On the conditions of the solvability of boundary value problems for a single high-order equation with variable coefficients

Abstract

A Dirichlet-type problem is studied for an equation of even order with variable coefficients. A criterion for the uniqueness of a solution is given. The solution is built in the form of a Fourier series. When justifying the convergence of the series, the problem of small denominators arises. Sufficient conditions for the separability of the denominator from zero are obtained. It is shown that the solvability of the problem is influenced not only by the dimension of the rectangle, but also by the orders of the specified derivatives at the lower boundary of the rectangle.