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 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 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.