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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 148, pp. 1--6.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/148\hfil Trigonometric series]
{Trigonometric series adapted for the study of Neumann boundary-value
problems of\\ Lam\'e systems}

\author[B. Merouani, N. Liazidi \hfil EJDE-2017/148\hfilneg]
{Boubakeur Merouani, Nabil Liazidi}

\address{Boubakeur Merouani \newline
Applied Mathematics Laboratory
(LaMA), University of Setif 1, Algeria}
\email{mermathsb@hotmail.fr}

\address{Nabil Liazidi \newline
 Applied Mathematics Laboratory (LaMA),
University of Setif 1, Algeria}
\email{n.liazidi@outlook.fr}

\thanks{Submitted  March 21, 2017. Published June 23, 2017.}
\subjclass[2010]{35C09, 35C10, 35J57, 35Q70}
\keywords{Sector; crack; singularity; Lam\'e; trigonometric series; Airy function}

\begin{abstract}
 In this article, we study the solutions to Neumann boundary-value problems
 of Lam\'e system in a sectorial domains. We  study directly this
 problem, by using  trigonometric series, without going through the
 Airy functions. Results using the Airy function are given in \cite{t1}.
\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}
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}

We are interested in the study of functions $u$ belonging to the Sobolev
space $(H^{1}( S) )^2$, and that are solutions to Lam\'e type
system
\begin{equation}
\begin{gathered}
Lu=\Delta u+\nu _0\nabla ( \operatorname{div}u) =0,\quad \text{in }S \\
\sigma (u)\cdot\eta =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 Lam\'e constants, with $\lambda \geq 0$, $\mu >0$,
$\nu $ is a real number $( 0< \nu < \frac{1}{2}) $
called Poisson coefficient, and
\begin{equation*}
\sigma (u)=
\begin{pmatrix}
\sigma _{11}( u) & \sigma _{12}( u) \\
\sigma _{12}( u) & \sigma _{22}( u)
\end{pmatrix}\,.
\end{equation*}

Here, the components of the stress tensor $\sigma (u)$ are given by Hooke's
law
\begin{equation*}
\sigma _{ij}=\mu (\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}
}{\partial x_{i}})+\lambda (\nabla .u)\delta _{ij},\quad i,j=1,2.
\end{equation*}

We shall analyze the solutions $u$ of this problem which can be written in a
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 =\alpha ^2\sin ^2\omega ,\text{ }\operatorname{Re}\alpha
> 0  \label{e1.5}
\end{equation}
For further studies of the set $E$, see, for example,  Lozi and
Merouani \cite{l1,m3}.

It is well known that the problem \eqref{e1.3} is reduced to the problem of
Dirichlet for the bilaplacian of the Airy function. In this work, we
study directly  problem \eqref{e1.3} and find the results by Tcha-Kondor
\cite{t1} concerning the Airy function. The relations between the
Airy function and the stress tensor $\sigma (u) $ are given in  \cite{g2} by
\begin{equation}
\sigma _{11}( u) =\frac{\partial ^2H}{\partial ^2x_{2}},\quad
\sigma _{22}( u) =\frac{\partial ^2H}{\partial ^2x_1}, \quad
\sigma _{12}( u) =-\frac{\partial ^2H}{\partial x_1\partial x_{2}},  \label{e1.6}
\end{equation}

