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Limits of Almgren quasiminimal sets. (English) Zbl 1090.49025
Beckner, William (ed.) et al., Harmonic analysis at Mount Holyoke. Proceedings of an AMS-IMS-SIAM joint summer research conference, Mount Holyoke College, South Hadley, MA, USA, June 25–July 5, 2001. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-2903-3/pbk). Contemp. Math. 320, 119-145 (2003).
Let \(\Omega\subset {\mathbb R}^n\) be open, and let \(E\subset \Omega\) be a nonempty closed subset of \(\Omega\). Let \(M\geq 1\) and \(\delta\in(0,+\infty]\) be given. \(E\) is called an \((\Omega,M,\delta)\)-quasiminimal set of dimension \(d\) if \[ H^d(E\cap B)<\infty \quad \text{for any closed ball }B\subset\Omega \] and \[ H^d(E\cap W_f)\leq MH^d(f(E\cap W_f)) \qquad (\text{here } W_f=\{x\in\Omega;\;f(x)\not= x\}), \] for every Lipschitz map \(f: \Omega\to\Omega\) satisfying certain conditions involving the parameter \(\delta\). In this paper, the author shows that Hausdorff limits of Almgren quasiminimal sets are quasiminimal with the same constants, which generalizes a result of L. Ambrosio, N. Fusco and J. E. Hutchinson [Calc. Var. Partial Differ. Equ. 16, No. 2, 187–215 (2003; Zbl 1047.49015)].
For the entire collection see [Zbl 1013.00026].

MSC:
49Q20 Variational problems in a geometric measure-theoretic setting
28A75 Length, area, volume, other geometric measure theory
42B35 Function spaces arising in harmonic analysis
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