On De Giorgi Conjecture in Dimension $N \geq 9$
| dc.creator | del Pino, Manuel | |
| dc.creator | Kowalczyk, Mike | |
| dc.creator | Wei, Juncheng | |
| dc.date | 2008-06-19 | |
| dc.date | 2009-03-27 | |
| dc.date.accessioned | 2026-07-07T12:56:39Z | |
| dc.date.available | 2026-07-07T12:56:39Z | |
| dc.description | A celebrated conjecture due to De Giorgi states that any bounded solution of the equation $Δu + (1-u^2) u = 0 \hbox{in} \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 $Γ$ which is not a hyperplane, found by Bombieri, De Giorgi and Giusti in $\R^N$, $N\ge 9$, we prove that for any small $α>0$ there is a bounded solution $u_α(y)$ 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 $Γ_α:= α^{-1}Γ$. This solution constitutes a counterexample to De Giorgi conjecture for $N\ge 9$. | |
| dc.description | 67 pages | |
| dc.identifier | https://arxiv.org/abs/0806.3141 | |
| dc.identifier | http://arxiv.org/abs/0806.3141 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/224655 | |
| dc.subject | Analysis of PDEs | |
| dc.subject | Differential Geometry | |
| dc.subject | 35J25, 35J20, 35B33, 35B40 | |
| dc.title | On De Giorgi Conjecture in Dimension $N \geq 9$ | |
| dc.type | text |