The local structure of the free boundary in the fractional obstacle problem

Building upon the recent results in [M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower-dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 2018, 1, 125-184] we provide a thorough description of the free boundary for the solutions to the fractional obstacle problem in Rn+1 with obstacle function φ (suitably smooth and decaying fast at infinity) up to sets of null Hn-1 measure. In particular, if φ is analytic, the problem reduces to the zero obstacle case dealt with in [M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower-dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 2018, 1, 125-184] and therefore we retrieve the same results: Local finiteness of the (n-1)-dimensional Minkowski content of the free boundary (and thus of its Hausdorff measure), Hn-1-rectifiability of the free boundary, classification of the frequencies and of the blowups up to a set of Hausdorff dimension at most (n-2) in the free boundary. Instead, if φ ∈ Ck+1(Rn), k ≥ 2, similar results hold only for distinguished subsets of points in the free boundary where the order of contact of the solution with the obstacle function φ is less than k + 1.