\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{graphicx}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 238, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/238\hfil Coefficients of singularities]
{Coefficients of singularities for a simply
supported plate problems in plane sectors}

\author[B. Merouani, R. Boufenouche \hfil EJDE-2013/238\hfilneg]
{Boubakeur Merouani, Razika Boufenouche} 

\address{Boubakeur Merouani \newline
Department of Mathemaics, Univ. F. Abbas-S\'etif 1, Algeria}
\email{mermathsb@hotmail.fr}

\address{Razika Boufenouche \newline
Department of Mathemaics, Univ. Jijel, Algeria}
\email{r\_boufenouche@yahoo.fr}

\thanks{Submitted May 13, 2013. Published October 24, 2013.}
\subjclass[2000]{35B40, 35B65, 35C20}
\keywords{Crack sector; singularity; bilaplacian; solution series}

\begin{abstract}
 This article represents the solution to a plate problem in a plane 
 sector that is simple supported, as a series.
 By using appropriate  Green's functions, we  establish a biorthogonality  
 relation between the terms of the series, which allows us
 to calculate the coefficients.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

Let $S$ be the truncated plane sector of angle $\omega \leq 2\pi $,
and radius $\rho $ ($\rho $ is positive and fixed) defined by:
\begin{equation}
S=\{ ( r\cos \theta ,r\sin \theta ) \in\mathbb{R}^2,0< r< \rho ,\;
0< \theta < \omega \}  \label{e1.1}
\end{equation}
and $\Sigma $ the circular boundary part 
\begin{equation}
\Sigma =\{ ( \rho \cos \theta ,\rho \sin \theta ) \in
\mathbb{R} ^2,0< \theta < \omega \}.  \label{e1.2}
\end{equation}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig1}
\end{center}
\caption{}
\label{fig1}
\end{figure}

We are interested in the study of  a function $u$, belonging to the Sobolev
space $H^2( S) $, and being  the solution of
\begin{equation}
\begin{gathered}
\Delta ^2u=0,\quad \text{in }S \\
u=Mu=0,\quad \text{for }\theta =0,\omega,
\end{gathered}  \label{e1.3}
\end{equation}
where the operator $M$ represents the bending moment and is defined as
\begin{equation}
Mu=\nu \Delta u+( 1-\nu ) ( \partial _1^2u
n_1^2+2\partial _{12}^2u n_1n_2+\partial _2^2u
n_2^2) .  \label{e1.4}
\end{equation}
Here $\nu $ is a real number $( 0< \nu < 1/2) $ called
Poisson coefficient and $n=(n_1,n_2)$ is the unit outward normal vector
to $\Gamma _0$  and $\Gamma _1$ (See Figure \ref{fig1}).

The boundary conditions $u=0$ and $Mu=0$, for $\theta =0$, $\theta =\omega $ 
mean that the plate is simply supported.

This type of boundary conditions arises in problems of linear or non linear
vibrations of thin imperfect plates. See for example \cite[pages 5,6]{c1}
and the references therein.

We show that the solutions $u$ of this  problems can be written as a
series of the form
\begin{equation} 
u( r,\theta ) =\sum_{\alpha \in E} c_{\alpha}r^{\alpha }\phi _{\alpha }( \theta ). 
 \label{e1.5}
\end{equation}
Here $E$ stands for the set of solutions of the equation in a complex
variable $\alpha$:
\begin{equation}
\sin ^2( \alpha -1) \omega =\sin ^2\omega ,\quad\operatorname{Re}
\alpha > 1  \label{e1.6}
\end{equation}
For further studies of the set $E$, see for example Blum and Rannacher
\cite{b1},  and Grisvard \cite{g1}.

We will compute the coefficients $c_{\alpha }$ in
\eqref{e1.5}. This sort of calculations have already been done by
Tcha-Kondor \cite{t1} for the Dirichlet's boundary conditions, and
by  Chikouche-Aibeche \cite{c2} for the Neumann's boundary
conditions. These authors have established, thanks to the Green's formula, a
relation of biorthorgonality between any two functions $\phi _{\alpha}$,
which allows them to calculate the coefficients
$c_{\alpha }$. We follow the same approach.
Thus we need to write the appropriate Green formula for 
the domain $S$.
Using this formula, we establish a relation of biorthorgonality between
the functions $\phi _{\alpha}$.

