Abstract

We consider the Allen-Cahn equationΔ u + u (1 - u2) = 0 in RN . A celebrated conjecture by E. De Giorgi (1978) states that if u is a bounded solution to this problem such that ∂xN u > 0, then the level sets {u = λ}, λ ∈ R, must be hyperplanes at least if N ≤ 8. We construct a family of solutions which shows that this statement does not hold true for N ≥ 9. To cite this article: M. del Pino et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008). © 2008 Académie des sciences.