Regularity Results for Nonlocal Minimal Surfaces

In this note, we present some recent results in the study of nonlocal minimal surfaces. The notion of nonlocal minimal surface was introduced by Caffarelli, Roquejoffre, and Savin, they are boundaries of sets which minimize the nonlocal (or fractional) perimeter. In the last years, much interest has been devoted to the study of their regularity properties. Similarly to the classical local setting, a crucial ingredient in the study of regularity is the classification of minimal cones. In the nonlocal setting, only partial results are available, dealing mainly with the low-dimensional case. We describe the main achievements in the field, focusing in particular on the difference with respect to the classical theory and in the difficulties which arise due to the nonlocal character of the problem.