A direct approach to Plateau's problem in any codimension

G. De Philippis; A. De Rosa; F. Ghiraldin

doi:10.1016/j.aim.2015.10.007

A direct approach to Plateau's problem in any codimension

G. De Philippis, A. De Rosa, F. Ghiraldin

Mathematics

Research output: Contribution to journal › Article › peer-review

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 d-rectifiable closed subsets of Rn: 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 (d-1).

Original language	English (US)
Pages (from-to)	59-80
Number of pages	22
Journal	Advances in Mathematics
Volume	288
DOIs	https://doi.org/10.1016/j.aim.2015.10.007
State	Published - Jan 22 2016

Keywords

Geometric measure theory
Plateau's problem
Rectifiable sets

ASJC Scopus subject areas

Mathematics(all)

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10.1016/j.aim.2015.10.007

Cite this

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title = "A direct approach to Plateau's problem in any codimension",

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 d-rectifiable closed subsets of Rn: 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 (d-1).",

keywords = "Geometric measure theory, Plateau's problem, Rectifiable sets",

author = "{De Philippis}, G. and {De Rosa}, A. and F. Ghiraldin",

note = "Funding Information: The authors are grateful to Camillo De Lellis, Francesco Maggi and Emanuele Spadaro for many interesting comments and suggestions. This work has been supported by ERC 306247 Regularity of area-minimizing currents and by SNF 146349 Calculus of variations and fluid dynamics. Publisher Copyright: {\textcopyright} 2015 Elsevier Inc.",

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AU - De Rosa, A.

AU - Ghiraldin, F.

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N2 - 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 d-rectifiable closed subsets of Rn: 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 (d-1).

AB - 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 d-rectifiable closed subsets of Rn: 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 (d-1).

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A direct approach to Plateau's problem in any codimension

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Keywords

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