\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2008(2008), No. 145, pp. 1--13.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2008 Texas State University - San Marcos.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2008/145\hfil Stability and approximations] {Stability and approximations of eigenvalues and eigenfunctions for the Neumann Laplacian, part I} \author[R. Banuelos, M. M. H. Pang\hfil EJDE-2008/145\hfilneg] {Rodrigo Ba\~nuelos, Michael M. H. Pang} % in alphabetical order \address{Rodrigo Ba\~nuelos \newline Department of Mathematics\\ Purdue University\\ West Lafayette, IN 47906, USA} \email{banuelos@math.purdue.edu} \address{Michael M. H. Pang \newline Department of Mathematics\\ University of Missouri \\ Columbia, MO 65211, USA} \email{pangm@math.missouri.edu} \thanks{Submitted February 4, 2008. Published October 24, 2008.} \thanks{R. Banuelos was partially supported by grant 0603701-DMS from the NSF} \subjclass[2000]{35P05, 35P15} \keywords{Stability; approximations; Neumann eigenvalues and eigenfunctions} \begin{abstract} We investigate stability and approximation properties of the lowest nonzero eigenvalue and corresponding eigenfunction of the Neumann Laplacian on domains satisfying a heat kernel bound condition. The results and proofs in this paper will be used and extended in a sequel paper to obtain stability results for domains in $\mathbb{R}^2$ with a snowflake type boundary. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{hypothesis}[theorem]{Hypothesis} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} The goal of this paper and its sequel \cite{P3} is to prove stability results for the smallest positive Neumann eigenvalue and its associated eigenfunctions of domains in $\mathbb{R}^2$ with a snowflake type fractal boundary. In particular, our goal is that our results should apply to the Koch snowflake domain and its usual sequence of approximating polygons from inside. Suppose the Neumann Laplacian $-\Delta _\Omega \geq 0$ defined on a domain $\Omega$ in $\mathbb{R}^d$ has discrete spectrum. The numerical computation of its eigenvalues and eigenfunctions often assumes that if $\Omega$ is replaced by an approximating domain with polygonal or piecewise smooth boundary, then the eigenvalues and eigenfunctions will not change too much. This continuous dependence of the Neumann eigenvalues and eigenfunctions on the domain, however, is not obvious. Moreover, it is known that even if $\Omega$ has smooth boundary, the spectrum of its Neumann Laplacian does not necessarily remain discrete under ``small" perturbations (see \cite{HSS,HKP}). Therefore the approximating domain, apart from being ``close" to $\Omega$, must also satisfy some ``regularity conditions". Burenkov and Davies \cite{BD} studied this problem when $\Omega$ and its approximating domain both have a boundary satisfying a uniform Holder condition, and they obtained explicit estimates for the change in the Neumann eigenvalues. More recently, Renka \cite{R}, Benjai \cite{B}, and Neuberger, Sieben and Swift \cite{NSS} have numerically computed the Neumann eigenvalues and eigenfunctions of the Koch snowflake domain. However, the boundary of the Koch snowflake domain does not satisfy a uniform Holder condition. This motivates us to prove stability results for the Neumann eigenvalues and eigenfunctions starting from a different set of assumptions. In this paper we mainly consider the case when the lowest positive Neumann eigenvalue has multiplicity 1. In \cite{P3} we shall extend these results, and their ideas of proof, to the case when the lowest positive Neumann eigenvalue has multiplicity at least 2, and show that these results apply to the Koch snowflake domain and its usual sequence of approximating polygons from inside. Let $\Omega$ be a bounded domain in $\mathbb{R}^d$ and, for all sufficiently small $\delta >0$, say $0<\delta <\delta_0$, let $\Omega_\delta$ be a subdomain of $\Omega$ satisfying \begin{equation}\label{eq:1.1} \partial \Omega _\delta \subseteq \{x\in \Omega : \mathop{\rm dist} (x,\partial \Omega)\leq \delta \}. \end{equation} Let $P^\Omega_t (x,y)$ and $P^{\Omega_\delta}_t (x,y)$ be the heat kernels corresponding to the semigroups generated by $-\Delta _\Omega$ and $-\Delta _{\Omega_\delta}$, respectively. \begin{hypothesis}\label{hyp} \rm Our main assumption on $\Omega$ and $\Omega_{\delta}$ is that there exist $c_0\geq 1$ and $N>0$ such that, for all $0 0$ (see \cite[p.77]{D1}). \end{remark} Under Hypothesis \ref{hyp}, $-\Delta _\Omega$ and $-\Delta _{\Omega_\delta}$ have compact resolvent (see \cite[p. 61]{D1}). We let $0<\mu_2 \leq \mu_3\leq \dots$ be the eigenvalues of $-\Delta_\Omega$, counting multiplicity, and let $\varphi_2,\varphi_3,\varphi_4,\dots$ be the eigenfunction associated to $\mu_2,\mu_3,\mu_4, \dots$ respectively. We assume that $|\Omega|^{-1/2},\varphi_2,\varphi_3,\dots$ form a complete orthonormal system on $L^2(\Omega)$. We let $0<\mu^\delta_2\leq \mu^\delta_3 \leq \dots $ and $\varphi^\delta_2,\varphi^\delta_3,\varphi^\delta_4,\dots $ be the corresponding quantities for the Neumann Laplacian $-\Delta_{\Omega _\delta}$ on $\Omega_\delta$. \begin{theorem}\label{thm:1.1} Suppose $\Omega$ and $\Omega_\delta$ satisfy Hypothesis \ref{hyp}. Then \begin{equation}\label{eq:1.4} \lim_{\delta \downarrow 0} \mu^\delta_2 =\mu_2. \end{equation} \end{theorem} \begin{theorem}\label{thm:1.2} Suppose $\Omega$ and $\Omega_\delta$ satisfy Hypothesis \ref{hyp}. If $\mu_2$ has multiplicity $1$, then there exists $\delta_1>0$ such that \begin{equation}\label{eq:1.5} \mu^\delta_3 \geq \mu_2 +\delta_1 \end{equation} for all $0<\delta <\delta_1$. Hence, from \eqref{eq:1.4} and \eqref{eq:1.5}, $\mu^\delta_2$ has multiplicity $1$ for all $0<\delta <\delta_1$. \end{theorem} \begin{theorem}\label{thm:1.3} Suppose $\Omega$ and $\Omega_\delta$ satisfy Hypothesis \ref{hyp} and assumed that $\mu_2$ has multiplicity $1$. If $\Omega '$ is a subdomain of $\Omega$ such that $\overline{\Omega '}\subseteq \Omega$, then \begin{equation}\label{eq:1.6} \lim_{\delta\downarrow 0} \sup_{z\in \overline{\Omega '}} |\varphi ^\delta_2 (z)-\varphi_2 (z)|=0. \end{equation} \end{theorem} \begin{remark} \rm (i) Using Theorem~\ref{thm:1.2} in the following example, one can show that multiplicity 2 of $\mu _2$ is not stable under small perturbations. Let $\Omega (t)$, $0\leq t\leq 1$, be a continuous family of convex deformations from a long thin rectangle to a square. That is, \begin{itemize} \item[(a)] $\Omega (t)$ is a convex domain for $0\leq t\leq 1$, \item[(b)] $\Omega (0)$ is a long thin rectangle and $\Omega (1)$ is a square. \end{itemize} We can assume that $\Omega (t)$ is symmetric with respect to the $x$ and $y$ axes for all $t \in [0, 1]$. For each $t\in [0,1]$, let $\mu (t)$ be the smallest non-zero Neumann eigenvalue of $\Omega (t)$. Then $\mu (0)$ and $\mu (1)$ have multiplicity $1$ and $2$, respectively. So we can let $$ t_0=\inf \{t\in [0,1]:\mu (t) \text{has multiplicity 2}\}. $$ Let $\{t_n\}_{n=1}^{\infty}$ be a decreasing sequence of numbers in $[0, 1]$ such that $t_n \downarrow t_0$ as $n \to \infty$ and that $\mu(t_n) $ has multiplicity 2 for all $n = 1, 2, 3, \dots $. Since $\mathop{\rm dist}(\partial \Omega (t_0), \partial \Omega (t_n))\to0$ as $n\to\infty$ and since the domains $\Omega (t)$ are convex and symmetric with respect to the $x$ and $y$ axes, for each $n=1, 2, 3, \dots $ we can let $D(t_n)$ be an dilation of $\Omega (t_n)$ such that \begin{itemize} \item[(c)] $D(t_n) \subseteq \Omega (t_0)$, \item[(d)] $\mathop{\rm dist}(\partial \Omega (t_0), \partial D(t_n)) \to 0$ as $ n \to \infty$. \end{itemize} Let $\lambda _n$ be the smallest nonzero Neumann eigenvalue of $D(t_n)$. Then, since $D(t_n)$ is a dilation of $\Omega(t_n)$, the multiplicity of $\lambda _n$ is the same as that of $\mu(t_n)$; i.e., $\lambda_n$ has multiplicity 2 for all $n=1,2,3,\dots$. Then, by Theorem \ref{thm:1.2}, $\mu (t_0)$ must have multiplicity $2$. In particular, $t_0>0$. Let $\{s_n\}_{n=1}^{\infty}$ be an increasing sequence on $[0, 1]$ such that $s_n \uparrow t_0$ as $n \to \infty$. Then, just as for $\{t_n\}_{n=1}^{\infty}$ before, we can let $D(s_n)$ be a dilation of $\Omega(s_n)$ satisfying \begin{itemize} \item[(e)] $D(s_n) \subseteq \Omega(t_0)$, \item[(f)] $\mathop{\rm dist}(\partial \Omega(t_0), \partial D(s_n)) \to 0$ as $n\to \infty$. \end{itemize} Let $\zeta_n$ be the smallest nonzero Neumann eigenvalue of $D(s_n)$. Then $\zeta_n$ has the same multiplicity as that of $\mu(s_n)$ since $D(s_n)$ is a dilation of $\Omega (s_n)$. But, by the definition of $t_0$, $\zeta _n$ has multiplicity 1 for all $n=1,2,3, \dots$. \smallskip \noindent (ii) Theorems \ref{thm:1.1}, \ref{thm:1.2}, and \ref{thm:1.3} will be extended to the case when $\mu_2$ has multiplicity at least 2 in \cite{P3}. With an additional inductive argument, it is possible to extend these results to all higher Neumann eigenvalues and eigenfunctions and to more general elliptic operators, including some non-uniformly elliptic operators. We plan to return to these issues in a later paper. \smallskip \noindent (iii) We mention that spectral stability results for the Dirichlet Laplacian are much more extensive than those for the Neumann Laplacian. Sharp rates for the convergence of Dirichlet eigenvalues and eigenfunctions can be found in \cite{P2} and \cite{D3}. We refer the readers to the excellent article \cite{BLLC} for a recent survey of spectral stability results for the Dirichlet and Neumann Laplacians and for more general elliptic operators. \end{remark} \section{Proofs of Theorems~\ref{thm:1.1}, \ref{thm:1.2} and \ref{thm:1.3}} \begin{lemma}[{see \cite[Lemma 4.10]{P1}}]\label{lem:2.1} Let $\Sigma$ be a domain in $\mathbb{R}^d$, let $u$ be a solution of the parabolic equation \begin{equation*} \frac{\partial u}{\partial t}-\omega^{-1} \sum^d_{i,j=1} \big\{ \frac{\partial }{\partial x_i} \big(a_{ij}\frac{\partial u}{\partial x_j}\big)\big\} =0 \end{equation*} in $\sum \times (\tau _1,\tau _2)$, where $\omega$ and $\{ a_{ij}\}$ satisfy \begin{equation*} \begin{gathered} 0 <\lambda^{-1}\leq \{a_{ij}(x)\}\leq \lambda <\infty\\ 0 <\lambda^{-1}\leq \omega (x)\leq \lambda <\infty \end{gathered} \quad (x\in \Sigma) \end{equation*} for some $\lambda \geq 1$. Let $\Sigma'$ be a subdomain of $\Sigma$ and suppose that \begin{equation*} \mathop{\rm dist} (\Sigma ',\partial \Sigma)>\eta \quad\text{and}\quad t_1-\tau _1\geq \eta^2. \end{equation*} Then \begin{equation*} |u(x,t)-u(y,s)|\leq A[ |x-y|+(t-s)^{1/2}\}^\alpha \end{equation*} for all $x,y\in \Sigma'$ and $t,s\in [t_1,\tau _2]$, where $\alpha $ depends only on $d$ and $\lambda$ and \begin{equation*} A=\big( \frac{4}{\eta }\big)^\alpha \theta \end{equation*} where $\theta$ is the oscillation of $u$ in $\Sigma \times (\tau _1,\tau _2)$. \end{lemma} \begin{lemma}[\cite{BC}]\label{lem:2.2} Let $\Omega$ and $\Omega_\delta $, $0<\delta \leq \delta_0$, be as described in Section 1. Let $T^\Omega_t$ and $T^{\Omega_\delta}_t$ be the semigroups generated by the Neumann Laplacians $-\Delta _\Omega$ and $-\Delta_{\Omega_\delta}$ on $\Omega$ and $\Omega_\delta$, respectively. Then, for all $f\in L^\infty (\Omega)$ and compact subset $K\subseteq \Omega$, we have \begin{equation*} \lim_{\delta \downarrow 0}T_t^{\Omega_\delta} (f1_{\Omega_\delta})(x) =T^\Omega_t f(x)\quad (\text{a.e. }x\in K) \end{equation*} \end{lemma} \begin{proposition}\label{prop:2.3} For all $t_0\in (0,1]$ and all $x_0,y_0\in \Omega$, we have \begin{equation*} \lim_{\delta \downarrow 0} P^{\Omega_\delta}_{t_0}(x_0,y_0) =P^\Omega_{t_0}(x_0,y_0). \end{equation*} \end{proposition} \begin{proof} Applying Lemma \ref{lem:2.1} with \begin{gather*} \Sigma =\Omega,\quad \tau _1=\frac{1}{4}t_0,\quad \tau _2=1,\\ u(x,t)=P^\Omega_t(x,y_0),\quad \lambda =1,\quad \omega (x)\equiv 1,\\ \Sigma '=B\Big( x_0,\frac{1}{4}\mathop{\rm dist} (x_0,\partial \Omega)\Big),\quad t_1=\frac{1}{2}t_0,\\ \eta =\min \big\{\frac{1}{4}\mathop{\rm dist} (x_0,\partial \Omega), \frac{1}{2} t_0^{1/2}\big\}, \end{gather*} we obtain, for all $t \in (t_1 , \tau _2 )$ and $x \in B( x_0,\frac{1}{4}\mathop{\rm dist} (x_0,\partial \Omega))$, \begin{equation}\label{eq:2.1} |P^\Omega _t(x,y_0)-P^\Omega_t (x_0,y_0)|\leq A|x - x_0|^\alpha \end{equation} where $\alpha \in (0,1]$ depends only on $d$, and $A>0$ depends only on $d$, $\mathop{\rm dist} (x_0,\partial \Omega)$, $t_0$, $N$ and $c_0$ in \eqref{eq:1.2}. Similarly, we deduce that \begin{equation}\label{eq:2.2} |P^{\Omega_\delta}_t (x,y_0)-P^{\Omega _\delta}_t (x_0,y_0)| \leq A|x - x_0|^\alpha, \end{equation} where $\alpha $ and $A$ in \eqref{eq:2.2} have the same values as in \eqref{eq:2.1} for all $\delta >0$ satisfying \begin{equation*} 0<\delta <\min \big\{ \delta_0, \frac{1}{2}\mathop{\rm dist} (x_0,\partial \Omega ) \big\}. \end{equation*} For all $00$ be given. Then we can first fix $r\in \left( 0,\frac{1}{4}\mathop{\rm dist} (x_0,\partial \Omega)\right)$ such that \begin{equation}\label{eq:2.8} 00$ such that, for all $00$, let $f_\delta \in L^\infty (\Omega_\delta)$ such that \begin{equation*} \| f_\delta \|_\infty \leq M. \end{equation*} Then, for all $0 < t \leq 1$, \begin{equation*} \| T^\Omega_t (E_\delta f_\delta) -E_\delta (T^{\Omega_\delta}_t f_\delta ) \|_{L^2(\Omega)} \to 0 \end{equation*} as $\delta \downarrow 0$. \end{proposition} \begin{proof} Let $t\in (0,1]$ and $\epsilon \in (0,1)$ be fixed. Choose $\delta_3 >0$ sufficiently small so that \begin{equation}\label{eq:2.10} 2c_0t^{-\frac{N}{2}}M|\Omega \backslash \Omega_{\delta_3}| \leq \frac{\epsilon }{2}\end{equation} and \begin{equation}\label{eq:2.11} 2c_0t^{-\frac{N}{2}}M|\Omega ||\Omega \backslash \Omega_{\delta _3}|^{1/2}\leq \frac{\epsilon }{2}. \end{equation} Then, for all $\delta \in (0,\delta _3]$ and $x\in \Omega_{\delta_3}$, we have \begin{equation}\label{eq:2.12} \begin{aligned} &(T^\Omega_t E_\delta f_\delta -E_\delta T^{\Omega _\delta}_t f_\delta )(x)\\ & =\int_{\Omega_\delta} [P^\Omega_t(x,y)-P^{\Omega_\delta}_t (x,y)]f_\delta (y)dy\\ & =\Big( \int_{\Omega_\delta \backslash \Omega_{\delta_3}} +\int_{\Omega_{\delta_3}}\Big) \big[P^\Omega_t(x,y)-P^{\Omega_\delta}_t (x,y)\big] f_\delta (y)dy. \end{aligned} \end{equation} By \eqref{eq:1.2}, \eqref{eq:1.3} and \eqref{eq:2.10}, we have \begin{equation}\label{eq:2.13} \big| \int_{\Omega_\delta \backslash \Omega_{\delta_3}} [P^\Omega_t (x,y)-P^{\Omega_\delta}_t (x,y)]f_\delta (y)dy\big| \leq 2c_0t^{-\frac{N}{2}} M|\Omega \backslash \Omega_{\delta_3}| \leq \frac{\epsilon }{2}. \end{equation} By \eqref{eq:1.2}, \eqref{eq:1.3} and Proposition~\ref{prop:2.3}, there exists $\delta_4=\delta_4(\epsilon ,x)>0$ such that \begin{equation}\label{eq:2.14} \big|\int_{\Omega_{\delta_3}} [P^\Omega _t (x,y) -P^{\Omega_\delta}_t (x,y)]f_\delta (y)dy\big| \leq \frac{\epsilon }{2} \end{equation} for all $\delta \in (0,\delta_4)$. Therefore \eqref{eq:2.12}, \eqref{eq:2.13} and \eqref{eq:2.14} imply that \begin{equation}\label{eq:2.15} (T^\Omega_t E_\delta f_\delta -E_\delta T^{\Omega_\delta }_t f_\delta )(x) \to 0\quad \text{as } \delta \downarrow 0 \end{equation} for all $x\in \Omega_{\delta_3}$. Since, by \eqref{eq:1.2} and \eqref{eq:1.3}, \begin{equation*} \| T^\Omega _t E_\delta f_\delta -E_\delta T^{\Omega_\delta}_t f_\delta \|_\infty \leq 2c_0t^{-N/2} M|\Omega|, \end{equation*} there exists $\delta_5=\delta_5 (\epsilon )>0$ such that \begin{equation}\label{eq:2.16} \| R_{\delta_3} (T^\Omega _t E_\delta f_\delta -E_\delta T^{\Omega_\delta}_t f_\delta )\|^2_{L^2(\Omega_{\delta_3})} \leq \frac{\epsilon^2}{4} \end{equation} for all $\delta \in (0,\delta_5)$. Also, by \eqref{eq:2.11}, we have \begin{equation}\label{eq:2.17} \int_{\Omega \backslash \Omega_{\delta_3}} |(T^\Omega_t E_\delta f_\delta -E_\delta T^{\Omega_\delta }_t f_\delta )(x)|^2dx \leq (2c_0t^{-\frac{N}{2}}M|\Omega|)^2|\Omega \backslash \Omega_{\delta_3}|\leq \frac{\epsilon^2}{4} \end{equation} for all $\delta \in (0,\delta_3)$. The proposition now follows from \eqref{eq:2.16} and \eqref{eq:2.17}. \end{proof} \begin{proof}[Proof of Theorem \ref{thm:1.1}] Let $\epsilon \in (0,1)$ be given. For all sufficiently small $\delta >0$, let \begin{equation*} \beta_1 (\delta )=|\Omega_\delta |^{-1} \int_{\Omega_\delta} \varphi_2 (x)dx. \end{equation*} Taking inner products and norms in $L^2 (\Omega_\delta)$, we get \begin{align*} e^{-\mu^\delta_2 t} &\geq \|R_\delta (\varphi_2 -\beta_1 (\delta) )\|^{-2}_2 \langle T^{\Omega_\delta}_t R_\delta (\varphi_2 -\beta_1 (\delta)),R_\delta (\varphi_2-\beta_1 (\delta))\rangle\\ &= \| R_\delta (\varphi_2-\beta_1 (\delta))\|^{-2}_2 \{ \langle T^{\Omega_\delta}_t R_\delta \varphi_2, R_\delta \varphi_2\rangle -2 \langle R_\delta \varphi_2, \beta_1 (\delta )1_{\Omega_\delta}\rangle +\beta_1 (\delta)^2|\Omega_\delta |\}. \end{align*} So for $0<\delta <\delta_6$, we have \begin{equation} \label{eq:2.18} \begin{split} e^{-\mu^\delta _2 t} &\geq \|R_\delta (\varphi_2-\beta_1(\delta))\|^{-2}_2 \{\langle T^{\Omega_\delta}_t (R_\delta \varphi_2 -(R_\delta \varphi_2 )1_{\Omega_{\delta_6}})\\ &\quad +T^{\Omega_\delta}_t ((R_\delta \varphi_2 )1_{\Omega_{\delta_6}}),(R_\delta \varphi_2-(R_\delta \varphi_2)1_{\Omega_{\delta_6}})\\ &\quad +(R_\delta \varphi_2)1_{\Omega_{\delta_6}}\rangle -2 \langle R_\delta \varphi_2,\,\,\beta _1 (\delta)1_{\Omega_\delta}\rangle +\beta_1(\delta)^2|\Omega_\delta |\}\\ &= \| R_\delta (\varphi_2 -\beta_1 (\delta))\|^{-2}_2 \{ \langle T^{\Omega_\delta}_t (R_\delta \varphi_2 -(R_\delta \varphi_2 )1_{\Omega_{\delta_6}}),\\ &\quad (R_\delta \varphi_2 -(R_\delta \varphi_2)1_{\Omega_{\delta_6}} )\rangle +2\langle T^{\Omega_\delta}_t (R_\delta \varphi_2 -(R_\delta \varphi_2)1_{\Omega_{\delta_6}}),\\ &\quad (R_\delta \varphi_2 )1_{\Omega_{\delta_6}}\rangle +\langle T^{\Omega_\delta}_t (R_\delta \varphi_2)1_{\Omega_{\delta_6}},(R_\delta \varphi_2)1_{\Omega_{\delta_6}}\rangle\\ &\quad -2\langle R_\delta \varphi_2 ,\,\,\beta_1 (\delta )1_{\Omega_{\delta_6}}\rangle + \beta_1 (\delta)^2|\Omega_\delta|\}.