59 problems

Sirakov's equal-frequency uniqueness conjecture. In this equal-frequency setting, this is the unique positive solution.

Consider the generalized $\alpha$-SQG equation and express Hölder regularity through $\Lambda^{-1}\theta\in C^\gamma$. Let $\gamma_U(\alpha)$ denote the threshold for uniqueness in…

For a cluster enclosing $n$ regions of prescribed areas $a_1,\ldots,a_n$, call it minimizing if it has least perimeter among clusters enclosing and separating regions with those ar…

Let $U_{\Phi}$ be the set of interior slopes. A non-$\mathcal L$-invariant gradient Gibbs measure $\mu\in\mathcal P(\Omega,\mathcal F^{\tau})$ has approximate slope $u$ if, for eve…

Let a viscosity solution be a 2D Euler solution obtained as a zero-viscosity limit of 2D Navier–Stokes solutions, and call it dissipative when $Z_h(\omega)\geq 0$ for convex $h$. U…

Let $({\cal M},g_{\mu\nu})$ be a space satisfying the geometric and asymptotic conditions specified in the source for the Riemannian Schwarzschild and Taub-bolt instantons. Uniquen…

Let $G$ be a countably infinite graph of maximum degree … , the hard-core lattice gas on $G$ has a unique Gibbs measure whenever … for all vertices $x$; perhaps this remains true w…

Let $Z_{G,q}(\beta)$ denote the partition function of the anti-ferromagnetic $q$-state Potts model on a graph $G$, and let the tree-uniqueness threshold be the range of inverse tem…

Let $q$ be the number of Potts spins, let $\beta$ be the inverse temperature, and let $\beta_c(q,\Delta)$ be … Consider the anti-ferromagnetic Potts Gibbs distribution on the infin…

Kerr uniqueness conjecture. The Riemannian version of the Kerr solution should be the unique AF instanton.

Let $\mathcal{E}$ be the energy functional for spherical ferromagnets, let $H_{0,2}$ be the profile class used for the axisymmetric construction, and let $\kappa\geq 4$. Uniqueness…

Let $\boldsymbol{u}:\mathbb R^+\times\mathbb R\to\mathbb R^2$ be an $\mathbf L^\infty$ entropy solution of a $2\times 2$ system of conservation laws … with flux…

Consider the generalized $\alpha$-SQG equation on $\mathbb{R}^2$ with velocity recovered from the temperature by the $\alpha$-Biot–Savart law. Let $s_U$ denote the threshold for un…

Berman–Berndtsson's conjecture. Any solution is radial, and consequently the solution is unique. This conjecture concerns uniqueness for complex Monge–Ampère mean field equations o…

Lewin's conjecture. The corresponding classification and uniqueness results should generalize to the system for every $N\geq 3$: normalized solutions should necessarily satisfy…

Let $b$ and $\sigma$ be the drift and diffusion coefficients of the degenerate stochastic differential equation, and let $a:=\sigma\sigma^*/2$. For an initial measure $\nu$, consid…

Nonuniqueness conjecture. For $d\geq 2$, uniqueness is not guaranteed in general for the constrained approximate optimal transport map problem.

Let $N\geq 3$, let $s\in(0,2)$, and let $U$ be the unique solution defined in the paper, associated with the Hardy–Sobolev problem on the half-space $\mathbb{R}^N_+$. A critical po…

Uniqueness conjecture. Uniqueness holds across the entire range

Let $\Omega\subset\mathbb{R}^N$ be the domain in the Lane–Emden problem … with $1<p<2^*-1$, where $2^*$ is the critical Sobolev exponent. Assume that $\Omega$ is convex. Convex-dom…

Let $\Omega$ be a bounded smooth convex domain in $\mathbb R^N$, with $N\geq 2$, and let $\lambda\in\mathbb R$ and $p>1$. Consider the Dirichlet problem … A positive solution means…

Euclidean Black Hole Uniqueness Conjecture. Every non-flat AF instanton belongs to the Kerr family.

The uniqueness conjecture. Then $u=u_3$. The inequalities need not be uniform in $z$.

Further-criteria conjecture. Other general criteria guaranteeing the uniqueness of equilibria remain to be discovered.

A pitchfork is the complete translating soliton obtained by reflecting a fundamental graph over the strip $(0,w)\times\mathbb{R}$, where $w\geq\pi$, with the pitchfork boundary val…