Existence, uniqueness and regularity for solutions of the conical diffraction problem

This paper is devoted to the analysis of two Helmholtz equations in R-2 cou pled via quasiperiodic transmission conditions on a set of piecewise smooth interfaces. The solution of this system is quasiperiodic in one direction and satisfies outgoing wave conditions with respect to the other direction. It is shown that Maxwell's equations for the diffraction of a time-harmoni c oblique incident plane wave by periodic interfaces can be reduced to prob lems of this kind. The analysis is based on a strongly elliptic variational formulation of the differential problem in a bounded periodic cell involvi ng nonlocal boundary operators. We obtain existence and uniqueness results for solutions corresponding to electromagnetic fields with locally finite e nergy. Special attention is paid to the regularity and leading asymptotics of solutions near the edges of the interface.