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

Abstract

AbstractGiven a function φ and s ∈ (0, 1), we will study the solutions of the following obstacle problem: u ≥ φ in ℝn, (−▵)su ≥ 0 in ℝn, (−▵)su(x) = 0 for those x such that u(x) > φ(x), lim|x| → + ∞ u(x) = 0. We show that when φ is C1, s or smoother, the solution u is in the space C1, α for every α < s. In the case where the contact set {u = φ} is convex, we prove the optimal regularity result u ∈ C1, s. When φ is only C1, β for a β < s, we prove that our solution u is C1, α for every α < β. © 2006 Wiley Periodicals, Inc.