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 17 Dec 2018


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

A correction to this article is available online at https:/doi.org10.1007s00205-018-1273-x.


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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