Compact imbeddings in electromagnetism with interfaces between classical materials and meta-materials

Lucas Chesnel (1) , Patrick Ciarlet (1)

1 POEMS - Propagation des Ondes : Étude Mathématique et Simulation

In a meta-material, 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 meta-materials with negative electric permittivity, in a two-dimensional bounded domain Ω. We study the imbedding of the space of electric fields into L²(Ω)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 à la Birman-Solomyak. With the help of the Mellin transform, we prove that this singular part belongs to Hˢ(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.

Mots clés: Maxwell's equations; interface problem; meta-material; compact imbedding; regularity of fields; Mellin transform

Domaines: Analyse fonctionnelle [math.FA]

Lucas Chesnel, Patrick Ciarlet. Compact imbeddings in electromagnetism with interfaces between classical materials and meta-materials. 2011. ⟨hal-00602904⟩