text:
\section{Introduction}
\label{sec:1}

The main goal in this paper is to study the spectral properties of the
Neumann-Laplacian on the periodic singularly perturbed periodic quasi-cylinder
\(\Omega(\epsilon)\):
\begin{eqnarray}\label{form1}
-\Delta u^\epsilon(x)&=&\lambda^\epsilon u^\epsilon(x),\ x\in \Omega(\epsilon),\\
\label{form2}
\partial_n u^\epsilon(x)&=&0,\ x\in \partial\Omega(\epsilon).
\end{eqnarray}
This spectral problem should be interpreted as a singular perturbation of the
same problem in the straight cylinder \(\Omega=\omega\times{\mathbb{R}}\subset
{\mathbb{R}}^3\).

The quasi-cylinder \(\Omega(\epsilon)\) is a periodic set which depends on a
small parameter \(\epsilon>0\). It is obtained from the straight cylinder
\(\Omega\) by a periodic nucleation of small voids with diameter of order
\(\epsilon\) (see fig. 1a and fig 1b, respectively).
We remark that without lost of generality we have reduced the period to one by
rescaling.

\begin{figure}
\begin{center}
\resizebox{6cm}{4cm}{\includegraphics{VERYnewFEDOR1.eps}}
\caption{A cylinder and a quasi-cylinder with periodic family of voids}
\end{center}
\end{figure}

It is known (e.g. \cite{Wilcox}) that the spectrum \(\sigma\) of the
Neumann-Laplacian in the straight cylinder \(\Omega\) implies the continuous
spectrum \(\sigma_c\) which is the closed positive real semi-axis
\(\overline{\mathbb{R}_{+}}\), and the point spectrum is empty. In particular,
for any real \(\mu>0\) the functions, or \textit{oscillating waves},
\begin{equation}\label{wave1}
e^{\pm i\sqrt{\mu}z},
\end{equation}
give rise to a singular Weyl sequence of the problem operator \(A\) at
the point \(\mu\in \overline{\mathbb{R}_{+}}\) (cf. \S 9.1 in \cite{BiSo}).

The structure of the spectrum \(\sigma(\epsilon)\) in the perturbed quasi-cylinder
\(\Omega(\epsilon)\) is much more complicated. Owing to the Gel'fand transform
\cite{Gel} (see Section 2)
and the Floquet-Bloch theory (see e.g. \cite{Kuchbook, NaPl})
the spectrum \(\sigma(\epsilon)\) of the Neumann-Laplacian is
endowed with the band-gap structure. Namely, the essential spectrum
\(\sigma(\epsilon)=\sigma_e(\epsilon)\) is a union of closed connected segments
\(\Upsilon_n(\epsilon)\subset \overline{\mathbb{R}_{+}}\):
\begin{equation*}
\displaystyle\sigma(\epsilon)=\bigcup_{n\in \mathbb{N}}\Upsilon_n(\epsilon).
\end{equation*}
We obtain \textit{a spectral gap} in the spectrum if there exist
an interval in the positive real semi-axis which is free of the spectrum, see
the left part of
fig. 2. However, when the segments overlap each other (see the right part of
fig.2), no spectral gap opens.

Here we note that the essential spectrum \(\sigma_e(\epsilon)\) coincides with
the continuous spectrum provided none of the segments \(\Upsilon_n(\epsilon)\)
collapses to a single point, which becomes an eigenvalue of the operator
\(A(\epsilon)\) with infinite multiplicity, and belongs to the point spectrum
while the discrete spectrum is still empty. A negative answer to this
possibility is still an open question and the authors can
only prove the following: \textit{For any \(t>0\) there exists \(\epsilon_t>0\)
such that}
\(\sigma_p(\epsilon)\cap [0,t]=\emptyset\) when \(\epsilon\in (0,\epsilon_t]\).
We do not pay any attention to this incomplete result and therefore always speak
in this paper about the
essential spectrum.

\begin{figure}
\begin{center}\resizebox{8cm}{1cm}{\includegraphics{fedor2.eps}}\end{center}
\caption{The gap between the first and second bands,  while a gap between the second
and the fourth band is covered by the third one}
\end{figure}

In the literature the existence of spectral gaps is mainly investigated for
periodic media which are infinite in all directions while the coefficients of
differential operators, both in scalar and matrix case, are usually assumed
to be contrasting (see e.g. \cite{Fig,Fri,Gre,HemLin,Zhi} for scalar problems and \cite{Fil,na461} for systems).
Much less
results are obtained for periodic waveguides, which are infinite in one
direction only.  In this case spectral gaps ought to be opened by varying
the shape of the periodicity cell only which is in accordance with the results
in the known engineering practise.

There are two useful approaches to fulfil the goal. The first one utilises the
asymptotic analysis of the spectrum for the associated model problem in the
periodicity cell, like we do in the  present paper. This approach has been
realised for the Dirichlet-Laplacian in papers
\cite{Yoshitomi,FriSol,NaMaNo,CaNaPe} and others. We emphasise that our results
for the Neumann-Laplacian are completely new and require different techniques.

The second approach is based on the application of sharp parameter-dependent
estimates provided by Korn's inequalities, usually weighted and anisotropic,
see e.g. \cite{na435,na433,na454}  and others.
On one hand, this approach does not require a precise description of the dependence
of the cell
shape perturbation on the small geometric parameter, but, in contrast to the first
approach, one can only prove the existence of one or several spectral
gaps without any information on the position and width of the gaps.

Combining of the above-mentioned approaches one could get much more
elaborated results by using their advantages. However, we cannot examine here
the whole spectrum and to our knowledge there does not exist any results of this
type. Here indeed we discuss only the first spectral gap.

To prove or to disprove the existence of a gap between the segments
\(\Upsilon_1(\epsilon)\) and \(\Upsilon_2(\epsilon)\) (cf. fig. 2), we employ
the method of compound asymptotic expansions
\cite{MaNaPl} to the associated model spectral problems in the periodicity cell
\begin{equation}\label{cell}
\varpi_{\epsilon}=\varpi\setminus \overline{\theta_\epsilon^0},\quad
\varpi=\omega\times (0,1)
\end{equation}
of the quasi-cylinder (\ref{quasi}). The straight cylinder
with the small void \(\theta^0_\epsilon\) (see the two-dimensional dummy in fig.
3b) is nothing but a good example of a domain with the \textit{singularly
perturbed} boundary
\(\partial\varpi_{\epsilon}=\partial\varpi\cup\partial\theta_\epsilon^0\). The
asymptotic analysis of the eigenvalues for the Laplace operator have been
developed at a great extend in
\cite{Na61,Osava1,Gad,Na108,Na240,Na406,na416} and many others.
However, the dependence of the model
problem on the dual variable \(\eta\in[0,2\pi)\) of the Gel'fand transform
brings a serious complication to both, the formal asymptotic procedures and the
justification since, instead of a single spectral problem, we have to deal with
a family of spectral problems dependent on the real parameter \(\eta\in
[0,2\pi)\) (see Section 3 and Section 4 for more details).

\begin{figure}
\begin{center}\resizebox{6cm}{2.5cm}{\includegraphics{fedor3.eps}}\end{center}
\caption{The two-dimensional dummy of the periodicity cell}
\end{figure}

\section{Preliminaries and notations}
\label{seq:2}

\subsection{The quasi-cylinder and the Gel'fand transform}
\label{subseq:2-1}
The purpose of this section is to explain what do we mean by a
\textit{periodic quasi-cylinder}. There are several ways to describe it; but
here we start with the straight cylinder which is the cartesian product of a
bounded  Lipschitz domain \(\omega\subset\mathbb{R}^2\) and the real line:
\(\Omega=\omega\times\mathbb{R}\).

Consider now  a fixed bounded smooth domain \(\theta\) with the coordinate
origin \(\xi=0\) in its interior. We introduce the family of sets
\(\theta^j_{\epsilon}\) by defining
\[\theta^j_{\epsilon}=\{x=(y_1,y_2,z):\ (y_1,y_2,z-j)\in \theta^0_\epsilon\},
\theta^0_{\epsilon}=\{x: \xi=\epsilon^{-1}(x-x^0)\in \theta\},\]
where \(x^0\in\varpi=\omega\times (0,1)\).
Note that the diameter of the set \(\theta^j_{\epsilon}\) is of order
\(\epsilon\).

The quasi-cylinder \(\Omega(\epsilon)\) is a periodic set which depends
on a small parameter \(\epsilon>0\) and it will be obtained
from the straight cylinder \(\Omega\) excluding the sets
\(\overline{\theta^j_\epsilon}\)
from it (see fig. 1a and fig 1b, respectively).
In other words, the quasi-cylinder can be written as
\begin{equation}\label{quasi}
\Omega(\epsilon)=\Omega\setminus\bigcup_{j\in\mathbb{Z}}\overline{
\theta^j_\epsilon}.
\end{equation}

We call the set \(\varpi(\epsilon)=\varpi\setminus \theta^0_\epsilon\)
in \eqref{cell} as the
periodicity cell of the quasi-cylinder. Note that the cylinder \(\Omega\) can be
viewed as a quasi-cylinder with the straight periodicity cell \(\varpi\) as in (\ref{cell}).

We recall here the definition of the Gel'fand transform \cite{Gel}:
\begin{equation}
\label{Gelfand}
v(y,z)\mapsto\,V(y,z,\eta)=\frac{1}{\sqrt{2\pi}}\sum_{j\in \mathbb{Z}}
e^{-i\eta j}v(y,z+j),
\end{equation}
where \((y,z)\in \Omega(\epsilon)\) on the left, \(\eta\in[0,2\pi)\) and
\((y,z)\in\varpi(\epsilon)\) on the right. As it is well known this operator is
an isometric isomorphism from \(L^2(\Omega(\epsilon))\) onto
\(L^2(0,2\pi;L^2(\varpi(\epsilon)))\) and an isomorphism
between the Sobolev spaces
\(H^1(\Omega(\epsilon))\) and \(L^2(0,2\pi;H^1_\eta(\varpi(\epsilon)))\)
(see e.g. \cite{NaPl,Kuchbook}).
Here \(L^2(0,2\pi;L^2(\varpi(\epsilon)))\) consists of
\(L^2(\varpi(\epsilon))\)-valued (complex) \(L^2\)-functions on \([0,2\pi]\).
Similarly, the space \(L^2(0,2\pi;H^1_\eta(\varpi(\epsilon)))\) contains those
\(H^1_\eta(\omega(\epsilon))\)-valued (complex) \(L^2\)-functions on
\([0,2\pi]\). Finally we have denoted by \(H^1_\eta(\varpi(\epsilon))\) the space
of Sobolev functions \(f\) which are quasi-periodic in the \(z\)-variable,
i.e. such that function \((y,z)\mapsto e^{-i\eta z}f(y,z)\)
is 1-periodic in \(z\).

\subsection{The relationship between waves and Floquet waves in the straight
cylinder}
\label{subseq:2-2}

Before going into the asymptotic analysis of the spectral problem in the
periodic quasi-cylinder we will study the relationship of the standard wave
solutions and Floquet waves for the Neumann-Laplacian in the cylinder
\(\Omega\). They are the solutions of the differential equation
\begin{equation}\label{form3}
-\Delta_x u(x)=\lambda u(x),\ x\in\Omega\,,
\end{equation}
supplied with the Neumann conditions on the boundary
\begin{equation}\label{form4}
\partial_n u(x)=0,\ x\in\partial\Omega\,.
\end{equation}

Applying the Gel'fand transform to (\ref{form3}) and (\ref{form4}) we obtain
the model problem in the periodicity cell \(\varpi\) with the dual Gel'fand
parameter  \(\eta\in[0,2\pi)\), namely, the differential equation
\begin{equation}\label{form5}
-\Delta_x U(x,\eta)=\Lambda(\eta)U(x,\eta),\
(y,z)\in\varpi,
\end{equation}
the Neumann condition on the lateral boundary of \(\gamma=\partial\varpi\)
\begin{equation}\label{form6}
\partial_n U(x;\eta)=0,\ x\in \gamma,
\end{equation}
and the quasi-periodicity conditions on the opposite ends of the cylindrical cell
\(\varpi=\omega\times (0,1)\):
\begin{equation}\label{form7}
U(y,0)=e^{-i\eta}U(y,1),\ \partial_z U(y,0)=e^{-i\eta}\partial_z U(y,1),\
y\in\omega.
\end{equation}

Let \(\Lambda_p\) be an eigenvalue and let \(U_p\) be the
corresponding eigenfunction of the model spectral problem
\eqref{form4}-\eqref{form7},
then the Floquet wave can be written as
\begin{equation}
\label{Floq}
\Omega\ni (y,z)\mapsto e^{i\eta [z]}U_p(y,z-[z])
\end{equation}
where \([z]\in\mathbb{Z}\)
stands for the entire part of the real number \(z\). Notice that function
\eqref{Floq}
is smooth due to the quasi-periodicity conditions \eqref{form7} and it becomes a
solution of the homogeneous problem (\ref{form3}) with the spectral
parameter \(\lambda=\Lambda_p\).

Any function of the type (\ref{Floq}) must be obtained from the
standard waves
\begin{equation}
\label{wave}
(y,z)\mapsto e^{i\zeta z}V_q(y)
\end{equation}
in the cylinder \(\Omega\), where \(\zeta\in \mathbb{R}\) is the dual Fourier
variable and \(V_q\) is an eigenfunction of the model problem
\begin{equation}
\label{model-problem}
-\Delta_y V(y)=MV(y), \quad y\in\omega\,,
\quad \partial_nV(y)=0, y\in \partial\omega
\end{equation}
on the cross-section \(\omega\). Notice that the
wave solution (\ref{wave}) with \(q=1\) and \(V_1(y)=const\) corresponds to the
eigenvalue \(M_1=0\) and coincides with the wave solution (\ref{wave1}) where
\begin{equation*}
\mu=\zeta^2
\end{equation*}
stands for the parameter in the Helmholtz operator \(\Delta_x+\mu\). The main
difference between formulae (\ref{Floq}) and (\ref{wave}) follows from the
role played by the parameters \(\eta\in[0,2\pi)\) and \(\zeta\in\mathbb{R}\).

We write the wave solution \eqref{wave} as follows:
\begin{equation}
\label{relation1}
e^{i\zeta z}V_q(y)
=e^{i(\zeta-2 \pi m )[z]}
e^{i(\zeta-2 \pi m)(z-[z])}e^{2\pi m iz}V_q(y)
\end{equation}
where \(m=[(2\pi)^{-1}\zeta]\) is the maximal integer such that \(2\pi m\leq
\zeta\).
Now we denote the first multiplier on the right of \eqref{relation1} by
\(e^{i\eta[z]}\) and the remaining part by \(U_p(y,z-[z])\) so that
\begin{eqnarray}
\label{rel2}
U_p(y,z)&=&e^{i\eta z}e^{2\pi miz}V_q(y),\\
\eta&=&\zeta-2\pi m\in [0,2\pi). \nonumber
\end{eqnarray}
Since \(e^{2\pi mi z}\) is 1-periodic in \(z\), we see that the function
\(U_p\)
satisfies the quasi-periodicity conditions \eqref{form7}. In other words,
\eqref{relation1}
implies the Floquet representation \eqref{Floq} of the wave solution
\eqref{wave}
with the ingredients \eqref{rel2}.

