Problems for partial differential equations with nonlocal conditions. Metric approach to the problem of small denominators

Abstract

UDC 517.95A survey of works of the authors and their disciples devoted to the investigation of problems with nonlocal conditions with respect to a selected variable in cylindrical domains is presented. These problems are considered for linear equations and systems of partial differential equations that, in general, are ill posed in the Hadamard sense and whose solvability in certain scales of functional spaces is established for almost all (with respect to Lebesgue measure) vectors composed of the coefficients of the problem and the parameters of the domain.

An integral-boundary value problem for a partial differential equation of second order

“…In Section 3, the conditions for unique solvability of boundary value problems with integral condition (17), (18) and (19), (20) are established. For fixed v(t, x) , w(t, x), u(t, x) in each step of the algorithm we solve the boundary value problems with integral condition (15), (12) and (16), (11). For fixedλ(x) ,μ(t), λ(x) , µ(t), we solve the Goursat problem (4), (6).…”

“…Thus, we have a closed system of equations (4)-(6), (15), (12), (16), and (11) for determining unknown…”

“…boundary value problems (15), (12) and (16), (11). The unknowns are both v(t, x), w(t, x), u(t, x), andμ(t) ,…”

“…The constructed algorithm consists of two parts: we solve the boundary value problems with integral condition for the ordinary differential equations (15), (12) and (16), (11) in the first part, and we solve the Goursat problem for hyperbolic equations with functional parameters in the second part.…”

Multipoint nonlocal problem for factorized equation with dependent coefficients in conditions

“…To solve this problem we use the metric approach [5] to estimations from below of small denominators.…”