde u=\tilde f_\nu(y',0),\quad
(y',0)\in f^{(i)}(G_i\cap \partial G),
 \quad \nu=1,\dots,m, \label{e18}
\end{equation}
where $\tilde L_\nu$ are operators defined in (\ref{e17}), and for which additionally
\begin{equation}
\frac {\partial^j\tilde u(y',0,\lambda)}{\partial y_r^j}=0, \quad j\ne m_\nu, \quad
 j=0,\dots,\max\{m_\nu\}-1,\quad (y',0)\in f^{(i)}(G_i\cap \partial G). \label{e19}
\end{equation}
Consider for $(y',0)\in f^{(i)}(G_i\cap \partial G)$ the functions
\begin{equation}\begin{aligned}
\varphi_j(y'):=&0, \quad j\ne m_\nu, \quad j=0,\dots,\max\{m_\nu\}-1,\\
\varphi_{m_\nu}(y'):=&\Big(c_{\nu }(y',0)\Big)^{-1}
\Big[\tilde f_\nu(y',0)\\
&-\sum_{|\alpha|\le m_\nu,|\alpha_r|<m_\nu }c_{\nu \alpha}(y',0)
\frac {\partial^{\alpha}\varphi_{\alpha_r}(y')}
{\partial y_1^{\alpha_1} \cdots \partial y_{r-1}^{\alpha_{r-1}} }
-\sum_{p=0}^{m_\nu -1}
\tilde K_{\nu p}\varphi_p(y') \Big].\end{aligned}\label{e20}
\end{equation}
Since $W_q^s=(W_q^{s_0},W_q^{s_1})_{\theta,q}$, $s>0$ is not
 an integer, $s_0, s_1$ are integers,
$0<\theta<1$, and $s=(1-\theta)s_0+\theta s_1$,  by virtue of the
interpolation theorem \cite{h1} (see, e.g., \cite[1.16.4]{t1}) and
condition 1 of Theorem 2.4, the operators
$K_{\nu p}$ from $W_q^{k-p-\frac 1q}(\partial G)$ into
$W_q^{k-m_\nu-\frac 1q}(\partial G)$, for $m_\nu+1\le k\le \ell$, are compact.
Then, from conditions 1 and 3 of Theorem 2.4 and (\ref{e20}) it follows that
$\varphi_j \in W_q^{\ell-j-\frac 1q}(f^{(i)}(G_i\cap\partial G)$.

Let $ \eta_i(x) \in C^\infty(\mathbb{R}^r)$, $i=1,\dots,N$, and
$\mathop{\rm supp}\eta_i \subset G_i$, $\eta_i(x)=1$,
$x\in \mathop{\rm supp}\theta_i$.
Consider the function
\begin{align}
u(x,\lambda):=& \overline {\mathbb{R}}(\lambda) (\varphi_0,\dots,\varphi_{\tilde m-1})
(x,\lambda) \nonumber \\
:=&\sum_{i=1}^N  \eta_i(x)\mathbb{R}(\lambda)(\theta_i((f^{(i)})^{-1}(y',0))\varphi_0,
\label{e21} \\
&\dots, \theta_i((f^{(i)})^{-1}(y',0))\varphi_{\tilde m-1})(f^{(i)}(x),\lambda), \nonumber
\end{align} 
where $\tilde m=\max\{ m_\nu \} +1$, for $x\in G$ (where $ \eta_i(x)
\mathbb{R}\{ \quad\}(f^{(i)}(x),\lambda)=0$ outside of $G_i$).
By virtue of Corollary \ref{coro3}, for function (\ref{e21}) we have
\begin{equation}\begin{aligned}
\frac {\partial^j \tilde u(y',0,\lambda)} {\partial y_r^j}=&\frac {\partial^j  u((f^{(i)})^{-1}(y',0),\lambda)}
{\partial y_r^j} \\
=&\sum_{i=1}^N \theta_i((f^{(i)})^{-1}(y',0))\varphi_{j}
=\varphi_j(y'), \; y'\in \mathbb{R}^{r-1}, \, j=0,\dots,\tilde m-1,
\end{aligned}\label{e22}
\end{equation}
where $\varphi_j(y')$ is defined in (\ref{e20}). From (\ref{e20}) and (\ref{e22}) we get
(\ref{e18}) and (\ref{e15}).
Since the mapping $f^{(i)}$ is a diffeomorphism of class $C^\ell$ then,
by virtue of Corollary \ref{coro3}, function (\ref{e21})   satisfies  estimate (\ref{e16}).
 \hfill$\Box$

