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A direct approach to Plateau's problem in any codimension


De Philippis, Guido; De Rosa, Antonio; Ghiraldin, Francesco (2016). A direct approach to Plateau's problem in any codimension. Advances in Mathematics, 288:59-80.

Abstract

This paper proposes a direct approach to solve the Plateau's problem in codimension higher than one. The problem is formulated as the minimization of the Hausdorff measure among a family of $\mathit{d}$ -rectifiable closed subsets of $\mathbb{R}^{\mathit{n}}$: following the previous work [13], the existence result is obtained by a compactness principle valid under fairly general assumptions on the class of competitors. Such class is then specified to give meaning to boundary conditions. We also show that the obtained minimizers are regular up to a set of dimension less than ($\mathit{d}$−1).

Abstract

This paper proposes a direct approach to solve the Plateau's problem in codimension higher than one. The problem is formulated as the minimization of the Hausdorff measure among a family of $\mathit{d}$ -rectifiable closed subsets of $\mathbb{R}^{\mathit{n}}$: following the previous work [13], the existence result is obtained by a compactness principle valid under fairly general assumptions on the class of competitors. Such class is then specified to give meaning to boundary conditions. We also show that the obtained minimizers are regular up to a set of dimension less than ($\mathit{d}$−1).

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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:January 2016
Deposited On:12 Jan 2017 10:38
Last Modified:19 Aug 2018 06:41
Publisher:Elsevier
ISSN:0001-8708
OA Status:Green
Publisher DOI:https://doi.org/10.1016/j.aim.2015.10.007
Project Information:
  • : FunderFP7
  • : Grant ID306247
  • : Project TitleRAM - Regularity theory for area minimizing currents
  • : FunderSNSF
  • : Grant ID200020_146349
  • : Project TitleCalculus of variations and fluid dynamics

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