On the dirichlet problem in a plane domain with a cut

In the paper, a solvability of a model elliptic pseudo-differential equation in a plane domain with a cut along a ray is studied. Solution is sought in the Sobolev–Slobodetskii space. Using a special factorization for elliptic symbol one writes out a general solution for the equation in a domain with cut sector; this solution includes an arbitrary function. Using the Diriclet condition one reduces finding this function to solution of a system of one-dimensional linear integral equations. Further, one studies a behavior of these equations when the size if sector tends to zero, and the sectors transforms into a ray. As a result one obtains a certain integral equation, and a unique solvability of the equation is equivalent to a solvability of the Dirichlet problem in a plane domain with cut ray.