\begin{theorem} \label{thm5}
Let the following conditions be satisfied:
\begin{enumerate}
\item $b_{\nu \alpha}\in C^{\ell-m_\nu }(\overline G)$,  operators $K_{\nu p}$
 from $W_q^{m_\nu-p}(\partial G)$ into $L_q(\partial G)$ and
from $W_q^{\ell-p}(\partial G)$ into $W_q^{\ell-m_\nu}(\partial G)$
 are compact, where $\ell\ge\max\{ m_\nu \} +1$, $q\in(1,\infty),$  \ $\partial G \in
C^\ell$.
\item System (\ref{e1}) is normal.
\end{enumerate}
Then, for integer  $k \in [0,\ell]$,
\begin{equation}
\overline{W_q^{\ell}(G;L_\nu u=0,\nu= 1,\dots,m)} \Big|_{W_q^k(G)}=
W_q^k(G;L_\nu u=0,m_\nu\le k-1). \label{e23}
\end{equation}
\end{theorem}

\paragraph{Proof}
For $k=0$, (\ref{e23}) follows from the known embedding
$\overline{C_0^{\infty}(G)} \Big|_{L_q(G)}=L_q(G)$. Let $k\ge 1$.
Obviously,
\begin{equation}
\overline{W_q^{\ell}(G;L_\nu u=0,\nu=1,\dots,m)} \Big|_{W_q^k(G)}
\subset W_q^k(G;L_\nu u=0,m_\nu\le k-1). \label{e24}
\end{equation}
Indeed, let $u_n \in W_q^{\ell}(G;L_\nu u=0,\nu=1,\dots,m)$ and let
$$
\lim_{n\to\infty}\|u_{n}-u\|_{W_{q}^k(G)}=0.
$$
It is proved in Theorem \ref{thm4} that from condition 1 of Theorem 2.5 it follows that operators
$K_{\nu p}$ from $W_q^{k-p-\frac 1q}(\partial G)$ into
$W_q^{k-m_\nu-\frac 1q}(\partial G)$, for $m_\nu+1\le k\le \ell$, are  compact.
Then, by virtue of \cite[Theorem 4.7.1]{t1},  for $m_\nu\le k-1$ we have

\begin{equation}\begin{aligned}
\lim_{n\to\infty}\|L_\nu u_{n}-L_\nu u\|_{W_{q}^{k-m_\nu-\frac 1q} (\partial G)}
\le& C\lim_{n\to\infty}\| u_{n}- u\|_{W_{q}^{k-\frac 1q} (\partial G)}\\
\le& C\lim_{n\to\infty}\|u_{n}-u\|_{W_{q}^k(G)}=0,\quad m_\nu\le k-1.
\end{aligned}\label{e25}
\end{equation}
Thus, $L_\nu u=0$, $m_\nu\le k-1$,  since $L_\nu u_{n}=0$.

Now, we show the inverse inclusion
\begin{equation}
W_q^k(G;L_\nu u=0,m_\nu\le k-1)\subset
\overline{W_q^{\ell}(G;L_\nu u=0,\nu=1,\dots,m)} \Big|_{W_q^k(G)}.\label{e26}
\end{equation}
Let $u\in W_q^k(G;L_\nu u=0,m_\nu\le k-1)$.
Then, there exists a sequence of functions $v_n(x)\in C^\infty(G)$,
$n=1,\dots,\infty$,  such that
\begin{equation}
\lim_{n\to\infty}\|v_n-u\|_{W_{q}^{k}(G)}=0. \label{e27}
\end{equation}
 From (\ref{e27}) and (\ref{e25}) it follows that
\begin{equation}
\lim_{n\to\infty}\|L_\nu v_n\|_{W_{q}^{k-m_\nu-\frac 1q} (\partial G)}
=\|L_\nu u\|_{W_{q}^{k-m_\nu-\frac 1q} (\partial G)}=0, \ \ \ m_\nu\le k-1. \label{e28}
\end{equation}
By virtue of  Theorem \ref{thm4} (for $\tilde \ell=k$, $\lambda=\lambda_0$), there exists a
solution $w_n \in W_q^{\ell}(G)$ of the  system
\begin{equation}
L_\nu w_n=-L_\nu v_{n}, \quad m_\nu\le k-1,\label{e29}
\end{equation}
and
\begin{equation}
\|w_n\|_{W_q^{k}(G)}\le
C \sum_{m_\nu \le k-1} \|L_\nu v_n  \|_{W_q^{k-m_\nu -\frac 1q}(\partial G)}.\label{e30}
\end{equation}
Then, from (\ref{e28}) and (\ref{e30}) it follows that
\begin{equation}
\lim_{n\to\infty} \|w_n\|_{W_q^{k}(G)}=0. \label{e31}
\end{equation}
Let $\lambda_n$ be a sequence
tending to $\infty$, if with respect to $n$
\begin{equation}
\|L_\nu(v_n+w_n)\|_{W_q^{\ell-m_\nu -\frac 1q}(\partial G)}\le C, 
\; m_\nu \ge k;   \label{e32}
\end{equation}
and
\begin{equation}\lambda_n=\max_{m_\nu \ge k}\|L_\nu (v_n+w_n)\|_{W_q^{\ell-m_\nu 
-\frac 1q}(\partial G)}^\delta, \label{e33}
\end{equation}
 where $\delta>q$,  if 
$ \|L_\nu (v_n+w_n)\|_{W_q^{\ell-m_\nu -\frac 1q}(\partial G)}$
is not a bounded sequence at least for one $m_\nu \ge k$.

