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\ 

\vskip+1cm

\begin{center}
{\size{25}{74pt}\selectfont \bf 	\sc ABSTRACTS}  \\[1.5cm]
{\size{14}{44pt}\selectfont \bf 	DIFFERENTIAL EQUATIONS} \\[2mm]
{\size{14}{44pt}\selectfont \bf		AND} \\[2mm]
{\size{14}{44pt}\selectfont \bf	MATHEMATICAL PHYSICS}
\end{center}
\vskip+1cm
{\footnotesize
\begin{gather*}
\hskip-4cm\begin{picture}(12,3)
	\put(0,3){\special{em:graph demph98c.pcx}}
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}
\vskip+4cm

\begin{center}
	{\large \sc Symposium Dedicated to the 95th Birthday \\[1mm]
		Anniversary of Academician V. Kupradze }\\[1cm]
		November 3--6, 1998 
\end{center}

\vfill 

\begin{center}
	{\large Tbilisi\; 1998 }
\end{center}


\newpage 

\thispagestyle{empty}

This is a collection of abstracts of plenary and section talks
submitted to the Program Committee of the
Symposium ``Differential Equations and Mathematical
Physics'' (Tbilisi, Georgia, November 3--6, 1998).

All abstracts except the plenary talks 
are assorted to the following sections:

\begin{itemize}
\item  Differential Equations;

\item Mathematical Physics.
\end{itemize}

Abstracts are reproduced as presented by the authors.

\vskip+3cm 

\begin{tabbing}
\ \ \ Program Committee:\ \  \=I. Kiguradze (Chairman)\\
\>R. Bantsuri\\
\>T. Burchuladze\\
\>R. Duduchava \\[1cm]
\ \ \ \ \ Scientific Secretary\ \>N. Partsvania
\end{tabbing}



\vfill

\qquad {\bf http://www.rmi.acnet.ge/DEMPh}
\vskip+1cm


\qquad \copyright \;{\bf A. Razmadze Mathematical Institute}
\ 

\newpage 

\setcounter{page}{1}

\ 

\vskip+4cm

\thispagestyle{empty}

\begin{center}
{\size{20}{74pt}\selectfont \bf P L E N A R Y\;\; T A L K S}
\end{center}

\newpage 

\setcounter{page}{3}
\begin{center}
	{\bf R. Bantsuri} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		A. Razmadze Mathematical Institute,
		Georgian Academy of Sciences \\ 
		Tbilisi, Georgia} \\[0.5cm]
	{\bf PLANE ELASTICITY PROBLEMS FOR INFINITE DOMAINS WITH UNKNOWN AND 
		PARTLY UNKNOWN CONTOURS}
\end{center}

\bigskip

	Mixed problems of the plane elasticity are considered for infinite 
planes with two holes or with periodically disposed holes of two types. 
The contours of the holes are sets consisting of unknown curves and 
given segments. The tangent strain is equal to zero on the whole contour, 
while the normal strain on the unknown part of the contour and 
the displacement on the given segments are constant. 
To find the unknown part of the contour, it is additionally requested
the constancy of the tangent normal component on it. Effective methods
for solution of these problems are elaborated.

\newpage 

\begin{center}
	{\bf M. Basheleishvili} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		I. Vekua Institute of Applied Mathematics, 
		Tbilisi State University \\
		Tbilisi, Georgia} \\[0.5cm]
	{\bf RESULTS OBTAINED BY V. KUPRADZE IN THE TWO-DIMENSIONAL 
		ELASTICITY THEORY AND THEIR GENERALIZATIONS}
\end{center}

\bigskip

	Results obtained by V.Kupradze in two-dimensional problems 
of elastostatics for anisotropic bodies will be discussed. 
His approach based on the potential theory and the theory of singular 
(Fredholm) integral equations applied to the basic boundary value problems 
(BVPs) will be described in detail. 
For the basic BVPs (the first BVP -- the displacement vector is given 
on the boundary, the second BVP -- the stress vector is given on the boundary,
the mixed BVP -- the displacement vector is given on one part of the boundary 
and the stress vector is prescribed on the remainig part) 
V.~Kupradze created and justified a very efficient and simple numerical 
method which will be treated as well.

	The above mentioned results have been extended to various directions 
by many authors. In the talk these generalizations will be presented 
in detail.

\newpage 

\begin{center}
	{\bf R. Duduchava} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		A. Razmadze Mathematical Institute, 
		Georgian Academy of Sciences \\
		Tbilisi, Georgia} \\[0.5cm]
	{\bf ASYMPTOTICS FOR CRACK AND MIXED TYPE BOUNDARY VALUE 
           PROBLEMS -- THE WIENER--HOPF METHOD}
\end{center}

\bigskip


	Crack--type boundary value problems (BVPs) deal with 3D anisotropic 
elastic bodies which have a crack of the form of an arbitrary 2D smooth 
surface with a smooth boundary; on each side of the crack surface either 
the {\sc Dirichlet} type condition (i.e., the displacement vector) or the 
{\sc Neumann} type condition (i.e., the stress vector) is given. 

	We get a mixed type problem if, e.g., two 3D bodies are considered: 
one is bounded and the other is its exterior complement. 
On one part of the common smooth surface the bodies have the full contact, 
while on the complementary part there is a crack. 

	The method is implemented in four steps.

	I. Applying the potential method, the above--mentioned 
BVPs are reduced to boundary integral (pseudodifferential) 
equations on an open surface which coinsides, e.g., 
with the crack surface.  

	II. Extending the theorem on factorization, the full asymptotic 
expansion of a solution to PsDE in the vicinity of the boundary 
of the manifold ($L_p$--theory) is obtained including an explicit description 
of the exponents and of the logarithmic terms of the expansion. 

	III. Applying the established asymptotics of the densities 
(see Step II), a detailed spatial asymptotics of
the solution to BVP is derived. Explicit dependences between plane and 
spatial coefficients are found. 

	IV. The obtained information is used to establish dependences 
between stress and displacement fields on crack faces and crack prolongations,
which are extremely important for developing fracture criteria for
elastic materials with cracks.
 
	Most of results are shared with O. Chkadua, D. Natroshvili,
A. M. S\"{a}ndig, E. Shargorodsky and W. L. Wendland.

\newpage

\begin{center}
	{\bf I. Kiguradze} \\%[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		A. Razmadze Mathematical Institute,
		Georgian Academy of Sciences \\
		Tbilisi, Georgia} \\%[0.5cm]
	{\bf ON KNESER PROBLEM FOR HIGHER ORDER NONLINEAR ORDINARY 
		DIFFERENTIAL EQUATIONS}
\end{center}

\smallskip

	Consider the differential equation 
\setcounter{equation}{0}
%	(1)
\begin{equation}
	u^{(n)}=f(t,u,\dots,u^{(n-1)}), \tag{1}
\end{equation}
where $n\geq 2$, and $f:\,]0,+\infty[\,\times\,R^n\to R$ is a continuous 
function such that  
%	(2)
\begin{gather}
	f(t,0,\dots,0)=0,\;\;\;(-1)^nf(t,x_1,\dots,x_n)\geq 0\;\;
		\text{for}\;\;t>0, \notag  \\
\ \hskip+4cm	(-1)^{i-1} x_i\geq 0 \;\;(i=1,\dots,n). \tag{2}
\end{gather}
A non-negative solution $u:\,]0,+\infty[\,\to R_+$ of the equation (1) 
is called a {\it Kneser} solution if it satisfies the inequalities 
$(-1)^i u^{(i)}(t)\geq 0$ for $t>0$ $(i=0,\dots,n-1)$.

	Obviously, each Kneser solution $u$ has a finite or infinite right 
limit $u(0+)$. If $u(0+)=+\infty$, then the Kneser solution will be called 
{\it explosive}. 

\vskip+0.2cm
%	Theorem 1.
{\bf Theorem 1.\/} 
{\em Let, along with $(2)$, the condition 
\begin{gather*}
	|f(t,x_1,\dots,x_n)|\leq g(t)h(x_1,t|x_2|,\dots,t^{n-1}|x_n|) \\
\hskip+4cm 	\text{for}\;\;0<t<a, \;\;\;
	(-1)^{i-1} x_i\geq 0 \;\;\;(i=1,\dots,n) 
\end{gather*}
be fulfilled, where $a$ is a sufficiently small positive number, 
the functions $g:\,]0,a]\to R_+$ and $h:R_+^n\to R_+$ are continuous and 
$\int_0^a \tau^{n-1}g(\tau)\,d\tau<+\infty$.
Then for any $c_0\in R_+$ the equation $(1)$ has at least 
one Kneser solution satisfying the  initial condition $u(0+)=c_0$.
}
\vskip+0.2cm

%	Theorem 2.
{\bf Theorem 2.\/} 
{\em Let, along with $(2)$, the condition 
\begin{gather*}
	g_0(t)x_1^\lb\leq (-1)^n f(t,x_1,\dots,x_n) \leq 
		g(t)h(x_1,t|x_2|,\dots,t^{n-1}|x_n|) \\
\hskip+4cm 	\text{for}\;\;0<t<a, \;\;\;
	(-1)^{i-1} x_i\geq 0 \;\;\;(i=1,\dots,n) 
\end{gather*}
be fulfilled, where $a\in \,]0,+\infty[$, $\lb\in \,]1,+\infty[$, 
the functions $g_0:\,]0,a]\to R_+$,  \linebreak  
 $g:\,]0,a]\to R_+$ and 
$h:R_+^n\to R_+$ are continuous and 
$$	0<\int_0^t \tau^{n-1} g_0(\tau)d\tau \leq 
		\int_0^t \tau^{n-1} g(\tau)d\tau<+\infty \;\;
			\text{for}\;\;0<t\leq a. 		$$
Then the equation $(1)$ has at least one non-negative explosive 
Kneser solution. 
}
\vskip+0.2cm

	Global two-sided estimates of bounded and explosive 
Kneser solutions are established.

%	Acknowledgement
\section*{Acknowledgement}

	This work was supported by INTAS Grant 96-1060.

\newpage

\begin{center}
	{\bf D. Natroshvili} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		Georgian Technical University \\
		Tbilisi, Georgia} \\[0.5cm]
	{\bf DIRECT AND INVERSE FLUID-STRUCTURE INTERACTION PROBLEMS \\
		(GENERAL ANISOTROPIC CASE)}
\end{center}

\bigskip
   
	Three-dimensional mathematical problem of interaction between 
elastic and scalar oscillation fields are investigated in a general 
anisotropic case. 
   
	In the simplest physical situation, this interaction problem can be
described as follows. An acoustic wave propagates in an anisotropic 
homogeneous fluid domain of infinite extent where a bounded anisotropic 
elastic structure is immersed. The direct problem is to determine 
the scattered acoustic pressure in the fluid domain and the three-dimensional 
elastic field (the displacement field) in the anisotropic structure.

	The corresponding mathematical transmission problem is studied by 
the boundary integral equations method. The non-local approach based on the
variational formulation of the problem is considered as well 
(if the anisotropic elastic body in question is inhomogeneous).

	The special analysis is given for the wave scattering problems. 
The solvability of the non-homogeneous problem with physical transmission
data is proved for arbitrary oscillation frequencies. 

	These results are then applied to prove the uniqueness theorem 
in the inverse fluid-structure scattering problem. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5

\newpage 
\setcounter{page}{9}
\ 

\vskip+4cm
\thispagestyle{empty}

\begin{center}
{\size{20}{74pt}\selectfont S E C T I O N \;\;I}\\[2cm]
{\size{20}{74pt}\selectfont \bf DIFFERENTIAL EQUATIONS}
\end{center}

\newpage 

\setcounter{page}{11}
\begin{center}
	{\bf M. Ashordia} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		Sukhumi Branch of Tbilisi State University \\
		Tbilisi, Georgia} \\[0.5cm]
	{\bf ON A CRITERION OF CONVERGENCE OF DIFFERENCE SCHEMES 
		FOR LINEAR BOUNDARY VALUE PROBLEMS FOR SYSTEMS OF ORDINARY 
		DIFFERENTIAL EQUATIONS}
\end{center}

\bigskip

\setcounter{equation}{0}
	Let an integrable matrix-function $P: [a,b]\to \Bbb{R}^{n\times n}$, 
an integrable vector-function $q: [a,b]\to \Bbb{R}^{n\times n}$, 
a constant vector $c_0\in \Bbb{R}^n$ and a linear continuous operator $l$ 
acting from the space of all vector-functions $x: [a,b]\to \Bbb{R}^n$ 
of bounded variation with the norm 
$\|x\|_{{}_S}=\sup\{\|x(t)\|: \;t\in [a,b]\}$ into $\Bbb{R}^n$ be such that
the linear boundary value problem
%	(1),(2)
\begin{align}
	\frac{dx(t)}{dt} & =P(t)x(t)+q(t), \\
	l(x) & =c_0
\end{align}
has a unique solution $x^0$.

	Along with the problem (1), (2), consider the difference scheme
%	(1_m),(2_m)
\begin{gather}
	y(k)-y(k-1)= \notag \\
	=\frac{1}{m}\,\big(G_{1m}(k)y(k)+G_{2m}(k)y(k-1)+
		g_m(k)\big) \;\;\; (\,k\in \Bbb{N}_m\,),	\tag{1$_m$} \\
	L_m(y)=\gamma_m	\tag{2$_m$}
\end{gather}
$(m=2,3\ldots)$, where $\Bbb{N}_m=\{1,\dots,m\}$, 
$G_{jm}: \Bbb{N}_m\to \Bbb{R}^{n\times n}$ $(j=1,2)$, 
$g_m: \Bbb{N}_m\to \Bbb{R}^n$, $\gamma_m\in \Bbb{R}^n$, 
and $L_m$ is a linear continuous operator with respect to the norm 
$\|y\|_m=\max\{\|y(k)\|:\;k=0,\dots,m\}$. 

	Necessary and sufficient conditions are given for the sequence 
of boundary value problems $(1_m)$, $(2_m)$ $(m=2,3,\ldots)$ to have a unique
solution $y_m$ for any sufficiently large $m$ and 
$$	\lim_{m\to +\infty} \max\Big\{\Big\|y_m(k)-
		x^0\Big(a+\frac{b-a}{m}\,k\Big)\Big\|: \;\; 
			k=0,\dots,m\Big\}=0.		$$

%	Acknowledgement
\section*{Acknowledgement}

	This work was supported by INTAS Grant 96-1060.

\newpage

\begin{center}
	{\bf R. Bochorishvili} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		I. Vekua Institute of Applied Mathematics, 
		Tbilisi State University \\
		Tbilisi, Georgia} \\[0.5cm]
	{\bf ON A LOGARITHMIC METHOD WITH APPLICATION TO SYMMETRIC HYPERBOLIC 
		SYSTEMS OF NONLINEAR CONSERVATION LAWS}
\end{center}

\bigskip

	We present a new method for investigating time dependent PDEs. 
In ge\-ne\-ral features, this method can be formulated 
as the following 4 steps.

	1. Development of a semidiscrete scheme for the time dependent 
PDE under consideration.

	2. Selection of suitable norms.
	
	3. Derivation of apriori estimates in selected norms.

	4. Analysis on the basis of the obtained apriori estimates and 
conclusions for PDE under consideration.

	It should be emphasized that the logarithimic norm is a key 
in performing steps 2 and 3. That is why we call the method logarithmic. 
Note that it is impossible to select a norm and have the corresponding 
estimates in the case of arbitrarily constructed semidiscrete schemes. 
So in order to apply successfully the method, it is necessary to keep 
in mind possibilities offered by the logarithmic norm when performing step 1. 
Thus from this viewpoint the approach can be useful for constructing 
stable numerical schemes as well. 

	Capabilities of the developed approach is demonstrated by means 
of its application to symmetric hyperbolic systems 
of nonlinear conservation laws.  

