Content-Type: multipart/mixed; boundary="-------------0310070814125" This is a multi-part message in MIME format. ---------------0310070814125 Content-Type: text/plain; name="03-459.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-459.comments" 32 pages ---------------0310070814125 Content-Type: text/plain; name="03-459.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-459.keywords" Eigenvalues, Weil asymptotic formula, Laplace operator, variational method ---------------0310070814125 Content-Type: application/x-tex; name="Netr_10-03.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Netr_10-03.tex" % ------------------------------------------------------------------------ % AMS-LaTeX Paper ******************************************************** % **** ----------------------------------------------------------------------- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass[12pt]{amsart} %\usepackage{srcltx} % SRC Specials \usepackage{amssymb,amsthm,amsmath,a4} %----------------------------------- \headheight=6.15pt \textheight=215truemm \textwidth=138truemm \oddsidemargin=0in \evensidemargin=0in \topmargin=0in % ------------------------------------------------------------------------ % Over-full v-boxes on even pages are due to the \v{c} in author's name \vfuzz2pt % Don't report over-full v-boxes if over-edge is small \hfuzz2pt % Don't report over-full h-boxes if over-edge is small % THEOREMS --------------------------------------------------------------- \newtheorem{THM}{{\!}}[section] \newtheorem{THMX}{{\!}} \renewcommand{\theTHMX}{} % not numbered % \newtheorem{thm}{Theorem}[section] \newtheorem{theorem}[thm]{Theorem} \newtheorem{corollary}[thm]{Corollary} \newtheorem{lemma}[thm]{Lemma} \newtheorem{proposition}[thm]{Proposition} \theoremstyle{definition} \newtheorem{definition}[thm]{Definition} \newtheorem{example}[thm]{Example} \newtheorem{condition}[thm]{Condition} \theoremstyle{remark} \newtheorem{remark}[thm]{Remark} \numberwithin{equation}{section} %\setcounter{section}{0} %\setcounter{page}{1} %\setcounter{equation}{0} %%%%%%%%%%%%%%%%%%%%%%%% \font\sy=cmsy9 scaled\magstep1 \font\cyr=wncyr10 scaled\magstep1 \font\cyrm=wncyr9 scaled\magstep0 \font\rc=eurm10 scaled\magstep1 %%%%%%%%%%%%%%%%%% % MATH ------------------------------------------------------------------- %author's definition ----------------------------------------------------- \renewcommand\a{\alpha} \renewcommand\b{\beta} \newcommand\g{\gamma}\newcommand\G{\Gamma} \renewcommand\d{\delta}\newcommand\D{\Delta} \renewcommand\l{\lambda}\newcommand{\Lam}{\Lambda} \newcommand\n{\nabla} \renewcommand\o{\omega}\renewcommand\O{\Omega} \newcommand{\eps}{{\varepsilon}} \def\var{\varphi} \def\sig{\sigma} \def\vark{\varkappa} \def\CC{{\mathcal C}} \def\IC{{\mathcal I}} \def\JC{{\mathcal J}} \def\KC{{\mathcal K}} \def\LC{{\mathcal L}} \def\MC{{\mathcal M}} \def\NC{{\mathcal N}} \def\OC{{\mathcal O}} \def\PC{{\mathcal P}} \def\XC{{\mathcal X}} \def\YC{{\mathcal Y}} \def\VC{{\mathcal V}} \def\DR{{\mathrm D}} \def\NR{{\mathrm N}} \def\Lip{\mathrm{Lip}} \def\lip{\mathrm{lip}} \def\Osc{\mathrm{Osc}\,} \def\MB{\mathbf M} \def\PB{\mathbf P} \def\VB{\mathbf V} \def\kB{\mathbf k} \def\bR{\mathrm b} \def\dR{\mathrm d} \def\eR{\mathrm e} %%%%%%%%%%%%%%%%%%%%% \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\C}{\mathbb{C}} \newcommand{\Z}{\mathbb{Z}} %%%%%%%%%%%%%%%%%%% %\def\appendixname{{\bf Appendix}} %\newtheorem{theorem}{\bf Theorem}[section] %%%%%%%%%%%%%%%%%%% \newcommand{\supp}{\operatorname{supp}} \newcommand{\diam}{\operatorname{diam}} \newcommand{\codim}{\operatorname{codim}} \newcommand{\dist}{\operatorname{dist}} \renewcommand{\Re}{\operatorname{Re}} \renewcommand{\Im}{\operatorname{Im}} \def\leq {\leqslant} \def\geq {\geqslant} \begin{document} \title[Weil formula]{Weil asymptotic formula for the Laplacian on domains with rough boundaries} \author[Netrusov]{Yu. Netrusov\,$^1$} \author[Safarov]{Yu. Safarov} \address {School of Mathematics, University of Bristol, Bristol BS8 1TW, UK} \email{Y.Netrusov@bris.ac.uk} \address {Department of Mathematics, King's College, London WC2R 2LS, UK} \email{ysafarov@mth.kcl.ac.uk} \date{October 2003} \subjclass{35P20, 35J20} \footnote{Research supported by EPSRC grant GR/A00249/01} \begin{abstract} We study asymptotic distribution of eigenvalues of the Laplacian on a bounded domain in $\,\R^n$. Our main results include an explicit remainder estimate in the Weyl formula for the Dirichlet Laplacian on an arbitrary bounded domain, sufficient conditions for the validity of the Weyl formula for the Neumann Laplacian on a domain with continuous boundary in terms of smoothness of the boundary and a remainder estimate in this formula. In particular, we show that the Weyl formula holds true for the Neumann Laplacian on a $\,\Lip_\alpha$-domain whenever $\,(d-1)/\alpha2\,$. If the log-Sobolev inequality holds on $\,\Omega\,$ then $\,N_\NR(\Omega,\lambda)\,$ is exponentially bounded \cite{Ma}. For domains $\,\Omega\,$ with sufficiently smooth boundaries, (\ref{0.1}) is true for the both functions $\,N_\DR\,$ and $\,N_\NR\,$ and the remainder (i.e., the right hand side) is $\,O(\lambda^{d-1})\,$ \cite{Iv1}, \cite{Se}. The proof is based on the study of propagation of singularities for the corresponding evolution equation (see \cite{Iv3} or \cite{SV}). If $\,\Omega\,$ has a rough boundary then the propagation of singularities near $\,\partial\Omega\,$ cannot be effectively described and one has to invoke the variational technique. Let $\,\Omega_\delta^\bR\,$ and $\,\Omega_\delta^\eR\,$ be the internal and external $\,\delta$-neighbourhoods of $\,\partial\Omega\,$ respectively. The classical variational proof of the Weyl formula involves covering the domain by a finite collection of disjoint cubes $\,\{Q_j\}_{j\in\JC}\,$ and using the Dirichlet--Neumann bracketing. It is convenient to assume that $\,\{Q_j\}_{j\in\JC}\,$ is the subset of the family of Whitney cubes covering $\,\Omega\bigcup\Omega_\delta^\eR\,$ (see Theorem~\ref{T3.3}), which consists of the cubes $\,Q_j\,$ such that $\,Q_j\bigcap\Omega\ne\emptyset\,$. In view of the Rayleigh--Ritz variational formula, we have the estimates $\,\sum_{j\in\JC_0}N_\DR(Q_j,\lambda)\leq N_\DR(\Omega,\lambda)\leq \sum_{j\in\JC}N_\NR(Q_j,\lambda)\,$, where $\,\{Q_j\}_{j\in\JC_0}\,$ is the set of cubes $\,Q_j\,$ lying inside $\,\Omega\,$. If $\,\mu_d(\partial\Omega)=0\,$ then, estimating $\,N_\DR(Q_j,\lambda)\,$ and $\,N_\NR(Q_j,\lambda)\,$ for each $\,j\,$ and taking $\,\delta=\lambda^{-1}\,$, we obtain (\ref{0.1}) and (\ref{0.2}) for the Dirichlet Laplacian. It is possible to get rid of the condition $\,\mu_d(\partial\Omega)=0\,$ but this requires additional arguments. Similarly, the Rayleigh--Ritz formula implies that \newline $\,\sum_{j\in\JC_0}N_\DR(Q_j,\lambda)\leq N_\NR(\Omega,\lambda)\leq\sum_{j\in\JC_{m\delta}}N_\NR(Q_j,\lambda)+ N_\NR(\bigcup_{j\in\JC\setminus\JC_{m\delta}}Q_j\bigcap\Omega,\lambda)\,$, where $\,\{Q_j\}_{j\in\JC_{m\delta}}\,$ is the set of cubes lying inside $\,\Omega\setminus\Omega_{m\delta}^\bR\,$. If for some $\,m\in\N\,$ and all sufficiently small positive $\,\delta\,$ there exist uniformly bounded extension operators from the Sobolev space $\,W^{1,2}(\Omega_{m\delta}^\bR)\,$ to $\,W^{1,2}(\Omega_{m\delta}^\bR\bigcup\Omega_\delta^\eR)\,$ then $\,N_\NR(\bigcup_{j\in\JC\setminus\JC_{m\delta}}Q_j\bigcap\Omega,\lambda)\leq N_\NR(\bigcup_{j\in\JC\setminus\JC_{m\delta}}Q_j,C\lambda) =\sum_{j\in\JC\setminus\JC_{m\delta}}N_\NR(Q_j,C\lambda)\,$, where $\,C\,$ is a sufficiently large constant. If, in addition, $\,\mu_d(\partial\Omega)=0\,$ then, estimating the counting functions on the cubes and taking $\,\delta=\lambda^{-1}\,$, we obtain (\ref{0.1}) and (\ref{0.2}) for $\,N_\NR(\Omega,\lambda)\,$. However, the known extension theorems require certain regularity conditions on the boundary (for instance, it is sufficient to assume that $\,\partial\Omega\,$ belongs to the Lipschitz class or satisfies the cone condition). Domains with very irregular boundaries do not have the $\,W^{1,2}$-extension property, in which case the above scheme does not work the Neumann Laplacian. To the best of our knowledge, in all papers devoted to the Weyl formula for $\,N_\NR(\Omega,\lambda)\,$ the authors either implicitly assumed that the domain has the $\,W^{1,2}$-extension property or directly applied a suitable extension theorem. The main aim of this paper is to introduce a different technique which does not use an extension theorem. Instead of disjoint cubes, we cover the domain $\,\Omega\,$ by a family of relatively simple sets $\,S_m\subset\Omega\,$. For each of these sets the counting function $\,N(S_m,\lambda)\,$ can be effectively estimated from below and above. The sets $\,S_m\,$ may overlap but, under certain conditions on $\,\Omega\,$, the multiplicity of their intersection does not exceed a constant depending only on the dimension $\,d\,$. This allows us to apply the Dirichlet--Neumann bracketing and obtain the Weyl asymptotic formula with a remainder estimate for the Neumann Laplacian on domains without the extension property (Theorem \ref{T1.3}). The remainder term in this formula may well be of higher order than the first term. Then our asymptotic formula turns into an estimate for $\,N_\NR(\Omega,\lambda)\,$. In particular, this may happen if $\,\Omega\in\Lip_\alpha\,$, that is, if $\,\partial\Omega\,$ coincides with the subgraph of a $\,\Lip_\alpha$-function in a neighbourhood of each boundary point. We prove that $\,N_\NR(\Omega,\lambda)-C_{d,W}\,\mu_d(\Omega)\,\lambda^d= O(\lambda^{(d-1)/\alpha})\,$ whenever $\,\Omega\in\Lip_\alpha\,$ and $\,\alpha\in(0,1)\,$ (Corollary \ref{C1.6}) and that this estimate is order sharp (Theorem \ref{T1.10}). If $\,(d-1)/\alphab\}\,, \end{align*} $\,\Osc(f,\Omega'):=\frac12\,\bigl(\sup\limits_{x\in\Omega'}f(x) -\inf\limits_{x\in\Omega'} f(x)\bigr)\,$ and $\,|f|_\alpha\ :=\ \sup\limits_{x,\,y\in\Omega'}\frac{|f(x)-f(y)|}{|x-y|^\alpha}\,$. \item[$\bullet$] $\,Q_a^{(n)}\,$ is the open $\,n$-dimensional cube with edges of length $a$ parallel to the coordinate axes. If the size or the dimension of the cube $Q_a^{(n)}$ is not important for our purposes or evident from the context then we shall omit the corresponding index $a$ or $n$. However, we shall always be assuming that the cube is open and that its edges are parallel to the coordinate axes. \item[$\bullet$] $\Lip_\alpha$ is the space of functions $f$ on a cube $Q$ such that $\,|f|_\alpha<\infty\,$ and $\lip_\alpha$ is the closure of $\,\Lip_1\,$ in $\,\Lip_\alpha\,$ with respect to the seminorm $\,|\cdot|_\alpha\,$. \end{enumerate} \begin{definition}\label{D1.1} Given a bounded function $f$ on the cube $Q^{(n)}$ and $\delta>0$, we shall denote by $\,\VC_\delta(f,Q^{(n)})\,$ the maximal number of disjoint cubes $Q^{(n)}(i)\subset Q^{(n)}$ such that $\Osc(f,Q^{(n)}(i))\geq \delta$ for each $i$. If $\Osc(f,Q^{(n)})<\delta$ then we define $\,\VC_\delta(f,Q^{(n)}):=1\,$. \end{definition} \begin{definition}\label{D1.2} If $\,\tau\,$ is a positive nondecreasing function on $\,(0,+\infty)\,$, let $\,BV_{\tau,\infty}(Q)\,$ be the space spanned by all continuous functions $\,f\,$ on $\,\overline{Q}\,$ such that $\,\VC_{1/t}(f,Q)\leq\tau(t)\,$ for all $t>0\,$. \end{definition} We shall briefly discuss the relation between $\,BV_{\tau,\infty}(Q)$ and known function spaces in Subsection~\ref{S5.3}. Let $\,X\,$ be a space of continuous real-valued functions defined on a cube $\,Q^{(d-1)}\,$. We shall say that $\Omega$ belongs to the class $X$ and write $\Omega\in X$ if for each $z\in\partial\Omega$ there exists a neighbourhood $\,\OC_z\,$ of the point $\,z\,$, a linear orthogonal map $U:\R^d\to\R^d$, a cube $\,Q_a^{(d-1)}\subset Q^{(d-1)}\,$, a function $\,f\in X\,$ and $\,b\in\R\,$ such that $\,U(\OC_z\bigcap\Omega)=\{x\in G_{f,\,b}\;|\; x'\in Q_a^{(d-1)}\}\,$. Since $\,\partial\Omega\,$ is compact, for every bounded set $\Omega\in BV_{\tau,\infty}$ there exists a finite collection of domains $\,\Omega_l\subset\Omega\,$, $\,l\in\LC\,$, such that \begin{enumerate} \item[(a)] $\partial\Omega\subset\bigcup_{l\in\LC}\overline{\Omega_l}\,$; \item[(b)] for each $l$ we have $\,U_l(\Omega_l)=G_{f_l,\,b_l}\,$, where $\,U_l:\R^d\to\R^d\,$ is a linear orthogonal map, $\,f_l\in BV_{\tau,\infty}(Q_{a_l}^{(d-1)})\,$ and $\,b_l<\inf f_l\,$; \item[(c)] $\,a_l\leq D_\Omega\,$ and $\,\sup f_l-b_l\leq D_\Omega\,$ for all $l\in\LC$. \end{enumerate} Let us fix such a collection $\,\{\Omega_l\}_{l\in\LC}\,$ and denote $\,n_\Omega:=\#\LC\,$ and $$ C_{\Omega,\,\tau}\ :=\ \sum_{l\in\LC}\sup_{t>0} \left(\VC_{1/t}(f_l,Q_{a_l}^{(d-1)})/\tau(t)\right)\,. $$ Let $\,\delta_\Omega\,$ be the largest positive number such that $\,\Omega_{\delta_\Omega}^\bR\subset\bigcup_{l\in\LC}\Omega_l\,$, $\,\delta_\Omega\leq\sqrt{d}\,a_l\,$ and $\,2\delta_\Omega\leq\inf f_l-b_l\,$ for all $l\in\LC$. \subsection{Main results}\label{S1.2} Throughout the paper we shall denote by $C_d$ various constants depending only on the dimension $d$. Constants appearing in the most important estimates are numbered by an additional lower index; in our opinion, this makes our proofs more transparent. Their precise (but not necessarily best possible) values are given in Section 6. \begin{theorem}\label{T1.3} If $\,\Omega\in BV_{\tau,\infty}\,$ and $\,\lambda\geq\delta_\Omega^{-1}\,$ then \begin{multline}\label{1.1} |\,N_\NR(\Omega,\lambda)-C_{d,W}\,\mu_d(\Omega)\,\lambda^d\,|\\ \leq\ C_{d,9}\,C_{\Omega,\tau}\,n_\Omega^{1/2}\lambda \int_{(2D_\Omega)^{-1}}^{C_\Omega\,\lambda}t^{-2}\,\tau(t)\,\dR t\;+\;C_{d,10}\,n_\Omega\,\lambda^{d-1} \int_0^{C_\Omega\,\lambda} \mu_d(\Omega_{t^{-1}}^\bR)\,\dR t\,, \end{multline} where $\,C_\Omega:=4\,C_{d,8}\,n_\Omega^{1/2}\,$. If, in addition, $\,\Omega\subset\R^2\,$ then there exists a positive constant $\,c\,$ independent of $\Omega$ such that \begin{multline}\label{1.2} |\,N_\NR(\Omega,\lambda)-(4\pi)^{-1}\mu_2(\Omega)\,\lambda^2\,|\ \leq\ c\,C_{\Omega,\tau}\,\tau(c\,n_\Omega^{1/2}\lambda)\\ +\;c\,n_\Omega\,\lambda\,\left(D_\Omega+ \int_0^{c\,n_\Omega^{1/2}\lambda} \mu_2(\Omega_{t^{-1}}^\bR)\,\dR t\right),\qquad\forall\lambda\geq \delta_\Omega^{-1}\,. \end{multline} \end{theorem} \begin{remark}\label{R1.4} For each continuous function $\,f\,$ on a closed cube there exists a positive nondecreasing function $\,\tau\,$ such that $\,f\in BV_{\tau,\infty}\,$. Therefore Theorem~\ref{T1.3} allows one to obtain an estimate of the form (\ref{1.1}) for every domain $\,\Omega\in C\,$. In particular, this implies the following well known result: if $\,\Omega\in C\,$ then the essential spectrum of the Neumann Laplacian on $\,\Omega\,$ is empty. \end{remark} The next two corollaries are simple consequences of Theorem~\ref{T1.3}. \begin{corollary}\label{C1.5} If $\,\Omega\in BV_{\tau,\infty}\,$ then there exists a constant $C_\Omega$ such that \begin{multline}\label{1.3} |\,N_\NR(\Omega,\lambda)-C_{d,W}\,\mu_d(\Omega)\,\lambda^d\,|\\ \leq\ C_\Omega\,\lambda^{d-1} \int_{C_\Omega^{-1}}^{C_\Omega\lambda}\left(t^{-1}+t^{-d}\,\tau(t)\right)\dR t\,, \qquad\forall\lambda\geq C_\Omega\,. \end{multline} \end{corollary} \begin{corollary}\label{C1.6} If $\,\alpha\in(0,1)$ and $\Omega\in\Lip_\alpha$ then \begin{equation}\label{1.4} N_\NR(\Omega,\lambda)\ =\ C_{d,W}\,\mu_d(\Omega)\,\lambda^d \;+\;O\left(\lambda^{(d-1)/\alpha}\right),\qquad\lambda\to+\infty. \end{equation} If $\,\alpha\in(0,1)$ and $\Omega\in\lip_\alpha$ then \begin{equation}\label{1.5} N_\NR(\Omega,\lambda)\ =\ C_{d,W}\,\mu_d(\Omega)\,\lambda^d \;+\;o\left(\lambda^{(d-1)/\alpha}\right),\qquad\lambda\to+\infty. \end{equation} \end{corollary} \begin{remark}\label{R1.7} If $\,\alpha\leq1-d^{-1}\,$ then the asymptotic formula (\ref{1.4}) turns into the estimate $\,N_\NR(\Omega,\lambda)=O\left(\lambda^{(d-1)/\alpha}\right)$. Similarly, if $\,\alpha<1-d^{-1}\,$ then (\ref{1.5}) takes the form $\,N_\NR(\Omega,\lambda)=o\left(\lambda^{(d-1)/\alpha}\right)$. \end{remark} The following estimates for the Dirichlet Laplacian are much simpler. The inequality (\ref{1.6}) seems to be new but results of this type are known to experts. Corollary~\ref{C1.9} is an immediate consequence of Theorem \ref{T1.8}; (\ref{1.7}) also follows from (\ref{0.2}). \begin{theorem}\label{T1.8} For all $\,\lambda>0\,$ we have \begin{equation}\label{1.6} |\,N_\DR(\Omega,\lambda)-C_{d,W}\,\mu_d(\Omega)\,\lambda^d\,|\ \leq\ C_{d,11}\,\lambda^{d-1} \int_0^\lambda\mu_d(\Omega_{t^{-1}}^\bR)\,\dR t\,. \end{equation} \end{theorem} \begin{corollary}\label{C1.9} If $\,\alpha\in(0,1)$ and $\Omega\in\Lip_\alpha$ then \begin{equation}\label{1.7} N_\DR(\Omega,\lambda)\ =\ C_{d,W}\,\mu_d(\Omega)\,\lambda^d \;+\;O\left(\lambda^{d-\alpha}\right),\qquad\lambda\to+\infty. \end{equation} If $\,\alpha\in(0,1)$ and $\Omega\in\lip_\alpha$ then \begin{equation}\label{1.8} N_\DR(\Omega,\lambda)\ =\ C_{d,W}\,\mu_d(\Omega)\,\lambda^d \;+\;o\left(\lambda^{d-\alpha}\right),\qquad\lambda\to+\infty. \end{equation} \end{corollary} Note that $\,(d-1)/\alpha>d-\alpha\,$ whenever $\,\alpha\in(0,1)\,$. Therefore the remainder estimate in Corollary~\ref{C1.9} is better than that in Corollary~\ref{C1.6}. The following theorem shows that the asymptotic formulae (\ref{1.4}) and (\ref{1.5}) are order sharp. \begin{theorem}\label{T1.10} Let $\alpha\in(0,1)$. Then \begin{enumerate} \item[(1)] there exist a bounded domain $\Omega\in\Lip_\alpha$ and a positive constant $C_\Omega$ such that $\;N_\NR(\Omega,\lambda)\geq C_{d,W}\,\mu_d(\Omega)\,\lambda^d+C_\Omega^{-1}\,\lambda^{(d-1)/\alpha}\;$ for all $\,\lambda>C_\Omega\,$; \item[(2)] for each nonnegative function $\phi$ on $(0,+\infty)$ vanishing at $+\infty$ there exist a bounded domain $\Omega\in\lip_\alpha$ and a positive constant $C_{\phi,\Omega}$ such that $\;N_\NR(\Omega,\lambda)\geq C_{d,W}\,\mu_d(\Omega)\,\lambda^d+ C_{\phi,\Omega}^{-1}\,\phi(\lambda)\,\lambda^{(d-1)/\alpha}\;$ for all $\,\lambda>C_{\phi,\Omega}\,$. \end{enumerate} \end{theorem} \begin{remark}\label{R1.11} In \cite{BD} the authors proved that \begin{equation}\label{1.9} 01\}\,$; if $-\Delta _\NR$ has at least two eigenvalues lying below its essential spectrum (or the essential spectrum is empty) then $\,\lambda_{1,\NR}(\Omega)\,$ coincides with the smallest nonzero eigenvalue of the operator $\,\sqrt{-\Delta _\NR}\,$. By the spectral theorem, we have $\,\lambda_{1,\NR}(\Omega)\geq\lambda\,$ if and only if $\,\int_\Omega|u(x)|^2\,\dR x\leq \lambda^{-2}\int_\Omega|\nabla u(x)|^2\,\dR x\,$ for all functions $\,u\in W^{1,2}(\Omega)\,$ such that $\,\int_\Omega u(x)\,\dR x=0\,$. Note that $\,\int_\Omega|u(x)|^2\,\dR x\leq\int_\Omega|u(x)-c|^2\,\dR x\,$ for all $c\in\C$ whenever $\int_\Omega u(x)\,\dR x=0$. \end{remark} \begin{definition}\label{D2.5} Denote by $\PB(\delta)$ the set of all rectangles with edges parallel to the coordinate axes, such that the length of the maximal edge does not exceed $\delta\,$. If $f$ is a continuous function on $\overline{Q^{(d-1)}}$, let $\VB(\delta,f)$ be the class of domains $V\subset G_f$ which can be represented in the form $\,V=G_{f,\,b}(Q_c^{(d-1)})\,$, where $\,Q_c^{(d-1)}\subset Q^{(d-1)}\,$, $\,c\leq\delta\,$, $\,b=\inf f-\delta\,$ and $\,\Osc(f,Q_c^{(d-1)})\leq\delta/2\,$. We shall write $\,V\in\VB(\delta)\,$ if $\,V\in\VB(\delta,f)\,$ for some continuous function $f$. Finally, let $\,\MB(\delta)\,$ be the class of open sets $\,M\subset\R^d\,$ such that $\,M\subset Q_\delta^{(d)}\,$ for some cube $\,Q_\delta^{(d)}\,$. \end{definition} \begin{lemma}\label{L2.6} Let $\,\delta\,$ be an arbitrary positive number. \begin{enumerate} \item[(1)] If $\,P\in\PB(\delta)$ then $N_\NR(P,\lambda)=1\,$ for all $\,\lambda\leq\pi\delta^{-1}$. \item[(2)] If $V\in\VB(\delta)$ then $N_\NR(V,\lambda)=1\,$ for all $\lambda\leq(1+2\pi^{-2})^{-1/2}\delta^{-1}$. \item[(3)] If $\,M\in\MB(\delta)\,$, $\,M\subset Q_\delta^{(d)}\,$ and $\,\Upsilon:=\partial M\bigcap Q_\delta^{(d)}\,$ then we have $\,N_{\NR,\DR}(M,\Upsilon,\lambda)\leq1\,$ for all $\,\lambda\leq\pi\delta^{-1}$ and $\,N_{\NR,\DR}(M,\Upsilon,\lambda)=0\,$ for all $\,\lambda\leq (2^{-1}-2^{-1}\delta^{-d}\mu_d(M))^{1/2}\,\pi\delta^{-1}\,$. \end{enumerate} \end{lemma} \begin{proof} If $\,P\,$ is a rectangle then $\,\lambda_{1,\NR}=\pi\,a^{-1}\,$, where $\,a\,$ is the length of its maximal edge. This implies (1). Assume now that $\,V\in\VB(\delta,f)\,$, where $\,f\,$ is a continuous function on $\,\overline{Q_c^{(d-1)}}\,$ and denote $\,b:=\inf f-\delta\,$ and $\,P:=Q_c^{(d-1)}\times(b,b+\delta)\,$. Clearly, $\,P\in\PB(\delta)\,$. Let $\,u\in W^{1,2}(V)\,$ and $\,c'_u\,$ the average of $\,u\,$ over $\,P\,$. If $\,r\in[b,b+\delta]\,$ and $\,s\in[b+\delta,f(x')]\,$ then, by Jensen's inequality, $$ |u(x',s)-u(x',r)|^2\ =\ |\int_r^s\partial_t\,u(x',t)\,\dR t\,|^2\ \leq\ (s-r)\int_b^{f(x')}|\partial_t\,u(x',t)|^2\,\dR t\,. $$ Since $\,\int_b^{b+\delta}\int_{b+\delta}^f(s-r)\,\dR s\,\dR r= (\delta/2)\,(f-b-\delta)\,(f-b)\,$ and $$ 0\leq f-b-\delta=f-\inf f\leq2\,\Osc(f,Q_c^{(d-1)})\leq\delta\,, $$ we