\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 47, pp. 1--54.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/47\hfil Regularity for a clamped grid equation]
{Regularity for a clamped grid equation $u_{xxxx}+u_{yyyy}=f $
on a domain with a corner}

\author[T. Gerasimov, G. Sweers\hfil EJDE-2009/47\hfilneg]
{Tymofiy Gerasimov, Guido Sweers}  % in alphabetical order

\address{Tymofiy Gerasimov \newline
DIAM-EWI, Delft University of Technology, PO box 5031, 2600 GA
Delft, The Netherlands}
\email{t.gerasimov@tudelft.nl}

\address{Guido Sweers \newline
MI, Universit\"{a}t zu K\"{o}ln, D 50931 Cologne, Germany}
\email{gsweers@math.uni-koeln.de}

\thanks{Submitted December 10, 2008. Published April 2, 2009.}
\subjclass[2000]{35J40, 46E35, 35P30}
\keywords{Nonisotropic; fourth order PDE;  domain with corner;
\hfill\break\indent
 clamped grid; weighted Sobolev space; regularity}

\begin{abstract}
 The operator $L=\frac{\partial ^{4}}{\partial x^{4}}
 +\frac{\partial ^{4}}{\partial y^{4}}$ appears in a model for the
 vertical displacement of a two-dimensional grid that consists of
 two perpendicular sets of elastic fibers or rods. We are interested
 in the behaviour of such a grid that is clamped at the boundary and
 more specifically near a corner of the domain.
 Kondratiev supplied the appropriate setting in the sense of Sobolev
 type spaces tailored to find the optimal regularity. Inspired by
 the Laplacian and the Bilaplacian models one expect, except maybe for
 some special angles that the optimal regularity improves when angle
 decreases. For the homogeneous Dirichlet problem with this special
 non-isotropic fourth order operator such a result does not hold true.
 We will show the existence of an interval
 $( \frac{1}{2}\pi ,\omega _{\star })$,
 $\omega _{\star }/\pi \approx 0.528\dots$
 (in degrees $\omega _{\star }\approx 95.1\dots^{\circ} $),
 in which the optimal regularity improves with increasing opening angle.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\tableofcontents

\section{Introduction}

\subsection{The model}

A model for small deformations of a thin isotropic elastic plate is $%
u_{xxxx}+2u_{xxyy}+u_{yyyy}=f$. Here $f$ is a force density and $u$ is the
vertical displacement of a plate; the model neglects the influence of
horizontal deviations. Non-isotropic elastic plates are still modeled by
fourth order differential equations but the coefficients in front of the
derivatives of $u$ may vary. Two interesting extreme cases are $L_1=\tfrac{%
\partial ^{4}}{\partial x^{4}} +\tfrac{\partial ^{4}}{\partial y^{4}}$ and $%
L_2=\frac{1}{2}\tfrac{\partial ^{4}}{\partial x^{4}}+3\tfrac{\partial ^{4} }{%
\partial x^2\partial y^2}+\frac{1}{2}\tfrac{\partial ^{4}}{\partial y^{4}}$.
One may think of these operators as of the operators appearing in the model
of an elastic medium consisting of two sets of intertwined (not glued)
perpendicular fibers: $\tfrac{\partial ^{4}}{\partial x^{4}}+\tfrac{
\partial ^{4}}{\partial y^{4}}$ for fibers running in cartesian directions
(Figure \ref{figure1}, left). The differential operator is not rotation
invariant. For a diagonal grid the rotation of $\frac{1}{4}\pi $ transforms $%
L_1$ into $L_2$ (Figure \ref{figure1}, right). We will call such medium
\emph{a grid}.

We should mention that sets of fibers are connected such that the vertical
positions coincide but there is no connection that forces a torsion in the
fibers. Such torsion would occur if the fibers are glued or imbedded in a
softer medium. For those models see \cite{NSS}. The appropriate linearized
model in that last situation would contain mixed fourth order derivatives.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.44\textwidth]{fig1a} % Grid(alligned)
\includegraphics[width=0.44\textwidth]{fig1b} % Grid(digaonal)
\end{center}
\caption{A fragment of a rectangular grid with aligned and diagonal fibers.}
\label{figure1}
\end{figure}

A first place where operator $L_1$ appears is J. II. Bernoulli's paper \cite%
{Ber}. He assumed that it was the appropriate model for an isotropic plate.
It was soon dismissed as a model for such a plate, since indeed it failed to
have rotational symmetry.

\subsection{The setting}

We will focus on $L_1$ supplied with homogeneous Dirichlet boundary
conditions. This problem, which we call `\emph{a clamped grid'}, is as
follows:
\begin{equation}
\begin{gathered} u_{xxxx}+u_{yyyy}=f \quad\text{in }\Omega , \\
u=\frac{\partial u}{\partial \nu }=0\quad\text{on }\partial \Omega .
\end{gathered}  \label{0}
\end{equation}
Here $\Omega \subset \mathbb{R}^2$ is open and bounded, and $\nu $ is the
unit outward normal vector on $\partial \Omega $. The boundary conditions in
(\ref{0}) correspond to the clamped situation meaning that the vertical
position and the angle are fixed to be $0$ at the boundary.

One verifies directly that the operator $L_1=\tfrac{\partial ^{4}}{ \partial
x^{4}}+\tfrac{\partial ^{4}}{\partial y^{4}}$ is elliptic in $\overline{%
\Omega }$. One may also prove, if the normal is well-defined, that the
boundary value problem (\ref{0}) is regular elliptic. Indeed, the Dirichlet
problem which fixes the zero and first order derivatives at the boundary, is
regular elliptic for any fourth order uniformly elliptic operator. Hence,
under the assumption that $\Omega $ is bounded and $\partial \Omega \in
C^{\infty }$ the full classical regularity result (see e.g. \cite{LM}) for
problem (\ref{0}) can be used to find for $k\geq 0$ and $p\in 1,\infty ) $:
\begin{equation}
\text{if $f\in W^{k,p}(\Omega )$ then $u\in W^{k+4,p}(\Omega )$}.
\label{2.0}
\end{equation}

If $\Omega $ in (\ref{0}) has a piecewise smooth boundary $\partial \Omega $
with, say, one angular point, the result (\ref{2.0}) in general does not
apply. Instead, one may use the theory developed by Kondratiev \cite{K}.
This theory provides the appropriate treatment of problem (\ref{0}) by
employing the weighted Sobolev space $V_{\beta }^{k,p}(\Omega )$ (see
Definition \ref{Def9}), where $k\geq 0$ is the differentiability index and $%
\beta \in \mathbb{R}$ characterizes the powerlike growth of the solution
near the angular point. Within the framework of the Kondratiev spaces $%
V_{\beta }^{k,p}(\Omega )$ the regularity result ``analogous'' to (\ref{2.0}%
) will then be as follows. There is a countable set of functions $\{ u_{j}\}
_{j\in \mathbb{N}}$ such that for all $k\in \mathbb{N}$:
\begin{equation}
\text{if $f\in V_{\beta }^{k,p}(\Omega )$ then $u=w+%
\sum_{j=1}^{J_{k}}c_{j}u_{j}$ with $w\in V_{\beta }^{k+4,p}(\Omega )$.}
\label{2}
\end{equation}
We will restrict our formulations to $p=2$.

Partial differential equations on domains with corners have obtained a lot
of attention in the literature. After the seminal paper by Kondatiev \cite{K}
many authors of which we would like to mention Kozlov, Maz'ya, Rossmann \cite%
{KMR, KMR2}, Grisvard \cite{Gr}, Dauge \cite{Dauge}, Costabel and Dauge \cite%
{Costabel}, Nazarov and Plamenevsky \cite{NPl} have contributed. For
applications in elasticity theory we refer to Leguillon and Sanchez-Palencia
\cite{Leg}, Blum and Rannacher \cite{Blum}. A recent paper of Kawohl and
Sweers \cite{KSw} concerned the positivity question for the operators $L_1$
and $L_2$ in a rectangular domain for hinged boundary conditions.

\subsection{The target}

In this paper, we will focus particularly on the optimal regularity for the
boundary value problem which depends on the opening angle of the corner. For
the sake of a simple presentation, we will consider (\ref{0}) in a domain $%
\Omega \subset \mathbb{R}^2$ which has one corner in $0\in \partial \Omega $%
\ with opening angle $\omega \in ( 0,2\pi] $. A more appropriate formulation
of the problem should read as:
\begin{equation}
\begin{gathered} u_{xxxx}+u_{yyyy}=f \quad\text{in } \Omega , \\
u=0\quad\text{on }\partial \Omega , \\ \frac{\partial u}{\partial \nu
}=0\quad\text{on }\partial \Omega \backslash \{0\}, \end{gathered}  \label{1}
\end{equation}
with prescribed growth behaviour near $0$.

To be more precise in the description of a domain $\Omega $, we assume the
following condition.

\begin{condition} \label{Condition0} \rm
The domain $\Omega $ has a smooth boundary except at
$(x,y)=0$, and is such that in the vicinity of $0$ it locally coincides with
a sector. In other words,

\begin{enumerate}
\item $\partial \Omega \backslash \{0\}$ is $C^{\infty }$,

\item there exists $\varepsilon >0,\omega \in ( 0,2\pi] :\Omega \cap
B_{\varepsilon }(0)=\mathcal{K}_{\omega }\cap B_{\varepsilon }(0)$,
\end{enumerate}
where $B_{\varepsilon }(0)=\{(x,y):| (x,y)| <\varepsilon\}$ is the open
ball of radius $\varepsilon $ centered at $(x,y)=0$ and
$\mathcal{K}_{\omega }$\ an infinite sector with an opening angle $\omega $:
\begin{equation}
\mathcal{K}_{\omega }=\left\{ (r\cos (\theta ),r\sin (\theta )):\ 0<r<\infty
\text{ and }0<\theta <\omega \right\} .  \label{Komega}
\end{equation}
\end{condition}

In Figure \ref{figure2} some domains $\Omega $ which satisfy the condition
above are sketched.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.44\textwidth]{fig2a}
%{Figure01.pdf} {Figure02.pdf}
\includegraphics[width=0.44\textwidth]{fig2b}
\end{center}
\caption{Examples for $\Omega $}
\label{figure2}
\end{figure}

For the elliptic problem one might roughly distinguish between papers that
focus on the general theory and those papers that explicitly study in detail
the results for one special model. If one chooses a special fourth order
model then it usually has the biharmonic operator in the differential
equation. For the biharmonic problem of the type \eqref{1}\ the optimal
regularity due to the corner of $\Omega $ `improves' when the opening angle $%
\omega $ decreases. In fact Kondratiev in \cite[page 210]{K} states that

\begin{quotation}
`` \dots and to obtain the theorems about the differential properties of
solution. We do this for the number of concrete equations in \S\ 5. In
particular, it is derived that the differential properties of the solution
are getting better when the cone opening decreases.''
\end{quotation}

One of the peculiar results for the present clamped grid problem is that
this does not apply for the whole range $0$ to $2\pi $. We will show that
there is an interval $( \frac{1}{2}\pi ,\omega _{\star }) $, with $\omega
_{\star }/\pi \approx 0.528\dots$ (in degrees $\omega _{\star }\approx
95.1\dots^{\circ }$), where the optimal regularity \emph{increases} with
increasing $\omega $. This is outlined in the table below. The actual curve
that displays the connection between $\omega $ and $\lambda $, a parameter
for the differential properties, is obtained numerically. The discretization
is chosen fine enough such that analytical estimates show that the numerical
errors are so small that they do not destroy the structure.

\begin{table}[ht]
\renewcommand{\arraystretch}{1.5}
\par
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
operator $L$ in \eqref{1} & opening angle $\omega $ &
\parbox{42mm}{
regularity of the solution $u$ to \eqref{1}
in dependence of $\omega $} \\ \hline
$\Delta ^2$ & $( 0,2\pi] $ & decreases \\ \hline
$\frac{\partial ^{4}}{\partial x^{4}}+\frac{\partial ^{4}}{\partial y^{4}}$
&
\parbox{30mm}{\begin{center}
$(0,\frac{1}{2}\pi]$, $[\omega _{\star },2\pi]$\\[2pt]
$[ \frac{1}{2}\pi ,\omega _{\star }]$
\end{center} } &
\parbox{42mm}{\begin{center}
decreases \\
increases\end{center}} \\ \hline
\end{tabular}%
\end{center}
\caption{Optimal regularity of the homogeneous Dirichlet problem for $\Delta
^2$ and $\tfrac{\partial ^{4}}{\partial x^{4}}+\tfrac{ \partial ^{4}}{%
\partial y^{4}}$.}
\label{table1}
\end{table}

For a graph displaying relation between $\omega $ and $\lambda $ see Figure %
\ref{figure3}. In Figure \ref{figure6} one finds a more detailed view. The
lowest value of the appearing $\lambda $ is a measure for the regularity.
See Figure \ref{figure9}.

\subsection{The lineup}

The paper is divided into 5 sections and several appendices. We start in
Section \ref{ExUn} by recalling the results for existence and uniqueness of
a weak solution $u$ to problem \eqref{1}. In Section \ref{WSob} the weighted
Sobolev spaces $V_{\beta }^{l,2}(\Omega )$ are introduced. Section \ref%
{SingSol} studies the homogeneous problem \eqref{1} in the infinite cone $%
\mathcal{K}_{\omega }$. We derive (almost explicitly) a countable set of
functions $\{ u_{j}\} _{j\in \mathbb{N}}$ solving this problem. They will
contribute in Section \ref{Reg} to the regularity statement for $u$ of type (%
\ref{2}). We address the Kondratiev theory in order to give the asymptotic
representation for the solution $u$ to \eqref{1} in terms of $\{ u_{j}\}
_{j\in \mathbb{N}}$.

The first appendix recalls imbedding results for $W^{k,2}(\Omega )$ and $%
V_{\beta }^{l,2}(\Omega )$ based on a Hardy inequality. The other appendices
contain computational and numerical results. The elaborate third appendix
confirms that indeed the errors in the numerical results are small enough.
This appendix also contains an explicit version of the Morse Theorem, which
is necessary for an analytical error bound that confirms the numerical
results.

\section{Existence and uniqueness\label{ExUn}}

For the present so-called clamped boundary conditions existence of an
appropriate weak solution can be obtained in a standard way even when the
corner is not convex. Let us recall the arguments for the existence of a
weak solution to problem \eqref{1}. The function space for these weak
solutions is
\begin{equation*}
{\mathaccent"7017 W}\text{}^{2,2}(\Omega )=\overline{C_{c}^{\infty }(\Omega )%
}^{\| .\| _{W^{2,2}(\Omega )}}.
\end{equation*}
where $C_{c}^{\infty }(\Omega )$ is the space of infinitely smooth functions
with compact support in $\Omega $.

\begin{definition}\label{Def4} \rm
A function $\tilde{u}\in {\mathaccent"7017 W}$$^{2,2}(\Omega )$
is a weak solution of the boundary value problem \eqref{1} with
$f\in L^2(\Omega )$, if
\begin{equation}
\int_{\Omega }\left( \tilde{u}_{xx}\varphi _{xx}+\tilde{u}
_{yy}\varphi _{yy}-f\varphi \right) dx\,dy=0\quad \text{for all }
\varphi \in {\mathaccent"7017 W}\text{}^{2,2}(\Omega ).  \label{3.2}
\end{equation}
\end{definition}

\begin{theorem}\label{Weak solution}
Suppose $f\in L^2(\Omega )$. Then a weak solution of
the boundary value problem \eqref{1} in the sense of Definition \ref{Def4}
exists. Moreover, this solution is unique.
\end{theorem}

\begin{proof}
The proof uses the variational formulation of the problem \eqref{1}, namely,
\begin{equation}
\text{Minimize: \ }E(u) =\int_{\Omega }\left( \tfrac{1}{ 2}\left(
u_{xx}^2+u_{yy}^2\right) -fu\right) dx\,dy\quad \text{on }{\mathaccent"7017 W%
}\text{}^{2,2}(\Omega ).  \label{300}
\end{equation}
This functional is coercive: For $u\in C_0^{\infty }( \bar{\Omega}) $ it
follows from $u=u_{x}=0$ on $\partial \Omega $ that one finds by a Poincar%
\'{e} inequality:
\begin{equation}
\int_{\Omega }u^2dx\,dy\leq C\int_{\Omega }u_{x}^2dx\,dy\leq C^2\int_{\Omega
}u_{xx}^2dx\,dy  \label{poincare}
\end{equation}
and a similar result for $x$ replaced by $y$. For the mixed second
derivative the clamped boundary conditions allow an integration by parts
such that
\begin{equation}
\int_{\Omega }u_{xy}^2dx\,dy=\int_{\Omega }u_{xx}u_{yy}dx\,dy\leq \tfrac{1}{2%
} \int_{\Omega }\left( u_{xx}^2+u_{yy}^2\right) dx\,dy.  \label{mixed}
\end{equation}
By a density argument (\ref{poincare}) and (\ref{mixed}) hold for $u\in {%
\mathaccent"7017 W}{}^{2,2}(\Omega )$. Hence $\| u\|_{W^{2,2}(\Omega )}\to
\infty $ implies $E(u) \to \infty $. A quadratic functional that is coercive
is even strictly convex and hence has at most one minimizer. This minimizer
exists since $u\mapsto E(u) $ is weakly lower semicontinuous. The integral
form of the Euler-Lagrange equation that the minimizer satisfies, defines
this minimizer as a weak solution. Moreover, since a weak solution is a
critical point of $E$ defined in (\ref{300}) and since the critical point is
unique, so is the weak solution.
\end{proof}

\begin{remark} \rm
For $u\in {\mathaccent"7017 W}^{2,2}(\Omega )$ we have just shown that
$\| u\| _{W^{2,2}(\Omega )}\leq C\int_{\Omega }(
u_{xx}^2+u_{yy}^2) dx\,dy$. For the hinged grid, that is
$u\in W^{2,2}(\Omega )\cap {\mathaccent"7017 W}^{1,2}(\Omega )$
a Poincar\'{e} inequality still yields (\ref{poincare}). Indeed,
for $u=0$ on $\partial \Omega $ there exists on every line $y=c$ that
intersects $\Omega $ an
$x_{c} $ with $(x_{c},c)\in \Omega $ and $u_{x}(x_{c},c)=0$ and starting from
this point one proves the second inequality in (\ref{poincare}). The real
problem is (\ref{mixed}). Indeed, this estimate does not hold on domains
with non-convex corners for $u\in W^{2,2}(\Omega )\cap {\mathaccent"7017 W}${}
$^{1,2}(\Omega )$.
\end{remark}

\section{Kondratiev's weighted Sobolev spaces\label{WSob}}

Due to Kondratiev \cite{K}, one of the appropriate functional spaces for the
boundary value problems of the type \eqref{1} are the weighted Sobolev space
$V_{\beta }^{l,2}$. Such spaces can be defined in different ways: either via
the set of the square-integrable weighted weak derivatives in $\Omega $ (see
\cite{K, Gr}), or via the completion of the set of infinitely differentiable
on $\Omega $ functions with bounded support in $\Omega $, with respect to a
certain norm (see \cite{KMR, NSw}).

In our case $\Omega \subset \mathbb{R}^2$ is open, bounded, and has a corner
in $0\in \partial \Omega $. It also holds that $\partial \Omega \backslash
\{0\}$ is smooth, and that $\Omega \cap B_{\varepsilon }(0)= \mathcal{K}%
_{\omega }\cap B_{\varepsilon }(0)$, where $B_{\varepsilon }(0)$ is a ball
of radius $\varepsilon >0$ and $\mathcal{K}_{\omega }$\ is an infinite
sector with an opening angle $\omega \in ( 0,2\pi) $. These weighted spaces
are as follows:

\begin{definition}\label{Def9} \rm
Let $l\in \{ 0,1,2,\dots \} $ and $\beta \in \mathbb{R}$. Then
$V_{\beta }^{l,2}(\Omega )$ is defined as a completion:
\begin{gather}
V_{\beta }^{l,2}(\Omega ) = \overline{C_{c}^{\infty }\big( \overline{
\Omega }\backslash \{0\}\big) }^{\| \cdot \| }\quad \text{with}
\label{pre50} \\
\| u\| :=\| u\| _{V_{\beta }^{l,2}(\Omega
)} =\Big( \sum_{| \alpha | =0}^{l}\int_{\Omega
}( x^2+y^2) ^{\beta -l+| \alpha |
}| D^{\alpha }u| ^2dx\,dy\Big) ^{1/2},
\label{50}
\end{gather}
where
\begin{equation*}
C_{c}^{\infty }\left( \overline{\Omega }\backslash \{0\}\right) :=\left\{
u\in C_{c}^{\infty }\left( \overline{\Omega }\right) :\text{\rm support}
(u)\subset \overline{\Omega }\backslash B_{\varepsilon }(0)\right\} .
\end{equation*}
\end{definition}

The space $V_{\beta }^{l,2}(\Omega )$ consists of all functions $u:\Omega
\to \mathbb{R}$ such that for each multiindex $\alpha =( \alpha_1,\alpha _2)
$ with $| \alpha | \leq l$, $D^{\alpha }u=\frac{\partial ^{| \alpha | }u}{%
\partial x^{\alpha _1}\partial y^{\alpha _2}}$ exists in the weak sense and $%
r^{\beta -l+| \alpha | }D^{\alpha }u\in L^2(\Omega )$. Here $r=( x^2+y^2)
^{1/2}$.

Straightforward from the definition of the norm the following continuous
imbeddings hold (see \cite[Section 6.2, lemma 6.2.1]{KMR}):
\begin{equation}
V_{\beta _2}^{l_2,2}(\Omega )\subset V_{\beta _1}^{l_1,2}(\Omega ) \quad%
\text{if }l_2\geq l_1\geq 0, \; \beta _2-l_2\leq \beta _1-l_1.  \label{66}
\end{equation}

To have the appropriate space for zero Dirichlet boundary conditions in
problem \eqref{1} we also define the corresponding space.

\begin{definition} \rm
For $l\in \{ 0,1,2,\dots\} $ and $\beta \in \mathbb{R}$, set
\begin{equation}
{\mathaccent"7017 V}_{\beta }^{l,2}(\Omega )=\overline{C_{c}^{\infty
}\left( \Omega \right) }^{\| \cdot \| },  \label{51}
\end{equation}
with $\| \cdot \| $ as the norm (\ref{50}) and
$C_{c}^{\infty }\left( \Omega \right) :=\left\{ u\in C_{c}^{\infty }\left(
\bar{\Omega}\right) :\mathop{\rm support}(u)\subset \Omega \right\} $.
\end{definition}

\begin{remark} \rm
For $u\in {\mathaccent"7017 V}${}$_{\beta }^{l,2}(\Omega )$ one finds
$D^{\alpha }u=0$ on $\partial \Omega $ for
$| \alpha | \leq \ell -1$ where $D^{\alpha }u=0$ is understood in the
sense of traces.
\end{remark}

\section{Homogeneous problem in an infinite sector, singular solutions}

\label{SingSol}

The first step in order to improve the regularity of a weak solution is to
consider the homogeneous problem in an infinite cone:
\begin{equation}
\begin{gathered} u_{xxxx}+u_{yyyy}=0 \quad\text{in } \mathcal{K}_{\omega },
\\ u=\frac{\partial u}{\partial \nu }=0\quad\text{on }\partial
\mathcal{K}_{\omega}\backslash \{0\}. \end{gathered}  \label{3}
\end{equation}
Here $\mathcal{K}_{\omega }$ is as in (\ref{Komega}). We will derive almost
explicit formula's for power type solutions to (\ref{3}).

\subsection{Reduced problem\label{Reduced problem}}

The reduced problem for (\ref{3}) is obtained in the following way. By
Kondratiev \cite{K} one should consider the power type solutions of (\ref{3}%
):
\begin{equation}
u=r^{\lambda +1}\Phi (\theta ),  \label{3.10}
\end{equation}
with $x=r\cos (\theta )$ and $y=r\sin (\theta )$. Here $\lambda \in \mathbb{C%
} $ and $\Phi :[ 0,\omega] \to \mathbb{R}$.

We insert $u$ from (\ref{3.10}) into problem (\ref{3}) and find
\begin{equation*}
\left( \tfrac{\partial ^{4}}{\partial x^{4}}+\tfrac{\partial ^{4}}{\partial
y^{4}}\right) r^{\lambda +1}\Phi (\theta )=r^{\lambda -3}\mathcal{L}\left(
\theta ,\tfrac{d}{d\theta },\lambda \right) \Phi (\theta ),
\end{equation*}
with
\begin{equation}
\begin{aligned} \mathcal{L}\left( \theta ,\tfrac{d}{d\theta },\lambda
\right) &=\tfrac{3}{4} \left( 1+\tfrac{1}{3}\cos (4\theta )\right)
\tfrac{d^{4}}{d\theta ^{4}} +\left( \lambda -2\right) \sin (4\theta
)\tfrac{d^{3}}{d\theta ^{3}}+ \\ &\quad +\tfrac{3}{2}\left( \lambda
^2-1-\left( \lambda ^2-4\lambda -\tfrac{7}{ 3}\right) \cos (4\theta )\right)
\tfrac{d^2}{d\theta ^2}+ \\ &\quad +\left( -\lambda ^{3}+6\lambda
^2-7\lambda -2\right) \sin (4\theta ) \tfrac{d}{d\theta }+ \\ &\quad
+\tfrac{3}{4}\left( \lambda ^{4}-2\lambda ^2+1+\tfrac{1}{3}\left( \lambda
^{4}-8\lambda ^{3}+14\lambda ^2+8\lambda -15\right) \cos (4\theta )\right) .
\label{5} \end{aligned}
\end{equation}
Then we obtain a $\lambda $-dependent boundary value problem for $\Phi $:
\begin{equation}
\begin{gathered} \mathcal{L}\left( \theta ,\tfrac{d}{d\theta },\lambda
\right) \Phi =0 \quad\text{in }(0,\omega ), \\ \Phi =\tfrac{d\Phi }{d\theta
}=0\quad\text{on }\partial (0,\omega ). \end{gathered}  \label{4}
\end{equation}

\begin{remark} \label{rmk4.1} \rm
The nonlinear eigenvalue problem \eqref{4} appears by a Mellin
transformation:
\begin{equation*}
\Phi (\theta )=(\mathcal{M}u)(\lambda )=\int_0^{\infty }r^{-\lambda
-2}u(r,\theta )dr.
\end{equation*}
\end{remark}

So, the reduced problem for (\ref{3}) we mentioned above is problem \eqref{4}%
. Before we start analyzing it, let us fix some basic notions.

\begin{definition}\label{Def5} \rm
Every number $\lambda _0\in \mathbb{C}$, such that there
exists a nonzero function $\Phi _0$ satisfying \eqref{4}, is said to be an
eigenvalue of problem \eqref{4}, while $\Phi _0\in C^{4}[ 0,\omega
] $ is called its eigenfunction. Such pairs $\left( \lambda _0,\Phi
_0\right) $ are called solutions to problem \eqref{4}.

If $( \lambda _0,\Phi _0) $ solves \eqref{4} and if $\Phi
_1 $ is a nonzero function that solves
\begin{equation}
\begin{gathered}
\mathcal{L}(\lambda _0)\Phi _1+\mathcal{L}'(\lambda _0)\Phi
_0=0\quad\text{in }(0,\omega ), \\
\Phi =\tfrac{d\Phi }{d\theta }=0\quad\text{on }\partial (0,\omega ),
\end{gathered}  \label{4bis}
\end{equation}
then $\Phi _1$ is a generalized eigenfunction (of order $1$) for
\eqref{4} with eigenvalue $\lambda _0$.
\end{definition}

\begin{remark}
Similarly, one may define generalized eigenfunctions of higher order.
\end{remark}

The following holds for \eqref{4}.

\begin{lemma}\label{lemma2}
Let $\theta \in (0,\omega )$, $\omega \leq 2\pi $. For every
fixed $\lambda \notin \{ \pm 1,0\} $ in \eqref{4}, let us set
\begin{gather*}
\varphi _1(\theta )=\left( \cos (\theta )+\tau _1\sin (\theta )\right)
^{\lambda +1}, \quad
\varphi _2(\theta )=\left( \cos (\theta )+\tau _2\sin
(\theta )\right) ^{\lambda +1}, \\
\varphi _{3}(\theta )=\left( \cos (\theta )-\tau _1\sin (\theta )\right)
^{\lambda +1}, \quad \varphi _{4}(\theta )=\left( \cos (\theta )
-\tau _2\sin (\theta )\right) ^{\lambda +1},
\end{gather*}
where $\tau _1=\frac{\sqrt{2}}{2}(1+i) $, $\tau _2=\frac{
\sqrt{2}}{2}(1-i) $ and $i=\sqrt{-1}$.

The set $S_{\lambda }:=\{ \varphi _{m}\} _{m=1}^{4}$ is a
fundamental system of solutions to the equation
$\mathcal{L}\left( \theta ,\tfrac{\partial }{\partial \theta },\lambda \right) \Phi =0$ on
$(0,\omega )$.
\end{lemma}

\begin{proof}
The derivation of $\varphi _{m}$, $m=1,\dots,4$ in $S_{\lambda }$ is rather
technical and we refer to Appendix \ref{AppendixC}. There we also compute
the Wronskian:
\begin{equation*}
W\left( \varphi _1(\theta) ,\varphi _2(\theta) ,\varphi _{3}(\theta)
,\varphi _{4}(\theta) \right) =16\left( \lambda +1\right) ^{3}\lambda
^2\left( \lambda -1\right) \left( \cos ^{4}(\theta )+\sin ^{4}(\theta)
\right) ^{\lambda -2}.
\end{equation*}
It is non-zero on $\theta \in (0,2\pi ]$ except for $\lambda \in \left\{ \pm
1,0\right\} $. Hence, for every fixed $\lambda \notin \left\{ \pm
1,0\right\} $ the set $\{ \varphi _{m}\} _{m=1}^{4}$ consists of four linear
independent functions on $(0,\omega )$, $\omega \leq 2\pi $.
\end{proof}

\begin{lemma}\label{lemma3}
In the particular cases $\lambda \in \left\{ \pm 1,0\right\} $
in \eqref{4}, one finds the following fundamental systems:
\begin{gather*}
S_{-1}=\{ 1,\arctan ( \cos (2\theta )) ,
\mathrm{arctanh}( \tfrac{\sqrt{2}}{2}\sin (2\theta )) ,
\varphi _{4}(\theta )\} , \\
S_0=\left\{ \sin (\theta ),\text{ }\cos (\theta ),\text{ }\varphi
_{3}(\theta ),\text{ }\varphi _{4}(\theta )\right\} , \\
S_1=\left\{ 1,\text{ }\sin (2\theta ),\text{ }\cos (2\theta ),\text{ }
\varphi _{4}(\theta )\right\} ,
\end{gather*}
where the explicit formulas for $\varphi _{4}\in S_{-1}$,
$\left\{ \varphi_{3},\varphi _{4}\right\} \in S_0$ and
$\varphi _{4}\in S_1$\ are given
in Appendix \ref{AppendixC}.
\end{lemma}

\begin{proof}
The fundamental systems $S_{-1},S_0,S_1$ are given in Appendix \ref%
{AppendixC}. By straightforward computations one finds that for every above $%
S_{\lambda }$, $\lambda \in \left\{ \pm 1,0\right\} $ the corresponding
Wronskian $W$ is proportional to $\left( \cos ^{4}(\theta )+\sin
^{4}(\theta) \right) ^{\lambda -2}$, $\lambda \in \left\{ \pm 1,0\right\} $
and hence is nonzero on $\theta \in (0,2\pi ]$.
\end{proof}

In terms of the fundamental systems $S$ we have $\Phi $ that solves $%
\mathcal{L}\left( \theta ,\tfrac{\partial }{\partial \theta },\lambda
\right) \Phi =0$ as
\begin{equation*}
\Phi (\theta )=\sum_{m=1}^{4}b_{m}\varphi _{m}(\theta ),
\end{equation*}
where $b_{m}\in \mathbb{C}$. Inserting this expression into the boundary
conditions of problem \eqref{4}, we find a homogeneous system of four
equations in the unknowns $\left\{ b_{m}\right\} _{m=1}^{4}$ reading as
\begin{equation*}
Ab:=%
\begin{pmatrix}
\varphi _1(0)) & \varphi _2(0)) & \varphi _{3}(0)) & \varphi _{4}(0)) \\
\varphi _1^{\prime }(0)) & \varphi _2^{\prime }(0)) & \varphi _{3}^{\prime
}(0)) & \varphi _{4}^{\prime }(0)) \\
\varphi _1(\omega) & \varphi _2(\omega) & \varphi _{3}(\omega) & \varphi
_{4}(\omega) \\
\varphi _1^{\prime }(\omega) & \varphi _2^{\prime }(\omega) & \varphi
_{3}^{\prime }(\omega) & \varphi _{4}^{\prime }(\omega)%
\end{pmatrix}
\begin{pmatrix}
b_1 \\
b_2 \\
b_{3} \\
b_{4}%
\end{pmatrix}
=0,
\end{equation*}
where $\omega \in ( 0,2\pi] $. It admits non-trivial solutions for $\{
b_{m}\} _{m=1}^{4}$ if and only if $\det (A)=0$. Hence, the eigenvalues $%
\lambda $ of problem \eqref{4} in sense of Definition \ref{Def5} will be
completely determined by the characteristic equation $\det(A)=0$.

