*Accepted Paper*

**Inserted:** 14 mar 2019

**Last Updated:** 19 oct 2020

**Journal:** Advances in Calculus of Variations

**Year:** 2019

**Abstract:**

Building upon the recent results in {FoSp17} we provide a thorough description of the free boundary for solutions to the fractional obstacle problem in $\mathbb{R}^{n+1}$ with obstacle function $\varphi$ (suitably smooth and decaying fast at infinity) up to sets of null $\mathcal{H}^{n-1}$ measure. In particular, if $\varphi$ is analytic, the problem reduces to the zero obstacle case dealt with in {FoSp17} 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 blow-ups up to a set of Hausdorff dimension at most $(n-2)$ in the free boundary. \end{itemize}

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 $\varphi$ is less than $k+1$.

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