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 $k+1$.

中文翻译：

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分数障碍问题中自由边界的局部结构

基于 \cite{FoSp17} 中的最新结果，我们提供了 $\mathbb{R}^{n+1}$ 中带有障碍函数 $\varphi$（适当平滑和在无穷远处快速衰减）直到空 $\mathcal{H}^{n-1}$ 度量集。特别是，如果 $\varphi$ 是解析的，问题就归结为 \cite{FoSp17} 中处理的零障碍情况，因此我们检索到相同的结果：（i）$(n-1)$-的局部有限性自由边界的维度 Minkowski 内容（以及其 Hausdorff 测度），（ii）$\mathcal{H}^{n-1}$-自由边界的可修正性，（iii）频率的分类直到集合Hausdorff 维至多 $(n-2)$ 和 $\mathcal{H}^{n-1}$ 的爆炸分类几乎每个自由边界点。反而，