Relationship (\ref{rel2}) helps us to list all eigenpairs
\(\{\Lambda_p,U_p\}\) of the model problem in the straight cell \(\varpi\) in
terms of the eigenvalues
\begin{equation}
\label{Meig}
0=M_1<M_2\leq M_3\leq\ldots
\end{equation}
and eigenfunctions \(\{V_1,\,V_2,\,V_3,\dots\}\) of the model problem
\eqref{model-problem}.
Namely, there holds the relations
\begin{equation}
\label{eigrel}
\Lambda_{q,m}(\eta)=M_q+(\eta+2\pi m)^2,\quad q\in{\mathbb{N}},\, m\in\mathbb{Z}.
\end{equation}
We emphasise that the eigenvalues \(\Lambda_{q,m}(\eta)\) are now enumerated
with two indexes so that it is necessary to reorder them to get the monotone increasing
unbounded sequence
\begin{equation}
\label{seq-Lambda}
\Lambda_1(\eta)\leq \Lambda_2(\eta)\leq\cdots\leq \Lambda_k(\eta)\leq \cdots.
\end{equation}

The lowest dispersion curve (\ref{eigrel}) in fig. 4a,
corresponding to the smallest eigenvalue \(M_1=0\) of the Neumann problem in
\(\omega\), divides into an infinite family of smooth finite curves which by
above formulae are joined to the lattice in fig 4b.  In the sequel we are
mainly interested in the couple of lowest curves in the lattice which are
redrawn in fig 5.a,
\begin{equation}
\label{dispcurve}
\Lambda^0_+(\eta):=\Lambda_{0,0}(\eta)=\eta^2,\
\Lambda^0_{-}(\eta):=\Lambda_{0,-1}(\eta)=(\eta-2\pi)^2 .
\end{equation}
We shall use the notation (\ref{dispcurve}) throughout the paper.

\begin{figure}
\begin{center}\resizebox{8cm}{3cm}{\includegraphics{fedor4.eps}}\end{center}
\caption{The  lowest dispersion curve and the lattice frame made from it}
\end{figure}

The variational formulation of problem (\ref{form5})-(\ref{form7}) reads as
\begin{equation}
\label{vari-form3}
(\nabla_xU,\nabla_x V)_{\varpi}=\Lambda(\eta) (U,V)_{\varpi}\,,
V\in H^1_{\eta}(\varpi)\,.
\end{equation}
Since for any real parameter \(\eta\) the bilinear form on the left-hand side
of (\ref{vari-form3}) is symmetric, closed and positive, we can associate with it
a self-adjoint positive operator in the
Lebesque space \(L^2(\varpi)\) \cite[Thm. 10.1.2]{BiSo}. Moreover, the compact
embedding \(H^1(\varpi)\subset L^2(\varpi)\) and Theorems 10.1.5 and 10.2.2 in
\cite{BiSo} ensure that indeed the spectrum of this operator is discrete and
forms
an unbounded non-negative sequence (\ref{seq-Lambda}) which is constructed from
(\ref{Meig}).

\subsection{Perturbation of the dispersion curves}
\label{subseq:14}

In the perturbed periodicity cell \(\varpi_{\epsilon}\) the both dispersion
curves (\ref{dispcurve}) experience a perturbation and, in principle, there can
occur two different situations pictured in fig. 5.b. and 5.c. First, the
dispersion curves \([0,2\pi) \ni \eta\to \Lambda_{\pm}^\epsilon(\eta)\) may
still intersect so that the spectral segments \(\Upsilon_1(\epsilon)\) and
\(\Upsilon_2(\epsilon)\) touch each other and the gap between them is closed.
The second possibility is that the graphs  of the functions
\begin{equation*}
\eta\mapsto\Lambda^\epsilon_{1}(\eta),\,
\eta\mapsto \Lambda^\epsilon_{2}(\eta)
\end{equation*}
do not intersect and the area overshadowed in fig. 5c projects into a gap
between segments \(\Upsilon_1(\epsilon)\) and \(\Upsilon_2(\epsilon)\). In the
sequel we will describe when the gap appears and indicate asymptotically its
position and width.

\begin{figure}
\begin{center}\resizebox{8cm}{3cm}{\includegraphics{fedor5.eps}}\end{center}
\caption{Unperturbed (a), intersecting (b) and disjoint (c) curves}
\end{figure}

Before proceeding with the problem statement, the formal asymptotic analysis and
the justification, we note that the gap opens only under the condition
\begin{equation}
\label{cond1}
M_1+\pi^2<M_2.
\end{equation}
Otherwise, the segment \(\Upsilon_2(\epsilon)\), which corresponds to the
perturbed eigenvalues in (\ref{eigrel}) with \(q=2\) and \(m=0,-1\) covers the
gap opened between the segments \(\Upsilon_1(\epsilon)\) and
\(\Upsilon_3(\epsilon)\) (see fig.6) which are now generated by eigenvalues in
(\ref{eigrel}) with \(q=1\) and \(m=0,-1\).

\begin{figure}
\begin{center}\resizebox{7cm}{3cm}{\includegraphics{fedor6.eps}}\end{center}
\caption{Overlapping bands. The dotted lines correspond to the perturbation of
the lowest curves in fig. 5.a.}
\end{figure}

However, the rescaling performed in the very beginning rescales
also eigenvalues. To see this, assume that the period \(l\)
of the void nucleation is bigger than
one. Now the rescaling diminishes the size of the cross-section, i.e. the change
of the variables
\((y,z)\to(y',z')=(l^{-1}y,l^{-1}z)\)
yields the shrunken cross-section
\[\omega'=\{y'\in \mathbb{R}^2:\ ly'\in\omega\}.\]
Simultaneously, eigenvalues of the model problem in \(\omega\) changes into
the eigenvalues \(M_1'=0\) and \(M_2'=l^2M_2\) of the model problem in the
shrunken cross-section \(\omega'\). By choosing \(l\) large enough we fulfil
the condition (\ref{cond1}) for \(M_1'\). In other words, posing the small voids
\(\theta_{\epsilon}^{0}\) with a sufficiently large period may open
a gap in the essential spectrum, where as if the period of the voids is
sufficiently small first triple of the spectral bands surely overlap.

Assuming condition (\ref{cond1}) we study the spectral problem
\eqref{form1}-\eqref{form2}. Applying the Gel'fand transform
(\ref{Gelfand}) to problem (\ref{form1})-(\ref{form2}),
we obtain the model spectral problem in the periodicity cell
\(\varpi_{\epsilon}\) depending on \(\eta\in [0,2\pi)\):
\begin{eqnarray}\label{spctrlprob1}
-\Delta U^\epsilon(y,z)&=&\Lambda^\epsilon(\eta)U^\epsilon(y,z),\
(y,z)\in\varpi_{\epsilon}\,,\\
\label{bndrycond}
\partial_n U^\epsilon(x)&=&0,\ x\in
\gamma\cup\partial\theta(\epsilon)\,,\\
\label{percond}
U^\epsilon(y,0)=e^{-i\eta}U^\epsilon(y,1),&&\partial_z
U^\epsilon(y,0)=e^{-i\eta}\partial_z
U^\epsilon(y,1),\
y\in\gamma_0.
\end{eqnarray}

As above, using Theorem 10.1.2 \cite{BiSo} we associate with these spectral
problems a family of unbounded positive self-adjoint operators
\(T^\epsilon(\eta)\) on \(L^2(\varpi_{\epsilon})\) with discrete spectra, i.e.,
the eigenvalue sequences
\[0\leq \Lambda^\epsilon_{1}(\eta)\leq\Lambda^\epsilon_{2}(\eta)\leq \cdots\leq
\Lambda^\epsilon_{k}(\eta)\leq \cdots\rightarrow\infty.\]
The immediate objective becomes to construct asymptotics for these
eigenvalues as \(\epsilon\to+0\).

\subsection{Mixed boundary value problem}

Although all the calculations are presented for the
Neumann conditions on the boundary of the quasi-cylinder \(\Omega(\epsilon)\), we
also discuss slight modifications, which are needed to adapt our
calculations to the mixed boundary value problem
\begin{eqnarray}\label{dir-1}
-\Delta u^\epsilon(x)&=&\lambda^\epsilon u^\epsilon(x),\
x\in \Omega(\epsilon)\,,\nonumber\\
\partial_n u^\epsilon(x)&=&0,\ x\in
\partial\Omega(\epsilon)\setminus \partial \Omega=\cup_{j\in\mathbb{Z}}\partial
\theta^j_\epsilon\,,\\
u^\epsilon(x)&=&0, x\in \partial \Omega\,.\nonumber
\end{eqnarray}
We emphasise that the
asymptotic procedure of \cite{MaNaPl} employed here is not sensitive to the type
of boundary conditions on the lateral part of the boundary
\(\partial\omega\times (0,1)\) of the periodicity cell is employed.
However, the change of the boundary condition from
Neumann
to Dirichlet on the surfaces of small voids crucially simplifies the
asymptotic analysis\footnote[1]{This does not hold in the two-dimensional
case, where the existence of a gap is still fully an open question.}. In this
way our paper becomes much more technical than \cite{NaMaNo} and \cite{CaNaPe},
where the Dirichlet problem was considered in the regularly and singularly
perturbed periodicity cells. The particular conclusions on the geometrical
characteristics of the gap are also different.

\section{The formal asymptotic procedure}
\label{sec: 3}

In this section we present a calculation which furnishes an asymptotic
formula for the first two eigenvalues (\ref{dispcurve}).
In order to simplify the presentation we start with considering the case when the eigenvalues for
the model problem are simple, e.g. when the Gel'fand dual variable \(\eta\) is
not equal to \(\pi\). Notice that the dispersion curves in fig. 5a
intersect each other just at \(\eta=\pi\), and we examine this situation
in the end of this section.

In Sections 3.3 and 3.4 we give detailed calculations of the
boundary layers and a regular terms which lead to a direct formula for the main
correction term in the asymptotics of the eigenvalues.
With this asymptotic formula we are then able
to consider also the case \(\eta\approx \pi\) although at \(\eta=\pi\) where the model
problem has a multiple eigenvalue. As a result, in Section 3.5 we derive a
system of two linear algebraic equations from which we can find out the
perturbation terms in the asymptotics of the eigenvalues when
\(\eta\) approximately equal to \(\pi\).

To make the necessary conclusion for the case \(\eta\approx \pi\), we will apply
the trick proposed in \cite{NaMaNo} and introduce the additional deviation
parameter \(\psi\) in the representation
\begin{equation}
\label{psi}
\eta=\pi+\psi\epsilon^3\,.
\end{equation}
With the help of this new variable \(\psi\in[-\psi_0,\psi_0]\) we can
write the asymptotic formulae at the \(c\epsilon^3\)-neighbourhood of the point
\(\eta=\pi\).

We emphasise here the following difference in the notation: In Sections
(\ref{subsec: 31})-(\ref{subsec:34}) we set
\begin{eqnarray}
 \label{eta1}
\Lambda_+^{\epsilon}(\eta)&=&\Lambda_1^{\epsilon}(\eta),\,\Lambda_{-}^{\epsilon}(\eta)=\Lambda_2^{\epsilon}(\eta),\,
\eta\in [0,\pi),\\
\Lambda_+^{\epsilon}(\eta)&=&\Lambda_2^{\epsilon}(\eta),\,\Lambda_{-}^{\epsilon}(\eta)=\Lambda_1^{\epsilon}(\eta),\,
\eta\in (\pi,2\pi),\nonumber
\end{eqnarray}
while in Section (\ref{subsec:35})
\begin{equation}\label{eta2}
\Lambda_+^{\epsilon}(\eta)=\Lambda_2^{\epsilon}(\eta),\,\Lambda_{-}^{\epsilon}(\eta)=\Lambda_1^{\epsilon}(\eta),\,
\eta=\pi.
\end{equation}
The evident difference in the notations (\ref{eta1}) and (\ref{eta2}) helps to keep similar
asymptotic formulae in both cases.

Calculations in Sections \ref{subsec: 31}-\ref{subsec:34} are given for the Neumann problem
(\ref{form1})-(\ref{form2}) in such a way that they can easily
adopted for the mixed boundary value problem (\ref{dir-1}).
Namely, the eigenpair of the problem (\ref{model-problem}) \(\{M_1=0\), \(V_1=const\}\)  must be replaced
by the first eigenpair \(\{M_1, V_1\}\) of the Dirichlet problem
\begin{equation}
\label{S4}
-\Delta_y V(y)=M V(y),\, y\in \omega,\qquad V(y)=0, \, y\in\partial \omega\,.
\end{equation}
Notice that the eigenvalue \(M_1\) is positive and simple while the corresponding eigenfunction \(V_1\)
can be chosen positive inside the cross-section \(\omega\).

\subsection{Asymptotic expansions}
\label{subsec: 31}

To describe the asymptotic behaviour of two first eigenvalues
\(\Lambda^\epsilon_{\pm}(\eta)\),
we accept the following \textit{ansatz}:
\begin{equation}\label{ansatz1}
\Lambda^\epsilon_\pm(\eta)=\Lambda^0_\pm(\eta)+\epsilon^3\Lambda'_\pm(\eta)+
\widetilde{\Lambda}^\epsilon_\pm(\eta),
\end{equation}
where \(\Lambda^0_\pm(\eta)\) is the eigenvalue (\ref{dispcurve}) of problem
(\ref{form3})-(\ref{form5}),
\(\Lambda'_\pm(\eta)\) is a correction term and \(\widetilde{\Lambda}
^\epsilon_\pm(\eta)\) a small remainder to be evaluated and estimated.

According to general asymptotic method in domains with singular
perturbations of the boundaries (see e.g. \cite{MaNaPl}) the
corresponding asymptotic ansatz for eigenfunctions reads as follows
\begin{eqnarray}
\label{ansatz2}
U^{\epsilon}_\pm(x;\eta)&=&U^0_\pm(x;\eta)+\epsilon
\chi(x)W^1_\pm(\epsilon^{-1}(x-x^0);\eta)\nonumber\\
&+& \epsilon^2\chi(x)W^2_\pm(\epsilon^{-1}(x-x^0);\eta)
+ \epsilon^3U'_\pm(x;\eta)+\widetilde{U}^\epsilon_\pm(x;\eta).
\end{eqnarray}
The functions \(W^j_\pm\) are of boundary layer type and \(\chi\) is a smooth
cut-off function which equals one in a neighbourhood of the point \(x^0\) and
vanishes in the vicinity of the boundary \(\partial\varpi\). The functions
\(U^0_\pm\) are the first two eigenfunctions of the unperturbed problem (cf. formula (\ref{rel2})), namely
\begin{equation}
\label{eigen2}
U^0_{+}(x;\eta)=e^{i\eta z}V_1(y), \quad
U^0_{-}(x;\eta)=e^{i(\eta-2\pi) z}V_1(y),
\end{equation}
where \(V_1\) is the first
eigenfunction of the Laplace operator on the cross-section, normalised
in \(L^2(\varpi)\). Clearly, both the functions \eqref{eigen2}
meets the quasi-periodicity conditions \eqref{form7}.
Note that in the case of the Neumann boundary conditions on
\(\partial \Omega\) the function \(V_1\) is just a constant. One readily
verifies that since \(M_1=0\) is the first eigenvalue of the Laplacian on the
cross-section in the Neumann case, the first couple of eigenvalues of the
unperturbed problem take the form (\ref{dispcurve}).

