Scattering of time-harmonic electromagnetic plane waves by perfectly conducting diffraction gratings

Consider the scattering of time-harmonic electromagnetic plane waves by a doubly periodic surface in R 3. The medium above the surface is supposed to be homogeneous and isotropic with a constant dielectric coefficient, while the material below is perfectly conducting. This paper is concerned with the existence of quasiperiodic solutions for any frequency. Based on an equivalent variational formulation established by the mortar technique of Nitsche, we verify the existence of solutions for a broad class of incident waves including plane waves. The only assumption is that the grating profile is a Lipschitz biperiodic surface. Note that the solvability result of the present paper covers the resonance case where Rayleigh frequencies are allowed. Finally, non-uniqueness examples are presented in the resonance case and in the case of TE or TM polarization for classical gratings.