We deduce the following four cases:
\begin{equation}
\det (A):=%
\begin{cases}
P(\omega ,\lambda ) & \text{when } \lambda \notin \{ \pm 1,0\}, \\
P_{-1}(\omega ) & \text{when } \lambda =-1, \\
P_0(\omega ) & \text{when } \lambda =0, \\
P_1(\omega ) & \text{when } \lambda =1.%
\end{cases}
\label{6}
\end{equation}
The explicit formulas for $P$ reads as
\begin{equation}
\begin{aligned} P(\omega ,\lambda) &=\left( 1-\tfrac{\sqrt{2}}{2}\sin
(2\omega )\right) ^{\lambda }+\left( 1+\tfrac{\sqrt{2}}{2}\sin (2\omega
)\right) ^{\lambda } \\ &\quad +\left( \tfrac{1}{2}+\tfrac{1}{2}\cos
^2(2\omega )\right) ^{\frac{1}{2} \lambda }\Big[ 2\cos \left( \lambda \left(
\arctan \left( \tfrac{\sqrt{2}}{2 }\tan (2\omega )\right) +\ell \pi \right)
\right) \\ &\quad -4\cos \left( \lambda \arctan \left( \tan ^2(\omega
)\right) \right) \Big] , \end{aligned}  \label{6.1}
\end{equation}
where
\begin{gather*}
\ell =0\quad\text{if }\omega \in ( 0,\frac{1}{4}\pi ] ,\quad \ell =1\quad%
\text{if }\omega \in ( \frac{1}{4}\pi ,\frac{3}{4}\pi ], \\
\ell =2\quad\text{if }\omega \in ( \frac{3}{4}\pi ,\frac{5}{4}\pi ], \quad
\ell =3\quad\text{if }\omega \in ( \frac{5}{4}\pi ,\frac{7}{4}\pi ], \\
\ell =4\quad\text{if }\omega \in ( \frac{7}{4}\pi ,2\pi ] .
\end{gather*}
In particular, for $\omega \in \{ \frac{1}{2}\pi ,\pi ,\frac{3}{2}\pi
,2\pi\} $ in (\ref{6.1}) we have
\begin{gather*}
P( \tfrac{1}{2}\pi ,\lambda ) = 2+2\cos (\pi \lambda )-4\cos ( \tfrac{1}{2}%
\pi \lambda ) , \\
P( \pi ,\lambda ) =-4+4\cos ^2(\pi \lambda ), \\
P( \tfrac{3}{2}\pi ,\lambda ) = 8\cos ^{3}(\pi \lambda )-6\cos (\pi \lambda
)-4\cos ( \tfrac{1}{2}\pi \lambda ) +2, \\
P( 2\pi ,\lambda ) =16\cos ^{4}(\pi \lambda )-16\cos ^2(\pi \lambda ).
\end{gather*}
Formulas for $P_{-1},P_0,P_1$ in (\ref{6}) are available in Appendix \ref%
{AppendixC}.

\subsection{Analysis of the eigenvalues $\protect\lambda $}

To describe the eigenvalues $\lambda $ of \eqref{4} for a fixed $\omega $
and, what is more important, their behavior in dependence on $\omega $, we
analyze the equation $\det (A)=0$ on the interval $\omega \in( 0,2\pi] $.

First, we find that the equations $P_{-1}(\omega )=0$ and $P_1(\omega )=0$
have identical solutions on $( 0,2\pi] $, that are denoted $\omega \in \{
\pi ,\omega _0,2\pi \} $. The approximation $\omega _0/\pi \approx
1.424\dots $ (in degrees $\omega _0\approx 256.25\dots^{\circ }$) is
obtained by the Maple 9.5 package.\ Equation $P_0(\omega )=0$ has no
solutions on $\omega \in (0,2\pi ]$. Hence, $\lambda \in \{ \pm 1\} $ are
the eigenvalues of \eqref{4} for the above values of $\omega $, while $%
\lambda =0$ is not an eigenvalue of \eqref{4}.

Now we consider $P(\omega ,\lambda )=0$ on $\omega \in ( 0,2\pi] $; here $P$
is given by (\ref{6.1}). We note that for every $\lambda \in \mathbb{C}
\backslash \{ \pm 1,0\} $ it holds that
\begin{equation*}
P(\omega ,-\lambda )=( \tfrac{3}{4}+\tfrac{1}{4}\cos (4\omega )) ^{-\lambda
}P(\omega ,\lambda ),
\end{equation*}
that is, the solutions $\lambda $ of $P(\omega ,\lambda )=0$ are symmetric
with respect to the $\omega $-axis. It is immediate that if $\lambda $ is an
eigenvalue then so is $\overline{\lambda }$. It is convenient\ to introduce
the following notation.

\begin{notation} \label{Notation1} \rm
For every fixed $\omega \in ( 0,2\pi] $ we
write $\{ \lambda _{j}\} _{j=1}^{\infty }$ for the collection of
the eigenvalues of problem \eqref{4} in the sense of Definition \ref{Def5},
which have positive real part $\mathop{\rm Re}(\lambda) >0$ and are
ordered by increasing real part.
\end{notation}

The complete set of eigenvalues to problem \eqref{4} will then read as $\{
-\lambda _{j},\lambda _{j}\} _{j=1}^{\infty }$. Now the following lemma can
be formulated.

\begin{lemma}\label{lemma4} \rm
Let $\mathcal{L}$ be the operator given by (\ref{5}).

\begin{itemize}
\item For every fixed $\omega \in (0,2\pi ]\backslash \left\{ \pi ,\omega
_0,2\pi \right\} $ the set $\{ \lambda _{j}\} _{j=1}^{\infty }$
from Notation \ref{Notation1} is given by
\begin{equation*}
\{ \lambda _{j}\} _{j=1}^{\infty }=\left\{ \lambda \in \mathbb{C}
:\mathop{\rm Re}(\lambda) \in \mathbb{R}^{+}\backslash
\{1\},\text{ \ }P(\omega ,\lambda )=0\right\} .
\end{equation*}

\item For every fixed $\omega \in \left\{ \pi ,\omega _0,2\pi \right\} $
the set $\{ \lambda _{j}\} _{j=1}^{\infty }$ from Notation \ref
{Notation1} is given by
\begin{equation*}
\{ \lambda _{j}\} _{j=1}^{\infty }=\left\{ \lambda \in \mathbb{C}
:\mathop{\rm Re}(\lambda) \in \mathbb{R}^{+}\backslash
\{1\},\text{ \ }P(\omega ,\lambda )=0\right\} \cup \{1\}.
\end{equation*}
\end{itemize}
Here $\omega _0$ is a solution of $P_1(\omega )=0$ on
$\omega \in (\pi,2\pi )$ with the approximation
$\omega _0/\pi \approx 1.424\dots$ (in degrees
$\omega _0\approx 256.25\dots^{\circ }$).
\end{lemma}

\subsection{Intermezzo: a comparison with $\Delta ^2$}

Let the grid-operator $\tfrac{\partial ^{4}}{\partial x^{4}}+\tfrac{\partial
^{4}}{\partial y^{4}}$ in problems \eqref{1}, (\ref{3}) be replaced by the
bilaplacian $\Delta ^2=\tfrac{\partial ^{4}}{\partial x^{4}}+2\tfrac{
\partial ^{4}}{\partial x^2\partial y^2}+\tfrac{\partial ^{4}}{\partial y^{4}%
}$. We recall some results for that operator, in particular, the eigenvalues
$\{ \lambda _{j}\} _{j=1}^{\infty }$ of the corresponding reduced problem.
We will compare them to those given in Lemma \ref{lemma4}.

So, for $\Delta ^2$ in (\ref{3}) the reduced problem of the type \eqref{4}
has an operator $\mathcal{L}$ reading as (see e.g. \cite[page 88]{Gr}):
\begin{equation}
\mathcal{L}\left( \theta ,\tfrac{d}{d\theta },\lambda \right) =\tfrac{d^{4}}{
d\theta ^{4}}+2\left( \lambda ^2+1\right) \tfrac{d^2}{d\theta ^2} +\left(
\lambda ^{4}-2\lambda ^2+1\right) .  \label{8.0}
\end{equation}
Proceeding as above one obtains that the corresponding determinants (see
\cite[page 89]{Gr} or \cite[page 561]{Blum}) are the following:
\begin{equation}
\det (A):=%
\begin{cases}
\sin ^2(\lambda \omega )-\lambda ^2\sin ^2(\omega ) & \text{when } \lambda
\notin \{ \pm 1,0\} , \\
\sin ^2(\omega )-\omega ^2 & \text{when } \lambda =0, \\
\sin (\omega )\left( \sin (\omega )-\omega \cos (\omega )\right) & \text{%
when } \lambda \in \{ \pm 1\} .%
\end{cases}
\label{8.1}
\end{equation}

Note that for every $\lambda \in \mathbb{C} \backslash \{ \pm 1,0\} $ the
function $\sin ^2(\lambda \omega )-\lambda ^2\sin ^2(\omega )$ is even with
respect to $\omega $ and hence the Notation \ref{Notation1} is applicable
here. Analysis of\ $\det (A)=0$\ with $\det (A)$ as in (\ref{8.1}) enables
to formulate the analog of Lemma \ref{lemma4}. Namely,

\begin{lemma}\label{lemma4.1}
Let $\mathcal{L}$ be the operator given by (\ref{8.0}).

\begin{itemize}
\item For every fixed $\omega \in (0,2\pi ]\backslash \left\{ \pi ,\omega
_0,2\pi \right\} $ the set $\{ \lambda _{j}\} _{j=1}^{\infty }$
from Notation \ref{Notation1} is given by
\begin{equation*}
\{ \lambda _{j}\} _{j=1}^{\infty }=\left\{ \lambda \in \mathbb{C}
:\mathop{\rm Re}(\lambda) \in \mathbb{R}^{+}\backslash
\{1\}:\sin ^2(\lambda \omega )-\lambda ^2\sin ^2(\omega
)=0\right\} .
\end{equation*}

\item For every fixed $\omega \in \left\{ \pi ,\omega _0,2\pi \right\} $
the set $\{ \lambda _{j}\} _{j=1}^{\infty }$ from Notation \ref
{Notation1} is given by
\begin{equation*}
\{ \lambda _{j}\} _{j=1}^{\infty }=\left\{ \lambda \in \mathbb{C}
:\mathop{\rm Re}(\lambda) \in \mathbb{R}^{+}\backslash
\{1\}:\sin ^2(\lambda \omega )-\lambda ^2\sin ^2(\omega
)=0\right\} \cup \{1\}.
\end{equation*}
\end{itemize}
Here $\omega _0$ is a solution of $\tan (\omega )=\omega $ on $\omega \in
(\pi ,2\pi )$ with the approximation $\omega _0/\pi \approx 1.430\dots$ (in
degrees $\omega _0\approx 257.45\dots^{\circ }$).
\end{lemma}

\subsection{Analysis of the eigenvalues $\protect\lambda $ (continued)}

Let $(\omega ,\lambda) $ be the pair that solves the equations of Lemmas \ref%
{lemma4} and \ref{lemma4.1}. In Figure \ref{figure3} we plot the pairs $%
\left( \omega ,\mathop{\rm Re}(\lambda) \right) $\ inside the region $\left(
\omega ,\mathop{\rm Re}(\lambda) \right) \in \left( 0;2\pi \right] \times %
\left[ 0,7.200\right] $.

\begin{remark} \label{Remark10}\rm
The numerical computations are performed with the Maple 9.5
package in the following way: at a first cycle for every $\omega _{n}=\frac{
21}{180}\pi +\frac{1}{60}\pi n$, $n=0,\dots,113$ we compute the entries of the
set $\{ \lambda _{j}\} _{j=1}^{N}$. Here, $N$ is determined by
the condition: $\mathop{\rm Re}\left( \lambda _{N}\right) \leq 7.200$ and
$\mathop{\rm Re}\left( \lambda _{N+1}\right) >7.200$. The points $(\omega
,\lambda )$ where $\lambda _{j}$ transits from the complex plane to the real
one or vice-versa are solutions to the system $P(\omega ,\lambda )=0$ and
$\frac{\partial P}{\partial \lambda }(\omega ,\lambda )=0$ (the justification
for the second condition will be discussed in Lemma \ref{lemma8}).
\end{remark}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.44\textwidth]{fig3a}
% Figure1(Clamped).pdf} Figure2(Clamped).pdf}
\includegraphics[width=0.44\textwidth]{fig3b}
\end{center}
\caption{Some first eigenvalues $\protect\lambda _{j}$ in $(\protect\omega ,
\mathop{\rm Re}(\protect\lambda)) \in ( 0,2\protect\pi ] \times [ 0,7.200] $
of problem \eqref{4}, where $\mathcal{L}$ is related respectively to $\frac{%
\partial ^{4}}{\partial x^{4}}+\frac{\partial ^{4}}{\partial y^{4}}$ (on the
top) and $\Delta ^2$ (on the bottom). Dashed lines depict the real part of
those $\protect\lambda_{j}\in \mathbb{C} $, solid lines are for purely real $%
\protect\lambda _{j}$; the vertical thin lines mark out values $\left\{\frac{%
1}{2}\protect\pi ,\protect\pi ,\frac{3}{2}\protect\pi ,2\protect\pi \right\}
$ on $\protect\omega$-axis.}
\label{figure3}
\end{figure}

In Figure \ref{figure3} one sees the difference in the behavior of the
eigenvalues in the corresponding cases. In particular, in the top plot (the
case $L=\tfrac{\partial ^{4}}{\partial x^{4}}+\tfrac{\partial ^{4}}{\partial
y^{4}}$) there are the loops and the ellipses in the vicinities of $\omega
\in \left\{ \frac{1}{2}\pi ,\frac{3}{2}\pi \right\} $ (we inclose them in
the rectangles). The bottom plot (the case $L=\Delta ^2$) looks much simpler
near the same region. As mentioned, the contribution of the first eigenvalue
$\lambda _1$ to the regularity of the solution $u$ to our problem \eqref{1}
is the most essential. So, it is important for us to know the dependence of
the eigenvalues ${\lambda }$ on the opening angle $\omega $. In this sense,
the region $\left( \omega ,\mathop{\rm Re}(\lambda) \right) \in V$ (Figure %
\ref{figure3}, top) seems to be the most interesting part and the model one.
One observes that inside $V$ the graph of the implicit function ${P(}${$%
\omega $}${{{,}\lambda )=0}}$ looks like a deformed $8$-shaped curve. So, if
one proves that everywhere in $V$, ${P(}${$\omega $}${{{,}\lambda )=0}}$
allows its local parametrization in {$\omega \mapsto \lambda =\psi ({\omega }%
)$} or $\lambda \mapsto \omega =\varphi (\lambda )$, then the bottom part of
this graph is ${\lambda }_1$ and there is a subset of the this bottom part
where ${\lambda }_1$ as a function of $\omega $ increases with increasing $%
\omega $.

\subsubsection{Behavior of ${\protect\lambda }$ in $V$\label{Behavior}}

So let us fix the open rectangular domain $V=\{ (\omega ,\lambda) :[ \frac{70%
}{180}\pi ,\frac{110}{180}\pi] \times[ 2.900,5.100]\}$, the function $P{\in C%
}^{\infty }(V,\mathbb{R} ) $ is given by (\ref{6.1}) with $\ell =1$:
\begin{equation}
\begin{aligned} P(\omega ,\lambda) &=\left( 1-\tfrac{\sqrt{2}}{2}\sin
(2\omega )\right) ^{\lambda }+\left( 1+\tfrac{\sqrt{2}}{2}\sin (2\omega
)\right) ^{\lambda }& \\ &\quad +\left( \tfrac{1}{2}+\tfrac{1}{2}\cos
^2(2\omega )\right) ^{\frac{1}{2} \lambda }\Big[ 2\cos \left( \lambda \left(
\arctan \left( \tfrac{\sqrt{2}}{2 }\tan (2\omega )\right) +\pi \right)
\right) \\ &\quad -4\cos \left( \lambda \arctan \left( \tan ^2(\omega
)\right) \right) \Big] .& \end{aligned}  \label{9}
\end{equation}
and set
\begin{equation}
\Gamma {:=\{{({\omega ,}\lambda )\in V:P({\omega ,}\lambda )=0\},}}
\label{10}
\end{equation}
as a zero level set of $P$ in $V$.

\begin{remark}\label{Remark11}\rm
To plot the set $\Gamma $ we perform the
computations to ${{P({\omega ,}\lambda )=0}}$ in $V$ in the spirit of Remark
\ref{Remark10}.
\end{remark}

In particular, for $\omega =\frac{1}{2}\pi $ being set in (\ref{9}) we
obtain ${{P\left( {{\tfrac{1}{2}\pi {,}\lambda }}\right) =}}2+2\cos (\pi
\lambda )-4\cos \left( \tfrac{1}{2}\pi \lambda \right) $. The equation ${{\
P\left( {{\tfrac{1}{2}\pi {,}\lambda }}\right) =0}}$\ admits exact solutions
for $\lambda $ in the interval $\left( 2.900,5.100\right) $, namely, ${\
\lambda }\in \left\{ {3},{{{4},{5}}}\right\} $. This yields the points
\begin{equation*}
\big( \frac{1}{2}\pi ,3\big) =:c_1, \quad \big( \frac{1}{2}\pi ,4\big) =:a,
\quad \big( \frac{1}{2}\pi ,5\big) =:c_{4},
\end{equation*}
of $\Gamma $. It also holds straightforwardly that $\frac{\partial P}{
\partial \omega }\left( c_1\right) =\frac{\partial P}{\partial \omega }
\left( c_{4}\right) =0$ and hence one may guess that horizontal tangents to
the set $\Gamma $ exist at those points (in Lemma \ref{lemma7} this
situation will be discussed in details for the point $c_1$). For $a$ we find
directly that $\frac{\partial P}{\partial \omega }(a) = \frac{\partial P}{%
\partial \lambda }(a) =0$ and hence more detailed analysis is required.
Additionally to $c_1,c_{4}$, we will also specify four other points of the
set $\Gamma $. Denoted as $c_2,c_{3},c_{5},c_{6}$, they are defined by the
system $P(\omega ,\lambda )=0$ and $\frac{\partial P}{\partial \lambda }%
(\omega ,\lambda )=0$. The latter condition (we will justify it in Lemma \ref%
{lemma8} for the point $c_2$) gives us a hint that vertical tangents to $%
\Gamma $ exist at those points. The approximations for the coordinates of $%
c_{i}$, $i=1,\dots,6$ are listed in the table and we plot the level set $%
\Gamma $ in Figure \ref{figure4}.

\begin{table}[ht]
\renewcommand{\arraystretch}{1.3}
\par
\begin{center}
\begin{tabular}{|c|c|c|l|}
\hline
Point of $\Gamma $ & Coordinates $\left( \omega /\pi ,\text{ }\lambda
\right) $ & $\omega $ in degrees & property of $\Gamma $ at $c_{k}$ \\
\hline\hline
$c_1$ & $(\frac{1}{2}, 3)$ & $90^{\circ }$ & horizontal tangent \\ \hline
$c_2$ & $(0.528\dots, 3.220\dots)$ & $\approx 95.1\dots^{\circ }$ & vertical
tangent \\ \hline
$c_{3}$ & $(0.591\dots, 4.291\dots)$ & $\approx 106.4\dots^{\circ }$ &
vertical tangent \\ \hline
$c_{4}$ & $(\frac{1}{2}, 5)$ & $90^{\circ }$ & horizontal tangent \\ \hline
$c_{5}$ & $(0.477\dots, 4.746\dots)$ & $\approx 85.96\dots^{\circ }$ &
vertical tangent \\ \hline
$c_{6}$ & $(0.412\dots, 3.655\dots)$ & $\approx 74.2\dots^{\circ }$ &
vertical tangent \\ \hline
\end{tabular}%
\end{center}
\caption{Approximations for the points of the level set $\Gamma $.}
\label{table2}
\end{table}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig4} % {Figure3.pdf}
\end{center}
\caption{The level set $\Gamma $ (solid line) in $V$.}
\label{figure4}
\end{figure}

As we mention in Remark \ref{Remark11}, the set $\Gamma $ as in \eqref{10}
was found by means of numerical computations. In order to show that the plot
of $\Gamma $ is adequate, we study the implicit function ${{P({\omega ,}
\lambda )=0}}$ in $V$ analytically. It is done in several steps.

The first lemma studies ${{P({\omega ,}\lambda )=0}}$ in the vicinity of the
point
\begin{equation}
a=(\tfrac{1}{2}\pi ,4) \in \Gamma .  \label{10.1}
\end{equation}

\begin{lemma}\label{lemma5}
Let $U=I\times J\subset V$ be the closed rectangle with $I=
\left[ \tfrac{88}{180}\pi ,\tfrac{92}{180}\pi \right] $,
$J=[3.940,4.060] $ and let point $a\in U$ be as in (\ref{10.1}). The set
$\Gamma $ given by \eqref{10} consists of two smooth branches passing through
$a$. Their tangents at $a$ are $\lambda =4$ and $\lambda =-\tfrac{16\sqrt{2}
}{\pi }\omega +4$.
\end{lemma}

\begin{proof}
Let $DP$ stand for the gradient vector and $D^2P$ is the Hessian matrix. For
the given $a$ we already know that $DP(a)=0$. We also find
\begin{equation*}
\tfrac{\partial ^2P}{\partial \omega ^2}(a)=0, \quad \tfrac{\partial ^2P}{%
\partial \omega \partial \lambda }(a)=-8\sqrt{2}\pi,\quad \tfrac{\partial ^2P%
}{\partial \lambda ^2}(a)=-\pi ^2.
\end{equation*}
That is, $\det D^2P(a)=-128\pi ^2$ and by Proposition \ref{Proposition.ap.D}
and remark \ref{remark2,ap.D} (Appendix \ref{AppendixD}) it holds that
\begin{equation}
P(\omega ,\lambda )=-\tfrac{1}{2}h_2(\omega ,\lambda) \left( 16\sqrt{2}%
h_1(\omega ,\lambda) +\pi h_2(\omega ,\lambda) \right) \quad\text{on }U,
\label{11.0}
\end{equation}
where $h_1,h_2\in C^{\infty }\left( U,\mathbb{R}\right) $ are given by
almost explicit formulas in (\ref{D18}), (\ref{D19}) in the same lemma. We
also have that $h_1(a)=h_2(a)=0$ and
\begin{gather}
\tfrac{\partial h_1}{\partial \omega }(a)=1, \quad \tfrac{\partial h_1}{%
\partial \lambda }(a)=0,  \label{12} \\
\tfrac{\partial h_2}{\partial \omega }(a)=0, \quad \tfrac{\partial h_2}{%
\partial \lambda }(a)=1.  \label{13}
\end{gather}
Due to (\ref{11.0}) we deduce that in $U$:
\begin{equation}
P(\omega ,\lambda )=0 \quad\text{if and only if}\quad h_2(\omega ,\lambda)
=0 \text{ or }16\sqrt{2}h_1(\omega ,\lambda) +\pi h_2(\omega ,\lambda) =0.
\label{11}
\end{equation}
By applying the Implicit Function Theorem to the functions $h_2(\omega
,\lambda) =0$ and $16\sqrt{2}h_1(\omega ,\lambda) +\pi h_2(\omega ,\lambda)
=0$ in $U$ one finds a parametrization $\omega \mapsto \lambda =\eta ({%
\omega })$ for each of these implicit functions. Indeed:

(1) For $h_2(\omega ,\lambda) =0$ it is shown in Lemma \ref{lemma3,ap.D}
(Appendix \ref{AppendixD}) that
\begin{equation*}
\tfrac{\partial h_2}{\partial \lambda }(\omega ,\lambda) >0 \quad\text{on }U,
\end{equation*}
and hence there exists $\eta _1:I\to J$, $\eta _1\in {C}^{\infty}(I)$ such
that
\begin{equation*}
h_2(\omega ,\eta _1(\omega) )=0,
\end{equation*}
and
\begin{equation*}
\eta _1^{\prime }(\omega) =-\tfrac{\partial h_2}{\partial \omega }(\omega
,\eta _1(\omega) )\big[ \tfrac{\partial h_2 }{\partial \lambda }(\omega
,\eta _1(\omega) )\big] ^{-1},
\end{equation*}
for all $\omega \in I$. We have that $\eta _1( \frac{1}{2}\pi ) =4$ and due
to (\ref{13}) we find $\eta _1^{\prime }( \tfrac{1}{2}\pi) =0$. Hence, there
is a smooth branch of $\Gamma $ in $U$ passing through $a$, which is given
by $\lambda =\eta _1(\omega) $ with the tangent $\lambda =4$.

(2) For $16\sqrt{2}h_1(\omega ,\lambda) +\pi h_2(\omega ,\lambda) =0$ it is
shown in Lemma \ref{lemma4,ap.D} (Appendix \ref{AppendixD}) that
\begin{equation*}
16\sqrt{2}\tfrac{\partial h_1}{\partial \lambda }(\omega ,\lambda) +\pi
\tfrac{\partial h_2}{\partial \lambda }(\omega ,\lambda) >0 \quad\text{on }U,
\end{equation*}
and hence there exists $\eta _2:\tilde{I}\to J$, $\eta _2\in {C} ^{\infty }(%
\tilde{I})$, where $\tilde{I}\subset I$, such that
\begin{equation*}
16\sqrt{2}h_1\left( \omega ,\eta _2(\omega) \right) +\pi h_2\left( \omega
,\eta _2(\omega) \right) =0,
\end{equation*}
and
\begin{equation*}
\eta _2^{\prime }(\omega) =-\frac{16\sqrt{2}\tfrac{\partial h_1}{\partial
\omega }(\omega ,\eta _2(\omega) )+\pi \tfrac{ \partial h_2}{\partial \omega
}(\omega ,\eta _2(\omega) )}{ 16\sqrt{2}\tfrac{\partial h_1}{\partial
\lambda }(\omega , \eta _2(\omega) )+\pi \tfrac{\partial h_2}{\partial
\lambda }(\omega ,\eta _2(\omega) )},
\end{equation*}
for all $\omega \in \tilde{I}$. We have that $\eta _2( \frac{1}{2}\pi ) =4$
and due to (\ref{12}) and (\ref{13}) we obtain
\begin{equation*}
\eta _2^{\prime }( \tfrac{1}{2}\pi) =-\tfrac{16\sqrt{2}}{\pi }.
\end{equation*}
Hence, there is another smooth branch of $\Gamma $ in $U$ passing through $a$
and given by $\lambda =\eta _2(\omega) $. The tangent is $\lambda =-\tfrac{16%
\sqrt{2}}{\pi }\omega +4$.
\end{proof}

The next lemma studies ${{P({\omega ,}\lambda )=0}}$ locally in $V$ but away
from the point $a$.

\begin{lemma} \label{lemma6}
Let
\begin{gather*}
H_1=\{ (\omega ,\lambda) :[ \tfrac{84}{180}\pi ,
\tfrac{90}{180}\pi ] \times [ 4.030,4.970] \} ,\\
H_2=\{ (\omega ,\lambda) :[ \tfrac{87}{180}\pi ,
\tfrac{101}{180}\pi ] \times [ 4.750,5.100] \} ,\\
H_{3}=\{ (\omega ,\lambda) :[ \tfrac{100}{180}\pi ,
\tfrac{108}{180}\pi ] \times [ 4.000,4.850] \} ,\\
H_{4}=\{ (\omega ,\lambda) :[ \tfrac{91}{180}\pi ,
\tfrac{102}{180}\pi ] \times [ 3.950,4.100] \} ,\\
H_{5}=\{ (\omega ,\lambda) :[ \tfrac{90}{180}\pi ,
\tfrac{96}{180}\pi ] \times [ 3.030,3.970] \} ,\\
H_{6}=\{ (\omega ,\lambda) :[ \tfrac{79}{180}\pi ,
\tfrac{94}{180}\pi ] \times [ 2.900,3.230] \} , \\
H_{7}=\{ (\omega ,\lambda) :[ \tfrac{72}{180}\pi ,
\tfrac{80}{180}\pi ] \times [ 3.150,4.000] \} ,\\
H_{8}=\{ (\omega ,\lambda) :[ \tfrac{78}{180}\pi ,
\tfrac{89}{180}\pi ] \times [ 3.900,4.050] \} ,
\end{gather*}
and $U$ be as in Lemma \ref{lemma5}. Then $\cup _{j=1}^{8}H_{j}$ covers the
set $\Gamma $ in $V$ (see Figure \ref{figure5}) and in each $H_{j}$ the
following holds:
\begin{center} \renewcommand{\arraystretch}{1.5}
\begin{tabular}{|c|c|c|}
\hline
\rm Rectangle & \rm Property in $H_{j}$ & \rm The set $\Gamma $
in $H_{j}$ is given by \\ \hline\hline
$H_{2k-1}$ & $\frac{\partial P}{\partial \omega }
(\omega ,\lambda )\neq 0$ & $\omega =\phi _{2k-1}(\lambda ):\phi _{2k-1}\in
C^{\infty }(J_{2k-1})$ \\ \hline
$H_{2k}$ & $\frac{\partial P}{\partial \lambda }
(\omega ,\lambda )\neq 0$ & $\lambda =\psi _{2k}({\omega }):\psi _{2k}\in
C^{\infty }(I_{2k})$ \\ \hline
\end{tabular}
\renewcommand{\arraystretch}{1}
\end{center}
Here $k=1,\dots,4$.
\end{lemma}

\begin{proof}
In Claims \ref{Claim5,ap.D} -- \ref{Claim12,ap.D} of Appendix \ref{AppendixD}
we constructed the rectangles $H_{j}\subset V$, $j=1,\dots,8$ such that the
results of the second column in a table above hold. In Figure \ref{figure5}
we sketched the covering of the set $\Gamma $ in $V$ with the rectangles $%
H_{j}$, $j=1,\dots,8$.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig5} %{Figure(Cover1).pdf}
\end{center}
\caption{For lemma \protect\ref{lemma6}.}
\label{figure5}
\end{figure}

Due to result of the second column we can apply the Implicit Function
Theorem to the function ${{P({\omega ,}\lambda )=0}}$ in every $H_{j}$, $%
j=1,\dots,8$ in order to obtain $\omega =\phi _{2k-1}(\lambda )$ or ${%
\lambda } ${$=\psi _{2k}({\omega })$, }$k=1,\dots,4$. By assumption $P{\in C}%
^{\infty }(V,\mathbb{R})$ and hence $\phi ,\psi $ are $C^{\infty }$ on the
corresponding intervals $J,I$.
\end{proof}

Based on the results of the two lemmas above, we arrive at the following
result.

\begin{proposition}\label{Proposition1}
The set $\Gamma$ given by \eqref{10} is an $8$-shaped
curve. That is, there exists an open set $\tilde{V}\supset [ -1,1]^2$
\ and a $C^{\infty }$-diffeomorphism $S:V\to \tilde{V}$ such that
\begin{equation*}
S\left( \Gamma \right) =\left\{ (\sin (2t),\sin (t)),\text{ }0\leq t<2\pi
\right\} .
\end{equation*}
\end{proposition}

Henceforth, we will call the set $\Gamma $ a curve (having one
self-intersection point) which means that every part of the set $\Gamma $ is
locally parametrizable in {$\omega $} or $\lambda $.

