On De Giorgi's conjecture in dimension N≥9
Manuel del Pino, Michał Kowalczyk, Juncheng Wei · Annals of Mathematics · 2011
A celebrated conjecture due to De Giorgi states that any bounded solution of the equation ∆u + (1 -u 2 )u = 0 in R N with ∂y N 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.Starting from a minimal graph Γ which is not a hyperplane, found by Bombieri, De Giorgi and Giusti in R N , N ≥ 9, we prove that for any small α > 0 there is a bounded solution uα(y) with ∂y N uα > 0, which resembles tanh Ä t √ 2 ä , where t = t(y) denotes a choice of signed distance to the blown-up minimal graph Γα := α -1 Γ.This solution is a counterexample to De Giorgi's conjecture for N ≥ 9.