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Rectifiability of the free boundary for varifolds. (English) Zbl 1494.49029

Given a compact domain \(M\subseteq\mathbb{R}^n\) with smooth boundary \(\partial M\), a \(k\)-dimensional varifold \(V\) is said to have free boundary at \(\partial M\) provided that there is a \(\|V\|\)-integrable (generalized) “mean curvature” vector \(X\in L^1(M,\|V\|)\) yielding the following first variation formula for any vector field \(X\) tangent to \(\partial M\) (see the precise Definition 2.4 in the paper under review): \[ \int_{G_k(M)}\operatorname{div}_{S} X(x) \,dV(x,S)=-\int_M \langle X,H\rangle \,d\|V\|.\tag{1} \] In case \(V\) is associated to a smooth \(k\)-dimensional surface, then \(H\) is its main curvature vector and the latter formula expresses that one has \(\partial \Sigma\subseteq\partial M\) and that \(\Sigma\) meets \(\partial M\) orthogonally.
When \(\Sigma\subseteq M\) is a smooth \(k\)-dimensional surface with smooth boundary \(\partial \Sigma\) meeting \(\partial M\) orthogonally, the first-variation formula easily upper bound estimates on the Hausdorff measure \(\mathcal{H}^{k-1}(\partial \Sigma)\) of the form: \[ \mathcal{H}^{k-1}(\partial \Sigma)\leq c \left(\frac{\mathcal{H}^k(\Sigma)}{R(M)}+\int_\Sigma |H|\,d\mathcal{H}^k\right),\tag{2} \] where \(R(M)\) stands for the minimal radius of curvature of \(\partial M\), and a bound that can also be localized in balls centered on \(\partial M\). Simple proofs of such bounds heavily depend on the fact that \(\Sigma\) and its boundary are assumed to be smooth.
In the paper under review, the author investigates the possibility of obtaining similar estimates for a general varifold with free boundary.
Refining a result stated by M. Grüter and J. Jost [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 13, 129–169 (1986; Zbl 0615.49018)] and by N. Edelen [J. Reine Angew. Math. 758, 95–137 (2020; Zbl 1433.53124)], the author proves that if a varifold \(V\) with free boundary at \(\partial M\) enjoys the first variation formula (2), then it has bounded first variation; he also obtains bounds on the “boundary part” \(\sigma_V\) of its first variation in the spirit of (2) – see Theorem 1.1 in the paper under review. Rectifiability results on \(\sigma_V\) are also obtained under suitable assumptions – see Theorem 1.2 sqq. there.

MSC:

49Q15 Geometric measure and integration theory, integral and normal currents in optimization
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
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References:

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[17] E-MAIL: ldemasi@sissa.it
[18] KEY WORDS AND PHRASES: Varifolds, free boundary, rectifiability, Hausdorff dimension, density set, monotonicity formula.
[19] MATHEMATICS SUBJECT CLASSIFICATION: 49Q15 (53A07). Received: April 12, 2021.
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