Nonlocal minimal surfaces: interior regularity, quantitative estimates   and boundary stickiness
  We consider surfaces which minimize a nonlocal perimeter functional and we discuss their interior regularity and rigidity properties, in a quantitative and qualitative way, and their (perhaps rather surprising) boundary behavior. We present at least a sketch of the proofs of these results, in a way that aims to be as elementary and self contained as possible. 
Serena Dipierro and Enrico Valdinoci
math.AP