\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{epic} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 100, pp. 1--54.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2011 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2011/100\hfil Stability and approximations of eigenvalues] {Stability and approximations of eigenvalues and eigenfunctions of the Neumann Laplacian, part 3} \author[M. M. H. Pang\hfil EJDE-2011/100\hfilneg] {Michael M. H. Pang} \address{Michael M. H. Pang \newline Department of Mathematics\\ University of Missouri-Columbia\\ Columbia, MO 65211, USA} \email{pangm@missouri.edu} \thanks{Submitted November 5, 2010. Published August 7, 2011.} \subjclass[2000]{35P05, 35P15} \keywords{Stability; approximations; Neumann eigenvalues and eigenfunctions} \begin{abstract} This article is a sequel to two earlier articles (one of them written jointly with R. Banuelos) on stability results for the Neumann eigenvalues and eigenfunctions of domains in $\mathbb{R}^2$ with a snowflake type fractal boundary. In particular we want our results to be applicable to the Koch snowflake domain. In the two earlier papers we assumed that a domain $\Omega\subseteq\mathbb{R}^2$ which has a snowflake type boundary is approximated by a family of subdomains and that the Neumann heat kernel of $\Omega$ and those of its approximating subdomains satisfy a uniform bound for all sufficiently small $t>0$. The purpose of this paper is to extend the results in the two earlier papers to the situations where the approximating domains are not necessarily subdomains of $\Omega$. We then apply our results to the Koch snowflake domain when it is approximated from outside by a decreasing sequence of polygons. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{proposition}[theorem]{Proposition} \allowdisplaybreaks \section{Introduction} \label{sec:1} This paper is a sequel to the papers \cite{BP,P2}. The goal of these three papers is to prove stability results for the Neumann eigenvalues and eigenfunctions of domains in $\mathbb{R}^2$ with a snowflake type fractal boundary. In particular we want our results to be applicable to the Koch snowflake domain. In \cite{BP} and \cite{P2} we assumed that a domain $\Omega\subseteq\mathbb{R}^2$ which has a snowflake type boundary is approximated by a family of subdomains and that the Neumann heat kernel of $\Omega$ and those of its approximating subdomains satisfy a uniform bound for all sufficiently small $t>0$ (see Hypothesis 1.1 of \cite{BP} and \cite{P2}). The referee of \cite{BP} asked whether stability results similar to those in \cite{BP} and \cite{P2} are still true if the approximating domains of $\Omega$ are not necessarily subdomains of $\Omega$ and whether the proofs in \cite{BP} and \cite{P2} can be extended to those situations. If the results and methods in \cite{BP} and \cite {P2} can be extended to those situations, then they can be applied to domains, such as the Koch snowflake domain, which can be approximated by a familiar decreasing sequence of polygons from outside. The method in \cite{BP,P2} can be extended to situations when the approximating domains are not necessarily subdomains of $\Omega$, but not in a straight forward manner. The purpose of this paper is to work out such an extension and to apply it to the Koch snowflake domain when it is approximated from outside by a decreasing sequence of polygons. To state our results we first fix notation. Let $\Omega\subseteq\mathbb{R}^N$, $N\geq2$, be a bounded Sobolev extension domain. Let $\epsilon_0>0$ be sufficiently small, depending on $\Omega$. For each $\epsilon\in(0,\epsilon_0]$, let $\Omega_{\epsilon}$, $\Omega^{\epsilon}$ and $\Omega(\epsilon)$ be bounded Sobolev extension domains in $\mathbb{R}^N$ satisfying the following assumptions: \begin{equation} \label{eq:1.1} \begin{gathered} \Omega_{\epsilon} \supseteq\{ x\in\Omega :\operatorname{dist} (x,\partial\Omega)>\epsilon\},\\ \Omega^{\epsilon} \subseteq\{x\in\mathbb{R}^N:\operatorname{dist} (x,\Omega)<\epsilon\},\\ \Omega_{\epsilon} \subseteq\Omega(\epsilon)\subseteq\Omega^{\epsilon}. \end{gathered} \end{equation} We shall assume that \begin{equation} \Omega_{\epsilon_1}\supseteq\Omega_{\epsilon_2}\quad \text{if }0<\epsilon_1\leq\epsilon_2\leq\epsilon_0\label{eq:1.2}\end{equation} and that \begin{equation} \Omega^{\epsilon_1}\subseteq\Omega^{\epsilon_2}\quad\text{if }0<\epsilon_1\leq\epsilon_2\leq\epsilon_0.\label{eq:1.3} \end{equation} Let $-\Delta_{\epsilon}$, $-\Delta$, $-\Delta^{\epsilon}$, $-\Delta(\epsilon)$ be the Neumann Laplacian defined on $\Omega_{\epsilon}$, $\Omega$, $\Omega^{\epsilon}$ and $\Omega(\epsilon)$, respectively, and let $P_{\epsilon}(t,x,y)$, $P(t,x,y)$, $P^{\epsilon}(t,x,y)$ and $P(\epsilon)(t,x,y)$ be the heat kernel of $e^{-\Delta_{\epsilon}t}$, $e^{-\Delta t}$, $e^{-\Delta^{\epsilon}t}$ and $e^{-\Delta(\epsilon)t}$, respectively. We shall assume that there exists a positive continuous function $c:(0,1]\to(0,\infty)$ such that for all $0<\epsilon\leq\epsilon_0$ and all $00$, let $\mathcal{H}_{\epsilon}$ and $\mathcal{H}^{\epsilon}$ be Hilbert spaces satisfying the following assumptions: \begin{itemize} \item [(A1)] If $0<\epsilon_2\leq\epsilon_1$, then $\mathcal{H}^{\epsilon_2}\subseteq\mathcal{H}^{\epsilon_1}$. \item [(A2)] If $0<\epsilon_2\leq\epsilon_1$, then $\mathcal{H}_{\epsilon_2}\supseteq\mathcal{H}_{\epsilon_1}$. \item [(A3)] $\cap_{\epsilon>0}\mathcal{H}^{\epsilon}=\mathcal{H} =\cup_{\epsilon>0}\mathcal{H}_{\epsilon}$. \item [(A4)] For all $f\in\mathcal{H}$ we have \[ \|f-P_{\mathcal{H}, \mathcal{H}_{\epsilon}}f\|_{\mathcal{H}}\to0 \quad \text{as }\epsilon\downarrow0, \] where $\|\cdot\|_{\mathcal{H}}$ denotes the norm in $\mathcal{H}$. \item [(A5)] If $\epsilon_1>0$ and if $f\in\mathcal{H}^{\epsilon_1}$, then \[ \|P_{\mathcal{H}^{\epsilon_1}, \mathcal{H}} f-P_{\mathcal{H}^{\epsilon_1}, \mathcal{H}^{\epsilon}}f \|_{\mathcal{H}^{\epsilon}}\to0 \quad \text{as }\epsilon\downarrow0. \] \end{itemize} For all sufficiently small $\epsilon>0$ let $\mathcal{A}(\epsilon)$ and $\mathcal{B}(\epsilon)$ be Hilbert spaces satisfying the following assumptions: \begin{itemize} \item [(A6)] $\mathcal{H}_{\epsilon}\subseteq\mathcal{B} (\epsilon)\subseteq\mathcal{H}\cap\mathcal{A}(\epsilon) \subseteq\mathcal{A}(\epsilon)\subseteq\mathcal{H}^{\epsilon}$, \item [(A7)] For all $f\in\mathcal{H}$ we have \[ \|f-P_{\mathcal{H}, \mathcal{B}(\epsilon)}f\|_{\mathcal{H}}\to0 \quad \text{as }\epsilon\downarrow0. \] \end{itemize} We assume that for all sufficiently small $\epsilon>0$ there exists a closed subspace $\mathcal{C}(\epsilon)$ of $\mathcal{A}(\epsilon)$ satisfying the following assumptions: \begin{itemize} \item [(A8)] $\mathcal{C}(\epsilon)\subseteq\mathcal{H}^{\bot} \mathcal{H}^{\epsilon}$. \end{itemize} \begin{lemma}\label{lem:2.1} If $\epsilon_1>0$, then for all $f\in\mathcal{H}^{\epsilon_1}$, we have \[ \|P_{\mathcal{H}^{\epsilon_1},\mathcal{C}(\epsilon)}f \|_{\mathcal{H}^{\epsilon_1}}\to0 \quad \text{as }\epsilon\downarrow0. \] \end{lemma} \begin{proof} Let $f\in\mathcal{H}^{\epsilon_1}$. Then \[ \begin{split} &\|P_{\mathcal{H}^{\bot}\mathcal{H}^{\epsilon}, \mathcal{C}(\epsilon)}(P_{\mathcal{H}^{\epsilon}, \mathcal{H}^{\bot}\mathcal{H}^{\epsilon}}(P_{\mathcal{H}^{\epsilon_1}, \mathcal{H}^{\epsilon}}f))\|_{\mathcal{H}^{\epsilon_1}}\\ &= \|P_{\mathcal{H}^{\bot}\mathcal{H}^{\epsilon}, \mathcal{C}(\epsilon)}[(I_{\mathcal{H}^{\epsilon}}-P_{\mathcal{H}^{\epsilon}, \mathcal{H}})(P_{\mathcal{H}^{\epsilon_1}, \mathcal{H}^{\epsilon}}f)]\|_{\mathcal{H}^{\epsilon_1}}\\ &\leq \|P_{\mathcal{H}^{\bot}\mathcal{H}^{\epsilon}, \mathcal{C}(\epsilon)}\|\|P_{\mathcal{H}^{\epsilon_1}, \mathcal{H}^{\epsilon}}f-P_{\mathcal{H}^{\epsilon _1}, \mathcal{H}}\, f \|_{\mathcal{H}^{\epsilon_1}}\\ &\leq \|P_{\mathcal{H}^{\epsilon_1}, \mathcal{H}^{\epsilon}}f-P_{\mathcal{H}^{\epsilon_1}, \mathcal{H}}\, f\|_{\mathcal{H}^{\epsilon_1}} \to 0\quad \text{as }\epsilon\downarrow0. \end{split} \] \end{proof} We assume that for all sufficiently small $\epsilon>0$ there exists a closed subspace $\mathcal{D}(\epsilon)$ of $\mathcal{A}(\epsilon)$ satisfying the following assumptions: \begin{itemize} \item [(A9)] $\mathcal{A}(\epsilon)=\mathcal{B}(\epsilon)\oplus\mathcal{C}(\epsilon)\oplus\mathcal{D}(\epsilon)$, where $\oplus$ denotes orthogonal direct sum. \item [(A10)] If $\epsilon_1>0$, then, for all $f\in\mathcal{H}^{\epsilon_1}$, \[ \|P_{\mathcal{H}^{\epsilon_1}, \mathcal{D}(\epsilon)}f\|_{\mathcal{H}^{\epsilon_1}}\to0 \quad \text{as }\epsilon\downarrow0.\] \end{itemize} For all sufficiently small $\epsilon>0$ let $Q^{\epsilon}$ and $Q_{\epsilon}$ be non-negative closed quadratic forms with domains $\operatorname{Dom}(Q^{\epsilon})\subseteq\mathcal{H}^{\epsilon}$ and $\operatorname{Dom}(Q_{\epsilon})\subseteq\mathcal{H}_{\epsilon}$, respectively. Let $Q$ be a non-negative closed quadratic form with domain $\operatorname{Dom}(Q)\subseteq\mathcal{H}$. We assume that $Q$, $Q^{\epsilon}$ and $Q_{\epsilon}$ satisfy the following assumptions: \begin{itemize} \item [(A11)] For all sufficiently small $\epsilon > 0$, we have \begin{itemize} \item [(i)] $\operatorname{Dom}(Q^{\epsilon})$ is dense in $\mathcal{H}^{\epsilon}$, \item [(ii)] $\operatorname{Dom}(Q_{\epsilon})$ is dense in $\mathcal{H}_{\epsilon}$, \item [(iii)] $\operatorname{Dom}(Q)$ is dense in $\mathcal{H}$. \end{itemize} \item [(A12)] For $0<\epsilon_2\leq\epsilon_1$ we have \begin{itemize} \item [(i)] $P_{\mathcal{H}^{\epsilon_1}, \mathcal{H}^{\epsilon_2}}(\operatorname{Dom}(Q^{\epsilon_1}))=\operatorname{Dom}(Q^{\epsilon_2})$, \item [(ii)] $P_{\mathcal{H}^{\epsilon_1}, \mathcal{H}}(\operatorname{Dom}(Q^{\epsilon_1}))=\operatorname{Dom}(Q)$. \end{itemize} \item [(A13)] If $\epsilon_1>0$, then, for all sufficiently small $\epsilon>0$, we have \[ P_{\mathcal{H}^{\epsilon_1}, \mathcal{H}_{\epsilon}}(\operatorname{Dom}(Q^{\epsilon_1}))=\operatorname{Dom}(Q_{\epsilon}).\] \item [{(A14)}] For $0<\epsilon_2\leq\epsilon_1$ we have \begin{itemize} \item [{(i)}] $P_{\mathcal{H}_{\epsilon_2,}\mathcal{H}_{\epsilon_1}}(\operatorname{Dom}(Q_{\epsilon_2}))=\operatorname{Dom}(Q_{\epsilon_1})$, \item [{(ii)}] $P_{\mathcal{H}, \mathcal{H}_{\epsilon_1}}(\operatorname{Dom}(Q))=\operatorname{Dom}(Q_{\epsilon_1})$. \end{itemize} \end{itemize} \begin{definition}\label{def:2.2} \rm Let $\epsilon_0>0$ be fixed. For $0<\epsilon\leq\epsilon_0$ let $\hat{Q}^{\epsilon}$ be the quadratic form with domain \[ \operatorname{Dom}(\hat{Q}^{\epsilon})=\operatorname{Dom}(Q^{\epsilon})\oplus (\mathcal{H}^{\epsilon})^{\bot}\mathcal{H}^{\epsilon_0} \] and, if $f,g\in\operatorname{Dom}(Q^{\epsilon})$ and $h,i\in(\mathcal{H}^{\epsilon})^{\bot}\mathcal{H}^{\epsilon_0}$, we define $\hat{Q}^{\epsilon}(f\oplus h,g\oplus i)$ by \[ \hat{Q}^{\epsilon}(f+h,g+i) =Q^{\epsilon}(f,g) =Q^{\epsilon}(P_{\mathcal{H}^{\epsilon_0}, \mathcal{H}^{\epsilon}}(f+h),P_{\mathcal{H}^{\epsilon_0}, \mathcal{H}^{\epsilon}}(g+i)). \] \end{definition} Similarly we write $\hat{Q}$ and $\hat{Q}_{\epsilon}$ for the quadratic forms with domains \begin{gather*} \operatorname{Dom}(\hat{Q}) =\operatorname{Dom}(Q)\oplus\mathcal{H}^{\bot}\mathcal{H}^{\epsilon_0},\\ \operatorname{Dom}(\hat{Q}_{\epsilon}) =\operatorname{Dom}(Q)\oplus(\mathcal{H}_{\epsilon})^{\bot}\mathcal{H}^{\epsilon_0}, \end{gather*} respectively, and, for all $f,g\in\mathcal{H}^{\epsilon_0}$, we define $\hat{Q}(f,g)$ and $\hat{Q}_{\epsilon}(f,g)$ by \begin{gather*} \hat{Q}(f,g) =Q(P_{\mathcal{H}^{\epsilon_0}, \mathcal{H}}\, f,P_{\mathcal{H}^{\epsilon_0},\mathcal{H}}\, g),\\ \hat{Q}_{\epsilon}(f,g) =Q_{\epsilon}(P_{\mathcal{H}^{\epsilon_0}, \mathcal{H}_{\epsilon}}f,P_{\mathcal{H}^{\epsilon_0}, \mathcal{H}_{\epsilon}}g), \end{gather*} respectively. We assume that these quadratic forms satisfy the following assumptions: \begin{itemize} \item [{(A15)}] If $0<\epsilon_2\leq\epsilon_1\leq\epsilon_0$, then, for all $f\in\mathcal{H}^{\epsilon_1}$, we have \begin{itemize} \item [{(i)}] $Q^{\epsilon_1}(f,f)\geq Q^{\epsilon_2}(P_{\mathcal{H}^{\epsilon_1}, \mathcal{H}^{\epsilon_2}}f, P_{\mathcal{H}^{\epsilon_1}, \mathcal{H}^{\epsilon_2}}f)$, \item [{(ii)}] $Q^{\epsilon_1}(f,f)\geq Q(P_{\mathcal{H}^{\epsilon_1}, \mathcal{H}}\, f, P_{\mathcal{H}^{\epsilon_1}, \mathcal{H}}\, f)$. \end{itemize} \item [{(A16)}] For all $\epsilon_1,\epsilon_2\in(0,\epsilon_0]$ and all $f\in\mathcal{H}^{\epsilon_1}$, we have \[\ Q^{\epsilon_1}(f,f)\geq Q_{\epsilon_2}(P_{\mathcal{H}^{\epsilon_1}, \mathcal{H}_{\epsilon_2}}f, P_{\mathcal{H}^{\epsilon_1},\mathcal{H}_{\epsilon_2}}f).\] \item [{(A17)}] If $0<\epsilon_2\leq\epsilon_1\leq\epsilon_0$, then, for all $f\in\mathcal{H}_{\epsilon_2}$, we have \[ Q_{\epsilon_2}(f,f)\geq Q_{\epsilon_1}(P_{\mathcal{H}_{\epsilon_2}, \mathcal{H}_{\epsilon_1}}f, P_{\mathcal{H}_{\epsilon_2}, \mathcal{H}_{\epsilon_1}}f).\] \item [{(A18)}] For all $0<\epsilon\leq\epsilon_0$ and all $f\in\mathcal{H}$, we have \[ Q(f,f)\geq Q_{\epsilon}(P_{\mathcal{H}, \mathcal{H}_{\epsilon}}f, P_{\mathcal{H}, \mathcal{H}_{\epsilon}}f). \] \item [{(A19)}] For all $f\in\mathcal{H}$ we have \[ Q(f,f)=\lim_{\epsilon\downarrow0}Q_{\epsilon}(P_{\mathcal{H}, \mathcal{H}_{\epsilon}}f, P_{\mathcal{H}, \mathcal{H}_{\epsilon}}f). \] \item [{(A20)}] For all $f\in\operatorname{Dom}(Q^{\epsilon_0})$ we have \[ Q(P_{\mathcal{H}^{\epsilon_0},\mathcal{H}}\, f, P_{\mathcal{H}^{\epsilon_0},\mathcal{H}}\, f)=\lim_{\epsilon\downarrow0}Q^{\epsilon}(P_{\mathcal{H}^{\epsilon_0}, \mathcal{H}^{\epsilon}}f, P_{\mathcal{H}^{\epsilon_0}, \mathcal{H}^{\epsilon}}f). \] \end{itemize} \begin{definition}\label{def:2.3} \rm For $0<\epsilon\leq\epsilon_0$ let $H_{\epsilon}\geq0$ be the self-adjoint operator associated to $Q_{\epsilon}$ with domain $D(H_{\epsilon})\subseteq\mathcal{H}_{\epsilon}$. Similarly, let $H^{\epsilon}\geq0$ and $H\geq0$ be the self-adjoint operators associated to $Q^{\epsilon}$ and $Q$, respectively, with domains $D(H^{\epsilon})\subseteq\mathcal{H}^{\epsilon}$ and $D(H)\subseteq\mathcal{H}$. \end{definition} Assumptions (A11)--(A18) imply that we have an increasing family of non-negative quadratic forms: \begin{equation} \dots\leq\hat{Q}_{\epsilon_1}\leq\dots\leq\hat{Q}_{\epsilon_2}\leq\dots\leq\hat{Q}\leq\dots\leq\hat{Q}^{\epsilon_3}\leq\dots\leq\hat{Q}^{\epsilon_4}\leq\dots\label{eq:2.1}\end{equation} where \begin{equation} 0<\epsilon_2\leq\epsilon_1\leq\epsilon_0\quad \text{and }\quad 0<\epsilon_3\leq\epsilon_4\leq\epsilon_0.\label{eq:2.2} \end{equation} So by \cite[Theorem 4.17]{D1} we have, for all $\lambda>0$, \begin{align*} \dots & \geq(\lambda+H_{\epsilon_1})^{-1}\oplus\lambda^{-1} \geq\dots\geq(\lambda+H_{\epsilon_2})^{-1}\oplus\lambda^{-1}\geq\dots\\ \dots & \geq(\lambda+H)^{-1}\oplus\lambda^{-1}\geq\dots \geq(\lambda+H^{\epsilon_3})^{-1}\oplus\lambda^{-1}\geq\dots\\ \dots & \geq(\lambda+H^{\epsilon_4})^{-1}\oplus\lambda^{-1} \geq\dots \end{align*} if $\epsilon_1,\epsilon_2,\epsilon_3,\epsilon_4$ satisfy \eqref{eq:2.2}, where $(\lambda+H_{\epsilon_1})^{-1}\oplus\lambda^{-1}$ is the operator defined on $\mathcal{H}^{\epsilon_0}=\mathcal{H}_{\epsilon_1} \oplus(\mathcal{H}_{\epsilon_1})^{\bot}\mathcal{H}^{\epsilon_0}$ by \[ [(\lambda+H_{\epsilon_1})^{-1}\oplus\lambda^{-1}](f+g) =(\lambda+H_{\epsilon_1})^{-1}f+\lambda^{-1}g \] for all $f\in\mathcal{H}_{\epsilon_1}$ and all $g\in(\mathcal{H}_{\epsilon_1})^{\bot}\mathcal{H}^{\epsilon_0}$. Similarly the operators $(\lambda+H)^{-1}\oplus\lambda^{-1}$ and $(\lambda+H^{\epsilon})^{-1}\oplus\lambda^{-1}$ are defined on $\mathcal{H}^{\epsilon_0}=\mathcal{H}\oplus\mathcal{H}^{\bot} \mathcal{H}^{\epsilon_0}$ and $\mathcal{H}^{\epsilon_0}=\mathcal{H}^{\epsilon}\oplus (\mathcal{H}^{\epsilon})^{\bot}\mathcal{H}^{\epsilon_0}$, respectively. For $0<\epsilon\leq\epsilon_0$ let $Q(\epsilon)$ be a closed non-negative quadratic form with domain $\operatorname{Dom}(Q(\epsilon))\subseteq\mathcal{A}(\epsilon)$ satisfying the following assumptions: \begin{itemize} \item [{(A21)}] $\operatorname{Dom}(Q(\epsilon))$ is dense in $\mathcal{A}(\epsilon)$. \item [{(A22)}] For $0<\epsilon\leq\epsilon_0$ we have \begin{itemize} \item [{(i)}] $P_{\mathcal{H}^{\epsilon}, \mathcal{A}(\epsilon)}(\operatorname{Dom}(Q^{\epsilon}))=\operatorname{Dom}(Q(\epsilon))$, \item [{(ii)}] $P_{\mathcal{A}(\epsilon), \mathcal{H}_{\epsilon}}(\operatorname{Dom}(Q(\epsilon)))=\operatorname{Dom}(Q_{\epsilon})$. \end{itemize} \item[(A23)] If $0<\epsilon\leq\epsilon_0$, then, for all $f\in\mathcal{H}^{\epsilon}$, we have\[ Q^{\epsilon}(f,f)\geq Q(\epsilon)(P_{\mathcal{H}^{\epsilon}, \mathcal{A}(\epsilon)}f,P_{\mathcal{H}^{\epsilon}, \mathcal{A}(\epsilon)}f),\] and, for all $g\in\mathcal{A}(\epsilon)$, we have\[ Q(\epsilon)(g,g)\geq Q_{\epsilon}(P_{\mathcal{A}(\epsilon), \mathcal{H}_{\epsilon}}g,P_{\mathcal{A}(\epsilon), \mathcal{H}_{\epsilon}}g).