have $$ \int_b^{g(x')}\int_{g(x')}^{f(x')}|u(x',s)-u(x',r)|^2\,\dR s\,\dR r\\ \leq\ \delta^3\int_b^{f(x')}|\partial_t\,u(x',t)|^2\,\dR t\,. $$ In view of Remark~\ref{R2.4} and (1), we also have \begin{equation}\label{2.5} \int_P|u(x)-c'_u|^2\,\dR x\ \leq\ \pi^{-2}\,\delta^2\int_P|\nabla u(x)|^2\,\dR x. \end{equation} Integrating the inequality $$ |u(x',s)-c'_u|^2\ \leq\ (1+\gamma)\,|u(x',r)-c'_u|^2+(1+\gamma^{-1})\,|u(x',s)-u(x',r)|^2 $$ over $\,r\in[b,b+\delta]\,$, $\,s\in[b+\delta,f(x')]\,$ and $\,x'\in\Omega'\,$ and applying these two estimates, we obtain \begin{multline*} \delta\int_{V\setminus P}|u(x)-c'_u|^2\,\dR x\ \leq (1+\gamma)\,\pi^{-2}\,\delta^3 \int_P|\nabla u(x)|^2\,\dR x\\ +(1+\gamma^{-1})\,\delta^3 \int_V|\partial_{x_d}u(x)|^2\,\dR x \end{multline*} for all $\,\gamma>0\,$. Dividing both sides by $\,\delta\,$ and substituting $\,\gamma=\pi^2\,$, we see that $\,\int_{V\setminus P}|u(x)-c'_u|^2\,\dR x\,$ is estimated by $\,(1+\pi^{-2})\,\delta^2\int_V|\nabla u(x)|^2\,\dR x\,$. Now (2) follows from (\ref{2.5}) and Remark~\ref{R2.4}. In order to prove (3), let us consider a function $\,u\in W^{1,2}(M)\,$ which vanishes near $\,\Upsilon\,$ and extend it by zero to the whole cube $\,Q_\delta^{(d)}\,$. Since $\,u\in W^{1,2}(Q_\delta^{(d)})\,$, (1) implies the first inequality (3). If $\,c_u\,$ is the average of $\,u\,$ over $\,Q_\delta^{(d)}\,$ then \begin{equation}\label{2.6} \int_M|c_u|^2\,\dR x\ \leq\ \mu_d(M)\,\delta^{-d}\left(\int_M|c_u|^2\,\dR x +\int_{Q_\delta^{(d)}}|u(x)-c_u|^2\,\dR x\right). \end{equation} Therefore Remark~\ref{R2.4} and (1) imply that \begin{multline*} \int_M|u(x)|^2\,\dR x\ \leq\ 2\int_{Q_\delta^{(d)}}|u(x)-c_u|^2\,\dR x+2\int_M|c_u|^2\,\dR x\\ \leq\ 2\left(1+\mu_d(M)\,\delta^{-d} \left(1-\mu_d(M)\,\delta^{-d}\right)^{-1}\right)\int_{Q_\delta^{(d)}} |u(x)-c_u|^2\,\dR x\\ \leq\ 2\,\pi^{-2}\,\delta^2\left(1-\mu_d(M)\,\delta^{-d}\right)^{-1} \int_M|\nabla u(x)|^2\,\dR x\,. \end{multline*} The second identity (3) follows from the above inequality and the Rayleigh--Ritz formula. \end{proof} \begin{remark}\label{R2.7} The second estimate in Lemma~\ref{L2.6}(3) is sufficient for our purposes but is very rough. One can obtain a much more precise result in terms of capacities (see \cite{M2}, Chapter 10, Section 1). \end{remark} \begin{lemma}\label{L2.8} Let $\,\delta>0\,$. Then for all $\,\lambda>0\,$ we have $$ -\,C_{d,1}\left((\delta\lambda)^{d-1}+1\right)\ \leq\ N(Q_\delta^{(d)},\lambda)-C_{d,W}\,(\delta\lambda)^d\ \leq\ C_{d,1}\left((\delta\lambda)^{d-1}+1\right). $$ \end{lemma} \begin{proof} Changing variables $\tilde x=\delta\,x$, we see that \begin{equation}\label{2.7} N(\Omega,\delta\lambda)\ =\ N(\delta\Omega,\lambda)\,,\quad\text{where}\quad \delta\Omega:=\{x\in\R^d\,|\,\delta^{-1}x\in\Omega\}\,. \end{equation} Therefore it is sufficient to prove the required estimates only for $\,\delta=1\,$. If $\,\Omega=\Omega'\times\Omega''\,$, $\,\Upsilon'\subset\partial\Omega'\,$ and $\,\Upsilon''\subset\partial\Omega''\,$ then, separating variables, we obtain \begin{equation}\label{2.8} N_{\NR,\DR}(\Omega,\Upsilon,\lambda)\ =\ \int N_{\NR,\DR}\left(\Omega',\Upsilon',\sqrt{\lambda^2-\mu^2}\right)\,\dR N_{\NR,\DR}(\Omega'',\Upsilon'',\mu)\,, \end{equation} where $\,\Upsilon=(\Upsilon'\times\partial\Omega'')\bigcup (\partial\Omega'\times\Upsilon'')\,$ and the right hand side is a Stieltjes integral. Using (\ref{2.8}), explicit formulae for the counting functions on the unit interval and the identities \begin{equation}\label{2.9} \int_0^\lambda(\lambda^2-\mu^2)^{n/2}\,\dR\mu \ =\ \lambda^{n+1}\,\omega_{n+1}\,(2\,\omega_n)^{-1}\,,\qquad \forall n=1,2,\ldots, \end{equation} one can easily prove the required inequality by induction in $\,d\,$. \end{proof} \begin{remark}\label{R2.9} Lemma~\ref{L2.8} is an immediate consequence of well known results on spectral asymptotics in domains with piecewise smooth boundaries (see, for example, \cite{Iv2} or \cite{F}); a similar result holds true for higher order elliptic operators and operators with variable coefficients \cite{V}. We have given an independent proof in order to find the explicit constant $\,C_{d,1}\,$. \end{remark} \section{Properties of domains and their partitions}\label{S3} \subsection{Besicovitch's and Whitney's theorems}\label{S3.1} We shall use the following version of Besicovitch's theorem. \begin{theorem}\label{T3.1} There are two constants $\,\CC_n\geq1\,$ and $\,\hat\CC_n\geq1\,$ depending only on the dimension $n$, such that for every compact set $\,K\subset\R^n\,$ and every positive function $\rho$ on $K$ one can find a finite subset $\,\YC\subset K\,$ and a family of cubes $\,\{Q_{\rho(y)}^{(n)}[y]\}_{y\in\YC}\,$ centred on $y$, which satisfy the following conditions: \begin{enumerate} \item[(1)] $K\subset\bigcup_{y\in\YC}Q_{\rho(y)}^{(n)}[y]\,$, \item[(2)] $\aleph\{K\bigcap Q_{\rho(y)}^{(n)}[y]\}_{y\in\YC}\leq\CC_n\,$; \item[(3)] there exists a subset $\hat\YC\subset\YC$ such that $\,\#\YC\leq \hat\CC_n(\#\hat\YC)\,$ and the cubes $\{Q_{\rho(y)}^{(n)}[y]\}_{y\in\hat{\YC}}$ are mutually disjoint. \end{enumerate} \end{theorem} Theorem \ref{T3.1} is proved in the same way as Besicovitch's theorem in \cite{G}, Chapter 1. \begin{corollary}\label{C3.2} Let $f$ be a continuous function on the closure $\overline{Q^{(d-1)}}$. Then for every $\eps>0$ there exists a finite family of cubes $\{Q^{(d-1)}(x)\}_{x\in\XC}$ such that \begin{enumerate} \item[(1)] $\bigcup_{x\in\XC}\overline{Q^{(d-1)}}(x)=\overline{Q^{(d-1)}}$; \item[(2)] $\aleph\{Q^{(d-1)}(x)\}\leq C_{d,2}$; \item[(3)] $\#\XC\leq C_{d,3}\,\VC_\eps(f,\overline{Q^{(d-1)}})$; \item[(4)] $\Osc(f,Q^{(d-1)}(x))\leq\eps$ for each $x\in\XC$. \end{enumerate} \end{corollary} \begin{proof} Without loss of generality we can assume that $Q^{(d-1)}=(-1,1)^{d-1}$ and $\Osc(f,Q^{(d-1)})>\eps$. Let us denote by $\,Q_t^{(d-1)}[y]\,$ the cube of the size $\,t\,$ centred on $\,y\,$, define $$ \rho(y):=\inf\{t>0\;|\;\Osc(f,Q^{(d-1)}\bigcap Q_t^{(d-1)}[y])=\eps\}\,,\qquad y\in\overline{Q^{(d-1)}}\,, $$ apply Besicovitch's theorem to the set $K=\overline{Q^{(d-1)}}$ and find the sets $\YC$ and $\hat\YC$. If $y\in\YC$, denote $\,P^{(d-1)}[y]:=Q^{(d-1)}\bigcap Q_{\rho(y)}^{(d-1)}[y]\,$ and assume that $$ P^{(d-1)}[y]\ =\ (a_1(y),b_1(y))\times(a_2(y),b_2(y)) \times\dots\times(a_{d-1}(y),b_{d-1}(y))\,, $$ where $\,-1\leq a_j(y)-1$ and $b_j(y)<1$ then $a'_j(y)=a_j(y)$ and $b'_j(y)=b_j(y)$; \item[(+1)] if $b_j(y)=1$ then $a'_j(y)=b_j(y)-c(y)$ and $b'_j(y)=1$. \end{enumerate} Let us consider the set $\Sigma=\{-1,0,1\}^{d-1}$ of all $\,(d-1)$-dimensional vectors $\sigma=(\sigma_1,\ldots,\sigma_{d-1})$ with entries $\sigma_j$ equal to $\,-1$, $0$ or $1$. Denote by $\hat\YC_\sigma$ the set of points $y\in\hat\YC$ such that $a_j(y)$ and $b_j(y)$ satisfy the condition ($\sigma_j$) for all $j=1,\ldots,d-1$. Since $\,\aleph\{P^{(d-1)}[y]\}_{y\in\hat\YC}=1$, for each $\sigma\in\Sigma$ the cubes $\{Q'(y)\}_{y\in\hat\YC_\sigma}=1$ are mutually disjoint. Therefore $\#\hat\YC_\sigma\leq \VC_\eps(f,\overline{Q^{(d-1)}})$ for all $\sigma\in\Sigma$ (see Definition~\ref{D1.1}) and, consequently, $\,\#\hat\YC\leq(\#\Sigma)\,\VC_\eps(f,\overline{Q^{(d-1)}}) \leq3^{d-1}\,\VC_\eps(f,\overline{Q^{(d-1)}})\,$. This estimate and Theorem~\ref{T3.1}(3) imply that $\,\#\YC\leq3^{d-1}\,\hat\CC_{d-1}\,\VC_\eps(f,\overline{Q^{(d-1)}})\,$. Since $\,\YC\subset\overline{Q^{(d-1)}}\,$, we have $\,1/2\leq(b_j(y)-a_j(y))^{-1}(b_k(y)-a_k(y))\leq2\,$ for all $\,j,k=1,\ldots,d-1\,$ and $\,y\in\YC\,$. Using this inequality, one can easily show by induction in $\,d\,$ that every rectangle $\,P^{(d-1)}[y]\,$ coincides with the union of a finite collection of cubes $\,\{Q^{(d-1)}(x)\}_{x\in\XC_y}\,$ such that $\,\#\XC_y\leq2^{d-1}\,$ and $\,\aleph\{Q^{(d-1)}(x)\}_{x\in\XC_y}\leq2^{d-1}\,$. Let $\,\XC:=\bigcup_{y\in\YC}\XC_y\,$. In view of the first two conditions of Theorem~\ref{T3.1}, the family $\,\{Q^{(d-1)}(x)\}_{x\in\XC}\,$ satisfies (1) and (2). The upper bound $\,\#\YC\leq3^{d-1}\,\hat\CC_{d-1}\,\VC_\eps(f,\overline{Q^{(d-1)}})\,$ implies (3). Finally, since $\,\Osc(f,P^{(d-1)}[y])=\eps\,$ and $\,Q^{(d-1)}(x)\subset P^{(d-1)}[y]\,$ whenever $\,x\in\XC_y\,$ , we have (4). \end{proof} The following theorem is due to Whitney. It can be found, for example, in \cite{St}, Chapter VI, or \cite{G}, Chapter 1. \begin{theorem}\label{T3.3} There exists a countable family of mutually disjoint cubes $\,\{Q_{2^{-i}}^{(d)}(i,n)\}_{n\in\NC(i)\,,\,i\in\IC}\,$ such that $\,\overline{\Omega}=\bigcup_{i\in\IC}\bigcup_{n\in\NC_i} \overline{Q_{2^{-i}}^{(d)}(i,n)}\,$ and \begin{equation}\label{3.1} Q_{2^{-i}}^{(d)}(i,n)\subset\{x\in\Omega\;|\; \sqrt{d}\,2^{-i}\leq\dist(x,\partial\Omega)\leq4\sqrt{d}\,2^{-i}\}\,. \end{equation} Here $\IC$ is a subset of $\Z$ and $\NC_i$ are some finite index sets. \end{theorem} \subsection{Auxiliary results}\label{S3.2} In this subsection we shall prove several technical results concerning domains $G_{f,\,b}\,$. \begin{lemma}\label{L3.4} Let $f$ be a continuous function defined on the closure $\overline{Q_a^{(d-1)}}$. Then for every $\delta>0$ and $m\in\Z_+$ there exists a finite family of cubes $\{Q^{(d-1)}(k)\}_{k\in\KC_m}$ such that \begin{enumerate} \item[(1)] $\bigcup_{k\in\KC_m}\overline{Q^{(d-1)}(k)}=\overline{Q_a^{(d-1)}}$; \item[(2)] $Q^{(d-1)}(k)\in\PB(\delta)$ for all $k\in\KC_m$; \item[(3)] $\,\aleph\{Q^{(d-1)}(k)\}_{k\in\KC_m} \leq C_{d,2}$; \item[(4)] $\Osc(f,Q^{(d-1)}(k))\leq2^{m-1}\delta$ for all $k\in \KC_m$; \item[(5)] $\#\{k\in\KC_m\;|\;\mu_{d-1}(Q^{(d-1)}(k))\leq2^{1-d}\,\delta^{d-1}\}\leq C_{d,3}\,\VC_{2^{m-1}\delta}(f,Q_a^{(d-1)})\,$. \end{enumerate} \end{lemma} \begin{proof} Let $\{Q^{(d-1)}(x)\}_{x\in\XC}$ be a family of cubes satisfying the conditions of Corollary~\ref{C3.2} with $\eps=2^{m-1}\delta$. Assume that $Q^{(d-1)}(x)=Q_{a_x}^{(d-1)}\,$ with some $a_x>0$ and denote by $\XC_\delta$ the set of all indices $x\in\XC$ such that $a_x\leq\delta$. For each $x\in\XC\setminus\XC_\delta$, we choose a positive integer $m_x$ such that $a_x/m_x\in(\delta/2,\delta]$ and split the closed cube $\overline{Q^{(d-1)}(x)}$ into the union of $m_x^{d-1}$ congruent closed cubes $\overline{Q_{a_x/m_x}^{(d-1)}(x,j)}$, $j=1,\ldots,m_x^{d-1}$. Let $Q_{a_x/m_x}^{(d-1)}(x,j)$ be the corresponding disjoint open cubes and $$ \{Q^{(d-1)}(k)\}_{k\in\KC}:=\{Q^{(d-1)}(k)\}_{x\in\XC_\delta} \bigcup\{Q_{a_x/m_x}^{(d-1)}(x,j)\}_{x\in\XC\setminus\XC_\delta, \,j=1,\ldots,m_x^{d-1}}\,. $$ Then (2) holds true and (1), (3), (4) and (5) follow from