Volianska I. Boundary value problems for partial differential equations in the plane domains

The thesis deals with boundary-value problems with local and nonlocal two-point and local multi-point conditions by time variable and certain conditions (periodicity, Dirichlet, infinite behavior) by spatial variable for linear partial differential 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 coefficients of the series that represent the solutions of the problems can be arbitrarily small for an infinite number of members of a series and this causes the divergence of the series in the corresponding functional spaces. The dissertation identifies classes of linear homogeneous partial differential equations and classes of boundary conditions (local and nonlocal in time variable) for which 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 differential 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 differential 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 differential-operator equation in a complex domain is investigated. The necessary and sufficient conditions for uniqueness and sufficient 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 coefficients of the equations and for the parameters of the boundary conditions are described, the fulfillment 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 problems, that based on the method of selection of the dominant additive in their evelopment are installed. The methods of two-sided estimations of values of functions 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 differential equations and system of such equations and also in the study of specific problems of practice which are modeled by considered problems.