\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {2004-Fez conference on Differential Equations and Mechanics \newline \em Electronic Journal of Differential Equations}, Conference 11, 2004, pp. 23--32.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2004 Texas State University - San Marcos.} \vspace{9mm}} \setcounter{page}{23} \begin{document} \title[\hfilneg EJDE/Conf/11 \hfil Heat equation on unbounded domains] {On the well-posedness of the heat equation on unbounded domains} \author[W. Arendt, S. Lalaoui R.\hfil EJDE/Conf/11 \hfilneg] {Wolfgang Arendt, Soumia Lalaoui Rhali} % in alphabetical order \address{Wolfgang Arendt \hfill\break Universit\"{a}t Ulm \\ Angewandte Analysis\\ D-89069 Ulm, Germany} \email{arendt@mathematik.uni-ulm.de} \address{Soumia Lalaoui Rhali\hfill\break Facult\'{e} Polydisciplinaire de Taza\\ Universit\'{e} Sidi Mohamed Ben Abdellah\\ B.P: 1223 Taza\\ Morocco} \email {slalaoui@ucam.ac.ma} \date{} \thanks{Published October 15, 2004.} \thanks{The second author gratefully acknowledges his support by the DAAD} \subjclass[2000]{35K05, 35K20, 47D06} \keywords{Unbounded domains; heat equation; Dirichlet problem; \hfill\break\indent resolvent positive operators} \begin{abstract} This work concerns the well-posedness of the heat equation in an unbounded open domain, under basic regularity assumptions on this domain. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{example}[theorem]{Example} \newtheorem{definition}[theorem]{Definition} \section{Introduction} Let $\Omega$ be an open set of $\mathbb{R}^{n}$ with boundary $\Gamma=\partial\Omega$ and consider the problem % $P_{\tau }(u_{0},\varphi)$, $$\label{Ptau} \begin{gathered} u'(t)=\Delta u(t), \quad t\in [ 0,\tau ] \\ u(t)\big|_{\Gamma }=\varphi (t), \quad t\in [ 0,\tau ] \\ u(0)=u_{0}, \end{gathered}$$ where $u_{0}\in C(\overline{\Omega })$, $\varphi \in C([0,\tau ];C(\Gamma))$, $\tau >0$. The aim of this work is to study the well-posedness of \eqref{Ptau} when $\Omega$ is unbounded. The case where $\Omega$ is bounded has been studied in \cite[Chapter 6]{ar}, and sufficient conditions on the initial data $u_{0}$ and the boundary condition $\varphi$ are given to show that the problem \eqref{Ptau} is well-posed in $C(([0,\tau ];C(\overline{\Omega }))$ whenever $\Omega$ is regular (See definition \ref{1.1}). We point out here that the regularity assumption is equivalent when $\Omega$ is bounded to that the Dirichlet problem, \eqref{Dp}, $$\label{Dp} \begin{gathered} u\in C(\overline{\Omega }) \\ \Delta u=0\quad \text{in }\mathcal{D}(\Omega )' \\ u\big|_{\Gamma}=\phi, \end{gathered}$$ has for all $\phi \in C(\Gamma )$ a classical solution $u$, that means that $u$ is a solution of \eqref{Dp} and $u\in C^{2}(\Omega )$. (See \cite{dl}, \cite{gt} for instance). The situation is more complicated when $\Omega$ is unbounded since one must take into account the condition at infinity that the solution of \eqref{Dp} satisfies (see Theorem \ref{1.2}), and the choice of the space $X\subset C(\overline{\Omega })$ in which the solution $u(t)$ of \eqref{Ptau} belongs will be imposed by this condition at infinity and then by the choice of the unbounded regular open set $\Omega$. In this work, the unbounded set is taken in the case $n=1$ as an interval of $\mathbb{R}$ and as the exterior of a ball of $\mathbb{R}^{n}$ for $n\geq 2$. When $n=2$, we deal with the heat equation with homogeneous boundary conditions and for $n\geq 3$, the heat equation with inhomogeneous boundary conditions is studied for an exterior domain. The organization of this work is as follows: In Section 2, we recall some preliminaries results