\subsubsection{Eigenvalue $\protect\lambda _1$ as the bottom part of $\Gamma
$}

The curve $\Gamma $ in a rectangle $V$ combines the graphs of the first four
eigenvalues $\lambda _1,\dots,\lambda _{4}$ of the boundary value problem ( %
\ref{4}) as functions of $\omega $ as far as they are real.\ Here we focus
on the eigenvalue $\lambda _1$ which is a bottom part of $\Gamma $ (the
segment $c_{6}c_1c_2\subset \Gamma $ in Figure \ref{figure4}). In
particular, we prove that as a function of $\omega $ the eigenvalue $\lambda
_1=\lambda _1(\omega )$ increases between the points $c_1,c_2$ (the
approximations for their coordinates are given in Table \ref{table2}). The
situation is illustrated by Figure \ref{figure6}.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig6} %{Figure5.pdf}
\end{center}
\caption{Increase of $\protect\lambda _1$ between $c_1 $ and $c_2$}
\label{figure6}
\end{figure}

To prove this result, we follow the approach used in Lemmas \ref{lemma5} and %
\ref{lemma6}. To be more precise, we fix two rectangles $\{ H_0,H_{\star }\}
\subset V$ such that $H_0\cap H_{\star }\ne \emptyset $ and $H_0\cup
H_{\star }$ covers the part of $\Gamma $ containing the segment $c_1c_2$
(see Figure \ref{figure7}). We parameterize $\Gamma $ in $H_0,H_{\star }$\
as {$\omega \mapsto \lambda =\psi ({\omega })$} and $\lambda \mapsto \omega
=\varphi (\lambda )$, respectively, and study the properties of these
parametrizations (convexity-concavity, extremum points, the intervals of
increase-decrease). This will enable to gain the information about $c_1c_2$.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.44\textwidth]{fig7a} \quad % {Figure(Cover2).pdf}
\includegraphics[width=0.44\textwidth]{fig7b} % {Figure(Cover3).pdf}
\end{center}
\caption{The rectangles $H_0,H_{\star }$ from lemmas \protect\ref{lemma7}
and \protect\ref{lemma8}, respectively (on the left); the enlarged view (on
the right).}
\label{figure7}
\end{figure}

\begin{lemma} \label{lemma7}
Let $H_0=I_0\times J_0\subset V$ be the closed rectangle
with $I_0=\left[ \tfrac{84}{180}\pi ,\tfrac{94}{180}\pi \right] $ and
$J_0=\left[ 2.960,3.060\right] $. It holds that $\Gamma $ in $H_0$ is
given by $\lambda =\psi (\omega )$, $\psi \in C^{\infty }(\omega _{\alpha
},\omega _{\beta })$, $(\omega _{\alpha },\omega _{\beta })\subset I_0$
and is such that it attains its minimum on $\left( \omega _{\alpha },\omega
_{\beta }\right) $ at $\omega =\omega _0=\frac{1}{2}\pi $ and increases
monotonically on $\left( \omega _0,\omega _{\beta }\right) $. Here $\omega
_{\alpha },\omega _{\beta }$ are the solutions to the equation $P(\omega
,3.060)=0$ on $\omega \in \left( \frac{84}{180}\pi ,\frac{1}{2}\pi \right) $
and on $\omega \in \left( \frac{1}{2}\pi ,\frac{94}{180}\pi \right) $,
respectively, with $P$ given by (\ref{9}).
\end{lemma}

\begin{proof}
By Lemma \ref{lemma6} we know that
\begin{equation}
P(\omega ,\lambda )=0\quad\text{if and only if } P\left( \omega ,\psi
(\omega) \right) =0 \text{ in }H_{6},  \label{16}
\end{equation}
and if we take the rectangle $H_0$ defined as in lemma above, then due to $%
H_0\subset H_{6}$, (\ref{16}) will also hold in $H_0$. Moreover, we also set
$H_0$ in such a way that its top boundary intersects $\Gamma $ at two
points, meaning that we find two solutions of $P(\omega ,3.060)=0$ with $P$
as in (\ref{9}). We name these two solutions $\omega _{\alpha
},\omega_{\beta }$.

Hence, we deduce that $\Gamma $ in $H_0$ is given by $\lambda =\psi (\omega
) $, $\psi \in C^{\infty }(\omega _{\alpha },\omega _{\beta })$ and
satisfies $\psi (\omega _{\alpha })=\psi (\omega _{\beta })=3.060$. Due to
condition
\begin{equation*}
\psi (\omega _{\alpha })=\psi (\omega _{\beta }),
\end{equation*}
by Rolle's theorem there exists $\omega _0\in ( \omega _{\alpha },\omega
_{\beta }) $ such that $\psi ^{\prime }(\omega _0)=0$.

Since $P\left( \omega _0,\psi \left( \omega _0\right) \right) =0$ and due to
\begin{equation*}
\psi ^{\prime }(\omega )=-\tfrac{\partial P}{\partial \omega }(\omega ,\psi
(\omega) )[ \tfrac{\partial P}{\partial \lambda }(\omega ,\psi (\omega) )]
^{-1},
\end{equation*}
we solve the system $P(\omega ,\lambda )=0$ and $\frac{\partial P}{\partial
\omega }(\omega ,\lambda )=0$ in $H_0$ in order to find $\omega _0$. Its
solution is a point $c_1=( \frac{1}{2}\pi ,3) $ and hence
\begin{equation*}
\omega _0=\tfrac{1}{2}\pi .
\end{equation*}
We deduce that $\lambda =\psi (\omega )$ attains its local extremum at $%
\omega =\omega _0$.

Next we show that $\lambda =\psi (\omega )$ has a minimum at $\omega =\omega
_0$ on $(\omega _{\alpha },\omega _{\beta })$. For this purpose we consider
a function $G\in C^{\infty }( H_0,\mathbb{R}) $ such that
\begin{equation}
G\left( \omega ,\psi (\omega )\right) =\psi ^{\prime \prime }(\omega ).
\label{14}
\end{equation}
For an explicit formula for $G$ see Appendix \ref{AppendixD}. In Claim \ref%
{Claim13,ap.D} of this Appendix we show that
\begin{equation}
G(\omega ,\lambda) >0\quad\text{on }H_0.  \label{15}
\end{equation}
This condition together with (\ref{14}) yields
\begin{equation*}
G\left( \omega ,\psi (\omega )\right) =\psi ^{\prime \prime }(\omega )>0\quad%
\text{on }(\omega _{\alpha }, \omega _{\beta }),
\end{equation*}
meaning that $\lambda =\psi (\omega )$ is convex on $(\omega
_{\alpha},\omega _{\beta })$.

The result is that $\lambda =\psi (\omega )$ attains its minimum on $(\omega
_{\alpha },\omega _{\beta })$ at $\omega =\omega _0=\frac{1}{2}\pi $ and
increases monotonically on the interval $\omega \in \left( \omega _0,\omega
_{\beta }\right) $.
\end{proof}

We also have the following result.

\begin{lemma}\label{lemma8}
Let $H_{\star }=I_{\star }\times J_{\star }\subset V$ be the
closed rectangle with $I_{\star }=\left[ \tfrac{93.5}{180}\pi ,\tfrac{95.5}{
180}\pi \right] $ and $J_{\star }=\left[ 3.030,3.600\right] $. It holds that
$\Gamma $ in $H_{\star }$ is given by $\omega =\varphi (\lambda )$, $\varphi
\in C^{\infty }(\lambda _{\gamma },\lambda _{\delta })$, $(\lambda _{\gamma
},\lambda _{\delta })\subset J_{\star }$ and is such that it attains its
maximum on $\left( \lambda _{\gamma },\lambda _{\delta }\right) $ at
$\lambda =\lambda _{\star }\approx 3.220\dots$ and increases monotonically on
the interval $\left( \lambda _{\gamma },\lambda _{\star }\right) $. Here
$\lambda _{\gamma },\lambda _{\delta }$ are the solutions to the equation
$P\left( \tfrac{93.5}{180}\pi ,\lambda \right) =0$ on $\lambda \in \left(
3.030,3.100\right) $ and on $\lambda \in \left( 3.500,3.600\right) $,
respectively. Also, $\lambda _{\star }$ is the solution to the system
$P(\omega ,\lambda )=0$ and $\frac{\partial P}{\partial \lambda }(\omega
,\lambda )=0$ on $\lambda \in \left( \lambda _{\gamma },\lambda _{\delta
}\right) $; $P$ given by (\ref{9}).
\end{lemma}

\begin{proof}
By Lemma \ref{lemma6} we know that
\begin{equation}
P(\omega ,\lambda )=0\quad\text{if and only if}\quad P\left( \varphi
(\lambda ),\lambda \right) =0 \text{ in }H_{5},  \label{17}
\end{equation}
and if we take the rectangle $H_{\star }$ defined as in lemma above, then
due to $H_{\star }\subset H_{5}$, (\ref{17}) will also hold in $H_{\star }$.
Moreover, we also set $H_{\star }$ in such a way that its left boundary
intersects $\Gamma $ at two points, meaning we find two solutions of $%
P\left( \tfrac{93.5}{180}\pi ,\lambda \right) =0$ with $P$ as in (\ref{9}).
We name these two solutions $\lambda _{\gamma },\lambda _{\delta }$.

Hence, we deduce that $\Gamma $ in $H_{\star }$ is given by $\omega =\varphi
(\lambda )$, $\varphi \in C^{\infty }(\lambda _{\gamma },\lambda _{\delta })$
and satisfies $\varphi (\lambda _{\gamma })=\varphi (\lambda _{\delta })=
\tfrac{93.5}{180}\pi $. Due to condition
\begin{equation*}
\varphi (\lambda _{\gamma })=\varphi (\lambda _{\delta }),
\end{equation*}
by Rolle's theorem there exists $\lambda _{\star }\in \left( \lambda
_{\gamma },\lambda _{\delta }\right) $ such that $\varphi ^{\prime }(\lambda
_{\star })=0$.

Since $P\left( \varphi (\lambda _{\star }),\lambda _{\star }\right) =0$ and
due to
\begin{equation*}
\varphi ^{\prime }(\lambda )=-\tfrac{\partial P}{\partial \lambda }(\varphi
(\lambda ),\lambda )[ \tfrac{\partial P}{\partial \omega }(\varphi (\lambda
),\lambda )] ^{-1},
\end{equation*}
we solve the system $P(\omega ,\lambda )=0$ and $\frac{\partial P}{\partial
\lambda }(\omega ,\lambda )=0$ in $H_{\star }$ in order to find $\lambda
_{\star }$. Its solution is a point $c_2=\big( \tilde{\omega},\tilde{ \lambda%
}\big) $, where $\tilde{\omega}/\pi \approx 0.528\dots$ and $\tilde{ \lambda}%
\approx 3.220\dots$. Hence,
\begin{equation*}
\lambda _{\star }\approx 3.220\dots\,.
\end{equation*}
We deduce that $\omega =\varphi (\lambda )$ attains its local extremum at $%
\lambda =\lambda _{\star }$.

Next we show that $\omega =\varphi (\lambda )$ has a maximum at $\lambda
=\lambda _{\star }$ on $(\lambda _{\gamma },\lambda _{\delta })$. For this
purpose we consider a function $F\in C^{\infty }\left( H_{\star },\mathbb{R}
\right) $ such that
\begin{equation}
F\left( \varphi (\lambda ),\lambda \right) =\varphi ^{\prime \prime
}(\lambda ).  \label{18}
\end{equation}
For explicit formula for $F$ see Appendix \ref{AppendixD}. In Claim \ref%
{Claim14,ap.D} of this Appendix we show that
\begin{equation}
\begin{tabular}{ccc}
$F(\omega ,\lambda) <0\quad\text{on }H_{\star }.$ &  &
\end{tabular}
\label{19}
\end{equation}
This condition together with (\ref{18}) yields
\begin{equation*}
F\left( \varphi (\lambda ),\lambda \right) =\varphi ^{\prime \prime
}(\lambda )<0 \quad\text{on }(\lambda _{\gamma },\lambda _{\delta }),
\end{equation*}
meaning that $\omega =\varphi (\lambda )$ is concave on $(\lambda _{\gamma
},\lambda _{\delta })$. The result is that $\omega =\varphi (\lambda )$
attains its maximum on $\left( \lambda _{\gamma },\lambda _{\delta }\right) $
at $\lambda =\lambda _{\star }\approx 3.220\dots$ and increases
monotonically on the interval $\lambda \in \left( \lambda _{\gamma },\lambda
_{\star }\right) $.
\end{proof}

\begin{theorem}\label{Increase}
As a function of $\omega $ the first eigenvalue
$\lambda_1=\lambda _1(\omega )$ of the boundary value problem \eqref{4}
increases on $\omega \in \left( \frac{1}{2}\pi ,\omega _{\star }\right) $.
Here $\omega _{\star }/\pi \approx 0.528\dots$ (in degrees $\omega _{\star
}\approx 95.1\dots^{\circ }$) and $\lambda _{\star }\approx 3.220\dots$ .
\end{theorem}

\subsection{The multiplicities of $\{ \protect\lambda _{j}\}_{j=1}^{\infty }$
and the structure of a singular solution}

Here we proceed with the qualitative analysis of the eigenvalues $\{ \lambda
_{j}\} _{j=1}^{\infty }$ of problem \eqref{4}.

\begin{definition} \rm
Let $\omega \in ( 0,2\pi] $\ be fixed. The eigenvalue $\lambda
_{j}$, $j\in \mathbb{N} ^{+}$ of problem \eqref{4} is said to have an
algebraic multiplicity\ $\kappa ^{(j)}\geq 1$, if the following holds:
\begin{equation*}
P(\omega ,\lambda _{j})=0, \quad
\tfrac{dP}{d\lambda }(\omega ,\lambda _{j})=0, \quad\dots,\quad
\tfrac{d^{\kappa ^{(j)}-1}P}{d\lambda ^{\kappa
^{(j)}-1}}(\omega ,\lambda _{j})=0, \quad
\tfrac{d^{\kappa ^{(j)}}P}{d\lambda^{\kappa ^{(j)}}}(\omega ,\lambda _{j})
\neq 0.
\end{equation*}
\end{definition}

Based on the numerical approximations for some first eigenvalues $\lambda
_{j}$, $j\in \mathbb{N}^{+}$ depicted in Figure \ref{figure3} (the top one)
and partly by our derivations (namely, the existence of the solution to the
system $P(\omega ,\lambda )=\frac{\partial P}{\partial \lambda }(\omega
,\lambda )=0$ in Lemma \ref{lemma8}) we believe that the maximal algebraic
multiplicity of a certain $\lambda _{j}$ of problem \eqref{4} is at most $2$
. Indeed, generically $3$ curves never intersect at one point, meaning that
geometrically the algebraic multiplicity will always be at most $2$.

\begin{definition} \rm
The eigenvalue $\lambda _{j}$, $j\in \mathbb{N}^{+}$ of problem \eqref{4}
is said to have a geometric multiplicity $I^{(j)}\geq 1$, if the number of
linearly independent eigenfunctions $\Phi $ equals $I^{(j)}$.
\end{definition}

For given $\lambda _{j}$, $j\in \mathbb{N}^{+}$ of problem \eqref{4} the
three cases occur:

1. $\kappa ^{(j)}=I^{(j)}=1$ one finds a solution $(\lambda _{j},\Phi
_0^{(j)})$ of \eqref{4} and then the solution of (\ref{3}) reads:
\begin{equation}
u_0^{(j)}=r^{\lambda _{j}+1}\Phi _0^{(j)}(\theta );  \label{26}
\end{equation}

2. $\kappa ^{(j)}=2$, $I^{(j)}=1$ one finds a solution $(\lambda _{j},\Phi
_0^{(j)})$ of \eqref{4} and a generalized solution $(\lambda _{j},\Phi
_1^{(j)})$, with $\Phi _1^{(j)}$ found from the equation
\begin{equation*}
\mathcal{L}(\lambda _{j})\Phi _1^{(j)}+\mathcal{L}^{\prime }(\lambda
_{j})\Phi _0^{(j)}=0,
\end{equation*}
where $\mathcal{L}(\lambda )$ is given by (\ref{5}) and $\mathcal{L}^{\prime
}(\lambda )=\frac{d}{d\lambda }\mathcal{L}(\lambda )$. Then we have two
solutions of (\ref{3}):
\begin{equation}
u_0^{(j)}=r^{\lambda _{j}+1}\Phi _0^{(j)}(\theta )\text{ \ \ and \ \ }
u_1^{(j)}=r^{\lambda _{j}+1}\left( \Phi _1^{(j)}(\theta )+\log (r)\Phi
_0^{(j)}(\theta )\right) ;  \label{27}
\end{equation}

3. $\kappa ^{(j)}=I^{(j)}=2$ one finds two solutions $(\lambda _{j},\Phi
_0^{(j)})$, $(\lambda _{j},\Phi _1^{(j)})$ of \eqref{4}, where $\Phi
_0^{(j)} $ and $\Phi _1^{(j)}$ are linearly independent on $\theta \in
(0,\omega )$ and then we again have two solutions of (\ref{3}):
\begin{equation}
u_0^{(j)}=r^{\lambda _{j}+1}\Phi _0^{(j)}(\theta )\text{ \ \ and \ \ }
u_1^{(j)}=r^{\lambda _{j}+1}\Phi _1^{(j)}(\theta ).  \label{28}
\end{equation}

Let us note that for an opening angle $\omega \in \{ \frac{1}{2}\pi ,\pi ,%
\frac{3}{2}\pi ,2\pi \} $ of the sector $\mathcal{K}_{\omega }$ one can find
the eigenvalues $\{ \lambda _{j}\} _{j}^{\infty }$ of the problem \eqref{4}
explicitly. Moreover, if $\omega \in \left\{ \frac{1 }{2}\pi ,\pi \right\} $%
, then for every given $\lambda _{j}$ one can compute explicitly the
corresponding eigenfunctions $\Phi _{q}^{(j)}$, $q=0,\dots,\kappa ^{(j)}-1$,
whereas if $\omega \in \left\{ \frac{3}{2}\pi ,2\pi \right\} $, the
eigenfunctions $\Phi _{q}^{(j)}$ can be computed explicitly only for some $%
\lambda _{j}$. Thus, in Appendix \ref{AppendixE} we bring the formulas of
some first functions $\Phi _{q}^{(j)}$ (if computable) and the respective
solutions $u_{q}^{(j)}$ to (\ref{3}).

These functions $r^{\lambda +1}\Phi (\theta )$ and $r^{\lambda +1}\log
(r)\Phi (\theta )$ determine the bands for the regularity in Kondratiev's
theory. Details are found in the next section.

\section{Regularity results\label{Reg}}

In this subsection we will give the regularity result for the boundary value
problem \eqref{1} under consideration. In order to do this we refer to the
key theorem of the Kondratiev theory (see e.g. \cite[Theorem 6.4.1]{KMR}).
The general version adapted to our problem \eqref{1} will read as:

\begin{theorem}[Kondratiev] \label{Theorem1}
Let $u\in V_{\beta _1}^{l_1,2}(\Omega )$ with $l_1\in
\mathbb{N}$, $\beta _1\in \mathbb{R}$ be a solution of the elliptic
boundary value problem \eqref{1}.

Suppose that $f\in V_{\beta _2}^{l_2,2}(\Omega )$, where $l_2\in
\mathbb{N}$, $\beta _2\in \mathbb{R}$ and such that
$l_1-\beta _1<l_2-\beta _2+4$.
If no eigenvalue $\lambda _{j}$ of problem \eqref{4} lies on the lines
\begin{equation*}
\mathop{\rm Re}(\lambda )=l_1-\beta _1-2, \quad
\mathop{\rm Re}(\lambda )=l_2-\beta _2+2,
\end{equation*}
while the strip
\begin{equation*}
l_1-\beta _1-2<\mathop{\rm Re}(\lambda )<l_2-\beta _2+2,
\end{equation*}
contains the eigenvalues $\lambda _{n},\lambda _{n+1}\dots,\lambda _{n+N}$,
then $u$ has the representation
\begin{equation}
u=w+\chi (r)\sum_{j=n}^{n+N}\sum_{q=0}^{\kappa
^{(j)}-1}c_{q}^{(j)}u_{q}^{(j)},  \label{62}
\end{equation}
where $w\in V_{\beta _2}^{l_2+4,2}(\Omega )$,
$\chi \in C_0^{\infty }\left[ 0,\varepsilon \right) $
is a cut-off function such that $\chi (r)=1$
in the neighborhood of $r=0$, $\kappa ^{(j)}\leq 2$ is the algebraic
multiplicity of $\lambda _{j}$ and $u_{q}^{(j)}$ are the solutions of the
problem (\ref{3}) in $\mathcal{K}_{\omega }$ given by formulas
\eqref{26}, \eqref{27}, \eqref{28}.
\end{theorem}

Let us recall that by Theorem \ref{Weak solution} there exists a unique weak
solution $u\in {\mathaccent"7017 W}$$^{2,2}(\Omega )$ of \eqref{1} with $%
f\in L^2(\Omega )$. Since $L^2(\Omega )=V_0^{0,2}(\Omega )$ and\ by
Corollary \ref{coro-imbed} one has ${\mathaccent"7017 W}^{2,2}(\Omega )= {%
\mathaccent"7017 V}${}$_0^{2,2}(\Omega )\subset V_0^{2,2}(\Omega )$, we
conclude that for $f\in V_0^{0,2}(\Omega )$ we have $u\in V_0^{2,2}(\Omega )$%
.

Then assuming more regularity for $f\in V_0^{0,2}(\Omega )$ we apply Theorem %
\ref{Theorem1} to our problem \eqref{1}. Using Lemma \ref{lemma10} we may
consider three different cases:
\begin{equation*}
f\in
\begin{cases}
V_{\beta }^{k,2}(\Omega ), & k\geq 0,\;\beta \geq k, \\
{\mathaccent"7017 W}{}^{k,2}(\Omega ), & k\geq 1, \\
W^{k,2}(\Omega ), & k\geq 0,%
\end{cases}%
\end{equation*}
and obtain the following result (in order to describe all three cases and
also for the convenience we arranged this result as a table):

\begin{theorem}\label{Theorem2}
Let $f\in L^2(\Omega )$ and let $u\in {\mathaccent"7017 W}{}^{2,2}(\Omega )$
 be a weak solution to \eqref{1}.

\begin{center} \renewcommand{\arraystretch}{1.5}
\begin{tabular}{|c|c|c|c|}
\hline
 \rm $f$ is in &\hspace{-1mm} $V_{\beta }^{k,2}(\Omega
)$, $k\geq 0$, $\beta \geq k$\hspace{-2mm}& ${\mathaccent"7017 W}^{k,2}(\Omega )$,
$k\geq 1$ & $W^{k,2}(\Omega )$, $k\geq 0$ \\ \hline
\parbox{21.5mm}{\rm no eigenvalue $\lambda _{j}$ of \eqref{4} lies on the lines}
& $\begin{cases}
\mathop{\rm Re}(\lambda )=0, \\
\mathop{\rm Re}(\lambda )=k-\beta +2
\end{cases}$\rule[-16pt]{0pt}{38pt}
& $\begin{cases}
\mathop{\rm Re}(\lambda )=0, \\
\mathop{\rm Re}(\lambda )=k+2
\end{cases} $\hspace{-2mm}
& $\begin{cases}
\mathop{\rm Re}(\lambda )=0, \\
\mathop{\rm Re}(\lambda )=2
\end{cases} $ \\ \hline
\rm while the strip
&\hspace{-1mm}$0<\mathop{\rm Re}(\lambda )<k-\beta +2 $\hspace{-2mm}%
&\hspace{-.5mm}$0<\mathop{\rm Re}(\lambda )<k+2$\hspace{-2mm}& $0<\mathop{\rm Re}(\lambda )<2$ \\ \hline
\rm contains
& \multicolumn{3}{|c|}{$\lambda _1,\lambda _2,\dots,\lambda _{N}$} \\ \hline
& \multicolumn{3}{|c|}{$u=w+\chi (r)\sum_{j=1}^{N}\sum_{q=0}^{\kappa
^{(j)}-1}c_{q}^{(j)}u_{q}^{(j)}$\rule[-7pt]{0pt}{22pt}} \\ \hline
\rm where $w$ is in
& $V_{\beta }^{k+4,2}(\Omega )$
& $W^{k+4,2}(\Omega )$
& \hspace{-1mm}$W^{k+4,2}(\Omega ,| x| ^{2k}d\mu )$\hspace{-1mm} \\ \hline
\end{tabular}
\renewcommand{\arraystretch}{1}
\end{center}
and $\chi \in C_0^{\infty }\left[ 0,\varepsilon \right) $ is a cut-off
function such that $\chi =1$ in the neighborhood of $r=0$, $\kappa
^{(j)}\leq 2$ is the algebraic multiplicity of $\lambda _{j}$ and
$u_{q}^{(j)}$ are the solutions of the problem (\ref{3}) in $\mathcal{K}
_{\omega }$ given by formulas \eqref{26}, \eqref{27}, \eqref{28}.
\end{theorem}

\begin{remark} \rm
The first column gives the optimal regularity in the sense of Kondratiev's
spaces. The second and third column shows two corollaries with the more
commonly used Sobolev spaces. Away from the corner also these results are
optimal. Of course the optimal regularity near a corner can not be stated
using just the standard Sobolev spaces $W^{\ell ,2}(\Omega )$.
\end{remark}

\begin{proof}[Proof of Theorem {Theorem2}]
The first column is just a representation of the previous theorem in the
case that $f\in L^2(\Omega )$ and a weak solution $u\in {\mathaccent"7017 W}
$$^{2,2}(\Omega )$ is known to exist. Additional regularity in the sense of
Kondratiev's weighted Sobolev spaces for $f$ implies the representation as
stated for the solution $u$. In the second and third column the most common
special cases are listed independently. For the second column one uses the
imbeddings
\begin{equation*}
{\mathaccent"7017 W}{}^{k,2}(\Omega )\subset V_0^{k,2}(\Omega )\text{ and }
V_0^{k+4,2}(\Omega )\subset W^{k+4,2}(\Omega ),
\end{equation*}
and for the third
\begin{equation*}
W^{k,2}(\Omega )\subset V_{k}^{k,2}(\Omega )\text{ and }V_{k}^{k+4,2}(\Omega
)\subset W^{k+4,2}(\Omega ,| x| ^{2k}d\mu ).
\end{equation*}
\end{proof}

As the last step of our analysis we derive the regularity in $\Omega $ for
the singular part $\sum_{j=1}^{N}\sum_{q=0}^{\kappa
^{(j)}-1}c_{q}^{(j)}u_{q}^{(j)}$ of the solution $u$ given in Theorem \ref%
{Theorem2}.

