Flatness results for nonlocal phase transitions

We consider a nonlocal version of the Allen-Cahn equation, which models phase transitions problems. In the classical setting, the connection between the Allen-Cahn equation and the classification of entire minimal surfaces is well known and motivates a celebrated conjecture by E. De Giorgi on the one-dimensional symmetry of bounded monotone solutions to the (classical) Allen-Cahn equation up to dimension 8. In this work, we present some recent results in the study of the nonlocal analogue of this phase transition problem. In particular we describe the results obtained in several contributions where the classification of certain entire bounded solutions to the fractional Allen-Cahn equation has been obtained. Moreover we describe the connection between the fractional Allen-Cahn equation and the fractional perimeter functional, and we present also some results in the classifications of nonlocal minimal surfaces.