lying between the regularity property for unbounded sets and the well-posedness of the Dirichlet problem \eqref{Dp}. We also recall some existence results for Cauchy problems with resolvent positive operators. We present in Section 3 our main result, the method of proof consists on reformulating \eqref{Ptau} as a Cauchy problem with the Poisson operator. Section 4 is devoted to the study of the well-posedness of this Cauchy problem, we first show that the Poisson operator has a positive resolvent in $X\times C(\Gamma )$. Using results of Section 2, we then show the well-posedness of \eqref{Ptau}. \section{Preliminaries} \subsection*{The Dirichlet Problem} Let $\Omega$ be an open set of $\mathbb{R}^{n}$ with boundary $\Gamma =\partial \Omega$. \begin{definition}[\cite{dl}] \label{1.1} \rm (a) Let $z\in \Gamma$. we say that $z$ is a regular boundary point of $\Omega$ if there exists $r>0$, and $w\in C(\overline{\Omega \cap B(z,r)})$ such that \begin{gather*} \Delta w\leq 0, \quad \text{in }\mathcal{D}(\Omega \cap B(z,r))' \\ w(x)>0, \quad x\in (\Omega \cap B(z,r))\backslash \{z\} \\ w(z)=0\,. \end{gather*} Then the function $w$ is called a barrier. \\ (b) We say that $\Omega$ is regular if all boundary points are regular. \end{definition} This regularity property is related to the Dirichlet problem \eqref{Dp} as follows. \begin{theorem}[\cite{dl}] \label{1.2} Let $\Omega$ be an unbounded set, not dense in $\mathbb{R}^{n}(n\geq 2)$ with boundary $\Gamma$. Then the following two assertions are equivalent: \\ (i) For every continuous $\phi$ with compact support in $\Gamma$, there exists a classical solution of \eqref{Dp} satisfying the following null condition at infinity \begin{itemize} \item[(NC)] There exists $h$ harmonic on $\Omega$ such that $h\in C(\overline{\Omega })$, with $h(x)>0$ for $| x|$ large so that $\lim_{|x| \to +\infty }\frac{u(x)}{h(x)}=0.$ \end{itemize} (ii) All boundary points of $\Omega$ are regular. \end{theorem} \begin{example}[\cite{dl}]\label{1.3} \rm Let $n\in \mathbb{N}^{\ast }$.\newline (a) Case of an interval of $\mathbb{R}$. Let $\Omega _{1}=]1,\infty[$, then $\Omega _{1}$ is regular and for all $\phi \in \mathbb{R}$ and all $c\in \mathbb{R}$, there exists a unique classical solution of \eqref{Dp} satisfying the condition at infinity: \begin{equation*} \lim_{x\rightarrow +\infty }\frac{u(x)}{x}=c. \end{equation*} (b) Case of the exterior of a ball of $\mathbb{R}^{n}$, $n\geq 2$. Let $\Omega _{n}=\mathbb{R}^{n}\setminus \overline{B(0,1)}$, then $\Omega _{n}$ is regular and given $u$ a bounded classical solution of \eqref{Dp}, then $% c=\lim_{\left| x\right| \rightarrow \infty }u(x)$ exists and \begin{equation*} u(x)=(1-\frac{1}{{|x|}^{n-2}})c+\frac{1}{{\sigma }_{n}}\int_{\partial B}% \frac{{|x|}^{2}-1}{{|t-x|}^{n}}\phi (t)d\gamma (t) \end{equation*} is a classical solution of \eqref{Dp}, with $\sigma _{n}$ being the total surface area of the unit sphere in $\mathbb{R}^{n}$. Conversely, the function $u$ given by the last formula is a classical solution of \eqref{Dp}% . Moreover, one has:\newline If $n=2$, then for all $\phi \in C(\partial B)$, there exists a unique classical solution of \eqref{Dp} satisfying the condition at infinity: \begin{equation*} u\text{ is bounded on }\Omega . \end{equation*} This solution will have a limit at infinity which is imposed by the giving $\phi$: $$\lim_{| x| \to \infty }u(x)=\frac{1}{2\pi }\int_{\partial B}\phi (t)d\gamma (t). \label{0}$$ If $n\geq 3$, then for all $\phi \in C(\partial B)$ and all $c\in \mathbb{R}$, there exists a unique classical solution of \eqref{Dp} satisfying the condition at infinity: \begin{equation*} \lim_{| x| \to \infty }u(x)=c. \end{equation*} \end{example} Note that a solution $u$ of \eqref{Dp} satisfying the null condition at infinity $(\mathbf{NC})$ does not necessarily satisfy: \begin{equation*} \lim_{| x| \to \infty }u(x)=0. \end{equation*} This remains true for the exterior of a compact set of $\mathbb{R}^{n}$, $n\geq 3$. \begin{proposition}[\cite{dl}]\label{1.4} Let $K$ be a compact set of $\mathbb{R}^{n}$, $n\geq 3$ with boundary $\Gamma$. If $\Omega =\mathbb{R}^{n}\setminus K$ is regular, then for all $\phi \in C(\Gamma)$, there exists a unique classical solution of \eqref{Dp} satisfying the condition at infinity: \begin{equation*} \lim_{| x| \to \infty }u(x)=0. \end{equation*} \end{proposition} \subsection*{Cauchy Problem} Let $X$ be a Banach space and consider the inhomogeneous Cauchy Problem: %(ACP_{f}) $$\label{ACPf} \begin{gathered} u'(t)=Au(t)+f(t),\quad t\in [ 0,\tau ] \\ u(0)=u_{0}, \end{gathered}$$ where $u_{0}\in X$ and $f\in C([0,\tau ];X)$. \begin{definition} \label{1.5} \rm A mild solution of $(ACP_{f})$ is a function $u\in C([0,\tau ];X)$ such that $\int_{0}^{t}u(s)ds\in D(A)$ and for all $t\in [ 0,\tau ]$, \begin{equation*} u(t)=u_{0}+A\int_{0}^{t}u(s)ds+\int_{0}^{t}f(s)ds\,. \end{equation*} \end{definition} We recall now some results on resolvent positive operators and Cauchy problems, we refer to \cite[Chapter 3]{ar}, for more details. \begin{theorem}[\cite{ar}] \label{1.6} Let $A$ be a resolvent positive operator on a Banach lattice $X$, that means, there exists $w\in \mathbb{R}$ such that $(w,\infty )\subset \rho (A)$ and $R(\lambda ,A)\geq 0$ for all $\lambda >w$. \\ (i) Let $u_{0}\in D(A)$,$\;f_{0}\in X$ such that $Au_{0}+f_{0}\in \overline{D(A)}$. Let $f(t)=f_{0}+\int_{0}^{t}f'(s)ds$ where $f'\in L^{1}((0,\tau );X)$. Then $(ACP_{f})$ has a unique mild solution. \\ (ii) Let $f\in C([0,\tau ];X_{+})$, $u_{0}\in X_{+}$ and let $u$ be a mild solution of $(ACP_{f})$. Then $u(t)\geq 0$ for all $t\in [ 0,\tau ]$. \end{theorem} Define now the Gaussian semigroup $(G(t)) _{t\geq 0}$ on the space $C_{0}(\mathbb{R}^{n})$ of all continuous functions vanishing at infinity by: \begin{equation*} G(t)f(x)=(4\pi t) ^{-n/2}\int_{\mathbb{R}^{n}}f(x-y)e^{-|y|^{2}/(4t)}dy, \quad t>0,x\in \mathbb{R}^{n},f\in C_{0}(\mathbb{R}^{n}). \end{equation*} \begin{theorem}[\cite{ar}] \label{1.7} The family $(G(t)) _{t\geq 0}$ defines a bounded holomorphic $C_{0}-$semigroup of angle $\frac{\pi }{2}$ on $C_{0}( \mathbb{R}^{n})$. Its generator is the Laplacian $\Delta _{G}$ on $C_{0}( \mathbb{R}^{n})$ with maximal domain; i.e., \begin{gather*} D(\Delta _{G})=\{ f\in C_{0}(\mathbb{R}^{n}),\;\Delta f\in C_{0}( \mathbb{R}^{n})\} , \\ \Delta _{G}f=\Delta f, \end{gather*} here one identifies $C_{0}(\mathbb{R}^{n})$ with a subspace of $\mathcal{D}(\mathbb{R}^{n})'$. \end{theorem} \begin{proposition}[\cite{ar}]\label{1.8} Let $A$ be the generator of a bounded $C_{0}$-group $(U(t)) _{t\in \mathbb{R}}$ on $X$. Then $A^{2}$ generates a bounded holomorphic $C_{0}$-semigroup $(T(t)) _{t\geq 0}$ of angle $\frac{\pi }{2}$ on $X$. Moreover, for $t>0$, \begin{equation*} T(t)=(4\pi t)^{-1/2}\int_{\mathbb{R}}e^{-| y| ^{2}/(4t)} U(y)\,dy. \end{equation*} \end{proposition} \section{Main result} We consider the problem \eqref{Ptau} with $\Omega$ presenting the cases in Example \ref{1.3} and Proposition \ref{1.4}. Since in the case $n=2$, the condition at $\inf$inity (\ref{0}) is imposed by the boundary function, we restrict our study of \eqref{Ptau} for $n=2$ to the case where $\varphi =0.$ \begin{theorem} \label{1.9} Let $n\in \mathbb{N}$. \textbf{Case $n=1$:} Let $\Omega _{1}=] 1,+\infty [$ with boundary $\Gamma _{1}=\{ 1\}$ and denote by $\big(C_{\infty }(\overline{\Omega }_{1});\| .