\newpage 

\begin{center}
	{\bf A. Gagnidze} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		I. Javakhishvili Tbilisi State University \\ 
		Tbilisi, Georgia} \\[0.5cm]
	{\bf THEOREMS OF THE UNIQUENESS OF SOLUTIONS 
		OF BOUNDARY VALUE PROBLEMS FOR EVOLUTIONARY EQUATIONS} 
\end{center}

\bigskip

	A survey of the results on the uniqueness of solutions of initial
boundary value problems for parabolic equations and systems is given 
in the present report. The history of this problem takes its origin from the 
1930s and is reflected in the works of Tikhonov, Holmgren, 
Teklund and others. Therein theorems on the uniqueness of solutions 
of the Cauchy problem for the heat equation in classes of growing functions 
have been formulated for the first time. 
Subsequently, the interest to these problems increased after appearance 
of the well-known papers by Petrovski\v{\i}. 

	A large number of works, in particular for second order equations 
are devoted to investigation of this problem. 
Very interesting and important results have been obtained by such authors 
as Ole\v{\i}nik, Radkevich, Iosifyan, Kamynin, Khimchenko, Ladyzhenskaya, 
Smirnova, Landis, Zhi\-to\-mir\-ski\v{\i}, Ivasishen, Eidelman, Shishkov, 
Gagnidze, Solonnikov, Marinov, Kalashnikov, and others. 
For the last years, in Georgia, the results obtained 
in this direction by Gagnidze, Gugulashvili and Maisuradze are 
of great interest. 

\newpage

\begin{center}
	{\bf K. Gelashvili} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		I. Javakhishvili Tbilisi State University \\ 
		Tbilisi, Georgia} \\[0.5cm]
	{\bf ON INTEGRAL REPRESENTATION OF SOLUTION OF THE CAUCHY PROBLEM}
\end{center}				

\bigskip

\setcounter{equation}{0}
	For the Cauchy problem \(\frac{dx}{dt}={f_{t}}(x), x(t_0)= x_0,\) 
the problem of representation of solution as a field function 
is considered.

	Let \(V\!\subset \! R^2, M\) be a monoid with compatible algebraic 
and limiting structures (i.e., the binary operation $\ast$ is continuous),
\( \Sigma\) denote the set of all partitions of the form
\(\sigma=\{0=s_0\leq \xi_1\leq s_1\leq \cdots \leq \xi_n\leq s_n=1\} 
(s_0<\cdots<s_n)\) directed in a usual way.

\setcounter{defi}{0}
%	Definition 1.
\begin{defi} 
We say that \(g\in M\) is the integral of the function 
\(h:V \rightarrow M\) from \(a\) to $b$ $(a,b\in R)$ and write 
$g=^{(\ast)}\!\int_{a}^{b}h(\tau,d\tau),$ if for some $\sigma_0\in \Sigma$ 
the directedness 
\[	\big\{h(a+\xi_n(b-a),\Delta s_n(b-a))\ast\cdots\ast
		h(a+\xi_1(b-a),\Delta s_1(b-a))\big\}_{\sigma_0\leq 
			\sigma \in  \Sigma}		\] 
is defined correctly and converges to $g$.
\end{defi}		

%	Definition 2. 
\begin{defi} 
We say that the field $(t,x)\mapsto f_{t}(x)$ has Property $E$, 
if $(t,x)\mapsto f_{t}(x)$ maps $[a,b]\times R^r$ into $R^r$, 
$-\infty<a,b<\infty, \big\{f_{t}(x)\big\}_{t\in[a,b]}$ is a continuous family 
for every $x\in R^r$ and there exists $k\geq 0$ such that 
\[	\Vert f_{t}(x_1)-f_{t}(x_2)\Vert \leq k\Vert x_1-x_2\Vert,
		\forall t\in [a,b], \forall x_1,x_2\in R^r.		\]
\end{defi}

	$C_{Lip}(R^r)$ denotes the set of Lipschitz's mappings 
$f:R^r\rightarrow R^r$. $C_{Lip}(R^r)$ is provided with compatible limiting 
and monoid (created by the composition $\circ$ and the identical mapping 
$I:R^r\rightarrow R^r$) structures.

%	Theorem.
{\bf Theorem.\/}
{\em Let $(t,x)\mapsto f_{t}(x)$ have Property $E$. 
Then for every $(t_0,t,x)\in(a,b)^{2}\times R^r$ there exists 
$^{(\circ)}\!\int_{t_0}^{t}(I+d\tau f_\tau)$, 
and $\psi(t)=\big(^{(\circ)}\!\int_{t_0}^{t}(I+d\tau f_\tau)\big)(x)$
is a solution of the equation  \(\frac{dx}{dt}={f_{t}}(x)\)
with the initial condition $\psi (t_0)=x_0$. }

\newpage

\begin{center}
	{\bf J. Gvazava} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		A. Razmadze Mathematical Institute, 
		Georgian Academy of Sciences \\
		Tbilisi, Georgia} \\[0.5cm]
	{\bf ON BEHAVIOUR OF SOLUTIONS OF A NONLINEAR SINGULAR CAUCHY PROBLEM 
		NEAR THE BOUNDARY OF DOMAINS OF THEIR REGULARITY}
\end{center}

\bigskip

\setcounter{equation}{0}
	We consider the quasilinear mixed type equation 
%	(A)
\begin{equation}
	(shp)^2u_{tt}-2\big[(shp)(shq)ch(p+q)\big]u_{xt}+
		(shq)^2u_{xx}=0,		\tag{$A$}
\end{equation}
where $p$ and $q$ denote the first order derivatives of the solution
in $x$ and $t$, respectively. The class of hyperbolic solutions of $(A)$ 
is defined by the inequality $pq(p+q)\neq 0$. 
The characteristic roots $\xi_1=(e^p-1)(1-e^{-1})^{-1}$, 
$\xi_2=(e^{-p}-1)(e^q-1)^{-1}$ at the same time are riemannian invariants 
for $(A)$. In the half-plane $t\geq 0$, we consider the Cauchy problem 
with the data
%	(B)
\begin{equation}
	u\big|_{t=0}=\tau(x), \;\;\; u_t\big|_{t=0}=\nu(x), \;\; 
		x\in [a,b],	\tag{$B$}
\end{equation}
where the given functions $\tau,\,\nu\in C^2[a,b]$ guarantee the 
hyperbolity of $(A)$ on the finite or unbounded support of data. 
Characteristic curves of $(A)$ under the conditions $(B)$ are represented 
by the one-parametric families of straight lines 
$t-\xi_k(\alpha)(x-\alpha)=0$, $k=1,2$, $\alpha\in [a,b]$, 
$\xi_k(\alpha)=\xi_k[\tau'(\alpha),\nu(\alpha)]$. 
The set of the points of their intersection cover the domain of influence 
of the data $(B)$. The smoothness of initial perturbations doesn't guarantee 
not only the regularity of solutions in this domain, 
but their singlevaluedness as well. 

\vskip+0.2cm
{\bf Theorem.\/}
{\em If 
\vskip+0.2cm

	$1)$ \centerline{$\tau'(x)\nu(x)[\tau'(x)+\nu(x)]\neq 0$,}
\vskip+0.2cm

	$2)$ \centerline{$\xi_2^2(x)+\xi_1(a)[(b-x)\xi_2(x)]'>0, \;\;\; 
x\in [a,b]$,} 
\vskip+0.2cm

	$3)$ \centerline{$\xi_1^2(x)+\xi_2(b)[(x-a)\xi_1(x)]'>0, \;\;\; 
x\in [a,x_0)$} 	
\vskip+0.2cm

\noindent and changes the sign at the point $x=x_0$, then the boundary 
of the regularity domain of the solution of $(A)$, $(B)$ contains 
the characteristic $t-\xi_2(b)(x-b)=0$. 
Each point of the interval $x<x_0$ of this straight line 
is essentially singular. }

\newpage

\begin{center}
	{\bf T. Jangveladze} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		I. Vekua Institute of Applied Mathematics,
		Tbilisi State University\\
		Tbilisi, Georgia} \\[0.5cm]
	{\bf  ON A NONLINEAR INTEGRO-DIFFERENTIAL EQUATION}
\end{center}

\smallskip

\setcounter{equation}{0}
In the cylinder $\Omega\times(0,\infty)$ of the variables $x_1, x_2,\dots, x_n,t$,
where $\Omega\subset R^n$ is a bounded domain with the sufficiently smooth 
boundary 
$\partial \Omega$, we consider the initial boundary value problem
\begin{gather}
	\frac{\partial U}{\partial t}-\sum\limits^{n}_{i=1}
		\frac{\partial}{\partial x_i}\bigg[a\bigg(
		\int\limits_0^t\vert\nabla U\vert^2 d\tau\bigg)
		\frac{\partial U}{\partial x_i}\bigg]=0, \\
	U(x,t)=0, \;\; (x,t)\in\partial\Omega\times(0,\infty), \\
	U(x,0)=U_0(x), \;\; x\in\Omega.
\end{gather}
Here $a=a(s)$ and $U_0=U_0(x)$ are given functions, 
$\nabla=(\partial/\partial x_1,\dots, \partial/\partial x_n)$.

The equation (1) is a generalization of equations arising in mathematical
modelling of the process of penetrating of the electromagnetic field in a 
substance whose coefficient of electroconductivity depends on
the temerature.

The main purpose of our work is to study the asymptotic behaviour 
as $t\rightarrow \infty$ of the solution of the probllem (1)--(3). 
If $a(s)\ge a_0=const>0$
and $U_0\in L_2(\Omega)$, it is not difficult to obtain the a priori estimate
$$
	\Vert U(\cdot, t)\Vert_{L_2(\Omega)}\le e^{-ct}\Vert U_0\Vert_{L_2(\Omega)},
	c=const>0.
$$

The main results are the following theorems.
%\vskip+0.2cm

{\bf Theorem 1.} {\em 
If $a(s)\ge a_0=const>0$, $a'(s)\ge 0$, $a''(s)\le 0$,
then the solution of the problem $(1)$--$(3)$ admits the estimates
\begin{equation}
	\int\limits_0^\infty \int\limits_\Omega 
		\vert\nabla U\vert^2 dxdt\le C, \;\;\;\;
	\int\limits_0^\infty\bigg\vert\frac{d}{dt} \int\limits_\Omega
		\vert\nabla U\vert^2 dx \bigg\vert dt\le C.
\end{equation}}
%\vskip+0.2cm

From (4) we get $\Vert U(\cdot, t)\Vert_{W_2^1(\Omega)}\rightarrow 0$ as
$t\rightarrow\infty$.

For the one-dimensional $(n=1)$ analogue the equation (1) with the 
boundary conditions
\begin{equation}
	U(0, t)=0,\;\;\;\; U(1,t)=1,
\end{equation}
we have
%\vskip+0.2cm

{\bf Theorem 2.} {\em 
If $a(s)=(a+s)^p$, $0<p\le 1$, $U_0(0)=0$, $U_0(1)=1$,
$U_0\in W_2^2(0, 1)$, then the solution of the one-dimensional problem 
$(1)$, $(3)$, $(5)$ admits the estimates}
$$	\frac{\partial U}{\partial x}=1+O(t^{-1-p}), \;\;\;\;
	\frac{\partial U}{\partial t}=O(t^{-1}), \;\;\;\;t\ge 1.	$$

\newpage

\begin{center}
	{\bf G. Kharatishvili, T. Tadumadze, and N. Gorgodze} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		Institute of Cybernetics, Georgian Academy of Sciences \\
		Tbilisi, Georgia \\
		I. Vekua Institute of Applied Mathematics, 
		Tbilisi State University \\
		Tbilisi, Georgia} \\[0.5cm]
	{\bf CONTINUOUS DEPENDENCE AND DIFFERENTIABILITY WITH RESPECT 
		TO INITIAL DATA AND RIGHT-HAND SIDE SOLUTIONS 
		OF DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENT} 
\end{center}

\bigskip

	Non-linear differential equations with variable delay and 
quasi-linear differential equations of neutral type are considered, 
when at initial moment of time the value of initial function, 
generaly speaking, do not coincide with the initial value of the trajectory 
(discontinuity at initial moment). 
The theorems on continuity of the Cauchy problem solution with respect 
to initial data and right-hand side are proved. 
The perturbations of the initial data i.e. of the initial function and 
initial values (initial moment, initial value of trajectory) 
are small in the uniform and Euclidean norm, respectively. 
The pertrurbation of the right-hand side of equation is small 
in the integral sense. Representation formulas for the differential 
of solution are obtained, when pertrurbations are small in the Euclidean 
topology. If the effect of discontinuity at the initial moment influences 
upon the right-hand side of the equation, then, as distinct from the early 
obtained formula, representation formulas for the differential contain 
a new term.

	Finally we note that the results obtained in this work play an 
important role when investigating optimal control problems 
with deviating argument. 

\newpage

\begin{center}
	{\bf S. Kharibegashvili} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		A. Razmadze Mathematical Institute, 
		Georgian Academy of Sciences \\
		Tbilisi, Georgia} \\[0.5cm]
	{\bf ON THE SOLVABILITY OF A MULTIDIMENSIONAL BOUNDARY VALUE PROBLEM 
		FOR A HIGHER ORDER NON-STRICTLY HYPERBOLIC EQUATION }
\end{center}

\bigskip

\setcounter{equation}{0}
	In the space $R^{n+1}$ of the variables $t$ and $x=(x_1,\dots,x_n)$,  
let us consider the fourth order hyperbolic equation 
%	(1)
\begin{equation}
	\square^2u+L(\partial)u=F,
\end{equation}
where $\square\equiv \frac{\partial^2}{\partial t^2}-
\sum\limits_{i=1}^n \frac{\partial^2}{\partial x_i^2}$, $n\geq 2$, 
$L(\partial)$ is a linear third order partial differential operator 
with constant coefficients, 
$\partial\equiv (\frac{\partial}{\partial t},\frac{\partial}{\partial x_1},
\dots,\frac{\partial}{\partial x_n})$, $F$ is a given function and 
$u$ is an unknown function.

	Denote by $D: t>|x|$ the conic domain bounded by the characteristic 
cone $S_1:\,t=|x|$. Let $P_0(t_0,x_0)$ be an arbitrary point of $D$ and 
$S_2:\,|t-t_0|=-|x-x_0|$ be the light cone of the past with the vertex at
$P_0$. Denote by $D_0:\,|x|<t<t_0-|x-x_0|$ the domain bounded by the 
characteristic cones $S_1$ and $S_2$. Let $S=\partial D_0\cap S_1$.

	Consider the following boundary value problem for the equation (1):
find in the domain $D_0$ a solution $u(t,x)$ of the equation (1) 
satisfying the bo\-un\-da\-ry conditions
%	(2)
\begin{equation}
	u\big|_{S_0}=0, \;\;\; \frac{\partial u}{\partial n}\Big|_{S_0}=0,
\end{equation}
where $\frac{\partial}{\partial n}$ is the derivative along the outer 
normal to $S_0$.

\vskip+0.2cm
{\bf Theorem.\/} 
{\em Let $L(\xi)=l(\xi)(\xi_0^2-\xi_1^2-\cdots -\xi_n^2)+L_0(\xi)$, 
$\xi=(\xi_0,\xi_1,\dots,\xi_n)\in R^{n+1}$, where $l(\xi)$ is an arbitrary
linear form and $L_0(\xi)$ is an arbitrary polynomial of second order. 
Then for any $F\in \overset{\circ}{W}{}_2^2(D_0)$ there exists a unique 
solution of the problem $(1)$, $(2)$ in the space $W_2^4(D_0)$. }

\newpage

\begin{center}
	{\bf T. Kiguradze} \\%[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		I. Javakhishvili Tbilisi State University \\
		Tbilisi, Georgia} \\%[0.5cm]
	{\bf ON BOUNDED AND TIME-PERIODIC SOLUTIONS OF NONLINEAR 
		WAVE EQUATIONS}
\end{center}

%\bigskip

\setcounter{equation}{0}
	In the strip $D_a = R \times [0,a]$, the nonlinear wave equation
%	(1)
\begin{equation}
	u_{tt} - u_{xx} + g_1(u) u_t  + g(t,x,u) = 0
\end{equation}
is considered. It is assumed that $g_1:R \to R$ and $g: D_a \times R \to R$
are continuously differentiable functions and for any $\rho > 0$
$$	\sup \big\{|g(t,x,z)|: \;\; (t,x)\in \cd_a, \;\; 
		|z| \leq \rho \big\}<+\infty.		$$
By a solution of the equation (1) we understand a classical solution.

