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

created by focardi on 14 Mar 2019

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Submitted Paper

Inserted: 14 mar 2019
Last Updated: 14 mar 2019

Year: 2019

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 up to a set of Hausdorff dimension at most $(n-2)$ and classification of the blow-ups at $\mathcal{H}^{n-1}$ almost every free boundary point.

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 $k+1$.