Calculus of Variations and Geometric Measure Theory
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M. Focardi - E. Spadaro

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 19 Sep 2018


Accepted Paper

Inserted: 6 mar 2017
Last Updated: 19 sep 2018

Journal: ARMA
Year: 2017


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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