Published Paper
Inserted: 11 mar 2021
Last Updated: 24 mar 2026
Journal: Indiana Univ. Math. J.
Volume: 70
Number: 6
Pages: 2603--2651
Year: 2021
Doi: 10.1512/iumj.2021.70.9401
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.
Keywords: Rectifiability, Hausdorff dimension, varifolds, monotonicity formula, free boundary
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