Electromagnetic wave scattering on rough surface embedded by bounded open cavities

The paper is devoted to the electromagnetic wave scattering by bounded open cavities embedded in the infinite rough surface. The medium in the cavities is inhomogeneous. The propagation of the electromagnetic wave is ruled by the time-harmonic Maxwell’s equations ∇ ×E − iωμH = 0, ∇ ×H + iωɛE = 0 in Ω∪R+3. The imaginary part of the dielectric coefficient ɛ is assumed nonnegative. To search for the solution of the equation, we define the functional space in the whole space by extending the functions to be zero outside of the cavity Ω, which make it possible to find the transparent boundary condition and define the boundary operator. And then we use the Hodge-decomposition to restrict the solution in some weighted curl space. After reducing the problem to its variational form, it’s possible to use the Lax–Milgram Theorem and the Fredholm alternative to obtain the unique solution of the equation.