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Consider Poisson equation $\nabla \cdot (\sigma(x)\nabla u)=0$ in a domain $D$, where $\sigma(x)$ is the spatially dependent conductivity. On the boundary we have $2$ electrodes $E_1$ and $E_2$ (Dirichlet BC $u=0$ on $E_1$ and $u=1$ on $E_2$). And the rest of the boundary is insulating material $du/d\vec n=0$ (Neumann BC). The electrodes do not have any contact impedance. This would be an ordinary Ohmic resistor with resistance R.

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 $\sigma(x)$

I am looking for proof or disproof that $R_J$ is minimized when $J$ is obtained by solving the Poisson equation. I am also looking for some references on this idea or concept.

Consider Poisson equation $\nabla \cdot (\sigma(x)\nabla u)=0$ in a domain $D$, where $\sigma(x)$ is the spatially dependent conductivity. On the boundary we have $2$ electrodes $E_1$ and $E_2$ (Dirichlet BC $u=0$ on $E_1$ and $u=1$ on $E_2$). And the rest of the boundary is insulating material $du/d\vec n=0$ (Neumann BC). The electrodes do not have any contact impedance. This would be an ordinary Ohmic resistor with resistance R.

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 $\sigma(x)$

I am looking for proof or disproof to the conjecture that $R_J$ is minimized when $J$ is obtained by solving the Poisson equation. I am also looking for some references on this idea or concept.

Consider Poisson equation $\nabla \cdot (\sigma(x)\nabla u)=0$ in a domain $D$, where $\sigma(x)$ is the spatially dependent conductivity. On the boundary we have $2$ electrodes $E_1$ and $E_2$ (Dirichlet BC $u=0$ on $E_1$ and $u=1$ on $E_2$). And the rest of the boundary is insulating material $du/d\vec n=0$ (Neumann BC). The electrodes do not have any contact impedance. This would be an ordinary Ohmic resistor with resistance R.

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 $E_2$ is 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 thickness dt2, and we take the limit as dt2 goes to zero aswell. The resistance of each resistor of thickness $dt*dt2$ is computed from $\sigma(x)$

I am looking for proof or disproof that $R_J$ is minimized when $J$ is obtained by solving the Poisson equation. I am also looking for some references on this idea or concept.

Consider Poisson equation $\nabla \cdot (\sigma(x)\nabla u)=0$ in a domain $D$, where $\sigma(x)$ is the spatially dependent conductivity. On the boundary we have $2$ electrodes $E_1$ and $E_2$ (Dirichlet BC $u=0$ on $E_1$ and $u=1$ on $E_2$). And the rest of the boundary is insulating material $du/d\vec n=0$ (Neumann BC). The electrodes do not have any contact impedance. This would be an ordinary Ohmic resistor with resistance R.

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 $\sigma(x)$

I am looking for proof or disproof that $R_J$ is minimized when $J$ is obtained by solving the Poisson equation. I am also looking for some references on this idea or concept.

Consider Poisson equation $\nabla \cdot (\sigma(x)\nabla u)=0$ in a domain $D$, where $\sigma(x)$ is the spatially dependent conductivity. On the boundary we have $2$ electrodes $E_1$ and $E_2$ (Dirichlet BC $u=0$ on $E_1$ and $u=1$ on $E_2$). And the rest of the boundary is insulating material $du/d\vec n=0$ (Neumann BC). The electrodes do not have any contact impedance. This would be an ordinary Ohmic resistor with resistance R.

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 $E_2$ is 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 thickness dt2, and we take the limit as dt2 goes to zero aswell. The resistance of each resistor of thickness $dt*dt2$ is computed from $\sigma(x)$

I am looking for proof or disproof that $R_J$ is minimized when $J$ is obtained by solving the Poisson equation.

Consider Poisson equation $\nabla \cdot (\sigma(x)\nabla u)=0$ in a domain $D$, where $\sigma(x)$ is the spatially dependent conductivity. On the boundary we have $2$ electrodes $E_1$ and $E_2$ (Dirichlet BC $u=0$ on $E_1$ and $u=1$ on $E_2$). And the rest of the boundary is insulating material $du/d\vec n=0$ (Neumann BC). The electrodes do not have any contact impedance. This would be an ordinary Ohmic resistor with resistance R.

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 $E_2$ is 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 thickness dt2, and we take the limit as dt2 goes to zero aswell. The resistance of each resistor of thickness $dt*dt2$ is computed from $\sigma(x)$

I am looking for proof or disproof that $R_J$ is minimized when $J$ is obtained by solving the Poisson equation. I am also looking for some references on this idea or concept.