12 problems

Let $u$ be a bounded solution of the Allen–Cahn equation … Assume that $u$ is monotone in one direction, namely $u_{x_n}>0$. De Giorgi's conjecture. If $n\leq 8$, then $u$ is one-d…

Finite index De Giorgi conjecture. If $4\leq n\leq 7$, then $u$ is one-dimensional.

Del Pino–Kowalczyk–Wei conjecture. Outside a large ball, each level set of $u$ has finitely many components, each asymptotic either to a plane or to a catenoid; after a rotation of…

Finite index implies finite ends. For $n\geq 3$, every finite Morse index solution has finitely many ends.

Wang–Wei classification conjecture. If $n\leq 7$, then $u$ is one-dimensional: either $u\equiv \pm1$, or

Stable Allen–Cahn regularity conjecture. If $n \leq 7$, then the curvatures of the level sets of $u_{\varepsilon_j}$ in these interfacial regions are uniformly bounded along the se…

Let $\phi: \mathbb{R}^{n-1} \times \mathbb{R}^+ \to \mathbb{R}$. Suppose that $\phi(s,0) \equiv 0$. Let $f: \mathbb{R}^{n-1} \to \mathbb{R}$, and suppose … Here $g$ is the one-dime…

Nonlocal De Giorgi conjecture. Is $u$ one-dimensional, at least under suitable restrictions on $\alpha$ and $n$? This is the fractional analogue of De Giorgi's one-dimensional symm…

Projected Wiener dynamics conjecture. For $T_\varepsilon\leq c\varepsilon\eta_\varepsilon^{-1}\wedge\tau$, one expects

Let $u:\mathbb{R}^n\to[-1,1]$ be a solution of the Allen–Cahn equation in the whole of $\mathbb{R}^n$ such that … uniformly with respect to…

Let $u_N$ be the solution to the regularized problem, and let $\|\cdot\|_s$ denote the $H^s(\mathbb{T}^2)$ norm. For every $t>0$, the Sobolev-norm limit conjecture. … This predicts…

Let $u_N$ be the regularizations of the two-dimensional stochastic Allen–Cahn equation, and let $\langle\cdot,\cdot\rangle$ denote the $L^2$-duality pairing. For every $t>0$ and ev…