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

Abstract

Abstract 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 ℝ n + 1 {\mathbb{R}^{n+1}} with obstacle function φ (suitably smooth and decaying fast at infinity) up to sets of null ℋ n - 1 {{\mathcal{H}}^{n-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: (i) local finiteness of the ( n - 1 ) {(n-1)} -dimensional Minkowski content of the free boundary (and thus of its Hausdorff measure), (ii) ℋ n - 1 {{\mathcal{H}}^{n-1}} -rectifiability of the free boundary, (iii) classification of the frequencies and of the blowups up to a set of Hausdorff dimension at most ( n - 2 ) {(n-2)} in the free boundary. Instead, if φ ∈ C k + 1 ⁢ ( ℝ n ) {\varphi\in C^{k+1}(\mathbb{R}^{n})} , k ≥ 2 {k\geq 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 {k+1} .