On the conditions for the solvability of boundary-value problems for a high-order equation with a discontinuous coefficient

TL;DR

This paper establishes necessary and sufficient conditions for the uniqueness and existence of solutions to a high-order boundary value problem with discontinuous coefficients, addressing the 'small denominators' issue and providing specific examples.

Contribution

It introduces new criteria for the solvability of high-order boundary value problems with discontinuous coefficients, including conditions to avoid the 'small denominators' problem.

Findings

Necessary and sufficient conditions for solution uniqueness.

Conditions for the separability of 'small denominators' from zero.

An example of a non-solvable boundary value problem using Fourier method.

Abstract

The paper considers a boundary value problem for the high-order Lavrent'ev-Bitsadze equation. Necessary and sufficient conditions for the uniqueness of the solution are found. When substantiating the existence, the problem of "small denominators" arises. Sufficient conditions for the separability of the "small denominator" from zero are found. An example of a boundary value problem not solvable by the Fourier method is given, in the cases rectangle with integer side measurements.