T-coercivity for scalar interface problems between dielectrics and metamaterials

Some electromagnetic materials have, in a given frequency range, an effective dielectric permittivity and/or a magnetic permeability which are real-valued negative coefficients when dissipation is neglected. They are usually called metamaterials. We study a scalar transmission problem between a classical dielectric material and a metamaterial, set in an open, bounded subset of R, with d = 2, 3. Our aim is to characterize occurences where the problem is well-posed within the Fredholm (or coercive + compact) framework. For that, we build some criteria, based on the geometry of the interface between the dielectric and the metamaterial. The proofs combine simple geometrical arguments with the approach of T -coercivity, introduced by the first and third authors and co-worker. Furthermore, the use of localization techniques allows us to derive well-posedness under conditions that involve the knowledge of the coefficients only near the interface. When the coefficients are piecewise constant, we establish the optimality of the criteria. 1991 Mathematics Subject Classification. 35Q60,35Q61,35J20. The dates will be set by the publisher. Introduction In electromagnetism, one can model materials that exhibit real-valued strictly negative electric permittivity and/or magnetic permeability, within given frequency ranges. These so-called metamaterials, or left-handed materials, raise unusual questions. Among others, in a domain Ω of R (d = 2, 3), divided into a classical dielectric material and a metamaterial, proving the existence of electromagnetic fields, and computing them, is a challenging issue (see for instance [11, 19, 21, 23, 24]). For example, let us consider a problem in a twodimensional domain, set in the time-harmonic regime with pulsation ω > 0. Then, the transmission problems