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

We provide a thorough description of the free boundary for the lower dimensional obstacle problem in $R^{n+1}$ % with $0$ obstacle imposed on a hyperplane up to sets of null $cH^{n-1}$ measure. In particular, we prove % the following results: egin{itemize} item[(i)] local finiteness of the $(n-1)$-dimensional Hausdorff measure of the free boundary, item[(ii)] $cH^{n-1}$-rectifiability of the free boundary, item[(iii)] classification of the frequencies up to a set of Hausdorff dimension at most $(n-2)$ and classification of the blow-ups at $cH^{n-1}$ almost every free boundary point. end{itemize}