Corollary~\ref{C3.2}(1), Corollary~\ref{C3.2}(2), Corollary~\ref{C3.2}(4) and Corollary~\ref{C3.2}(3) respectively. \end{proof} \begin{theorem}\label{T3.5} Let $f$ be a continuous function on $\overline{Q_a^{(d-1)}}\,$, $\,\delta\in(0,\sqrt{d}\,a]\,$ and $\,b\in[-\infty,\,\inf f-2\delta]\,$. Then there exist countable families of sets $\{P_j\}_{j\in\JC}$ and $\{V_k\}_{k\in\KC}$ satisfying the following conditions: \begin{enumerate} \item[(1)] $P_j\subset G_{f,b}$ and $P_j\in\PB(\delta)$ for all $j\in\JC$; \item[(2)] $V_k\subset G_{f,b}$ and $V_k\in\VB(\delta,f)$ for all $k\in\KC$; \item[(3)] $\aleph\{P_j\}\leq3C_{d,2}+1$ and $\,\aleph\{V_k\}\leq C_{d,2}$; \item[(4)] $G_{f,b}\subset\bigcup_{j\in\JC,\,k\in\KC} \left(\overline{P_j}\bigcup\overline{V_k}\right)$; \item[(5)] $\#\{k\in\KC\;|\;\mu_d(V_k)\leq 2^{1-d}\,\delta^d\}\leq C_{d,3}\,\VC_{\delta/2}(f,Q_a^{(d-1)})\,$ and \newline $\#\{j\in\JC\;|\;\mu_d(P_j)\leq(2\sqrt{d})^{-d}\,\delta^d\}\leq C_{d,3}\sum_{m=0}^{m_\delta}2^m\,\VC_{2^{m-1}\delta}(f,Q_a^{(d-1)})\,$, \newline where $\,m_\delta:=\min\,\{m\in\Z_+\;|\;2^{m-1}\delta\geq \Osc(f,Q_a^{(d-1)})\}\,$. \end{enumerate} \end{theorem} \begin{proof} Let $\{Q^{(d-1)}(k)\}_{k\in\KC_m}$ be the same families of cubes as in Lemma~\ref{L3.4}, $c_k:=\inf_{x\in Q^{(d-1)}(k)}f(x)$, $\,b_k=c_k-\delta\,$, $V_k:=G_{f,b_k}(Q^{(d-1)}(k))$ and $$ P_{m,k,n}:=Q^{(d-1)}(k)\times(c_k-n\delta,c_k-n\delta+\delta)\,, $$ where $\,k\in\bigcup_m\KC_m\,$ and $n\in\Z_+$. Denote $\NC_m:=\{2^m+1,\ldots,2^m+2^{m+1}\}\,$. Lemma~\ref{L3.4}(4) implies that \begin{equation} \bigcup_{k\in\KC_m,n\in\NC_m}P_{m,k,n}\ \subset\ \{x\in G_f\;|\;2^m\delta\leq f(x')-x_d\leq 2^{m+2}\delta\}\,,\label{3.2} \end{equation} for all $m=0,1,\ldots,m_\delta$. Let $\;\KC:=\KC_0\,$, $\,\JC_*:=\bigcup_{m=0}^{m_\delta}\KC_m\times\NC_m\;$ and $\,\{P_j\}_{j_*\in\JC_*}:= \bigcup_{m=0}^{m_\delta}\{P_{m,k,n}\}_{k\in\KC_m,n\in\NC_m}\,$. Assume that $x\in G_f$. If $f(x')-x_d\leq2\delta$ then, by Lemma~\ref{L3.4}(1), we have $x\in\bigcup_{k\in\KC}\left(\overline{V_k}\bigcup\overline{P_{0,k,2}}\right)\,$. If $f(x')-x_d>2^{m_\delta+1}\delta$ then $$ \dist(x,\Gamma_f)\geq f(x')-x_d-2\,\Osc(f,Q_a^{(d-1)})\geq f(x')-x_d-2^{m_\delta}\delta>2^{m_\delta}\delta\geq\delta. $$ Finally, if $2\delta\leq f(x')-x_d\leq 2^{m_\delta+1}\delta$ then $2^{m+1}\delta\leq f(x')-x_d\leq2^{m+1}\delta+2^m\delta$ for some nonnegative integer $m\leq m_\delta$ and, in view of Lemma~\ref{L3.4}(1) and Lemma~\ref{L3.4}(4), we have $x\in\bigcup_{k\in\KC_m,n\in\NC_m}P_{m,k,n}$. Therefore \begin{equation}\label{3.3} \{x\in G_f\;|\;\dist(x,\Gamma_f)\leq\delta\}\subset\bigcup_{j_*\in\JC_*,\,k\in\KC} \left(\overline{P_{j_*}}\bigcup\overline{V_k}\right)\,. \end{equation} Let us choose a constant $\,c\in(\delta/(2\sqrt{d}),\delta/\sqrt{d}]\,$ in such a way that $\,a/c\in\N\,$ and split the set $\,\overline{Q_a^{(d-1)}}\times[b,+\infty)\,$ into the union of congruent closed cubes $\,\overline{Q_c^{(d-1)}(i)}\,$ whose interiors $\,Q_c^{(d-1)}(i)\,$ are mutually disjoint. Let $\,\{P_j\}_{j\in\JC}\,$ be the collection of all the rectangles $\,P_{j_*}\,$ and all the cubes $\,Q_c^{(d-1)}(i)\,$ which are contained in $\,G_{f,b}\,$. Then (1) and (2) are obvious. The second inequality (3) and (5) follow from the corresponding statements of Lemma~\ref{L3.4}. The first inequality (3) is a consequence of (\ref{3.2}), Lemma~\ref{L3.4}(3) and the identity $\aleph\left\{[2^m,2^{m+2}]\right\}_{i\in\Z_+}=3$. It remains to prove (4). Let $\,x\in G_f\,$. If $\,\dist(x,\Gamma_f)\leq\delta\,$ then, by (\ref{3.3}), either $\,x\in\overline{V_k}\,$ for some $\,k\in\KC\,$ or $\,x\in \overline{P_{j^*}}\,$ for some $\,j^*\in\JC^*\,$. Since $\,P_{j_*}\in\PB(\delta)\,$ and $\,b\leq\inf f-2\delta\,$, in the latter case $\,P_{j_*}\subset G_{f,b}\,$. If $\,\dist(x,\Gamma_f)>\delta\,$ then the cube $\,Q_c^{(d-1)}(i)\,$, whose closure contains $\,x\,$, is a subset of $\,G_{f,b}\,$ because its diameter does not exceed $\,\delta\,$. Therefore (4) holds true. \end{proof} In the two dimensional case we also have the following, more precise result. \begin{theorem}\label{T3.6} Let the conditions of Theorem~{\rm\ref{T3.5}} be fulfilled and $\,d=2\,$. Then there exists countable families of sets $\,\{P_j\}_{j\in\JC}\,$ and $\,\{V_k\}_{k\in\KC}\,$ such that \begin{enumerate} \item[(1)] $P_j\subset G_{f,b}$ and $P_j\in\PB(\delta)$ for all $j\in\JC$; \item[(2)] $V_k\subset G_{f,b}$ and $V_k\in\VB(\delta,f)$ for all $k\in\KC$; \item[(3)] $\aleph\left(\{P_j\}_{j\in\JC}\bigcup\{V_k\}_{k\in\KC}\right)\leq2$; \item[(4)] $G_{f,b}\subset\bigcup_{j\in\JC,\,k\in\KC} \left(\overline{P_j}\bigcup\overline{V_k}\right)$; \item[(5)] $\#\{k\in\KC\;|\;\mu_2(V_k)\leq\delta^2/2\}\leq \VC_{\delta/2}(f,Q_a^{(1)})\,$ and \newline $\#\{j\in\JC\;|\;\mu_2(P_j)\leq\delta^2/8\}\leq 6\,\VC_{\delta/2}(f,Q_a^{(1)})+12a/\delta\,$. \end{enumerate} \end{theorem} \begin{proof} In the two dimensional case we do not need Besicovitch's theorem because the `cube' $\,Q^{(1)}_a\,$ coincides with an interval of the form $\,(b,b+a)\,$. Given $\,\eps>0\,$, one can easily construct a finite family $\,\{Q^{(1)}(x)\}_{x\in\XC}\,$ of disjoint subintervals $\,Q^{(1)}(x)\in(a,a+b)\,$ satisfying the conditions (1)--(4) of Corollary~\ref{C3.2} with $\,C_{d,2}=C_{d,3}=1\,$. Therefore Lemma~\ref{L3.4} remains valid if we substitute $\,C_{d,2}=C_{d,3}=1\,$. Let $\,k\in\KC:=\XC\,$ and $\,b_k\,$, $\,Q^{(1)}(k)\,$ and $\,V_k=G_{f,b_k}(Q^{(1)}(k))\,$ be the same as in the proof of Theorem~\ref{T3.5}. By the above, the first inequality in Theorem~\ref{T3.5}(5) holds true with $\,C_{d,3}=1\,$. Therefore $\,\#\KC\leq\VC_{\delta/2}(f,Q_a^{(1)})+2a/\delta\,$ (the second term is the maximal number of intervals $\,Q^{(1)}(k)\,$ whose length exceeds $\,\delta/2\,$). Let $\,V_f:=\bigcup_{k\in\KC}V_k\,$. The set $\,G_f\setminus V_f\,$ is a polygon with edges parallel to coordinate axes which has at most $\,2\,\VC_{\delta/2}(f,Q_a^{(1)})\,$ vertices lying on the horizontal lines $\,\{x\;|\;x_1\in Q_a^{(1)}\,,\,x_2=b_k\}\,$. Let us choose a constant $\,c\in(\delta/2,\delta]\,$ in such a way that $\,a/c\in\N\,$ and split the interval $\,Q^{(1)}_a\,$ into the union of $\,a/c\,$ intervals $\,(a_l,a_{l+1})\,$ of length $\,c\,$; if $\,a<\delta\,$ then we take $\,(a_1,a_2):=Q^{(1)}_a\,$. Denote $$ \KC'_l\ :=\ \{k\in\KC\;|\;[a_{l-2},a_{l+3}]\bigcap \overline{Q^{(1)}(k)}\ne\emptyset\}\,,\quad b_{k,\,l}:=\min_{k\in\KC'_l}b_k\,, $$ and $\,P_{k,\,l}:=(a_l,a_{l+1})\times(b_k,b'_k)\,$ where $\,b'_k:=\min\{b_{k'}\;|\;b_{k'}>b_k,\,k'\in\KC'_l\}\,$; we assume that $\,P_{k,\,l}:=\emptyset\,$ whenever $\,b_k=\max\{b_{k'}\;|\;k'\in\KC'_l\}\,$. We have $\,\dist(x,\Gamma_f)>\delta\,$ whenever $\,x_1\in[a_l,a_{l+1}]\,$ and $\,x_2\delta_0\,$. If $\,x\in\Omega_l\bigcap\Omega_{\delta_0}^\bR\,$ then, by Theorem~\ref{T3.5}(4), we have $\,x\in\bigcup_{j\in\JC(l),\,k\in\KC(l)}\left(\overline{P_j} \bigcup\overline{V_k}\right)$. In this case $\,x\in\bigcup_{j\in\JC'(l),\,k\in\KC(l)}\left(\overline{P_j} \bigcup\overline{V_k}\right)$ because $\,\diam P_j\leq\sqrt{d}\,\delta\,$. Therefore $\Omega_{\delta_0}^\bR\,$ is a subset of $\,\bigcup_{j\in\JC,\,k\in\KC} \left(\overline{P_j}\bigcup\overline{V_k}\right)$. The estimates $\,\sup_{x\in V_k}\dist(x,\partial\Omega)\leq\sqrt{d}\,\delta\,$ and $\,\diam P_j\leq\sqrt{d}\,\delta\,$ imply the second inclusion (4). In order to prove (5), let us denote by $M_\delta$ the smallest positive integer such that $\,2^{M_\delta-1}\delta\geq D_\Omega\,$. By Theorem~\ref{T3.5}(5), we have $$ \#\{j\in\bigcup_{l\in\LC}\JC(l)\;|\;\mu_d(P_j)\leq 2^{1-d}\,\delta^d\}\leq C_{d,3}\,C_{\Omega,\,\tau}\sum_{m=0}^{M_\delta}2^m\,\tau({(2^{m-1}\delta})^{-1})\,. $$ Since $2^{M_\delta-1}\delta\leq2D_\Omega\,$, applying Lemma~\ref{L3.7} with $\,a=(2D_\Omega)^{-1}\delta\,$, $\,b=2\,$ and $\,h(t)=t^{-1}\,\tau(\delta^{-1}t)\,$, we obtain $$ \#\{j\in\bigcup_{l\in\LC}\JC(l)\;|\;\mu_d(P_j)\leq 2^{1-d}\,\delta^d\}\ \leq\ 4\,C_{d,3}\,C_{\Omega,\,\tau}\,\delta^{-1} \int_{(2D_\Omega)^{-1}}^{4/\delta}t^{-2}\,\tau(t)\,\dR t\,. $$ Now the second estimate (5) follows from the first inequality (3) and the second inclusion (4). Similarly, the first estimate (5) is a consequence of the second inequality (3), the second inclusion (4) and the first inequality in Theorem~\ref{T3.5}(5). \end{proof} \begin{corollary}\label{C3.9} Let $\Omega\in BV_{\tau,\infty}$ and $\,\Omega\in\R^2\,$. Then for each $\delta\in(0,\delta_\Omega]$ there exist families of sets $\{P_j\}_{j\in\JC}$ and $\{V_k\}_{k\in\KC}$ satisfying the conditions {\rm (1), (2)} and {\rm(4)} of Corollary~{\rm\ref{C3.8}} such that \begin{enumerate} \item[(3$'$)] $\aleph\left(\{P_j\}_{j\in\JC} \bigcup\{V_k\}_{k\in\KC}\right)\leq2\,n_\Omega\,$; \item[(5$'$)] $\,\#\KC\leq C_{\Omega,\tau}\,\tau(2/\delta)+2\,n_\Omega\, \delta^{-2}\,\mu_2(\Omega_{\delta_1}^\bR)\,$ and \newline $\, \#\JC\ \leq\ 6\,C_{\Omega,\tau}\,\tau(2/\delta)\;+\;12\,D_\Omega/\delta\;+\;16\,n_\Omega\, \delta^{-2}\,\mu_2(\Omega_{\delta_1}^\bR)\,$. \end{enumerate} \end{corollary} \begin{proof} The corollary is proved in the same way as Corollary~\ref{C3.8}, with the use of Theorem~\ref{T3.6} instead of Theorem~\ref{T3.5}. \end{proof} Our proof of Theorem \ref{T1.8} is based on the following simple lemma. \begin{lemma}\label{L3.10} Let $\,\Omega\,$ be an arbitrary domain. Then for every $\delta>0$ there exists a family of sets $\,\{M_k\}_{k\in\KC}\,$ satisfying the following conditions: \begin{enumerate} \item[(1)] $\,M_k\subset\Omega\,$ and $M_k\in\MB(\delta)$ for each $k\in\KC\,$; \item[(2)] $\aleph\{M_j\}=1\,$; \item[(3)] $\Omega_{\delta_0}^\bR\ \subset\bigcup\limits_{k\in\KC} \overline{M_k}\ \subset\ \Omega_{\delta_1}^\bR\,$, where $\delta_0:=\delta/\sqrt{d}$ and $\delta_1:=\sqrt{d}\,\delta+\delta/\sqrt{d}\,$. \end{enumerate} \end{lemma} \begin{proof} Consider an arbitrary cover of $\,\R^d\,$ by closed cubes $\,\overline{Q_\delta^{(d)}(k)}\,$ with disjoint interiors $\,Q_\delta^{(d)}(k)\,$ and define $\,\{M_k\}_{k\in\KC}:=\{\Omega\bigcap Q_\delta^{(d)}(k)\}_{k\in\KC}\,$, where $\,\KC\,$ the set of indices $\,k\,$ such that $\,\Omega_{\delta_0}^\bR\bigcap Q_\delta^{(d)}(k)\ne\emptyset\,$. \end{proof} \section{Spectral asymptotics}\label{S4} \subsection{Estimates of the counting function}\label{S4.1} In this section we shall always assume that $\,\delta_0:=\delta/\sqrt{d}\,$, $\,\delta_1:=\sqrt{d}\,\delta+\delta/\sqrt{d}\,$ and denote \begin{equation}\label{4.1} R_\Omega(\lambda,\delta_1)\ :=\ 3\,(4\sqrt{d})\,C_{d,1}\int_{\delta_1}^\infty \left(s^{-1}\lambda^{d-1}+s^{-d}\right)\,\dR(\mu_d(\Omega_s^\bR))\,, \end{equation} where $\,\int\left(s^{-1}\lambda^{d-1}+s^{-d}\right)\,\dR(\mu_d(\Omega_s^\bR))\,$ is understood as a Stieltjes integral. \begin{theorem}\label{T4.1} If $\,\Omega\in\R^d\,$ is an arbitrary domain and $\,\delta>0\,$ then \begin{equation}\label{4.2} N(\Omega,\lambda)-C_{d,W}\,\mu_d(\Omega)\,\lambda^d\ \geq\ -\,R_\Omega(\lambda,\delta_1)- C_{d,W}\,\mu_d(\Omega_{4\delta_1}^\bR)\,\lambda^d\,,\quad\forall\lambda>0\,, \end{equation} and \begin{equation}\label{4.3} N_\DR(\Omega,\lambda)-C_{d,W}\,\mu_d(\Omega)\,\lambda^d\ \leq\ R_\Omega(\lambda,\delta_1)\;+\;((4d)^d+2)\,\delta^{-d}\,\mu_d(\Omega_{4\delta_1}^\bR) \end{equation} for all $\,\lambda\leq\delta^{-1}\,$. If $\,\Omega\in BV_{\tau,\infty}\,$ and $\delta\in(0,\delta_\Omega]$ then \begin{multline}\label{4.4} N_\NR(\Omega,\lambda)-C_{d,W}\,\mu_d(\Omega)\,\lambda^d\ \leq\ R_\Omega(\lambda,\delta_1)\;+\;(4d)^d\,\delta^{-d}\,\mu_d(\Omega_{4\delta_1}^\bR)\\ +\;C_{d,6}\,n_\Omega\,\delta^{-d}\,\mu_d(\Omega_{\delta_1}^\bR)\;+\;8\,C_{d,3}\, C_{\Omega,\,\tau}\,\delta^{-1} \int_{(2D_\Omega)^{-1}}^{4/\delta}t^{-2}\,\tau(t)\,\dR t \end{multline} for all $\,\lambda\leq\min\{1,C^{1/2}_{d,9}\,n_\Omega^{-1/2}\}\,\delta^{-1}\,$. \end{theorem} \begin{proof} Let $\,Q_{2^{-i}}^{(d)}(i,n)\,$ be the Whitney cubes introduced in Theorem~\ref{T3.3}, $$ \IC_\delta^-:=\{i\in\IC\;|\;\sqrt{d}\,2^{-i}\leq\delta_0/4\}\,,\quad \IC_\delta^+:= \{i\in\IC\;|\;\sqrt{d}\,2^{-i}>\delta_1\}\,, $$ $\,\IC_\delta^0:=\IC\setminus(\IC_\delta^+\bigcup\IC_\delta^-)\,$ and $\,\Omega_\delta^\sigma:=\bigcup_{i\in\IC_\delta^\sigma} \bigcup_{n\in\NC_i}Q_{2^{-i}}^{(d)}(i,n)$, where $\,\sigma=+\,$, $\,\sigma=0\,$ or $\,\sigma=-\,$. The set $\,\Omega_\delta^\sigma\,$ are mutually disjoint and $\,\overline{\Omega}=\overline{\Omega_\delta^+}\bigcup\overline{\Omega_\delta^0} \bigcup\overline{\Omega_\delta^-}\,$. By virtue of (\ref{3.1}), \begin{equation}\label{4.5} \Omega_\delta^-\subset\Omega_{\delta_0}^\bR\,,\quad \Omega_\delta^0\subset\Omega_{4\delta_1}^\bR\setminus\Omega_{\delta_0/4}^\bR\,,\quad \Omega\setminus\Omega_{4\delta_1}^\bR\subset\Omega_\delta^+ \subset\Omega\setminus\Omega_{\delta_1}^\bR\,. \end{equation} and \begin{equation}\label{4.6} \#\NC_i\ \leq\ 2^{i\,d}\left(\mu_d(\Omega_{4\sqrt{d}\,2^{-i}}^\bR) -\mu_d(\Omega_{\sqrt{d}\,2^{-i}}^\bR)\right)\,,\quad\forall i\in\IC\,. \end{equation} In view of the second inclusion (\ref{4.5}), we have \begin{equation}\label{4.7} \sum_{i\in\IC_\delta^0}\#\NC_i\ \leq\ (4\sqrt{d}\,\delta_0^{-1})^d\,\mu_d(\Omega_{4\delta_1}^\bR)\ =\ (4d)^d\,\delta^{-d}\,\mu_d(\Omega_{4\delta_1}^\bR)\,. \end{equation} Since $\,\aleph\{\,[\sqrt{d}\,2^{-i},4\sqrt{d}\,2^{-i}]\,\}_{i\in\Z}=3\,$ and $\,\Omega_s^\bR=\Omega_{D_\Omega}^\bR\,$ for all $\,s\geq D_\Omega\,$, the inequalities (\ref{4.6}) imply that \begin{equation}\label{4.8} \sum_{i\in\IC_\delta^+}((2^i)^{1-d}\lambda^{d-1}+1)\,\#\NC_i\ \leq\ 3\,(4\sqrt{d})\int_{\delta_1}^\infty (s^{-1}\lambda^{d-1}+s^{-d})\,\dR(\mu_d(\Omega_s^\bR)) \end{equation} for all $\,\lambda>0\,$. By Lemma~\ref{L2.1}, \begin{multline}\label{4.9} N(\Omega,\lambda) -C_{d,W}\,\mu_d(\Omega)\,\lambda^d\\ \geq\ -\,C_{d,W}\,\mu_d(\Omega\setminus\Omega_\delta^+)\,\lambda^d\, +\,\left(N_\DR(\Omega_\delta^+,\lambda) -C_{d,W}\,\mu_d(\Omega_\delta^+)\,\lambda^d\right)\,, \end{multline} \begin{multline}\label{4.10} N_\DR(\Omega,\lambda) -C_{d,W}\,\mu_d(\Omega)\,\lambda^d\\ \leq\ N_{\NR,\DR}(\Omega\setminus\Omega_\delta^+,\partial\Omega,\lambda) \,+\,\left(N_\NR(\Omega_\delta^+,\lambda) -C_{d,W}\,\mu_d(\Omega_\delta^+)\,\lambda^d\right). \end{multline} and \begin{multline}\label{4.11} N_\NR(\Omega,\lambda) -C_{d,W}\,\mu_d(\Omega)\,\lambda^d\\ \leq\ N_\NR(\Omega\setminus\Omega_\delta^+,\lambda) \,+\,\left(N_\NR(\Omega_\delta^+,\lambda) -C_{d,W}\,\mu_d(\Omega_\delta^+)\,\lambda^d\right) \end{multline} Lemma~\ref{L2.1} implies that \begin{multline*} \sum_{n\in\NC_i,\,i\in\IC_\delta^+} \left(N_\DR(Q_{2^{-i}}^{(d)}(i,n),\lambda)-C_{d,W}\, (2^{-i}\lambda)^d\right)\ \leq\ N(\Omega_\delta^+,\lambda)-C_{d,W}\, \mu_d(\Omega_\delta^+)\,\lambda^d\\ \leq\sum_{n\in\NC_i,\,i\in\IC_\delta^+} \left(N_\NR(Q_{2^{-i}}^{(d)}(i,n),\lambda)-C_{d,W}\, (2^{-i}\lambda)^d\right). \end{multline*} In view of Lemma~\ref{L2.8}, the right and left hand sides are estimated from below and above by $\,\pm\,C_{d,1} \sum_{i\in\IC_\delta^+}\left((2^i)^{1-d}\lambda^{d-1}+1\right)\#\NC_i$. Therefore, by (\ref{4.8}), \begin{equation}\label{4.12} |\,N(\Omega_\delta^+,\lambda) -C_{d,W}\,\mu_d(\Omega_\delta^+)\,\lambda^d\,|\ \leq\ R_\Omega(\lambda,\delta_1)\,,\qquad\forall\lambda>0\,. \end{equation} Since $\,\Omega\setminus\Omega_{4\delta_1}^\bR\subset\Omega_\delta^+\,$, the lower bound (\ref{4.2}) is an immediate consequence of (\ref{4.9}) and (\ref{4.12}). Assume that $\,\lambda\leq\delta^{-1}\,$. Let $\,\{M_k\}_{k\in\KC}\,$ be the family of sets introduced in Lemma~\ref{L3.10} and $$ \{S_m\}_{m\in\MC_\DR}\ :=\ \{Q_{2^{-i}}^{(d)}(i,n)\}_{n\in\NC_j,\,i\in\IC_\delta^0} \bigcup\{M_k\}_{k\in\KC}\,. $$ Lemma~\ref{L3.10}(3) and (\ref{4.5}) imply that $\,\bigcup_{m\in\MC_\DR}S_m=\Omega\setminus\Omega_\delta^+\,$. In view of Lemma~\ref{L3.10}(2), we have $\,\aleph\{S_m\}_{m\in\MC_\DR}\leq2\,$. Consequently, by Lemma~\ref{L2.2}, $$ N_{\NR,\DR}(\Omega\setminus\Omega_\delta^+,\partial\Omega,\lambda)\ \leq\ \sum_{m\in\MC_\DR}N_{\NR,\DR}(S_m,\Upsilon_m,\sqrt{2}\,\lambda)\,, $$ where $\,\Upsilon_m=\partial S_m\bigcap\partial\Omega\,$. Since each set $\,S_m\,$ belongs either to $\,\PB(d^{-1/2}\delta_1)\,$ or to $\,\MB(\delta)\,$, Lemma~\ref{L2.6} implies that $\,N_\NR(S_m,\Upsilon_m,\sqrt{2}\,\lambda)\leq1\,$. Moreover, if $\,S_m\in\MB(\delta)\,$ then, in view of Lemma~\ref{L2.6}(3), $\,N_\NR(S_m,\Upsilon_m,\sqrt{2}\lambda)>0\,$ only if $\,\mu_d(S_m)\geq\delta^d-4\pi^{-2}\,\delta^{d+2}\lambda^2\,$. By Lemma~\ref{L3.10}(3), the number of set $\,M\in\{M_k\}_{k\in\KC}\,$ satisfying this estimate does not exceed $$ \left(1-4\pi^{-2}\,\delta^2\lambda^2\right)^{-1} \delta^{-d}\,\mu_d(\Omega_{\delta_1}^\bR) \ \leq\ 2\,\delta^{-d}\,\mu_d(\Omega_{\delta_1}^\bR) $$ Taking into account (\ref{4.7}), we obtain $$ N_{\NR,\DR}(\Omega\setminus\Omega_\delta^+,\partial\Omega,\lambda)\ \leq\ (4d)^d\,\delta^{-d}\,\mu_d(\Omega_{4\delta_1}^\bR)+ 2\,\delta^{-d}\,\mu_d(\Omega_{\delta_1}^\bR)\,. $$ This estimate, (\ref{4.10}) and (\ref{4.12}) imply (\ref{4.3}). In order to prove (\ref{4.4}), let us consider the family of sets $\{P_j\}_{j\in\JC}$ and $\{V_k\}_{k\in\KC}$ constructed in Corollary~\ref{C3.8} and define $$ \{S_m\}_{m\in\MC_\NR}\ :=\ \{Q_{2^{-i}\delta}^{(d)}(i,n)\}_{n\in\NC_j,\,i\in\IC_\delta^0} \bigcup\{P_j\}_{j\in\JC}\bigcup\{V_k\}_{k\in\KC}\,. $$ Corollary~\ref{C3.8}(4) and (\ref{4.5}) imply that $\,\bigcup_{m\in\MC}S_m=\Omega\setminus\Omega_\delta^+\,$. In view of Corollary~\ref{C3.8}(3), we have $\,\aleph\{S_m\}_{m\in\MC}\leq n_\Omega\,C_{d,4}^2\,$. Consequently, by Lemma~\ref{L2.2}, $$ N_\NR(\Omega\setminus\Omega_\delta^+,\lambda)\ \leq\ \sum_{m\in\MC_\NR}N_\NR(S_m,n_\Omega^{1/2}\,C_{d,4}\,\lambda)\,. $$ Since each set $\,S_m\,$ belongs either to $\,\VB(\delta)\,$ or to $\,\PB(d^{-1/2}\delta_1)\,$, Lemma~\ref{L2.6} implies that $\,N_\NR(S_m,n_\Omega^{1/2}\,C_{d,4}\,\lambda)=1\,$ whenever $\,n_\Omega^{1/2}\,C_{d,4}\,\lambda\leq C_{d,5}\,\delta^{-1}\,$. Estimating $\,\#\MC\,$ with the use of (\ref{4.7}) and Corollary~\ref{C3.8}(5) and applying the inequalities $$ (\delta/4)\,\tau(\delta/2)\ =\ \tau(\delta/2)\int_{2/\delta}^{4/\delta}t^{-2}\,\dR t\ \leq\ \int_{2/\delta}^{4/\delta}t^{-2}\,\tau(t)\,\dR t\ \leq\ \int_{(2D_\Omega)^{-1}}^{4/\delta}t^{-2}\,\tau(t)\,\dR t\,, $$ we see that \begin{multline}\label{4.13} N_\NR(\Omega\setminus\Omega_\delta^+,\lambda)\ \leq\ 8\,C_{d,3}\, C_{\Omega,\,\tau}\,\delta^{-1} \int_{(2D_\Omega)^{-1}}^{4/\delta}t^{-2}\,\tau(t)\,\dR t\\ +\;(4d/\delta)^d\,\mu_d(\Omega_{4\delta_1}^\bR)\;+\; C_{d,6}\,n_\Omega\,\delta^{-d}\,\mu_d(\Omega_{\delta_1}^\bR) \end{multline} for all $\,\lambda\leq C_{d,7}\,n_\Omega^{-1/2}\,\delta^{-1}\,$. Now (\ref{4.4}) follows from (\ref{4.11}) and (\ref{4.12}). \end{proof} \subsection{Two dimensional domains}\label{S4.2} If $d=2$, $\,\tau(t)=t\,$ and $\,\delta\asymp\lambda^{-1}\,$ then the first term on the right hand side of (\ref{4.13}) coincides with $\,c\,\lambda\,\log\lambda\,$, where $c$ is some constant. On the other hand, for two dimensional domains with smooth boundaries we have $\,N_\NR(\Omega_{\lambda^{-1}}^\bR,\lambda)\sim\lambda\,$ as $\lambda\to\infty$ (see, for example, \cite{SV}). The following lemma gives a refined estimate for $\,N_\NR(\Omega\setminus\Omega_\delta^+,\lambda)\,$, which does not contain the logarithmic factor. \begin{lemma}\label{L4.2} Let $\,\Omega\subset\R^2\,$, $\,\Omega\in BV_{\tau,\infty}\,$, $\,\delta\in(0,\delta_\Omega]\,$ and $\,\Omega_\delta^+\,$ be defined as in Subsection {\rm\ref{S4.1}}. Then for all $\,\lambda\leq\frac{\sqrt{2}}{3}\,n_\Omega^{-1/2}\delta^{-1}\,$ we have \begin{equation}\label{4.14} N_\NR(\Omega\setminus\Omega_\delta^+,\lambda)\ \leq\ 7\,C_{\Omega,\tau}\,\tau(2/\delta)+(64+18\,n_\Omega)\, \delta^{-2}\,\mu_2(\Omega_{4\delta_1}^\bR)+12\,D_\Omega/\delta\,. \end{equation} \end{lemma} \begin{proof} Applying the same arguments as in the proof of Theorem~\ref{T4.1} but using Corollary~\ref{C3.9} instead of Corollary~\ref{C3.8}, one obtains (\ref{4.14}) instead of (\ref{4.13}). \end{proof} \subsection{Proof of Theorems \ref{T1.3}, \ref{T1.8} and Corollary~\ref{C1.5}}\label{S4.3} Integrating by parts in the Stieltjes integral and changing variables $\,s=t^{-1}\,$, we obtain \begin{multline}\label{4.15} \int_\eps^\infty (s^{-1}\lambda^{d-1}+s^{-d})\,\dR(\mu_d(\Omega_s^\bR)) \;+\;(\eps^{-1}\lambda^{d-1}+\eps^{-d})\,\mu_d(\Omega_\eps^\bR)\\ = \int_0^{\eps^{-1}}(\lambda^{d-1}+d\,t^{d-1})\, \mu_d(\Omega_{t^{-1}}^\bR)\,\dR t\,,\qquad\forall\eps>0\,. \end{multline} Therefore $\,\left((4\delta_1)^{-1}\lambda^{d-1}+(4\delta_1)^{-d}\right) \mu_d(\Omega_{4\delta_1}^\bR)\leq(\lambda^{d-1}+d\,\delta_1^{1-d}) \int_0^{\delta_1^{-1}}\,\mu_d(\Omega_{t^{-1}}^\bR)\,\dR t\,$ and $\,\int_{\delta_1}^\infty (s^{-1}\lambda^{d-1}+s^{-d})\,\dR(\mu_d(\Omega_s^\bR))\leq (\lambda^{d-1}+d\,\delta_1^{1-d}) \int_0^{\delta_1^{-1}}\,\mu_d(\Omega_{t^{-1}}^\bR)\,\dR t\,$. Applying these inequalities and the estimates (\ref{4.2})--(\ref{4.4}) with $\,\delta_1^{-1}=\lambda\,$ or $\,\delta^{-1}=C_{d,8}\,n_\Omega^{1/2}\,\lambda\,$, we obtain (\ref{1.1}) and (\ref{1.6}). The estimate (\ref{1.2}) is proved in the same manner, using (\ref{4.14}) instead