\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: $$S=\{ ( r\cos \theta ,r\sin \theta ) \in\mathbb{R}^2,0< r< \rho ,\; 0< \theta < \omega \} \label{e1.1}$$ and $\Sigma$ the circular boundary part $$\Sigma =\{ ( \rho \cos \theta ,\rho \sin \theta ) \in \mathbb{R} ^2,0< \theta < \omega \}. \label{e1.2}$$ \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 $$\begin{gathered} \Delta ^2u=0,\quad \text{in }S \\ u=Mu=0,\quad \text{for }\theta =0,\omega, \end{gathered} \label{e1.3}$$ where the operator $M$ represents the bending moment and is defined as $$Mu=\nu \Delta u+( 1-\nu ) ( \partial _1^2u n_1^2+2\partial _{12}^2u n_1n_2+\partial _2^2u n_2^2) . \label{e1.4}$$ 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 $$u( r,\theta ) =\sum_{\alpha \in E} c_{\alpha}r^{\alpha }\phi _{\alpha }( \theta ). \label{e1.5}$$ Here $E$ stands for the set of solutions of the equation in a complex variable $\alpha$: $$\sin ^2( \alpha -1) \omega =\sin ^2\omega ,\quad\operatorname{Re} \alpha > 1 \label{e1.6}$$ 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{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{theorem} \begin{proof} We use the Green formula $$\int_S ( v\Delta ^2u-u\Delta ^2v) dx =\int_\Gamma \big[ ( uNv+\frac{\partial u}{\partial n}Mv) -( vNu+\frac{\partial v}{\partial n}Mu) \big] \,d\sigma , \label{e2.5}$$ 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 $$\int_\Sigma \big[ ( uNv+\frac{\partial u}{\partial n} Mv) -( vNu+\frac{\partial v}{\partial n}Mu) d\sigma \big] =0 \label{e2.6}$$ On $\Sigma$, for the function $u_{\alpha }=r^{\alpha }\phi _{\alpha}$, we have $$\label{e2.7} \begin{gathered} \frac{\partial u_{\alpha }}{\partial n} =\frac{\partial u_{\alpha }}{\partial r}=\alpha r^{\alpha -1}\phi _{\alpha }, \\ Mu_{\alpha } = r^{\alpha -2}\big\{ [ \alpha ^2-( 1-\nu ) \alpha ] \phi _{\alpha }+\nu \phi _{\alpha }''\big\} , \\ Nu_{\alpha } = r^{\alpha -3}\big\{ -\alpha ^2( \alpha -2) \phi _{\alpha }+[ ( \nu -2) \alpha +( 3-\nu ) ] \phi _{\alpha }''\big\} . \end{gathered}$$ 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 $$\int_0^\omega \phi _{\alpha }\phi _{\beta}''d\theta =\int_0^\omega \phi _{\alpha }''\phi _{\beta }d\theta \label{e2.8}$$ which is obtained by a double integration by parts: $$\int_0^\omega \phi _{\alpha }\phi _{\beta}''d\theta =\int_0^\omega \phi _{\alpha }''\phi _{\beta }d\theta +[ \phi _{\alpha },\phi _{\beta }'] _0^{\omega }-[ \phi _{\alpha }',\phi _{\beta }] _0^{\omega }, \label{e2.9}$$ 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 $$[ \phi _{\alpha },\phi _{\beta }'] _0^{\omega }- [ \phi _{\alpha }',\phi _{\beta }] _0^{\omega }=0, \label{e2.10}$$ and $\alpha \neq \beta$, then $$[ \phi _{\alpha },\phi _{\beta }] =\int_0^\omega \big\{ [ ( \alpha ^2-2\alpha ) \phi _{\alpha }+\phi _{\alpha }''] \overline{\phi _{\beta }}+[ ( \overline{\beta }^2-2\overline{\beta }) \overline{\phi _{\beta }}+\overline{\phi _{\beta }''}] \phi _{\alpha }\big\}\, d\theta =0 \label{e2.11}$$ \end{corollary} \begin{remark} \label{rmk2.4} \rm For $u_{\alpha }=r^{\alpha }\phi _{\alpha }$ we have $$\Delta u_{\alpha }-\frac{2}{r}\frac{\partial u_{\alpha }}{\partial