\end{split}\end{equation} But \begin{equation}\begin{split}\label{eq:2.19} &\langle T^{\Omega_\delta}_t ((R_\delta \varphi_2 )1_{\Omega_{\delta_6}}),(R_\delta \varphi_2)1_{\Omega_{\delta_6}}\rangle\\ & =\langle T^{\Omega_\delta}_t ((R_\delta \varphi_2)1_{\Omega_{\delta_6}})-R_\delta T^\Omega_t (\varphi_2 1_{\Omega_{\delta_6}}),(R_\delta \varphi_2)1_{\Omega_{\delta_6}}\rangle\\ &\quad + \langle R_\delta T^\Omega_t (\varphi_2 1_{\Omega_{\delta_6}}),(R_\delta \varphi_2)1_{\Omega_{\delta_6}}\rangle\\ & = \langle T^{\Omega_\delta}_t ((R_\delta \varphi_2 )1_{\Omega_{\delta_6}})-R_\delta T^\Omega_t (\varphi_2 1_{\Omega_{\delta _6}}),(R_\delta \varphi_2)1_{\Omega_{\delta_6}}\rangle\\ &\quad +\langle R_\delta T^\Omega_t (\varphi_2 1_{ \Omega_{\delta_6}}) - R_{\delta}T^{\Omega }_t \varphi _2 + R_{\delta } T^{\Omega }_t \varphi _2,\,\, (R_\delta \varphi_2) 1_{\Omega_{\delta_6}} - R_{\delta }\varphi _2 + R_{\delta }\varphi _2\rangle \\ & = \langle T^{\Omega_\delta}_t ((R_\delta \varphi_2 )1_{\Omega_{\delta_6}})-R_\delta T^\Omega_t (\varphi_2 1_{\Omega_{\delta _6}}),(R_\delta \varphi_2)1_{\Omega_{\delta_6}}\rangle\\ &\quad +\langle R_\delta T^\Omega_t (\varphi_2 1_{\Omega \backslash \Omega_{\delta_6}}),R_\delta (\varphi_2 1_{\Omega \backslash \Omega_{\delta_6}})\rangle -\langle R_\delta T^\Omega_t (\varphi_2 1_{\Omega\backslash\Omega_{\delta_6}}),R_\delta \varphi_2\rangle\\ &\quad -\langle R_\delta T^\Omega_t\varphi_2,R_\delta (\varphi_21_{\Omega \backslash \Omega_\delta})\rangle + \langle R_\delta T^\Omega_t \varphi_2,R_\delta \varphi_2\rangle. \end{split}\end{equation} From \eqref{eq:2.18} and \eqref{eq:2.19} we obtain \begin{equation}\begin{split}\label{eq:2.20} e^{-\mu^\delta_2t} &\geq \| R_\delta (\varphi_2-\beta_1 (\delta ))\|^{-2}_2 \Big\{ \langle T^{\Omega_\delta}_t R_\delta (\varphi_2 1_{\Omega \backslash\Omega_{\delta_6}}),R_\delta (\varphi_2 1_{\Omega\backslash\Omega_{\delta_6}})\rangle\\ &\quad + 2\langle T^{\Omega_\delta}_t R_\delta (\varphi_2 1_{\Omega \backslash\Omega_{\delta_6}}),R_\delta (\varphi_2 1_{\Omega_{\delta_6}})\rangle\\ &\quad -2\langle R_\delta \varphi_2,\beta_1 (\delta)1_{\Omega_{\delta_6}}\rangle +\beta_1 (\delta)^2|\Omega_\delta |\\ &\quad +\langle T^{\Omega_\delta}_t ((R_\delta \varphi_2 )1_{\Omega_{\delta_6}})-R_\delta T^\Omega_t (\varphi_2 1_{\Omega_{\delta_6}}),(R_\delta \varphi_2)1_{\Omega_{\delta_6}}\rangle\\ &\quad + \langle R_\delta T^\Omega_t (\varphi_21_{\Omega \backslash\Omega_{\delta_6}}),R_\delta (\varphi_2 1_{\Omega \backslash\Omega_{\delta_6}})\rangle - \langle R_\delta T^\Omega_t (\varphi_21_{\Omega \backslash\Omega_{\delta_6}}),R_\delta \varphi_2\rangle\\ &\quad -e^{-\mu_2t}\langle R_\delta \varphi_2,R_\delta (\varphi_2 1_{\Omega \backslash\Omega_{\delta_6}})\rangle +e^{-\mu_2t}\langle R_\delta \varphi_2,R_\delta \varphi_2 \rangle \Big\}\\ & = A\{ B_1+B_2-B_3+B_4+B_5 +B_6-B_7-B_8+B_9\}.\end{split}\end{equation} Since $\varphi_2 $ is orthogonal to $1$ in $L^2(\Omega)$, we have \begin{equation}\label{eq:2.21} \lim_{\delta \downarrow 0} \beta_1 (\delta)=0. \end{equation} Hence \begin{equation}\label{eq:2.22} \lim_{\delta \downarrow 0}A=1,\quad \lim_{\delta \downarrow 0}B_3=0,\quad \lim_{\delta \downarrow 0}B_4=0. \end{equation} Since \begin{equation*} \| \varphi_2 \|_\infty=e^{\mu_2t}\|T^\Omega_t \varphi_2\|_\infty \leq e^{\mu_2 t}c^{1/2}_0t^{-\frac{N}{4}}\quad (00$ sufficiently small so that \begin{equation*} \| \varphi_2 1_{\Omega \backslash \Omega _{\delta_6}}\|_{L^2(\Omega)}\leq \frac{\epsilon}{12}.\end{equation*} Then we have, for $0<\delta <\delta_6$, \begin{equation}\label{eq:2.23} |B_1|\leq \frac{\epsilon }{12},\quad |B_2|\leq \frac{\epsilon }{6},\quad |B_6|\leq \frac{\epsilon}{12},\quad |B_7|\leq \frac{\epsilon}{12},\quad |B_8|\leq \frac{\epsilon}{12}, \end{equation} and \begin{equation}\label{eq:2.24} B_9 =e^{-\mu_2t}\Big\{ \int_\Omega \varphi_2 (x)^2dx -\int_{\Omega \backslash\Omega_\delta} \varphi_2 (x)^2dx\Big\} = e^{-\mu_2t}-B_{10} \end{equation} where \begin{equation}\label{eq:2.25} 0\leq B_{10}=e^{-\mu_2t}\int_{\Omega \backslash\Omega_\delta} \varphi_2 (x)^2 dx\leq \frac{\epsilon^2}{144}<\frac{\epsilon}{12}. \end{equation} By Proposition \ref{prop:2.4} we have \begin{equation}\label{eq:2.26} \lim_{\delta \downarrow 0}B_5=0. \end{equation} Thus, by \eqref{eq:2.22} and \eqref{eq:2.26}, there exists $\delta_7\in (0,\delta_6]$ such that, for all $0<\delta <\delta_7$, \begin{equation}\label{eq:2.27} |B_3|\leq \frac{\epsilon }{12},\quad |B_4|\leq \frac{\epsilon}{12},\quad |B_5|\leq \frac{\epsilon}{12},\quad |A-1|\leq \frac{\epsilon}{12}. \end{equation} Then, by \eqref{eq:2.20}, \eqref{eq:2.23}, \eqref{eq:2.24}, \eqref{eq:2.25} and \eqref{eq:2.27}, we have \begin{equation}\label{eq:2.28} e^{\mu^\delta_2 t}\geq e^{-\mu_2t}-\epsilon\quad (0<\delta <\delta_7). \end{equation} We