TY - JOUR

T1 - Oscillating behaviour of the spectrum for a plasmonic problem in a domain with a rounded corner

AU - Chesnel, Lucas

AU - Claeys, Xavier

AU - Nazarov, Sergei A.

PY - 2018/7/1

Y1 - 2018/7/1

N2 - 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 [L. Chesnel, X. Claeys and S.A. Nazarov, Asymp. Anal. 88 (2014) 43-74], we highlighted an unusual instability phenomenon for the source term problem associated with (P): for certain configurations, when the interface between the subdomains 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.

AB - 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 [L. Chesnel, X. Claeys and S.A. Nazarov, Asymp. Anal. 88 (2014) 43-74], we highlighted an unusual instability phenomenon for the source term problem associated with (P): for certain configurations, when the interface between the subdomains 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.

KW - Asymptotic analysis

KW - Corner

KW - Metamaterial

KW - Negative materials

KW - Plasmonic

KW - Sign-changing coefficients

UR - http://www.scopus.com/inward/record.url?scp=85052548646&partnerID=8YFLogxK

U2 - 10.1051/m2an/2016080

DO - 10.1051/m2an/2016080

M3 - Article

AN - SCOPUS:85052548646

VL - 52

SP - 1285

EP - 1313

JO - ESAIM: Mathematical Modelling and Numerical Analysis

JF - ESAIM: Mathematical Modelling and Numerical Analysis

SN - 0764-583X

IS - 4

ER -