Regularity of flat level sets in phase transitions

Abstract

We consider local minimizers of the Ginzburg-Landau energy functional \[\int \frac{1}{2}|\nabla u|^2 + \frac{1}{4}(1-u^2)^2dx\] and prove that, if the $0$ level set is included in a flat cylinder then, in the interior, it is included in a flatter cylinder. As a consequence we prove a conjecture of De Giorgi which states that level sets of global solutions of \[\triangle u=u^3-u\] such that \[\quad |u|\le 1, \quad \partial_n u>0, \quad \lim_{x_n \to \pm \infty}u(x’,x_n)=\pm 1\] are hyperplanes in dimension $n \le 8$.