On the De Giorgi conjecture in half spaces

In 1978 E. De Giorgi conjectured that the only bounded solutions of $\mathrm{\Delta }u=u^3-u$ in the entire space ${\mathbb{R}}^n$ , with strictly positive derivative in one direction, are one dimentional. At least if $n\le 8$. This conjecture has been given a considerable attention in the recent years. And a weak version of it was recently proven by O. Savin. In this talk we will introduce the conjecture, discuss some of its history and show that it isn't true if ${\mathbb{R}}^n$ is changed to the upper half space $\mathbb{R}{}_{+}^n$.