We will adapt the technique used in \cite{c2,m4,m5,t1}, for the
bilaplacian and for the Dirichlet's boundary conditions for the Lam\'e's
system. Here, we treat problem of Lam\'e system in sector for Neumann
boundary-value. As in \cite{m4}, we establish, thanks to the Betti formula
instead of Green formula, a relation of orthogonality between the functions
$v_{\alpha }$ and $v_{\beta }$ allowing us to compute the coefficients of 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 $r^{\alpha }(v_{1,\alpha }( \theta ),v_{2,\alpha }( \theta ) )$
in problem \eqref{e1.3}  and
using the change of variables
\begin{equation}
\begin{gathered}
w_1( \theta ) =\cos \theta v_1( \theta ) +\sin
\theta v_{2}( \theta ), \\
w_{2}( \theta ) =-\sin \theta v_1( \theta ) +\cos
\theta v_{2}( \theta )
\end{gathered} \label{e2.1}
\end{equation}
 leads us the system
\begin{equation}
\begin{gathered}
w_1''( \theta ) +( \nu _0+1)
( \alpha ^2-1) w_1( \theta ) +(\nu _0(
\alpha -1) -2)w_{2}'( \theta ) =0, \\
( \nu _0+1) w_{2}''( \theta )
+( \alpha ^2-1) w_{2}( \theta ) +(\nu _0(
\alpha +1) +2)w_1'( \theta ) =0, \\
w_1'( 0) +( \alpha -1) w_{2}(0) =0, \\
(\nu _0( \alpha +1) +1)w_1( 0)   +( \nu_0+1) w_{2}'( 0) =0; \\
\begin{aligned}
&((\nu _0( \alpha +1) +1)w_1( \omega ) +( \nu
_0+1) w_{2}'( \omega ) )\sin \omega\\
& + (w_1'( \omega ) +( \alpha -1)
w_{2}( \omega ) )\cos \omega =0,
\end{aligned} \\
\begin{aligned}
&((\nu _0( \alpha +1) +1)w_1( \omega ) +( \nu
_0+1) w_{2}'( \omega ) )\cos \omega \\
&- (w_1'( \omega ) +( \alpha -1)w_{2}( \omega ) )\sin \omega =0,
\end{aligned}
\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} 2v_0\alpha \cos ( \alpha -2) \theta -2(v_0(\alpha +2)+2)\cos
\alpha \theta \\
-2v_0\alpha \sin ( \alpha -2) \theta +2(v_0\alpha -2)\sin
\alpha \theta
\end{pmatrix}
\label{e2.3}
\end{equation}
and
\begin{equation}
\psi _{\alpha }( \theta )
= \begin{pmatrix}
2v_0\alpha \sin ( \alpha -2) \theta -2(v_0\alpha +2)\sin
\alpha \theta \\
2v_0\alpha \cos ( \alpha -2) \theta -2(v_0(\alpha -2)-2)\cos
\alpha \theta
\end{pmatrix},
 \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's 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 $\eta =
\begin{pmatrix}
\cos \theta  \\
\sin \theta
\end{pmatrix}$
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 the Betti's formula we obtain
\begin{equation}
\int_{\Sigma} [ v\sigma ( u) \cdot\eta -u\sigma
( v) \cdot\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 =\mu r^{\alpha -1}M_{\alpha ,v_0}(w_{\alpha })
\label{e2.8}
\end{equation}
with $M_{\alpha ,v_0}(w_{\alpha })$ being the matrix
\begin{equation*}
\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}
\end{equation*}
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 solutions of $( 2.2) $ with
$\alpha $ and $\beta $ solutions 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}

The above corollary follows by substituting \eqref{e2.9} in \eqref{e2.5},
to obtain \eqref{e2.10}.

For $\omega _{\alpha }=r^{\alpha }\varphi _{\alpha }$, we define the operator
\begin{equation*}
T\omega _{\alpha }=r^{\alpha -1}
\begin{pmatrix}
(v_0+1)\omega _{1,\alpha } \\
\omega _{2,\alpha }
\end{pmatrix}.
\end{equation*}


\begin{corollary} \label{coro3}
From Corollary \ref{coro2}, if $\alpha \neq \overline{\beta }$, we
have
\begin{equation}
\int_{\Sigma}  ( T\omega _{\alpha }\cdot \overline{\omega
_{\beta }}+\omega _{\alpha }.T\overline{\omega _{\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}  ( T\omega _{\alpha }\text{.}\overline{
\omega _{\beta }}+\omega _{\alpha }.T\overline{\omega _{\beta }})
d\sigma =2r^{\alpha +\beta -1}\int ((v_0+1)\omega _{1,\alpha }\omega _{1,
\overline{\beta }}+\omega _{2,\alpha }\omega _{2,\overline{\beta }})d\theta
=0.
\end{equation*}
\end{proof}

\begin{corollary} \label{coro4}
 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\cdot \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 into account the definition of the operator $T$ we have
\begin{align*}
&\int_{\Sigma}  ( Tu\cdot \overline{u_{\beta }}+u\cdot 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\cdot \overline{u_{\beta }}
+u\cdot 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{coro4} results from this
last equality.
\end{proof}

The technique we develop for the study of the trigonometric series is based
on Theorem \ref{thm1} and Corollary \ref{coro4}.
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 computations

