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 ℝ^2 and ℝ^3 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.