The stability conjecture for Allen–Cahn equations

Let $u$ be a bounded, stable solution of the Allen–Cahn equation

$$-\Delta u=u-u^3$$

in $\mathbb{R}^n$. A level set of $u$ is a set of the form $\{x\in\mathbb{R}^n:u(x)=c\}$.

Stability conjecture. If $n\leq 7$, then all level sets of $u$ are hyperplanes.

This is the stability version of De Giorgi's conjecture. The conjecture remains open for dimensions $3\leq n\leq 7$; the source notes that related results establish it in some lower-dimensional or additional-hypothesis settings, while counterexamples occur in dimensions $n\geq 9$.