Junction of ferroelectric thin cylinders

In this paper, starting from a non-convex and nonlocal $3D-$variational model for the electric polarization in a ferroelectric material, and using an asymptotic process based on dimensional reduction, we analyze junction phenomena for two vertical joined ferroelectric thin cilynders with differents small thicknesses. We obtain three variational models for thin wire joined to a thin film, depending on the limit of the ratio between these two thicknesses.