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.
Keywords:equation of mixed type, singular coefficient, Dirichlet problem, Keldysh problem,
survey, uniqueness, orthogonal series, small denominators, bounds, existence, stability.
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