In the case of a crack domain  $(\omega =2\pi) $ this relation reduces
to the simple one obtained by Tcha-Kondor. This enables us, in this
particular situation, to find an explicit formula for the coefficients
$c_{\beta }$.

\section{Separation of variables}

Replacing $u$ by $r^{\alpha }\phi _{\alpha }( \theta ) $ in
problem \eqref{e1.3}  leads us to the boundary value problem
\begin{gather}
\phi _{\alpha }^{( 4) }( \theta ) +( 2\alpha
^2-4\alpha +4) \phi _{\alpha }^{( 2) }( \theta
) +\alpha ^2( \alpha -2) ^2\phi _{\alpha }( \theta ) =0 , \label{e2.1}
\\
\phi _{\alpha }^{( 2) }( \theta ) +[ \nu \alpha
^2+( 1-\nu ) \alpha ] \phi _{\alpha }( \theta
) =0,\quad \theta =0,\quad \theta =\omega  \label{e2.2}
\\
\phi _{\alpha }( \theta ) =0, \quad \theta =0, \quad \theta
=\omega  \label{e2.3}
\end{gather}
A relation similar to the orthogonality obtained for the biharmonic operator
is given, in the following theorem.

\begin{theorem} \label{thm2.1}
Let $\phi _{\alpha }$ and $\phi _{\beta }$ be solutions of
\eqref{e2.1} with $\alpha $ and $\beta $ solutions of \eqref{e1.6}.
Then, for $\alpha \neq \beta $, one has
\begin{equation}
\begin{aligned}
{}[ \phi _{\alpha },\phi _{\beta }]
&=\int_0^\omega \big[ ( \alpha ^2-2\alpha ) \phi _{\alpha }
-\frac{\nu ( \alpha +\overline{\beta }) +( 3-\nu ) -2\alpha }{\alpha -
\overline{\beta }}\phi _{\alpha }''\big] \overline{\phi
_{\beta }} \\
&\quad +\big[ ( \overline{\beta }^2-2\overline{\beta }) \overline{
\phi _{\beta }}-\frac{\nu ( \alpha +\overline{\beta }) +(
3-\nu ) -2\overline{\beta }}{\alpha -\overline{\beta }}\overline{\phi
_{\beta }''}\big] \phi _{\alpha }
\, d\theta =0.
\end{aligned}\label{e2.4}
\end{equation}
\end{theorem}

\begin{proof}
We use the Green formula
\begin{equation}
\int_S  ( v\Delta ^2u-u\Delta ^2v) dx
=\int_\Gamma  \big[ ( uNv+\frac{\partial u}{\partial n}Mv)
-( vNu+\frac{\partial v}{\partial n}Mu) \big] \,d\sigma ,
\label{e2.5}
\end{equation}
where
\begin{equation*}
Nu=-\frac{\partial \Delta u}{\partial n}+( 1-\nu ) (
\partial _1^2u n_1n_2-\partial _{12}^2u (
n_1^2-n_2^2) +\partial _2^2u n_1n_2),
\end{equation*}
and $\Gamma$ is the boundary of $S$. For two functions $u,v$
which are solutions of \eqref{e1.3}, using the Green's formula we obtain
\begin{equation}
\int_\Sigma \big[ ( uNv+\frac{\partial u}{\partial n}
Mv) -( vNu+\frac{\partial v}{\partial n}Mu) d\sigma \big]
=0  \label{e2.6}
\end{equation}
On $\Sigma$,  for the function
$u_{\alpha }=r^{\alpha }\phi _{\alpha}$, we have
\begin{equation} \label{e2.7}
\begin{gathered}
\frac{\partial u_{\alpha }}{\partial n}
 =\frac{\partial u_{\alpha }}{\partial r}=\alpha r^{\alpha -1}\phi _{\alpha },
 \\
Mu_{\alpha } = r^{\alpha -2}\big\{ [ \alpha ^2-( 1-\nu )
\alpha ] \phi _{\alpha }+\nu \phi _{\alpha }''\big\} ,
 \\
Nu_{\alpha } = r^{\alpha -3}\big\{ -\alpha ^2( \alpha -2)
\phi _{\alpha }+[ ( \nu -2) \alpha +( 3-\nu )
] \phi _{\alpha }''\big\} .
\end{gathered}
\end{equation}
The  results follow from the application of formula \eqref{e2.6}
to the biharmonic functions $u_{\alpha }=r^{\alpha }\phi _{\alpha }$ and
$u_{\beta }=r^{\overline{\beta }}\overline{\phi _{\beta }}$, and by using
relations \eqref{e2.7}.
\end{proof}

