The Dirichlet problem for inhomogeneous mixed-type equation with Lavrent'ev-Bitsadze operator

Abstract

The first boundary value problem for a mixed type equation with Lavrent'ev-Bitsadze operator in a rectangular domain is studied. We show that well-posedness of the problem depends substantially on the ratio of the sides of the rectangle from the hyperbolic part of the mixed domain. A criterion for uniqueness of a solution is established. The solution is constructed as the Fourier series. The justification of uniform convergence of the series leads to the problem of small denominators. In this regard, we give estimates for small denominators to be separated from zero, the corresponding asymptotic formulas are obtained. These estimates are applied to show the convergence of the series in the class of regular solutions of this equation. Estimates for stability of the solution with respect to given boundary functions and the right-hand side are established.