Analyse spectrale et singularités d'un problème de transmission non coercif

A non elliptic spectral problem related to the analysis of superconducting micro-strip lines

“…This result is obtained thanks to a perturbation method, using the Dirichlet to Neumann operator. Let us emphasize here that in [3], more precise results about self-adjointness in non elliptic transmission problems have been obtained in the paticular case (and simpler compared to the one studied here) where the exterior of the strip is supposed to be homogeneous (see Rem. 2.5).…”

“…This particular case, and more generally the question of selfadjointness of non elliptic transmission operators through a regular or polygonal interface between two media has been investigated in [3]. For regular interfaces, it is proved in [3] that the problem is self-adjoint if and only if ε S /ε A = −1. For polygonal interfaces, it is proved that for some negative values of the contrast ε S /ε A constituting an interval containing −1, the operator A is not self-adjoint.…”

“…For the upper ones, the results of [3] can be applied, while for the lower ones, one needs to generalize the method proposed in [3] to the case of a three-media interface. Nevertheless, note that [3] already shows that our operator A is not self-adjoint (at least) for ε S ∈ I π/2 = [−3, −1/3], because of the singularities generated by the upper corners. Consequently, the constant c in Theorem 2.4 satisfies c > 3.…”

T-Coercivity for the Maxwell Problem with Sign-Changing Coefficients

“…Let us consider the operator for the electric eld A N (ω) : (8). The previous identity allows us to write…”

T-coercivity for scalar interface problems between dielectrics and metamaterials

“…With the help of integral equations, it was first proven in [8] by Costabel and Stephan that, when the interface Σ is smooth (of C 2 -class), problem (P) is well-posed in the Fredholm sense if, and only if, the contrast κ σ := σ 2 /σ 1 is different from −1. Second, the influence of corners over the interface was specifically studied in [4] (see also [9,22]). The authors proved that, when there is a single corner with a right angle on the interface, problem (P), with a right-hand side f in L 2 (Ω), is not well-posed in the Fredholm sense if, and only if, κ σ ∈ [−3; −1/3] (similar results can be obtained for any value of the angle).…”