Generalized impedance boundary condition at high frequency for a domain with thin layer: the circular case

Consider a conducting disk surrounded by a thin dielectric layer submitted to an electric field at the pulsation omega. The conductivity of the layer grows like omega(1-gamma), gamma is an element of [0,1], when the pulsation w tends to infinity. Using a pseudodifferential approach on the torus, we build an equivalent boundary condition with the help of an appropriate factorization of Helmholtz operator in the layer. This generalized impedance condition approximates the thin membrane in the high frequency limit for small thickness of the layer. L-2-error estimates are given and we illustrate our results with numerical simulations. This work extends, in the circular geometry, previous works of Lafitte and Lebeau (Lafitte O. Lebeau G. 1993, Equations de Maxwell et operateur d'impedance sur le bord Wan obstacle convexe absorbant. Comptes Rendus de l' Academic dis Science, Paris, Serie I, Mathematiques, 316(11), 1177-1182); (Lafitte O.D., 1999, Diffraction in the high frequency regime by a thin layer of dielectric material. 1. The equivalent impedance boundary condition. SIAM Journal on Applied Mathematics, 59(3), 1028-1052 (electronic)) in which gamma identically equals zero.