Two-scale analysis for very rough thin layers. An explicit characterization of the polarization tensor

We study the behaviour of the steady-state voltage potential in a material composed of a two-dimensional object surrounded by a very rough thin layer and embedded in an ambient medium. The roughness of the layer is described by a quasi $\eps$--periodic function, $\eps$ being a small parameter, while the mean thickness of the layer is of magnitude $\eps^\beta$, where $\beta\in(0,1)$. Using the two-scale analysis, we replace the very rough thin layer by appropriate transmission conditions on the boundary of the object, which lead to an explicit characterization of the polarization tensor of Vogelius and Capdeboscq (ESAIM:M2AN. 2003; 37:159-173). This paper extends the previous works Poignard (Math. Meth. App. Sci. 2009; 32:435-453) and Ciuperca \textsl{et al.} (Research report INRIA RR-6812), in which $\beta\geq1$