of (\ref{4.13}). Finally, since $\,\int_a^bt^{-2}\,\tau(t)\,\dR t\leq b^{d-2}\int_a^bt^{-d}\,\tau(t)\,\dR t\,$, (\ref{1.3}) is a consequence of (\ref{1.1}) and the following lemma. \begin{lemma}\label{L4.3} If $\Omega\in BV_{\tau,\infty}$ then $$ \mu_d(\Omega_\eps^\bR)\ \leq C_{d,2}\,3^d\,n_\Omega\,D_\Omega^{d-1}\eps+ C_{d,3}\,3^d\,C_{\Omega,\,\tau}\,\eps^d\,\tau(\eps^{-1})\,, \qquad\forall\eps>0\,. $$ \end{lemma} \begin{proof} Assume first that $f$ is a continuous function on the closed cube $\overline{Q_a^{(d-1)}}$. Let $\{Q^{(d-1)}(x)\}_{x\in\XC}$ be the same family of cubes as in Corollary~\ref{C3.2}, $\Gamma_f(x):=\{z\in\Gamma_f\;|\;z'\in Q^{(d-1)}(x)\}$ and $\XC_\eps:=\{x\in\XC\,|\,Q^{(d-1)}(x)\in\PB(\eps)\}$. If $\,\dist(y,\Gamma_f)\leq\eps\,$ then $\dist(y,\Gamma_f(x))\leq\eps\,$ for some $x\in\XC$. Therefore \begin{multline*} \mu_d\left(\{y\in Q_a^{(d-1)}\;|\;\dist(y,\Gamma_f)\leq\eps\}\right)\\ \leq\ \sum_{x\in\XC}\mu_d\left(\{y\in Q_a^{(d-1)}\;|\;\dist(y,\Gamma_f(x))\leq\eps\}\right). \end{multline*} The set $\,\{y\in Q_a^{(d-1)}\;|\;\dist(y,\Gamma_f(x))\leq\eps\}\,$ lies in the $\,\eps$-neighbourhood of the rectangle $\,Q^{(d-1)}(x)\times\left(\inf_{z\in Q^{(d-1)}(x)}f(z)\,,\,\sup_{z\in Q^{(d-1)}(x)}f(z)\right)\,$. In view of Corollary~\ref{C3.2}(4), the measure of this $\,\eps$-neighbourhood does not exceed $\,3\eps\,(a_x+2\eps)^{d-1}\,$, where $a_x$ is the length of the edge of $Q^{(d-1)}(x)$. Therefore $$ \mu_d\left(\{y\in Q_a^{(d-1)}\;|\;\dist(y,\Gamma_f)\leq\eps\}\right) \leq3^d\,\eps^d\,(\#\XC_\eps)+\sum_{x\in\XC\setminus\XC_\eps}3^d\, \eps\,a_x^{d-1}\,. $$ Now the obvious inequality $ \sum_{x\in\XC}a_x^{d-1}\leq a^{d-1}\,\aleph\{Q^{(d-1)}(x)\}_{x\in\XC} $ and Corollary~\ref{C3.2}(3) imply that $$ \mu_d(\{y\in\R^d\;|\;\dist(y,\Gamma_f)\leq\eps\})\ \leq\ C_{d,2}\,3^d\eps\,a^{d-1}+C_{d,3}\,3^d\eps^d\,\VC_\eps(f,Q_a^{(d-1)})\,. $$ Since $\,\Omega_\eps^\bR=\bigcup_{l\in\LC} \{x\in\Omega\,|\,\dist(x,\Gamma_{f_l})\leq\eps\}$, where $\,f_l\,$ are the functions introduced in Subsection~\ref{S1.1}, the lemma follows from this inequality. \end{proof} \subsection{Proof of Corollaries~\ref{C1.6} and \ref{C1.9}}\label{S4.4} Let $\,\Omega\in\Lip_\alpha\,$, $\,f_l\,$ be the functions introduced in Subsection {\rm\ref{S1.1}} and $\,|\Omega|_\alpha:=\max_l|f_l|_\alpha\,$, where $\,|\cdot|_\alpha\,$ is the seminorm defined in Subsection \ref{S1.1}. If $\,x\in G_{f_l}\,$ and $\,\dist(x,(y',f_l(y'))\leq\delta\,$ then \begin{equation}\label{4.16} f_l(x')-x_d\ \leq\ |x_d-f_l(y')|+|f_l(y')-f_l(x')|\ \leq\ \delta+\delta^\alpha\,|f_l|_\alpha\,. \end{equation} Therefore $\,\{x\in G_{f_l}\;|\;\dist(x,\Gamma_{f_l})\leq\delta\}\subset\{x\in G_{g_l}\;|\; f_l(x')-x_d\leq\delta+\delta^\alpha\,|f_l|_\alpha\}\,$ and, consequently $\,\mu_d(\{x\in G_{f_l}\;|\;\dist(x,\Gamma_{f_l})\leq\delta\})\leq a^{d-1}\,(\delta+\delta^\alpha\,|f_l|_\alpha)\,$. This immediately implies the following lemma. \begin{lemma}\label{L4.4} If $\,\Omega\in\Lip_\alpha\,$ and $\,\delta\leq\delta_\Omega\,$ then $\,\mu_d(\Omega_\delta^\bR)\leq n_\Omega\,D_\Omega^{d-1}\,(\delta+\delta^\alpha\,|\Omega|_\alpha)\,$. \end{lemma} If $\,Q_c^{(d-1)}\subset Q_{a_l}^{(d-1)}\,$ then $\,\diam Q_c^{(d-1)}=d^{1/2}\,c\,$ and \begin{equation}\label{4.17} 2\,\Osc(f,Q_c^{(d-1)})\ \leq\ \sup_{x',y'\in Q_c^{(d-1)}}|f_l(x')-f_l(y')|\ \leq\ d^{\alpha/2}\,c^\alpha\,|f|_\alpha\,. \end{equation} Therefore $\,c^{d-1}\geq d^{(1-d)/2}\,|f|_\alpha^{(1-d)/\alpha}\,\delta^{(d-1)/\alpha}\,$ whenever $\,\Osc(f,Q_c^{(d-1)})\geq\delta/2\,$ and, consequently, \begin{equation}\label{4.18} \VC_{\delta/2}(f,Q_a^{(d-1)})\ \leq\ d^{(d-1)/2}\,a^{d-1}\,|f|_\alpha^{(d-1)/\alpha}\,\delta^{(1-d)/\alpha}+1\,. \end{equation} The inequality (\ref{4.18}) implies the following result. \begin{lemma}\label{L4.5} If $\,\Omega\in\Lip_\alpha\,$ and \begin{equation}\label{4.19} \tau(t)\ =\ 2^{(1-d)/\alpha}\,d^{(d-1)/2}\,D_\Omega^{d-1}\, |\Omega|_\alpha^{(d-1)/\alpha}\,t^{(d-1)/\alpha}+1 \end{equation} then $\,\Omega\in BV_{\infty,\tau}\,$ and $\,C_{\Omega,\tau}\leq n_\Omega \,$. \end{lemma} Clearly, (\ref{1.4}) follows from (\ref{1.1}) and Lemma~\ref{L4.5}. Similarly, (\ref{1.6}) and Lemma~\ref{L4.4} imply (\ref{1.7}). It remains to prove (\ref{1.5}) and (\ref{1.8}). Assume that $\,\Omega\in\lip_\alpha\,$. Then for each $\,\eps>0\,$ we can find functions $\,f_{l,\,1}^{(\eps)}\in\Lip_1\,$ and $\,f_{l,\,2}^{(\eps)}\in\Lip_\alpha\,$ such that $\,f_l=f_{l,\,1}^{(\eps)}+f_{l,\,2}^{(\eps)}\,$ and $\,|f_{l,\,2}^{(\eps)}|_\alpha\leq\eps\,$. Obviously, $\,\VC_\delta(f_{l,\,1}^{(\eps)}+f_{l,\,2}^{(\eps)},Q)\ \leq\ \VC_{\delta/2}(f_{l,\,1}^{(\eps)},Q)\;+\; \VC_{\delta/2}(f_{l,\,2}^{(\eps)},Q)\,$. Therefore (\ref{4.18}) implies that \begin{multline}\label{4.20} \VC_\delta(f_l,Q_{a_l}^{(d-1)})\ \leq\ d^{(d-1)/2}\,D_\Omega^{d-1}\left(\eps^{(d-1)/\alpha}\,\delta^{(1-d)/\alpha} +C_\eps^{d-1}\,\delta^{1-d}\right)+2\\ \leq\ \eps^{(d-1)/\alpha}\,\tau_\eps(\delta^{-1})\,, \end{multline} where $\,C_\eps:=\max_l\,|f_{l,\,2}^{(\eps)}|_1\,$, and $$ \tau_\eps(t)\ :=\ d^{(d-1)/2}\,D_\Omega^{d-1}\left(t^{(d-1)/\alpha} +C_{\eps,\Omega}\,\eps^{(1-d)/\alpha}\,t^{d-1}\right)+2\,\eps^{(1-d)/\alpha}\,. $$ We also have $$ |f_l(x')-f_l(y')|\leq\eps\,|x'-y'|^\alpha+|f_{l,\,2}^{(\eps)}|_1\,|x'-y'|\,, \qquad\forall x',y'\in Q_{a_l}^{(d-1)}\,. $$ Therefore, instead of (\ref{4.16}), we obtain $\,f_l(x')-x_d\leq \delta+\delta\,|f_{l,\,2}^{(\eps)}|_1+\delta^\alpha\,\eps\,$. This implies that $\,\mu_d(\Omega_\delta^\bR)\leq n_\Omega\,D_\Omega^{d-1}\,(\delta+C_{\eps}\,\delta+\delta^\alpha\,\eps)\,$ whenever $\,\delta\leq\delta_\Omega\,$. In view of (\ref{4.20}), we have $\,\Omega\in BV_{\infty,\tau_\eps}\,$ and $\,C_{\Omega,\tau_\eps}\leq\eps^{(d-1)/\alpha}\,n_\Omega\,$. Choosing a sufficiently large constant $\,C\,$ and applying (\ref{4.2})--(\ref{4.4}) with $\,\delta=C\,\lambda^{-1}\,$ and $\,\tau=\tau_\eps\,$, we see that \begin{align*} |\,N_\NR(\Omega,\lambda)-C_{d,W}\,\mu_d(\Omega)\,\lambda^d\,|\ &\leq\ \eps^{(d-1)/\alpha}\,C'_\Omega\,\lambda^{(d-1)/\alpha} +C'_{\Omega,\eps}\,\lambda^{d-1}\,,\\ |\,N_\DR(\Omega,\lambda)-C_{d,W}\,\mu_d(\Omega)\,\lambda^d\,|\ &\leq\ \eps\,C'_\Omega\,\lambda^{d-\alpha} +C'_{\Omega,\eps}\,\lambda^{d-1} \end{align*} for all $\,\lambda>1\,$, where $\,C'_\Omega\,$ is a constant depending only on the domain $\,\Omega\,$ and $\,C'_{\Omega,\eps}\,$ is a constant depending on $\,\Omega\,$ and $\,\eps\,$. Since $\,\eps\,$ can be made arbitrarily small, these inequalities imply (\ref{1.5}) and (\ref{1.8}). \qed \subsection{Proof of Theorem~\ref{T1.10}}\label{S4.5} Let $\,Q_1^{(d-1)}=(0,1)^{d-1}\,$, $\,\alpha\in(0,1)\,$ and $\,p\,$ be a sufficiently large positive integer. In particular, we shall be assuming that $\,p\geq\max\{\alpha^{-1},(1-\alpha)^{-1}\}\,$ and, consequently, \begin{equation}\label{4.21} 2^{1-\alpha p}\leq1\,,\quad\left(1-2^{-\alpha p}\right)^{-1}\ \leq\ 2\,,\quad \left(1-2^{(1-\alpha)\,p}\right)^{-1}\ \leq\ 2 \end{equation} and \begin{equation}\label{4.22} \left(2^{(1-\alpha)\,(n+1)p}-1\right)\left(2^{(1-\alpha)\, p}-1\right)^{-1}\ \leq\ 2^{1+(1-\alpha)\,np}\,,\qquad\forall n=1,2,\ldots \end{equation} Given $\,j\in\Z_+\,$, let us denote by $\,\KC_j\,$ the set of nonnegative integer vectors $\,\kB=(k_1,\ldots,k_{d-1})\in\Z_+^{d-1}\,$ such that $\,\max_i\,k_i\leq2^{jp}-1\,$ and consider the $\,(d-1)$-dimensional cubes $$ Q(j,\kB)\ :=\ \{x'\in\R^{d-1}\;|\;2^{jp}x'-\kB\in Q_1^{(d-1)} \}\,,\qquad\kB\in\KC_j\,. $$ with edges of length $\,{2^{-jp}}\,$. For each fixed $\,j\in\Z_+\,$ and $\,\kB\in\KC_j\,$ the cubes $\,Q(j,\kB)\,$ are disjoint and $\,\overline{Q_1^{(d-1)}}= \bigcup_{\kB\in\KC_j}\overline{Q(j,\kB)}\,$. Let $\,\psi\in\Lip_1\,$ be a nonnegative Lipschitz function on $\,Q_1^{(d-1)}\,$ vanishing on the boundary $\,\partial Q_1^{(d-1)}\,$, $\,a_\psi:=\sup\psi\,$ and $\,b_{\psi,p}:=\sqrt{d}\,2^{3-(1-\alpha)p}\,(|\psi|_1+a_\psi)\,$. We shall be assuming that $\,p\,$ is large enough so that $\,a_\psi>b_{\psi,p}\,$. Let us extend $\,\psi\,$ by 0 to the whole space $\,\R^{d-1}\,$ and define $$ g_j(x')\ :=\ \sum_{\kB\in\KC_j} \psi(2^{jp}x'-\kB)\,,\qquad f_n(x')\ :=\ \sum_{j=0}^n\eps_j\,2^{-\alpha\,jp}\,g_j(x') $$ and $\,f(x'):=\lim_{n\to\infty}f_n(x')=\sum_{j=0}^\infty\eps_j\,2^{-\alpha jp}\,g_j(x')\,$, where $\,\{\eps_j\}\,$ is a nonincreasing sequence such that $\,\eps_j\in[0,1]\,$ and \begin{equation}\label{4.23} 2^{(1-\alpha)\left([j/2]-j\right)p}\ \leq\ \eps_{[j/2]}\ \leq\ 2\,\eps_j\,,\qquad\forall j=1,2,\ldots \end{equation} Note that the condition (\ref{4.23}) is fulfilled whenever $\,\{\eps_j\}\,$ is a sufficiently slowly decreasing sequence. \begin{lemma}\label{L4.6} We have \begin{enumerate} \item[(1)] $\,g_j=0\,$ on $\,\partial Q(j,\kB)\,$ for all $\,\kB\in\KC_n\,$ and $\,j\geq n\,$; \item[(2)] $\,0\leq f(x')-f_n(x')\leq2\,\eps_{n+1}\,2^{-\alpha\,(n+1)p}\,a_\psi \leq\eps_{n+1}\,2^{-\alpha\,np}\,a_\psi\,$; \item[(3)] $\,|f_n|_\beta\leq2^{1+(\beta-\alpha)np}\,(|\psi|_1+a_\psi)\,$ for all $\,\beta\in[\alpha,1]\,$; \item[(4)] $\,f\in\Lip_\alpha\,$ and $\,|f|_\alpha\leq2\,(|\psi|_1+a_\psi)\,$; \item[(5)] $\,f\in\lip_\alpha\,$ whenever $\,\eps_j\to0\,$ as $\,j\to\infty\,$; \item[(6)] $\,2\,\Osc(f_{n-1},Q(n,\kB))\leq\eps_n\,2^{-\alpha\,np}\,b_{\psi,p}\,$ for all $\,\kB\in\KC_n\,$. \end{enumerate} \end{lemma} \begin{proof} (1) is obvious and (2) immediately follows from (\ref{4.21}). In order to prove (3), let us fix $\,\beta\in[\alpha,1]\,$, denote $\,n':=\max\{j\;|\;2^{-jp}\geq|x'-y'|\}\,$, $\,n'':=\min\{n,n'\}\,$ and estimate \begin{multline*} \sum_{j=0}^n\frac{|g_j(x')-g_j(y')|}{2^{\alpha jp}\,|x'-y'|^\beta}\ =\ \sum_{j=0}^{n''}\frac{|g_j(x')-g_j(y')|}{2^{\alpha jp}\,|x'-y'|^\beta}\;+\; \sum_{j=n''+1}^n\frac{|g_j(x')-g_j(y')|}{2^{\alpha jp}\,|x'-y'|^\beta}\\ \leq\ |\psi|_1\sum_{j=0}^{n''}2^{(1-\alpha)jp}\,|x'-y'|^{1-\beta} \;+\;a_\psi\sum_{j=n''+1}^n2^{-\alpha jp}\,|x'-y'|^{-\beta}\,. \end{multline*} In view of (\ref{4.22}), the first term on the right hand side is estimated by $\,|\psi|_1\sum_{j=0}^{n''}2^{(1-\alpha)jp+(1-\beta)np}\leq 2^{1+(\beta-\alpha)np}|\psi|_1\,$. If $\,n\leq n'\,$ then the second term on the right hand side vanishes; if $\,n>n'\,$ then, by (\ref{4.21}), it does not exceed $\,2\,a_\psi\,2^{-\alpha(n''+1)p}|x'-y'|^{-\beta} \leq2\,a_\psi\,2^{(\beta-\alpha)(n''+1)p}\leq2^{1+(\beta-\alpha)np}a_\psi\,$. Thus, \begin{equation}\label{4.24} \sum_{j=0}^n\frac{|g_j(x')-g_j(y')|}{2^{\alpha jp}\,|x'-y'|^\beta}\ \leq\ 2^{1+(\beta-\alpha)np}\,(|\psi|_1+a_\psi)\,. \end{equation} This estimate immediately implies (3) and (4). The inclusion (5) is also a consequence of (\ref{4.24}) because $\,|f-f_n|_\alpha\leq\eps_{n+1}\,\sup_{x',y'} \sum_{j=0}^\infty\frac{|g_j(x')-g_j(y')|}{2^{\alpha jp}\,|x'-y'|^\alpha}\,$. Finally, in view of (\ref{4.23}) and (\ref{4.24}), we have \begin{equation}\label{4.25} (|\psi|_1+a_\psi)^{-1}|f_j|_1\ \leq\ 2^{1+(1-\alpha)[j/2]p}\;+\;\eps_{[j/2]}\,2^{1+(1-\alpha)jp}\ \leq\ \eps_j\,2^{3+(1-\alpha)jp}\,. \end{equation} Since $\,\diam Q(n,\kB)=\sqrt{d}\,2^{-np}\,$, (\ref{4.25}) with $\,j=n-1\,$ implies (6). \end{proof} Let $\,\Omega:=G_{f,\,0}\,$, $\,\Omega_{n,\,\kB}:=\{x\in\Omega\;|\;x'\in Q(n,\kB)\,,\, x_d\in(f_{n-1}(x'),f(x'))\}\,$, $\,\Upsilon_{n\,,\kB}:=\partial\Omega_{n,\,\kB}\setminus\partial\Omega\,$ and $\,\Omega_{n-1}\,$ be the interior of $\,\Omega\setminus\left(\bigcup_{\kB\in\KC_n}\Omega_{n,\,\kB}\right)\,$. Denote $\,a_{n,\,\kB}:=\sup\limits_{x'\in Q(n,\kB)}f_{n-1}(x')\,$ and consider the function $$ u_{n,\,\kB}(x)\ :=\ \begin{cases} \sin\left(2^{\alpha\,np}(x_d-a_{n,\,\kB})/\eps_n\right)\,, &x_d\geq a_{n-1,\,\kB}\,,\\ 0\,,&x_dd-\alpha>d-1\,$. Therefore taking $\,\eps_0=\eps_1=\dots=1\,$, we obtain a domain satisfying the conditions of Theorem~\ref{T1.10}(1). If $\,\phi\,$ is a nonnegative function on $(0,+\infty)$ and $\,\phi(\lambda)\to0\,$ as $\,\lambda\to\infty\,$ then we can choose a sequence $\,\eps_n\,$ converging to zero and satisfying (\ref{4.23}) in such a way that the function $\,\phi(\lambda)\,\lambda^{(d-1)/\alpha}\,$ and the last two terms in (\ref{4.26}) are estimated by $\,(c_{\psi,p}\,2^{\alpha p})^{(1-d)/\alpha}\,\eps_{n+1}^{(d-1)/\alpha}\,\lambda^{(d-1)/\alpha}\,$ for all $\,\lambda\in\left[c_{\psi,p}\,\eps_n^{-1}\,2^{\alpha np},\,c_{\psi,p}\,\eps_{n+1}^{-1}\,2^{\alpha(n+1)p}\right)\,$ and all sufficiently large $\,n\,$. In view of Lemma~\ref{L4.6}(5), this proves Theorem~\ref{T1.10}(2). \qed \section{Remarks and generalisations}\label{S5} \subsection{Poincar\'e inequality}\label{S5.1} According to the Poincar\'e inequality, \begin{equation}\label{5.1} \int_\Omega |u|^2\,\dR x\leq c_\Omega\int _\Omega |\nabla u|^2\,\dR x\quad\text{whenever}\ u\in W^{1,2}(\Omega)\ \text{and}\ \int_\Omega u\,\dR x=0, \end{equation} where $\,c_\Omega\,$ is a positive constant. By Remark~\ref{R2.4}, the Poincar\'e inequality (\ref{5.1}) on a domain $\,\Omega\,$ holds true if and only if the zero eigenvalue of the Neumann Laplacian is isolated and $\,c_\Omega\geq\lambda_{1,\NR}^{-2}(\Omega)\,$. \begin{lemma}\label{L5.1} Let $\,\Omega\,$ satisfies {\rm(\ref{5.1})} and $\,\tilde\Omega\subset\R^d\,$. If there exist an invertible map $F:\Omega\to\tilde\Omega$ and a constant $\,C_F\,$ such that $\,|F(x)-F(y)|\leq C_F\,|x-y|\,$ for all $\,x,y\in\Omega\,$ and $\,|F^{-1}(x)-F^{-1}(y)|\leq C_F\,|x-y|\,$ for all $x,y\in\tilde\Omega$ then $\,\tilde\Omega\,$ also satisfies {\rm(\ref{5.1})} with a positive constant $\,c_{\tilde\Omega}=C_d\,C_F^{2d+2}c_\Omega\,$. \end{lemma} \begin{proof} Let $v\in W^{1,2}(\tilde\Omega)$, $u(x):=v(F^{-1}(x))$ and $c_u:=\int_\Omega u(x)\,\dR x$. Under the conditions of the lemma the maps $F$ and $F^{-1}$ are differentiable almost everywhere. Changing variables and estimating the Jacobians, we obtain $$ \int_{\tilde\Omega}|v(y)-c_u|^2\,\dR y\ \leq\ C_d\,C_F^d\int_\Omega |u(x)-c_u|^2\,\dR x $$ and $$ \int_{\tilde\Omega} |\nabla v(y)|^2\,\dR y\ \geq\ C_d\,C_F^{-d-2}\int_\Omega|\nabla u(x)|^2\,\dR x\,. $$ These two estimates and the Poincar\'e inequality (\ref{5.1}) imply that $$ \int_{\tilde\Omega}|v(y)|^2\,\dR y\ \leq\ \int_{\tilde\Omega}|v(y)-c_u|^2\,\dR y\ \leq\ C_d\,C_F^{2d+2}c_\Omega\int_{\tilde\Omega} |\nabla v(y)|^2\,\dR y $$ whenever $\int_{\tilde\Omega}v\,\dR y=0$. \end{proof} Lemma~\ref{L5.1} allows one to extend Theorem~\ref{T1.3} to more general domains. \begin{theorem}\label{T5.2} Assume that there exists a finite collection of domains $\Omega_l\subset\Omega$ such that \begin{enumerate} \item[(a)] $\partial\Omega\subset\bigcup_l\overline{\Omega_l}\,$; \item[(b$'$)] for each $l$ there exist an invertible map $F_l:\R^d\to\R^d$ satisfying the conditions of Lemma~{\rm\ref{L5.1}} such that $F_l(\Omega_l)=G_{f_l,\,b_l}\,$, where $\,f_l\in BV_{\tau,\infty}(Q_{a_l}^{(d-1)})\,$ and $\,b_l<\inf f_l\,$; \item[(c)] $\,a_l\leq D_\Omega\,$ and $\,\sup f_l-b_l\leq D_\Omega\,$ for all $l\in\LC$. \end{enumerate} Then {\rm(\ref{1.1})} holds true. \end{theorem} \begin{proof} Let $\,C_{F_l}\,$ be the constant introduced in Lemma~\ref{L5.1} and $\,C:=\max_lC_{F_l}\,$. Under conditions of the theorem, Corollary~\ref{C3.8} remains valid if we replace $\,U_l\,$ with $\,F_l\,$ and take $\,\delta_n:=C^{-1}\,\delta_n\,$. Since (\ref{5.1}) is equivalent to the identity $\,N_\NR(\Omega,c_\Omega^{-2})=1\,$, Lemma~\ref{L2.6} and Lemma~\ref{L5.1} imply that $\,N_\NR(S_m,\lambda)=1\,$ for all $\,\lambda\leq c'_\Omega\,\delta^{-1}\,$, where $\,S_m\,$ are the same sets as in the proof of Theorem~\ref{T4.1} and $\,c'_\Omega\,$ is a constant depending on the domain $\,\Omega\,$. Therefore, using the same arguments as in Subsection~\ref{S4.1}, we obtain the estimates (\ref{4.2}) and (\ref{4.4}) with some other constants (which may depend on $\,\Omega\,$). In the same way as in Subsection~\ref{S4.3}, these estimates imply (\ref{1.1}). \end{proof} The following example shows that Theorem~\ref{T5.2} is not just a formal generalization of Theorem~\ref{T1.3}. \begin{example}\label{E5.3} Let $\,f\,$ be a nowhere differentiable $\,\Lip_\alpha$-function on the interval $\,[0,1]\,$. Assume that $\,f>1\,$ and consider the domain $$ \Omega\ :=\ \{(\varphi,r)\in\R^2\;|\;\varphi\in(0,1)\,,\,1\delta^{-1}\,, $$ for all $\,\delta>0\,$ (see Remark~\ref{R2.9}). \item[(iv)] Instead of Lemma~\ref{L2.6} we have the following result. \end{enumerate} \begin{lemma}\label{L5.4} There exists a constant $\,c_A\,$ depending only on the operator $A$ and the dimension $d$ such that the following statements hold true. \begin{enumerate} \item[(1)] If $\,P\in\PB(\delta)$ then $N_\NR(P,\lambda)=\dim\PC_m\,$ for all $\,\lambda\leq c_A\,\delta^{-1}$. \item[(2)] If $\,V\in\VB(\delta)$ then $N_\NR(V,\lambda)=\dim\PC_m\,$ for all $\,\lambda\leq c_A\,\delta^{-1}$. \item[(3)] If $\,M\in\MB(\delta)\,$, $\,M\subset Q_\delta^{(d)}\,$ and $\,\Upsilon:=\partial M\bigcap Q_\delta^{(d)}\,$ then we have $\,N_{\NR,\DR}(M,\Upsilon,\lambda)\leq\dim\PC_m\,$ for all $\,\lambda\leq c_A\,\delta^{-1}\,$ and $\,N_{\NR,\DR}(M,\Upsilon,\lambda)=0\,$ for all $\,\lambda\leq (1-c_A^{-1}\,\delta^{-d}\mu_d(M))_+^{1/(2m)}\,c_A\,\delta^{-1}\,$. \end{enumerate} \end{lemma} \begin{proof} We shall denote by $\,C\,$ various constants depending only on $A$ and $d$. Since $\,A(\xi)\leq C\,\sum_{j=1}^d\,\xi_j^{2m}\,$, it is sufficient to prove the lemma assuming that $\,A(D_x)=A_m(D_x):=\sum_{j=1}^d\,D_{x_j}^{2m}\,$. Then (1) is easily obtained by separation of variables. If $\,u\in W^{m,2}(Q_\delta^{(d)})\,$, $\,u\equiv0\,$ outside $\,M\,$ and $\,p_u\,$ is the projection of $\,u\,$ onto the subspace $\,\PC_m(M)\,$ then \begin{multline*} \int_M|p_u|^2\,\dR x\ \leq\ \mu_d(M)\sup_{x\in Q_\delta^{(d)}}|p_u(x)|^2\ \leq\ C\,\mu_d(M)\,\delta^{-d}\int_{Q_\delta^{(d)}}|p_u|^2\,\dR x\\ =\ C\,\mu_d(M)\,\delta^{-d}\left(\int_M|p_u|^2\,\dR x+\int_{Q_\delta^{(d)}}|u-p_u|^2\,\dR x\right)\,. \end{multline*} Applying (ii) and this estimate instead of Remark~\ref{R2.4} and (\ref{2.6}), we obtain (3) in the same way as Lemma~\ref{L2.6}(3). In order to prove (2), let us assume that $\,V=G_{f,\,b}(Q_c^{(d-1)})\,$ with $\,c\leq\delta\,$, $\,b=\inf f-\delta$ and $\,\Osc f\leq\delta/2\,$ and consider a function $\,u\in W^{m,2}(V)\,$. Let $\,p_{u;\,r,\,k}(x')\,$ be the projection of the function $\,\partial_{x_d}^k\,u(x',r)\in L^2(Q_c^{(d-1)})\,$ onto the subspace $\,\PC_{m-k}(Q_c^{(d-1)})\,$, $\,p_{u;\,r}(x):= \sum_{k=0}^{m-1}\,\frac1{k!}\,(x_d-r)^k\;p_{u;\,r,\,k}(x')\,$ and $\,v_r(x):=\sum_{k=0}^{m-1}\,\frac1{k!}\,(x_d-r)^k\,\partial_{x_d}^ku(x',r)\,$, where $\,r\in[b,b+\delta]\,$ and $\,x_d\in[b,f(x')]\,$. We have \begin{equation}\label{5.2} |u(x)-p_{u;\,r}(x)|^2\ \leq\ 2\,|u(x)-v_r(x)|^2+2\,|v_r(x)-p_{u;\,r}(x)|^2\,. \end{equation} Since $\,|x_d-b|\leq2\delta\,$, Jensen's inequality implies that \begin{multline*} |u(x)-v_r(x)|^2\ =\ ((m-1)!)^{-2}\,|\int_r^{x_d}(x_d-t)^{m-1}\,\partial_{x_d}^mu(x',t)\,\dR t\,|^2\\ \leq\ ((m-1)!)^{-2}\,|x_d-r|\,\int_r^{x_d}(x_d-t)^{2m-2}\,|\partial_{x_d}^mu(x',t)|^2\,\dR t\\ \leq\ ((m-1)!)^{-2}\,(2\delta)^{2m-1}\int_b^{f(x')}|\partial_{x_d}^mu(x)|^2\,\dR x_d\,. \end{multline*} In view of (ii) and (1), we also have $$ \int_{Q_c^{(d-1)}}|\partial_{x_d}^k\,u(x)-p_{u;\,r,\,k}(x')|^2\,\dR x'\ \leq\ C\,\delta^{2m-2k}\,Q_{A'_{m-k}}[\partial_{x_d}^k\,u(x)] $$ for all $\,k=0,\ldots,m-1\,$, where $\,A'_{m-k}(D_{x'}):=\sum_{j=1}^{d-1}\,D_{x_j}^{2m-2k}\,$ and $\,Q_{A'_{m-k}}\,$ is the quadratic form of $\,A'_{m-k}\,$ with domain $\,W^{m-k,\,2}(Q_c^{(d-1)})\,$. Therefore, integrating (\ref{5.2}) over $\,r\in[b,b+\delta]\,$, $\,x_d\in[b,f(x')]\,$, $\,x'\in Q_c^{(d-1)}$ and estimating $\,|x_d-r|\leq2\delta\,$, we obtain \begin{multline*} \delta^{-1}\int_b^{b+\delta}\int|u(x)-p_{u;\,r}(x)|^2\,\dR x\,\dR r\\ \leq\ C\,\delta^{2m}\int_V|\partial_{x_d}^mu(x)|^2\,\dR x\ +C\,\delta^{2m}\sum_{k=0}^{m-1}\,\sum_{|\alpha|=m}\int_P |\partial_x^\alpha u(x)|^2\,\dR x\,, \end{multline*} where $\,P=Q_c^{(d-1)}\times(b,b+\delta)\,$. Since the $\,L_2$-norms of the mixed derivatives $\,\partial_x^\alpha u(x)\,$ on a rectangle are estimated by the $\,L_2$-norms of the derivatives $\,\partial_{x_j}^m\,$, this estimate and (ii) imply (2). \end{proof} Applying the same arguments as in Section~\ref{S4} and using (iii) and Lemma~\ref{L5.4}, we obtain the following result. \begin{theorem}\label{T5.5} Let $\,A\,$ be a homogeneous nonnegative elliptic differential operator of order $\,2m\,$ with real constant coefficients. If $\,N_\NR(\lambda,\Omega)\,$ and $\,N_\DR(\lambda,\Omega)\,$ denote the number of eigenvalues of the corresponding self-adjoint operator lying below $\,\lambda^{2m}\,$ then Theorems {\rm\ref{T1.3}}, {\rm\ref{T1.8}} and Corollaries {\rm\ref{C1.5}}, {\rm\ref{C1.6}}, {\rm\ref{C1.9}} holds true with $\,C_{d,W}:=C_{A,W}\,$. \end{theorem} \subsection{Other function spaces}\label{S5.3} Let $\,B^\alpha_{p,q}\,$ be the Besov space and $\,BV_{\beta,\infty}:=BV_{\tau_\beta,\infty}\,$ where $\,\tau_\beta(t)=(t^\beta+1)\,$ and $\beta\in(0,+\infty)$. Lemma~\ref{L4.5} implies that $\,B^\alpha_{\infty,\infty}=\Lip_\alpha\subset BV_{(d-1)/\alpha,\infty}\,$. Estimating the norm of the embedding $\,B_{p,\infty}^\alpha(Q_a^{(d-1)})\hookrightarrow C(Q_a^{(d-1)})\,$ for $\,\alpha p> d-1\,$ and $\,a>0\,$, one can also show that $\,B_{p,\infty}^\alpha\subset BV_{(d-1)/\alpha,\infty}\,$ whenever $\,\alpha p>d-1\,$. \subsection{Open problems} \subsubsection{The spaces $BV_{\tau,\infty}$}\label{S5.3.1} The space $\,BV_{\beta,\infty}\,$ or $\,BV_{\tau,\infty}$ (under certain conditions on the function $\tau$) is a Banach space with respect to an appropriate norm. Similar spaces have been considered in the dimension one, but we could not find references in the multidimensional case. It would be interesting to find a more constructive description of these spaces and to investigate their properties. \subsubsection{More general domains}\label{S5.3.2} The crucial point in our proof of Theorem~\ref{T1.3} is the construction of the families $\,\{S_m\}_\MC\,$ such that \begin{enumerate} \item[(i)] $\,\Omega_\delta^\bR\subset\bigcup_mS_m\subset\Omega\,$, \item[(ii)] $\,\aleph\{S_m\}_\MC\leq C\,$, \item[(iii)] $\,N_\NR(S_m,\lambda)\leq C'\,$ whenever $\,\lambda\leq C''\delta^{-1}\,$, \end{enumerate} where $\,C\,$, $\,C'\,$ and $\,C''\,$ are some constants independent of $\,\delta\in\R_+\,$. The remainder estimate in the Weyl formula for the Neumann Laplacian depends on the behaviour of $\,\#\MC\,$ as $\,\delta\to0\,$. In this paper we were assuming that $\,\Omega\,$ is the union of subgraphs of continuous functions, used Lemma~\ref{L2.6} in order to prove (iii) and applied Corollary~\ref{C3.2} in order to estimate $\,\aleph\{S_m\}\,$ and $\,\#\MC\,$. Theorem~\ref{T3.1} allows one to construct families of open sets $\,S_m\,$ satisfying (i)--(iii) for many other domains $\,\Omega\,$. It should be possible to find less restrictive sufficient conditions which guarantee the existence of such families and imply an asymptotic formula for $\,N_\NR(\Omega,\lambda)\,$. \subsubsection{Operators with variable coefficients}\label{S5.3.3} Our main goal was to estimate the contribution of $\,\partial\Omega\,$ to the Weyl formula. In the interior part of $\,\Omega\,$ we used the old fashioned variational technique based on the Whitney decomposition and Dirichlet--Neumann bracketing. There are much more advanced methods of studying the asymptotic behaviour of the spectral function at the interior points (see the monographs \cite{Iv3}, \cite{SV} or the recent papers \cite{BI}, \cite{Iv4}), which are applicable to operators with variable coefficients. Freezing the coefficients at an arbitrary point $\,x\in S_m\,$, we see that (iii) remains valid for a uniformly elliptic operator $\,A\,$ with variable coefficients, provided that the corresponding quadratic form is homogeneous, the coefficients are uniformly continuous, $\,\delta\,$ is sufficiently small and $\,\diam S_m\leq c\,\delta\,$ with some constant $\,c\,$ independent of $\,\delta\,$. Using this observation and applying a more powerful technique in the interior of $\,\Omega\,$, one can try to extend our results to operators with variable coefficients. \subsubsection{Reminder estimate for the Dirichlet Laplacian}\label{S5.3.4} It is not difficult to construct a bounded domain $\,\Omega\,$ such that $\,\lim_{\delta\to0}\,|\delta^{-\alpha}\,\mu_d(\Omega_\delta^\bR)|=C'\,$ and \begin{equation}\label{5.3} N_\DR(\Omega,\lambda) -C_{d,W}\,\mu_d(\Omega)\,\lambda^d\ \geq\ -\,C^{-1}\,\lambda^{d-\alpha}\,,\qquad\forall\lambda>C\,, \end{equation} where $C$ and $C'$ are some positive constants. For example, it can be done by considering a cube with a sequence of `cracks' converging to the outer boundary, which get denser as the outer boundary is approached (similar domains were studied in \cite{LV} and \cite{MV}). For such a domain the estimate (\ref{1.7}) is order sharp. It would be interesting find a domain $\,\Omega\in\Lip_\alpha\,$ satisfying (\ref{5.3}) (cf. Theorem \ref{T1.10}). Note that in the known examples disproving the so-called Berry conjecture (see, for instance, \cite{BLe} or \cite{LV}) the domain does not belong to the class $\,\Lip_\alpha\,$. \section{Constants} Throughout the paper $\,C_{d,W}\,$ is the Weyl constant (see Subsection~\ref{S1.1}), $$ C_{d,1}:=\sum_{n=0}^{d-1}\frac{n!\,(d-n)!}{d!}\,C_{n,W}\,,\qquad C_{0,W}:=1\,, $$ $\,C_{d,2}=2^{d-1}\,\CC_{d-1}\,$ and $\,C_{d,3}=6^{d-1}\,\hat\CC_{d-1}\,$ where $\,\CC_{d-1}\,$ and $\,\hat\CC_{d-1}\,$ are the constants introduced in Theorem~\ref{T3.1}, $$ C_{d,4}:=(4\,C_{d,2}+2)^{1/2}\,,\quad C_{d,5}:=\min\left\{(1+2\pi^{-2})^{-1/2},\,\pi(1+d^{-1})^{-1}\right\}\,, $$ $$ C_{d,6}:=2^{d-1}\,C_{d,2}+(3\,C_{d,2}+1)\,(2\sqrt{d})^d\,,\quad C_{d,7}:=C_{d,4}^{-1}\,C_{d,5}\,, $$ $$ C_{d,8}:=\max\{1,C_{d,7}^{-1/2}\}\,,\quad C_{d,9}:=8\,C_{d,3}\,C_{d,8}\,, $$ $$ C_{d,10}:=(d+1)\left(12\sqrt{d}\,C_{d,1}+4\,C_{d,W} +(4^dd^d+C_{d,6})\,(4d^{1/2}+4d^{-1/2})^d\right), $$ $$ C_{d,11}:=(d+1)\left(12\sqrt{d}\,C_{d,1}+4\,C_{d,W} +(4^dd^d+2)\,(4d^{1/2}+4d^{-1/2})^d\right). $$ \begin{remark} If $\rho$ is continuous then Theorem~\ref{T3.1} holds true with $\,\CC_n=2^n\,$ and $\,\hat\CC_n=4^n\,$ (see \cite{G}). Since the function $\rho$ in the proof of Corollary~\ref{C3.2} is continuous, all our results remain valid for $\,C_{d,2}=4^{d-1}\,$ and $\,C_{d,3}=24^{d-1}$. \end{remark} \begin{thebibliography}{MMM} \bibitem[BD]{BD} V. Burenkov and E.B. Davies. {\it Spectral stability of the Neumann Laplacian,} J. Differential Equations {\bf 186} (2002), no. 2, 485--508. \bibitem[BI]{BI} M. Bronstein and V. Ivrii. {\it Sharp spectral asymptotics for operators with irregular coefficients. I. Pushing the limits,} Comm. Partial Differential Equations {\bf 28} (2003), no. 1-2, 83--102. \bibitem[BLe]{BLe} M. van den Berg and M. Levitin. {\it Functions of Weierstrass type and spectral asymptotics for iterated sets,} Quart. J. Math. Oxford Ser. {\bf 47} (1996), no. 188, 493--509. \bibitem[BLi]{BLi} M. van den Berg and M. Lianantonakis. {\it Asymptotics for the spectrum of the Dirichlet Laplacian on horn-shaped regions}. Indiana Univ. Math. J. 50 (2001), no. 1, 299--333. \bibitem[BS]{BS} M.S. Birman and M.Z. Solomyak. {\it The principal term of spectral asymptotics for ``non-smooth'' elliptic problems,} Funktsional. Anal. i Prilozhen. {\bf 4:4} (1970), 1--13 (Russian), English transl. in Functional Anal. Appl. {\bf 4} (1971). \bibitem[F]{F} B. Fedosov. {\it Asymptotic formulae for the eigenvalues of the Laplace operator in the case of a polygonal domain} (Russian), Dokl. Akad. Nauk SSSR {\bf 151} (1963), 786--789. \bibitem[G]{G} M. de Guzm\'an. {\it Differentiation of integrals in $\R^n\,$,} Springer Verlag, 1975 (Lecture Notes in Mathematics, v. 481). \bibitem[HSS]{HSS} R. Hempel, L. Seco and B. Simon. {\it The essential spectrum of Neumann Laplacians on some bounded singular domains,} J. Funct. Anal. {\bf 102} (1991), no. 2, 448--483. \bibitem[Iv1]{Iv1} V. Ivrii. {\it On the second term of the spectral asymptotics for the Laplace-Beltrami operator on manifolds with boundary,} Funktsional. Anal. i Prilozhen. {\bf 14} (1980), no. 2, 25--34. English transl. in Functional Anal. Appl. {\bf 14} (1980), 98--106. \bibitem[Iv2]{Iv2} \bysame. {\it The asymptotic Weyl formula for the Laplace-Beltrami operator in Riemannian polyhedra and domains with conical singularities of the boundary,} Dokl. Akad. Nauk SSSR 288 (1986), 35--38 (Russian). \bibitem[Iv3]{Iv3} \bysame. {\it Microlocal Analysis and Precise Spectral Asymptotics,} Springer-Verlag, SMM, 1998. \bibitem[Iv4]{Iv4} \bysame. {\it Sharp spectral asymptotics for operators with irregular coefficients. II. Domains with boundaries and degenerations,} Comm. Partial Differential Equations {\bf 28} (2003), no. 1-2, 103--128. \bibitem[LV]{LV} M. Levitin and D. Vassiliev. {\it Spectral asymptotics, renewal theorem, and the Berry conjecture for a class of fractals,} Proc. London Math. Soc. {\bf 72} (1996), 188--214. \bibitem[M1]{M1} V. Maz'ya. {\sl On Neumann's problem for domains with irregular boundaries,} Siberian Math. J. {\bf 9} (1968), 990--1012. \bibitem[M2]{M2} V. Maz'ya. {\sl Sobolev spaces,} Leningrad University, Leningrad, 1985. English translation in {\sl Springer Series in Soviet Mathematics,} Springer-Verlag, Berlin, 1985. \bibitem[Ma]{Ma} C. Mason. {\sl Log-Sobolev inequalities and regions with exterior exponential cusps,} J. Funct. Anal. 198 (2003), 341--360. \bibitem[Me]{Me} G. M\'etivier. {\it Valeurs propres de problèmes aux limites elliptiques irr\'eguli\`eres} (French), Bull. Soc. Math. France Suppl. M\'em. {\bf 51--52} (1977), 125--219. \bibitem[Mi]{Mi} Y. Miyazaki. {\it A sharp asymptotic remainder estimates for the eigenvalues of operators associated with strongly elliptic sesquilinear forms,} Japan. J. Math. {\bf 15} (1989), no. 1, 65--97. \bibitem[MV]{MV} S. Molchanov and B. Vainberg. {\it On spectral asymptotics for domains with fractal boundaries of cabbage type,} Math. Phys. Anal. Geom. {\bf 1} (1998), no. 2, 145--170. \bibitem[RS-N]{RS-N} F. Riesz and B. Sz.-Nagy. {\it Le\c{c}ons d'analyse fonctionnelle,} (French) Académie des Sciences de Hongrie, Akadémiai Kiadó, Budapest, 1952. English translation: {\it Functional analysis,} Dover Publications Inc., New York, 1990. \bibitem[Se]{Se} R. Seeley. {\it An estimate near the boundary for the spectral function of the Laplace operator,} Amer. J. Math. {\bf 102} (1980), no. 5, 869--902. \bibitem[St]{St} E. Stein. {\it Singular integrals and differentiability properties of functions,} Princeton University Press, Princeton, 1970. \bibitem[Sa]{Sa} Yu. Safarov. {\it Fourier Tauberian Theorems and applications,} J. Funct. Anal. {\bf 185} (2001), 111--128. \bibitem[Si]{Si} B. Simon. {\it The Neumann Laplacian of a jelly roll,} Proc. Amer. Math. Soc. {\bf 114} (1992), no. 3, 783--785. \bibitem[SV]{SV} Yu. Safarov and D. Vassiliev. {\it The asymptotic distribution of eigenvalues of partial differential operators,} American Mathematical Society, 1996. \bibitem[V]{V} D. Vassiliev. {\it Two-term asymptotic behavior of the spectrum of a boundary value problem in the case of a piecewise smooth boundary,} Dokl. Akad. Nauk SSSR {\bf 286} (1986), no. 5, 1043--1046. English translation: Soviet Math. Dokl. {\bf 33} (1986), no. 1, 227--230. \bibitem[W]{W} B.L. van der Waerden. {\it Ein einfaches Beispiel einer nichtdifferenzierbaren stetigen Funktion,} Math. Zeitschr. {\bf 32} (1930), 474--475. \bibitem[Z]{Z} L. Zielinski. {\it Asymptotic distribution of eigenvalues for elliptic boundary value problems,} Asymptot. Anal. {\bf 16} (1998), no. 3-4, 181--201. \end{thebibliography} \end{document} ---------------0310070814125--