\] \end{itemize} \begin{definition}\label{def:2.4} \rm For $0<\epsilon\leq\epsilon_0$ we define the quadratic form $\hat{Q}(\epsilon)$, with domain \[ \operatorname{Dom}(\hat{Q}(\epsilon))=\operatorname{Dom}(Q(\epsilon))\oplus\mathcal{A} (\epsilon)^{\bot}\mathcal{H}^{\epsilon_0} \subseteq\mathcal{H}^{\epsilon_0} \] by \[ \hat{Q}(\epsilon)(f,g)=Q(\epsilon)(P_{\mathcal{H}^{\epsilon_0}, \mathcal{A}(\epsilon)}f,P_{\mathcal{H}^{\epsilon_0}, \mathcal{A}(\epsilon)}g) \] for all $f,g\in\mathcal{H}^{\epsilon_0}$. We let $H(\epsilon)\geq0$ be the self-adjoint operator associated to $Q(\epsilon)$ with domain $D(H(\epsilon))\subseteq\mathcal{A}(\epsilon)$. \end{definition} Assumption (A23) implies that if $0<\epsilon \leq \epsilon_0$, then \begin{equation}\label{eq:2.3} \hat{Q}_\epsilon \leq \hat{Q}(\epsilon)\leq \hat{Q}^\epsilon \end{equation} and hence, by \cite[Theorem 4.17]{D1}, \begin{equation}\label{eq:2.4} (\lambda +H_\epsilon )^{-1}\oplus \lambda^{-1} \geq (\lambda +H (\epsilon ))^{-1} \oplus \lambda^{-1} \geq (\lambda +H^\epsilon )^{-1}\oplus \lambda ^{-1} \end{equation} for all $\lambda >0$, where $(\lambda +H(\epsilon ))^{-1} \oplus \lambda^{-1}$ is the operator defined on $\mathcal{H}^{\epsilon_0}=\mathcal{A}(\epsilon) \oplus \mathcal{A}(\epsilon)^\bot \mathcal{H}^{\epsilon_0}$ by \[ ((\lambda +H(\epsilon ))^{-1}\oplus \lambda^{-1})(f+g) =(\lambda +H(\epsilon))^{-1} f+\lambda^{-1}g \] for all $f\in \mathcal{A}(\epsilon)$ and $g\in \mathcal{A}(\epsilon )^\bot \mathcal{H}^{\epsilon_0}$. \begin{proposition}[{\cite[Theorem 4.32]{D1}}] \label{prop:2.5} Let $K _n\geq 0$ be an increasing sequence of non-negative self-adjoint operators with domains in a Hilbert space $\mathcal{U}$. Put \[ \mathcal{E}=\cap_n D(K^{1/2}_n) \] and let $\hat{\mathcal{U}}$ be the closure of $\mathcal{E}$. Then there exists a self-adjoint operator $K\geq 0$ with domain $D(K)\subseteq \hat{\mathcal{U}}$ such that its associated quadratic form domain equal $\mathcal{E}$ and that \[ \langle K^{1/2} f,K^{1/2} f\rangle = \lim_{n\to \infty} \langle K^{1/2} _n f, K^{1/2} _n f\rangle \quad (f\in \mathcal{E}). \] Moreover \[ \lim_{n\to \infty} \{ \sup_{0\leq t\leq a} \| e^{-K_nt} f-e^{-Kt} f\| \} =0 \] for all $a\geq 0$ and $f\in \hat{\mathcal{U}}$. Hence for all $\lambda >0$ we have \[ \| (\lambda +K_n)^{-1} f-(\lambda +K)^{-1} f\| \to 0 \quad \text{as } n\to \infty \] for all $f\in \hat{\mathcal{U}}$. \end{proposition} \begin{definition}\label{def:2.6} \rm For $0<\epsilon \leq \epsilon_0$ we let $\hat{H}_{\epsilon} $ and $\hat{H}^\epsilon$ be the operators with domains $D(\hat{H}_\epsilon )=D(H_\epsilon )\oplus (\mathcal{H}_\epsilon )^\bot \mathcal{H}^{\epsilon_0}$ and $D(\hat{H}^\epsilon)=D(H^\epsilon )\oplus (\mathcal{H}^\epsilon )^\bot \mathcal{H}^{\epsilon_0}$, respectively, defined by \[ \hat{H}_\epsilon (f+g)=H_\epsilon f =H_\epsilon P_{\mathcal{H}^{\epsilon_0} , \mathcal{H}_\epsilon }(f+g) \] for all $f+g\in D(H_\epsilon )\oplus (\mathcal{H}_\epsilon )^\bot \mathcal{H}^{\epsilon _0}$, and \[ \hat{H}^\epsilon (f+g) =H^\epsilon f=H^\epsilon P_{\mathcal{H}^{\epsilon _0}, \mathcal{H}^\epsilon }(f+g) \] for all $f+ g\in D(H^\epsilon )\oplus (\mathcal{H}^\epsilon )^\bot \mathcal{H}^{\epsilon_0}$. Similarly we write $\hat{H}$ to denote the operator with domain $D(\hat{H})=D(H)\oplus \mathcal{H}^\bot \mathcal{H}^{\epsilon_0}$ defined by \[ \hat{H} (f+g) =\hat{H} f=\hat{H} P_{\mathcal{H}^{\epsilon _0} , \mathcal{H}} \,(f+g) \] for all $f+g\in D(H)\oplus \mathcal{H}^\bot \mathcal{H}^{\epsilon_0}$. \end{definition} We also write $\hat{\hat{H}}_\epsilon$ for the operator with domain $D(\hat{\hat{H}}_\epsilon )=D(H_\epsilon )\oplus (\mathcal{H}_\epsilon) ^\bot \mathcal{H}$ defined by \[ \hat{\hat{H}}_\epsilon (f+g)=H_\epsilon f=H_\epsilon P_{\mathcal{H}, \mathcal{H}_\epsilon }(f+g) \] for all $f+g\in D(H_\epsilon )\oplus (\mathcal{H}_\epsilon )^\bot \mathcal{H}$. \begin{lemma}\label{lem:2.7} We have \begin{enumerate} \item[(i)] $\lim_{\epsilon \downarrow 0} \{ \sup_{0\leq t\leq a} \| e^{-\hat{\hat{H}}_\epsilon t} f-e^{-Ht}f\| _{\mathcal{H}}\}=0$ for all $f\in \mathcal{H}$ and $a\geq 0$. Also \[ \lim_{\epsilon \downarrow 0} \| (\lambda +\hat{\hat{H}}_\epsilon )^{-1} f-(\lambda +H)^{-1} f\| _{\mathcal{H}}=0 \] for all $f\in \mathcal{H}$ and $a\geq 0$. \item[(ii)] $\lim_{\epsilon \downarrow 0} \{ \sup_{0\leq t\leq a} \| e^{-\hat{H}_\epsilon t} f -e^{-\hat{H}t} f\| _{\mathcal{H}^{\epsilon_0}}\}=0$ for all $f\in \mathcal{H}^{\epsilon_0}$ and $a\geq 0$. Also \[ \lim_{\epsilon\downarrow 0} \| (\lambda +\hat{H}_\epsilon ) ^{-1} f-(\lambda +\hat{H})^{-1} f\| _{\mathcal{H}^{\epsilon_0}}=0 \] for all $f\in \mathcal{H}^{\epsilon_0}$ and $\lambda >0$. \end{enumerate} \end{lemma} \begin{proof} To prove (i) we apply Proposition \ref{prop:2.5} with $\mathcal{U}=\mathcal{H}$, $K_n = \hat{\hat{\mathcal{H_\varepsilon}}}$ and then use Assumptions (A17), (A18) and (A19). Similarly, to prove (ii) we apply Proposition 2.5 with $\mathcal{U}=\mathcal{H}^{\epsilon_0}$ and $K_n = \hat{\mathcal{H_\varepsilon}}$, and then use Assumptions (A17), (A18) and (A19). \end{proof} \begin{definition}\label{def:2.8}\rm Let $\mathcal{U}$ be a Hilbert space and let $Q\geq 0$ be a closed quadratic form with domain $\operatorname{Dom} (Q)\subseteq \mathcal{U}$. (Note that $\operatorname{Dom} (Q)$ is not necessarily dense in $\mathcal{U}$.) Let $H\geq 0$ be the self-adjoint operator associated to $Q$ with domain $D(H)\subseteq \overline{\operatorname{Dom} (Q)}$. If $\phi :\mathbb{R}\to \mathbb{R}$ is a bounded measurable function, then we define the bounded operator $\phi (H)$ on $\mathcal{U}=\overline{\operatorname{Dom} (Q)} \oplus ((\overline{\operatorname{Dom} (Q)})^\bot \mathcal{U})$ by \begin{equation}\label{eq:2.5} \phi (Q) (f+g)=\phi (H)f=\phi (H)(P_{\mathcal{U}, \overline{\operatorname{Dom} (Q)}} (f+g))\end{equation} for all $f\in \overline{\operatorname{Dom} (Q)}$ and $g\in (\overline{\operatorname{Dom}(Q)})^\bot \mathcal{U}$. \end{definition} Similarly, on $\mathcal{U}=\overline{\operatorname{Dom} (Q)} \oplus ((\overline{\operatorname{Dom} (Q)})^\bot \mathcal{U})$, we define the bounded operator $[\phi (Q)]_M$ by \begin{equation} \label{eq:2.6} \begin{split} [\phi (Q)]_M(f+g)&= \phi (H)f+g\\ &=\phi (H)P_{\mathcal{U}, \overline{\operatorname{Dom} (Q)}}\,(f+g) +P_{\mathcal{U}, \overline{(\operatorname{Dom} (Q))}^\bot \mathcal{U}}(f+g) \end{split} \end{equation} for all $f\in \overline{\operatorname{Dom} (Q)}$ and $g\in (\overline{\operatorname{Dom} (Q)})^\bot \mathcal{U}$. In both \eqref{eq:2.5} and \eqref{eq:2.6}, $\phi (H)$ is the bounded operator on $\overline{\operatorname{Dom} (Q)}$ defined using the spectral theorem. \begin{definition}\label{def:2.9} \rm Let $\mathcal{U}$ be a Hilbert space and for $n=1,2,3,\dots$ let $Q_n\geq 0$ be a closed quadratic form with domain $\operatorname{Dom} (Q_n)\subseteq \mathcal{U}$. ($\operatorname{Dom} (Q_n)$ is not necessarily dense in $\mathcal{U}$.) Let $Q\geq 0$ be a closed quadratic form with domain in $\mathcal{U}$. ($\operatorname{Dom} (Q)$ is not necessarily dense in $\mathcal{U}$.) We say that $Q_n$ converges to $Q$ in the strong resolvent sense $(\operatorname{srs} )$ if for some $\lambda >0$ we have \[ \lim_{n\to \infty} (\lambda+Q_n)^{-1}f=(\lambda +Q)^{-1}f\quad (f\in \mathcal{U}). \] \end{definition} \begin{lemma}\label{lem:2.10} Let $\mathcal{U}$, $Q_n$ and $Q$ be as in Definitions~\ref{def:2.8} and \ref{def:2.9}. Let $P_n$ be the orthogonal projection of $\mathcal{U}$ onto $\overline{\operatorname{Dom} (Q_n)}$. Suppose that for all $f\in \mathcal{U}$ we have \[ \| P_n f-f\| \to 0 \quad \text{as } n\to \infty. \] Suppose also that $\overline{\operatorname{Dom} (Q)}=\mathcal{U}$. Then $Q_n\overset{\operatorname{srs}}{\longrightarrow} Q$ as $n\to \infty$ is equivalent to \[ [(\lambda +Q_n)^{-1} ]_M \,f\to [(\lambda +Q)^{-1}]_M f \quad \text{as } n\to \infty \] for some $\lambda >0$ and for all $f\in \mathcal{U}$. \end{lemma} The proof of this lemma is obvious. \begin{proposition}[{\cite[Theorem~1.2.3]{D2}}]\label{prop:2.11} Let $K_n\geq 0$, $n=1,2,3,\dots $, and $K\geq 0$ be self-adjoint operators with domains in a Hilbert space $\mathcal{U}$. Suppose that \[ K_1\geq K_2 \geq \dots \geq K_n \geq K_{n+1}\geq \dots \geq K \] and that their associated quadratic forms satisfy \[ \langle K^{1/2} f, K^{1/2} f\rangle =\lim_{n\to \infty} \langle K^{1/2}_n f, K^{1/2}_n f\rangle \] for all $f$ in a form core of $K$. Then $K_n$ converges to $K$ in the strong resolvent sense. \end{proposition} \begin{definition}\label{def:2.12} \rm We let $\mathcal{C}$ be the subspace of $\operatorname{Dom} (\hat{Q})$ defined by \[ \mathcal{C}=\cup_{0<\epsilon \leq \epsilon_0} \operatorname{Dom} (Q^\epsilon )\oplus (\mathcal{H}^\epsilon )^\bot \mathcal{H}^{\epsilon_0}.\] (Note that $\mathcal{C}$ is a subspace of $\operatorname{Dom} (\hat{Q})$ by Assumption~(A15).) \end{definition} \begin{lemma}\label{lem:2.13} $\mathcal{C}$ is a form core of $\hat{Q}$. \end{lemma} \begin{proof} We first recall that, by Assumption~(A12), \[ \operatorname{Dom} (Q)=P_{\mathcal{H}^\epsilon , \mathcal{H}} (\operatorname{Dom} (Q^\epsilon ))\quad (0<\epsilon \leq \epsilon_0). \] Let $f=g+h\in \operatorname{Dom} (\hat{Q})$, where $g\in \operatorname{Dom} (Q)$ and $h\in \mathcal{H}^\bot \mathcal{H}^{\epsilon_0}$. Let $\alpha \in \operatorname{Dom} (Q^{\epsilon_0})$ such that \[ P_{\mathcal{H}^{\epsilon_0},\mathcal{H}}\alpha =g. \] For $0<\epsilon \leq \epsilon_0$ let \begin{align*} g_\epsilon &=P_{\mathcal{H}^{\epsilon_0}, \mathcal{H}^\epsilon}\alpha =P_{\mathcal{H}^\epsilon , \mathcal{H}}\, g_\epsilon +P_{\mathcal{H}^\epsilon , \mathcal{H}^\bot \mathcal{H}^\epsilon} g_\epsilon\\ &=P_{\mathcal{H}^{\epsilon_0},\mathcal{H}} \alpha +P_{\mathcal{H}^\epsilon ,\mathcal{H}^\bot \mathcal{H}^\epsilon} g_\epsilon \\ &=g+P_{\mathcal{H}^\epsilon , \mathcal{H}^\bot \mathcal{H}^\epsilon} g_\epsilon \end{align*} and let \[ h_\epsilon =P_{\mathcal{H}^\bot \mathcal{H}^{\epsilon_0} , (\mathcal{H}^\epsilon )^\bot \mathcal{H}^{\epsilon _0}} h \] and let \begin{equation} \label{eq:2.7} \begin{split} f_\epsilon &= g_\epsilon +h_\epsilon \\ &= P_{\mathcal{H}^{\epsilon_0}, \mathcal{H}} \alpha +P_{\mathcal{H}^\epsilon ,\mathcal{H}^\bot \mathcal{H}^\epsilon } g_\epsilon +h_\epsilon \\ &= g+P_{\mathcal{H}^\epsilon ,\mathcal{H}^\bot \mathcal{H}^\epsilon } g_\epsilon +h_\epsilon . \end{split} \end{equation} Then, by (A12), $f_\epsilon \in \operatorname{Dom} (Q^\epsilon )\oplus (\mathcal{H}^\epsilon )^\bot \mathcal{H}^{\epsilon_0}$. Since \begin{align*} h&=P_{\mathcal{H}^\bot \mathcal{H}^{\epsilon_0} ,(\mathcal{H}^\epsilon )^\bot \mathcal{H}^{\epsilon_0}} h+ (I_{\mathcal{H}^\bot \mathcal{H}^{\epsilon_0}} -P_{\mathcal{H}^\bot \mathcal{H}^{\epsilon_0}, \,(\mathcal{H}^\epsilon )^\bot \mathcal{H}^{\epsilon_0}} )h\\ &= h_\epsilon +(I_{\mathcal{H}^\bot \mathcal{H}^{\epsilon_0}} -P_{\mathcal{H}^\bot \mathcal{H}^{\epsilon_0} , (\mathcal{H}^\epsilon)^\bot \mathcal{H}^{\epsilon _0}})h, \end{align*} we have \begin{equation} \label{eq:2.8} \begin{split} f-f_\epsilon &= g+h-(g+P_{\mathcal{H}^\epsilon , \mathcal{H}^\bot \mathcal{H}^\epsilon} g_\epsilon +h_\epsilon )\\ &=g+h_\epsilon +(I_{\mathcal{H}^\bot \mathcal{H}^{\epsilon _0}} -P_{\mathcal{H}^\bot \mathcal{H}^{\epsilon_0} , (\mathcal{H}^\epsilon)^\bot \mathcal{H}^{\epsilon_0}})h - (g+P_{\mathcal{H}^\epsilon , \mathcal{H}^\bot \mathcal{H}^\epsilon} g_\epsilon +h_\epsilon )\\ &= (I_{\mathcal{H}^\bot \mathcal{H}^{\epsilon_0}} -P_{\mathcal{H}^\bot \mathcal{H}^{\epsilon_0}, (\mathcal{H}^\epsilon )^\bot \mathcal{H}^{\epsilon_0}} )h-P_{\mathcal{H}^\epsilon , \mathcal{H}^\bot \mathcal{H}^\epsilon} g_\epsilon\\ &= (I_{\mathcal{H}^\bot \mathcal{H}^{\epsilon_0}}-P_{\mathcal{H}^\bot \mathcal{H}^{\epsilon_0}, (\mathcal{H}^\epsilon )^\bot \mathcal{H}^{\epsilon_0}})h-P_{\mathcal{H}^{\epsilon_0}, \mathcal{H}^\bot \mathcal{H}^\epsilon} \alpha\\ &=(I_{\mathcal{H}^\epsilon} -P_{\mathcal{H}^\epsilon , \mathcal{H}} )P_{\mathcal{H}^{\epsilon_0}, \mathcal{H}^\epsilon } h - (I_{\mathcal{H}^\epsilon } -P_{\mathcal{H}^\epsilon , \mathcal{H} } )P_{\mathcal{H}^{\epsilon_0} , \mathcal{H}^\epsilon } \alpha\\ & \to 0 \quad \text{as } \epsilon \downarrow 0 \quad \text{(by (A5))}.\end{split}\end{equation} Also, by~\eqref{eq:2.8}, \begin{align*} \hat{Q} (f-f_\epsilon ,f-f_\epsilon ) &=\hat{Q} (P_{\mathcal{H}^\epsilon, \, \mathcal{H}^\bot \mathcal{H}^\epsilon} P_{\mathcal{H}^{\epsilon_0} , \mathcal{H}^\epsilon} (h-\alpha), P_{\mathcal{H}^\epsilon , \mathcal{H}^\bot \mathcal{H}^\epsilon} P_{\mathcal{H}^{\epsilon_0} , \mathcal{H}^\epsilon} (h-\alpha ))\\ &=0 \end{align*} since $P_{\mathcal{H}^\epsilon , \mathcal{H}^\bot \mathcal{H}^\epsilon } P_{\mathcal{H}^{\epsilon _0} , \mathcal{H}^\epsilon} (h-\alpha )\in \mathcal{H}^\bot \mathcal{H}^{\epsilon_0}$. Also, by Assumption~(A12), it is not difficult to show that $\mathcal{C}$ is closed under addition and scalar multiplication. Hence $\mathcal{C}$ is a form core of $\hat{Q}$. \end{proof} \begin{theorem}\label{thm:2.14} We have $\hat{H}^\epsilon \overset{\operatorname{srs}}{\longrightarrow} \hat{H}$ as $\epsilon \downarrow 0$. \end{theorem} \begin{proof} Let $\delta \in (0,\epsilon _0]$ and let $f\in \operatorname{Dom} (Q^\delta )\oplus (\mathcal{H}^{\delta})^\bot \mathcal{H}^{\epsilon_0}$. Then, for $0<\epsilon<\delta$, we have, by (A5), \[ P_{\mathcal{H}^{\epsilon_0} , \mathcal{H}^\epsilon} f =P_{\mathcal{H}^{\epsilon_0} , \mathcal{H}^\epsilon }f -P_{\mathcal{H}^{\epsilon_0} ,\mathcal{H}} f+P_{\mathcal{H}^{\epsilon_0} ,\mathcal{H}} f \to P_{\mathcal{H}^{\epsilon_0} , \mathcal{H} } f \quad \text{as } \epsilon\downarrow 0. \] Hence, for $0<\epsilon <\delta$, we have, by (A20), \begin{align*} \hat{Q}^\epsilon (f,f)&= Q^\epsilon (P_{\mathcal{H}^{\epsilon_0} , \mathcal{H}^\epsilon } f,P_{\mathcal{H}^{\epsilon_0} , \mathcal{H}^\epsilon } f)\\ &\to Q(P_{\mathcal{H}^{\epsilon_0} ,\mathcal{H}} f,P_{\mathcal{H}^{\epsilon_0} ,\mathcal{H} } f) = \hat{Q} (f,f)\quad \text{as } \epsilon \downarrow 0. \end{align*} Thus for all $f\in \mathcal{C}$ we have \begin{equation}\label{eq:2.9} \hat{Q}(f,f)=\lim_{\epsilon \downarrow 0} \hat{Q}^\epsilon (f,f). \end{equation} The theorem now follows from Proposition~\ref{prop:2.11} together with \eqref{eq:2.1}, \eqref{eq:2.9} and Lemma~\ref{lem:2.13}. \end{proof} \begin{definition}\label{def:2.15} \rm For $0<\epsilon \leq \epsilon_0$ we let $\hat{H}(\epsilon )$ be the operator with domain $D(\hat{H}(\epsilon ))=D(H(\epsilon ))\oplus \mathcal{A} (\epsilon )^\bot \mathcal{H}^{\epsilon _0}$ defined by \[ \hat{H} (\epsilon )(f+g)=H(\epsilon )f =H(\epsilon )P_{\mathcal{H}^{\epsilon_0} , \mathcal{A}(\epsilon)} (f+g) \] for all $f\in D(H(\epsilon))$ and $g \in \mathcal{A}(\epsilon )^\bot \mathcal{H}^{\epsilon_0}$. Thus $\hat{H}(\epsilon )\geq 0$ is the self-adjoint operator associated to the quadratic form $\hat{Q}(\epsilon )$ defined in Definition~\ref{def:2.4}. By \eqref{eq:2.4} we have \[ (\lambda +\hat{H}_\epsilon )^{-1} \geq (\lambda +\hat{H}(\epsilon ))^{-1}\geq (\lambda +\hat{H}^\epsilon )^{-1} \] for all $\lambda >0$ and $0<\epsilon \leq \epsilon _0$; i.e., \begin{equation}\label{eq:2.10} \langle (\lambda +\hat{H}_\epsilon )^{-1} f, f\rangle \geq \langle (\lambda +\hat{H}(\epsilon ))^{-1} f, f\rangle \geq \langle (\lambda +\hat{H}^\epsilon )^{-1} f, f\rangle \end{equation} for all $f\in \mathcal{H}^{\epsilon _0}$, $\lambda >0$ and $0<\epsilon \leq \epsilon _0$. \end{definition} \begin{lemma}\label{lem:2.16} We have \[ \langle (\lambda +\hat{H})^{-1} f,f\rangle =\lim_{\epsilon \downarrow 0} \langle (\lambda +\hat{H}(\epsilon ))^{-1} f,f\rangle \] for all $f\in \mathcal{H}^{\epsilon _0}$ and $\lambda >0$. \end{lemma} \begin{proof} This lemma follows from the second inequality of Lemma~\ref{lem:2.7}(ii), Theorem~\ref{thm:2.14} and \eqref{eq:2.10}. \end{proof} \begin{theorem}\label{thm:2.17} For all $\lambda >0$ we have \[ (\lambda +\hat{H})^{-1} f=\lim_{\epsilon\downarrow 0} (\lambda+\hat{H}(\epsilon))^{-1} f \] for all $f\in \mathcal{H}^{\epsilon_0}$. Hence for all $a>0$ and $f\in\mathcal{H}^{\epsilon_0}$ we have \[ \lim_{\epsilon\downarrow 0} \{ \sup_{0\leq t\leq a} \| e^{-\hat{H}(\epsilon )t} f-e ^{-\hat{H}t} f\|_{\mathcal{H}^{\epsilon_0}} \} =0. \] \end{theorem} \begin{proof} By Lemma~\ref{lem:2.7}(ii), we have, for all $\lambda >0$, \[ (\lambda +\hat{H})^{-1} f=\lim_{\epsilon\downarrow 0} (\lambda +\hat{H}_\epsilon)^{-1} f\quad (f\in \mathcal{H}^{\epsilon_0}). \] This is equivalent to having \begin{equation}\label{eq:2.11} \lim_{\epsilon \downarrow 0} \{ \sup_{0\leq t\leq a} \| e^{-\hat{H}_\epsilon t} f-e^{-\hat{H} t} f\| _{\mathcal{H}^{\epsilon_0}}\}=0\quad (f\in \mathcal{H}^{\epsilon_0} ) \end{equation} for all $a>0$ (see, for example \cite[Theorem 3.17]{D1}). Similarly, Theorem~\ref{thm:2.14} is equivalent to \begin{equation}\label{eq:2.12} \lim_{\epsilon\downarrow 0} \{\sup_{0\leq t\leq a} \| e^{-\hat{H}^\epsilon t} f-e^{-\hat{H}t} f\|_{\mathcal{H}^{\epsilon _0}} \}=0\quad (f\in \mathcal{H}^{\epsilon_0}) \end{equation} for all $a>0$. Since, for $\lambda >0$, we have \begin{gather*} (\lambda +\hat{H}_\epsilon )^{-1/2} f =\int^\infty_0 \frac{1}{\sqrt{\pi t}}e^{-\lambda t} e^{-\hat{H}_\epsilon t} f\,dt\quad (f\in \mathcal{H}^{\epsilon_0}),\\ (\lambda+\hat{H}^\epsilon)^{-1/2} f = \int^\infty_0 \frac{1}{\sqrt{\pi t}} e^{-\lambda t} e^{-\hat{H}^\epsilon t} f\,dt \quad (f\in \mathcal{H}^{\epsilon _0} ), \end{gather*} we have, from \eqref{eq:2.11} and \eqref{eq:2.12}, \begin{equation}\label{eq:2.13} \lim_{\epsilon \downarrow 0} (\lambda +\hat{H}_\epsilon )^{-1/2} f =\lim_{\epsilon \downarrow 0 }(\lambda +\hat{H}^\epsilon )^{-1/2} f =(\lambda +\hat{H})^{-1/2}f \end{equation} for all $\lambda >0$ and $f\in \mathcal{H}^{\epsilon_0}$. Since, for all $\lambda >0$, \[ (\lambda +\hat{H}_\epsilon )^{-1} \geq (\lambda +\hat{H}(\epsilon))^{-1}\geq (\lambda +\hat{H}^\epsilon )^{-1}, \] we have \[ (\lambda +\hat{H}_\epsilon )^{-1/2} \geq (\lambda +\hat{H}(\epsilon))^{-1/2} \geq (\lambda +\hat{H}^\epsilon )^{-1/2} \] for all $\lambda >0$ (see, for example, \cite[Lemma 