\| _{C_{\infty } (\overline{\Omega }_{1})}\big)$ the Banach space \begin{equation*} C_{\infty }(\overline{\Omega }_{1}):=\big\{ u\in C([ 1,+\infty [ ),\lim\limits_{x\rightarrow +\infty }\frac{u(x)}{x}\text{ exists}\big\} \end{equation*} with the norm $\| u\| _{C_{\infty }(\overline{\Omega } _{1})}=\max_{x\in [ 1,\infty [ }| u(x)/x|$. Then for all $u_{0}\in$ $C_{\infty }(\overline{\Omega }_{1})$ and all $\varphi \in C([ 0,\tau ] )$ such that $u_{0}(1)=\varphi (0),$ there exists a unique mild solution $u\in C([0,\tau ];C_{\infty }(\overline{\Omega }_{1}))$ of the problem $$\label{Ptau1} \begin{gathered} u_{t}(t,x)=u''(t,x), \quad t\in [ 0,\tau ],\; x\in ] 1,+\infty [ \\ u(t,1)=\varphi (t), \quad t\in [ 0,\tau ] \\ u(0,x)=u_{0}(x). \end{gathered}$$ % \textbf{Case $n=2$:} Let $\Omega _{2}=\mathbb{R}^{2}\setminus B(0,1)$ with boundary $\Gamma _{2}=\partial B$ and set $$C_{\infty }(\overline{\Omega}_{2}):=\{ u\in C(\overline{\Omega }_{2}), \text{ }u\big|_{ \Gamma _{2}}=0 \text{ and }\lim_{| x| \to +\infty }u(x)=0\}$$ with the supremum norm$\| u\| _{C_{\infty }(\overline{\Omega } _{2})}=\max_{x\in \Omega _{2}}| u(x)|$. Then for all $u_{0}\in$ $C_{\infty }(\overline{\Omega }_{2})$, there exists a unique mild solution $u\in C([0,\tau ];C_{\infty }(\overline{\Omega }_{2}))$ of the problem $$\label{Ptau2} \begin{gathered} u'(t)=\Delta u(t), \quad t\in [ 0,\tau ] \\ u\big|_{ \Gamma _{2}}=0, \\ u(0)=u_{0}. \end{gathered}$$ \textbf{Case $n\geq 3$:} Let $\Omega _{n}=\mathbb{R}^{n}\setminus B(0,1)$ or more generally $\Omega _{n}=\mathbb{R}^{n}\setminus K$ with $K$ being a compact set of $\mathbb{R}^{n}$ with boundary $\Gamma _{n}$, and set $$C_{\infty }(\overline{\Omega }_{n}):=\{ u\in C(\overline{\Omega }_{n}), \;\lim_{| x| \to +\infty }u(x)=0\}$$ with the supremum norm. If $\Omega _{n}$ is regular, then for all $u_{0}\in C_{\infty }(\overline{\Omega }_{n})$ and all $\varphi \in C([ 0,\tau] ;C(\Gamma _{n}))$ such that $u_0\big|_{\Gamma_{n}}=\varphi (0)$, there exists a unique mild solution $u\in C([0,\tau];C_{\infty }(\overline{\Omega }_{n}))$ of the problem $$\label{Ptaun} \begin{gathered} u'(t)=\Delta u(t), \quad t\in [ 0,\tau ] \\ u(t)\big|_{ \Gamma _{n}}=\varphi (t), \quad t\in [ 0,\tau ] \\ u(0)=u_{0}. \end{gathered}$$ \end{theorem} Let $\Omega _{n}$, $n\geq 1$ be defined as in Theorem \ref{1.9} and define the operator $\Delta _{\max }^{n}$ on $C_{\infty }(\overline{\Omega }_{n})$ as follows \begin{gather*} D(\Delta _{\max }^{n})=\{ u\in C_{\infty }(\overline{\Omega } _{n}),\Delta u\in C_{\infty }(\overline{\Omega }_{n})\} \\ \Delta _{\max }^{n}u=\Delta u \quad \text{in }\mathcal{D}(\Omega_{n})'. \end{gather*} We mean by mild solution of \eqref{Ptaun} a function $u\in C([0,\tau ];C_{\infty }(\overline{\Omega }_{n}))$ such that $\int_{0}^{t}u(s)ds\in D(\Delta _{\max }^{n})$ and for all $t\in [ 0,\tau]$, \begin{gather*} u(t)=u_{0}+\Delta \int_{0}^{t}u(s)ds\quad\text{in }\mathcal{D}(\Omega_{n})' \\ u(t)\big|_{\Gamma _{n}}=\varphi (t). \end{gather*} To prove Theorem \ref{1.9}, we will reformulate the problem \eqref{Ptaun} as an inhomogeneous Cauchy problem with resolvent positive operator. \section{Inhomogeneous Cauchy Problem} Define for $n\geq 1$ the Poisson operators $A_{n}$ with domain $D(A_{n})=D(\Delta _{\max }^{n})\times \{ 0\}$ by \begin{gather*} A_{1}(u,0)=(\Delta u,-u(1)), \\ A_{2}(u,0)=(\Delta u,0), \\ A_{n}(u,0)=(\Delta u,-u\big|_{\Gamma _{n}}),\quad n\geq 3, \end{gather*} and consider the Cauchy problem $$\label{ACPn} \begin{gathered} U'(t)=A_{n}U(t)+\Phi _{n}(t),\quad t\in [0,\tau ] \\ U(0)=U_{0}, \end{gathered}$$ where $U_{0}=(u_{0},0)$, $u_{0}\in C_{\infty }(\overline{\Omega}_{n})$ is the initial data , $\Phi _{2}=(0,0)$ and for $n\neq 2$, $\Phi _{n}(t)=(0,\varphi (t)), \varphi \in C([0,\tau ];C(\Gamma _{n}))$ is the boundary condition. \begin{proposition} \label{1.10} Let $n\geq 1$ and $U\in C([ 0,\tau ] ;C_{\infty }(\overline{\Omega }_{n})\times C(\Gamma _{n}))$. Then $U$ is a mild solution of \eqref{ACPn} if and only if $U(t)=(u(t),0)$ where $u\in C([0,\tau ];C_{\infty }(\overline{ \Omega }_{n}))$ is the mild solution of \eqref{Ptau}. \end{proposition} The proof is immediate from the definition of $A_{n}$ and the fact that $\overline{D(A_{n})}=C_{\infty }(\overline{\Omega }_{n})\times \{0\}$. To show the well-posedness of \eqref{ACPn}, we first prove that $A_{n}$ is a resolvent positive operator. \begin{theorem} \label{1.11} Let $\lambda >0$, if $n\neq 2$ then for all $(f,\phi) \in C_{\infty }(\overline{\Omega }_{n})\times C(\Gamma _{n})$ there exists a unique function $u\in D(\Delta _{\max }^{n})$ such that $$\begin{gathered} (\lambda -\Delta )u=f\quad\text{in }\mathcal{D}(\Omega _{n})'\\ u\big|_{\Gamma _{n}}=\phi . \end{gathered} \label{a1}$$ Moreover, if $f\leq 0$, $\phi \leq 0$, then $u\leq 0$. If $n=2$, then for all $f\in C_{\infty }(\overline{\Omega }_{2})$, there exists a unique function $u\in D(\Delta _{\max }^{2})$ such that $$\begin{gathered} (\lambda -\Delta )u=f\;\;\quad\text{in }\mathcal{D}(\Omega _{2})'\\ u\big|_{ \Gamma _{2}}=0. \end{gathered} \label{a2}$$ Moreover, if $f\leq 0$, then $u\leq 0$. \end{theorem} \begin{proof} (1) \textbf{Existence.} (a) Case $n=1:$ \newline Set $C_{\infty }(\mathbb{R})=\{ f\in C(\mathbb{R}),\;\lim_{x \to -\infty }f(x)=\text{0 and }\lim_{x\to +\infty } \frac{f(x)}{x}\text{ exists}\}$ and define on $C_{\infty }(\mathbb{R})$ the translation group \begin{equation*} T(t)f(x)=f(x-t),\quad t\in \mathbb{R},\;x\in \mathbb{R.} \end{equation*} Then $(T(t))_{t\in \mathbb{R}}$ is a $C_{0}-$group with generator $A_{T}$ defined by \begin{gather*} D(A_{T})=\{ f\in C_{\infty }(\mathbb{R}),f'\in C_{\infty }( \mathbb{R})\} \\ A_{T}f=f'. \end{gather*} It follows from Proposition \ref{1.8} that $A_{T}^{2}$ generates a $C_{0}-$semigroup $(G(t))_{t\geq 0}$ which is the Gaussian semigroup: \begin{equation*} G(t)f(x)=(4\pi t)^{-1/2}\int_{\mathbb{R}}e^{-| y|^{2}/(4t)}f(x-y)dy. \end{equation*} Moreover, $G(t)C_{\infty }(\overline{\Omega }_{1})\subset C_{\infty }( \overline{\Omega }_{1})$ for all $t\geq 0$. Let $\lambda >0$ and $(f,\phi ) \in C_{\infty }(\overline{\Omega }_{1})\times \mathbb{R}$, and take \begin{gather*} v_{0}(x)=\int_{0}^{+\infty }e^{-\lambda t}G(t)f(x)dt, \\ v(x)=(\phi -v_{0}(1))e^{-\sqrt{\lambda }(x-1)}, \end{gather*} Then $u=v+v_{0}$ is a solution of (\ref{a1}). \\ (b) Case $n\geq 2$: Let $f\in C_{\infty }(\overline{\Omega }_{n})$. Then $f$ can be extended to $C_{0}(\mathbb{R}^{n})$. Since the Gaussian semigroup generates an holomorphic $C_{0}-$semigroup on $C_{0}(\mathbb{R}^{n})$, we get that \begin{equation*} v_{0}(x)=\int_{0}^{+\infty }e^{-\lambda t}G(t)f(x)dt, \end{equation*} is a solution of $$(\lambda -\Delta )v=f\text{ for all }\lambda >0. \label{a3}$$ Moreover, if $f\in C_{\infty }(\overline{\Omega }_{n})$, then $v_{0}\in C_{\infty }(\overline{\Omega }_{n})$. \\ If $n=2$, then $v_{0}$ is a solution of $(\lambda -A_{2})(u_{0},0)=(f,0).