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

Abstract

Building upon the recent results in \cite{FoSp17} we provide a thorough description of the free boundary for 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 \cite{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. Instead, if $\varphi\in C^{k+1}(\mathbb{R}^n)$, $k\geq 2$, similar results hold only for a distinguished subset of points in the free boundary where the order of contact of the solution and the obstacle is less than $k+1$.