The direct electromagnetic scattering problem from an imperfectly conducting cylinder at oblique incidence

We consider the scattering of a time-harmonic electromagnetic wave by an imperfectly conducting infinite cylinder at oblique incidence. We assume that the cylinder is embedded in an inhomogeneous medium, but both the cylinder and the medium are uniform along the axis of the cylinder. Since the x components and y components of electric field and magnetic field can be expressed in terms of their z components if the cylinder is parallel to the z axis, we can derive from Maxwell's equations and the Leontovich impedance boundary condition that our scattering problem is modeled as a boundary value problem for their z components with oblique boundary condition. Using Rellich's lemma, the uniqueness of solutions to the boundary value problem is justified. To show the existence of its solution, the Lax-Phillips method is used. The key point for that is to prove the solvability of the associated oblique derivative problem in a bounded domain consisting of two boundaries which are the boundary of the cross-section of the cylinder with coupled oblique boundary condition and that of a domain containing this cross-section with Dirichlet boundary condition.