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 $\phi$ (suitably smooth and decaying fast at infinity) up to sets of null ${\mathcal{H}}^{n-1}$ measure. In particular, if $\phi$ 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 $\phi \in C^{k+1}({\mathbb{R}}^n)$, $k\ge 2$, 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 $k+1$.