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\begin{center}
ON SOME GENERALISATION OF NON-LOCAL BOUNDARY VALUE PROBLEMS FOR ELLIPTIC 
EQUATIONS
\end{center}

\vspace*{0.3cm}
\centerline{\it{N. Gordeziani, E. Gordeziani}}

\vspace*{0.3cm}
\centerline{\it I. Javakhishvili Tbilisi State University}

\vspace*{0.3cm}
\par In the present work there are considered the following problems:
\par  {\bf Problem 1.} Consider a bounded domain $\Omega$, with the boundary
$\Gamma$. Let $\Omega_i,(i=\overline{1,m})$ be domains with boundaries
$\Gamma_i$, such that each of the following domains is placed strictly inside
of the proceeding ones. In addition, $\Gamma_i$ represents a diffeomorfic
image of $\Gamma, X^{(i)}\in\Gamma_i, X\in\Gamma, I_i(X)=X^{(i)}$. $\Gamma$ and
$\Gamma_i$ are Liapunov surfaces.
\par Let $L$ be uniform elliptic operator of the following type:
$$
L\equiv\sum\limits^n_{i,k=1}a_{i,k}(x)\frac{\partial^2}{\partial x_i
\partial x_k}+\sum\limits^n_{i=1}b_i(x)\frac{\partial}{\partial x_i}+c(x),
$$
 where $a_{i,k}(x), b_i(x), c(x) (c(x)\le 0)$ are prescribed functions.
\par We have to find the regular solution of the following equation:
\begin{equation}
Lu(x)=F(x),\;\; x\in\Omega,
\end{equation}
satisfying the following non-local conditions
\begin{equation}
u(X)=\sum\limits^m_{i=1}q_iu(X^{(i)})+\Phi (X),
\end{equation}
where $\Phi, F$ are prescribed functions, $F$ is defined on $\Omega, \Phi$ -
on $\Gamma, q_i(i=\overline{1,m})$ are prescribed constants,
$\sum\limits^m_{i=1}q_i=q_0.$
\par We assume, that $a_{i,k}, b_i, c, F$ and $\Phi$ are such, that there
exists the regular solution of Dirichlet problem for the equation (1).
\par In order to solve the problem there was suggested the following iteration
procedure:
\begin{equation}
\begin{array}{l}
Lu^{(k+1)}(x)=F(x),\\
u^{k+1}(X)=\sum\limits^m_{i=1}q_iu{(k)}(X^{(i)})+\Phi(X),\;\; k=0,1,...\\
u^{(0)}(X^{(0)})=0.
\end{array}
\end{equation}
\par {\bf Theorem 1.} {\it If $\bigg|\sum\limits^m_{i=1}q_i\bigg|=q_0<1$, in
addition, either all $q_i\le 0$ or $q_i>0$, then there exists the unique
regular solution of the problem {\rm (1)-(2), (3)} iteration procedure converges with
speed of geometrical progression and the following estimation is valid:
$\max\limits_{\bar\Omega}|(x)-u^{(k)}(x)|\le cq_0^k, \forall k$,
where $c$ is the constant, which does not depend on $u(x)$ and $u^{(k)}(x)$.}
\par  {\bf Problem 2.} In this case domain $\Omega$ is kernel with $\Gamma$ boundary 
and $a$ radius. Respectively, $\Omega_i(i=\overline{0,m})$ are concentrated 
kernels with $\Gamma_i$ boundaries and $a_i$ radiuses, $a>a_0>a_1>...>a_m>0.$ 
The centre of coordinate system is placed in the centre of kernels. Here we 
bring in the  prescribed  functions: $F(r,\theta,\phi)\in C^1(\Omega)$ and 
$f(\theta,\phi)\in C^2(\Gamma)$.
\par We have to find the regular solution of the equation 
\begin{equation}
\Delta u(x)=0, \;\; x\in\Omega,
\end{equation}
satisfying the followig generalisation of before known non-local boundary 
conditions,
\begin{equation}