\section{Complete case study of the crack}

To simplify calculations, we decompose every solution $u$ of \eqref{e1.3} in
two parts with respect to $\theta $
\begin{equation*}
u=\mathfrak{U}_1+\mathfrak{U}_{2}.
\end{equation*}

\subsection{Study of the first part}
The first part is the expression $\varphi _{\alpha }$ and is given by \eqref{e2.3},
where
\[
E=\{\frac{k}{2},\; k\in \mathbb{N} ^{\ast }\}
\]
 because $\omega =2\pi$.
After some calculation, we obtain
\begin{equation*}
[ \varphi _{\alpha },\varphi _{\alpha }]
=4[v_0^2(v_0+2)\alpha ^2+(v_0(\alpha +2)+2)^2(1+v_0)+(v_0\alpha
-2)^2] \pi \rho ^{2\alpha -1}\neq 0.
\end{equation*}
We define the sub-sector
\begin{equation*}
S_{\rho _0}=S\cap \{( r\cos \theta ,r\sin \theta ) \in \mathbb{R}
^2,\quad r< \rho _0\},\quad \rho _0<\rho .
\end{equation*}
We define the traces on $\Sigma $,
\begin{equation*}
\mathfrak{U}_1= \xi _1\in ( \tilde{H}^{1/2} (\Sigma)) ^2, \quad
T\mathfrak{U}_1=\phi_1\in ( H^{1/2}(\Sigma )) ^2.
\end{equation*}
Let
\begin{equation}
c_{\alpha }=A_{\alpha ,v_0}\int_0^{2\pi }( \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) ( \rho _0,\theta ) d\theta ,
\label{e3.1}
\end{equation}
with
\begin{equation*}
A_{\alpha ,v_0}
=\frac{\rho _0^{-\alpha }}{8\pi [ v_0^2(v_0+2)
\alpha ^2+(1+v_0)(v_0(\alpha +2)+2)^2+(v_0\alpha -2)^2] }\,.
\end{equation*}

\begin{corollary} \label{coro7}
If $\mathfrak{U}_1$ is a 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{coro4}.

(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 a exists positive $\varepsilon $,
sufficiently small such that the solution of problem \eqref{e1.3} is written as
\begin{equation*}
\mathfrak{U}_1=\sum_{\alpha \in E} K_{\alpha }r^{\alpha}\varphi _{\alpha },
\end{equation*}
which  converges for $r< \varepsilon$. Then 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} \rm
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
\begin{equation*}
E=\{ \frac{k}{2},\; k\in \mathbb{N}^{\ast }\}
\end{equation*}
because $\omega =2\pi$. After some calculations, we obtain
\begin{equation*}
[ \psi _{\alpha },\psi _{\alpha }]
=4[v_0^2(v_0+2)\alpha ^2+(v_0+1)(v_0\alpha +2)^2+(v_0(\alpha
-2)-2)^2] \pi \rho ^{2\alpha -1}\neq 0
\end{equation*}
We define the following trace of $\Sigma $,
\begin{equation*}
\mathfrak{U}_{2}=\xi _{2}\in ( \tilde{H}^{1/2}(\Sigma)) ^2,\quad
T\mathfrak{U}_{2}=\phi _{2}\in (\tilde{H}^{1/2}(\Sigma )) ^2.
\end{equation*}
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\Big) ( \rho _0,\theta ) d\theta .
\label{e3.3}
\end{equation}
with
\begin{equation*}
B_{\alpha ,v_0}
=\frac{\rho _0^{-\alpha }}{8\pi  [
v_0^2(v_0+2)\alpha ^2+(v_0+1)(v_0\alpha +2)^2+(v_0(\alpha
-2)-2)^2] }\,.
\end{equation*}

\begin{corollary} \label{coro9}
If $\mathfrak{U}_{2}$ is solution of  problem \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} \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}

\begin{remark} \rm
Using the formulas \eqref{e1.6}, we find the results concerning the Airy functions.
\end{remark}


Equation \eqref{e2.8} in \cite{m5} is false (Typing error). We have to write $\mu
r^{\alpha -1}M_{\alpha ,v_0}(w_{\alpha })$ and not $\frac{1}{\mu }
r^{\alpha -1}M_{\alpha ,v_0}(w_{\alpha })$.


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