\begin{remark} \label{rmk2.2} \rm
The relation \eqref{e2.4} between the functions $\phi _{\alpha }$
and $\phi _{\beta }$ is similar to the relation of biorthorgonality obtained
when the functions $\phi _{\alpha }$ and $\phi _{\beta }$
satisfying \eqref{e2.1} with the Dirichlet boundary conditions
$\phi _{\alpha }=\phi_{\alpha }'=\phi _{\beta }
=\phi _{\beta }'=0$ for $\theta =0$ and $\theta =\omega$.
In this latter case, the relation is given by
\begin{equation}
\int_0^\omega \phi _{\alpha }\phi _{\beta}''d\theta
=\int_0^\omega \phi _{\alpha }''\phi _{\beta }d\theta  \label{e2.8}
\end{equation}
which is obtained by a double integration by parts:
\begin{equation}
\int_0^\omega \phi _{\alpha }\phi _{\beta}''d\theta
=\int_0^\omega \phi _{\alpha }''\phi _{\beta }d\theta
+[ \phi _{\alpha },\phi _{\beta }'] _0^{\omega }-[ \phi _{\alpha
}',\phi _{\beta }] _0^{\omega },  \label{e2.9}
\end{equation}
and using the Dirichlet's boundary conditions.
\end{remark}

The following corollary is an immediate consequence of remark \ref{rmk2.2}.


\begin{corollary} \label{coro2.3}
Let $\phi _{\alpha }$ and $\phi _{\beta }$ be solutions of
\eqref{e2.1} with $\alpha $ and $\beta $ solutions of \eqref{e2.6}.
Suppose in addition that
\begin{equation}
[ \phi _{\alpha },\phi _{\beta }'] _0^{\omega }-
[ \phi _{\alpha }',\phi _{\beta }] _0^{\omega }=0,
\label{e2.10}
\end{equation}
and $\alpha \neq \beta $, then
\begin{equation}
[ \phi _{\alpha },\phi _{\beta }]
=\int_0^\omega \big\{ [ ( \alpha ^2-2\alpha ) \phi _{\alpha
}+\phi _{\alpha }''] \overline{\phi _{\beta }}+[
( \overline{\beta }^2-2\overline{\beta }) \overline{\phi
_{\beta }}+\overline{\phi _{\beta }''}] \phi
_{\alpha }\big\}\, d\theta =0  \label{e2.11}
\end{equation}
\end{corollary}

\begin{remark} \label{rmk2.4} \rm
For $u_{\alpha }=r^{\alpha }\phi _{\alpha }$ we have
\begin{equation}
\Delta u_{\alpha }-\frac{2}{r}\frac{\partial u_{\alpha }}{\partial r}
=r^{\alpha -2}[ ( \alpha ^2-2\alpha ) \phi _{\alpha }+\phi
_{\alpha }''].  \label{e2.12}
\end{equation}
\end{remark}


Let $P$ be the operator $P=\Delta -\frac{2}{r}\frac{\partial }{\partial r}$.
From the corollary \ref{coro2.3} and Remark \ref{rmk2.4}, we
deduce the following result.

