30 problems

Berestycki–Caffarelli–Nirenberg conjecture. If $\mathbb{R}^n\setminus\overline{\Omega}$ is connected, then the existence of a bounded solution implies that $\Omega$ is a ball, a ha…

Let $\Omega \subset \mathbb{R}^d$ be a bounded domain, and consider the overdetermined elliptic problem referred to in the paper as equation, whose solution has constant forcing an…

Let $\Omega$ be a bounded $C^{2,\epsilon}$ domain in $\mathbb{R}^{N+1}$, with $\epsilon\in(0,1)$ and connected boundary. Let $u$ be a nontrivial solution of the overdetermined eige…

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 $C$ be a convex set that is not a ball, let $g$ be a generic non-constant function on $\partial C$, and let $\mathcal{QS}(f,g)$ denote the corresponding overdetermined free bou…

Classification conjecture for Serrin bands. Any Serrin band in $\mathbb{R}^2$ is, up to similarity, one of the periodic bands $\Omega_\tau$, with $\tau\in(0,1]$.

Let $(\mathbb{S}^2,g)$ be the standard round two-sphere, and let $\Omega\subset\mathbb{S}^2$ be a smooth domain. Suppose there is a nontrivial function $u$ and a number $\mu>0$ suc…

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.

BCN Conjecture. If $\Omega$ is a smooth domain with connected complement $\Omega^c$, then $\Omega$ is either a ball, a half-space, a generalized cylinder…

Let $\omega\subset\mathbb R^{N-1}$ be a bounded domain, let $\mathcal C_\omega$ be the cylinder over $\omega$, and let $\Omega^*$ be an optimal domain for the energy…

Let $\omega\subset\mathbb R^{N-1}$ be a bounded domain and let $\mathcal C_\omega$ be the cylinder over $\omega$. For a prescribed large volume $c$, let $\Omega^*$ be an optimal do…

Let $\Omega\subset\mathbb{R}^2$ be a smooth bounded domain with disconnected boundary. Consider the overdetermined Neumann problem … and suppose that the solution is locally consta…

Berenstein conjecture. [The supplied candidate statement ends here and does not include the conjectured conclusion.]

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.

Let $D$ be a domain with sufficiently regular boundary, and consider its equilibrium distribution on the boundary, meaning the charge distribution associated with the electrostatic…

Let $N\geqslant 3$ be a fixed integer, and consider polygons with $N$ sides in the setting of Theorems 2 and 3, concerning the relevant Riesz-type inequalities and overdetermined p…

Let $\Omega\subseteq\mathbb{S}^{n+1}$ be a domain with smooth connected boundary. Suppose there is a positive function $u\in C^2(\overline{\Omega})$ and a function…

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 $0$.

The question is whether the rigidity conclusion of Theorem remains valid when the cone is not assumed to be convex, but the domain $\Omega$ is assumed to be starshaped with respect…