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Abstract. This paper focus on the properties of boundary layers in periodic homogenization in rectangular domains which are either fixed or have an oscillating boundary. Such boundary layers are highly oscillating near the boundary and decay exponentially fast in the interior to a non-zero limit that we call boundary layer tail. The influence of these boundary layer tails on interior error estimates is emphasized. They mainly have two effects (at first order with respect to the period ε): first, they add a dispersive term to the homogenized equation, and second, they yield an effective Fourier boundary condition.Résumé. Cet article est consacréà l'étude des couches limites en homogénéisation périodique dans des domaines rectangulaires qui, soit sont fixes, soit possèdent une frontière oscillante. Ces couches limites sont très oscillantes près du bord et décroissent exponentiellement viteà l'intérieur vers une limite non nulle que nous appelons queue de couche limite. L'influence de ces queues de couches limites pour l'obtention d'estimations d'erreur intérieures est miseà jour. Elles ont pour effets principaux (au premier ordre en ε, la période) : premièrement, d'ajouter un terme dispersif dans l'équation homogénéisée, deuxièmement, de conduireà une condition aux limites effective de type Fourier.

In this paper we derive a hierarchy of models for electrical conduction in a biological tissue, which is represented by a periodic array of period e of conducting phases surrounded by dielectric shells of thickness epsilon eta included in a conductive matrix. Such a hierarchy will be obtained from the Maxwell equations by means of a concentration process eta-->0, followed by a homogenization limit with respect to epsilon. These models are then compared with regard to their physical meaning and mathematical issues

We study a problem set in a finely mixed periodic medium, modelling electrical conduction in biological tissues. The unknown electric potential solves standard elliptic equations set in different conductive regions (the intracellular and extracellular spaces), separated by a dielectric surface (the cell membranes), which exhibits both a capacitive and a nonlinear conductive behaviour. Accordingly, dynamical conditions prevail on the membranes, so that the dependence of the solution on the time variable t is not only of parametric character. As the spatial period of the medium goes to zero, the electric potential approaches in a suitable sense a homogenization limit u0, which keeps the prescribed boundary data, and solves the equation [Formula: see text]. This is an elliptic equation containing a term depending on the history of the gradient of u0; the matrices B0, A1 in it depend on the microstructure of the medium. More exactly, we have that, in the limit, the current is still divergence-free, but it depends on the history of the potential gradient, so that memory effects explicitly appear. The limiting equation also contains a term ℱ keeping trace of the initial data.