On De Giorgi’s conjecture in dimension $N\ge 9$

Abstract

A celebrated conjecture due to De Giorgi states that any bounded solution of the equation $\Delta u + (1-u^2) u = 0 \hbox{in} \mathbb{R}^N $ with $\partial_{y_N}u >0$ must be such that its level sets $\{u=\lambda\}$ are all hyperplanes, at least for dimension $N\le 8$. A counterexample for $N\ge 9$ has long been believed to exist. Starting from a minimal graph $\Gamma$ which is not a hyperplane, found by Bombieri, De Giorgi and Giusti in $\Bbb{R}^N$, $N\ge 9$, we prove that for any small $\alpha >0$ there is a bounded solution $u_\alpha(y)$ with $\partial_{y_N}u_\alpha >0$, which resembles $ \tanh \left ( \frac t{\sqrt{2}}\right ) $, where $t=t(y)$ denotes a choice of signed distance to the blown-up minimal graph $\Gamma_\alpha := \alpha^{-1}\Gamma$. This solution is a counterexample to De Giorgi’s conjecture for $N\ge 9$.