We establish a partial rectifiability result for the free boundary of a $k$-varifold $V$. Namely, we first refine a theorem of Gr\"uter and Jost by showing that the first variation of a general varifold with free boundary is a Radon measure. Next we show that if the mean curvature $H$ of $V$ is in $L^p$ for some $p \in [1,k]$, then the set of points where the $k$-density of $V$ does not exist or is infinite has Hausdorff dimension at most $k-p$. We use this result to prove, under suitable assumptions, that the part of the first variation of $V$ with positive and finite $(k-1)$-density is $(k-1)$-rectifiable.

Rectifiability of the free boundary for varifolds / De Masi, Luigi. - In: INDIANA UNIVERSITY MATHEMATICS JOURNAL. - ISSN 0022-2518. - 70:6(2021), pp. 2603-2651. [10.1512/iumj.2021.70.9401]

### Rectifiability of the free boundary for varifolds

#### Abstract

We establish a partial rectifiability result for the free boundary of a $k$-varifold $V$. Namely, we first refine a theorem of Gr\"uter and Jost by showing that the first variation of a general varifold with free boundary is a Radon measure. Next we show that if the mean curvature $H$ of $V$ is in $L^p$ for some $p \in [1,k]$, then the set of points where the $k$-density of $V$ does not exist or is infinite has Hausdorff dimension at most $k-p$. We use this result to prove, under suitable assumptions, that the part of the first variation of $V$ with positive and finite $(k-1)$-density is $(k-1)$-rectifiable.
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2021
70
6
2603
2651
https://arxiv.org/abs/2010.08723
De Masi, Luigi
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/129550