$ \\ If $n\geq 3$, it remains to show that there exists a solution of $$\begin{gathered} (\lambda -\Delta )v=0, \quad\text{in }\mathcal{D}(\Omega_{n})'\\ v\big|_{ \Gamma _{n}}=\phi -v_0\big|_{ \Gamma _{n}}=:\psi . \end{gathered} \label{a4}$$ Let $\Omega _{nk}=\Omega _{n}\cap B(0,R_{k})\;$where $(R_{k}) _{k\geq 1}$ is an increasing sequence of positif reals such that $R_{k}\to \infty$ as $k\to \infty \;$and consider the following problem on $C(\overline{\Omega }_{nk})$. $$\begin{gathered} (\lambda -\Delta )v_{k}=0 \quad\text{in }\mathcal{D}(\Omega_{nk})' \\ v_k\big|_{ \Gamma _{k}}=0\quad \text{on }\Gamma_{k}=\partial B(0,R_{k}) \\ v_k\big|_{ \Gamma _{n}}=\psi . \end{gathered} \label{a5}$$ Since $\Omega _{n}$ is regular, $\Omega _{nk}$ is regular and it follows from \cite{lsw}, \cite{st} that (\ref{a5}) has a solution$v_{k}\in C(\overline{ \Omega }_{nk})$. Our aim now is to show that the sequence $(v_{k}) _{k\geq 1}$ converges to the solution of (\ref{a4}), for that, we use the following maximum principle due to \cite{ar}. \begin{theorem}[Maximum Principle for distributional solutions] \label{1.12} Let $\Omega _{0}$ be a bounded open set of $\mathbb{R}^{n}$ with boundary $\Gamma$. Let $M\geq 0$, $\lambda \geq 0$, $u\in C(\overline{\Omega }_{0})$ such that\newline $(i)$ $\lambda u-\Delta u\leq 0$, in $\mathcal{D}(\Omega _{0})'$\newline $(ii)$ $u\big|_{ \Gamma }\leq M$,\newline Then $u\leq M$ on $\overline{\Omega }_{0}$. \end{theorem} Without loss of generality, we can assume that $\psi \geq0$.\newline \textbf{Claim 1}: $(v_{k}) _{k\geq 1}$ is an increasing bounded sequence. Indeed, by applying the Maximum principle in $\Omega _{nk}$ to $v_{k}$ and $v_{k}-v_{k+1}$ respectively, we obtain$:$\begin{equation*} 0\leq v_{k}\leq \| \psi \| , \end{equation*} and \begin{gather*} (\lambda -\Delta )(v_{k}-v_{k+1})=0,\;\;\text{in }\mathcal{D}(\Omega _{nk})' \\ (v_{k}-v_{k+1})\big|_{ \Gamma _{k}}=-v_{k+1}\leq 0, \\ (v_{k}-v_{k+1})\big|_{ \Gamma_{n}}=0. \end{gather*} Hence $v_{k}\leq v_{k+1}\text{ in }\Omega _{nk}$. \noindent\textbf{Claim 2:} Let $v=\lim_{k\to \infty }v_{k}$, then $v\in C_{\infty }(\overline{\Omega }_{n})$. Indeed, denote by $w_{k}\;$the solution of the problem \begin{gather*} \Delta w_{k}=0, \quad\text{in }\mathcal{D}(\Omega_{nk})' \\ w_k\big|_{ \Gamma _{k}}=0, \\ w_k\big|_{\Gamma _{n}}=\psi . \end{gather*} Then $w_{k}\geq 0$. Define the Poisson operator $B_{k}$ on $C(\overline{\Omega }_{nk})\times C(\Gamma _{n}\cup \Gamma _{k})$ by \begin{gather*} D(B_{k})=\{ w\in C(\overline{\Omega }_{nk}),\Delta w\in C(\overline{ \Omega }_{nk})\} \times \{0\}, \\ B_{k}(w,0)=(\Delta w,-(w\big|_{ \Gamma _{n}},w\big|_{ \Gamma _{k}})). \end{gather*} Since $\Omega _{nk}$ is regular, we deduce from \cite[Chapter 6]{ar}, that $B_{k}$ is a resolvent positive operator and then $$(w_{k},0) = R(\lambda ,B_{k})(\lambda w_{k},(\psi ,0)) \geq R(\lambda ,B_{k})(0,(\psi ,0)) = (v_{k},0). \label{a6}$$ On the other hand, it follows from Proposition \ref{1.4} that for all $\Phi \in C(\Gamma _{n})$, the Dirichlet problem \eqref{Dp}(with $\phi=\Phi$) has a unique solution $w$ satisfying the condition at infinity $\lim_{| x| \to \infty }w(x)=0$. Moreover, if $\Phi \leq 0$, then $w\leq 0.$ Indeed, let $\varepsilon >0$, since $w\in C^{1}(\Omega _{n})$, there exists $\Omega _{0}\subset \subset \Omega _{n}$ such that $\mathop{\rm supp}(w-\varepsilon ) ^{+}\subset \Omega _{0}$, Thus $(w-\varepsilon) ^{+}\in H_{0}^{1}(\Omega _{0})$ and $\int_{\{ w>\varepsilon \} }| \nabla w| ^{2}=0$. Hence, \begin{equation*} w\leq \varepsilon . \end{equation*} Denote by $w_{0}$ the solution of \eqref{Dp} (with $\phi=\psi$) vanishing to zero at infinity$,$ then \begin{gather*} \Delta (w_{k}-w_{0})=0,\;\;\quad\text{in }\mathcal{D}(\Omega _{nk})'\\ (w_{k}-w_{0})\big|_{ \Gamma _{k}}=-w_{0\mid \Gamma _{k}}\leq 0, \\ (w_{k}-w_{0})\big|_{ \Gamma _{n}}=0. \end{gather*} Theorem \ref{1.12} and (\ref{a6}) imply that $0\leq v_{k}\leq w_{k}\leq w_{0}$. Hence \begin{equation*} \lim_{| x| \to \infty }v(x)=\lim_{| x|\to \infty }w_{0}(x)=0. \end{equation*} Finally, $u=v_{0}+v$ is a solution of (\ref{a1}). \\ (2)\textbf{Positivity and Uniqueness.