On Ventcel-type transmission conditions for a Helmholtz problem with a non-uniform thin layer

We study the asymptotic behaviour of the electric field in the transverse magnetic (TM) mode, propagating in a material composed of a two-dimensional object surrounded by a thin layer of non-constant thickness and embedded in an ambient medium. Using an asymptotic expansion of the solution \(u^{\varepsilon }\) $u^{\epsilon }$ to the Helmholtz problem, we derive Ventcel-type transmission conditions on the limit interface \(\Gamma \) $\mathrm{\Gamma }$ modelling the effect of the thin layer with accuracy up to \(O(\varepsilon ^{2})\) $O({\epsilon }^2)$ .