\alpha\frac{\partial u}{\partial r}\bigg|_{r=a}+\beta u\bigg|_{r=a}=
\sum\limits^m_{i-0}\alpha_iu\bigg|_{r=a_i}+\frac{\gamma}{a-a_0}
\int\limits^a_{a_0}udr+f,
\end{equation}
where $u(x)\in C^2(\Omega)\bigcap C^1(\bar\Omega)$, 
\ $\Delta$ is the Laplace operator written in spherical coordinates,
\ $\alpha,\beta,\alpha_i$ and $\gamma$ are prescribed constants.
\par {\bf Theorem 2.} {\it If $\beta\ge 0$ and $\beta>\sum\limits^m_{i=0}\alpha_i+\gamma$,
then there exists the unique regular solution of the problem {\rm(4)-(5)}, it is 
unique and can be written in spherical functions:
$$
\begin{array}{l}
u(x)=\frac{1}{\beta-\sum\limits_0^m\alpha_i-\gamma}Y_0+
\sum\limits^{\infty}_{n=1}\left(\frac{r}{a}\right)^n
\left[\alpha\frac{n}{a}+\beta-\sum\limits^m_0\alpha_i
\left(\frac{a_i}{a}\right)^n-\right.\\
\left.-\frac{\gamma}{(n+1)(a-a_0)}
\left[a-a_0\left(\frac{a_0}{a}\right)^n\right]\right]^{-1}Y_n,
\end{array}
$$
if $\beta=\sum\limits_{n=1}^m\alpha_i-\gamma$ and $\int\limits_0^{2\pi}
\int\limits_0^\pi f\sin\theta d\theta d\varphi=0,$ solution is defined with 
precision of any constsnt,
$$
\begin{array}{l}
u(x)=\sum\limits^{\infty}_{n=1}\left(\frac{r}{a}\right)^n
\left[\alpha\frac{n}{a}+\beta-\sum\limits^m_0\alpha_i
\left(\frac{a_i}{a}\right)^n-\right.\\
\left.\frac{\gamma}{(n+1)(a-a_0)}\left[a-a_0\left(\frac{a_0}{a}\right)^n
\right]\right]^{-1}Y_n+c,
\end{array}
$$
where $c$ is an arbitrary constant, $Y_n(\theta,\varphi)$ is a spherical 
harmonic which can be expressed by Legandre's goint polynomials and Fourier 
coefficients.}
\par {\bf Problem 3.} In this case domain $\Omega$ is cylinder, $\Gamma^{(1)}_0$ 
and $\Gamma^{(2)}_0$ are upper side and bottom boundaries, $Y$ is a side 
boundary, with $a$ radius. Respectively, $\Omega_i$ are consentrated 
cylinders, with $a_i$ radiuses $\Gamma_i$ side boundaries; The heights of these 
cylinders equal to each other, it is $l$.
\par We have to find the regular solution of Laplace equation (4), satisfying 
(5) non-local condition, in addition 
\begin{equation}
u\bigg|_{\Gamma^{(1)}_0}=u\bigg|_{\Gamma^{(2)}_0}=0.
\end{equation}
\par {\bf Theorem 3.} {\it If $\beta\ge 0$ and $\beta>\sum\limits^m_0\alpha_i+\gamma$, then 
there exists the unique regular solution of the problem {\rm (4)-(5)-(6)} and it can 
be expressed by Bessell functions:
$$
\begin{array}{l}
\displaystyle
u(x)=\sum\limits^\infty_{n=0}\sum\limits^\infty_{k=1}
\left[\frac{\alpha}{2}\frac{I_{n-1}\left(\frac{\pi k}{l}a\right)-
I_2{n+1}\left(\frac{\pi k}{l}a\right)}{I_n\left(\frac{\pi k}{l}a\right)}+
\beta-\right.\\
\\ \ 
\displaystyle
\left.-\sum\limits_{i=0}^m\alpha_i\frac{I_n(a_i)}{I_n(a)}-\gamma
\frac{I_n^0\left(\frac{\pi k}{l}(a-a_0)\right)}{I_n(a)}\right]^{-1}Y_{nk},
\end{array}
$$
where 
$$\displaystyle I_n^0\left(\frac{\pi k}{l}(a-a_0)\right)=
\sum\limits_{\nu=0}^{\infty}\frac{1}{\gamma(\nu+1)\gamma(\nu+n+1)(2\nu+n+1)}
\left(\frac{\pi k}{l}(a-a_0)\right)^{2\nu+n},$$
 $Y_{nk}$ are expressed by 
Fourier coeficients, $I_n$ are modified functions of Bessell. }

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