\subsection{Regularity for the singular part of $u$}

The first term of the summation $\sum_{j=1}^{N}\sum_{q=0}^{%
\kappa^{(j)}-1}c_{q}^{(j)}u_{q}^{(j)}$ in the solution $u$ defines the
regularity of the whole sum. From formulas \eqref{26}, \eqref{27}, \eqref{28}
we know that depending on the algebraic multiplicity of $\lambda _1$ it
reads as
\begin{equation*}
r^{\lambda _1+1}\Phi (\theta) \quad\text{or}\quad r^{\lambda_1+1}\left( \Psi
(\theta )+\log (r)\Phi (\theta )\right) ,
\end{equation*}
where $\lambda _1\in \mathbb{C} $ is the first eigenvalue of \eqref{4} such
that $\mathop{\rm Re}(\lambda _1)>0 $ and $\Phi (\theta) ,\Psi (\theta )\in
C^{\infty }[ 0,\omega] $, $0<\omega <2\pi $.

\begin{lemma}\label{lemma18}
Let $\Phi (\theta) \in C^{\infty }[0,\omega] $ be nontrivial and
$0<\omega <2\pi $. Let also $\lambda
\in \mathbb{C} \backslash \mathbb{Z}$ with $\mathop{\rm Re}(\lambda )>0$.
Suppose that $k\in \{ 0,1,2,3,\dots\} $. Then the following are
equivalent:
\begin{enumerate}
\item $r^{\lambda +1}\Phi (\theta) \in W^{k,2}(\Omega )$,

\item $r^{\lambda +1}\log (r)\Phi (\theta) \in W^{k,2}(\Omega) $,

\item $\mathop{\rm Re}(\lambda )+1>k-1$.
\end{enumerate}
\end{lemma}

\begin{proof}
If $\lambda $ is not an integer, then the first item and second items are
equivalent with an integrability condition for the $k^{\text{th}}$
-derivative that reads as $2\mathop{\rm Re}( \lambda +1-k) +1>-1$.
\end{proof}

\begin{remark} \rm
To restrict the already heavy technical aspects we have not
considered (weighted) Sobolev spaces with non-integer coefficients $k$ and 
H\"{o}lder spaces. A similar result will hold for $k$ is noninteger.
Concerning H\"{o}lder spaces:
\begin{equation*}
r^{\lambda +1}\Phi (\theta) \in C^{k,\gamma }(\bar{\Omega})\quad
\text{for }\mathop{\rm Re}(\lambda )+1\geq k+\gamma \text{ with }
k\in \mathbb{N},\gamma \in [ 0,1) .
\end{equation*}
For the second function it holds that
\begin{equation*}
r^{\lambda +1}\log (r)\Phi (\theta) \in C^{k,\gamma }(\bar{
\Omega})\quad \text{for }\mathop{\rm Re}(\lambda )+1>k+\gamma \text{ with }k\in
\mathbb{N},\gamma \in [ 0,1) .
\end{equation*}
\end{remark}

A useful consequence of the above lemma is that for every fixed $\lambda \in
\mathbb{C} $ with $\mathop{\rm Re}(\lambda) >0$ we deduce
\begin{equation}
\left\{ r^{\lambda +1}\Phi (\theta) , r^{\lambda +1}\log (r)\Phi (\theta)
\right\} \in W^{\lceil \mathop{\rm Re}(\lambda) \rceil +1,2}(\Omega ),
\label{68}
\end{equation}
where $\lceil \cdot \rceil $ stands for the ceiling function (defined as $%
\lceil x\rceil =\min \{ n\in \mathbb{Z} :x\leq n\} $).

\begin{remark} \rm
In the particular cases, namely, $\omega =\frac{1}{2}\pi $ and
$\omega =\pi $
we know that each term $u_{q}^{(j)}$ of the singular part
$\sum_{j=1}^{N}\sum_{q=0}^{\kappa ^{(j)}-1}c_{q}^{(j)}u_{q}^{(j)}$ is a
polynomial in $x,y$ of order $\lambda _{j}+1$
(see Appendix \ref{AppendixE}). That is, for every $\lambda _{j}$,
$j\in \mathbb{N}^{+}$ we have
\begin{equation}
r^{\lambda _{j}+1}\Phi ^{(j)}(\theta) =P_{\lambda
_{j}+1}(x,y)\in C^{\infty }\left( \overline{\Omega }\right) .  \label{69}
\end{equation}
For non-polynomials the result in Lemma \ref{lemma18} even holds for
$\lambda \in \mathbb{N}$.
\end{remark}

Now, in order to use the result of Lemma \ref{lemma18}, we proceed with
Figure \ref{figure8} where we plot the $\mathop{\rm Re}(\lambda _1) $ as a
function of the opening angle $\omega $ on the interval $\omega \in (0,2\pi
] $. The two cases are compared: the plots of $\mathop{\rm Re} (\lambda _1) $
of the boundary value problem \eqref{4}\ for $\mathcal{L}$ related to $L=%
\tfrac{\partial ^{4}}{\partial x^{4}}+\tfrac{ \partial ^{4}}{\partial y^{4}}$
and $L=\Delta ^2$. In Figure \ref{figure9} we split the plot of Figure \ref%
{figure8} into two: a plot of $\mathop{\rm Re} (\lambda _1) $ on $\omega \in
(0,\pi ]$ and $\omega \in [\pi ,2\pi] $.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig8} % Figure(FirstEigenvalue)
\end{center}
\caption{The plot of eigenvalue $\protect\lambda _1$ in $\left(\protect%
\omega ,\mathop{\rm Re}(\protect\lambda) \right) \in ( 0,2\protect\pi]
\times \left[ 0,7.200\right] $ of problem ( \protect\ref{4}). For $\mathcal{L%
}$ related to $\frac{\partial ^{4}}{\partial x^{4}}+\frac{\partial ^{4}}{%
\partial y^{4}}$, $\protect\lambda _1$ is represented by the red line and
for $\mathcal{L}$ related to $\Delta ^2$, by the blue line. Dashed lines
depict the real part of $\protect\lambda _1\in \mathbb{C} $, solid lines are
for purely real $\protect\lambda _1$; the vertical lines mark out $\left\{%
\frac{1}{2}\protect\pi , \protect\pi , \frac{3}{2}\protect\pi ,2\protect\pi %
\right\}$ on $\protect\omega $-axis.}
\label{figure8}
\end{figure}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig9a} \\[0pt]
\includegraphics[width=0.6\textwidth]{fig9b}
% Figure(FirstEigenvalue2).pdf}
\end{center}
\caption{The plot of Figure \protect\ref{figure8} rescaled. The curves for
the grid-operator and the bilaplacian intersect four times. Three of these
points (we obviously exclude $\protect\omega =\protect\pi $) seem to be
special: It looks like the curves intersect at $\protect\omega =\frac{3}{4}%
\protect\pi $, $\protect\omega =\frac{5}{4} \protect\pi $ and $\protect%
\omega =\frac{7}{4}\protect\pi $ (the intersection points are marked by
cross). The numerical approximation shows however that the first values $%
\protect\lambda _{1,Grid}$ and $\protect\lambda _{1,Bilaplace}$ at those
points only coincide up to three digits.}
\label{figure9}
\end{figure}

Based on the numerical approximations to $\lambda _1$ and partly on the
analytical estimates for $\lambda _1$ we conclude the following.

\begin{claim}\label{Claim2} \rm
$(0,2\pi ]\ni \omega \mapsto \mathop{\rm Re}(\lambda _1(\omega
)) $ is a continuous function and
\begin{gather*}
\text{for } \omega \in (0,\omega _1)/\{ \frac{1}{2}\pi\}:
\mathop{\rm Re}(\lambda _1)>3,\quad
\text{for } \omega =\frac{1}{2}\pi : \mathop{\rm Re}(\lambda _1)=3, \\
\text{for } \omega \in [ \omega _1,\omega _2): 3\geq \mathrm{Re
}(\lambda _1)>2, \quad
\text{for } \omega \in [ \omega _2,\pi ): 2\geq \mathop{\rm Re}
(\lambda _1)>1,\\
\text{for } \omega \in [ \pi ,2\pi ]: 1\geq \mathop{\rm Re}(\lambda
_1)\geq \frac{1}{2}.
\end{gather*}
Here $\omega _1,\omega _2$ are respectively the solutions of $P(\omega
,3+i\xi )=0$ on $\omega \in \left( \frac{1}{2}\pi ,\frac{120}{180}\pi
\right) $ and of $P(\omega ,2+i\xi )=0$ on $\omega \in \left( \frac{2}{3}\pi
,\frac{3}{4}\pi \right) $, where $P$ as in formula (\ref{6.1}) for $\ell =1$
. The approximation are $\omega _1/\pi \approx 0.555\dots$ (in degrees
$\omega _1\approx 99.9\dots^{\circ }$) and $\omega _2/\pi \approx 0.720\dots$
(in degrees $\omega _2\approx 129.7\dots^{\circ }$).
\end{claim}

\subsection{Consequences}

For the numerical results from Claim \ref{Claim2}, it holds by Theorem \ref%
{Theorem2} that:

\begin{corollary}\label{Corollary100}
Let $u\in {\mathaccent"7017 W}^{2,2}(\Omega )$ be a
weak solution of problem \eqref{1} with $f\in L^2(\Omega )$. Then
\begin{gather*}
\text{for } \omega \in ( 0,\omega _2) : u\in W^{4,2}(\Omega),\quad
\text{for }\omega \in (\omega _2,\pi ): u\in W^{3,2}(\Omega ),\\
\text{for }\omega =\pi : u\in W^{4,2}(\Omega ),\quad
\text{for }\omega \in (\pi ,2\pi ]: u\in W^{2,2}(\Omega ).
\end{gather*}
Here $\omega _2$ is as in Claim \ref{Claim2}.
\end{corollary}

\begin{remark} \rm
For the opening angle $\omega =\omega _2$ we have
$\mathop{\rm Re}(\lambda _1) =2$ and hence Theorem \ref{Theorem2} does not apply.
Nevertheless, assuming $f\in L^2(\Omega )$ to be more regular, e.g. in
$V_0^{1,2}(\Omega )$ or ${\mathaccent"7017 W}^{1,2}(\Omega )$, we may
show that
\begin{equation*}
\text{for }\omega =\omega _2: u\in W^{3,2}(\Omega ).
\end{equation*}
\end{remark}

\begin{proof}
By Theorem \ref{Theorem2} if $f\in L^2(\Omega )$, the solution $u$ of
problem \eqref{1} reads as
\begin{equation}
u=w+\chi (r)\sum_{0<\lambda _{j}<2}\sum_{q=0}^{\kappa
^{(j)}-1}c_{q}^{(j)}u_{q}^{(j)},  \label{70}
\end{equation}
with $w\in W^{4,2}(\Omega )$. Due to \ref{Claim2} we see that the sum in ( %
\ref{70}) has no terms when $\omega \in (0,\omega _2) $ and hence for $%
\omega \in (0,\omega _2) $ we have $u\in W^{4,2}(\Omega )$.

For $\omega \in (\omega _2,\pi )\cup (\pi ,2\pi ]$ the first term of sum in (%
\ref{70}), depending on the algebraic multiplicity of $\lambda _1$, reads as
$u_0^{(1)}=r^{\lambda _1+1}\Phi _0^{(1)}(\theta )$, or as a linear
combination of $u_0^{(1)}=r^{\lambda _1+1}\Phi _0^{(1)}(\theta ) $\ and\ $%
u_1^{(1)}=r^{\lambda _1+1}\left( \Phi _1^{(1)}(\theta )+\log (r)\Phi
_0^{(1)}(\theta )\right) $. By (\ref{68}) we have that $\left\{
u_0^{(1)},u_1^{(1)}\right\} \in W^{\lceil \mathop{\rm Re}(\lambda) \rceil
+1,2}(\Omega )$, where due to Claim \ref{Claim2} we may deduce that $\lceil %
\mathop{\rm Re}(\lambda _1) \rceil +1=3$, when $\omega \in (\omega _2,\pi )$
and $\lceil \mathop{\rm Re}(\lambda _1) \rceil +1=2$, when $\omega \in (\pi
,2\pi ]$. This results in $u\in W^{3,2}(\Omega )$ for $\omega \in (\omega
_2,\pi )$ and $u\in W^{2,2}(\Omega )$ for $\omega \in (\pi ,2\pi ]$.

Finally, for $\omega =\pi $ due to (\ref{69}) the singular part is of $%
C^{\infty }( \overline{\Omega }) $ and hence $u\in W^{4,2}(\Omega )$ in this
case.
\end{proof}

\section{Comparing (weighted) Sobolev spaces}

\subsection{One-dimensional Hardy-type inequalities}

\begin{lemma}[A higher order one-dimensional Hardy inequality]
\label{lemma1,ap.F}
Let $w$ be a function in $C_0^{\infty }[ x_1,x_2] $. For
every $k\geq 1$ it holds that
\begin{equation}
\int_{x_1}^{x_2}\left( \tfrac{w(x)}{(x-x_1) ^{k}}
\right) ^2dx=\tfrac{4^{k}}{\left( 2k-1\right) ^2\left( 2k-3\right)
^2\dots3^21^2}\int_{x_1}^{x_2}\left( w^{(k)}(x)\right) ^2dx.
\label{F2}
\end{equation}
\end{lemma}

\begin{proof}
It holds straightforwardly that
\begin{align*}
&\int_{x_1}^{x_2}\left( \tfrac{w(x)}{(x-x_1) ^{k}}\right)^2dx \\
&= \frac{1}{1-2k} \left[ \left( w(x)\right) ^2(x-x_1) ^{1-2k}\right] |
_{x_1}^{x_2}+\frac{2}{2k-1} \int_{x_1}^{x_2}w(x)w^{\prime }(x)(x-x_1)
^{1-2k}dx \\
&\leq \frac{2}{2k-1}\Big( \int_{x_1}^{x_2} \left( \tfrac{w(x)}{(x-x_1) ^{k}}%
\right) ^2dx\Big) ^{1/2} \Big(\int_{x_1}^{x_2}\left( \tfrac{w^{\prime }(x)}{%
(x-x_1)^{k-1}}\right) ^2dx \Big) ^{1/2}
\end{align*}
and the first step in the proof of (\ref{F2}) follows. Repeating the
argument for $w^{\prime }$ and $k-1$ etc. will give the result.
\end{proof}

\begin{remark} \rm
Since ${\mathaccent"7017 W}^{k,2}(x_1,x_2)$, $k\geq 1$ is the closure
of $C_0^{\infty }[ x_1,x_2] $ in the $W^{k,2}$-norm, one
can use the results of Lemma \ref{lemma1,ap.F} for every
$w\in {\mathaccent"7017 W}^{k,2}(x_1,x_2)$, $k\geq 1$.
\end{remark}

\subsection{Imbeddings}

As mentioned e.g. in \cite[page 240]{K} or \cite[Chapter 7, summary]{KMR},
the family of weighted spaces $V_{\beta }^{l,2}$ does not contain the
ordinary Sobolev spaces without weight. More precisely: $W^{k,2}\notin
\left\{ V_{\beta }^{l,2}\right\} _{l,\beta }$ for $k\geq 1$. We will prove
the imbedding results for bounded $\Omega $ that satisfy Condition \ref%
{Condition0}.

\begin{lemma} \label{lemma10}
Let $\beta \in \mathbb{R}$ and $l\in \left\{ 0,1,2,\dots
\right\} $. Then the following holds:
\begin{itemize}
\item[(a)]  $V_{\beta }^{l,2}(\Omega )\subset W^{l,2}(\Omega )$
if and only if $\beta \leq 0$,
\item[(b)] $ W^{l,2}(\Omega )\subset V_{\beta }^{l,2}(\Omega )$
if and only if $\beta \geq l$,
\item[(c)] ${\mathaccent"7017 V}_{\beta }^{l,2}(\Omega )\subset
{\mathaccent"7017 W}^{l,2}(\Omega )$ if and only if $
\beta \leq 0$,
\item[(d)] ${\mathaccent"7017 W}^{l,2}(\Omega )\subset
{\mathaccent"7017 V}_{\beta }^{l,2}(\Omega )$ if and only if $\beta \geq 0$.
\end{itemize}
\end{lemma}

\begin{corollary}\label{coro-imbed}
For $l\in \{0,1,2,\dots\}$ one has
\begin{equation*}
{\mathaccent"7017 W}{}^{l,2}(\Omega )
= {\mathaccent"7017 V}_0^{l,2}(\Omega ).
\end{equation*}
\end{corollary}

\begin{proof}[Proof of Lemma \protect\ref{lemma10}]
Let $\Omega $ be as in Condition \ref{Condition0} and $\Omega \subset
B_{M}(0)$, where $B_{M}(0)$ is an open ball of radius $M>0$. The statement
in a) goes as follows: for $(x,y)\in \Omega $ one has $0\leq r\leq M$\ and
hence $r^{2\left( \beta -l+| \alpha | \right) }\geq M^{2\left( \beta -l+|
\alpha | \right) }$ if and only if $\beta -l+| \alpha | \leq 0$. Since $%
0\leq | \alpha | \leq l$, we obtain $\beta \leq 0$. This enables us to have
the estimate
\begin{align*}
\| u\| _{V_{\beta }^{l,2}(\Omega )} &=\Big( \sum_{|\alpha |
=0}^{l}\int_{\Omega }r^{2( \beta -l+| \alpha |) }| D^{\alpha }u|^2dx\,dy%
\Big) ^{1/2} \\
&\geq \Big( \sum_{| \alpha| =0}^{l}\int_{\Omega }M^{2( \beta -l+| \alpha |)
}| D^{\alpha }u| ^2dx\,dy\Big) ^{1/2} \\
&\geq \min (1,M^{\beta -l})\Big( \sum_{| \alpha |=0}^{l}\int_{\Omega }|
D^{\alpha }u| ^2dx\,dy\Big) ^{1/2} \\
&=\min (1,M^{\beta -l})\| u\|_{W^{l,2}(\Omega )},
\end{align*}
which is the result in (a).

To prove the statement in (b) we notice that $r^{2( \beta -l+|\alpha |)
}\leq M^{2( \beta -l+| \alpha|) }$ if and only if $\beta -l+| \alpha | \geq 0
$. Due to $0\leq | \alpha | \leq l$, we obtain $\beta \geq l$ and then the
estimate holds
\begin{align*}
\| u\| _{V_{\beta }^{l,2}(\Omega )} &=\Big( \sum_{| \alpha |
=0}^{l}\int_{\Omega }r^{2\left( \beta -l+| \alpha | \right) }| D^{\alpha
}u|^2dx\,dy\Big) ^{1/2} \\
&\leq \Big( \sum_{| \alpha | =0}^{l}\int_{\Omega }M^{2\left( \beta -l+|
\alpha | \right) }| D^{\alpha }u| ^2dx\,dy\Big) ^{1/2} \\
&\leq \max (1,M^{\beta -l})\Big( \sum_{| \alpha | =0}^{l}\int_{\Omega }|
D^{\alpha }u| ^2dx\,dy\Big) ^{1/2} \\
&=\max (1,M^{\beta -l})\| u\|_{W^{l,2}(\Omega )}.
\end{align*}
This is the result in (b).

To prove the statements in (c) and (d) we set $\theta =\frac{1}{2} \omega $
where the opening angle $\omega \in (0,2\pi) $. We also use the fact that
for our domain there exists $c>0$ such that $r>c\rho (x,y)$, where $\rho $
denotes the distance from a point $(x,y)$ on the lines
\begin{equation*}
\ell :y=\tan (\theta) x+\tau ,
\end{equation*}
with $\tau \in \mathbb{R}$ to the point $(x_1,y_1)\in \partial \Omega $. In
particular, it holds that
\begin{equation*}
\rho ^2=(x-x_1) ^2(1+\tan ^2(\theta)) .
\end{equation*}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.65\textwidth]{fig10}% Figure(Hardy)
\end{center}
\caption{Domain $\Omega $ with a concave corner $\protect\omega $
intersected by $\ell $}
\label{figure10}
\end{figure}

We may integrate along the lines $\ell $ and use the one-dimensional
Hardy-inequality to find that there exist $\tilde{C}_{l}\in \mathbb{R}^{+}$
with
\begin{equation}
\| u\| _{V_0^{l,2}(\Omega )}\leq \tilde{C}_{l}\| u\| _{W^{l,2}(\Omega
)}\quad \text{ for all }u\in C_{c}^{\infty } (\Omega) .  \label{F.3}
\end{equation}
On the other hand, using the same trick as in proof of a) we find $C_{l}\in
\mathbb{R}^{+}$ such that
\begin{equation}
C_{l}\| u\| _{W^{l,2}(\Omega )}\leq \| u\| _{V_0^{l,2}(\Omega )}\quad \text{%
for all }u\in C_{c}^{\infty } (\Omega) .  \label{F.4}
\end{equation}
Estimates (\ref{F.3}), (\ref{F.4}) yield
\begin{equation*}
{\mathaccent"7017 W}^{l,2}(\Omega )={\mathaccent"7017 V}_0^{l,2} (\Omega).
\end{equation*}
Due to imbedding ${\mathaccent"7017 V}_{\beta _1}^{l,2}(\Omega )\subset {%
\mathaccent"7017 V}_0^{l,2}(\Omega )\subset {\mathaccent"7017 V}_{\beta
_2}^{l,2}(\Omega )$ when $\beta _1\leq 0\leq \beta _2$ one obtains the
result in (c) and (d).
\end{proof}

\section{A fundamental system of solutions\label{AppendixC}}

\subsection{Derivation of system $S_{\protect\lambda }$}

Let us find the fundamental set of solutions to equation
\begin{equation}
\mathcal{L}( \theta ,\tfrac{d}{d\theta },\lambda ) \Phi =0,  \label{C1}
\end{equation}
with $\mathcal{L}( \theta ,\tfrac{d}{d\theta },\lambda) $ as in formula (\ref%
{5}). For this $\mathcal{L}$ it seems to be hard to derive a set of
functions solving (\ref{C1}) explicitly. The following approach applies in
this case.

For $L=\tfrac{\partial ^{4}}{\partial x^{4}}+\tfrac{\partial ^{4}}{\partial
y^{4}}$ we find $L\left( r^{\lambda +1}\Phi \right) =r^{\lambda -3}\mathcal{%
L }\left( \theta ,\tfrac{d}{d\theta },\lambda \right) \Phi $ and hence
instead of $\mathcal{L}\left( \theta ,\tfrac{d}{d\theta },\lambda \right)
\Phi =0$ we may consider the equation
\begin{equation}
L\left( r^{\lambda +1}\Phi \right) =0.  \label{C2}
\end{equation}
Operator $L$ admits the decomposition
\begin{equation*}
L=\tfrac{\partial ^{4}}{\partial x^{4}}+\tfrac{\partial ^{4}}{\partial y^{4}}
=\prod_{p=1}^2\left( \tfrac{\partial }{\partial y}-\tau _{p}\tfrac{ \partial
}{\partial x}\right) \prod_{p=1}^2\left( \tfrac{\partial }{ \partial y}+\tau
_{p}\tfrac{\partial }{\partial x}\right) ,
\end{equation*}
with $\tau _1=\frac{\sqrt{2}}{2}(1+i) $, $\tau _2=\frac{\sqrt{2}}{2}(1-i) $
and hence every function of the form $F(x\pm \tau _{p}y)$ solves (\ref{C2}).
Therefore, we have that
\begin{equation*}
r^{\lambda +1}\Phi (\theta )=\sum_{p=1}^2c_{p}f_{p}(x+\tau
_{p}y)+c_{p+2}f_{p+2}(x-\tau _{p}y),
\end{equation*}
and after translation $\{ f_{p},f_{p+2}\} _{p=1}^2$ into polar coordinates
we set
\begin{gather*}
f_{p}\left( r\cos (\theta )+\tau _{p}r\sin (\theta )\right) :=\left( r\cos
(\theta )+\tau _{p}r\sin (\theta )\right) ^{\lambda +1}, \\
f_{p+2}\left( r\cos (\theta )-\tau _{p}r\sin (\theta )\right) :=\left( r\cos
(\theta )-\tau _{p}r\sin (\theta )\right) ^{\lambda +1},
\end{gather*}
So, the set of functions
\begin{equation}
\varphi _1(\theta )=\left( \cos (\theta )+\tau _1\sin (\theta )\right)
^{\lambda +1},\quad \varphi _2(\theta )=\left( \cos (\theta )+\tau _2\sin
(\theta )\right) ^{\lambda +1},  \label{C-1}
\end{equation}
\begin{equation}
\varphi _{3}(\theta )=\left( \cos (\theta )-\tau _1\sin (\theta )\right)
^{\lambda +1},\quad \varphi _{4}(\theta )=\left( \cos (\theta )-\tau _2\sin
(\theta )\right) ^{\lambda +1},  \label{C-2}
\end{equation}
is a set of solutions to (\ref{C1}). The Wronskian for $\{ \varphi _{m}\}
_{m=1}^{4}$ reads as
\begin{equation}
W\left( \varphi _1(\theta ),\varphi _2(\theta ),\varphi _{3}(\theta
),\varphi _{4}(\theta )\right) =\det
\begin{pmatrix}
\varphi _1(\theta ) & \varphi _2(\theta ) & \varphi _{3}(\theta ) & \varphi
_{4}(\theta ) \\
\varphi _1^{\prime }(\theta ) & \varphi _2^{\prime }(\theta ) & \varphi
_{3}^{\prime }(\theta ) & \varphi _{4}^{\prime }(\theta ) \\
\varphi _1^{\prime \prime }(\theta ) & \varphi _2^{\prime \prime }(\theta )
& \varphi _{3}^{\prime \prime }(\theta ) & \varphi _{4}^{\prime \prime
}(\theta ) \\
\varphi _1^{\prime \prime \prime }(\theta ) & \varphi _2^{\prime \prime
\prime }(\theta ) & \varphi _{3}^{\prime \prime \prime }(\theta ) & \varphi
_{4}^{\prime \prime \prime }(\theta )%
\end{pmatrix}%
,  \label{C0}
\end{equation}
and by straightforward computations one finds
\begin{equation*}
W=16\left( \lambda +1\right) ^{3}\lambda ^2\left( \lambda -1\right) \left(
\cos ^{4}(\theta )+\sin ^{4}(\theta) \right) ^{\lambda -2},
\end{equation*}
which is non-zero on $\theta \in (0,2\pi ]$ except for $\lambda \in \{\pm
1,0\} $. Hence, except for these values $\{ \varphi _{m}\} _{m=1}^{4}$ given
in (\ref{C-1}), (\ref{C-2}) is a fundamental system of solutions to (\ref{C1}%
).

\subsection{Derivation of systems $S_{-1}$, $S_0$, $S_1$}

Here we find the fundamental systems of solutions to equation $\mathcal{L}
\left( \theta ,\tfrac{\partial }{\partial \theta },\lambda \right) \Phi =0$
when $\lambda \in \left\{ \pm 1,0\right\} $. We will go into details in
solving the corresponding equation for every $\lambda \in \left\{ \pm
1,0\right\} $.

\subsubsection{Case $\protect\lambda =-1$}

For $\lambda =-1$ the equation (\ref{C1}) reads as
\begin{equation}
\tfrac{1}{4}\left( 3+\cos (4\theta )\right) \Phi ^{\prime \prime \prime
\prime }-3\sin (4\theta )\Phi ^{\prime \prime \prime }+\left( 3-11\cos
(4\theta )\right) \Phi ^{\prime \prime }+12\sin (4\theta )\Phi ^{\prime }=0.
\label{C3}
\end{equation}
First we set $\Phi (\theta )=\int F(\theta )d\theta $ and obtain the
equation for $F$:
\begin{equation*}
\tfrac{1}{4}\left( 3+\cos (4\theta )\right) F^{\prime \prime \prime }-3\sin
(4\theta )F^{\prime \prime }+\left( 3-11\cos (4\theta )\right) F^{\prime
}+12\sin (4\theta )F=0.
\end{equation*}
The first integral of the above equation reads as
\begin{equation*}
\tfrac{1}{4}\left( 3+\cos (4\theta )\right) F^{\prime \prime }-2\sin
(4\theta )F^{\prime }+3(1-\cos (4\theta ))F=c_0.
\end{equation*}
We use the change of variables $F(\theta )=\left( 3+\cos (4\theta )\right)
^{-1}G(\theta )$ and get
\begin{equation*}
G^{\prime \prime }+4G=4c_0.
\end{equation*}
Solution of the last equation reads as
\begin{equation*}
G(\theta )=c_1\sin (2\theta )+c_2\cos (2\theta )+c_0.
\end{equation*}
and then
\begin{equation*}
F(\theta )=c_1\tfrac{\sin (2\theta )}{3+\cos (4\theta )}+c_2\tfrac{\cos
(2\theta )}{3+\cos (4\theta )}+c_{3}\tfrac{1}{3+\cos (4\theta )}.
\end{equation*}
As a result, $\Phi $ that solves (\ref{C3}) will read as
\begin{equation*}
\Phi (\theta )=A_1+A_2\int \tfrac{\sin (2\theta )}{3+\cos (4\theta )}
d\theta +A_{3}\int \tfrac{\cos (2\theta )}{3+\cos (4\theta )}d\theta
+A_{4}\int \tfrac{1}{3+\cos (4\theta )}d\theta ,
\end{equation*}
and then the candidates that may form the fundamental system of solutions to
(\ref{C3}) will be the following:
\begin{gather*}
\varphi _1(\theta )=1, \\
\varphi _2(\theta )=-4\int \tfrac{\sin (2\theta )}{3+\cos (4\theta )}
d\theta =\arctan \left( \cos (2\theta )\right) , \\
\varphi _{3}(\theta )=4\sqrt{2}\int \tfrac{\cos (2\theta )}{3+\cos (4\theta
) }d\theta =\mathrm{arctanh}\left( \tfrac{\sqrt{2}}{2}\sin (2\theta )\right)
, \\
\varphi _{4}(\theta )=4\sqrt{2}\int \tfrac{1}{3+\cos (4\theta )}d\theta =%
\begin{cases}
\arctan \left( \frac{\sqrt{2}}{2}\tan (2\theta )\right) +\ell \pi , & \theta
\in \left( \frac{2\ell -1}{4}\pi ,\frac{2\ell +1}{4}\pi \right), \\
2\theta & \theta =\frac{2\ell +1}{4}\pi ,%
\end{cases}%
\end{gather*}
with $\ell =0,\dots,4$. Formula for $\varphi _{4}$ is given on the interval $%
\theta \in \left( -\frac{1}{4}\pi ,2\pi +\frac{1}{4}\pi \right) $ in order
to have the concise explicit form.

The Wronskian $W$ of $\varphi _1,\dots,\varphi _{4}$ is proportional to $%
\left( \cos ^{4}(\theta )+\sin ^{4}(\theta) \right) ^{-3}$ and is non-zero
on $\theta \in (0,2\pi ]$. Hence, $\{ \varphi _{m}\} _{m=1}^{4}$ defined as
above is a fundamental system of solutions to (\ref{C3}).

\subsubsection{Case $\protect\lambda =0$}

For $\lambda =0$ the equation (\ref{C1}) reads as:
\begin{equation}
\begin{aligned} &\tfrac{1}{4}\left( 3+\cos (4\theta )\right) \Phi ''''-2\sin
(4\theta )\Phi '''+\tfrac{1}{2}\left( 3-7\cos (4\theta )\right) \Phi ''\\
&-2\sin (4\theta )\Phi' +\tfrac{3}{4}\left( 1-5\cos (4\theta )\right) \Phi
=0, \end{aligned}  \label{C8}
\end{equation}
and can be split as follows:
\begin{equation*}
\left( \tfrac{d^2}{d\theta ^2}+1\right) \left( \tfrac{1}{4}\left( 3+\cos
(4\theta )\right) \left( \tfrac{d^2}{d\theta ^2}+1\right) \right) \Phi =0.
\end{equation*}
So, $\Phi $ solves
\begin{equation*}
\Phi ^{\prime \prime }+\Phi =A\tfrac{\sin (\theta )}{3+\cos (4\theta )}+B
\tfrac{\cos (\theta )}{3+\cos (4\theta )},
\end{equation*}
and after integrating this equation we obtain
\begin{align*}
&\Phi (\theta ) \\
&=A_1\sin (\theta )+A_2\cos (\theta )+A_{3}\Big( \tfrac{1}{ 2}\sin (\theta
)\arctan \left( \cos (2\theta )\right) +4\cos (\theta )\int_0^{\theta }%
\tfrac{\sin ^2(y)}{3+\cos (4y)}dy\Big) \\
&\quad +A_{4}\Big( \tfrac{1}{2}\cos (\theta )\arctan \left( \cos (2\theta
)\right) +4\sin (\theta )\int_0^{\theta }\tfrac{\cos ^2(y)}{3+\cos (4y)} dy%
\Big).
\end{align*}
%\label{C10}
The candidates that may form the fundamental system of solutions to (\ref{C8}%
) will be the following:
\begin{gather*}
\varphi _1(\theta )=\sin (\theta ), \quad \varphi _2(\theta )=\cos (\theta ),
\\
\varphi _{3}(\theta )=\tfrac{1}{2}\sin (\theta )\arctan \left( \cos (2\theta
)\right) +4\cos (\theta )\int_0^{\theta }\tfrac{\sin ^2(y)}{ 3+\cos (4y)}dy,
\\
\varphi _{4}(\theta )=\tfrac{1}{2}\cos (\theta )\arctan \left( \cos (2\theta
)\right) +4\sin (\theta )\int_0^{\theta }\tfrac{\cos ^2(y)}{ 3+\cos (4y)}dy.
\end{gather*}
The Wronskian $W$ of $\varphi _1,\dots,\varphi _{4}$ is proportional to $%
\left( \cos ^{4}(\theta )+\sin ^{4}(\theta) \right) ^{-2}$ and is non-zero
on $\theta \in (0,2\pi ]$. Hence, $\{ \varphi _{m}\} _{m=1}^{4}$ defined as
above is a fundamental system of solutions to (\ref{C8}).

\subsubsection{Case $\protect\lambda =1$}

For $\lambda =1$ the equation (\ref{C1}) reads as:
\begin{equation}
\tfrac{1}{4}\left( 3+\cos (4\theta )\right) \Phi ^{\prime \prime \prime
\prime }-\sin (4\theta )\Phi ^{\prime \prime \prime }+\left( 3+\cos (4\theta
)\right) \Phi ^{\prime \prime }-4\sin (4\theta )\Phi ^{\prime }=0.
\label{C11}
\end{equation}
We set $\Phi (\theta )=\int F(\theta )d\theta $ and obtain the equation for $%
F$:
\begin{equation*}
\tfrac{1}{4}\left( 3+\cos (4\theta )\right) F^{\prime \prime \prime }-\sin
(4\theta )F^{\prime \prime }+\left( 3+\cos (4\theta )\right) F^{\prime
}-4\sin (4\theta )F=0.
\end{equation*}
It holds that
\begin{gather*}
\left( 3+\cos (4\theta )\right) F^{\prime \prime \prime }-4\sin (4\theta
)F^{\prime \prime }=-2g(\theta ), \\
\left( 3+\cos (4\theta )\right) F^{\prime }-4\sin (4\theta )F=\frac{1}{2}
g(\theta ).
\end{gather*}
and we obtain, respectively,
\begin{equation*}
F^{\prime \prime }(\theta )=-4\tfrac{\int g(\theta )d\theta +C_1}{3+\cos
(4\theta )}, \quad F(\theta )=\tfrac{\int g(\theta )d\theta +C_2}{%
3+\cos(4\theta )}.
\end{equation*}
Comparing the expressions for $F^{\prime \prime }(\theta )$ and $F(\theta )$
we deduce that $F$ solves
\begin{equation}
F^{\prime \prime }+4F=\dfrac{c_0}{3+\cos (4\theta )}.  \label{C12}
\end{equation}
The solution of (\ref{C12}) reads as
\begin{align*}
F(\theta )&=c_1\sin (2\theta )+c_2\cos (2\theta )+c_0\Big( \tfrac{1}{4} \cos
(2\theta )\arctan \left( \cos (2\theta )\right) \\
&\quad +\tfrac{\sqrt{2}}{8} \sin (2\theta )\mathrm{arctanh}\big( \tfrac{%
\sqrt{2}}{2}\sin (2\theta )\big) \Big) ,
\end{align*}
which being integrated yields
\begin{equation}  \label{C13}
\begin{aligned} \Phi (\theta )&=A_1+A_2\cos (2\theta )+A_{3}\sin (2\theta)\\
&\quad +A_{4}\int_0^{\theta }\left( \cos (2y)\arctan \left( \cos (2y)\right)
+\tfrac{\sqrt{2}}{2}\sin (2y)\mathrm{arctanh}\left( \tfrac{\sqrt{2}}{2 }\sin
(2y)\right) \right) dy \end{aligned}
\end{equation}
The candidates that may form the fundamental system of solutions to (\ref%
{C11}) will be the following:
\begin{gather*}
\varphi _1(\theta )=1, \quad \varphi _2(\theta )=\sin (2\theta ), \quad
\varphi _{3}(\theta )=\cos (2\theta ), \\
\varphi _{4}(\theta )=\int_0^{\theta }\left( \cos (2y)\arctan \left( \cos
(2y)\right) +\tfrac{\sqrt{2}}{2}\sin (2y)\mathrm{arctanh} \left(\tfrac{\sqrt{%
2}}{2}\sin (2y)\right) \right) dy.
\end{gather*}
The Wronskian $W$ of $\varphi _1,\dots,\varphi _{4}$ is proportional to $%
\left( \cos ^{4}(\theta )+\sin ^{4}(\theta) \right) ^{-1}$ and is non-zero
on $\theta \in (0,2\pi ]$. Hence, $\{ \varphi _{m}\} _{m=1}^{4}$ defined as
above is a fundamental system of solutions to (\ref{C8}).

