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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 203, pp. 1--5.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/203\hfil A nonlocal boundary problem]
{A nonlocal boundary problem for the Laplace operator in a half disk}

\author[G. A. Besbaev, I. Orazov, M. A. Sadybekov \hfil EJDE-2014/203\hfilneg]
{Gani A. Besbaev, Isabek Orazov, Makhmud A. Sadybekov}  % in alphabetical order

\address{Gani A. Besbaev \newline
Faculty of Information technology,
Auezov South Kazakhstan state University, \newline
 Shymkent, Kazakhstan}
 \email{besbaev@mail.ru}

\address{Isabek Orazov \newline
The Natural-Pedagogical faculty,
Auezov South Kazakhstan state University, \newline
Shymkent, Kazakhstan} 
\email{i\_orazov@mail.ru}

\address{Makhmud A. Sadybekov \newline
Institute of Mathematics and Mathematical Modeling,
Almaty, Kazakhstan}
\email{makhmud-s@mail.ru}

\thanks{Submitted July 11, 2014. Published September 30, 2014.}
\subjclass[2000]{33C10, 34B30, 35P10}
\keywords{Laplace equation; basis; eigenfunctions; \hfill\break\indent
nonlocal boundary value problem}

\begin{abstract}
 In the present work we investigate the nonlocal boundary problem for
 the Laplace equation in a half disk. The difference of this problem
 is the impossibility of direct applying of the Fourier method
 (separation of variables). Because the corresponding spectral problem
 for the ordinary differential equation has the system of eigenfunctions
 not forming a basis. Based on these eigenfunctions there is constructed
 a special system of functions that already forms the basis.
 This is used for solving of the nonlocal boundary equation. The existence
 and the uniqueness of the classical solution of the problem are proved.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\allowdisplaybreaks

\section{Formulation of the problem}

Our goal is to find a function $u(r,\theta)\in C^{0}(\bar{D})\cap
C^2(D)$ satisfying  equation
\begin{equation} \label{eq1}
\Delta u=0
\end{equation}
in $D$,  with the boundary conditions
\begin{gather}\label{eq2}
u(1,\theta)=f(\theta),\quad 0\leq\theta\leq\pi, \\
\label{eq3}
u(r,0)=0,\quad r\in[0,1], \\
\label{eq4}
\frac{\partial u}{\partial\theta}(r,0)=\frac{\partial
u}{\partial\theta}(r,\pi)+\alpha u(r,\pi),\quad r\in(0,1)
\end{gather}
 where $D=\{(r,\theta):0<r<1, 0<\theta<\pi\}$; $\alpha>0$;
 $f(\theta)\in C^2[0,\pi]$, $f(0)=0$, $f'(0)=f'(\pi)+\alpha f(\pi)$.

Problem \eqref{eq1}--\eqref{eq4} with $\alpha=0$ was considered
in  \cite{MA,MA1} for the Laplace equation, and in
 \cite{ma,amb} for the Helmholtz equation. 
The existence and the uniqueness of the solution of the problem are
proved by applying the method of separation of variables and proving
the basis of the special function systems of the Samarskii-Ionkin
type in  $L_{p}$. In contrast to these papers in case of $\alpha\neq
0$ it is impossible to use directly the Fourier method of the
separation of the variables. Because the corresponding spectral
problem for the ordinary differential equation has the system of
eigenfunctions not forming a basis.

\section{Uniqueness of the solution}

\begin{theorem}\label{th1}
The solution of  problem \eqref{eq1}--\eqref{eq4} is unique.
\end{theorem}

\begin{proof} 
Suppose that there exist two functions
$u_{1}(r,\theta)$ and $u_{2}(r,\theta)$ satisfying the conditions of
the problem \eqref{eq1} - \eqref{eq4}. We show that the function
$u(r,\theta)=u_{1}(r,\theta)-u_{2}(r,\theta)$ is equal to $0$.

Consider the function
$$
U(r,\theta)=u(r,\theta)+u(r,\pi-\theta)
$$
 in $D_{1}=\{(r,\theta):0<r<1$, $0<\theta<\pi/2\}$. It is
easy to see that
\begin{gather*}
\Delta U=0;\\
\frac{\partial U}{\partial\theta}(r,\pi/2)=0;\\
\frac{\partial U}{\partial\theta}(r,0)=\alpha U(r,0) \quad\text{for }0<r<1;\\
U(1,\theta)=0 \quad\text{for } 0\leq\theta\leq\pi/2.
\end{gather*}
Since $\alpha>0$, it follows that $U=0$ in  $\bar{D_{1}}$ by the maximum
principle and the Zaremba-Giraud principle \cite[p. 26]{bits}
 for the Laplace equation. This means that
$u(r,\theta)=-u(r,\pi-\theta)$, in particular $u(r,0)=u(r,\pi)=0$ at
$r\in[0,1]$. The equality $u(r,\theta)=0$ in $\bar{D}$ follows from
the uniqueness of the solution of the Dirichlet problem for the
Laplace equation. The proof of the theorem is complete.
\end{proof}