\subsection{The boundary layers}
\label{subsec: 32}

The boundary layer \(W^1_\pm\) depends on the  ``fast''
variables (``stretched`` coordinates)
\[\xi=(\xi_1,\xi_2,\xi_3)=\epsilon^{-1}(x-x^0)\]
and is needed to compensate for that the main regular term \(U^0_\pm\)
in the expansion does not fulfil the Neumann type boundary condition
\[\partial_{n(x)}u(x)=0,\, x\in \partial\theta_\epsilon.\]
By the Taylor formula applied to the function \(U^0_{\pm}\) in the ''slow``
variables \(x\), we have
\[\partial_{n(\xi)} U^0_{\pm}(x;\eta)=
n(\xi)\cdot \nabla_xU^0_{\pm}(x^0)+O(\epsilon),\,x\in\partial \theta_\epsilon.\]
Thus \(W^1_\pm\) ought to be a solution of the exterior Neumann problem
\begin{eqnarray}\label{form10}
\Delta_\xi W^1_\pm(\xi)&=&0,\ \xi\in\mathbb{R}^3\setminus \bar{\theta}\,,\\
\partial_{n(\xi)} W^1_\pm(\xi)&:=&G^1_{\pm}(\xi)=-n(\xi)\cdot
\nabla_xU^0_{\pm}(x^0;\eta),\
\xi\in\partial\theta.\nonumber
\end{eqnarray}
Since \(\int_{\partial \theta}n_k(\xi)\,ds(\xi)=0\) for \(k=1,2,3,\)
the function \(G^1_{\pm}\) satisfies the orthogonality condition
\[\int_{\partial\theta}G^1_{\pm}(\xi)\,ds_\xi=0.\]
Thus, the problem (\ref{form10}) admits a solution which decays faster at
infinity than the fundamental solution \(\Phi(\xi)=(4\pi|\xi|)^{-1}\)
of the Laplace operator in \({\mathbb R}^3\). Moreover, the solution has the
asymptotic behaviour
\begin{equation}\label{form11}
W^1_\pm(\xi)= \sum_{j,k=1}^{3}Q_{jk}\frac{\partial\Phi}{\partial
\xi_k}(\xi) \frac{\partial U^0_{\pm}}{\partial x_j}(x^0;\eta)+O(|\xi|^{-3})
\end{equation}
where \(Q=(Q_{jk})_{j,k=1,2,3}\) is the matrix of the
virtual mass for the set \(\overline{\theta}\subset\mathbb{R}^3\),
which is symmetric and negatively definite for any subset \(\theta\) with a positive volume
(see \cite[Appendix G]{PolyaSzego}).

\subparagraph{Remark 3.1} As shown in \cite[Appendix G]{PolyaSzego} (see also \cite{MaNaPl}) the representation
\begin{equation}
 \label{QQ}
Q=-\mathbb{I}_3\textrm{meas}_3(\theta)+\widetilde Q
\end{equation}
is valid, where \(\mathbb{I}_3\) is the \(3\times 3\) identity matrix and \(\widetilde Q\) is non-positive.
The latter is still negative definite in the case \(\textrm{meas}_3(\theta)\neq 0\);
but degenerates for a straight crack.

In order to find the appropriate boundary value problem for
the second boundary layer term \(W^2_\pm\) we take into account a quadratic
polynomial in the Taylor formula for \(U_\pm^0\):
\[
\partial_a U^0_\pm(x;\eta)=\sum_{j=1}^3 a_j\frac{\partial U^0_\pm}{\partial
x_j}(x^0;\eta)+
\frac{1}{2}\sum_{j,k=1}^3\partial_a(x_jx_k)
\frac{\partial^2U^0_\pm}{\partial x_j\partial x_k}(x^0;\eta)+O(|x-x^0|^2).
\]

In this way, we conclude that the
second boundary layer term \(W^2_\pm\) should satisfy the following
exterior Neumann problem:
\begin{eqnarray*}\label{form12}
-\Delta_\xi W^2_\pm(\xi)&=& 0,\
\xi\in \Xi=
\mathbb{R}^3\setminus \overline{\theta},\\
\partial_{n(\xi)} W^2_\pm(\xi)&=&-\frac12\sum_{j,k=1}^{3}
\frac{\partial(\xi_j\xi_k)}{\partial n(\xi)}
\frac{\partial^2U^0_\pm}{\partial x_j\partial x_k}(x^0;\eta),\
\xi\in\partial\theta.\nonumber
\end{eqnarray*}
The function \(W^2_\pm\) admits the representation
\begin{equation}
\label{number}
W^2_\pm(\xi)= c_{\pm}\rho^{-1}+O(\rho^{-2}), \quad \rho=|\xi|\to \infty\,.
\end{equation}
To calculate the coefficient \(c_\pm\) in (\ref{number}) we use the
Green formula in the domain \(\Xi_R=B_R(0)\setminus \overline{\theta}\)
where \(B_R(0)\) is a ball of radius \(R\) centred at \(\xi=0\) to
obtain
\begin{eqnarray}
\label{calc-c-1}
0=\int_{\Xi_R}\Delta_\xi W^2_\pm(\xi)\,d\xi
&=& \int_{\partial \Xi_R}\partial_{n(\xi)}W^2_{\pm}(\xi)ds_\xi\nonumber\\
&=&\int_{\mathbb{S}_R}\frac{\partial W^2_{\pm}}{\partial \rho}(\xi)ds_\xi
+\int_{\partial\theta}\partial_{n(\xi)}W^2_{\pm}
(\xi)ds_\xi\nonumber\\
&=&\int_{\mathbb{S}_R}\frac{\partial W^2_\pm}{\partial\rho}(\xi)\, ds_\xi\\
&-&\frac12\sum_{j,k=1}^3\frac{\partial^2 U^0_\pm}{\partial x_j\partial x_k}(x^0)
\int_{\partial\theta}
\frac{\partial(\xi_j\xi_k)}{\partial n(\xi)}\,ds_\xi \nonumber\,.
\end{eqnarray}
Notice that on the sphere \(\mathbb{S}_R=\partial B_R(0)\) we have
\(\partial_{n(\xi)}=\partial_\rho\).

Applying the asymptotic expansion (\ref{number}), we see that in
(\ref{calc-c-1})
\begin{equation*}
\label{4pic}
\lim_{R\to+\infty}\int_{\mathbb{S}_R}\frac{\partial
W^2_\pm}{\partial\rho}(\xi)\, ds_\xi
= \lim_{R\to+\infty}\int_{\mathbb{S}_R}
\left(-\frac{c_\pm}{\rho^2}+O(\rho^{-3})\right)\,ds_\xi
= -4\pi c_{\pm}\,.
\end{equation*}
Using the Green formula again but now inside \(\theta\), note that the normal
becomes inward then, we derive the relation
\begin{eqnarray*}
\label{I_3}
&-&\frac{1}{2}\sum_{j,k=1}^3\frac{\partial^2 U^0_\pm}{\partial x_j\partial
x_k}(x^0)\int_{\partial\theta}
\partial_{n(\xi)}(\xi_j\xi_k)\,ds_\xi\nonumber\\
&=&\frac{1}{2}\sum_{j,k=1}^3\frac{\partial^2 U^0_\pm}{\partial x_j\partial
x_k}(x^0)\int_{\theta}
\Delta(\xi_j\xi_k)\,d\xi
=\sum_{j=1}^3\frac{\partial^2 U^0_\pm}{\partial x_j^2}(x^0)\textrm{meas}_3(\theta)\\
&=&\Delta U^0_\pm(x^0)\textrm{meas}_3(\theta)=-
\Lambda_\pm^0(\eta) U^0_\pm(x^0)\textrm{meas}_3(\theta)\,.\nonumber
\end{eqnarray*}
In the last equality we also have used the fact that \(U^0_\pm\) is
the eigenfunction of the unperturbed problem corresponding to the eigenvalue
\(\Lambda^0_{\pm}(\eta)\) ( see (\ref{form5})).

Collecting the above results we finally
obtain the formula
\begin{equation}
\label{calc-c-4}
4\pi c_\pm=-\Lambda^0_{\pm}(\eta) U^0_\pm(x^0)\textrm{meas}_3\theta.
\end{equation}

\subsection{The regular correction}
\label{subsec:33}

For the calculation of the main correction term \(\Lambda_{\pm}'(\eta)\)
in the asymptotic expansion of \(\Lambda^\epsilon_\pm(\eta)\) we substitute the
asymptotic ans\"atze (\ref{ansatz1}) and (\ref{ansatz2}) into the spectral
problem (\ref{spctrlprob1})-(\ref{bndrycond}). By (\ref{form11}), (\ref{number}) and (\ref{calc-c-4})
the boundary layer terms \(W^1_\pm(\epsilon^{-1}(x-x^0);\eta)\) and
\(W^2_\pm(\epsilon^{-1}(x-x^0);\eta)\) have the following behaviour:
\begin{eqnarray}
\label{W1x}
W^1_\pm\left(\frac{1}{\epsilon}(x-x^0);\eta\right)&=&
\epsilon^2\sum_{j,k=1}^{3} Q_{jk} \frac{\partial \Phi}{\partial x_k}(x-x^0)\frac{\partial U^0_\pm}{\partial
x_j}(x^0;\eta)\\
&+&O(\epsilon^3|x-x^0|^{-3}),\nonumber
\end{eqnarray}
\begin{equation}
\label{W2x}
W^2_\pm(\epsilon^{-1}(x-x^0);\eta)=\epsilon c_\pm |x-x^0|^{-1}
+O(\epsilon^2|x-x^0|^{-2})\,.
\end{equation}
Hence we notice that in (\ref{ansatz2}) the contribution of the boundary layer
terms written in the ''slow`` variables looks as follows:
\begin{equation}
\label{eps4}
\epsilon^1W^1_\pm(\epsilon^{-1}(x-x^0);\eta)+\epsilon^2W^2_\pm(\epsilon^{-1}
(x-x^0);\eta)=\epsilon^3w_\pm(x;\eta)+O(\epsilon^4|x-x^0|^{-3})\,.
\end{equation}
In other words, the main regular correction is of order \(\epsilon^3\).
We write the term \(w_\pm(x)\) as a sum
\begin{equation}\label{www1}w_\pm(x)=w^1_\pm(x)+w^2_\pm(x),\end{equation}
where
\begin{eqnarray}
\label{w_1}
w^1_\pm(x)=
-\sum_{j,k=1}^3Q_{jk}\frac{\partial U^0_{\pm}}{\partial
x_j}(x^0)\frac{x_k-x_k^0}{4\pi|x-x^0|^3},
\,
w^2_\pm(x)=\frac{c_{\pm}}{|x-x^0|}.
\end{eqnarray}

Now substituting the asymptotic ansatz (\ref{ansatz2}) for \(U^\epsilon_{\pm}\)
in the equation (\ref{spctrlprob1}) we get,
using that \(U^0_\pm\) is the solution of unperturbed problem,
\begin{eqnarray*}
-\Delta_x U^\epsilon_{\pm}(x;\eta)&=&
\Lambda^0_{\pm}(\eta)U^0_{\pm}(x;\eta)-
\epsilon^3\Delta_x(\chi(x)w_\pm(x;\eta)+U'_{\pm}(x;\eta))\\
&&+O(\epsilon^4|x-x^0|^{-3}).
\end{eqnarray*}
On the other hand, since \(\Lambda^\epsilon_\pm(\eta)\) is assumed to be an
eigenvalue, we also have the relationship
\begin{eqnarray*}
-\Delta_x U^\epsilon_{\pm}(x;\eta)&=&
(\Lambda^0_\pm(\eta)+\epsilon^3\Lambda'_\pm(\eta))
(U^0_\pm(x;\eta)+\epsilon^3(\chi(x)w_\pm(x)+U'_{\pm}(x;\eta)))\\
&&+O(\epsilon^4|x-x^0|).
\end{eqnarray*}
Thus, setting the coefficients on \(\epsilon^3\) equal, we obtain the
following equation:
\begin{eqnarray}
\label{u'}
-\left(\Delta_x+\Lambda^0_{\pm}(\eta)\right)U'_\pm(x;\eta)
=\Lambda'_\pm(\eta)U^0_\pm(x;\eta)\\
+\left(\Delta_x+\Lambda^0_{\pm}(\eta)\right)
\left(\chi(x)w_\pm(x;\eta)\right), \quad x\in \varpi \nonumber
\end{eqnarray}
with the boundary and quasi-periodicity conditions
\begin{eqnarray}
\label{u'_bound}
\partial_n U'_{\pm}(x;\eta)&=&0,\, x\in \gamma,\\
\label{u'_bound_per}
U'_{\pm} (y,0; \eta)&=&e^{-i\eta}U'_{\pm} (y,1;\eta),\, y\in\gamma_0,\\
\partial_z U'_{\pm}(y,0;\eta)&=&e^{-i\eta}\partial_z U'_{\pm} (y,1; \eta),\, y\in\gamma_0.\nonumber
\end{eqnarray}

By the Fredholm alternative the boundary
value problem (\ref{u'})-(\ref{u'_bound_per}) has a solution \(U'_\pm\)
if and only if the right hand side
of the equation (\ref{u'}) is orthogonal to the eigenfunction \(U^0_\pm\)
in \(L^2(\varpi)\). The normalisation \(\left\|U^0_\pm;L^2(\varpi)\right\|=1\)
then yields the expression for the correction term
\begin{eqnarray}
\label{l'}
\Lambda'_\pm(\eta)&=&\Lambda'_\pm(\eta)\int_{\varpi}
\overline{U^0_\pm(x;\eta)}U^0_\pm(x;\eta)\,dx\\
&=&-\int_\varpi \overline{U^0_\pm(x;\eta)}
\left(\Delta_x+\Lambda^0_\pm(\eta)\right)
\left(\chi(x)w_\pm(x;\eta)\right)\,dx\nonumber\,.
\end{eqnarray}
Recalling that the functions in (\ref{w_1}) are harmonic, we have
\[
 (\Delta_x+\Lambda^0_\pm(\eta))(\chi(x)w_\pm(x;\eta))=O(|x-x^0|^{-2}).
\]
Hence the last integral in (\ref{l'}) converges absolutely, and therefore it can be
calculated as a limit of an integral over the set \(\varpi\setminus
B_r(x^0)\) as \(r\to +0\). Since the cut-off
function \(\chi\) vanishes at the boundary of \(\varpi\) we obtain by the Green
formula that
\begin{equation}
\label{L}
\Lambda'_\pm(\eta)=
-\lim_{r\to +0}\int_{{\mathbb S}_r}\left(
\overline{U^0_\pm(x;\eta)}\partial_{n(x)}
w_\pm(x;\eta)-w_\pm(x;\eta)\overline{\partial_{n(x)} U^0_\pm(x;\eta) }\right)\,ds.
\end{equation}

Substituting \(w_\pm\) in (\ref{L}) we get
\[\Lambda'_\pm(\eta)=K_1+K_2+K_3+K_4\]
where
\begin{eqnarray*}
K_j&=&-\lim_{r\to
0}\int_{\mathbb{S}_r(x^0)}\overline{U^0_\pm(x;\eta)}\partial_{n(x)} w^j_\pm(x;\eta)
ds_x,\,j=1,2,\\
K_{j+2}&=&\lim_{r\to 0}\int_{\mathbb{S}_r(x^0)}w^j_\pm(x;\eta)
\overline{\partial_{n(x)} U^0_\pm(x;\eta)}ds_x,\,j=1,2.
\end{eqnarray*}

