### Abstract

A celebrated conjecture due to De Giorgi states that any bounded so-lution of the equation Δu + (1-u^{2})u = 0 in R{double-struck}^{N} with ∂_{yN} u > 0 must be such that its level sets {u=λ} are all hyperplanes, at least for dimension N ≤ 8. A counterexample for N ≥ 9 has long been believed to exist. Start-ing from a minimal graph Γ which is not a hyperplane, found by Bombieri, De Giorgi and Giusti in R{double-struck}^{N}, N ≥ 9, we prove that for any small α > 0 there is a bounded solution u_{α}(y) with ∂_{yN} u_{α} > 0, which resembles tanh(t/√2(, where t = t(y) denotes a choice of signed distance to the blown-up mini-mal graph Γ_{α} := α^{-1}Γ. This solution is a counterexample to De Giorgi's conjecture for N ≥ 9.

Original language | English |
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Pages (from-to) | 1485-1569 |

Number of pages | 85 |

Journal | Annals of Mathematics |

Volume | 174 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Nov 2011 |

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

## Cite this

*Annals of Mathematics*,

*174*(3), 1485-1569. https://doi.org/10.4007/annals.2011.174.3.3