Fonctionne avec Indico
v3.2.9
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.