On the De Giorgi conjecture in half spaces

In 1978 E. De Giorgi conjectured that the only bounded solutions of $\Delta u = u^3 - u$ 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 $\mathbb R{^n _+}$.