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

Abstract Given a function φ and s ∈ (0, 1), we will study the solutions of the following obstacle problem: u ≥ φ in ℝ n , (−▵) s u ≥ 0 in ℝ n , (−▵) s u ( x ) = 0 for those x such that u ( x ) > φ( x ), 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 where 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. © 2006 Wiley Periodicals, Inc.