On De Giorgi's conjecture and beyond

We consider the problem of existence of entire solutions to the Allen-Cahn equation Δu + u - u³ = 0 in ℝ^N, usually regarded as a prototype for the modeling of phase transition phenomena. In particular, exploiting the link between the Allen-Cahn equation and minimal surface theory in dimensions N ≥ 9, we find a solution, u, with ∂_xNu > 0, such that its level sets are close to a nonplanar, minimal, entire graph. This counterexample provides a negative answer to a celebrated question by Ennio de Giorgi [De Giorgi E (1979) Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978), 131-188, Pitagora, Bologna]. Our results suggest parallels of De Giorgi's conjecture for finite Morse index solutions in two and three dimensions and suggest a possible program of classification of all entire solutions.