next prove the reverse inequality of \eqref{eq:2.28}. For all $0<\delta <\delta_6$ we have \begin{equation}\label{eq:2.29} \begin{split} e^{-\mu_2 t} &\geq \langle T^\Omega_t E_\delta \varphi^\delta_2 ,E_\delta \varphi^\delta_2\rangle\\ &= \langle T^\Omega_t E_\delta [(\varphi^\delta_2 -\varphi^\delta_2 1_{\Omega_{\delta_6}} )+\varphi^\delta _21_{\Omega_{\delta _6}}], E_\delta [(\varphi^\delta_2 -\varphi^\delta_2 1_{\Omega_{\delta_6}})+\varphi^\delta _2 1_{\Omega _{\delta_6}}]\rangle\\ &= \langle T^\Omega_t E_\delta [\varphi^\delta_2 1_{\Omega \backslash\Omega_{\delta_6}}],E_\delta [\varphi^\delta_21_{\Omega\backslash\Omega_{\delta_6}}]\rangle +2\langle T^\Omega_t E_\delta [\varphi^\delta _2 1_{\Omega \backslash \Omega_{\delta_6}}],E_\delta [\varphi^\delta_2 1_{\Omega_{\delta_6}}]\rangle\\ &\quad +\langle T^\Omega_t E_\delta [\varphi^\delta_2 1_{\Omega_{\delta_6}}] ,E_\delta[\varphi^\delta_2 1_{\Omega_{\delta_6}}]\rangle\\ &= \langle T^\Omega_t E_\delta [\varphi^\delta_2 1_{\Omega \backslash \Omega_{\delta_6}}],E_\delta [\varphi^\delta_2 1_{\Omega \backslash\Omega_{\delta_6}}]\rangle +2\langle T^\Omega_t E_\delta [\varphi^\delta_2 1_{\Omega\backslash \Omega_{\delta_6}}],E_\delta[\varphi^\delta_21_{\Omega_{\delta_6}}]\rangle\\ &\quad +\langle T^\Omega_t E_\delta[\varphi^\delta_21_{\Omega_{\delta_6}} ]-E_\delta T^{\Omega _\delta}_t (\varphi^\delta_2 1_{\Omega_{\delta_6}}),\,E_\delta [\varphi^\delta_2 1_{\Omega_{\delta_6}}]\rangle\\ &\quad +\langle E_\delta T^{\Omega_\delta}_t (\varphi^\delta_2 1_{\Omega_{\delta_6}}),E_\delta [\varphi^\delta_2 1_{\Omega_{\delta_6}}]\rangle.\end{split}\end{equation} But \begin{equation}\label{eq:2.30}\begin{split} &\langle E_\delta T^{\Omega_\delta}_t (\varphi^\delta_2 1_{\Omega_{\delta_6}}), E_\delta (\varphi^\delta_21_{\Omega_{\delta_6}})\rangle\\ &=\langle T^{\Omega_\delta}_t (\varphi^\delta_2 1_{\Omega_{\delta_6}}),\varphi^\delta_2 1_{\Omega_{\delta_6}}\rangle_{L^2(\Omega_\delta)}\\ &= \langle T^{\Omega_\delta}_t \varphi^\delta_2 -T^{\Omega_\delta}_t (\varphi^\delta_2 1_{\Omega_\delta \backslash \Omega_{\delta_6}}),\varphi^\delta_2-\varphi^\delta_2 1_{\Omega_\delta\backslash\Omega_{\delta_6}}\rangle_{L^2(\Omega_\delta)}\\ &= e^{-\mu^\delta_2 t}-2e^{-\mu^\delta_2 t}\langle \varphi^\delta_2 1_{\Omega_\delta \backslash\Omega_{\delta_6}},\varphi^\delta_2 \rangle _{L^2(\Omega_\delta)}\\ &\quad+\langle T^{\Omega_\delta}_t (\varphi^\delta_2 1_{\Omega_\delta \backslash \Omega_{\delta_6}}),\varphi^\delta_2 1_{\Omega_\delta \backslash \Omega_{\delta_6}}\rangle_{L^2(\Omega_\delta)}. \end{split} \end{equation} From \eqref{eq:2.29} and \eqref{eq:2.30} we have, for $0<\delta <\delta_6$, \begin{equation}\begin{split}\label{eq:2.31} e^{-\mu_2 t} &\geq \langle T^\Omega_t E_\delta [\varphi^\delta_2 1_{\Omega\backslash\Omega_{\delta_6}}],E_\delta[\varphi^\delta_2 1_{\Omega \backslash \Omega_{\delta_6}}]\rangle\\ &\quad +2\langle T^\Omega_t E_\delta [\varphi^\delta_2 1_{\Omega \backslash \Omega _{\delta_6}}], E_\delta [\varphi^\delta_2 1_{\Omega _{\delta_6}}]\rangle\\ &\quad +\langle T^\Omega _t E_\delta [\varphi^\delta_2 1_{\Omega_{\delta_6}}]-E_\delta T^{\Omega_\delta}_t (\varphi^\delta_2 1_{\Omega_{\delta_6}}),E_\delta [\varphi^\delta_2 1_{\Omega_{\delta_6}}]\rangle\\ &\quad -2e^{-\mu^\delta_2t}\langle \varphi^\delta_2 1_{\Omega_\delta \backslash \Omega_{\delta_6}},\varphi^\delta_2\rangle_{L^2(\Omega_\delta)}\\ &\quad + \langle T^{\Omega_\delta}_t (\varphi^\delta_2 1_{\Omega_\delta \backslash\Omega_{\delta_6}}),\varphi^\delta_2 1_{\Omega_\delta \backslash \Omega_{\delta_6}}\rangle_{L^2(\Omega_\delta)} +e^{-\mu^\delta_2 t}\\ &= C_1+C_2 +C_3 -C_4+C_5+e^{-\mu^\delta_2 t}.\end{split} \end{equation} We now need the following estimate from \cite{W}: \begin{equation}\label{eq:2.32} \mu^\delta_2 \leq p^{2}_{d/2, 1}\pi^{d/2} \Gamma (\frac{d}{2}+1)^{-1}|\Omega_\delta |^{-\frac{2}{d}} \end{equation} where $p_{\nu, k}$ denotes the kth positive zero of the derivative of $x^{1 - \nu}J_{\nu}(x)$ and $J_{\nu}(x)$ is the standard Bessel function of the first kind of order $\nu$. From \eqref{eq:1.3} and \eqref{eq:2.32} we obtain, for $0 0$, \begin{equation}\label{eq:2.33} \|\varphi^\delta_2\|_\infty=e^{\mu^\delta_2 t}\|T^{\Omega_\delta}_t \varphi^\delta_2\|_\infty \leq e^{\mu^\delta_2t}c^{1/2}_0t^{-\frac{N}{4}} =c^{1/2}_0 c^t_1 t^{-N/4}\quad (00$, we have \begin{equation*}\begin{split} e^{-\mu^\delta_2 t}\langle \varphi_3 ,E_\delta \varphi^\delta _2\rangle &=\langle \varphi_3 ,E_\delta T^{\Omega_\delta}_t \varphi^\delta_2\rangle\\ & =\langle \varphi_3,E_\delta T^{\Omega_\delta}_t \varphi^\delta_2-T^\Omega_t E_\delta \varphi^\delta_2\rangle +\langle \varphi_3,T^\Omega_t E_\delta \varphi^\delta_2\rangle\\ & =\langle \varphi_3,E_\delta T^{\Omega_\delta}_t \varphi^\delta_2-T^\Omega E_\delta\varphi^\delta_2\rangle +e^{-\mu_3t}\langle \varphi_3,E_\delta \varphi^\delta_2\rangle.