	For the equation (1) the following problems on bounded and 
time-periodic solutions
% 	(2),(3)
\begin{gather}
\ \hskip-0.7cm	\sup_{(t,x) \in \cd_a} \big\{|u_t(t,x)|+|u_x(t,x)|\big\}<+\infty, 
		\;\;\; u(t,0) = u(t,a) = 0  \for  t \in \br, \\
\ \hskip-0.7cm	u(t+\om,x)=u(t,x), \;\;\; u(t,0) = u(t,a) = 0  \for  (t,x)\in \cd_a
\end{gather}
are studied and sufficient conditions of their solvability are established.
In par\-ti\-cu\-lar, the following theorems are true.

	By $G_1$ denote the function $G_1(z)=\int_0^z g_1(\xi)\,d\xi$.

\setcounter{th}{0}
%	Theorem 1
\begin{th}
Let there exist positive constants $\delta$, $\eta$ and 
$\gamma \in (0,1)$ such that
\begin{gather*}
	|g_1(z)| \geq \delta  \for  z \in \br, \\
	|4g(t,x,z) - g_1(z) G_1(z) | \leq
        	\frac{\gamma \delta^2}{1-\exp(- 2 a)} |z| + \eta
		           \for (t,x,z)\in \cd_a\times \br.
\end{gather*}
Then the problem $(1),(2)$ is solvable. Moreover,
if in addition $g$ is $\om$-periodic in the first argument, 
then the problem $(1),(3)$ is also solvable.
\end{th}

%	Theorem 2
\begin{th}
Let $g(t+\om,x,z) \equiv g(t,x,z)$, $\inf \{ |g_1(z)|: z \in \br \} > 0$ and
there exist a continuous function $ g_0 : \br \to \br$  such that
\begin{gather*}
	\liminf_{|z| \to +\infty} g_0(z) \sgn(z)  > 0, \\
	\lim_{|z| \to +\infty} \frac{ g(t,x,z) - g_0(z) }{ h(z) } = 0 \;\;
		\mbox{uniformly with respect to} \;\;  (t,x) \in \cd_a,
\end{gather*}
where $h(z) = \min \{ |g_0(z)|, |g_0(z)g_1(z)G_1(z)|^{\frac{1}{2}} \}$.
Then the problem $(1),(2)$ has at least one solution.
\end{th}

%	Acknowledgement
\section*{Acknowledgement}

	This work was supported by INTAS Grant 96-1060.

\newpage

\begin{center}
	{\bf L. Kokilashvili} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		I. Javakhishvili Tbilisi State University \\ 
		Tbilisi, Georgia} \\[0.5cm]
	{\bf ON A TWO-POINT NON-LINEAR BOUNDARY VALUE PROBLEM}
\end{center}

\bigskip

\setcounter{equation}{0}
	We consider the following two-point boundary value problem 
%	(1),(2)
\begin{gather}
	u^{(n)}=\sum_{k=1}^{n-1} p_k(t)u^{(k)}+
		f(t,u,u^{^{\prime}},\dots,u^{(n-1)}), \\
\begin{gathered}
	u^{(i)}(a)=\varphi _{1i}(u(a),u^{^{\prime}}(a),\dots,
		u^{(n-1)}(a)) \quad (i=0,\dots,n_0-1), \\ 
	u^{(j)}(b)=\varphi _{1j}(u(a),u^{^{\prime}}(a),\dots,
		u^{(n-1)}(a)) \quad (j=0,\dots,n-n_0-1), 
\end{gathered}
\end{gather}
where $n\geq 2$, $-\infty<a<b<+\infty$, $n_0$ is the integer part 
of $\frac{n}{2}$, each of the functions $p_k: [a,b]\rightarrow \bold{R}$ 
for $k\in \{1,\dots,n-1\} $ is continuously differentiable up to order $k-1$ 
inclusively, the functions $f:[a,b]\times \bold{R}^n\rightarrow \bold{R}$, 
$\varphi_{1i}: \bold{R}^n\rightarrow \bold{R}$ $(i=0,\dots,n_0-1)$
and $\varphi _{2j}: \bold{R}^n\rightarrow \bold{R}$ $(j=0,\dots,n-n_0-1)$ 
are continuous.

	Moreover, let there exist continuous functions 
$\alpha_0: [a,b]\rightarrow \bold{R}$, 
$\alpha_i: [a,b]\rightarrow \bold{R}_{+}$ $(i=1,\dots,n_0-1)$, 
$\alpha: [a,b]\rightarrow \bold{R}_{+}$ and 
$h_0: [a,b]\times \bold{R}_{+}^{n_0}\rightarrow \bold{R}_{+}$, 
and constants $c_i\geq 0$ and $\lambda_i\in [0,1]$, $i\in \{1,2\}$, 
such that the inequalities
\begin{gather*}
	(-1)^{n-n_0-1}f(t,x_0,x_1,\dots,x_{n-1})\sgn x_0\geq 
		-\sum_{i=0}^{n_0-1} \alpha _i(t)|x_i|-\alpha(t), \\ 
	\big|f(t,x_0,x_1,\dots,x_{n-1})\big|\leq 
		h\big(t,|x_0|,|x_1|,\dots,|x_{n_0-1}|\big) 
\end{gather*}
hold on $[a,b]\times \bold{R}^n$, and the conditions
\begin{gather*}
	\big|\varphi_1(u(a),u^{^{\prime }}(a),\dots,u^{(n-1)}(a))\big|\leq 
		c_1\Big(1+\sum_{k=n_0}^{n-1} |u^{(k)}(a)|\Big)^{-\lambda _1} \\
\ \hskip+6cm (i=0,\dots,n_0-1), \\ 
	\big|\varphi_2(u(b),u^{^{\prime }}(b),\dots,u^{(n-1)}(b))\big|\leq 
		c_2\Big(1+\sum_{k=n-n_0}^{n-1} |u^{(k)}(b)|\Big)^{-\lambda 2} \\
\ \hskip+6cm (j=0,\dots,n-n_0-1) 
\end{gather*}
are fulfilled on $\bold{R}^n$.

	It is established that under some restrictions on the functions 
$\alpha$, $h$ and $p_k$ $(k=1,\dots,n-1)$ the problem (1), (2) is solvable. 
The problem of unique solvability of (1), (2) is also investigated.

\newpage

\begin{center}
	{\bf V. Kokilashvili} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		A. Razmadze Mathematical Institute, 
		Georgian Academy of Sciences \\
		Tbilisi, Georgia} \\[0.5cm]
	{\bf SOME INTEGRAL TRANSFORMS IN WEIGHTED SPACES}
\end{center}

\bigskip
	
In the report some integral transforms are considered arising in airfoil
equations, two-dimensional scattering of accustic waves by thin obatacles and
two-dimensional viscous fluid flow. 
The main goal of the report is to present author's recent results concerning
mapping properties of these transforms in weighted Banach function spaces. 
Multidimensional analogs of the above mentioned problems will be 
discussed as well.

\newpage

\begin{center}
	{\bf R. Koplatadze} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		A. Razmadze Mathematical Institute, 
			Georgian Academy of Sciences \\
		Tbilisi, Georgia} \\[0.3cm]
	{\bf AN INTEGRAL ANALOGUE OF STURM'S COMPARISON THEOREM}
\end{center}

\smallskip

\setcounter{equation}{0}
Consider the linear ordinary differential equations
%	(1)-(2)
\begin{align}
	u''+p(t)u & =0, \\
	v''+q(t)v & =0,
\end{align}
where
%	(3)
\begin{equation}
	p,q\in L([a,b];R_+).
\end{equation}

The following comparison theorem is valid
%\vskip+0.1cm

{\bf Theorem 1.} {\em 
Let the condition $(3)$ be fulfilled, $v$ be a solution of $(2)$
satisfying $v(t_1)=v'(t_0)=v(t_1)=0$, $v(t)\neq 0$ for 
$t\in \,]t_1,t_2[$\, $(t_1<t_0<t_2)$. Let, moreover, the condition
$$	\int_{t_0}^t[p(s)-q(s)]\,ds\,(t-t_0)\geq 0 \quad 
		\forall t\in [t_1,t_2]		$$
be fulfilled. Then for any nontrivial solution $u$ of $(1)$ either
$	u(t_1)=u'(t_0)=u(t_2)=0		$
or there is $t_*\in ]t_1,t_2[$ such that $u(t_*)=0$.
Moreover, if 
$$	\int_{t_1}^{t_2}[p(s)-q(s)]\,ds>0, 	$$
then the second condition necessarily holds.}
%\vskip+0.1cm

Theorem 1 generalizes Sturm's well-known comparison theorem under 
the conditions (3). It enables one to obtain new sufficient conditions 
for the oscillation of the equation (1), 
as well as for the solvability of certain boundary value problems.
%\vskip+0.1cm

{\bf Theorem 2.} {\em Let $p\in L([a,b];R_{+})$ and
%	(4)
\begin{equation}
	\int_{\frac{a+b}{2}}^t \Big[p(s)-\frac{\pi^2}{(b-a)^2}\Big]\,ds\,
		\Big(t-\frac{a+b}{2}\Big)\geq 0 \;\;\;
		\forall t\in [a,b].
\end{equation}
Then any solution of $(1)$ has at least one zero on $[a,b]$.}
%\vskip+0.1cm

%{\bf Theorem 3.} {\em 
% Let $p\in L_{loc}(R_+;R)$, a sequence $[a_k,b_k]_{k=1}^{+\infty}$ 
%of segments exists such that $a_k<b_k\leq a_{k+1}$, 
%$p:[a_k,b_k]\to R_+$ $(k=1,2,\dots)$, $\lim\limits_{k\to+\infty}=+\infty$ and
%$$	\int_{\frac{a_k+b_k}{2}}^t \Big[p(s)-\frac{\pi^2}{(b_k-a_k)^2}\Big]\,ds\,
%		\Big(t-\frac{a_k+b_k}{2}\Big)\geq 0 \;\;
%		\forall t\in [a_k,b_k]\;\;(k=1,2,\dots).	$$
%Then the equation $(1)$ is oscillatory.}
%\vskip+0.1cm

Consider the following boundary value problem for the equation (1).
%	(5)
\begin{equation}
	u(a)=u(b)=0.	
\end{equation}

{\bf Theorem 3.} {\em
 Let $p\in L([a,b];R_+)$, the condition $(4)$ be fulfilled and
$p(t)\leq \frac{4\pi^2}{(b-a)^2}$, $t\in [a,b],$
$	\int_a^b \big[\frac{4\pi^2}{(b-a)^2}-p(t)\big]dt>0,\;\;
		\int_a^b\big[p(t)-\frac{\pi^2}{(b-a)^2}\big]dt>0.	$ 
Then the problem $(1)$, $(5)$ has only the zero solution.}

\newpage

\begin{center}
	{\bf G. Kvinikadze} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		A. Razmadze Mathematical Institute, 
			Georgian Academy of Sciences \\
		Tbilisi, Georgia} \\[0.5cm]
	{\bf ON PROPER SOLUTIONS OF LINEAR ADVANCED DIFFERENTIAL EQUATIONS}
\end{center}

\bigskip

\setcounter{equation}{0}
We consider the differential equation
%	(1)
\begin{equation}
	u^{(n)}(t)=p(t)u(\tau(t)),
\end{equation}
where $p\in L_{loc}(R_+;R_+)$, $\tau(t)\geq t$ for $t\in R_+$.
\vskip+0.1cm

{\bf Definition.} A solution $u$ of (1) is said to be proper if it is
defined in a neighbourhood of $+\infty$. 
\vskip+0.1cm

Below sufficient conditions are given for (1) to have a proper
solution $u:[t_0,+\infty[\to R$ satisfying 
%	(2)
\begin{equation}
	u^{(i)}(t)>0 \;\;\;\text{for}\;\;\;t\geq t_0\;\;\;
		(i=0,\dots,n-1).
\end{equation}
\vskip+0.1cm

{\bf Theorem 1.} 
{\em Let $q\in ]0,+\infty[$,
\begin{gather*}
	\limsup_{t\to+\infty}
		 (\tau(t)-t)=\Delta<+\infty, \;\;\;
	\inf\big\{\lambda^{-n}e^{\lambda}:\lambda\in ]0,+\infty[\big\}\leq q.
\end{gather*}
Let, moreover,
$$	\limsup_{t\to +\infty} e^{-\lambda_0t}\int_0^t e^{\lambda_0s}p(s)\,ds<
		\frac{1}{q\lambda_0},	$$
where $\lambda_0$ is a root of $\lambda^{-n}e^{\lambda}=q$.
Then $(1)$ has a proper solution satisfying~$(2)$.}
\vskip+0.1cm

{\bf Theorem 2.} {\em Let $q\in ]0,+\infty[$,
\begin{gather*}
	 \limsup_{t\to+\infty}
		 \frac{\tau(t)}{t}=\alpha<+\infty, \quad
	\inf\bigg\{\frac{\alpha^\lambda}{\prod\limits_{i=0}^{n-1}(\lambda-i)}:
		\lambda\in ]n-1,+\infty[\bigg\}\leq q.
\end{gather*}
Let, moreover,
$$	\limsup_{t\to +\infty} t^{n-\lambda_0-1}\int_0^t s^{\lambda_0}p(s)\,ds<
		\frac{1}{q(\lambda_0+1-n)},	$$
where $\lambda_0$ is a root of 
$\alpha^\lambda=q\prod\limits_{i=0}^{n-1}(\lambda-i)$.
Then $(1)$ has a proper solution satisfying $(2)$.}
\vskip+0.1cm

{\em Remark.}
 If in the hypotheses of Theorems 1 and 2 the additional condition 
$\int\limits^{+\infty}t^{n-1}p(t)\,dt=+\infty$
is fulfilled, then the solution $u$ satisfies the condition
$\lim\limits_{t\to+\infty} u^{(n-1)}(t)=+\infty$.

\newpage

\begin{center}
	{\bf M. Mania and A. Novikov} \\%[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		A. Razmadze Mathematical Institute,
		Georgian Academy of Sciences \\
		Tbilisi, Georgia \\
		The University of Newcastle \\
		Australia} \\%[0.5cm]
	{\bf A NONLINEAR PARABOLIC EQUATION FOR THE MAXIMAL HEDGING 
		PROBABILITY} 
\end{center}

\smallskip

\setcounter{equation}{0}
	We consider a security market model consisting of a stock and a bond, 
where the stock price process $S=(S_t,t\in[0,T])$ is determined by SDE
$$	dS_t=\mu(t,S_t)dt+\sigma(t,S_t)dW_t, \;\;\; S_0>0, \;\;\; 
		t\in[0,T],		$$
and for simplicity we assume that the bond price is constant.
Here $W$ is a standard Wiener process defined on a 
probability space $(\Omega, F,P)$.

	Suppose that: the function $\mu:[0,T]\times R\to R$ is measurable,
$\sigma:[0,T]\times R\to R$ is continuous and there exist constants 
$\lambda>0$, $C>0$ such that $|\mu(t,x)|\le Cx$, 
$\lambda x\le |\sigma(t,x)|\le Cx$, $\forall t\le T, x\in R_+$.

