%0 Journal Article
%T Boundary layer correctors and generalized polarization tensor for periodic rough thin layers. A review for the conductivity problem
%A Poignard Clair
%J ESAIM : Proceedings
%D 2012
%I EDP Sciences
%R 10.1051/proc/201237004
%X We study the behaviour of the steady-state voltage potential in a material composed of a two-dimensional object surrounded by a rough thin layer and embedded in an ambient medium. The roughness of the layer is supposed to be ¦Å¦Á¨Cperiodic, ¦Å being the magnitude of the mean thickness of the layer, and ¦Á a positive parameter describing the degree of roughness. For ¦Å tending to zero, we determine the appropriate boundary layer correctors which lead to approximate transmission conditions equivalent to the effect of the rough thin layer. We also provide an explicit characterization of the polarization tensor as defined by Capdeboscq and Vogelius in [9]. The present paper revisits the previous works of the author [11, 13, 16, 17], and it also provides new results for the very rough case ¦Á > 1. Dans cet article, nous consid¨¦rons le probl¨¨me de conduction dans un domaine bidimensionnel compos¨¦ d¡¯une fine membrane rugueuse entourant un domaine conducteur, le tout plong¨¦ dans un milieu ambiant de conductivit¨¦ diff¨¦rente. La rugosit¨¦ de la membrane est suppos¨¦e ¦Å¦Á¨Cp¨¦riodique, ¦Å ¨¦tant l¡¯¨¦paisseur moyenne de la membrane, et ¦Á un param¨¨tre positif d¨¦crivant le degr¨¦ de rugosit¨¦. Nous d¨¦terminons des correcteurs de couche limite conduisant ¨¤ la construction de conditions de transmission approch¨¦es lorsque le param¨¨tre ¦Å tend vers zero. Nous donnons aussi une caract¨¦risation explicite du tenseur de polarisation d¨¦fini par Capdeboscq and Vogelius dans [9]. Cet article revisite des r¨¦sultats pr¨¦c¨¦dents de l¡¯auteur obtenus dans [11, 13, 16, 17], et pr¨¦sente de nouveaux r¨¦sultats pour le cas tr¨¨s rugueux ¦Á > 1.
%U http://dx.doi.org/10.1051/proc/201237004