\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 $$\mathcal{M}=\begin{pmatrix} a_{11}\Delta & a_{12}\Delta\\ a_{21}\Delta & a_{22}\Delta \end{pmatrix} \label{e1.1}$$ 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 $$\mathcal{M}=\begin{pmatrix} a_{11}\Delta & \dots & a_{1n}\Delta\\ \vdots & \vdots & \vdots\\ a_{n1}\Delta & \dots & a_{nn}\Delta \end{pmatrix} \label{e1.2}$$ 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 $$A(x,D)=\begin{pmatrix} A_{11}(x,D) & A_{12}(x,D)\\ A_{21}(x,D) & A_{22}(x,D) \end{pmatrix} \label{e1.3}$$ where every element $A_{hl}$ is a symmetric second order differential operator given by $$A_{hl}(x,D)u=-{\sum_{j,k=1}^N}\frac{\partial}{\partial x_{j} }\Big( a_{jk}(x)\frac{\partial u}{\partial x_{k}}\Big) \label{e1.4}$$ 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 $$A(x,D)=\begin{pmatrix} A_{11}(x,D) & A_{12}(x,D)\\ A_{21}(x,D) & A_{22}(x,D) \end{pmatrix} \label{e2.1}$$ where $$A_{hl}(x,D)u=-{\sum_{j,k=1}^N}\frac{\partial}{\partial x_{j}}\Big( a_{jk}^{hl}(x)\frac{\partial u}{\partial x_{k}}\Big)\quad x\in \overline{\Omega},\; h,l=1,2 \label{e2.2}$$ under the following assumptions: \begin{itemize} \item[(H1)] The operators $A_{hl}$ $(h, l=1, 2)$ are symmetric; i.e., $$a_{kj}^{hl}(x)=a_{jk}^{hl}(x),\quad x\in\overline{\Omega}, \text{ for all } j,\; k=1,\dots N \label{e2.3}$$ \item[(H2)] The operators $A_{hl}$ $(h=1,2)$ are regular; i.e., $$a_{jk}^{hl}(x)\in C^1( \overline{\Omega};\mathbb{R}) ,\quad h,l=1,2\text{ and }j,k=1,\dots N \label{e2.4}$$ \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 $x\in\Omega$, $${\sum_{j,k=1}^N}a_{jk}^{mm}(x)\xi_{j}\xi_{k}\geq\mu{\sum _{j=1}^N}\xi_{j}^2=\mu| \xi| ^2,\text{ }m=1\text{ or }m=2 \label{e2.5}$$ \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 $x\in\Omega$ $${\sum_{j,k=1}^N}a_{jk}^{ll}(x)\xi_{j}\xi_{k}\geq0, \label{e2.6}$$ \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 $x\in\Omega$ $${\sum_{j,k=1}^N}( a_{jk}^{12}+a_{jk}^{21}) (x)\xi_{j} \xi_{k}\geq0. \label{e2.7}$$ \end{itemize} We give now some definitions which will be used in the sequel. We define the operator $A$ with domain $$D( A) =( H^2(\Omega)\cap H_0^1(\Omega)) ^2, \label{e2.8}$$ as $$Au\equiv \begin{pmatrix} A_{11} & A_{12}\\ A_{21} & A_{22} \end{pmatrix} u=A(x,D)u\equiv \begin{pmatrix} A_{11}(x,D)u_{1}+A_{12}(x,D)u_{2}\\ A_{21}(x,D)u_{1}+A_{22}(x,D)u_{2} \end{pmatrix} \label{e2.9}$$ 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 $$D( A_{hl}) =H^2(\Omega)\cap H_0^1(\Omega) \label{e2.10}$$ and defined by $$A_{hl}u=A_{hl}(x,D)u \label{e2.11}$$ 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 $u=(u_{1},\dots,u_{n})\in L_{n}^p(\Omega)$: $$\varphi(u)={\sum_{j=1}^n}\langle u_{j},vj\rangle \label{e2.12}$$ 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 $$J(x)=\{ x^{\ast}\in X^{\ast}: \langle x^{\ast},x\rangle =\| x\| ^2=\| x^{\ast }\| ^2 \} \label{e2.13}$$ \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 $$\lambda\| u\| \leq\| ( \lambda I+A)u\| \label{e3.1}$$ \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 $$\langle Au,u^{\ast}\rangle =\langle A_{11}u_{l},u_{1}^{\ast }\rangle +\langle A_{22}u_{2},u_{2}^{\ast}\rangle +\langle A_{12}u_{2},u_{1}^{\ast}\rangle +\langle A_{21} u_{1},u_{2}^{\ast}\rangle \label{e3.2}$$ 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 $$\langle A_{hh}u_{h},u_{h}^{\ast}\rangle = \int_{\Omega} \sum_{j,k=1}^N a_{jk}^{hh}(x)(\alpha_{hk}\alpha_{hj}+\beta_{hk}\beta_{hj})dx,\quad h=1,2\,. \label{e3.3}$$ Also, integrating by parts we have \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} and \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} Then substituting \eqref{e3.3}--\eqref{e3.5} into \eqref{e3.2} yields \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} Set $$| \alpha_{h}| ^2= \sum_{j=1}^N \int_{\Omega} \alpha_{hj}^2dx, \quad \ | \beta_{h}| ^2=\sum_{j=1}^N\int_{\Omega} \beta_{hj}^2dx,\quad h=1,\text{ }2 \label{e3.7}$$ 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 $$\operatorname{Re}\langle Au,u^{\ast}\rangle \geq2\mu( \sum_{h=1}^2 | \alpha_{h}| ^2+ \sum_{h=1}^2 | \beta_{h}| ^2) \geq0 \label{e3.8}$$ 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 $$A^{\ast}=\begin{pmatrix} A_{11}^{\ast} & A_{21}^{\ast}\\ A_{12}^{\ast} & A_{22}^{\ast} \end{pmatrix} \label{e3.9}$$ 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 $u\in D(A)$, $$\lambda\| u\| \leq\| ( \lambda I+A) ^{\ast}u\| \label{e3.10}$$ 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 $$\| ( \lambda I+A) ^{-1}\| \leq\frac{1}{\lambda },\quad \text{for all }\lambda>0 \label{e3.11}$$ 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 $$\mathcal{N}(A)=\{ \langle x^{\ast},Ax\rangle : x\in D(A),\; x^{\ast}\in X^{\ast},\; \langle x^{\ast},x\rangle =\| x\| =\| x^{\ast}\| =1\} \label{e3.12}$$ If we put $$| a_{jk}^{hl}(x)| \leq M, \quad \text{for all }h, l=1,2\text{ and } j,k=1,\dots,N \label{e3.13}$$ we get from \eqref{e3.6} that the imaginary part of $\langle Au,u^{\ast}\rangle$ $$| \operatorname{Im}\langle Au,u^{\ast}\rangle | \leq M\Big(\sum_{h=1}^2 | \alpha_{h}| ^2+ \sum_{h=1}^2 | \beta_{h}| ^2\Big) \label{e3.14}$$ and hence from \eqref{e3.8} and \eqref{e3.14}, we find that $$\frac{| \operatorname{Im}\langle Au,u^{\ast}\rangle | }{| \operatorname{Re}\langle Au,u^{\ast }\rangle | }\leq\frac{M}{2\mu} \label{e3.15}$$ 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 $$\mathcal{S}_{\theta}=\{ \lambda: | \arg\lambda| <\pi-\theta\} \label{e3.16}$$ 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 $$\| ( \lambda I+A) ^{-1}\| \leq\frac{1} {C_{\theta}| \lambda| },\quad \text{for all }\lambda \in\mathcal{S}_{\theta} \label{e3.17}$$ 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. 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