De Giorgi conjecture for the Allen–Cahn equation in dimensions 4–8

Establish one-dimensional symmetry of bounded entire solutions u to the Allen–Cahn equation Δu = u^3 − u in ℝ^n for 4 ≤ n ≤ 8 under the standard monotonicity assumption ∂_{x_n}u > 0, i.e., prove that the level sets of u are hyperplanes without imposing additional assumptions such as odd symmetry or graphical level sets.

Background

The paper recalls the classical De Giorgi conjecture for the single Allen–Cahn equation and summarizes known results: it is proved in ℝ2 and ℝ3 and known to fail in dimension ≥9 due to constructions based on Simons’ cone. In the intermediate dimensions, only partial results are known under extra assumptions.

Within this context, the authors explicitly note that for the single-equation case the full conjecture remains unresolved in dimensions 4 through 8 unless additional hypotheses are imposed, highlighting a long-standing gap in the theory of phase transitions and elliptic PDE symmetry results.

References

In dimensions \mathbb{R}{4}-\mathbb{R}{8}, the validity of the De Giorgi conjecture remains open, except under additional assumptions.

— Phase transitions in two-component Bose-Einstein condensates (I): The De Giorgi conjecture for the local problem in $\mathbb{R}^{3}$  (2509.19124 - Wu et al., 23 Sep 2025) in Subsection "Classical theory in a special case α=2"

De Giorgi's conjecture, proposed in 1978, is a phase transition analogue of the Bernstein problem for minimal graphs. Let $u:\mathbb Rn\to(-1,1)$ be a classical solution of eq:allen-cahn satisfying $\partial_{x_n}u>0$. If $n\leq8$, then $u$ is one-dimensional: there exist $e\in\mathbb S{n-1}$ and $\phi:\mathbb R\to(-1,1)$ such that

u(x)=\phi(e\cdot x).

eq:allen-cahn:

−Δu+W′(u)=0,W(t)=14(1−t2)2.-\Delta u+W'(u)=0,\qquad W(t)=\tfrac14(1-t^2)^2.

— Stable solutions of the Allen--Cahn equation in dimension three are one-dimensional  (2609.30194 - Chan et al., 24 Sep 2026) in Section 1, subsection “De Giorgi's conjecture and its stable version”