\,\, \left\{ \begin{array}{l} \displaystyle- \frac{\partial}{\partial x} \left( \bar{A}_\perp \frac{\partial \bar{\phi}}{\partial x}\right)   = \bar{f}(x) \,,\quad \text{in} \quad \Omega_x\,, \\[0mm]  \displaystyle \bar{\phi} = 0 \quad \text{on} \quad \partial \Omega_x \,