Homogenization of a nonlocal electrostatic equation

We find the effective (homogenized) properties of a composite (a heterogeneous material) supplied with spatially nonlocal constitutive relations. We homogenize an electrostatic equation in a periodic setting. The current density is given as a spatial convolution of the electric field with a conductivity kernel. It turns out that the homogenized equation also has a nonlocal constitutive relation if we do not scale the non-localness. However, if we decrease the neighborhood which influence the current density simultaneously as we make the fine structure scale finer and finer then we obtain a constitutive relation which is local.