\begin{corollary} \label{coro2.5}
Under the hypotheses of corollary \ref{coro2.3}, if $\alpha \neq
\beta $, we have
\begin{equation}
\int_\Sigma( Pu_{\alpha }\cdot \overline{u_{\beta }}
+u_{\alpha }\cdot P\overline{u_{\beta }}) d\sigma =0.  \label{e2.13}
\end{equation}
\end{corollary}

\section{Formula for the coefficients in the crack case}

The crack case $(\omega =2\pi)$ is an important one, among
singular domains, in the applications. Moreover in this case
the solutions of \eqref{e2.6} are explicitly known and we have
\begin{equation*}
E=\{ \frac{k}{2},k\in\mathbb{N},\,k> 2\}
\end{equation*}
and these roots are of multiplicity 2.

In this framework we assume that the solution $u$ admits the
representation
\begin{equation} \label{e3.1}
\begin{gathered}
u =\sum_{\alpha \in E} ( c_{\alpha }u_{\alpha }+d_{\alpha }v_{\alpha }) ,\quad
E=\{ \frac{k}{2},\, k\in \mathbb{N} ,\, k> 2\} ,  \\
u_{\alpha } =r^{\alpha }\phi _{\alpha }( \theta ) ,\quad
v_{\alpha }=r^{\alpha }\psi _{\alpha }( \theta )
\end{gathered}
\end{equation}
the solutions  $\phi _{\alpha }$ and $\psi _{\alpha }$, in terms of $\theta $,
are the odd functions:
\begin{gather}
\phi _{\alpha }( \theta ) =\sin ( \alpha -2) \theta , \label{e3.2}\\
\psi _{\alpha }( \theta ) =\sin \alpha \theta .  \label{e3.3}
\end{gather}
and since $\alpha =k/2$, we obtain
\begin{equation}
\phi _{\alpha }( 0) =\phi _{\alpha }( \omega ) =\psi
_{\alpha }( 0) =\psi _{\alpha }( \omega ) =0
\label{e3.4}
\end{equation}
and thus
\begin{gather*}
[ \phi _{\alpha },\phi _{\beta }']_0^{\omega }
=[ \phi _{\alpha }',\phi _{\beta }]_0^{\omega }=0, \quad
[ \psi _{\alpha },\psi _{\beta }']_0^{\omega }
=[ \psi _{\alpha }',\psi _{\beta }]_0^{\omega }=0, \\
[ \phi _{\alpha },\psi _{\beta }'] _0^{\omega }=
[ \phi _{\alpha }',\psi _{\beta }] _0^{\omega }=0.
\end{gather*}

From here comes the idea of decomposing the solution $u$ of \eqref{e1.3}
 into two parts as  follows:
\begin{equation} \label{e3.5}
\begin{gathered}
u = w_1+w_2,  \\
w_i =\sum_{\alpha \in E_i}
( c_{\alpha }u_{\alpha }+d_{\alpha }v_{\alpha }) ,\quad i=1,2,   \\
E_1 = \{ 2m,\,m> 1\} ,\quad E_2=\{ 2m+1,\, 2m> 1\} .
\end{gathered}
\end{equation}

