What is the appropriate 3D model?

Problem 1: Homogenization of a Periodic Conducting medium

Solve

$$\nabla^2 \phi = 0 \nonumber$$

in $\Omega$, subject to

$$n\cdot {1\over r_c} \nabla \phi = I_m(z,x)\nonumber$$

on the boundary $\partial \Omega_m$, where $I_m(z,x)$ is periodic in $z = {x\over \epsilon}$.

Using multiple scale analysis, we find

$$\phi ={1\over \epsilon} \Phi(x) + W(z) \cdot T^{-1}\nabla_x \Phi(x) + O(\epsilon \Phi). \nonumber$$

where $\Phi$ satisfies the equation

$$\nabla \cdot (\sigma \nabla \Phi) = -{\epsilon\over V}\int_{\partial \Omega_m}I_m(z,x) dS_z\nonumber$$

and $T$ is a coordinate transformation matrix.