} Let $(f,\phi ) \in C_{\infty }(\overline{\Omega }_{n})\times C(\Gamma _{n})$ such that $f\leq 0$, $\phi \leq 0$ and $u$ a solution of (\ref{a1}).\\ Case $n=1$: Since in that case $u\in C^{2}(\Omega _{1})\cap C(\overline{\Omega }_{1})$, we apply the Phragm\{e}n-Lindel\"{o}f principle to deduce that $u\leq 0$ whenever $f\leq 0$, $\phi \leq 0$. (See \cite[Chapter 2]{pw}). By applying this maximum principle to $u$ and $-u$ respectively when $f=0$, we get uniqueness. \newline Case $n\geq 2$: Since $u\in D(\Delta _{\max }^{n}),$ we get $u\in C^{1}(\Omega _{n})$. Let $\Omega _{0}\subset \subset \Omega _{n}$ such that $\mathop{\rm supp}(u-\varepsilon ) ^{+}\subset \Omega _{0}$, $\varepsilon >0$. then $(u-\varepsilon ) ^{+}\in H_{0}^{1}(\Omega _{0})$ and \begin{align*} \int f(u-\varepsilon ) ^{+} & = \lambda \int u(u-\varepsilon ) ^{+}+\int \nabla u\nabla (u-\varepsilon )^{+} \\ & = \lambda \int (u-\varepsilon ) (u-\varepsilon ) ^{+}+\varepsilon \lambda \int (u-\varepsilon ) ^{+} +\int_{\{ u>\varepsilon \} }| \nabla u| ^{2} \\ & \leq 0. \end{align*} Hence $u\leq \varepsilon$. \end{proof} We are now in position to show the well-posedness of the Cauchy problem \eqref{ACPn}. If $n\neq 2,$ let $\varphi \in W^{1,1}((0,\tau );C(\Gamma _{n}))$ and $U_{0}=(u_{0},0)\in D(A_{n})=D(\Delta _{\max }^{n})\times \{0\}$, then \begin{equation*} A_{n}U_{0}+\Phi _{n}(0)=(\Delta u_{0},-u_{0\mid \Gamma _{n}}+\varphi (0)). \end{equation*} Hence $A_{n}U_{0}+\Phi _{n}(0)\in \overline{D(A_{n})}=C_{\infty }(\overline{ \Omega }_{n})\times \{0\}$ if and only if $$u_0\big|_{\Gamma _{n}}=\varphi (0). \label{a7}$$ Assumption (\ref{a7}) becomes trivial in the case $n=2$ since we have assumed $\varphi =0$. $\newline$On the other hand, it follows from Theorem \ref{1.11} that $A_{n}$ is a resolvent positive operator. Hence, by applying Theorems \ref{1.6} we obtain the following result. \begin{proposition} \label{1.13} Let $n\in \mathbb{N}$.\newline \textbf{Case $n=1$:} Let $\Omega _{1}=] 1,+\infty [.$ Then for all $u_{0}\in D(\Delta _{\max }^{1})$ and all $\varphi \in W^{1,1}((0,\tau ))$ such that $u_{0}(1)=\varphi (0)$, there exists a unique mild solution of \eqref{ACPn} with $n=1$. \newline \textbf{Case $n=2$:} Let $\Omega _{2}=\mathbb{R}^{2}\setminus B(0,1)$. Then for all $u_{0}\in$ $D(\Delta _{\max }^{2})$, there exists a unique mild solution of \eqref{ACPn} with $n=2$. \newline \textbf{Case $n\geq 3$:} Let $\Omega _{n}=\mathbb{R}^{n}\setminus K$ with boundary $\Gamma _{n}$, $K$ being a compact set of $\mathbb{R}^{n}$. If $\Omega _{n}$ is regular, then for all $u_{0}\in$ $D(\Delta _{\max }^{n})$ and all $\varphi \in W^{1,1}((0,\tau );C(\Gamma _{n}))$ such that $u_0\big|_{ \Gamma_{n}}=\varphi (0)$, there exists a unique mild solution of \eqref{ACPn}. \end{proposition} The proof of Theorem \ref{1.9} will be complete by combining Theorem \ref {1.10} and the following result. \begin{proposition} \label{1.14} (i) Let $u_{0}\in C_{\infty }(\overline{\Omega }_{1})$ and $\varphi \in C([0,\tau ])$ such that $u_{0}(1)=\varphi (0)$, then there exists a unique mild solution of \eqref{ACPn} with $n=1$.\newline (ii) Let $u_{0}\in C_{\infty }(\overline{\Omega }_{2})$, then there exists a unique mild solution of \eqref{ACPn} with $n=2$.