*Accepted Paper*

**Inserted:** 6 mar 2017

**Last Updated:** 19 sep 2018

**Journal:** ARMA

**Year:** 2017

**Abstract:**

We provide a thorough description of the free boundary for the lower dimensional obstacle problem in $\mathbb{R}^{n+1}$ up to sets of null $\mathcal{H}^{n-1}$ measure. In particular, we prove

(i) local finiteness of the $(n-1)$-dimensional Hausdorff measure of the free boundary;

(ii) $\mathcal{H}^{n-1}$-rectifiability of the free boundary,

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

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