We establish a partial rectifiability result for the free boundary of a k-varifold V. Namely, we first refine a theorem of Gruter 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
De Masi, Luigi
2021-01-01
Abstract
We establish a partial rectifiability result for the free boundary of a k-varifold V. Namely, we first refine a theorem of Gruter 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.File | Dimensione | Formato | |
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