4.19]{D1}); i.e., for all $0<\epsilon \leq \epsilon_0$, $\lambda >0$ and $f\in \mathcal{H}^{\epsilon_0}$, we have \begin{equation}\label{eq:2.14} \langle (\lambda +\hat{H}_\epsilon )^{-1/2} f,f\rangle \geq \langle (\lambda +\hat{H}(\epsilon ))^{-1/2} f,f\rangle \geq \langle (\lambda +\hat{H}^\epsilon )^{-1/2} f,f\rangle. \end{equation} Hence, from \eqref{eq:2.13} and \eqref{eq:2.14}, we have \begin{equation}\label{eq:2.15} \langle (\lambda +\hat{H})^{-1/2} f,f\rangle =\lim_{\epsilon \downarrow 0} \langle (\lambda +\hat{H} (\epsilon ))^{-1/2} f,f \rangle \end{equation} for all $\lambda >0$ and $f\in \mathcal{H}^{\epsilon_0}$. The polarization identity (see, for example, \cite[p.103]{D1}) and \eqref{eq:2.15} imply that \begin{equation}\label{eq:2.16} \langle (\lambda +\hat{H})^{-1/2} f,g\rangle =\lim_{\epsilon \downarrow 0} \langle (\lambda +H(\epsilon ))^{-1/2} f,g\rangle \end{equation} for all $\lambda >0$ and $f,g\in \mathcal{H}^{\epsilon_0}$. We now need the following result. \begin{proposition}[{See \cite[Problem 4.11]{D1}}] \label{prop:2.18} Let $\mathcal{U}$ be a Hilbert space and let $f,f_n\in \mathcal{U}$ for $n=1,2,3,\dots$. Suppose that \[ \langle f, g\rangle =\lim_{n\to \infty} \langle f_n, g\rangle \quad (g\in \mathcal{U}). \] Then \[ \lim_{n\to \infty} \| f_n-f\|=0\quad \text{if and only if}\quad \lim_{n\to \infty} \| f_n\|=\| f\|. \] \end{proposition} By Lemma~\ref{lem:2.16}, we have \begin{equation}\label{eq:2.17} \lim_{\epsilon \downarrow 0} \| (\lambda +\hat{H}(\epsilon )) ^{-1/2} f\|_{\mathcal{H}^{\epsilon _0}} =\| (\lambda +\hat{H})^{-1/2} f\| _{\mathcal{H}^{\epsilon _0}} \end{equation} for all $\lambda >0$ and $f\in \mathcal{H}^{\epsilon _0}$. Proposition~\ref{prop:2.18} together with \eqref{eq:2.16} and \eqref{eq:2.17} imply that \begin{equation}\label{eq:2.18} (\lambda +\hat{H})^{-1/2} f=\lim_{\epsilon \downarrow 0} (\lambda +\hat{H}(\epsilon ))^{-1/2} f \end{equation} for all $\lambda >0$ and $f\in \mathcal{H}^{\epsilon_0}$. Hence \begin{align*} &(\lambda +\hat{H}(\epsilon ))^{-1} f-(\lambda +\hat{H})^{-1} f\\ &= (\lambda +\hat{H}(\epsilon ))^{-1/2} [(\lambda +\hat{H}(\epsilon ))^{-1/2} f-(\lambda +\hat{H})^{-1/2} f]\\ &\quad + (\lambda +\hat{H}(\epsilon ))^{-1/2} (\lambda +\hat{H})^{-1/2} f-(\lambda +\hat{H})^{-1/2} (\lambda +\hat{H})^{-1/2} f \to 0 \quad \text{as } \epsilon \downarrow 0 \end{align*} for all $\lambda >0$ and $f\in \mathcal{H}^{\epsilon _0}$. The strong convergence of $e^{-\hat{H}(\epsilon )t} $ to $e^{-\hat{H}t}$ now follows from \cite[Theorem 3.17]{D1}. \end{proof} We next impose more assumptions on the operators $H$, $H^\epsilon $, $H_\epsilon $ and $H(\epsilon )$, $0<\epsilon \leq \epsilon_0$: \begin{itemize} \item[(A24)] $H$, $H^\epsilon $, $H_\epsilon$, $H(\epsilon )$, $0<\epsilon \leq \epsilon_0$, have compact resolvents in the Hilbert spaces $\mathcal{H}$, $\mathcal{H}^\epsilon $, $\mathcal{H}_\epsilon $ and $\mathcal{A}(\epsilon )$, respectively. \item[(A25)] $0\in Sp (H)$, $0\in Sp (H^\epsilon )$, $0\in Sp (H_\epsilon )$ and $0\in Sp (H(\epsilon ))$, $0<\epsilon \leq \epsilon _0$. \end{itemize} \begin{definition}\label{def:2.19} \rm We shall write $\{ \mu_i\}^\infty_{i=1}$ for the eigenvalues of $H$, where $\{ \mu_i\}^\infty_{i=1}$ is a non-decreasing sequence and the eigenvalues are listed repeatedly according to multiplicity. Similarly, for $0<\epsilon \leq \epsilon _0$, we shall write $\{ \mu^\epsilon _i\}^\infty_{i=1}$, $\{ \mu _{i,\epsilon }\}^\infty _{i=1}$, and $\{ \mu_i (\epsilon )\}^\infty_{i=1}$ for the eigenvalues of $H^\epsilon$, $H_\epsilon $ and $H(\epsilon )$, respectively. Thus, by (A25), we have \[ 0=\mu_1=\mu^\epsilon _1 =\mu_{1,\epsilon } =\mu_1 (\epsilon )\quad (0<\epsilon \leq \epsilon _0). \] We shall also write $\{ \varphi_i\}^\infty_{i=1}$, $\{ \varphi^\epsilon _i\}^\infty_{i=1}$, $\{\varphi_{i,\epsilon }\}^\infty_{i=1}$ and $\{ \varphi_i(\epsilon)\}^\infty_{i=1}$ for the corresponding normalized eigenvectors of $H$, $H^\epsilon$, $H_\epsilon$ and $H(\epsilon)$, respectively. We shall also assume that $\{ \varphi_i\}^\infty_{i=1}$, $\{ \varphi _i^\epsilon \}^\infty_{i=1}$, $\{ \varphi_{i,\epsilon }\}^\infty_{i=1}$ and $\{ \varphi_i(\epsilon )\}^\infty_{i=1}$ are complete orthonormal systems in their respective Hilbert spaces $\mathcal{H}$, $\mathcal{H}^\epsilon$, $\mathcal{H}_\epsilon$ and $\mathcal{H}(\epsilon )$. \end{definition} \begin{itemize} \item[(A26)] $\mu_1$, $\mu_1^\epsilon$, $\mu_{1,\epsilon }$, $\mu_1(\epsilon)$, $0<\epsilon \leq \epsilon_0$, all have multiplicity $1$. \item[(A27)] For $0<\epsilon \leq \epsilon_0$, we assume that $P_{\mathcal{H}^{\epsilon_0}, \mathcal{H}} \varphi^{\epsilon_0}_1$, $P_{\mathcal{H}^{\epsilon_0}, \mathcal{H}^\epsilon} \varphi^{\epsilon_0}_1$, $P_{\mathcal{H}^{\epsilon_0}, \mathcal{H}_\epsilon}\varphi^{\epsilon_0}_1$ and $P_{\mathcal{H}^{\epsilon_0}, \mathcal{A}(\epsilon)} \varphi_1^{\epsilon_0}$ are eigenvectors of $H$, $H^\epsilon$, $H_\epsilon$ and $H(\epsilon)$, respectively, associated to the eigenvalue $0=\mu _1= \mu_1^\epsilon =\mu_{1,\epsilon}=\mu_1(\epsilon)$. In fact we assume that $\varphi_1$, $\varphi_1^\epsilon $, $\varphi_{1,\epsilon}$ and $\varphi_1(\epsilon)$ are chosen so that \begin{gather*} \varphi_1 =\| P_{\mathcal{H}^{\epsilon_0} ,\mathcal{H}} \varphi_1^{\epsilon _0} \|_\mathcal{H}^{-1} P_{\mathcal{H}^{\epsilon_0}, \mathcal{H}} \varphi^{\epsilon_0}_1,\\ \varphi^\epsilon_1 = \|P_{\mathcal{H}^{\epsilon_0}, \mathcal{H}^\epsilon} \varphi_1^{\epsilon _0} \|^{-1}_{\mathcal{H}^\epsilon} P_{\mathcal{H}^{\epsilon_0}, \mathcal{H}^\epsilon} \varphi_1^{\epsilon_0},\\ \varphi_{1,\epsilon }= \| P_{\mathcal{H}^{\epsilon_0} , \mathcal{H}_\epsilon} \varphi_1^{\epsilon_0} \| ^{-1}_{\mathcal{H}_\epsilon} P_{\mathcal{H}^{\epsilon_0}, \, \mathcal{H}_\epsilon} \varphi^{\epsilon_0}_1,\\ \varphi_1 (\epsilon)=\|P_{\mathcal{H}^{\epsilon_0}, \mathcal{A}(\epsilon)} \varphi_1^{\epsilon_0} \|^{-1}_{\mathcal{A} (\epsilon)}P_{\mathcal{H}^{\epsilon_0} , \mathcal{A}(\epsilon)} \varphi_1^{\epsilon_0}. \end{gather*} \item[(A28)] For all $00$, there exists $\epsilon _1\in (0,\epsilon _0]$ such that \begin{equation}\label{eq:2.27} \mu_2 (\epsilon )t\leq \mu_2 t+\delta \end{equation} for all $0<\epsilon \leq \epsilon _1$. We next prove the reverse inequality of \eqref{eq:2.27}. For all $0<\epsilon \leq \epsilon_0$ let \begin{equation}\label{eq:2.28} \gamma _1 (\epsilon )=\langle P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} \varphi_2 (\epsilon ), \varphi_1\rangle_{\mathcal{H}}. \end{equation} Then \begin{equation}\begin{split}\label{eq:2.29} e^{-\mu_2 t}&\geq \|P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} \varphi_2 (\epsilon )-\gamma_1 (\epsilon )\varphi_1\|^{-2}_{\mathcal{H}}\\ &\quad \times \langle e^{-Ht} (P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} \varphi_2 (\epsilon ) -\gamma_1(\epsilon )\varphi_1), P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} \varphi_2 (\epsilon )-\gamma_1 (\epsilon )\varphi_1\rangle_{\mathcal{H}}\\ &=\| P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} \varphi_2 (\epsilon ) -\gamma_1 (\epsilon )\varphi_1 \|^{-2}_{\mathcal{H}}\\ &\quad \times \{ \langle e^{-Ht}P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} \varphi_2 (\epsilon ),P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} \varphi_2 (\epsilon )\rangle_{\mathcal{H}}\\ &\quad -2\gamma_1 (\epsilon )\langle e^{-Ht}P_{\mathcal{A}(\epsilon ), \mathcal{B} (\epsilon )} \varphi_2 (\epsilon ),\varphi_1\rangle_{\mathcal{H}} +\gamma_1 (\epsilon )^2\}\\ &=\| P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} \varphi_2(\epsilon )-\gamma_1 (\epsilon )\varphi_1\|^{-2}_{\mathcal{H}}\\ &\quad\times \{ \langle e^{-Ht}P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} \varphi_2 (\epsilon ),P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )}\varphi_2 (\epsilon )\rangle_{\mathcal{H}}-\gamma_1 (\epsilon )^2\}.\end{split}\end{equation} Consider the term \begin{equation}\begin{split}\label{eq:2.30} &\langle e^{-Ht} P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} \varphi_2 (\epsilon ),P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} \varphi_2(\epsilon )\rangle_{\mathcal{H}}\\ &= \langle e^{-Ht} P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} \varphi_2 (\epsilon )-P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} e^{-H(\epsilon )t} \varphi_2 (\epsilon ), P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} \varphi_2 (\epsilon )\rangle_{\mathcal{H}}\\ &\quad +\langle P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )}e^{-H(\epsilon )t} \varphi_2 (\epsilon ),P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} \varphi_2(\epsilon )\rangle _{\mathcal{H}}\\ &= \langle e^{-Ht}P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} \varphi_2 (\epsilon ) -P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} e^{-H(\epsilon )t} \varphi_2 (\epsilon ) ,P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} \varphi_2 (\epsilon )\rangle_{\mathcal{H}}\\ &\quad + e^{-\mu_2 (\epsilon )t} \langle P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} \varphi_2 (\epsilon ), P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} \varphi_2 (\epsilon )\rangle_{\mathcal{H}}. \end{split}\end{equation} Therefore, by (A28), \eqref{eq:2.27} and \eqref{eq:2.30}, for all $\delta >0$ there exists $\epsilon _1 \in (0,\epsilon _0]$ such that \begin{equation}\label{eq:2.31} \langle e^{-Ht} P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} \varphi_2 (\epsilon ) ,P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} \varphi_2 (\epsilon )\rangle_{\mathcal{H}} \geq e^{-\mu_2 (\epsilon )t-\delta } \end{equation} for all $\epsilon \in (0,\epsilon _1]$. We next consider the term $\gamma_1 (\epsilon )$ defined in \eqref{eq:2.28}: \begin{equation} \label{eq:2.32} \gamma_1 (\epsilon ) =\langle P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} \varphi_2 (\epsilon ),\varphi_1\rangle _{\mathcal{H}} =\langle P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} \varphi_2 (\epsilon ),P_{\mathcal{H}, \mathcal{B}(\epsilon )} \varphi_1 +P_{\mathcal{H}, \mathcal{B}(\epsilon ) ^\bot \mathcal{H}} \varphi_1 \rangle_{\mathcal{H}}. \end{equation} But, by (A27), \begin{equation}\begin{split}\label{eq:2.33} P_{\mathcal{H}, \mathcal{B}(\epsilon )} \varphi_1 &=\| P_{\mathcal{H}^{\epsilon _0}, \mathcal{H}}\varphi_1^{\epsilon _0} \|^{-1}_{\mathcal{H}} P_{\mathcal{H}, \mathcal{B}(\epsilon )} P_{\mathcal{H}^{\epsilon _0}, \mathcal{H}}\varphi^{\epsilon _0}_1\\ &=\| P_{\mathcal{H}^{\epsilon _0}, \mathcal{H}}\varphi_1^{\epsilon _0} \|^{-1}_{\mathcal{H}} P_{\mathcal{H}^{\epsilon _0}, \mathcal{B}(\epsilon)} \varphi^{\epsilon _0}_1\\ &=\| P_{\mathcal{H}^{\epsilon _0}, \mathcal{H}}\varphi^{\epsilon _0}_1 \|^{-1}_{\mathcal{H}} P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} P_{\mathcal{H}^{\epsilon _0}, \mathcal{A}(\epsilon )} \varphi^{\epsilon _0}_1\\ &= \| P_{\mathcal{H}^{\epsilon _0}, \mathcal{H}} \varphi^{\epsilon _0}_1\|^{-1}_{\mathcal{H}}\| P_{\mathcal{H}^{\epsilon_0}, \mathcal{A}(\epsilon )} \varphi^{\epsilon _0}_1 \| _{\mathcal{A}(\epsilon )} P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} \varphi_1 (\epsilon ). \end{split}\end{equation} Hence, from \eqref{eq:2.32} and \eqref{eq:2.33}, \begin{equation}\begin{split} \label{eq:2.34} \gamma_1(\epsilon )&=\langle P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} \varphi_2 (\epsilon ),\|P_{\mathcal{H}^{\epsilon _0}, \mathcal{H}}\varphi_1^{\epsilon_0}\|^{-1}_{\mathcal{H}} \| P_{\mathcal{H}^{\epsilon _0} , \mathcal{A}(\epsilon )}\varphi_1^{\epsilon_0} \|_{\mathcal{A}(\epsilon)}\\ &\quad \times P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} \varphi_1 (\epsilon) +P_{\mathcal{H}, \mathcal{B}(\epsilon)^\bot \mathcal{H}}\varphi_1\rangle _{\mathcal{H}}\\ &=\langle P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon)} \varphi_2 (\epsilon),\|P_{\mathcal{H}^{\epsilon_0}, \mathcal{H}}\varphi^{\epsilon_0}_1\|^{-1}_{\mathcal{H}}\| P_{\mathcal{H}^{\epsilon_0}, \mathcal{A}(\epsilon)}\varphi^{\epsilon_0}_1\|_{\mathcal{A}(\epsilon)}\\ &\quad \times P_{\mathcal{A}(\epsilon), \mathcal{B}(\epsilon)} \varphi_1(\epsilon)\rangle _{\mathcal{A}(\epsilon )} +\langle P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )}\varphi_2 (\epsilon),P_{\mathcal{H}, \mathcal{B}(\epsilon)^\bot \mathcal{H}}\varphi_1\rangle_{\mathcal{H}}\\ &= \langle {P_{\mathcal{A}(\epsilon), \mathcal{B}(\epsilon)}}\varphi_2 (\epsilon) ,\| P_{\mathcal{H}^{\epsilon_0}, \mathcal{H}}\varphi^{\epsilon_0}_1\|^{-1}_{\mathcal{H}} \| P_{\mathcal{H}^{\epsilon_0}, \mathcal{A}(\epsilon)} \varphi^{\epsilon_0}_1\|_{\mathcal{A}(\epsilon)}\\ &\quad \times P_{\mathcal{A}(\epsilon), \mathcal{B}(\epsilon)} \varphi_1(\epsilon )\rangle _{\mathcal{A}(\epsilon)}. \end{split}\end{equation} We show that the last line of \eqref{eq:2.34} tends to $0$ as $\epsilon\downarrow 0$: We have \begin{equation}\begin{split}\label{eq:2.35} 0&= \langle \varphi_2(\epsilon),\varphi_1 (\epsilon)\rangle _{\mathcal{A}(\epsilon)}\\ &=\langle P_{\mathcal{A}(\epsilon), \mathcal{B}(\epsilon)} \varphi_2 (\epsilon),P_{\mathcal{A}(\epsilon), \mathcal{B}(\epsilon)} \varphi_1 (\epsilon)\rangle_{\mathcal{B}(\epsilon)}\\ &\quad + \langle P_{\mathcal{A}(\epsilon ), \mathcal{C}(\epsilon)}\varphi_2 (\epsilon),P_{\mathcal{A}(\epsilon), \mathcal{C}(\epsilon)} \varphi_1(\epsilon)\rangle _{\mathcal{C}(\epsilon)}\\ &\quad +\langle P_{\mathcal{A}(\epsilon ), \mathcal{D}(\epsilon)} \varphi_2 (\epsilon), P_{\mathcal{A}(\epsilon), \mathcal{D}(\epsilon)} \varphi_1(\epsilon)\rangle_{\mathcal{D}(\epsilon)}. \end{split}\end{equation} Since, by (A28), \begin{equation}\label{eq:2.36} \lim_{\epsilon \downarrow 0}\|P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon)} \varphi_n (\epsilon)\|_{\mathcal{B} (\epsilon )}=1,\end{equation} we have \begin{equation}\label{eq:2.37} \lim_{\epsilon \downarrow 0} \| P_{\mathcal{A}(\epsilon), \mathcal{C}(\epsilon)} \varphi_n (\epsilon)\|_{\mathcal{C}(\epsilon)} =\lim_{\epsilon\downarrow 0}\| P_{\mathcal{A}(\epsilon), \mathcal{D}(\epsilon)} \varphi_n(\epsilon)\|_{\mathcal{D}(\epsilon)}=0. \end{equation} From \eqref{eq:2.35}, \eqref{eq:2.36} and \eqref{eq:2.37} we obtain \begin{equation}\label{eq:2.38} \lim_{\epsilon\downarrow 0} \langle P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon)} \varphi_2 (\epsilon),P_{\mathcal{A}(\epsilon), \mathcal{B}(\epsilon)}\varphi_1 (\epsilon)\rangle_{\mathcal{B} (\epsilon)}=0.\end{equation} Hence, by \eqref{eq:2.34}, \eqref{eq:2.38} and \eqref{eq:2.23}, we have \begin{equation}\label{eq:2.39} \lim_{\epsilon \downarrow 0} \gamma_1 (\epsilon)=0. \end{equation} Thus, by \eqref{eq:2.39} and (A28), we have \begin{equation}\begin{split}\label{eq:2.40} &\| P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} \varphi_2 (\epsilon) -\gamma_1 (\epsilon )\varphi_1 \|^2_{\mathcal{H}}\\ &= \| P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon)} \varphi_2 (\epsilon)-\gamma_1 (\epsilon) P_{\mathcal{H}, \mathcal{B}(\epsilon)} \varphi_1 -\gamma_1 (\epsilon)P_{\mathcal{H}, \mathcal{B}(\epsilon)^\bot \mathcal{H}}\varphi_1 \|_{\mathcal{H}}^2\\ &= \| P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon)} \varphi_2 (\epsilon) -\gamma_1 (\epsilon )P_{\mathcal{H}, \mathcal{B}(\epsilon)} \varphi_1 \|^2_{\mathcal{B}(\epsilon)} +\gamma_1 (\epsilon )^2 \|P_{\mathcal{H}, \mathcal{B}(\epsilon )^\bot \mathcal{H}}\varphi_1 \|^2_{\mathcal{H}}\\ & \to 1\quad \text{as } \epsilon\downarrow 0. \end{split}\end{equation} Combining \eqref{eq:2.29}, \eqref{eq:2.31}, \eqref{eq:2.39} and \eqref{eq:2.40}, we see that for all $\delta >0$, there exists $\epsilon _1 \in (0,\epsilon_0]$ such that \begin{equation}\label{eq:2.41} \mu_2 t\leq \mu_2 (\epsilon )t+\delta \end{equation} for all $\epsilon \in (0,\epsilon_1]$. The theorem now follows from \eqref{eq:2.27} and \eqref{eq:2.41}. \end{proof} \begin{definition}\label{def:2.21}\rm We now define the sequence $\{k_i\}^\infty_{i=1}$ of positive integers as follows: Suppose we list the eigenvalues $\{ \mu_n\}^\infty_{n=1}$ of $H$ in a way reflecting their multiplicities. Then the positive integers $k_i$ are defined by: \begin{equation}\begin{split}\label{eq:2.42} 0&=\mu_1 <\mu_2 =\mu_3 =\dots =\mu_{k_2}<\mu_{k_2+1}=\dots =\mu_{k_3}\\ &< \mu_{k_3+1}=\dots =\mu_{k_4}<\mu_{k_4+1}=\dots . \end{split}\end{equation} We also define $k_1=1$. \end{definition} \begin{lemma}\label{lem:2.22} Let $p\geq 1$ be an integer and let $i$ be an integer satisfying \[ k_p+10$ and a strictly decreasing sequence $\{\epsilon_m\}^\infty_{m=1}$ of positive numbers such that $\epsilon_m\downarrow 0$ as $m\to \infty$, and that \[ \mu_i(\epsilon _m)\geq \mu_i+\eta \quad (m=1,2,3,\dots ). \] For $j=1,2,3,\dots $ we regard $P_{\mathcal{H},\mathcal{B}(\epsilon)} \varphi_j$ as a vector in $\mathcal{A}(\epsilon )$ and write \begin{equation}\label{eq:2.45} P_{\mathcal{H}, \mathcal{B}(\epsilon)} \varphi_j =\sum^\infty_{\ell=1} a_{j,\ell} (\epsilon )\varphi_{\ell} (\epsilon ). \end{equation} Then, for all $00$ such that for all sufficiently small $\epsilon>0$ we have \[ \mu_{k_p+1} (\epsilon)\geq \mu_{k_p}+\eta. \] \end{lemma} \begin{proof} For $i=1,2,3,\dots$ let \begin{equation}\label{eq:2.54} P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon)} \varphi_i (\epsilon) =\sum^\infty_{\ell=1} b_{i,\ell} (\epsilon)\varphi_\ell , \end{equation} regarding $P_{\mathcal{A}(\epsilon),\mathcal{B}(\epsilon)} \varphi_i(\epsilon)$ as a vector in $\mathcal{H}$. Suppose the