The term \(K_1\) is computed by a direct integration:
\begin{eqnarray*}
K_1&=&\sum_{j,k=1}^3Q_{jk}\frac{\partial U^0_{\pm}}{\partial x_j}(x^0;\eta)
\lim_{r\to +0}
\int_{{\mathbb S}_r(0)}
\overline{U^0_\pm(x^0+x;\eta)}
\partial_{n(x)}\frac{x_k}{4\pi|x|^3}\,ds_x\\
&=&\sum_{j,k=1}^3Q_{jk}\frac{\partial U^0_{\pm}}{\partial x_j}(x^0;\eta)
\lim_{r\to +0}
\int\limits_{{\mathbb S}_r(0)}
\overline{U^0_\pm(x^0+x;\eta)}
\left(\frac{2x_k}{4\pi r^4}\right) \,ds_x\\
&=&
\frac{2}{3}(\nabla_xU_\pm^0(x^0;\eta))^\top Q \nabla_x
\overline{U_\pm^0(x^0;\eta)}\,.
\end{eqnarray*}

Since
\(\partial_{n(x)} w^2_\pm(x;\eta)=-\partial_\rho \left(c_{\pm}\rho^{-1}\right)+O(\rho^{-3})\)
on \( {\mathbb S}_r(x^0)\) (see (\ref{number}) we have
\begin{multline*}
K_2=\lim_{r\to 0}\int_{{\mathbb S}_r(x^0)}
\overline{U^0_\pm(x;\eta)}\frac{\partial}{\partial\rho}
\left(\frac{c_{\pm}}{|x|}\right) \,d\varphi\\
=-\lim_{r\to 0}\int_{ {\mathbb S}_r(x^0)}
\overline{U^0_\pm(x;\eta)}c_{\pm}r^{-2}d\varphi
=-4\pi c_\pm \overline{U^0_\pm(x^0;\eta)}.
\end{multline*}

For the computation of \(K_3\) we write
\begin{equation*}
w^1_\pm(x)\overline{\partial_{n(x)} U^0_\pm(x;\eta)}
=
\sum_{j,k=1}^3Q_{jk}\frac{\partial U^0_\pm}{\partial x_j}(x^0;\eta)
\frac{x_k-x_k^0}{4\pi|x-x^0|^4} (x-x^0)\overline{\nabla_xU^0_\pm(x;\eta)}\,.
\end{equation*}
This yields us
\begin{eqnarray*}
K_3&=&
\sum_{j,k=1}^3Q_{jk}\frac{\partial U^0_\pm}{\partial x_j}(x^0;\eta)
\lim_{\rho\to 0}\int\limits_{ {\mathbb S}_\rho(0)}
\frac{x_k^2}{4\pi|x|^4}\overline{\frac{\partial U^0_\pm}{\partial
x_k}(x^0+x;\eta)}\,ds_x\\
&=&
\frac{1}{3}\nabla_xU^0_\pm(x^0;\eta)^\top Q
\overline{\nabla_xU^0_\pm(x^0;\eta)}\,.
\end{eqnarray*}
Finally, it is easy to see that
\begin{equation*}
K_4=-\lim_{\rho\to +0}
\int_{{\mathbb S}_\rho(x^0)}
\frac{c}{|x-x^0|}\overline{\partial_{n(x)} U^0_\pm(x;\eta)}\,ds=0\,.
\end{equation*}
Collecting the previous results we finally obtain the
formula for the main asymptotic correction term
\[\Lambda'_\pm(\eta)=F_{\pm}(\eta)\]
where
\begin{eqnarray*}
\label{maincorr}
F_{\pm}(\eta)&=&
\nabla_xU^0_\pm(x^0;\eta)^\top Q \overline{\nabla_xU^0_\pm(x^0;\eta)}
\\
&+&\left(M_1 +(\eta-\pi\pm\pi)^2\right)\left|U^0_\pm(x^0;\eta)\right|^2\textrm{meas}_3(\theta)\,.
\end{eqnarray*}

The remainder \(\widetilde{\Lambda}^\epsilon_\pm(\eta)\) in the asymptotic
expansion \eqref{ansatz1} will be estimated in \(\S 4\).
Note that \(F_+(\pi)=F_-(\pi)\) and denote it by \(F_0\), so that
\begin{equation}
\label{F_0}
F_0=\nabla_xU^0_\pm(x^0;\pi)^\top Q \overline{\nabla_xU^0_\pm(x^0;\pi)}
+\left(M_1 +\pi^2\right)\left|U^0_\pm(x^0;\pi)\right|^2\textrm{meas}_3(\theta).
\end{equation}

\subsection{Perturbation: \(\eta\approx\pi\)}
\label{subsec:34}
In this section we will study the behaviour of the eigenvalues
\(\Lambda^\epsilon_\pm(\eta)\) when \(\eta\) is close to \(\pi\).  For the unperturbed
problem, i.e. when \(\epsilon=0\), the eigenvalue
\(\Lambda_\pm^0(\pi)=M_1+\pi^2\)
has multiplicity two. Due to this change of the multiplicity our analysis
becomes slightly different. However, we may use the previous calculations.

In the vicinity of \(\eta=\pi\) we apply the formula \eqref{psi}.
In this case the asymptotic ans\"atze
(\ref{ansatz1}) and (\ref{ansatz2}) are still valid but their entries
depend on the deviation parameter \(\psi\). Moreover the main regular part
of the expansion becomes the linear combination of the eigenfunctions
\(U^0_{+}\) and \(U^0_{-}\):
\begin{eqnarray*}
\mathbf{U}^0_\pm(x;\pi,\psi)
&=&a_1^{\pm}(\psi)U^0_+(x;\pi)+a_2^\pm(\psi)U^0_-(x;\pi)\nonumber\\
&=&\left(a_1^\pm(\psi)e^{\pi i z}+
a_2^\pm(\psi)e^{-\pi i z}\right)V_1(y)
\end{eqnarray*}
where the coefficient vector
\(\mathbf{a}^\pm(\psi)=(a_1^\pm(\psi),a_2^\pm(\psi))^\top\)
is to be determined. Without loss of generality we may assume that
\(|\mathbf{a}^\pm|=1\).
By the same argument as in the previous section, we have
the boundary layers \(\mathbf{W}^1\)and \(\mathbf{W}^2\) as
in (\ref{W1x}) and (\ref{W2x}). However, in this case they
are the linear combinations
\begin{eqnarray}\label{WWW}
\mathbf{W}^j(\xi;\pi,\psi)&=&a^\pm_1(\psi)W^j_{+}(\xi;\pi)+a^\pm_2(\psi)W^j_{-}(\xi;\pi),\,j=1,2.
\end{eqnarray}
Then equation (\ref{u'}) takes the form
\begin{eqnarray}\label{u'psi}
(-\Delta_x&-&\Lambda^0(\pi))\mathbf{U}'_\pm(x;\pi,\psi)=
\Lambda'(\pi,\psi)\mathbf{U}^0_\pm(x;\pi,\psi)\nonumber\\
&-&\left(-\Delta_x- (M_1+\pi^2)\right)\left(\chi(x)\mathbf{w}(x;\pi,\psi)\right),\
x\in\varpi\setminus x^0,
\nonumber
\end{eqnarray}
where the \(\mathbf{w}(x;\pi,\psi)\) is calculated for the boundary layer terms (\ref{WWW}) according to the
formulae (\ref{eps4}) and (\ref{www1}).
The boundary and quasi-periodicity conditions (\ref{u'_bound}) and
(\ref{u'_bound_per})
turn into following:
\begin{eqnarray}
\label{u'psi_bound}
\partial_n \mathbf{U}'_{\pm}(x;\pi,\psi)&=&0, \quad x\in \gamma,\\
\label{u'psi_bound_per}
\mathbf{U}'_{\pm} (y,0;\pi,\psi)&=&e^{-i\pi}\mathbf{U}'_{\pm} (y,1;\pi,\psi)-i\psi e^{-i\pi}
\mathbf{U}^0_\pm(y,1;\pi,\psi),
y\in\gamma_0 \\
\partial_z \mathbf{U}'_{\pm} (y,0;\pi,\psi)&=&
e^{-i\pi}\partial_z \mathbf{U}'_{\pm} (y,1;\pi,\psi)-i\psi e^{-i \pi}\partial_z
\mathbf{U}^0_\pm(y,1;\pi,\psi),
 y\in\gamma_0.\nonumber
\end{eqnarray}
Note that the extra terms on the
right-hand side of \eqref{u'psi_bound_per} is due to
the original quasi-periodicity conditions \eqref{percond}
and the Taylor formula
\[e^{-i(\pi+\psi\epsilon^3)}=e^{-i\pi}(1-i\psi\epsilon^3+O(\epsilon^6)),\,
\epsilon\to 0.\]
To find \(\Lambda'(\pi,\psi)\) we multiply equation \eqref{u'psi} by
\(\overline{U^0_+(x;\pi)}\)
and integrate over \(\varpi\)
\begin{eqnarray}
\label{59}
\int_\varpi
\overline{U^0_+(x;\pi)}(-\Delta_x&-&\Lambda^0(\pi))\mathbf{U}'_\pm(x;\pi,\psi)\,dx=
\Lambda'(\pi,\psi)a_1^\pm (\psi) \\
&-&\int_\varpi\overline{U^0_+(x;\pi)}\left(-\Delta_x-
(M_1+\pi^2)\right)(\chi(x)\mathbf{w}(x;\pi,\psi))\,dx.\nonumber
\end{eqnarray}
The integral on the right can be calculated in a similar way as in the previous
section and it will provide the formula
\begin{eqnarray}
&-&\int_\varpi \overline{U^0_+(x;\pi)}\left(-\Delta_x-
(M_1+\pi^2)\right)\left(\chi(x)\mathbf{w}(x;\pi,\psi)\right)\,dx
=a_1^\pm(\psi)F_0+ \nonumber\\
&+&
a_2^\pm(\psi)\left(\nabla_xU^0_-(x^0;\pi)^\top Q\overline{\nabla_xU^0_+(x^0;\pi)}
-(M_1+\pi^2)U^0_-(x^0;\pi)\overline{U^0_+(x^0;\pi)}\right)\nonumber\\
&=&a_1^\pm(\psi)F_0+a_2^\pm(\psi)F_1,\nonumber
\end{eqnarray}
where \(F_0\) is given in \eqref{F_0} and
\begin{eqnarray}
\label{F_1}
F_1&=&\nabla_xU^0_-(x^0;\pi)^\top Q\overline{\nabla_xU^0_+(x^0;\pi)}\\
&+&(M_1+\pi^2)U^0_-(x^0;\pi)\overline{U^0_+(x^0;\pi)}\textrm{meas}_3(\theta).\nonumber
\end{eqnarray}
The integral on the left hand side of \eqref{59} is calculated by means of the Green
formula
\begin{eqnarray*}
&&\int_\varpi
\overline{U^0_+(x;\pi)}(-\Delta_x-\Lambda^0(\pi))\mathbf{U}'_\pm(x;\pi,\psi)\,dx\nonumber \\
&&=\int_{\partial\varpi}(\mathbf{U}'_\pm(x;\pi,\psi)\overline{\partial_n U^0_{+}(x;\pi)}-
\partial_n \mathbf{U}'_\pm(x;\pi,\psi)\overline{U^0_{+}(x;\pi)})\,ds_x=:I\,.
\end{eqnarray*}
For calculating this surface integral we take into account the Neumann conditions
\eqref{u'psi_bound}
for \(\mathbf{U}'_\pm\) and \eqref{form6} for \(U^0_+\) on \(\gamma\),
the quasi-periodicity conditions \eqref{form7} and \eqref{u'psi_bound_per} to obtain
\begin{eqnarray}
I=&&\int_\gamma \big(-\overline{\partial_z U^0_+(y,0;\pi)} \mathbf{U}'_\pm(y,0;\pi,\psi)
+\overline{U^0_+(y,0;\pi)}\partial_z \mathbf{U}'_\pm(y,0;\pi,\psi)\big) dy \nonumber\\
+&&\int_\gamma \big(
\overline{\partial_z U^0_+(y,1;\pi)} \mathbf{U}'_\pm(y,1;\pi,\psi)-
\overline{U^0_+(y,0;\pi)}\partial_z \mathbf{U}'_\pm(y,1;\pi,\psi)\big) dy \nonumber\\
=&& 2\pi\psi a_1^\pm(\psi) \int_\gamma |V_1(y)|^2\,dy =2\pi\psi
a_1^\pm(\psi).\nonumber
\end{eqnarray}
In last equality we have used the normalisation condition for \(V_1\) and the trivial
equality
\(\|U^0_+;L^2(\varpi)\|=\|V_1;L^2(\gamma)\|\).

Thus the first equation for \(\Lambda'(\pi,\psi)\) takes form
\begin{equation*}
\label{Lambda'-first}
a_1^\pm(\psi)(F_0-2\pi\psi)+a_2^\pm(\psi)F_1=\Lambda'(\pi,\psi)a_1^\pm(\psi).
\end{equation*}
Applying the same procedure for $U^0_-$, we get the second equation
\begin{equation*}
\label{Lambda'-second}
-a_1^\pm(\psi)\overline{F_1}
+a_2^\pm(\psi)(F_0+2\pi\psi)=\Lambda'(\pi,\psi)a_2^\pm(\psi).
\end{equation*}
Finally, we have noticed that the coefficients \(a_1^\pm(\psi)\) and \(a_2^\pm(\psi)\) satisfy the
system of algebraic equations written in the matrix form
\begin{equation}
\label{sys a}
\left(\begin{matrix}
-2\pi\psi+F_0& F_1\\
-\overline{F_1}&2\pi \psi+F_0
\end{matrix}
\right)
\left(
\begin{matrix}
a_1^\pm(\psi)\\a_2^\pm(\psi)
\end{matrix}
\right)
=\Lambda'_\pm(\pi;\psi)
\left(
\begin{matrix}
a_1^\pm(\psi)\\a_2^\pm(\psi)
\end{matrix}
\right) .
\end{equation}
In other words, the main asymptotic corrections to the eigenvalues
\(\Lambda^0_\pm(\eta)\)
are eigenvalues of the matrix on the left-hand side of (\ref{sys a}).
At the end a simple computation provides us the formula
\begin{equation}
\label{L'psi}
\Lambda'_\pm(\pi,\psi)=F_0 \pm\sqrt{4\pi^2\psi^2+|F_1|^2}\,,
\end{equation}
where \(F_0\) and \(F_1\) are given in \eqref{F_0} and \eqref{F_1}.
We emphasise that in the case \(F_1\ne 0\) the graphs of the eigenvalues
\(\Lambda^\epsilon_\pm(\pi+\psi\epsilon^3)\) given by \eqref{ansatz1}
and \eqref{L'psi} look like the ones in fig. 5c.