\section{Forming the basis}

If solutions to \eqref{eq1} satisfying the conditions
\eqref{eq3}, \eqref{eq4} are sought in the form 
$$
u(r,\theta)=R(r)\varphi(\theta),
$$
then $R(r)=r^{\sqrt{\lambda}}$, $\operatorname{Re}\sqrt{\lambda}\geq0$,
and for the function $\varphi(\theta)$ we have the spectral problem
\begin{equation}\label{eq5}
\begin{gathered}
-\varphi''(\theta)=\lambda\varphi(\theta),\quad 0<\theta<\pi;\\
\varphi(0)=0,\quad \varphi'(0)=\varphi'(\pi)+\alpha\varphi(\pi).
\end{gathered}
\end{equation}

This problem has two groups of eigenvalues. All the eigenvalues are
simple and the corresponding system of eigenfunctions does not form
the basis in $L_{2}(0,\pi)$ \cite{LL}. However, in  \cite{mok} a
special system of functions is built based of these eigenfunctions
which  forms the basis. This fact was applied for the
solution of the nonlocal initial-boundary problem for the heat
equation. In  \cite{OS} one family of problems simulating the
determination of the temperature and density of heat sources from
given values of the initial and final temperature is similarly
considered.

Let us present the necessary facts from  \cite{mok}. 
Problem \eqref{eq5} has two groups of eigenvalues
$\lambda^{(1)}_k=(2k)^2$, $k=1,2,\dots$,
$\lambda^{(2)}_k=(2\beta_k)^2$, $k=0,1,2,\dots $.
Herein $\beta_k$ are roots of the equation
$tg\beta=\alpha/2\beta$, $\beta>0$, they satisfy the inequalities
$k<\beta_k<k+1/2$, $k=0,1,2,\dots $, and two-side estimates are carried
out for $\delta_k=\beta_k-k$ where $k$ is large enough,
\begin{equation}\label{eq6}
\frac{\alpha}{2k}\big(1-\frac{1}{2k}\big)
<\delta_k<\frac{\alpha}{2k}\big(1+\frac{1}{2k}\big).
\end{equation}

The eigenfunctions of the problem \eqref{eq5} have the form
$$
\varphi^{(1)}_k(\theta)=\sin(2k\theta),\quad
k=1,2,\dots;\; \varphi^{(2)}_k(x)=\sin (2\beta_k\theta), \;
k=0,1,2,\dots .
$$
This system is almost normed but does not form even an ordinary
basis in  $L_{2}(0,\pi)$. The additional system constructed from the
previous one
\begin{gather*}
\varphi_0(\theta)=(2\beta_0)^{-1}\varphi^{(2)}_0(\theta),\\
\varphi_{2k}(\theta)=\varphi^{(1)}_k(\theta),\\
\varphi_{2k-1}(\theta)=(\varphi^{(2)}_k(\theta)
 -\varphi^{(1)}_k(\theta))(2\delta_k)^{-1},\quad k=1,2,\dots
\end{gather*}
is a Riesz basis in $L_{2}(0,\pi)$. Biorthogonal to it, is
the system
\begin{gather*}
\psi_0(\theta)=2\beta_0\psi^{(2)}_0(\theta),\\
\psi_{2k}(\theta)=\psi^{(2)}_k(\theta)+\psi^{(1)}_k(\theta),\\
\psi_{2k-1}(\theta)=2\delta_k\psi^{(2)}_k(\theta),~k=1,2,\dots.
\end{gather*}
This system is constructed from the eigenfunctions
\begin{gather*}
\psi^{(1)}_k(\theta)=C^{(1)}_k\cos(2k\theta+\gamma_k),\quad k=1,2,\dots ,\\
\psi^{(2)}_k(\theta)=C^{(2)}_k\cos(\beta_k(1-2\theta)),\quad k=0,1,2,\dots.
\end{gather*}
of the problem conjugated to \eqref{eq5}. The constants
$C^{(j)}_k$ are taken from the biorthogonal relations
$\big(\varphi^{(j)}_k,\psi^{(j)}_k\big)=1$, $j=1,2$.