\subsection*{Calculation of $c_{\beta }$ and $d_{\beta }$}

From the expressions of $\phi _{\alpha }$, $\psi _{\alpha }$ one
easily sees that:
\begin{equation} \label{e3.6}
\begin{gathered}
\text{if $\alpha \in E_1$,  then
$\phi _{\alpha }'(0) =\phi _{\alpha }'( \omega )$ and
$\psi _{\alpha }'( 0) =\psi _{\alpha }'( \omega ) $},\\
\text{if $\alpha \in E_2$, then
$\phi _{\alpha }'(0) =-\phi _{\alpha }'( \omega )$ and
$\psi _{\alpha }'( 0) =-\psi _{\alpha }'( \omega )$}.
\end{gathered}
\end{equation}
Equations \eqref{e3.4} and \eqref{e3.6} allow us to apply
corollary \ref{coro2.5}  to functions $u_{\alpha }$ and $u_{\beta }$
(resp. $u_{\alpha },v_{\beta }$  and $v_{\alpha },v_{\beta}$)
and get the relations:
\begin{equation} \label{e3.7}
\begin{gathered}
\int_\sigma ( Pw_i\cdot u_{\beta }+w_i\cdot Pu_{\beta }) d\sigma
=2c_{\beta }\int_\sigma ( u_{\beta }\cdot Pu_{\beta }) d\sigma
+d_{\beta }\int_\sigma ( Pv_{\beta }\text{.}u_{\beta }
+v_{\beta }\cdot Pu_{\beta }) d\sigma ,
 \\
\int_\sigma ( Pw_i\cdot v_{\beta }+w_i\cdot Pv_{\beta}) d\sigma
=c_{\beta }\int_\sigma ( Pu_{\beta }
\cdot v_{\beta }+u_{\beta }\cdot Pv_{\beta }) d\sigma +2d_{\beta }
\int_\sigma ( Pv_{\beta }\cdot v_{\beta }) d\sigma .
\end{gathered}
\end{equation}
By direct calculations we obtain
\begin{equation} \label{e3.8}
\begin{gathered}
\int_\sigma ( Pu_{\beta }\cdot v_{\beta }+u_{\beta
}\cdot Pv_{\beta }) d\sigma =0,
\\
\int_\sigma ( u_{\beta }\cdot Pu_{\beta }) d\sigma
=( \beta -2) \omega \rho ^{2\beta -1}
\\
\int_\sigma ( Pv_{\beta }.v_{\beta }) d\sigma
=-\beta \omega \rho ^{2\beta -1}
\end{gathered}
\end{equation}
and from this we get  our main the result.

\begin{theorem} \label{thm3.1}
Let $u$ be a the solution of \eqref{e1.3} written in the form
\begin{equation}
u=w_1+w_2  \label{e3.9}
\end{equation}
where
\begin{equation}
w_i=\sum_{\alpha \in E_i} ( c_{\alpha }u_{\alpha
}+d_{\alpha }v_{\alpha }) , i=1,2  \label{e3.10}
\end{equation}
Suppose that the series that gives $w_i$ is uniformly convergent in $S$.
Then for any $\alpha \in E_i$, $i=1,2$ we have
\begin{equation} \label{3.11}
\begin{gathered}
c_{\alpha } = \frac{\rho ^{1-2\alpha }}{2( \alpha -2) \omega }
\int_\sigma ( Pw_i\cdot u_{\alpha }+w_i\cdot Pu_{\alpha
}) d\sigma   \\
d_{\alpha }  = \frac{-\rho ^{1-2\alpha }}{2\alpha \omega }
\int_\Sigma ( Pu_i\cdot v_{\alpha }+w_i\cdot Pv_{\alpha }) d\sigma
\end{gathered}
\end{equation}
\end{theorem}

\begin{remark} \label{rmk3.2} \rm
Let $\zeta \in H^{3/2}( \Sigma ) \cap H_0^{1}(\Sigma ) $ be the trace
of $u$ on $\Sigma $ and $\chi \in H^{-1/2}( \Sigma ) $ the trace of
$Pu$ on $\Sigma $.
\end{remark}

If $u$ is regular in order that $\zeta \in H^{4}( ] 0,2\pi [) $ and
 $\chi \in H^2( ] 0,2\pi [ ) $, then we
have a uniform convergence of the series in $\overline{S_{\rho _0}}$ for
all $\rho _0< \rho $, see \cite{t1}.

\subsection{Independence of the coefficients}

\begin{proposition} \label{prop3.3}
The coefficients $c_{\beta }$ (resp $d_{\beta }$)  are
independent of $\rho $.
\end{proposition}

