10 problems

Finite-time nontriviality conjecture. For every $m\in\mathbb{N}$, this problem admits nontrivial solutions, meaning solutions whose domains are not Euclidean balls.

Exterior-domain rigidity conjecture. Complements of closed balls are the only exterior domains whose solution satisfies the discrete-time overdetermined boundary condition.

Let $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain, let $\nu$ denote the unit outward normal on $\partial\Omega$, and let $\beta>0$. Consider the overdetermined system ……

Anisotropic Serrin-type conjecture for rough domains. Wulff shapes are the unique such sets satisfying the overdetermined system.

Let $m\in\mathbb{N}$ and let $\Omega_k$, $k\in\{1,\dots,m\}$, be as in the introduction. Consider the multi-phase overdetermined problem described by problem … admits a solution…

Souam's conjecture. If $\Omega\subset S^2$ is a sufficiently regular Schiffer domain, then $\Omega$ is either a geodesic disk or a round symmetric annulus.

Elliptic ordering conjecture. The solutions of the outer problem form an elliptically ordered family: if $|\Omega_1|<|\Omega_2|$, then

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 p…

Conductivity convergence conjecture. For fixed $V_0$ and $D$, the solution $\Omega$ of the outer problem converges to a ball as $\sigma_c\to 1$.

Small-core convergence conjecture. For fixed $V_0$, the solution $\Omega$ of the outer problem converges to a ball as the diameter of $D$ tends to $0$.