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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 181, pp. 1--6.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/181\hfil Trigonometric series]
{Trigonometric series adapted for the study of Dirichlet boundary-value
problems of Lam\'e systems}

\author[B. Merouani, R. Boufenouche \hfil EJDE-2015/181\hfilneg]
{Boubakeur Merouani, Razika Boufenouche}

\address{Boubakeur Merouani \newline
Department of Mathemaics,
Univ. Setif 1, Algeria}
\email{mermathsb@hotmail.fr}

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

\thanks{Submitted February 6, 2015. Published July 1, 2015.}
\subjclass[2010]{35B40, 35B65, 35C20}
\keywords{Sector; crack; singularity; Lam\'e system; series}

\begin{abstract}
 Several authors have used trigonometric series for describing the solutions
 to elliptic equations in a plane sector; for example, the study of the
 biharmonic operator with different boundary conditions, can be found in
 \cite{c2,m4,t1}. The main goal of this article is to adapt those techniques
 for the study of Lam\'e systems in a sector.
\end{abstract}

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

\section{Introduction}

Let $S$ be the truncated plane sector of angle $\omega \leq 2\pi ,$ and
positive radius $\rho $:
\begin{equation}
S=\{ ( r\cos \theta ,r\sin \theta ) \in\mathbb{R}^2,\,
0< r< \rho ,\; 0< \theta < \omega \}\,.  \label{e1.1}
\end{equation}
Let $\Sigma $ be 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}
We are interested in the study of functions $u$ belonging to the Sobolev
space $[ H^1( S) ] ^2$, and that are solutions to the Lam\'e type system
\begin{equation}
\begin{gathered}
Lu=\Delta u+\nu _0\nabla ( \operatorname{div}u) =0\quad \text{in }S \\
u=0\quad \text{for }\theta =0,\, \omega\,,
\end{gathered} \label{e1.3}
\end{equation}
where
\begin{equation*}
\nu _0=( 1-2\nu ) ^{-1}=\frac{\lambda +\mu }{\mu },
\end{equation*}
$\lambda $, $\mu $ are the Lam\'e constants, with $\lambda \geq 0$,
$\mu >0$ and $\nu $ is a real number ($0< \nu < 1/2)$ called
Poisson coefficient.

We shall analyze the solutions $u$ of this problem which can be written in
series of the form:
\begin{equation}
u( r,\theta ) =\sum_{\alpha \in E} c_{\alpha
}r^{\alpha }v_{\alpha }( \theta ) .  \label{e1.4}
\end{equation}
Here $E$ stands for the set of solutions of the equation in a complex
variable $\alpha ( \nu _0) $,
\begin{equation}
\sin ^2\alpha \omega =\Big( \frac{\nu _0}{\nu _0+2}\text{ }\alpha
\big) ^2\sin ^2\omega ,\quad \operatorname{Re}\alpha \succ 0  \label{e1.5}
\end{equation}
For further studies of the set $E$, see, for example Lozi \cite{l1}.

We will adapt the technique used in \cite{c2,m4,t1}, for the bilaplacian.
The novelty is that, here, we treat a system of PDE's instead of the equations
studied before. For this purpose we introduce a Betti formula instead of a
Green formula.
To compute the coefficients of the the singularities
which can occur in the solutions, this technique is easier and more direct
than the classical one used  in \cite{g2}.

 We will focus on the important case of the crack, i.e. $\omega =2\pi $. The calculations in that case are more explicit and give the known
results for the Laplacian as just a particular case.