lemma is false. Then there exists a strictly decreasing sequence of positive numbers $\{ \epsilon_m\}^\infty_{m=1}$ such that $\epsilon_m\downarrow 0$ as $m\to \infty$ and that \begin{equation}\label{eq:2.55} \mu_{k_p+1}(\epsilon _m)\to \mu_{k_p} \quad \text{as } m\to \infty. \end{equation} Then, for all $00$ there exists $\epsilon _1\in (0,\epsilon _0]$ such that \begin{equation}\label{eq:2.74} e^{-\mu_{k_p+1}(\epsilon) t}\geq e^{-\mu_{k_p+1}t}-\delta \end{equation} for all $\epsilon \in (0,\epsilon_1]$. Next we prove the reverse inequality of \eqref{eq:2.74}. For $i=1,2,3,\dots$ and $\epsilon \in (0,\epsilon_0]$, let \begin{equation}\label{eq:2.75} P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} \varphi_i(\epsilon )=\sum^\infty_{\ell=1} \gamma_{i,\ell} (\epsilon )\varphi_\ell \in \mathcal{B}(\epsilon )\subseteq \mathcal{H} \end{equation} and let \begin{equation}\label{eq:2.76} g_{k_p+1} (\epsilon )=P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon)} \varphi_{k_p+1} (\epsilon ) -\sum^{k_p}_{\ell=1} \gamma _{k_p+1,\ell}(\epsilon )\varphi_\ell . \end{equation} Then %\label{eq:2.77} \begin{align} &e^{-\mu_{k_p+1}t} \nonumber\\ &\geq \| g_{k_p+1} (\epsilon )\| _{\mathcal{H}}^{-2} \langle e^{-Ht}g_{k_p+1}(\epsilon ),g_{k_p+1} (\epsilon ) \rangle _{\mathcal{H}} \nonumber\\ &=\| g_{k_p+1} (\epsilon )\|^{-2}_{\mathcal{H}}\{\langle e^{-Ht} P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon)}\varphi_{k_p+1} (\epsilon ), P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon)} \varphi_{k_p+1}(\epsilon )\rangle_{\mathcal{H}} \nonumber\\ &\quad -2\Big\langle e^{-Ht}P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon) } \varphi_{k_p+1} (\epsilon ), \sum^{k_p}_{\ell =1} \gamma _{k_p+1,\ell} (\epsilon )\varphi_\ell \Big\rangle _{\mathcal{H}} \nonumber\\ &\quad + \Big\langle e^{-Ht} \sum^{k_p}_{\ell =1} \gamma_{k_{p+1, \ell}} (\epsilon )\varphi_\ell , \sum^{k_p}_{\ell=1}\gamma_{k_p+1,\ell}(\epsilon )\varphi_\ell \Big\rangle_{\mathcal{H}}\Big\} \nonumber\\ &=\| g_{k_p+1} (\epsilon ) \|^{-2}_{\mathcal{H}} \Big\{ \langle e^{-Ht} P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} \varphi_{k_p+1}(\epsilon ) -P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} e^{-H(\epsilon )t} \varphi_{k_p+1} (\epsilon ), \nonumber\\ & \quad P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} \varphi_{k_p+1} (\epsilon )\rangle _{\mathcal{H}} +\langle P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} e^{-H(\epsilon )t} \varphi_{k_p+1} (\epsilon ), \nonumber\\ & \quad P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} \varphi_{k_p+1} (\epsilon )\rangle _{\mathcal{H}} -2\sum^{k_p}_{\ell =1} e^{-\mu _\ell t} \gamma_{k_p+1, \ell} (\epsilon )^2 \nonumber \\ &\quad +\sum_{\ell=1}^{k_p} e^{-\mu _\ell t} \gamma_{k_p+1,\ell } (\epsilon )^2\Big\} \label{eq:2.77} \\ &= \| g_{k_p+1} (\epsilon )\|^{-2}_{\mathcal{H}} \Big\{ \langle e^{-Ht} P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} \varphi_{k_p+1} (\epsilon ) -P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} e^{-H(\epsilon )t} \varphi_{k_p+1} (\epsilon ), \nonumber\\ & \quad P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} \varphi_{k_p+1} (\epsilon )\rangle_{\mathcal{H}} + e^{-\mu_{k_p+1}(\epsilon )t} \sum^\infty_{\ell=1} \gamma_{k_p+1,\ell}(\epsilon )^2 \nonumber\\ & \quad - \sum^{k_p}_{\ell =1} e^{-\mu _\ell t} \gamma _{k_p+1,\ell } (\epsilon )^2\Big\}. \nonumber \end{align} Now, by (A28), \begin{equation}\begin{split}\label{eq:2.78} &\| e^{-Ht} P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} \varphi_{k_p+1}(\epsilon ) -P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} e^{-H(\epsilon )t} \varphi_{k_p+1} (\epsilon )\|^2_{\mathcal{H}}\\ &= \sum^{\infty}_{\ell =1} \gamma_{k_p+1,\ell }(\epsilon )^2 (e ^{-\mu_\ell t} - e ^{-\mu _{k_p+1} (\epsilon )t} )^2 \to 0 \quad \text{as } \epsilon \downarrow 0, \end{split}\end{equation} in particular we have \begin{equation}\label{eq:2.79} \lim_{\epsilon \downarrow 0} \sum^{k_p}_{\ell =1} \gamma _{k_p+1,\ell} (\epsilon )^2 (e^{-\mu_\ell t} - e^{-\mu_{k_p+1}(\epsilon )t} )^2=0. \end{equation} But, by Lemma~\ref{lem:2.23}, there exists $\eta >0$ such that for all sufficiently small $\epsilon >0$ we have \begin{equation}\label{eq:2.80} \mu_{k_p+1} (\epsilon )\geq \mu_{k_p}+\eta . \end{equation} Thus, from \eqref{eq:2.79} and \eqref{eq:2.80}, we have \begin{equation}\label{eq:2.81} \lim_{\epsilon \downarrow 0} \sum^{k_p}_{\ell =1} \gamma_{k_p+1,\ell } (\epsilon)^2=0.\end{equation} Hence, by (A28), \eqref{eq:2.81} and \eqref{eq:2.76}, we obtain \begin{equation}\label{eq:2.82} \lim_{\epsilon \downarrow 0} \| g_{k_p+1} (\epsilon )\|_{\mathcal{H}}=1.\end{equation} Therefore, by \eqref{eq:2.75}, \eqref{eq:2.77}, \eqref{eq:2.81}, \eqref{eq:2.82} and (A28), given any $\delta >0$, there exists $\epsilon_1 \in (0,\epsilon _0]$ such that \begin{equation}\label{eq:2.83} e^{-\mu_{k_p+1}t}\geq e^{-\mu_{k_p+1}(\epsilon )t}-\delta \end{equation} for all $\epsilon \in (0,\epsilon_1]$. The lemma now follows from \eqref{eq:2.74} and \eqref{eq:2.83}. \end{proof} \begin{theorem}\label{thm:2.25} For all $i=1,2,3,\dots$, we have $ \lim_{\epsilon \downarrow 0} \mu_i(\epsilon )=\mu_i$. \end{theorem} The above theorem follows from Theorem~\ref{thm:2.20}, and Lemmas~\ref{lem:2.22} and \ref{lem:2.24}. \begin{theorem}\label{thm:2.26} For all $j=1,2,3,\dots $ and $\epsilon \in (0,\epsilon _0]$ let \begin{equation}\label{eq:2.84} P_{\mathcal{H}, \mathcal{B}(\epsilon)}\varphi_j =\sum^\infty_{\ell =1} a_{j,\ell} (\epsilon )\varphi_\ell (\epsilon ) \in \mathcal{B}(\epsilon )\subseteq \mathcal{A}(\epsilon ). \end{equation} Let $p\geq 1$ be an integer. For $i=k_p+1,\dots$, $k_{p+1}$ and $\epsilon \in (0,\epsilon_0]$ let \begin{equation}\label{eq:2.85} \hat{\psi}_i (\epsilon )=P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )}\Big(\sum^{k_{p+1}}_{\ell =k_p+1} a_{i,\ell} (\epsilon )\varphi_\ell (\epsilon )\Big) \end{equation} and let \begin{equation}\label{eq:2.86} \psi_{i}(\epsilon )=\|\hat{\psi}_i(\epsilon ) \|^{-1}_{\mathcal{B}(\epsilon )} \hat{\psi}_i (\epsilon ). \end{equation} Then for each $i=k_p+1,\dots ,k_{p+1}$ we have \begin{equation}\label{eq:2.87} \lim_{\epsilon \downarrow 0} \| \varphi_i -\psi _i (\epsilon )\|_{\mathcal{H}} =0. \end{equation} \end{theorem} \begin{proof} For $i=k_p+1,\dots ,k_{p+1}$ and $\epsilon \in (0,\epsilon_0]$ we have \begin{equation}\begin{split}\label{eq:2.88} &\| \varphi_i-\psi_i(\epsilon )\|_{\mathcal{H}}\\ &\leq \| \varphi_i-P_{\mathcal{H}, \mathcal{B}(\epsilon )} \varphi_i \| _{\mathcal{H}}+ \| P_{\mathcal{H}, \mathcal{B}(\epsilon )} \varphi_i -\hat{\psi }_i (\epsilon )\|_{\mathcal{B}(\epsilon )} + \| \hat{\psi }_i(\epsilon )-\psi _i (\epsilon ) \|_{\mathcal{B}(\epsilon )}\\ &\leq \| \varphi_i -P_{\mathcal{H}, \mathcal{B}(\epsilon )} \varphi_i \| _{\mathcal{H}} +\big\| P_{\mathcal{H}, \mathcal{B}(\epsilon )} \varphi_i-\sum^{k_{p+1}} _{\ell =k_p+1} a_{i,\ell} (\epsilon ) \varphi_\ell (\epsilon ) \big\|_{\mathcal{A}(\epsilon )}\\ &\quad + \big\| \sum^{k_{p+1}}_{\ell =k_p+1} a_{i, \ell} (\epsilon )[\varphi_{\ell}(\epsilon )-P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon)} \varphi_\ell (\epsilon )] \big\|_{\mathcal{A}(\epsilon )} + \| \hat{\psi}_i(\epsilon )-\psi_i (\epsilon ) \|_{\mathcal{B}(\epsilon )}. \end{split}\end{equation} Consider the term \[ \big\| P_{\mathcal{H}, \mathcal{B}(\epsilon )} \varphi_i -\sum^{k_{p+1}}_{\ell =k_p+1} a_{i, \ell } (\epsilon ) \varphi_\ell (\epsilon )\big\|_{\mathcal{A}(\epsilon )} \] in \eqref{eq:2.88}. We have \begin{equation}\begin{split}\label{eq:2.89} & e^{-H(\epsilon )t} P_{\mathcal{H}, \mathcal{B}(\epsilon )} \varphi_i -P_{\mathcal{H}, \mathcal{B}(\epsilon )} e^{-Ht}\varphi_i\\ &= \Big( \sum^{k_p}_{\ell=1} +\sum^{k_{p+1}}_{\ell =k_p+1} +\sum^{\infty}_{\ell =k_{p+1}+1}\Big) a_{i,\ell } (\epsilon )[e^{-\mu_\ell (\epsilon )t} -e ^{-\mu_it}]\varphi_\ell(\epsilon ). \end{split}\end{equation} By (A28) and the orthogonality of $\{ \varphi_\ell (\epsilon )\}^\infty_{\ell =1}$, each of the three sums in \eqref{eq:2.89} approaches $0$ as $\epsilon \downarrow 0$. Hence, together with Theorem~\ref{thm:2.25}, we have, for $i=k_p+1,\dots ,k_{p+1}$, \begin{equation}\label{eq:2.90} \Big( \sum^{k_p}_{\ell =1} +\sum^\infty_{\ell=k_{p+1}+1}\Big) a_{i,\ell}(\epsilon )^2\to 0 \quad \text{as } \epsilon \downarrow 0. \end{equation} Thus, for $i=k_p+1,\dots ,k_{p+1}$, \begin{equation}\begin{split}\label{eq:2.91} &\lim_{\epsilon \downarrow 0} \big\|P_{\mathcal{H}, \mathcal{B}(\epsilon )} \varphi_i-\sum^{k_{p+1}}_{\ell =k_p+1} a_{i,\ell } (\epsilon ) \varphi_\ell (\epsilon )\big\|^2_{\mathcal{A}(\epsilon)}\\ &= \lim_{\epsilon \downarrow 0} \Big( \sum^{k_p}_{\ell =1} +\sum^{\infty}_{\ell =k_{p+1}+1} \Big) a_{i,\ell }(\epsilon )^2=0. \end{split}\end{equation} By (A9) and (A28) we have \[ \lim_{\epsilon \downarrow 0} \| \varphi_\ell (\epsilon ) -P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} \varphi_\ell (\epsilon )\|_{\mathcal{A}(\epsilon )}=0 \] for all $\ell =1,2,3,\dots$. Thus \begin{equation}\label{eq:2.92} \lim_{\epsilon \downarrow 0} \big\| \sum^{k_{p+1}}_{\ell =k_p+1} a_{i,\ell } (\epsilon ) [\varphi_\ell (\epsilon ) -P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} \varphi_\ell (\epsilon )]\big\|_{\mathcal{A}(\epsilon )}=0. \end{equation} By (A7) and \eqref{eq:2.91} we have, for $i=k_p+1,\dots ,k_{p+1}$, \begin{equation}\label{eq:2.93} \lim_{\epsilon \downarrow 0} \sum^{k_{p+1}}_{\ell =k_p+1} a_{i,\ell }(\epsilon )^2=1. \end{equation} Thus, for $i=k_p+1,\dots , k_{p+1}$, \eqref{eq:2.85}, \eqref{eq:2.92} and \eqref{eq:2.93} imply \begin{equation}\begin{split}\label{eq:2.94} \lim_{\epsilon \downarrow 0} \| \hat{\psi}_i (\epsilon )\|_{\mathcal{B}(\epsilon)} &= \lim_{\epsilon \downarrow 0} \| \hat{\psi}_i(\epsilon ) \|_{\mathcal{A}(\epsilon)}\\ &=\lim_{\epsilon \downarrow 0} \big\| \Big( \hat{\psi}_i (\epsilon ) -\sum^{k_{p+1}}_{\ell =k_p+1} a_{i,\ell} (\epsilon )\varphi_\ell (\epsilon )\Big) +\sum^{k_{p+1}}_{\ell =k_p+1} a_{i,\ell } (\epsilon )\varphi_\ell (\epsilon )\big\|_{\mathcal{A}(\epsilon )}\\ &=\lim_{\epsilon \downarrow 0} \big\| \sum^{k_{p+1}}_{\ell =k_p+1} a_{i,\ell } (\epsilon )\varphi_\ell (\epsilon )\big\| _{\mathcal{A}(\epsilon )}\\ &=\lim_{\epsilon \downarrow 0} \Big\{ \sum^{k_{p+1}}_{\ell =k_p+1} a_{i,\ell } (\epsilon )^2 \Big\}^{1/2} =1. \end{split}\end{equation} Therefore, \begin{equation} \label{eq:2.95} \lim_{\epsilon \downarrow 0}\| \hat{\psi}_i(\epsilon ) -\psi_i (\epsilon )\|_{\mathcal{B}(\epsilon )} =\lim_{\epsilon \downarrow 0} |1-\| \hat{\psi}_i (\epsilon ) \|^{-1}_{\mathcal{B}(\epsilon )} |\|\hat{\psi}_i (\epsilon ) \|_{\mathcal{B}(\epsilon )} =0. \end{equation} The theorem now follows from \eqref{eq:2.88}, (A7), \eqref{eq:2.91}, \eqref{eq:2.92} and \eqref{eq:2.95}. \end{proof} \section{Application to Neumann Laplacians on domains in $\mathbb{R}^N$}\label{sec:3} The purpose of this section is to show that the assumptions (A1)--(A28) in Section~\ref{sec:2} are all satisfied when applying the abstract theory in Section~\ref{sec:2} to the situation studied in this section. For our application, it will be easy to check that (A1)--(A27) are satisfied. So we shall show that (A28) holds for our application. Throughout this section we let $\Omega \subseteq \mathbb{R}^N$ be a bounded Sobolev extension domain. Fix a sufficiently small $\epsilon _0>0$. For each $\epsilon \in (0,\epsilon _o]$ let $\Omega_\epsilon $, $\Omega^\epsilon$ and $\Omega (\epsilon )$ be bounded Sobolev extension domains in $\mathbb{R}^N$ satisfying \begin{equation}\label{eq:3.1} \begin{gathered} \Omega_\epsilon \supseteq \{ x \in \Omega :\operatorname{dist} (x ,\partial \Omega )>\epsilon \},\\ \Omega ^\epsilon \subseteq \{ x \in \mathbb{R}^N:\operatorname{dist} (x ,\Omega )<\epsilon \},\\ \Omega _\epsilon \subseteq \Omega (\epsilon )\subseteq \Omega^\epsilon . \end{gathered}\end{equation} We shall assume that $\{ \Omega _\epsilon \}_{0<\epsilon \leq \epsilon_0}$ is a decreasing family of domains in the sense that \begin{equation}\label{eq:3.2} \Omega_{\epsilon_1}\supseteq \Omega_{\epsilon _2}\quad \text{if } 0<\epsilon_1 \leq \epsilon _2. \end{equation} Similarly we shall assume that $\{\Omega^\epsilon \}_{0<\epsilon \leq \epsilon _0}$ is an increasing family of domains in the sense that \begin{equation}\label{eq:3.3} \Omega^{\epsilon _1} \subseteq \Omega^{\epsilon _2}\quad \text{if }0<\epsilon _1\leq \epsilon_2. \end{equation} We shall apply the abstract theory in Section~\ref{sec:2} by putting: \begin{equation} \begin{gathered} \mathcal{H}^\epsilon =L^2(\Omega^\epsilon ),\quad \mathcal{H}_\epsilon =L^2(\Omega_\epsilon ),\quad \mathcal{H}(\epsilon )=L^2(\Omega (\epsilon ))=\mathcal{A}(\epsilon ), \\ \mathcal{B} (\epsilon )=L^2(\Omega \cap \Omega (\epsilon )),\quad \mathcal{C}(\epsilon )=L^2 (\Omega (\epsilon) \backslash \Omega ),\quad \mathcal{D}(\epsilon )=\{ 0\}. \end{gathered}\label{eq:3.4} \end{equation} Let $-\Delta _\epsilon $, $-\Delta $, $-\Delta ^\epsilon$, $-\Delta (\epsilon )$ be the Neumann Laplacian defined on $\Omega_\epsilon $, $\Omega$, $\Omega^\epsilon$ and $\Omega (\epsilon )$, respectively. When applying the abstract theory in Section~\ref{sec:2} we shall put \begin{equation}\label{eq:3.5} H_\epsilon =-\Delta_\epsilon ,\quad H=-\Delta ,\quad H^\epsilon =-\Delta^\epsilon,\quad H(\epsilon )=-\Delta (\epsilon ). \end{equation} We shall write $P_\epsilon (t,x,y)$, $P(t,x,y)$, $P^\epsilon (t,x,y)$ and $P(\epsilon )(t,x,y)$ for the heat kernel of $e^{\Delta _\epsilon t}$, $e^{\Delta t}$, $e ^{\Delta ^\epsilon t}$ and $e^{\Delta (\epsilon )t}$, respectively. We shall assume that there exists a positive continuous functions $c:(0,1]\to (0,\infty)$ such that \begin{equation} \begin{gathered} P_\epsilon (t,x,y)\leq c(t) \quad (x,y\in \Omega_\epsilon ),\\ P(t,x,y)\leq c(t) \quad (x,y\in \Omega),\\ P^\epsilon (t,x,y)\leq c(t) \quad (x,y\in \Omega^\epsilon )\\ P(\epsilon )(t,x,y) \leq c(t) \quad (x,y\in \Omega (\epsilon )) \end{gathered}\label{eq:3.6} \end{equation} for all $0<\epsilon \leq \epsilon_0$ and all $0\eta$ and $ t_1-\tau _1 \geq \eta^2$. Then \[ |u(x ,t)-u(y,s)|\leq A[|x-y|+|t-s|^{1/2}]^{\alpha} \] for all $x,y\in \Sigma '$ and $t,s\in [t_1,\tau_2)$, where $\alpha\in (0,1]$ depends only on $d$ and $\lambda$, and \[ A=\big( \frac{4}{\eta}\big)^\alpha \theta \] where $\theta$ is the oscillation of $u$ in $\Sigma \times (\tau _1,\tau_2)$. \end{proposition} \begin{theorem}\label{thm:3.2} We have $\lim_{\epsilon \downarrow 0} P(\epsilon )(t,x,y)=P(t,x,y)$ for all $t\in (0,1]$ and $x,y\in \Omega$. \end{theorem} \begin{proof} Suppose, for a contradiction, that for some $t_0\in (0,1]$ and some $x_0,y_0\in \Sigma $ we have \begin{equation}\label{eq:3.7} P(\epsilon )(t_o,x_0,y_0) \not\to P(t_o,x_o,y_o)\quad \text{as } \epsilon \downarrow 0. \end{equation} Then there exist $c_1\geq 1$ and a decreasing sequence $\{\epsilon_n\}^\infty_{n=1}$ of positive numbers such that $\epsilon _n\downarrow 0$ as $n\to \infty$ and that \begin{equation}\label{eq:3.8} c^{-1}_1\leq |P(\epsilon _n)(t_0,x_0,y_0)-P(t_0,x_0,y_0)|\quad (n=1,2,3,\dots ). \end{equation} Applying Proposition~\ref{prop:3.1} with \begin{equation} \label{eq:3.9} \begin{gathered} \Sigma =B\big( x_0, \frac{5}{8} \operatorname{dist} (x_0,\partial \Omega)\big),\quad \Sigma '=B\big( x_0, \frac{1}{8} \operatorname{dist} (x_0,\partial \Omega )\big),\\ u(t,x)=P(t,x,y_0),\quad \lambda =1,\quad \tau_1 =\frac{1}{4} t_0,\quad \tau_2=1,\quad t_1=\frac{1}{2}t_0,\\ \eta =\min\big\{ \frac{3}{8} \operatorname{dist} (x_0,\partial \Omega), \frac{1}{2} t^{1/2}_0\big\}, \end{gathered} \end{equation} we obtain, for all $s,t\in (t_1,\tau _2) =(t_0/2,1)$ and all $x \in B(x_0, \operatorname{dist} (x_0,\partial \Omega)/8)$, \begin{equation}\label{eq:3.10} |P(t,x,y_0)-P(s,x_0,y_0)|\leq A[|x-x_0|+|t-s|^{1/2}]^\alpha \end{equation} where $\alpha \in (0,1]$ depends only on $N$ and \begin{equation}\label{eq:3.11} A=\big(\frac{4}{\eta}\big)^\alpha \theta\end{equation} where \begin{equation}\label{eq:3.12} \theta=\sup_{\frac{1}{4}t_0\leq t\leq 1} c(t). \end{equation} (Hence $A$ depends only on $N$, $\operatorname{dist} (x_0,\partial \Omega)$ and $t_0$.) We may assume that, for all $n=1,2,3,\dots$, we have \[ 0<\epsilon_n<\min \{ \frac{3}{8}\operatorname{dist} (x_0,\partial \Omega), \frac{3}{8} \operatorname{dist} (y_0,\partial \Omega)\}. \] By a similar argument we deduce that \begin{equation}\label{eq:3.13} |P(\epsilon_n)(t,x,y_0)-P(\epsilon_n)(s,x_0,y_0) | \leq A[|x-x_0|+|t-s|^{1/2} ]^\alpha \end{equation} for all $s, t\in (t_1,\tau_2) =( t_0/2,1)$, all $x\in B(x_0, \operatorname{dist} (x_0,\partial \Omega)/8)$ and all $n=1,2,3,\dots $, and where $\alpha $ and $A$ in \eqref{eq:3.13} have the same values as those in \eqref{eq:3.10}. Let \begin{equation}\label{eq:3.14} R=\min\big\{ (4Ac_1)^{-\frac{1}{\alpha}},\frac{1}{8} \operatorname{dist} (x_0,\partial \Omega )\big\}. \end{equation} Then, for all $x\in B (x_0,R)$, $t\in (t_0/2,1)$ and $n=1,2,3,\dots $, we have \begin{gather}\label{eq:3.15} |P(t,x,y_0)-P(t,x_0,y_0)|\leq (4c_1)^{-1},\\ \label{eq:3.16} |P(\epsilon_n)(t,x,y_0)-P(\epsilon _n)(t,x_0,y_0)|\leq (4c_1)^{-1}. \end{gather} For $x\in B(x_0,R)$ and $n=1,2,3,\dots $ we have \begin{equation}\begin{split}\label{eq:3.17} &|P(\epsilon _n)(t_0,x_0,y_0)-P(t_0,x_0,y_0)|\\ &\leq |P(\epsilon_n) (t_0,x_0,y_0)-P(\epsilon_n)(t_0,x,y_0)| +|P(\epsilon _n)(t_0,x,y_0)-P(t_0,x,y_0)|\\ &\quad +|P(t_0,x,y_0)-P(t_0,x_0,y_0)|. \end{split}\end{equation} So, by \eqref{eq:3.8}, \eqref{eq:3.15}, \eqref{eq:3.16} and \eqref{eq:3.17}, we obtain \begin{equation}\label{eq:3.18} \frac{1}{2} c^{-1}_1 \leq |P(\epsilon _n)(t_0,x,y_0)-P(t_0,x,y_0)| \end{equation} for all $x\in B(x_0,R)$ and $n=1,2,3,\dots $. Integrating \eqref{eq:3.18} over $B(x_0,R)$ we obtain \begin{equation}\begin{split}\label{eq:3.19} &\frac{1}{2} c^{-1}_1 |B(x_0,R)|\\ &\leq \Big|\int_{\Omega }P(t_0 ,x,y_0)1_{B(x_0,R)} (x) \,dx -\int_{\Omega (\epsilon_n)} P(\epsilon _n)(t_0,x,y_0) 1_{B(x_0,R)}(x)\, dx\Big|. \end{split} \end{equation} Put \[ u(t,y)=\int_\Omega P(t,x,y)1_{B(x_0,R)} (x)\, dx \] and, for $n=1,2,3,\dots $, put \[ u_n(t,y)=\int_{\Omega (\epsilon _n)} P(\epsilon _n)(t,x,y)1_{B(x_0,R)} (x)\, dx. \] Then $u(t,y)$ and $u_n(t,y)$ satisfy the parabolic equations \begin{gather*} \frac{\partial u}{\partial t} =\Delta u \quad\text{in } (0,1)\times \Omega,\\ \frac{\partial u_n}{\partial t} =\Delta u_n \quad \text{in } (0,1)\times \Omega (\epsilon _n), \end{gather*} respectively. So we can apply the parabolic Harnack inequality (Proposition~\ref{prop:3.1}) to $u(t,y)$ and $u_n(t,y)$ and, as in \eqref{eq:3.10} and \eqref{eq:3.13}, obtain \begin{equation}\label{eq:3.20} |u(t,y)-u(s,y_0)|\leq \tilde{A}[|y-y_0| +|t-s|^{1/2}|^{\tilde{\alpha}} \end{equation} for all $y\in B(y_0,\operatorname{dist} (y_0,\partial\Omega)/8)$ and $t,s\in (t_0/2,1)$, and \begin{equation}\label{eq:3.21} |u_n(t,y)-u_n(s,y_0)|\leq \tilde{A}[|y-y_0|+|t-s|^{1/2}] ^{\tilde{\alpha}} \end{equation} for all $y\in B(y_0,\operatorname{dist} (y_0,\partial \Omega )/8)$ and $t,s\in (t_0/2,1)$ where $\tilde{\alpha}\in (0,1]$ depends only on $N$ and \[ \tilde{A}=\big(\frac{4}{\tilde{\eta}} \big)^{\tilde{\alpha}} \tilde{\theta}\leq \big(\frac{4}{\tilde{\eta}}\big)^{\tilde{\alpha}} \] where \[ \tilde{\eta} = \min\{\frac{3}{8}\operatorname{dist} (y_0,\partial \Omega ), \frac{1}{2}t_0^{1/2}\} \] and \begin{align*} \tilde{\theta} &=\sup \big\{ \frac{1}{4} t_0\leq t\leq 1, |y-y_0| \leq \frac{5}{8} \operatorname{dist} (y_0,\partial \Omega): u(t,y)\big\}\\ &\leq \sup \big\{ \frac{1}{4} t_0 \leq t\leq 1, \,|y-y_0| \leq \frac{5}{8} \operatorname{dist} (y_0,\partial \Omega ): \int_\Omega P(t,x,y)\,dx\big\} \leq 1. \end{align*} (Hence $\tilde{A}$ depends only on $N$, $t_0$ and $\operatorname{dist}(y_0,\partial \Omega)$.) Let \[ \tilde{R} =\min\big\{ \big[ \frac{1}{8} |B(x_0,R) |c^{-1}_1 \tilde{A}^{-1} \big]^{\frac{1}{\tilde{\alpha}}} , \frac{1}{8} \operatorname{dist} (y_0,\partial \Omega )\big\}. \] Then, by \eqref{eq:3.20} and \eqref{eq:3.21}, \begin{gather}\label{eq:3.22} |u(t_0,y)-u(t_0,y_0)|\leq \frac{1}{8} |B(x_0,R)|c^{-1}_1,\\ \label{eq:3.23} |u_n (t_0,y)-u_n(t_0,y_0)|\leq \frac{1}{8} |B(x_0,R)|c^{-1}_1 \end{gather} for all $y\in B(y_0,\tilde{R}$). Thus, for all $y\in B(y_0,\tilde{R})$, we have \begin{equation}\begin{split}\label{eq:3.24} &|u_n(t_0,y_0)-u(t_0,y_0)|\\ &\leq |u_n(t_0,y_0)-u_n(t_0,y)|+|u_n(t_0,y)-u(t_0,y)| + |u(t_0,y)-u(t_0,y_0)|\\ &\leq \frac{1}{4} |B(x_0,R)|c^{-1}_1 + |u_n(t_0,y)-u(t_0,y)|. \end{split}\end{equation} So, by \eqref{eq:3.19} and \eqref{eq:3.24}, we have \begin{equation}\label{eq:3.25} \frac{1}{4} |B(x_0,R)|c^{-1}_1 \leq |u_n(t_0,y)-u(t_0,y)| \end{equation} for all $y\in B(y_0,\tilde{R})$. But \[ u_n(t_0,y)= \int_{\Omega (\epsilon _n)} P(\epsilon _n) (t_0,x,y)1_{B(x_0,R)} (x)\,dx =(e^{-H(\epsilon _n)t_0} \, 1_{B(x_0,R)})(y) \] and \[ u(t_0,y)= \int_\Omega P(t_0,x,y)1_{B(x_0,R)} (x)\,dx =(e^{-Ht_0} \, 1_{B(x_0,R)})(y). \] Thus \eqref{eq:3.25} implies \begin{align*} &\int_{B(y_0,\tilde{R})} |(e^{-H(\epsilon _n)t_0} 1_{B(x_0,R)} )(y)-(e^{-Ht_0}1_{B(x_0,R)})(y)|^2\,dy\\ & \geq \frac{1}{16} c^{-2}_1 |B(x_0,R)|^2 |B(y_0,\tilde{R})|, \end{align*} hence, for all $n=1,2,3,\dots$, \begin{equation}\begin{split}\label{eq:3.26} &\|e^{-H(\epsilon _n)t_0} 1_{B(x_0,R)} - e^{-Ht_0} 1_{B(x_0,R)} \|^2_{L^2(B(y_0,\tilde{R}))}\\ &\geq \frac{1}{16} c^{-2}_1 |B(x_0,R)|^2|B(y_0,\tilde{R})|. \end{split}\end{equation} Let $f\in L^2(\Omega^{\epsilon_0})$ be the function defined by \[ f(y)=\begin{cases} 1 &\text{if } |y-x_0|0$ we can first choose $\epsilon_1\in (0,\epsilon_0]$ such that \[ c(1)^{1/2} e^{\lambda_n(B)/2} |\Omega \backslash \Omega _{\epsilon_1} |2c(t)\leq \delta /2, \] then, by Theorem~\ref{thm:3.2}, we can choose $\epsilon_2\in (0,\epsilon_1]$ such that \[ c(1)^{1/2}e^{\lambda_n (B)/2} \int_{\Omega_{\epsilon _1}} |P(t,x,y)-P(\epsilon )(t,x,y)|\, dy\leq \delta/2 \] for all $\epsilon \in (0,\epsilon_2]$. Thus for all $t\in (0,1]$, $n=1,2,3,$ $\dots $ and $x\in \Omega$, we have \begin{equation}\label{eq:3.38} \lim_{\epsilon \downarrow 0} F_{t,n,\epsilon} (x)=0. \end{equation} From \eqref{eq:3.37} we see that \begin{equation}\label{eq:3.39} F_{t,n,\epsilon} (x)\leq [c(1)^{1/2} e^{\lambda_n(B)/2} 5c(t) |\Omega |]^2 \end{equation} for all $t\in (0,1]$, $n=1,2,3,\dots$, $\epsilon \in (0,\epsilon_0]$ and $x\in \Omega$. Therefore, by \eqref{eq:3.38}, \eqref{eq:3.39} and the dominated convergence theorem, we have \begin{equation}\begin{split} \label{eq:3.40} & \lim_{\epsilon \downarrow 0} \int_{\Omega \cap \Omega (\epsilon)} |(P_{\mathcal{H}, \mathcal{B}(\epsilon)} e^{-Ht} \varphi_n - e^{-H(\epsilon )t}P_{\mathcal{H}, \mathcal{B} (\epsilon)} \varphi_n)(x)|^2\, dx\\ &= \lim_{\epsilon\downarrow 0} \int_\Omega F_{t,n,\epsilon} (x)\, dx = 0 \end{split}\end{equation} for all $t\in (0,1]$ and $n=1,2,3,\dots$. So \begin{equation}\begin{split}\label{eq:3.41} & \| P_{\mathcal{H}, \mathcal{B}(\epsilon)} e^{-Ht} \varphi_n -e^{-H(\epsilon )t}P_{\mathcal{H}, \mathcal{B}(\epsilon)} \varphi_n\|^2_{\mathcal{A}(\epsilon)}\\ &= \int_{\Omega (\epsilon)} |(P_{\mathcal{H}, \mathcal{B}(\epsilon)} e^{-Ht}\varphi_n- e^{-H(\epsilon)t} P_{\mathcal{H}, \mathcal{B}(\epsilon)} \varphi_n)(x)|^2\, dx\\ &= \int_{\Omega (\epsilon )\backslash \Omega} |(P_{\mathcal{H}, \mathcal{B}(\epsilon )} e^{-Ht} \varphi_n- e^{-H(\epsilon )t} P_{\mathcal{H}, \mathcal{B}(\epsilon)} \varphi_n)(x)|^2\, dx\\ &\quad + \int_{\Omega (\epsilon)\cap \Omega} |(P_{\mathcal{H}, \mathcal{B}(\epsilon)} e^{-Ht}\varphi_n - e^{-H(\epsilon )t} P_{\mathcal{H}, \mathcal{B}(\epsilon )} \varphi_n )(x)|^2\, dx\\ &\leq |\Omega (\epsilon )\backslash \Omega |\{ |\Omega \cap \Omega (\epsilon) |c(t)c(1)^{1/2}e^{\lambda_n (B)/2}\}^2\\ &\quad +\int_{\Omega (\epsilon )\cap \Omega} |(P_{\mathcal{H}, \mathcal{B}(\epsilon)} e^{-Ht} \varphi_n- e^{-H(\epsilon )t} P_{\mathcal{H}, \mathcal{B}(\epsilon)} \varphi_n )(x)|^2\, dx\\ &\to 0 \quad \text{as } \epsilon \downarrow 0 \end{split}\end{equation} where we have used \eqref{eq:3.27}, \eqref{eq:3.33} and \eqref{eq:3.40}. Hence the first equality in (A28) holds in this application. We next consider the second equality in (A28). For $x\in \Omega$, $t\in (0,1]$ and $n=1,2,3,\dots $ we have \begin{equation}\begin{split} & (e^{-Ht} P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon )} \varphi_n (\epsilon) -P_{\mathcal{A}(\epsilon), \mathcal{B}(\epsilon)} e^{-H(\epsilon )t} \varphi_n (\epsilon) )(x)\\ &= \int_{\Omega \cap \Omega (\epsilon)} P(t,x,y)\varphi_n (\epsilon) (y)\, dy-1_{\Omega \cap \Omega (\epsilon)} (x) f_{t,n,\epsilon} (x) \end{split}\end{equation} where \begin{equation}\label{eq:3.43} f_{t,n,\epsilon} (x)=\begin{cases} 0 &\text{if } x\in \Omega \backslash \Omega (\epsilon)\\ \int_{\Omega (\epsilon)} P(\epsilon) (t,x,y)\varphi_n (\epsilon)(y)\, dy&\text{if } x\in \Omega \cap \Omega (\epsilon). \end{cases}\end{equation} So for $x\in \Omega \backslash \Omega (\epsilon )$, $t\in (0,1]$, $n=1,2,3,\dots$ and $\epsilon \in (0,\epsilon_0]$, by \eqref{eq:3.30}, we have \begin{equation}\begin{split}\label{eq:3.44} & |(e^{-Ht} P_{\mathcal{A}(\epsilon), \mathcal{B}(\epsilon)} \varphi_n (\epsilon) -P_{\mathcal{A}(\epsilon), \mathcal{B}(\epsilon)} e^{-H(\epsilon )t} \varphi_n (\epsilon)) (x)|\\ &= \Big|\int_{\Omega \cap \Omega (\epsilon)} P(t,x,y)\varphi_n(\epsilon )(y)\, dy\Big|\\ &\leq |\Omega \cap \Omega (\epsilon)|c(t)c(1)^{1/2} e^{\lambda_n(B)/2}. \end{split}\end{equation} For $t\in (0,1]$, $n=1,2,3,\dots $ and $\epsilon \in (0,\epsilon_0]$ we define $G_{t,n,\epsilon}: \Omega \to \mathbb{R}$ by \begin{equation}\label{eq:3.45} G_{t,n,\epsilon} (x)=\begin{cases} 0 &\text{if } x\in \Omega \backslash \Omega (\epsilon)\\ |(e^{-Ht} P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon)} \varphi_n (\epsilon)-P_{\mathcal{A}(\epsilon), \mathcal{B}(\epsilon)} e^{-H(\epsilon )t} \varphi_n(\epsilon ))(x)| &\text{if } x\in \Omega \cap \Omega (\epsilon). \end{cases} \end{equation} If $0<\epsilon \leq \epsilon_1\leq \epsilon_0$ and $x\in \Omega \cap \Omega (\epsilon)$, then \begin{equation}\begin{split}\label{eq:3.46} G_{t,n,\epsilon} (x) &= \Big| \int_{\Omega \cap \Omega (\epsilon)} [P(t,x,y)-P(\epsilon )(t,x,y)]\varphi_n(\epsilon) (y)\, dy\\ &\quad - \int_{\Omega (\epsilon)\backslash \Omega} P(\epsilon )(t,x,y)\varphi_n (\epsilon )(y)\, dy\Big|\\ &\leq \Big| \int_{(\Omega \cap \Omega (\epsilon))\backslash \Omega_{\epsilon_1}} [P(t,x,y)-P(\epsilon)(t,x,y)] \varphi_n (\epsilon)(y)\, dy\Big|\\ &\quad + \Big| \int_{\Omega_{\epsilon_1}} [P(t,x,y) -P(\epsilon) (t,x,y)]\varphi_n (\epsilon) (y)\, dy\Big|\\ &\quad + \Big| \int_{\Omega (\epsilon )\backslash \Omega} P(\epsilon )(t,x,y)\varphi_n (\epsilon )(y) \, dy\Big|\\ &\leq c(1)^{1/2} e^{\lambda_n(B)/2} \{ |\Omega^{\epsilon_1} \backslash \Omega_{\epsilon_1} |3c(t)\\ &\quad +\int_{\Omega_{\epsilon_1}} |P(t,x,y) -P(\epsilon )(t,x,y)|\, dy\}. \end{split}\end{equation} Thus, for fixed $t\in (0,1]$, $n=1,2,3,\dots$ and $x\in \Omega$, given any $\delta >0$ we can first choose $\epsilon_1\in (0,\epsilon_0]$ such that \[ c(1)^{1/2} e^{\lambda_n (B)/2} |\Omega^{\epsilon_1} \backslash \Omega_{\epsilon_1} |3c(t) \leq \delta /2, \] then, by Theorem~\ref{eq:3.2}, we can find $\epsilon_2\in (0,\epsilon_1]$ such that \[ c(1)^{1/2} e^{\lambda_n(B)/2} \int_{\Omega _{\epsilon_1}} |P(t,x,y)-P(\epsilon )(t,x,y)|\, dy\leq \delta /2 \] for all $\epsilon \in (0,\epsilon_2]$. Therefore, for all $t\in (0,1]$, $n=1,2,3,\dots $ and $x\in \Omega$, we have \begin{equation}\label{eq:3.47} \lim_{\epsilon\downarrow 0} G_{t,n,\epsilon} (x)=0. \end{equation} Also, from \eqref{eq:3.46}, we have, for all $t\in (0,1]$, $n=1,2,3,\dots$, $\epsilon \in (0,\epsilon_0]$ and $x\in\Omega$, \begin{equation}\label{eq:3.48} G_{t,n,\epsilon} (x)\leq c(1)^{1/2}e^{\lambda_n(B)/2} 5c(t) |\Omega^{\epsilon_0}|. \end{equation} Hence we have \begin{equation}\begin{split}\label{eq:3.49} & \lim_{\epsilon \downarrow 0} \int_{\Omega \cap \Omega (\epsilon)} |(e^{-Ht} P_{\mathcal{A}(\epsilon), \mathcal{B}(\epsilon)} \varphi_n (\epsilon)-P_{\mathcal{A}(\epsilon), \mathcal{B}(\epsilon)} e^{-H(\epsilon)t} \varphi_n (\epsilon))(x)|^2\, dx\\ &= \lim_{\epsilon\downarrow 0} \int_\Omega G_{t,n,\epsilon} (x)^2 \, dx = 0 \end{split}\end{equation} for all $t\in (0,1]$ and $n=1,2,3,\dots$, using \eqref{eq:3.45}, \eqref{eq:3.47}, \eqref{eq:3.48} and the dominated convergence theorem. So, for all $t\in (0,1]$ and $n=1,2,3,\dots$, we have \begin{align*} & \| e^{-Ht} P_{\mathcal{A}(\epsilon), \mathcal{B}(\epsilon)} \varphi_n(\epsilon) -P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon)} e^{-H(\epsilon)t} \varphi_n(\epsilon)\|^2_{\mathcal{H}}\\ &= \int_\Omega |(e^{-Ht} P_{\mathcal{A}(\epsilon), \mathcal{B}(\epsilon)}\varphi_n(\epsilon) -P_{\mathcal{A}(\epsilon), \mathcal{B}(\epsilon)}e^{-H(\epsilon)t} \varphi_n(\epsilon))(x)|^2\, dx\\ &= \Big( \int_{\Omega \backslash \Omega (\epsilon)} +\int_{\Omega \cap \Omega (\epsilon)} \Big) |(e ^{-Ht} P_{\mathcal{A}(\epsilon), \mathcal{B}(\epsilon)} \varphi_n (\epsilon) -P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon)} e^{-H(\epsilon )t} \varphi_n(\epsilon)) (x)|^2 \, dx\\ &\leq |\Omega \backslash \Omega (\epsilon)|(|\Omega \cap \Omega (\epsilon)|c(t) c(1)^{1/2} e^{\lambda_n (B)/2})^2\\ &\quad +\int_{\Omega \cap \Omega (\epsilon)} |(e^{-Ht} P_{\mathcal{A}(\epsilon), \mathcal{B}(\epsilon)} \varphi_n (\epsilon) -P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon) }e^{-H(\epsilon)t} \varphi_n(\epsilon))(x)|^2\, dx\\ &\to 0 \quad \text{as } \epsilon \downarrow 0 \end{align*} where we have used \eqref{eq:3.44} and \eqref{eq:3.49}. Hence the second equality of (A28) holds in this application. Finally we consider the third equality in (A28). For $\epsilon \in (0,\epsilon_0]$ and $n=1,2,3,\dots $ we have \[ \varphi_n (\epsilon)=1_{\Omega (\epsilon)\backslash \Omega} \varphi_n (\epsilon) +1_{\Omega (\epsilon)\cap \Omega} \varphi_n (\epsilon). \] By \eqref{eq:3.30} we have \[ \int_{\Omega (\epsilon)\backslash \Omega} |\varphi _n(\epsilon )(x)|^2 \, dx \leq |\Omega^\epsilon \backslash \Omega |c(1) e^{\lambda_n (B)} \to 0 \quad \text{as }\epsilon \downarrow 0. \] Since $\| \varphi_n(\epsilon) \|^2_{\mathcal{A}(\epsilon)} =1$, we must have \[ \| P_{\mathcal{A}(\epsilon ), \mathcal{B}(\epsilon)} \varphi_n (\epsilon) \|^2_{\mathcal{H}} =\int_{\Omega \cap \Omega (\epsilon )} |\varphi_n(\epsilon )(x)|^2 \, dx \to 1\quad \text{as } \epsilon \downarrow 0, \] hence the third equality of (A28) holds in this application. \begin{theorem}\label{thm:3.3} We use the notation in Section~\ref{sec:2}. In particular, we shall use the notation in Definition~\ref{def:2.21} and Theorem~\ref{thm:2.26}. Let $K$ be a compact subset of $\Omega$. Then we have \begin{equation}\label{eq:3.50} \lim_{\epsilon \downarrow 0} \big\{ \sup_{x\in K} |\varphi_i (x)-\psi_i (\epsilon) (x)|\big\}=0 \end{equation} for all $i=1,2,3,\dots $. \end{theorem} \begin{proof} We need to consider only $i\geq 2$. Let $p\geq 1$ be an integer and let \[ k_p+1\le i\leq k_{p+1}. \] We assume, for a contradiction, that \eqref{eq:3.50} is false. Then there exist $\delta >0$ and a decreasing sequence $\{ \epsilon_m\}^\infty_{m=1}$ in $( 0,\epsilon_0/2]$, with $\lim_{m\to \infty} \epsilon_m=0$, and a sequence of points $\{ x_m\}^\infty_{m=1}$ in $K$ such that \begin{equation}\label{eq:3.51} |\varphi_i (x_m)-\psi_i(\epsilon_m)(x_m)|\geq \delta \quad (m=1,2,3,\dots ). \end{equation} We can choose $\hat{\epsilon}\in (0,\min \{1,\epsilon_0\})$ such that \[ D=\{ x\in \Omega :\operatorname{dist} (x,\partial \Omega)> \hat{\epsilon}\}\supseteq K \] and that, by Theorem~\ref{thm:2.25}, \begin{equation}\label{eq:3.52} |\mu_\ell (\epsilon)-\mu_i|\leq 1 \end{equation} for all $\ell =k_p+1,\dots , k_{p+1}$ and $\epsilon \in (0,\hat{\epsilon}]$. Applying the parabolic Harnack inequality, Proposition~\ref{prop:3.1}, with $\Sigma =\Omega$ or $\Sigma =\Omega (\epsilon)$ $(0<\epsilon \leq \frac{1}{2} \hat{\epsilon})$, $\Sigma '=D$, $\omega =1$, $a_{ij}=\delta_{ij}$, $\tau _1=1$, $\tau _2=2$, $t_1=5/4$, $\eta =\frac{1}{2}\hat{\epsilon}$ and \begin{equation}\label{eq:3.53} u(x,t)=e^{-\mu_it}\varphi_i (x)\quad (10$ depends only on $N$. But this contradicts \eqref{eq:2.89} of Theorem~\ref{thm:2.26}. Therefore \eqref{eq:3.50} must hold and the theorem is proved. \end{proof} \section{Application to Koch snowflake}\label{sec:4} In this section we let $\Omega \subseteq \mathbb{R}^2$ be the Koch snowflake. Let $\{ \Omega _{\rm in}(n) \}^\infty_{n=1}$ be the usual sequence of polygons approximating $\Omega$ from inside, with $\Omega_{\rm in}(1)$ being an equilateral triangle. Let $\{\Omega_{\rm out} (n)\}^\infty_{n=1}$ be the usual sequence of polygons approximating $\Omega$ from outside, with $\Omega_{\rm out} (1)$ being a regular hexagon. We first recall the definition of $(\epsilon ,\delta)$-domains (see \cite{J}): \begin{definition}\label{def:4.1} \rm Let $D$ be a domain in $\mathbb{R}^d$ and let $\epsilon >0$ and $0<\delta\leq \infty$. We say that $D$ is an $(\epsilon ,\delta)$-domain if for any two distinct points $p_1$, $p_2\in D$ with $|p_1-p_2|\leq \delta$, there exists a rectifiable path $\Gamma \subseteq D$ joining $p_1$ to $p_2$ satisfying the following conditions: \begin{itemize} \item[(i)] length $(\Gamma )\leq \epsilon^{-1} |p_1-p_2|$, \item[(ii)] for all $p\in \Gamma$ we have \begin{equation}\label{eq:4.1} \operatorname{dist} (p,\partial D)\geq \epsilon |p_1-p_2|^{-1} |p-p_1||p-p_2|. \end{equation} \end{itemize} We note that if $D$ is an $(\epsilon ,\delta)$-domain, then any dilation of $D$ is also on $(\epsilon, \delta)$-domain. \end{definition} We shall need the following result. \begin{proposition}[{\cite[Proposition 3.2]{P2}}]\label{prop:4.2} There exist $\hat{\epsilon}, \hat{\delta}>0$, independent of $n$, such that $\Omega \times \mathbb{R}$ and $\Omega_{\rm in}(n)\times \mathbb{R}$, $n=1,2,3,\dots $, are all $(\hat{\epsilon},\hat{\delta })$-domains in $\mathbb{R}^3$. \end{proposition} Our main result in this section is as follows. \begin{figure}[ht] \begin{center} \setlength{\unitlength}{5mm} \begin{picture}(18,14)(0,0) \scriptsize \drawline(0,5.196)(1,3.464)(3,3.464)(2,1.732)(3,0)(5,0)(6,1.73)(7,0)(9,0)(10,1.732)% (9,3.464)(11,3.464)(12,5.196)(13,3.464)(15,3.464)(16,5.196)% (15,6.938)(17,6.938)(18,8.66)(17,10.392)(15,10.392)(16,12.124)(15,13,856)% (13,13.856)(12,12.124)(11,13.856)(9,13.856)(8,12,124)(9,10.392)(7,10.392)% (6,8.66)(5,10.392)(3,10.392)(2,8.66)(3,6.938)(1,6.938)(0,5.196) \dashline{0.2}(3,3.464)(5,3.464)(6,1.732)(7,3.364)(9,3.464)(8,5.196)(10,8.66)% (9,10.392)(11,10.293)(12,12.124)(13,10.392)(15,10.392)(14,8.66)(15,6.938)% (13,6.938)(12,5.196)(11,6.938)(7,6.938)(6,8.66)(5,6.938)(3,6.938)% (4,5.196)(3,3.464) \end{picture} \end{center} \caption{The polygon $S$ in the proof of Theorem \ref{thm:4.3}} \label{fig1} \end{figure} \begin{theorem}\label{thm:4.3} There exist $\check{\epsilon},\check{\delta} \in (0,\infty)$, independent of $n$, such that $\Omega _{\rm out} (n)\times \mathbb{R}$ is an $(\check{\epsilon },\check{\delta})$-domain in $\mathbb{R}^3$ for all $n=1,2,3,\dots$. \end{theorem} \begin{proof} Fix $n\in \mathbb{N}$ and let $(x_1, y_1, z_1)$, $(x_2,y_2,z_2)\in \Omega_{\rm out} (n)\times \mathbb{R}$. By Proposition~\ref{prop:4.2} we see that it suffices to consider only the following two cases: \begin{itemize} \item[Case 1] Both $(x_1,y_1)$ and $(x_2,y_2)$ are in $\Omega_{\rm out}(n)\backslash \Omega_{\rm in} (n+1)$, \item[Case 2] $(x_1,y_1)\in \Omega_{\rm out}(n)\backslash \Omega _{\rm in} (n+1)$ but $(x_2,y_2)\in \Omega_{\rm in}(n+1)$. \end{itemize} Let $S$ be the polygon in Figure~\ref{fig1}. Then, since $S\times \mathbb{R}$ is a Lipschitz domain, there exist $\tilde{\epsilon},\tilde{\delta }>0$ such that $S\times \mathbb{R}$ is an $(\tilde{\epsilon}, \tilde{\delta})$-domain in $\mathbb{R}^3$. Therefore, since any dilation, translation, or rotation of $S\times \mathbb{R}$ is also an $(\tilde{\epsilon},\tilde{\delta})$-domain, we shall assume that $(x_1,y_1)$ and $(x_2,y_2)$ are not both inside a domain $R\subseteq \Omega _{\rm out} (n)$ that is obtained by a finite sequence of dilations, translations, and rotations of $S$ and that some of the edges of $R$ are also edges of $\Omega_{\rm out}(n)$. This assumption implies that \begin{equation}\label{eq:4.2} |(x_1,y_1)-(x_2,y_2)|\geq 3L_{\rm in}(n+1) \end{equation} where $L_{\rm in}(n+1)$ denotes the length of each side of the polygon $\Omega_{\rm in}(n+1)$. For every edge of $\Omega_{\rm out} (n)$ there corresponds two edges of $\Omega_{\rm in}(n+1)$ as shown in Figure~\ref{fig2}. \begin{figure}[ht] \begin{center} \setlength{\unitlength}{1mm} \begin{picture}(120,40)(0,0) \scriptsize \put(0,0){\line(3,4){30}} \put(30,40){\line(3,-4){30}} \put(60,0){\line(1,0){60}} \dashline{1}(30,40)(120,0) \dottedline{1}(30,40)(30,0) \put(27,39){$v$} \put(18,16.5){$a(x_1,y_1)$} \put(29.3,19.4){$\bullet$} \put(46,28){$\bullet$} \put(46,25){$(x_1,y_1)$} \qbezier(30,20)(40,20)(47,29) \put(37,19){$\uparrow$} \put(33,16){$\Gamma(x_1,y_1)$} \end{picture} \end{center} \caption{Edges: $--$ of $\Omega_{\rm out}(n)$ and --- of $\Omega_{\rm in}(n+1)$} \label{fig2} \end{figure} Referring to Figure~\ref{fig2}, suppose $(x_1,y_1)\in \Omega_{\rm out}(n)\backslash \Omega_{\rm in}(n+1)$ and suppose $(x_1,y_1)$ is within a distance of $\frac{1}{2} \cos (\frac{\pi}{6}) L_{\rm in} (n+1) =\frac{\sqrt{3}}{4} L_{\rm in}(n+1)$ from an acute vertex $v$ of $\Omega_{\rm in}(n+1)$. Then we let $a(x_1,y_1)$ be the point on the angle bisector of $\Omega_{\rm in}(n+1)$ at $v$ that is of the same distance from $v$ as $(x_1,y_1)$ is from $v$. The arc of the circle, centered at $v$ and with radius $|v-(x_1,y_1)|$, starting at $a(x_1,y_1)$ and ending at $(x_1,y_1)$ will be denoted by $\Gamma (x_1,y_1)$, see Figure~\ref{fig2}. \begin{figure}[htb] \begin{center} \setlength{\unitlength}{1mm} \begin{picture}(120,40)(0,0) \scriptsize \put(0,0){\line(3,4){30}} \put(30,40){\line(3,-4){30}} \put(60,0){\line(1,0){60}} \dashline{1}(30,40)(120,0) \dottedline{1}(30,40)(30,0) \put(27,39){$v$} \put(-3,-1){$u$} \put(56,-1){$w$} \put(17,9.5){$a(x_1,y_1)$} \put(29,12.4){$\bullet$} \put(30,13){\line(1,0){31}} \put(60,12.4){$\bullet$} \put(59,9.5){$(x_1,y_1)$} \put(35,9.5){$\Gamma(x_1,y_1)$} \end{picture} \end{center} \caption{Edges: $--$ of $\Omega_{\rm out}(n)$ and --- of $\Omega_{\rm in}(n+1)$} \label{fig3} \end{figure} Referring to Figure~\ref{fig3}, suppose $(x_1,y_1)\in \Omega_{\rm out}(n)\backslash \Omega_{\rm in}(n+1)$ and suppose $(x_1,y_1)$ is not within a distance of $\frac{1}{2} \cos (\pi/6)L_{\rm in}(n+1)=\frac{\sqrt{3}}{4} L_{\rm in}(n+1)$ from any acute vertex of $\Omega_{\rm in}(n+1)$. Then we let $a(x_1,y_1)$ be the center of the triangle with vertices $u$, $v$ and $w$. The straight line segment joining $a(x_1,y_1)$ to $(x_1,y_1)$ will be denoted by $\Gamma (x_1,y_1)$, see Figure~\ref{fig3}. Note that in either case we have \begin{equation}\label{eq:4.3} \operatorname{length} (\Gamma (x_1,y_1))\leq 3L_{\rm in} (n+1). \end{equation} We shall assume that $n$ is sufficiently large so that \begin{equation}\label{eq:4.4} L_{\rm in}(n+1)\leq \hat{\delta}/9. \end{equation} Then if \begin{equation}\label{eq:4.5} |(x_1,y_1,z_1)-(x_2,y_2,z_2)|\leq \hat{\delta}/3, \end{equation} then, by \eqref{eq:4.3}, \begin{equation}\begin{split}\label{eq:4.6} &|(a(x_1,y_1),z_1)-(a(x_2,y_2),z_2)|\\ &\leq |(a(x_1,y_1),z_1)-(x_1,y_1,z_1)| +|(x_1,y_1,z_1)-(x_2,y_2,z_2)|\\ &\quad +|(x_2,y_2,z_2)-(a(x_2,y_2),z_2)|\\ &\leq \operatorname{length}(\Gamma (x_1,y_1))+\hat{\delta}/3 +\operatorname{length} (\Gamma (x_2,y_2)) \leq \hat{\delta}. \end{split}\end{equation} We first present the proof for Case 1. We shall divide the proof for this case into a number of subcases: \subsection*{Case 1(i)} Here we assume that both $(x_1,y_1)$ and $(x_2,y_2)$ are in $\Omega _{\rm out}(n)\backslash \Omega_{\rm in} (n+1)$, that $(x_1,y_1)$ and $(x_2,y_2)$ are not both inside a region $R\subseteq \Omega_{\rm out} (n)$ which is obtained from the polygon $S$ in Figure~\ref{fig1} by a finite sequence of dilations and isometries and that some of the edges of $R$ are also edges of $\Omega_{\rm out} (n)$, that both $(x_1,y_1)$ and $(x_2,y_2)$ are within a distance of $\frac{1}{2}\cos (\pi/6)L_{\rm in}(n+1) =\frac{\sqrt{3}}{4} L_{\rm in}(n+1)$ from some acute vertices $v_1$ and $v_2$, respectively, of $\Omega_{\rm in}(n+1)$, and that \eqref{eq:4.4} and \eqref{eq:4.5} hold. Since $a(x_1,y_1)$, $a(x_2,y_2)\in \Omega_{\rm in}(n+1)$ and \eqref{eq:4.5} holds, Proposition~\ref{prop:4.2} and \eqref{eq:4.6} imply that there exists a rectifiable path $\Gamma \subseteq \Omega_{\rm in}(n+1)\times \mathbb{R}$ joining $(a(x_1,y_1),z_1)$ to $(a(x_2,y_2),z_2)$ and satisfying: \begin{itemize} \item[(A)] $\operatorname{length}(\Gamma)\leq \hat{\epsilon}^{-1}|(a(x_1,y_1),z_1)-(a(x_2,y_2) ,z_2)|$ \item[(B)] for all $p\in \Gamma$ we have \begin{equation} \label{eq:4.7} \begin{split} \operatorname{dist} (p,\partial \Omega_{\rm in}(n+1)\times \mathbb{R}) &\geq \hat{\epsilon }\, |(a(x_1,y_1),z_1)-(a(x_2,y_2),z_2)|^{-1}\\ &\quad\times |p-(a(x_1,y_1),z_1)| |p-(a(x_2,y_2),z_2)|. \end{split}\end{equation} \end{itemize} Now, by \eqref{eq:4.2}, for $i=1,2$, we have \begin{equation} \label{eq:4.8} \operatorname{length} (\Gamma (x_i,y_i)) \leq \frac{\sqrt{3}}{4} L_{\rm in}(n+1)\frac{\pi}{3} \leq \frac{\sqrt{3}\pi}{36} |(x_1,y_1,z_1)-(x_2,y_2,z_2)|. \end{equation} Also we have, by (A) and \eqref{eq:4.8}, \begin{equation}\begin{split}\label{eq:4.9} \operatorname{length} (\Gamma ) &\leq \hat{\epsilon}^{-1} |(a(x_1,y_1),z_1)-(a(x_2,y_2),z_2)|\\ &\leq \hat{\epsilon }^{-1} \big\{ |(a(x_1,y_1),z_1)-(x_1,y_1,z_1)| +|(x_1,y_1,z_1)-(x_2,y_2,z_2)|\\ &\quad +|(x_2,y_2,z_2)-(a(x_2,y_2),z_2)|\big\}\\ & \leq \hat{\epsilon}^{-1} \big( \frac{\sqrt{3}\pi}{18} +1\big) |(x_1,y_1,z_1)-(x_2,y_2,z_2)|. \end{split}\end{equation} In the reverse direction we have, by \eqref{eq:4.8}, \begin{equation}\begin{split}\label{eq:4.10} &|(x_1,y_1,z_1)-(x_2,y_2,z_2)|\\ &\leq |(x_1,y_1,z_1)-(a(x_1,y_1),z_1)|+|(a(x_1,y_1),z_1)-(a(x_2,y_2),z_2)|\\ &\quad +|(a(x_2,y_2),z_2)-(x_2,y_2,z_2)|\\ &\leq \frac{\sqrt{3}\pi}{18} |(x_1,y_1,z_1)-(x_2,y_2,z_2)| +|(a(x_1,y_1),z_1)-(a(x_2,y_2) ,z_2)| \end{split}\end{equation} and hence \begin{equation}\label{eq:4.11} |(x_1,y_1,z_1)-(x_2,y_2,z_2)|\leq \big( 1-\frac{\sqrt{3}\pi}{18} \big)^{-1} |(a(x_1,y_1),z_1)-(a(x_2,y_2),z_2)|. \end{equation} Let $p\in \Gamma$ and suppose that \begin{equation}\label{eq:4.12} |p-(a(x_i,y_i),z_i)|\geq \frac{1}{2}\operatorname{dist} (a(x_i,y_i), \partial \Omega_{\rm in} (n+1)) \end{equation} for $i=1,2$. Then, referring to Figure~\ref{fig2}, \begin{equation}\begin{split}\label{eq:4.13} |p-(a(x_i,y_i),z_i)|&\geq \frac{1}{2}\sin \big(\frac{\pi}{6}\big) |a(x_i,y_i)-v_i|\\ &=\frac{3}{4\pi} |a(x_i,y_i)-v_i|\frac{\pi}{3}\\ &\geq \frac{3}{4\pi}|(a(x_i,y_i),z_i)-(x_i,y_i,z_i)| \end{split}\end{equation} for $i=1,2,$. Hence \begin{equation}\begin{split}\label{eq:4.14} |p-(x_i,y_i,z_i)| &\leq |p-(a(x_i,y_i),z_i)| + |(a(x_i,y_i),z_i)-(x_i,y_i,z_i)|\\ &\leq \big( 1+\frac{4\pi}{3}\big) |p-(a(x_i,y_i),z_i)| \end{split}\end{equation} for $i=1,2$. Combining \eqref{eq:4.7}, \eqref{eq:4.9} and \eqref{eq:4.14} we get \begin{equation}\begin{split}\label{eq:4.15} & \operatorname{dist} (p,\partial \Omega_{\rm out} (n)\times \mathbb{R})\\ &\geq \operatorname{dist} (p,\partial \Omega_{\rm in} (n+1)\times \mathbb{R})\\ &\geq \hat{\epsilon} \, |(a(x_1,y_1),z_1)-(a(x_2,y_2),z_2)|^{-1} |p-(a(x_1,y_1),z_1)|\\ &\quad \times |p-(a(x_2,y_2),z_2)|\\ &\geq \hat{\epsilon } \big(\frac{\sqrt{3}\pi}{18} +1\big)^{-1} \big( 1+\frac{4\pi}{3}\big)^{-2} |(x_1,y_1,z_1)-(x_2,y_2,z_2)|^{-1}\\ &\quad\times |p-(x_1,y_1,z_1)||p-(x_2,y_2,z_2)|. \end{split}\end{equation} Next let $p\in \Gamma$ and suppose that \begin{equation}\label{eq:4.16} |p-(a(x_1,y_1),z_1)|<\frac{1}{2}\operatorname{dist} (a(x_1,y_1),\partial \Omega_{\rm in} (n+1)). \end{equation} Referring to Figure~\ref{fig2}, we have, by \eqref{eq:4.16}, \begin{equation}\begin{split}\label{eq:4.17} |p-(x_1,y_1,z_1)|&\leq |p-(a(x_1,y_1),z_1)| +|(a(x_1,y_1),z_1)-(x_1,y_1,z_1)|\\ &\leq \big(\frac{1}{4}+\frac{\pi}{3}\big)|a(x_1,y_1)-v_1|. \end{split}\end{equation} So, by \eqref{eq:4.16} and \eqref{eq:4.17}, \begin{equation}\begin{split} \label{eq:4.18} \operatorname{dist} (p,\partial \Omega_{\rm out} (n)\times \mathbb{R}) &\geq \operatorname{dist} (p,\partial \Omega_{\rm in}(n+1)\times \mathbb{R})\\ &\geq \frac{1}{2} \operatorname{dist} (a(x_1,y_1),\partial \Omega_{\rm in} (n+1))\\ &=\frac{1}{4} |a(x_1,y_1)-v_1|\\ &\geq \frac{1}{4} \big( \frac{1}{4} +\frac{\pi}{3}\big)^{-1} |p-(x_1,y_1,z_1)|. \end{split}\end{equation} Also, by \eqref{eq:4.16}, \eqref{eq:4.9}, \eqref{eq:4.8}, \eqref{eq:4.2}, \begin{equation}\begin{split}\label{eq:4.19} & |p-(x_2,y_2,z_2)|\\ &\leq |p-(a(x_1,y_1),z_1)| + |(a(x_1,y_1),z_1)-(a(x_2,y_2),z_2)|\\ &\quad +|(a(x_2,y_2),z_2)-(x_2,y_2,z_2)|\\ &\leq \frac{1}{4} |a(x_1,y_1)-v_1| +\big(\frac{\sqrt{3}\pi}{18} +1\big) |(x_1,y_1,z_1)-(x_2,y_2,z_2)|\\ &\quad +\frac{\sqrt{3}\pi}{36} |(x_1,y_1,z_1)-(x_2,y_2,z_2)|\\ &\leq \frac{\sqrt{3}}{16} L_{\rm in}(n+1) +\big(\frac{\sqrt{3}\pi}{12} +1\big) |(x_1,y_1,z_1)-(x_2,y_2,z_2)|\\ &\leq \frac{\sqrt{3}}{48} |(x_1,y_1)-(x_2,y_2)| + \big( \frac{\sqrt{3}\pi}{12} +1\big) |(x_1,y_1,z_1)-(x_2,y_2,z_2)|\\ &\leq \big(\frac{\sqrt{3}}{48} +\frac{\sqrt{3}\pi}{12} +1\big) |(x_1,y_1,z_1)-(x_2,y_2,z_2)|. \end{split}\end{equation} Combining \eqref{eq:4.18} and \eqref{eq:4.19} we obtain \begin{equation}\begin{split}\label{eq:4.20} &\operatorname{dist} (p,\partial \Omega_{\rm out} (n)\times \mathbb{R})\\ &\geq \frac{1}{4} \big( \frac{1}{4} +\frac{\pi}{3}\big)^{-1} \big( \frac{\sqrt{3}}{48} +\frac{\sqrt{3}\pi}{12} +1\big)^{-1} |(x_1,y_1,z_1)-(x_2,y_2,z_2)|^{-1}\\ &\quad \times |p-(x_1,y_1,z_1)||p-(x_2,y_2,z_2)|. \end{split}\end{equation} Now let $p=(x,y,z_1)\in \Gamma (x_1,y_1)\times \{z_1\}$ and let \begin{equation}\label{eq:4.21} k_1=\inf \{ b^{-1}\sin b:00. \end{equation} \begin{figure}[ht] \begin{center} \setlength{\unitlength}{1mm} \begin{picture}(120,40)(0,0) \scriptsize \put(0,0){\line(3,4){30}} \put(30,40){\line(3,-4){30}} \put(60,0){\line(1,0){60}} \dashline{1}(30,40)(120,0) \dottedline{1}(30,40)(30,0) \put(26,40){$v_1$} \put(18,16.5){$a(x_1,y_1)$} \put(29.3,19.4){$\bullet$} \put(46.7,28.6){$\bullet$} \put(48,34){$(x_1,y_1)$} \put(48.4,30.7){$\swarrow$} \qbezier[20](30,20)(40,20)(47,29) \put(44,25.9){$\bullet$} \dottedline{1}(30,40)(75,0) \dottedline{1}(30,40)(95,0) \put(44.5,23){$(x,y)$} \qbezier(61.5,12.2)(65,13)(66,18) \put(65.5,12.5){$b$} \qbezier(66,18)(69,19)(70,22) \put(70,18){$a$} \end{picture} \end{center} \caption{Edges: $--$ of $\Omega_{\rm out}(n)$ and --- of $\Omega_{\rm in}(n+1)$} \label{fig4} \end{figure} Referring to Figure~\ref{fig4} we have \begin{equation}\begin{split}\label{eq:4.22} \operatorname{dist} (p, \partial \Omega_{\rm out} (n)\times \mathbb{R}) &= \operatorname{dist} ((x,y),\partial \Omega_{\rm out} (n))\\ &= |(x_1,y_1)-v_1|\sin (a+b)\\ &= |(x_1,y_1)-v_1|b(b^{-1}\sin (a+b))\\ &\geq |(x_1,y_1)-v_1|b(b^{-1}\sin b)\\ &\geq |(x_1,y_1)-v_1|bk_1\\ &\geq k_1 |(x_1,y_1)-(x,y)|\\ &= k_1|(x_1,y_1,z_1)-(x,y,z_1)|. \end{split}\end{equation} Also we have, for $p=(x,y,z_1) \in \Gamma (x_1,y_1)\times \{ z_1\}$, \begin{equation}\begin{split}\label{eq:4.23} |p-(x_2,y_2,z_2)| &\leq |p-(x_1,y_1,z_1)|+|(x_1,y_1,z_1)-(x_2,y_2,z_2)|\\ &\leq \frac{\pi\sqrt{3}}{12} L_{\rm in} (n+1)+|(x_1,y_1,z_1)-(x_2,y_2,z_2)|\\ &\leq \big( \frac{\pi \sqrt{3}}{36} +1\big) |(x_1,y_1,z_1)- (x_2,y_2,z_2)| \end{split}\end{equation} where we have used \eqref{eq:4.2}. Thus, combining \eqref{eq:4.22} and \eqref{eq:4.23}, we have, for all $p=(x,y,z_1)\in \Gamma (x_1,y_1)\times \{z_1\}$, \begin{equation}\begin{split}\label{eq:4.24} \operatorname{dist} (p,\partial \Omega_{\rm out} (n)\times \mathbb{R}) &\geq \Big( \frac{\pi\sqrt{3}}{36} +1\Big)^{-1} k_1 |(x_1,y_1,z_1)-(x_2,y_2,z_1)|^{-1}\\ &\quad\times |p-(x_1,y_1,z_1)||p-(x_2,y_2,z_2)|. \end{split}\end{equation} Similarly, we have, for all $p=(x,y,z_2)\in \Gamma (x_2,y_2)\times \{ z_2\}$, \begin{equation}\begin{split}\label{eq:4.25} \operatorname{dist} (p,\partial \Omega_{\rm out} (n)\times \mathbb{R}) &\geq \Big( \frac{\pi \sqrt{3}}{36} +1\Big)^{-1} k_1 |(x_1,y_1,z_1)-(x_2,y_2,z_2)|\\ &\quad\times |p-(x_1,y_1,z_1)||p-(x_2,y_2,z_2)|. \end{split}\end{equation} From \eqref{eq:4.5}, \eqref{eq:4.6}, \eqref{eq:4.8}, \eqref{eq:4.9}, \eqref{eq:4.15}, \eqref{eq:4.20}, \eqref{eq:4.24} and \eqref{eq:4.25}, we see that if $(x_1, y_1, z_1)$ and $(x_2,y_2,z_2)$ satisfy the assumptions of Case 1(i) and if \begin{equation}\label{eq:4.26} |(x_1,y_1,z_1)-(x_2,y_2,z_2)|\leq \hat{\delta}/3, \end{equation} then there exists a path \begin{equation}\label{eq:4.27} \tilde{\Gamma} =(\Gamma (x_1,y_1)\times \{ z_1\})+\Gamma +(\Gamma (x_2,y_2)\times \{z_2\}) \end{equation} joining $(x_1,y_1,z_1)$ to $(x_2,y_2,z_2)$ satisfying \begin{equation}\label{eq:4.28} \operatorname{length}(\tilde{\Gamma}) \leq \big[ \frac{\sqrt{3}\pi}{18} +\hat{\epsilon } \big( \frac{\sqrt{3}\pi}{18} +1\big)\big] |(x_1,y_1,z_1)-(x_2,y_2,z_2)| \end{equation} and for all $p\in \tilde{\Gamma}$ we have \begin{equation}\begin{split}\label{eq:4.29} &\operatorname{dist}(p,\partial \Omega_{\rm out}(n)\times \mathbb{R})\\ &\geq \epsilon_1 |(x_1,y_1,z_1)-(x_2,y_2,z_2)|^{-1}|p -(x_1,y_1,z_1)||p-(x_2,y_2,z_2)| \end{split}\end{equation} where \begin{equation}\begin{split}\label{eq:4.30} \epsilon_1 &= \min \Big\{ \hat{\epsilon} \Big( \frac{\sqrt{3}\pi}{18} +1\Big)^{-1} \Big( 1+\frac{4\pi}{3}\Big)^{-2},\\ &\quad \frac{1}{4} \Big( \frac{1}{4} +\frac{\pi}{3}\Big)^{-1} \Big( \frac{\sqrt{3}}{48} +\frac{\sqrt{3}\pi}{12}+1\Big)^{-1}, k_1 \Big( \frac{\sqrt{3}\pi}{36} +1\Big)^{-1} \Big\}. \end{split}\end{equation} \subsection*{Case 1(ii)} Here we assume that both $(x_1,y_1)$ and $(x_2,y_2)$ are in $\Omega _{\rm out} (n)\backslash \Omega_{\rm in} (n+1)$, that $(x_1,y_1)$ and $(x_2,y_2)$ are not both inside a region $R\subseteq \Omega_{\rm out} (n)$ which is obtained from the polygon $S$ in Figure~\ref{fig1} by a finite sequence of dilations and isometries and that some of the edges of $R$ are also edges of $\Omega _{\rm out }(n)$, that both $(x_1,y_1)$ and $(x_2,y_2)$ are not within a distance of $\frac{1}{2}\cos (\pi /6) L_{\rm in} (n+1) =\frac{\sqrt{3}}{4} L_{\rm in} (n+1)$ from any acute vertex of $\Omega_{\rm in}(n+1)$, and that \eqref{eq:4.4} and \eqref{eq:4.5} hold. \begin{figure}[ht] \begin{center} \setlength{\unitlength}{1mm} \begin{picture}(120,50)(0,-10) \scriptsize \put(0,0){\line(3,4){30}} \put(30,40){\line(3,-4){30}} \put(60,0){\line(1,0){60}} \dashline{1}(30,40)(120,0) \put(120,0){\line(-3,-4){10}} \dottedline{1}(30,40)(30,0) \put(29.4,39){$\bullet$} \put(26.8,39){$v$} \put(29,39){$\bullet$} \put(-1,-1){$\bullet$} \put(-3,-1){$u$} \put(59.4,-.8){$\bullet$} \put(56.5,-1){$w$} \put(119,-.8){$\bullet$} \put(121,-1){$\sigma$} \dottedline{1}(30,13)(40,35) \dottedline{1}(30,13)(50.5,30) \put(50.4,29.9){$\bullet$} \put(52,32){$\alpha$} \qbezier[20](30,13)(47,15)(50.6,30) \dottedline{1}(30,13)(61.6,3) \put(55.8,4){$\bullet$} \put(53.5,3){$\tau$} \put(60.8,2.4){$\bullet$} \put(63.5,2){$\eta$} \dottedline{1}(54,13)(60,26) \qbezier(56.6,13)(56.6,14.8)(55,15.5) \put(57,14.2){$\theta$} \dottedline{1}(60,0)(70,22) \put(69.5,21.3){$\bullet$} \put(71.5,23){$\gamma$} \put(30,13){\line(1,0){31}} \put(29,12.3){$\bullet$} \put(19,10){$a(x_1,y_1)$} \put(60,12.3){$\bullet$} \put(55,10){$(x_1,y_1)$} \qbezier(35,11.5)(36.3,16)(32.5,18) \put(36.5,14.5){$\beta_2$} \qbezier(37,18.5)(36,20.2)(33.6,21.5) \put(36.5,21.5){$\beta_1$} \end{picture} \end{center} \caption{Edges: $--$ of $\Omega_{\rm out}(n)$ and --- of $\Omega_{\rm in}(n+1)$} \label{fig5} \end{figure} In Figure~\ref{fig5}, let $\theta$ be the angle between the line segment $\overline{(x_1,y_1),a(x_1,y_1)}$ joining $(x_1,y_1)$ to $a(x_1,y_1)$ and the line perpendicular to the line segment $\overline{v,\sigma}$ joining $v$ to $\sigma$. Let $\alpha$ be the point on $\overline{v,\sigma}$ such that the length of $\overline{v,\alpha}$ is $\frac{1}{2}\cos (\pi/6)L_{\rm in}(n+1)$. Let $\gamma $ be the midpoint of $\overline{v,\sigma}$ and let $\eta $ be a point on $\overline{\gamma, w}$ such that \begin{equation}\label{eq:4.31} \operatorname{length}(\overline{\gamma ,\eta}) =\frac{3}{4}\operatorname{length}(\overline{\gamma, w}). \end{equation} Let $\beta_1$ and $\beta_2$ be the angles between the line perpendicular to $\overline{v,\sigma}$ and the line segments $\overline{a(x_1,y_1),\alpha}$ and $\overline{a(x_1,y_1),\eta}$, respectively. Let $\tau$ be the point of intersection of the line segments $\overline{v,w}$ and $\overline{a(x_1,y_1),\eta}$. If \begin{equation}\label{eq:4.32} \beta _1\leq \theta \leq \beta_2, \end{equation} then, for all $(x,y)\in \overline{a(x_1,y_1),(x_1,y_1)}$, we have \begin{equation}\label{eq:4.33} \operatorname{dist} ((x,y),\partial \Omega_{\rm out}(n)) \geq (\cos \theta) |(x,y)-(x_1,y_1)| \geq (\cos \beta_1)|(x,y)-(x_1,y_1)|. \end{equation} If \begin{equation}\label{eq:4.34} \beta_2 < \theta < \frac{\pi}{2}; \end{equation} i.e., if $(x_1,y_1)\in \Delta (\tau, \eta ,w)$, where $\Delta (\tau ,\eta ,w)$ denotes the triangle with vertices $\tau$, $\eta$ and $w$, then, for all $(x,y)\in\overline{a(x_1,y_1),(x_1,y_1)}$, we have \begin{equation}\begin{split}\label{eq:4.35} \operatorname{dist} ((x,y) ,\partial \Omega _{\rm out} (n)) &\geq \operatorname{dist} (\Delta (a(x_1,y_1),\eta ,w),\partial \Omega_{\rm out} (n))\\ &= |\eta -\gamma |\\ &= 3 |\eta -w|\\ &= k_2 |a(x_1,y_1)-\eta | \\ &\geq k_2 |(x,y)-(x_1,y_1)| \end{split} \end{equation} for some $k_2>0$ independent of $n$, $(x_1,y_1)$ and $(x_2,y_2)$. Combining \eqref{eq:4.33} and \eqref{eq:4.35} we have, for all $p=(x,y,z_1)\in \Gamma (x_1,y_1)\times \{ z_1\}$, \begin{equation}\label{eq:4.36} \operatorname{dist} (p,\partial \Omega _{\rm out} (n)\times \mathbb{R}) \geq \min \{ \cos \beta_1,k_2 \} |p-(x_1,y_1,z_1)|. \end{equation} Also, by \eqref{eq:4.2}, for all $p=(x,y,z_1)\in \Gamma (x_1,y_1)\times \{z_1\}$, we have \begin{equation}\begin{split}\label{eq:4.37} |p-(x_2,y_2,z_2)| &\leq |p-(x_1,y_1,z_1)|+|(x_1,y_1,z_1)-(x_2,y_2,z_2)|\\ &\leq L_{\rm in} (n+1) +|(x_1,y_1,z_1)-(x_2,y_2,z_2)|\\ &\leq \frac{4}{3} |(x_1,y_1,z_1)-(x_2,y_2,z_2)|. \end{split}\end{equation} Thus combining \eqref{eq:4.36} and \eqref{eq:4.37} we have, for all $p=(x,y,z_1)\in \Gamma (x_1,y_1)\times \{z_1\}$, \begin{equation}\begin{split}\label{eq:4.38} \operatorname{dist} (p,\partial \Omega_{\rm out} (n)\times \mathbb{R}) &\geq \frac{3}{4} \min \{\cos \beta_1 ,k_2\} |(x_1,y_1,z_1)-(x_2,y_2,z_2)|^{-1}\\ &\quad \times |p-(x_1,y_1,z_1)| |p-(x_2,y_2,z_2)|. \end{split}\end{equation} Similarly, for all $p=(x,y,z_2)\in\Gamma (x_2,y_2)\times \{ z_2\}$, we have \begin{equation}\begin{split}\label{eq:4.39} \operatorname{dist} (p,\partial \Omega_{\rm out} (n)\times \mathbb{R}) &\geq \frac{3}{4} \min \{\cos \beta_1,k_2\} |(x_1,y_1,z_1)-(x_2,y_2,z_2)|^{-1}\\ & \quad \times |p-(x_1,y_1,z_1) | |p-(x_2,y_2,z_2)|. \end{split}\end{equation} Referring to Figure~\ref{fig5} we have, by \eqref{eq:4.2}, \begin{equation}\begin{split}\label{eq:4.40} \operatorname{length}(\Gamma (x_1,y_1)\times \{ z_1\}) &=\operatorname{length}(\Gamma (x_1,y_1))\\ &\leq L_{\rm in} (n+1)\\ &\leq \frac{1}{3}|(x_1,y_1,z_1)-(x_2,y_2,z_2)|. \end{split}\end{equation} Similarly we have \begin{equation}\label{eq:4.41} \operatorname{length}(\Gamma (x_2,y_2)\times \{ z_2\}) \leq \frac{1}{3} |(x_1,y_1,z_1)-(x_2,y_2,z_2)|. \end{equation} So, from \eqref{eq:4.39} and \eqref{eq:4.40}, we have \begin{equation}\begin{split}\label{eq:4.42} \operatorname{length}(\Gamma ) &\leq \hat{\epsilon}^{-1} |(a(x_1,y_1),z_1)-(a(x_2,y_2),z_2)|\\ &\leq \hat{\epsilon }^{-1} \{ |(a(x_1,y_1),z_1)-(x_1,y_1,z_1)|+|(x_1,y_1,z_1)-(x_2,y_2,z_2)|\\ &\quad + |(x_2,y_2,z_2)-(a(x_2,y_2),z_2)|\\ &\leq \frac{5}{3} \hat{\epsilon}^{-1} |(x_1,y_1,z_1)-(x_2,y_2,z_2)|. \end{split}\end{equation} Now let $p\in \Gamma$ and suppose that \begin{equation}\label{eq:4.43} |p-(a(x_i,y_i),z_i)|\geq \frac{1}{2}\operatorname{dist} (a(x_i,y_i), \partial \Omega_{\rm in} (n+1)) \end{equation} for $i=1,2$. Then, referring to Figure~\ref{fig3}, we have \begin{equation}\label{eq:4.44} |p-(a(x_i,y_i),z_i)|\geq \frac{1}{4\sqrt{3}} L_{\rm in} (n+1) \geq \frac{1}{4\sqrt{3}} |(a(x_i,y_i),z_i)-(x_i,y_i,z_i)| \end{equation} for $i=1,2$. Hence \begin{equation}\begin{split}\label{eq:4.45} |p-(x_i,y_i,z_i)|&\leq |p-(a(x_i,y_i),z_i)| + |(a(x_i,y_i),z_i)-(x_i,y_i,z_i)|\\ & \leq (1+4\sqrt{3})|p-(a(x_i,y_i),z_i)| \end{split}\end{equation} for $i=1,2$. Thus, by \eqref{eq:4.42}, \begin{equation}\begin{split}\label{eq:4.46} \operatorname{dist} (p,\partial \Omega_{\rm out}(n)\times \mathbb{R}) &\geq \operatorname{dist} (p,\partial \Omega_{\rm in} (n+1)\times \mathbb{R})\\ &\geq \hat{\epsilon} |(a(x_1,y_1),z_1)-(a(x_2,y_2),z_2)|^{-1}\\ &\times |p-(a(x_1,y_1),z_1)||p-(a(x_2,y_2),z_2)|\\ &\geq \frac{3}{5} (1+4\sqrt{3})^{-2} \hat{\epsilon} |(x_1,y_1,z_1)-(x_2,y_2,z_2)|^{-1}\\ &\quad \times |p-(x_1,y_1,z_1)| |p-(x_2,y_2,z_2)|. \end{split}\end{equation} Next let $p\in \Gamma$ and suppose that \[ |p-(a(x_1,y_1),z_1) |<\frac{1}{2} \operatorname{dist} (a(x_1,y_1), \partial \Omega_{\rm in}(n+1)). \] Then, referring to Figure~\ref{fig3}, \begin{equation}\begin{split}\label{eq:4.47} \operatorname{dist} (p,\partial \Omega_{\rm out} (n)\times \mathbb{R}) &\geq \operatorname{dist} (p,\partial \Omega_{\rm in} (n+1)\times \mathbb{R})\\ &\geq \frac{1}{2}\operatorname{dist} ((a(x_1,y_1),z_1),\partial \Omega_{\rm in} (n+1)\times \mathbb{R})\\ &= \frac{1}{4\sqrt{3}} L_{\rm in} (n+1) \end{split}\end{equation} and \begin{equation}\begin{split}\label{eq:4.48} |p-(x_1,y_1,z_1)|&\leq |p-(a(x_1,y_1),z_1)|+|(a(x_1,y_1),z_1) -(x_1,y_1,z_1)|\\ &\leq \frac{1}{2} \operatorname{dist} (a(x_1,y_1),\partial \Omega_{\rm in} (n+1))+L_{\rm in} (n+1)\\ &= \big(\frac{1}{4\sqrt{3}}+1\big) L_{\rm in} (n+1), \end{split}\end{equation} hence, from \eqref{eq:4.47} and \eqref{eq:4.48}, \begin{equation}\label{eq:4.49} \operatorname{dist} (p,\partial \Omega_{\rm out} (n)\times \mathbb{R})\geq \frac{1}{4\sqrt{3}} \big( \frac{1}{4\sqrt{3}}+1\big)^{-1} |p-(x_1,y_1,z_1)|. \end{equation} Also, by \eqref{eq:4.48} and \eqref{eq:4.2}, we have \begin{equation}\begin{split}\label{eq:4.50} |p-(x_2,y_2,z_2)| &\leq |p-(x_1,y_1,z_1)|+|(x_1,y_1,z_1)-(x_2,y_2,z_2)|\\ &\leq \big[ \big(\frac{1}{4\sqrt{3}} +1\big) 3^{-1} +1\big] |(x_1,y_1,z_1)-(x_2,y_2,z_2)|. \end{split}\end{equation} Combining \eqref{eq:4.49} and \eqref{eq:4.50} we obtain \begin{equation}\begin{split}\label{eq:4.51} &\operatorname{dist} (p,\partial \Omega_{\rm out} (n)\times \mathbb{R})\\ &\geq \frac{1}{4\sqrt{3}} \Big( \frac{1}{4\sqrt{3}} +1\Big) ^{-1} \Big[ \big( \frac{1}{4\sqrt{3}}+1\big) 3^{-1} +1\Big]^{-1} |(x_1,y_1,z_1) -(x_2,y_2,z_2)|^{-1}\\ &\quad \times |p-(x_1,y_1,z_1)||p-(x_2,y_2,z_2)|. \end{split}\end{equation} By symmetry, if $p\in \Gamma$ and if \[ |p-(a(x_2,y_2),z_2)|<\frac{1}{2}\operatorname{dist} (a(x_2,y_2), \partial \Omega_{\rm in} (n+1)), \] then we have \begin{equation}\begin{split}\label{eq:4.52} &\operatorname{dist} (p,\partial \Omega_{\rm out} (n)\times \mathbb{R})\\ &\geq \frac{1}{4\sqrt{3}} \left( \frac{1}{4\sqrt{3}} +1\right)^{-1} \left[ \left( \frac{1}{4\sqrt{3}}+1\right) 3^{-1} +1\right]^{-1} |(x_1,y_1,z_1)-(x_2,y_2,z_2)|^{-1}\\ &\quad \times |p-(x_1,y_1,z_1)||p-(x_2,y_2,z_2)|. \end{split}\end{equation} Thus from \eqref{eq:4.5}, \eqref{eq:4.6}, \eqref{eq:4.38}, \eqref{eq:4.39}, \eqref{eq:4.40}, \eqref{eq:4.41}, \eqref{eq:4.42}, \eqref{eq:4.46}, \eqref{eq:4.51} and \eqref{eq:4.52}, we see that if $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ satisfy the assumptions of Case~1(ii) and if \[ |(x_1,y_1,z_1)-(x_2,y_2,z_2)|\leq \hat{\delta}/3, \] then there exists a path \begin{equation}\label{eq:4.53} \tilde{\Gamma} =(\Gamma (x_1,y_1)\times \{ z_1\})+\Gamma +(\Gamma (x_2,y_2)\times \{ z_2\}) \end{equation} joining $(x_1,y_1,z_1)$ to $(x_2,y_2,z_2)$ satisfying \begin{equation}\label{eq:4.54} \operatorname{length}(\tilde{\Gamma})\leq \big( \frac{2}{3} +\frac{5}{3} \hat{\epsilon}^{-1} \big) |(x_1,y_1,z_1)-(x_2,y_2,z_2)| \end{equation} and for all $p\in \tilde{\Gamma}$ we have \begin{equation}\begin{split}\label{eq:4.55} & \operatorname{dist} (p,\partial \Omega_{\rm out} (n)\times \mathbb{R})\\ &\geq \epsilon_2 |(x_1,y_1,z_1)-(x_2,y_2,z_2)|^{-1} |p -(x_1,y_1,z_1) | |p-(x_2,y_2,z_2)| \end{split}\end{equation} where \begin{equation}\begin{split}\label{eq:4.56} \epsilon_2 &=\min \Big\{ \frac{3}{4}\cos \beta_1 , \frac{3}{4} k_2, \frac{3}{5} (1+4\sqrt{3})^{-2} \hat{\epsilon},\\ &\quad \frac{1}{4\sqrt{3}} \Big( \frac{1}{4\sqrt{3}} +1\Big)^{-1} \Big[ \big( \frac{1}{4\sqrt{3}} +1\big) 3^{-1} +1\Big]^{-1}\Big\}. \end{split}\end{equation} \subsection*{Case 1(iii)} Here we assume that both $(x_1,y_1)$ and $(x_2,y_2)$ are in $\Omega_{\rm out} (n)\backslash \Omega_{\rm in} (n+1)$, that $(x_1,y_1)$ and $(x_2,y_2)$ are not both inside a region $R\subseteq \Omega_{\rm out} (n)$ which is obtained from the polygon $S$ in Figure~\ref{fig1} by a finite sequence of dilations and isometries and that some of the edges of $R$ are also edges of $\Omega_{\rm out} (n)$, that $(x_1,y_1)$ is not within a distance of $\frac{1}{2}\cos (\pi/6)L_{\rm in} (n+1)$ from any acute vertex of $\Omega_{\rm in} (n+1)$, that $(x_2,y_2)$ is within a distance of $\frac{1}{2} \cos (\pi/6)L_{\rm in} (n+1)$ from an acute vertex $v_2$ of $\Omega_{\rm in} (n+1)$, and that \eqref{eq:4.4} and \eqref{eq:4.5} hold. In this case, by \eqref{eq:4.2}, we have \begin{equation}\begin{split}\label{eq:4.57} \operatorname{length}(\Gamma (x_2,y_2)\times \{ z_1\}) &\leq \frac{\pi}{3} |(x_2,y_2,z_2)-v_2|\\ &\leq \frac{\pi \sqrt{3}}{12} L_{\rm in} (n+1)\\ &\leq \frac{\pi\sqrt{3}}{36} |(x_1,y_1,z_1)-(x_2,y_2,z_2)| \end{split}\end{equation} and, referring to Figure~\ref{fig5}, \begin{equation}\begin{split}\label{eq:4.58} \operatorname{length}(\Gamma (x_1,y_1)\times \{ z_1\}) &= |a(x_1,y_1)-(x_1,y_1)|\\ &\leq L_{\rm in} (n+1)\\ &\leq \frac{1}{3} |(x_1,y_1,z_1)-(x_2,y_2,z_2)|. \end{split}\end{equation} Hence \begin{equation}\begin{split}\label{eq:4.59} \operatorname{length}(\Gamma ) &\leq \hat{\epsilon}^{-1} |(a(x_1,y_1),z_1) -(a(x_2,y_2),z_2)|\\ &\leq \hat{\epsilon}^{-1} \{ | (a(x_1,y_1),z_1) -(x_1,y_1,z_1)| + |(x_1,y_1,z_1)-(x_2,y_2,z_2)|\\ &\quad + |(x_2,y_2,z_2)-(a(x_2,y_2),z_2)|\}\\ &\leq \hat{\epsilon}^{-1} \big( \frac{4}{3} +\frac{\pi\sqrt{3}}{36} \big) |(x_1,y_1,z_1) -(x_2,y_2,z_2)|. \end{split}\end{equation} From \eqref{eq:4.57} and \eqref{eq:4.58} we have \begin{align*} &|(x_1,y_1,z_1)-(x_2,y_2,z_2)|\\ &\leq |(x_1,y_1,z_1)-(a(x_1,y_1),z_1)|+|(a(x_1,y_1),z_1) -(a(x_2,y_2),z_2)|\\ &\quad + |(a(x_2,y_2),z_2)-(x_2,y_2,z_2)|\\ &\leq \frac{1}{3} |(x_1,y_1,z_1)-(x_2,y_2,z_2)| + |(a(x_1,y_1),z_1)-(a(x_2,y_2),z_2)|\\ &\quad + \frac{\pi \sqrt{3}}{36} |(x_1,y_1,z_1)-(x_2,y_2,z_2)|, \end{align*} hence \begin{equation}\label{eq:4.60} |(x_1,y_1,z_1)-(x_2,y_2,z_2)| \leq \Big( \frac{2}{3}-\frac{\pi\sqrt{3}}{36} \Big)^{-1} |(a(x_1,y_1),z_1)-(a(x_2,y_2),z_2)|. \end{equation} Let $p\in \Gamma$ and suppose that \[ |p-(a(x_i,y_i),z_i)|\geq \frac{1}{2} \operatorname{dist} (a(x_i,y_i), \partial \Omega _{\rm in} (n+1)) \] for $i=1,2$. Then \[ |p-(a(x_1,y_1),z_1)|\geq \frac{1}{4\sqrt{3}} L_{\rm in} (n+1) \geq \frac{1}{4\sqrt{3}} |(a(x_1,y_1),z_1)-(x_1,y_1,z_1)| \] and so \begin{equation}\begin{split}\label{eq:4.61} |p-(x_1,y_1,z_1)|&\leq |p-(a(x_1,y_1),z_1)| +|(a(x_1,y_1),z_1)-(x_1,y_1,z_1)|\\ &\leq (1+4\sqrt{3})|p-(a(x_1,y_1),z_1)|. \end{split}\end{equation} Also \begin{align*} |p-(a(x_2,y_2),z_2)|&\geq \frac{1}{2}\sin (\pi/6) |a(x_2,y_2)-v_2|\\ &=\frac{3}{4\pi} |a(x_2,y_2)-v_2|\frac{\pi}{3}\\ &\geq \frac{3}{4\pi} |(a(x_2,y_2),z_2)-(x_2,y_2,z_2)|, \end{align*} and hence \begin{equation}\begin{split}\label{eq:4.62} |p-(x_2,y_2,z_2)|&\leq |p-(a(x_2,y_2),z_2)| + |(a(x_2,y_2),z_2)-(x_2,y_2,z_2)|\\ &\leq \big( 1+\frac{4\pi}{3} \big) |p-(a(x_2,y_2),z_2)|. \end{split}\end{equation} Thus, by \eqref{eq:4.59}, \eqref{eq:4.61} and \eqref{eq:4.62}, \begin{equation}\begin{split} \label{eq:4.63} &\operatorname{dist} (p,\partial \Omega_{\rm out} (n)\times \mathbb{R})\\ &\geq \operatorname{dist} (p,\partial \Omega_{\rm in} (n+1)\times \mathbb{R})\\ &\geq \hat{\epsilon} |(a(x_1,y_1),z_1)-(a(x_2,y_2),z_2)|^{-1} |p-(a(x_1,y_1),z_1)|\\ &\quad\times |p-(a(x_2,y_2),z_2)|\\ &\geq \hat{\epsilon} \big( \frac{4}{3}+\frac{\pi\sqrt{3}}{36} \big)^{-1} (1+4\sqrt{3} )^{-1} \Big( 1+\frac{4\pi}{3}\Big)^{-1}\\ &\quad \times |(x_1,y_1,z_1)-(x_2,y_2,z_2)|^{-1} |p-(x_1,y_1,z_1)| |p-(x_2,y_2,z_2)|. \end{split}\end{equation} Next let $p\in \Gamma$ and suppose that \[ |p-(a(x_1,y_1),z_1)|<\frac{1}{2}\operatorname{dist} (a(x_1,y_1), \partial\Omega_{\rm in} (n+1)). \] Then \eqref{eq:4.47}, \eqref{eq:4.48}, \eqref{eq:4.49} and \eqref{eq:4.50} still hold. Hence we have \begin{equation}\begin{split}\label{eq:4.64} &\operatorname{dist} (p,\partial \Omega_{\rm out} (n)\times \mathbb{R})\\ &\geq \frac{1}{4\sqrt{3}} \Big( \frac{1}{4\sqrt{3}}+1\Big)^{-1} \Big[ \big( \frac{1}{4\sqrt{3}} +1\big) 3^{-1} +1\Big]^{-1}\\ &\quad\times |(x_1,y_1,z_1)-(x_2,y_2,z_2)|^{-1} |p-(x_1,y_1,z_1)| |p-(x_2,y_2,z_2)|. \end{split}\end{equation} Now let $p\in \Gamma$ and suppose that \[ |p-(a(x_2,y_2),z_2)|<\frac{1}{2}\operatorname{dist} (a(x_2,y_2), \partial \Omega_{\rm in}(n+1)). \] Then \begin{equation}\begin{split}\label{eq:4.65} |p-(x_2,y_2,z_2)|&\leq |p-(a(x_2,y_2),z_2)| + |(a(x_2,y_2),z_2)-(x_2,y_2,z_2)|\\ &\leq \big( \frac{1}{4} +\frac{\pi}{3}\big) |a(x_2,y_2)-v_2|. \end{split}\end{equation} So \begin{equation}\begin{split}\label{eq:4.66} \operatorname{dist} (p,\partial \Omega_{\rm out} (n)\times \mathbb{R}) &\geq \operatorname{dist} (p,\partial \Omega_{\rm in} (n+1)\times \mathbb{R})\\ &\geq \frac{1}{2} \operatorname{dist} (a(x_2,y_2),\partial \Omega_{\rm in} (n+1))\\ &= \frac{1}{4} |a(x_2,y_2)-v_2|\\ &\geq \frac{1}{4} \Big( \frac{1}{4}+\frac{\pi}{3}\Big)^{-1} |p-(x_2,y_2,z_2)|. \end{split}\end{equation} Also we have, by \eqref{eq:4.59}, \eqref{eq:4.58} and \eqref{eq:4.2}, \begin{align} |p-(x_1,y_1,z_1)| &\leq |p-(a(x_2,y_2),z_2)|+|(a(x_2,y_2),z_2)-(a(x_1,y_1),z_1)| \nonumber\\ &\quad + |(a(x_1,y_1),z_1)-(x_1,y_1,z_1)| \nonumber \\ &\leq \frac{1}{4} |a(x_2,y_2)-v_2| +\big( \frac{4}{3} +\frac{\pi\sqrt{3}}{36} \big) |(x_1,y_1,z_1)-(x_2,y_2,z_2)| \nonumber\\ &\quad +\frac{1}{3} |(x_1,y_1,z_1)-(x_2,y_2,z_2)| \label{eq:4.67} \\ &\leq \frac{\sqrt{3}}{16} L_{\rm in} (n+1) +\big( \frac{5}{3}+\frac{\pi\sqrt{3}}{36}\big) |(x_1,y_1,z_1)-(x_2,y_2,z_2)| \nonumber \\ &\leq \big( \frac{\sqrt{3}}{48} +\frac{5}{3} +\frac{\pi\sqrt{3}}{36} \big) |(x_1,y_1,z_1)-(x_2,y_2,z_2)|. \nonumber \end{align} Combining \eqref{eq:4.66} and \eqref{eq:4.67}, we have \begin{equation}\begin{split}\label{eq:4.68} &\operatorname{dist} (p,\partial \Omega_{\rm out} (n)\times \mathbb{R})\\ &\geq \frac{1}{4} \Big( \frac{1}{4} +\frac{\pi}{3} \Big)^{-1} \Big( \frac{\sqrt{3}}{48} +\frac{5}{3} +\frac{\pi \sqrt{3}}{36} \big)^{-1} |(x_1,y_1,z_1)-(x_2,y_2,z_2)|^{-1}\\ &\quad \times |p-(x_1,y_1,z_1)| |p-(x_2,y_2,z_2)|. \end{split}\end{equation} Now let $p\in \Gamma (x_2,y_2)\times \{ z_2\}$. Let $k_1>0$ be the constant defined in \eqref{eq:4.21}. Then calculations similar to those in \eqref{eq:4.22} give \begin{equation}\label{eq:4.69} \operatorname{dist} (p,\partial \Omega_{\rm out} (n)\times \mathbb{R}) \geq k_1 |p-(x_2,y_2,z_2)|. \end{equation} Also, by \eqref{eq:4.2}, \begin{equation}\begin{split}\label{eq:4.70} |p-(x_1,y_1,z_1)| &\leq |p-(x_2,y_2,z_2)|+|(x_2,y_2,z_2)-(x_1,y_1,z_1)|\\ &\leq \frac{\pi \sqrt{3}}{12} L_{\rm in} (n+1)+|(x_1,y_1,z_1)-(x_2,y_2,z_2)|\\ &\leq \big(\frac{\pi\sqrt{3}}{36}+1\big) |(x_1,y_1,z_1)-(x_2,y_2,z_2)|. \end{split}\end{equation} Combining \eqref{eq:4.69} and \eqref{eq:4.70} we obtain \begin{equation}\begin{split}\label{eq:4.71} \operatorname{dist} (p,\partial \Omega_{\rm out} (n)\times \mathbb{R}) &\geq k_1 \Big( \frac{\pi\sqrt{3}}{36} +1\Big)^{-1} |(x_1,y_1,z_1) -(x_2,y_2,z_2)|^{-1}\\ &\times |p-(x_1,y_1,z_1)| |p-(x_2,y_2,z_2)|. \end{split}\end{equation} Let $p\in \Gamma (x_1,y_1)\times \{ z_1\}$. Then \eqref{eq:4.36} and \eqref{eq:4.37}, and their proofs, still hold. Thus we have \begin{equation}\begin{split} \label{eq:4.72} \operatorname{dist} (p,\partial \Omega_{\rm out} (n)\times \mathbb{R}) &\geq \frac{3}{4} \min \{\cos \beta _1,k_2\} |(x_1,y_1,z_1)-(x_2,y_2,z_2)|^{-1}\\ &\quad \times |p-(x_1,y_1,z_1)||p-(x_2,y_2,z_2)|, \end{split}\end{equation} where $\beta_1$ and $k_2$ are constants described in Case~1(ii). Thus from \eqref{eq:4.5}, \eqref{eq:4.6}, \eqref{eq:4.57}, \eqref{eq:4.58}, \eqref{eq:4.59}, \eqref{eq:4.63}, \eqref{eq:4.64}, \eqref{eq:4.68}, \eqref{eq:4.71} and \eqref{eq:4.78}, we see that if $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ satisfy the assumptions of Case~1(iii) and if \[ |(x_1,y_1,z_1)-(x_2,y_2,z_2)|\leq \hat{\delta }/3, \] then there exists a path \begin{equation}\label{eq:4.73} \tilde{\Gamma} =(\Gamma (x_1,y_1)\times \{z_1\})+\Gamma +(\Gamma (x_2,y_2)\times \{ z_2\}) \end{equation} joining $(x_1,y_1,z_1)$ to $(x_2,y_2,z_2)$ satisfying \begin{equation} \label{eq:4.74} \operatorname{length}(\tilde{\Gamma}) \leq \big[ \frac{1}{3} +\frac{\pi\sqrt{3}}{36} +\hat{\epsilon }^{-1} \big( \frac{4}{3}+\frac{\pi\sqrt{3}}{36} \big) \big] |(x_1,y_1,z_1) -(x_2,y_2,z_2)| \end{equation} and for all $p\in \tilde{\Gamma}$ we have \begin{equation}\begin{split}\label{eq:4.75} &\operatorname{dist} (p,\partial \Omega_{\rm out} (n)\times \mathbb{R})\\ &\geq \epsilon_3 |(x_1,y_1,z_1)-(x_2,y_2,z_2)|^{-1} |p-(x_1,y_1,z_1)| |p-(x_2,y_2,z_2)| \end{split}\end{equation} where \begin{equation}\begin{split}\label{eq:4.76} \epsilon_3 &= \min \Big\{ \hat{\epsilon} \Big( \frac{4}{3} +\frac{\pi\sqrt{3}}{36} \Big)^{-1} (1+4\sqrt{3})^{-1} \Big( 