Oscillating behaviour of the spectrum for a plasmonic problem in a domain with a rounded corner

Lucas Chesnel (1, 2) , Xavier Claeys (3, 4) , Sergei A. Nazarov (5)

1 DeFI - Shape reconstruction and identification
2 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
3 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
4 ALPINES - Algorithms and parallel tools for integrated numerical simulations
5 SPbU - St Petersburg State University

We investigate the eigenvalue problem −div(σ∇u) = λu (P) in a 2D domain Ω divided into two regions Ω±. We are interested in situations where σ takes positive values on Ω+ and negative ones on Ω−. Such problems appear in time harmonic electromagnetics in the modeling of plasmonic technologies. In a recent work [15], we highlighted an unusual instability phenomenon for the source term problem associated with (P): for certain configurations, when the interface between the subdo-mains Ω± presents a rounded corner, the solution may depend critically on the value of the rounding parameter. In the present article, we explain this property studying the eigenvalue problem (P). We provide an asymptotic expansion of the eigenvalues and prove error estimates. We establish an oscillatory behaviour of the eigenvalues as the rounding parameter of the corner tends to zero. We end the paper illustrating this phenomenon with numerical experiments.

Mots clés: corner; negative materials; plasmonic; metamaterial; asymptotic analysis; sign-changing coefficients

Domaines: Equations aux dérivées partielles [math.AP]

Lucas Chesnel, Xavier Claeys, Sergei A. Nazarov. Oscillating behaviour of the spectrum for a plasmonic problem in a domain with a rounded corner. ESAIM: Mathematical Modelling and Numerical Analysis, 2018, ⟨10.1051/m2an/2016080⟩. ⟨hal-01240977v2⟩