\section{Separation of variables}

Replacing $u$ by $r^{\alpha }v_{\alpha }( \theta ) =r^{\alpha
}( v_{1,\alpha }( \theta ) ,v_{2,\alpha }( \theta
) ) $ in the problem \eqref{e1.3} and using the change
of variables
\begin{equation}
\begin{gathered}
w_{1,\alpha }( \theta ) =\cos \theta v_{1,\alpha }( \theta
) +\sin \theta v_{2,\alpha }( \theta ), \\
w_{2,\alpha }( \theta ) =-\sin \theta v_{1,\alpha }( \theta
) +\cos \theta v_{2,\alpha }( \theta )
\end{gathered} \label{e2.1}
\end{equation}
 leads us  the system
\begin{equation}
\begin{gathered}
w_{1,\alpha }''( \theta ) +( \nu
_0+1) ( \alpha ^2-1) w_{1,\alpha }( \theta )
+(\nu _0( \alpha -1) -2)w_{2,\alpha }'( \theta
) =0; \\
( \nu _0+1) w_{2,\alpha }''( \theta
) +( \alpha ^2-1) w_{2,\alpha }( \theta )
+(\nu _0( \alpha +1) +2)w_{1,\alpha }'( \theta
) =0. \\
w_{1,\alpha }( 0) =w_{2,\alpha }( 0) =0. \\
\cos \omega w_{1,\alpha }( \omega ) -\sin \omega w_{2,\alpha
}( \omega ) =\sin \omega w_{1,\alpha }( \omega ) +\cos
\omega w_{2,\alpha }( \omega ) =0.
\end{gathered}  \label{e2.2}
\end{equation}
By Merouani \cite{m2}, the solutions of \eqref{e2.2} are linear combination of
the functions
\begin{equation}
\varphi _{\alpha }( \theta )
= \begin{pmatrix} 2\alpha v_0[ \cos ( \alpha -2) \theta -\cos \alpha \theta] \\
 -2\alpha v_0\sin ( \alpha -2) \theta +2(v_0(\alpha -2)-4)\sin
\alpha \theta
\end{pmatrix} \label{e2.3}
\end{equation}
and
\begin{equation}
\psi _{\alpha }( \theta )
= \begin{pmatrix} 2\alpha v_0\sin ( \alpha -2) \theta -2(v_0(\alpha +2)+4)\sin \alpha \theta \\
 2\alpha v_0[ \cos ( \alpha -2) \theta -\cos \alpha \theta]
\end{pmatrix}
\quad 0< \theta < \omega ,
\label{e2.4}
\end{equation}
A relationship, similar to classical orthogonality, for this system is given
by the following theorem.

\begin{theorem} \label{thm1}
Let $w_{\alpha }=( w_{1,\alpha },w_{2,\alpha }) $ and
 $w_{\beta }=( w_{1,\beta },w_{2,\beta }) $ be solutions of
\eqref{e2.2}  with $\alpha $ and $\beta $ solutions of \eqref{e1.5}.
Then, for $\beta \neq \overline{\alpha }$, we have
\begin{equation}
\begin{aligned}
&[ w_{\alpha },w_{\beta }]\\
&= \int_0^\omega \Big[ \big[ \frac{1}{(
\overline{\beta }-\alpha ) }\nu _0( w_{2,\alpha }',w_{1,\alpha }') +( (v_0+1)w_{1,\alpha
},w_{2,\alpha }) \big]
\begin{pmatrix}
\overline{w}_{1,\beta } \\
\overline{w}_{2,\beta }
\end{pmatrix}
\Big] d\theta =0
\end{aligned} \label{e2.5}
\end{equation}
\end{theorem}

\begin{proof}
We shall use  Betti formula
\begin{equation}
\int_S ( vLu-uLv) dx=\int_\Gamma [
v\sigma ( u) \cdot\eta -u\sigma ( v) \cdot\eta ] d\sigma
\label{e2.6}
\end{equation}
where
\begin{equation*}
\sigma ( u) =
\begin{pmatrix}
\sigma _{11}( u) & \sigma _{12}( u) \\
\sigma _{12}( u) & \sigma _{22}( u)
\end{pmatrix}
\end{equation*}
is the tensor of stress, $\eta =
\begin{pmatrix}
\cos \theta \\
\sin \theta
\end{pmatrix}
$ is the outward unit vector normal to $\Sigma ,$ and $\Gamma $ is the
boundary of $S$. For two functions $u,v$ which are solutions of \eqref{e1.3},
using  Betti's formula we obtain
\begin{equation}
\int_{\Sigma } [ v\sigma ( u) .\eta -u\sigma
( v) .\eta ] d\sigma =0  \label{e2.7}
\end{equation}
on $\Sigma ,$ for the function $u=r^{\alpha }\varphi _{\alpha }$, taking
account of the change of variables \eqref{e2.1}, we have
\begin{equation}
\sigma ( u) \cdot\eta =\frac{r^{\alpha -1}}{\mu }M_{\alpha
,v_0}(w_{\alpha })  \label{e2.8}
\end{equation}
with
$M_{\alpha ,v_0}(w_{\alpha })$ being the matrix
\[
\begin{pmatrix}
((v_0-1)w_{2,\alpha }'+( \alpha ( v_0+1)
+(v_0-1)) w_{1,\alpha })\cos \theta -(w_{1,\alpha }'+( \alpha -1) w_{2,\alpha })\sin \theta \\
(w_{1,\alpha }'+( \alpha -1) w_{2,\alpha })\cos
\theta +((v_0-1)w_{2,\alpha }'+( \alpha (
v_0+1) +(v_0-1)) w_{1,\alpha })\sin \theta
\end{pmatrix}
\]
The results follow from the application of formula \eqref{e2.7} to the functions
$u=r^{\alpha }\varphi _{\alpha }$ and $u=r^{\beta }\psi _{\beta }$, and by
using relation \eqref{e2.8}.
\end{proof}