\begin{proof}
Let us prove that the derivative of $c_{\beta }$ with respect to $\rho $ is
zero. Observing the expression of $c_{\beta }$ in Theorem \ref{thm3.1},
we just have to prove that the derivative, with respect to $\rho $,
of
\begin{equation}
\gamma _{\beta }=\rho ^{1-2\beta }\int_\sigma ( Pw_i
\cdot u_{\beta }+w_i\cdot Pu_{\beta }) d\sigma .  \label{e3.12}
\end{equation}
vanishes.
By derivation with respect to $r$ we have
\begin{equation} \label{e3.13}
\begin{aligned}
\gamma _{\beta }'
&=\int_0^\omega \Big\{
 \frac{\partial }{\partial r}( \Delta w_i) r^{2-\beta }\phi
 _{\beta }+\big[ ( 2-\beta ) \Delta w_i-2\frac{\partial ^2w_i}{\partial r^2}
+( \beta ^2-2) \frac{1}{r}\frac{
 \partial w_i}{\partial r}\big] r^{1-\beta }\phi _{\beta } \\
&\quad +\frac{\partial w_i}{\partial r}r^{-\beta }\phi _{\beta }''
-\beta w_ir^{-1-\beta }\big[ ( \beta ^2-2\beta ) \phi
_{\beta }+\phi _{\beta }''\big]
\Big\}\, d\theta . 
\end{aligned} 
\end{equation}
On $\Sigma $, we have
\[
\frac{\partial }{\partial r}( \Delta w_i) =-Nw_i+(
1-\upsilon ) \big[ \frac{1}{r^{3}}\frac{\partial ^2w_i}{\partial
\theta ^2}-\frac{1}{r^2}\frac{\partial ^{3}w_i}{\partial r\partial
\theta ^2}\big] ,
\]
and
\begin{equation}
( 2-\beta ) \Delta w_i-2\frac{\partial ^2w_i}{\partial r^2
}=-\beta Mw_i+[ 2-( 1-\upsilon ) \beta ] [
\frac{1}{r}\frac{\partial w_i}{\partial r}+\frac{1}{r^2}\frac{\partial
^2w_i}{\partial \theta ^2}] .  \label{e3.14}
\end{equation}
Using these formulas in the expression of $\gamma _{\beta }'$
we obtain
\begin{equation} \label{e3.15}
\begin{aligned}
&\gamma _{\beta }'\\
&=-\int_0^\omega ( \beta Mw_ir^{1-\beta }\phi _{\beta }
 +Nw_ir^{2-\beta }\phi _{\beta}) d\theta   \\
&\quad +\int_0^\omega \big\{ [ ( \beta
^2-( 1-\upsilon ) \beta ) \phi _{\beta }+\phi _{\beta
}''] \frac{\partial u_i}{\partial r}-(
1-\upsilon ) \frac{\partial ^{3}u_i}{\partial r\partial \theta ^2}
\phi _{\beta }\big\} r^{-\beta }d\theta   \\
&\quad +\int_0^\omega \big\{ [ 2-( 1-\upsilon
) ( \beta -1) ] \frac{\partial ^2w_i}{\partial
\theta ^2}\phi _{\beta }-\beta w_i[ ( \beta ^2-2\beta
) \phi _{\beta }+\phi _{\beta }''] \big\}
r^{-1-\beta }d\theta .
\end{aligned}
\end{equation}
By a double integration by parts, we verify that
\begin{gather}
\int_0^\omega \frac{\partial ^2w_i}{\partial
\theta ^2}\phi _{\beta }d\theta =\int_0^\omega
w_i\phi _{\beta }''d\theta  \label{e3.16}
\\
\int_0^\omega \frac{\partial ^{3}w_i}{\partial
r\partial \theta ^2}\phi _{\beta }d\theta 
=\int_0^\omega \frac{\partial w_i}{\partial r}\phi _{\beta }''d\theta  \label{e3.17}
\end{gather}
Using \eqref{e3.15}--\eqref{e3.17} in the expression of
$ \gamma _{\beta }'$ and putting the $\rho ^{1-2\beta }$, we
obtain
\begin{equation} \label{e3.18}
\begin{aligned}
\gamma _{\beta }'
&= -\rho ^{1-2\beta }\int_0\omega ( \beta Mw_ir^{\beta -1}\phi _{\beta }
+Nw_ir^{\beta}\phi _{\beta }) \rho d\theta  \\
&\quad +\rho ^{1-2\beta }\int_0^\omega [ (
\beta ^2-( 1-\upsilon ) \beta ) \phi _{\beta }+\upsilon
\phi _{\beta }''] r^{\beta -2}\frac{\partial w_i}{
\partial r}\rho d\theta   \\
&\quad +\rho ^{1-2\beta }\int_0^\omega \big\{ [
-\beta ^2( \beta -2) ] \phi _{\beta }+[ -(
2-\upsilon ) \beta +( 3-\upsilon ) ] \phi _{\beta
}''\big\} r^{\beta -3}w_i\rho d\theta .
\end{aligned}
\end{equation}
Taking into account of \eqref{e2.7}, whose expressions appear
explicitly in $\gamma _{\beta }'$, we obtain
\begin{equation} \label{e3.19}
\gamma _{\beta }'
=\rho ^{1-2\beta }\int_\Sigma -
\big[ \big( u_{\beta }Nw_i+\frac{\partial u_{\beta }}{\partial n}
Mw_i\big) -\big( u_iNu_{\beta }+\frac{\partial w_i}{\partial n}
Mu_{\beta }\big) \big]\, d\sigma =0.
\end{equation}
We follow the same analysis to prove the independence of $d_{\beta }$ with
respect to $\rho $.
\end{proof}

