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Hölder regularity of two-dimensional almost-minimal sets in \(\mathbb R^n\). (English) Zbl 1213.49051

Summary: We give a different and probably more elementary proof of a good part of Jean Taylor’s regularity theorem for Almgren almost-minimal sets of dimension 2 in \(\mathbb R^3\). We use this opportunity to settle some details about almost-minimal sets, extend a part of Taylor’s result to almost-minimal sets of dimension 2 in \(\mathbb R^n\), and give the expected characterization of the closed sets \(E\) of dimension 2 in \(\mathbb R^3\) that are minimal, in the sense that \(H^2(E\setminus F)\leq H^2 (F\setminus E)\) for every closed set \(F\) such that there is a bounded set \(B\) so that \(F=E\) out of \(B\) and \(F\) separates points of \(\mathbb R^3\setminus B\) that \(E\) separates.

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
49N60 Regularity of solutions in optimal control
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