A counterexample to a conjecture by De Giorgi in large dimensions

Abstract

We consider the Allen-Cahn equation Delta u + u(1 - u(2)) = 0 in R-N.

A celebrated conjecture by E. De Giorgi (1978) states that if u is it bounded Solution to this problem Such that partial derivative(xN) u > 0, then the level sets {u =lambda}, lambda is an element of R, must be hyperplanes at least if N <= 8. We construct a family of solutions Which shows that this statement does not hold true for N >= 9.