C2,α regularity of flat free boundaries for the thin one-phase problem

We prove C-2,C-alpha regularity of sufficiently flat free boundaries, for the thin one-phase problem in which the free boundary occurs on a lower dimensional subspace. This problem appears also as a model of a one-phase free boundary problem in the context of the fractional Laplacian (-Delta)(1/2). (c) 2012 Elsevier Inc. All rights reserved.