Information on the thesis

The thesis deals with boundary-value problems with local and nonlocal two-point
and local multi-point conditions by time variable and certain conditions (periodi-
city, Dirichlet, innite behavior) by spatial variable for linear partial dierential
equations in the limited and unlimited two-dimensional domains. In general, such
problems are conditionally correct, because the solvability of the problems is related
with the problem of small denominators. This problem is that the denominators of
the coecients of the series that represent the solutions of the problems can be
arbitrarily small for an innite number of members of a series and this causes the
divergence of the series in the corresponding functional spaces.
The dissertation identies classes of linear homogeneous partial dierential equa-
tions and classes of boundary conditions (local and nonlocal in time variable) for whi-
ch the problem of small denominators is absent. One spatial variable is characteristic
of these problems.
In the paper, reviews the works, which connected with the dissertation research
is conducted and the directions of consideration of problems that remain unstudied
are resulted. Sobolev spaces and the spaces of exponential type for functions with
one spatial variable are described.
The conditions for the unambiguous solvability of the Dirichlet problem for the
partial dierential equation, that considered in the rectangle, and also for the local
two-point problem for the high order equation, that studied in the unlimited strip,
are received.
The conditions for correct solvability of nonlocal boundary value problems for
linear partial dierential equation in the plane (bounded and unbounded by spatial
variable) domains are established. A boundary value problem with nonlocal two-
point conditions by a time variable for dierential-operator equation in a complex
domain is investigated.
The necessary and sucient conditions for uniqueness and sucient conditions
for the existence of a solution of multipoint problems in the spaces of exponential
type are founded.
Constructive formulas for solutions of the problems in the form of Fourier series
(or integrals) or Laurent series are obtained.
The conditions for the coecients of the equations and for the parameters of
the boundary conditions are described, the fulllment of which avoids the problem
of small denominators and ensures the correct solvability of these problems in the
plane domains.
The asymptotic estimates below for the characteristic determinants of the pro-
blems, that based on the method of selection of the dominant additive in their
development are installed. The methods of two-sided estimations of values of functi-
ons of the roots (characteristic polynomials), which arising at the construction of
solutions of problems, are developed.
The results of the thesis are of theoretical importance. They can be used in further
researches of the boundary-value problems for the partial dierential equations and
system of such equations and also in the study of specic problems of practice which
are modeled by considered problems.
Key words: partial dierential equation, boundary-value problem, two-dimen-
sional domain, multi-point conditions, nonlocal conditions, Dirichlet-type condi-
tions, characteristic polynomial, discriminant of a polynomial, resultant of two
polynomials, Fourier series, Laurent series, Fourier integral.