The Helmholtz equation in a plane with different conditions posed on different sides of the cuts

A boundary problem for the Helmholtz equation outside cuts in a plane is studied. The Dirichlet condition is posed on one side of each cut while the impedance boundary condition is posed on the other side. The existence and uniquness of the boundary problem solution are proved. An integral representation of the solution in terms of potentials is obtained. The densities in potentials are found by solving a system of Fredholm integral equations of the second kind which is uniquely solvable.