Regularity in a one-phase free boundary problem for the fractional   Laplacian
  For a one-phase free boundary problem involving a fractional Laplacian, we prove that "flat free boundaries" are $C^{1,\alpha}$. We recover the regularity results of Caffarelli for viscosity solutions of the classical Bernoulli-type free boundary problem with the standard Laplacian. 
Daniela De Silva, Jean-Michel Roquejoffre
math.AP