Conductivity Equation

Combining the steady-state Continuity Equation $\nabla \cdot \mathbf{J}=0$ with Ohm’s law $\mathbf{J}=-\gamma \nabla u$ gives the conductivity equation

$$\nabla \cdot (\gamma \nabla u)=0\text{in }\Omega ,$$

for the electric potential $u:\Omega \to \mathbb{R}$ in a bounded Lipschitz domain $\Omega \subset {\mathbb{R}}^n$ ($n=2,3$). The conductivity $\gamma \in L^{\infty }(\Omega )$ is assumed bounded above and below,

$$0<{\gamma }_{\min }\le \gamma (x)\le {\gamma }_{\max }<\infty \text{a.e.},$$

which makes the operator uniformly elliptic. For constant $\gamma$ the equation reduces to the Laplace equation $\Delta u=0$.

On its own the equation has infinitely many solutions. A unique solution needs a boundary condition: either the voltage (Dirichlet Problem) or the current (Neumann Problem). Physically realistic electrodes lead to the Robin-type conditions of the Complete Electrode Model.

The weak form is derived in Weak Formulation of the Conductivity Equation. Existence and uniqueness follow from the Lax-Milgram Theorem.

References

1. L. C. Evans (2010). Partial Differential Equations, 2nd ed. AMS Graduate Studies in Mathematics 19. doi:10.1090/gsm/019
2. L. Borcea (2002). Electrical impedance tomography. Inverse Problems 18(6), R99–R136. doi:10.1088/0266-5611/18/6/201