	Consider an option with the payoff $f(S_T)$, 
where $f=(f(x),x\in R_+)$ is a nonnegative measurable function.
Denote by $\Pi (t,x,f)$ a collection of self-financing strategies 
(hedges) such that $X^\pi_T=x+\int_t^T\pi_udS_u\ge f_T$, a.s..
The number $C_t(f)=\inf \{x:\Pi (T,x,f)\neq \emptyset \}$ is called 
the fair price of the option at the time $t$.
Under the present assumption there exists a unique equivalent 
(to $P$) martingale measure $P^*$ for the stock price process, 
the market model is complete and there is a minimal hedging strategy 
$\pi^*$ such that $X^{\pi^*}_T=f(S_T)$ a.s.. Besides,
$$	C_t(f)=C(t,S_t), \;\;\; \text{where} \;\;\; 
		C(t,y)=E^*(f(S_T)|S_t=y).		$$

	If the initial capital is less than the fair price $C(0,S_0)$ 
of the option, then it is impossible to hedge the financial obligation 
$f(S_T)$ with probability 1 and the problem is to calculate the maximal 
probability of hedging and to construct the corresponding hedging strategy. 
This problem in different settings and cases was studied by Karatzas, 
F\"ollmer, Novikov. We consider the maximization problem
$P^\pi(x)=P(X^\pi_T=f(S_T)|X^\pi_0=x)\to\max_{\pi\in\Pi}$.

	Let $P(x):=\sup_{\pi\in\Pi}P^\pi(x)$ and introduce the value function 
$$	\tilde{V}(t,x,y):=\esssup_{\pi\in\Pi(t)}P\{X_T^\pi=f(S_T)|
		X^\pi_t=x, S_t=y\}.		$$

	It is evident that $P(x)=\tilde{V}(0,x,S_0)$.

\vskip+0.2cm
%	Theorem.
{\bf Theorem.\/} 
{\em $\tilde{V}(t,x,y)$ is the unique solution of the equation
$$	2\sigma^2(t)V_tV_{xx}+2\mu(t)\sigma^2(t)(V_yV_{xx}-V_xV_{xy})+
		\sigma^4(t)V_{yy}V_{xx}-\sigma^2(t)V_{xy}=\mu^2(t)V_x 	  $$
subject to the boundary conditions: $($along with the inequality $V_{xx}<0)$
$$	V(T,x,y)=I(x=f(y)), \;\;\; V(t,0,y)=P(f(S_T)=0), \;\;\; 
		V(t,C(t,y),y)=1. 		$$
Moreover, the optimal hedge is given by 
$\pi_t^* =-\frac{\mu(t)V_x+\sigma^2(t)V_{xy}}{\sigma^2(t)V_{xx}}$. }

\newpage


\begin{center}
	{\bf S. Mukhigulashvili} \\%[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		A. Razmadze Mathematical Institute, 
		Georgian Academy of Sciences \\
		Tbilisi, Georgia} \\[0.1cm]
	{\bf ON SOLVABILITY OF TWO-POINT BOUNDARY VALUE PROBLEMS FOR SECOND 
                ORDER NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS}
\end{center}

\smallskip

\setcounter{equation}{0}         
	For the functional differential equation
%	(1)
\begin{equation}
	u''(t)=p(t)u(\tau(t))+q(u)(t)     \tag{$1$}
\end{equation}
we consider the following boundary value problems
%	(2_1)-(2_2)
\begin{gather}
	u(a)=0,\;\;\;u(b)=0,    \tag{$2_1$}\\
 	u(a)=0,\;\;\;u'(b)=0.   \tag{$2_2$}
\end{gather}

	We imply that $p\in L([a,b])$, $\tau:[a,b]\to[a,b]$ is measurable, 
$q:C([a,b])\to L([a,b])$ is a continuous operator and has a summable majorant 
on each bounded ball from $C([a,b])$. Besides we imply that for any 
$u\in C([a,b])$ and almost every $t\in[a,b]$ the inequality 
%	(3)
\begin{equation}
	q(u)(t)\sign u(t)\geq -\delta(t) \tag{3}
\end{equation}
is satisfied, where $\delta:[a,b]\to [0,+\infty[$ is a summable function.

\vskip+0.2cm
%	Theorem 1.
{\bf Theorem 1.\/}
{\em Let $i\in\{1,2\}$, $p(t)\geq 0$ for $a\leq t\leq b$ and 
%	(4_i)
\begin{equation}
	\int_a^bp(s)|\tau(s)-s|ds<4.  \tag{$4_i$}
\end{equation}
Then the problem $(1)$, $(2_i)$ has at least one solution. }
\vskip+0.2cm

%	Theorem 2. 
{\bf Theorem 2.\/} 
{\em Let $i\in\{1,2\}$, $p(t)\geq 0$ for $a\leq t\leq b$ and 
%	(5_i)
\begin{equation}
	\int_a^bp(s)ds<\frac{16}{i(b-a)}. \tag{$5_i$}
\end{equation}
Then the problem $(1)$, $(2_i)$ has at least one solution.}
\vskip+0.2cm

%       Theorem 3.
{\bf Theorem 3.\/} 	
{\em Let $i\in\{1,2\}$ and there exist $\beta\in[0,1]$ such that
%	(6_i)
\begin{equation}
	\int_a^b|p(s)|(\tau(s)-a)^{\beta}(b-\tau(s))^{\beta(2-i)}ds<
		\frac{[4^{1-\beta}(b-a)^{\beta}]^{2-i}}{(b-a)^{1-\beta}}
			\,. \tag{$6_i$}
\end{equation}
Then the problem $(1)$, $(2_i)$ has at least one solution.}
\vskip+0.2cm

	The condition $(5_1)$, ($(6_1)$) is optimal, 
because for any $\varepsilon>0$, an operator $q$ can be found such that if
$$	\int_a^bp(s)ds=\frac{16}{b-a}+\varepsilon\;\;\;
		\bigg(\,\int_a^b|p(s)|ds=\frac{4}{b-a}+\varepsilon\,\bigg), $$
then the problem (1), ($2_1$) has no solution. 

\newpage

\begin{center}
	{\bf E. Obolashvili} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		A. Razmadze Mathematical Institute, 
		Georgian Academy of Sciences \\
		Tbilisi, Georgia} \\[0.5cm]
	{\bf GENERALIZED BELTRAMI EQUATION IN CLIFFORD ANALYSIS} 
\end{center}

\bigskip

\setcounter{equation}{0}
	Consider the equation
%	(1)
\begin{equation}
	\overline{\partial}u+q_1\partial u+
		q_2\overline{\partial}\overline{u}=0, 
\end{equation}
where $\overline{\partial}$ is a modification of the Dirac operator, 
$\partial$ is its conjugate,
$$	u(x)=\sum_A u_{{}_A}(x)l_{{}_A}, \;\;\; x=\sum_0^n x_kl_k, \;\;\; 
		q_k(x)=\sum_0^n q_{kj}(x)l_j, \;\; k=1,2,		$$
$l_{{}_A}$ is the basis of the Clifford spaces $R_{(n)}$, $R_{(n,n-1)}$ 
$(n\geq 1)$, $A=(\alpha_1\cdots\alpha_k)$, 
$1\leq \alpha_1<\alpha_2<\cdots<\alpha_k\leq n$, 
$u_{{}_A}(x)$ and $q_{kj}(x)$ are real functions. 
In the space of one complex variable $R_{(1)}$, this equation, known 
as a generalization of the classical Beltrami equation, was considered only 
as elliptic with the condition $|q_1|+|q_2|<1$. 
Using the properties of these spaces, we can prove

\vskip+0.2cm
%	Theorem 1.
{\bf Theorem.\/} 
{\bf 1.} {\em In the space $R_{(n)}$ the equation $(1)$ will be elliptic if 
$$	a) \;\; |q_1|+|q_2|<1 \;\;\; \text{or} \;\;\; 
		b) \;\; \big|\,|q_1|-|q_2|\,\big|>1,		$$
will be hyperbolic if
$$	c) \;\; \big|\,|q_1|-|q_2|\,\big|<1 \;\;\; \text{and} \;\;\; 
		|q_1|+|q_2|>1,		$$
will be parabolic if 
$$	d) \;\; |q_1|+|q_2|=1 \;\;\; \text{or} \;\;\; 
		\big|\,|q_1|-|q_2|\,\big|=1.		$$
Note that the case $b)$ can be reduced to the case $a)$. }

	{\bf 2.} {\em In the space $R_{(n,n-1)}$ the equation $(1)$ will be 
hyperbolic with conditions $a)$ and $b)$, will be elliptic with $c)$ and
will be parabolic with $d)$. }
\vskip+0.2cm

	We can also prove that the multidimensional Beltrami equation 
$$	\overline{\partial}u+q\partial u=0, \;\;\; |q|\neq 1		$$
with a constant vector $q$ can be reduced to the equation 
$\overline{\partial}u=0$. 

	That is why in the space $R_{(3,2)}$ it will be related to 
the Dirac equations of relativity quantum mechanics and the Maxwell equations 
of electromagnetic fields, and in the space $R_{(2)}$ it will be related 
to the Moisil-Theodorescu equations. Corresponding initial and boundary value
problems can be solved. 

\newpage

\begin{center}
	{\bf N. Partsvania} \\%[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		A. Razmadze Mathematical Institute, 
		Georgian Academy of Sciences \\
		Tbilisi, Georgia} \\%[0.3cm]
	{\bf OSCILLATION AND NONOSCILLATION CRITERIA FOR TWO-DIMENSIONAL 
		LINEAR DIFFERENTIAL SYSTEMS WITH DEVIATING ARGUMENTS}
\end{center}

\smallskip

\setcounter{equation}{0}
        Consider the system
%	(1)
\begin{equation}
\begin{aligned}
	u_1'(t) & =p(t)u_2(\sigma(t)), \\
 	u_2'(t) & =-q(t)u_1(\tau(t)),
\end{aligned}
\end{equation}
where $p,q\in L_{loc}(\Bbb{R}_{+};\Bbb{R}_{+})$ and 
$\tau,\sigma\in C(\Bbb{R}_{+};\Bbb{R}_{+})$ are such that
\begin{gather*}
	\sigma(t)\leq t \;\;\; \text{for} \;\;\; t\in \Bbb{R}_{+}, \\
	\lim_{t\to +\infty} \tau(t)=+\infty, \;\;\; 
		\lim_{t\to +\infty} \sigma(t)=+\infty, \\
	\int_0^{+\infty} p(t)\,dt=+\infty.
\end{gather*}

	A proper solution $(u_1,u_2)$ of the system (1) is said to be 
{\em oscillatory} if both $u_1$ and $u_2$ have sequences of zeros tending to
infinity. If there exists $t_0\in \Bbb{R}_{+}$ such that $u_1(t)u_2(t)\neq 0$ 
for $t\geq t_0$, then $(u_1,u_2)$ is said to be {\em nonoscillatory}.

\vskip+0.2cm
%	Theorem 1.
{\bf Theorem 1.\/} 
{\em Let for some $\lambda\in \,]0,1[$ the inequality
$$	\lmp_{t\to +\infty} \, h^{-\lambda}(t)
		 \int_0^t p(s) \int_s^{+\infty} q(\xi)H(\xi)
			h^\lambda(\xi)\,d\xi\,ds> 
		\frac{(\lambda-2)^2}{4\lambda(1-\lambda)}	$$
be fulfilled, where
%	(2)
\begin{gather}
	h(t)=\int_0^t p(s)\,ds, \\
	H(t)=\begin{cases} 
		1 & \text{for} \;\;\; \tau(t)\geq t, \\
		\displaystyle \frac{h(\tau(t))}{h(t)} & \text{for} \;\;\; 
			\tau(t)<t.
	     \end{cases} \notag 
\end{gather}
Then every proper solution of the system $(1)$ is oscillatory. }
\vskip+0.2cm

%	Theorem 2.
{\bf Theorem 2.\/} 
{\em Let for some $\lambda\in \,]0,1[$\,,
$$	\int^{+\infty} q(s)h^\lambda(\tau(s))\,ds<+\infty	$$
and 
$$	\lmp_{t\to +\infty} \, h^{-\lambda}(t)
		 \int_0^t p(s) \int_{\sigma(s)}^{+\infty} q(\xi)
			h^\lambda(\tau(\xi))\,d\xi\,ds<1,	$$
where the function $h$ is defined by $(2)$. 
Then the system $(1)$ has a proper non\-oscil\-la\-to\-ry solution. }


\newpage

\begin{center}
	{\bf Z. Sokhadze} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		A. Tsereteli Kutaisi State University\\
		Kutaisi, Georgia} \\[0.5cm]
	{\bf ON THE SOLVABILITY OF THE WEIGHTED INITIAL VALUE PROBLEM 
		FOR EVOLUTION SINGULAR FUNCTIONAL DIFFERENTIAL EQUATIONS 
		OF HIGHER ORDERS}
\end{center}

\bigskip

\setcounter{equation}{0}
	In the present paper on the basis of the results obtained 
in [1, 2] optimal, in a certain sence, sufficient conditions for 
the solvability of the weighted initial value problem
%	(1),(2)
\begin{gather}
	u^{(n)}(t)=f(u)(t), \\
	\lim_{t\to a} \frac{u^{(k)}(t)}{h^{(k)}(t)}=0 \;\;\; (k=0,\dots,n-1)
\end{gather}
are established, where $f\in C^{n-1}([a,b];\Bbb{R}^m)\to L_{loc}(]a,b];\Bbb{R}^m)$ 
is a continuous Volterra operator and $h: [a,b]\to [0,+\infty[$ 
is an $(n-1)$-times continuously differentiable function such that
%	(3)
\begin{equation}
	h^{(k)}(a)=0 \;\; (k=0,\dots,n-2), \;\;\; 
		h^{(n-1)}(t)>0 \;\; \text{for} \;\; a<t\leq b.
\end{equation}

	A particular case of (1) is the vector differential equation 
with delay
%	(4)
\begin{gather}
\ \hskip-0.5cm	u^{(n)}(t)= \notag \\
\ \hskip-0.5cm	=f_0\big(t,u(\tau_{10}(t)),...,u^{(n-1)}(\tau_{1\,n-1}(t)),...,
		u(\tau_{l0}(t)),...,u^{(n-1)}(\tau_{l\,n-1}(t))\big),
\end{gather}
where $f_0: \,]a,b]\times \Bbb{R}^{lmn}\to \Bbb{R}^m$ satisfies the 
local Carath\'{e}odory conditions, and $\tau_{ik}: [a,b]\to [a,b]$ 
are measurable functions such that
$$	\tau_{ik}(t)\leq t \;\; \text{for} \;\; a\leq t\leq b \;\;\; 
		(i=1,\dots,l; \;\; k=0,\dots,n-1).		$$

\vskip+0.8cm
\centerline{\sc References}
\vskip+0.4cm

1. I. Kiguradze and Z. Sokhadze, On the Cauchy problem for 
evolution singular functional differential equations. (Russian)
{\em Differentsial'nye Uravneniya} {\bf 33}(1997), No. 1, 48--59.

2. I. Kiguradze and Z. Sokhadze, On the global solvability 
of the Cauchy problem for singular functional differential equations.
{\em Georgian Math. J.} {\bf 4}(1997), No. 4, 355-372.

\newpage

\begin{center}
	{\bf T. Surguladze} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		Kutaisi Technical University \\
		Kutaisi, Georgia} \\[0.5cm]
	{\bf THE LIMITING AMPLITUDE PRINCIPLE FOR HYPERBOLIC 
		DIFFERENTIAL EQUATIONS WITH PERIODIC COEFFICIENTS}
\end{center}

\bigskip

\setcounter{equation}{0}
	We study the asymptotic behavior as $t\to \infty$ and 
$x\in [a,b]$ for solutions of the following problem
%	(1)
\begin{equation}
\begin{gathered}
	u_{tt}-u_{xx}+q(x)u=f(x)e^{-i\omega t}, \;\;\; 
		x\in R^1, \;\;\; t>0, \\ 
	u\big|_{t=0}=0, \;\;\; u_t\big|_{t=0}=0, \;\; x\in R^1,
\end{gathered}
\end{equation}
where $q(x)$ is piecewise continuous and periodic with the period $1$;
$f(x)\in C_0^\infty(R^1)$, $\sup f\subset [0,1]$; $\omega\geq 0$ is a real
number; $b>0$ is any fixed constant.