 Apply Theorem \ref{thm4} (for $\tilde \ell=\ell$, $\lambda=\lambda_n$)
 to the system
\begin{equation}\begin{gathered}
L_\nu g_n=0, \quad m_\nu\le k-1,\\
L_\nu g_n=-L_\nu (v_n+w_n) \quad m_\nu \ge k.
\end{gathered} \label{e34}
\end{equation}
Then there exists a  solution $g_n(x)$ of (\ref{e34}) on $W_{q}^{\ell}(G)$ and 
for this solution, as $n\to \infty$,
\begin{align*}
\lambda_{n}^{\ell-k}\|g_n\|_{W_{q}^{k}(G)}
\le &C\sum_{\ m_\nu \ge k}\Big(\lambda_{n}^{\ell-m_\nu-\frac 1q}
\|L_\nu (v_n+w_n)\|_{L_{q}(\partial G)}\\
&+\|L_\nu (v_n+w_n)\|_{W_{q}^{\ell-m_\nu -\frac 1q}(\partial G)}\Big)\,.
\end{align*}
 From this and (\ref{e32}),(\ref{e33}) we have
\begin{equation}
\begin{aligned}
\|g_n\|_{W_{q}^{k}(G)}&\le C\sum_{m_\nu \ge k}
(\lambda_{n}^{\ell-m_\nu-\frac 1q+\frac 1\delta}
+\lambda_n^{\frac 1\delta})\lambda_n^{-\ell+k}\\
&\le C\sum_{m_\nu \ge k}
(\lambda_{n}^{k-m_\nu-\frac 1q+\frac 1\delta}+\lambda_n^{-\ell+k+\frac 1\delta}). 
\end{aligned}\label{e35}
\end{equation}
Since $\delta>q$,   (\ref{e35}) and (\ref{e32}),(\ref{e33}) imply
\begin{equation}
\lim_{n\to\infty}\|g_n\|_{W_{q}^{k}(G)}=0. \label{e36}
\end{equation}
Now, it is easy to see that for the sequence of functions
$u_n=v_{n}+w_n+g_n\in {W_{q}^{\ell}(G)}$ the  relations
\begin{align}
&L_\nu u_n=0,\quad  \nu=1,\dots,m,\label{e37}\\
&\lim_{n\to\infty}\|u_n-u\|_{W_{q}^{k}(G)}=0 \label{e38}
\end{align}
hold. Namely, (\ref{e37}) follows from (\ref{e29}) and (\ref{e34}), and (\ref{e38}) follows from (\ref{e27}),
 (\ref{e31}), and (\ref{e36}). So, the inverse inclusion (\ref{e26}) has also been proved.
 From (\ref{e24}) and (\ref{e26}) it follows (\ref{e23}). \hfill$\Box$


\section {Appendix}

\begin{lemma}[{\cite[Lemma 2.5.3 and formula 2.5.3/11]{t1}}]
\label{lmA1}
Let $1<q<\infty$,  $c_{r-1} \|(1+|x|^2)^{-\frac r2}\|_{L_1(\mathbb{R}^{r-1})}=1$
 and
$$
\Big(P(t)u\Big)(x)=c_{r-1}\int_{\mathbb{R}^{r-1}} \frac t{(|x-y|^2+t^2)^{r/2}}
u(y)dy, \quad 0<t<\infty, \; u \in L_q(\mathbb{R}^{r-1}).
$$
If additionally $P(0)=I$, then $P(t)$ is a holomorphic  semigroup in the space
$L_q(\mathbb{R}^{r-1})$. If $\Lambda$ is a corresponding infinitesimal 
operator for $P(t)$, then
$$
\Lambda^{2m}u=(-1)^m \Delta^m u, \quad  
D(\Lambda^{m})=W_q^{m}(\mathbb{R}^{r-1}),
\quad  m=1,2,\dots.
$$
\end{lemma}

The estimate of the  semigroup is a central point in the theory
of  differential-operator equations.

