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Partial regularity for mass-minimizing currents in Hilbert spaces


Ambrosio, Luigi; De Lellis, Camillo; Schmidt, Thomas (2015). Partial regularity for mass-minimizing currents in Hilbert spaces. Journal für die Reine und Angewandte Mathematik, 0(0):Epub ahead of print.

Abstract

Recently, the theory of currents and the existence theory for Plateau's problem have been extended to the case of finite-dimensional currents in infinite-dimensional manifolds or even metric spaces; see [Acta Math. 185 (2000), 1–80] (and also [Proc. Lond. Math. Soc. (3) 106 (2013), 1121–1142], [Adv. Calc. Var. 7 (2014), 227–240] for the most recent developments). In this paper, in the case when the ambient space is Hilbert, we provide the first partial regularity result, in a dense open set of the support, for n-dimensional integral currents which locally minimize the mass. Our proof follows with minor variants [Indiana Univ. Math. J. 31 (1982), 415–434], implementing Lipschitz approximation and harmonic approximation without indirect arguments and with estimates which depend only on the dimension n and not on codimension or dimension of the target space.

Recently, the theory of currents and the existence theory for Plateau's problem have been extended to the case of finite-dimensional currents in infinite-dimensional manifolds or even metric spaces; see [Acta Math. 185 (2000), 1–80] (and also [Proc. Lond. Math. Soc. (3) 106 (2013), 1121–1142], [Adv. Calc. Var. 7 (2014), 227–240] for the most recent developments). In this paper, in the case when the ambient space is Hilbert, we provide the first partial regularity result, in a dense open set of the support, for n-dimensional integral currents which locally minimize the mass. Our proof follows with minor variants [Indiana Univ. Math. J. 31 (1982), 415–434], implementing Lipschitz approximation and harmonic approximation without indirect arguments and with estimates which depend only on the dimension n and not on codimension or dimension of the target space.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:6 June 2015
Deposited On:14 Jan 2016 12:27
Last Modified:06 Oct 2016 01:00
Publisher:De Gruyter
ISSN:0075-4102
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:https://doi.org/10.1515/crelle-2015-0011
Permanent URL: https://doi.org/10.5167/uzh-111059

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