\subsection{Final formulas}
\label{subsec:35}

For the Neumann problem (\ref{form1})-(\ref{form2}) we have \(M_1=0\),
\(V_1(y)=\textrm{meas}_2(\omega)^{-\frac12}\) and
\[U^0_{\pm}(x;\pi)=e^{\pm i\pi z}(\textrm{meas}_2(\omega))^{-\frac12}.\]
Substituting these into the formulae (\ref{F_0}) and (\ref{F_1}) we obtain
\begin{equation*}
F_0=\pi^2 \frac{Q_{33}+\textrm{meas}_3(\theta)}{\textrm{meas}_2(\omega)},\,
|F_1|=\pi^2 \frac{-Q_{33}+\textrm{meas}_3(\theta)}{\textrm{meas}_2(\omega)}.
\end{equation*}
By Remark 3.1 it follows that \(-Q_{33}>\textrm{meas}_3(\theta)\) and hence
\begin{equation}
 \label{oulu1}
F_0<0,\,|F_1|>0.
\end{equation}

However, for the mixed boundary value problem (\ref{dir-1})
\[
U^0_{\pm}(x;\pi)=e^{\pm i\pi z}V_1(y),
\]
where \(\{M_1,V_1\}\) is the first eigenpair of the Dirichlet problem (\ref{S4}).
In this case the coefficients \(F_0\) and \(F_1\) are
\begin{eqnarray}\label{oulu2}
F_0&=&\nabla_y V_1(y^0)^\top Q'\nabla_y V_1(y^0)+(M_1+\pi^2(Q_{33}+\textrm{meas}_3(\theta))|V_1(y^0)|^2\\
F_1&=&e^{-2\pi i z_0}
\big[\nabla_y V_1(y^0)^\top Q'\nabla_y V_1(y^0)+
(M_1-\pi^2(Q_{33}-\textrm{meas}_3(\theta))|V_1(y^0)|^2 \big]\nonumber\\
&-&i2\pi e^{-2\pi i z_0}\{Q_{31}\frac{\partial V_1}{\partial y_1}(y^0)+
Q_{32}\frac{\partial V_1}{\partial y_2}(y^0)\}V_1(y^0)\nonumber
\end{eqnarray}
where \(Q'\) is the \(2\times 2\) upper left block of the
virtual mass matrix \(Q\) for the set \(\overline{\theta}\subset
\mathbb{R}^3\).

\section{Justification of the asymptotics and existence of a spectral gap}
\label{sec:4}

In this section we will first estimate the difference
\(\Lambda_\pm^\epsilon(\eta)-\Lambda_\pm^0(\eta)\)
outside the \(C\epsilon^{\frac72}\)-neighbourhood of the point \(\eta=\pi\) and
show that the perturbed eigenvalues \(\Lambda_\pm^\epsilon(\eta)\) are close to
the eigenvalues of the unperturbed problem. This is a rather standard step in
the asymptotic analysis of eigenvalues in singularly perturbed domains.
The second step is more elaborated.

In order to prove the opening of the spectral gap we have to derive approriate estimates of the remainder
in the asymptotic expansion of \(\Lambda_\pm^\epsilon(\eta)\) constructed in section 3
inside the \(C\epsilon^{\frac72}\)-neighbourhood of the point \(\eta=\pi\).
To this end, apply the following classical
lemma on ''near eigenvalues and eigenvectors`` (see
\cite{ViLu} and also, e.g., \cite[Ch. 5]{BiSo}).

\begin{lemma}
\label{nearEig}
Let \({\cal B}\) be a selfadjoint, positive, and compact operator in Hilbert
space \({\cal H}\) with the inner product \((\cdot,\cdot)_{\cal H}\). If there
exists a number \(b>0\) and an element \({\cal W}\in {\cal H}\) such that
\(\|{\cal W}\|_{\cal H}=1\) and \(\|{\cal BW}-b{\cal W}; {\cal H}
\|=\tau\in(0,b)\), then the segment \([b-\tau,b+\tau]\) contains at least one
eigenvalue of \({\cal B}\).
\end{lemma}

\subsection{\(\eta\) far from \(\pi\)}
\label{subsec:41}

In this section we will prove the following theorem
\begin{theorem}
For any \(\alpha> 0\) there exist \(C,\,C(\alpha)>0\) and \(\epsilon_\alpha>0\), such
that for all \(\epsilon\in[0,\epsilon_\alpha)\)
\begin{equation*}
\label{NN}
\Lambda_k^0(\eta)-C_\alpha\epsilon^{3-\alpha}\leq\Lambda_k^\epsilon(\eta)\leq
\Lambda_k^0(\eta)+C\epsilon^3 ,\,k=1,2.
\end{equation*}
\end{theorem}

The following lemma is a direct
consequence of the one-dimensional Hardy inequality:
if \(u(b)=0\), then
\begin{equation*}
\label{hardy}
\int_a^b |u(r)|^2\,dr\leq 4\int_a^b  |\partial_r u(r)|^2r^2\,dr \,.
\end{equation*}

\begin{lemma}
There exists a positive constant \(C=C(\theta, \varpi,x^0)\) such that the
following inequality holds:
\[
\left\|(r+\epsilon)^{-1}u;L^2(\varpi_\epsilon)\right\|^2\leq
C\left(\left\|\nabla_x
u;L^2(\varpi_\epsilon)\right\|^2
+\|u;L^2(\varpi\setminus\mathcal{U})\|^2\right)\,,
\]
where $r=|x-x_0|$ and \(\mathcal{U}\subset \varpi \) is a fixed
neighbourhood of the point \(x^0\), independent of \(\epsilon\).
\end{lemma}

We also need an a priori estimate for the eigenfunctions
of the problem \eqref{spctrlprob1}-\eqref{percond} in the singularly
perturbed domain \(\varpi_\epsilon\). The method developed in \cite{MaNaPl} gives
such estimates in weighted Kondratiev norms
\begin{equation}
\label{norm1}
\left\|u^\epsilon;V_{\beta}^l(\varpi_\epsilon)\right\|^2=
\sum_{k=0}^l
\int_{\varpi_\epsilon}r^{2(\beta-l+k)}
\left|\nabla_x^ku^\epsilon(x)\right|^2\, dx\,.
\end{equation}
and the step-weighted norms
\begin{equation}
\label{norm2}
\left\|U^\epsilon;V_{\beta,0}^2(\varpi_\epsilon)\right\|=
\left(\|\nabla_xU^\epsilon; V^1_\beta(\varpi_\epsilon)\|^2+
\|U^\epsilon;V^0_{\beta-1}(\varpi_\epsilon)\|^2\right)^{1/2}.
\end{equation}
For the Neumann problem, the latter norm provides asymptotically
sharp estimates and we formulate this estimate with references
to the general result in \cite[Ch. 4 and 5]{MaNaPl} and paper \cite{na235}
where the detailed explanation of the method is given for a concrete
boundary value problem.

\begin{lemma}
Let \(U^\epsilon\) be an eigenfunction of problem
\eqref{spctrlprob1}-\eqref{percond} with the eigenvalue \(\Lambda^\epsilon\).
For any \(\beta\in (-\frac{1}{2},\frac12)\) the inequality
\[
\left\|U^\epsilon; V^2_{\beta,0}(\varpi_\epsilon)\right\|\leq
C_\beta\Lambda_\epsilon
\left\|U^\epsilon;L^2(\varpi_\epsilon)\right\|
\]
is valid while a constant \(C_\beta\) is independent of \(U^\epsilon\)
and \(\epsilon\in (0,\epsilon_0],\, \epsilon_0>0\). At the same time
\(C_\beta\to +\infty\) as \(\beta\to \pm \frac{1}{2}\).
\end{lemma}

\textbf{Proof of Theorem 4.2}:

According to the max-min principle (cf. \cite[Thm. 10.2.2]{BiSo})
\begin{equation}
\Lambda_k^\epsilon(\eta)=\sup_{E_k} \inf_{u\in E_k}\frac{\|\nabla_x
u;L^2(\varpi_\epsilon)\|^2}
{\|u;L^2(\varpi_\epsilon)\|^2}\,,
\end{equation}
where supremum is taken over all subspaces
\(E_k\subset H^1_{\eta}(\varpi_\epsilon)\) with co-dimension \(k-1\).
Every such \(E_k\) contains the linear combination of
eigenfunctions \(v=a_1U_1^0+a_2U_2^0+\ldots+ a_k U_k^0\) with \(|a_1|^2+\cdot+|a_k|^2=1\), so in the case
\(c_k\epsilon^3\leq 1\) we conclude
\begin{eqnarray*}
\Lambda_k^\epsilon(\eta)&&\leq
\frac{\|\nabla_x v;L^2(\varpi_\epsilon)\|^2}
{\|v;L^2(\varpi_\epsilon)\|^2}\leq
\frac{\|\nabla_x v;L^2(\varpi)\|^2}
{\|v;L^2(\varpi)\|^2-\|v;L^2(\theta_\epsilon)\|^2}
\\
&&\leq\frac{
|a_1|^2\Lambda^0_1(\eta)+|a_2|^2\Lambda^0_2(\eta)+\ldots+|a_k|^2\Lambda^0_k(\eta)}
{|a_1|^2+|a_2|^2+\ldots+|a_k|^2-c_k\epsilon^3}\leq
\frac{\Lambda_k^0(\eta)}{1-c_k\epsilon^3}\\
&\leq& \Lambda^0_k(\eta)(1+C_k\epsilon^3).
\end{eqnarray*}

Next we give a proof of the first inequality of the statement for
\(\Lambda^\epsilon_1\). By the minimum principle \cite[Thm. 10.2.1]{BiSo}, we have
\[
\Lambda_1^0(\eta)=
\inf_{v\in H^1_{\eta}(\varpi)}
\frac{\left\|\nabla_x v;L^2(\varpi)\right\|^2 }{\left\|v;L^2(\varpi)\right\|^2}.
\]
Replacing $v$ by the eigenfunction \(u_1^\epsilon\) normalised to 1 in
\(L^2(\varpi_\epsilon)\)  is not possible and we construct an extension
of function $u_1^\epsilon$ as follows.
Consider the function \(w(\epsilon^{-1}(x-x^0))=u_1^\epsilon(x)\) which can be
written as
\[w(\xi)=w^\perp(\xi)+w_0,\]
where \(w^\perp$ is orthogonal to 1 in \(L^2(\partial \theta)\). Since
\(\partial\theta\) is smooth,
function \(w^\perp\) can be extended to a function
\(\widetilde{w}^\perp\in H^1(\theta)\) by some fixed
continuous linear extension operator such that the inequality
\begin{eqnarray}\label{ww}
\|\nabla_\xi\widetilde{w}^\perp;L^2(\theta)\|&\leq&C\|w^\perp;H^1(B_R\setminus\theta)\|\\
&\leq&C\|\nabla_\xi w^\perp;L^2(B_R\setminus\theta)\|=C\|\nabla_\xi w;L^2(B_R\setminus\theta)\|\nonumber
\end{eqnarray}
is valid. Setting
\(\widetilde{u}_1^\epsilon(x):=\widetilde{w}^\perp(\epsilon^{-1}(x-x^0))+w_0\) we derive from (\ref{ww})
\begin{equation*}\label{uu}
\|\nabla_x\widetilde{u}^\epsilon_1;L^2(\theta_\epsilon)\|\leq C
\|\nabla_x u^\epsilon_1;L^2(B_{R\epsilon}\setminus\theta_\epsilon)\|.
\end{equation*}
Now we have
\begin{eqnarray*}\label{inequality}
\Lambda_1^0(\eta)&\leq&
\frac{\left\|\nabla_x u_1^\epsilon;L^2(\varpi_\epsilon)\right\|^2+
\left\|\nabla_x
\widetilde{u}_1^\epsilon;L^2(\theta_\epsilon)\right\|^2}
{\left\|u_1^\epsilon;L^2(\varpi_\epsilon)\right\|^2+
\left\|\widetilde{u}_1^\epsilon;L^2(\theta_\epsilon)\right\|^2}\nonumber\\
&\leq&{\Lambda_1^\epsilon(\eta)+
\left\|\nabla_x\widetilde{u}_1^\epsilon;L^2(\theta_\epsilon)\right\|^2}\\
&\leq&
\Lambda_1^\epsilon(\eta) + C \left\|\nabla_x
u_1^\epsilon;L^2(B_{R\epsilon}\setminus\theta_\epsilon)\right\|^2\nonumber
\end{eqnarray*}
Owing to Lemma 4.4 and the definitions (\ref{norm1}) and (\ref{norm2}) we have
\begin{eqnarray*}
\|\nabla_x u^\epsilon_1;L^2(B_{R\epsilon)}\setminus\theta_\epsilon)\|^2&\leq&
c\epsilon^{2(1-\beta)}\|\tau^{\beta-1}\nabla_x u^\epsilon_1;L^2(B_{R\epsilon}\setminus\theta_\epsilon)\|^2\\
&\leq& c\epsilon^{2(1-\beta)}\|u^\epsilon_1;V_{\beta,0}^2(\varpi_\epsilon)\|^2,
\end{eqnarray*}
where we can take \(\beta=-\frac12+\frac\alpha2\in(-\frac12,\frac12)\) and obtain the first estimate. The
second inequality for the eigenvalues \(\Lambda_k^\epsilon(\eta)\) is proved in
a similar manner but with an application of the max-min principle.

\subsection{\(\eta\approx\pi\)}
\label{subseq:42}

Theorem 4.2 ensures the following statement: For any
small positive \(\eta_0\) and \(m_0\) one can find \(\epsilon_0=\epsilon(\eta_0,m_0)>0\)
such that for all \(\epsilon\in [0,\epsilon_0)\) and
\(\eta\in(\pi-\eta_0,\pi+\eta_0)\) the interval
\((M_1+(\pi-\eta_0)^2-m_0,M_1+(\pi-\eta_0)^2+m_0)\) contains either two simple
eigenvalues, or one eigenvalue with multiplicity two.

The immediate objective becomes to justify the asymptotics of
the eigenvalues constructed in Section 3.

In Hilbert space \({\cal H}^\epsilon(\eta):=H^1_\eta(\varpi_\epsilon)\)
with scalar product
\[
(u,v)_{H^1_\eta(\varpi_\epsilon)}=(\nabla_x u,\nabla_x v)_{\varpi_\epsilon}
\]
we define the operator
\(\mathcal{B}^\epsilon(\eta)\) by the formula
\begin{equation}\label{BB}
(\mathcal{B}^\epsilon(\eta)u,v)_{{\cal H}^\epsilon(\eta)}=(u,v)_{
\varpi_\epsilon},\, u,v\in {\cal H}^\epsilon(\eta)\,.
\end{equation}
The operator is self-adjoint, positive and compact. Thus the spectrum
of \(\mathcal{B}^\epsilon(\eta)\) consists of the point
$\beta_0=0$ implying the essential spectrum and of the positive decreasing sequence of
eigenvalues
\begin{equation}\label{seq}
\beta_1^\epsilon(\eta)\geq \beta_2^\epsilon(\eta)\geq \ldots\geq
\beta_j^\epsilon(\eta)\geq\ldots
\end{equation}
The variational formulation of the spectral problem (\ref{spctrlprob1})
\begin{equation*}
(\nabla U^\epsilon,\nabla V)_{\varpi^\epsilon}=\Lambda^\epsilon(\eta)(U^\epsilon,
V)_{\varpi_\epsilon},\,
V\in {\cal H}^\epsilon(\eta),
\end{equation*}
gives the following relationship between the eigenvalues of the operator
\(\mathcal{B}^\epsilon(\eta)\) and the boundary value problem
(\ref{spctrlprob1})-(\ref{percond})
\begin{equation}\label{beta}
\Lambda_j^\epsilon(\eta)=\beta_j^\epsilon(\eta)^{-1}\,.
\end{equation}
The relationship (\ref{beta}) turns the sequence (\ref{seq}) into the sequence (\ref{seq-Lambda}).

