Wang–Wei classification conjecture for stable Allen–Cahn solutions

Let $u: \mathbb{R}^n \to [-1,1]$ be a stable solution of the Allen–Cahn equation with $\varepsilon=1$,

$$-\Delta u + W'(u)=0 \quad \text{in } \mathbb{R}^n.$$

Assume that $u$ has bounded energy density, meaning that $\mathbf{M}_\infty(u):=\lim_{r\to\infty}\mathbf{M}_r(u)<\infty$, where $\mathbf{M}_r(u)=r^{1-n}\mathcal{E}^1(u,B_r)$.

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

$$u(x)=\tanh\left(\frac{e\cdot x-s_0}{\sqrt{2}}\right)$$

for some unit vector $e\in\mathbb{S}^{n-1}$ and $s_0\in\mathbb{R}$.

This classification is described as a conjectural reduction of the stable phase-transition regularity problem. A positive resolution in a dimension would imply the corresponding regularity result and, through cited work of Wang–Wei and Chodosh–Mantoulidis, has implications for multiplicity-one and Morse-index conjectures in Allen–Cahn min–max theory.