\newline (iii) Assume that $\Omega _{n}=\mathbb{R}^{n}\setminus K$ is regular and let $u_{0}\in C_{\infty }(\overline{\Omega }_{n})$ and $\varphi \in C([0,\tau ];C(\Gamma _{n}))$ such that $u_0\big|_{ \Gamma n}=\varphi (0)$, then there exists a unique mild solution of \eqref{ACPn}. \end{proposition} \begin{proof} Choose $u_{0_{k}}\in D(\Delta _{\max }^{n})$ such that $\,u_{0_{k}} \to u_{0}$ as $k\to \infty$ in $C_{\infty }(\overline{ \Omega }_{n})$. Choose $\varphi _{k}\in W^{1,1}((0,\tau);C(\Gamma _{n}))$ such that $\varphi _{k}(0)=u_{0_k}\big|_{\Gamma _{n}}$ and $\varphi _{k}\to \varphi$as $k\to \infty$ in $C([0,\tau ];C(\Gamma_{n} ))$. By applying Proposition \ref{1.13} and Theorem \ref{1.10}, we deduce that there exists a unique mild solution $u_{k}\in C_{\infty }(\overline{\Omega }_{n})$ of $P_{\tau }(u_{0_{k}},\varphi _{k})$. We can show that $$\| u_{k}\| _{C([0,\tau ];C_{\infty }(\overline{\Omega }_{n}))}\leq \max \{ \| \varphi _{k}\| _{C([0,\tau ];C(\Gamma _{n}))},\| u_{0k}\| _{C_{\infty }(\overline{\Omega }_{n})}\} .$$ where $$\left\| \varphi _{k}\right\| _{C([0,\tau ];C(\Gamma _{n}))}=\sup_{0\leq t\leq \tau}\left\| \varphi _{k}(t)\right\| _{C(\Gamma _{n})}$$ $$\left\| u_{k}\right\| _{E_{\infty }(\overline{\Omega }_{n})}=\sup_{0\leq t\leq \tau}\left\| u_{k}(t)\right\| _{E_{\infty }(\overline{\Omega }_{n})}.$$ Hence $(u_{k})_{k\geq 1}$ is a Cauchy sequence in $C([0,\tau ];C_{\infty }(\overline{\Omega }_{n}))$. Let $u=\lim_{k\to \infty }u_{k}$, then $\int_{0}^{t}u(s)ds=\lim_{k\to \infty }\int_{0}^{t}u_{k}(s)ds \in D(\Delta _{\max }^{n})$ and \begin{gather*} u(t)=u_{0}+\Delta \int_{0}^{t}u(s)ds\quad\text{in }\mathcal{D}(\Omega _{n})' \\ u(t)\big|_{ \Gamma _{n}}=\lim_{k\to \infty }\varphi _{k}(t)=\varphi (t). \end{gather*} for all $t\in [ 0,\tau ]$. \end{proof} \begin{thebibliography}{99} \bibitem{ar1} W. Arendt: Resolvent positive operator and integrated semigroups, Proc. London Math. Soc, 54, (1987), 321-349. \bibitem{ar} W. Arendt, C.J.K. Batty, M. Hieber, F. Neubrander: Vector valued Laplace transforms and Cauchy Problems. Monographs in Mathematics. Birkh\"{a}user Verlag Basel. 2001. \bibitem{ar5} W. Arendt, Ph. B\'{e}nilan, Wiener regularity and heat semigroups on spaces of continuous functions, Progress in Nonlinear Differential Equations and Applications. Escher, Simonett, eds., Birkh\"{a} user, Basel (1998), 29-49. \bibitem{b} H. Brezis: Analyse Fonctionnelle Masson, Paris 1983. \bibitem{dl} R. Dautray, J.L. Lions: Mathematical Analysis and Numerical Methods for Science and Technology. Springer Verlag 1999. \bibitem{ds} G. Da Prato, E. Sinestrari: Differential operators with non-dense domain. Annali Scoula Normale Superiore Pisa 14 (1987), 285-344. \bibitem{ev} L.C. Evans: Partial Differential Equations. Amer. Math. Soc., Providence, Rhode Island 1998. \bibitem{gr} G. Greiner: Perturbing the boundary conditions of a generator. Huston J. Math. 13 (1987) 213-229. \bibitem{gt} D. Gilbarg, N.S. Trudinger: Elliptic Partial Differential Equations of Second Order. Springer, Berlin 1977. \bibitem{lsw} W. Littman, G. Stampacchia, H.F.Weinberger: Regular points for elliptic equations with discontinous coefficients. Ann. Scoula Norm.Sup. Pisa 17 (1963), 43-77. \bibitem{pa} A. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin 1983. \bibitem{pw} M.H. Protter, H.F. Weinberger: Maximum Principle in Differential Equations. Prentice-Hall Partial Differential Equations Series. 1967. \bibitem{st} G. Stampacchia: Le probl\{e}me de Dirichlet pour les \'{e} quations elliptiques du second ordre \`{a} coeifficients discontinus. Ann. Inst. Fourier Grenoble 15 (1965) 189-256. \bibitem{th} H.R. Thieme: Remarks on resolvent positive operators and their perturbation. Discrete and Continous Dynamical Systems 4 (1998) 73-90. \end{thebibliography} \end{document}