Calculus of Variations and Geometric Measure Theory
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G. De Philippis - A. De Rosa - F. Ghiraldin

A direct approach to Plateau's problem in any codimension

created by dephilipp on 28 Jan 2015
modified by derosa on 22 Apr 2016


Published Paper

Inserted: 28 jan 2015
Last Updated: 22 apr 2016

Journal: Adv. in Math.
Volume: 288
Pages: 59–80
Year: 2015
Doi: 10.1016/j.aim.2015.10.007
Links: Adv. in Math.


This paper aims to propose 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 $\mathbb{R}^n$: following the previous work $[$DGM14$]$ 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: Plateau's problem; Geometric measure theory; Rectifiable sets


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