We consider the problem of existence of entire solutions to the Allen-Cahn equation Δu + u - u3 = 0 in ℝN, 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.
|Number of pages||6|
|Journal||Proceedings of the National Academy of Sciences of the United States of America|
|Publication status||Published - 1 May 2012|
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