Stable solutions of the Allen-Cahn equation in dimension 8 and minimal cones.
Résumé
For all n \geq 1, we are interested in bounded solutions of the Allen-Cahn equation \Delta u+ u− u^3 = 0 which are defined in all R^{n+1} and whose zero set is asymptotic to a given minimal cone. In particular, in dimension n + 1 \geq 8, we prove the existence of stable solutions of the Allen-Cahn equation whose zero sets are not hyperplanes.