\subsection{The explicit formulas for $P_{-1}$, $P_0$, $P_1$,}

For $\lambda \in \{ \pm 1,0\} $ we obtain:
\begin{gather*}
P_{-1}(\omega) =-\tfrac{32}{\pi }\sin (2\omega )\int_0^{\omega }\tfrac{\sin
^2(\theta )}{3+\cos (4\theta )} d\theta +\left( 1-\cos (2\omega )\right)
\left( 1-\tfrac{4}{\pi }\arctan \left( \cos (2\omega )\right) \right) , \\
\begin{aligned} P_0(\omega) &=\pi \arctan \left( \cos (2\omega )\right)
-2\arctan ^2\left( \cos (2\omega )\right) \\ &\quad +128\int_0^{\omega }
\tfrac{\sin ^2(\theta )}{3+\cos (4\theta )}d\theta \int_0^{\omega
}\tfrac{\cos ^2(\theta )}{3+\cos (4\theta )}d\theta , \end{aligned} \\
P_1(\omega) =-\sin (2\omega )\left( \sin (2\omega )-\tfrac{8}{ \pi }\varphi
_{4}(\omega )\right) +\left( 1-\cos (2\omega )\right) \left( \cos (2\omega )-%
\tfrac{4}{\pi }\varphi _{4}^{\prime }(\omega )\right) ,
\end{gather*}
where $\varphi _{4}(\theta )$ is given by corresponding formula from the
case $\lambda =1$.

\section{Analytical tools for the numerical computation\label{AppendixD}}

\subsection{Implicit function and discretization}

Consider a rectangle $U=[a,b]\times [ c,d]$. For $n,m\in \mathbb{N}^{+} $
and $i=0,\dots,n$, $j=0,\dots,m$ we set
\begin{equation*}
\begin{tabular}{cc}
$x_{i}=a+i\Delta x,\quad y_{j}=c+j\Delta y$, &
\end{tabular}%
\end{equation*}
where $\Delta x=\frac{b-a}{n}$, $\Delta y=\frac{d-c}{m}$.

Let $F\in C^{1}(U,\mathbb{R})$ such that $F(x_{i},y_{j})>0$ for all $%
i=0,\dots,n$ and $j=0,\dots,m$. The question to resolve is how fine should
we take the\ discretization of $U$ in order to be sure that $F>0$ on $U$.
The following result holds.

\begin{lemma}
\label{lemma1,ap.D}Suppose$\min_{(x_{i},y_{j})\in U}
F(x_{i},y_{j})>0$. If
\begin{equation}
\max \left\{ \Delta x,\Delta y\right\} \leq \sqrt{2}
\dfrac{\min_{(x_{i},y_{j})\in U} F(x_{i},y_{j})}
{\sup_{U}|DF(x,y)| },  \label{D10}
\end{equation}
then $F$ is strictly positive on $U$.
\end{lemma}

\begin{proof}
For every $(x,y)\in U$ there is $(x_{i},y_{j})$ with $| x-x_{i}| \leq \frac{1%
}{2}\Delta x$ and $| y-y_{j}| \leq \frac{1}{2}\Delta y$. By the mean value
theorem there exists $(\xi _1,\xi _2)\in [ (x,y) ,(x_{i},y_{j})]$ such that
\begin{equation*}
F(x,y)=F(x_{i},y_{j})+DF(\xi _1,\xi _2)\cdot (x-x_{i},y-y_{j}).
\end{equation*}
The following chain of estimates then holds
\begin{align*}
F(x,y) & = F(x_{i},y_{j})+DF(\xi _1,\xi _2)\cdot (x-x_{i},y-y_{j}) \\
& \geq \min_{(x_{i},y_{j})\in U} F(x_{i},y_{j}) -\sup_{U} | DF(x,y)| |
(x-x_{i},y-y_{j})| \\
& \geq \min_{(x_{i},y_{j})\in U} F(x_{i},y_{j}) -\frac{\sqrt{2}}{2}\sup_{U}|
DF(x,y)| \max \left\{ \Delta x,\Delta y\right\} .
\end{align*}
This last expression is positive if (\ref{D10}) holds.
\end{proof}

\subsection{A version of the Morse theorem}

Let $V\subset \mathbb{R}^2$\ be open and bounded, $F\in C^{\infty }(V,%
\mathbb{R})$. For the gradient of $F$ we use ${D}F$ and ${D}^2F$ is the
Hessian matrix.

\begin{definition} \rm
A point $a\in V$ is said to be a critical point of $F$ if ${D}F(a)=0$.
Moreover, the critical point $a\in V$ is said to be non-degenerate
 if $\det {D}^2F(a)\neq 0$.
\end{definition}

To study the level set $\Gamma $ defined in subsection \ref{Behavior} we
need the Morse theorem. The original version of the theorem reads as (see
\cite{Pal}):

\begin{quotation}
Let $V$ be a Banach space, $O$ a convex neighborhood of the origin in $V$
and $f:O\to \mathbb{R}$ a $C^{k+2}$ function ($k\geq 1$) having the origin
as a non-degenerate critical point, with $f(0)=0$. Then there is a
neighborhood $U$ of the origin and a $C^{k}$ diffeomorphism $\phi
:U\to O$ with $\phi (0)=0$ and $D\phi (0)=I$, the identity map of $V, $ such
that for $x\in U$, $f(\phi (x))=\frac{1}{2}(D^2f(0)x,x) $.
\end{quotation}

Below we give our formulation of the theorem. This formulation is more
convenient for our purposes. We will give a constructive proof that allows
us to find an explicit neighborhood of a critical point where the diffeomorphism exists.

\begin{theorem}\label{th1,ap.D}
Let $V\subset \mathbb{R}^2$\ be open and bounded, $F\in
C^{\infty }(V,\mathbb{R})$. Suppose $a=(a_1,a_2) \in V$ is a
non-degenerate critical point of $F$. There exists a neighborhood
$W_{a}\subset V$ of $a$ and a $C^{\infty }$-diffeomorphism
$h:W_{a}\to U_0$, where $U_0\subset \mathbb{R}^2$\ is a
neighborhood\ of $0$, such that $F$ in $W_{a}$ is representable as:
\begin{equation}
F(x)=F{(a)+}h(x)\left( \tfrac{1}{2}{D}^2F(a)\right) h(x)^{T},  \label{D1}
\end{equation}
where $T$ stands for a transposition.
Moreover, the neighborhood $W_{a}$ is fixed by
\begin{equation*}
W_{a}\subset \left\{ x\in V:\det B(x)\geq 0\text{ \ \ and \ }
b_{11}(x)+2\sqrt{\det B(x)}+b_{22}(x)>0\right\} .
\end{equation*}
where
\begin{equation*}
B(x) =\begin{pmatrix}
b_{11}(x) & b_{12}(x) \\
b_{21}(x) & b_{22}(x)
\end{pmatrix} =\left( \tfrac{1}{2}D^2F(a)\right)
^{-1}\int_0^{1}\int_0^{1}sD^2F\left( a+ts\left(
x-a\right) \right) dt\,ds.
\end{equation*}
\end{theorem}

\begin{proof}
Let $a\in V$ be such that ${D}F(a)=0$ and $\det {D}^2F(a)\neq 0$. First, for
every $x\in V$ we have
\begin{equation*}
F(x)-F(a)= {F}\left( a+s(x-a)\right) | _0^{1}=\int_0^{1}\tfrac{d}{ds}{F}%
\left( a+s(x-a)\right) ds,
\end{equation*}
and due to $\tfrac{d}{ds}{F}\left( a+s(x-a)\right) =D{F}\left(
a+s(x-a)\right) (x-a)^{T}$ it will follow that
\begin{equation*}
F(x)=F{(a)+}\int_0^{1}D{F}\left( a+s(x-a)\right) ds(x-a)^{T}.
\end{equation*}
Analogously, we obtain
\begin{equation*}
DF(x)=DF(a){+}\int_0^{1}D^2F\left( a+t(x-a)\right) dt(x-a)^{T},
\end{equation*}
where by assumption $DF(a)=0$. As a result, for every $x\in V$, $F$ is
representable in terms of $D^2F$ as:
\begin{equation*}
F(x)=F{(a)+}(x-a)\int_0^{1}\int_0^{1}sD^2F\left( a+ts(x-a)\right)
dt\,ds(x-a)^{T},
\end{equation*}
or shortly
\begin{equation}
F(x)=F{(a)+}(x-a)K(x)(x-a)^{T}.  \label{D1.1}
\end{equation}
Here $K(x)=\int_0^{1}\int_0^{1}sD^2F\left( a+ts(x-a)\right) dt\,ds$ is a
symmetric matrix. With this definition, $K(a)=\frac{1}{2}D^2F(a) $ is
symmetric and invertible ($\det {D}^2F(a)\neq 0$ by assumption).

Let us bring some intermediate results.

(1) For every $x\in V$, there exists matrix $B$ such that
\begin{equation}
K(x)=K(a)B(x).  \label{D1.2}
\end{equation}
Indeed, since $K(a)$ is invertible, the matrix $B(x)=K(a)^{-1}K(x)$ is
well-defined. We write
\begin{equation*}
B(x)=%
\begin{pmatrix}
b_{11}(x) & b_{12}(x) \\
b_{21}(x) & b_{22}(x)%
\end{pmatrix}%
,
\end{equation*}
Since $F\in C^{\infty }(V,\mathbb{R})$, so are $b_{ij}$, $i,j=1,2$. Note
that $B(a)=I$.

(2) Since $x\mapsto B(x)$ is $C^{\infty }$ in a neighborhood of $a$ and $%
B(a)=I$, $B(x)$ is positive definite in a neighborhood of $a$, and hence
allows a square root. In particular, it holds that
\begin{equation}
C(x)=\sqrt{B(x)}:=\tfrac{1}{2\pi i}\oint_{\gamma }\sqrt{z}\left(
Iz-B(x)\right) ^{-1}dz,  \label{D1.11}
\end{equation}
where $\gamma $ is a Jordan curve in $\mathbb{C} $ which goes around the
eigenvalues $\lambda _1,\lambda _2\in \mathbb{C}$ of $B(x)$ and does not
intersect $\mathop{\rm Re}(z)\leq 0$, $\mathrm{Im}(z)=0$.

One may check that $C$, defined as follows
\begin{equation*}
C(x)=%
\begin{pmatrix}
\tfrac{b_{11}(x)+\sqrt{\det B(x)}}{\sqrt{b_{11}(x)+2\sqrt{\det B(x)}
+b_{22}(x)}} & \tfrac{b_{12}(x)}{\sqrt{b_{11}(x)+2\sqrt{\det B(x)}+b_{22}(x)}
} \\
\tfrac{b_{21}(x)}{\sqrt{b_{11}(x)+2\sqrt{\det B(x)}+b_{22}(x)}} & \tfrac{
b_{22}(x)+\sqrt{\det B(x)}}{\sqrt{b_{11}(x)+2\sqrt{\det B(x)}+b_{22}(x)}}%
\end{pmatrix}%
,
\end{equation*}
is indeed such that
\begin{equation}
C(x)^2=B(x).  \label{D1.20}
\end{equation}
With this definition, $C(a)=I$ and $C(a)^2=I=B(a)$ as required. Also, matrix
$C$ is well defined when
\begin{equation}
\det B(x)\geq 0, \quad b_{11}(x)+2\sqrt{\det B(x)}+b_{22}(x)>0.  \label{D1.6}
\end{equation}

(3) For those $x$ one finds
\begin{align*}
K(a)B(x) & = K(x)=K(x)K(a)^{-1}K(a) \\
&=K(x)^{T}\left( K(a)^{-1}\right)^{T}K(a) \\
& = \left( K(a)^{-1}K(x)\right) ^{T}K(a)=B^{T}(x)K(a).
\end{align*}
Due to this equality, we deduce the following
\begin{equation*}
\left( Iz-B(x)\right) K(a)^{-1}=K(a)^{-1}\left( Iz-B(x)\right) ^{T},
\end{equation*}
and hence
\begin{align*}
K(a)\left( Iz-B(x)\right) ^{-1} & = \left( \left( Iz-B(x)\right)
K(a)^{-1}\right) ^{-1}=\left( K(a)^{-1}\left( Iz-B(x)\right) ^{T}\right)
^{-1} \\
& = \left( \left( Iz-B(x)\right) ^{T}\right) ^{-1}K(a)=\left( \left(
Iz-B(x)\right) ^{-1}\right) ^{T}K(a).
\end{align*}
Applying the integration (\ref{D1.11}) to the last identity we find
\begin{equation}
K(a)C(x)=C(x)^{T}K(a).  \label{D1.21}
\end{equation}
Combining (\ref{D1.2}), (\ref{D1.20}) and (\ref{D1.21}) we have
\begin{equation*}
K(x)=K(a)B(x)=K(a)C(x)^2=C(x)^{T}K(a)C(x),
\end{equation*}
and therefore (\ref{D1.1}) for those $x$ results in
\begin{equation}
F(x)=F(a)+F{(a)+}h(x)K(a)h(x)^{T},  \label{D1.7}
\end{equation}
where $K(a)=\tfrac{1}{2}{D}^2F(a)$ and
\begin{equation*}
h(x)^{T}=C(x)(x-a)^{T}.
\end{equation*}
Note that by (\ref{D1.6}) the representation for $F$ in (\ref{D1.7}) holds
on a set $W_{a}\subset V$ which is star-shaped with respect to $a$ and such
that
\begin{equation}
W_{a}\subset \big\{ x\in V:\det B(x)\geq 0\text{ and } b_{11}(x)+2\sqrt{\det
B(x)}+b_{22}(x)>0\big\} .  \label{D21}
\end{equation}
\end{proof}

\begin{remark} \rm
For each pair $(F,a) $ one can obtain an explicit estimate for
$W_{a}$ in (\ref{D21}). We will do this in the next subsection for the pair
we are interested in.
\end{remark}

\subsection{The Morse Theorem applied}

Let $P$ be the function given by formula (\ref{9}) and which is defined on
\begin{equation*}
V=\left\{ (\omega ,\lambda) :\left[ \tfrac{70}{180}\pi ,\tfrac{ 110}{180}\pi %
\right] \times \left[ 2.900,5.100\right] \right\} .
\end{equation*}
Let us recall it here:
\begin{equation}
\begin{aligned} P(\omega ,\lambda) &=\left( 1-\tfrac{\sqrt{2}}{2}\sin
(2\omega )\right) ^{\lambda }+\left( 1+\tfrac{\sqrt{2}}{2}\sin (2\omega
)\right) ^{\lambda } \\ &\quad +\left( \tfrac{1}{2}+\tfrac{1}{2}\cos
^2(2\omega )\right) ^{\frac{1}{2} \lambda }\Big[ 2\cos \left( \lambda \left(
\arctan \left( \tfrac{\sqrt{2}}{2 }\tan (2\omega )\right) +\pi \right)
\right) \\ &\quad -4\cos \left( \lambda \arctan \left( \tan ^2(\omega
)\right) \right) \Big] . \end{aligned}  \label{D2}
\end{equation}
The point $a=( \tfrac{1}{2}\pi ,4) $ is such that $P(a)=0$ and $DP(a)=0$.
Theorem \ref{th1,ap.D} gives us the tool to study $P$ in the vicinity of $a$%
. In particular, the following holds.

\begin{proposition}\label{Proposition.ap.D}
Let $a$ be as above. There is a closed ball
$W_{R}(a)\subset V$ of a radius $R$ centered at $a$ such that on $W_{R}(a)$
we have:
\begin{equation}
P(\omega ,\lambda )=-\tfrac{1}{2}h_2(\omega ,\lambda) \left(
16\sqrt{2}h_1(\omega ,\lambda) +\pi h_2(\omega ,\lambda) \right) .  \label{D3.1}
\end{equation}
Here $h_1,h_2\in C^{\infty }\left( W_{R}(a),\mathbb{R}\right) $ are
given by:
\begin{gather}
h_1(\omega ,\lambda) =\left( \omega -\tfrac{1}{2}\pi \right)
c_{11}(\omega ,\lambda) +\left( \lambda -4\right) c_{12}(\omega ,\lambda) ,  \label{D18}
\\
h_2(\omega ,\lambda) =\left( \omega -\tfrac{1}{2}\pi \right)
c_{21}(\omega ,\lambda) +\left( \lambda -4\right) c_{22}(\omega ,\lambda) ,  \label{D19}
\end{gather}
with $c_{ij}\in C^{\infty }\left( W_{R}(a),\mathbb{R}\right) $, $i,j=1,2$
are the entries of matrix $C$:
\begin{equation}
C(\omega ,\lambda) =\begin{pmatrix}
c_{11} & c_{12} \\
c_{21} & c_{22}
\end{pmatrix} (\omega ,\lambda)
=\begin{pmatrix}
\tfrac{b_{11}+\sqrt{\det B}}{\sqrt{b_{11}+2\sqrt{\det B}+b_{22}}} & \tfrac{
b_{12}}{\sqrt{b_{11}+2\sqrt{\det B}+b_{22}}} \\
\tfrac{b_{21}}{\sqrt{b_{11}+2\sqrt{\det B}+b_{22}}} & \tfrac{b_{22}+\sqrt{
\det B}}{\sqrt{b_{11}+2\sqrt{\det B}+b_{22}}}
\end{pmatrix}
 (\omega ,\lambda) ,  \label{D20}
\end{equation}
while $b_{ij}\in C^{\infty }(V,\mathbb{R})$, $i,j=1,2$ are as follows
\begin{equation}
\begin{aligned}
B(\omega ,\lambda)
&=\begin{pmatrix}
b_{11} & b_{12} \\
b_{21} & b_{22}
\end{pmatrix} (\omega ,\lambda) \\
&=\left( \tfrac{1}{2}D^2P(a)\right)
^{-1}\int_0^{1}\int_0^{1}sD^2P\left( a+ts\left( (\omega ,\lambda) -a\right)
 \right) dt\,ds.
\end{aligned}
 \label{D15}
\end{equation}
Note that $B(a)=I$ and $C(a)=I$.
The ball $W_{R}(a)$ is fixed by
\begin{equation}
W_{R}(a):=\left\{ (\omega ,\lambda) \in V:| \left(
\omega -\tfrac{1}{2}\pi ,\lambda -4\right) | \leq R\right\} ,
\label{D3.2}
\end{equation}
with $R=-\tfrac{1}{120}\pi ^2-\tfrac{\sqrt{2}}{18}\pi +\tfrac{1}{120}\pi
\sqrt{\pi ^2+\tfrac{40}{3}\sqrt{2}\pi +\tfrac{1568}{9}}\approx 0.078\dots$
(In $\omega $-direction we have that $R\approx 4.5\dots^{\circ }$).
\end{proposition}

\subsubsection{Computational results I}

It is straightforward for $a=\left( \frac{1}{2}\pi ,4\right) $ that
\begin{equation}
\begin{tabular}{ccc}
$\tfrac{\partial ^2P}{\partial \omega ^2}(a)=0,\quad \tfrac{\partial ^2P }{%
\partial \omega \partial \lambda }(a)=-8\sqrt{2}\pi ,\quad \tfrac{\partial
^2P}{\partial \lambda ^2}(a)=-\pi ^2$, &  &
\end{tabular}
\label{D5}
\end{equation}
and hence
\begin{equation}
\det D^2P(a)=-128\pi ^2.  \label{D5.1}
\end{equation}
To simplify the notation, we use $x$ instead of $(\omega ,\lambda) $ when $%
(\omega ,\lambda) $ stands for a argument. Now let us bring two alternatives
representations for the entries of matrix $B$ given by (\ref{D15}), which we
will use later on.\medskip

\noindent\textit{Representation I.} 
We will need the explicit formula for the coefficients $b_{ij}$, $i,j=1,2$.
Let us find them in a straightforward way from (\ref{D15}). We write down
the integral term $\int_0^{1}\int _0^{1}sD^2P\left( a+ts\left( x-a\right)
\right) dt\,ds$ in (\ref{D15}) as follows
\begin{equation}
\int_0^{1}\int_0^{1}sD^2P\left( a+ts\left( x-a\right) \right) dt\,ds=%
\begin{pmatrix}
r_1(x) & r_2(x) \\
r_2(x) & r_{3}(x)%
\end{pmatrix}
,  \label{D1.45}
\end{equation}
where $r_{j}$, $j=1,2,3$ read as:
\begin{gather}
r_1(x)=\int_0^{1}\int_0^{1}s\tfrac{\partial ^2P}{ \partial \omega ^2}%
(a+ts(x-a))dt\,ds,  \label{D4.1} \\
r_2(x)=\int_0^{1}\int_0^{1}s\tfrac{\partial ^2P}{ \partial \omega \partial
\lambda }(a+ts(x-a))dt\,ds,  \label{D4.2} \\
r_{3}(x)=\int_0^{1}\int_0^{1}s\tfrac{\partial ^2P}{ \partial \lambda ^2}%
(a+ts(x-a))dt\,ds.  \label{D4.3}
\end{gather}
Then the entries $b_{ij}$, $i,j=1,2$ of matrix $B$ in terms of $r_{j}$, $%
j=1,2,3$ and due to (\ref{D5}), (\ref{D5.1}) will read:
\begin{gather}
b_{11}(x)=\tfrac{2}{\det {D}^2P(a)}\left( \tfrac{\partial ^2P}{\partial
\lambda ^2}(a)r_1(x)-\tfrac{\partial ^2P}{\partial \omega \partial \lambda }%
(a)r_2(x)\right) =\tfrac{1}{64}r_1(x)-\tfrac{\sqrt{2}}{8\pi } r_2(x),
\label{D2.1} \\
b_{12}(x)=\tfrac{2}{\det {D}^2P(a)}\left( \tfrac{\partial ^2P}{\partial
\lambda ^2}(a)r_2(x)-\tfrac{\partial ^2P}{\partial \omega \partial \lambda }%
(a)r_{3}(x)\right) =\tfrac{1}{64}r_2(x)-\tfrac{\sqrt{2}}{8\pi } r_{3}(x),
\label{D2.2} \\
b_{21}(x)=\tfrac{2}{\det {D}^2P(a)}\left( \tfrac{\partial ^2P}{\partial
\omega ^2}(a)r_2(x)-\tfrac{\partial ^2P}{\partial \omega \partial \lambda }%
(a)r_1(x)\right) =-\tfrac{\sqrt{2}}{8\pi }r_1(x),  \label{D2.3} \\
b_{22}(x)=\tfrac{2}{\det {D}^2P(a)}\left( \tfrac{\partial ^2P}{\partial
\omega ^2}(a)r_{3}(x)-\tfrac{\partial ^2P}{\partial \omega \partial \lambda }%
(a)r_2(x)\right) =-\tfrac{\sqrt{2}}{8\pi }r_2(x).  \label{D2.4}
\end{gather}

\noindent\textit{Representation II.}
On the other hand, let us obtain for $r_{j}$, $j=1,2,3$ the following
representations:
\begin{equation*}
r_1(x)=\tfrac{1}{2}\tfrac{\partial ^2P}{\partial \omega ^2} (a)+q_1(x),
\quad r_2(x)=\tfrac{1}{2}\tfrac{\partial ^2P}{\partial \omega \partial
\lambda }(a)+q_2(x), \quad r_{3}(x)=\tfrac{1}{2}\tfrac{\partial ^2P}{%
\partial \lambda ^2} (a)+q_{3}(x),
\end{equation*}
where
\begin{gather}
\begin{aligned} q_1(x)&=\int_0^{1}\int_0^{1}\int_0^{1}ts^2 \left(
\tfrac{\partial ^{3}P}{\partial \omega ^{3}}(a+ts\sigma (x-a)),\tfrac{
\partial ^{3}P}{\partial \omega ^2\partial \lambda }(a+ts\sigma
(x-a))\right) \\ &\quad\times (x-a)^{T}d\sigma dt\,ds, \end{aligned}
\label{D1.46} \\
\begin{aligned} q_2(x)&=\int_0^{1}\int_0^{1}\int_0^{1}ts^2 \left(
\tfrac{\partial ^{3}P}{\partial \omega ^2\partial \lambda } (a+ts\sigma
(x-a)),\tfrac{\partial ^{3}P}{\partial \omega \partial \lambda
^2}(a+ts\sigma (x-a))\right)\\ &\quad\times (x-a)^{T}d\sigma dt\,ds,
\end{aligned}  \label{D1.47} \\
\begin{aligned} q_{3}(x) &=\int_0^{1}\int_0^{1}\int_0^{1}ts^2 \left(
\tfrac{\partial ^{3}P}{\partial \omega \partial \lambda ^2} (a+ts\sigma
(x-a)),\tfrac{\partial ^{3}P}{\partial \lambda ^{3}}(a+ts\sigma
(x-a))\right) \\ &\quad\times(x-a)^{T}d\sigma dt\,ds. \end{aligned}
\label{D1.48}
\end{gather}
This will yield another representation formulas for $b_{ij}$, $i,j=1,2$ of
matrix $B$, namely,
\begin{gather}
b_{11}(x)=1+\tfrac{2}{\det {D}^2P(a)}\left( \tfrac{\partial ^2P}{ \partial
\lambda ^2}(a)q_1(x)-\tfrac{\partial ^2P}{\partial \omega \partial \lambda }%
(a)q_2(x)\right) =1+\tfrac{1}{64}q_1(x)-\tfrac{\sqrt{2} }{8\pi }q_2(x),
\label{D29} \\
b_{12}(x)=\tfrac{2}{\det {D}^2P(a)}\left( \tfrac{\partial ^2P}{\partial
\lambda ^2}(a)q_2(x)-\tfrac{\partial ^2P}{\partial \omega \partial \lambda }%
(a)q_{3}(x)\right) =\tfrac{1}{64}q_2(x)-\tfrac{\sqrt{2}}{8\pi } q_{3}(x),
\label{D30} \\
b_{21}(x)=\tfrac{2}{\det {D}^2P(a)}\left( \tfrac{\partial ^2P}{\partial
\omega ^2}(a)q_2(x)-\tfrac{\partial ^2P}{\partial \omega \partial \lambda }%
(a)q_1(x)\right) =-\tfrac{\sqrt{2}}{8\pi }q_1(x),  \label{D31} \\
b_{22}(x)=1+\tfrac{2}{\det {D}^2P(a)}\left( \tfrac{\partial ^2P}{ \partial
\omega ^2}(a)q_{3}(x)-\tfrac{\partial ^2P}{\partial \omega \partial \lambda }%
(a)q_2(x)\right) =1-\tfrac{\sqrt{2}}{8\pi }q_2(x).  \label{D32}
\end{gather}
This representation, together with the estimates from above for $| q_{j}| $,
$j=1,2,3$ on $V$ given below, will be useful in the proof of Proposition \ref%
{Proposition.ap.D}.\medskip

\noindent\textit{Estimates for $| q_{j}| $, $j=1,2,3$ on $V$.} 
Let $q_{j}$, $j=1,2,3$ be given by (\ref{D1.46}) -- (\ref{D1.48}). We will
need the estimates for $| q_{j}| $, $j=1,2,3$ on $V$. Let us also use the
notations $\tfrac{\partial ^{3}P}{\partial (\omega ,\lambda) ^{\alpha _{j}}},%
\tfrac{\partial ^{3}P}{\partial (\omega ,\lambda) ^{\beta _{j}}}$ for the
corresponding differentiations in each $q_{j}$, $j=1,2,3$. For example for $%
q_1$ it will be $\tfrac{\partial ^{3}P}{\partial (\omega ,\lambda) ^{\alpha
_{j}}}=\tfrac{\partial ^{3}P}{\partial \omega ^{3}}$ and $\tfrac{\partial
^{3}P}{\partial (\omega ,\lambda) ^{\beta _{j}}}=\tfrac{ \partial ^{3}P}{%
\partial \omega ^2\partial \lambda }$, etc. For every $q_{j}$, $j=1,2,3$ we
then in general have:
\begin{equation}
\begin{aligned} | q_{j}(x)| &=\big|\int_0^{1}\int_0^{1}\int_0^{1}ts^2\left(
\tfrac{ \partial ^{3}P}{\partial (\omega ,\lambda) ^{\alpha _{j}}}
(a+ts\sigma (x-a)),\tfrac{\partial ^{3}P}{\partial (\omega ,\lambda) ^{\beta
_{j}}}(a+ts\sigma (x-a))\right) \\ &\quad\times(x-a)^{T}d\sigma dt\,ds\big|
\\ &\leq \int_0^{1}\int_0^{1}\int_0^{1}ts^2\sup_{x\in V}| \left(
\tfrac{\partial ^{3}P}{\partial (\omega ,\lambda) ^{\alpha
_{j}}}(x),\tfrac{\partial ^{3}P}{\partial (\omega ,\lambda) ^{\beta
_{j}}}(x)\right) | d\sigma dt\,ds| x-a| \\ &=\tfrac{1}{6}\sup_{x\in V}|
\left( \tfrac{\partial ^{3}P}{\partial (\omega ,\lambda) ^{\alpha
_{j}}}(x),\tfrac{\partial ^{3}P}{ \partial (\omega ,\lambda) ^{\beta
_{j}}}(x)\right) | | x-a| \leq M_{j}| x-a| . \end{aligned}  \label{D6}
\end{equation}
where $M_{j}$ is an upper bound for function $\tfrac{1}{6}| \left(\tfrac{%
\partial ^{3}P}{\partial (\omega ,\lambda) ^{\alpha _{j}} }(x),\tfrac{%
\partial ^{3}P}{\partial (\omega ,\lambda) ^{\beta _{j}}}(x)\right) | $ on $%
V $. In particular, one shows by explicit and tedious computations that for $%
M_1=180$, $M_2=75$, $M_{3}=30$, the estimate (\ref{D6}) for the
corresponding $| q_{j}| $, $j=1,2,3$ holds true. We depict the results in
Table \ref{table1.ap.D}.

\begin{table}[ht]
\renewcommand{\arraystretch}{1.5}
\par
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
$| q_1(x)| <180| x-a|$ & $| q_2(x)| <75| x-a| $ & $| q_{3}(x)| <30| x-a| $
\\ \hline
\end{tabular}%
\end{center}
\caption{Estimates from above for $| q_{j}| $, $j=1,2,3$ on $V$}
\label{table1.ap.D}
\end{table}

Now we proceed with a proof of Proposition \ref{Proposition.ap.D}.