If the function $f(\theta)$ is in $C^2[0,\pi]$ and satisfies the
boundary conditions of  problem \eqref{eq5}, then its Fourier
series by the system ${\varphi_k(\theta)}$ converges uniformly. We
can calculate that
\begin{equation}\label{eq7}
\begin{gathered}
\varphi''_0(\theta)=-\lambda^{(2)}_0(\theta),~\varphi''_{2k}(\theta)
=-\lambda^{(1)}_k\varphi_{2k}(\theta),\\
\varphi''_{2k-1}(\theta)=-\lambda^{(2)}_k\varphi_{2k-1}(\theta)
-\frac{\lambda^{(2)}_k-\lambda^{(1)}_k}{2\delta_k}\varphi_{2k}(\theta).
\end{gathered}
\end{equation}

\section{Construction of the formal solution to the problem}

Considering section 3, we can write any solution of 
\eqref{eq1}--\eqref{eq4} in the form of a biorthogonal series
\begin{equation}\label{eq8}
u(r,\theta)=\sum^{\infty}_{k=0}R_k(r)\varphi_k(\theta),
\end{equation}
 where
$R_k(r)=(u(r,\cdot)$ and
$\psi_k(\cdot))\equiv\int^{\pi}_0u(r,\theta)\psi_k(\theta)d\theta$.
Functions \eqref{eq8} satisfy the boundary
conditions \eqref{eq3} and \eqref{eq4}.

Substituting \eqref{eq8} in  \eqref{eq1} and the boundary
conditions \eqref{eq2}, taking into account \eqref{eq7}, for finding
unknown functions $R_k(r)$ we obtain the following problems
\begin{equation}\label{eq9}
\begin{gathered}
r^2R''_0(r)+rR'_0(r)-\lambda^{(2)}_0R_0(r)=0,\\\\
r^2R''_{2k-1}(r)+rR'_{2k-1}(r)-\lambda^{(2)}_kR_{2k-1}(r)=0,\\\\
r^2R''_{2k}(r)+rR'_{2k}(r)-\lambda^{(1)}_kR_{2k}(r)
=\frac{\lambda^{(2)}_k-\lambda^{(1)}_k}{2\delta_k}R_{2k-1}(r),
\end{gathered}
\end{equation}
 with the boundary conditions $R_k(1)=f_k$, where
$f_k$ are the Fourier coefficients of the expansion of the
function $f(\theta)$ into the biorthogonal series by
${\varphi_k(\theta)}$.

The regular solution of \eqref{eq9} exists, is unique and can be
written in the explicit form
\begin{equation}\label{eq10}
\begin{gathered}
R_0(r)=f_0r^{\sqrt{\lambda^{(2)}_0}},~R_{2k-1}(r)
 =f_{2k-1}r^{\sqrt{\lambda^{(2)}_k}},\\
R_{2k}(r)=f_{2k}r^{\sqrt{\lambda^{(1)}_k}}+f_{2k-1}
\frac{r^{\sqrt{\lambda^{(2)}_k}}-r^{\sqrt{\lambda^{(1)}_k}}}{2\delta_k}.
\end{gathered}
\end{equation}
Substituting \eqref{eq10} in \eqref{eq8}, we obtain a formal
solution
\begin{equation}\label{eq11}
\begin{aligned}
u(r,\theta)
&=f_0\frac{r^{2\beta_0}}{2\beta_0}\sin(2\beta_0\theta)
 +\sum^{\infty}_{k=1}f_{2k-1}\frac{r^{2k}}{2\delta_k}[r^{2\delta_k}
 \sin(2(k+\delta_k)\theta)-\sin(2k\theta)]\\
&\quad +\sum^{\infty}_{k=1}f_{2k}r^{2k}\sin(2k\theta).
\end{aligned}
\end{equation}

\section{Main Theorem} 
Our main result reads as follows.

\begin{theorem}\label{th2}
If $f(\theta)\in C^2[0,\pi]$, $f(0)=0$, $f'(0)=f'(\pi)+\alpha f(\pi)$,
then there exists a unique classical solution 
$u(r,\theta)\in C^{0}(\bar{D})\cap C^2(D)$ of  problem \eqref{eq1}--\eqref{eq4}.
\end{theorem}

\begin{proof} 
The uniqueness of the classical solution of the
problem follows from Theorem \ref{th1}. The formal solution of the
problem is shown in the form of \eqref{eq11}. To make sure
that these functions are really the desired solutions we need to
verify the applicability of the superposition principle. For it we
need to show the convergence of the series, the possibility of
termwise differentiation, and to prove the continuity of these
functions on the boundary of the half-disk.

