Partial regularity for mass-minimizing currents in Hilbert spaces
type: Item
instance of: scholarly article
title: Partial regularity for mass-minimizing currents in Hilbert spaces
zbMATH Open document ID: 1391.30067
author: Luigi Ambrosio
author: Camillo De Lellis
author: Thomas Schmidt
published in: Journal für die Reine und Angewandte Mathematik
publication date: 2018-01-09T00:00:00Z
full work available at URL: Hilbert regularity.pdf
review text: This paper investigates finite-dimensional integral currents in infinite-dimensional Hilbert spaces. Hence the framework is the one looking at currents as linear functionals on certain spaces of Lipschitz functions. The main result is that for locally mass-minimizing currents, the regular set is dense in the support of the current. A current \(T\in \mathbf I_n(\mathcal H)\) is locally mass-minimizing in an open set \(\Omega\subset \mathcal H\) if \[ \mathbf M(T)\leq \mathbf M(T+R) \] whenever \(R\in \mathbf I_n(\mathcal H)\), \(\partial R=0\) and \(\mathrm{dist}(\mathrm{spt} R,\mathcal H\setminus \Omega)>0\). Roughly speaking, the regular set of a current is the collection of points in the support of the current such that it is represented near each point by a \(C^1\) graph with some multiplicity. After giving an introduction and stating preliminaries, the authors first show a result about the extension of a Lipschitz function to a harmonic function. This is followed by a regularity result concerning representing a current locally by a Hölder continuous function. The proof spans over the next two sections. In the first of them, Lipschitz approximations are considered. The main result in the second section is about the excess decay. Finally, the following section ends the proof, using, however, a result proven in the appendix. The first part of the appendix looks at Cartesian products of currents and also at a retraction of a current on graphs. Finally, the paper ends with providing a monotonicity formula used earlier and some density-type result.
reviewed by: Thomas Zürcher
Mathematics Subject Classification ID: 30L99
Mathematics Subject Classification ID: 58A25
Mathematics Subject Classification ID: 49Q15
zbMATH DE Number: 6823299
zbMATH Keywords: currents in infinite-dimensional Hilbert spaces
zbMATH Keywords: regular sets
MaRDI profile type: Publication
Wikidata QID: Q102417222
OpenAlex ID: W2564141900
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DOI: 10.1515/CRELLE-2015-0011
description: scientific article; zbMATH DE number 6823299