1+\frac{4\pi}{3}\Big)^{-1}, \\ &\quad \frac{1}{4\sqrt{3}} \Big( \frac{1}{4\sqrt{3}} +1\Big)^{-1} \Big[ \big( \frac{1}{4\sqrt{3}} +1 \big) 3^{-1} +1\Big]^{-1},\\ &\quad \frac{1}{4} \Big( \frac{1}{4} +\frac{\pi}{3} \Big)^{-1} \Big( \frac{\sqrt{3}}{48}+\frac{5}{3} +\frac{\pi\sqrt{3}}{36}\Big)^{-1}, k_1\left( \frac{\pi \sqrt{3}}{36} +1\right)^{-1},\\ &\quad \frac{3}{4} \cos \beta_1, \frac{3}{4} k_2\Big\}. \end{split}\end{equation} \subsection*{Case 2(i)} Here we assume that $(x_1,y_1)\in \Omega_{\rm out} (n)\backslash \Omega_{\rm in} (n+1)$ and $(x_2,y_2)\in \Omega_{\rm in} (n+1)$, that $(x_1,y_1)$ and $(x_2,y_2)$ are not both inside a region $R\subseteq \Omega_{\rm out} (n)$ which is obtained from the polygon $S$ in Figure~\ref{fig1} by a finite sequence of dilations and isometries and that some of the edges of $R$ are also edges of $\Omega_{\rm out} (n)$, that $(x_1,y_1)$ is within a distance of $\frac{1}{2}\cos (\pi/6) L_{\rm in} (n+1)$ from an acute vertex $v_1$ of $\Omega_{\rm in} (n+1)$, and that \eqref{eq:4.4} and \eqref{eq:4.5} hold. By Proposition~\ref{prop:4.2} there exists a path $\Gamma \subseteq \Omega_{\rm in} (n+1)\times \mathbb{R}$ joining $(a(x_1,y_1),z_1)$ to $(x_2,y_2,z_2)$ satisfying \begin{itemize} \item[(C)] $\operatorname{length}(\Gamma )\leq \hat{\epsilon}^{-1} |(a(x_1,y_1),z_1)-(x_2,y_2,z_2)|$, \item[(D)] for all $p\in \Gamma$ we have \begin{equation}\begin{split}\label{eq:4.77} &\operatorname{dist} (p, \partial \Omega_{\rm in} (n+1) \times \mathbb{R})\\ &\geq \hat{\epsilon} |(a(x_1,y_1),z_1)-(x_2,y_2,z_2)|^{-1} |p-(a(x_1,y_1),z_1)| |p-(x_2,y_2,z_2)|. \end{split}\end{equation} \end{itemize} By \eqref{eq:4.2}, we have \begin{equation}\begin{split}\label{eq:4.78} &|(a(x_1,y_1),z_1)-(x_2,y_2,z_2)|\\ &\leq |(a(x_1,y_1),z_1)-(x_1,y_1,z_1)|+|(x_1,y_1,z_1)-(x_2,y_2,z_2)|\\ &\leq \frac{\pi\sqrt{3}}{12} L_{\rm in} (n+1)+|(x_1,y_1,z_1)-(x_2,y_2,z_2)|\\ &\leq \big( \frac{\pi\sqrt{3}}{36} +1\big) |(x_1,y_1,z_1)-(x_2,y_2,z_2)|\end{split}\end{equation} and \begin{align*} &|(x_1,y_1,z_1)-(x_2,y_2,z_2)|\\ &\leq |(x_1,y_1,z_1)-(a(x_1,y_1),z_1)|+|(a (x_1,y_1),z_1) - (x_2,y_2,z_2)|\\ &\leq \frac{\pi\sqrt{3}}{36} |(x_1,y_1,z_1) -(x_2,y_2 ,z_2)| + |(a(x_1,y_1),z_1)-(x_2,y_2,z_2)| \end{align*} and hence \begin{equation}\label{eq:4.79} |(x_1,y_1,z_1) - (x_2,y_2,z_2)|\leq \Big( 1-\frac{\pi\sqrt{3}}{36} \Big)^{-1} |(a(x_1,y_1),z_1)-(x_2,y_2,z_2)|. \end{equation} Thus \begin{equation}\begin{split}\label{eq:4.80} \operatorname{length}(\Gamma ) &\leq \hat{\epsilon}^{-1} |(a(x_1,y_1),z_1) -(x_2,y_2,z_2)|\\ &\leq \hat{\epsilon}^{-1} \big(\frac{\pi\sqrt{3}}{36} +1\big) |(x_1,y_1,z_1)-(x_2,y_2,z_2)| \end{split}\end{equation} and \begin{equation}\begin{split}\label{eq:4.81} \operatorname{length}(\Gamma (x_1,y_1)\times \{ z_1\}) &=\operatorname{length}(\Gamma (x_1,y_1))\\ &\leq \frac{\pi \sqrt{3}}{12} L_{\rm in} (n+1)\\ &\leq \frac{\pi \sqrt{3}}{36} |(x_1,y_1,z_1)-(x_2,y_2,z_2)|. \end{split}\end{equation} Let $p=(x,y,z_1) \in \Gamma (x_1,y_1)\times \{ z_1\}$. Let $k_1>0$ be the constant defined by \eqref{eq:4.21}. Then \eqref{eq:4.22} and \eqref{eq:4.23} still hold, and hence we have \begin{equation}\begin{split} \label{eq:4.82} \operatorname{dist} (p,\partial \Omega_{\rm out} (n)\times \mathbb{R}) &\geq k_1\Big( \frac{\pi\sqrt{3}}{36} +1\Big)^{-1} |(x_1,y_1,z_1)-(x_2,y_2,z_2)|^{-1}\\ &\times |p-(x_1,y_1,z_1)| |p-(x_2,y_2,z_2)|. \end{split}\end{equation} Let $p\in \Gamma$ and suppose that \[ |p-(a(x_1,y_1),z_1)| \geq \frac{1}{2} \operatorname{dist} (a(x_1,y_1), \partial \Omega _{\rm in} (n+1)). \] Then \begin{align*} |p-(a(x_1,y_1),z_1)|&\geq \frac{1}{2}\sin (\pi/6) |a(x_1,y_1)-v_1|\\ &=\frac{3}{4\pi} \left( \frac{\pi}{3} |a(x_1,y_1)-v_1|\right)\\ &\geq \frac{3}{4\pi} |(a(x_1,y_1),z_1)-(x_1,y_1,z_1)|, \end{align*} hence \begin{equation}\begin{split}\label{eq:4.83} |p-(x_1,y_1,z_1) |&\leq |p-(a(x_1,y_1),z_1)| + |(a(x_1,y_1),z_1)-(x_1,y_1,z_1)|\\ &\leq \big( 1+\frac{4\pi}{3} \big) |p-(a(x_1,y_1),z_1)|. \end{split}\end{equation} Combining \eqref{eq:4.80} and \eqref{eq:4.83} we have \begin{equation}\begin{split}\label{eq:4.84} &\operatorname{dist} (p,\partial \Omega_{\rm out} (n)\times \mathbb{R})\\ &\geq \operatorname{dist} (p,\partial \Omega_{\rm in} (n+1)\times \mathbb{R})\\ &\geq \hat{\epsilon} |(a(x_1,y_1),z_1) -(x_2,y_2,z_2)|^{-1} |p-(a(x_1,y_1),z_1)| |p-(x_2,y_2,z_2)|\\ &\geq \hat{\epsilon}\Big( \frac{\pi \sqrt{3}}{36} +1\Big)^{-1} \Big( 1+\frac{4\pi}{3}\Big)^{-1} |(x_1,y_1,z_1)-(x_2,y_2,z_2)|^{-1}\\ &\times |p-(x_1,y_1,z_1)| |p-(x_2,y_2,z_2)|. \end{split}\end{equation} Next let $p\in \Gamma$ and suppose that \[ |p-(a(x_1,y_1),z_1)|<\frac{1}{2} \operatorname{dist} (a(x_1,y_1), \partial \Omega_{\rm in} (n+1)).\] Then \begin{equation}\begin{split} \label{eq:4.85} |p-(x_1,y_1,z_1)|&\leq |p-(a(x_1,y_1),z_1) |+|(a(x_1,y_1),z_1 ) -(x_1,y_1,z_1)|\\ &\leq \frac{1}{4} |a(x_1,y_1)-v_1|+\frac{\pi}{3} |a(x_1,y_1)-v_1|\\ &= \big( \frac{1}{4} +\frac{\pi}{3} \big) |a(x_1,y_1)-v_1|, \end{split}\end{equation} and so \begin{equation}\begin{split}\label{eq:4.86} \operatorname{dist} (p,\partial \Omega_{\rm out} (n)\times \mathbb{R}) &\geq \operatorname{dist} (p,\partial \Omega_{\rm in} (n+1)\times \mathbb{R})\\ &\geq \frac{1}{2} \operatorname{dist} (a(x_1,y_1),\partial \Omega_{\rm in} (n+1))\\ &= \frac{1}{4} |a(x_1,y_1)-v_1|\\ &\geq \frac{1}{4} \big( \frac{1}{4} +\frac{\pi}{3}\big)^{-1} |p-(x_1,y_1,z_1)|. \end{split}\end{equation} Also we have, by \eqref{eq:4.80} and \eqref{eq:4.2}, \begin{equation}\begin{split}\label{eq:4.87} &|p-(x_2,y_2,z_2)|\\ &\leq |p-(a(x_1,y_1),z_1)| + |(a(x_1,y_1),z_1) -(x_2,y_2,z_2)|\\ &\leq \frac{1}{4} |a(x_1,y_1)-v_1 | +\big( \frac{\pi\sqrt{3}}{36} +1\big) |(x_1,y_1,z_1)- (x_2,y_2,z_2)|\\ &\leq \frac{\sqrt{3}}{16} L_{\rm in} (n+1) +\big( \frac{\pi\sqrt{3}}{36} +1\big) |(x_1,y_1,z_1)-(x_2,y_2,z_2)|\\ &\leq \big( \frac{\sqrt{3}}{48} +\frac{\pi\sqrt{3}}{36} +1\big) |(x_1,y_1,z_1)-(x_2,y_2,z_2)|. \end{split}\end{equation} Combining \eqref{eq:4.86} and \eqref{eq:4.87} we obtain \begin{equation}\begin{split}\label{eq:4.88} &\operatorname{dist} (p,\partial \Omega_{\rm out} (n)\times \mathbb{R})\\ &\geq \frac{1}{4} \Big( \frac{1}{4} +\frac{\pi}{3} \Big)^{-1} \Big( \frac{\sqrt{3}}{48} +\frac{\pi\sqrt{3}}{36} +1\Big)^{-1} |(x_1,y_1,z_1)-(x_2,y_2,z_2)|^{-1}\\ &\quad \times |p-(x_1,y_1,z_1)||p-(x_2,y_2,z_2)|. \end{split}\end{equation} Thus from \eqref{eq:4.5}, \eqref{eq:4.6}, \eqref{eq:4.80}, \eqref{eq:4.81}, \eqref{eq:4.82}, \eqref{eq:4.84}, and \eqref{eq:4.88}, we see that if $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ satisfy the assumptions of Case~2(i) and if \[ |(x_1,y_1,z_1)-(x_2,y_2,z_2)|\leq \hat{\delta}/3, \] then there exists a path \begin{equation}\label{eq:4.89} \tilde{\Gamma} =(\Gamma (x_1,y_1)\times \{z_1\} )+\Gamma \end{equation} joining $(x_1,y_1,z_1)$ to $(x_2,y_2,z_2)$ satisfying \begin{equation}\label{eq:4.90} \operatorname{length}(\tilde{\Gamma}) \leq \Big( \frac{\pi\sqrt{3}}{36} +\hat{\epsilon}^{-1} \big( \frac{\pi\sqrt{3}}{36} +1\big)\Big) |(x_1,y_1,z_1)-(x_2,y_2,z_2)| \end{equation} and for all $p\in \tilde{\Gamma}$ we have \begin{equation}\begin{split} \label{eq:4.91} &\operatorname{dist} (p,\partial \Omega_{\rm out} (n)\times \mathbb{R})\\ &\geq \epsilon_4 |(x_1,y_1,z_1)-(x_2,y_2,z_2)|^{-1} |p-(x_1,y_1,z_1) ||p-(x_2,y_2,z_2)| \end{split}\end{equation} where \begin{equation}\begin{split} \epsilon_4 &= \min \Big\{ k_1 \Big( \frac{\pi \sqrt{3}}{36}+1\Big)^{-1}, \hat{\epsilon} \Big( \frac{\pi \sqrt{3}}{36} +1\Big)^{-1} \Big( 1+\frac{4\pi}{3} \Big)^{-1},\\ &\quad \frac{1}{4} \Big( \frac{1}{4} +\frac{\pi}{3}\Big)^{-1} \Big( \frac{\sqrt{3}}{48} +\frac{\pi \sqrt{3}}{36} +1\Big)^{-1} \Big\}. \end{split}\end{equation} \subsection*{Case 2(ii)} Here we assume that $(x_1,y_1)\in \Omega_{\rm out}(n)\backslash \Omega _{\rm in} (n+1)$ and $(x_2,y_2)\in \Omega_{\rm in} (n+1)$, that $(x_1,y_1)$ and $(x_2,y_2)$ are not both inside a region $R\subseteq \Omega_{\rm out} (n)$ which is obtained from the polygon $S$ in Figure~\ref{fig1} by a finite sequence of dilations and isometries and that some of the edges of $R$ are also edges of $\Omega_{\rm out} (n)$, that $(x_1,y_1)$ is not within a distance of $\frac{1}{2}\cos (\pi/6)L_{\rm in} (n+1)$ from any acute vertex of $\Omega_{\rm in} (n+1)$, and that \eqref{eq:4.4} and \eqref{eq:4.5} holds. Referring to Figure~\ref{fig5}, we see that in this case \eqref{eq:4.31}-\eqref{eq:4.38} and \eqref{eq:4.40} still hold, and so, for all $p\in \Gamma (x_1,y_1)\times \{ z_1\}$, we have \begin{equation}\begin{split}\label{eq:4.93} \operatorname{dist} (p,\partial \Omega_{\rm out} (n)\times \mathbb{R}) &\geq \frac{3}{4} \min\{\cos \beta_1 ,k_2\} |(x_1,y_1,z_1) - (x_2,y_2,z_2)|^{-1}\\ &\quad \times |p-(x_1,y_1,z_1)| |p-(x_2,y_2,z_2)|, \end{split}\end{equation} and \begin{equation}\label{eq:4.94} \operatorname{length}(\Gamma (x_1,y_1)\times \{z_1\} ) \leq \frac{1}{3} |(x_1,y_1,z_1)-(x_2,y_2,z_2)|.\end{equation} Hence \begin{equation}\begin{split}\label{eq:4.95} &\operatorname{length}(\Gamma)\\ &\leq \hat{\epsilon}^{-1} |(a(x_1,y_1),z_1)-(x_2,y_2,z_2)|\\ &\leq \hat{\epsilon}^{-1} \{ |(a(x_1,y_1),z_1)-(x_1,y_1,z_1)| +|(x_1,y_1,z_1)- (x_2,y_2,z_2)|\}\\ &\leq \frac{4}{3} \hat{\epsilon}^{-1} |(x_1,y_1,z_1) -(x_2,y_2,z_2)|. \end{split}\end{equation} Let $p\in \Gamma$ and suppose that \[ |p-(a(x_1,y_1),z_1)|\geq \frac{1}{2} \operatorname{dist} (a(x_1,y_1), \partial \Omega_{\rm in} (n+1)). \] Then \begin{equation} \label{eq:4.96} |p-(a(x_1,y_1),z_1)|\geq \frac{1}{4\sqrt{3}} L_{\rm in} (n+1) \geq \frac{1}{4\sqrt{3}} |(a(x_1,y_1),z_1) -(x_1,y_1,z_1)|. \end{equation} Thus \begin{equation}\begin{split}\label{eq:4.97} |p-(x_1,y_1,z_1)|&\leq |p-(a(x_1,y_1),z_1)| +|(a(x_1,y_1),z_1)-(x_1,y_1,z_1)|\\ &\leq (1+4\sqrt{3})|p-(a(x_1,y_1),z_1)|. \end{split}\end{equation} Also, from \eqref{eq:4.2} and Figure~\ref{fig5}, \begin{equation}\begin{split}\label{eq:4.98} &|(a(x_1,y_1),z_1) -(x_2,y_2,z_2)|\\ &\leq |(a(x_1,y_1),z_1)-(x_1,y_1,z_1)| + |(x_1, y_1, z_1) - (x_2, y_2, z_2)|\\ &\leq L_{\rm in} (n+1) +|(x_1,y_1,z_1)-(x_2,y_2,z_2)|\\ &\leq \frac{4}{3} |(x_1,y_1,z_1) -(x_2,y_2,z_2)|. \end{split}\end{equation} Hence we have, from \eqref{eq:4.97} and \eqref{eq:4.98}, \begin{equation}\begin{split}\label{eq:4.99} \operatorname{dist} (p,\partial \Omega_{\rm out} (n)\times \mathbb{R}) &\geq \operatorname{dist} (p,\partial \Omega_{\rm in} (n+1)\times \mathbb{R})\\ &\geq \hat{\epsilon} |(a(x_1,y_1),z_1)-(x_2,y_2,z_2)|^{-1} |p-(a(x_1,y_1),z_1)|\\ &\times |p-(x_2,y_2,z_2)|\\ &\geq \hat{\epsilon} \frac{3}{4} (1+4\sqrt{3} )^{-1} |(x_1,y_1,z_1)-(x_2,y_2,z_2)|^{-1}\\ &\times |p-(x_1,y_1,z_1)| |p-(x_2,y_2,z_2)|. \end{split}\end{equation} Now let $p\in \Gamma$ and suppose that \[ |p-(a(x_1,y_1) ,z_1)|<\frac{1}{2} \operatorname{dist} (a(x_1,y_1), \partial \Omega_{\rm in} (n+1)). \] Then \begin{equation}\begin{split}\label{eq:4.100} \operatorname{dist} (p,\partial \Omega_{\rm out} (n)\times \mathbb{R}) &\geq \operatorname{dist} (p,\partial \Omega_{\rm in} (n+1)\times \mathbb{R})\\ &\geq \frac{1}{2} \operatorname{dist} ((a(x_1,y_1),z_1),\partial \Omega_{\rm in} (n+1)\times \mathbb{R})\\ &= \frac{1}{4\sqrt{3}} L_{\rm in} (n+1) \end{split}\end{equation} and, referring to Figure~\ref{fig5}, \begin{equation}\begin{split}\label{eq:4.101} |p-(x_1,y_1,z_1)|&\leq |p-(a(x_1,y_1),z_1)| + |(a(x_1,y_1),z_1) -(x_1,y_1,z_1)|\\ &\leq \big( \frac{1}{4\sqrt{3}} +1\big) L_{\rm in} (n+1). \end{split}\end{equation} Hence, combining \eqref{eq:4.100} and \eqref{eq:4.101}, we get \begin{equation}\label{eq:4.102} \operatorname{dist} (p,\partial \Omega_{\rm out} (n)\times \mathbb{R}) \geq \frac{1}{4\sqrt{3}} \Big(\frac{1}{4\sqrt{3}} +1\Big)^{-1} |p-(x_1,y_1,z_1)|. \end{equation} Also, by \eqref{eq:4.2} and \eqref{eq:4.101}, \begin{equation}\begin{split}\label{eq:4.103} |p-(x_2,y_2,z_2)| &\leq |p-(x_1,y_1,z_1)|+|(x_1,y_1,z_1)-(x_2,y_2,z_2)|\\ &\leq \big( \frac{1}{4\sqrt{3}}+1\big) L_{\rm in} (n+1)+|(x_1,y_1,z_1)-(x_2,y_2,z_2)|\\ &\leq \big( \frac{1}{12\sqrt{3}}+\frac{4}{3}\big) |(x_1,y_1,z_1)-(x_2,y_2,z_2)|. \end{split}\end{equation} Combining \eqref{eq:4.102} and \eqref{eq:4.103} we obtain \begin{equation}\begin{split}\label{eq:4.104} &\operatorname{dist} (p,\partial \Omega_{\rm out} (n)\times \mathbb{R})\\ &\geq \frac{1}{4\sqrt{3}} \Big( \frac{1}{4\sqrt{3}}+1\big)^{-1} \Big( \frac{1}{12\sqrt{3}} +\frac{4}{3}\Big)^{-1} |(x_1,y_1,z_1)-(x_2,y_2,z_2)|^{-1}\\ &\quad \times |p-(x_1,y_1,z_1)| |p-(x_2,y_2,z_2)|. \end{split}\end{equation} Thus from \eqref{eq:4.5}, \eqref{eq:4.6}, \eqref{eq:4.93}, \eqref{eq:4.94}, \eqref{eq:4.95}, \eqref{eq:4.99}, and \eqref{eq:4.104}, we see that if $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ satisfy the assumptions of Case~2(ii) and if \[ |(x_1,y_1,z_1)-(x_2,y_2,z_2)|\leq \hat{\delta}/3, \] then there exists a path \begin{equation}\label{eq:4.105} \tilde{\Gamma} =(\Gamma (x_1,y_1)\times \{ z_1\})+\Gamma \end{equation} joining $(x_1,y_1,z_1)$ to $(x_2,y_2,z_2)$ satisfying \begin{equation}\label{eq:4.106} \operatorname{length}(\tilde{\Gamma}) \leq \big( \frac{1}{3} +\frac{4}{3} \hat{\epsilon}^{-1}\big) |(x_1,y_1,z_1)-(x_2,y_2,z_2)| \end{equation} and for all $p\in \tilde{\Gamma}$ we have \begin{equation}\begin{split}\label{eq:4.107} &\operatorname{dist} (p,\partial \Omega_{\rm out} (n)\times \mathbb{R})\\ &\geq \epsilon_5 |(x_1,y_1,z_1)-(x_2,y_2,z_2)|^{-1} |p-(x_1,y_1,z_1)| |p-(x_2,y_2,z_2)| \end{split}\end{equation} where \begin{equation}\begin{split}\label{eq:4.108} \epsilon_5&= \min\Big\{ \frac{3}{4}\cos \beta_1, \frac{3}{4} k_2, \frac{3}{4} \big(1 +4\sqrt{3}\big)^{-1} \hat{\epsilon},\\ &\quad \frac{1}{4\sqrt{3}} \Big( \frac{1}{4\sqrt{3}} +1\Big)^{-1} \Big( \frac{1}{12\sqrt{3}} +\frac{4}{3}\Big)^{-1} \Big\}. \end{split}\end{equation} To summarize: Cases~1(i),(ii),(iii), 2(i),(ii), exhaust all possibilities of at least one of $(x_1,y_1,z_1)$ or $(x_2,y_x,z_2)$ is in $(\Omega_{\rm out} (n)\times \mathbb{R})\backslash (\Omega_{\rm in} (n+1)\times \mathbb{R})$ with $(x_1,y_1)$ and $(x_2,y_2)$ not both contained in a region $R\subseteq \Omega _{\rm out} (n)$ which is obtained from the polygon $S$ in Figure~\ref{fig1} by a finite sequence of dilations and isometries and that at least one of the edges of $R$ is also an edge of $\Omega_{\rm out} (n)$. Let \begin{equation}\label{eq:4.109} \delta_6 =\hat{\delta} /3 \end{equation} and \begin{equation}\begin{split}\label{eq:4.110} \epsilon_6 &= \min \Big\{ \Big[ \frac{\sqrt{3}\pi}{18} +\hat{\epsilon} \big( \frac{\sqrt{3}\pi}{18} +1\big)\Big]^{-1}, \Big( \frac{2}{3} +\frac{5}{3}\hat{\epsilon}^{-1} \Big)^{-1}, \\ &\quad \Big[\frac{1}{3} +\frac{\pi\sqrt{3}}{36} +\hat{\epsilon}^{-1} \big( \frac{4}{3} +\frac{\pi\sqrt{3}}{36} \big) \Big]^{-1} ,\Big[ \frac{\pi\sqrt{3}}{36} +\hat{\epsilon}^{-1} \Big(\frac{\pi\sqrt{3}}{36} +1\Big)\Big]^{-1},\\ &\quad \Big( \frac{1}{3} +\frac{4}{3} \hat{\epsilon}^{-1}\Big)^{-1}, \epsilon_1, \epsilon_2, \, \epsilon_3, \epsilon_4, \epsilon_5\Big\}. \end{split}\end{equation} Then we have proved that in each of these cases, if \begin{equation}\label{eq:4.111} |(x_1,y_1,z_1)-(x_2,y_2,z_2)|\leq \delta _6, \end{equation} then there exists a path $\tilde{\Gamma} \subseteq \Omega_{\rm out} (n)\times \mathbb{R}$ joining $(x_1,y_1,z_1)$ to $(x_2,y_2,z_2)$ satisfying \begin{equation}\label{eq:4.112} \operatorname{length}(\tilde{\Gamma}) \leq \epsilon_6^{-1} |(x_1,y_1,z_1) -(x_2,y_2,z_2)| \end{equation} and for all $p\in \tilde{\Gamma}$ we have \begin{equation}\begin{split}\label{eq:4.113} &\operatorname{dist} (p,\partial \Omega_{\rm out} (n)\times \mathbb{R})\\ &\geq \epsilon_6 |(x_1,y_1,z_1)-(x_2,y_2,z_2)|^{-1} |p-(x_1,y_1,z_1)| |p-(x_2,y_2,z_2)|. \end{split}\end{equation} This together with Proposition~\ref{prop:4.2} and the fact $S\times \mathbb{R}$ is a Lipschitz domain, and thus an $(\epsilon_7,\delta_7)$-domain in $\mathbb{R}^3$ for some $\epsilon_7,\delta_7>0$, complete the proof of Theorem~\ref{thm:4.3}. \end{proof} We finish this section by giving the proof of Theorem~\ref{thm:1.3}. We shall need the following results: \begin{proposition}[{see \cite[Theorem 1]{J}}]\label{prop:4.4} Let $D\in \mathbb{R}^d$ be an $(\epsilon,\delta)$-domain. Suppose $k\in \{1,2,3,\dots \}$ and $1\leq p\leq \infty$. Then there exists a bounded extension operator $\Lambda_{k,p}:W^{k,p} (D)\to W^{k,p} (\mathbb{R}^d)$ such that \[ (\Lambda_{k,p} f)|_D \,=f\quad (f\in W^{k,p}(D)). \] Moreover, the norm $\| \Lambda_{k,p}\|$ depends only on $\epsilon $, $\delta$, $k$, $p$ and the dimension $d$. \end{proposition} \begin{proposition}[{see \cite[p.47]{D2}}] \label{prop:4.5} Suppose $D\subseteq \mathbb{R}^d$ is a domain such that for some $p\in [1,d)$ there exists a bounded extension operator $\Lambda_{1,p}:W^{1,p} (D)\to W^{1,p} (\mathbb{R}^d)$ satisfying \[ (\Lambda_{1,p} f)|_D\,=f\quad (f\in W^{1,p}(D)). \] Let $q$ be defined by $\frac{1}{q}=\frac{1}{p}-\frac{1}{d}$. Then there exists $c=c(d)\geq 1$ such that \[ \| f\|_q\leq c\|\Lambda_{1,p} \| \{ \| \nabla f\|^p_p +\| f\|^p_p\} ^{\frac{1}{p}}\quad (f\in W^{1,p}(D)). \] \end{proposition} \begin{proposition} [{see \cite[Theorem 2.4.2, Corollaries 2.4.3,and 2.2.8]{D2}}] \label{prop:4.6} Let $D\subseteq \mathbb{R}^d$, $d\geq 3$, be a domain. Suppose there exists $c_1\geq 1$ such that \[ \| f\|_{\frac{2d}{d-2}} \leq c_1 \{ \|\nabla f\|^2_2 +\| f\|^2_2\} ^{1/2} \quad (f\in W^{1,2} (D)). \] Then there exists $c_2\geq 1$, depending only on $c_1$ and $d$, such that \[ P^D(t,x,y)\leq c_2t^{-d/2}\quad (0