r} =r^{\alpha -2}[ ( \alpha ^2-2\alpha ) \phi _{\alpha }+\phi _{\alpha }'']. \label{e2.12}$$ \end{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 $$\int_\Sigma( Pu_{\alpha }\cdot \overline{u_{\beta }} +u_{\alpha }\cdot P\overline{u_{\beta }}) d\sigma =0. \label{e2.13}$$ \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 $$\label{e3.1} \begin{gathered} u =\sum_{\alpha \in E} ( c_{\alpha }u_{\alpha }+d_{\alpha }v_{\alpha }) ,\quad E=\{ \frac{k}{2},\, k\in \mathbb{N} ,\, k> 2\} , \\ u_{\alpha } =r^{\alpha }\phi _{\alpha }( \theta ) ,\quad v_{\alpha }=r^{\alpha }\psi _{\alpha }( \theta ) \end{gathered}$$ 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 $$\phi _{\alpha }( 0) =\phi _{\alpha }( \omega ) =\psi _{\alpha }( 0) =\psi _{\alpha }( \omega ) =0 \label{e3.4}$$ 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: $$\label{e3.5} \begin{gathered} u = w_1+w_2, \\ w_i =\sum_{\alpha \in E_i} ( c_{\alpha }u_{\alpha }+d_{\alpha }v_{\alpha }) ,\quad i=1,2, \\ E_1 = \{ 2m,\,m> 1\} ,\quad E_2=\{ 2m+1,\, 2m> 1\} . \end{gathered}$$ \subsection*{Calculation of $c_{\beta }$ and $d_{\beta }$} From the expressions of $\phi _{\alpha }$, $\psi _{\alpha }$ one easily sees that: $$\label{e3.6} \begin{gathered} \text{if \alpha \in E_1, then \phi _{\alpha }'(0) =\phi _{\alpha }'( \omega ) and \psi _{\alpha }'( 0) =\psi _{\alpha }'( \omega ) },\\ \text{if \alpha \in E_2, then \phi _{\alpha }'(0) =-\phi _{\alpha }'( \omega ) and \psi _{\alpha }'( 0) =-\psi _{\alpha }'( \omega )}. \end{gathered}$$ 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: $$\label{e3.7} \begin{gathered} \int_\sigma ( Pw_i\cdot u_{\beta }+w_i\cdot Pu_{\beta }) d\sigma =2c_{\beta }\int_\sigma ( u_{\beta }\cdot Pu_{\beta }) d\sigma +d_{\beta }\int_\sigma ( Pv_{\beta }\text{.}u_{\beta } +v_{\beta }\cdot Pu_{\beta }) d\sigma , \\ \int_\sigma ( Pw_i\cdot v_{\beta }+w_i\cdot Pv_{\beta}) d\sigma =c_{\beta }\int_\sigma ( Pu_{\beta } \cdot v_{\beta }+u_{\beta }\cdot Pv_{\beta }) d\sigma +2d_{\beta } \int_\sigma ( Pv_{\beta }\cdot v_{\beta }) d\sigma . \end{gathered}$$ By direct calculations we obtain $$\label{e3.8} \begin{gathered} \int_\sigma ( Pu_{\beta }\cdot v_{\beta }+u_{\beta }\cdot Pv_{\beta }) d\sigma =0, \\ \int_\sigma ( u_{\beta }\cdot Pu_{\beta }) d\sigma =( \beta -2) \omega \rho ^{2\beta -1} \\ \int_\sigma ( Pv_{\beta }.v_{\beta }) d\sigma =-\beta \omega \rho ^{2\beta -1} \end{gathered}$$ 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 $$u=w_1+w_2 \label{e3.9}$$ where $$w_i=\sum_{\alpha \in E_i} ( c_{\alpha }u_{\alpha }+d_{\alpha }v_{\alpha }) , i=1,2 \label{e3.10}$$ 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 $$\label{3.11} \begin{gathered} c_{\alpha } = \frac{\rho ^{1-2\alpha }}{2( \alpha -2) \omega } \int_\sigma ( Pw_i\cdot u_{\alpha }+w_i\cdot Pu_{\alpha }) d\sigma \\ d_{\alpha } = \frac{-\rho ^{1-2\alpha }}{2\alpha \omega } \int_\Sigma ( Pu_i\cdot v_{\alpha }+w_i\cdot Pv_{\alpha }) d\sigma \end{gathered}$$ \end{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 $$\gamma _{\beta }=\rho ^{1-2\beta }\int_\sigma ( Pw_i \cdot u_{\beta }+w_i\cdot Pu_{\beta }) d\sigma . \label{e3.12}$$ vanishes. By