\begin{corollary} \label{coro2}
Let $w_{\alpha }$ and $w_{\beta }$ be solution of \eqref{e2.2}  with
$ \alpha $ and $\beta $ solution of \eqref{e1.5}. Suppose in addition
that
\begin{equation}
\int_0^\omega ( w_{2,\alpha }',w_{1,\alpha }')
\begin{pmatrix}
\overline{w}_{1,\beta } \\
\overline{w}_{2,\beta }
\end{pmatrix}
=0  \label{e2.9}
\end{equation}
and $\alpha \neq \overline{\beta }$, then
\begin{equation}
[ w_{\alpha },w_{\beta }] =\int_0^\omega
[ ( (v_0+1)w_{1,\alpha },w_{2,\alpha }) ]
\begin{pmatrix}
\overline{w}_{1,\beta } \\
\overline{w}_{2,\beta }
\end{pmatrix}
d\theta =0  \label{e2.10}
\end{equation}
\end{corollary}

\begin{proof}
Substituting \eqref{e2.9} in \eqref{e2.5}, we obtain \eqref{e2.10}.
\end{proof}

\begin{remark} \label{rmk3} \rm
For $w_{\alpha }=r^{\alpha }( w_{1,\alpha },w_{2,\alpha }) $, we
define the  operator
\begin{equation*}
Tw_{\alpha }=r^{\alpha -1}
\begin{pmatrix}
(v_0+1)w_{1,\alpha } \\
w_{2,\alpha }
\end{pmatrix}
\end{equation*}
\end{remark}

\begin{corollary} \label{coro4}
From corollary \ref{coro2},  if $\alpha \neq \overline{\beta }$, we
have
\begin{equation}
\int_{\Sigma } ( Tw_{\alpha } \overline{w_{\beta }}
+w_{\alpha }T\overline{w_{\beta }}) d\sigma =0.  \label{e2.11}
\end{equation}
\end{corollary}

\begin{proof}
From the definition of the operator $T$ and Corollary \ref{coro2} we have
\begin{equation*}
\int_\Sigma ( Tw_{\alpha }\text{.}\overline{w_{\beta }}
+w_{\alpha }T\overline{w_{\beta }}) d\sigma =2r^{\alpha +\beta -1}
\int_0^\omega ((v_0+1)w_{1,\alpha }w_{1,\overline{
\beta }}+w_{2,\alpha }w_{2,\overline{\beta }})d\theta =0.
\end{equation*}
\end{proof}

\begin{corollary} \label{coro5}
Suppose that $u=\sum_{\alpha \in E} c_{\alpha }r^{\alpha
}\varphi _{\alpha }$ is uniformly convergent in $\overline{S}$.
If $[\varphi _{\overline{\beta }},\varphi _{\beta }] \neq 0$ then
\begin{equation*}
c_{\overline{\beta }}=\frac{1}{2}\rho ^{-2\overline{\beta }+1}\frac{
\int_\Sigma ( Tu \overline{u_{\beta }}+u T\overline{
u_{\beta }}) d\sigma }{[ \varphi _{\overline{\beta }},\varphi
_{\beta }] }.
\end{equation*}
\end{corollary}