\subsection{Convergence of the series}

We first  write $c_{\alpha }$\ and $d_{\alpha }$ in the form
\begin{equation}
c_{\alpha }=I_i\rho ^{-\alpha },\quad 
d_{\alpha }=J_i\rho ^{-\alpha } \label{e3.20}
\end{equation}
with
\begin{equation} \label{e3.21}
\begin{gathered}
I_i=\frac{\rho }{2\omega ( \alpha -2) }\int_\sigma
( Pw_i\phi _{\alpha }+w_i\rho ^{-2}[ ( \alpha
^2-2\alpha ) \phi _{\alpha }+\phi _{\alpha }'']
) d\sigma ,
\\
J_i=\frac{-\rho }{2\omega \alpha }\int_\sigma (
Pw_i\psi _{\alpha }+w_i\rho ^{-2}[ ( \alpha ^2-2\alpha
) \psi _{\alpha }+\psi _{\alpha }''] )
d\sigma.
\end{gathered}
\end{equation}
The solution $u$ of \eqref{e1.3} is then written as
\begin{gather}
u=w_1+w_2  \label{e3.22} \\
w_i=\sum_{\alpha \in E_i} [ ( \frac{r}{\rho }
) ^{\alpha }I_i\phi _{\alpha }+( \frac{r}{\rho })
^{\alpha }J_i\psi _{\alpha }] , \quad i=1,2  \label{e3.23}
\end{gather}
and we have the following result.

\begin{theorem} \label{thm3.4}
The series \eqref{e3.23} converges uniformly in 
$\overline{S_{\rho_0}}$ for all $\rho _0< \rho $.
\end{theorem}