We now apply Lemma 4.1 to the operator (\ref{BB}). To this end we set
\begin{eqnarray}\label{AA2}
b_{\pm}&=&(\ell_\pm)^{-1}=(\Lambda_\pm^0(\pi)+\epsilon^3\Lambda'_{\pm}(\pi,
\psi))^{-1},\\
\mathcal{V}_\pm(x)&=&\|\mathcal{U}_\pm;
{\cal H}^\epsilon(\pi+\psi\epsilon^3)\|^{-1}\mathcal{U}_\pm(x),\nonumber
\end{eqnarray}
where
\begin{eqnarray}\label{AA1}
\mathcal{U}_\pm(x)&=&X_\epsilon(x)\mathbf{U}^0_\pm(x;\pi,\psi)+
(1-X_\epsilon(x))\mathbf{P}^0_\pm(x^0;\pi,\psi) \\
&+&\chi_\theta(x)\mathcal{W}_\pm(x;\pi,\psi)+\epsilon^3X_\epsilon(x)\mathbf{U}'_\pm(x;\pi,\psi)
+\epsilon^6 \mathbf{U}''(x;\psi),\nonumber
\end{eqnarray}
and
\begin{eqnarray*}
\mathbf{P}^0_\pm(x;\pi,\psi)&=&\mathbf{U}^0_\pm(x^0;\pi,\psi)+
\sum_{j=1}^3x_j\frac{\partial \mathbf{U}^0_\pm}{\partial x_j}(x^0;\pi,\psi)\\
&+&\frac{1}{2}\sum_{j,k=1}^3x_jx_k\frac{\partial^2 \mathbf{U}^0_\pm}{\partial
x_jx_k}(x^0;\pi,\psi),\\
\mathcal{W}_\pm(x;\pi,\psi)&=&\epsilon
\mathbf{W}^1_\pm(\epsilon^{-1}(x-x^0);\pi,\psi)+\epsilon^2\mathbf{W}^2_\pm(\epsilon^{-1}
(x-x^0);\pi,\psi)\,.
\end{eqnarray*}
Here the cut-off function \(X_\epsilon\) is defined by
\[X_\epsilon(x)=1-\chi_\theta(\epsilon^{-1}(x-x^0))\]
where \(\chi_\theta\in C_0^\infty(\mathbb{R}^3)\)
is such that $\chi_\theta=1$ in the neighbourhood of the set
\(\overline{\theta}\). The term \(\mathbf{U}''\in H^1(\varpi)\) in the sum is added
to satisfy the quasi-periodicity condition (\ref{percond}) with \(\eta=\pi^2+\epsilon^3\psi\).
Due to the definition of the functions $W^1_\pm$, $W^2_\pm$, \(X_\epsilon\),
\(U''\) and
\(\chi_\theta\) the function \(\mathcal{V}_\pm\) belongs to space
\(\mathcal{H}^\epsilon(\pi+\psi\epsilon^3)\).

To use Lemma 4.1 we have to verify that
\(b_\pm=(\ell_\pm)^{-1}\) and
\(\left\|U^0_\pm;{\mathcal{H}^\epsilon(\pi+\psi\epsilon^3)}\right\|\)
are separated from zero, and then we have to estimate
\[\tau_\pm=\left\|\mathcal{BW}_\pm-b\mathcal{W}_\pm;
{\mathcal{H}^\epsilon(\pi+\psi\epsilon^3)}\right\| \]
for \(\epsilon>0\) small enough. We proceed with the following assertion

\begin{lemma}
For all \(\epsilon>0\) there holds the inequality
\begin{equation}
\label{U}
\left|\left\|\mathcal{U}_\pm;{\mathcal{H}^\epsilon(\pi+\psi\epsilon^3)}
\right\|
-(M_1+\pi^2)^{1/2}|\mathbf{a}^\pm|_{{\mathbb C}^2}\right|\leq
c\epsilon^{3/2}(1+\epsilon^{3/2}\psi).
\end{equation}
\end{lemma}

\begin{proof} Since
\(\mathbf{U}^0_\pm(x;\pi,\psi)=a_1^\pm(\psi) U^0_{+}(x)+a_2^\pm(\psi) U^0_{-}(x)\),
\(|\mathbf{a}^\pm|_{\mathbb{C}^2}=1\), is
a solution of the problem (\ref{form5})-(\ref{form7}) for
\(\eta=\pi+\psi\epsilon^3\), we have
\begin{equation*}
\left\|\nabla_x \mathbf{U}_\pm^0;L^2(\varpi)\right\|^2
=(M_1+\pi^2).
\end{equation*}
The expression on left-hand side of the inequality (\ref{U}) is estimated by the
sum of the expressions
\begin{eqnarray*}
&&\left\|\nabla_x \mathbf{U}_\pm^0;{L^2(\theta_\epsilon)}\right\|,\\
&&\left\|(1-X_\epsilon)(\mathbf{U}^0_\pm-\mathbf{P}^0_\pm;
{{\cal
H}^\epsilon(\pi+\psi\epsilon^3)}\right\|,
\left\|\chi_\theta \mathbf{W}_\pm;{{\cal H}^\epsilon(\pi+\psi\epsilon^3)}\right\|, \\
&&
\left\|\epsilon^3X_\epsilon \mathbf{U}'_\pm;{{\cal
H}^\epsilon(\pi+\psi\epsilon^3)}\right\|,
\left\|\epsilon^6\mathbf{U}'';{{\cal H}^\epsilon(\pi+\psi\epsilon^3)}\right\|.
\end{eqnarray*}

To process the first two norms it is sufficient to observe
that the supports of the functions
\(\nabla_x X_\epsilon\) and \(1-X_\epsilon\) are contained in a ball of radius
\(O(\epsilon)\), where the Taylor formula
\begin{equation}\label{Taylor}
\left|\mathbf{U}^0_\pm(x;\pi,\psi)-\mathbf{P}^0_\pm(x;\pi,\psi)\right|\leq
c\left|x-x^0\right|^3\leq C\epsilon^3\,
\end{equation}
is valid.

For the estimation of the norm
\(\left\|\chi_\theta \mathbf{W}_\pm;{{\cal H}^\epsilon(\pi+\psi\epsilon^3)}\right\|\)
we have to take into account the behaviour of the function $\mathbf{W}_\pm$ at infinity.
A direct calculations leads to the estimate
\begin{eqnarray*}
\left\|\chi_\theta \mathbf{W}_\pm;{\cal H}^\epsilon(\pi+\psi \epsilon^3)\right\|^2& \leq&
c\epsilon^6 \left(\int_{c\epsilon}^C (r^{-6}+(1+|\psi|\epsilon^3)^2
r^{-4})r^2\,dr \right)
|\mathbf{a}^\pm|^2\leq\\
&\leq& c\epsilon^{3}(1+\epsilon^{3/2}|\psi|)^2.
\end{eqnarray*}
Finally, using that \(\mathbf{U}'_\pm,\mathbf{U}''\in H^1(\varpi)\), we get the bound
for the last two norms which proves the statement.
\end{proof}

Under the restriction \(\epsilon^{3/2}|\psi|\in(0,\psi_0]\) we conclude
inequalities
\begin{equation*}
\ell_\pm\geq c_\ell>0, \qquad
\left\|{\cal U}_\pm;{{\cal H}^\epsilon(\pi+\psi\epsilon^3)}\right\|\geq c_{\cal
U}>0\,,
\end{equation*}
with some positive constants \(c_\ell\) and \(c_{\cal U}\), depending on
\(\psi_0\), \(\varpi\) and \(\theta\) only.

For the estimation of \(\tau\) we use the formula for the norm
\begin{eqnarray*}
\tau&=&\left\|\mathcal{B}^\epsilon(\pi+\psi\epsilon^3)\mathcal{V}_\pm-
\ell_\pm\mathcal{V}_\pm;\mathcal{H}^\epsilon(\pi+\psi\epsilon^3)\right\|\\
&=& \max_{Z}\left|\left({\cal B}^\epsilon(\pi+\psi\epsilon^3)
\mathcal{V}_\pm-\ell_\pm\mathcal{V}_\pm,
Z\right)_{{\cal H}^\epsilon(\pi+\psi\epsilon^3)}\right|\\
&\leq& c_\ell^{-1}c_{\mathcal{U}}^{-1} \max|T_\pm(Z)|\,.
\end{eqnarray*}
Here the maximum is calculated over all functions
\(Z\in {\cal H}^\epsilon(\pi+\psi\epsilon^3)\)
with the unit norm, and \(T_\pm\) stands for
\begin{eqnarray*}
\label{T}
T_\pm(Z)&=&\left(\nabla_x {\cal U}_\pm, \nabla_x Z\right)_{L^2(\varpi_\epsilon)^3}\\
&-&(\Lambda_\pm^0+\epsilon^3\Lambda'_\pm(\pi,\psi))({\cal
U}_\pm,Z)_{L^2(\varpi_\epsilon)}.\nonumber
\end{eqnarray*}
\(T_\pm(Z)\) can be now represented as a sum of the expressions
\begin{eqnarray}\label{ZZZ}
S^{(1)}_\pm(Z)&=&(-(\Delta_x+\Lambda^0_\pm)\mathbf{U}^0_\pm,X_\epsilon Z)_{\varpi_\epsilon}=0,\nonumber\\
S_\pm^{(2)}(Z)&=&(-[\Delta_x,X_\epsilon](\mathbf{U}^0_\pm-\mathbf{P}^0_\pm),Z)_{\varpi_\epsilon},\nonumber\\
S_\pm^{(3)}(Z)&=&\epsilon(-\Delta_x \mathbf{W}^1_\pm,\chi_\theta Z)_{\varpi_\epsilon}-
(\epsilon\partial_n \mathbf{W}^1_\pm-\nabla_x\mathbf{U}^0_\pm(x^0;\pi,\psi)^\top n,Z )_{\partial\theta_\epsilon}\\
&+&\epsilon^2(-\Delta_x\mathbf{W}^2_\pm,\chi_\theta Z)_{\varpi_\epsilon}-
(\epsilon^2\partial_n \mathbf{W}^2_\pm,Z)_{\partial\theta_\epsilon}\nonumber\\
&-&\frac12\big(\sum_{j,k=1}^3\partial_n(x_jx_k)
\frac{\partial^2 \mathbf{U}^0_\pm}{\partial x_j\partial x_k}(x^0;\pi,\psi),Z\big)_{\partial\theta_\epsilon}\nonumber\\
S_\pm^{(4)}(Z)&=&(-[\Delta_x,\chi_\theta](\epsilon \mathbf{W}^1_\pm+
\epsilon^2\mathbf{W}^2_\pm),Z)_{\varpi_\epsilon}\nonumber\\
&-&(\Lambda^0_\pm+
\epsilon^3\Lambda'_\pm)(\epsilon \mathbf{W}^1_\pm+
\epsilon^2\mathbf{W}^2_\pm,\chi_\theta Z)_{\varpi_\epsilon}\nonumber\\
S^{(5)}_\pm(Z)&=&\epsilon^3 (-(\Delta_x+\Lambda^0_\pm+\Lambda'_\pm)\mathbf{U}'_\pm+
\mathcal{F}_1,Z)_{\varpi\epsilon}\nonumber\\
S^{(6)}_\pm(Z)&=&\epsilon^3(-[\Delta_x,X_\epsilon]\mathbf{U}'_\pm,Z)_{\varpi_\epsilon}
-\epsilon^6\Lambda'_\pm(\mathbf{U}'_\pm,X_\epsilon Z)_{\varpi_\epsilon}
+\epsilon^6(\nabla_x\mathbf{U}'',\nabla_xZ)_{\varpi_\epsilon}\nonumber\\
&-&\epsilon^6(\Lambda^0_\pm+\epsilon^3\Lambda'_\pm)(\mathbf{U}'',Z)_{\varpi_\epsilon}\nonumber
\end{eqnarray}

Here \(\mathcal{F}_1\)  denotes
the last term in equation (\ref{u'psi}).
The expressions (\ref{ZZZ}) can be
divided into three types:
\begin{itemize}
\item the terms which appears due to the multiplication the
function \(U^0_\pm\) by the cut-off function,
\item the terms which stem from the asymptotic boundary layers and
\item finally, from terms which are asymptotic smooth corrections of \(U'_\pm\).
\end{itemize}

To estimate the first type of terms it is enough to observe that the supports of
\(\nabla_x
X_\epsilon\) and \(1-X_\epsilon\) are included in a ball of radius
\(O(\epsilon)\),
and to apply the Taylor formula (\ref{Taylor}) in this ball. Hence the H\"older
inequality gives us the upper bound
\begin{equation*}
c\epsilon^{7/2}\|Z;{H^1(\varpi_\epsilon)}\|
\end{equation*}
for first type terms.

Components of the third type, except for the surface integrals, are evaluated in the
same manner owing to the additional multiplier \(\epsilon^3\).
The surface integral
\(\left(\partial_n \mathbf{U}'_\pm,Z\right)_{\partial \theta_\epsilon}\)
can be estimated by the trace inequality and the asymptotic behaviour of the function
\(\mathbf{U}'_\pm\) in the vicinity of the point \(x^0\)
which provides us the point wise estimate \(|\mathbf{U}'_\pm(x)|\leq c\log|x-x^0|\).

Now it is suffices to estimate the contribution coming from the asymptotic boundary
layer terms. For this assessment we use the main parts of the functions
\(\mathbf{W}^1_\pm\) and \(\mathbf{W}^2_\pm\). Let us denote by
\[\epsilon \mathbf{W}^1_\pm(\xi;\pi,\psi)+\epsilon^2\mathbf{W}^2_\pm (\xi;\pi,\psi)=\epsilon^3
\mathbf{w}^0_\pm(x)+\widetilde{\mathbf{w}}_\pm(x).\]
The asymptotic behaviour of \(\mathbf{w}^0\) in the neighbourhood of \(x^0\) is
\(|x-x^0|^{-2}\) and \(\widetilde{\mathbf{w}}_\pm(x)=O(\epsilon^4)\). Thus by Lemma 4.3,
or more precisely by the Hardy-type inequality,

\begin{eqnarray*}
&&\left|\Lambda^0_\pm(\pi)(\chi\widetilde{\mathbf{w}}_\pm,Z)_{\varpi_\epsilon}
+\epsilon^3\Lambda'_\pm(\pi,\psi)(\chi
\mathbf{w}^0_\pm,Z)_{\varpi_\epsilon}\right|\nonumber\\
&\leq&
c\epsilon^{7/2}\||x-x^0|^{-1}Z;{L_2(\varpi_\epsilon)}
\| \\
&\leq&
c\epsilon^{7/2}\|Z;{H^1(\varpi_\epsilon)}\|.\nonumber
\end{eqnarray*}

The evaluation of the remaining terms shall be made at the expense of an
additional factor \(\epsilon^3\), or due to the replacement of \(\mathbf{w}^0_\pm\) by
\(\widetilde{\mathbf{w}}_\pm\) leads to an improvement in the estimates.

