The large-volume uniqueness conjecture for the outer problem with high-conductivity core

Let $D$ be the core, let $V_0$ prescribe the volume of the outer domain, and let $\sigma_c$ be the core conductivity. The outer problem seeks a solution $\Omega$ containing $D$.

Large-volume uniqueness conjecture. If $\sigma_c>1$, then there exists a threshold $V^*\geq |D|$ such that for all $V_0>V^*$, the outer problem has a unique solution $\Omega$. In particular, if $D$ is not a ball, then $V^*>V_0$ and the boundaries $\partial D$ and $\partial\Omega$ touch in the limit as $V_0\to V^*$.

The paper presents global existence and uniqueness as an unresolved difficulty and gives this conjecture as one of three possible goals for comparison or subsolution methods. The asserted global result is therefore open.