	Our main result is the following

%	Theorem.
{\bf Theorem.\/} 
{\em Let the Hill operator $H_0$ be positive, $q\neq \const$ and the number
$\omega^2$ do not lie on the boundary of the spectrum of the operator $H_0$.
Then the solution of the problem $(1)$ has the following form
$$	u(x,t)=-ie^{-i\omega t} \zeta(x,\omega)+
		\frac{1}{\sqrt{t}}\,\big[u_1(x,t)+u_2(x,t)\big]+\nu(x,t),  $$
where $u_1(x,t)$, $u_2(x,t)$ are known functions, 
the function $\zeta(x,\omega)$ is determined by the help of the limiting 
amplitude principle and the function $\nu(x,t)$ admits the estimate
$$	|\nu(x,t)|\leq \frac{C(b)}{t}\,\|f\|_{{}_{L^2}}		$$
as $t\to \infty$ and $x\in [a,b]$.
}

\newpage

\begin{center}
	{\bf Z. Tsintsadze} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		I. Javakhishvili Tbilisi State University \\
		Tbilisi, Georgia} \\[0.5cm]
	{\bf THE MAXIMUM PRINCIPLE FOR QUASILINEAR SYSTEMS WITH A DEVIATING
		ARGUMENT OF A NEUTRAL TYPE IN THE PRESENCE 
		OF MIXED RESTRICTIONS}
\end{center}

\bigskip

\setcounter{equation}{0}
	We consider the optimal control problem 
%	(1)
\begin{equation}
	I=h\left( x\left( T\right) \right) \longrightarrow \min  \label{f1}
\end{equation}
under the restrictions 
%	(2),(3),(4)
\begin{gather}
	\dot{x}\left( t\right) =
		f\left( x\left( t\right) ,x\left( t-\Theta \right) ,
	\dot{x}\left( t-\Theta \right) ,u\left( t\right) \right), \label{f2} \\
	g\left( x\left( t\right) ,x\left( t-\Theta \right) ,\dot{x}\left( t-\Theta
		\right) ,u\left( t\right) \right) \leq 0,  \label{f3} \\
	x\left( t\right) =\varphi \left( t\right) ,\qquad -\Theta \leq t\leq 0,
			\label{f4} 
\end{gather}
where $\Theta $ is a fixed positive number, the scalar function $h$ is 
continuously differentiable, $x(t)\in W_{1,1}^{n}[-\Theta,T]$, 
$u(t)\in L_{1}^{r}[0,T]$, (see [1], $f=(f^1,\dots,f^n)$, $g=(g^1,\dots,g^m)$. 
Note that the conditions (\ref{f2}), (\ref{f3}) are fulfilled almost 
everywhere on $[0,T] $ and the restrictions (\ref{f3}) satisfy the standard
conditions of general position (see [2]).

	In the problem under consideration the initial function 
$\varphi(t)\in W_{1,1}^n[-\Theta,0]$ is a priori fixed and the moment 
$T$ is assumed to be known.

	If the functions $h$, $f^i$, $i=\overline{1,n}$ and $g^j$, 
$j=\overline{1,m}$, which are continuously differentiable with respect 
to their arguments, satisfy the conditions of convexity (see [1]) and 
the vector functions $f$ and $g$ are linear with respect to the control
parameters, then we can reduce the problem (\ref{f1})--(\ref{f4}) 
to an extremal problem, for which an analog of Pontryagin's maximum 
principle can be proved using the scheme from [2].

\vskip+0.8cm
\centerline{\sc References}
\vskip+0.4cm

1. A. D. Ioffe and V. M. Tikhomirov, Theory of extremal problems. 
(Russian), {\em ``Nauka'', Moscow}, 1974.

2. Z. A. Tsintsadze, The Lagrange principle of taking restrictions off and 
its application to linear optimal control problems in the presence 
of mixed restrictions and delays.  {\em Mem. Differential Equations 
Math. Physics} {\bf 11}(1997), 105--128.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newpage
\setcounter{page}{31}
\ 

\vskip+4cm
\thispagestyle{empty}
\begin{center}
{\size{20}{74pt}\selectfont S E C T I O N \;\;II}\\[2cm]
{\size{20}{74pt}\selectfont \bf MATHEMATICAL PHYSICS}
\end{center}

\newpage

\setcounter{page}{33}
\begin{center}
	{\bf J. Bezhuashvili and L. Giorgashvili} \\[0.1cm]
	{\size{8}{14pt}\selectfont \bf 
		Georgian Technical University \\
		Tbilisi, Georgia} \\[0.3cm]
	{\bf ANALOGUES OF POISSON'S FORMULA FOR PROBLEMS 
		OF STATIC THERMODIFFUSION} 
\end{center}

\bigskip

\setcounter{equation}{0}
	The system of homogeneous equations of elastic thermodiffusion 
has the form 
%	(1)
\begin{equation}
\begin{gathered}
	\mu\Delta u+(\lambda+\mu)\grad\dv u-
		\gamma_1\grad u_4-\gamma_2\grad u_5=0, \\
	\Delta u_j=0, \;\;\; j=4,5,
\end{gathered}
\end{equation}
where $u=(u_1,u_2,u_3)$ is the vector of displacement, 
$u_4$ is the deviation of temperature, $u_5$ is the ``chemical potential'' 
of the solid, $\lambda$, $\mu$, $\gamma_1$, $\gamma_2$ are elastic constants.

	Let $\Omega^{+}$ stand for the ball bounded by the sphere 
$\partial\Omega$ of the radius $R$, 
$\Omega^{-}=R^3\backslash \overline{\Omega}^{+}$.
Consider the following problem: find in the region $\Omega^{+}$ 
($\Omega^{-}$) a regular solution of the system (1) which satisfies on 
$\partial\Omega$ the following boundary condition:
$$	(I)^{\pm}: \;\;\; [u(z)]^{\pm}=f(z), \;\;\; 
		[u_j(z)]^{\pm}=f_j(z), \;\; j=4,5,		$$
where $f=(f_1,f_2,f_3)$ and $f_j(z)$, $j=1,2,\dots,5$, are given functions. 
The solution of the problem $(I)^{+}$ is given by the formula 
\begin{align*}
	u(x) & =\frac{1}{4\pi R} \int_{\partial\Omega} 
		M^{+}(x,y)F(y)\,d_yS, \\
	u_j(x) & =\frac{1}{4\pi R} \int_{\partial\Omega} 
		\frac{R^2-r^2}{|x-y|^3}\,f_j(y)\,d_yS, \;\;\; j=4,5,	
\end{align*}
where
\allowdisplaybreaks{
\begin{gather*}
	M^{+}(x,y)=\|M_{ij}^{+}(x,y)\|_{{}_{3\times 5}}, \\
	M_{ij}^{+}(x,y)=(1-\delta_{4j})(1-\delta_{5j})
		\Big[\delta_{ij}\,\frac{R^2-r^2}{|x-y|^3}+\beta(R^2-r^2)\,
			\frac{\partial^2\Phi(x,y)}{\partial x_i\partial x_j}
					\Big]- \\
	-\frac{\beta(\gamma_1\delta_{4j}+\gamma_2\delta_{5j})}{\lambda+\mu}\,
		(R^2-r^2)\,\frac{\partial}{\partial x_i}\,
			\widetilde{\Phi}(x,y), \;\;\; 
				i=1,2,3, \;\; j=1,2,\dots,5, \\
	F=(f,f_4,f_5), \\
	\Phi(x,y)=\int_0^1 \Big[\frac{R^2-r^2t^2}
			{(R^2-2Rrt\cos\gamma+r^2t^2)^{3/2}}-
		\frac{1}{R}-\frac{3rt\cos\gamma}{R^2}\Big]\,
			\frac{dt}{t^{1+\alpha}}\,, \\
	\widetilde{\Phi}(x,y)=\int_0^1 \frac{R^2-r^2t^2}
		{(R^2-2Rrt\cos\gamma+r^2t^2)^{3/2}}\,\frac{dt}{t^\alpha}\,, \\
	\alpha=(\lambda+2\mu)/(\lambda+3\mu)<1, \;\;\; 
		\beta=(\lambda+\mu)/(2(\lambda+3\mu)).
\end{gather*}}
\!\!The solution of the problem $(I)^{-}$ has an analogous form.

\newpage

\begin{center}
	{\bf T. Burchuladze and T. Buchukuri} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		A. Razmadze Mathematical Institute, 
		Georgian Academy of Sciences \\
		Tbilisi, Georgia} \\[0.5cm]
	{\bf BOUNDARY VALUE PROBLEMS OF GENERALIZED ELECTROELASTICITY}
\end{center}

\bigskip

\setcounter{equation}{0}
	The paper deals with R. A. Toupin's model of generalized 
elec\-tro\-elas\-ti\-ci\-ty. In this model, unlike the classical 
electroelasticity, the existence of a polarization vector 
is taken into account.

	Boundary value problems of dynamics and pseudo-oscillation 
for homogeneous unisotropic electroelastic media are studied when 
on the whole boundary either the displacement vector and 
the electric potential or the vector of mechanical stresses and 
boundary charges are given. For these problems, the theorems of existence and 
uniqueness are proved and properties of solutions are established. 

\newpage

\begin{center}
	{\bf T. Burchuladze and R. Rukhadze} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		A. Razmadze Mathematical Institute, 
		Georgian Academy of Sciences \\
		Tbilisi, Georgia \\
		Georgian Technical University \\
		Tbilisi, Georgia} \\[0.5cm]
	{\bf ASYMPTOTIC DISTRIBUTION OF EIGENELEMENTS 
		OF THE BASIC TWO-DIMENSIONAL BOUNDARY-CONTACT OSCILLATION 
		PROBLEMS OF THE CLASSICAL AND OF THE COUPLE-STRESS 
		ELASTICITY THEORY}
\end{center}

\bigskip

	The basic two-dimensional boundary-contact oscillation problems 
of the classical and the couple-stress elasticity theory are considered 
for a piece\-wi\-se-homogeneous isotropic media.

	Using the Carleman method and the Tauberian theorem (Ikehara), 
formulas of asymptotic distribution of the eigenvalues and 
the eigenfunctions are established. 

\newpage

\begin{center}
	{\bf O. Chkadua} \\[0.2cm]
		{\size{8}{14pt}\selectfont \bf 
	A. Razmadze Mathematical Institute, Georgian Academy of Sciences \\
		Tbilisi, Georgia} \\[0.5cm]
	{\bf SOME BOUNDARY-CONTACT PROBLEMS OF THE COUPLE-STRESS ELASTICITY}
\end{center}

\bigskip

	Boundary-contact problems of the couple-stress elasticity are 
investigated for anisotropic homogeneous media with contact on a part of 
the boundary. Here, one medium has the smooth boundary and 
the other medium has the boundary containing a closed cuspidal edge 
(the corresponding bihedral angle is equal to $2\pi$). 

	Theorems on the existence and uniqueness of solutions 
of boundary-contact problems in Besov and Bessel potential spaces 
are obtained by using the potential theory and the general theory 
of pseudodifferential equations on a manifold with a boundary. 
The smoothness of solutions is studied in closed domains occupied by elastic 
media. Asymptotic properties of solutions are also studied.

\newpage

\begin{center}
	{\bf A. Dzhishkariani} \\%[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		A. Razmadze Mathematical Institute,
		Georgian Academy of Sciences \\ 
		Tbilisi, Georgia} \\%[0.5cm]
	{\bf ON THE CONVERGENCE OF RESIDUALS FOR PROJECTIVE AND 
		PROJECTIVE-ITERATIVE METHODS IN ELLIPTIC PROBLEMS}
\end{center}

%\bigskip

\setcounter{equation}{0}
	Approximate projective methods of the solution of elliptic 
bo\-un\-da\-ry value problems take their origin in the works of Ritz 
(J.f.d. reine und angewandte Math., 1908, 135, H. 1) and Galerkin 
(Vestnik inzhenerov 1915, 19).

	Elliptic linear problems are discribed by the operator equations
%	(1),(2)
\begin{gather}
	Au=f, \;\;\; u\in D(A), \;\;\; f\in H,	\tag{1} \\
	Au+Ku=f, \;\;\; u\in D(A), \;\;\; f\in H,	\tag{2}
\end{gather}
where $A$ is a self-adjoint positive definite operator in the Hilbert space 
$H$, $D(K)\supset D(A)$.

	The expressions $\delta_n\equiv Au_n-f$, $\delta_n\equiv Au_n+Ku_n-f$ 
are called residuals, where $u_n$ is the approximate solution. 

	The first works devoted to the problems of the convergence 
of re\-si\-duals to zero, as $n\to \infty$, are due to Mikhlin 
(DAN SSSR, 1956, 106, No. 3) and Vainikko (DAN SSSR, 1968, 179, No. 5). 
In their works the coordinate system is formed by 
the eigen elements of another 
operator $B$ which is similar to $A$. The convergence of the residual $Au_h-f$
to zero, as $h\to 0$, in the method of finite elements has been proved 
by the author under
certain conditions (DAN SSSR, 1980, 254, No. 5).

	The projective-iterative method has been suggested by Porter and 
Stirling (IMA J. Numer. Analys., 1993, 13).

	The basic results of the author:

	i) let the operator $I+KA^{-1}$ be continuously invertible in $H$, 
the operator $KA^{-1/2}$ be bounded in $H$, the exact solution of the problem 
(2) be $u\in D(A^\alpha)$, $\alpha\geq 1$, and let in the Bubnov--Galerkin 
method the coordinate system be formed by the 
eigenelements of an operator $B$ similar to $A$. 
Then for the residual 
of the projective-iterative method when $l$ is fixed and $n\to \infty$, 
the estimate
%	(3)
\begin{equation}
	\|\delta_{n,l}\|_{{}_H}\equiv \big\|A\widetilde{u}_{n,l}+
			K\widetilde{u}_{n,l}-f\|_{{}_H}=
		o\big(\lambda n^{-(\frac{l+1}{2}+\alpha-1)}\big)    \tag{3}
\end{equation}
is valid, where $\widetilde{u}_{n,l}$ is the approximate solution by the
projective-iterative method, $\lambda_n$ is the $n$-th eigen value 
of the operator $B$, as $\lambda_n\to \infty$, $n\to \infty$ and 
$l$ is the number of projective-iterative cycles, $\widetilde{u}_{n,-1}=u_n$;

	ii) under the same conditions for $u\in D(A)$, 
for the method of finite elements with iterations, the estimate 
%	(4)
\begin{equation}
	\|\delta_{h,l}\|_{{}_H}=\big\|A\widetilde{u}_{h,l}+
			K\widetilde{u}_{h,l}-f\|_{{}_H}=
		O\big(\|A^{1/2}\Pi^hA^{-1}\|^l\|A^{1/2}\Pi^hu\|\big)   \tag{4}
\end{equation}
is fulfilled, where $\Pi^h\equiv I-\Pi_h$, $\Pi_h$ is an orthoprojector 
in $H_A$.

\newpage

\begin{center}
	{\bf R. Gachechiladze and O. Maisaia } \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
	        A. Razmadze Mathematical Institute, 
		Georgian Academy of Sciences \\
		Tbilisi, Georgia} \\[0.5cm]
	{\bf PROBLEMS OF STATICS AND DYNAMICS OF THE COUPLE-STRESS ELASTICITY 
		WITH REGARD FOR FRICTION} 
\end{center}

\bigskip

\setcounter{equation}{0}
	Static and dynamic problems  of the couple-srtress elasticity are  
investigated for  non-homogeneous anisotropic media, when frictions are 
prescribed on a part of the boundary or on the entire boundary. 
In particular, in the case of statics we consider the following problem: 
Suppose that $\Om\sbs \bR^3$ is a bounded region with the boundary $\Gm$, 
$\Gm\in C^\infty$, and let $\Gm=\Gm_1\cup \Gm_2$, $\Gm_1\cap \Gm_2=\vnth$.

