Compact imbeddings in electromagnetism with interfaces between classical materials and metamaterials

In a metamaterial, the electric permittivity and/or the magnetic permeability can be negative in given frequency ranges. We investigate the solution of the time-harmonic Maxwell equations in a composite material, made up of classical materials, and metamaterials with negative electric permittivity, in a two-dimensional bounded domain O. We study the imbedding of the space of electric fields into L²(ωa)2. In particular, we extend the famous result of Weber, proving that it is compact. This result is obtained by studying the regularity of the fields. We first isolate their most singular part, using a decomposition ́a la Birman and Solomyak. With the help of the Mellin transform, we prove that this singular part belongs to H ^s(ω)2 for some s > 0. Finally, we show that the compact imbedding result holds as soon as no ratio of permittivities between two adjacent materials is equal to -1.