Asymptotic behaviour of thin ferroelectric materials

We discuss the behaviour of a model of ferroelectric material represented by a thin cylinder with small thickness ν > 0. This model is described by a couple of Maxwell equations satisfied by the electromagnetic field (H,E) and electric polarization field P . We give a complete description of the limit model as ν → 0 in the linear case when (E,H) and P satisfy a SilverMüller type boundary condition. When the potential is nonlinear and P satisfies the boundary condition P × n = 0 we prove the strong convergence of the polarization field which allows to give the description of the behaviour of the nonlinear problem when the thickness ν tends to 0. We observe that the behaviour is very sensitive to the choice of the boundary conditions satisfied by the different fields.