\documentclass[twoside]{article} \usepackage{amssymb,amsmath} \pagestyle{myheadings} \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] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \renewcommand{\theequation}{\thesection.\arabic{equation}} \catcode@=11 \@addtoreset{equation}{section} \catcode@=12 \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 $$L_{\nu }u=\sum_{|\alpha|=m_\nu }b_{\nu \alpha}(x')D^\alpha u(x')+ \sum_{p=0}^{m_\nu -1}K_{\nu p}\frac {\partial^p u(x')}{\partial n^p}, \quad x'\in\partial G, \ \nu =1,\dots,m, \label{e1}$$ 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, $0<\theta<1$, $10$ 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 $$\frac {\partial^j u_j(y',0,\lambda)} {\partial y_r^j} =\varphi_j(y'), \quad y'\in \mathbb{R}^{r-1}, \label{e2}$$ such that the following estimate holds $$\sum_{k=0}^{\tilde\ell} \lambda^{\tilde\ell-k} \|u_j\|_{W_q^k(\mathbb{R}^r_+)} \le C\Bigl(\|\varphi_j\|_{W_q^{\tilde\ell-j-\frac 1q}(\mathbb{R}^{r-1})} +\lambda^{\tilde\ell-j-\frac 1q} \|\varphi_j\|_{L_q(\mathbb{R}^{r-1})} \Bigr), \label{e3}$$ 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 $$u_j(y_r,\lambda)=\text{e}^{-y_r(A+\lambda^2 I)^{1/2}}g_j, \label{e4}$$ 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, $$u_j(y_r,\lambda)=(-1)^j \text{e}^{-y_r(A+\lambda^2 I)^{1/2}} (A+\lambda^2 I)^{-j/2}\varphi_j. \label{e5}$$ 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 $$\lambda^{(\ell-k)q} \|u_j\|^q_{W_q^k(\mathbb{R}^r_+)} \le C\Bigl(\|\varphi_j\|_{(E,E(A^\ell))_{\frac {\ell-j}{2\ell}-\frac 1{2\ell q},q}}^q +\lambda^{(\ell-j)q- 1} \|\varphi_j\|^q_E \Bigr). \label{e6}$$ Since $$(E,E(A^\ell))_{\frac {\ell-j}{2\ell}-\frac 1{2\ell q},q}= (L_q(\mathbb{R}^{r-1}),W_q^{2\ell}(\mathbb{R}^{r-1}))_{\frac {\ell-j}{2\ell}-\frac 1{2\ell q},q}= W_q^{\ell-j-\frac 1q}(\mathbb{R}^{r-1}), \label{e7}$$ 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 $$\frac {\partial^j u(y',0,\lambda)} {\partial y_r^j}=\varphi_j(y'), \quad y'\in \mathbb{R}^{r-1}, \; j=0,\dots,m-1,\label{e8}$$ such that the following estimate holds $$\sum_{k=0}^{\tilde\ell} \lambda^{\tilde\ell-k} \|u\|_{W_q^k(\mathbb{R}^r_+)} \le C\sum_{j=0}^{m-1} \Bigl(\|\varphi_j\|_{W_q^{\tilde\ell-j-\frac 1q}(\mathbb{R}^{r-1})} +\lambda^{\tilde\ell-j-\frac 1q} \|\varphi_j\|_{L_q(\mathbb{R}^{r-1})} \Bigr), \label{e9}$$ where $m\le \tilde \ell\le \ell$. \end{theorem} \paragraph{Proof} By virtue of Lions and Magenes \cite[Theorem 1.3.2]{l1}, if $$U_j(y_r,\lambda)=\sum_{p=1}^m c_{pj}u_j(py_r,\lambda), \label{e10}$$ where $$u_j(py_r,\lambda)=(-1)^j \text{e}^{-py_r(A+\lambda^2 I)^{1/2}} (A+\lambda^2 I)^{-\frac j2}\varphi_j, \label{e11}$$ the operator $A$ is defined in the proof of Lemma \ref{lm1}, and the complex numbers $c_{pj}$ satisfy the systems $$\sum_{p=1}^m p^k c_{pj}=\begin{cases} 0, & k\ne j, \\ 1, & k=j, \end{cases} \quad k=0,\dots,m-1, \label{e12}$$ then $$\frac {\partial^k U_j(0,\lambda)} {\partial y_r^k}=\begin{cases} 0, & k\ne j, \\ \varphi_j, & k=j. \end{cases} \label{e13}$$ Consequently, the function $$u(y_r,\lambda)=\sum_{j=0}^{m-1}U_j(y_r,\lambda) =\sum_{j=0}^{m-1}\sum_{p=1}^m c_{pj}u_j(py_r,\lambda) \label{e14}$$ 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 $$L_\nu u(x',\lambda)=f_\nu(x'), \quad x'\in \partial G, \; \nu=1,\dots,m, \label{e15}$$ where $L_\nu$ are defined in (\ref{e1}), such that the following estimate holds $$\sum_{k=0}^{\tilde\ell} \lambda^{\tilde\ell-k} \|u\|_{W_q^k(G)} \le C\sum_{\nu=1}^{m} \Bigl(\|f_\nu\|_{W_q^{\tilde\ell-m_\nu-\frac 1q}(\partial G)} +\lambda^{\tilde\ell-m_\nu-\frac 1q} \|f_\nu\|_{L_q(\partial G)} \Bigr), \label{e16}$$ 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|0,$satisfying the relations $$\tilde L_\nu \tilde u=\tilde f_\nu(y',0),\quad (y',0)\in f^{(i)}(G_i\cap \partial G), \quad \nu=1,\dots,m, \label{e18}$$ where$\tilde L_\nu$are operators defined in (\ref{e17}), and for which additionally $$\frac {\partial^j\tilde u(y',0,\lambda)}{\partial y_r^j}=0, \quad j\ne m_\nu, \quad j=0,\dots,\max\{m_\nu\}-1,\quad (y',0)\in f^{(i)}(G_i\cap \partial G). \label{e19}$$ Consider for$(y',0)\in f^{(i)}(G_i\cap \partial G)the functions \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|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{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} 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 $k \in [0,\ell]$, $$\overline{W_q^{\ell}(G;L_\nu u=0,\nu= 1,\dots,m)} \Big|_{W_q^k(G)}= W_q^k(G;L_\nu u=0,m_\nu\le k-1). \label{e23}$$ \end{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, $$\overline{W_q^{\ell}(G;L_\nu u=0,\nu=1,\dots,m)} \Big|_{W_q^k(G)} \subset W_q^k(G;L_\nu u=0,m_\nu\le k-1). \label{e24}$$ 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{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} Thus, $L_\nu u=0$, $m_\nu\le k-1$, since $L_\nu u_{n}=0$. Now, we show the inverse inclusion $$W_q^k(G;L_\nu u=0,m_\nu\le k-1)\subset \overline{W_q^{\ell}(G;L_\nu u=0,\nu=1,\dots,m)} \Big|_{W_q^k(G)}.\label{e26}$$ 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 $$\lim_{n\to\infty}\|v_n-u\|_{W_{q}^{k}(G)}=0. \label{e27}$$ From (\ref{e27}) and (\ref{e25}) it follows that $$\lim_{n\to\infty}\|L_\nu v_n\|_{W_{q}^{k-m_\nu-\frac 1q} (\partial G)} =\|L_\nu u\|_{W_{q}^{k-m_\nu-\frac 1q} (\partial G)}=0, \ \ \ m_\nu\le k-1. \label{e28}$$ 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 $$L_\nu w_n=-L_\nu v_{n}, \quad m_\nu\le k-1,\label{e29}$$ and $$\|w_n\|_{W_q^{k}(G)}\le C \sum_{m_\nu \le k-1} \|L_\nu v_n \|_{W_q^{k-m_\nu -\frac 1q}(\partial G)}.\label{e30}$$ Then, from (\ref{e28}) and (\ref{e30}) it follows that $$\lim_{n\to\infty} \|w_n\|_{W_q^{k}(G)}=0. \label{e31}$$ Let $\lambda_n$ be a sequence tending to $\infty$, if with respect to $n$ $$\|L_\nu(v_n+w_n)\|_{W_q^{\ell-m_\nu -\frac 1q}(\partial G)}\le C, \; m_\nu \ge k; \label{e32}$$ and $$\lambda_n=\max_{m_\nu \ge k}\|L_\nu (v_n+w_n)\|_{W_q^{\ell-m_\nu -\frac 1q}(\partial G)}^\delta, \label{e33}$$ 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 $$\begin{gathered} L_\nu g_n=0, \quad m_\nu\le k-1,\\ L_\nu g_n=-L_\nu (v_n+w_n) \quad m_\nu \ge k. \end{gathered} \label{e34}$$ 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{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} Since $\delta>q$, (\ref{e35}) and (\ref{e32}),(\ref{e33}) imply $$\lim_{n\to\infty}\|g_n\|_{W_{q}^{k}(G)}=0. \label{e36}$$ 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 $1\pi/2$ 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}. \begin{thebibliography}{00} \frenchspacing \bibitem{d1} Dore, G. and Yakubov, S., Semigroup estimates and noncoercive boundary value problems, {\it Semigroup Forum}, {\bf 60} (2000), 93--121. \bibitem{f1} Fattorini, H. O., {\it The Cauchy Problem,}'' Addison-Wesley, Reading, Mass., 1983. \bibitem{g1} Grisvard, P., Caracterization de quelques espaces d'interpolation, {\it Arch. Rat. Mech. Anal.}, {\bf 25} (1967), 40--63. \bibitem{h1} Hayakawa, K., Interpolation by the real method preserves compactness of operators, {\it J. Math. Soc. Japan}, 21 (1969), 189--199. \bibitem{l1} Lions, J. L. and Magenes, E., {\it Non-Homogeneous Boundary Value Problems and Applications,}" v.I, Springer-Verlag, Berlin, 1972. \bibitem{s1} Seeley, R., Interpolation in $L^p$ with boundary conditions, {\it Studia Math.}, {\bf 44} (1972), 47--60. \bibitem{t1} Triebel, H., {\it Interpolation Theory. Function Spaces. Differential Operators,}" North-Holland, Amsterdam, 1978. \bibitem{y1} Yakubov, S. and Yakubov, Ya., {\it Differential-Operator Equations. Ordinary and Partial Differential Equations,}" Chapman and Hall/CRC, Boca Raton, 2000, p. 568. \end{thebibliography} \noindent\textsc{Sasun Yakubov} \\ Department of Mathematics, University of Haifa, \\ Haifa 31905, Israel \\ e-mail: rsmaf06@mathcs.haifa.ac.il \end{document}