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

## TL;DR

This paper thoroughly analyzes the structure and measure-theoretic properties of the free boundary in the lower dimensional obstacle problem, providing classifications and rectifiability results.

## Contribution

It offers a detailed description of the free boundary, proving its local finiteness, rectifiability, and classifying blow-ups and frequencies for almost every free boundary point.

## Key findings

- Finite $(n-1)$-dimensional Hausdorff measure of the free boundary
- Rectifiability of the free boundary
- Classification of blow-ups and frequencies

## 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.