\begin{proof}
For $u=\sum_{\alpha \in E} c_{\alpha }r^{\alpha }\varphi _{\alpha}$ and taking account the definition of the operator $T$ we have
\begin{align*}
&\int_\Sigma ( Tu\text{.}\overline{u_{\beta }}+u.T\overline{
u_{\beta }}) d\sigma \\
&=\int_0^\omega \Big( \Big( \sum_{\alpha \in E}
 c_{\alpha }r^{\alpha -1}
\begin{pmatrix}
(v_0+1)\varphi _{1,\alpha } \\
\varphi _{2,\alpha }
\end{pmatrix}
\Big) r^{\overline{\beta }}\varphi _{^{\overline{\beta }}} \\
&\quad +\Big(\sum_{\alpha \in E} c_{\alpha }r^{\alpha }\varphi _{\alpha
}\Big) r^{\overline{\beta }-1}
\begin{pmatrix}
(v_0+1)\varphi _{1,\overline{\beta }} \\
\varphi _{2,\overline{\beta }}
\end{pmatrix}
\Big) d\theta \\
&=\sum_{\alpha \in E} c_{\alpha }r^{\overline{\beta }+\alpha -1}
\int_0^\omega \Big(
\begin{pmatrix}
(v_0+1)\varphi _{1,\alpha } \\
\varphi _{2,\alpha }
\end{pmatrix}
\varphi _{^{\overline{\beta }}}+\varphi _{\alpha }
\begin{pmatrix}
(v_0+1)\varphi _{1,\overline{\beta }} \\
\varphi _{2,\overline{\beta }}
\end{pmatrix}
\Big) d\theta .
\end{align*}
From Corollary \ref{coro2}, if $\alpha \neq \overline{\beta }$, then
\begin{equation*}
\int_\Sigma ( Tu \overline{u_{\beta }}+u T\overline{
u_{\beta }}) d\sigma =2C\overline{_{\beta }}[ \varphi _{\overline{
\beta }},\varphi _{\beta }] \rho ^{2\overline{\beta }-1}.
\end{equation*}
Expression $c_{\overline{\beta }}$ of Corollary \ref{coro5} results from this last
equality.
\end{proof}

\begin{remark} \label{rmk6} \rm
The technique we develop for the study of the trigonometric series is based
on Theorem \ref{thm1} and Corollary \ref{coro5}. To illustrate this, we study the following
trigonometric series in the particular case of the crack ($\omega =2\pi $),
which is an important case of singular domains. The explicit knowledge of
the roots of \eqref{e1.5} simplifies the computations.
\end{remark}

\section{Complete case study of the crack}

To simplify the calculations, we decompose every solution $u$ of \eqref{e1.3}
into two parts with respect to $\theta $
\begin{equation*}
u=\mathfrak{U}_1+\mathfrak{U}_2.
\end{equation*}

\subsection{Study of first part}
The first part is the expression $\varphi _{\alpha }$ and is given by
\eqref{e2.3} where
\begin{equation*}
E=\{ \frac{k}{2},\, k\in \mathbb{N}^{\ast }\} \quad
\text{ because }\omega =2\pi .
\end{equation*}
After some calculation, we obtain that
\begin{equation*}
[ \varphi _{\beta },\varphi _{\beta }]
=[ \beta^2v_0^2(2v_0+3)+4( v_0(\beta -2)-4) ^2] \pi
\rho ^{2\beta -1} \neq 0\,.
\end{equation*}

Define the sub-sector
\begin{equation*}
S_{\rho _0}=S\cap \{( r\cos \theta ,r\sin \theta ) \in\mathbb{R}^2,r< \rho _0\},\rho _0<\rho .
\end{equation*}
We define the  traces on $\Sigma $,
\begin{equation*}
\mathfrak{U}_1=\xi_1\in \big( \tilde{H}^{1/2}(\Sigma
)\big) ^2\quad \text{and} \quad
T\mathfrak{U}_1=\phi_1\in \big(
\tilde{H}^{1/2}(\Sigma )\big) ^2.
\end{equation*}
Let
\begin{equation}
c_{\alpha }=A_{\alpha ,v_0}\int_0^{2\pi }\Big( \xi _1
\begin{pmatrix}
(v_0+1)\varphi _{1,\alpha } \\
\varphi _{2,\alpha }
\end{pmatrix}
+\rho _0
\begin{pmatrix}
\varphi _{1,\alpha } \\
\varphi _{2,\alpha }
\end{pmatrix}
\phi _1\Big) ( \rho _0,\theta ) d\theta
\label{e3.1}
\end{equation}
with
\begin{equation*}
A_{\alpha ,v_0}=\frac{\rho _0^{-\alpha }}{2[ \alpha
^2v_0^2(2v_0+3)+4( v_0(\alpha -2)-4) ^2] \pi }
\end{equation*}

\begin{corollary} \label{coro7}
If $\mathfrak{U}_1$ is solution of \eqref{e1.3}, then
\begin{equation}
\mathfrak{U}_1=\sum_{\alpha \in E} c_{\alpha }r^{\alpha
}\varphi _{\alpha }  \label{e3.2}
\end{equation}
where $c_{\alpha }$ is given by \eqref{e3.1}. The series converges uniformly in
$\overline{S}_{\rho _0}$ for all $\rho _0<\rho $. Moreover \eqref{e3.2}
converges globally in $( H^1(S_{\rho })) ^2$, if
$\alpha ^{3/2}c_{\alpha }\rho ^{\alpha }\in l^2$.
\end{corollary}

\begin{proof}
(i) if \eqref{e3.2} occurs, then $c_{\alpha }$ is expressed by \eqref{e3.1}
 under Corollary \ref{coro5}.

