Speaker
Description
(Dimitrios Gazoulis - National and Kapodistrian University of Athens)
In this talk we study the level sets of solutions of the Allen-Cahn equation and we prove local minimality of the zero level set with respect to certain perimeter functional with density. This provides a direct relationship between phase transition type problems and minimal surfaces with some weight. In particular, we establish that if u is a solution to the Allen-Cahn equation that satisfy $u_{x_n}>0$ and $\lim_{x_n \to \pm \infty} u(x',x_n)=\pm1$, then the zero level set of $u$ locally minimizes a perimeter type functional. As an application, we estublish the De Giorgi conjecture, proved by O. Savin, by reducing it to a Bernstein type result for anisotropic perimeter functionals obtained by L. Simon, thus directly linking it to the geometric problem.