\vskip+0.2cm
%	Problem.
{\bf Problem.\/}
Let $\cG\in (L^2(\Om))^6$, $f\in L^\infty(\Gm_2)$, 
$|f|\geq m>0$ almost everywhere on $\Gm_2$, 
$\Phi\in (H_{1/2}(\Gm_1))^6$ and $\vf\in (L^\infty(\Gm_2))^3$. Find 
$U\in (H_1(\Om))^6$ which is a weak solution of the equation
$$	\cM(x,\pa) U(x)+\cG(x)=0, \quad x\in \Om,	$$
the restriction on $\Gm_2$ of tangential components of force stress are 
functions of the class $(L^\infty(\Gm_2))^3$, and
\begin{gather*}
	U\big|_{\Gm_1}=\Phi, \;\;\;(\sg U)_n\big|_{\Gm_2}=f, \;\;\;
		\mu U\big|_{\Gm_2}=\vf, \;\;\;
		|(\sg U)_\top|_{\Gm_2}<g \Ra u_\top=0, \\
	|(\sg U)_\top|_{\Gm_2}=g \Ra \exists \lb\geq 0:
		u_\top=-\lb(\sg U)_\top,
\end{gather*}
where $\cM(x,\pa)$ is the matrix differential operator of statics 
of the couple-stress elasticity, $\sg U$ is the vector of force stress, 
$\mu U$ is a couple-stress vector; 
by $a_n$ and $a_\top$ are denoted respectively the normal and 
the tangential component of the vector $a\in \bR^3$, 
and $g=\cF|(\sg U)_n|$; $\cF$ is a measurable bounded function 
on $\Gm_2$, $\cF(x)\geq\cF_0>0$.
\vskip+0.2cm

	The cases $\mes \Gm_1>0$ and $\Gm_1=\vnth$ are considered separately.
In the latter case, sufficient conditions for the existence 
of a solution are written out. 
An analogous problem is investigated in the dynamic case.

\newpage

\begin{center}
	{\bf D. Gordeziani and G. Avalishvili} \\%[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		I. Vekua Institute of Applied Mathematics, 
		Tbilisi State University \\
		Tbilisi, Georgia} \\%[0.3cm]
	{\bf ON THE INVESTIGATION OF NONLINEAR MODELS OF THE THEORY 
		OF MIXTURES} 
\end{center}

%\smallskip

\setcounter{equation}{0}
	We consider the homogeneous, isotropic elastic mixture 
of two components
%	(1)
\begin{equation}
\begin{array}{l}
	\displaystyle\sum\limits _{k=1}^3 
		\frac{\partial \sigma_{ki}^1}{\partial x_k}(u^1,u^2)-
			\pi_i(u^1,u^2)=-\rho_1 F_i(x),\\
	\displaystyle\sum\limits _{k=1}^3 
		\frac{\partial \sigma_{ki}^2}{\partial x_k}(u^1,u^2)+
			\pi_i(u^1,u^2)=-\rho_2 G_i(x),
\end{array} \;\;\;	x\in \Omega,
\end{equation}
where $i=\overline{1,3}$, $u^1$, $u^2$ are the displacement vectors 
of the first and second components respectively, 
$F=(F_1,F_2,F_3)$, $G=(G_1,G_2,G_3)$ are body forces. 
The components of the partial stresses $\sigma_{ki}^1$, $\sigma_{ki}^2$ and 
the diffusive force $\pi_i$ are of the following form 
$$	\begin{array}{c}
	\vspace{0.7mm}
		\displaystyle \sigma_{ki}^1=1/2(-\alpha_2+
			\lambda_1 e_{pp}^1+\lambda_3 e_{qq}^2)
			\delta_{ki}+\mu_1 e_{ki}^1+\mu_3 e_{ki}^2-
			\lambda_5 h_{ki},\\ \vspace{0.5mm}
		\displaystyle \sigma_{ki}^2=1/2(\alpha_2+
			\lambda_4 e_{pp}^1+\lambda_2 e_{qq}^2)
			\delta_{ki}+\mu_3 e_{ki}^1+\mu_2 e_{ki}^2+
			\lambda_5 h_{ki},\\ %\vspace{0.5mm}
		\displaystyle \pi_i=\frac{\alpha_2 \rho_2}{\rho}
			\frac{\partial e_{pp}^1}{\partial x_i}+
			\frac{\alpha_2 \rho_1}{\rho}
			\frac{\partial e_{pp}^2}{\partial x_i},
				\;\; \rho=\rho_1+\rho_2,
	\end{array}			$$
where the tensor indices take the values 1, 2, 3 and are subject to the 
summation convention, $\alpha_2$, $\lambda_i(i=\overline{1,5})$, 
$\mu_j(j=\overline{1,3})$ are material constants, $\rho_1$, $\rho_2$ 
are the massdensities of the two solids. We have denoted 
$$	\begin{array}{c}
	\vspace{0.5mm}
		\displaystyle e_{ki}^s=e_{ki}(u^s)=\frac{1}{2} \Big( 
			\frac{\partial u_i^s}{\partial x_k}
			+\frac {\partial u_k^s}{\partial x_i}
			+\frac{\partial u_m^s}{\partial x_k}
			\frac{\partial u_m^s}{\partial x_i}\Big), \;\; 
				s=1,2, \\ 
		\displaystyle h_{ki}=\frac{1}{2} \Big(
			\frac{\partial u_i^1}{\partial x_k}-
			\frac{\partial u_k^1}{\partial x_i}+
			\frac{\partial u_k^2}{\partial x_i}-
			\frac{\partial u_i^2}{\partial x_k}+ 
			\frac{\partial u_m^1}{\partial x_k}
			\frac{\partial u_m^2}{\partial x_i}-
			\frac{\partial u_m^1}{\partial x_i}
			\frac{\partial u_m^2}{\partial x_k}\Big).
	\end{array}			$$    

	For the system (1) the following theorem is true.

\vskip+0.2cm
%	Theorem.
{\bf Theorem.} 
{\em Let $\Omega\subset{\Bbb R}$ be a bounded region with the boundary 
$\Gamma=\partial\Omega$ of the class ${\Bbb C}^2$. If 
$$	\begin{array}{c}
		\displaystyle\mu_1>0,\;\mu_3^2<\mu_1\mu_2,\;
			\lambda_5 \le 0,\;\lambda_1
			+\frac{2}{3}\mu_1-\frac{\rho_2}{\rho}\alpha_2>0,\\
		\displaystyle \Big(\lambda_3+\frac{2}{3}\mu_3-
			\frac{\rho_1}{\rho}\alpha_2 \Big)^2<
			\Big(\lambda_1+\frac{2}{3}\mu_1-
			\frac{\rho_2}{\rho}\alpha_2 \Big)     
			\Big(\lambda_2+\frac{2}{3}\mu_2+
			\frac{\rho_1}{\rho}\alpha_2 \Big),
	\end{array}		$$
then there exist neighbourhoods $F$ and $U$ of $\bf 0$ in the spaces 
$[L^2(\Omega)]^6$ and 
$V(\Omega)=\{u|\;u\in [H^2(\Omega)]^6,\;u|_{\Gamma}=0\}$ 
respectively, such that for each $f\in F$ the boundary-value problem 
$Au=f$ has the unique solution $u\in U$, $u=(u^1,u^2)$, 
where $A$ is the operator, appearing is in the left-hand part 
of the system $(1)$. }

\newpage

\begin{center}
	{\bf G. Jaiani} \\%[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		I. Vekua Institute of Applied Mathematics, 
		Tbilisi State University \\
		Tbilisi, Georgia} \\%[0.5cm]
	{\bf ON RELATION  OF A MATHEMATICAL MODEL OF BARS 
		TO THE THREE-DIMENSIONAL ELASTICITY THEORY}
\end{center}

%\smallskip

\setcounter{equation}{0}
	Consider $ (N_3,N_2)$ -- approximation of a model of bars given 
in [1, 2]. Further the notation from those works will be used. 

	Let the width and the thickness of the bar be given  by 
$2h_i(x_1)=h_0^ix_1^{\alpha_i}$, $i=2,3$ $(h_0^i=\const>0$, 
$\alpha_i=\const\ge 0)$, respectively.

%\vskip+0.2cm
{\bf Problem 1.\/} 
Find bounded weighted double moments
${\mathop {v_j}\limits^{n_3,n_2}}(x_1)\in\linebreak  C^3(]0,l[)\cap C(]0,l])$,
$j=1,2,3$, $n_i=\overline{0,N_i}$, $i=2,3,$ of the displacement vector 
components $u_j(x_1,x_2,x_3)$ under the boundary conditions
%	(1),(2)
\begin{gather}
	\begin{array}{r}
		{\mathop {v_j}\limits^{n_3,n_2}}(0)=	
			{\mathop {\varphi_j^0}\limits^{n_3,n_2}},\;\;
			j=1,2,3,\;\;n_i=\overline{0,N_i},\;\;
			i=2,3,\;\;\text{if} \\
		(2N_2+1)\alpha _2+(2N_3+1)\alpha _3<1;
	\end{array} \\
	{\mathop {v_j}\limits^{n_3,n_2}}(\ell)=
		{\mathop {\varphi_j^\ell}\limits^{n_3,n_2}},\;\;j=1,2,3,\;\;
			n_i=\overline{0,N_i},\;\;i=2,3,
\end{gather}
provided the lateral boundaries 
$(x_i=\mathop {h_i}\limits^{\hspace{-0.3ex}(\pm)}(x_1)$, $i=2,3$, $0<x_1<l)$ 
of the bar are arbitrarily loaded. 
The functions ${\mathop {\varphi_j^0}\limits^{n_3,n_2}}$, 
${\mathop {\varphi_j^\ell}\limits^{n_3,n_2}}$ 
and the loadings $\mathop {f_j^i}\limits^{\hspace{-0.3ex}(\pm)}$ are given.
%\vskip+0.2cm

%	STATEMENT. 
{\bf Statement.\/}
Problem 1 is correct in the sense of Hadamard. 

	If we consider the limit as $N_i \to +\infty$, $i=2,3,$ then from (1) 
we have 
$$	\alpha_i<\frac{1}{2N_i+1},\;\; \text{and}\;\;
		\lim _{N_i\to+\infty}\alpha_i=0,\;\;i=2,3.	$$
Thus Problem 1 tends to the following three-dimensional model

%\vskip+0.2cm
{\bf Problem 2.\/}
Find $ u_i(x_1,x_2,x_3)\in C^3(v)\cap \left\{\!\!\begin{array}{l}
		C(\bar V)\;\;{\rm if}\;\alpha_i=0,\; i=2,3,\\
		C(\bar V \backslash \{0,x_2,x_3\})\;{\rm if}\;\alpha_i>0
			\end{array} \right. $ 
subject to the boundary conditions
\begin{gather*}
	u_j(0,x_2,x_3)=\varphi_j^0(x_2,x_3),\;\;j=1,2,3,\;\;
		\text{if}\;\;\alpha_i=0,\;\; i=2,3; \\
	u_j(\ell,x_2,x_3)=\varphi_j^\ell(x_2,x_3)\;\;j=1,2,3, \\
	X_{ij}(x_1,x_2,x_3)\Big|_{x_i=
		\mathop {h_i}\limits^{\hspace{-0.3ex}(\pm)}(x_1)}=
		\mathop {f_j^i}\limits^{\hspace{-0.3ex}(\pm)}(x_1,x_{5-i}),
		\;\;j=1,2,3,\;\;i=2,3.
\end{gather*}

\vskip+0.2cm
\centerline{\sc References}
%\vskip+0.2cm

1. G. Jaiani, On a model of a bar with variable thickness. 
{\em Bull. TICMI} {\bf 2}(1998), 36--40.

2. On a mathematical model of a bar with variable rectangular 
cross-section. {\em Universitaet Potsdam, Institut fuer Mathematik, 
Preprint $98/21$, August}, 1998.


\newpage

\begin{center}
	{\bf G. Kapanadze} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		I. Javakhishvili Tbilisi State University \\
		Tbilisi, Georgia} \\[0.5cm]
	{\bf ON A PROBLEM OF BENDING OF A DOUBLE-CONNECTED PLATE
		BOUNDED BY BROKEN LINES}
\end{center}

\bigskip


	The problem of bending of a plate bounded by two convex polygons 
is studied in the case where piecewise constant normal bending moments 
act at the boundary of the plate, or the angles of rotation 
of the boundary points are given beforehand. 
A function mapping conformally the given region onto an annulus 
is constructed, and the problem of finding the unknown bending of the plate 
is reduced to the Riemann--Hilbert problem for the annulus. 
This method allows one to solve the problem efficiently (in quadratures). 

\newpage

\begin{center}
	{\bf G. Khatiashvili} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		N. Muskhelishvili Institute of Computational Mathematics \\
		Georgian Academy of Sciences \\
		Tbilisi, Georgia} \\[0.5cm]
	{\bf SOLUTION OF SOME BOUNDARY VALUE PROBLEMS FOR NONHOMOGENEOUS 
		ELASTIC BODIES} 
\end{center}

\bigskip

	For an elliptic ring composed of different isotropic materials, 
the Dirichlet and Neuman problems are considered. 
The indicated problems are also studied in the case where a
composed elliptic ring is strengthened with ani\-sot\-ro\-pic elliptic kernel 
or the anisotropic infinite environment is stren\-gthe\-ned with an 
isotropic composed elliptic ring.

	Some boundary problems for binary isotropic mixtures 
are also con\-si\-de\-red.

\newpage

\begin{center}
	{\bf N. Khatiashvili} \\%[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		I. Vekua Institute of Applied Mathematics, 
		Tbilisi State University \\
		Tbilisi, Georgia} \\%[0.3cm]
	{\bf ON A SINGULAR INTEGRAL EQUATION WITH DOUBLY PERIODIC KERNEL}
\end{center}
%\smallskip

\setcounter{equation}{0}
	In Hydrodynamics, mainly in the theory of hydroturbines, 
compressors and pumps, the lattice theory is widely used [1]. 
This theory contains boundary value problems for piecewise analytic functions 
in the infinite domains cut along the countable number of segments [1]. 
On the other hand, this problems are closely connected with singular integral 
equations with doubly periodic kernels [2].

	Consider the following integral equation
%	(1)
$$	\frac{1}{\pi i}\int_0^a u(t)\cdot K(t,x)dt=if(x), \;\;
		K(t,x)=\zeta(t-x)-\zeta(t+x),\;\;x\in [0,a], \eqno{(1)}	  $$
where $a$ and $\omega$, $0<a<\omega$, are constants, $\zeta$ is 
Weierstrass's ``$\zeta$-function'' for the periods  $2\omega$ and $2ih$, [3]. 
When $t$ varies on the whole complex plane and $z=x+iy$, 
$K(t,z)$ is a doubly periodic kernel with respect to $t$, 
with periods $2\omega$ and $2ih$, 
$f(x)\in H[0,a]$, ($f(0)=0$, $H$ is the H\"{o}lder class) is a given function
and $u(x)$ is an unknown function, satisfying 
$$	\int_0^a u(t)dt=0.  \eqno{(2)}		$$

%\vskip+0.2cm
%	Theorem.
{\bf Theorem.\/} 
{\em In the class $H[0,a]$ there exists a unique solution 
of the equation $(1)$ satisfying the condition $(2)$, and this solution is 
of the form
$$	u(x)=\frac{\sqrt{X(x)}}{\pi }\cdot\int_0^a\frac{f(t)}
		{\sqrt{X(t)}}\cdot K(t,x)dt, \;\;	
		X(z)=C\cdot\frac{\sigma(z-a)\sigma(z+a)}{\sigma^2(z)}, $$
where $C$ is a constant, $\sigma(z)$ is Weierstrass's ``$\sigma$-function''
for the periods $2\omega$ and $ih$, $\sqrt X(z)$ is the branch for
which $\sqrt X(z)\ge 0$, when $X(z)\ge 0$. 