Collecting obtained inequalities we obtain that \(\tau\leq c \epsilon^{7/2}\).
Thus by Lemma 4.4 the operator
\({\cal}B^\epsilon(\pi+\psi\epsilon^3)\)
has just two eigenvalues \(\beta_\pm(\pi+\psi\epsilon^3)\) in the interval
\([\ell_\pm^{-1}- c\epsilon^{3/2},\ell_\pm^{-1}+ c\epsilon^{3/2}]\).

Now the relationship (\ref{beta}) of the spectral parameters of the operator \({\mathcal{B}}\)
gives the result.

\begin{theorem}
There exist \(C>0\), \(\epsilon_0>0\) and \(\psi_0>0\) such that for all
\(\epsilon\in(0,\epsilon_0]\)
under the condition \(|\psi|\epsilon^{3/2}<\psi_0\) the problem
(\ref{form3})-(\ref{form5}) with \(\eta=\psi+\pi\epsilon^3\) has two eigenvalues
satisfying the inequality
\begin{equation*}
\left|\Lambda^\epsilon_\pm(\pi+\psi\epsilon^3)-M_1-\pi^2-\epsilon^3\Lambda'_{\pm
}(\pi,
\psi)\right|\leq C \epsilon^{7/2}\,,
\end{equation*}
where \(\Lambda'_\pm\) is taken from (\ref{L'psi}).
\end{theorem}

\subsection{The detection of the gap.}

To detect a gap in the spectrum we apply the results of previous computations.
First, if \(|\eta-\pi|> \psi_0\epsilon^{3/2}\) with \(\psi_0>0\), then by means of
Theorem 4.2 we will show that the interval
\begin{equation}
\label{interval}
(M_1+\pi^2+(F_0-|F_1|)\epsilon^3 +c\epsilon^{7/2},
M_1+\pi^2+(F_0+|F_1|)\epsilon^3-c\epsilon^{7/2})
\end{equation}
is free of spectrum. Otherwise, in the case $|\eta-\pi|\leq
\psi_0\epsilon^{3/2}$
we make use of much more elaborated asymptotic result of Theorem 4.6 to
show that in the interval (\ref{interval}) there is no eigenvalues.
This provides us the existence of the  spectral gap.

To realise this scheme, we first observe that, if with some \(\psi_0>0\)
\[\eta\in (0, \pi-\psi_0\epsilon^{3/2}]\cup[\pi+\psi_0\epsilon^{3/2},2\pi)\,, \]
then by Theorem 4.2
\begin{equation}\label{one}
\Lambda_1^\epsilon (\eta)\leq \Lambda^0_1(\eta)+C\epsilon^3
\leq M_1 +\pi^2-2\pi|\psi_0\epsilon^{3/2}|+\psi_0^2\epsilon^3+C\epsilon^3
\end{equation}
and choosing \(\delta\in (0,\frac12]\) in Theorem 4.2
\begin{equation}\label{two}
\Lambda_2^\epsilon (\eta)\geq
\Lambda_2^0 (\eta)-C\epsilon^{3-\alpha}\geq  M_1
+\pi^2+2\pi|\psi_0\epsilon^{3/2}|+\psi_0^2\epsilon^3-C\epsilon^{3-\alpha}\,.
\end{equation}
Because of the terms \(\pm2\pi \psi_0\epsilon^{\frac32}\) on the right hand side of
(\ref{one}) and (\ref{two}) we may choose \(\epsilon>0\) so small that
\[
\Lambda_1^\epsilon (\eta)<M_1+\pi^2+(F_0-|F_1|)\epsilon^3 +c\epsilon^{7/2}
\]
and
\[
\Lambda_2^\epsilon (\eta)>M_1+\pi^2+(F_0+|F_1|) \epsilon^3 -c\epsilon^{7/2} \,,
\]
and therefore the interval (\ref{interval}) does not include any eigenvalues.

If \(\eta=\pi+ \psi\epsilon^3 \in [\pi-\psi_0\epsilon^{3/2},
\pi+\psi_0\epsilon^{3/2}] \), then by Theorem 4.6
we have

\begin{eqnarray}
\Lambda_1^\epsilon (\eta)&\leq& M_1 +\pi^2+\epsilon^3
\left(F_0 -\sqrt{|F_1|^2+4\pi^2\psi^2}\right) +c\epsilon^{7/2}\\
&\leq&
M_1+\pi^2+(F_0-|F_1|)\epsilon^3 +c\epsilon^{7/2}\nonumber
\end{eqnarray}
and
\begin{eqnarray}
\Lambda_{2}^\epsilon (\eta)&\geq& M_1 +\pi^2+\epsilon^3
\left(F_0 +\sqrt{|F_1|^2+4\pi^2\psi^2}\right) -c\epsilon^{7/2}\\
&\geq&
M_1+\pi^2 +(F_0+|F_1|)\epsilon^3  - c\epsilon^{7/2}\nonumber
\end{eqnarray}
so interval (\ref{interval}) also is free of eigenvalues and we can formulate the
main result of the paper.

\begin{theorem} Assume that \(|F_1|>0\). Then
there exist positive numbers \(\epsilon_0\) and \(c_0\) such
that,
for \(\epsilon \in (0, \epsilon_0]\),
the essential spectrum of the problem \eqref{form1}-\eqref{form2} or
\eqref{dir-1} has a gap of length
\(\ell(\epsilon)\),
\[
\left|\ell(\epsilon)-2|F_1| \epsilon^3\right|<2 c_0 \epsilon^{7/2}
\]
located just after the first segment \(\Upsilon_1(\epsilon)\). Moreover, the gap includes the interval
\[
(M_1+\pi^2 +(F_0-|F_1|)\epsilon^3-c_0\epsilon^{7/2},
M_1+\pi^2+(F_0+|F_1|)\epsilon^3+c_0\epsilon^{7/2})\,.
\]
\end{theorem}

\subsection{Asymptotic formulae for the position and size of the spectral gap}

We proceed with discussing the Neumann problem (\ref{form1})-(\ref{form2}).
Based on the relations (\ref{oulu1}) we first conclude that the condition \(|F_1|>0\) of Theorem 4.7
is always fulfilled, so that the gap
\(I(\epsilon)=(A_{-}(\epsilon),A_{+}(\epsilon))\)
opens in the spectrum. Moreover, its length is
\[
 2\pi^2\epsilon^3\frac{\textrm{meas}_3(\theta)-Q_{33}}{\textrm{meas}_2(\omega)}+O(\epsilon^\frac72),
\]
where the endpoints admit the asymptotic form
\begin{eqnarray*}
A_{-}(\epsilon)&=&\pi^2+2\pi^2\epsilon^3\frac{Q_{33}}{\textrm{meas}_2(\omega)}+O(\epsilon^\frac72),\\
A_{+}(\epsilon)&=&\pi^2+2\pi^2\epsilon^3\frac{\textrm{meas}_3(\theta)}{\textrm{meas}_2(\omega)}+O(\epsilon^\frac72).
\end{eqnarray*}
Since \(Q_{33}<0\) (see Remark 3.1), the gap covers the point \(\pi^2\).

For the mixed boundary value problem (\ref{dir-1}) let \(y^0\) be the maximum point of
the positive eigenfunction \(V_1\).
Then \(\nabla_y V_1(y^0)=0\) and the formula (\ref{oulu2}) leads to a
similar simple situation as for the Neumann problem, because then the number
\begin{eqnarray*}
 e^{2\pi i z_0}F_1&=&
\nabla_y V_1(y^0)^\top Q'\nabla_y V_1(y^0)+
(M_1+\pi^2(\textrm{meas}_3(\theta)-Q_{33}))|V_1(y^0)|^2\\
&-&2\pi i\big(Q_{31}V_1(y^0)\frac{\partial V_1(y^0)}{\partial y_1}+
Q_{32}V_1(y^0)\frac{\partial V_1(y^0)}{\partial y_2}\big)\\
&=&(M_1+\pi^2(\textrm{meas}_3(\theta)-Q_{33})|V_1(y^0)|^2
\end{eqnarray*}
is for surely positive. The interval \(I(\epsilon)\) has the endpoints
\begin{eqnarray*}
A_{-}(\epsilon)&=&M_1+\pi^2+2\pi^2\epsilon^3Q_{33}|V_1(y^0)|^2+O(\epsilon^\frac72),\\
A_{+}(\epsilon)&=&M_1+\pi^2+2\epsilon^3(M_1+\pi^2\textrm{meas}_3(\theta))|V_1(y^0)|^2+O(\epsilon^\frac72)
\end{eqnarray*}
and its length is
\[
2\epsilon^3(M_1+\pi^2(\textrm{meas}_3(\theta)-Q_{33}))|V_1(y^0)|^2+O(\epsilon^\frac72).
\]
The point \(M_1+\pi^2\) is contained in the interval \(I(\epsilon)\).

Theorem 4.7 ensures the opening of the in the case \(|F_1|>0\). However, in the mixed boundary value
problem varying the position of the point \(y^0\)
in \(\omega\) we may come across the relation \(F_1=0\). In this case the asymptotic
formulae obtained here do not allow us to conclude the existence of the gap in the spectrum.

To show that there exist points for which \(|F_1|=0\), we assume that \(Q_{12}=Q_{13}=0\),
for example due to the central symmetry of the set \(\theta\). Then the expression
\[ e^{2\pi i z_0}F_1=
\nabla_y V_1(y^0)^\top Q'\nabla_y V_1(y^0)+
(M_1+\pi^2(\textrm{meas}_3(\theta)-Q_{33}))|V_1(y^0)|^2\]
is real. Moreover,if \(y^0\) is the maximum point of \(V_1\), then the expression is positive;
but it becomes negative when \(y^0\) locates close to the boundary \(\partial \omega\)
where the homogeneous Dirichlet condition is imposed.
This is due to the fact that the second term, which is positive, diminishes, while the first term remains
strictly negative all the time according to Remark 3.1, because the normal derivative
\(\partial_n V_1\)  is negative everywhere on \(\partial \omega\). Hence there must be a subset
in \(\omega\) where \(F_1=0\). To prove or disprove the existence
of a spectral gap one needs to construct higher order terms in the asymptotic ans\"atze.

\subsection*{Acknowledgements} The first author was supported by the Chebyshev Laboratory
under RF government grant 11.G34.31.0026. The second author was supported
by Russian Foundation on Basic Research grant 09-01-00759.

redpajama_set_name: RedPajamaArXiv


White Marble Kitchen Top - 62 Marvellous Marble Kitchens That Spell Luxury (Part III). 62 Marvellous Marble Kitchens That Spell Luxury (Part II). white marble granite countertops. white kitchen calacatta marble. white kitchen marble splashback.
Published at Monday, February 19th 2018, 09:17:14 AM by Francesca Rizzo. Kitchen Backsplashes. If you can't pick a single color or even a single pattern to work with, then don't. Be creative and incorporate all the different elements that you find beautiful into a backsplash design that defines you and your home. Take this one for instance. It combines travertine tiles into two different patterns, each with its own role.
Published at Friday, February 16th 2018, 08:36:45 AM by Francesca Rizzo. Kitchen. Green kitchen cabinets at the bottom and white ones at the top…what a chic combination, especially when mixed with all those accent pieces. We really like the bright red accents in this kitchen and the way everything is in perfect sync. Those wooden compartments at the top are a really cool feature too and the stainless steel appliances don't look out of place either.
Published at Saturday, June 09th 2018, 22:25:37 PM. Kitchen By Bettine Roux Roux. The kitchen is one of the most exciting spaces in a home and one that requires a lot of attention to details during the design and planning process. Choosing a layout is a very important step. You might not be able to make this decision from the start so try first to get your priorities in order to make a list of all the elements that you want to include in the design and ways to do it. You should also pick a style that you want to use…this can also help you find the ideal layout later on.


SKU: SC LS472. Categories: Pendants, Red Coral Sponge.
Red Sponge Coral Heart Pendant set in Sterling Silver. A nice, simple, yet elegant pendant idea for a loved one. Dont forget to take a look at our red sponge coral earrings to match.
Red Sponge Coral has been used in jewellery for thousands of years. The sponge coral is the base of an already dead coral colony. Coral (especially blue and red) fall under strong nature conversation rules. For this reason we only sell sponge coral jewellery. After the coral colony has died the sponge coral rises to the surface of the ocean and are collected by fishermen.


MSU Lifeguards are trained and regularly tested in performing all the skills required by the American Red Cross.
MSU Swim Instructors are certified through Swim Lesson University. Swim Lesson University is nationally recognized by USA Swimming and the World Aquatic Babies and Children Network.
We are currently accepting applications for lifeguards and swim instructors for the Spring and Summer seasons of 2019! If you are interested in becoming part of our team, stop by the pool and speak to the Aquatics Coordinator or fill out an application in the Membership Services Office at the front of the building.


text:
Cap Metro Plan Could Bring Bus Service Every 15 Minutes, Seven Days A Week
Pavel Mezihorak for KUT
The Capital Metro board has approved a major overhaul of its bus system, affecting more than half of all routes in the area. The vote was 6 to 2.
While some buses will run more often, other routes will be eliminated. The changes will go into effect June 3.
Cap Metro has held a handful of public meetings and webinars, as well as posted information at bus stops, since September.
KUT's Syeda Hasan reports
But some regular riders, like Veronica Nelson, didn't know that proposed changes were on the table. Nelson catches the Route 5 bus on Speedway Avenue to commute from her apartment to her job downtown. Under the new plan, the 5 would be moved off Speedway and onto Lamar Boulevard.
Cap Metro notes that riders have others options nearby, like the Route 7 bus on Duval Street. But for Nelson, that means a longer walk.
"If I go take the 7, there's no guarantee that I'm going to be able to catch it on time because I have to walk over to it," she says. "But also it makes, I guess, more popular stops through campus. It generally takes longer."
The plan proposes eliminating 13 bus routes.
Credit Courtesy of Capital Metro
The plan also proposes eliminating 13 bus routes, two of which would not be replaced with any alternative service. Todd Hemingson, Cap Metro's executive vice president of planning and development, says the goal is to make the system more efficient and grid-like, with an emphasis on more frequent service.
"By that, we mean routes that operate every 15 minutes, or better, essentially all day long, seven days a week," Hemingson says, "and that's a pretty big change for us."
He says the changes would mean fewer buses twisting and turning through neighborhood streets. According to Cap Metro's own analysis, most people would still live a short walk away from a stop.
"What we found is that 97.6 percent of current riders will still be within a five-minute walk of service if the board approves the plan," Hemingson says.
What's not certain is how the new plan could affect MetroAccess, an on-demand ride service for people with disabilities. Cap Metro's latest reports show that every weekday, more than 2,000 people use the service. Hemingson says his team will introduce a plan that would incorporate those services, though he can't disclose the details just yet.
We're going to propose something that we think will address most, if not all, of the concerns we've heard," he says.
This post has been updated to reflect the board's vote.
Cap Metro
12th & Chicon Bus Stop Will Showcase East Austin's African-American History
By Kate Groetzinger • May 3, 2017
The bus stop at the southeast corner of 12th Street and Chicon once featured vibrant plaques commemorating the history of East Austin. But the plaques have fallen into disrepair since being installed in 2003. Now, Capital Metro wants to redo the bus stop to honor the area's African-American legacy.
Cap Metro Board Approves Redesign of Austin's Public Transit System
By Syeda Hasan • Feb 27, 2017
The Capital Metro board of directors on Monday unanimously approved a plan to overhaul its transit service. Cap Metro says Connections 2025 is designed to improve rider experience by creating a 24/7 transit system and expanding service.

redpajama_set_name: RedPajamaCommonCrawl


On the eve of the two-hour "Desperate Housewives" finale, the stars of the hit ABC series proved just how deep their off-screen feud is with their Wisteria Lane neighbor Teri Hatcher, reports Celebuzz.
"Housewives" Eva Longoria, Marcia Cross, Felicity Huffman and Vanessa Williams teamed up to provide the crew with a parting gift -- but one name was left off the card -- Hatcher's.
"The girls don't get along with Teri so they organized this and left her out," a production source told Celebuzz.
When asked about the feud brewing with Hatcher, Eva Longoria said, "Teri has always done her own thing. No drama... Don't read into the card."
Actress Teri Hatcher was a no-show at the series wrap party in late April.


