AbstractBuilding 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 {\mathbb{R}^{n+1}} with obstacle function
φ (suitably smooth and decaying fast at infinity) up to sets of null {{\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)}-dimensional Minkowski content of the free boundary (and thus of its Hausdorff measure),(ii){{\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)} in the free boundary.Instead, if {\varphi\in C^{k+1}(\mathbb{R}^{n})}, {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}.
Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem
