We would like to present some results on the surfaces which

minimize the nonlocal perimeter functional.

These objects naturally arise in the study of the interfaces of phase

transitions, when the particles exhibit long-range interactions or when

boundary effects are present.

In addition, these nonlocal minimal surfaces

have concrete applications in several areas, such as image processing and

mathematical biology.

We will present some results concerning interior regularity and rigidity of nonlocal minimal surfaces, with some quantitative estimates and some

qualitative descriptions in several examples. We will describe also the

boundary behavior of these objects.