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} up to sets of null H^{n−1} measure. In particular, we prove (i) local finiteness of the (n−1)-dimensional Hausdorff measure of the free boundary, (ii) 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 H^{n−1} almost every free boundary point.