A soothing aromatic blend made 100% from natural extract. Made of Frankincense, Lemon, Lavender and few other oils that has an aroma that your family is bound to love.
Exploring should be easy when you're a little baby. Laughing and having fun should be the work of the day.
Made of Frankincense, Lemon, Lavender and few other oils that has an aroma that your family is bound to love.


Budgeting just isn't restricted to family or huge business discipline, It must also be thought-about earlier than touring. Budget will give you an thought about all your spending. So, it is possible for you to to determine how much to spend on a particular space. People usually neglect to say some essential issues while preparing a journey budget. Here are some things to remember once you funds your trip.


The back of a photo, including the clipped article and date it was published.
You can think you know a place, and then you fall down a rabbit hole and realize you don't know so much. I read a few books recently that made me feel a whole new excitement and disdain for Chicago history. Someone asked me what books I was reading, probably because I was spouting pointed opinions, but I read these books in a short time, and it transformed me. These are the books about Chicago you should read if you want to know how things got to be unfair, difficult, and dangerous, particularly for Black and brown communities. I'm interested in schools, the south and west sides, and young people, in-particular, and they are the lens I look through when I think about the power dynamics and history of Chicago. There are a few that just came out that I'm excited about. Dr Elizabeth Todd Breland's upcoming book is high my to-read list.
Danns, Dionne. Desegregating Chicago's Public Schools: Policy Implementation, Politics, and Protest, 1965-1985 (2014) With the dual impetus of Brown v. Board of Education in 1954 and the Civil Rights Act of 1964, Chicago, like so many other cities, began the process of desegregating its public school system. What resulted was a unique study in the implementation and transformation of public policy, as the city dealt with and pushed back against directives and lawsuits from both the state and federal governments. In this book, Dionne Danns provides the story of how public policy on this historic topic was formed by stakeholders at all levels, from superintendents to parents to state and federal officials, and how politics and stakeholder perceptions and protests determined outcomes for the school system.
Diamond, Andrew J Mean Streets: Chicago Youths and the Everyday Struggle for Empowerment in the Multiracial City, 1908-1969 (2009) Mean Streets focuses on the streets, parks, schools, and commercial venues of Chicago from the era of the 1919 race riot to the civil rights battles of the 1960s to cast a new light on street gangs and to place youths at the center of the twentieth-century American experience. Andrew J. Diamond breaks new ground by showing that teens and young adults stood at the vanguard of grassroots mobilizations in working-class Chicago, playing key roles in the formation of racial identities as they defended neighborhood boundaries. Drawing from a wide range of sources to capture the experiences of young Mexicans, Puerto Ricans, African Americans, Italians, Poles, and others in the multiracial city, Diamond argues that Chicago youths gained a sense of themselves in opposition to others.
Satter, Beryl, Family Properties: How the Struggle Over Race and Real Estate Transformed Chicago and Urban America (2010) In this powerful book, Beryl Satter identifies the true causes of the city's black slums and the ruin of urban neighborhoods throughout the country: not, as some have argued, black pathology, the culture of poverty, or white flight, but a widespread and institutionalized system of legal and financial exploitation. In Satter's riveting account of a city in crisis, unscrupulous lawyers, slumlords, and speculators are pitched against religious reformers, community organizers, and an impassioned attorney who launched a crusade against the profiteers—the author's father, Mark J. Satter. At the heart of the struggle stand the black migrants who, having left the South with its legacy of sharecropping, suddenly find themselves caught in a new kind of debt peonage. Satter shows the interlocking forces at work in their oppression: the discriminatory practices of the banking industry; the federal policies that created the country's shameful "dual housing market"; the economic anxieties that fueled white violence; and the tempting profits to be made by preying on the city's most vulnerable population.
Fernandez, Lilia. Brown in the Windy City: Mexicans and Puerto Ricans in Postwar Chicago (2014) JUST READ IT. None of the blurbs I found did it justice.
Innis-Jimenez, Michael. Steel Barrio: The Great Mexican Migration to South Chicago, 1915-1940 (2013) Examining how the fortunes of Mexicans in South Chicago were linked to the environment they helped to build, Steel Barrio offers new insights into how and why Mexican Americans created community. This book investigates the years between the World Wars, the period that witnessed the first, massive influx of Mexicans into Chicago. South Chicago Mexicans lived in a neighborhood whose literal and figurative boundaries were defined by steel mills, which dominated economic life for Mexican immigrants. Yet while the mills provided jobs for Mexican men, they were neither the center of community life nor the source of collective identity. Steel Barrio argues that the Mexican immigrant and Mexican American men and women who came to South Chicago created physical and imagined community not only to defend against the ever-present social, political, and economic harassment and discrimination, but to grow in a foreign, polluted environment.
Chicanas of 18th Street: Narratives of a Movement from Latino Chicago (Latinos in Chicago and Midwest) edited by Leonard Ramirez (2011) Overflowing with powerful testimonies of six female community activists who have lived and worked in the Pilsen neighborhood of Chicago, Chicanas of 18th Street reveals the convictions and approaches of those organizing for social reform. In chronicling a pivotal moment in the history of community activism in Chicago, the women discuss how education, immigration, religion, identity, and acculturation affected the Chicano movement. Chicanas of 18th Street underscores the hierarchies of race, gender, and class while stressing the interplay of individual and collective values in the development of community reform.
Wilkerson, Isabel. The Warmth of Other Suns: The Epic Story of America's Great Migration, 2011. In this epic, beautifully written masterwork, Pulitzer Prize–winning author Isabel Wilkerson chronicles one of the great untold stories of American history: the decades-long migration of black citizens who fled the South for northern and western cities, in search of a better life.


Helping your dog keep cool during the summer protects him from heatstroke.
Except for the expulsion of a little moisture from their paw pads, dogs don't sweat to cool down the way people do. Sizzling summer weather makes them vulnerable to life-threatening heat stroke and dehydration. Products to help your dog chill out are available at pet stores, but you can make your own for little or no money by using things you may have around the house.
Your cooling pad will consist of a coolant layer sandwiched between two layers of foam and enclosed in a fabric cover. A pillowcase works nicely for small dogs; for larger dogs, you can repurpose bedding or a sleeping bag and cut it to the desired size. A sewing machine is useful but not essential for attaching layers of fabric -- a sturdy hand-stitched seam, snaps or fabric adhesive are all acceptable alternatives. Cut two pieces of medium density foam about an inch thick and insert them into your fabric envelope. The cooling pad should lie flat, like a mattress -- curled-up edges mean that the foam needs a further trim. Slide the coolants between the foam layers, secure the opening with snaps or Velcro, and you've got an easily laundered cooling pad suitable for indoor or outdoor use. When used outside, be sure to put it in a shady spot.
Refrigerant-filled ice packs are available in many shapes and sizes, including flexible blankets. After freezing, the gel inside thaws much more slowly than ice, thereby cooling everything in its immediate proximity for a longer duration. However, if you use ice packs that attract condensation, you'll need to take steps to prevent fabric from becoming soggy by wrapping them in plastic bags or foil. Using either condensation-proof "no sweat" coolants or old-fashioned hot water bottles eliminates the need for waterproof coverings. When hot water bottles are filled with ordinary tap water and frozen, their natural rubber or PVC exteriors deflect condensation while resisting equalization between the cold inner and hot ambient temperatures.
A vest with side pockets to hold coolants against your dog's body is another hedge against overheating. The size and shape of your dog will influence your choice of style. Cargo pants have ready-made pockets perfect for holding ice packs and easily can be repurposed to fashion vests for larger dogs. If you don't have an old pair, check a thrift shop. You don't need to purchase a sewing pattern to make a dog's vest -- many are available free on sewing websites. To make a rough template of your own, lay a sheet of newspaper over your dog's back and trace the outline of the vest in felt pen, leaving generous seam allowances. You can fine-tune the fit after the fabric is cut so too much is preferable to too little.
Nontoxic polymer water-absorbing crystals that expand and turn into gel after being soaked in water are available in craft stores. When the gel is encased in cotton, the fabric absorbs the water, creating a cooling effect as it gradually evaporates. To make a simple version, cut two strips of cotton in a length and width appropriate for the size of your dog's neck. Stitch them, right sides together, leaving one end of the tube open. Turn this fabric tube right side out. Add the quantity recommended by the manufacturer and stitch the final end closed with the crystals inside the defined area. Soak it until the gel is fully saturated and tie it around your dog's neck. While the crystals are pet safe, be sure your pooch doesn't snack on the tie.
SewingSupport.com: Free Dog Clothes Patterns - Over 100 designs!
Psychology Today: Canine Corner: Do Dogs Sweat?


text:
Netflix Revolution: Streaming Movies While on Vacation
Doug Parker
Netflix Revolution
Talking about the major revolution that Netflix conducted in the media industry does not do justice to the innovations it has introduced to its viewers. The history of the company's development is so thickly punctuated by game-changing breakthroughs that it should be credited with not one, but several revolutions.
Everything Netflix did over the course of two decades – be it delivering a content catalog to customers' homes, launching videos on demand, hosting its business on a cloud, or pioneering video technology – is known to have unleashed innovatory forces that changed the way we think about entertainment.
More and more viewers worldwide now subscribe to Netflix, enjoying its wide variety of programming and often avoiding its geo-blockage with this list of tips for optimizing their Netflix experience.
Earlier Endeavors
To appreciate the revolutionary spirit of Netflix's ideas, we need to look at each of them from a historical vantage point lest the magnitude of its latest achievements dwarfs its earlier accomplishments. However uncomplicated it may seem now, the idea that set the company's business in motion in 1998 was decisively ground-breaking. Back then, having implemented a subscription-based business model, Netflix started delivering DVDs to people's houses via US Postal Service.
To attract viewers' attention to the movies they could order, Netflix created their online catalog. This was an unprecedented move in those times. Before Netflix struck upon the idea to design the web-based list of movies, people had only leafed through film catalogs in stores – like Blockbuster. The advantage of this web catalog of films was immediately clear: regardless of their physical location, viewers could order any movie existing in Netflix's library. In-store rentals, by contrast, restricted customers to the store's arsenal of DVDs. The web catalog also spared people the trouble of getting dressed, and going outside, just to fetch a movie.
Movie on Demand and Unlimited Rentals
Driven to boost rental rates, Netflix soon introduced another fresh detail into its business. Customers were invited to wait in line for the next movie they wanted to watch. The creation of the waiting line had a stimulating effect upon viewers: they were motivated to return a rented DVD sooner to overtake those who lined up to order the same film.
Along with the waiting line, Netflix then created a movie recommendation system, Cinewatch, unobtrusively advising customers on what to view next. With their interest in a new film or show aroused, subscribers were thought to be more willing to order another DVD. A few years later, Netflix came up with yet another ingenious idea and launched an Unlimited Movie Rental program.
The new subscription model offered people unlimited movie rentals and allowed them to order as many as four DVDs simultaneously at a fee of fewer than twenty dollars a month. The goal of this program was to strengthen customers' loyalty and retention. A content recommendation system, Cinematch, was soon invented. Likewise, Cinematch was designed to keep customers irresistibly attracted to Netflix's library.
Technological developments made an inexorable demand on Netflix to keep pace with the times and, in 2007, inspired it to start its streaming services known as the Watch Now program. Targeted at power users with broadband internet connections, the program required them to have a 1mbps internet connection to watch films and a 3mbps internet connection to stream DVD-quality movies.
For less than twenty dollars, viewers could access 18 hours of streaming content, receiving it through a browser applet that they needed to install in advance. Netflix's subsequent, slightly more expensive, plan allowed customers to enjoy unlimited streaming and choose any movie among a thousand available videos. An unqualified success, the Watch Now initiative increased the number of Netflix's subscribers by 18 percent, almost doubled its income in comparison to previous years, and laid the foundation to a new cultural phenomenon christened as "binge-watching".
Video Player and the Cloud
The next challenge the company took up was the development of a video player that could be connected to a television, through which movies could be streamed over the internet. This is when Microsoft came to Netflix's assistance. In collaboration with Microsoft, Netflix developed a native app for the game console Microsoft Xbox 360 that enabled people to watch movies on their television through their game console. Although working in tandem with Microsoft opened a market of twelve million Xbox live members and, as a result, brought Netflix's revenue to 1 billion for the first time in its history, this partnership still was not the peak of the company's development. Other pathbreaking discoveries soon followed.
In 2010, due to some technical setbacks in its system, Netflix decided to host its business logic on the cloud. This emergency decision later proved to be another successful milestone in the company's growth. By building a solid cloud infrastructure, Netflix could expand its services to more than 190 countries.
After signing several content-licensing deals with television studios and, with time, becoming a major revenue source for their channels, Netflix initiated its own original programming – issuing its first original series titled House of Cards. Netflix Originals liberated the company from its dependence on various movie and television studios and gave it leverage over its supply chain of content providers. The decision to start its own movie series was based on strict analytics.
When pondering over what series to launch, Netflix factored in such details as the popularity of the genre, director, and actor, and diligently analyzed people's computing responses to the 1990 BBC miniseries on which Netflix's analysts thought to base their new drama. Having scientifically ascertained that the majority of people would gladly watch a political thriller directed by David Fincher and starring Kevin Spacey, Netflix commissioned two seasons of the series for 100 million dollars. The House of Cards was destined to continue for 13 seasons and generated not only a loyal audience but also imitators of Netflix's daring idea. Amazon Prime Instant Video and Hotstar immediately followed suit and began their own original series in the hope of recapitulating Netflix's resounding success.
The ending of Netflix's revolutionary enterprises is nowhere in sight. With more and more people subscribing to Netflix and practicing marathon-viewing, Netflix will surely continue catering to their tastes and wishes, probably soon changing our entertainment world out of all recognition.
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As you are all aware, Queensland is currently in the midst of a natural disaster not seen in more than 100 years.
Media reports suggest that donations of clothing, food, etc are no longer required. However, for the emergency assistance and rebuilding they need CASH!
I, personally, have donated this week and intend to do so every pay day for the next few fortnights. We need our fellow Australian's to dig deep and help us recover from this terrible event. I will also be donating any left over funds (after operating costs) for this site at the end of next week.
Think about the amount of money you may have saved by viewing this site and consider donating some of that to the appeal.
Please dig deep and support your fellow Australians.