\begin{proof}
Set
\begin{equation} \label{e3.24}
\begin{aligned}
H_{i,\alpha }
&=\int_0^\omega ( Pu_i\phi _{\alpha }+u_i\rho ^{-2}
 [ ( \alpha ^2-2\alpha ) \phi _{\alpha }+\phi _{\alpha }''] ) d\theta \\
&=\int_0^\omega Pu_i\phi _{\alpha }d\theta
+( \alpha ^2-2\alpha ) \rho ^{-2}\int_0^\omega 
u_i\phi _{\alpha }d\theta +\rho ^{-2}
\int_0^\omega u_i\phi _{\alpha }''d\theta
\end{aligned}
\end{equation}
We show that $H_{i,\alpha }$ is  $1/\alpha $ times by bounded
term, for $\alpha $ large enough. According to \eqref{e3.17}, we have
\begin{equation}
\int_0^\omega u_i\phi _{\alpha }''d\theta
=\int_0^\omega u_i''\phi _{\alpha }d\theta  \label{e3.25}
\end{equation}
Replacing $\phi _{\alpha }$ by its expression and integrating by parts we obtain
\begin{equation}
\int_0^\omega u_i''\phi _{\alpha
}d\theta =\frac{1}{\alpha }\Big[ \frac{-\alpha }{( \alpha -2) }
\int_0^\omega u_i'''\cos
( \alpha -2) \theta d\theta \Big]  \label{e3.26}
\end{equation}
On the other hand, by a triple integration by parts, we have
\begin{equation}
( \alpha ^2-2\alpha ) \int_0^\omega
u_i\phi _{\alpha }d\theta
=\frac{1}{\alpha }\Big[ \frac{\alpha ^2}{
( \alpha -2) ^2}\int_0^\omega
u_i'''\cos ( \alpha -2) \theta d\theta
\Big]  \label{e3.28}
\end{equation}
Also, integrating by parts, we obtain
\begin{equation}
\int_0^\omega ( \Delta u_i-\frac{2}{r}\frac{
\partial u_i}{\partial r}) \phi _{\alpha }d\theta
=\frac{1}{\alpha }
\Big[ \frac{-\alpha }{( \alpha -2) }\
 int_0^\omega ( \frac{\partial }{\partial \theta }( \Delta u_i)
-\frac{2}{r}\frac{\partial ^2u_i}{\partial r\partial \theta })
\cos ( \alpha -2) \theta d\theta \Big]  \label{e3.29}
\end{equation}
Then, we deduce the existence of a constant $C_0$\ such that:
\begin{equation}
\vert H_{i,\alpha }\vert \leq \frac{C_0}{\alpha }  \label{e3.30}
\end{equation}
Using this last inequality and the fact that $\phi _{\alpha }$ is bounded
as well as the term $1/(2\omega ( \alpha -2) )$ for large $
\alpha $ we deduce the existence of a constant $C$ such that
\begin{equation}
\big| \sum_{\alpha \in E_i} c_{\alpha }r^{\alpha }\phi_{\alpha }\big|
\leq \sum_{\alpha \in E_i} \frac{C}{
\alpha }( \frac{r}{\rho }) ^{\alpha }  \label{e3.31}
\end{equation}
which converges uniformly in $\overline{S_{\rho _0}}$ for
$\rho _0< \rho $. Convergence of  $\sum_{\alpha \in E_i}
d_{\alpha }r^{\alpha }\psi _{\alpha }$ is proved by the same way.
\end{proof}

\begin{thebibliography}{0}

\bibitem{b1} H. Blum, R. Rannacher;
\emph{On the boundary value problem of the
biharmonic operator on domains with angular corners}, Maths. Methods
Appl. Sci. 2 (1980), no. 4, 556-581.

\bibitem{c1} C\'{e}dric Camier;
\emph{Mod\'{e}lisation et \'{e}tude num\'{e}rique de
vibrations non-lin\'{e}aire de plaques circulaires minces imparfaites,
Application aux cymbales}. Th\`{e}se
Pr\'{e}sent\'{e}e et soutenue publiquement le 2 f\'{e}vrier 2009 pour
l'otention du Docteur de l'Ecole Polytechnique.

\bibitem{c2} W. Chikouche, A. Aibeche;
\emph{Coefficients of singularities of the
biharmonic problem of Neumann type: case of the crack}. IJMMS 2003: 5,
305-313, Hindawi Publishing Corp.

\bibitem{g1} P. Grisvard;
\emph{Singularities in boundary values problems}, Recherches
en Math\'{e}matiques Appliqu\'{e}es, Vol. 22, Masson, Paris, 1992 (in French).

\bibitem{t1} O. Tcha-Kondor;
\emph{Nouvelles s\'{e}ries trigonom\'{e}triques adapt
\'{e}es \`{a} l'\'{e}tude de probl\`{e}mes aux limites pour l'\'{e}quation
biharmonique.\'{E}tude du cas de la fissure} [New trigonometric series
adapted to the study of boundary value problems for the biharmonic equation.
The cas of the crack], C. R. Acad. Sci. Paris s\'{e}r. I, Math. 315(1992),
no. 5, 541-544 (in French).

\end{thebibliography}

\end{document}