The first boundary-value problem for an equation of mixed type with a singular coefficient

Abstract

We study the first boundary-value problem in a rectangle for an equation of mixed type with a singular coefficient. We establish a criterion for the uniqueness of solutions and construct the solution as the sum of a series in the system of eigenfunctions of a one-dimensional eigenvalue problem. Justifying the uniform convergence of the series encounters a problem of small denominators. To deal with this we obtain bounds for the separation of the small denominators from zero along with the corresponding asymptotic results. These bounds enable us to justify the convergence of the series in the class of regular solutions of the equation.