\subsubsection{Checking the range for Morse}

\begin{proof}[Proof of Proposition \protect\ref{Proposition.ap.D}]
Representation (\ref{D3.1}) is a consequence of Theorem \ref{th1,ap.D}. It
is straightforward for $a=\left( \frac{1}{2}\pi ,4\right) $ that $P(a)=0$.
We also find that $DP(a)=0$, meaning $a$ is a critical point of $P$. Due to
( \ref{D5.1}) we conclude that $a$ is a non-degenerate critical point of $P$.

By Theorem \ref{th1,ap.D} in a vicinity $W_{a}\subset V$ of $a$ which is
defined in (\ref{D21}), we obtain
\begin{equation}
\begin{aligned} P(x)&=\left( h_1(x),h_2(x)\right) \cdot
\tfrac{1}{2}{D}^2P(a)\cdot \left( h_1(x),h_2(x)\right) ^{T} \\
&=-\tfrac{1}{2}h_2(x) \left( 16\sqrt{2}h_1(x) +\pi h_2(x) \right) ,
\end{aligned}  \label{D36}
\end{equation}
where $h_1,h_2\in C^{\infty }\left( W_{a},\mathbb{R}\right) $. Their
explicit formulas read as (\ref{D18}), (\ref{D19}).

We show that $W_{a}$ in our case can be taken as a closed ball $W_{R}(a)$
centered at $a$ of radius $R$ and the numerical approximation for its range
is given by (\ref{D3.2}). We will do this in two steps.

(1) Let us solve the inequality $\det B(x)\geq 0$ on $V$. Due to (\ref{D29})
-- (\ref{D32}) we will get
\begin{align*}
\det B(x)&=b_{11}(x)b_{22}(x)-b_{12}(x)b_{21}(x) \\
&=1-\tfrac{\sqrt{2}}{4\pi }q_2(x)+\tfrac{1}{64}q_1(x)-\tfrac{1}{32\pi ^2 }%
q_1(x)q_{3}(x)+\tfrac{1}{32\pi ^2}q_2^2(x) \\
&\geq 1-\tfrac{\sqrt{2}}{4\pi }| q_2(x)| -\tfrac{1}{64} | q_1(x)| -\tfrac{1}{%
32\pi ^2}| q_1(x)| | q_{3}(x)| \geq \dots
\end{align*}
we use estimates for $| q_1(x)| $, $|q_2(x)| $ and $| q_{3}(x)| $ from Table %
\ref{table1.ap.D} to get
\begin{equation*}
\dots\geq 1-\tfrac{75\sqrt{2}}{4\pi }| x-a| -\tfrac{180}{64} | x-a| -\tfrac{%
5400}{32\pi ^2}| x-a|^2.
\end{equation*}
The above expression is positive for all $x\in V$ such that $| x-a| \leq R_1$%
, with
\begin{equation*}
R_1=-\tfrac{1}{120}\pi ^2-\tfrac{\sqrt{2}}{18}\pi +\tfrac{1}{120}\pi \sqrt{%
\pi ^2+\tfrac{40}{3}\sqrt{2}\pi +\tfrac{1568}{9}}.
\end{equation*}
The numerical approximation is $R_1\approx 0.078\dots$ . Hence, the first
estimate for a range of $W_{a}$ is $|x-a| \leq R_1$.

(2) Let us solve $b_{11}(x)+2\sqrt{\det B(x)}+b_{22}(x)>0$ on $V$. We have
\begin{equation*}
b_{11}(x)+2\sqrt{\det B(x)}+b_{22}(x)\geq b_{11}(x)+b_{22}(x)=\dots
\end{equation*}
due to formulas (\ref{D29}), (\ref{D32}) we obtain
\begin{equation*}
2-\tfrac{\sqrt{2}}{4\pi }q_2(x)+\tfrac{1}{64}q_1(x)\geq 2-\tfrac{\sqrt{2 }}{%
4\pi }| q_2(x)| -\tfrac{1}{64}| q_1(x)| \geq \dots
\end{equation*}
we use the estimates for $| q_1(x)| $ and $| q_2(x)| $\ from Table \ref%
{table1.ap.D}\ to get
\begin{equation*}
\dots\geq 2-\tfrac{75\sqrt{2}}{4\pi }| x-a| -\tfrac{180}{64} | x-a| .
\end{equation*}
The above expression is strictly positive for all $x\in V$ such that
\begin{equation*}
\begin{tabular}{ccc}
$| x-a| <R_2$ & with & $R_2=\frac{32\pi }{\left( 300 \sqrt{2}+45\pi \right) }
$,%
\end{tabular}%
\end{equation*}
and this is the second estimate for $W_{a}$.

Comparing the approximations to $R_1\approx 0.078\dots$ and $R_2\approx
0.178\dots$ we set $R:=R_1$ and $W_{a}:=W_{R}(a)=\{ x\in V:| x-a| \leq R\} $%
. Result (\ref{D3.2}) follows.
\end{proof}

\begin{remark}\label{remark2,ap.D} \rm
Let the rectangle $U\subset W_{R}(a)$ containing the
point $a=\left( \frac{1}{2}\pi ,4\right) $ be defined as follows:
\begin{equation}
U:=\big\{ (\omega ,\lambda) :\left[ \tfrac{1}{2}\pi -\tfrac{2}{
180}\pi ,\tfrac{1}{2}\pi +\tfrac{2}{180}\pi \right] \times [
4-0.060,4+0.060] \big\} .  \label{D100}
\end{equation}
Proposition \ref{Proposition.ap.D} holds true for the given $U$.
\end{remark}

\subsection{On the insecting curves from Morse}

Here we consider $h_2(\omega ,\lambda) =0$ and $16\sqrt{2} h_1(\omega
,\lambda) +\pi h_2(\omega ,\lambda) =0$ in $U$ with $h_{i}$, $i=1,2$ as in
Proposition \ref{Proposition.ap.D}. Consider in $U$ given by (\ref{D100})
the two implicit functions:
\begin{equation}
h_2(\omega ,\lambda) =0,  \label{D22}
\end{equation}
\begin{equation}
16\sqrt{2}h_1(\omega ,\lambda) +\pi h_2(\omega ,\lambda) =0.  \label{D23}
\end{equation}
At $(\omega ,\lambda) =a$ it holds that $h_1(a)=h_2(a) =0$; that is,
\begin{equation*}
h_2(a) =0,
\end{equation*}
\begin{equation*}
16\sqrt{2}h_1(a) +\pi h_2(a) =0.
\end{equation*}
Below, by means of Lemma \ref{lemma1,ap.D} and some numerical computations,
we will show that the following holds on $U$:
\begin{gather*}
\tfrac{\partial h_2}{\partial \lambda }(\omega ,\lambda) >0, \\
16\sqrt{2}\tfrac{\partial h_1}{\partial \lambda }(\omega ,\lambda) +\pi
\tfrac{\partial h_2}{\partial \lambda }(\omega ,\lambda) >0,
\end{gather*}
and hence we can apply the Implicit Function Theorem proving that every
function (\ref{D22}) and (\ref{D23}) allows its local parametrization $%
\omega \mapsto \lambda (\omega )$ in $U$. This fact is used in Lemma \ref%
{lemma5}. Now some preparatory technical steps are required.

\subsubsection{Computational results II}

\noindent\textit{Upper bounds for $| r_{j}| $, $j=1,2,3$ on $U$.} 
Let $r_{j}$, $j=1,2,3$ be given by (\ref{D4.1})--(\ref{D4.3}). We will find
the upper bounds for $| r_{j}| $, $j=1,2,3$ on $U$. Setting again $\tfrac{%
\partial ^2P}{\partial (\omega ,\lambda) ^{\alpha _{j}}}$ for the
corresponding differentiations in each $r_{j}$, $j=1,2,3$, we in general
deduce that
\begin{align*}
| r_{j}(x)| &=|\int_0^{1}\int_0^{1}s\tfrac{\partial ^2P}{\partial (\omega
,\lambda) ^{\alpha _{j}}}(a+ts(x-a))dt\,ds| \\
&\leq \int_0^{1}\int_0^{1}s\sup_{x\in U}| \tfrac{ \partial ^2P}{\partial
(\omega ,\lambda) ^{\alpha _{j}}} (x)| dt\,ds \\
&=\tfrac{1}{2}\sup_{x\in U}| \tfrac{\partial ^2P}{\partial (\omega ,\lambda)
^{\alpha _{j}}}(x)| \leq Q_{j},
\end{align*}
where $Q_{j}$, $j=1,2,3$ is an upper bound for the function $\tfrac{1}{2}|
\tfrac{\partial ^2P}{\partial (\omega ,\lambda) ^{\alpha _{j}}}(x)| $ on $U$.

In an analogous way we find the upper bounds for $| \frac{\partial r_{j}}{%
\partial \omega }| , | \frac{\partial r_{j}}{\partial \lambda }| $ and $|
\frac{\partial ^2r_{j}}{\partial \omega \partial \lambda }| , | \frac{%
\partial^2r_{j}}{\partial \lambda ^2}| $, $j=1,2,3$ on $U$, we will need
later on. Explicit bounds are given in Table \ref{table2.ap.D}. Note that we
skip the derivatives $\frac{\partial ^2r_{j}}{\partial \omega ^2}$ since
there will be no need for them.
\begin{table}[ht]
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
$| r_1(x)| <5$ &
\parbox{43mm}{$|\frac{\partial r_1}{\partial \omega }(x)| <43$, 
$| \frac{\partial r_1}{\partial \lambda }(x)| <25$} &
\parbox{48mm}{
$| \frac{\partial ^2r_1}{\partial \omega \partial \lambda }(x)| <175$, 
$| \frac{\partial ^2r_1}{\partial \lambda ^2}(x)|<65$} \rule[-2.5mm]{0mm}{6.5mm}\\ \hline\hline
$| r_2(x)| <19$ &
\parbox{43mm}{
$| \frac{\partial r_2}{\partial \omega }(x)| <25$, 
$| \frac{\partial r_2}{\partial \lambda }(x)| <6$} &
\parbox{47mm}{$| \frac{\partial ^2r_2}{\partial \omega \partial \lambda }
(x)| <65$, 
$| \frac{\partial ^2r_2}{\partial \lambda ^2}(x)|<33$} \rule[-2.5mm]{0mm}{6.5mm}\\ \hline\hline
$| r_{3}(x)| <6$ &
\parbox{43mm}{
$| \frac{\partial r_{3}}{\partial \omega }(x)| <6$, 
$| \frac{\partial r_{3}}{\partial \lambda }(x)| <4$} &
\parbox{47mm}{
$| \frac{\partial ^2r_{3}}{\partial \omega \partial \lambda }(x)| <33$, 
$| \frac{\partial ^2r_{3}}{\partial \lambda ^2}(x)|<16$
} \rule[-2.5mm]{0mm}{6.5mm}\\ \hline
\end{tabular}%
\end{center}
\caption{Estimates from above for the absolute value of $r_{j}$, $j=1,2,3$
and some higher order derivatives on $U$}
\label{table2.ap.D}
\end{table}

\noindent\textit{Upper bounds for $| q_{j}| $, $j=1,2,3$\ on $U$.} 
Let $q_{j}$, $j=1,2,3$ by given by (\ref{D1.46}) -- (\ref{D1.48}). Earlier
we found the estimates for $| q_{j}| $, $j=1,2,3$ on $V$ of the type $|
q_{j}| \leq M_{j}| x-a| $, $j=1,2,3$ (see Table \ref{table1.ap.D}). Here, we
will obtain the constants which are the upper bounds for $| q_{j}| $, $%
j=1,2,3$ on $U$.

Setting $\tfrac{\partial ^{3}P}{\partial (\omega ,\lambda) ^{\alpha _{j}}},%
\tfrac{\partial ^{3}P}{\partial (\omega , \lambda) ^{\beta _{j}}}$ for the
corresponding differentiations in each $q_{j} $, $j=1,2,3$, analogously to (%
\ref{D6}), we have that
\begin{equation}
\begin{aligned} | q_{j}(x)| &=| \int_0^{1}\int_0^{1}\int_0^{1}ts^2\left(
\tfrac{ \partial ^{3}P}{\partial (\omega ,\lambda) ^{\alpha _{j}}}
(a+ts\sigma (x-a)),\tfrac{\partial ^{3}P}{\partial (\omega ,\lambda) ^{\beta
_{j}}}(a+ts\sigma (x-a))\right) \\ &\quad \times (x-a)^{T}d\sigma dt\,ds| \\
&\leq \tfrac{1}{6}\sup_{x\in U}| \left( \tfrac{\partial ^{3}P}{ \partial
(\omega ,\lambda) ^{\alpha _{j}}}(x),\tfrac{\partial ^{3}P}{\partial (\omega
,\lambda) ^{\beta _{j}}}(x)\right) | \max_{U}| x-a| \leq M_{j}\max_{U}|x-a|
, \end{aligned}  \label{D99}
\end{equation}
where $M_{j}$ is an upper bound for the function $\tfrac{1}{6}|\left( \tfrac{%
\partial ^{3}P}{\partial (\omega ,\lambda) ^{\alpha _{j}}}(x), \tfrac{%
\partial ^{3}P}{\partial (\omega ,\lambda) ^{\beta _{j}}}(x)\right) |$ on $U
$. In particular, it holds that
\begin{equation}
\begin{aligned} \max_{U}| x-a|& =| x-a|\big| _{\partial U}\\ &= \sqrt{\left(
\omega -\tfrac{1}{2} \pi \right) ^2+\left( \lambda -4\right) ^2} \big|
_{(\omega ,\lambda ) =( \tfrac{1}{2}\pi -\tfrac{2}{180}\pi ,4-0.060)}\\
&=\sqrt{( \tfrac{1}{90}\pi ) ^2+0.060^2}, \end{aligned}  \label{D104}
\end{equation}
and
\begin{equation*}
M_1=44, \quad M_2=27, \quad M_{3}=11.
\end{equation*}
The explicit upper bounds are given in Table \ref{table3.ap.D}.

\begin{table}[ht]
\renewcommand{\arraystretch}{1.5}
\par
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
$| q_1(x)| <3.1$ & $| q_2(x)| <1.9$ & $| q_{3}(x)| <0.8$ \\ \hline
\end{tabular}%
\end{center}
\caption{Estimates from above for $| q_{j}| $, $j=1,2,3$ on $U$}
\label{table3.ap.D}
\end{table}

\noindent\textit{Lower bounds for $F(x)=\det B(x)$ and $G(x)=b_{11}(x)+2\protect%
\sqrt{\det B(x)}+b_{22}(x)$ on $U$.} 
Let us set
\begin{equation}
\begin{gathered} F(x)=\det B(x), \\ G(x)=b_{11}(x)+2\sqrt{\det
B(x)}+b_{22}(x), \end{gathered}  \label{D102}
\end{equation}
$b_{ij}$, $i,j=1,2$ are given in (\ref{D29})--(\ref{D32}) and find the lower
bounds for $F$ and $G$ on $U$. By construction of the ball $W_{R}(a)\supset
U $ from Proposition \ref{Proposition.ap.D} we know that $F\geq 0$ and $G>0$
on $W_{R}(a)$. More precisely, for every $x\in W_{R}(a)$ (and hence for
every $x\in U$) it holds that
\begin{gather*}
F(x)\geq 1-\tfrac{\sqrt{2}}{4\pi }| q_2(x)| -\tfrac{1}{ 64}| q_1(x)| -\tfrac{%
1}{32\pi ^2}| q_1(x)| | q_{3}(x)| , \\
G(x)\geq 2-\tfrac{\sqrt{2}}{4\pi }| q_2(x)| -\tfrac{1}{ 64}| q_1(x)| .
\end{gather*}
Due to results of Table \ref{table3.ap.D} we finally obtain that on $U$,
\begin{equation}
\begin{gathered} F(x)\geq 0.730\dots>0.7, \\ G(x)\geq 1.738\dots>1.7.
\end{gathered}  \label{D101}
\end{equation}

\noindent\textit{Upper bounds for $| b_{ij}| $, $i,j=1,2$ on $U $.}
Let $b_{ij}$, $i,j=1,2$ be given by (\ref{D2.1}) -- (\ref{D2.4}), namely,
\begin{gather*}
b_{11}(x)=\tfrac{1}{64}r_1(x)-\tfrac{\sqrt{2}}{8\pi }r_2(x), \\
b_{12}(x)=\tfrac{1}{64}r_2(x)-\tfrac{\sqrt{2}}{8\pi }r_{3}(x), \\
b_{21}(x)=-\tfrac{\sqrt{2}}{8\pi }r_1(x), \quad b_{22}(x)=-\tfrac{\sqrt{2}}{%
8\pi }r_2(x),
\end{gather*}
with $r_{j}$, $j=1,2,3$ as in (\ref{D4.1})--(\ref{D4.3}).

Using the results of Table \ref{table2.ap.D} we will find the following
upper bounds for the absolute values of $b_{ij}$, $i,j=1,2$ and some higher
order derivatives on $U$ (see Table \ref{table4.ap.D}).

\begin{table}[ht]
\renewcommand{\arraystretch}{1.5}
\par
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
$| b_{11}(x)| <1.2$ &
\parbox{47mm}{
$| \frac{\partial b_{11}}{\partial \omega }(x)| <2.1$,
$| \frac{\partial b_{11}}{\partial \lambda }(x)| <0.8$} &
\parbox{49mm}{
$| \frac{\partial ^2b_{11}}{\partial \omega \partial \lambda }(x)| <6.4$,
$| \frac{\partial ^2b_{11}}{\partial \lambda ^2}(x)|<2.9$} \\ \hline
$| b_{12}(x)| <0.7$ &
\parbox{47mm}{
$| \frac{\partial b_{12}}{\partial \omega }(x)| <0.8$,
$| \frac{\partial b_{12}}{\partial \lambda }(x)| <0.4$} &
\parbox{49mm}{
$| \frac{\partial ^2b_{12}}{\partial \omega \partial \lambda }(x)| <2.9$,
$| \frac{\partial ^2b_{12}}{\partial \lambda ^2}(x)|<1.5$ } \\ \hline
$| b_{21}(x)| <0.3$ &
\parbox{47mm}{
$| \frac{\partial b_{21}}{\partial \omega }(x)| <2.5$,
$| \frac{\partial b_{21}}{\partial \lambda }(x)| <1.5$} &
\parbox{49mm}{
$| \frac{\partial ^2b_{21}}{\partial \omega \partial \lambda }(x)| <9.9$,
$| \frac{\partial ^2b_{21}}{\partial \lambda ^2}(x)|<3.7$} \\ \hline
$| b_{22}(x)| <1.1$ &
\parbox{47mm}{
$| \frac{\partial b_{22}}{\partial \omega }(x)| <1.5$,
$| \frac{\partial b_{22}}{\partial \lambda }(x)| <0.4$} &
\parbox{49mm}{
$| \frac{\partial ^2b_{22}}{\partial \omega \partial \lambda }(x)| <3.7$,
$| \frac{\partial ^2b_{22}}{\partial \lambda ^2}(x)|<1.9$} \\ \hline
\end{tabular}%
\end{center}
\caption{Estimates from above for the absolute value of $b_{ij}$, $i,j=1,2$
and some higher order derivatives on $U$}
\label{table4.ap.D}
\end{table}

\noindent\textit{Upper bounds for $F(x)=\det B(x)$ and $G(x)=b_{11}(x)+2\protect%
\sqrt{\det B(x)}+b_{22}(x)$ on $U$.} 
Recall that $F$ and $G$ are given by (\ref{D102}). They are positive
functions on $U$ with lower bounds as in (\ref{D101}). Here we find their
upper bounds together with the upper bounds for some higher order
derivatives.

In particular, in order to obtain the estimates for $F$, $| \frac{\partial F%
}{\partial \omega }| , | \frac{\partial F}{\partial \lambda }| $ and $|
\frac{\partial ^2F}{\partial \omega \partial \lambda }| , | \frac{\partial^2F%
}{\partial \lambda ^2}| $ on $U$, we use the results of Table \ref%
{table4.ap.D}. The estimates found are presented in the first row of Table %
\ref{table5.ap.D}.

To find the estimates for $| \frac{\partial G}{\partial \omega }| ,| \frac{%
\partial G}{\partial \lambda } | $ and $| \frac{\partial ^2G}{\partial
\omega \partial \lambda }| ,| \frac{\partial ^2G}{\partial \lambda ^2} | $
we use

\begin{itemize}
\item the lower bound for $F$on $U$, namely, $F(x)>0.7$,

\item the results of Table \ref{table4.ap.D} and

\item the results from the first row of Table \ref{table5.ap.D}.
\end{itemize}

E.g., for $| \tfrac{\partial G}{\partial \omega }| $ we will have that on $U$
\begin{align*}
| \tfrac{\partial G}{\partial \omega }(x)| &=| \tfrac{\partial b_{11}}{%
\partial \omega }(x)+\tfrac{\tfrac{\partial F}{ \partial \omega }(x)}{\sqrt{%
F(x)}}+\tfrac{\partial b_{22}}{\partial \omega } (x)| \\
&\leq \sup_{U}| \tfrac{\partial b_{11}}{ \partial \omega }(x)| +\tfrac{%
\sup_{U}| \tfrac{ \partial F}{\partial \omega }(x)| }{\inf_{U}\sqrt{F(x)}}
+\sup_{U}| \tfrac{\partial b_{22}}{\partial \omega } (x)| \\
& <2.1+\tfrac{6.1}{\sqrt{0.7}}+1.5\approx 10.891\dots<11.
\end{align*}
The other estimates on $U$ for the derivatives of $G$ listed above are
obtained in an analogous way and presented in Table \ref{table5.ap.D}.

\begin{table}[ht]
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
$F(x)<1.53$ &
\parbox{24mm}{
$| \frac{\partial F}{\partial \omega }(x)| <6.10$, \hfill
$| \frac{\partial F}{\partial \lambda }(x)| <2.53$} &
\parbox{29mm}{
$| \frac{\partial ^2F}{\partial \omega \partial \lambda }(x)| <23.52$,  \hfill
$| \frac{\partial ^2F}{\partial \lambda ^2}(x)| <10.35$}\rule[-14pt]{0pt}{34pt} \\ \hline
$-$ &
\parbox{22mm}{$| \frac{\partial G}{\partial \omega }(x)| <11$,  \hfill
$| \frac{\partial G}{\partial \lambda }(x)| <4.3$} &
\parbox{27mm}{
$| \frac{\partial ^2G}{\partial \omega \partial \lambda }(x)| <51.4$,  \hfill
$| \frac{\partial ^2G}{\partial \lambda ^2}(x)| <22.7$}\rule[-14pt]{0pt}{34pt} \\ \hline
\end{tabular}%
\end{center}
\caption{Estimates from above for $F$ and $G$ on $U$}
\label{table5.ap.D}
\end{table}

\noindent\textit{Upper bounds for $| c_{ij}| $, $i,j=1,2$ on $U $.} 
Let $c_{ij}$, $i,j=1,2$ be given by given by (\ref{D20}). It is convenient
for further computations to set
\begin{equation}
c_{ij}(x)=\tfrac{b_{ij}(x)+A\sqrt{F(x)}}{\sqrt{G(x)}},  \label{D103}
\end{equation}
where $F$ and $G$ are as in (\ref{D102}) and
\begin{equation*}
A=%
\begin{cases}
1 & \text{if }(i,j)=\{ (1,1),\text{ }(2,2)\} , \\
0 & \text{if }(i,j)=\{ (1,2),\text{ }(2,1)\}.%
\end{cases}%
\end{equation*}
To obtain the estimates for $| \frac{\partial c_{ij}}{ \partial \omega }| ,|
\frac{\partial c_{ij}}{\partial \lambda }| $ and $| \frac{\partial ^2c_{ij}}{%
\partial \omega \partial \lambda }| ,| \frac{\partial ^2c_{ij}}{ \partial
\lambda ^2}| $, $i,j=1,2$ on $U$ we use

\begin{itemize}
\item the lower bound for $F$ and $G$ on $U$, defined by formula (\ref{D101}%
),

\item the results of Table \ref{table4.ap.D} and

\item the results of Table \ref{table5.ap.D}.
\end{itemize}

E.g., the estimate for $| \frac{\partial c_{ij}}{\partial \lambda } | $ on $%
U $ is found in the following way:
\begin{align*}
| \tfrac{\partial c_{ij}}{\partial \lambda }(x)| &=\big| \tfrac{\tfrac{%
\partial b_{ij}}{\partial \lambda }(x)}{\sqrt{G(x)} }+\tfrac{1}{2}A\tfrac{%
\tfrac{\partial F}{\partial \lambda }(x)}{\sqrt{G(x)} \sqrt{F(x)}}-\tfrac{1}{%
2}\tfrac{\tfrac{\partial G}{\partial \lambda }(x)}{ G^{3/2}(x)}\left(
b_{ij}(x)+A\sqrt{F(x)}\right) \big| \\
&\leq \big| \tfrac{\sup_{U}| \tfrac{\partial b_{ij}}{ \partial \lambda }(x)|
}{\inf_{U}\sqrt{G(x)}}+\tfrac{1}{ 2}A\tfrac{\sup_{U}| \tfrac{\partial F}{%
\partial \lambda } (x)| }{\inf_{U}\left( \sqrt{G(x)}\sqrt{F(x)}\right) } \\
&\quad +\tfrac{1}{2}\tfrac{\sup_{U}| \tfrac{\partial G}{ \partial \lambda }%
(x)| }{\inf_{U}G^{3/2}(x)}\left( \sup_{U}| b_{ij}(x)| +A\sup_{U} \sqrt{F(x)}%
\right) \big| .
\end{align*}
The other estimates on $U$ for the derivatives of $c_{ij}$, $i,j=1,2$ listed
are obtained in an analogous way and listed in Table \ref{table6.ap.D} below.

\begin{table}[ht]
\renewcommand{\arraystretch}{1.5}
\par
\begin{center}
\begin{tabular}{|c|c|}
\hline
$| \frac{\partial c_{11}}{\partial \lambda }(x)| <4.2$ &
\parbox{27mm}{
$| \frac{\partial ^2c_{11}}{\partial \omega \partial \lambda }
(x)| <99.4$
$| \frac{\partial ^2c_{11}}{\partial \lambda ^2}(x)|<35$} \\ \hline
\parbox{24mm}{
$| \frac{\partial c_{12}}{\partial \omega }(x)| <2.4$
$| \frac{\partial c_{12}}{\partial \lambda }(x)| <1$} &
\parbox{27mm}{
$| \frac{\partial ^2c_{12}}{\partial \omega \partial \lambda }
(x)| <18.7$
$| \frac{\partial ^2c_{12}}{\partial \lambda ^2}(x)|<8.1$} \\ \hline
$| \frac{\partial c_{21}}{\partial \lambda }(x)| <1.5$ &
\parbox{27mm}{
$| \frac{\partial ^2c_{21}}{\partial \omega \partial \lambda }
(x)| <20.1$
$| \frac{\partial ^2c_{21}}{\partial \lambda ^2}(x)|<8.4$} \\ \hline
\parbox{24mm}{
$| \frac{\partial c_{22}}{\partial \omega }(x)| <9.8$
$| \frac{\partial c_{22}}{\partial \lambda }(x)| <3.8$} &
\parbox{27mm}{
$| \frac{\partial ^2c_{22}}{\partial \omega \partial \lambda }(x)| <94.4$
$| \frac{\partial ^2c_{22}}{\partial \lambda ^2}(x)|<38.7$} \\ \hline
\end{tabular}%
\end{center}
\caption{Estimates from above for the absolute value of some higher order
derivatives of $c_{ij}$, $i,j=1,2$ on $U$}
\label{table6.ap.D}
\end{table}

\noindent\textit{Upper bounds for $| h_{i}| $, $i=1,2$ on $U$.} 
Let $h_{i}$, $i=1,2$ be given by formulas (\ref{D18}), (\ref{D19}). First we
compute the following derivatives of $h_{i}$, $i=1,2$ we will need below:
\begin{gather}
\tfrac{\partial h_1}{\partial \lambda }(x) =c_{12}(x) +\left( \tfrac{%
\partial c_{11}}{\partial \lambda }(x) , \tfrac{\partial c_{12}}{\partial
\lambda }(x) \right) \left( x-a\right) ^{T},  \label{D24} \\
\tfrac{\partial ^2h_1}{\partial \omega \partial \lambda }(x) =\tfrac{%
\partial c_{12}}{\partial \omega }(x)+\tfrac{\partial c_{11}}{ \partial
\lambda }(x)+\left( \tfrac{\partial ^2c_{11}}{\partial \omega \partial
\lambda }(x) ,\tfrac{\partial ^2c_{12}}{\partial \omega \partial \lambda }%
(x) \right) \left( x-a\right) ^{T},  \label{D25} \\
\tfrac{\partial ^2h_1}{\partial \lambda ^2}(x) =2\tfrac{ \partial c_{12}}{%
\partial \lambda }(x)+\left( \tfrac{\partial ^2c_{11}}{ \partial \lambda ^2}%
(x) ,\tfrac{\partial ^2c_{12}}{\partial \lambda ^2}(x) \right) \left(
x-a\right) ^{T},  \label{D25.1} \\
\tfrac{\partial h_2}{\partial \lambda }(x) =c_{22}(x) +\left( \tfrac{%
\partial c_{21}}{\partial \lambda }(x) , \tfrac{\partial c_{22}}{\partial
\lambda }(x) \right) \left( x-a\right) ^{T},  \label{D26} \\
\tfrac{\partial ^2h_2}{\partial \omega \partial \lambda }(x) =\tfrac{%
\partial c_{22}}{\partial \omega }(x)+\tfrac{\partial c_{21}}{ \partial
\lambda }(x)+\left( \tfrac{\partial ^2c_{21}}{\partial \omega \partial
\lambda }(x) ,\tfrac{\partial ^2c_{22}}{\partial \omega \partial \lambda }%
(x) \right) \left( x-a\right) ^{T},  \label{D27} \\
\tfrac{\partial ^2h_2}{\partial \lambda ^2}(x) =2\tfrac{ \partial c_{22}}{%
\partial \lambda }(x)+\left( \tfrac{\partial ^2c_{21}}{ \partial \lambda ^2}%
(x) ,\tfrac{\partial ^2c_{22}}{\partial \lambda ^2}(x) \right) \left(
x-a\right) ^{T},  \label{D27.1}
\end{gather}
where $c_{ij}$, $i,j=1,2$ are as in (\ref{D103}) and $a=\left( \frac{1}{2}%
\pi ,4\right) $.

