Nonlocal stable De Giorgi conjecture

Let $0<s<1$ and let $u$ be a stable solution of the fractional Allen–Cahn equation

$$(-\Delta)^s u=f(u),\qquad |u|<1\quad\text{in }\mathbb{R}^d.$$

Here, stability means

$$\int_{\mathbb{R}^d}\left((-\Delta)^s v+f'(u)v\right)v\ge 0\quad\text{for all }v\in C^2_0(\mathbb{R}^d).$$

Nonlocal stable De Giorgi conjecture. If $d\le 7$, then $u$ is a $1$-D solution. Stable solutions include local minimizers and monotone stationary solutions, and the conjecture concerns symmetry of stable entire solutions. Its resolution status is not established in the supplied text.