	Also the following condition must be fulfilled 
$\int_0^a f(t)/\sqrt{X(t)}\cdot dt=0$. }
\vskip+0.2cm

	The  theorem is proved by using Villa's formula for the stripe [1, 3]
and properties of singular integrals [4].

\vskip+0.2cm
\centerline{\sc References}
%\vskip+0.2cm

1. A. E. Sedov, Plane problems of hydrodynamics and aerodynamics. (Russian)
{\em Nauka, Moscow}, 1980.

2. N. G. Khatiashvili, On Dirichlet problem for Laplace equation for stripe 
with cuts. (Russian) {\em University Press., Tbilisi}, 1985.

3. Dr. Werner von Koppenfels und Dr. Friedmann Stallmann, 
Praxis der Konformen Abbildung. {\em Berlin}, 1959.

4. N. I. Muskhelishvili, Singular Integral Equations. {\em Holland}, 1953.

\newpage

\begin{center}
	{\bf G. Khuskivadze and V. Paatashvili} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		A. Razmadze Mathematical Institute, 
		Georgian Academy of Sciences \\
		Tbilisi, Georgia} \\[0.5cm]
	{\bf AN APPLICATION OF THE LINEAR CONJUGATION PROBLEM TO STUDY 
		OF PROPERTIES OF CONFORMAL MAPPINGS} 
\end{center}

\bigskip

\setcounter{equation}{0}
	It is shown that one can establish new properties 
of conformal mappings of the unit circle onto simply connected domains 
with non-smooth bo\-un\-da\-ries using solutions 
of boundary value problems for analytic functions. 
In particular, the following assertion is proved. 
Let $D$ be a simply connected domain bounded by a piecewise smooth curve with
the interior angles $\pi\nu_k$ $(0\leq \nu_k\leq 2)$ at the angular points
$t_k$ $(k=\overline{1,n})$. Let, moreover, $z=z(w)$ be a conformal mapping 
of the unit circle onto $D$. Then
$$	z'(w)=\prod_{k=1}^n (w-c_k)^{\nu_k-1}z_0(w), \;\;\; 
		t_k=z(c_k), \;\;\; z_0^{\pm1}\in \underset{p>1}{\cap} H^p,  $$
where $H^p$ is the Hardy class.

\newpage

\begin{center}
	{\bf S. Kukudzhanov} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		A. Razmadze Mathematical Institute, 
		Georgian Academy of Sciences \\
		Tbilisi, Georgia} \\[0.5cm]
	{\bf STABILITY OF ORTHOTROPIC SHELLS OF ROTATION 
		CLOSE TO CYLINDRICAL ONES WITH ELASTIC FILLER UNDER TORSION}
\end{center}

\bigskip			

	The stability of orthotropic shells of rotation, 
which by their form are close to cylindrical ones with an elastic filler, 
is investigated under the action of torques applied to the shell ends. 
The filler is assumed to be a solid elastic body contacting 
(over the outer surface) with the shell. Both the long and the midlength 
shells are considered. The filler is assumed to be of slipping type.
The investigation is performed by solving the equations of stability 
of the theory of shells and also the equations of equilibrium of the theory 
of elasticity. The shells of positive and negative Gaussian curvature are 
considered. For orthotropic shells of rotation, whose generating midsurface 
is described by a parabolic function, we obtained formulas in 
dimensionless form and the universal curves of dependence of the critical 
shearing stress on the elastic parameters of orthotropy, 
the dimensionless rigidity of the elastic filler and 
on the dimensionless amplitude of deviation from the cylindrical form. 
Deviations of order of thickness are considered. It is established that in the
isotropic case the critical stresses under small values of rigidity of the
elastic filler may essentially differ for the shells under consideration, 
whereas this difference becomes insignificant as the rigidity of the elastic 
filler increases. In the orthotropic case, different degrees of the influence 
of the elastic parameters on the critical loading are revealed for the shells 
of positive and negative Gaussian curvature. The conditions are obtained 
under which one may neglect the influence of the boundary conditions 
on the critical loading.

\newpage

\begin{center}
	{\bf V. Kukudzhanov and D. Schneiderman} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		Institute for Problems in Mechanics, 
		Russian Academy of Sciences \\ 
		Moscow, Russia} \\[0.5cm]
	{\bf NUMERICAL SOLUTION OF ELASTIC PROBLEMS OF THE NON-AXIALLY 
		SYMMETRIC DEFORMATION OF BODIES OF REVOLUTION 
		WITH NON-REGULAR BOUNDARIES}
\end{center}

\bigskip

     A numerical method of solving 3-D problems involving elastic isotropic
bodies of revolution under a non-axially symmetric load is developed.
It is the Ritz method in which an expansion in a system of trigonometric 
functions along a peripheral direction and a two-dimensional finite-element
approximation with respect to the coordinates in the meridian plane are used.
Singular functions determining the solution near edge lines or conic points
are added to the finite-element approximation.

     The analysis of a test problem shows that by using the singular function
we increase the accuracy of the computation of the displacements near the
singularity. However, the addition of the singular functions to the 
finite-element approximation leads to ill-conditioned stiffness matrix.
The bordering techniques of Faddeev and Faddeeva are used to avoid this
problem. To calculate the singular function factor with sufficient accuracy
the fine subdivision of a region is demanded. This is due to the fact that
when the matrix is ill conditioned, the solution depends drastically on the
errors of the numerical integration resulting from the calculation of the
matrix coefficients. These errors are reduced when the region subdivision
is more fine.

     The method is used for solving the problem of the deformation of a
half-space with a blind hole under the load applied to the lateral surface
of the hole. The problem simulates the process of boring a hole in a body
which is used to determine the residual stresses in it.

\newpage

\begin{center}
	{\bf T. Meunargia} \\%[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		I. Vekua Institute of Applied mathematics, 
		Tbilisi State University \\
		Tbilisi, Georgia} \\[0.3cm]
	{\bf STRESS CONCENTRATIONS IN THE NON-LINEAR THEORY OF PLATES} 
\end{center}

\smallskip

\setcounter{equation}{0}
	In the present report three dimensional problems of the non-linear
elasticity theory are reduced to two dimensional problems of non-linear
plates by means of I. Vekua method. Then we consider the case of geometrical
non-linear plate for approximate $N=0$, which have the following complex form:
$$	\left\{	\begin{array}{l}
		%\displaystyle 
		\frac{\partial}{\partial z}
			[T_{11}-T_{22}+i(T_{12}+T_{21})]+
				\frac{\partial}{\partial{\bar z}}
			[T_{11}-T_{22}+i(T_{12}+T_{21})]+F_+=0,\\
		%\displaystyle 
		\frac{\partial T_{+}}{\partial z}+
			\frac{\partial{\overline T}_{+}}
				{\partial{\bar z}}+F_3=0 \;\;\;\;\; 
		\Big(z=x_1+ix_2,\;\;\frac{\partial}{\partial z}=\frac{1}{2}
		\Big(\frac{\partial}{\partial x_1}-
			i\frac{\partial}{\partial x_2}\Big)\Big),
		\end{array}	\right.		$$
where
$$	\left\{ \begin{array}{l}
		%\displaystyle 
		T_{11}-T_{22}+i(T_{12}+T_{21})=
			(\sigma_{11}-\sigma_{22}+2i\sigma_{12})\Big(1+
				\frac{\partial u_+}{\partial z}\Big)+ \\
			\ \hskip+2cm +(\sigma_{11}+\sigma_{22})
				\frac{\partial u_+}{\partial{\bar z}},\\
		%\displaystyle 
		T_{11}+T_{22}+i(T_{12}-T_{21})=
			(\sigma_{11}+\sigma_{22})\Big(1+
				\frac{\partial u_+}{\partial z}\Big)+ \\
			\ \hskip+2cm  + (\sigma_{11}-\sigma_{22}-2i\sigma_{12})
				\frac{\partial u_+}{\partial{\bar z}},\\
		%\displaystyle 
		T_{+}=T_{13}+iT_{23}=\sigma_++
			(\sigma_{11}-\sigma_{22}+2i\sigma_{12})
				\frac{\partial u_3}{\partial z}+
			(\sigma_{11}+\sigma_{22})
				\frac{\partial u_3}{\partial{\bar z}}.
		\end{array}		\right.			$$
Here
$$	\left\{ \begin{array}{l}
		%\displaystyle 
		\sigma_{11}-\sigma_{22}+2i\sigma_{12}=
			4\mu\Big(\frac{\partial u_+}{\partial{\bar z}}
				+\frac{\partial u_+}{\partial{\bar z}}
			\frac{\partial{\bar u}_+}{\partial{\bar z}}\Big),\\
		%\displaystyle 
		\sigma_{11}+\sigma_{22}=
			2(\lambda+\mu)\Big(\theta+
				\frac{\partial u_+}{\partial z}
			\frac{\partial{\bar u}_+}{\partial{\bar z}}+
				\frac{\partial{\bar u}_+}{\partial z}
			\frac{\partial u_+}{\partial{\bar z}}\Big),\\
		%\displaystyle 
			\sigma_{+}=
			2\mu\frac{\partial u_3}{\partial{\bar z}} \;\;\;
			\Big(\theta=\frac{\partial u_+}{\partial z}+
				\frac{\partial{\bar u}_+}{\partial{\bar z}}
					\;\;u_+=u_1+iu_2\Big).
		\end{array}	\right.				$$
\par The boundary condition can be written as follows:
$$	\left\{	\begin{array}{l}
		%\displaystyle 
		T_{(ll)}+iT_{(ls)}=
			\frac{1}{2}\Big\{T_{11}+T_{22}+i(T_{12}-T_{21})+ \\
			\ \hskip+2cm +[T_{11}-T_{22}+i(T_{12}+T_{21})]
				\Big(\frac{d{\bar z}}{ds}\Big)^2\Big\},\\
		%\displaystyle 
		T_{(ls)}=-Im\Big(T_{+}\frac{d{\bar z}}{ds}\Big).
		\end{array}		\right.		$$

	We solve problems of concentration of stresses around holes by means 
of the small parameter method.

\vskip+0.4cm
\centerline{\sc References}
\vskip+0.2cm

1. I. N. Vekua, Shell theory: General methods of constraction. 
{\em Nauka, Moscow}, 1982.

2. P. Ciarlet, Mathematical elasticity, vol. I. (Russian) 
{\em Mir, Moscow,} 1992. 

\newpage

\begin{center}
	{\bf Sh. Mzhavanadze} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		Engineering-Economical Institute \\
		Poti, Georgia} \\[0.5cm]
	{\bf THE REVERSE AND HALF-REVERSE PROBLEMS OF ELASTICITY FOR PLANES 
		WEAKENED BY SYMMETRICALLY POSED HOLES}
\end{center}

\bigskip

	The paper deals with problems of a certain class for the 
elastic plane weakened by symmetrically posed holes with unknown and
partly unknown contours. The tangent strain is zero on the contour of the hole
while the normal strain on the unknown part of the hole and the normal
displacement on the known part are constant. It is requested to find the
unknown part of the contour and the distribution of the strains for the case
where the tangent normal strain is constant.

	Using methods of the theory of analytical functions, 
a method of solving is given for the problems mentioned above. 
Solutions of some concrete problems are given and the sought for contours 
are constructed.

\newpage

\begin{center}
	{\bf J. Peradze} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		I. Javakhishvili Tbilisi State University \\
		Tbilisi, Georgia} \\[0.5cm]
	{\bf ON SOME PROPERTIES OF  TIMOSHENKO'S SYSTEM}
\end{center}

\bigskip

\setcounter{equation}{0}
	The paper deals with the problem of solvability of the nonlinear 
system of the plates theory
\begin{gather*}
	\frac{\partial N_i}{\partial x_i}+
		\frac{\partial T}{\partial x_{j}}+p_i=0, \\
	\frac{\partial Q_1}{\partial x_1}+\frac{\partial Q_2}{\partial x_2}+
			\frac{\partial}{\partial x_1}
		\Big(N_1\frac{\partial w}{\partial x_1}+
			T\frac{\partial w}{\partial x_2}\Big)+
		\frac{\partial}{\partial x_2}
			\Big(T\frac{\partial w}{\partial x_1}+
			N_2\frac{\partial w}{\partial x_2}\Big)+q=0, \\
	\frac{\partial M_i}{\partial x_i}+
		\frac{\partial H}{\partial x_j}-Q_i=0.
\end{gather*}
Here
\begin{gather*}
	N_i=\frac{Eh}{1-\nu^2}\bigg\{\frac{\partial u_i}{\partial x_i}+
		{1\over 2}\Big(\frac{\partial w}{\partial x_i}\Big)^2+
			\nu\Big[\frac{\partial u_j}{\partial x_j}+
		{1\over 2}\Big(\frac{\partial w}{\partial x_j}\Big)^2
			\Big]\bigg\}, \\
	T=\frac{Eh}{2(1+\nu)}\Big(\frac{\partial u_1}{\partial x_2}+
		\frac{\partial u_2}{\partial x_1}+
			\frac{\partial w}{\partial x_1}
				\frac{\partial w}{\partial x_2}\Big), \;\;\; 
		Q_i=k_0^2\frac{Eh}{2(1+\nu)}
			\Big(\frac{\partial w}{\partial x_i}+\psi_i\Big), \\
	M_i=\frac{Eh^3}{12(1-\nu^2)}\Big(\frac{\partial\psi_i}{\partial x_i}+
		\nu\frac{\partial\psi_j}{\partial x_j}\Big), \\
	H=\frac{Eh^3}{24(1+\nu)}\Big(\frac{\partial\psi_1}{\partial x_2}+
		\nu\frac{\partial\psi_2}{\partial x_1}\Big), 
\end{gather*}
$u_i(x)$, $w(x)$, $\psi_i(x)$ are unknown functions satisfying certain 
boundary conditions, $p_i(x)$, $q(x)$ are given functions, 
$x=(x_1,x_2)$, $i,j=1,2$, $i\neq j$, $E$, $h$, $\nu$, $k_0$ are known 
positive parametrs and $\nu<1$. 

	The results are also given for Timoshenko's one-dimensional model.


\newpage

\begin{center}
	{\bf A. Saginashvili} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		A. Razmadze Mathematical Institute, 
		Georgian Academy of Sciences \\
		Tbilisi, Georgia} \\[0.5cm]
	{\bf ON GENERALIZED RIEMANN PROBLEM} 
\end{center}

\bigskip

\setcounter{equation}{0}
	Let $D^{+}$ and $D^{-}$ be inner and outer domains with the boundary 
$\Gamma$, where $\Gamma$ is a simple, closed Carleson curve such that 
at each point $t\in \Gamma$ it is homeomorphic to some logarithmic star. 
$L_p(\Gamma,\omega)$, $1<p<\infty$, is the weighted Lebesgue space with 
the Muckenhoupt weight.

\vskip+0.2cm
{\bf Generalized Riemann problem:\/} 
look for an analytic function $\Phi(z)$ which can be represented as a 
Caushy integral with a density from $L_p(\Gamma,\omega)$ and 
whose boundary values $\Phi^{+}(t)$ from $D^{+}$ and 
$\Phi^{-}(t)$ from $D^{-}$ coupled by the following condition:
$$	\Phi^{+}(t)=a(t)\Phi^{-}(t)+b(t)\overline{\Phi^{-}(t)}+f(t), \;\;\; 
		t\in \Gamma,		$$
where $a,\,b\in C(\Gamma)$, $f\in L_p(\Gamma,\omega)$.
\vskip+0.2cm

	For each $t\in \Gamma$ local parameters $0<\alpha(t)<2\pi$, 
$-\infty<\delta(t)<+\infty$, $0<\eta_1(t)\leq \eta_2(t)<1$ are defined. 
In particular: if $\Gamma$ is a piecewise smooth curve, 
then $\alpha(t)$ is the angular measure at the point $t$ and 
$\delta(t)=0$; if $\omega$ is a Khvedelidze weight, 
$\omega(\tau)=|\tau-t|^\beta$, $\tau\in \Gamma$, 
$-\frac{1}{p}<\beta<1-\frac{1}{p}$, 
then $\eta_1(t)=\eta_2(t)=\beta+\frac{1}{p}$. 

