Minimality of level sets in phase transitions

Abstract

In this work we study the level sets of entire solutions of the Allen-Cahn equation and we prove minimality of the zero level set with respect to a certain perimeter functional with density. This provides a direct relationship between phase transition type problems and minimal surfaces with some weight. In addition, we obtain that the zero level set of entire solutions of the Allen-Cahn equation has zero mean curvature. As an application, we establish the De Giorgi conjecture in it's original statement, without the limiting assumption of O. Savin, by reducing it to classical Bernstein type results for minimal graphs, thus directly linking it to the geometric problem.