On critical points of the relative fractional perimeter

We study the localization of sets with constant nonlocal mean curvature and prescribed small volume in a bounded open set with smooth boundary, proving that they are sufficiently close to critical points of a suitable non-local potential. We then consider the fractional perimeter in half-spaces. We prove the existence of a minimizer under fixed volume constraint, showing some of its properties such as smoothness and symmetry, being a graph in the xN -direction, and characterizing its intersection with the hyperplane {xN = 0}.