The possibility of differentiating the series \eqref{eq11} any
number of times at  $r<1$ is an obvious consequence of the
convergence of power series and two-sided estimates \eqref{eq6} for
$\delta_k$. Let us justify the uniform convergence of the series
\eqref{eq8} at $r\leq1$. For this we use the sign of the uniform
convergence of Weierstrass. By direct calculation it is easy to see
that the series \eqref{eq11} is majorized by the series
$C_{1}(|f_0|+|f_{1}|+|f_{2}|+\dots )$. This series converges
\cite{mok} due to the requirements of the theorem imposed on
$f(\theta)$. Since all the terms of the series \eqref{eq11} are
continuous functions, then the function $u(r,\theta)$  is continuous
in the boundary domain $\bar{D}$. The proof is complete.
\end{proof}

\section{Conjugated problem: existence and uniqueness of the solution}

Let us now formulate a problem conjugated to
\eqref{eq1}-\eqref{eq4}. 
We look for a function $v(r,\theta)\in C^{0}(\bar{D})\cap C^2(D)$ 
satisfying the equation
\begin{equation}\label{eq12}
\Delta v=0
\end{equation}
in $D$ with the boundary conditions
\begin{gather}\label{eq13}
v(1,\theta)=g(\theta),\quad 0\leq\theta\leq\pi,\\
\label{eq14}
v(r,0)=v(r,\pi),\quad r\in[0,1], \\
\label{eq15}
\frac{\partial v}{\partial\theta}(r,\pi)+\alpha
v(r,\pi)=0,\quad r\in(0,1),
\end{gather}
 where $g(\theta)\in C^2[0,\pi]$, $g(0)=g(\pi)$, $g'(\pi)+\alpha g (\pi)=0$.

We can easily verify the conjugacy of the problems
\eqref{eq1}--\eqref{eq4} and \eqref{eq12}--\eqref{eq15} by direct
calculation. The uniqueness of the solution of  problem
\eqref{eq12}--\eqref{eq15} follows from the maximum principle and the
Zaremba-Giraud principle \cite[p. 26]{bits} for the Laplace
equation. The existence of the solution and its representation in
the form of a biorthogonal series can be proved similar to Theorem
\ref{th2}. Let us show this result without the proof.

\begin{theorem}\label{th3}
If $g(\theta)\in C^2[0,\pi]$,$g(0)=g(\pi)$, 
$g'(\pi)+\alpha g(\pi)=0$, then there exists a unique classical solution
$v(r,\theta)\in C^{0}(\bar{D})\cap C^2(D)$ of problem
\eqref{eq12}-\eqref{eq15}.
\end{theorem}

\begin{thebibliography}{00}

\bibitem{bits} A. V. Bitsadze;
\emph{Nekotorye klassy uravnenii v chastnykh proizvodnykh}.
Nauka, Moscow, 1981. (In Russian)

\bibitem{LL} P. Lang, J. Locker;
\emph{Spectral theory of two-point differential operators
determined by $D^2$ II. Analysis of case}. Journal of Mathematical
Analysis and Applications, 146 (1) (1990), pp. 148-191.

\bibitem{MA} E. I. Moiseev, V. E. Ambartsumyan;
\emph{On the solvability of nonlocal boundary value problem with the
equality of flows at the part of the boundary and conjugated to its
problem}. Differential Equations, 46(5) (2010), pp. 718-725.

\bibitem{MA1} E. I. Moiseev, V. E. Ambartsumyan;
\emph{On the solvability of nonlocal boundary value problem with the
equality of flows at the part of the boundary and conjugated to its
problem}. Differential Equations, 46(6) (2010), pp. 892-895.

\bibitem{ma} E. I. Moiseev, V. E. Ambartsumyan;
\emph{Solvability of some nonlocal boundary value problems for the
Helmholtz equation in a half-disk}. Doklady Mathematics, 82 (1)
(2010), pp. 621-624.

\bibitem{amb} E. I. Moiseev, V. E. Ambartsumyan;
\emph{On the solvability of nonlocal boundary value problem for the
Helmholtz equation with the equality of flows at the part of the
boundary and its conjugated problem}. Integral Transforms and
Special Functions, 21 (12) (2010), pp. 897-906.

\bibitem{mok} A. Y. Mokin;
\emph{On a family of initial-boundary value problems for the heat
equation}. Differential Equations, 45 (1) (2009), pp. 126-141.

\bibitem{OS} I. Orazov, M. A. Sadybekov;
\emph{One nonlocal problem of determination of the temperature and
density of heat sources}. Russian Mathematics (Iz. VUZ), 56 (2)
(2012), pp. 60-64.

\end{thebibliography}

\end{document}