This talk will be devoted to introduce the Neumann problem for the fractional Laplacian, in contrast with the Dirichlet problem, focusing on recent results concerning the boundary regularity. First, we are motivating the problem through a probabilistic interpretation. Next, we are giving basic properties of the problem. Finally, we are going to present some boundary regularity results, as well as the main ideas for the proofs.
This is a joint work with Alessandro Audrito (ETH-Zurich) and Xavier Ros-Oton (ICREA-UB).