derivation with respect to $r$ we have \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} 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 $$( 2-\beta ) \Delta w_i-2\frac{\partial ^2w_i}{\partial r^2 }=-\beta Mw_i+[ 2-( 1-\upsilon ) \beta ] [ \frac{1}{r}\frac{\partial w_i}{\partial r}+\frac{1}{r^2}\frac{\partial ^2w_i}{\partial \theta ^2}] . \label{e3.14}$$ Using these formulas in the expression of $\gamma _{\beta }'$ we obtain \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} 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 \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} Taking into account of \eqref{e2.7}, whose expressions appear explicitly in $\gamma _{\beta }'$, we obtain $$\label{e3.19} \gamma _{\beta }' =\rho ^{1-2\beta }\int_\Sigma - \big[ \big( u_{\beta }Nw_i+\frac{\partial u_{\beta }}{\partial n} Mw_i\big) -\big( u_iNu_{\beta }+\frac{\partial w_i}{\partial n} Mu_{\beta }\big) \big]\, d\sigma =0.$$ 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 $$c_{\alpha }=I_i\rho ^{-\alpha },\quad d_{\alpha }=J_i\rho ^{-\alpha } \label{e3.20}$$ with $$\label{e3.21} \begin{gathered} I_i=\frac{\rho }{2\omega ( \alpha -2) }\int_\sigma ( Pw_i\phi _{\alpha }+w_i\rho ^{-2}[ ( \alpha ^2-2\alpha ) \phi _{\alpha }+\phi _{\alpha }''] ) d\sigma , \\ J_i=\frac{-\rho }{2\omega \alpha }\int_\sigma ( Pw_i\psi _{\alpha }+w_i\rho ^{-2}[ ( \alpha ^2-2\alpha ) \psi _{\alpha }+\psi _{\alpha }''] ) d\sigma. \end{gathered}$$ 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 \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} We show that $H_{i,\alpha }$ is $1/\alpha$ times by bounded term, for $\alpha$ large enough. According to \eqref{e3.17}, we have $$\int_0^\omega u_i\phi _{\alpha }''d\theta =\int_0^\omega u_i''\phi _{\alpha }d\theta \label{e3.25}$$ Replacing $\phi _{\alpha }$ by its expression and integrating by parts we obtain $$\int_0^\omega u_i''\phi _{\alpha }d\theta =\frac{1}{\alpha }\Big[ \frac{-\alpha }{( \alpha -2) } \int_0^\omega u_i'''\cos ( \alpha -2) \theta d\theta \Big] \label{e3.26}$$ On the other hand, by a triple integration by parts, we have $$( \alpha ^2-2\alpha ) \int_0^\omega u_i\phi _{\alpha }d\theta =\frac{1}{\alpha }\Big[ \frac{\alpha ^2}{ ( \alpha -2) ^2}\int_0^\omega u_i'''\cos ( \alpha -2) \theta d\theta \Big] \label{e3.28}$$ Also, integrating by parts, we obtain $$\int_0^\omega ( \Delta u_i-\frac{2}{r}\frac{ \partial u_i}{\partial r}) \phi _{\alpha }d\theta =\frac{1}{\alpha } \Big[ \frac{-\alpha }{( \alpha -2) }\ int_0^\omega ( \frac{\partial }{\partial \theta }( \Delta u_i) -\frac{2}{r}\frac{\partial ^2u_i}{\partial r\partial \theta }) \cos ( \alpha -2) \theta d\theta \Big] \label{e3.29}$$ Then, we deduce the existence of a constant $C_0$\ such that: $$\vert H_{i,\alpha }\vert \leq \frac{C_0}{\alpha } \label{e3.30}$$ 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 $$\big| \sum_{\alpha \in E_i} c_{\alpha }r^{\alpha }\phi_{\alpha }\big| \leq \sum_{\alpha \in E_i} \frac{C}{ \alpha }( \frac{r}{\rho }) ^{\alpha } \label{e3.31}$$ 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. 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