On De Giorgi Conjecture in Dimension $N\ge 9$

Abstract

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

67 pages

References in corpus (1)

• The Toda system and multiple-end solutions of autonomous planar elliptic problems

Cited by in corpus (2)

• Qualitative properties of saddle-shaped solutions to bistable diffusion equations
• Nonexistence of nonconstant global minimizers with limit at $\infty$ of semilinear elliptic equations in all of $R^N$