On De Giorgi's conjecture in dimensions 4 and 5 (Q1394582)

The authors develop an approach for establishing in some important cases, a conjecture made by De Giorgi more than 20 years ago. Conjecture: Suppose that \(u\) is an entire solution of the equation \[ \Delta u+u-u^3=0, \quad |u|=1, \qquad x=(x',x_n)\in\mathbb R^n \] satisfying \(\frac{\partial u}{\partial x_n}>0\), \(x\in\mathbb R^n\). Then, at least for \(n\leq 8\), the level sets of \(u\) must be hyperplanes. The main message of the authors is that De Giorgi's conjecture is true in dimensions \(n=4,5\) provided that the solutions are also assumed to satisfy an anti-symmetry condition.