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

Given a function phi and s is an element of (0, 1), we will study the solutions of the following obstacle problem: center dot u > phi in R-n, center dot (-Delta)(s)u >= 0 in R-n, center dot (-Delta)(s)u (x) = 0 for those x such that u (x) > phi(x), center dot lim(vertical bar x vertical bar ->+infinity) u(x) = 0. We show that when phi is C-1,C-s or,smoother, the solution u is in the space C-1,C-alpha for every alpha < s. In the case where the contact set {u = phi} is convex, we prove the optimal regularity result u is an element of C-1,C-s. When phi is only C-1,C-beta for a beta < s, we prove that our solution u is C-1,C-alpha for every ce < 6. (c) 2006 Wiley Periodicals, Inc.