A one-phase problem for the fractional Laplacian: regularity of flat   free boundaries
  We consider a one-phase free boundary problem involving a fractional Laplacian $(-\Delta)^\alpha$, $0<\alpha <1,$ and we prove that ``flat free boundaries" are $C^{1,\gamma}$. We thus extend the known result for the case $\alpha=1/2.$ 
Daniela De Silva, Ovidiu Savin, Yannick Sire
math.AP