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on De Giorgi's conjecture in dimension eight

The statement

Let $u : \mathbb{R}^8 \to \mathbb{R}$ be a function. We say that $u$ is a bounded classical solution to the Allen–Cahn equation if $u$ is twice continuously differentiable, there exists $C > 0$ such that $|u(x)| \leq C$ for all $x \in \mathbb{R}^8$ , and $\Delta u(x) + u(x) - u(x)^3 = 0$ for all $x \in \mathbb{R}^8$ , where $\Delta u$ denotes the Laplacian of $u$ . We say that $u$ has positive first partial derivative if $\partial_1 u(x) > 0$ for all $x \in \mathbb{R}^8$ . We say that $u$ has hyperplane level sets if for every $y$ in the range of $u$ , there exists an affine subspace $S \subseteq \mathbb{R}^8$ of codimension one such that the level set $u^{-1}(\{y\}) = \{x \in \mathbb{R}^8 : u(x) = y\}$ equals $S$ . Prove that if $u$ is a bounded classical solution to the Allen–Cahn equation and has positive first partial derivative, then $u$ has hyperplane level sets.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.