Using the results of Table \ref{table6.ap.D} we find the estimates for $|
\frac{\partial ^2h_{i}}{\partial \omega \partial \lambda } | ,| \frac{%
\partial ^2h_{i}}{\partial \lambda ^2} | $, $i=1,2$ on $U$. E.g., for $|
\frac{\partial ^2h_1}{\partial \omega \partial \lambda }| $ it holds that on
$U$:
\begin{align*}
| \tfrac{\partial ^2h_1}{\partial \omega \partial \lambda }(x)| &=| \tfrac{%
\partial c_{12}}{\partial \omega }(x)+ \tfrac{\partial c_{11}}{\partial
\lambda }(x)+\left( \tfrac{\partial ^2c_{11}}{\partial \omega \partial
\lambda }(x) ,\tfrac{ \partial ^2c_{12}}{\partial \omega \partial \lambda }%
(x) \right) \left( x-a\right) ^{T}| \\
&\leq \sup_{U}| \tfrac{\partial c_{12}}{\partial \omega } (x)| +\sup_{U}|
\tfrac{\partial c_{11}}{\partial \lambda }(x)| \\
&\quad +\sup_{U}| \left( \tfrac{\partial ^2c_{11}}{\partial \omega \partial
\lambda }(x) ,\tfrac{\partial ^2c_{12}}{\partial \omega \partial \lambda }
(x) \right) | \max_{U}| x-a| <\dots
\end{align*}
for $\max_{U}| x-a| $ see formula (\ref{D104}) and then
\begin{equation*}
\dots<2.4+4.2+\sqrt{99.4^2+18.7^2}\sqrt{\left( \tfrac{1}{90}\pi \right)
^2+0.060^2}\approx 13.621\dots<13.7.
\end{equation*}
Analogously, the other estimates on $U$ are obtained (see Table \ref%
{table7.ap.D}).

\begin{table}[ht]
\renewcommand{\arraystretch}{1.5}
\par
\begin{center}
\begin{tabular}{|c|c|}
\hline
$| \frac{\partial ^2h_1}{\partial \omega \partial \lambda }(x)| <13.7$ & $|
\frac{\partial ^2h_2}{\partial \omega \partial \lambda }(x)| <18$ \\ \hline
$| \frac{\partial ^2h_1}{\partial \lambda ^2}(x)|<4.5$ & $| \frac{\partial
^2h_2}{\partial \lambda ^2}(x)| <10.4$ \\ \hline
\end{tabular}%
\end{center}
\caption{Estimates from above for the absolute value of some higher order
derivatives of $h_{i}$, $i=1,2$ on $U$}
\label{table7.ap.D}
\end{table}

\subsubsection{Strict positivity of the functions $\frac{\partial h_2}{%
\partial \protect\lambda }(\protect\omega ,\protect\lambda) $ and $16\protect%
\sqrt{2}\frac{\partial h_1}{\partial \protect\lambda } (\protect\omega ,%
\protect\lambda) +\protect\pi \frac{\partial h_2}{\partial \protect\lambda }(%
\protect\omega ,\protect\lambda) $ on $U$}

Let us fix the notations which are common for two lemmas.

\begin{notation}\label{Notation1,ap.D} \rm
Let $U$ be as in (\ref{D100}), namely,
\begin{equation*}
U:=\left\{ (\omega ,\lambda) :\left[ \tfrac{88}{180}\pi ,\tfrac{
92}{180}\pi \right] \times \left[ 3.940,4.060\right] \right\} .
\end{equation*}
By $\{ (\omega _{i},\lambda _{j})\} _{\substack{ i=0,\dots,n  \\
j=0,\dots,m}}$ we mean a discretization of $U$ which is defined as follows
\begin{equation*}
\omega _{i}=\tfrac{88}{180}\pi +i\Delta \omega , \quad
\lambda_{j}=3.94+j\Delta \lambda ,
\end{equation*}
with
\begin{equation*}
\Delta \omega =\tfrac{\tfrac{4}{180}\pi }{n}, \quad \Delta \lambda =
\tfrac{0.12}{m}.
\end{equation*}
\end{notation}

Then we deduce the following two results.

\begin{lemma}\label{lemma3,ap.D}
It holds that
\begin{equation}
\frac{\partial h_2}{\partial \lambda }(\omega ,\lambda) >0\quad\text{on }U.
\label{D33}
\end{equation}
\end{lemma}

\begin{proof}
Fix $n=m=2$ and consider the discretization $\{ (\omega _{i},\lambda _{j})\}
_{i,j}$ of $U$ given by Notation \ref{Notation1,ap.D}. By straightforward
computations it holds that
\begin{equation}
\tfrac{\partial h_2}{\partial \lambda }(\omega _{i},\lambda _{j})>0,
\label{D105}
\end{equation}
for all $i=0,\dots,2$, $j=0,\dots,2$ and, moreover,
\begin{equation}
\min_{(\omega _{i},\lambda _{j})\in U} \tfrac{\partial h_2}{ \partial
\lambda }(\omega _{i},\lambda _{j})=\tfrac{\partial h_2}{\partial \lambda }%
(\omega _0,\lambda _0)\approx 0.952\dots\,.  \label{D106}
\end{equation}
Next to this, we compute
\begin{equation*}
a_1=\max \big\{ \Delta \omega ,\Delta \lambda \big\} =\max \big\{ \tfrac{%
\tfrac{4}{180}\pi }{n},\tfrac{0.12}{m}\big\} =\tfrac{0.12}{2}=0.060,
\end{equation*}
and, by taking into account the results of Table \ref{table7.ap.D} and (\ref%
{D106}), we also find
\begin{equation*}
a_2=\sqrt{2}\dfrac{\min_{(\omega _{i},\lambda _{j})\in U} \tfrac{ \partial
h_2}{\partial \lambda }(\omega _{i},\lambda _{j})} {\sup_{U} | \left( \tfrac{%
\partial ^2h_2}{\partial \omega \partial \lambda }(\omega ,\lambda ),\tfrac{%
\partial ^2h_2}{\partial \lambda ^2} (\omega ,\lambda )\right) | }\approx
0.065\dots\,.
\end{equation*}
Since $a_1<a_2$, by Lemma \ref{lemma1,ap.D} we conclude that the constructed
discretization $\left\{ (\omega _{i},\lambda _{j})\right\} _{\substack{ %
i=0,\dots,2  \\ j=0,\dots,2}}$ of $U$ to be appropriate in the sense that
condition (\ref{D105}) yields a strict positivity of $\frac{ \partial h_2}{%
\partial \lambda }(\omega ,\lambda) $ on $U$.
\end{proof}

\begin{lemma}\label{lemma4,ap.D}
Let $U$ be as in (\ref{D100}). It holds that
\begin{equation}
16\sqrt{2}\frac{\partial h_1}{\partial \lambda }(\omega ,\lambda)
+\pi \frac{\partial h_2}{\partial \lambda }(\omega ,\lambda) >0
\quad\text{on }U.
\label{D60}
\end{equation}
\end{lemma}

\begin{proof}
Fix $n=7$, $m=12$ and consider the discretization $\left\{ (\omega
_{i},\lambda _{j})\right\} _{i,j}$ of $U$ given by Notation \ref%
{Notation1,ap.D}. By straightforward computations it holds that
\begin{equation}
16\sqrt{2}\tfrac{\partial h_1}{\partial \lambda }(\omega _{i},\lambda
_{j})+\pi \tfrac{\partial h_2}{\partial \lambda }(\omega _{i},\lambda
_{j})>0,  \label{D107}
\end{equation}
for all $i=0,\dots,7$, $j=0,\dots,12$ and, moreover,
\begin{equation}
\begin{aligned} &\min_{(\omega _{i},\lambda _{j})\in U} \left(
16\sqrt{2}\tfrac{ \partial h_1}{\partial \lambda }(\omega _{i},\lambda
_{j})+\pi \tfrac{ \partial h_2}{\partial \lambda }(\omega _{i},\lambda
_{j})\right)\\ &=16\sqrt{2}\tfrac{\partial h_1}{\partial \lambda }(\omega
_0,\lambda _0) +\pi \tfrac{\partial h_2}{\partial \lambda }(\omega
_0,\lambda _0)\approx 2.936\dots\,. \end{aligned}  \label{D108}
\end{equation}
Next to this, we compute
\begin{equation*}
b_1=\max \big\{ \Delta \omega ,\Delta \lambda \big\} =\max\big\{ \tfrac{%
\tfrac{4}{180}\pi }{n},\tfrac{0.12}{m}\big\} =\tfrac{0.12}{12}=0.010,
\end{equation*}
and, by taking into account the results of Table \ref{table7.ap.D} and (\ref%
{D108}), we also find
\begin{align*}
b_2&=\sqrt{2}\dfrac{\min_{(\omega _{i},\lambda _{j})\in U} \left( 16\sqrt{2}%
\tfrac{\partial h_1}{\partial \lambda }(\omega _{i},\lambda _{j})+\pi \tfrac{%
\partial h_2}{\partial \lambda }(\omega _{i},\lambda _{j})\right) }{%
\sup_{U}| \left( 16\sqrt{2}\tfrac{ \partial ^2h_1}{\partial \omega \partial
\lambda }(\omega ,\lambda )+\pi \tfrac{\partial ^2h_2}{\partial \omega
\partial \lambda }(\omega ,\lambda ),16\sqrt{2}\tfrac{\partial ^2h_1}{%
\partial \lambda ^2} (\omega ,\lambda )+\pi \tfrac{\partial ^2h_2}{\partial
\lambda ^2} (\omega ,\lambda )\right) | } \\
&\approx 0.011\dots\,.
\end{align*}

Since $b_1<b_2$, by Lemma \ref{lemma1,ap.D} we conclude that the constructed
discretization $\{ (\omega _{i},\lambda _{j})\}_{i,j}$ of $U$ to be
appropriate in the sense that condition (\ref{D107}) yields a strict
positivity of $16\sqrt{2} \frac{\partial h_1}{\partial \lambda }(\omega
,\lambda) +\pi \frac{\partial h_2}{\partial \lambda }(\omega ,\lambda) $ on $%
U$.
\end{proof}

\subsection{On $P(\protect\omega ,\protect\lambda) =0$ in $V$ away from $a=(
\frac{1}{2}\protect\pi ,4)$}

Recall here function $P$ defined on $V=\left\{ (\omega ,\lambda) :[ \frac{70%
}{180}\pi ,\frac{110}{180}\pi ] \times [2.900,5.100] \right\} $:
\begin{equation}
\begin{aligned} P(\omega ,\lambda) &=\left( 1-\tfrac{\sqrt{2}}{2}\sin
(2\omega )\right) ^{\lambda }+\left( 1+\tfrac{\sqrt{2}}{2}\sin (2\omega
)\right) ^{\lambda }\\ &\quad +\left( \tfrac{1}{2}+\tfrac{1}{2}\cos
^2(2\omega )\right) ^{\frac{1}{2} \lambda }\Big[ 2\cos \left( \lambda \left(
\arctan \left( \tfrac{\sqrt{2}}{2 }\tan (2\omega )\right) +\pi \right)
\right)\\ &\quad -4\cos \left( \lambda \arctan \left( \tan ^2(\omega
)\right) \right) \Big] . \end{aligned}  \label{D109}
\end{equation}

\subsubsection{Set of Claims I}

In a set of claims below we describe some properties of the first
derivatives $\frac{\partial P}{\partial \omega }$ and $\frac{\partial P}{%
\partial \lambda }$\ on $V$ away from the point $a=\left( \frac{1}{2}%
\pi,4\right) $ which are used in Lemma \ref{lemma5}.

\begin{claim}\label{Claim5,ap.D}\rm
Let $H_1\subset V$ be
$H_1=\left\{ (\omega ,\lambda) :\left[ \tfrac{84}{180}\pi ,\tfrac{90}{180}\pi
\right] \times \left[ 4.030,4.970\right] \right\}$.
It holds that $\frac{\partial P}{\partial \omega }(\omega ,\lambda) <0$
 on $H_1$.
\end{claim}

\begin{proof}
First we find the following estimates:
\begin{equation}
| \frac{\partial ^2P}{\partial \omega ^2}(\omega ,\lambda) | <162,\quad |
\tfrac{\partial ^2P}{\partial \omega \partial \lambda }(\omega ,\lambda )|
<45\quad \text{on }H_1.  \label{D110}
\end{equation}
Then we fix $n=14$, $m=120$ and consider the discretization $\{
(\omega_{i},\lambda _{j})\} _{i,j}$ of $H_1 $ such that
\begin{equation*}
\omega _{i}=\tfrac{84}{180}\pi +i\Delta \omega , \quad \lambda
_{j}=4.030+j\Delta \lambda ,
\end{equation*}
with
\begin{equation*}
\Delta \omega =\tfrac{\tfrac{6}{180}\pi }{n}, \quad \Delta \lambda =\tfrac{%
0.94}{m}.
\end{equation*}
By straightforward computations it holds that
\begin{equation}
-\tfrac{\partial P}{\partial \omega }(\omega _{i},\lambda _{j})>0,
\label{D111}
\end{equation}
for all $i=0,\dots,14$, $j=0,\dots,120$ and, moreover,
\begin{equation}
\min_{(\omega _{i},\lambda _{j})\in H_1} \left( -\tfrac{\partial P}{\partial
\omega }(\omega _{i},\lambda _{j})\right) =-\tfrac{\partial P}{ \partial
\omega }(\omega _{9},\lambda _0)\approx 1.022\dots\,.  \label{D112}
\end{equation}
Next to this, we compute
\begin{equation*}
c_1=\max \left\{ \Delta \omega ,\Delta \lambda \right\} =\max \big\{ \tfrac{%
\tfrac{6}{180}\pi }{n},\tfrac{0.94}{m}\big\} =\tfrac{0.94}{120}\approx
0.00783\dots\,,
\end{equation*}
and, by taking into account (\ref{D110}) and (\ref{D112}), we also find
\begin{equation*}
c_2=\sqrt{2}\dfrac{\min_{(\omega _{i},\lambda _{j})\in H_1} \left( -\tfrac{%
\partial P}{\partial \omega }(\omega _{i},\lambda _{j})\right) }{\sup_{H_1} %
\big| \big( \tfrac{\partial ^2P }{\partial \omega ^2}(\omega ,\lambda ),%
\tfrac{\partial ^2P}{\partial \omega \partial \lambda }(\omega ,\lambda )%
\big) \big| }\approx 0.00860\dots\,.
\end{equation*}

Since $c_1<c_2$, by Lemma \ref{lemma1,ap.D} we conclude that the constructed
discretization $\left\{ (\omega _{i},\lambda _{j})\right\} _{\substack{ %
i=0,\dots,14  \\ j=0,\dots,120}}$ of $H_1$ to be appropriate\ in the sense
that condition (\ref{D111}) yields a strict positivity of $-\tfrac{ \partial
P}{\partial \omega }(\omega ,\lambda )$ on $H_1$, or, in other words, $%
\tfrac{\partial P}{\partial \omega }(\omega ,\lambda )<0$ on $H_1$.
\end{proof}

\begin{claim}\label{Claim6,ap.D} \rm
Let $H_2\subset V$ be
$H_2=\left\{ (\omega ,\lambda) :\left[ \tfrac{87}{180}\pi ,
\tfrac{101}{180}\pi \right] \times \left[ 4.750,5.100\right] \right\}$.
It holds that $\frac{\partial P}{\partial \lambda }(\omega ,\lambda) >0$
 on $H_2$.
\end{claim}

\begin{proof}
First we find the following estimates:
\begin{equation}
\begin{tabular}{llll}
$| \tfrac{\partial ^2P}{\partial \omega \partial \lambda }(\omega ,\lambda)
| <48,\quad | \tfrac{\partial ^2P }{\partial \lambda ^2}(\omega ,\lambda )|
<25\quad\text{on }H_2.$ &  &  &
\end{tabular}
\label{D120}
\end{equation}
Then we fix $n=40$, $m=55$ and consider the discretization $\left\{
(\omega_{i},\lambda _{j})\right\} _{i,j}$, $i=0,\dots,40$, $j=0,\dots,55$ of
$H_2 $ such that
\begin{equation*}
\omega _{i}=\tfrac{87}{180}\pi +i\Delta \omega , \quad
\lambda_{j}=4.750+j\Delta \lambda ,
\end{equation*}
with
\begin{equation*}
\Delta \omega =\tfrac{\tfrac{14}{180}\pi }{n}, \quad \Delta \lambda = \tfrac{%
0.35}{m}.
\end{equation*}
By straightforward computations it holds that
\begin{equation}
\tfrac{\partial P}{\partial \lambda }(\omega _{i},\lambda _{j})>0,
\label{D121}
\end{equation}
for all $i=0,\dots,40$, $j=0,\dots,55$ and, moreover,
\begin{equation}
\min_{(\omega _{i},\lambda _{j})\in H_2} \tfrac{\partial P}{ \partial
\lambda }(\omega _{i},\lambda _{j})=\tfrac{\partial P}{\partial \lambda }%
(\omega _0,\lambda _0)\approx 0.245\dots\,.  \label{D122}
\end{equation}
Next to this, we compute
\begin{equation*}
c_1=\max \left\{ \Delta \omega ,\Delta \lambda \right\} =\max \big\{ \tfrac{%
\tfrac{14}{180}\pi }{n},\tfrac{0.35}{m}\big\} =\tfrac{0.35}{ 55}\approx
0.00636\dots\,,
\end{equation*}
and, by taking into account (\ref{D120}) and (\ref{D122}), we also find
\begin{equation*}
c_2=\sqrt{2}\dfrac{\min_{(\omega _{i},\lambda _{j})\in H_2} \tfrac{\partial P%
}{\partial \lambda }(\omega _{i},\lambda _{j})} {\sup_{H_2}\big| \big(
\tfrac{\partial ^2P}{\partial \omega \partial \lambda }(\omega ,\lambda ),%
\tfrac{\partial ^2P}{\partial \lambda ^2}(\omega ,\lambda )\big) \big| }%
\approx 0.00641\dots\,.
\end{equation*}
Since $c_1<c_2$, by Lemma \ref{lemma1,ap.D} we conclude that the constructed
discretization $\left\{ (\omega _{i},\lambda _{j})\right\} _{\substack{ %
i=0,\dots,40  \\ j=0,\dots,55}}$ of $H_2$ to be appropriate in the sense
that condition (\ref{D121}) yields a strict positivity of $\tfrac{\partial P%
}{\partial \lambda }(\omega ,\lambda )$ on $H_2$.
\end{proof}

\begin{claim}\label{Claim7,ap.D}\rm
Let $H_{3}\subset V$ be
$H_{3}=\left\{ (\omega ,\lambda) :\left[ \tfrac{100}{180}\pi ,
\tfrac{108}{180}\pi \right] \times \left[ 4.000,4.850\right] \right\}$.
It holds that $\frac{\partial P}{\partial \omega }(\omega ,\lambda) >0$
on $H_{3}$.
\end{claim}

\begin{proof}
First we find the following estimates:
\begin{equation}
| \frac{\partial ^2P}{\partial \omega ^2}(\omega ,\lambda) | <166, \quad |
\tfrac{\partial ^2P}{\partial \omega \partial \lambda }(\omega ,\lambda )|
<37\quad\text{on }H_{3}.  \label{D130}
\end{equation}
Then we fix $n=10$, $m=60$ and consider the discretization $\left\{
(\omega_{i},\lambda _{j})\right\} _{i,j}$ of $H_{3} $ such that
\begin{equation*}
\omega _{i}=\tfrac{100}{180}\pi +i\Delta \omega , \quad
\lambda_{j}=4.000+j\Delta \lambda ,
\end{equation*}
with
\begin{equation*}
\Delta \omega =\tfrac{\tfrac{8}{180}\pi }{n},\quad \Delta \lambda = \tfrac{%
0.85}{m}.
\end{equation*}
By straightforward computations it holds that
\begin{equation}
\tfrac{\partial P}{\partial \omega }(\omega _{i},\lambda _{j})>0,
\label{D131}
\end{equation}
for all $i=0,\dots,10$, $j=0,\dots,60$ and, moreover,
\begin{equation}
\min_{(\omega _{i},\lambda _{j})\in H_{3}} \tfrac{\partial P}{ \partial
\omega }(\omega _{i},\lambda _{j})=\tfrac{\partial P}{\partial \omega }%
(\omega _0,\lambda _{11})\approx 1.885\dots\,.  \label{D132}
\end{equation}
Next to this, we compute
\begin{equation*}
c_1=\max \left\{ \Delta \omega ,\Delta \lambda \right\} =\max \big\{ \tfrac{%
\tfrac{8}{180}\pi }{n},\tfrac{0.85}{m}\big\} =\tfrac{0.85}{60}\approx
0.0142\dots\,,
\end{equation*}
and, by taking into account (\ref{D130}) and (\ref{D132}), we also find
\begin{equation*}
c_2=\sqrt{2}\dfrac{\min_{(\omega _{i},\lambda _{j})\in H_{3}} \tfrac{%
\partial P}{\partial \omega }(\omega _{i},\lambda _{j})} {\sup_{H_{3}}\big| %
\big( \tfrac{\partial ^2P}{\partial \omega ^2} (\omega ,\lambda ),\tfrac{%
\partial ^2P}{\partial \omega \partial \lambda } (\omega ,\lambda )\big) %
\big| }\approx 0.0157\dots\,.
\end{equation*}

Since $c_1<c_2$, by Lemma \ref{lemma1,ap.D} we conclude that the constructed
discretization $\left\{ (\omega _{i},\lambda _{j})\right\} _{\substack{ %
i=0,\dots,10  \\ j=0,\dots,60}}$ of $H_{3}$ to be appropriate\ in the sense
that condition (\ref{D131}) yields a strict positivity of $\tfrac{ \partial P%
}{\partial \omega }(\omega ,\lambda )$ on $H_{3}$.
\end{proof}

\begin{claim}\label{Claim8,ap.D} \rm
Let $H_{4}\subset V$ be
$H_{4}=\{ (\omega ,\lambda) :[ \tfrac{91}{180}\pi ,\tfrac{102}{180}\pi]
\times [ 3.950,4.100] \} $.
It holds that $\frac{\partial P}{\partial \lambda }(\omega ,\lambda) <0$
on $H_{4}$.
\end{claim}

\begin{proof}
First we find the following estimates:
\begin{equation}
| \tfrac{\partial ^2P}{\partial \omega \partial \lambda }(\omega ,\lambda) |
<38, \quad | \tfrac{\partial ^2P }{\partial \lambda ^2}(\omega ,\lambda )|
<11 \quad\text{on }H_{4}.  \label{D140}
\end{equation}
Then we fix $n=50$, $m=36$ and consider the discretization $\left\{ (\omega
_{i},\lambda _{j})\right\} _{i,j}$, $i=0,\dots,50$, $j=0,\dots,36$ of $H_{4}$
such that
\begin{equation*}
\omega _{i}=\tfrac{91}{180}\pi +i\Delta \omega , \quad
\lambda_{j}=3.950+j\Delta \lambda ,
\end{equation*}
with
\begin{equation*}
\Delta \omega =\tfrac{\tfrac{11}{180}\pi }{n}, \quad \Delta \lambda =\tfrac{%
0.15}{m}.
\end{equation*}
By straightforward computations it holds that
\begin{equation}
-\tfrac{\partial P}{\partial \lambda }(\omega _{i},\lambda _{j})>0,
\label{D141}
\end{equation}
for all $i=0,\dots,50$, $j=0,\dots,36$ and, moreover,
\begin{equation}
\min_{(\omega _{i},\lambda _{j})\in H_{4}} \left( -\tfrac{\partial P}{%
\partial \lambda }(\omega _{i},\lambda _{j})\right) =-\tfrac{\partial P}{
\partial \lambda }(\omega _0,\lambda _0)\approx 0.118\dots\,.  \label{D142}
\end{equation}
Next to this, we compute
\begin{equation*}
c_1=\max \left\{ \Delta \omega ,\Delta \lambda \right\} =\max \big\{ \tfrac{%
\tfrac{11}{180}\pi }{n},\tfrac{0.15}{m}\big\} =\tfrac{0.15}{36}%
=0.00416\dots\,,
\end{equation*}
and, by taking into account (\ref{D140}) and (\ref{D142}), we also find
\begin{equation*}
c_2=\sqrt{2}\dfrac{\min_{(\omega _{i},\lambda _{j})\in H_{4}} \left( -\tfrac{%
\partial P}{\partial \lambda }(\omega _{i},\lambda _{j})\right) }{%
\sup_{H_{4}} \big| \big( \tfrac{\partial ^2P }{\partial \omega \partial
\lambda }(\omega ,\lambda ),\tfrac{\partial ^2P }{\partial \lambda ^2}%
(\omega ,\lambda )\big) \big| }\approx 0.00423\dots\,.
\end{equation*}

Since $c_1<c_2$, by Lemma \ref{lemma1,ap.D} we conclude that the constructed
discretization $\left\{ (\omega _{i},\lambda _{j})\right\} _{\substack{ %
i=0,\dots,50  \\ j=0,\dots,36}}$ of $H_{4}$ to be appropriate\ in the sense
that condition (\ref{D141}) yields a strict positivity of $-\tfrac{ \partial
P}{\partial \lambda }(\omega ,\lambda )$ on $H_{4}$, or, in other words $%
\tfrac{\partial P}{\partial \lambda }(\omega ,\lambda )<0$ on $H_{4}$.
\end{proof}

\begin{claim}\label{Claim9,ap.D}
Let $H_{5}\subset V$ be
$H_{5}=\left\{ (\omega ,\lambda) :\left[ \tfrac{90}{180}\pi ,
\tfrac{96}{180}\pi \right] \times \left[ 3.030,3.970\right] \right\}$.
It holds that $\frac{\partial P}{\partial \omega }(\omega ,\lambda) >0$
on $H_{5}$.
\end{claim}

\begin{proof}
First we find the following estimates:
\begin{equation}
| \frac{\partial ^2P}{\partial \omega ^2}(\omega ,\lambda) | <105, \quad |
\tfrac{\partial ^2P}{\partial \omega \partial \lambda }(\omega ,\lambda )|
<37\quad\text{on } H_{5}.  \label{D150}
\end{equation}
Then we fix $n=11$, $m=94$ and consider the discretization $\left\{
(\omega_{i},\lambda _{j})\right\} _{i,j}$, $i=0,\dots,11$, $j=0,\dots,94$ of
$H_{5} $ such that
\begin{equation*}
\omega _{i}=\tfrac{90}{180}\pi +i\Delta \omega , \quad
\lambda_{j}=3.030+j\Delta \lambda ,
\end{equation*}
with
\begin{equation*}
\Delta \omega =\tfrac{\tfrac{6}{180}\pi }{n}, \quad \Delta \lambda = \tfrac{%
0.94}{m}.
\end{equation*}
By straightforward computations it holds that
\begin{equation}
\tfrac{\partial P}{\partial \omega }(\omega _{i},\lambda _{j})>0,
\label{D151}
\end{equation}
for all $i=0,\dots,11$, $j=0,\dots,94$ and, moreover,
\begin{equation}
\min_{(\omega _{i},\lambda _{j})\in H_{5}} \tfrac{\partial P}{\partial
\omega }(\omega _{i},\lambda _{j}) =\tfrac{\partial P}{\partial \omega }%
(\omega _0,\lambda _0) \approx 0.807\dots\,.  \label{D152}
\end{equation}
Next to this, we compute
\begin{equation*}
c_1=\max \left\{ \Delta \omega ,\Delta \lambda \right\} =\max \big\{ \tfrac{%
\tfrac{6}{180}\pi }{n},\tfrac{0.94}{m}\big\} =\tfrac{0.94}{94}\approx 0.0100,
\end{equation*}
and, by taking into account (\ref{D150}) and (\ref{D152}), we also find
\begin{equation*}
c_2=\sqrt{2}\dfrac{\min_{(\omega _{i},\lambda _{j})\in H_{5}} \tfrac{%
\partial P}{\partial \omega }(\omega _{i},\lambda _{j})} {\sup_{H_{5}} \big| %
\big( \tfrac{\partial ^2P}{\partial \omega ^2} (\omega ,\lambda ),\tfrac{%
\partial ^2P}{\partial \omega \partial \lambda } (\omega ,\lambda )\big) %
\big| }\approx 0.0102\dots\,.
\end{equation*}

Since $c_1<c_2$, by Lemma \ref{lemma1,ap.D} we conclude that the constructed
discretization $\left\{ (\omega _{i},\lambda _{j})\right\} _{\substack{ %
i=0,\dots,11  \\ j=0,\dots,94}}$ of $H_{5}$ to be appropriate\ in the sense
that condition (\ref{D151}) yields a strict positivity of $\tfrac{ \partial P%
}{\partial \omega }(\omega ,\lambda )$ on $H_{5}$.
\end{proof}

\begin{claim} \label{Claim10,ap.D} \rm
Let $H_{6}\subset V$ be  $H_{6}=\left\{ (\omega ,\lambda) :\left[ \tfrac{79}{180}\pi ,\tfrac{94}{180}\pi
\right] \times \left[ 2.900,3.230\right] \right\}$.

