On the regularity of the Non-dynamic Parabolic Fractional Obstacle   Problem
  In the class of the so called non-dynamic Fractional Obstacle Problems of parabolic type, it is shown how to obtain higher regularity as well as optimal regularity of the space derivatives of the solution. Furthermore, at free boundary points of positive parabolic density, it is proven that the time derivative of the solution is H\"{o}lder continuous. Finally, at regular free boundary points, space-time regularity of the corresponding free boundary is obtained for any fraction $s\in(0,1)$. 
Ioannis Athanasopoulos, Luis Caffarelli, Emmanouil Milakis
math.AP