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

created by focardi on 06 Mar 2017
modified on 17 Dec 2018

[BibTeX]

Published Paper

Inserted: 6 mar 2017
Last Updated: 17 dec 2018

Journal: Archive for Rational Mechanics and Analysis
Volume: 230
Pages: 125--184
Year: 2018
Notes:

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.