Flatness results for nonlocal phase transitions in low dimensions.

We present some recent results in the study of the fractional Allen-Cahn equation. In particular, we are interested in the analogue, for the fractional case, of a well known De Giorgi conjecture about one-dimensional symmetry of bounded monotone solutions. In dimension n=2 and for any fractional power 0<s<1 of the Laplacian, the conjecture is known to be true. In this seminar, we will address the 3-dimensional case. Depending wheter s is below or above 1/2, we need to exploit different techniques and ingredients in the proof of the one-dimensional symmetry. In particular, when s<1/2, some properties of the so-called nonlocal minimal surfaces, will play a crucial role. This talk is based on several papers in collaboration with X. Cabré, J. Serra, and E. Valdinoci.