Information on the thesis

The thesis deals with the nonlocal boundary-value problems for the partial dif-
ferential equations and systems of such equations with the differential operator
B = (B₁, . . . , B_p ), where B_j ≡ z_j ∂/∂z_j, j = 1, . . . , p, which acts on the functions
of complex spatial variables z₁ , . . . , z_p.
А criterion for the unique solvability of these problems and а sufficient conditions
for the existence of its solutions are established in Sobolev scale of spaces of
functions of several complex variables, in the spaces of functions, which are
Dirichlet-Taylor series with fixed spectrum, and in the Hilbert Hörmander spaces
forming a refined Sobolev scale of spaces.
The proof of the solvability of the problem for the partial differential equation
with weakly nonlinear right-hand side is carried out within the Nash–Moser iterative
scheme. In this scheme, the important point is the construction of estimates of the
norms of inverse linearized operators in the appropriate spaces in each iteration.
These problems, in general, are conditionally correct, and their solvability related
to the problem of small denominators. By using the metric approach, we prove the
theorems on lower estimates of small denominators appearing in the construction
of solutions of the analyzed problems. They imply the unique solvability of the
problems for almost all (with respect to the Lebesgue measure) vectors formed by
the coefficients of the equations and the parameter of nonlocal conditions.
The results of the thesis are of theoretical importance. They can be used in
further researches of the nonlocal boundary-value problems for the partial differenti-
al equations and system of such equations and also in the study of specific problems
of practice which are modeled by considered problems.
Key words: partial differential equation, nonlocal problem, small denominators,
metric estimation, Nash–Moser iterative scheme, complex variables.