\end{split}\end{equation*} Thus \begin{equation}\label{eq:2.37} (e^{-\mu^\delta_2 t}-e^{-\mu_3t})\langle \varphi_3,E_\delta \varphi^\delta_2\rangle =\langle \varphi_3,E_\delta T^{\Omega_\delta}_t\varphi^\delta_2-T^\Omega_t E_\delta \varphi^\delta_2\rangle. \end{equation} Let $\epsilon \in (0,1)$ be given. Then \eqref{eq:1.4} and Proposition~\ref{prop:2.4} imply that for any $t\in (0,1]$ there exists $\delta_9>0$ such that, for all $0<\delta <\delta_9$, we have \begin{equation}\label{eq:2.38} |\langle \varphi_3,E_\delta T^{\Omega_\delta}_t \varphi^\delta_2 -T^\Omega_t E_\delta\varphi^\delta_2\rangle | \leq \frac{1}{2}(e^{-\mu_2t}-e^{-\mu_3t})\epsilon \end{equation} and \begin{equation}\label{eq:2.39} 0<\frac{1}{2} (e^{-\mu_2t}-e^{-\mu_3t}) \leq e^{-\mu^\delta_2 t}-e^{-\mu_3 t}. \end{equation} Therefore, by \eqref{eq:2.37}, \eqref{eq:2.38} and \eqref{eq:2.39}, we have \begin{equation*} |\langle \varphi_3,E_\delta \varphi^\delta_2\rangle | \leq \epsilon \quad (0<\delta <\delta_9). \end{equation*} This proves the proposition. \end{proof} \begin{proof}[Proof of Theorem \ref{thm:1.2}] Suppose \eqref{eq:1.5} is false. Let $\{\epsilon _k \}^\infty_{k=1}$ be a decreasing sequence of positive numbers such that $\lim_{k\to \infty } \epsilon_k=0$ and, by \eqref{eq:1.4}, \begin{equation}\label{eq:2.40} \lim_{k\to \infty} \mu^{\epsilon_k}_3 =\mu_2.\end{equation} Let \begin{gather*} E_{\epsilon_k}\varphi^{\epsilon_k}_3=a_1 (k)|\Omega|^{-1/2} +a_2 (k)\varphi_2 +\sum^\infty_{\ell=3} a_\ell (k) \varphi_\ell, \\ E_{\epsilon_k}\varphi^{\epsilon_k}_2=b_1(k)|\Omega|^{-1/2} +b_2 (k)\varphi_2 +\sum^\infty_{\ell=3} b_\ell (k)\varphi_\ell . \end{gather*} Then \begin{equation}\label{eq:2.41} a_1 (k)=\int_\Omega E_{\epsilon_k}\varphi^{\epsilon_k}_{3} |\Omega |^{-1/2}dx=\int_{\Omega_{\epsilon_k}}\varphi^{\epsilon_k}_3 dx|\Omega|^{-1/2}=0. \end{equation} We next want to show that \begin{equation}\label{eq:2.42} \big\|\sum^\infty_{\ell =3}a_\ell (k)\varphi_\ell \big\| _2 \to 0 \quad \text{as } k\to \infty. \end{equation} Let $t\in (0,1]$ and consider \begin{equation}\begin{split}\label{eq:2.43} &T^\Omega_t E_{\epsilon_k}\varphi^{\epsilon_k}_3 -E_{\epsilon_k}T^{\Omega_{\epsilon_k}}_t \varphi^{\epsilon_k}_3\\ &=T^\Omega_t E_{\epsilon_k}\varphi^{\epsilon_k}_3 -e^{-\mu_3^{\epsilon_k}t}E_{\epsilon_k}\varphi^{\epsilon_k}_3\\ &= (e^{-\mu_2t}-e^{-\mu_3^{\epsilon_k}t})a_2 (k)\varphi_2 +\sum^\infty_{\ell =3} a_\ell(k)(e^{-\mu_\ell t}-e^{-\mu_3^{\epsilon_k}t})\varphi_\ell\\ &=(e^{-\mu_2t}-e^{-\mu^{\epsilon_k}_3t})a_2 (k)\varphi_2+\sum^\infty_{\ell =3}a_\ell(k)(e^{-\mu_\ell t}-e^{-\mu_2t})\varphi_\ell\\ &\quad +\sum^\infty_{\ell =3} a_\ell(k)(e^{-\mu_2t} -e^{-\mu_3^{\epsilon_k}t})\varphi_\ell . \end{split}\end{equation} Now \begin{equation}\begin{split}\label{eq:2.44} \big\| \sum^\infty_{\ell =3} a_k(\ell ) (e^{-\mu_{\ell}t}-e^{-\mu_2t})\varphi_\ell \big\|^2_2 &=\sum^\infty_{\ell =3}a_k(\ell )^2(e^{-\mu_\ell t}-e^{-\mu_2t})^2\\ &\geq (e^{-\mu_2t}-e^{-\mu_3t})^2\sum^\infty_{\ell =3}a_\ell(k)^2\\ &= (e^{-\mu_2t}-e^{-\mu_3t})^2 \big\|\sum^\infty_{\ell =3}a_\ell (k)\varphi_\ell\big\|^2_2. \end{split} \end{equation} By Proposition~\ref{prop:2.4} we have \begin{equation}\label{eq:2.45} \| T^\Omega_t E_{\epsilon_k}\varphi^{\epsilon_k}_3 -E_{\epsilon_k}T^{\Omega_{\epsilon_k}}_t \varphi^{\epsilon_k}_3\|_2\to 0 \quad \text{as } k\to \infty. \end{equation} Thus, by \eqref{eq:2.40}, \eqref{eq:2.41}, \eqref{eq:2.43} and \eqref{eq:2.45}, we obtain \begin{equation}\label{eq:2.46} \lim_{k\to \infty} \big\| \sum^\infty_{\ell =3}a_\ell (k)(e^{-\mu_{\ell}t} -e^{-\mu_2t})\varphi_\ell \big\|^2_2=0. \end{equation} So \eqref{eq:2.42} follows from \eqref{eq:2.44} and \eqref{eq:2.46}. By a similar argument we can show that \begin{equation}\label{eq:2.47} b_1(k)=0 \end{equation} and \begin{equation}\label{eq:2.48} \lim_{k\to \infty}\big\|\sum^\infty_{\ell =3} b_\ell (k)\varphi_\ell \big\|_2=0. \end{equation} Since \eqref{eq:2.41}, \eqref{eq:2.42}, \eqref{eq:2.47} and \eqref{eq:2.48} imply that \begin{equation*} \lim_{k\to \infty} a_2 (k)=\lim_{k\to \infty} b_2 (k)=1, \end{equation*} we have \begin{equation*}\begin{split} 0&= \langle \varphi^{\epsilon_k}_3, \varphi_2^{\epsilon _k} \rangle _{L^2(\Omega_{\epsilon_k})}\\ &= \langle E_{\epsilon_k}\varphi^{\epsilon_k}_3 ,E_{\epsilon_k}\varphi^{\epsilon_k}_2\rangle _{L^2(\Omega)}\\ &= \Big\langle a_2 (k)\varphi_2 +\sum^\infty_{\ell =3} a_\ell (k)\varphi_\ell , b_2 (k)\varphi_2 +\sum^\infty_{\ell =3} b_\ell (k)\varphi_\ell \Big\rangle_{L^2(\Omega)}\\ &=a_2 (k)b_2 (k)+\Big\langle \sum^\infty_{\ell =3} a_\ell (k)\varphi_\ell , \sum^\infty_{\ell =3} b_\ell (k)\varphi_\ell \Big\rangle _{L^2(\Omega)} \to 1 \quad\text{as } k\to \infty \end{split} \end{equation*} which gives a contradiction. Thus \eqref{eq:1.5} holds. \end{proof} \begin{proof}[Proof of Theorem \ref{thm:1.3}] By \eqref{eq:1.4} there exists $\delta_{10}\in ( 0,\frac{1}{2})$ such that \begin{equation*} \mu^\delta_2<2\mu_2\quad (0<\delta <\delta_{10}) \end{equation*} and that \begin{equation*} D=\{ x\in \Omega :\mathop{\rm dist} (x,\partial \Omega) > \delta_{10}\} \supseteq \overline{\Omega '}. \end{equation*} Applying Lemma \ref{lem:2.1} with $\Sigma =\Omega$ or $\Sigma = \Omega_{\delta}$ for $0 < \delta < \frac{1}{2}\delta_{10}$, $\Sigma'=D$, $\omega=1$, $a_{ij}=\delta_{ij}$, $\tau _1=1$, $\tau _2=2$, $t_1=\frac{3}{2}$, $\eta = \frac{1}{2}\delta_{10}$ and \begin{equation*} u(x,t)=e^{-\mu^\delta_2 t}\varphi_2^\delta (x) \end{equation*} for $0<\delta <\frac{1}{2}\delta_{10}$, or \begin{equation*} u(x,t)=e^{-\mu_2t}\varphi_2 (x), \end{equation*} we see that there exists $\alpha >0$ such that \begin{gather}\label{eq:2.49} |\varphi^\delta_2 (x)-\varphi^\delta_2 (y)|\leq B|x-y|^\alpha,\\ \label{eq:2.50} |\varphi_2 (x)-\varphi_2(y)|\leq B|x-y|^\alpha \end{gather} for all $x,y\in D$ and $0<\delta <\frac{1}{2}\delta_{10}$, where, by \eqref{eq:1.2}, \eqref{eq:1.3}, \eqref{eq:1.4} and \eqref{eq:2.33}, we can assume that $\delta_{10}\in ( 0,\frac{1}{2})$ is sufficiently small that \begin{equation*} B=(8/\delta_{10})^\alpha 2c^{1/2}_0 c_1e^{4\mu_{2}}. \end{equation*} Let \begin{equation*} E_\delta \varphi^\delta_2=b_2(\delta)\varphi_2 +\sum^\infty_{\ell =3} b_\ell (\delta)\varphi_\ell \quad (0<\delta<\delta_{10}). \end{equation*} Then, as in the proof of Theorem \ref{thm:1.2}, we have $\lim_{\delta \downarrow 0} b_2 (\delta)=1$ and \begin{equation*} \lim_{\delta \downarrow 0} \big\|\sum^\infty_{\ell =3} b_\ell (\delta )\varphi_\ell \big\|_2 =0. \end{equation*} Thus \begin{equation}\label{eq:2.51} \| \varphi_2 -E_\delta \varphi^\delta_2 \|_2 \to 0\quad \text{as } \delta \downarrow 0.\end{equation} Let \begin{equation*} r=\mathop{\rm dist} (\overline{\Omega '},\partial D). \end{equation*} Suppose that \eqref{eq:1.6} is false. Then there exist $\epsilon >0$, a decreasing sequence of positive numbers $\{ \eta_k\}^\infty_{k=1}$ and a sequence of points $\{ z_k\}^\infty_{k=1}$ in $\overline{\Omega '}$ such that $\lim_{k\to \infty}\eta _k=0$ and \begin{equation}\label{eq:2.52} |\varphi_2 ^{\eta_k}(z_k)-\varphi_2 (z_k)|\geq \epsilon \quad (k=1,2,3,\dots ). \end{equation} Then for all $w\in D$ satisfying \begin{equation*} |w-z_k|\leq \min \big\{ r,\big( \frac{\epsilon}{6}\big)^{1/\alpha} B^{-\frac{1}{\alpha}}\big\} \end{equation*} we have, by \eqref{eq:2.49} and \eqref{eq:2.50}, \begin{gather}\label{eq:2.53} |\varphi^{\eta_k}_2 (z_k)-\varphi^{\eta_k}_2 (w)| \leq \frac{\epsilon}{6},\\ \label{eq:2.54} |\varphi_2 (z_k)-\varphi_2 (w)|\leq \frac{\epsilon}{6}, \end{gather} hence, from \eqref{eq:2.52}, \eqref{eq:2.53} and \eqref{eq:2.54}, we have \begin{equation*} |\varphi_2^{\eta_k}(w)-\varphi_2 (w)|\geq \frac{2\epsilon }{3}. \end{equation*} Let $R=\min \big\{ r,(\frac{\epsilon }{6})^{1/\alpha}B^{-\frac{1}{\alpha}}\big\}$. Then \begin{equation}\label{eq:2.55} \int_{B(z_k,R)} |\varphi^{\eta_k}_2-\varphi_2|^2dx \geq \frac{4\epsilon^2}{9}c_2 R^d>0 \end{equation} for all $k=1,2,3,\dots $, where $c_2>0$ depends only on $d$. But \eqref{eq:2.55} contradicts \eqref{eq:2.51}, hence \eqref{eq:1.6} holds. \end{proof} \subsection*{Acknowledgments} We want to thank Mark Ashbaugh and Chris Burdzy for many useful discussions on the topic of this paper and for pointing several relevant references to us. We also thank the referee for many useful comments which improved both the content and presentation of the paper. \begin{thebibliography}{00} \bibitem{B} L. Banjai; Eigenfrequencies of fractal drums, J. Computational and Applied Math. 198 (2007), 1-18. \bibitem{BC} K. Burdzy and Z. Q. Chen; Weak convergence of reflected Brownian Motion, Electronic Comm. Probab. 3 (1998), 29-33. \bibitem{BD} V. I. Burenkov and E. B. Davies; Spectral stability of the Neumann Laplacian, J. Diff. Eqns. 186 (2002), 485-508. \bibitem{BLLC} V. 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