The Helmholtz equation with different impedance boundary conditions on different sides of slits in a plane

The boundary value problem for the Helmholtz equation is studied outside slits in a plane. The impedance boundary conditions are specified on the slits. In general, the impedance conditions may be different at different sides of each slit. In a particular case, the impedance conditions may be the same on both sides of each slit. We prove that the classical solution to the problem exists, and it is unique. We obtain the integral representation for a solution to the problem in the form of potentials, the densities in which are uniquely determined from the uniquely solvable system of the Fredholm integral equations of the second kind and index zero.