For $\varphi\in (0,\pi)$ we denote by $\Sigma_\varphi$ the closed
sector $\{ \lambda \in \mathbb{C}:  |\arg \lambda|\le \varphi \,\}\cup \{0\}$.

Let $A$ be a closed, densely defined operator in a complex Banach space
$E$. We say that $A$ is of  type $\varphi$ with bound $L$ if for every
$\lambda \in \Sigma_\varphi$ the operator $A + \lambda I$ is invertible
with bounded inverse and
$$
\|(A + \lambda I)^{-1}\| \le \frac{L}{|\lambda|+1} \,.
$$

We note that, in particular, an operator of type $\varphi$ is invertible.
Moreover, an operator of type $\varphi$ with $\varphi>\pi/2$ is the
inverse of the generator of a holomorphic semigroup whose norm decays
exponentially at infinity (see Fattorini \cite[Theorem 4.2.2]{f1}).
If $A$ is of type $\varphi$ with bound $L$, then for every
$\lambda\in\Sigma_\varphi$,
$$
\|A(A + \lambda I)^{-1}\| = \|I - \lambda(A + \lambda I)^{-1}\| \le
1 + \frac{L|\lambda|}{|\lambda|+1} \le 1+L \,.
$$

We remark that, in view of the properties of the resolvent operator, if
$A + \lambda I$ is invertible for $\lambda\in [0,\infty)$ and
$\sup_{\lambda\in [0,\infty)}\|(\lambda+1)(A + \lambda I)^{-1}\|<
\infty$,
then there exists $\varphi>0$ such that $A$ is of type $\varphi$.



\begin{lemma} [{\cite[Lemma 5.4.2/6]{y1}}] \label{lmA2} 
 Let $A$ be a closed, densely defined operator in a Banach space $E$  and
$$
\| R(\lambda,A) \| \le C(1+|\lambda|)^{-1}, \quad |\arg \lambda|\ge 
\pi-\varphi,
$$
where $R(\lambda,A):=(\lambda I-A)^{-1}$ is the  resolvent
of the  operator $A$ and $ 0\le \varphi <\pi$.
Then,
\begin{enumerate}
\item[a)]
  For $|\arg\lambda| \le\varphi$,  $\alpha \in \mathbb{R}$  there exist
fractional powers $A^{\alpha}$ and $(A+\lambda I)^{\alpha}$
for  $0\le \alpha \le \beta$ with
$$
\| A^{\alpha} (A+\lambda I)^{-\beta} \| \le C (1+|\lambda|)^{\alpha-\beta},
\quad  |\arg\lambda|\le\varphi ;
$$

\item[b)]
 For $|\arg\lambda| \le\varphi$ there exists the semigroup
$\hbox{\rm e}^{-x (A+\lambda I)^{1/2} }$, which is  holomorphic for
$x>0$ and strongly continuous for $x\ge 0$; moreover,  for 
$\alpha \in \mathbb{R}$ and for some $\omega>0$,
$$
\| (A+\lambda I)^{\alpha}\text{e}^{-x (A+\lambda I)^{1/2} } \|
\le C \hbox{\rm e}^{-\omega x |\lambda| ^{1/2} },
\quad  \ x \ge x_0>0, \; |\arg\lambda|\le\varphi.
$$
\end{enumerate}
\end{lemma}

\begin{theorem}[{ \cite[Theorem 5.4.2/1]{y1}}] \label{thmA3} 
Let $E$ be a complex Banach space, $A$ be a  closed operator in $E$ of
type $\varphi$ with bound $L$.
Moreover, let $m$ be a positive integer, $p \in (1,\infty)$ and
$\alpha \in (\frac{1}{2p},m+\frac{1}{2p})$.
Then, there exists $C$  (depending only on $L$, $\varphi$, $m$,
$\alpha$ and $p)$ such that for every
$u \in (E,E(A^m))_{\frac{\alpha}{m}-\frac{1}{2mp},p}$ and
$\lambda \in \Sigma_\varphi$,
$$
\int_{0}^{\infty} \Bigl \|(A + \lambda I)^\alpha
\hbox{\rm e}^{-x(A + \lambda I)^{\frac 12}} u \Bigr \|^p dx \\
\le C\Bigl(\|u\|_{(E,E(A^m))_{\frac{\alpha}{m}-\frac{1}{2mp},p}}^p
+|\lambda|^{p\alpha-\frac{1}{2}} \|u\|^p \Bigr) \,.
$$
\end{theorem}
The proof of this theorem can be found in \cite{d1}.



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\end{thebibliography}

\noindent\textsc{Sasun Yakubov} \\
Department of Mathematics, University of Haifa, \\
Haifa 31905, Israel \\
e-mail:  rsmaf06@mathcs.haifa.ac.il
\end{document}