It holds that $\frac{\partial P}{\partial \lambda }(\omega ,\lambda) <0$
on $H_{6}$.
\end{claim}

\begin{proof}
First we find the following estimates:
\begin{equation}
| \tfrac{\partial ^2P}{\partial \omega \partial \lambda }(\omega ,\lambda) |
<33, \quad | \tfrac{\partial ^2P }{\partial \lambda ^2}(\omega ,\lambda )|
<21\quad\text{on }H_{6}.  \label{D160}
\end{equation}
Then we fix $n=33$, $m=41$ and consider the discretization $\left\{
(\omega_{i},\lambda _{j})\right\} _{i,j}$, $i=0,\dots,33$, $j=0,\dots,41$ of
$H_{6} $ such that
\begin{equation*}
\omega _{i}=\tfrac{79}{180}\pi +i\Delta \omega , \quad
\lambda_{j}=2.900+j\Delta \lambda ,
\end{equation*}
with
\begin{equation*}
\Delta \omega =\tfrac{\tfrac{15}{180}\pi }{n}, \quad \Delta \lambda = \tfrac{%
0.33}{m}.
\end{equation*}
By straightforward computations it holds that
\begin{equation}
-\tfrac{\partial P}{\partial \lambda }(\omega _{i},\lambda _{j})>0,
\label{D161}
\end{equation}
for all $i=0,\dots,33$, $j=0,\dots,41$ and, moreover,
\begin{equation}
\min_{(\omega _{i},\lambda _{j})\in H_{6}} \left( -\tfrac{\partial P}{%
\partial \lambda }(\omega _{i},\lambda _{j})\right) =-\tfrac{\partial P}{
\partial \lambda }(\omega _{33},\lambda _{41}) \approx 0.227\dots\,.
\label{D162}
\end{equation}
Next to this, we compute
\begin{equation*}
c_1=\max \left\{ \Delta \omega ,\Delta \lambda \right\} =\max \big\{ \tfrac{%
\tfrac{15}{180}\pi }{n},\tfrac{0.33}{m}\big\} =\tfrac{0.33}{ 41}%
=0.00805\dots\,,
\end{equation*}
and, by taking into account (\ref{D160}) and (\ref{D162}), we also find
\begin{equation*}
c_2=\sqrt{2}\dfrac{\min_{(\omega _{i},\lambda _{j})\in H_{6}} \left( -\tfrac{%
\partial P}{\partial \lambda }(\omega _{i},\lambda _{j})\right) }{%
\sup_{H_{6}} \big| \big( \tfrac{\partial ^2P }{\partial \omega \partial
\lambda }(\omega ,\lambda ),\tfrac{\partial ^2P }{\partial \lambda ^2}%
(\omega ,\lambda )\big) \big| }\approx 0.00820\dots\,.
\end{equation*}

Since $c_1<c_2$, by Lemma \ref{lemma1,ap.D} we conclude that the constructed
discretization $\left\{ (\omega _{i},\lambda _{j})\right\} _{\substack{ %
i=0,\dots,33  \\ j=0,\dots,41}}$ of $H_{6}$ to be appropriate\ in the sense
that condition (\ref{D161}) yields a strict positivity of $-\tfrac{ \partial
P}{\partial \lambda }(\omega ,\lambda )$ on $H_{6}$, or, in other words $%
\tfrac{\partial P}{\partial \lambda }(\omega ,\lambda )<0$ on $H_{6}$.
\end{proof}

\begin{claim}\label{Claim11,ap.D} \rm
Let $H_{7}\subset V$ be  $H_{7}=\left\{ (\omega ,\lambda) :\left[ \tfrac{72}{180}\pi ,\tfrac{80}{180}\pi
\right] \times \left[ 3.150,4.000\right] \right\}$.
It holds that $\frac{\partial P}{\partial \omega }(\omega ,\lambda) <0$
on $H_{7}$.
\end{claim}

\begin{proof}
First we find the following estimates:
\begin{equation}
| \frac{\partial ^2P}{\partial \omega ^2}(\omega ,\lambda) | <115, \quad |
\tfrac{\partial ^2P}{ \partial \omega \partial \lambda }(\omega ,\lambda )|
<21\quad \text{on }H_{7}.  \label{D170}
\end{equation}
Then we fix $n=5$, $m=30$ and consider the discretization $\left\{
(\omega_{i},\lambda _{j})\right\} _{i,j}$, $i=0,\dots,5$, $j=0,\dots,30$ of $%
H_{7} $ such that
\begin{equation*}
\omega _{i}=\tfrac{72}{180}\pi +i\Delta \omega , \quad
\lambda_{j}=3.150+j\Delta \lambda ,
\end{equation*}
with
\begin{equation*}
\Delta \omega =\tfrac{\tfrac{8}{180}\pi }{n}, \quad \Delta \lambda =\tfrac{%
0.85}{m}.
\end{equation*}

By straightforward computations it holds that
\begin{equation}
-\tfrac{\partial P}{\partial \omega }(\omega _{i},\lambda _{j})>0,
\label{D171}
\end{equation}
for all $i=0,\dots,5$, $j=0,\dots,30$ and, moreover,
\begin{equation}
\min_{(\omega _{i},\lambda _{j})\in H_{7}} \left( -\tfrac{\partial P}{%
\partial \omega }(\omega _{i},\lambda _{j})\right) =-\tfrac{\partial P}{
\partial \omega }(\omega _{5},\lambda _{27})\approx 2.663\dots\,.
\label{D172}
\end{equation}
Next to this, we compute
\begin{equation*}
c_1=\max \left\{ \Delta \omega ,\Delta \lambda \right\} =\max \big\{ \tfrac{%
\tfrac{8}{180}\pi }{n},\tfrac{0.85}{m}\big\} =\tfrac{0.85}{ 30}\approx
0.0283\dots\,,
\end{equation*}
and, by taking into account (\ref{D170}) and (\ref{D172}), we also find
\begin{equation*}
c_2=\sqrt{2}\dfrac{\min_{(\omega _{i},\lambda _{j})\in H_{7}} \left( -\tfrac{%
\partial P}{\partial \omega }(\omega _{i},\lambda _{j})\right) }{\sup_{H_{7}}%
\big| \big( \tfrac{\partial ^2P }{\partial \omega ^2}(\omega ,\lambda ),%
\tfrac{\partial ^2P}{\partial \omega \partial \lambda }(\omega ,\lambda )%
\big) \big| }\approx 0.0322\dots\,.
\end{equation*}

Since $c_1<c_2$, by Lemma \ref{lemma1,ap.D} we conclude that the constructed
discretization $\left\{ (\omega _{i},\lambda _{j})\right\} _{\substack{ %
i=0,\dots,5  \\ j=0,\dots,30}}$ of $H_{7}$ to be appropriate\ in the sense
that condition (\ref{D171}) yields a strict positivity of $-\tfrac{ \partial
P}{\partial \omega }(\omega ,\lambda )$ on $H_{7}$, or, in other words, $%
\tfrac{\partial P}{\partial \omega }(\omega ,\lambda )<0$ on $H_{7}$.
\end{proof}

\begin{claim}\label{Claim12,ap.D}
Let $H_{8}\subset V$ be  $H_{8}=\left\{ (\omega ,\lambda) :\left[ \tfrac{78}{180}\pi ,\tfrac{89}{180}\pi
\right] \times \left[ 3.900,4.050\right] \right\} $.
It holds that $\frac{\partial P}{\partial \lambda }(\omega ,\lambda) >0$
on $H_{8}$.
\end{claim}

\begin{proof}
First we find the following estimates:
\begin{equation}
| \tfrac{\partial ^2P}{\partial \omega \partial \lambda }(\omega ,\lambda) |
<36, \quad | \tfrac{\partial ^2P }{\partial \lambda ^2}(\omega ,\lambda )|
<10 \quad\text{on }H_{8}.  \label{D180}
\end{equation}
Then we fix $n=40$, $m=30$ and consider the discretization $\{
(\omega_{i},\lambda _{j})\} _{i,j}$, $i=0,\dots,40$, $j=0,\dots,30$ of $%
H_{8} $ such that
\begin{equation*}
\omega _{i}=\tfrac{78}{180}\pi +i\Delta \omega , \quad
\lambda_{j}=3.900+j\Delta \lambda ,
\end{equation*}
with
\begin{equation*}
\Delta \omega =\tfrac{\tfrac{11}{180}\pi }{n}, \quad \Delta \lambda =\tfrac{%
0.15}{m}.
\end{equation*}
By straightforward computations it holds that
\begin{equation}
\tfrac{\partial P}{\partial \lambda }(\omega _{i},\lambda _{j})>0,
\label{D181}
\end{equation}
for all $i=0,\dots,40$, $j=0,\dots,30$ and, moreover,
\begin{equation}
\min_{(\omega _{i},\lambda _{j})\in H_{8}} \tfrac{\partial P}{\partial
\lambda }(\omega _{i},\lambda _{j}) =\tfrac{\partial P}{\partial\lambda }%
(\omega _{40},\lambda _{30}) \approx 0.134\dots\,.  \label{D182}
\end{equation}
Next to this, we compute
\begin{equation*}
c_1=\max \left\{ \Delta \omega ,\Delta \lambda \right\} =\max \big\{ \tfrac{%
\tfrac{11}{180}\pi }{n},\tfrac{0.15}{m}\big\} =\tfrac{0.15}{30}=0.00500,
\end{equation*}
and, by taking into account (\ref{D180}) and (\ref{D182}), we also find
\begin{equation*}
c_2=\sqrt{2}\dfrac{\min_{(\omega _{i},\lambda _{j})\in H_{8}} \tfrac{%
\partial P}{\partial \lambda }(\omega _{i},\lambda _{j})} {\sup_{H_{8}}{\sup
}\big| \big( \tfrac{\partial ^2P}{\partial \omega \partial \lambda }(\omega
,\lambda ),\tfrac{\partial ^2P}{\partial \lambda ^2}(\omega ,\lambda )\big) %
\big| }\approx 0.00508\dots\,.
\end{equation*}

Since $c_1<c_2$, by Lemma \ref{lemma1,ap.D} we conclude that the constructed
discretization $\left\{ (\omega _{i},\lambda _{j})\right\} _{\substack{ %
i=0,\dots,40  \\ j=0,\dots,30}}$ of $H_{8}$ to be appropriate\ in the sense
that condition (\ref{D181}) yields a strict positivity of $\tfrac{ \partial P%
}{\partial \lambda }(\omega ,\lambda )$ on $H_{8}$.
\end{proof}

\subsubsection{Set of Claims II}

Let $H_0\subset V$ be as follows
\begin{equation*}
H_0=\left\{ (\omega ,\lambda) :\left[ \tfrac{84}{180}\pi , \tfrac{94}{180}%
\pi \right] \times \left[ 2.960,3.060\right] \right\} .
\end{equation*}
Consider a function $G:H_0\to \mathbb{R}$ given by
\begin{align*}
G(\omega ,\lambda ) &=-\tfrac{\partial ^2P}{\partial \omega ^2}(\omega
,\lambda )\left[ \tfrac{\partial P}{\partial \lambda }(\omega ,\lambda ) %
\right] ^{-1}+2\tfrac{\partial ^2P}{\partial \omega \partial \lambda }
(\omega ,\lambda )\tfrac{\partial P}{\partial \omega }(\omega ,\lambda ) %
\left[ \tfrac{\partial P}{\partial \lambda }(\omega ,\lambda )\right] ^{-2}
\\
&\quad - \tfrac{\partial ^2P}{\partial \lambda ^2}(\omega ,\lambda )\left[
\tfrac{ \partial P}{\partial \omega }(\omega ,\lambda )\right] ^2\left[
\tfrac{ \partial P}{\partial \lambda }(\omega ,\lambda )\right] ^{-3},
\end{align*}
where $P$ is as in (\ref{D109}).

\begin{claim} \label{Claim13,ap.D} \rm
It holds that $G(\omega ,\lambda )>0$ on $H_0$.
\end{claim}

\begin{proof}
First we find the following estimates:
\begin{equation}
| \tfrac{\partial G}{\partial \omega }(\omega ,\lambda) | <25000, \quad |
\tfrac{\partial G}{\partial \lambda }(\omega ,\lambda )| <14000 \quad\text{%
on }H_0.  \label{D190}
\end{equation}
Then we fix $n=600$, $m=300$ and consider the discretization $\left\{(\omega
_{i},\lambda _{j})\right\} _{i,j}$, $i=0,\dots,600$, $j=0,\dots,300$ of $H_0$
such that
\begin{equation*}
\omega _{i}=\tfrac{84}{180}\pi +i\Delta \omega , \quad
\lambda_{j}=2.960+j\Delta \lambda ,
\end{equation*}
with
\begin{equation*}
\Delta \omega =\tfrac{\tfrac{10}{180}\pi }{n},\quad \Delta \lambda = \tfrac{%
0.1}{m}.
\end{equation*}
By straightforward computations it holds that
\begin{equation}
G(\omega _{i},\lambda _{j})>0,  \label{D191}
\end{equation}
for all $i=0,\dots,600$, $j=0,\dots,300$ and, moreover,
\begin{equation}
\min_{(\omega _{i},\lambda _{j})\in H_0} G(\omega _{i},\lambda
_{j})=G(\omega _0,\lambda _0)\approx 8.380\dots\,.  \label{D192}
\end{equation}
Next to this, we compute
\begin{equation*}
c_1=\max \left\{ \Delta \omega ,\Delta \lambda \right\} =\max \big\{ \tfrac{%
\tfrac{10}{180}\pi }{n},\tfrac{0.1}{m}\big\} =\tfrac{0.1}{ 300}\approx
0.000(3),
\end{equation*}
and, by taking into account (\ref{D190}) and (\ref{D192}), we also find
\begin{equation*}
c_2=\sqrt{2}\dfrac{\min_{(\omega _{i},\lambda _{j})\in H_0} G(\omega
_{i},\lambda _{j})}{\sup_{H_0}\big| \big( \tfrac{ \partial G}{\partial
\omega }(\omega ,\lambda ),\tfrac{\partial G}{\partial \lambda }(\omega
,\lambda )\big) \big| }\approx 0.000414\dots\,.
\end{equation*}

Since $c_1<c_2$, by Lemma \ref{lemma1,ap.D} we conclude that the constructed
discretization $\left\{ (\omega _{i},\lambda _{j})\right\} _{i,j}$ of $H_0$
to be appropriate in the sense that condition (\ref{D191}) yields a strict
positivity of $G(\omega ,\lambda )$ on $H_0$.
\end{proof}

Let $H_{\star }\subset V$ be as follows
\begin{equation*}
H_{\star }=\left\{ (\omega ,\lambda) :\left[ \tfrac{93.5}{180} \pi ,\tfrac{%
95.5}{180}\pi \right] \times \left[ 3.030,3.600\right] \right\} .
\end{equation*}
Consider a function $F:H_{\star }\to \mathbb{R}$ given by
\begin{align*}
F(\omega ,\lambda )&=-\tfrac{\partial ^2P}{\partial \lambda ^2}(\omega
,\lambda )\left[ \tfrac{\partial P}{\partial \omega }(\omega ,\lambda ) %
\right] ^{-1}+2\tfrac{\partial ^2P}{\partial \omega \partial \lambda }
(\omega ,\lambda )\tfrac{\partial P}{\partial \lambda }(\omega ,\lambda ) %
\left[ \tfrac{\partial P}{\partial \omega }(\omega ,\lambda )\right] ^{-2} \\
&\quad - \tfrac{\partial ^2P}{\partial \omega ^2}(\omega ,\lambda )\left[
\tfrac{ \partial P}{\partial \lambda }(\omega ,\lambda )\right] ^2\left[
\tfrac{ \partial P}{\partial \omega }(\omega ,\lambda )\right] ^{-3},
\end{align*}
where $P$ is as in (\ref{D109}).

\begin{claim} \label{Claim14,ap.D}\rm
It holds that $F(\omega ,\lambda )<0$ on $H_{\star }$.
\end{claim}

\begin{proof}
First we find the following estimates:
\begin{equation}
| \tfrac{\partial F}{\partial \omega }(\omega ,\lambda) | <180,\quad |
\tfrac{\partial F}{\partial \lambda }(\omega ,\lambda )| <80\quad\text{on }%
H_{\star }.  \label{D200}
\end{equation}
Then we fix $n=70$, $m=1100$ and consider the discretization $\left\{
(\omega _{i},\lambda _{j})\right\} _{i,j}$, $i=0,\dots,70$, $j=0,\dots,1100$
of $H_{\star }$ such that
\begin{equation*}
\omega _{i}=\tfrac{93.5}{180}\pi +i\Delta \omega ,\quad \lambda
_{j}=3.030+j\Delta \lambda ,
\end{equation*}
with
\begin{equation*}
\Delta \omega =\tfrac{\tfrac{2}{180}\pi }{n},\quad \Delta \lambda = \tfrac{%
0.57}{m}.
\end{equation*}
By straightforward computations it holds that
\begin{equation}
-F(\omega _{i},\lambda _{j})>0,  \label{D201}
\end{equation}
for all $i=0,\dots,70$, $j=0,\dots,1100$ and, moreover,
\begin{equation}
\min_{(\omega _{i},\lambda _{j})\in H_{\star }} \left( -F(\omega
_{i},\lambda _{j})\right) =-F(\omega _{70},\lambda _{1100})\approx
0.0773\dots \,.  \label{D202}
\end{equation}
Next to this, we compute
\begin{equation*}
c_1=\max \left\{ \Delta \omega ,\Delta \lambda \right\} =\max \big\{ \tfrac{%
\tfrac{2}{180}\pi }{n},\tfrac{0.57}{m}\big\} =\tfrac{0.57}{ 1100}%
=0.000518\dots,
\end{equation*}
and, by taking into account (\ref{D200}) and (\ref{D202}), we also find
\begin{equation*}
c_2=\sqrt{2}\dfrac{\min_{(\omega _{i},\lambda _{j})\in H_{\star }} \left(
-F(\omega _{i},\lambda _{j})\right) } {\sup_{H_{\star }}\big| \big( \tfrac{%
\partial F}{\partial \omega }(\omega ,\lambda ),\tfrac{\partial F}{\partial
\lambda }(\omega ,\lambda )\big) \big| }\approx 0.000555\dots\,.
\end{equation*}

Since $c_1<c_2$, by Lemma \ref{lemma1,ap.D} we conclude that the constructed
discretization $\left\{ (\omega _{i},\lambda _{j})\right\} _{i,j}$ of $%
H_{\star }$ to be appropriate\ in the sense that condition (\ref{D161})
yields a strict positivity of $-F(\omega ,\lambda )$ on $H_{\star }$, or, in
other words $F(\omega ,\lambda )<0$ on $H_{\star }$.
\end{proof}

\section{Explicit formulas to the homogeneous problem in the cone when $%
\protect\omega \in \left\{ \frac{1}{2}\protect\pi ,\protect\pi ,\frac{3}{2}
\protect\pi ,2\protect\pi \right\} $}

\label{AppendixE}

Here we give the explicit solutions to the homogeneous problem (\ref{3}) in $%
\mathcal{K}_{\omega }$ when $\omega \in \left\{ \frac{1}{2}\pi ,\pi ,\frac{3
}{2}\pi ,2\pi \right\} $.

\subsection{Case $\protect\omega =\frac{1}{2}\protect\pi $}

The eigenvalues $\lambda $ in this case are determined by the characteristic
equation (see subsection \ref{Reduced problem}):
\begin{equation*}
2+2\cos (\pi \lambda )-4\cos \left( \tfrac{1}{2}\pi \lambda \right) =0,
\quad \lambda \notin \left\{ 0,\pm 1\right\} .
\end{equation*}
The set of positive solutions of the above equation reads as:
\begin{equation*}
\{ \lambda _{j}\} _{j=1}^{\infty }=\left\{ \lambda _{3j-2},\text{ }\lambda
_{3j-1},\text{ }\lambda _{3j}\right\} _{j=1}^{\infty }=\left\{ -1+4j,\text{ }%
4j,\text{ }1+4j\right\} _{j=1}^{\infty }.
\end{equation*}
Here every values $\lambda _{3j-2},\lambda _{3j}$ has algebraic and
geometric multiplicity $1$, while $\lambda _{3j-1}$ has algebraic and
geometric multiplicity $2$.

\begin{table}[ht]
\renewcommand{\arraystretch}{1.5}
\par
\begin{center}
{\footnotesize
\begin{tabular}{|c|c|c|c|}
\hline
$j$ & $\lambda _{j}$ & $\kappa ^{(j)}$ & $u_{q}^{(j)}=r^{\lambda _{j}+1}\Phi
_{q}^{(j)}(\theta )$ \\ \hline\hline
$1$ & $3$ & $1$ & $x^{2}y^{2}$ \\
2-3 & $4$ & $2$ & $%
\begin{cases}
x^{3}y^{2} \\
x^{2}y^{3}%
\end{cases}%
$ \\
$4$ & $5$ & $1$ & $x^{3}y^{3}$ \\ \hline
$5$ & $7$ & $1$ & $x^{6}y^{2}-x^{2}y^{6}$ \\
6-7 & $8$ & $2$ & $%
\begin{cases}
x^{7}y^{2}-\frac{7}{3}x^{3}y^{6} \\
x^{2}y^{7}-\frac{7}{3}x^{6}y%
\end{cases}%
$ \\
$8$ & $9$ & $1$ & $x^{7}y^{3}-x^{3}y^{7}$ \\ \hline
$9$ & $11$ & $1$ & $x^{10}y^{2}-14x^{6}y^{6}+x^{2}y^{10}$ \\
10-11 & $12$ & $2$ & $%
\begin{cases}
x^{11}y^{2}-22x^{7}y^{6}+\frac{11}{3}x^{3}y^{10} \\
x^{2}y^{11}-22x^{6}y^{7}+\frac{11}{3}x^{10}y^{3}%
\end{cases}%
$ \\
$12$ & $13$ & $1$ & $x^{11}y^{3}-\frac{66}{7}x^{7}y^{7}+x^{3}y^{11}$ \\
\hline
\multicolumn{4}{|c|}{etc.} \\ \hline
\end{tabular}
}
\end{center}
\caption{The first three groups of solutions $u_{q}^{(j)}$, $q=0,\dots ,%
\protect\kappa ^{(j)}-1$ of problem (\protect\ref{3}) in $\mathcal{K}_{%
\protect\omega }$ of measure $\protect\omega =\frac{1}{2}\protect\pi $.}
\label{table10}
\end{table}

In the table above we see that the functions that solve the homogeneous
problem (\ref{3}) in $\mathcal{K}_{\omega }$\ of measure $\omega =\frac{1}{2}
\pi $ are given by polynomials in $x$ and $y$, which makes a difference to
the case when the operator of the problem is the bilaplacian $\Delta ^2$.

\subsection{Case $\protect\omega =\protect\pi $.}

The eigenvalues $\lambda $ in this case are determined by the characteristic
equation (see subsection \ref{Reduced problem}):
\begin{equation*}
\sin ^2(\pi \lambda )=0,\quad \lambda \neq \pm 1,
\end{equation*}
plus the values $\lambda =\pm 1$, which are determined by the conditions $%
P_{-1}(\pi )=P_1(\pi )=0$. Hence, the set of positive solutions reads as
\begin{equation*}
\{ \lambda _{j}\} _{j=1}^{\infty }=j,
\end{equation*}
where $\lambda _1=1$ has algebraic and geometric multiplicity $1$, while $%
\lambda _{j}$ for $j\geq 2$ has algebraic and geometric multiplicity $2$.

\begin{table}[ht]
\renewcommand{\arraystretch}{1.5}
\par
\begin{center}
{\footnotesize
\begin{tabular}{|c|c|c|c|}
\hline
$j$ & $\lambda _{j}$ & $\kappa ^{(j)}$ & $u_{q}^{(j)}=r^{\lambda _{j}+1}\Phi
_{q}^{(j)}(\theta )$ \\ \hline\hline
$1$ & $1$ & $1$ & $y^{2}$ \\ \hline
$2-3$ & $2$ & $2$ &
\begin{tabular}{l}
$xy^{2}$ \\
$y^{3}$%
\end{tabular}
\\ \hline
$4-5$ & $3$ & $2$ &
\begin{tabular}{l}
$x^{2}y^{2}$ \\
$xy^{3}$%
\end{tabular}
\\ \hline
$6-7$ & $4$ & $2$ &
\begin{tabular}{l}
$x^{3}y^{2}$ \\
$x^{2}y^{3}$%
\end{tabular}
\\ \hline
\multicolumn{4}{|c|}{etc.} \\ \hline
\end{tabular}
}
\end{center}
\caption{Some first solutions $u_{q}^{(j)}$, $q=0,\dots ,\protect\kappa %
^{(j)}-1$ of problem (\protect\ref{3}) in
$\mathcal{K}_{\protect\omega }$ of measure $\protect\omega
=\protect\pi $.} \label{table11}
\end{table}

Here, when the opening angle $\omega $ of $\mathcal{K}_{\omega }$ is $\pi $,
as well as in the case $\omega =\frac{1}{2}\pi $, the solutions (\ref{3}) in
$\mathcal{K}_{\omega }$\ are given by polynomials in $x$ and $y$. We can not
achieve this when the operator $L$ of the problem is the bilaplacian $\Delta
^2$.

\subsection{Case $\protect\omega =\frac{3}{2}\protect\pi $.}

The characteristic equation $P\left( \tfrac{3}{2}\pi ,\lambda \right) =0$, $%
\lambda \neq \pm 1$, where $P$ is given in subsection \ref{Reduced problem},
can be factorized and the eigenvalues $\lambda $ are determined by the
system:
\begin{gather*}
\cos \left( \tfrac{1}{2}\pi \lambda \right) =0,\quad \lambda \neq \pm 1, \\
\cos \left( \tfrac{1}{2}\pi \lambda \right) =1,\quad \lambda \neq \pm 1, \\
\cos ^{4}\left( \tfrac{1}{2}\pi \lambda \right) +\cos ^{3}\left( \tfrac{1}{2}
\pi \lambda \right) -\tfrac{1}{2}\cos ^2\left( \tfrac{1}{2}\pi \lambda
\right) -\tfrac{1}{2}\cos \left( \tfrac{1}{2}\pi \lambda \right) +\tfrac{1}{
16}=0,\quad \lambda \neq \pm 1.
\end{gather*}
The solutions with positive real part of the system above read,
respectively,
\begin{gather*}
\left\{ \lambda _{n_1}\right\} _{n_1=1}^{\infty }=\left\{ 1+2n_1\right\}
_{n_1=1}^{\infty },\quad \left\{ \lambda _{n_2}\right\} _{n_2=1}^{\infty
}=\left\{ 4n_2\right\} _{n_2=1}^{\infty }, \\
\left\{ \lambda _{n_{3}}\right\} _{n_{3}=1}^{\infty }=\left\{
-1+2n_{3}+\left( -1\right) ^{n_{3}}\left( 1-\mu _1\right) \right\}
_{n_{3}=1}^{\infty }, \\
\left\{ \lambda _{n_{4}}\right\} _{n_{4}=1}^{\infty }=\left\{
-1+2n_{4}+\left( -1\right) ^{n_{4}}\left( 1-\mu _2\right) \right\}
_{n_{4}=1}^{\infty }, \\
\left\{ \lambda _{n_{5}}\right\} _{n_{5}=1}^{\infty }=\left\{
-1+2n_{5}+\left( -1\right) ^{n_{5}+1}\left( 1-\gamma _1\right) \pm i\gamma
_2\right\} _{n_{5}=1}^{\infty },
\end{gather*}
where $\mu _1,\mu _2$ are the first two positive solutions of the equation
\begin{equation}
s^{4}+s^{3}-\tfrac{1}{2}s^2-\tfrac{1}{2}s+\tfrac{1}{16}=0,  \label{E1}
\end{equation}
with $s=\cos \left( \tfrac{1}{2}\pi \mu \right) $, while $(\gamma _1,\gamma
_2)$\ is the first positive solution of (\ref{E1}) with $s=-\cos \left(
\tfrac{1}{2}\pi \gamma _1\right) \cosh \left( \tfrac{1}{2} \pi \gamma
_2\right) +i\sin \left( \tfrac{1}{2}\pi \gamma _1\right) \sinh \left( \tfrac{%
1}{2}\pi \gamma _2\right) $. The numerical approximations (up to three
digits) are the following: $\mu _1\approx 0.536\dots$, $\mu _2\approx
0.926\dots$ and $\gamma _1\approx 0.345\dots$, $\gamma _2\approx 0.179\dots$.

Note also that every $\lambda _{n_2}=4n_2$, $n_2=1,2.3,\dots$ has algebraic
and geometric multiplicity $2$, while every $\lambda _{n_{k}}$, for each $%
k=1,3,4,5$ has algebraic and geometric multiplicity $1$. The set $\{ \lambda
_{j}\} _{j=1}^{\infty }$ is the combination of the found sets above.

\begin{table}[ht]
\renewcommand{\arraystretch}{1.5}
\par
\begin{center}
{\footnotesize
\begin{tabular}{|c|c|c|c|c|}
\hline
$j$ & $\lambda _{j}$ & $\kappa ^{(j)}$ & $\Phi _{q}^{(j)}(\theta ) $ & $%
u_{q}^{(j)}=r^{\lambda _{j}+1}\Phi _{q}^{(j)}(\theta )$ \\ \hline\hline
1,2,3-4,5-6 & \multicolumn{4}{|l|}{$\approx 0.536\ldots , \approx
0.926\dots, \approx 1.655\ldots\pm i0.179\dots, \approx 2.345\ldots\pm
i0.179\ldots $} \\ \hline
$7$ & $3$ & $1$ & $\sin ^2(\theta )\cos ^2(\theta ) $ & $x^2y^2$ \\ \hline
$8,9$ & \multicolumn{4}{|l|}{$\approx 3.074\ldots, \approx 3.464\ldots $} \\
\hline
10-11 & $4$ & $2$ & $%
\begin{cases}
\sin ^2(\theta )\cos ^{3}(\theta ) \\
\cos ^2(\theta )\sin ^{3}(\theta )%
\end{cases}%
$ & $%
\begin{cases}
x^{3}y^2 \\
x^2y^{3}%
\end{cases}%
$ \\ \hline
$12,13$ & \multicolumn{4}{|l|}{$\approx 4.536\ldots , \approx 4.926\ldots , $%
} \\ \hline
$14$ & $5$ & $1$ & $\sin ^{3}(\theta )\cos ^{3}(\theta ) $ & $x^{3}y^{3}$ \\
\hline
15-16,17-18 & \multicolumn{4}{|l|}{$\approx 5.655\ldots\pm i0.179\ldots ,
\approx 6.345\ldots\pm i0.179\ldots ,$} \\ \hline
$19$ & $7$ & $1$ & $\sin ^2(\theta )\cos ^2(\theta )\left( \cos ^{4}(\theta
)-\sin ^{4}(\theta )\right) $ & $x^{6}y^2-x^2y^{6}$ \\ \hline
$20,21$ & \multicolumn{4}{|l|}{$\approx 7.074\ldots , \approx 7.464\ldots $}
\\ \hline
$22-23$ & $8$ & $2$ & $%
\begin{cases}
\sin ^2(\theta )\cos ^{3}(\theta )\left( \cos ^{4}(\theta )-\frac{7}{3} \sin
^{4}(\theta )\right) \\
\cos ^2(\theta )\sin ^{3}(\theta )\left( \sin ^{4}(\theta )-\frac{7}{3} \cos
^{4}(\theta )\right)%
\end{cases}%
$ & $%
\begin{cases}
x^{7}y^2-\frac{7}{3}x^{3}y^{6} \\
x^2y^{7}-\frac{7}{3}x^{6}y%
\end{cases}%
$ \\ \hline
\multicolumn{5}{|c|}{etc.} \\ \hline
\end{tabular}
}
\end{center}
\caption{The first solutions $(\protect\lambda _{j},\Phi_{q}^{(j)}) $, $%
q=0,\dots,\protect\kappa ^{(j)}-1$ of (\protect\ref{4}) and the solutions $%
u_{q}^{(j)}$ of (\protect\ref{3}) in $\mathcal{K}_{\protect\omega }$ of
measure $\protect\omega =\frac{3}{2}\protect\pi $. The situation without
explicit formula is marked by `` \ldots ''.}
\label{table12}
\end{table}

\subsection{Case $\protect\omega =2\protect\pi $.}


\begin{table}[ht]
\renewcommand{\arraystretch}{1.5}
\par
\begin{center}
{\footnotesize
\begin{tabular}{|c|c|c|c|c|}
\hline
$j$ & $\lambda _{j}$ & $\kappa ^{(j)}$ & $\Phi _{q}^{(j)}(\theta ) $ & $%
u_{q}^{(j)}=r^{\lambda _{j}+1}\Phi _{q}^{(j)}(\theta )$ \\ \hline\hline
$1$ & $\frac{1}{2}$ & $2$ & $\ldots $ & $\ldots $ \\ \hline
$2$ & $1$ & $1$ & $\sin ^2(\theta )$ & $y^2$ \\ \hline
$3$ & $\frac{3}{2}$ & $2$ & $\ldots $ & $\ldots $ \\ \hline
4-5 & $2$ & $2$ & $%
\begin{cases}
\sin ^2(\theta )\cos (\theta ) \\
\sin ^{3}(\theta )%
\end{cases}%
$ &
\begin{tabular}{l}
$xy^2$ \\
$y^{3}$%
\end{tabular}
\\ \hline
$6$ & $\frac{5}{2}$ & $2$ & $\ldots $ & $\ldots $ \\ \hline
7-8 & $3$ & $2$ & $%
\begin{cases}
\sin ^2(\theta )\cos ^2(\theta ) \\
\sin ^{3}(\theta )\cos (\theta )%
\end{cases}%
$ &
\begin{tabular}{l}
$x^2y^2$ \\
$xy^{3}$%
\end{tabular}
\\ \hline
$9$ & $\frac{7}{2}$ & $2$ & $\ldots $ & $\ldots $ \\ \hline
10-11 & $4$ & $2$ & $%
\begin{cases}
\sin ^2(\theta )\cos ^{3}(\theta ) \\
\sin ^{3}(\theta )\cos ^2(\theta )%
\end{cases}
$ &
\begin{tabular}{l}
$x^{3}y^2$ \\
$x^2y^{3}$%
\end{tabular}
\\ \hline
\multicolumn{5}{|c|}{etc.} \\ \hline
\end{tabular}
}
\end{center}
\caption{Some first solutions $(\protect\lambda _{j},\Phi _{q}^{(j)})$, $%
q=0,\dots,\protect\kappa ^{(j)}-1$ of (\protect\ref{4}) and the solutions $%
u_{q}^{(j)}$ of (\protect\ref{3}) in $\mathcal{K}_{\protect\omega }$ of
measure $\protect\omega =2 \protect\pi $. The situation when the explicit
formula unavailable is marked by ``\ldots''.}
\label{table13}
\end{table}

The eigenvalues $\lambda $ in this case are determined by the characteristic
equation (see subsection \ref{Reduced problem}):
\begin{equation*}
\cos ^{4}(\pi \lambda )-\cos ^2(\pi \lambda )=0,\quad \lambda \neq \pm 1,
\end{equation*}
plus the values $\lambda =\pm 1$, which are determined by the conditions $%
P_{-1}(2\pi )=P_1(2\pi )=0$. The positive solutions are given by the set
\begin{equation*}
\{ \lambda _{j}\} _{j=1}^{\infty }=\tfrac{1}{2}j,
\end{equation*}
where $\lambda _2=1$ has algebraic and geometric multiplicity $1$, while $%
\lambda _{j}$ for $j=1$ and $j\geq 3$ has algebraic and geometric
multiplicity $2$.

\subsection*{Acknowledgements}

We thank the anonymous referee for his careful reading of the manuscript.

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