Calculus of Variations and Geometric Measure Theory

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 11 Jan 2022


Published Paper

Inserted: 28 jan 2015
Last Updated: 11 jan 2022

Journal: Adv. in Math.
Volume: 288
Pages: 59–80
Year: 2016
Doi: 10.1016/j.aim.2015.10.007

ArXiv: 1501.07109 PDF
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 \cite{DelGhiMag} 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