(ii) if $\mathfrak{U}_1$ is solution of \eqref{e1.3} and $c_{\alpha }$ given
by \eqref{e3.1} then $c_{\alpha }=\circ (\alpha \rho _0^{-\alpha })$. This
implies the uniform convergence of the series in $\overline{S}_{\rho _0}$
towards some $ W_1$ satisfying \eqref{e1.3}.

From  Grisvard-Geymonat \cite{g1}, there exists a positive $\varepsilon$,
sufficiently small such that the solution of \eqref{e1.3}
is written as
\[
\mathfrak{U}_1=\sum_{\alpha \in E}
K_{\alpha }r^{\alpha }\varphi _{\alpha }
\]
 which converges for $r< \varepsilon$.
Theorem \ref{thm1} implies that $K_{\alpha }=c_{\alpha }$ therefore
${W}_1$ and $\mathfrak{U}_1$ coincide in $S_{\varepsilon }$. They coincide
in $S_{\rho _0}$ since they are real analytic.
\end{proof}

\begin{remark} \label{rmk8}
If $\xi _1$ belongs to the space $( H^2(]0,2\pi[ ) ) ^2$
and $\phi _1$ to $(H^1(] 0,2\pi[ ) ) ^2$, then
$ c_{\alpha }=\circ (\alpha \rho _0^{-\alpha })$ and we have uniform
convergence of the series in $\overline{S}_{\rho _0}$ for all $\rho
_0\leq \rho $.
\end{remark}

\subsection{Study of the second part}

The second part is the expression $\psi _{\alpha }$  given by \eqref{e2.4}
 where
\[
E=\{ \frac{k}{2},\, k\in \mathbb{N}^{\ast }\}
\]
because $\omega =2\pi$.
After some calculations, we obtain 
\[
[ \psi _{\alpha },\psi _{\alpha }]
=[ (v_0+3)\alpha ^2v_0^2+4(v_0+1)(v_0(\alpha +2)+4)^2] \pi \rho ^{2\alpha
-1} \neq 0\,.
\]
 We define the following trace on $\Sigma $,
\[
\mathfrak{U}_2=\xi _2\in ( \tilde{H}^{1/2}(\Sigma )) ^2
\quad \text{and}\quad
T\mathfrak{U}_2=\phi _2\in ( \tilde{H}^{1/2}(\Sigma )) ^2.
\]
Let
\begin{equation}
d_{\alpha }=B\alpha ,v_0\int_0^{2\pi }
\Big( \xi _1 \begin{pmatrix}
(v_0+1)\psi _{1,\alpha } \\
\psi _{2,\alpha }
\end{pmatrix}
+\rho _0
\begin{pmatrix}
\psi _{1,\alpha } \\
\psi _{2,\alpha }
\end{pmatrix}
\phi _1) ( \rho _0,\theta ) d\theta .
\label{e3.3}
\end{equation}
 with
\begin{equation*}
B_{\alpha ,v_0}=\frac{\rho _0^{-\alpha }}{2[ (v_0+3)\alpha
^2v_0^2+4(v_0+1)(v_0(\alpha +2)+4)^2] \pi }
\end{equation*}

\begin{corollary} \label{coro9}
If $\mathfrak{U}_2$ is solution of  \eqref{e1.3} then
\begin{equation}
\mathfrak{U}_2=\sum_{\alpha \in E} d_{\alpha }r^{\alpha }\psi
_{\alpha }  \label{e3.4}
\end{equation}
where $d_{\alpha }$ is given by \eqref{e3.3}. The series converges uniformly in
 $ \overline{S}_{\rho _0}$ for all $\rho _0<\rho $. Moreover \eqref{e3.4}
converges globally in $( H^1(S_{\rho })) ^2$, if
$\alpha ^{3/2}d_{\alpha }\rho ^{\alpha }\in l^2$.
\end{corollary}

\begin{remark} \label{rmk10} \rm
For $v_0=0$ we obtain the trigonometric series for the Laplace equation in
a sector. This is compatible with \eqref{e1.3} with $v_0=0$.
\end{remark}

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\end{document}