The Maxwell system in waveguides with several cylindrical outlets to infinity and nonhomogeneous anisotropic filling

Abstract:

A waveguide occupies a domain $G$ in $\mathbb {R}^3$ with several cylindrical outlets to infinity; the boundary $\partial G$ is assumed to be smooth. The dielectric $\varepsilon$ and magnetic $\mu$ permittivities are matrix-valued functions smooth and positive definite in $\overline {G}$. At every cylindrical outlet, the matrices $\varepsilon$ and $\mu$ tend, at infinity, to limit matrices independent of the axial variable. The limit matrices can be arbitrary smooth and positive definite matrix-valued functions of the transverse coordinates in the corresponding cylinder. In such a waveguide, the stationary Maxwell system with perfectly conducting boundary conditions and a real spectral parameter is considered. In the presence of charges and currents, the corresponding boundary value problem with radiation conditions turns out to be well posed. A unitary scattering matrix is also defined. The Maxwell system is extended to an elliptic system. The results for the Maxwell system are derived from those obtained for the elliptic problem.