On the measure and the structure of the free boundary of the lower dimensional obstacle problem

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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