\vskip+0.2cm
%	Theorem.
{\bf Theorem.\/} 
{\em General Riemann problem is Fredholm if and only if
$$	|a^2(t)|-|b^2(t)|\Big(
		\frac{sh[(\pi-\alpha(t))(\xi+i(-\delta(t)\xi+\eta))]}
			{sh\pi(\xi+i(-\delta(t)\xi+\eta))}\Big)^2\neq 0  $$ 
for all $t\in \Gamma$, $-\infty\leq \xi\leq +\infty$, 
$\eta_1(t)\leq \eta\leq \eta_2(t)$. }


\newpage

\begin{center}
	{\bf Z. Samsonia} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		N. Muskhelishvili Institute of Computational Mathematics \\
		Georgian Academy of Sciences \\
		Tbilisi, Georgia} \\[0.5cm]
	{\bf ON A BOUNDARY PROBLEM FOR ELLIPTIC SYSTEMS AND 
		ITS APPLICATION IN THE MAPPING PRACTICE}
\end{center}

\bigskip

	A uniformly elliptic first order linear system with coefficients
of certain smoothness is considered in a finite $n$-connected ($n\ge 2$) 
domain bounded with piecewise analytic contours. The so-called mixed boundary
problem is solved for such systems.

	It is shown that the solution of the above problem makes it possible 
to construct related quasiconformal mappings. For practical reasons the case
is especially considered where the domain is either doubly or triply connected
and is mapped to concrete canonical domains.

\newpage

\begin{center}
	{\bf J. Sanikidze} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		N. Muskhelishvili Institute of Computational Mathematics \\
		Georgian Academy of Sciences \\
		Tbilisi, Georgia} \\[0.5cm]
	{\bf ON APPROXIMATE SOLUTION OF SOME SINGULAR INTEGRAL EQUATIONS}
\end{center}

\bigskip

	As is known, one of the convenient methods of investigation 
of bo\-un\-da\-ry value problems of mathematical physics and elasticity 
theory is the application of Cauchy type integrals, 
which reduces the original problems to certain boundary integral equations. 
For their solution the schemes of approximation of singular integrals prove 
to be quite effective.

	In the present report questions of numerical solution of a number
of concrete boundary value problems are considered, 
based on representation of the solution in the potential form $\RRe \Phi(z)$, 
$\Phi(z)={1\over\pi}\int\limits_L{\varphi(t) \over t-z}dt$ 
(where $z$ belongs to the considered domain in the complex plane, and
$\varphi$ is an unknown real function).

\newpage

\begin{center}
	{\bf N. Shavlakadze} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		A. Razmadze Mathematical Institute,
		Georgian Academy of Sciences \\ 
		Tbilisi, Georgia} \\[0.5cm]
	{\bf THE BENDING PROBLEM FOR THE FINITE PLATE WITH 
		THE ELASTIC INCLUSION}
\end{center}

\bigskip

\setcounter{equation}{0}
	The problem of the isotropic, circle formed plate's bending 
is considered when it contains the thin elastic inclusion of variable 
rigidity and the plate's contour is bound rigidly.

	The existence of the inclusion implies the skip of the across force: \linebreak 
$<\!N_y\!>=\mu(x)$, $|x|<a$, while plate's bending, the angle of
rotation and the moment of bending are continuous when going over the segment
of the inclusion. The equilibrium equation of the inclusion has the following
form: 
%	(1)
\begin{gather}
	\frac{d^2}{dx^2}\,D_0(x)\,\frac{d^2}{dx^2}\,\omega_0(x)=-\mu(x), 
			\;\;\; |x|<a, \\
	\big[D_0(x)\omega_0''(x)\big]\,\Big|_{x=\pm a}=0, \;\;\; 
		\big[D_0(x)\omega_0''(x)\big]'\,\Big|_{x=\pm a}=0, \notag 
\end{gather}
where $D_0(x)$ is the inclusion rigidity and $\omega_0(x)$ is its bending.

	Using the methods of the theory of analitical functions and 
taking into account the contact conditions, we get the following 
integro-differential equation with respect to the sought for 
function $\mu(x)$:
%	(2)
\begin{gather}
	\nu(x)-\frac{D_0(x)}{8\pi D}\int_{-a}^a \frac{v'(t)\,dt}{t-x}+
		\frac{D_0(x)}{8\pi D}\int_{-a}^a R(x,t)v'(t)\,dt= \notag \\
	=D_0(x)f(x), \;\;\; |x|<a,
\end{gather}
where $v''(x)\equiv \mu(x)$, $v(\pm a)=0$, $v'(\pm a)=0$, $D$ is the
cilindrical rigidity of the plate, $R(x,t)$ is the regular kernel in the 
square $-a\leq (x,t)\leq a$. The characteristic part of the equation (2) 
under certain conditions can be reversed and reduced a Fredholm's 
integral equation of the second kind.

\newpage

\begin{center}
	{\bf D. Shulaia} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		I. Vekua Institute of Applied Mathematics, 
		Tbilisi State University \\
		Tbilisi, Georgia} \\[0.5cm]
	{\bf HILBERT--SHMIDT TYPE THEOREMS IN THE LINEAR MULTIGROUP 
		TRANSPORT THEORY}
\end{center}

\bigskip

\setcounter{equation}{0}
	As it is well known, the Fourier transformation of the solution 
of the linear multigroup transport equations satisfy the following system 
of inhomogenous integral equations
%	(A)
\begin{equation}
	(\nu I - \mu l)\widetilde{\phi}=
		\nu\int_{-1}^{+1} g\widetilde{\phi} d\mu'+f, \;\;\;
			\mu\in(-1;+1),	\tag{$A$}
\end{equation}
where $\nu$ is a parameter, $ l $ is a diagonal matrix, $g(\mu;\mu')\in H$
(the H\"{o}lder class) and $f(\mu)\in H^*$ (the Muskhelishvili class) 
are square and column matrices respectively.

	Using a result from [1], we study the possibility of representation 
of the solution of $(A)$ by regular and singular eigenfunctions 
of the homogeneous system (characteristic system) corresponding to $(A)$. 
Namely, the following theorem is true:

\vskip+0.2cm
{\bf Theorem.\/}
{\em If $ \nu \notin\{\nu_i\}\cup[-1;+1]$, then the unique solution 
of $(A)$ can be represented in the form 
\begin{align*}
	\widetilde{\phi}_{\nu}(\mu) & =
		\sum\limits_i\frac{\nu_i}{\nu-\nu_i}\phi_{\nu_i}(\mu)
			\frac{1}{N(\nu_i)}\int_{-1}^{+1}\phi_{\nu_i}(\mu')
				f(\mu')d\mu'+ \\
	& +\int_{-1}^{+1}\frac{t}{\nu-t}\phi_t(\mu)N^{-1}(t)
		\int_{-1}^{+1}\phi_t(\mu')f(\mu')d\mu'dt, 
\end{align*}
where $\phi_{\nu _i}^* $ are the regular eigenfunctions and 
$\phi_{\nu}^* $ are the singular ones of the adjoint characteristic system, 
the superscript $T$ means transpose, $N(\nu_i )$ and $N(\nu)$ 
are the normalisation coefficient and matrix, respectively. }

\vskip+0.8cm
\centerline{\sc References}
\vskip+0.4cm

1. D. Shulaia and O. Tskhadaia, Seminar of I. Vekua Institute of Applied 
Mathematics Reports. {\em Tbilisy University Press, Tbilisi} {\bf 23}(1997). 

\newpage

\begin{center}
	{\bf K. Svanadze} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		A. Tsereteli Kutaisi State University\\
		Kutaisi, Georgia} \\[0.5cm]
	{\bf ON SOLUTIONS OF TWO-DIMENSIONAL BOUNDARY VALUE PROBLEMS 
		OF STABLE OSCILLATIONS OF THE THEORY OF ELASTIC MIXTURES}
\end{center}

\bigskip

\setcounter{equation}{0}
	In the work the basic boundary value problems of the linear theory 
of two isotropic similar mixtures are investigated. 
The equations are of the form 
\begin{align*}
	a_1\Delta u'+b_1\grad\dv u'+c\Delta u''+d\grad\dv u''+
		\rho_1\sigma^2u' & =\Psi', \\
	c\Delta u'+d\grad\dv u'+a_2\Delta u''+b_2\grad\dv u''+
		\rho_2\sigma^2u' & =\Psi'',
\end{align*}
where $\Delta$ is the Laplace operator, $\rho_1$ and $\rho_2$ are the partial
densities, $\Psi'=\{\Psi_1',\Psi_2'\}$ and $\Psi''=\{\Psi_1'',\Psi_2''\}$ 
are the mass forces, $u'=\{u_1',u_2'\}$ and $u'=\{u_1'',u_2''\}$ 
are partial displacements depending on the variables $x_1$ and $x_2$,
$\sigma^2$ is the oscillation frequency, $a_1$, $a_2$, $b_1$, $b_2$, $c$, $d$ 
are constants which charaterise mechanical properties of the elastic mixture 
in question. They satisfy certain conditions.

	With the aid of the potential method and the theory of singular 
integral equations, existence and uniqueness theorems are proved for basic 
external boundary value problems of stable oscillations in the case where 
either the displacement or the generalized stress vector is known 
on the boundary.


\newpage

\begin{center}
	{\bf M. Svanadze} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		I. Vekua Institute of Applied Mathematics, 
		Tbilisi State University \\
		Tbilisi, Georgia} \\[0.5cm]
	{\bf THE CONNECTION BETWEEN PLANE WAVES AND EIGENFREQUENCIES
		IN THE THEORY OF TWO-COMPONENT ELASTIC MIXTURES}
\end{center}

\bigskip

	Existence theorems of eigenfrequencies of interior homogeneous
boundary value problems of steady oscillations in the theory of 
two-com\-po\-nent elastic mixtures are proved. Some basic properties of wave
numbers are treated. The connection between plane waves and
eigenfrequencies is established.


\newpage

\begin{center}
	{\bf T. Vashakmadze} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		I. Vekua Institute of Applied Mathematics, 
		Tbilisi State University \\
		Tbilisi, Georgia} \\[0.5cm]
	{\bf ON REFINED THEORIES OF ELASTIC SHELLS}
\end{center}

\bigskip

	On the STAMM-94 [1] we formulated some unsolved mathematical problems 
of nonlinear solid mechanics. Below we continue to adduce some new results 
which substantiate the possibility of research and 
solving of the corresponding mathematical problems.

	1. A class of refined theories in Reissner -- von Karman sense is 
constructed with control parameters for elastic shallow shells and 
geometrical and physical simplifying assumptions. 
Same problems will be discussed also for non-shallow cases.

	{\bf 2.\/} Numerical processes and characterizing data of a design 
of some problems for refined theories of elastic plates with boundary layers
will be adduced.

\vskip+0.8cm
\centerline{\sc References}
\vskip+0.4cm

1. T. Vashakmadze, Some mathematical problems of the theory of nonlinear 
elasticity. {\em Trends in Appl. of Math. to Mech. Pitman. Monographs $\&$ 
Surv. Pure and Appl. Math.} {\bf 77}(1995) {\em Longman}: 348--357. 

\newpage

\begin{center}
	{\bf D. Zeragia} \\[0.2cm]
	{\size{8}{14pt}\selectfont \bf 
		N. Muskhelishvili Institute of Computational Mathematics \\
		Georgian Academy of Sciences \\
		Tbilisi, Georgia} \\[0.5cm]
	{\bf ON THE BIFURCATION PROBLEM OF A STRAIGHT ROD UNDER TORSION}
\end{center}

\bigskip

	We investigate the problem on bifurcation of equalibrium forms 
of the axis of a rod which is immovably fastened at one end, 
has a free cylindrical fixing at another one and is loaded 
by the torsion moment $H$.

	The problem is described by a boundary-value problem for a system 
of three non-linear ordinary differential equations dependent 
on five real parameters, $H$ being the main one. 
The corresponding branching equation is two-dimensional with 
a small real parameter.

	It turns out that bifurcation points not always appear; 
we establish conditions when this is the case.

\newpage
\setcounter{page}{59}

\twocolumn[\begin{center}\large\bf Author index \\ \ \\ \ \end{center}]

Ashordia, M.,		\quad {11}	%	1

Avalishvili, G.,		\quad {39}	%	3

Bantsuri, R.,		\quad {3}	%	3

Basheleishvili, M.,		\quad {4}	%	3

Bezhuashvili, Yu.,	\quad {33}	%	3

Bochorishvili, R.,	\quad {12}	%	2

Buchukuri, T.,		\quad {34}	%	3

Burchuladze, T.,		\quad {34, 35}	%	3

Chkadua, O.,		\quad {36}	%	3

Duduchava, R.,		\quad {5}	%	3

Dzhishkariani, A.,	\quad {37}	%	3

Gachechiladze, R.,	\quad {38}	%	3

Gagnidze, A.,	\quad {13}	%	3

Gelashvili, K.,	\quad {14}	%	3

Giorgashvili, L.,	\quad {33}	%	3

Gordeziani, D., 		\quad {39}	%	3

Gorgodze, N., 		\quad {17}

Gvazava, J.,		\quad {15}	%	2

Jaiani, G.,		\quad {40}	%	23

Jangveladze, T.,	\quad {16}

Kapanadze, G.,		\quad {41}	%	3

Kharatishvili, G.,	\quad {17}	%	1

Kharibegashvili, S.,	\quad {18}	%	2

Khatiashvili, G.,	\quad {42}	%	3

Khatiashvili, N.,	\quad {43}	%	3

Khuskivadze, G., 	\quad {44}

Kiguradze, I.,		\quad {6}	%	1

Kiguradze, T.,		\quad {19}	%	2

Kokilashvili, L.,	\quad {20}	%	1

Kokilashvili, V.,	\quad {21}	%	2

Koplatadze, R.,		\quad {22}	%	1

Kukudzhanov, S.,		\quad {45}	%	3

Kukudzhanov, V.,		\quad {46}	%	3

Kvinikadze, G.,		\quad {23}	%	1

Maisaia, O.,		\quad {38}	%	3

Mania, M.,		\quad {24}	%	2

Meunargia, T.,		\quad {47}	%	3

Mukhigulashvili, S.,	\quad {25}

Mzhavanadze, Sh.,	\quad {48}

Natroshvili, D.,		\quad {7}	%	3

Novikov, A., 		\quad {24}	%	1

Obolashvili, E.,		\quad {26}	%	2

Paatashvili, V.,		\quad {44}	%	2

Partsvania, N.,		\quad {27}	%	1

Peradze, J.,		\quad {49}	%	3

Rukhadze, R.,		\quad {35}	%	3

Saginashvili, A.,	\quad {50}	%	3

Samsonia, Z.,		\quad {51}	%	3

Sanikidze, J.,		\quad {52}	%	3

Schneiderman, D.,		\quad {46}	%	3

Shavlakadze, N.,		\quad {53}	%	3

Shulaia, D.,		\quad {54}	%	3

Sokhadze, Z.,		\quad {28}	%	1

Surguladze, T.,		\quad {29}	%	3

Svanadze, K.,		\quad {55}	%	3

Svanadze, M.,		\quad {56}	%	3

Tadumadze, T.,		\quad {17}	%	1

Tsintsadze, Z.,		\quad {30}	%	1

Vashakmadze, T.,		\quad {57}	%	3

Zeragia, D.,		\quad {58}	%	3



\end{document}
