Regularity of the obstacle problem for a fractional power of the laplace operator

Given a function ϕ and s ∈ (0, 1), we will study the solutions of the following obstacle problem

1. u ≥ ϕ in R n
2. (−4) su ≥ 0 in R n
3. (−4) su(x) = 0 for those x such that u(x) > ϕ(x)
4. lim|x|→+∞ u(x) = 0 We show that when ϕ is C 1,s or smoother, the solution u is in the space C 1,α for every α < s. In the case that the contact set {u = ϕ} is convex, we prove the optimal regularity result u ∈ C 1,s. When ϕ is only C 1,β for a β < s, we prove that our solution u is C 1,α for every α < β.