Consider a domain $D$, where $\sigma(x)$ is the spatially dependent "conductivity". On the boundary we have $2$ "electrodes" $E_1$ and $E_2$ where matter flows in and out. The rest of the boundary is insulating material $J\cdot\vec n=0$ (Neumann BC).
To each divergence-free vector field $J(x)$ on $D$ which flow strictly from $E_1$ to $E_2$ we can assign a "resistance" $R_J$ as follows, because we can insert non-conducting walls along the streamlines, we can consider the streamlines from $E_1$ to $E_2$ as resistors of thickness dt, then $R_J$ is defined as the resistance of them all in parallel (so that would be $1/(1/R_1+1/R_2....))$, in the limit as dt goes to 0. Each of the streamline resistors is defined as a series connection of resistors (So thats $R_a+R_b...$) with length dt2, and we take the limit as dt2 goes to zero aswell. The resistance of each resistor of thickness $dt*dt2$ is computed from $\sigma(x)$
Let $K(x)$ be the $J(x)$ with smallest $R_J$.
What differential equation governs $K(x)$? (besides $\nabla \cdot K=0) $
Edit: This is probably incorrect because du is not constant, streamlines can come closer along the flow for instance, $R_J=1/(\int_{E_1} 1/(\int_{C_u} (1/\sigma(x))ds))du$
Where the u integral is along the length of the electrode, and $C_u$ is the streamline starting at u.