# 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 $n \le 8$.

## Authors

Ovidiu Savin

Department of Mathematics
Columbia University
New York, NY 10027
United States